id	sid	tid	token	lemma	pos
ejpam-4352	1	1	european	european	PROPN
ejpam-4352	1	2	journal	journal	PROPN
ejpam-4352	1	3	of	of	ADP
ejpam-4352	1	4	pure	pure	ADJ
ejpam-4352	1	5	and	and	CCONJ
ejpam-4352	1	6	applied	apply	VERB
ejpam-4352	1	7	mathematics	mathematic	NOUN
ejpam-4352	1	8	vol	vol	NOUN
ejpam-4352	1	9	.	.	PROPN
ejpam-4352	2	1	15	15	NUM
ejpam-4352	2	2	,	,	PUNCT
ejpam-4352	2	3	no	no	INTJ
ejpam-4352	2	4	.	.	NOUN
ejpam-4352	2	5	2	2	NUM
ejpam-4352	2	6	,	,	PUNCT
ejpam-4352	2	7	2022	2022	NUM
ejpam-4352	2	8	,	,	PUNCT
ejpam-4352	2	9	342	342	NUM
ejpam-4352	2	10	-	-	SYM
ejpam-4352	2	11	353	353	NUM
ejpam-4352	2	12	issn	issn	PROPN
ejpam-4352	2	13	1307	1307	NUM
ejpam-4352	2	14	-	-	SYM
ejpam-4352	2	15	5543	5543	NUM
ejpam-4352	2	16	–	–	PUNCT
ejpam-4352	2	17	ejpam.com	ejpam.com	X
ejpam-4352	2	18	published	publish	VERB
ejpam-4352	2	19	by	by	ADP
ejpam-4352	2	20	new	new	PROPN
ejpam-4352	2	21	york	york	PROPN
ejpam-4352	2	22	business	business	PROPN
ejpam-4352	3	1	global	global	PROPN
ejpam-4352	3	2	a	a	DET
ejpam-4352	3	3	variant	variant	NOUN
ejpam-4352	3	4	of	of	ADP
ejpam-4352	3	5	hop	hop	NOUN
ejpam-4352	3	6	domination	domination	NOUN
ejpam-4352	3	7	in	in	ADP
ejpam-4352	3	8	graphs	graph	NOUN
ejpam-4352	3	9	sergio	sergio	PROPN
ejpam-4352	3	10	r.	r.	PROPN
ejpam-4352	3	11	canoy	canoy	PROPN
ejpam-4352	3	12	,	,	PUNCT
ejpam-4352	3	13	jr.1	jr.1	PROPN
ejpam-4352	3	14	,	,	PUNCT
ejpam-4352	4	1	gemma	gemma	PROPN
ejpam-4352	4	2	p.	p.	PROPN
ejpam-4352	4	3	salasalan2,∗	salasalan2,∗	PROPN
ejpam-4352	4	4	1	1	NUM
ejpam-4352	4	5	department	department	NOUN
ejpam-4352	4	6	of	of	ADP
ejpam-4352	4	7	mathematics	mathematic	NOUN
ejpam-4352	4	8	and	and	CCONJ
ejpam-4352	4	9	statistics	statistic	NOUN
ejpam-4352	4	10	,	,	PUNCT
ejpam-4352	4	11	college	college	NOUN
ejpam-4352	4	12	of	of	ADP
ejpam-4352	4	13	science	science	NOUN
ejpam-4352	4	14	and	and	CCONJ
ejpam-4352	4	15	mathematics	mathematic	NOUN
ejpam-4352	4	16	,	,	PUNCT
ejpam-4352	4	17	center	center	NOUN
ejpam-4352	4	18	for	for	ADP
ejpam-4352	4	19	graph	graph	NOUN
ejpam-4352	4	20	theory	theory	NOUN
ejpam-4352	4	21	,	,	PUNCT
ejpam-4352	4	22	algebra	algebra	NOUN
ejpam-4352	4	23	and	and	CCONJ
ejpam-4352	4	24	analysis	analysis	NOUN
ejpam-4352	4	25	-	-	PUNCT
ejpam-4352	4	26	prism	prism	NOUN
ejpam-4352	4	27	,	,	PUNCT
ejpam-4352	4	28	msu	msu	PROPN
ejpam-4352	4	29	-	-	PUNCT
ejpam-4352	4	30	iligan	iligan	PROPN
ejpam-4352	4	31	institute	institute	PROPN
ejpam-4352	4	32	of	of	ADP
ejpam-4352	4	33	technology	technology	PROPN
ejpam-4352	4	34	,	,	PUNCT
ejpam-4352	4	35	9200	9200	NUM
ejpam-4352	4	36	iligan	iligan	ADJ
ejpam-4352	4	37	city	city	NOUN
ejpam-4352	4	38	,	,	PUNCT
ejpam-4352	4	39	philippines	philippines	PROPN
ejpam-4352	4	40	2	2	NUM
ejpam-4352	4	41	department	department	NOUN
ejpam-4352	4	42	of	of	ADP
ejpam-4352	4	43	arts	art	NOUN
ejpam-4352	4	44	and	and	CCONJ
ejpam-4352	4	45	sciences	sciences	PROPN
ejpam-4352	4	46	,	,	PUNCT
ejpam-4352	4	47	institute	institute	NOUN
ejpam-4352	4	48	of	of	ADP
ejpam-4352	4	49	arts	art	NOUN
ejpam-4352	4	50	and	and	CCONJ
ejpam-4352	4	51	sciences	sciences	PROPN
ejpam-4352	4	52	,	,	PUNCT
ejpam-4352	4	53	davao	davao	PROPN
ejpam-4352	4	54	del	del	PROPN
ejpam-4352	4	55	sur	sur	PROPN
ejpam-4352	4	56	state	state	PROPN
ejpam-4352	4	57	college	college	PROPN
ejpam-4352	4	58	,	,	PUNCT
ejpam-4352	4	59	matti	matti	PROPN
ejpam-4352	4	60	,	,	PUNCT
ejpam-4352	4	61	digos	digos	PROPN
ejpam-4352	4	62	,	,	PUNCT
ejpam-4352	4	63	davao	davao	PROPN
ejpam-4352	4	64	del	del	PROPN
ejpam-4352	4	65	sur	sur	PROPN
ejpam-4352	4	66	,	,	PUNCT
ejpam-4352	4	67	philippines	philippine	NOUN
ejpam-4352	4	68	abstract	abstract	ADJ
ejpam-4352	4	69	.	.	PUNCT
ejpam-4352	5	1	let	let	VERB
ejpam-4352	5	2	g	g	PRON
ejpam-4352	5	3	be	be	AUX
ejpam-4352	5	4	a	a	DET
ejpam-4352	5	5	connected	connected	ADJ
ejpam-4352	5	6	graph	graph	NOUN
ejpam-4352	5	7	with	with	ADP
ejpam-4352	5	8	vertex	vertex	NOUN
ejpam-4352	5	9	and	and	CCONJ
ejpam-4352	5	10	edge	edge	NOUN
ejpam-4352	5	11	sets	set	NOUN
ejpam-4352	5	12	v	v	ADP
ejpam-4352	5	13	(	(	PUNCT
ejpam-4352	5	14	g	g	NOUN
ejpam-4352	5	15	)	)	PUNCT
ejpam-4352	5	16	and	and	CCONJ
ejpam-4352	5	17	e(g	e(g	PROPN
ejpam-4352	5	18	)	)	PUNCT
ejpam-4352	5	19	,	,	PUNCT
ejpam-4352	5	20	respectively	respectively	ADV
ejpam-4352	5	21	.	.	PUNCT
ejpam-4352	6	1	a	a	DET
ejpam-4352	6	2	set	set	NOUN
ejpam-4352	6	3	s	s	NOUN
ejpam-4352	6	4	⊆	⊆	NUM
ejpam-4352	6	5	v	v	NOUN
ejpam-4352	6	6	(	(	PUNCT
ejpam-4352	6	7	g	g	NOUN
ejpam-4352	6	8	)	)	PUNCT
ejpam-4352	6	9	is	be	AUX
ejpam-4352	6	10	a	a	DET
ejpam-4352	6	11	hop	hop	NOUN
ejpam-4352	6	12	dominating	dominating	NOUN
ejpam-4352	6	13	set	set	NOUN
ejpam-4352	6	14	of	of	ADP
ejpam-4352	6	15	g	g	PROPN
ejpam-4352	6	16	if	if	SCONJ
ejpam-4352	6	17	for	for	ADP
ejpam-4352	6	18	each	each	PRON
ejpam-4352	6	19	v	v	NUM
ejpam-4352	6	20	∈	∈	PROPN
ejpam-4352	6	21	v	v	NOUN
ejpam-4352	6	22	(	(	PUNCT
ejpam-4352	6	23	g	g	NOUN
ejpam-4352	6	24	)	)	PUNCT
ejpam-4352	6	25	\	\	PROPN
ejpam-4352	7	1	s	s	X
ejpam-4352	7	2	,	,	PUNCT
ejpam-4352	7	3	there	there	PRON
ejpam-4352	7	4	exists	exist	VERB
ejpam-4352	7	5	w	w	PROPN
ejpam-4352	7	6	∈	∈	PROPN
ejpam-4352	7	7	s	s	VERB
ejpam-4352	8	1	such	such	ADJ
ejpam-4352	8	2	that	that	PRON
ejpam-4352	8	3	dg(v	dg(v	ADJ
ejpam-4352	8	4	,	,	PUNCT
ejpam-4352	8	5	w	w	NOUN
ejpam-4352	8	6	)	)	PUNCT
ejpam-4352	8	7	=	=	SYM
ejpam-4352	8	8	2	2	X
ejpam-4352	8	9	.	.	X
ejpam-4352	8	10	a	a	DET
ejpam-4352	8	11	set	set	NOUN
ejpam-4352	8	12	s	s	NOUN
ejpam-4352	8	13	⊆	⊆	NUM
ejpam-4352	8	14	v	v	NOUN
ejpam-4352	8	15	(	(	PUNCT
ejpam-4352	8	16	g	g	NOUN
ejpam-4352	8	17	)	)	PUNCT
ejpam-4352	8	18	is	be	AUX
ejpam-4352	8	19	a	a	DET
ejpam-4352	8	20	super	super	ADV
ejpam-4352	8	21	hop	hop	NOUN
ejpam-4352	8	22	dominating	dominating	NOUN
ejpam-4352	8	23	set	set	NOUN
ejpam-4352	8	24	if	if	SCONJ
ejpam-4352	8	25	ehpng(v	ehpng(v	PROPN
ejpam-4352	8	26	,	,	PUNCT
ejpam-4352	8	27	v	v	NOUN
ejpam-4352	8	28	(	(	PUNCT
ejpam-4352	8	29	g	g	NOUN
ejpam-4352	8	30	)	)	PUNCT
ejpam-4352	8	31	\	\	PROPN
ejpam-4352	9	1	s	s	X
ejpam-4352	9	2	)	)	PUNCT
ejpam-4352	9	3	̸=	̸=	PROPN
ejpam-4352	9	4	∅	∅	NOUN
ejpam-4352	9	5	for	for	ADP
ejpam-4352	9	6	each	each	PRON
ejpam-4352	9	7	v	v	NUM
ejpam-4352	9	8	∈	∈	PROPN
ejpam-4352	9	9	v	v	NOUN
ejpam-4352	9	10	(	(	PUNCT
ejpam-4352	9	11	g	g	NOUN
ejpam-4352	9	12	)	)	PUNCT
ejpam-4352	9	13	\	\	PROPN
ejpam-4352	10	1	s	s	X
ejpam-4352	10	2	,	,	PUNCT
ejpam-4352	10	3	where	where	SCONJ
ejpam-4352	10	4	ehpng(v	ehpng(v	X
ejpam-4352	10	5	,	,	PUNCT
ejpam-4352	10	6	v	v	NOUN
ejpam-4352	10	7	(	(	PUNCT
ejpam-4352	10	8	g	g	NOUN
ejpam-4352	10	9	)	)	PUNCT
ejpam-4352	10	10	\	\	PART
ejpam-4352	11	1	s	s	X
ejpam-4352	11	2	)	)	PUNCT
ejpam-4352	11	3	is	be	AUX
ejpam-4352	11	4	the	the	DET
ejpam-4352	11	5	set	set	NOUN
ejpam-4352	11	6	containing	contain	VERB
ejpam-4352	11	7	all	all	DET
ejpam-4352	11	8	the	the	DET
ejpam-4352	11	9	external	external	ADJ
ejpam-4352	11	10	hop	hop	NOUN
ejpam-4352	11	11	private	private	ADJ
ejpam-4352	11	12	neighbors	neighbor	NOUN
ejpam-4352	11	13	of	of	ADP
ejpam-4352	11	14	v	v	NOUN
ejpam-4352	11	15	with	with	ADP
ejpam-4352	11	16	respect	respect	NOUN
ejpam-4352	11	17	to	to	ADP
ejpam-4352	11	18	v	v	NOUN
ejpam-4352	11	19	(	(	PUNCT
ejpam-4352	11	20	g	g	NOUN
ejpam-4352	11	21	)	)	PUNCT
ejpam-4352	11	22	\	\	PUNCT
ejpam-4352	12	1	s.	s.	PROPN
ejpam-4352	12	2	the	the	DET
ejpam-4352	12	3	minimum	minimum	ADJ
ejpam-4352	12	4	cardinality	cardinality	NOUN
ejpam-4352	12	5	of	of	ADP
ejpam-4352	12	6	a	a	DET
ejpam-4352	12	7	super	super	ADV
ejpam-4352	12	8	hop	hop	NOUN
ejpam-4352	12	9	dominating	dominating	NOUN
ejpam-4352	12	10	set	set	NOUN
ejpam-4352	12	11	of	of	ADP
ejpam-4352	12	12	g	g	NOUN
ejpam-4352	12	13	,	,	PUNCT
ejpam-4352	12	14	denoted	denote	VERB
ejpam-4352	12	15	by	by	ADP
ejpam-4352	12	16	γs	γs	NUM
ejpam-4352	12	17	h(g	h(g	NOUN
ejpam-4352	12	18	)	)	PUNCT
ejpam-4352	12	19	,	,	PUNCT
ejpam-4352	12	20	is	be	AUX
ejpam-4352	12	21	called	call	VERB
ejpam-4352	12	22	the	the	DET
ejpam-4352	12	23	super	super	PROPN
ejpam-4352	12	24	hop	hop	NOUN
ejpam-4352	12	25	domination	domination	NOUN
ejpam-4352	12	26	number	number	NOUN
ejpam-4352	12	27	of	of	ADP
ejpam-4352	12	28	g.	g.	PROPN
ejpam-4352	12	29	in	in	ADP
ejpam-4352	12	30	this	this	DET
ejpam-4352	12	31	paper	paper	NOUN
ejpam-4352	12	32	,	,	PUNCT
ejpam-4352	12	33	we	we	PRON
ejpam-4352	12	34	investigate	investigate	VERB
ejpam-4352	12	35	the	the	DET
ejpam-4352	12	36	concept	concept	NOUN
ejpam-4352	12	37	and	and	CCONJ
ejpam-4352	12	38	study	study	VERB
ejpam-4352	12	39	it	it	PRON
ejpam-4352	12	40	for	for	ADP
ejpam-4352	12	41	graphs	graph	NOUN
ejpam-4352	12	42	resulting	result	VERB
ejpam-4352	12	43	from	from	ADP
ejpam-4352	12	44	some	some	DET
ejpam-4352	12	45	binary	binary	ADJ
ejpam-4352	12	46	operations	operation	NOUN
ejpam-4352	12	47	.	.	PUNCT
ejpam-4352	13	1	specifically	specifically	ADV
ejpam-4352	13	2	,	,	PUNCT
ejpam-4352	13	3	we	we	PRON
ejpam-4352	13	4	characterize	characterize	VERB
ejpam-4352	13	5	the	the	DET
ejpam-4352	13	6	super	super	ADJ
ejpam-4352	13	7	hop	hop	NOUN
ejpam-4352	13	8	dominating	dominating	NOUN
ejpam-4352	13	9	sets	set	NOUN
ejpam-4352	13	10	in	in	ADP
ejpam-4352	13	11	the	the	DET
ejpam-4352	13	12	join	join	NOUN
ejpam-4352	13	13	,	,	PUNCT
ejpam-4352	13	14	and	and	CCONJ
ejpam-4352	13	15	lexicographic	lexicographic	ADJ
ejpam-4352	13	16	products	product	NOUN
ejpam-4352	13	17	of	of	ADP
ejpam-4352	13	18	graphs	graph	NOUN
ejpam-4352	13	19	,	,	PUNCT
ejpam-4352	13	20	and	and	CCONJ
ejpam-4352	13	21	determine	determine	VERB
ejpam-4352	13	22	bounds	bound	NOUN
ejpam-4352	13	23	of	of	ADP
ejpam-4352	13	24	the	the	DET
ejpam-4352	13	25	super	super	PROPN
ejpam-4352	13	26	hop	hop	NOUN
ejpam-4352	13	27	domination	domination	NOUN
ejpam-4352	13	28	number	number	NOUN
ejpam-4352	13	29	of	of	ADP
ejpam-4352	13	30	each	each	PRON
ejpam-4352	13	31	of	of	ADP
ejpam-4352	13	32	these	these	DET
ejpam-4352	13	33	graphs	graph	NOUN
ejpam-4352	13	34	.	.	PUNCT
ejpam-4352	14	1	2020	2020	NUM
ejpam-4352	14	2	mathematics	mathematic	NOUN
ejpam-4352	14	3	subject	subject	NOUN
ejpam-4352	14	4	classifications	classification	NOUN
ejpam-4352	14	5	:	:	PUNCT
ejpam-4352	14	6	05c69	05c69	X
ejpam-4352	14	7	key	key	ADJ
ejpam-4352	14	8	words	word	NOUN
ejpam-4352	14	9	and	and	CCONJ
ejpam-4352	14	10	phrases	phrase	NOUN
ejpam-4352	14	11	:	:	PUNCT
ejpam-4352	14	12	hop	hop	NOUN
ejpam-4352	14	13	domination	domination	NOUN
ejpam-4352	14	14	,	,	PUNCT
ejpam-4352	14	15	super	sup	ADJ
ejpam-4352	14	16	hop	hop	PROPN
ejpam-4352	14	17	domination	domination	NOUN
ejpam-4352	14	18	,	,	PUNCT
ejpam-4352	14	19	complement	complement	NOUN
ejpam-4352	14	20	-	-	PUNCT
ejpam-4352	14	21	super	super	ADJ
ejpam-4352	14	22	domination	domination	NOUN
ejpam-4352	14	23	,	,	PUNCT
ejpam-4352	14	24	join	join	NOUN
ejpam-4352	14	25	,	,	PUNCT
ejpam-4352	14	26	lexicographic	lexicographic	ADJ
ejpam-4352	14	27	product	product	NOUN
ejpam-4352	14	28	1	1	NUM
ejpam-4352	14	29	.	.	PUNCT
ejpam-4352	15	1	introduction	introduction	NOUN
ejpam-4352	15	2	super	super	ADJ
ejpam-4352	15	3	domination	domination	NOUN
ejpam-4352	15	4	in	in	ADP
ejpam-4352	15	5	a	a	DET
ejpam-4352	15	6	graph	graph	NOUN
ejpam-4352	15	7	was	be	AUX
ejpam-4352	15	8	first	first	ADV
ejpam-4352	15	9	introduced	introduce	VERB
ejpam-4352	15	10	and	and	CCONJ
ejpam-4352	15	11	studied	study	VERB
ejpam-4352	15	12	by	by	ADP
ejpam-4352	15	13	lemanska	lemanska	PROPN
ejpam-4352	15	14	et	et	PROPN
ejpam-4352	15	15	al	al	PROPN
ejpam-4352	15	16	.	.	PUNCT
ejpam-4352	16	1	in	in	ADP
ejpam-4352	16	2	[	[	X
ejpam-4352	16	3	8	8	NUM
ejpam-4352	16	4	]	]	PUNCT
ejpam-4352	16	5	.	.	PUNCT
ejpam-4352	17	1	this	this	DET
ejpam-4352	17	2	concept	concept	NOUN
ejpam-4352	17	3	uses	use	VERB
ejpam-4352	17	4	the	the	DET
ejpam-4352	17	5	concept	concept	NOUN
ejpam-4352	17	6	of	of	ADP
ejpam-4352	17	7	external	external	ADJ
ejpam-4352	17	8	private	private	ADJ
ejpam-4352	17	9	neighbor	neighbor	NOUN
ejpam-4352	17	10	of	of	ADP
ejpam-4352	17	11	a	a	DET
ejpam-4352	17	12	vertex	vertex	NOUN
ejpam-4352	17	13	in	in	ADP
ejpam-4352	17	14	some	some	DET
ejpam-4352	17	15	subset	subset	NOUN
ejpam-4352	17	16	of	of	ADP
ejpam-4352	17	17	the	the	DET
ejpam-4352	17	18	vertex	vertex	NOUN
ejpam-4352	17	19	set	set	NOUN
ejpam-4352	17	20	of	of	ADP
ejpam-4352	17	21	a	a	DET
ejpam-4352	17	22	graph	graph	NOUN
ejpam-4352	17	23	.	.	PUNCT
ejpam-4352	18	1	dettlaff	dettlaff	VERB
ejpam-4352	18	2	et	et	PROPN
ejpam-4352	18	3	al	al	PROPN
ejpam-4352	18	4	.	.	PUNCT
ejpam-4352	19	1	in	in	ADP
ejpam-4352	19	2	[	[	X
ejpam-4352	19	3	3	3	NUM
ejpam-4352	19	4	]	]	X
ejpam-4352	19	5	determined	determine	VERB
ejpam-4352	19	6	the	the	DET
ejpam-4352	19	7	super	super	ADJ
ejpam-4352	19	8	domination	domination	NOUN
ejpam-4352	19	9	number	number	NOUN
ejpam-4352	19	10	of	of	ADP
ejpam-4352	19	11	lexicographic	lexicographic	ADJ
ejpam-4352	19	12	products	product	NOUN
ejpam-4352	19	13	of	of	ADP
ejpam-4352	19	14	graphs	graph	NOUN
ejpam-4352	19	15	.	.	PUNCT
ejpam-4352	20	1	also	also	ADV
ejpam-4352	20	2	,	,	PUNCT
ejpam-4352	20	3	dettlaff	dettlaff	VERB
ejpam-4352	20	4	in	in	ADP
ejpam-4352	20	5	[	[	X
ejpam-4352	20	6	2	2	NUM
ejpam-4352	20	7	]	]	PUNCT
ejpam-4352	20	8	determined	determine	VERB
ejpam-4352	20	9	some	some	DET
ejpam-4352	20	10	values	value	NOUN
ejpam-4352	20	11	and	and	CCONJ
ejpam-4352	20	12	bounds	bound	NOUN
ejpam-4352	20	13	for	for	ADP
ejpam-4352	20	14	the	the	DET
ejpam-4352	20	15	super	super	ADJ
ejpam-4352	20	16	domination	domination	NOUN
ejpam-4352	20	17	number	number	NOUN
ejpam-4352	20	18	of	of	ADP
ejpam-4352	20	19	some	some	DET
ejpam-4352	20	20	cartesian	cartesian	ADJ
ejpam-4352	20	21	products	product	NOUN
ejpam-4352	20	22	of	of	ADP
ejpam-4352	20	23	graphs	graph	NOUN
ejpam-4352	20	24	.	.	PUNCT
ejpam-4352	21	1	paraico	paraico	NOUN
ejpam-4352	21	2	and	and	CCONJ
ejpam-4352	21	3	canoy	canoy	ADJ
ejpam-4352	21	4	in	in	ADP
ejpam-4352	21	5	[	[	X
ejpam-4352	21	6	13	13	NUM
ejpam-4352	21	7	]	]	PUNCT
ejpam-4352	21	8	characterized	characterize	VERB
ejpam-4352	21	9	the	the	DET
ejpam-4352	21	10	super	super	ADJ
ejpam-4352	21	11	dominating	dominating	NOUN
ejpam-4352	21	12	sets	set	NOUN
ejpam-4352	21	13	in	in	ADP
ejpam-4352	21	14	the	the	DET
ejpam-4352	21	15	lexicographic	lexicographic	NOUN
ejpam-4352	21	16	and	and	CCONJ
ejpam-4352	21	17	the	the	DET
ejpam-4352	21	18	cartesian	cartesian	ADJ
ejpam-4352	21	19	products	product	NOUN
ejpam-4352	21	20	of	of	ADP
ejpam-4352	21	21	graphs	graph	NOUN
ejpam-4352	21	22	and	and	CCONJ
ejpam-4352	21	23	obtained	obtain	VERB
ejpam-4352	21	24	bounds	bound	NOUN
ejpam-4352	21	25	for	for	ADP
ejpam-4352	21	26	the	the	DET
ejpam-4352	21	27	super	super	ADJ
ejpam-4352	21	28	domination	domination	NOUN
ejpam-4352	21	29	numbers	number	NOUN
ejpam-4352	21	30	of	of	ADP
ejpam-4352	21	31	these	these	DET
ejpam-4352	21	32	graphs	graph	NOUN
ejpam-4352	21	33	.	.	PUNCT
ejpam-4352	22	1	recently	recently	ADV
ejpam-4352	22	2	,	,	PUNCT
ejpam-4352	22	3	natarajan	natarajan	PROPN
ejpam-4352	22	4	and	and	CCONJ
ejpam-4352	22	5	ayyaswamy	ayyaswamy	PROPN
ejpam-4352	23	1	[	[	X
ejpam-4352	23	2	11	11	NUM
ejpam-4352	23	3	]	]	PUNCT
ejpam-4352	23	4	introduced	introduce	VERB
ejpam-4352	23	5	and	and	CCONJ
ejpam-4352	23	6	studied	study	VERB
ejpam-4352	23	7	the	the	DET
ejpam-4352	23	8	concept	concept	NOUN
ejpam-4352	23	9	of	of	ADP
ejpam-4352	23	10	hop	hop	NOUN
ejpam-4352	23	11	domination	domination	NOUN
ejpam-4352	23	12	in	in	ADP
ejpam-4352	23	13	a	a	DET
ejpam-4352	23	14	graph	graph	NOUN
ejpam-4352	23	15	.	.	PUNCT
ejpam-4352	24	1	ayyaswamy	ayyaswamy	PROPN
ejpam-4352	24	2	et	et	PROPN
ejpam-4352	24	3	al	al	PROPN
ejpam-4352	24	4	.	.	PUNCT
ejpam-4352	25	1	in	in	ADP
ejpam-4352	25	2	[	[	X
ejpam-4352	25	3	1	1	X
ejpam-4352	25	4	]	]	PUNCT
ejpam-4352	25	5	also	also	ADV
ejpam-4352	25	6	investigated	investigate	VERB
ejpam-4352	25	7	the	the	DET
ejpam-4352	25	8	concept	concept	NOUN
ejpam-4352	25	9	and	and	CCONJ
ejpam-4352	25	10	obtained	obtain	VERB
ejpam-4352	25	11	∗corresponding	∗corresponde	VERB
ejpam-4352	25	12	author	author	NOUN
ejpam-4352	25	13	.	.	PUNCT
ejpam-4352	26	1	doi	doi	NOUN
ejpam-4352	26	2	:	:	PUNCT
ejpam-4352	26	3	https://doi.org/10.29020/nybg.ejpam.v15i2.4352	https://doi.org/10.29020/nybg.ejpam.v15i2.4352	ADJ
ejpam-4352	26	4	email	email	NOUN
ejpam-4352	26	5	addresses	address	NOUN
ejpam-4352	26	6	:	:	PUNCT
ejpam-4352	26	7	sergio.canoy@g.msuiit.edu.ph	sergio.canoy@g.msuiit.edu.ph	PROPN
ejpam-4352	26	8	(	(	PUNCT
ejpam-4352	26	9	s.	s.	PROPN
ejpam-4352	26	10	canoy	canoy	PROPN
ejpam-4352	26	11	,	,	PUNCT
ejpam-4352	26	12	jr	jr	PROPN
ejpam-4352	26	13	.	.	PUNCT
ejpam-4352	26	14	)	)	PUNCT
ejpam-4352	27	1	gemma.salasalan@dssc.edu.ph	gemma.salasalan@dssc.edu.ph	PROPN
ejpam-4352	27	2	(	(	PUNCT
ejpam-4352	27	3	g.	g.	NOUN
ejpam-4352	27	4	salasalan	salasalan	NOUN
ejpam-4352	27	5	)	)	PUNCT
ejpam-4352	27	6	https://www.ejpam.com	https://www.ejpam.com	NOUN
ejpam-4352	28	1	342	342	NUM
ejpam-4352	29	1	©	©	ADP
ejpam-4352	29	2	2022	2022	NUM
ejpam-4352	29	3	ejpam	ejpam	VERB
ejpam-4352	29	4	all	all	DET
ejpam-4352	29	5	rights	right	NOUN
ejpam-4352	29	6	reserved	reserve	VERB
ejpam-4352	29	7	.	.	PUNCT
ejpam-4352	30	1	s.	s.	PROPN
ejpam-4352	30	2	canoy	canoy	PROPN
ejpam-4352	30	3	,	,	PUNCT
ejpam-4352	30	4	jr	jr	PROPN
ejpam-4352	30	5	.	.	PROPN
ejpam-4352	30	6	,	,	PUNCT
ejpam-4352	30	7	g.	g.	PROPN
ejpam-4352	30	8	salasalan	salasalan	PROPN
ejpam-4352	30	9	/	/	SYM
ejpam-4352	30	10	eur	eur	PROPN
ejpam-4352	30	11	.	.	PUNCT
ejpam-4352	31	1	j.	j.	PROPN
ejpam-4352	31	2	pure	pure	PROPN
ejpam-4352	31	3	appl	appl	PROPN
ejpam-4352	31	4	.	.	PROPN
ejpam-4352	31	5	math	math	PROPN
ejpam-4352	31	6	,	,	PUNCT
ejpam-4352	31	7	15	15	NUM
ejpam-4352	31	8	(	(	PUNCT
ejpam-4352	31	9	2	2	NUM
ejpam-4352	31	10	)	)	PUNCT
ejpam-4352	31	11	(	(	PUNCT
ejpam-4352	31	12	2022	2022	NUM
ejpam-4352	31	13	)	)	PUNCT
ejpam-4352	31	14	,	,	PUNCT
ejpam-4352	31	15	342	342	NUM
ejpam-4352	31	16	-	-	SYM
ejpam-4352	31	17	353	353	NUM
ejpam-4352	31	18	343	343	NUM
ejpam-4352	31	19	bounds	bound	NOUN
ejpam-4352	31	20	of	of	ADP
ejpam-4352	31	21	the	the	DET
ejpam-4352	31	22	hop	hop	NOUN
ejpam-4352	31	23	domination	domination	NOUN
ejpam-4352	31	24	number	number	NOUN
ejpam-4352	31	25	of	of	ADP
ejpam-4352	31	26	some	some	DET
ejpam-4352	31	27	graphs	graph	NOUN
ejpam-4352	31	28	.	.	PUNCT
ejpam-4352	32	1	the	the	DET
ejpam-4352	32	2	concept	concept	NOUN
ejpam-4352	32	3	and	and	CCONJ
ejpam-4352	32	4	some	some	PRON
ejpam-4352	32	5	of	of	ADP
ejpam-4352	32	6	its	its	PRON
ejpam-4352	32	7	variations	variation	NOUN
ejpam-4352	32	8	are	be	AUX
ejpam-4352	32	9	also	also	ADV
ejpam-4352	32	10	studied	study	VERB
ejpam-4352	32	11	in	in	ADP
ejpam-4352	32	12	[	[	X
ejpam-4352	32	13	5	5	NUM
ejpam-4352	32	14	]	]	PUNCT
ejpam-4352	32	15	,	,	PUNCT
ejpam-4352	32	16	[	[	X
ejpam-4352	32	17	6	6	NUM
ejpam-4352	32	18	]	]	PUNCT
ejpam-4352	32	19	,	,	PUNCT
ejpam-4352	32	20	[	[	X
ejpam-4352	32	21	7	7	NUM
ejpam-4352	32	22	]	]	PUNCT
ejpam-4352	32	23	,	,	PUNCT
ejpam-4352	32	24	[	[	X
ejpam-4352	32	25	10	10	NUM
ejpam-4352	32	26	]	]	PUNCT
ejpam-4352	32	27	,	,	PUNCT
ejpam-4352	32	28	[	[	X
ejpam-4352	32	29	9	9	NUM
ejpam-4352	32	30	]	]	PUNCT
ejpam-4352	32	31	,	,	PUNCT
ejpam-4352	32	32	[	[	X
ejpam-4352	32	33	12	12	NUM
ejpam-4352	32	34	]	]	PUNCT
ejpam-4352	32	35	,	,	PUNCT
ejpam-4352	32	36	and	and	CCONJ
ejpam-4352	32	37	[	[	X
ejpam-4352	32	38	14	14	NUM
ejpam-4352	32	39	]	]	PUNCT
ejpam-4352	32	40	.	.	PUNCT
ejpam-4352	33	1	motivated	motivate	VERB
ejpam-4352	33	2	by	by	ADP
ejpam-4352	33	3	these	these	DET
ejpam-4352	33	4	previous	previous	ADJ
ejpam-4352	33	5	studies	study	NOUN
ejpam-4352	33	6	and	and	CCONJ
ejpam-4352	33	7	,	,	PUNCT
ejpam-4352	33	8	in	in	ADP
ejpam-4352	33	9	particular	particular	ADJ
ejpam-4352	33	10	,	,	PUNCT
ejpam-4352	33	11	[	[	X
ejpam-4352	33	12	3	3	NUM
ejpam-4352	33	13	]	]	PUNCT
ejpam-4352	33	14	,	,	PUNCT
ejpam-4352	33	15	[	[	X
ejpam-4352	33	16	8	8	NUM
ejpam-4352	33	17	]	]	PUNCT
ejpam-4352	33	18	,	,	PUNCT
ejpam-4352	33	19	and	and	CCONJ
ejpam-4352	33	20	[	[	X
ejpam-4352	33	21	13	13	NUM
ejpam-4352	33	22	]	]	PUNCT
ejpam-4352	33	23	,	,	PUNCT
ejpam-4352	33	24	we	we	PRON
ejpam-4352	33	25	introduce	introduce	VERB
ejpam-4352	33	26	herein	herein	ADJ
ejpam-4352	33	27	the	the	DET
ejpam-4352	33	28	concept	concept	NOUN
ejpam-4352	33	29	of	of	ADP
ejpam-4352	33	30	super	super	ADJ
ejpam-4352	33	31	hop	hop	NOUN
ejpam-4352	33	32	domination	domination	NOUN
ejpam-4352	33	33	and	and	CCONJ
ejpam-4352	33	34	investigate	investigate	VERB
ejpam-4352	33	35	it	it	PRON
ejpam-4352	33	36	for	for	ADP
ejpam-4352	33	37	some	some	DET
ejpam-4352	33	38	graphs	graph	NOUN
ejpam-4352	33	39	and	and	CCONJ
ejpam-4352	33	40	graphs	graph	NOUN
ejpam-4352	33	41	resulting	result	VERB
ejpam-4352	33	42	from	from	ADP
ejpam-4352	33	43	the	the	DET
ejpam-4352	33	44	join	join	NOUN
ejpam-4352	33	45	,	,	PUNCT
ejpam-4352	33	46	and	and	CCONJ
ejpam-4352	33	47	lexicographic	lexicographic	ADJ
ejpam-4352	33	48	product	product	NOUN
ejpam-4352	33	49	of	of	ADP
ejpam-4352	33	50	two	two	NUM
ejpam-4352	33	51	graphs	graph	NOUN
ejpam-4352	33	52	.	.	PUNCT
ejpam-4352	34	1	2	2	X
ejpam-4352	34	2	.	.	X
ejpam-4352	34	3	terminology	terminology	NOUN
ejpam-4352	34	4	and	and	CCONJ
ejpam-4352	34	5	notation	notation	NOUN
ejpam-4352	34	6	let	let	VERB
ejpam-4352	34	7	g	g	NOUN
ejpam-4352	34	8	=	=	SYM
ejpam-4352	34	9	(	(	PUNCT
ejpam-4352	34	10	v	v	NOUN
ejpam-4352	34	11	(	(	PUNCT
ejpam-4352	34	12	g	g	NOUN
ejpam-4352	34	13	)	)	PUNCT
ejpam-4352	34	14	,	,	PUNCT
ejpam-4352	34	15	e(g	e(g	PROPN
ejpam-4352	34	16	)	)	PUNCT
ejpam-4352	34	17	)	)	PUNCT
ejpam-4352	34	18	be	be	AUX
ejpam-4352	34	19	a	a	DET
ejpam-4352	34	20	connected	connected	ADJ
ejpam-4352	34	21	graph	graph	NOUN
ejpam-4352	34	22	and	and	CCONJ
ejpam-4352	34	23	let	let	VERB
ejpam-4352	34	24	v	v	NUM
ejpam-4352	34	25	∈	∈	PROPN
ejpam-4352	34	26	v	v	NOUN
ejpam-4352	34	27	(	(	PUNCT
ejpam-4352	34	28	g	g	NOUN
ejpam-4352	34	29	)	)	PUNCT
ejpam-4352	34	30	.	.	PUNCT
ejpam-4352	35	1	the	the	DET
ejpam-4352	35	2	open	open	ADJ
ejpam-4352	35	3	neighborhood	neighborhood	NOUN
ejpam-4352	35	4	of	of	ADP
ejpam-4352	35	5	v	v	NOUN
ejpam-4352	35	6	is	be	AUX
ejpam-4352	35	7	the	the	DET
ejpam-4352	35	8	set	set	NOUN
ejpam-4352	35	9	ng(v	ng(v	PUNCT
ejpam-4352	35	10	)	)	PUNCT
ejpam-4352	35	11	=	=	PRON
ejpam-4352	36	1	{	{	PUNCT
ejpam-4352	36	2	z	z	NOUN
ejpam-4352	36	3	∈	∈	PROPN
ejpam-4352	36	4	v	v	NOUN
ejpam-4352	36	5	(	(	PUNCT
ejpam-4352	36	6	g	g	NOUN
ejpam-4352	36	7	)	)	PUNCT
ejpam-4352	36	8	:	:	PUNCT
ejpam-4352	36	9	vz	vz	PROPN
ejpam-4352	36	10	∈	∈	PROPN
ejpam-4352	36	11	e(g	e(g	PROPN
ejpam-4352	36	12	)	)	PUNCT
ejpam-4352	36	13	}	}	PUNCT
ejpam-4352	36	14	and	and	CCONJ
ejpam-4352	36	15	its	its	PRON
ejpam-4352	36	16	closed	closed	ADJ
ejpam-4352	36	17	neighborhood	neighborhood	NOUN
ejpam-4352	36	18	is	be	AUX
ejpam-4352	36	19	ng[v	ng[v	ADJ
ejpam-4352	36	20	]	]	X
ejpam-4352	36	21	=	=	SYM
ejpam-4352	36	22	ng(v	ng(v	X
ejpam-4352	36	23	)	)	PUNCT
ejpam-4352	36	24	∪	∪	ADP
ejpam-4352	36	25	{	{	PUNCT
ejpam-4352	36	26	v	v	NOUN
ejpam-4352	36	27	}	}	PUNCT
ejpam-4352	36	28	.	.	PUNCT
ejpam-4352	37	1	the	the	DET
ejpam-4352	37	2	open	open	ADJ
ejpam-4352	37	3	hop	hop	NOUN
ejpam-4352	37	4	neighborhood	neighborhood	NOUN
ejpam-4352	37	5	of	of	ADP
ejpam-4352	37	6	v	v	NOUN
ejpam-4352	37	7	is	be	AUX
ejpam-4352	37	8	the	the	DET
ejpam-4352	37	9	set	set	ADJ
ejpam-4352	37	10	n2	n2	ADJ
ejpam-4352	37	11	g(v	g(v	PROPN
ejpam-4352	37	12	)	)	PUNCT
ejpam-4352	37	13	=	=	PRON
ejpam-4352	37	14	{	{	PUNCT
ejpam-4352	37	15	u	u	NOUN
ejpam-4352	37	16	∈	∈	PROPN
ejpam-4352	37	17	v	v	NOUN
ejpam-4352	37	18	(	(	PUNCT
ejpam-4352	37	19	g	g	NOUN
ejpam-4352	37	20	)	)	PUNCT
ejpam-4352	37	21	:	:	PUNCT
ejpam-4352	37	22	dg(u	dg(u	X
ejpam-4352	37	23	,	,	PUNCT
ejpam-4352	37	24	v	v	NOUN
ejpam-4352	37	25	)	)	PUNCT
ejpam-4352	37	26	=	=	SYM
ejpam-4352	37	27	2	2	X
ejpam-4352	37	28	}	}	PUNCT
ejpam-4352	37	29	,	,	PUNCT
ejpam-4352	37	30	where	where	SCONJ
ejpam-4352	37	31	dg(u	dg(u	X
ejpam-4352	37	32	,	,	PUNCT
ejpam-4352	37	33	v	v	NOUN
ejpam-4352	37	34	)	)	PUNCT
ejpam-4352	37	35	denotes	denote	VERB
ejpam-4352	37	36	the	the	DET
ejpam-4352	37	37	distance	distance	NOUN
ejpam-4352	37	38	between	between	ADP
ejpam-4352	37	39	vertices	vertex	NOUN
ejpam-4352	37	40	u	u	NOUN
ejpam-4352	37	41	and	and	CCONJ
ejpam-4352	37	42	v	v	NOUN
ejpam-4352	37	43	in	in	ADP
ejpam-4352	37	44	g.	g.	PROPN
ejpam-4352	37	45	the	the	DET
ejpam-4352	37	46	closed	close	VERB
ejpam-4352	37	47	hop	hop	NOUN
ejpam-4352	37	48	neighborhood	neighborhood	NOUN
ejpam-4352	37	49	of	of	ADP
ejpam-4352	37	50	v	v	NOUN
ejpam-4352	37	51	is	be	AUX
ejpam-4352	37	52	n2	n2	ADJ
ejpam-4352	37	53	g[v	g[v	NOUN
ejpam-4352	37	54	]	]	X
ejpam-4352	37	55	=	=	SYM
ejpam-4352	37	56	n2	n2	ADJ
ejpam-4352	37	57	g(v	g(v	PROPN
ejpam-4352	37	58	)	)	PUNCT
ejpam-4352	37	59	∪	∪	ADP
ejpam-4352	37	60	{	{	PUNCT
ejpam-4352	37	61	v	v	NOUN
ejpam-4352	37	62	}	}	PUNCT
ejpam-4352	37	63	.	.	PUNCT
ejpam-4352	38	1	the	the	DET
ejpam-4352	38	2	open	open	ADJ
ejpam-4352	38	3	hop	hop	NOUN
ejpam-4352	38	4	neighborhood	neighborhood	NOUN
ejpam-4352	38	5	of	of	ADP
ejpam-4352	38	6	a	a	DET
ejpam-4352	38	7	⊆	⊆	NUM
ejpam-4352	38	8	v	v	NOUN
ejpam-4352	38	9	(	(	PUNCT
ejpam-4352	38	10	g	g	NOUN
ejpam-4352	38	11	)	)	PUNCT
ejpam-4352	38	12	is	be	AUX
ejpam-4352	38	13	the	the	DET
ejpam-4352	38	14	set	set	ADJ
ejpam-4352	38	15	n2	n2	NOUN
ejpam-4352	38	16	g(a	g(a	PROPN
ejpam-4352	38	17	)	)	PUNCT
ejpam-4352	39	1	=	=	PUNCT
ejpam-4352	39	2	∪v∈an	∪v∈an	NOUN
ejpam-4352	39	3	2	2	NUM
ejpam-4352	39	4	g(v	g(v	PROPN
ejpam-4352	39	5	)	)	PUNCT
ejpam-4352	39	6	and	and	CCONJ
ejpam-4352	39	7	its	its	PRON
ejpam-4352	39	8	closed	closed	ADJ
ejpam-4352	39	9	hop	hop	NOUN
ejpam-4352	39	10	neighborhood	neighborhood	NOUN
ejpam-4352	39	11	is	be	AUX
ejpam-4352	39	12	n2	n2	ADJ
ejpam-4352	39	13	g[a	g[a	NOUN
ejpam-4352	39	14	]	]	X
ejpam-4352	39	15	=	=	PUNCT
ejpam-4352	39	16	a	a	DET
ejpam-4352	39	17	∪n2	∪n2	NOUN
ejpam-4352	39	18	g(a	g(a	PROPN
ejpam-4352	39	19	)	)	PUNCT
ejpam-4352	39	20	.	.	PUNCT
ejpam-4352	40	1	a	a	DET
ejpam-4352	40	2	set	set	NOUN
ejpam-4352	40	3	s	s	NOUN
ejpam-4352	40	4	⊆	⊆	NUM
ejpam-4352	40	5	v	v	NOUN
ejpam-4352	40	6	(	(	PUNCT
ejpam-4352	40	7	g	g	NOUN
ejpam-4352	40	8	)	)	PUNCT
ejpam-4352	40	9	is	be	AUX
ejpam-4352	40	10	a	a	DET
ejpam-4352	40	11	dominating	dominating	NOUN
ejpam-4352	40	12	set	set	NOUN
ejpam-4352	40	13	(	(	PUNCT
ejpam-4352	40	14	hop	hop	NOUN
ejpam-4352	40	15	dominating	dominating	NOUN
ejpam-4352	40	16	set	set	NOUN
ejpam-4352	40	17	)	)	PUNCT
ejpam-4352	40	18	of	of	ADP
ejpam-4352	40	19	a	a	DET
ejpam-4352	40	20	graph	graph	NOUN
ejpam-4352	40	21	g	g	NOUN
ejpam-4352	40	22	if	if	SCONJ
ejpam-4352	40	23	for	for	ADP
ejpam-4352	40	24	each	each	PRON
ejpam-4352	40	25	v	v	NUM
ejpam-4352	40	26	∈	∈	PROPN
ejpam-4352	40	27	v	v	NOUN
ejpam-4352	40	28	(	(	PUNCT
ejpam-4352	40	29	g	g	NOUN
ejpam-4352	40	30	)	)	PUNCT
ejpam-4352	40	31	\	\	PROPN
ejpam-4352	41	1	s	s	X
ejpam-4352	41	2	,	,	PUNCT
ejpam-4352	41	3	there	there	PRON
ejpam-4352	41	4	exists	exist	VERB
ejpam-4352	41	5	w	w	PROPN
ejpam-4352	41	6	∈	∈	PROPN
ejpam-4352	41	7	s	s	VERB
ejpam-4352	41	8	such	such	ADJ
ejpam-4352	41	9	that	that	DET
ejpam-4352	41	10	dg(v	dg(v	ADJ
ejpam-4352	41	11	,	,	PUNCT
ejpam-4352	41	12	w	w	NOUN
ejpam-4352	41	13	)	)	PUNCT
ejpam-4352	41	14	=	=	SYM
ejpam-4352	41	15	1	1	NUM
ejpam-4352	41	16	(	(	PUNCT
ejpam-4352	41	17	resp	resp	NOUN
ejpam-4352	41	18	.	.	PUNCT
ejpam-4352	42	1	dg(v	dg(v	PUNCT
ejpam-4352	42	2	,	,	PUNCT
ejpam-4352	42	3	w	w	NOUN
ejpam-4352	42	4	)	)	PUNCT
ejpam-4352	42	5	=	=	SYM
ejpam-4352	42	6	2	2	NUM
ejpam-4352	42	7	)	)	PUNCT
ejpam-4352	42	8	.	.	PUNCT
ejpam-4352	43	1	the	the	DET
ejpam-4352	43	2	smallest	small	ADJ
ejpam-4352	43	3	cardinality	cardinality	NOUN
ejpam-4352	43	4	of	of	ADP
ejpam-4352	43	5	a	a	DET
ejpam-4352	43	6	dominating	dominating	NOUN
ejpam-4352	43	7	(	(	PUNCT
ejpam-4352	43	8	resp	resp	NOUN
ejpam-4352	43	9	.	.	PUNCT
ejpam-4352	44	1	hop	hop	PROPN
ejpam-4352	44	2	dominating	dominating	NOUN
ejpam-4352	44	3	)	)	PUNCT
ejpam-4352	45	1	set	set	NOUN
ejpam-4352	45	2	of	of	ADP
ejpam-4352	45	3	g	g	NOUN
ejpam-4352	45	4	,	,	PUNCT
ejpam-4352	45	5	denoted	denote	VERB
ejpam-4352	45	6	by	by	ADP
ejpam-4352	45	7	γ(g	γ(g	PROPN
ejpam-4352	45	8	)	)	PUNCT
ejpam-4352	45	9	(	(	PUNCT
ejpam-4352	45	10	resp	resp	NOUN
ejpam-4352	45	11	.	.	PUNCT
ejpam-4352	45	12	γh(g	γh(g	NOUN
ejpam-4352	45	13	)	)	PUNCT
ejpam-4352	45	14	)	)	PUNCT
ejpam-4352	45	15	,	,	PUNCT
ejpam-4352	45	16	is	be	AUX
ejpam-4352	45	17	called	call	VERB
ejpam-4352	45	18	the	the	DET
ejpam-4352	45	19	domination	domination	NOUN
ejpam-4352	45	20	number	number	NOUN
ejpam-4352	45	21	(	(	PUNCT
ejpam-4352	45	22	resp	resp	NOUN
ejpam-4352	45	23	.	.	PUNCT
ejpam-4352	46	1	hop	hop	PROPN
ejpam-4352	46	2	domination	domination	NOUN
ejpam-4352	46	3	number	number	NOUN
ejpam-4352	46	4	)	)	PUNCT
ejpam-4352	46	5	of	of	ADP
ejpam-4352	46	6	g.	g.	PROPN
ejpam-4352	46	7	let	let	VERB
ejpam-4352	46	8	s	s	PRON
ejpam-4352	46	9	be	be	AUX
ejpam-4352	46	10	a	a	DET
ejpam-4352	46	11	subset	subset	NOUN
ejpam-4352	46	12	of	of	ADP
ejpam-4352	46	13	v	v	NOUN
ejpam-4352	46	14	(	(	PUNCT
ejpam-4352	46	15	g	g	NOUN
ejpam-4352	46	16	)	)	PUNCT
ejpam-4352	46	17	and	and	CCONJ
ejpam-4352	46	18	let	let	VERB
ejpam-4352	46	19	v	v	X
ejpam-4352	46	20	∈	∈	VERB
ejpam-4352	46	21	s.	s.	PROPN
ejpam-4352	46	22	a	a	DET
ejpam-4352	46	23	vertex	vertex	NOUN
ejpam-4352	46	24	w	w	PROPN
ejpam-4352	46	25	∈	∈	PROPN
ejpam-4352	46	26	v	v	ADP
ejpam-4352	46	27	(	(	PUNCT
ejpam-4352	46	28	g	g	NOUN
ejpam-4352	46	29	)	)	PUNCT
ejpam-4352	46	30	\	\	PROPN
ejpam-4352	47	1	s	s	PART
ejpam-4352	47	2	is	be	AUX
ejpam-4352	47	3	an	an	DET
ejpam-4352	47	4	external	external	ADJ
ejpam-4352	47	5	private	private	ADJ
ejpam-4352	47	6	neighbor	neighbor	NOUN
ejpam-4352	47	7	(	(	PUNCT
ejpam-4352	47	8	external	external	ADJ
ejpam-4352	47	9	hop	hop	NOUN
ejpam-4352	47	10	private	private	ADJ
ejpam-4352	47	11	neighbor	neighbor	NOUN
ejpam-4352	47	12	)	)	PUNCT
ejpam-4352	47	13	of	of	ADP
ejpam-4352	47	14	v	v	NOUN
ejpam-4352	47	15	with	with	ADP
ejpam-4352	47	16	respect	respect	NOUN
ejpam-4352	47	17	to	to	ADP
ejpam-4352	47	18	s	s	PRON
ejpam-4352	47	19	if	if	SCONJ
ejpam-4352	47	20	ng(w)∩s	ng(w)∩s	PROPN
ejpam-4352	47	21	=	=	PUNCT
ejpam-4352	47	22	{	{	PUNCT
ejpam-4352	47	23	v	v	NOUN
ejpam-4352	47	24	}	}	PUNCT
ejpam-4352	47	25	(	(	PUNCT
ejpam-4352	47	26	resp	resp	NOUN
ejpam-4352	47	27	.	.	PUNCT
ejpam-4352	48	1	n2	n2	PROPN
ejpam-4352	48	2	g(w)∩s	g(w)∩s	PROPN
ejpam-4352	49	1	=	=	PUNCT
ejpam-4352	49	2	{	{	PUNCT
ejpam-4352	49	3	v	v	NOUN
ejpam-4352	49	4	}	}	PUNCT
ejpam-4352	49	5	)	)	PUNCT
ejpam-4352	49	6	.	.	PUNCT
ejpam-4352	50	1	the	the	DET
ejpam-4352	50	2	set	set	NOUN
ejpam-4352	50	3	containing	contain	VERB
ejpam-4352	50	4	all	all	DET
ejpam-4352	50	5	the	the	DET
ejpam-4352	50	6	external	external	ADJ
ejpam-4352	50	7	private	private	ADJ
ejpam-4352	50	8	neighbors	neighbor	NOUN
ejpam-4352	50	9	(	(	PUNCT
ejpam-4352	50	10	resp	resp	NOUN
ejpam-4352	50	11	.	.	PUNCT
ejpam-4352	51	1	hop	hop	PROPN
ejpam-4352	51	2	private	private	ADJ
ejpam-4352	51	3	)	)	PUNCT
ejpam-4352	51	4	neighbors	neighbor	NOUN
ejpam-4352	51	5	of	of	ADP
ejpam-4352	51	6	v	v	NOUN
ejpam-4352	51	7	with	with	ADP
ejpam-4352	51	8	respect	respect	NOUN
ejpam-4352	51	9	to	to	ADP
ejpam-4352	51	10	s	s	PRON
ejpam-4352	51	11	is	be	AUX
ejpam-4352	51	12	denoted	denote	VERB
ejpam-4352	51	13	by	by	ADP
ejpam-4352	51	14	epng(v	epng(v	PROPN
ejpam-4352	51	15	,	,	PUNCT
ejpam-4352	51	16	s	s	PART
ejpam-4352	51	17	)	)	PUNCT
ejpam-4352	51	18	(	(	PUNCT
ejpam-4352	51	19	resp	resp	NOUN
ejpam-4352	51	20	.	.	PUNCT
ejpam-4352	52	1	ehpng(v	ehpng(v	ADP
ejpam-4352	52	2	,	,	PUNCT
ejpam-4352	52	3	s	s	NOUN
ejpam-4352	52	4	)	)	PUNCT
ejpam-4352	52	5	)	)	PUNCT
ejpam-4352	52	6	.	.	PUNCT
ejpam-4352	53	1	a	a	DET
ejpam-4352	53	2	set	set	NOUN
ejpam-4352	53	3	s	s	NOUN
ejpam-4352	53	4	⊆	⊆	NUM
ejpam-4352	53	5	v	v	NOUN
ejpam-4352	53	6	(	(	PUNCT
ejpam-4352	53	7	g	g	NOUN
ejpam-4352	53	8	)	)	PUNCT
ejpam-4352	53	9	is	be	AUX
ejpam-4352	53	10	called	call	VERB
ejpam-4352	53	11	a	a	DET
ejpam-4352	53	12	super	super	ADJ
ejpam-4352	53	13	dominating	dominating	NOUN
ejpam-4352	53	14	set	set	NOUN
ejpam-4352	53	15	(	(	PUNCT
ejpam-4352	53	16	resp	resp	NOUN
ejpam-4352	53	17	.	.	PUNCT
ejpam-4352	54	1	super	super	ADJ
ejpam-4352	54	2	hop	hop	PROPN
ejpam-4352	54	3	dominating	dominating	NOUN
ejpam-4352	54	4	set	set	NOUN
ejpam-4352	54	5	)	)	PUNCT
ejpam-4352	54	6	if	if	SCONJ
ejpam-4352	54	7	epng(v	epng(v	PROPN
ejpam-4352	54	8	,	,	PUNCT
ejpam-4352	54	9	v	v	NOUN
ejpam-4352	54	10	(	(	PUNCT
ejpam-4352	54	11	g)\s	g)\s	NOUN
ejpam-4352	54	12	)	)	PUNCT
ejpam-4352	54	13	̸=	̸=	PROPN
ejpam-4352	54	14	∅	∅	NOUN
ejpam-4352	54	15	(	(	PUNCT
ejpam-4352	54	16	resp	resp	NOUN
ejpam-4352	54	17	.	.	PUNCT
ejpam-4352	55	1	ehpng(v	ehpng(v	ADP
ejpam-4352	55	2	,	,	PUNCT
ejpam-4352	55	3	v	v	INTJ
ejpam-4352	55	4	(	(	PUNCT
ejpam-4352	55	5	g)\s	g)\s	NOUN
ejpam-4352	55	6	)	)	PUNCT
ejpam-4352	55	7	̸=	̸=	PROPN
ejpam-4352	55	8	∅	∅	NOUN
ejpam-4352	55	9	)	)	PUNCT
ejpam-4352	55	10	for	for	ADP
ejpam-4352	55	11	each	each	DET
ejpam-4352	55	12	v	v	NUM
ejpam-4352	55	13	∈	∈	NOUN
ejpam-4352	55	14	v	v	NOUN
ejpam-4352	55	15	(	(	PUNCT
ejpam-4352	55	16	g)\s	g)\s	NOUN
ejpam-4352	55	17	.	.	PUNCT
ejpam-4352	56	1	the	the	DET
ejpam-4352	56	2	smallest	small	ADJ
ejpam-4352	56	3	cardinality	cardinality	NOUN
ejpam-4352	56	4	of	of	ADP
ejpam-4352	56	5	a	a	DET
ejpam-4352	56	6	super	super	ADJ
ejpam-4352	56	7	dominating	dominating	NOUN
ejpam-4352	56	8	(	(	PUNCT
ejpam-4352	56	9	resp	resp	NOUN
ejpam-4352	56	10	.	.	PUNCT
ejpam-4352	57	1	super	super	ADJ
ejpam-4352	57	2	hop	hop	NOUN
ejpam-4352	57	3	dominating	dominating	NOUN
ejpam-4352	57	4	)	)	PUNCT
ejpam-4352	58	1	set	set	NOUN
ejpam-4352	58	2	of	of	ADP
ejpam-4352	58	3	g	g	NOUN
ejpam-4352	58	4	,	,	PUNCT
ejpam-4352	58	5	denoted	denote	VERB
ejpam-4352	58	6	by	by	ADP
ejpam-4352	58	7	γs(g	γs(g	NOUN
ejpam-4352	58	8	)	)	PUNCT
ejpam-4352	58	9	(	(	PUNCT
ejpam-4352	58	10	resp	resp	NOUN
ejpam-4352	58	11	.	.	PUNCT
ejpam-4352	58	12	γsh(g	γsh(g	NOUN
ejpam-4352	58	13	)	)	PUNCT
ejpam-4352	58	14	)	)	PUNCT
ejpam-4352	59	1	,	,	PUNCT
ejpam-4352	59	2	is	be	AUX
ejpam-4352	59	3	called	call	VERB
ejpam-4352	59	4	the	the	DET
ejpam-4352	59	5	super	super	ADJ
ejpam-4352	59	6	domination	domination	NOUN
ejpam-4352	59	7	number	number	NOUN
ejpam-4352	59	8	(	(	PUNCT
ejpam-4352	59	9	resp	resp	NOUN
ejpam-4352	59	10	.	.	PUNCT
ejpam-4352	60	1	super	super	ADJ
ejpam-4352	60	2	hop	hop	PROPN
ejpam-4352	60	3	domination	domination	NOUN
ejpam-4352	60	4	number	number	NOUN
ejpam-4352	60	5	)	)	PUNCT
ejpam-4352	60	6	of	of	ADP
ejpam-4352	60	7	g.	g.	PROPN
ejpam-4352	60	8	any	any	DET
ejpam-4352	60	9	super	super	ADJ
