id	sid	tid	token	lemma	pos
ejpam-6849	1	1	european	european	PROPN
ejpam-6849	1	2	journal	journal	PROPN
ejpam-6849	1	3	of	of	ADP
ejpam-6849	1	4	pure	pure	ADJ
ejpam-6849	1	5	and	and	CCONJ
ejpam-6849	1	6	applied	applied	ADJ
ejpam-6849	1	7	mathematics	mathematic	NOUN
ejpam-6849	1	8	2025	2025	NUM
ejpam-6849	1	9	,	,	PUNCT
ejpam-6849	1	10	vol	vol	NOUN
ejpam-6849	1	11	.	.	PROPN
ejpam-6849	1	12	18	18	NUM
ejpam-6849	1	13	,	,	PUNCT
ejpam-6849	1	14	issue	issue	NOUN
ejpam-6849	1	15	4	4	NUM
ejpam-6849	1	16	,	,	PUNCT
ejpam-6849	1	17	article	article	NOUN
ejpam-6849	1	18	number	number	NOUN
ejpam-6849	1	19	6849	6849	NUM
ejpam-6849	1	20	issn	issn	PROPN
ejpam-6849	1	21	1307	1307	NUM
ejpam-6849	1	22	-	-	SYM
ejpam-6849	1	23	5543	5543	NUM
ejpam-6849	1	24	–	–	PUNCT
ejpam-6849	1	25	ejpam.com	ejpam.com	X
ejpam-6849	1	26	published	publish	VERB
ejpam-6849	1	27	by	by	ADP
ejpam-6849	1	28	new	new	PROPN
ejpam-6849	1	29	york	york	PROPN
ejpam-6849	1	30	business	business	PROPN
ejpam-6849	1	31	global	global	ADJ
ejpam-6849	1	32	vertex	vertex	NOUN
ejpam-6849	1	33	-	-	PUNCT
ejpam-6849	1	34	edge	edge	NOUN
ejpam-6849	1	35	dominating	dominating	NOUN
ejpam-6849	1	36	sets	set	NOUN
ejpam-6849	1	37	of	of	ADP
ejpam-6849	1	38	some	some	DET
ejpam-6849	1	39	graphs	graph	NOUN
ejpam-6849	1	40	under	under	ADP
ejpam-6849	1	41	binary	binary	ADJ
ejpam-6849	1	42	operations	operation	NOUN
ejpam-6849	1	43	jerry	jerry	PROPN
ejpam-6849	1	44	tayab1,2,∗	tayab1,2,∗	PROPN
ejpam-6849	1	45	,	,	PUNCT
ejpam-6849	1	46	ferdinand	ferdinand	PROPN
ejpam-6849	2	1	p.	p.	PROPN
ejpam-6849	3	1	jamil1,2	jamil1,2	PROPN
ejpam-6849	3	2	,	,	PUNCT
ejpam-6849	3	3	imelda	imelda	PROPN
ejpam-6849	3	4	s.	s.	PROPN
ejpam-6849	3	5	aniversario1,2	aniversario1,2	PROPN
ejpam-6849	3	6	1	1	NUM
ejpam-6849	3	7	department	department	NOUN
ejpam-6849	3	8	of	of	ADP
ejpam-6849	3	9	mathematics	mathematic	NOUN
ejpam-6849	3	10	and	and	CCONJ
ejpam-6849	3	11	statistics	statistic	NOUN
ejpam-6849	3	12	,	,	PUNCT
ejpam-6849	3	13	college	college	NOUN
ejpam-6849	3	14	of	of	ADP
ejpam-6849	3	15	science	science	NOUN
ejpam-6849	3	16	and	and	CCONJ
ejpam-6849	3	17	mathematics	mathematic	NOUN
ejpam-6849	3	18	,	,	PUNCT
ejpam-6849	3	19	mindanao	mindanao	PROPN
ejpam-6849	3	20	state	state	PROPN
ejpam-6849	3	21	university	university	PROPN
ejpam-6849	3	22	-	-	PUNCT
ejpam-6849	3	23	iligan	iligan	PROPN
ejpam-6849	3	24	institute	institute	PROPN
ejpam-6849	3	25	of	of	ADP
ejpam-6849	3	26	technology	technology	PROPN
ejpam-6849	3	27	,	,	PUNCT
ejpam-6849	3	28	9200	9200	NUM
ejpam-6849	3	29	iligan	iligan	ADJ
ejpam-6849	3	30	city	city	NOUN
ejpam-6849	3	31	,	,	PUNCT
ejpam-6849	3	32	philippines	philippine	NOUN
ejpam-6849	3	33	2	2	NUM
ejpam-6849	3	34	center	center	NOUN
ejpam-6849	3	35	for	for	ADP
ejpam-6849	3	36	mathematical	mathematical	ADJ
ejpam-6849	3	37	and	and	CCONJ
ejpam-6849	3	38	theoretical	theoretical	ADJ
ejpam-6849	3	39	physical	physical	ADJ
ejpam-6849	3	40	sciences	science	NOUN
ejpam-6849	3	41	,	,	PUNCT
ejpam-6849	3	42	premier	premier	PROPN
ejpam-6849	3	43	research	research	PROPN
ejpam-6849	3	44	institute	institute	PROPN
ejpam-6849	3	45	of	of	ADP
ejpam-6849	3	46	science	science	NOUN
ejpam-6849	3	47	and	and	CCONJ
ejpam-6849	3	48	mathematics	mathematic	NOUN
ejpam-6849	3	49	,	,	PUNCT
ejpam-6849	3	50	mindanao	mindanao	PROPN
ejpam-6849	3	51	state	state	PROPN
ejpam-6849	3	52	university	university	PROPN
ejpam-6849	3	53	-	-	PUNCT
ejpam-6849	3	54	iligan	iligan	PROPN
ejpam-6849	3	55	institute	institute	PROPN
ejpam-6849	3	56	of	of	ADP
ejpam-6849	3	57	technology	technology	PROPN
ejpam-6849	3	58	,	,	PUNCT
ejpam-6849	3	59	9200	9200	NUM
ejpam-6849	3	60	iligan	iligan	ADJ
ejpam-6849	3	61	city	city	NOUN
ejpam-6849	3	62	,	,	PUNCT
ejpam-6849	3	63	philippines	philippine	NOUN
ejpam-6849	3	64	abstract	abstract	ADJ
ejpam-6849	3	65	.	.	PUNCT
ejpam-6849	4	1	given	give	VERB
ejpam-6849	4	2	a	a	DET
ejpam-6849	4	3	simple	simple	ADJ
ejpam-6849	4	4	undirected	undirected	ADJ
ejpam-6849	4	5	graph	graph	NOUN
ejpam-6849	4	6	g	g	PROPN
ejpam-6849	4	7	=	=	PUNCT
ejpam-6849	4	8	(	(	PUNCT
ejpam-6849	4	9	v	v	NOUN
ejpam-6849	4	10	(	(	PUNCT
ejpam-6849	4	11	g	g	NOUN
ejpam-6849	4	12	)	)	PUNCT
ejpam-6849	4	13	,	,	PUNCT
ejpam-6849	4	14	e(g	e(g	PROPN
ejpam-6849	4	15	)	)	PUNCT
ejpam-6849	4	16	)	)	PUNCT
ejpam-6849	4	17	,	,	PUNCT
ejpam-6849	4	18	a	a	DET
ejpam-6849	4	19	vertex	vertex	NOUN
ejpam-6849	4	20	u	u	NOUN
ejpam-6849	4	21	∈	∈	PROPN
ejpam-6849	4	22	v	v	ADP
ejpam-6849	4	23	(	(	PUNCT
ejpam-6849	4	24	g	g	NOUN
ejpam-6849	4	25	)	)	PUNCT
ejpam-6849	4	26	vertex	vertex	NOUN
ejpam-6849	4	27	-	-	PUNCT
ejpam-6849	4	28	edge	edge	NOUN
ejpam-6849	4	29	dominates	dominate	VERB
ejpam-6849	4	30	the	the	DET
ejpam-6849	4	31	edge	edge	NOUN
ejpam-6849	4	32	xy	xy	PROPN
ejpam-6849	4	33	∈	∈	PROPN
ejpam-6849	4	34	e(g	e(g	PROPN
ejpam-6849	4	35	)	)	PUNCT
ejpam-6849	4	36	if	if	SCONJ
ejpam-6849	4	37	one	one	NUM
ejpam-6849	4	38	of	of	ADP
ejpam-6849	4	39	the	the	DET
ejpam-6849	4	40	following	follow	VERB
ejpam-6849	4	41	holds	hold	VERB
ejpam-6849	4	42	:	:	PUNCT
ejpam-6849	4	43	(	(	PUNCT
ejpam-6849	4	44	1	1	X
ejpam-6849	4	45	)	)	PUNCT
ejpam-6849	4	46	u	u	NOUN
ejpam-6849	4	47	=	=	NOUN
ejpam-6849	4	48	x	x	X
ejpam-6849	4	49	or	or	CCONJ
ejpam-6849	4	50	u	u	X
ejpam-6849	4	51	=	=	PROPN
ejpam-6849	4	52	y	y	PROPN
ejpam-6849	4	53	,	,	PUNCT
ejpam-6849	4	54	(	(	PUNCT
ejpam-6849	4	55	2	2	X
ejpam-6849	4	56	)	)	PUNCT
ejpam-6849	4	57	ux	ux	NOUN
ejpam-6849	4	58	∈	∈	PROPN
ejpam-6849	4	59	e(g	e(g	PROPN
ejpam-6849	4	60	)	)	PUNCT
ejpam-6849	4	61	or	or	CCONJ
ejpam-6849	4	62	uy	uy	PROPN
ejpam-6849	4	63	∈	∈	PROPN
ejpam-6849	4	64	e(g	e(g	PROPN
ejpam-6849	4	65	)	)	PUNCT
ejpam-6849	4	66	.	.	PUNCT
ejpam-6849	5	1	a	a	DET
ejpam-6849	5	2	subset	subset	NOUN
ejpam-6849	5	3	s	s	VERB
ejpam-6849	5	4	⊆	⊆	NUM
ejpam-6849	5	5	v	v	NOUN
ejpam-6849	5	6	(	(	PUNCT
ejpam-6849	5	7	g	g	NOUN
ejpam-6849	5	8	)	)	PUNCT
ejpam-6849	5	9	is	be	AUX
ejpam-6849	5	10	a	a	DET
ejpam-6849	5	11	vertex	vertex	NOUN
ejpam-6849	5	12	-	-	PUNCT
ejpam-6849	5	13	edge	edge	NOUN
ejpam-6849	5	14	dominating	dominating	NOUN
ejpam-6849	5	15	set	set	NOUN
ejpam-6849	5	16	of	of	ADP
ejpam-6849	5	17	g	g	PROPN
ejpam-6849	5	18	if	if	SCONJ
ejpam-6849	5	19	for	for	ADP
ejpam-6849	5	20	each	each	DET
ejpam-6849	5	21	xy	xy	PROPN
ejpam-6849	5	22	∈	∈	PROPN
ejpam-6849	5	23	e(g	e(g	PROPN
ejpam-6849	5	24	)	)	PUNCT
ejpam-6849	5	25	,	,	PUNCT
ejpam-6849	5	26	there	there	PRON
ejpam-6849	5	27	exists	exist	VERB
ejpam-6849	5	28	u	u	PROPN
ejpam-6849	5	29	∈	∈	PROPN
ejpam-6849	5	30	s	s	VERB
ejpam-6849	5	31	such	such	ADJ
ejpam-6849	5	32	that	that	SCONJ
ejpam-6849	5	33	u	u	PROPN
ejpam-6849	5	34	vertex	vertex	NOUN
ejpam-6849	5	35	-	-	PUNCT
ejpam-6849	5	36	edge	edge	NOUN
ejpam-6849	5	37	dominates	dominate	VERB
ejpam-6849	5	38	xy	xy	PROPN
ejpam-6849	5	39	.	.	PUNCT
ejpam-6849	6	1	a	a	DET
ejpam-6849	6	2	vertex	vertex	NOUN
ejpam-6849	6	3	-	-	PUNCT
ejpam-6849	6	4	edge	edge	NOUN
ejpam-6849	6	5	dominating	dominating	NOUN
ejpam-6849	6	6	set	set	NOUN
ejpam-6849	6	7	s	s	PROPN
ejpam-6849	6	8	⊆	⊆	NUM
ejpam-6849	6	9	v	v	NOUN
ejpam-6849	6	10	(	(	PUNCT
ejpam-6849	6	11	g	g	NOUN
ejpam-6849	6	12	)	)	PUNCT
ejpam-6849	6	13	is	be	AUX
ejpam-6849	6	14	a	a	DET
ejpam-6849	6	15	total	total	ADJ
ejpam-6849	6	16	vertex	vertex	NOUN
ejpam-6849	6	17	-	-	PUNCT
ejpam-6849	6	18	edge	edge	NOUN
ejpam-6849	6	19	dominating	dominating	NOUN
ejpam-6849	6	20	set	set	VERB
ejpam-6849	6	21	if	if	SCONJ
ejpam-6849	6	22	for	for	ADP
ejpam-6849	6	23	each	each	DET
ejpam-6849	6	24	u	u	PROPN
ejpam-6849	6	25	∈	∈	PROPN
ejpam-6849	6	26	s	s	PART
ejpam-6849	6	27	,	,	PUNCT
ejpam-6849	6	28	there	there	PRON
ejpam-6849	6	29	exists	exist	VERB
ejpam-6849	6	30	v	v	ADP
ejpam-6849	6	31	∈	∈	PROPN
ejpam-6849	6	32	s	s	NOUN
ejpam-6849	6	33	for	for	ADP
ejpam-6849	6	34	which	which	PRON
ejpam-6849	6	35	uv	uv	NOUN
ejpam-6849	6	36	∈	∈	PROPN
ejpam-6849	6	37	e(g	e(g	PROPN
ejpam-6849	6	38	)	)	PUNCT
ejpam-6849	6	39	.	.	PUNCT
ejpam-6849	7	1	the	the	DET
ejpam-6849	7	2	minimum	minimum	ADJ
ejpam-6849	7	3	cardinality	cardinality	NOUN
ejpam-6849	7	4	of	of	ADP
ejpam-6849	7	5	a	a	DET
ejpam-6849	7	6	vertex	vertex	NOUN
ejpam-6849	7	7	-	-	PUNCT
ejpam-6849	7	8	edge	edge	NOUN
ejpam-6849	7	9	(	(	PUNCT
ejpam-6849	7	10	resp	resp	NOUN
ejpam-6849	7	11	.	.	PUNCT
ejpam-6849	8	1	total	total	ADJ
ejpam-6849	8	2	vertex	vertex	NOUN
ejpam-6849	8	3	-	-	PUNCT
ejpam-6849	8	4	edge	edge	NOUN
ejpam-6849	8	5	)	)	PUNCT
ejpam-6849	8	6	dominating	dominating	NOUN
ejpam-6849	8	7	set	set	NOUN
ejpam-6849	8	8	of	of	ADP
ejpam-6849	8	9	g	g	PROPN
ejpam-6849	8	10	is	be	AUX
ejpam-6849	8	11	the	the	DET
ejpam-6849	8	12	vertex	vertex	NOUN
ejpam-6849	8	13	-	-	PUNCT
ejpam-6849	8	14	edge	edge	NOUN
ejpam-6849	8	15	domination	domination	NOUN
ejpam-6849	8	16	number	number	NOUN
ejpam-6849	8	17	(	(	PUNCT
ejpam-6849	8	18	resp	resp	NOUN
ejpam-6849	8	19	.	.	PUNCT
ejpam-6849	9	1	total	total	ADJ
ejpam-6849	9	2	vertex	vertex	NOUN
ejpam-6849	9	3	-	-	PUNCT
ejpam-6849	9	4	edge	edge	NOUN
ejpam-6849	9	5	domination	domination	NOUN
ejpam-6849	9	6	number	number	NOUN
ejpam-6849	9	7	)	)	PUNCT
ejpam-6849	9	8	of	of	ADP
ejpam-6849	9	9	g.	g.	PROPN
ejpam-6849	9	10	this	this	DET
ejpam-6849	9	11	paper	paper	NOUN
ejpam-6849	9	12	investigates	investigate	VERB
ejpam-6849	9	13	the	the	DET
ejpam-6849	9	14	vertex	vertex	NOUN
ejpam-6849	9	15	-	-	PUNCT
ejpam-6849	9	16	edge	edge	NOUN
ejpam-6849	9	17	domination	domination	NOUN
ejpam-6849	9	18	and	and	CCONJ
ejpam-6849	9	19	total	total	ADJ
ejpam-6849	9	20	vertex	vertex	NOUN
ejpam-6849	9	21	-	-	PUNCT
ejpam-6849	9	22	edge	edge	NOUN
ejpam-6849	9	23	domination	domination	NOUN
ejpam-6849	9	24	in	in	ADP
ejpam-6849	9	25	the	the	DET
ejpam-6849	9	26	join	join	NOUN
ejpam-6849	9	27	,	,	PUNCT
ejpam-6849	9	28	corona	corona	PROPN
ejpam-6849	9	29	,	,	PUNCT
ejpam-6849	9	30	lexicographic	lexicographic	ADJ
ejpam-6849	9	31	product	product	NOUN
ejpam-6849	9	32	,	,	PUNCT
ejpam-6849	9	33	complementary	complementary	ADJ
ejpam-6849	9	34	prism	prism	NOUN
ejpam-6849	9	35	and	and	CCONJ
ejpam-6849	9	36	edge	edge	NOUN
ejpam-6849	9	37	corona	corona	NOUN
ejpam-6849	9	38	of	of	ADP
ejpam-6849	9	39	graphs	graph	NOUN
ejpam-6849	9	40	.	.	PUNCT
ejpam-6849	10	1	it	it	PRON
ejpam-6849	10	2	provides	provide	VERB
ejpam-6849	10	3	complete	complete	ADJ
ejpam-6849	10	4	characterizations	characterization	NOUN
ejpam-6849	10	5	of	of	ADP
ejpam-6849	10	6	both	both	CCONJ
ejpam-6849	10	7	the	the	DET
ejpam-6849	10	8	vertex	vertex	NOUN
ejpam-6849	10	9	-	-	PUNCT
ejpam-6849	10	10	edge	edge	NOUN
ejpam-6849	10	11	dominating	dominating	NOUN
ejpam-6849	10	12	sets	set	NOUN
ejpam-6849	10	13	and	and	CCONJ
ejpam-6849	10	14	total	total	ADJ
ejpam-6849	10	15	vertex	vertex	NOUN
ejpam-6849	10	16	-	-	PUNCT
ejpam-6849	10	17	edge	edge	NOUN
ejpam-6849	10	18	dominating	dominating	NOUN
ejpam-6849	10	19	sets	set	NOUN
ejpam-6849	10	20	in	in	ADP
ejpam-6849	10	21	these	these	DET
ejpam-6849	10	22	families	family	NOUN
ejpam-6849	10	23	of	of	ADP
ejpam-6849	10	24	graphs	graph	NOUN
ejpam-6849	10	25	,	,	PUNCT
ejpam-6849	10	26	and	and	CCONJ
ejpam-6849	10	27	establishes	establish	VERB
ejpam-6849	10	28	sharp	sharp	ADJ
ejpam-6849	10	29	bounds	bound	NOUN
ejpam-6849	10	30	,	,	PUNCT
ejpam-6849	10	31	if	if	SCONJ
ejpam-6849	10	32	not	not	PART
ejpam-6849	10	33	the	the	DET
ejpam-6849	10	34	exact	exact	ADJ
ejpam-6849	10	35	values	value	NOUN
ejpam-6849	10	36	,	,	PUNCT
ejpam-6849	10	37	for	for	ADP
ejpam-6849	10	38	their	their	PRON
ejpam-6849	10	39	respective	respective	ADJ
ejpam-6849	10	40	vertex	vertex	NOUN
ejpam-6849	10	41	-	-	PUNCT
ejpam-6849	10	42	edge	edge	NOUN
ejpam-6849	10	43	domination	domination	NOUN
ejpam-6849	10	44	and	and	CCONJ
ejpam-6849	10	45	total	total	ADJ
ejpam-6849	10	46	vertex	vertex	NOUN
ejpam-6849	10	47	-	-	PUNCT
ejpam-6849	10	48	edge	edge	NOUN
ejpam-6849	10	49	domination	domination	NOUN
ejpam-6849	10	50	numbers	number	NOUN
ejpam-6849	10	51	.	.	PUNCT
ejpam-6849	11	1	2020	2020	NUM
ejpam-6849	11	2	mathematics	mathematic	NOUN
ejpam-6849	11	3	subject	subject	NOUN
ejpam-6849	11	4	classifications	classification	NOUN
ejpam-6849	11	5	:	:	PUNCT
ejpam-6849	11	6	05c69	05c69	X
ejpam-6849	11	7	key	key	ADJ
ejpam-6849	11	8	words	word	NOUN
ejpam-6849	11	9	and	and	CCONJ
ejpam-6849	11	10	phrases	phrase	NOUN
ejpam-6849	11	11	:	:	PUNCT
ejpam-6849	11	12	domination	domination	NOUN
ejpam-6849	11	13	,	,	PUNCT
ejpam-6849	11	14	vertex	vertex	NOUN
ejpam-6849	11	15	-	-	PUNCT
ejpam-6849	11	16	edge	edge	NOUN
ejpam-6849	11	17	domination	domination	NOUN
ejpam-6849	11	18	,	,	PUNCT
ejpam-6849	11	19	total	total	ADJ
ejpam-6849	11	20	vertex	vertex	NOUN
ejpam-6849	11	21	-	-	PUNCT
ejpam-6849	11	22	edge	edge	NOUN
ejpam-6849	11	23	domination	domination	NOUN
ejpam-6849	11	24	1	1	NUM
ejpam-6849	11	25	.	.	PUNCT
ejpam-6849	12	1	introduction	introduction	NOUN
ejpam-6849	12	2	vertex	vertex	NOUN
ejpam-6849	12	3	-	-	PUNCT
ejpam-6849	12	4	edge	edge	NOUN
ejpam-6849	12	5	domination	domination	NOUN
ejpam-6849	12	6	in	in	ADP
ejpam-6849	12	7	graphs	graph	NOUN
ejpam-6849	12	8	was	be	AUX
ejpam-6849	12	9	first	first	ADV
ejpam-6849	12	10	introduced	introduce	VERB
ejpam-6849	12	11	by	by	ADP
ejpam-6849	12	12	peters	peters	PROPN
ejpam-6849	12	13	[	[	X
ejpam-6849	12	14	1	1	X
ejpam-6849	12	15	]	]	PUNCT
ejpam-6849	12	16	in	in	ADP
ejpam-6849	12	17	1986	1986	NUM
ejpam-6849	12	18	,	,	PUNCT
ejpam-6849	12	19	and	and	CCONJ
ejpam-6849	12	20	is	be	AUX
ejpam-6849	12	21	a	a	DET
ejpam-6849	12	22	graph	graph	NOUN
ejpam-6849	12	23	protection	protection	NOUN
ejpam-6849	12	24	strategy	strategy	NOUN
ejpam-6849	12	25	which	which	PRON
ejpam-6849	12	26	basically	basically	ADV
ejpam-6849	12	27	came	come	VERB
ejpam-6849	12	28	from	from	ADP
ejpam-6849	12	29	the	the	DET
ejpam-6849	12	30	marriage	marriage	NOUN
ejpam-6849	12	31	of	of	ADP
ejpam-6849	12	32	the	the	DET
ejpam-6849	12	33	two	two	NUM
ejpam-6849	12	34	concepts	concept	NOUN
ejpam-6849	12	35	,	,	PUNCT
ejpam-6849	12	36	namely	namely	ADV
ejpam-6849	12	37	the	the	DET
ejpam-6849	12	38	domination	domination	NOUN
ejpam-6849	12	39	(	(	PUNCT
ejpam-6849	12	40	about	about	ADP
ejpam-6849	12	41	static	static	ADJ
ejpam-6849	12	42	positioning	positioning	NOUN
ejpam-6849	12	43	of	of	ADP
ejpam-6849	12	44	guards	guard	NOUN
ejpam-6849	12	45	which	which	PRON
ejpam-6849	12	46	protect	protect	VERB
ejpam-6849	12	47	the	the	DET
ejpam-6849	12	48	vertices	vertex	NOUN
ejpam-6849	12	49	)	)	PUNCT
ejpam-6849	12	50	and	and	CCONJ
ejpam-6849	12	51	vertex	vertex	NOUN
ejpam-6849	12	52	covering	covering	NOUN
ejpam-6849	12	53	(	(	PUNCT
ejpam-6849	12	54	a	a	DET
ejpam-6849	12	55	static	static	ADJ
ejpam-6849	12	56	positioning	positioning	NOUN
ejpam-6849	12	57	of	of	ADP
ejpam-6849	12	58	guards	guard	NOUN
ejpam-6849	12	59	which	which	PRON
ejpam-6849	12	60	protect	protect	VERB
ejpam-6849	12	61	the	the	DET
ejpam-6849	12	62	edges	edge	NOUN
ejpam-6849	12	63	)	)	PUNCT
ejpam-6849	12	64	of	of	ADP
ejpam-6849	12	65	graph	graph	NOUN
ejpam-6849	12	66	.	.	PUNCT
ejpam-6849	13	1	the	the	DET
ejpam-6849	13	2	importance	importance	NOUN
ejpam-6849	13	3	of	of	ADP
ejpam-6849	13	4	vertex	vertex	NOUN
ejpam-6849	13	5	-	-	PUNCT
ejpam-6849	13	6	edge	edge	NOUN
ejpam-6849	13	7	domination	domination	NOUN
ejpam-6849	13	8	is	be	AUX
ejpam-6849	13	9	best	well	ADV
ejpam-6849	13	10	illustrated	illustrate	VERB
ejpam-6849	13	11	by	by	ADP
ejpam-6849	13	12	the	the	DET
ejpam-6849	13	13	so	so	ADV
ejpam-6849	13	14	called	call	VERB
ejpam-6849	13	15	“	"	PUNCT
ejpam-6849	13	16	searchlight	searchlight	NOUN
ejpam-6849	13	17	problem	problem	NOUN
ejpam-6849	13	18	”	"	PUNCT
ejpam-6849	13	19	(	(	PUNCT
ejpam-6849	13	20	see	see	VERB
ejpam-6849	13	21	[	[	X
ejpam-6849	13	22	2	2	NUM
ejpam-6849	13	23	]	]	PUNCT
ejpam-6849	13	24	,	,	PUNCT
ejpam-6849	13	25	[	[	X
ejpam-6849	13	26	3	3	NUM
ejpam-6849	13	27	]	]	PUNCT
ejpam-6849	13	28	,	,	PUNCT
ejpam-6849	13	29	[	[	X
ejpam-6849	13	30	4	4	NUM
ejpam-6849	13	31	]	]	SYM
ejpam-6849	13	32	)	)	PUNCT
ejpam-6849	13	33	which	which	PRON
ejpam-6849	13	34	,	,	PUNCT
ejpam-6849	13	35	inspired	inspire	VERB
ejpam-6849	13	36	by	by	ADP
ejpam-6849	13	37	the	the	DET
ejpam-6849	13	38	famous	famous	ADJ
ejpam-6849	13	39	art	art	NOUN
ejpam-6849	13	40	gallery	gallery	NOUN
ejpam-6849	13	41	problem	problem	NOUN
ejpam-6849	13	42	,	,	PUNCT
ejpam-6849	13	43	attempts	attempt	VERB
ejpam-6849	13	44	to	to	ADP
ejpam-6849	13	45	∗corresponding	∗corresponde	VERB
ejpam-6849	13	46	author	author	NOUN
ejpam-6849	13	47	.	.	PUNCT
ejpam-6849	14	1	doi	doi	NOUN
ejpam-6849	14	2	:	:	PUNCT
ejpam-6849	14	3	https://doi.org/10.29020/nybg.ejpam.v18i4.6849	https://doi.org/10.29020/nybg.ejpam.v18i4.6849	NOUN
ejpam-6849	14	4	email	email	NOUN
ejpam-6849	14	5	addresses	address	NOUN
ejpam-6849	14	6	:	:	PUNCT
ejpam-6849	14	7	jerry.tayab@g.msuiit.edu.ph	jerry.tayab@g.msuiit.edu.ph	PROPN
ejpam-6849	14	8	(	(	PUNCT
ejpam-6849	14	9	j.	j.	PROPN
ejpam-6849	14	10	tayab	tayab	PROPN
ejpam-6849	14	11	)	)	PUNCT
ejpam-6849	14	12	,	,	PUNCT
ejpam-6849	14	13	ferdinand.jamil@g.msuiit.edu.ph	ferdinand.jamil@g.msuiit.edu.ph	PROPN
ejpam-6849	14	14	(	(	PUNCT
ejpam-6849	14	15	f.	f.	PROPN
ejpam-6849	14	16	jamil	jamil	PROPN
ejpam-6849	14	17	)	)	PUNCT
ejpam-6849	14	18	,	,	PUNCT
ejpam-6849	14	19	imelda.aniversario@g.msuiit.edu.ph	imelda.aniversario@g.msuiit.edu.ph	PROPN
ejpam-6849	14	20	(	(	PUNCT
ejpam-6849	14	21	i.	i.	PROPN
ejpam-6849	14	22	aniversario	aniversario	PROPN
ejpam-6849	14	23	)	)	PUNCT
ejpam-6849	14	24	https://www.ejpam.com	https://www.ejpam.com	X
ejpam-6849	15	1	1	1	NUM
ejpam-6849	15	2	copyright	copyright	NOUN
ejpam-6849	15	3	:	:	PUNCT
ejpam-6849	15	4	©	©	PROPN
ejpam-6849	15	5	2025	2025	NUM
ejpam-6849	15	6	the	the	DET
ejpam-6849	15	7	author(s	author(s	NOUN
ejpam-6849	15	8	)	)	PUNCT
ejpam-6849	15	9	.	.	PUNCT
ejpam-6849	16	1	(	(	PUNCT
ejpam-6849	16	2	cc	cc	NOUN
ejpam-6849	16	3	by	by	ADP
ejpam-6849	16	4	-	-	PUNCT
ejpam-6849	16	5	nc	nc	PROPN
ejpam-6849	16	6	4.0	4.0	NUM
ejpam-6849	16	7	)	)	PUNCT
ejpam-6849	16	8	j.	j.	PROPN
ejpam-6849	16	9	tayab	tayab	PROPN
ejpam-6849	16	10	,	,	PUNCT
ejpam-6849	16	11	f.	f.	PROPN
ejpam-6849	16	12	p.	p.	PROPN
ejpam-6849	16	13	jamil	jamil	PROPN
ejpam-6849	16	14	,	,	PUNCT
ejpam-6849	16	15	i.	i.	PROPN
ejpam-6849	16	16	aniversario	aniversario	PROPN
ejpam-6849	16	17	/	/	SYM
ejpam-6849	16	18	eur	eur	PROPN
ejpam-6849	16	19	.	.	PUNCT
ejpam-6849	17	1	j.	j.	PROPN
ejpam-6849	17	2	pure	pure	PROPN
ejpam-6849	17	3	appl	appl	PROPN
ejpam-6849	17	4	.	.	PROPN
ejpam-6849	17	5	math	math	PROPN
ejpam-6849	17	6	,	,	PUNCT
ejpam-6849	17	7	18	18	NUM
ejpam-6849	17	8	(	(	PUNCT
ejpam-6849	17	9	4	4	NUM
ejpam-6849	17	10	)	)	PUNCT
ejpam-6849	17	11	(	(	PUNCT
ejpam-6849	17	12	2025	2025	NUM
ejpam-6849	17	13	)	)	PUNCT
ejpam-6849	17	14	,	,	PUNCT
ejpam-6849	17	15	6849	6849	NUM
ejpam-6849	17	16	2	2	NUM
ejpam-6849	17	17	of	of	ADP
ejpam-6849	17	18	19	19	NUM
ejpam-6849	17	19	use	use	NOUN
ejpam-6849	17	20	searchlights	searchlight	NOUN
ejpam-6849	17	21	to	to	PART
ejpam-6849	17	22	find	find	VERB
ejpam-6849	17	23	an	an	DET
ejpam-6849	17	24	intruder	intruder	NOUN
ejpam-6849	17	25	in	in	ADP
ejpam-6849	17	26	a	a	DET
ejpam-6849	17	27	graph	graph	NOUN
ejpam-6849	17	28	.	.	PUNCT
ejpam-6849	18	1	in	in	ADP
ejpam-6849	18	2	this	this	DET
ejpam-6849	18	3	case	case	NOUN
ejpam-6849	18	4	,	,	PUNCT
ejpam-6849	18	5	the	the	DET
ejpam-6849	18	6	guards	guard	NOUN
ejpam-6849	18	7	,	,	PUNCT
ejpam-6849	18	8	each	each	PRON
ejpam-6849	18	9	of	of	ADP
ejpam-6849	18	10	whom	whom	PRON
ejpam-6849	18	11	holds	hold	VERB
ejpam-6849	18	12	a	a	DET
ejpam-6849	18	13	searchlight	searchlight	NOUN
ejpam-6849	18	14	,	,	PUNCT
ejpam-6849	18	15	must	must	AUX
ejpam-6849	18	16	shine	shine	VERB
ejpam-6849	18	17	a	a	DET
ejpam-6849	18	18	searchlight	searchlight	NOUN
ejpam-6849	18	19	down	down	ADP
ejpam-6849	18	20	some	some	DET
ejpam-6849	18	21	edge	edge	NOUN
ejpam-6849	18	22	where	where	SCONJ
ejpam-6849	18	23	they	they	PRON
ejpam-6849	18	24	think	think	VERB
ejpam-6849	18	25	there	there	PRON
ejpam-6849	18	26	might	might	AUX
ejpam-6849	18	27	be	be	AUX
ejpam-6849	18	28	an	an	DET
ejpam-6849	18	29	intruder	intruder	NOUN
ejpam-6849	18	30	.	.	PUNCT
ejpam-6849	19	1	vertex	vertex	NOUN
ejpam-6849	19	2	-	-	PUNCT
ejpam-6849	19	3	edge	edge	NOUN
ejpam-6849	19	4	domination	domination	NOUN
ejpam-6849	19	5	as	as	ADV
ejpam-6849	19	6	well	well	ADV
ejpam-6849	19	7	as	as	ADP
ejpam-6849	19	8	its	its	PRON
ejpam-6849	19	9	variant	variant	NOUN
ejpam-6849	19	10	,	,	PUNCT
ejpam-6849	19	11	the	the	DET
ejpam-6849	19	12	total	total	ADJ
ejpam-6849	19	13	vertex	vertex	NOUN
ejpam-6849	19	14	-	-	PUNCT
ejpam-6849	19	15	edge	edge	NOUN
ejpam-6849	19	16	domination	domination	NOUN
ejpam-6849	19	17	,	,	PUNCT
ejpam-6849	19	18	is	be	AUX
ejpam-6849	19	19	very	very	ADV
ejpam-6849	19	20	-	-	PUNCT
ejpam-6849	19	21	well	well	NOUN
ejpam-6849	19	22	studied	study	VERB
ejpam-6849	19	23	in	in	ADP
ejpam-6849	19	24	trees	tree	NOUN
ejpam-6849	19	25	(	(	PUNCT
ejpam-6849	19	26	see	see	VERB
ejpam-6849	19	27	[	[	X
ejpam-6849	19	28	5	5	NUM
ejpam-6849	19	29	]	]	PUNCT
ejpam-6849	19	30	,	,	PUNCT
ejpam-6849	20	1	[	[	X
ejpam-6849	20	2	6	6	NUM
ejpam-6849	20	3	]	]	PUNCT
ejpam-6849	20	4	,	,	PUNCT
ejpam-6849	20	5	[	[	X
ejpam-6849	20	6	7	7	NUM
ejpam-6849	20	7	]	]	PUNCT
ejpam-6849	20	8	,	,	PUNCT
ejpam-6849	20	9	[	[	X
ejpam-6849	20	10	8	8	NUM
ejpam-6849	20	11	]	]	PUNCT
ejpam-6849	20	12	,	,	PUNCT
ejpam-6849	20	13	[	[	X
ejpam-6849	20	14	9	9	NUM
ejpam-6849	20	15	]	]	NUM
ejpam-6849	20	16	)	)	PUNCT
ejpam-6849	20	17	,	,	PUNCT
ejpam-6849	20	18	in	in	ADP
ejpam-6849	20	19	some	some	DET
ejpam-6849	20	20	special	special	ADJ
ejpam-6849	20	21	graphs	graph	NOUN
ejpam-6849	20	22	(	(	PUNCT
ejpam-6849	20	23	see	see	VERB
ejpam-6849	20	24	[	[	X
ejpam-6849	20	25	8	8	NUM
ejpam-6849	20	26	]	]	PUNCT
ejpam-6849	20	27	,	,	PUNCT
ejpam-6849	20	28	[	[	X
ejpam-6849	20	29	1	1	NUM
ejpam-6849	20	30	]	]	NUM
ejpam-6849	20	31	)	)	PUNCT
ejpam-6849	20	32	,	,	PUNCT
ejpam-6849	20	33	in	in	ADP
ejpam-6849	20	34	cubic	cubic	ADJ
ejpam-6849	20	35	graphs	graph	NOUN
ejpam-6849	20	36	and	and	CCONJ
ejpam-6849	20	37	grids	grid	NOUN
ejpam-6849	20	38	(	(	PUNCT
ejpam-6849	20	39	see	see	VERB
ejpam-6849	20	40	[	[	X
ejpam-6849	20	41	10	10	NUM
ejpam-6849	20	42	]	]	PUNCT
ejpam-6849	20	43	,	,	PUNCT
ejpam-6849	20	44	[	[	X
ejpam-6849	20	45	7	7	NUM
ejpam-6849	20	46	]	]	NUM
ejpam-6849	20	47	)	)	PUNCT
ejpam-6849	20	48	,	,	PUNCT
ejpam-6849	20	49	in	in	ADP
ejpam-6849	20	50	connected	connected	ADJ
ejpam-6849	20	51	c5	c5	PROPN
ejpam-6849	20	52	-	-	PUNCT
ejpam-6849	20	53	free	free	ADJ
ejpam-6849	20	54	graphs	graph	NOUN
ejpam-6849	20	55	and	and	CCONJ
ejpam-6849	20	56	connected	connect	VERB
ejpam-6849	20	57	k1,k	k1,k	PROPN
ejpam-6849	20	58	free	free	ADJ
ejpam-6849	20	59	graphs	graph	NOUN
ejpam-6849	20	60	(	(	PUNCT
ejpam-6849	20	61	see	see	VERB
ejpam-6849	20	62	[	[	X
ejpam-6849	20	63	11	11	NUM
ejpam-6849	20	64	]	]	NUM
ejpam-6849	20	65	)	)	PUNCT
ejpam-6849	20	66	.	.	PUNCT
ejpam-6849	21	1	peters	peters	PROPN
ejpam-6849	21	2	also	also	ADV
ejpam-6849	21	3	dealt	deal	VERB
ejpam-6849	21	4	with	with	ADP
ejpam-6849	21	5	the	the	DET
ejpam-6849	21	6	complexity	complexity	NOUN
ejpam-6849	21	7	problems	problem	NOUN
ejpam-6849	21	8	of	of	ADP
ejpam-6849	21	9	the	the	DET
ejpam-6849	21	10	parameter	parameter	NOUN
ejpam-6849	21	11	in	in	ADP
ejpam-6849	21	12	[	[	X
ejpam-6849	21	13	1	1	NUM
ejpam-6849	21	14	]	]	PUNCT
ejpam-6849	21	15	.	.	PUNCT
ejpam-6849	22	1	in	in	ADP
ejpam-6849	22	2	the	the	DET
ejpam-6849	22	3	present	present	ADJ
ejpam-6849	22	4	paper	paper	NOUN
ejpam-6849	22	5	,	,	PUNCT
ejpam-6849	22	6	we	we	PRON
ejpam-6849	22	7	investigate	investigate	VERB
ejpam-6849	22	8	the	the	DET
ejpam-6849	22	9	verter	verter	NOUN
ejpam-6849	22	10	-	-	PUNCT
ejpam-6849	22	11	edge	edge	NOUN
ejpam-6849	22	12	and	and	CCONJ
ejpam-6849	22	13	total	total	ADJ
ejpam-6849	22	14	vertex	vertex	NOUN
ejpam-6849	22	15	-	-	PUNCT
ejpam-6849	22	16	edge	edge	NOUN
ejpam-6849	22	17	domination	domination	NOUN
ejpam-6849	22	18	in	in	ADP
ejpam-6849	22	19	the	the	DET
ejpam-6849	22	20	join	join	NOUN
ejpam-6849	22	21	,	,	PUNCT
ejpam-6849	22	22	corona	corona	PROPN
ejpam-6849	22	23	,	,	PUNCT
ejpam-6849	22	24	lexicographic	lexicographic	ADJ
ejpam-6849	22	25	product	product	NOUN
ejpam-6849	22	26	and	and	CCONJ
ejpam-6849	22	27	edge	edge	NOUN
ejpam-6849	22	28	corona	corona	NOUN
ejpam-6849	22	29	of	of	ADP
ejpam-6849	22	30	graphs	graph	NOUN
ejpam-6849	22	31	.	.	PUNCT
ejpam-6849	23	1	all	all	PRON
ejpam-6849	23	2	throughout	throughout	ADP
ejpam-6849	23	3	this	this	DET
ejpam-6849	23	4	paper	paper	NOUN
ejpam-6849	23	5	,	,	PUNCT
ejpam-6849	23	6	we	we	PRON
ejpam-6849	23	7	consider	consider	VERB
ejpam-6849	23	8	only	only	ADV
ejpam-6849	23	9	graphs	graph	NOUN
ejpam-6849	23	10	which	which	PRON
ejpam-6849	23	11	are	be	AUX
ejpam-6849	23	12	simple	simple	ADJ
ejpam-6849	23	13	,	,	PUNCT
ejpam-6849	23	14	finite	finite	ADJ
ejpam-6849	23	15	and	and	CCONJ
ejpam-6849	23	16	undirected	undirected	ADJ
ejpam-6849	23	17	.	.	PUNCT
ejpam-6849	24	1	given	give	VERB
ejpam-6849	24	2	a	a	DET
ejpam-6849	24	3	graph	graph	NOUN
ejpam-6849	24	4	g	g	NOUN
ejpam-6849	24	5	=	=	PUNCT
ejpam-6849	24	6	(	(	PUNCT
ejpam-6849	24	7	v	v	NOUN
ejpam-6849	24	8	(	(	PUNCT
ejpam-6849	24	9	g	g	NOUN
ejpam-6849	24	10	)	)	PUNCT
ejpam-6849	24	11	,	,	PUNCT
ejpam-6849	24	12	e(g	e(g	PROPN
ejpam-6849	24	13	)	)	PUNCT
ejpam-6849	24	14	)	)	PUNCT
ejpam-6849	24	15	,	,	PUNCT
ejpam-6849	24	16	we	we	PRON
ejpam-6849	24	17	call	call	VERB
ejpam-6849	24	18	v	v	ADP
ejpam-6849	24	19	(	(	PUNCT
ejpam-6849	24	20	g	g	NOUN
ejpam-6849	24	21	)	)	PUNCT
ejpam-6849	24	22	the	the	DET
ejpam-6849	24	23	vertex	vertex	NOUN
ejpam-6849	24	24	set	set	NOUN
ejpam-6849	24	25	of	of	ADP
ejpam-6849	24	26	g	g	PROPN
ejpam-6849	24	27	and	and	CCONJ
ejpam-6849	24	28	e(g	e(g	PROPN
ejpam-6849	24	29	)	)	PUNCT
ejpam-6849	24	30	its	its	PRON
ejpam-6849	24	31	edge	edge	NOUN
ejpam-6849	24	32	set	set	NOUN
ejpam-6849	24	33	.	.	PUNCT
ejpam-6849	25	1	the	the	DET
ejpam-6849	25	2	cardinality	cardinality	PROPN
ejpam-6849	25	3	|v	|v	PROPN
ejpam-6849	25	4	(	(	PUNCT
ejpam-6849	25	5	g)|	g)|	NOUN
ejpam-6849	25	6	of	of	ADP
ejpam-6849	25	7	v	v	NOUN
ejpam-6849	25	8	(	(	PUNCT
ejpam-6849	25	9	g	g	NOUN
ejpam-6849	25	10	)	)	PUNCT
ejpam-6849	25	11	is	be	AUX
ejpam-6849	25	12	the	the	DET
ejpam-6849	25	13	order	order	NOUN
ejpam-6849	25	14	of	of	ADP
ejpam-6849	25	15	g.	g.	PROPN
ejpam-6849	25	16	if	if	SCONJ
ejpam-6849	25	17	e(g	e(g	PROPN
ejpam-6849	25	18	)	)	PUNCT
ejpam-6849	26	1	=	=	PUNCT
ejpam-6849	26	2	∅	∅	NOUN
ejpam-6849	26	3	,	,	PUNCT
ejpam-6849	26	4	then	then	ADV
ejpam-6849	26	5	g	g	PROPN
ejpam-6849	26	6	is	be	AUX
ejpam-6849	26	7	an	an	DET
ejpam-6849	26	8	empty	empty	ADJ
ejpam-6849	26	9	graph	graph	NOUN
ejpam-6849	26	10	.	.	PUNCT
ejpam-6849	27	1	all	all	DET
ejpam-6849	27	2	terminologies	terminology	NOUN
ejpam-6849	27	3	used	use	VERB
ejpam-6849	27	4	here	here	ADV
ejpam-6849	27	5	which	which	PRON
ejpam-6849	27	6	are	be	AUX
ejpam-6849	27	7	not	not	PART
ejpam-6849	27	8	being	be	AUX
ejpam-6849	27	9	defined	define	VERB
ejpam-6849	27	10	are	be	AUX
ejpam-6849	27	11	adapted	adapt	VERB
ejpam-6849	27	12	from	from	ADP
ejpam-6849	27	13	[	[	X
ejpam-6849	27	14	12	12	NUM
ejpam-6849	27	15	]	]	PUNCT
ejpam-6849	27	16	.	.	PUNCT
ejpam-6849	28	1	given	give	VERB
ejpam-6849	28	2	a	a	DET
ejpam-6849	28	3	graph	graph	NOUN
ejpam-6849	28	4	g	g	NOUN
ejpam-6849	28	5	,	,	PUNCT
ejpam-6849	28	6	g	g	PROPN
ejpam-6849	28	7	−	−	PROPN
ejpam-6849	28	8	v	v	NOUN
ejpam-6849	28	9	refers	refer	VERB
ejpam-6849	28	10	to	to	ADP
ejpam-6849	28	11	the	the	DET
ejpam-6849	28	12	resulting	result	VERB
ejpam-6849	28	13	graph	graph	NOUN
ejpam-6849	28	14	after	after	ADP
ejpam-6849	28	15	removing	remove	VERB
ejpam-6849	28	16	vertex	vertex	NOUN
ejpam-6849	28	17	v	v	NOUN
ejpam-6849	28	18	and	and	CCONJ
ejpam-6849	28	19	all	all	DET
ejpam-6849	28	20	incident	incident	NOUN
ejpam-6849	28	21	edges	edge	NOUN
ejpam-6849	28	22	from	from	ADP
ejpam-6849	28	23	g.	g.	PROPN
ejpam-6849	28	24	if	if	SCONJ
ejpam-6849	28	25	the	the	DET
ejpam-6849	28	26	removal	removal	NOUN
ejpam-6849	28	27	of	of	ADP
ejpam-6849	28	28	vertex	vertex	NOUN
ejpam-6849	28	29	v	v	NOUN
ejpam-6849	28	30	from	from	ADP
ejpam-6849	28	31	g	g	PROPN
ejpam-6849	28	32	increases	increase	VERB
ejpam-6849	28	33	the	the	DET
ejpam-6849	28	34	number	number	NOUN
ejpam-6849	28	35	of	of	ADP
ejpam-6849	28	36	components	component	NOUN
ejpam-6849	28	37	,	,	PUNCT
ejpam-6849	28	38	i.e.	i.e.	X
ejpam-6849	28	39	,	,	PUNCT
ejpam-6849	28	40	g	g	PROPN
ejpam-6849	28	41	−	−	PROPN
ejpam-6849	28	42	v	v	NOUN
ejpam-6849	28	43	has	have	VERB
ejpam-6849	28	44	more	more	ADJ
ejpam-6849	28	45	components	component	NOUN
ejpam-6849	28	46	than	than	ADP
ejpam-6849	28	47	g	g	NOUN
ejpam-6849	28	48	,	,	PUNCT
ejpam-6849	28	49	then	then	ADV
ejpam-6849	28	50	v	v	NOUN
ejpam-6849	28	51	is	be	AUX
ejpam-6849	28	52	called	call	VERB
ejpam-6849	28	53	a	a	DET
ejpam-6849	28	54	cut	cut	ADJ
ejpam-6849	28	55	vertex	vertex	NOUN
ejpam-6849	28	56	of	of	ADP
ejpam-6849	28	57	g.	g.	PROPN
ejpam-6849	28	58	for	for	ADP
ejpam-6849	28	59	s	s	PROPN
ejpam-6849	28	60	⊆	⊆	NUM
ejpam-6849	28	61	v	v	NOUN
ejpam-6849	28	62	(	(	PUNCT
ejpam-6849	28	63	g	g	NOUN
ejpam-6849	28	64	)	)	PUNCT
ejpam-6849	28	65	,	,	PUNCT
ejpam-6849	28	66	⟨s⟩	⟨s⟩	PROPN
ejpam-6849	28	67	is	be	AUX
ejpam-6849	28	68	the	the	DET
ejpam-6849	28	69	induced	induced	ADJ
ejpam-6849	28	70	subgraph	subgraph	NOUN
ejpam-6849	28	71	of	of	ADP
ejpam-6849	28	72	g	g	PROPN
ejpam-6849	28	73	with	with	ADP
ejpam-6849	28	74	vertex	vertex	NOUN
ejpam-6849	28	75	set	set	NOUN
ejpam-6849	28	76	s	s	PART
ejpam-6849	28	77	and	and	CCONJ
ejpam-6849	28	78	edge	edge	NOUN
ejpam-6849	28	79	set	set	NOUN
ejpam-6849	28	80	{	{	PUNCT
ejpam-6849	28	81	xy	xy	PROPN
ejpam-6849	28	82	∈	∈	PROPN
ejpam-6849	28	83	e(g	e(g	PROPN
ejpam-6849	28	84	)	)	PUNCT
ejpam-6849	28	85	:	:	PUNCT
ejpam-6849	29	1	x	x	X
ejpam-6849	29	2	,	,	PUNCT
ejpam-6849	29	3	y	y	PROPN
ejpam-6849	29	4	∈	∈	PROPN
ejpam-6849	29	5	s	s	PART
ejpam-6849	29	6	}	}	PUNCT
ejpam-6849	29	7	.	.	PUNCT
ejpam-6849	30	1	let	let	VERB
ejpam-6849	30	2	g	g	NOUN
ejpam-6849	30	3	and	and	CCONJ
ejpam-6849	30	4	h	h	NOUN
ejpam-6849	30	5	be	be	AUX
ejpam-6849	30	6	disjoint	disjoint	NOUN
ejpam-6849	30	7	graphs	graph	NOUN
ejpam-6849	30	8	.	.	PUNCT
ejpam-6849	31	1	the	the	DET
ejpam-6849	31	2	join	join	NOUN
ejpam-6849	31	3	of	of	ADP
ejpam-6849	31	4	g	g	PROPN
ejpam-6849	31	5	and	and	CCONJ
ejpam-6849	31	6	h	h	NOUN
ejpam-6849	31	7	is	be	AUX
ejpam-6849	31	8	the	the	DET
ejpam-6849	31	9	graph	graph	NOUN
ejpam-6849	31	10	g	g	NOUN
ejpam-6849	31	11	+	+	CCONJ
ejpam-6849	31	12	h	h	NOUN
ejpam-6849	31	13	with	with	ADP
ejpam-6849	31	14	vertex	vertex	NOUN
ejpam-6849	31	15	set	set	VERB
ejpam-6849	31	16	v	v	NOUN
ejpam-6849	31	17	(	(	PUNCT
ejpam-6849	31	18	g	g	NOUN
ejpam-6849	31	19	)	)	PUNCT
ejpam-6849	31	20	∪	∪	NOUN
ejpam-6849	31	21	v	v	NOUN
ejpam-6849	31	22	(	(	PUNCT
ejpam-6849	31	23	h	h	NOUN
ejpam-6849	31	24	)	)	PUNCT
ejpam-6849	31	25	and	and	CCONJ
ejpam-6849	31	26	edge	edge	VERB
ejpam-6849	31	27	set	set	VERB
ejpam-6849	31	28	e(g	e(g	NOUN
ejpam-6849	31	29	)	)	PUNCT
ejpam-6849	31	30	∪	∪	ADP
ejpam-6849	31	31	e(h	e(h	PROPN
ejpam-6849	31	32	)	)	PUNCT
ejpam-6849	31	33	∪	∪	NOUN
ejpam-6849	31	34	{	{	PUNCT
ejpam-6849	31	35	uv	uv	NOUN
ejpam-6849	31	36	:	:	PUNCT
ejpam-6849	31	37	u	u	PROPN
ejpam-6849	31	38	∈	∈	PROPN
ejpam-6849	31	39	v	v	ADP
ejpam-6849	31	40	(	(	PUNCT
ejpam-6849	31	41	g	g	NOUN
ejpam-6849	31	42	)	)	PUNCT
ejpam-6849	31	43	,	,	PUNCT
ejpam-6849	31	44	v	v	X
ejpam-6849	31	45	∈	∈	PROPN
ejpam-6849	31	46	v	v	NOUN
ejpam-6849	31	47	(	(	PUNCT
ejpam-6849	31	48	h	h	NOUN
ejpam-6849	31	49	)	)	PUNCT
ejpam-6849	31	50	}	}	PUNCT
ejpam-6849	31	51	.	.	PUNCT
ejpam-6849	32	1	the	the	DET
ejpam-6849	32	2	corona	corona	NOUN
ejpam-6849	32	3	of	of	ADP
ejpam-6849	32	4	g	g	PROPN
ejpam-6849	32	5	and	and	CCONJ
ejpam-6849	32	6	h	h	NOUN
ejpam-6849	32	7	is	be	AUX
ejpam-6849	32	8	the	the	DET
ejpam-6849	32	9	graph	graph	NOUN
ejpam-6849	32	10	g	g	PROPN
ejpam-6849	32	11	◦	◦	NOUN
ejpam-6849	32	12	h	h	NOUN
ejpam-6849	32	13	obtained	obtain	VERB
ejpam-6849	32	14	by	by	ADP
ejpam-6849	32	15	taking	take	VERB
ejpam-6849	32	16	one	one	NUM
ejpam-6849	32	17	copy	copy	NOUN
ejpam-6849	32	18	of	of	ADP
ejpam-6849	32	19	g	g	PROPN
ejpam-6849	32	20	and	and	CCONJ
ejpam-6849	32	21	|v	|v	PROPN
ejpam-6849	32	22	(	(	PUNCT
ejpam-6849	32	23	g)|	g)|	NOUN
ejpam-6849	32	24	copies	copy	NOUN
ejpam-6849	32	25	of	of	ADP
ejpam-6849	32	26	h	h	NOUN
ejpam-6849	32	27	,	,	PUNCT
ejpam-6849	32	28	and	and	CCONJ
ejpam-6849	32	29	then	then	ADV
ejpam-6849	32	30	joining	join	VERB
ejpam-6849	32	31	the	the	DET
ejpam-6849	32	32	ith	ith	PROPN
ejpam-6849	32	33	vertex	vertex	NOUN
ejpam-6849	32	34	of	of	ADP
ejpam-6849	32	35	g	g	NOUN
ejpam-6849	32	36	to	to	ADP
ejpam-6849	32	37	every	every	DET
ejpam-6849	32	38	vertex	vertex	NOUN
ejpam-6849	32	39	in	in	ADP
ejpam-6849	32	40	the	the	DET
ejpam-6849	32	41	ith	ith	PROPN
ejpam-6849	32	42	copy	copy	NOUN
ejpam-6849	32	43	of	of	ADP
ejpam-6849	32	44	h.	h.	PROPN
ejpam-6849	32	45	in	in	ADP
ejpam-6849	32	46	particular	particular	ADJ
ejpam-6849	32	47	,	,	PUNCT
ejpam-6849	32	48	we	we	PRON
ejpam-6849	32	49	call	call	VERB
ejpam-6849	32	50	g	g	PROPN
ejpam-6849	32	51	◦	◦	NOUN
ejpam-6849	32	52	k1	k1	X
ejpam-6849	32	53	the	the	DET
ejpam-6849	32	54	corona	corona	NOUN
ejpam-6849	32	55	of	of	ADP
ejpam-6849	32	56	g	g	PROPN
ejpam-6849	32	57	,	,	PUNCT
ejpam-6849	32	58	and	and	CCONJ
ejpam-6849	32	59	write	write	VERB
ejpam-6849	32	60	cor(g	cor(g	PROPN
ejpam-6849	32	61	)	)	PUNCT
ejpam-6849	33	1	=	=	SYM
ejpam-6849	33	2	g	g	PROPN
ejpam-6849	33	3	◦	◦	NOUN
ejpam-6849	33	4	k1	k1	NOUN
ejpam-6849	33	5	.	.	PUNCT
ejpam-6849	34	1	the	the	DET
ejpam-6849	34	2	edge	edge	NOUN
ejpam-6849	34	3	corona	corona	NOUN
ejpam-6849	34	4	of	of	ADP
ejpam-6849	34	5	g	g	PROPN
ejpam-6849	34	6	and	and	CCONJ
ejpam-6849	34	7	h	h	NOUN
ejpam-6849	34	8	is	be	AUX
ejpam-6849	34	9	the	the	DET
ejpam-6849	34	10	graph	graph	NOUN
ejpam-6849	34	11	g	g	PROPN
ejpam-6849	34	12	⋄	⋄	PROPN
ejpam-6849	34	13	h	h	NOUN
ejpam-6849	34	14	obtained	obtain	VERB
ejpam-6849	34	15	by	by	ADP
ejpam-6849	34	16	taking	take	VERB
ejpam-6849	34	17	one	one	NUM
ejpam-6849	34	18	copy	copy	NOUN
ejpam-6849	34	19	of	of	ADP
ejpam-6849	34	20	g	g	PROPN
ejpam-6849	34	21	and	and	CCONJ
ejpam-6849	34	22	|e(g)|	|e(g)|	ADJ
ejpam-6849	34	23	copies	copy	NOUN
ejpam-6849	34	24	of	of	ADP
ejpam-6849	34	25	h	h	NOUN
ejpam-6849	34	26	and	and	CCONJ
ejpam-6849	34	27	joining	join	VERB
ejpam-6849	34	28	each	each	PRON
ejpam-6849	34	29	of	of	ADP
ejpam-6849	34	30	the	the	DET
ejpam-6849	34	31	end	end	NOUN
ejpam-6849	34	32	vertices	vertice	VERB
ejpam-6849	34	33	u	u	NOUN
ejpam-6849	34	34	and	and	CCONJ
ejpam-6849	34	35	v	v	NOUN
ejpam-6849	34	36	of	of	ADP
ejpam-6849	34	37	each	each	DET
ejpam-6849	34	38	edge	edge	NOUN
ejpam-6849	34	39	uv	uv	NOUN
ejpam-6849	34	40	of	of	ADP
ejpam-6849	34	41	g	g	NOUN
ejpam-6849	34	42	to	to	ADP
ejpam-6849	34	43	every	every	DET
ejpam-6849	34	44	vertex	vertex	NOUN
ejpam-6849	34	45	of	of	ADP
ejpam-6849	34	46	the	the	DET
ejpam-6849	34	47	copy	copy	NOUN
ejpam-6849	34	48	huv	huv	PROPN
ejpam-6849	34	49	of	of	ADP
ejpam-6849	34	50	h.	h.	PROPN
ejpam-6849	34	51	the	the	DET
ejpam-6849	34	52	composition	composition	NOUN
ejpam-6849	34	53	(	(	PUNCT
ejpam-6849	34	54	or	or	CCONJ
ejpam-6849	34	55	lexicographic	lexicographic	ADJ
ejpam-6849	34	56	product	product	NOUN
ejpam-6849	34	57	)	)	PUNCT
ejpam-6849	34	58	of	of	ADP
ejpam-6849	34	59	g	g	PROPN
ejpam-6849	34	60	and	and	CCONJ
ejpam-6849	34	61	h	h	NOUN
ejpam-6849	34	62	is	be	AUX
ejpam-6849	34	63	the	the	DET
ejpam-6849	34	64	graph	graph	NOUN
ejpam-6849	34	65	g[h	g[h	PROPN
ejpam-6849	34	66	]	]	PUNCT
ejpam-6849	34	67	with	with	ADP
ejpam-6849	34	68	v	v	NOUN
ejpam-6849	34	69	(	(	PUNCT
ejpam-6849	34	70	g[h	g[h	PROPN
ejpam-6849	34	71	]	]	PUNCT
ejpam-6849	34	72	)	)	PUNCT
ejpam-6849	35	1	=	=	SYM
ejpam-6849	35	2	v	v	X
ejpam-6849	35	3	(	(	PUNCT
ejpam-6849	35	4	g	g	NOUN
ejpam-6849	35	5	)	)	PUNCT
ejpam-6849	35	6	×	×	NOUN
ejpam-6849	35	7	v	v	NOUN
ejpam-6849	35	8	(	(	PUNCT
ejpam-6849	35	9	h	h	NOUN
ejpam-6849	35	10	)	)	PUNCT
ejpam-6849	35	11	and	and	CCONJ
ejpam-6849	35	12	(	(	PUNCT
ejpam-6849	35	13	u	u	NOUN
ejpam-6849	35	14	,	,	PUNCT
ejpam-6849	35	15	v)(u′	v)(u′	NOUN
ejpam-6849	35	16	,	,	PUNCT
ejpam-6849	35	17	v′	v′	NOUN
ejpam-6849	35	18	)	)	PUNCT
ejpam-6849	35	19	∈	∈	NOUN
ejpam-6849	35	20	e(g[h	e(g[h	NOUN
ejpam-6849	35	21	]	]	PUNCT
ejpam-6849	35	22	)	)	PUNCT
ejpam-6849	35	23	if	if	SCONJ
ejpam-6849	35	24	and	and	CCONJ
ejpam-6849	35	25	only	only	ADV
ejpam-6849	35	26	if	if	SCONJ
ejpam-6849	35	27	either	either	CCONJ
ejpam-6849	35	28	uu′	uu′	PROPN
ejpam-6849	35	29	∈	∈	PROPN
ejpam-6849	35	30	e(g	e(g	PROPN
ejpam-6849	35	31	)	)	PUNCT
ejpam-6849	35	32	or	or	CCONJ
ejpam-6849	35	33	u	u	X
ejpam-6849	35	34	=	=	PUNCT
ejpam-6849	35	35	u′	u′	PROPN
ejpam-6849	35	36	and	and	CCONJ
ejpam-6849	35	37	vv′	vv′	NOUN
ejpam-6849	35	38	∈	∈	PROPN
ejpam-6849	35	39	e(h	e(h	PROPN
ejpam-6849	35	40	)	)	PUNCT
ejpam-6849	35	41	.	.	PUNCT
ejpam-6849	36	1	in	in	ADP
ejpam-6849	36	2	any	any	PRON
ejpam-6849	36	3	of	of	ADP
ejpam-6849	36	4	these	these	DET
ejpam-6849	36	5	graphs	graph	NOUN
ejpam-6849	36	6	,	,	PUNCT
ejpam-6849	36	7	g	g	PROPN
ejpam-6849	36	8	and	and	CCONJ
ejpam-6849	36	9	h	h	NOUN
ejpam-6849	36	10	are	be	AUX
ejpam-6849	36	11	referred	refer	VERB
ejpam-6849	36	12	to	to	ADP
ejpam-6849	36	13	as	as	ADP
ejpam-6849	36	14	their	their	PRON
ejpam-6849	36	15	basic	basic	ADJ
ejpam-6849	36	16	component	component	NOUN
ejpam-6849	36	17	graphs	graph	NOUN
ejpam-6849	36	18	.	.	PUNCT
ejpam-6849	37	1	the	the	DET
ejpam-6849	37	2	complementary	complementary	ADJ
ejpam-6849	37	3	prism	prism	NOUN
ejpam-6849	37	4	gg	gg	PROPN
ejpam-6849	37	5	is	be	AUX
ejpam-6849	37	6	formed	form	VERB
ejpam-6849	37	7	from	from	ADP
ejpam-6849	37	8	g	g	PROPN
ejpam-6849	37	9	and	and	CCONJ
ejpam-6849	37	10	its	its	PRON
ejpam-6849	37	11	complement	complement	NOUN
ejpam-6849	37	12	g	g	NOUN
ejpam-6849	37	13	by	by	ADP
ejpam-6849	37	14	adding	add	VERB
ejpam-6849	37	15	a	a	DET
ejpam-6849	37	16	perfect	perfect	ADJ
ejpam-6849	37	17	matching	matching	NOUN
ejpam-6849	37	18	between	between	ADP
ejpam-6849	37	19	corresponding	corresponding	ADJ
ejpam-6849	37	20	vertices	vertex	NOUN
ejpam-6849	37	21	of	of	ADP
ejpam-6849	37	22	g	g	PROPN
ejpam-6849	37	23	and	and	CCONJ
ejpam-6849	37	24	g.	g.	PROPN
ejpam-6849	37	25	if	if	SCONJ
ejpam-6849	37	26	for	for	ADP
ejpam-6849	37	27	each	each	DET
ejpam-6849	37	28	v	v	NUM
ejpam-6849	37	29	∈	∈	PROPN
ejpam-6849	37	30	v	v	NOUN
ejpam-6849	37	31	(	(	PUNCT
ejpam-6849	37	32	g	g	NOUN
ejpam-6849	37	33	)	)	PUNCT
ejpam-6849	37	34	,	,	PUNCT
ejpam-6849	37	35	v	v	NOUN
ejpam-6849	37	36	is	be	AUX
ejpam-6849	37	37	the	the	DET
ejpam-6849	37	38	vertex	vertex	NOUN
ejpam-6849	37	39	in	in	ADP
ejpam-6849	37	40	g	g	PROPN
ejpam-6849	37	41	corresponding	correspond	VERB
ejpam-6849	37	42	to	to	ADP
ejpam-6849	37	43	v	v	NOUN
ejpam-6849	37	44	,	,	PUNCT
ejpam-6849	37	45	then	then	ADV
ejpam-6849	37	46	gg	gg	PROPN
ejpam-6849	37	47	is	be	AUX
ejpam-6849	37	48	formed	form	VERB
ejpam-6849	37	49	by	by	ADP
ejpam-6849	37	50	adding	add	VERB
ejpam-6849	37	51	the	the	DET
ejpam-6849	37	52	edge	edge	NOUN
ejpam-6849	37	53	vv	vv	NOUN
ejpam-6849	37	54	for	for	ADP
ejpam-6849	37	55	every	every	DET
ejpam-6849	37	56	v	v	NUM
ejpam-6849	37	57	∈	∈	NOUN
ejpam-6849	37	58	v	v	NOUN
ejpam-6849	37	59	(	(	PUNCT
ejpam-6849	37	60	g	g	NOUN
ejpam-6849	37	61	)	)	PUNCT
ejpam-6849	37	62	.	.	PUNCT
ejpam-6849	38	1	for	for	ADP
ejpam-6849	38	2	vertices	vertex	NOUN
ejpam-6849	38	3	u	u	NOUN
ejpam-6849	38	4	and	and	CCONJ
ejpam-6849	38	5	v	v	NOUN
ejpam-6849	38	6	of	of	ADP
ejpam-6849	38	7	a	a	DET
ejpam-6849	38	8	graph	graph	NOUN
ejpam-6849	38	9	g	g	NOUN
ejpam-6849	38	10	,	,	PUNCT
ejpam-6849	38	11	a	a	DET
ejpam-6849	38	12	u	u	NOUN
ejpam-6849	38	13	-	-	NOUN
ejpam-6849	38	14	v	v	ADJ
ejpam-6849	38	15	geodesic	geodesic	NOUN
ejpam-6849	38	16	is	be	AUX
ejpam-6849	38	17	any	any	DET
ejpam-6849	38	18	shortest	short	ADJ
ejpam-6849	38	19	path	path	NOUN
ejpam-6849	38	20	in	in	ADP
ejpam-6849	38	21	g	g	NOUN
ejpam-6849	38	22	joining	join	VERB
ejpam-6849	38	23	u	u	NOUN
ejpam-6849	38	24	and	and	CCONJ
ejpam-6849	38	25	v.	v.	ADP
ejpam-6849	38	26	the	the	DET
ejpam-6849	38	27	length	length	NOUN
ejpam-6849	38	28	of	of	ADP
ejpam-6849	38	29	a	a	DET
ejpam-6849	38	30	u	u	NOUN
ejpam-6849	38	31	-	-	NOUN
ejpam-6849	38	32	v	v	ADJ
ejpam-6849	38	33	geodesic	geodesic	NOUN
ejpam-6849	38	34	is	be	AUX
ejpam-6849	38	35	the	the	DET
ejpam-6849	38	36	distance	distance	NOUN
ejpam-6849	38	37	between	between	ADP
ejpam-6849	38	38	u	u	NOUN
ejpam-6849	38	39	and	and	CCONJ
ejpam-6849	38	40	v	v	NOUN
ejpam-6849	38	41	,	,	PUNCT
ejpam-6849	38	42	and	and	CCONJ
ejpam-6849	38	43	is	be	AUX
ejpam-6849	38	44	denoted	denote	VERB
ejpam-6849	38	45	by	by	ADP
ejpam-6849	38	46	dg(u	dg(u	NOUN
ejpam-6849	38	47	,	,	PUNCT
ejpam-6849	38	48	v	v	NOUN
ejpam-6849	38	49	)	)	PUNCT
ejpam-6849	38	50	.	.	PUNCT
ejpam-6849	39	1	the	the	DET
ejpam-6849	39	2	eccentricity	eccentricity	NOUN
ejpam-6849	39	3	of	of	ADP
ejpam-6849	39	4	v	v	NOUN
ejpam-6849	39	5	refers	refer	VERB
ejpam-6849	39	6	to	to	ADP
ejpam-6849	39	7	the	the	DET
ejpam-6849	39	8	quantity	quantity	NOUN
ejpam-6849	39	9	e(v	e(v	NOUN
ejpam-6849	39	10	)	)	PUNCT
ejpam-6849	40	1	=	=	SYM
ejpam-6849	40	2	max{dg(u	max{dg(u	X
ejpam-6849	40	3	,	,	PUNCT
ejpam-6849	40	4	v	v	NOUN
ejpam-6849	40	5	)	)	PUNCT
ejpam-6849	40	6	:	:	PUNCT
ejpam-6849	41	1	v	v	X
ejpam-6849	41	2	∈	∈	PROPN
ejpam-6849	41	3	v	v	NOUN
ejpam-6849	41	4	(	(	PUNCT
ejpam-6849	41	5	g	g	NOUN
ejpam-6849	41	6	)	)	PUNCT
ejpam-6849	41	7	}	}	PUNCT
ejpam-6849	41	8	.	.	PUNCT
ejpam-6849	42	1	customarily	customarily	ADV
ejpam-6849	42	2	,	,	PUNCT
ejpam-6849	42	3	diam(g	diam(g	NOUN
ejpam-6849	42	4	)	)	PUNCT
ejpam-6849	42	5	=	=	SYM
ejpam-6849	42	6	max{e(v	max{e(v	PROPN
ejpam-6849	42	7	)	)	PUNCT
ejpam-6849	42	8	:	:	PUNCT
ejpam-6849	43	1	v	v	X
ejpam-6849	43	2	∈	∈	PROPN
ejpam-6849	43	3	v	v	NOUN
ejpam-6849	43	4	(	(	PUNCT
ejpam-6849	43	5	g	g	NOUN
ejpam-6849	43	6	)	)	PUNCT
ejpam-6849	43	7	}	}	PUNCT
ejpam-6849	43	8	.	.	PUNCT
ejpam-6849	44	1	in	in	ADP
ejpam-6849	44	2	this	this	DET
ejpam-6849	44	3	paper	paper	NOUN
ejpam-6849	44	4	,	,	PUNCT
ejpam-6849	44	5	we	we	PRON
ejpam-6849	44	6	write	write	VERB
ejpam-6849	44	7	e(g	e(g	NOUN
ejpam-6849	44	8	)	)	PUNCT
ejpam-6849	45	1	=	=	PUNCT
ejpam-6849	45	2	min{e(v	min{e(v	PROPN
ejpam-6849	45	3	)	)	PUNCT
ejpam-6849	45	4	:	:	PUNCT
ejpam-6849	46	1	v	v	X
ejpam-6849	46	2	∈	∈	PROPN
ejpam-6849	46	3	v	v	NOUN
ejpam-6849	46	4	(	(	PUNCT
ejpam-6849	46	5	g	g	NOUN
ejpam-6849	46	6	)	)	PUNCT
ejpam-6849	46	7	}	}	PUNCT
ejpam-6849	46	8	.	.	PUNCT
ejpam-6849	47	1	vertices	vertice	VERB
ejpam-6849	47	2	u	u	NOUN
ejpam-6849	47	3	and	and	CCONJ
ejpam-6849	47	4	v	v	NOUN
ejpam-6849	47	5	of	of	ADP
ejpam-6849	47	6	a	a	DET
ejpam-6849	47	7	graph	graph	NOUN
ejpam-6849	47	8	g	g	NOUN
ejpam-6849	47	9	are	be	AUX
ejpam-6849	47	10	neighbors	neighbor	NOUN
ejpam-6849	47	11	if	if	SCONJ
ejpam-6849	47	12	uv	uv	PROPN
ejpam-6849	47	13	∈	∈	PROPN
ejpam-6849	47	14	e(g	e(g	PROPN
ejpam-6849	47	15	)	)	PUNCT
ejpam-6849	47	16	.	.	PUNCT
ejpam-6849	48	1	the	the	DET
ejpam-6849	48	2	open	open	ADJ
ejpam-6849	48	3	neighborhood	neighborhood	NOUN
ejpam-6849	48	4	of	of	ADP
ejpam-6849	48	5	v	v	NOUN
ejpam-6849	48	6	refers	refer	VERB
ejpam-6849	48	7	to	to	ADP
ejpam-6849	48	8	the	the	DET
ejpam-6849	48	9	set	set	NOUN
ejpam-6849	48	10	ng(v	ng(v	PUNCT
ejpam-6849	48	11	)	)	PUNCT
ejpam-6849	48	12	consisting	consist	VERB
ejpam-6849	48	13	of	of	ADP
ejpam-6849	48	14	all	all	DET
ejpam-6849	48	15	neighbors	neighbor	NOUN
ejpam-6849	48	16	of	of	ADP
ejpam-6849	48	17	v.	v.	ADV
ejpam-6849	48	18	if	if	SCONJ
ejpam-6849	48	19	ng(v	ng(v	NOUN
ejpam-6849	48	20	)	)	PUNCT
ejpam-6849	48	21	=	=	SYM
ejpam-6849	48	22	∅	∅	NOUN
ejpam-6849	48	23	,	,	PUNCT
ejpam-6849	48	24	then	then	ADV
ejpam-6849	48	25	v	v	NOUN
ejpam-6849	48	26	is	be	AUX
ejpam-6849	48	27	an	an	DET
ejpam-6849	48	28	isolated	isolated	ADJ
ejpam-6849	48	29	vertex	vertex	NOUN
ejpam-6849	48	30	.	.	PUNCT
ejpam-6849	49	1	the	the	DET
ejpam-6849	49	2	degree	degree	NOUN
ejpam-6849	49	3	of	of	ADP
ejpam-6849	49	4	v	v	NOUN
ejpam-6849	49	5	refers	refer	VERB
ejpam-6849	49	6	to	to	ADP
ejpam-6849	49	7	the	the	DET
ejpam-6849	49	8	cardinality	cardinality	NOUN
ejpam-6849	49	9	|ng(v)|	|ng(v)|	NOUN
ejpam-6849	49	10	of	of	ADP
ejpam-6849	49	11	the	the	DET
ejpam-6849	49	12	open	open	ADJ
ejpam-6849	49	13	neighborhood	neighborhood	NOUN
ejpam-6849	49	14	of	of	ADP
ejpam-6849	49	15	v	v	NOUN
ejpam-6849	49	16	,	,	PUNCT
ejpam-6849	49	17	and	and	CCONJ
ejpam-6849	49	18	δ(g	δ(g	PUNCT
ejpam-6849	49	19	)	)	PUNCT
ejpam-6849	49	20	is	be	AUX
ejpam-6849	49	21	the	the	DET
ejpam-6849	49	22	minimum	minimum	ADJ
ejpam-6849	49	23	degree	degree	NOUN
ejpam-6849	49	24	of	of	ADP
ejpam-6849	49	25	a	a	DET
ejpam-6849	49	26	vertex	vertex	NOUN
ejpam-6849	49	27	of	of	ADP
ejpam-6849	49	28	g.	g.	PROPN
ejpam-6849	49	29	the	the	DET
ejpam-6849	49	30	closed	close	VERB
ejpam-6849	49	31	neighborhood	neighborhood	NOUN
ejpam-6849	49	32	of	of	ADP
ejpam-6849	49	33	v	v	NOUN
ejpam-6849	49	34	is	be	AUX
ejpam-6849	49	35	the	the	DET
ejpam-6849	49	36	set	set	NOUN
ejpam-6849	49	37	ng[v	ng[v	NOUN
ejpam-6849	49	38	]	]	X
ejpam-6849	49	39	=	=	SYM
ejpam-6849	49	40	ng(v	ng(v	X
ejpam-6849	49	41	)	)	PUNCT
ejpam-6849	49	42	∪	∪	ADP
ejpam-6849	49	43	{	{	PUNCT
ejpam-6849	49	44	v	v	NOUN
ejpam-6849	49	45	}	}	PUNCT
ejpam-6849	49	46	.	.	PUNCT
ejpam-6849	50	1	customarily	customarily	ADV
ejpam-6849	50	2	,	,	PUNCT
ejpam-6849	50	3	for	for	ADP
ejpam-6849	50	4	s	s	PROPN
ejpam-6849	50	5	⊆	⊆	NUM
ejpam-6849	50	6	v	v	NOUN
ejpam-6849	50	7	(	(	PUNCT
ejpam-6849	50	8	g	g	NOUN
ejpam-6849	50	9	)	)	PUNCT
ejpam-6849	50	10	,	,	PUNCT
ejpam-6849	50	11	ng(s	ng(s	NUM
ejpam-6849	50	12	)	)	PUNCT
ejpam-6849	50	13	=	=	SYM
ejpam-6849	50	14	∪v∈sng(v	∪v∈sng(v	PROPN
ejpam-6849	50	15	)	)	PUNCT
ejpam-6849	50	16	and	and	CCONJ
ejpam-6849	50	17	j.	j.	PROPN
ejpam-6849	50	18	tayab	tayab	PROPN
ejpam-6849	50	19	,	,	PUNCT
ejpam-6849	50	20	f.	f.	PROPN
ejpam-6849	50	21	p.	p.	PROPN
ejpam-6849	50	22	jamil	jamil	PROPN
ejpam-6849	50	23	,	,	PUNCT
ejpam-6849	50	24	i.	i.	PROPN
ejpam-6849	50	25	aniversario	aniversario	PROPN
ejpam-6849	50	26	/	/	SYM
ejpam-6849	50	27	eur	eur	PROPN
ejpam-6849	50	28	.	.	PUNCT
ejpam-6849	51	1	j.	j.	PROPN
ejpam-6849	51	2	pure	pure	PROPN
ejpam-6849	51	3	appl	appl	PROPN
ejpam-6849	51	4	.	.	PROPN
ejpam-6849	51	5	math	math	PROPN
ejpam-6849	51	6	,	,	PUNCT
ejpam-6849	51	7	18	18	NUM
ejpam-6849	51	8	(	(	PUNCT
ejpam-6849	51	9	4	4	NUM
ejpam-6849	51	10	)	)	PUNCT
ejpam-6849	51	11	(	(	PUNCT
ejpam-6849	51	12	2025	2025	NUM
ejpam-6849	51	13	)	)	PUNCT
ejpam-6849	51	14	,	,	PUNCT
ejpam-6849	51	15	6849	6849	NUM
ejpam-6849	51	16	3	3	NUM
ejpam-6849	51	17	of	of	ADP
ejpam-6849	51	18	19	19	NUM
ejpam-6849	51	19	ng[s	ng[	NOUN
ejpam-6849	51	20	]	]	PUNCT
ejpam-6849	51	21	=	=	SYM
ejpam-6849	51	22	∪v∈sng[v	∪v∈sng[v	NOUN
ejpam-6849	51	23	]	]	PUNCT
ejpam-6849	51	24	.	.	PUNCT
ejpam-6849	52	1	a	a	DET
ejpam-6849	52	2	subset	subset	NOUN
ejpam-6849	52	3	s	s	VERB
ejpam-6849	52	4	⊆	⊆	NUM
ejpam-6849	52	5	v	v	NOUN
ejpam-6849	52	6	(	(	PUNCT
ejpam-6849	52	7	g	g	NOUN
ejpam-6849	52	8	)	)	PUNCT
ejpam-6849	52	9	is	be	AUX
ejpam-6849	52	10	a	a	DET
ejpam-6849	52	11	dominating	dominating	NOUN
ejpam-6849	52	12	set	set	NOUN
ejpam-6849	52	13	of	of	ADP
ejpam-6849	52	14	g	g	PROPN
ejpam-6849	52	15	if	if	SCONJ
ejpam-6849	52	16	ng[s	ng[	NOUN
ejpam-6849	52	17	]	]	PUNCT
ejpam-6849	52	18	=	=	SYM
ejpam-6849	52	19	v	v	NOUN
ejpam-6849	52	20	(	(	PUNCT
ejpam-6849	52	21	g	g	NOUN
ejpam-6849	52	22	)	)	PUNCT
ejpam-6849	52	23	.	.	PUNCT
ejpam-6849	53	1	a	a	DET
ejpam-6849	53	2	dominating	dominating	NOUN
ejpam-6849	53	3	set	set	NOUN
ejpam-6849	53	4	s	s	NOUN
ejpam-6849	53	5	of	of	ADP
ejpam-6849	53	6	g	g	PROPN
ejpam-6849	53	7	is	be	AUX
ejpam-6849	53	8	a	a	DET
ejpam-6849	53	9	total	total	ADJ
ejpam-6849	53	10	dominating	dominating	NOUN
ejpam-6849	53	11	set	set	NOUN
ejpam-6849	53	12	if	if	SCONJ
ejpam-6849	53	13	s	s	VERB
ejpam-6849	53	14	⊆	⊆	NUM
ejpam-6849	53	15	ng(s	ng(s	NUM
ejpam-6849	53	16	)	)	PUNCT
ejpam-6849	53	17	.	.	PUNCT
ejpam-6849	54	1	the	the	DET
ejpam-6849	54	2	minimum	minimum	PROPN
ejpam-6849	54	3	cardinality	cardinality	PROPN
ejpam-6849	54	4	γ(g	γ(g	PROPN
ejpam-6849	54	5	)	)	PUNCT
ejpam-6849	54	6	of	of	ADP
ejpam-6849	54	7	a	a	DET
ejpam-6849	54	8	dominating	dominating	NOUN
ejpam-6849	54	9	set	set	NOUN
ejpam-6849	54	10	of	of	ADP
ejpam-6849	54	11	g	g	PROPN
ejpam-6849	54	12	is	be	AUX
ejpam-6849	54	13	the	the	DET
ejpam-6849	54	14	domination	domination	NOUN
ejpam-6849	54	15	number	number	NOUN
ejpam-6849	54	16	of	of	ADP
ejpam-6849	54	17	g.	g.	PROPN
ejpam-6849	54	18	the	the	DET
ejpam-6849	54	19	minimum	minimum	ADJ
ejpam-6849	54	20	cardinality	cardinality	NOUN
ejpam-6849	54	21	γt(g	γt(g	PUNCT
ejpam-6849	54	22	)	)	PUNCT
ejpam-6849	54	23	of	of	ADP
ejpam-6849	54	24	a	a	DET
ejpam-6849	54	25	total	total	ADJ
ejpam-6849	54	26	dominating	dominating	NOUN
ejpam-6849	54	27	set	set	NOUN
ejpam-6849	54	28	is	be	AUX
ejpam-6849	54	29	the	the	DET
ejpam-6849	54	30	total	total	ADJ
ejpam-6849	54	31	domination	domination	NOUN
ejpam-6849	54	32	number	number	NOUN
ejpam-6849	54	33	of	of	ADP
ejpam-6849	54	34	g.	g.	PROPN
ejpam-6849	54	35	a	a	DET
ejpam-6849	54	36	dominating	dominating	NOUN
ejpam-6849	54	37	(	(	PUNCT
ejpam-6849	54	38	resp	resp	NOUN
ejpam-6849	54	39	.	.	PUNCT
ejpam-6849	55	1	total	total	ADJ
ejpam-6849	55	2	dominating	dominating	NOUN
ejpam-6849	55	3	)	)	PUNCT
ejpam-6849	55	4	set	set	NOUN
ejpam-6849	55	5	of	of	ADP
ejpam-6849	55	6	cardinality	cardinality	PROPN
ejpam-6849	55	7	γ(g	γ(g	PROPN
ejpam-6849	55	8	)	)	PUNCT
ejpam-6849	55	9	(	(	PUNCT
ejpam-6849	55	10	resp	resp	NOUN
ejpam-6849	55	11	.	.	PUNCT
ejpam-6849	55	12	γt(g	γt(g	PUNCT
ejpam-6849	55	13	)	)	PUNCT
ejpam-6849	55	14	)	)	PUNCT
ejpam-6849	55	15	is	be	AUX
ejpam-6849	55	16	called	call	VERB
ejpam-6849	55	17	a	a	DET
ejpam-6849	55	18	γ	γ	NOUN
ejpam-6849	55	19	-	-	PUNCT
ejpam-6849	55	20	set	set	ADJ
ejpam-6849	55	21	(	(	PUNCT
ejpam-6849	55	22	resp	resp	NOUN
ejpam-6849	55	23	.	.	PUNCT
ejpam-6849	56	1	γt	γt	NOUN
ejpam-6849	56	2	-	-	PUNCT
ejpam-6849	56	3	set	set	NOUN
ejpam-6849	56	4	of	of	ADP
ejpam-6849	56	5	g.	g.	PROPN
ejpam-6849	56	6	the	the	DET
ejpam-6849	56	7	reader	reader	NOUN
ejpam-6849	56	8	is	be	AUX
ejpam-6849	56	9	referred	refer	VERB
ejpam-6849	56	10	to	to	ADP
ejpam-6849	56	11	[	[	X
ejpam-6849	56	12	13	13	NUM
ejpam-6849	56	13	]	]	PUNCT
ejpam-6849	56	14	,	,	PUNCT
ejpam-6849	57	1	[	[	X
ejpam-6849	57	2	14	14	NUM
ejpam-6849	57	3	]	]	PUNCT
ejpam-6849	57	4	,	,	PUNCT
ejpam-6849	57	5	[	[	X
ejpam-6849	57	6	15	15	NUM
ejpam-6849	57	7	]	]	PUNCT
ejpam-6849	57	8	,	,	PUNCT
ejpam-6849	57	9	[	[	X
ejpam-6849	57	10	16	16	NUM
ejpam-6849	57	11	]	]	PUNCT
ejpam-6849	57	12	,	,	PUNCT
ejpam-6849	58	1	[	[	X
ejpam-6849	58	2	17	17	NUM
ejpam-6849	58	3	]	]	PUNCT
ejpam-6849	58	4	,	,	PUNCT
ejpam-6849	58	5	and	and	CCONJ
ejpam-6849	58	6	[	[	X
ejpam-6849	58	7	18	18	NUM
ejpam-6849	58	8	]	]	PUNCT
ejpam-6849	58	9	for	for	ADP
ejpam-6849	58	10	the	the	DET
ejpam-6849	58	11	history	history	NOUN
ejpam-6849	58	12	,	,	PUNCT
ejpam-6849	58	13	fundamental	fundamental	ADJ
ejpam-6849	58	14	concepts	concept	NOUN
ejpam-6849	58	15	and	and	CCONJ
ejpam-6849	58	16	recent	recent	ADJ
ejpam-6849	58	17	developments	development	NOUN
ejpam-6849	58	18	of	of	ADP
ejpam-6849	58	19	domination	domination	NOUN
ejpam-6849	58	20	in	in	ADP
ejpam-6849	58	21	graphs	graph	NOUN
ejpam-6849	58	22	as	as	ADV
ejpam-6849	58	23	well	well	ADV
ejpam-6849	58	24	as	as	ADP
ejpam-6849	58	25	its	its	PRON
ejpam-6849	58	26	various	various	ADJ
ejpam-6849	58	27	applications	application	NOUN
ejpam-6849	58	28	.	.	PUNCT
ejpam-6849	59	1	vertex	vertex	NOUN
ejpam-6849	59	2	u	u	PROPN
ejpam-6849	59	3	of	of	ADP
ejpam-6849	59	4	g	g	PROPN
ejpam-6849	59	5	is	be	AUX
ejpam-6849	59	6	said	say	VERB
ejpam-6849	59	7	to	to	ADP
ejpam-6849	59	8	vertex	vertex	NOUN
ejpam-6849	59	9	-	-	PUNCT
ejpam-6849	59	10	edge	edge	NOUN
ejpam-6849	59	11	dominate	dominate	NOUN
ejpam-6849	59	12	(	(	PUNCT
ejpam-6849	59	13	or	or	CCONJ
ejpam-6849	59	14	ve	ve	VERB
ejpam-6849	59	15	-	-	PUNCT
ejpam-6849	59	16	dominate	dominate	VERB
ejpam-6849	59	17	)	)	PUNCT
ejpam-6849	59	18	edge	edge	NOUN
ejpam-6849	59	19	xy	xy	PROPN
ejpam-6849	59	20	∈	∈	PROPN
ejpam-6849	59	21	e(g	e(g	PROPN
ejpam-6849	59	22	)	)	PUNCT
ejpam-6849	59	23	if	if	SCONJ
ejpam-6849	59	24	one	one	NUM
ejpam-6849	59	25	of	of	ADP
ejpam-6849	59	26	the	the	DET
ejpam-6849	59	27	following	follow	VERB
ejpam-6849	59	28	holds	hold	VERB
ejpam-6849	59	29	:	:	PUNCT
ejpam-6849	59	30	•	•	NUM
ejpam-6849	59	31	u	u	NOUN
ejpam-6849	59	32	=	=	NOUN
ejpam-6849	59	33	x	x	X
ejpam-6849	59	34	or	or	CCONJ
ejpam-6849	59	35	u	u	X
ejpam-6849	59	36	=	=	PROPN
ejpam-6849	59	37	y	y	PROPN
ejpam-6849	59	38	;	;	PUNCT
ejpam-6849	59	39	•	•	NUM
ejpam-6849	59	40	ux	ux	PROPN
ejpam-6849	59	41	∈	∈	PROPN
ejpam-6849	59	42	e(g	e(g	PROPN
ejpam-6849	59	43	)	)	PUNCT
ejpam-6849	59	44	or	or	CCONJ
ejpam-6849	59	45	uy	uy	PROPN
ejpam-6849	59	46	∈	∈	PROPN
ejpam-6849	59	47	e(g	e(g	PROPN
ejpam-6849	59	48	)	)	PUNCT
ejpam-6849	59	49	.	.	PUNCT
ejpam-6849	60	1	a	a	DET
ejpam-6849	60	2	subset	subset	NOUN
ejpam-6849	60	3	s	s	VERB
ejpam-6849	60	4	⊆	⊆	NUM
ejpam-6849	60	5	v	v	NOUN
ejpam-6849	60	6	(	(	PUNCT
ejpam-6849	60	7	g	g	NOUN
ejpam-6849	60	8	)	)	PUNCT
ejpam-6849	60	9	is	be	AUX
ejpam-6849	60	10	a	a	DET
ejpam-6849	60	11	ve	ve	NOUN
ejpam-6849	60	12	-	-	PUNCT
ejpam-6849	60	13	dominating	dominate	VERB
ejpam-6849	60	14	set	set	NOUN
ejpam-6849	60	15	of	of	ADP
ejpam-6849	60	16	g	g	PROPN
ejpam-6849	60	17	if	if	SCONJ
ejpam-6849	60	18	for	for	ADP
ejpam-6849	60	19	every	every	DET
ejpam-6849	60	20	xy	xy	PROPN
ejpam-6849	60	21	∈	∈	PROPN
ejpam-6849	60	22	e(g	e(g	PROPN
ejpam-6849	60	23	)	)	PUNCT
ejpam-6849	60	24	,	,	PUNCT
ejpam-6849	60	25	there	there	PRON
ejpam-6849	60	26	exists	exist	VERB
ejpam-6849	60	27	u	u	PROPN
ejpam-6849	60	28	∈	∈	PROPN
ejpam-6849	60	29	s	s	X
ejpam-6849	60	30	for	for	ADP
ejpam-6849	60	31	which	which	PRON
ejpam-6849	60	32	u	u	NOUN
ejpam-6849	60	33	ve	ve	VERB
ejpam-6849	60	34	-	-	VERB
ejpam-6849	60	35	dominates	dominate	VERB
ejpam-6849	60	36	xy	xy	PROPN
ejpam-6849	60	37	.	.	PUNCT
ejpam-6849	61	1	a	a	DET
ejpam-6849	61	2	ve	ve	NOUN
ejpam-6849	61	3	-	-	PUNCT
ejpam-6849	61	4	dominating	dominate	VERB
ejpam-6849	61	5	set	set	NOUN
ejpam-6849	61	6	is	be	AUX
ejpam-6849	61	7	a	a	DET
ejpam-6849	61	8	total	total	ADJ
ejpam-6849	61	9	ve	ve	NOUN
ejpam-6849	61	10	-	-	PUNCT
ejpam-6849	61	11	dominating	dominating	NOUN
ejpam-6849	61	12	set	set	NOUN
ejpam-6849	61	13	if	if	SCONJ
ejpam-6849	61	14	,	,	PUNCT
ejpam-6849	61	15	in	in	ADP
ejpam-6849	61	16	addition	addition	NOUN
ejpam-6849	61	17	,	,	PUNCT
ejpam-6849	61	18	⟨s⟩	⟨s⟩	PROPN
ejpam-6849	61	19	has	have	VERB
ejpam-6849	61	20	no	no	DET
ejpam-6849	61	21	isolated	isolated	ADJ
ejpam-6849	61	22	vertex	vertex	NOUN
ejpam-6849	61	23	.	.	PUNCT
ejpam-6849	62	1	the	the	DET
ejpam-6849	62	2	minimum	minimum	ADJ
ejpam-6849	62	3	cardinality	cardinality	NOUN
ejpam-6849	62	4	of	of	ADP
ejpam-6849	62	5	a	a	DET
ejpam-6849	62	6	ve	ve	NOUN
ejpam-6849	62	7	-	-	PUNCT
ejpam-6849	62	8	dominating	dominate	VERB
ejpam-6849	62	9	set	set	NOUN
ejpam-6849	62	10	is	be	AUX
ejpam-6849	62	11	the	the	DET
ejpam-6849	62	12	ve	ve	NOUN
ejpam-6849	62	13	-	-	PUNCT
ejpam-6849	62	14	domination	domination	NOUN
ejpam-6849	62	15	number	number	NOUN
ejpam-6849	62	16	of	of	ADP
ejpam-6849	62	17	g.	g.	PROPN
ejpam-6849	62	18	similarly	similarly	ADV
ejpam-6849	62	19	,	,	PUNCT
ejpam-6849	62	20	the	the	DET
ejpam-6849	62	21	minimum	minimum	ADJ
ejpam-6849	62	22	cardinality	cardinality	NOUN
ejpam-6849	62	23	of	of	ADP
ejpam-6849	62	24	a	a	DET
ejpam-6849	62	25	total	total	ADJ
ejpam-6849	62	26	ve	ve	NOUN
ejpam-6849	62	27	-	-	PUNCT
ejpam-6849	62	28	dominating	dominate	VERB
ejpam-6849	62	29	set	set	NOUN
ejpam-6849	62	30	is	be	AUX
ejpam-6849	62	31	the	the	DET
ejpam-6849	62	32	total	total	ADJ
ejpam-6849	62	33	ve	ve	NOUN
ejpam-6849	62	34	-	-	PUNCT
ejpam-6849	62	35	domination	domination	NOUN
ejpam-6849	62	36	number	number	NOUN
ejpam-6849	62	37	of	of	ADP
ejpam-6849	62	38	g.	g.	NOUN
ejpam-6849	62	39	we	we	PRON
ejpam-6849	62	40	use	use	VERB
ejpam-6849	62	41	the	the	DET
ejpam-6849	62	42	symbols	symbol	NOUN
ejpam-6849	62	43	γve(g	γve(g	PROPN
ejpam-6849	62	44	)	)	PUNCT
ejpam-6849	62	45	and	and	CCONJ
ejpam-6849	62	46	γt	γt	PROPN
ejpam-6849	62	47	ve(g	ve(g	PROPN
ejpam-6849	62	48	)	)	PUNCT
ejpam-6849	62	49	to	to	PART
ejpam-6849	62	50	refer	refer	VERB
ejpam-6849	62	51	to	to	ADP
ejpam-6849	62	52	the	the	DET
ejpam-6849	62	53	ve	ve	NOUN
ejpam-6849	62	54	-	-	PUNCT
ejpam-6849	62	55	domination	domination	NOUN
ejpam-6849	62	56	number	number	NOUN
ejpam-6849	62	57	and	and	CCONJ
ejpam-6849	62	58	total	total	ADJ
ejpam-6849	62	59	ve	ve	NOUN
ejpam-6849	62	60	-	-	PUNCT
ejpam-6849	62	61	domination	domination	NOUN
ejpam-6849	62	62	number	number	NOUN
ejpam-6849	62	63	of	of	ADP
ejpam-6849	62	64	g	g	NOUN
ejpam-6849	62	65	,	,	PUNCT
ejpam-6849	62	66	respectively	respectively	ADV
ejpam-6849	62	67	.	.	PUNCT
ejpam-6849	63	1	we	we	PRON
ejpam-6849	63	2	also	also	ADV
ejpam-6849	63	3	use	use	VERB
ejpam-6849	63	4	the	the	DET
ejpam-6849	63	5	terms	term	NOUN
ejpam-6849	63	6	γve	γve	ADV
