id	sid	tid	token	lemma	pos
ejpam-4914	1	1	european	european	PROPN
ejpam-4914	1	2	journal	journal	PROPN
ejpam-4914	1	3	of	of	ADP
ejpam-4914	1	4	pure	pure	ADJ
ejpam-4914	1	5	and	and	CCONJ
ejpam-4914	1	6	applied	apply	VERB
ejpam-4914	1	7	mathematics	mathematic	NOUN
ejpam-4914	1	8	vol	vol	NOUN
ejpam-4914	1	9	.	.	PUNCT
ejpam-4914	2	1	16	16	NUM
ejpam-4914	2	2	,	,	PUNCT
ejpam-4914	2	3	no	no	INTJ
ejpam-4914	2	4	.	.	NOUN
ejpam-4914	2	5	4	4	NUM
ejpam-4914	2	6	,	,	PUNCT
ejpam-4914	2	7	2023	2023	NUM
ejpam-4914	2	8	,	,	PUNCT
ejpam-4914	2	9	2431	2431	NUM
ejpam-4914	2	10	-	-	SYM
ejpam-4914	2	11	2449	2449	NUM
ejpam-4914	2	12	issn	issn	PROPN
ejpam-4914	2	13	1307	1307	NUM
ejpam-4914	2	14	-	-	SYM
ejpam-4914	2	15	5543	5543	NUM
ejpam-4914	2	16	–	–	PUNCT
ejpam-4914	3	1	ejpam.com	ejpam.com	X
ejpam-4914	3	2	published	publish	VERB
ejpam-4914	3	3	by	by	ADP
ejpam-4914	3	4	new	new	PROPN
ejpam-4914	3	5	york	york	PROPN
ejpam-4914	3	6	business	business	PROPN
ejpam-4914	3	7	global	global	PROPN
ejpam-4914	3	8	hop	hop	PROPN
ejpam-4914	3	9	italian	italian	ADJ
ejpam-4914	3	10	domination	domination	NOUN
ejpam-4914	3	11	in	in	ADP
ejpam-4914	3	12	graphs	graph	NOUN
ejpam-4914	3	13	sergio	sergio	PROPN
ejpam-4914	3	14	r.	r.	PROPN
ejpam-4914	3	15	canoy	canoy	PROPN
ejpam-4914	3	16	,	,	PUNCT
ejpam-4914	3	17	jr.1,2	jr.1,2	PROPN
ejpam-4914	3	18	,	,	PUNCT
ejpam-4914	3	19	ferdinand	ferdinand	PROPN
ejpam-4914	3	20	p.	p.	PROPN
ejpam-4914	4	1	jamil1,2	jamil1,2	PROPN
ejpam-4914	4	2	and	and	CCONJ
ejpam-4914	4	3	sheila	sheila	PROPN
ejpam-4914	4	4	m.	m.	PROPN
ejpam-4914	4	5	menchavez1,2,∗	menchavez1,2,∗	PROPN
ejpam-4914	4	6	1	1	NUM
ejpam-4914	4	7	department	department	NOUN
ejpam-4914	4	8	of	of	ADP
ejpam-4914	4	9	mathematics	mathematic	NOUN
ejpam-4914	4	10	and	and	CCONJ
ejpam-4914	4	11	statistics	statistic	NOUN
ejpam-4914	4	12	,	,	PUNCT
ejpam-4914	4	13	college	college	NOUN
ejpam-4914	4	14	of	of	ADP
ejpam-4914	4	15	science	science	NOUN
ejpam-4914	4	16	and	and	CCONJ
ejpam-4914	4	17	mathematics	mathematic	NOUN
ejpam-4914	4	18	,	,	PUNCT
ejpam-4914	4	19	mindanao	mindanao	PROPN
ejpam-4914	4	20	state	state	PROPN
ejpam-4914	4	21	university	university	PROPN
ejpam-4914	4	22	-	-	PUNCT
ejpam-4914	4	23	iligan	iligan	PROPN
ejpam-4914	4	24	institute	institute	PROPN
ejpam-4914	4	25	of	of	ADP
ejpam-4914	4	26	technology	technology	PROPN
ejpam-4914	4	27	,	,	PUNCT
ejpam-4914	4	28	9200	9200	NUM
ejpam-4914	4	29	iligan	iligan	ADJ
ejpam-4914	4	30	city	city	NOUN
ejpam-4914	4	31	,	,	PUNCT
ejpam-4914	4	32	philippines	philippine	NOUN
ejpam-4914	4	33	2	2	NUM
ejpam-4914	4	34	center	center	NOUN
ejpam-4914	4	35	for	for	ADP
ejpam-4914	4	36	graph	graph	NOUN
ejpam-4914	4	37	theory	theory	NOUN
ejpam-4914	4	38	,	,	PUNCT
ejpam-4914	4	39	algebra	algebra	NOUN
ejpam-4914	4	40	and	and	CCONJ
ejpam-4914	4	41	analysis	analysis	NOUN
ejpam-4914	4	42	,	,	PUNCT
ejpam-4914	4	43	premier	premier	PROPN
ejpam-4914	4	44	research	research	PROPN
ejpam-4914	4	45	institute	institute	PROPN
ejpam-4914	4	46	of	of	ADP
ejpam-4914	4	47	science	science	NOUN
ejpam-4914	4	48	and	and	CCONJ
ejpam-4914	4	49	mathematics	mathematics	PROPN
ejpam-4914	4	50	(	(	PUNCT
ejpam-4914	4	51	prism	prism	NOUN
ejpam-4914	4	52	)	)	PUNCT
ejpam-4914	4	53	,	,	PUNCT
ejpam-4914	4	54	mindanao	mindanao	PROPN
ejpam-4914	4	55	state	state	PROPN
ejpam-4914	4	56	university	university	PROPN
ejpam-4914	4	57	iligan	iligan	PROPN
ejpam-4914	4	58	institute	institute	PROPN
ejpam-4914	4	59	of	of	ADP
ejpam-4914	4	60	technology	technology	PROPN
ejpam-4914	4	61	,	,	PUNCT
ejpam-4914	4	62	9200	9200	NUM
ejpam-4914	4	63	iligan	iligan	ADJ
ejpam-4914	4	64	city	city	NOUN
ejpam-4914	4	65	,	,	PUNCT
ejpam-4914	4	66	philippines	philippine	NOUN
ejpam-4914	4	67	abstract	abstract	ADJ
ejpam-4914	4	68	.	.	PUNCT
ejpam-4914	5	1	given	give	VERB
ejpam-4914	5	2	a	a	DET
ejpam-4914	5	3	simple	simple	ADJ
ejpam-4914	5	4	graph	graph	NOUN
ejpam-4914	5	5	g	g	PROPN
ejpam-4914	5	6	=	=	PUNCT
ejpam-4914	5	7	(	(	PUNCT
ejpam-4914	5	8	v	v	NOUN
ejpam-4914	5	9	(	(	PUNCT
ejpam-4914	5	10	g	g	NOUN
ejpam-4914	5	11	)	)	PUNCT
ejpam-4914	5	12	,	,	PUNCT
ejpam-4914	5	13	e(g	e(g	PROPN
ejpam-4914	5	14	)	)	PUNCT
ejpam-4914	5	15	)	)	PUNCT
ejpam-4914	5	16	,	,	PUNCT
ejpam-4914	5	17	a	a	DET
ejpam-4914	5	18	function	function	NOUN
ejpam-4914	5	19	f	f	NOUN
ejpam-4914	5	20	:	:	PUNCT
ejpam-4914	5	21	v	v	X
ejpam-4914	5	22	(	(	PUNCT
ejpam-4914	5	23	g	g	NOUN
ejpam-4914	5	24	)	)	PUNCT
ejpam-4914	5	25	→	→	SYM
ejpam-4914	5	26	{	{	PUNCT
ejpam-4914	5	27	0	0	NUM
ejpam-4914	5	28	,	,	PUNCT
ejpam-4914	5	29	1	1	NUM
ejpam-4914	5	30	,	,	PUNCT
ejpam-4914	5	31	2	2	NUM
ejpam-4914	5	32	}	}	PUNCT
ejpam-4914	5	33	is	be	AUX
ejpam-4914	5	34	a	a	DET
ejpam-4914	5	35	hop	hop	NOUN
ejpam-4914	5	36	italian	italian	ADJ
ejpam-4914	5	37	dominating	dominating	NOUN
ejpam-4914	5	38	function	function	NOUN
ejpam-4914	5	39	if	if	SCONJ
ejpam-4914	5	40	for	for	ADP
ejpam-4914	5	41	every	every	DET
ejpam-4914	5	42	vertex	vertex	NOUN
ejpam-4914	5	43	v	v	NOUN
ejpam-4914	5	44	with	with	ADP
ejpam-4914	5	45	f(v	f(v	NOUN
ejpam-4914	5	46	)	)	PUNCT
ejpam-4914	6	1	=	=	PUNCT
ejpam-4914	6	2	0	0	NUM
ejpam-4914	7	1	there	there	PRON
ejpam-4914	7	2	exists	exist	VERB
ejpam-4914	7	3	a	a	DET
ejpam-4914	7	4	vertex	vertex	NOUN
ejpam-4914	7	5	u	u	NOUN
ejpam-4914	7	6	with	with	ADP
ejpam-4914	7	7	f(u	f(u	PROPN
ejpam-4914	7	8	)	)	PUNCT
ejpam-4914	7	9	=	=	SYM
ejpam-4914	7	10	2	2	NUM
ejpam-4914	7	11	for	for	ADP
ejpam-4914	7	12	which	which	PRON
ejpam-4914	7	13	u	u	NOUN
ejpam-4914	7	14	and	and	CCONJ
ejpam-4914	7	15	v	v	NOUN
ejpam-4914	7	16	are	be	AUX
ejpam-4914	7	17	of	of	ADP
ejpam-4914	7	18	distance	distance	NOUN
ejpam-4914	7	19	2	2	NUM
ejpam-4914	7	20	from	from	ADP
ejpam-4914	7	21	each	each	DET
ejpam-4914	7	22	other	other	ADJ
ejpam-4914	7	23	or	or	CCONJ
ejpam-4914	7	24	there	there	ADV
ejpam-4914	7	25	exist	exist	VERB
ejpam-4914	7	26	two	two	NUM
ejpam-4914	7	27	vertices	vertex	NOUN
ejpam-4914	7	28	w	w	NOUN
ejpam-4914	7	29	and	and	CCONJ
ejpam-4914	7	30	z	z	NOUN
ejpam-4914	7	31	for	for	ADP
ejpam-4914	7	32	which	which	PRON
ejpam-4914	7	33	f(w	f(w	VERB
ejpam-4914	7	34	)	)	PUNCT
ejpam-4914	7	35	=	=	SYM
ejpam-4914	7	36	1	1	NUM
ejpam-4914	7	37	=	=	SYM
ejpam-4914	7	38	f(z	f(z	PROPN
ejpam-4914	7	39	)	)	PUNCT
ejpam-4914	7	40	and	and	CCONJ
ejpam-4914	7	41	each	each	PRON
ejpam-4914	7	42	of	of	ADP
ejpam-4914	7	43	w	w	PROPN
ejpam-4914	7	44	and	and	CCONJ
ejpam-4914	7	45	z	z	NOUN
ejpam-4914	7	46	is	be	AUX
ejpam-4914	7	47	of	of	ADP
ejpam-4914	7	48	distance	distance	NOUN
ejpam-4914	7	49	2	2	NUM
ejpam-4914	7	50	from	from	ADP
ejpam-4914	7	51	v.	v.	ADP
ejpam-4914	7	52	the	the	DET
ejpam-4914	7	53	minimum	minimum	ADJ
ejpam-4914	7	54	weight∑	weight∑	PROPN
ejpam-4914	7	55	v∈v	v∈v	NOUN
ejpam-4914	7	56	(	(	PUNCT
ejpam-4914	7	57	g	g	NOUN
ejpam-4914	7	58	)	)	PUNCT
ejpam-4914	7	59	f(v	f(v	NOUN
ejpam-4914	7	60	)	)	PUNCT
ejpam-4914	7	61	of	of	ADP
ejpam-4914	7	62	a	a	DET
ejpam-4914	7	63	hop	hop	NOUN
ejpam-4914	7	64	italian	italian	ADJ
ejpam-4914	7	65	dominating	dominating	NOUN
ejpam-4914	7	66	function	function	NOUN
ejpam-4914	7	67	is	be	AUX
ejpam-4914	7	68	the	the	DET
ejpam-4914	7	69	hop	hop	NOUN
ejpam-4914	7	70	italian	italian	ADJ
ejpam-4914	7	71	domination	domination	NOUN
ejpam-4914	7	72	number	number	NOUN
ejpam-4914	7	73	of	of	ADP
ejpam-4914	7	74	g	g	NOUN
ejpam-4914	7	75	,	,	PUNCT
ejpam-4914	7	76	and	and	CCONJ
ejpam-4914	7	77	is	be	AUX
ejpam-4914	7	78	denoted	denote	VERB
ejpam-4914	7	79	by	by	ADP
ejpam-4914	7	80	γhi(g	γhi(g	PROPN
ejpam-4914	7	81	)	)	PUNCT
ejpam-4914	7	82	.	.	PUNCT
ejpam-4914	8	1	in	in	ADP
ejpam-4914	8	2	this	this	DET
ejpam-4914	8	3	paper	paper	NOUN
ejpam-4914	8	4	,	,	PUNCT
ejpam-4914	8	5	we	we	PRON
ejpam-4914	8	6	initiate	initiate	VERB
ejpam-4914	8	7	the	the	DET
ejpam-4914	8	8	study	study	NOUN
ejpam-4914	8	9	of	of	ADP
ejpam-4914	8	10	the	the	DET
ejpam-4914	8	11	hop	hop	NOUN
ejpam-4914	8	12	italian	italian	ADJ
ejpam-4914	8	13	domination	domination	NOUN
ejpam-4914	8	14	.	.	PUNCT
ejpam-4914	9	1	first	first	ADV
ejpam-4914	9	2	,	,	PUNCT
ejpam-4914	9	3	we	we	PRON
ejpam-4914	9	4	establish	establish	VERB
ejpam-4914	9	5	some	some	DET
ejpam-4914	9	6	properties	property	NOUN
ejpam-4914	9	7	of	of	ADP
ejpam-4914	9	8	the	the	DET
ejpam-4914	9	9	the	the	DET
ejpam-4914	9	10	hop	hop	NOUN
ejpam-4914	9	11	italian	italian	ADJ
ejpam-4914	9	12	dominating	dominating	NOUN
ejpam-4914	9	13	function	function	NOUN
ejpam-4914	9	14	and	and	CCONJ
ejpam-4914	9	15	characterize	characterize	VERB
ejpam-4914	9	16	graphs	graph	NOUN
ejpam-4914	9	17	g	g	NOUN
ejpam-4914	9	18	with	with	ADP
ejpam-4914	9	19	smaller	small	ADJ
ejpam-4914	9	20	values	value	NOUN
ejpam-4914	9	21	for	for	ADP
ejpam-4914	9	22	γhi(g	γhi(g	PROPN
ejpam-4914	9	23	)	)	PUNCT
ejpam-4914	9	24	.	.	PUNCT
ejpam-4914	10	1	next	next	ADV
ejpam-4914	10	2	,	,	PUNCT
ejpam-4914	10	3	we	we	PRON
ejpam-4914	10	4	explore	explore	VERB
ejpam-4914	10	5	the	the	DET
ejpam-4914	10	6	relationships	relationship	NOUN
ejpam-4914	10	7	of	of	ADP
ejpam-4914	10	8	the	the	DET
ejpam-4914	10	9	hop	hop	NOUN
ejpam-4914	10	10	italian	italian	ADJ
ejpam-4914	10	11	domination	domination	NOUN
ejpam-4914	10	12	number	number	NOUN
ejpam-4914	10	13	with	with	ADP
ejpam-4914	10	14	closely	closely	ADV
ejpam-4914	10	15	related	relate	VERB
ejpam-4914	10	16	concepts	concept	NOUN
ejpam-4914	10	17	,	,	PUNCT
ejpam-4914	10	18	particularly	particularly	ADV
ejpam-4914	10	19	the	the	DET
ejpam-4914	10	20	hop	hop	NOUN
ejpam-4914	10	21	roman	roman	ADJ
ejpam-4914	10	22	domination	domination	NOUN
ejpam-4914	10	23	number	number	NOUN
ejpam-4914	10	24	and	and	CCONJ
ejpam-4914	10	25	the	the	DET
ejpam-4914	10	26	2hop	2hop	NUM
ejpam-4914	10	27	domination	domination	NOUN
ejpam-4914	10	28	number	number	NOUN
ejpam-4914	10	29	.	.	PUNCT
ejpam-4914	11	1	finally	finally	ADV
ejpam-4914	11	2	,	,	PUNCT
ejpam-4914	11	3	we	we	PRON
ejpam-4914	11	4	investigate	investigate	VERB
ejpam-4914	11	5	the	the	DET
ejpam-4914	11	6	hop	hop	NOUN
ejpam-4914	11	7	italian	italian	ADJ
ejpam-4914	11	8	domination	domination	NOUN
ejpam-4914	11	9	in	in	ADP
ejpam-4914	11	10	the	the	DET
ejpam-4914	11	11	complementary	complementary	ADJ
ejpam-4914	11	12	prism	prism	NOUN
ejpam-4914	11	13	,	,	PUNCT
ejpam-4914	11	14	join	join	NOUN
ejpam-4914	11	15	,	,	PUNCT
ejpam-4914	11	16	corona	corona	NOUN
ejpam-4914	11	17	and	and	CCONJ
ejpam-4914	11	18	lexicographic	lexicographic	ADJ
ejpam-4914	11	19	product	product	NOUN
ejpam-4914	11	20	of	of	ADP
ejpam-4914	11	21	graphs	graph	NOUN
ejpam-4914	11	22	.	.	PUNCT
ejpam-4914	12	1	2020	2020	NUM
ejpam-4914	12	2	mathematics	mathematic	NOUN
ejpam-4914	12	3	subject	subject	NOUN
ejpam-4914	12	4	classifications	classification	NOUN
ejpam-4914	12	5	:	:	PUNCT
ejpam-4914	12	6	05c69	05c69	X
ejpam-4914	12	7	key	key	ADJ
ejpam-4914	12	8	words	word	NOUN
ejpam-4914	12	9	and	and	CCONJ
ejpam-4914	12	10	phrases	phrase	NOUN
ejpam-4914	12	11	:	:	PUNCT
ejpam-4914	12	12	hop	hop	NOUN
ejpam-4914	12	13	italian	italian	ADJ
ejpam-4914	12	14	dominating	dominating	NOUN
ejpam-4914	12	15	function	function	NOUN
ejpam-4914	12	16	,	,	PUNCT
ejpam-4914	12	17	hop	hop	NOUN
ejpam-4914	12	18	italian	italian	ADJ
ejpam-4914	12	19	domination	domination	NOUN
ejpam-4914	12	20	number	number	NOUN
ejpam-4914	12	21	1	1	NUM
ejpam-4914	12	22	.	.	PUNCT
ejpam-4914	12	23	introduction	introduction	NOUN
ejpam-4914	12	24	the	the	DET
ejpam-4914	12	25	history	history	NOUN
ejpam-4914	12	26	of	of	ADP
ejpam-4914	12	27	the	the	DET
ejpam-4914	12	28	roman	roman	ADJ
ejpam-4914	12	29	domination	domination	NOUN
ejpam-4914	12	30	in	in	ADP
ejpam-4914	12	31	graphs	graph	NOUN
ejpam-4914	12	32	can	can	AUX
ejpam-4914	12	33	be	be	AUX
ejpam-4914	12	34	traced	trace	VERB
ejpam-4914	12	35	back	back	ADV
ejpam-4914	12	36	to	to	ADP
ejpam-4914	12	37	the	the	DET
ejpam-4914	12	38	military	military	ADJ
ejpam-4914	12	39	strategy	strategy	NOUN
ejpam-4914	12	40	adapted	adapt	VERB
ejpam-4914	12	41	by	by	ADP
ejpam-4914	12	42	constantine	constantine	PROPN
ejpam-4914	12	43	the	the	DET
ejpam-4914	12	44	great	great	ADJ
ejpam-4914	12	45	(	(	PUNCT
ejpam-4914	12	46	emperor	emperor	NOUN
ejpam-4914	12	47	of	of	ADP
ejpam-4914	12	48	rome	rome	PROPN
ejpam-4914	12	49	)	)	PUNCT
ejpam-4914	12	50	during	during	ADP
ejpam-4914	12	51	the	the	DET
ejpam-4914	12	52	fourth	fourth	ADJ
ejpam-4914	12	53	century	century	NOUN
ejpam-4914	12	54	ad	ad	NOUN
ejpam-4914	12	55	(	(	PUNCT
ejpam-4914	12	56	see	see	VERB
ejpam-4914	12	57	[	[	X
ejpam-4914	12	58	23	23	NUM
ejpam-4914	12	59	,	,	PUNCT
ejpam-4914	12	60	26	26	NUM
ejpam-4914	12	61	]	]	PUNCT
ejpam-4914	12	62	)	)	PUNCT
ejpam-4914	12	63	.	.	PUNCT
ejpam-4914	13	1	in	in	ADP
ejpam-4914	13	2	order	order	NOUN
ejpam-4914	13	3	to	to	PART
ejpam-4914	13	4	defend	defend	VERB
ejpam-4914	13	5	his	his	PRON
ejpam-4914	13	6	cities	city	NOUN
ejpam-4914	13	7	constantine	constantine	PROPN
ejpam-4914	13	8	issued	issue	VERB
ejpam-4914	13	9	a	a	DET
ejpam-4914	13	10	decree	decree	NOUN
ejpam-4914	13	11	that	that	PRON
ejpam-4914	13	12	any	any	DET
ejpam-4914	13	13	city	city	NOUN
ejpam-4914	13	14	without	without	ADP
ejpam-4914	13	15	a	a	DET
ejpam-4914	13	16	legion	legion	NOUN
ejpam-4914	13	17	stationed	station	VERB
ejpam-4914	13	18	to	to	PART
ejpam-4914	13	19	secure	secure	VERB
ejpam-4914	13	20	it	it	PRON
ejpam-4914	13	21	must	must	AUX
ejpam-4914	13	22	neighbor	neighbor	VERB
ejpam-4914	13	23	another	another	DET
ejpam-4914	13	24	city	city	NOUN
ejpam-4914	13	25	having	have	VERB
ejpam-4914	13	26	two	two	NUM
ejpam-4914	13	27	stationed	station	VERB
ejpam-4914	13	28	legions	legion	NOUN
ejpam-4914	13	29	.	.	PUNCT
ejpam-4914	14	1	if	if	SCONJ
ejpam-4914	14	2	the	the	DET
ejpam-4914	14	3	first	first	ADJ
ejpam-4914	14	4	were	be	AUX
ejpam-4914	14	5	attacked	attack	VERB
ejpam-4914	14	6	,	,	PUNCT
ejpam-4914	14	7	then	then	ADV
ejpam-4914	14	8	the	the	DET
ejpam-4914	14	9	second	second	NOUN
ejpam-4914	14	10	could	could	AUX
ejpam-4914	14	11	deploy	deploy	VERB
ejpam-4914	14	12	a	a	DET
ejpam-4914	14	13	legion	legion	NOUN
ejpam-4914	14	14	to	to	PART
ejpam-4914	14	15	protect	protect	VERB
ejpam-4914	14	16	it	it	PRON
ejpam-4914	14	17	without	without	ADP
ejpam-4914	14	18	becoming	become	VERB
ejpam-4914	14	19	vulnerable	vulnerable	ADJ
ejpam-4914	14	20	itself	itself	PRON
ejpam-4914	14	21	.	.	PUNCT
ejpam-4914	15	1	it	it	PRON
ejpam-4914	15	2	is	be	AUX
ejpam-4914	15	3	called	call	VERB
ejpam-4914	15	4	defense	defense	NOUN
ejpam-4914	15	5	-	-	PUNCT
ejpam-4914	15	6	in	in	ADP
ejpam-4914	15	7	-	-	PUNCT
ejpam-4914	15	8	depth	depth	NOUN
ejpam-4914	15	9	strategy	strategy	NOUN
ejpam-4914	15	10	,	,	PUNCT
ejpam-4914	15	11	which	which	PRON
ejpam-4914	15	12	used	use	VERB
ejpam-4914	15	13	only	only	ADV
ejpam-4914	15	14	four	four	NUM
ejpam-4914	15	15	field	field	NOUN
ejpam-4914	15	16	armies	army	NOUN
ejpam-4914	15	17	(	(	PUNCT
ejpam-4914	15	18	fa	fa	NOUN
ejpam-4914	15	19	)	)	PUNCT
ejpam-4914	15	20	available	available	ADJ
ejpam-4914	15	21	for	for	SCONJ
ejpam-4914	15	22	deployment	deployment	NOUN
ejpam-4914	15	23	to	to	PART
ejpam-4914	15	24	defend	defend	VERB
ejpam-4914	15	25	a	a	DET
ejpam-4914	15	26	total	total	NOUN
ejpam-4914	15	27	of	of	ADP
ejpam-4914	15	28	eight	eight	NUM
ejpam-4914	15	29	regions	region	NOUN
ejpam-4914	15	30	.	.	PUNCT
ejpam-4914	16	1	roman	roman	ADJ
ejpam-4914	16	2	domination	domination	NOUN
ejpam-4914	16	3	as	as	ADP
ejpam-4914	16	4	a	a	DET
ejpam-4914	16	5	mathematical	mathematical	ADJ
ejpam-4914	16	6	concept	concept	NOUN
ejpam-4914	16	7	was	be	AUX
ejpam-4914	16	8	introduced	introduce	VERB
ejpam-4914	16	9	by	by	ADP
ejpam-4914	16	10	cockayne	cockayne	PROPN
ejpam-4914	16	11	,	,	PUNCT
ejpam-4914	16	12	dreyer	dreyer	PROPN
ejpam-4914	16	13	,	,	PUNCT
ejpam-4914	16	14	s.m	s.m	PROPN
ejpam-4914	16	15	.	.	PROPN
ejpam-4914	16	16	hedetniemi	hedetniemi	PROPN
ejpam-4914	16	17	and	and	CCONJ
ejpam-4914	16	18	s.t	s.t	PROPN
ejpam-4914	16	19	.	.	PROPN
ejpam-4914	16	20	hedetniemi	hedetniemi	PROPN
ejpam-4914	17	1	[	[	X
ejpam-4914	17	2	9	9	NUM
ejpam-4914	17	3	]	]	PUNCT
ejpam-4914	17	4	in	in	ADP
ejpam-4914	17	5	2004	2004	NUM
ejpam-4914	17	6	.	.	PUNCT
ejpam-4914	18	1	thereafter	thereafter	ADV
ejpam-4914	18	2	,	,	PUNCT
ejpam-4914	18	3	it	it	PRON
ejpam-4914	18	4	has	have	AUX
ejpam-4914	18	5	become	become	VERB
ejpam-4914	18	6	an	an	DET
ejpam-4914	18	7	active	active	ADJ
ejpam-4914	18	8	∗corresponding	∗corresponding	NOUN
ejpam-4914	18	9	author	author	NOUN
ejpam-4914	18	10	.	.	PUNCT
ejpam-4914	19	1	doi	doi	NOUN
ejpam-4914	19	2	:	:	PUNCT
ejpam-4914	19	3	https://doi.org/10.29020/nybg.ejpam.v16i4.4914	https://doi.org/10.29020/nybg.ejpam.v16i4.4914	NUM
ejpam-4914	19	4	email	email	NOUN
ejpam-4914	19	5	addresses	address	NOUN
ejpam-4914	19	6	:	:	PUNCT
ejpam-4914	19	7	sergio.canoy@g.msuiit.edu.ph	sergio.canoy@g.msuiit.edu.ph	PROPN
ejpam-4914	19	8	(	(	PUNCT
ejpam-4914	19	9	s.r.jr	s.r.jr	NOUN
ejpam-4914	19	10	.	.	PUNCT
ejpam-4914	20	1	canoy	canoy	ADJ
ejpam-4914	20	2	)	)	PUNCT
ejpam-4914	20	3	,	,	PUNCT
ejpam-4914	20	4	ferdinand.jamil@g.msuiit.edu.ph	ferdinand.jamil@g.msuiit.edu.ph	PROPN
ejpam-4914	20	5	(	(	PUNCT
ejpam-4914	20	6	f.	f.	PROPN
ejpam-4914	20	7	jamil	jamil	PROPN
ejpam-4914	20	8	)	)	PUNCT
ejpam-4914	20	9	,	,	PUNCT
ejpam-4914	20	10	sheila.menchavez@g.msuiit.edu.ph	sheila.menchavez@g.msuiit.edu.ph	PROPN
ejpam-4914	20	11	(	(	PUNCT
ejpam-4914	20	12	s.	s.	PROPN
ejpam-4914	20	13	menchavez	menchavez	PROPN
ejpam-4914	20	14	)	)	PUNCT
ejpam-4914	20	15	https://www.ejpam.com	https://www.ejpam.com	X
ejpam-4914	20	16	2431	2431	NUM
ejpam-4914	21	1	©	©	ADP
ejpam-4914	21	2	2023	2023	NUM
ejpam-4914	21	3	ejpam	ejpam	NOUN
ejpam-4914	21	4	all	all	DET
ejpam-4914	21	5	rights	right	NOUN
ejpam-4914	21	6	reserved	reserve	VERB
ejpam-4914	21	7	.	.	PUNCT
ejpam-4914	22	1	s.r	s.r	PROPN
ejpam-4914	22	2	.	.	PROPN
ejpam-4914	22	3	jr	jr	PROPN
ejpam-4914	22	4	.	.	PROPN
ejpam-4914	22	5	canoy	canoy	PROPN
ejpam-4914	22	6	,	,	PUNCT
ejpam-4914	22	7	f.p	f.p	PROPN
ejpam-4914	22	8	.	.	PROPN
ejpam-4914	22	9	jamil	jamil	PROPN
ejpam-4914	22	10	and	and	CCONJ
ejpam-4914	22	11	s.m	s.m	PROPN
ejpam-4914	22	12	.	.	PROPN
ejpam-4914	22	13	menchavez	menchavez	PROPN
ejpam-4914	22	14	/	/	PUNCT
ejpam-4914	22	15	eur	eur	PROPN
ejpam-4914	22	16	.	.	PUNCT
ejpam-4914	23	1	j.	j.	PROPN
ejpam-4914	23	2	pure	pure	PROPN
ejpam-4914	23	3	appl	appl	PROPN
ejpam-4914	23	4	.	.	PROPN
ejpam-4914	23	5	math	math	PROPN
ejpam-4914	23	6	,	,	PUNCT
ejpam-4914	23	7	16	16	NUM
ejpam-4914	23	8	(	(	PUNCT
ejpam-4914	23	9	4	4	NUM
ejpam-4914	23	10	)	)	PUNCT
ejpam-4914	23	11	(	(	PUNCT
ejpam-4914	23	12	2023	2023	NUM
ejpam-4914	23	13	)	)	PUNCT
ejpam-4914	23	14	,	,	PUNCT
ejpam-4914	23	15	2431	2431	NUM
ejpam-4914	23	16	-	-	SYM
ejpam-4914	23	17	2449	2449	NUM
ejpam-4914	23	18	2432	2432	NUM
ejpam-4914	23	19	research	research	NOUN
ejpam-4914	23	20	area	area	NOUN
ejpam-4914	23	21	(	(	PUNCT
ejpam-4914	23	22	see	see	VERB
ejpam-4914	23	23	[	[	X
ejpam-4914	23	24	1	1	NUM
ejpam-4914	23	25	,	,	PUNCT
ejpam-4914	23	26	6	6	NUM
ejpam-4914	23	27	,	,	PUNCT
ejpam-4914	23	28	12	12	NUM
ejpam-4914	23	29	,	,	PUNCT
ejpam-4914	23	30	18	18	NUM
ejpam-4914	23	31	,	,	PUNCT
ejpam-4914	23	32	19	19	NUM
ejpam-4914	23	33	,	,	PUNCT
ejpam-4914	23	34	21	21	NUM
ejpam-4914	23	35	,	,	PUNCT
ejpam-4914	23	36	22	22	NUM
ejpam-4914	23	37	,	,	PUNCT
ejpam-4914	23	38	25	25	NUM
ejpam-4914	23	39	,	,	PUNCT
ejpam-4914	23	40	27	27	NUM
ejpam-4914	23	41	]	]	PUNCT
ejpam-4914	23	42	)	)	PUNCT
ejpam-4914	23	43	.	.	PUNCT
ejpam-4914	24	1	it	it	PRON
ejpam-4914	24	2	models	model	VERB
ejpam-4914	24	3	many	many	ADJ
ejpam-4914	24	4	facility	facility	NOUN
ejpam-4914	24	5	location	location	NOUN
ejpam-4914	24	6	problems	problem	NOUN
ejpam-4914	24	7	(	(	PUNCT
ejpam-4914	24	8	see	see	VERB
ejpam-4914	24	9	[	[	X
ejpam-4914	24	10	7	7	NUM
ejpam-4914	24	11	]	]	NUM
ejpam-4914	24	12	)	)	PUNCT
ejpam-4914	24	13	,	,	PUNCT
ejpam-4914	24	14	where	where	SCONJ
ejpam-4914	24	15	f(v	f(v	NOUN
ejpam-4914	24	16	)	)	PUNCT
ejpam-4914	24	17	is	be	AUX
ejpam-4914	24	18	viewed	view	VERB
ejpam-4914	24	19	as	as	ADP
ejpam-4914	24	20	cost	cost	NOUN
ejpam-4914	24	21	function	function	NOUN
ejpam-4914	24	22	.	.	PUNCT
ejpam-4914	25	1	units	unit	NOUN
ejpam-4914	25	2	with	with	ADP
ejpam-4914	25	3	cost	cost	NOUN
ejpam-4914	25	4	2	2	NUM
ejpam-4914	25	5	may	may	AUX
ejpam-4914	25	6	be	be	AUX
ejpam-4914	25	7	able	able	ADJ
ejpam-4914	25	8	to	to	PART
ejpam-4914	25	9	serve	serve	VERB
ejpam-4914	25	10	neighboring	neighboring	NOUN
ejpam-4914	25	11	locations	location	NOUN
ejpam-4914	25	12	,	,	PUNCT
ejpam-4914	25	13	while	while	SCONJ
ejpam-4914	25	14	units	unit	NOUN
ejpam-4914	25	15	with	with	ADP
ejpam-4914	25	16	costs	cost	NOUN
ejpam-4914	25	17	1	1	NUM
ejpam-4914	25	18	can	can	AUX
ejpam-4914	25	19	serve	serve	VERB
ejpam-4914	25	20	only	only	ADV
ejpam-4914	25	21	their	their	PRON
ejpam-4914	25	22	own	own	ADJ
ejpam-4914	25	23	location	location	NOUN
ejpam-4914	25	24	.	.	PUNCT
ejpam-4914	26	1	in	in	ADP
ejpam-4914	26	2	a	a	DET
ejpam-4914	26	3	communication	communication	NOUN
ejpam-4914	26	4	network	network	NOUN
ejpam-4914	26	5	,	,	PUNCT
ejpam-4914	26	6	f(v	f(v	PROPN
ejpam-4914	26	7	)	)	PUNCT
ejpam-4914	26	8	=	=	SYM
ejpam-4914	26	9	2	2	NUM
ejpam-4914	26	10	is	be	AUX
ejpam-4914	26	11	assigned	assign	VERB
ejpam-4914	26	12	to	to	ADP