ejpam-4352	60	10	dominating	dominating	NOUN
ejpam-4352	60	11	(	(	PUNCT
ejpam-4352	60	12	resp	resp	NOUN
ejpam-4352	60	13	.	.	PUNCT
ejpam-4352	61	1	super	super	ADJ
ejpam-4352	61	2	hop	hop	NOUN
ejpam-4352	61	3	dominating	dominating	NOUN
ejpam-4352	61	4	)	)	PUNCT
ejpam-4352	62	1	set	set	NOUN
ejpam-4352	62	2	of	of	ADP
ejpam-4352	62	3	g	g	NOUN
ejpam-4352	62	4	with	with	ADP
ejpam-4352	62	5	cardinality	cardinality	NOUN
ejpam-4352	62	6	γs(g	γs(g	PUNCT
ejpam-4352	62	7	)	)	PUNCT
ejpam-4352	62	8	(	(	PUNCT
ejpam-4352	62	9	resp	resp	NOUN
ejpam-4352	62	10	.	.	PUNCT
ejpam-4352	62	11	γsh(g	γsh(g	NOUN
ejpam-4352	62	12	)	)	PUNCT
ejpam-4352	62	13	)	)	PUNCT
ejpam-4352	62	14	is	be	AUX
ejpam-4352	62	15	called	call	VERB
ejpam-4352	62	16	a	a	DET
ejpam-4352	62	17	γs	γs	NOUN
ejpam-4352	62	18	-	-	PUNCT
ejpam-4352	62	19	set	set	VERB
ejpam-4352	62	20	(	(	PUNCT
ejpam-4352	62	21	resp	resp	NOUN
ejpam-4352	62	22	.	.	PUNCT
ejpam-4352	63	1	γ	γ	PROPN
ejpam-4352	63	2	s	s	VERB
ejpam-4352	63	3	h	h	NOUN
ejpam-4352	63	4	-	-	PUNCT
ejpam-4352	63	5	set	set	NOUN
ejpam-4352	63	6	)	)	PUNCT
ejpam-4352	63	7	of	of	ADP
ejpam-4352	63	8	g.	g.	PROPN
ejpam-4352	63	9	a	a	DET
ejpam-4352	63	10	subset	subset	NOUN
ejpam-4352	63	11	d	d	NOUN
ejpam-4352	63	12	of	of	ADP
ejpam-4352	63	13	v	v	NOUN
ejpam-4352	63	14	(	(	PUNCT
ejpam-4352	63	15	g	g	NOUN
ejpam-4352	63	16	)	)	PUNCT
ejpam-4352	63	17	is	be	AUX
ejpam-4352	63	18	a	a	DET
ejpam-4352	63	19	complement	complement	NOUN
ejpam-4352	63	20	-	-	PUNCT
ejpam-4352	63	21	super	super	ADJ
ejpam-4352	63	22	dominating	dominating	NOUN
ejpam-4352	63	23	set	set	NOUN
ejpam-4352	63	24	of	of	ADP
ejpam-4352	63	25	g	g	PROPN
ejpam-4352	63	26	(	(	PUNCT
ejpam-4352	63	27	super	super	ADJ
ejpam-4352	63	28	dominating	dominating	NOUN
ejpam-4352	63	29	set	set	NOUN
ejpam-4352	63	30	of	of	ADP
ejpam-4352	63	31	g	g	NOUN
ejpam-4352	63	32	)	)	PUNCT
ejpam-4352	63	33	if	if	SCONJ
ejpam-4352	63	34	for	for	ADP
ejpam-4352	63	35	each	each	PRON
ejpam-4352	63	36	v	v	NUM
ejpam-4352	63	37	∈	∈	PROPN
ejpam-4352	63	38	v	v	NOUN
ejpam-4352	63	39	(	(	PUNCT
ejpam-4352	63	40	g	g	NOUN
ejpam-4352	63	41	)	)	PUNCT
ejpam-4352	63	42	\	\	PUNCT
ejpam-4352	64	1	d	d	X
ejpam-4352	64	2	,	,	PUNCT
ejpam-4352	64	3	there	there	PRON
ejpam-4352	64	4	exists	exist	VERB
ejpam-4352	64	5	w	w	PROPN
ejpam-4352	64	6	∈	∈	PROPN
ejpam-4352	64	7	d	d	SYM
ejpam-4352	64	8	\	\	NOUN
ejpam-4352	64	9	ng(v	ng(v	PUNCT
ejpam-4352	64	10	)	)	PUNCT
ejpam-4352	64	11	=	=	SYM
ejpam-4352	64	12	d	d	NOUN
ejpam-4352	64	13	∩	∩	NOUN
ejpam-4352	64	14	ng(v	ng(v	NOUN
ejpam-4352	64	15	)	)	PUNCT
ejpam-4352	64	16	such	such	ADJ
ejpam-4352	64	17	that	that	SCONJ
ejpam-4352	64	18	[	[	X
ejpam-4352	64	19	v	v	X
ejpam-4352	64	20	(	(	PUNCT
ejpam-4352	64	21	g	g	NOUN
ejpam-4352	64	22	)	)	PUNCT
ejpam-4352	64	23	\	\	NOUN
ejpam-4352	64	24	ng(w	ng(w	NOUN
ejpam-4352	64	25	)	)	PUNCT
ejpam-4352	64	26	]	]	PUNCT
ejpam-4352	65	1	∩	∩	NOUN
ejpam-4352	66	1	[	[	X
ejpam-4352	66	2	v	v	X
ejpam-4352	66	3	(	(	PUNCT
ejpam-4352	66	4	g	g	NOUN
ejpam-4352	66	5	)	)	PUNCT
ejpam-4352	66	6	\	\	PUNCT
ejpam-4352	67	1	d	d	X
ejpam-4352	67	2	]	]	X
ejpam-4352	67	3	=	=	SYM
ejpam-4352	67	4	ng(w	ng(w	NOUN
ejpam-4352	67	5	)	)	PUNCT
ejpam-4352	67	6	∩	∩	NOUN
ejpam-4352	68	1	[	[	X
ejpam-4352	68	2	v	v	X
ejpam-4352	68	3	(	(	PUNCT
ejpam-4352	68	4	g	g	NOUN
ejpam-4352	68	5	)	)	PUNCT
ejpam-4352	68	6	\	\	PUNCT
ejpam-4352	69	1	d	d	X
ejpam-4352	69	2	]	]	X
ejpam-4352	69	3	=	=	PUNCT
ejpam-4352	69	4	{	{	PUNCT
ejpam-4352	69	5	v	v	NOUN
ejpam-4352	69	6	}	}	PUNCT
ejpam-4352	69	7	.	.	PUNCT
ejpam-4352	70	1	the	the	DET
ejpam-4352	70	2	smallest	small	ADJ
ejpam-4352	70	3	cardinality	cardinality	NOUN
ejpam-4352	70	4	of	of	ADP
ejpam-4352	70	5	a	a	DET
ejpam-4352	70	6	complement	complement	NOUN
ejpam-4352	70	7	-	-	PUNCT
ejpam-4352	70	8	super	super	ADJ
ejpam-4352	70	9	dominating	dominating	NOUN
ejpam-4352	70	10	set	set	NOUN
ejpam-4352	70	11	of	of	ADP
ejpam-4352	70	12	g	g	NOUN
ejpam-4352	70	13	,	,	PUNCT
ejpam-4352	70	14	denoted	denote	VERB
ejpam-4352	70	15	by	by	ADP
ejpam-4352	70	16	γcs(g	γcs(g	PROPN
ejpam-4352	70	17	)	)	PUNCT
ejpam-4352	70	18	=	=	SYM
ejpam-4352	70	19	γs(g	γs(g	NUM
ejpam-4352	70	20	)	)	PUNCT
ejpam-4352	70	21	,	,	PUNCT
ejpam-4352	70	22	is	be	AUX
ejpam-4352	70	23	called	call	VERB
ejpam-4352	70	24	the	the	DET
ejpam-4352	70	25	complement	complement	NOUN
ejpam-4352	70	26	-	-	PUNCT
ejpam-4352	70	27	super	super	ADJ
ejpam-4352	70	28	domination	domination	NOUN
ejpam-4352	70	29	number	number	NOUN
ejpam-4352	70	30	of	of	ADP
ejpam-4352	70	31	g	g	PROPN
ejpam-4352	70	32	(	(	PUNCT
ejpam-4352	70	33	super	super	ADJ
ejpam-4352	70	34	domination	domination	NOUN
ejpam-4352	70	35	number	number	NOUN
ejpam-4352	70	36	of	of	ADP
ejpam-4352	70	37	g	g	NOUN
ejpam-4352	70	38	)	)	PUNCT
ejpam-4352	70	39	.	.	PUNCT
ejpam-4352	71	1	any	any	DET
ejpam-4352	71	2	complement	complement	NOUN
ejpam-4352	71	3	-	-	PUNCT
ejpam-4352	71	4	super	super	ADJ
ejpam-4352	71	5	dominating	dominating	NOUN
ejpam-4352	71	6	set	set	NOUN
ejpam-4352	71	7	of	of	ADP
ejpam-4352	71	8	g	g	PROPN
ejpam-4352	71	9	with	with	ADP
ejpam-4352	71	10	cardinality	cardinality	NOUN
ejpam-4352	71	11	equal	equal	ADJ
ejpam-4352	71	12	to	to	ADP
ejpam-4352	71	13	γcs(g	γcs(g	NUM
ejpam-4352	71	14	)	)	PUNCT
ejpam-4352	71	15	is	be	AUX
ejpam-4352	71	16	called	call	VERB
ejpam-4352	71	17	a	a	DET
ejpam-4352	71	18	γcs	γcs	NOUN
ejpam-4352	71	19	-	-	PUNCT
ejpam-4352	71	20	set	set	NOUN
ejpam-4352	71	21	.	.	PUNCT
ejpam-4352	72	1	3	3	X
ejpam-4352	72	2	.	.	X
ejpam-4352	72	3	results	result	NOUN
ejpam-4352	72	4	remark	remark	VERB
ejpam-4352	72	5	1	1	NUM
ejpam-4352	72	6	.	.	PUNCT
ejpam-4352	73	1	γsh(kn	γsh(kn	VERB
ejpam-4352	73	2	)	)	PUNCT
ejpam-4352	74	1	=	=	SYM
ejpam-4352	74	2	n	n	CCONJ
ejpam-4352	74	3	for	for	ADP
ejpam-4352	74	4	all	all	DET
ejpam-4352	74	5	n	n	PRON
ejpam-4352	74	6	≥	≥	NUM
ejpam-4352	74	7	1	1	NUM
ejpam-4352	74	8	.	.	PUNCT
ejpam-4352	74	9	proposition	proposition	NOUN
ejpam-4352	74	10	1	1	NUM
ejpam-4352	74	11	.	.	PUNCT
ejpam-4352	75	1	let	let	VERB
ejpam-4352	75	2	g	g	PRON
ejpam-4352	75	3	be	be	AUX
ejpam-4352	75	4	a	a	DET
ejpam-4352	75	5	graph	graph	NOUN
ejpam-4352	75	6	of	of	ADP
ejpam-4352	75	7	order	order	NOUN
ejpam-4352	75	8	n	n	PRON
ejpam-4352	75	9	≥	≥	NOUN
ejpam-4352	75	10	1	1	NUM
ejpam-4352	75	11	.	.	PUNCT
ejpam-4352	76	1	then	then	ADV
ejpam-4352	76	2	max{γh(g	max{γh(g	NOUN
ejpam-4352	76	3	)	)	PUNCT
ejpam-4352	76	4	,	,	PUNCT
ejpam-4352	76	5	⌈n2	⌈n2	NOUN
ejpam-4352	76	6	⌉	⌉	NOUN
ejpam-4352	76	7	}	}	PUNCT
ejpam-4352	76	8	≤	≤	ADJ
ejpam-4352	76	9	γsh(g	γsh(g	NOUN
ejpam-4352	76	10	)	)	PUNCT
ejpam-4352	76	11	≤	≤	NUM
ejpam-4352	76	12	n.	n.	NOUN
ejpam-4352	76	13	proof	proof	NOUN
ejpam-4352	76	14	.	.	PUNCT
ejpam-4352	77	1	since	since	SCONJ
ejpam-4352	77	2	every	every	DET
ejpam-4352	77	3	super	super	ADJ
ejpam-4352	77	4	hop	hop	NOUN
ejpam-4352	77	5	dominating	dominating	NOUN
ejpam-4352	77	6	set	set	NOUN
ejpam-4352	77	7	is	be	AUX
ejpam-4352	77	8	hop	hop	NOUN
ejpam-4352	77	9	dominating	dominating	NOUN
ejpam-4352	77	10	,	,	PUNCT
ejpam-4352	77	11	it	it	PRON
ejpam-4352	77	12	follows	follow	VERB
ejpam-4352	77	13	that	that	PRON
ejpam-4352	77	14	γh(g	γh(g	ADP
ejpam-4352	77	15	)	)	PUNCT
ejpam-4352	77	16	≤	≤	NUM
ejpam-4352	77	17	γsh(g	γsh(g	NOUN
ejpam-4352	77	18	)	)	PUNCT
ejpam-4352	77	19	.	.	PUNCT
ejpam-4352	78	1	now	now	ADV
ejpam-4352	78	2	let	let	VERB
ejpam-4352	78	3	s	s	PRON
ejpam-4352	78	4	be	be	AUX
ejpam-4352	78	5	a	a	DET
ejpam-4352	78	6	γsh	γsh	NOUN
ejpam-4352	78	7	-	-	PUNCT
ejpam-4352	78	8	set	set	NOUN
ejpam-4352	78	9	of	of	ADP
ejpam-4352	78	10	g.	g.	PROPN
ejpam-4352	78	11	then	then	ADV
ejpam-4352	78	12	,	,	PUNCT
ejpam-4352	78	13	by	by	ADP
ejpam-4352	78	14	definition	definition	NOUN
ejpam-4352	78	15	of	of	ADP
ejpam-4352	78	16	super	super	ADJ
ejpam-4352	78	17	hop	hop	NOUN
ejpam-4352	78	18	dominating	dominating	NOUN
ejpam-4352	78	19	set	set	NOUN
ejpam-4352	78	20	,	,	PUNCT
ejpam-4352	78	21	s.	s.	PROPN
ejpam-4352	78	22	canoy	canoy	PROPN
ejpam-4352	78	23	,	,	PUNCT
ejpam-4352	78	24	jr	jr	PROPN
ejpam-4352	78	25	.	.	PROPN
ejpam-4352	78	26	,	,	PUNCT
ejpam-4352	78	27	g.	g.	PROPN
ejpam-4352	78	28	salasalan	salasalan	PROPN
ejpam-4352	78	29	/	/	SYM
ejpam-4352	78	30	eur	eur	PROPN
ejpam-4352	78	31	.	.	PUNCT
ejpam-4352	79	1	j.	j.	PROPN
ejpam-4352	79	2	pure	pure	PROPN
ejpam-4352	79	3	appl	appl	PROPN
ejpam-4352	79	4	.	.	PROPN
ejpam-4352	79	5	math	math	PROPN
ejpam-4352	79	6	,	,	PUNCT
ejpam-4352	79	7	15	15	NUM
ejpam-4352	79	8	(	(	PUNCT
ejpam-4352	79	9	2	2	NUM
ejpam-4352	79	10	)	)	PUNCT
ejpam-4352	79	11	(	(	PUNCT
ejpam-4352	79	12	2022	2022	NUM
ejpam-4352	79	13	)	)	PUNCT
ejpam-4352	79	14	,	,	PUNCT
ejpam-4352	79	15	342	342	NUM
ejpam-4352	79	16	-	-	SYM
ejpam-4352	79	17	353	353	NUM
ejpam-4352	79	18	344	344	NUM
ejpam-4352	79	19	|s|	|s|	PROPN
ejpam-4352	79	20	≥	≥	PROPN
ejpam-4352	79	21	|v	|v	X
ejpam-4352	79	22	(	(	PUNCT
ejpam-4352	79	23	g	g	NOUN
ejpam-4352	79	24	)	)	PUNCT
ejpam-4352	79	25	\	\	NOUN
ejpam-4352	80	1	s|	s|	PROPN
ejpam-4352	80	2	.	.	PUNCT
ejpam-4352	81	1	this	this	PRON
ejpam-4352	81	2	implies	imply	VERB
ejpam-4352	81	3	that	that	SCONJ
ejpam-4352	81	4	γsh(g	γsh(g	NOUN
ejpam-4352	81	5	)	)	PUNCT
ejpam-4352	81	6	=	=	SYM
ejpam-4352	81	7	|s|	|s|	PROPN
ejpam-4352	81	8	≥	≥	NOUN
ejpam-4352	81	9	⌈n2	⌈n2	NOUN
ejpam-4352	81	10	⌉.	⌉.	ADV
ejpam-4352	81	11	moreover	moreover	ADV
ejpam-4352	81	12	,	,	PUNCT
ejpam-4352	81	13	since	since	SCONJ
ejpam-4352	81	14	v	v	NOUN
ejpam-4352	81	15	(	(	PUNCT
ejpam-4352	81	16	g	g	NOUN
ejpam-4352	81	17	)	)	PUNCT
ejpam-4352	81	18	is	be	AUX
ejpam-4352	81	19	a	a	DET
ejpam-4352	81	20	super	super	ADV
ejpam-4352	81	21	hop	hop	NOUN
ejpam-4352	81	22	dominating	dominating	NOUN
ejpam-4352	81	23	set	set	NOUN
ejpam-4352	81	24	,	,	PUNCT
ejpam-4352	81	25	we	we	PRON
ejpam-4352	81	26	have	have	VERB
ejpam-4352	81	27	⌈n2	⌈n2	NOUN
ejpam-4352	81	28	⌉	⌉	SYM
ejpam-4352	82	1	≤	≤	ADJ
ejpam-4352	82	2	γsh(g	γsh(g	NOUN
ejpam-4352	82	3	)	)	PUNCT
ejpam-4352	82	4	≤	≤	NUM
ejpam-4352	82	5	n.	n.	NOUN
ejpam-4352	82	6	therefore	therefore	ADV
ejpam-4352	82	7	,	,	PUNCT
ejpam-4352	82	8	the	the	DET
ejpam-4352	82	9	assertion	assertion	NOUN
ejpam-4352	82	10	holds	hold	VERB
ejpam-4352	82	11	.	.	PUNCT
ejpam-4352	83	1	theorem	theorem	NOUN
ejpam-4352	83	2	1	1	NUM
ejpam-4352	83	3	.	.	PUNCT
ejpam-4352	84	1	let	let	VERB
ejpam-4352	84	2	g	g	PRON
ejpam-4352	84	3	be	be	AUX
ejpam-4352	84	4	a	a	DET
ejpam-4352	84	5	graph	graph	NOUN
ejpam-4352	84	6	of	of	ADP
ejpam-4352	84	7	order	order	NOUN
ejpam-4352	84	8	n	n	PRON
ejpam-4352	84	9	≥	≥	NOUN
ejpam-4352	84	10	1	1	NUM
ejpam-4352	84	11	.	.	PUNCT
ejpam-4352	85	1	then	then	ADV
ejpam-4352	85	2	γsh(g	γsh(g	NOUN
ejpam-4352	85	3	)	)	PUNCT
ejpam-4352	85	4	=	=	SYM
ejpam-4352	86	1	n	n	NOUN
ejpam-4352	86	2	if	if	SCONJ
ejpam-4352	86	3	and	and	CCONJ
ejpam-4352	86	4	only	only	ADV
ejpam-4352	86	5	if	if	SCONJ
ejpam-4352	86	6	each	each	DET
ejpam-4352	86	7	component	component	NOUN
ejpam-4352	86	8	c	c	NOUN
ejpam-4352	86	9	of	of	ADP
ejpam-4352	86	10	g	g	PROPN
ejpam-4352	86	11	is	be	AUX
ejpam-4352	86	12	a	a	DET
ejpam-4352	86	13	complete	complete	ADJ
ejpam-4352	86	14	graph	graph	NOUN
ejpam-4352	86	15	.	.	PUNCT
ejpam-4352	87	1	proof	proof	NOUN
ejpam-4352	87	2	.	.	PUNCT
ejpam-4352	88	1	suppose	suppose	VERB
ejpam-4352	88	2	that	that	SCONJ
ejpam-4352	88	3	γsh(g	γsh(g	NOUN
ejpam-4352	88	4	)	)	PUNCT
ejpam-4352	88	5	=	=	SYM
ejpam-4352	88	6	n.	n.	NOUN
ejpam-4352	88	7	suppose	suppose	VERB
ejpam-4352	88	8	further	far	ADV
ejpam-4352	88	9	that	that	SCONJ
ejpam-4352	88	10	there	there	PRON
ejpam-4352	88	11	exists	exist	VERB
ejpam-4352	88	12	a	a	DET
ejpam-4352	88	13	component	component	NOUN
ejpam-4352	88	14	c	c	NOUN
ejpam-4352	88	15	of	of	ADP
ejpam-4352	88	16	g	g	PROPN
ejpam-4352	88	17	such	such	ADJ
ejpam-4352	88	18	that	that	SCONJ
ejpam-4352	88	19	c	c	PROPN
ejpam-4352	88	20	is	be	AUX
ejpam-4352	88	21	not	not	PART
ejpam-4352	88	22	a	a	DET
ejpam-4352	88	23	complete	complete	ADJ
ejpam-4352	88	24	graph	graph	NOUN
ejpam-4352	88	25	.	.	PUNCT
ejpam-4352	89	1	then	then	ADV
ejpam-4352	89	2	there	there	PRON
ejpam-4352	89	3	exist	exist	VERB
ejpam-4352	89	4	x	x	NOUN
ejpam-4352	89	5	,	,	PUNCT
ejpam-4352	89	6	y	y	PROPN
ejpam-4352	89	7	∈	∈	PROPN
ejpam-4352	89	8	v	v	NOUN
ejpam-4352	89	9	(	(	PUNCT
ejpam-4352	89	10	c	c	NOUN
ejpam-4352	89	11	)	)	PUNCT
ejpam-4352	89	12	such	such	ADJ
ejpam-4352	89	13	that	that	DET
ejpam-4352	89	14	dc(x	dc(x	NOUN
ejpam-4352	89	15	,	,	PUNCT
ejpam-4352	89	16	y	y	NOUN
ejpam-4352	89	17	)	)	PUNCT
ejpam-4352	89	18	=	=	SYM
ejpam-4352	90	1	dg(x	dg(x	X
ejpam-4352	90	2	,	,	PUNCT
ejpam-4352	90	3	y	y	NOUN
ejpam-4352	90	4	)	)	PUNCT
ejpam-4352	90	5	=	=	SYM
ejpam-4352	91	1	2	2	X
ejpam-4352	91	2	.	.	PUNCT
ejpam-4352	92	1	this	this	PRON
ejpam-4352	92	2	implies	imply	VERB
ejpam-4352	92	3	that	that	SCONJ
ejpam-4352	92	4	s	s	VERB
ejpam-4352	92	5	=	=	SYM
ejpam-4352	92	6	v	v	ADJ
ejpam-4352	92	7	(	(	PUNCT
ejpam-4352	92	8	g	g	NOUN
ejpam-4352	92	9	)	)	PUNCT
ejpam-4352	92	10	\	\	NOUN
ejpam-4352	92	11	{	{	PUNCT
ejpam-4352	92	12	x	x	NOUN
ejpam-4352	92	13	}	}	PUNCT
ejpam-4352	92	14	is	be	AUX
ejpam-4352	92	15	a	a	DET
ejpam-4352	92	16	super	super	ADV
ejpam-4352	92	17	hop	hop	NOUN
ejpam-4352	92	18	dominating	dominating	NOUN
ejpam-4352	92	19	set	set	NOUN
ejpam-4352	92	20	of	of	ADP
ejpam-4352	92	21	g.	g.	PROPN
ejpam-4352	92	22	hence	hence	ADV
ejpam-4352	92	23	,	,	PUNCT
ejpam-4352	92	24	γsh(g	γsh(g	SYM
ejpam-4352	92	25	)	)	PUNCT
ejpam-4352	92	26	≤	≤	NUM
ejpam-4352	92	27	|s|	|s|	PROPN
ejpam-4352	92	28	=	=	PUNCT
ejpam-4352	92	29	n	n	CCONJ
ejpam-4352	92	30	−	−	PROPN
ejpam-4352	92	31	1	1	NUM
ejpam-4352	92	32	,	,	PUNCT
ejpam-4352	92	33	contrary	contrary	ADV
ejpam-4352	92	34	to	to	ADP
ejpam-4352	92	35	the	the	DET
ejpam-4352	92	36	assumption	assumption	NOUN
ejpam-4352	92	37	that	that	SCONJ
ejpam-4352	92	38	γsh(g	γsh(g	X
ejpam-4352	92	39	)	)	PUNCT
ejpam-4352	92	40	=	=	SYM
ejpam-4352	92	41	n.	n.	PROPN
ejpam-4352	92	42	thus	thus	ADV
ejpam-4352	92	43	,	,	PUNCT
ejpam-4352	92	44	each	each	DET
ejpam-4352	92	45	component	component	NOUN
ejpam-4352	92	46	of	of	ADP
ejpam-4352	92	47	g	g	PROPN
ejpam-4352	92	48	is	be	AUX
ejpam-4352	92	49	a	a	DET
ejpam-4352	92	50	complete	complete	ADJ
ejpam-4352	92	51	graph	graph	NOUN
ejpam-4352	92	52	.	.	PUNCT
ejpam-4352	93	1	for	for	ADP
ejpam-4352	93	2	the	the	DET
ejpam-4352	93	3	converse	converse	NOUN
ejpam-4352	93	4	,	,	PUNCT
ejpam-4352	93	5	suppose	suppose	VERB
ejpam-4352	93	6	that	that	SCONJ
ejpam-4352	93	7	each	each	DET
ejpam-4352	93	8	component	component	NOUN
ejpam-4352	93	9	of	of	ADP
ejpam-4352	93	10	g	g	PROPN
ejpam-4352	93	11	is	be	AUX
ejpam-4352	93	12	a	a	DET
ejpam-4352	93	13	complete	complete	ADJ
ejpam-4352	93	14	graph	graph	NOUN
ejpam-4352	93	15	.	.	PUNCT
ejpam-4352	94	1	let	let	VERB
ejpam-4352	94	2	s	s	PRON
ejpam-4352	94	3	be	be	AUX
ejpam-4352	94	4	a	a	DET
ejpam-4352	94	5	γsh	γsh	NOUN
ejpam-4352	94	6	-	-	PUNCT
ejpam-4352	94	7	set	set	NOUN
ejpam-4352	94	8	of	of	ADP
ejpam-4352	94	9	g	g	NOUN
ejpam-4352	94	10	and	and	CCONJ
ejpam-4352	94	11	suppose	suppose	VERB
ejpam-4352	94	12	that	that	SCONJ
ejpam-4352	94	13	s	s	VERB
ejpam-4352	94	14	̸=	̸=	PROPN
ejpam-4352	94	15	v	v	NOUN
ejpam-4352	94	16	(	(	PUNCT
ejpam-4352	94	17	g	g	NOUN
ejpam-4352	94	18	)	)	PUNCT
ejpam-4352	94	19	.	.	PUNCT
ejpam-4352	95	1	let	let	VERB
ejpam-4352	95	2	v	v	NUM
ejpam-4352	95	3	∈	∈	PROPN
ejpam-4352	95	4	v	v	NOUN
ejpam-4352	95	5	(	(	PUNCT
ejpam-4352	95	6	g	g	NOUN
ejpam-4352	95	7	)	)	PUNCT
ejpam-4352	95	8	\	\	PROPN
ejpam-4352	95	9	s	s	PART
ejpam-4352	95	10	and	and	CCONJ
ejpam-4352	95	11	let	let	VERB
ejpam-4352	95	12	c	c	NOUN
ejpam-4352	95	13	be	be	AUX
ejpam-4352	95	14	the	the	DET
ejpam-4352	95	15	component	component	NOUN
ejpam-4352	95	16	of	of	ADP
ejpam-4352	95	17	g	g	NOUN
ejpam-4352	95	18	with	with	ADP
ejpam-4352	95	19	v	v	NUM
ejpam-4352	95	20	∈	∈	PROPN
ejpam-4352	95	21	v	v	NOUN
ejpam-4352	95	22	(	(	PUNCT
ejpam-4352	95	23	c	c	NOUN
ejpam-4352	95	24	)	)	PUNCT
ejpam-4352	95	25	.	.	PUNCT
ejpam-4352	96	1	let	let	VERB
ejpam-4352	96	2	w	w	NOUN
ejpam-4352	96	3	∈	∈	PROPN
ejpam-4352	96	4	ehpng(v	ehpng(v	PROPN
ejpam-4352	96	5	,	,	PUNCT
ejpam-4352	96	6	v	v	PROPN
ejpam-4352	96	7	(	(	PUNCT
ejpam-4352	96	8	g	g	NOUN
ejpam-4352	96	9	)	)	PUNCT
ejpam-4352	96	10	\	\	PROPN
ejpam-4352	97	1	s	s	X
ejpam-4352	97	2	)	)	PUNCT
ejpam-4352	97	3	.	.	PUNCT
ejpam-4352	98	1	then	then	ADV
ejpam-4352	98	2	w	w	PROPN
ejpam-4352	98	3	∈	∈	PROPN
ejpam-4352	98	4	v	v	ADP
ejpam-4352	98	5	(	(	PUNCT
ejpam-4352	98	6	c	c	NOUN
ejpam-4352	98	7	)	)	PUNCT
ejpam-4352	98	8	and	and	CCONJ
ejpam-4352	98	9	dg(v	dg(v	X
ejpam-4352	98	10	,	,	PUNCT
ejpam-4352	98	11	w	w	NOUN
ejpam-4352	98	12	)	)	PUNCT
ejpam-4352	98	13	=	=	SYM
ejpam-4352	98	14	2	2	NUM
ejpam-4352	98	15	,	,	PUNCT
ejpam-4352	98	16	contrary	contrary	ADV
ejpam-4352	98	17	to	to	ADP
ejpam-4352	98	18	the	the	DET
ejpam-4352	98	19	assumption	assumption	NOUN
ejpam-4352	98	20	that	that	SCONJ
ejpam-4352	98	21	c	c	PROPN
ejpam-4352	98	22	is	be	AUX
ejpam-4352	98	23	a	a	DET
ejpam-4352	98	24	complete	complete	ADJ
ejpam-4352	98	25	graph	graph	NOUN
ejpam-4352	98	26	.	.	PUNCT
ejpam-4352	99	1	thus	thus	ADV
ejpam-4352	99	2	,	,	PUNCT
ejpam-4352	99	3	s	s	VERB
ejpam-4352	99	4	=	=	SYM
ejpam-4352	99	5	v	v	X
ejpam-4352	99	6	(	(	PUNCT
ejpam-4352	99	7	g	g	NOUN
ejpam-4352	99	8	)	)	PUNCT
ejpam-4352	99	9	,	,	PUNCT
ejpam-4352	99	10	showing	show	VERB
ejpam-4352	99	11	that	that	SCONJ
ejpam-4352	99	12	γsh(g	γsh(g	NOUN
ejpam-4352	99	13	)	)	PUNCT
ejpam-4352	99	14	=	=	VERB
ejpam-4352	99	15	n.	n.	NOUN
ejpam-4352	99	16	the	the	DET
ejpam-4352	99	17	next	next	ADJ
ejpam-4352	99	18	results	result	NOUN
ejpam-4352	99	19	are	be	AUX
ejpam-4352	99	20	consequences	consequence	NOUN
ejpam-4352	99	21	of	of	ADP
ejpam-4352	99	22	theorem	theorem	ADJ
ejpam-4352	99	23	1	1	NUM
ejpam-4352	99	24	.	.	PUNCT
ejpam-4352	99	25	corollary	corollary	ADJ
ejpam-4352	99	26	1	1	NUM
ejpam-4352	99	27	.	.	PUNCT
ejpam-4352	100	1	let	let	VERB
ejpam-4352	100	2	g	g	PRON
ejpam-4352	100	3	be	be	AUX
ejpam-4352	100	4	a	a	DET
ejpam-4352	100	5	connected	connected	ADJ
ejpam-4352	100	6	graph	graph	NOUN
ejpam-4352	100	7	of	of	ADP
ejpam-4352	100	8	order	order	NOUN
ejpam-4352	100	9	n.	n.	NOUN
ejpam-4352	100	10	then	then	ADV
ejpam-4352	100	11	γsh(g	γsh(g	NOUN
ejpam-4352	100	12	)	)	PUNCT
ejpam-4352	100	13	=	=	SYM
ejpam-4352	101	1	n	n	NOUN
ejpam-4352	101	2	if	if	SCONJ
ejpam-4352	101	3	and	and	CCONJ
ejpam-4352	101	4	only	only	ADV
ejpam-4352	101	5	if	if	SCONJ
ejpam-4352	101	6	g	g	PROPN
ejpam-4352	101	7	=	=	PROPN
ejpam-4352	101	8	kn	kn	PROPN
ejpam-4352	101	9	.	.	PUNCT
ejpam-4352	101	10	corollary	corollary	PROPN
ejpam-4352	101	11	2	2	NUM
ejpam-4352	101	12	.	.	PUNCT
ejpam-4352	102	1	let	let	VERB
ejpam-4352	102	2	g	g	PRON
ejpam-4352	102	3	be	be	AUX
ejpam-4352	102	4	a	a	DET
ejpam-4352	102	5	connected	connected	ADJ
ejpam-4352	102	6	non	non	ADJ
ejpam-4352	102	7	-	-	ADJ
ejpam-4352	102	8	complete	complete	ADJ
ejpam-4352	102	9	graph	graph	NOUN
ejpam-4352	102	10	of	of	ADP
ejpam-4352	102	11	order	order	NOUN
ejpam-4352	102	12	n.	n.	NOUN
ejpam-4352	102	13	then	then	ADV
ejpam-4352	102	14	γsh(g	γsh(g	NOUN
ejpam-4352	102	15	)	)	PUNCT
ejpam-4352	102	16	≤	≤	NUM
ejpam-4352	102	17	n−	n−	NOUN
ejpam-4352	102	18	1	1	NUM
ejpam-4352	102	19	.	.	PUNCT
ejpam-4352	103	1	corollary	corollary	ADJ
ejpam-4352	103	2	3	3	X
ejpam-4352	103	3	.	.	PUNCT
ejpam-4352	104	1	if	if	SCONJ
ejpam-4352	104	2	g	g	PROPN
ejpam-4352	104	3	is	be	AUX
ejpam-4352	104	4	the	the	DET
ejpam-4352	104	5	complete	complete	ADJ
ejpam-4352	104	6	graph	graph	NOUN
ejpam-4352	104	7	of	of	ADP
ejpam-4352	104	8	order	order	NOUN
ejpam-4352	104	9	n	n	CCONJ
ejpam-4352	104	10	,	,	PUNCT
ejpam-4352	104	11	then	then	ADV
ejpam-4352	104	12	γsh(g	γsh(g	NOUN
ejpam-4352	104	13	)	)	PUNCT
ejpam-4352	104	14	+	+	NUM
ejpam-4352	104	15	γsh(g	γsh(g	X
ejpam-4352	104	16	)	)	PUNCT
ejpam-4352	104	17	=	=	SYM
ejpam-4352	104	18	2n	2n	NUM
ejpam-4352	104	19	and	and	CCONJ
ejpam-4352	104	20	γsh(g).γsh(g	γsh(g).γsh(g	NOUN
ejpam-4352	104	21	)	)	PUNCT
ejpam-4352	104	22	=	=	SYM
ejpam-4352	104	23	n2	n2	NOUN
ejpam-4352	104	24	.	.	PUNCT
ejpam-4352	104	25	theorem	theorem	NOUN
ejpam-4352	104	26	2	2	NUM
ejpam-4352	104	27	.	.	PUNCT
ejpam-4352	104	28	let	let	VERB
ejpam-4352	104	29	g	g	PRON
ejpam-4352	104	30	be	be	AUX
ejpam-4352	104	31	a	a	DET
ejpam-4352	104	32	connected	connected	ADJ
ejpam-4352	104	33	non	non	ADJ
ejpam-4352	104	34	-	-	ADJ
ejpam-4352	104	35	complete	complete	ADJ
ejpam-4352	104	36	graph	graph	NOUN
ejpam-4352	104	37	of	of	ADP
ejpam-4352	104	38	oder	oder	PROPN
ejpam-4352	104	39	n.	n.	PROPN
ejpam-4352	104	40	then	then	ADV
ejpam-4352	104	41	(	(	PUNCT
ejpam-4352	104	42	i	i	NOUN
ejpam-4352	104	43	)	)	PUNCT
ejpam-4352	104	44	n	n	CCONJ
ejpam-4352	104	45	≤	≤	NOUN
ejpam-4352	104	46	γsh(g	γsh(g	NOUN
ejpam-4352	104	47	)	)	PUNCT
ejpam-4352	105	1	+	+	NUM
ejpam-4352	105	2	γsh(g	γsh(g	NOUN
ejpam-4352	105	3	)	)	PUNCT
ejpam-4352	105	4	≤	≤	NOUN
ejpam-4352	106	1	2n−	2n−	NUM
ejpam-4352	106	2	1	1	NUM
ejpam-4352	106	3	and	and	CCONJ
ejpam-4352	106	4	(	(	PUNCT
ejpam-4352	106	5	ii	ii	NOUN
ejpam-4352	106	6	)	)	PUNCT
ejpam-4352	106	7	n2	n2	NOUN
ejpam-4352	106	8	4	4	NUM
ejpam-4352	106	9	≤	≤	NUM
ejpam-4352	106	10	γsh(g).γsh(g	γsh(g).γsh(g	NOUN
ejpam-4352	106	11	)	)	PUNCT
ejpam-4352	106	12	≤	≤	NOUN
ejpam-4352	106	13	n2	n2	NOUN
ejpam-4352	106	14	−	−	PROPN
ejpam-4352	106	15	n.	n.	NOUN
ejpam-4352	106	16	proof	proof	NOUN
ejpam-4352	106	17	.	.	PUNCT
ejpam-4352	107	1	by	by	ADP
ejpam-4352	107	2	corollary	corollary	ADJ
ejpam-4352	107	3	2	2	NUM
ejpam-4352	107	4	,	,	PUNCT
ejpam-4352	107	5	γsh(g	γsh(g	NOUN
ejpam-4352	107	6	)	)	PUNCT
ejpam-4352	107	7	≤	≤	NOUN
ejpam-4352	107	8	n	n	CCONJ
ejpam-4352	107	9	−	−	PROPN
ejpam-4352	107	10	1	1	NUM
ejpam-4352	107	11	.	.	PUNCT
ejpam-4352	108	1	also	also	ADV
ejpam-4352	108	2	,	,	PUNCT
ejpam-4352	108	3	by	by	ADP
ejpam-4352	108	4	proposition	proposition	NOUN
ejpam-4352	108	5	1	1	NUM
ejpam-4352	108	6	,	,	PUNCT
ejpam-4352	108	7	γsh(g	γsh(g	NOUN
ejpam-4352	108	8	)	)	PUNCT
ejpam-4352	108	9	≤	≤	PUNCT
ejpam-4352	108	10	n.	n.	NOUN
ejpam-4352	108	11	these	these	PRON
ejpam-4352	108	12	imply	imply	VERB
ejpam-4352	108	13	that	that	SCONJ
ejpam-4352	108	14	γsh(g	γsh(g	NOUN
ejpam-4352	108	15	)	)	PUNCT
ejpam-4352	108	16	+	+	NUM
ejpam-4352	108	17	γsh(g	γsh(g	NOUN
ejpam-4352	108	18	)	)	PUNCT
ejpam-4352	108	19	≤	≤	NOUN
ejpam-4352	108	20	(	(	PUNCT
ejpam-4352	108	21	n−	n−	NOUN
ejpam-4352	108	22	1	1	NUM
ejpam-4352	108	23	)	)	PUNCT
ejpam-4352	108	24	+	+	NUM
ejpam-4352	108	25	n	n	NOUN
ejpam-4352	108	26	=	=	SYM
ejpam-4352	108	27	2n−	2n−	PROPN
ejpam-4352	108	28	1	1	NUM
ejpam-4352	108	29	and	and	CCONJ
ejpam-4352	108	30	γsh(g).γsh(g	γsh(g).γsh(g	NOUN
ejpam-4352	108	31	)	)	PUNCT
ejpam-4352	108	32	≤	≤	NOUN
ejpam-4352	108	33	(	(	PUNCT
ejpam-4352	108	34	n−	n−	NOUN
ejpam-4352	108	35	1)n	1)n	X
ejpam-4352	108	36	=	=	SYM
ejpam-4352	108	37	n2	n2	PROPN
ejpam-4352	108	38	−	−	PROPN
ejpam-4352	108	39	n.	n.	NOUN
ejpam-4352	108	40	the	the	DET
ejpam-4352	108	41	left	left	ADJ
ejpam-4352	108	42	inequalities	inequality	NOUN
ejpam-4352	108	43	follow	follow	VERB
ejpam-4352	108	44	from	from	ADP
ejpam-4352	108	45	proposition	proposition	NOUN
ejpam-4352	108	46	1	1	NUM
ejpam-4352	108	47	.	.	PUNCT
ejpam-4352	108	48	note	note	VERB
ejpam-4352	108	49	that	that	SCONJ
ejpam-4352	108	50	the	the	DET
ejpam-4352	108	51	upper	upper	ADJ
ejpam-4352	108	52	bounds	bound	NOUN
ejpam-4352	108	53	in	in	ADP
ejpam-4352	108	54	theorem	theorem	ADJ
ejpam-4352	108	55	2	2	NUM
ejpam-4352	108	56	are	be	AUX
ejpam-4352	108	57	tight	tight	ADJ
ejpam-4352	108	58	.	.	PUNCT
ejpam-4352	109	1	indeed	indeed	ADV
ejpam-4352	109	2	,	,	PUNCT
ejpam-4352	109	3	if	if	SCONJ
ejpam-4352	109	4	g	g	PROPN
ejpam-4352	109	5	=	=	SYM
ejpam-4352	109	6	k1,n−1	k1,n−1	PROPN
ejpam-4352	109	7	,	,	PUNCT
ejpam-4352	109	8	then	then	ADV
ejpam-4352	109	9	g	g	PROPN
ejpam-4352	109	10	=	=	PROPN
ejpam-4352	109	11	k1∪kn−1	k1∪kn−1	PROPN
ejpam-4352	109	12	.	.	PUNCT
ejpam-4352	110	1	it	it	PRON
ejpam-4352	110	2	is	be	AUX
ejpam-4352	110	3	easy	easy	ADJ
ejpam-4352	110	4	to	to	PART
ejpam-4352	110	5	show	show	VERB
ejpam-4352	110	6	that	that	SCONJ
ejpam-4352	110	7	γsh(g	γsh(g	NOUN
ejpam-4352	110	8	)	)	PUNCT
ejpam-4352	110	9	=	=	SYM
ejpam-4352	111	1	n−1	n−1	PROPN
ejpam-4352	111	2	.	.	PUNCT
ejpam-4352	111	3	by	by	ADP
ejpam-4352	111	4	theorem	theorem	NOUN
ejpam-4352	111	5	1	1	NUM
ejpam-4352	111	6	,	,	PUNCT
ejpam-4352	111	7	γsh(g	γsh(g	NOUN
ejpam-4352	111	8	)	)	PUNCT
ejpam-4352	111	9	=	=	SYM
ejpam-4352	111	10	n.	n.	NOUN
ejpam-4352	111	11	hence	hence	ADV
ejpam-4352	111	12	,	,	PUNCT
ejpam-4352	111	13	γsh(g)+	γsh(g)+	PROPN
ejpam-4352	111	14	γsh(g	γsh(g	PROPN
ejpam-4352	111	15	)	)	PUNCT
ejpam-4352	111	16	=	=	PUNCT
ejpam-4352	112	1	2n−	2n−	NUM
ejpam-4352	112	2	1	1	NUM
ejpam-4352	112	3	and	and	CCONJ
ejpam-4352	112	4	γsh(g).γsh(g	γsh(g).γsh(g	NOUN
ejpam-4352	112	5	)	)	PUNCT
ejpam-4352	112	6	=	=	SYM
ejpam-4352	112	7	n2−n	n2−n	PROPN
ejpam-4352	112	8	.	.	PUNCT
ejpam-4352	113	1	the	the	DET
ejpam-4352	113	2	lower	low	ADJ
ejpam-4352	113	3	bounds	bound	NOUN
ejpam-4352	113	4	are	be	AUX
ejpam-4352	113	5	also	also	ADV
ejpam-4352	113	6	attainable	attainable	ADJ
ejpam-4352	113	7	.	.	PUNCT
ejpam-4352	114	1	it	it	PRON
ejpam-4352	114	2	can	can	AUX
ejpam-4352	114	3	be	be	AUX
ejpam-4352	114	4	verified	verify	VERB
ejpam-4352	114	5	that	that	SCONJ
ejpam-4352	114	6	γsh(p4	γsh(p4	NOUN
ejpam-4352	114	7	)	)	PUNCT
ejpam-4352	114	8	+	+	CCONJ
ejpam-4352	114	9	γsh(p	γsh(p	NOUN
ejpam-4352	114	10	4	4	NUM
ejpam-4352	114	11	)	)	PUNCT
ejpam-4352	114	12	=	=	SYM
ejpam-4352	114	13	4	4	NUM
ejpam-4352	114	14	and	and	CCONJ
ejpam-4352	114	15	γsh(p4).γ	γsh(p4).γ	PROPN
ejpam-4352	114	16	s	s	PART
ejpam-4352	114	17	h(p	h(p	PROPN
ejpam-4352	114	18	4	4	NUM
ejpam-4352	114	19	)	)	PUNCT
ejpam-4352	114	20	=	=	SYM
ejpam-4352	115	1	16	16	NUM
ejpam-4352	115	2	4	4	NUM
ejpam-4352	115	3	=	=	SYM
ejpam-4352	115	4	4	4	NUM
ejpam-4352	115	5	.	.	PUNCT
ejpam-4352	116	1	the	the	DET
ejpam-4352	116	2	join	join	NOUN
ejpam-4352	116	3	of	of	ADP
ejpam-4352	116	4	graphs	graph	NOUN
ejpam-4352	116	5	g	g	NOUN
ejpam-4352	116	6	and	and	CCONJ
ejpam-4352	116	7	h	h	NOUN
ejpam-4352	116	8	is	be	AUX
ejpam-4352	116	9	the	the	DET
ejpam-4352	116	10	graph	graph	NOUN
ejpam-4352	116	11	g+h	g+h	PROPN
ejpam-4352	116	12	with	with	ADP
ejpam-4352	116	13	vertex	vertex	NOUN
ejpam-4352	116	14	set	set	VERB
ejpam-4352	116	15	v	v	NOUN
ejpam-4352	116	16	(	(	PUNCT
ejpam-4352	116	17	g+h	g+h	NOUN
ejpam-4352	116	18	)	)	PUNCT
ejpam-4352	116	19	=	=	SYM
ejpam-4352	116	20	v	v	X
ejpam-4352	116	21	(	(	PUNCT
ejpam-4352	116	22	g	g	NOUN
ejpam-4352	116	23	)	)	PUNCT
ejpam-4352	116	24	∪	∪	NOUN
ejpam-4352	116	25	v	v	NOUN
ejpam-4352	116	26	(	(	PUNCT
ejpam-4352	116	27	h	h	NOUN
ejpam-4352	116	28	)	)	PUNCT
ejpam-4352	116	29	and	and	CCONJ
ejpam-4352	116	30	edge	edge	NOUN
ejpam-4352	116	31	set	set	VERB
ejpam-4352	116	32	e(g+h	e(g+h	NUM
ejpam-4352	116	33	)	)	PUNCT
ejpam-4352	116	34	=	=	SYM
ejpam-4352	116	35	e(g	e(g	NOUN
ejpam-4352	116	36	)	)	PUNCT
ejpam-4352	116	37	∪	∪	ADP
ejpam-4352	116	38	e(h	e(h	PROPN
ejpam-4352	116	39	)	)	PUNCT
ejpam-4352	116	40	∪	∪	NOUN
ejpam-4352	116	41	{	{	PUNCT
ejpam-4352	116	42	uv	uv	NOUN
ejpam-4352	116	43	:	:	PUNCT
ejpam-4352	116	44	u	u	PROPN
ejpam-4352	116	45	∈	∈	PROPN
ejpam-4352	116	46	v	v	ADP
ejpam-4352	116	47	(	(	PUNCT
ejpam-4352	116	48	g	g	NOUN
ejpam-4352	116	49	)	)	PUNCT
ejpam-4352	116	50	and	and	CCONJ
ejpam-4352	116	51	v	v	ADP
ejpam-4352	116	52	∈	∈	PROPN
ejpam-4352	116	53	v	v	NOUN
ejpam-4352	116	54	(	(	PUNCT
ejpam-4352	116	55	h	h	NOUN
ejpam-4352	116	56	)	)	PUNCT
ejpam-4352	116	57	}	}	PUNCT
ejpam-4352	116	58	.	.	PUNCT
ejpam-4352	117	1	theorem	theorem	NOUN
ejpam-4352	117	2	3	3	X
ejpam-4352	117	3	.	.	PUNCT
ejpam-4352	118	1	let	let	VERB
ejpam-4352	118	2	g	g	NOUN
ejpam-4352	119	1	and	and	CCONJ
ejpam-4352	119	2	h	h	NOUN
ejpam-4352	119	3	be	be	VERB
ejpam-4352	119	4	any	any	DET
ejpam-4352	119	5	two	two	NUM
ejpam-4352	119	6	graphs	graph	NOUN
ejpam-4352	119	7	.	.	PUNCT
ejpam-4352	120	1	a	a	DET
ejpam-4352	120	2	subset	subset	NOUN
ejpam-4352	120	3	s	s	X
ejpam-4352	120	4	of	of	ADP
ejpam-4352	120	5	v	v	NOUN
ejpam-4352	120	6	(	(	PUNCT
ejpam-4352	120	7	g	g	PROPN
ejpam-4352	120	8	+	+	NOUN
ejpam-4352	120	9	h	h	NOUN
ejpam-4352	120	10	)	)	PUNCT
ejpam-4352	120	11	is	be	AUX
ejpam-4352	120	12	a	a	DET
ejpam-4352	120	13	super	super	ADV
ejpam-4352	120	14	hop	hop	NOUN
ejpam-4352	120	15	dominating	dominating	NOUN
ejpam-4352	120	16	set	set	NOUN
ejpam-4352	120	17	of	of	ADP
ejpam-4352	120	18	g+h	g+h	PROPN
ejpam-4352	121	1	if	if	SCONJ
ejpam-4352	121	2	and	and	CCONJ
ejpam-4352	121	3	only	only	ADV
ejpam-4352	121	4	if	if	SCONJ
ejpam-4352	121	5	s	s	X
ejpam-4352	121	6	=	=	PUNCT
ejpam-4352	121	7	sg∪sh	sg∪sh	PROPN
ejpam-4352	121	8	where	where	SCONJ
ejpam-4352	121	9	sg	sg	PROPN
ejpam-4352	121	10	and	and	CCONJ
ejpam-4352	121	11	sh	sh	PROPN
ejpam-4352	121	12	are	be	AUX
ejpam-4352	121	13	complement	complement	NOUN
ejpam-4352	121	14	-	-	PUNCT
ejpam-4352	121	15	super	super	ADJ
ejpam-4352	121	16	dominating	dominating	NOUN
ejpam-4352	121	17	sets	set	NOUN
ejpam-4352	121	18	of	of	ADP
ejpam-4352	121	19	g	g	PROPN
ejpam-4352	121	20	and	and	CCONJ
ejpam-4352	121	21	h	h	NOUN
ejpam-4352	121	22	,	,	PUNCT
ejpam-4352	121	23	respectively	respectively	ADV
ejpam-4352	121	24	.	.	PUNCT
ejpam-4352	122	1	s.	s.	PROPN
ejpam-4352	122	2	canoy	canoy	PROPN
ejpam-4352	122	3	,	,	PUNCT
ejpam-4352	122	4	jr	jr	PROPN
ejpam-4352	122	5	.	.	PROPN
ejpam-4352	122	6	,	,	PUNCT
ejpam-4352	122	7	g.	g.	PROPN
ejpam-4352	122	8	salasalan	salasalan	PROPN
ejpam-4352	122	9	/	/	SYM
ejpam-4352	122	10	eur	eur	PROPN
ejpam-4352	122	11	.	.	PUNCT
ejpam-4352	123	1	j.	j.	PROPN
ejpam-4352	123	2	pure	pure	PROPN
ejpam-4352	123	3	appl	appl	PROPN
ejpam-4352	123	4	.	.	PROPN
ejpam-4352	123	5	math	math	PROPN
ejpam-4352	123	6	,	,	PUNCT
ejpam-4352	123	7	15	15	NUM
ejpam-4352	123	8	(	(	PUNCT
ejpam-4352	123	9	2	2	NUM
ejpam-4352	123	10	)	)	PUNCT
ejpam-4352	123	11	(	(	PUNCT
ejpam-4352	123	12	2022	2022	NUM
ejpam-4352	123	13	)	)	PUNCT
ejpam-4352	123	14	,	,	PUNCT
ejpam-4352	123	15	342	342	NUM
ejpam-4352	123	16	-	-	SYM
ejpam-4352	123	17	353	353	NUM
ejpam-4352	123	18	345	345	NUM
ejpam-4352	123	19	proof	proof	NOUN
ejpam-4352	123	20	.	.	PUNCT
ejpam-4352	123	21	suppose	suppose	VERB
ejpam-4352	123	22	that	that	SCONJ
ejpam-4352	123	23	s	s	VERB
ejpam-4352	123	24	is	be	AUX
ejpam-4352	123	25	a	a	DET
ejpam-4352	123	26	super	super	ADV
ejpam-4352	123	27	hop	hop	NOUN
ejpam-4352	123	28	dominating	dominating	NOUN
ejpam-4352	123	29	set	set	NOUN
ejpam-4352	123	30	of	of	ADP
ejpam-4352	123	31	g	g	PROPN
ejpam-4352	123	32	+	+	PROPN
ejpam-4352	123	33	h.	h.	PROPN
ejpam-4352	123	34	let	let	VERB
ejpam-4352	123	35	sg	sg	ADP
ejpam-4352	123	36	=	=	SYM
ejpam-4352	123	37	s	s	PART
ejpam-4352	123	38	∩	∩	ADJ
ejpam-4352	123	39	v	v	X
ejpam-4352	123	40	(	(	PUNCT
ejpam-4352	123	41	g	g	NOUN
ejpam-4352	123	42	)	)	PUNCT
ejpam-4352	123	43	and	and	CCONJ
ejpam-4352	123	44	sh	sh	INTJ
ejpam-4352	123	45	=	=	SYM
ejpam-4352	123	46	s	s	PROPN
ejpam-4352	123	47	∩	∩	ADJ
ejpam-4352	123	48	v	v	ADJ
ejpam-4352	123	49	(	(	PUNCT
ejpam-4352	123	50	h	h	NOUN
ejpam-4352	123	51	)	)	PUNCT
ejpam-4352	123	52	.	.	PUNCT
ejpam-4352	124	1	let	let	VERB
ejpam-4352	124	2	v	v	NUM
ejpam-4352	124	3	∈	∈	PROPN
ejpam-4352	124	4	v	v	NOUN
ejpam-4352	124	5	(	(	PUNCT
ejpam-4352	124	6	g	g	NOUN
ejpam-4352	124	7	)	)	PUNCT
ejpam-4352	124	8	\	\	PROPN
ejpam-4352	124	9	sg	sg	PROPN
ejpam-4352	124	10	.	.	PUNCT
ejpam-4352	125	1	since	since	SCONJ
ejpam-4352	125	2	s	s	PROPN
ejpam-4352	125	3	is	be	AUX
ejpam-4352	125	4	a	a	DET
ejpam-4352	125	5	super	super	ADV
ejpam-4352	125	6	hop	hop	NOUN
ejpam-4352	125	7	dominating	dominating	NOUN
ejpam-4352	125	8	set	set	NOUN
ejpam-4352	125	9	of	of	ADP
ejpam-4352	125	10	g+h	g+h	PROPN
ejpam-4352	125	11	,	,	PUNCT
ejpam-4352	125	12	there	there	PRON
ejpam-4352	125	13	exists	exist	VERB
ejpam-4352	125	14	w	w	PROPN
ejpam-4352	125	15	∈	∈	PROPN
ejpam-4352	125	16	ehpn(v	ehpn(v	PROPN
ejpam-4352	125	17	,	,	PUNCT
ejpam-4352	125	18	v	v	PROPN
ejpam-4352	125	19	(	(	PUNCT
ejpam-4352	125	20	g+h	g+h	NOUN
ejpam-4352	125	21	)	)	PUNCT
ejpam-4352	125	22	\	\	PROPN
ejpam-4352	126	1	s	s	X
ejpam-4352	126	2	)	)	PUNCT
ejpam-4352	126	3	.	.	PUNCT
ejpam-4352	127	1	since	since	SCONJ
ejpam-4352	127	2	v	v	NOUN
ejpam-4352	127	3	(	(	PUNCT
ejpam-4352	127	4	h	h	NOUN
ejpam-4352	127	5	)	)	PUNCT
ejpam-4352	127	6	⊆	⊆	NUM
ejpam-4352	127	7	ng+h(v	ng+h(v	NOUN
ejpam-4352	127	8	)	)	PUNCT
ejpam-4352	127	9	,	,	PUNCT
ejpam-4352	127	10	it	it	PRON
ejpam-4352	127	11	follows	follow	VERB
ejpam-4352	127	12	that	that	SCONJ
ejpam-4352	127	13	w	w	PROPN
ejpam-4352	127	14	∈	∈	PROPN
ejpam-4352	127	15	sg	sg	ADP
ejpam-4352	127	16	\ng(v	\ng(v	NOUN
ejpam-4352	127	17	)	)	PUNCT
ejpam-4352	127	18	and	and	CCONJ
ejpam-4352	127	19	n2	n2	ADJ
ejpam-4352	127	20	g+h(w	g+h(w	NOUN
ejpam-4352	127	21	)	)	PUNCT
ejpam-4352	127	22	∩	∩	NOUN
ejpam-4352	127	23	(	(	PUNCT
ejpam-4352	127	24	v	v	NOUN
ejpam-4352	127	25	(	(	PUNCT
ejpam-4352	127	26	g+h	g+h	NOUN
ejpam-4352	127	27	)	)	PUNCT
ejpam-4352	127	28	\	\	PROPN
ejpam-4352	128	1	s	s	X
ejpam-4352	128	2	)	)	PUNCT
ejpam-4352	128	3	=	=	PUNCT
ejpam-4352	129	1	[	[	X
ejpam-4352	129	2	v	v	X
ejpam-4352	129	3	(	(	PUNCT
ejpam-4352	129	4	g	g	NOUN
ejpam-4352	129	5	)	)	PUNCT
ejpam-4352	129	6	\ng(w	\ng(w	PUNCT
ejpam-4352	129	7	)	)	PUNCT
ejpam-4352	129	8	]	]	PUNCT
ejpam-4352	130	1	∩	∩	NOUN
ejpam-4352	130	2	[	[	X
ejpam-4352	130	3	v	v	X
ejpam-4352	130	4	(	(	PUNCT
ejpam-4352	130	5	g	g	NOUN
ejpam-4352	130	6	)	)	PUNCT
ejpam-4352	130	7	\	\	PUNCT
ejpam-4352	131	1	sg	sg	PROPN
ejpam-4352	131	2	]	]	X
ejpam-4352	131	3	=	=	PUNCT
ejpam-4352	131	4	{	{	PUNCT
ejpam-4352	131	5	v	v	NOUN
ejpam-4352	131	6	}	}	PUNCT
ejpam-4352	131	7	.	.	PUNCT
ejpam-4352	132	1	this	this	PRON
ejpam-4352	132	2	shows	show	VERB
ejpam-4352	132	3	that	that	SCONJ
ejpam-4352	132	4	sg	sg	PROPN
ejpam-4352	132	5	is	be	AUX
ejpam-4352	132	6	a	a	DET
ejpam-4352	132	7	complement	complement	NOUN
ejpam-4352	132	8	-	-	PUNCT
ejpam-4352	132	9	super	super	ADJ
ejpam-4352	132	10	dominating	dominating	NOUN
ejpam-4352	132	11	set	set	NOUN
ejpam-4352	132	12	of	of	ADP
ejpam-4352	132	13	g.	g.	PROPN
ejpam-4352	132	14	similarly	similarly	ADV
ejpam-4352	132	15	,	,	PUNCT
ejpam-4352	132	16	sh	sh	PROPN
ejpam-4352	132	17	is	be	AUX
ejpam-4352	132	18	a	a	DET
ejpam-4352	132	19	complement	complement	NOUN
ejpam-4352	132	20	-	-	PUNCT
ejpam-4352	132	21	super	super	ADJ
ejpam-4352	132	22	dominating	dominating	NOUN
ejpam-4352	132	23	set	set	NOUN
ejpam-4352	132	24	of	of	ADP
ejpam-4352	132	25	h.	h.	PROPN
ejpam-4352	132	26	for	for	ADP
ejpam-4352	132	27	the	the	DET
ejpam-4352	132	28	converse	converse	NOUN
ejpam-4352	132	29	,	,	PUNCT