ejpam-6849	63	7	-	-	PUNCT
ejpam-6849	63	8	set	set	ADJ
ejpam-6849	63	9	(	(	PUNCT
ejpam-6849	63	10	resp	resp	NOUN
ejpam-6849	63	11	.	.	PUNCT
ejpam-6849	64	1	γt	γt	ADP
ejpam-6849	64	2	ve	ve	PROPN
ejpam-6849	64	3	-	-	PUNCT
ejpam-6849	64	4	set	set	NOUN
ejpam-6849	64	5	)	)	PUNCT
ejpam-6849	65	1	to	to	PART
ejpam-6849	65	2	refer	refer	VERB
ejpam-6849	65	3	to	to	ADP
ejpam-6849	65	4	any	any	DET
ejpam-6849	65	5	ve	ve	NOUN
ejpam-6849	65	6	-	-	PUNCT
ejpam-6849	65	7	dominating	dominate	VERB
ejpam-6849	65	8	(	(	PUNCT
ejpam-6849	65	9	resp	resp	NOUN
ejpam-6849	65	10	.	.	PUNCT
ejpam-6849	66	1	total	total	ADJ
ejpam-6849	66	2	ve	ve	PROPN
ejpam-6849	66	3	-	-	PUNCT
ejpam-6849	66	4	dominating	dominating	NOUN
ejpam-6849	66	5	)	)	PUNCT
ejpam-6849	66	6	set	set	NOUN
ejpam-6849	66	7	of	of	ADP
ejpam-6849	66	8	cardinality	cardinality	PROPN
ejpam-6849	66	9	γve(g	γve(g	PROPN
ejpam-6849	66	10	)	)	PUNCT
ejpam-6849	66	11	(	(	PUNCT
ejpam-6849	66	12	resp	resp	NOUN
ejpam-6849	66	13	.	.	PUNCT
ejpam-6849	67	1	γt	γt	PROPN
ejpam-6849	67	2	ve(g	ve(g	PROPN
ejpam-6849	67	3	)	)	PUNCT
ejpam-6849	67	4	)	)	PUNCT
ejpam-6849	67	5	.	.	PUNCT
ejpam-6849	68	1	for	for	ADP
ejpam-6849	68	2	every	every	DET
ejpam-6849	68	3	nontrivial	nontrivial	ADJ
ejpam-6849	68	4	connected	connect	VERB
ejpam-6849	68	5	graph	graph	NOUN
ejpam-6849	68	6	g	g	PROPN
ejpam-6849	68	7	,	,	PUNCT
ejpam-6849	68	8	γve(g	γve(g	PROPN
ejpam-6849	68	9	)	)	PUNCT
ejpam-6849	68	10	≤	≤	NUM
ejpam-6849	68	11	γt	γt	ADP
ejpam-6849	68	12	ve(g	ve(g	PROPN
ejpam-6849	68	13	)	)	PUNCT
ejpam-6849	68	14	≤	≤	NOUN
ejpam-6849	68	15	2γve(g)[11	2γve(g)[11	NUM
ejpam-6849	68	16	]	]	SYM
ejpam-6849	68	17	.	.	PUNCT
ejpam-6849	69	1	2	2	X
ejpam-6849	69	2	.	.	X
ejpam-6849	69	3	preliminary	preliminary	ADJ
ejpam-6849	69	4	results	result	NOUN
ejpam-6849	69	5	the	the	DET
ejpam-6849	69	6	following	follow	VERB
ejpam-6849	69	7	formalizes	formalize	VERB
ejpam-6849	69	8	the	the	DET
ejpam-6849	69	9	alternative	alternative	ADJ
ejpam-6849	69	10	definition	definition	NOUN
ejpam-6849	69	11	given	give	VERB
ejpam-6849	69	12	by	by	ADP
ejpam-6849	69	13	w.	w.	PROPN
ejpam-6849	69	14	klostermeyer	klostermeyer	PROPN
ejpam-6849	69	15	et	et	PROPN
ejpam-6849	69	16	al	al	PROPN
ejpam-6849	69	17	.	.	PUNCT
ejpam-6849	70	1	in	in	ADP
ejpam-6849	70	2	[	[	X
ejpam-6849	70	3	10	10	NUM
ejpam-6849	70	4	]	]	PUNCT
ejpam-6849	70	5	.	.	PUNCT
ejpam-6849	71	1	proposition	proposition	NOUN
ejpam-6849	71	2	1	1	NUM
ejpam-6849	71	3	.	.	PUNCT
ejpam-6849	71	4	s	s	PART
ejpam-6849	71	5	⊆	⊆	NUM
ejpam-6849	71	6	v	v	NOUN
ejpam-6849	71	7	(	(	PUNCT
ejpam-6849	71	8	g	g	NOUN
ejpam-6849	71	9	)	)	PUNCT
ejpam-6849	71	10	is	be	AUX
ejpam-6849	71	11	a	a	DET
ejpam-6849	71	12	ve	ve	NOUN
ejpam-6849	71	13	-	-	PUNCT
ejpam-6849	71	14	dominating	dominate	VERB
ejpam-6849	71	15	set	set	NOUN
ejpam-6849	71	16	of	of	ADP
ejpam-6849	71	17	g	g	PROPN
ejpam-6849	71	18	if	if	SCONJ
ejpam-6849	72	1	and	and	CCONJ
ejpam-6849	72	2	only	only	ADV
ejpam-6849	72	3	if	if	SCONJ
ejpam-6849	72	4	v	v	INTJ
ejpam-6849	72	5	(	(	PUNCT
ejpam-6849	72	6	g	g	NOUN
ejpam-6849	72	7	)	)	PUNCT
ejpam-6849	72	8	\	\	NOUN
ejpam-6849	72	9	ng(s	ng(s	NUM
ejpam-6849	72	10	)	)	PUNCT
ejpam-6849	72	11	is	be	AUX
ejpam-6849	72	12	either	either	CCONJ
ejpam-6849	72	13	empty	empty	ADJ
ejpam-6849	72	14	or	or	CCONJ
ejpam-6849	72	15	a	a	DET
ejpam-6849	72	16	nonempty	nonempty	ADJ
ejpam-6849	72	17	independent	independent	ADJ
ejpam-6849	72	18	subset	subset	NOUN
ejpam-6849	72	19	of	of	ADP
ejpam-6849	72	20	v	v	NOUN
ejpam-6849	72	21	(	(	PUNCT
ejpam-6849	72	22	g	g	NOUN
ejpam-6849	72	23	)	)	PUNCT
ejpam-6849	72	24	.	.	PUNCT
ejpam-6849	73	1	proof	proof	NOUN
ejpam-6849	73	2	.	.	PUNCT
ejpam-6849	74	1	first	first	ADV
ejpam-6849	74	2	,	,	PUNCT
ejpam-6849	74	3	let	let	VERB
ejpam-6849	74	4	s	s	PRON
ejpam-6849	74	5	be	be	AUX
ejpam-6849	74	6	a	a	DET
ejpam-6849	74	7	ve	ve	NOUN
ejpam-6849	74	8	-	-	PUNCT
ejpam-6849	74	9	dominating	dominate	VERB
ejpam-6849	74	10	set	set	NOUN
ejpam-6849	74	11	of	of	ADP
ejpam-6849	74	12	g.	g.	PROPN
ejpam-6849	75	1	if	if	SCONJ
ejpam-6849	75	2	v	v	X
ejpam-6849	75	3	(	(	PUNCT
ejpam-6849	75	4	g	g	NOUN
ejpam-6849	75	5	)	)	PUNCT
ejpam-6849	75	6	\	\	NOUN
ejpam-6849	75	7	ng(s	ng(s	NUM
ejpam-6849	75	8	)	)	PUNCT
ejpam-6849	75	9	=	=	NOUN
ejpam-6849	75	10	∅	∅	NOUN
ejpam-6849	75	11	,	,	PUNCT
ejpam-6849	75	12	then	then	ADV
ejpam-6849	75	13	we	we	PRON
ejpam-6849	75	14	are	be	AUX
ejpam-6849	75	15	done	do	VERB
ejpam-6849	75	16	.	.	PUNCT
ejpam-6849	76	1	suppose	suppose	VERB
ejpam-6849	76	2	that	that	SCONJ
ejpam-6849	76	3	v	v	INTJ
ejpam-6849	76	4	(	(	PUNCT
ejpam-6849	76	5	g	g	NOUN
ejpam-6849	76	6	)	)	PUNCT
ejpam-6849	76	7	\	\	NOUN
ejpam-6849	76	8	ng(s	ng(s	NUM
ejpam-6849	76	9	)	)	PUNCT
ejpam-6849	76	10	̸=	̸=	PROPN
ejpam-6849	76	11	∅	∅	NOUN
ejpam-6849	76	12	and	and	CCONJ
ejpam-6849	76	13	let	let	VERB
ejpam-6849	76	14	x	x	PRON
ejpam-6849	76	15	,	,	PUNCT
ejpam-6849	76	16	y	y	PROPN
ejpam-6849	76	17	∈	∈	PROPN
ejpam-6849	76	18	v	v	ADP
ejpam-6849	76	19	(	(	PUNCT
ejpam-6849	76	20	g	g	NOUN
ejpam-6849	76	21	)	)	PUNCT
ejpam-6849	76	22	\	\	NOUN
ejpam-6849	76	23	ng(s	ng(s	NUM
ejpam-6849	76	24	)	)	PUNCT
ejpam-6849	76	25	.	.	PUNCT
ejpam-6849	77	1	suppose	suppose	VERB
ejpam-6849	77	2	that	that	SCONJ
ejpam-6849	77	3	xy	xy	PROPN
ejpam-6849	77	4	∈	∈	PROPN
ejpam-6849	77	5	e(g	e(g	PROPN
ejpam-6849	77	6	)	)	PUNCT
ejpam-6849	77	7	.	.	PUNCT
ejpam-6849	78	1	then	then	ADV
ejpam-6849	78	2	there	there	PRON
ejpam-6849	78	3	exists	exist	VERB
ejpam-6849	78	4	u	u	PROPN
ejpam-6849	78	5	∈	∈	PROPN
ejpam-6849	78	6	s	s	X
ejpam-6849	78	7	for	for	ADP
ejpam-6849	78	8	which	which	PRON
ejpam-6849	78	9	u	u	NOUN
ejpam-6849	78	10	ve	ve	VERB
ejpam-6849	78	11	-	-	VERB
ejpam-6849	78	12	dominates	dominate	VERB
ejpam-6849	78	13	xy	xy	NOUN
ejpam-6849	78	14	.	.	PUNCT
ejpam-6849	79	1	if	if	SCONJ
ejpam-6849	79	2	u	u	PRON
ejpam-6849	79	3	=	=	NOUN
ejpam-6849	79	4	x	x	PROPN
ejpam-6849	79	5	(	(	PUNCT
ejpam-6849	79	6	resp	resp	NOUN
ejpam-6849	79	7	.	.	PUNCT
ejpam-6849	80	1	u	u	NOUN
ejpam-6849	80	2	=	=	PROPN
ejpam-6849	80	3	y	y	PROPN
ejpam-6849	80	4	)	)	PUNCT
ejpam-6849	80	5	,	,	PUNCT
ejpam-6849	80	6	then	then	ADV
ejpam-6849	80	7	y	y	PROPN
ejpam-6849	80	8	∈	∈	PROPN
ejpam-6849	80	9	ng(s	ng(s	PRON
ejpam-6849	80	10	)	)	PUNCT
ejpam-6849	80	11	(	(	PUNCT
ejpam-6849	80	12	resp	resp	NOUN
ejpam-6849	80	13	.	.	PUNCT
ejpam-6849	81	1	x	x	PUNCT
ejpam-6849	81	2	∈	∈	PROPN
ejpam-6849	81	3	ng(s	ng(s	PRON
ejpam-6849	81	4	)	)	PUNCT
ejpam-6849	81	5	)	)	PUNCT
ejpam-6849	81	6	,	,	PUNCT
ejpam-6849	81	7	a	a	DET
ejpam-6849	81	8	contradiction	contradiction	NOUN
ejpam-6849	81	9	.	.	PUNCT
ejpam-6849	82	1	however	however	ADV
ejpam-6849	82	2	,	,	PUNCT
ejpam-6849	82	3	if	if	SCONJ
ejpam-6849	82	4	ux	ux	PROPN
ejpam-6849	82	5	∈	∈	PROPN
ejpam-6849	82	6	e(g	e(g	PROPN
ejpam-6849	82	7	)	)	PUNCT
ejpam-6849	82	8	(	(	PUNCT
ejpam-6849	82	9	resp	resp	NOUN
ejpam-6849	82	10	.	.	PUNCT
ejpam-6849	83	1	uy	uy	PROPN
ejpam-6849	83	2	∈	∈	PROPN
ejpam-6849	83	3	e(g	e(g	PROPN
ejpam-6849	83	4	)	)	PUNCT
ejpam-6849	83	5	)	)	PUNCT
ejpam-6849	83	6	,	,	PUNCT
ejpam-6849	83	7	then	then	ADV
ejpam-6849	83	8	x	x	X
ejpam-6849	83	9	∈	∈	PROPN
ejpam-6849	83	10	ng(s	ng(s	PRON
ejpam-6849	83	11	)	)	PUNCT
ejpam-6849	83	12	(	(	PUNCT
ejpam-6849	83	13	resp	resp	NOUN
ejpam-6849	83	14	.	.	PUNCT
ejpam-6849	84	1	y	y	PROPN
ejpam-6849	84	2	∈	∈	PROPN
ejpam-6849	84	3	ng(s	ng(s	PRON
ejpam-6849	84	4	)	)	PUNCT
ejpam-6849	84	5	)	)	PUNCT
ejpam-6849	84	6	,	,	PUNCT
ejpam-6849	84	7	a	a	DET
ejpam-6849	84	8	contradiction	contradiction	NOUN
ejpam-6849	84	9	.	.	PUNCT
ejpam-6849	85	1	since	since	SCONJ
ejpam-6849	85	2	x	x	PROPN
ejpam-6849	85	3	and	and	CCONJ
ejpam-6849	85	4	y	y	PROPN
ejpam-6849	85	5	are	be	AUX
ejpam-6849	85	6	arbitrary	arbitrary	ADJ
ejpam-6849	85	7	,	,	PUNCT
ejpam-6849	85	8	v	v	ADJ
ejpam-6849	85	9	(	(	PUNCT
ejpam-6849	85	10	g	g	NOUN
ejpam-6849	85	11	)	)	PUNCT
ejpam-6849	85	12	\	\	NOUN
ejpam-6849	85	13	ng(s	ng(s	NUM
ejpam-6849	85	14	)	)	PUNCT
ejpam-6849	85	15	is	be	AUX
ejpam-6849	85	16	an	an	DET
ejpam-6849	85	17	independent	independent	ADJ
ejpam-6849	85	18	set	set	NOUN
ejpam-6849	85	19	.	.	PUNCT
ejpam-6849	86	1	conversely	conversely	ADV
ejpam-6849	86	2	,	,	PUNCT
ejpam-6849	86	3	first	first	ADV
ejpam-6849	86	4	suppose	suppose	VERB
ejpam-6849	86	5	that	that	SCONJ
ejpam-6849	86	6	v	v	X
ejpam-6849	86	7	(	(	PUNCT
ejpam-6849	86	8	g	g	NOUN
ejpam-6849	86	9	)	)	PUNCT
ejpam-6849	86	10	\	\	NOUN
ejpam-6849	86	11	ng(s	ng(s	NUM
ejpam-6849	86	12	)	)	PUNCT
ejpam-6849	86	13	=	=	VERB
ejpam-6849	86	14	∅.	∅.	AUX
ejpam-6849	86	15	let	let	VERB
ejpam-6849	86	16	xy	xy	PROPN
ejpam-6849	86	17	∈	∈	PROPN
ejpam-6849	86	18	e(g	e(g	PROPN
ejpam-6849	86	19	)	)	PUNCT
ejpam-6849	86	20	.	.	PUNCT
ejpam-6849	87	1	in	in	ADP
ejpam-6849	87	2	particular	particular	ADJ
ejpam-6849	87	3	,	,	PUNCT
ejpam-6849	87	4	since	since	SCONJ
ejpam-6849	87	5	v	v	NOUN
ejpam-6849	87	6	(	(	PUNCT
ejpam-6849	87	7	g	g	NOUN
ejpam-6849	87	8	)	)	PUNCT
ejpam-6849	87	9	=	=	PUNCT
ejpam-6849	87	10	ng(s	ng(s	NUM
ejpam-6849	87	11	)	)	PUNCT
ejpam-6849	87	12	,	,	PUNCT
ejpam-6849	87	13	there	there	PRON
ejpam-6849	87	14	exists	exist	VERB
ejpam-6849	87	15	u	u	PROPN
ejpam-6849	87	16	∈	∈	PROPN
ejpam-6849	87	17	s	s	X
ejpam-6849	87	18	for	for	ADP
ejpam-6849	87	19	which	which	PRON
ejpam-6849	87	20	ux	ux	PROPN
ejpam-6849	87	21	∈	∈	PROPN
ejpam-6849	87	22	e(g	e(g	PROPN
ejpam-6849	87	23	)	)	PUNCT
ejpam-6849	87	24	.	.	PUNCT
ejpam-6849	88	1	observe	observe	VERB
ejpam-6849	88	2	that	that	SCONJ
ejpam-6849	88	3	u	u	NOUN
ejpam-6849	88	4	,	,	PUNCT
ejpam-6849	88	5	and	and	CCONJ
ejpam-6849	88	6	thus	thus	ADV
ejpam-6849	88	7	s	s	PART
ejpam-6849	88	8	,	,	PUNCT
ejpam-6849	88	9	ve	ve	VERB
ejpam-6849	88	10	-	-	PUNCT
ejpam-6849	88	11	dominates	dominate	VERB
ejpam-6849	88	12	xy	xy	NOUN
ejpam-6849	88	13	.	.	PUNCT
ejpam-6849	89	1	next	next	ADV
ejpam-6849	89	2	,	,	PUNCT
ejpam-6849	89	3	suppose	suppose	VERB
ejpam-6849	89	4	that	that	SCONJ
ejpam-6849	89	5	v	v	X
ejpam-6849	89	6	(	(	PUNCT
ejpam-6849	89	7	g	g	NOUN
ejpam-6849	89	8	)	)	PUNCT
ejpam-6849	89	9	\	\	NOUN
ejpam-6849	89	10	ng(s	ng(s	NUM
ejpam-6849	89	11	)	)	PUNCT
ejpam-6849	89	12	is	be	AUX
ejpam-6849	89	13	a	a	DET
ejpam-6849	89	14	nonempty	nonempty	ADJ
ejpam-6849	89	15	independent	independent	ADJ
ejpam-6849	89	16	set	set	NOUN
ejpam-6849	89	17	,	,	PUNCT
ejpam-6849	89	18	and	and	CCONJ
ejpam-6849	89	19	j.	j.	PROPN
ejpam-6849	89	20	tayab	tayab	PROPN
ejpam-6849	89	21	,	,	PUNCT
ejpam-6849	89	22	f.	f.	PROPN
ejpam-6849	89	23	p.	p.	PROPN
ejpam-6849	89	24	jamil	jamil	PROPN
ejpam-6849	89	25	,	,	PUNCT
ejpam-6849	89	26	i.	i.	PROPN
ejpam-6849	89	27	aniversario	aniversario	PROPN
ejpam-6849	89	28	/	/	SYM
ejpam-6849	89	29	eur	eur	PROPN
ejpam-6849	89	30	.	.	PUNCT
ejpam-6849	90	1	j.	j.	PROPN
ejpam-6849	90	2	pure	pure	PROPN
ejpam-6849	90	3	appl	appl	PROPN
ejpam-6849	90	4	.	.	PROPN
ejpam-6849	90	5	math	math	PROPN
ejpam-6849	90	6	,	,	PUNCT
ejpam-6849	90	7	18	18	NUM
ejpam-6849	90	8	(	(	PUNCT
ejpam-6849	90	9	4	4	NUM
ejpam-6849	90	10	)	)	PUNCT
ejpam-6849	90	11	(	(	PUNCT
ejpam-6849	90	12	2025	2025	NUM
ejpam-6849	90	13	)	)	PUNCT
ejpam-6849	90	14	,	,	PUNCT
ejpam-6849	90	15	6849	6849	NUM
ejpam-6849	90	16	4	4	NUM
ejpam-6849	90	17	of	of	ADP
ejpam-6849	90	18	19	19	NUM
ejpam-6849	90	19	let	let	VERB
ejpam-6849	90	20	xy	xy	PROPN
ejpam-6849	90	21	∈	∈	PROPN
ejpam-6849	90	22	e(g	e(g	PROPN
ejpam-6849	90	23	)	)	PUNCT
ejpam-6849	90	24	.	.	PUNCT
ejpam-6849	91	1	suppose	suppose	VERB
ejpam-6849	91	2	that	that	SCONJ
ejpam-6849	91	3	s	s	AUX
ejpam-6849	91	4	does	do	AUX
ejpam-6849	91	5	not	not	PART
ejpam-6849	91	6	ve	ve	AUX
ejpam-6849	91	7	-	-	VERB
ejpam-6849	91	8	dominate	dominate	VERB
ejpam-6849	91	9	xy	xy	NOUN
ejpam-6849	91	10	.	.	PUNCT
ejpam-6849	92	1	then	then	ADV
ejpam-6849	92	2	x	x	X
ejpam-6849	92	3	/∈	/∈	PUNCT
ejpam-6849	92	4	ng(s	ng(s	NUM
ejpam-6849	92	5	)	)	PUNCT
ejpam-6849	92	6	and	and	CCONJ
ejpam-6849	92	7	y	y	PROPN
ejpam-6849	92	8	/∈	/∈	PROPN
ejpam-6849	92	9	ng(s	ng(s	NUM
ejpam-6849	92	10	)	)	PUNCT
ejpam-6849	92	11	.	.	PUNCT
ejpam-6849	93	1	since	since	SCONJ
ejpam-6849	93	2	v	v	NOUN
ejpam-6849	93	3	(	(	PUNCT
ejpam-6849	93	4	g	g	NOUN
ejpam-6849	93	5	)	)	PUNCT
ejpam-6849	93	6	\	\	NOUN
ejpam-6849	93	7	ng(s	ng(s	NUM
ejpam-6849	93	8	)	)	PUNCT
ejpam-6849	93	9	is	be	AUX
ejpam-6849	93	10	an	an	DET
ejpam-6849	93	11	independent	independent	ADJ
ejpam-6849	93	12	set	set	NOUN
ejpam-6849	93	13	,	,	PUNCT
ejpam-6849	93	14	xy	xy	PROPN
ejpam-6849	93	15	/∈	/∈	PUNCT
ejpam-6849	94	1	e(g	e(g	PROPN
ejpam-6849	94	2	)	)	PUNCT
ejpam-6849	94	3	,	,	PUNCT
ejpam-6849	94	4	a	a	DET
ejpam-6849	94	5	contradiction	contradiction	NOUN
ejpam-6849	94	6	.	.	PUNCT
ejpam-6849	95	1	■	■	PUNCT
ejpam-6849	95	2	if	if	SCONJ
ejpam-6849	95	3	g	g	PROPN
ejpam-6849	95	4	=	=	SYM
ejpam-6849	95	5	kn	kn	PROPN
ejpam-6849	95	6	,	,	PUNCT
ejpam-6849	95	7	then	then	ADV
ejpam-6849	95	8	every	every	DET
ejpam-6849	95	9	s	s	PROPN
ejpam-6849	95	10	⊆	⊆	NUM
ejpam-6849	95	11	v	v	NOUN
ejpam-6849	95	12	(	(	PUNCT
ejpam-6849	95	13	g	g	NOUN
ejpam-6849	95	14	)	)	PUNCT
ejpam-6849	95	15	is	be	AUX
ejpam-6849	95	16	a	a	DET
ejpam-6849	95	17	ve	ve	NOUN
ejpam-6849	95	18	-	-	PUNCT
ejpam-6849	95	19	dominating	dominate	VERB
ejpam-6849	95	20	set	set	NOUN
ejpam-6849	95	21	of	of	ADP
ejpam-6849	95	22	g.	g.	PROPN
ejpam-6849	95	23	hence	hence	ADV
ejpam-6849	95	24	,	,	PUNCT
ejpam-6849	95	25	γve(g	γve(g	PROPN
ejpam-6849	95	26	)	)	PUNCT
ejpam-6849	96	1	=	=	SYM
ejpam-6849	96	2	0	0	PUNCT
ejpam-6849	97	1	if	if	SCONJ
ejpam-6849	97	2	and	and	CCONJ
ejpam-6849	97	3	only	only	ADV
ejpam-6849	97	4	if	if	SCONJ
ejpam-6849	97	5	g	g	PROPN
ejpam-6849	97	6	=	=	PROPN
ejpam-6849	97	7	kn	kn	PROPN
ejpam-6849	97	8	.	.	PUNCT
ejpam-6849	98	1	if	if	SCONJ
ejpam-6849	98	2	g	g	PROPN
ejpam-6849	98	3	=	=	PROPN
ejpam-6849	98	4	kn	kn	PROPN
ejpam-6849	98	5	or	or	CCONJ
ejpam-6849	98	6	g	g	PROPN
ejpam-6849	98	7	is	be	AUX
ejpam-6849	98	8	the	the	DET
ejpam-6849	98	9	complete	complete	ADJ
ejpam-6849	98	10	multipartite	multipartite	NOUN
ejpam-6849	98	11	kn1,n2,	kn1,n2,	NOUN
ejpam-6849	98	12	...	...	PUNCT
ejpam-6849	98	13	,nk	,nk	PUNCT
ejpam-6849	98	14	,	,	PUNCT
ejpam-6849	98	15	then	then	ADV
ejpam-6849	98	16	s	s	VERB
ejpam-6849	98	17	is	be	AUX
ejpam-6849	98	18	a	a	DET
ejpam-6849	98	19	ve	ve	NOUN
ejpam-6849	98	20	-	-	PUNCT
ejpam-6849	98	21	dominating	dominate	VERB
ejpam-6849	98	22	set	set	NOUN
ejpam-6849	98	23	of	of	ADP
ejpam-6849	98	24	g	g	NOUN
ejpam-6849	98	25	for	for	ADP
ejpam-6849	98	26	every	every	DET
ejpam-6849	98	27	nonempty	nonempty	NOUN
ejpam-6849	98	28	s	s	VERB
ejpam-6849	98	29	⊆	⊆	NUM
ejpam-6849	98	30	v	v	NOUN
ejpam-6849	98	31	(	(	PUNCT
ejpam-6849	98	32	g	g	NOUN
ejpam-6849	98	33	)	)	PUNCT
ejpam-6849	98	34	.	.	PUNCT
ejpam-6849	99	1	proposition	proposition	NOUN
ejpam-6849	99	2	2	2	NUM
ejpam-6849	99	3	.	.	PUNCT
ejpam-6849	100	1	[	[	X
ejpam-6849	100	2	19	19	NUM
ejpam-6849	100	3	]	]	PUNCT
ejpam-6849	100	4	for	for	ADP
ejpam-6849	100	5	the	the	DET
ejpam-6849	100	6	complete	complete	ADJ
ejpam-6849	100	7	graph	graph	NOUN
ejpam-6849	100	8	kn	kn	PROPN
ejpam-6849	100	9	,	,	PUNCT
ejpam-6849	100	10	the	the	DET
ejpam-6849	100	11	complete	complete	ADJ
ejpam-6849	100	12	bipartite	bipartite	PROPN
ejpam-6849	100	13	km	km	PROPN
ejpam-6849	100	14	,	,	PUNCT
ejpam-6849	100	15	n	n	CCONJ
ejpam-6849	100	16	,	,	PUNCT
ejpam-6849	100	17	complete	complete	ADJ
ejpam-6849	100	18	r	r	NOUN
ejpam-6849	100	19	-	-	ADJ
ejpam-6849	100	20	partite	partite	ADJ
ejpam-6849	100	21	kn1,n2,	kn1,n2,	NOUN
ejpam-6849	100	22	...	...	PUNCT
ejpam-6849	100	23	,nr	,nr	PUNCT
ejpam-6849	100	24	,	,	PUNCT
ejpam-6849	100	25	path	path	NOUN
ejpam-6849	100	26	pn	pn	PROPN
ejpam-6849	100	27	and	and	CCONJ
ejpam-6849	100	28	cycle	cycle	NOUN
ejpam-6849	100	29	cn	cn	PROPN
ejpam-6849	100	30	,	,	PUNCT
ejpam-6849	100	31	we	we	PRON
ejpam-6849	100	32	have	have	VERB
ejpam-6849	100	33	:	:	PUNCT
ejpam-6849	100	34	(	(	PUNCT
ejpam-6849	100	35	i	i	NOUN
ejpam-6849	100	36	)	)	PUNCT
ejpam-6849	100	37	γve(kn	γve(kn	NOUN
ejpam-6849	100	38	)	)	PUNCT
ejpam-6849	101	1	=	=	SYM
ejpam-6849	101	2	γve(km	γve(km	NOUN
ejpam-6849	101	3	,	,	PUNCT
ejpam-6849	101	4	n	n	CCONJ
ejpam-6849	101	5	)	)	PUNCT
ejpam-6849	101	6	=	=	SYM
ejpam-6849	101	7	γve(kn1,n2,	γve(kn1,n2,	PROPN
ejpam-6849	101	8	...	...	PUNCT
ejpam-6849	101	9	,nr	,nr	PUNCT
ejpam-6849	101	10	)	)	PUNCT
ejpam-6849	102	1	=	=	SYM
ejpam-6849	102	2	1	1	NUM
ejpam-6849	102	3	;	;	PUNCT
ejpam-6849	102	4	(	(	PUNCT
ejpam-6849	102	5	ii	ii	NOUN
ejpam-6849	102	6	)	)	PUNCT
ejpam-6849	102	7	γve(pn	γve(pn	NUM
ejpam-6849	102	8	)	)	PUNCT
ejpam-6849	102	9	=	=	PRON
ejpam-6849	102	10	⌊n+2	⌊n+2	NUM
ejpam-6849	102	11	4	4	NUM
ejpam-6849	102	12	⌋	⌋	ADJ
ejpam-6849	102	13	;	;	PUNCT
ejpam-6849	102	14	(	(	PUNCT
ejpam-6849	102	15	iii	iii	X
ejpam-6849	102	16	)	)	PUNCT
ejpam-6849	102	17	γve(cn	γve(cn	NUM
ejpam-6849	102	18	)	)	PUNCT
ejpam-6849	103	1	=	=	PRON
ejpam-6849	103	2	⌊n+3	⌊n+3	AUX
ejpam-6849	103	3	4	4	NUM
ejpam-6849	103	4	⌋.	⌋.	NOUN
ejpam-6849	103	5	proposition	proposition	NOUN
ejpam-6849	103	6	3	3	NUM
ejpam-6849	103	7	.	.	PUNCT
ejpam-6849	104	1	[	[	X
ejpam-6849	104	2	1	1	X
ejpam-6849	104	3	]	]	PUNCT
ejpam-6849	104	4	for	for	ADP
ejpam-6849	104	5	any	any	DET
ejpam-6849	104	6	graph	graph	NOUN
ejpam-6849	104	7	of	of	ADP
ejpam-6849	104	8	order	order	NOUN
ejpam-6849	104	9	n	n	CCONJ
ejpam-6849	104	10	,	,	PUNCT
ejpam-6849	104	11	γve(g	γve(g	PROPN
ejpam-6849	104	12	)	)	PUNCT
ejpam-6849	104	13	≤	≤	NOUN
ejpam-6849	104	14	n	n	DET
ejpam-6849	104	15	2	2	NUM
ejpam-6849	104	16	.	.	PUNCT
ejpam-6849	105	1	observation	observation	NOUN
ejpam-6849	105	2	1	1	NUM
ejpam-6849	105	3	.	.	PUNCT
ejpam-6849	106	1	let	let	VERB
ejpam-6849	106	2	g	g	PRON
ejpam-6849	106	3	be	be	AUX
ejpam-6849	106	4	a	a	DET
ejpam-6849	106	5	disconnected	disconnected	ADJ
ejpam-6849	106	6	graph	graph	NOUN
ejpam-6849	106	7	with	with	ADP
ejpam-6849	106	8	nontrivial	nontrivial	ADJ
ejpam-6849	106	9	components	component	NOUN
ejpam-6849	106	10	g1	g1	PROPN
ejpam-6849	106	11	,	,	PUNCT
ejpam-6849	106	12	g2	g2	PROPN
ejpam-6849	106	13	,	,	PUNCT
ejpam-6849	106	14	.	.	PUNCT
ejpam-6849	106	15	.	.	PUNCT
ejpam-6849	107	1	.	.	PUNCT
ejpam-6849	108	1	,	,	PUNCT
ejpam-6849	109	1	gn	gn	PROPN
ejpam-6849	109	2	.	.	PUNCT
ejpam-6849	109	3	then	then	ADV
ejpam-6849	109	4	s	s	VERB
ejpam-6849	109	5	⊆	⊆	NUM
ejpam-6849	109	6	v	v	NOUN
ejpam-6849	109	7	(	(	PUNCT
ejpam-6849	109	8	g	g	NOUN
ejpam-6849	109	9	)	)	PUNCT
ejpam-6849	109	10	is	be	AUX
ejpam-6849	109	11	a	a	DET
ejpam-6849	109	12	(	(	PUNCT
ejpam-6849	109	13	total	total	NOUN
ejpam-6849	109	14	)	)	PUNCT
ejpam-6849	109	15	ve	ve	AUX
ejpam-6849	109	16	-	-	PUNCT
ejpam-6849	109	17	dominating	dominate	VERB
ejpam-6849	109	18	set	set	NOUN
ejpam-6849	109	19	of	of	ADP
ejpam-6849	109	20	g	g	PROPN
ejpam-6849	109	21	if	if	SCONJ
ejpam-6849	110	1	and	and	CCONJ
ejpam-6849	110	2	only	only	ADV
ejpam-6849	110	3	if	if	SCONJ
ejpam-6849	110	4	s	s	ADP
ejpam-6849	110	5	∩	∩	ADJ
ejpam-6849	110	6	v	v	X
ejpam-6849	110	7	(	(	PUNCT
ejpam-6849	110	8	gk	gk	NOUN
ejpam-6849	110	9	)	)	PUNCT
ejpam-6849	110	10	is	be	AUX
ejpam-6849	110	11	a	a	DET
ejpam-6849	110	12	(	(	PUNCT
ejpam-6849	110	13	total	total	NOUN
ejpam-6849	110	14	)	)	PUNCT
ejpam-6849	110	15	ve	ve	AUX
ejpam-6849	110	16	-	-	PUNCT
ejpam-6849	110	17	dominating	dominate	VERB
ejpam-6849	110	18	set	set	NOUN
ejpam-6849	110	19	of	of	ADP
ejpam-6849	110	20	gk	gk	PROPN
ejpam-6849	110	21	for	for	ADP
ejpam-6849	110	22	each	each	PRON
ejpam-6849	110	23	k	k	NOUN
ejpam-6849	110	24	=	=	SYM
ejpam-6849	110	25	1	1	NUM
ejpam-6849	110	26	,	,	PUNCT
ejpam-6849	110	27	2	2	NUM
ejpam-6849	110	28	,	,	PUNCT
ejpam-6849	110	29	.	.	PUNCT
ejpam-6849	110	30	.	.	PUNCT
ejpam-6849	111	1	.	.	PUNCT
ejpam-6849	112	1	,	,	PUNCT
ejpam-6849	112	2	n.	n.	NOUN
ejpam-6849	112	3	in	in	ADP
ejpam-6849	112	4	particular	particular	ADJ
ejpam-6849	112	5	,	,	PUNCT
ejpam-6849	112	6	γve(g	γve(g	PROPN
ejpam-6849	112	7	)	)	PUNCT
ejpam-6849	113	1	=	=	SYM
ejpam-6849	113	2	n∑	n∑	NOUN
ejpam-6849	113	3	k=1	k=1	PROPN
ejpam-6849	113	4	γve(gk	γve(gk	NOUN
ejpam-6849	113	5	)	)	PUNCT
ejpam-6849	113	6	and	and	CCONJ
ejpam-6849	113	7	γt	γt	PROPN
ejpam-6849	113	8	ve(g	ve(g	PROPN
ejpam-6849	113	9	)	)	PUNCT
ejpam-6849	114	1	=	=	PUNCT
ejpam-6849	115	1	n∑	n∑	NOUN
ejpam-6849	115	2	k=1	k=1	ADP
ejpam-6849	115	3	γt	γt	NOUN
ejpam-6849	115	4	ve(gk	ve(gk	NOUN
ejpam-6849	115	5	)	)	PUNCT
ejpam-6849	115	6	.	.	PUNCT
ejpam-6849	116	1	proposition	proposition	NOUN
ejpam-6849	116	2	4	4	NUM
ejpam-6849	116	3	.	.	PUNCT
ejpam-6849	117	1	let	let	VERB
ejpam-6849	117	2	g	g	NOUN
ejpam-6849	117	3	be	be	AUX
ejpam-6849	117	4	any	any	DET
ejpam-6849	117	5	graph	graph	NOUN
ejpam-6849	117	6	of	of	ADP
ejpam-6849	117	7	order	order	NOUN
ejpam-6849	117	8	n.	n.	NOUN
ejpam-6849	117	9	then	then	ADV
ejpam-6849	117	10	(	(	PUNCT
ejpam-6849	117	11	i	i	NOUN
ejpam-6849	117	12	)	)	PUNCT
ejpam-6849	117	13	γve(g	γve(g	PROPN
ejpam-6849	117	14	)	)	PUNCT
ejpam-6849	118	1	=	=	SYM
ejpam-6849	118	2	n	n	CCONJ
ejpam-6849	118	3	−	−	PROPN
ejpam-6849	118	4	1	1	NUM
ejpam-6849	118	5	if	if	SCONJ
ejpam-6849	118	6	and	and	CCONJ
ejpam-6849	118	7	only	only	ADV
ejpam-6849	118	8	if	if	SCONJ
ejpam-6849	118	9	g	g	NOUN
ejpam-6849	118	10	=	=	SYM
ejpam-6849	118	11	{	{	PUNCT
ejpam-6849	118	12	k1	k1	NOUN
ejpam-6849	118	13	,	,	PUNCT
ejpam-6849	118	14	p2	p2	PROPN
ejpam-6849	118	15	}	}	PUNCT
ejpam-6849	118	16	.	.	PUNCT
ejpam-6849	119	1	(	(	PUNCT
ejpam-6849	119	2	ii	ii	X
ejpam-6849	119	3	)	)	PUNCT
ejpam-6849	119	4	γve(g	γve(g	PROPN
ejpam-6849	119	5	)	)	PUNCT
ejpam-6849	120	1	=	=	PUNCT
ejpam-6849	120	2	1	1	NUM
ejpam-6849	120	3	if	if	SCONJ
ejpam-6849	120	4	and	and	CCONJ
ejpam-6849	120	5	only	only	ADV
ejpam-6849	120	6	if	if	SCONJ
ejpam-6849	120	7	g	g	PROPN
ejpam-6849	120	8	has	have	VERB
ejpam-6849	120	9	exactly	exactly	ADV
ejpam-6849	120	10	one	one	NUM
ejpam-6849	120	11	nontrivial	nontrivial	ADJ
ejpam-6849	120	12	component	component	NOUN
ejpam-6849	120	13	g′	g′	NOUN
ejpam-6849	120	14	,	,	PUNCT
ejpam-6849	120	15	where	where	SCONJ
ejpam-6849	120	16	there	there	PRON
ejpam-6849	120	17	exists	exist	VERB
ejpam-6849	120	18	a	a	DET
ejpam-6849	120	19	vertex	vertex	NOUN
ejpam-6849	120	20	v	v	ADP
ejpam-6849	120	21	∈	∈	PROPN
ejpam-6849	120	22	v	v	NOUN
ejpam-6849	120	23	(	(	PUNCT
ejpam-6849	120	24	g′	g′	NOUN
ejpam-6849	120	25	)	)	PUNCT
ejpam-6849	120	26	such	such	ADJ
ejpam-6849	120	27	that	that	PRON
ejpam-6849	120	28	dg(v	dg(v	NOUN
ejpam-6849	120	29	,	,	PUNCT
ejpam-6849	120	30	x	x	X
ejpam-6849	120	31	)	)	PUNCT
ejpam-6849	120	32	=	=	SYM
ejpam-6849	120	33	1	1	NUM
ejpam-6849	120	34	or	or	CCONJ
ejpam-6849	120	35	dg(v	dg(v	PROPN
ejpam-6849	120	36	,	,	PUNCT
ejpam-6849	120	37	y	y	NOUN
ejpam-6849	120	38	)	)	PUNCT
ejpam-6849	120	39	=	=	SYM
ejpam-6849	120	40	1	1	NUM
ejpam-6849	120	41	for	for	ADP
ejpam-6849	120	42	every	every	DET
ejpam-6849	120	43	xy	xy	PROPN
ejpam-6849	120	44	∈	∈	PROPN
ejpam-6849	120	45	e(g	e(g	PROPN
ejpam-6849	120	46	)	)	PUNCT
ejpam-6849	120	47	.	.	PUNCT
ejpam-6849	121	1	proof	proof	NOUN
ejpam-6849	121	2	.	.	PUNCT
ejpam-6849	122	1	the	the	DET
ejpam-6849	122	2	case	case	NOUN
ejpam-6849	122	3	where	where	SCONJ
ejpam-6849	122	4	g	g	PROPN
ejpam-6849	122	5	is	be	AUX
ejpam-6849	122	6	nonempty	nonempty	ADV
ejpam-6849	122	7	follows	follow	VERB
ejpam-6849	122	8	directly	directly	ADV
ejpam-6849	122	9	from	from	ADP
ejpam-6849	122	10	proposition	proposition	NOUN
ejpam-6849	122	11	3	3	NUM
ejpam-6849	122	12	.	.	PUNCT
ejpam-6849	122	13	suppose	suppose	VERB
ejpam-6849	122	14	g	g	PROPN
ejpam-6849	122	15	is	be	AUX
ejpam-6849	122	16	an	an	DET
ejpam-6849	122	17	empty	empty	ADJ
ejpam-6849	122	18	graph	graph	NOUN
ejpam-6849	122	19	.	.	PUNCT
ejpam-6849	123	1	then	then	ADV
ejpam-6849	123	2	γve(g	γve(g	NUM
ejpam-6849	123	3	)	)	PUNCT
ejpam-6849	124	1	=	=	SYM
ejpam-6849	124	2	0	0	X
ejpam-6849	124	3	.	.	PUNCT
ejpam-6849	125	1	if	if	SCONJ
ejpam-6849	125	2	γve(g	γve(g	PROPN
ejpam-6849	125	3	)	)	PUNCT
ejpam-6849	126	1	=	=	SYM
ejpam-6849	126	2	n	n	CCONJ
ejpam-6849	126	3	−	−	PROPN
ejpam-6849	126	4	1	1	NUM
ejpam-6849	126	5	,	,	PUNCT
ejpam-6849	126	6	then	then	ADV
ejpam-6849	126	7	n	n	NOUN
ejpam-6849	126	8	=	=	SYM
ejpam-6849	126	9	1	1	X
ejpam-6849	126	10	.	.	PUNCT
ejpam-6849	127	1	conversely	conversely	ADV
ejpam-6849	127	2	,	,	PUNCT
ejpam-6849	127	3	if	if	SCONJ
ejpam-6849	127	4	n	n	NOUN
ejpam-6849	127	5	=	=	SYM
ejpam-6849	127	6	1	1	NUM
ejpam-6849	127	7	,	,	PUNCT
ejpam-6849	127	8	then	then	ADV
ejpam-6849	127	9	γve(g	γve(g	NUM
ejpam-6849	127	10	)	)	PUNCT
ejpam-6849	128	1	=	=	SYM
ejpam-6849	128	2	n	n	CCONJ
ejpam-6849	128	3	−	−	NOUN
ejpam-6849	128	4	1	1	NUM
ejpam-6849	128	5	.	.	PUNCT
ejpam-6849	129	1	thus	thus	ADV
ejpam-6849	129	2	,	,	PUNCT
ejpam-6849	129	3	(	(	PUNCT
ejpam-6849	129	4	i	i	NOUN
ejpam-6849	129	5	)	)	PUNCT
ejpam-6849	129	6	holds	hold	VERB
ejpam-6849	129	7	.	.	PUNCT
ejpam-6849	130	1	to	to	PART
ejpam-6849	130	2	prove	prove	VERB
ejpam-6849	130	3	(	(	PUNCT
ejpam-6849	130	4	ii	ii	NOUN
ejpam-6849	130	5	)	)	PUNCT
ejpam-6849	130	6	,	,	PUNCT
ejpam-6849	130	7	first	first	ADV
ejpam-6849	130	8	,	,	PUNCT
ejpam-6849	130	9	suppose	suppose	VERB
ejpam-6849	130	10	that	that	SCONJ
ejpam-6849	130	11	γve(g	γve(g	PROPN
ejpam-6849	130	12	)	)	PUNCT
ejpam-6849	131	1	=	=	SYM
ejpam-6849	131	2	1	1	X
ejpam-6849	131	3	.	.	PUNCT
ejpam-6849	132	1	in	in	ADP
ejpam-6849	132	2	view	view	NOUN
ejpam-6849	132	3	of	of	ADP
ejpam-6849	132	4	observation	observation	NOUN
ejpam-6849	132	5	1	1	NUM
ejpam-6849	132	6	,	,	PUNCT
ejpam-6849	132	7	g	g	PROPN
ejpam-6849	132	8	has	have	VERB
ejpam-6849	132	9	exactly	exactly	ADV
ejpam-6849	132	10	one	one	NUM
ejpam-6849	132	11	nontrivial	nontrivial	ADJ
ejpam-6849	132	12	component	component	NOUN
ejpam-6849	132	13	g′	g′	NOUN
ejpam-6849	132	14	and	and	CCONJ
ejpam-6849	132	15	γve(g′	γve(g′	NOUN
ejpam-6849	132	16	)	)	PUNCT
ejpam-6849	133	1	=	=	SYM
ejpam-6849	133	2	γve(g	γve(g	PROPN
ejpam-6849	133	3	)	)	PUNCT
ejpam-6849	134	1	=	=	SYM
ejpam-6849	135	1	1	1	X
ejpam-6849	135	2	.	.	PUNCT
ejpam-6849	135	3	let	let	VERB
ejpam-6849	135	4	s	s	VERB
ejpam-6849	135	5	=	=	PUNCT
ejpam-6849	135	6	{	{	PUNCT
ejpam-6849	135	7	v	v	NOUN
ejpam-6849	135	8	}	}	PUNCT
ejpam-6849	135	9	,	,	PUNCT
ejpam-6849	135	10	where	where	SCONJ
ejpam-6849	135	11	v	v	X
ejpam-6849	135	12	∈	∈	PROPN
ejpam-6849	135	13	v	v	NOUN
ejpam-6849	135	14	(	(	PUNCT
ejpam-6849	135	15	g′	g′	NOUN
ejpam-6849	135	16	)	)	PUNCT
ejpam-6849	135	17	for	for	ADP
ejpam-6849	135	18	which	which	PRON
ejpam-6849	135	19	v	v	AUX
ejpam-6849	135	20	ve	ve	VERB
ejpam-6849	135	21	-	-	PUNCT
ejpam-6849	135	22	dominates	dominate	VERB
ejpam-6849	135	23	every	every	DET
ejpam-6849	135	24	xy	xy	PROPN
ejpam-6849	135	25	∈	∈	PROPN
ejpam-6849	135	26	e(g	e(g	PROPN
ejpam-6849	135	27	)	)	PUNCT
ejpam-6849	135	28	.	.	PUNCT
ejpam-6849	136	1	let	let	VERB
ejpam-6849	136	2	xy	xy	PROPN
ejpam-6849	136	3	∈	∈	PROPN
ejpam-6849	136	4	e(g	e(g	PROPN
ejpam-6849	136	5	)	)	PUNCT
ejpam-6849	136	6	.	.	PUNCT
ejpam-6849	137	1	then	then	ADV
ejpam-6849	137	2	xy	xy	PROPN
ejpam-6849	137	3	∈	∈	PROPN
ejpam-6849	137	4	e(g′	e(g′	NUM
ejpam-6849	137	5	)	)	PUNCT
ejpam-6849	137	6	.	.	PUNCT
ejpam-6849	138	1	if	if	SCONJ
ejpam-6849	138	2	x	x	X
ejpam-6849	138	3	=	=	SYM
ejpam-6849	138	4	v	v	X
ejpam-6849	138	5	(	(	PUNCT
ejpam-6849	138	6	resp	resp	NOUN
ejpam-6849	138	7	.	.	PUNCT
ejpam-6849	139	1	y	y	PROPN
ejpam-6849	139	2	=	=	SYM
ejpam-6849	139	3	v	v	NOUN
ejpam-6849	139	4	)	)	PUNCT
ejpam-6849	139	5	,	,	PUNCT
ejpam-6849	139	6	then	then	ADV
ejpam-6849	139	7	dg(v	dg(v	PUNCT
ejpam-6849	139	8	,	,	PUNCT
ejpam-6849	139	9	y	y	NOUN
ejpam-6849	139	10	)	)	PUNCT
ejpam-6849	140	1	=	=	SYM
ejpam-6849	140	2	dg′(v	dg′(v	PROPN
ejpam-6849	140	3	,	,	PUNCT
ejpam-6849	140	4	y	y	NOUN
ejpam-6849	140	5	)	)	PUNCT
ejpam-6849	140	6	=	=	SYM
ejpam-6849	140	7	1	1	NUM
ejpam-6849	140	8	(	(	PUNCT
ejpam-6849	140	9	resp	resp	NOUN
ejpam-6849	140	10	.	.	PUNCT
ejpam-6849	141	1	dg(v	dg(v	X
ejpam-6849	141	2	,	,	PUNCT
ejpam-6849	141	3	x	x	X
ejpam-6849	141	4	)	)	PUNCT
ejpam-6849	142	1	=	=	SYM
ejpam-6849	142	2	dg′(v	dg′(v	ADV
ejpam-6849	142	3	,	,	PUNCT
ejpam-6849	142	4	x	x	NOUN
ejpam-6849	142	5	)	)	PUNCT
ejpam-6849	142	6	=	=	SYM
ejpam-6849	142	7	1	1	NUM
ejpam-6849	142	8	)	)	PUNCT
ejpam-6849	142	9	.	.	PUNCT
ejpam-6849	143	1	suppose	suppose	VERB
ejpam-6849	143	2	that	that	SCONJ
ejpam-6849	143	3	x	x	PROPN
ejpam-6849	143	4	̸=	̸=	PROPN
ejpam-6849	143	5	v	v	NOUN
ejpam-6849	143	6	and	and	CCONJ
ejpam-6849	143	7	y	y	NOUN
ejpam-6849	143	8	=	=	PROPN
ejpam-6849	143	9	̸	̸	NUM
ejpam-6849	144	1	v.	v.	CCONJ
ejpam-6849	144	2	since	since	SCONJ
ejpam-6849	144	3	v	v	NUM
ejpam-6849	144	4	ve	ve	VERB
ejpam-6849	144	5	-	-	PUNCT
ejpam-6849	144	6	dominates	dominate	VERB
ejpam-6849	144	7	xy	xy	NOUN
ejpam-6849	144	8	,	,	PUNCT
ejpam-6849	144	9	dg(v	dg(v	X
ejpam-6849	144	10	,	,	PUNCT
ejpam-6849	144	11	x	x	NOUN
ejpam-6849	144	12	)	)	PUNCT
ejpam-6849	145	1	=	=	SYM
ejpam-6849	145	2	dg′(v	dg′(v	ADV
ejpam-6849	145	3	,	,	PUNCT
ejpam-6849	145	4	x	x	NOUN
ejpam-6849	145	5	)	)	PUNCT
ejpam-6849	145	6	=	=	SYM
ejpam-6849	145	7	1	1	NUM
ejpam-6849	145	8	or	or	CCONJ
ejpam-6849	145	9	dg(v	dg(v	PROPN
ejpam-6849	145	10	,	,	PUNCT
ejpam-6849	145	11	y	y	NOUN
ejpam-6849	145	12	)	)	PUNCT
ejpam-6849	145	13	=	=	SYM
ejpam-6849	146	1	dg′(v	dg′(v	PROPN
ejpam-6849	146	2	,	,	PUNCT
ejpam-6849	146	3	y	y	NOUN
ejpam-6849	146	4	)	)	PUNCT
ejpam-6849	146	5	=	=	SYM
ejpam-6849	147	1	1	1	X
ejpam-6849	147	2	.	.	PUNCT
ejpam-6849	148	1	next	next	ADV
ejpam-6849	148	2	,	,	PUNCT
ejpam-6849	148	3	for	for	ADP
ejpam-6849	148	4	the	the	DET
ejpam-6849	148	5	converse	converse	NOUN
ejpam-6849	148	6	,	,	PUNCT
ejpam-6849	148	7	let	let	VERB
ejpam-6849	148	8	g	g	NOUN
ejpam-6849	148	9	have	have	VERB
ejpam-6849	148	10	exactly	exactly	ADV
ejpam-6849	148	11	one	one	NUM
ejpam-6849	148	12	nontrivial	nontrivial	ADJ
ejpam-6849	148	13	component	component	NOUN
ejpam-6849	148	14	g′	g′	NOUN
ejpam-6849	148	15	,	,	PUNCT
ejpam-6849	148	16	where	where	SCONJ
ejpam-6849	148	17	there	there	PRON
ejpam-6849	148	18	exists	exist	VERB
ejpam-6849	148	19	a	a	DET
ejpam-6849	148	20	vertex	vertex	NOUN
ejpam-6849	148	21	v	v	ADP
ejpam-6849	148	22	∈	∈	PROPN
ejpam-6849	148	23	v	v	NOUN
ejpam-6849	148	24	(	(	PUNCT
ejpam-6849	148	25	g′	g′	NOUN
ejpam-6849	148	26	)	)	PUNCT
ejpam-6849	148	27	such	such	ADJ
ejpam-6849	148	28	that	that	PRON
ejpam-6849	148	29	dg(v	dg(v	NOUN
ejpam-6849	148	30	,	,	PUNCT
ejpam-6849	148	31	x	x	X
ejpam-6849	148	32	)	)	PUNCT
ejpam-6849	148	33	=	=	SYM
ejpam-6849	149	1	dg′(v	dg′(v	ADV
ejpam-6849	149	2	,	,	PUNCT
ejpam-6849	149	3	x	x	NOUN
ejpam-6849	149	4	)	)	PUNCT
ejpam-6849	149	5	=	=	SYM
ejpam-6849	149	6	1	1	NUM
ejpam-6849	149	7	or	or	CCONJ
ejpam-6849	149	8	dg(v	dg(v	PROPN
ejpam-6849	149	9	,	,	PUNCT
ejpam-6849	149	10	y	y	NOUN
ejpam-6849	149	11	)	)	PUNCT
ejpam-6849	150	1	=	=	SYM
ejpam-6849	150	2	dg′(v	dg′(v	PROPN
ejpam-6849	150	3	,	,	PUNCT
ejpam-6849	150	4	y	y	NOUN
ejpam-6849	150	5	)	)	PUNCT
ejpam-6849	150	6	=	=	SYM
ejpam-6849	150	7	1	1	NUM
ejpam-6849	150	8	for	for	ADP
ejpam-6849	150	9	every	every	DET
ejpam-6849	150	10	xy	xy	PROPN
ejpam-6849	150	11	∈	∈	PROPN
ejpam-6849	150	12	e(g	e(g	PROPN
ejpam-6849	150	13	)	)	PUNCT
ejpam-6849	150	14	.	.	PUNCT
ejpam-6849	151	1	it	it	PRON
ejpam-6849	151	2	is	be	AUX
ejpam-6849	151	3	worth	worth	ADJ
ejpam-6849	151	4	noting	note	VERB
ejpam-6849	151	5	that	that	SCONJ
ejpam-6849	151	6	xy	xy	PROPN
ejpam-6849	151	7	∈	∈	PROPN
ejpam-6849	151	8	e(g	e(g	PROPN
ejpam-6849	151	9	)	)	PUNCT
ejpam-6849	152	1	if	if	SCONJ
ejpam-6849	152	2	and	and	CCONJ
ejpam-6849	152	3	only	only	ADV
ejpam-6849	152	4	if	if	SCONJ
ejpam-6849	152	5	xy	xy	PROPN
ejpam-6849	152	6	∈	∈	PROPN
ejpam-6849	152	7	e(g′	e(g′	NUM
ejpam-6849	152	8	)	)	PUNCT
ejpam-6849	152	9	.	.	PUNCT
ejpam-6849	153	1	let	let	VERB
ejpam-6849	153	2	x	x	PRON
ejpam-6849	153	3	,	,	PUNCT
ejpam-6849	153	4	y	y	PROPN
ejpam-6849	153	5	∈	∈	PROPN
ejpam-6849	153	6	v	v	ADP
ejpam-6849	153	7	(	(	PUNCT
ejpam-6849	153	8	g	g	NOUN
ejpam-6849	153	9	)	)	PUNCT
ejpam-6849	153	10	\	\	NOUN
ejpam-6849	153	11	ng(v	ng(v	PUNCT
ejpam-6849	153	12	)	)	PUNCT
ejpam-6849	153	13	with	with	ADP
ejpam-6849	153	14	x	x	X
ejpam-6849	154	1	=	=	NOUN
ejpam-6849	154	2	̸	̸	ADJ
ejpam-6849	154	3	y.	y.	NOUN
ejpam-6849	154	4	suppose	suppose	VERB
ejpam-6849	154	5	that	that	SCONJ
ejpam-6849	154	6	xy	xy	PROPN
ejpam-6849	154	7	∈	∈	PROPN
ejpam-6849	154	8	e(g	e(g	PROPN
ejpam-6849	154	9	)	)	PUNCT
ejpam-6849	154	10	.	.	PUNCT
ejpam-6849	155	1	then	then	ADV
ejpam-6849	155	2	,	,	PUNCT
ejpam-6849	155	3	by	by	ADP
ejpam-6849	155	4	the	the	DET
ejpam-6849	155	5	assumption	assumption	NOUN
ejpam-6849	155	6	,	,	PUNCT
ejpam-6849	155	7	dg(v	dg(v	X
ejpam-6849	155	8	,	,	PUNCT
ejpam-6849	155	9	x	x	NOUN
ejpam-6849	155	10	)	)	PUNCT
ejpam-6849	155	11	=	=	SYM
ejpam-6849	155	12	dg′(v	dg′(v	ADV
ejpam-6849	155	13	,	,	PUNCT
ejpam-6849	155	14	x	x	NOUN
ejpam-6849	155	15	)	)	PUNCT
ejpam-6849	155	16	=	=	SYM
ejpam-6849	155	17	1	1	NUM
ejpam-6849	155	18	or	or	CCONJ
ejpam-6849	155	19	dg(v	dg(v	PROPN
ejpam-6849	155	20	,	,	PUNCT
ejpam-6849	155	21	y	y	NOUN
ejpam-6849	155	22	)	)	PUNCT
ejpam-6849	155	23	=	=	SYM
ejpam-6849	155	24	dg′(v	dg′(v	PROPN
ejpam-6849	155	25	,	,	PUNCT
ejpam-6849	155	26	y	y	NOUN
ejpam-6849	155	27	)	)	PUNCT
ejpam-6849	155	28	=	=	SYM
ejpam-6849	155	29	1	1	X
ejpam-6849	155	30	.	.	X
ejpam-6849	155	31	that	that	PRON
ejpam-6849	155	32	is	be	AUX
ejpam-6849	155	33	,	,	PUNCT
ejpam-6849	155	34	x	x	SYM
ejpam-6849	155	35	∈	∈	NOUN
ejpam-6849	155	36	ng(v	ng(v	PUNCT
ejpam-6849	155	37	)	)	PUNCT
ejpam-6849	155	38	or	or	CCONJ
ejpam-6849	155	39	y	y	PROPN
ejpam-6849	155	40	∈	∈	PROPN
ejpam-6849	155	41	ng(v	ng(v	NOUN
ejpam-6849	155	42	)	)	PUNCT
ejpam-6849	155	43	,	,	PUNCT
ejpam-6849	155	44	a	a	DET
ejpam-6849	155	45	contradiction	contradiction	NOUN
ejpam-6849	155	46	.	.	PUNCT
ejpam-6849	156	1	therefore	therefore	ADV
ejpam-6849	156	2	,	,	PUNCT
ejpam-6849	156	3	v	v	X
ejpam-6849	156	4	(	(	PUNCT
ejpam-6849	156	5	g	g	NOUN
ejpam-6849	156	6	)	)	PUNCT
ejpam-6849	156	7	\	\	NOUN
ejpam-6849	156	8	ng(v	ng(v	PUNCT
ejpam-6849	156	9	)	)	PUNCT
ejpam-6849	156	10	is	be	AUX
ejpam-6849	156	11	an	an	DET
ejpam-6849	156	12	independent	independent	ADJ
ejpam-6849	156	13	set	set	NOUN
ejpam-6849	156	14	.	.	PUNCT
ejpam-6849	157	1	by	by	ADP
ejpam-6849	157	2	proposition	proposition	NOUN
ejpam-6849	157	3	1	1	NUM
ejpam-6849	157	4	,	,	PUNCT
ejpam-6849	157	5	s	s	VERB
ejpam-6849	157	6	=	=	PUNCT
ejpam-6849	157	7	{	{	PUNCT
ejpam-6849	157	8	v	v	NOUN
ejpam-6849	157	9	}	}	PUNCT
ejpam-6849	157	10	is	be	AUX
ejpam-6849	157	11	a	a	DET
ejpam-6849	157	12	ve	ve	NOUN
ejpam-6849	157	13	-	-	PUNCT
ejpam-6849	157	14	dominating	dominate	VERB
ejpam-6849	157	15	set	set	NOUN
ejpam-6849	157	16	of	of	ADP
ejpam-6849	157	17	g.	g.	PROPN
ejpam-6849	157	18	consequently	consequently	ADV
ejpam-6849	157	19	,	,	PUNCT
ejpam-6849	157	20	γve(g	γve(g	PROPN
ejpam-6849	157	21	)	)	PUNCT
ejpam-6849	157	22	=	=	SYM
ejpam-6849	157	23	|s|	|s|	NOUN
ejpam-6849	157	24	=	=	SYM
ejpam-6849	157	25	1	1	NUM
ejpam-6849	157	26	.	.	PUNCT
ejpam-6849	158	1	■	■	PUNCT
ejpam-6849	158	2	j.	j.	PROPN
ejpam-6849	158	3	tayab	tayab	PROPN
ejpam-6849	158	4	,	,	PUNCT
ejpam-6849	158	5	f.	f.	PROPN
ejpam-6849	158	6	p.	p.	PROPN
ejpam-6849	158	7	jamil	jamil	PROPN
ejpam-6849	158	8	,	,	PUNCT
ejpam-6849	158	9	i.	i.	PROPN
ejpam-6849	158	10	aniversario	aniversario	PROPN
ejpam-6849	158	11	/	/	SYM
ejpam-6849	158	12	eur	eur	PROPN
ejpam-6849	158	13	.	.	PUNCT
ejpam-6849	159	1	j.	j.	PROPN
ejpam-6849	159	2	pure	pure	PROPN
ejpam-6849	159	3	appl	appl	PROPN
ejpam-6849	159	4	.	.	PROPN
ejpam-6849	159	5	math	math	PROPN
ejpam-6849	159	6	,	,	PUNCT
ejpam-6849	159	7	18	18	NUM
ejpam-6849	159	8	(	(	PUNCT
ejpam-6849	159	9	4	4	NUM
ejpam-6849	159	10	)	)	PUNCT
ejpam-6849	159	11	(	(	PUNCT
ejpam-6849	159	12	2025	2025	NUM
ejpam-6849	159	13	)	)	PUNCT
ejpam-6849	159	14	,	,	PUNCT
ejpam-6849	159	15	6849	6849	NUM
ejpam-6849	159	16	5	5	NUM
ejpam-6849	159	17	of	of	ADP
ejpam-6849	159	18	19	19	NUM
ejpam-6849	159	19	observation	observation	NOUN
ejpam-6849	159	20	2	2	NUM
ejpam-6849	159	21	.	.	PUNCT
ejpam-6849	160	1	a	a	DET
ejpam-6849	160	2	graph	graph	NOUN
ejpam-6849	160	3	g	g	PROPN
ejpam-6849	160	4	admits	admit	VERB
ejpam-6849	160	5	a	a	DET
ejpam-6849	160	6	total	total	ADJ
ejpam-6849	160	7	ve	ve	NOUN
ejpam-6849	160	8	-	-	PUNCT
ejpam-6849	160	9	dominating	dominating	NOUN
ejpam-6849	160	10	set	set	NOUN
ejpam-6849	160	11	if	if	SCONJ
ejpam-6849	160	12	and	and	CCONJ
ejpam-6849	160	13	only	only	ADV
ejpam-6849	160	14	if	if	SCONJ
ejpam-6849	160	15	g	g	PROPN
ejpam-6849	160	16	is	be	AUX
ejpam-6849	160	17	not	not	PART
ejpam-6849	160	18	an	an	DET
ejpam-6849	160	19	empty	empty	ADJ
ejpam-6849	160	20	graph	graph	NOUN
ejpam-6849	160	21	.	.	PUNCT
ejpam-6849	161	1	observation	observation	NOUN
ejpam-6849	161	2	3	3	NUM
ejpam-6849	161	3	.	.	PUNCT
ejpam-6849	162	1	for	for	ADP
ejpam-6849	162	2	the	the	DET
ejpam-6849	162	3	complete	complete	ADJ
ejpam-6849	162	4	graph	graph	NOUN
ejpam-6849	162	5	kn	kn	PROPN
ejpam-6849	162	6	(	(	PUNCT
ejpam-6849	162	7	n	n	CCONJ
ejpam-6849	162	8	≥	≥	NOUN
ejpam-6849	162	9	2	2	NUM
ejpam-6849	162	10	)	)	PUNCT
ejpam-6849	162	11	,	,	PUNCT
ejpam-6849	162	12	the	the	DET
ejpam-6849	162	13	complete	complete	ADJ
ejpam-6849	162	14	bipartite	bipartite	PROPN
ejpam-6849	162	15	km	km	PROPN
ejpam-6849	162	16	,	,	PUNCT
ejpam-6849	162	17	n	n	CCONJ
ejpam-6849	162	18	,	,	PUNCT
ejpam-6849	162	19	complete	complete	ADJ
ejpam-6849	162	20	r	r	NOUN
ejpam-6849	162	21	-	-	ADJ
ejpam-6849	162	22	partite	partite	ADJ
ejpam-6849	162	23	kn1,n2,	kn1,n2,	NOUN
ejpam-6849	162	24	...	...	PUNCT
ejpam-6849	162	25	,nr	,nr	PUNCT
ejpam-6849	162	26	,	,	PUNCT
ejpam-6849	162	27	path	path	NOUN
ejpam-6849	162	28	pn	pn	PROPN
ejpam-6849	162	29	and	and	CCONJ
ejpam-6849	162	30	cycle	cycle	NOUN
ejpam-6849	162	31	cn	cn	PROPN
ejpam-6849	162	32	,	,	PUNCT
ejpam-6849	162	33	we	we	PRON
ejpam-6849	162	34	have	have	AUX
ejpam-6849	162	35	:	:	PUNCT
ejpam-6849	162	36	(	(	PUNCT
ejpam-6849	162	37	i	i	NOUN
ejpam-6849	162	38	)	)	PUNCT
ejpam-6849	162	39	γt	γt	PROPN
ejpam-6849	162	40	ve(kn	ve(kn	PROPN
ejpam-6849	162	41	)	)	PUNCT
ejpam-6849	163	1	=	=	PUNCT
ejpam-6849	163	2	γt	γt	PROPN
ejpam-6849	163	3	ve(km	ve(km	PROPN
ejpam-6849	163	4	,	,	PUNCT
ejpam-6849	163	5	n	n	CCONJ
ejpam-6849	163	6	)	)	PUNCT
ejpam-6849	163	7	=	=	SYM
ejpam-6849	163	8	γt	γt	PROPN
ejpam-6849	163	9	ve(kn1,n2,	ve(kn1,n2,	PROPN
ejpam-6849	163	10	...	...	PUNCT
ejpam-6849	163	11	,nr	,nr	PUNCT
ejpam-6849	163	12	)	)	PUNCT
ejpam-6849	163	13	=	=	SYM
ejpam-6849	163	14	2	2	NUM
ejpam-6849	163	15	;	;	PUNCT
ejpam-6849	163	16	(	(	PUNCT
ejpam-6849	163	17	ii	ii	NOUN
ejpam-6849	163	18	)	)	PUNCT
ejpam-6849	163	19	for	for	ADP
ejpam-6849	163	20	n	n	X
ejpam-6849	163	21	≥	≥	NUM
ejpam-6849	163	22	2	2	NUM
ejpam-6849	163	23	,	,	PUNCT
ejpam-6849	163	24	γt	γt	NOUN
ejpam-6849	163	25	ve(pn	ve(pn	NOUN
ejpam-6849	163	26	)	)	PUNCT
ejpam-6849	163	27	=	=	PUNCT
ejpam-6849	163	28			NUM
ejpam-6849	163	29	2	2	NUM
ejpam-6849	163	30	,	,	PUNCT
ejpam-6849	163	31	if	if	SCONJ
ejpam-6849	163	32	n	n	NOUN
ejpam-6849	163	33	=	=	SYM
ejpam-6849	163	34	2	2	NUM
ejpam-6849	163	35	,	,	PUNCT
ejpam-6849	163	36	3	3	NUM
ejpam-6849	163	37	+	+	CCONJ
ejpam-6849	163	38	2⌊n−3	2⌊n−3	NUM
ejpam-6849	163	39	5	5	NUM
ejpam-6849	163	40	⌋	⌋	NOUN
ejpam-6849	163	41	,	,	PUNCT
ejpam-6849	163	42	if	if	SCONJ
ejpam-6849	163	43	n	n	PRON
ejpam-6849	163	44	̸=	̸=	PROPN
ejpam-6849	163	45	2	2	NUM
ejpam-6849	163	46	but	but	CCONJ
ejpam-6849	163	47	n	n	CCONJ
ejpam-6849	163	48	≡	≡	PROPN
ejpam-6849	163	49	2	2	NUM
ejpam-6849	163	50	mod	mod	NOUN
ejpam-6849	163	51	5	5	NUM
ejpam-6849	163	52	,	,	PUNCT
ejpam-6849	163	53	2	2	NUM
ejpam-6849	163	54	+	+	CCONJ
ejpam-6849	163	55	2⌊n−3	2⌊n−3	NUM
ejpam-6849	163	56	5	5	NUM
ejpam-6849	163	57	⌋	⌋	NOUN
ejpam-6849	163	58	,	,	PUNCT
ejpam-6849	163	59	if	if	SCONJ
ejpam-6849	163	60	n	n	PRON
ejpam-6849	163	61	≡	≡	PROPN
ejpam-6849	163	62	0	0	NUM
ejpam-6849	163	63	,	,	PUNCT
ejpam-6849	163	64	1	1	NUM
ejpam-6849	163	65	,	,	PUNCT
ejpam-6849	163	66	3	3	NUM
ejpam-6849	163	67	,	,	PUNCT
ejpam-6849	163	68	4	4	NUM
ejpam-6849	163	69	mod	mod	NOUN
ejpam-6849	163	70	5	5	NUM
ejpam-6849	163	71	.	.	PUNCT
ejpam-6849	163	72	(	(	PUNCT
ejpam-6849	163	73	iii	iii	NOUN
ejpam-6849	163	74	)	)	PUNCT
ejpam-6849	163	75	for	for	ADP
ejpam-6849	163	76	n	n	X
ejpam-6849	163	77	≥	≥	NUM
ejpam-6849	163	78	3	3	NUM
ejpam-6849	163	79	,	,	PUNCT
ejpam-6849	163	80	γt	γt	NOUN
ejpam-6849	163	81	ve(cn	ve(cn	NOUN
ejpam-6849	163	82	)	)	PUNCT
ejpam-6849	163	83	=	=	PUNCT
ejpam-6849	163	84			NUM
ejpam-6849	163	85	2	2	NUM
ejpam-6849	163	86	,	,	PUNCT
ejpam-6849	163	87	if	if	SCONJ
ejpam-6849	163	88	n	n	ADV
ejpam-6849	163	89	∈	∈	PROPN
ejpam-6849	163	90	{	{	PUNCT
ejpam-6849	163	91	3	3	NUM
ejpam-6849	163	92	,	,	PUNCT
ejpam-6849	163	93	4	4	NUM
ejpam-6849	163	94	,	,	PUNCT
ejpam-6849	163	95	5	5	NUM
ejpam-6849	163	96	}	}	PUNCT
ejpam-6849	163	97	,	,	PUNCT
ejpam-6849	163	98	3	3	X
ejpam-6849	163	99	,	,	PUNCT
ejpam-6849	163	100	if	if	SCONJ
ejpam-6849	163	101	n	n	NOUN
ejpam-6849	163	102	=	=	SYM
ejpam-6849	163	103	6	6	NUM
ejpam-6849	163	104	,	,	PUNCT
ejpam-6849	163	105	5	5	NUM
ejpam-6849	163	106	+	+	NUM
ejpam-6849	163	107	2⌊n−7	2⌊n−7	NUM
ejpam-6849	163	108	5	5	NUM
ejpam-6849	163	109	⌋	⌋	NOUN
ejpam-6849	163	110	,	,	PUNCT
ejpam-6849	163	111	if	if	SCONJ
ejpam-6849	163	112	n	n	PRON
ejpam-6849	163	113	≥	≥	VERB
ejpam-6849	163	114	7	7	NUM
ejpam-6849	163	115	and	and	CCONJ
ejpam-6849	163	116	n	n	PRON
ejpam-6849	163	117	≡	≡	PROPN
ejpam-6849	163	118	1	1	NUM
ejpam-6849	163	119	mod	mod	NOUN
ejpam-6849	163	120	5	5	NUM
ejpam-6849	163	121	,	,	PUNCT
ejpam-6849	163	122	4	4	NUM
ejpam-6849	163	123	+	+	NUM
ejpam-6849	163	124	2⌊n−7	2⌊n−7	NUM
ejpam-6849	163	125	5	5	NUM
ejpam-6849	163	126	⌋	⌋	NOUN
ejpam-6849	163	127	,	,	PUNCT
ejpam-6849	163	128	if	if	SCONJ
ejpam-6849	163	129	n	n	PRON
ejpam-6849	163	130	≥	≥	VERB
ejpam-6849	163	131	7	7	NUM
ejpam-6849	163	132	and	and	CCONJ
ejpam-6849	163	133	n	n	PRON
ejpam-6849	163	134	≡	≡	PROPN
ejpam-6849	163	135	0	0	NUM
ejpam-6849	163	136	,	,	PUNCT
ejpam-6849	163	137	2	2	NUM
ejpam-6849	163	138	,	,	PUNCT
ejpam-6849	163	139	3	3	NUM
ejpam-6849	163	140	,	,	PUNCT
ejpam-6849	163	141	4	4	NUM
ejpam-6849	163	142	mod	mod	NOUN
ejpam-6849	163	143	5	5	NUM
ejpam-6849	163	144	.	.	PUNCT
ejpam-6849	163	145	proposition	proposition	NOUN
ejpam-6849	163	146	5	5	NUM
ejpam-6849	163	147	.	.	PUNCT
ejpam-6849	164	1	let	let	VERB
ejpam-6849	164	2	g	g	PRON
ejpam-6849	164	3	be	be	AUX
ejpam-6849	164	4	a	a	DET
ejpam-6849	164	5	nonempty	nonempty	ADJ
ejpam-6849	164	6	graph	graph	NOUN
ejpam-6849	164	7	of	of	ADP
ejpam-6849	164	8	order	order	NOUN
ejpam-6849	164	9	n.	n.	NOUN
ejpam-6849	164	10	then	then	ADV
ejpam-6849	164	11	(	(	PUNCT
ejpam-6849	164	12	i	i	NOUN
ejpam-6849	164	13	)	)	PUNCT
ejpam-6849	164	14	γt	γt	PROPN
ejpam-6849	164	15	ve(g	ve(g	PROPN
ejpam-6849	164	16	)	)	PUNCT
ejpam-6849	165	1	=	=	SYM
ejpam-6849	166	1	n	n	NOUN
ejpam-6849	166	2	if	if	SCONJ
ejpam-6849	166	3	and	and	CCONJ
ejpam-6849	166	4	only	only	ADV
ejpam-6849	166	5	if	if	SCONJ
ejpam-6849	166	6	g	g	PROPN
ejpam-6849	166	7	is	be	AUX
ejpam-6849	166	8	the	the	DET
ejpam-6849	166	9	(	(	PUNCT
ejpam-6849	166	10	disjoint	disjoint	NOUN
ejpam-6849	166	11	)	)	PUNCT
ejpam-6849	166	12	union	union	NOUN
ejpam-6849	166	13	of	of	ADP
ejpam-6849	166	14	copies	copy	NOUN
ejpam-6849	166	15	of	of	ADP
ejpam-6849	166	16	k2	k2	NOUN
ejpam-6849	166	17	;	;	PUNCT
ejpam-6849	166	18	(	(	PUNCT
ejpam-6849	166	19	ii	ii	NOUN
ejpam-6849	166	20	)	)	PUNCT
ejpam-6849	166	21	γt	γt	NOUN
ejpam-6849	166	22	ve(g	ve(g	PROPN
ejpam-6849	166	23	)	)	PUNCT
ejpam-6849	166	24	=	=	SYM
ejpam-6849	167	1	n	n	CCONJ
ejpam-6849	167	2	−	−	NUM
ejpam-6849	167	3	1	1	NUM
ejpam-6849	167	4	if	if	SCONJ
ejpam-6849	167	5	and	and	CCONJ
ejpam-6849	167	6	only	only	ADV
ejpam-6849	167	7	if	if	SCONJ
ejpam-6849	167	8	g	g	PROPN
ejpam-6849	167	9	is	be	AUX
ejpam-6849	167	10	one	one	NUM
ejpam-6849	167	11	of	of	ADP
ejpam-6849	167	12	the	the	DET
ejpam-6849	167	13	following	following	NOUN
ejpam-6849	167	14	:	:	PUNCT
ejpam-6849	167	15	p3	p3	PROPN
ejpam-6849	167	16	,	,	PUNCT
ejpam-6849	167	17	k3	k3	VERB
ejpam-6849	167	18	,	,	PUNCT
ejpam-6849	167	19	the	the	DET
ejpam-6849	167	20	(	(	PUNCT
ejpam-6849	167	21	disjoint	disjoint	PROPN
ejpam-6849	167	22	)	)	PUNCT
ejpam-6849	167	23	union	union	NOUN
ejpam-6849	167	24	of	of	ADP
ejpam-6849	167	25	k1	k1	PROPN
ejpam-6849	167	26	and	and	CCONJ
ejpam-6849	167	27	copies	copy	NOUN
ejpam-6849	167	28	of	of	ADP
ejpam-6849	167	29	k2	k2	NOUN
ejpam-6849	167	30	,	,	PUNCT
ejpam-6849	167	31	the	the	DET
ejpam-6849	167	32	(	(	PUNCT
ejpam-6849	167	33	disjoint	disjoint	PROPN
ejpam-6849	167	34	)	)	PUNCT
ejpam-6849	167	35	union	union	NOUN
ejpam-6849	167	36	of	of	ADP
ejpam-6849	167	37	p3	p3	PROPN
ejpam-6849	167	38	and	and	CCONJ
ejpam-6849	167	39	copies	copy	NOUN
ejpam-6849	167	40	of	of	ADP
ejpam-6849	167	41	k2	k2	NOUN
ejpam-6849	167	42	,	,	PUNCT
ejpam-6849	167	43	the	the	DET
ejpam-6849	167	44	(	(	PUNCT
ejpam-6849	167	45	disjoint	disjoint	NOUN
ejpam-6849	167	46	)	)	PUNCT
ejpam-6849	167	47	union	union	NOUN
ejpam-6849	167	48	of	of	ADP
ejpam-6849	167	49	k3	k3	PROPN
ejpam-6849	167	50	and	and	CCONJ
ejpam-6849	167	51	copies	copy	NOUN
ejpam-6849	167	52	of	of	ADP
ejpam-6849	167	53	k2	k2	NOUN
ejpam-6849	167	54	;	;	PUNCT
ejpam-6849	167	55	(	(	PUNCT
ejpam-6849	167	56	iii	iii	X
ejpam-6849	167	57	)	)	PUNCT
ejpam-6849	167	58	γt	γt	NOUN
ejpam-6849	167	59	ve(g	ve(g	PROPN
ejpam-6849	167	60	)	)	PUNCT
ejpam-6849	168	1	=	=	SYM
ejpam-6849	168	2	2	2	NUM
ejpam-6849	168	3	if	if	SCONJ
ejpam-6849	168	4	and	and	CCONJ
ejpam-6849	168	5	only	only	ADV
ejpam-6849	168	6	if	if	SCONJ
ejpam-6849	168	7	there	there	PRON
ejpam-6849	168	8	exists	exist	VERB
ejpam-6849	168	9	uv	uv	PROPN
ejpam-6849	168	10	∈	∈	PROPN
ejpam-6849	168	11	e(g	e(g	PROPN
ejpam-6849	168	12	)	)	PUNCT
ejpam-6849	168	13	such	such	ADJ
ejpam-6849	168	14	that	that	PRON
ejpam-6849	168	15	for	for	ADP
ejpam-6849	168	16	every	every	DET
ejpam-6849	168	17	xy	xy	PROPN
ejpam-6849	168	18	∈	∈	PROPN
ejpam-6849	168	19	e(g)\{uv	e(g)\{uv	PROPN
ejpam-6849	168	20	}	}	PUNCT
ejpam-6849	168	21	,	,	PUNCT
ejpam-6849	168	22	e(g	e(g	PROPN
ejpam-6849	168	23	)	)	PUNCT
ejpam-6849	168	24	∩	∩	NOUN
ejpam-6849	168	25	{	{	PUNCT
ejpam-6849	168	26	ux	ux	PROPN
ejpam-6849	168	27	,	,	PUNCT
ejpam-6849	168	28	vx	vx	PROPN
ejpam-6849	168	29	,	,	PUNCT
ejpam-6849	168	30	uy	uy	PROPN
ejpam-6849	168	31	,	,	PUNCT
ejpam-6849	168	32	vy	vy	PROPN
ejpam-6849	168	33	}	}	PUNCT
ejpam-6849	168	34	̸=	̸=	PROPN
ejpam-6849	168	35	∅.	∅.	ADP
ejpam-6849	168	36	proof	proof	NOUN
ejpam-6849	168	37	.	.	PUNCT
ejpam-6849	169	1	suppose	suppose	VERB
ejpam-6849	169	2	that	that	SCONJ
ejpam-6849	169	3	γt	γt	PROPN
ejpam-6849	169	4	ve(g	ve(g	PROPN
ejpam-6849	169	5	)	)	PUNCT
ejpam-6849	169	6	=	=	SYM
ejpam-6849	170	1	n	n	CCONJ
ejpam-6849	170	2	,	,	PUNCT
ejpam-6849	171	1	and	and	CCONJ
ejpam-6849	171	2	let	let	VERB
ejpam-6849	171	3	k	k	PRON
ejpam-6849	171	4	be	be	AUX
ejpam-6849	171	5	a	a	DET
ejpam-6849	171	6	nontrivial	nontrivial	ADJ
ejpam-6849	171	7	component	component	NOUN
ejpam-6849	171	8	of	of	ADP
ejpam-6849	171	9	g.	g.	PROPN
ejpam-6849	171	10	suppose	suppose	VERB
ejpam-6849	171	11	that	that	SCONJ
ejpam-6849	171	12	k	k	PROPN
ejpam-6849	171	13	̸=	̸=	PROPN
ejpam-6849	171	14	k2	k2	PROPN
ejpam-6849	171	15	.	.	PUNCT
ejpam-6849	172	1	pick	pick	VERB
ejpam-6849	172	2	a	a	DET
ejpam-6849	172	3	non	non	ADJ
ejpam-6849	172	4	-	-	ADJ
ejpam-6849	172	5	cut	cut	ADJ
ejpam-6849	172	6	vertex	vertex	NOUN
ejpam-6849	172	7	x	x	SYM
ejpam-6849	172	8	∈	∈	PROPN
ejpam-6849	172	9	v	v	X
ejpam-6849	172	10	(	(	PUNCT
ejpam-6849	172	11	k	k	NOUN
ejpam-6849	172	12	)	)	PUNCT
ejpam-6849	172	13	.	.	PUNCT
ejpam-6849	173	1	since	since	SCONJ
ejpam-6849	173	2	s	s	PART
ejpam-6849	173	3	=	=	SYM
ejpam-6849	173	4	v	v	PROPN
ejpam-6849	173	5	(	(	PUNCT
ejpam-6849	173	6	k	k	NOUN
ejpam-6849	173	7	)	)	PUNCT
ejpam-6849	173	8	\	\	NOUN
ejpam-6849	173	9	{	{	PUNCT
ejpam-6849	173	10	x	x	NOUN
ejpam-6849	173	11	}	}	PUNCT
ejpam-6849	173	12	is	be	AUX
ejpam-6849	173	13	a	a	DET
ejpam-6849	173	14	total	total	ADJ
ejpam-6849	173	15	ve	ve	NOUN
ejpam-6849	173	16	-	-	PUNCT
ejpam-6849	173	17	set	set	NOUN
ejpam-6849	173	18	of	of	ADP
ejpam-6849	173	19	k	k	PROPN
ejpam-6849	173	20	,	,	PUNCT
ejpam-6849	173	21	γt	γt	NOUN
ejpam-6849	173	22	ve(k	ve(k	NOUN
ejpam-6849	173	23	)	)	PUNCT
ejpam-6849	173	24	≤	≤	NOUN
ejpam-6849	173	25	|v	|v	X
ejpam-6849	173	26	(	(	PUNCT
ejpam-6849	173	27	k)|	k)|	NOUN
ejpam-6849	173	28	−	−	PROPN
ejpam-6849	173	29	1	1	NUM
ejpam-6849	173	30	.	.	PUNCT
ejpam-6849	174	1	by	by	ADP
ejpam-6849	174	2	observation	observation	NOUN
ejpam-6849	174	3	1	1	NUM
ejpam-6849	174	4	,	,	PUNCT
ejpam-6849	174	5	γt	γt	NOUN
ejpam-6849	174	6	ve(g	ve(g	NOUN
ejpam-6849	174	7	)	)	PUNCT
ejpam-6849	174	8	≤	≤	NOUN
ejpam-6849	175	1	n	n	CCONJ
ejpam-6849	175	2	−	−	PROPN
ejpam-6849	175	3	1	1	NUM
ejpam-6849	175	4	,	,	PUNCT
ejpam-6849	175	5	a	a	DET
ejpam-6849	175	6	contradiction	contradiction	NOUN
ejpam-6849	175	7	.	.	PUNCT
ejpam-6849	176	1	thus	thus	ADV
ejpam-6849	176	2	,	,	PUNCT
ejpam-6849	176	3	g	g	PROPN
ejpam-6849	176	4	is	be	AUX
ejpam-6849	176	5	the	the	DET
ejpam-6849	176	6	union	union	NOUN
ejpam-6849	176	7	of	of	ADP
ejpam-6849	176	8	copies	copy	NOUN
ejpam-6849	176	9	of	of	ADP
ejpam-6849	176	10	k2	k2	NOUN
ejpam-6849	176	11	.	.	PUNCT
ejpam-6849	177	1	the	the	DET
ejpam-6849	177	2	converse	converse	NOUN
ejpam-6849	177	3	is	be	AUX
ejpam-6849	177	4	clear	clear	ADJ
ejpam-6849	177	5	.	.	PUNCT
ejpam-6849	178	1	thus	thus	ADV
ejpam-6849	178	2	,	,	PUNCT
ejpam-6849	178	3	(	(	PUNCT
ejpam-6849	178	4	i	i	NOUN
ejpam-6849	178	5	)	)	PUNCT
ejpam-6849	178	6	holds	hold	VERB
ejpam-6849	178	7	.	.	PUNCT
ejpam-6849	178	8	suppose	suppose	VERB
ejpam-6849	178	9	that	that	SCONJ
ejpam-6849	178	10	γt	γt	PROPN
ejpam-6849	178	11	ve(g	ve(g	PROPN
ejpam-6849	178	12	)	)	PUNCT
ejpam-6849	179	1	=	=	PUNCT
ejpam-6849	179	2	n−	n−	NOUN
ejpam-6849	179	3	1	1	NUM
ejpam-6849	179	4	,	,	PUNCT
ejpam-6849	179	5	and	and	CCONJ
ejpam-6849	179	6	let	let	VERB
ejpam-6849	179	7	x	x	SYM
ejpam-6849	179	8	∈	∈	PROPN
ejpam-6849	179	9	v	v	X
ejpam-6849	179	10	(	(	PUNCT
ejpam-6849	179	11	g	g	NOUN
ejpam-6849	179	12	)	)	PUNCT
ejpam-6849	179	13	such	such	ADJ
ejpam-6849	179	14	that	that	PRON
ejpam-6849	179	15	s	s	PART
ejpam-6849	179	16	=	=	X
ejpam-6849	179	17	v	v	X
ejpam-6849	179	18	(	(	PUNCT
ejpam-6849	179	19	g	g	NOUN
ejpam-6849	179	20	)	)	PUNCT
ejpam-6849	179	21	\{x	\{x	NOUN
ejpam-6849	179	22	}	}	PUNCT
ejpam-6849	179	23	is	be	AUX
ejpam-6849	179	24	a	a	DET
ejpam-6849	179	25	γt	γt	NOUN
ejpam-6849	179	26	ve	ve	NOUN
ejpam-6849	179	27	-	-	PUNCT
ejpam-6849	179	28	set	set	NOUN
ejpam-6849	179	29	of	of	ADP
ejpam-6849	179	30	g.	g.	PROPN
ejpam-6849	179	31	let	let	VERB
ejpam-6849	179	32	c	c	NOUN
ejpam-6849	179	33	be	be	AUX
ejpam-6849	179	34	the	the	DET
ejpam-6849	179	35	component	component	NOUN
ejpam-6849	179	36	of	of	ADP
ejpam-6849	179	37	g	g	NOUN
ejpam-6849	179	38	for	for	ADP
ejpam-6849	179	39	which	which	PRON
ejpam-6849	179	40	x	x	SYM
ejpam-6849	179	41	∈	∈	PROPN
ejpam-6849	179	42	v	v	X
ejpam-6849	179	43	(	(	PUNCT
ejpam-6849	179	44	c	c	NOUN
ejpam-6849	179	45	)	)	PUNCT
ejpam-6849	179	46	.	.	PUNCT
ejpam-6849	180	1	put	put	VERB
ejpam-6849	180	2	sc	sc	PROPN
ejpam-6849	180	3	=	=	SYM
ejpam-6849	180	4	s	s	PROPN
ejpam-6849	180	5	∩	∩	ADJ
ejpam-6849	180	6	v	v	NOUN
ejpam-6849	180	7	(	(	PUNCT
ejpam-6849	180	8	c	c	NOUN
ejpam-6849	180	9	)	)	PUNCT
ejpam-6849	180	10	=	=	NOUN
ejpam-6849	180	11	v	v	X
ejpam-6849	180	12	(	(	PUNCT
ejpam-6849	180	13	c	c	NOUN
ejpam-6849	180	14	)	)	PUNCT
ejpam-6849	180	15	\	\	NOUN
ejpam-6849	180	16	{	{	PUNCT
ejpam-6849	180	17	x	x	NOUN
ejpam-6849	180	18	}	}	PUNCT
ejpam-6849	180	19	.	.	PUNCT
ejpam-6849	181	1	in	in	ADP
ejpam-6849	181	2	view	view	NOUN
ejpam-6849	181	3	of	of	ADP
ejpam-6849	181	4	observation	observation	NOUN
ejpam-6849	181	5	1	1	NUM
ejpam-6849	181	6	,	,	PUNCT
ejpam-6849	181	7	sc	sc	PROPN
ejpam-6849	181	8	is	be	AUX
ejpam-6849	181	9	a	a	DET
ejpam-6849	181	10	γt	γt	NOUN
ejpam-6849	181	11	ve	ve	NOUN
ejpam-6849	181	12	-	-	PUNCT
ejpam-6849	181	13	set	set	NOUN
ejpam-6849	181	14	of	of	ADP
ejpam-6849	181	15	c.	c.	NOUN
ejpam-6849	181	16	suppose	suppose	VERB
ejpam-6849	181	17	that	that	SCONJ
ejpam-6849	181	18	|v	|v	PROPN
ejpam-6849	181	19	(	(	PUNCT
ejpam-6849	181	20	c)|	c)|	PROPN
ejpam-6849	181	21	≥	≥	NUM
ejpam-6849	181	22	4	4	NUM
ejpam-6849	181	23	,	,	PUNCT
ejpam-6849	181	24	and	and	CCONJ
ejpam-6849	181	25	let	let	VERB
ejpam-6849	181	26	y	y	PROPN
ejpam-6849	181	27	∈	∈	PROPN
ejpam-6849	181	28	v	v	X
ejpam-6849	181	29	(	(	PUNCT
ejpam-6849	181	30	c	c	NOUN
ejpam-6849	181	31	)	)	PUNCT
ejpam-6849	181	32	∩	∩	NOUN
ejpam-6849	181	33	nc(x	nc(x	ADV
ejpam-6849	181	34	)	)	PUNCT
ejpam-6849	181	35	.	.	PUNCT
ejpam-6849	182	1	since	since	SCONJ
ejpam-6849	182	2	sc	sc	PROPN
ejpam-6849	182	3	is	be	AUX
ejpam-6849	182	4	a	a	DET
ejpam-6849	182	5	total	total	ADJ
ejpam-6849	182	6	ve	ve	NOUN
ejpam-6849	182	7	-	-	PUNCT
ejpam-6849	182	8	dominating	dominating	NOUN
ejpam-6849	182	9	set	set	NOUN
ejpam-6849	182	10	,	,	PUNCT
ejpam-6849	182	11	there	there	PRON
ejpam-6849	182	12	exists	exist	VERB
ejpam-6849	182	13	q	q	PROPN
ejpam-6849	182	14	∈	∈	PROPN
ejpam-6849	182	15	sc	sc	PROPN
ejpam-6849	182	16	∩	∩	NOUN
ejpam-6849	182	17	ng(y	ng(y	NOUN
ejpam-6849	182	18	)	)	PUNCT
ejpam-6849	182	19	(	(	PUNCT
ejpam-6849	182	20	hence	hence	ADV
ejpam-6849	182	21	,	,	PUNCT
ejpam-6849	182	22	|nc(y	|nc(y	PRON
ejpam-6849	182	23	)	)	PUNCT
ejpam-6849	182	24	\	\	NOUN
ejpam-6849	182	25	{	{	PUNCT
ejpam-6849	182	26	x}|	x}|	X
ejpam-6849	182	27	≥	≥	NUM
ejpam-6849	182	28	1	1	NUM
ejpam-6849	182	29	)	)	PUNCT
ejpam-6849	182	30	.	.	PUNCT
ejpam-6849	183	1	suppose	suppose	VERB
ejpam-6849	183	2	⟨sc	⟨sc	PROPN
ejpam-6849	183	3	\	\	PROPN
ejpam-6849	183	4	{	{	PUNCT
ejpam-6849	183	5	y}⟩	y}⟩	PROPN
ejpam-6849	183	6	has	have	VERB
ejpam-6849	183	7	no	no	DET
ejpam-6849	183	8	isolated	isolated	ADJ
ejpam-6849	183	9	vertex	vertex	NOUN
ejpam-6849	183	10	.	.	PUNCT
ejpam-6849	184	1	let	let	VERB
ejpam-6849	184	2	w	w	PROPN
ejpam-6849	184	3	∈	∈	PROPN
ejpam-6849	184	4	sc	sc	PROPN
ejpam-6849	184	5	\	\	PROPN
ejpam-6849	184	6	{	{	PUNCT
ejpam-6849	184	7	y	y	NOUN
ejpam-6849	184	8	}	}	PUNCT
ejpam-6849	184	9	and	and	CCONJ
ejpam-6849	184	10	let	let	VERB
ejpam-6849	184	11	p	p	PRON
ejpam-6849	184	12	∈	∈	PROPN
ejpam-6849	184	13	v	v	NOUN
ejpam-6849	184	14	(	(	PUNCT
ejpam-6849	184	15	c	c	NOUN
ejpam-6849	184	16	)	)	PUNCT
ejpam-6849	184	17	∩	∩	NOUN
ejpam-6849	184	18	nc(w	nc(w	NOUN
ejpam-6849	184	19	)	)	PUNCT
ejpam-6849	184	20	.	.	PUNCT
ejpam-6849	185	1	then	then	ADV
ejpam-6849	185	2	w	w	PROPN
ejpam-6849	185	3	ve	ve	VERB
ejpam-6849	185	4	-	-	PUNCT
ejpam-6849	185	5	dominates	dominate	VERB
ejpam-6849	185	6	pw	pw	PROPN
ejpam-6849	185	7	.	.	PUNCT
ejpam-6849	186	1	since	since	SCONJ
ejpam-6849	186	2	w	w	PROPN
ejpam-6849	186	3	was	be	AUX
ejpam-6849	186	4	arbitrarily	arbitrarily	ADV
ejpam-6849	186	5	chosen	choose	VERB
ejpam-6849	186	6	,	,	PUNCT
ejpam-6849	186	7	it	it	PRON
ejpam-6849	186	8	follows	follow	VERB
ejpam-6849	186	9	that	that	SCONJ
ejpam-6849	186	10	sc	sc	PROPN
ejpam-6849	186	11	\	\	PROPN
ejpam-6849	186	12	{	{	PUNCT
ejpam-6849	186	13	y	y	NOUN
ejpam-6849	186	14	}	}	PUNCT
ejpam-6849	186	15	is	be	AUX
ejpam-6849	186	16	a	a	DET
ejpam-6849	186	17	ve	ve	NOUN
ejpam-6849	186	18	-	-	PUNCT
ejpam-6849	186	19	dominating	dominate	VERB
ejpam-6849	186	20	set	set	NOUN
ejpam-6849	186	21	of	of	ADP
ejpam-6849	186	22	c.	c.	PROPN
ejpam-6849	186	23	moreover	moreover	ADV
ejpam-6849	186	24	,	,	PUNCT
ejpam-6849	186	25	since	since	SCONJ
ejpam-6849	186	26	⟨sc	⟨sc	PROPN
ejpam-6849	186	27	\	\	PROPN
ejpam-6849	186	28	{	{	PUNCT
ejpam-6849	186	29	y}⟩	y}⟩	PROPN
ejpam-6849	186	30	has	have	VERB
ejpam-6849	186	31	no	no	DET
ejpam-6849	186	32	isolated	isolated	ADJ
ejpam-6849	186	33	vertex	vertex	NOUN
ejpam-6849	186	34	,	,	PUNCT
ejpam-6849	186	35	sc	sc	PROPN
ejpam-6849	186	36	\	\	PROPN
ejpam-6849	186	37	{	{	PUNCT
ejpam-6849	186	38	y	y	NOUN
ejpam-6849	186	39	}	}	PUNCT
ejpam-6849	186	40	is	be	AUX
ejpam-6849	186	41	a	a	DET
ejpam-6849	186	42	total	total	ADJ
ejpam-6849	186	43	ve	ve	NOUN
ejpam-6849	186	44	-	-	PUNCT
ejpam-6849	186	45	dominating	dominate	VERB
ejpam-6849	186	46	set	set	NOUN
ejpam-6849	186	47	of	of	ADP
ejpam-6849	186	48	c	c	PROPN
ejpam-6849	186	49	,	,	PUNCT
ejpam-6849	186	50	a	a	DET
ejpam-6849	186	51	contradiction	contradiction	NOUN
ejpam-6849	186	52	.	.	PUNCT
ejpam-6849	187	1	next	next	ADV
ejpam-6849	187	2	,	,	PUNCT
ejpam-6849	187	3	suppose	suppose	VERB
ejpam-6849	187	4	⟨sc	⟨sc	PROPN
ejpam-6849	187	5	\	\	PROPN
ejpam-6849	187	6	{	{	PUNCT
ejpam-6849	187	7	y}⟩	y}⟩	PROPN
ejpam-6849	187	8	has	have	VERB
ejpam-6849	187	9	an	an	DET
ejpam-6849	187	10	isolated	isolated	ADJ
ejpam-6849	187	11	vertex	vertex	NOUN
ejpam-6849	187	12	,	,	PUNCT
ejpam-6849	187	13	say	say	VERB
ejpam-6849	187	14	p.	p.	NOUN
ejpam-6849	187	15	since	since	SCONJ
ejpam-6849	187	16	⟨sc⟩	⟨sc⟩	NOUN
ejpam-6849	187	17	has	have	VERB
ejpam-6849	187	18	no	no	DET
ejpam-6849	187	19	isolated	isolated	ADJ
ejpam-6849	187	20	vertex	vertex	NOUN
ejpam-6849	187	21	,	,	PUNCT
ejpam-6849	187	22	it	it	PRON
ejpam-6849	187	23	follows	follow	VERB
ejpam-6849	187	24	that	that	SCONJ
ejpam-6849	187	25	yp	yp	PROPN
ejpam-6849	187	26	∈	∈	PROPN
ejpam-6849	187	27	e(g	e(g	PROPN
ejpam-6849	187	28	)	)	PUNCT
ejpam-6849	187	29	.	.	PUNCT
ejpam-6849	188	1	note	note	VERB