ejpam-4914	26	13	locations	location	NOUN
ejpam-4914	26	14	where	where	SCONJ
ejpam-4914	26	15	we	we	PRON
ejpam-4914	26	16	install	install	VERB
ejpam-4914	26	17	wireless	wireless	ADJ
ejpam-4914	26	18	hubs	hub	NOUN
ejpam-4914	26	19	which	which	PRON
ejpam-4914	26	20	are	be	AUX
ejpam-4914	26	21	more	more	ADV
ejpam-4914	26	22	expensive	expensive	ADJ
ejpam-4914	26	23	but	but	CCONJ
ejpam-4914	26	24	can	can	AUX
ejpam-4914	26	25	serve	serve	VERB
ejpam-4914	26	26	neighboring	neighboring	NOUN
ejpam-4914	26	27	locations	location	NOUN
ejpam-4914	26	28	,	,	PUNCT
ejpam-4914	26	29	while	while	SCONJ
ejpam-4914	26	30	f(v	f(v	NOUN
ejpam-4914	26	31	)	)	PUNCT
ejpam-4914	27	1	=	=	SYM
ejpam-4914	27	2	1	1	NUM
ejpam-4914	27	3	is	be	AUX
ejpam-4914	27	4	assigned	assign	VERB
ejpam-4914	27	5	to	to	ADP
ejpam-4914	27	6	locations	location	NOUN
ejpam-4914	27	7	where	where	SCONJ
ejpam-4914	27	8	we	we	PRON
ejpam-4914	27	9	install	install	VERB
ejpam-4914	27	10	wired	wire	VERB
ejpam-4914	27	11	hubs	hub	NOUN
ejpam-4914	27	12	which	which	PRON
ejpam-4914	27	13	function	function	VERB
ejpam-4914	27	14	at	at	ADP
ejpam-4914	27	15	low	low	ADJ
ejpam-4914	27	16	-	-	PUNCT
ejpam-4914	27	17	range	range	NOUN
ejpam-4914	27	18	but	but	CCONJ
ejpam-4914	27	19	are	be	AUX
ejpam-4914	27	20	cheaper	cheap	ADJ
ejpam-4914	27	21	.	.	PUNCT
ejpam-4914	28	1	in	in	ADP
ejpam-4914	28	2	2016	2016	NUM
ejpam-4914	28	3	,	,	PUNCT
ejpam-4914	28	4	the	the	DET
ejpam-4914	28	5	roman	roman	ADJ
ejpam-4914	28	6	2	2	NUM
ejpam-4914	28	7	-	-	PUNCT
ejpam-4914	28	8	domination	domination	NOUN
ejpam-4914	28	9	was	be	AUX
ejpam-4914	28	10	introduced	introduce	VERB
ejpam-4914	28	11	by	by	ADP
ejpam-4914	28	12	chellali	chellali	PROPN
ejpam-4914	28	13	,	,	PUNCT
ejpam-4914	28	14	haynes	hayne	NOUN
ejpam-4914	28	15	,	,	PUNCT
ejpam-4914	28	16	hedetniemi	hedetniemi	ADV
ejpam-4914	28	17	and	and	CCONJ
ejpam-4914	28	18	mcrae	mcrae	PROPN
ejpam-4914	29	1	[	[	X
ejpam-4914	29	2	8	8	NUM
ejpam-4914	29	3	]	]	PUNCT
ejpam-4914	29	4	.	.	PUNCT
ejpam-4914	30	1	it	it	PRON
ejpam-4914	30	2	is	be	AUX
ejpam-4914	30	3	also	also	ADV
ejpam-4914	30	4	called	call	VERB
ejpam-4914	30	5	italian	italian	ADJ
ejpam-4914	30	6	domination	domination	NOUN
ejpam-4914	30	7	.	.	PUNCT
ejpam-4914	31	1	a	a	DET
ejpam-4914	31	2	function	function	NOUN
ejpam-4914	31	3	f	f	NOUN
ejpam-4914	31	4	:	:	PUNCT
ejpam-4914	31	5	v	v	X
ejpam-4914	31	6	(	(	PUNCT
ejpam-4914	31	7	g	g	NOUN
ejpam-4914	31	8	)	)	PUNCT
ejpam-4914	31	9	→	→	SYM
ejpam-4914	31	10	{	{	PUNCT
ejpam-4914	31	11	0	0	NUM
ejpam-4914	31	12	,	,	PUNCT
ejpam-4914	31	13	1	1	NUM
ejpam-4914	31	14	,	,	PUNCT
ejpam-4914	31	15	2	2	NUM
ejpam-4914	31	16	}	}	PUNCT
ejpam-4914	31	17	is	be	AUX
ejpam-4914	31	18	an	an	DET
ejpam-4914	31	19	italian	italian	ADJ
ejpam-4914	31	20	dominating	dominating	NOUN
ejpam-4914	31	21	function	function	NOUN
ejpam-4914	31	22	provided	provide	VERB
ejpam-4914	31	23	for	for	ADP
ejpam-4914	31	24	every	every	DET
ejpam-4914	31	25	vertex	vertex	NOUN
ejpam-4914	31	26	v	v	NOUN
ejpam-4914	31	27	with	with	ADP
ejpam-4914	31	28	f(v	f(v	NOUN
ejpam-4914	31	29	)	)	PUNCT
ejpam-4914	32	1	=	=	SYM
ejpam-4914	32	2	0	0	NUM
ejpam-4914	32	3	we	we	PRON
ejpam-4914	32	4	have∑	have∑	VERB
ejpam-4914	32	5	x∈ng(v	x∈ng(v	PROPN
ejpam-4914	32	6	)	)	PUNCT
ejpam-4914	32	7	f(x	f(x	PROPN
ejpam-4914	32	8	)	)	PUNCT
ejpam-4914	32	9	≥	≥	NOUN
ejpam-4914	32	10	2	2	NUM
ejpam-4914	32	11	,	,	PUNCT
ejpam-4914	32	12	where	where	SCONJ
ejpam-4914	32	13	n(v	n(v	PROPN
ejpam-4914	32	14	)	)	PUNCT
ejpam-4914	32	15	is	be	AUX
ejpam-4914	32	16	the	the	DET
ejpam-4914	32	17	set	set	NOUN
ejpam-4914	32	18	of	of	ADP
ejpam-4914	32	19	all	all	DET
ejpam-4914	32	20	vertices	vertex	NOUN
ejpam-4914	32	21	adjacent	adjacent	ADJ
ejpam-4914	32	22	to	to	ADP
ejpam-4914	32	23	v.	v.	VERB
ejpam-4914	32	24	apparently	apparently	ADV
ejpam-4914	32	25	,	,	PUNCT
ejpam-4914	32	26	a	a	DET
ejpam-4914	32	27	roman	roman	ADJ
ejpam-4914	32	28	dominating	dominating	NOUN
ejpam-4914	32	29	function	function	NOUN
ejpam-4914	32	30	is	be	AUX
ejpam-4914	32	31	an	an	DET
ejpam-4914	32	32	italian	italian	ADJ
ejpam-4914	32	33	dominating	dominating	NOUN
ejpam-4914	32	34	function	function	NOUN
ejpam-4914	32	35	.	.	PUNCT
ejpam-4914	33	1	the	the	DET
ejpam-4914	33	2	italian	italian	ADJ
ejpam-4914	33	3	domination	domination	NOUN
ejpam-4914	33	4	number	number	NOUN
ejpam-4914	33	5	is	be	AUX
ejpam-4914	33	6	the	the	DET
ejpam-4914	33	7	minimum	minimum	ADJ
ejpam-4914	33	8	weight	weight	NOUN
ejpam-4914	33	9	of	of	ADP
ejpam-4914	33	10	an	an	DET
ejpam-4914	33	11	italian	italian	ADJ
ejpam-4914	33	12	dominating	dominating	NOUN
ejpam-4914	33	13	function	function	NOUN
ejpam-4914	33	14	.	.	PUNCT
ejpam-4914	34	1	excellent	excellent	ADJ
ejpam-4914	34	2	references	reference	NOUN
ejpam-4914	34	3	for	for	ADP
ejpam-4914	34	4	italian	italian	ADJ
ejpam-4914	34	5	domination	domination	NOUN
ejpam-4914	34	6	include	include	VERB
ejpam-4914	34	7	[	[	X
ejpam-4914	34	8	8	8	NUM
ejpam-4914	34	9	,	,	PUNCT
ejpam-4914	34	10	20	20	NUM
ejpam-4914	34	11	]	]	PUNCT
ejpam-4914	34	12	.	.	PUNCT
ejpam-4914	35	1	in	in	ADP
ejpam-4914	35	2	2017	2017	NUM
ejpam-4914	35	3	,	,	PUNCT
ejpam-4914	35	4	shabani	shabani	PROPN
ejpam-4914	36	1	[	[	X
ejpam-4914	36	2	24	24	NUM
ejpam-4914	36	3	]	]	PUNCT
ejpam-4914	36	4	introduced	introduce	VERB
ejpam-4914	36	5	the	the	DET
ejpam-4914	36	6	hop	hop	NOUN
ejpam-4914	36	7	roman	roman	ADJ
ejpam-4914	36	8	domination	domination	NOUN
ejpam-4914	36	9	.	.	PUNCT
ejpam-4914	37	1	a	a	DET
ejpam-4914	37	2	hop	hop	NOUN
ejpam-4914	37	3	roman	roman	ADJ
ejpam-4914	37	4	dominating	dominating	NOUN
ejpam-4914	37	5	function	function	NOUN
ejpam-4914	37	6	on	on	ADP
ejpam-4914	37	7	g	g	PROPN
ejpam-4914	37	8	is	be	AUX
ejpam-4914	37	9	a	a	DET
ejpam-4914	37	10	function	function	NOUN
ejpam-4914	37	11	f	f	NOUN
ejpam-4914	37	12	:	:	PUNCT
ejpam-4914	37	13	v	v	X
ejpam-4914	37	14	(	(	PUNCT
ejpam-4914	37	15	g	g	NOUN
ejpam-4914	37	16	)	)	PUNCT
ejpam-4914	37	17	→	→	SYM
ejpam-4914	37	18	{	{	PUNCT
ejpam-4914	37	19	0	0	NUM
ejpam-4914	37	20	,	,	PUNCT
ejpam-4914	37	21	1	1	NUM
ejpam-4914	37	22	,	,	PUNCT
ejpam-4914	37	23	2	2	NUM
ejpam-4914	37	24	}	}	PUNCT
ejpam-4914	37	25	satisfying	satisfy	VERB
ejpam-4914	37	26	the	the	DET
ejpam-4914	37	27	property	property	NOUN
ejpam-4914	37	28	that	that	PRON
ejpam-4914	37	29	for	for	ADP
ejpam-4914	37	30	every	every	DET
ejpam-4914	37	31	vertex	vertex	NOUN
ejpam-4914	37	32	v	v	NOUN
ejpam-4914	37	33	of	of	ADP
ejpam-4914	37	34	g	g	NOUN
ejpam-4914	37	35	with	with	ADP
ejpam-4914	37	36	f(v	f(v	NOUN
ejpam-4914	37	37	)	)	PUNCT
ejpam-4914	38	1	=	=	PUNCT
ejpam-4914	38	2	0	0	X
ejpam-4914	39	1	there	there	PRON
ejpam-4914	39	2	is	be	VERB
ejpam-4914	39	3	a	a	DET
ejpam-4914	39	4	vertex	vertex	NOUN
ejpam-4914	39	5	u	u	NOUN
ejpam-4914	39	6	with	with	ADP
ejpam-4914	39	7	f(u	f(u	PROPN
ejpam-4914	39	8	)	)	PUNCT
ejpam-4914	39	9	=	=	SYM
ejpam-4914	39	10	2	2	NUM
ejpam-4914	39	11	for	for	ADP
ejpam-4914	39	12	which	which	PRON
ejpam-4914	39	13	the	the	DET
ejpam-4914	39	14	distance	distance	NOUN
ejpam-4914	39	15	dg(u	dg(u	X
ejpam-4914	39	16	,	,	PUNCT
ejpam-4914	39	17	v	v	NOUN
ejpam-4914	39	18	)	)	PUNCT
ejpam-4914	39	19	between	between	ADP
ejpam-4914	39	20	u	u	NOUN
ejpam-4914	39	21	and	and	CCONJ
ejpam-4914	39	22	v	v	NOUN
ejpam-4914	39	23	is	be	AUX
ejpam-4914	39	24	2	2	NUM
ejpam-4914	39	25	.	.	PUNCT
ejpam-4914	40	1	it	it	PRON
ejpam-4914	40	2	was	be	AUX
ejpam-4914	40	3	largely	largely	ADV
ejpam-4914	40	4	motivated	motivate	VERB
ejpam-4914	40	5	by	by	ADP
ejpam-4914	40	6	the	the	DET
ejpam-4914	40	7	concept	concept	NOUN
ejpam-4914	40	8	of	of	ADP
ejpam-4914	40	9	hop	hop	NOUN
ejpam-4914	40	10	domination	domination	NOUN
ejpam-4914	40	11	which	which	PRON
ejpam-4914	40	12	is	be	AUX
ejpam-4914	40	13	relatively	relatively	ADV
ejpam-4914	40	14	well	well	ADV
ejpam-4914	40	15	-	-	PUNCT
ejpam-4914	40	16	known	know	VERB
ejpam-4914	40	17	to	to	PART
ejpam-4914	40	18	have	have	VERB
ejpam-4914	40	19	a	a	DET
ejpam-4914	40	20	wide	wide	ADJ
ejpam-4914	40	21	range	range	NOUN
ejpam-4914	40	22	of	of	ADP
ejpam-4914	40	23	applications	application	NOUN
ejpam-4914	40	24	in	in	ADP
ejpam-4914	40	25	social	social	ADJ
ejpam-4914	40	26	network	network	NOUN
ejpam-4914	40	27	.	.	PUNCT
ejpam-4914	41	1	hop	hop	PROPN
ejpam-4914	41	2	roman	roman	ADJ
ejpam-4914	41	3	domination	domination	NOUN
ejpam-4914	41	4	in	in	ADP
ejpam-4914	41	5	graphs	graph	NOUN
ejpam-4914	41	6	was	be	AUX
ejpam-4914	41	7	further	far	ADV
ejpam-4914	41	8	studied	study	VERB
ejpam-4914	41	9	in	in	ADP
ejpam-4914	41	10	[	[	X
ejpam-4914	41	11	21	21	NUM
ejpam-4914	41	12	,	,	PUNCT
ejpam-4914	41	13	22	22	NUM
ejpam-4914	41	14	]	]	PUNCT
ejpam-4914	41	15	.	.	PUNCT
ejpam-4914	42	1	this	this	DET
ejpam-4914	42	2	present	present	ADJ
ejpam-4914	42	3	paper	paper	NOUN
ejpam-4914	42	4	intends	intend	VERB
ejpam-4914	42	5	to	to	PART
ejpam-4914	42	6	introduce	introduce	VERB
ejpam-4914	42	7	and	and	CCONJ
ejpam-4914	42	8	initiate	initiate	VERB
ejpam-4914	42	9	the	the	DET
ejpam-4914	42	10	study	study	NOUN
ejpam-4914	42	11	of	of	ADP
ejpam-4914	42	12	the	the	DET
ejpam-4914	42	13	hop	hop	NOUN
ejpam-4914	42	14	italian	italian	ADJ
ejpam-4914	42	15	domination	domination	NOUN
ejpam-4914	42	16	.	.	PUNCT
ejpam-4914	43	1	we	we	PRON
ejpam-4914	43	2	will	will	AUX
ejpam-4914	43	3	establish	establish	VERB
ejpam-4914	43	4	some	some	PRON
ejpam-4914	43	5	of	of	ADP
ejpam-4914	43	6	its	its	PRON
ejpam-4914	43	7	properties	property	NOUN
ejpam-4914	43	8	and	and	CCONJ
ejpam-4914	43	9	make	make	VERB
ejpam-4914	43	10	characterizations	characterization	NOUN
ejpam-4914	43	11	for	for	ADP
ejpam-4914	43	12	some	some	DET
ejpam-4914	43	13	special	special	ADJ
ejpam-4914	43	14	graphs	graph	NOUN
ejpam-4914	43	15	.	.	PUNCT
ejpam-4914	44	1	we	we	PRON
ejpam-4914	44	2	will	will	AUX
ejpam-4914	44	3	explore	explore	VERB
ejpam-4914	44	4	its	its	PRON
ejpam-4914	44	5	relationships	relationship	NOUN
ejpam-4914	44	6	with	with	ADP
ejpam-4914	44	7	the	the	DET
ejpam-4914	44	8	hop	hop	NOUN
ejpam-4914	44	9	roman	roman	ADJ
ejpam-4914	44	10	domination	domination	NOUN
ejpam-4914	44	11	and	and	CCONJ
ejpam-4914	44	12	other	other	ADJ
ejpam-4914	44	13	related	relate	VERB
ejpam-4914	44	14	hop	hop	NOUN
ejpam-4914	44	15	domination	domination	NOUN
ejpam-4914	44	16	concepts	concept	NOUN
ejpam-4914	44	17	.	.	PUNCT
ejpam-4914	45	1	finally	finally	ADV
ejpam-4914	45	2	,	,	PUNCT
ejpam-4914	45	3	we	we	PRON
ejpam-4914	45	4	will	will	AUX
ejpam-4914	45	5	investigate	investigate	VERB
ejpam-4914	45	6	the	the	DET
ejpam-4914	45	7	hop	hop	NOUN
ejpam-4914	45	8	italian	italian	ADJ
ejpam-4914	45	9	domination	domination	NOUN
ejpam-4914	45	10	in	in	ADP
ejpam-4914	45	11	graphs	graph	NOUN
ejpam-4914	45	12	under	under	ADP
ejpam-4914	45	13	some	some	DET
ejpam-4914	45	14	binary	binary	ADJ
ejpam-4914	45	15	operations	operation	NOUN
ejpam-4914	45	16	.	.	PUNCT
ejpam-4914	46	1	all	all	PRON
ejpam-4914	46	2	throughout	throughout	ADP
ejpam-4914	46	3	this	this	DET
ejpam-4914	46	4	paper	paper	NOUN
ejpam-4914	46	5	,	,	PUNCT
ejpam-4914	46	6	we	we	PRON
ejpam-4914	46	7	consider	consider	VERB
ejpam-4914	46	8	only	only	ADV
ejpam-4914	46	9	graphs	graph	NOUN
ejpam-4914	46	10	which	which	PRON
ejpam-4914	46	11	are	be	AUX
ejpam-4914	46	12	simple	simple	ADJ
ejpam-4914	46	13	,	,	PUNCT
ejpam-4914	46	14	finite	finite	ADJ
ejpam-4914	46	15	and	and	CCONJ
ejpam-4914	46	16	undirected	undirected	ADJ
ejpam-4914	46	17	.	.	PUNCT
ejpam-4914	47	1	given	give	VERB
ejpam-4914	47	2	a	a	DET
ejpam-4914	47	3	graph	graph	NOUN
ejpam-4914	47	4	g	g	NOUN
ejpam-4914	47	5	=	=	PUNCT
ejpam-4914	47	6	(	(	PUNCT
ejpam-4914	47	7	v	v	NOUN
ejpam-4914	47	8	(	(	PUNCT
ejpam-4914	47	9	g	g	NOUN
ejpam-4914	47	10	)	)	PUNCT
ejpam-4914	47	11	,	,	PUNCT
ejpam-4914	47	12	e(g	e(g	PROPN
ejpam-4914	47	13	)	)	PUNCT
ejpam-4914	47	14	)	)	PUNCT
ejpam-4914	47	15	,	,	PUNCT
ejpam-4914	47	16	we	we	PRON
ejpam-4914	47	17	call	call	VERB
ejpam-4914	47	18	v	v	ADP
ejpam-4914	47	19	(	(	PUNCT
ejpam-4914	47	20	g	g	NOUN
ejpam-4914	47	21	)	)	PUNCT
ejpam-4914	47	22	the	the	DET
ejpam-4914	47	23	vertex	vertex	NOUN
ejpam-4914	47	24	set	set	NOUN
ejpam-4914	47	25	of	of	ADP
ejpam-4914	47	26	g	g	PROPN
ejpam-4914	47	27	and	and	CCONJ
ejpam-4914	47	28	e(g	e(g	PROPN
ejpam-4914	47	29	)	)	PUNCT
ejpam-4914	47	30	its	its	PRON
ejpam-4914	47	31	edge	edge	NOUN
ejpam-4914	47	32	set	set	NOUN
ejpam-4914	47	33	.	.	PUNCT
ejpam-4914	48	1	the	the	DET
ejpam-4914	48	2	cardinality	cardinality	PROPN
ejpam-4914	48	3	|v	|v	PROPN
ejpam-4914	48	4	(	(	PUNCT
ejpam-4914	48	5	g)|	g)|	NOUN
ejpam-4914	48	6	of	of	ADP
ejpam-4914	48	7	v	v	NOUN
ejpam-4914	48	8	(	(	PUNCT
ejpam-4914	48	9	g	g	NOUN
ejpam-4914	48	10	)	)	PUNCT
ejpam-4914	48	11	is	be	AUX
ejpam-4914	48	12	the	the	DET
ejpam-4914	48	13	order	order	NOUN
ejpam-4914	48	14	of	of	ADP
ejpam-4914	48	15	g.	g.	PROPN
ejpam-4914	48	16	all	all	DET
ejpam-4914	48	17	terminologies	terminology	NOUN
ejpam-4914	48	18	used	use	VERB
ejpam-4914	48	19	here	here	ADV
ejpam-4914	48	20	which	which	PRON
ejpam-4914	48	21	are	be	AUX
ejpam-4914	48	22	not	not	PART
ejpam-4914	48	23	being	be	AUX
ejpam-4914	48	24	defined	define	VERB
ejpam-4914	48	25	are	be	AUX
ejpam-4914	48	26	adapted	adapt	VERB
ejpam-4914	48	27	from	from	ADP
ejpam-4914	48	28	[	[	X
ejpam-4914	48	29	3	3	NUM
ejpam-4914	48	30	]	]	PUNCT
ejpam-4914	48	31	.	.	PUNCT
ejpam-4914	49	1	let	let	VERB
ejpam-4914	49	2	g	g	NOUN
ejpam-4914	49	3	and	and	CCONJ
ejpam-4914	49	4	h	h	NOUN
ejpam-4914	49	5	be	be	AUX
ejpam-4914	49	6	disjoint	disjoint	NOUN
ejpam-4914	49	7	graphs	graph	NOUN
ejpam-4914	49	8	.	.	PUNCT
ejpam-4914	50	1	the	the	DET
ejpam-4914	50	2	complementary	complementary	ADJ
ejpam-4914	50	3	prism	prism	NOUN
ejpam-4914	50	4	gg	gg	PROPN
ejpam-4914	50	5	is	be	AUX
ejpam-4914	50	6	formed	form	VERB
ejpam-4914	50	7	from	from	ADP
ejpam-4914	50	8	g	g	PROPN
ejpam-4914	50	9	and	and	CCONJ
ejpam-4914	50	10	its	its	PRON
ejpam-4914	50	11	complement	complement	NOUN
ejpam-4914	50	12	g	g	NOUN
ejpam-4914	50	13	by	by	ADP
ejpam-4914	50	14	adding	add	VERB
ejpam-4914	50	15	a	a	DET
ejpam-4914	50	16	perfect	perfect	ADJ
ejpam-4914	50	17	matching	matching	NOUN
ejpam-4914	50	18	between	between	ADP
ejpam-4914	50	19	corresponding	corresponding	ADJ
ejpam-4914	50	20	vertices	vertex	NOUN
ejpam-4914	50	21	of	of	ADP
ejpam-4914	50	22	g	g	PROPN
ejpam-4914	50	23	and	and	CCONJ
ejpam-4914	50	24	g.	g.	PROPN
ejpam-4914	50	25	if	if	SCONJ
ejpam-4914	50	26	for	for	ADP
ejpam-4914	50	27	each	each	DET
ejpam-4914	50	28	v	v	NUM
ejpam-4914	50	29	∈	∈	PROPN
ejpam-4914	50	30	v	v	NOUN
ejpam-4914	50	31	(	(	PUNCT
ejpam-4914	50	32	g	g	NOUN
ejpam-4914	50	33	)	)	PUNCT
ejpam-4914	50	34	,	,	PUNCT
ejpam-4914	50	35	v	v	NOUN
ejpam-4914	50	36	is	be	AUX
ejpam-4914	50	37	the	the	DET
ejpam-4914	50	38	vertex	vertex	NOUN
ejpam-4914	50	39	in	in	ADP
ejpam-4914	50	40	g	g	PROPN
ejpam-4914	50	41	corresponding	correspond	VERB
ejpam-4914	50	42	to	to	ADP
ejpam-4914	50	43	v	v	NOUN
ejpam-4914	50	44	,	,	PUNCT
ejpam-4914	50	45	then	then	ADV
ejpam-4914	50	46	gg	gg	PROPN
ejpam-4914	50	47	is	be	AUX
ejpam-4914	50	48	formed	form	VERB
ejpam-4914	50	49	by	by	ADP
ejpam-4914	50	50	adding	add	VERB
ejpam-4914	50	51	the	the	DET
ejpam-4914	50	52	edge	edge	NOUN
ejpam-4914	50	53	vv	vv	NOUN
ejpam-4914	50	54	for	for	ADP
ejpam-4914	50	55	every	every	DET
ejpam-4914	50	56	v	v	NUM
ejpam-4914	50	57	∈	∈	NOUN
ejpam-4914	50	58	v	v	NOUN
ejpam-4914	50	59	(	(	PUNCT
ejpam-4914	50	60	g	g	NOUN
ejpam-4914	50	61	)	)	PUNCT
ejpam-4914	50	62	.	.	PUNCT
ejpam-4914	51	1	the	the	DET
ejpam-4914	51	2	corona	corona	NOUN
ejpam-4914	51	3	of	of	ADP
ejpam-4914	51	4	g	g	PROPN
ejpam-4914	51	5	and	and	CCONJ
ejpam-4914	51	6	h	h	NOUN
ejpam-4914	51	7	is	be	AUX
ejpam-4914	51	8	the	the	DET
ejpam-4914	51	9	graph	graph	NOUN
ejpam-4914	51	10	g	g	PROPN
ejpam-4914	51	11	◦	◦	NOUN
ejpam-4914	51	12	h	h	NOUN
ejpam-4914	51	13	obtained	obtain	VERB
ejpam-4914	51	14	by	by	ADP
ejpam-4914	51	15	taking	take	VERB
ejpam-4914	51	16	one	one	NUM
ejpam-4914	51	17	copy	copy	NOUN
ejpam-4914	51	18	of	of	ADP
ejpam-4914	51	19	g	g	PROPN
ejpam-4914	51	20	and	and	CCONJ
ejpam-4914	51	21	|v	|v	PROPN
ejpam-4914	51	22	(	(	PUNCT
ejpam-4914	51	23	g)|	g)|	NOUN
ejpam-4914	51	24	copies	copy	NOUN
ejpam-4914	51	25	of	of	ADP
ejpam-4914	51	26	h	h	NOUN
ejpam-4914	51	27	,	,	PUNCT
ejpam-4914	51	28	and	and	CCONJ
ejpam-4914	51	29	then	then	ADV
ejpam-4914	51	30	joining	join	VERB
ejpam-4914	51	31	the	the	DET
ejpam-4914	51	32	ith	ith	PROPN
ejpam-4914	51	33	vertex	vertex	NOUN
ejpam-4914	51	34	of	of	ADP
ejpam-4914	51	35	g	g	NOUN
ejpam-4914	51	36	to	to	ADP
ejpam-4914	51	37	every	every	DET
ejpam-4914	51	38	vertex	vertex	NOUN
ejpam-4914	51	39	in	in	ADP
ejpam-4914	51	40	the	the	DET
ejpam-4914	51	41	ith	ith	PROPN
ejpam-4914	51	42	copy	copy	NOUN
ejpam-4914	51	43	of	of	ADP
ejpam-4914	51	44	h.	h.	PROPN
ejpam-4914	51	45	in	in	ADP
ejpam-4914	51	46	particular	particular	ADJ
ejpam-4914	51	47	,	,	PUNCT
ejpam-4914	51	48	we	we	PRON
ejpam-4914	51	49	call	call	VERB
ejpam-4914	51	50	g	g	PROPN
ejpam-4914	51	51	◦	◦	NOUN
ejpam-4914	51	52	k1	k1	NOUN
ejpam-4914	51	53	the	the	DET
ejpam-4914	51	54	corona	corona	NOUN
ejpam-4914	51	55	of	of	ADP
ejpam-4914	51	56	g	g	PROPN
ejpam-4914	51	57	,	,	PUNCT
ejpam-4914	51	58	and	and	CCONJ
ejpam-4914	51	59	write	write	VERB
ejpam-4914	51	60	cor(g	cor(g	PROPN
ejpam-4914	51	61	)	)	PUNCT
ejpam-4914	52	1	=	=	SYM
ejpam-4914	52	2	g	g	PROPN
ejpam-4914	52	3	◦	◦	NOUN
ejpam-4914	52	4	k1	k1	NOUN
ejpam-4914	52	5	.	.	PUNCT
ejpam-4914	53	1	the	the	DET
ejpam-4914	53	2	composition	composition	NOUN
ejpam-4914	53	3	(	(	PUNCT
ejpam-4914	53	4	or	or	CCONJ
ejpam-4914	53	5	lexicographic	lexicographic	ADJ
ejpam-4914	53	6	product	product	NOUN
ejpam-4914	53	7	)	)	PUNCT
ejpam-4914	53	8	of	of	ADP
ejpam-4914	53	9	g	g	PROPN
ejpam-4914	53	10	and	and	CCONJ
ejpam-4914	53	11	h	h	NOUN
ejpam-4914	53	12	is	be	AUX
ejpam-4914	53	13	the	the	DET
ejpam-4914	53	14	graph	graph	NOUN
ejpam-4914	53	15	g[h	g[h	PROPN
ejpam-4914	53	16	]	]	PUNCT
ejpam-4914	53	17	with	with	ADP
ejpam-4914	53	18	v	v	NOUN
ejpam-4914	53	19	(	(	PUNCT
ejpam-4914	53	20	g[h	g[h	PROPN
ejpam-4914	53	21	]	]	PUNCT
ejpam-4914	53	22	)	)	PUNCT
ejpam-4914	53	23	=	=	SYM
ejpam-4914	53	24	v	v	X
ejpam-4914	53	25	(	(	PUNCT
ejpam-4914	53	26	g	g	NOUN
ejpam-4914	53	27	)	)	PUNCT
ejpam-4914	53	28	×	×	NOUN
ejpam-4914	53	29	v	v	NOUN
ejpam-4914	53	30	(	(	PUNCT
ejpam-4914	53	31	h	h	NOUN
ejpam-4914	53	32	)	)	PUNCT
ejpam-4914	53	33	and	and	CCONJ
ejpam-4914	53	34	(	(	PUNCT
ejpam-4914	53	35	u	u	NOUN
ejpam-4914	53	36	,	,	PUNCT
ejpam-4914	53	37	v)(u′	v)(u′	NOUN
ejpam-4914	53	38	,	,	PUNCT
ejpam-4914	53	39	v′	v′	NOUN
ejpam-4914	53	40	)	)	PUNCT
ejpam-4914	53	41	∈	∈	NOUN
ejpam-4914	53	42	e(g[h	e(g[h	NOUN
ejpam-4914	53	43	]	]	PUNCT
ejpam-4914	53	44	)	)	PUNCT
ejpam-4914	53	45	if	if	SCONJ
ejpam-4914	53	46	and	and	CCONJ
ejpam-4914	53	47	only	only	ADV
ejpam-4914	53	48	if	if	SCONJ
ejpam-4914	53	49	either	either	CCONJ
ejpam-4914	53	50	uu′	uu′	PROPN
ejpam-4914	53	51	∈	∈	PROPN
ejpam-4914	53	52	e(g	e(g	PROPN
ejpam-4914	53	53	)	)	PUNCT
ejpam-4914	53	54	or	or	CCONJ
ejpam-4914	53	55	u	u	X
ejpam-4914	53	56	=	=	PUNCT
ejpam-4914	53	57	u′	u′	PROPN
ejpam-4914	53	58	and	and	CCONJ
ejpam-4914	53	59	vv′	vv′	NOUN
ejpam-4914	53	60	∈	∈	PROPN
ejpam-4914	53	61	e(h	e(h	PROPN
ejpam-4914	53	62	)	)	PUNCT
ejpam-4914	53	63	.	.	PUNCT
ejpam-4914	54	1	in	in	ADP
ejpam-4914	54	2	any	any	PRON
ejpam-4914	54	3	of	of	ADP
ejpam-4914	54	4	these	these	DET
ejpam-4914	54	5	graphs	graph	NOUN
ejpam-4914	54	6	,	,	PUNCT
ejpam-4914	54	7	g	g	PROPN
ejpam-4914	54	8	and	and	CCONJ
ejpam-4914	54	9	h	h	NOUN
ejpam-4914	54	10	are	be	AUX
ejpam-4914	54	11	referred	refer	VERB
ejpam-4914	54	12	to	to	ADP
ejpam-4914	54	13	as	as	ADP
ejpam-4914	54	14	their	their	PRON
ejpam-4914	54	15	basic	basic	ADJ
ejpam-4914	54	16	component	component	NOUN
ejpam-4914	54	17	graphs	graph	NOUN
ejpam-4914	54	18	.	.	PUNCT
ejpam-4914	55	1	for	for	ADP
ejpam-4914	55	2	vertices	vertex	NOUN
ejpam-4914	55	3	u	u	NOUN
ejpam-4914	55	4	and	and	CCONJ
ejpam-4914	55	5	v	v	NOUN
ejpam-4914	55	6	of	of	ADP
ejpam-4914	55	7	a	a	DET
ejpam-4914	55	8	graph	graph	NOUN
ejpam-4914	55	9	g	g	NOUN
ejpam-4914	55	10	,	,	PUNCT
ejpam-4914	55	11	a	a	DET
ejpam-4914	55	12	u	u	NOUN
ejpam-4914	55	13	-	-	NOUN
ejpam-4914	55	14	v	v	ADJ
ejpam-4914	55	15	geodesic	geodesic	NOUN
ejpam-4914	55	16	is	be	AUX
ejpam-4914	55	17	any	any	DET
ejpam-4914	55	18	shortest	short	ADJ
ejpam-4914	55	19	path	path	NOUN
ejpam-4914	55	20	in	in	ADP
ejpam-4914	55	21	g	g	NOUN
ejpam-4914	55	22	joining	join	VERB
ejpam-4914	55	23	u	u	NOUN
ejpam-4914	55	24	and	and	CCONJ
ejpam-4914	55	25	v.	v.	ADP
ejpam-4914	55	26	the	the	DET
ejpam-4914	55	27	length	length	NOUN
ejpam-4914	55	28	of	of	ADP
ejpam-4914	55	29	a	a	DET
ejpam-4914	55	30	u	u	NOUN
ejpam-4914	55	31	-	-	NOUN
ejpam-4914	55	32	v	v	ADJ
ejpam-4914	55	33	geodesic	geodesic	NOUN
ejpam-4914	55	34	is	be	AUX
ejpam-4914	55	35	the	the	DET
ejpam-4914	55	36	distance	distance	NOUN
ejpam-4914	55	37	between	between	ADP
ejpam-4914	55	38	u	u	NOUN
ejpam-4914	55	39	and	and	CCONJ
ejpam-4914	55	40	v	v	NOUN
ejpam-4914	55	41	,	,	PUNCT
ejpam-4914	55	42	and	and	CCONJ
ejpam-4914	55	43	is	be	AUX
ejpam-4914	55	44	denoted	denote	VERB
ejpam-4914	55	45	by	by	ADP
ejpam-4914	55	46	dg(u	dg(u	NOUN
ejpam-4914	55	47	,	,	PUNCT