ejpam-4352	132	30	suppose	suppose	VERB
ejpam-4352	132	31	that	that	SCONJ
ejpam-4352	132	32	s	s	VERB
ejpam-4352	132	33	=	=	PUNCT
ejpam-4352	132	34	sg	sg	X
ejpam-4352	132	35	∪	∪	ADJ
ejpam-4352	132	36	sh	sh	PROPN
ejpam-4352	132	37	,	,	PUNCT
ejpam-4352	132	38	where	where	SCONJ
ejpam-4352	132	39	sg	sg	PROPN
ejpam-4352	132	40	and	and	CCONJ
ejpam-4352	132	41	sh	sh	PROPN
ejpam-4352	132	42	are	be	AUX
ejpam-4352	132	43	complementsuper	complementsuper	NOUN
ejpam-4352	132	44	dominating	dominating	NOUN
ejpam-4352	132	45	sets	set	NOUN
ejpam-4352	132	46	of	of	ADP
ejpam-4352	132	47	g	g	PROPN
ejpam-4352	132	48	and	and	CCONJ
ejpam-4352	132	49	h	h	NOUN
ejpam-4352	132	50	,	,	PUNCT
ejpam-4352	132	51	respectively	respectively	ADV
ejpam-4352	132	52	.	.	PUNCT
ejpam-4352	133	1	let	let	VERB
ejpam-4352	133	2	v	v	NUM
ejpam-4352	133	3	∈	∈	PROPN
ejpam-4352	133	4	v	v	NOUN
ejpam-4352	133	5	(	(	PUNCT
ejpam-4352	133	6	g	g	PROPN
ejpam-4352	133	7	+	+	NOUN
ejpam-4352	133	8	h	h	NOUN
ejpam-4352	133	9	)	)	PUNCT
ejpam-4352	133	10	\	\	PUNCT
ejpam-4352	134	1	s.	s.	PROPN
ejpam-4352	135	1	if	if	SCONJ
ejpam-4352	135	2	v	v	NUM
ejpam-4352	135	3	∈	∈	PROPN
ejpam-4352	135	4	v	v	NOUN
ejpam-4352	135	5	(	(	PUNCT
ejpam-4352	135	6	g	g	NOUN
ejpam-4352	135	7	)	)	PUNCT
ejpam-4352	135	8	,	,	PUNCT
ejpam-4352	135	9	then	then	ADV
ejpam-4352	135	10	v	v	X
ejpam-4352	135	11	∈	∈	PROPN
ejpam-4352	135	12	v	v	NOUN
ejpam-4352	135	13	(	(	PUNCT
ejpam-4352	135	14	g	g	NOUN
ejpam-4352	135	15	)	)	PUNCT
ejpam-4352	135	16	\	\	PROPN
ejpam-4352	135	17	sg	sg	PROPN
ejpam-4352	135	18	.	.	PUNCT
ejpam-4352	136	1	since	since	SCONJ
ejpam-4352	136	2	sg	sg	PROPN
ejpam-4352	136	3	is	be	AUX
ejpam-4352	136	4	a	a	DET
ejpam-4352	136	5	complement	complement	NOUN
ejpam-4352	136	6	-	-	PUNCT
ejpam-4352	136	7	super	super	ADJ
ejpam-4352	136	8	dominating	dominating	NOUN
ejpam-4352	136	9	set	set	NOUN
ejpam-4352	136	10	of	of	ADP
ejpam-4352	136	11	g	g	NOUN
ejpam-4352	136	12	,	,	PUNCT
ejpam-4352	136	13	there	there	PRON
ejpam-4352	136	14	exists	exist	VERB
ejpam-4352	136	15	w	w	PROPN
ejpam-4352	136	16	∈	∈	PROPN
ejpam-4352	136	17	sg\ng(v	sg\ng(v	NOUN
ejpam-4352	136	18	)	)	PUNCT
ejpam-4352	136	19	such	such	ADJ
ejpam-4352	136	20	that	that	SCONJ
ejpam-4352	136	21	[	[	X
ejpam-4352	136	22	v	v	X
ejpam-4352	136	23	(	(	PUNCT
ejpam-4352	136	24	g)\ng(w)]∩[v	g)\ng(w)]∩[v	NOUN
ejpam-4352	136	25	(	(	PUNCT
ejpam-4352	136	26	g)\sg	g)\sg	PROPN
ejpam-4352	136	27	]	]	PUNCT
ejpam-4352	136	28	=	=	SYM
ejpam-4352	136	29	n2	n2	PROPN
ejpam-4352	136	30	g+h(w)∩(v	g+h(w)∩(v	PROPN
ejpam-4352	136	31	(	(	PUNCT
ejpam-4352	136	32	g+h)\s	g+h)\s	NUM
ejpam-4352	136	33	)	)	PUNCT
ejpam-4352	136	34	=	=	PRON
ejpam-4352	136	35	{	{	PUNCT
ejpam-4352	136	36	v	v	NOUN
ejpam-4352	136	37	}	}	PUNCT
ejpam-4352	136	38	.	.	PUNCT
ejpam-4352	137	1	this	this	PRON
ejpam-4352	137	2	implies	imply	VERB
ejpam-4352	137	3	that	that	SCONJ
ejpam-4352	137	4	w	w	PROPN
ejpam-4352	137	5	∈	∈	PROPN
ejpam-4352	137	6	ehpn(v	ehpn(v	PROPN
ejpam-4352	137	7	,	,	PUNCT
ejpam-4352	137	8	v	v	INTJ
ejpam-4352	137	9	(	(	PUNCT
ejpam-4352	137	10	g	g	PROPN
ejpam-4352	137	11	+	+	NOUN
ejpam-4352	137	12	h	h	NOUN
ejpam-4352	137	13	)	)	PUNCT
ejpam-4352	137	14	\	\	PROPN
ejpam-4352	138	1	s	s	X
ejpam-4352	138	2	)	)	PUNCT
ejpam-4352	138	3	.	.	PUNCT
ejpam-4352	139	1	similarly	similarly	ADV
ejpam-4352	139	2	,	,	PUNCT
ejpam-4352	139	3	if	if	SCONJ
ejpam-4352	139	4	v	v	NUM
ejpam-4352	139	5	∈	∈	PROPN
ejpam-4352	139	6	v	v	NOUN
ejpam-4352	139	7	(	(	PUNCT
ejpam-4352	139	8	g	g	NOUN
ejpam-4352	139	9	)	)	PUNCT
ejpam-4352	139	10	\	\	PUNCT
ejpam-4352	140	1	sh	sh	INTJ
ejpam-4352	140	2	,	,	PUNCT
ejpam-4352	140	3	then	then	ADV
ejpam-4352	140	4	there	there	PRON
ejpam-4352	140	5	exists	exist	VERB
ejpam-4352	140	6	z	z	PROPN
ejpam-4352	140	7	∈	∈	PROPN
ejpam-4352	140	8	ehpn(v	ehpn(v	PROPN
ejpam-4352	140	9	,	,	PUNCT
ejpam-4352	140	10	v	v	INTJ
ejpam-4352	140	11	(	(	PUNCT
ejpam-4352	140	12	g	g	PROPN
ejpam-4352	140	13	+	+	NOUN
ejpam-4352	140	14	h	h	NOUN
ejpam-4352	140	15	)	)	PUNCT
ejpam-4352	140	16	\	\	PROPN
ejpam-4352	141	1	s	s	X
ejpam-4352	141	2	)	)	PUNCT
ejpam-4352	141	3	.	.	PUNCT
ejpam-4352	142	1	this	this	PRON
ejpam-4352	142	2	shows	show	VERB
ejpam-4352	142	3	that	that	SCONJ
ejpam-4352	142	4	s	s	VERB
ejpam-4352	142	5	is	be	AUX
ejpam-4352	142	6	a	a	DET
ejpam-4352	142	7	super	super	ADV
ejpam-4352	142	8	hop	hop	NOUN
ejpam-4352	142	9	dominating	dominating	NOUN
ejpam-4352	142	10	set	set	NOUN
ejpam-4352	142	11	of	of	ADP
ejpam-4352	142	12	g+h	g+h	PROPN
ejpam-4352	142	13	.	.	PUNCT
ejpam-4352	143	1	theorem	theorem	ADJ
ejpam-4352	143	2	4	4	NUM
ejpam-4352	143	3	.	.	PUNCT
ejpam-4352	144	1	let	let	VERB
ejpam-4352	144	2	g	g	PRON
ejpam-4352	144	3	be	be	AUX
ejpam-4352	144	4	a	a	DET
ejpam-4352	144	5	graph	graph	NOUN
ejpam-4352	144	6	of	of	ADP
ejpam-4352	144	7	order	order	NOUN
ejpam-4352	144	8	n.	n.	NOUN
ejpam-4352	144	9	then	then	ADV
ejpam-4352	144	10	⌈n2	⌈n2	VERB
ejpam-4352	144	11	⌉	⌉	SYM
ejpam-4352	144	12	≤	≤	NUM
ejpam-4352	144	13	γcs(g	γcs(g	NUM
ejpam-4352	144	14	)	)	PUNCT
ejpam-4352	144	15	≤	≤	NOUN
ejpam-4352	144	16	n.	n.	NOUN
ejpam-4352	144	17	moreover	moreover	ADV
ejpam-4352	144	18	,	,	PUNCT
ejpam-4352	144	19	(	(	PUNCT
ejpam-4352	144	20	i	i	NOUN
ejpam-4352	144	21	)	)	PUNCT
ejpam-4352	144	22	γcs(g	γcs(g	X
ejpam-4352	144	23	)	)	PUNCT
ejpam-4352	145	1	=	=	SYM
ejpam-4352	145	2	n	n	NOUN
ejpam-4352	145	3	if	if	SCONJ
ejpam-4352	145	4	and	and	CCONJ
ejpam-4352	145	5	only	only	ADV
ejpam-4352	145	6	if	if	SCONJ
ejpam-4352	145	7	g	g	PROPN
ejpam-4352	145	8	=	=	SYM
ejpam-4352	145	9	kn	kn	PROPN
ejpam-4352	145	10	;	;	PUNCT
ejpam-4352	145	11	and	and	CCONJ
ejpam-4352	145	12	(	(	PUNCT
ejpam-4352	145	13	ii	ii	NOUN
ejpam-4352	145	14	)	)	PUNCT
ejpam-4352	145	15	for	for	ADP
ejpam-4352	145	16	n	n	X
ejpam-4352	145	17	≥	≥	NOUN
ejpam-4352	145	18	4	4	NUM
ejpam-4352	145	19	and	and	CCONJ
ejpam-4352	145	20	even	even	ADV
ejpam-4352	145	21	,	,	PUNCT
ejpam-4352	145	22	we	we	PRON
ejpam-4352	145	23	have	have	VERB
ejpam-4352	145	24	γcs(g	γcs(g	NUM
ejpam-4352	145	25	)	)	PUNCT
ejpam-4352	145	26	=	=	SYM
ejpam-4352	146	1	n	n	DET
ejpam-4352	146	2	2	2	NUM
ejpam-4352	146	3	if	if	SCONJ
ejpam-4352	146	4	and	and	CCONJ
ejpam-4352	146	5	only	only	ADV
ejpam-4352	146	6	if	if	SCONJ
ejpam-4352	146	7	g	g	PROPN
ejpam-4352	146	8	has	have	VERB
ejpam-4352	146	9	an	an	DET
ejpam-4352	146	10	(	(	PUNCT
ejpam-4352	146	11	n2	n2	ADJ
ejpam-4352	146	12	−	−	PROPN
ejpam-4352	146	13	1)-regular	1)-regular	NUM
ejpam-4352	146	14	bipartite	bipartite	PROPN
ejpam-4352	146	15	subgraph	subgraph	NOUN
ejpam-4352	146	16	h	h	PROPN
ejpam-4352	146	17	with	with	ADP
ejpam-4352	146	18	partite	partite	ADJ
ejpam-4352	146	19	sets	set	NOUN
ejpam-4352	146	20	a	a	PRON
ejpam-4352	146	21	and	and	CCONJ
ejpam-4352	146	22	b	b	NOUN
ejpam-4352	146	23	such	such	ADJ
ejpam-4352	146	24	that	that	DET
ejpam-4352	146	25	v	v	NOUN
ejpam-4352	146	26	(	(	PUNCT
ejpam-4352	146	27	g	g	NOUN
ejpam-4352	146	28	)	)	PUNCT
ejpam-4352	146	29	=	=	PUNCT
ejpam-4352	146	30	a	a	DET
ejpam-4352	146	31	∪	∪	X
ejpam-4352	146	32	b	b	NOUN
ejpam-4352	146	33	=	=	SYM
ejpam-4352	146	34	v	v	PROPN
ejpam-4352	146	35	(	(	PUNCT
ejpam-4352	146	36	h	h	NOUN
ejpam-4352	146	37	)	)	PUNCT
ejpam-4352	146	38	,	,	PUNCT
ejpam-4352	146	39	|a|	|a|	PROPN
ejpam-4352	146	40	=	=	SYM
ejpam-4352	146	41	|b|	|b|	PROPN
ejpam-4352	146	42	=	=	SYM
ejpam-4352	146	43	n	n	PRON
ejpam-4352	146	44	2	2	NUM
ejpam-4352	146	45	and	and	CCONJ
ejpam-4352	146	46	e(g	e(g	NOUN
ejpam-4352	146	47	)	)	PUNCT
ejpam-4352	147	1	=	=	SYM
ejpam-4352	147	2	e(⟨a⟩)∪e(⟨b⟩)∪e(h	e(⟨a⟩)∪e(⟨b⟩)∪e(h	PROPN
ejpam-4352	147	3	)	)	PUNCT
ejpam-4352	147	4	,	,	PUNCT
ejpam-4352	147	5	where	where	SCONJ
ejpam-4352	147	6	⟨a⟩	⟨a⟩	PROPN
ejpam-4352	147	7	is	be	AUX
ejpam-4352	147	8	the	the	DET
ejpam-4352	147	9	graph	graph	NOUN
ejpam-4352	147	10	induced	induce	VERB
ejpam-4352	147	11	by	by	ADP
ejpam-4352	147	12	a.	a.	NOUN
ejpam-4352	147	13	proof	proof	NOUN
ejpam-4352	147	14	.	.	PUNCT
ejpam-4352	148	1	let	let	VERB
ejpam-4352	148	2	s	s	PRON
ejpam-4352	148	3	be	be	AUX
ejpam-4352	148	4	a	a	DET
ejpam-4352	148	5	complement	complement	NOUN
ejpam-4352	148	6	-	-	PUNCT
ejpam-4352	148	7	super	super	ADJ
ejpam-4352	148	8	dominating	dominating	NOUN
ejpam-4352	148	9	set	set	NOUN
ejpam-4352	148	10	of	of	ADP
ejpam-4352	148	11	g.	g.	PROPN
ejpam-4352	148	12	by	by	ADP
ejpam-4352	148	13	definition	definition	NOUN
ejpam-4352	148	14	,	,	PUNCT
ejpam-4352	148	15	|s|	|s|	PROPN
ejpam-4352	148	16	≥	≥	NOUN
ejpam-4352	148	17	|v	|v	PROPN
ejpam-4352	148	18	(	(	PUNCT
ejpam-4352	148	19	g)\	g)\	PRON
ejpam-4352	148	20	s|	s|	VERB
ejpam-4352	148	21	=	=	PUNCT
ejpam-4352	148	22	n−	n−	NOUN
ejpam-4352	148	23	|s|	|s|	NOUN
ejpam-4352	148	24	.	.	PUNCT
ejpam-4352	149	1	hence	hence	ADV
ejpam-4352	149	2	,	,	PUNCT
ejpam-4352	149	3	|s|	|s|	PROPN
ejpam-4352	149	4	≥	≥	NOUN
ejpam-4352	149	5	n	n	CCONJ
ejpam-4352	149	6	2	2	NUM
ejpam-4352	149	7	,	,	PUNCT
ejpam-4352	149	8	showing	show	VERB
ejpam-4352	149	9	that	that	SCONJ
ejpam-4352	149	10	⌈n2	⌈n2	NOUN
ejpam-4352	149	11	⌉	⌉	X
ejpam-4352	149	12	≤	≤	NUM
ejpam-4352	149	13	γcs(g	γcs(g	NUM
ejpam-4352	149	14	)	)	PUNCT
ejpam-4352	149	15	≤	≤	NOUN
ejpam-4352	149	16	n.	n.	NOUN
ejpam-4352	149	17	for	for	ADP
ejpam-4352	149	18	(	(	PUNCT
ejpam-4352	149	19	i	i	NOUN
ejpam-4352	149	20	)	)	PUNCT
ejpam-4352	149	21	,	,	PUNCT
ejpam-4352	149	22	suppose	suppose	VERB
ejpam-4352	149	23	that	that	SCONJ
ejpam-4352	149	24	γcs(g	γcs(g	X
ejpam-4352	149	25	)	)	PUNCT
ejpam-4352	149	26	=	=	VERB
ejpam-4352	149	27	n.	n.	NOUN
ejpam-4352	149	28	since	since	SCONJ
ejpam-4352	149	29	γcs(g	γcs(g	PROPN
ejpam-4352	149	30	)	)	PUNCT
ejpam-4352	149	31	=	=	SYM
ejpam-4352	149	32	γs(g	γs(g	NUM
ejpam-4352	149	33	)	)	PUNCT
ejpam-4352	149	34	,	,	PUNCT
ejpam-4352	149	35	it	it	PRON
ejpam-4352	149	36	follows	follow	VERB
ejpam-4352	149	37	that	that	SCONJ
ejpam-4352	149	38	g	g	PROPN
ejpam-4352	149	39	=	=	PROPN
ejpam-4352	149	40	kn	kn	PROPN
ejpam-4352	149	41	.	.	PUNCT
ejpam-4352	150	1	hence	hence	ADV
ejpam-4352	150	2	,	,	PUNCT
ejpam-4352	150	3	g	g	PROPN
ejpam-4352	150	4	=	=	SYM
ejpam-4352	150	5	kn	kn	PROPN
ejpam-4352	150	6	.	.	PUNCT
ejpam-4352	151	1	the	the	DET
ejpam-4352	151	2	converse	converse	NOUN
ejpam-4352	151	3	is	be	AUX
ejpam-4352	151	4	clear	clear	ADJ
ejpam-4352	151	5	.	.	PUNCT
ejpam-4352	152	1	to	to	PART
ejpam-4352	152	2	show	show	VERB
ejpam-4352	152	3	(	(	PUNCT
ejpam-4352	152	4	ii	ii	NOUN
ejpam-4352	152	5	)	)	PUNCT
ejpam-4352	152	6	,	,	PUNCT
ejpam-4352	152	7	suppose	suppose	VERB
ejpam-4352	152	8	first	first	ADV
ejpam-4352	152	9	that	that	SCONJ
ejpam-4352	152	10	γcs(g	γcs(g	X
ejpam-4352	152	11	)	)	PUNCT
ejpam-4352	152	12	=	=	SYM
ejpam-4352	153	1	n	n	DET
ejpam-4352	153	2	2	2	NUM
ejpam-4352	153	3	.	.	PUNCT
ejpam-4352	154	1	let	let	VERB
ejpam-4352	154	2	s	s	PRON
ejpam-4352	154	3	be	be	AUX
ejpam-4352	154	4	an	an	DET
ejpam-4352	154	5	γcs	γcs	NOUN
ejpam-4352	154	6	-	-	PUNCT
ejpam-4352	154	7	set	set	NOUN
ejpam-4352	154	8	of	of	ADP
ejpam-4352	154	9	g.	g.	PROPN
ejpam-4352	154	10	then	then	ADV
ejpam-4352	154	11	|s|	|s|	PROPN
ejpam-4352	154	12	=	=	SYM
ejpam-4352	154	13	n	n	PRON
ejpam-4352	154	14	2	2	NUM
ejpam-4352	154	15	.	.	PUNCT
ejpam-4352	155	1	let	let	VERB
ejpam-4352	155	2	v	v	NUM
ejpam-4352	155	3	∈	∈	PROPN
ejpam-4352	155	4	v	v	NOUN
ejpam-4352	155	5	(	(	PUNCT
ejpam-4352	155	6	g	g	NOUN
ejpam-4352	155	7	)	)	PUNCT
ejpam-4352	155	8	\	\	PUNCT
ejpam-4352	156	1	s.	s.	PROPN
ejpam-4352	156	2	then	then	ADV
ejpam-4352	156	3	there	there	PRON
ejpam-4352	156	4	exists	exist	VERB
ejpam-4352	156	5	xv	xv	PROPN
ejpam-4352	156	6	∈	∈	PROPN
ejpam-4352	156	7	s	s	PART
ejpam-4352	156	8	\ng(v	\ng(v	NOUN
ejpam-4352	156	9	)	)	PUNCT
ejpam-4352	156	10	such	such	ADJ
ejpam-4352	156	11	that	that	SCONJ
ejpam-4352	156	12	[	[	X
ejpam-4352	156	13	v	v	X
ejpam-4352	156	14	(	(	PUNCT
ejpam-4352	156	15	g	g	NOUN
ejpam-4352	156	16	)	)	PUNCT
ejpam-4352	156	17	\ng(xv	\ng(xv	NUM
ejpam-4352	156	18	)	)	PUNCT
ejpam-4352	156	19	]	]	PUNCT
ejpam-4352	156	20	∩	∩	NOUN
ejpam-4352	156	21	(	(	PUNCT
ejpam-4352	156	22	v	v	NOUN
ejpam-4352	156	23	(	(	PUNCT
ejpam-4352	156	24	g	g	NOUN
ejpam-4352	156	25	)	)	PUNCT
ejpam-4352	156	26	\	\	PROPN
ejpam-4352	157	1	s	s	X
ejpam-4352	157	2	)	)	PUNCT
ejpam-4352	157	3	=	=	SYM
ejpam-4352	157	4	{	{	PUNCT
ejpam-4352	157	5	v	v	NOUN
ejpam-4352	157	6	}	}	PUNCT
ejpam-4352	157	7	.	.	PUNCT
ejpam-4352	158	1	since	since	SCONJ
ejpam-4352	158	2	|v	|v	PROPN
ejpam-4352	158	3	(	(	PUNCT
ejpam-4352	158	4	g	g	NOUN
ejpam-4352	158	5	)	)	PUNCT
ejpam-4352	158	6	\	\	NOUN
ejpam-4352	158	7	s|	s|	NOUN
ejpam-4352	158	8	=	=	SYM
ejpam-4352	158	9	n	n	CCONJ
ejpam-4352	158	10	2	2	NUM
ejpam-4352	158	11	and	and	CCONJ
ejpam-4352	158	12	s	s	VERB
ejpam-4352	158	13	is	be	AUX
ejpam-4352	158	14	a	a	DET
ejpam-4352	158	15	complement	complement	NOUN
ejpam-4352	158	16	-	-	PUNCT
ejpam-4352	158	17	super	super	ADJ
ejpam-4352	158	18	dominating	dominating	NOUN
ejpam-4352	158	19	set	set	NOUN
ejpam-4352	158	20	of	of	ADP
ejpam-4352	158	21	g	g	PROPN
ejpam-4352	158	22	,	,	PUNCT
ejpam-4352	158	23	vy	vy	PROPN
ejpam-4352	158	24	∈	∈	PROPN
ejpam-4352	158	25	e(g	e(g	PROPN
ejpam-4352	158	26	)	)	PUNCT
ejpam-4352	158	27	for	for	ADP
ejpam-4352	158	28	all	all	DET
ejpam-4352	158	29	y	y	PROPN
ejpam-4352	158	30	∈	∈	PROPN
ejpam-4352	158	31	s	s	PART
ejpam-4352	158	32	\	\	X
ejpam-4352	158	33	{	{	PUNCT
ejpam-4352	158	34	xv	xv	PROPN
ejpam-4352	158	35	}	}	PUNCT
ejpam-4352	158	36	.	.	PUNCT
ejpam-4352	159	1	thus	thus	ADV
ejpam-4352	159	2	,	,	PUNCT
ejpam-4352	159	3	ng(v	ng(v	NOUN
ejpam-4352	159	4	)	)	PUNCT
ejpam-4352	159	5	∩	∩	NOUN
ejpam-4352	159	6	s	s	PART
ejpam-4352	159	7	=	=	SYM
ejpam-4352	159	8	s	s	PART
ejpam-4352	159	9	\	\	X
ejpam-4352	159	10	{	{	PUNCT
ejpam-4352	159	11	xv	xv	PROPN
ejpam-4352	159	12	}	}	PUNCT
ejpam-4352	159	13	.	.	PUNCT
ejpam-4352	160	1	let	let	VERB
ejpam-4352	160	2	u	u	NOUN
ejpam-4352	160	3	,	,	PUNCT
ejpam-4352	160	4	v	v	PROPN
ejpam-4352	160	5	∈	∈	PROPN
ejpam-4352	160	6	v	v	NOUN
ejpam-4352	160	7	(	(	PUNCT
ejpam-4352	160	8	g	g	NOUN
ejpam-4352	160	9	)	)	PUNCT
ejpam-4352	160	10	\	\	PROPN
ejpam-4352	161	1	s	s	PART
ejpam-4352	161	2	with	with	ADP
ejpam-4352	161	3	u	u	NOUN
ejpam-4352	161	4	̸=	̸=	PROPN
ejpam-4352	161	5	v.	v.	ADP
ejpam-4352	161	6	since	since	SCONJ
ejpam-4352	161	7	ng(xv	ng(xv	ADJ
ejpam-4352	161	8	)	)	PUNCT
ejpam-4352	161	9	∩	∩	NOUN
ejpam-4352	161	10	(	(	PUNCT
ejpam-4352	161	11	v	v	NOUN
ejpam-4352	161	12	(	(	PUNCT
ejpam-4352	161	13	g	g	NOUN
ejpam-4352	161	14	)	)	PUNCT
ejpam-4352	161	15	\	\	PROPN
ejpam-4352	162	1	s	s	X
ejpam-4352	162	2	)	)	PUNCT
ejpam-4352	162	3	=	=	SYM
ejpam-4352	162	4	v	v	NOUN
ejpam-4352	162	5	(	(	PUNCT
ejpam-4352	162	6	g	g	NOUN
ejpam-4352	162	7	)	)	PUNCT
ejpam-4352	162	8	\	\	PUNCT
ejpam-4352	163	1	(	(	PUNCT
ejpam-4352	163	2	s	s	NOUN
ejpam-4352	163	3	∪	∪	X
ejpam-4352	163	4	{	{	PUNCT
ejpam-4352	163	5	v	v	NOUN
ejpam-4352	163	6	}	}	PUNCT
ejpam-4352	163	7	)	)	PUNCT
ejpam-4352	163	8	,	,	PUNCT
ejpam-4352	163	9	it	it	PRON
ejpam-4352	163	10	follows	follow	VERB
ejpam-4352	163	11	that	that	SCONJ
ejpam-4352	163	12	u	u	PROPN
ejpam-4352	163	13	∈	∈	PROPN
ejpam-4352	163	14	ng(xv	ng(xv	PROPN
ejpam-4352	163	15	)	)	PUNCT
ejpam-4352	163	16	.	.	PUNCT
ejpam-4352	164	1	since	since	SCONJ
ejpam-4352	164	2	u	u	PROPN
ejpam-4352	164	3	/∈	/∈	PROPN
ejpam-4352	164	4	ng(xu	ng(xu	PROPN
ejpam-4352	164	5	)	)	PUNCT
ejpam-4352	164	6	,	,	PUNCT
ejpam-4352	164	7	xu	xu	PROPN
ejpam-4352	164	8	̸=	̸=	PROPN
ejpam-4352	164	9	xv	xv	PROPN
ejpam-4352	164	10	.	.	PUNCT
ejpam-4352	165	1	hence	hence	ADV
ejpam-4352	165	2	,	,	PUNCT
ejpam-4352	165	3	s	s	PART
ejpam-4352	165	4	=	=	PUNCT
ejpam-4352	165	5	{	{	PUNCT
ejpam-4352	165	6	xv	xv	X
ejpam-4352	165	7	:	:	PUNCT
ejpam-4352	165	8	v	v	NUM
ejpam-4352	165	9	∈	∈	PROPN
ejpam-4352	165	10	v	v	NOUN
ejpam-4352	165	11	(	(	PUNCT
ejpam-4352	165	12	g	g	NOUN
ejpam-4352	165	13	)	)	PUNCT
ejpam-4352	165	14	\	\	PUNCT
ejpam-4352	166	1	s	s	X
ejpam-4352	166	2	}	}	PUNCT
ejpam-4352	166	3	.	.	PUNCT
ejpam-4352	167	1	let	let	VERB
ejpam-4352	167	2	a	a	DET
ejpam-4352	167	3	=	=	SYM
ejpam-4352	167	4	s	s	X
ejpam-4352	167	5	and	and	CCONJ
ejpam-4352	167	6	b	b	X
ejpam-4352	167	7	=	=	SYM
ejpam-4352	167	8	v	v	X
ejpam-4352	167	9	(	(	PUNCT
ejpam-4352	167	10	g	g	NOUN
ejpam-4352	167	11	)	)	PUNCT
ejpam-4352	167	12	\	\	PUNCT
ejpam-4352	168	1	s.	s.	PROPN
ejpam-4352	168	2	consider	consider	VERB
ejpam-4352	168	3	the	the	DET
ejpam-4352	168	4	bipartite	bipartite	PROPN
ejpam-4352	168	5	graph	graph	NOUN
ejpam-4352	168	6	h	h	NOUN
ejpam-4352	168	7	with	with	ADP
ejpam-4352	168	8	partite	partite	ADJ
ejpam-4352	168	9	sets	set	NOUN
ejpam-4352	168	10	a	a	PRON
ejpam-4352	168	11	and	and	CCONJ
ejpam-4352	168	12	b.	b.	PROPN
ejpam-4352	168	13	then	then	ADV
ejpam-4352	168	14	|a|	|a|	PROPN
ejpam-4352	168	15	=	=	SYM
ejpam-4352	168	16	|b|	|b|	PROPN
ejpam-4352	168	17	=	=	PUNCT
ejpam-4352	168	18	n	n	PRON
ejpam-4352	168	19	2	2	NUM
ejpam-4352	168	20	.	.	PUNCT
ejpam-4352	169	1	since	since	SCONJ
ejpam-4352	169	2	ng(v	ng(v	NOUN
ejpam-4352	169	3	)	)	PUNCT
ejpam-4352	169	4	∩	∩	NOUN
ejpam-4352	169	5	s	s	PART
ejpam-4352	169	6	=	=	SYM
ejpam-4352	169	7	s	s	PART
ejpam-4352	169	8	\	\	X
ejpam-4352	169	9	{	{	PUNCT
ejpam-4352	169	10	xv	xv	PROPN
ejpam-4352	169	11	}	}	PUNCT
ejpam-4352	169	12	for	for	ADP
ejpam-4352	169	13	each	each	DET
ejpam-4352	169	14	v	v	ADP
ejpam-4352	169	15	∈	∈	PROPN
ejpam-4352	169	16	b	b	NOUN
ejpam-4352	169	17	,	,	PUNCT
ejpam-4352	169	18	it	it	PRON
ejpam-4352	169	19	follows	follow	VERB
ejpam-4352	169	20	that	that	SCONJ
ejpam-4352	169	21	degh(v	degh(v	NOUN
ejpam-4352	169	22	)	)	PUNCT
ejpam-4352	169	23	=	=	SYM
ejpam-4352	170	1	n	n	PRON
ejpam-4352	170	2	2	2	NUM
ejpam-4352	170	3	−	−	NOUN
ejpam-4352	170	4	1	1	NUM
ejpam-4352	170	5	for	for	ADP
ejpam-4352	170	6	each	each	DET
ejpam-4352	170	7	v	v	PROPN
ejpam-4352	170	8	∈	∈	PROPN
ejpam-4352	170	9	b.	b.	NOUN
ejpam-4352	170	10	also	also	ADV
ejpam-4352	170	11	,	,	PUNCT
ejpam-4352	170	12	since	since	SCONJ
ejpam-4352	170	13	[	[	X
ejpam-4352	170	14	v	v	X
ejpam-4352	170	15	(	(	PUNCT
ejpam-4352	170	16	g	g	NOUN
ejpam-4352	170	17	)	)	PUNCT
ejpam-4352	170	18	\ng(xv)]∩	\ng(xv)]∩	NOUN
ejpam-4352	170	19	(	(	PUNCT
ejpam-4352	170	20	v	v	NOUN
ejpam-4352	170	21	(	(	PUNCT
ejpam-4352	170	22	g	g	NOUN
ejpam-4352	170	23	)	)	PUNCT
ejpam-4352	170	24	\s	\s	NOUN
ejpam-4352	170	25	)	)	PUNCT
ejpam-4352	170	26	=	=	PRON
ejpam-4352	170	27	{	{	PUNCT
ejpam-4352	170	28	v	v	NOUN
ejpam-4352	170	29	}	}	PUNCT
ejpam-4352	170	30	for	for	ADP
ejpam-4352	170	31	each	each	DET
ejpam-4352	170	32	v	v	ADP
ejpam-4352	170	33	∈	∈	PROPN
ejpam-4352	170	34	b	b	PROPN
ejpam-4352	170	35	,	,	PUNCT
ejpam-4352	170	36	degh(xv	degh(xv	PROPN
ejpam-4352	170	37	)	)	PUNCT
ejpam-4352	170	38	=	=	SYM
ejpam-4352	170	39	n	n	PRON
ejpam-4352	170	40	2	2	NUM
ejpam-4352	170	41	−	−	NOUN
ejpam-4352	170	42	1	1	NUM
ejpam-4352	170	43	for	for	ADP
ejpam-4352	170	44	each	each	DET
ejpam-4352	170	45	xv	xv	PROPN
ejpam-4352	170	46	∈	∈	PROPN
ejpam-4352	170	47	a.	a.	NOUN
ejpam-4352	170	48	thus	thus	ADV
ejpam-4352	170	49	,	,	PUNCT
ejpam-4352	170	50	h	h	PROPN
ejpam-4352	170	51	is	be	AUX
ejpam-4352	170	52	an	an	DET
ejpam-4352	170	53	(	(	PUNCT
ejpam-4352	170	54	n2	n2	ADJ
ejpam-4352	170	55	−1)-regular	−1)-regular	ADJ
ejpam-4352	170	56	graph	graph	NOUN
ejpam-4352	170	57	.	.	PUNCT
ejpam-4352	171	1	moreover	moreover	ADV
ejpam-4352	171	2	,	,	PUNCT
ejpam-4352	171	3	e(g	e(g	PROPN
ejpam-4352	171	4	)	)	PUNCT
ejpam-4352	171	5	=	=	SYM
ejpam-4352	171	6	e(⟨a⟩)∪e(⟨b⟩)∪e(h	e(⟨a⟩)∪e(⟨b⟩)∪e(h	PROPN
ejpam-4352	171	7	)	)	PUNCT
ejpam-4352	171	8	.	.	PUNCT
ejpam-4352	172	1	for	for	ADP
ejpam-4352	172	2	the	the	DET
ejpam-4352	172	3	converse	converse	NOUN
ejpam-4352	172	4	,	,	PUNCT
ejpam-4352	172	5	suppose	suppose	VERB
ejpam-4352	172	6	that	that	SCONJ
ejpam-4352	172	7	g	g	PROPN
ejpam-4352	172	8	has	have	VERB
ejpam-4352	172	9	the	the	DET
ejpam-4352	172	10	given	give	VERB
ejpam-4352	172	11	property	property	NOUN
ejpam-4352	172	12	.	.	PUNCT
ejpam-4352	173	1	let	let	VERB
ejpam-4352	173	2	v	v	NUM
ejpam-4352	173	3	∈	∈	PROPN
ejpam-4352	173	4	b	b	NOUN
ejpam-4352	173	5	=	=	SYM
ejpam-4352	173	6	v	v	PROPN
ejpam-4352	173	7	(	(	PUNCT
ejpam-4352	173	8	g	g	NOUN
ejpam-4352	173	9	)	)	PUNCT
ejpam-4352	173	10	\	\	NOUN
ejpam-4352	173	11	a.	a.	NOUN
ejpam-4352	173	12	then	then	ADV
ejpam-4352	173	13	by	by	ADP
ejpam-4352	173	14	assumption	assumption	NOUN
ejpam-4352	173	15	,	,	PUNCT
ejpam-4352	173	16	degh(v	degh(v	NOUN
ejpam-4352	173	17	)	)	PUNCT
ejpam-4352	173	18	=	=	SYM
ejpam-4352	174	1	n	n	PRON
ejpam-4352	174	2	2	2	NUM
ejpam-4352	174	3	−	−	NUM
ejpam-4352	174	4	1	1	NUM
ejpam-4352	174	5	.	.	PUNCT
ejpam-4352	175	1	this	this	PRON
ejpam-4352	175	2	implies	imply	VERB
ejpam-4352	175	3	that	that	SCONJ
ejpam-4352	175	4	there	there	PRON
ejpam-4352	175	5	exists	exist	VERB
ejpam-4352	175	6	w	w	PROPN
ejpam-4352	175	7	∈	∈	PROPN
ejpam-4352	175	8	a	a	DET
ejpam-4352	175	9	such	such	ADJ
ejpam-4352	175	10	[	[	X
ejpam-4352	175	11	v	v	X
ejpam-4352	175	12	(	(	PUNCT
ejpam-4352	175	13	h	h	NOUN
ejpam-4352	175	14	)	)	PUNCT
ejpam-4352	175	15	\	\	NOUN
ejpam-4352	175	16	nh(w	nh(w	PUNCT
ejpam-4352	175	17	)	)	PUNCT
ejpam-4352	175	18	]	]	PUNCT
ejpam-4352	175	19	∩	∩	PROPN
ejpam-4352	175	20	b	b	X
ejpam-4352	175	21	=	=	SYM
ejpam-4352	175	22	{	{	PUNCT
ejpam-4352	175	23	v	v	NOUN
ejpam-4352	175	24	}	}	PUNCT
ejpam-4352	175	25	.	.	PUNCT
ejpam-4352	176	1	since	since	SCONJ
ejpam-4352	176	2	e(g	e(g	PROPN
ejpam-4352	176	3	)	)	PUNCT
ejpam-4352	176	4	=	=	SYM
ejpam-4352	176	5	e(⟨a⟩	e(⟨a⟩	NOUN
ejpam-4352	176	6	)	)	PUNCT
ejpam-4352	176	7	∪	∪	ADP
ejpam-4352	176	8	e(⟨b⟩	e(⟨b⟩	NOUN
ejpam-4352	176	9	)	)	PUNCT
ejpam-4352	176	10	∪	∪	ADP
ejpam-4352	176	11	e(h	e(h	PROPN
ejpam-4352	176	12	)	)	PUNCT
ejpam-4352	176	13	,	,	PUNCT
ejpam-4352	176	14	it	it	PRON
ejpam-4352	176	15	follows	follow	VERB
ejpam-4352	176	16	that	that	SCONJ
ejpam-4352	176	17	s.	s.	PROPN
ejpam-4352	176	18	canoy	canoy	PROPN
ejpam-4352	176	19	,	,	PUNCT
ejpam-4352	176	20	jr	jr	PROPN
ejpam-4352	176	21	.	.	PROPN
ejpam-4352	176	22	,	,	PUNCT
ejpam-4352	176	23	g.	g.	PROPN
ejpam-4352	176	24	salasalan	salasalan	PROPN
ejpam-4352	176	25	/	/	SYM
ejpam-4352	176	26	eur	eur	PROPN
ejpam-4352	176	27	.	.	PUNCT
ejpam-4352	177	1	j.	j.	PROPN
ejpam-4352	177	2	pure	pure	PROPN
ejpam-4352	177	3	appl	appl	PROPN
ejpam-4352	177	4	.	.	PROPN
ejpam-4352	177	5	math	math	PROPN
ejpam-4352	177	6	,	,	PUNCT
ejpam-4352	177	7	15	15	NUM
ejpam-4352	177	8	(	(	PUNCT
ejpam-4352	177	9	2	2	NUM
ejpam-4352	177	10	)	)	PUNCT
ejpam-4352	177	11	(	(	PUNCT
ejpam-4352	177	12	2022	2022	NUM
ejpam-4352	177	13	)	)	PUNCT
ejpam-4352	177	14	,	,	PUNCT
ejpam-4352	177	15	342	342	NUM
ejpam-4352	177	16	-	-	SYM
ejpam-4352	177	17	353	353	NUM
ejpam-4352	177	18	346	346	NUM
ejpam-4352	177	19	[	[	X
ejpam-4352	177	20	v	v	X
ejpam-4352	177	21	(	(	PUNCT
ejpam-4352	177	22	g	g	NOUN
ejpam-4352	177	23	)	)	PUNCT
ejpam-4352	177	24	\ng(w	\ng(w	PUNCT
ejpam-4352	177	25	)	)	PUNCT
ejpam-4352	177	26	]	]	PUNCT
ejpam-4352	178	1	∩	∩	PROPN
ejpam-4352	178	2	b	b	X
ejpam-4352	178	3	=	=	SYM
ejpam-4352	179	1	[	[	X
ejpam-4352	179	2	v	v	X
ejpam-4352	179	3	(	(	PUNCT
ejpam-4352	179	4	h	h	NOUN
ejpam-4352	179	5	)	)	PUNCT
ejpam-4352	179	6	\nh(w	\nh(w	PROPN
ejpam-4352	179	7	)	)	PUNCT
ejpam-4352	179	8	]	]	PUNCT
ejpam-4352	180	1	∩	∩	PROPN
ejpam-4352	180	2	b	b	X
ejpam-4352	180	3	=	=	SYM
ejpam-4352	180	4	{	{	PUNCT
ejpam-4352	180	5	v	v	NOUN
ejpam-4352	180	6	}	}	PUNCT
ejpam-4352	180	7	.	.	PUNCT
ejpam-4352	181	1	this	this	PRON
ejpam-4352	181	2	shows	show	VERB
ejpam-4352	181	3	that	that	SCONJ
ejpam-4352	181	4	a	a	PRON
ejpam-4352	181	5	is	be	AUX
ejpam-4352	181	6	a	a	DET
ejpam-4352	181	7	complementsuper	complementsuper	NOUN
ejpam-4352	181	8	dominating	dominating	NOUN
ejpam-4352	181	9	set	set	NOUN
ejpam-4352	181	10	of	of	ADP
ejpam-4352	181	11	g.	g.	PROPN
ejpam-4352	181	12	therefore	therefore	ADV
ejpam-4352	181	13	n	n	ADV
ejpam-4352	181	14	2	2	NUM
ejpam-4352	181	15	≤	≤	NUM
ejpam-4352	181	16	γcs(g	γcs(g	NUM
ejpam-4352	181	17	)	)	PUNCT
ejpam-4352	181	18	≤	≤	NUM
ejpam-4352	181	19	|a|	|a|	PROPN
ejpam-4352	181	20	=	=	PROPN
ejpam-4352	181	21	n	n	PRON
ejpam-4352	181	22	2	2	NUM
ejpam-4352	181	23	,	,	PUNCT
ejpam-4352	181	24	i.e.	i.e.	X
ejpam-4352	181	25	,	,	PUNCT
ejpam-4352	181	26	γcs(g	γcs(g	X
ejpam-4352	181	27	)	)	PUNCT
ejpam-4352	181	28	=	=	SYM
ejpam-4352	181	29	n	n	PRON
ejpam-4352	181	30	2	2	NUM
ejpam-4352	181	31	.	.	PUNCT
ejpam-4352	182	1	theorem	theorem	NOUN
ejpam-4352	182	2	5	5	NUM
ejpam-4352	182	3	.	.	PUNCT
ejpam-4352	183	1	let	let	VERB
ejpam-4352	183	2	g	g	PRON
ejpam-4352	183	3	be	be	AUX
ejpam-4352	183	4	a	a	DET
ejpam-4352	183	5	graph	graph	NOUN
ejpam-4352	183	6	of	of	ADP
ejpam-4352	183	7	order	order	NOUN
ejpam-4352	183	8	n.	n.	NOUN
ejpam-4352	184	1	then	then	ADV
ejpam-4352	184	2	(	(	PUNCT
ejpam-4352	184	3	i	i	NOUN
ejpam-4352	184	4	)	)	PUNCT
ejpam-4352	184	5	γcs(g	γcs(g	X
ejpam-4352	184	6	)	)	PUNCT
ejpam-4352	184	7	=	=	SYM
ejpam-4352	184	8	1	1	NUM
ejpam-4352	184	9	if	if	SCONJ
ejpam-4352	184	10	and	and	CCONJ
ejpam-4352	184	11	only	only	ADV
ejpam-4352	184	12	if	if	SCONJ
ejpam-4352	184	13	g	g	PROPN
ejpam-4352	184	14	=	=	SYM
ejpam-4352	184	15	k1	k1	PROPN
ejpam-4352	184	16	or	or	CCONJ
ejpam-4352	184	17	g	g	NOUN
ejpam-4352	184	18	=	=	SYM
ejpam-4352	184	19	k2	k2	PROPN
ejpam-4352	184	20	.	.	PUNCT
ejpam-4352	185	1	(	(	PUNCT
ejpam-4352	185	2	ii	ii	NOUN
ejpam-4352	185	3	)	)	PUNCT
ejpam-4352	185	4	γcs(g	γcs(g	PROPN
ejpam-4352	185	5	)	)	PUNCT
ejpam-4352	185	6	=	=	SYM
ejpam-4352	185	7	2	2	NUM
ejpam-4352	185	8	if	if	SCONJ
ejpam-4352	185	9	and	and	CCONJ
ejpam-4352	185	10	only	only	ADV
ejpam-4352	185	11	if	if	SCONJ
ejpam-4352	185	12	g	g	PROPN
ejpam-4352	185	13	∈	∈	PROPN
ejpam-4352	185	14	{	{	PUNCT
ejpam-4352	185	15	k2	k2	PROPN
ejpam-4352	185	16	,	,	PUNCT
ejpam-4352	185	17	p3,k2	p3,k2	PROPN
ejpam-4352	185	18	∪k1,k3	∪k1,k3	PROPN
ejpam-4352	185	19	,	,	PUNCT
ejpam-4352	185	20	p4	p4	ADJ
ejpam-4352	185	21	,	,	PUNCT
ejpam-4352	185	22	c4,k2	c4,k2	PROPN
ejpam-4352	185	23	∪k2	∪k2	X
ejpam-4352	185	24	}	}	PUNCT
ejpam-4352	185	25	.	.	PUNCT
ejpam-4352	186	1	proof	proof	NOUN
ejpam-4352	186	2	.	.	PUNCT
ejpam-4352	187	1	(	(	PUNCT
ejpam-4352	187	2	i	i	NOUN
ejpam-4352	187	3	)	)	PUNCT
ejpam-4352	187	4	suppose	suppose	VERB
ejpam-4352	187	5	γcs(g	γcs(g	X
ejpam-4352	187	6	)	)	PUNCT
ejpam-4352	187	7	=	=	SYM
ejpam-4352	187	8	1	1	NUM
ejpam-4352	187	9	and	and	CCONJ
ejpam-4352	187	10	let	let	VERB
ejpam-4352	187	11	s	s	PRON
ejpam-4352	187	12	=	=	NOUN
ejpam-4352	187	13	{	{	PUNCT
ejpam-4352	187	14	v	v	NOUN
ejpam-4352	187	15	}	}	PUNCT
ejpam-4352	187	16	be	be	AUX
ejpam-4352	187	17	a	a	DET
ejpam-4352	187	18	γcs	γcs	NOUN
ejpam-4352	187	19	-	-	PUNCT
ejpam-4352	187	20	set	set	NOUN
ejpam-4352	187	21	of	of	ADP
ejpam-4352	187	22	g.	g.	PROPN
ejpam-4352	187	23	by	by	ADP
ejpam-4352	187	24	theorem	theorem	NOUN
ejpam-4352	187	25	4	4	NUM
ejpam-4352	187	26	,	,	PUNCT
ejpam-4352	187	27	1	1	NUM
ejpam-4352	187	28	≤	≤	NUM
ejpam-4352	187	29	n	n	CCONJ
ejpam-4352	187	30	≤	≤	NOUN
ejpam-4352	187	31	2	2	NUM
ejpam-4352	187	32	.	.	PUNCT
ejpam-4352	188	1	if	if	SCONJ
ejpam-4352	188	2	n	n	NOUN
ejpam-4352	188	3	=	=	SYM
ejpam-4352	188	4	1	1	NUM
ejpam-4352	188	5	,	,	PUNCT
ejpam-4352	188	6	then	then	ADV
ejpam-4352	188	7	g	g	PROPN
ejpam-4352	188	8	=	=	PROPN
ejpam-4352	188	9	k1	k1	PROPN
ejpam-4352	188	10	.	.	PUNCT
ejpam-4352	188	11	suppose	suppose	VERB
ejpam-4352	188	12	n	n	PROPN
ejpam-4352	188	13	=	=	SYM
ejpam-4352	188	14	2	2	NUM
ejpam-4352	188	15	and	and	CCONJ
ejpam-4352	188	16	let	let	VERB
ejpam-4352	188	17	w	w	NOUN
ejpam-4352	188	18	∈	∈	PROPN
ejpam-4352	188	19	v	v	ADP
ejpam-4352	188	20	g	g	NOUN
ejpam-4352	188	21	)	)	PUNCT
ejpam-4352	188	22	\	\	NOUN
ejpam-4352	188	23	{	{	PUNCT
ejpam-4352	188	24	v	v	NOUN
ejpam-4352	188	25	}	}	PUNCT
ejpam-4352	188	26	.	.	PUNCT
ejpam-4352	189	1	since	since	SCONJ
ejpam-4352	189	2	s	s	PROPN
ejpam-4352	189	3	is	be	AUX
ejpam-4352	189	4	a	a	DET
ejpam-4352	189	5	complement	complement	NOUN
ejpam-4352	189	6	-	-	PUNCT
ejpam-4352	189	7	super	super	ADJ
ejpam-4352	189	8	dominating	dominating	NOUN
ejpam-4352	189	9	set	set	NOUN
ejpam-4352	189	10	of	of	ADP
ejpam-4352	189	11	g	g	PROPN
ejpam-4352	189	12	,	,	PUNCT
ejpam-4352	189	13	vw	vw	PROPN
ejpam-4352	189	14	/∈	/∈	PUNCT
ejpam-4352	189	15	e(g	e(g	PROPN
ejpam-4352	189	16	)	)	PUNCT
ejpam-4352	189	17	.	.	PUNCT
ejpam-4352	190	1	it	it	PRON
ejpam-4352	190	2	follows	follow	VERB
ejpam-4352	190	3	that	that	SCONJ
ejpam-4352	190	4	g	g	PROPN
ejpam-4352	190	5	=	=	SYM
ejpam-4352	190	6	k2	k2	PROPN
ejpam-4352	190	7	.	.	PUNCT
ejpam-4352	191	1	the	the	DET
ejpam-4352	191	2	converse	converse	NOUN
ejpam-4352	191	3	is	be	AUX
ejpam-4352	191	4	clear	clear	ADJ
ejpam-4352	191	5	.	.	PUNCT
ejpam-4352	192	1	(	(	PUNCT
ejpam-4352	192	2	ii	ii	NOUN
ejpam-4352	192	3	)	)	PUNCT
ejpam-4352	192	4	suppose	suppose	VERB
ejpam-4352	192	5	γcs(g	γcs(g	X
ejpam-4352	192	6	)	)	PUNCT
ejpam-4352	192	7	=	=	SYM
ejpam-4352	192	8	2	2	X
ejpam-4352	192	9	.	.	PUNCT
ejpam-4352	192	10	then	then	ADV
ejpam-4352	192	11	2	2	NUM
ejpam-4352	192	12	≤	≤	NOUN
ejpam-4352	192	13	n	n	PRON
ejpam-4352	192	14	≤	≤	NOUN
ejpam-4352	192	15	4	4	NUM
ejpam-4352	192	16	by	by	ADP
ejpam-4352	192	17	theorem	theorem	NOUN
ejpam-4352	192	18	4	4	NUM
ejpam-4352	192	19	.	.	PUNCT
ejpam-4352	193	1	let	let	VERB
ejpam-4352	193	2	s	s	VERB
ejpam-4352	193	3	=	=	X
ejpam-4352	193	4	{	{	PUNCT
ejpam-4352	193	5	a	a	PRON
ejpam-4352	193	6	,	,	PUNCT
ejpam-4352	193	7	b	b	AUX
ejpam-4352	193	8	}	}	PUNCT
ejpam-4352	193	9	be	be	AUX
ejpam-4352	193	10	a	a	DET
ejpam-4352	193	11	γcs	γcs	NOUN
ejpam-4352	193	12	-	-	PUNCT
ejpam-4352	193	13	set	set	NOUN
ejpam-4352	193	14	of	of	ADP
ejpam-4352	193	15	g.	g.	PROPN
ejpam-4352	193	16	if	if	SCONJ
ejpam-4352	193	17	n	n	PROPN
ejpam-4352	193	18	=	=	SYM
ejpam-4352	193	19	2	2	NUM
ejpam-4352	193	20	,	,	PUNCT
ejpam-4352	193	21	then	then	ADV
ejpam-4352	193	22	g	g	PROPN
ejpam-4352	193	23	=	=	PROPN
ejpam-4352	193	24	k2	k2	PROPN
ejpam-4352	193	25	since	since	SCONJ
ejpam-4352	193	26	γcs(k2	γcs(k2	NOUN
ejpam-4352	193	27	)	)	PUNCT
ejpam-4352	193	28	=	=	SYM
ejpam-4352	194	1	1	1	X
ejpam-4352	194	2	.	.	PUNCT
ejpam-4352	194	3	suppose	suppose	VERB
ejpam-4352	194	4	n	n	PROPN
ejpam-4352	194	5	=	=	SYM
ejpam-4352	194	6	3	3	NUM
ejpam-4352	194	7	and	and	CCONJ
ejpam-4352	194	8	let	let	VERB
ejpam-4352	194	9	c	c	NOUN
ejpam-4352	194	10	∈	∈	PROPN
ejpam-4352	194	11	v	v	NOUN
ejpam-4352	194	12	(	(	PUNCT
ejpam-4352	194	13	g)\s	g)\s	NOUN
ejpam-4352	194	14	.	.	PUNCT
ejpam-4352	195	1	we	we	PRON
ejpam-4352	195	2	may	may	AUX
ejpam-4352	195	3	assume	assume	VERB
ejpam-4352	195	4	that	that	SCONJ
ejpam-4352	196	1	[	[	X
ejpam-4352	196	2	v	v	X
ejpam-4352	196	3	(	(	PUNCT
ejpam-4352	196	4	g)\ng(a)]∩[v	g)\ng(a)]∩[v	NOUN
ejpam-4352	196	5	(	(	PUNCT
ejpam-4352	196	6	g)\s	g)\s	NOUN
ejpam-4352	196	7	]	]	X
ejpam-4352	196	8	=	=	PUNCT
ejpam-4352	196	9	{	{	PUNCT
ejpam-4352	196	10	c	c	NOUN
ejpam-4352	196	11	}	}	PUNCT
ejpam-4352	196	12	.	.	PUNCT
ejpam-4352	197	1	if	if	SCONJ
ejpam-4352	197	2	ab	ab	PROPN
ejpam-4352	197	3	∈	∈	PROPN
ejpam-4352	197	4	e(g	e(g	PROPN
ejpam-4352	197	5	)	)	PUNCT
ejpam-4352	197	6	,	,	PUNCT
ejpam-4352	197	7	then	then	ADV
ejpam-4352	197	8	g	g	PROPN
ejpam-4352	197	9	=	=	PROPN
ejpam-4352	197	10	p3	p3	PROPN
ejpam-4352	197	11	or	or	CCONJ
ejpam-4352	197	12	g	g	NOUN
ejpam-4352	197	13	=	=	SYM
ejpam-4352	197	14	k2∪k1	k2∪k1	PROPN
ejpam-4352	197	15	.	.	PUNCT
ejpam-4352	198	1	if	if	SCONJ
ejpam-4352	198	2	ab	ab	PROPN
ejpam-4352	198	3	/∈	/∈	PUNCT
ejpam-4352	198	4	e(g	e(g	PROPN
ejpam-4352	198	5	)	)	PUNCT
ejpam-4352	198	6	,	,	PUNCT
ejpam-4352	198	7	then	then	ADV
ejpam-4352	198	8	g	g	PROPN
ejpam-4352	198	9	=	=	PUNCT
ejpam-4352	198	10	k3	k3	X
ejpam-4352	198	11	or	or	CCONJ
ejpam-4352	198	12	g	g	NOUN
ejpam-4352	198	13	=	=	PROPN
ejpam-4352	198	14	k2	k2	PROPN
ejpam-4352	198	15	∪k1	∪k1	PROPN
ejpam-4352	198	16	.	.	PUNCT
ejpam-4352	199	1	finally	finally	ADV
ejpam-4352	199	2	,	,	PUNCT
ejpam-4352	199	3	let	let	VERB
ejpam-4352	199	4	n	n	X
ejpam-4352	199	5	=	=	SYM
ejpam-4352	199	6	4	4	NUM
ejpam-4352	199	7	and	and	CCONJ
ejpam-4352	199	8	let	let	VERB
ejpam-4352	199	9	v	v	NOUN
ejpam-4352	199	10	(	(	PUNCT
ejpam-4352	199	11	g	g	NOUN
ejpam-4352	199	12	)	)	PUNCT
ejpam-4352	199	13	=	=	NOUN
ejpam-4352	199	14	{	{	PUNCT
ejpam-4352	199	15	a	a	PRON
ejpam-4352	199	16	,	,	PUNCT
ejpam-4352	199	17	b	b	NOUN
ejpam-4352	199	18	,	,	PUNCT
ejpam-4352	199	19	c	c	NOUN
ejpam-4352	199	20	,	,	PUNCT
ejpam-4352	199	21	d	d	NOUN
ejpam-4352	199	22	}	}	PUNCT
ejpam-4352	199	23	.	.	PUNCT
ejpam-4352	200	1	since	since	SCONJ
ejpam-4352	200	2	s	s	PROPN
ejpam-4352	200	3	is	be	AUX
ejpam-4352	200	4	a	a	DET
ejpam-4352	200	5	γcs	γcs	NOUN
ejpam-4352	200	6	-	-	PUNCT
ejpam-4352	200	7	set	set	NOUN
ejpam-4352	200	8	,	,	PUNCT
ejpam-4352	200	9	we	we	PRON
ejpam-4352	200	10	may	may	AUX
ejpam-4352	200	11	assume	assume	VERB
ejpam-4352	200	12	that	that	SCONJ
ejpam-4352	200	13	[	[	X
ejpam-4352	200	14	v	v	X
ejpam-4352	200	15	(	(	PUNCT
ejpam-4352	200	16	g	g	NOUN
ejpam-4352	200	17	)	)	PUNCT
ejpam-4352	200	18	\ng(a	\ng(a	PROPN
ejpam-4352	200	19	)	)	PUNCT
ejpam-4352	200	20	]	]	PUNCT
ejpam-4352	200	21	∩	∩	NOUN
ejpam-4352	201	1	[	[	X
ejpam-4352	201	2	v	v	X
ejpam-4352	201	3	(	(	PUNCT
ejpam-4352	201	4	g	g	NOUN
ejpam-4352	201	5	)	)	PUNCT
ejpam-4352	201	6	\	\	PUNCT
ejpam-4352	202	1	s	s	X
ejpam-4352	202	2	]	]	X
ejpam-4352	202	3	=	=	X
ejpam-4352	202	4	{	{	PUNCT
ejpam-4352	202	5	c	c	NOUN
ejpam-4352	202	6	}	}	PUNCT
ejpam-4352	202	7	and	and	CCONJ
ejpam-4352	202	8	[	[	X
ejpam-4352	202	9	v	v	X
ejpam-4352	202	10	(	(	PUNCT
ejpam-4352	202	11	g	g	NOUN
ejpam-4352	202	12	)	)	PUNCT
ejpam-4352	202	13	\	\	NOUN
ejpam-4352	202	14	ng(b	ng(b	NOUN
ejpam-4352	202	15	)	)	PUNCT
ejpam-4352	202	16	]	]	PUNCT
ejpam-4352	203	1	∩	∩	NOUN
ejpam-4352	204	1	[	[	X
ejpam-4352	204	2	v	v	X
ejpam-4352	204	3	(	(	PUNCT
ejpam-4352	204	4	g	g	NOUN
ejpam-4352	204	5	)	)	PUNCT
ejpam-4352	204	6	\	\	PUNCT
ejpam-4352	205	1	s	s	X
ejpam-4352	205	2	]	]	X
ejpam-4352	205	3	=	=	PUNCT
ejpam-4352	205	4	{	{	PUNCT
ejpam-4352	205	5	d	d	NOUN
ejpam-4352	205	6	}	}	PUNCT
ejpam-4352	205	7	.	.	PUNCT
ejpam-4352	206	1	then	then	ADV
ejpam-4352	206	2	ad	ad	NOUN
ejpam-4352	206	3	,	,	PUNCT
ejpam-4352	206	4	bc	bc	PROPN
ejpam-4352	206	5	∈	∈	PROPN
ejpam-4352	206	6	e(g	e(g	PROPN
ejpam-4352	206	7	)	)	PUNCT
ejpam-4352	206	8	.	.	PUNCT
ejpam-4352	206	9	suppose	suppose	VERB
ejpam-4352	206	10	ab	ab	PROPN
ejpam-4352	206	11	∈	∈	PROPN
ejpam-4352	206	12	e(g	e(g	PROPN
ejpam-4352	206	13	)	)	PUNCT
ejpam-4352	206	14	.	.	PUNCT
ejpam-4352	207	1	if	if	SCONJ
ejpam-4352	207	2	cd	cd	PROPN
ejpam-4352	207	3	∈	∈	PROPN
ejpam-4352	207	4	e(g	e(g	PROPN
ejpam-4352	207	5	)	)	PUNCT
ejpam-4352	207	6	,	,	PUNCT
ejpam-4352	207	7	then	then	ADV
ejpam-4352	207	8	g	g	PROPN
ejpam-4352	207	9	=	=	PROPN
ejpam-4352	207	10	c4	c4	NOUN