ejpam-6849	188	2	that	that	SCONJ
ejpam-6849	188	3	⟨sc	⟨sc	PROPN
ejpam-6849	188	4	\{p}⟩	\{p}⟩	PROPN
ejpam-6849	188	5	has	have	VERB
ejpam-6849	188	6	no	no	DET
ejpam-6849	188	7	isolated	isolated	ADJ
ejpam-6849	188	8	vertex	vertex	NOUN
ejpam-6849	188	9	because	because	SCONJ
ejpam-6849	188	10	⟨sc⟩	⟨sc⟩	NOUN
ejpam-6849	188	11	has	have	VERB
ejpam-6849	188	12	no	no	DET
ejpam-6849	188	13	isolated	isolated	ADJ
ejpam-6849	188	14	vertex	vertex	NOUN
ejpam-6849	188	15	and	and	CCONJ
ejpam-6849	188	16	p	p	NOUN
ejpam-6849	188	17	∈	∈	PROPN
ejpam-6849	188	18	end(g	end(g	PROPN
ejpam-6849	188	19	)	)	PUNCT
ejpam-6849	188	20	.	.	PUNCT
ejpam-6849	189	1	it	it	PRON
ejpam-6849	189	2	is	be	AUX
ejpam-6849	189	3	easy	easy	ADJ
ejpam-6849	189	4	to	to	PART
ejpam-6849	189	5	verify	verify	VERB
ejpam-6849	189	6	(	(	PUNCT
ejpam-6849	189	7	following	follow	VERB
ejpam-6849	189	8	n	n	PRON
ejpam-6849	189	9	earlier	early	ADJ
ejpam-6849	189	10	argument	argument	NOUN
ejpam-6849	189	11	)	)	PUNCT
ejpam-6849	189	12	that	that	SCONJ
ejpam-6849	189	13	⟨sc	⟨sc	PROPN
ejpam-6849	189	14	\	\	PROPN
ejpam-6849	189	15	{	{	PUNCT
ejpam-6849	189	16	p}⟩	p}⟩	PROPN
ejpam-6849	189	17	is	be	AUX
ejpam-6849	189	18	a	a	DET
ejpam-6849	189	19	ve	ve	NOUN
ejpam-6849	189	20	-	-	PUNCT
ejpam-6849	189	21	dominating	dominate	VERB
ejpam-6849	189	22	set	set	NOUN
ejpam-6849	189	23	of	of	ADP
ejpam-6849	189	24	c.	c.	PROPN
ejpam-6849	189	25	again	again	ADV
ejpam-6849	189	26	,	,	PUNCT
ejpam-6849	189	27	this	this	PRON
ejpam-6849	189	28	gives	give	VERB
ejpam-6849	189	29	a	a	DET
ejpam-6849	189	30	contradiction	contradiction	NOUN
ejpam-6849	189	31	.	.	PUNCT
ejpam-6849	190	1	therefore	therefore	ADV
ejpam-6849	190	2	,	,	PUNCT
ejpam-6849	190	3	|v	|v	PROPN
ejpam-6849	190	4	(	(	PUNCT
ejpam-6849	190	5	c)|	c)|	PROPN
ejpam-6849	190	6	≤	≤	NOUN
ejpam-6849	190	7	3	3	X
ejpam-6849	190	8	.	.	PUNCT
ejpam-6849	190	9	j.	j.	PROPN
ejpam-6849	190	10	tayab	tayab	PROPN
ejpam-6849	190	11	,	,	PUNCT
ejpam-6849	190	12	f.	f.	PROPN
ejpam-6849	190	13	p.	p.	PROPN
ejpam-6849	190	14	jamil	jamil	PROPN
ejpam-6849	190	15	,	,	PUNCT
ejpam-6849	190	16	i.	i.	PROPN
ejpam-6849	190	17	aniversario	aniversario	PROPN
ejpam-6849	190	18	/	/	SYM
ejpam-6849	190	19	eur	eur	PROPN
ejpam-6849	190	20	.	.	PUNCT
ejpam-6849	191	1	j.	j.	PROPN
ejpam-6849	191	2	pure	pure	PROPN
ejpam-6849	191	3	appl	appl	PROPN
ejpam-6849	191	4	.	.	PROPN
ejpam-6849	191	5	math	math	PROPN
ejpam-6849	191	6	,	,	PUNCT
ejpam-6849	191	7	18	18	NUM
ejpam-6849	191	8	(	(	PUNCT
ejpam-6849	191	9	4	4	NUM
ejpam-6849	191	10	)	)	PUNCT
ejpam-6849	191	11	(	(	PUNCT
ejpam-6849	191	12	2025	2025	NUM
ejpam-6849	191	13	)	)	PUNCT
ejpam-6849	191	14	,	,	PUNCT
ejpam-6849	191	15	6849	6849	NUM
ejpam-6849	191	16	6	6	NUM
ejpam-6849	191	17	of	of	ADP
ejpam-6849	191	18	19	19	NUM
ejpam-6849	191	19	moreover	moreover	ADV
ejpam-6849	192	1	,	,	PUNCT
ejpam-6849	192	2	the	the	DET
ejpam-6849	192	3	definition	definition	NOUN
ejpam-6849	192	4	of	of	ADP
ejpam-6849	192	5	c	c	PROPN
ejpam-6849	192	6	implies	imply	VERB
ejpam-6849	192	7	that	that	SCONJ
ejpam-6849	192	8	c	c	PROPN
ejpam-6849	192	9	̸=	̸=	PROPN
ejpam-6849	192	10	k2	k2	NOUN
ejpam-6849	192	11	.	.	PUNCT
ejpam-6849	193	1	if	if	SCONJ
ejpam-6849	193	2	g	g	PROPN
ejpam-6849	193	3	=	=	SYM
ejpam-6849	193	4	c	c	PROPN
ejpam-6849	193	5	(	(	PUNCT
ejpam-6849	193	6	connected	connect	VERB
ejpam-6849	193	7	)	)	PUNCT
ejpam-6849	193	8	,	,	PUNCT
ejpam-6849	193	9	then	then	ADV
ejpam-6849	193	10	either	either	CCONJ
ejpam-6849	193	11	c	c	PROPN
ejpam-6849	193	12	=	=	SYM
ejpam-6849	193	13	p3	p3	PROPN
ejpam-6849	193	14	or	or	CCONJ
ejpam-6849	193	15	c	c	NOUN
ejpam-6849	193	16	=	=	SYM
ejpam-6849	193	17	k3	k3	PROPN
ejpam-6849	193	18	.	.	PUNCT
ejpam-6849	194	1	otherwise	otherwise	ADV
ejpam-6849	194	2	,	,	PUNCT
ejpam-6849	194	3	(	(	PUNCT
ejpam-6849	194	4	i	i	NOUN
ejpam-6849	194	5	)	)	PUNCT
ejpam-6849	194	6	implies	imply	VERB
ejpam-6849	194	7	that	that	SCONJ
ejpam-6849	194	8	g	g	PROPN
ejpam-6849	194	9	is	be	AUX
ejpam-6849	194	10	the	the	DET
ejpam-6849	194	11	disjoint	disjoint	PROPN
ejpam-6849	194	12	union	union	NOUN
ejpam-6849	194	13	of	of	ADP
ejpam-6849	194	14	copies	copy	NOUN
ejpam-6849	194	15	of	of	ADP
ejpam-6849	194	16	k2	k2	NOUN
ejpam-6849	194	17	and	and	CCONJ
ejpam-6849	194	18	exactly	exactly	ADV
ejpam-6849	194	19	one	one	NUM
ejpam-6849	194	20	of	of	ADP
ejpam-6849	194	21	the	the	DET
ejpam-6849	194	22	following	following	NOUN
ejpam-6849	194	23	:	:	PUNCT
ejpam-6849	194	24	k1	k1	NOUN
ejpam-6849	194	25	,	,	PUNCT
ejpam-6849	194	26	p3	p3	PROPN
ejpam-6849	194	27	and	and	CCONJ
ejpam-6849	194	28	k3	k3	PROPN
ejpam-6849	194	29	.	.	PUNCT
ejpam-6849	195	1	the	the	DET
ejpam-6849	195	2	converse	converse	NOUN
ejpam-6849	195	3	of	of	ADP
ejpam-6849	195	4	(	(	PUNCT
ejpam-6849	195	5	ii	ii	NOUN
ejpam-6849	195	6	)	)	PUNCT
ejpam-6849	195	7	is	be	AUX
ejpam-6849	195	8	clear	clear	ADJ
ejpam-6849	195	9	.	.	PUNCT
ejpam-6849	196	1	finally	finally	ADV
ejpam-6849	196	2	,	,	PUNCT
ejpam-6849	196	3	suppose	suppose	VERB
ejpam-6849	196	4	that	that	SCONJ
ejpam-6849	196	5	γt	γt	PROPN
ejpam-6849	196	6	ve(g	ve(g	PROPN
ejpam-6849	196	7	)	)	PUNCT
ejpam-6849	196	8	=	=	SYM
ejpam-6849	196	9	2	2	NUM
ejpam-6849	196	10	,	,	PUNCT
ejpam-6849	196	11	and	and	CCONJ
ejpam-6849	196	12	let	let	VERB
ejpam-6849	196	13	s	s	PRON
ejpam-6849	196	14	=	=	PUNCT
ejpam-6849	196	15	{	{	PUNCT
ejpam-6849	196	16	u	u	NOUN
ejpam-6849	196	17	,	,	PUNCT
ejpam-6849	196	18	v	v	NOUN
ejpam-6849	196	19	}	}	PUNCT
ejpam-6849	196	20	be	be	AUX
ejpam-6849	196	21	a	a	DET
ejpam-6849	196	22	γt	γt	NOUN
ejpam-6849	196	23	ve	ve	NOUN
ejpam-6849	196	24	-	-	PUNCT
ejpam-6849	196	25	set	set	NOUN
ejpam-6849	196	26	of	of	ADP
ejpam-6849	196	27	g.	g.	PROPN
ejpam-6849	196	28	let	let	VERB
ejpam-6849	196	29	xy	xy	PROPN
ejpam-6849	196	30	∈	∈	PROPN
ejpam-6849	196	31	e(g	e(g	PROPN
ejpam-6849	196	32	)	)	PUNCT
ejpam-6849	196	33	\	\	PUNCT
ejpam-6849	197	1	{	{	PUNCT
ejpam-6849	197	2	uv	uv	NOUN
ejpam-6849	197	3	}	}	PUNCT
ejpam-6849	197	4	and	and	CCONJ
ejpam-6849	197	5	put	put	VERB
ejpam-6849	197	6	t	t	NOUN
ejpam-6849	197	7	=	=	SYM
ejpam-6849	197	8	{	{	PUNCT
ejpam-6849	197	9	ux	ux	PROPN
ejpam-6849	197	10	,	,	PUNCT
ejpam-6849	197	11	vx	vx	PROPN
ejpam-6849	197	12	,	,	PUNCT
ejpam-6849	197	13	uy	uy	PROPN
ejpam-6849	197	14	,	,	PUNCT
ejpam-6849	197	15	vy	vy	NOUN
ejpam-6849	197	16	}	}	PUNCT
ejpam-6849	197	17	.	.	PUNCT
ejpam-6849	198	1	if	if	SCONJ
ejpam-6849	198	2	uv	uv	NOUN
ejpam-6849	198	3	and	and	CCONJ
ejpam-6849	198	4	xy	xy	PROPN
ejpam-6849	198	5	have	have	VERB
ejpam-6849	198	6	a	a	DET
ejpam-6849	198	7	common	common	ADJ
ejpam-6849	198	8	vertex	vertex	NOUN
ejpam-6849	198	9	,	,	PUNCT
ejpam-6849	198	10	then	then	ADV
ejpam-6849	198	11	t	t	PROPN
ejpam-6849	198	12	∩	∩	PROPN
ejpam-6849	198	13	e(g	e(g	PROPN
ejpam-6849	198	14	)	)	PUNCT
ejpam-6849	199	1	̸=	̸=	PROPN
ejpam-6849	199	2	∅.	∅.	ADV
ejpam-6849	199	3	suppose	suppose	VERB
ejpam-6849	199	4	otherwise	otherwise	ADV
ejpam-6849	199	5	.	.	PUNCT
ejpam-6849	200	1	assume	assume	VERB
ejpam-6849	200	2	wlog	wlog	NOUN
ejpam-6849	200	3	that	that	SCONJ
ejpam-6849	200	4	u	u	PROPN
ejpam-6849	200	5	ve	ve	VERB
ejpam-6849	200	6	-	-	VERB
ejpam-6849	200	7	dominates	dominate	VERB
ejpam-6849	200	8	xy	xy	PROPN
ejpam-6849	200	9	.	.	PUNCT
ejpam-6849	201	1	then	then	ADV
ejpam-6849	201	2	either	either	CCONJ
ejpam-6849	201	3	ux	ux	PROPN
ejpam-6849	201	4	∈	∈	PROPN
ejpam-6849	201	5	e(g	e(g	PROPN
ejpam-6849	201	6	)	)	PUNCT
ejpam-6849	201	7	or	or	CCONJ
ejpam-6849	201	8	uy	uy	PROPN
ejpam-6849	201	9	∈	∈	PROPN
ejpam-6849	201	10	e(g	e(g	PROPN
ejpam-6849	201	11	)	)	PUNCT
ejpam-6849	201	12	.	.	PUNCT
ejpam-6849	202	1	in	in	ADP
ejpam-6849	202	2	any	any	DET
ejpam-6849	202	3	case	case	NOUN
ejpam-6849	202	4	,	,	PUNCT
ejpam-6849	202	5	t	t	PROPN
ejpam-6849	202	6	∩	∩	PROPN
ejpam-6849	202	7	e(g	e(g	PROPN
ejpam-6849	202	8	)	)	PUNCT
ejpam-6849	203	1	̸=	̸=	PROPN
ejpam-6849	203	2	∅.	∅.	ADP
ejpam-6849	203	3	the	the	DET
ejpam-6849	203	4	converse	converse	NOUN
ejpam-6849	203	5	is	be	AUX
ejpam-6849	203	6	straightforward	straightforward	ADJ
ejpam-6849	203	7	.	.	PUNCT
ejpam-6849	204	1	■	■	PUNCT
ejpam-6849	204	2	proposition	proposition	NOUN
ejpam-6849	204	3	6	6	NUM
ejpam-6849	204	4	.	.	PUNCT
ejpam-6849	205	1	for	for	ADP
ejpam-6849	205	2	every	every	DET
ejpam-6849	205	3	positive	positive	ADJ
ejpam-6849	205	4	integers	integer	NOUN
ejpam-6849	205	5	a	a	PRON
ejpam-6849	205	6	and	and	CCONJ
ejpam-6849	205	7	b	b	NOUN
ejpam-6849	205	8	with	with	ADP
ejpam-6849	205	9	2	2	NUM
ejpam-6849	205	10	≤	≤	NOUN
ejpam-6849	205	11	a	a	DET
ejpam-6849	205	12	≤	≤	NUM
ejpam-6849	205	13	b	b	NOUN
ejpam-6849	205	14	≤	≤	NUM
ejpam-6849	205	15	2a	2a	NUM
ejpam-6849	205	16	,	,	PUNCT
ejpam-6849	205	17	there	there	PRON
ejpam-6849	205	18	exists	exist	VERB
ejpam-6849	205	19	a	a	DET
ejpam-6849	205	20	connected	connected	ADJ
ejpam-6849	205	21	graph	graph	NOUN
ejpam-6849	205	22	g	g	NOUN
ejpam-6849	205	23	for	for	ADP
ejpam-6849	205	24	which	which	PRON
ejpam-6849	205	25	γve(g	γve(g	PROPN
ejpam-6849	205	26	)	)	PUNCT
ejpam-6849	206	1	=	=	SYM
ejpam-6849	206	2	a	a	PROPN
ejpam-6849	206	3	and	and	CCONJ
ejpam-6849	206	4	γt	γt	NOUN
ejpam-6849	206	5	ve(g	ve(g	PROPN
ejpam-6849	206	6	)	)	PUNCT
ejpam-6849	207	1	=	=	SYM
ejpam-6849	207	2	b.	b.	PROPN
ejpam-6849	207	3	proof	proof	NOUN
ejpam-6849	207	4	.	.	PUNCT
ejpam-6849	208	1	suppose	suppose	VERB
ejpam-6849	208	2	that	that	SCONJ
ejpam-6849	208	3	a	a	DET
ejpam-6849	208	4	=	=	X
ejpam-6849	208	5	b.	b.	PROPN
ejpam-6849	208	6	for	for	ADP
ejpam-6849	208	7	each	each	DET
ejpam-6849	208	8	k	k	PROPN
ejpam-6849	208	9	∈	∈	PROPN
ejpam-6849	208	10	{	{	PUNCT
ejpam-6849	208	11	1	1	NUM
ejpam-6849	208	12	,	,	PUNCT
ejpam-6849	208	13	2	2	NUM
ejpam-6849	208	14	,	,	PUNCT
ejpam-6849	208	15	.	.	PUNCT
ejpam-6849	208	16	.	.	PUNCT
ejpam-6849	209	1	.	.	PUNCT
ejpam-6849	210	1	,	,	PUNCT
ejpam-6849	210	2	a	a	X
ejpam-6849	210	3	}	}	PUNCT
ejpam-6849	210	4	,	,	PUNCT
ejpam-6849	210	5	let	let	VERB
ejpam-6849	210	6	[	[	X
ejpam-6849	210	7	x	x	X
ejpam-6849	210	8	=	=	SYM
ejpam-6849	210	9	xk	xk	PROPN
ejpam-6849	210	10	1	1	NUM
ejpam-6849	210	11	,	,	PUNCT
ejpam-6849	210	12	xk	xk	PROPN
ejpam-6849	210	13	2	2	NUM
ejpam-6849	210	14	,	,	PUNCT
ejpam-6849	210	15	xk	xk	X
ejpam-6849	210	16	3	3	NUM
ejpam-6849	210	17	,	,	PUNCT
ejpam-6849	210	18	xk	xk	X
ejpam-6849	210	19	4	4	X
ejpam-6849	210	20	]	]	PUNCT
ejpam-6849	210	21	denote	denote	VERB
ejpam-6849	210	22	the	the	DET
ejpam-6849	210	23	kth	kth	PROPN
ejpam-6849	210	24	copy	copy	NOUN
ejpam-6849	210	25	of	of	ADP
ejpam-6849	210	26	p4	p4	ADJ
ejpam-6849	210	27	.	.	PUNCT
ejpam-6849	211	1	take	take	VERB
ejpam-6849	211	2	g	g	NOUN
ejpam-6849	211	3	=	=	SYM
ejpam-6849	211	4	g1	g1	PROPN
ejpam-6849	211	5	,	,	PUNCT
ejpam-6849	211	6	where	where	SCONJ
ejpam-6849	211	7	g1	g1	PROPN
ejpam-6849	211	8	is	be	AUX
ejpam-6849	211	9	the	the	DET
ejpam-6849	211	10	graph	graph	NOUN
ejpam-6849	211	11	provided	provide	VERB
ejpam-6849	211	12	in	in	ADP
ejpam-6849	211	13	figure	figure	NOUN
ejpam-6849	211	14	1	1	NUM
ejpam-6849	211	15	obtained	obtain	VERB
ejpam-6849	211	16	by	by	ADP
ejpam-6849	211	17	connecting	connect	VERB
ejpam-6849	211	18	these	these	DET
ejpam-6849	211	19	a	a	DET
ejpam-6849	211	20	copies	copy	NOUN
ejpam-6849	211	21	of	of	ADP
ejpam-6849	211	22	p4	p4	NOUN
ejpam-6849	211	23	by	by	ADP
ejpam-6849	211	24	having	have	VERB
ejpam-6849	211	25	x1	x1	PROPN
ejpam-6849	211	26	1	1	NUM
ejpam-6849	211	27	=	=	SYM
ejpam-6849	211	28	x2	x2	NOUN
ejpam-6849	211	29	1	1	NUM
ejpam-6849	211	30	=	=	SYM
ejpam-6849	211	31	·	·	PUNCT
ejpam-6849	211	32	·	·	PUNCT
ejpam-6849	211	33	·	·	PUNCT
ejpam-6849	212	1	=	=	PUNCT
ejpam-6849	212	2	xa	xa	PROPN
ejpam-6849	212	3	1	1	NUM
ejpam-6849	212	4	and	and	CCONJ
ejpam-6849	212	5	through	through	ADP
ejpam-6849	212	6	the	the	DET
ejpam-6849	212	7	path	path	NOUN
ejpam-6849	213	1	[	[	X
ejpam-6849	213	2	x1	x1	PROPN
ejpam-6849	213	3	3	3	NUM
ejpam-6849	213	4	,	,	PUNCT
ejpam-6849	213	5	x2	x2	PROPN
ejpam-6849	213	6	3	3	NUM
ejpam-6849	213	7	,	,	PUNCT
ejpam-6849	213	8	.	.	PUNCT
ejpam-6849	213	9	.	.	PUNCT
ejpam-6849	213	10	.	.	PUNCT
ejpam-6849	214	1	,	,	PUNCT
ejpam-6849	214	2	xa	xa	PROPN
ejpam-6849	214	3	3].the	3].the	DET
ejpam-6849	214	4	set	set	NOUN
ejpam-6849	214	5	s	s	X
ejpam-6849	214	6	=	=	PUNCT
ejpam-6849	214	7	{	{	PUNCT
ejpam-6849	214	8	x1	x1	PROPN
ejpam-6849	214	9	3	3	NUM
ejpam-6849	214	10	,	,	PUNCT
ejpam-6849	214	11	x2	x2	PROPN
ejpam-6849	214	12	3	3	NUM
ejpam-6849	214	13	,	,	PUNCT
ejpam-6849	214	14	.	.	PUNCT
ejpam-6849	214	15	.	.	PUNCT
ejpam-6849	215	1	.	.	PUNCT
ejpam-6849	216	1	,	,	PUNCT
ejpam-6849	216	2	xa	xa	PROPN
ejpam-6849	216	3	3	3	X
ejpam-6849	216	4	}	}	PUNCT
ejpam-6849	216	5	is	be	AUX
ejpam-6849	216	6	both	both	CCONJ
ejpam-6849	216	7	a	a	DET
ejpam-6849	216	8	γve	γve	NOUN
ejpam-6849	216	9	-	-	PUNCT
ejpam-6849	216	10	set	set	VERB
ejpam-6849	216	11	and	and	CCONJ
ejpam-6849	216	12	a	a	DET
ejpam-6849	216	13	γt	γt	NOUN
ejpam-6849	216	14	ve	ve	NOUN
ejpam-6849	216	15	-	-	PUNCT
ejpam-6849	216	16	set	set	NOUN
ejpam-6849	216	17	of	of	ADP
ejpam-6849	216	18	g.	g.	PROPN
ejpam-6849	216	19	thus	thus	ADV
ejpam-6849	216	20	,	,	PUNCT
ejpam-6849	216	21	....................................	....................................	PUNCT
ejpam-6849	216	22	....................................	....................................	PUNCT
ejpam-6849	217	1	....................................	....................................	PUNCT
ejpam-6849	218	1	....................................	....................................	PUNCT
ejpam-6849	219	1	•	•	NUM
ejpam-6849	219	2	•	•	NUM
ejpam-6849	219	3	•	•	NOUN
ejpam-6849	219	4	•	•	NUM
ejpam-6849	219	5	....................................	....................................	PUNCT
ejpam-6849	219	6	....................................	....................................	PUNCT
ejpam-6849	219	7	....................................	....................................	PUNCT
ejpam-6849	219	8	....................................	....................................	PUNCT
ejpam-6849	219	9	....................................	....................................	PUNCT
ejpam-6849	219	10	...................	...................	PUNCT
ejpam-6849	219	11	..................	..................	PUNCT
ejpam-6849	219	12	..................	..................	PUNCT
ejpam-6849	219	13	.......	.......	PUNCT
ejpam-6849	219	14	............	............	PUNCT
ejpam-6849	219	15	...........	...........	PUNCT
ejpam-6849	219	16	...........	...........	PUNCT
ejpam-6849	219	17	...........	...........	PUNCT
ejpam-6849	219	18	...........	...........	PUNCT
ejpam-6849	219	19	...........	...........	PUNCT
ejpam-6849	219	20	........	........	PUNCT
ejpam-6849	219	21	..............................................................	..............................................................	PUNCT
ejpam-6849	219	22	...........................................................................	...........................................................................	PUNCT
ejpam-6849	219	23	............................................................................	............................................................................	PUNCT
ejpam-6849	219	24	............................................................................	............................................................................	PUNCT
ejpam-6849	219	25	............................................................................	............................................................................	PUNCT
ejpam-6849	219	26	............................................................................	............................................................................	PUNCT
ejpam-6849	219	27	....................................	....................................	PUNCT
ejpam-6849	219	28	....................................	....................................	PUNCT
ejpam-6849	219	29	....................................	....................................	PUNCT
ejpam-6849	219	30	....................................	....................................	PUNCT
ejpam-6849	219	31	...................	...................	PUNCT
ejpam-6849	219	32	..................	..................	PUNCT
ejpam-6849	219	33	..................	..................	PUNCT
ejpam-6849	219	34	.......	.......	PUNCT
ejpam-6849	219	35	..............................................................	..............................................................	PUNCT
ejpam-6849	219	36	............	............	PUNCT
ejpam-6849	219	37	...........	...........	PUNCT
ejpam-6849	219	38	...........	...........	PUNCT
ejpam-6849	219	39	...........	...........	PUNCT
ejpam-6849	219	40	...........	...........	PUNCT
ejpam-6849	219	41	...........	...........	PUNCT
ejpam-6849	219	42	...........	...........	PUNCT
ejpam-6849	219	43	.........	.........	PUNCT
ejpam-6849	219	44	.......................................................................................	.......................................................................................	PUNCT
ejpam-6849	219	45	...	...	PUNCT
ejpam-6849	219	46	...	...	PUNCT
ejpam-6849	219	47	...	...	PUNCT
ejpam-6849	219	48	.........	.........	PUNCT
ejpam-6849	219	49	........	........	PUNCT
ejpam-6849	219	50	........	........	PUNCT
ejpam-6849	219	51	........	........	PUNCT
ejpam-6849	219	52	........	........	PUNCT
ejpam-6849	219	53	........	........	PUNCT
ejpam-6849	220	1	.....	.....	PUNCT
ejpam-6849	220	2	.........	.........	PUNCT
ejpam-6849	220	3	........	........	PUNCT
ejpam-6849	220	4	........	........	PUNCT
ejpam-6849	220	5	........	........	PUNCT
ejpam-6849	220	6	........	........	PUNCT
ejpam-6849	220	7	........	........	PUNCT
ejpam-6849	221	1	.....	.....	PUNCT
ejpam-6849	221	2	.........	.........	PUNCT
ejpam-6849	221	3	........	........	PUNCT
ejpam-6849	221	4	........	........	PUNCT
ejpam-6849	222	1	....	....	PUNCT
ejpam-6849	222	2	.........	.........	PUNCT
ejpam-6849	222	3	........	........	PUNCT
ejpam-6849	222	4	........	........	PUNCT
ejpam-6849	223	1	....	....	PUNCT
ejpam-6849	223	2	g1	g1	PROPN
ejpam-6849	223	3	x	x	PUNCT
ejpam-6849	224	1	x1	x1	PROPN
ejpam-6849	224	2	3	3	NUM
ejpam-6849	224	3	x2	x2	NOUN
ejpam-6849	224	4	3	3	NUM
ejpam-6849	224	5	xa−1	xa−1	PROPN
ejpam-6849	224	6	3	3	NUM
ejpam-6849	224	7	xa	xa	PROPN
ejpam-6849	224	8	3	3	NUM
ejpam-6849	224	9	....................................	....................................	PUNCT
ejpam-6849	224	10	....................................	....................................	PUNCT
ejpam-6849	224	11	....................................	....................................	PUNCT
ejpam-6849	224	12	....................................	....................................	PUNCT
ejpam-6849	224	13	....................................	....................................	PUNCT
ejpam-6849	224	14	....................................	....................................	PUNCT
ejpam-6849	224	15	....................................	....................................	PUNCT
ejpam-6849	224	16	....................................	....................................	PUNCT
ejpam-6849	224	17	....................................	....................................	PUNCT
ejpam-6849	224	18	....................................	....................................	PUNCT
ejpam-6849	224	19	....................................	....................................	PUNCT
ejpam-6849	224	20	....................................	....................................	PUNCT
ejpam-6849	224	21	....................................	....................................	PUNCT
ejpam-6849	224	22	....................................	....................................	PUNCT
ejpam-6849	224	23	..............................................................	..............................................................	PUNCT
ejpam-6849	224	24	....................................	....................................	PUNCT
ejpam-6849	224	25	..............................................................	..............................................................	PUNCT
ejpam-6849	224	26	....................................	....................................	PUNCT
ejpam-6849	224	27	............................................................................	............................................................................	PUNCT
ejpam-6849	224	28	................................................................................................................	................................................................................................................	PUNCT
ejpam-6849	224	29	........................................................................	........................................................................	PUNCT
ejpam-6849	224	30	...................	...................	PUNCT
ejpam-6849	224	31	..................	..................	PUNCT
ejpam-6849	224	32	..................	..................	PUNCT
ejpam-6849	224	33	.......	.......	PUNCT
ejpam-6849	224	34	...	...	PUNCT
ejpam-6849	224	35	.................................	.................................	PUNCT
ejpam-6849	224	36	...................	...................	PUNCT
ejpam-6849	224	37	..................	..................	PUNCT
ejpam-6849	224	38	..................	..................	PUNCT
ejpam-6849	224	39	.......	.......	PUNCT
ejpam-6849	224	40	...	...	PUNCT
ejpam-6849	224	41	.................................	.................................	PUNCT
ejpam-6849	224	42	............................................................................	............................................................................	PUNCT
ejpam-6849	224	43	................................................................................................................	................................................................................................................	PUNCT
ejpam-6849	224	44	........................................................................	........................................................................	PUNCT
ejpam-6849	224	45	............	............	PUNCT
ejpam-6849	224	46	...........	...........	PUNCT
ejpam-6849	224	47	...........	...........	PUNCT
ejpam-6849	224	48	...........	...........	PUNCT
ejpam-6849	224	49	...........	...........	PUNCT
ejpam-6849	224	50	...........	...........	PUNCT
ejpam-6849	224	51	...........	...........	PUNCT
ejpam-6849	224	52	.........	.........	PUNCT
ejpam-6849	224	53	....................................	....................................	PUNCT
ejpam-6849	224	54	............	............	PUNCT
ejpam-6849	224	55	...........	...........	PUNCT
ejpam-6849	224	56	...........	...........	PUNCT
ejpam-6849	224	57	...........	...........	PUNCT
ejpam-6849	224	58	...........	...........	PUNCT
ejpam-6849	224	59	...........	...........	PUNCT
ejpam-6849	224	60	........	........	PUNCT
ejpam-6849	224	61	....................................	....................................	PUNCT
ejpam-6849	224	62	............................................................................	............................................................................	PUNCT
ejpam-6849	224	63	................................................................................................................	................................................................................................................	PUNCT
ejpam-6849	224	64	........................................................................	........................................................................	PUNCT
ejpam-6849	224	65	....................................	....................................	PUNCT
ejpam-6849	224	66	....................................	....................................	PUNCT
ejpam-6849	224	67	....................................	....................................	PUNCT
ejpam-6849	224	68	....................................	....................................	PUNCT
ejpam-6849	225	1	•	•	NUM
ejpam-6849	225	2	•	•	NUM
ejpam-6849	225	3	•	•	NOUN
ejpam-6849	225	4	•	•	NUM
ejpam-6849	225	5	....................................	....................................	PUNCT
ejpam-6849	225	6	....................................	....................................	PUNCT
ejpam-6849	225	7	....................................	....................................	PUNCT
ejpam-6849	225	8	....................................	....................................	PUNCT
ejpam-6849	225	9	....................................	....................................	PUNCT
ejpam-6849	225	10	...................	...................	PUNCT
ejpam-6849	225	11	..................	..................	PUNCT
ejpam-6849	225	12	..................	..................	PUNCT
ejpam-6849	225	13	.......	.......	PUNCT
ejpam-6849	225	14	............	............	PUNCT
ejpam-6849	225	15	...........	...........	PUNCT
ejpam-6849	225	16	...........	...........	PUNCT
ejpam-6849	225	17	...........	...........	PUNCT
ejpam-6849	225	18	...........	...........	PUNCT
ejpam-6849	225	19	...........	...........	PUNCT
ejpam-6849	225	20	........	........	PUNCT
ejpam-6849	225	21	..............................................................	..............................................................	PUNCT
ejpam-6849	225	22	...........................................................................	...........................................................................	PUNCT
ejpam-6849	225	23	............................................................................	............................................................................	PUNCT
ejpam-6849	225	24	............................................................................	............................................................................	PUNCT
ejpam-6849	225	25	............................................................................	............................................................................	PUNCT
ejpam-6849	225	26	............................................................................	............................................................................	PUNCT
ejpam-6849	225	27	....................................	....................................	PUNCT
ejpam-6849	225	28	....................................	....................................	PUNCT
ejpam-6849	225	29	....................................	....................................	PUNCT
ejpam-6849	225	30	....................................	....................................	PUNCT
ejpam-6849	225	31	...................	...................	PUNCT
ejpam-6849	225	32	..................	..................	PUNCT
ejpam-6849	225	33	..................	..................	PUNCT
ejpam-6849	225	34	.......	.......	PUNCT
ejpam-6849	225	35	..............................................................	..............................................................	PUNCT
ejpam-6849	225	36	............	............	PUNCT
ejpam-6849	225	37	...........	...........	PUNCT
ejpam-6849	225	38	...........	...........	PUNCT
ejpam-6849	225	39	...........	...........	PUNCT
ejpam-6849	225	40	...........	...........	PUNCT
ejpam-6849	225	41	...........	...........	PUNCT
ejpam-6849	225	42	...........	...........	PUNCT
ejpam-6849	225	43	.........	.........	PUNCT
ejpam-6849	225	44	.......................................................................................	.......................................................................................	PUNCT
ejpam-6849	225	45	...	...	PUNCT
ejpam-6849	225	46	...	...	PUNCT
ejpam-6849	225	47	...	...	PUNCT
ejpam-6849	225	48	.........	.........	PUNCT
ejpam-6849	225	49	........	........	PUNCT
ejpam-6849	225	50	........	........	PUNCT
ejpam-6849	225	51	........	........	PUNCT
ejpam-6849	225	52	........	........	PUNCT
ejpam-6849	225	53	........	........	PUNCT
ejpam-6849	226	1	.....	.....	PUNCT
ejpam-6849	226	2	.........	.........	PUNCT
ejpam-6849	226	3	........	........	PUNCT
ejpam-6849	226	4	........	........	PUNCT
ejpam-6849	226	5	........	........	PUNCT
ejpam-6849	226	6	........	........	PUNCT
ejpam-6849	226	7	........	........	PUNCT
ejpam-6849	227	1	.....	.....	PUNCT
ejpam-6849	227	2	.........	.........	PUNCT
ejpam-6849	227	3	........	........	PUNCT
ejpam-6849	227	4	........	........	PUNCT
ejpam-6849	228	1	....	....	PUNCT
ejpam-6849	228	2	.........	.........	PUNCT
ejpam-6849	228	3	........	........	PUNCT
ejpam-6849	228	4	........	........	PUNCT
ejpam-6849	229	1	....	....	PUNCT
ejpam-6849	230	1	g2	g2	PROPN
ejpam-6849	230	2	x	x	PUNCT
ejpam-6849	231	1	x1	x1	PROPN
ejpam-6849	231	2	3	3	NUM
ejpam-6849	231	3	x2	x2	PROPN
ejpam-6849	231	4	3	3	NUM
ejpam-6849	231	5	xa−k−1	xa−k−1	NUM
ejpam-6849	231	6	3	3	NUM
ejpam-6849	231	7	xa−k	xa−k	PROPN
ejpam-6849	231	8	3	3	NUM
ejpam-6849	231	9	...	...	PUNCT
ejpam-6849	231	10	...	...	PUNCT
ejpam-6849	231	11	...	...	PUNCT
ejpam-6849	231	12	••	••	NOUN
ejpam-6849	231	13	••	••	NOUN
ejpam-6849	231	14	••	••	NOUN
ejpam-6849	231	15	y1z1	y1z1	PROPN
ejpam-6849	231	16	y2z2	y2z2	PROPN
ejpam-6849	231	17	ykzk	ykzk	ADJ
ejpam-6849	231	18	figure	figure	NOUN
ejpam-6849	231	19	1	1	NUM
ejpam-6849	231	20	:	:	PUNCT
ejpam-6849	231	21	graphs	graph	NOUN
ejpam-6849	231	22	satisfying	satisfy	VERB
ejpam-6849	231	23	the	the	DET
ejpam-6849	231	24	conditions	condition	NOUN
ejpam-6849	231	25	in	in	ADP
ejpam-6849	231	26	proposition	proposition	NOUN
ejpam-6849	231	27	6	6	NUM
ejpam-6849	231	28	γve(g	γve(g	NOUN
ejpam-6849	231	29	)	)	PUNCT
ejpam-6849	232	1	=	=	SYM
ejpam-6849	232	2	γt	γt	PROPN
ejpam-6849	232	3	ve(g	ve(g	PROPN
ejpam-6849	232	4	)	)	PUNCT
ejpam-6849	233	1	=	=	PUNCT
ejpam-6849	233	2	a.	a.	NOUN
ejpam-6849	233	3	now	now	ADV
ejpam-6849	233	4	,	,	PUNCT
ejpam-6849	233	5	let	let	VERB
ejpam-6849	233	6	b	b	NOUN
ejpam-6849	233	7	=	=	PUNCT
ejpam-6849	233	8	a	a	PROPN
ejpam-6849	234	1	+	+	X
ejpam-6849	234	2	k	k	NOUN
ejpam-6849	234	3	,	,	PUNCT
ejpam-6849	234	4	where	where	SCONJ
ejpam-6849	234	5	1	1	NUM
ejpam-6849	234	6	≤	≤	NUM
ejpam-6849	234	7	k	k	X
ejpam-6849	234	8	≤	≤	PROPN
ejpam-6849	234	9	a.	a.	NOUN
ejpam-6849	234	10	construct	construct	VERB
ejpam-6849	234	11	a	a	DET
ejpam-6849	234	12	graph	graph	NOUN
ejpam-6849	234	13	g1	g1	NOUN
ejpam-6849	234	14	as	as	ADP
ejpam-6849	234	15	above	above	ADV
ejpam-6849	234	16	but	but	CCONJ
ejpam-6849	234	17	using	use	VERB
ejpam-6849	234	18	only	only	ADV
ejpam-6849	234	19	a	a	DET
ejpam-6849	234	20	−	−	NOUN
ejpam-6849	234	21	k	k	PROPN
ejpam-6849	234	22	copies	copy	NOUN
ejpam-6849	234	23	of	of	ADP
ejpam-6849	234	24	p4	p4	ADJ
ejpam-6849	234	25	.	.	PUNCT
ejpam-6849	235	1	obtain	obtain	VERB
ejpam-6849	235	2	g	g	NOUN
ejpam-6849	235	3	as	as	ADP
ejpam-6849	235	4	the	the	DET
ejpam-6849	235	5	graph	graph	NOUN
ejpam-6849	235	6	g2	g2	PROPN
ejpam-6849	235	7	in	in	ADP
ejpam-6849	235	8	figure	figure	NOUN
ejpam-6849	235	9	1	1	NUM
ejpam-6849	235	10	obtained	obtain	VERB
ejpam-6849	235	11	by	by	ADP
ejpam-6849	235	12	connecting	connect	VERB
ejpam-6849	235	13	to	to	AUX
ejpam-6849	235	14	g1	g1	VERB
ejpam-6849	235	15	k	k	PROPN
ejpam-6849	235	16	copies	copy	NOUN
ejpam-6849	235	17	of	of	ADP
ejpam-6849	235	18	p5	p5	ADJ
ejpam-6849	235	19	using	use	VERB
ejpam-6849	235	20	vertex	vertex	NOUN
ejpam-6849	235	21	x.	x.	NOUN
ejpam-6849	236	1	then	then	ADV
ejpam-6849	236	2	s	s	VERB
ejpam-6849	236	3	=	=	PUNCT
ejpam-6849	236	4	{	{	PUNCT
ejpam-6849	236	5	x1	x1	PROPN
ejpam-6849	236	6	3	3	NUM
ejpam-6849	236	7	,	,	PUNCT
ejpam-6849	236	8	x2	x2	PROPN
ejpam-6849	236	9	3	3	NUM
ejpam-6849	236	10	,	,	PUNCT
ejpam-6849	236	11	.	.	PUNCT
ejpam-6849	236	12	.	.	PUNCT
ejpam-6849	236	13	.	.	PUNCT
ejpam-6849	237	1	,	,	PUNCT
ejpam-6849	237	2	xa−k	xa−k	PROPN
ejpam-6849	237	3	3	3	NUM
ejpam-6849	237	4	}	}	PUNCT
ejpam-6849	237	5	∪	∪	NOUN
ejpam-6849	237	6	{	{	PUNCT
ejpam-6849	237	7	y1	y1	NOUN
ejpam-6849	237	8	,	,	PUNCT
ejpam-6849	237	9	y2	y2	PROPN
ejpam-6849	237	10	,	,	PUNCT
ejpam-6849	237	11	.	.	PUNCT
ejpam-6849	237	12	.	.	PUNCT
ejpam-6849	237	13	.	.	PUNCT
ejpam-6849	238	1	,	,	PUNCT
ejpam-6849	238	2	yk	yk	PROPN
ejpam-6849	238	3	}	}	PUNCT
ejpam-6849	238	4	is	be	AUX
ejpam-6849	238	5	a	a	DET
ejpam-6849	238	6	γve	γve	NOUN
ejpam-6849	238	7	-	-	PUNCT
ejpam-6849	238	8	set	set	NOUN
ejpam-6849	238	9	of	of	ADP
ejpam-6849	238	10	g.	g.	PROPN
ejpam-6849	238	11	also	also	ADV
ejpam-6849	238	12	,	,	PUNCT
ejpam-6849	238	13	s	s	VERB
ejpam-6849	238	14	∪	∪	X
ejpam-6849	238	15	{	{	PUNCT
ejpam-6849	238	16	z1	z1	ADJ
ejpam-6849	238	17	,	,	PUNCT
ejpam-6849	238	18	z2	z2	PROPN
ejpam-6849	238	19	,	,	PUNCT
ejpam-6849	238	20	.	.	PUNCT
ejpam-6849	238	21	.	.	PUNCT
ejpam-6849	238	22	.	.	PUNCT
ejpam-6849	239	1	,	,	PUNCT
ejpam-6849	239	2	zk	zk	PROPN
ejpam-6849	239	3	}	}	PUNCT
ejpam-6849	239	4	is	be	AUX
ejpam-6849	239	5	a	a	DET
ejpam-6849	239	6	γt	γt	NOUN
ejpam-6849	239	7	ve	ve	NOUN
ejpam-6849	239	8	-	-	PUNCT
ejpam-6849	239	9	set	set	NOUN
ejpam-6849	239	10	of	of	ADP
ejpam-6849	239	11	g.	g.	PROPN
ejpam-6849	239	12	for	for	ADP
ejpam-6849	239	13	this	this	DET
ejpam-6849	239	14	g	g	PROPN
ejpam-6849	239	15	,	,	PUNCT
ejpam-6849	239	16	γve(g	γve(g	PROPN
ejpam-6849	239	17	)	)	PUNCT
ejpam-6849	240	1	=	=	PRON
ejpam-6849	240	2	(	(	PUNCT
ejpam-6849	240	3	a	a	DET
ejpam-6849	240	4	−	−	PROPN
ejpam-6849	240	5	k	k	NOUN
ejpam-6849	240	6	)	)	PUNCT
ejpam-6849	241	1	+	+	CCONJ
ejpam-6849	241	2	k	k	X
ejpam-6849	241	3	=	=	PUNCT
ejpam-6849	241	4	a	a	PROPN
ejpam-6849	241	5	and	and	CCONJ
ejpam-6849	241	6	γt	γt	NOUN
ejpam-6849	241	7	ve(g	ve(g	PROPN
ejpam-6849	241	8	)	)	PUNCT
ejpam-6849	242	1	=	=	PUNCT
ejpam-6849	243	1	a	a	DET
ejpam-6849	243	2	+	+	X
ejpam-6849	243	3	k	k	PROPN
ejpam-6849	243	4	=	=	PUNCT
ejpam-6849	243	5	b.	b.	PROPN
ejpam-6849	244	1	■	■	PUNCT
ejpam-6849	244	2	corollary	corollary	ADJ
ejpam-6849	244	3	1	1	NUM
ejpam-6849	244	4	.	.	PUNCT
ejpam-6849	245	1	the	the	DET
ejpam-6849	245	2	difference	difference	NOUN
ejpam-6849	245	3	γt	γt	X
ejpam-6849	245	4	ve(g	ve(g	PROPN
ejpam-6849	245	5	)	)	PUNCT
ejpam-6849	246	1	−	−	PROPN
ejpam-6849	246	2	γve(g	γve(g	PROPN
ejpam-6849	246	3	)	)	PUNCT
ejpam-6849	246	4	can	can	AUX
ejpam-6849	246	5	be	be	AUX
ejpam-6849	246	6	made	make	VERB
ejpam-6849	246	7	arbitrarily	arbitrarily	ADV
ejpam-6849	246	8	large	large	ADJ
ejpam-6849	246	9	.	.	PUNCT
ejpam-6849	247	1	proof	proof	NOUN
ejpam-6849	247	2	.	.	PUNCT
ejpam-6849	248	1	let	let	VERB
ejpam-6849	248	2	k	k	PRON
ejpam-6849	248	3	be	be	AUX
ejpam-6849	248	4	any	any	DET
ejpam-6849	248	5	positive	positive	ADJ
ejpam-6849	248	6	integer	integer	NOUN
ejpam-6849	248	7	.	.	PUNCT
ejpam-6849	249	1	choose	choose	VERB
ejpam-6849	249	2	an	an	DET
ejpam-6849	249	3	integer	integer	NOUN
ejpam-6849	249	4	a	a	DET
ejpam-6849	249	5	≥	≥	NOUN
ejpam-6849	249	6	k	k	NOUN
ejpam-6849	249	7	and	and	CCONJ
ejpam-6849	249	8	put	put	VERB
ejpam-6849	249	9	b	b	NOUN
ejpam-6849	249	10	=	=	PUNCT
ejpam-6849	249	11	a	a	DET
ejpam-6849	249	12	+	+	X
ejpam-6849	249	13	k.	k.	NOUN
ejpam-6849	249	14	by	by	ADP
ejpam-6849	249	15	proposition	proposition	NOUN
ejpam-6849	249	16	6	6	NUM
ejpam-6849	249	17	,	,	PUNCT
ejpam-6849	249	18	there	there	PRON
ejpam-6849	249	19	exists	exist	VERB
ejpam-6849	249	20	a	a	DET
ejpam-6849	249	21	connected	connected	ADJ
ejpam-6849	249	22	graph	graph	NOUN
ejpam-6849	249	23	g	g	NOUN
ejpam-6849	249	24	for	for	ADP
ejpam-6849	249	25	which	which	PRON
ejpam-6849	249	26	γt	γt	NOUN
ejpam-6849	249	27	ve(g	ve(g	PROPN
ejpam-6849	249	28	)	)	PUNCT
ejpam-6849	250	1	−	−	PROPN
ejpam-6849	250	2	γve(g	γve(g	PROPN
ejpam-6849	250	3	)	)	PUNCT
ejpam-6849	251	1	=	=	SYM
ejpam-6849	251	2	b	b	X
ejpam-6849	251	3	−	−	PROPN
ejpam-6849	251	4	a	a	PROPN
ejpam-6849	251	5	=	=	X
ejpam-6849	251	6	k.	k.	PROPN
ejpam-6849	251	7	since	since	SCONJ
ejpam-6849	251	8	k	k	PROPN
ejpam-6849	251	9	is	be	AUX
ejpam-6849	251	10	arbitrary	arbitrary	ADJ
ejpam-6849	251	11	,	,	PUNCT
ejpam-6849	251	12	the	the	DET
ejpam-6849	251	13	conclusion	conclusion	NOUN
ejpam-6849	251	14	follows	follow	VERB
ejpam-6849	251	15	.	.	PUNCT
ejpam-6849	252	1	■	■	PUNCT
ejpam-6849	252	2	3	3	X
ejpam-6849	252	3	.	.	X
ejpam-6849	252	4	on	on	ADP
ejpam-6849	252	5	families	family	NOUN
ejpam-6849	252	6	of	of	ADP
ejpam-6849	252	7	graphs	graph	NOUN
ejpam-6849	252	8	under	under	ADP
ejpam-6849	252	9	binary	binary	ADJ
ejpam-6849	252	10	operations	operation	NOUN
ejpam-6849	252	11	proposition	proposition	NOUN
ejpam-6849	252	12	7	7	NUM
ejpam-6849	252	13	.	.	PUNCT
ejpam-6849	252	14	let	let	VERB
ejpam-6849	252	15	g	g	NOUN
ejpam-6849	252	16	and	and	CCONJ
ejpam-6849	252	17	h	h	NOUN
ejpam-6849	252	18	be	be	VERB
ejpam-6849	252	19	any	any	DET
ejpam-6849	252	20	graphs	graph	NOUN
ejpam-6849	252	21	,	,	PUNCT
ejpam-6849	252	22	and	and	CCONJ
ejpam-6849	252	23	let	let	VERB
ejpam-6849	252	24	s	s	PRON
ejpam-6849	252	25	⊆	⊆	NUM
ejpam-6849	252	26	(	(	PUNCT
ejpam-6849	252	27	g	g	PROPN
ejpam-6849	252	28	+	+	NOUN
ejpam-6849	252	29	h	h	NOUN
ejpam-6849	252	30	)	)	PUNCT
ejpam-6849	252	31	with	with	ADP
ejpam-6849	252	32	s	s	VERB
ejpam-6849	252	33	̸=	̸=	PROPN
ejpam-6849	252	34	∅.	∅.	NOUN
ejpam-6849	252	35	then	then	ADV
ejpam-6849	252	36	s	s	VERB
ejpam-6849	252	37	is	be	AUX
ejpam-6849	252	38	a	a	DET
ejpam-6849	252	39	(	(	PUNCT
ejpam-6849	252	40	total	total	NOUN
ejpam-6849	252	41	)	)	PUNCT
ejpam-6849	252	42	ve	ve	AUX
ejpam-6849	252	43	-	-	PUNCT
ejpam-6849	252	44	dominating	dominate	VERB
ejpam-6849	252	45	set	set	NOUN
ejpam-6849	252	46	of	of	ADP
ejpam-6849	252	47	g	g	PROPN
ejpam-6849	253	1	+	+	CCONJ
ejpam-6849	253	2	h	h	NOUN
ejpam-6849	253	3	if	if	SCONJ
ejpam-6849	253	4	and	and	CCONJ
ejpam-6849	253	5	only	only	ADV
ejpam-6849	253	6	if	if	SCONJ
ejpam-6849	253	7	one	one	NUM
ejpam-6849	253	8	of	of	ADP
ejpam-6849	253	9	the	the	DET
ejpam-6849	253	10	following	follow	VERB
ejpam-6849	253	11	holds	hold	NOUN
ejpam-6849	253	12	:	:	PUNCT
ejpam-6849	253	13	j.	j.	PROPN
ejpam-6849	253	14	tayab	tayab	PROPN
ejpam-6849	253	15	,	,	PUNCT
ejpam-6849	253	16	f.	f.	PROPN
ejpam-6849	253	17	p.	p.	PROPN
ejpam-6849	253	18	jamil	jamil	PROPN
ejpam-6849	253	19	,	,	PUNCT
ejpam-6849	253	20	i.	i.	PROPN
ejpam-6849	253	21	aniversario	aniversario	PROPN
ejpam-6849	253	22	/	/	SYM
ejpam-6849	253	23	eur	eur	PROPN
ejpam-6849	253	24	.	.	PUNCT
ejpam-6849	254	1	j.	j.	PROPN
ejpam-6849	254	2	pure	pure	PROPN
ejpam-6849	254	3	appl	appl	PROPN
ejpam-6849	254	4	.	.	PROPN
ejpam-6849	254	5	math	math	PROPN
ejpam-6849	254	6	,	,	PUNCT
ejpam-6849	254	7	18	18	NUM
ejpam-6849	254	8	(	(	PUNCT
ejpam-6849	254	9	4	4	NUM
ejpam-6849	254	10	)	)	PUNCT
ejpam-6849	254	11	(	(	PUNCT
ejpam-6849	254	12	2025	2025	NUM
ejpam-6849	254	13	)	)	PUNCT
ejpam-6849	254	14	,	,	PUNCT
ejpam-6849	254	15	6849	6849	NUM
ejpam-6849	254	16	7	7	NUM
ejpam-6849	254	17	of	of	ADP
ejpam-6849	254	18	19	19	NUM
ejpam-6849	254	19	(	(	PUNCT
ejpam-6849	254	20	i	i	NOUN
ejpam-6849	254	21	)	)	PUNCT
ejpam-6849	254	22	s	s	VERB
ejpam-6849	254	23	⊆	⊆	NUM
ejpam-6849	254	24	v	v	NOUN
ejpam-6849	254	25	(	(	PUNCT
ejpam-6849	254	26	g	g	NOUN
ejpam-6849	254	27	)	)	PUNCT
ejpam-6849	254	28	and	and	CCONJ
ejpam-6849	254	29	s	s	VERB
ejpam-6849	254	30	is	be	AUX
ejpam-6849	254	31	a	a	DET
ejpam-6849	254	32	(	(	PUNCT
ejpam-6849	254	33	total	total	NOUN
ejpam-6849	254	34	)	)	PUNCT
ejpam-6849	254	35	ve	ve	AUX
ejpam-6849	254	36	-	-	PUNCT
ejpam-6849	254	37	dominating	dominate	VERB
ejpam-6849	254	38	set	set	NOUN
ejpam-6849	254	39	of	of	ADP
ejpam-6849	254	40	g	g	NOUN
ejpam-6849	254	41	;	;	PUNCT
ejpam-6849	254	42	(	(	PUNCT
ejpam-6849	254	43	ii	ii	NOUN
ejpam-6849	254	44	)	)	PUNCT
ejpam-6849	254	45	s	s	PART
ejpam-6849	254	46	⊆	⊆	NUM
ejpam-6849	254	47	v	v	NOUN
ejpam-6849	254	48	(	(	PUNCT
ejpam-6849	254	49	h	h	NOUN
ejpam-6849	254	50	)	)	PUNCT
ejpam-6849	254	51	and	and	CCONJ
ejpam-6849	254	52	s	s	VERB
ejpam-6849	254	53	is	be	AUX
ejpam-6849	254	54	a	a	DET
ejpam-6849	254	55	(	(	PUNCT
ejpam-6849	254	56	total	total	NOUN
ejpam-6849	254	57	)	)	PUNCT
ejpam-6849	254	58	ve	ve	AUX
ejpam-6849	254	59	-	-	PUNCT
ejpam-6849	254	60	dominating	dominate	VERB
ejpam-6849	254	61	set	set	NOUN
ejpam-6849	254	62	of	of	ADP
ejpam-6849	254	63	h	h	NOUN
ejpam-6849	254	64	;	;	PUNCT
ejpam-6849	254	65	(	(	PUNCT
ejpam-6849	254	66	iii	iii	X
ejpam-6849	254	67	)	)	PUNCT
ejpam-6849	254	68	s	s	PART
ejpam-6849	254	69	∩	∩	ADJ
ejpam-6849	254	70	v	v	ADJ
ejpam-6849	254	71	(	(	PUNCT
ejpam-6849	254	72	g	g	NOUN
ejpam-6849	254	73	)	)	PUNCT
ejpam-6849	254	74	̸=	̸=	PROPN
ejpam-6849	254	75	∅	∅	NOUN
ejpam-6849	254	76	and	and	CCONJ
ejpam-6849	254	77	s	s	VERB
ejpam-6849	254	78	∩	∩	ADJ
ejpam-6849	254	79	v	v	ADJ
ejpam-6849	254	80	(	(	PUNCT
ejpam-6849	254	81	h	h	NOUN
ejpam-6849	254	82	)	)	PUNCT
ejpam-6849	254	83	̸=	̸=	PROPN
ejpam-6849	254	84	∅.	∅.	PRON
ejpam-6849	254	85	proof	proof	NOUN
ejpam-6849	254	86	.	.	PUNCT
ejpam-6849	255	1	assume	assume	VERB
ejpam-6849	255	2	s	s	PRON
ejpam-6849	255	3	is	be	AUX
ejpam-6849	255	4	a	a	DET
ejpam-6849	255	5	(	(	PUNCT
ejpam-6849	255	6	total	total	NOUN
ejpam-6849	255	7	)	)	PUNCT
ejpam-6849	255	8	ve	ve	AUX
ejpam-6849	255	9	-	-	PUNCT
ejpam-6849	255	10	dominating	dominate	VERB
ejpam-6849	255	11	set	set	NOUN
ejpam-6849	255	12	of	of	ADP
ejpam-6849	255	13	g	g	PROPN
ejpam-6849	255	14	+	+	CCONJ
ejpam-6849	255	15	h.	h.	PROPN
ejpam-6849	255	16	suppose	suppose	VERB
ejpam-6849	255	17	that	that	SCONJ
ejpam-6849	255	18	s	s	VERB
ejpam-6849	255	19	⊆	⊆	NUM
ejpam-6849	255	20	v	v	NOUN
ejpam-6849	255	21	(	(	PUNCT
ejpam-6849	255	22	g	g	NOUN
ejpam-6849	255	23	)	)	PUNCT
ejpam-6849	255	24	.	.	PUNCT
ejpam-6849	256	1	if	if	SCONJ
ejpam-6849	256	2	s	s	VERB
ejpam-6849	256	3	=	=	SYM
ejpam-6849	256	4	v	v	X
ejpam-6849	256	5	(	(	PUNCT
ejpam-6849	256	6	g	g	NOUN
ejpam-6849	256	7	)	)	PUNCT
ejpam-6849	256	8	,	,	PUNCT
ejpam-6849	256	9	then	then	ADV
ejpam-6849	256	10	we	we	PRON
ejpam-6849	256	11	are	be	AUX
ejpam-6849	256	12	done	do	VERB
ejpam-6849	256	13	.	.	PUNCT
ejpam-6849	257	1	suppose	suppose	VERB
ejpam-6849	257	2	that	that	SCONJ
ejpam-6849	257	3	s	s	VERB
ejpam-6849	257	4	̸=	̸=	PROPN
ejpam-6849	257	5	v	v	NOUN
ejpam-6849	257	6	(	(	PUNCT
ejpam-6849	257	7	g	g	NOUN
ejpam-6849	257	8	)	)	PUNCT
ejpam-6849	257	9	.	.	PUNCT
ejpam-6849	258	1	since	since	SCONJ
ejpam-6849	258	2	v	v	NOUN
ejpam-6849	258	3	(	(	PUNCT
ejpam-6849	258	4	g	g	NOUN
ejpam-6849	258	5	)	)	PUNCT
ejpam-6849	258	6	\	\	NOUN
ejpam-6849	258	7	ng(s	ng(s	NUM
ejpam-6849	258	8	)	)	PUNCT
ejpam-6849	258	9	=	=	SYM
ejpam-6849	258	10	v	v	X
ejpam-6849	258	11	(	(	PUNCT
ejpam-6849	258	12	g	g	PROPN
ejpam-6849	258	13	+	+	NOUN
ejpam-6849	258	14	h	h	NOUN
ejpam-6849	258	15	)	)	PUNCT
ejpam-6849	258	16	\	\	NOUN
ejpam-6849	258	17	ng+h(s	ng+h(s	PROPN
ejpam-6849	258	18	)	)	PUNCT
ejpam-6849	258	19	,	,	PUNCT
ejpam-6849	258	20	(	(	PUNCT
ejpam-6849	258	21	1	1	X
ejpam-6849	258	22	)	)	PUNCT
ejpam-6849	258	23	v	v	NOUN
ejpam-6849	258	24	(	(	PUNCT
ejpam-6849	258	25	g	g	NOUN
ejpam-6849	258	26	)	)	PUNCT
ejpam-6849	258	27	\	\	NOUN
ejpam-6849	258	28	ng(s	ng(s	NUM
ejpam-6849	258	29	)	)	PUNCT
ejpam-6849	258	30	is	be	AUX
ejpam-6849	258	31	an	an	DET
ejpam-6849	258	32	independent	independent	ADJ
ejpam-6849	258	33	set	set	NOUN
ejpam-6849	258	34	.	.	PUNCT
ejpam-6849	259	1	by	by	ADP
ejpam-6849	259	2	proposition	proposition	NOUN
ejpam-6849	259	3	1	1	NUM
ejpam-6849	259	4	,	,	PUNCT
ejpam-6849	259	5	s	s	VERB
ejpam-6849	259	6	is	be	AUX
ejpam-6849	259	7	a	a	DET
ejpam-6849	259	8	(	(	PUNCT
ejpam-6849	259	9	total	total	NOUN
ejpam-6849	259	10	)	)	PUNCT
ejpam-6849	259	11	ve	ve	AUX
ejpam-6849	259	12	-	-	PUNCT
ejpam-6849	259	13	dominating	dominate	VERB
ejpam-6849	259	14	set	set	NOUN
ejpam-6849	259	15	of	of	ADP
ejpam-6849	259	16	g	g	NOUN
ejpam-6849	259	17	,	,	PUNCT
ejpam-6849	259	18	and	and	CCONJ
ejpam-6849	259	19	(	(	PUNCT
ejpam-6849	259	20	i	i	NOUN
ejpam-6849	259	21	)	)	PUNCT
ejpam-6849	259	22	holds	hold	VERB
ejpam-6849	259	23	.	.	PUNCT
ejpam-6849	260	1	similarly	similarly	ADV
ejpam-6849	260	2	,	,	PUNCT
ejpam-6849	260	3	if	if	SCONJ
ejpam-6849	260	4	s	s	VERB
ejpam-6849	260	5	⊆	⊆	NUM
ejpam-6849	260	6	v	v	NOUN
ejpam-6849	260	7	(	(	PUNCT
ejpam-6849	260	8	h	h	NOUN
ejpam-6849	260	9	)	)	PUNCT
ejpam-6849	260	10	,	,	PUNCT
ejpam-6849	260	11	then	then	ADV
ejpam-6849	260	12	(	(	PUNCT
ejpam-6849	260	13	ii	ii	NOUN
ejpam-6849	260	14	)	)	PUNCT
ejpam-6849	260	15	holds	hold	VERB
ejpam-6849	260	16	.	.	PUNCT
ejpam-6849	261	1	if	if	SCONJ
ejpam-6849	261	2	both	both	PRON
ejpam-6849	261	3	(	(	PUNCT
ejpam-6849	261	4	i	i	NOUN
ejpam-6849	261	5	)	)	PUNCT
ejpam-6849	261	6	and	and	CCONJ
ejpam-6849	261	7	(	(	PUNCT
ejpam-6849	261	8	ii	ii	NOUN
ejpam-6849	261	9	)	)	PUNCT
ejpam-6849	261	10	do	do	AUX
ejpam-6849	261	11	not	not	PART
ejpam-6849	261	12	hold	hold	VERB
ejpam-6849	261	13	,	,	PUNCT
ejpam-6849	261	14	then	then	ADV
ejpam-6849	261	15	(	(	PUNCT
ejpam-6849	261	16	iii	iii	NOUN
ejpam-6849	261	17	)	)	PUNCT
ejpam-6849	261	18	holds	hold	VERB
ejpam-6849	261	19	.	.	PUNCT
ejpam-6849	262	1	conversely	conversely	ADV
ejpam-6849	262	2	,	,	PUNCT
ejpam-6849	262	3	if	if	SCONJ
ejpam-6849	262	4	(	(	PUNCT
ejpam-6849	262	5	i	i	NOUN
ejpam-6849	262	6	)	)	PUNCT
ejpam-6849	262	7	holds	hold	VERB
ejpam-6849	262	8	,	,	PUNCT
ejpam-6849	262	9	then	then	ADV
ejpam-6849	262	10	equation	equation	NOUN
ejpam-6849	262	11	(	(	PUNCT
ejpam-6849	262	12	1	1	NUM
ejpam-6849	262	13	)	)	PUNCT
ejpam-6849	262	14	implies	imply	VERB
ejpam-6849	262	15	that	that	SCONJ
ejpam-6849	262	16	v	v	X
ejpam-6849	262	17	(	(	PUNCT
ejpam-6849	262	18	g	g	PROPN
ejpam-6849	262	19	+	+	NOUN
ejpam-6849	262	20	h	h	NOUN
ejpam-6849	262	21	)	)	PUNCT
ejpam-6849	262	22	\	\	NOUN
ejpam-6849	262	23	ng+h(s	ng+h(s	PROPN
ejpam-6849	262	24	)	)	PUNCT
ejpam-6849	262	25	is	be	AUX
ejpam-6849	262	26	an	an	DET
ejpam-6849	262	27	independent	independent	ADJ
ejpam-6849	262	28	set	set	NOUN
ejpam-6849	262	29	.	.	PUNCT
ejpam-6849	263	1	consequently	consequently	ADV
ejpam-6849	263	2	,	,	PUNCT
ejpam-6849	263	3	s	s	VERB
ejpam-6849	263	4	is	be	AUX
ejpam-6849	263	5	a	a	DET
ejpam-6849	263	6	(	(	PUNCT
ejpam-6849	263	7	total	total	NOUN
ejpam-6849	263	8	)	)	PUNCT
ejpam-6849	263	9	ve	ve	AUX
ejpam-6849	263	10	-	-	PUNCT
ejpam-6849	263	11	dominating	dominate	VERB
ejpam-6849	263	12	set	set	NOUN
ejpam-6849	263	13	of	of	ADP
ejpam-6849	263	14	g	g	PROPN
ejpam-6849	263	15	+	+	CCONJ
ejpam-6849	263	16	h.	h.	NOUN
ejpam-6849	263	17	the	the	DET
ejpam-6849	263	18	same	same	ADJ
ejpam-6849	263	19	conclusion	conclusion	NOUN
ejpam-6849	263	20	is	be	AUX
ejpam-6849	263	21	attained	attain	VERB
ejpam-6849	263	22	if	if	SCONJ
ejpam-6849	263	23	(	(	PUNCT
ejpam-6849	263	24	ii	ii	NOUN
ejpam-6849	263	25	)	)	PUNCT
ejpam-6849	263	26	holds	hold	VERB
ejpam-6849	263	27	.	.	PUNCT
ejpam-6849	264	1	now	now	ADV
ejpam-6849	264	2	,	,	PUNCT
ejpam-6849	264	3	suppose	suppose	VERB
ejpam-6849	264	4	that	that	SCONJ
ejpam-6849	264	5	(	(	PUNCT
ejpam-6849	264	6	iii	iii	NOUN
ejpam-6849	264	7	)	)	PUNCT
ejpam-6849	264	8	holds	hold	VERB
ejpam-6849	264	9	for	for	ADP
ejpam-6849	264	10	s.	s.	PROPN
ejpam-6849	264	11	since	since	SCONJ
ejpam-6849	264	12	v	v	PROPN
ejpam-6849	264	13	(	(	PUNCT
ejpam-6849	264	14	g	g	PROPN
ejpam-6849	264	15	+	+	NOUN
ejpam-6849	264	16	h	h	NOUN
ejpam-6849	264	17	)	)	PUNCT
ejpam-6849	264	18	\	\	NOUN
ejpam-6849	264	19	ng+h(s	ng+h(s	PROPN
ejpam-6849	264	20	)	)	PUNCT
ejpam-6849	264	21	=	=	NOUN
ejpam-6849	264	22	∅	∅	NOUN
ejpam-6849	264	23	,	,	PUNCT
ejpam-6849	264	24	s	s	PART
ejpam-6849	264	25	is	be	AUX
ejpam-6849	264	26	a	a	DET
ejpam-6849	264	27	(	(	PUNCT
ejpam-6849	264	28	total	total	NOUN
ejpam-6849	264	29	)	)	PUNCT
ejpam-6849	264	30	ve	ve	AUX
ejpam-6849	264	31	-	-	PUNCT
ejpam-6849	264	32	dominating	dominate	VERB
ejpam-6849	264	33	set	set	NOUN
ejpam-6849	264	34	of	of	ADP
ejpam-6849	264	35	g	g	PROPN
ejpam-6849	264	36	+	+	CCONJ
ejpam-6849	264	37	h	h	NOUN
ejpam-6849	264	38	by	by	ADP
ejpam-6849	264	39	proposition	proposition	NOUN
ejpam-6849	264	40	1	1	NUM
ejpam-6849	264	41	.	.	PUNCT
ejpam-6849	265	1	■	■	PUNCT
ejpam-6849	265	2	corollary	corollary	ADJ
ejpam-6849	265	3	2	2	NUM
ejpam-6849	265	4	.	.	PUNCT
ejpam-6849	266	1	let	let	VERB
ejpam-6849	266	2	g	g	NOUN
ejpam-6849	266	3	and	and	CCONJ
ejpam-6849	266	4	h	h	NOUN
ejpam-6849	266	5	be	be	VERB
ejpam-6849	266	6	any	any	DET
ejpam-6849	266	7	graphs	graph	NOUN
ejpam-6849	266	8	.	.	PUNCT
ejpam-6849	267	1	then	then	ADV
ejpam-6849	267	2	γt	γt	PROPN
ejpam-6849	267	3	ve(g	ve(g	PROPN
ejpam-6849	267	4	+	+	CCONJ
ejpam-6849	267	5	h	h	NOUN
ejpam-6849	267	6	)	)	PUNCT
ejpam-6849	267	7	=	=	SYM
ejpam-6849	267	8	2	2	NUM
ejpam-6849	267	9	,	,	PUNCT
ejpam-6849	267	10	and	and	CCONJ
ejpam-6849	267	11	γve(g	γve(g	PROPN
ejpam-6849	268	1	+	+	CCONJ
ejpam-6849	268	2	h	h	X
ejpam-6849	268	3	)	)	PUNCT
ejpam-6849	268	4	=	=	NOUN
ejpam-6849	268	5	{	{	PUNCT
ejpam-6849	268	6	1	1	NUM
ejpam-6849	268	7	,	,	PUNCT
ejpam-6849	268	8	if	if	SCONJ
ejpam-6849	268	9	g	g	PROPN
ejpam-6849	268	10	or	or	CCONJ
ejpam-6849	268	11	h	h	NOUN
ejpam-6849	268	12	is	be	AUX
ejpam-6849	268	13	empty	empty	ADJ
ejpam-6849	268	14	;	;	PUNCT
ejpam-6849	268	15	min{γve(g	min{γve(g	PROPN
ejpam-6849	268	16	)	)	PUNCT
ejpam-6849	268	17	,	,	PUNCT
ejpam-6849	268	18	γve(h	γve(h	PROPN
ejpam-6849	268	19	)	)	PUNCT
ejpam-6849	268	20	,	,	PUNCT
ejpam-6849	268	21	2	2	NUM
ejpam-6849	268	22	}	}	PUNCT
ejpam-6849	268	23	,	,	PUNCT
ejpam-6849	268	24	else	else	ADV
ejpam-6849	268	25	.	.	PUNCT
ejpam-6849	269	1	in	in	ADP
ejpam-6849	269	2	particular	particular	ADJ
ejpam-6849	269	3	,	,	PUNCT
ejpam-6849	269	4	γve(km	γve(km	NOUN
ejpam-6849	269	5	,	,	PUNCT
ejpam-6849	269	6	n	n	CCONJ
ejpam-6849	269	7	)	)	PUNCT
ejpam-6849	269	8	=	=	SYM
ejpam-6849	269	9	1	1	NUM
ejpam-6849	269	10	and	and	CCONJ
ejpam-6849	269	11	γt	γt	PROPN
ejpam-6849	269	12	ve(km	ve(km	PROPN
ejpam-6849	269	13	,	,	PUNCT
ejpam-6849	269	14	n	n	CCONJ
ejpam-6849	269	15	)	)	PUNCT
ejpam-6849	269	16	=	=	SYM
ejpam-6849	269	17	2	2	NUM
ejpam-6849	269	18	for	for	ADP
ejpam-6849	269	19	all	all	DET
ejpam-6849	269	20	m	m	NOUN
ejpam-6849	269	21	,	,	PUNCT
ejpam-6849	269	22	n	n	PRON
ejpam-6849	269	23	≥	≥	NOUN
ejpam-6849	269	24	1	1	NUM
ejpam-6849	269	25	.	.	PUNCT
ejpam-6849	270	1	proposition	proposition	NOUN
ejpam-6849	270	2	8	8	NUM
ejpam-6849	270	3	.	.	PUNCT
ejpam-6849	271	1	let	let	VERB
ejpam-6849	271	2	g	g	PRON
ejpam-6849	271	3	be	be	AUX
ejpam-6849	271	4	a	a	DET
ejpam-6849	271	5	connected	connected	ADJ
ejpam-6849	271	6	graph	graph	NOUN
ejpam-6849	271	7	and	and	CCONJ
ejpam-6849	271	8	h	h	NOUN
ejpam-6849	271	9	be	be	AUX
ejpam-6849	271	10	a	a	DET
ejpam-6849	271	11	nonempty	nonempty	ADJ
ejpam-6849	271	12	graph	graph	NOUN
ejpam-6849	271	13	.	.	PUNCT
ejpam-6849	272	1	then	then	ADV
ejpam-6849	272	2	s	s	VERB
ejpam-6849	272	3	⊆	⊆	NUM
ejpam-6849	272	4	v	v	NOUN
ejpam-6849	272	5	(	(	PUNCT
ejpam-6849	272	6	g	g	PROPN
ejpam-6849	272	7	◦	◦	NOUN
ejpam-6849	272	8	h	h	NOUN
ejpam-6849	272	9	)	)	PUNCT
ejpam-6849	272	10	is	be	AUX
ejpam-6849	272	11	a	a	DET
ejpam-6849	272	12	ve	ve	NOUN
ejpam-6849	272	13	-	-	PUNCT
ejpam-6849	272	14	dominating	dominate	VERB
ejpam-6849	272	15	set	set	NOUN
ejpam-6849	272	16	of	of	ADP
ejpam-6849	272	17	g	g	PROPN
ejpam-6849	272	18	◦	◦	NOUN
ejpam-6849	272	19	h	h	NOUN
ejpam-6849	273	1	if	if	SCONJ
ejpam-6849	274	1	and	and	CCONJ
ejpam-6849	274	2	only	only	ADV
ejpam-6849	274	3	if	if	SCONJ
ejpam-6849	274	4	s	s	VERB
ejpam-6849	274	5	=	=	NOUN
ejpam-6849	274	6	a	a	DET
ejpam-6849	274	7	∪	∪	X
ejpam-6849	274	8	(	(	PUNCT
ejpam-6849	274	9	∪v∈v	∪v∈v	X
ejpam-6849	274	10	(	(	PUNCT
ejpam-6849	274	11	g)sv	g)sv	PROPN
ejpam-6849	274	12	)	)	PUNCT
ejpam-6849	274	13	,	,	PUNCT
ejpam-6849	274	14	(	(	PUNCT
ejpam-6849	274	15	2	2	X
ejpam-6849	274	16	)	)	PUNCT
ejpam-6849	274	17	where	where	SCONJ
ejpam-6849	274	18	a	a	DET
ejpam-6849	274	19	⊆	⊆	NUM
ejpam-6849	274	20	v	v	NOUN
ejpam-6849	274	21	(	(	PUNCT
ejpam-6849	274	22	g	g	NOUN
ejpam-6849	274	23	)	)	PUNCT
ejpam-6849	274	24	and	and	CCONJ
ejpam-6849	274	25	sv	sv	X
ejpam-6849	274	26	⊆	⊆	NUM
ejpam-6849	274	27	v	v	X
ejpam-6849	274	28	(	(	PUNCT
ejpam-6849	274	29	hv	hv	PROPN
ejpam-6849	274	30	)	)	PUNCT
ejpam-6849	274	31	for	for	ADP
ejpam-6849	274	32	each	each	DET
ejpam-6849	274	33	v	v	NUM
ejpam-6849	274	34	∈	∈	PROPN
ejpam-6849	274	35	v	v	NOUN
ejpam-6849	274	36	(	(	PUNCT
ejpam-6849	274	37	g	g	NOUN
ejpam-6849	274	38	)	)	PUNCT
ejpam-6849	274	39	such	such	ADJ
ejpam-6849	274	40	that	that	SCONJ
ejpam-6849	274	41	sv	sv	PROPN
ejpam-6849	274	42	is	be	AUX
ejpam-6849	274	43	a	a	DET
ejpam-6849	274	44	ve	ve	NOUN
ejpam-6849	274	45	-	-	PUNCT
ejpam-6849	274	46	dominating	dominate	VERB
ejpam-6849	274	47	set	set	NOUN
ejpam-6849	274	48	of	of	ADP
ejpam-6849	274	49	hv	hv	PROPN
ejpam-6849	274	50	for	for	ADP
ejpam-6849	274	51	all	all	DET
ejpam-6849	274	52	v	v	ADP
ejpam-6849	274	53	∈	∈	NUM
ejpam-6849	274	54	v	v	NOUN
ejpam-6849	274	55	(	(	PUNCT
ejpam-6849	274	56	g	g	NOUN
ejpam-6849	274	57	)	)	PUNCT
ejpam-6849	274	58	\	\	NOUN
ejpam-6849	274	59	a.	a.	NOUN
ejpam-6849	274	60	proof	proof	NOUN
ejpam-6849	274	61	.	.	PUNCT
ejpam-6849	275	1	first	first	ADV
ejpam-6849	275	2	,	,	PUNCT
ejpam-6849	275	3	assume	assume	VERB
ejpam-6849	275	4	that	that	SCONJ
ejpam-6849	275	5	s	s	VERB
ejpam-6849	275	6	is	be	AUX
ejpam-6849	275	7	a	a	DET
ejpam-6849	275	8	ve	ve	NOUN
ejpam-6849	275	9	-	-	PUNCT
ejpam-6849	275	10	dominating	dominate	VERB
ejpam-6849	275	11	set	set	NOUN
ejpam-6849	275	12	of	of	ADP
ejpam-6849	275	13	g	g	PROPN
ejpam-6849	275	14	◦	◦	PROPN
ejpam-6849	275	15	h.	h.	PROPN
ejpam-6849	275	16	put	put	VERB
ejpam-6849	275	17	a	a	DET
ejpam-6849	275	18	=	=	SYM
ejpam-6849	275	19	s	s	NOUN
ejpam-6849	275	20	∩	∩	ADJ
ejpam-6849	275	21	v	v	X
ejpam-6849	275	22	(	(	PUNCT
ejpam-6849	275	23	g	g	NOUN
ejpam-6849	275	24	)	)	PUNCT
ejpam-6849	275	25	and	and	CCONJ
ejpam-6849	275	26	sv	sv	X
ejpam-6849	275	27	=	=	SYM
ejpam-6849	275	28	s	s	PROPN
ejpam-6849	275	29	∩	∩	ADJ
ejpam-6849	275	30	v	v	X
ejpam-6849	275	31	(	(	PUNCT
ejpam-6849	275	32	hv	hv	PROPN
ejpam-6849	275	33	)	)	PUNCT
ejpam-6849	275	34	for	for	ADP
ejpam-6849	275	35	each	each	DET
ejpam-6849	275	36	v	v	NUM
ejpam-6849	275	37	∈	∈	PROPN
ejpam-6849	275	38	v	v	NOUN
ejpam-6849	275	39	(	(	PUNCT
ejpam-6849	275	40	g	g	NOUN
ejpam-6849	275	41	)	)	PUNCT
ejpam-6849	275	42	.	.	PUNCT
ejpam-6849	276	1	then	then	ADV
ejpam-6849	276	2	s	s	VERB
ejpam-6849	276	3	satisfies	satisfie	NOUN
ejpam-6849	276	4	equation	equation	NOUN
ejpam-6849	276	5	1	1	NUM
ejpam-6849	276	6	.	.	PUNCT
ejpam-6849	277	1	let	let	VERB
ejpam-6849	277	2	v	v	NUM
ejpam-6849	277	3	∈	∈	PROPN
ejpam-6849	277	4	v	v	NOUN
ejpam-6849	277	5	(	(	PUNCT
ejpam-6849	277	6	g	g	NOUN
ejpam-6849	277	7	)	)	PUNCT
ejpam-6849	277	8	\	\	PROPN
ejpam-6849	278	1	a	a	PRON
ejpam-6849	278	2	,	,	PUNCT
ejpam-6849	278	3	and	and	CCONJ
ejpam-6849	278	4	xy	xy	PROPN
ejpam-6849	278	5	∈	∈	PROPN
ejpam-6849	278	6	e(hv	e(hv	PROPN
ejpam-6849	278	7	)	)	PUNCT
ejpam-6849	278	8	.	.	PUNCT
ejpam-6849	279	1	since	since	SCONJ
ejpam-6849	279	2	s	s	PROPN
ejpam-6849	279	3	is	be	AUX
ejpam-6849	279	4	a	a	DET
ejpam-6849	279	5	ve	ve	NOUN
ejpam-6849	279	6	-	-	PUNCT
ejpam-6849	279	7	dominating	dominating	NOUN
ejpam-6849	279	8	set	set	NOUN
ejpam-6849	279	9	,	,	PUNCT
ejpam-6849	279	10	there	there	PRON
ejpam-6849	279	11	exists	exist	VERB
ejpam-6849	279	12	u	u	PROPN
ejpam-6849	279	13	∈	∈	PROPN
ejpam-6849	279	14	s	s	VERB
ejpam-6849	279	15	such	such	ADJ
ejpam-6849	279	16	that	that	SCONJ
ejpam-6849	279	17	u	u	PROPN
ejpam-6849	279	18	ve	ve	VERB
ejpam-6849	279	19	-	-	VERB
ejpam-6849	279	20	dominates	dominate	VERB
ejpam-6849	279	21	xy	xy	PROPN
ejpam-6849	279	22	.	.	PUNCT
ejpam-6849	280	1	since	since	SCONJ
ejpam-6849	280	2	v	v	NUM
ejpam-6849	280	3	/∈	/∈	SYM
ejpam-6849	280	4	s	s	X
ejpam-6849	280	5	,	,	PUNCT
ejpam-6849	280	6	u	u	PROPN
ejpam-6849	280	7	∈	∈	PROPN
ejpam-6849	280	8	sv	sv	PROPN
ejpam-6849	280	9	.	.	PUNCT
ejpam-6849	280	10	accordingly	accordingly	ADV
ejpam-6849	280	11	,	,	PUNCT
ejpam-6849	280	12	sv	sv	PROPN
ejpam-6849	280	13	is	be	AUX
ejpam-6849	280	14	a	a	DET
ejpam-6849	280	15	ve	ve	NOUN
ejpam-6849	280	16	-	-	PUNCT
ejpam-6849	280	17	dominating	dominate	VERB
ejpam-6849	280	18	set	set	NOUN
ejpam-6849	280	19	of	of	ADP
ejpam-6849	280	20	hv	hv	PROPN
ejpam-6849	280	21	.	.	PUNCT
ejpam-6849	281	1	conversely	conversely	ADV
ejpam-6849	281	2	,	,	PUNCT
ejpam-6849	281	3	suppose	suppose	VERB
ejpam-6849	281	4	s	s	NOUN
ejpam-6849	281	5	is	be	AUX
ejpam-6849	281	6	as	as	SCONJ
ejpam-6849	281	7	given	give	VERB
ejpam-6849	281	8	in	in	ADP
ejpam-6849	281	9	equation	equation	NOUN
ejpam-6849	281	10	1	1	NUM
ejpam-6849	281	11	such	such	ADJ
ejpam-6849	281	12	that	that	SCONJ
ejpam-6849	281	13	sv	sv	PROPN
ejpam-6849	281	14	is	be	AUX
ejpam-6849	281	15	a	a	DET
ejpam-6849	281	16	ve	ve	NOUN
ejpam-6849	281	17	-	-	PUNCT
ejpam-6849	281	18	dominating	dominate	VERB
ejpam-6849	281	19	set	set	NOUN
ejpam-6849	281	20	of	of	ADP
ejpam-6849	281	21	hv	hv	PROPN
ejpam-6849	281	22	for	for	SCONJ
ejpam-6849	281	23	all	all	PRON
ejpam-6849	281	24	v	v	ADP
ejpam-6849	281	25	∈	∈	NUM
ejpam-6849	281	26	v	v	NOUN
ejpam-6849	281	27	(	(	PUNCT
ejpam-6849	281	28	g	g	NOUN
ejpam-6849	281	29	)	)	PUNCT
ejpam-6849	281	30	\	\	NOUN
ejpam-6849	281	31	a.	a.	NOUN
ejpam-6849	281	32	let	let	VERB
ejpam-6849	281	33	xy	xy	PROPN
ejpam-6849	281	34	∈	∈	PROPN
ejpam-6849	281	35	e(g	e(g	PROPN
ejpam-6849	281	36	◦	◦	PROPN