ejpam-4914	55	48	v	v	NOUN
ejpam-4914	55	49	)	)	PUNCT
ejpam-4914	55	50	.	.	PUNCT
ejpam-4914	56	1	s.r	s.r	PROPN
ejpam-4914	56	2	.	.	PROPN
ejpam-4914	56	3	jr	jr	PROPN
ejpam-4914	56	4	.	.	PROPN
ejpam-4914	56	5	canoy	canoy	PROPN
ejpam-4914	56	6	,	,	PUNCT
ejpam-4914	56	7	f.p	f.p	PROPN
ejpam-4914	56	8	.	.	PROPN
ejpam-4914	56	9	jamil	jamil	PROPN
ejpam-4914	56	10	and	and	CCONJ
ejpam-4914	56	11	s.m	s.m	PROPN
ejpam-4914	56	12	.	.	PROPN
ejpam-4914	56	13	menchavez	menchavez	PROPN
ejpam-4914	56	14	/	/	PUNCT
ejpam-4914	56	15	eur	eur	PROPN
ejpam-4914	56	16	.	.	PUNCT
ejpam-4914	57	1	j.	j.	PROPN
ejpam-4914	57	2	pure	pure	PROPN
ejpam-4914	57	3	appl	appl	PROPN
ejpam-4914	57	4	.	.	PROPN
ejpam-4914	57	5	math	math	PROPN
ejpam-4914	57	6	,	,	PUNCT
ejpam-4914	57	7	16	16	NUM
ejpam-4914	57	8	(	(	PUNCT
ejpam-4914	57	9	4	4	NUM
ejpam-4914	57	10	)	)	PUNCT
ejpam-4914	57	11	(	(	PUNCT
ejpam-4914	57	12	2023	2023	NUM
ejpam-4914	57	13	)	)	PUNCT
ejpam-4914	57	14	,	,	PUNCT
ejpam-4914	57	15	2431	2431	NUM
ejpam-4914	57	16	-	-	SYM
ejpam-4914	57	17	2449	2449	NUM
ejpam-4914	57	18	2433	2433	NUM
ejpam-4914	57	19	the	the	DET
ejpam-4914	57	20	eccentricity	eccentricity	NOUN
ejpam-4914	57	21	of	of	ADP
ejpam-4914	57	22	v	v	NOUN
ejpam-4914	57	23	refers	refer	VERB
ejpam-4914	57	24	to	to	ADP
ejpam-4914	57	25	the	the	DET
ejpam-4914	57	26	quantity	quantity	NOUN
ejpam-4914	57	27	e(v	e(v	NOUN
ejpam-4914	57	28	)	)	PUNCT
ejpam-4914	57	29	=	=	SYM
ejpam-4914	57	30	max{dg(u	max{dg(u	X
ejpam-4914	57	31	,	,	PUNCT
ejpam-4914	57	32	v	v	NOUN
ejpam-4914	57	33	)	)	PUNCT
ejpam-4914	57	34	:	:	PUNCT
ejpam-4914	57	35	v	v	X
ejpam-4914	57	36	∈	∈	PROPN
ejpam-4914	57	37	v	v	NOUN
ejpam-4914	57	38	(	(	PUNCT
ejpam-4914	57	39	g	g	NOUN
ejpam-4914	57	40	)	)	PUNCT
ejpam-4914	57	41	}	}	PUNCT
ejpam-4914	57	42	.	.	PUNCT
ejpam-4914	58	1	customarily	customarily	ADV
ejpam-4914	58	2	,	,	PUNCT
ejpam-4914	58	3	diam(g	diam(g	NOUN
ejpam-4914	58	4	)	)	PUNCT
ejpam-4914	58	5	=	=	SYM
ejpam-4914	58	6	max{e(v	max{e(v	PROPN
ejpam-4914	58	7	)	)	PUNCT
ejpam-4914	58	8	:	:	PUNCT
ejpam-4914	59	1	v	v	X
ejpam-4914	59	2	∈	∈	PROPN
ejpam-4914	59	3	v	v	NOUN
ejpam-4914	59	4	(	(	PUNCT
ejpam-4914	59	5	g	g	NOUN
ejpam-4914	59	6	)	)	PUNCT
ejpam-4914	59	7	}	}	PUNCT
ejpam-4914	59	8	.	.	PUNCT
ejpam-4914	60	1	in	in	ADP
ejpam-4914	60	2	this	this	DET
ejpam-4914	60	3	paper	paper	NOUN
ejpam-4914	60	4	,	,	PUNCT
ejpam-4914	60	5	we	we	PRON
ejpam-4914	60	6	write	write	VERB
ejpam-4914	60	7	e(g	e(g	NOUN
ejpam-4914	60	8	)	)	PUNCT
ejpam-4914	61	1	=	=	PUNCT
ejpam-4914	61	2	min{e(v	min{e(v	PROPN
ejpam-4914	61	3	)	)	PUNCT
ejpam-4914	61	4	:	:	PUNCT
ejpam-4914	62	1	v	v	X
ejpam-4914	62	2	∈	∈	PROPN
ejpam-4914	62	3	v	v	NOUN
ejpam-4914	62	4	(	(	PUNCT
ejpam-4914	62	5	g	g	NOUN
ejpam-4914	62	6	)	)	PUNCT
ejpam-4914	62	7	}	}	PUNCT
ejpam-4914	62	8	.	.	PUNCT
ejpam-4914	63	1	vertices	vertice	VERB
ejpam-4914	63	2	u	u	NOUN
ejpam-4914	63	3	and	and	CCONJ
ejpam-4914	63	4	v	v	NOUN
ejpam-4914	63	5	of	of	ADP
ejpam-4914	63	6	a	a	DET
ejpam-4914	63	7	graph	graph	NOUN
ejpam-4914	63	8	g	g	NOUN
ejpam-4914	63	9	are	be	AUX
ejpam-4914	63	10	neighbors	neighbor	NOUN
ejpam-4914	63	11	if	if	SCONJ
ejpam-4914	63	12	uv	uv	PROPN
ejpam-4914	63	13	∈	∈	PROPN
ejpam-4914	63	14	e(g	e(g	PROPN
ejpam-4914	63	15	)	)	PUNCT
ejpam-4914	63	16	.	.	PUNCT
ejpam-4914	64	1	the	the	DET
ejpam-4914	64	2	open	open	ADJ
ejpam-4914	64	3	neighborhood	neighborhood	NOUN
ejpam-4914	64	4	of	of	ADP
ejpam-4914	64	5	v	v	NOUN
ejpam-4914	64	6	refers	refer	VERB
ejpam-4914	64	7	to	to	ADP
ejpam-4914	64	8	the	the	DET
ejpam-4914	64	9	set	set	NOUN
ejpam-4914	64	10	ng(v	ng(v	PUNCT
ejpam-4914	64	11	)	)	PUNCT
ejpam-4914	64	12	consisting	consist	VERB
ejpam-4914	64	13	of	of	ADP
ejpam-4914	64	14	all	all	DET
ejpam-4914	64	15	neighbors	neighbor	NOUN
ejpam-4914	64	16	of	of	ADP
ejpam-4914	64	17	v.	v.	ADP
ejpam-4914	64	18	the	the	DET
ejpam-4914	64	19	degree	degree	NOUN
ejpam-4914	64	20	of	of	ADP
ejpam-4914	64	21	v	v	NOUN
ejpam-4914	64	22	refers	refer	VERB
ejpam-4914	64	23	to	to	ADP
ejpam-4914	64	24	the	the	DET
ejpam-4914	64	25	cardinality	cardinality	NOUN
ejpam-4914	64	26	|ng(v)|	|ng(v)|	NOUN
ejpam-4914	64	27	of	of	ADP
ejpam-4914	64	28	the	the	DET
ejpam-4914	64	29	open	open	ADJ
ejpam-4914	64	30	neighborhood	neighborhood	NOUN
ejpam-4914	64	31	of	of	ADP
ejpam-4914	64	32	v	v	NOUN
ejpam-4914	64	33	,	,	PUNCT
ejpam-4914	64	34	and	and	CCONJ
ejpam-4914	64	35	δ(g	δ(g	PUNCT
ejpam-4914	64	36	)	)	PUNCT
ejpam-4914	64	37	is	be	AUX
ejpam-4914	64	38	the	the	DET
ejpam-4914	64	39	minimum	minimum	ADJ
ejpam-4914	64	40	degree	degree	NOUN
ejpam-4914	64	41	of	of	ADP
ejpam-4914	64	42	a	a	DET
ejpam-4914	64	43	vertex	vertex	NOUN
ejpam-4914	64	44	of	of	ADP
ejpam-4914	64	45	g.	g.	PROPN
ejpam-4914	64	46	the	the	DET
ejpam-4914	64	47	closed	close	VERB
ejpam-4914	64	48	neighborhood	neighborhood	NOUN
ejpam-4914	64	49	of	of	ADP
ejpam-4914	64	50	v	v	NOUN
ejpam-4914	64	51	is	be	AUX
ejpam-4914	64	52	the	the	DET
ejpam-4914	64	53	set	set	NOUN
ejpam-4914	64	54	ng[v	ng[v	NOUN
ejpam-4914	64	55	]	]	X
ejpam-4914	64	56	=	=	SYM
ejpam-4914	64	57	ng(v	ng(v	X
ejpam-4914	64	58	)	)	PUNCT
ejpam-4914	64	59	∪	∪	ADP
ejpam-4914	64	60	{	{	PUNCT
ejpam-4914	64	61	v	v	NOUN
ejpam-4914	64	62	}	}	PUNCT
ejpam-4914	64	63	.	.	PUNCT
ejpam-4914	65	1	customarily	customarily	ADV
ejpam-4914	65	2	,	,	PUNCT
ejpam-4914	65	3	for	for	ADP
ejpam-4914	65	4	s	s	PROPN
ejpam-4914	65	5	⊆	⊆	NUM
ejpam-4914	65	6	v	v	NOUN
ejpam-4914	65	7	(	(	PUNCT
ejpam-4914	65	8	g	g	NOUN
ejpam-4914	65	9	)	)	PUNCT
ejpam-4914	65	10	,	,	PUNCT
ejpam-4914	65	11	ng(s	ng(s	NUM
ejpam-4914	65	12	)	)	PUNCT
ejpam-4914	65	13	=	=	SYM
ejpam-4914	65	14	∪v∈sng(v	∪v∈sng(v	PROPN
ejpam-4914	65	15	)	)	PUNCT
ejpam-4914	65	16	and	and	CCONJ
ejpam-4914	65	17	ng[s	ng[s	PROPN
ejpam-4914	65	18	]	]	PUNCT
ejpam-4914	65	19	=	=	SYM
ejpam-4914	65	20	∪v∈sng[v	∪v∈sng[v	X
ejpam-4914	65	21	]	]	PUNCT
ejpam-4914	65	22	.	.	PUNCT
ejpam-4914	66	1	a	a	DET
ejpam-4914	66	2	subset	subset	NOUN
ejpam-4914	66	3	s	s	VERB
ejpam-4914	66	4	⊆	⊆	NUM
ejpam-4914	66	5	v	v	NOUN
ejpam-4914	66	6	(	(	PUNCT
ejpam-4914	66	7	g	g	NOUN
ejpam-4914	66	8	)	)	PUNCT
ejpam-4914	66	9	is	be	AUX
ejpam-4914	66	10	a	a	DET
ejpam-4914	66	11	dominating	dominating	NOUN
ejpam-4914	66	12	set	set	NOUN
ejpam-4914	66	13	of	of	ADP
ejpam-4914	66	14	g	g	PROPN
ejpam-4914	66	15	if	if	SCONJ
ejpam-4914	66	16	ng[s	ng[	NOUN
ejpam-4914	66	17	]	]	PUNCT
ejpam-4914	66	18	=	=	SYM
ejpam-4914	66	19	v	v	NOUN
ejpam-4914	66	20	(	(	PUNCT
ejpam-4914	66	21	g	g	NOUN
ejpam-4914	66	22	)	)	PUNCT
ejpam-4914	66	23	.	.	PUNCT
ejpam-4914	67	1	the	the	DET
ejpam-4914	67	2	minimum	minimum	PROPN
ejpam-4914	67	3	cardinality	cardinality	PROPN
ejpam-4914	67	4	γ(g	γ(g	PROPN
ejpam-4914	67	5	)	)	PUNCT
ejpam-4914	67	6	of	of	ADP
ejpam-4914	67	7	a	a	DET
ejpam-4914	67	8	dominating	dominating	NOUN
ejpam-4914	67	9	set	set	NOUN
ejpam-4914	67	10	of	of	ADP
ejpam-4914	67	11	g	g	PROPN
ejpam-4914	67	12	is	be	AUX
ejpam-4914	67	13	the	the	DET
ejpam-4914	67	14	domination	domination	NOUN
ejpam-4914	67	15	number	number	NOUN
ejpam-4914	67	16	of	of	ADP
ejpam-4914	67	17	g.	g.	PROPN
ejpam-4914	67	18	a	a	DET
ejpam-4914	67	19	dominating	dominating	NOUN
ejpam-4914	67	20	set	set	NOUN
ejpam-4914	67	21	of	of	ADP
ejpam-4914	67	22	cardinality	cardinality	PROPN
ejpam-4914	67	23	γ(g	γ(g	PROPN
ejpam-4914	67	24	)	)	PUNCT
ejpam-4914	67	25	is	be	AUX
ejpam-4914	67	26	called	call	VERB
ejpam-4914	67	27	a	a	DET
ejpam-4914	67	28	γ	γ	NOUN
ejpam-4914	67	29	-	-	PUNCT
ejpam-4914	67	30	set	set	NOUN
ejpam-4914	67	31	of	of	ADP
ejpam-4914	67	32	g.	g.	PROPN
ejpam-4914	67	33	the	the	DET
ejpam-4914	67	34	reader	reader	NOUN
ejpam-4914	67	35	is	be	AUX
ejpam-4914	67	36	referred	refer	VERB
ejpam-4914	67	37	to	to	ADP
ejpam-4914	67	38	[	[	X
ejpam-4914	67	39	2	2	NUM
ejpam-4914	67	40	,	,	PUNCT
ejpam-4914	67	41	10	10	NUM
ejpam-4914	67	42	,	,	PUNCT
ejpam-4914	67	43	11	11	NUM
ejpam-4914	67	44	,	,	PUNCT
ejpam-4914	67	45	13	13	NUM
ejpam-4914	67	46	,	,	PUNCT
ejpam-4914	67	47	16	16	NUM
ejpam-4914	67	48	,	,	PUNCT
ejpam-4914	67	49	17	17	NUM
ejpam-4914	67	50	]	]	PUNCT
ejpam-4914	67	51	for	for	ADP
ejpam-4914	67	52	the	the	DET
ejpam-4914	67	53	history	history	NOUN
ejpam-4914	67	54	,	,	PUNCT
ejpam-4914	67	55	fundamental	fundamental	ADJ
ejpam-4914	67	56	concepts	concept	NOUN
ejpam-4914	67	57	and	and	CCONJ
ejpam-4914	67	58	recent	recent	ADJ
ejpam-4914	67	59	developments	development	NOUN
ejpam-4914	67	60	of	of	ADP
ejpam-4914	67	61	domination	domination	NOUN
ejpam-4914	67	62	in	in	ADP
ejpam-4914	67	63	graphs	graph	NOUN
ejpam-4914	67	64	as	as	ADV
ejpam-4914	67	65	well	well	ADV
ejpam-4914	67	66	as	as	ADP
ejpam-4914	67	67	its	its	PRON
ejpam-4914	67	68	various	various	ADJ
ejpam-4914	67	69	applications	application	NOUN
ejpam-4914	67	70	.	.	PUNCT
ejpam-4914	68	1	a	a	DET
ejpam-4914	68	2	set	set	NOUN
ejpam-4914	68	3	s	s	NOUN
ejpam-4914	68	4	⊆	⊆	NUM
ejpam-4914	68	5	v	v	NOUN
ejpam-4914	68	6	(	(	PUNCT
ejpam-4914	68	7	g	g	NOUN
ejpam-4914	68	8	)	)	PUNCT
ejpam-4914	68	9	is	be	AUX
ejpam-4914	68	10	a	a	DET
ejpam-4914	68	11	pointwise	pointwise	ADJ
ejpam-4914	68	12	nondominating	nondominate	VERB
ejpam-4914	68	13	set	set	NOUN
ejpam-4914	68	14	of	of	ADP
ejpam-4914	68	15	g	g	PROPN
ejpam-4914	68	16	(	(	PUNCT
ejpam-4914	68	17	or	or	CCONJ
ejpam-4914	68	18	pnd	pnd	NOUN
ejpam-4914	68	19	-	-	PUNCT
ejpam-4914	68	20	set	set	NOUN
ejpam-4914	68	21	of	of	ADP
ejpam-4914	68	22	g	g	NOUN
ejpam-4914	68	23	)	)	PUNCT
ejpam-4914	68	24	if	if	SCONJ
ejpam-4914	68	25	for	for	ADP
ejpam-4914	68	26	each	each	DET
ejpam-4914	68	27	v	v	NUM
ejpam-4914	68	28	∈	∈	NOUN
ejpam-4914	68	29	v	v	NOUN
ejpam-4914	68	30	(	(	PUNCT
ejpam-4914	68	31	g)\s	g)\s	NOUN
ejpam-4914	68	32	,	,	PUNCT
ejpam-4914	68	33	there	there	PRON
ejpam-4914	68	34	exists	exist	VERB
ejpam-4914	68	35	u	u	PROPN
ejpam-4914	68	36	∈	∈	PROPN
ejpam-4914	68	37	s	s	VERB
ejpam-4914	68	38	such	such	ADJ
ejpam-4914	68	39	that	that	DET
ejpam-4914	68	40	uv	uv	NOUN
ejpam-4914	68	41	/∈	/∈	PUNCT
ejpam-4914	68	42	e(g	e(g	PROPN
ejpam-4914	68	43	)	)	PUNCT
ejpam-4914	68	44	.	.	PUNCT
ejpam-4914	69	1	the	the	DET
ejpam-4914	69	2	smallest	small	ADJ
ejpam-4914	69	3	cardinality	cardinality	NOUN
ejpam-4914	69	4	of	of	ADP
ejpam-4914	69	5	a	a	DET
ejpam-4914	69	6	pointwise	pointwise	ADJ
ejpam-4914	69	7	nondominating	nondominate	VERB
ejpam-4914	69	8	set	set	NOUN
ejpam-4914	69	9	of	of	ADP
ejpam-4914	69	10	g	g	NOUN
ejpam-4914	69	11	,	,	PUNCT
ejpam-4914	69	12	denoted	denote	VERB
ejpam-4914	69	13	by	by	ADP
ejpam-4914	69	14	pnd(g	pnd(g	PROPN
ejpam-4914	69	15	)	)	PUNCT
ejpam-4914	69	16	,	,	PUNCT
ejpam-4914	69	17	is	be	AUX
ejpam-4914	69	18	called	call	VERB
ejpam-4914	69	19	the	the	DET
ejpam-4914	69	20	pointwise	pointwise	PROPN
ejpam-4914	69	21	nondomination	nondomination	NOUN
ejpam-4914	69	22	number	number	NOUN
ejpam-4914	69	23	of	of	ADP
ejpam-4914	69	24	g.	g.	PROPN
ejpam-4914	69	25	any	any	DET
ejpam-4914	69	26	point	point	NOUN
ejpam-4914	69	27	-	-	PUNCT
ejpam-4914	69	28	wise	wise	ADV
ejpam-4914	69	29	nondominating	nondominate	VERB
ejpam-4914	69	30	(	(	PUNCT
ejpam-4914	69	31	resp	resp	NOUN
ejpam-4914	69	32	.	.	PUNCT
ejpam-4914	70	1	dominating	dominate	VERB
ejpam-4914	70	2	pointwise	pointwise	PROPN
ejpam-4914	70	3	nondominating	nondominate	VERB
ejpam-4914	70	4	)	)	PUNCT
ejpam-4914	70	5	set	set	NOUN
ejpam-4914	70	6	s	s	PRON
ejpam-4914	70	7	of	of	ADP
ejpam-4914	70	8	g	g	NOUN
ejpam-4914	70	9	of	of	ADP
ejpam-4914	70	10	cardinality	cardinality	PROPN
ejpam-4914	70	11	|s|	|s|	PROPN
ejpam-4914	70	12	=	=	SYM
ejpam-4914	70	13	pnd(g	pnd(g	PROPN
ejpam-4914	70	14	)	)	PUNCT
ejpam-4914	70	15	(	(	PUNCT
ejpam-4914	70	16	resp	resp	NOUN
ejpam-4914	70	17	.	.	PUNCT
ejpam-4914	71	1	|s|	|s|	PROPN
ejpam-4914	71	2	=	=	SYM
ejpam-4914	71	3	γpnd(g	γpnd(g	PROPN
ejpam-4914	71	4	)	)	PUNCT
ejpam-4914	71	5	)	)	PUNCT
ejpam-4914	72	1	,	,	PUNCT
ejpam-4914	72	2	is	be	AUX
ejpam-4914	72	3	called	call	VERB
ejpam-4914	72	4	a	a	DET
ejpam-4914	72	5	pnd	pnd	NOUN
ejpam-4914	72	6	-	-	PUNCT
ejpam-4914	72	7	set	set	VERB
ejpam-4914	72	8	(	(	PUNCT
ejpam-4914	72	9	resp	resp	NOUN
ejpam-4914	72	10	.	.	PUNCT
ejpam-4914	73	1	γpnd	γpnd	NOUN
ejpam-4914	73	2	-	-	PUNCT
ejpam-4914	73	3	set	set	NOUN
ejpam-4914	73	4	)	)	PUNCT
ejpam-4914	73	5	of	of	ADP
ejpam-4914	73	6	g.	g.	PROPN
ejpam-4914	73	7	pnd	pnd	NOUN
ejpam-4914	73	8	-	-	PUNCT
ejpam-4914	73	9	sets	set	NOUN
ejpam-4914	73	10	are	be	AUX
ejpam-4914	73	11	introduced	introduce	VERB
ejpam-4914	73	12	and	and	CCONJ
ejpam-4914	73	13	discussed	discuss	VERB
ejpam-4914	73	14	in	in	ADP
ejpam-4914	73	15	[	[	X
ejpam-4914	73	16	5	5	NUM
ejpam-4914	73	17	]	]	PUNCT
ejpam-4914	73	18	.	.	PUNCT
ejpam-4914	74	1	a	a	DET
ejpam-4914	74	2	subset	subset	NOUN
ejpam-4914	74	3	s	s	X
ejpam-4914	74	4	of	of	ADP
ejpam-4914	74	5	v	v	NOUN
ejpam-4914	74	6	(	(	PUNCT
ejpam-4914	74	7	g	g	NOUN
ejpam-4914	74	8	)	)	PUNCT
ejpam-4914	74	9	is	be	AUX
ejpam-4914	74	10	a	a	DET
ejpam-4914	74	11	hop	hop	NOUN
ejpam-4914	74	12	dominating	dominating	NOUN
ejpam-4914	74	13	set	set	NOUN
ejpam-4914	74	14	of	of	ADP
ejpam-4914	74	15	g	g	PROPN
ejpam-4914	74	16	if	if	SCONJ
ejpam-4914	74	17	for	for	ADP
ejpam-4914	74	18	each	each	PRON
ejpam-4914	74	19	v	v	NUM
ejpam-4914	74	20	∈	∈	PROPN
ejpam-4914	74	21	v	v	NOUN
ejpam-4914	74	22	(	(	PUNCT
ejpam-4914	74	23	g	g	NOUN
ejpam-4914	74	24	)	)	PUNCT
ejpam-4914	74	25	\	\	PROPN
ejpam-4914	75	1	s	s	X
ejpam-4914	75	2	,	,	PUNCT
ejpam-4914	75	3	there	there	PRON
ejpam-4914	75	4	exists	exist	VERB
ejpam-4914	75	5	u	u	PROPN
ejpam-4914	75	6	∈	∈	PROPN
ejpam-4914	75	7	s	s	X
ejpam-4914	75	8	for	for	ADP
ejpam-4914	75	9	which	which	PRON
ejpam-4914	75	10	dg(u	dg(u	ADJ
ejpam-4914	75	11	,	,	PUNCT
ejpam-4914	75	12	v	v	NOUN
ejpam-4914	75	13	)	)	PUNCT
ejpam-4914	76	1	=	=	SYM
ejpam-4914	76	2	2	2	X
ejpam-4914	76	3	.	.	PUNCT
ejpam-4914	77	1	the	the	DET
ejpam-4914	77	2	minimum	minimum	ADJ
ejpam-4914	77	3	cardinality	cardinality	NOUN
ejpam-4914	77	4	of	of	ADP
ejpam-4914	77	5	a	a	DET
ejpam-4914	77	6	hop	hop	NOUN
ejpam-4914	77	7	dominating	dominating	NOUN
ejpam-4914	77	8	set	set	NOUN
ejpam-4914	77	9	is	be	AUX
ejpam-4914	77	10	called	call	VERB
ejpam-4914	77	11	the	the	DET
ejpam-4914	77	12	hop	hop	NOUN
ejpam-4914	77	13	domination	domination	NOUN
ejpam-4914	77	14	number	number	NOUN
ejpam-4914	77	15	of	of	ADP
ejpam-4914	77	16	g	g	NOUN
ejpam-4914	77	17	,	,	PUNCT
ejpam-4914	77	18	and	and	CCONJ
ejpam-4914	77	19	is	be	AUX
ejpam-4914	77	20	denoted	denote	VERB
ejpam-4914	77	21	by	by	ADP
ejpam-4914	77	22	γh(g	γh(g	NOUN
ejpam-4914	77	23	)	)	PUNCT
ejpam-4914	77	24	.	.	PUNCT
ejpam-4914	78	1	any	any	DET
ejpam-4914	78	2	hop	hop	NOUN
ejpam-4914	78	3	dominating	dominating	NOUN
ejpam-4914	78	4	set	set	NOUN
ejpam-4914	78	5	of	of	ADP
ejpam-4914	78	6	cardinality	cardinality	NOUN
ejpam-4914	78	7	γh(g	γh(g	PUNCT
ejpam-4914	78	8	)	)	PUNCT
ejpam-4914	78	9	is	be	AUX
ejpam-4914	78	10	called	call	VERB
ejpam-4914	78	11	γh	γh	ADV
ejpam-4914	78	12	-	-	PUNCT
ejpam-4914	78	13	set	set	NOUN
ejpam-4914	78	14	of	of	ADP
ejpam-4914	78	15	g.	g.	PROPN
ejpam-4914	78	16	good	good	ADJ
ejpam-4914	78	17	references	reference	NOUN
ejpam-4914	78	18	on	on	ADP
ejpam-4914	78	19	hop	hop	NOUN
ejpam-4914	78	20	domination	domination	NOUN
ejpam-4914	78	21	include	include	VERB
ejpam-4914	78	22	[	[	X
ejpam-4914	78	23	4	4	NUM
ejpam-4914	78	24	,	,	PUNCT
ejpam-4914	78	25	5	5	NUM
ejpam-4914	78	26	,	,	PUNCT
ejpam-4914	78	27	15	15	NUM
ejpam-4914	78	28	]	]	PUNCT
ejpam-4914	78	29	.	.	PUNCT
ejpam-4914	79	1	at	at	ADP
ejpam-4914	79	2	times	time	NOUN
ejpam-4914	79	3	we	we	PRON
ejpam-4914	79	4	write	write	VERB
ejpam-4914	79	5	s	s	PRON
ejpam-4914	79	6	∈	∈	NOUN
ejpam-4914	79	7	hd(g	hd(g	NOUN
ejpam-4914	79	8	)	)	PUNCT
ejpam-4914	79	9	to	to	PART
ejpam-4914	79	10	mean	mean	VERB
ejpam-4914	79	11	that	that	SCONJ
ejpam-4914	79	12	s	s	VERB
ejpam-4914	79	13	is	be	AUX
ejpam-4914	79	14	a	a	DET
ejpam-4914	79	15	hop	hop	NOUN
ejpam-4914	79	16	dominating	dominating	NOUN
ejpam-4914	79	17	set	set	NOUN
ejpam-4914	79	18	of	of	ADP
ejpam-4914	79	19	g.	g.	PROPN
ejpam-4914	79	20	for	for	ADP
ejpam-4914	79	21	a	a	DET
ejpam-4914	79	22	vertex	vertex	NOUN
ejpam-4914	79	23	v	v	NOUN
ejpam-4914	79	24	of	of	ADP
ejpam-4914	79	25	a	a	DET
ejpam-4914	79	26	connected	connected	ADJ
ejpam-4914	79	27	graph	graph	NOUN
ejpam-4914	79	28	g	g	NOUN
ejpam-4914	79	29	,	,	PUNCT
ejpam-4914	79	30	ng(v	ng(v	PUNCT
ejpam-4914	79	31	,	,	PUNCT
ejpam-4914	79	32	2	2	X
ejpam-4914	79	33	)	)	PUNCT
ejpam-4914	79	34	=	=	PRON
ejpam-4914	79	35	{	{	PUNCT
ejpam-4914	79	36	u	u	NOUN
ejpam-4914	79	37	∈	∈	PROPN
ejpam-4914	79	38	v	v	NOUN
ejpam-4914	79	39	(	(	PUNCT
ejpam-4914	79	40	g	g	NOUN
ejpam-4914	79	41	)	)	PUNCT
ejpam-4914	79	42	:	:	PUNCT
ejpam-4914	79	43	dg(u	dg(u	X
ejpam-4914	79	44	,	,	PUNCT
ejpam-4914	79	45	v	v	NOUN
ejpam-4914	79	46	)	)	PUNCT
ejpam-4914	79	47	=	=	SYM
ejpam-4914	79	48	2	2	NUM
ejpam-4914	79	49	}	}	PUNCT
ejpam-4914	79	50	.	.	PUNCT
ejpam-4914	80	1	each	each	DET
ejpam-4914	80	2	element	element	NOUN
ejpam-4914	80	3	of	of	ADP
ejpam-4914	80	4	ng(v	ng(v	NOUN
ejpam-4914	80	5	,	,	PUNCT
ejpam-4914	80	6	2	2	NUM
ejpam-4914	80	7	)	)	PUNCT
ejpam-4914	80	8	is	be	AUX
ejpam-4914	80	9	called	call	VERB
ejpam-4914	80	10	a	a	DET
ejpam-4914	80	11	hop	hop	NOUN
ejpam-4914	80	12	-	-	PUNCT
ejpam-4914	80	13	neighbor	neighbor	NOUN
ejpam-4914	80	14	of	of	ADP
ejpam-4914	80	15	v.	v.	ADP
ejpam-4914	80	16	for	for	ADP
ejpam-4914	80	17	s	s	PROPN
ejpam-4914	80	18	⊆	⊆	NUM
ejpam-4914	80	19	v	v	NOUN
ejpam-4914	80	20	(	(	PUNCT
ejpam-4914	80	21	g	g	NOUN
ejpam-4914	80	22	)	)	PUNCT
ejpam-4914	80	23	,	,	PUNCT
ejpam-4914	80	24	ng(s	ng(s	CCONJ
ejpam-4914	80	25	,	,	PUNCT
ejpam-4914	80	26	2	2	X
ejpam-4914	80	27	)	)	PUNCT
ejpam-4914	80	28	=	=	SYM
ejpam-4914	80	29	∪v∈sng(v	∪v∈sng(v	PROPN
ejpam-4914	80	30	,	,	PUNCT
ejpam-4914	80	31	2	2	NUM
ejpam-4914	80	32	)	)	PUNCT
ejpam-4914	80	33	and	and	CCONJ
ejpam-4914	80	34	ng[s	ng[s	PROPN
ejpam-4914	80	35	,	,	PUNCT
ejpam-4914	80	36	2	2	NUM
ejpam-4914	80	37	]	]	PUNCT
ejpam-4914	80	38	=	=	SYM
ejpam-4914	80	39	ng(s	ng(s	NOUN
ejpam-4914	80	40	,	,	PUNCT
ejpam-4914	80	41	2)∪s	2)∪s	NUM
ejpam-4914	80	42	.	.	PUNCT
ejpam-4914	81	1	precisely	precisely	ADV
ejpam-4914	81	2	,	,	PUNCT
ejpam-4914	81	3	s	s	VERB
ejpam-4914	81	4	is	be	AUX
ejpam-4914	81	5	a	a	DET
ejpam-4914	81	6	hop	hop	NOUN
ejpam-4914	81	7	dominating	dominating	NOUN
ejpam-4914	81	8	set	set	NOUN
ejpam-4914	81	9	if	if	SCONJ
ejpam-4914	81	10	and	and	CCONJ
ejpam-4914	81	11	only	only	ADV
ejpam-4914	81	12	if	if	SCONJ
ejpam-4914	81	13	ng[s	ng[	NOUN
ejpam-4914	81	14	,	,	PUNCT
ejpam-4914	81	15	2	2	NUM
ejpam-4914	81	16	]	]	PUNCT
ejpam-4914	81	17	=	=	SYM
ejpam-4914	81	18	v	v	NOUN
ejpam-4914	81	19	(	(	PUNCT
ejpam-4914	81	20	g	g	NOUN
ejpam-4914	81	21	)	)	PUNCT
ejpam-4914	81	22	.	.	PUNCT
ejpam-4914	82	1	a	a	DET
ejpam-4914	82	2	subset	subset	NOUN
ejpam-4914	82	3	s	s	X
ejpam-4914	82	4	of	of	ADP
ejpam-4914	82	5	v	v	NOUN
ejpam-4914	82	6	(	(	PUNCT
ejpam-4914	82	7	g	g	NOUN
ejpam-4914	82	8	)	)	PUNCT
ejpam-4914	82	9	is	be	AUX
ejpam-4914	82	10	a	a	DET
ejpam-4914	82	11	2	2	NUM
ejpam-4914	82	12	-	-	PUNCT
ejpam-4914	82	13	hop	hop	NOUN
ejpam-4914	82	14	dominating	dominating	NOUN
ejpam-4914	82	15	set	set	NOUN
ejpam-4914	82	16	of	of	ADP
ejpam-4914	82	17	g	g	PROPN
ejpam-4914	82	18	if	if	SCONJ
ejpam-4914	82	19	for	for	ADP
ejpam-4914	82	20	each	each	DET
ejpam-4914	82	21	v	v	NUM
ejpam-4914	82	22	∈	∈	NOUN
ejpam-4914	82	23	v	v	NOUN
ejpam-4914	82	24	(	(	PUNCT
ejpam-4914	82	25	g)\s	g)\s	NOUN
ejpam-4914	82	26	,	,	PUNCT
ejpam-4914	82	27	there	there	PRON
ejpam-4914	82	28	exist	exist	VERB
ejpam-4914	82	29	distinct	distinct	ADJ
ejpam-4914	82	30	vertices	vertex	NOUN
ejpam-4914	82	31	u	u	NOUN
ejpam-4914	82	32	,	,	PUNCT
ejpam-4914	82	33	w	w	PROPN
ejpam-4914	82	34	∈	∈	PROPN
ejpam-4914	82	35	s	s	X
ejpam-4914	82	36	for	for	ADP
ejpam-4914	82	37	which	which	PRON
ejpam-4914	82	38	dg(u	dg(u	ADJ
ejpam-4914	82	39	,	,	PUNCT
ejpam-4914	82	40	v	v	NOUN
ejpam-4914	82	41	)	)	PUNCT
ejpam-4914	82	42	=	=	SYM
ejpam-4914	82	43	2	2	NUM
ejpam-4914	82	44	=	=	SYM
ejpam-4914	82	45	dg(w	dg(w	X
ejpam-4914	82	46	,	,	PUNCT
ejpam-4914	82	47	v	v	NOUN
ejpam-4914	82	48	)	)	PUNCT
ejpam-4914	82	49	.	.	PUNCT
ejpam-4914	83	1	the	the	DET
ejpam-4914	83	2	minimum	minimum	ADJ
ejpam-4914	83	3	cardinality	cardinality	NOUN
ejpam-4914	83	4	of	of	ADP
ejpam-4914	83	5	a	a	DET
ejpam-4914	83	6	2	2	NUM
ejpam-4914	83	7	-	-	PUNCT
ejpam-4914	83	8	hop	hop	NOUN
ejpam-4914	83	9	dominating	dominating	NOUN
ejpam-4914	83	10	set	set	NOUN
ejpam-4914	83	11	of	of	ADP
ejpam-4914	83	12	g	g	PROPN
ejpam-4914	83	13	is	be	AUX
ejpam-4914	83	14	the	the	DET
ejpam-4914	83	15	2	2	NUM
ejpam-4914	83	16	-	-	PUNCT
ejpam-4914	83	17	hop	hop	NOUN
ejpam-4914	83	18	domination	domination	NOUN
ejpam-4914	83	19	number	number	NOUN
ejpam-4914	83	20	of	of	ADP
ejpam-4914	83	21	g	g	NOUN
ejpam-4914	83	22	,	,	PUNCT
ejpam-4914	83	23	denoted	denote	VERB
ejpam-4914	83	24	by	by	ADP
ejpam-4914	83	25	γ2h(g	γ2h(g	NOUN
ejpam-4914	83	26	)	)	PUNCT
ejpam-4914	83	27	.	.	PUNCT
ejpam-4914	84	1	a	a	DET
ejpam-4914	84	2	comprehensive	comprehensive	ADJ
ejpam-4914	84	3	study	study	NOUN
ejpam-4914	84	4	on	on	ADP
ejpam-4914	84	5	2	2	NUM
ejpam-4914	84	6	-	-	PUNCT
ejpam-4914	84	7	hop	hop	NOUN
ejpam-4914	84	8	domination	domination	NOUN
ejpam-4914	84	9	is	be	AUX