ejpam-4352	207	11	.	.	PUNCT
ejpam-4352	208	1	if	if	SCONJ
ejpam-4352	208	2	cd	cd	PROPN
ejpam-4352	208	3	/∈	/∈	PUNCT
ejpam-4352	208	4	e(g	e(g	PROPN
ejpam-4352	208	5	)	)	PUNCT
ejpam-4352	208	6	,	,	PUNCT
ejpam-4352	208	7	then	then	ADV
ejpam-4352	208	8	g	g	PROPN
ejpam-4352	208	9	=	=	SYM
ejpam-4352	208	10	p4	p4	ADJ
ejpam-4352	208	11	.	.	PUNCT
ejpam-4352	209	1	next	next	ADV
ejpam-4352	209	2	,	,	PUNCT
ejpam-4352	209	3	suppose	suppose	VERB
ejpam-4352	209	4	that	that	SCONJ
ejpam-4352	209	5	ab	ab	PROPN
ejpam-4352	209	6	/∈	/∈	PUNCT
ejpam-4352	209	7	e(g	e(g	PROPN
ejpam-4352	209	8	)	)	PUNCT
ejpam-4352	209	9	.	.	PUNCT
ejpam-4352	210	1	if	if	SCONJ
ejpam-4352	210	2	cd	cd	PROPN
ejpam-4352	210	3	∈	∈	PROPN
ejpam-4352	210	4	e(g	e(g	PROPN
ejpam-4352	210	5	)	)	PUNCT
ejpam-4352	210	6	,	,	PUNCT
ejpam-4352	210	7	then	then	ADV
ejpam-4352	210	8	g	g	PROPN
ejpam-4352	210	9	=	=	SYM
ejpam-4352	210	10	p4	p4	ADJ
ejpam-4352	210	11	.	.	PUNCT
ejpam-4352	211	1	otherwise	otherwise	ADV
ejpam-4352	211	2	,	,	PUNCT
ejpam-4352	211	3	g	g	PROPN
ejpam-4352	211	4	=	=	SYM
ejpam-4352	211	5	k2∪k2	k2∪k2	PROPN
ejpam-4352	211	6	.	.	PUNCT
ejpam-4352	212	1	therefore	therefore	ADV
ejpam-4352	212	2	,	,	PUNCT
ejpam-4352	212	3	g	g	PROPN
ejpam-4352	212	4	∈	∈	PROPN
ejpam-4352	212	5	{	{	PUNCT
ejpam-4352	212	6	k2	k2	NOUN
ejpam-4352	212	7	,	,	PUNCT
ejpam-4352	212	8	p3,k2∪k1,k3	p3,k2∪k1,k3	NOUN
ejpam-4352	212	9	,	,	PUNCT
ejpam-4352	212	10	p4	p4	ADJ
ejpam-4352	212	11	,	,	PUNCT
ejpam-4352	212	12	c4,k2∪k2	c4,k2∪k2	PROPN
ejpam-4352	212	13	}	}	PUNCT
ejpam-4352	212	14	.	.	PUNCT
ejpam-4352	213	1	the	the	DET
ejpam-4352	213	2	converse	converse	NOUN
ejpam-4352	213	3	is	be	AUX
ejpam-4352	213	4	clear	clear	ADJ
ejpam-4352	213	5	.	.	PUNCT
ejpam-4352	214	1	theorem	theorem	ADJ
ejpam-4352	214	2	6	6	NUM
ejpam-4352	214	3	.	.	PUNCT
ejpam-4352	215	1	let	let	VERB
ejpam-4352	215	2	n	n	PRON
ejpam-4352	215	3	be	be	AUX
ejpam-4352	215	4	a	a	DET
ejpam-4352	215	5	positive	positive	ADJ
ejpam-4352	215	6	integer	integer	NOUN
ejpam-4352	215	7	.	.	PUNCT
ejpam-4352	216	1	then	then	ADV
ejpam-4352	216	2	γcs(pn	γcs(pn	X
ejpam-4352	216	3	)	)	PUNCT
ejpam-4352	216	4	=	=	SYM
ejpam-4352	217	1			NOUN
ejpam-4352	217	2	n	n	ADV
ejpam-4352	217	3	if	if	SCONJ
ejpam-4352	217	4	n	n	NOUN
ejpam-4352	217	5	=	=	SYM
ejpam-4352	217	6	1	1	NUM
ejpam-4352	217	7	,	,	PUNCT
ejpam-4352	217	8	2	2	NUM
ejpam-4352	217	9	2	2	NUM
ejpam-4352	217	10	if	if	SCONJ
ejpam-4352	217	11	n	n	NOUN
ejpam-4352	217	12	=	=	SYM
ejpam-4352	217	13	3	3	NUM
ejpam-4352	217	14	n−	n−	NOUN
ejpam-4352	217	15	2	2	NUM
ejpam-4352	217	16	if	if	SCONJ
ejpam-4352	217	17	n	n	PRON
ejpam-4352	217	18	≥	≥	NOUN
ejpam-4352	217	19	4	4	NUM
ejpam-4352	217	20	.	.	PUNCT
ejpam-4352	217	21	proof	proof	NOUN
ejpam-4352	217	22	.	.	PUNCT
ejpam-4352	218	1	let	let	VERB
ejpam-4352	218	2	pn	pn	VERB
ejpam-4352	218	3	=	=	PUNCT
ejpam-4352	219	1	[	[	X
ejpam-4352	219	2	v1	v1	NOUN
ejpam-4352	219	3	,	,	PUNCT
ejpam-4352	219	4	v2	v2	PROPN
ejpam-4352	219	5	,	,	PUNCT
ejpam-4352	219	6	·	·	PUNCT
ejpam-4352	219	7	·	·	PUNCT
ejpam-4352	219	8	·	·	PUNCT
ejpam-4352	219	9	,	,	PUNCT
ejpam-4352	219	10	vn	vn	X
ejpam-4352	219	11	]	]	PUNCT
ejpam-4352	219	12	.	.	PUNCT
ejpam-4352	220	1	clearly	clearly	ADV
ejpam-4352	220	2	,	,	PUNCT
ejpam-4352	220	3	γcs(p1	γcs(p1	PROPN
ejpam-4352	220	4	)	)	PUNCT
ejpam-4352	220	5	=	=	SYM
ejpam-4352	220	6	1	1	NUM
ejpam-4352	220	7	,	,	PUNCT
ejpam-4352	220	8	γcs(p2	γcs(p2	ADJ
ejpam-4352	220	9	)	)	PUNCT
ejpam-4352	220	10	=	=	SYM
ejpam-4352	220	11	2	2	NUM
ejpam-4352	220	12	,	,	PUNCT
ejpam-4352	220	13	and	and	CCONJ
ejpam-4352	220	14	γcs(p3	γcs(p3	PROPN
ejpam-4352	220	15	)	)	PUNCT
ejpam-4352	221	1	=	=	SYM
ejpam-4352	221	2	2	2	X
ejpam-4352	221	3	.	.	X
ejpam-4352	221	4	suppose	suppose	VERB
ejpam-4352	221	5	n	n	PRON
ejpam-4352	221	6	≥	≥	NUM
ejpam-4352	221	7	4	4	NUM
ejpam-4352	221	8	.	.	PUNCT
ejpam-4352	222	1	since	since	SCONJ
ejpam-4352	222	2	s0	s0	PROPN
ejpam-4352	222	3	=	=	SYM
ejpam-4352	222	4	v	v	PROPN
ejpam-4352	222	5	(	(	PUNCT
ejpam-4352	222	6	pn	pn	NOUN
ejpam-4352	222	7	)	)	PUNCT
ejpam-4352	222	8	\	\	NOUN
ejpam-4352	222	9	{	{	PUNCT
ejpam-4352	222	10	v1	v1	PROPN
ejpam-4352	222	11	,	,	PUNCT
ejpam-4352	222	12	vn	vn	PROPN
ejpam-4352	222	13	}	}	PUNCT
ejpam-4352	222	14	is	be	AUX
ejpam-4352	222	15	a	a	DET
ejpam-4352	222	16	complement	complement	NOUN
ejpam-4352	222	17	-	-	PUNCT
ejpam-4352	222	18	super	super	ADJ
ejpam-4352	222	19	dominating	dominating	NOUN
ejpam-4352	222	20	set	set	NOUN
ejpam-4352	222	21	of	of	ADP
ejpam-4352	222	22	pn	pn	PROPN
ejpam-4352	222	23	,	,	PUNCT
ejpam-4352	222	24	it	it	PRON
ejpam-4352	222	25	follows	follow	VERB
ejpam-4352	222	26	that	that	SCONJ
ejpam-4352	222	27	γcs(pn	γcs(pn	X
ejpam-4352	222	28	)	)	PUNCT
ejpam-4352	222	29	≤	≤	NOUN
ejpam-4352	222	30	n	n	CCONJ
ejpam-4352	222	31	−	−	PROPN
ejpam-4352	222	32	2	2	NUM
ejpam-4352	222	33	.	.	PUNCT
ejpam-4352	222	34	suppose	suppose	VERB
ejpam-4352	222	35	γcs(pn	γcs(pn	NOUN
ejpam-4352	222	36	)	)	PUNCT
ejpam-4352	222	37	<	<	X
ejpam-4352	223	1	n	n	CCONJ
ejpam-4352	223	2	−	−	NOUN
ejpam-4352	224	1	2	2	X
ejpam-4352	224	2	.	.	PUNCT
ejpam-4352	225	1	let	let	VERB
ejpam-4352	225	2	s	s	PRON
ejpam-4352	225	3	be	be	AUX
ejpam-4352	225	4	a	a	DET
ejpam-4352	225	5	γcs	γcs	NOUN
ejpam-4352	225	6	-	-	PUNCT
ejpam-4352	225	7	set	set	NOUN
ejpam-4352	225	8	of	of	ADP
ejpam-4352	225	9	pn	pn	PROPN
ejpam-4352	225	10	and	and	CCONJ
ejpam-4352	225	11	let	let	VERB
ejpam-4352	225	12	v	v	ADP
ejpam-4352	225	13	,	,	PUNCT
ejpam-4352	225	14	w	w	NOUN
ejpam-4352	225	15	,	,	PUNCT
ejpam-4352	225	16	and	and	CCONJ
ejpam-4352	225	17	x	x	ADJ
ejpam-4352	225	18	be	be	AUX
ejpam-4352	225	19	distinct	distinct	ADJ
ejpam-4352	225	20	elements	element	NOUN
ejpam-4352	225	21	of	of	ADP
ejpam-4352	225	22	v	v	NOUN
ejpam-4352	225	23	(	(	PUNCT
ejpam-4352	225	24	pn	pn	NOUN
ejpam-4352	225	25	)	)	PUNCT
ejpam-4352	225	26	\	\	PROPN
ejpam-4352	226	1	s.	s.	PROPN
ejpam-4352	226	2	we	we	PRON
ejpam-4352	226	3	may	may	AUX
ejpam-4352	226	4	assume	assume	VERB
ejpam-4352	226	5	that	that	SCONJ
ejpam-4352	226	6	v	v	NOUN
ejpam-4352	226	7	=	=	SYM
ejpam-4352	226	8	vj	vj	INTJ
ejpam-4352	226	9	,	,	PUNCT
ejpam-4352	226	10	w	w	PROPN
ejpam-4352	226	11	=	=	SYM
ejpam-4352	226	12	vr	vr	PROPN
ejpam-4352	226	13	,	,	PUNCT
ejpam-4352	226	14	and	and	CCONJ
ejpam-4352	226	15	x	x	X
ejpam-4352	226	16	=	=	NOUN
ejpam-4352	226	17	vs	vs	ADP
ejpam-4352	226	18	where	where	SCONJ
ejpam-4352	226	19	1	1	NUM
ejpam-4352	226	20	≤	≤	NUM
ejpam-4352	226	21	j	j	NOUN
ejpam-4352	226	22	<	<	X
ejpam-4352	226	23	r	r	X
ejpam-4352	226	24	<	<	X
ejpam-4352	226	25	s	s	PART
ejpam-4352	226	26	≤	≤	NOUN
ejpam-4352	226	27	n.	n.	NOUN
ejpam-4352	226	28	since	since	SCONJ
ejpam-4352	226	29	s	s	PROPN
ejpam-4352	226	30	is	be	AUX
ejpam-4352	226	31	a	a	DET
ejpam-4352	226	32	complement	complement	NOUN
ejpam-4352	226	33	-	-	PUNCT
ejpam-4352	226	34	super	super	ADJ
ejpam-4352	226	35	dominating	dominating	NOUN
ejpam-4352	226	36	set	set	NOUN
ejpam-4352	226	37	of	of	ADP
ejpam-4352	226	38	pn	pn	PROPN
ejpam-4352	226	39	and	and	CCONJ
ejpam-4352	226	40	w	w	PROPN
ejpam-4352	226	41	∈	∈	PROPN
ejpam-4352	226	42	v	v	ADP
ejpam-4352	226	43	(	(	PUNCT
ejpam-4352	226	44	pn	pn	NOUN
ejpam-4352	226	45	)	)	PUNCT
ejpam-4352	226	46	\s	\	NOUN
ejpam-4352	226	47	,	,	PUNCT
ejpam-4352	226	48	there	there	PRON
ejpam-4352	226	49	exists	exist	VERB
ejpam-4352	226	50	p	p	PROPN
ejpam-4352	226	51	∈	∈	PROPN
ejpam-4352	226	52	s	s	VERB
ejpam-4352	226	53	such	such	ADJ
ejpam-4352	226	54	that	that	SCONJ
ejpam-4352	226	55	[	[	X
ejpam-4352	226	56	v	v	X
ejpam-4352	226	57	(	(	PUNCT
ejpam-4352	226	58	pn	pn	NOUN
ejpam-4352	226	59	)	)	PUNCT
ejpam-4352	226	60	\npn(p)]∩	\npn(p)]∩	NOUN
ejpam-4352	226	61	[	[	X
ejpam-4352	226	62	v	v	X
ejpam-4352	226	63	(	(	PUNCT
ejpam-4352	226	64	pn	pn	NOUN
ejpam-4352	226	65	)	)	PUNCT
ejpam-4352	226	66	\s	\s	NOUN
ejpam-4352	226	67	]	]	PUNCT
ejpam-4352	227	1	=	=	PUNCT
ejpam-4352	227	2	{	{	PUNCT
ejpam-4352	227	3	w	w	NOUN
ejpam-4352	227	4	}	}	PUNCT
ejpam-4352	227	5	.	.	PUNCT
ejpam-4352	228	1	this	this	PRON
ejpam-4352	228	2	implies	imply	VERB
ejpam-4352	228	3	that	that	SCONJ
ejpam-4352	228	4	pv	pv	ADV
ejpam-4352	228	5	,	,	PUNCT
ejpam-4352	228	6	px	px	PROPN
ejpam-4352	228	7	∈	∈	PROPN
ejpam-4352	228	8	e(pn	e(pn	NUM
ejpam-4352	228	9	)	)	PUNCT
ejpam-4352	228	10	.	.	PUNCT
ejpam-4352	229	1	since	since	SCONJ
ejpam-4352	229	2	j	j	PROPN
ejpam-4352	229	3	<	<	X
ejpam-4352	229	4	s	s	PROPN
ejpam-4352	229	5	,	,	PUNCT
ejpam-4352	229	6	it	it	PRON
ejpam-4352	229	7	follows	follow	VERB
ejpam-4352	229	8	that	that	SCONJ
ejpam-4352	229	9	p	p	NOUN
ejpam-4352	229	10	=	=	PUNCT
ejpam-4352	229	11	vj+1	vj+1	PROPN
ejpam-4352	229	12	and	and	CCONJ
ejpam-4352	229	13	s	s	NOUN
ejpam-4352	229	14	=	=	PUNCT
ejpam-4352	229	15	j+2	j+2	PROPN
ejpam-4352	229	16	.	.	PUNCT
ejpam-4352	230	1	this	this	PRON
ejpam-4352	230	2	,	,	PUNCT
ejpam-4352	230	3	however	however	ADV
ejpam-4352	230	4	,	,	PUNCT
ejpam-4352	230	5	would	would	AUX
ejpam-4352	230	6	imply	imply	VERB
ejpam-4352	230	7	that	that	SCONJ
ejpam-4352	230	8	r	r	NOUN
ejpam-4352	230	9	=	=	SYM
ejpam-4352	230	10	j+1	j+1	NOUN
ejpam-4352	230	11	(	(	PUNCT
ejpam-4352	230	12	i.e.	i.e.	X
ejpam-4352	230	13	,	,	PUNCT
ejpam-4352	230	14	w	w	PROPN
ejpam-4352	230	15	=	=	SYM
ejpam-4352	230	16	vr	vr	NOUN
ejpam-4352	230	17	=	=	SYM
ejpam-4352	230	18	p	p	PROPN
ejpam-4352	230	19	because	because	SCONJ
ejpam-4352	230	20	j	j	PROPN
ejpam-4352	230	21	<	<	X
ejpam-4352	230	22	r	r	X
ejpam-4352	230	23	<	<	X
ejpam-4352	230	24	s	s	NOUN
ejpam-4352	230	25	)	)	PUNCT
ejpam-4352	230	26	,	,	PUNCT
ejpam-4352	230	27	a	a	DET
ejpam-4352	230	28	contradiction	contradiction	NOUN
ejpam-4352	230	29	.	.	PUNCT
ejpam-4352	231	1	therefore	therefore	ADV
ejpam-4352	231	2	,	,	PUNCT
ejpam-4352	231	3	γcs(pn	γcs(pn	NUM
ejpam-4352	231	4	)	)	PUNCT
ejpam-4352	231	5	=	=	SYM
ejpam-4352	231	6	n−	n−	NOUN
ejpam-4352	231	7	2	2	NUM
ejpam-4352	231	8	.	.	PUNCT
ejpam-4352	231	9	theorem	theorem	VERB
ejpam-4352	231	10	7	7	NUM
ejpam-4352	231	11	.	.	PUNCT
ejpam-4352	232	1	let	let	VERB
ejpam-4352	232	2	n	n	PRON
ejpam-4352	232	3	be	be	AUX
ejpam-4352	232	4	a	a	DET
ejpam-4352	232	5	positive	positive	ADJ
ejpam-4352	232	6	integer	integer	NOUN
ejpam-4352	232	7	and	and	CCONJ
ejpam-4352	232	8	n	n	PRON
ejpam-4352	232	9	≥	≥	NOUN
ejpam-4352	232	10	3	3	NUM
ejpam-4352	232	11	.	.	PUNCT
ejpam-4352	233	1	then	then	ADV
ejpam-4352	233	2	γcs(cn	γcs(cn	NUM
ejpam-4352	233	3	)	)	PUNCT
ejpam-4352	234	1	=	=	PRON
ejpam-4352	234	2	{	{	PUNCT
ejpam-4352	234	3	3	3	NUM
ejpam-4352	234	4	if	if	SCONJ
ejpam-4352	234	5	n	n	X
ejpam-4352	234	6	=	=	SYM
ejpam-4352	234	7	3	3	NUM
ejpam-4352	234	8	,	,	PUNCT
ejpam-4352	234	9	6	6	NUM
ejpam-4352	234	10	n−	n−	NOUN
ejpam-4352	234	11	2	2	NUM
ejpam-4352	234	12	if	if	SCONJ
ejpam-4352	234	13	n	n	NOUN
ejpam-4352	234	14	=	=	SYM
ejpam-4352	234	15	4	4	NUM
ejpam-4352	234	16	,	,	PUNCT
ejpam-4352	234	17	5	5	NUM
ejpam-4352	234	18	and	and	CCONJ
ejpam-4352	234	19	n	n	PRON
ejpam-4352	234	20	≥	≥	NOUN
ejpam-4352	234	21	7	7	NUM
ejpam-4352	234	22	.	.	PUNCT
ejpam-4352	235	1	s.	s.	PROPN
ejpam-4352	235	2	canoy	canoy	PROPN
ejpam-4352	235	3	,	,	PUNCT
ejpam-4352	235	4	jr	jr	PROPN
ejpam-4352	235	5	.	.	PROPN
ejpam-4352	235	6	,	,	PUNCT
ejpam-4352	235	7	g.	g.	PROPN
ejpam-4352	235	8	salasalan	salasalan	PROPN
ejpam-4352	235	9	/	/	SYM
ejpam-4352	235	10	eur	eur	PROPN
ejpam-4352	235	11	.	.	PUNCT
ejpam-4352	236	1	j.	j.	PROPN
ejpam-4352	236	2	pure	pure	PROPN
ejpam-4352	236	3	appl	appl	PROPN
ejpam-4352	236	4	.	.	PROPN
ejpam-4352	236	5	math	math	PROPN
ejpam-4352	236	6	,	,	PUNCT
ejpam-4352	236	7	15	15	NUM
ejpam-4352	236	8	(	(	PUNCT
ejpam-4352	236	9	2	2	NUM
ejpam-4352	236	10	)	)	PUNCT
ejpam-4352	236	11	(	(	PUNCT
ejpam-4352	236	12	2022	2022	NUM
ejpam-4352	236	13	)	)	PUNCT
ejpam-4352	236	14	,	,	PUNCT
ejpam-4352	236	15	342	342	NUM
ejpam-4352	236	16	-	-	SYM
ejpam-4352	236	17	353	353	NUM
ejpam-4352	236	18	347	347	NUM
ejpam-4352	236	19	proof	proof	NOUN
ejpam-4352	236	20	.	.	PUNCT
ejpam-4352	237	1	let	let	VERB
ejpam-4352	237	2	cn	cn	PROPN
ejpam-4352	237	3	=	=	PUNCT
ejpam-4352	238	1	[	[	X
ejpam-4352	238	2	v1	v1	NOUN
ejpam-4352	238	3	,	,	PUNCT
ejpam-4352	238	4	v2	v2	PROPN
ejpam-4352	238	5	,	,	PUNCT
ejpam-4352	238	6	·	·	PUNCT
ejpam-4352	238	7	·	·	PUNCT
ejpam-4352	238	8	·	·	PUNCT
ejpam-4352	238	9	,	,	PUNCT
ejpam-4352	238	10	vn	vn	X
ejpam-4352	238	11	,	,	PUNCT
ejpam-4352	238	12	v1	v1	PROPN
ejpam-4352	238	13	]	]	PUNCT
ejpam-4352	238	14	.	.	PUNCT
ejpam-4352	239	1	now	now	ADV
ejpam-4352	239	2	γcs(c3	γcs(c3	NOUN
ejpam-4352	239	3	)	)	PUNCT
ejpam-4352	239	4	=	=	SYM
ejpam-4352	239	5	3	3	NUM
ejpam-4352	239	6	by	by	ADP
ejpam-4352	239	7	theorem	theorem	NOUN
ejpam-4352	239	8	4	4	NUM
ejpam-4352	239	9	.	.	PUNCT
ejpam-4352	239	10	suppose	suppose	VERB
ejpam-4352	239	11	4	4	NUM
ejpam-4352	239	12	≤	≤	NUM
ejpam-4352	239	13	n	n	CCONJ
ejpam-4352	239	14	≤	≤	NOUN
ejpam-4352	239	15	5	5	NUM
ejpam-4352	239	16	.	.	PUNCT
ejpam-4352	240	1	it	it	PRON
ejpam-4352	240	2	can	can	AUX
ejpam-4352	240	3	be	be	AUX
ejpam-4352	240	4	verified	verify	VERB
ejpam-4352	240	5	easily	easily	ADV
ejpam-4352	240	6	that	that	SCONJ
ejpam-4352	240	7	s0	s0	NOUN
ejpam-4352	240	8	=	=	SYM
ejpam-4352	240	9	v	v	PROPN
ejpam-4352	240	10	(	(	PUNCT
ejpam-4352	240	11	cn	cn	PROPN
ejpam-4352	240	12	)	)	PUNCT
ejpam-4352	240	13	\	\	PROPN
ejpam-4352	240	14	{	{	PUNCT
ejpam-4352	240	15	v1	v1	NOUN
ejpam-4352	240	16	,	,	PUNCT
ejpam-4352	240	17	v2	v2	PROPN
ejpam-4352	240	18	}	}	PUNCT
ejpam-4352	240	19	is	be	AUX
ejpam-4352	240	20	a	a	DET
ejpam-4352	240	21	γcs	γcs	NOUN
ejpam-4352	240	22	-	-	PUNCT
ejpam-4352	240	23	set	set	NOUN
ejpam-4352	240	24	of	of	ADP
ejpam-4352	240	25	cn	cn	PROPN
ejpam-4352	240	26	.	.	PUNCT
ejpam-4352	241	1	thus	thus	ADV
ejpam-4352	241	2	,	,	PUNCT
ejpam-4352	241	3	γcs(cn	γcs(cn	NUM
ejpam-4352	241	4	)	)	PUNCT
ejpam-4352	242	1	=	=	PUNCT
ejpam-4352	242	2	n−	n−	NOUN
ejpam-4352	242	3	2	2	NUM
ejpam-4352	242	4	.	.	PUNCT
ejpam-4352	243	1	next	next	ADV
ejpam-4352	243	2	,	,	PUNCT
ejpam-4352	243	3	suppose	suppose	VERB
ejpam-4352	243	4	that	that	SCONJ
ejpam-4352	243	5	n	n	PROPN
ejpam-4352	243	6	=	=	SYM
ejpam-4352	243	7	6	6	NUM
ejpam-4352	243	8	.	.	PUNCT
ejpam-4352	244	1	since	since	SCONJ
ejpam-4352	244	2	s1	s1	PROPN
ejpam-4352	244	3	=	=	PROPN
ejpam-4352	244	4	v	v	PROPN
ejpam-4352	244	5	(	(	PUNCT
ejpam-4352	244	6	c6	c6	PROPN
ejpam-4352	244	7	)	)	PUNCT
ejpam-4352	244	8	\	\	PROPN
ejpam-4352	244	9	{	{	PUNCT
ejpam-4352	244	10	v1	v1	PROPN
ejpam-4352	244	11	,	,	PUNCT
ejpam-4352	244	12	v3	v3	PROPN
ejpam-4352	244	13	,	,	PUNCT
ejpam-4352	244	14	v5	v5	PROPN
ejpam-4352	244	15	}	}	PUNCT
ejpam-4352	244	16	=	=	SYM
ejpam-4352	244	17	{	{	PUNCT
ejpam-4352	244	18	v2	v2	PROPN
ejpam-4352	244	19	,	,	PUNCT
ejpam-4352	244	20	v4	v4	PROPN
ejpam-4352	244	21	,	,	PUNCT
ejpam-4352	244	22	v6	v6	PROPN
ejpam-4352	244	23	}	}	PUNCT
ejpam-4352	244	24	is	be	AUX
ejpam-4352	244	25	a	a	DET
ejpam-4352	244	26	complement	complement	NOUN
ejpam-4352	244	27	-	-	PUNCT
ejpam-4352	244	28	super	super	ADJ
ejpam-4352	244	29	dominating	dominating	NOUN
ejpam-4352	244	30	set	set	NOUN
ejpam-4352	244	31	of	of	ADP
ejpam-4352	244	32	cn	cn	PROPN
ejpam-4352	244	33	,	,	PUNCT
ejpam-4352	244	34	γcs(c6	γcs(c6	PROPN
ejpam-4352	244	35	)	)	PUNCT
ejpam-4352	244	36	=	=	SYM
ejpam-4352	244	37	3	3	NUM
ejpam-4352	244	38	by	by	ADP
ejpam-4352	244	39	theorem	theorem	NOUN
ejpam-4352	244	40	4	4	NUM
ejpam-4352	244	41	.	.	PUNCT
ejpam-4352	244	42	lastly	lastly	ADV
ejpam-4352	244	43	,	,	PUNCT
ejpam-4352	244	44	suppose	suppose	VERB
ejpam-4352	244	45	that	that	SCONJ
ejpam-4352	244	46	n	n	PROPN
ejpam-4352	244	47	≥	≥	NUM
ejpam-4352	244	48	7	7	NUM
ejpam-4352	244	49	.	.	PUNCT
ejpam-4352	245	1	the	the	DET
ejpam-4352	245	2	set	set	NOUN
ejpam-4352	245	3	s∗	s∗	PROPN
ejpam-4352	245	4	=	=	SYM
ejpam-4352	245	5	v	v	PROPN
ejpam-4352	245	6	(	(	PUNCT
ejpam-4352	245	7	cn	cn	PROPN
ejpam-4352	245	8	)	)	PUNCT
ejpam-4352	245	9	\	\	PROPN
ejpam-4352	245	10	{	{	PUNCT
ejpam-4352	245	11	v1	v1	NOUN
ejpam-4352	245	12	,	,	PUNCT
ejpam-4352	245	13	v2	v2	NOUN
ejpam-4352	245	14	}	}	PUNCT
ejpam-4352	245	15	=	=	SYM
ejpam-4352	245	16	{	{	PUNCT
ejpam-4352	245	17	v3	v3	PROPN
ejpam-4352	245	18	,	,	PUNCT
ejpam-4352	245	19	v4	v4	PROPN
ejpam-4352	245	20	,	,	PUNCT
ejpam-4352	245	21	·	·	PUNCT
ejpam-4352	245	22	·	·	PUNCT
ejpam-4352	245	23	·	·	PUNCT
ejpam-4352	245	24	,	,	PUNCT
ejpam-4352	245	25	vn	vn	PROPN
ejpam-4352	245	26	}	}	PUNCT
ejpam-4352	245	27	is	be	AUX
ejpam-4352	245	28	a	a	DET
ejpam-4352	245	29	complement	complement	NOUN
ejpam-4352	245	30	-	-	PUNCT
ejpam-4352	245	31	super	super	ADJ
ejpam-4352	245	32	dominating	dominating	NOUN
ejpam-4352	245	33	set	set	NOUN
ejpam-4352	245	34	of	of	ADP
ejpam-4352	245	35	cn	cn	PROPN
ejpam-4352	245	36	.	.	PUNCT
ejpam-4352	246	1	consequently	consequently	ADV
ejpam-4352	246	2	,	,	PUNCT
ejpam-4352	246	3	γcs(cn	γcs(cn	PROPN
ejpam-4352	246	4	)	)	PUNCT
ejpam-4352	246	5	≤	≤	PUNCT
ejpam-4352	247	1	n−2	n−2	PROPN
ejpam-4352	247	2	.	.	PUNCT
ejpam-4352	248	1	let	let	VERB
ejpam-4352	248	2	s	s	PRON
ejpam-4352	248	3	be	be	AUX
ejpam-4352	248	4	a	a	DET
ejpam-4352	248	5	γcs	γcs	NOUN
ejpam-4352	248	6	-	-	PUNCT
ejpam-4352	248	7	set	set	NOUN
ejpam-4352	248	8	of	of	ADP
ejpam-4352	248	9	cn	cn	PROPN
ejpam-4352	248	10	and	and	CCONJ
ejpam-4352	248	11	suppose	suppose	VERB
ejpam-4352	248	12	|s|	|s|	PROPN
ejpam-4352	248	13	<	<	X
ejpam-4352	248	14	n−2	n−2	PROPN
ejpam-4352	248	15	.	.	PUNCT
ejpam-4352	249	1	let	let	VERB
ejpam-4352	249	2	a	a	DET
ejpam-4352	249	3	,	,	PUNCT
ejpam-4352	249	4	b	b	NOUN
ejpam-4352	249	5	,	,	PUNCT
ejpam-4352	249	6	and	and	CCONJ
ejpam-4352	249	7	c	c	PROPN
ejpam-4352	249	8	be	be	AUX
ejpam-4352	249	9	distinct	distinct	ADJ
ejpam-4352	249	10	elements	element	NOUN
ejpam-4352	249	11	of	of	ADP
ejpam-4352	249	12	v	v	NOUN
ejpam-4352	249	13	(	(	PUNCT
ejpam-4352	249	14	cn)\s	cn)\s	NOUN
ejpam-4352	249	15	.	.	PUNCT
ejpam-4352	250	1	we	we	PRON
ejpam-4352	250	2	may	may	AUX
ejpam-4352	250	3	assume	assume	VERB
ejpam-4352	250	4	that	that	SCONJ
ejpam-4352	250	5	a	a	DET
ejpam-4352	250	6	=	=	SYM
ejpam-4352	250	7	v1	v1	NOUN
ejpam-4352	250	8	,	,	PUNCT
ejpam-4352	250	9	b	b	X
ejpam-4352	250	10	=	=	SYM
ejpam-4352	250	11	vm	vm	PROPN
ejpam-4352	250	12	,	,	PUNCT
ejpam-4352	250	13	and	and	CCONJ
ejpam-4352	250	14	c	c	X
ejpam-4352	250	15	=	=	SYM
ejpam-4352	250	16	vr	vr	NOUN
ejpam-4352	250	17	where	where	SCONJ
ejpam-4352	250	18	1	1	NUM
ejpam-4352	250	19	<	<	X
ejpam-4352	250	20	m	m	VERB
ejpam-4352	250	21	<	<	X
ejpam-4352	250	22	r	r	NOUN
ejpam-4352	250	23	≤	≤	NUM
ejpam-4352	250	24	n.	n.	NOUN
ejpam-4352	250	25	since	since	SCONJ
ejpam-4352	250	26	s	s	PROPN
ejpam-4352	250	27	is	be	AUX
ejpam-4352	250	28	a	a	DET
ejpam-4352	250	29	complement	complement	NOUN
ejpam-4352	250	30	-	-	PUNCT
ejpam-4352	250	31	super	super	ADJ
ejpam-4352	250	32	dominating	dominating	NOUN
ejpam-4352	250	33	set	set	NOUN
ejpam-4352	250	34	of	of	ADP
ejpam-4352	250	35	cn	cn	PROPN
ejpam-4352	250	36	and	and	CCONJ
ejpam-4352	250	37	c	c	NOUN
ejpam-4352	250	38	∈	∈	PROPN
ejpam-4352	250	39	v	v	NOUN
ejpam-4352	250	40	(	(	PUNCT
ejpam-4352	250	41	cn)\s	cn)\s	NOUN
ejpam-4352	250	42	,	,	PUNCT
ejpam-4352	250	43	there	there	PRON
ejpam-4352	250	44	exists	exist	VERB
ejpam-4352	250	45	y	y	PROPN
ejpam-4352	250	46	∈	∈	PROPN
ejpam-4352	250	47	s	s	VERB
ejpam-4352	250	48	such	such	ADJ
ejpam-4352	250	49	that	that	SCONJ
ejpam-4352	250	50	[	[	X
ejpam-4352	250	51	v	v	X
ejpam-4352	250	52	(	(	PUNCT
ejpam-4352	250	53	cn)\ncn(y)]∩	cn)\ncn(y)]∩	NOUN
ejpam-4352	250	54	[	[	X
ejpam-4352	250	55	v	v	X
ejpam-4352	250	56	(	(	PUNCT
ejpam-4352	250	57	cn)\s	cn)\s	NOUN
ejpam-4352	250	58	]	]	X
ejpam-4352	251	1	=	=	X
ejpam-4352	251	2	{	{	PUNCT
ejpam-4352	251	3	c	c	NOUN
ejpam-4352	251	4	}	}	PUNCT
ejpam-4352	251	5	.	.	PUNCT
ejpam-4352	252	1	this	this	PRON
ejpam-4352	252	2	implies	imply	VERB
ejpam-4352	252	3	that	that	SCONJ
ejpam-4352	252	4	ay	ay	NOUN
ejpam-4352	252	5	,	,	PUNCT
ejpam-4352	252	6	yb	yb	PROPN
ejpam-4352	252	7	∈	∈	PROPN
ejpam-4352	252	8	e(cn	e(cn	NOUN
ejpam-4352	252	9	)	)	PUNCT
ejpam-4352	252	10	.	.	PUNCT
ejpam-4352	253	1	with	with	ADP
ejpam-4352	253	2	the	the	DET
ejpam-4352	253	3	assumption	assumption	NOUN
ejpam-4352	253	4	that	that	SCONJ
ejpam-4352	253	5	a	a	DET
ejpam-4352	253	6	=	=	X
ejpam-4352	253	7	v1	v1	NOUN
ejpam-4352	253	8	and	and	CCONJ
ejpam-4352	253	9	m	m	NOUN
ejpam-4352	253	10	<	<	X
ejpam-4352	253	11	r	r	NOUN
ejpam-4352	253	12	,	,	PUNCT
ejpam-4352	253	13	we	we	PRON
ejpam-4352	253	14	find	find	VERB
ejpam-4352	253	15	that	that	SCONJ
ejpam-4352	253	16	y	y	PROPN
ejpam-4352	253	17	=	=	PUNCT
ejpam-4352	253	18	v2	v2	PROPN
ejpam-4352	253	19	and	and	CCONJ
ejpam-4352	253	20	b	b	NOUN
ejpam-4352	253	21	=	=	PROPN
ejpam-4352	253	22	v3	v3	PROPN
ejpam-4352	253	23	.	.	PUNCT
ejpam-4352	254	1	applying	apply	VERB
ejpam-4352	254	2	the	the	DET
ejpam-4352	254	3	same	same	ADJ
ejpam-4352	254	4	argument	argument	NOUN
ejpam-4352	254	5	to	to	ADP
ejpam-4352	254	6	a	a	DET
ejpam-4352	254	7	∈	∈	PROPN
ejpam-4352	254	8	v	v	NOUN
ejpam-4352	254	9	(	(	PUNCT
ejpam-4352	254	10	cn	cn	PROPN
ejpam-4352	254	11	)	)	PUNCT
ejpam-4352	254	12	\	\	PROPN
ejpam-4352	255	1	s	s	X
ejpam-4352	255	2	,	,	PUNCT
ejpam-4352	255	3	we	we	PRON
ejpam-4352	255	4	find	find	VERB
ejpam-4352	255	5	that	that	SCONJ
ejpam-4352	255	6	there	there	PRON
ejpam-4352	255	7	exists	exist	VERB
ejpam-4352	255	8	z	z	PROPN
ejpam-4352	255	9	∈	∈	PROPN
ejpam-4352	255	10	s	s	VERB
ejpam-4352	255	11	such	such	ADJ
ejpam-4352	255	12	that	that	SCONJ
ejpam-4352	255	13	[	[	X
ejpam-4352	255	14	v	v	X
ejpam-4352	255	15	(	(	PUNCT
ejpam-4352	255	16	cn	cn	PROPN
ejpam-4352	255	17	)	)	PUNCT
ejpam-4352	255	18	\ncn(z)]∩	\ncn(z)]∩	NOUN
ejpam-4352	256	1	[	[	X
ejpam-4352	256	2	v	v	X
ejpam-4352	256	3	(	(	PUNCT
ejpam-4352	256	4	cn	cn	NOUN
ejpam-4352	256	5	)	)	PUNCT
ejpam-4352	256	6	\s	\s	NOUN
ejpam-4352	256	7	]	]	PUNCT
ejpam-4352	257	1	=	=	SYM
ejpam-4352	257	2	{	{	PUNCT
ejpam-4352	257	3	a	a	NOUN
ejpam-4352	257	4	}	}	PUNCT
ejpam-4352	257	5	.	.	PUNCT
ejpam-4352	258	1	hence	hence	ADV
ejpam-4352	258	2	,	,	PUNCT
ejpam-4352	258	3	bz	bz	PROPN
ejpam-4352	258	4	,	,	PUNCT
ejpam-4352	258	5	zc	zc	PROPN
ejpam-4352	258	6	∈	∈	PROPN
ejpam-4352	258	7	e(cn	e(cn	NOUN
ejpam-4352	258	8	)	)	PUNCT
ejpam-4352	258	9	,	,	PUNCT
ejpam-4352	258	10	implying	imply	VERB
ejpam-4352	258	11	that	that	SCONJ
ejpam-4352	258	12	z	z	NOUN
ejpam-4352	258	13	=	=	PUNCT
ejpam-4352	258	14	v4	v4	NOUN
ejpam-4352	258	15	and	and	CCONJ
ejpam-4352	258	16	c	c	NOUN
ejpam-4352	258	17	=	=	SYM
ejpam-4352	258	18	v5	v5	PROPN
ejpam-4352	258	19	.	.	PUNCT
ejpam-4352	259	1	finally	finally	ADV
ejpam-4352	259	2	,	,	PUNCT
ejpam-4352	259	3	for	for	ADP
ejpam-4352	259	4	b	b	PROPN
ejpam-4352	259	5	∈	∈	PROPN
ejpam-4352	259	6	v	v	ADP
ejpam-4352	259	7	(	(	PUNCT
ejpam-4352	259	8	cn	cn	PROPN
ejpam-4352	259	9	)	)	PUNCT
ejpam-4352	259	10	\	\	PROPN
ejpam-4352	260	1	s	s	X
ejpam-4352	260	2	,	,	PUNCT
ejpam-4352	260	3	there	there	PRON
ejpam-4352	260	4	also	also	ADV
ejpam-4352	260	5	exists	exist	VERB
ejpam-4352	260	6	q	q	PROPN
ejpam-4352	260	7	∈	∈	PROPN
ejpam-4352	260	8	s	s	VERB
ejpam-4352	260	9	such	such	ADJ
ejpam-4352	260	10	that	that	SCONJ
ejpam-4352	260	11	[	[	X
ejpam-4352	260	12	v	v	X
ejpam-4352	260	13	(	(	PUNCT
ejpam-4352	260	14	cn	cn	PROPN
ejpam-4352	260	15	)	)	PUNCT
ejpam-4352	260	16	\ncn(q	\ncn(q	PROPN
ejpam-4352	260	17	)	)	PUNCT
ejpam-4352	260	18	]	]	PUNCT
ejpam-4352	260	19	∩	∩	NOUN
ejpam-4352	261	1	[	[	X
ejpam-4352	261	2	v	v	X
ejpam-4352	261	3	(	(	PUNCT
ejpam-4352	261	4	cn	cn	PROPN
ejpam-4352	261	5	)	)	PUNCT
ejpam-4352	261	6	\	\	PROPN
ejpam-4352	261	7	s	s	X
ejpam-4352	261	8	]	]	X
ejpam-4352	261	9	=	=	PUNCT
ejpam-4352	261	10	{	{	PUNCT
ejpam-4352	261	11	b	b	NOUN
ejpam-4352	261	12	}	}	PUNCT
ejpam-4352	261	13	.	.	PUNCT
ejpam-4352	262	1	it	it	PRON
ejpam-4352	262	2	follows	follow	VERB
ejpam-4352	262	3	that	that	SCONJ
ejpam-4352	262	4	qv1	qv1	ADV
ejpam-4352	262	5	,	,	PUNCT
ejpam-4352	262	6	qv5	qv5	PROPN
ejpam-4352	262	7	∈	∈	PROPN
ejpam-4352	262	8	e(cn	e(cn	NOUN
ejpam-4352	262	9	)	)	PUNCT
ejpam-4352	262	10	.	.	PUNCT
ejpam-4352	263	1	since	since	SCONJ
ejpam-4352	263	2	n	n	PROPN
ejpam-4352	263	3	≥	≥	NOUN
ejpam-4352	263	4	7	7	NUM
ejpam-4352	263	5	,	,	PUNCT
ejpam-4352	263	6	no	no	DET
ejpam-4352	263	7	such	such	ADJ
ejpam-4352	263	8	vertex	vertex	NOUN
ejpam-4352	263	9	q	q	NOUN
ejpam-4352	263	10	of	of	ADP
ejpam-4352	263	11	cn	cn	PROPN
ejpam-4352	263	12	exists	exist	VERB
ejpam-4352	263	13	.	.	PUNCT
ejpam-4352	264	1	consequently	consequently	ADV
ejpam-4352	264	2	,	,	PUNCT
ejpam-4352	264	3	s	s	VERB
ejpam-4352	264	4	is	be	AUX
ejpam-4352	264	5	not	not	PART
ejpam-4352	264	6	a	a	DET
ejpam-4352	264	7	complement	complement	NOUN
ejpam-4352	264	8	-	-	PUNCT
ejpam-4352	264	9	super	super	ADJ
ejpam-4352	264	10	dominating	dominating	NOUN
ejpam-4352	264	11	set	set	NOUN
ejpam-4352	264	12	of	of	ADP
ejpam-4352	264	13	cn	cn	PROPN
ejpam-4352	264	14	,	,	PUNCT
ejpam-4352	264	15	a	a	DET
ejpam-4352	264	16	contradiction	contradiction	NOUN
ejpam-4352	264	17	.	.	PUNCT
ejpam-4352	265	1	therefore	therefore	ADV
ejpam-4352	265	2	,	,	PUNCT
ejpam-4352	265	3	γcs(cn	γcs(cn	PROPN
ejpam-4352	265	4	)	)	PUNCT
ejpam-4352	266	1	=	=	PUNCT
ejpam-4352	266	2	n−	n−	NOUN
ejpam-4352	266	3	2	2	NUM
ejpam-4352	266	4	.	.	PUNCT
ejpam-4352	266	5	theorem	theorem	NOUN
ejpam-4352	266	6	8	8	NUM
ejpam-4352	266	7	.	.	PUNCT
ejpam-4352	267	1	let	let	VERB
ejpam-4352	267	2	g	g	PRON
ejpam-4352	267	3	be	be	AUX
ejpam-4352	267	4	a	a	DET
ejpam-4352	267	5	graph	graph	NOUN
ejpam-4352	267	6	of	of	ADP
ejpam-4352	267	7	order	order	NOUN
ejpam-4352	267	8	n	n	PRON
ejpam-4352	267	9	≥	≥	NOUN
ejpam-4352	267	10	2	2	NUM
ejpam-4352	267	11	.	.	PUNCT
ejpam-4352	268	1	if	if	SCONJ
ejpam-4352	268	2	s	s	PROPN
ejpam-4352	268	3	is	be	AUX
ejpam-4352	268	4	a	a	DET
ejpam-4352	268	5	complement	complement	NOUN
ejpam-4352	268	6	-	-	PUNCT
ejpam-4352	268	7	super	super	ADJ
ejpam-4352	268	8	dominating	dominating	NOUN
ejpam-4352	268	9	set	set	NOUN
ejpam-4352	268	10	of	of	ADP
ejpam-4352	268	11	g	g	PROPN
ejpam-4352	268	12	,	,	PUNCT
ejpam-4352	268	13	then	then	ADV
ejpam-4352	268	14	v	v	X
ejpam-4352	268	15	(	(	PUNCT
ejpam-4352	268	16	g	g	NOUN
ejpam-4352	268	17	)	)	PUNCT
ejpam-4352	268	18	\	\	PUNCT
ejpam-4352	269	1	s	s	PART
ejpam-4352	269	2	contains	contain	VERB
ejpam-4352	269	3	at	at	ADP
ejpam-4352	269	4	most	most	ADV
ejpam-4352	269	5	a	a	DET
ejpam-4352	269	6	single	single	ADJ
ejpam-4352	269	7	isolated	isolated	ADJ
ejpam-4352	269	8	vertex	vertex	NOUN
ejpam-4352	269	9	of	of	ADP
ejpam-4352	269	10	g.	g.	PROPN
ejpam-4352	269	11	in	in	ADP
ejpam-4352	269	12	particular	particular	ADJ
ejpam-4352	269	13	,	,	PUNCT
ejpam-4352	269	14	if	if	SCONJ
ejpam-4352	269	15	g	g	PROPN
ejpam-4352	269	16	=	=	SYM
ejpam-4352	269	17	kn	kn	PROPN
ejpam-4352	269	18	,	,	PUNCT
ejpam-4352	269	19	then	then	ADV
ejpam-4352	269	20	s	s	VERB
ejpam-4352	269	21	=	=	SYM
ejpam-4352	269	22	v	v	PROPN
ejpam-4352	269	23	(	(	PUNCT
ejpam-4352	269	24	g)\{v	g)\{v	PROPN
ejpam-4352	269	25	}	}	PUNCT
ejpam-4352	269	26	is	be	AUX
ejpam-4352	269	27	a	a	DET
ejpam-4352	269	28	γcs	γcs	NOUN
ejpam-4352	269	29	-	-	PUNCT
ejpam-4352	269	30	set	set	NOUN
ejpam-4352	269	31	of	of	ADP
ejpam-4352	269	32	g	g	NOUN
ejpam-4352	269	33	for	for	ADP
ejpam-4352	269	34	each	each	DET
ejpam-4352	269	35	v	v	NUM
ejpam-4352	269	36	∈	∈	PROPN
ejpam-4352	269	37	v	v	NOUN
ejpam-4352	269	38	(	(	PUNCT
ejpam-4352	269	39	g	g	NOUN
ejpam-4352	269	40	)	)	PUNCT
ejpam-4352	269	41	,	,	PUNCT
ejpam-4352	269	42	that	that	ADV
ejpam-4352	269	43	is	is	ADV
ejpam-4352	269	44	,	,	PUNCT
ejpam-4352	269	45	γcs(g	γcs(g	X
ejpam-4352	269	46	)	)	PUNCT
ejpam-4352	269	47	=	=	SYM
ejpam-4352	269	48	n−1	n−1	PROPN
ejpam-4352	269	49	.	.	PUNCT
ejpam-4352	269	50	proof	proof	NOUN
ejpam-4352	269	51	.	.	PUNCT
ejpam-4352	270	1	if	if	SCONJ
ejpam-4352	270	2	s	s	VERB
ejpam-4352	270	3	=	=	SYM
ejpam-4352	270	4	v	v	X
ejpam-4352	270	5	(	(	PUNCT
ejpam-4352	270	6	g	g	NOUN
ejpam-4352	270	7	)	)	PUNCT
ejpam-4352	270	8	,	,	PUNCT
ejpam-4352	270	9	then	then	ADV
ejpam-4352	270	10	we	we	PRON
ejpam-4352	270	11	are	be	AUX
ejpam-4352	270	12	done	do	VERB
ejpam-4352	270	13	.	.	PUNCT
ejpam-4352	271	1	suppose	suppose	VERB
ejpam-4352	271	2	s	s	VERB
ejpam-4352	271	3	̸=	̸=	PROPN
ejpam-4352	271	4	v	v	NOUN
ejpam-4352	271	5	(	(	PUNCT
ejpam-4352	271	6	g	g	NOUN
ejpam-4352	271	7	)	)	PUNCT
ejpam-4352	271	8	and	and	CCONJ
ejpam-4352	271	9	let	let	VERB
ejpam-4352	271	10	v	v	NUM
ejpam-4352	271	11	∈	∈	PROPN
ejpam-4352	271	12	v	v	NOUN
ejpam-4352	271	13	(	(	PUNCT
ejpam-4352	271	14	g	g	NOUN
ejpam-4352	271	15	)	)	PUNCT
ejpam-4352	271	16	\	\	PUNCT
ejpam-4352	272	1	s.	s.	PROPN
ejpam-4352	272	2	then	then	ADV
ejpam-4352	272	3	there	there	PRON
ejpam-4352	272	4	exists	exist	VERB
ejpam-4352	272	5	w	w	PROPN
ejpam-4352	272	6	∈	∈	PROPN
ejpam-4352	272	7	s	s	VERB
ejpam-4352	272	8	such	such	ADJ
ejpam-4352	272	9	that	that	SCONJ
ejpam-4352	272	10	[	[	X
ejpam-4352	272	11	v	v	X
ejpam-4352	272	12	(	(	PUNCT
ejpam-4352	272	13	g	g	NOUN
ejpam-4352	272	14	)	)	PUNCT
ejpam-4352	272	15	\	\	NOUN
ejpam-4352	272	16	ng(w	ng(w	NOUN
ejpam-4352	272	17	)	)	PUNCT
ejpam-4352	272	18	]	]	PUNCT
ejpam-4352	273	1	∩	∩	NOUN
ejpam-4352	274	1	[	[	X
ejpam-4352	274	2	v	v	X
ejpam-4352	274	3	(	(	PUNCT
ejpam-4352	274	4	g	g	NOUN
ejpam-4352	274	5	)	)	PUNCT
ejpam-4352	274	6	\	\	PUNCT
ejpam-4352	275	1	s	s	X
ejpam-4352	275	2	]	]	X
ejpam-4352	275	3	=	=	PUNCT
ejpam-4352	275	4	{	{	PUNCT
ejpam-4352	275	5	v	v	NOUN
ejpam-4352	275	6	}	}	PUNCT
ejpam-4352	275	7	.	.	PUNCT
ejpam-4352	276	1	hence	hence	ADV
ejpam-4352	276	2	,	,	PUNCT
ejpam-4352	276	3	if	if	SCONJ
ejpam-4352	276	4	v	v	NOUN
ejpam-4352	276	5	is	be	AUX
ejpam-4352	276	6	an	an	DET
ejpam-4352	276	7	isolated	isolated	ADJ
ejpam-4352	276	8	vertex	vertex	NOUN
ejpam-4352	276	9	of	of	ADP
ejpam-4352	276	10	g	g	NOUN
ejpam-4352	276	11	,	,	PUNCT
ejpam-4352	276	12	then	then	ADV
ejpam-4352	276	13	there	there	PRON
ejpam-4352	276	14	can	can	AUX
ejpam-4352	276	15	be	be	AUX
ejpam-4352	276	16	no	no	DET
ejpam-4352	276	17	other	other	ADJ
ejpam-4352	276	18	isolated	isolate	VERB
ejpam-4352	276	19	vertex	vertex	NOUN
ejpam-4352	276	20	in	in	ADP
ejpam-4352	276	21	v	v	NOUN
ejpam-4352	276	22	(	(	PUNCT
ejpam-4352	276	23	g	g	NOUN
ejpam-4352	276	24	)	)	PUNCT
ejpam-4352	276	25	\	\	PUNCT
ejpam-4352	277	1	s.	s.	PROPN
ejpam-4352	277	2	in	in	ADP
ejpam-4352	277	3	other	other	ADJ
ejpam-4352	277	4	words	word	NOUN
ejpam-4352	277	5	,	,	PUNCT
ejpam-4352	277	6	v	v	NOUN
ejpam-4352	277	7	(	(	PUNCT
ejpam-4352	277	8	g	g	NOUN
ejpam-4352	277	9	)	)	PUNCT
ejpam-4352	277	10	\	\	PUNCT
ejpam-4352	278	1	s	s	PART
ejpam-4352	278	2	contains	contain	VERB
ejpam-4352	278	3	at	at	ADP
ejpam-4352	278	4	most	most	ADV
ejpam-4352	278	5	a	a	DET
ejpam-4352	278	6	single	single	ADJ
ejpam-4352	278	7	isolated	isolated	ADJ
ejpam-4352	278	8	vertex	vertex	NOUN
ejpam-4352	278	9	of	of	ADP
ejpam-4352	278	10	g.	g.	PROPN
ejpam-4352	278	11	suppose	suppose	VERB
ejpam-4352	278	12	now	now	ADV
ejpam-4352	278	13	that	that	SCONJ
ejpam-4352	278	14	g	g	PROPN
ejpam-4352	278	15	=	=	PROPN
ejpam-4352	278	16	kn	kn	PROPN
ejpam-4352	278	17	and	and	CCONJ
ejpam-4352	278	18	let	let	VERB
ejpam-4352	278	19	v	v	NUM
ejpam-4352	278	20	∈	∈	PROPN
ejpam-4352	278	21	v	v	NOUN
ejpam-4352	278	22	(	(	PUNCT
ejpam-4352	278	23	g	g	NOUN
ejpam-4352	278	24	)	)	PUNCT
ejpam-4352	278	25	.	.	PUNCT
ejpam-4352	279	1	then	then	ADV
ejpam-4352	279	2	clearly	clearly	ADV
ejpam-4352	279	3	,	,	PUNCT
ejpam-4352	279	4	s	s	VERB
ejpam-4352	279	5	=	=	SYM
ejpam-4352	279	6	v	v	X
ejpam-4352	279	7	(	(	PUNCT
ejpam-4352	279	8	g	g	NOUN
ejpam-4352	279	9	)	)	PUNCT
ejpam-4352	279	10	\	\	NOUN
ejpam-4352	279	11	{	{	PUNCT
ejpam-4352	279	12	v	v	NOUN
ejpam-4352	279	13	}	}	PUNCT
ejpam-4352	279	14	is	be	AUX
ejpam-4352	279	15	a	a	DET
ejpam-4352	279	16	complement	complement	NOUN
ejpam-4352	279	17	-	-	PUNCT
ejpam-4352	279	18	super	super	ADJ
ejpam-4352	279	19	dominating	dominating	NOUN
ejpam-4352	279	20	set	set	NOUN
ejpam-4352	279	21	of	of	ADP
ejpam-4352	279	22	g.	g.	PROPN
ejpam-4352	279	23	since	since	SCONJ
ejpam-4352	279	24	it	it	PRON
ejpam-4352	279	25	is	be	AUX
ejpam-4352	279	26	not	not	PART
ejpam-4352	279	27	possible	possible	ADJ
ejpam-4352	279	28	for	for	ADP
ejpam-4352	279	29	v	v	NOUN
ejpam-4352	279	30	(	(	PUNCT
ejpam-4352	279	31	g	g	NOUN
ejpam-4352	279	32	)	)	PUNCT
ejpam-4352	279	33	\	\	PROPN
ejpam-4352	280	1	s	s	VERB
ejpam-4352	280	2	to	to	PART
ejpam-4352	280	3	contain	contain	VERB
ejpam-4352	280	4	more	more	ADJ
ejpam-4352	280	5	than	than	ADP
ejpam-4352	280	6	one	one	NUM
ejpam-4352	280	7	isolated	isolated	ADJ
ejpam-4352	280	8	vertices	vertex	NOUN
ejpam-4352	280	9	,	,	PUNCT
ejpam-4352	280	10	it	it	PRON
ejpam-4352	280	11	follows	follow	VERB
ejpam-4352	280	12	that	that	SCONJ
ejpam-4352	280	13	s	s	VERB
ejpam-4352	280	14	is	be	AUX
ejpam-4352	280	15	a	a	DET
ejpam-4352	280	16	γcs	γcs	NOUN
ejpam-4352	280	17	-	-	PUNCT
ejpam-4352	280	18	set	set	NOUN
ejpam-4352	280	19	of	of	ADP
ejpam-4352	280	20	g.	g.	PROPN
ejpam-4352	280	21	thus	thus	ADV
ejpam-4352	280	22	,	,	PUNCT
ejpam-4352	280	23	γcs(g	γcs(g	X
ejpam-4352	280	24	)	)	PUNCT
ejpam-4352	280	25	=	=	SYM
ejpam-4352	281	1	n−1	n−1	PROPN
ejpam-4352	281	2	.	.	PUNCT
ejpam-4352	281	3	theorem	theorem	VERB
ejpam-4352	281	4	9	9	NUM
ejpam-4352	281	5	.	.	PUNCT
ejpam-4352	282	1	let	let	VERB
ejpam-4352	282	2	g1	g1	PROPN
ejpam-4352	282	3	,	,	PUNCT
ejpam-4352	282	4	g2	g2	PROPN
ejpam-4352	282	5	,	,	PUNCT
ejpam-4352	282	6	·	·	PUNCT
ejpam-4352	282	7	·	·	PUNCT
ejpam-4352	282	8	·	·	PUNCT
ejpam-4352	282	9	,	,	PUNCT
ejpam-4352	282	10	gn	gn	X
ejpam-4352	282	11	be	be	AUX
ejpam-4352	282	12	the	the	DET
ejpam-4352	282	13	components	component	NOUN
ejpam-4352	282	14	of	of	ADP
ejpam-4352	282	15	a	a	DET
ejpam-4352	282	16	graph	graph	NOUN
ejpam-4352	282	17	g	g	NOUN
ejpam-4352	282	18	,	,	PUNCT
ejpam-4352	282	19	where	where	SCONJ
ejpam-4352	282	20	n	n	PRON
ejpam-4352	282	21	≥	≥	NOUN
ejpam-4352	282	22	2	2	NUM
ejpam-4352	282	23	.	.	PUNCT
ejpam-4352	283	1	then	then	ADV
ejpam-4352	283	2	s	s	VERB
ejpam-4352	283	3	⊆	⊆	NUM
ejpam-4352	283	4	v	v	NOUN
ejpam-4352	283	5	(	(	PUNCT
ejpam-4352	283	6	g	g	NOUN
ejpam-4352	283	7	)	)	PUNCT
ejpam-4352	283	8	is	be	AUX
ejpam-4352	283	9	a	a	DET
ejpam-4352	283	10	complement	complement	NOUN
ejpam-4352	283	11	-	-	PUNCT
ejpam-4352	283	12	super	super	ADJ
ejpam-4352	283	13	dominating	dominating	NOUN
ejpam-4352	283	14	set	set	NOUN
ejpam-4352	283	15	of	of	ADP
ejpam-4352	283	16	g	g	PROPN
ejpam-4352	283	17	if	if	SCONJ
ejpam-4352	284	1	and	and	CCONJ
ejpam-4352	284	2	only	only	ADV
ejpam-4352	284	3	if	if	SCONJ
ejpam-4352	284	4	one	one	NUM
ejpam-4352	284	5	of	of	ADP
ejpam-4352	284	6	the	the	DET
ejpam-4352	284	7	following	follow	VERB
ejpam-4352	284	8	holds	hold	VERB
ejpam-4352	284	9	:	:	PUNCT
ejpam-4352	284	10	(	(	PUNCT