ejpam-6849	281	37	h	h	NOUN
ejpam-6849	281	38	)	)	PUNCT
ejpam-6849	281	39	.	.	PUNCT
ejpam-6849	282	1	we	we	PRON
ejpam-6849	282	2	consider	consider	VERB
ejpam-6849	282	3	the	the	DET
ejpam-6849	282	4	following	follow	VERB
ejpam-6849	282	5	cases	case	NOUN
ejpam-6849	282	6	:	:	PUNCT
ejpam-6849	282	7	case	case	NOUN
ejpam-6849	282	8	1	1	NUM
ejpam-6849	282	9	:	:	PUNCT
ejpam-6849	282	10	x	x	SYM
ejpam-6849	282	11	∈	∈	NOUN
ejpam-6849	282	12	v	v	ADP
ejpam-6849	282	13	(	(	PUNCT
ejpam-6849	282	14	g	g	NOUN
ejpam-6849	282	15	)	)	PUNCT
ejpam-6849	282	16	or	or	CCONJ
ejpam-6849	282	17	y	y	PROPN
ejpam-6849	282	18	∈	∈	PROPN
ejpam-6849	282	19	v	v	ADP
ejpam-6849	282	20	(	(	PUNCT
ejpam-6849	282	21	g	g	NOUN
ejpam-6849	282	22	)	)	PUNCT
ejpam-6849	282	23	wlog	wlog	NOUN
ejpam-6849	282	24	assume	assume	VERB
ejpam-6849	282	25	x	x	SYM
ejpam-6849	282	26	∈	∈	PROPN
ejpam-6849	282	27	v	v	X
ejpam-6849	282	28	(	(	PUNCT
ejpam-6849	282	29	g	g	NOUN
ejpam-6849	282	30	)	)	PUNCT
ejpam-6849	282	31	.	.	PUNCT
ejpam-6849	283	1	if	if	SCONJ
ejpam-6849	283	2	x	x	SYM
ejpam-6849	283	3	∈	∈	PROPN
ejpam-6849	283	4	a	a	PRON
ejpam-6849	283	5	,	,	PUNCT
ejpam-6849	283	6	then	then	ADV
ejpam-6849	283	7	s	s	AUX
ejpam-6849	283	8	ve	ve	VERB
ejpam-6849	283	9	-	-	PUNCT
ejpam-6849	283	10	dominates	dominate	VERB
ejpam-6849	283	11	xy	xy	PRON
ejpam-6849	283	12	.	.	PUNCT
ejpam-6849	283	13	suppose	suppose	VERB
ejpam-6849	283	14	that	that	SCONJ
ejpam-6849	283	15	x	x	SYM
ejpam-6849	283	16	/∈	/∈	PUNCT
ejpam-6849	283	17	a.	a.	NOUN
ejpam-6849	283	18	since	since	SCONJ
ejpam-6849	283	19	h	h	PROPN
ejpam-6849	283	20	is	be	AUX
ejpam-6849	283	21	nonempty	nonempty	ADJ
ejpam-6849	283	22	,	,	PUNCT
ejpam-6849	283	23	hx	hx	PROPN
ejpam-6849	283	24	contains	contain	VERB
ejpam-6849	283	25	an	an	DET
ejpam-6849	283	26	edge	edge	NOUN
ejpam-6849	283	27	ab	ab	PROPN
ejpam-6849	283	28	.	.	PUNCT
ejpam-6849	284	1	because	because	SCONJ
ejpam-6849	284	2	x	x	PROPN
ejpam-6849	284	3	/∈	/∈	PROPN
ejpam-6849	284	4	s	s	X
ejpam-6849	284	5	,	,	PUNCT
ejpam-6849	284	6	there	there	PRON
ejpam-6849	284	7	exists	exist	VERB
ejpam-6849	284	8	u	u	PROPN
ejpam-6849	284	9	∈	∈	PROPN
ejpam-6849	284	10	sx	sx	PROPN
ejpam-6849	284	11	such	such	ADJ
ejpam-6849	284	12	that	that	SCONJ
ejpam-6849	284	13	u	u	PROPN
ejpam-6849	284	14	ve	ve	AUX
ejpam-6849	284	15	-	-	PUNCT
ejpam-6849	284	16	dominates	dominate	VERB
ejpam-6849	284	17	ab	ab	PROPN
ejpam-6849	284	18	.	.	PUNCT
ejpam-6849	285	1	since	since	SCONJ
ejpam-6849	285	2	ux	ux	PROPN
ejpam-6849	285	3	∈	∈	PROPN
ejpam-6849	285	4	e(g	e(g	PROPN
ejpam-6849	285	5	◦	◦	PROPN
ejpam-6849	285	6	h	h	NOUN
ejpam-6849	285	7	)	)	PUNCT
ejpam-6849	285	8	,	,	PUNCT
ejpam-6849	285	9	u	u	NOUN
ejpam-6849	285	10	and	and	CCONJ
ejpam-6849	285	11	therefore	therefore	ADV
ejpam-6849	285	12	,	,	PUNCT
ejpam-6849	285	13	s	s	AUX
ejpam-6849	285	14	ve	ve	AUX
ejpam-6849	285	15	-	-	PUNCT
ejpam-6849	285	16	dominates	dominate	VERB
ejpam-6849	285	17	xy	xy	PROPN
ejpam-6849	285	18	.	.	PUNCT
ejpam-6849	286	1	j.	j.	PROPN
ejpam-6849	286	2	tayab	tayab	PROPN
ejpam-6849	286	3	,	,	PUNCT
ejpam-6849	286	4	f.	f.	PROPN
ejpam-6849	286	5	p.	p.	PROPN
ejpam-6849	286	6	jamil	jamil	PROPN
ejpam-6849	286	7	,	,	PUNCT
ejpam-6849	286	8	i.	i.	PROPN
ejpam-6849	286	9	aniversario	aniversario	PROPN
ejpam-6849	286	10	/	/	SYM
ejpam-6849	286	11	eur	eur	PROPN
ejpam-6849	286	12	.	.	PUNCT
ejpam-6849	287	1	j.	j.	PROPN
ejpam-6849	287	2	pure	pure	PROPN
ejpam-6849	287	3	appl	appl	PROPN
ejpam-6849	287	4	.	.	PROPN
ejpam-6849	287	5	math	math	PROPN
ejpam-6849	287	6	,	,	PUNCT
ejpam-6849	287	7	18	18	NUM
ejpam-6849	287	8	(	(	PUNCT
ejpam-6849	287	9	4	4	NUM
ejpam-6849	287	10	)	)	PUNCT
ejpam-6849	287	11	(	(	PUNCT
ejpam-6849	287	12	2025	2025	NUM
ejpam-6849	287	13	)	)	PUNCT
ejpam-6849	287	14	,	,	PUNCT
ejpam-6849	287	15	6849	6849	NUM
ejpam-6849	287	16	8	8	NUM
ejpam-6849	287	17	of	of	ADP
ejpam-6849	287	18	19	19	NUM
ejpam-6849	287	19	case	case	NOUN
ejpam-6849	287	20	2	2	NUM
ejpam-6849	287	21	:	:	PUNCT
ejpam-6849	287	22	x	x	X
ejpam-6849	287	23	/∈	/∈	PUNCT
ejpam-6849	288	1	v	v	INTJ
ejpam-6849	288	2	(	(	PUNCT
ejpam-6849	288	3	g	g	NOUN
ejpam-6849	288	4	)	)	PUNCT
ejpam-6849	288	5	and	and	CCONJ
ejpam-6849	288	6	y	y	PROPN
ejpam-6849	288	7	/∈	/∈	PUNCT
ejpam-6849	289	1	v	v	INTJ
ejpam-6849	289	2	(	(	PUNCT
ejpam-6849	289	3	g	g	NOUN
ejpam-6849	289	4	)	)	PUNCT
ejpam-6849	289	5	there	there	PRON
ejpam-6849	289	6	exists	exist	VERB
ejpam-6849	289	7	v	v	ADP
ejpam-6849	289	8	∈	∈	PROPN
ejpam-6849	289	9	v	v	NOUN
ejpam-6849	289	10	(	(	PUNCT
ejpam-6849	289	11	g	g	NOUN
ejpam-6849	289	12	)	)	PUNCT
ejpam-6849	289	13	for	for	ADP
ejpam-6849	289	14	which	which	PRON
ejpam-6849	289	15	xy	xy	PROPN
ejpam-6849	289	16	∈	∈	PROPN
ejpam-6849	289	17	e(hv	e(hv	PROPN
ejpam-6849	289	18	)	)	PUNCT
ejpam-6849	289	19	.	.	PUNCT
ejpam-6849	290	1	if	if	SCONJ
ejpam-6849	290	2	v	v	NUM
ejpam-6849	290	3	∈	∈	PROPN
ejpam-6849	290	4	a	a	PRON
ejpam-6849	290	5	,	,	PUNCT
ejpam-6849	290	6	then	then	ADV
ejpam-6849	290	7	v	v	NOUN
ejpam-6849	290	8	,	,	PUNCT
ejpam-6849	290	9	and	and	CCONJ
ejpam-6849	290	10	therefore	therefore	ADV
ejpam-6849	290	11	s	s	PROPN
ejpam-6849	290	12	,	,	PUNCT
ejpam-6849	290	13	ve	ve	VERB
ejpam-6849	290	14	-	-	PUNCT
ejpam-6849	290	15	dominates	dominate	VERB
ejpam-6849	290	16	xy	xy	PRON
ejpam-6849	290	17	.	.	PUNCT
ejpam-6849	290	18	suppose	suppose	VERB
ejpam-6849	290	19	that	that	SCONJ
ejpam-6849	290	20	v	v	X
ejpam-6849	290	21	/∈	/∈	PUNCT
ejpam-6849	290	22	a.	a.	NOUN
ejpam-6849	291	1	then	then	ADV
ejpam-6849	291	2	sv	sv	INTJ
ejpam-6849	291	3	ve	ve	AUX
ejpam-6849	291	4	-	-	PUNCT
ejpam-6849	291	5	dominates	dominate	VERB
ejpam-6849	291	6	xy	xy	NOUN
ejpam-6849	291	7	by	by	ADP
ejpam-6849	291	8	the	the	DET
ejpam-6849	291	9	assumption	assumption	NOUN
ejpam-6849	291	10	.	.	PUNCT
ejpam-6849	292	1	thus	thus	ADV
ejpam-6849	292	2	,	,	PUNCT
ejpam-6849	292	3	s	s	AUX
ejpam-6849	292	4	ve	ve	VERB
ejpam-6849	292	5	-	-	PUNCT
ejpam-6849	292	6	dominates	dominate	VERB
ejpam-6849	292	7	xy	xy	PROPN
ejpam-6849	292	8	.	.	PUNCT
ejpam-6849	293	1	since	since	SCONJ
ejpam-6849	293	2	xy	xy	PROPN
ejpam-6849	293	3	is	be	AUX
ejpam-6849	293	4	arbitrary	arbitrary	ADJ
ejpam-6849	293	5	,	,	PUNCT
ejpam-6849	293	6	s	s	PART
ejpam-6849	293	7	is	be	AUX
ejpam-6849	293	8	a	a	DET
ejpam-6849	293	9	ve	ve	NOUN
ejpam-6849	293	10	-	-	PUNCT
ejpam-6849	293	11	dominating	dominate	VERB
ejpam-6849	293	12	set	set	NOUN
ejpam-6849	293	13	of	of	ADP
ejpam-6849	293	14	g	g	PROPN
ejpam-6849	293	15	◦	◦	PROPN
ejpam-6849	293	16	h.	h.	PROPN
ejpam-6849	294	1	■	■	PUNCT
ejpam-6849	294	2	corollary	corollary	ADJ
ejpam-6849	294	3	3	3	X
ejpam-6849	294	4	.	.	PUNCT
ejpam-6849	295	1	let	let	VERB
ejpam-6849	295	2	g	g	PRON
ejpam-6849	295	3	be	be	AUX
ejpam-6849	295	4	a	a	DET
ejpam-6849	295	5	connected	connected	ADJ
ejpam-6849	295	6	graph	graph	NOUN
ejpam-6849	295	7	of	of	ADP
ejpam-6849	295	8	order	order	NOUN
ejpam-6849	295	9	n	n	PRON
ejpam-6849	295	10	≥	≥	NOUN
ejpam-6849	295	11	2	2	NUM
ejpam-6849	295	12	and	and	CCONJ
ejpam-6849	295	13	h	h	NOUN
ejpam-6849	295	14	any	any	DET
ejpam-6849	295	15	graph	graph	NOUN
ejpam-6849	295	16	.	.	PUNCT
ejpam-6849	296	1	(	(	PUNCT
ejpam-6849	296	2	i	i	NOUN
ejpam-6849	296	3	)	)	PUNCT
ejpam-6849	296	4	if	if	SCONJ
ejpam-6849	296	5	h	h	NOUN
ejpam-6849	296	6	is	be	AUX
ejpam-6849	296	7	an	an	DET
ejpam-6849	296	8	empty	empty	ADJ
ejpam-6849	296	9	graph	graph	NOUN
ejpam-6849	296	10	,	,	PUNCT
ejpam-6849	296	11	then	then	ADV
ejpam-6849	296	12	γve(g	γve(g	VERB
ejpam-6849	296	13	◦	◦	NOUN
ejpam-6849	296	14	h	h	NOUN
ejpam-6849	296	15	)	)	PUNCT
ejpam-6849	296	16	=	=	PUNCT
ejpam-6849	296	17	γ(g	γ(g	PROPN
ejpam-6849	296	18	)	)	PUNCT
ejpam-6849	296	19	and	and	CCONJ
ejpam-6849	296	20	γt	γt	NOUN
ejpam-6849	296	21	ve(g	ve(g	NOUN
ejpam-6849	296	22	◦	◦	PROPN
ejpam-6849	296	23	h	h	NOUN
ejpam-6849	296	24	)	)	PUNCT
ejpam-6849	296	25	=	=	NUM
ejpam-6849	296	26	γt(g	γt(g	NUM
ejpam-6849	296	27	)	)	PUNCT
ejpam-6849	296	28	.	.	PUNCT
ejpam-6849	297	1	(	(	PUNCT
ejpam-6849	297	2	ii	ii	NOUN
ejpam-6849	297	3	)	)	PUNCT
ejpam-6849	297	4	if	if	SCONJ
ejpam-6849	297	5	h	h	NOUN
ejpam-6849	297	6	is	be	AUX
ejpam-6849	297	7	a	a	DET
ejpam-6849	297	8	nonempty	nonempty	ADJ
ejpam-6849	297	9	graph	graph	NOUN
ejpam-6849	297	10	,	,	PUNCT
ejpam-6849	297	11	then	then	ADV
ejpam-6849	297	12	γve(g	γve(g	VERB
ejpam-6849	297	13	◦	◦	NOUN
ejpam-6849	297	14	h	h	NOUN
ejpam-6849	297	15	)	)	PUNCT
ejpam-6849	297	16	=	=	NOUN
ejpam-6849	297	17	γt	γt	NOUN
ejpam-6849	297	18	ve(g	ve(g	NOUN
ejpam-6849	297	19	◦	◦	PROPN
ejpam-6849	297	20	h	h	NOUN
ejpam-6849	297	21	)	)	PUNCT
ejpam-6849	297	22	=	=	SYM
ejpam-6849	297	23	n.	n.	NOUN
ejpam-6849	297	24	proof	proof	NOUN
ejpam-6849	297	25	.	.	PUNCT
ejpam-6849	298	1	let	let	VERB
ejpam-6849	298	2	h	h	PRON
ejpam-6849	298	3	be	be	AUX
ejpam-6849	298	4	an	an	DET
ejpam-6849	298	5	empty	empty	ADJ
ejpam-6849	298	6	graph	graph	NOUN
ejpam-6849	298	7	.	.	PUNCT
ejpam-6849	299	1	let	let	VERB
ejpam-6849	299	2	s	s	PRON
ejpam-6849	299	3	⊆	⊆	NUM
ejpam-6849	299	4	v	v	NOUN
ejpam-6849	299	5	(	(	PUNCT
ejpam-6849	299	6	g	g	NOUN
ejpam-6849	299	7	)	)	PUNCT
ejpam-6849	299	8	be	be	AUX
ejpam-6849	299	9	a	a	DET
ejpam-6849	299	10	(	(	PUNCT
ejpam-6849	299	11	total	total	ADJ
ejpam-6849	299	12	)	)	PUNCT
ejpam-6849	299	13	dominating	dominating	NOUN
ejpam-6849	299	14	set	set	NOUN
ejpam-6849	299	15	of	of	ADP
ejpam-6849	299	16	g.	g.	PROPN
ejpam-6849	299	17	we	we	PRON
ejpam-6849	299	18	claim	claim	VERB
ejpam-6849	299	19	that	that	SCONJ
ejpam-6849	299	20	s	s	VERB
ejpam-6849	299	21	is	be	AUX
ejpam-6849	299	22	a	a	DET
ejpam-6849	299	23	(	(	PUNCT
ejpam-6849	299	24	total	total	NOUN
ejpam-6849	299	25	)	)	PUNCT
ejpam-6849	299	26	ve	ve	AUX
ejpam-6849	299	27	-	-	PUNCT
ejpam-6849	299	28	dominating	dominate	VERB
ejpam-6849	299	29	set	set	NOUN
ejpam-6849	299	30	of	of	ADP
ejpam-6849	299	31	g	g	PROPN
ejpam-6849	299	32	◦	◦	PROPN
ejpam-6849	299	33	h.	h.	PROPN
ejpam-6849	300	1	let	let	VERB
ejpam-6849	300	2	xy	xy	PROPN
ejpam-6849	300	3	∈	∈	PROPN
ejpam-6849	300	4	e(g	e(g	PROPN
ejpam-6849	300	5	◦	◦	PROPN
ejpam-6849	300	6	h	h	NOUN
ejpam-6849	300	7	)	)	PUNCT
ejpam-6849	300	8	.	.	PUNCT
ejpam-6849	301	1	assume	assume	VERB
ejpam-6849	301	2	wlog	wlog	NOUN
ejpam-6849	301	3	that	that	SCONJ
ejpam-6849	301	4	x	x	PUNCT
ejpam-6849	301	5	∈	∈	NOUN
ejpam-6849	301	6	v	v	X
ejpam-6849	301	7	(	(	PUNCT
ejpam-6849	301	8	g	g	NOUN
ejpam-6849	301	9	)	)	PUNCT
ejpam-6849	301	10	.	.	PUNCT
ejpam-6849	302	1	if	if	SCONJ
ejpam-6849	302	2	x	x	SYM
ejpam-6849	302	3	∈	∈	PROPN
ejpam-6849	302	4	s	s	PROPN
ejpam-6849	302	5	,	,	PUNCT
ejpam-6849	302	6	then	then	ADV
ejpam-6849	302	7	s	s	AUX
ejpam-6849	302	8	ve	ve	VERB
ejpam-6849	302	9	-	-	PUNCT
ejpam-6849	302	10	dominates	dominate	VERB
ejpam-6849	302	11	xy	xy	PRON
ejpam-6849	302	12	.	.	PUNCT
ejpam-6849	302	13	suppose	suppose	VERB
ejpam-6849	302	14	that	that	SCONJ
ejpam-6849	302	15	x	x	PROPN
ejpam-6849	302	16	/∈	/∈	PROPN
ejpam-6849	302	17	s.	s.	PROPN
ejpam-6849	302	18	since	since	SCONJ
ejpam-6849	302	19	s	s	PROPN
ejpam-6849	302	20	is	be	AUX
ejpam-6849	302	21	a	a	DET
ejpam-6849	302	22	dominating	dominating	NOUN
ejpam-6849	302	23	set	set	NOUN
ejpam-6849	302	24	of	of	ADP
ejpam-6849	302	25	g	g	NOUN
ejpam-6849	302	26	,	,	PUNCT
ejpam-6849	302	27	there	there	PRON
ejpam-6849	302	28	exists	exist	VERB
ejpam-6849	302	29	u	u	PROPN
ejpam-6849	302	30	∈	∈	PROPN
ejpam-6849	302	31	s	s	X
ejpam-6849	302	32	for	for	ADP
ejpam-6849	302	33	which	which	PRON
ejpam-6849	302	34	ux	ux	PROPN
ejpam-6849	302	35	∈	∈	PROPN
ejpam-6849	302	36	e(g	e(g	PROPN
ejpam-6849	302	37	)	)	PUNCT
ejpam-6849	302	38	.	.	PUNCT
ejpam-6849	303	1	this	this	PRON
ejpam-6849	303	2	means	mean	VERB
ejpam-6849	303	3	u	u	NOUN
ejpam-6849	303	4	,	,	PUNCT
ejpam-6849	303	5	hence	hence	ADV
ejpam-6849	303	6	s	s	PROPN
ejpam-6849	303	7	,	,	PUNCT
ejpam-6849	303	8	ve	ve	VERB
ejpam-6849	303	9	-	-	PUNCT
ejpam-6849	303	10	dominates	dominate	VERB
ejpam-6849	303	11	xy	xy	PROPN
ejpam-6849	303	12	.	.	PUNCT
ejpam-6849	304	1	therefore	therefore	ADV
ejpam-6849	304	2	,	,	PUNCT
ejpam-6849	304	3	s	s	VERB
ejpam-6849	304	4	is	be	AUX
ejpam-6849	304	5	a	a	DET
ejpam-6849	304	6	(	(	PUNCT
ejpam-6849	304	7	total	total	NOUN
ejpam-6849	304	8	)	)	PUNCT
ejpam-6849	304	9	ve	ve	AUX
ejpam-6849	304	10	-	-	PUNCT
ejpam-6849	304	11	dominating	dominate	VERB
ejpam-6849	304	12	set	set	NOUN
ejpam-6849	304	13	of	of	ADP
ejpam-6849	304	14	g	g	PROPN
ejpam-6849	304	15	◦	◦	PROPN
ejpam-6849	304	16	h.	h.	NOUN
ejpam-6849	304	17	since	since	SCONJ
ejpam-6849	304	18	s	s	PROPN
ejpam-6849	304	19	is	be	AUX
ejpam-6849	304	20	arbitrary	arbitrary	ADJ
ejpam-6849	304	21	,	,	PUNCT
ejpam-6849	304	22	γve(g	γve(g	VERB
ejpam-6849	304	23	◦	◦	NOUN
ejpam-6849	304	24	h	h	NOUN
ejpam-6849	304	25	)	)	PUNCT
ejpam-6849	304	26	≤	≤	PROPN
ejpam-6849	304	27	γ(g	γ(g	PROPN
ejpam-6849	304	28	)	)	PUNCT
ejpam-6849	304	29	and	and	CCONJ
ejpam-6849	304	30	γt	γt	NOUN
ejpam-6849	304	31	ve(g	ve(g	NOUN
ejpam-6849	304	32	◦	◦	PROPN
ejpam-6849	304	33	h	h	NOUN
ejpam-6849	304	34	)	)	PUNCT
ejpam-6849	304	35	≤	≤	NOUN
ejpam-6849	304	36	γt(g	γt(g	PUNCT
ejpam-6849	304	37	)	)	PUNCT
ejpam-6849	304	38	.	.	PUNCT
ejpam-6849	305	1	to	to	PART
ejpam-6849	305	2	get	get	VERB
ejpam-6849	305	3	the	the	DET
ejpam-6849	305	4	other	other	ADJ
ejpam-6849	305	5	inequalities	inequality	NOUN
ejpam-6849	305	6	,	,	PUNCT
ejpam-6849	305	7	first	first	ADV
ejpam-6849	305	8	let	let	VERB
ejpam-6849	305	9	s	s	PRON
ejpam-6849	305	10	⊆	⊆	NUM
ejpam-6849	305	11	v	v	NOUN
ejpam-6849	305	12	(	(	PUNCT
ejpam-6849	305	13	g	g	PROPN
ejpam-6849	305	14	◦	◦	NOUN
ejpam-6849	305	15	h	h	NOUN
ejpam-6849	305	16	)	)	PUNCT
ejpam-6849	305	17	be	be	VERB
ejpam-6849	305	18	a	a	DET
ejpam-6849	305	19	γve	γve	NOUN
ejpam-6849	305	20	-	-	PUNCT
ejpam-6849	305	21	set	set	NOUN
ejpam-6849	305	22	of	of	ADP
ejpam-6849	305	23	g	g	PROPN
ejpam-6849	305	24	◦	◦	NOUN
ejpam-6849	305	25	h.	h.	NOUN
ejpam-6849	305	26	put	put	VERB
ejpam-6849	305	27	a	a	DET
ejpam-6849	305	28	=	=	X
ejpam-6849	305	29	s	s	NOUN
ejpam-6849	305	30	∩v	∩v	NOUN
ejpam-6849	305	31	(	(	PUNCT
ejpam-6849	305	32	g	g	NOUN
ejpam-6849	305	33	)	)	PUNCT
ejpam-6849	305	34	and	and	CCONJ
ejpam-6849	305	35	b	b	X
ejpam-6849	305	36	=	=	PRON
ejpam-6849	305	37	{	{	PUNCT
ejpam-6849	305	38	v	v	NUM
ejpam-6849	305	39	∈	∈	NOUN
ejpam-6849	305	40	v	v	NOUN
ejpam-6849	305	41	(	(	PUNCT
ejpam-6849	305	42	g	g	NOUN
ejpam-6849	305	43	)	)	PUNCT
ejpam-6849	305	44	:	:	PUNCT
ejpam-6849	305	45	s	s	VERB
ejpam-6849	305	46	∩	∩	ADJ
ejpam-6849	305	47	v	v	X
ejpam-6849	305	48	(	(	PUNCT
ejpam-6849	305	49	hv	hv	NOUN
ejpam-6849	305	50	)	)	PUNCT
ejpam-6849	305	51	̸=	̸=	PROPN
ejpam-6849	305	52	∅	∅	NOUN
ejpam-6849	305	53	}	}	PUNCT
ejpam-6849	305	54	.	.	PUNCT
ejpam-6849	306	1	define	define	VERB
ejpam-6849	306	2	s∗	s∗	PROPN
ejpam-6849	306	3	=	=	PUNCT
ejpam-6849	306	4	a	a	DET
ejpam-6849	306	5	∪	∪	X
ejpam-6849	306	6	b.	b.	NOUN
ejpam-6849	306	7	then	then	ADV
ejpam-6849	306	8	|s∗|	|s∗|	NUM
ejpam-6849	306	9	≤	≤	NUM
ejpam-6849	306	10	|s|	|s|	PROPN
ejpam-6849	306	11	=	=	SYM
ejpam-6849	306	12	γve(g	γve(g	PROPN
ejpam-6849	306	13	◦	◦	NOUN
ejpam-6849	306	14	h	h	NOUN
ejpam-6849	306	15	)	)	PUNCT
ejpam-6849	306	16	.	.	PUNCT
ejpam-6849	307	1	we	we	PRON
ejpam-6849	307	2	claim	claim	VERB
ejpam-6849	307	3	that	that	SCONJ
ejpam-6849	307	4	s∗	s∗	PROPN
ejpam-6849	307	5	is	be	AUX
ejpam-6849	307	6	a	a	DET
ejpam-6849	307	7	dominating	dominating	NOUN
ejpam-6849	307	8	set	set	NOUN
ejpam-6849	307	9	of	of	ADP
ejpam-6849	307	10	g.	g.	PROPN
ejpam-6849	307	11	let	let	VERB
ejpam-6849	307	12	v	v	NUM
ejpam-6849	307	13	∈	∈	PROPN
ejpam-6849	307	14	v	v	NOUN
ejpam-6849	307	15	(	(	PUNCT
ejpam-6849	307	16	g)\s∗.	g)\s∗.	ADJ
ejpam-6849	307	17	pick	pick	VERB
ejpam-6849	307	18	u	u	PROPN
ejpam-6849	307	19	∈	∈	PROPN
ejpam-6849	307	20	v	v	X
ejpam-6849	307	21	(	(	PUNCT
ejpam-6849	307	22	hv	hv	PROPN
ejpam-6849	307	23	)	)	PUNCT
ejpam-6849	307	24	.	.	PUNCT
ejpam-6849	308	1	there	there	PRON
ejpam-6849	308	2	exists	exist	VERB
ejpam-6849	308	3	w	w	PROPN
ejpam-6849	308	4	∈	∈	PROPN
ejpam-6849	308	5	s	s	VERB
ejpam-6849	308	6	such	such	ADJ
ejpam-6849	308	7	that	that	SCONJ
ejpam-6849	308	8	w	w	PROPN
ejpam-6849	308	9	ve	ve	VERB
ejpam-6849	308	10	-	-	PUNCT
ejpam-6849	308	11	dominates	dominate	VERB
ejpam-6849	308	12	uv	uv	NOUN
ejpam-6849	308	13	.	.	PUNCT
ejpam-6849	309	1	since	since	SCONJ
ejpam-6849	309	2	s	s	PROPN
ejpam-6849	309	3	∩	∩	ADJ
ejpam-6849	309	4	v	v	X
ejpam-6849	309	5	(	(	PUNCT
ejpam-6849	309	6	hv	hv	NOUN
ejpam-6849	309	7	)	)	PUNCT
ejpam-6849	309	8	=	=	NOUN
ejpam-6849	310	1	∅	∅	NOUN
ejpam-6849	310	2	,	,	PUNCT
ejpam-6849	310	3	w	w	PROPN
ejpam-6849	310	4	∈	∈	PROPN
ejpam-6849	310	5	a.	a.	NOUN
ejpam-6849	310	6	thus	thus	ADV
ejpam-6849	310	7	,	,	PUNCT
ejpam-6849	310	8	w	w	PROPN
ejpam-6849	310	9	∈	∈	PROPN
ejpam-6849	310	10	s∗	s∗	PROPN
ejpam-6849	310	11	∩	∩	NOUN
ejpam-6849	310	12	ng(v	ng(v	NUM
ejpam-6849	310	13	)	)	PUNCT
ejpam-6849	310	14	.	.	PUNCT
ejpam-6849	311	1	this	this	PRON
ejpam-6849	311	2	means	mean	VERB
ejpam-6849	311	3	that	that	SCONJ
ejpam-6849	311	4	s∗	s∗	PROPN
ejpam-6849	311	5	is	be	AUX
ejpam-6849	311	6	a	a	DET
ejpam-6849	311	7	dominating	dominating	NOUN
ejpam-6849	311	8	set	set	NOUN
ejpam-6849	311	9	of	of	ADP
ejpam-6849	311	10	g.	g.	PROPN
ejpam-6849	311	11	therefore	therefore	ADV
ejpam-6849	311	12	,	,	PUNCT
ejpam-6849	311	13	γ(g	γ(g	PROPN
ejpam-6849	311	14	)	)	PUNCT
ejpam-6849	311	15	≤	≤	PUNCT
ejpam-6849	312	1	γve(g	γve(g	PUNCT
ejpam-6849	312	2	◦	◦	NOUN
ejpam-6849	312	3	h	h	NOUN
ejpam-6849	312	4	)	)	PUNCT
ejpam-6849	312	5	.	.	PUNCT
ejpam-6849	313	1	next	next	ADV
ejpam-6849	313	2	,	,	PUNCT
ejpam-6849	313	3	suppose	suppose	VERB
ejpam-6849	313	4	that	that	SCONJ
ejpam-6849	313	5	s	s	VERB
ejpam-6849	313	6	is	be	AUX
ejpam-6849	313	7	a	a	DET
ejpam-6849	313	8	γt	γt	ADJ
ejpam-6849	313	9	ve	ve	NOUN
ejpam-6849	313	10	-	-	PUNCT
ejpam-6849	313	11	set	set	NOUN
ejpam-6849	313	12	of	of	ADP
ejpam-6849	313	13	g	g	PROPN
ejpam-6849	313	14	◦	◦	PROPN
ejpam-6849	313	15	h.	h.	PROPN
ejpam-6849	313	16	let	let	VERB
ejpam-6849	313	17	a	a	DET
ejpam-6849	313	18	=	=	X
ejpam-6849	313	19	s	s	NOUN
ejpam-6849	313	20	∩	∩	ADJ
ejpam-6849	313	21	v	v	X
ejpam-6849	313	22	(	(	PUNCT
ejpam-6849	313	23	g	g	NOUN
ejpam-6849	313	24	)	)	PUNCT
ejpam-6849	313	25	and	and	CCONJ
ejpam-6849	313	26	b	b	X
ejpam-6849	313	27	=	=	SYM
ejpam-6849	313	28	{	{	PUNCT
ejpam-6849	313	29	v	v	NOUN
ejpam-6849	313	30	∈	∈	PROPN
ejpam-6849	313	31	a	a	DET
ejpam-6849	313	32	:	:	PUNCT
ejpam-6849	313	33	s	s	NOUN
ejpam-6849	313	34	∩	∩	ADJ
ejpam-6849	313	35	v	v	X
ejpam-6849	313	36	(	(	PUNCT
ejpam-6849	313	37	hv	hv	NOUN
ejpam-6849	313	38	)	)	PUNCT
ejpam-6849	313	39	̸=	̸=	PROPN
ejpam-6849	313	40	∅	∅	NOUN
ejpam-6849	313	41	}	}	PUNCT
ejpam-6849	313	42	.	.	PUNCT
ejpam-6849	314	1	for	for	ADP
ejpam-6849	314	2	each	each	DET
ejpam-6849	314	3	v	v	NUM
ejpam-6849	314	4	∈	∈	PROPN
ejpam-6849	314	5	b	b	NOUN
ejpam-6849	314	6	,	,	PUNCT
ejpam-6849	314	7	choose	choose	VERB
ejpam-6849	314	8	uv	uv	NOUN
ejpam-6849	314	9	∈	∈	PROPN
ejpam-6849	314	10	ng(v	ng(v	NOUN
ejpam-6849	314	11	)	)	PUNCT
ejpam-6849	314	12	.	.	PUNCT
ejpam-6849	315	1	define	define	VERB
ejpam-6849	315	2	s∗	s∗	PROPN
ejpam-6849	315	3	=	=	PUNCT
ejpam-6849	315	4	a	a	DET
ejpam-6849	315	5	∪	∪	X
ejpam-6849	315	6	{	{	PUNCT
ejpam-6849	315	7	uv	uv	NOUN
ejpam-6849	315	8	:	:	PUNCT
ejpam-6849	315	9	v	v	NUM
ejpam-6849	315	10	∈	∈	PROPN
ejpam-6849	315	11	b	b	NOUN
ejpam-6849	315	12	}	}	PUNCT
ejpam-6849	315	13	.	.	PUNCT
ejpam-6849	316	1	let	let	VERB
ejpam-6849	316	2	v	v	NUM
ejpam-6849	316	3	∈	∈	PROPN
ejpam-6849	316	4	v	v	NOUN
ejpam-6849	316	5	(	(	PUNCT
ejpam-6849	316	6	g	g	NOUN
ejpam-6849	316	7	)	)	PUNCT
ejpam-6849	316	8	\	\	NOUN
ejpam-6849	316	9	s∗	s∗	PROPN
ejpam-6849	316	10	and	and	CCONJ
ejpam-6849	316	11	let	let	VERB
ejpam-6849	316	12	u	u	PRON
ejpam-6849	316	13	∈	∈	PROPN
ejpam-6849	316	14	v	v	X
ejpam-6849	316	15	(	(	PUNCT
ejpam-6849	316	16	hv	hv	PROPN
ejpam-6849	316	17	)	)	PUNCT
ejpam-6849	316	18	.	.	PUNCT
ejpam-6849	317	1	there	there	PRON
ejpam-6849	317	2	exists	exist	VERB
ejpam-6849	317	3	w	w	PROPN
ejpam-6849	317	4	∈	∈	PROPN
ejpam-6849	317	5	s	s	VERB
ejpam-6849	317	6	such	such	ADJ
ejpam-6849	317	7	that	that	SCONJ
ejpam-6849	317	8	w	w	PROPN
ejpam-6849	317	9	ve	ve	VERB
ejpam-6849	317	10	-	-	PUNCT
ejpam-6849	317	11	dominates	dominate	VERB
ejpam-6849	317	12	uv	uv	NOUN
ejpam-6849	317	13	.	.	PUNCT
ejpam-6849	318	1	since	since	SCONJ
ejpam-6849	318	2	s	s	PROPN
ejpam-6849	318	3	is	be	AUX
ejpam-6849	318	4	a	a	DET
ejpam-6849	318	5	total	total	ADJ
ejpam-6849	318	6	ve	ve	NOUN
ejpam-6849	318	7	-	-	PUNCT
ejpam-6849	318	8	dominating	dominate	VERB
ejpam-6849	318	9	set	set	NOUN
ejpam-6849	318	10	and	and	CCONJ
ejpam-6849	318	11	v	v	ADP
ejpam-6849	318	12	/∈	/∈	SYM
ejpam-6849	319	1	s	s	X
ejpam-6849	319	2	,	,	PUNCT
ejpam-6849	319	3	u	u	PROPN
ejpam-6849	319	4	/∈	/∈	PROPN
ejpam-6849	319	5	s.	s.	PROPN
ejpam-6849	319	6	hence	hence	ADV
ejpam-6849	319	7	,	,	PUNCT
ejpam-6849	319	8	w	w	PROPN
ejpam-6849	319	9	∈	∈	PROPN
ejpam-6849	319	10	a	a	DET
ejpam-6849	319	11	showing	showing	NOUN
ejpam-6849	319	12	that	that	PRON
ejpam-6849	319	13	s∗	s∗	PROPN
ejpam-6849	319	14	is	be	AUX
ejpam-6849	319	15	a	a	DET
ejpam-6849	319	16	dominating	dominating	NOUN
ejpam-6849	319	17	set	set	NOUN
ejpam-6849	319	18	of	of	ADP
ejpam-6849	319	19	g.	g.	PROPN
ejpam-6849	319	20	let	let	VERB
ejpam-6849	319	21	w	w	NOUN
ejpam-6849	319	22	∈	∈	PROPN
ejpam-6849	319	23	s∗.	s∗.	ADJ
ejpam-6849	319	24	if	if	SCONJ
ejpam-6849	319	25	w	w	PROPN
ejpam-6849	319	26	/∈	/∈	PROPN
ejpam-6849	319	27	a	a	NOUN
ejpam-6849	319	28	,	,	PUNCT
ejpam-6849	319	29	then	then	ADV
ejpam-6849	319	30	w	w	PROPN
ejpam-6849	319	31	=	=	PUNCT
ejpam-6849	319	32	uv	uv	NOUN
ejpam-6849	319	33	for	for	ADP
ejpam-6849	319	34	some	some	PRON
ejpam-6849	319	35	v	v	ADP
ejpam-6849	319	36	∈	∈	PROPN
ejpam-6849	319	37	b	b	NOUN
ejpam-6849	319	38	⊆	⊆	NUM
ejpam-6849	319	39	a.	a.	NOUN
ejpam-6849	319	40	note	note	NOUN
ejpam-6849	319	41	here	here	ADV
ejpam-6849	319	42	that	that	SCONJ
ejpam-6849	319	43	v	v	X
ejpam-6849	319	44	∈	∈	PROPN
ejpam-6849	319	45	s∗	s∗	PROPN
ejpam-6849	319	46	and	and	CCONJ
ejpam-6849	319	47	wv	wv	PROPN
ejpam-6849	319	48	∈	∈	PROPN
ejpam-6849	319	49	e(g	e(g	PROPN
ejpam-6849	319	50	)	)	PUNCT
ejpam-6849	319	51	.	.	PUNCT
ejpam-6849	320	1	suppose	suppose	VERB
ejpam-6849	320	2	that	that	SCONJ
ejpam-6849	320	3	w	w	PROPN
ejpam-6849	320	4	∈	∈	PROPN
ejpam-6849	320	5	a.	a.	NOUN
ejpam-6849	320	6	if	if	SCONJ
ejpam-6849	320	7	s	s	VERB
ejpam-6849	320	8	∩	∩	ADJ
ejpam-6849	320	9	v	v	X
ejpam-6849	320	10	(	(	PUNCT
ejpam-6849	320	11	hw	hw	NOUN
ejpam-6849	320	12	)	)	PUNCT
ejpam-6849	321	1	=	=	NOUN
ejpam-6849	321	2	∅	∅	NOUN
ejpam-6849	321	3	,	,	PUNCT
ejpam-6849	321	4	then	then	ADV
ejpam-6849	321	5	since	since	SCONJ
ejpam-6849	321	6	s	s	NOUN
ejpam-6849	321	7	is	be	AUX
ejpam-6849	321	8	a	a	DET
ejpam-6849	321	9	total	total	ADJ
ejpam-6849	321	10	ve	ve	NOUN
ejpam-6849	321	11	-	-	PUNCT
ejpam-6849	321	12	dominating	dominating	NOUN
ejpam-6849	321	13	set	set	NOUN
ejpam-6849	321	14	,	,	PUNCT
ejpam-6849	321	15	there	there	PRON
ejpam-6849	321	16	exists	exist	VERB
ejpam-6849	321	17	z	z	PROPN
ejpam-6849	321	18	∈	∈	PROPN
ejpam-6849	321	19	a	a	DET
ejpam-6849	321	20	∩	∩	NOUN
ejpam-6849	321	21	ng(w	ng(w	NOUN
ejpam-6849	321	22	)	)	PUNCT
ejpam-6849	321	23	.	.	PUNCT
ejpam-6849	322	1	if	if	SCONJ
ejpam-6849	322	2	s	s	ADP
ejpam-6849	322	3	∩	∩	ADJ
ejpam-6849	322	4	v	v	X
ejpam-6849	322	5	(	(	PUNCT
ejpam-6849	322	6	hw	hw	NOUN
ejpam-6849	322	7	)	)	PUNCT
ejpam-6849	322	8	̸=	̸=	NOUN
ejpam-6849	322	9	∅	∅	NOUN
ejpam-6849	322	10	,	,	PUNCT
ejpam-6849	322	11	then	then	ADV
ejpam-6849	322	12	w	w	PROPN
ejpam-6849	322	13	∈	∈	PROPN
ejpam-6849	322	14	b	b	PROPN
ejpam-6849	322	15	and	and	CCONJ
ejpam-6849	322	16	uw	uw	PROPN
ejpam-6849	322	17	∈	∈	PROPN
ejpam-6849	322	18	s∗	s∗	PROPN
ejpam-6849	322	19	∩	∩	NOUN
ejpam-6849	322	20	ng(w	ng(w	NOUN
ejpam-6849	322	21	)	)	PUNCT
ejpam-6849	322	22	.	.	PUNCT
ejpam-6849	323	1	this	this	PRON
ejpam-6849	323	2	completely	completely	ADV
ejpam-6849	323	3	shows	show	VERB
ejpam-6849	323	4	that	that	SCONJ
ejpam-6849	323	5	s∗	s∗	PROPN
ejpam-6849	323	6	is	be	AUX
ejpam-6849	323	7	a	a	DET
ejpam-6849	323	8	total	total	ADJ
ejpam-6849	323	9	dominating	dominating	NOUN
ejpam-6849	323	10	set	set	NOUN
ejpam-6849	323	11	of	of	ADP
ejpam-6849	323	12	g.	g.	PROPN
ejpam-6849	323	13	thus	thus	ADV
ejpam-6849	323	14	,	,	PUNCT
ejpam-6849	323	15	γt(g	γt(g	PUNCT
ejpam-6849	323	16	)	)	PUNCT
ejpam-6849	323	17	≤	≤	NOUN
ejpam-6849	323	18	|s∗|	|s∗|	NUM
ejpam-6849	323	19	≤	≤	NUM
ejpam-6849	323	20	|s|	|s|	PROPN
ejpam-6849	323	21	=	=	PUNCT
ejpam-6849	323	22	γt	γt	NOUN
ejpam-6849	323	23	ve(g	ve(g	NOUN
ejpam-6849	323	24	◦	◦	PROPN
ejpam-6849	323	25	h	h	NOUN
ejpam-6849	323	26	)	)	PUNCT
ejpam-6849	323	27	.	.	PUNCT
ejpam-6849	324	1	this	this	PRON
ejpam-6849	324	2	proves	prove	VERB
ejpam-6849	324	3	(	(	PUNCT
ejpam-6849	324	4	i	i	NOUN
ejpam-6849	324	5	)	)	PUNCT
ejpam-6849	324	6	.	.	PUNCT
ejpam-6849	325	1	now	now	ADV
ejpam-6849	325	2	,	,	PUNCT
ejpam-6849	325	3	we	we	PRON
ejpam-6849	325	4	prove	prove	VERB
ejpam-6849	325	5	(	(	PUNCT
ejpam-6849	325	6	ii	ii	NOUN
ejpam-6849	325	7	)	)	PUNCT
ejpam-6849	325	8	.	.	PUNCT
ejpam-6849	326	1	by	by	ADP
ejpam-6849	326	2	proposition	proposition	NOUN
ejpam-6849	326	3	8	8	NUM
ejpam-6849	326	4	,	,	PUNCT
ejpam-6849	326	5	v	v	NOUN
ejpam-6849	326	6	(	(	PUNCT
ejpam-6849	326	7	g	g	NOUN
ejpam-6849	326	8	)	)	PUNCT
ejpam-6849	326	9	is	be	AUX
ejpam-6849	326	10	a	a	DET
ejpam-6849	326	11	ve	ve	NOUN
ejpam-6849	326	12	-	-	PUNCT
ejpam-6849	326	13	dominating	dominate	VERB
ejpam-6849	326	14	set	set	NOUN
ejpam-6849	326	15	,	,	PUNCT
ejpam-6849	326	16	and	and	CCONJ
ejpam-6849	326	17	therefore	therefore	ADV
ejpam-6849	326	18	a	a	DET
ejpam-6849	326	19	total	total	ADJ
ejpam-6849	326	20	ve	ve	NOUN
ejpam-6849	326	21	-	-	PUNCT
ejpam-6849	326	22	dominating	dominate	VERB
ejpam-6849	326	23	set	set	NOUN
ejpam-6849	326	24	,	,	PUNCT
ejpam-6849	326	25	of	of	ADP
ejpam-6849	326	26	g	g	PROPN
ejpam-6849	326	27	◦	◦	PROPN
ejpam-6849	326	28	h.	h.	PROPN
ejpam-6849	327	1	thus	thus	ADV
ejpam-6849	327	2	,	,	PUNCT
ejpam-6849	327	3	γve(g	γve(g	PUNCT
ejpam-6849	327	4	◦	◦	NOUN
ejpam-6849	327	5	h	h	NOUN
ejpam-6849	327	6	)	)	PUNCT
ejpam-6849	327	7	≤	≤	NOUN
ejpam-6849	327	8	n	n	CCONJ
ejpam-6849	327	9	and	and	CCONJ
ejpam-6849	327	10	γt	γt	ADJ
ejpam-6849	327	11	ve(g	ve(g	NOUN
ejpam-6849	327	12	◦	◦	PROPN
ejpam-6849	327	13	h	h	NOUN
ejpam-6849	327	14	)	)	PUNCT
ejpam-6849	327	15	≤	≤	PROPN
ejpam-6849	327	16	n.	n.	PROPN
ejpam-6849	327	17	j.	j.	PROPN
ejpam-6849	327	18	tayab	tayab	PROPN
ejpam-6849	327	19	,	,	PUNCT
ejpam-6849	327	20	f.	f.	PROPN
ejpam-6849	327	21	p.	p.	PROPN
ejpam-6849	327	22	jamil	jamil	PROPN
ejpam-6849	327	23	,	,	PUNCT
ejpam-6849	327	24	i.	i.	PROPN
ejpam-6849	327	25	aniversario	aniversario	PROPN
ejpam-6849	327	26	/	/	SYM
ejpam-6849	327	27	eur	eur	PROPN
ejpam-6849	327	28	.	.	PUNCT
ejpam-6849	328	1	j.	j.	PROPN
ejpam-6849	328	2	pure	pure	PROPN
ejpam-6849	328	3	appl	appl	PROPN
ejpam-6849	328	4	.	.	PROPN
ejpam-6849	328	5	math	math	PROPN
ejpam-6849	328	6	,	,	PUNCT
ejpam-6849	328	7	18	18	NUM
ejpam-6849	328	8	(	(	PUNCT
ejpam-6849	328	9	4	4	NUM
ejpam-6849	328	10	)	)	PUNCT
ejpam-6849	328	11	(	(	PUNCT
ejpam-6849	328	12	2025	2025	NUM
ejpam-6849	328	13	)	)	PUNCT
ejpam-6849	328	14	,	,	PUNCT
ejpam-6849	328	15	6849	6849	NUM
ejpam-6849	328	16	9	9	NUM
ejpam-6849	328	17	of	of	ADP
ejpam-6849	328	18	19	19	NUM
ejpam-6849	328	19	let	let	VERB
ejpam-6849	328	20	s	s	PRON
ejpam-6849	328	21	⊆	⊆	NUM
ejpam-6849	328	22	v	v	NOUN
ejpam-6849	328	23	(	(	PUNCT
ejpam-6849	328	24	g	g	PROPN
ejpam-6849	328	25	◦	◦	NOUN
ejpam-6849	328	26	h	h	NOUN
ejpam-6849	328	27	)	)	PUNCT
ejpam-6849	328	28	be	be	VERB
ejpam-6849	328	29	a	a	DET
ejpam-6849	328	30	γve	γve	NOUN
ejpam-6849	328	31	-	-	PUNCT
ejpam-6849	328	32	set	set	NOUN
ejpam-6849	328	33	of	of	ADP
ejpam-6849	328	34	g	g	PROPN
ejpam-6849	328	35	◦	◦	PROPN
ejpam-6849	328	36	h.	h.	NOUN
ejpam-6849	328	37	by	by	ADP
ejpam-6849	328	38	proposition	proposition	NOUN
ejpam-6849	328	39	8	8	NUM
ejpam-6849	328	40	,	,	PUNCT
ejpam-6849	328	41	s	s	VERB
ejpam-6849	328	42	=	=	PUNCT
ejpam-6849	328	43	a	a	DET
ejpam-6849	328	44	∪	∪	X
ejpam-6849	328	45	(	(	PUNCT
ejpam-6849	328	46	∪v∈v	∪v∈v	X
ejpam-6849	328	47	(	(	PUNCT
ejpam-6849	328	48	g)sv	g)sv	PROPN
ejpam-6849	328	49	)	)	PUNCT
ejpam-6849	328	50	,	,	PUNCT
ejpam-6849	328	51	where	where	SCONJ
ejpam-6849	328	52	a	a	DET
ejpam-6849	328	53	⊆	⊆	NUM
ejpam-6849	328	54	v	v	NOUN
ejpam-6849	328	55	(	(	PUNCT
ejpam-6849	328	56	g	g	NOUN
ejpam-6849	328	57	)	)	PUNCT
ejpam-6849	328	58	and	and	CCONJ
ejpam-6849	328	59	sv	sv	X
ejpam-6849	328	60	⊆	⊆	NUM
ejpam-6849	328	61	v	v	X
ejpam-6849	328	62	(	(	PUNCT
ejpam-6849	328	63	hv	hv	PROPN
ejpam-6849	328	64	)	)	PUNCT
ejpam-6849	328	65	for	for	ADP
ejpam-6849	328	66	each	each	DET
ejpam-6849	328	67	v	v	NUM
ejpam-6849	328	68	∈	∈	PROPN
ejpam-6849	328	69	v	v	NOUN
ejpam-6849	328	70	(	(	PUNCT
ejpam-6849	328	71	g	g	NOUN
ejpam-6849	328	72	)	)	PUNCT
ejpam-6849	328	73	such	such	ADJ
ejpam-6849	328	74	that	that	SCONJ
ejpam-6849	328	75	sv	sv	PROPN
ejpam-6849	328	76	is	be	AUX
ejpam-6849	328	77	a	a	DET
ejpam-6849	328	78	ve	ve	NOUN
ejpam-6849	328	79	-	-	PUNCT
ejpam-6849	328	80	dominating	dominate	VERB
ejpam-6849	328	81	set	set	NOUN
ejpam-6849	328	82	of	of	ADP
ejpam-6849	328	83	hv	hv	PROPN
ejpam-6849	328	84	for	for	ADP
ejpam-6849	328	85	all	all	PRON
ejpam-6849	328	86	v	v	ADP
ejpam-6849	328	87	∈	∈	NUM
ejpam-6849	328	88	v	v	NOUN
ejpam-6849	328	89	(	(	PUNCT
ejpam-6849	328	90	g	g	NOUN
ejpam-6849	328	91	)	)	PUNCT
ejpam-6849	328	92	\	\	NOUN
ejpam-6849	328	93	a.	a.	NOUN
ejpam-6849	328	94	thus	thus	ADV
ejpam-6849	328	95	,	,	PUNCT
ejpam-6849	328	96	γt	γt	NOUN
ejpam-6849	328	97	ve(g	ve(g	NOUN
ejpam-6849	328	98	◦	◦	PROPN
ejpam-6849	328	99	h	h	NOUN
ejpam-6849	328	100	)	)	PUNCT
ejpam-6849	328	101	≥	≥	NOUN
ejpam-6849	328	102	γve(g	γve(g	PUNCT
ejpam-6849	328	103	◦	◦	NOUN
ejpam-6849	328	104	h	h	NOUN
ejpam-6849	328	105	)	)	PUNCT
ejpam-6849	328	106	=	=	PUNCT
ejpam-6849	328	107	|s|	|s|	PROPN
ejpam-6849	328	108	≥	≥	PROPN
ejpam-6849	328	109	|a|	|a|	PROPN
ejpam-6849	328	110	+	+	PROPN
ejpam-6849	328	111	∑	∑	ADV
ejpam-6849	328	112	v∈v	v∈v	NOUN
ejpam-6849	328	113	(	(	PUNCT
ejpam-6849	328	114	g)\a	g)\a	VERB
ejpam-6849	328	115	|sv|	|sv|	PROPN
ejpam-6849	328	116	≥	≥	NOUN
ejpam-6849	328	117	|v	|v	X
ejpam-6849	328	118	(	(	PUNCT
ejpam-6849	328	119	g)|	g)|	NOUN
ejpam-6849	328	120	=	=	PUNCT
ejpam-6849	328	121	n.	n.	NOUN
ejpam-6849	328	122	■	■	NOUN
ejpam-6849	328	123	proposition	proposition	NOUN
ejpam-6849	328	124	9	9	NUM
ejpam-6849	328	125	.	.	PUNCT
ejpam-6849	329	1	let	let	VERB
ejpam-6849	329	2	g	g	PRON
ejpam-6849	329	3	be	be	AUX
ejpam-6849	329	4	a	a	DET
ejpam-6849	329	5	graph	graph	NOUN
ejpam-6849	329	6	of	of	ADP
ejpam-6849	329	7	order	order	NOUN
ejpam-6849	329	8	n.	n.	NOUN
ejpam-6849	329	9	then	then	ADV
ejpam-6849	329	10	(	(	PUNCT
ejpam-6849	329	11	i	i	NOUN
ejpam-6849	329	12	)	)	PUNCT
ejpam-6849	329	13	γve(gg	γve(gg	NOUN
ejpam-6849	329	14	)	)	PUNCT
ejpam-6849	330	1	=	=	SYM
ejpam-6849	330	2	1	1	NUM
ejpam-6849	330	3	if	if	SCONJ
ejpam-6849	330	4	and	and	CCONJ
ejpam-6849	330	5	only	only	ADV
ejpam-6849	330	6	if	if	SCONJ
ejpam-6849	330	7	g	g	PROPN
ejpam-6849	330	8	∈	∈	PROPN
ejpam-6849	330	9	{	{	PUNCT
ejpam-6849	330	10	kn	kn	PROPN
ejpam-6849	330	11	,	,	PUNCT
ejpam-6849	330	12	kn	kn	PROPN
ejpam-6849	330	13	}	}	PUNCT
ejpam-6849	330	14	;	;	PUNCT
ejpam-6849	330	15	(	(	PUNCT
ejpam-6849	330	16	ii	ii	NOUN
ejpam-6849	330	17	)	)	PUNCT
ejpam-6849	330	18	γt	γt	PROPN
ejpam-6849	330	19	ve(gg	ve(gg	PROPN
ejpam-6849	330	20	)	)	PUNCT
ejpam-6849	330	21	=	=	SYM
ejpam-6849	330	22	2	2	NUM
ejpam-6849	330	23	if	if	SCONJ
ejpam-6849	330	24	and	and	CCONJ
ejpam-6849	330	25	only	only	ADV
ejpam-6849	330	26	if	if	SCONJ
ejpam-6849	330	27	one	one	NUM
ejpam-6849	330	28	of	of	ADP
ejpam-6849	330	29	the	the	DET
ejpam-6849	330	30	following	following	NOUN
ejpam-6849	330	31	holds	hold	VERB
ejpam-6849	330	32	:	:	PUNCT
ejpam-6849	330	33	(	(	PUNCT
ejpam-6849	330	34	a	a	X
ejpam-6849	330	35	)	)	PUNCT
ejpam-6849	330	36	there	there	PRON
ejpam-6849	330	37	exists	exist	VERB
ejpam-6849	330	38	v	v	ADP
ejpam-6849	330	39	∈	∈	PROPN
ejpam-6849	330	40	v	v	NOUN
ejpam-6849	330	41	(	(	PUNCT
ejpam-6849	330	42	g	g	NOUN
ejpam-6849	330	43	)	)	PUNCT
ejpam-6849	330	44	for	for	ADP
ejpam-6849	330	45	which	which	PRON
ejpam-6849	330	46	{	{	PUNCT
ejpam-6849	330	47	v	v	NOUN
ejpam-6849	330	48	}	}	PUNCT
ejpam-6849	330	49	and	and	CCONJ
ejpam-6849	330	50	{	{	PUNCT
ejpam-6849	330	51	v	v	NOUN
ejpam-6849	330	52	}	}	PUNCT
ejpam-6849	330	53	are	be	AUX
ejpam-6849	330	54	ve	ve	AUX
ejpam-6849	330	55	-	-	PUNCT
ejpam-6849	330	56	dominating	dominating	NOUN
ejpam-6849	330	57	sets	set	NOUN
ejpam-6849	330	58	of	of	ADP
ejpam-6849	330	59	g	g	NOUN
ejpam-6849	330	60	and	and	CCONJ
ejpam-6849	330	61	g	g	NOUN
ejpam-6849	330	62	,	,	PUNCT
ejpam-6849	330	63	respectively	respectively	ADV
ejpam-6849	330	64	.	.	PUNCT
ejpam-6849	331	1	(	(	PUNCT
ejpam-6849	331	2	b	b	X
ejpam-6849	331	3	)	)	PUNCT
ejpam-6849	331	4	g	g	NOUN
ejpam-6849	331	5	has	have	VERB
ejpam-6849	331	6	a	a	DET
ejpam-6849	331	7	γt	γt	NOUN
ejpam-6849	331	8	-	-	ADJ
ejpam-6849	331	9	set	set	ADJ
ejpam-6849	331	10	{	{	PUNCT
ejpam-6849	331	11	u	u	NOUN
ejpam-6849	331	12	,	,	PUNCT
ejpam-6849	331	13	v	v	NOUN
ejpam-6849	331	14	}	}	PUNCT
ejpam-6849	331	15	such	such	ADJ
ejpam-6849	331	16	that	that	SCONJ
ejpam-6849	331	17	xy	xy	PROPN
ejpam-6849	331	18	∈	∈	PROPN
ejpam-6849	331	19	e(g	e(g	PROPN
ejpam-6849	331	20	)	)	PUNCT
ejpam-6849	331	21	for	for	ADP
ejpam-6849	331	22	all	all	DET
ejpam-6849	331	23	x	x	NOUN
ejpam-6849	331	24	,	,	PUNCT
ejpam-6849	331	25	y	y	PROPN
ejpam-6849	331	26	∈	∈	PROPN
ejpam-6849	331	27	v	v	ADP
ejpam-6849	331	28	(	(	PUNCT
ejpam-6849	331	29	g	g	NOUN
ejpam-6849	331	30	)	)	PUNCT
ejpam-6849	331	31	\	\	NOUN
ejpam-6849	331	32	{	{	PUNCT
ejpam-6849	331	33	u	u	NOUN
ejpam-6849	331	34	,	,	PUNCT
ejpam-6849	331	35	v	v	NOUN
ejpam-6849	331	36	}	}	PUNCT
ejpam-6849	331	37	.	.	PUNCT
ejpam-6849	332	1	(	(	PUNCT
ejpam-6849	332	2	c	c	X
ejpam-6849	332	3	)	)	PUNCT
ejpam-6849	332	4	g	g	NOUN
ejpam-6849	332	5	has	have	AUX
ejpam-6849	332	6	a	a	DET
ejpam-6849	332	7	γt	γt	NOUN
ejpam-6849	332	8	-	-	ADJ
ejpam-6849	332	9	set	set	ADJ
ejpam-6849	332	10	{	{	PUNCT
ejpam-6849	332	11	u	u	NOUN
ejpam-6849	332	12	,	,	PUNCT
ejpam-6849	332	13	v	v	NOUN
ejpam-6849	332	14	}	}	PUNCT
ejpam-6849	332	15	such	such	ADJ
ejpam-6849	332	16	that	that	SCONJ
ejpam-6849	332	17	xy	xy	PROPN
ejpam-6849	332	18	∈	∈	PROPN
ejpam-6849	332	19	e(g	e(g	PROPN
ejpam-6849	332	20	)	)	PUNCT
ejpam-6849	332	21	for	for	ADP
ejpam-6849	332	22	all	all	DET
ejpam-6849	332	23	x	x	NOUN
ejpam-6849	332	24	,	,	PUNCT
ejpam-6849	332	25	y	y	PROPN
ejpam-6849	332	26	∈	∈	PROPN
ejpam-6849	332	27	v	v	ADP
ejpam-6849	332	28	(	(	PUNCT
ejpam-6849	332	29	g	g	NOUN
ejpam-6849	332	30	)	)	PUNCT
ejpam-6849	332	31	\	\	NOUN
ejpam-6849	332	32	{	{	PUNCT
ejpam-6849	332	33	u	u	NOUN
ejpam-6849	332	34	,	,	PUNCT
ejpam-6849	332	35	v	v	NOUN
ejpam-6849	332	36	}	}	PUNCT
ejpam-6849	332	37	;	;	PUNCT
ejpam-6849	332	38	(	(	PUNCT
ejpam-6849	332	39	iii	iii	X
ejpam-6849	332	40	)	)	PUNCT
ejpam-6849	332	41	γt	γt	NOUN
ejpam-6849	332	42	ve(gg	ve(gg	NOUN
ejpam-6849	332	43	)	)	PUNCT
ejpam-6849	332	44	̸=	̸=	PROPN
ejpam-6849	332	45	2n	2n	NUM
ejpam-6849	333	1	−	−	ADP
ejpam-6849	334	1	1	1	NUM
ejpam-6849	334	2	;	;	PUNCT
ejpam-6849	334	3	and	and	CCONJ
ejpam-6849	334	4	(	(	PUNCT
ejpam-6849	334	5	iv	iv	X
ejpam-6849	334	6	)	)	PUNCT
ejpam-6849	334	7	γt	γt	X
ejpam-6849	334	8	ve(gg	ve(gg	PROPN
ejpam-6849	334	9	)	)	PUNCT
ejpam-6849	335	1	=	=	SYM
ejpam-6849	335	2	2n	2n	NUM
ejpam-6849	335	3	if	if	SCONJ
ejpam-6849	335	4	and	and	CCONJ
ejpam-6849	335	5	only	only	ADV
ejpam-6849	335	6	if	if	SCONJ
ejpam-6849	335	7	g	g	PROPN
ejpam-6849	335	8	=	=	SYM
ejpam-6849	335	9	k1	k1	PROPN
ejpam-6849	335	10	.	.	PUNCT
ejpam-6849	336	1	proof	proof	NOUN
ejpam-6849	336	2	.	.	PUNCT
ejpam-6849	337	1	if	if	SCONJ
ejpam-6849	337	2	g	g	PROPN
ejpam-6849	337	3	=	=	PROPN
ejpam-6849	337	4	kn	kn	PROPN
ejpam-6849	337	5	and	and	CCONJ
ejpam-6849	337	6	v	v	ADP
ejpam-6849	337	7	∈	∈	PROPN
ejpam-6849	337	8	v	v	NOUN
ejpam-6849	337	9	(	(	PUNCT
ejpam-6849	337	10	g	g	NOUN
ejpam-6849	337	11	)	)	PUNCT
ejpam-6849	337	12	,	,	PUNCT
ejpam-6849	337	13	then	then	ADV
ejpam-6849	337	14	dgg(x	dgg(x	PRON
ejpam-6849	337	15	,	,	PUNCT
ejpam-6849	337	16	v	v	NOUN
ejpam-6849	337	17	)	)	PUNCT
ejpam-6849	337	18	=	=	SYM
ejpam-6849	337	19	1	1	NUM
ejpam-6849	337	20	or	or	CCONJ
ejpam-6849	337	21	dgg(y	dgg(y	PROPN
ejpam-6849	337	22	,	,	PUNCT
ejpam-6849	337	23	v	v	NOUN
ejpam-6849	337	24	)	)	PUNCT
ejpam-6849	337	25	=	=	SYM
ejpam-6849	337	26	1	1	NUM
ejpam-6849	337	27	for	for	ADP
ejpam-6849	337	28	each	each	DET
ejpam-6849	337	29	xy	xy	PROPN
ejpam-6849	337	30	∈	∈	PROPN
ejpam-6849	337	31	e(gg	e(gg	PROPN
ejpam-6849	337	32	)	)	PUNCT
ejpam-6849	337	33	.	.	PUNCT
ejpam-6849	338	1	by	by	ADP
ejpam-6849	338	2	proposition	proposition	NOUN
ejpam-6849	338	3	4	4	NUM
ejpam-6849	338	4	,	,	PUNCT
ejpam-6849	338	5	γve(gg	γve(gg	NOUN
ejpam-6849	338	6	)	)	PUNCT
ejpam-6849	338	7	=	=	SYM
ejpam-6849	339	1	1	1	X
ejpam-6849	339	2	.	.	PUNCT
ejpam-6849	340	1	in	in	ADP
ejpam-6849	340	2	case	case	NOUN
ejpam-6849	340	3	g	g	PROPN
ejpam-6849	340	4	=	=	SYM
ejpam-6849	340	5	kn	kn	PROPN
ejpam-6849	340	6	,	,	PUNCT
ejpam-6849	340	7	we	we	PRON
ejpam-6849	340	8	use	use	VERB
ejpam-6849	340	9	the	the	DET
ejpam-6849	340	10	same	same	ADJ
ejpam-6849	340	11	argument	argument	NOUN
ejpam-6849	340	12	on	on	ADP
ejpam-6849	340	13	g	g	PROPN
ejpam-6849	340	14	=	=	PROPN
ejpam-6849	340	15	kn	kn	PROPN
ejpam-6849	340	16	.	.	PUNCT
ejpam-6849	341	1	conversely	conversely	ADV
ejpam-6849	341	2	,	,	PUNCT
ejpam-6849	341	3	suppose	suppose	VERB
ejpam-6849	341	4	that	that	SCONJ
ejpam-6849	341	5	γve(gg	γve(gg	NOUN
ejpam-6849	341	6	)	)	PUNCT
ejpam-6849	341	7	=	=	SYM
ejpam-6849	342	1	1	1	X
ejpam-6849	342	2	.	.	X
ejpam-6849	342	3	assume	assume	VERB
ejpam-6849	342	4	g	g	PROPN
ejpam-6849	342	5	/∈	/∈	PUNCT
ejpam-6849	342	6	{	{	PUNCT
ejpam-6849	342	7	kn	kn	PROPN
ejpam-6849	342	8	,	,	PUNCT
ejpam-6849	342	9	kn	kn	PROPN
ejpam-6849	342	10	}	}	PUNCT
ejpam-6849	342	11	.	.	PUNCT
ejpam-6849	343	1	let	let	VERB
ejpam-6849	343	2	v	v	NUM
ejpam-6849	343	3	∈	∈	PROPN
ejpam-6849	343	4	v	v	NOUN
ejpam-6849	343	5	(	(	PUNCT
ejpam-6849	343	6	gg	gg	NOUN
ejpam-6849	343	7	)	)	PUNCT
ejpam-6849	343	8	such	such	ADJ
ejpam-6849	343	9	that	that	SCONJ
ejpam-6849	343	10	{	{	PUNCT
ejpam-6849	343	11	v	v	NOUN
ejpam-6849	343	12	}	}	PUNCT
ejpam-6849	343	13	is	be	AUX
ejpam-6849	343	14	a	a	DET
ejpam-6849	343	15	ve	ve	NOUN
ejpam-6849	343	16	-	-	PUNCT
ejpam-6849	343	17	dominating	dominate	VERB
ejpam-6849	343	18	set	set	NOUN
ejpam-6849	343	19	of	of	ADP
ejpam-6849	343	20	gg	gg	PROPN
ejpam-6849	343	21	.	.	PUNCT
ejpam-6849	344	1	wlog	wlog	PROPN
ejpam-6849	344	2	assume	assume	VERB
ejpam-6849	344	3	that	that	SCONJ
ejpam-6849	344	4	v	v	X
ejpam-6849	344	5	∈	∈	PROPN
ejpam-6849	344	6	v	v	NOUN
ejpam-6849	344	7	(	(	PUNCT
ejpam-6849	344	8	g	g	NOUN
ejpam-6849	344	9	)	)	PUNCT
ejpam-6849	344	10	.	.	PUNCT
ejpam-6849	345	1	we	we	PRON
ejpam-6849	345	2	consider	consider	VERB
ejpam-6849	345	3	the	the	DET
ejpam-6849	345	4	following	follow	VERB
ejpam-6849	345	5	cases	case	NOUN
ejpam-6849	345	6	:	:	PUNCT
ejpam-6849	345	7	case	case	NOUN
ejpam-6849	345	8	1	1	NUM
ejpam-6849	345	9	:	:	PUNCT
ejpam-6849	345	10	suppose	suppose	VERB
ejpam-6849	345	11	that	that	SCONJ
ejpam-6849	345	12	dg(u	dg(u	ADJ
ejpam-6849	345	13	,	,	PUNCT
ejpam-6849	345	14	v	v	NOUN
ejpam-6849	345	15	)	)	PUNCT
ejpam-6849	345	16	=	=	SYM
ejpam-6849	345	17	1	1	NUM
ejpam-6849	345	18	for	for	ADP
ejpam-6849	345	19	all	all	DET
ejpam-6849	345	20	u	u	NOUN
ejpam-6849	345	21	∈	∈	PROPN
ejpam-6849	345	22	v	v	NOUN
ejpam-6849	345	23	(	(	PUNCT
ejpam-6849	345	24	g	g	NOUN
ejpam-6849	345	25	)	)	PUNCT
ejpam-6849	345	26	\	\	NOUN
ejpam-6849	345	27	{	{	PUNCT
ejpam-6849	345	28	v	v	NOUN
ejpam-6849	345	29	}	}	PUNCT
ejpam-6849	345	30	.	.	PUNCT
ejpam-6849	346	1	since	since	SCONJ
ejpam-6849	346	2	g	g	PROPN
ejpam-6849	346	3	̸=	̸=	PROPN
ejpam-6849	346	4	kn	kn	PROPN
ejpam-6849	346	5	,	,	PUNCT
ejpam-6849	346	6	there	there	PRON
ejpam-6849	346	7	exist	exist	VERB
ejpam-6849	346	8	x	x	NOUN
ejpam-6849	346	9	,	,	PUNCT
ejpam-6849	346	10	y	y	PROPN
ejpam-6849	346	11	∈	∈	PROPN
ejpam-6849	346	12	v	v	ADP
ejpam-6849	346	13	(	(	PUNCT
ejpam-6849	346	14	g	g	NOUN
ejpam-6849	346	15	)	)	PUNCT
ejpam-6849	346	16	for	for	ADP
ejpam-6849	346	17	which	which	PRON
ejpam-6849	346	18	dg(x	dg(x	X
ejpam-6849	346	19	,	,	PUNCT
ejpam-6849	346	20	y	y	NOUN
ejpam-6849	346	21	)	)	PUNCT
ejpam-6849	346	22	=	=	SYM
ejpam-6849	346	23	2	2	X
ejpam-6849	346	24	.	.	X
ejpam-6849	346	25	observe	observe	VERB
ejpam-6849	346	26	that	that	SCONJ
ejpam-6849	346	27	x	x	PUNCT
ejpam-6849	346	28	y	y	PROPN
ejpam-6849	346	29	∈	∈	PROPN
ejpam-6849	346	30	e(gg	e(gg	PROPN
ejpam-6849	346	31	)	)	PUNCT
ejpam-6849	346	32	and	and	CCONJ
ejpam-6849	346	33	v	v	NOUN
ejpam-6849	346	34	does	do	AUX
ejpam-6849	346	35	not	not	PART
ejpam-6849	346	36	ve	ve	AUX
ejpam-6849	346	37	-	-	PUNCT
ejpam-6849	346	38	dominate	dominate	VERB
ejpam-6849	346	39	x	x	SYM
ejpam-6849	346	40	y	y	PROPN
ejpam-6849	346	41	,	,	PUNCT
ejpam-6849	346	42	a	a	DET
ejpam-6849	346	43	contradiction	contradiction	NOUN
ejpam-6849	346	44	.	.	PUNCT
ejpam-6849	347	1	case	case	NOUN
ejpam-6849	347	2	2	2	NUM
ejpam-6849	347	3	:	:	PUNCT
ejpam-6849	347	4	suppose	suppose	VERB
ejpam-6849	347	5	that	that	SCONJ
ejpam-6849	347	6	dg(u	dg(u	ADJ
ejpam-6849	347	7	,	,	PUNCT
ejpam-6849	347	8	v	v	NOUN
ejpam-6849	347	9	)	)	PUNCT
ejpam-6849	347	10	=	=	SYM
ejpam-6849	347	11	2	2	NUM
ejpam-6849	347	12	for	for	ADP
ejpam-6849	347	13	some	some	DET
ejpam-6849	347	14	u	u	NOUN
ejpam-6849	347	15	∈	∈	PROPN
ejpam-6849	347	16	v	v	NOUN
ejpam-6849	347	17	(	(	PUNCT
ejpam-6849	347	18	g	g	NOUN
ejpam-6849	347	19	)	)	PUNCT
ejpam-6849	347	20	.	.	PUNCT
ejpam-6849	348	1	in	in	ADP
ejpam-6849	348	2	this	this	DET
ejpam-6849	348	3	case	case	NOUN
ejpam-6849	348	4	,	,	PUNCT
ejpam-6849	348	5	it	it	PRON
ejpam-6849	348	6	is	be	AUX
ejpam-6849	348	7	easy	easy	ADJ
ejpam-6849	348	8	to	to	PART
ejpam-6849	348	9	see	see	VERB
ejpam-6849	348	10	that	that	SCONJ
ejpam-6849	348	11	v	v	NOUN
ejpam-6849	348	12	does	do	AUX
ejpam-6849	348	13	not	not	PART
ejpam-6849	348	14	ve	ve	AUX
ejpam-6849	348	15	-	-	PUNCT
ejpam-6849	348	16	dominate	dominate	VERB
ejpam-6849	348	17	uu	uu	PRON
ejpam-6849	348	18	∈	∈	PROPN
ejpam-6849	348	19	e(gg	e(gg	PROPN
ejpam-6849	348	20	)	)	PUNCT
ejpam-6849	348	21	,	,	PUNCT
ejpam-6849	348	22	a	a	DET
ejpam-6849	348	23	contradiction	contradiction	NOUN
ejpam-6849	348	24	.	.	PUNCT
ejpam-6849	349	1	the	the	DET
ejpam-6849	349	2	above	above	ADJ
ejpam-6849	349	3	contradictions	contradiction	NOUN
ejpam-6849	349	4	imply	imply	VERB
ejpam-6849	349	5	that	that	SCONJ
ejpam-6849	349	6	g	g	PROPN
ejpam-6849	349	7	∈	∈	PROPN
ejpam-6849	349	8	{	{	PUNCT
ejpam-6849	349	9	kn	kn	PROPN
ejpam-6849	349	10	,	,	PUNCT
ejpam-6849	349	11	kn	kn	PROPN
ejpam-6849	349	12	}	}	PUNCT
ejpam-6849	349	13	.	.	PUNCT
ejpam-6849	350	1	this	this	PRON
ejpam-6849	350	2	proves	prove	VERB
ejpam-6849	350	3	(	(	PUNCT
ejpam-6849	350	4	i	i	NOUN
ejpam-6849	350	5	)	)	PUNCT
ejpam-6849	350	6	.	.	PUNCT
ejpam-6849	351	1	suppose	suppose	VERB
ejpam-6849	351	2	that	that	SCONJ
ejpam-6849	351	3	γt	γt	PROPN
ejpam-6849	351	4	ve(gg	ve(gg	PROPN
ejpam-6849	351	5	)	)	PUNCT
ejpam-6849	351	6	=	=	SYM
ejpam-6849	351	7	2	2	NUM
ejpam-6849	351	8	,	,	PUNCT
ejpam-6849	351	9	and	and	CCONJ
ejpam-6849	351	10	let	let	VERB
ejpam-6849	351	11	s	s	PRON
ejpam-6849	351	12	=	=	PUNCT
ejpam-6849	351	13	{	{	PUNCT
ejpam-6849	351	14	u	u	NOUN
ejpam-6849	351	15	,	,	PUNCT
ejpam-6849	351	16	v	v	NOUN
ejpam-6849	351	17	}	}	PUNCT
ejpam-6849	351	18	be	be	AUX
ejpam-6849	351	19	a	a	DET
ejpam-6849	351	20	γt	γt	NOUN
ejpam-6849	351	21	ve	ve	NOUN
ejpam-6849	351	22	-	-	PUNCT
ejpam-6849	351	23	set	set	NOUN
ejpam-6849	351	24	of	of	ADP
ejpam-6849	351	25	gg	gg	PROPN
ejpam-6849	351	26	.	.	PUNCT
ejpam-6849	351	27	suppose	suppose	VERB
ejpam-6849	351	28	that	that	SCONJ
ejpam-6849	351	29	u	u	PROPN
ejpam-6849	351	30	=	=	PUNCT
ejpam-6849	351	31	v.	v.	CCONJ
ejpam-6849	351	32	assume	assume	VERB
ejpam-6849	351	33	v	v	ADP
ejpam-6849	351	34	∈	∈	PROPN
ejpam-6849	351	35	v	v	NOUN
ejpam-6849	351	36	(	(	PUNCT
ejpam-6849	351	37	g	g	NOUN
ejpam-6849	351	38	)	)	PUNCT
ejpam-6849	351	39	.	.	PUNCT
ejpam-6849	352	1	let	let	VERB
ejpam-6849	352	2	xy	xy	PROPN
ejpam-6849	352	3	∈	∈	PROPN
ejpam-6849	352	4	e(g	e(g	PROPN
ejpam-6849	352	5	)	)	PUNCT
ejpam-6849	352	6	.	.	PUNCT
ejpam-6849	353	1	if	if	SCONJ
ejpam-6849	353	2	x	x	X
ejpam-6849	353	3	=	=	SYM
ejpam-6849	353	4	v	v	NOUN
ejpam-6849	353	5	or	or	CCONJ
ejpam-6849	353	6	y	y	PROPN
ejpam-6849	353	7	=	=	PUNCT
ejpam-6849	353	8	v	v	PROPN
ejpam-6849	353	9	,	,	PUNCT
ejpam-6849	353	10	then	then	ADV
ejpam-6849	353	11	v	v	X
ejpam-6849	353	12	ve	ve	VERB
ejpam-6849	353	13	-	-	PUNCT
ejpam-6849	353	14	dominates	dominate	VERB
ejpam-6849	353	15	xy	xy	PRON
ejpam-6849	353	16	.	.	PUNCT
ejpam-6849	353	17	suppose	suppose	VERB
ejpam-6849	353	18	that	that	SCONJ
ejpam-6849	353	19	x	x	PROPN
ejpam-6849	353	20	̸=	̸=	PROPN
ejpam-6849	353	21	v	v	NOUN
ejpam-6849	353	22	and	and	CCONJ
ejpam-6849	353	23	y	y	NOUN
ejpam-6849	353	24	=	=	PROPN
ejpam-6849	353	25	̸	̸	PUNCT
ejpam-6849	353	26	v.	v.	CCONJ
ejpam-6849	353	27	since	since	SCONJ
ejpam-6849	353	28	s	s	PART
ejpam-6849	353	29	ve	ve	VERB
ejpam-6849	353	30	-	-	PUNCT
ejpam-6849	353	31	dominates	dominate	VERB
ejpam-6849	353	32	xy	xy	PROPN
ejpam-6849	353	33	,	,	PUNCT
ejpam-6849	353	34	v	v	PRON
ejpam-6849	353	35	ve	ve	VERB
ejpam-6849	353	36	-	-	PUNCT
ejpam-6849	353	37	dominates	dominate	VERB
ejpam-6849	353	38	xy	xy	PROPN
ejpam-6849	353	39	.	.	PUNCT
ejpam-6849	354	1	similarly	similarly	ADV
ejpam-6849	354	2	,	,	PUNCT
ejpam-6849	354	3	if	if	SCONJ
ejpam-6849	354	4	xy	xy	PROPN
ejpam-6849	354	5	∈	∈	PROPN
ejpam-6849	354	6	e(g	e(g	PROPN
ejpam-6849	354	7	)	)	PUNCT
ejpam-6849	354	8	,	,	PUNCT
ejpam-6849	354	9	then	then	ADV
ejpam-6849	354	10	v	v	AUX
ejpam-6849	354	11	ve	ve	VERB
ejpam-6849	354	12	-	-	PUNCT
ejpam-6849	354	13	dominates	dominate	VERB
ejpam-6849	354	14	xy	xy	PROPN
ejpam-6849	354	15	.	.	PUNCT
ejpam-6849	355	1	this	this	PRON
ejpam-6849	355	2	proves	prove	VERB
ejpam-6849	355	3	(	(	PUNCT
ejpam-6849	355	4	ii)(a	ii)(a	PROPN
ejpam-6849	355	5	)	)	PUNCT
ejpam-6849	355	6	.	.	PUNCT
ejpam-6849	356	1	suppose	suppose	VERB
ejpam-6849	356	2	that	that	SCONJ
ejpam-6849	356	3	s	s	VERB
ejpam-6849	356	4	⊆	⊆	NUM
ejpam-6849	356	5	v	v	NOUN
ejpam-6849	356	6	(	(	PUNCT
ejpam-6849	356	7	g	g	NOUN
ejpam-6849	356	8	)	)	PUNCT
ejpam-6849	356	9	.	.	PUNCT
ejpam-6849	357	1	first	first	ADV
ejpam-6849	357	2	,	,	PUNCT
ejpam-6849	357	3	we	we	PRON
ejpam-6849	357	4	claim	claim	VERB
ejpam-6849	357	5	that	that	SCONJ
ejpam-6849	357	6	s	s	VERB
ejpam-6849	357	7	is	be	AUX
ejpam-6849	357	8	a	a	DET
ejpam-6849	357	9	total	total	ADJ
ejpam-6849	357	10	dominating	dominating	NOUN
ejpam-6849	357	11	set	set	NOUN
ejpam-6849	357	12	of	of	ADP
ejpam-6849	357	13	g.	g.	PROPN
ejpam-6849	357	14	let	let	VERB
ejpam-6849	357	15	x	x	SYM
ejpam-6849	357	16	∈	∈	PROPN
ejpam-6849	357	17	v	v	X
ejpam-6849	357	18	(	(	PUNCT
ejpam-6849	357	19	g)\s	g)\s	NOUN
ejpam-6849	357	20	.	.	PUNCT
ejpam-6849	358	1	since	since	SCONJ
ejpam-6849	358	2	u	u	NOUN
ejpam-6849	358	3	or	or	CCONJ
ejpam-6849	358	4	v	v	ADP
ejpam-6849	358	5	ve	ve	VERB
ejpam-6849	358	6	-	-	PUNCT
ejpam-6849	358	7	dominates	dominate	VERB
ejpam-6849	358	8	xx	xx	NUM
ejpam-6849	358	9	,	,	PUNCT
ejpam-6849	358	10	ux	ux	PROPN
ejpam-6849	358	11	∈	∈	PROPN
ejpam-6849	358	12	e(g	e(g	PROPN
ejpam-6849	358	13	)	)	PUNCT
ejpam-6849	358	14	or	or	CCONJ
ejpam-6849	358	15	vx	vx	PROPN
ejpam-6849	358	16	∈	∈	PROPN
ejpam-6849	358	17	e(g	e(g	PROPN
ejpam-6849	358	18	)	)	PUNCT
ejpam-6849	358	19	.	.	PUNCT
ejpam-6849	359	1	thus	thus	ADV
ejpam-6849	359	2	,	,	PUNCT
ejpam-6849	359	3	s	s	VERB
ejpam-6849	359	4	is	be	AUX
ejpam-6849	359	5	a	a	DET
ejpam-6849	359	6	dominating	dominating	NOUN
ejpam-6849	359	7	set	set	NOUN
ejpam-6849	359	8	,	,	PUNCT
ejpam-6849	359	9	hence	hence	ADV
ejpam-6849	359	10	a	a	DET
ejpam-6849	359	11	total	total	ADJ
ejpam-6849	359	12	dominating	dominating	NOUN
ejpam-6849	359	13	set	set	NOUN
ejpam-6849	359	14	,	,	PUNCT
ejpam-6849	359	15	of	of	ADP
ejpam-6849	359	16	g.	g.	PROPN
ejpam-6849	359	17	next	next	ADV
ejpam-6849	359	18	,	,	PUNCT
ejpam-6849	359	19	let	let	VERB
ejpam-6849	359	20	x	x	PRON
ejpam-6849	359	21	,	,	PUNCT
ejpam-6849	359	22	y	y	PROPN
ejpam-6849	359	23	∈	∈	PROPN
ejpam-6849	359	24	v	v	ADP
ejpam-6849	359	25	(	(	PUNCT
ejpam-6849	359	26	g	g	NOUN
ejpam-6849	359	27	)	)	PUNCT
ejpam-6849	359	28	\	\	PUNCT
ejpam-6849	360	1	s.	s.	PROPN
ejpam-6849	360	2	suppose	suppose	VERB
ejpam-6849	360	3	that	that	SCONJ
ejpam-6849	360	4	x	x	PUNCT
ejpam-6849	360	5	y	y	PROPN
ejpam-6849	360	6	∈	∈	PROPN
ejpam-6849	360	7	e(g	e(g	PROPN
ejpam-6849	360	8	)	)	PUNCT
ejpam-6849	360	9	.	.	PUNCT
ejpam-6849	361	1	then	then	ADV
ejpam-6849	361	2	u	u	NOUN
ejpam-6849	361	3	or	or	CCONJ
ejpam-6849	361	4	v	v	ADP
ejpam-6849	361	5	ve	ve	VERB
ejpam-6849	361	6	-	-	PUNCT
ejpam-6849	361	7	dominates	dominate	VERB
ejpam-6849	361	8	x	x	PART
ejpam-6849	361	9	y.	y.	NOUN
ejpam-6849	361	10	this	this	PRON
ejpam-6849	361	11	is	be	AUX
ejpam-6849	361	12	impossible	impossible	ADJ
ejpam-6849	361	13	since	since	SCONJ
ejpam-6849	361	14	min{dgg(u	min{dgg(u	PROPN
ejpam-6849	361	15	,	,	PUNCT
ejpam-6849	361	16	x	x	NOUN
ejpam-6849	361	17	)	)	PUNCT
ejpam-6849	361	18	,	,	PUNCT
ejpam-6849	361	19	dgg(u	dgg(u	PROPN
ejpam-6849	361	20	,	,	PUNCT
ejpam-6849	361	21	y	y	PROPN
ejpam-6849	361	22	)	)	PUNCT
ejpam-6849	361	23	,	,	PUNCT
ejpam-6849	361	24	dgg(v	dgg(v	PROPN
ejpam-6849	361	25	,	,	PUNCT
ejpam-6849	361	26	x	x	NOUN
ejpam-6849	361	27	)	)	PUNCT
ejpam-6849	361	28	,	,	PUNCT
ejpam-6849	361	29	dgg(v	dgg(v	PROPN
ejpam-6849	361	30	,	,	PUNCT
ejpam-6849	361	31	y	y	NOUN
ejpam-6849	361	32	)	)	PUNCT
ejpam-6849	361	33	}	}	PUNCT
ejpam-6849	361	34	≥	≥	NOUN
ejpam-6849	361	35	2	2	NUM
ejpam-6849	361	36	.	.	PUNCT
ejpam-6849	362	1	thus	thus	ADV
ejpam-6849	362	2	,	,	PUNCT
ejpam-6849	362	3	x	x	PUNCT
ejpam-6849	362	4	y	y	PROPN
ejpam-6849	362	5	/∈	/∈	PUNCT
ejpam-6849	362	6	e(g	e(g	PROPN
ejpam-6849	362	7	)	)	PUNCT
ejpam-6849	362	8	and	and	CCONJ
ejpam-6849	362	9	(	(	PUNCT
ejpam-6849	362	10	ii)(b	ii)(b	ADJ
ejpam-6849	362	11	)	)	PUNCT
ejpam-6849	362	12	holds	hold	VERB
ejpam-6849	362	13	.	.	PUNCT
ejpam-6849	363	1	similarly	similarly	ADV
ejpam-6849	363	2	,	,	PUNCT
ejpam-6849	363	3	if	if	SCONJ
ejpam-6849	363	4	s	s	VERB
ejpam-6849	363	5	⊆	⊆	NUM
ejpam-6849	363	6	v	v	NOUN
ejpam-6849	363	7	(	(	PUNCT
ejpam-6849	363	8	g	g	NOUN
ejpam-6849	363	9	)	)	PUNCT
ejpam-6849	363	10	,	,	PUNCT
ejpam-6849	363	11	then	then	ADV
ejpam-6849	363	12	(	(	PUNCT
ejpam-6849	363	13	ii)(c	ii)(c	PROPN
ejpam-6849	363	14	)	)	PUNCT
ejpam-6849	363	15	holds	hold	VERB
ejpam-6849	363	16	.	.	PUNCT
ejpam-6849	364	1	j.	j.	PROPN
ejpam-6849	364	2	tayab	tayab	PROPN
ejpam-6849	364	3	,	,	PUNCT
ejpam-6849	364	4	f.	f.	PROPN
ejpam-6849	364	5	p.	p.	PROPN
ejpam-6849	364	6	jamil	jamil	PROPN
ejpam-6849	364	7	,	,	PUNCT
ejpam-6849	364	8	i.	i.	PROPN
ejpam-6849	364	9	aniversario	aniversario	PROPN
ejpam-6849	364	10	/	/	SYM
ejpam-6849	364	11	eur	eur	PROPN
ejpam-6849	364	12	.	.	PUNCT
ejpam-6849	365	1	j.	j.	PROPN
ejpam-6849	365	2	pure	pure	PROPN
ejpam-6849	365	3	appl	appl	PROPN
ejpam-6849	365	4	.	.	PROPN
ejpam-6849	365	5	math	math	PROPN
ejpam-6849	365	6	,	,	PUNCT
ejpam-6849	365	7	18	18	NUM
ejpam-6849	365	8	(	(	PUNCT
ejpam-6849	365	9	4	4	NUM
ejpam-6849	365	10	)	)	PUNCT
ejpam-6849	365	11	(	(	PUNCT
ejpam-6849	365	12	2025	2025	NUM
ejpam-6849	365	13	)	)	PUNCT
ejpam-6849	365	14	,	,	PUNCT
ejpam-6849	365	15	6849	6849	NUM
ejpam-6849	365	16	10	10	NUM
ejpam-6849	365	17	of	of	ADP
ejpam-6849	365	18	19	19	NUM
ejpam-6849	365	19	assume	assume	VERB
ejpam-6849	365	20	that	that	SCONJ
ejpam-6849	365	21	(	(	PUNCT
ejpam-6849	365	22	ii)(a	ii)(a	NOUN
ejpam-6849	365	23	)	)	PUNCT
ejpam-6849	365	24	holds	hold	VERB
ejpam-6849	365	25	for	for	ADP
ejpam-6849	365	26	g.	g.	NOUN
ejpam-6849	365	27	let	let	VERB
ejpam-6849	365	28	xy	xy	PROPN
ejpam-6849	365	29	∈	∈	PROPN
ejpam-6849	365	30	e(gg	e(gg	PROPN
ejpam-6849	365	31	)	)	PUNCT
ejpam-6849	365	32	\	\	NOUN
ejpam-6849	365	33	{	{	PUNCT
ejpam-6849	365	34	vv	vv	NOUN
ejpam-6849	365	35	}	}	PUNCT
ejpam-6849	365	36	.	.	PUNCT
ejpam-6849	366	1	if	if	SCONJ
ejpam-6849	366	2	x	x	SYM
ejpam-6849	366	3	∈	∈	PROPN
ejpam-6849	366	4	{	{	PUNCT
ejpam-6849	366	5	v	v	NOUN
ejpam-6849	366	6	,	,	PUNCT
ejpam-6849	366	7	v	v	NOUN
ejpam-6849	366	8	}	}	PUNCT
ejpam-6849	366	9	or	or	CCONJ
ejpam-6849	366	10	y	y	PROPN
ejpam-6849	366	11	∈	∈	PROPN
ejpam-6849	366	12	{	{	PUNCT
ejpam-6849	366	13	v	v	NOUN
ejpam-6849	366	14	,	,	PUNCT
ejpam-6849	366	15	v	v	NOUN
ejpam-6849	366	16	}	}	PUNCT
ejpam-6849	366	17	,	,	PUNCT
ejpam-6849	366	18	then	then	ADV
ejpam-6849	366	19	e(gg	e(gg	NUM
ejpam-6849	366	20	)	)	PUNCT
ejpam-6849	366	21	∩	∩	NOUN
ejpam-6849	366	22	{	{	PUNCT
ejpam-6849	366	23	xv	xv	PROPN
ejpam-6849	366	24	,	,	PUNCT
ejpam-6849	366	25	xv	xv	PROPN
ejpam-6849	366	26	,	,	PUNCT
ejpam-6849	366	27	yv	yv	PROPN
ejpam-6849	366	28	,	,	PUNCT
ejpam-6849	366	29	yv	yv	NOUN
ejpam-6849	366	30	}	}	PUNCT
ejpam-6849	366	31	=	=	NOUN
ejpam-6849	366	32	̸	̸	PUNCT
ejpam-6849	366	33	∅.	∅.	ADV
ejpam-6849	366	34	suppose	suppose	VERB
ejpam-6849	366	35	that	that	SCONJ
ejpam-6849	366	36	{	{	PUNCT
ejpam-6849	366	37	x	x	NOUN
ejpam-6849	366	38	,	,	PUNCT
ejpam-6849	366	39	y	y	NOUN
ejpam-6849	366	40	}	}	PUNCT
ejpam-6849	366	41	∩	∩	ADJ
ejpam-6849	366	42	{	{	PUNCT
ejpam-6849	366	43	v	v	NOUN
ejpam-6849	366	44	,	,	PUNCT
ejpam-6849	366	45	v	v	NOUN
ejpam-6849	366	46	}	}	PUNCT
ejpam-6849	366	47	=	=	PUNCT
ejpam-6849	366	48	∅.	∅.	NOUN
ejpam-6849	366	49	if	if	SCONJ
ejpam-6849	366	50	xy	xy	PROPN
ejpam-6849	366	51	∈	∈	PROPN
ejpam-6849	366	52	e(g	e(g	PROPN
ejpam-6849	366	53	)	)	PUNCT
ejpam-6849	366	54	,	,	PUNCT
ejpam-6849	366	55	then	then	ADV
ejpam-6849	366	56	v	v	AUX
ejpam-6849	366	57	ve	ve	VERB
ejpam-6849	366	58	-	-	PUNCT
ejpam-6849	366	59	dominates	dominate	VERB
ejpam-6849	366	60	xy	xy	PROPN
ejpam-6849	366	61	.	.	PUNCT
ejpam-6849	367	1	thus	thus	ADV
ejpam-6849	367	2	,	,	PUNCT
ejpam-6849	367	3	e(gg	e(gg	PROPN
ejpam-6849	367	4	)	)	PUNCT
ejpam-6849	367	5	∩	∩	NOUN
ejpam-6849	367	6	{	{	PUNCT
ejpam-6849	367	7	vx	vx	PROPN
ejpam-6849	367	8	,	,	PUNCT
ejpam-6849	367	9	vy	vy	NOUN
ejpam-6849	367	10	}	}	PUNCT
ejpam-6849	367	11	̸=	̸=	PROPN
ejpam-6849	367	12	∅.	∅.	ADV
ejpam-6849	367	13	similarly	similarly	ADV
ejpam-6849	367	14	,	,	PUNCT
ejpam-6849	367	15	if	if	SCONJ
ejpam-6849	367	16	xy	xy	PROPN
ejpam-6849	367	17	∈	∈	PROPN
ejpam-6849	367	18	e(g	e(g	PROPN
ejpam-6849	367	19	)	)	PUNCT
ejpam-6849	367	20	,	,	PUNCT
ejpam-6849	367	21	then	then	ADV
ejpam-6849	367	22	e(gg	e(gg	NUM
ejpam-6849	367	23	)	)	PUNCT
ejpam-6849	367	24	∩	∩	NOUN
ejpam-6849	367	25	{	{	PUNCT
ejpam-6849	367	26	xv	xv	PROPN
ejpam-6849	367	27	,	,	PUNCT
ejpam-6849	367	28	yv	yv	PROPN
ejpam-6849	367	29	}	}	PUNCT
ejpam-6849	367	30	̸=	̸=	PROPN
ejpam-6849	367	31	∅.	∅.	ADP
ejpam-6849	367	32	now	now	ADV
ejpam-6849	367	33	,	,	PUNCT
ejpam-6849	367	34	suppose	suppose	VERB
ejpam-6849	367	35	that	that	SCONJ
ejpam-6849	367	36	y	y	PROPN
ejpam-6849	367	37	=	=	PUNCT
ejpam-6849	367	38	x	x	PUNCT
ejpam-6849	367	39	and	and	CCONJ
ejpam-6849	367	40	assume	assume	VERB
ejpam-6849	367	41	x	x	SYM
ejpam-6849	367	42	∈	∈	PROPN
ejpam-6849	367	43	v	v	X
ejpam-6849	367	44	(	(	PUNCT
ejpam-6849	367	45	g	g	NOUN
ejpam-6849	367	46	)	)	PUNCT
ejpam-6849	367	47	.	.	PUNCT
ejpam-6849	368	1	then	then	ADV
ejpam-6849	368	2	either	either	CCONJ
ejpam-6849	368	3	vx	vx	PROPN
ejpam-6849	368	4	∈	∈	PROPN
ejpam-6849	368	5	e(g	e(g	PROPN
ejpam-6849	368	6	)	)	PUNCT
ejpam-6849	368	7	or	or	CCONJ
ejpam-6849	368	8	v	v	NOUN
ejpam-6849	368	9	x	x	SYM
ejpam-6849	368	10	∈	∈	PROPN
ejpam-6849	368	11	e(g	e(g	PROPN
ejpam-6849	368	12	)	)	PUNCT
ejpam-6849	368	13	.	.	PUNCT
ejpam-6849	369	1	thus	thus	ADV
ejpam-6849	369	2	,	,	PUNCT
ejpam-6849	369	3	e(gg	e(gg	PROPN
ejpam-6849	369	4	)	)	PUNCT
ejpam-6849	369	5	∩	∩	NOUN
ejpam-6849	369	6	{	{	PUNCT
ejpam-6849	369	7	xv	xv	PROPN
ejpam-6849	369	8	,	,	PUNCT
ejpam-6849	369	9	yv	yv	PROPN
ejpam-6849	369	10	}	}	PUNCT
ejpam-6849	369	11	̸=	̸=	PROPN
ejpam-6849	369	12	∅.	∅.	ADV
ejpam-6849	369	13	by	by	ADP
ejpam-6849	369	14	proposition	proposition	NOUN
ejpam-6849	369	15	5	5	NUM
ejpam-6849	369	16	,	,	PUNCT
ejpam-6849	369	17	γt	γt	X
ejpam-6849	369	18	ve(gg	ve(gg	NOUN
ejpam-6849	369	19	)	)	PUNCT
ejpam-6849	369	20	=	=	SYM
ejpam-6849	370	1	2	2	X
ejpam-6849	370	2	.	.	PUNCT
ejpam-6849	371	1	next	next	ADV
ejpam-6849	371	2	,	,	PUNCT
ejpam-6849	371	3	assume	assume	VERB
ejpam-6849	371	4	that	that	SCONJ
ejpam-6849	371	5	(	(	PUNCT
ejpam-6849	371	6	ii)(b	ii)(b	ADJ
ejpam-6849	371	7	)	)	PUNCT
ejpam-6849	371	8	holds	hold	VERB
ejpam-6849	371	9	.	.	PUNCT
ejpam-6849	372	1	let	let	VERB
ejpam-6849	372	2	xy	xy	PROPN
ejpam-6849	372	3	∈	∈	PROPN
ejpam-6849	372	4	e(gg	e(gg	PROPN
ejpam-6849	372	5	)	)	PUNCT
ejpam-6849	372	6	\	\	NOUN
ejpam-6849	372	7	{	{	PUNCT
ejpam-6849	372	8	uv	uv	NOUN
ejpam-6849	372	9	}	}	PUNCT
ejpam-6849	372	10	.	.	PUNCT
ejpam-6849	373	1	if	if	SCONJ
ejpam-6849	373	2	x	x	SYM
ejpam-6849	373	3	∈	∈	PROPN