ejpam-4914	84	10	given	give	VERB
ejpam-4914	84	11	in	in	ADP
ejpam-4914	84	12	[	[	X
ejpam-4914	84	13	14	14	NUM
ejpam-4914	84	14	]	]	SYM
ejpam-4914	84	15	,	,	PUNCT
ejpam-4914	84	16	where	where	SCONJ
ejpam-4914	84	17	2	2	NUM
ejpam-4914	84	18	-	-	PUNCT
ejpam-4914	84	19	hop	hop	NOUN
ejpam-4914	84	20	domination	domination	NOUN
ejpam-4914	84	21	is	be	AUX
ejpam-4914	84	22	referred	refer	VERB
ejpam-4914	84	23	to	to	ADP
ejpam-4914	84	24	as	as	ADV
ejpam-4914	84	25	double	double	ADJ
ejpam-4914	84	26	hop	hop	NOUN
ejpam-4914	84	27	domination	domination	NOUN
ejpam-4914	84	28	.	.	PUNCT
ejpam-4914	85	1	here	here	ADV
ejpam-4914	85	2	we	we	PRON
ejpam-4914	85	3	also	also	ADV
ejpam-4914	85	4	write	write	VERB
ejpam-4914	85	5	s	s	PRON
ejpam-4914	85	6	∈	∈	ADJ
ejpam-4914	85	7	2	2	NUM
ejpam-4914	85	8	-	-	PUNCT
ejpam-4914	85	9	hd(g	hd(g	NOUN
ejpam-4914	85	10	)	)	PUNCT
ejpam-4914	85	11	to	to	PART
ejpam-4914	85	12	mean	mean	VERB
ejpam-4914	85	13	that	that	SCONJ
ejpam-4914	85	14	s	s	VERB
ejpam-4914	85	15	is	be	AUX
ejpam-4914	85	16	a	a	DET
ejpam-4914	85	17	2	2	NUM
ejpam-4914	85	18	-	-	PUNCT
ejpam-4914	85	19	hop	hop	NOUN
ejpam-4914	85	20	dominating	dominating	NOUN
ejpam-4914	85	21	set	set	NOUN
ejpam-4914	85	22	of	of	ADP
ejpam-4914	85	23	g	g	PROPN
ejpam-4914	85	24	a	a	DET
ejpam-4914	85	25	set	set	NOUN
ejpam-4914	85	26	s	s	NOUN
ejpam-4914	85	27	⊆	⊆	NUM
ejpam-4914	85	28	v	v	NOUN
ejpam-4914	85	29	(	(	PUNCT
ejpam-4914	85	30	g	g	NOUN
ejpam-4914	85	31	)	)	PUNCT
ejpam-4914	85	32	is	be	AUX
ejpam-4914	85	33	a	a	DET
ejpam-4914	85	34	(	(	PUNCT
ejpam-4914	85	35	1	1	NUM
ejpam-4914	85	36	,	,	PUNCT
ejpam-4914	85	37	2)∗-dominating	2)∗-dominate	VERB
ejpam-4914	85	38	set	set	NOUN
ejpam-4914	85	39	of	of	ADP
ejpam-4914	85	40	g	g	PROPN
ejpam-4914	85	41	(	(	PUNCT
ejpam-4914	85	42	resp	resp	NOUN
ejpam-4914	85	43	.	.	PUNCT
ejpam-4914	86	1	(	(	PUNCT
ejpam-4914	86	2	1	1	NUM
ejpam-4914	86	3	,	,	PUNCT
ejpam-4914	86	4	2)∗-total	2)∗-total	ADJ
ejpam-4914	86	5	dominating	dominating	NOUN
ejpam-4914	86	6	set	set	NOUN
ejpam-4914	86	7	)	)	PUNCT
ejpam-4914	86	8	if	if	SCONJ
ejpam-4914	86	9	it	it	PRON
ejpam-4914	86	10	is	be	AUX
ejpam-4914	86	11	both	both	CCONJ
ejpam-4914	86	12	a	a	DET
ejpam-4914	86	13	dominating	dominating	NOUN
ejpam-4914	86	14	(	(	PUNCT
ejpam-4914	86	15	resp	resp	NOUN
ejpam-4914	86	16	.	.	PUNCT
ejpam-4914	87	1	a	a	DET
ejpam-4914	87	2	total	total	ADJ
ejpam-4914	87	3	dominating	dominating	NOUN
ejpam-4914	87	4	)	)	PUNCT
ejpam-4914	87	5	set	set	NOUN
ejpam-4914	87	6	and	and	CCONJ
ejpam-4914	87	7	a	a	DET
ejpam-4914	87	8	hop	hop	NOUN
ejpam-4914	87	9	dominating	dominating	NOUN
ejpam-4914	87	10	set	set	NOUN
ejpam-4914	87	11	of	of	ADP
ejpam-4914	87	12	g.	g.	PROPN
ejpam-4914	87	13	the	the	DET
ejpam-4914	87	14	smallest	small	ADJ
ejpam-4914	87	15	cardinality	cardinality	NOUN
ejpam-4914	87	16	of	of	ADP
ejpam-4914	87	17	a	a	DET
ejpam-4914	87	18	(	(	PUNCT
ejpam-4914	87	19	1	1	NUM
ejpam-4914	87	20	,	,	PUNCT
ejpam-4914	87	21	2)∗	2)∗	NOUN
ejpam-4914	87	22	-dominating	-dominating	NOUN
ejpam-4914	87	23	(	(	PUNCT
ejpam-4914	87	24	resp	resp	NOUN
ejpam-4914	87	25	.	.	PUNCT
ejpam-4914	88	1	(	(	PUNCT
ejpam-4914	88	2	1	1	NUM
ejpam-4914	88	3	,	,	PUNCT
ejpam-4914	88	4	2)∗-total	2)∗-total	ADJ
ejpam-4914	88	5	dominating	dominating	NOUN
ejpam-4914	88	6	)	)	PUNCT
ejpam-4914	88	7	set	set	NOUN
ejpam-4914	88	8	of	of	ADP
ejpam-4914	88	9	g	g	NOUN
ejpam-4914	88	10	,	,	PUNCT
ejpam-4914	88	11	denoted	denote	VERB
ejpam-4914	88	12	by	by	ADP
ejpam-4914	88	13	γ∗1,2(g	γ∗1,2(g	NOUN
ejpam-4914	88	14	)	)	PUNCT
ejpam-4914	88	15	(	(	PUNCT
ejpam-4914	88	16	resp.γ∗t1,2(g	resp.γ∗t1,2(g	NOUN
ejpam-4914	88	17	)	)	PUNCT
ejpam-4914	88	18	)	)	PUNCT
ejpam-4914	88	19	is	be	AUX
ejpam-4914	88	20	called	call	VERB
ejpam-4914	88	21	the	the	DET
ejpam-4914	88	22	(	(	PUNCT
ejpam-4914	88	23	1	1	NUM
ejpam-4914	88	24	,	,	PUNCT
ejpam-4914	88	25	2)∗	2)∗	NOUN
ejpam-4914	88	26	-domination	-domination	NOUN
ejpam-4914	88	27	number	number	NOUN
ejpam-4914	88	28	(	(	PUNCT
ejpam-4914	88	29	resp	resp	NOUN
ejpam-4914	88	30	.	.	PUNCT
ejpam-4914	89	1	(	(	PUNCT
ejpam-4914	89	2	1	1	NUM
ejpam-4914	89	3	,	,	PUNCT
ejpam-4914	89	4	2)∗total	2)∗total	NUM
ejpam-4914	89	5	domination	domination	NOUN
ejpam-4914	89	6	number	number	NOUN
ejpam-4914	89	7	)	)	PUNCT
ejpam-4914	89	8	of	of	ADP
ejpam-4914	89	9	g.	g.	PROPN
ejpam-4914	89	10	a	a	PRON
ejpam-4914	89	11	(	(	PUNCT
ejpam-4914	89	12	1	1	NUM
ejpam-4914	89	13	,	,	PUNCT
ejpam-4914	89	14	2)∗	2)∗	NOUN
ejpam-4914	89	15	-dominating	-dominating	NOUN
ejpam-4914	89	16	(	(	PUNCT
ejpam-4914	89	17	resp	resp	NOUN
ejpam-4914	89	18	.	.	PUNCT
ejpam-4914	90	1	(	(	PUNCT
ejpam-4914	90	2	1	1	NUM
ejpam-4914	90	3	,	,	PUNCT
ejpam-4914	90	4	2)∗	2)∗	NOUN
ejpam-4914	90	5	-total	-total	ADJ
ejpam-4914	90	6	dominating	dominating	NOUN
ejpam-4914	90	7	)	)	PUNCT
ejpam-4914	90	8	set	set	VERB
ejpam-4914	90	9	s	s	PRON
ejpam-4914	90	10	with	with	ADP
ejpam-4914	90	11	|s|	|s|	PROPN
ejpam-4914	90	12	=	=	PUNCT
ejpam-4914	90	13	γ∗1,2(g	γ∗1,2(g	NOUN
ejpam-4914	90	14	)	)	PUNCT
ejpam-4914	90	15	(	(	PUNCT
ejpam-4914	90	16	resp	resp	NOUN
ejpam-4914	90	17	.	.	PUNCT
ejpam-4914	91	1	|s|	|s|	PROPN
ejpam-4914	91	2	=	=	SYM
ejpam-4914	91	3	γ∗t1,2(g	γ∗t1,2(g	PROPN
ejpam-4914	91	4	)	)	PUNCT
ejpam-4914	91	5	)	)	PUNCT
ejpam-4914	91	6	is	be	AUX
ejpam-4914	91	7	called	call	VERB
ejpam-4914	91	8	a	a	DET
ejpam-4914	91	9	γ∗1,2	γ∗1,2	NOUN
ejpam-4914	91	10	-	-	PUNCT
ejpam-4914	91	11	set	set	VERB
ejpam-4914	91	12	(	(	PUNCT
ejpam-4914	91	13	resp	resp	NOUN
ejpam-4914	91	14	.	.	PUNCT
ejpam-4914	92	1	γ∗t1,2	γ∗t1,2	ADJ
ejpam-4914	92	2	-	-	PUNCT
ejpam-4914	92	3	set	set	NOUN
ejpam-4914	92	4	)	)	PUNCT
ejpam-4914	92	5	of	of	ADP
ejpam-4914	92	6	g.	g.	PROPN
ejpam-4914	92	7	the	the	DET
ejpam-4914	92	8	concept	concept	NOUN
ejpam-4914	92	9	of	of	ADP
ejpam-4914	92	10	(	(	PUNCT
ejpam-4914	92	11	1	1	NUM
ejpam-4914	92	12	,	,	PUNCT
ejpam-4914	92	13	2)∗-domination	2)∗-domination	NOUN
ejpam-4914	92	14	(	(	PUNCT
ejpam-4914	92	15	a	a	DET
ejpam-4914	92	16	variation	variation	NOUN
ejpam-4914	92	17	of	of	ADP
ejpam-4914	92	18	(	(	PUNCT
ejpam-4914	92	19	1	1	NUM
ejpam-4914	92	20	,	,	PUNCT
ejpam-4914	92	21	2)-domination	2)-domination	NUM
ejpam-4914	92	22	)	)	PUNCT
ejpam-4914	92	23	is	be	AUX
ejpam-4914	92	24	introduced	introduce	VERB
ejpam-4914	92	25	in	in	ADP
ejpam-4914	92	26	[	[	X
ejpam-4914	92	27	4	4	NUM
ejpam-4914	92	28	]	]	PUNCT
ejpam-4914	92	29	.	.	PUNCT
ejpam-4914	93	1	a	a	DET
ejpam-4914	93	2	function	function	NOUN
ejpam-4914	93	3	f	f	NOUN
ejpam-4914	93	4	:	:	PUNCT
ejpam-4914	93	5	v	v	X
ejpam-4914	93	6	(	(	PUNCT
ejpam-4914	93	7	g	g	NOUN
ejpam-4914	93	8	)	)	PUNCT
ejpam-4914	93	9	→	→	SYM
ejpam-4914	93	10	{	{	PUNCT
ejpam-4914	93	11	0	0	NUM
ejpam-4914	93	12	,	,	PUNCT
ejpam-4914	93	13	1	1	NUM
ejpam-4914	93	14	,	,	PUNCT
ejpam-4914	93	15	2	2	NUM
ejpam-4914	93	16	}	}	PUNCT
ejpam-4914	93	17	is	be	AUX
ejpam-4914	93	18	a	a	DET
ejpam-4914	93	19	hop	hop	NOUN
ejpam-4914	93	20	roman	roman	ADJ
ejpam-4914	93	21	dominating	dominating	NOUN
ejpam-4914	93	22	function	function	NOUN
ejpam-4914	93	23	of	of	ADP
ejpam-4914	93	24	g	g	PROPN
ejpam-4914	93	25	if	if	SCONJ
ejpam-4914	93	26	for	for	ADP
ejpam-4914	93	27	each	each	DET
ejpam-4914	93	28	v	v	NUM
ejpam-4914	93	29	∈	∈	PROPN
ejpam-4914	93	30	v	v	NOUN
ejpam-4914	93	31	(	(	PUNCT
ejpam-4914	93	32	g	g	NOUN
ejpam-4914	93	33	)	)	PUNCT
ejpam-4914	93	34	with	with	ADP
ejpam-4914	93	35	f(v	f(v	NOUN
ejpam-4914	93	36	)	)	PUNCT
ejpam-4914	94	1	=	=	SYM
ejpam-4914	94	2	0	0	NUM
ejpam-4914	95	1	there	there	PRON
ejpam-4914	95	2	exists	exist	VERB
ejpam-4914	95	3	u	u	PROPN
ejpam-4914	95	4	∈	∈	PROPN
ejpam-4914	95	5	v	v	ADP
ejpam-4914	95	6	(	(	PUNCT
ejpam-4914	95	7	g	g	NOUN
ejpam-4914	95	8	)	)	PUNCT
ejpam-4914	95	9	for	for	ADP
ejpam-4914	95	10	which	which	PRON
ejpam-4914	95	11	dg(u	dg(u	X
ejpam-4914	95	12	,	,	PUNCT
ejpam-4914	95	13	v	v	NOUN
ejpam-4914	95	14	)	)	PUNCT
ejpam-4914	95	15	=	=	SYM
ejpam-4914	95	16	2	2	NUM
ejpam-4914	95	17	and	and	CCONJ
ejpam-4914	95	18	f(u	f(u	PROPN
ejpam-4914	95	19	)	)	PUNCT
ejpam-4914	95	20	=	=	SYM
ejpam-4914	95	21	2	2	X
ejpam-4914	95	22	.	.	PUNCT
ejpam-4914	95	23	the	the	DET
ejpam-4914	95	24	s.r	s.r	PROPN
ejpam-4914	95	25	.	.	PROPN
ejpam-4914	95	26	jr	jr	PROPN
ejpam-4914	95	27	.	.	PROPN
ejpam-4914	95	28	canoy	canoy	PROPN
ejpam-4914	95	29	,	,	PUNCT
ejpam-4914	95	30	f.p	f.p	PROPN
ejpam-4914	95	31	.	.	PROPN
ejpam-4914	95	32	jamil	jamil	PROPN
ejpam-4914	95	33	and	and	CCONJ
ejpam-4914	95	34	s.m	s.m	PROPN
ejpam-4914	95	35	.	.	PROPN
ejpam-4914	95	36	menchavez	menchavez	PROPN
ejpam-4914	95	37	/	/	PUNCT
ejpam-4914	95	38	eur	eur	PROPN
ejpam-4914	95	39	.	.	PUNCT
ejpam-4914	96	1	j.	j.	PROPN
ejpam-4914	96	2	pure	pure	PROPN
ejpam-4914	96	3	appl	appl	PROPN
ejpam-4914	96	4	.	.	PROPN
ejpam-4914	96	5	math	math	PROPN
ejpam-4914	96	6	,	,	PUNCT
ejpam-4914	96	7	16	16	NUM
ejpam-4914	96	8	(	(	PUNCT
ejpam-4914	96	9	4	4	NUM
ejpam-4914	96	10	)	)	PUNCT
ejpam-4914	96	11	(	(	PUNCT
ejpam-4914	96	12	2023	2023	NUM
ejpam-4914	96	13	)	)	PUNCT
ejpam-4914	96	14	,	,	PUNCT
ejpam-4914	96	15	2431	2431	NUM
ejpam-4914	96	16	-	-	SYM
ejpam-4914	96	17	2449	2449	NUM
ejpam-4914	96	18	2434	2434	NUM
ejpam-4914	96	19	sum	sum	NOUN
ejpam-4914	96	20	ωg(f	ωg(f	PRON
ejpam-4914	96	21	)	)	PUNCT
ejpam-4914	96	22	=	=	SYM
ejpam-4914	96	23	∑	∑	PUNCT
ejpam-4914	96	24	v∈v	v∈v	PROPN
ejpam-4914	96	25	(	(	PUNCT
ejpam-4914	96	26	g	g	NOUN
ejpam-4914	96	27	)	)	PUNCT
ejpam-4914	96	28	f(v	f(v	NOUN
ejpam-4914	96	29	)	)	PUNCT
ejpam-4914	96	30	is	be	AUX
ejpam-4914	96	31	the	the	DET
ejpam-4914	96	32	weight	weight	NOUN
ejpam-4914	96	33	of	of	ADP
ejpam-4914	96	34	f	f	PROPN
ejpam-4914	96	35	in	in	ADP
ejpam-4914	96	36	g.	g.	PROPN
ejpam-4914	96	37	the	the	DET
ejpam-4914	96	38	minimum	minimum	ADJ
ejpam-4914	96	39	weight	weight	NOUN
ejpam-4914	96	40	of	of	ADP
ejpam-4914	96	41	a	a	DET
ejpam-4914	96	42	hop	hop	NOUN
ejpam-4914	96	43	roman	roman	ADJ
ejpam-4914	96	44	dominating	dominating	NOUN
ejpam-4914	96	45	function	function	NOUN
ejpam-4914	96	46	of	of	ADP
ejpam-4914	96	47	g	g	PROPN
ejpam-4914	96	48	is	be	AUX
ejpam-4914	96	49	the	the	DET
ejpam-4914	96	50	hop	hop	NOUN
ejpam-4914	96	51	roman	roman	ADJ
ejpam-4914	96	52	domination	domination	NOUN
ejpam-4914	96	53	number	number	NOUN
ejpam-4914	96	54	of	of	ADP
ejpam-4914	96	55	g	g	NOUN
ejpam-4914	96	56	,	,	PUNCT
ejpam-4914	96	57	and	and	CCONJ
ejpam-4914	96	58	is	be	AUX
ejpam-4914	96	59	denoted	denote	VERB
ejpam-4914	96	60	by	by	ADP
ejpam-4914	96	61	γhr(g	γhr(g	PROPN
ejpam-4914	96	62	)	)	PUNCT
ejpam-4914	96	63	.	.	PUNCT
ejpam-4914	97	1	1.1	1.1	NUM
ejpam-4914	97	2	.	.	PUNCT
ejpam-4914	98	1	some	some	DET
ejpam-4914	98	2	known	know	VERB
ejpam-4914	98	3	results	result	NOUN
ejpam-4914	98	4	theorem	theorem	VERB
ejpam-4914	98	5	1.1	1.1	NUM
ejpam-4914	98	6	.	.	PUNCT
ejpam-4914	99	1	[	[	X
ejpam-4914	99	2	24	24	NUM
ejpam-4914	99	3	]	]	PUNCT
ejpam-4914	99	4	for	for	ADP
ejpam-4914	99	5	any	any	DET
ejpam-4914	99	6	graph	graph	NOUN
ejpam-4914	99	7	g	g	NOUN
ejpam-4914	99	8	,	,	PUNCT
ejpam-4914	99	9	γhr(g	γhr(g	PROPN
ejpam-4914	99	10	)	)	PUNCT
ejpam-4914	99	11	≤	≤	NOUN
ejpam-4914	99	12	2γh(g	2γh(g	NOUN
ejpam-4914	99	13	)	)	PUNCT
ejpam-4914	99	14	.	.	PUNCT
ejpam-4914	100	1	observation	observation	NOUN
ejpam-4914	100	2	1.2	1.2	NUM
ejpam-4914	100	3	.	.	PUNCT
ejpam-4914	101	1	(	(	PUNCT
ejpam-4914	101	2	i	i	NOUN
ejpam-4914	101	3	)	)	PUNCT
ejpam-4914	101	4	γhr(pn	γhr(pn	NOUN
ejpam-4914	101	5	)	)	PUNCT
ejpam-4914	101	6	=	=	NOUN
ejpam-4914	101	7	{	{	PUNCT
ejpam-4914	101	8	4k	4k	NOUN
ejpam-4914	101	9	+	+	CCONJ
ejpam-4914	101	10	r	r	NOUN
ejpam-4914	101	11	,	,	PUNCT
ejpam-4914	101	12	if	if	SCONJ
ejpam-4914	101	13	n	n	NOUN
ejpam-4914	101	14	=	=	SYM
ejpam-4914	101	15	6k	6k	NOUN
ejpam-4914	101	16	+	+	CCONJ
ejpam-4914	101	17	r	r	NOUN
ejpam-4914	101	18	;	;	PUNCT
ejpam-4914	101	19	0	0	NUM
ejpam-4914	101	20	≤	≤	NUM
ejpam-4914	101	21	r	r	NOUN
ejpam-4914	101	22	≤	≤	NUM
ejpam-4914	101	23	3	3	NUM
ejpam-4914	101	24	;	;	PUNCT
ejpam-4914	101	25	k	k	X
ejpam-4914	101	26	≥	≥	X
ejpam-4914	101	27	0	0	NUM
ejpam-4914	101	28	4k	4k	NUM
ejpam-4914	101	29	+	+	CCONJ
ejpam-4914	101	30	4	4	NUM
ejpam-4914	101	31	,	,	PUNCT
ejpam-4914	101	32	if	if	SCONJ
ejpam-4914	101	33	n	n	NOUN
ejpam-4914	101	34	=	=	SYM
ejpam-4914	101	35	6k	6k	NOUN
ejpam-4914	102	1	+	+	CCONJ
ejpam-4914	102	2	r	r	NOUN
ejpam-4914	102	3	;	;	PUNCT
ejpam-4914	102	4	4	4	NUM
ejpam-4914	102	5	≤	≤	NOUN
ejpam-4914	102	6	r	r	NOUN
ejpam-4914	102	7	≤	≤	NUM
ejpam-4914	102	8	5	5	NUM
ejpam-4914	102	9	;	;	PUNCT
ejpam-4914	102	10	k	k	X
ejpam-4914	102	11	≥	≥	X
ejpam-4914	102	12	0	0	NUM
ejpam-4914	102	13	,	,	PUNCT
ejpam-4914	102	14	and	and	CCONJ
ejpam-4914	102	15	(	(	PUNCT
ejpam-4914	102	16	ii	ii	NOUN
ejpam-4914	102	17	)	)	PUNCT
ejpam-4914	102	18	γhr(cn	γhr(cn	NOUN
ejpam-4914	102	19	)	)	PUNCT
ejpam-4914	102	20	=	=	PUNCT
ejpam-4914	102	21			NOUN
ejpam-4914	102	22	3	3	NUM
ejpam-4914	102	23	,	,	PUNCT
ejpam-4914	102	24	if	if	SCONJ
ejpam-4914	102	25	n	n	NOUN
ejpam-4914	102	26	=	=	SYM
ejpam-4914	102	27	3	3	NUM
ejpam-4914	102	28	;	;	PUNCT
ejpam-4914	102	29	4	4	NUM
ejpam-4914	102	30	,	,	PUNCT
ejpam-4914	102	31	if	if	SCONJ
ejpam-4914	102	32	n	n	NOUN
ejpam-4914	102	33	=	=	SYM
ejpam-4914	102	34	4	4	NUM
ejpam-4914	102	35	,	,	PUNCT
ejpam-4914	102	36	5	5	NUM
ejpam-4914	102	37	;	;	PUNCT
ejpam-4914	102	38	4k	4k	NUM
ejpam-4914	102	39	+	+	CCONJ
ejpam-4914	102	40	r	r	NOUN
ejpam-4914	102	41	,	,	PUNCT
ejpam-4914	102	42	if	if	SCONJ
ejpam-4914	102	43	n	n	NOUN
ejpam-4914	102	44	=	=	SYM
ejpam-4914	102	45	6k	6k	NOUN
ejpam-4914	102	46	+	+	CCONJ
ejpam-4914	102	47	r	r	NOUN
ejpam-4914	102	48	;	;	PUNCT
ejpam-4914	102	49	0	0	NUM
ejpam-4914	102	50	≤	≤	NUM
ejpam-4914	102	51	r	r	NOUN
ejpam-4914	102	52	≤	≤	NUM
ejpam-4914	102	53	3	3	NUM
ejpam-4914	102	54	;	;	PUNCT
ejpam-4914	102	55	k	k	X
ejpam-4914	102	56	≥	≥	NUM
ejpam-4914	102	57	1	1	NUM
ejpam-4914	102	58	4k	4k	NUM
ejpam-4914	102	59	+	+	CCONJ
ejpam-4914	102	60	4	4	NUM
ejpam-4914	102	61	,	,	PUNCT
ejpam-4914	102	62	if	if	SCONJ
ejpam-4914	102	63	n	n	NOUN
ejpam-4914	102	64	=	=	SYM
ejpam-4914	102	65	6k	6k	NOUN
ejpam-4914	103	1	+	+	CCONJ
ejpam-4914	103	2	r	r	NOUN
ejpam-4914	103	3	;	;	PUNCT
ejpam-4914	103	4	4	4	NUM
ejpam-4914	103	5	≤	≤	NOUN
ejpam-4914	103	6	r	r	NOUN
ejpam-4914	103	7	≤	≤	NUM
ejpam-4914	103	8	5	5	NUM
ejpam-4914	103	9	;	;	PUNCT
ejpam-4914	103	10	k	k	X
ejpam-4914	103	11	≥	≥	NUM
ejpam-4914	103	12	1	1	NUM
ejpam-4914	103	13	.	.	NOUN
ejpam-4914	103	14	2	2	NUM
ejpam-4914	103	15	.	.	X
ejpam-4914	103	16	hop	hop	PROPN
ejpam-4914	103	17	italian	italian	ADJ
ejpam-4914	103	18	domination	domination	NOUN
ejpam-4914	103	19	a	a	DET
ejpam-4914	103	20	function	function	NOUN
ejpam-4914	104	1	f	f	NOUN
ejpam-4914	104	2	:	:	PUNCT
ejpam-4914	104	3	v	v	X
ejpam-4914	104	4	(	(	PUNCT
ejpam-4914	104	5	g	g	NOUN
ejpam-4914	104	6	)	)	PUNCT
ejpam-4914	104	7	→	→	SYM
ejpam-4914	104	8	{	{	PUNCT
ejpam-4914	104	9	0	0	NUM
ejpam-4914	104	10	,	,	PUNCT
ejpam-4914	104	11	1	1	NUM
ejpam-4914	104	12	,	,	PUNCT
ejpam-4914	104	13	2	2	NUM
ejpam-4914	104	14	}	}	PUNCT
ejpam-4914	104	15	is	be	AUX
ejpam-4914	104	16	a	a	DET
ejpam-4914	104	17	hop	hop	NOUN
ejpam-4914	104	18	italian	italian	ADJ
ejpam-4914	104	19	dominating	dominating	NOUN
ejpam-4914	104	20	function	function	NOUN
ejpam-4914	104	21	(	(	PUNCT
ejpam-4914	104	22	or	or	CCONJ
ejpam-4914	104	23	hid	hide	VERB
ejpam-4914	104	24	-	-	PUNCT
ejpam-4914	104	25	function	function	NOUN
ejpam-4914	104	26	)	)	PUNCT
ejpam-4914	104	27	of	of	ADP
ejpam-4914	104	28	g	g	PROPN
ejpam-4914	104	29	if	if	SCONJ
ejpam-4914	104	30	for	for	ADP
ejpam-4914	104	31	each	each	PRON
ejpam-4914	104	32	v	v	NUM
ejpam-4914	104	33	∈	∈	PROPN
ejpam-4914	104	34	v	v	NOUN
ejpam-4914	104	35	(	(	PUNCT
ejpam-4914	104	36	g	g	NOUN
ejpam-4914	104	37	)	)	PUNCT
ejpam-4914	104	38	with	with	ADP
ejpam-4914	104	39	f(v	f(v	NOUN
ejpam-4914	104	40	)	)	PUNCT
ejpam-4914	104	41	=	=	SYM
ejpam-4914	104	42	0	0	NUM
ejpam-4914	104	43	,	,	PUNCT
ejpam-4914	104	44	∑	∑	ADV
ejpam-4914	104	45	x∈ng(v;2	x∈ng(v;2	NUM
ejpam-4914	104	46	)	)	PUNCT
ejpam-4914	104	47	f(x	f(x	PROPN
ejpam-4914	104	48	)	)	PUNCT
ejpam-4914	104	49	≥	≥	NOUN
ejpam-4914	105	1	2	2	NUM
ejpam-4914	105	2	.	.	PUNCT
ejpam-4914	105	3	more	more	ADV
ejpam-4914	105	4	precisely	precisely	ADV
ejpam-4914	105	5	,	,	PUNCT
ejpam-4914	105	6	f	f	PROPN
ejpam-4914	105	7	is	be	AUX
ejpam-4914	105	8	an	an	DET
ejpam-4914	105	9	hid	hid	NOUN
ejpam-4914	105	10	-	-	PUNCT
ejpam-4914	105	11	function	function	NOUN
ejpam-4914	105	12	of	of	ADP
ejpam-4914	105	13	g	g	PROPN
ejpam-4914	105	14	if	if	SCONJ
ejpam-4914	106	1	and	and	CCONJ
ejpam-4914	106	2	only	only	ADV
ejpam-4914	106	3	if	if	SCONJ
ejpam-4914	106	4	at	at	ADV
ejpam-4914	106	5	least	least	ADJ
ejpam-4914	106	6	one	one	NUM
ejpam-4914	106	7	of	of	ADP
ejpam-4914	106	8	the	the	DET
ejpam-4914	106	9	following	following	NOUN
ejpam-4914	106	10	holds	hold	VERB
ejpam-4914	106	11	for	for	ADP
ejpam-4914	106	12	each	each	DET
ejpam-4914	106	13	v	v	NUM
ejpam-4914	106	14	∈	∈	PROPN
ejpam-4914	106	15	v	v	NOUN
ejpam-4914	106	16	(	(	PUNCT
ejpam-4914	106	17	g	g	NOUN
ejpam-4914	106	18	)	)	PUNCT
ejpam-4914	106	19	with	with	ADP
ejpam-4914	106	20	f(v	f(v	NOUN
ejpam-4914	106	21	)	)	PUNCT
ejpam-4914	106	22	=	=	SYM
ejpam-4914	107	1	0	0	NUM
ejpam-4914	107	2	:	:	PUNCT
ejpam-4914	107	3	(	(	PUNCT
ejpam-4914	107	4	i	i	NOUN
ejpam-4914	107	5	)	)	PUNCT
ejpam-4914	107	6	there	there	PRON
ejpam-4914	107	7	exists	exist	VERB
ejpam-4914	107	8	u	u	PROPN
ejpam-4914	107	9	∈	∈	PROPN
ejpam-4914	107	10	v	v	ADP
ejpam-4914	107	11	(	(	PUNCT
ejpam-4914	107	12	g	g	NOUN
ejpam-4914	107	13	)	)	PUNCT
ejpam-4914	107	14	for	for	ADP
ejpam-4914	107	15	which	which	PRON
ejpam-4914	107	16	f(u	f(u	PROPN
ejpam-4914	107	17	)	)	PUNCT
ejpam-4914	107	18	=	=	SYM
ejpam-4914	107	19	2	2	NUM
ejpam-4914	107	20	and	and	CCONJ
ejpam-4914	107	21	dg(u	dg(u	NOUN
ejpam-4914	107	22	,	,	PUNCT
ejpam-4914	107	23	v	v	NOUN
ejpam-4914	107	24	)	)	PUNCT
ejpam-4914	107	25	=	=	SYM
ejpam-4914	107	26	2	2	NUM
ejpam-4914	107	27	;	;	PUNCT
ejpam-4914	107	28	(	(	PUNCT
ejpam-4914	107	29	ii	ii	NOUN
ejpam-4914	107	30	)	)	PUNCT
ejpam-4914	107	31	there	there	PRON
ejpam-4914	107	32	exist	exist	VERB
ejpam-4914	107	33	distinct	distinct	ADJ
ejpam-4914	107	34	u	u	NOUN
ejpam-4914	107	35	,	,	PUNCT
ejpam-4914	107	36	w	w	PROPN
ejpam-4914	107	37	∈	∈	PROPN
ejpam-4914	107	38	v	v	ADP
ejpam-4914	107	39	(	(	PUNCT
ejpam-4914	107	40	g	g	NOUN
ejpam-4914	107	41	)	)	PUNCT
ejpam-4914	107	42	for	for	ADP
ejpam-4914	107	43	which	which	PRON
ejpam-4914	107	44	f(u	f(u	PROPN
ejpam-4914	107	45	)	)	PUNCT
ejpam-4914	107	46	=	=	SYM
ejpam-4914	107	47	1	1	NUM
ejpam-4914	107	48	=	=	PUNCT
ejpam-4914	107	49	f(w	f(w	PROPN
ejpam-4914	107	50	)	)	PUNCT
ejpam-4914	107	51	and	and	CCONJ
ejpam-4914	107	52	dg(u	dg(u	X
ejpam-4914	107	53	,	,	PUNCT
ejpam-4914	107	54	v	v	NOUN
ejpam-4914	107	55	)	)	PUNCT
ejpam-4914	107	56	=	=	SYM
ejpam-4914	107	57	2	2	NUM
ejpam-4914	107	58	=	=	SYM
ejpam-4914	107	59	dg(w	dg(w	X
ejpam-4914	107	60	,	,	PUNCT
ejpam-4914	107	61	v	v	NOUN
ejpam-4914	107	62	)	)	PUNCT
ejpam-4914	107	63	.	.	PUNCT
ejpam-4914	108	1	the	the	DET
ejpam-4914	108	2	minimum	minimum	ADJ
ejpam-4914	108	3	weight	weight	NOUN
ejpam-4914	108	4	∑	∑	PUNCT
ejpam-4914	108	5	v∈v	v∈v	NOUN
ejpam-4914	108	6	(	(	PUNCT
ejpam-4914	108	7	g	g	NOUN
ejpam-4914	108	8	)	)	PUNCT
ejpam-4914	108	9	f(v	f(v	NOUN
ejpam-4914	108	10	)	)	PUNCT
ejpam-4914	108	11	of	of	ADP
ejpam-4914	108	12	an	an	DET
ejpam-4914	108	13	hid	hid	NOUN
ejpam-4914	108	14	-	-	PUNCT
ejpam-4914	108	15	function	function	NOUN
ejpam-4914	108	16	of	of	ADP
ejpam-4914	108	17	g	g	PROPN
ejpam-4914	108	18	is	be	AUX
ejpam-4914	108	19	the	the	DET
ejpam-4914	108	20	hop	hop	NOUN
ejpam-4914	108	21	italian	italian	ADJ
ejpam-4914	108	22	domination	domination	NOUN
ejpam-4914	108	23	number	number	NOUN
ejpam-4914	108	24	of	of	ADP
ejpam-4914	108	25	g	g	NOUN
ejpam-4914	108	26	,	,	PUNCT
ejpam-4914	108	27	and	and	CCONJ
ejpam-4914	108	28	is	be	AUX
ejpam-4914	108	29	denoted	denote	VERB
ejpam-4914	108	30	by	by	ADP
ejpam-4914	108	31	γhi(g	γhi(g	PROPN
ejpam-4914	108	32	)	)	PUNCT
ejpam-4914	108	33	.	.	PUNCT
ejpam-4914	109	1	if	if	SCONJ
ejpam-4914	109	2	hid(g	hid(g	PROPN
ejpam-4914	109	3	)	)	PUNCT
ejpam-4914	109	4	denotes	denote	VERB
ejpam-4914	109	5	the	the	DET
ejpam-4914	109	6	collection	collection	NOUN
ejpam-4914	109	7	of	of	ADP
ejpam-4914	109	8	all	all	DET
ejpam-4914	109	9	hidfunctions	hidfunction	NOUN
ejpam-4914	109	10	of	of	ADP
ejpam-4914	109	11	g	g	NOUN
ejpam-4914	109	12	,	,	PUNCT
ejpam-4914	109	13	then	then	ADV
ejpam-4914	109	14	γhi(g	γhi(g	PROPN
ejpam-4914	109	15	)	)	PUNCT
ejpam-4914	110	1	=	=	SYM
ejpam-4914	110	2	min{ωg(f	min{ωg(f	PROPN
ejpam-4914	110	3	)	)	PUNCT
ejpam-4914	110	4	:	:	PUNCT
ejpam-4914	111	1	f	f	PROPN
ejpam-4914	111	2	∈	∈	PROPN
ejpam-4914	111	3	hid(g	hid(g	PROPN
ejpam-4914	111	4	)	)	PUNCT
ejpam-4914	111	5	}	}	PUNCT
ejpam-4914	111	6	.	.	PUNCT
ejpam-4914	112	1	a	a	DET
ejpam-4914	112	2	hop	hop	NOUN
ejpam-4914	112	3	italian	italian	ADJ
ejpam-4914	112	4	dominating	dominating	NOUN
ejpam-4914	112	5	function	function	NOUN
ejpam-4914	112	6	f	f	PROPN
ejpam-4914	112	7	of	of	ADP
ejpam-4914	112	8	g	g	PROPN
ejpam-4914	112	9	with	with	ADP
ejpam-4914	112	10	ωg(f	ωg(f	NOUN
ejpam-4914	112	11	)	)	PUNCT
ejpam-4914	112	12	=	=	SYM