ejpam-4352	284	11	i	i	NOUN
ejpam-4352	284	12	)	)	PUNCT
ejpam-4352	284	13	s	s	PART
ejpam-4352	285	1	=	=	X
ejpam-4352	285	2	d	d	X
ejpam-4352	285	3	∪	∪	X
ejpam-4352	285	4	[	[	X
ejpam-4352	285	5	∪j	∪j	X
ejpam-4352	285	6	̸=kv	̸=kv	PROPN
ejpam-4352	285	7	(	(	PUNCT
ejpam-4352	285	8	gj	gj	PROPN
ejpam-4352	285	9	)	)	PUNCT
ejpam-4352	285	10	]	]	PUNCT
ejpam-4352	285	11	for	for	ADP
ejpam-4352	285	12	some	some	DET
ejpam-4352	285	13	k	k	PROPN
ejpam-4352	285	14	≤	≤	NOUN
ejpam-4352	285	15	n	n	CCONJ
ejpam-4352	285	16	and	and	CCONJ
ejpam-4352	285	17	for	for	ADP
ejpam-4352	285	18	some	some	DET
ejpam-4352	285	19	complement	complement	NOUN
ejpam-4352	285	20	-	-	PUNCT
ejpam-4352	285	21	super	super	ADV
ejpam-4352	285	22	dominating	dominating	NOUN
ejpam-4352	285	23	set	set	NOUN
ejpam-4352	285	24	d	d	PROPN
ejpam-4352	285	25	of	of	ADP
ejpam-4352	285	26	gk	gk	PROPN
ejpam-4352	285	27	.	.	PUNCT
ejpam-4352	285	28	(	(	PUNCT
ejpam-4352	285	29	ii	ii	NOUN
ejpam-4352	285	30	)	)	PUNCT
ejpam-4352	285	31	s	s	PART
ejpam-4352	285	32	=	=	SYM
ejpam-4352	285	33	v	v	X
ejpam-4352	285	34	(	(	PUNCT
ejpam-4352	285	35	g	g	NOUN
ejpam-4352	285	36	)	)	PUNCT
ejpam-4352	285	37	\	\	NOUN
ejpam-4352	285	38	{	{	PUNCT
ejpam-4352	285	39	v	v	NOUN
ejpam-4352	285	40	}	}	PUNCT
ejpam-4352	285	41	for	for	ADP
ejpam-4352	285	42	some	some	DET
ejpam-4352	285	43	isolated	isolate	VERB
ejpam-4352	285	44	vertex	vertex	NOUN
ejpam-4352	285	45	v	v	NOUN
ejpam-4352	285	46	or	or	CCONJ
ejpam-4352	285	47	s	s	NOUN
ejpam-4352	285	48	=	=	PUNCT
ejpam-4352	286	1	[	[	X
ejpam-4352	286	2	v	v	X
ejpam-4352	286	3	(	(	PUNCT
ejpam-4352	286	4	gk	gk	PROPN
ejpam-4352	286	5	)	)	PUNCT
ejpam-4352	286	6	\	\	PROPN
ejpam-4352	286	7	{	{	PUNCT
ejpam-4352	286	8	v	v	NOUN
ejpam-4352	286	9	}	}	PUNCT
ejpam-4352	286	10	]	]	PUNCT
ejpam-4352	286	11	∪	∪	ADP
ejpam-4352	286	12	[	[	PUNCT
ejpam-4352	286	13	∪j	∪j	X
ejpam-4352	286	14	̸=kv	̸=kv	PROPN
ejpam-4352	286	15	(	(	PUNCT
ejpam-4352	286	16	gj	gj	PROPN
ejpam-4352	286	17	)	)	PUNCT
ejpam-4352	286	18	]	]	PUNCT
ejpam-4352	286	19	for	for	ADP
ejpam-4352	286	20	some	some	DET
ejpam-4352	286	21	non	non	ADJ
ejpam-4352	286	22	-	-	ADJ
ejpam-4352	286	23	trivial	trivial	ADJ
ejpam-4352	286	24	component	component	NOUN
ejpam-4352	286	25	gk	gk	NOUN
ejpam-4352	286	26	and	and	CCONJ
ejpam-4352	286	27	for	for	ADP
ejpam-4352	286	28	some	some	DET
ejpam-4352	286	29	v	v	NUM
ejpam-4352	286	30	∈	∈	PROPN
ejpam-4352	286	31	v	v	NOUN
ejpam-4352	286	32	(	(	PUNCT
ejpam-4352	286	33	gk	gk	PROPN
ejpam-4352	286	34	)	)	PUNCT
ejpam-4352	286	35	.	.	PUNCT
ejpam-4352	287	1	(	(	PUNCT
ejpam-4352	287	2	iii	iii	X
ejpam-4352	287	3	)	)	PUNCT
ejpam-4352	287	4	s	s	PART
ejpam-4352	288	1	=	=	PUNCT
ejpam-4352	289	1	[	[	X
ejpam-4352	289	2	v	v	X
ejpam-4352	289	3	(	(	PUNCT
ejpam-4352	289	4	gk)\{v}]∪[v	gk)\{v}]∪[v	PROPN
ejpam-4352	289	5	(	(	PUNCT
ejpam-4352	289	6	gr)\{w}]∪	gr)\{w}]∪	PROPN
ejpam-4352	289	7	[	[	PUNCT
ejpam-4352	289	8	⋃	⋃	PROPN
ejpam-4352	289	9	j	j	PROPN
ejpam-4352	289	10	̸=k	̸=k	PROPN
ejpam-4352	289	11	,	,	PUNCT
ejpam-4352	289	12	r	r	NOUN
ejpam-4352	289	13	v	v	NOUN
ejpam-4352	289	14	(	(	PUNCT
ejpam-4352	289	15	gj	gj	NOUN
ejpam-4352	289	16	)	)	PUNCT
ejpam-4352	289	17	]	]	PUNCT
ejpam-4352	289	18	for	for	ADP
ejpam-4352	289	19	distinct	distinct	ADJ
ejpam-4352	289	20	non	non	ADJ
ejpam-4352	289	21	-	-	ADJ
ejpam-4352	289	22	trivial	trivial	ADJ
ejpam-4352	289	23	components	component	NOUN
ejpam-4352	289	24	gk	gk	PROPN
ejpam-4352	289	25	and	and	CCONJ
ejpam-4352	289	26	gr	gr	PROPN
ejpam-4352	289	27	and	and	CCONJ
ejpam-4352	289	28	for	for	ADP
ejpam-4352	289	29	some	some	DET
ejpam-4352	289	30	v	v	NUM
ejpam-4352	289	31	∈	∈	PROPN
ejpam-4352	289	32	v	v	NOUN
ejpam-4352	289	33	(	(	PUNCT
ejpam-4352	289	34	gk	gk	PROPN
ejpam-4352	289	35	)	)	PUNCT
ejpam-4352	289	36	and	and	CCONJ
ejpam-4352	289	37	w	w	PROPN
ejpam-4352	289	38	∈	∈	PROPN
ejpam-4352	289	39	v	v	ADP
ejpam-4352	289	40	(	(	PUNCT
ejpam-4352	289	41	gr	gr	NOUN
ejpam-4352	289	42	)	)	PUNCT
ejpam-4352	289	43	.	.	PUNCT
ejpam-4352	290	1	proof	proof	NOUN
ejpam-4352	290	2	.	.	PUNCT
ejpam-4352	291	1	suppose	suppose	VERB
ejpam-4352	291	2	s	s	PRON
ejpam-4352	291	3	is	be	AUX
ejpam-4352	291	4	a	a	DET
ejpam-4352	291	5	complement	complement	NOUN
ejpam-4352	291	6	-	-	PUNCT
ejpam-4352	291	7	super	super	ADJ
ejpam-4352	291	8	dominating	dominating	NOUN
ejpam-4352	291	9	set	set	NOUN
ejpam-4352	291	10	of	of	ADP
ejpam-4352	291	11	g.	g.	PROPN
ejpam-4352	291	12	if	if	SCONJ
ejpam-4352	291	13	s	s	VERB
ejpam-4352	291	14	=	=	SYM
ejpam-4352	291	15	v	v	X
ejpam-4352	291	16	(	(	PUNCT
ejpam-4352	291	17	g	g	NOUN
ejpam-4352	291	18	)	)	PUNCT
ejpam-4352	291	19	,	,	PUNCT
ejpam-4352	291	20	then	then	ADV
ejpam-4352	291	21	we	we	PRON
ejpam-4352	291	22	may	may	AUX
ejpam-4352	291	23	take	take	VERB
ejpam-4352	291	24	d	d	NOUN
ejpam-4352	291	25	=	=	SYM
ejpam-4352	291	26	v	v	PROPN
ejpam-4352	291	27	(	(	PUNCT
ejpam-4352	291	28	g1	g1	PROPN
ejpam-4352	291	29	)	)	PUNCT
ejpam-4352	291	30	which	which	PRON
ejpam-4352	291	31	is	be	AUX
ejpam-4352	291	32	a	a	DET
ejpam-4352	291	33	complement	complement	NOUN
ejpam-4352	291	34	-	-	PUNCT
ejpam-4352	291	35	super	super	ADJ
ejpam-4352	291	36	dominating	dominating	NOUN
ejpam-4352	291	37	set	set	NOUN
ejpam-4352	291	38	of	of	ADP
ejpam-4352	291	39	g1	g1	NOUN
ejpam-4352	291	40	.	.	PUNCT
ejpam-4352	292	1	hence	hence	ADV
ejpam-4352	292	2	,	,	PUNCT
ejpam-4352	292	3	s	s	PROPN
ejpam-4352	292	4	of	of	ADP
ejpam-4352	292	5	type	type	NOUN
ejpam-4352	292	6	(	(	PUNCT
ejpam-4352	292	7	i	i	NOUN
ejpam-4352	292	8	)	)	PUNCT
ejpam-4352	292	9	.	.	PUNCT
ejpam-4352	293	1	suppose	suppose	VERB
ejpam-4352	293	2	s	s	VERB
ejpam-4352	293	3	̸=	̸=	PROPN
ejpam-4352	293	4	v	v	NOUN
ejpam-4352	293	5	(	(	PUNCT
ejpam-4352	293	6	g	g	NOUN
ejpam-4352	293	7	)	)	PUNCT
ejpam-4352	293	8	.	.	PUNCT
ejpam-4352	294	1	let	let	VERB
ejpam-4352	294	2	v	v	NUM
ejpam-4352	294	3	∈	∈	PROPN
ejpam-4352	294	4	v	v	NOUN
ejpam-4352	294	5	(	(	PUNCT
ejpam-4352	294	6	g	g	NOUN
ejpam-4352	294	7	)	)	PUNCT
ejpam-4352	294	8	\	\	PROPN
ejpam-4352	294	9	s	s	PART
ejpam-4352	294	10	and	and	CCONJ
ejpam-4352	294	11	let	let	VERB
ejpam-4352	294	12	k	k	PROPN
ejpam-4352	294	13	∈	∈	PROPN
ejpam-4352	294	14	{	{	PUNCT
ejpam-4352	294	15	1	1	NUM
ejpam-4352	294	16	,	,	PUNCT
ejpam-4352	294	17	2	2	NUM
ejpam-4352	294	18	,	,	PUNCT
ejpam-4352	294	19	·	·	PUNCT
ejpam-4352	294	20	·	·	PUNCT
ejpam-4352	294	21	·	·	PUNCT
ejpam-4352	294	22	,	,	PUNCT
ejpam-4352	294	23	n	n	CCONJ
ejpam-4352	294	24	}	}	PUNCT
ejpam-4352	294	25	such	such	ADJ
ejpam-4352	294	26	that	that	PRON
ejpam-4352	294	27	v	v	NUM
ejpam-4352	294	28	∈	∈	PROPN
ejpam-4352	294	29	v	v	NOUN
ejpam-4352	294	30	(	(	PUNCT
ejpam-4352	294	31	gk	gk	PROPN
ejpam-4352	294	32	)	)	PUNCT
ejpam-4352	294	33	.	.	PUNCT
ejpam-4352	295	1	s.	s.	PROPN
ejpam-4352	295	2	canoy	canoy	PROPN
ejpam-4352	295	3	,	,	PUNCT
ejpam-4352	295	4	jr	jr	PROPN
ejpam-4352	295	5	.	.	PROPN
ejpam-4352	295	6	,	,	PUNCT
ejpam-4352	295	7	g.	g.	PROPN
ejpam-4352	295	8	salasalan	salasalan	PROPN
ejpam-4352	295	9	/	/	SYM
ejpam-4352	295	10	eur	eur	PROPN
ejpam-4352	295	11	.	.	PUNCT
ejpam-4352	296	1	j.	j.	PROPN
ejpam-4352	296	2	pure	pure	PROPN
ejpam-4352	296	3	appl	appl	PROPN
ejpam-4352	296	4	.	.	PROPN
ejpam-4352	296	5	math	math	PROPN
ejpam-4352	296	6	,	,	PUNCT
ejpam-4352	296	7	15	15	NUM
ejpam-4352	296	8	(	(	PUNCT
ejpam-4352	296	9	2	2	NUM
ejpam-4352	296	10	)	)	PUNCT
ejpam-4352	296	11	(	(	PUNCT
ejpam-4352	296	12	2022	2022	NUM
ejpam-4352	296	13	)	)	PUNCT
ejpam-4352	296	14	,	,	PUNCT
ejpam-4352	296	15	342	342	NUM
ejpam-4352	296	16	-	-	SYM
ejpam-4352	296	17	353	353	NUM
ejpam-4352	296	18	348	348	NUM
ejpam-4352	296	19	since	since	SCONJ
ejpam-4352	296	20	s	s	NOUN
ejpam-4352	296	21	is	be	AUX
ejpam-4352	296	22	a	a	DET
ejpam-4352	296	23	complement	complement	NOUN
ejpam-4352	296	24	-	-	PUNCT
ejpam-4352	296	25	super	super	ADJ
ejpam-4352	296	26	dominating	dominating	NOUN
ejpam-4352	296	27	set	set	NOUN
ejpam-4352	296	28	of	of	ADP
ejpam-4352	296	29	g	g	NOUN
ejpam-4352	296	30	,	,	PUNCT
ejpam-4352	296	31	there	there	PRON
ejpam-4352	296	32	exists	exist	VERB
ejpam-4352	296	33	z	z	PROPN
ejpam-4352	296	34	∈	∈	PROPN
ejpam-4352	296	35	s	s	VERB
ejpam-4352	297	1	such	such	ADJ
ejpam-4352	297	2	that	that	SCONJ
ejpam-4352	297	3	[	[	X
ejpam-4352	297	4	v	v	X
ejpam-4352	297	5	(	(	PUNCT
ejpam-4352	297	6	g	g	NOUN
ejpam-4352	297	7	)	)	PUNCT
ejpam-4352	297	8	\	\	NOUN
ejpam-4352	297	9	ng(z	ng(z	PROPN
ejpam-4352	297	10	)	)	PUNCT
ejpam-4352	297	11	]	]	PUNCT
ejpam-4352	297	12	∩	∩	NOUN
ejpam-4352	298	1	[	[	X
ejpam-4352	298	2	v	v	X
ejpam-4352	298	3	(	(	PUNCT
ejpam-4352	298	4	g	g	NOUN
ejpam-4352	298	5	)	)	PUNCT
ejpam-4352	298	6	\	\	PUNCT
ejpam-4352	299	1	s	s	X
ejpam-4352	299	2	]	]	X
ejpam-4352	299	3	=	=	PUNCT
ejpam-4352	299	4	{	{	PUNCT
ejpam-4352	299	5	v	v	NOUN
ejpam-4352	299	6	}	}	PUNCT
ejpam-4352	299	7	.	.	PUNCT
ejpam-4352	300	1	suppose	suppose	VERB
ejpam-4352	300	2	first	first	ADV
ejpam-4352	300	3	that	that	SCONJ
ejpam-4352	300	4	z	z	PROPN
ejpam-4352	300	5	∈	∈	PROPN
ejpam-4352	300	6	v	v	ADP
ejpam-4352	300	7	(	(	PUNCT
ejpam-4352	300	8	gk	gk	PROPN
ejpam-4352	300	9	)	)	PUNCT
ejpam-4352	300	10	.	.	PUNCT
ejpam-4352	301	1	then	then	ADV
ejpam-4352	301	2	clearly	clearly	ADV
ejpam-4352	301	3	,	,	PUNCT
ejpam-4352	301	4	∪j	∪j	PROPN
ejpam-4352	301	5	̸=kv	̸=kv	PROPN
ejpam-4352	301	6	(	(	PUNCT
ejpam-4352	301	7	gj	gj	PROPN
ejpam-4352	301	8	)	)	PUNCT
ejpam-4352	301	9	⊆	⊆	NUM
ejpam-4352	301	10	s.	s.	PROPN
ejpam-4352	301	11	let	let	VERB
ejpam-4352	301	12	d	d	NOUN
ejpam-4352	301	13	=	=	SYM
ejpam-4352	301	14	s∩v	s∩v	PROPN
ejpam-4352	301	15	(	(	PUNCT
ejpam-4352	301	16	gk	gk	PROPN
ejpam-4352	301	17	)	)	PUNCT
ejpam-4352	301	18	.	.	PUNCT
ejpam-4352	302	1	then	then	ADV
ejpam-4352	302	2	[	[	X
ejpam-4352	302	3	v	v	X
ejpam-4352	302	4	(	(	PUNCT
ejpam-4352	302	5	gk)\ngk	gk)\ngk	PROPN
ejpam-4352	302	6	(	(	PUNCT
ejpam-4352	302	7	z)]∩[v	z)]∩[v	PROPN
ejpam-4352	302	8	(	(	PUNCT
ejpam-4352	302	9	gk)\d	gk)\d	NOUN
ejpam-4352	302	10	]	]	X
ejpam-4352	302	11	=	=	PUNCT
ejpam-4352	302	12	{	{	PUNCT
ejpam-4352	302	13	v	v	NOUN
ejpam-4352	302	14	}	}	PUNCT
ejpam-4352	302	15	.	.	PUNCT
ejpam-4352	303	1	thus	thus	ADV
ejpam-4352	303	2	,	,	PUNCT
ejpam-4352	303	3	if	if	SCONJ
ejpam-4352	303	4	d	d	PROPN
ejpam-4352	303	5	=	=	SYM
ejpam-4352	303	6	v	v	PROPN
ejpam-4352	303	7	(	(	PUNCT
ejpam-4352	303	8	gk)\{v	gk)\{v	NOUN
ejpam-4352	303	9	}	}	PUNCT
ejpam-4352	303	10	,	,	PUNCT
ejpam-4352	303	11	then	then	ADV
ejpam-4352	303	12	d	d	PROPN
ejpam-4352	303	13	is	be	AUX
ejpam-4352	303	14	a	a	DET
ejpam-4352	303	15	complement	complement	NOUN
ejpam-4352	303	16	-	-	PUNCT
ejpam-4352	303	17	super	super	ADJ
ejpam-4352	303	18	dominating	dominating	NOUN
ejpam-4352	303	19	set	set	NOUN
ejpam-4352	303	20	of	of	ADP
ejpam-4352	303	21	gk	gk	PROPN
ejpam-4352	303	22	.	.	PROPN
ejpam-4352	303	23	suppose	suppose	VERB
ejpam-4352	304	1	d	d	X
ejpam-4352	304	2	̸=	̸=	PROPN
ejpam-4352	304	3	v	v	PROPN
ejpam-4352	304	4	(	(	PUNCT
ejpam-4352	304	5	gk	gk	PROPN
ejpam-4352	304	6	)	)	PUNCT
ejpam-4352	304	7	\	\	PROPN
ejpam-4352	304	8	{	{	PUNCT
ejpam-4352	304	9	v	v	NOUN
ejpam-4352	304	10	}	}	PUNCT
ejpam-4352	304	11	and	and	CCONJ
ejpam-4352	304	12	let	let	VERB
ejpam-4352	304	13	x	x	SYM
ejpam-4352	304	14	∈	∈	PROPN
ejpam-4352	304	15	v	v	ADP
ejpam-4352	304	16	(	(	PUNCT
ejpam-4352	304	17	gk)\(d∪{v	gk)\(d∪{v	NOUN
ejpam-4352	304	18	}	}	PUNCT
ejpam-4352	304	19	)	)	PUNCT
ejpam-4352	304	20	.	.	PUNCT
ejpam-4352	305	1	then	then	ADV
ejpam-4352	305	2	there	there	PRON
ejpam-4352	305	3	exists	exist	VERB
ejpam-4352	305	4	y	y	PROPN
ejpam-4352	305	5	∈	∈	PROPN
ejpam-4352	305	6	s	s	VERB
ejpam-4352	305	7	such	such	ADJ
ejpam-4352	305	8	that	that	SCONJ
ejpam-4352	305	9	[	[	X
ejpam-4352	305	10	v	v	X
ejpam-4352	305	11	(	(	PUNCT
ejpam-4352	305	12	g)\ng(y)]∩[v	g)\ng(y)]∩[v	NOUN
ejpam-4352	305	13	(	(	PUNCT
ejpam-4352	305	14	g)\s	g)\s	NOUN
ejpam-4352	305	15	]	]	PUNCT
ejpam-4352	305	16	=	=	PUNCT
ejpam-4352	305	17	{	{	PUNCT
ejpam-4352	305	18	x	x	NOUN
ejpam-4352	305	19	}	}	PUNCT
ejpam-4352	305	20	.	.	PUNCT
ejpam-4352	306	1	since	since	SCONJ
ejpam-4352	306	2	x	x	PROPN
ejpam-4352	306	3	̸=	̸=	PROPN
ejpam-4352	306	4	v	v	NOUN
ejpam-4352	306	5	,	,	PUNCT
ejpam-4352	306	6	it	it	PRON
ejpam-4352	306	7	follows	follow	VERB
ejpam-4352	306	8	that	that	SCONJ
ejpam-4352	307	1	y	y	PROPN
ejpam-4352	307	2	∈	∈	PROPN
ejpam-4352	307	3	d	d	NOUN
ejpam-4352	307	4	and	and	CCONJ
ejpam-4352	307	5	[	[	X
ejpam-4352	307	6	v	v	X
ejpam-4352	307	7	(	(	PUNCT
ejpam-4352	307	8	gk	gk	PROPN
ejpam-4352	307	9	)	)	PUNCT
ejpam-4352	307	10	\	\	PROPN
ejpam-4352	307	11	ngk	ngk	PROPN
ejpam-4352	307	12	(	(	PUNCT
ejpam-4352	307	13	y	y	NOUN
ejpam-4352	307	14	)	)	PUNCT
ejpam-4352	307	15	]	]	PUNCT
ejpam-4352	308	1	∩	∩	NOUN
ejpam-4352	308	2	[	[	X
ejpam-4352	308	3	v	v	X
ejpam-4352	308	4	(	(	PUNCT
ejpam-4352	308	5	gk	gk	NOUN
ejpam-4352	308	6	)	)	PUNCT
ejpam-4352	308	7	\d	\d	NOUN
ejpam-4352	308	8	]	]	X
ejpam-4352	308	9	=	=	PUNCT
ejpam-4352	308	10	{	{	PUNCT
ejpam-4352	308	11	x	x	NOUN
ejpam-4352	308	12	}	}	PUNCT
ejpam-4352	308	13	.	.	PUNCT
ejpam-4352	309	1	thus	thus	ADV
ejpam-4352	309	2	,	,	PUNCT
ejpam-4352	309	3	d	d	PRON
ejpam-4352	309	4	is	be	AUX
ejpam-4352	309	5	a	a	DET
ejpam-4352	309	6	complement	complement	NOUN
ejpam-4352	309	7	-	-	PUNCT
ejpam-4352	309	8	super	super	ADJ
ejpam-4352	309	9	dominating	dominating	NOUN
ejpam-4352	309	10	set	set	NOUN
ejpam-4352	309	11	of	of	ADP
ejpam-4352	309	12	gk	gk	PROPN
ejpam-4352	309	13	,	,	PUNCT
ejpam-4352	309	14	showing	show	VERB
ejpam-4352	309	15	that	that	SCONJ
ejpam-4352	309	16	(	(	PUNCT
ejpam-4352	309	17	i	i	NOUN
ejpam-4352	309	18	)	)	PUNCT
ejpam-4352	309	19	holds	hold	VERB
ejpam-4352	309	20	.	.	PUNCT
ejpam-4352	310	1	next	next	ADV
ejpam-4352	310	2	,	,	PUNCT
ejpam-4352	310	3	suppose	suppose	VERB
ejpam-4352	310	4	that	that	SCONJ
ejpam-4352	310	5	z	z	PROPN
ejpam-4352	310	6	∈	∈	PROPN
ejpam-4352	310	7	v	v	ADP
ejpam-4352	310	8	(	(	PUNCT
ejpam-4352	310	9	gr	gr	NOUN
ejpam-4352	310	10	)	)	PUNCT
ejpam-4352	310	11	for	for	ADP
ejpam-4352	310	12	some	some	DET
ejpam-4352	310	13	r	r	NOUN
ejpam-4352	310	14	≤	≤	NOUN
ejpam-4352	310	15	n	n	CCONJ
ejpam-4352	310	16	with	with	ADP
ejpam-4352	310	17	r	r	NOUN
ejpam-4352	310	18	̸=	̸=	PROPN
ejpam-4352	310	19	k	k	NOUN
ejpam-4352	310	20	and	and	CCONJ
ejpam-4352	310	21	let	let	VERB
ejpam-4352	310	22	d∗	d∗	NOUN
ejpam-4352	310	23	=	=	SYM
ejpam-4352	310	24	s	s	NOUN
ejpam-4352	310	25	∩	∩	ADJ
ejpam-4352	310	26	v	v	X
ejpam-4352	310	27	(	(	PUNCT
ejpam-4352	310	28	gr	gr	NOUN
ejpam-4352	310	29	)	)	PUNCT
ejpam-4352	310	30	.	.	PUNCT
ejpam-4352	311	1	note	note	VERB
ejpam-4352	311	2	that	that	SCONJ
ejpam-4352	311	3	if	if	SCONJ
ejpam-4352	311	4	gk	gk	PROPN
ejpam-4352	311	5	̸=	̸=	PROPN
ejpam-4352	311	6	⟨v⟩	⟨v⟩	PROPN
ejpam-4352	311	7	,	,	PUNCT
ejpam-4352	311	8	then	then	ADV
ejpam-4352	311	9	[	[	X
ejpam-4352	311	10	v	v	X
ejpam-4352	311	11	(	(	PUNCT
ejpam-4352	311	12	gk)\{v}]∪	gk)\{v}]∪	PROPN
ejpam-4352	311	13	[	[	X
ejpam-4352	311	14	∪j	∪j	X
ejpam-4352	311	15	̸=k	̸=k	PROPN
ejpam-4352	311	16	,	,	PUNCT
ejpam-4352	311	17	rv	rv	PROPN
ejpam-4352	311	18	(	(	PUNCT
ejpam-4352	311	19	gj	gj	PROPN
ejpam-4352	311	20	)	)	PUNCT
ejpam-4352	311	21	]	]	PUNCT
ejpam-4352	311	22	⊆	⊆	NUM
ejpam-4352	311	23	s.	s.	PROPN
ejpam-4352	311	24	suppose	suppose	VERB
ejpam-4352	311	25	first	first	ADV
ejpam-4352	311	26	that	that	SCONJ
ejpam-4352	311	27	d∗	d∗	NOUN
ejpam-4352	311	28	=	=	SYM
ejpam-4352	311	29	v	v	NOUN
ejpam-4352	311	30	(	(	PUNCT
ejpam-4352	311	31	gr	gr	NOUN
ejpam-4352	311	32	)	)	PUNCT
ejpam-4352	311	33	.	.	PUNCT
ejpam-4352	312	1	if	if	SCONJ
ejpam-4352	312	2	gk	gk	PROPN
ejpam-4352	312	3	=	=	SYM
ejpam-4352	312	4	⟨v⟩	⟨v⟩	PROPN
ejpam-4352	312	5	,	,	PUNCT
ejpam-4352	312	6	then	then	ADV
ejpam-4352	312	7	s	s	VERB
ejpam-4352	312	8	=	=	SYM
ejpam-4352	312	9	v	v	PROPN
ejpam-4352	312	10	(	(	PUNCT
ejpam-4352	312	11	g	g	NOUN
ejpam-4352	312	12	)	)	PUNCT
ejpam-4352	312	13	\	\	NOUN
ejpam-4352	312	14	{	{	PUNCT
ejpam-4352	312	15	v	v	NOUN
ejpam-4352	312	16	}	}	PUNCT
ejpam-4352	312	17	.	.	PUNCT
ejpam-4352	313	1	otherwise	otherwise	ADV
ejpam-4352	313	2	,	,	PUNCT
ejpam-4352	313	3	s	s	VERB
ejpam-4352	313	4	=	=	PUNCT
ejpam-4352	314	1	[	[	X
ejpam-4352	314	2	v	v	X
ejpam-4352	314	3	(	(	PUNCT
ejpam-4352	314	4	gk	gk	PROPN
ejpam-4352	314	5	)	)	PUNCT
ejpam-4352	314	6	\	\	PROPN
ejpam-4352	314	7	{	{	PUNCT
ejpam-4352	314	8	v	v	NOUN
ejpam-4352	314	9	}	}	PUNCT
ejpam-4352	314	10	]	]	PUNCT
ejpam-4352	314	11	∪	∪	ADP
ejpam-4352	314	12	[	[	PUNCT
ejpam-4352	314	13	∪j	∪j	X
ejpam-4352	314	14	̸=kv	̸=kv	PROPN
ejpam-4352	314	15	(	(	PUNCT
ejpam-4352	314	16	gj	gj	PROPN
ejpam-4352	314	17	)	)	PUNCT
ejpam-4352	314	18	]	]	PUNCT
ejpam-4352	314	19	.	.	PUNCT
ejpam-4352	315	1	hence	hence	ADV
ejpam-4352	315	2	,	,	PUNCT
ejpam-4352	315	3	s	s	X
ejpam-4352	315	4	is	be	AUX
ejpam-4352	315	5	of	of	ADP
ejpam-4352	315	6	type	type	NOUN
ejpam-4352	315	7	(	(	PUNCT
ejpam-4352	315	8	ii	ii	NOUN
ejpam-4352	315	9	)	)	PUNCT
ejpam-4352	315	10	.	.	PUNCT
ejpam-4352	316	1	next	next	ADV
ejpam-4352	316	2	,	,	PUNCT
ejpam-4352	316	3	suppose	suppose	VERB
ejpam-4352	316	4	that	that	SCONJ
ejpam-4352	316	5	d∗	d∗	VERB
ejpam-4352	316	6	̸=	̸=	PROPN
ejpam-4352	316	7	v	v	NOUN
ejpam-4352	316	8	(	(	PUNCT
ejpam-4352	316	9	gr	gr	NOUN
ejpam-4352	316	10	)	)	PUNCT
ejpam-4352	316	11	.	.	PUNCT
ejpam-4352	317	1	let	let	VERB
ejpam-4352	317	2	w	w	NOUN
ejpam-4352	317	3	∈	∈	PROPN
ejpam-4352	317	4	v	v	ADP
ejpam-4352	317	5	(	(	PUNCT
ejpam-4352	317	6	gr	gr	NOUN
ejpam-4352	317	7	)	)	PUNCT
ejpam-4352	317	8	\d∗.	\d∗.	NOUN
ejpam-4352	317	9	then	then	ADV
ejpam-4352	317	10	z	z	PROPN
ejpam-4352	317	11	∈	∈	PROPN
ejpam-4352	317	12	ng(w	ng(w	NOUN
ejpam-4352	317	13	)	)	PUNCT
ejpam-4352	317	14	.	.	PUNCT
ejpam-4352	318	1	let	let	VERB
ejpam-4352	318	2	p	p	PRON
ejpam-4352	318	3	∈	∈	PROPN
ejpam-4352	318	4	s	s	VERB
ejpam-4352	318	5	such	such	ADJ
ejpam-4352	318	6	that	that	SCONJ
ejpam-4352	318	7	[	[	X
ejpam-4352	318	8	v	v	X
ejpam-4352	318	9	(	(	PUNCT
ejpam-4352	318	10	g	g	NOUN
ejpam-4352	318	11	)	)	PUNCT
ejpam-4352	318	12	\ng(p	\ng(p	NOUN
ejpam-4352	318	13	)	)	PUNCT
ejpam-4352	318	14	]	]	PUNCT
ejpam-4352	318	15	∩	∩	NOUN
ejpam-4352	319	1	[	[	X
ejpam-4352	319	2	v	v	X
ejpam-4352	319	3	(	(	PUNCT
ejpam-4352	319	4	g	g	NOUN
ejpam-4352	319	5	)	)	PUNCT
ejpam-4352	319	6	\	\	PUNCT
ejpam-4352	320	1	s	s	X
ejpam-4352	320	2	]	]	X
ejpam-4352	320	3	=	=	PUNCT
ejpam-4352	320	4	{	{	PUNCT
ejpam-4352	320	5	w	w	NOUN
ejpam-4352	320	6	}	}	PUNCT
ejpam-4352	320	7	.	.	PUNCT
ejpam-4352	321	1	since	since	SCONJ
ejpam-4352	321	2	v	v	NUM
ejpam-4352	321	3	∈	∈	PROPN
ejpam-4352	321	4	[	[	X
ejpam-4352	321	5	v	v	X
ejpam-4352	321	6	(	(	PUNCT
ejpam-4352	321	7	g	g	NOUN
ejpam-4352	321	8	)	)	PUNCT
ejpam-4352	321	9	\	\	PUNCT
ejpam-4352	322	1	s	s	X
ejpam-4352	322	2	]	]	X
ejpam-4352	322	3	∩	∩	ADJ
ejpam-4352	322	4	v	v	X
ejpam-4352	322	5	(	(	PUNCT
ejpam-4352	322	6	gk	gk	PROPN
ejpam-4352	322	7	)	)	PUNCT
ejpam-4352	322	8	)	)	PUNCT
ejpam-4352	323	1	]	]	PUNCT
ejpam-4352	323	2	,	,	PUNCT
ejpam-4352	323	3	p	p	X
ejpam-4352	323	4	/∈	/∈	PROPN
ejpam-4352	324	1	∪j	∪j	PROPN
ejpam-4352	324	2	̸=kv	̸=kv	PROPN
ejpam-4352	324	3	(	(	PUNCT
ejpam-4352	324	4	gj	gj	PROPN
ejpam-4352	324	5	)	)	PUNCT
ejpam-4352	324	6	.	.	PUNCT
ejpam-4352	325	1	hence	hence	ADV
ejpam-4352	325	2	,	,	PUNCT
ejpam-4352	325	3	p	p	PROPN
ejpam-4352	325	4	∈	∈	PROPN
ejpam-4352	325	5	v	v	ADP
ejpam-4352	325	6	(	(	PUNCT
ejpam-4352	325	7	gk	gk	NOUN
ejpam-4352	325	8	)	)	PUNCT
ejpam-4352	325	9	∩	∩	PROPN
ejpam-4352	325	10	ngk	ngk	PROPN
ejpam-4352	325	11	(	(	PUNCT
ejpam-4352	325	12	v	v	NOUN
ejpam-4352	325	13	)	)	PUNCT
ejpam-4352	325	14	.	.	PUNCT
ejpam-4352	326	1	moreover	moreover	ADV
ejpam-4352	326	2	,	,	PUNCT
ejpam-4352	326	3	|v	|v	PROPN
ejpam-4352	326	4	(	(	PUNCT
ejpam-4352	326	5	gr	gr	NOUN
ejpam-4352	326	6	)	)	PUNCT
ejpam-4352	326	7	\	\	NOUN
ejpam-4352	326	8	d∗|	d∗|	NOUN
ejpam-4352	326	9	=	=	SYM
ejpam-4352	326	10	1	1	NUM
ejpam-4352	326	11	,	,	PUNCT
ejpam-4352	326	12	that	that	ADV
ejpam-4352	326	13	is	is	ADV
ejpam-4352	326	14	,	,	PUNCT
ejpam-4352	326	15	d∗	d∗	PROPN
ejpam-4352	326	16	=	=	SYM
ejpam-4352	326	17	v	v	PROPN
ejpam-4352	326	18	(	(	PUNCT
ejpam-4352	326	19	gr	gr	NOUN
ejpam-4352	326	20	)	)	PUNCT
ejpam-4352	326	21	\	\	NOUN
ejpam-4352	326	22	{	{	PUNCT
ejpam-4352	326	23	w	w	NOUN
ejpam-4352	326	24	}	}	PUNCT
ejpam-4352	326	25	.	.	PUNCT
ejpam-4352	327	1	consequently	consequently	ADV
ejpam-4352	327	2	,	,	PUNCT
ejpam-4352	327	3	s	s	VERB
ejpam-4352	327	4	=	=	PUNCT
ejpam-4352	328	1	[	[	X
ejpam-4352	328	2	v	v	X
ejpam-4352	328	3	(	(	PUNCT
ejpam-4352	328	4	gk)\{v}]∪	gk)\{v}]∪	PROPN
ejpam-4352	328	5	[	[	X
ejpam-4352	328	6	v	v	X
ejpam-4352	328	7	(	(	PUNCT
ejpam-4352	328	8	gr)\{w}]∪	gr)\{w}]∪	PROPN
ejpam-4352	328	9	[	[	X
ejpam-4352	328	10	∪j	∪j	X
ejpam-4352	328	11	̸=r	̸=r	PROPN
ejpam-4352	328	12	,	,	PUNCT
ejpam-4352	328	13	kv	kv	PROPN
ejpam-4352	328	14	(	(	PUNCT
ejpam-4352	328	15	gj	gj	PROPN
ejpam-4352	328	16	)	)	PUNCT
ejpam-4352	328	17	]	]	PUNCT
ejpam-4352	328	18	,	,	PUNCT
ejpam-4352	328	19	showing	show	VERB
ejpam-4352	328	20	that	that	SCONJ
ejpam-4352	328	21	(	(	PUNCT
ejpam-4352	328	22	iii	iii	NOUN
ejpam-4352	328	23	)	)	PUNCT
ejpam-4352	328	24	holds	hold	VERB
ejpam-4352	328	25	.	.	PUNCT
ejpam-4352	329	1	for	for	ADP
ejpam-4352	329	2	the	the	DET
ejpam-4352	329	3	converse	converse	NOUN
ejpam-4352	329	4	,	,	PUNCT
ejpam-4352	329	5	suppose	suppose	VERB
ejpam-4352	329	6	first	first	ADV
ejpam-4352	329	7	that	that	SCONJ
ejpam-4352	329	8	s	s	VERB
ejpam-4352	329	9	is	be	AUX
ejpam-4352	329	10	of	of	ADP
ejpam-4352	329	11	type	type	NOUN
ejpam-4352	329	12	(	(	PUNCT
ejpam-4352	329	13	i	i	NOUN
ejpam-4352	329	14	)	)	PUNCT
ejpam-4352	329	15	,	,	PUNCT
ejpam-4352	329	16	that	that	ADV
ejpam-4352	329	17	is	be	AUX
ejpam-4352	329	18	,	,	PUNCT
ejpam-4352	329	19	s	s	PART
ejpam-4352	329	20	=	=	X
ejpam-4352	329	21	d	d	AUX
ejpam-4352	329	22	∪	∪	X
ejpam-4352	329	23	[	[	X
ejpam-4352	329	24	∪j	∪j	X
ejpam-4352	329	25	̸=kv	̸=kv	PROPN
ejpam-4352	329	26	(	(	PUNCT
ejpam-4352	329	27	gj	gj	PROPN
ejpam-4352	329	28	)	)	PUNCT
ejpam-4352	329	29	]	]	PUNCT
ejpam-4352	329	30	for	for	ADP
ejpam-4352	329	31	some	some	DET
ejpam-4352	329	32	k	k	PROPN
ejpam-4352	329	33	≤	≤	NOUN
ejpam-4352	329	34	n	n	CCONJ
ejpam-4352	329	35	and	and	CCONJ
ejpam-4352	329	36	for	for	ADP
ejpam-4352	329	37	some	some	DET
ejpam-4352	329	38	complement	complement	NOUN
ejpam-4352	329	39	-	-	PUNCT
ejpam-4352	329	40	super	super	ADV
ejpam-4352	329	41	dominating	dominating	NOUN
ejpam-4352	329	42	set	set	NOUN
ejpam-4352	329	43	d	d	PROPN
ejpam-4352	329	44	of	of	ADP
ejpam-4352	329	45	gk	gk	PROPN
ejpam-4352	329	46	.	.	PUNCT
ejpam-4352	330	1	let	let	VERB
ejpam-4352	330	2	v	v	NUM
ejpam-4352	330	3	∈	∈	PROPN
ejpam-4352	330	4	v	v	NOUN
ejpam-4352	330	5	(	(	PUNCT
ejpam-4352	330	6	g	g	NOUN
ejpam-4352	330	7	)	)	PUNCT
ejpam-4352	330	8	\	\	PUNCT
ejpam-4352	331	1	s.	s.	PROPN
ejpam-4352	331	2	then	then	ADV
ejpam-4352	331	3	v	v	ADP
ejpam-4352	331	4	∈	∈	PROPN
ejpam-4352	331	5	v	v	NOUN
ejpam-4352	331	6	(	(	PUNCT
ejpam-4352	331	7	gk	gk	NOUN
ejpam-4352	331	8	)	)	PUNCT
ejpam-4352	331	9	\d	\d	NOUN
ejpam-4352	331	10	.	.	PUNCT
ejpam-4352	332	1	since	since	SCONJ
ejpam-4352	332	2	d	d	PROPN
ejpam-4352	332	3	is	be	AUX
ejpam-4352	332	4	a	a	DET
ejpam-4352	332	5	complement	complement	NOUN
ejpam-4352	332	6	-	-	PUNCT
ejpam-4352	332	7	super	super	ADJ
ejpam-4352	332	8	dominating	dominating	NOUN
ejpam-4352	332	9	set	set	NOUN
ejpam-4352	332	10	of	of	ADP
ejpam-4352	332	11	gk	gk	PROPN
ejpam-4352	332	12	,	,	PUNCT
ejpam-4352	332	13	there	there	PRON
ejpam-4352	332	14	exists	exist	VERB
ejpam-4352	332	15	z	z	NOUN
ejpam-4352	332	16	∈	∈	PROPN
ejpam-4352	333	1	d	d	ADP
ejpam-4352	333	2	such	such	ADJ
ejpam-4352	333	3	that	that	SCONJ
ejpam-4352	333	4	[	[	X
ejpam-4352	333	5	v	v	X
ejpam-4352	333	6	(	(	PUNCT
ejpam-4352	333	7	gk	gk	NOUN
ejpam-4352	333	8	)	)	PUNCT
ejpam-4352	333	9	\ngk	\ngk	NOUN
ejpam-4352	333	10	(	(	PUNCT
ejpam-4352	333	11	z	z	NOUN
ejpam-4352	333	12	)	)	PUNCT
ejpam-4352	333	13	]	]	PUNCT
ejpam-4352	333	14	∩	∩	NOUN
ejpam-4352	333	15	[	[	X
ejpam-4352	333	16	v	v	X
ejpam-4352	333	17	(	(	PUNCT
ejpam-4352	333	18	gk	gk	NOUN
ejpam-4352	333	19	)	)	PUNCT
ejpam-4352	333	20	\d	\d	NOUN
ejpam-4352	333	21	]	]	X
ejpam-4352	333	22	=	=	PUNCT
ejpam-4352	333	23	{	{	PUNCT
ejpam-4352	333	24	v	v	NOUN
ejpam-4352	333	25	}	}	PUNCT
ejpam-4352	333	26	.	.	PUNCT
ejpam-4352	334	1	it	it	PRON
ejpam-4352	334	2	follows	follow	VERB
ejpam-4352	334	3	that	that	SCONJ
ejpam-4352	335	1	[	[	X
ejpam-4352	335	2	v	v	X
ejpam-4352	335	3	(	(	PUNCT
ejpam-4352	335	4	g	g	NOUN
ejpam-4352	335	5	)	)	PUNCT
ejpam-4352	335	6	\ng(z	\ng(z	NOUN
ejpam-4352	335	7	)	)	PUNCT
ejpam-4352	335	8	]	]	PUNCT
ejpam-4352	335	9	∩	∩	NOUN
ejpam-4352	335	10	[	[	X
ejpam-4352	335	11	v	v	X
ejpam-4352	335	12	(	(	PUNCT
ejpam-4352	335	13	g	g	NOUN
ejpam-4352	335	14	)	)	PUNCT
ejpam-4352	335	15	\	\	PUNCT
ejpam-4352	336	1	s	s	X
ejpam-4352	336	2	]	]	X
ejpam-4352	336	3	=	=	PUNCT
ejpam-4352	336	4	{	{	PUNCT
ejpam-4352	336	5	v	v	NOUN
ejpam-4352	336	6	}	}	PUNCT
ejpam-4352	336	7	,	,	PUNCT
ejpam-4352	336	8	showing	show	VERB
ejpam-4352	336	9	that	that	SCONJ
ejpam-4352	336	10	s	s	VERB
ejpam-4352	336	11	is	be	AUX
ejpam-4352	336	12	a	a	DET
ejpam-4352	336	13	complement	complement	NOUN
ejpam-4352	336	14	-	-	PUNCT
ejpam-4352	336	15	super	super	ADJ
ejpam-4352	336	16	dominating	dominating	NOUN
ejpam-4352	336	17	set	set	NOUN
ejpam-4352	336	18	of	of	ADP
ejpam-4352	336	19	g.	g.	PROPN
ejpam-4352	336	20	clearly	clearly	ADV
ejpam-4352	336	21	,	,	PUNCT
ejpam-4352	336	22	if	if	SCONJ
ejpam-4352	336	23	s	s	NOUN
ejpam-4352	336	24	is	be	AUX
ejpam-4352	336	25	of	of	ADP
ejpam-4352	336	26	type	type	NOUN
ejpam-4352	336	27	(	(	PUNCT
ejpam-4352	336	28	ii	ii	NOUN
ejpam-4352	336	29	)	)	PUNCT
ejpam-4352	336	30	,	,	PUNCT
ejpam-4352	336	31	then	then	ADV
ejpam-4352	336	32	s	s	VERB
ejpam-4352	336	33	is	be	AUX
ejpam-4352	336	34	a	a	DET
ejpam-4352	336	35	complement	complement	NOUN
ejpam-4352	336	36	-	-	PUNCT
ejpam-4352	336	37	super	super	ADJ
ejpam-4352	336	38	dominating	dominating	NOUN
ejpam-4352	336	39	set	set	NOUN
ejpam-4352	336	40	of	of	ADP
ejpam-4352	336	41	g.	g.	PROPN
ejpam-4352	336	42	finally	finally	ADV
ejpam-4352	336	43	,	,	PUNCT
ejpam-4352	336	44	suppose	suppose	VERB
ejpam-4352	336	45	that	that	SCONJ
ejpam-4352	336	46	s	s	VERB
ejpam-4352	336	47	=	=	PUNCT
ejpam-4352	337	1	[	[	X
ejpam-4352	337	2	v	v	X
ejpam-4352	337	3	(	(	PUNCT
ejpam-4352	337	4	gk	gk	PROPN
ejpam-4352	337	5	)	)	PUNCT
ejpam-4352	337	6	\	\	PROPN
ejpam-4352	337	7	{	{	PUNCT
ejpam-4352	337	8	v	v	NOUN
ejpam-4352	337	9	}	}	PUNCT
ejpam-4352	337	10	]	]	PUNCT
ejpam-4352	337	11	∪	∪	ADP
ejpam-4352	337	12	[	[	X
ejpam-4352	337	13	v	v	X
ejpam-4352	337	14	(	(	PUNCT
ejpam-4352	337	15	gr	gr	NOUN
ejpam-4352	337	16	)	)	PUNCT
ejpam-4352	337	17	\	\	NOUN
ejpam-4352	337	18	{	{	PUNCT
ejpam-4352	337	19	w	w	NOUN
ejpam-4352	337	20	}	}	PUNCT
ejpam-4352	337	21	]	]	PUNCT
ejpam-4352	337	22	∪	∪	ADP
ejpam-4352	337	23	[	[	PUNCT
ejpam-4352	337	24	⋃	⋃	PROPN
ejpam-4352	337	25	j	j	PROPN
ejpam-4352	337	26	̸=k	̸=k	PROPN
ejpam-4352	337	27	,	,	PUNCT
ejpam-4352	337	28	r	r	NOUN
ejpam-4352	337	29	v	v	NOUN
ejpam-4352	337	30	(	(	PUNCT
ejpam-4352	337	31	gj	gj	NOUN
ejpam-4352	337	32	)	)	PUNCT
ejpam-4352	337	33	]	]	PUNCT
ejpam-4352	337	34	for	for	ADP
ejpam-4352	337	35	some	some	DET
ejpam-4352	337	36	k	k	NOUN
ejpam-4352	337	37	,	,	PUNCT
ejpam-4352	337	38	r	r	NOUN
ejpam-4352	337	39	≤	≤	NUM
ejpam-4352	337	40	n	n	CCONJ
ejpam-4352	337	41	,	,	PUNCT
ejpam-4352	337	42	where	where	SCONJ
ejpam-4352	337	43	v	v	X
ejpam-4352	337	44	∈	∈	PROPN
ejpam-4352	337	45	v	v	NOUN
ejpam-4352	337	46	(	(	PUNCT
ejpam-4352	337	47	gk	gk	PROPN
ejpam-4352	337	48	)	)	PUNCT
ejpam-4352	337	49	,	,	PUNCT
ejpam-4352	337	50	w	w	PROPN
ejpam-4352	337	51	∈	∈	PROPN
ejpam-4352	337	52	v	v	ADP
ejpam-4352	337	53	(	(	PUNCT
ejpam-4352	337	54	gr	gr	NOUN
ejpam-4352	337	55	)	)	PUNCT
ejpam-4352	337	56	,	,	PUNCT
ejpam-4352	337	57	and	and	CCONJ
ejpam-4352	337	58	both	both	PRON
ejpam-4352	337	59	[	[	X
ejpam-4352	337	60	v	v	X
ejpam-4352	337	61	(	(	PUNCT
ejpam-4352	337	62	gk	gk	PROPN
ejpam-4352	337	63	)	)	PUNCT
ejpam-4352	337	64	\	\	PROPN
ejpam-4352	337	65	{	{	PUNCT
ejpam-4352	337	66	v	v	NOUN
ejpam-4352	337	67	}	}	PUNCT
ejpam-4352	337	68	]	]	PUNCT
ejpam-4352	337	69	and	and	CCONJ
ejpam-4352	337	70	[	[	X
ejpam-4352	337	71	v	v	X
ejpam-4352	337	72	(	(	PUNCT
ejpam-4352	337	73	gr	gr	NOUN
ejpam-4352	337	74	)	)	PUNCT
ejpam-4352	337	75	\	\	NOUN
ejpam-4352	337	76	{	{	PUNCT
ejpam-4352	337	77	w	w	NOUN
ejpam-4352	337	78	}	}	PUNCT
ejpam-4352	337	79	]	]	PUNCT
ejpam-4352	337	80	are	be	AUX
ejpam-4352	337	81	non	non	ADJ
ejpam-4352	337	82	-	-	ADJ
ejpam-4352	337	83	empty	empty	ADJ
ejpam-4352	337	84	sets	set	NOUN
ejpam-4352	337	85	.	.	PUNCT
ejpam-4352	338	1	pick	pick	VERB
ejpam-4352	338	2	p	p	PROPN
ejpam-4352	338	3	∈	∈	PROPN
ejpam-4352	338	4	v	v	ADP
ejpam-4352	338	5	(	(	PUNCT
ejpam-4352	338	6	gk	gk	NOUN
ejpam-4352	338	7	)	)	PUNCT
ejpam-4352	338	8	∩ngk	∩ngk	PROPN
ejpam-4352	338	9	(	(	PUNCT
ejpam-4352	338	10	v	v	NOUN
ejpam-4352	338	11	)	)	PUNCT
ejpam-4352	338	12	and	and	CCONJ
ejpam-4352	338	13	q	q	PROPN
ejpam-4352	338	14	∈	∈	PROPN
ejpam-4352	338	15	v	v	ADP
ejpam-4352	338	16	(	(	PUNCT
ejpam-4352	338	17	gr	gr	NOUN
ejpam-4352	338	18	)	)	PUNCT
ejpam-4352	338	19	∩ngr(w	∩ngr(w	NOUN
ejpam-4352	338	20	)	)	PUNCT
ejpam-4352	338	21	.	.	PUNCT
ejpam-4352	339	1	then	then	ADV
ejpam-4352	339	2	[	[	X
ejpam-4352	339	3	v	v	X
ejpam-4352	339	4	(	(	PUNCT
ejpam-4352	339	5	g	g	NOUN
ejpam-4352	339	6	)	)	PUNCT
ejpam-4352	339	7	\ng(p	\ng(p	NOUN
ejpam-4352	339	8	)	)	PUNCT
ejpam-4352	339	9	]	]	PUNCT
ejpam-4352	339	10	∩	∩	NOUN
ejpam-4352	339	11	[	[	X
ejpam-4352	339	12	v	v	X
ejpam-4352	339	13	(	(	PUNCT
ejpam-4352	339	14	g	g	NOUN
ejpam-4352	339	15	)	)	PUNCT
ejpam-4352	339	16	\	\	PUNCT
ejpam-4352	340	1	s	s	X
ejpam-4352	340	2	]	]	X
ejpam-4352	340	3	=	=	PUNCT
ejpam-4352	340	4	{	{	PUNCT
ejpam-4352	340	5	w	w	NOUN
ejpam-4352	340	6	}	}	PUNCT
ejpam-4352	340	7	and	and	CCONJ
ejpam-4352	340	8	[	[	X
ejpam-4352	340	9	v	v	X
ejpam-4352	340	10	(	(	PUNCT
ejpam-4352	340	11	g)\ng(q)]∩	g)\ng(q)]∩	NOUN
ejpam-4352	340	12	[	[	X
ejpam-4352	340	13	v	v	X
ejpam-4352	340	14	(	(	PUNCT
ejpam-4352	340	15	g)\s	g)\s	NOUN
ejpam-4352	340	16	]	]	PUNCT
ejpam-4352	340	17	=	=	PUNCT
ejpam-4352	340	18	{	{	PUNCT
ejpam-4352	340	19	v	v	NOUN
ejpam-4352	340	20	}	}	PUNCT
ejpam-4352	340	21	,	,	PUNCT
ejpam-4352	340	22	showing	show	VERB
ejpam-4352	340	23	that	that	SCONJ
ejpam-4352	340	24	s	s	VERB
ejpam-4352	340	25	is	be	AUX
ejpam-4352	340	26	a	a	DET
ejpam-4352	340	27	complement	complement	NOUN
ejpam-4352	340	28	-	-	PUNCT
ejpam-4352	340	29	super	super	ADJ
ejpam-4352	340	30	dominating	dominating	NOUN
ejpam-4352	340	31	set	set	NOUN
ejpam-4352	340	32	of	of	ADP
ejpam-4352	340	33	g.	g.	PROPN
ejpam-4352	340	34	theorem	theorem	VERB
ejpam-4352	340	35	10	10	NUM
ejpam-4352	340	36	.	.	PUNCT
ejpam-4352	341	1	let	let	VERB
ejpam-4352	341	2	g1	g1	PROPN
ejpam-4352	341	3	,	,	PUNCT
ejpam-4352	341	4	g2	g2	PROPN
ejpam-4352	341	5	,	,	PUNCT
ejpam-4352	341	6	·	·	PUNCT
ejpam-4352	341	7	·	·	PUNCT
ejpam-4352	341	8	·	·	PUNCT
ejpam-4352	341	9	,	,	PUNCT
ejpam-4352	341	10	gk	gk	PROPN
ejpam-4352	341	11	,	,	PUNCT
ejpam-4352	341	12	where	where	SCONJ
ejpam-4352	341	13	k	k	PROPN
ejpam-4352	341	14	≥	≥	NUM
ejpam-4352	341	15	2	2	NUM
ejpam-4352	341	16	,	,	PUNCT
ejpam-4352	341	17	be	be	AUX
ejpam-4352	341	18	the	the	DET
ejpam-4352	341	19	components	component	NOUN
ejpam-4352	341	20	of	of	ADP
ejpam-4352	341	21	a	a	DET
ejpam-4352	341	22	graph	graph	NOUN
ejpam-4352	341	23	g	g	NOUN
ejpam-4352	341	24	of	of	ADP
ejpam-4352	341	25	order	order	NOUN
ejpam-4352	341	26	n.	n.	NOUN
ejpam-4352	341	27	then	then	ADV
ejpam-4352	341	28	each	each	PRON
ejpam-4352	341	29	of	of	ADP
ejpam-4352	341	30	the	the	DET
ejpam-4352	341	31	following	following	ADJ
ejpam-4352	341	32	statements	statement	NOUN
ejpam-4352	341	33	holds	hold	VERB
ejpam-4352	341	34	.	.	PUNCT
ejpam-4352	342	1	(	(	PUNCT
ejpam-4352	342	2	i	i	NOUN
ejpam-4352	342	3	)	)	PUNCT
ejpam-4352	342	4	γcs(g	γcs(g	X
ejpam-4352	342	5	)	)	PUNCT
ejpam-4352	343	1	=	=	SYM
ejpam-4352	343	2	n−1	n−1	PROPN
ejpam-4352	343	3	if	if	SCONJ
ejpam-4352	343	4	and	and	CCONJ
ejpam-4352	343	5	only	only	ADV
ejpam-4352	343	6	if	if	SCONJ
ejpam-4352	343	7	g	g	PROPN
ejpam-4352	343	8	=	=	VERB
ejpam-4352	343	9	kn	kn	PROPN
ejpam-4352	343	10	or	or	CCONJ
ejpam-4352	343	11	g	g	PROPN
ejpam-4352	343	12	has	have	VERB
ejpam-4352	343	13	exactly	exactly	ADV
ejpam-4352	343	14	a	a	DET
ejpam-4352	343	15	single	single	ADJ
ejpam-4352	343	16	non	non	ADJ
ejpam-4352	343	17	-	-	ADJ
ejpam-4352	343	18	trivial	trivial	ADJ
ejpam-4352	343	19	component	component	NOUN
ejpam-4352	343	20	gs	gs	NOUN
ejpam-4352	343	21	and	and	CCONJ
ejpam-4352	343	22	|v	|v	PROPN
ejpam-4352	343	23	(	(	PUNCT
ejpam-4352	343	24	gs)|	gs)|	PROPN
ejpam-4352	343	25	−	−	PROPN
ejpam-4352	343	26	1	1	NUM
ejpam-4352	343	27	≤	≤	NUM
ejpam-4352	343	28	γcs(gs	γcs(gs	NOUN
ejpam-4352	343	29	)	)	PUNCT
ejpam-4352	343	30	≤	≤	PUNCT
ejpam-4352	343	31	|v	|v	X
ejpam-4352	343	32	(	(	PUNCT
ejpam-4352	343	33	gs)|	gs)|	PROPN
ejpam-4352	343	34	.	.	PUNCT
ejpam-4352	344	1	(	(	PUNCT
ejpam-4352	344	2	ii	ii	NOUN
ejpam-4352	344	3	)	)	PUNCT
ejpam-4352	344	4	if	if	SCONJ
ejpam-4352	344	5	g	g	PROPN
ejpam-4352	344	6	has	have	VERB
ejpam-4352	344	7	at	at	ADV
ejpam-4352	344	8	least	least	ADV
ejpam-4352	344	9	two	two	NUM
ejpam-4352	344	10	non	non	ADJ
ejpam-4352	344	11	-	-	ADJ
ejpam-4352	344	12	trivial	trivial	ADJ
ejpam-4352	344	13	components	component	NOUN
ejpam-4352	344	14	,	,	PUNCT
ejpam-4352	344	15	then	then	ADV
ejpam-4352	344	16	γcs(g	γcs(g	X
ejpam-4352	344	17	)	)	PUNCT
ejpam-4352	344	18	=	=	SYM
ejpam-4352	344	19	min{n−	min{n−	PROPN
ejpam-4352	344	20	2	2	NUM
ejpam-4352	344	21	,	,	PUNCT
ejpam-4352	344	22	n−	n−	NOUN
ejpam-4352	344	23	ηg	ηg	ADV
ejpam-4352	344	24	}	}	PUNCT
ejpam-4352	344	25	,	,	PUNCT
ejpam-4352	344	26	where	where	SCONJ
ejpam-4352	344	27	ηg	ηg	ADV
ejpam-4352	344	28	=	=	PUNCT
ejpam-4352	344	29	max{|v	max{|v	PROPN
ejpam-4352	344	30	(	(	PUNCT
ejpam-4352	344	31	gj)|	gj)|	NOUN
ejpam-4352	344	32	−	−	NOUN
ejpam-4352	344	33	γcs(gj	γcs(gj	NOUN
ejpam-4352	344	34	)	)	PUNCT
ejpam-4352	344	35	:	:	PUNCT
ejpam-4352	345	1	j	j	X
ejpam-4352	345	2	=	=	SYM
ejpam-4352	345	3	1	1	NUM
ejpam-4352	345	4	,	,	PUNCT
ejpam-4352	345	5	2	2	NUM
ejpam-4352	345	6	,	,	PUNCT
ejpam-4352	345	7	·	·	PUNCT
ejpam-4352	345	8	·	·	PUNCT
ejpam-4352	345	9	·	·	PUNCT
ejpam-4352	345	10	,	,	PUNCT
ejpam-4352	345	11	k	k	NOUN
ejpam-4352	345	12	}	}	PUNCT
ejpam-4352	345	13	.	.	PUNCT
ejpam-4352	346	1	proof	proof	NOUN
ejpam-4352	346	2	.	.	PUNCT
ejpam-4352	347	1	(	(	PUNCT
ejpam-4352	347	2	i	i	NOUN
ejpam-4352	347	3	)	)	PUNCT