ejpam-6849	373	4	{	{	PUNCT
ejpam-6849	373	5	u	u	NOUN
ejpam-6849	373	6	,	,	PUNCT
ejpam-6849	373	7	v	v	NOUN
ejpam-6849	373	8	}	}	PUNCT
ejpam-6849	373	9	or	or	CCONJ
ejpam-6849	373	10	y	y	PROPN
ejpam-6849	373	11	∈	∈	PROPN
ejpam-6849	373	12	{	{	PUNCT
ejpam-6849	373	13	u	u	NOUN
ejpam-6849	373	14	,	,	PUNCT
ejpam-6849	373	15	v	v	NOUN
ejpam-6849	373	16	}	}	PUNCT
ejpam-6849	373	17	,	,	PUNCT
ejpam-6849	373	18	then	then	ADV
ejpam-6849	373	19	e(gg	e(gg	PROPN
ejpam-6849	373	20	)	)	PUNCT
ejpam-6849	373	21	∩	∩	NOUN
ejpam-6849	373	22	{	{	PUNCT
ejpam-6849	373	23	ux	ux	PROPN
ejpam-6849	373	24	,	,	PUNCT
ejpam-6849	373	25	uy	uy	PROPN
ejpam-6849	373	26	,	,	PUNCT
ejpam-6849	373	27	vx	vx	PROPN
ejpam-6849	373	28	,	,	PUNCT
ejpam-6849	373	29	vy	vy	NOUN
ejpam-6849	373	30	}	}	PUNCT
ejpam-6849	373	31	=	=	PUNCT
ejpam-6849	373	32	̸	̸	PUNCT
ejpam-6849	373	33	∅.	∅.	ADV
ejpam-6849	373	34	suppose	suppose	VERB
ejpam-6849	373	35	that	that	SCONJ
ejpam-6849	373	36	{	{	PUNCT
ejpam-6849	373	37	x	x	NOUN
ejpam-6849	373	38	,	,	PUNCT
ejpam-6849	373	39	y	y	NOUN
ejpam-6849	373	40	}	}	PUNCT
ejpam-6849	373	41	∩	∩	NOUN
ejpam-6849	373	42	{	{	PUNCT
ejpam-6849	373	43	u	u	NOUN
ejpam-6849	373	44	,	,	PUNCT
ejpam-6849	373	45	v	v	NOUN
ejpam-6849	373	46	}	}	PUNCT
ejpam-6849	373	47	=	=	PUNCT
ejpam-6849	373	48	∅.	∅.	VERB
ejpam-6849	373	49	if	if	SCONJ
ejpam-6849	373	50	x	x	PROPN
ejpam-6849	373	51	∈	∈	PROPN
ejpam-6849	373	52	v	v	X
ejpam-6849	373	53	(	(	PUNCT
ejpam-6849	373	54	g	g	NOUN
ejpam-6849	373	55	)	)	PUNCT
ejpam-6849	373	56	,	,	PUNCT
ejpam-6849	373	57	then	then	ADV
ejpam-6849	373	58	since	since	SCONJ
ejpam-6849	373	59	{	{	PUNCT
ejpam-6849	373	60	u	u	NOUN
ejpam-6849	373	61	,	,	PUNCT
ejpam-6849	373	62	v	v	NOUN
ejpam-6849	373	63	}	}	PUNCT
ejpam-6849	373	64	is	be	AUX
ejpam-6849	373	65	a	a	DET
ejpam-6849	373	66	dominating	dominating	NOUN
ejpam-6849	373	67	set	set	NOUN
ejpam-6849	373	68	of	of	ADP
ejpam-6849	373	69	g	g	NOUN
ejpam-6849	373	70	,	,	PUNCT
ejpam-6849	373	71	e(gg	e(gg	PROPN
ejpam-6849	373	72	)	)	PUNCT
ejpam-6849	373	73	∩	∩	NOUN
ejpam-6849	373	74	{	{	PUNCT
ejpam-6849	373	75	ux	ux	PROPN
ejpam-6849	373	76	,	,	PUNCT
ejpam-6849	373	77	vx	vx	AUX
ejpam-6849	373	78	}	}	PUNCT
ejpam-6849	373	79	̸=	̸=	PROPN
ejpam-6849	373	80	∅.	∅.	ADV
ejpam-6849	373	81	suppose	suppose	VERB
ejpam-6849	373	82	that	that	SCONJ
ejpam-6849	373	83	x	x	NOUN
ejpam-6849	373	84	,	,	PUNCT
ejpam-6849	373	85	y	y	PROPN
ejpam-6849	373	86	∈	∈	PROPN
ejpam-6849	373	87	v	v	NOUN
ejpam-6849	373	88	(	(	PUNCT
ejpam-6849	373	89	g	g	NOUN
ejpam-6849	373	90	)	)	PUNCT
ejpam-6849	373	91	.	.	PUNCT
ejpam-6849	374	1	then	then	ADV
ejpam-6849	374	2	(	(	PUNCT
ejpam-6849	374	3	exactly	exactly	ADV
ejpam-6849	374	4	)	)	PUNCT
ejpam-6849	374	5	one	one	NUM
ejpam-6849	374	6	of	of	ADP
ejpam-6849	374	7	the	the	DET
ejpam-6849	374	8	following	follow	VERB
ejpam-6849	374	9	is	be	AUX
ejpam-6849	374	10	true	true	ADJ
ejpam-6849	374	11	:	:	PUNCT
ejpam-6849	374	12	x	x	SYM
ejpam-6849	374	13	=	=	SYM
ejpam-6849	374	14	v	v	PROPN
ejpam-6849	374	15	,	,	PUNCT
ejpam-6849	374	16	y	y	PROPN
ejpam-6849	374	17	=	=	SYM
ejpam-6849	374	18	v	v	PROPN
ejpam-6849	374	19	,	,	PUNCT
ejpam-6849	374	20	x	x	PUNCT
ejpam-6849	374	21	=	=	SYM
ejpam-6849	374	22	u	u	NOUN
ejpam-6849	374	23	,	,	PUNCT
ejpam-6849	374	24	y	y	PROPN
ejpam-6849	374	25	=	=	PUNCT
ejpam-6849	374	26	u.	u.	NOUN
ejpam-6849	374	27	in	in	ADP
ejpam-6849	374	28	any	any	DET
ejpam-6849	374	29	case	case	NOUN
ejpam-6849	374	30	,	,	PUNCT
ejpam-6849	374	31	e(gg	e(gg	PROPN
ejpam-6849	374	32	)	)	PUNCT
ejpam-6849	374	33	∩	∩	NOUN
ejpam-6849	374	34	{	{	PUNCT
ejpam-6849	374	35	ux	ux	PROPN
ejpam-6849	374	36	,	,	PUNCT
ejpam-6849	374	37	uy	uy	PROPN
ejpam-6849	374	38	,	,	PUNCT
ejpam-6849	374	39	vx	vx	PROPN
ejpam-6849	374	40	,	,	PUNCT
ejpam-6849	374	41	vy	vy	NOUN
ejpam-6849	374	42	}	}	PUNCT
ejpam-6849	374	43	=	=	SYM
ejpam-6849	374	44	̸	̸	ADV
ejpam-6849	374	45	∅.	∅.	NOUN
ejpam-6849	374	46	by	by	ADP
ejpam-6849	374	47	proposition	proposition	NOUN
ejpam-6849	374	48	5	5	NUM
ejpam-6849	374	49	,	,	PUNCT
ejpam-6849	374	50	γt	γt	X
ejpam-6849	374	51	ve(gg	ve(gg	NOUN
ejpam-6849	374	52	)	)	PUNCT
ejpam-6849	374	53	=	=	SYM
ejpam-6849	375	1	2	2	X
ejpam-6849	375	2	.	.	X
ejpam-6849	375	3	similarly	similarly	ADV
ejpam-6849	375	4	,	,	PUNCT
ejpam-6849	375	5	if	if	SCONJ
ejpam-6849	375	6	(	(	PUNCT
ejpam-6849	375	7	ii)(c	ii)(c	PROPN
ejpam-6849	375	8	)	)	PUNCT
ejpam-6849	375	9	holds	hold	VERB
ejpam-6849	375	10	,	,	PUNCT
ejpam-6849	375	11	then	then	ADV
ejpam-6849	375	12	γt	γt	PROPN
ejpam-6849	375	13	ve(gg	ve(gg	NOUN
ejpam-6849	375	14	)	)	PUNCT
ejpam-6849	375	15	=	=	SYM
ejpam-6849	376	1	2	2	X
ejpam-6849	376	2	.	.	PUNCT
ejpam-6849	376	3	statements	statement	NOUN
ejpam-6849	376	4	(	(	PUNCT
ejpam-6849	376	5	iii	iii	NOUN
ejpam-6849	376	6	)	)	PUNCT
ejpam-6849	376	7	and	and	CCONJ
ejpam-6849	376	8	(	(	PUNCT
ejpam-6849	376	9	iv	iv	X
ejpam-6849	376	10	)	)	PUNCT
ejpam-6849	376	11	immediately	immediately	ADV
ejpam-6849	376	12	follow	follow	VERB
ejpam-6849	376	13	from	from	ADP
ejpam-6849	376	14	proposition	proposition	NOUN
ejpam-6849	376	15	5(ii	5(ii	NUM
ejpam-6849	376	16	)	)	PUNCT
ejpam-6849	376	17	and	and	CCONJ
ejpam-6849	376	18	proposition	proposition	NOUN
ejpam-6849	376	19	5(i	5(i	NUM
ejpam-6849	376	20	)	)	PUNCT
ejpam-6849	376	21	,	,	PUNCT
ejpam-6849	376	22	respectively	respectively	ADV
ejpam-6849	376	23	.	.	PUNCT
ejpam-6849	377	1	■	■	PUNCT
ejpam-6849	377	2	graphs	graph	VERB
ejpam-6849	377	3	g	g	NOUN
ejpam-6849	377	4	in	in	ADP
ejpam-6849	377	5	proposition	proposition	NOUN
ejpam-6849	377	6	9(ii	9(ii	NUM
ejpam-6849	377	7	)	)	PUNCT
ejpam-6849	377	8	need	need	AUX
ejpam-6849	377	9	not	not	PART
ejpam-6849	377	10	have	have	VERB
ejpam-6849	377	11	to	to	PART
ejpam-6849	377	12	be	be	AUX
ejpam-6849	377	13	complete	complete	ADJ
ejpam-6849	377	14	.	.	PUNCT
ejpam-6849	378	1	graphs	graph	NOUN
ejpam-6849	378	2	g1	g1	NOUN
ejpam-6849	378	3	=	=	PUNCT
ejpam-6849	378	4	k1,3	k1,3	PROPN
ejpam-6849	378	5	and	and	CCONJ
ejpam-6849	378	6	g2	g2	PROPN
ejpam-6849	378	7	=	=	PUNCT
ejpam-6849	378	8	c4	c4	PROPN
ejpam-6849	378	9	in	in	ADP
ejpam-6849	378	10	figure	figure	NOUN
ejpam-6849	378	11	2	2	NUM
ejpam-6849	378	12	are	be	AUX
ejpam-6849	378	13	examples	example	NOUN
ejpam-6849	378	14	for	for	ADP
ejpam-6849	378	15	statements	statement	NOUN
ejpam-6849	378	16	(	(	PUNCT
ejpam-6849	378	17	ii)(a	ii)(a	PROPN
ejpam-6849	378	18	)	)	PUNCT
ejpam-6849	378	19	and	and	CCONJ
ejpam-6849	378	20	(	(	PUNCT
ejpam-6849	378	21	ii)(b	ii)(b	PROPN
ejpam-6849	378	22	)	)	PUNCT
ejpam-6849	378	23	,	,	PUNCT
ejpam-6849	378	24	respectively	respectively	ADV
ejpam-6849	378	25	.	.	PUNCT
ejpam-6849	379	1	in	in	ADP
ejpam-6849	379	2	each	each	DET
ejpam-6849	379	3	case	case	NOUN
ejpam-6849	379	4	,	,	PUNCT
ejpam-6849	379	5	γt	γt	X
ejpam-6849	379	6	ve(gg	ve(gg	NOUN
ejpam-6849	379	7	)	)	PUNCT
ejpam-6849	379	8	=	=	SYM
ejpam-6849	380	1	2	2	X
ejpam-6849	380	2	.	.	PUNCT
ejpam-6849	380	3	....................................	....................................	PUNCT
ejpam-6849	380	4	....................................	....................................	PUNCT
ejpam-6849	380	5	....................................	....................................	PUNCT
ejpam-6849	380	6	....................................	....................................	PUNCT
ejpam-6849	380	7	....................................	....................................	PUNCT
ejpam-6849	380	8	....................................	....................................	PUNCT
ejpam-6849	380	9	....................................	....................................	PUNCT
ejpam-6849	380	10	..............................................................................................................................................................................................................................................................................................................................................................................................................................	..............................................................................................................................................................................................................................................................................................................................................................................................................................	PUNCT
ejpam-6849	380	11	............................................................................................................................................................................................................................................................	............................................................................................................................................................................................................................................................	PUNCT
ejpam-6849	380	12	..........................	..........................	PUNCT
ejpam-6849	380	13	.........................	.........................	PUNCT
ejpam-6849	380	14	.........................	.........................	PUNCT
ejpam-6849	380	15	.........................	.........................	PUNCT
ejpam-6849	380	16	.........................	.........................	PUNCT
ejpam-6849	380	17	..........................	..........................	PUNCT
ejpam-6849	380	18	.........................	.........................	PUNCT
ejpam-6849	380	19	.........................	.........................	PUNCT
ejpam-6849	380	20	.........................	.........................	PUNCT
ejpam-6849	380	21	.........................	.........................	PUNCT
ejpam-6849	380	22	....................................	....................................	PUNCT
ejpam-6849	380	23	.................................................................................................................................................	.................................................................................................................................................	PUNCT
ejpam-6849	380	24	....................................	....................................	PUNCT
ejpam-6849	380	25	............................................................................................................................................................................................................................................................	............................................................................................................................................................................................................................................................	PUNCT
ejpam-6849	381	1	....................................	....................................	PUNCT
ejpam-6849	381	2	....................................	....................................	PUNCT
ejpam-6849	382	1	•	•	NUM
ejpam-6849	382	2	•	•	NOUN
ejpam-6849	382	3	.........	.........	PUNCT
ejpam-6849	382	4	........	........	PUNCT
ejpam-6849	382	5	........	........	PUNCT
ejpam-6849	382	6	........	........	PUNCT
ejpam-6849	382	7	........	........	PUNCT
ejpam-6849	382	8	........	........	PUNCT
ejpam-6849	382	9	........	........	PUNCT
ejpam-6849	382	10	........	........	PUNCT
ejpam-6849	382	11	........	........	PUNCT
ejpam-6849	382	12	........	........	PUNCT
ejpam-6849	382	13	........	........	PUNCT
ejpam-6849	382	14	........	........	PUNCT
ejpam-6849	382	15	........	........	PUNCT
ejpam-6849	382	16	........	........	PUNCT
ejpam-6849	382	17	......	......	PUNCT
ejpam-6849	382	18	.........	.........	PUNCT
ejpam-6849	382	19	........	........	PUNCT
ejpam-6849	382	20	........	........	PUNCT
ejpam-6849	382	21	........	........	PUNCT
ejpam-6849	382	22	........	........	PUNCT
ejpam-6849	382	23	........	........	PUNCT
ejpam-6849	382	24	........	........	PUNCT
ejpam-6849	382	25	........	........	PUNCT
ejpam-6849	382	26	........	........	PUNCT
ejpam-6849	382	27	........	........	PUNCT
ejpam-6849	382	28	........	........	PUNCT
ejpam-6849	382	29	........	........	PUNCT
ejpam-6849	382	30	........	........	PUNCT
ejpam-6849	382	31	........	........	PUNCT
ejpam-6849	382	32	......	......	PUNCT
ejpam-6849	382	33	.........	.........	PUNCT
ejpam-6849	382	34	........	........	PUNCT
ejpam-6849	382	35	........	........	PUNCT
ejpam-6849	382	36	........	........	PUNCT
ejpam-6849	382	37	........	........	PUNCT
ejpam-6849	382	38	........	........	PUNCT
ejpam-6849	382	39	........	........	PUNCT
ejpam-6849	382	40	........	........	PUNCT
ejpam-6849	382	41	........	........	PUNCT
ejpam-6849	382	42	........	........	PUNCT
ejpam-6849	382	43	........	........	PUNCT
ejpam-6849	382	44	........	........	PUNCT
ejpam-6849	382	45	........	........	PUNCT
ejpam-6849	382	46	........	........	PUNCT
ejpam-6849	382	47	......	......	PUNCT
ejpam-6849	382	48	.........	.........	PUNCT
ejpam-6849	382	49	........	........	PUNCT
ejpam-6849	382	50	........	........	PUNCT
ejpam-6849	382	51	........	........	PUNCT
ejpam-6849	382	52	........	........	PUNCT
ejpam-6849	382	53	........	........	PUNCT
ejpam-6849	382	54	........	........	PUNCT
ejpam-6849	382	55	........	........	PUNCT
ejpam-6849	382	56	........	........	PUNCT
ejpam-6849	382	57	........	........	PUNCT
ejpam-6849	382	58	........	........	PUNCT
ejpam-6849	382	59	........	........	PUNCT
ejpam-6849	382	60	........	........	PUNCT
ejpam-6849	382	61	........	........	PUNCT
ejpam-6849	382	62	......	......	PUNCT
ejpam-6849	383	1	v	v	NUM
ejpam-6849	383	2	u	u	NOUN
ejpam-6849	383	3	g1	g1	PROPN
ejpam-6849	383	4	:	:	PUNCT
ejpam-6849	383	5	g1	g1	PROPN
ejpam-6849	383	6	:	:	PUNCT
ejpam-6849	383	7	....................................	....................................	PUNCT
ejpam-6849	383	8	....................................	....................................	PUNCT
ejpam-6849	383	9	....................................	....................................	PUNCT
ejpam-6849	383	10	....................................	....................................	PUNCT
ejpam-6849	383	11	....................................	....................................	PUNCT
ejpam-6849	383	12	....................................	....................................	PUNCT
ejpam-6849	383	13	....................................	....................................	PUNCT
ejpam-6849	383	14	..............................................................	..............................................................	PUNCT
ejpam-6849	383	15	.........................	.........................	PUNCT
ejpam-6849	383	16	.........................	.........................	PUNCT
ejpam-6849	383	17	.........................	.........................	PUNCT
ejpam-6849	383	18	.........................	.........................	PUNCT
ejpam-6849	383	19	....................................	....................................	PUNCT
ejpam-6849	383	20	............................................................................................................................................................................................................................................................	............................................................................................................................................................................................................................................................	PUNCT
ejpam-6849	383	21	....................................	....................................	PUNCT
ejpam-6849	383	22	..........................	..........................	PUNCT
ejpam-6849	383	23	.........................	.........................	PUNCT
ejpam-6849	383	24	.........................	.........................	PUNCT
ejpam-6849	383	25	.........................	.........................	PUNCT
ejpam-6849	383	26	.........................	.........................	PUNCT
ejpam-6849	383	27	................................................................................................................................................................................................................................................................................................	................................................................................................................................................................................................................................................................................................	PUNCT
ejpam-6849	383	28	....................................	....................................	PUNCT
ejpam-6849	383	29	....................................	....................................	PUNCT
ejpam-6849	383	30	..........................................................................................................................................................................................................................................................................................................................................................................................	..........................................................................................................................................................................................................................................................................................................................................................................................	PUNCT
ejpam-6849	383	31	.................................................................................................................................................	.................................................................................................................................................	PUNCT
ejpam-6849	383	32	.........	.........	PUNCT
ejpam-6849	383	33	........	........	PUNCT
ejpam-6849	383	34	........	........	PUNCT
ejpam-6849	383	35	........	........	PUNCT
ejpam-6849	383	36	........	........	PUNCT
ejpam-6849	383	37	........	........	PUNCT
ejpam-6849	383	38	........	........	PUNCT
ejpam-6849	383	39	........	........	PUNCT
ejpam-6849	383	40	........	........	PUNCT
ejpam-6849	383	41	........	........	PUNCT
ejpam-6849	383	42	........	........	PUNCT
ejpam-6849	383	43	........	........	PUNCT
ejpam-6849	383	44	........	........	PUNCT
ejpam-6849	383	45	........	........	PUNCT
ejpam-6849	383	46	......	......	PUNCT
ejpam-6849	383	47	.........	.........	PUNCT
ejpam-6849	383	48	........	........	PUNCT
ejpam-6849	383	49	........	........	PUNCT
ejpam-6849	383	50	........	........	PUNCT
ejpam-6849	383	51	........	........	PUNCT
ejpam-6849	383	52	........	........	PUNCT
ejpam-6849	383	53	........	........	PUNCT
ejpam-6849	383	54	........	........	PUNCT
ejpam-6849	383	55	........	........	PUNCT
ejpam-6849	383	56	........	........	PUNCT
ejpam-6849	383	57	........	........	PUNCT
ejpam-6849	383	58	........	........	PUNCT
ejpam-6849	383	59	........	........	PUNCT
ejpam-6849	383	60	........	........	PUNCT
ejpam-6849	383	61	......	......	PUNCT
ejpam-6849	383	62	.........	.........	PUNCT
ejpam-6849	383	63	........	........	PUNCT
ejpam-6849	383	64	........	........	PUNCT
ejpam-6849	383	65	........	........	PUNCT
ejpam-6849	383	66	........	........	PUNCT
ejpam-6849	383	67	........	........	PUNCT
ejpam-6849	383	68	........	........	PUNCT
ejpam-6849	383	69	........	........	PUNCT
ejpam-6849	383	70	........	........	PUNCT
ejpam-6849	383	71	........	........	PUNCT
ejpam-6849	383	72	........	........	PUNCT
ejpam-6849	383	73	........	........	PUNCT
ejpam-6849	383	74	........	........	PUNCT
ejpam-6849	383	75	........	........	PUNCT
ejpam-6849	383	76	......	......	PUNCT
ejpam-6849	383	77	.........	.........	PUNCT
ejpam-6849	383	78	........	........	PUNCT
ejpam-6849	383	79	........	........	PUNCT
ejpam-6849	383	80	........	........	PUNCT
ejpam-6849	383	81	........	........	PUNCT
ejpam-6849	383	82	........	........	PUNCT
ejpam-6849	383	83	........	........	PUNCT
ejpam-6849	383	84	........	........	PUNCT
ejpam-6849	383	85	........	........	PUNCT
ejpam-6849	383	86	........	........	PUNCT
ejpam-6849	383	87	........	........	PUNCT
ejpam-6849	383	88	........	........	PUNCT
ejpam-6849	383	89	........	........	PUNCT
ejpam-6849	383	90	........	........	PUNCT
ejpam-6849	384	1	......	......	PUNCT
ejpam-6849	385	1	•	•	NUM
ejpam-6849	385	2	•	•	NUM
ejpam-6849	385	3	g2	g2	PROPN
ejpam-6849	385	4	:	:	PUNCT
ejpam-6849	386	1	g2	g2	PROPN
ejpam-6849	386	2	:	:	PUNCT
ejpam-6849	386	3	u	u	NOUN
ejpam-6849	386	4	v	v	NUM
ejpam-6849	386	5	figure	figure	NOUN
ejpam-6849	386	6	2	2	NUM
ejpam-6849	386	7	:	:	PUNCT
ejpam-6849	386	8	examples	example	NOUN
ejpam-6849	386	9	of	of	ADP
ejpam-6849	386	10	a	a	DET
ejpam-6849	386	11	noncomplete	noncomplete	ADJ
ejpam-6849	386	12	graph	graph	NOUN
ejpam-6849	386	13	g	g	NOUN
ejpam-6849	386	14	for	for	ADP
ejpam-6849	386	15	which	which	PRON
ejpam-6849	386	16	γt	γt	ADP
ejpam-6849	386	17	ve(gg	ve(gg	PROPN
ejpam-6849	386	18	)	)	PUNCT
ejpam-6849	386	19	=	=	SYM
ejpam-6849	387	1	2	2	X
ejpam-6849	387	2	.	.	X
ejpam-6849	387	3	proposition	proposition	NOUN
ejpam-6849	387	4	10	10	NUM
ejpam-6849	387	5	.	.	PUNCT
ejpam-6849	388	1	for	for	ADP
ejpam-6849	388	2	any	any	DET
ejpam-6849	388	3	graph	graph	NOUN
ejpam-6849	388	4	g	g	NOUN
ejpam-6849	388	5	,	,	PUNCT
ejpam-6849	388	6	(	(	PUNCT
ejpam-6849	388	7	i	i	NOUN
ejpam-6849	388	8	)	)	PUNCT
ejpam-6849	388	9	γve(gg	γve(gg	NOUN
ejpam-6849	388	10	)	)	PUNCT
ejpam-6849	388	11	≤	≤	NOUN
ejpam-6849	388	12	min{γ(g	min{γ(g	PROPN
ejpam-6849	388	13	)	)	PUNCT
ejpam-6849	389	1	+	+	NUM
ejpam-6849	390	1	γve(g	γve(g	NOUN
ejpam-6849	390	2	)	)	PUNCT
ejpam-6849	390	3	,	,	PUNCT
ejpam-6849	390	4	γ(g	γ(g	PROPN
ejpam-6849	390	5	)	)	PUNCT
ejpam-6849	390	6	+	+	NUM
ejpam-6849	391	1	γve(g	γve(g	NOUN
ejpam-6849	391	2	)	)	PUNCT
ejpam-6849	391	3	}	}	PUNCT
ejpam-6849	391	4	;	;	PUNCT
ejpam-6849	391	5	(	(	PUNCT
ejpam-6849	391	6	ii	ii	NOUN
ejpam-6849	391	7	)	)	PUNCT
ejpam-6849	391	8	provided	provide	VERB
ejpam-6849	391	9	g	g	PROPN
ejpam-6849	391	10	and	and	CCONJ
ejpam-6849	391	11	g	g	PROPN
ejpam-6849	391	12	have	have	VERB
ejpam-6849	391	13	no	no	DET
ejpam-6849	391	14	isolated	isolate	VERB
ejpam-6849	391	15	vertices	vertex	NOUN
ejpam-6849	391	16	,	,	PUNCT
ejpam-6849	391	17	γt	γt	X
ejpam-6849	391	18	ve(gg	ve(gg	NOUN
ejpam-6849	391	19	)	)	PUNCT
ejpam-6849	391	20	≤	≤	NUM
ejpam-6849	391	21	min{γt(g	min{γt(g	NOUN
ejpam-6849	391	22	)	)	PUNCT
ejpam-6849	392	1	+	+	CCONJ
ejpam-6849	392	2	γt	γt	NOUN
ejpam-6849	392	3	ve(g	ve(g	NOUN
ejpam-6849	392	4	)	)	PUNCT
ejpam-6849	392	5	,	,	PUNCT
ejpam-6849	392	6	γt(g	γt(g	PUNCT
ejpam-6849	392	7	)	)	PUNCT
ejpam-6849	393	1	+	+	CCONJ
ejpam-6849	393	2	γt	γt	NOUN
ejpam-6849	393	3	ve(g	ve(g	NOUN
ejpam-6849	393	4	)	)	PUNCT
ejpam-6849	393	5	}	}	PUNCT
ejpam-6849	393	6	.	.	PUNCT
ejpam-6849	394	1	proof	proof	NOUN
ejpam-6849	394	2	.	.	PUNCT
ejpam-6849	395	1	let	let	VERB
ejpam-6849	395	2	s	s	PRON
ejpam-6849	395	3	⊆	⊆	NUM
ejpam-6849	395	4	v	v	NOUN
ejpam-6849	395	5	(	(	PUNCT
ejpam-6849	395	6	g	g	NOUN
ejpam-6849	395	7	)	)	PUNCT
ejpam-6849	395	8	be	be	AUX
ejpam-6849	395	9	a	a	DET
ejpam-6849	395	10	γ	γ	NOUN
ejpam-6849	395	11	-	-	PUNCT
ejpam-6849	395	12	set	set	NOUN
ejpam-6849	395	13	of	of	ADP
ejpam-6849	395	14	g	g	NOUN
ejpam-6849	395	15	and	and	CCONJ
ejpam-6849	395	16	s∗	s∗	VERB
ejpam-6849	395	17	⊆	⊆	NUM
ejpam-6849	395	18	v	v	NOUN
ejpam-6849	395	19	(	(	PUNCT
ejpam-6849	395	20	g	g	NOUN
ejpam-6849	395	21	)	)	PUNCT
ejpam-6849	395	22	a	a	DET
ejpam-6849	395	23	γve	γve	NOUN
ejpam-6849	395	24	-	-	PUNCT
ejpam-6849	395	25	set	set	NOUN
ejpam-6849	395	26	of	of	ADP
ejpam-6849	395	27	g.	g.	PROPN
ejpam-6849	395	28	then	then	ADV
ejpam-6849	395	29	s	s	VERB
ejpam-6849	395	30	∪	∪	ADJ
ejpam-6849	395	31	s∗	s∗	PROPN
ejpam-6849	395	32	ve	ve	AUX
ejpam-6849	395	33	-	-	PUNCT
ejpam-6849	395	34	dominates	dominate	VERB
ejpam-6849	395	35	v	v	NOUN
ejpam-6849	395	36	(	(	PUNCT
ejpam-6849	395	37	g	g	NOUN
ejpam-6849	395	38	)	)	PUNCT
ejpam-6849	395	39	∪	∪	NOUN
ejpam-6849	395	40	v	v	NOUN
ejpam-6849	395	41	(	(	PUNCT
ejpam-6849	395	42	g	g	NOUN
ejpam-6849	395	43	)	)	PUNCT
ejpam-6849	395	44	.	.	PUNCT
ejpam-6849	396	1	let	let	VERB
ejpam-6849	396	2	u	u	PRON
ejpam-6849	396	3	∈	∈	PROPN
ejpam-6849	396	4	v	v	ADP
ejpam-6849	396	5	(	(	PUNCT
ejpam-6849	396	6	g	g	NOUN
ejpam-6849	396	7	)	)	PUNCT
ejpam-6849	396	8	.	.	PUNCT
ejpam-6849	397	1	there	there	PRON
ejpam-6849	397	2	exists	exist	VERB
ejpam-6849	397	3	v	v	ADP
ejpam-6849	397	4	∈	∈	PROPN
ejpam-6849	397	5	s	s	VERB
ejpam-6849	397	6	such	such	ADJ
ejpam-6849	397	7	that	that	SCONJ
ejpam-6849	397	8	u	u	PROPN
ejpam-6849	397	9	∈	∈	PROPN
ejpam-6849	397	10	ng[u	ng[u	PROPN
ejpam-6849	397	11	]	]	X
ejpam-6849	397	12	.	.	PUNCT
ejpam-6849	398	1	if	if	SCONJ
ejpam-6849	398	2	u	u	PROPN
ejpam-6849	398	3	=	=	PROPN
ejpam-6849	398	4	v	v	NOUN
ejpam-6849	398	5	,	,	PUNCT
ejpam-6849	398	6	then	then	ADV
ejpam-6849	398	7	v	v	X
ejpam-6849	398	8	ve	ve	VERB
ejpam-6849	398	9	-	-	PUNCT
ejpam-6849	398	10	dominates	dominate	VERB
ejpam-6849	398	11	uu	uu	ADJ
ejpam-6849	398	12	.	.	PUNCT
ejpam-6849	399	1	if	if	SCONJ
ejpam-6849	399	2	v	v	ADP
ejpam-6849	399	3	=	=	NOUN
ejpam-6849	399	4	̸	̸	NUM
ejpam-6849	399	5	u	u	NOUN
ejpam-6849	399	6	,	,	PUNCT
ejpam-6849	399	7	then	then	ADV
ejpam-6849	399	8	uv	uv	PROPN
ejpam-6849	399	9	∈	∈	PROPN
ejpam-6849	399	10	e(g	e(g	PROPN
ejpam-6849	399	11	)	)	PUNCT
ejpam-6849	399	12	and	and	CCONJ
ejpam-6849	399	13	v	v	AUX
ejpam-6849	399	14	ve	ve	VERB
ejpam-6849	399	15	-	-	PUNCT
ejpam-6849	399	16	dominates	dominate	VERB
ejpam-6849	399	17	uu	uu	PROPN
ejpam-6849	399	18	.	.	PUNCT
ejpam-6849	400	1	in	in	ADP
ejpam-6849	400	2	any	any	DET
ejpam-6849	400	3	case	case	NOUN
ejpam-6849	400	4	,	,	PUNCT
ejpam-6849	400	5	s	s	NOUN
ejpam-6849	400	6	∪	∪	ADJ
ejpam-6849	400	7	s∗	s∗	PROPN
ejpam-6849	400	8	ve	ve	VERB
ejpam-6849	400	9	-	-	PUNCT
ejpam-6849	400	10	dominates	dominate	VERB
ejpam-6849	400	11	uu	uu	PROPN
ejpam-6849	400	12	.	.	PUNCT
ejpam-6849	401	1	since	since	SCONJ
ejpam-6849	401	2	u	u	NOUN
ejpam-6849	401	3	is	be	AUX
ejpam-6849	401	4	arbitrary	arbitrary	ADJ
ejpam-6849	401	5	,	,	PUNCT
ejpam-6849	401	6	s	s	NOUN
ejpam-6849	401	7	∪	∪	ADJ
ejpam-6849	401	8	s∗	s∗	PROPN
ejpam-6849	401	9	is	be	AUX
ejpam-6849	401	10	a	a	DET
ejpam-6849	401	11	ve	ve	NOUN
ejpam-6849	401	12	-	-	PUNCT
ejpam-6849	401	13	dominating	dominate	VERB
ejpam-6849	401	14	set	set	NOUN
ejpam-6849	401	15	of	of	ADP
ejpam-6849	401	16	gg	gg	PROPN
ejpam-6849	401	17	.	.	PUNCT
ejpam-6849	402	1	hence	hence	ADV
ejpam-6849	402	2	,	,	PUNCT
ejpam-6849	402	3	γve(gg	γve(gg	NOUN
ejpam-6849	402	4	)	)	PUNCT
ejpam-6849	402	5	≤	≤	NUM
ejpam-6849	402	6	|s|	|s|	VERB
ejpam-6849	402	7	+	+	CCONJ
ejpam-6849	402	8	|s∗|	|s∗|	NUM
ejpam-6849	402	9	=	=	SYM
ejpam-6849	402	10	γ(g	γ(g	PROPN
ejpam-6849	402	11	)	)	PUNCT
ejpam-6849	402	12	+	+	NUM
ejpam-6849	403	1	γve(g	γve(g	NOUN
ejpam-6849	403	2	)	)	PUNCT
ejpam-6849	403	3	.	.	PUNCT
ejpam-6849	404	1	similarly	similarly	ADV
ejpam-6849	404	2	,	,	PUNCT
ejpam-6849	404	3	γve(gg	γve(gg	NOUN
ejpam-6849	404	4	)	)	PUNCT
ejpam-6849	404	5	≤	≤	PROPN
ejpam-6849	404	6	γ(g	γ(g	PROPN
ejpam-6849	404	7	)	)	PUNCT
ejpam-6849	405	1	+	+	NUM
ejpam-6849	406	1	γve(g	γve(g	NOUN
ejpam-6849	406	2	)	)	PUNCT
ejpam-6849	406	3	.	.	PUNCT
ejpam-6849	407	1	this	this	PRON
ejpam-6849	407	2	proves	prove	VERB
ejpam-6849	407	3	(	(	PUNCT
ejpam-6849	407	4	i	i	NOUN
ejpam-6849	407	5	)	)	PUNCT
ejpam-6849	407	6	.	.	PUNCT
ejpam-6849	408	1	the	the	DET
ejpam-6849	408	2	inequality	inequality	NOUN
ejpam-6849	408	3	in	in	ADP
ejpam-6849	408	4	(	(	PUNCT
ejpam-6849	408	5	ii	ii	NOUN
ejpam-6849	408	6	)	)	PUNCT
ejpam-6849	408	7	is	be	AUX
ejpam-6849	408	8	proved	prove	VERB
ejpam-6849	408	9	similarly	similarly	ADV
ejpam-6849	408	10	.	.	PUNCT
ejpam-6849	409	1	■	■	PUNCT
ejpam-6849	409	2	j.	j.	PROPN
ejpam-6849	409	3	tayab	tayab	PROPN
ejpam-6849	409	4	,	,	PUNCT
ejpam-6849	409	5	f.	f.	PROPN
ejpam-6849	409	6	p.	p.	PROPN
ejpam-6849	409	7	jamil	jamil	PROPN
ejpam-6849	409	8	,	,	PUNCT
ejpam-6849	409	9	i.	i.	PROPN
ejpam-6849	409	10	aniversario	aniversario	PROPN
ejpam-6849	409	11	/	/	SYM
ejpam-6849	409	12	eur	eur	PROPN
ejpam-6849	409	13	.	.	PUNCT
ejpam-6849	410	1	j.	j.	PROPN
ejpam-6849	410	2	pure	pure	PROPN
ejpam-6849	410	3	appl	appl	PROPN
ejpam-6849	410	4	.	.	PROPN
ejpam-6849	410	5	math	math	PROPN
ejpam-6849	410	6	,	,	PUNCT
ejpam-6849	410	7	18	18	NUM
ejpam-6849	410	8	(	(	PUNCT
ejpam-6849	410	9	4	4	NUM
ejpam-6849	410	10	)	)	PUNCT
ejpam-6849	410	11	(	(	PUNCT
ejpam-6849	410	12	2025	2025	NUM
ejpam-6849	410	13	)	)	PUNCT
ejpam-6849	410	14	,	,	PUNCT
ejpam-6849	410	15	6849	6849	NUM
ejpam-6849	410	16	11	11	NUM
ejpam-6849	410	17	of	of	ADP
ejpam-6849	410	18	19	19	NUM
ejpam-6849	410	19	proposition	proposition	NOUN
ejpam-6849	410	20	11	11	NUM
ejpam-6849	410	21	.	.	PUNCT
ejpam-6849	411	1	let	let	VERB
ejpam-6849	411	2	g	g	PRON
ejpam-6849	411	3	be	be	AUX
ejpam-6849	411	4	a	a	DET
ejpam-6849	411	5	graph	graph	NOUN
ejpam-6849	411	6	with	with	ADP
ejpam-6849	411	7	one	one	NUM
ejpam-6849	411	8	isolated	isolate	VERB
ejpam-6849	411	9	vertex	vertex	NOUN
ejpam-6849	411	10	v.	v.	ADP
ejpam-6849	411	11	then	then	ADV
ejpam-6849	411	12	γve(gg	γve(gg	X
ejpam-6849	411	13	)	)	PUNCT
ejpam-6849	411	14	≤	≤	NOUN
ejpam-6849	411	15	1	1	NUM
ejpam-6849	412	1	+	+	NUM
ejpam-6849	412	2	γve(g	γve(g	VERB
ejpam-6849	412	3	−	−	NUM
ejpam-6849	412	4	v	v	NOUN
ejpam-6849	412	5	)	)	PUNCT
ejpam-6849	412	6	(	(	PUNCT
ejpam-6849	412	7	3	3	X
ejpam-6849	412	8	)	)	PUNCT
ejpam-6849	412	9	and	and	CCONJ
ejpam-6849	412	10	γt	γt	ADP
ejpam-6849	412	11	ve(gg	ve(gg	NOUN
ejpam-6849	412	12	)	)	PUNCT
ejpam-6849	412	13	≤	≤	NUM
ejpam-6849	412	14	2	2	NUM
ejpam-6849	413	1	+	+	CCONJ
ejpam-6849	413	2	γt	γt	NOUN
ejpam-6849	413	3	ve(g	ve(g	PUNCT
ejpam-6849	413	4	−	−	PROPN
ejpam-6849	413	5	v	v	NOUN
ejpam-6849	413	6	)	)	PUNCT
ejpam-6849	413	7	.	.	PUNCT
ejpam-6849	414	1	(	(	PUNCT
ejpam-6849	414	2	4	4	X
ejpam-6849	414	3	)	)	PUNCT
ejpam-6849	414	4	equality	equality	NOUN
ejpam-6849	414	5	in	in	ADP
ejpam-6849	414	6	equation	equation	NOUN
ejpam-6849	414	7	3	3	NUM
ejpam-6849	414	8	is	be	AUX
ejpam-6849	414	9	attained	attain	VERB
ejpam-6849	414	10	if	if	SCONJ
ejpam-6849	414	11	gg	gg	PROPN
ejpam-6849	414	12	has	have	VERB
ejpam-6849	414	13	a	a	DET
ejpam-6849	414	14	γve	γve	NOUN
ejpam-6849	414	15	-	-	PUNCT
ejpam-6849	414	16	set	set	NOUN
ejpam-6849	414	17	which	which	PRON
ejpam-6849	414	18	contains	contain	VERB
ejpam-6849	414	19	v.	v.	ADV
ejpam-6849	414	20	under	under	ADP
ejpam-6849	414	21	the	the	DET
ejpam-6849	414	22	same	same	ADJ
ejpam-6849	414	23	condition	condition	NOUN
ejpam-6849	414	24	1	1	NUM
ejpam-6849	415	1	+	+	NUM
ejpam-6849	415	2	γve(g	γve(g	VERB
ejpam-6849	415	3	−	−	NUM
ejpam-6849	415	4	v	v	NOUN
ejpam-6849	415	5	)	)	PUNCT
ejpam-6849	415	6	≤	≤	NOUN
ejpam-6849	415	7	γt	γt	ADP
ejpam-6849	415	8	ve(gg	ve(gg	PROPN
ejpam-6849	415	9	)	)	PUNCT
ejpam-6849	415	10	,	,	PUNCT
ejpam-6849	415	11	(	(	PUNCT
ejpam-6849	415	12	5	5	NUM
ejpam-6849	415	13	)	)	PUNCT
ejpam-6849	415	14	and	and	CCONJ
ejpam-6849	415	15	this	this	DET
ejpam-6849	415	16	lower	lower	ADV
ejpam-6849	415	17	bound	bind	VERB
ejpam-6849	415	18	is	be	AUX
ejpam-6849	415	19	sharp	sharp	ADJ
ejpam-6849	415	20	.	.	PUNCT
ejpam-6849	416	1	proof	proof	NOUN
ejpam-6849	416	2	.	.	PUNCT
ejpam-6849	417	1	equation	equation	NOUN
ejpam-6849	417	2	3	3	NUM
ejpam-6849	417	3	follows	follow	VERB
ejpam-6849	417	4	from	from	ADP
ejpam-6849	417	5	proposition	proposition	NOUN
ejpam-6849	417	6	10	10	NUM
ejpam-6849	417	7	.	.	PUNCT
ejpam-6849	418	1	equation	equation	NOUN
ejpam-6849	418	2	4	4	NUM
ejpam-6849	418	3	follows	follow	VERB
ejpam-6849	418	4	from	from	ADP
ejpam-6849	418	5	the	the	DET
ejpam-6849	418	6	fact	fact	NOUN
ejpam-6849	418	7	that	that	SCONJ
ejpam-6849	418	8	if	if	SCONJ
ejpam-6849	418	9	s	s	VERB
ejpam-6849	418	10	⊆	⊆	NUM
ejpam-6849	418	11	v	v	NOUN
ejpam-6849	418	12	(	(	PUNCT
ejpam-6849	418	13	g	g	PROPN
ejpam-6849	418	14	−	−	PROPN
ejpam-6849	418	15	v	v	NOUN
ejpam-6849	418	16	)	)	PUNCT
ejpam-6849	418	17	is	be	AUX
ejpam-6849	418	18	a	a	DET
ejpam-6849	418	19	γt	γt	NOUN
ejpam-6849	418	20	ve	ve	NOUN
ejpam-6849	418	21	-	-	PUNCT
ejpam-6849	418	22	set	set	NOUN
ejpam-6849	418	23	of	of	ADP
ejpam-6849	418	24	g	g	PROPN
ejpam-6849	418	25	−	−	PROPN
ejpam-6849	418	26	v	v	NOUN
ejpam-6849	418	27	,	,	PUNCT
ejpam-6849	418	28	then	then	ADV
ejpam-6849	418	29	s	s	VERB
ejpam-6849	418	30	∪	∪	X
ejpam-6849	418	31	{	{	PUNCT
ejpam-6849	418	32	v	v	NOUN
ejpam-6849	418	33	,	,	PUNCT
ejpam-6849	418	34	v	v	NOUN
ejpam-6849	418	35	}	}	PUNCT
ejpam-6849	418	36	is	be	AUX
ejpam-6849	418	37	a	a	DET
ejpam-6849	418	38	total	total	ADJ
ejpam-6849	418	39	ve	ve	NOUN
ejpam-6849	418	40	-	-	PUNCT
ejpam-6849	418	41	dominating	dominate	VERB
ejpam-6849	418	42	set	set	NOUN
ejpam-6849	418	43	of	of	ADP
ejpam-6849	418	44	gg	gg	PROPN
ejpam-6849	418	45	.	.	PUNCT
ejpam-6849	419	1	let	let	VERB
ejpam-6849	419	2	s	s	PRON
ejpam-6849	419	3	⊆	⊆	NUM
ejpam-6849	419	4	v	v	NOUN
ejpam-6849	419	5	(	(	PUNCT
ejpam-6849	419	6	gg	gg	NOUN
ejpam-6849	419	7	)	)	PUNCT
ejpam-6849	419	8	be	be	AUX
ejpam-6849	419	9	a	a	DET
ejpam-6849	419	10	γve	γve	NOUN
ejpam-6849	419	11	-	-	PUNCT
ejpam-6849	419	12	set	set	NOUN
ejpam-6849	419	13	of	of	ADP
ejpam-6849	419	14	gg	gg	NOUN
ejpam-6849	419	15	with	with	ADP
ejpam-6849	419	16	v	v	PROPN
ejpam-6849	419	17	∈	∈	NOUN
ejpam-6849	419	18	s.	s.	PROPN
ejpam-6849	419	19	let	let	VERB
ejpam-6849	419	20	s∗	s∗	PROPN
ejpam-6849	419	21	=	=	SYM
ejpam-6849	419	22	{	{	PUNCT
ejpam-6849	419	23	w	w	PROPN
ejpam-6849	419	24	∈	∈	PROPN
ejpam-6849	419	25	v	v	NOUN
ejpam-6849	419	26	(	(	PUNCT
ejpam-6849	419	27	g	g	PROPN
ejpam-6849	419	28	−	−	PROPN
ejpam-6849	419	29	v	v	NOUN
ejpam-6849	419	30	)	)	PUNCT
ejpam-6849	419	31	:	:	PUNCT
ejpam-6849	419	32	w	w	X
ejpam-6849	419	33	∈	∈	PROPN
ejpam-6849	419	34	s	s	PART
ejpam-6849	419	35	}	}	PUNCT
ejpam-6849	419	36	,	,	PUNCT
ejpam-6849	419	37	and	and	CCONJ
ejpam-6849	419	38	put	put	VERB
ejpam-6849	419	39	t	t	NOUN
ejpam-6849	419	40	=	=	PUNCT
ejpam-6849	419	41	s∗	s∗	PROPN
ejpam-6849	419	42	∪	∪	X
ejpam-6849	419	43	(	(	PUNCT
ejpam-6849	419	44	s	s	X
ejpam-6849	419	45	∩	∩	ADJ
ejpam-6849	419	46	v	v	NOUN
ejpam-6849	419	47	(	(	PUNCT
ejpam-6849	419	48	g	g	PROPN
ejpam-6849	419	49	−	−	PROPN
ejpam-6849	419	50	v	v	NOUN
ejpam-6849	419	51	)	)	PUNCT
ejpam-6849	419	52	)	)	PUNCT
ejpam-6849	419	53	.	.	PUNCT
ejpam-6849	420	1	let	let	VERB
ejpam-6849	420	2	xy	xy	PROPN
ejpam-6849	420	3	∈	∈	PROPN
ejpam-6849	420	4	e(g	e(g	PROPN
ejpam-6849	420	5	−	−	PROPN
ejpam-6849	420	6	v	v	NOUN
ejpam-6849	420	7	)	)	PUNCT
ejpam-6849	420	8	.	.	PUNCT
ejpam-6849	421	1	there	there	PRON
ejpam-6849	421	2	exists	exist	VERB
ejpam-6849	421	3	w	w	PROPN
ejpam-6849	421	4	∈	∈	PROPN
ejpam-6849	421	5	s	s	X
ejpam-6849	421	6	for	for	ADP
ejpam-6849	421	7	which	which	PRON
ejpam-6849	421	8	w	w	PROPN
ejpam-6849	421	9	ve	ve	VERB
ejpam-6849	421	10	-	-	PUNCT
ejpam-6849	421	11	dominates	dominate	VERB
ejpam-6849	421	12	xy	xy	NOUN
ejpam-6849	421	13	.	.	PUNCT
ejpam-6849	422	1	if	if	SCONJ
ejpam-6849	422	2	w	w	PROPN
ejpam-6849	422	3	∈	∈	PROPN
ejpam-6849	422	4	v	v	NOUN
ejpam-6849	422	5	(	(	PUNCT
ejpam-6849	422	6	g	g	PROPN
ejpam-6849	422	7	−	−	PROPN
ejpam-6849	422	8	v	v	NOUN
ejpam-6849	422	9	)	)	PUNCT
ejpam-6849	422	10	,	,	PUNCT
ejpam-6849	422	11	then	then	ADV
ejpam-6849	422	12	t	t	PROPN
ejpam-6849	422	13	ve	ve	AUX
ejpam-6849	422	14	-	-	PUNCT
ejpam-6849	422	15	dominates	dominate	VERB
ejpam-6849	422	16	xy	xy	PRON
ejpam-6849	422	17	.	.	PUNCT
ejpam-6849	422	18	suppose	suppose	VERB
ejpam-6849	422	19	that	that	SCONJ
ejpam-6849	422	20	w	w	PROPN
ejpam-6849	422	21	∈	∈	PROPN
ejpam-6849	422	22	v	v	ADP
ejpam-6849	422	23	(	(	PUNCT
ejpam-6849	422	24	g	g	NOUN
ejpam-6849	422	25	)	)	PUNCT
ejpam-6849	422	26	.	.	PUNCT
ejpam-6849	423	1	then	then	ADV
ejpam-6849	423	2	w	w	PROPN
ejpam-6849	423	3	∈	∈	PROPN
ejpam-6849	423	4	s∗	s∗	PROPN
ejpam-6849	423	5	and	and	CCONJ
ejpam-6849	423	6	either	either	ADV
ejpam-6849	423	7	w	w	X
ejpam-6849	423	8	=	=	PUNCT
ejpam-6849	423	9	x	x	X
ejpam-6849	423	10	or	or	CCONJ
ejpam-6849	423	11	w	w	PROPN
ejpam-6849	423	12	=	=	SYM
ejpam-6849	423	13	y.	y.	NOUN
ejpam-6849	423	14	in	in	ADP
ejpam-6849	423	15	either	either	DET
ejpam-6849	423	16	case	case	NOUN
ejpam-6849	423	17	,	,	PUNCT
ejpam-6849	423	18	t	t	PROPN
ejpam-6849	423	19	ve	ve	AUX
ejpam-6849	423	20	-	-	PUNCT
ejpam-6849	423	21	dominates	dominate	VERB
ejpam-6849	423	22	xy	xy	PROPN
ejpam-6849	423	23	.	.	PUNCT
ejpam-6849	424	1	since	since	SCONJ
ejpam-6849	424	2	xy	xy	PROPN
ejpam-6849	424	3	is	be	AUX
ejpam-6849	424	4	arbitrary	arbitrary	ADJ
ejpam-6849	424	5	,	,	PUNCT
ejpam-6849	424	6	t	t	PROPN
ejpam-6849	424	7	is	be	AUX
ejpam-6849	424	8	a	a	DET
ejpam-6849	424	9	ve	ve	NOUN
ejpam-6849	424	10	-	-	PUNCT
ejpam-6849	424	11	dominating	dominate	VERB
ejpam-6849	424	12	set	set	NOUN
ejpam-6849	424	13	of	of	ADP
ejpam-6849	424	14	g	g	PROPN
ejpam-6849	424	15	−	−	PROPN
ejpam-6849	424	16	v.	v.	ADP
ejpam-6849	424	17	hence	hence	ADV
ejpam-6849	424	18	,	,	PUNCT
ejpam-6849	424	19	γve(gg	γve(gg	NOUN
ejpam-6849	424	20	)	)	PUNCT
ejpam-6849	425	1	=	=	SYM
ejpam-6849	425	2	|s|	|s|	NOUN
ejpam-6849	425	3	≥	≥	NOUN
ejpam-6849	425	4	1	1	NUM
ejpam-6849	425	5	+	+	CCONJ
ejpam-6849	425	6	|t	|t	VERB
ejpam-6849	425	7	|	|	ADV
ejpam-6849	425	8	≥	≥	NOUN
ejpam-6849	425	9	1	1	NUM
ejpam-6849	425	10	+	+	CCONJ
ejpam-6849	425	11	γve(g	γve(g	VERB
ejpam-6849	426	1	−	−	NUM
ejpam-6849	426	2	v	v	NOUN
ejpam-6849	426	3	)	)	PUNCT
ejpam-6849	426	4	.	.	PUNCT
ejpam-6849	427	1	if	if	SCONJ
ejpam-6849	427	2	s	s	NOUN
ejpam-6849	427	3	were	be	AUX
ejpam-6849	427	4	taken	take	VERB
ejpam-6849	427	5	as	as	ADP
ejpam-6849	427	6	a	a	DET
ejpam-6849	427	7	γt	γt	NOUN
ejpam-6849	427	8	ve	ve	NOUN
ejpam-6849	427	9	-	-	PUNCT
ejpam-6849	427	10	set	set	NOUN
ejpam-6849	427	11	of	of	ADP
ejpam-6849	427	12	gg	gg	PROPN
ejpam-6849	427	13	,	,	PUNCT
ejpam-6849	427	14	then	then	ADV
ejpam-6849	427	15	the	the	DET
ejpam-6849	427	16	above	above	ADJ
ejpam-6849	427	17	argument	argument	NOUN
ejpam-6849	427	18	implies	imply	VERB
ejpam-6849	427	19	that	that	SCONJ
ejpam-6849	427	20	γt	γt	PROPN
ejpam-6849	427	21	ve(gg	ve(gg	PROPN
ejpam-6849	427	22	)	)	PUNCT
ejpam-6849	427	23	=	=	PUNCT
ejpam-6849	427	24	|s|	|s|	NOUN
ejpam-6849	427	25	≥	≥	NOUN
ejpam-6849	427	26	1	1	NUM
ejpam-6849	427	27	+	+	CCONJ
ejpam-6849	427	28	|t	|t	VERB
ejpam-6849	427	29	|	|	ADV
ejpam-6849	427	30	≥	≥	NOUN
ejpam-6849	427	31	1	1	NUM
ejpam-6849	427	32	+	+	CCONJ
ejpam-6849	427	33	γve(g	γve(g	VERB
ejpam-6849	428	1	−	−	NUM
ejpam-6849	428	2	v	v	NOUN
ejpam-6849	428	3	)	)	PUNCT
ejpam-6849	428	4	.	.	PUNCT
ejpam-6849	429	1	if	if	SCONJ
ejpam-6849	429	2	,	,	PUNCT
ejpam-6849	429	3	in	in	ADP
ejpam-6849	429	4	particular	particular	ADJ
ejpam-6849	429	5	,	,	PUNCT
ejpam-6849	429	6	g	g	PROPN
ejpam-6849	429	7	is	be	AUX
ejpam-6849	429	8	the	the	DET
ejpam-6849	429	9	graph	graph	NOUN
ejpam-6849	429	10	g1	g1	NOUN
ejpam-6849	429	11	=	=	SYM
ejpam-6849	429	12	k1	k1	PROPN
ejpam-6849	429	13	∪	∪	PROPN
ejpam-6849	429	14	p3	p3	PROPN
ejpam-6849	429	15	in	in	ADP
ejpam-6849	429	16	figure	figure	NOUN
ejpam-6849	429	17	2(a	2(a	NUM
ejpam-6849	429	18	)	)	PUNCT
ejpam-6849	429	19	,	,	PUNCT
ejpam-6849	429	20	then	then	ADV
ejpam-6849	429	21	γt	γt	PROPN
ejpam-6849	429	22	ve(gg	ve(gg	NOUN
ejpam-6849	429	23	)	)	PUNCT
ejpam-6849	429	24	=	=	SYM
ejpam-6849	430	1	2	2	NUM
ejpam-6849	430	2	=	=	SYM
ejpam-6849	430	3	1	1	NUM
ejpam-6849	430	4	+	+	NUM
ejpam-6849	430	5	γve(g	γve(g	PROPN
ejpam-6849	430	6	)	)	PUNCT
ejpam-6849	430	7	,	,	PUNCT
ejpam-6849	430	8	showing	show	VERB
ejpam-6849	430	9	that	that	SCONJ
ejpam-6849	430	10	the	the	DET
ejpam-6849	430	11	lower	lower	ADV
ejpam-6849	430	12	bound	bind	VERB
ejpam-6849	430	13	in	in	ADP
ejpam-6849	430	14	equation	equation	NOUN
ejpam-6849	430	15	5	5	NUM
ejpam-6849	430	16	is	be	AUX
ejpam-6849	430	17	sharp	sharp	ADJ
ejpam-6849	430	18	.	.	PUNCT
ejpam-6849	431	1	■	■	PUNCT
ejpam-6849	431	2	strict	strict	ADJ
ejpam-6849	431	3	inequality	inequality	NOUN
ejpam-6849	431	4	can	can	AUX
ejpam-6849	431	5	be	be	AUX
ejpam-6849	431	6	attained	attain	VERB
ejpam-6849	431	7	in	in	ADP
ejpam-6849	431	8	proposition	proposition	NOUN
ejpam-6849	431	9	11	11	NUM
ejpam-6849	431	10	.	.	PUNCT
ejpam-6849	432	1	consider	consider	VERB
ejpam-6849	432	2	the	the	DET
ejpam-6849	432	3	graph	graph	NOUN
ejpam-6849	432	4	g2	g2	PROPN
ejpam-6849	432	5	=	=	SYM
ejpam-6849	432	6	k1	k1	PROPN
ejpam-6849	432	7	∪	∪	X
ejpam-6849	432	8	(	(	PUNCT
ejpam-6849	432	9	k2	k2	ADJ
ejpam-6849	432	10	∪	∪	ADJ
ejpam-6849	432	11	k2	k2	NOUN
ejpam-6849	432	12	)	)	PUNCT
ejpam-6849	432	13	.	.	PUNCT
ejpam-6849	433	1	then	then	ADV
ejpam-6849	433	2	g2	g2	PROPN
ejpam-6849	433	3	is	be	AUX
ejpam-6849	433	4	the	the	DET
ejpam-6849	433	5	wheel	wheel	NOUN
ejpam-6849	433	6	k1	k1	NOUN
ejpam-6849	433	7	+	+	CCONJ
ejpam-6849	433	8	c4	c4	NOUN
ejpam-6849	433	9	,	,	PUNCT
ejpam-6849	433	10	and	and	CCONJ
ejpam-6849	433	11	g2g2	g2g2	PROPN
ejpam-6849	433	12	is	be	AUX
ejpam-6849	433	13	the	the	DET
ejpam-6849	433	14	graph	graph	NOUN
ejpam-6849	433	15	in	in	ADP
ejpam-6849	433	16	figure	figure	NOUN
ejpam-6849	433	17	3(b).observe	3(b).observe	NUM
ejpam-6849	433	18	that	that	PRON
ejpam-6849	433	19	γve(gg	γve(gg	NOUN
ejpam-6849	433	20	)	)	PUNCT
ejpam-6849	433	21	=	=	SYM
ejpam-6849	433	22	2	2	NUM
ejpam-6849	433	23	and	and	CCONJ
ejpam-6849	433	24	is	be	AUX
ejpam-6849	433	25	determined	determine	VERB
ejpam-6849	433	26	by	by	ADP
ejpam-6849	433	27	the	the	DET
ejpam-6849	433	28	ve	ve	PROPN
ejpam-6849	433	29	-	-	PUNCT
ejpam-6849	433	30	dominating	dominate	VERB
ejpam-6849	433	31	set	set	NOUN
ejpam-6849	433	32	s	s	PART
ejpam-6849	433	33	=	=	PUNCT
ejpam-6849	433	34	{	{	PUNCT
ejpam-6849	433	35	x	x	PROPN
ejpam-6849	433	36	,	,	PUNCT
ejpam-6849	433	37	y	y	NOUN
ejpam-6849	433	38	}	}	PUNCT
ejpam-6849	433	39	of	of	ADP
ejpam-6849	433	40	gg	gg	PROPN
ejpam-6849	433	41	.	.	PUNCT
ejpam-6849	434	1	note	note	NOUN
ejpam-6849	434	2	,	,	PUNCT
ejpam-6849	434	3	on	on	ADP
ejpam-6849	434	4	the	the	DET
ejpam-6849	434	5	other	other	ADJ
ejpam-6849	434	6	hand	hand	NOUN
ejpam-6849	434	7	,	,	PUNCT
ejpam-6849	434	8	that	that	SCONJ
ejpam-6849	434	9	1	1	NUM
ejpam-6849	434	10	+	+	NUM
ejpam-6849	434	11	γve(g	γve(g	VERB
ejpam-6849	435	1	−	−	NUM
ejpam-6849	435	2	v	v	NOUN
ejpam-6849	435	3	)	)	PUNCT
ejpam-6849	435	4	=	=	SYM
ejpam-6849	436	1	3	3	NUM
ejpam-6849	436	2	,	,	PUNCT
ejpam-6849	436	3	where	where	SCONJ
ejpam-6849	436	4	v	v	NOUN
ejpam-6849	436	5	is	be	AUX
ejpam-6849	436	6	the	the	DET
ejpam-6849	436	7	isolated	isolated	ADJ
ejpam-6849	436	8	vertex	vertex	NOUN
ejpam-6849	436	9	of	of	ADP
ejpam-6849	436	10	g.	g.	PROPN
ejpam-6849	436	11	j.	j.	PROPN
ejpam-6849	436	12	tayab	tayab	PROPN
ejpam-6849	436	13	,	,	PUNCT
ejpam-6849	436	14	f.	f.	PROPN
ejpam-6849	436	15	p.	p.	PROPN
ejpam-6849	436	16	jamil	jamil	PROPN
ejpam-6849	436	17	,	,	PUNCT
ejpam-6849	436	18	i.	i.	PROPN
ejpam-6849	436	19	aniversario	aniversario	PROPN
ejpam-6849	436	20	/	/	SYM
ejpam-6849	436	21	eur	eur	PROPN
ejpam-6849	436	22	.	.	PUNCT
ejpam-6849	437	1	j.	j.	PROPN
ejpam-6849	437	2	pure	pure	PROPN
ejpam-6849	437	3	appl	appl	PROPN
ejpam-6849	437	4	.	.	PROPN
ejpam-6849	437	5	math	math	PROPN
ejpam-6849	437	6	,	,	PUNCT
ejpam-6849	437	7	18	18	NUM
ejpam-6849	437	8	(	(	PUNCT
ejpam-6849	437	9	4	4	NUM
ejpam-6849	437	10	)	)	PUNCT
ejpam-6849	437	11	(	(	PUNCT
ejpam-6849	437	12	2025	2025	NUM
ejpam-6849	437	13	)	)	PUNCT
ejpam-6849	437	14	,	,	PUNCT
ejpam-6849	437	15	6849	6849	NUM
ejpam-6849	437	16	12	12	NUM
ejpam-6849	437	17	of	of	ADP
ejpam-6849	437	18	19	19	NUM
ejpam-6849	437	19	....................................	....................................	PUNCT
ejpam-6849	437	20	....................................	....................................	PUNCT
ejpam-6849	438	1	....................................	....................................	PUNCT
ejpam-6849	438	2	....................................	....................................	PUNCT
ejpam-6849	439	1	....................................	....................................	PUNCT
ejpam-6849	439	2	....................................	....................................	PUNCT
ejpam-6849	440	1	....................................	....................................	PUNCT
ejpam-6849	440	2	....................................	....................................	PUNCT
ejpam-6849	441	1	...................	...................	PUNCT
ejpam-6849	441	2	..................	..................	PUNCT
ejpam-6849	442	1	..................	..................	PUNCT
ejpam-6849	442	2	..................	..................	PUNCT
ejpam-6849	442	3	.............	.............	PUNCT
ejpam-6849	443	1	....................................	....................................	PUNCT
ejpam-6849	443	2	.................................................................................................................................................................................................................................................	.................................................................................................................................................................................................................................................	PUNCT
ejpam-6849	444	1	....................................	....................................	PUNCT
ejpam-6849	444	2	.......................................................................................................................	.......................................................................................................................	PUNCT
ejpam-6849	445	1	.......................................................................................................................................................................	.......................................................................................................................................................................	PUNCT
ejpam-6849	445	2	........................................	........................................	PUNCT
ejpam-6849	445	3	........................................	........................................	PUNCT
ejpam-6849	445	4	........................................	........................................	PUNCT
ejpam-6849	445	5	........................................	........................................	PUNCT
ejpam-6849	445	6	........	........	PUNCT
ejpam-6849	445	7	....................................................................	....................................................................	PUNCT
ejpam-6849	445	8	...........................................................................................................................................................................................................................................................................	...........................................................................................................................................................................................................................................................................	PUNCT
ejpam-6849	445	9	.........	.........	PUNCT
ejpam-6849	445	10	........	........	PUNCT
ejpam-6849	445	11	........	........	PUNCT
ejpam-6849	445	12	........	........	PUNCT
ejpam-6849	445	13	........	........	PUNCT
ejpam-6849	445	14	........	........	PUNCT
ejpam-6849	445	15	........	........	PUNCT
ejpam-6849	445	16	........	........	PUNCT
ejpam-6849	445	17	........	........	PUNCT
ejpam-6849	445	18	........	........	PUNCT
ejpam-6849	445	19	........	........	PUNCT
ejpam-6849	445	20	........	........	PUNCT
ejpam-6849	445	21	........	........	PUNCT
ejpam-6849	445	22	........	........	PUNCT
ejpam-6849	446	1	......	......	PUNCT
ejpam-6849	446	2	.........	.........	PUNCT
ejpam-6849	446	3	........	........	PUNCT
ejpam-6849	446	4	........	........	PUNCT
ejpam-6849	446	5	........	........	PUNCT
ejpam-6849	446	6	........	........	PUNCT
ejpam-6849	446	7	........	........	PUNCT
ejpam-6849	446	8	........	........	PUNCT
ejpam-6849	446	9	........	........	PUNCT
ejpam-6849	446	10	........	........	PUNCT
ejpam-6849	446	11	........	........	PUNCT
ejpam-6849	446	12	........	........	PUNCT
ejpam-6849	446	13	........	........	PUNCT
ejpam-6849	446	14	........	........	PUNCT
ejpam-6849	446	15	........	........	PUNCT
ejpam-6849	447	1	......	......	PUNCT
ejpam-6849	447	2	.........	.........	PUNCT
ejpam-6849	447	3	........	........	PUNCT
ejpam-6849	447	4	........	........	PUNCT
ejpam-6849	447	5	........	........	PUNCT
ejpam-6849	447	6	........	........	PUNCT
ejpam-6849	447	7	........	........	PUNCT
ejpam-6849	447	8	........	........	PUNCT
ejpam-6849	447	9	........	........	PUNCT
ejpam-6849	447	10	........	........	PUNCT
ejpam-6849	447	11	........	........	PUNCT
ejpam-6849	447	12	........	........	PUNCT
ejpam-6849	447	13	........	........	PUNCT
ejpam-6849	447	14	........	........	PUNCT
ejpam-6849	447	15	........	........	PUNCT
ejpam-6849	448	1	......	......	PUNCT
ejpam-6849	448	2	.........	.........	PUNCT
ejpam-6849	448	3	........	........	PUNCT
ejpam-6849	448	4	........	........	PUNCT
ejpam-6849	448	5	........	........	PUNCT
ejpam-6849	448	6	........	........	PUNCT
ejpam-6849	448	7	........	........	PUNCT
ejpam-6849	448	8	........	........	PUNCT
ejpam-6849	448	9	........	........	PUNCT
ejpam-6849	448	10	........	........	PUNCT
ejpam-6849	448	11	........	........	PUNCT
ejpam-6849	448	12	........	........	PUNCT
ejpam-6849	448	13	........	........	PUNCT
ejpam-6849	448	14	........	........	PUNCT
ejpam-6849	449	1	........	........	PUNCT
ejpam-6849	450	1	......	......	PUNCT
ejpam-6849	451	1	v	v	NUM
ejpam-6849	451	2	w	w	PROPN
ejpam-6849	451	3	•	•	NOUN
ejpam-6849	451	4	•g1	•g1	NOUN
ejpam-6849	451	5	:	:	PUNCT
ejpam-6849	451	6	g1	g1	PROPN
ejpam-6849	451	7	:	:	PUNCT
ejpam-6849	451	8	(	(	PUNCT
ejpam-6849	451	9	a	a	X
ejpam-6849	451	10	)	)	PUNCT
ejpam-6849	451	11	....................................	....................................	PUNCT
ejpam-6849	451	12	....................................	....................................	PUNCT
ejpam-6849	452	1	....................................	....................................	PUNCT
ejpam-6849	452	2	....................................	....................................	PUNCT
ejpam-6849	453	1	....................................	....................................	PUNCT
ejpam-6849	453	2	....................................	....................................	PUNCT
ejpam-6849	454	1	....................................	....................................	PUNCT
ejpam-6849	454	2	....................................	....................................	PUNCT
ejpam-6849	455	1	..........................	..........................	PUNCT
ejpam-6849	455	2	.........................	.........................	PUNCT
ejpam-6849	456	1	.........................	.........................	PUNCT
ejpam-6849	456	2	.........................	.........................	PUNCT
ejpam-6849	456	3	.........................	.........................	PUNCT
ejpam-6849	457	1	....................................	....................................	PUNCT
ejpam-6849	457	2	............................................................................................................................................................................................................................................................	............................................................................................................................................................................................................................................................	PUNCT
ejpam-6849	458	1	....................................	....................................	PUNCT
ejpam-6849	458	2	..........................	..........................	PUNCT
ejpam-6849	459	1	.........................	.........................	PUNCT
ejpam-6849	459	2	.........................	.........................	PUNCT
ejpam-6849	459	3	.........................	.........................	PUNCT
ejpam-6849	459	4	.........................	.........................	PUNCT
ejpam-6849	459	5	................................................................................................................................................................................................................................................................................................	................................................................................................................................................................................................................................................................................................	PUNCT
ejpam-6849	460	1	....................................	....................................	PUNCT
ejpam-6849	460	2	....................................	....................................	PUNCT
ejpam-6849	461	1	.........	.........	PUNCT
ejpam-6849	461	2	........	........	PUNCT
ejpam-6849	461	3	........	........	PUNCT
ejpam-6849	461	4	........	........	PUNCT
ejpam-6849	461	5	........	........	PUNCT
ejpam-6849	461	6	........	........	PUNCT
ejpam-6849	461	7	........	........	PUNCT
ejpam-6849	461	8	........	........	PUNCT
ejpam-6849	461	9	........	........	PUNCT
ejpam-6849	461	10	........	........	PUNCT
ejpam-6849	461	11	........	........	PUNCT
ejpam-6849	461	12	........	........	PUNCT
ejpam-6849	461	13	........	........	PUNCT
ejpam-6849	461	14	........	........	PUNCT
ejpam-6849	462	1	......	......	PUNCT
ejpam-6849	462	2	.........	.........	PUNCT
ejpam-6849	462	3	........	........	PUNCT
ejpam-6849	462	4	........	........	PUNCT
ejpam-6849	462	5	........	........	PUNCT
ejpam-6849	462	6	........	........	PUNCT
ejpam-6849	462	7	........	........	PUNCT
ejpam-6849	462	8	........	........	PUNCT
ejpam-6849	462	9	........	........	PUNCT
ejpam-6849	462	10	........	........	PUNCT
ejpam-6849	462	11	........	........	PUNCT
ejpam-6849	462	12	........	........	PUNCT
ejpam-6849	462	13	........	........	PUNCT
ejpam-6849	462	14	........	........	PUNCT
ejpam-6849	462	15	........	........	PUNCT
ejpam-6849	463	1	......	......	PUNCT
ejpam-6849	463	2	.........	.........	PUNCT
ejpam-6849	463	3	........	........	PUNCT
ejpam-6849	463	4	........	........	PUNCT
ejpam-6849	463	5	........	........	PUNCT
ejpam-6849	463	6	........	........	PUNCT
ejpam-6849	463	7	........	........	PUNCT
ejpam-6849	463	8	........	........	PUNCT
ejpam-6849	463	9	........	........	PUNCT
ejpam-6849	463	10	........	........	PUNCT
ejpam-6849	463	11	........	........	PUNCT
ejpam-6849	463	12	........	........	PUNCT
ejpam-6849	463	13	........	........	PUNCT
ejpam-6849	463	14	........	........	PUNCT
ejpam-6849	463	15	........	........	PUNCT
ejpam-6849	464	1	......	......	PUNCT
ejpam-6849	464	2	.........	.........	PUNCT
ejpam-6849	464	3	........	........	PUNCT
ejpam-6849	464	4	........	........	PUNCT
ejpam-6849	464	5	........	........	PUNCT
ejpam-6849	464	6	........	........	PUNCT
ejpam-6849	464	7	........	........	PUNCT
ejpam-6849	464	8	........	........	PUNCT
ejpam-6849	464	9	........	........	PUNCT
ejpam-6849	464	10	........	........	PUNCT
ejpam-6849	464	11	........	........	PUNCT
ejpam-6849	464	12	........	........	PUNCT
ejpam-6849	464	13	........	........	PUNCT
ejpam-6849	464	14	........	........	PUNCT
ejpam-6849	464	15	........	........	PUNCT
ejpam-6849	465	1	......	......	PUNCT
ejpam-6849	465	2	..........................................................................................................................................................................................................................................................................................................................................................................................	..........................................................................................................................................................................................................................................................................................................................................................................................	PUNCT
ejpam-6849	466	1	.................................................................................................................................................	.................................................................................................................................................	PUNCT
ejpam-6849	466	2	....................................	....................................	PUNCT
ejpam-6849	467	1	....................................	....................................	PUNCT
ejpam-6849	467	2	.............	.............	PUNCT