ejpam-4914	112	13	γhi(g	γhi(g	PROPN
ejpam-4914	112	14	)	)	PUNCT
ejpam-4914	112	15	is	be	AUX
ejpam-4914	112	16	called	call	VERB
ejpam-4914	112	17	γhi	γhi	ADJ
ejpam-4914	112	18	-	-	PUNCT
ejpam-4914	112	19	function	function	NOUN
ejpam-4914	112	20	of	of	ADP
ejpam-4914	112	21	g.	g.	PROPN
ejpam-4914	112	22	as	as	ADP
ejpam-4914	112	23	usual	usual	ADJ
ejpam-4914	112	24	,	,	PUNCT
ejpam-4914	112	25	for	for	ADP
ejpam-4914	112	26	f	f	PROPN
ejpam-4914	112	27	:	:	PUNCT
ejpam-4914	112	28	v	v	X
ejpam-4914	112	29	(	(	PUNCT
ejpam-4914	112	30	g	g	NOUN
ejpam-4914	112	31	)	)	PUNCT
ejpam-4914	112	32	→	→	SYM
ejpam-4914	112	33	{	{	PUNCT
ejpam-4914	112	34	0	0	NUM
ejpam-4914	112	35	,	,	PUNCT
ejpam-4914	112	36	1	1	NUM
ejpam-4914	112	37	,	,	PUNCT
ejpam-4914	112	38	2	2	X
ejpam-4914	112	39	}	}	PUNCT
ejpam-4914	112	40	we	we	PRON
ejpam-4914	112	41	write	write	VERB
ejpam-4914	112	42	f	f	PROPN
ejpam-4914	112	43	=	=	SYM
ejpam-4914	112	44	(	(	PUNCT
ejpam-4914	112	45	v0	v0	PROPN
ejpam-4914	112	46	,	,	PUNCT
ejpam-4914	112	47	v1	v1	NOUN
ejpam-4914	112	48	,	,	PUNCT
ejpam-4914	112	49	v2	v2	PROPN
ejpam-4914	112	50	)	)	PUNCT
ejpam-4914	112	51	,	,	PUNCT
ejpam-4914	112	52	where	where	SCONJ
ejpam-4914	112	53	vk	vk	VERB
ejpam-4914	112	54	=	=	SYM
ejpam-4914	112	55	{	{	PUNCT
ejpam-4914	112	56	v	v	NUM
ejpam-4914	112	57	∈	∈	NOUN
ejpam-4914	112	58	v	v	NOUN
ejpam-4914	112	59	(	(	PUNCT
ejpam-4914	112	60	g	g	NOUN
ejpam-4914	112	61	)	)	PUNCT
ejpam-4914	112	62	:	:	PUNCT
ejpam-4914	112	63	f(v	f(v	NOUN
ejpam-4914	112	64	)	)	PUNCT
ejpam-4914	113	1	=	=	SYM
ejpam-4914	113	2	k	k	NOUN
ejpam-4914	113	3	}	}	PUNCT
ejpam-4914	113	4	for	for	ADP
ejpam-4914	113	5	each	each	DET
ejpam-4914	113	6	k	k	PROPN
ejpam-4914	113	7	∈	∈	PROPN
ejpam-4914	113	8	{	{	PUNCT
ejpam-4914	113	9	0	0	NUM
ejpam-4914	113	10	,	,	PUNCT
ejpam-4914	113	11	1	1	NUM
ejpam-4914	113	12	,	,	PUNCT
ejpam-4914	113	13	2	2	NUM
ejpam-4914	113	14	}	}	PUNCT
ejpam-4914	113	15	.	.	PUNCT
ejpam-4914	114	1	thus	thus	ADV
ejpam-4914	114	2	,	,	PUNCT
ejpam-4914	114	3	f	f	PROPN
ejpam-4914	114	4	=	=	SYM
ejpam-4914	114	5	(	(	PUNCT
ejpam-4914	114	6	v0	v0	PROPN
ejpam-4914	114	7	,	,	PUNCT
ejpam-4914	114	8	v1	v1	NOUN
ejpam-4914	114	9	,	,	PUNCT
ejpam-4914	114	10	v2	v2	NOUN
ejpam-4914	114	11	)	)	PUNCT
ejpam-4914	114	12	∈	∈	PROPN
ejpam-4914	114	13	hid(g	hid(g	PROPN
ejpam-4914	114	14	)	)	PUNCT
ejpam-4914	115	1	if	if	SCONJ
ejpam-4914	115	2	and	and	CCONJ
ejpam-4914	115	3	only	only	ADV
ejpam-4914	115	4	if	if	SCONJ
ejpam-4914	115	5	for	for	ADP
ejpam-4914	115	6	each	each	DET
ejpam-4914	115	7	v	v	ADP
ejpam-4914	115	8	∈	∈	PROPN
ejpam-4914	115	9	v0	v0	NOUN
ejpam-4914	115	10	,	,	PUNCT
ejpam-4914	115	11	v2	v2	PROPN
ejpam-4914	115	12	∩ng(v	∩ng(v	PROPN
ejpam-4914	115	13	,	,	PUNCT
ejpam-4914	115	14	2	2	NUM
ejpam-4914	115	15	)	)	PUNCT
ejpam-4914	115	16	̸=	̸=	PROPN
ejpam-4914	115	17	∅	∅	NOUN
ejpam-4914	115	18	or	or	CCONJ
ejpam-4914	115	19	|v1	|v1	VERB
ejpam-4914	115	20	∩ng(v)|	∩ng(v)|	PROPN
ejpam-4914	115	21	≥	≥	NUM
ejpam-4914	115	22	2	2	NUM
ejpam-4914	115	23	.	.	PUNCT
ejpam-4914	116	1	if	if	SCONJ
ejpam-4914	116	2	f	f	PROPN
ejpam-4914	116	3	=	=	SYM
ejpam-4914	116	4	(	(	PUNCT
ejpam-4914	116	5	v0	v0	PROPN
ejpam-4914	116	6	,	,	PUNCT
ejpam-4914	116	7	v1	v1	NOUN
ejpam-4914	116	8	,	,	PUNCT
ejpam-4914	116	9	v2	v2	PROPN
ejpam-4914	116	10	)	)	PUNCT
ejpam-4914	116	11	is	be	AUX
ejpam-4914	116	12	a	a	DET
ejpam-4914	116	13	γhi	γhi	ADJ
ejpam-4914	116	14	-function	-function	NOUN
ejpam-4914	116	15	of	of	ADP
ejpam-4914	116	16	g	g	NOUN
ejpam-4914	116	17	,	,	PUNCT
ejpam-4914	116	18	then	then	ADV
ejpam-4914	116	19	v1	v1	VERB
ejpam-4914	116	20	∪	∪	ADJ
ejpam-4914	116	21	v2	v2	PROPN
ejpam-4914	116	22	is	be	AUX
ejpam-4914	116	23	a	a	DET
ejpam-4914	116	24	hop	hop	NOUN
ejpam-4914	116	25	dominating	dominating	NOUN
ejpam-4914	116	26	set	set	NOUN
ejpam-4914	116	27	of	of	ADP
ejpam-4914	116	28	g	g	NOUN
ejpam-4914	116	29	so	so	SCONJ
ejpam-4914	116	30	that	that	SCONJ
ejpam-4914	116	31	γh(g	γh(g	ADP
ejpam-4914	116	32	)	)	PUNCT
ejpam-4914	116	33	≤	≤	NOUN
ejpam-4914	116	34	|v1	|v1	CCONJ
ejpam-4914	116	35	∪	∪	X
ejpam-4914	116	36	v2|	v2|	ADJ
ejpam-4914	116	37	≤	≤	NOUN
ejpam-4914	116	38	ωg(f	ωg(f	NOUN
ejpam-4914	116	39	)	)	PUNCT
ejpam-4914	116	40	=	=	SYM
ejpam-4914	116	41	γhi(g	γhi(g	PROPN
ejpam-4914	116	42	)	)	PUNCT
ejpam-4914	116	43	.	.	PUNCT
ejpam-4914	117	1	now	now	ADV
ejpam-4914	117	2	observe	observe	VERB
ejpam-4914	117	3	that	that	SCONJ
ejpam-4914	117	4	if	if	SCONJ
ejpam-4914	117	5	s	s	VERB
ejpam-4914	117	6	⊆	⊆	NUM
ejpam-4914	117	7	v	v	NOUN
ejpam-4914	117	8	(	(	PUNCT
ejpam-4914	117	9	g	g	NOUN
ejpam-4914	117	10	)	)	PUNCT
ejpam-4914	117	11	is	be	AUX
ejpam-4914	117	12	a	a	DET
ejpam-4914	117	13	γ2h	γ2h	NOUN
ejpam-4914	117	14	-	-	PUNCT
ejpam-4914	117	15	set	set	NOUN
ejpam-4914	117	16	of	of	ADP
ejpam-4914	117	17	g	g	NOUN
ejpam-4914	117	18	,	,	PUNCT
ejpam-4914	117	19	then	then	ADV
ejpam-4914	117	20	f	f	PROPN
ejpam-4914	117	21	=	=	PUNCT
ejpam-4914	117	22	(	(	PUNCT
ejpam-4914	117	23	v	v	NOUN
ejpam-4914	117	24	(	(	PUNCT
ejpam-4914	117	25	g	g	NOUN
ejpam-4914	117	26	)	)	PUNCT
ejpam-4914	117	27	\	\	PROPN
ejpam-4914	118	1	s	s	PROPN
ejpam-4914	118	2	,	,	PUNCT
ejpam-4914	118	3	s,∅	s,∅	NOUN
ejpam-4914	118	4	)	)	PUNCT
ejpam-4914	118	5	∈	∈	PROPN
ejpam-4914	118	6	hid(g	hid(g	PROPN
ejpam-4914	118	7	)	)	PUNCT
ejpam-4914	118	8	.	.	PUNCT
ejpam-4914	119	1	thus	thus	ADV
ejpam-4914	119	2	,	,	PUNCT
ejpam-4914	119	3	γhi(g	γhi(g	PROPN
ejpam-4914	119	4	)	)	PUNCT
ejpam-4914	119	5	≤	≤	NUM
ejpam-4914	119	6	|s|	|s|	PROPN
ejpam-4914	119	7	=	=	SYM
ejpam-4914	119	8	γ2h(g	γ2h(g	NOUN
ejpam-4914	119	9	)	)	PUNCT
ejpam-4914	119	10	.	.	PUNCT
ejpam-4914	120	1	moreover	moreover	ADV
ejpam-4914	120	2	,	,	PUNCT
ejpam-4914	120	3	since	since	SCONJ
ejpam-4914	120	4	a	a	DET
ejpam-4914	120	5	hop	hop	NOUN
ejpam-4914	120	6	roman	roman	ADJ
ejpam-4914	120	7	dominating	dominating	NOUN
ejpam-4914	120	8	function	function	NOUN
ejpam-4914	120	9	is	be	AUX
ejpam-4914	120	10	a	a	DET
ejpam-4914	120	11	hop	hop	NOUN
ejpam-4914	120	12	italian	italian	ADJ
ejpam-4914	120	13	dominating	dominating	NOUN
ejpam-4914	120	14	function	function	NOUN
ejpam-4914	120	15	,	,	PUNCT
ejpam-4914	120	16	γhi(g	γhi(g	PROPN
ejpam-4914	120	17	)	)	PUNCT
ejpam-4914	120	18	≤	≤	NOUN
ejpam-4914	120	19	min{γhr(g	min{γhr(g	PROPN
ejpam-4914	120	20	)	)	PUNCT
ejpam-4914	120	21	,	,	PUNCT
ejpam-4914	120	22	γ2h(g	γ2h(g	PROPN
ejpam-4914	120	23	)	)	PUNCT
ejpam-4914	120	24	}	}	PUNCT
ejpam-4914	120	25	.	.	PUNCT
ejpam-4914	121	1	(	(	PUNCT
ejpam-4914	121	2	1	1	X
ejpam-4914	121	3	)	)	PUNCT
ejpam-4914	121	4	s.r	s.r	PROPN
ejpam-4914	121	5	.	.	PROPN
ejpam-4914	121	6	jr	jr	PROPN
ejpam-4914	121	7	.	.	PROPN
ejpam-4914	121	8	canoy	canoy	PROPN
ejpam-4914	121	9	,	,	PUNCT
ejpam-4914	121	10	f.p	f.p	PROPN
ejpam-4914	121	11	.	.	PROPN
ejpam-4914	121	12	jamil	jamil	PROPN
ejpam-4914	121	13	and	and	CCONJ
ejpam-4914	121	14	s.m	s.m	PROPN
ejpam-4914	121	15	.	.	PROPN
ejpam-4914	121	16	menchavez	menchavez	PROPN
ejpam-4914	121	17	/	/	PUNCT
ejpam-4914	121	18	eur	eur	PROPN
ejpam-4914	121	19	.	.	PUNCT
ejpam-4914	122	1	j.	j.	PROPN
ejpam-4914	122	2	pure	pure	PROPN
ejpam-4914	122	3	appl	appl	PROPN
ejpam-4914	122	4	.	.	PROPN
ejpam-4914	122	5	math	math	PROPN
ejpam-4914	122	6	,	,	PUNCT
ejpam-4914	122	7	16	16	NUM
ejpam-4914	122	8	(	(	PUNCT
ejpam-4914	122	9	4	4	NUM
ejpam-4914	122	10	)	)	PUNCT
ejpam-4914	122	11	(	(	PUNCT
ejpam-4914	122	12	2023	2023	NUM
ejpam-4914	122	13	)	)	PUNCT
ejpam-4914	122	14	,	,	PUNCT
ejpam-4914	122	15	2431	2431	NUM
ejpam-4914	122	16	-	-	SYM
ejpam-4914	122	17	2449	2449	NUM
ejpam-4914	122	18	2435	2435	NUM
ejpam-4914	122	19	let	let	VERB
ejpam-4914	122	20	g	g	NOUN
ejpam-4914	122	21	be	be	AUX
ejpam-4914	122	22	the	the	DET
ejpam-4914	122	23	graph	graph	NOUN
ejpam-4914	122	24	in	in	ADP
ejpam-4914	122	25	figure	figure	NOUN
ejpam-4914	122	26	1	1	NUM
ejpam-4914	122	27	obtained	obtain	VERB
ejpam-4914	122	28	by	by	ADP
ejpam-4914	122	29	joining	join	VERB
ejpam-4914	122	30	two	two	NUM
ejpam-4914	122	31	copies	copy	NOUN
ejpam-4914	122	32	of	of	ADP
ejpam-4914	122	33	p5	p5	NOUN
ejpam-4914	122	34	,	,	PUNCT
ejpam-4914	122	35	say	say	VERB
ejpam-4914	123	1	[	[	X
ejpam-4914	123	2	x1	x1	ADJ
ejpam-4914	123	3	,	,	PUNCT
ejpam-4914	123	4	x2	x2	PROPN
ejpam-4914	123	5	,	,	PUNCT
ejpam-4914	123	6	x3	x3	PROPN
ejpam-4914	123	7	,	,	PUNCT
ejpam-4914	123	8	x4	x4	PROPN
ejpam-4914	123	9	,	,	PUNCT
ejpam-4914	123	10	x5	x5	NOUN
ejpam-4914	123	11	]	]	PUNCT
ejpam-4914	123	12	and	and	CCONJ
ejpam-4914	123	13	[	[	X
ejpam-4914	123	14	y1	y1	INTJ
ejpam-4914	123	15	,	,	PUNCT
ejpam-4914	123	16	y2	y2	PROPN
ejpam-4914	123	17	,	,	PUNCT
ejpam-4914	123	18	y3	y3	PROPN
ejpam-4914	123	19	,	,	PUNCT
ejpam-4914	123	20	y4	y4	PROPN
ejpam-4914	123	21	,	,	PUNCT
ejpam-4914	123	22	y5	y5	PROPN
ejpam-4914	123	23	]	]	PUNCT
ejpam-4914	123	24	,	,	PUNCT
ejpam-4914	123	25	using	use	VERB
ejpam-4914	123	26	the	the	DET
ejpam-4914	123	27	edges	edge	NOUN
ejpam-4914	123	28	x1y1	x1y1	X
ejpam-4914	123	29	,	,	PUNCT
ejpam-4914	123	30	x3y3	x3y3	NUM
ejpam-4914	123	31	and	and	CCONJ
ejpam-4914	123	32	x5	x5	PROPN
ejpam-4914	123	33	,	,	PUNCT
ejpam-4914	123	34	y5	y5	PROPN
ejpam-4914	123	35	.	.	PUNCT
ejpam-4914	124	1	then	then	ADV
ejpam-4914	124	2	γhi(g	γhi(g	PROPN
ejpam-4914	124	3	)	)	PUNCT
ejpam-4914	125	1	=	=	SYM
ejpam-4914	125	2	γhr(g	γhr(g	PROPN
ejpam-4914	125	3	)	)	PUNCT
ejpam-4914	125	4	=	=	SYM
ejpam-4914	125	5	....................................................................................................	....................................................................................................	PUNCT
ejpam-4914	125	6	....................................................................................................	....................................................................................................	PUNCT
ejpam-4914	125	7	....................................................................................................	....................................................................................................	PUNCT
ejpam-4914	125	8	....................................................................................................	....................................................................................................	PUNCT
ejpam-4914	125	9	....................................	....................................	PUNCT
ejpam-4914	125	10	....................................................................................................	....................................................................................................	PUNCT
ejpam-4914	125	11	....................................................................................................	....................................................................................................	PUNCT
ejpam-4914	125	12	....................................................................................................	....................................................................................................	PUNCT
ejpam-4914	125	13	....................................................................................................	....................................................................................................	PUNCT
ejpam-4914	126	1	....................................	....................................	PUNCT
ejpam-4914	127	1	•	•	NUM
ejpam-4914	127	2	•	•	NUM
ejpam-4914	127	3	•	•	NUM
ejpam-4914	127	4	•	•	NUM
ejpam-4914	127	5	•	•	NUM
ejpam-4914	127	6	•	•	NUM
ejpam-4914	127	7	•	•	NUM
ejpam-4914	127	8	•	•	NUM
ejpam-4914	127	9	•	•	NOUN
ejpam-4914	127	10	•	•	NOUN
ejpam-4914	127	11	.........	.........	PUNCT
ejpam-4914	127	12	........	........	PUNCT
ejpam-4914	127	13	........	........	PUNCT
ejpam-4914	127	14	........	........	PUNCT
ejpam-4914	127	15	........	........	PUNCT
ejpam-4914	127	16	........	........	PUNCT
ejpam-4914	127	17	........	........	PUNCT
ejpam-4914	127	18	........	........	PUNCT
ejpam-4914	127	19	........	........	PUNCT
ejpam-4914	127	20	........	........	PUNCT
ejpam-4914	127	21	........	........	PUNCT
ejpam-4914	127	22	........	........	PUNCT
ejpam-4914	128	1	.....	.....	PUNCT
ejpam-4914	128	2	.........	.........	PUNCT
ejpam-4914	128	3	........	........	PUNCT
ejpam-4914	128	4	........	........	PUNCT
ejpam-4914	128	5	........	........	PUNCT
ejpam-4914	128	6	........	........	PUNCT
ejpam-4914	128	7	........	........	PUNCT
ejpam-4914	128	8	........	........	PUNCT
ejpam-4914	128	9	........	........	PUNCT
ejpam-4914	128	10	........	........	PUNCT
ejpam-4914	128	11	........	........	PUNCT
ejpam-4914	128	12	........	........	PUNCT
ejpam-4914	128	13	........	........	PUNCT
ejpam-4914	129	1	.....	.....	PUNCT
ejpam-4914	129	2	.........	.........	PUNCT
ejpam-4914	129	3	........	........	PUNCT
ejpam-4914	129	4	........	........	PUNCT
ejpam-4914	129	5	........	........	PUNCT
ejpam-4914	129	6	........	........	PUNCT
ejpam-4914	129	7	........	........	PUNCT
ejpam-4914	129	8	........	........	PUNCT
ejpam-4914	129	9	........	........	PUNCT
ejpam-4914	129	10	........	........	PUNCT
ejpam-4914	129	11	........	........	PUNCT
ejpam-4914	129	12	........	........	PUNCT
ejpam-4914	129	13	........	........	PUNCT
ejpam-4914	130	1	.....	.....	PUNCT
ejpam-4914	131	1	x1	x1	NUM
ejpam-4914	132	1	x2	x2	NOUN
ejpam-4914	132	2	x3	x3	PROPN
ejpam-4914	133	1	x4	x4	PROPN
ejpam-4914	133	2	x5	x5	PROPN
ejpam-4914	133	3	y1	y1	ADJ
ejpam-4914	133	4	y2	y2	NOUN
ejpam-4914	133	5	y3	y3	NOUN
ejpam-4914	133	6	y4	y4	ADJ
ejpam-4914	133	7	y5	y5	NOUN
ejpam-4914	133	8	figure	figure	NOUN
ejpam-4914	133	9	1	1	NUM
ejpam-4914	133	10	:	:	PUNCT
ejpam-4914	133	11	a	a	DET
ejpam-4914	133	12	graph	graph	NOUN
ejpam-4914	133	13	g	g	NOUN
ejpam-4914	133	14	with	with	ADP
ejpam-4914	133	15	γhi(g	γhi(g	PROPN
ejpam-4914	133	16	)	)	PUNCT
ejpam-4914	134	1	=	=	SYM
ejpam-4914	134	2	γhr(g	γhr(g	PROPN
ejpam-4914	134	3	)	)	PUNCT
ejpam-4914	134	4	4	4	NUM
ejpam-4914	134	5	<	<	SYM
ejpam-4914	134	6	6	6	NUM
ejpam-4914	134	7	=	=	SYM
ejpam-4914	134	8	γ2h(g	γ2h(g	NOUN
ejpam-4914	134	9	)	)	PUNCT
ejpam-4914	134	10	.	.	PUNCT
ejpam-4914	135	1	in	in	ADP
ejpam-4914	135	2	this	this	DET
ejpam-4914	135	3	case	case	NOUN
ejpam-4914	135	4	,	,	PUNCT
ejpam-4914	135	5	γhr(g	γhr(g	PROPN
ejpam-4914	135	6	)	)	PUNCT
ejpam-4914	135	7	=	=	SYM
ejpam-4914	135	8	γhi(g	γhi(g	PROPN
ejpam-4914	135	9	)	)	PUNCT
ejpam-4914	135	10	is	be	AUX
ejpam-4914	135	11	determined	determine	VERB
ejpam-4914	135	12	by	by	ADP
ejpam-4914	135	13	the	the	DET
ejpam-4914	135	14	function	function	NOUN
ejpam-4914	135	15	f	f	PROPN
ejpam-4914	135	16	=	=	SYM
ejpam-4914	135	17	(	(	PUNCT
ejpam-4914	135	18	v	v	NOUN
ejpam-4914	135	19	(	(	PUNCT
ejpam-4914	135	20	g	g	NOUN
ejpam-4914	135	21	)	)	PUNCT
ejpam-4914	135	22	\	\	NOUN
ejpam-4914	135	23	{	{	PUNCT
ejpam-4914	135	24	x3	x3	PROPN
ejpam-4914	135	25	,	,	PUNCT
ejpam-4914	135	26	y3},∅	y3},∅	PROPN
ejpam-4914	135	27	,	,	PUNCT
ejpam-4914	135	28	{	{	PUNCT
ejpam-4914	135	29	x3	x3	ADJ
ejpam-4914	135	30	,	,	PUNCT
ejpam-4914	135	31	y3	y3	NOUN
ejpam-4914	135	32	}	}	PUNCT
ejpam-4914	135	33	)	)	PUNCT
ejpam-4914	135	34	.	.	PUNCT
ejpam-4914	136	1	on	on	ADP
ejpam-4914	136	2	the	the	DET
ejpam-4914	136	3	other	other	ADJ
ejpam-4914	136	4	hand	hand	NOUN
ejpam-4914	136	5	,	,	PUNCT
ejpam-4914	136	6	if	if	SCONJ
ejpam-4914	136	7	g	g	PROPN
ejpam-4914	136	8	=	=	SYM
ejpam-4914	136	9	c5	c5	PROPN
ejpam-4914	136	10	,	,	PUNCT
ejpam-4914	136	11	then	then	ADV
ejpam-4914	136	12	γhi(g	γhi(g	PROPN
ejpam-4914	136	13	)	)	PUNCT
ejpam-4914	136	14	=	=	SYM
ejpam-4914	136	15	γ2h(g	γ2h(g	PROPN
ejpam-4914	136	16	)	)	PUNCT
ejpam-4914	136	17	=	=	SYM
ejpam-4914	136	18	3	3	NUM
ejpam-4914	136	19	<	<	SYM
ejpam-4914	136	20	4	4	NUM
ejpam-4914	136	21	=	=	SYM
ejpam-4914	136	22	γhr(g	γhr(g	NOUN
ejpam-4914	136	23	)	)	PUNCT
ejpam-4914	136	24	.	.	PUNCT
ejpam-4914	137	1	however	however	ADV
ejpam-4914	137	2	,	,	PUNCT
ejpam-4914	137	3	if	if	SCONJ
ejpam-4914	137	4	g	g	PROPN
ejpam-4914	137	5	=	=	VERB
ejpam-4914	137	6	kp	kp	PROPN
ejpam-4914	137	7	(	(	PUNCT
ejpam-4914	137	8	the	the	DET
ejpam-4914	137	9	complete	complete	ADJ
ejpam-4914	137	10	graph	graph	NOUN
ejpam-4914	137	11	on	on	ADP
ejpam-4914	137	12	p	p	NOUN
ejpam-4914	137	13	vertices	vertex	NOUN
ejpam-4914	137	14	)	)	PUNCT
ejpam-4914	137	15	,	,	PUNCT
ejpam-4914	137	16	then	then	ADV
ejpam-4914	137	17	γhi(g	γhi(g	NUM
ejpam-4914	137	18	)	)	PUNCT
ejpam-4914	137	19	=	=	SYM
ejpam-4914	137	20	γhr(g	γhr(g	PROPN
ejpam-4914	137	21	)	)	PUNCT
ejpam-4914	137	22	=	=	SYM
ejpam-4914	137	23	γ2h(g	γ2h(g	PROPN
ejpam-4914	137	24	)	)	PUNCT
ejpam-4914	137	25	=	=	SYM
ejpam-4914	138	1	p.	p.	NOUN
ejpam-4914	138	2	observation	observation	NOUN
ejpam-4914	138	3	2.1	2.1	NUM
ejpam-4914	138	4	.	.	PUNCT
ejpam-4914	139	1	let	let	VERB
ejpam-4914	139	2	g	g	NOUN
ejpam-4914	139	3	be	be	AUX
ejpam-4914	139	4	any	any	DET
ejpam-4914	139	5	graph	graph	NOUN
ejpam-4914	139	6	.	.	PUNCT
ejpam-4914	140	1	then	then	ADV
ejpam-4914	140	2	(	(	PUNCT
ejpam-4914	140	3	i	i	NOUN
ejpam-4914	140	4	)	)	PUNCT
ejpam-4914	140	5	γhi(g	γhi(g	PROPN
ejpam-4914	140	6	)	)	PUNCT
ejpam-4914	140	7	=	=	SYM
ejpam-4914	140	8	γhr(g	γhr(g	PROPN
ejpam-4914	140	9	)	)	PUNCT
ejpam-4914	140	10	if	if	SCONJ
ejpam-4914	140	11	and	and	CCONJ
ejpam-4914	140	12	only	only	ADV
ejpam-4914	140	13	if	if	SCONJ
ejpam-4914	140	14	g	g	PROPN
ejpam-4914	140	15	has	have	VERB
ejpam-4914	140	16	a	a	DET
ejpam-4914	140	17	γhi	γhi	ADJ
ejpam-4914	140	18	-	-	PUNCT
ejpam-4914	140	19	function	function	NOUN
ejpam-4914	140	20	that	that	PRON
ejpam-4914	140	21	is	be	AUX
ejpam-4914	140	22	a	a	DET
ejpam-4914	140	23	hop	hop	NOUN
ejpam-4914	140	24	roman	roman	ADJ
ejpam-4914	140	25	dominating	dominating	NOUN
ejpam-4914	140	26	function	function	NOUN
ejpam-4914	140	27	of	of	ADP
ejpam-4914	140	28	g	g	NOUN
ejpam-4914	140	29	;	;	PUNCT
ejpam-4914	140	30	(	(	PUNCT
ejpam-4914	140	31	ii	ii	NOUN
ejpam-4914	140	32	)	)	PUNCT
ejpam-4914	140	33	γhi(g	γhi(g	PROPN
ejpam-4914	140	34	)	)	PUNCT
ejpam-4914	141	1	=	=	SYM
ejpam-4914	141	2	γ2h(g	γ2h(g	NOUN
ejpam-4914	141	3	)	)	PUNCT
ejpam-4914	141	4	if	if	SCONJ
ejpam-4914	141	5	and	and	CCONJ
ejpam-4914	141	6	only	only	ADV
ejpam-4914	141	7	if	if	SCONJ
ejpam-4914	141	8	g	g	PROPN
ejpam-4914	141	9	has	have	VERB
ejpam-4914	141	10	a	a	DET
ejpam-4914	141	11	γhi	γhi	ADJ
ejpam-4914	141	12	-	-	PUNCT
ejpam-4914	141	13	function	function	NOUN
ejpam-4914	141	14	(	(	PUNCT
ejpam-4914	141	15	v0	v0	NOUN
ejpam-4914	141	16	,	,	PUNCT
ejpam-4914	141	17	v1	v1	NOUN
ejpam-4914	141	18	,	,	PUNCT
ejpam-4914	141	19	v0	v0	NOUN
ejpam-4914	141	20	)	)	PUNCT
ejpam-4914	141	21	for	for	ADP
ejpam-4914	141	22	which	which	PRON
ejpam-4914	141	23	v2	v2	NOUN
ejpam-4914	141	24	=	=	PUNCT
ejpam-4914	141	25	∅.	∅.	PRON
ejpam-4914	141	26	observation	observation	NOUN
ejpam-4914	141	27	2.2	2.2	NUM
ejpam-4914	141	28	.	.	PUNCT
ejpam-4914	142	1	on	on	ADP
ejpam-4914	142	2	paths	path	NOUN
ejpam-4914	142	3	,	,	PUNCT
ejpam-4914	142	4	cycles	cycle	NOUN
ejpam-4914	142	5	and	and	CCONJ
ejpam-4914	142	6	complete	complete	ADJ
ejpam-4914	142	7	bipartite	bipartite	NOUN
ejpam-4914	142	8	graphs	graph	NOUN
ejpam-4914	142	9	:	:	PUNCT
ejpam-4914	142	10	(	(	PUNCT
ejpam-4914	142	11	i	i	NOUN
ejpam-4914	142	12	)	)	PUNCT
ejpam-4914	142	13	γhi(pn	γhi(pn	NOUN
ejpam-4914	142	14	)	)	PUNCT
ejpam-4914	142	15	=	=	PUNCT
ejpam-4914	142	16			NOUN
ejpam-4914	142	17	2	2	NUM
ejpam-4914	142	18	,	,	PUNCT
ejpam-4914	142	19	if	if	SCONJ
ejpam-4914	142	20	n	n	NOUN
ejpam-4914	142	21	=	=	SYM
ejpam-4914	142	22	2	2	NUM
ejpam-4914	142	23	;	;	PUNCT
ejpam-4914	142	24	3	3	NUM
ejpam-4914	142	25	,	,	PUNCT
ejpam-4914	142	26	if	if	SCONJ
ejpam-4914	142	27	n	n	NOUN
ejpam-4914	142	28	=	=	SYM
ejpam-4914	142	29	3	3	NUM
ejpam-4914	142	30	;	;	PUNCT
ejpam-4914	142	31	2k	2k	NUM
ejpam-4914	142	32	+	+	CCONJ
ejpam-4914	142	33	2	2	NUM
ejpam-4914	142	34	,	,	PUNCT
ejpam-4914	142	35	if	if	SCONJ
ejpam-4914	142	36	n	n	NOUN
ejpam-4914	142	37	=	=	SYM
ejpam-4914	142	38	4k	4k	NOUN
ejpam-4914	142	39	+	+	CCONJ
ejpam-4914	142	40	r	r	NOUN
ejpam-4914	142	41	with	with	ADP
ejpam-4914	142	42	0	0	NUM
ejpam-4914	142	43	≤	≤	NUM
ejpam-4914	142	44	r	r	NOUN
ejpam-4914	142	45	≤	≤	NUM
ejpam-4914	142	46	2	2	NUM
ejpam-4914	142	47	;	;	PUNCT
ejpam-4914	142	48	k	k	X
ejpam-4914	142	49	≥	≥	NUM
ejpam-4914	142	50	1	1	NUM
ejpam-4914	142	51	;	;	PUNCT
ejpam-4914	142	52	2k	2k	NUM
ejpam-4914	142	53	+	+	CCONJ
ejpam-4914	142	54	3	3	NUM
ejpam-4914	142	55	,	,	PUNCT
ejpam-4914	142	56	if	if	SCONJ
ejpam-4914	142	57	n	n	NOUN
ejpam-4914	142	58	=	=	SYM
ejpam-4914	142	59	4k	4k	NOUN
ejpam-4914	142	60	+	+	NOUN
ejpam-4914	142	61	3	3	NUM
ejpam-4914	142	62	;	;	PUNCT
ejpam-4914	142	63	k	k	X
ejpam-4914	142	64	≥	≥	NUM
ejpam-4914	142	65	1	1	NUM
ejpam-4914	142	66	.	.	PUNCT
ejpam-4914	142	67	(	(	PUNCT
ejpam-4914	142	68	ii	ii	NOUN
ejpam-4914	142	69	)	)	PUNCT
ejpam-4914	142	70	γhi(cn	γhi(cn	NOUN
ejpam-4914	142	71	)	)	PUNCT
ejpam-4914	142	72	=	=	PUNCT
ejpam-4914	142	73			NOUN
ejpam-4914	142	74	3	3	NUM
ejpam-4914	142	75	,	,	PUNCT
ejpam-4914	142	76	if	if	SCONJ
ejpam-4914	142	77	n	n	NOUN
ejpam-4914	142	78	=	=	SYM
ejpam-4914	142	79	3	3	NUM
ejpam-4914	142	80	,	,	PUNCT
ejpam-4914	142	81	5	5	NUM
ejpam-4914	142	82	;	;	PUNCT