ejpam-4352	347	4	suppose	suppose	VERB
ejpam-4352	347	5	γcs(g	γcs(g	X
ejpam-4352	347	6	)	)	PUNCT
ejpam-4352	347	7	=	=	SYM
ejpam-4352	348	1	n	n	CCONJ
ejpam-4352	348	2	−	−	NOUN
ejpam-4352	349	1	1	1	X
ejpam-4352	349	2	.	.	PUNCT
ejpam-4352	349	3	suppose	suppose	VERB
ejpam-4352	349	4	g	g	PROPN
ejpam-4352	349	5	̸=	̸=	PROPN
ejpam-4352	349	6	kn	kn	PROPN
ejpam-4352	349	7	and	and	CCONJ
ejpam-4352	349	8	assume	assume	VERB
ejpam-4352	349	9	that	that	SCONJ
ejpam-4352	349	10	g	g	PROPN
ejpam-4352	349	11	has	have	VERB
ejpam-4352	349	12	two	two	NUM
ejpam-4352	349	13	non	non	ADJ
ejpam-4352	349	14	-	-	ADJ
ejpam-4352	349	15	trivial	trivial	ADJ
ejpam-4352	349	16	components	component	NOUN
ejpam-4352	349	17	,	,	PUNCT
ejpam-4352	349	18	say	say	VERB
ejpam-4352	349	19	gm	gm	PROPN
ejpam-4352	349	20	and	and	CCONJ
ejpam-4352	349	21	gr	gr	PROPN
ejpam-4352	349	22	.	.	PROPN
ejpam-4352	349	23	pick	pick	VERB
ejpam-4352	349	24	v	v	NUM
ejpam-4352	349	25	∈	∈	PROPN
ejpam-4352	349	26	v	v	NOUN
ejpam-4352	349	27	(	(	PUNCT
ejpam-4352	349	28	gm	gm	PROPN
ejpam-4352	349	29	)	)	PUNCT
ejpam-4352	349	30	and	and	CCONJ
ejpam-4352	349	31	w	w	PROPN
ejpam-4352	349	32	∈	∈	PROPN
ejpam-4352	349	33	v	v	ADP
ejpam-4352	349	34	(	(	PUNCT
ejpam-4352	349	35	gr	gr	NOUN
ejpam-4352	349	36	)	)	PUNCT
ejpam-4352	349	37	.	.	PUNCT
ejpam-4352	350	1	then	then	ADV
ejpam-4352	350	2	s∗	s∗	PROPN
ejpam-4352	350	3	=	=	PUNCT
ejpam-4352	351	1	[	[	X
ejpam-4352	351	2	v	v	X
ejpam-4352	351	3	(	(	PUNCT
ejpam-4352	351	4	gk	gk	PROPN
ejpam-4352	351	5	)	)	PUNCT
ejpam-4352	351	6	\	\	PROPN
ejpam-4352	351	7	{	{	PUNCT
ejpam-4352	351	8	v	v	NOUN
ejpam-4352	351	9	}	}	PUNCT
ejpam-4352	351	10	]	]	PUNCT
ejpam-4352	351	11	∪	∪	ADP
ejpam-4352	351	12	[	[	X
ejpam-4352	351	13	v	v	X
ejpam-4352	351	14	(	(	PUNCT
ejpam-4352	351	15	gr	gr	NOUN
ejpam-4352	351	16	)	)	PUNCT
ejpam-4352	351	17	\	\	NOUN
ejpam-4352	351	18	{	{	PUNCT
ejpam-4352	351	19	w	w	NOUN
ejpam-4352	351	20	}	}	PUNCT
ejpam-4352	351	21	]	]	PUNCT
ejpam-4352	351	22	∪	∪	ADP
ejpam-4352	351	23	[	[	X
ejpam-4352	351	24	∪j	∪j	X
ejpam-4352	351	25	̸=k	̸=k	PROPN
ejpam-4352	351	26	,	,	PUNCT
ejpam-4352	351	27	rv	rv	PROPN
ejpam-4352	351	28	(	(	PUNCT
ejpam-4352	351	29	gj	gj	PROPN
ejpam-4352	351	30	)	)	PUNCT
ejpam-4352	351	31	]	]	PUNCT
ejpam-4352	351	32	is	be	AUX
ejpam-4352	351	33	a	a	DET
ejpam-4352	351	34	complement	complement	NOUN
ejpam-4352	351	35	-	-	PUNCT
ejpam-4352	351	36	super	super	ADJ
ejpam-4352	351	37	dominating	dominating	NOUN
ejpam-4352	351	38	set	set	NOUN
ejpam-4352	351	39	of	of	ADP
ejpam-4352	351	40	g	g	NOUN
ejpam-4352	351	41	by	by	ADP
ejpam-4352	351	42	theorem	theorem	NOUN
ejpam-4352	351	43	9(iii	9(iii	NUM
ejpam-4352	351	44	)	)	PUNCT
ejpam-4352	351	45	.	.	PUNCT
ejpam-4352	352	1	hence	hence	ADV
ejpam-4352	352	2	,	,	PUNCT
ejpam-4352	352	3	γcs(g	γcs(g	X
ejpam-4352	352	4	)	)	PUNCT
ejpam-4352	352	5	≤	≤	NOUN
ejpam-4352	352	6	|s∗|	|s∗|	NUM
ejpam-4352	352	7	=	=	SYM
ejpam-4352	352	8	n	n	PRON
ejpam-4352	352	9	−	−	PROPN
ejpam-4352	352	10	2	2	NUM
ejpam-4352	352	11	,	,	PUNCT
ejpam-4352	352	12	a	a	DET
ejpam-4352	352	13	contradiction	contradiction	NOUN
ejpam-4352	352	14	.	.	PUNCT
ejpam-4352	353	1	since	since	SCONJ
ejpam-4352	353	2	g	g	PROPN
ejpam-4352	353	3	̸=	̸=	PROPN
ejpam-4352	353	4	kn	kn	PROPN
ejpam-4352	353	5	,	,	PUNCT
ejpam-4352	353	6	g	g	PROPN
ejpam-4352	353	7	has	have	VERB
ejpam-4352	353	8	exactly	exactly	ADV
ejpam-4352	353	9	a	a	DET
ejpam-4352	353	10	single	single	ADJ
ejpam-4352	353	11	non	non	ADJ
ejpam-4352	353	12	-	-	ADJ
ejpam-4352	353	13	trivial	trivial	ADJ
ejpam-4352	353	14	component	component	NOUN
ejpam-4352	353	15	,	,	PUNCT
ejpam-4352	353	16	say	say	VERB
ejpam-4352	353	17	gs	gs	INTJ
ejpam-4352	353	18	.	.	PUNCT
ejpam-4352	354	1	let	let	VERB
ejpam-4352	354	2	d	d	PRON
ejpam-4352	354	3	be	be	AUX
ejpam-4352	354	4	a	a	DET
ejpam-4352	354	5	γcs	γcs	NOUN
ejpam-4352	354	6	-	-	PUNCT
ejpam-4352	354	7	set	set	NOUN
ejpam-4352	354	8	of	of	ADP
ejpam-4352	354	9	gs	gs	NOUN
ejpam-4352	354	10	.	.	PUNCT
ejpam-4352	355	1	then	then	ADV
ejpam-4352	355	2	s	s	VERB
ejpam-4352	355	3	=	=	X
ejpam-4352	355	4	d	d	X
ejpam-4352	355	5	∪	∪	X
ejpam-4352	355	6	[	[	X
ejpam-4352	355	7	∪j	∪j	X
ejpam-4352	355	8	̸=kv	̸=kv	PROPN
ejpam-4352	355	9	(	(	PUNCT
ejpam-4352	355	10	gj	gj	PROPN
ejpam-4352	355	11	)	)	PUNCT
ejpam-4352	355	12	]	]	PUNCT
ejpam-4352	355	13	is	be	AUX
ejpam-4352	355	14	a	a	DET
ejpam-4352	355	15	complement	complement	NOUN
ejpam-4352	355	16	-	-	PUNCT
ejpam-4352	355	17	super	super	ADJ
ejpam-4352	355	18	dominating	dominating	NOUN
ejpam-4352	355	19	set	set	NOUN
ejpam-4352	355	20	of	of	ADP
ejpam-4352	355	21	g	g	NOUN
ejpam-4352	355	22	by	by	ADP
ejpam-4352	355	23	theorem	theorem	NOUN
ejpam-4352	355	24	9(i	9(i	NUM
ejpam-4352	355	25	)	)	PUNCT
ejpam-4352	355	26	.	.	PUNCT
ejpam-4352	356	1	since	since	SCONJ
ejpam-4352	356	2	γcs(g	γcs(g	NUM
ejpam-4352	356	3	)	)	PUNCT
ejpam-4352	356	4	=	=	PUNCT
ejpam-4352	356	5	n−	n−	NOUN
ejpam-4352	356	6	1	1	NUM
ejpam-4352	356	7	≤	≤	NUM
ejpam-4352	356	8	|s|	|s|	PROPN
ejpam-4352	356	9	,	,	PUNCT
ejpam-4352	356	10	|v	|v	X
ejpam-4352	356	11	(	(	PUNCT
ejpam-4352	356	12	gs)|	gs)|	PROPN
ejpam-4352	356	13	−	−	PROPN
ejpam-4352	356	14	1	1	NUM
ejpam-4352	356	15	≤	≤	NUM
ejpam-4352	356	16	|d|	|d|	PROPN
ejpam-4352	356	17	=	=	SYM
ejpam-4352	356	18	γcs(gs	γcs(gs	PROPN
ejpam-4352	356	19	)	)	PUNCT
ejpam-4352	356	20	≤	≤	NOUN
ejpam-4352	356	21	|v	|v	X
ejpam-4352	356	22	(	(	PUNCT
ejpam-4352	356	23	gs)|	gs)|	PROPN
ejpam-4352	356	24	.	.	PUNCT
ejpam-4352	357	1	s.	s.	PROPN
ejpam-4352	357	2	canoy	canoy	PROPN
ejpam-4352	357	3	,	,	PUNCT
ejpam-4352	357	4	jr	jr	PROPN
ejpam-4352	357	5	.	.	PROPN
ejpam-4352	357	6	,	,	PUNCT
ejpam-4352	357	7	g.	g.	PROPN
ejpam-4352	357	8	salasalan	salasalan	PROPN
ejpam-4352	357	9	/	/	SYM
ejpam-4352	357	10	eur	eur	PROPN
ejpam-4352	357	11	.	.	PUNCT
ejpam-4352	358	1	j.	j.	PROPN
ejpam-4352	358	2	pure	pure	PROPN
ejpam-4352	358	3	appl	appl	PROPN
ejpam-4352	358	4	.	.	PROPN
ejpam-4352	358	5	math	math	PROPN
ejpam-4352	358	6	,	,	PUNCT
ejpam-4352	358	7	15	15	NUM
ejpam-4352	358	8	(	(	PUNCT
ejpam-4352	358	9	2	2	NUM
ejpam-4352	358	10	)	)	PUNCT
ejpam-4352	358	11	(	(	PUNCT
ejpam-4352	358	12	2022	2022	NUM
ejpam-4352	358	13	)	)	PUNCT
ejpam-4352	358	14	,	,	PUNCT
ejpam-4352	358	15	342	342	NUM
ejpam-4352	358	16	-	-	SYM
ejpam-4352	358	17	353	353	NUM
ejpam-4352	358	18	349	349	NUM
ejpam-4352	358	19	for	for	ADP
ejpam-4352	358	20	the	the	DET
ejpam-4352	358	21	converse	converse	NOUN
ejpam-4352	358	22	,	,	PUNCT
ejpam-4352	358	23	suppose	suppose	VERB
ejpam-4352	358	24	first	first	ADV
ejpam-4352	358	25	that	that	SCONJ
ejpam-4352	358	26	g	g	PROPN
ejpam-4352	358	27	=	=	PROPN
ejpam-4352	358	28	kn	kn	PROPN
ejpam-4352	358	29	.	.	PUNCT
ejpam-4352	359	1	then	then	ADV
ejpam-4352	359	2	γcs(g	γcs(g	X
ejpam-4352	359	3	)	)	PUNCT
ejpam-4352	359	4	=	=	SYM
ejpam-4352	359	5	n−1	n−1	PROPN
ejpam-4352	359	6	by	by	ADP
ejpam-4352	359	7	theorem	theorem	NOUN
ejpam-4352	359	8	8	8	NUM
ejpam-4352	359	9	.	.	PUNCT
ejpam-4352	360	1	next	next	ADV
ejpam-4352	360	2	,	,	PUNCT
ejpam-4352	360	3	suppose	suppose	VERB
ejpam-4352	360	4	g	g	PROPN
ejpam-4352	360	5	has	have	VERB
ejpam-4352	360	6	a	a	DET
ejpam-4352	360	7	single	single	ADJ
ejpam-4352	360	8	non	non	ADJ
ejpam-4352	360	9	-	-	ADJ
ejpam-4352	360	10	trivial	trivial	ADJ
ejpam-4352	360	11	component	component	NOUN
ejpam-4352	360	12	gs	gs	NOUN
ejpam-4352	360	13	and	and	CCONJ
ejpam-4352	360	14	|v	|v	PROPN
ejpam-4352	360	15	(	(	PUNCT
ejpam-4352	360	16	gs)|	gs)|	PROPN
ejpam-4352	360	17	−	−	PROPN
ejpam-4352	360	18	1	1	NUM
ejpam-4352	360	19	≤	≤	NUM
ejpam-4352	360	20	γcs(gs	γcs(gs	NOUN
ejpam-4352	360	21	)	)	PUNCT
ejpam-4352	360	22	≤	≤	PUNCT
ejpam-4352	360	23	|v	|v	PROPN
ejpam-4352	360	24	(	(	PUNCT
ejpam-4352	360	25	gs)|	gs)|	PROPN
ejpam-4352	360	26	.	.	PUNCT
ejpam-4352	361	1	since	since	SCONJ
ejpam-4352	361	2	v	v	NOUN
ejpam-4352	361	3	(	(	PUNCT
ejpam-4352	361	4	g	g	NOUN
ejpam-4352	361	5	)	)	PUNCT
ejpam-4352	361	6	\	\	NOUN
ejpam-4352	361	7	{	{	PUNCT
ejpam-4352	361	8	x	x	NOUN
ejpam-4352	361	9	}	}	PUNCT
ejpam-4352	361	10	is	be	AUX
ejpam-4352	361	11	a	a	DET
ejpam-4352	361	12	complement	complement	NOUN
ejpam-4352	361	13	-	-	PUNCT
ejpam-4352	361	14	super	super	ADJ
ejpam-4352	361	15	dominating	dominating	NOUN
ejpam-4352	361	16	set	set	NOUN
ejpam-4352	361	17	of	of	ADP
ejpam-4352	361	18	g	g	PROPN
ejpam-4352	361	19	for	for	ADP
ejpam-4352	361	20	each	each	DET
ejpam-4352	361	21	isolated	isolate	VERB
ejpam-4352	361	22	vertex	vertex	NOUN
ejpam-4352	361	23	x	x	NOUN
ejpam-4352	361	24	,	,	PUNCT
ejpam-4352	361	25	it	it	PRON
ejpam-4352	361	26	follows	follow	VERB
ejpam-4352	361	27	that	that	SCONJ
ejpam-4352	361	28	γcs(g	γcs(g	X
ejpam-4352	361	29	)	)	PUNCT
ejpam-4352	361	30	≤	≤	NOUN
ejpam-4352	361	31	n	n	CCONJ
ejpam-4352	361	32	−	−	PROPN
ejpam-4352	361	33	1	1	NUM
ejpam-4352	361	34	.	.	PUNCT
ejpam-4352	362	1	let	let	VERB
ejpam-4352	362	2	s	s	PRON
ejpam-4352	362	3	be	be	AUX
ejpam-4352	362	4	a	a	DET
ejpam-4352	362	5	γcs	γcs	NOUN
ejpam-4352	362	6	-	-	PUNCT
ejpam-4352	362	7	set	set	NOUN
ejpam-4352	362	8	of	of	ADP
ejpam-4352	362	9	g.	g.	PROPN
ejpam-4352	362	10	if	if	SCONJ
ejpam-4352	362	11	γcs(gs	γcs(gs	NOUN
ejpam-4352	362	12	)	)	PUNCT
ejpam-4352	362	13	=	=	SYM
ejpam-4352	362	14	|v	|v	PROPN
ejpam-4352	362	15	(	(	PUNCT
ejpam-4352	362	16	gs)|	gs)|	PROPN
ejpam-4352	362	17	,	,	PUNCT
ejpam-4352	362	18	then	then	ADV
ejpam-4352	362	19	s	s	X
ejpam-4352	362	20	of	of	ADP
ejpam-4352	362	21	type	type	NOUN
ejpam-4352	362	22	(	(	PUNCT
ejpam-4352	362	23	ii	ii	NOUN
ejpam-4352	362	24	)	)	PUNCT
ejpam-4352	362	25	by	by	ADP
ejpam-4352	362	26	theorem	theorem	NOUN
ejpam-4352	362	27	9	9	NUM
ejpam-4352	362	28	.	.	PUNCT
ejpam-4352	362	29	hence	hence	ADV
ejpam-4352	362	30	,	,	PUNCT
ejpam-4352	362	31	γcs(g	γcs(g	X
ejpam-4352	362	32	)	)	PUNCT
ejpam-4352	362	33	=	=	SYM
ejpam-4352	362	34	n	n	CCONJ
ejpam-4352	362	35	−	−	NOUN
ejpam-4352	363	1	1	1	X
ejpam-4352	363	2	.	.	PUNCT
ejpam-4352	364	1	if	if	SCONJ
ejpam-4352	364	2	γcs(gs	γcs(gs	NOUN
ejpam-4352	364	3	)	)	PUNCT
ejpam-4352	364	4	=	=	SYM
ejpam-4352	364	5	|v	|v	PROPN
ejpam-4352	364	6	(	(	PUNCT
ejpam-4352	364	7	gs)|	gs)|	PROPN
ejpam-4352	364	8	−	−	PROPN
ejpam-4352	364	9	1	1	NUM
ejpam-4352	364	10	,	,	PUNCT
ejpam-4352	364	11	then	then	ADV
ejpam-4352	364	12	s	s	VERB
ejpam-4352	364	13	=	=	SYM
ejpam-4352	364	14	d∪	d∪	PROPN
ejpam-4352	365	1	[	[	X
ejpam-4352	365	2	∪j	∪j	X
ejpam-4352	365	3	̸=sv	̸=sv	PROPN
ejpam-4352	365	4	(	(	PUNCT
ejpam-4352	365	5	gj	gj	NOUN
ejpam-4352	365	6	)	)	PUNCT
ejpam-4352	365	7	]	]	PUNCT
ejpam-4352	365	8	where	where	SCONJ
ejpam-4352	365	9	d	d	NOUN
ejpam-4352	365	10	is	be	AUX
ejpam-4352	365	11	γcs	γcs	NOUN
ejpam-4352	365	12	-	-	PUNCT
ejpam-4352	365	13	set	set	NOUN
ejpam-4352	365	14	of	of	ADP
ejpam-4352	365	15	gs	gs	NOUN
ejpam-4352	365	16	or	or	CCONJ
ejpam-4352	365	17	s	s	NOUN
ejpam-4352	365	18	is	be	AUX
ejpam-4352	365	19	of	of	ADP
ejpam-4352	365	20	type	type	NOUN
ejpam-4352	365	21	(	(	PUNCT
ejpam-4352	365	22	ii	ii	NOUN
ejpam-4352	365	23	)	)	PUNCT
ejpam-4352	365	24	by	by	ADP
ejpam-4352	365	25	theorem	theorem	NOUN
ejpam-4352	365	26	9	9	NUM
ejpam-4352	365	27	.	.	PUNCT
ejpam-4352	365	28	in	in	ADP
ejpam-4352	365	29	either	either	DET
ejpam-4352	365	30	case	case	NOUN
ejpam-4352	365	31	,	,	PUNCT
ejpam-4352	365	32	γcs(g	γcs(g	X
ejpam-4352	365	33	)	)	PUNCT
ejpam-4352	365	34	=	=	SYM
ejpam-4352	365	35	n−	n−	NOUN
ejpam-4352	365	36	1	1	NUM
ejpam-4352	365	37	.	.	PUNCT
ejpam-4352	365	38	(	(	PUNCT
ejpam-4352	365	39	ii	ii	NOUN
ejpam-4352	365	40	)	)	PUNCT
ejpam-4352	365	41	let	let	VERB
ejpam-4352	365	42	gm	gm	PROPN
ejpam-4352	365	43	and	and	CCONJ
ejpam-4352	365	44	gr	gr	PROPN
ejpam-4352	365	45	be	be	AUX
ejpam-4352	365	46	non	non	ADJ
ejpam-4352	365	47	-	-	ADJ
ejpam-4352	365	48	trivial	trivial	ADJ
ejpam-4352	365	49	components	component	NOUN
ejpam-4352	365	50	and	and	CCONJ
ejpam-4352	365	51	pick	pick	VERB
ejpam-4352	365	52	v	v	NUM
ejpam-4352	365	53	∈	∈	PROPN
ejpam-4352	365	54	v	v	NOUN
ejpam-4352	365	55	(	(	PUNCT
ejpam-4352	365	56	gm	gm	PROPN
ejpam-4352	365	57	)	)	PUNCT
ejpam-4352	365	58	and	and	CCONJ
ejpam-4352	365	59	w	w	PROPN
ejpam-4352	365	60	∈	∈	PROPN
ejpam-4352	365	61	v	v	ADP
ejpam-4352	365	62	(	(	PUNCT
ejpam-4352	365	63	gr	gr	NOUN
ejpam-4352	365	64	)	)	PUNCT
ejpam-4352	365	65	.	.	PUNCT
ejpam-4352	366	1	then	then	ADV
ejpam-4352	366	2	s′	s′	ADJ
ejpam-4352	366	3	=	=	PUNCT
ejpam-4352	367	1	[	[	X
ejpam-4352	367	2	v	v	X
ejpam-4352	367	3	(	(	PUNCT
ejpam-4352	367	4	gm)\{v}]∪[v	gm)\{v}]∪[v	PROPN
ejpam-4352	367	5	(	(	PUNCT
ejpam-4352	367	6	gr)\{w}]∪	gr)\{w}]∪	PROPN
ejpam-4352	367	7	[	[	PUNCT
ejpam-4352	367	8	⋃	⋃	PROPN
ejpam-4352	367	9	j	j	PROPN
ejpam-4352	367	10	̸=k	̸=k	PROPN
ejpam-4352	367	11	,	,	PUNCT
ejpam-4352	367	12	r	r	NOUN
ejpam-4352	367	13	v	v	NOUN
ejpam-4352	367	14	(	(	PUNCT
ejpam-4352	367	15	gj	gj	NOUN
ejpam-4352	367	16	)	)	PUNCT
ejpam-4352	367	17	]	]	PUNCT
ejpam-4352	367	18	is	be	AUX
ejpam-4352	367	19	a	a	DET
ejpam-4352	367	20	complement	complement	NOUN
ejpam-4352	367	21	-	-	PUNCT
ejpam-4352	367	22	super	super	ADJ
ejpam-4352	367	23	dominating	dominating	NOUN
ejpam-4352	367	24	set	set	NOUN
ejpam-4352	367	25	of	of	ADP
ejpam-4352	367	26	g	g	NOUN
ejpam-4352	367	27	by	by	ADP
ejpam-4352	367	28	theorem	theorem	NOUN
ejpam-4352	367	29	9(iii	9(iii	NUM
ejpam-4352	367	30	)	)	PUNCT
ejpam-4352	367	31	.	.	PUNCT
ejpam-4352	368	1	it	it	PRON
ejpam-4352	368	2	follows	follow	VERB
ejpam-4352	368	3	that	that	SCONJ
ejpam-4352	368	4	γcs(g	γcs(g	X
ejpam-4352	368	5	)	)	PUNCT
ejpam-4352	368	6	≤	≤	NUM
ejpam-4352	368	7	n−	n−	NOUN
ejpam-4352	368	8	2	2	NUM
ejpam-4352	368	9	.	.	PUNCT
ejpam-4352	369	1	let	let	VERB
ejpam-4352	369	2	s	s	PRON
ejpam-4352	369	3	be	be	AUX
ejpam-4352	369	4	a	a	DET
ejpam-4352	369	5	γcs	γcs	NOUN
ejpam-4352	369	6	-	-	PUNCT
ejpam-4352	369	7	set	set	NOUN
ejpam-4352	369	8	of	of	ADP
ejpam-4352	369	9	g.	g.	PROPN
ejpam-4352	369	10	since	since	SCONJ
ejpam-4352	369	11	every	every	DET
ejpam-4352	369	12	complement	complement	NOUN
ejpam-4352	369	13	-	-	PUNCT
ejpam-4352	369	14	super	super	ADJ
ejpam-4352	369	15	dominating	dominating	NOUN
ejpam-4352	369	16	set	set	NOUN
ejpam-4352	369	17	of	of	ADP
ejpam-4352	369	18	type	type	NOUN
ejpam-4352	369	19	(	(	PUNCT
ejpam-4352	369	20	ii	ii	NOUN
ejpam-4352	369	21	)	)	PUNCT
ejpam-4352	369	22	(	(	PUNCT
ejpam-4352	369	23	see	see	VERB
ejpam-4352	369	24	theorem	theorem	NOUN
ejpam-4352	369	25	9	9	NUM
ejpam-4352	369	26	)	)	PUNCT
ejpam-4352	369	27	is	be	AUX
ejpam-4352	369	28	of	of	ADP
ejpam-4352	369	29	cardinality	cardinality	PROPN
ejpam-4352	369	30	n−1	n−1	PROPN
ejpam-4352	369	31	,	,	PUNCT
ejpam-4352	369	32	it	it	PRON
ejpam-4352	369	33	follows	follow	VERB
ejpam-4352	369	34	that	that	SCONJ
ejpam-4352	369	35	s	s	VERB
ejpam-4352	369	36	is	be	AUX
ejpam-4352	369	37	of	of	ADP
ejpam-4352	369	38	type	type	NOUN
ejpam-4352	369	39	(	(	PUNCT
ejpam-4352	369	40	i	i	NOUN
ejpam-4352	369	41	)	)	PUNCT
ejpam-4352	369	42	or	or	CCONJ
ejpam-4352	369	43	type	type	NOUN
ejpam-4352	369	44	(	(	PUNCT
ejpam-4352	369	45	iii	iii	NOUN
ejpam-4352	369	46	)	)	PUNCT
ejpam-4352	369	47	.	.	PUNCT
ejpam-4352	370	1	if	if	SCONJ
ejpam-4352	370	2	s	s	NOUN
ejpam-4352	370	3	is	be	AUX
ejpam-4352	370	4	of	of	ADP
ejpam-4352	370	5	type	type	NOUN
ejpam-4352	370	6	(	(	PUNCT
ejpam-4352	370	7	iii	iii	NOUN
ejpam-4352	370	8	)	)	PUNCT
ejpam-4352	370	9	,	,	PUNCT
ejpam-4352	370	10	then	then	ADV
ejpam-4352	370	11	γcs(g	γcs(g	X
ejpam-4352	370	12	)	)	PUNCT
ejpam-4352	370	13	=	=	SYM
ejpam-4352	370	14	n−2	n−2	PROPN
ejpam-4352	370	15	.	.	PUNCT
ejpam-4352	371	1	if	if	SCONJ
ejpam-4352	371	2	s	s	NOUN
ejpam-4352	371	3	is	be	AUX
ejpam-4352	371	4	of	of	ADP
ejpam-4352	371	5	type	type	NOUN
ejpam-4352	371	6	(	(	PUNCT
ejpam-4352	371	7	i	i	NOUN
ejpam-4352	371	8	)	)	PUNCT
ejpam-4352	371	9	,	,	PUNCT
ejpam-4352	371	10	then	then	ADV
ejpam-4352	371	11	there	there	PRON
ejpam-4352	371	12	exists	exist	VERB
ejpam-4352	371	13	a	a	DET
ejpam-4352	371	14	(	(	PUNCT
ejpam-4352	371	15	non	non	ADJ
ejpam-4352	371	16	-	-	ADJ
ejpam-4352	371	17	trivial	trivial	ADJ
ejpam-4352	371	18	component	component	NOUN
ejpam-4352	371	19	)	)	PUNCT
ejpam-4352	371	20	gs	gs	ADP
ejpam-4352	371	21	such	such	ADJ
ejpam-4352	371	22	that	that	PRON
ejpam-4352	371	23	s	s	PART
ejpam-4352	371	24	=	=	X
ejpam-4352	371	25	ds	ds	ADJ
ejpam-4352	371	26	∪	∪	X
ejpam-4352	371	27	[	[	X
ejpam-4352	371	28	∪j	∪j	X
ejpam-4352	371	29	̸=sv	̸=sv	PROPN
ejpam-4352	371	30	(	(	PUNCT
ejpam-4352	371	31	gj	gj	NOUN
ejpam-4352	371	32	)	)	PUNCT
ejpam-4352	371	33	]	]	PUNCT
ejpam-4352	371	34	,	,	PUNCT
ejpam-4352	371	35	where	where	SCONJ
ejpam-4352	371	36	ds	ds	NOUN
ejpam-4352	371	37	is	be	AUX
ejpam-4352	371	38	a	a	DET
ejpam-4352	371	39	γcs	γcs	NOUN
ejpam-4352	371	40	-	-	PUNCT
ejpam-4352	371	41	set	set	NOUN
ejpam-4352	371	42	of	of	ADP
ejpam-4352	371	43	gs	gs	NOUN
ejpam-4352	371	44	.	.	PUNCT
ejpam-4352	372	1	in	in	ADP
ejpam-4352	372	2	this	this	DET
ejpam-4352	372	3	case	case	NOUN
ejpam-4352	372	4	,	,	PUNCT
ejpam-4352	372	5	γcs(g	γcs(g	X
ejpam-4352	372	6	)	)	PUNCT
ejpam-4352	372	7	=	=	SYM
ejpam-4352	372	8	γcs(gs	γcs(gs	NOUN
ejpam-4352	372	9	)	)	PUNCT
ejpam-4352	373	1	+	+	CCONJ
ejpam-4352	373	2	∑	∑	PROPN
ejpam-4352	373	3	j	j	PROPN
ejpam-4352	373	4	̸=s	̸=s	PROPN
ejpam-4352	373	5	|v	|v	X
ejpam-4352	373	6	(	(	PUNCT
ejpam-4352	373	7	gj)|	gj)|	NOUN
ejpam-4352	373	8	.	.	PUNCT
ejpam-4352	373	9	hence	hence	ADV
ejpam-4352	373	10	,	,	PUNCT
ejpam-4352	373	11	γcs(g	γcs(g	PROPN
ejpam-4352	373	12	)	)	PUNCT
ejpam-4352	373	13	+	+	CCONJ
ejpam-4352	373	14	|v	|v	X
ejpam-4352	373	15	(	(	PUNCT
ejpam-4352	373	16	gs)|	gs)|	PROPN
ejpam-4352	373	17	−	−	PROPN
ejpam-4352	373	18	γcs(gs	γcs(gs	PROPN
ejpam-4352	373	19	)	)	PUNCT
ejpam-4352	373	20	=	=	SYM
ejpam-4352	373	21	n	n	CCONJ
ejpam-4352	373	22	,	,	PUNCT
ejpam-4352	373	23	that	that	ADV
ejpam-4352	373	24	is	is	ADV
ejpam-4352	373	25	,	,	PUNCT
ejpam-4352	373	26	γcs(g	γcs(g	X
ejpam-4352	373	27	)	)	PUNCT
ejpam-4352	373	28	=	=	SYM
ejpam-4352	374	1	n	n	PRON
ejpam-4352	374	2	−	−	PROPN
ejpam-4352	375	1	[	[	X
ejpam-4352	375	2	|v	|v	X
ejpam-4352	375	3	(	(	PUNCT
ejpam-4352	375	4	gs)|	gs)|	PROPN
ejpam-4352	375	5	−	−	PROPN
ejpam-4352	375	6	γcs(gs	γcs(gs	PROPN
ejpam-4352	375	7	)	)	PUNCT
ejpam-4352	375	8	]	]	PUNCT
ejpam-4352	375	9	.	.	PUNCT
ejpam-4352	376	1	clearly	clearly	ADV
ejpam-4352	376	2	,	,	PUNCT
ejpam-4352	376	3	ηg	ηg	PRON
ejpam-4352	376	4	=	=	ADJ
ejpam-4352	376	5	|v	|v	X
ejpam-4352	376	6	(	(	PUNCT
ejpam-4352	376	7	gs)|	gs)|	PROPN
ejpam-4352	376	8	−	−	PROPN
ejpam-4352	376	9	γcs(gs	γcs(gs	PROPN
ejpam-4352	376	10	)	)	PUNCT
ejpam-4352	376	11	.	.	PUNCT
ejpam-4352	377	1	accordingly	accordingly	ADV
ejpam-4352	377	2	,	,	PUNCT
ejpam-4352	377	3	γcs(g	γcs(g	X
ejpam-4352	377	4	)	)	PUNCT
ejpam-4352	377	5	=	=	SYM
ejpam-4352	377	6	min{n−	min{n−	PROPN
ejpam-4352	377	7	2	2	NUM
ejpam-4352	377	8	,	,	PUNCT
ejpam-4352	377	9	n−	n−	NOUN
ejpam-4352	377	10	ηg	ηg	ADV
ejpam-4352	377	11	}	}	PUNCT
ejpam-4352	377	12	.	.	PUNCT
ejpam-4352	378	1	the	the	DET
ejpam-4352	378	2	next	next	ADJ
ejpam-4352	378	3	result	result	NOUN
ejpam-4352	378	4	is	be	AUX
ejpam-4352	378	5	a	a	DET
ejpam-4352	378	6	consequence	consequence	NOUN
ejpam-4352	378	7	of	of	ADP
ejpam-4352	378	8	theorem	theorem	ADJ
ejpam-4352	378	9	3	3	NUM
ejpam-4352	378	10	,	,	PUNCT
ejpam-4352	378	11	theorem	theorem	VERB
ejpam-4352	378	12	4	4	NUM
ejpam-4352	378	13	,	,	PUNCT
ejpam-4352	378	14	and	and	CCONJ
ejpam-4352	378	15	theorem	theorem	VERB
ejpam-4352	378	16	10	10	NUM
ejpam-4352	378	17	.	.	PUNCT
ejpam-4352	378	18	corollary	corollary	ADJ
ejpam-4352	378	19	4	4	NUM
ejpam-4352	378	20	.	.	PUNCT
ejpam-4352	379	1	let	let	VERB
ejpam-4352	379	2	g	g	NOUN
ejpam-4352	379	3	and	and	CCONJ
ejpam-4352	379	4	h	h	NOUN
ejpam-4352	379	5	be	be	VERB
ejpam-4352	379	6	any	any	DET
ejpam-4352	379	7	two	two	NUM
ejpam-4352	379	8	graphs	graph	NOUN
ejpam-4352	379	9	of	of	ADP
ejpam-4352	379	10	orders	order	NOUN
ejpam-4352	379	11	m	m	VERB
ejpam-4352	379	12	and	and	CCONJ
ejpam-4352	379	13	n	n	CCONJ
ejpam-4352	379	14	,	,	PUNCT
ejpam-4352	379	15	respectively	respectively	ADV
ejpam-4352	379	16	.	.	PUNCT
ejpam-4352	380	1	then	then	ADV
ejpam-4352	380	2	γsh(g+h	γsh(g+h	NOUN
ejpam-4352	380	3	)	)	PUNCT
ejpam-4352	380	4	=	=	SYM
ejpam-4352	381	1	γcs(g	γcs(g	X
ejpam-4352	381	2	)	)	PUNCT
ejpam-4352	381	3	+	+	CCONJ
ejpam-4352	381	4	γcs(h	γcs(h	PROPN
ejpam-4352	381	5	)	)	PUNCT
ejpam-4352	381	6	.	.	PUNCT
ejpam-4352	382	1	in	in	ADP
ejpam-4352	382	2	particular	particular	ADJ
ejpam-4352	382	3	,	,	PUNCT
ejpam-4352	382	4	(	(	PUNCT
ejpam-4352	382	5	i	i	NOUN
ejpam-4352	382	6	)	)	PUNCT
ejpam-4352	382	7	γsh(g+h	γsh(g+h	NOUN
ejpam-4352	382	8	)	)	PUNCT
ejpam-4352	383	1	=	=	PUNCT
ejpam-4352	384	1	m+	m+	NUM
ejpam-4352	384	2	n	n	NOUN
ejpam-4352	384	3	if	if	SCONJ
ejpam-4352	384	4	g	g	PROPN
ejpam-4352	384	5	and	and	CCONJ
ejpam-4352	384	6	h	h	NOUN
ejpam-4352	384	7	are	be	AUX
ejpam-4352	384	8	complete	complete	ADJ
ejpam-4352	384	9	;	;	PUNCT
ejpam-4352	384	10	(	(	PUNCT
ejpam-4352	384	11	ii	ii	NOUN
ejpam-4352	384	12	)	)	PUNCT
ejpam-4352	384	13	γsh(g+h	γsh(g+h	NOUN
ejpam-4352	384	14	)	)	PUNCT
ejpam-4352	385	1	=	=	SYM
ejpam-4352	386	1	m+	m+	NUM
ejpam-4352	386	2	γcs(h	γcs(h	PROPN
ejpam-4352	386	3	)	)	PUNCT
ejpam-4352	386	4	if	if	SCONJ
ejpam-4352	386	5	g	g	PROPN
ejpam-4352	386	6	=	=	SYM
ejpam-4352	386	7	km	km	PROPN
ejpam-4352	386	8	;	;	PUNCT
ejpam-4352	386	9	and	and	CCONJ
ejpam-4352	386	10	(	(	PUNCT
ejpam-4352	386	11	iii	iii	NOUN
ejpam-4352	386	12	)	)	PUNCT
ejpam-4352	386	13	γsh(g+h	γsh(g+h	NOUN
ejpam-4352	386	14	)	)	PUNCT
ejpam-4352	386	15	=	=	PUNCT
ejpam-4352	387	1	m+	m+	NUM
ejpam-4352	387	2	n−	n−	NOUN
ejpam-4352	387	3	2	2	NUM
ejpam-4352	387	4	if	if	SCONJ
ejpam-4352	387	5	g	g	NOUN
ejpam-4352	387	6	=	=	NOUN
ejpam-4352	387	7	km	km	PROPN
ejpam-4352	387	8	and	and	CCONJ
ejpam-4352	387	9	h	h	NOUN
ejpam-4352	388	1	=	=	PROPN
ejpam-4352	388	2	kn	kn	PROPN
ejpam-4352	388	3	for	for	ADP
ejpam-4352	388	4	m	m	PROPN
ejpam-4352	388	5	,	,	PUNCT
ejpam-4352	388	6	n	n	PRON
ejpam-4352	388	7	≥	≥	NOUN
ejpam-4352	388	8	2	2	NUM
ejpam-4352	388	9	.	.	PUNCT
ejpam-4352	389	1	the	the	DET
ejpam-4352	389	2	lexicographic	lexicographic	ADJ
ejpam-4352	389	3	product	product	NOUN
ejpam-4352	389	4	of	of	ADP
ejpam-4352	389	5	graphs	graph	NOUN
ejpam-4352	389	6	g	g	PROPN
ejpam-4352	389	7	and	and	CCONJ
ejpam-4352	389	8	h	h	NOUN
ejpam-4352	389	9	,	,	PUNCT
ejpam-4352	389	10	denoted	denote	VERB
ejpam-4352	389	11	by	by	ADP
ejpam-4352	389	12	g[h	g[h	NOUN
ejpam-4352	389	13	]	]	PUNCT
ejpam-4352	389	14	,	,	PUNCT
ejpam-4352	389	15	is	be	AUX
ejpam-4352	389	16	the	the	DET
ejpam-4352	389	17	graph	graph	NOUN
ejpam-4352	389	18	with	with	ADP
ejpam-4352	389	19	vertex	vertex	NOUN
ejpam-4352	389	20	set	set	VERB
ejpam-4352	389	21	v	v	NOUN
ejpam-4352	389	22	(	(	PUNCT
ejpam-4352	389	23	g[h	g[h	PROPN
ejpam-4352	389	24	]	]	PUNCT
ejpam-4352	389	25	)	)	PUNCT
ejpam-4352	389	26	=	=	SYM
ejpam-4352	389	27	v	v	X
ejpam-4352	389	28	(	(	PUNCT
ejpam-4352	389	29	g)×	g)×	NOUN
ejpam-4352	389	30	v	v	NOUN
ejpam-4352	389	31	(	(	PUNCT
ejpam-4352	389	32	h	h	NOUN
ejpam-4352	389	33	)	)	PUNCT
ejpam-4352	389	34	such	such	ADJ
ejpam-4352	389	35	that	that	SCONJ
ejpam-4352	389	36	(	(	PUNCT
ejpam-4352	389	37	v	v	NOUN
ejpam-4352	389	38	,	,	PUNCT
ejpam-4352	389	39	a)(u	a)(u	ADJ
ejpam-4352	389	40	,	,	PUNCT
ejpam-4352	389	41	b	b	X
ejpam-4352	389	42	)	)	PUNCT
ejpam-4352	389	43	∈	∈	NOUN
ejpam-4352	389	44	e(g[h	e(g[h	NOUN
ejpam-4352	389	45	]	]	PUNCT
ejpam-4352	389	46	)	)	PUNCT
ejpam-4352	389	47	if	if	SCONJ
ejpam-4352	389	48	and	and	CCONJ
ejpam-4352	389	49	only	only	ADV
ejpam-4352	389	50	if	if	SCONJ
ejpam-4352	389	51	either	either	DET
ejpam-4352	389	52	uv	uv	PROPN
ejpam-4352	389	53	∈	∈	PROPN
ejpam-4352	389	54	e(g	e(g	PROPN
ejpam-4352	389	55	)	)	PUNCT
ejpam-4352	389	56	or	or	CCONJ
ejpam-4352	389	57	u	u	X
ejpam-4352	389	58	=	=	PROPN
ejpam-4352	389	59	v	v	PROPN
ejpam-4352	389	60	and	and	CCONJ
ejpam-4352	389	61	ab	ab	PROPN
ejpam-4352	389	62	∈	∈	PROPN
ejpam-4352	389	63	e(h	e(h	PROPN
ejpam-4352	389	64	)	)	PUNCT
ejpam-4352	389	65	.	.	PUNCT
ejpam-4352	390	1	note	note	VERB
ejpam-4352	390	2	that	that	SCONJ
ejpam-4352	390	3	every	every	DET
ejpam-4352	390	4	non	non	ADJ
ejpam-4352	390	5	-	-	ADJ
ejpam-4352	390	6	empty	empty	ADJ
ejpam-4352	390	7	subset	subset	NOUN
ejpam-4352	390	8	c	c	NOUN
ejpam-4352	390	9	of	of	ADP
ejpam-4352	390	10	v	v	PROPN
ejpam-4352	390	11	(	(	PUNCT
ejpam-4352	390	12	g	g	NOUN
ejpam-4352	390	13	)	)	PUNCT
ejpam-4352	390	14	×	×	NOUN
ejpam-4352	390	15	v	v	NOUN
ejpam-4352	390	16	(	(	PUNCT
ejpam-4352	390	17	h	h	NOUN
ejpam-4352	390	18	)	)	PUNCT
ejpam-4352	390	19	can	can	AUX
ejpam-4352	390	20	be	be	AUX
ejpam-4352	390	21	expressed	express	VERB
ejpam-4352	390	22	as	as	ADP
ejpam-4352	390	23	c	c	X
ejpam-4352	390	24	=	=	SYM
ejpam-4352	390	25	∪x∈s	∪x∈s	PROPN
ejpam-4352	391	1	[	[	X
ejpam-4352	391	2	{	{	PUNCT
ejpam-4352	391	3	x	x	NOUN
ejpam-4352	391	4	}	}	PUNCT
ejpam-4352	391	5	×	×	PROPN
ejpam-4352	391	6	tx	tx	PROPN
ejpam-4352	391	7	]	]	X
ejpam-4352	391	8	,	,	PUNCT
ejpam-4352	391	9	where	where	SCONJ
ejpam-4352	391	10	s	s	VERB
ejpam-4352	391	11	⊆	⊆	NUM
ejpam-4352	391	12	v	v	NOUN
ejpam-4352	391	13	(	(	PUNCT
ejpam-4352	391	14	g	g	NOUN
ejpam-4352	391	15	)	)	PUNCT
ejpam-4352	391	16	and	and	CCONJ
ejpam-4352	391	17	tx	tx	VERB
ejpam-4352	391	18	⊆	⊆	NUM
ejpam-4352	391	19	v	v	NOUN
ejpam-4352	391	20	(	(	PUNCT
ejpam-4352	391	21	h	h	NOUN
ejpam-4352	391	22	)	)	PUNCT
ejpam-4352	391	23	for	for	ADP
ejpam-4352	391	24	each	each	DET
ejpam-4352	391	25	x	x	PROPN
ejpam-4352	391	26	∈	∈	PROPN
ejpam-4352	391	27	s.	s.	PROPN
ejpam-4352	391	28	theorem	theorem	VERB
ejpam-4352	391	29	11	11	NUM
ejpam-4352	391	30	.	.	PUNCT
ejpam-4352	392	1	let	let	VERB
ejpam-4352	392	2	g	g	NOUN
ejpam-4352	393	1	and	and	CCONJ
ejpam-4352	393	2	h	h	NOUN
ejpam-4352	393	3	be	be	VERB
ejpam-4352	393	4	any	any	DET
ejpam-4352	393	5	connected	connected	ADJ
ejpam-4352	393	6	non	non	ADJ
ejpam-4352	393	7	-	-	ADJ
ejpam-4352	393	8	trivial	trivial	ADJ
ejpam-4352	393	9	graphs	graph	NOUN
ejpam-4352	393	10	.	.	PUNCT
ejpam-4352	394	1	then	then	ADV
ejpam-4352	394	2	c	c	NOUN
ejpam-4352	394	3	=	=	PUNCT
ejpam-4352	394	4	⋃	⋃	PROPN
ejpam-4352	394	5	x∈s	x∈s	NOUN
ejpam-4352	395	1	[	[	X
ejpam-4352	395	2	{	{	PUNCT
ejpam-4352	395	3	x}×tx	x}×tx	X
ejpam-4352	395	4	]	]	X
ejpam-4352	395	5	,	,	PUNCT
ejpam-4352	395	6	where	where	SCONJ
ejpam-4352	395	7	s	s	VERB
ejpam-4352	395	8	⊆	⊆	NUM
ejpam-4352	395	9	v	v	NOUN
ejpam-4352	395	10	(	(	PUNCT
ejpam-4352	395	11	g	g	NOUN
ejpam-4352	395	12	)	)	PUNCT
ejpam-4352	395	13	and	and	CCONJ
ejpam-4352	395	14	tx	tx	VERB
ejpam-4352	395	15	⊆	⊆	NUM
ejpam-4352	395	16	v	v	NOUN
ejpam-4352	395	17	(	(	PUNCT
ejpam-4352	395	18	h	h	NOUN
ejpam-4352	395	19	)	)	PUNCT
ejpam-4352	395	20	for	for	ADP
ejpam-4352	395	21	each	each	DET
ejpam-4352	395	22	x	x	SYM
ejpam-4352	395	23	∈	∈	PROPN
ejpam-4352	395	24	s	s	NOUN
ejpam-4352	395	25	,	,	PUNCT
ejpam-4352	395	26	is	be	AUX
ejpam-4352	395	27	a	a	DET
ejpam-4352	395	28	super	super	ADV
ejpam-4352	395	29	hop	hop	NOUN
ejpam-4352	395	30	dominating	dominating	NOUN
ejpam-4352	395	31	set	set	NOUN
ejpam-4352	395	32	of	of	ADP
ejpam-4352	395	33	g[h	g[h	PROPN
ejpam-4352	395	34	]	]	PUNCT
ejpam-4352	395	35	if	if	SCONJ
ejpam-4352	395	36	and	and	CCONJ
ejpam-4352	395	37	only	only	ADV
ejpam-4352	395	38	if	if	SCONJ
ejpam-4352	395	39	the	the	DET
ejpam-4352	395	40	following	follow	VERB
ejpam-4352	395	41	statements	statement	NOUN
ejpam-4352	395	42	hold	hold	VERB
ejpam-4352	395	43	.	.	PUNCT
ejpam-4352	396	1	(	(	PUNCT
ejpam-4352	396	2	i	i	NOUN
ejpam-4352	396	3	)	)	PUNCT
ejpam-4352	396	4	s	s	PART
ejpam-4352	396	5	=	=	SYM
ejpam-4352	396	6	v	v	NOUN
ejpam-4352	396	7	(	(	PUNCT
ejpam-4352	396	8	g	g	NOUN
ejpam-4352	396	9	)	)	PUNCT
ejpam-4352	396	10	.	.	PUNCT
ejpam-4352	397	1	(	(	PUNCT
ejpam-4352	397	2	ii	ii	NOUN
ejpam-4352	397	3	)	)	PUNCT
ejpam-4352	397	4	for	for	ADP
ejpam-4352	397	5	each	each	DET
ejpam-4352	397	6	x	x	SYM
ejpam-4352	397	7	∈	∈	PROPN
ejpam-4352	397	8	s	s	VERB
ejpam-4352	397	9	with	with	ADP
ejpam-4352	397	10	|v	|v	PROPN
ejpam-4352	397	11	(	(	PUNCT
ejpam-4352	397	12	h	h	NOUN
ejpam-4352	397	13	)	)	PUNCT
ejpam-4352	397	14	\	\	PUNCT
ejpam-4352	397	15	tx|	tx|	PROPN
ejpam-4352	397	16	≥	≥	NUM
ejpam-4352	397	17	2	2	NUM
ejpam-4352	397	18	,	,	PUNCT
ejpam-4352	397	19	the	the	DET
ejpam-4352	397	20	following	follow	VERB
ejpam-4352	397	21	conditions	condition	NOUN
ejpam-4352	397	22	hold	hold	VERB
ejpam-4352	397	23	:	:	PUNCT
ejpam-4352	397	24	(	(	PUNCT
ejpam-4352	397	25	a	a	X
ejpam-4352	397	26	)	)	PUNCT
ejpam-4352	397	27	tx	tx	PROPN
ejpam-4352	397	28	is	be	AUX
ejpam-4352	397	29	a	a	DET
ejpam-4352	397	30	complement	complement	NOUN
ejpam-4352	397	31	-	-	PUNCT
ejpam-4352	397	32	super	super	ADJ
ejpam-4352	397	33	dominating	dominating	NOUN
ejpam-4352	397	34	set	set	NOUN
ejpam-4352	397	35	of	of	ADP
ejpam-4352	397	36	h	h	NOUN
ejpam-4352	397	37	,	,	PUNCT
ejpam-4352	397	38	and	and	CCONJ
ejpam-4352	397	39	s.	s.	PROPN
ejpam-4352	397	40	canoy	canoy	PROPN
ejpam-4352	397	41	,	,	PUNCT
ejpam-4352	397	42	jr	jr	PROPN
ejpam-4352	397	43	.	.	PROPN
ejpam-4352	397	44	,	,	PUNCT
ejpam-4352	397	45	g.	g.	PROPN
ejpam-4352	397	46	salasalan	salasalan	PROPN
ejpam-4352	397	47	/	/	SYM
ejpam-4352	397	48	eur	eur	PROPN
ejpam-4352	397	49	.	.	PUNCT
ejpam-4352	398	1	j.	j.	PROPN
ejpam-4352	398	2	pure	pure	PROPN
ejpam-4352	398	3	appl	appl	PROPN
ejpam-4352	398	4	.	.	PROPN
ejpam-4352	398	5	math	math	PROPN
ejpam-4352	398	6	,	,	PUNCT
ejpam-4352	398	7	15	15	NUM
ejpam-4352	398	8	(	(	PUNCT
ejpam-4352	398	9	2	2	NUM
ejpam-4352	398	10	)	)	PUNCT
ejpam-4352	398	11	(	(	PUNCT
ejpam-4352	398	12	2022	2022	NUM
ejpam-4352	398	13	)	)	PUNCT
ejpam-4352	398	14	,	,	PUNCT
ejpam-4352	398	15	342	342	NUM
ejpam-4352	398	16	-	-	SYM
ejpam-4352	398	17	353	353	NUM
ejpam-4352	398	18	350	350	NUM
ejpam-4352	398	19	(	(	PUNCT
ejpam-4352	398	20	b	b	NOUN
ejpam-4352	398	21	)	)	PUNCT
ejpam-4352	398	22	ty	ty	NOUN
ejpam-4352	399	1	=	=	NOUN
ejpam-4352	399	2	v	v	PROPN
ejpam-4352	399	3	(	(	PUNCT
ejpam-4352	399	4	h	h	NOUN
ejpam-4352	399	5	)	)	PUNCT
ejpam-4352	399	6	for	for	ADP
ejpam-4352	399	7	all	all	DET
ejpam-4352	399	8	y	y	PROPN
ejpam-4352	399	9	∈	∈	PROPN
ejpam-4352	399	10	n2	n2	NOUN
ejpam-4352	399	11	g(x	g(x	PROPN
ejpam-4352	399	12	)	)	PUNCT
ejpam-4352	399	13	.	.	PUNCT
ejpam-4352	400	1	(	(	PUNCT
ejpam-4352	400	2	iii	iii	X
ejpam-4352	400	3	)	)	PUNCT
ejpam-4352	400	4	for	for	ADP
ejpam-4352	400	5	each	each	DET
ejpam-4352	400	6	x	x	SYM
ejpam-4352	400	7	∈	∈	PROPN
ejpam-4352	400	8	v	v	ADP
ejpam-4352	400	9	(	(	PUNCT
ejpam-4352	400	10	g	g	NOUN
ejpam-4352	400	11	)	)	PUNCT
ejpam-4352	400	12	with	with	ADP
ejpam-4352	400	13	|v	|v	PROPN
ejpam-4352	400	14	(	(	PUNCT
ejpam-4352	400	15	h	h	NOUN
ejpam-4352	400	16	)	)	PUNCT
ejpam-4352	400	17	\	\	PUNCT
ejpam-4352	400	18	tx|	tx|	PROPN
ejpam-4352	400	19	=	=	SYM
ejpam-4352	400	20	1	1	NUM
ejpam-4352	400	21	,	,	PUNCT
ejpam-4352	400	22	at	at	ADV
ejpam-4352	400	23	least	least	ADJ
ejpam-4352	400	24	one	one	NUM
ejpam-4352	400	25	of	of	ADP
ejpam-4352	400	26	the	the	DET
ejpam-4352	400	27	following	follow	VERB
ejpam-4352	400	28	conditions	condition	NOUN
ejpam-4352	400	29	holds	hold	VERB
ejpam-4352	400	30	:	:	PUNCT
ejpam-4352	400	31	(	(	PUNCT
ejpam-4352	400	32	a	a	X
ejpam-4352	400	33	)	)	PUNCT
ejpam-4352	400	34	tx	tx	PROPN
ejpam-4352	400	35	is	be	AUX
ejpam-4352	400	36	a	a	DET
ejpam-4352	400	37	complement	complement	NOUN
ejpam-4352	400	38	-	-	PUNCT
ejpam-4352	400	39	super	super	ADJ
ejpam-4352	400	40	dominating	dominating	NOUN
ejpam-4352	400	41	set	set	NOUN
ejpam-4352	400	42	of	of	ADP
ejpam-4352	400	43	h	h	NOUN
ejpam-4352	400	44	and	and	CCONJ
ejpam-4352	400	45	ty	ty	INTJ
ejpam-4352	400	46	=	=	NOUN
ejpam-4352	400	47	v	v	NOUN
ejpam-4352	400	48	(	(	PUNCT
ejpam-4352	400	49	h	h	NOUN
ejpam-4352	400	50	)	)	PUNCT
ejpam-4352	400	51	for	for	ADP
ejpam-4352	400	52	all	all	DET
ejpam-4352	400	53	y	y	PROPN
ejpam-4352	400	54	∈	∈	PROPN
ejpam-4352	400	55	n2	n2	NOUN
ejpam-4352	400	56	g(x	g(x	PROPN
ejpam-4352	400	57	)	)	PUNCT
ejpam-4352	400	58	.	.	PUNCT
ejpam-4352	401	1	(	(	PUNCT
ejpam-4352	401	2	b	b	X
ejpam-4352	401	3	)	)	PUNCT
ejpam-4352	401	4	there	there	PRON
ejpam-4352	401	5	exist	exist	VERB
ejpam-4352	401	6	z	z	PROPN
ejpam-4352	401	7	∈	∈	PROPN
ejpam-4352	401	8	n2	n2	NOUN
ejpam-4352	401	9	g(x	g(x	PROPN
ejpam-4352	401	10	)	)	PUNCT
ejpam-4352	401	11	and	and	CCONJ
ejpam-4352	401	12	t	t	PROPN
ejpam-4352	401	13	∈	∈	PROPN
ejpam-4352	401	14	tz	tz	NOUN
ejpam-4352	401	15	such	such	ADJ
ejpam-4352	402	1	that	that	SCONJ
ejpam-4352	402	2	tw	tw	VERB
ejpam-4352	402	3	=	=	SYM
ejpam-4352	402	4	v	v	PROPN
ejpam-4352	402	5	(	(	PUNCT
ejpam-4352	402	6	h	h	NOUN
ejpam-4352	402	7	)	)	PUNCT
ejpam-4352	402	8	for	for	ADP
ejpam-4352	402	9	all	all	DET
ejpam-4352	402	10	w	w	PROPN
ejpam-4352	402	11	∈	∈	PROPN
ejpam-4352	402	12	n2	n2	NOUN
ejpam-4352	402	13	g(z)\{x	g(z)\{x	PROPN
ejpam-4352	402	14	}	}	PUNCT
ejpam-4352	402	15	and	and	CCONJ
ejpam-4352	402	16	v	v	NOUN
ejpam-4352	402	17	(	(	PUNCT
ejpam-4352	402	18	h	h	NOUN
ejpam-4352	402	19	)	)	PUNCT
ejpam-4352	402	20	\	\	PUNCT
ejpam-4352	402	21	tz	tz	PROPN
ejpam-4352	402	22	⊆	⊆	NUM
ejpam-4352	402	23	nh(t	nh(t	NUM
ejpam-4352	402	24	)	)	PUNCT
ejpam-4352	402	25	,	,	PUNCT
ejpam-4352	402	26	where	where	SCONJ
ejpam-4352	402	27	|v	|v	PROPN
ejpam-4352	402	28	(	(	PUNCT
ejpam-4352	402	29	h	h	NOUN
ejpam-4352	402	30	)	)	PUNCT
ejpam-4352	402	31	\	\	PUNCT
ejpam-4352	402	32	tz|	tz|	ADP
ejpam-4352	402	33	≤	≤	NUM
ejpam-4352	402	34	1	1	NUM
ejpam-4352	402	35	.	.	PUNCT
ejpam-4352	402	36	proof	proof	NOUN
ejpam-4352	402	37	.	.	PUNCT
ejpam-4352	403	1	suppose	suppose	VERB
ejpam-4352	403	2	c	c	NOUN
ejpam-4352	403	3	=	=	SYM
ejpam-4352	403	4	∪x∈s({x	∪x∈s({x	PROPN
ejpam-4352	403	5	}	}	PUNCT
ejpam-4352	403	6	×	×	NOUN
ejpam-4352	403	7	tx	tx	PROPN
ejpam-4352	403	8	)	)	PUNCT
ejpam-4352	403	9	is	be	AUX
ejpam-4352	403	10	a	a	DET
ejpam-4352	403	11	super	super	ADV
ejpam-4352	403	12	hop	hop	NOUN
ejpam-4352	403	13	dominating	dominating	NOUN
ejpam-4352	403	14	set	set	NOUN
ejpam-4352	403	15	of	of	ADP
ejpam-4352	403	16	g[h	g[h	PROPN
ejpam-4352	403	17	]	]	PUNCT
ejpam-4352	403	18	.	.	PUNCT
ejpam-4352	404	1	suppose	suppose	VERB
ejpam-4352	404	2	there	there	PRON
ejpam-4352	404	3	exists	exist	VERB
ejpam-4352	404	4	x	x	X
ejpam-4352	404	5	∈	∈	PROPN
ejpam-4352	404	6	v	v	X
ejpam-4352	404	7	(	(	PUNCT
ejpam-4352	404	8	g	g	NOUN
ejpam-4352	404	9	)	)	PUNCT
ejpam-4352	404	10	\	\	PROPN
ejpam-4352	404	11	s	s	PART
ejpam-4352	404	12	and	and	CCONJ
ejpam-4352	404	13	let	let	VERB
ejpam-4352	404	14	a	a	DET
ejpam-4352	404	15	∈	∈	PROPN
ejpam-4352	404	16	v	v	NOUN
ejpam-4352	404	17	(	(	PUNCT
ejpam-4352	404	18	h	h	NOUN
ejpam-4352	404	19	)	)	PUNCT
ejpam-4352	404	20	.	.	PUNCT
ejpam-4352	405	1	since	since	SCONJ
ejpam-4352	405	2	h	h	PROPN