ejpam-6849	467	3	.............	.............	PUNCT
ejpam-6849	468	1	..............	..............	PUNCT
ejpam-6849	468	2	..............	..............	PUNCT
ejpam-6849	468	3	...............	...............	PUNCT
ejpam-6849	468	4	................	................	PUNCT
ejpam-6849	468	5	.................	.................	PUNCT
ejpam-6849	468	6	...................	...................	PUNCT
ejpam-6849	468	7	......................	......................	PUNCT
ejpam-6849	468	8	..............................	..............................	PUNCT
ejpam-6849	468	9	......................................................................................................................................................................................................................................................................................	......................................................................................................................................................................................................................................................................................	PUNCT
ejpam-6849	468	10	..................	..................	PUNCT
ejpam-6849	469	1	...................	...................	PUNCT
ejpam-6849	469	2	....................	....................	PUNCT
ejpam-6849	470	1	.....................	.....................	PUNCT
ejpam-6849	470	2	......................	......................	PUNCT
ejpam-6849	470	3	.......................	.......................	PUNCT
ejpam-6849	470	4	.........................	.........................	PUNCT
ejpam-6849	470	5	............................	............................	PUNCT
ejpam-6849	470	6	.................................	.................................	PUNCT
ejpam-6849	471	1	.............................................	.............................................	PUNCT
ejpam-6849	471	2	.....................................................................................................................................................................................................................................................................................................................................................	.....................................................................................................................................................................................................................................................................................................................................................	PUNCT
ejpam-6849	472	1	................................................	................................................	PUNCT
ejpam-6849	472	2	................................................	................................................	PUNCT
ejpam-6849	473	1	................................................	................................................	PUNCT
ejpam-6849	473	2	................................................	................................................	PUNCT
ejpam-6849	473	3	...........	...........	PUNCT
ejpam-6849	473	4	.......................................................................................................................	.......................................................................................................................	PUNCT
ejpam-6849	473	5	.........	.........	PUNCT
ejpam-6849	473	6	........	........	PUNCT
ejpam-6849	473	7	........	........	PUNCT
ejpam-6849	473	8	........	........	PUNCT
ejpam-6849	473	9	........	........	PUNCT
ejpam-6849	473	10	........	........	PUNCT
ejpam-6849	473	11	........	........	PUNCT
ejpam-6849	473	12	........	........	PUNCT
ejpam-6849	473	13	........	........	PUNCT
ejpam-6849	473	14	........	........	PUNCT
ejpam-6849	473	15	........	........	PUNCT
ejpam-6849	473	16	........	........	PUNCT
ejpam-6849	473	17	........	........	PUNCT
ejpam-6849	473	18	........	........	PUNCT
ejpam-6849	474	1	......	......	PUNCT
ejpam-6849	474	2	g2	g2	PROPN
ejpam-6849	474	3	:	:	PUNCT
ejpam-6849	475	1	g2	g2	PROPN
ejpam-6849	475	2	:	:	PUNCT
ejpam-6849	475	3	•	•	NUM
ejpam-6849	475	4	•	•	NOUN
ejpam-6849	475	5	x	x	SYM
ejpam-6849	475	6	y	y	PROPN
ejpam-6849	475	7	v	v	X
ejpam-6849	475	8	(	(	PUNCT
ejpam-6849	475	9	b	b	NOUN
ejpam-6849	475	10	)	)	PUNCT
ejpam-6849	475	11	figure	figure	NOUN
ejpam-6849	475	12	3	3	NUM
ejpam-6849	475	13	:	:	PUNCT
ejpam-6849	475	14	examples	example	NOUN
ejpam-6849	475	15	of	of	ADP
ejpam-6849	475	16	graphs	graph	NOUN
ejpam-6849	475	17	illustrating	illustrate	VERB
ejpam-6849	475	18	proposition	proposition	NOUN
ejpam-6849	475	19	11	11	NUM
ejpam-6849	475	20	.	.	PUNCT
ejpam-6849	476	1	proposition	proposition	NOUN
ejpam-6849	476	2	12	12	NUM
ejpam-6849	476	3	.	.	PUNCT
ejpam-6849	477	1	let	let	VERB
ejpam-6849	477	2	g	g	NOUN
ejpam-6849	477	3	and	and	CCONJ
ejpam-6849	477	4	h	h	NOUN
ejpam-6849	477	5	be	be	AUX
ejpam-6849	477	6	connected	connect	VERB
ejpam-6849	477	7	graphs	graph	NOUN
ejpam-6849	477	8	with	with	ADP
ejpam-6849	477	9	h	h	NOUN
ejpam-6849	477	10	nontrivial	nontrivial	NOUN
ejpam-6849	477	11	,	,	PUNCT
ejpam-6849	477	12	and	and	CCONJ
ejpam-6849	477	13	c	c	PROPN
ejpam-6849	477	14	⊆	⊆	NUM
ejpam-6849	477	15	v	v	NOUN
ejpam-6849	477	16	(	(	PUNCT
ejpam-6849	477	17	g[h	g[h	PROPN
ejpam-6849	477	18	]	]	PUNCT
ejpam-6849	477	19	)	)	PUNCT
ejpam-6849	477	20	.	.	PUNCT
ejpam-6849	478	1	then	then	ADV
ejpam-6849	478	2	c	c	PROPN
ejpam-6849	478	3	is	be	AUX
ejpam-6849	478	4	a	a	DET
ejpam-6849	478	5	ve	ve	NOUN
ejpam-6849	478	6	-	-	PUNCT
ejpam-6849	478	7	dominating	dominate	VERB
ejpam-6849	478	8	set	set	NOUN
ejpam-6849	478	9	of	of	ADP
ejpam-6849	478	10	g[h	g[h	NOUN
ejpam-6849	478	11	]	]	PUNCT
ejpam-6849	478	12	if	if	SCONJ
ejpam-6849	478	13	and	and	CCONJ
ejpam-6849	478	14	only	only	ADV
ejpam-6849	478	15	if	if	SCONJ
ejpam-6849	478	16	c	c	PROPN
ejpam-6849	478	17	=	=	SYM
ejpam-6849	478	18	∪x∈s	∪x∈s	PROPN
ejpam-6849	478	19	(	(	PUNCT
ejpam-6849	478	20	{	{	PUNCT
ejpam-6849	478	21	x	x	NOUN
ejpam-6849	478	22	}	}	PUNCT
ejpam-6849	478	23	×	×	PROPN
ejpam-6849	478	24	tx	tx	PROPN
ejpam-6849	478	25	)	)	PUNCT
ejpam-6849	478	26	,	,	PUNCT
ejpam-6849	478	27	(	(	PUNCT
ejpam-6849	478	28	6	6	NUM
ejpam-6849	478	29	)	)	PUNCT
ejpam-6849	478	30	where	where	SCONJ
ejpam-6849	478	31	s	s	VERB
ejpam-6849	478	32	⊆	⊆	NUM
ejpam-6849	478	33	v	v	NOUN
ejpam-6849	478	34	(	(	PUNCT
ejpam-6849	478	35	g	g	NOUN
ejpam-6849	478	36	)	)	PUNCT
ejpam-6849	478	37	and	and	CCONJ
ejpam-6849	478	38	tx	tx	VERB
ejpam-6849	478	39	⊆	⊆	NUM
ejpam-6849	478	40	v	v	NOUN
ejpam-6849	478	41	(	(	PUNCT
ejpam-6849	478	42	h	h	NOUN
ejpam-6849	478	43	)	)	PUNCT
ejpam-6849	478	44	for	for	ADP
ejpam-6849	478	45	each	each	DET
ejpam-6849	478	46	x	x	SYM
ejpam-6849	478	47	∈	∈	PROPN
ejpam-6849	478	48	s	s	VERB
ejpam-6849	478	49	such	such	ADJ
ejpam-6849	478	50	that	that	SCONJ
ejpam-6849	478	51	each	each	PRON
ejpam-6849	478	52	of	of	ADP
ejpam-6849	478	53	the	the	DET
ejpam-6849	478	54	following	follow	VERB
ejpam-6849	478	55	holds	hold	VERB
ejpam-6849	478	56	:	:	PUNCT
ejpam-6849	478	57	(	(	PUNCT
ejpam-6849	478	58	i	i	NOUN
ejpam-6849	478	59	)	)	PUNCT
ejpam-6849	478	60	s	s	VERB
ejpam-6849	478	61	is	be	AUX
ejpam-6849	478	62	a	a	DET
ejpam-6849	478	63	dominating	dominating	NOUN
ejpam-6849	478	64	set	set	NOUN
ejpam-6849	478	65	of	of	ADP
ejpam-6849	478	66	g	g	NOUN
ejpam-6849	478	67	;	;	PUNCT
ejpam-6849	478	68	(	(	PUNCT
ejpam-6849	478	69	ii	ii	NOUN
ejpam-6849	478	70	)	)	PUNCT
ejpam-6849	478	71	tx	tx	PROPN
ejpam-6849	478	72	is	be	AUX
ejpam-6849	478	73	a	a	DET
ejpam-6849	478	74	ve	ve	NOUN
ejpam-6849	478	75	-	-	PUNCT
ejpam-6849	478	76	dominating	dominate	VERB
ejpam-6849	478	77	set	set	NOUN
ejpam-6849	478	78	of	of	ADP
ejpam-6849	478	79	h	h	NOUN
ejpam-6849	478	80	for	for	ADP
ejpam-6849	478	81	each	each	DET
ejpam-6849	478	82	x	x	SYM
ejpam-6849	478	83	∈	∈	PROPN
ejpam-6849	478	84	s	s	PART
ejpam-6849	478	85	\	\	NOUN
ejpam-6849	478	86	ng(s	ng(s	NUM
ejpam-6849	478	87	)	)	PUNCT
ejpam-6849	478	88	.	.	PUNCT
ejpam-6849	479	1	proof	proof	NOUN
ejpam-6849	479	2	.	.	PUNCT
ejpam-6849	480	1	first	first	ADV
ejpam-6849	480	2	,	,	PUNCT
ejpam-6849	480	3	assume	assume	VERB
ejpam-6849	480	4	that	that	SCONJ
ejpam-6849	480	5	c	c	PROPN
ejpam-6849	480	6	is	be	AUX
ejpam-6849	480	7	a	a	DET
ejpam-6849	480	8	ve	ve	NOUN
ejpam-6849	480	9	-	-	PUNCT
ejpam-6849	480	10	dominating	dominate	VERB
ejpam-6849	480	11	set	set	NOUN
ejpam-6849	480	12	of	of	ADP
ejpam-6849	480	13	g[h	g[h	NOUN
ejpam-6849	480	14	]	]	PUNCT
ejpam-6849	480	15	.	.	PUNCT
ejpam-6849	481	1	let	let	VERB
ejpam-6849	481	2	s	s	PRON
ejpam-6849	481	3	be	be	AUX
ejpam-6849	481	4	the	the	DET
ejpam-6849	481	5	g	g	NOUN
ejpam-6849	481	6	-	-	PUNCT
ejpam-6849	481	7	projection	projection	NOUN
ejpam-6849	481	8	cg	cg	NOUN
ejpam-6849	481	9	of	of	ADP
ejpam-6849	481	10	c	c	PROPN
ejpam-6849	481	11	,	,	PUNCT
ejpam-6849	481	12	i.e.	i.e.	X
ejpam-6849	481	13	,	,	PUNCT
ejpam-6849	481	14	s	s	NOUN
ejpam-6849	481	15	=	=	NOUN
ejpam-6849	481	16	cg	cg	NOUN
ejpam-6849	481	17	=	=	SYM
ejpam-6849	481	18	{	{	PUNCT
ejpam-6849	481	19	x	x	PROPN
ejpam-6849	481	20	∈	∈	PROPN
ejpam-6849	481	21	v	v	NOUN
ejpam-6849	481	22	(	(	PUNCT
ejpam-6849	481	23	g	g	NOUN
ejpam-6849	481	24	)	)	PUNCT
ejpam-6849	481	25	:	:	PUNCT
ejpam-6849	481	26	(	(	PUNCT
ejpam-6849	481	27	x	x	X
ejpam-6849	481	28	,	,	PUNCT
ejpam-6849	481	29	y	y	NOUN
ejpam-6849	481	30	)	)	PUNCT
ejpam-6849	481	31	∈	∈	PROPN
ejpam-6849	481	32	c	c	NOUN
ejpam-6849	481	33	for	for	ADP
ejpam-6849	481	34	some	some	DET
ejpam-6849	481	35	y	y	PROPN
ejpam-6849	481	36	∈	∈	PROPN
ejpam-6849	481	37	v	v	ADP
ejpam-6849	481	38	(	(	PUNCT
ejpam-6849	481	39	h	h	NOUN
ejpam-6849	481	40	)	)	PUNCT
ejpam-6849	481	41	}	}	PUNCT
ejpam-6849	481	42	.	.	PUNCT
ejpam-6849	482	1	then	then	ADV
ejpam-6849	482	2	c	c	X
ejpam-6849	482	3	=	=	SYM
ejpam-6849	482	4	∪x∈s	∪x∈s	PROPN
ejpam-6849	482	5	(	(	PUNCT
ejpam-6849	482	6	{	{	PUNCT
ejpam-6849	482	7	x	x	NOUN
ejpam-6849	482	8	}	}	PUNCT
ejpam-6849	482	9	×	×	PROPN
ejpam-6849	482	10	tx	tx	PROPN
ejpam-6849	482	11	)	)	PUNCT
ejpam-6849	482	12	,	,	PUNCT
ejpam-6849	482	13	where	where	SCONJ
ejpam-6849	482	14	tx	tx	VERB
ejpam-6849	482	15	=	=	PUNCT
ejpam-6849	482	16	{	{	PUNCT
ejpam-6849	482	17	y	y	PROPN
ejpam-6849	482	18	∈	∈	PROPN
ejpam-6849	482	19	v	v	PROPN
ejpam-6849	482	20	(	(	PUNCT
ejpam-6849	482	21	h	h	NOUN
ejpam-6849	482	22	)	)	PUNCT
ejpam-6849	482	23	:	:	PUNCT
ejpam-6849	482	24	(	(	PUNCT
ejpam-6849	482	25	x	x	X
ejpam-6849	482	26	,	,	PUNCT
ejpam-6849	482	27	y	y	NOUN
ejpam-6849	482	28	)	)	PUNCT
ejpam-6849	482	29	∈	∈	PROPN
ejpam-6849	482	30	c	c	NOUN
ejpam-6849	482	31	}	}	PUNCT
ejpam-6849	482	32	for	for	ADP
ejpam-6849	482	33	each	each	DET
ejpam-6849	482	34	x	x	SYM
ejpam-6849	482	35	∈	∈	PROPN
ejpam-6849	482	36	s.	s.	PROPN
ejpam-6849	482	37	we	we	PRON
ejpam-6849	482	38	claim	claim	VERB
ejpam-6849	482	39	that	that	SCONJ
ejpam-6849	482	40	s	s	VERB
ejpam-6849	482	41	is	be	AUX
ejpam-6849	482	42	a	a	DET
ejpam-6849	482	43	dominating	dominating	NOUN
ejpam-6849	482	44	set	set	NOUN
ejpam-6849	482	45	of	of	ADP
ejpam-6849	482	46	g.	g.	PROPN
ejpam-6849	482	47	suppose	suppose	VERB
ejpam-6849	482	48	not	not	PART
ejpam-6849	482	49	,	,	PUNCT
ejpam-6849	482	50	and	and	CCONJ
ejpam-6849	482	51	let	let	VERB
ejpam-6849	482	52	x	x	SYM
ejpam-6849	482	53	∈	∈	PROPN
ejpam-6849	482	54	v	v	X
ejpam-6849	482	55	(	(	PUNCT
ejpam-6849	482	56	g	g	NOUN
ejpam-6849	482	57	)	)	PUNCT
ejpam-6849	482	58	\	\	PUNCT
ejpam-6849	482	59	ng[s	ng[s	PROPN
ejpam-6849	482	60	]	]	PUNCT
ejpam-6849	482	61	.	.	PUNCT
ejpam-6849	483	1	let	let	VERB
ejpam-6849	483	2	wz	wz	AUX
ejpam-6849	483	3	∈	∈	VERB
ejpam-6849	483	4	e(h	e(h	PROPN
ejpam-6849	483	5	)	)	PUNCT
ejpam-6849	483	6	.	.	PUNCT
ejpam-6849	484	1	then	then	ADV
ejpam-6849	484	2	(	(	PUNCT
ejpam-6849	484	3	x	x	X
ejpam-6849	484	4	,	,	PUNCT
ejpam-6849	484	5	w)(x	w)(x	PROPN
ejpam-6849	484	6	,	,	PUNCT
ejpam-6849	484	7	z	z	NOUN
ejpam-6849	484	8	)	)	PUNCT
ejpam-6849	484	9	∈	∈	NOUN
ejpam-6849	484	10	e(g[h	e(g[h	NOUN
ejpam-6849	484	11	]	]	PUNCT
ejpam-6849	484	12	)	)	PUNCT
ejpam-6849	484	13	and	and	CCONJ
ejpam-6849	484	14	there	there	PRON
ejpam-6849	484	15	exists	exist	VERB
ejpam-6849	484	16	(	(	PUNCT
ejpam-6849	484	17	u	u	NOUN
ejpam-6849	484	18	,	,	PUNCT
ejpam-6849	484	19	v	v	NOUN
ejpam-6849	484	20	)	)	PUNCT
ejpam-6849	484	21	∈	∈	PROPN
ejpam-6849	484	22	c	c	NOUN
ejpam-6849	484	23	such	such	ADJ
ejpam-6849	484	24	that	that	DET
ejpam-6849	484	25	(	(	PUNCT
ejpam-6849	484	26	u	u	NOUN
ejpam-6849	484	27	,	,	PUNCT
ejpam-6849	484	28	v	v	NOUN
ejpam-6849	484	29	)	)	PUNCT
ejpam-6849	484	30	ve	ve	VERB
ejpam-6849	484	31	-	-	PUNCT
ejpam-6849	484	32	dominates	dominate	VERB
ejpam-6849	484	33	(	(	PUNCT
ejpam-6849	484	34	x	x	X
ejpam-6849	484	35	,	,	PUNCT
ejpam-6849	484	36	w)(x	w)(x	PROPN
ejpam-6849	484	37	,	,	PUNCT
ejpam-6849	484	38	z	z	NOUN
ejpam-6849	484	39	)	)	PUNCT
ejpam-6849	484	40	.	.	PUNCT
ejpam-6849	485	1	since	since	SCONJ
ejpam-6849	485	2	x	x	PROPN
ejpam-6849	485	3	/∈	/∈	PROPN
ejpam-6849	485	4	s	s	X
ejpam-6849	485	5	,	,	PUNCT
ejpam-6849	485	6	u	u	PROPN
ejpam-6849	485	7	̸=	̸=	PROPN
ejpam-6849	485	8	x.	x.	PUNCT
ejpam-6849	485	9	thus	thus	ADV
ejpam-6849	485	10	,	,	PUNCT
ejpam-6849	485	11	ux	ux	PROPN
ejpam-6849	485	12	∈	∈	PROPN
ejpam-6849	485	13	e(g	e(g	PROPN
ejpam-6849	485	14	)	)	PUNCT
ejpam-6849	485	15	,	,	PUNCT
ejpam-6849	485	16	a	a	DET
ejpam-6849	485	17	contradiction	contradiction	NOUN
ejpam-6849	485	18	.	.	PUNCT
ejpam-6849	486	1	this	this	PRON
ejpam-6849	486	2	means	mean	VERB
ejpam-6849	486	3	that	that	SCONJ
ejpam-6849	486	4	v	v	X
ejpam-6849	486	5	(	(	PUNCT
ejpam-6849	486	6	g	g	NOUN
ejpam-6849	486	7	)	)	PUNCT
ejpam-6849	486	8	=	=	PUNCT
ejpam-6849	486	9	ng[s	ng[s	PROPN
ejpam-6849	486	10	]	]	PUNCT
ejpam-6849	486	11	,	,	PUNCT
ejpam-6849	486	12	and	and	CCONJ
ejpam-6849	486	13	the	the	DET
ejpam-6849	486	14	claim	claim	NOUN
ejpam-6849	486	15	is	be	AUX
ejpam-6849	486	16	done	do	VERB
ejpam-6849	486	17	.	.	PUNCT
ejpam-6849	487	1	therefore	therefore	ADV
ejpam-6849	487	2	,	,	PUNCT
ejpam-6849	487	3	(	(	PUNCT
ejpam-6849	487	4	i	i	NOUN
ejpam-6849	487	5	)	)	PUNCT
ejpam-6849	487	6	holds	hold	VERB
ejpam-6849	487	7	.	.	PUNCT
ejpam-6849	488	1	now	now	ADV
ejpam-6849	488	2	,	,	PUNCT
ejpam-6849	488	3	to	to	PART
ejpam-6849	488	4	prove	prove	VERB
ejpam-6849	488	5	(	(	PUNCT
ejpam-6849	488	6	ii	ii	NOUN
ejpam-6849	488	7	)	)	PUNCT
ejpam-6849	488	8	,	,	PUNCT
ejpam-6849	488	9	let	let	VERB
ejpam-6849	488	10	x	x	PUNCT
ejpam-6849	488	11	∈	∈	PROPN
ejpam-6849	488	12	s	s	PART
ejpam-6849	488	13	\	\	NOUN
ejpam-6849	488	14	ng(s	ng(s	NUM
ejpam-6849	488	15	)	)	PUNCT
ejpam-6849	488	16	.	.	PUNCT
ejpam-6849	489	1	let	let	VERB
ejpam-6849	489	2	wz	wz	AUX
ejpam-6849	489	3	∈	∈	VERB
ejpam-6849	489	4	e(h	e(h	PROPN
ejpam-6849	489	5	)	)	PUNCT
ejpam-6849	489	6	.	.	PUNCT
ejpam-6849	490	1	then	then	ADV
ejpam-6849	490	2	(	(	PUNCT
ejpam-6849	490	3	x	x	X
ejpam-6849	490	4	,	,	PUNCT
ejpam-6849	490	5	w)(x	w)(x	PROPN
ejpam-6849	490	6	,	,	PUNCT
ejpam-6849	490	7	z	z	NOUN
ejpam-6849	490	8	)	)	PUNCT
ejpam-6849	490	9	∈	∈	NOUN
ejpam-6849	490	10	e(g[h	e(g[h	NOUN
ejpam-6849	490	11	]	]	PUNCT
ejpam-6849	490	12	)	)	PUNCT
ejpam-6849	490	13	and	and	CCONJ
ejpam-6849	490	14	there	there	PRON
ejpam-6849	490	15	exists	exist	VERB
ejpam-6849	490	16	(	(	PUNCT
ejpam-6849	490	17	u	u	NOUN
ejpam-6849	490	18	,	,	PUNCT
ejpam-6849	490	19	v	v	NOUN
ejpam-6849	490	20	)	)	PUNCT
ejpam-6849	490	21	∈	∈	PROPN
ejpam-6849	490	22	c	c	NOUN
ejpam-6849	490	23	for	for	ADP
ejpam-6849	490	24	which	which	PRON
ejpam-6849	490	25	(	(	PUNCT
ejpam-6849	490	26	u	u	NOUN
ejpam-6849	490	27	,	,	PUNCT
ejpam-6849	490	28	v	v	NOUN
ejpam-6849	490	29	)	)	PUNCT
ejpam-6849	490	30	ve	ve	VERB
ejpam-6849	490	31	-	-	PUNCT
ejpam-6849	490	32	dominates	dominate	VERB
ejpam-6849	490	33	(	(	PUNCT
ejpam-6849	490	34	x	x	X
ejpam-6849	490	35	,	,	PUNCT
ejpam-6849	490	36	w)(x	w)(x	PROPN
ejpam-6849	490	37	,	,	PUNCT
ejpam-6849	490	38	z	z	NOUN
ejpam-6849	490	39	)	)	PUNCT
ejpam-6849	490	40	in	in	ADP
ejpam-6849	490	41	g[h	g[h	NOUN
ejpam-6849	490	42	]	]	PUNCT
ejpam-6849	490	43	.	.	PUNCT
ejpam-6849	491	1	if	if	SCONJ
ejpam-6849	491	2	(	(	PUNCT
ejpam-6849	491	3	u	u	NOUN
ejpam-6849	491	4	,	,	PUNCT
ejpam-6849	491	5	v	v	NOUN
ejpam-6849	491	6	)	)	PUNCT
ejpam-6849	491	7	=	=	SYM
ejpam-6849	491	8	(	(	PUNCT
ejpam-6849	491	9	x	x	X
ejpam-6849	491	10	,	,	PUNCT
ejpam-6849	491	11	w	w	PROPN
ejpam-6849	491	12	)	)	PUNCT
ejpam-6849	491	13	,	,	PUNCT
ejpam-6849	491	14	then	then	ADV
ejpam-6849	491	15	u	u	X
ejpam-6849	491	16	=	=	NOUN
ejpam-6849	491	17	x	x	X
ejpam-6849	491	18	so	so	ADV
ejpam-6849	491	19	that	that	PRON
ejpam-6849	491	20	v	v	X
ejpam-6849	491	21	=	=	SYM
ejpam-6849	491	22	w	w	PROPN
ejpam-6849	491	23	∈	∈	PROPN
ejpam-6849	491	24	tx	tx	PROPN
ejpam-6849	491	25	and	and	CCONJ
ejpam-6849	491	26	tx	tx	VERB
ejpam-6849	491	27	ve	ve	VERB
ejpam-6849	491	28	-	-	PUNCT
ejpam-6849	491	29	dominates	dominate	VERB
ejpam-6849	491	30	wz	wz	PROPN
ejpam-6849	491	31	.	.	PUNCT
ejpam-6849	492	1	similarly	similarly	ADV
ejpam-6849	492	2	,	,	PUNCT
ejpam-6849	492	3	if	if	SCONJ
ejpam-6849	492	4	(	(	PUNCT
ejpam-6849	492	5	u	u	NOUN
ejpam-6849	492	6	,	,	PUNCT
ejpam-6849	492	7	v	v	NOUN
ejpam-6849	492	8	)	)	PUNCT
ejpam-6849	492	9	=	=	SYM
ejpam-6849	492	10	(	(	PUNCT
ejpam-6849	492	11	x	x	X
ejpam-6849	492	12	,	,	PUNCT
ejpam-6849	492	13	z	z	NOUN
ejpam-6849	492	14	)	)	PUNCT
ejpam-6849	492	15	,	,	PUNCT
ejpam-6849	492	16	then	then	ADV
ejpam-6849	492	17	tx	tx	AUX
ejpam-6849	492	18	ve	ve	VERB
ejpam-6849	492	19	-	-	PUNCT
ejpam-6849	492	20	dominates	dominate	VERB
ejpam-6849	492	21	wz	wz	PROPN
ejpam-6849	492	22	.	.	PUNCT
ejpam-6849	492	23	suppose	suppose	VERB
ejpam-6849	492	24	that	that	SCONJ
ejpam-6849	492	25	(	(	PUNCT
ejpam-6849	492	26	u	u	NOUN
ejpam-6849	492	27	,	,	PUNCT
ejpam-6849	492	28	v)(x	v)(x	NOUN
ejpam-6849	492	29	,	,	PUNCT
ejpam-6849	492	30	w	w	NOUN
ejpam-6849	492	31	)	)	PUNCT
ejpam-6849	492	32	∈	∈	NOUN
ejpam-6849	492	33	e(g[h	e(g[h	NOUN
ejpam-6849	492	34	]	]	PUNCT
ejpam-6849	492	35	)	)	PUNCT
ejpam-6849	492	36	.	.	PUNCT
ejpam-6849	493	1	since	since	SCONJ
ejpam-6849	493	2	x	x	PROPN
ejpam-6849	493	3	/∈	/∈	PROPN
ejpam-6849	493	4	ng(s	ng(s	NUM
ejpam-6849	493	5	)	)	PUNCT
ejpam-6849	493	6	,	,	PUNCT
ejpam-6849	493	7	u	u	NOUN
ejpam-6849	493	8	=	=	PUNCT
ejpam-6849	493	9	x	x	X
ejpam-6849	493	10	and	and	CCONJ
ejpam-6849	493	11	vw	vw	PROPN
ejpam-6849	493	12	∈	∈	PROPN
ejpam-6849	493	13	e(h	e(h	PROPN
ejpam-6849	493	14	)	)	PUNCT
ejpam-6849	493	15	.	.	PUNCT
ejpam-6849	494	1	this	this	PRON
ejpam-6849	494	2	means	mean	VERB
ejpam-6849	494	3	that	that	SCONJ
ejpam-6849	494	4	v	v	NOUN
ejpam-6849	494	5	,	,	PUNCT
ejpam-6849	494	6	and	and	CCONJ
ejpam-6849	494	7	hence	hence	ADV
ejpam-6849	494	8	tx	tx	PROPN
ejpam-6849	494	9	,	,	PUNCT
ejpam-6849	494	10	ve	ve	VERB
ejpam-6849	494	11	-	-	PUNCT
ejpam-6849	494	12	dominates	dominate	VERB
ejpam-6849	494	13	wz	wz	PROPN
ejpam-6849	494	14	.	.	PUNCT
ejpam-6849	495	1	similarly	similarly	ADV
ejpam-6849	495	2	,	,	PUNCT
ejpam-6849	495	3	if	if	SCONJ
ejpam-6849	495	4	(	(	PUNCT
ejpam-6849	495	5	u	u	NOUN
ejpam-6849	495	6	,	,	PUNCT
ejpam-6849	495	7	v)(x	v)(x	NOUN
ejpam-6849	495	8	,	,	PUNCT
ejpam-6849	495	9	z	z	NOUN
ejpam-6849	495	10	)	)	PUNCT
ejpam-6849	495	11	∈	∈	NOUN
ejpam-6849	495	12	e(g[h	e(g[h	NOUN
ejpam-6849	495	13	]	]	PUNCT
ejpam-6849	495	14	)	)	PUNCT
ejpam-6849	495	15	,	,	PUNCT
ejpam-6849	495	16	then	then	ADV
ejpam-6849	495	17	tx	tx	AUX
ejpam-6849	495	18	ve	ve	AUX
ejpam-6849	495	19	-	-	PUNCT
ejpam-6849	495	20	dominates	dominate	VERB
ejpam-6849	495	21	wz	wz	PROPN
ejpam-6849	495	22	.	.	PUNCT
ejpam-6849	495	23	accordingly	accordingly	ADV
ejpam-6849	495	24	,	,	PUNCT
ejpam-6849	495	25	tx	tx	PROPN
ejpam-6849	495	26	is	be	AUX
ejpam-6849	495	27	a	a	DET
ejpam-6849	495	28	ve	ve	NOUN
ejpam-6849	495	29	-	-	PUNCT
ejpam-6849	495	30	dominating	dominate	VERB
ejpam-6849	495	31	set	set	NOUN
ejpam-6849	495	32	of	of	ADP
ejpam-6849	495	33	h.	h.	PROPN
ejpam-6849	495	34	next	next	ADV
ejpam-6849	495	35	,	,	PUNCT
ejpam-6849	495	36	assume	assume	VERB
ejpam-6849	495	37	that	that	SCONJ
ejpam-6849	495	38	c	c	PROPN
ejpam-6849	495	39	satisfies	satisfy	VERB
ejpam-6849	495	40	equation	equation	NOUN
ejpam-6849	495	41	6	6	NUM
ejpam-6849	495	42	where	where	SCONJ
ejpam-6849	495	43	s	s	VERB
ejpam-6849	495	44	⊆	⊆	NUM
ejpam-6849	495	45	v	v	NOUN
ejpam-6849	495	46	(	(	PUNCT
ejpam-6849	495	47	g	g	NOUN
ejpam-6849	495	48	)	)	PUNCT
ejpam-6849	495	49	and	and	CCONJ
ejpam-6849	495	50	tx	tx	VERB
ejpam-6849	495	51	⊆	⊆	NUM
ejpam-6849	495	52	v	v	NOUN
ejpam-6849	495	53	(	(	PUNCT
ejpam-6849	495	54	h	h	NOUN
ejpam-6849	495	55	)	)	PUNCT
ejpam-6849	495	56	satisfying	satisfy	VERB
ejpam-6849	495	57	conditions	condition	NOUN
ejpam-6849	495	58	(	(	PUNCT
ejpam-6849	495	59	i	i	NOUN
ejpam-6849	495	60	)	)	PUNCT
ejpam-6849	495	61	and	and	CCONJ
ejpam-6849	495	62	(	(	PUNCT
ejpam-6849	495	63	ii	ii	NOUN
ejpam-6849	495	64	)	)	PUNCT
ejpam-6849	495	65	,	,	PUNCT
ejpam-6849	495	66	respectively	respectively	ADV
ejpam-6849	495	67	.	.	PUNCT
ejpam-6849	496	1	note	note	VERB
ejpam-6849	496	2	first	first	ADV
ejpam-6849	496	3	that	that	SCONJ
ejpam-6849	496	4	since	since	SCONJ
ejpam-6849	496	5	s	s	NOUN
ejpam-6849	496	6	is	be	AUX
ejpam-6849	496	7	a	a	DET
ejpam-6849	496	8	dominating	dominating	NOUN
ejpam-6849	496	9	set	set	NOUN
ejpam-6849	496	10	,	,	PUNCT
ejpam-6849	496	11	s	s	PART
ejpam-6849	496	12	is	be	AUX
ejpam-6849	496	13	a	a	DET
ejpam-6849	496	14	ve	ve	NOUN
ejpam-6849	496	15	-	-	PUNCT
ejpam-6849	496	16	dominating	dominate	VERB
ejpam-6849	496	17	set	set	NOUN
ejpam-6849	496	18	of	of	ADP
ejpam-6849	496	19	g.	g.	PROPN
ejpam-6849	497	1	let	let	VERB
ejpam-6849	497	2	(	(	PUNCT
ejpam-6849	497	3	x	x	NOUN
ejpam-6849	497	4	,	,	PUNCT
ejpam-6849	497	5	y)(w	y)(w	PROPN
ejpam-6849	497	6	,	,	PUNCT
ejpam-6849	497	7	z	z	NOUN
ejpam-6849	497	8	)	)	PUNCT
ejpam-6849	497	9	∈	∈	NOUN
ejpam-6849	497	10	e(g[h	e(g[h	NOUN
ejpam-6849	497	11	]	]	PUNCT
ejpam-6849	497	12	)	)	PUNCT
ejpam-6849	497	13	.	.	PUNCT
ejpam-6849	498	1	we	we	PRON
ejpam-6849	498	2	consider	consider	VERB
ejpam-6849	498	3	the	the	DET
ejpam-6849	498	4	following	follow	VERB
ejpam-6849	498	5	cases	case	NOUN
ejpam-6849	498	6	:	:	PUNCT
ejpam-6849	498	7	case	case	NOUN
ejpam-6849	498	8	1	1	NUM
ejpam-6849	498	9	:	:	PUNCT
ejpam-6849	498	10	suppose	suppose	VERB
ejpam-6849	498	11	that	that	SCONJ
ejpam-6849	498	12	xw	xw	PROPN
ejpam-6849	498	13	∈	∈	PROPN
ejpam-6849	498	14	e(g	e(g	PROPN
ejpam-6849	498	15	)	)	PUNCT
ejpam-6849	498	16	.	.	PUNCT
ejpam-6849	499	1	by	by	ADP
ejpam-6849	499	2	a	a	DET
ejpam-6849	499	3	ve	ve	NOUN
ejpam-6849	499	4	-	-	PUNCT
ejpam-6849	499	5	dominating	dominating	NOUN
ejpam-6849	499	6	set	set	NOUN
ejpam-6849	499	7	,	,	PUNCT
ejpam-6849	499	8	there	there	PRON
ejpam-6849	499	9	exists	exist	VERB
ejpam-6849	499	10	u	u	PROPN
ejpam-6849	499	11	∈	∈	PROPN
ejpam-6849	499	12	s	s	VERB
ejpam-6849	499	13	such	such	ADJ
ejpam-6849	499	14	that	that	SCONJ
ejpam-6849	499	15	u	u	PROPN
ejpam-6849	499	16	ve	ve	AUX
ejpam-6849	499	17	-	-	PUNCT
ejpam-6849	499	18	dominates	dominate	VERB
ejpam-6849	499	19	xw	xw	PROPN
ejpam-6849	499	20	.	.	PUNCT
ejpam-6849	500	1	pick	pick	VERB
ejpam-6849	500	2	v	v	ADP
ejpam-6849	500	3	∈	∈	PROPN
ejpam-6849	500	4	tu	tu	PROPN
ejpam-6849	500	5	.	.	PUNCT
ejpam-6849	501	1	if	if	SCONJ
ejpam-6849	501	2	u	u	PRON
ejpam-6849	501	3	=	=	NOUN
ejpam-6849	501	4	x	x	PROPN
ejpam-6849	501	5	(	(	PUNCT
ejpam-6849	501	6	resp	resp	NOUN
ejpam-6849	501	7	.	.	PUNCT
ejpam-6849	502	1	u	u	NOUN
ejpam-6849	502	2	=	=	PROPN
ejpam-6849	502	3	w	w	PROPN
ejpam-6849	502	4	)	)	PUNCT
ejpam-6849	502	5	,	,	PUNCT
ejpam-6849	502	6	then	then	ADV
ejpam-6849	502	7	(	(	PUNCT
ejpam-6849	502	8	u	u	NOUN
ejpam-6849	502	9	,	,	PUNCT
ejpam-6849	502	10	v)(w	v)(w	NOUN
ejpam-6849	502	11	,	,	PUNCT
ejpam-6849	502	12	z	z	NOUN
ejpam-6849	502	13	)	)	PUNCT
ejpam-6849	502	14	∈	∈	NOUN
ejpam-6849	502	15	e(g[h	e(g[h	NOUN
ejpam-6849	502	16	]	]	PUNCT
ejpam-6849	502	17	)	)	PUNCT
ejpam-6849	502	18	(	(	PUNCT
ejpam-6849	502	19	resp	resp	NOUN
ejpam-6849	502	20	.	.	PUNCT
ejpam-6849	503	1	(	(	PUNCT
ejpam-6849	503	2	u	u	NOUN
ejpam-6849	503	3	,	,	PUNCT
ejpam-6849	503	4	v)(x	v)(x	NOUN
ejpam-6849	503	5	,	,	PUNCT
ejpam-6849	503	6	y	y	NOUN
ejpam-6849	503	7	)	)	PUNCT
ejpam-6849	503	8	∈	∈	NOUN
ejpam-6849	503	9	e(g[h	e(g[h	NOUN
ejpam-6849	503	10	]	]	PUNCT
ejpam-6849	503	11	)	)	PUNCT
ejpam-6849	503	12	)	)	PUNCT
ejpam-6849	503	13	.	.	PUNCT
ejpam-6849	504	1	this	this	PRON
ejpam-6849	504	2	means	mean	VERB
ejpam-6849	504	3	that	that	SCONJ
ejpam-6849	504	4	(	(	PUNCT
ejpam-6849	504	5	u	u	NOUN
ejpam-6849	504	6	,	,	PUNCT
ejpam-6849	504	7	v	v	NOUN
ejpam-6849	504	8	)	)	PUNCT
ejpam-6849	504	9	,	,	PUNCT
ejpam-6849	504	10	and	and	CCONJ
ejpam-6849	504	11	hence	hence	ADV
ejpam-6849	504	12	c	c	AUX
ejpam-6849	504	13	,	,	PUNCT
ejpam-6849	504	14	ve	ve	VERB
ejpam-6849	504	15	-	-	PUNCT
ejpam-6849	504	16	dominates	dominate	VERB
ejpam-6849	504	17	(	(	PUNCT
ejpam-6849	504	18	x	x	NOUN
ejpam-6849	504	19	,	,	PUNCT
ejpam-6849	504	20	y)(w	y)(w	PROPN
ejpam-6849	504	21	,	,	PUNCT
ejpam-6849	504	22	z	z	NOUN
ejpam-6849	504	23	)	)	PUNCT
ejpam-6849	504	24	.	.	PUNCT
ejpam-6849	505	1	on	on	ADP
ejpam-6849	505	2	the	the	DET
ejpam-6849	505	3	other	other	ADJ
ejpam-6849	505	4	hand	hand	NOUN
ejpam-6849	505	5	,	,	PUNCT
ejpam-6849	505	6	if	if	SCONJ
ejpam-6849	505	7	ux	ux	PROPN
ejpam-6849	505	8	∈	∈	PROPN
ejpam-6849	505	9	e(g	e(g	PROPN
ejpam-6849	505	10	)	)	PUNCT
ejpam-6849	505	11	(	(	PUNCT
ejpam-6849	505	12	resp	resp	NOUN
ejpam-6849	505	13	.	.	PUNCT
ejpam-6849	506	1	uw	uw	PROPN
ejpam-6849	506	2	∈	∈	PROPN
ejpam-6849	506	3	e(g	e(g	PROPN
ejpam-6849	506	4	)	)	PUNCT
ejpam-6849	506	5	)	)	PUNCT
ejpam-6849	506	6	,	,	PUNCT
ejpam-6849	506	7	then	then	ADV
ejpam-6849	506	8	(	(	PUNCT
ejpam-6849	506	9	u	u	NOUN
ejpam-6849	506	10	,	,	PUNCT
ejpam-6849	506	11	v)(x	v)(x	NOUN
ejpam-6849	506	12	,	,	PUNCT
ejpam-6849	506	13	y	y	NOUN
ejpam-6849	506	14	)	)	PUNCT
ejpam-6849	506	15	∈	∈	NOUN
ejpam-6849	506	16	e(g[h	e(g[h	NOUN
ejpam-6849	506	17	]	]	PUNCT
ejpam-6849	506	18	)	)	PUNCT
ejpam-6849	506	19	(	(	PUNCT
ejpam-6849	506	20	resp	resp	NOUN
ejpam-6849	506	21	.	.	PUNCT
ejpam-6849	507	1	(	(	PUNCT
ejpam-6849	507	2	u	u	NOUN
ejpam-6849	507	3	,	,	PUNCT
ejpam-6849	507	4	v)(w	v)(w	NOUN
ejpam-6849	507	5	,	,	PUNCT
ejpam-6849	507	6	z	z	NOUN
ejpam-6849	507	7	)	)	PUNCT
ejpam-6849	507	8	∈	∈	NOUN
ejpam-6849	507	9	e(g[h	e(g[h	NOUN
ejpam-6849	507	10	]	]	PUNCT
ejpam-6849	507	11	)	)	PUNCT
ejpam-6849	507	12	)	)	PUNCT
ejpam-6849	507	13	.	.	PUNCT
ejpam-6849	508	1	thus	thus	ADV
ejpam-6849	508	2	,	,	PUNCT
ejpam-6849	508	3	(	(	PUNCT
ejpam-6849	508	4	u	u	NOUN
ejpam-6849	508	5	,	,	PUNCT
ejpam-6849	508	6	v	v	NOUN
ejpam-6849	508	7	)	)	PUNCT
ejpam-6849	508	8	,	,	PUNCT
ejpam-6849	508	9	and	and	CCONJ
ejpam-6849	508	10	hence	hence	ADV
ejpam-6849	508	11	,	,	PUNCT
ejpam-6849	508	12	c	c	PROPN
ejpam-6849	508	13	ve	ve	VERB
ejpam-6849	508	14	-	-	PUNCT
ejpam-6849	508	15	dominates	dominate	VERB
ejpam-6849	508	16	(	(	PUNCT
ejpam-6849	508	17	x	x	NOUN
ejpam-6849	508	18	,	,	PUNCT
ejpam-6849	508	19	y)(w	y)(w	PROPN
ejpam-6849	508	20	,	,	PUNCT
ejpam-6849	508	21	z	z	NOUN
ejpam-6849	508	22	)	)	PUNCT
ejpam-6849	508	23	.	.	PUNCT
ejpam-6849	509	1	j.	j.	PROPN
ejpam-6849	509	2	tayab	tayab	PROPN
ejpam-6849	509	3	,	,	PUNCT
ejpam-6849	509	4	f.	f.	PROPN
ejpam-6849	509	5	p.	p.	PROPN
ejpam-6849	509	6	jamil	jamil	PROPN
ejpam-6849	509	7	,	,	PUNCT
ejpam-6849	509	8	i.	i.	PROPN
ejpam-6849	509	9	aniversario	aniversario	PROPN
ejpam-6849	509	10	/	/	SYM
ejpam-6849	509	11	eur	eur	PROPN
ejpam-6849	509	12	.	.	PUNCT
ejpam-6849	510	1	j.	j.	PROPN
ejpam-6849	510	2	pure	pure	PROPN
ejpam-6849	510	3	appl	appl	PROPN
ejpam-6849	510	4	.	.	PROPN
ejpam-6849	510	5	math	math	PROPN
ejpam-6849	510	6	,	,	PUNCT
ejpam-6849	510	7	18	18	NUM
ejpam-6849	510	8	(	(	PUNCT
ejpam-6849	510	9	4	4	NUM
ejpam-6849	510	10	)	)	PUNCT
ejpam-6849	510	11	(	(	PUNCT
ejpam-6849	510	12	2025	2025	NUM
ejpam-6849	510	13	)	)	PUNCT
ejpam-6849	510	14	,	,	PUNCT
ejpam-6849	510	15	6849	6849	NUM
ejpam-6849	510	16	13	13	NUM
ejpam-6849	510	17	of	of	ADP
ejpam-6849	510	18	19	19	NUM
ejpam-6849	510	19	case	case	NOUN
ejpam-6849	510	20	2	2	NUM
ejpam-6849	510	21	:	:	PUNCT
ejpam-6849	510	22	now	now	ADV
ejpam-6849	510	23	suppose	suppose	VERB
ejpam-6849	510	24	that	that	SCONJ
ejpam-6849	510	25	x	x	PROPN
ejpam-6849	510	26	=	=	SYM
ejpam-6849	510	27	w	w	PROPN
ejpam-6849	510	28	and	and	CCONJ
ejpam-6849	510	29	yz	yz	PROPN
ejpam-6849	510	30	∈	∈	PROPN
ejpam-6849	510	31	e(h	e(h	PROPN
ejpam-6849	510	32	)	)	PUNCT
ejpam-6849	510	33	.	.	PUNCT
ejpam-6849	511	1	if	if	SCONJ
ejpam-6849	511	2	x	x	SYM
ejpam-6849	511	3	∈	∈	PROPN
ejpam-6849	511	4	ng(s	ng(s	PRON
ejpam-6849	511	5	)	)	PUNCT
ejpam-6849	511	6	and	and	CCONJ
ejpam-6849	511	7	u	u	PROPN
ejpam-6849	511	8	∈	∈	PROPN
ejpam-6849	511	9	s	s	PART
ejpam-6849	511	10	∩	∩	NOUN
ejpam-6849	511	11	ng(x	ng(x	NUM
ejpam-6849	511	12	)	)	PUNCT
ejpam-6849	511	13	,	,	PUNCT
ejpam-6849	511	14	pick	pick	VERB
ejpam-6849	511	15	v	v	ADP
ejpam-6849	511	16	∈	∈	PROPN
ejpam-6849	511	17	tu	tu	PROPN
ejpam-6849	511	18	.	.	PUNCT
ejpam-6849	512	1	then	then	ADV
ejpam-6849	512	2	(	(	PUNCT
ejpam-6849	512	3	u	u	NOUN
ejpam-6849	512	4	,	,	PUNCT
ejpam-6849	512	5	v	v	NOUN
ejpam-6849	512	6	)	)	PUNCT
ejpam-6849	512	7	∈	∈	PROPN
ejpam-6849	512	8	c	c	PROPN
ejpam-6849	512	9	and	and	CCONJ
ejpam-6849	512	10	(	(	PUNCT
ejpam-6849	512	11	u	u	NOUN
ejpam-6849	512	12	,	,	PUNCT
ejpam-6849	512	13	v)(x	v)(x	NOUN
ejpam-6849	512	14	,	,	PUNCT
ejpam-6849	512	15	y	y	NOUN
ejpam-6849	512	16	)	)	PUNCT
ejpam-6849	512	17	∈	∈	NOUN
ejpam-6849	512	18	e(g[h	e(g[h	NOUN
ejpam-6849	512	19	]	]	PUNCT
ejpam-6849	512	20	)	)	PUNCT
ejpam-6849	512	21	.	.	PUNCT
ejpam-6849	513	1	this	this	PRON
ejpam-6849	513	2	means	mean	VERB
ejpam-6849	513	3	that	that	SCONJ
ejpam-6849	513	4	(	(	PUNCT
ejpam-6849	513	5	u	u	NOUN
ejpam-6849	513	6	,	,	PUNCT
ejpam-6849	513	7	v	v	NOUN
ejpam-6849	513	8	)	)	PUNCT
ejpam-6849	513	9	,	,	PUNCT
ejpam-6849	513	10	and	and	CCONJ
ejpam-6849	513	11	hence	hence	ADV
ejpam-6849	513	12	c	c	AUX
ejpam-6849	513	13	,	,	PUNCT
ejpam-6849	513	14	ve	ve	VERB
ejpam-6849	513	15	-	-	PUNCT
ejpam-6849	513	16	dominates	dominate	VERB
ejpam-6849	513	17	(	(	PUNCT
ejpam-6849	513	18	x	x	NOUN
ejpam-6849	513	19	,	,	PUNCT
ejpam-6849	513	20	y)(w	y)(w	PROPN
ejpam-6849	513	21	,	,	PUNCT
ejpam-6849	513	22	z	z	NOUN
ejpam-6849	513	23	)	)	PUNCT
ejpam-6849	513	24	.	.	PUNCT
ejpam-6849	514	1	suppose	suppose	VERB
ejpam-6849	514	2	,	,	PUNCT
ejpam-6849	514	3	on	on	ADP
ejpam-6849	514	4	the	the	DET
ejpam-6849	514	5	other	other	ADJ
ejpam-6849	514	6	hand	hand	NOUN
ejpam-6849	514	7	,	,	PUNCT
ejpam-6849	514	8	that	that	SCONJ
ejpam-6849	514	9	x	x	X
ejpam-6849	514	10	/∈	/∈	NOUN
ejpam-6849	514	11	ng(s	ng(s	NUM
ejpam-6849	514	12	)	)	PUNCT
ejpam-6849	514	13	.	.	PUNCT
ejpam-6849	515	1	since	since	SCONJ
ejpam-6849	515	2	s	s	PROPN
ejpam-6849	515	3	is	be	AUX
ejpam-6849	515	4	a	a	DET
ejpam-6849	515	5	dominating	dominating	NOUN
ejpam-6849	515	6	set	set	NOUN
ejpam-6849	515	7	,	,	PUNCT
ejpam-6849	515	8	x	x	PUNCT
ejpam-6849	515	9	∈	∈	PROPN
ejpam-6849	515	10	s	s	PART
ejpam-6849	515	11	\	\	NOUN
ejpam-6849	515	12	ng(s	ng(s	NUM
ejpam-6849	515	13	)	)	PUNCT
ejpam-6849	515	14	.	.	PUNCT
ejpam-6849	516	1	by	by	ADP
ejpam-6849	516	2	condition	condition	NOUN
ejpam-6849	516	3	(	(	PUNCT
ejpam-6849	516	4	ii	ii	NOUN
ejpam-6849	516	5	)	)	PUNCT
ejpam-6849	516	6	,	,	PUNCT
ejpam-6849	516	7	there	there	PRON
ejpam-6849	516	8	exists	exist	VERB
ejpam-6849	516	9	v	v	ADP
ejpam-6849	516	10	∈	∈	PROPN
ejpam-6849	516	11	tx	tx	ADP
ejpam-6849	516	12	such	such	ADJ
ejpam-6849	516	13	that	that	PRON
ejpam-6849	516	14	v	v	PROPN
ejpam-6849	516	15	ve	ve	VERB
ejpam-6849	516	16	-	-	PUNCT
ejpam-6849	516	17	dominates	dominate	VERB
ejpam-6849	516	18	yz	yz	PROPN
ejpam-6849	516	19	.	.	PUNCT
ejpam-6849	517	1	this	this	PRON
ejpam-6849	517	2	means	mean	VERB
ejpam-6849	517	3	that	that	SCONJ
ejpam-6849	517	4	(	(	PUNCT
ejpam-6849	517	5	x	x	NOUN
ejpam-6849	517	6	,	,	PUNCT
ejpam-6849	517	7	v	v	NOUN
ejpam-6849	517	8	)	)	PUNCT
ejpam-6849	517	9	∈	∈	PROPN
ejpam-6849	517	10	c	c	NOUN
ejpam-6849	517	11	and	and	CCONJ
ejpam-6849	517	12	(	(	PUNCT
ejpam-6849	517	13	x	x	NOUN
ejpam-6849	517	14	,	,	PUNCT
ejpam-6849	517	15	v	v	NOUN
ejpam-6849	517	16	)	)	PUNCT
ejpam-6849	517	17	,	,	PUNCT
ejpam-6849	517	18	and	and	CCONJ
ejpam-6849	517	19	hence	hence	ADV
ejpam-6849	517	20	c	c	AUX
ejpam-6849	517	21	,	,	PUNCT
ejpam-6849	517	22	ve	ve	VERB
ejpam-6849	517	23	-	-	PUNCT
ejpam-6849	517	24	dominates	dominate	VERB
ejpam-6849	517	25	(	(	PUNCT
ejpam-6849	517	26	x	x	NOUN
ejpam-6849	517	27	,	,	PUNCT
ejpam-6849	517	28	y)(w	y)(w	PROPN
ejpam-6849	517	29	,	,	PUNCT
ejpam-6849	517	30	z	z	NOUN
ejpam-6849	517	31	)	)	PUNCT
ejpam-6849	517	32	.	.	PUNCT
ejpam-6849	518	1	accordingly	accordingly	ADV
ejpam-6849	518	2	,	,	PUNCT
ejpam-6849	518	3	c	c	PROPN
ejpam-6849	518	4	is	be	AUX
ejpam-6849	518	5	a	a	DET
ejpam-6849	518	6	ve	ve	NOUN
ejpam-6849	518	7	-	-	PUNCT
ejpam-6849	518	8	dominating	dominate	VERB
ejpam-6849	518	9	set	set	NOUN
ejpam-6849	518	10	of	of	ADP
ejpam-6849	518	11	g[h	g[h	NOUN
ejpam-6849	518	12	]	]	PUNCT
ejpam-6849	518	13	.	.	PUNCT
ejpam-6849	519	1	■	■	PUNCT
ejpam-6849	519	2	lemma	lemma	PROPN
ejpam-6849	519	3	1	1	NUM
ejpam-6849	519	4	.	.	PUNCT
ejpam-6849	520	1	[	[	X
ejpam-6849	520	2	13	13	NUM
ejpam-6849	520	3	,	,	PUNCT
ejpam-6849	520	4	14	14	NUM
ejpam-6849	520	5	]	]	PUNCT
ejpam-6849	520	6	if	if	SCONJ
ejpam-6849	520	7	s	s	PROPN
ejpam-6849	520	8	is	be	AUX
ejpam-6849	520	9	a	a	DET
ejpam-6849	520	10	dominating	dominating	NOUN
ejpam-6849	520	11	set	set	NOUN
ejpam-6849	520	12	of	of	ADP
ejpam-6849	520	13	a	a	DET
ejpam-6849	520	14	graph	graph	NOUN
ejpam-6849	520	15	g	g	NOUN
ejpam-6849	520	16	without	without	ADP
ejpam-6849	520	17	isolated	isolated	ADJ
ejpam-6849	520	18	vertices	vertex	NOUN
ejpam-6849	520	19	,	,	PUNCT
ejpam-6849	520	20	then	then	ADV
ejpam-6849	520	21	γt(g	γt(g	PUNCT
ejpam-6849	520	22	)	)	PUNCT
ejpam-6849	520	23	≤	≤	NUM
ejpam-6849	520	24	|s	|s	PROPN
ejpam-6849	520	25	∩	∩	NOUN
ejpam-6849	520	26	ng(s)|	ng(s)|	PUNCT
ejpam-6849	520	27	+	+	NUM
ejpam-6849	520	28	2|s	2|s	PROPN
ejpam-6849	520	29	\	\	NOUN
ejpam-6849	520	30	ng(s)|	ng(s)|	PROPN
ejpam-6849	520	31	.	.	PUNCT
ejpam-6849	520	32	consequently	consequently	ADV
ejpam-6849	520	33	,	,	PUNCT
ejpam-6849	520	34	γt(g	γt(g	PUNCT
ejpam-6849	520	35	)	)	PUNCT
ejpam-6849	520	36	≤	≤	NUM
ejpam-6849	520	37	2γ(g	2γ(g	NUM
ejpam-6849	520	38	)	)	PUNCT
ejpam-6849	520	39	.	.	PUNCT
ejpam-6849	521	1	corollary	corollary	ADJ
ejpam-6849	521	2	4	4	NUM
ejpam-6849	521	3	.	.	PUNCT
ejpam-6849	522	1	let	let	VERB
ejpam-6849	522	2	g	g	NOUN
ejpam-6849	522	3	and	and	CCONJ
ejpam-6849	522	4	h	h	NOUN
ejpam-6849	522	5	be	be	AUX
ejpam-6849	522	6	connected	connect	VERB
ejpam-6849	522	7	graphs	graph	NOUN
ejpam-6849	522	8	with	with	ADP
ejpam-6849	522	9	h	h	NOUN
ejpam-6849	522	10	nontrivial	nontrivial	NOUN
ejpam-6849	522	11	.	.	PUNCT
ejpam-6849	523	1	(	(	PUNCT
ejpam-6849	523	2	i	i	NOUN
ejpam-6849	523	3	)	)	PUNCT
ejpam-6849	523	4	if	if	SCONJ
ejpam-6849	523	5	γve(h	γve(h	PROPN
ejpam-6849	523	6	)	)	PUNCT
ejpam-6849	523	7	=	=	SYM
ejpam-6849	523	8	1	1	NUM
ejpam-6849	523	9	,	,	PUNCT
ejpam-6849	523	10	then	then	ADV
ejpam-6849	523	11	γve(g[h	γve(g[h	NUM
ejpam-6849	523	12	]	]	PUNCT
ejpam-6849	523	13	)	)	PUNCT
ejpam-6849	523	14	=	=	PUNCT
ejpam-6849	523	15	γ(g	γ(g	PROPN
ejpam-6849	523	16	)	)	PUNCT
ejpam-6849	523	17	.	.	PUNCT
ejpam-6849	524	1	(	(	PUNCT
ejpam-6849	524	2	ii	ii	NOUN
ejpam-6849	524	3	)	)	PUNCT
ejpam-6849	524	4	if	if	SCONJ
ejpam-6849	524	5	γve(h	γve(h	PROPN
ejpam-6849	524	6	)	)	PUNCT
ejpam-6849	524	7	≥	≥	NOUN
ejpam-6849	524	8	2	2	NUM
ejpam-6849	524	9	,	,	PUNCT
ejpam-6849	524	10	then	then	ADV
ejpam-6849	524	11	γve(g[h	γve(g[h	NUM
ejpam-6849	524	12	]	]	PUNCT
ejpam-6849	524	13	)	)	PUNCT
ejpam-6849	524	14	=	=	SYM
ejpam-6849	524	15	γt(g	γt(g	NUM
ejpam-6849	524	16	)	)	PUNCT
ejpam-6849	524	17	.	.	PUNCT
ejpam-6849	525	1	proof	proof	NOUN
ejpam-6849	525	2	.	.	PUNCT
ejpam-6849	526	1	let	let	VERB
ejpam-6849	526	2	s	s	PRON
ejpam-6849	526	3	⊆	⊆	NUM
ejpam-6849	526	4	v	v	NOUN
ejpam-6849	526	5	(	(	PUNCT
ejpam-6849	526	6	g	g	NOUN
ejpam-6849	526	7	)	)	PUNCT
ejpam-6849	526	8	be	be	AUX
ejpam-6849	526	9	a	a	DET
ejpam-6849	526	10	γ	γ	NOUN
ejpam-6849	526	11	-	-	PUNCT
ejpam-6849	526	12	set	set	VERB
ejpam-6849	526	13	and	and	CCONJ
ejpam-6849	526	14	v	v	ADP
ejpam-6849	526	15	∈	∈	PROPN
ejpam-6849	526	16	v	v	NOUN
ejpam-6849	526	17	(	(	PUNCT
ejpam-6849	526	18	h	h	NOUN
ejpam-6849	526	19	)	)	PUNCT
ejpam-6849	526	20	for	for	ADP
ejpam-6849	526	21	which	which	PRON
ejpam-6849	526	22	{	{	PUNCT
ejpam-6849	526	23	v	v	NOUN
ejpam-6849	526	24	}	}	PUNCT
ejpam-6849	526	25	is	be	AUX
ejpam-6849	526	26	a	a	DET
ejpam-6849	526	27	ve	ve	NOUN
ejpam-6849	526	28	-	-	PUNCT
ejpam-6849	526	29	dominating	dominate	VERB
ejpam-6849	526	30	set	set	NOUN
ejpam-6849	526	31	of	of	ADP
ejpam-6849	526	32	h.	h.	PROPN
ejpam-6849	526	33	put	put	VERB
ejpam-6849	526	34	tx	tx	PROPN
ejpam-6849	526	35	=	=	SYM
ejpam-6849	526	36	{	{	PUNCT
ejpam-6849	526	37	v	v	NOUN
ejpam-6849	526	38	}	}	PUNCT
ejpam-6849	526	39	for	for	ADP
ejpam-6849	526	40	all	all	DET
ejpam-6849	526	41	x	x	SYM
ejpam-6849	526	42	∈	∈	PROPN
ejpam-6849	526	43	s.	s.	PROPN
ejpam-6849	526	44	by	by	ADP
ejpam-6849	526	45	proposition	proposition	NOUN
ejpam-6849	526	46	12	12	NUM
ejpam-6849	526	47	,	,	PUNCT
ejpam-6849	526	48	s	s	PART
ejpam-6849	526	49	×	×	NOUN
ejpam-6849	526	50	{	{	PUNCT
ejpam-6849	526	51	v	v	NOUN
ejpam-6849	526	52	}	}	PUNCT
ejpam-6849	526	53	=	=	SYM
ejpam-6849	526	54	∪x∈s	∪x∈s	PROPN
ejpam-6849	526	55	(	(	PUNCT
ejpam-6849	526	56	{	{	PUNCT
ejpam-6849	526	57	x	x	NOUN
ejpam-6849	526	58	}	}	PUNCT
ejpam-6849	526	59	×	×	PROPN
ejpam-6849	526	60	tx	tx	PROPN
ejpam-6849	526	61	)	)	PUNCT
ejpam-6849	526	62	is	be	AUX
ejpam-6849	526	63	a	a	DET
ejpam-6849	526	64	ve	ve	NOUN
ejpam-6849	526	65	-	-	PUNCT
ejpam-6849	526	66	dominating	dominate	VERB
ejpam-6849	526	67	set	set	NOUN
ejpam-6849	526	68	of	of	ADP
ejpam-6849	526	69	g[h	g[h	PROPN
ejpam-6849	526	70	]	]	X
ejpam-6849	526	71	yielding	yield	VERB
ejpam-6849	526	72	γve(g[h	γve(g[h	NUM
ejpam-6849	526	73	]	]	PUNCT
ejpam-6849	526	74	)	)	PUNCT
ejpam-6849	526	75	≤	≤	NUM
ejpam-6849	526	76	|s|	|s|	PROPN
ejpam-6849	526	77	=	=	SYM
ejpam-6849	526	78	γ(g	γ(g	PROPN
ejpam-6849	526	79	)	)	PUNCT
ejpam-6849	526	80	.	.	PUNCT
ejpam-6849	527	1	for	for	ADP
ejpam-6849	527	2	the	the	DET
ejpam-6849	527	3	other	other	ADJ
ejpam-6849	527	4	inequality	inequality	NOUN
ejpam-6849	527	5	,	,	PUNCT
ejpam-6849	527	6	note	note	NOUN
ejpam-6849	527	7	from	from	ADP
ejpam-6849	527	8	proposition	proposition	NOUN
ejpam-6849	527	9	12	12	NUM
ejpam-6849	527	10	that	that	SCONJ
ejpam-6849	527	11	every	every	DET
ejpam-6849	527	12	ve	ve	NOUN
ejpam-6849	527	13	-	-	PUNCT
ejpam-6849	527	14	dominating	dominate	VERB
ejpam-6849	527	15	set	set	NOUN
ejpam-6849	527	16	of	of	ADP
ejpam-6849	527	17	g[h	g[h	PROPN
ejpam-6849	527	18	]	]	PUNCT
ejpam-6849	527	19	is	be	AUX
ejpam-6849	527	20	of	of	ADP
ejpam-6849	527	21	the	the	DET
ejpam-6849	527	22	form	form	NOUN
ejpam-6849	527	23	c	c	NOUN
ejpam-6849	527	24	=	=	SYM
ejpam-6849	527	25	∪x∈s	∪x∈s	PROPN
ejpam-6849	527	26	(	(	PUNCT
ejpam-6849	527	27	{	{	PUNCT
ejpam-6849	527	28	x	x	NOUN
ejpam-6849	527	29	}	}	PUNCT
ejpam-6849	527	30	×	×	PROPN
ejpam-6849	527	31	tx	tx	PROPN
ejpam-6849	527	32	)	)	PUNCT
ejpam-6849	527	33	,	,	PUNCT
ejpam-6849	527	34	where	where	SCONJ
ejpam-6849	527	35	s	s	VERB
ejpam-6849	527	36	⊆	⊆	NUM
ejpam-6849	527	37	v	v	NOUN
ejpam-6849	527	38	(	(	PUNCT
ejpam-6849	527	39	g	g	NOUN
ejpam-6849	527	40	)	)	PUNCT
ejpam-6849	527	41	is	be	AUX
ejpam-6849	527	42	a	a	DET
ejpam-6849	527	43	dominating	dominating	NOUN
ejpam-6849	527	44	set	set	NOUN
ejpam-6849	527	45	of	of	ADP
ejpam-6849	527	46	g	g	PROPN
ejpam-6849	527	47	and	and	CCONJ
ejpam-6849	527	48	tx	tx	VERB
ejpam-6849	527	49	⊆	⊆	NUM
ejpam-6849	527	50	v	v	NOUN
ejpam-6849	527	51	(	(	PUNCT
ejpam-6849	527	52	h	h	NOUN
ejpam-6849	527	53	)	)	PUNCT
ejpam-6849	527	54	is	be	AUX
ejpam-6849	527	55	a	a	DET
ejpam-6849	527	56	ve	ve	NOUN
ejpam-6849	527	57	-	-	PUNCT
ejpam-6849	527	58	dominating	dominate	VERB
ejpam-6849	527	59	set	set	NOUN
ejpam-6849	527	60	of	of	ADP
ejpam-6849	527	61	h	h	NOUN
ejpam-6849	527	62	for	for	ADP
ejpam-6849	527	63	each	each	DET
ejpam-6849	527	64	x	x	SYM
ejpam-6849	527	65	∈	∈	PROPN
ejpam-6849	527	66	s	s	PART
ejpam-6849	527	67	\	\	NOUN
ejpam-6849	527	68	ng(s	ng(s	NUM
ejpam-6849	527	69	)	)	PUNCT
ejpam-6849	527	70	.	.	PUNCT
ejpam-6849	528	1	for	for	ADP
ejpam-6849	528	2	any	any	DET
ejpam-6849	528	3	such	such	ADJ
ejpam-6849	528	4	c	c	NOUN
ejpam-6849	528	5	,	,	PUNCT
ejpam-6849	528	6	we	we	PRON
ejpam-6849	528	7	have	have	VERB
ejpam-6849	528	8	|c|	|c|	PROPN
ejpam-6849	528	9	=	=	PUNCT
ejpam-6849	528	10	∑	∑	PROPN
ejpam-6849	528	11	x∈s	x∈s	PROPN
ejpam-6849	528	12	|tx|	|tx|	PROPN
ejpam-6849	528	13	≥	≥	NUM
ejpam-6849	528	14	|s|	|s|	PROPN
ejpam-6849	528	15	=	=	SYM
ejpam-6849	528	16	γ(g	γ(g	PROPN
ejpam-6849	528	17	)	)	PUNCT
ejpam-6849	528	18	.	.	PUNCT
ejpam-6849	529	1	therefore	therefore	ADV
ejpam-6849	529	2	,	,	PUNCT
ejpam-6849	529	3	γve(g[h	γve(g[h	NUM
ejpam-6849	529	4	]	]	PUNCT
ejpam-6849	529	5	)	)	PUNCT
ejpam-6849	529	6	≥	≥	PROPN
ejpam-6849	529	7	γ(g	γ(g	PROPN
ejpam-6849	529	8	)	)	PUNCT
ejpam-6849	529	9	.	.	PUNCT
ejpam-6849	530	1	this	this	PRON
ejpam-6849	530	2	proves	prove	VERB
ejpam-6849	530	3	(	(	PUNCT
ejpam-6849	530	4	i	i	NOUN
ejpam-6849	530	5	)	)	PUNCT
ejpam-6849	530	6	.	.	PUNCT
ejpam-6849	531	1	now	now	ADV
ejpam-6849	531	2	we	we	PRON
ejpam-6849	531	3	prove	prove	VERB
ejpam-6849	531	4	(	(	PUNCT
ejpam-6849	531	5	ii	ii	NOUN
ejpam-6849	531	6	)	)	PUNCT
ejpam-6849	531	7	using	use	VERB
ejpam-6849	531	8	proposition	proposition	NOUN
ejpam-6849	531	9	12	12	NUM
ejpam-6849	531	10	.	.	PUNCT
ejpam-6849	532	1	for	for	ADP
ejpam-6849	532	2	all	all	DET
ejpam-6849	532	3	total	total	ADJ
ejpam-6849	532	4	dominating	dominating	NOUN
ejpam-6849	532	5	sets	set	NOUN
ejpam-6849	532	6	s	s	NOUN
ejpam-6849	532	7	of	of	ADP
ejpam-6849	532	8	g	g	NOUN
ejpam-6849	532	9	and	and	CCONJ
ejpam-6849	532	10	v	v	ADP
ejpam-6849	532	11	∈	∈	PROPN
ejpam-6849	532	12	v	v	NOUN
ejpam-6849	532	13	(	(	PUNCT
ejpam-6849	532	14	h	h	NOUN
ejpam-6849	532	15	)	)	PUNCT
ejpam-6849	532	16	,	,	PUNCT
ejpam-6849	532	17	since	since	SCONJ
ejpam-6849	532	18	s	s	NOUN
ejpam-6849	532	19	\	\	NOUN
ejpam-6849	532	20	ng(s	ng(s	NUM
ejpam-6849	532	21	)	)	PUNCT
ejpam-6849	532	22	=	=	NOUN
ejpam-6849	532	23	∅	∅	NOUN
ejpam-6849	532	24	,	,	PUNCT
ejpam-6849	532	25	c	c	PROPN
ejpam-6849	532	26	=	=	SYM
ejpam-6849	532	27	s	s	PROPN
ejpam-6849	532	28	×	×	NOUN
ejpam-6849	532	29	{	{	PUNCT
ejpam-6849	532	30	v	v	NOUN
ejpam-6849	532	31	}	}	PUNCT
ejpam-6849	532	32	=	=	SYM
ejpam-6849	532	33	∪x∈s	∪x∈s	PROPN
ejpam-6849	532	34	(	(	PUNCT
ejpam-6849	532	35	{	{	PUNCT
ejpam-6849	532	36	x	x	NOUN
ejpam-6849	532	37	}	}	PUNCT
ejpam-6849	532	38	×	×	PROPN
ejpam-6849	532	39	{	{	PUNCT
ejpam-6849	532	40	v	v	NOUN
ejpam-6849	532	41	}	}	PUNCT
ejpam-6849	532	42	)	)	PUNCT
ejpam-6849	532	43	is	be	AUX
ejpam-6849	532	44	a	a	DET
ejpam-6849	532	45	ve	ve	NOUN
ejpam-6849	532	46	-	-	PUNCT
ejpam-6849	532	47	dominating	dominate	VERB
ejpam-6849	532	48	set	set	NOUN
ejpam-6849	532	49	of	of	ADP
ejpam-6849	532	50	g[h	g[h	NOUN
ejpam-6849	532	51	]	]	PUNCT
ejpam-6849	532	52	so	so	SCONJ
ejpam-6849	532	53	that	that	SCONJ
ejpam-6849	532	54	γve(g[h	γve(g[h	NUM
ejpam-6849	532	55	]	]	PUNCT
ejpam-6849	532	56	)	)	PUNCT
ejpam-6849	532	57	≤	≤	NUM
ejpam-6849	532	58	|s|	|s|	PROPN
ejpam-6849	532	59	=	=	PUNCT
ejpam-6849	532	60	γt(g	γt(g	NUM
ejpam-6849	532	61	)	)	PUNCT
ejpam-6849	532	62	.	.	PUNCT
ejpam-6849	533	1	now	now	ADV
ejpam-6849	533	2	,	,	PUNCT
ejpam-6849	533	3	let	let	VERB
ejpam-6849	533	4	c	c	NOUN
ejpam-6849	533	5	=	=	SYM
ejpam-6849	533	6	∪x∈s	∪x∈s	PROPN
ejpam-6849	533	7	(	(	PUNCT
ejpam-6849	533	8	{	{	PUNCT
ejpam-6849	533	9	x	x	NOUN
ejpam-6849	533	10	}	}	PUNCT
ejpam-6849	533	11	×	×	PROPN
ejpam-6849	533	12	tx	tx	PROPN
ejpam-6849	533	13	)	)	PUNCT
ejpam-6849	533	14	,	,	PUNCT
ejpam-6849	533	15	where	where	SCONJ
ejpam-6849	533	16	s	s	VERB
ejpam-6849	533	17	⊆	⊆	NUM
ejpam-6849	533	18	v	v	NOUN
ejpam-6849	533	19	(	(	PUNCT
ejpam-6849	533	20	g	g	NOUN
ejpam-6849	533	21	)	)	PUNCT
ejpam-6849	533	22	and	and	CCONJ
ejpam-6849	533	23	tx	tx	VERB
ejpam-6849	533	24	⊆	⊆	NUM
ejpam-6849	533	25	v	v	NOUN
ejpam-6849	533	26	(	(	PUNCT
ejpam-6849	533	27	h	h	NOUN
ejpam-6849	533	28	)	)	PUNCT
ejpam-6849	533	29	,	,	PUNCT
ejpam-6849	533	30	be	be	AUX
ejpam-6849	533	31	a	a	DET
ejpam-6849	533	32	ve	ve	NOUN
ejpam-6849	533	33	-	-	PUNCT
ejpam-6849	533	34	dominating	dominate	VERB
ejpam-6849	533	35	set	set	NOUN
ejpam-6849	533	36	of	of	ADP
ejpam-6849	533	37	g[h	g[h	NOUN
ejpam-6849	533	38	]	]	PUNCT
ejpam-6849	533	39	.	.	PUNCT
ejpam-6849	534	1	necessarily	necessarily	ADV
ejpam-6849	534	2	,	,	PUNCT
ejpam-6849	534	3	s	s	VERB
ejpam-6849	534	4	is	be	AUX
ejpam-6849	534	5	a	a	DET
ejpam-6849	534	6	dominating	dominating	NOUN
ejpam-6849	534	7	set	set	NOUN
ejpam-6849	534	8	of	of	ADP
ejpam-6849	534	9	g	g	PROPN
ejpam-6849	534	10	and	and	CCONJ
ejpam-6849	534	11	tx	tx	PROPN
ejpam-6849	534	12	is	be	AUX
ejpam-6849	534	13	a	a	DET
ejpam-6849	534	14	ve	ve	NOUN
ejpam-6849	534	15	-	-	PUNCT
ejpam-6849	534	16	dominating	dominate	VERB
ejpam-6849	534	17	set	set	NOUN
ejpam-6849	534	18	of	of	ADP
ejpam-6849	534	19	h	h	NOUN
ejpam-6849	534	20	for	for	ADP
ejpam-6849	534	21	each	each	DET
ejpam-6849	534	22	x	x	SYM
ejpam-6849	534	23	∈	∈	PROPN
ejpam-6849	534	24	s	s	PART
ejpam-6849	534	25	\	\	NOUN
ejpam-6849	534	26	ng(s	ng(s	NUM
ejpam-6849	534	27	)	)	PUNCT
ejpam-6849	534	28	.	.	PUNCT
ejpam-6849	535	1	by	by	ADP
ejpam-6849	535	2	lemma	lemma	PROPN
ejpam-6849	535	3	1	1	NUM
ejpam-6849	535	4	,	,	PUNCT
ejpam-6849	535	5	|c|	|c|	PROPN
ejpam-6849	535	6	=	=	PUNCT
ejpam-6849	535	7	∑	∑	PROPN
ejpam-6849	535	8	x∈s	x∈s	PROPN
ejpam-6849	535	9	|tx|	|tx|	PROPN
ejpam-6849	535	10	≥	≥	NUM
ejpam-6849	535	11	2|s|	2|s|	NUM
ejpam-6849	535	12	≥	≥	NOUN
ejpam-6849	535	13	2γ(g	2γ(g	NUM
ejpam-6849	535	14	)	)	PUNCT
ejpam-6849	535	15	≥	≥	NOUN
ejpam-6849	535	16	γt(g	γt(g	NUM
ejpam-6849	535	17	)	)	PUNCT
ejpam-6849	535	18	.	.	PUNCT
ejpam-6849	536	1	since	since	SCONJ
ejpam-6849	536	2	c	c	PROPN
ejpam-6849	536	3	is	be	AUX
ejpam-6849	536	4	arbitrary	arbitrary	ADJ
ejpam-6849	536	5	,	,	PUNCT
ejpam-6849	536	6	γve(g[h	γve(g[h	NUM
ejpam-6849	536	7	]	]	PUNCT
ejpam-6849	536	8	)	)	PUNCT
ejpam-6849	536	9	≥	≥	NOUN
ejpam-6849	536	10	γt(g	γt(g	NUM
ejpam-6849	536	11	)	)	PUNCT
ejpam-6849	536	12	.	.	PUNCT
ejpam-6849	537	1	■	■	PUNCT
ejpam-6849	537	2	the	the	DET
ejpam-6849	537	3	following	follow	VERB
ejpam-6849	537	4	immediately	immediately	ADV
ejpam-6849	537	5	follows	follow	VERB
ejpam-6849	537	6	from	from	ADP
ejpam-6849	537	7	proposition	proposition	NOUN
ejpam-6849	537	8	12	12	NUM
ejpam-6849	537	9	j.	j.	PROPN
ejpam-6849	537	10	tayab	tayab	PROPN
ejpam-6849	537	11	,	,	PUNCT
ejpam-6849	537	12	f.	f.	PROPN
ejpam-6849	537	13	p.	p.	PROPN
ejpam-6849	537	14	jamil	jamil	PROPN
ejpam-6849	537	15	,	,	PUNCT
ejpam-6849	537	16	i.	i.	PROPN
ejpam-6849	537	17	aniversario	aniversario	PROPN
ejpam-6849	537	18	/	/	SYM
ejpam-6849	537	19	eur	eur	PROPN
ejpam-6849	537	20	.	.	PUNCT
ejpam-6849	538	1	j.	j.	PROPN
ejpam-6849	538	2	pure	pure	PROPN
ejpam-6849	538	3	appl	appl	PROPN
ejpam-6849	538	4	.	.	PROPN
ejpam-6849	538	5	math	math	PROPN
ejpam-6849	538	6	,	,	PUNCT
ejpam-6849	538	7	18	18	NUM
ejpam-6849	538	8	(	(	PUNCT
ejpam-6849	538	9	4	4	NUM
ejpam-6849	538	10	)	)	PUNCT
ejpam-6849	538	11	(	(	PUNCT
ejpam-6849	538	12	2025	2025	NUM
ejpam-6849	538	13	)	)	PUNCT
ejpam-6849	538	14	,	,	PUNCT
ejpam-6849	538	15	6849	6849	NUM
ejpam-6849	538	16	14	14	NUM
ejpam-6849	538	17	of	of	ADP
ejpam-6849	538	18	19	19	NUM
ejpam-6849	538	19	proposition	proposition	NOUN
ejpam-6849	538	20	13	13	NUM
ejpam-6849	538	21	.	.	PUNCT
ejpam-6849	539	1	let	let	VERB
ejpam-6849	539	2	g	g	NOUN
ejpam-6849	539	3	and	and	CCONJ
ejpam-6849	539	4	h	h	NOUN
ejpam-6849	539	5	be	be	AUX
ejpam-6849	539	6	connected	connect	VERB
ejpam-6849	539	7	nontrivial	nontrivial	ADJ
ejpam-6849	539	8	graphs	graph	NOUN
ejpam-6849	539	9	,	,	PUNCT
ejpam-6849	539	10	and	and	CCONJ
ejpam-6849	539	11	let	let	VERB
ejpam-6849	539	12	c	c	PROPN
ejpam-6849	539	13	⊆	⊆	NUM
ejpam-6849	539	14	v	v	NOUN
ejpam-6849	539	15	(	(	PUNCT
ejpam-6849	539	16	g[h	g[h	PROPN
ejpam-6849	539	17	]	]	PUNCT
ejpam-6849	539	18	)	)	PUNCT
ejpam-6849	539	19	.	.	PUNCT
ejpam-6849	540	1	then	then	ADV
ejpam-6849	540	2	c	c	PROPN
ejpam-6849	540	3	is	be	AUX
ejpam-6849	540	4	a	a	DET
ejpam-6849	540	5	total	total	ADJ
ejpam-6849	540	6	ve	ve	NOUN
ejpam-6849	540	7	-	-	PUNCT
ejpam-6849	540	8	dominating	dominate	VERB
ejpam-6849	540	9	set	set	NOUN
ejpam-6849	540	10	of	of	ADP
ejpam-6849	540	11	g[h	g[h	NOUN
ejpam-6849	540	12	]	]	PUNCT
ejpam-6849	540	13	if	if	SCONJ
ejpam-6849	540	14	and	and	CCONJ
ejpam-6849	540	15	only	only	ADV
ejpam-6849	540	16	if	if	SCONJ
ejpam-6849	540	17	c	c	PROPN
ejpam-6849	540	18	=	=	SYM
ejpam-6849	540	19	∪x∈s	∪x∈s	PROPN
ejpam-6849	540	20	(	(	PUNCT
ejpam-6849	540	21	{	{	PUNCT
ejpam-6849	540	22	x	x	NOUN
ejpam-6849	540	23	}	}	PUNCT
ejpam-6849	540	24	×	×	PROPN
ejpam-6849	540	25	tx	tx	PROPN
ejpam-6849	540	26	)	)	PUNCT
ejpam-6849	540	27	,	,	PUNCT
ejpam-6849	540	28	(	(	PUNCT
ejpam-6849	540	29	7	7	X
ejpam-6849	540	30	)	)	PUNCT
ejpam-6849	540	31	where	where	SCONJ
ejpam-6849	540	32	s	s	VERB
ejpam-6849	540	33	⊆	⊆	NUM
ejpam-6849	540	34	v	v	NOUN
ejpam-6849	540	35	(	(	PUNCT
ejpam-6849	540	36	g	g	NOUN
ejpam-6849	540	37	)	)	PUNCT
ejpam-6849	540	38	and	and	CCONJ
ejpam-6849	540	39	tx	tx	VERB
ejpam-6849	540	40	⊆	⊆	NUM
ejpam-6849	540	41	v	v	NOUN
ejpam-6849	540	42	(	(	PUNCT
ejpam-6849	540	43	h	h	NOUN
ejpam-6849	540	44	)	)	PUNCT
ejpam-6849	540	45	for	for	SCONJ
ejpam-6849	540	46	each	each	DET
ejpam-6849	540	47	x	x	SYM
ejpam-6849	540	48	∈	∈	PROPN
ejpam-6849	540	49	s	s	VERB
ejpam-6849	540	50	such	such	ADJ
ejpam-6849	540	51	that	that	DET
ejpam-6849	540	52	one	one	NUM
ejpam-6849	540	53	of	of	ADP
ejpam-6849	540	54	the	the	DET
ejpam-6849	540	55	following	follow	VERB
ejpam-6849	540	56	holds	hold	VERB
ejpam-6849	540	57	:	:	PUNCT
ejpam-6849	540	58	(	(	PUNCT
ejpam-6849	540	59	i	i	NOUN
ejpam-6849	540	60	)	)	PUNCT
ejpam-6849	540	61	s	s	AUX
ejpam-6849	540	62	is	be	AUX
ejpam-6849	540	63	a	a	DET
ejpam-6849	540	64	total	total	ADJ
ejpam-6849	540	65	dominating	dominating	NOUN
ejpam-6849	540	66	set	set	NOUN
ejpam-6849	540	67	of	of	ADP
ejpam-6849	540	68	g	g	NOUN
ejpam-6849	540	69	;	;	PUNCT
ejpam-6849	540	70	(	(	PUNCT
ejpam-6849	540	71	ii	ii	NOUN
ejpam-6849	540	72	)	)	PUNCT
ejpam-6849	540	73	each	each	PRON
ejpam-6849	540	74	of	of	ADP
ejpam-6849	540	75	the	the	DET
ejpam-6849	540	76	following	follow	VERB
ejpam-6849	540	77	holds	hold	VERB
ejpam-6849	540	78	:	:	PUNCT
ejpam-6849	540	79	(	(	PUNCT
ejpam-6849	540	80	a	a	X
ejpam-6849	540	81	)	)	PUNCT
ejpam-6849	540	82	s	s	VERB
ejpam-6849	540	83	is	be	AUX
ejpam-6849	540	84	a	a	DET
ejpam-6849	540	85	dominating	dominating	NOUN
ejpam-6849	540	86	set	set	NOUN
ejpam-6849	540	87	of	of	ADP
ejpam-6849	540	88	g	g	NOUN
ejpam-6849	540	89	;	;	PUNCT
ejpam-6849	540	90	and	and	CCONJ
ejpam-6849	540	91	(	(	PUNCT
ejpam-6849	540	92	b	b	X
ejpam-6849	540	93	)	)	PUNCT
ejpam-6849	540	94	tx	tx	PROPN
ejpam-6849	540	95	is	be	AUX
ejpam-6849	540	96	a	a	DET
ejpam-6849	540	97	total	total	ADJ
ejpam-6849	540	98	ve	ve	NOUN
ejpam-6849	540	99	-	-	PUNCT
ejpam-6849	540	100	dominating	dominate	VERB
ejpam-6849	540	101	set	set	NOUN
ejpam-6849	540	102	of	of	ADP
ejpam-6849	540	103	h	h	NOUN
ejpam-6849	540	104	for	for	ADP
ejpam-6849	540	105	each	each	DET
ejpam-6849	540	106	x	x	SYM
ejpam-6849	540	107	∈	∈	PROPN
ejpam-6849	540	108	s	s	PART
ejpam-6849	540	109	\	\	NOUN
ejpam-6849	540	110	ng(s	ng(s	NUM
ejpam-6849	540	111	)	)	PUNCT
ejpam-6849	540	112	.	.	PUNCT
ejpam-6849	541	1	corollary	corollary	ADJ
ejpam-6849	541	2	5	5	NUM
ejpam-6849	541	3	.	.	PUNCT
ejpam-6849	542	1	if	if	SCONJ
ejpam-6849	542	2	g	g	PROPN
ejpam-6849	542	3	and	and	CCONJ
ejpam-6849	542	4	h	h	NOUN
ejpam-6849	542	5	are	be	AUX
ejpam-6849	542	6	connected	connect	VERB
ejpam-6849	542	7	nontrivial	nontrivial	ADJ
ejpam-6849	542	8	graphs	graph	NOUN
ejpam-6849	542	9	,	,	PUNCT
ejpam-6849	542	10	then	then	ADV
ejpam-6849	542	11	γt	γt	ADP
ejpam-6849	542	12	ve(g[h	ve(g[h	NOUN
ejpam-6849	542	13	]	]	X
ejpam-6849	542	14	)	)	PUNCT
ejpam-6849	542	15	=	=	SYM
ejpam-6849	542	16	γt(g	γt(g	NUM
ejpam-6849	542	17	)	)	PUNCT
ejpam-6849	542	18	.	.	PUNCT
ejpam-6849	543	1	proof	proof	NOUN
ejpam-6849	543	2	.	.	PUNCT
ejpam-6849	544	1	by	by	ADP
ejpam-6849	544	2	proposition	proposition	NOUN
ejpam-6849	544	3	13	13	NUM
ejpam-6849	544	4	,	,	PUNCT
ejpam-6849	544	5	s	s	AUX
ejpam-6849	544	6	×	×	PROPN
ejpam-6849	544	7	{	{	PUNCT
ejpam-6849	544	8	y	y	NOUN
ejpam-6849	544	9	}	}	PUNCT
ejpam-6849	544	10	is	be	AUX
ejpam-6849	544	11	a	a	DET
ejpam-6849	544	12	total	total	ADJ
ejpam-6849	544	13	ve	ve	NOUN
ejpam-6849	544	14	-	-	PUNCT
ejpam-6849	544	15	dominating	dominate	VERB
ejpam-6849	544	16	set	set	NOUN
ejpam-6849	544	17	of	of	ADP
ejpam-6849	544	18	g[h	g[h	PROPN
ejpam-6849	544	19	]	]	PUNCT
ejpam-6849	544	20	for	for	ADP
ejpam-6849	544	21	all	all	DET
ejpam-6849	544	22	total	total	ADJ
ejpam-6849	544	23	dominating	dominating	NOUN
ejpam-6849	544	24	sets	set	NOUN
ejpam-6849	544	25	s	s	PART
ejpam-6849	544	26	⊆	⊆	NUM
ejpam-6849	544	27	v	v	NOUN
ejpam-6849	544	28	(	(	PUNCT
ejpam-6849	544	29	g	g	NOUN
ejpam-6849	544	30	)	)	PUNCT
ejpam-6849	544	31	of	of	ADP
ejpam-6849	544	32	g	g	PROPN
ejpam-6849	544	33	and	and	CCONJ
ejpam-6849	544	34	all	all	DET
ejpam-6849	544	35	y	y	PROPN
ejpam-6849	544	36	∈	∈	PROPN
ejpam-6849	544	37	v	v	ADP
ejpam-6849	544	38	(	(	PUNCT
ejpam-6849	544	39	h	h	NOUN
ejpam-6849	544	40	)	)	PUNCT
ejpam-6849	544	41	.	.	PUNCT
ejpam-6849	545	1	therefore	therefore	ADV
ejpam-6849	545	2	,	,	PUNCT
ejpam-6849	545	3	γt	γt	NOUN
ejpam-6849	545	4	ve(g[h	ve(g[h	NOUN
ejpam-6849	545	5	]	]	X
ejpam-6849	545	6	)	)	PUNCT
ejpam-6849	545	7	≤	≤	NOUN
ejpam-6849	545	8	γt(g	γt(g	PUNCT
ejpam-6849	545	9	)	)	PUNCT
ejpam-6849	545	10	.	.	PUNCT
ejpam-6849	546	1	now	now	ADV
ejpam-6849	546	2	,	,	PUNCT
ejpam-6849	546	3	let	let	VERB
ejpam-6849	546	4	c	c	NOUN
ejpam-6849	546	5	=	=	SYM
ejpam-6849	546	6	∪x∈s	∪x∈s	PROPN
ejpam-6849	546	7	(	(	PUNCT
ejpam-6849	546	8	{	{	PUNCT
ejpam-6849	546	9	x	x	NOUN
ejpam-6849	546	10	}	}	PUNCT
ejpam-6849	546	11	×	×	PROPN
ejpam-6849	546	12	tx	tx	PROPN
ejpam-6849	546	13	)	)	PUNCT
ejpam-6849	546	14	⊆	⊆	NUM
ejpam-6849	546	15	v	v	NOUN
ejpam-6849	546	16	(	(	PUNCT
ejpam-6849	546	17	g[h	g[h	PROPN
ejpam-6849	546	18	]	]	PUNCT
ejpam-6849	546	19	)	)	PUNCT
ejpam-6849	546	20	be	be	VERB
ejpam-6849	546	21	a	a	DET
ejpam-6849	546	22	γt	γt	NOUN
ejpam-6849	546	23	ve	ve	NOUN
ejpam-6849	546	24	-	-	PUNCT
ejpam-6849	546	25	set	set	NOUN
ejpam-6849	546	26	of	of	ADP
ejpam-6849	546	27	g[h	g[h	NOUN
ejpam-6849	546	28	]	]	PUNCT
ejpam-6849	546	29	.	.	PUNCT
ejpam-6849	547	1	if	if	SCONJ
ejpam-6849	547	2	s	s	PROPN
ejpam-6849	547	3	is	be	AUX
ejpam-6849	547	4	a	a	DET
ejpam-6849	547	5	total	total	ADJ
ejpam-6849	547	6	dominating	dominating	NOUN
ejpam-6849	547	7	set	set	NOUN
ejpam-6849	547	8	of	of	ADP
ejpam-6849	547	9	g	g	NOUN
ejpam-6849	547	10	,	,	PUNCT
ejpam-6849	547	11	then	then	ADV
ejpam-6849	547	12	|c|	|c|	PROPN
ejpam-6849	547	13	≥	≥	NUM
ejpam-6849	547	14	|s|	|s|	VERB
ejpam-6849	547	15	≥	≥	NOUN
ejpam-6849	547	16	γt(g	γt(g	NUM
ejpam-6849	547	17	)	)	PUNCT
ejpam-6849	547	18	.	.	PUNCT
ejpam-6849	548	1	otherwise	otherwise	ADV
ejpam-6849	548	2	,	,	PUNCT
ejpam-6849	548	3	lemma	lemma	PROPN
ejpam-6849	548	4	1	1	NUM
ejpam-6849	548	5	and	and	CCONJ
ejpam-6849	548	6	proposition	proposition	NOUN
ejpam-6849	548	7	13	13	NUM
ejpam-6849	548	8	imply	imply	VERB
ejpam-6849	548	9	that	that	DET
ejpam-6849	548	10	|c|	|c|	PROPN
ejpam-6849	548	11	=	=	SYM
ejpam-6849	548	12	∑	∑	NOUN
ejpam-6849	548	13	x∈s∩ng(s	x∈s∩ng(s	NOUN
ejpam-6849	548	14	)	)	PUNCT
ejpam-6849	548	15	|tx|	|tx|	NOUN
ejpam-6849	548	16	+	+	CCONJ
ejpam-6849	548	17	∑	∑	PUNCT
ejpam-6849	548	18	x∈s\ng(s	x∈s\ng(s	NUM
ejpam-6849	548	19	)	)	PUNCT
ejpam-6849	548	20	|tx|	|tx|	PROPN
ejpam-6849	548	21	≥	≥	PROPN
ejpam-6849	548	22	|s	|s	PROPN
ejpam-6849	548	23	∩	∩	NOUN
ejpam-6849	548	24	ng(s)|	ng(s)|	PUNCT
ejpam-6849	549	1	+	+	NUM
ejpam-6849	549	2	2|s	2|s	NUM
ejpam-6849	549	3	\	\	NOUN
ejpam-6849	549	4	ng(s)|	ng(s)|	X
ejpam-6849	549	5	≥	≥	X
ejpam-6849	549	6	γt(g	γt(g	NUM
ejpam-6849	549	7	)	)	PUNCT
ejpam-6849	549	8	.	.	PUNCT
ejpam-6849	550	1	in	in	ADP
ejpam-6849	550	2	any	any	DET
ejpam-6849	550	3	case	case	NOUN
ejpam-6849	550	4	,	,	PUNCT
ejpam-6849	550	5	γt	γt	NOUN
ejpam-6849	550	6	ve(g[h	ve(g[h	NOUN
ejpam-6849	550	7	]	]	PUNCT
ejpam-6849	550	8	)	)	PUNCT
ejpam-6849	550	9	≥	≥	NOUN
ejpam-6849	550	10	γt(g	γt(g	NUM
ejpam-6849	550	11	)	)	PUNCT
ejpam-6849	550	12	.	.	PUNCT
ejpam-6849	551	1	■	■	PUNCT
ejpam-6849	551	2	for	for	ADP
ejpam-6849	551	3	each	each	DET
ejpam-6849	551	4	uv	uv	PROPN
ejpam-6849	551	5	∈	∈	PROPN
ejpam-6849	551	6	e(g	e(g	PROPN
ejpam-6849	551	7	)	)	PUNCT
ejpam-6849	551	8	,	,	PUNCT
ejpam-6849	551	9	huv	huv	PROPN
ejpam-6849	551	10	denotes	denote	VERB
ejpam-6849	551	11	that	that	PRON
ejpam-6849	551	12	copy	copy	VERB