ejpam-4914	142	83	4	4	NUM
ejpam-4914	142	84	,	,	PUNCT
ejpam-4914	142	85	if	if	SCONJ
ejpam-4914	142	86	n	n	NOUN
ejpam-4914	142	87	=	=	SYM
ejpam-4914	142	88	4	4	NUM
ejpam-4914	142	89	;	;	PUNCT
ejpam-4914	142	90	2k	2k	NUM
ejpam-4914	142	91	+	+	CCONJ
ejpam-4914	142	92	2	2	NUM
ejpam-4914	142	93	,	,	PUNCT
ejpam-4914	142	94	if	if	SCONJ
ejpam-4914	142	95	n	n	NOUN
ejpam-4914	142	96	=	=	SYM
ejpam-4914	142	97	4k	4k	NOUN
ejpam-4914	143	1	+	+	NOUN
ejpam-4914	143	2	2	2	NUM
ejpam-4914	143	3	+	+	CCONJ
ejpam-4914	143	4	r	r	NOUN
ejpam-4914	143	5	with	with	ADP
ejpam-4914	143	6	0	0	NUM
ejpam-4914	143	7	≤	≤	NUM
ejpam-4914	143	8	r	r	NOUN
ejpam-4914	143	9	≤	≤	NUM
ejpam-4914	143	10	2	2	NUM
ejpam-4914	143	11	;	;	PUNCT
ejpam-4914	143	12	k	k	X
ejpam-4914	143	13	≥	≥	NUM
ejpam-4914	143	14	1	1	NUM
ejpam-4914	143	15	;	;	PUNCT
ejpam-4914	143	16	2k	2k	NUM
ejpam-4914	143	17	+	+	CCONJ
ejpam-4914	143	18	3	3	NUM
ejpam-4914	143	19	,	,	PUNCT
ejpam-4914	143	20	if	if	SCONJ
ejpam-4914	143	21	n	n	NOUN
ejpam-4914	143	22	=	=	SYM
ejpam-4914	143	23	4k	4k	NOUN
ejpam-4914	143	24	+	+	NOUN
ejpam-4914	143	25	5	5	NUM
ejpam-4914	143	26	;	;	PUNCT
ejpam-4914	143	27	k	k	X
ejpam-4914	143	28	≥	≥	NUM
ejpam-4914	143	29	1	1	NUM
ejpam-4914	143	30	.	.	PUNCT
ejpam-4914	143	31	(	(	PUNCT
ejpam-4914	143	32	iii	iii	NOUN
ejpam-4914	143	33	)	)	PUNCT
ejpam-4914	143	34	γhi(km	γhi(km	ADJ
ejpam-4914	143	35	,	,	PUNCT
ejpam-4914	143	36	n	n	CCONJ
ejpam-4914	143	37	)	)	PUNCT
ejpam-4914	143	38	=	=	SYM
ejpam-4914	144	1			NOUN
ejpam-4914	144	2	2	2	NUM
ejpam-4914	144	3	,	,	PUNCT
ejpam-4914	144	4	if	if	SCONJ
ejpam-4914	144	5	m	m	VERB
ejpam-4914	144	6	=	=	SYM
ejpam-4914	144	7	n	n	NOUN
ejpam-4914	144	8	=	=	SYM
ejpam-4914	144	9	1	1	NUM
ejpam-4914	144	10	;	;	PUNCT
ejpam-4914	144	11	3	3	NUM
ejpam-4914	144	12	,	,	PUNCT
ejpam-4914	144	13	if	if	SCONJ
ejpam-4914	144	14	m	m	VERB
ejpam-4914	144	15	=	=	SYM
ejpam-4914	144	16	1	1	NUM
ejpam-4914	144	17	(	(	PUNCT
ejpam-4914	144	18	resp	resp	NOUN
ejpam-4914	144	19	.	.	PUNCT
ejpam-4914	145	1	n	n	NOUN
ejpam-4914	145	2	=	=	NOUN
ejpam-4914	145	3	1	1	NUM
ejpam-4914	145	4	)	)	PUNCT
ejpam-4914	145	5	and	and	CCONJ
ejpam-4914	145	6	n	n	PRON
ejpam-4914	145	7	≥	≥	NUM
ejpam-4914	145	8	2	2	NUM
ejpam-4914	145	9	(	(	PUNCT
ejpam-4914	145	10	resp	resp	PROPN
ejpam-4914	145	11	m	m	PROPN
ejpam-4914	145	12	≥	≥	NOUN
ejpam-4914	145	13	2	2	NUM
ejpam-4914	145	14	)	)	PUNCT
ejpam-4914	145	15	;	;	PUNCT
ejpam-4914	145	16	4	4	NUM
ejpam-4914	145	17	,	,	PUNCT
ejpam-4914	145	18	if	if	SCONJ
ejpam-4914	145	19	m	m	PROPN
ejpam-4914	145	20	≥	≥	NOUN
ejpam-4914	145	21	2	2	NUM
ejpam-4914	145	22	and	and	CCONJ
ejpam-4914	145	23	n	n	PRON
ejpam-4914	145	24	≥	≥	NOUN
ejpam-4914	145	25	2	2	NUM
ejpam-4914	145	26	2.1	2.1	NUM
ejpam-4914	145	27	.	.	PUNCT
ejpam-4914	146	1	some	some	DET
ejpam-4914	146	2	properties	property	NOUN
ejpam-4914	146	3	and	and	CCONJ
ejpam-4914	146	4	graphs	graph	NOUN
ejpam-4914	146	5	with	with	ADP
ejpam-4914	146	6	small	small	ADJ
ejpam-4914	146	7	values	value	NOUN
ejpam-4914	146	8	of	of	ADP
ejpam-4914	146	9	γhi	γhi	NOUN
ejpam-4914	146	10	let	let	VERB
ejpam-4914	146	11	f	f	PROPN
ejpam-4914	146	12	=	=	SYM
ejpam-4914	146	13	(	(	PUNCT
ejpam-4914	146	14	v0	v0	PROPN
ejpam-4914	146	15	,	,	PUNCT
ejpam-4914	146	16	v1	v1	NOUN
ejpam-4914	146	17	,	,	PUNCT
ejpam-4914	146	18	v2	v2	PROPN
ejpam-4914	146	19	)	)	PUNCT
ejpam-4914	146	20	be	be	AUX
ejpam-4914	146	21	a	a	DET
ejpam-4914	146	22	γhi	γhi	ADJ
ejpam-4914	146	23	-function	-function	NOUN
ejpam-4914	146	24	of	of	ADP
ejpam-4914	146	25	g.	g.	NOUN
ejpam-4914	146	26	a	a	DET
ejpam-4914	146	27	vertex	vertex	NOUN
ejpam-4914	146	28	w	w	PROPN
ejpam-4914	146	29	∈	∈	PROPN
ejpam-4914	146	30	v0	v0	NOUN
ejpam-4914	146	31	is	be	AUX
ejpam-4914	146	32	an	an	DET
ejpam-4914	146	33	italian	italian	ADJ
ejpam-4914	146	34	private	private	ADJ
ejpam-4914	146	35	hop	hop	NOUN
ejpam-4914	146	36	-	-	PUNCT
ejpam-4914	146	37	neighbor	neighbor	NOUN
ejpam-4914	146	38	of	of	ADP
ejpam-4914	146	39	v	v	NOUN
ejpam-4914	146	40	∈	∈	NOUN
ejpam-4914	146	41	v1	v1	NOUN
ejpam-4914	146	42	∪	∪	NOUN
ejpam-4914	146	43	v2	v2	PROPN
ejpam-4914	146	44	under	under	ADP
ejpam-4914	146	45	f	f	PROPN
ejpam-4914	146	46	provided	provide	VERB
ejpam-4914	146	47	∑	∑	PROPN
ejpam-4914	146	48	u∈ng(w,2)\{v	u∈ng(w,2)\{v	PROPN
ejpam-4914	146	49	}	}	PUNCT
ejpam-4914	146	50	f(u	f(u	PROPN
ejpam-4914	146	51	)	)	PUNCT
ejpam-4914	146	52	<	<	X
ejpam-4914	147	1	2	2	X
ejpam-4914	147	2	.	.	PUNCT
ejpam-4914	147	3	if	if	SCONJ
ejpam-4914	147	4	no	no	DET
ejpam-4914	147	5	confusion	confusion	NOUN
ejpam-4914	147	6	arises	arise	VERB
ejpam-4914	147	7	,	,	PUNCT
ejpam-4914	147	8	instead	instead	ADV
ejpam-4914	147	9	of	of	ADP
ejpam-4914	147	10	saying	say	VERB
ejpam-4914	147	11	itaian	itaian	ADJ
ejpam-4914	147	12	private	private	ADJ
ejpam-4914	147	13	hop	hop	NOUN
ejpam-4914	147	14	-	-	PUNCT
ejpam-4914	147	15	neighbor	neighbor	NOUN
ejpam-4914	147	16	of	of	ADP
ejpam-4914	147	17	v	v	NOUN
ejpam-4914	147	18	under	under	ADP
ejpam-4914	147	19	f	f	PROPN
ejpam-4914	147	20	,	,	PUNCT
ejpam-4914	147	21	we	we	PRON
ejpam-4914	147	22	simply	simply	ADV
ejpam-4914	147	23	say	say	VERB
ejpam-4914	147	24	italian	italian	ADJ
ejpam-4914	147	25	private	private	ADJ
ejpam-4914	147	26	hop	hop	NOUN
ejpam-4914	147	27	-	-	PUNCT
ejpam-4914	147	28	neighbor	neighbor	NOUN
ejpam-4914	147	29	of	of	ADP
ejpam-4914	147	30	v.	v.	PROPN
ejpam-4914	147	31	s.r	s.r	PROPN
ejpam-4914	147	32	.	.	PROPN
ejpam-4914	147	33	jr	jr	PROPN
ejpam-4914	147	34	.	.	PROPN
ejpam-4914	147	35	canoy	canoy	PROPN
ejpam-4914	147	36	,	,	PUNCT
ejpam-4914	147	37	f.p	f.p	PROPN
ejpam-4914	147	38	.	.	PROPN
ejpam-4914	147	39	jamil	jamil	PROPN
ejpam-4914	147	40	and	and	CCONJ
ejpam-4914	147	41	s.m	s.m	PROPN
ejpam-4914	147	42	.	.	PROPN
ejpam-4914	147	43	menchavez	menchavez	PROPN
ejpam-4914	147	44	/	/	PUNCT
ejpam-4914	147	45	eur	eur	PROPN
ejpam-4914	147	46	.	.	PUNCT
ejpam-4914	148	1	j.	j.	PROPN
ejpam-4914	148	2	pure	pure	PROPN
ejpam-4914	148	3	appl	appl	PROPN
ejpam-4914	148	4	.	.	PROPN
ejpam-4914	148	5	math	math	PROPN
ejpam-4914	148	6	,	,	PUNCT
ejpam-4914	148	7	16	16	NUM
ejpam-4914	148	8	(	(	PUNCT
ejpam-4914	148	9	4	4	NUM
ejpam-4914	148	10	)	)	PUNCT
ejpam-4914	148	11	(	(	PUNCT
ejpam-4914	148	12	2023	2023	NUM
ejpam-4914	148	13	)	)	PUNCT
ejpam-4914	148	14	,	,	PUNCT
ejpam-4914	148	15	2431	2431	NUM
ejpam-4914	148	16	-	-	SYM
ejpam-4914	148	17	2449	2449	NUM
ejpam-4914	148	18	2436	2436	NUM
ejpam-4914	148	19	observe	observe	VERB
ejpam-4914	148	20	that	that	SCONJ
ejpam-4914	148	21	the	the	DET
ejpam-4914	148	22	function	function	NOUN
ejpam-4914	148	23	given	give	VERB
ejpam-4914	148	24	by	by	ADP
ejpam-4914	148	25	f(x	f(x	PROPN
ejpam-4914	148	26	)	)	PUNCT
ejpam-4914	148	27	=	=	SYM
ejpam-4914	148	28	1	1	NUM
ejpam-4914	148	29	for	for	ADP
ejpam-4914	148	30	all	all	PRON
ejpam-4914	148	31	x	x	SYM
ejpam-4914	148	32	∈	∈	NOUN
ejpam-4914	148	33	v	v	NOUN
ejpam-4914	148	34	(	(	PUNCT
ejpam-4914	148	35	g	g	NOUN
ejpam-4914	148	36	)	)	PUNCT
ejpam-4914	148	37	is	be	AUX
ejpam-4914	148	38	a	a	DET
ejpam-4914	148	39	γhi	γhi	ADJ
ejpam-4914	148	40	-function	-function	NOUN
ejpam-4914	148	41	of	of	ADP
ejpam-4914	148	42	g	g	NOUN
ejpam-4914	148	43	=	=	PUNCT
ejpam-4914	148	44	kp	kp	PROPN
ejpam-4914	148	45	.	.	PUNCT
ejpam-4914	149	1	in	in	ADP
ejpam-4914	149	2	this	this	DET
ejpam-4914	149	3	case	case	NOUN
ejpam-4914	149	4	,	,	PUNCT
ejpam-4914	149	5	v2	v2	NOUN
ejpam-4914	149	6	=	=	SYM
ejpam-4914	149	7	∅	∅	NOUN
ejpam-4914	149	8	,	,	PUNCT
ejpam-4914	149	9	and	and	CCONJ
ejpam-4914	149	10	such	such	ADJ
ejpam-4914	149	11	is	be	AUX
ejpam-4914	149	12	a	a	DET
ejpam-4914	149	13	particular	particular	ADJ
ejpam-4914	149	14	case	case	NOUN
ejpam-4914	149	15	of	of	ADP
ejpam-4914	149	16	the	the	DET
ejpam-4914	149	17	following	follow	VERB
ejpam-4914	149	18	proposition	proposition	NOUN
ejpam-4914	149	19	.	.	PUNCT
ejpam-4914	150	1	proposition	proposition	NOUN
ejpam-4914	150	2	2.3	2.3	NUM
ejpam-4914	150	3	.	.	PUNCT
ejpam-4914	151	1	for	for	ADP
ejpam-4914	151	2	every	every	DET
ejpam-4914	151	3	graph	graph	NOUN
ejpam-4914	151	4	g	g	NOUN
ejpam-4914	151	5	,	,	PUNCT
ejpam-4914	151	6	there	there	PRON
ejpam-4914	151	7	exists	exist	VERB
ejpam-4914	151	8	a	a	DET
ejpam-4914	151	9	γhi	γhi	ADJ
ejpam-4914	151	10	-	-	PUNCT
ejpam-4914	151	11	function	function	NOUN
ejpam-4914	151	12	f	f	NOUN
ejpam-4914	151	13	=	=	SYM
ejpam-4914	151	14	(	(	PUNCT
ejpam-4914	151	15	v0	v0	PROPN
ejpam-4914	151	16	,	,	PUNCT
ejpam-4914	151	17	v1	v1	NOUN
ejpam-4914	151	18	,	,	PUNCT
ejpam-4914	151	19	v2	v2	NOUN
ejpam-4914	151	20	)	)	PUNCT
ejpam-4914	151	21	such	such	ADJ
ejpam-4914	151	22	that	that	SCONJ
ejpam-4914	151	23	either	either	CCONJ
ejpam-4914	151	24	v2	v2	NOUN
ejpam-4914	151	25	=	=	SYM
ejpam-4914	151	26	∅	∅	NOUN
ejpam-4914	151	27	or	or	CCONJ
ejpam-4914	151	28	v2	v2	VERB
ejpam-4914	151	29	̸=	̸=	PROPN
ejpam-4914	151	30	∅	∅	NOUN
ejpam-4914	151	31	and	and	CCONJ
ejpam-4914	151	32	v	v	NOUN
ejpam-4914	151	33	has	have	VERB
ejpam-4914	151	34	at	at	ADV
ejpam-4914	151	35	least	least	ADV
ejpam-4914	151	36	three	three	NUM
ejpam-4914	151	37	italian	italian	ADJ
ejpam-4914	151	38	private	private	ADJ
ejpam-4914	151	39	hop	hop	NOUN
ejpam-4914	151	40	-	-	PUNCT
ejpam-4914	151	41	neighbors	neighbor	NOUN
ejpam-4914	151	42	for	for	ADP
ejpam-4914	151	43	each	each	DET
ejpam-4914	151	44	v	v	NUM
ejpam-4914	151	45	∈	∈	PROPN
ejpam-4914	151	46	v2	v2	NOUN
ejpam-4914	151	47	.	.	PUNCT
ejpam-4914	152	1	proof	proof	NOUN
ejpam-4914	152	2	:	:	PUNCT
ejpam-4914	152	3	let	let	VERB
ejpam-4914	152	4	f	f	PROPN
ejpam-4914	152	5	=	=	SYM
ejpam-4914	152	6	(	(	PUNCT
ejpam-4914	152	7	v0	v0	PROPN
ejpam-4914	152	8	,	,	PUNCT
ejpam-4914	152	9	v1	v1	NOUN
ejpam-4914	152	10	,	,	PUNCT
ejpam-4914	152	11	v2	v2	PROPN
ejpam-4914	152	12	)	)	PUNCT
ejpam-4914	152	13	be	be	AUX
ejpam-4914	152	14	a	a	DET
ejpam-4914	152	15	γhi	γhi	ADJ
ejpam-4914	152	16	-function	-function	NOUN
ejpam-4914	152	17	of	of	ADP
ejpam-4914	152	18	g	g	NOUN
ejpam-4914	152	19	with	with	ADP
ejpam-4914	152	20	a	a	DET
ejpam-4914	152	21	minimum	minimum	ADJ
ejpam-4914	152	22	|v2|	|v2|	NOUN
ejpam-4914	152	23	.	.	PUNCT
ejpam-4914	153	1	if	if	SCONJ
ejpam-4914	153	2	v2	v2	NOUN
ejpam-4914	153	3	=	=	NOUN
ejpam-4914	153	4	∅	∅	NOUN
ejpam-4914	153	5	,	,	PUNCT
ejpam-4914	153	6	then	then	ADV
ejpam-4914	153	7	the	the	DET
ejpam-4914	153	8	proposition	proposition	NOUN
ejpam-4914	153	9	holds	hold	VERB
ejpam-4914	153	10	.	.	PUNCT
ejpam-4914	153	11	suppose	suppose	VERB
ejpam-4914	153	12	that	that	SCONJ
ejpam-4914	153	13	v2	v2	PROPN
ejpam-4914	153	14	̸=	̸=	PROPN
ejpam-4914	153	15	∅	∅	NOUN
ejpam-4914	153	16	,	,	PUNCT
ejpam-4914	153	17	and	and	CCONJ
ejpam-4914	153	18	let	let	VERB
ejpam-4914	153	19	v	v	NUM
ejpam-4914	153	20	∈	∈	PROPN
ejpam-4914	153	21	v2	v2	NOUN
ejpam-4914	153	22	.	.	PUNCT
ejpam-4914	154	1	we	we	PRON
ejpam-4914	154	2	claim	claim	VERB
ejpam-4914	154	3	that	that	SCONJ
ejpam-4914	154	4	v	v	NOUN
ejpam-4914	154	5	has	have	VERB
ejpam-4914	154	6	at	at	ADV
ejpam-4914	154	7	least	least	ADV
ejpam-4914	154	8	three	three	NUM
ejpam-4914	154	9	italian	italian	ADJ
ejpam-4914	154	10	private	private	ADJ
ejpam-4914	154	11	hop	hop	NOUN
ejpam-4914	154	12	-	-	PUNCT
ejpam-4914	154	13	neighbors	neighbor	NOUN
ejpam-4914	154	14	.	.	PUNCT
ejpam-4914	155	1	first	first	ADV
ejpam-4914	155	2	,	,	PUNCT
ejpam-4914	155	3	note	note	VERB
ejpam-4914	155	4	that	that	SCONJ
ejpam-4914	155	5	if	if	SCONJ
ejpam-4914	155	6	v	v	NOUN
ejpam-4914	155	7	has	have	VERB
ejpam-4914	155	8	no	no	DET
ejpam-4914	155	9	italian	italian	ADJ
ejpam-4914	155	10	private	private	ADJ
ejpam-4914	155	11	hop	hop	NOUN
ejpam-4914	155	12	-	-	PUNCT
ejpam-4914	155	13	neighbor	neighbor	NOUN
ejpam-4914	155	14	in	in	ADP
ejpam-4914	155	15	v0	v0	PROPN
ejpam-4914	155	16	,	,	PUNCT
ejpam-4914	155	17	then	then	ADV
ejpam-4914	155	18	g	g	PROPN
ejpam-4914	155	19	=	=	SYM
ejpam-4914	155	20	(	(	PUNCT
ejpam-4914	155	21	v0	v0	NOUN
ejpam-4914	155	22	,	,	PUNCT
ejpam-4914	155	23	v1	v1	NOUN
ejpam-4914	155	24	∪	∪	ADJ
ejpam-4914	155	25	{	{	PUNCT
ejpam-4914	155	26	v	v	NOUN
ejpam-4914	155	27	}	}	PUNCT
ejpam-4914	155	28	,	,	PUNCT
ejpam-4914	155	29	v2	v2	PROPN
ejpam-4914	155	30	\	\	PROPN
ejpam-4914	155	31	{	{	PUNCT
ejpam-4914	155	32	v	v	NOUN
ejpam-4914	155	33	}	}	PUNCT
ejpam-4914	155	34	)	)	PUNCT
ejpam-4914	155	35	∈	∈	PROPN
ejpam-4914	155	36	hid(g	hid(g	PROPN
ejpam-4914	155	37	)	)	PUNCT
ejpam-4914	155	38	with	with	ADP
ejpam-4914	155	39	ωg(g	ωg(g	NOUN
ejpam-4914	155	40	)	)	PUNCT
ejpam-4914	155	41	<	<	X
ejpam-4914	155	42	ωg(f	ωg(f	NUM
ejpam-4914	155	43	)	)	PUNCT
ejpam-4914	155	44	,	,	PUNCT
ejpam-4914	155	45	a	a	DET
ejpam-4914	155	46	contradiction	contradiction	NOUN
ejpam-4914	155	47	.	.	PUNCT
ejpam-4914	156	1	next	next	ADV
ejpam-4914	156	2	,	,	PUNCT
ejpam-4914	156	3	suppose	suppose	VERB
ejpam-4914	156	4	that	that	SCONJ
ejpam-4914	156	5	v	v	NOUN
ejpam-4914	156	6	has	have	VERB
ejpam-4914	156	7	exactly	exactly	ADV
ejpam-4914	156	8	one	one	NUM
ejpam-4914	156	9	italian	italian	ADJ
ejpam-4914	156	10	private	private	ADJ
ejpam-4914	156	11	hop	hop	NOUN
ejpam-4914	156	12	-	-	PUNCT
ejpam-4914	156	13	neighbor	neighbor	NOUN
ejpam-4914	156	14	w	w	PROPN
ejpam-4914	156	15	∈	∈	PROPN
ejpam-4914	156	16	v0	v0	NOUN
ejpam-4914	156	17	.	.	PUNCT
ejpam-4914	157	1	if	if	SCONJ
ejpam-4914	157	2	∑	∑	PROPN
ejpam-4914	157	3	u∈ng(w,2)\{v	u∈ng(w,2)\{v	ADJ
ejpam-4914	157	4	}	}	PUNCT
ejpam-4914	157	5	f(u	f(u	PROPN
ejpam-4914	157	6	)	)	PUNCT
ejpam-4914	157	7	=	=	SYM
ejpam-4914	157	8	0	0	NUM
ejpam-4914	157	9	,	,	PUNCT
ejpam-4914	157	10	then	then	ADV
ejpam-4914	157	11	g	g	PROPN
ejpam-4914	157	12	=	=	PUNCT
ejpam-4914	157	13	(	(	PUNCT
ejpam-4914	157	14	v	v	NOUN
ejpam-4914	157	15	∗	∗	NOUN
ejpam-4914	157	16	0	0	NUM
ejpam-4914	157	17	,	,	PUNCT
ejpam-4914	157	18	v	v	NOUN
ejpam-4914	157	19	∗	∗	NOUN
ejpam-4914	157	20	1	1	NUM
ejpam-4914	157	21	,	,	PUNCT
ejpam-4914	157	22	v	v	NOUN
ejpam-4914	157	23	∗	∗	X
ejpam-4914	157	24	2	2	NUM
ejpam-4914	157	25	)	)	PUNCT
ejpam-4914	157	26	∈	∈	PROPN
ejpam-4914	157	27	hid(g	hid(g	PROPN
ejpam-4914	157	28	)	)	PUNCT
ejpam-4914	157	29	with	with	ADP
ejpam-4914	157	30	ωg(g	ωg(g	NOUN
ejpam-4914	157	31	)	)	PUNCT
ejpam-4914	157	32	=	=	NOUN
ejpam-4914	157	33	ωg(f	ωg(f	NOUN
ejpam-4914	157	34	)	)	PUNCT
ejpam-4914	157	35	,	,	PUNCT
ejpam-4914	157	36	where	where	SCONJ
ejpam-4914	157	37	v	v	NOUN
ejpam-4914	157	38	∗	∗	NOUN
ejpam-4914	157	39	0	0	NUM
ejpam-4914	158	1	=	=	SYM
ejpam-4914	158	2	v0	v0	NOUN
ejpam-4914	158	3	\	\	PUNCT
ejpam-4914	158	4	{	{	PUNCT
ejpam-4914	158	5	w	w	NOUN
ejpam-4914	158	6	}	}	PUNCT
ejpam-4914	158	7	,	,	PUNCT
ejpam-4914	158	8	v	v	NOUN
ejpam-4914	158	9	∗	∗	NOUN
ejpam-4914	158	10	1	1	NUM
ejpam-4914	158	11	=	=	SYM
ejpam-4914	158	12	v1	v1	NOUN
ejpam-4914	158	13	∪	∪	X
ejpam-4914	158	14	{	{	PUNCT
ejpam-4914	158	15	w	w	PROPN
ejpam-4914	158	16	,	,	PUNCT
ejpam-4914	158	17	v	v	NOUN
ejpam-4914	158	18	}	}	PUNCT
ejpam-4914	158	19	and	and	CCONJ
ejpam-4914	158	20	v	v	ADP
ejpam-4914	158	21	∗	∗	NOUN
ejpam-4914	158	22	2	2	NUM
ejpam-4914	158	23	=	=	SYM
ejpam-4914	158	24	v2	v2	PROPN
ejpam-4914	158	25	\	\	NOUN
ejpam-4914	158	26	{	{	PUNCT
ejpam-4914	158	27	v	v	NOUN
ejpam-4914	158	28	}	}	PUNCT
ejpam-4914	158	29	.	.	PUNCT
ejpam-4914	159	1	since	since	SCONJ
ejpam-4914	159	2	|v	|v	PROPN
ejpam-4914	159	3	∗	∗	NOUN
ejpam-4914	159	4	2	2	NUM
ejpam-4914	159	5	|	|	ADV
ejpam-4914	159	6	<	<	X
ejpam-4914	159	7	|v2|	|v2|	NOUN
ejpam-4914	159	8	,	,	PUNCT
ejpam-4914	159	9	this	this	PRON
ejpam-4914	159	10	is	be	AUX
ejpam-4914	159	11	a	a	DET
ejpam-4914	159	12	contradiction	contradiction	NOUN
ejpam-4914	159	13	to	to	ADP
ejpam-4914	159	14	the	the	DET
ejpam-4914	159	15	choice	choice	NOUN
ejpam-4914	159	16	of	of	ADP
ejpam-4914	159	17	f	f	PROPN
ejpam-4914	159	18	.	.	PUNCT
ejpam-4914	160	1	on	on	ADP
ejpam-4914	160	2	the	the	DET
ejpam-4914	160	3	other	other	ADJ
ejpam-4914	160	4	hand	hand	NOUN
ejpam-4914	160	5	,	,	PUNCT
ejpam-4914	160	6	if	if	SCONJ
ejpam-4914	160	7	∑	∑	ADV
ejpam-4914	160	8	u∈ng(w,2)\{v	u∈ng(w,2)\{v	ADJ
ejpam-4914	160	9	}	}	PUNCT
ejpam-4914	160	10	f(u	f(u	PROPN
ejpam-4914	160	11	)	)	PUNCT
ejpam-4914	160	12	=	=	SYM
ejpam-4914	160	13	1	1	NUM
ejpam-4914	160	14	,	,	PUNCT
ejpam-4914	160	15	then	then	ADV
ejpam-4914	160	16	g	g	PROPN
ejpam-4914	160	17	=	=	SYM
ejpam-4914	160	18	(	(	PUNCT
ejpam-4914	160	19	v0	v0	NOUN
ejpam-4914	160	20	,	,	PUNCT
ejpam-4914	160	21	v1	v1	NOUN
ejpam-4914	160	22	∪	∪	ADJ
ejpam-4914	160	23	{	{	PUNCT
ejpam-4914	160	24	v	v	NOUN
ejpam-4914	160	25	}	}	PUNCT
ejpam-4914	160	26	,	,	PUNCT
ejpam-4914	160	27	v2	v2	PROPN
ejpam-4914	160	28	\	\	PROPN
ejpam-4914	160	29	{	{	PUNCT
ejpam-4914	160	30	v	v	NOUN
ejpam-4914	160	31	}	}	PUNCT
ejpam-4914	160	32	)	)	PUNCT
ejpam-4914	160	33	∈	∈	PROPN
ejpam-4914	160	34	hid(g	hid(g	PROPN
ejpam-4914	160	35	)	)	PUNCT
ejpam-4914	160	36	with	with	ADP
ejpam-4914	160	37	ωg(g	ωg(g	NOUN
ejpam-4914	160	38	)	)	PUNCT
ejpam-4914	160	39	<	<	X
ejpam-4914	160	40	ωg(f	ωg(f	NUM
ejpam-4914	160	41	)	)	PUNCT
ejpam-4914	160	42	,	,	PUNCT
ejpam-4914	160	43	a	a	DET
ejpam-4914	160	44	contradiction	contradiction	NOUN
ejpam-4914	160	45	.	.	PUNCT
ejpam-4914	161	1	finally	finally	ADV
ejpam-4914	161	2	,	,	PUNCT
ejpam-4914	161	3	suppose	suppose	VERB
ejpam-4914	161	4	that	that	SCONJ
ejpam-4914	161	5	v	v	NOUN
ejpam-4914	161	6	has	have	VERB
ejpam-4914	161	7	exactly	exactly	ADV
ejpam-4914	161	8	two	two	NUM
ejpam-4914	161	9	neighbors	neighbor	NOUN
ejpam-4914	161	10	w	w	NOUN
ejpam-4914	161	11	and	and	CCONJ
ejpam-4914	161	12	z	z	PROPN
ejpam-4914	161	13	in	in	ADP
ejpam-4914	161	14	v0	v0	NOUN
ejpam-4914	161	15	.	.	PUNCT
ejpam-4914	162	1	exactly	exactly	ADV
ejpam-4914	162	2	one	one	NUM
ejpam-4914	162	3	of	of	ADP
ejpam-4914	162	4	the	the	DET
ejpam-4914	162	5	following	following	NOUN
ejpam-4914	162	6	holds	hold	VERB
ejpam-4914	162	7	:	:	PUNCT
ejpam-4914	162	8	(	(	PUNCT
ejpam-4914	162	9	a	a	X
ejpam-4914	162	10	)	)	PUNCT
ejpam-4914	162	11	∑	∑	PROPN
ejpam-4914	162	12	u∈ng(w,2)\{v	u∈ng(w,2)\{v	PROPN
ejpam-4914	162	13	}	}	PUNCT
ejpam-4914	162	14	f(u	f(u	PROPN
ejpam-4914	162	15	)	)	PUNCT
ejpam-4914	162	16	=	=	SYM
ejpam-4914	162	17	0	0	NUM
ejpam-4914	162	18	and	and	CCONJ
ejpam-4914	162	19	∑	∑	ADV
ejpam-4914	162	20	u∈ng(z,2)\{v	u∈ng(z,2)\{v	ADJ
ejpam-4914	162	21	}	}	PUNCT
ejpam-4914	162	22	f(u	f(u	PROPN
ejpam-4914	162	23	)	)	PUNCT
ejpam-4914	162	24	=	=	SYM
ejpam-4914	162	25	0	0	NUM
ejpam-4914	162	26	;	;	PUNCT
ejpam-4914	162	27	(	(	PUNCT
ejpam-4914	162	28	b	b	X
ejpam-4914	162	29	)	)	PUNCT
ejpam-4914	162	30	∑	∑	ADV
ejpam-4914	162	31	u∈ng(w,2)\{v	u∈ng(w,2)\{v	PROPN
ejpam-4914	162	32	}	}	PUNCT
ejpam-4914	162	33	f(u	f(u	PROPN
ejpam-4914	162	34	)	)	PUNCT
ejpam-4914	162	35	=	=	SYM
ejpam-4914	162	36	1	1	NUM
ejpam-4914	162	37	and	and	CCONJ
ejpam-4914	162	38	∑	∑	ADV
ejpam-4914	162	39	u∈ng(z,2)\{v	u∈ng(z,2)\{v	ADJ
ejpam-4914	162	40	}	}	PUNCT
ejpam-4914	162	41	f(u	f(u	PROPN
ejpam-4914	162	42	)	)	PUNCT
ejpam-4914	162	43	=	=	SYM
ejpam-4914	162	44	1	1	NUM
ejpam-4914	162	45	;	;	PUNCT
ejpam-4914	162	46	(	(	PUNCT
ejpam-4914	162	47	c	c	X
ejpam-4914	162	48	)	)	PUNCT
ejpam-4914	162	49	∑	∑	ADV
ejpam-4914	162	50	u∈ng(w,2)\{v	u∈ng(w,2)\{v	PROPN
ejpam-4914	162	51	}	}	PUNCT
ejpam-4914	162	52	f(u	f(u	PROPN
ejpam-4914	162	53	)	)	PUNCT
ejpam-4914	162	54	=	=	SYM
ejpam-4914	162	55	0	0	NUM
ejpam-4914	162	56	and	and	CCONJ
ejpam-4914	162	57	∑	∑	ADV
ejpam-4914	162	58	u∈ng(z,2)\{v	u∈ng(z,2)\{v	ADJ
ejpam-4914	162	59	}	}	PUNCT
ejpam-4914	162	60	f(u	f(u	PROPN
ejpam-4914	162	61	)	)	PUNCT
ejpam-4914	162	62	=	=	SYM
ejpam-4914	162	63	1	1	NUM
ejpam-4914	162	64	;	;	PUNCT
ejpam-4914	162	65	and	and	CCONJ
ejpam-4914	162	66	(	(	PUNCT
ejpam-4914	162	67	d	d	X
ejpam-4914	162	68	)	)	PUNCT
ejpam-4914	162	69	∑	∑	ADV
ejpam-4914	162	70	u∈ng(w,2)\{v	u∈ng(w,2)\{v	PROPN
ejpam-4914	162	71	}	}	PUNCT
ejpam-4914	162	72	f(u	f(u	PROPN
ejpam-4914	162	73	)	)	PUNCT
ejpam-4914	162	74	=	=	SYM
ejpam-4914	162	75	1	1	NUM
ejpam-4914	162	76	and	and	CCONJ
ejpam-4914	162	77	∑	∑	ADV
ejpam-4914	162	78	u∈ng(z,2)\{v	u∈ng(z,2)\{v	ADJ
ejpam-4914	162	79	}	}	PUNCT
ejpam-4914	162	80	f(u	f(u	PROPN
ejpam-4914	162	81	)	)	PUNCT
ejpam-4914	162	82	=	=	SYM
ejpam-4914	163	1	0	0	X
ejpam-4914	163	2	.	.	PUNCT
ejpam-4914	163	3	suppose	suppose	VERB
ejpam-4914	163	4	that	that	SCONJ
ejpam-4914	163	5	(	(	PUNCT
ejpam-4914	163	6	a	a	X
ejpam-4914	163	7	)	)	PUNCT
ejpam-4914	163	8	holds	hold	VERB
ejpam-4914	163	9	for	for	ADP
ejpam-4914	163	10	f	f	PROPN
ejpam-4914	163	11	.	.	PUNCT
ejpam-4914	164	1	put	put	VERB
ejpam-4914	164	2	v	v	NUM
ejpam-4914	164	3	∗	∗	NOUN
ejpam-4914	164	4	0	0	NUM
ejpam-4914	165	1	=	=	SYM
ejpam-4914	165	2	{	{	PUNCT
ejpam-4914	165	3	v	v	NOUN
ejpam-4914	165	4	}	}	PUNCT
ejpam-4914	165	5	∪	∪	NOUN
ejpam-4914	165	6	(	(	PUNCT
ejpam-4914	165	7	v0	v0	NOUN
ejpam-4914	165	8	\	\	PUNCT
ejpam-4914	165	9	{	{	PUNCT
ejpam-4914	165	10	w	w	PROPN
ejpam-4914	165	11	,	,	PUNCT
ejpam-4914	165	12	z	z	NOUN
ejpam-4914	165	13	}	}	PUNCT
ejpam-4914	165	14	)	)	PUNCT
ejpam-4914	165	15	,	,	PUNCT
ejpam-4914	165	16	v	v	X
ejpam-4914	165	17	∗	∗	NOUN
ejpam-4914	165	18	1	1	NUM
ejpam-4914	165	19	=	=	SYM
ejpam-4914	165	20	v1	v1	NOUN
ejpam-4914	165	21	∪	∪	X
ejpam-4914	165	22	{	{	PUNCT
ejpam-4914	165	23	w	w	PROPN
ejpam-4914	165	24	,	,	PUNCT
ejpam-4914	165	25	z	z	NOUN
ejpam-4914	165	26	}	}	PUNCT