ejpam-4352	405	3	is	be	AUX
ejpam-4352	405	4	non	non	ADJ
ejpam-4352	405	5	-	-	ADJ
ejpam-4352	405	6	trivial	trivial	ADJ
ejpam-4352	405	7	,	,	PUNCT
ejpam-4352	405	8	it	it	PRON
ejpam-4352	405	9	follows	follow	VERB
ejpam-4352	405	10	that	that	SCONJ
ejpam-4352	405	11	ehpng[h	ehpng[h	PROPN
ejpam-4352	405	12	]	]	X
ejpam-4352	405	13	(	(	PUNCT
ejpam-4352	405	14	(	(	PUNCT
ejpam-4352	405	15	x	x	NOUN
ejpam-4352	405	16	,	,	PUNCT
ejpam-4352	405	17	a	a	PRON
ejpam-4352	405	18	)	)	PUNCT
ejpam-4352	405	19	,	,	PUNCT
ejpam-4352	405	20	v	v	NOUN
ejpam-4352	405	21	(	(	PUNCT
ejpam-4352	405	22	g[h	g[h	PROPN
ejpam-4352	405	23	]	]	PUNCT
ejpam-4352	405	24	)	)	PUNCT
ejpam-4352	405	25	\	\	PUNCT
ejpam-4352	406	1	c	c	X
ejpam-4352	406	2	)	)	PUNCT
ejpam-4352	406	3	=	=	NOUN
ejpam-4352	406	4	∅	∅	NOUN
ejpam-4352	406	5	,	,	PUNCT
ejpam-4352	406	6	contrary	contrary	ADJ
ejpam-4352	406	7	to	to	ADP
ejpam-4352	406	8	our	our	PRON
ejpam-4352	406	9	assumption	assumption	NOUN
ejpam-4352	406	10	that	that	SCONJ
ejpam-4352	406	11	c	c	PROPN
ejpam-4352	406	12	is	be	AUX
ejpam-4352	406	13	a	a	DET
ejpam-4352	406	14	super	super	ADV
ejpam-4352	406	15	hop	hop	NOUN
ejpam-4352	406	16	dominating	dominating	NOUN
ejpam-4352	406	17	set	set	NOUN
ejpam-4352	406	18	.	.	PUNCT
ejpam-4352	407	1	thus	thus	ADV
ejpam-4352	407	2	,	,	PUNCT
ejpam-4352	407	3	s	s	VERB
ejpam-4352	407	4	=	=	SYM
ejpam-4352	407	5	v	v	X
ejpam-4352	407	6	(	(	PUNCT
ejpam-4352	407	7	g	g	NOUN
ejpam-4352	407	8	)	)	PUNCT
ejpam-4352	407	9	,	,	PUNCT
ejpam-4352	407	10	showing	show	VERB
ejpam-4352	407	11	that	that	SCONJ
ejpam-4352	407	12	(	(	PUNCT
ejpam-4352	407	13	i	i	NOUN
ejpam-4352	407	14	)	)	PUNCT
ejpam-4352	407	15	holds	hold	VERB
ejpam-4352	407	16	.	.	PUNCT
ejpam-4352	408	1	now	now	ADV
ejpam-4352	408	2	let	let	VERB
ejpam-4352	408	3	x	x	X
ejpam-4352	408	4	∈	∈	PROPN
ejpam-4352	408	5	s	s	PART
ejpam-4352	408	6	=	=	X
ejpam-4352	408	7	v	v	ADJ
ejpam-4352	408	8	(	(	PUNCT
ejpam-4352	408	9	g	g	NOUN
ejpam-4352	408	10	)	)	PUNCT
ejpam-4352	408	11	with	with	ADP
ejpam-4352	408	12	|v	|v	PROPN
ejpam-4352	408	13	(	(	PUNCT
ejpam-4352	408	14	h	h	NOUN
ejpam-4352	408	15	)	)	PUNCT
ejpam-4352	408	16	\	\	PUNCT
ejpam-4352	408	17	tx|	tx|	PROPN
ejpam-4352	408	18	≥	≥	NUM
ejpam-4352	408	19	2	2	NUM
ejpam-4352	408	20	and	and	CCONJ
ejpam-4352	408	21	let	let	VERB
ejpam-4352	408	22	p	p	PRON
ejpam-4352	408	23	∈	∈	PROPN
ejpam-4352	408	24	v	v	ADP
ejpam-4352	408	25	(	(	PUNCT
ejpam-4352	408	26	h	h	NOUN
ejpam-4352	408	27	)	)	PUNCT
ejpam-4352	408	28	\	\	PROPN
ejpam-4352	409	1	tx	tx	PROPN
ejpam-4352	409	2	.	.	PUNCT
ejpam-4352	410	1	since	since	SCONJ
ejpam-4352	410	2	c	c	PROPN
ejpam-4352	410	3	is	be	AUX
ejpam-4352	410	4	a	a	DET
ejpam-4352	410	5	super	super	ADV
ejpam-4352	410	6	hop	hop	NOUN
ejpam-4352	410	7	dominating	dominating	NOUN
ejpam-4352	410	8	set	set	NOUN
ejpam-4352	410	9	of	of	ADP
ejpam-4352	410	10	g[h	g[h	PROPN
ejpam-4352	410	11	]	]	PUNCT
ejpam-4352	410	12	,	,	PUNCT
ejpam-4352	410	13	ehpng[h]((x	ehpng[h]((x	PROPN
ejpam-4352	410	14	,	,	PUNCT
ejpam-4352	410	15	p	p	NOUN
ejpam-4352	410	16	)	)	PUNCT
ejpam-4352	410	17	,	,	PUNCT
ejpam-4352	410	18	v	v	NOUN
ejpam-4352	410	19	(	(	PUNCT
ejpam-4352	410	20	g[h	g[h	PROPN
ejpam-4352	410	21	]	]	PUNCT
ejpam-4352	410	22	)	)	PUNCT
ejpam-4352	410	23	\	\	PUNCT
ejpam-4352	411	1	c	c	X
ejpam-4352	411	2	)	)	PUNCT
ejpam-4352	411	3	̸=	̸=	PROPN
ejpam-4352	411	4	∅.	∅.	ADV
ejpam-4352	411	5	let	let	VERB
ejpam-4352	411	6	(	(	PUNCT
ejpam-4352	411	7	y	y	NOUN
ejpam-4352	411	8	,	,	PUNCT
ejpam-4352	411	9	q	q	NOUN
ejpam-4352	411	10	)	)	PUNCT
ejpam-4352	411	11	∈	∈	PROPN
ejpam-4352	412	1	ehpng[h]((x	ehpng[h]((x	ADP
ejpam-4352	412	2	,	,	PUNCT
ejpam-4352	412	3	p	p	NOUN
ejpam-4352	412	4	)	)	PUNCT
ejpam-4352	412	5	,	,	PUNCT
ejpam-4352	412	6	v	v	NOUN
ejpam-4352	412	7	(	(	PUNCT
ejpam-4352	412	8	g[h	g[h	PROPN
ejpam-4352	412	9	]	]	PUNCT
ejpam-4352	412	10	)	)	PUNCT
ejpam-4352	412	11	\	\	PUNCT
ejpam-4352	413	1	c	c	X
ejpam-4352	413	2	)	)	PUNCT
ejpam-4352	413	3	.	.	PUNCT
ejpam-4352	414	1	by	by	ADP
ejpam-4352	414	2	assumption	assumption	NOUN
ejpam-4352	414	3	,	,	PUNCT
ejpam-4352	414	4	y	y	PROPN
ejpam-4352	414	5	/∈	/∈	PUNCT
ejpam-4352	414	6	n2	n2	PROPN
ejpam-4352	414	7	g(x	g(x	PROPN
ejpam-4352	414	8	)	)	PUNCT
ejpam-4352	414	9	.	.	PUNCT
ejpam-4352	415	1	hence	hence	ADV
ejpam-4352	415	2	,	,	PUNCT
ejpam-4352	415	3	y	y	PROPN
ejpam-4352	415	4	=	=	PUNCT
ejpam-4352	415	5	x	x	PROPN
ejpam-4352	415	6	and	and	CCONJ
ejpam-4352	415	7	q	q	PROPN
ejpam-4352	415	8	∈	∈	PROPN
ejpam-4352	415	9	tx	tx	PROPN
ejpam-4352	415	10	\	\	PROPN
ejpam-4352	415	11	nh(p	nh(p	PROPN
ejpam-4352	415	12	)	)	PUNCT
ejpam-4352	415	13	.	.	PUNCT
ejpam-4352	416	1	moreover	moreover	ADV
ejpam-4352	416	2	,	,	PUNCT
ejpam-4352	416	3	since	since	SCONJ
ejpam-4352	416	4	(	(	PUNCT
ejpam-4352	416	5	x	x	NOUN
ejpam-4352	416	6	,	,	PUNCT
ejpam-4352	416	7	q	q	NOUN
ejpam-4352	416	8	)	)	PUNCT
ejpam-4352	416	9	∈	∈	PROPN
ejpam-4352	416	10	ehpng[h]((x	ehpng[h]((x	ADP
ejpam-4352	416	11	,	,	PUNCT
ejpam-4352	416	12	p	p	NOUN
ejpam-4352	416	13	)	)	PUNCT
ejpam-4352	416	14	,	,	PUNCT
ejpam-4352	416	15	v	v	NOUN
ejpam-4352	416	16	(	(	PUNCT
ejpam-4352	416	17	g[h	g[h	PROPN
ejpam-4352	416	18	]	]	PUNCT
ejpam-4352	416	19	)	)	PUNCT
ejpam-4352	416	20	\	\	PUNCT
ejpam-4352	416	21	c	c	X
ejpam-4352	416	22	)	)	PUNCT
ejpam-4352	416	23	,	,	PUNCT
ejpam-4352	416	24	[	[	X
ejpam-4352	416	25	v	v	X
ejpam-4352	416	26	(	(	PUNCT
ejpam-4352	416	27	h	h	NOUN
ejpam-4352	416	28	)	)	PUNCT
ejpam-4352	416	29	\	\	NOUN
ejpam-4352	416	30	nh(q	nh(q	PROPN
ejpam-4352	416	31	)	)	PUNCT
ejpam-4352	416	32	]	]	PUNCT
ejpam-4352	416	33	∩	∩	NOUN
ejpam-4352	416	34	[	[	X
ejpam-4352	416	35	v	v	X
ejpam-4352	416	36	(	(	PUNCT
ejpam-4352	416	37	h	h	NOUN
ejpam-4352	416	38	)	)	PUNCT
ejpam-4352	416	39	\	\	PUNCT
ejpam-4352	417	1	tx	tx	PROPN
ejpam-4352	417	2	]	]	X
ejpam-4352	417	3	=	=	X
ejpam-4352	417	4	{	{	PUNCT
ejpam-4352	417	5	p	p	X
ejpam-4352	417	6	}	}	PUNCT
ejpam-4352	417	7	and	and	CCONJ
ejpam-4352	417	8	ty	ty	INTJ
ejpam-4352	417	9	=	=	NOUN
ejpam-4352	417	10	v	v	NOUN
ejpam-4352	417	11	(	(	PUNCT
ejpam-4352	417	12	h	h	NOUN
ejpam-4352	417	13	)	)	PUNCT
ejpam-4352	417	14	for	for	ADP
ejpam-4352	417	15	all	all	DET
ejpam-4352	417	16	y	y	PROPN
ejpam-4352	417	17	∈	∈	PROPN
ejpam-4352	417	18	s	s	VERB
ejpam-4352	417	19	∩n2	∩n2	NOUN
ejpam-4352	417	20	g(x	g(x	PROPN
ejpam-4352	417	21	)	)	PUNCT
ejpam-4352	417	22	,	,	PUNCT
ejpam-4352	417	23	showing	show	VERB
ejpam-4352	417	24	that	that	SCONJ
ejpam-4352	417	25	(	(	PUNCT
ejpam-4352	417	26	a	a	X
ejpam-4352	417	27	)	)	PUNCT
ejpam-4352	417	28	and	and	CCONJ
ejpam-4352	417	29	(	(	PUNCT
ejpam-4352	417	30	b	b	NOUN
ejpam-4352	417	31	)	)	PUNCT
ejpam-4352	417	32	of	of	ADP
ejpam-4352	417	33	(	(	PUNCT
ejpam-4352	417	34	ii	ii	NOUN
ejpam-4352	417	35	)	)	PUNCT
ejpam-4352	417	36	hold	hold	VERB
ejpam-4352	417	37	.	.	PUNCT
ejpam-4352	418	1	finally	finally	ADV
ejpam-4352	418	2	,	,	PUNCT
ejpam-4352	418	3	let	let	VERB
ejpam-4352	418	4	x	x	PUNCT
ejpam-4352	418	5	∈	∈	NOUN
ejpam-4352	418	6	s	s	X
ejpam-4352	418	7	with	with	ADP
ejpam-4352	418	8	|v	|v	PROPN
ejpam-4352	418	9	(	(	PUNCT
ejpam-4352	418	10	h)\tx|	h)\tx|	X
ejpam-4352	418	11	=	=	SYM
ejpam-4352	418	12	1	1	NUM
ejpam-4352	418	13	and	and	CCONJ
ejpam-4352	418	14	let	let	VERB
ejpam-4352	418	15	p	p	PRON
ejpam-4352	418	16	∈	∈	PROPN
ejpam-4352	418	17	v	v	NOUN
ejpam-4352	418	18	(	(	PUNCT
ejpam-4352	418	19	h)\tx	h)\tx	PROPN
ejpam-4352	418	20	.	.	PROPN
ejpam-4352	418	21	suppose	suppose	VERB
ejpam-4352	418	22	that	that	SCONJ
ejpam-4352	418	23	(	(	PUNCT
ejpam-4352	418	24	a	a	X
ejpam-4352	418	25	)	)	PUNCT
ejpam-4352	418	26	of	of	ADP
ejpam-4352	418	27	(	(	PUNCT
ejpam-4352	418	28	iii	iii	NOUN
ejpam-4352	418	29	)	)	PUNCT
ejpam-4352	418	30	does	do	AUX
ejpam-4352	418	31	not	not	PART
ejpam-4352	418	32	hold	hold	VERB
ejpam-4352	418	33	.	.	PUNCT
ejpam-4352	419	1	since	since	SCONJ
ejpam-4352	419	2	c	c	PROPN
ejpam-4352	419	3	is	be	AUX
ejpam-4352	419	4	a	a	DET
ejpam-4352	419	5	super	super	ADV
ejpam-4352	419	6	hop	hop	NOUN
ejpam-4352	419	7	dominating	dominating	NOUN
ejpam-4352	419	8	set	set	NOUN
ejpam-4352	419	9	of	of	ADP
ejpam-4352	419	10	g[h	g[h	PROPN
ejpam-4352	419	11	]	]	PUNCT
ejpam-4352	419	12	and	and	CCONJ
ejpam-4352	419	13	(	(	PUNCT
ejpam-4352	419	14	x	x	X
ejpam-4352	419	15	,	,	PUNCT
ejpam-4352	419	16	p	p	NOUN
ejpam-4352	419	17	)	)	PUNCT
ejpam-4352	419	18	∈	∈	PROPN
ejpam-4352	419	19	v	v	NOUN
ejpam-4352	419	20	(	(	PUNCT
ejpam-4352	419	21	g[h])\c	g[h])\c	VERB
ejpam-4352	419	22	,	,	PUNCT
ejpam-4352	419	23	let	let	VERB
ejpam-4352	419	24	(	(	PUNCT
ejpam-4352	419	25	z	z	NOUN
ejpam-4352	419	26	,	,	PUNCT
ejpam-4352	419	27	t	t	PROPN
ejpam-4352	419	28	)	)	PUNCT
ejpam-4352	419	29	∈	∈	PROPN
ejpam-4352	420	1	ehpng[h]((x	ehpng[h]((x	ADP
ejpam-4352	420	2	,	,	PUNCT
ejpam-4352	420	3	p	p	NOUN
ejpam-4352	420	4	)	)	PUNCT
ejpam-4352	420	5	,	,	PUNCT
ejpam-4352	420	6	v	v	NOUN
ejpam-4352	420	7	(	(	PUNCT
ejpam-4352	420	8	g[h	g[h	PROPN
ejpam-4352	420	9	]	]	PUNCT
ejpam-4352	420	10	)	)	PUNCT
ejpam-4352	420	11	\	\	PUNCT
ejpam-4352	421	1	c	c	X
ejpam-4352	421	2	)	)	PUNCT
ejpam-4352	421	3	.	.	PUNCT
ejpam-4352	422	1	this	this	PRON
ejpam-4352	422	2	,	,	PUNCT
ejpam-4352	422	3	together	together	ADV
ejpam-4352	422	4	with	with	ADP
ejpam-4352	422	5	the	the	DET
ejpam-4352	422	6	assumption	assumption	NOUN
ejpam-4352	422	7	,	,	PUNCT
ejpam-4352	422	8	implies	imply	VERB
ejpam-4352	422	9	that	that	SCONJ
ejpam-4352	422	10	z	z	PROPN
ejpam-4352	422	11	∈	∈	PROPN
ejpam-4352	422	12	n2	n2	NOUN
ejpam-4352	422	13	g(x	g(x	PROPN
ejpam-4352	422	14	)	)	PUNCT
ejpam-4352	422	15	,	,	PUNCT
ejpam-4352	422	16	t	t	PROPN
ejpam-4352	422	17	∈	∈	PROPN
ejpam-4352	422	18	tz	tz	PROPN
ejpam-4352	422	19	,	,	PUNCT
ejpam-4352	422	20	tw	tw	NOUN
ejpam-4352	422	21	=	=	SYM
ejpam-4352	422	22	v	v	PROPN
ejpam-4352	422	23	(	(	PUNCT
ejpam-4352	422	24	h	h	NOUN
ejpam-4352	422	25	)	)	PUNCT
ejpam-4352	422	26	for	for	ADP
ejpam-4352	422	27	all	all	DET
ejpam-4352	422	28	w	w	PROPN
ejpam-4352	422	29	∈	∈	PROPN
ejpam-4352	422	30	n2	n2	NOUN
ejpam-4352	422	31	g(z	g(z	PROPN
ejpam-4352	422	32	)	)	PUNCT
ejpam-4352	422	33	\	\	NOUN
ejpam-4352	422	34	{	{	PUNCT
ejpam-4352	422	35	x	x	NOUN
ejpam-4352	422	36	}	}	PUNCT
ejpam-4352	422	37	and	and	CCONJ
ejpam-4352	422	38	v	v	NOUN
ejpam-4352	422	39	(	(	PUNCT
ejpam-4352	422	40	h	h	NOUN
ejpam-4352	422	41	)	)	PUNCT
ejpam-4352	422	42	\	\	PUNCT
ejpam-4352	422	43	tz	tz	PROPN
ejpam-4352	422	44	⊆	⊆	NUM
ejpam-4352	422	45	nh(t	nh(t	NUM
ejpam-4352	422	46	)	)	PUNCT
ejpam-4352	422	47	.	.	PUNCT
ejpam-4352	423	1	suppose	suppose	VERB
ejpam-4352	423	2	|v	|v	PROPN
ejpam-4352	423	3	(	(	PUNCT
ejpam-4352	423	4	h)\tz|	h)\tz|	ADP
ejpam-4352	423	5	>	>	SYM
ejpam-4352	423	6	1	1	NUM
ejpam-4352	423	7	,	,	PUNCT
ejpam-4352	423	8	say	say	VERB
ejpam-4352	423	9	c	c	NOUN
ejpam-4352	423	10	,	,	PUNCT
ejpam-4352	423	11	d	d	PROPN
ejpam-4352	423	12	∈	∈	PROPN
ejpam-4352	423	13	v	v	NOUN
ejpam-4352	423	14	(	(	PUNCT
ejpam-4352	423	15	h)\tz	h)\tz	ADP
ejpam-4352	423	16	where	where	SCONJ
ejpam-4352	423	17	c	c	PROPN
ejpam-4352	423	18	̸=	̸=	PROPN
ejpam-4352	423	19	d.	d.	PROPN
ejpam-4352	423	20	then	then	ADV
ejpam-4352	423	21	ehpng[h]((z	ehpng[h]((z	PROPN
ejpam-4352	423	22	,	,	PUNCT
ejpam-4352	423	23	c	c	NOUN
ejpam-4352	423	24	)	)	PUNCT
ejpam-4352	423	25	,	,	PUNCT
ejpam-4352	423	26	v	v	X
ejpam-4352	423	27	(	(	PUNCT
ejpam-4352	423	28	g[h])\c	g[h])\c	NOUN
ejpam-4352	423	29	)	)	PUNCT
ejpam-4352	423	30	=	=	SYM
ejpam-4352	423	31	∅	∅	NOUN
ejpam-4352	423	32	,	,	PUNCT
ejpam-4352	423	33	contrary	contrary	ADJ
ejpam-4352	423	34	to	to	ADP
ejpam-4352	423	35	the	the	DET
ejpam-4352	423	36	assumption	assumption	NOUN
ejpam-4352	423	37	that	that	SCONJ
ejpam-4352	423	38	c	c	PROPN
ejpam-4352	423	39	is	be	AUX
ejpam-4352	423	40	a	a	DET
ejpam-4352	423	41	super	super	ADV
ejpam-4352	423	42	hop	hop	NOUN
ejpam-4352	423	43	dominating	dominating	NOUN
ejpam-4352	423	44	set	set	NOUN
ejpam-4352	423	45	of	of	ADP
ejpam-4352	423	46	g[h	g[h	PROPN
ejpam-4352	423	47	]	]	PUNCT
ejpam-4352	423	48	.	.	PUNCT
ejpam-4352	424	1	therefore	therefore	ADV
ejpam-4352	424	2	,	,	PUNCT
ejpam-4352	424	3	|v	|v	PROPN
ejpam-4352	424	4	(	(	PUNCT
ejpam-4352	424	5	h	h	NOUN
ejpam-4352	424	6	)	)	PUNCT
ejpam-4352	424	7	\	\	PUNCT
ejpam-4352	425	1	tz|	tz|	ADP
ejpam-4352	425	2	≤	≤	NUM
ejpam-4352	425	3	1	1	NUM
ejpam-4352	425	4	,	,	PUNCT
ejpam-4352	425	5	showing	show	VERB
ejpam-4352	425	6	that	that	SCONJ
ejpam-4352	425	7	(	(	PUNCT
ejpam-4352	425	8	b	b	NOUN
ejpam-4352	425	9	)	)	PUNCT
ejpam-4352	425	10	of	of	ADP
ejpam-4352	425	11	(	(	PUNCT
ejpam-4352	425	12	iii	iii	NOUN
ejpam-4352	425	13	)	)	PUNCT
ejpam-4352	425	14	holds	hold	VERB
ejpam-4352	425	15	.	.	PUNCT
ejpam-4352	426	1	for	for	ADP
ejpam-4352	426	2	the	the	DET
ejpam-4352	426	3	converse	converse	NOUN
ejpam-4352	426	4	,	,	PUNCT
ejpam-4352	426	5	suppose	suppose	VERB
ejpam-4352	426	6	that	that	SCONJ
ejpam-4352	426	7	c	c	NOUN
ejpam-4352	426	8	=	=	SYM
ejpam-4352	426	9	∪x∈s({x	∪x∈s({x	PROPN
ejpam-4352	426	10	}	}	PUNCT
ejpam-4352	426	11	×	×	NOUN
ejpam-4352	426	12	tx	tx	PROPN
ejpam-4352	426	13	)	)	PUNCT
ejpam-4352	426	14	satisfies	satisfy	VERB
ejpam-4352	426	15	conditions	condition	NOUN
ejpam-4352	426	16	(	(	PUNCT
ejpam-4352	426	17	i	i	NOUN
ejpam-4352	426	18	)	)	PUNCT
ejpam-4352	426	19	,	,	PUNCT
ejpam-4352	426	20	(	(	PUNCT
ejpam-4352	426	21	ii	ii	NOUN
ejpam-4352	426	22	)	)	PUNCT
ejpam-4352	426	23	,	,	PUNCT
ejpam-4352	426	24	and	and	CCONJ
ejpam-4352	426	25	(	(	PUNCT
ejpam-4352	426	26	iii	iii	NOUN
ejpam-4352	426	27	)	)	PUNCT
ejpam-4352	426	28	.	.	PUNCT
ejpam-4352	427	1	let	let	VERB
ejpam-4352	427	2	(	(	PUNCT
ejpam-4352	427	3	x	x	X
ejpam-4352	427	4	,	,	PUNCT
ejpam-4352	427	5	p	p	NOUN
ejpam-4352	427	6	)	)	PUNCT
ejpam-4352	427	7	∈	∈	PROPN
ejpam-4352	427	8	v	v	NOUN
ejpam-4352	427	9	(	(	PUNCT
ejpam-4352	427	10	g[h])\c	g[h])\c	PROPN
ejpam-4352	427	11	.	.	PUNCT
ejpam-4352	428	1	since	since	SCONJ
ejpam-4352	428	2	s	s	PART
ejpam-4352	428	3	=	=	SYM
ejpam-4352	428	4	v	v	PROPN
ejpam-4352	428	5	(	(	PUNCT
ejpam-4352	428	6	g	g	NOUN
ejpam-4352	428	7	)	)	PUNCT
ejpam-4352	428	8	,	,	PUNCT
ejpam-4352	428	9	p	p	X
ejpam-4352	428	10	/∈	/∈	PUNCT
ejpam-4352	428	11	tx	tx	INTJ
ejpam-4352	428	12	.	.	PUNCT
ejpam-4352	429	1	if	if	SCONJ
ejpam-4352	429	2	|v	|v	PROPN
ejpam-4352	429	3	(	(	PUNCT
ejpam-4352	429	4	h)\tx|	h)\tx|	X
ejpam-4352	429	5	≥	≥	NOUN
ejpam-4352	429	6	2	2	NUM
ejpam-4352	429	7	,	,	PUNCT
ejpam-4352	429	8	then	then	ADV
ejpam-4352	429	9	by	by	ADP
ejpam-4352	429	10	condition	condition	NOUN
ejpam-4352	429	11	(	(	PUNCT
ejpam-4352	429	12	ii	ii	NOUN
ejpam-4352	429	13	)	)	PUNCT
ejpam-4352	429	14	,	,	PUNCT
ejpam-4352	429	15	there	there	PRON
ejpam-4352	429	16	exists	exist	VERB
ejpam-4352	429	17	q	q	PROPN
ejpam-4352	429	18	∈	∈	PROPN
ejpam-4352	429	19	ehpnh(p	ehpnh(p	PROPN
ejpam-4352	429	20	,	,	PUNCT
ejpam-4352	429	21	v	v	PROPN
ejpam-4352	429	22	(	(	PUNCT
ejpam-4352	429	23	h)\tx	h)\tx	NOUN
ejpam-4352	429	24	)	)	PUNCT
ejpam-4352	429	25	.	.	PUNCT
ejpam-4352	430	1	clearly	clearly	ADV
ejpam-4352	430	2	,	,	PUNCT
ejpam-4352	430	3	(	(	PUNCT
ejpam-4352	430	4	x	x	X
ejpam-4352	430	5	,	,	PUNCT
ejpam-4352	430	6	q	q	NOUN
ejpam-4352	430	7	)	)	PUNCT
ejpam-4352	430	8	∈	∈	PROPN
ejpam-4352	430	9	ehpng[h]((x	ehpng[h]((x	ADP
ejpam-4352	430	10	,	,	PUNCT
ejpam-4352	430	11	p	p	NOUN
ejpam-4352	430	12	)	)	PUNCT
ejpam-4352	430	13	,	,	PUNCT
ejpam-4352	430	14	v	v	NOUN
ejpam-4352	430	15	(	(	PUNCT
ejpam-4352	430	16	g[h])\	g[h])\	NOUN
ejpam-4352	430	17	c	c	NOUN
ejpam-4352	430	18	)	)	PUNCT
ejpam-4352	430	19	.	.	PUNCT
ejpam-4352	430	20	suppose	suppose	VERB
ejpam-4352	430	21	that	that	SCONJ
ejpam-4352	430	22	|v	|v	PROPN
ejpam-4352	430	23	(	(	PUNCT
ejpam-4352	430	24	h	h	NOUN
ejpam-4352	430	25	)	)	PUNCT
ejpam-4352	430	26	\	\	PUNCT
ejpam-4352	430	27	tx|	tx|	PROPN
ejpam-4352	430	28	=	=	PUNCT
ejpam-4352	430	29	1	1	X
ejpam-4352	430	30	.	.	PUNCT
ejpam-4352	430	31	then	then	ADV
ejpam-4352	430	32	by	by	ADP
ejpam-4352	430	33	(	(	PUNCT
ejpam-4352	430	34	iii	iii	NOUN
ejpam-4352	430	35	)	)	PUNCT
ejpam-4352	430	36	,	,	PUNCT
ejpam-4352	430	37	ehpng[h]((x	ehpng[h]((x	PROPN
ejpam-4352	430	38	,	,	PUNCT
ejpam-4352	430	39	p	p	NOUN
ejpam-4352	430	40	)	)	PUNCT
ejpam-4352	430	41	,	,	PUNCT
ejpam-4352	430	42	v	v	NOUN
ejpam-4352	430	43	(	(	PUNCT
ejpam-4352	430	44	g[h	g[h	PROPN
ejpam-4352	430	45	]	]	PUNCT
ejpam-4352	430	46	)	)	PUNCT
ejpam-4352	430	47	\	\	PUNCT
ejpam-4352	431	1	c	c	X
ejpam-4352	431	2	)	)	PUNCT
ejpam-4352	431	3	̸=	̸=	PROPN
ejpam-4352	431	4	∅.	∅.	ADV
ejpam-4352	431	5	accordingly	accordingly	ADV
ejpam-4352	431	6	,	,	PUNCT
ejpam-4352	431	7	c	c	PROPN
ejpam-4352	431	8	is	be	AUX
ejpam-4352	431	9	a	a	DET
ejpam-4352	431	10	super	super	ADV
ejpam-4352	431	11	hop	hop	NOUN
ejpam-4352	431	12	dominating	dominating	NOUN
ejpam-4352	431	13	set	set	NOUN
ejpam-4352	431	14	of	of	ADP
ejpam-4352	431	15	g[h	g[h	PROPN
ejpam-4352	431	16	]	]	PUNCT
ejpam-4352	431	17	.	.	PUNCT
ejpam-4352	432	1	let	let	VERB
ejpam-4352	432	2	g	g	PRON
ejpam-4352	432	3	be	be	AUX
ejpam-4352	432	4	a	a	DET
ejpam-4352	432	5	graph	graph	NOUN
ejpam-4352	432	6	.	.	PUNCT
ejpam-4352	433	1	a	a	DET
ejpam-4352	433	2	set	set	NOUN
ejpam-4352	433	3	s	s	NOUN
ejpam-4352	433	4	⊆	⊆	NUM
ejpam-4352	433	5	v	v	NOUN
ejpam-4352	433	6	(	(	PUNCT
ejpam-4352	433	7	g	g	NOUN
ejpam-4352	433	8	)	)	PUNCT
ejpam-4352	433	9	is	be	AUX
ejpam-4352	433	10	a	a	DET
ejpam-4352	433	11	hop	hop	NOUN
ejpam-4352	433	12	independent	independent	ADJ
ejpam-4352	433	13	set	set	NOUN
ejpam-4352	433	14	of	of	ADP
ejpam-4352	433	15	g	g	PROPN
ejpam-4352	433	16	if	if	SCONJ
ejpam-4352	433	17	dg(x	dg(x	NUM
ejpam-4352	433	18	,	,	PUNCT
ejpam-4352	433	19	y	y	NOUN
ejpam-4352	433	20	)	)	PUNCT
ejpam-4352	433	21	̸=	̸=	PROPN
ejpam-4352	433	22	2	2	NUM
ejpam-4352	433	23	for	for	ADP
ejpam-4352	433	24	any	any	DET
ejpam-4352	433	25	two	two	NUM
ejpam-4352	433	26	vertices	vertex	NOUN
ejpam-4352	433	27	x	x	X
ejpam-4352	433	28	,	,	PUNCT
ejpam-4352	433	29	y	y	PROPN
ejpam-4352	433	30	∈	∈	PROPN
ejpam-4352	433	31	s.	s.	PROPN
ejpam-4352	433	32	the	the	DET
ejpam-4352	433	33	hop	hop	PROPN
ejpam-4352	433	34	independence	independence	NOUN
ejpam-4352	433	35	number	number	NOUN
ejpam-4352	433	36	of	of	ADP
ejpam-4352	433	37	g	g	NOUN
ejpam-4352	433	38	,	,	PUNCT
ejpam-4352	433	39	denoted	denote	VERB
ejpam-4352	433	40	by	by	ADP
ejpam-4352	433	41	αh(g	αh(g	NOUN
ejpam-4352	433	42	)	)	PUNCT
ejpam-4352	433	43	,	,	PUNCT
ejpam-4352	433	44	is	be	AUX
ejpam-4352	433	45	the	the	DET
ejpam-4352	433	46	largest	large	ADJ
ejpam-4352	433	47	cardinality	cardinality	NOUN
ejpam-4352	433	48	of	of	ADP
ejpam-4352	433	49	a	a	DET
ejpam-4352	433	50	hop	hop	NOUN
ejpam-4352	433	51	independent	independent	ADJ
ejpam-4352	433	52	set	set	NOUN
ejpam-4352	433	53	of	of	ADP
ejpam-4352	433	54	g.	g.	PROPN
ejpam-4352	433	55	any	any	DET
ejpam-4352	433	56	hop	hop	NOUN
ejpam-4352	433	57	independent	independent	ADJ
ejpam-4352	433	58	set	set	NOUN
ejpam-4352	433	59	of	of	ADP
ejpam-4352	433	60	g	g	NOUN
ejpam-4352	433	61	with	with	ADP
ejpam-4352	433	62	cardinality	cardinality	NOUN
ejpam-4352	433	63	αh(g	αh(g	NOUN
ejpam-4352	433	64	)	)	PUNCT
ejpam-4352	433	65	is	be	AUX
ejpam-4352	433	66	called	call	VERB
ejpam-4352	433	67	an	an	DET
ejpam-4352	433	68	αh	αh	NOUN
ejpam-4352	433	69	-	-	PUNCT
ejpam-4352	433	70	set	set	NOUN
ejpam-4352	433	71	of	of	ADP
ejpam-4352	433	72	g.	g.	PROPN
ejpam-4352	433	73	the	the	DET
ejpam-4352	433	74	concept	concept	NOUN
ejpam-4352	433	75	has	have	AUX
ejpam-4352	433	76	been	be	AUX
ejpam-4352	433	77	introduced	introduce	VERB
ejpam-4352	433	78	and	and	CCONJ
ejpam-4352	433	79	studied	study	VERB
ejpam-4352	433	80	in	in	ADP
ejpam-4352	433	81	[	[	X
ejpam-4352	433	82	4	4	NUM
ejpam-4352	433	83	]	]	PUNCT
ejpam-4352	433	84	.	.	PUNCT
ejpam-4352	434	1	corollary	corollary	ADJ
ejpam-4352	434	2	5	5	NUM
ejpam-4352	434	3	.	.	PUNCT
ejpam-4352	435	1	let	let	VERB
ejpam-4352	435	2	g	g	NOUN
ejpam-4352	435	3	and	and	CCONJ
ejpam-4352	435	4	h	h	PROPN
ejpam-4352	435	5	be	be	VERB
ejpam-4352	435	6	non	non	ADJ
ejpam-4352	435	7	-	-	ADJ
ejpam-4352	435	8	trivial	trivial	ADJ
ejpam-4352	435	9	connected	connected	ADJ
ejpam-4352	435	10	graphs	graph	NOUN
ejpam-4352	435	11	of	of	ADP
ejpam-4352	435	12	orders	order	NOUN
ejpam-4352	435	13	m	m	VERB
ejpam-4352	435	14	and	and	CCONJ
ejpam-4352	435	15	n	n	CCONJ
ejpam-4352	435	16	,	,	PUNCT
ejpam-4352	435	17	respectively	respectively	ADV
ejpam-4352	435	18	.	.	PUNCT
ejpam-4352	436	1	then	then	ADV
ejpam-4352	436	2	γsh(g[h	γsh(g[h	NUM
ejpam-4352	436	3	]	]	NUM
ejpam-4352	436	4	)	)	PUNCT
ejpam-4352	436	5	≤	≤	NOUN
ejpam-4352	436	6	(	(	PUNCT
ejpam-4352	436	7	γcs(h)−n)αh(g)+mn	γcs(h)−n)αh(g)+mn	PROPN
ejpam-4352	436	8	.	.	PUNCT
ejpam-4352	437	1	moreover	moreover	ADV
ejpam-4352	437	2	,	,	PUNCT
ejpam-4352	437	3	if	if	SCONJ
ejpam-4352	437	4	γcs(h	γcs(h	PROPN
ejpam-4352	437	5	)	)	PUNCT
ejpam-4352	437	6	≤	≤	NOUN
ejpam-4352	437	7	n−2	n−2	PROPN
ejpam-4352	437	8	,	,	PUNCT
ejpam-4352	437	9	then	then	ADV
ejpam-4352	437	10	γsh(g[h	γsh(g[h	NUM
ejpam-4352	437	11	]	]	PUNCT
ejpam-4352	437	12	)	)	PUNCT
ejpam-4352	437	13	=	=	SYM
ejpam-4352	437	14	(	(	PUNCT
ejpam-4352	437	15	γcs(h)−	γcs(h)−	PROPN
ejpam-4352	437	16	n)αh(g	n)αh(g	PROPN
ejpam-4352	437	17	)	)	PUNCT
ejpam-4352	437	18	+	+	PROPN
ejpam-4352	437	19	mn	mn	PROPN
ejpam-4352	437	20	.	.	PUNCT
ejpam-4352	437	21	s.	s.	PROPN
ejpam-4352	437	22	canoy	canoy	PROPN
ejpam-4352	437	23	,	,	PUNCT
ejpam-4352	437	24	jr	jr	PROPN
ejpam-4352	437	25	.	.	PROPN
ejpam-4352	437	26	,	,	PUNCT
ejpam-4352	437	27	g.	g.	PROPN
ejpam-4352	437	28	salasalan	salasalan	PROPN
ejpam-4352	437	29	/	/	SYM
ejpam-4352	437	30	eur	eur	PROPN
ejpam-4352	437	31	.	.	PUNCT
ejpam-4352	438	1	j.	j.	PROPN
ejpam-4352	438	2	pure	pure	PROPN
ejpam-4352	438	3	appl	appl	PROPN
ejpam-4352	438	4	.	.	PROPN
ejpam-4352	438	5	math	math	PROPN
ejpam-4352	438	6	,	,	PUNCT
ejpam-4352	438	7	15	15	NUM
ejpam-4352	438	8	(	(	PUNCT
ejpam-4352	438	9	2	2	NUM
ejpam-4352	438	10	)	)	PUNCT
ejpam-4352	438	11	(	(	PUNCT
ejpam-4352	438	12	2022	2022	NUM
ejpam-4352	438	13	)	)	PUNCT
ejpam-4352	438	14	,	,	PUNCT
ejpam-4352	438	15	342	342	NUM
ejpam-4352	438	16	-	-	SYM
ejpam-4352	438	17	353	353	NUM
ejpam-4352	438	18	351	351	NUM
ejpam-4352	438	19	proof	proof	NOUN
ejpam-4352	438	20	.	.	PUNCT
ejpam-4352	439	1	let	let	VERB
ejpam-4352	439	2	a	a	PRON
ejpam-4352	439	3	be	be	AUX
ejpam-4352	439	4	a	a	DET
ejpam-4352	439	5	αh	αh	NOUN
ejpam-4352	439	6	-	-	PUNCT
ejpam-4352	439	7	set	set	NOUN
ejpam-4352	439	8	of	of	ADP
ejpam-4352	439	9	g	g	NOUN
ejpam-4352	439	10	and	and	CCONJ
ejpam-4352	439	11	let	let	VERB
ejpam-4352	439	12	d	d	PRON
ejpam-4352	439	13	be	be	AUX
ejpam-4352	439	14	a	a	DET
ejpam-4352	439	15	γcs	γcs	NOUN
ejpam-4352	439	16	-	-	PUNCT
ejpam-4352	439	17	set	set	NOUN
ejpam-4352	439	18	of	of	ADP
ejpam-4352	439	19	h.	h.	PROPN
ejpam-4352	439	20	set	set	VERB
ejpam-4352	439	21	tx	tx	PROPN
ejpam-4352	440	1	=	=	SYM
ejpam-4352	441	1	d	d	PROPN
ejpam-4352	441	2	for	for	ADP
ejpam-4352	441	3	each	each	DET
ejpam-4352	441	4	x	x	SYM
ejpam-4352	441	5	∈	∈	PROPN
ejpam-4352	441	6	a	a	PRON
ejpam-4352	441	7	and	and	CCONJ
ejpam-4352	441	8	tx	tx	PROPN
ejpam-4352	441	9	=	=	SYM
ejpam-4352	441	10	v	v	PROPN
ejpam-4352	441	11	(	(	PUNCT
ejpam-4352	441	12	h	h	NOUN
ejpam-4352	441	13	)	)	PUNCT
ejpam-4352	441	14	for	for	ADP
ejpam-4352	441	15	each	each	DET
ejpam-4352	441	16	x	x	SYM
ejpam-4352	441	17	∈	∈	PROPN
ejpam-4352	441	18	v	v	ADP
ejpam-4352	441	19	(	(	PUNCT
ejpam-4352	441	20	g	g	NOUN
ejpam-4352	441	21	)	)	PUNCT
ejpam-4352	441	22	\a	\a	ADJ
ejpam-4352	441	23	.	.	PUNCT
ejpam-4352	442	1	then	then	ADV
ejpam-4352	442	2	,	,	PUNCT
ejpam-4352	442	3	by	by	ADP
ejpam-4352	442	4	theorem	theorem	NOUN
ejpam-4352	442	5	11	11	NUM
ejpam-4352	442	6	,	,	PUNCT
ejpam-4352	442	7	c0	c0	NOUN
ejpam-4352	442	8	=	=	SYM
ejpam-4352	442	9	∪x∈v	∪x∈v	PROPN
ejpam-4352	442	10	(	(	PUNCT
ejpam-4352	442	11	g)({x}×	g)({x}×	NOUN
ejpam-4352	442	12	tx	tx	PROPN
ejpam-4352	442	13	)	)	PUNCT
ejpam-4352	442	14	=	=	SYM
ejpam-4352	442	15	(	(	PUNCT
ejpam-4352	442	16	a×d	a×d	ADV
ejpam-4352	442	17	)	)	PUNCT
ejpam-4352	442	18	∪	∪	ADP
ejpam-4352	442	19	[	[	X
ejpam-4352	442	20	(	(	PUNCT
ejpam-4352	442	21	v	v	NOUN
ejpam-4352	442	22	(	(	PUNCT
ejpam-4352	442	23	g	g	NOUN
ejpam-4352	442	24	)	)	PUNCT
ejpam-4352	442	25	\a)×	\a)×	NOUN
ejpam-4352	442	26	v	v	NOUN
ejpam-4352	442	27	(	(	PUNCT
ejpam-4352	442	28	h	h	NOUN
ejpam-4352	442	29	)	)	PUNCT
ejpam-4352	442	30	]	]	PUNCT
ejpam-4352	442	31	is	be	AUX
ejpam-4352	442	32	a	a	DET
ejpam-4352	442	33	super	super	ADV
ejpam-4352	442	34	hop	hop	NOUN
ejpam-4352	442	35	dominating	dominating	NOUN
ejpam-4352	442	36	set	set	NOUN
ejpam-4352	442	37	of	of	ADP
ejpam-4352	442	38	g[h	g[h	NOUN
ejpam-4352	442	39	]	]	PUNCT
ejpam-4352	442	40	.	.	PUNCT
ejpam-4352	443	1	hence	hence	ADV
ejpam-4352	443	2	,	,	PUNCT
ejpam-4352	443	3	γsh(g[h	γsh(g[h	NUM
ejpam-4352	443	4	]	]	PUNCT
ejpam-4352	443	5	)	)	PUNCT
ejpam-4352	443	6	≤	≤	NUM
ejpam-4352	443	7	|c0|	|c0|	NOUN
ejpam-4352	443	8	=	=	SYM
ejpam-4352	443	9	|a||d|+	|a||d|+	PROPN
ejpam-4352	443	10	(	(	PUNCT
ejpam-4352	443	11	m−	m−	PROPN
ejpam-4352	443	12	|a|)n	|a|)n	PROPN
ejpam-4352	443	13	=	=	SYM
ejpam-4352	443	14	αh(g)γcs(h	αh(g)γcs(h	PROPN
ejpam-4352	443	15	)	)	PUNCT
ejpam-4352	443	16	+	+	PUNCT
ejpam-4352	444	1	[	[	X
ejpam-4352	444	2	m−	m−	PROPN
ejpam-4352	444	3	αh(g)]n	αh(g)]n	ADV
ejpam-4352	444	4	=	=	SYM
ejpam-4352	444	5	(	(	PUNCT
ejpam-4352	444	6	γcs(h)−	γcs(h)−	PROPN
ejpam-4352	444	7	n)αh(g	n)αh(g	PROPN
ejpam-4352	444	8	)	)	PUNCT
ejpam-4352	444	9	+	+	PROPN
ejpam-4352	444	10	mn	mn	PROPN
ejpam-4352	444	11	.	.	PROPN
ejpam-4352	445	1	next	next	ADV
ejpam-4352	445	2	,	,	PUNCT
ejpam-4352	445	3	let	let	VERB
ejpam-4352	445	4	c	c	NOUN
ejpam-4352	445	5	=	=	SYM
ejpam-4352	445	6	∪x∈v	∪x∈v	X
ejpam-4352	445	7	(	(	PUNCT
ejpam-4352	445	8	g)({x	g)({x	NOUN
ejpam-4352	445	9	}	}	PUNCT
ejpam-4352	445	10	×	×	NOUN
ejpam-4352	445	11	tx	tx	PROPN
ejpam-4352	445	12	)	)	PUNCT
ejpam-4352	445	13	be	be	AUX
ejpam-4352	445	14	a	a	DET
ejpam-4352	445	15	γsh	γsh	NOUN
ejpam-4352	445	16	-	-	PUNCT
ejpam-4352	445	17	set	set	NOUN
ejpam-4352	445	18	of	of	ADP
ejpam-4352	445	19	g[h	g[h	NOUN
ejpam-4352	445	20	]	]	PUNCT
ejpam-4352	445	21	.	.	PUNCT
ejpam-4352	446	1	let	let	VERB
ejpam-4352	446	2	r	r	NOUN
ejpam-4352	446	3	=	=	PRON
ejpam-4352	446	4	{	{	PUNCT
ejpam-4352	446	5	v	v	NUM
ejpam-4352	446	6	∈	∈	NOUN
ejpam-4352	446	7	v	v	NOUN
ejpam-4352	446	8	(	(	PUNCT
ejpam-4352	446	9	g	g	NOUN
ejpam-4352	446	10	)	)	PUNCT
ejpam-4352	446	11	:	:	PUNCT
ejpam-4352	447	1	|v	|v	PROPN
ejpam-4352	447	2	(	(	PUNCT
ejpam-4352	447	3	h	h	NOUN
ejpam-4352	447	4	)	)	PUNCT
ejpam-4352	447	5	\	\	PUNCT
ejpam-4352	448	1	tv|	tv|	PROPN
ejpam-4352	448	2	≥	≥	NUM
ejpam-4352	448	3	2	2	NUM
ejpam-4352	448	4	}	}	PUNCT
ejpam-4352	448	5	.	.	PUNCT
ejpam-4352	449	1	since	since	SCONJ
ejpam-4352	449	2	c	c	PROPN
ejpam-4352	449	3	is	be	AUX
ejpam-4352	449	4	a	a	DET
ejpam-4352	449	5	super	super	ADV
ejpam-4352	449	6	hop	hop	NOUN
ejpam-4352	449	7	dominating	dominating	NOUN
ejpam-4352	449	8	set	set	NOUN
ejpam-4352	449	9	,	,	PUNCT
ejpam-4352	449	10	r	r	NOUN
ejpam-4352	449	11	must	must	AUX
ejpam-4352	449	12	be	be	AUX
ejpam-4352	449	13	a	a	DET
ejpam-4352	449	14	hop	hop	NOUN
ejpam-4352	449	15	independent	independent	ADJ
ejpam-4352	449	16	set	set	NOUN
ejpam-4352	449	17	of	of	ADP
ejpam-4352	449	18	g	g	NOUN
ejpam-4352	449	19	by	by	ADP
ejpam-4352	449	20	theorem	theorem	NOUN
ejpam-4352	449	21	11(ii	11(ii	NUM
ejpam-4352	449	22	)	)	PUNCT
ejpam-4352	449	23	.	.	PUNCT
ejpam-4352	450	1	also	also	ADV
ejpam-4352	450	2	,	,	PUNCT
ejpam-4352	450	3	since	since	SCONJ
ejpam-4352	450	4	c	c	PROPN
ejpam-4352	450	5	is	be	AUX
ejpam-4352	450	6	a	a	DET
ejpam-4352	450	7	γsh	γsh	NOUN
ejpam-4352	450	8	-	-	PUNCT
ejpam-4352	450	9	set	set	NOUN
ejpam-4352	450	10	of	of	ADP
ejpam-4352	450	11	g[h	g[h	PROPN
ejpam-4352	450	12	]	]	PUNCT
ejpam-4352	450	13	,	,	PUNCT
ejpam-4352	450	14	tv	tv	NOUN
ejpam-4352	450	15	is	be	AUX
ejpam-4352	450	16	a	a	DET
ejpam-4352	450	17	γcs	γcs	NOUN
ejpam-4352	450	18	-	-	PUNCT
ejpam-4352	450	19	set	set	NOUN
ejpam-4352	450	20	of	of	ADP
ejpam-4352	450	21	h	h	NOUN
ejpam-4352	450	22	for	for	ADP
ejpam-4352	450	23	each	each	DET
ejpam-4352	450	24	v	v	NOUN
ejpam-4352	450	25	∈	∈	PROPN
ejpam-4352	450	26	r.	r.	NOUN
ejpam-4352	450	27	now	now	ADV
ejpam-4352	450	28	let	let	VERB
ejpam-4352	450	29	u	u	PRON
ejpam-4352	450	30	∈	∈	PROPN
ejpam-4352	450	31	v	v	ADP
ejpam-4352	450	32	(	(	PUNCT
ejpam-4352	450	33	g	g	NOUN
ejpam-4352	450	34	)	)	PUNCT
ejpam-4352	450	35	\	\	NOUN
ejpam-4352	450	36	r	r	NOUN
ejpam-4352	450	37	and	and	CCONJ
ejpam-4352	450	38	suppose	suppose	VERB
ejpam-4352	450	39	that	that	SCONJ
ejpam-4352	450	40	|v	|v	PROPN
ejpam-4352	450	41	(	(	PUNCT
ejpam-4352	450	42	h	h	NOUN
ejpam-4352	450	43	)	)	PUNCT
ejpam-4352	450	44	\	\	PUNCT
ejpam-4352	450	45	tu|	tu|	ADP
ejpam-4352	450	46	=	=	SYM
ejpam-4352	450	47	1	1	X
ejpam-4352	450	48	.	.	PUNCT
ejpam-4352	450	49	by	by	ADP
ejpam-4352	450	50	theorem	theorem	NOUN
ejpam-4352	450	51	11(ii	11(ii	NUM
ejpam-4352	450	52	)	)	PUNCT
ejpam-4352	450	53	,	,	PUNCT
ejpam-4352	450	54	dg(u	dg(u	X
ejpam-4352	450	55	,	,	PUNCT
ejpam-4352	450	56	w	w	NOUN
ejpam-4352	450	57	)	)	PUNCT
ejpam-4352	450	58	̸=	̸=	PROPN
ejpam-4352	450	59	2	2	NUM
ejpam-4352	450	60	for	for	ADP
ejpam-4352	450	61	all	all	DET
ejpam-4352	450	62	w	w	PROPN
ejpam-4352	450	63	∈	∈	NOUN
ejpam-4352	450	64	r	r	NOUN
ejpam-4352	450	65	(	(	PUNCT
ejpam-4352	450	66	otherwise	otherwise	ADV
ejpam-4352	450	67	,	,	PUNCT
ejpam-4352	450	68	tu	tu	PROPN
ejpam-4352	450	69	=	=	SYM
ejpam-4352	450	70	v	v	PROPN
ejpam-4352	450	71	(	(	PUNCT
ejpam-4352	450	72	h	h	NOUN
ejpam-4352	450	73	)	)	PUNCT
ejpam-4352	450	74	,	,	PUNCT
ejpam-4352	450	75	a	a	DET
ejpam-4352	450	76	contradiction	contradiction	NOUN
ejpam-4352	450	77	)	)	PUNCT
ejpam-4352	450	78	.	.	PUNCT
ejpam-4352	451	1	suppose	suppose	VERB
ejpam-4352	451	2	that	that	SCONJ
ejpam-4352	451	3	tx	tx	PROPN
ejpam-4352	451	4	=	=	SYM
ejpam-4352	451	5	v	v	PROPN
ejpam-4352	451	6	(	(	PUNCT
ejpam-4352	451	7	h	h	NOUN
ejpam-4352	451	8	)	)	PUNCT
ejpam-4352	451	9	for	for	ADP
ejpam-4352	451	10	all	all	PRON
ejpam-4352	451	11	x	x	SYM
ejpam-4352	451	12	∈	∈	PROPN
ejpam-4352	451	13	n2	n2	NOUN
ejpam-4352	451	14	g(u	g(u	PROPN
ejpam-4352	451	15	)	)	PUNCT
ejpam-4352	451	16	.	.	PUNCT
ejpam-4352	452	1	replace	replace	VERB
ejpam-4352	452	2	tu	tu	PROPN
ejpam-4352	452	3	by	by	ADP
ejpam-4352	452	4	a	a	DET
ejpam-4352	452	5	γcs	γcs	NOUN
ejpam-4352	452	6	-	-	PUNCT
ejpam-4352	452	7	set	set	VERB
ejpam-4352	452	8	t	t	NOUN
ejpam-4352	452	9	′	′	NUM
ejpam-4352	452	10	u	u	PROPN
ejpam-4352	452	11	of	of	ADP
ejpam-4352	452	12	h.	h.	PROPN
ejpam-4352	452	13	then	then	ADV
ejpam-4352	452	14	the	the	DET
ejpam-4352	452	15	set	set	ADJ
ejpam-4352	452	16	c∗	c∗	NOUN
ejpam-4352	452	17	=	=	SYM
ejpam-4352	452	18	∪x∈v	∪x∈v	PROPN
ejpam-4352	452	19	(	(	PUNCT
ejpam-4352	452	20	g)\{u}({x	g)\{u}({x	PROPN
ejpam-4352	452	21	}	}	PUNCT
ejpam-4352	452	22	×	×	PROPN
ejpam-4352	452	23	tx	tx	PROPN
ejpam-4352	452	24	)	)	PUNCT
ejpam-4352	452	25	∪	∪	NOUN
ejpam-4352	452	26	(	(	PUNCT
ejpam-4352	452	27	{	{	PUNCT
ejpam-4352	452	28	u	u	NOUN
ejpam-4352	452	29	}	}	PUNCT
ejpam-4352	452	30	×	×	NOUN
ejpam-4352	452	31	t	t	NOUN
ejpam-4352	452	32	′	′	NUM
ejpam-4352	452	33	u	u	NOUN
ejpam-4352	452	34	)	)	PUNCT
ejpam-4352	452	35	is	be	AUX
ejpam-4352	452	36	a	a	DET
ejpam-4352	452	37	super	super	ADV
ejpam-4352	452	38	hop	hop	NOUN
ejpam-4352	452	39	dominating	dominating	NOUN
ejpam-4352	452	40	set	set	NOUN
ejpam-4352	452	41	of	of	ADP
ejpam-4352	452	42	g[h	g[h	PROPN
ejpam-4352	452	43	]	]	PUNCT
ejpam-4352	452	44	by	by	ADP
ejpam-4352	452	45	theorem	theorem	NOUN
ejpam-4352	452	46	11	11	NUM
ejpam-4352	452	47	.	.	PUNCT
ejpam-4352	453	1	moreover	moreover	ADV
ejpam-4352	453	2	,	,	PUNCT
ejpam-4352	453	3	|c∗|	|c∗|	VERB
ejpam-4352	453	4	<	<	X
ejpam-4352	453	5	|c|	|c|	PROPN
ejpam-4352	453	6	,	,	PUNCT
ejpam-4352	453	7	contrary	contrary	ADJ
ejpam-4352	453	8	to	to	ADP
ejpam-4352	453	9	the	the	DET
ejpam-4352	453	10	assumption	assumption	NOUN
ejpam-4352	453	11	that	that	SCONJ
ejpam-4352	453	12	c	c	PROPN
ejpam-4352	453	13	is	be	AUX
ejpam-4352	453	14	a	a	DET
ejpam-4352	453	15	γsh	γsh	NOUN
ejpam-4352	453	16	-	-	PUNCT
ejpam-4352	453	17	set	set	NOUN
ejpam-4352	453	18	of	of	ADP
ejpam-4352	453	19	g[h	g[h	NOUN
ejpam-4352	453	20	]	]	PUNCT
ejpam-4352	453	21	.	.	PUNCT
ejpam-4352	454	1	hence	hence	ADV
ejpam-4352	454	2	,	,	PUNCT
ejpam-4352	454	3	tx	tx	ADP
ejpam-4352	454	4	̸=	̸=	PROPN
ejpam-4352	454	5	v	v	PROPN
ejpam-4352	454	6	(	(	PUNCT
ejpam-4352	454	7	h	h	NOUN
ejpam-4352	454	8	)	)	PUNCT
ejpam-4352	454	9	for	for	ADP
ejpam-4352	454	10	some	some	DET
ejpam-4352	454	11	x	x	SYM
ejpam-4352	454	12	∈	∈	PROPN
ejpam-4352	454	13	n2	n2	NOUN
ejpam-4352	454	14	g(u	g(u	PROPN
ejpam-4352	454	15	)	)	PUNCT
ejpam-4352	454	16	.	.	PUNCT
ejpam-4352	455	1	it	it	PRON
ejpam-4352	455	2	follows	follow	VERB
ejpam-4352	455	3	from	from	ADP
ejpam-4352	455	4	theorem	theorem	ADJ
ejpam-4352	455	5	11(iii	11(iii	NUM
ejpam-4352	455	6	)	)	PUNCT
ejpam-4352	455	7	that	that	SCONJ
ejpam-4352	455	8	there	there	PRON
ejpam-4352	455	9	exists	exist	VERB
ejpam-4352	455	10	a	a	DET
ejpam-4352	455	11	z	z	NOUN
ejpam-4352	455	12	∈	∈	PROPN
ejpam-4352	455	13	n2	n2	NOUN
ejpam-4352	455	14	g(u	g(u	PROPN
ejpam-4352	455	15	)	)	PUNCT
ejpam-4352	455	16	and	and	CCONJ
ejpam-4352	455	17	t	t	PROPN
ejpam-4352	455	18	∈	∈	PROPN
ejpam-4352	455	19	tz	tz	NOUN
ejpam-4352	455	20	such	such	ADJ
ejpam-4352	455	21	that	that	SCONJ
ejpam-4352	455	22	tw	tw	VERB
ejpam-4352	455	23	=	=	SYM
ejpam-4352	455	24	v	v	PROPN
ejpam-4352	455	25	(	(	PUNCT
ejpam-4352	455	26	h	h	NOUN
ejpam-4352	455	27	)	)	PUNCT
ejpam-4352	455	28	for	for	ADP
ejpam-4352	455	29	all	all	DET
ejpam-4352	455	30	w	w	PROPN
ejpam-4352	455	31	∈	∈	PROPN
ejpam-4352	455	32	n2	n2	NOUN
ejpam-4352	455	33	g(z	g(z	PROPN
ejpam-4352	455	34	)	)	PUNCT
ejpam-4352	455	35	\	\	NOUN
ejpam-4352	455	36	{	{	PUNCT
ejpam-4352	455	37	u	u	NOUN
ejpam-4352	455	38	}	}	PUNCT
ejpam-4352	455	39	and	and	CCONJ
ejpam-4352	455	40	v	v	NOUN
ejpam-4352	455	41	(	(	PUNCT
ejpam-4352	455	42	h)\tz	h)\tz	PROPN
ejpam-4352	455	43	⊆	⊆	NUM
ejpam-4352	455	44	nh(t	nh(t	NUM
ejpam-4352	455	45	)	)	PUNCT
ejpam-4352	455	46	,	,	PUNCT
ejpam-4352	455	47	where	where	SCONJ
ejpam-4352	455	48	|v	|v	PROPN
ejpam-4352	455	49	(	(	PUNCT
ejpam-4352	455	50	h)\tz|	h)\tz|	ADP
ejpam-4352	455	51	≤	≤	NUM
ejpam-4352	455	52	1	1	NUM
ejpam-4352	455	53	.	.	PUNCT
ejpam-4352	456	1	replace	replace	VERB
ejpam-4352	456	2	tu	tu	PROPN
ejpam-4352	456	3	by	by	ADP
ejpam-4352	456	4	t	t	PROPN
ejpam-4352	456	5	′	′	NUM
ejpam-4352	456	6	u	u	NOUN
ejpam-4352	456	7	=	=	SYM
ejpam-4352	456	8	v	v	PROPN
ejpam-4352	456	9	(	(	PUNCT
ejpam-4352	456	10	h	h	NOUN
ejpam-4352	456	11	)	)	PUNCT
ejpam-4352	456	12	and	and	CCONJ