ejpam-6849	551	13	of	of	ADP
ejpam-6849	551	14	h	h	NOUN
ejpam-6849	551	15	being	be	AUX
ejpam-6849	551	16	joined	join	VERB
ejpam-6849	551	17	to	to	ADP
ejpam-6849	551	18	g	g	PROPN
ejpam-6849	551	19	through	through	ADP
ejpam-6849	551	20	edge	edge	NOUN
ejpam-6849	551	21	uv	uv	NOUN
ejpam-6849	551	22	.	.	PUNCT
ejpam-6849	552	1	we	we	PRON
ejpam-6849	552	2	also	also	ADV
ejpam-6849	552	3	write	write	VERB
ejpam-6849	552	4	huv	huv	PROPN
ejpam-6849	553	1	+	+	CCONJ
ejpam-6849	553	2	uv	uv	NOUN
ejpam-6849	553	3	to	to	PART
ejpam-6849	553	4	denote	denote	VERB
ejpam-6849	553	5	that	that	DET
ejpam-6849	553	6	subgraph	subgraph	NOUN
ejpam-6849	553	7	of	of	ADP
ejpam-6849	553	8	g	g	PROPN
ejpam-6849	553	9	⋄	⋄	PROPN
ejpam-6849	553	10	h	h	NOUN
ejpam-6849	553	11	induced	induce	VERB
ejpam-6849	553	12	by	by	ADP
ejpam-6849	553	13	v	v	PROPN
ejpam-6849	553	14	(	(	PUNCT
ejpam-6849	553	15	huv	huv	PROPN
ejpam-6849	553	16	)	)	PUNCT
ejpam-6849	553	17	∪	∪	NOUN
ejpam-6849	553	18	{	{	PUNCT
ejpam-6849	553	19	u	u	NOUN
ejpam-6849	553	20	,	,	PUNCT
ejpam-6849	553	21	v	v	NOUN
ejpam-6849	553	22	}	}	PUNCT
ejpam-6849	553	23	.	.	PUNCT
ejpam-6849	554	1	proposition	proposition	NOUN
ejpam-6849	554	2	14	14	NUM
ejpam-6849	554	3	.	.	PUNCT
ejpam-6849	555	1	let	let	VERB
ejpam-6849	555	2	g	g	PRON
ejpam-6849	555	3	be	be	AUX
ejpam-6849	555	4	a	a	DET
ejpam-6849	555	5	nontrivial	nontrivial	ADJ
ejpam-6849	555	6	connected	connect	VERB
ejpam-6849	555	7	graph	graph	NOUN
ejpam-6849	555	8	and	and	CCONJ
ejpam-6849	555	9	h	h	NOUN
ejpam-6849	555	10	be	be	AUX
ejpam-6849	555	11	any	any	DET
ejpam-6849	555	12	nonempty	nonempty	ADJ
ejpam-6849	555	13	graph	graph	NOUN
ejpam-6849	555	14	.	.	PUNCT
ejpam-6849	556	1	then	then	ADV
ejpam-6849	556	2	s	s	VERB
ejpam-6849	556	3	⊆	⊆	NUM
ejpam-6849	556	4	v	v	NOUN
ejpam-6849	556	5	(	(	PUNCT
ejpam-6849	556	6	g	g	PROPN
ejpam-6849	556	7	⋄	⋄	PROPN
ejpam-6849	556	8	h	h	NOUN
ejpam-6849	556	9	)	)	PUNCT
ejpam-6849	556	10	is	be	AUX
ejpam-6849	556	11	a	a	DET
ejpam-6849	556	12	ve	ve	NOUN
ejpam-6849	556	13	-	-	PUNCT
ejpam-6849	556	14	dominating	dominate	VERB
ejpam-6849	556	15	set	set	NOUN
ejpam-6849	556	16	of	of	ADP
ejpam-6849	556	17	g	g	PROPN
ejpam-6849	556	18	⋄	⋄	PROPN
ejpam-6849	556	19	h	h	NOUN
ejpam-6849	556	20	if	if	SCONJ
ejpam-6849	557	1	and	and	CCONJ
ejpam-6849	557	2	only	only	ADV
ejpam-6849	557	3	if	if	SCONJ
ejpam-6849	557	4	s	s	VERB
ejpam-6849	557	5	=	=	NOUN
ejpam-6849	557	6	a	a	DET
ejpam-6849	557	7	∪	∪	X
ejpam-6849	557	8	(	(	PUNCT
ejpam-6849	557	9	∪uv∈e(g)suv	∪uv∈e(g)suv	PROPN
ejpam-6849	557	10	)	)	PUNCT
ejpam-6849	557	11	,	,	PUNCT
ejpam-6849	557	12	(	(	PUNCT
ejpam-6849	557	13	8)	8)	NUM
ejpam-6849	557	14	where	where	SCONJ
ejpam-6849	557	15	a	a	DET
ejpam-6849	557	16	⊆	⊆	NUM
ejpam-6849	557	17	v	v	NOUN
ejpam-6849	557	18	(	(	PUNCT
ejpam-6849	557	19	g	g	NOUN
ejpam-6849	557	20	)	)	PUNCT
ejpam-6849	557	21	and	and	CCONJ
ejpam-6849	557	22	suv	suv	PROPN
ejpam-6849	557	23	⊆	⊆	NUM
ejpam-6849	557	24	v	v	NOUN
ejpam-6849	557	25	(	(	PUNCT
ejpam-6849	557	26	huv	huv	PROPN
ejpam-6849	557	27	)	)	PUNCT
ejpam-6849	557	28	for	for	ADP
ejpam-6849	557	29	each	each	DET
ejpam-6849	557	30	uv	uv	PROPN
ejpam-6849	557	31	∈	∈	PROPN
ejpam-6849	557	32	e(g	e(g	PROPN
ejpam-6849	557	33	)	)	PUNCT
ejpam-6849	557	34	such	such	ADJ
ejpam-6849	557	35	that	that	DET
ejpam-6849	557	36	suv	suv	PROPN
ejpam-6849	557	37	is	be	AUX
ejpam-6849	557	38	a	a	DET
ejpam-6849	557	39	ve	ve	NOUN
ejpam-6849	557	40	-	-	PUNCT
ejpam-6849	557	41	dominating	dominate	VERB
ejpam-6849	557	42	set	set	NOUN
ejpam-6849	557	43	of	of	ADP
ejpam-6849	557	44	huv	huv	PROPN
ejpam-6849	557	45	whenever	whenever	SCONJ
ejpam-6849	557	46	{	{	PUNCT
ejpam-6849	557	47	u	u	NOUN
ejpam-6849	557	48	,	,	PUNCT
ejpam-6849	557	49	v	v	NOUN
ejpam-6849	557	50	}	}	PUNCT
ejpam-6849	557	51	∩	∩	NOUN
ejpam-6849	557	52	a	a	DET
ejpam-6849	557	53	=	=	PUNCT
ejpam-6849	557	54	∅.	∅.	PROPN
ejpam-6849	557	55	j.	j.	PROPN
ejpam-6849	557	56	tayab	tayab	PROPN
ejpam-6849	557	57	,	,	PUNCT
ejpam-6849	557	58	f.	f.	PROPN
ejpam-6849	557	59	p.	p.	PROPN
ejpam-6849	557	60	jamil	jamil	PROPN
ejpam-6849	557	61	,	,	PUNCT
ejpam-6849	557	62	i.	i.	PROPN
ejpam-6849	557	63	aniversario	aniversario	PROPN
ejpam-6849	557	64	/	/	SYM
ejpam-6849	557	65	eur	eur	PROPN
ejpam-6849	557	66	.	.	PUNCT
ejpam-6849	558	1	j.	j.	PROPN
ejpam-6849	558	2	pure	pure	PROPN
ejpam-6849	558	3	appl	appl	PROPN
ejpam-6849	558	4	.	.	PROPN
ejpam-6849	558	5	math	math	PROPN
ejpam-6849	558	6	,	,	PUNCT
ejpam-6849	558	7	18	18	NUM
ejpam-6849	558	8	(	(	PUNCT
ejpam-6849	558	9	4	4	NUM
ejpam-6849	558	10	)	)	PUNCT
ejpam-6849	558	11	(	(	PUNCT
ejpam-6849	558	12	2025	2025	NUM
ejpam-6849	558	13	)	)	PUNCT
ejpam-6849	558	14	,	,	PUNCT
ejpam-6849	558	15	6849	6849	NUM
ejpam-6849	558	16	15	15	NUM
ejpam-6849	558	17	of	of	ADP
ejpam-6849	558	18	19	19	NUM
ejpam-6849	558	19	proof	proof	NOUN
ejpam-6849	558	20	.	.	PUNCT
ejpam-6849	559	1	let	let	VERB
ejpam-6849	559	2	s	s	PRON
ejpam-6849	559	3	⊆	⊆	NUM
ejpam-6849	559	4	v	v	NOUN
ejpam-6849	559	5	(	(	PUNCT
ejpam-6849	559	6	g	g	PROPN
ejpam-6849	559	7	⋄	⋄	PROPN
ejpam-6849	559	8	h	h	NOUN
ejpam-6849	559	9	)	)	PUNCT
ejpam-6849	559	10	be	be	VERB
ejpam-6849	559	11	a	a	DET
ejpam-6849	559	12	ve	ve	NOUN
ejpam-6849	559	13	-	-	PUNCT
ejpam-6849	559	14	dominating	dominate	VERB
ejpam-6849	559	15	set	set	NOUN
ejpam-6849	559	16	of	of	ADP
ejpam-6849	559	17	g	g	PROPN
ejpam-6849	559	18	⋄	⋄	PROPN
ejpam-6849	559	19	h.	h.	PROPN
ejpam-6849	559	20	put	put	VERB
ejpam-6849	559	21	a	a	DET
ejpam-6849	559	22	=	=	SYM
ejpam-6849	559	23	s	s	NOUN
ejpam-6849	559	24	∩	∩	ADJ
ejpam-6849	559	25	v	v	X
ejpam-6849	559	26	(	(	PUNCT
ejpam-6849	559	27	g	g	NOUN
ejpam-6849	559	28	)	)	PUNCT
ejpam-6849	559	29	and	and	CCONJ
ejpam-6849	559	30	suv	suv	PROPN
ejpam-6849	559	31	=	=	PROPN
ejpam-6849	559	32	s	s	PROPN
ejpam-6849	559	33	∩	∩	ADJ
ejpam-6849	559	34	v	v	X
ejpam-6849	559	35	(	(	PUNCT
ejpam-6849	559	36	huv	huv	PROPN
ejpam-6849	559	37	)	)	PUNCT
ejpam-6849	559	38	for	for	ADP
ejpam-6849	559	39	each	each	DET
ejpam-6849	559	40	uv	uv	PROPN
ejpam-6849	559	41	∈	∈	PROPN
ejpam-6849	559	42	e(g	e(g	PROPN
ejpam-6849	559	43	)	)	PUNCT
ejpam-6849	559	44	.	.	PUNCT
ejpam-6849	560	1	then	then	ADV
ejpam-6849	560	2	s	s	VERB
ejpam-6849	560	3	satisfies	satisfie	NOUN
ejpam-6849	560	4	equation	equation	NOUN
ejpam-6849	560	5	8	8	NUM
ejpam-6849	560	6	.	.	PUNCT
ejpam-6849	561	1	let	let	VERB
ejpam-6849	561	2	uv	uv	PRON
ejpam-6849	561	3	∈	∈	PROPN
ejpam-6849	561	4	e(g	e(g	PROPN
ejpam-6849	561	5	)	)	PUNCT
ejpam-6849	561	6	such	such	ADJ
ejpam-6849	561	7	that	that	SCONJ
ejpam-6849	561	8	{	{	PUNCT
ejpam-6849	561	9	u	u	NOUN
ejpam-6849	561	10	,	,	PUNCT
ejpam-6849	561	11	v	v	NOUN
ejpam-6849	561	12	}	}	PUNCT
ejpam-6849	561	13	∩	∩	NOUN
ejpam-6849	561	14	a	a	DET
ejpam-6849	561	15	=	=	SYM
ejpam-6849	561	16	∅	∅	NOUN
ejpam-6849	561	17	,	,	PUNCT
ejpam-6849	561	18	and	and	CCONJ
ejpam-6849	561	19	let	let	VERB
ejpam-6849	561	20	xy	xy	PROPN
ejpam-6849	561	21	∈	∈	PROPN
ejpam-6849	561	22	e(huv	e(huv	PROPN
ejpam-6849	561	23	)	)	PUNCT
ejpam-6849	561	24	.	.	PUNCT
ejpam-6849	562	1	since	since	SCONJ
ejpam-6849	562	2	s	s	PROPN
ejpam-6849	562	3	is	be	AUX
ejpam-6849	562	4	a	a	DET
ejpam-6849	562	5	ve	ve	NOUN
ejpam-6849	562	6	-	-	PUNCT
ejpam-6849	562	7	dominating	dominate	VERB
ejpam-6849	562	8	set	set	NOUN
ejpam-6849	562	9	of	of	ADP
ejpam-6849	562	10	g	g	PROPN
ejpam-6849	562	11	⋄	⋄	PROPN
ejpam-6849	562	12	h	h	NOUN
ejpam-6849	562	13	,	,	PUNCT
ejpam-6849	562	14	there	there	PRON
ejpam-6849	562	15	exists	exist	VERB
ejpam-6849	562	16	w	w	PROPN
ejpam-6849	562	17	∈	∈	PROPN
ejpam-6849	562	18	s	s	VERB
ejpam-6849	562	19	such	such	ADJ
ejpam-6849	562	20	that	that	SCONJ
ejpam-6849	562	21	w	w	PROPN
ejpam-6849	562	22	ve	ve	VERB
ejpam-6849	562	23	-	-	PUNCT
ejpam-6849	562	24	dominates	dominate	VERB
ejpam-6849	562	25	xy	xy	PROPN
ejpam-6849	562	26	.	.	PUNCT
ejpam-6849	563	1	since	since	SCONJ
ejpam-6849	563	2	{	{	PUNCT
ejpam-6849	563	3	u	u	NOUN
ejpam-6849	563	4	,	,	PUNCT
ejpam-6849	563	5	v	v	NOUN
ejpam-6849	563	6	}	}	PUNCT
ejpam-6849	563	7	∩	∩	NOUN
ejpam-6849	563	8	s	s	PART
ejpam-6849	563	9	=	=	SYM
ejpam-6849	563	10	∅	∅	NOUN
ejpam-6849	563	11	,	,	PUNCT
ejpam-6849	563	12	w	w	PROPN
ejpam-6849	563	13	∈	∈	PROPN
ejpam-6849	563	14	suv	suv	NOUN
ejpam-6849	563	15	.	.	PUNCT
ejpam-6849	564	1	thus	thus	ADV
ejpam-6849	564	2	,	,	PUNCT
ejpam-6849	564	3	suv	suv	PROPN
ejpam-6849	564	4	is	be	AUX
ejpam-6849	564	5	a	a	DET
ejpam-6849	564	6	ve	ve	NOUN
ejpam-6849	564	7	-	-	PUNCT
ejpam-6849	564	8	dominating	dominate	VERB
ejpam-6849	564	9	set	set	NOUN
ejpam-6849	564	10	of	of	ADP
ejpam-6849	564	11	huv	huv	PROPN
ejpam-6849	564	12	.	.	PUNCT
ejpam-6849	565	1	conversely	conversely	ADV
ejpam-6849	565	2	,	,	PUNCT
ejpam-6849	565	3	suppose	suppose	VERB
ejpam-6849	565	4	that	that	SCONJ
ejpam-6849	565	5	the	the	DET
ejpam-6849	565	6	set	set	NOUN
ejpam-6849	565	7	s	s	X
ejpam-6849	565	8	in	in	ADP
ejpam-6849	565	9	equation	equation	NOUN
ejpam-6849	565	10	8	8	NUM
ejpam-6849	565	11	has	have	VERB
ejpam-6849	565	12	the	the	DET
ejpam-6849	565	13	property	property	NOUN
ejpam-6849	565	14	that	that	PRON
ejpam-6849	565	15	suv	suv	PROPN
ejpam-6849	565	16	is	be	AUX
ejpam-6849	565	17	a	a	DET
ejpam-6849	565	18	vedominating	vedominate	VERB
ejpam-6849	565	19	set	set	NOUN
ejpam-6849	565	20	of	of	ADP
ejpam-6849	565	21	huv	huv	PROPN
ejpam-6849	565	22	for	for	ADP
ejpam-6849	565	23	each	each	DET
ejpam-6849	565	24	uv	uv	PROPN
ejpam-6849	565	25	∈	∈	PROPN
ejpam-6849	565	26	e(g	e(g	PROPN
ejpam-6849	565	27	)	)	PUNCT
ejpam-6849	565	28	with	with	ADP
ejpam-6849	565	29	{	{	PUNCT
ejpam-6849	565	30	u	u	NOUN
ejpam-6849	565	31	,	,	PUNCT
ejpam-6849	565	32	v	v	NOUN
ejpam-6849	565	33	}	}	PUNCT
ejpam-6849	565	34	∩	∩	NOUN
ejpam-6849	565	35	a	a	PRON
ejpam-6849	565	36	=	=	PUNCT
ejpam-6849	565	37	∅.	∅.	AUX
ejpam-6849	565	38	let	let	VERB
ejpam-6849	565	39	xy	xy	PROPN
ejpam-6849	565	40	∈	∈	PROPN
ejpam-6849	565	41	e(g	e(g	PROPN
ejpam-6849	565	42	⋄	⋄	PROPN
ejpam-6849	565	43	h	h	NOUN
ejpam-6849	565	44	)	)	PUNCT
ejpam-6849	565	45	and	and	CCONJ
ejpam-6849	565	46	let	let	VERB
ejpam-6849	565	47	uv	uv	PRON
ejpam-6849	565	48	∈	∈	PROPN
ejpam-6849	565	49	e(g	e(g	PROPN
ejpam-6849	565	50	)	)	PUNCT
ejpam-6849	565	51	such	such	ADJ
ejpam-6849	565	52	that	that	SCONJ
ejpam-6849	565	53	xy	xy	PROPN
ejpam-6849	565	54	∈	∈	PROPN
ejpam-6849	565	55	e(huv	e(huv	PROPN
ejpam-6849	565	56	+	+	CCONJ
ejpam-6849	565	57	uv	uv	NOUN
ejpam-6849	565	58	)	)	PUNCT
ejpam-6849	565	59	.	.	PUNCT
ejpam-6849	566	1	first	first	ADV
ejpam-6849	566	2	,	,	PUNCT
ejpam-6849	566	3	suppose	suppose	VERB
ejpam-6849	566	4	that	that	SCONJ
ejpam-6849	566	5	{	{	PUNCT
ejpam-6849	566	6	u	u	NOUN
ejpam-6849	566	7	,	,	PUNCT
ejpam-6849	566	8	v	v	NOUN
ejpam-6849	566	9	}	}	PUNCT
ejpam-6849	566	10	∩	∩	NOUN
ejpam-6849	566	11	a	a	DET
ejpam-6849	566	12	=	=	SYM
ejpam-6849	566	13	∅.	∅.	PROPN
ejpam-6849	566	14	then	then	ADV
ejpam-6849	566	15	suv	suv	PROPN
ejpam-6849	566	16	is	be	AUX
ejpam-6849	566	17	a	a	DET
ejpam-6849	566	18	ve	ve	NOUN
ejpam-6849	566	19	-	-	PUNCT
ejpam-6849	566	20	dominating	dominate	VERB
ejpam-6849	566	21	set	set	NOUN
ejpam-6849	566	22	of	of	ADP
ejpam-6849	566	23	huv	huv	PROPN
ejpam-6849	566	24	.	.	PUNCT
ejpam-6849	567	1	if	if	SCONJ
ejpam-6849	567	2	{	{	PUNCT
ejpam-6849	567	3	x	x	NOUN
ejpam-6849	567	4	,	,	PUNCT
ejpam-6849	567	5	y	y	NOUN
ejpam-6849	567	6	}	}	PUNCT
ejpam-6849	567	7	∩	∩	NOUN
ejpam-6849	567	8	{	{	PUNCT
ejpam-6849	567	9	u	u	NOUN
ejpam-6849	567	10	,	,	PUNCT
ejpam-6849	567	11	v	v	NOUN
ejpam-6849	567	12	}	}	PUNCT
ejpam-6849	567	13	̸=	̸=	PROPN
ejpam-6849	567	14	∅	∅	NOUN
ejpam-6849	567	15	,	,	PUNCT
ejpam-6849	567	16	then	then	ADV
ejpam-6849	567	17	since	since	SCONJ
ejpam-6849	567	18	{	{	PUNCT
ejpam-6849	567	19	zu	zu	ADV
ejpam-6849	567	20	,	,	PUNCT
ejpam-6849	567	21	zv	zv	NOUN
ejpam-6849	567	22	}	}	PUNCT
ejpam-6849	567	23	⊆	⊆	NUM
ejpam-6849	567	24	e(g	e(g	PROPN
ejpam-6849	567	25	⋄	⋄	PROPN
ejpam-6849	567	26	h	h	NOUN
ejpam-6849	567	27	)	)	PUNCT
ejpam-6849	567	28	,	,	PUNCT
ejpam-6849	567	29	z	z	PROPN
ejpam-6849	567	30	ve	ve	AUX
ejpam-6849	567	31	-	-	PUNCT
ejpam-6849	567	32	dominates	dominate	VERB
ejpam-6849	567	33	xy	xy	PROPN
ejpam-6849	567	34	for	for	ADP
ejpam-6849	567	35	all	all	DET
ejpam-6849	567	36	z	z	PROPN
ejpam-6849	567	37	∈	∈	PROPN
ejpam-6849	567	38	suv	suv	NOUN
ejpam-6849	567	39	.	.	PUNCT
ejpam-6849	568	1	if	if	SCONJ
ejpam-6849	568	2	,	,	PUNCT
ejpam-6849	568	3	on	on	ADP
ejpam-6849	568	4	the	the	DET
ejpam-6849	568	5	other	other	ADJ
ejpam-6849	568	6	hand	hand	NOUN
ejpam-6849	568	7	,	,	PUNCT
ejpam-6849	568	8	{	{	PUNCT
ejpam-6849	568	9	x	x	NOUN
ejpam-6849	568	10	,	,	PUNCT
ejpam-6849	568	11	y	y	NOUN
ejpam-6849	568	12	}	}	PUNCT
ejpam-6849	568	13	∩	∩	NOUN
ejpam-6849	568	14	{	{	PUNCT
ejpam-6849	568	15	u	u	NOUN
ejpam-6849	568	16	,	,	PUNCT
ejpam-6849	568	17	v	v	NOUN
ejpam-6849	568	18	}	}	PUNCT
ejpam-6849	568	19	=	=	SYM
ejpam-6849	568	20	∅	∅	NOUN
ejpam-6849	568	21	,	,	PUNCT
ejpam-6849	568	22	then	then	ADV
ejpam-6849	568	23	xy	xy	PROPN
ejpam-6849	568	24	∈	∈	PROPN
ejpam-6849	568	25	e(huv	e(huv	PROPN
ejpam-6849	568	26	)	)	PUNCT
ejpam-6849	568	27	and	and	CCONJ
ejpam-6849	568	28	there	there	PRON
ejpam-6849	568	29	exists	exist	VERB
ejpam-6849	568	30	z	z	PROPN
ejpam-6849	568	31	∈	∈	PROPN
ejpam-6849	568	32	suv	suv	NOUN
ejpam-6849	568	33	such	such	ADJ
ejpam-6849	568	34	that	that	SCONJ
ejpam-6849	568	35	z	z	NOUN
ejpam-6849	568	36	ve	ve	VERB
ejpam-6849	568	37	-	-	PUNCT
ejpam-6849	568	38	dominates	dominate	VERB
ejpam-6849	568	39	xy	xy	PRON
ejpam-6849	568	40	.	.	PUNCT
ejpam-6849	569	1	in	in	ADP
ejpam-6849	569	2	this	this	DET
ejpam-6849	569	3	case	case	NOUN
ejpam-6849	569	4	,	,	PUNCT
ejpam-6849	569	5	s	s	AUX
ejpam-6849	569	6	ve	ve	VERB
ejpam-6849	569	7	-	-	PUNCT
ejpam-6849	569	8	dominates	dominate	VERB
ejpam-6849	569	9	xy	xy	NOUN
ejpam-6849	569	10	.	.	PUNCT
ejpam-6849	570	1	next	next	ADV
ejpam-6849	570	2	,	,	PUNCT
ejpam-6849	570	3	suppose	suppose	VERB
ejpam-6849	570	4	that	that	SCONJ
ejpam-6849	570	5	{	{	PUNCT
ejpam-6849	570	6	u	u	NOUN
ejpam-6849	570	7	,	,	PUNCT
ejpam-6849	570	8	v	v	NOUN
ejpam-6849	570	9	}	}	PUNCT
ejpam-6849	570	10	∩	∩	NOUN
ejpam-6849	570	11	a	a	DET
ejpam-6849	570	12	=	=	NOUN
ejpam-6849	570	13	̸	̸	ADV
ejpam-6849	570	14	∅.	∅.	PRON
ejpam-6849	570	15	wlog	wlog	NOUN
ejpam-6849	570	16	,	,	PUNCT
ejpam-6849	570	17	assume	assume	VERB
ejpam-6849	570	18	that	that	SCONJ
ejpam-6849	570	19	u	u	PROPN
ejpam-6849	570	20	∈	∈	PROPN
ejpam-6849	570	21	a.	a.	NOUN
ejpam-6849	570	22	if	if	SCONJ
ejpam-6849	570	23	x	x	PROPN
ejpam-6849	570	24	=	=	SYM
ejpam-6849	570	25	u	u	NOUN
ejpam-6849	570	26	,	,	PUNCT
ejpam-6849	570	27	then	then	ADV
ejpam-6849	570	28	u	u	NOUN
ejpam-6849	570	29	,	,	PUNCT
ejpam-6849	570	30	and	and	CCONJ
ejpam-6849	570	31	hence	hence	ADV
ejpam-6849	570	32	s	s	PROPN
ejpam-6849	570	33	,	,	PUNCT
ejpam-6849	570	34	ve	ve	VERB
ejpam-6849	570	35	-	-	PUNCT
ejpam-6849	570	36	dominates	dominate	VERB
ejpam-6849	570	37	xy	xy	NOUN
ejpam-6849	570	38	.	.	PUNCT
ejpam-6849	571	1	if	if	SCONJ
ejpam-6849	571	2	x	x	PRON
ejpam-6849	571	3	=	=	NOUN
ejpam-6849	571	4	̸	̸	NUM
ejpam-6849	571	5	u	u	NOUN
ejpam-6849	571	6	,	,	PUNCT
ejpam-6849	571	7	then	then	ADV
ejpam-6849	571	8	ux	ux	PROPN
ejpam-6849	571	9	∈	∈	PROPN
ejpam-6849	571	10	e(g	e(g	PROPN
ejpam-6849	571	11	⋄	⋄	PROPN
ejpam-6849	571	12	h	h	NOUN
ejpam-6849	571	13	)	)	PUNCT
ejpam-6849	571	14	and	and	CCONJ
ejpam-6849	571	15	so	so	ADV
ejpam-6849	571	16	u	u	NOUN
ejpam-6849	571	17	,	,	PUNCT
ejpam-6849	571	18	and	and	CCONJ
ejpam-6849	571	19	hence	hence	ADV
ejpam-6849	571	20	s	s	PROPN
ejpam-6849	571	21	,	,	PUNCT
ejpam-6849	571	22	ve	ve	VERB
ejpam-6849	571	23	-	-	PUNCT
ejpam-6849	571	24	dominates	dominate	VERB
ejpam-6849	571	25	xy	xy	PROPN
ejpam-6849	571	26	.	.	PUNCT
ejpam-6849	572	1	accordingly	accordingly	ADV
ejpam-6849	572	2	,	,	PUNCT
ejpam-6849	572	3	s	s	VERB
ejpam-6849	572	4	is	be	AUX
ejpam-6849	572	5	a	a	DET
ejpam-6849	572	6	ve	ve	NOUN
ejpam-6849	572	7	-	-	PUNCT
ejpam-6849	572	8	dominating	dominate	VERB
ejpam-6849	572	9	set	set	NOUN
ejpam-6849	572	10	of	of	ADP
ejpam-6849	572	11	g	g	PROPN
ejpam-6849	572	12	⋄	⋄	PROPN
ejpam-6849	572	13	h.	h.	PROPN
ejpam-6849	573	1	■	■	PUNCT
ejpam-6849	573	2	corollary	corollary	ADJ
ejpam-6849	573	3	6	6	NUM
ejpam-6849	573	4	.	.	PUNCT
ejpam-6849	574	1	let	let	VERB
ejpam-6849	574	2	g	g	PRON
ejpam-6849	574	3	be	be	AUX
ejpam-6849	574	4	a	a	DET
ejpam-6849	574	5	nontrivial	nontrivial	ADJ
ejpam-6849	574	6	connected	connect	VERB
ejpam-6849	574	7	graph	graph	NOUN
ejpam-6849	574	8	.	.	PUNCT
ejpam-6849	575	1	(	(	PUNCT
ejpam-6849	575	2	i	i	NOUN
ejpam-6849	575	3	)	)	PUNCT
ejpam-6849	575	4	if	if	SCONJ
ejpam-6849	575	5	h	h	NOUN
ejpam-6849	575	6	is	be	AUX
ejpam-6849	575	7	an	an	DET
ejpam-6849	575	8	empty	empty	ADJ
ejpam-6849	575	9	graph	graph	NOUN
ejpam-6849	575	10	,	,	PUNCT
ejpam-6849	575	11	then	then	ADV
ejpam-6849	575	12	γve(g	γve(g	PROPN
ejpam-6849	575	13	⋄	⋄	NOUN
ejpam-6849	575	14	h	h	NOUN
ejpam-6849	575	15	)	)	PUNCT
ejpam-6849	575	16	=	=	PUNCT
ejpam-6849	575	17	γ(g	γ(g	PROPN
ejpam-6849	575	18	)	)	PUNCT
ejpam-6849	575	19	.	.	PUNCT
ejpam-6849	576	1	(	(	PUNCT
ejpam-6849	576	2	ii	ii	NOUN
ejpam-6849	576	3	)	)	PUNCT
ejpam-6849	576	4	if	if	SCONJ
ejpam-6849	576	5	h	h	NOUN
ejpam-6849	576	6	is	be	AUX
ejpam-6849	576	7	a	a	DET
ejpam-6849	576	8	nonempty	nonempty	ADJ
ejpam-6849	576	9	graph	graph	NOUN
ejpam-6849	576	10	,	,	PUNCT
ejpam-6849	576	11	then	then	ADV
ejpam-6849	576	12	γve(g	γve(g	PROPN
ejpam-6849	576	13	⋄	⋄	NOUN
ejpam-6849	576	14	h	h	NOUN
ejpam-6849	576	15	)	)	PUNCT
ejpam-6849	577	1	=	=	NOUN
ejpam-6849	577	2	min{|s|	min{|s|	NOUN
ejpam-6849	577	3	:	:	PUNCT
ejpam-6849	577	4	s	s	VERB
ejpam-6849	577	5	⊆	⊆	NUM
ejpam-6849	577	6	v	v	NOUN
ejpam-6849	577	7	(	(	PUNCT
ejpam-6849	577	8	g	g	NOUN
ejpam-6849	577	9	)	)	PUNCT
ejpam-6849	577	10	and	and	CCONJ
ejpam-6849	577	11	ng(v	ng(v	NUM
ejpam-6849	577	12	(	(	PUNCT
ejpam-6849	577	13	g	g	NOUN
ejpam-6849	577	14	)	)	PUNCT
ejpam-6849	577	15	\	\	PROPN
ejpam-6849	578	1	s	s	X
ejpam-6849	578	2	)	)	PUNCT
ejpam-6849	578	3	⊆	⊆	NUM
ejpam-6849	578	4	s	s	NOUN
ejpam-6849	578	5	}	}	PUNCT
ejpam-6849	578	6	.	.	PUNCT
ejpam-6849	579	1	proof	proof	NOUN
ejpam-6849	579	2	.	.	PUNCT
ejpam-6849	580	1	suppose	suppose	VERB
ejpam-6849	580	2	that	that	SCONJ
ejpam-6849	580	3	h	h	NOUN
ejpam-6849	580	4	is	be	AUX
ejpam-6849	580	5	an	an	DET
ejpam-6849	580	6	empty	empty	ADJ
ejpam-6849	580	7	graph	graph	NOUN
ejpam-6849	580	8	.	.	PUNCT
ejpam-6849	581	1	let	let	VERB
ejpam-6849	581	2	s	s	PRON
ejpam-6849	581	3	⊆	⊆	NUM
ejpam-6849	581	4	v	v	NOUN
ejpam-6849	581	5	(	(	PUNCT
ejpam-6849	581	6	g	g	NOUN
ejpam-6849	581	7	)	)	PUNCT
ejpam-6849	581	8	be	be	AUX
ejpam-6849	581	9	a	a	DET
ejpam-6849	581	10	γ	γ	NOUN
ejpam-6849	581	11	-	-	PUNCT
ejpam-6849	581	12	set	set	NOUN
ejpam-6849	581	13	of	of	ADP
ejpam-6849	581	14	g.	g.	PROPN
ejpam-6849	581	15	let	let	VERB
ejpam-6849	581	16	xy	xy	PROPN
ejpam-6849	581	17	∈	∈	PROPN
ejpam-6849	581	18	e(g	e(g	PROPN
ejpam-6849	581	19	⋄	⋄	PROPN
ejpam-6849	581	20	h	h	NOUN
ejpam-6849	581	21	)	)	PUNCT
ejpam-6849	581	22	,	,	PUNCT
ejpam-6849	581	23	and	and	CCONJ
ejpam-6849	581	24	let	let	VERB
ejpam-6849	581	25	uv	uv	PRON
ejpam-6849	581	26	∈	∈	PROPN
ejpam-6849	581	27	e(g	e(g	PROPN
ejpam-6849	581	28	)	)	PUNCT
ejpam-6849	581	29	such	such	ADJ
ejpam-6849	581	30	that	that	SCONJ
ejpam-6849	581	31	xy	xy	PROPN
ejpam-6849	581	32	∈	∈	PROPN
ejpam-6849	581	33	e(huv	e(huv	PROPN
ejpam-6849	581	34	+	+	CCONJ
ejpam-6849	581	35	uv	uv	NOUN
ejpam-6849	581	36	)	)	PUNCT
ejpam-6849	581	37	.	.	PUNCT
ejpam-6849	582	1	if	if	SCONJ
ejpam-6849	582	2	u	u	PROPN
ejpam-6849	582	3	∈	∈	PROPN
ejpam-6849	582	4	s	s	X
ejpam-6849	582	5	(	(	PUNCT
ejpam-6849	582	6	resp	resp	NOUN
ejpam-6849	582	7	.	.	PUNCT
ejpam-6849	583	1	v	v	X
ejpam-6849	583	2	∈	∈	PROPN
ejpam-6849	583	3	s	s	NOUN
ejpam-6849	583	4	)	)	PUNCT
ejpam-6849	583	5	,	,	PUNCT
ejpam-6849	583	6	then	then	ADV
ejpam-6849	583	7	u	u	X
ejpam-6849	583	8	(	(	PUNCT
ejpam-6849	583	9	resp	resp	NOUN
ejpam-6849	583	10	.	.	PUNCT
ejpam-6849	584	1	v	v	X
ejpam-6849	584	2	)	)	PUNCT
ejpam-6849	584	3	ve	ve	VERB
ejpam-6849	584	4	-	-	PUNCT
ejpam-6849	584	5	dominates	dominate	VERB
ejpam-6849	584	6	xy	xy	PRON
ejpam-6849	584	7	.	.	PUNCT
ejpam-6849	584	8	suppose	suppose	VERB
ejpam-6849	584	9	that	that	SCONJ
ejpam-6849	584	10	u	u	PROPN
ejpam-6849	584	11	/∈	/∈	PROPN
ejpam-6849	584	12	s	s	NOUN
ejpam-6849	584	13	and	and	CCONJ
ejpam-6849	584	14	v	v	NOUN
ejpam-6849	584	15	/∈	/∈	PUNCT
ejpam-6849	584	16	s.	s.	PROPN
ejpam-6849	584	17	since	since	SCONJ
ejpam-6849	584	18	s	s	PROPN
ejpam-6849	584	19	is	be	AUX
ejpam-6849	584	20	a	a	DET
ejpam-6849	584	21	dominating	dominating	NOUN
ejpam-6849	584	22	set	set	NOUN
ejpam-6849	584	23	,	,	PUNCT
ejpam-6849	584	24	there	there	PRON
ejpam-6849	584	25	exist	exist	VERB
ejpam-6849	584	26	w	w	ADP
ejpam-6849	584	27	,	,	PUNCT
ejpam-6849	584	28	z	z	PROPN
ejpam-6849	584	29	∈	∈	PROPN
ejpam-6849	584	30	s	s	X
ejpam-6849	584	31	for	for	ADP
ejpam-6849	584	32	which	which	PRON
ejpam-6849	584	33	uw	uw	PROPN
ejpam-6849	584	34	∈	∈	PROPN
ejpam-6849	584	35	e(g	e(g	PROPN
ejpam-6849	584	36	)	)	PUNCT
ejpam-6849	584	37	and	and	CCONJ
ejpam-6849	584	38	yz	yz	PROPN
ejpam-6849	584	39	∈	∈	PROPN
ejpam-6849	584	40	e(g	e(g	PROPN
ejpam-6849	584	41	)	)	PUNCT
ejpam-6849	584	42	.	.	PUNCT
ejpam-6849	585	1	in	in	ADP
ejpam-6849	585	2	this	this	DET
ejpam-6849	585	3	case	case	NOUN
ejpam-6849	585	4	,	,	PUNCT
ejpam-6849	585	5	w	w	AUX
ejpam-6849	585	6	ve	ve	VERB
ejpam-6849	585	7	-	-	VERB
ejpam-6849	585	8	dominates	dominate	VERB
ejpam-6849	585	9	xy	xy	PROPN
ejpam-6849	585	10	or	or	CCONJ
ejpam-6849	585	11	z	z	PROPN
ejpam-6849	585	12	ve	ve	VERB
ejpam-6849	585	13	-	-	PUNCT
ejpam-6849	585	14	dominates	dominate	VERB
ejpam-6849	585	15	xy	xy	PROPN
ejpam-6849	585	16	.	.	PUNCT
ejpam-6849	586	1	thus	thus	ADV
ejpam-6849	586	2	,	,	PUNCT
ejpam-6849	586	3	s	s	VERB
ejpam-6849	586	4	is	be	AUX
ejpam-6849	586	5	a	a	DET
ejpam-6849	586	6	ve	ve	NOUN
ejpam-6849	586	7	-	-	PUNCT
ejpam-6849	586	8	dominating	dominate	VERB
ejpam-6849	586	9	set	set	NOUN
ejpam-6849	586	10	of	of	ADP
ejpam-6849	586	11	g	g	PROPN
ejpam-6849	586	12	⋄	⋄	PROPN
ejpam-6849	586	13	h.	h.	PROPN
ejpam-6849	586	14	consequently	consequently	ADV
ejpam-6849	586	15	,	,	PUNCT
ejpam-6849	586	16	γve(g	γve(g	PROPN
ejpam-6849	586	17	⋄	⋄	NOUN
ejpam-6849	586	18	h	h	NOUN
ejpam-6849	586	19	)	)	PUNCT
ejpam-6849	586	20	≤	≤	NUM
ejpam-6849	586	21	|s|	|s|	PROPN
ejpam-6849	586	22	=	=	SYM
ejpam-6849	586	23	γ(g	γ(g	PROPN
ejpam-6849	586	24	)	)	PUNCT
ejpam-6849	586	25	.	.	PUNCT
ejpam-6849	587	1	to	to	PART
ejpam-6849	587	2	get	get	VERB
ejpam-6849	587	3	the	the	DET
ejpam-6849	587	4	other	other	ADJ
ejpam-6849	587	5	inequality	inequality	NOUN
ejpam-6849	587	6	,	,	PUNCT
ejpam-6849	587	7	let	let	VERB
ejpam-6849	587	8	s	s	PRON
ejpam-6849	587	9	⊆	⊆	NUM
ejpam-6849	587	10	v	v	NOUN
ejpam-6849	587	11	(	(	PUNCT
ejpam-6849	587	12	g	g	PROPN
ejpam-6849	587	13	⋄	⋄	PROPN
ejpam-6849	587	14	h	h	NOUN
ejpam-6849	587	15	)	)	PUNCT
ejpam-6849	587	16	be	be	VERB
ejpam-6849	587	17	a	a	DET
ejpam-6849	587	18	ve	ve	NOUN
ejpam-6849	587	19	-	-	PUNCT
ejpam-6849	587	20	dominating	dominate	VERB
ejpam-6849	587	21	set	set	NOUN
ejpam-6849	587	22	of	of	ADP
ejpam-6849	587	23	g	g	PROPN
ejpam-6849	587	24	⋄	⋄	PROPN
ejpam-6849	587	25	h.	h.	PROPN
ejpam-6849	587	26	put	put	VERB
ejpam-6849	587	27	s1	s1	NOUN
ejpam-6849	587	28	=	=	SYM
ejpam-6849	587	29	s	s	PART
ejpam-6849	587	30	∩v	∩v	NOUN
ejpam-6849	587	31	(	(	PUNCT
ejpam-6849	587	32	g	g	NOUN
ejpam-6849	587	33	)	)	PUNCT
ejpam-6849	587	34	.	.	PUNCT
ejpam-6849	588	1	for	for	ADP
ejpam-6849	588	2	each	each	DET
ejpam-6849	588	3	uv	uv	PROPN
ejpam-6849	588	4	∈	∈	PROPN
ejpam-6849	588	5	e(g	e(g	PROPN
ejpam-6849	588	6	)	)	PUNCT
ejpam-6849	588	7	,	,	PUNCT
ejpam-6849	588	8	put	put	VERB
ejpam-6849	588	9	suv	suv	NOUN
ejpam-6849	588	10	=	=	VERB
ejpam-6849	588	11	s	s	PART
ejpam-6849	588	12	∩v	∩v	NOUN
ejpam-6849	588	13	(	(	PUNCT
ejpam-6849	588	14	huv	huv	PROPN
ejpam-6849	588	15	)	)	PUNCT
ejpam-6849	588	16	.	.	PUNCT
ejpam-6849	589	1	for	for	ADP
ejpam-6849	589	2	each	each	DET
ejpam-6849	589	3	uv	uv	PROPN
ejpam-6849	589	4	∈	∈	PROPN
ejpam-6849	589	5	e(g	e(g	PROPN
ejpam-6849	589	6	)	)	PUNCT
ejpam-6849	589	7	for	for	ADP
ejpam-6849	589	8	which	which	PRON
ejpam-6849	589	9	suv	suv	NOUN
ejpam-6849	589	10	̸=	̸=	PROPN
ejpam-6849	589	11	∅	∅	NOUN
ejpam-6849	589	12	,	,	PUNCT
ejpam-6849	589	13	put	put	VERB
ejpam-6849	589	14	either	either	CCONJ
ejpam-6849	589	15	xuv	xuv	PUNCT
ejpam-6849	589	16	=	=	SYM
ejpam-6849	589	17	u	u	NOUN
ejpam-6849	589	18	or	or	CCONJ
ejpam-6849	589	19	xuv	xuv	NOUN
ejpam-6849	589	20	=	=	X
ejpam-6849	590	1	v.	v.	ADP
ejpam-6849	590	2	define	define	VERB
ejpam-6849	590	3	s2	s2	NOUN
ejpam-6849	590	4	=	=	SYM
ejpam-6849	590	5	{	{	PUNCT
ejpam-6849	590	6	xuv	xuv	NOUN
ejpam-6849	590	7	:	:	PUNCT
ejpam-6849	590	8	uv	uv	PROPN
ejpam-6849	590	9	∈	∈	PROPN
ejpam-6849	590	10	e(g	e(g	PROPN
ejpam-6849	590	11	)	)	PUNCT
ejpam-6849	590	12	with	with	ADP
ejpam-6849	590	13	suv	suv	NOUN
ejpam-6849	590	14	=	=	SYM
ejpam-6849	590	15	̸	̸	NOUN
ejpam-6849	590	16	∅	∅	NOUN
ejpam-6849	590	17	}	}	PUNCT
ejpam-6849	590	18	and	and	CCONJ
ejpam-6849	590	19	define	define	VERB
ejpam-6849	590	20	s∗	s∗	PROPN
ejpam-6849	590	21	=	=	SYM
ejpam-6849	590	22	s1	s1	PROPN
ejpam-6849	590	23	∪	∪	X
ejpam-6849	590	24	s2	s2	PROPN
ejpam-6849	590	25	.	.	PUNCT
ejpam-6849	591	1	we	we	PRON
ejpam-6849	591	2	claim	claim	VERB
ejpam-6849	591	3	that	that	SCONJ
ejpam-6849	591	4	s∗	s∗	PROPN
ejpam-6849	591	5	is	be	AUX
ejpam-6849	591	6	a	a	DET
ejpam-6849	591	7	dominating	dominating	NOUN
ejpam-6849	591	8	set	set	NOUN
ejpam-6849	591	9	of	of	ADP
ejpam-6849	591	10	g.	g.	PROPN
ejpam-6849	591	11	let	let	VERB
ejpam-6849	591	12	z	z	PROPN
ejpam-6849	591	13	∈	∈	PROPN
ejpam-6849	591	14	v	v	ADP
ejpam-6849	591	15	(	(	PUNCT
ejpam-6849	591	16	g	g	NOUN
ejpam-6849	591	17	)	)	PUNCT
ejpam-6849	591	18	\	\	NOUN
ejpam-6849	591	19	s∗	s∗	PROPN
ejpam-6849	591	20	and	and	CCONJ
ejpam-6849	591	21	choose	choose	VERB
ejpam-6849	591	22	w	w	PROPN
ejpam-6849	591	23	∈	∈	PROPN
ejpam-6849	591	24	v	v	ADP
ejpam-6849	591	25	(	(	PUNCT
ejpam-6849	591	26	g	g	NOUN
ejpam-6849	591	27	)	)	PUNCT
ejpam-6849	591	28	such	such	ADJ
ejpam-6849	591	29	that	that	SCONJ
ejpam-6849	591	30	zw	zw	PROPN
ejpam-6849	591	31	∈	∈	PROPN
ejpam-6849	591	32	e(g	e(g	PROPN
ejpam-6849	591	33	)	)	PUNCT
ejpam-6849	591	34	.	.	PUNCT
ejpam-6849	592	1	if	if	SCONJ
ejpam-6849	592	2	szw	szw	VERB
ejpam-6849	592	3	̸=	̸=	PROPN
ejpam-6849	592	4	∅	∅	NOUN
ejpam-6849	592	5	,	,	PUNCT
ejpam-6849	592	6	then	then	ADV
ejpam-6849	592	7	since	since	SCONJ
ejpam-6849	592	8	z	z	PROPN
ejpam-6849	592	9	/∈	/∈	PUNCT
ejpam-6849	592	10	s2	s2	PROPN
ejpam-6849	592	11	,	,	PUNCT
ejpam-6849	592	12	w	w	PROPN
ejpam-6849	592	13	∈	∈	PROPN
ejpam-6849	592	14	s2	s2	PROPN
ejpam-6849	592	15	.	.	PUNCT
ejpam-6849	593	1	in	in	ADP
ejpam-6849	593	2	this	this	DET
ejpam-6849	593	3	case	case	NOUN
ejpam-6849	593	4	,	,	PUNCT
ejpam-6849	593	5	z	z	PROPN
ejpam-6849	593	6	∈	∈	PROPN
ejpam-6849	593	7	ng(s∗	ng(s∗	NUM
ejpam-6849	593	8	)	)	PUNCT
ejpam-6849	593	9	.	.	PUNCT
ejpam-6849	594	1	suppose	suppose	VERB
ejpam-6849	594	2	that	that	SCONJ
ejpam-6849	594	3	szw	szw	NOUN
ejpam-6849	594	4	=	=	PRON
ejpam-6849	594	5	∅.	∅.	AUX
ejpam-6849	594	6	pick	pick	VERB
ejpam-6849	594	7	t	t	PROPN
ejpam-6849	594	8	∈	∈	PROPN
ejpam-6849	594	9	v	v	PROPN
ejpam-6849	594	10	(	(	PUNCT
ejpam-6849	594	11	hzw	hzw	NOUN
ejpam-6849	594	12	)	)	PUNCT
ejpam-6849	594	13	.	.	PUNCT
ejpam-6849	595	1	there	there	PRON
ejpam-6849	595	2	exists	exist	VERB
ejpam-6849	595	3	u	u	PROPN
ejpam-6849	595	4	∈	∈	PROPN
ejpam-6849	595	5	s	s	X
ejpam-6849	595	6	for	for	ADP
ejpam-6849	595	7	which	which	PRON
ejpam-6849	595	8	u	u	NOUN
ejpam-6849	595	9	ve	ve	VERB
ejpam-6849	595	10	-	-	PUNCT
ejpam-6849	595	11	dominates	dominate	VERB
ejpam-6849	595	12	zt	zt	PROPN
ejpam-6849	595	13	.	.	PUNCT
ejpam-6849	596	1	if	if	SCONJ
ejpam-6849	596	2	u	u	PROPN
ejpam-6849	596	3	∈	∈	PROPN
ejpam-6849	596	4	v	v	X
ejpam-6849	596	5	(	(	PUNCT
ejpam-6849	596	6	g	g	NOUN
ejpam-6849	596	7	)	)	PUNCT
ejpam-6849	596	8	,	,	PUNCT
ejpam-6849	596	9	then	then	ADV
ejpam-6849	596	10	u	u	PROPN
ejpam-6849	596	11	∈	∈	PROPN
ejpam-6849	596	12	s1	s1	NOUN
ejpam-6849	596	13	and	and	CCONJ
ejpam-6849	596	14	uz	uz	PROPN
ejpam-6849	596	15	∈	∈	PROPN
ejpam-6849	596	16	e(g	e(g	PROPN
ejpam-6849	596	17	)	)	PUNCT
ejpam-6849	597	1	so	so	SCONJ
ejpam-6849	597	2	that	that	SCONJ
ejpam-6849	597	3	z	z	PROPN
ejpam-6849	597	4	∈	∈	PROPN
ejpam-6849	597	5	ng(s∗	ng(s∗	NUM
ejpam-6849	597	6	)	)	PUNCT
ejpam-6849	597	7	.	.	PUNCT
ejpam-6849	598	1	if	if	SCONJ
ejpam-6849	598	2	u	u	PROPN
ejpam-6849	598	3	/∈	/∈	VERB
ejpam-6849	598	4	v	v	INTJ
ejpam-6849	598	5	(	(	PUNCT
ejpam-6849	598	6	g	g	NOUN
ejpam-6849	598	7	)	)	PUNCT
ejpam-6849	598	8	,	,	PUNCT
ejpam-6849	598	9	then	then	ADV
ejpam-6849	598	10	u	u	PROPN
ejpam-6849	598	11	∈	∈	PROPN
ejpam-6849	598	12	svz	svz	NOUN
ejpam-6849	598	13	for	for	ADP
ejpam-6849	598	14	some	some	PRON
ejpam-6849	598	15	v	v	ADP
ejpam-6849	598	16	∈	∈	PROPN
ejpam-6849	598	17	ng(z	ng(z	NUM
ejpam-6849	598	18	)	)	PUNCT
ejpam-6849	598	19	.	.	PUNCT
ejpam-6849	599	1	then	then	ADV
ejpam-6849	599	2	xvz	xvz	PUNCT
ejpam-6849	600	1	=	=	SYM
ejpam-6849	600	2	v	v	PROPN
ejpam-6849	600	3	∈	∈	PROPN
ejpam-6849	600	4	s2	s2	NOUN
ejpam-6849	600	5	so	so	SCONJ
ejpam-6849	600	6	that	that	SCONJ
ejpam-6849	600	7	z	z	PROPN
ejpam-6849	600	8	∈	∈	PROPN
ejpam-6849	600	9	ng(s∗	ng(s∗	NUM
ejpam-6849	600	10	)	)	PUNCT
ejpam-6849	600	11	.	.	PUNCT
ejpam-6849	601	1	since	since	SCONJ
ejpam-6849	601	2	z	z	PROPN
ejpam-6849	601	3	is	be	AUX
ejpam-6849	601	4	arbitrary	arbitrary	ADJ
ejpam-6849	601	5	,	,	PUNCT
ejpam-6849	601	6	s∗	s∗	PROPN
ejpam-6849	601	7	is	be	AUX
ejpam-6849	601	8	a	a	DET
ejpam-6849	601	9	dominating	dominating	NOUN
ejpam-6849	601	10	set	set	NOUN
ejpam-6849	601	11	of	of	ADP
ejpam-6849	601	12	g.	g.	PROPN
ejpam-6849	601	13	hence	hence	ADV
ejpam-6849	601	14	,	,	PUNCT
ejpam-6849	601	15	γ(g	γ(g	PROPN
ejpam-6849	601	16	)	)	PUNCT
ejpam-6849	601	17	≤	≤	NOUN
ejpam-6849	601	18	|s∗|	|s∗|	NUM
ejpam-6849	601	19	≤	≤	NUM
ejpam-6849	601	20	|s|	|s|	PROPN
ejpam-6849	601	21	.	.	PUNCT
ejpam-6849	602	1	since	since	SCONJ
ejpam-6849	602	2	s	s	NOUN
ejpam-6849	602	3	is	be	AUX
ejpam-6849	602	4	arbitrary	arbitrary	ADJ
ejpam-6849	602	5	,	,	PUNCT
ejpam-6849	602	6	γ(g	γ(g	PROPN
ejpam-6849	602	7	)	)	PUNCT
ejpam-6849	602	8	≤	≤	PUNCT
ejpam-6849	603	1	γve(g	γve(g	PUNCT
ejpam-6849	603	2	⋄	⋄	NOUN
ejpam-6849	603	3	h	h	NOUN
ejpam-6849	603	4	)	)	PUNCT
ejpam-6849	603	5	.	.	PUNCT
ejpam-6849	604	1	this	this	PRON
ejpam-6849	604	2	completely	completely	ADV
ejpam-6849	604	3	proves	prove	VERB
ejpam-6849	604	4	(	(	PUNCT
ejpam-6849	604	5	i	i	NOUN
ejpam-6849	604	6	)	)	PUNCT
ejpam-6849	604	7	.	.	PUNCT
ejpam-6849	605	1	now	now	ADV
ejpam-6849	605	2	we	we	PRON
ejpam-6849	605	3	prove	prove	VERB
ejpam-6849	605	4	(	(	PUNCT
ejpam-6849	605	5	ii	ii	NOUN
ejpam-6849	605	6	)	)	PUNCT
ejpam-6849	605	7	.	.	PUNCT
ejpam-6849	606	1	note	note	VERB
ejpam-6849	606	2	first	first	ADV
ejpam-6849	606	3	that	that	SCONJ
ejpam-6849	606	4	if	if	SCONJ
ejpam-6849	606	5	s	s	VERB
ejpam-6849	606	6	=	=	SYM
ejpam-6849	606	7	v	v	X
ejpam-6849	606	8	(	(	PUNCT
ejpam-6849	606	9	g)\{x	g)\{x	PROPN
ejpam-6849	606	10	}	}	PUNCT
ejpam-6849	606	11	,	,	PUNCT
ejpam-6849	606	12	where	where	SCONJ
ejpam-6849	606	13	x	x	X
ejpam-6849	606	14	∈	∈	PROPN
ejpam-6849	606	15	v	v	X
ejpam-6849	606	16	(	(	PUNCT
ejpam-6849	606	17	g	g	NOUN
ejpam-6849	606	18	)	)	PUNCT
ejpam-6849	606	19	,	,	PUNCT
ejpam-6849	606	20	then	then	ADV
ejpam-6849	606	21	ng(v	ng(v	PUNCT
ejpam-6849	606	22	(	(	PUNCT
ejpam-6849	606	23	g)\	g)\	NOUN
ejpam-6849	606	24	s	s	PART
ejpam-6849	606	25	)	)	PUNCT
ejpam-6849	606	26	=	=	SYM
ejpam-6849	606	27	ng(x	ng(x	X
ejpam-6849	606	28	)	)	PUNCT
ejpam-6849	606	29	⊆	⊆	NUM
ejpam-6849	606	30	s.	s.	PROPN
ejpam-6849	606	31	put	put	VERB
ejpam-6849	606	32	α	α	NOUN
ejpam-6849	606	33	=	=	NOUN
ejpam-6849	606	34	min{|s|	min{|s|	NOUN
ejpam-6849	606	35	:	:	PUNCT
ejpam-6849	606	36	s	s	VERB
ejpam-6849	606	37	⊆	⊆	NUM
ejpam-6849	606	38	v	v	NOUN
ejpam-6849	606	39	(	(	PUNCT
ejpam-6849	606	40	g	g	NOUN
ejpam-6849	606	41	)	)	PUNCT
ejpam-6849	606	42	and	and	CCONJ
ejpam-6849	606	43	ng(v	ng(v	NUM
ejpam-6849	606	44	(	(	PUNCT
ejpam-6849	606	45	g	g	NOUN
ejpam-6849	606	46	)	)	PUNCT
ejpam-6849	606	47	\	\	PROPN
ejpam-6849	607	1	s	s	X
ejpam-6849	607	2	)	)	PUNCT
ejpam-6849	607	3	⊆	⊆	NUM
ejpam-6849	607	4	s	s	NOUN
ejpam-6849	607	5	}	}	PUNCT
ejpam-6849	607	6	.	.	PUNCT
ejpam-6849	608	1	let	let	VERB
ejpam-6849	608	2	s	s	PRON
ejpam-6849	608	3	⊆	⊆	NUM
ejpam-6849	608	4	v	v	NOUN
ejpam-6849	608	5	(	(	PUNCT
ejpam-6849	608	6	g	g	NOUN
ejpam-6849	608	7	)	)	PUNCT
ejpam-6849	608	8	j.	j.	PROPN
ejpam-6849	608	9	tayab	tayab	PROPN
ejpam-6849	608	10	,	,	PUNCT
ejpam-6849	608	11	f.	f.	PROPN
ejpam-6849	608	12	p.	p.	PROPN
ejpam-6849	608	13	jamil	jamil	PROPN
ejpam-6849	608	14	,	,	PUNCT
ejpam-6849	608	15	i.	i.	PROPN
ejpam-6849	608	16	aniversario	aniversario	PROPN
ejpam-6849	608	17	/	/	SYM
ejpam-6849	608	18	eur	eur	PROPN
ejpam-6849	608	19	.	.	PUNCT
ejpam-6849	609	1	j.	j.	PROPN
ejpam-6849	609	2	pure	pure	PROPN
ejpam-6849	609	3	appl	appl	PROPN
ejpam-6849	609	4	.	.	PROPN
ejpam-6849	609	5	math	math	PROPN
ejpam-6849	609	6	,	,	PUNCT
ejpam-6849	609	7	18	18	NUM
ejpam-6849	609	8	(	(	PUNCT
ejpam-6849	609	9	4	4	NUM
ejpam-6849	609	10	)	)	PUNCT
ejpam-6849	609	11	(	(	PUNCT
ejpam-6849	609	12	2025	2025	NUM
ejpam-6849	609	13	)	)	PUNCT
ejpam-6849	609	14	,	,	PUNCT
ejpam-6849	609	15	6849	6849	NUM
ejpam-6849	609	16	16	16	NUM
ejpam-6849	609	17	of	of	ADP
ejpam-6849	609	18	19	19	NUM
ejpam-6849	609	19	such	such	ADJ
ejpam-6849	609	20	that	that	PRON
ejpam-6849	609	21	ng(x	ng(x	NUM
ejpam-6849	609	22	)	)	PUNCT
ejpam-6849	610	1	⊆	⊆	NUM
ejpam-6849	610	2	s	s	NOUN
ejpam-6849	610	3	for	for	ADP
ejpam-6849	610	4	each	each	DET
ejpam-6849	610	5	x	x	SYM
ejpam-6849	610	6	∈	∈	PROPN
ejpam-6849	610	7	v	v	ADP
ejpam-6849	610	8	(	(	PUNCT
ejpam-6849	610	9	g	g	NOUN
ejpam-6849	610	10	)	)	PUNCT
ejpam-6849	610	11	\	\	PUNCT
ejpam-6849	611	1	s.	s.	PROPN
ejpam-6849	611	2	write	write	VERB
ejpam-6849	611	3	s	s	PROPN
ejpam-6849	611	4	=	=	PUNCT
ejpam-6849	611	5	a	a	DET
ejpam-6849	611	6	∪	∪	X
ejpam-6849	611	7	(	(	PUNCT
ejpam-6849	611	8	∪uv∈e(g)suv	∪uv∈e(g)suv	PROPN
ejpam-6849	611	9	)	)	PUNCT
ejpam-6849	611	10	,	,	PUNCT
ejpam-6849	611	11	where	where	SCONJ
ejpam-6849	611	12	a	a	PRON
ejpam-6849	611	13	=	=	X
ejpam-6849	611	14	s	s	X
ejpam-6849	611	15	and	and	CCONJ
ejpam-6849	611	16	suv	suv	NOUN
ejpam-6849	611	17	=	=	NOUN
ejpam-6849	611	18	∅	∅	NOUN
ejpam-6849	611	19	for	for	ADP
ejpam-6849	611	20	all	all	DET
ejpam-6849	611	21	uv	uv	PROPN
ejpam-6849	611	22	∈	∈	PROPN
ejpam-6849	611	23	e(g	e(g	PROPN
ejpam-6849	611	24	)	)	PUNCT
ejpam-6849	611	25	.	.	PUNCT
ejpam-6849	612	1	note	note	VERB
ejpam-6849	612	2	that	that	SCONJ
ejpam-6849	612	3	by	by	ADP
ejpam-6849	612	4	the	the	DET
ejpam-6849	612	5	definition	definition	NOUN
ejpam-6849	612	6	of	of	ADP
ejpam-6849	612	7	s	s	PROPN
ejpam-6849	612	8	,	,	PUNCT
ejpam-6849	612	9	{	{	PUNCT
ejpam-6849	612	10	u	u	NOUN
ejpam-6849	612	11	,	,	PUNCT
ejpam-6849	612	12	v	v	NOUN
ejpam-6849	612	13	}	}	PUNCT
ejpam-6849	612	14	∩	∩	NOUN
ejpam-6849	612	15	a	a	DET
ejpam-6849	612	16	̸=	̸=	PROPN
ejpam-6849	612	17	∅	∅	NOUN
ejpam-6849	612	18	for	for	ADP
ejpam-6849	612	19	all	all	DET
ejpam-6849	612	20	uv	uv	PROPN
ejpam-6849	612	21	∈	∈	PROPN
ejpam-6849	612	22	e(g	e(g	PROPN
ejpam-6849	612	23	)	)	PUNCT
ejpam-6849	612	24	.	.	PUNCT
ejpam-6849	613	1	by	by	ADP
ejpam-6849	613	2	proposition	proposition	NOUN
ejpam-6849	613	3	14	14	NUM
ejpam-6849	613	4	,	,	PUNCT
ejpam-6849	613	5	s	s	VERB
ejpam-6849	613	6	is	be	AUX
ejpam-6849	613	7	a	a	DET
ejpam-6849	613	8	ve	ve	NOUN
ejpam-6849	613	9	-	-	PUNCT
ejpam-6849	613	10	dominating	dominate	VERB
ejpam-6849	613	11	set	set	NOUN
ejpam-6849	613	12	of	of	ADP
ejpam-6849	613	13	g	g	PROPN
ejpam-6849	613	14	⋄	⋄	PROPN
ejpam-6849	613	15	h.	h.	PROPN
ejpam-6849	613	16	consequently	consequently	ADV
ejpam-6849	613	17	,	,	PUNCT
ejpam-6849	613	18	γve(g	γve(g	PROPN
ejpam-6849	613	19	⋄	⋄	NOUN
ejpam-6849	613	20	h	h	NOUN
ejpam-6849	613	21	)	)	PUNCT
ejpam-6849	613	22	≤	≤	NUM
ejpam-6849	613	23	|s|	|s|	PROPN
ejpam-6849	613	24	.	.	PUNCT
ejpam-6849	614	1	since	since	SCONJ
ejpam-6849	614	2	s	s	NOUN
ejpam-6849	614	3	is	be	AUX
ejpam-6849	614	4	arbitrary	arbitrary	ADJ
ejpam-6849	614	5	,	,	PUNCT
ejpam-6849	614	6	γve(g	γve(g	PROPN
ejpam-6849	614	7	⋄	⋄	NOUN
ejpam-6849	614	8	h	h	NOUN
ejpam-6849	614	9	)	)	PUNCT
ejpam-6849	614	10	≤	≤	NOUN
ejpam-6849	614	11	α	α	X
ejpam-6849	614	12	.	.	PUNCT
ejpam-6849	615	1	for	for	ADP
ejpam-6849	615	2	the	the	DET
ejpam-6849	615	3	other	other	ADJ
ejpam-6849	615	4	inequality	inequality	NOUN
ejpam-6849	615	5	,	,	PUNCT
ejpam-6849	615	6	let	let	VERB
ejpam-6849	615	7	s	s	PRON
ejpam-6849	615	8	⊆	⊆	NUM
ejpam-6849	615	9	v	v	NOUN
ejpam-6849	615	10	(	(	PUNCT
ejpam-6849	615	11	g	g	PROPN
ejpam-6849	615	12	⋄	⋄	PROPN
ejpam-6849	615	13	h	h	NOUN
ejpam-6849	615	14	)	)	PUNCT
ejpam-6849	615	15	be	be	AUX
ejpam-6849	615	16	a	a	DET
ejpam-6849	615	17	γve	γve	NOUN
ejpam-6849	615	18	-	-	PUNCT
ejpam-6849	615	19	set	set	NOUN
ejpam-6849	615	20	of	of	ADP
ejpam-6849	615	21	g	g	PROPN
ejpam-6849	615	22	⋄	⋄	PROPN
ejpam-6849	615	23	h.	h.	NOUN
ejpam-6849	615	24	by	by	ADP
ejpam-6849	615	25	proposition	proposition	NOUN
ejpam-6849	615	26	14	14	NUM
ejpam-6849	615	27	,	,	PUNCT
ejpam-6849	615	28	s	s	VERB
ejpam-6849	615	29	=	=	PUNCT
ejpam-6849	615	30	a	a	DET
ejpam-6849	615	31	∪	∪	X
ejpam-6849	615	32	(	(	PUNCT
ejpam-6849	615	33	∪uv∈e(g)suv	∪uv∈e(g)suv	PROPN
ejpam-6849	615	34	)	)	PUNCT
ejpam-6849	615	35	,	,	PUNCT
ejpam-6849	615	36	where	where	SCONJ
ejpam-6849	615	37	a	a	DET
ejpam-6849	615	38	=	=	SYM
ejpam-6849	615	39	s	s	NOUN
ejpam-6849	615	40	∩	∩	ADJ
ejpam-6849	615	41	v	v	X
ejpam-6849	615	42	(	(	PUNCT
ejpam-6849	615	43	g	g	NOUN
ejpam-6849	615	44	)	)	PUNCT
ejpam-6849	615	45	and	and	CCONJ
ejpam-6849	615	46	suv	suv	PROPN
ejpam-6849	615	47	⊆	⊆	NUM
ejpam-6849	615	48	v	v	NOUN
ejpam-6849	615	49	(	(	PUNCT
ejpam-6849	615	50	huv	huv	PROPN
ejpam-6849	615	51	)	)	PUNCT
ejpam-6849	615	52	with	with	ADP
ejpam-6849	615	53	suv	suv	PROPN
ejpam-6849	615	54	a	a	DET
ejpam-6849	615	55	ve	ve	NOUN
ejpam-6849	615	56	-	-	PUNCT
ejpam-6849	615	57	dominating	dominate	VERB
ejpam-6849	615	58	set	set	NOUN
ejpam-6849	615	59	of	of	ADP
ejpam-6849	615	60	huv	huv	PROPN
ejpam-6849	615	61	,	,	PUNCT
ejpam-6849	615	62	hence	hence	ADV
ejpam-6849	615	63	nonempty	nonempty	VERB
ejpam-6849	615	64	,	,	PUNCT
ejpam-6849	615	65	whenever	whenever	SCONJ
ejpam-6849	615	66	{	{	PUNCT
ejpam-6849	615	67	u	u	NOUN
ejpam-6849	615	68	,	,	PUNCT
ejpam-6849	615	69	v}∩a	v}∩a	ADJ
ejpam-6849	615	70	=	=	PUNCT
ejpam-6849	615	71	∅.	∅.	NOUN
ejpam-6849	615	72	for	for	ADP
ejpam-6849	615	73	each	each	DET
ejpam-6849	615	74	uv	uv	PROPN
ejpam-6849	615	75	∈	∈	PROPN
ejpam-6849	615	76	e(g	e(g	PROPN
ejpam-6849	615	77	)	)	PUNCT
ejpam-6849	615	78	with	with	ADP
ejpam-6849	615	79	{	{	PUNCT
ejpam-6849	615	80	u	u	NOUN
ejpam-6849	615	81	,	,	PUNCT
ejpam-6849	615	82	v}∩a	v}∩a	ADJ
ejpam-6849	615	83	=	=	NOUN
ejpam-6849	615	84	∅	∅	NOUN
ejpam-6849	615	85	,	,	PUNCT
ejpam-6849	615	86	put	put	VERB
ejpam-6849	615	87	either	either	CCONJ
ejpam-6849	615	88	xuv	xuv	PUNCT
ejpam-6849	615	89	=	=	SYM
ejpam-6849	615	90	u	u	NOUN
ejpam-6849	615	91	or	or	CCONJ
ejpam-6849	615	92	xuv	xuv	NOUN
ejpam-6849	615	93	=	=	X
ejpam-6849	616	1	v.	v.	ADP
ejpam-6849	616	2	define	define	VERB
ejpam-6849	616	3	b	b	NOUN
ejpam-6849	616	4	=	=	SYM
ejpam-6849	616	5	{	{	PUNCT
ejpam-6849	616	6	xuv	xuv	NOUN
ejpam-6849	616	7	:	:	PUNCT
ejpam-6849	616	8	uv	uv	PROPN
ejpam-6849	616	9	∈	∈	PROPN
ejpam-6849	616	10	e(g	e(g	PROPN
ejpam-6849	616	11	)	)	PUNCT
ejpam-6849	616	12	with	with	ADP
ejpam-6849	616	13	{	{	PUNCT
ejpam-6849	616	14	u	u	NOUN
ejpam-6849	616	15	,	,	PUNCT
ejpam-6849	616	16	v	v	NOUN
ejpam-6849	616	17	}	}	PUNCT
ejpam-6849	616	18	∩	∩	NOUN
ejpam-6849	616	19	a	a	DET
ejpam-6849	616	20	=	=	NOUN
ejpam-6849	616	21	∅	∅	NOUN
ejpam-6849	616	22	}	}	PUNCT
ejpam-6849	616	23	and	and	CCONJ
ejpam-6849	616	24	put	put	VERB
ejpam-6849	616	25	s∗	s∗	PROPN
ejpam-6849	616	26	=	=	PUNCT
ejpam-6849	616	27	a	a	DET
ejpam-6849	616	28	∪	∪	X
ejpam-6849	616	29	b.	b.	NOUN
ejpam-6849	616	30	it	it	PRON
ejpam-6849	616	31	is	be	AUX
ejpam-6849	616	32	worth	worth	ADJ
ejpam-6849	616	33	noting	note	VERB
ejpam-6849	616	34	that	that	SCONJ
ejpam-6849	616	35	a	a	DET
ejpam-6849	616	36	∩	∩	ADJ
ejpam-6849	616	37	b	b	NOUN
ejpam-6849	616	38	=	=	NOUN
ejpam-6849	616	39	∅	∅	NOUN
ejpam-6849	616	40	and	and	CCONJ
ejpam-6849	616	41	|s∗|	|s∗|	NUM
ejpam-6849	616	42	=	=	SYM
ejpam-6849	616	43	|a|	|a|	PROPN
ejpam-6849	616	44	+	+	PROPN
ejpam-6849	616	45	|b|	|b|	PROPN
ejpam-6849	616	46	≤	≤	PROPN
ejpam-6849	616	47	|a|	|a|	PROPN
ejpam-6849	616	48	+	+	CCONJ
ejpam-6849	616	49	∑	∑	NOUN
ejpam-6849	616	50	uv∈e(g	uv∈e(g	NOUN
ejpam-6849	616	51	)	)	PUNCT
ejpam-6849	616	52	|suv|	|suv|	NOUN
ejpam-6849	616	53	=	=	PUNCT
ejpam-6849	616	54	|s|	|s|	PROPN
ejpam-6849	616	55	=	=	SYM
ejpam-6849	616	56	γve(g	γve(g	PROPN
ejpam-6849	616	57	⋄	⋄	NOUN
ejpam-6849	616	58	h	h	NOUN
ejpam-6849	616	59	)	)	PUNCT
ejpam-6849	616	60	.	.	PUNCT
ejpam-6849	617	1	let	let	VERB
ejpam-6849	617	2	v	v	NUM
ejpam-6849	617	3	∈	∈	PROPN
ejpam-6849	617	4	v	v	NOUN
ejpam-6849	617	5	(	(	PUNCT
ejpam-6849	617	6	g	g	NOUN
ejpam-6849	617	7	)	)	PUNCT
ejpam-6849	617	8	\	\	NOUN
ejpam-6849	618	1	s∗	s∗	PROPN
ejpam-6849	618	2	,	,	PUNCT
ejpam-6849	618	3	and	and	CCONJ
ejpam-6849	618	4	let	let	VERB
ejpam-6849	618	5	u	u	PRON
ejpam-6849	618	6	∈	∈	PROPN
ejpam-6849	618	7	ng(v	ng(v	PUNCT
ejpam-6849	618	8	)	)	PUNCT
ejpam-6849	618	9	.	.	PUNCT
ejpam-6849	619	1	if	if	SCONJ
ejpam-6849	619	2	u	u	PROPN
ejpam-6849	619	3	/∈	/∈	VERB
ejpam-6849	619	4	s∗	s∗	PROPN
ejpam-6849	619	5	,	,	PUNCT
ejpam-6849	619	6	then	then	ADV
ejpam-6849	619	7	{	{	PUNCT
ejpam-6849	619	8	u	u	NOUN
ejpam-6849	619	9	,	,	PUNCT
ejpam-6849	619	10	v	v	NOUN
ejpam-6849	619	11	}	}	PUNCT
ejpam-6849	619	12	∩	∩	NOUN
ejpam-6849	619	13	a	a	DET
ejpam-6849	619	14	=	=	PUNCT
ejpam-6849	619	15	∅.	∅.	NOUN
ejpam-6849	619	16	this	this	PRON
ejpam-6849	619	17	means	mean	VERB
ejpam-6849	619	18	that	that	SCONJ
ejpam-6849	619	19	xuv	xuv	PUNCT
ejpam-6849	619	20	=	=	SYM
ejpam-6849	619	21	u	u	NOUN
ejpam-6849	619	22	or	or	CCONJ
ejpam-6849	619	23	xuv	xuv	NOUN
ejpam-6849	619	24	=	=	SYM
ejpam-6849	619	25	v	v	NOUN
ejpam-6849	619	26	,	,	PUNCT
ejpam-6849	619	27	which	which	PRON
ejpam-6849	619	28	is	be	AUX
ejpam-6849	619	29	impossible	impossible	ADJ
ejpam-6849	619	30	since	since	SCONJ
ejpam-6849	619	31	xuv	xuv	PROPN
ejpam-6849	619	32	∈	∈	PROPN
ejpam-6849	619	33	b	b	PROPN
ejpam-6849	620	1	⊆	⊆	NUM
ejpam-6849	620	2	s∗.	s∗.	ADJ
ejpam-6849	620	3	therefore	therefore	ADV
ejpam-6849	620	4	,	,	PUNCT
ejpam-6849	620	5	u	u	PROPN
ejpam-6849	620	6	∈	∈	PROPN
ejpam-6849	620	7	s∗.	s∗.	ADJ
ejpam-6849	620	8	since	since	SCONJ
ejpam-6849	620	9	u	u	NOUN
ejpam-6849	620	10	is	be	AUX
ejpam-6849	620	11	arbitrary	arbitrary	ADJ
ejpam-6849	620	12	,	,	PUNCT
ejpam-6849	620	13	ng(v	ng(v	PUNCT
ejpam-6849	620	14	)	)	PUNCT
ejpam-6849	620	15	⊆	⊆	NUM
ejpam-6849	620	16	s∗.	s∗.	ADJ
ejpam-6849	620	17	hence	hence	ADV
ejpam-6849	620	18	,	,	PUNCT
ejpam-6849	620	19	α	α	NOUN
ejpam-6849	620	20	≤	≤	NOUN
ejpam-6849	620	21	|s∗|	|s∗|	PRON
ejpam-6849	620	22	≤	≤	NUM
ejpam-6849	621	1	γve(g	γve(g	PUNCT
ejpam-6849	621	2	⋄	⋄	NOUN
ejpam-6849	621	3	h	h	NOUN
ejpam-6849	621	4	)	)	PUNCT
ejpam-6849	621	5	.	.	PUNCT
ejpam-6849	622	1	■	■	PUNCT
ejpam-6849	622	2	it	it	PRON
ejpam-6849	622	3	is	be	AUX
ejpam-6849	622	4	clear	clear	ADJ
ejpam-6849	622	5	from	from	ADP
ejpam-6849	622	6	corollary	corollary	ADJ
ejpam-6849	622	7	6(ii	6(ii	NOUN
ejpam-6849	622	8	)	)	PUNCT
ejpam-6849	622	9	that	that	SCONJ
ejpam-6849	622	10	if	if	SCONJ
ejpam-6849	622	11	g	g	PROPN
ejpam-6849	622	12	is	be	AUX
ejpam-6849	622	13	a	a	DET
ejpam-6849	622	14	connected	connected	ADJ
ejpam-6849	622	15	nontrivial	nontrivial	ADJ
ejpam-6849	622	16	graph	graph	NOUN
ejpam-6849	622	17	and	and	CCONJ
ejpam-6849	622	18	h	h	NOUN
ejpam-6849	622	19	is	be	AUX
ejpam-6849	622	20	a	a	DET
ejpam-6849	622	21	nonempty	nonempty	ADJ
ejpam-6849	622	22	graph	graph	NOUN
ejpam-6849	622	23	,	,	PUNCT
ejpam-6849	622	24	then	then	ADV
ejpam-6849	622	25	γ(g	γ(g	PROPN
ejpam-6849	622	26	)	)	PUNCT
ejpam-6849	622	27	≤	≤	PUNCT
ejpam-6849	623	1	γve(g	γve(g	PUNCT
ejpam-6849	623	2	⋄	⋄	NOUN
ejpam-6849	623	3	h	h	NOUN
ejpam-6849	623	4	)	)	PUNCT
ejpam-6849	623	5	≤	≤	NOUN
ejpam-6849	623	6	|v	|v	X
ejpam-6849	623	7	(	(	PUNCT
ejpam-6849	623	8	g)|	g)|	INTJ
ejpam-6849	623	9	−	−	NOUN
ejpam-6849	623	10	1	1	NUM
ejpam-6849	623	11	.	.	PUNCT
ejpam-6849	624	1	in	in	ADP
ejpam-6849	624	2	view	view	NOUN
ejpam-6849	624	3	of	of	ADP
ejpam-6849	624	4	proposition	proposition	NOUN
ejpam-6849	624	5	14	14	NUM
ejpam-6849	624	6	,	,	PUNCT
ejpam-6849	624	7	the	the	DET
ejpam-6849	624	8	following	following	NOUN
ejpam-6849	624	9	is	be	AUX
ejpam-6849	624	10	clear	clear	ADJ
ejpam-6849	624	11	.	.	PUNCT
ejpam-6849	625	1	proposition	proposition	NOUN
ejpam-6849	625	2	15	15	NUM
ejpam-6849	625	3	.	.	PUNCT
ejpam-6849	626	1	let	let	VERB
ejpam-6849	626	2	g	g	PRON
ejpam-6849	626	3	be	be	AUX
ejpam-6849	626	4	a	a	DET
ejpam-6849	626	5	nontrivial	nontrivial	ADJ
ejpam-6849	626	6	connected	connect	VERB
ejpam-6849	626	7	graph	graph	NOUN
ejpam-6849	626	8	and	and	CCONJ
ejpam-6849	626	9	h	h	NOUN
ejpam-6849	626	10	be	be	AUX
ejpam-6849	626	11	any	any	DET
ejpam-6849	626	12	nonempty	nonempty	ADJ
ejpam-6849	626	13	graph	graph	NOUN
ejpam-6849	626	14	.	.	PUNCT
ejpam-6849	627	1	then	then	ADV
ejpam-6849	627	2	s	s	VERB
ejpam-6849	627	3	⊆	⊆	NUM
ejpam-6849	627	4	v	v	NOUN
ejpam-6849	627	5	(	(	PUNCT
ejpam-6849	627	6	g	g	PROPN
ejpam-6849	627	7	⋄	⋄	PROPN
ejpam-6849	627	8	h	h	NOUN
ejpam-6849	627	9	)	)	PUNCT
ejpam-6849	627	10	is	be	AUX
ejpam-6849	627	11	a	a	DET
ejpam-6849	627	12	total	total	ADJ
ejpam-6849	627	13	ve	ve	NOUN
ejpam-6849	627	14	-	-	PUNCT
ejpam-6849	627	15	dominating	dominate	VERB
ejpam-6849	627	16	set	set	NOUN
ejpam-6849	627	17	of	of	ADP
ejpam-6849	627	18	g	g	PROPN
ejpam-6849	627	19	⋄	⋄	PROPN
ejpam-6849	627	20	h	h	NOUN
ejpam-6849	627	21	if	if	SCONJ
ejpam-6849	628	1	and	and	CCONJ
ejpam-6849	628	2	only	only	ADV
ejpam-6849	628	3	if	if	SCONJ
ejpam-6849	628	4	s	s	VERB
ejpam-6849	628	5	=	=	NOUN
ejpam-6849	628	6	a	a	DET
ejpam-6849	628	7	∪	∪	X
ejpam-6849	628	8	(	(	PUNCT
ejpam-6849	628	9	∪uv∈e(g)suv	∪uv∈e(g)suv	PROPN
ejpam-6849	628	10	)	)	PUNCT
ejpam-6849	628	11	,	,	PUNCT
ejpam-6849	628	12	(	(	PUNCT
ejpam-6849	628	13	9	9	X
ejpam-6849	628	14	)	)	PUNCT
ejpam-6849	628	15	where	where	SCONJ
ejpam-6849	628	16	a	a	DET
ejpam-6849	628	17	⊆	⊆	NUM
ejpam-6849	628	18	v	v	NOUN
ejpam-6849	628	19	(	(	PUNCT
ejpam-6849	628	20	g	g	NOUN
ejpam-6849	628	21	)	)	PUNCT
ejpam-6849	628	22	and	and	CCONJ
ejpam-6849	628	23	suv	suv	PROPN
ejpam-6849	628	24	⊆	⊆	NUM
ejpam-6849	628	25	v	v	NOUN
ejpam-6849	628	26	(	(	PUNCT
ejpam-6849	628	27	huv	huv	PROPN
ejpam-6849	628	28	)	)	PUNCT
ejpam-6849	628	29	for	for	ADP
ejpam-6849	628	30	each	each	DET
ejpam-6849	628	31	uv	uv	PROPN
ejpam-6849	628	32	∈	∈	PROPN
ejpam-6849	628	33	e(g	e(g	NOUN
ejpam-6849	628	34	)	)	PUNCT
ejpam-6849	628	35	satisfying	satisfy	VERB
ejpam-6849	628	36	the	the	DET
ejpam-6849	628	37	following	following	NOUN
ejpam-6849	628	38	:	:	PUNCT
ejpam-6849	628	39	(	(	PUNCT
ejpam-6849	628	40	i	i	NOUN
ejpam-6849	628	41	)	)	PUNCT
ejpam-6849	628	42	suv	suv	PROPN
ejpam-6849	628	43	is	be	AUX
ejpam-6849	628	44	a	a	DET
ejpam-6849	628	45	total	total	ADJ
ejpam-6849	628	46	ve	ve	NOUN
ejpam-6849	628	47	-	-	PUNCT
ejpam-6849	628	48	dominating	dominate	VERB
ejpam-6849	628	49	set	set	NOUN
ejpam-6849	628	50	of	of	ADP
ejpam-6849	628	51	huv	huv	PROPN
ejpam-6849	628	52	for	for	ADP
ejpam-6849	628	53	each	each	DET
ejpam-6849	628	54	uv	uv	PROPN
ejpam-6849	628	55	∈	∈	PROPN
ejpam-6849	628	56	e(g	e(g	PROPN
ejpam-6849	628	57	)	)	PUNCT
ejpam-6849	628	58	for	for	ADP
ejpam-6849	628	59	which	which	PRON
ejpam-6849	628	60	{	{	PUNCT
ejpam-6849	628	61	u	u	NOUN
ejpam-6849	628	62	,	,	PUNCT
ejpam-6849	628	63	v}∩a	v}∩a	VERB
ejpam-6849	628	64	=	=	PUNCT
ejpam-6849	628	65	∅.	∅.	PROPN
ejpam-6849	628	66	(	(	PUNCT
ejpam-6849	628	67	ii	ii	NOUN
ejpam-6849	628	68	)	)	PUNCT
ejpam-6849	628	69	a	a	DET
ejpam-6849	628	70	∩	∩	NOUN
ejpam-6849	628	71	ng(u	ng(u	NOUN
ejpam-6849	628	72	)	)	PUNCT
ejpam-6849	628	73	̸=	̸=	PROPN
ejpam-6849	628	74	∅	∅	NOUN
ejpam-6849	628	75	for	for	ADP
ejpam-6849	628	76	all	all	PRON
ejpam-6849	628	77	u	u	NOUN
ejpam-6849	628	78	∈	∈	PROPN
ejpam-6849	628	79	a	a	DET
ejpam-6849	628	80	for	for	ADP
ejpam-6849	628	81	which	which	PRON
ejpam-6849	628	82	suv	suv	NOUN
ejpam-6849	628	83	=	=	NOUN
ejpam-6849	628	84	∅	∅	NOUN
ejpam-6849	628	85	for	for	ADP
ejpam-6849	628	86	all	all	PRON
ejpam-6849	628	87	v	v	ADP
ejpam-6849	628	88	∈	∈	NOUN
ejpam-6849	628	89	ng(u	ng(u	NOUN
ejpam-6849	628	90	)	)	PUNCT
ejpam-6849	628	91	.	.	PUNCT
ejpam-6849	629	1	corollary	corollary	ADJ
ejpam-6849	629	2	7	7	NUM
ejpam-6849	629	3	.	.	PUNCT
ejpam-6849	630	1	let	let	VERB
ejpam-6849	630	2	g	g	PRON
ejpam-6849	630	3	be	be	AUX
ejpam-6849	630	4	a	a	DET
ejpam-6849	630	5	nontrivial	nontrivial	ADJ
ejpam-6849	630	6	connected	connect	VERB
ejpam-6849	630	7	graph	graph	NOUN
ejpam-6849	630	8	.	.	PUNCT
ejpam-6849	631	1	(	(	PUNCT
ejpam-6849	631	2	i	i	NOUN
ejpam-6849	631	3	)	)	PUNCT
ejpam-6849	631	4	if	if	SCONJ
ejpam-6849	631	5	h	h	NOUN
ejpam-6849	631	6	is	be	AUX
ejpam-6849	631	7	an	an	DET
ejpam-6849	631	8	empty	empty	ADJ
ejpam-6849	631	9	graph	graph	NOUN
ejpam-6849	631	10	,	,	PUNCT
ejpam-6849	631	11	then	then	ADV
ejpam-6849	631	12	γt	γt	PROPN
ejpam-6849	631	13	ve(g	ve(g	PROPN
ejpam-6849	631	14	⋄	⋄	PROPN
ejpam-6849	631	15	h	h	NOUN
ejpam-6849	631	16	)	)	PUNCT
ejpam-6849	631	17	=	=	SYM
ejpam-6849	631	18	γt(g	γt(g	NUM
ejpam-6849	631	19	)	)	PUNCT
ejpam-6849	631	20	.	.	PUNCT
ejpam-6849	632	1	(	(	PUNCT
ejpam-6849	632	2	ii	ii	NOUN
ejpam-6849	632	3	)	)	PUNCT
ejpam-6849	632	4	if	if	SCONJ
ejpam-6849	632	5	h	h	NOUN
ejpam-6849	632	6	is	be	AUX
ejpam-6849	632	7	a	a	DET
ejpam-6849	632	8	nonempty	nonempty	ADJ
ejpam-6849	632	9	graph	graph	NOUN
ejpam-6849	632	10	,	,	PUNCT
ejpam-6849	632	11	then	then	ADV
ejpam-6849	632	12	γt	γt	PROPN
ejpam-6849	632	13	ve(g	ve(g	PROPN
ejpam-6849	632	14	⋄	⋄	PROPN
ejpam-6849	632	15	h	h	NOUN
ejpam-6849	632	16	)	)	PUNCT
ejpam-6849	633	1	=	=	NOUN
ejpam-6849	633	2	min{|s|	min{|s|	NOUN
ejpam-6849	633	3	:	:	PUNCT
ejpam-6849	633	4	s	s	VERB
ejpam-6849	633	5	⊆	⊆	NUM
ejpam-6849	633	6	v	v	NOUN
ejpam-6849	633	7	(	(	PUNCT
ejpam-6849	633	8	g	g	NOUN
ejpam-6849	633	9	)	)	PUNCT
ejpam-6849	633	10	with	with	ADP
ejpam-6849	633	11	s	s	PROPN
ejpam-6849	633	12	⊆	⊆	NUM
ejpam-6849	633	13	ng(s	ng(s	NUM
ejpam-6849	633	14	)	)	PUNCT
ejpam-6849	633	15	and	and	CCONJ
ejpam-6849	633	16	ng(v	ng(v	NUM
ejpam-6849	633	17	(	(	PUNCT
ejpam-6849	633	18	g	g	NOUN
ejpam-6849	633	19	)	)	PUNCT
ejpam-6849	633	20	\	\	PROPN
ejpam-6849	634	1	s	s	X
ejpam-6849	634	2	)	)	PUNCT
ejpam-6849	634	3	⊆	⊆	NUM
ejpam-6849	634	4	s	s	NOUN
ejpam-6849	634	5	}	}	PUNCT
ejpam-6849	634	6	.	.	PUNCT
ejpam-6849	635	1	j.	j.	PROPN
ejpam-6849	635	2	tayab	tayab	PROPN
ejpam-6849	635	3	,	,	PUNCT
ejpam-6849	635	4	f.	f.	PROPN
ejpam-6849	635	5	p.	p.	PROPN
ejpam-6849	635	6	jamil	jamil	PROPN
ejpam-6849	635	7	,	,	PUNCT
ejpam-6849	635	8	i.	i.	PROPN
ejpam-6849	635	9	aniversario	aniversario	PROPN
ejpam-6849	635	10	/	/	SYM
ejpam-6849	635	11	eur	eur	PROPN
ejpam-6849	635	12	.	.	PUNCT
ejpam-6849	636	1	j.	j.	PROPN
ejpam-6849	636	2	pure	pure	PROPN
ejpam-6849	636	3	appl	appl	PROPN
ejpam-6849	636	4	.	.	PROPN
ejpam-6849	636	5	math	math	PROPN
ejpam-6849	636	6	,	,	PUNCT
ejpam-6849	636	7	18	18	NUM
ejpam-6849	636	8	(	(	PUNCT
ejpam-6849	636	9	4	4	NUM
ejpam-6849	636	10	)	)	PUNCT
ejpam-6849	636	11	(	(	PUNCT
ejpam-6849	636	12	2025	2025	NUM
ejpam-6849	636	13	)	)	PUNCT
ejpam-6849	636	14	,	,	PUNCT
ejpam-6849	636	15	6849	6849	NUM
ejpam-6849	636	16	17	17	NUM
ejpam-6849	636	17	of	of	ADP
ejpam-6849	636	18	19	19	NUM
ejpam-6849	636	19	proof	proof	NOUN
ejpam-6849	636	20	.	.	PUNCT
ejpam-6849	637	1	assume	assume	VERB
ejpam-6849	637	2	h	h	PROPN
ejpam-6849	637	3	is	be	AUX
ejpam-6849	637	4	an	an	DET
ejpam-6849	637	5	empty	empty	ADJ
ejpam-6849	637	6	graph	graph	NOUN
ejpam-6849	637	7	.	.	PUNCT
ejpam-6849	638	1	let	let	VERB
ejpam-6849	638	2	s	s	PRON
ejpam-6849	638	3	⊆	⊆	NUM
ejpam-6849	638	4	v	v	NOUN
ejpam-6849	638	5	(	(	PUNCT
ejpam-6849	638	6	g	g	NOUN
ejpam-6849	638	7	)	)	PUNCT
ejpam-6849	638	8	be	be	AUX
ejpam-6849	638	9	a	a	DET
ejpam-6849	638	10	γt	γt	NOUN
ejpam-6849	638	11	-	-	NOUN
ejpam-6849	638	12	set	set	NOUN
ejpam-6849	638	13	of	of	ADP
ejpam-6849	638	14	g.	g.	PROPN
ejpam-6849	638	15	since	since	SCONJ
ejpam-6849	638	16	s	s	PROPN
ejpam-6849	638	17	is	be	AUX
ejpam-6849	638	18	a	a	DET
ejpam-6849	638	19	dominating	dominating	NOUN
ejpam-6849	638	20	set	set	NOUN
ejpam-6849	638	21	of	of	ADP
ejpam-6849	638	22	g	g	NOUN
ejpam-6849	638	23	,	,	PUNCT
ejpam-6849	638	24	following	follow	VERB
ejpam-6849	638	25	the	the	DET
ejpam-6849	638	26	necessity	necessity	NOUN
ejpam-6849	638	27	proof	proof	NOUN
ejpam-6849	638	28	of	of	ADP
ejpam-6849	638	29	corollary	corollary	ADJ
ejpam-6849	638	30	6(i	6(i	NUM
ejpam-6849	638	31	)	)	PUNCT
ejpam-6849	638	32	will	will	AUX
ejpam-6849	638	33	show	show	VERB
ejpam-6849	638	34	that	that	SCONJ
ejpam-6849	638	35	s	s	VERB
ejpam-6849	638	36	is	be	AUX
ejpam-6849	638	37	a	a	DET
ejpam-6849	638	38	ve	ve	NOUN
ejpam-6849	638	39	-	-	PUNCT
ejpam-6849	638	40	dominating	dominate	VERB
ejpam-6849	638	41	set	set	NOUN
ejpam-6849	638	42	of	of	ADP
ejpam-6849	638	43	g	g	PROPN
ejpam-6849	638	44	⋄	⋄	PROPN
ejpam-6849	638	45	h.	h.	PROPN
ejpam-6849	638	46	consequently	consequently	ADV
ejpam-6849	638	47	,	,	PUNCT
ejpam-6849	638	48	s	s	VERB
ejpam-6849	638	49	is	be	AUX
ejpam-6849	638	50	a	a	DET
ejpam-6849	638	51	total	total	ADJ
ejpam-6849	638	52	ve	ve	NOUN
ejpam-6849	638	53	-	-	PUNCT
ejpam-6849	638	54	dominating	dominate	VERB
ejpam-6849	638	55	set	set	NOUN
ejpam-6849	638	56	of	of	ADP
ejpam-6849	638	57	g	g	PROPN
ejpam-6849	638	58	⋄	⋄	PROPN
ejpam-6849	638	59	h.	h.	PROPN
ejpam-6849	639	1	thus	thus	ADV
ejpam-6849	639	2	,	,	PUNCT
ejpam-6849	639	3	γt	γt	PROPN
ejpam-6849	639	4	ve(g	ve(g	PROPN
ejpam-6849	639	5	⋄	⋄	PROPN
ejpam-6849	639	6	h	h	PROPN
ejpam-6849	639	7	)	)	PUNCT
ejpam-6849	639	8	≤	≤	NUM
ejpam-6849	639	9	|s|	|s|	PROPN
ejpam-6849	639	10	=	=	PUNCT
ejpam-6849	639	11	γt(g	γt(g	NUM
ejpam-6849	639	12	)	)	PUNCT
ejpam-6849	639	13	.	.	PUNCT
ejpam-6849	640	1	to	to	PART
ejpam-6849	640	2	get	get	VERB
ejpam-6849	640	3	the	the	DET
ejpam-6849	640	4	other	other	ADJ
ejpam-6849	640	5	inequality	inequality	NOUN
ejpam-6849	640	6	,	,	PUNCT
ejpam-6849	640	7	let	let	VERB
ejpam-6849	640	8	s	s	PRON
ejpam-6849	640	9	⊆	⊆	NUM
ejpam-6849	640	10	v	v	NOUN
ejpam-6849	640	11	(	(	PUNCT
ejpam-6849	640	12	g	g	PROPN
ejpam-6849	640	13	⋄	⋄	PROPN
ejpam-6849	640	14	h	h	NOUN
ejpam-6849	640	15	)	)	PUNCT
ejpam-6849	640	16	be	be	VERB
ejpam-6849	640	17	a	a	DET
ejpam-6849	640	18	γt	γt	NOUN
ejpam-6849	640	19	ve	ve	NOUN
ejpam-6849	640	20	-	-	PUNCT
ejpam-6849	640	21	set	set	NOUN
ejpam-6849	640	22	of	of	ADP
ejpam-6849	640	23	g	g	PROPN
ejpam-6849	640	24	⋄	⋄	PROPN
ejpam-6849	640	25	h.	h.	NOUN
ejpam-6849	640	26	define	define	VERB
ejpam-6849	640	27	s1	s1	PROPN
ejpam-6849	640	28	=	=	SYM
ejpam-6849	640	29	s	s	PROPN
ejpam-6849	640	30	∩	∩	ADJ
ejpam-6849	640	31	v	v	X
ejpam-6849	640	32	(	(	PUNCT
ejpam-6849	640	33	g	g	NOUN
ejpam-6849	640	34	)	)	PUNCT
ejpam-6849	640	35	and	and	CCONJ
ejpam-6849	640	36	suv	suv	PROPN
ejpam-6849	640	37	=	=	PROPN
ejpam-6849	640	38	s	s	PROPN
ejpam-6849	640	39	∩	∩	ADJ
ejpam-6849	640	40	v	v	X
ejpam-6849	640	41	(	(	PUNCT
ejpam-6849	640	42	huv	huv	PROPN
ejpam-6849	640	43	)	)	PUNCT
ejpam-6849	640	44	for	for	ADP
ejpam-6849	640	45	each	each	DET
ejpam-6849	640	46	uv	uv	PROPN
ejpam-6849	640	47	∈	∈	PROPN
ejpam-6849	640	48	e(g	e(g	PROPN
ejpam-6849	640	49	)	)	PUNCT
ejpam-6849	640	50	.	.	PUNCT
ejpam-6849	641	1	note	note	VERB
ejpam-6849	641	2	that	that	SCONJ
ejpam-6849	641	3	since	since	SCONJ
ejpam-6849	641	4	s	s	NOUN
ejpam-6849	641	5	is	be	AUX
ejpam-6849	641	6	a	a	DET
ejpam-6849	641	7	total	total	ADJ
ejpam-6849	641	8	ve	ve	NOUN
ejpam-6849	641	9	-	-	PUNCT
ejpam-6849	641	10	dominating	dominate	VERB
ejpam-6849	641	11	set	set	NOUN
ejpam-6849	641	12	,	,	PUNCT
ejpam-6849	641	13	if	if	SCONJ
ejpam-6849	641	14	suv	suv	PROPN
ejpam-6849	641	15	̸=	̸=	PROPN
ejpam-6849	641	16	∅	∅	NOUN
ejpam-6849	641	17	,	,	PUNCT
ejpam-6849	641	18	then	then	ADV
ejpam-6849	641	19	u	u	PROPN
ejpam-6849	641	20	∈	∈	PROPN
ejpam-6849	641	21	s1	s1	NOUN
ejpam-6849	641	22	or	or	CCONJ
ejpam-6849	641	23	v	v	ADP
ejpam-6849	641	24	∈	∈	PROPN
ejpam-6849	641	25	s1	s1	NOUN
ejpam-6849	641	26	.	.	PUNCT
ejpam-6849	642	1	for	for	ADP
ejpam-6849	642	2	each	each	DET
ejpam-6849	642	3	uv	uv	PROPN
ejpam-6849	642	4	∈	∈	PROPN
ejpam-6849	642	5	e(g	e(g	PROPN
ejpam-6849	642	6	)	)	PUNCT
ejpam-6849	642	7	for	for	ADP
ejpam-6849	642	8	which	which	PRON
ejpam-6849	642	9	suv	suv	NOUN
ejpam-6849	642	10	̸=	̸=	PROPN
ejpam-6849	642	11	∅	∅	NOUN
ejpam-6849	642	12	,	,	PUNCT
ejpam-6849	642	13	write	write	VERB
ejpam-6849	642	14	u	u	NOUN
ejpam-6849	642	15	=	=	PUNCT
ejpam-6849	642	16	xuv	xuv	INTJ
ejpam-6849	642	17	whenever	whenever	SCONJ
ejpam-6849	642	18	u	u	NOUN