ejpam-4914	165	27	and	and	CCONJ
ejpam-4914	165	28	v	v	ADP
ejpam-4914	165	29	∗	∗	NOUN
ejpam-4914	165	30	2	2	NUM
ejpam-4914	165	31	=	=	SYM
ejpam-4914	165	32	v2	v2	PROPN
ejpam-4914	165	33	\	\	NOUN
ejpam-4914	165	34	{	{	PUNCT
ejpam-4914	165	35	v	v	NOUN
ejpam-4914	165	36	}	}	PUNCT
ejpam-4914	165	37	.	.	PUNCT
ejpam-4914	166	1	then	then	ADV
ejpam-4914	166	2	g	g	PROPN
ejpam-4914	166	3	=	=	PUNCT
ejpam-4914	166	4	(	(	PUNCT
ejpam-4914	166	5	v	v	NOUN
ejpam-4914	166	6	∗	∗	NOUN
ejpam-4914	166	7	0	0	NUM
ejpam-4914	166	8	,	,	PUNCT
ejpam-4914	166	9	v	v	NOUN
ejpam-4914	166	10	∗	∗	NOUN
ejpam-4914	166	11	1	1	NUM
ejpam-4914	166	12	,	,	PUNCT
ejpam-4914	166	13	v	v	NOUN
ejpam-4914	166	14	∗	∗	X
ejpam-4914	166	15	2	2	NUM
ejpam-4914	166	16	)	)	PUNCT
ejpam-4914	166	17	∈	∈	PROPN
ejpam-4914	166	18	hid(g	hid(g	PROPN
ejpam-4914	166	19	)	)	PUNCT
ejpam-4914	166	20	with	with	ADP
ejpam-4914	166	21	wg(g	wg(g	NOUN
ejpam-4914	166	22	)	)	PUNCT
ejpam-4914	166	23	=	=	SYM
ejpam-4914	166	24	wg(f	wg(f	NOUN
ejpam-4914	166	25	)	)	PUNCT
ejpam-4914	166	26	.	.	PUNCT
ejpam-4914	167	1	since	since	SCONJ
ejpam-4914	167	2	|v	|v	PROPN
ejpam-4914	167	3	∗	∗	NOUN
ejpam-4914	167	4	2	2	NUM
ejpam-4914	167	5	|	|	ADV
ejpam-4914	167	6	<	<	X
ejpam-4914	167	7	|v2|	|v2|	NOUN
ejpam-4914	167	8	,	,	PUNCT
ejpam-4914	167	9	this	this	PRON
ejpam-4914	167	10	is	be	AUX
ejpam-4914	167	11	a	a	DET
ejpam-4914	167	12	contradiction	contradiction	NOUN
ejpam-4914	167	13	to	to	ADP
ejpam-4914	167	14	the	the	DET
ejpam-4914	167	15	assumption	assumption	NOUN
ejpam-4914	167	16	of	of	ADP
ejpam-4914	167	17	f	f	PROPN
ejpam-4914	167	18	.	.	PUNCT
ejpam-4914	168	1	next	next	ADV
ejpam-4914	168	2	,	,	PUNCT
ejpam-4914	168	3	suppose	suppose	VERB
ejpam-4914	168	4	that	that	SCONJ
ejpam-4914	168	5	(	(	PUNCT
ejpam-4914	168	6	b	b	X
ejpam-4914	168	7	)	)	PUNCT
ejpam-4914	168	8	holds	hold	VERB
ejpam-4914	168	9	for	for	ADP
ejpam-4914	168	10	f	f	PROPN
ejpam-4914	168	11	.	.	PUNCT
ejpam-4914	169	1	in	in	ADP
ejpam-4914	169	2	this	this	DET
ejpam-4914	169	3	case	case	NOUN
ejpam-4914	169	4	,	,	PUNCT
ejpam-4914	169	5	define	define	VERB
ejpam-4914	169	6	v	v	ADP
ejpam-4914	169	7	∗	∗	NOUN
ejpam-4914	169	8	0	0	NUM
ejpam-4914	170	1	=	=	SYM
ejpam-4914	170	2	v0	v0	PROPN
ejpam-4914	170	3	,	,	PUNCT
ejpam-4914	170	4	v	v	NOUN
ejpam-4914	170	5	∗	∗	NOUN
ejpam-4914	170	6	1	1	NUM
ejpam-4914	170	7	=	=	SYM
ejpam-4914	170	8	v1	v1	NOUN
ejpam-4914	170	9	∪	∪	X
ejpam-4914	170	10	{	{	PUNCT
ejpam-4914	170	11	v	v	NOUN
ejpam-4914	170	12	}	}	PUNCT
ejpam-4914	170	13	and	and	CCONJ
ejpam-4914	170	14	v	v	ADP
ejpam-4914	170	15	∗	∗	NOUN
ejpam-4914	170	16	2	2	NUM
ejpam-4914	170	17	=	=	SYM
ejpam-4914	170	18	v2	v2	PROPN
ejpam-4914	170	19	\	\	NOUN
ejpam-4914	170	20	{	{	PUNCT
ejpam-4914	170	21	v	v	NOUN
ejpam-4914	170	22	}	}	PUNCT
ejpam-4914	170	23	.	.	PUNCT
ejpam-4914	171	1	then	then	ADV
ejpam-4914	171	2	g	g	PROPN
ejpam-4914	171	3	=	=	PUNCT
ejpam-4914	171	4	(	(	PUNCT
ejpam-4914	171	5	v	v	NOUN
ejpam-4914	171	6	∗	∗	NOUN
ejpam-4914	171	7	0	0	NUM
ejpam-4914	171	8	,	,	PUNCT
ejpam-4914	171	9	v	v	NOUN
ejpam-4914	171	10	∗	∗	NOUN
ejpam-4914	171	11	1	1	NUM
ejpam-4914	171	12	,	,	PUNCT
ejpam-4914	171	13	v	v	NOUN
ejpam-4914	171	14	∗	∗	X
ejpam-4914	171	15	2	2	NUM
ejpam-4914	171	16	)	)	PUNCT
ejpam-4914	171	17	∈	∈	PROPN
ejpam-4914	171	18	hid(g	hid(g	PROPN
ejpam-4914	171	19	)	)	PUNCT
ejpam-4914	171	20	with	with	ADP
ejpam-4914	171	21	wg(g	wg(g	NOUN
ejpam-4914	171	22	)	)	PUNCT
ejpam-4914	171	23	<	<	X
ejpam-4914	171	24	wg(f	wg(f	NOUN
ejpam-4914	171	25	)	)	PUNCT
ejpam-4914	171	26	,	,	PUNCT
ejpam-4914	171	27	a	a	DET
ejpam-4914	171	28	contradiction	contradiction	NOUN
ejpam-4914	171	29	.	.	PUNCT
ejpam-4914	172	1	next	next	ADV
ejpam-4914	172	2	,	,	PUNCT
ejpam-4914	172	3	suppose	suppose	VERB
ejpam-4914	172	4	that	that	SCONJ
ejpam-4914	172	5	(	(	PUNCT
ejpam-4914	172	6	c	c	X
ejpam-4914	172	7	)	)	PUNCT
ejpam-4914	172	8	holds	hold	VERB
ejpam-4914	172	9	for	for	ADP
ejpam-4914	172	10	f	f	PROPN
ejpam-4914	172	11	.	.	PUNCT
ejpam-4914	173	1	define	define	VERB
ejpam-4914	173	2	v	v	ADP
ejpam-4914	173	3	∗	∗	NOUN
ejpam-4914	173	4	0	0	NUM
ejpam-4914	174	1	=	=	SYM
ejpam-4914	174	2	v0	v0	NOUN
ejpam-4914	174	3	\	\	PUNCT
ejpam-4914	174	4	{	{	PUNCT
ejpam-4914	174	5	w	w	NOUN
ejpam-4914	174	6	}	}	PUNCT
ejpam-4914	174	7	,	,	PUNCT
ejpam-4914	174	8	v	v	NOUN
ejpam-4914	174	9	∗	∗	NOUN
ejpam-4914	174	10	1	1	NUM
ejpam-4914	174	11	=	=	SYM
ejpam-4914	174	12	v1	v1	NOUN
ejpam-4914	174	13	∪	∪	X
ejpam-4914	174	14	{	{	PUNCT
ejpam-4914	174	15	w	w	PROPN
ejpam-4914	174	16	,	,	PUNCT
ejpam-4914	174	17	v	v	NOUN
ejpam-4914	174	18	}	}	PUNCT
ejpam-4914	174	19	and	and	CCONJ
ejpam-4914	174	20	v	v	ADP
ejpam-4914	174	21	∗	∗	NOUN
ejpam-4914	174	22	2	2	NUM
ejpam-4914	174	23	=	=	SYM
ejpam-4914	174	24	v2	v2	PROPN
ejpam-4914	174	25	\	\	NOUN
ejpam-4914	174	26	{	{	PUNCT
ejpam-4914	174	27	v	v	NOUN
ejpam-4914	174	28	}	}	PUNCT
ejpam-4914	174	29	.	.	PUNCT
ejpam-4914	175	1	then	then	ADV
ejpam-4914	175	2	g	g	PROPN
ejpam-4914	175	3	=	=	PUNCT
ejpam-4914	175	4	(	(	PUNCT
ejpam-4914	175	5	v	v	NOUN
ejpam-4914	175	6	∗	∗	NOUN
ejpam-4914	175	7	0	0	NUM
ejpam-4914	175	8	,	,	PUNCT
ejpam-4914	175	9	v	v	NOUN
ejpam-4914	175	10	∗	∗	NOUN
ejpam-4914	175	11	1	1	NUM
ejpam-4914	175	12	,	,	PUNCT
ejpam-4914	175	13	v	v	NOUN
ejpam-4914	175	14	∗	∗	X
ejpam-4914	175	15	2	2	NUM
ejpam-4914	175	16	)	)	PUNCT
ejpam-4914	175	17	∈	∈	PROPN
ejpam-4914	175	18	hid(g	hid(g	PROPN
ejpam-4914	175	19	)	)	PUNCT
ejpam-4914	175	20	with	with	ADP
ejpam-4914	175	21	wg(g	wg(g	NOUN
ejpam-4914	175	22	)	)	PUNCT
ejpam-4914	175	23	=	=	SYM
ejpam-4914	175	24	wg(f	wg(f	NOUN
ejpam-4914	175	25	)	)	PUNCT
ejpam-4914	175	26	.	.	PUNCT
ejpam-4914	176	1	since	since	SCONJ
ejpam-4914	176	2	|v	|v	PROPN
ejpam-4914	176	3	∗	∗	NOUN
ejpam-4914	176	4	2	2	NUM
ejpam-4914	176	5	|	|	ADV
ejpam-4914	176	6	<	<	X
ejpam-4914	176	7	|v2|	|v2|	NOUN
ejpam-4914	176	8	,	,	PUNCT
ejpam-4914	176	9	this	this	PRON
ejpam-4914	176	10	is	be	AUX
ejpam-4914	176	11	a	a	DET
ejpam-4914	176	12	contradiction	contradiction	NOUN
ejpam-4914	176	13	.	.	PUNCT
ejpam-4914	177	1	similar	similar	ADJ
ejpam-4914	177	2	contradiction	contradiction	NOUN
ejpam-4914	177	3	is	be	AUX
ejpam-4914	177	4	attained	attain	VERB
ejpam-4914	177	5	if	if	SCONJ
ejpam-4914	177	6	(	(	PUNCT
ejpam-4914	177	7	d	d	X
ejpam-4914	177	8	)	)	PUNCT
ejpam-4914	177	9	holds	hold	VERB
ejpam-4914	177	10	for	for	ADP
ejpam-4914	177	11	f	f	PROPN
ejpam-4914	177	12	.	.	PUNCT
ejpam-4914	178	1	the	the	DET
ejpam-4914	178	2	above	above	ADJ
ejpam-4914	178	3	contradictions	contradiction	NOUN
ejpam-4914	178	4	imply	imply	VERB
ejpam-4914	178	5	that	that	SCONJ
ejpam-4914	178	6	v	v	NOUN
ejpam-4914	178	7	has	have	VERB
ejpam-4914	178	8	at	at	ADV
ejpam-4914	178	9	least	least	ADV
ejpam-4914	178	10	three	three	NUM
ejpam-4914	178	11	italian	italian	ADJ
ejpam-4914	178	12	private	private	ADJ
ejpam-4914	178	13	hop	hop	NOUN
ejpam-4914	178	14	-	-	PUNCT
ejpam-4914	178	15	neighbors	neighbor	NOUN
ejpam-4914	178	16	.	.	PUNCT
ejpam-4914	179	1	■	■	PUNCT
ejpam-4914	179	2	proposition	proposition	NOUN
ejpam-4914	179	3	2.4	2.4	NUM
ejpam-4914	179	4	.	.	PUNCT
ejpam-4914	180	1	let	let	VERB
ejpam-4914	180	2	g	g	PRON
ejpam-4914	180	3	be	be	AUX
ejpam-4914	180	4	a	a	DET
ejpam-4914	180	5	connected	connected	ADJ
ejpam-4914	180	6	graph	graph	NOUN
ejpam-4914	180	7	of	of	ADP
ejpam-4914	180	8	order	order	NOUN
ejpam-4914	180	9	n.	n.	NOUN
ejpam-4914	180	10	then	then	ADV
ejpam-4914	180	11	(	(	PUNCT
ejpam-4914	180	12	i	i	NOUN
ejpam-4914	180	13	)	)	PUNCT
ejpam-4914	181	1	γhi(g	γhi(g	PROPN
ejpam-4914	181	2	)	)	PUNCT
ejpam-4914	182	1	=	=	SYM
ejpam-4914	182	2	1	1	NUM
ejpam-4914	182	3	if	if	SCONJ
ejpam-4914	182	4	and	and	CCONJ
ejpam-4914	182	5	only	only	ADV
ejpam-4914	182	6	if	if	SCONJ
ejpam-4914	182	7	g	g	PROPN
ejpam-4914	182	8	=	=	SYM
ejpam-4914	182	9	k1	k1	PROPN
ejpam-4914	182	10	;	;	PUNCT
ejpam-4914	182	11	(	(	PUNCT
ejpam-4914	182	12	ii	ii	NOUN
ejpam-4914	182	13	)	)	PUNCT
ejpam-4914	182	14	γhi(g	γhi(g	PROPN
ejpam-4914	182	15	)	)	PUNCT
ejpam-4914	182	16	=	=	SYM
ejpam-4914	182	17	2	2	NUM
ejpam-4914	182	18	if	if	SCONJ
ejpam-4914	182	19	and	and	CCONJ
ejpam-4914	182	20	only	only	ADV
ejpam-4914	182	21	if	if	SCONJ
ejpam-4914	182	22	g	g	PROPN
ejpam-4914	182	23	=	=	SYM
ejpam-4914	182	24	k2	k2	PROPN
ejpam-4914	182	25	;	;	PUNCT
ejpam-4914	182	26	(	(	PUNCT
ejpam-4914	182	27	iii	iii	X
ejpam-4914	182	28	)	)	PUNCT
ejpam-4914	182	29	γhi(g	γhi(g	PROPN
ejpam-4914	182	30	)	)	PUNCT
ejpam-4914	183	1	=	=	SYM
ejpam-4914	183	2	3	3	NUM
ejpam-4914	183	3	if	if	SCONJ
ejpam-4914	183	4	and	and	CCONJ
ejpam-4914	183	5	only	only	ADV
ejpam-4914	183	6	if	if	SCONJ
ejpam-4914	183	7	γ2h(g	γ2h(g	NOUN
ejpam-4914	183	8	)	)	PUNCT
ejpam-4914	183	9	=	=	SYM
ejpam-4914	183	10	3	3	NUM
ejpam-4914	183	11	or	or	CCONJ
ejpam-4914	183	12	g	g	NOUN
ejpam-4914	183	13	=	=	PROPN
ejpam-4914	183	14	k1	k1	PROPN
ejpam-4914	183	15	+	+	CCONJ
ejpam-4914	183	16	(	(	PUNCT
ejpam-4914	183	17	k1	k1	PROPN
ejpam-4914	183	18	∪h	∪h	NUM
ejpam-4914	183	19	)	)	PUNCT
ejpam-4914	183	20	for	for	ADP
ejpam-4914	183	21	some	some	DET
ejpam-4914	183	22	graph	graph	NOUN
ejpam-4914	183	23	h	h	NOUN
ejpam-4914	183	24	of	of	ADP
ejpam-4914	183	25	order	order	NOUN
ejpam-4914	183	26	≥	≥	NOUN
ejpam-4914	183	27	3	3	X
ejpam-4914	183	28	.	.	PUNCT
ejpam-4914	184	1	s.r	s.r	PROPN
ejpam-4914	184	2	.	.	PROPN
ejpam-4914	184	3	jr	jr	PROPN
ejpam-4914	184	4	.	.	PROPN
ejpam-4914	184	5	canoy	canoy	PROPN
ejpam-4914	184	6	,	,	PUNCT
ejpam-4914	184	7	f.p	f.p	PROPN
ejpam-4914	184	8	.	.	PROPN
ejpam-4914	184	9	jamil	jamil	PROPN
ejpam-4914	184	10	and	and	CCONJ
ejpam-4914	184	11	s.m	s.m	PROPN
ejpam-4914	184	12	.	.	PROPN
ejpam-4914	184	13	menchavez	menchavez	PROPN
ejpam-4914	184	14	/	/	PUNCT
ejpam-4914	184	15	eur	eur	PROPN
ejpam-4914	184	16	.	.	PUNCT
ejpam-4914	185	1	j.	j.	PROPN
ejpam-4914	185	2	pure	pure	PROPN
ejpam-4914	185	3	appl	appl	PROPN
ejpam-4914	185	4	.	.	PROPN
ejpam-4914	185	5	math	math	PROPN
ejpam-4914	185	6	,	,	PUNCT
ejpam-4914	185	7	16	16	NUM
ejpam-4914	185	8	(	(	PUNCT
ejpam-4914	185	9	4	4	NUM
ejpam-4914	185	10	)	)	PUNCT
ejpam-4914	185	11	(	(	PUNCT
ejpam-4914	185	12	2023	2023	NUM
ejpam-4914	185	13	)	)	PUNCT
ejpam-4914	185	14	,	,	PUNCT
ejpam-4914	185	15	2431	2431	NUM
ejpam-4914	185	16	-	-	SYM
ejpam-4914	185	17	2449	2449	NUM
ejpam-4914	185	18	2437	2437	NUM
ejpam-4914	185	19	proof	proof	NOUN
ejpam-4914	185	20	:	:	PUNCT
ejpam-4914	185	21	for	for	ADP
ejpam-4914	185	22	(	(	PUNCT
ejpam-4914	185	23	i	i	NOUN
ejpam-4914	185	24	):	):	PUNCT
ejpam-4914	185	25	if	if	SCONJ
ejpam-4914	185	26	g	g	PROPN
ejpam-4914	185	27	=	=	SYM
ejpam-4914	185	28	k1	k1	PROPN
ejpam-4914	185	29	,	,	PUNCT
ejpam-4914	185	30	then	then	ADV
ejpam-4914	185	31	γhi(g	γhi(g	PROPN
ejpam-4914	185	32	)	)	PUNCT
ejpam-4914	186	1	=	=	SYM
ejpam-4914	186	2	1	1	X
ejpam-4914	186	3	.	.	PUNCT
ejpam-4914	187	1	conversely	conversely	ADV
ejpam-4914	187	2	,	,	PUNCT
ejpam-4914	187	3	if	if	SCONJ
ejpam-4914	187	4	γhi(g	γhi(g	PROPN
ejpam-4914	187	5	)	)	PUNCT
ejpam-4914	187	6	=	=	SYM
ejpam-4914	188	1	1	1	NUM
ejpam-4914	188	2	,	,	PUNCT
ejpam-4914	188	3	then	then	ADV
ejpam-4914	188	4	γh(g	γh(g	PUNCT
ejpam-4914	188	5	)	)	PUNCT
ejpam-4914	188	6	=	=	SYM
ejpam-4914	188	7	1	1	NUM
ejpam-4914	189	1	and	and	CCONJ
ejpam-4914	189	2	so	so	ADV
ejpam-4914	189	3	g	g	PROPN
ejpam-4914	189	4	=	=	PROPN
ejpam-4914	189	5	k1	k1	PROPN
ejpam-4914	189	6	.	.	PUNCT
ejpam-4914	190	1	for	for	ADP
ejpam-4914	190	2	(	(	PUNCT
ejpam-4914	190	3	ii	ii	NOUN
ejpam-4914	190	4	):	):	PUNCT
ejpam-4914	190	5	if	if	SCONJ
ejpam-4914	190	6	g	g	PROPN
ejpam-4914	190	7	=	=	SYM
ejpam-4914	190	8	k2	k2	PROPN
ejpam-4914	190	9	,	,	PUNCT
ejpam-4914	190	10	then	then	ADV
ejpam-4914	190	11	γhi(g	γhi(g	PROPN
ejpam-4914	190	12	)	)	PUNCT
ejpam-4914	190	13	=	=	SYM
ejpam-4914	190	14	2	2	X
ejpam-4914	190	15	.	.	X
ejpam-4914	190	16	assume	assume	VERB
ejpam-4914	190	17	that	that	SCONJ
ejpam-4914	190	18	γhi(g	γhi(g	ADV
ejpam-4914	190	19	)	)	PUNCT
ejpam-4914	191	1	=	=	SYM
ejpam-4914	192	1	2	2	X
ejpam-4914	192	2	.	.	PUNCT
ejpam-4914	192	3	by	by	ADP
ejpam-4914	192	4	proposition	proposition	NOUN
ejpam-4914	192	5	2.3	2.3	NUM
ejpam-4914	192	6	,	,	PUNCT
ejpam-4914	192	7	g	g	PROPN
ejpam-4914	192	8	has	have	VERB
ejpam-4914	192	9	a	a	DET
ejpam-4914	192	10	γhi	γhi	ADV
ejpam-4914	192	11	-function	-function	NOUN
ejpam-4914	192	12	f	f	NOUN
ejpam-4914	192	13	=	=	SYM
ejpam-4914	192	14	(	(	PUNCT
ejpam-4914	192	15	v0	v0	PROPN
ejpam-4914	192	16	,	,	PUNCT
ejpam-4914	192	17	v1	v1	NOUN
ejpam-4914	192	18	,	,	PUNCT
ejpam-4914	192	19	v2	v2	PROPN
ejpam-4914	192	20	)	)	PUNCT
ejpam-4914	192	21	for	for	ADP
ejpam-4914	192	22	which	which	PRON
ejpam-4914	192	23	either	either	CCONJ
ejpam-4914	192	24	v2	v2	NOUN
ejpam-4914	192	25	=	=	SYM
ejpam-4914	192	26	∅	∅	NOUN
ejpam-4914	192	27	or	or	CCONJ
ejpam-4914	192	28	v2	v2	VERB
ejpam-4914	192	29	̸=	̸=	PROPN
ejpam-4914	192	30	∅	∅	NOUN
ejpam-4914	192	31	and	and	CCONJ
ejpam-4914	192	32	each	each	DET
ejpam-4914	192	33	v	v	NOUN
ejpam-4914	192	34	∈	∈	PROPN
ejpam-4914	192	35	v2	v2	NOUN
ejpam-4914	192	36	has	have	VERB
ejpam-4914	192	37	at	at	ADV
ejpam-4914	192	38	least	least	ADV
ejpam-4914	192	39	3	3	NUM
ejpam-4914	192	40	private	private	ADJ
ejpam-4914	192	41	hop	hop	NOUN
ejpam-4914	192	42	-	-	PUNCT
ejpam-4914	192	43	neighbors	neighbor	NOUN
ejpam-4914	192	44	in	in	ADP
ejpam-4914	192	45	v0	v0	PROPN
ejpam-4914	192	46	.	.	PUNCT
ejpam-4914	193	1	suppose	suppose	VERB
ejpam-4914	193	2	that	that	SCONJ
ejpam-4914	193	3	v2	v2	PROPN
ejpam-4914	193	4	̸=	̸=	PROPN
ejpam-4914	193	5	∅.	∅.	ADV
ejpam-4914	193	6	let	let	VERB
ejpam-4914	193	7	v	v	ADP
ejpam-4914	193	8	∈	∈	PROPN
ejpam-4914	193	9	v2	v2	NOUN
ejpam-4914	193	10	and	and	CCONJ
ejpam-4914	193	11	let	let	VERB
ejpam-4914	193	12	u	u	PRON
ejpam-4914	193	13	∈	∈	PROPN
ejpam-4914	193	14	v0	v0	NOUN
ejpam-4914	193	15	be	be	AUX
ejpam-4914	193	16	a	a	DET
ejpam-4914	193	17	private	private	ADJ
ejpam-4914	193	18	hop	hop	NOUN
ejpam-4914	193	19	-	-	PUNCT
ejpam-4914	193	20	neighbor	neighbor	NOUN
ejpam-4914	193	21	of	of	ADP
ejpam-4914	193	22	v.	v.	ADV
ejpam-4914	193	23	then	then	ADV
ejpam-4914	193	24	there	there	PRON
ejpam-4914	193	25	exists	exist	VERB
ejpam-4914	193	26	a	a	DET
ejpam-4914	193	27	u	u	NOUN
ejpam-4914	193	28	-	-	NOUN
ejpam-4914	193	29	v	v	ADJ
ejpam-4914	193	30	geodesic	geodesic	NOUN
ejpam-4914	194	1	[	[	X
ejpam-4914	194	2	u	u	NOUN
ejpam-4914	194	3	,	,	PUNCT
ejpam-4914	194	4	w	w	PROPN
ejpam-4914	194	5	,	,	PUNCT
ejpam-4914	194	6	v	v	NOUN
ejpam-4914	194	7	]	]	X
ejpam-4914	194	8	in	in	ADP
ejpam-4914	194	9	g.	g.	PROPN
ejpam-4914	194	10	if	if	SCONJ
ejpam-4914	194	11	w	w	PROPN
ejpam-4914	194	12	∈	∈	PROPN
ejpam-4914	194	13	v1∪v2	v1∪v2	PROPN
ejpam-4914	194	14	,	,	PUNCT
ejpam-4914	194	15	then	then	ADV
ejpam-4914	194	16	wg(f	wg(f	PUNCT
ejpam-4914	194	17	)	)	PUNCT
ejpam-4914	194	18	≥	≥	NOUN
ejpam-4914	194	19	f(v	f(v	NOUN
ejpam-4914	194	20	)	)	PUNCT
ejpam-4914	195	1	+	+	NUM
ejpam-4914	195	2	f(w	f(w	PROPN
ejpam-4914	195	3	)	)	PUNCT
ejpam-4914	195	4	≥	≥	NOUN
ejpam-4914	195	5	3	3	NUM
ejpam-4914	195	6	.	.	PUNCT
ejpam-4914	196	1	if	if	SCONJ
ejpam-4914	196	2	w	w	PROPN
ejpam-4914	196	3	∈	∈	PROPN
ejpam-4914	196	4	v0	v0	NOUN
ejpam-4914	196	5	and	and	CCONJ
ejpam-4914	196	6	a	a	DET
ejpam-4914	196	7	∈	∈	PROPN
ejpam-4914	196	8	v2	v2	PROPN
ejpam-4914	196	9	\	\	NOUN
ejpam-4914	196	10	{	{	PUNCT
ejpam-4914	196	11	v	v	NOUN
ejpam-4914	196	12	}	}	PUNCT
ejpam-4914	196	13	for	for	ADP
ejpam-4914	196	14	which	which	PRON
ejpam-4914	196	15	dg(a	dg(a	X
ejpam-4914	196	16	,	,	PUNCT
ejpam-4914	196	17	w	w	NOUN
ejpam-4914	196	18	)	)	PUNCT
ejpam-4914	196	19	=	=	SYM
ejpam-4914	196	20	2	2	NUM
ejpam-4914	196	21	,	,	PUNCT
ejpam-4914	196	22	then	then	ADV
ejpam-4914	196	23	wg(f	wg(f	PUNCT
ejpam-4914	196	24	)	)	PUNCT
ejpam-4914	196	25	≥	≥	NOUN
ejpam-4914	196	26	f(v	f(v	NOUN
ejpam-4914	196	27	)	)	PUNCT
ejpam-4914	196	28	+	+	NUM
ejpam-4914	196	29	f(a	f(a	NOUN
ejpam-4914	196	30	)	)	PUNCT
ejpam-4914	196	31	=	=	SYM
ejpam-4914	197	1	3	3	X
ejpam-4914	197	2	.	.	PUNCT
ejpam-4914	197	3	either	either	DET
ejpam-4914	197	4	case	case	NOUN
ejpam-4914	197	5	is	be	AUX
ejpam-4914	197	6	a	a	DET
ejpam-4914	197	7	contradiction	contradiction	NOUN
ejpam-4914	197	8	.	.	PUNCT
ejpam-4914	198	1	thus	thus	ADV
ejpam-4914	198	2	,	,	PUNCT
ejpam-4914	198	3	v2	v2	NOUN
ejpam-4914	198	4	=	=	SYM
ejpam-4914	198	5	∅	∅	NOUN
ejpam-4914	198	6	and	and	CCONJ
ejpam-4914	198	7	|v1|	|v1|	NOUN
ejpam-4914	198	8	=	=	SYM
ejpam-4914	198	9	2	2	X
ejpam-4914	198	10	.	.	PUNCT
ejpam-4914	198	11	it	it	PRON
ejpam-4914	198	12	follows	follow	VERB
ejpam-4914	198	13	that	that	DET
ejpam-4914	198	14	v0	v0	NOUN
ejpam-4914	198	15	=	=	SYM
ejpam-4914	198	16	∅	∅	NOUN
ejpam-4914	198	17	and	and	CCONJ
ejpam-4914	198	18	|v	|v	PROPN
ejpam-4914	198	19	(	(	PUNCT
ejpam-4914	198	20	g)|	g)|	NOUN
ejpam-4914	198	21	=	=	PUNCT
ejpam-4914	198	22	|v1|	|v1|	NOUN
ejpam-4914	198	23	=	=	SYM
ejpam-4914	198	24	2	2	X
ejpam-4914	198	25	.	.	PUNCT
ejpam-4914	199	1	therefore	therefore	ADV
ejpam-4914	199	2	,	,	PUNCT
ejpam-4914	199	3	g	g	PROPN
ejpam-4914	199	4	=	=	SYM
ejpam-4914	199	5	k2	k2	PROPN
ejpam-4914	199	6	.	.	PUNCT
ejpam-4914	200	1	for	for	ADP
ejpam-4914	200	2	(	(	PUNCT
ejpam-4914	200	3	iii	iii	NOUN
ejpam-4914	200	4	):	):	PUNCT
ejpam-4914	200	5	if	if	SCONJ
ejpam-4914	200	6	g	g	PROPN
ejpam-4914	200	7	=	=	PROPN
ejpam-4914	200	8	k1	k1	PROPN
ejpam-4914	200	9	+	+	CCONJ
ejpam-4914	200	10	(	(	PUNCT
ejpam-4914	200	11	k1	k1	PROPN
ejpam-4914	200	12	∪h	∪h	NUM
ejpam-4914	200	13	)	)	PUNCT
ejpam-4914	200	14	for	for	ADP
ejpam-4914	200	15	some	some	DET
ejpam-4914	200	16	graph	graph	NOUN
ejpam-4914	200	17	h	h	NOUN
ejpam-4914	200	18	of	of	ADP
ejpam-4914	200	19	order	order	NOUN
ejpam-4914	200	20	≥	≥	NOUN
ejpam-4914	200	21	3	3	NUM
ejpam-4914	200	22	,	,	PUNCT
ejpam-4914	200	23	then	then	ADV
ejpam-4914	200	24	clearly	clearly	ADV
ejpam-4914	200	25	γhi(g	γhi(g	X
ejpam-4914	200	26	)	)	PUNCT
ejpam-4914	200	27	=	=	SYM
ejpam-4914	201	1	3	3	X
ejpam-4914	201	2	.	.	PUNCT
ejpam-4914	201	3	suppose	suppose	VERB
ejpam-4914	201	4	that	that	SCONJ
ejpam-4914	201	5	γ2h(g	γ2h(g	NOUN
ejpam-4914	201	6	)	)	PUNCT
ejpam-4914	201	7	=	=	SYM
ejpam-4914	202	1	3	3	X
ejpam-4914	202	2	.	.	PUNCT
ejpam-4914	202	3	then	then	ADV
ejpam-4914	202	4	g	g	PROPN
ejpam-4914	202	5	/∈	/∈	PUNCT
ejpam-4914	202	6	{	{	PUNCT
ejpam-4914	202	7	k1,k2	k1,k2	PROPN
ejpam-4914	202	8	}	}	PUNCT
ejpam-4914	202	9	.	.	PUNCT
ejpam-4914	203	1	by	by	ADP
ejpam-4914	203	2	(	(	PUNCT
ejpam-4914	203	3	ii	ii	NOUN
ejpam-4914	203	4	)	)	PUNCT
ejpam-4914	203	5	and	and	CCONJ
ejpam-4914	203	6	equation	equation	NOUN
ejpam-4914	203	7	1	1	NUM
ejpam-4914	203	8	,	,	PUNCT
ejpam-4914	203	9	γhi(g	γhi(g	PROPN
ejpam-4914	203	10	)	)	PUNCT
ejpam-4914	203	11	=	=	SYM
ejpam-4914	204	1	3	3	X
ejpam-4914	204	2	.	.	PUNCT
ejpam-4914	204	3	conversely	conversely	ADV
ejpam-4914	204	4	,	,	PUNCT
ejpam-4914	204	5	suppose	suppose	VERB
ejpam-4914	204	6	that	that	SCONJ
ejpam-4914	204	7	γhi(g	γhi(g	PROPN
ejpam-4914	204	8	)	)	PUNCT
ejpam-4914	204	9	=	=	SYM
ejpam-4914	205	1	3	3	NUM
ejpam-4914	205	2	,	,	PUNCT
ejpam-4914	205	3	and	and	CCONJ
ejpam-4914	205	4	let	let	VERB
ejpam-4914	205	5	f	f	PROPN
ejpam-4914	205	6	=	=	SYM
ejpam-4914	205	7	(	(	PUNCT
ejpam-4914	205	8	v0	v0	PROPN
ejpam-4914	205	9	,	,	PUNCT
ejpam-4914	205	10	v1	v1	NOUN
ejpam-4914	205	11	,	,	PUNCT
ejpam-4914	205	12	v2	v2	PROPN
ejpam-4914	205	13	)	)	PUNCT
ejpam-4914	205	14	be	be	AUX
ejpam-4914	205	15	a	a	DET
ejpam-4914	205	16	γhi	γhi	ADJ
ejpam-4914	205	17	-function	-function	NOUN
ejpam-4914	205	18	of	of	ADP
ejpam-4914	205	19	g	g	NOUN
ejpam-4914	205	20	such	such	ADJ
ejpam-4914	205	21	that	that	SCONJ
ejpam-4914	205	22	either	either	CCONJ
ejpam-4914	205	23	v2	v2	NOUN
ejpam-4914	205	24	=	=	SYM
ejpam-4914	205	25	∅	∅	NOUN
ejpam-4914	205	26	or	or	CCONJ
ejpam-4914	205	27	v2	v2	VERB
ejpam-4914	205	28	̸=	̸=	PROPN
ejpam-4914	205	29	∅	∅	NOUN
ejpam-4914	205	30	and	and	CCONJ
ejpam-4914	205	31	each	each	DET
ejpam-4914	205	32	v	v	NOUN
ejpam-4914	205	33	∈	∈	PROPN
ejpam-4914	205	34	v2	v2	NOUN
ejpam-4914	205	35	has	have	VERB
ejpam-4914	205	36	at	at	ADV
ejpam-4914	205	37	least	least	ADV
ejpam-4914	205	38	3	3	NUM
ejpam-4914	205	39	private	private	ADJ
ejpam-4914	205	40	hop	hop	NOUN
ejpam-4914	205	41	-	-	PUNCT
ejpam-4914	205	42	neighbors	neighbor	NOUN
ejpam-4914	205	43	in	in	ADP
ejpam-4914	205	44	v0	v0	PROPN
ejpam-4914	205	45	.	.	PUNCT
ejpam-4914	206	1	if	if	SCONJ
ejpam-4914	206	2	v2	v2	NOUN
ejpam-4914	206	3	=	=	NOUN
ejpam-4914	206	4	∅	∅	NOUN
ejpam-4914	206	5	,	,	PUNCT
ejpam-4914	206	6	then	then	ADV
ejpam-4914	206	7	by	by	ADP
ejpam-4914	206	8	γ2h(g	γ2h(g	NOUN
ejpam-4914	206	9	)	)	PUNCT
ejpam-4914	206	10	=	=	SYM
ejpam-4914	206	11	γhi(g	γhi(g	PROPN
ejpam-4914	206	12	)	)	PUNCT
ejpam-4914	206	13	=	=	SYM
ejpam-4914	206	14	3	3	NUM
ejpam-4914	206	15	by	by	ADP
ejpam-4914	206	16	observation	observation	NOUN
ejpam-4914	206	17	2.1(ii	2.1(ii	NUM
ejpam-4914	206	18	)	)	PUNCT
ejpam-4914	206	19	.	.	PUNCT
ejpam-4914	207	1	suppose	suppose	VERB
ejpam-4914	207	2	that	that	SCONJ
ejpam-4914	207	3	v2	v2	PROPN
ejpam-4914	207	4	̸=	̸=	PROPN
ejpam-4914	207	5	∅.	∅.	PRON
ejpam-4914	207	6	then	then	ADV
ejpam-4914	207	7	|v2|	|v2|	ADV
ejpam-4914	207	8	=	=	SYM
ejpam-4914	207	9	1	1	NUM
ejpam-4914	207	10	=	=	NOUN
ejpam-4914	207	11	|v1|	|v1|	NOUN