ejpam-4352	456	13	tz	tz	PROPN
ejpam-4352	456	14	by	by	ADP
ejpam-4352	456	15	a	a	DET
ejpam-4352	456	16	γcs	γcs	NOUN
ejpam-4352	456	17	-	-	PUNCT
ejpam-4352	456	18	set	set	VERB
ejpam-4352	456	19	t	t	NOUN
ejpam-4352	456	20	′	′	NUM
ejpam-4352	456	21	z	z	NOUN
ejpam-4352	456	22	of	of	ADP
ejpam-4352	456	23	h.	h.	PROPN
ejpam-4352	456	24	then	then	ADV
ejpam-4352	456	25	the	the	DET
ejpam-4352	456	26	set	set	ADJ
ejpam-4352	456	27	c1	c1	NOUN
ejpam-4352	456	28	=	=	SYM
ejpam-4352	456	29	∪x∈v	∪x∈v	PROPN
ejpam-4352	456	30	(	(	PUNCT
ejpam-4352	456	31	g)\{z	g)\{z	NOUN
ejpam-4352	456	32	,	,	PUNCT
ejpam-4352	456	33	u}({x}×tx)∪({u}×t	u}({x}×tx)∪({u}×t	PUNCT
ejpam-4352	456	34	′	′	NUM
ejpam-4352	457	1	u)∪({z}×t	u)∪({z}×t	NOUN
ejpam-4352	458	1	′	′	NUM
ejpam-4352	458	2	z	z	X
ejpam-4352	458	3	)	)	PUNCT
ejpam-4352	458	4	is	be	AUX
ejpam-4352	458	5	a	a	DET
ejpam-4352	458	6	super	super	ADV
ejpam-4352	458	7	hop	hop	NOUN
ejpam-4352	458	8	dominating	dominating	NOUN
ejpam-4352	458	9	set	set	NOUN
ejpam-4352	458	10	of	of	ADP
ejpam-4352	458	11	g[h	g[h	PROPN
ejpam-4352	458	12	]	]	PUNCT
ejpam-4352	458	13	by	by	ADP
ejpam-4352	458	14	theorem	theorem	NOUN
ejpam-4352	458	15	11	11	NUM
ejpam-4352	458	16	.	.	PUNCT
ejpam-4352	459	1	clearly	clearly	ADV
ejpam-4352	459	2	,	,	PUNCT
ejpam-4352	459	3	|t	|t	VERB
ejpam-4352	459	4	′	′	VERB
ejpam-4352	460	1	u|	u|	ADV
ejpam-4352	460	2	=	=	PUNCT
ejpam-4352	460	3	|tu|+	|tu|+	NOUN
ejpam-4352	460	4	1	1	NUM
ejpam-4352	460	5	.	.	PUNCT
ejpam-4352	461	1	if	if	SCONJ
ejpam-4352	461	2	γcs(h	γcs(h	PROPN
ejpam-4352	461	3	)	)	PUNCT
ejpam-4352	461	4	≤	≤	NUM
ejpam-4352	461	5	n−	n−	NOUN
ejpam-4352	461	6	3	3	NUM
ejpam-4352	461	7	,	,	PUNCT
ejpam-4352	461	8	then	then	ADV
ejpam-4352	461	9	|t	|t	VERB
ejpam-4352	461	10	′	′	NUM
ejpam-4352	462	1	z|	z|	CCONJ
ejpam-4352	462	2	<	<	X
ejpam-4352	462	3	|tz|	|tz|	NOUN
ejpam-4352	462	4	−	−	NUM
ejpam-4352	462	5	1	1	NUM
ejpam-4352	462	6	and	and	CCONJ
ejpam-4352	462	7	a	a	DET
ejpam-4352	462	8	straightforward	straightforward	ADJ
ejpam-4352	462	9	computation	computation	NOUN
ejpam-4352	462	10	shows	show	VERB
ejpam-4352	462	11	that	that	SCONJ
ejpam-4352	462	12	|c1|	|c1|	PROPN
ejpam-4352	462	13	<	<	X
ejpam-4352	462	14	|c|	|c|	PROPN
ejpam-4352	462	15	,	,	PUNCT
ejpam-4352	462	16	a	a	DET
ejpam-4352	462	17	contradiction	contradiction	NOUN
ejpam-4352	462	18	.	.	PUNCT
ejpam-4352	463	1	thus	thus	ADV
ejpam-4352	463	2	,	,	PUNCT
ejpam-4352	463	3	tu	tu	PROPN
ejpam-4352	463	4	=	=	SYM
ejpam-4352	463	5	v	v	PROPN
ejpam-4352	463	6	(	(	PUNCT
ejpam-4352	463	7	h	h	NOUN
ejpam-4352	463	8	)	)	PUNCT
ejpam-4352	463	9	for	for	ADP
ejpam-4352	463	10	all	all	PRON
ejpam-4352	463	11	u	u	PROPN
ejpam-4352	463	12	∈	∈	PROPN
ejpam-4352	463	13	v	v	NOUN
ejpam-4352	463	14	(	(	PUNCT
ejpam-4352	463	15	g	g	NOUN
ejpam-4352	463	16	)	)	PUNCT
ejpam-4352	463	17	\	\	PROPN
ejpam-4352	463	18	r.	r.	PROPN
ejpam-4352	463	19	since	since	SCONJ
ejpam-4352	463	20	c	c	PROPN
ejpam-4352	463	21	is	be	AUX
ejpam-4352	463	22	a	a	DET
ejpam-4352	463	23	γsh	γsh	NOUN
ejpam-4352	463	24	-	-	PUNCT
ejpam-4352	463	25	set	set	NOUN
ejpam-4352	463	26	of	of	ADP
ejpam-4352	463	27	g[h	g[h	PROPN
ejpam-4352	463	28	]	]	PUNCT
ejpam-4352	463	29	,	,	PUNCT
ejpam-4352	463	30	it	it	PRON
ejpam-4352	463	31	follows	follow	VERB
ejpam-4352	463	32	that	that	SCONJ
ejpam-4352	463	33	r	r	NOUN
ejpam-4352	463	34	is	be	AUX
ejpam-4352	463	35	a	a	DET
ejpam-4352	463	36	αh	αh	NOUN
ejpam-4352	463	37	-	-	PUNCT
ejpam-4352	463	38	set	set	NOUN
ejpam-4352	463	39	of	of	ADP
ejpam-4352	463	40	g.	g.	PROPN
ejpam-4352	463	41	this	this	PRON
ejpam-4352	463	42	implies	imply	VERB
ejpam-4352	463	43	that	that	SCONJ
ejpam-4352	463	44	γsh(g[h	γsh(g[h	NUM
ejpam-4352	463	45	]	]	PUNCT
ejpam-4352	463	46	)	)	PUNCT
ejpam-4352	463	47	=	=	SYM
ejpam-4352	463	48	|c|	|c|	PROPN
ejpam-4352	463	49	=	=	SYM
ejpam-4352	463	50	(	(	PUNCT
ejpam-4352	463	51	γcs(h	γcs(h	PROPN
ejpam-4352	463	52	)	)	PUNCT
ejpam-4352	463	53	−	−	ADP
ejpam-4352	463	54	n)αh(g	n)αh(g	NOUN
ejpam-4352	463	55	)	)	PUNCT
ejpam-4352	463	56	+	+	CCONJ
ejpam-4352	463	57	mn	mn	PROPN
ejpam-4352	463	58	.	.	PROPN
ejpam-4352	463	59	suppose	suppose	VERB
ejpam-4352	463	60	now	now	ADV
ejpam-4352	463	61	that	that	SCONJ
ejpam-4352	463	62	γcs(h	γcs(h	PROPN
ejpam-4352	463	63	)	)	PUNCT
ejpam-4352	463	64	=	=	SYM
ejpam-4352	464	1	n	n	CCONJ
ejpam-4352	464	2	−	−	PROPN
ejpam-4352	464	3	2	2	X
ejpam-4352	464	4	.	.	PUNCT
ejpam-4352	464	5	suppose	suppose	VERB
ejpam-4352	464	6	further	far	ADV
ejpam-4352	464	7	that	that	SCONJ
ejpam-4352	464	8	tz	tz	PROPN
ejpam-4352	464	9	=	=	SYM
ejpam-4352	464	10	v	v	PROPN
ejpam-4352	464	11	(	(	PUNCT
ejpam-4352	464	12	h	h	NOUN
ejpam-4352	464	13	)	)	PUNCT
ejpam-4352	464	14	.	.	PUNCT
ejpam-4352	465	1	then	then	ADV
ejpam-4352	465	2	|t	|t	VERB
ejpam-4352	465	3	′	′	NUM
ejpam-4352	466	1	z|	z|	CCONJ
ejpam-4352	466	2	<	<	X
ejpam-4352	466	3	|tz|	|tz|	NOUN
ejpam-4352	466	4	−	−	NUM
ejpam-4352	466	5	1	1	NUM
ejpam-4352	466	6	and	and	CCONJ
ejpam-4352	466	7	|c1|	|c1|	PROPN
ejpam-4352	466	8	<	<	X
ejpam-4352	466	9	|c|	|c|	PROPN
ejpam-4352	466	10	,	,	PUNCT
ejpam-4352	466	11	a	a	DET
ejpam-4352	466	12	contradiction	contradiction	NOUN
ejpam-4352	466	13	.	.	PUNCT
ejpam-4352	467	1	hence	hence	ADV
ejpam-4352	467	2	,	,	PUNCT
ejpam-4352	467	3	|tz|	|tz|	NOUN
ejpam-4352	467	4	=	=	SYM
ejpam-4352	467	5	n	n	CCONJ
ejpam-4352	467	6	−	−	NOUN
ejpam-4352	467	7	1	1	X
ejpam-4352	467	8	.	.	PUNCT
ejpam-4352	468	1	it	it	PRON
ejpam-4352	468	2	follows	follow	VERB
ejpam-4352	468	3	that	that	SCONJ
ejpam-4352	468	4	|t	|t	VERB
ejpam-4352	469	1	′	′	NUM
ejpam-4352	469	2	z|	z|	PRON
ejpam-4352	470	1	=	=	PUNCT
ejpam-4352	470	2	|tz|	|tz|	NOUN
ejpam-4352	470	3	−	−	NUM
ejpam-4352	470	4	1	1	NUM
ejpam-4352	470	5	and	and	CCONJ
ejpam-4352	470	6	|c1|	|c1|	PROPN
ejpam-4352	470	7	=	=	PROPN
ejpam-4352	470	8	|c|	|c|	PROPN
ejpam-4352	470	9	.	.	PUNCT
ejpam-4352	471	1	this	this	PRON
ejpam-4352	471	2	means	mean	VERB
ejpam-4352	471	3	that	that	SCONJ
ejpam-4352	471	4	in	in	ADP
ejpam-4352	471	5	the	the	DET
ejpam-4352	471	6	γsh	γsh	NOUN
ejpam-4352	471	7	-	-	PUNCT
ejpam-4352	471	8	set	set	VERB
ejpam-4352	471	9	c	c	NOUN
ejpam-4352	471	10	we	we	PRON
ejpam-4352	471	11	may	may	AUX
ejpam-4352	471	12	assume	assume	VERB
ejpam-4352	471	13	further	far	ADV
ejpam-4352	471	14	that	that	SCONJ
ejpam-4352	471	15	tu	tu	PROPN
ejpam-4352	471	16	=	=	SYM
ejpam-4352	471	17	v	v	PROPN
ejpam-4352	471	18	(	(	PUNCT
ejpam-4352	471	19	h	h	NOUN
ejpam-4352	471	20	)	)	PUNCT
ejpam-4352	471	21	for	for	ADP
ejpam-4352	471	22	each	each	DET
ejpam-4352	471	23	u	u	PROPN
ejpam-4352	471	24	∈	∈	PROPN
ejpam-4352	471	25	v	v	NOUN
ejpam-4352	471	26	(	(	PUNCT
ejpam-4352	471	27	g)\r	g)\r	NOUN
ejpam-4352	471	28	.	.	PUNCT
ejpam-4352	472	1	again	again	ADV
ejpam-4352	472	2	,	,	PUNCT
ejpam-4352	472	3	as	as	SCONJ
ejpam-4352	472	4	c	c	PROPN
ejpam-4352	472	5	is	be	AUX
ejpam-4352	472	6	a	a	DET
ejpam-4352	472	7	γsh	γsh	NOUN
ejpam-4352	472	8	-	-	PUNCT
ejpam-4352	472	9	set	set	NOUN
ejpam-4352	472	10	of	of	ADP
ejpam-4352	472	11	g[h	g[h	PROPN
ejpam-4352	472	12	]	]	PUNCT
ejpam-4352	472	13	,	,	PUNCT
ejpam-4352	472	14	r	r	NOUN
ejpam-4352	472	15	would	would	AUX
ejpam-4352	472	16	be	be	AUX
ejpam-4352	472	17	a	a	DET
ejpam-4352	472	18	αh	αh	NOUN
ejpam-4352	472	19	-	-	PUNCT
ejpam-4352	472	20	set	set	NOUN
ejpam-4352	472	21	of	of	ADP
ejpam-4352	472	22	g	g	NOUN
ejpam-4352	472	23	,	,	PUNCT
ejpam-4352	472	24	establishing	establish	VERB
ejpam-4352	472	25	the	the	DET
ejpam-4352	472	26	desired	desire	VERB
ejpam-4352	472	27	equality	equality	NOUN
ejpam-4352	472	28	.	.	PUNCT
ejpam-4352	473	1	we	we	PRON
ejpam-4352	473	2	point	point	VERB
ejpam-4352	473	3	out	out	ADP
ejpam-4352	473	4	that	that	SCONJ
ejpam-4352	473	5	the	the	DET
ejpam-4352	473	6	equality	equality	NOUN
ejpam-4352	473	7	in	in	ADP
ejpam-4352	473	8	corollary	corollary	ADJ
ejpam-4352	473	9	5	5	NUM
ejpam-4352	473	10	does	do	AUX
ejpam-4352	473	11	not	not	PART
ejpam-4352	473	12	necessarily	necessarily	ADV
ejpam-4352	473	13	hold	hold	VERB
ejpam-4352	473	14	if	if	SCONJ
ejpam-4352	473	15	γcs(h	γcs(h	PROPN
ejpam-4352	473	16	)	)	PUNCT
ejpam-4352	474	1	=	=	SYM
ejpam-4352	474	2	n−1	n−1	PROPN
ejpam-4352	474	3	,	,	PUNCT
ejpam-4352	474	4	where	where	SCONJ
ejpam-4352	474	5	n	n	X
ejpam-4352	474	6	=	=	SYM
ejpam-4352	474	7	|v	|v	PROPN
ejpam-4352	474	8	(	(	PUNCT
ejpam-4352	474	9	h)|	h)|	PROPN
ejpam-4352	474	10	.	.	PUNCT
ejpam-4352	475	1	to	to	PART
ejpam-4352	475	2	see	see	VERB
ejpam-4352	475	3	this	this	PRON
ejpam-4352	475	4	,	,	PUNCT
ejpam-4352	475	5	consider	consider	VERB
ejpam-4352	475	6	g	g	NOUN
ejpam-4352	475	7	=	=	SYM
ejpam-4352	475	8	p4	p4	ADJ
ejpam-4352	475	9	and	and	CCONJ
ejpam-4352	475	10	h	h	NOUN
ejpam-4352	475	11	=	=	PROPN
ejpam-4352	475	12	p3	p3	PROPN
ejpam-4352	475	13	.	.	PUNCT
ejpam-4352	476	1	then	then	ADV
ejpam-4352	476	2	αh(g	αh(g	NOUN
ejpam-4352	476	3	)	)	PUNCT
ejpam-4352	476	4	=	=	SYM
ejpam-4352	476	5	2	2	NUM
ejpam-4352	476	6	and	and	CCONJ
ejpam-4352	476	7	γcs(h	γcs(h	PROPN
ejpam-4352	476	8	)	)	PUNCT
ejpam-4352	476	9	=	=	PUNCT
ejpam-4352	477	1	2	2	X
ejpam-4352	477	2	.	.	X
ejpam-4352	477	3	it	it	PRON
ejpam-4352	477	4	can	can	AUX
ejpam-4352	477	5	be	be	AUX
ejpam-4352	477	6	verified	verify	VERB
ejpam-4352	477	7	easily	easily	ADV
ejpam-4352	477	8	that	that	SCONJ
ejpam-4352	477	9	γsh(g[h	γsh(g[h	NUM
ejpam-4352	477	10	]	]	PUNCT
ejpam-4352	477	11	)	)	PUNCT
ejpam-4352	477	12	=	=	SYM
ejpam-4352	478	1	8	8	NUM
ejpam-4352	478	2	<	<	SYM
ejpam-4352	478	3	10	10	NUM
ejpam-4352	478	4	=	=	SYM
ejpam-4352	478	5	(	(	PUNCT
ejpam-4352	478	6	2	2	NUM
ejpam-4352	478	7	−	−	NOUN
ejpam-4352	478	8	3)(2	3)(2	NUM
ejpam-4352	478	9	)	)	PUNCT
ejpam-4352	479	1	+	+	CCONJ
ejpam-4352	479	2	(	(	PUNCT
ejpam-4352	479	3	4)(3	4)(3	NUM
ejpam-4352	479	4	)	)	PUNCT
ejpam-4352	479	5	=	=	PUNCT
ejpam-4352	480	1	[	[	X
ejpam-4352	480	2	γcs(h)−	γcs(h)−	PROPN
ejpam-4352	480	3	3]αh(g	3]αh(g	NOUN
ejpam-4352	480	4	)	)	PUNCT
ejpam-4352	480	5	+	+	CCONJ
ejpam-4352	480	6	12	12	NUM
ejpam-4352	480	7	.	.	PUNCT
ejpam-4352	481	1	corollary	corollary	ADJ
ejpam-4352	481	2	6	6	NUM
ejpam-4352	481	3	.	.	PUNCT
ejpam-4352	482	1	let	let	VERB
ejpam-4352	482	2	h	h	PRON
ejpam-4352	482	3	be	be	AUX
ejpam-4352	482	4	a	a	DET
ejpam-4352	482	5	connected	connected	ADJ
ejpam-4352	482	6	graph	graph	NOUN
ejpam-4352	482	7	and	and	CCONJ
ejpam-4352	482	8	let	let	VERB
ejpam-4352	482	9	m	m	PRON
ejpam-4352	482	10	be	be	AUX
ejpam-4352	482	11	a	a	DET
ejpam-4352	482	12	positive	positive	ADJ
ejpam-4352	482	13	integer	integer	NOUN
ejpam-4352	482	14	.	.	PUNCT
ejpam-4352	483	1	then	then	ADV
ejpam-4352	483	2	γsh(km[h	γsh(km[h	ADP
ejpam-4352	483	3	]	]	PUNCT
ejpam-4352	483	4	)	)	PUNCT
ejpam-4352	483	5	=	=	PRON
ejpam-4352	483	6	{	{	PUNCT
ejpam-4352	483	7	γsh(h	γsh(h	PROPN
ejpam-4352	483	8	)	)	PUNCT
ejpam-4352	483	9	,	,	PUNCT
ejpam-4352	483	10	m	m	VERB
ejpam-4352	483	11	=	=	NOUN
ejpam-4352	483	12	1	1	NUM
ejpam-4352	483	13	m.γcs(h	m.γcs(h	NOUN
ejpam-4352	483	14	)	)	PUNCT
ejpam-4352	483	15	,	,	PUNCT
ejpam-4352	483	16	m	m	VERB
ejpam-4352	483	17	≥	≥	NOUN
ejpam-4352	483	18	2	2	NUM
ejpam-4352	483	19	.	.	PUNCT
ejpam-4352	483	20	proof	proof	NOUN
ejpam-4352	483	21	.	.	PUNCT
ejpam-4352	484	1	the	the	DET
ejpam-4352	484	2	result	result	NOUN
ejpam-4352	484	3	is	be	AUX
ejpam-4352	484	4	clear	clear	ADJ
ejpam-4352	484	5	if	if	SCONJ
ejpam-4352	484	6	m	m	NOUN
ejpam-4352	484	7	=	=	NOUN
ejpam-4352	484	8	1	1	X
ejpam-4352	484	9	.	.	PUNCT
ejpam-4352	484	10	suppose	suppose	VERB
ejpam-4352	484	11	m	m	PRON
ejpam-4352	484	12	≥	≥	NOUN
ejpam-4352	484	13	2	2	NUM
ejpam-4352	484	14	and	and	CCONJ
ejpam-4352	484	15	let	let	VERB
ejpam-4352	484	16	let	let	VERB
ejpam-4352	484	17	c	c	NOUN
ejpam-4352	484	18	=	=	SYM
ejpam-4352	484	19	∪x∈v	∪x∈v	X
ejpam-4352	484	20	(	(	PUNCT
ejpam-4352	484	21	km)({x	km)({x	PROPN
ejpam-4352	484	22	}	}	PUNCT
ejpam-4352	484	23	×	×	PROPN
ejpam-4352	484	24	tx	tx	PROPN
ejpam-4352	484	25	)	)	PUNCT
ejpam-4352	484	26	be	be	AUX
ejpam-4352	484	27	a	a	DET
ejpam-4352	484	28	γsh	γsh	NOUN
ejpam-4352	484	29	-	-	PUNCT
ejpam-4352	484	30	set	set	NOUN
ejpam-4352	484	31	of	of	ADP
ejpam-4352	484	32	km[h	km[h	PROPN
ejpam-4352	484	33	]	]	PUNCT
ejpam-4352	484	34	.	.	PUNCT
ejpam-4352	485	1	then	then	ADV
ejpam-4352	485	2	each	each	DET
ejpam-4352	485	3	tx	tx	PROPN
ejpam-4352	485	4	is	be	AUX
ejpam-4352	485	5	a	a	DET
ejpam-4352	485	6	complement	complement	NOUN
ejpam-4352	485	7	-	-	PUNCT
ejpam-4352	485	8	super	super	ADJ
ejpam-4352	485	9	dominating	dominating	NOUN
ejpam-4352	485	10	set	set	NOUN
ejpam-4352	485	11	of	of	ADP
ejpam-4352	485	12	h	h	NOUN
ejpam-4352	485	13	references	reference	NOUN
ejpam-4352	485	14	352	352	NUM
ejpam-4352	485	15	by	by	ADP
ejpam-4352	485	16	theorem	theorem	NOUN
ejpam-4352	485	17	11	11	NUM
ejpam-4352	485	18	.	.	PUNCT
ejpam-4352	486	1	in	in	ADP
ejpam-4352	486	2	particular	particular	ADJ
ejpam-4352	486	3	,	,	PUNCT
ejpam-4352	486	4	tx	tx	PROPN
ejpam-4352	486	5	is	be	AUX
ejpam-4352	486	6	a	a	DET
ejpam-4352	486	7	γcs	γcs	NOUN
ejpam-4352	486	8	-	-	PUNCT
ejpam-4352	486	9	set	set	NOUN
ejpam-4352	486	10	of	of	ADP
ejpam-4352	486	11	h	h	NOUN
ejpam-4352	486	12	for	for	ADP
ejpam-4352	486	13	all	all	PRON
ejpam-4352	486	14	x	x	SYM
ejpam-4352	486	15	∈	∈	PROPN
ejpam-4352	486	16	v	v	NOUN
ejpam-4352	486	17	(	(	PUNCT
ejpam-4352	486	18	km	km	PROPN
ejpam-4352	486	19	)	)	PUNCT
ejpam-4352	486	20	.	.	PUNCT
ejpam-4352	487	1	accordingly	accordingly	ADV
ejpam-4352	487	2	,	,	PUNCT
ejpam-4352	487	3	γsh(km[h	γsh(km[h	PROPN
ejpam-4352	487	4	]	]	PUNCT
ejpam-4352	487	5	)	)	PUNCT
ejpam-4352	488	1	=	=	SYM
ejpam-4352	488	2	|c|	|c|	PROPN
ejpam-4352	488	3	=	=	SYM
ejpam-4352	488	4	m.γcs(h	m.γcs(h	PROPN
ejpam-4352	488	5	)	)	PUNCT
ejpam-4352	488	6	.	.	PUNCT
ejpam-4352	489	1	conclusion	conclusion	NOUN
ejpam-4352	489	2	:	:	PUNCT
ejpam-4352	489	3	super	super	ADJ
ejpam-4352	489	4	hop	hop	PROPN
ejpam-4352	489	5	domination	domination	NOUN
ejpam-4352	489	6	,	,	PUNCT
ejpam-4352	489	7	a	a	DET
ejpam-4352	489	8	variant	variant	NOUN
ejpam-4352	489	9	of	of	ADP
ejpam-4352	489	10	hop	hop	NOUN
ejpam-4352	489	11	domination	domination	NOUN
ejpam-4352	489	12	,	,	PUNCT
ejpam-4352	489	13	has	have	AUX
ejpam-4352	489	14	been	be	AUX
ejpam-4352	489	15	introduced	introduce	VERB
ejpam-4352	489	16	and	and	CCONJ
ejpam-4352	489	17	studied	study	VERB
ejpam-4352	489	18	for	for	ADP
ejpam-4352	489	19	some	some	DET
ejpam-4352	489	20	graphs	graph	NOUN
ejpam-4352	489	21	and	and	CCONJ
ejpam-4352	489	22	graphs	graph	NOUN
ejpam-4352	489	23	resulting	result	VERB
ejpam-4352	489	24	from	from	ADP
ejpam-4352	489	25	the	the	DET
ejpam-4352	489	26	join	join	NOUN
ejpam-4352	489	27	and	and	CCONJ
ejpam-4352	489	28	lexicographic	lexicographic	ADJ
ejpam-4352	489	29	product	product	NOUN
ejpam-4352	489	30	of	of	ADP
ejpam-4352	489	31	two	two	NUM
ejpam-4352	489	32	graphs	graph	NOUN
ejpam-4352	489	33	.	.	PUNCT
ejpam-4352	490	1	in	in	ADP
ejpam-4352	490	2	the	the	DET
ejpam-4352	490	3	case	case	NOUN
ejpam-4352	490	4	of	of	ADP
ejpam-4352	490	5	the	the	DET
ejpam-4352	490	6	join	join	NOUN
ejpam-4352	490	7	of	of	ADP
ejpam-4352	490	8	graphs	graph	NOUN
ejpam-4352	490	9	,	,	PUNCT
ejpam-4352	490	10	the	the	DET
ejpam-4352	490	11	concept	concept	NOUN
ejpam-4352	490	12	of	of	ADP
ejpam-4352	490	13	complement	complement	NOUN
ejpam-4352	490	14	-	-	PUNCT
ejpam-4352	490	15	super	super	NOUN
ejpam-4352	490	16	domination	domination	NOUN
ejpam-4352	490	17	plays	play	VERB
ejpam-4352	490	18	a	a	DET
ejpam-4352	490	19	vital	vital	ADJ
ejpam-4352	490	20	role	role	NOUN
ejpam-4352	490	21	.	.	PUNCT
ejpam-4352	491	1	finding	find	VERB
ejpam-4352	491	2	the	the	DET
ejpam-4352	491	3	complement	complement	NOUN
ejpam-4352	491	4	-	-	PUNCT
ejpam-4352	491	5	super	super	ADJ
ejpam-4352	491	6	domination	domination	NOUN
ejpam-4352	491	7	number	number	NOUN
ejpam-4352	491	8	of	of	ADP
ejpam-4352	491	9	a	a	DET
ejpam-4352	491	10	graph	graph	NOUN
ejpam-4352	491	11	is	be	AUX
ejpam-4352	491	12	the	the	DET
ejpam-4352	491	13	same	same	ADJ
ejpam-4352	491	14	as	as	ADP
ejpam-4352	491	15	determining	determine	VERB
ejpam-4352	491	16	the	the	DET
ejpam-4352	491	17	super	super	ADJ
ejpam-4352	491	18	domination	domination	NOUN
ejpam-4352	491	19	number	number	NOUN
ejpam-4352	491	20	of	of	ADP
ejpam-4352	491	21	the	the	DET
ejpam-4352	491	22	complement	complement	NOUN
ejpam-4352	491	23	of	of	ADP
ejpam-4352	491	24	the	the	DET
ejpam-4352	491	25	graph	graph	NOUN
ejpam-4352	491	26	.	.	PUNCT
ejpam-4352	492	1	it	it	PRON
ejpam-4352	492	2	is	be	AUX
ejpam-4352	492	3	conjectured	conjecture	VERB
ejpam-4352	492	4	that	that	SCONJ
ejpam-4352	492	5	the	the	DET
ejpam-4352	492	6	problem	problem	NOUN
ejpam-4352	492	7	of	of	ADP
ejpam-4352	492	8	finding	find	VERB
ejpam-4352	492	9	a	a	DET
ejpam-4352	492	10	super	super	ADV
ejpam-4352	492	11	hop	hop	NOUN
ejpam-4352	492	12	dominating	dominating	NOUN
ejpam-4352	492	13	set	set	NOUN
ejpam-4352	492	14	is	be	AUX
ejpam-4352	492	15	not	not	PART
ejpam-4352	492	16	easy	easy	ADJ
ejpam-4352	492	17	,	,	PUNCT
ejpam-4352	492	18	that	that	ADV
ejpam-4352	492	19	is	is	ADV
ejpam-4352	492	20	,	,	PUNCT
ejpam-4352	492	21	np	np	INTJ
ejpam-4352	492	22	-	-	PUNCT
ejpam-4352	492	23	hard	hard	ADJ
ejpam-4352	492	24	(	(	PUNCT
ejpam-4352	492	25	np	np	INTJ
ejpam-4352	492	26	-	-	PUNCT
ejpam-4352	492	27	complete	complete	ADJ
ejpam-4352	492	28	)	)	PUNCT
ejpam-4352	492	29	.	.	PUNCT
ejpam-4352	493	1	it	it	PRON
ejpam-4352	493	2	is	be	AUX
ejpam-4352	493	3	recommended	recommend	VERB
ejpam-4352	493	4	that	that	SCONJ
ejpam-4352	493	5	some	some	DET
ejpam-4352	493	6	bounds	bound	NOUN
ejpam-4352	493	7	on	on	ADP
ejpam-4352	493	8	the	the	DET
ejpam-4352	493	9	super	super	PROPN
ejpam-4352	493	10	hop	hop	NOUN
ejpam-4352	493	11	domination	domination	NOUN
ejpam-4352	493	12	be	be	AUX
ejpam-4352	493	13	determined	determine	VERB
ejpam-4352	493	14	and	and	CCONJ
ejpam-4352	493	15	that	that	SCONJ
ejpam-4352	493	16	the	the	DET
ejpam-4352	493	17	parameter	parameter	NOUN
ejpam-4352	493	18	be	be	AUX
ejpam-4352	493	19	studied	study	VERB
ejpam-4352	493	20	for	for	ADP
ejpam-4352	493	21	other	other	ADJ
ejpam-4352	493	22	graphs	graph	NOUN
ejpam-4352	493	23	.	.	PUNCT
ejpam-4352	494	1	acknowledgements	acknowledgement	NOUN
ejpam-4352	494	2	the	the	DET
ejpam-4352	494	3	authors	author	NOUN
ejpam-4352	494	4	would	would	AUX
ejpam-4352	494	5	like	like	VERB
ejpam-4352	494	6	to	to	PART
ejpam-4352	494	7	thank	thank	VERB
ejpam-4352	494	8	the	the	DET
ejpam-4352	494	9	referees	referee	NOUN
ejpam-4352	494	10	for	for	ADP
ejpam-4352	494	11	their	their	PRON
ejpam-4352	494	12	comments	comment	NOUN
ejpam-4352	494	13	and	and	CCONJ
ejpam-4352	494	14	suggestions	suggestion	NOUN
ejpam-4352	494	15	which	which	PRON
ejpam-4352	494	16	helped	help	VERB
ejpam-4352	494	17	improve	improve	VERB
ejpam-4352	494	18	the	the	DET
ejpam-4352	494	19	paper	paper	NOUN
ejpam-4352	494	20	.	.	PUNCT
ejpam-4352	495	1	also	also	ADV
ejpam-4352	495	2	,	,	PUNCT
ejpam-4352	495	3	they	they	PRON
ejpam-4352	495	4	would	would	AUX
ejpam-4352	495	5	like	like	VERB
ejpam-4352	495	6	to	to	PART
ejpam-4352	495	7	extend	extend	VERB
ejpam-4352	495	8	their	their	PRON
ejpam-4352	495	9	thankfulness	thankfulness	NOUN
ejpam-4352	495	10	to	to	ADP
ejpam-4352	495	11	the	the	DET
ejpam-4352	495	12	department	department	PROPN
ejpam-4352	495	13	of	of	ADP
ejpam-4352	495	14	science	science	NOUN
ejpam-4352	495	15	and	and	CCONJ
ejpam-4352	495	16	technology	technology	NOUN
ejpam-4352	495	17	accelerated	accelerate	VERB
ejpam-4352	495	18	science	science	NOUN
ejpam-4352	495	19	and	and	CCONJ
ejpam-4352	495	20	technology	technology	NOUN
ejpam-4352	495	21	human	human	ADJ
ejpam-4352	495	22	resource	resource	NOUN
ejpam-4352	495	23	development	development	NOUN
ejpam-4352	495	24	program	program	NOUN
ejpam-4352	495	25	(	(	PUNCT
ejpam-4352	495	26	dost	dost	NOUN
ejpam-4352	495	27	-	-	PUNCT
ejpam-4352	495	28	asthrdp)-philippines	asthrdp)-philippine	NOUN
ejpam-4352	495	29	,	,	PUNCT
ejpam-4352	495	30	and	and	CCONJ
ejpam-4352	495	31	msu	msu	PROPN
ejpam-4352	495	32	-	-	PUNCT
ejpam-4352	495	33	iligan	iligan	PROPN
ejpam-4352	495	34	institute	institute	PROPN
ejpam-4352	495	35	of	of	ADP
ejpam-4352	495	36	technology	technology	NOUN
ejpam-4352	495	37	for	for	ADP
ejpam-4352	495	38	funding	fund	VERB
ejpam-4352	495	39	this	this	DET
ejpam-4352	495	40	research	research	NOUN
ejpam-4352	495	41	.	.	PUNCT
ejpam-4352	496	1	references	reference	NOUN
ejpam-4352	496	2	[	[	X
ejpam-4352	496	3	1	1	NUM
ejpam-4352	496	4	]	]	PUNCT
ejpam-4352	496	5	s.	s.	PROPN
ejpam-4352	496	6	ayyaswamy	ayyaswamy	PROPN
ejpam-4352	496	7	,	,	PUNCT
ejpam-4352	496	8	b.	b.	PROPN
ejpam-4352	496	9	krishnakumari	krishnakumari	PROPN
ejpam-4352	496	10	,	,	PUNCT
ejpam-4352	496	11	b.	b.	PROPN
ejpam-4352	496	12	natarjan	natarjan	PROPN
ejpam-4352	496	13	,	,	PUNCT
ejpam-4352	496	14	and	and	CCONJ
ejpam-4352	496	15	y.	y.	PROPN
ejpam-4352	496	16	venkatakrishnan	venkatakrishnan	PROPN
ejpam-4352	496	17	.	.	PUNCT
ejpam-4352	497	1	bounds	bound	NOUN
ejpam-4352	497	2	on	on	ADP
ejpam-4352	497	3	the	the	DET
ejpam-4352	497	4	hop	hop	NOUN
ejpam-4352	497	5	domination	domination	NOUN
ejpam-4352	497	6	number	number	NOUN
ejpam-4352	497	7	of	of	ADP
ejpam-4352	497	8	a	a	DET
ejpam-4352	497	9	tree	tree	NOUN
ejpam-4352	497	10	.	.	PUNCT
ejpam-4352	498	1	proceedings	proceeding	NOUN
ejpam-4352	498	2	-	-	PUNCT
ejpam-4352	498	3	mathematical	mathematical	ADJ
ejpam-4352	498	4	sciences	science	NOUN
ejpam-4352	498	5	,	,	PUNCT
ejpam-4352	498	6	125(4):449	125(4):449	NUM
ejpam-4352	498	7	–	–	PUNCT
ejpam-4352	498	8	455	455	NUM
ejpam-4352	498	9	,	,	PUNCT
ejpam-4352	498	10	2015	2015	NUM
ejpam-4352	498	11	.	.	PUNCT
ejpam-4352	499	1	[	[	X
ejpam-4352	499	2	2	2	NUM
ejpam-4352	499	3	]	]	PUNCT
ejpam-4352	499	4	m.	m.	NOUN
ejpam-4352	499	5	dettlaff	dettlaff	VERB
ejpam-4352	499	6	.	.	PUNCT
ejpam-4352	500	1	a	a	DET
ejpam-4352	500	2	note	note	NOUN
ejpam-4352	500	3	on	on	ADP
ejpam-4352	500	4	the	the	DET
ejpam-4352	500	5	super	super	ADJ
ejpam-4352	500	6	domination	domination	NOUN
ejpam-4352	500	7	number	number	NOUN
ejpam-4352	500	8	of	of	ADP
ejpam-4352	500	9	cartesian	cartesian	ADJ
ejpam-4352	500	10	product	product	NOUN
ejpam-4352	500	11	of	of	ADP
ejpam-4352	500	12	graphs	graph	NOUN
ejpam-4352	500	13	.	.	PUNCT
ejpam-4352	501	1	arxiv:1705.00928v1	arxiv:1705.00928v1	ADJ
ejpam-4352	501	2	,	,	PUNCT
ejpam-4352	501	3	2017	2017	NUM
ejpam-4352	501	4	.	.	PUNCT
ejpam-4352	502	1	[	[	X
ejpam-4352	502	2	3	3	X
ejpam-4352	502	3	]	]	X
ejpam-4352	502	4	m.	m.	NOUN
ejpam-4352	502	5	dettlaff	dettlaff	NOUN
ejpam-4352	502	6	,	,	PUNCT
ejpam-4352	502	7	m.	m.	NOUN
ejpam-4352	502	8	lemanska	lemanska	PROPN
ejpam-4352	502	9	,	,	PUNCT
ejpam-4352	502	10	j.a	j.a	PROPN
ejpam-4352	502	11	.	.	PROPN
ejpam-4352	502	12	rodriguez	rodriguez	PROPN
ejpam-4352	502	13	,	,	PUNCT
ejpam-4352	502	14	and	and	CCONJ
ejpam-4352	502	15	r.	r.	PROPN
ejpam-4352	502	16	zuazua	zuazua	PROPN
ejpam-4352	502	17	.	.	PUNCT
ejpam-4352	503	1	on	on	ADP
ejpam-4352	503	2	the	the	DET
ejpam-4352	503	3	super	super	ADJ
ejpam-4352	503	4	domination	domination	NOUN
ejpam-4352	503	5	number	number	NOUN
ejpam-4352	503	6	of	of	ADP
ejpam-4352	503	7	lexicographic	lexicographic	ADJ
ejpam-4352	503	8	product	product	NOUN
ejpam-4352	503	9	of	of	ADP
ejpam-4352	503	10	graphs	graph	NOUN
ejpam-4352	503	11	.	.	PUNCT
ejpam-4352	504	1	arxiv:1703.06034v1	arxiv:1703.06034v1	PROPN
ejpam-4352	504	2	,	,	PUNCT
ejpam-4352	504	3	2017	2017	NUM
ejpam-4352	504	4	.	.	PUNCT
ejpam-4352	505	1	[	[	X
ejpam-4352	505	2	4	4	X
ejpam-4352	505	3	]	]	PUNCT
ejpam-4352	505	4	j.	j.	PROPN
ejpam-4352	505	5	hassan	hassan	PROPN
ejpam-4352	505	6	,	,	PUNCT
ejpam-4352	505	7	s.	s.	PROPN
ejpam-4352	505	8	canoy	canoy	PROPN
ejpam-4352	505	9	jr	jr	PROPN
ejpam-4352	505	10	.	.	PROPN
ejpam-4352	505	11	,	,	PUNCT
ejpam-4352	505	12	and	and	CCONJ
ejpam-4352	505	13	a.	a.	PROPN
ejpam-4352	505	14	aradais	aradais	PROPN
ejpam-4352	505	15	.	.	PUNCT
ejpam-4352	506	1	hop	hop	PROPN
ejpam-4352	506	2	independent	independent	ADJ
ejpam-4352	506	3	sets	set	NOUN
ejpam-4352	506	4	in	in	ADP
ejpam-4352	506	5	a	a	DET
ejpam-4352	506	6	graph	graph	NOUN
ejpam-4352	506	7	.	.	PUNCT
ejpam-4352	507	1	european	european	ADJ
ejpam-4352	507	2	journal	journal	PROPN
ejpam-4352	507	3	of	of	ADP
ejpam-4352	507	4	pure	pure	ADJ
ejpam-4352	507	5	and	and	CCONJ
ejpam-4352	507	6	applied	applied	ADJ
ejpam-4352	507	7	mathematics	mathematic	NOUN
ejpam-4352	507	8	,	,	PUNCT
ejpam-4352	507	9	accepted	accept	VERB
ejpam-4352	507	10	,	,	PUNCT
ejpam-4352	507	11	2022	2022	NUM
ejpam-4352	507	12	.	.	PUNCT
ejpam-4352	508	1	[	[	X
ejpam-4352	508	2	5	5	NUM
ejpam-4352	508	3	]	]	PUNCT
ejpam-4352	508	4	m.	m.	NOUN
ejpam-4352	508	5	henning	henning	PROPN
ejpam-4352	508	6	and	and	CCONJ
ejpam-4352	508	7	n.	n.	PROPN
ejpam-4352	508	8	rad	rad	PROPN
ejpam-4352	508	9	.	.	PROPN
ejpam-4352	509	1	on	on	ADP
ejpam-4352	509	2	2	2	NUM
ejpam-4352	509	3	-	-	PUNCT
ejpam-4352	509	4	step	step	NOUN
ejpam-4352	509	5	and	and	CCONJ
ejpam-4352	509	6	hop	hop	NOUN
ejpam-4352	509	7	dominating	dominating	NOUN
ejpam-4352	509	8	sets	set	NOUN
ejpam-4352	509	9	in	in	ADP
ejpam-4352	509	10	graphs	graph	NOUN
ejpam-4352	509	11	.	.	PUNCT
ejpam-4352	510	1	graphs	graph	NOUN
ejpam-4352	510	2	and	and	CCONJ
ejpam-4352	510	3	combinatorics	combinatoric	NOUN
ejpam-4352	510	4	,	,	PUNCT
ejpam-4352	510	5	33(4):913–927	33(4):913–927	PROPN
ejpam-4352	510	6	,	,	PUNCT
ejpam-4352	510	7	2017	2017	NUM
ejpam-4352	510	8	.	.	PUNCT
ejpam-4352	511	1	[	[	X
ejpam-4352	511	2	6	6	X
ejpam-4352	511	3	]	]	PUNCT
ejpam-4352	511	4	s.	s.	PROPN
ejpam-4352	511	5	canoy	canoy	PROPN
ejpam-4352	511	6	jr	jr	PROPN
ejpam-4352	511	7	,	,	PUNCT
ejpam-4352	511	8	r.	r.	PROPN
ejpam-4352	511	9	mollejon	mollejon	NOUN
ejpam-4352	511	10	,	,	PUNCT
ejpam-4352	511	11	and	and	CCONJ
ejpam-4352	511	12	j.	j.	PROPN
ejpam-4352	511	13	g.	g.	PROPN
ejpam-4352	511	14	canoy	canoy	PROPN
ejpam-4352	511	15	.	.	PUNCT
ejpam-4352	512	1	hop	hop	PROPN
ejpam-4352	512	2	dominating	dominating	NOUN
ejpam-4352	512	3	sets	set	NOUN
ejpam-4352	512	4	in	in	ADP
ejpam-4352	512	5	graphs	graph	NOUN
ejpam-4352	512	6	under	under	ADP
ejpam-4352	512	7	binary	binary	ADJ
ejpam-4352	512	8	operations	operation	NOUN
ejpam-4352	512	9	.	.	PUNCT
ejpam-4352	513	1	eur	eur	PROPN
ejpam-4352	513	2	.	.	PUNCT
ejpam-4352	514	1	j.	j.	PROPN
ejpam-4352	514	2	pure	pure	PROPN
ejpam-4352	514	3	appl	appl	PROPN
ejpam-4352	514	4	.	.	PUNCT
ejpam-4352	514	5	math	math	PROPN
ejpam-4352	514	6	.	.	PUNCT
ejpam-4352	514	7	,	,	PUNCT
ejpam-4352	515	1	12(4):1455–1463	12(4):1455–1463	NUM
ejpam-4352	515	2	,	,	PUNCT
ejpam-4352	515	3	2019	2019	NUM
ejpam-4352	515	4	.	.	PUNCT
ejpam-4352	516	1	[	[	X
ejpam-4352	516	2	7	7	X
ejpam-4352	516	3	]	]	X
ejpam-4352	516	4	s.	s.	PROPN
ejpam-4352	516	5	canoy	canoy	PROPN
ejpam-4352	516	6	jr	jr	PROPN
ejpam-4352	516	7	and	and	CCONJ
ejpam-4352	516	8	g.	g.	PROPN
ejpam-4352	516	9	salasalan	salasalan	NOUN
ejpam-4352	516	10	.	.	PUNCT
ejpam-4352	517	1	locating	locate	VERB
ejpam-4352	517	2	-	-	PUNCT
ejpam-4352	517	3	hop	hop	NOUN
ejpam-4352	517	4	domination	domination	NOUN
ejpam-4352	517	5	in	in	ADP
ejpam-4352	517	6	graphs	graph	NOUN
ejpam-4352	517	7	.	.	PUNCT
ejpam-4352	518	1	kyungpook	kyungpook	PROPN
ejpam-4352	518	2	mathematical	mathematical	PROPN
ejpam-4352	518	3	journal	journal	PROPN
ejpam-4352	518	4	,	,	PUNCT
ejpam-4352	518	5	62:193–204	62:193–204	PROPN
ejpam-4352	518	6	,	,	PUNCT
ejpam-4352	518	7	2022	2022	NUM
ejpam-4352	518	8	.	.	PUNCT
ejpam-4352	519	1	references	reference	NOUN
ejpam-4352	519	2	353	353	NUM
ejpam-4352	520	1	[	[	X
ejpam-4352	520	2	8	8	NUM
ejpam-4352	520	3	]	]	X
ejpam-4352	520	4	m.	m.	NOUN
ejpam-4352	520	5	lemanska	lemanska	PROPN
ejpam-4352	520	6	,	,	PUNCT
ejpam-4352	520	7	v.	v.	ADP
ejpam-4352	520	8	swaminathan	swaminathan	ADV
ejpam-4352	520	9	,	,	PUNCT
ejpam-4352	520	10	y.b	y.b	PROPN
ejpam-4352	520	11	.	.	PROPN
ejpam-4352	520	12	venkatakrishnan	venkatakrishnan	NOUN
ejpam-4352	520	13	,	,	PUNCT
ejpam-4352	520	14	and	and	CCONJ
ejpam-4352	520	15	r.	r.	PROPN
ejpam-4352	520	16	zuazua	zuazua	PROPN
ejpam-4352	520	17	.	.	PUNCT
ejpam-4352	521	1	super	super	ADJ
ejpam-4352	521	2	dominating	dominating	NOUN
ejpam-4352	521	3	sets	set	NOUN
ejpam-4352	521	4	in	in	ADP
ejpam-4352	521	5	graphs	graph	NOUN
ejpam-4352	521	6	.	.	PUNCT
ejpam-4352	522	1	proceedings	proceeding	NOUN
ejpam-4352	522	2	of	of	ADP
ejpam-4352	522	3	the	the	DET
ejpam-4352	522	4	national	national	PROPN
ejpam-4352	522	5	academy	academy	PROPN
ejpam-4352	522	6	of	of	ADP
ejpam-4352	522	7	sciences	sciences	PROPN
ejpam-4352	522	8	,	,	PUNCT
ejpam-4352	522	9	india	india	PROPN
ejpam-4352	522	10	section	section	PROPN
ejpam-4352	522	11	a	a	PRON
ejpam-4352	522	12	:	:	PUNCT
ejpam-4352	522	13	physical	physical	ADJ
ejpam-4352	522	14	sciences	science	NOUN
ejpam-4352	522	15	,	,	PUNCT
ejpam-4352	522	16	85(3):353–357	85(3):353–357	NOUN
ejpam-4352	522	17	,	,	PUNCT
ejpam-4352	522	18	2015	2015	NUM
ejpam-4352	522	19	.	.	PUNCT
ejpam-4352	523	1	[	[	X
ejpam-4352	523	2	9	9	NUM
ejpam-4352	523	3	]	]	X
ejpam-4352	523	4	r.	r.	NOUN
ejpam-4352	523	5	mollejon	mollejon	NOUN
ejpam-4352	523	6	and	and	CCONJ
ejpam-4352	523	7	s.	s.	PROPN
ejpam-4352	523	8	canoy	canoy	PROPN
ejpam-4352	523	9	jr	jr	PROPN
ejpam-4352	523	10	.	.	PROPN
ejpam-4352	523	11	hop	hop	PROPN
ejpam-4352	523	12	dominating	dominating	NOUN
ejpam-4352	523	13	sets	set	NOUN
ejpam-4352	523	14	in	in	ADP
ejpam-4352	523	15	graphs	graph	NOUN
ejpam-4352	523	16	under	under	ADP
ejpam-4352	523	17	binary	binary	ADJ
ejpam-4352	523	18	operations	operation	NOUN
ejpam-4352	523	19	.	.	PUNCT
ejpam-4352	524	1	european	european	ADJ
ejpam-4352	524	2	journal	journal	PROPN
ejpam-4352	524	3	of	of	ADP
ejpam-4352	524	4	pure	pure	ADJ
ejpam-4352	524	5	and	and	CCONJ
ejpam-4352	524	6	applied	applied	ADJ
ejpam-4352	524	7	mathematics	mathematic	NOUN
ejpam-4352	524	8	,	,	PUNCT
ejpam-4352	524	9	12(4):1455–1463	12(4):1455–1463	NUM
ejpam-4352	524	10	,	,	PUNCT
ejpam-4352	524	11	2019	2019	NUM
ejpam-4352	524	12	.	.	PUNCT
ejpam-4352	525	1	[	[	X
ejpam-4352	525	2	10	10	NUM
ejpam-4352	525	3	]	]	X
ejpam-4352	525	4	r.	r.	NOUN
ejpam-4352	525	5	mollejon	mollejon	NOUN
ejpam-4352	525	6	and	and	CCONJ
ejpam-4352	525	7	s.	s.	PROPN
ejpam-4352	525	8	canoy	canoy	PROPN
ejpam-4352	525	9	jr	jr	PROPN
ejpam-4352	525	10	.	.	PROPN
ejpam-4352	525	11	double	double	ADJ
ejpam-4352	525	12	hop	hop	NOUN
ejpam-4352	525	13	dominating	dominating	NOUN
ejpam-4352	525	14	sets	set	NOUN
ejpam-4352	525	15	in	in	ADP
ejpam-4352	525	16	a	a	DET
ejpam-4352	525	17	graph	graph	NOUN
ejpam-4352	525	18	.	.	PUNCT
ejpam-4352	526	1	discrete	discrete	ADJ
ejpam-4352	526	2	mathematics	mathematic	NOUN
ejpam-4352	526	3	,	,	PUNCT
ejpam-4352	526	4	algorithms	algorithm	NOUN
ejpam-4352	526	5	and	and	CCONJ
ejpam-4352	526	6	applications	application	NOUN
ejpam-4352	526	7	,	,	PUNCT
ejpam-4352	526	8	https://doi.org/10.1142/s1793830921500579	https://doi.org/10.1142/s1793830921500579	NUM
ejpam-4352	526	9	,	,	PUNCT
ejpam-4352	526	10	2021	2021	NUM
ejpam-4352	526	11	.	.	PUNCT
ejpam-4352	527	1	[	[	X
ejpam-4352	527	2	11	11	NUM
ejpam-4352	527	3	]	]	X
ejpam-4352	527	4	c.	c.	PROPN
ejpam-4352	527	5	natarajan	natarajan	PROPN
ejpam-4352	527	6	and	and	CCONJ
ejpam-4352	527	7	s.	s.	PROPN
ejpam-4352	527	8	ayyaswamy	ayyaswamy	PROPN
ejpam-4352	527	9	.	.	PUNCT
ejpam-4352	528	1	hop	hop	PROPN
ejpam-4352	528	2	domination	domination	NOUN
ejpam-4352	528	3	in	in	ADP
ejpam-4352	528	4	graphs	graphs	PROPN
ejpam-4352	528	5	ii	ii	PROPN
ejpam-4352	528	6	.	.	PUNCT
ejpam-4352	528	7	versita	versita	PROPN
ejpam-4352	528	8	,	,	PUNCT
ejpam-4352	528	9	102(10):2393	102(10):2393	PROPN
ejpam-4352	528	10	–	–	PUNCT
ejpam-4352	528	11	2401	2401	NUM
ejpam-4352	528	12	,	,	PUNCT
ejpam-4352	528	13	2017	2017	NUM
ejpam-4352	528	14	.	.	PUNCT
ejpam-4352	529	1	[	[	X
ejpam-4352	529	2	12	12	NUM
ejpam-4352	529	3	]	]	X
ejpam-4352	529	4	y.	y.	PROPN
ejpam-4352	529	5	pabilona	pabilona	PROPN
ejpam-4352	529	6	and	and	CCONJ
ejpam-4352	529	7	h.	h.	PROPN
ejpam-4352	529	8	rara	rara	PROPN
ejpam-4352	529	9	.	.	PUNCT
ejpam-4352	530	1	connected	connect	VERB
ejpam-4352	530	2	hop	hop	NOUN
ejpam-4352	530	3	domination	domination	NOUN
ejpam-4352	530	4	in	in	ADP
ejpam-4352	530	5	graphs	graph	NOUN
ejpam-4352	530	6	under	under	ADP
ejpam-4352	530	7	some	some	DET
ejpam-4352	530	8	binary	binary	ADJ
ejpam-4352	530	9	operations	operation	NOUN
ejpam-4352	530	10	.	.	PUNCT
ejpam-4352	531	1	asian	asian	ADJ
ejpam-4352	531	2	-	-	PUNCT
ejpam-4352	531	3	european	european	ADJ
ejpam-4352	531	4	journal	journal	NOUN
ejpam-4352	531	5	of	of	ADP
ejpam-4352	531	6	mathematics	mathematics	PROPN
ejpam-4352	531	7	,	,	PUNCT
ejpam-4352	531	8	11(5):1850075–1–18500075–11	11(5):1850075–1–18500075–11	NUM
ejpam-4352	531	9	,	,	PUNCT
ejpam-4352	531	10	2018	2018	NUM
ejpam-4352	531	11	.	.	PUNCT
ejpam-4352	532	1	[	[	X
ejpam-4352	532	2	13	13	NUM
ejpam-4352	532	3	]	]	PUNCT
ejpam-4352	532	4	s.	s.	PROPN
ejpam-4352	532	5	paraico	paraico	PROPN
ejpam-4352	532	6	and	and	CCONJ
ejpam-4352	532	7	s.	s.	PROPN
ejpam-4352	532	8	canoy	canoy	PROPN
ejpam-4352	532	9	jr	jr	PROPN
ejpam-4352	532	10	.	.	PUNCT
ejpam-4352	532	11	super	super	ADJ
ejpam-4352	532	12	dominating	dominating	NOUN
ejpam-4352	532	13	sets	set	NOUN
ejpam-4352	532	14	in	in	ADP
ejpam-4352	532	15	some	some	DET
ejpam-4352	532	16	products	product	NOUN
ejpam-4352	532	17	of	of	ADP
ejpam-4352	532	18	graphs	graph	NOUN
ejpam-4352	532	19	.	.	PUNCT
ejpam-4352	533	1	far	far	PROPN
ejpam-4352	533	2	east	east	PROPN
ejpam-4352	533	3	journal	journal	PROPN
ejpam-4352	533	4	of	of	ADP
ejpam-4352	533	5	mathematical	mathematical	ADJ
ejpam-4352	533	6	sciences	sciences	PROPN
ejpam-4352	533	7	,	,	PUNCT
ejpam-4352	533	8	23(2):187–199	23(2):187–199	NUM
ejpam-4352	533	9	,	,	PUNCT
ejpam-4352	533	10	2015	2015	NUM
ejpam-4352	533	11	.	.	PUNCT
ejpam-4352	534	1	[	[	X
ejpam-4352	534	2	14	14	NUM
ejpam-4352	534	3	]	]	X
ejpam-4352	534	4	g.	g.	NOUN
ejpam-4352	534	5	salasalan	salasalan	NOUN
ejpam-4352	534	6	and	and	CCONJ
ejpam-4352	534	7	s.	s.	PROPN
ejpam-4352	534	8	canoy	canoy	PROPN
ejpam-4352	534	9	jr	jr	PROPN
ejpam-4352	534	10	.	.	PROPN
ejpam-4352	534	11	global	global	PROPN
ejpam-4352	534	12	hop	hop	PROPN
ejpam-4352	534	13	domination	domination	PROPN
ejpam-4352	534	14	numbers	number	NOUN
ejpam-4352	534	15	of	of	ADP
ejpam-4352	534	16	graphs	graph	NOUN
ejpam-4352	534	17	.	.	PUNCT
ejpam-4352	535	1	eur	eur	PROPN
ejpam-4352	535	2	.	.	PUNCT
ejpam-4352	536	1	j.	j.	PROPN
ejpam-4352	536	2	pure	pure	PROPN
ejpam-4352	536	3	appl	appl	PROPN
ejpam-4352	536	4	.	.	PUNCT
ejpam-4352	536	5	math	math	PROPN
ejpam-4352	536	6	.	.	PUNCT
ejpam-4352	536	7	,	,	PUNCT
ejpam-4352	536	8	14(1):112–125	14(1):112–125	NUM
ejpam-4352	536	9	,	,	PUNCT
ejpam-4352	536	10	2021	2021	NUM
ejpam-4352	536	11	.	.	PUNCT