ejpam-6849	642	19	/∈	/∈	PROPN
ejpam-6849	642	20	s1	s1	NOUN
ejpam-6849	642	21	and	and	CCONJ
ejpam-6849	642	22	write	write	VERB
ejpam-6849	642	23	v	v	NOUN
ejpam-6849	642	24	=	=	PUNCT
ejpam-6849	642	25	xuv	xuv	X
ejpam-6849	642	26	whenever	whenever	SCONJ
ejpam-6849	642	27	v	v	NUM
ejpam-6849	642	28	/∈	/∈	SYM
ejpam-6849	642	29	s1	s1	NOUN
ejpam-6849	642	30	.	.	PUNCT
ejpam-6849	643	1	put	put	VERB
ejpam-6849	643	2	s2	s2	NOUN
ejpam-6849	643	3	=	=	PRON
ejpam-6849	643	4	{	{	PUNCT
ejpam-6849	643	5	xuv	xuv	NOUN
ejpam-6849	643	6	:	:	PUNCT
ejpam-6849	643	7	uv	uv	PROPN
ejpam-6849	643	8	∈	∈	PROPN
ejpam-6849	643	9	e(g	e(g	PROPN
ejpam-6849	643	10	)	)	PUNCT
ejpam-6849	643	11	with	with	ADP
ejpam-6849	643	12	suv	suv	PROPN
ejpam-6849	643	13	̸=	̸=	PROPN
ejpam-6849	643	14	∅	∅	NOUN
ejpam-6849	643	15	}	}	PUNCT
ejpam-6849	643	16	.	.	PUNCT
ejpam-6849	644	1	let	let	VERB
ejpam-6849	644	2	s∗	s∗	PROPN
ejpam-6849	644	3	=	=	SYM
ejpam-6849	644	4	s1	s1	PROPN
ejpam-6849	644	5	∪	∪	X
ejpam-6849	644	6	s2	s2	PROPN
ejpam-6849	644	7	.	.	PUNCT
ejpam-6849	645	1	then	then	ADV
ejpam-6849	645	2	u	u	NOUN
ejpam-6849	645	3	,	,	PUNCT
ejpam-6849	645	4	v	v	PROPN
ejpam-6849	645	5	∈	∈	PROPN
ejpam-6849	645	6	s∗	s∗	NOUN
ejpam-6849	645	7	for	for	ADP
ejpam-6849	645	8	all	all	DET
ejpam-6849	645	9	uv	uv	PROPN
ejpam-6849	645	10	∈	∈	PROPN
ejpam-6849	645	11	e(g	e(g	PROPN
ejpam-6849	645	12	)	)	PUNCT
ejpam-6849	645	13	for	for	ADP
ejpam-6849	645	14	which	which	PRON
ejpam-6849	645	15	suv	suv	NOUN
ejpam-6849	645	16	̸=	̸=	PROPN
ejpam-6849	645	17	∅.	∅.	ADP
ejpam-6849	645	18	first	first	ADV
ejpam-6849	645	19	,	,	PUNCT
ejpam-6849	645	20	let	let	VERB
ejpam-6849	645	21	z	z	NOUN
ejpam-6849	645	22	∈	∈	PROPN
ejpam-6849	645	23	v	v	ADP
ejpam-6849	645	24	(	(	PUNCT
ejpam-6849	645	25	g	g	NOUN
ejpam-6849	645	26	)	)	PUNCT
ejpam-6849	645	27	\	\	VERB
ejpam-6849	646	1	s∗.	s∗.	ADJ
ejpam-6849	646	2	choose	choose	VERB
ejpam-6849	646	3	w	w	PROPN
ejpam-6849	646	4	∈	∈	PROPN
ejpam-6849	646	5	v	v	ADP
ejpam-6849	646	6	(	(	PUNCT
ejpam-6849	646	7	g	g	NOUN
ejpam-6849	646	8	)	)	PUNCT
ejpam-6849	646	9	such	such	ADJ
ejpam-6849	646	10	that	that	SCONJ
ejpam-6849	646	11	zw	zw	PROPN
ejpam-6849	646	12	∈	∈	PROPN
ejpam-6849	646	13	e(g	e(g	PROPN
ejpam-6849	646	14	)	)	PUNCT
ejpam-6849	646	15	.	.	PUNCT
ejpam-6849	647	1	if	if	SCONJ
ejpam-6849	647	2	w	w	PROPN
ejpam-6849	647	3	∈	∈	PROPN
ejpam-6849	647	4	s∗	s∗	PROPN
ejpam-6849	647	5	,	,	PUNCT
ejpam-6849	647	6	then	then	ADV
ejpam-6849	647	7	z	z	PROPN
ejpam-6849	647	8	∈	∈	PROPN
ejpam-6849	647	9	ng(s∗	ng(s∗	NUM
ejpam-6849	647	10	)	)	PUNCT
ejpam-6849	647	11	.	.	PUNCT
ejpam-6849	648	1	suppose	suppose	VERB
ejpam-6849	648	2	that	that	SCONJ
ejpam-6849	648	3	w	w	PROPN
ejpam-6849	648	4	/∈	/∈	INTJ
ejpam-6849	648	5	s∗.	s∗.	ADJ
ejpam-6849	648	6	since	since	SCONJ
ejpam-6849	648	7	s	s	PROPN
ejpam-6849	648	8	is	be	AUX
ejpam-6849	648	9	a	a	DET
ejpam-6849	648	10	ve	ve	NOUN
ejpam-6849	648	11	-	-	PUNCT
ejpam-6849	648	12	dominating	dominating	NOUN
ejpam-6849	648	13	set	set	NOUN
ejpam-6849	648	14	,	,	PUNCT
ejpam-6849	648	15	there	there	PRON
ejpam-6849	648	16	exists	exist	VERB
ejpam-6849	648	17	x	x	X
ejpam-6849	648	18	∈	∈	PROPN
ejpam-6849	648	19	s	s	VERB
ejpam-6849	648	20	such	such	ADJ
ejpam-6849	648	21	that	that	SCONJ
ejpam-6849	648	22	x	x	SYM
ejpam-6849	648	23	ve	ve	AUX
ejpam-6849	648	24	-	-	PUNCT
ejpam-6849	648	25	dominates	dominate	VERB
ejpam-6849	648	26	zw	zw	X
ejpam-6849	648	27	.	.	PUNCT
ejpam-6849	649	1	since	since	SCONJ
ejpam-6849	649	2	z	z	PROPN
ejpam-6849	649	3	/∈	/∈	PUNCT
ejpam-6849	649	4	s∗	s∗	PROPN
ejpam-6849	649	5	and	and	CCONJ
ejpam-6849	649	6	w	w	PROPN
ejpam-6849	649	7	/∈	/∈	PROPN
ejpam-6849	649	8	s∗	s∗	PROPN
ejpam-6849	649	9	,	,	PUNCT
ejpam-6849	649	10	x	x	SYM
ejpam-6849	649	11	∈	∈	PROPN
ejpam-6849	649	12	s1	s1	NOUN
ejpam-6849	649	13	.	.	PUNCT
ejpam-6849	650	1	this	this	PRON
ejpam-6849	650	2	means	mean	VERB
ejpam-6849	650	3	that	that	SCONJ
ejpam-6849	650	4	z	z	PROPN
ejpam-6849	650	5	∈	∈	PROPN
ejpam-6849	650	6	ng(s∗	ng(s∗	NUM
ejpam-6849	650	7	)	)	PUNCT
ejpam-6849	650	8	.	.	PUNCT
ejpam-6849	651	1	next	next	ADV
ejpam-6849	651	2	,	,	PUNCT
ejpam-6849	651	3	let	let	VERB
ejpam-6849	651	4	z	z	NOUN
ejpam-6849	651	5	∈	∈	PROPN
ejpam-6849	651	6	s∗.	s∗.	ADJ
ejpam-6849	651	7	if	if	SCONJ
ejpam-6849	651	8	z	z	PROPN
ejpam-6849	651	9	∈	∈	PROPN
ejpam-6849	651	10	s2	s2	PROPN
ejpam-6849	651	11	,	,	PUNCT
ejpam-6849	651	12	then	then	ADV
ejpam-6849	651	13	z	z	PROPN
ejpam-6849	651	14	∈	∈	PROPN
ejpam-6849	651	15	ng(s∗	ng(s∗	NUM
ejpam-6849	651	16	)	)	PUNCT
ejpam-6849	651	17	.	.	PUNCT
ejpam-6849	652	1	suppose	suppose	VERB
ejpam-6849	652	2	that	that	SCONJ
ejpam-6849	652	3	z	z	PROPN
ejpam-6849	652	4	∈	∈	PROPN
ejpam-6849	652	5	s1	s1	PROPN
ejpam-6849	652	6	.	.	PUNCT
ejpam-6849	653	1	since	since	SCONJ
ejpam-6849	653	2	s	s	PROPN
ejpam-6849	653	3	is	be	AUX
ejpam-6849	653	4	a	a	DET
ejpam-6849	653	5	total	total	ADJ
ejpam-6849	653	6	ve	ve	NOUN
ejpam-6849	653	7	-	-	PUNCT
ejpam-6849	653	8	dominating	dominating	NOUN
ejpam-6849	653	9	set	set	NOUN
ejpam-6849	653	10	,	,	PUNCT
ejpam-6849	653	11	there	there	PRON
ejpam-6849	653	12	exists	exist	VERB
ejpam-6849	653	13	x	x	X
ejpam-6849	653	14	∈	∈	PROPN
ejpam-6849	653	15	s	s	PART
ejpam-6849	653	16	∩	∩	NOUN
ejpam-6849	653	17	ng⋄h(z	ng⋄h(z	NOUN
ejpam-6849	653	18	)	)	PUNCT
ejpam-6849	653	19	.	.	PUNCT
ejpam-6849	654	1	if	if	SCONJ
ejpam-6849	654	2	x	x	SYM
ejpam-6849	654	3	∈	∈	PROPN
ejpam-6849	654	4	s1	s1	NOUN
ejpam-6849	654	5	,	,	PUNCT
ejpam-6849	654	6	then	then	ADV
ejpam-6849	654	7	z	z	PROPN
ejpam-6849	654	8	∈	∈	PROPN
ejpam-6849	654	9	ng(s∗	ng(s∗	NUM
ejpam-6849	654	10	)	)	PUNCT
ejpam-6849	654	11	.	.	PUNCT
ejpam-6849	655	1	if	if	SCONJ
ejpam-6849	655	2	,	,	PUNCT
ejpam-6849	655	3	on	on	ADP
ejpam-6849	655	4	the	the	DET
ejpam-6849	655	5	other	other	ADJ
ejpam-6849	655	6	hand	hand	NOUN
ejpam-6849	655	7	,	,	PUNCT
ejpam-6849	655	8	x	x	X
ejpam-6849	655	9	/∈	/∈	PUNCT
ejpam-6849	655	10	s1	s1	NOUN
ejpam-6849	655	11	and	and	CCONJ
ejpam-6849	655	12	w	w	NOUN
ejpam-6849	655	13	∈	∈	PROPN
ejpam-6849	655	14	v	v	ADP
ejpam-6849	655	15	(	(	PUNCT
ejpam-6849	655	16	g	g	NOUN
ejpam-6849	655	17	)	)	PUNCT
ejpam-6849	655	18	for	for	ADP
ejpam-6849	655	19	which	which	PRON
ejpam-6849	655	20	x	x	SYM
ejpam-6849	655	21	∈	∈	PROPN
ejpam-6849	655	22	szw	szw	NOUN
ejpam-6849	655	23	,	,	PUNCT
ejpam-6849	655	24	then	then	ADV
ejpam-6849	655	25	w	w	PROPN
ejpam-6849	655	26	=	=	SYM
ejpam-6849	655	27	xzw	xzw	PROPN
ejpam-6849	655	28	∈	∈	PROPN
ejpam-6849	655	29	s∗.	s∗.	INTJ
ejpam-6849	655	30	hence	hence	ADV
ejpam-6849	655	31	z	z	PROPN
ejpam-6849	655	32	∈	∈	PROPN
ejpam-6849	655	33	ng(s∗	ng(s∗	NUM
ejpam-6849	655	34	)	)	PUNCT
ejpam-6849	655	35	.	.	PUNCT
ejpam-6849	656	1	the	the	DET
ejpam-6849	656	2	above	above	ADJ
ejpam-6849	656	3	implies	imply	VERB
ejpam-6849	656	4	that	that	SCONJ
ejpam-6849	656	5	s∗	s∗	PROPN
ejpam-6849	656	6	is	be	AUX
ejpam-6849	656	7	a	a	DET
ejpam-6849	656	8	total	total	ADJ
ejpam-6849	656	9	dominating	dominating	NOUN
ejpam-6849	656	10	set	set	NOUN
ejpam-6849	656	11	of	of	ADP
ejpam-6849	656	12	g.	g.	PROPN
ejpam-6849	656	13	hence	hence	ADV
ejpam-6849	656	14	,	,	PUNCT
ejpam-6849	656	15	γt	γt	PROPN
ejpam-6849	656	16	ve(g	ve(g	PROPN
ejpam-6849	656	17	⋄	⋄	PROPN
ejpam-6849	656	18	h	h	NOUN
ejpam-6849	656	19	)	)	PUNCT
ejpam-6849	656	20	=	=	SYM
ejpam-6849	656	21	|s|	|s|	PROPN
ejpam-6849	656	22	≥	≥	NOUN
ejpam-6849	656	23	|s∗|	|s∗|	NUM
ejpam-6849	656	24	≥	≥	NOUN
ejpam-6849	656	25	γt(g	γt(g	NUM
ejpam-6849	656	26	)	)	PUNCT
ejpam-6849	656	27	.	.	PUNCT
ejpam-6849	657	1	this	this	PRON
ejpam-6849	657	2	proves	prove	VERB
ejpam-6849	657	3	(	(	PUNCT
ejpam-6849	657	4	i	i	NOUN
ejpam-6849	657	5	)	)	PUNCT
ejpam-6849	657	6	.	.	PUNCT
ejpam-6849	658	1	now	now	ADV
ejpam-6849	658	2	we	we	PRON
ejpam-6849	658	3	prove	prove	VERB
ejpam-6849	658	4	(	(	PUNCT
ejpam-6849	658	5	ii	ii	NOUN
ejpam-6849	658	6	)	)	PUNCT
ejpam-6849	658	7	.	.	PUNCT
ejpam-6849	659	1	put	put	VERB
ejpam-6849	659	2	α	α	NOUN
ejpam-6849	659	3	=	=	NOUN
ejpam-6849	659	4	min{|s|	min{|s|	NOUN
ejpam-6849	659	5	:	:	PUNCT
ejpam-6849	659	6	s	s	VERB
ejpam-6849	659	7	⊆	⊆	NUM
ejpam-6849	659	8	v	v	NOUN
ejpam-6849	659	9	(	(	PUNCT
ejpam-6849	659	10	g	g	NOUN
ejpam-6849	659	11	)	)	PUNCT
ejpam-6849	659	12	with	with	ADP
ejpam-6849	659	13	s	s	PROPN
ejpam-6849	659	14	⊆	⊆	NUM
ejpam-6849	659	15	ng(s	ng(s	NUM
ejpam-6849	659	16	)	)	PUNCT
ejpam-6849	659	17	and	and	CCONJ
ejpam-6849	659	18	ng(v	ng(v	PUNCT
ejpam-6849	659	19	(	(	PUNCT
ejpam-6849	659	20	g)\s	g)\s	NOUN
ejpam-6849	659	21	)	)	PUNCT
ejpam-6849	659	22	⊆	⊆	NUM
ejpam-6849	659	23	s	s	NOUN
ejpam-6849	659	24	}	}	PUNCT
ejpam-6849	659	25	.	.	PUNCT
ejpam-6849	660	1	first	first	ADV
ejpam-6849	660	2	,	,	PUNCT
ejpam-6849	660	3	note	note	VERB
ejpam-6849	660	4	that	that	SCONJ
ejpam-6849	660	5	if	if	SCONJ
ejpam-6849	660	6	s	s	VERB
ejpam-6849	660	7	⊆	⊆	NUM
ejpam-6849	660	8	v	v	NOUN
ejpam-6849	660	9	(	(	PUNCT
ejpam-6849	660	10	g	g	NOUN
ejpam-6849	660	11	)	)	PUNCT
ejpam-6849	660	12	such	such	ADJ
ejpam-6849	660	13	that	that	PRON
ejpam-6849	660	14	s	s	VERB
ejpam-6849	660	15	⊆	⊆	NUM
ejpam-6849	660	16	ng(s	ng(s	NUM
ejpam-6849	660	17	)	)	PUNCT
ejpam-6849	660	18	and	and	CCONJ
ejpam-6849	660	19	ng(v	ng(v	NUM
ejpam-6849	660	20	(	(	PUNCT
ejpam-6849	660	21	g	g	NOUN
ejpam-6849	660	22	)	)	PUNCT
ejpam-6849	660	23	\	\	PROPN
ejpam-6849	661	1	s	s	X
ejpam-6849	661	2	)	)	PUNCT
ejpam-6849	661	3	⊆	⊆	NUM
ejpam-6849	661	4	s	s	NOUN
ejpam-6849	661	5	,	,	PUNCT
ejpam-6849	661	6	then	then	ADV
ejpam-6849	661	7	following	follow	VERB
ejpam-6849	661	8	the	the	DET
ejpam-6849	661	9	argument	argument	NOUN
ejpam-6849	661	10	in	in	ADP
ejpam-6849	661	11	the	the	DET
ejpam-6849	661	12	necessity	necessity	NOUN
ejpam-6849	661	13	proof	proof	NOUN
ejpam-6849	661	14	of	of	ADP
ejpam-6849	661	15	corollary	corollary	ADJ
ejpam-6849	661	16	(	(	PUNCT
ejpam-6849	661	17	ii	ii	NOUN
ejpam-6849	661	18	)	)	PUNCT
ejpam-6849	661	19	will	will	AUX
ejpam-6849	661	20	show	show	VERB
ejpam-6849	661	21	that	that	SCONJ
ejpam-6849	661	22	s	s	VERB
ejpam-6849	661	23	is	be	AUX
ejpam-6849	661	24	a	a	DET
ejpam-6849	661	25	total	total	ADJ
ejpam-6849	661	26	ve	ve	NOUN
ejpam-6849	661	27	-	-	PUNCT
ejpam-6849	661	28	dominating	dominate	VERB
ejpam-6849	661	29	set	set	NOUN
ejpam-6849	661	30	of	of	ADP
ejpam-6849	661	31	g	g	PROPN
ejpam-6849	661	32	⋄	⋄	PROPN
ejpam-6849	661	33	h.	h.	PROPN
ejpam-6849	661	34	hence	hence	ADV
ejpam-6849	661	35	,	,	PUNCT
ejpam-6849	661	36	γt	γt	PROPN
ejpam-6849	661	37	ve(g	ve(g	PROPN
ejpam-6849	661	38	⋄	⋄	PROPN
ejpam-6849	661	39	h	h	PROPN
ejpam-6849	661	40	)	)	PUNCT
ejpam-6849	661	41	≤	≤	NOUN
ejpam-6849	662	1	α	α	X
ejpam-6849	662	2	.	.	PUNCT
ejpam-6849	663	1	next	next	ADV
ejpam-6849	663	2	,	,	PUNCT
ejpam-6849	663	3	let	let	VERB
ejpam-6849	663	4	s	s	PRON
ejpam-6849	663	5	⊆	⊆	NUM
ejpam-6849	663	6	v	v	NOUN
ejpam-6849	663	7	(	(	PUNCT
ejpam-6849	663	8	g	g	PROPN
ejpam-6849	663	9	⋄	⋄	PROPN
ejpam-6849	663	10	h	h	NOUN
ejpam-6849	663	11	)	)	PUNCT
ejpam-6849	663	12	be	be	VERB
ejpam-6849	663	13	a	a	DET
ejpam-6849	663	14	γt	γt	NOUN
ejpam-6849	663	15	ve	ve	NOUN
ejpam-6849	663	16	-	-	PUNCT
ejpam-6849	663	17	set	set	NOUN
ejpam-6849	663	18	of	of	ADP
ejpam-6849	663	19	g	g	PROPN
ejpam-6849	663	20	⋄	⋄	PROPN
ejpam-6849	663	21	h.	h.	NOUN
ejpam-6849	663	22	by	by	ADP
ejpam-6849	663	23	proposition	proposition	NOUN
ejpam-6849	663	24	15	15	NUM
ejpam-6849	663	25	,	,	PUNCT
ejpam-6849	663	26	s	s	VERB
ejpam-6849	663	27	=	=	PUNCT
ejpam-6849	663	28	a	a	DET
ejpam-6849	663	29	∪	∪	X
ejpam-6849	663	30	(	(	PUNCT
ejpam-6849	663	31	∪uv∈e(g)suv	∪uv∈e(g)suv	PROPN
ejpam-6849	663	32	)	)	PUNCT
ejpam-6849	663	33	,	,	PUNCT
ejpam-6849	663	34	where	where	SCONJ
ejpam-6849	663	35	a	a	DET
ejpam-6849	663	36	=	=	SYM
ejpam-6849	663	37	s	s	NOUN
ejpam-6849	663	38	∩	∩	ADJ
ejpam-6849	663	39	v	v	X
ejpam-6849	663	40	(	(	PUNCT
ejpam-6849	663	41	g	g	NOUN
ejpam-6849	663	42	)	)	PUNCT
ejpam-6849	663	43	and	and	CCONJ
ejpam-6849	663	44	suv	suv	PROPN
ejpam-6849	663	45	⊆	⊆	NUM
ejpam-6849	663	46	v	v	NOUN
ejpam-6849	663	47	(	(	PUNCT
ejpam-6849	663	48	huv	huv	PROPN
ejpam-6849	663	49	)	)	PUNCT
ejpam-6849	663	50	with	with	ADP
ejpam-6849	663	51	suv	suv	PROPN
ejpam-6849	663	52	a	a	DET
ejpam-6849	663	53	total	total	ADJ
ejpam-6849	663	54	ve	ve	NOUN
ejpam-6849	663	55	-	-	PUNCT
ejpam-6849	663	56	dominating	dominate	VERB
ejpam-6849	663	57	set	set	NOUN
ejpam-6849	663	58	of	of	ADP
ejpam-6849	663	59	huv	huv	PROPN
ejpam-6849	663	60	,	,	PUNCT
ejpam-6849	663	61	hence	hence	ADV
ejpam-6849	663	62	|suv|	|suv|	PROPN
ejpam-6849	663	63	≥	≥	NUM
ejpam-6849	663	64	2	2	NUM
ejpam-6849	663	65	,	,	PUNCT
ejpam-6849	663	66	whenever	whenever	SCONJ
ejpam-6849	663	67	{	{	PUNCT
ejpam-6849	663	68	u	u	NOUN
ejpam-6849	663	69	,	,	PUNCT
ejpam-6849	663	70	v	v	NOUN
ejpam-6849	663	71	}	}	PUNCT
ejpam-6849	663	72	∩	∩	NOUN
ejpam-6849	663	73	a	a	DET
ejpam-6849	663	74	=	=	X
ejpam-6849	663	75	∅.	∅.	NOUN
ejpam-6849	663	76	moreover	moreover	ADV
ejpam-6849	663	77	,	,	PUNCT
ejpam-6849	663	78	a	a	DET
ejpam-6849	663	79	∩	∩	NOUN
ejpam-6849	663	80	ng(u	ng(u	NOUN
ejpam-6849	663	81	)	)	PUNCT
ejpam-6849	663	82	̸=	̸=	PROPN
ejpam-6849	663	83	∅	∅	NOUN
ejpam-6849	663	84	for	for	ADP
ejpam-6849	663	85	all	all	PRON
ejpam-6849	663	86	u	u	NOUN
ejpam-6849	663	87	∈	∈	PROPN
ejpam-6849	663	88	a	a	DET
ejpam-6849	663	89	for	for	ADP
ejpam-6849	663	90	which	which	PRON
ejpam-6849	663	91	suv	suv	NOUN
ejpam-6849	663	92	=	=	NOUN
ejpam-6849	663	93	∅	∅	NOUN
ejpam-6849	663	94	for	for	ADP
ejpam-6849	663	95	all	all	PRON
ejpam-6849	663	96	v	v	ADP
ejpam-6849	663	97	∈	∈	NOUN
ejpam-6849	663	98	ng(u	ng(u	NOUN
ejpam-6849	663	99	)	)	PUNCT
ejpam-6849	663	100	.	.	PUNCT
ejpam-6849	664	1	for	for	ADP
ejpam-6849	664	2	each	each	DET
ejpam-6849	664	3	u	u	PROPN
ejpam-6849	664	4	∈	∈	PROPN
ejpam-6849	664	5	a	a	X
ejpam-6849	664	6	,	,	PUNCT
ejpam-6849	664	7	we	we	PRON
ejpam-6849	664	8	say	say	VERB
ejpam-6849	664	9	u	u	NOUN
ejpam-6849	664	10	has	have	VERB
ejpam-6849	664	11	the	the	DET
ejpam-6849	664	12	property	property	NOUN
ejpam-6849	664	13	pa	pa	PROPN
ejpam-6849	664	14	and	and	CCONJ
ejpam-6849	664	15	write	write	VERB
ejpam-6849	664	16	u	u	PROPN
ejpam-6849	664	17	∈	∈	PROPN
ejpam-6849	664	18	pa	pa	NOUN
ejpam-6849	664	19	if	if	SCONJ
ejpam-6849	664	20	there	there	PRON
ejpam-6849	664	21	exists	exist	VERB
ejpam-6849	664	22	v	v	ADP
ejpam-6849	664	23	∈	∈	PROPN
ejpam-6849	664	24	ng(u	ng(u	NOUN
ejpam-6849	664	25	)	)	PUNCT
ejpam-6849	664	26	such	such	ADJ
ejpam-6849	664	27	that	that	SCONJ
ejpam-6849	664	28	suv	suv	PROPN
ejpam-6849	664	29	̸=	̸=	PROPN
ejpam-6849	664	30	∅.	∅.	NOUN
ejpam-6849	664	31	for	for	ADP
ejpam-6849	664	32	each	each	DET
ejpam-6849	664	33	u	u	PROPN
ejpam-6849	664	34	∈	∈	PROPN
ejpam-6849	664	35	pa	pa	PROPN
ejpam-6849	664	36	,	,	PUNCT
ejpam-6849	664	37	we	we	PRON
ejpam-6849	664	38	pick	pick	VERB
ejpam-6849	664	39	one	one	NUM
ejpam-6849	664	40	such	such	ADJ
ejpam-6849	664	41	v	v	NOUN
ejpam-6849	664	42	and	and	CCONJ
ejpam-6849	664	43	write	write	VERB
ejpam-6849	664	44	v	v	NOUN
ejpam-6849	664	45	=	=	SYM
ejpam-6849	664	46	vu	vu	X
ejpam-6849	664	47	.	.	PUNCT
ejpam-6849	665	1	put	put	VERB
ejpam-6849	665	2	c	c	NOUN
ejpam-6849	665	3	=	=	SYM
ejpam-6849	665	4	{	{	PUNCT
ejpam-6849	665	5	vu	vu	X
ejpam-6849	665	6	:	:	PUNCT
ejpam-6849	665	7	u	u	PROPN
ejpam-6849	665	8	∈	∈	PROPN
ejpam-6849	665	9	pa	pa	PROPN
ejpam-6849	665	10	}	}	PUNCT
ejpam-6849	665	11	and	and	CCONJ
ejpam-6849	665	12	let	let	VERB
ejpam-6849	665	13	b	b	NOUN
ejpam-6849	665	14	=	=	PRON
ejpam-6849	665	15	{	{	PUNCT
ejpam-6849	665	16	u	u	NOUN
ejpam-6849	665	17	∈	∈	PROPN
ejpam-6849	665	18	v	v	NOUN
ejpam-6849	665	19	(	(	PUNCT
ejpam-6849	665	20	g	g	NOUN
ejpam-6849	665	21	)	)	PUNCT
ejpam-6849	665	22	\	\	PROPN
ejpam-6849	666	1	a	a	DET
ejpam-6849	666	2	:	:	PUNCT
ejpam-6849	666	3	∃v	∃v	PROPN
ejpam-6849	666	4	∈	∈	PROPN
ejpam-6849	666	5	v	v	NOUN
ejpam-6849	666	6	(	(	PUNCT
ejpam-6849	666	7	g	g	NOUN
ejpam-6849	666	8	)	)	PUNCT
ejpam-6849	666	9	\	\	PROPN
ejpam-6849	666	10	a	a	DET
ejpam-6849	666	11	for	for	ADP
ejpam-6849	666	12	which	which	PRON
ejpam-6849	666	13	uv	uv	NOUN
ejpam-6849	666	14	∈	∈	PROPN
ejpam-6849	666	15	e(g	e(g	PROPN
ejpam-6849	666	16	)	)	PUNCT
ejpam-6849	666	17	}	}	PUNCT
ejpam-6849	666	18	.	.	PUNCT
ejpam-6849	667	1	define	define	VERB
ejpam-6849	667	2	s∗	s∗	PROPN
ejpam-6849	667	3	=	=	PUNCT
ejpam-6849	667	4	a	a	DET
ejpam-6849	667	5	∪	∪	NOUN
ejpam-6849	667	6	b	b	NOUN
ejpam-6849	667	7	∪	∪	ADJ
ejpam-6849	667	8	c	c	NOUN
ejpam-6849	667	9	,	,	PUNCT
ejpam-6849	667	10	and	and	CCONJ
ejpam-6849	667	11	let	let	VERB
ejpam-6849	667	12	x	x	PUNCT
ejpam-6849	667	13	=	=	PRON
ejpam-6849	667	14	{	{	PUNCT
ejpam-6849	667	15	uv	uv	PROPN
ejpam-6849	667	16	∈	∈	PROPN
ejpam-6849	667	17	e(g	e(g	PROPN
ejpam-6849	667	18	)	)	PUNCT
ejpam-6849	667	19	:	:	PUNCT
ejpam-6849	667	20	{	{	PUNCT
ejpam-6849	667	21	u	u	NOUN
ejpam-6849	667	22	,	,	PUNCT
ejpam-6849	667	23	v	v	NOUN
ejpam-6849	667	24	}	}	PUNCT
ejpam-6849	667	25	∩	∩	NOUN
ejpam-6849	667	26	a	a	DET
ejpam-6849	667	27	=	=	SYM
ejpam-6849	667	28	∅	∅	NOUN
ejpam-6849	667	29	}	}	PUNCT
ejpam-6849	667	30	and	and	CCONJ
ejpam-6849	667	31	y	y	PROPN
ejpam-6849	667	32	=	=	PRON
ejpam-6849	667	33	{	{	PUNCT
ejpam-6849	667	34	uv	uv	PROPN
ejpam-6849	667	35	∈	∈	PROPN
ejpam-6849	667	36	e(g	e(g	PROPN
ejpam-6849	667	37	)	)	PUNCT
ejpam-6849	667	38	:	:	PUNCT
ejpam-6849	667	39	{	{	PUNCT
ejpam-6849	667	40	u	u	NOUN
ejpam-6849	667	41	,	,	PUNCT
ejpam-6849	667	42	v	v	NOUN
ejpam-6849	667	43	}	}	PUNCT
ejpam-6849	667	44	∩	∩	NOUN
ejpam-6849	667	45	a	a	DET
ejpam-6849	667	46	̸=	̸=	PROPN
ejpam-6849	667	47	∅	∅	NOUN
ejpam-6849	667	48	}	}	PUNCT
ejpam-6849	667	49	.	.	PUNCT
ejpam-6849	668	1	then	then	ADV
ejpam-6849	668	2	γt	γt	PROPN
ejpam-6849	668	3	ve(g	ve(g	PROPN
ejpam-6849	668	4	⋄	⋄	PROPN
ejpam-6849	668	5	h	h	NOUN
ejpam-6849	668	6	)	)	PUNCT
ejpam-6849	668	7	=	=	SYM
ejpam-6849	668	8	|s|	|s|	PROPN
ejpam-6849	668	9	=	=	PUNCT
ejpam-6849	668	10	|a|	|a|	PROPN
ejpam-6849	668	11	+	+	PROPN
ejpam-6849	668	12	∑	∑	NOUN
ejpam-6849	668	13	uv∈e(g	uv∈e(g	NOUN
ejpam-6849	668	14	)	)	PUNCT
ejpam-6849	669	1	|suv|	|suv|	PROPN
ejpam-6849	669	2	=	=	SYM
ejpam-6849	669	3	|a|	|a|	PROPN
ejpam-6849	669	4	+	+	CCONJ
ejpam-6849	669	5	∑	∑	PROPN
ejpam-6849	669	6	uv∈y	uv∈y	PROPN
ejpam-6849	669	7	|suv|	|suv|	PROPN
ejpam-6849	669	8	+	+	CCONJ
ejpam-6849	669	9	∑	∑	PROPN
ejpam-6849	669	10	uv∈x	uv∈x	INTJ
ejpam-6849	669	11	|suv|	|suv|	PROPN
ejpam-6849	669	12	≥	≥	NUM
ejpam-6849	669	13	|a	|a	VERB
ejpam-6849	669	14	∪	∪	ADP
ejpam-6849	669	15	c|	c|	PROPN
ejpam-6849	669	16	+	+	CCONJ
ejpam-6849	669	17	|b|	|b|	X
ejpam-6849	669	18	=	=	PUNCT
ejpam-6849	669	19	|s∗|	|s∗|	PROPN
ejpam-6849	669	20	.	.	PUNCT
ejpam-6849	670	1	let	let	VERB
ejpam-6849	670	2	u	u	PRON
ejpam-6849	670	3	∈	∈	PROPN
ejpam-6849	670	4	b.	b.	PROPN
ejpam-6849	670	5	then	then	ADV
ejpam-6849	670	6	there	there	PRON
ejpam-6849	670	7	exists	exist	VERB
ejpam-6849	670	8	v	v	ADP
ejpam-6849	670	9	∈	∈	PROPN
ejpam-6849	670	10	b	b	NOUN
ejpam-6849	670	11	such	such	ADJ
ejpam-6849	670	12	that	that	DET
ejpam-6849	670	13	uv	uv	PROPN
ejpam-6849	670	14	∈	∈	PROPN
ejpam-6849	670	15	e(g	e(g	PROPN
ejpam-6849	670	16	)	)	PUNCT
ejpam-6849	670	17	and	and	CCONJ
ejpam-6849	670	18	{	{	PUNCT
ejpam-6849	670	19	u	u	NOUN
ejpam-6849	670	20	,	,	PUNCT
ejpam-6849	670	21	v}∩a	v}∩a	ADJ
ejpam-6849	670	22	=	=	PUNCT
ejpam-6849	670	23	∅.	∅.	VERB
ejpam-6849	670	24	consequently	consequently	ADV
ejpam-6849	670	25	,	,	PUNCT
ejpam-6849	670	26	v	v	PROPN
ejpam-6849	670	27	∈	∈	NOUN
ejpam-6849	670	28	s∗	s∗	VERB
ejpam-6849	670	29	so	so	SCONJ
ejpam-6849	670	30	that	that	SCONJ
ejpam-6849	670	31	u	u	PROPN
ejpam-6849	670	32	∈	∈	NOUN
ejpam-6849	670	33	ng(s∗	ng(s∗	NUM
ejpam-6849	670	34	)	)	PUNCT
ejpam-6849	670	35	.	.	PUNCT
ejpam-6849	671	1	let	let	VERB
ejpam-6849	671	2	u	u	PRON
ejpam-6849	671	3	∈	∈	VERB
ejpam-6849	671	4	a.	a.	NOUN
ejpam-6849	671	5	if	if	SCONJ
ejpam-6849	671	6	u	u	PROPN
ejpam-6849	671	7	∈	∈	PROPN
ejpam-6849	671	8	pa	pa	PROPN
ejpam-6849	671	9	,	,	PUNCT
ejpam-6849	671	10	then	then	ADV
ejpam-6849	671	11	there	there	PRON
ejpam-6849	671	12	exists	exist	VERB
ejpam-6849	671	13	vu	vu	X
ejpam-6849	671	14	∈	∈	PROPN
ejpam-6849	671	15	c	c	PROPN
ejpam-6849	671	16	∩	∩	X
ejpam-6849	671	17	ng(u	ng(u	NOUN
ejpam-6849	671	18	)	)	PUNCT
ejpam-6849	671	19	.	.	PUNCT
ejpam-6849	672	1	j.	j.	PROPN
ejpam-6849	672	2	tayab	tayab	PROPN
ejpam-6849	672	3	,	,	PUNCT
ejpam-6849	672	4	f.	f.	PROPN
ejpam-6849	672	5	p.	p.	PROPN
ejpam-6849	672	6	jamil	jamil	PROPN
ejpam-6849	672	7	,	,	PUNCT
ejpam-6849	672	8	i.	i.	PROPN
ejpam-6849	672	9	aniversario	aniversario	PROPN
ejpam-6849	672	10	/	/	SYM
ejpam-6849	672	11	eur	eur	PROPN
ejpam-6849	672	12	.	.	PUNCT
ejpam-6849	673	1	j.	j.	PROPN
ejpam-6849	673	2	pure	pure	PROPN
ejpam-6849	673	3	appl	appl	PROPN
ejpam-6849	673	4	.	.	PROPN
ejpam-6849	673	5	math	math	PROPN
ejpam-6849	673	6	,	,	PUNCT
ejpam-6849	673	7	18	18	NUM
ejpam-6849	673	8	(	(	PUNCT
ejpam-6849	673	9	4	4	NUM
ejpam-6849	673	10	)	)	PUNCT
ejpam-6849	673	11	(	(	PUNCT
ejpam-6849	673	12	2025	2025	NUM
ejpam-6849	673	13	)	)	PUNCT
ejpam-6849	673	14	,	,	PUNCT
ejpam-6849	673	15	6849	6849	NUM
ejpam-6849	673	16	18	18	NUM
ejpam-6849	673	17	of	of	ADP
ejpam-6849	673	18	19	19	NUM
ejpam-6849	673	19	thus	thus	ADV
ejpam-6849	673	20	,	,	PUNCT
ejpam-6849	673	21	u	u	PROPN
ejpam-6849	673	22	∈	∈	PROPN
ejpam-6849	673	23	ng(s∗	ng(s∗	NUM
ejpam-6849	673	24	)	)	PUNCT
ejpam-6849	673	25	.	.	PUNCT
ejpam-6849	674	1	on	on	ADP
ejpam-6849	674	2	the	the	DET
ejpam-6849	674	3	other	other	ADJ
ejpam-6849	674	4	hand	hand	NOUN
ejpam-6849	674	5	,	,	PUNCT
ejpam-6849	674	6	if	if	SCONJ
ejpam-6849	674	7	u	u	PROPN
ejpam-6849	674	8	/∈	/∈	PROPN
ejpam-6849	674	9	pa	pa	PROPN
ejpam-6849	674	10	,	,	PUNCT
ejpam-6849	674	11	then	then	ADV
ejpam-6849	674	12	a	a	DET
ejpam-6849	674	13	∩	∩	ADJ
ejpam-6849	674	14	ng(u	ng(u	NOUN
ejpam-6849	674	15	)	)	PUNCT
ejpam-6849	674	16	̸=	̸=	PROPN
ejpam-6849	674	17	∅.	∅.	ADP
ejpam-6849	674	18	this	this	PRON
ejpam-6849	674	19	means	mean	VERB
ejpam-6849	674	20	u	u	PROPN
ejpam-6849	674	21	∈	∈	NOUN
ejpam-6849	674	22	ng(s∗	ng(s∗	NUM
ejpam-6849	674	23	)	)	PUNCT
ejpam-6849	674	24	.	.	PUNCT
ejpam-6849	675	1	now	now	ADV
ejpam-6849	675	2	,	,	PUNCT
ejpam-6849	675	3	let	let	VERB
ejpam-6849	675	4	u	u	PRON
ejpam-6849	675	5	∈	∈	PROPN
ejpam-6849	675	6	c.	c.	NOUN
ejpam-6849	675	7	then	then	ADV
ejpam-6849	675	8	u	u	X
ejpam-6849	675	9	=	=	X
ejpam-6849	675	10	ux	ux	PROPN
ejpam-6849	675	11	for	for	ADP
ejpam-6849	675	12	some	some	DET
ejpam-6849	675	13	x	x	SYM
ejpam-6849	675	14	∈	∈	PROPN
ejpam-6849	675	15	pa	pa	PROPN
ejpam-6849	675	16	and	and	CCONJ
ejpam-6849	675	17	so	so	ADV
ejpam-6849	675	18	u	u	PROPN
ejpam-6849	675	19	∈	∈	PROPN
ejpam-6849	675	20	ng(s∗	ng(s∗	NUM
ejpam-6849	675	21	)	)	PUNCT
ejpam-6849	675	22	.	.	PUNCT
ejpam-6849	676	1	all	all	DET
ejpam-6849	676	2	these	these	PRON
ejpam-6849	676	3	imply	imply	VERB
ejpam-6849	676	4	that	that	PRON
ejpam-6849	676	5	s∗	s∗	VERB
ejpam-6849	676	6	⊆	⊆	NUM
ejpam-6849	676	7	ng(s∗	ng(s∗	NUM
ejpam-6849	676	8	)	)	PUNCT
ejpam-6849	676	9	.	.	PUNCT
ejpam-6849	677	1	finally	finally	ADV
ejpam-6849	677	2	,	,	PUNCT
ejpam-6849	677	3	let	let	VERB
ejpam-6849	677	4	v	v	NUM
ejpam-6849	677	5	∈	∈	PROPN
ejpam-6849	677	6	v	v	NOUN
ejpam-6849	677	7	(	(	PUNCT
ejpam-6849	677	8	g	g	NOUN
ejpam-6849	677	9	)	)	PUNCT
ejpam-6849	677	10	\	\	NOUN
ejpam-6849	678	1	s∗	s∗	PROPN
ejpam-6849	678	2	,	,	PUNCT
ejpam-6849	678	3	and	and	CCONJ
ejpam-6849	678	4	let	let	VERB
ejpam-6849	678	5	u	u	PRON
ejpam-6849	678	6	∈	∈	PROPN
ejpam-6849	678	7	ng(v	ng(v	PUNCT
ejpam-6849	678	8	)	)	PUNCT
ejpam-6849	678	9	.	.	PUNCT
ejpam-6849	679	1	if	if	SCONJ
ejpam-6849	679	2	u	u	PROPN
ejpam-6849	679	3	/∈	/∈	VERB
ejpam-6849	679	4	s∗	s∗	PROPN
ejpam-6849	679	5	,	,	PUNCT
ejpam-6849	679	6	then	then	ADV
ejpam-6849	679	7	{	{	PUNCT
ejpam-6849	679	8	u	u	NOUN
ejpam-6849	679	9	,	,	PUNCT
ejpam-6849	679	10	v	v	NOUN
ejpam-6849	679	11	}	}	PUNCT
ejpam-6849	679	12	∩	∩	NOUN
ejpam-6849	679	13	a	a	DET
ejpam-6849	679	14	=	=	PUNCT
ejpam-6849	679	15	∅.	∅.	NOUN
ejpam-6849	679	16	this	this	PRON
ejpam-6849	679	17	means	mean	VERB
ejpam-6849	679	18	that	that	SCONJ
ejpam-6849	679	19	u	u	NOUN
ejpam-6849	679	20	,	,	PUNCT
ejpam-6849	679	21	v	v	PROPN
ejpam-6849	679	22	∈	∈	PROPN
ejpam-6849	679	23	b	b	NOUN
ejpam-6849	679	24	,	,	PUNCT
ejpam-6849	679	25	which	which	PRON
ejpam-6849	679	26	is	be	AUX
ejpam-6849	679	27	impossible	impossible	ADJ
ejpam-6849	679	28	.	.	PUNCT
ejpam-6849	680	1	therefore	therefore	ADV
ejpam-6849	680	2	,	,	PUNCT
ejpam-6849	680	3	u	u	PROPN
ejpam-6849	680	4	∈	∈	PROPN
ejpam-6849	680	5	s∗.	s∗.	ADJ
ejpam-6849	680	6	since	since	SCONJ
ejpam-6849	680	7	u	u	NOUN
ejpam-6849	680	8	is	be	AUX
ejpam-6849	680	9	arbitrary	arbitrary	ADJ
ejpam-6849	680	10	,	,	PUNCT
ejpam-6849	680	11	ng(v	ng(v	PUNCT
ejpam-6849	680	12	)	)	PUNCT
ejpam-6849	680	13	⊆	⊆	NUM
ejpam-6849	680	14	s∗.	s∗.	ADJ
ejpam-6849	680	15	hence	hence	ADV
ejpam-6849	680	16	,	,	PUNCT
ejpam-6849	680	17	α	α	PROPN
ejpam-6849	680	18	≤	≤	NOUN
ejpam-6849	680	19	|s∗|	|s∗|	PRON
ejpam-6849	680	20	≤	≤	NUM
ejpam-6849	680	21	γt	γt	ADP
ejpam-6849	680	22	ve(g	ve(g	PROPN
ejpam-6849	680	23	⋄	⋄	PROPN
ejpam-6849	680	24	h	h	PROPN
ejpam-6849	680	25	)	)	PUNCT
ejpam-6849	680	26	.	.	PUNCT
ejpam-6849	681	1	■	■	PUNCT
ejpam-6849	681	2	it	it	PRON
ejpam-6849	681	3	also	also	ADV
ejpam-6849	681	4	follows	follow	VERB
ejpam-6849	681	5	from	from	ADP
ejpam-6849	681	6	corollary7(ii	corollary7(ii	PROPN
ejpam-6849	681	7	)	)	PUNCT
ejpam-6849	681	8	that	that	SCONJ
ejpam-6849	681	9	if	if	SCONJ
ejpam-6849	681	10	g	g	PROPN
ejpam-6849	681	11	is	be	AUX
ejpam-6849	681	12	a	a	DET
ejpam-6849	681	13	connected	connected	ADJ
ejpam-6849	681	14	graph	graph	NOUN
ejpam-6849	681	15	of	of	ADP
ejpam-6849	681	16	order	order	NOUN
ejpam-6849	681	17	n	n	PRON
ejpam-6849	681	18	≥	≥	NOUN
ejpam-6849	681	19	3	3	NUM
ejpam-6849	681	20	and	and	CCONJ
ejpam-6849	681	21	h	h	NOUN
ejpam-6849	681	22	is	be	AUX
ejpam-6849	681	23	a	a	DET
ejpam-6849	681	24	nonempty	nonempty	ADJ
ejpam-6849	681	25	graph	graph	NOUN
ejpam-6849	681	26	,	,	PUNCT
ejpam-6849	681	27	then	then	ADV
ejpam-6849	681	28	γt(g	γt(g	PUNCT
ejpam-6849	681	29	)	)	PUNCT
ejpam-6849	681	30	≤	≤	NUM
ejpam-6849	681	31	γt	γt	ADP
ejpam-6849	681	32	ve(g	ve(g	PROPN
ejpam-6849	681	33	⋄	⋄	PROPN
ejpam-6849	681	34	h	h	NOUN
ejpam-6849	681	35	)	)	PUNCT
ejpam-6849	681	36	≤	≤	NOUN
ejpam-6849	681	37	n	n	CCONJ
ejpam-6849	681	38	−	−	PROPN
ejpam-6849	681	39	1	1	NUM
ejpam-6849	681	40	.	.	NOUN
ejpam-6849	681	41	4	4	NUM
ejpam-6849	681	42	.	.	X
ejpam-6849	681	43	conclusion	conclusion	NOUN
ejpam-6849	681	44	this	this	DET
ejpam-6849	681	45	study	study	NOUN
ejpam-6849	681	46	has	have	AUX
ejpam-6849	681	47	established	establish	VERB
ejpam-6849	681	48	several	several	ADJ
ejpam-6849	681	49	characterizations	characterization	NOUN
ejpam-6849	681	50	concerning	concern	VERB
ejpam-6849	681	51	the	the	DET
ejpam-6849	681	52	vertex	vertex	NOUN
ejpam-6849	681	53	-	-	PUNCT
ejpam-6849	681	54	edge	edge	NOUN
ejpam-6849	681	55	domination	domination	NOUN
ejpam-6849	681	56	and	and	CCONJ
ejpam-6849	681	57	total	total	ADJ
ejpam-6849	681	58	vertex	vertex	NOUN
ejpam-6849	681	59	-	-	PUNCT
ejpam-6849	681	60	edge	edge	NOUN
ejpam-6849	681	61	domination	domination	NOUN
ejpam-6849	681	62	in	in	ADP
ejpam-6849	681	63	graphs	graph	NOUN
ejpam-6849	681	64	under	under	ADP
ejpam-6849	681	65	various	various	ADJ
ejpam-6849	681	66	binary	binary	ADJ
ejpam-6849	681	67	operations	operation	NOUN
ejpam-6849	681	68	,	,	PUNCT
ejpam-6849	681	69	namely	namely	ADV
ejpam-6849	681	70	the	the	DET
ejpam-6849	681	71	join	join	NOUN
ejpam-6849	681	72	,	,	PUNCT
ejpam-6849	681	73	corona	corona	PROPN
ejpam-6849	681	74	,	,	PUNCT
ejpam-6849	681	75	complementary	complementary	ADJ
ejpam-6849	681	76	prism	prism	NOUN
ejpam-6849	681	77	,	,	PUNCT
ejpam-6849	681	78	lexicographic	lexicographic	ADJ
ejpam-6849	681	79	product	product	NOUN
ejpam-6849	681	80	,	,	PUNCT
ejpam-6849	681	81	and	and	CCONJ
ejpam-6849	681	82	edge	edge	NOUN
ejpam-6849	681	83	corona	corona	NOUN
ejpam-6849	681	84	.	.	PUNCT
ejpam-6849	682	1	each	each	DET
ejpam-6849	682	2	result	result	NOUN
ejpam-6849	682	3	provided	provide	VERB
ejpam-6849	682	4	necessary	necessary	ADJ
ejpam-6849	682	5	and	and	CCONJ
ejpam-6849	682	6	sufficient	sufficient	ADJ
ejpam-6849	682	7	conditions	condition	NOUN
ejpam-6849	682	8	for	for	ADP
ejpam-6849	682	9	a	a	DET
ejpam-6849	682	10	subset	subset	NOUN
ejpam-6849	682	11	of	of	ADP
ejpam-6849	682	12	vertices	vertex	NOUN
ejpam-6849	682	13	to	to	PART
ejpam-6849	682	14	be	be	AUX
ejpam-6849	682	15	a	a	DET
ejpam-6849	682	16	(	(	PUNCT
ejpam-6849	682	17	total	total	ADJ
ejpam-6849	682	18	)	)	PUNCT
ejpam-6849	682	19	vertex	vertex	NOUN
ejpam-6849	682	20	-	-	PUNCT
ejpam-6849	682	21	edge	edge	NOUN
ejpam-6849	682	22	dominating	dominating	NOUN
ejpam-6849	682	23	set	set	VERB
ejpam-6849	682	24	within	within	ADP
ejpam-6849	682	25	the	the	DET
ejpam-6849	682	26	resulting	result	VERB
ejpam-6849	682	27	graph	graph	NOUN
ejpam-6849	682	28	.	.	PUNCT
ejpam-6849	683	1	the	the	DET
ejpam-6849	683	2	results	result	NOUN
ejpam-6849	683	3	demonstrate	demonstrate	VERB
ejpam-6849	683	4	that	that	SCONJ
ejpam-6849	683	5	the	the	DET
ejpam-6849	683	6	(	(	PUNCT
ejpam-6849	683	7	total	total	ADJ
ejpam-6849	683	8	)	)	PUNCT
ejpam-6849	683	9	vertex	vertex	NOUN
ejpam-6849	683	10	-	-	PUNCT
ejpam-6849	683	11	edge	edge	NOUN
ejpam-6849	683	12	domination	domination	NOUN
ejpam-6849	683	13	behavior	behavior	NOUN
ejpam-6849	683	14	of	of	ADP
ejpam-6849	683	15	these	these	DET
ejpam-6849	683	16	resulting	result	VERB
ejpam-6849	683	17	graphs	graph	NOUN
ejpam-6849	683	18	is	be	AUX
ejpam-6849	683	19	fundamentally	fundamentally	ADV
ejpam-6849	683	20	determined	determine	VERB
ejpam-6849	683	21	by	by	ADP
ejpam-6849	683	22	the	the	DET
ejpam-6849	683	23	vertex	vertex	NOUN
ejpam-6849	683	24	-	-	PUNCT
ejpam-6849	683	25	edge	edge	NOUN
ejpam-6849	683	26	domination	domination	NOUN
ejpam-6849	683	27	properties	property	NOUN
ejpam-6849	683	28	of	of	ADP
ejpam-6849	683	29	their	their	PRON
ejpam-6849	683	30	component	component	NOUN
ejpam-6849	683	31	graphs	graph	NOUN
ejpam-6849	683	32	.	.	PUNCT
ejpam-6849	684	1	in	in	ADP
ejpam-6849	684	2	the	the	DET
ejpam-6849	684	3	join	join	NOUN
ejpam-6849	684	4	of	of	ADP
ejpam-6849	684	5	graphs	graph	NOUN
ejpam-6849	684	6	,	,	PUNCT
ejpam-6849	684	7	vertex	vertex	NOUN
ejpam-6849	684	8	-	-	PUNCT
ejpam-6849	684	9	edge	edge	NOUN
ejpam-6849	684	10	domination	domination	NOUN
ejpam-6849	684	11	depends	depend	VERB
ejpam-6849	684	12	on	on	ADP
ejpam-6849	684	13	the	the	DET
ejpam-6849	684	14	subset	subset	ADJ
ejpam-6849	684	15	relationships	relationship	NOUN
ejpam-6849	684	16	among	among	ADP
ejpam-6849	684	17	the	the	DET
ejpam-6849	684	18	component	component	NOUN
ejpam-6849	684	19	vertex	vertex	NOUN
ejpam-6849	684	20	sets	set	NOUN
ejpam-6849	684	21	;	;	PUNCT
ejpam-6849	684	22	in	in	ADP
ejpam-6849	684	23	the	the	DET
ejpam-6849	684	24	corona	corona	NOUN
ejpam-6849	684	25	and	and	CCONJ
ejpam-6849	684	26	edge	edge	NOUN
ejpam-6849	684	27	corona	corona	PROPN
ejpam-6849	684	28	,	,	PUNCT
ejpam-6849	684	29	the	the	DET
ejpam-6849	684	30	vertex	vertex	NOUN
ejpam-6849	684	31	-	-	PUNCT
ejpam-6849	684	32	edge	edge	NOUN
ejpam-6849	684	33	domination	domination	NOUN
ejpam-6849	684	34	properties	property	NOUN
ejpam-6849	684	35	are	be	AUX
ejpam-6849	684	36	inherited	inherit	VERB
ejpam-6849	684	37	through	through	ADP
ejpam-6849	684	38	the	the	DET
ejpam-6849	684	39	copies	copy	NOUN
ejpam-6849	684	40	of	of	ADP
ejpam-6849	684	41	the	the	DET
ejpam-6849	684	42	secondary	secondary	ADJ
ejpam-6849	684	43	graph	graph	NOUN
ejpam-6849	684	44	attached	attach	VERB
ejpam-6849	684	45	to	to	ADP
ejpam-6849	684	46	each	each	DET
ejpam-6849	684	47	vertex	vertex	NOUN
ejpam-6849	684	48	or	or	CCONJ
ejpam-6849	684	49	edge	edge	NOUN
ejpam-6849	684	50	of	of	ADP
ejpam-6849	684	51	the	the	DET
ejpam-6849	684	52	primary	primary	ADJ
ejpam-6849	684	53	graph	graph	NOUN
ejpam-6849	684	54	.	.	PUNCT
ejpam-6849	685	1	moreover	moreover	ADV
ejpam-6849	685	2	,	,	PUNCT
ejpam-6849	685	3	in	in	ADP
ejpam-6849	685	4	the	the	DET
ejpam-6849	685	5	complementary	complementary	ADJ
ejpam-6849	685	6	prism	prism	NOUN
ejpam-6849	685	7	,	,	PUNCT
ejpam-6849	685	8	specific	specific	ADJ
ejpam-6849	685	9	bounds	bound	NOUN
ejpam-6849	685	10	for	for	ADP
ejpam-6849	685	11	the	the	DET
ejpam-6849	685	12	vertex	vertex	NOUN
ejpam-6849	685	13	-	-	PUNCT
ejpam-6849	685	14	edge	edge	NOUN
ejpam-6849	685	15	domination	domination	NOUN
ejpam-6849	685	16	and	and	CCONJ
ejpam-6849	685	17	total	total	ADJ
ejpam-6849	685	18	vertex	vertex	NOUN
ejpam-6849	685	19	-	-	PUNCT
ejpam-6849	685	20	edge	edge	NOUN
ejpam-6849	685	21	domination	domination	NOUN
ejpam-6849	685	22	numbers	number	NOUN
ejpam-6849	685	23	were	be	AUX
ejpam-6849	685	24	established	establish	VERB
ejpam-6849	685	25	,	,	PUNCT
ejpam-6849	685	26	while	while	SCONJ
ejpam-6849	685	27	in	in	ADP
ejpam-6849	685	28	the	the	DET
ejpam-6849	685	29	lexicographic	lexicographic	ADJ
ejpam-6849	685	30	product	product	NOUN
ejpam-6849	685	31	,	,	PUNCT
ejpam-6849	685	32	the	the	DET
ejpam-6849	685	33	(	(	PUNCT
ejpam-6849	685	34	total	total	ADJ
ejpam-6849	685	35	)	)	PUNCT
ejpam-6849	685	36	vertex	vertex	NOUN
ejpam-6849	685	37	-	-	PUNCT
ejpam-6849	685	38	edge	edge	NOUN
ejpam-6849	685	39	dominating	dominating	NOUN
ejpam-6849	685	40	sets	set	NOUN
ejpam-6849	685	41	were	be	AUX
ejpam-6849	685	42	expressed	express	VERB
ejpam-6849	685	43	through	through	ADP
ejpam-6849	685	44	unions	union	NOUN
ejpam-6849	685	45	of	of	ADP
ejpam-6849	685	46	product	product	NOUN
ejpam-6849	685	47	sets	set	NOUN
ejpam-6849	685	48	derived	derive	VERB
ejpam-6849	685	49	from	from	ADP
ejpam-6849	685	50	the	the	DET
ejpam-6849	685	51	component	component	NOUN
ejpam-6849	685	52	graphs	graph	NOUN
ejpam-6849	685	53	.	.	PUNCT
ejpam-6849	686	1	acknowledgements	acknowledgement	NOUN
ejpam-6849	686	2	this	this	DET
ejpam-6849	686	3	research	research	NOUN
ejpam-6849	686	4	is	be	AUX
ejpam-6849	686	5	fully	fully	ADV
ejpam-6849	686	6	supported	support	VERB
ejpam-6849	686	7	by	by	ADP
ejpam-6849	686	8	the	the	DET
ejpam-6849	686	9	department	department	PROPN
ejpam-6849	686	10	of	of	ADP
ejpam-6849	686	11	science	science	NOUN
ejpam-6849	686	12	and	and	CCONJ
ejpam-6849	686	13	technology	technology	NOUN
ejpam-6849	686	14	accelerated	accelerate	VERB
ejpam-6849	686	15	science	science	NOUN
ejpam-6849	686	16	and	and	CCONJ
ejpam-6849	686	17	technology	technology	NOUN
ejpam-6849	686	18	human	human	ADJ
ejpam-6849	686	19	resource	resource	NOUN
ejpam-6849	686	20	development	development	NOUN
ejpam-6849	686	21	program	program	NOUN
ejpam-6849	686	22	(	(	PUNCT
ejpam-6849	686	23	dostasthrdp	dostasthrdp	PROPN
ejpam-6849	686	24	)	)	PUNCT
ejpam-6849	686	25	,	,	PUNCT
ejpam-6849	686	26	philippines	philippine	NOUN
ejpam-6849	686	27	,	,	PUNCT
ejpam-6849	686	28	and	and	CCONJ
ejpam-6849	686	29	the	the	DET
ejpam-6849	686	30	msu	msu	PROPN
ejpam-6849	686	31	-	-	PUNCT
ejpam-6849	686	32	iligan	iligan	PROPN
ejpam-6849	686	33	institute	institute	PROPN
ejpam-6849	686	34	of	of	ADP
ejpam-6849	686	35	technology	technology	NOUN
ejpam-6849	686	36	through	through	ADP
ejpam-6849	686	37	the	the	DET
ejpam-6849	686	38	office	office	NOUN
ejpam-6849	686	39	of	of	ADP
ejpam-6849	686	40	the	the	DET
ejpam-6849	686	41	vice	vice	NOUN
ejpam-6849	686	42	chancellor	chancellor	NOUN
ejpam-6849	686	43	for	for	ADP
ejpam-6849	686	44	research	research	NOUN
ejpam-6849	686	45	and	and	CCONJ
ejpam-6849	686	46	enterprise	enterprise	NOUN
ejpam-6849	686	47	(	(	PUNCT
ejpam-6849	686	48	ovcre	ovcre	NOUN
ejpam-6849	686	49	)	)	PUNCT
ejpam-6849	686	50	.	.	PUNCT
ejpam-6849	687	1	the	the	DET
ejpam-6849	687	2	authors	author	NOUN
ejpam-6849	687	3	would	would	AUX
ejpam-6849	687	4	like	like	VERB
ejpam-6849	687	5	to	to	PART
ejpam-6849	687	6	thank	thank	VERB
ejpam-6849	687	7	the	the	DET
ejpam-6849	687	8	reviewers	reviewer	NOUN
ejpam-6849	687	9	for	for	ADP
ejpam-6849	687	10	reading	read	VERB
ejpam-6849	687	11	and	and	CCONJ
ejpam-6849	687	12	recommending	recommend	VERB
ejpam-6849	687	13	invaluable	invaluable	ADJ
ejpam-6849	687	14	suggestions	suggestion	NOUN
ejpam-6849	687	15	to	to	ADP
ejpam-6849	687	16	the	the	DET
ejpam-6849	687	17	improvement	improvement	NOUN
ejpam-6849	687	18	of	of	ADP
ejpam-6849	687	19	the	the	DET
ejpam-6849	687	20	paper	paper	NOUN
ejpam-6849	687	21	.	.	PUNCT
ejpam-6849	688	1	references	reference	NOUN
ejpam-6849	688	2	[	[	X
ejpam-6849	688	3	1	1	NUM
ejpam-6849	688	4	]	]	X
ejpam-6849	688	5	k.w	k.w	PROPN
ejpam-6849	688	6	.	.	PROPN
ejpam-6849	688	7	peters	peters	PROPN
ejpam-6849	688	8	.	.	PUNCT
ejpam-6849	689	1	theoretical	theoretical	ADJ
ejpam-6849	689	2	and	and	CCONJ
ejpam-6849	689	3	algorithmic	algorithmic	ADJ
ejpam-6849	689	4	results	result	NOUN
ejpam-6849	689	5	on	on	ADP
ejpam-6849	689	6	domination	domination	NOUN
ejpam-6849	689	7	and	and	CCONJ
ejpam-6849	689	8	connectivity	connectivity	NOUN
ejpam-6849	689	9	.	.	PUNCT
ejpam-6849	690	1	phd	phd	NOUN
ejpam-6849	690	2	thesis	thesis	PROPN
ejpam-6849	690	3	,	,	PUNCT
ejpam-6849	690	4	clemson	clemson	NOUN
ejpam-6849	690	5	university	university	NOUN
ejpam-6849	690	6	,	,	PUNCT
ejpam-6849	690	7	1986	1986	NUM
ejpam-6849	690	8	.	.	PUNCT
ejpam-6849	691	1	[	[	X
ejpam-6849	691	2	2	2	NUM
ejpam-6849	691	3	]	]	PUNCT
ejpam-6849	691	4	d.	d.	PROPN
ejpam-6849	691	5	dereniowski	dereniowski	PROPN
ejpam-6849	691	6	,	,	PUNCT
ejpam-6849	691	7	h.	h.	PROPN
ejpam-6849	691	8	ono	ono	PROPN
ejpam-6849	691	9	,	,	PUNCT
ejpam-6849	691	10	i.	i.	PROPN
ejpam-6849	691	11	suzuki	suzuki	PROPN
ejpam-6849	691	12	,	,	PUNCT
ejpam-6849	691	13	l.	l.	PROPN
ejpam-6849	691	14	wrona	wrona	PROPN
ejpam-6849	691	15	,	,	PUNCT
ejpam-6849	691	16	m.	m.	NOUN
ejpam-6849	691	17	yamashita	yamashita	PROPN
ejpam-6849	691	18	,	,	PUNCT
ejpam-6849	691	19	and	and	CCONJ
ejpam-6849	691	20	p.	p.	PROPN
ejpam-6849	691	21	zylinski	zylinski	PROPN
ejpam-6849	691	22	.	.	PUNCT
ejpam-6849	692	1	the	the	DET
ejpam-6849	692	2	searchlight	searchlight	NOUN
ejpam-6849	692	3	problem	problem	NOUN
ejpam-6849	692	4	for	for	ADP
ejpam-6849	692	5	road	road	NOUN
ejpam-6849	692	6	networks	network	NOUN
ejpam-6849	692	7	.	.	PUNCT
ejpam-6849	693	1	theoretical	theoretical	ADJ
ejpam-6849	693	2	computer	computer	NOUN
ejpam-6849	693	3	science	science	NOUN
ejpam-6849	693	4	,	,	PUNCT
ejpam-6849	693	5	591:28–59	591:28–59	NUM
ejpam-6849	693	6	,	,	PUNCT
ejpam-6849	693	7	2015	2015	NUM
ejpam-6849	693	8	.	.	PUNCT
ejpam-6849	694	1	j.	j.	PROPN
ejpam-6849	694	2	tayab	tayab	PROPN
ejpam-6849	694	3	,	,	PUNCT
ejpam-6849	694	4	f.	f.	PROPN
ejpam-6849	694	5	p.	p.	PROPN
ejpam-6849	694	6	jamil	jamil	PROPN
ejpam-6849	694	7	,	,	PUNCT
ejpam-6849	694	8	i.	i.	PROPN
ejpam-6849	694	9	aniversario	aniversario	PROPN
ejpam-6849	694	10	/	/	SYM
ejpam-6849	694	11	eur	eur	PROPN
ejpam-6849	694	12	.	.	PUNCT
ejpam-6849	695	1	j.	j.	PROPN
ejpam-6849	695	2	pure	pure	PROPN
ejpam-6849	695	3	appl	appl	PROPN
ejpam-6849	695	4	.	.	PROPN
ejpam-6849	695	5	math	math	PROPN
ejpam-6849	695	6	,	,	PUNCT
ejpam-6849	695	7	18	18	NUM
ejpam-6849	695	8	(	(	PUNCT
ejpam-6849	695	9	4	4	NUM
ejpam-6849	695	10	)	)	PUNCT
ejpam-6849	695	11	(	(	PUNCT
ejpam-6849	695	12	2025	2025	NUM
ejpam-6849	695	13	)	)	PUNCT
ejpam-6849	695	14	,	,	PUNCT
ejpam-6849	695	15	6849	6849	NUM
ejpam-6849	695	16	19	19	NUM
ejpam-6849	695	17	of	of	ADP
ejpam-6849	695	18	19	19	NUM
ejpam-6849	695	19	[	[	SYM
ejpam-6849	695	20	3	3	NUM
ejpam-6849	695	21	]	]	X
ejpam-6849	695	22	w.f	w.f	PROPN
ejpam-6849	695	23	.	.	PROPN
ejpam-6849	695	24	klostermeyer	klostermeyer	PROPN
ejpam-6849	695	25	and	and	CCONJ
ejpam-6849	695	26	c.m	c.m	PROPN
ejpam-6849	695	27	.	.	PROPN
ejpam-6849	695	28	mynhardt	mynhardt	PROPN
ejpam-6849	695	29	.	.	PUNCT
ejpam-6849	696	1	edge	edge	NOUN
ejpam-6849	696	2	protection	protection	NOUN
ejpam-6849	696	3	in	in	ADP
ejpam-6849	696	4	graphs	graph	NOUN
ejpam-6849	696	5	.	.	PUNCT
ejpam-6849	697	1	australasian	australasian	ADJ
ejpam-6849	697	2	journal	journal	NOUN
ejpam-6849	697	3	of	of	ADP
ejpam-6849	697	4	combinatorics	combinatoric	NOUN
ejpam-6849	697	5	,	,	PUNCT
ejpam-6849	697	6	45:235–250	45:235–250	NUM
ejpam-6849	697	7	,	,	PUNCT
ejpam-6849	697	8	2009	2009	NUM
ejpam-6849	697	9	.	.	PUNCT
ejpam-6849	698	1	[	[	X
ejpam-6849	698	2	4	4	NUM
ejpam-6849	698	3	]	]	X
ejpam-6849	698	4	w.c.k	w.c.k	NOUN
ejpam-6849	698	5	.	.	PUNCT
ejpam-6849	698	6	yen	yen	NOUN
ejpam-6849	698	7	and	and	CCONJ
ejpam-6849	698	8	c.y	c.y	PROPN
ejpam-6849	698	9	.	.	PROPN
ejpam-6849	698	10	tang	tang	PROPN
ejpam-6849	698	11	.	.	PUNCT
ejpam-6849	699	1	an	an	DET
ejpam-6849	699	2	optimal	optimal	ADJ
ejpam-6849	699	3	algorithm	algorithm	NOUN
ejpam-6849	699	4	for	for	ADP
ejpam-6849	699	5	solving	solve	VERB
ejpam-6849	699	6	the	the	DET
ejpam-6849	699	7	searchlight	searchlight	NOUN
ejpam-6849	699	8	guarding	guard	VERB
ejpam-6849	699	9	problem	problem	NOUN
ejpam-6849	699	10	on	on	ADP
ejpam-6849	699	11	weight	weight	NOUN
ejpam-6849	699	12	two	two	NUM
ejpam-6849	699	13	-	-	PUNCT
ejpam-6849	699	14	terminal	terminal	NOUN
ejpam-6849	699	15	series	series	NOUN
ejpam-6849	699	16	-	-	PUNCT
ejpam-6849	699	17	parallel	parallel	ADJ
ejpam-6849	699	18	graphs	graph	NOUN
ejpam-6849	699	19	.	.	PUNCT
ejpam-6849	700	1	acta	acta	PROPN
ejpam-6849	700	2	informatica	informatica	PROPN
ejpam-6849	700	3	,	,	PUNCT
ejpam-6849	700	4	36:143–172	36:143–172	PROPN
ejpam-6849	700	5	,	,	PUNCT
ejpam-6849	700	6	1999	1999	NUM
ejpam-6849	700	7	.	.	PUNCT
ejpam-6849	701	1	[	[	X
ejpam-6849	701	2	5	5	X
ejpam-6849	701	3	]	]	PUNCT
ejpam-6849	701	4	h.	h.	NOUN
ejpam-6849	701	5	abdollahzadeh	abdollahzadeh	PROPN
ejpam-6849	701	6	ahangar	ahangar	NOUN
ejpam-6849	701	7	,	,	PUNCT
ejpam-6849	701	8	m.	m.	NOUN
ejpam-6849	701	9	chellali	chellali	PROPN
ejpam-6849	701	10	,	,	PUNCT
ejpam-6849	701	11	s.m	s.m	PROPN
ejpam-6849	701	12	.	.	PROPN
ejpam-6849	701	13	sheikholeslami	sheikholeslami	PROPN
ejpam-6849	701	14	,	,	PUNCT
ejpam-6849	701	15	m.	m.	NOUN
ejpam-6849	701	16	soroudi	soroudi	PROPN
ejpam-6849	701	17	,	,	PUNCT
ejpam-6849	701	18	and	and	CCONJ
ejpam-6849	701	19	l.	l.	PROPN
ejpam-6849	701	20	volkmann	volkmann	PROPN
ejpam-6849	701	21	.	.	PUNCT
ejpam-6849	702	1	total	total	ADJ
ejpam-6849	702	2	vertex	vertex	NOUN
ejpam-6849	702	3	-	-	PUNCT
ejpam-6849	702	4	edge	edge	NOUN
ejpam-6849	702	5	domination	domination	NOUN
ejpam-6849	702	6	in	in	ADP
ejpam-6849	702	7	trees	tree	NOUN
ejpam-6849	702	8	.	.	PUNCT
ejpam-6849	703	1	acta	acta	PROPN
ejpam-6849	703	2	math	math	PROPN
ejpam-6849	703	3	.	.	PUNCT
ejpam-6849	704	1	univ	univ	PROPN
ejpam-6849	704	2	.	.	PUNCT
ejpam-6849	704	3	comenianae	comenianae	PROPN
ejpam-6849	704	4	,	,	PUNCT
ejpam-6849	704	5	90(2):127	90(2):127	NUM
ejpam-6849	704	6	–	–	PUNCT
ejpam-6849	704	7	143	143	NUM
ejpam-6849	704	8	,	,	PUNCT
ejpam-6849	704	9	2021	2021	NUM
ejpam-6849	704	10	.	.	PUNCT
ejpam-6849	705	1	[	[	X
ejpam-6849	705	2	6	6	NUM
ejpam-6849	705	3	]	]	PUNCT
ejpam-6849	705	4	b.	b.	PROPN
ejpam-6849	705	5	krishnakumari	krishnakumari	PROPN
ejpam-6849	705	6	,	,	PUNCT
ejpam-6849	705	7	y.b	y.b	PROPN
ejpam-6849	705	8	.	.	PROPN
ejpam-6849	705	9	venkatakrishnan	venkatakrishnan	NOUN
ejpam-6849	705	10	,	,	PUNCT
ejpam-6849	705	11	and	and	CCONJ
ejpam-6849	705	12	m.	m.	PROPN
ejpam-6849	705	13	krzywkowski	krzywkowski	PROPN
ejpam-6849	705	14	.	.	PUNCT
ejpam-6849	706	1	bounds	bound	NOUN
ejpam-6849	706	2	on	on	ADP
ejpam-6849	706	3	the	the	DET
ejpam-6849	706	4	vertexedge	vertexedge	NOUN
ejpam-6849	706	5	domination	domination	NOUN
ejpam-6849	706	6	number	number	NOUN
ejpam-6849	706	7	of	of	ADP
ejpam-6849	706	8	a	a	DET
ejpam-6849	706	9	tree	tree	NOUN
ejpam-6849	706	10	.	.	PUNCT
ejpam-6849	707	1	comptes	compte	VERB
ejpam-6849	707	2	rendus	rendus	PROPN
ejpam-6849	707	3	mathã	mathã	PROPN
ejpam-6849	707	4	©	©	PROPN
ejpam-6849	707	5	matique	matique	ADJ
ejpam-6849	707	6	,	,	PUNCT
ejpam-6849	707	7	352:363–366	352:363–366	NUM
ejpam-6849	707	8	,	,	PUNCT
ejpam-6849	707	9	2014	2014	NUM
ejpam-6849	707	10	.	.	PUNCT
ejpam-6849	708	1	[	[	X
ejpam-6849	708	2	7	7	X
ejpam-6849	708	3	]	]	X
ejpam-6849	708	4	j.r	j.r	PROPN
ejpam-6849	708	5	.	.	PROPN
ejpam-6849	708	6	lewis	lewis	PROPN
ejpam-6849	708	7	.	.	PUNCT
ejpam-6849	708	8	vertex	vertex	NOUN
ejpam-6849	708	9	-	-	PUNCT
ejpam-6849	708	10	edge	edge	NOUN
ejpam-6849	708	11	and	and	CCONJ
ejpam-6849	708	12	edge	edge	NOUN
ejpam-6849	708	13	-	-	PUNCT
ejpam-6849	708	14	vertex	vertex	NOUN
ejpam-6849	708	15	domination	domination	NOUN
ejpam-6849	708	16	in	in	ADP
ejpam-6849	708	17	graphs	graph	NOUN
ejpam-6849	708	18	.	.	PUNCT
ejpam-6849	709	1	phd	phd	NOUN
ejpam-6849	709	2	thesis	thesis	PROPN
ejpam-6849	709	3	,	,	PUNCT
ejpam-6849	709	4	clemson	clemson	NOUN
ejpam-6849	709	5	university	university	NOUN
ejpam-6849	709	6	,	,	PUNCT
ejpam-6849	709	7	2007	2007	NUM
ejpam-6849	709	8	.	.	PUNCT
ejpam-6849	710	1	[	[	X
ejpam-6849	710	2	8	8	NUM
ejpam-6849	710	3	]	]	X
ejpam-6849	710	4	j.r	j.r	PROPN
ejpam-6849	710	5	.	.	PROPN
ejpam-6849	710	6	lewis	lewis	PROPN
ejpam-6849	710	7	,	,	PUNCT
ejpam-6849	710	8	s.t	s.t	PROPN
ejpam-6849	710	9	.	.	PROPN
ejpam-6849	710	10	hedetniemi	hedetniemi	PROPN
ejpam-6849	710	11	,	,	PUNCT
ejpam-6849	710	12	t.w	t.w	PROPN
ejpam-6849	710	13	.	.	PROPN
ejpam-6849	710	14	haynes	haynes	PROPN
ejpam-6849	710	15	,	,	PUNCT
ejpam-6849	710	16	and	and	CCONJ
ejpam-6849	710	17	g.h	g.h	PROPN
ejpam-6849	710	18	.	.	PROPN
ejpam-6849	710	19	fricke	fricke	PROPN
ejpam-6849	710	20	.	.	PUNCT
ejpam-6849	710	21	vertex	vertex	NOUN
ejpam-6849	710	22	-	-	PUNCT
ejpam-6849	710	23	edge	edge	NOUN
ejpam-6849	710	24	domination	domination	NOUN
ejpam-6849	710	25	.	.	PUNCT
ejpam-6849	711	1	utilitas	utilitas	PROPN
ejpam-6849	711	2	mathematica	mathematica	PROPN
ejpam-6849	711	3	,	,	PUNCT
ejpam-6849	711	4	81:193–213	81:193–213	PROPN
ejpam-6849	711	5	,	,	PUNCT
ejpam-6849	711	6	2010	2010	NUM
ejpam-6849	711	7	.	.	PUNCT
ejpam-6849	712	1	[	[	X
ejpam-6849	712	2	9	9	NUM
ejpam-6849	712	3	]	]	X
ejpam-6849	712	4	y.b	y.b	PROPN
ejpam-6849	712	5	.	.	PROPN
ejpam-6849	712	6	venkatakrishnan	venkatakrishnan	PROPN
ejpam-6849	712	7	,	,	PUNCT
ejpam-6849	712	8	c.	c.	PROPN
ejpam-6849	712	9	natarajan	natarajan	PROPN
ejpam-6849	712	10	,	,	PUNCT
ejpam-6849	712	11	and	and	CCONJ
ejpam-6849	712	12	g.	g.	PROPN
ejpam-6849	712	13	sathiamoorthy	sathiamoorthy	PROPN
ejpam-6849	712	14	.	.	PUNCT
ejpam-6849	713	1	vertex	vertex	NOUN
ejpam-6849	713	2	-	-	PUNCT
ejpam-6849	713	3	edge	edge	NOUN
ejpam-6849	713	4	and	and	CCONJ
ejpam-6849	713	5	connected	connected	ADJ
ejpam-6849	713	6	domination	domination	NOUN
ejpam-6849	713	7	numbers	number	NOUN
ejpam-6849	713	8	of	of	ADP
ejpam-6849	713	9	a	a	DET
ejpam-6849	713	10	tree	tree	NOUN
ejpam-6849	713	11	.	.	PUNCT
ejpam-6849	714	1	international	international	ADJ
ejpam-6849	714	2	journal	journal	NOUN
ejpam-6849	714	3	of	of	ADP
ejpam-6849	714	4	pure	pure	ADJ
ejpam-6849	714	5	and	and	CCONJ
ejpam-6849	714	6	applied	applied	ADJ
ejpam-6849	714	7	mathematics	mathematic	NOUN
ejpam-6849	714	8	,	,	PUNCT
ejpam-6849	714	9	119:103–111	119:103–111	NUM
ejpam-6849	714	10	,	,	PUNCT
ejpam-6849	714	11	2018	2018	NUM
ejpam-6849	714	12	.	.	PUNCT
ejpam-6849	715	1	[	[	X
ejpam-6849	715	2	10	10	NUM
ejpam-6849	715	3	]	]	X
ejpam-6849	715	4	w.f	w.f	PROPN
ejpam-6849	715	5	.	.	PROPN
ejpam-6849	715	6	klostermeyer	klostermeyer	PROPN
ejpam-6849	715	7	,	,	PUNCT
ejpam-6849	715	8	m.-e	m.-e	PROPN
ejpam-6849	715	9	.	.	PUNCT
ejpam-6849	716	1	messinger	messinger	PROPN
ejpam-6849	716	2	,	,	PUNCT
ejpam-6849	716	3	and	and	CCONJ
ejpam-6849	716	4	a.	a.	PROPN
ejpam-6849	716	5	yeo	yeo	PROPN
ejpam-6849	716	6	.	.	PROPN
ejpam-6849	716	7	dominating	dominate	VERB
ejpam-6849	716	8	vertex	vertex	NOUN
ejpam-6849	716	9	covers	cover	VERB
ejpam-6849	716	10	:	:	PUNCT
ejpam-6849	716	11	the	the	DET
ejpam-6849	716	12	vertex	vertex	NOUN
ejpam-6849	716	13	-	-	PUNCT
ejpam-6849	716	14	domination	domination	NOUN
ejpam-6849	716	15	problem	problem	NOUN
ejpam-6849	716	16	.	.	PUNCT
ejpam-6849	717	1	discussiones	discussione	NOUN
ejpam-6849	717	2	mathematicae	mathematicae	PROPN
ejpam-6849	717	3	graph	graph	NOUN
ejpam-6849	717	4	theory	theory	NOUN
ejpam-6849	717	5	,	,	PUNCT
ejpam-6849	717	6	41:123–134	41:123–134	PROPN
ejpam-6849	717	7	,	,	PUNCT
ejpam-6849	717	8	2021	2021	NUM
ejpam-6849	717	9	.	.	PUNCT
ejpam-6849	718	1	[	[	X
ejpam-6849	718	2	11	11	NUM
ejpam-6849	718	3	]	]	X
ejpam-6849	718	4	r.	r.	PROPN
ejpam-6849	718	5	boutrig	boutrig	NOUN
ejpam-6849	718	6	,	,	PUNCT
ejpam-6849	718	7	m.	m.	NOUN
ejpam-6849	718	8	chellali	chellali	PROPN
ejpam-6849	718	9	,	,	PUNCT
ejpam-6849	718	10	t.w	t.w	PROPN
ejpam-6849	718	11	.	.	PROPN
ejpam-6849	718	12	haynes	haynes	PROPN
ejpam-6849	718	13	,	,	PUNCT
ejpam-6849	718	14	and	and	CCONJ
ejpam-6849	718	15	s.t	s.t	PROPN
ejpam-6849	718	16	.	.	PROPN
ejpam-6849	718	17	hedetniemi	hedetniemi	PROPN
ejpam-6849	718	18	.	.	PUNCT
ejpam-6849	719	1	vertex	vertex	NOUN
ejpam-6849	719	2	-	-	PUNCT
ejpam-6849	719	3	edge	edge	NOUN
ejpam-6849	719	4	domination	domination	NOUN
ejpam-6849	719	5	in	in	ADP
ejpam-6849	719	6	graphs	graph	NOUN
ejpam-6849	719	7	.	.	PUNCT
ejpam-6849	720	1	aequationes	aequatione	NOUN
ejpam-6849	720	2	mathematicae	mathematicae	PROPN
ejpam-6849	720	3	,	,	PUNCT
ejpam-6849	720	4	90:355–366	90:355–366	PROPN
ejpam-6849	720	5	,	,	PUNCT
ejpam-6849	720	6	2016	2016	NUM
ejpam-6849	720	7	.	.	PUNCT
ejpam-6849	721	1	[	[	X
ejpam-6849	721	2	12	12	NUM
ejpam-6849	721	3	]	]	X
ejpam-6849	721	4	f.	f.	PROPN
ejpam-6849	721	5	buckley	buckley	PROPN
ejpam-6849	721	6	and	and	CCONJ
ejpam-6849	721	7	f.	f.	PROPN
ejpam-6849	721	8	harary	harary	PROPN
ejpam-6849	721	9	.	.	PUNCT
ejpam-6849	722	1	distance	distance	NOUN
ejpam-6849	722	2	in	in	ADP
ejpam-6849	722	3	graphs	graph	NOUN
ejpam-6849	722	4	.	.	PUNCT
ejpam-6849	723	1	addison	addison	PROPN
ejpam-6849	723	2	-	-	PUNCT
ejpam-6849	723	3	wesley	wesley	PROPN
ejpam-6849	723	4	,	,	PUNCT
ejpam-6849	723	5	redwood	redwood	NOUN
ejpam-6849	723	6	city	city	NOUN
ejpam-6849	723	7	,	,	PUNCT
ejpam-6849	723	8	ca	ca	NOUN
ejpam-6849	723	9	,	,	PUNCT
ejpam-6849	723	10	1990	1990	NUM
ejpam-6849	723	11	.	.	PUNCT
ejpam-6849	724	1	[	[	X
ejpam-6849	724	2	13	13	NUM
ejpam-6849	724	3	]	]	X
ejpam-6849	724	4	c.	c.	PROPN
ejpam-6849	724	5	armada	armada	PROPN
ejpam-6849	724	6	,	,	PUNCT
ejpam-6849	724	7	s.r	s.r	PROPN
ejpam-6849	724	8	.	.	PROPN
ejpam-6849	724	9	canoy	canoy	PROPN
ejpam-6849	724	10	jr	jr	PROPN
ejpam-6849	724	11	.	.	PROPN
ejpam-6849	724	12	,	,	PUNCT
ejpam-6849	724	13	and	and	CCONJ
ejpam-6849	724	14	c.	c.	PROPN
ejpam-6849	724	15	go	go	VERB
ejpam-6849	724	16	.	.	PUNCT
ejpam-6849	725	1	forcing	force	VERB
ejpam-6849	725	2	domination	domination	NOUN
ejpam-6849	725	3	numbers	number	NOUN
ejpam-6849	725	4	of	of	ADP
ejpam-6849	725	5	graphs	graph	NOUN
ejpam-6849	725	6	under	under	ADP
ejpam-6849	725	7	some	some	DET
ejpam-6849	725	8	binary	binary	ADJ
ejpam-6849	725	9	operations	operation	NOUN
ejpam-6849	725	10	.	.	PUNCT
ejpam-6849	726	1	advances	advance	NOUN
ejpam-6849	726	2	and	and	CCONJ
ejpam-6849	726	3	applications	application	NOUN
ejpam-6849	726	4	in	in	ADP
ejpam-6849	726	5	discrete	discrete	ADJ
ejpam-6849	726	6	mathematics	mathematic	NOUN
ejpam-6849	726	7	,	,	PUNCT
ejpam-6849	726	8	19(3):213–228	19(3):213–228	NUM
ejpam-6849	726	9	,	,	PUNCT
ejpam-6849	726	10	2018	2018	NUM
ejpam-6849	726	11	.	.	PUNCT
ejpam-6849	727	1	[	[	X
ejpam-6849	727	2	14	14	NUM
ejpam-6849	727	3	]	]	X
ejpam-6849	727	4	c.	c.	PROPN
ejpam-6849	727	5	armada	armada	PROPN
ejpam-6849	727	6	,	,	PUNCT
ejpam-6849	727	7	s.r	s.r	PROPN
ejpam-6849	727	8	.	.	PROPN
ejpam-6849	727	9	canoy	canoy	PROPN
ejpam-6849	727	10	jr	jr	PROPN
ejpam-6849	727	11	.	.	PROPN
ejpam-6849	727	12	,	,	PUNCT
ejpam-6849	727	13	and	and	CCONJ
ejpam-6849	727	14	c.	c.	PROPN
ejpam-6849	727	15	go	go	VERB
ejpam-6849	727	16	.	.	PUNCT
ejpam-6849	728	1	forcing	force	VERB
ejpam-6849	728	2	subsets	subset	NOUN
ejpam-6849	728	3	for	for	ADP
ejpam-6849	728	4	γc	γc	NOUN
ejpam-6849	728	5	-	-	PUNCT
ejpam-6849	728	6	sets	set	NOUN
ejpam-6849	728	7	and	and	CCONJ
ejpam-6849	728	8	γt	γt	NOUN
ejpam-6849	728	9	-	-	NOUN
ejpam-6849	728	10	sets	set	NOUN
ejpam-6849	728	11	in	in	ADP
ejpam-6849	728	12	the	the	DET
ejpam-6849	728	13	lexicographic	lexicographic	ADJ
ejpam-6849	728	14	product	product	NOUN
ejpam-6849	728	15	of	of	ADP
ejpam-6849	728	16	graphs	graph	NOUN
ejpam-6849	728	17	.	.	PUNCT
ejpam-6849	729	1	european	european	ADJ
ejpam-6849	729	2	journal	journal	PROPN
ejpam-6849	729	3	of	of	ADP
ejpam-6849	729	4	pure	pure	ADJ
ejpam-6849	729	5	and	and	CCONJ
ejpam-6849	729	6	applied	applied	ADJ
ejpam-6849	729	7	mathematics	mathematic	NOUN
ejpam-6849	729	8	,	,	PUNCT
ejpam-6849	729	9	12(4):1779	12(4):1779	NUM
ejpam-6849	729	10	,	,	PUNCT
ejpam-6849	729	11	2019	2019	NUM
ejpam-6849	729	12	.	.	PUNCT
ejpam-6849	730	1	[	[	X
ejpam-6849	730	2	15	15	NUM
ejpam-6849	730	3	]	]	X
ejpam-6849	730	4	c.	c.	PROPN
ejpam-6849	730	5	berge	berge	PROPN
ejpam-6849	730	6	.	.	PUNCT
ejpam-6849	731	1	thã	thã	PROPN
ejpam-6849	731	2	©	©	PROPN
ejpam-6849	731	3	orie	orie	PROPN
ejpam-6849	731	4	des	des	PROPN
ejpam-6849	731	5	graphes	graphes	PROPN
ejpam-6849	731	6	et	et	PROPN
ejpam-6849	731	7	ses	ses	PROPN
ejpam-6849	731	8	applications	application	NOUN
ejpam-6849	731	9	.	.	PUNCT
ejpam-6849	732	1	dunod	dunod	PROPN
ejpam-6849	732	2	,	,	PUNCT
ejpam-6849	732	3	paris	paris	PROPN
ejpam-6849	732	4	,	,	PUNCT
ejpam-6849	732	5	1958	1958	NUM
ejpam-6849	732	6	.	.	PUNCT
ejpam-6849	733	1	english	english	ADJ
ejpam-6849	733	2	translation	translation	NOUN
ejpam-6849	733	3	:	:	PUNCT
ejpam-6849	733	4	the	the	DET
ejpam-6849	733	5	theory	theory	NOUN
ejpam-6849	733	6	of	of	ADP
ejpam-6849	733	7	graphs	graph	NOUN
ejpam-6849	733	8	and	and	CCONJ
ejpam-6849	733	9	its	its	PRON
ejpam-6849	733	10	applications	application	NOUN
ejpam-6849	733	11	,	,	PUNCT
ejpam-6849	733	12	methuen	methuen	PROPN
ejpam-6849	733	13	(	(	PUNCT
ejpam-6849	733	14	london	london	PROPN
ejpam-6849	733	15	)	)	PUNCT
ejpam-6849	733	16	and	and	CCONJ
ejpam-6849	733	17	wiley	wiley	PROPN
ejpam-6849	733	18	(	(	PUNCT
ejpam-6849	733	19	new	new	PROPN
ejpam-6849	733	20	york	york	PROPN
ejpam-6849	733	21	)	)	PUNCT
ejpam-6849	733	22	,	,	PUNCT
ejpam-6849	733	23	1962	1962	NUM
ejpam-6849	733	24	.	.	PUNCT
ejpam-6849	734	1	[	[	X
ejpam-6849	734	2	16	16	NUM
ejpam-6849	734	3	]	]	X
ejpam-6849	734	4	e.	e.	PROPN
ejpam-6849	734	5	cockayne	cockayne	PROPN
ejpam-6849	734	6	and	and	CCONJ
ejpam-6849	734	7	s.	s.	PROPN
ejpam-6849	734	8	hedetniemi	hedetniemi	PROPN
ejpam-6849	734	9	.	.	PUNCT
ejpam-6849	735	1	towards	towards	ADP
ejpam-6849	735	2	a	a	DET
ejpam-6849	735	3	theory	theory	NOUN
ejpam-6849	735	4	of	of	ADP
ejpam-6849	735	5	domination	domination	NOUN
ejpam-6849	735	6	in	in	ADP
ejpam-6849	735	7	graphs	graph	NOUN
ejpam-6849	735	8	.	.	PUNCT
ejpam-6849	736	1	networks	network	NOUN
ejpam-6849	736	2	,	,	PUNCT
ejpam-6849	736	3	7(3):247–261	7(3):247–261	NUM
ejpam-6849	736	4	,	,	PUNCT
ejpam-6849	736	5	1977	1977	NUM
ejpam-6849	736	6	.	.	PUNCT
ejpam-6849	737	1	[	[	X
ejpam-6849	737	2	17	17	NUM
ejpam-6849	737	3	]	]	X
ejpam-6849	737	4	e.	e.	PROPN
ejpam-6849	737	5	cockayne	cockayne	PROPN
ejpam-6849	737	6	,	,	PUNCT
ejpam-6849	737	7	r.m	r.m	PROPN
ejpam-6849	737	8	.	.	PROPN
ejpam-6849	737	9	dawes	dawes	PROPN
ejpam-6849	737	10	,	,	PUNCT
ejpam-6849	737	11	and	and	CCONJ
ejpam-6849	737	12	s.t	s.t	PROPN
ejpam-6849	737	13	.	.	PROPN
ejpam-6849	737	14	hedetniemi	hedetniemi	PROPN
ejpam-6849	737	15	.	.	PUNCT
ejpam-6849	738	1	total	total	ADJ
ejpam-6849	738	2	domination	domination	NOUN
ejpam-6849	738	3	in	in	ADP
ejpam-6849	738	4	graphs	graph	NOUN
ejpam-6849	738	5	.	.	PUNCT
ejpam-6849	739	1	networks	network	NOUN
ejpam-6849	739	2	,	,	PUNCT
ejpam-6849	739	3	10(3):211–219	10(3):211–219	NUM
ejpam-6849	739	4	,	,	PUNCT
ejpam-6849	739	5	2006	2006	NUM
ejpam-6849	739	6	.	.	PUNCT
ejpam-6849	740	1	[	[	X
ejpam-6849	740	2	18	18	NUM
ejpam-6849	740	3	]	]	X
ejpam-6849	740	4	m.	m.	NOUN
ejpam-6849	740	5	henning	henning	PROPN
ejpam-6849	740	6	and	and	CCONJ
ejpam-6849	740	7	a.	a.	PROPN
ejpam-6849	740	8	yeo	yeo	PROPN
ejpam-6849	740	9	.	.	PROPN
ejpam-6849	741	1	total	total	ADJ
ejpam-6849	741	2	domination	domination	NOUN
ejpam-6849	741	3	in	in	ADP
ejpam-6849	741	4	graphs	graph	NOUN
ejpam-6849	741	5	.	.	PUNCT
ejpam-6849	742	1	springer	springer	NOUN
ejpam-6849	742	2	,	,	PUNCT
ejpam-6849	742	3	new	new	PROPN
ejpam-6849	742	4	york	york	PROPN
ejpam-6849	742	5	,	,	PUNCT
ejpam-6849	742	6	2013	2013	NUM
ejpam-6849	742	7	.	.	PUNCT
ejpam-6849	743	1	[	[	X
ejpam-6849	743	2	19	19	NUM
ejpam-6849	743	3	]	]	X
ejpam-6849	743	4	w.f	w.f	PROPN
ejpam-6849	743	5	.	.	PROPN
ejpam-6849	743	6	klostermeyer	klostermeyer	PROPN
ejpam-6849	743	7	and	and	CCONJ
ejpam-6849	743	8	c.m	c.m	PROPN
ejpam-6849	743	9	.	.	PROPN
ejpam-6849	743	10	mynhardt	mynhardt	PROPN
ejpam-6849	743	11	.	.	PUNCT
ejpam-6849	744	1	protecting	protect	VERB
ejpam-6849	744	2	a	a	DET
ejpam-6849	744	3	graph	graph	NOUN
ejpam-6849	744	4	with	with	ADP
ejpam-6849	744	5	mobile	mobile	ADJ
ejpam-6849	744	6	guards	guard	NOUN
ejpam-6849	744	7	.	.	PUNCT
ejpam-6849	745	1	applied	apply	VERB
ejpam-6849	745	2	analysis	analysis	NOUN
ejpam-6849	745	3	and	and	CCONJ
ejpam-6849	745	4	discrete	discrete	ADJ
ejpam-6849	745	5	mathematics	mathematic	NOUN
ejpam-6849	745	6	,	,	PUNCT
ejpam-6849	745	7	10:1–29	10:1–29	NUM
ejpam-6849	745	8	,	,	PUNCT
ejpam-6849	745	9	2016	2016	NUM
ejpam-6849	745	10	.	.	PUNCT