ejpam-4914	207	12	,	,	PUNCT
ejpam-4914	207	13	say	say	VERB
ejpam-4914	207	14	v2	v2	NOUN
ejpam-4914	207	15	=	=	SYM
ejpam-4914	207	16	{	{	PUNCT
ejpam-4914	207	17	v	v	NOUN
ejpam-4914	207	18	}	}	PUNCT
ejpam-4914	207	19	and	and	CCONJ
ejpam-4914	207	20	v1	v1	VERB
ejpam-4914	207	21	=	=	SYM
ejpam-4914	207	22	{	{	PUNCT
ejpam-4914	207	23	u	u	NOUN
ejpam-4914	207	24	}	}	PUNCT
ejpam-4914	207	25	.	.	PUNCT
ejpam-4914	208	1	by	by	ADP
ejpam-4914	208	2	proposition	proposition	NOUN
ejpam-4914	208	3	2.3	2.3	NUM
ejpam-4914	208	4	,	,	PUNCT
ejpam-4914	208	5	v	v	NOUN
ejpam-4914	208	6	has	have	VERB
ejpam-4914	208	7	at	at	ADV
ejpam-4914	208	8	least	least	ADV
ejpam-4914	208	9	3	3	NUM
ejpam-4914	208	10	italian	italian	ADJ
ejpam-4914	208	11	private	private	ADJ
ejpam-4914	208	12	hop	hop	NOUN
ejpam-4914	208	13	-	-	PUNCT
ejpam-4914	208	14	neighbors	neighbor	NOUN
ejpam-4914	208	15	in	in	ADP
ejpam-4914	208	16	v0	v0	PROPN
ejpam-4914	208	17	.	.	PUNCT
ejpam-4914	209	1	thus	thus	ADV
ejpam-4914	209	2	,	,	PUNCT
ejpam-4914	209	3	g	g	PROPN
ejpam-4914	209	4	=	=	PUNCT
ejpam-4914	209	5	⟨{u}⟩	⟨{u}⟩	NOUN
ejpam-4914	209	6	+	+	CCONJ
ejpam-4914	209	7	(	(	PUNCT
ejpam-4914	209	8	⟨{v}⟩	⟨{v}⟩	ADJ
ejpam-4914	209	9	∪h	∪h	NUM
ejpam-4914	209	10	)	)	PUNCT
ejpam-4914	209	11	=	=	SYM
ejpam-4914	209	12	k1	k1	NOUN
ejpam-4914	209	13	+	+	CCONJ
ejpam-4914	209	14	(	(	PUNCT
ejpam-4914	209	15	k1	k1	PROPN
ejpam-4914	209	16	∪h	∪h	NUM
ejpam-4914	209	17	)	)	PUNCT
ejpam-4914	209	18	,	,	PUNCT
ejpam-4914	209	19	where	where	SCONJ
ejpam-4914	209	20	h	h	NOUN
ejpam-4914	209	21	=	=	NOUN
ejpam-4914	209	22	⟨v0⟩	⟨v0⟩	NOUN
ejpam-4914	209	23	of	of	ADP
ejpam-4914	209	24	order	order	NOUN
ejpam-4914	209	25	≥	≥	NOUN
ejpam-4914	209	26	3	3	NUM
ejpam-4914	209	27	.	.	PUNCT
ejpam-4914	210	1	■	■	PUNCT
ejpam-4914	210	2	it	it	PRON
ejpam-4914	210	3	is	be	AUX
ejpam-4914	210	4	worth	worth	ADJ
ejpam-4914	210	5	noting	note	VERB
ejpam-4914	210	6	that	that	SCONJ
ejpam-4914	210	7	the	the	DET
ejpam-4914	210	8	family	family	NOUN
ejpam-4914	210	9	of	of	ADP
ejpam-4914	210	10	graphs	graph	NOUN
ejpam-4914	210	11	g	g	PROPN
ejpam-4914	210	12	for	for	ADP
ejpam-4914	210	13	which	which	PRON
ejpam-4914	210	14	γ2h(g	γ2h(g	VERB
ejpam-4914	210	15	)	)	PUNCT
ejpam-4914	210	16	=	=	SYM
ejpam-4914	210	17	γhi(g	γhi(g	PROPN
ejpam-4914	210	18	)	)	PUNCT
ejpam-4914	210	19	=	=	SYM
ejpam-4914	210	20	3	3	NUM
ejpam-4914	210	21	includes	include	VERB
ejpam-4914	210	22	p3	p3	PROPN
ejpam-4914	210	23	,	,	PUNCT
ejpam-4914	210	24	k3	k3	PROPN
ejpam-4914	210	25	,	,	PUNCT
ejpam-4914	210	26	c5	c5	PROPN
ejpam-4914	210	27	,	,	PUNCT
ejpam-4914	210	28	k1	k1	NOUN
ejpam-4914	210	29	-	-	PUNCT
ejpam-4914	210	30	gluing	gluing	NOUN
ejpam-4914	210	31	of	of	ADP
ejpam-4914	210	32	c5	c5	PROPN
ejpam-4914	210	33	and	and	CCONJ
ejpam-4914	210	34	k2	k2	ADJ
ejpam-4914	210	35	,	,	PUNCT
ejpam-4914	210	36	k2	k2	NOUN
ejpam-4914	210	37	-	-	PUNCT
ejpam-4914	210	38	gluing	gluing	NOUN
ejpam-4914	210	39	of	of	ADP
ejpam-4914	210	40	c5	c5	PROPN
ejpam-4914	210	41	and	and	CCONJ
ejpam-4914	210	42	k2	k2	PROPN
ejpam-4914	210	43	;	;	PUNCT
ejpam-4914	210	44	graph	graph	NOUN
ejpam-4914	210	45	g	g	NOUN
ejpam-4914	210	46	containing	contain	VERB
ejpam-4914	210	47	h	h	NOUN
ejpam-4914	210	48	=	=	PRON
ejpam-4914	210	49	k3	k3	VERB
ejpam-4914	210	50	such	such	ADJ
ejpam-4914	210	51	that	that	SCONJ
ejpam-4914	210	52	each	each	DET
ejpam-4914	210	53	v	v	NUM
ejpam-4914	210	54	∈	∈	NOUN
ejpam-4914	210	55	v	v	NOUN
ejpam-4914	210	56	(	(	PUNCT
ejpam-4914	210	57	g)\v	g)\v	NOUN
ejpam-4914	210	58	(	(	PUNCT
ejpam-4914	210	59	h	h	NOUN
ejpam-4914	210	60	)	)	PUNCT
ejpam-4914	210	61	is	be	AUX
ejpam-4914	210	62	adjacent	adjacent	ADJ
ejpam-4914	210	63	to	to	ADP
ejpam-4914	210	64	(	(	PUNCT
ejpam-4914	210	65	exactly	exactly	ADV
ejpam-4914	210	66	)	)	PUNCT
ejpam-4914	210	67	one	one	NUM
ejpam-4914	210	68	vertex	vertex	NOUN
ejpam-4914	210	69	of	of	ADP
ejpam-4914	210	70	h	h	NOUN
ejpam-4914	210	71	(	(	PUNCT
ejpam-4914	210	72	may	may	AUX
ejpam-4914	210	73	be	be	AUX
ejpam-4914	210	74	viewed	view	VERB
ejpam-4914	210	75	as	as	ADP
ejpam-4914	210	76	one	one	NUM
ejpam-4914	210	77	generate	generate	NOUN
ejpam-4914	210	78	by	by	ADP
ejpam-4914	210	79	a	a	DET
ejpam-4914	210	80	triangle	triangle	NOUN
ejpam-4914	210	81	k3	k3	VERB
ejpam-4914	210	82	;	;	PUNCT
ejpam-4914	210	83	the	the	DET
ejpam-4914	210	84	graph	graph	NOUN
ejpam-4914	210	85	g1	g1	NOUN
ejpam-4914	210	86	in	in	ADP
ejpam-4914	210	87	figure	figure	NOUN
ejpam-4914	210	88	2	2	NUM
ejpam-4914	210	89	(	(	PUNCT
ejpam-4914	210	90	may	may	AUX
ejpam-4914	210	91	be	be	AUX
ejpam-4914	210	92	viewed	view	VERB
ejpam-4914	210	93	as	as	ADP
ejpam-4914	210	94	one	one	NUM
ejpam-4914	210	95	generated	generate	VERB
ejpam-4914	210	96	by	by	ADP
ejpam-4914	210	97	mutually	mutually	ADV
ejpam-4914	210	98	nonadjacent	nonadjacent	ADJ
ejpam-4914	210	99	x	x	NOUN
ejpam-4914	210	100	,	,	PUNCT
ejpam-4914	210	101	y	y	PROPN
ejpam-4914	210	102	and	and	CCONJ
ejpam-4914	210	103	z	z	PROPN
ejpam-4914	210	104	)	)	PUNCT
ejpam-4914	210	105	;	;	PUNCT
ejpam-4914	210	106	and	and	CCONJ
ejpam-4914	210	107	the	the	DET
ejpam-4914	210	108	graph	graph	NOUN
ejpam-4914	210	109	g2	g2	PROPN
ejpam-4914	210	110	in	in	ADP
ejpam-4914	210	111	figure	figure	NOUN
ejpam-4914	210	112	2	2	NUM
ejpam-4914	210	113	(	(	PUNCT
ejpam-4914	210	114	may	may	AUX
ejpam-4914	210	115	be	be	AUX
ejpam-4914	210	116	viewed	view	VERB
ejpam-4914	210	117	as	as	ADP
ejpam-4914	210	118	one	one	NUM
ejpam-4914	210	119	generated	generate	VERB
ejpam-4914	210	120	by	by	ADP
ejpam-4914	210	121	path	path	NOUN
ejpam-4914	211	1	[	[	X
ejpam-4914	211	2	x	x	X
ejpam-4914	211	3	,	,	PUNCT
ejpam-4914	211	4	y	y	PROPN
ejpam-4914	211	5	,	,	PUNCT
ejpam-4914	211	6	z	z	NOUN
ejpam-4914	211	7	]	]	X
ejpam-4914	211	8	)	)	PUNCT
ejpam-4914	211	9	.	.	PUNCT
ejpam-4914	212	1	•	•	NUM
ejpam-4914	212	2	•	•	NUM
ejpam-4914	212	3	•	•	NUM
ejpam-4914	212	4	•	•	NUM
ejpam-4914	212	5	•	•	NUM
ejpam-4914	212	6	•	•	NUM
ejpam-4914	212	7	•	•	NUM
ejpam-4914	212	8	•	•	NOUN
ejpam-4914	212	9	•	•	NOUN
ejpam-4914	212	10	............................................................................................................................	............................................................................................................................	PUNCT
ejpam-4914	212	11	....................................	....................................	PUNCT
ejpam-4914	212	12	...............	...............	PUNCT
ejpam-4914	212	13	..............	..............	PUNCT
ejpam-4914	212	14	..............	..............	PUNCT
ejpam-4914	212	15	..............	..............	PUNCT
ejpam-4914	212	16	..............	..............	PUNCT
ejpam-4914	212	17	..............	..............	PUNCT
ejpam-4914	212	18	..............	..............	PUNCT
ejpam-4914	212	19	..............	..............	PUNCT
ejpam-4914	212	20	...........	...........	PUNCT
ejpam-4914	212	21	....................................	....................................	PUNCT
ejpam-4914	212	22	.........	.........	PUNCT
ejpam-4914	212	23	........	........	PUNCT
ejpam-4914	212	24	........	........	PUNCT
ejpam-4914	212	25	........	........	PUNCT
ejpam-4914	212	26	........	........	PUNCT
ejpam-4914	212	27	........	........	PUNCT
ejpam-4914	212	28	........	........	PUNCT
ejpam-4914	212	29	.......	.......	PUNCT
ejpam-4914	212	30	....................................	....................................	PUNCT
ejpam-4914	212	31	............................................................................................................................	............................................................................................................................	PUNCT
ejpam-4914	212	32	....................................	....................................	PUNCT
ejpam-4914	212	33	...............	...............	PUNCT
ejpam-4914	212	34	..............	..............	PUNCT
ejpam-4914	212	35	..............	..............	PUNCT
ejpam-4914	212	36	..............	..............	PUNCT
ejpam-4914	212	37	..............	..............	PUNCT
ejpam-4914	212	38	..............	..............	PUNCT
ejpam-4914	212	39	..............	..............	PUNCT
ejpam-4914	212	40	..............	..............	PUNCT
ejpam-4914	212	41	...........	...........	PUNCT
ejpam-4914	212	42	....................................	....................................	PUNCT
ejpam-4914	212	43	.........	.........	PUNCT
ejpam-4914	212	44	........	........	PUNCT
ejpam-4914	212	45	........	........	PUNCT
ejpam-4914	212	46	........	........	PUNCT
ejpam-4914	212	47	........	........	PUNCT
ejpam-4914	212	48	........	........	PUNCT
ejpam-4914	212	49	........	........	PUNCT
ejpam-4914	212	50	.......	.......	PUNCT
ejpam-4914	212	51	....................................	....................................	PUNCT
ejpam-4914	213	1	....................................	....................................	PUNCT
ejpam-4914	213	2	............................................................................................................	............................................................................................................	PUNCT
ejpam-4914	214	1	....................................	....................................	PUNCT
ejpam-4914	214	2	..........	..........	PUNCT
ejpam-4914	215	1	.........	.........	PUNCT
ejpam-4914	215	2	.........	.........	PUNCT
ejpam-4914	216	1	.........	.........	PUNCT
ejpam-4914	216	2	.........	.........	PUNCT
ejpam-4914	217	1	.........	.........	PUNCT
ejpam-4914	217	2	.........	.........	PUNCT
ejpam-4914	218	1	.........	.........	PUNCT
ejpam-4914	218	2	.........	.........	PUNCT
ejpam-4914	219	1	.........	.........	PUNCT
ejpam-4914	219	2	.........	.........	PUNCT
ejpam-4914	219	3	........	........	PUNCT
ejpam-4914	220	1	....................................	....................................	PUNCT
ejpam-4914	220	2	....................................	....................................	PUNCT
ejpam-4914	220	3	...........................................................................................................................................	...........................................................................................................................................	PUNCT
ejpam-4914	221	1	....................................	....................................	PUNCT
ejpam-4914	221	2	............	............	PUNCT
ejpam-4914	221	3	...........	...........	PUNCT
ejpam-4914	221	4	...........	...........	PUNCT
ejpam-4914	221	5	...........	...........	PUNCT
ejpam-4914	221	6	...........	...........	PUNCT
ejpam-4914	221	7	...........	...........	PUNCT
ejpam-4914	221	8	...........	...........	PUNCT
ejpam-4914	221	9	...........	...........	PUNCT
ejpam-4914	221	10	......	......	PUNCT
ejpam-4914	221	11	....................................	....................................	PUNCT
ejpam-4914	222	1	....................................	....................................	PUNCT
ejpam-4914	222	2	...........................................................................................................................................	...........................................................................................................................................	PUNCT
ejpam-4914	223	1	....................................	....................................	PUNCT
ejpam-4914	223	2	...............................................................................................	...............................................................................................	PUNCT
ejpam-4914	224	1	....................................	....................................	PUNCT
ejpam-4914	224	2	....................................	....................................	PUNCT
ejpam-4914	225	1	x	x	PUNCT
ejpam-4914	225	2	y	y	PROPN
ejpam-4914	225	3	z	z	PROPN
ejpam-4914	225	4	g1	g1	PROPN
ejpam-4914	225	5	•	•	ADV
ejpam-4914	225	6	•	•	NOUN
ejpam-4914	225	7	•	•	NOUN
ejpam-4914	225	8	•	•	NUM
ejpam-4914	225	9	•	•	NUM
ejpam-4914	225	10	•	•	NOUN
ejpam-4914	225	11	•	•	NOUN
ejpam-4914	225	12	............................................................................................................................	............................................................................................................................	PUNCT
ejpam-4914	225	13	....................................	....................................	PUNCT
ejpam-4914	225	14	...............	...............	PUNCT
ejpam-4914	225	15	..............	..............	PUNCT
ejpam-4914	225	16	..............	..............	PUNCT
ejpam-4914	225	17	..............	..............	PUNCT
ejpam-4914	225	18	..............	..............	PUNCT
ejpam-4914	225	19	..............	..............	PUNCT
ejpam-4914	225	20	..............	..............	PUNCT
ejpam-4914	225	21	..............	..............	PUNCT
ejpam-4914	225	22	...........	...........	PUNCT
ejpam-4914	225	23	....................................	....................................	PUNCT
ejpam-4914	225	24	.........	.........	PUNCT
ejpam-4914	225	25	........	........	PUNCT
ejpam-4914	225	26	........	........	PUNCT
ejpam-4914	225	27	........	........	PUNCT
ejpam-4914	225	28	........	........	PUNCT
ejpam-4914	225	29	........	........	PUNCT
ejpam-4914	225	30	........	........	PUNCT
ejpam-4914	225	31	.......	.......	PUNCT
ejpam-4914	225	32	....................................	....................................	PUNCT
ejpam-4914	225	33	............................................................................................................................	............................................................................................................................	PUNCT
ejpam-4914	225	34	....................................	....................................	PUNCT
ejpam-4914	225	35	...............	...............	PUNCT
ejpam-4914	225	36	..............	..............	PUNCT
ejpam-4914	225	37	..............	..............	PUNCT
ejpam-4914	225	38	..............	..............	PUNCT
ejpam-4914	225	39	..............	..............	PUNCT
ejpam-4914	225	40	..............	..............	PUNCT
ejpam-4914	225	41	..............	..............	PUNCT
ejpam-4914	225	42	..............	..............	PUNCT
ejpam-4914	225	43	...........	...........	PUNCT
ejpam-4914	225	44	....................................	....................................	PUNCT
ejpam-4914	225	45	.........	.........	PUNCT
ejpam-4914	225	46	........	........	PUNCT
ejpam-4914	225	47	........	........	PUNCT
ejpam-4914	225	48	........	........	PUNCT
ejpam-4914	225	49	........	........	PUNCT
ejpam-4914	225	50	........	........	PUNCT
ejpam-4914	225	51	........	........	PUNCT
ejpam-4914	225	52	.......	.......	PUNCT
ejpam-4914	225	53	....................................	....................................	PUNCT
ejpam-4914	225	54	......................................	......................................	PUNCT
ejpam-4914	225	55	.........................	.........................	PUNCT
ejpam-4914	225	56	.........................	.........................	PUNCT
ejpam-4914	225	57	.........................	.........................	PUNCT
ejpam-4914	225	58	.........................	.........................	PUNCT
ejpam-4914	225	59	......	......	PUNCT
ejpam-4914	225	60	....................................	....................................	PUNCT
ejpam-4914	226	1	..........................	..........................	PUNCT
ejpam-4914	226	2	.........................	.........................	PUNCT
ejpam-4914	226	3	.........................	.........................	PUNCT
ejpam-4914	226	4	.........................	.........................	PUNCT
ejpam-4914	226	5	.......	.......	PUNCT
ejpam-4914	226	6	....................................	....................................	PUNCT
ejpam-4914	227	1	....................................	....................................	PUNCT
ejpam-4914	228	1	x	x	PUNCT
ejpam-4914	228	2	y	y	PROPN
ejpam-4914	228	3	z	z	PROPN
ejpam-4914	228	4	g2	g2	PROPN
ejpam-4914	228	5	..........................................................................................................................................	..........................................................................................................................................	PUNCT
ejpam-4914	229	1	....................................	....................................	PUNCT
ejpam-4914	229	2	.........	.........	PUNCT
ejpam-4914	229	3	........	........	PUNCT
ejpam-4914	229	4	........	........	PUNCT
ejpam-4914	229	5	........	........	PUNCT
ejpam-4914	229	6	........	........	PUNCT
ejpam-4914	229	7	........	........	PUNCT
ejpam-4914	229	8	........	........	PUNCT
ejpam-4914	229	9	.......	.......	PUNCT
ejpam-4914	229	10	....................................	....................................	PUNCT
ejpam-4914	229	11	.........	.........	PUNCT
ejpam-4914	229	12	........	........	PUNCT
ejpam-4914	229	13	........	........	PUNCT
ejpam-4914	229	14	........	........	PUNCT
ejpam-4914	229	15	........	........	PUNCT
ejpam-4914	229	16	........	........	PUNCT
ejpam-4914	229	17	........	........	PUNCT
ejpam-4914	229	18	.......	.......	PUNCT
ejpam-4914	229	19	....................................	....................................	PUNCT
ejpam-4914	229	20	...............	...............	PUNCT
ejpam-4914	229	21	..............	..............	PUNCT
ejpam-4914	229	22	..............	..............	PUNCT
ejpam-4914	230	1	..............	..............	PUNCT
ejpam-4914	230	2	..............	..............	PUNCT
ejpam-4914	231	1	..............	..............	PUNCT
ejpam-4914	231	2	..............	..............	PUNCT
ejpam-4914	232	1	..............	..............	PUNCT
ejpam-4914	232	2	...........	...........	PUNCT
ejpam-4914	232	3	....................................	....................................	PUNCT
ejpam-4914	233	1	..........................................................................................................................................	..........................................................................................................................................	PUNCT
ejpam-4914	234	1	....................................	....................................	PUNCT
ejpam-4914	235	1	•	•	NUM
ejpam-4914	235	2	•	•	NUM
ejpam-4914	235	3	•	•	NUM
ejpam-4914	235	4	•	•	NOUN
ejpam-4914	235	5	•	•	NOUN
ejpam-4914	235	6	..................................................................................................................................................................................................................................	..................................................................................................................................................................................................................................	PUNCT
ejpam-4914	235	7	...............	...............	PUNCT
ejpam-4914	235	8	..............	..............	PUNCT
ejpam-4914	235	9	..............	..............	PUNCT
ejpam-4914	235	10	..............	..............	PUNCT
ejpam-4914	235	11	..............	..............	PUNCT
ejpam-4914	235	12	..............	..............	PUNCT
ejpam-4914	235	13	..............	..............	PUNCT
ejpam-4914	235	14	..............	..............	PUNCT
ejpam-4914	236	1	...........	...........	PUNCT
ejpam-4914	236	2	21	21	NUM
ejpam-4914	236	3	0	0	NUM
ejpam-4914	236	4	0	0	NUM
ejpam-4914	236	5	0	0	NUM
ejpam-4914	236	6	g3	g3	NOUN
ejpam-4914	236	7	figure	figure	NOUN
ejpam-4914	236	8	2	2	NUM
ejpam-4914	236	9	:	:	PUNCT
ejpam-4914	236	10	examples	example	NOUN
ejpam-4914	236	11	of	of	ADP
ejpam-4914	236	12	graphs	graph	NOUN
ejpam-4914	236	13	g	g	PROPN
ejpam-4914	236	14	described	describe	VERB
ejpam-4914	236	15	in	in	ADP
ejpam-4914	236	16	proposition	proposition	NOUN
ejpam-4914	236	17	2.4(iii	2.4(iii	NUM
ejpam-4914	236	18	)	)	PUNCT
ejpam-4914	236	19	with	with	ADP
ejpam-4914	236	20	γhi(g	γhi(g	PROPN
ejpam-4914	236	21	)	)	PUNCT
ejpam-4914	237	1	=	=	SYM
ejpam-4914	237	2	3	3	NUM
ejpam-4914	237	3	graph	graph	NOUN
ejpam-4914	237	4	g3	g3	NOUN
ejpam-4914	237	5	in	in	ADP
ejpam-4914	237	6	figure	figure	NOUN
ejpam-4914	237	7	2	2	NUM
ejpam-4914	237	8	shows	show	VERB
ejpam-4914	237	9	an	an	DET
ejpam-4914	237	10	example	example	NOUN
ejpam-4914	237	11	of	of	ADP
ejpam-4914	237	12	a	a	DET
ejpam-4914	237	13	graph	graph	NOUN
ejpam-4914	237	14	g	g	NOUN
ejpam-4914	237	15	with	with	ADP
ejpam-4914	237	16	γ2h(g	γ2h(g	NOUN
ejpam-4914	237	17	)	)	PUNCT
ejpam-4914	237	18	̸=	̸=	PROPN
ejpam-4914	237	19	3	3	NUM
ejpam-4914	237	20	=	=	SYM
ejpam-4914	237	21	γhi(g	γhi(g	PROPN
ejpam-4914	237	22	)	)	PUNCT
ejpam-4914	237	23	.	.	PUNCT
ejpam-4914	238	1	proposition	proposition	NOUN
ejpam-4914	238	2	2.5	2.5	NUM
ejpam-4914	238	3	.	.	PUNCT
ejpam-4914	239	1	(	(	PUNCT
ejpam-4914	239	2	i	i	NOUN
ejpam-4914	239	3	)	)	PUNCT
ejpam-4914	239	4	for	for	ADP
ejpam-4914	239	5	every	every	DET
ejpam-4914	239	6	nonnegative	nonnegative	ADJ
ejpam-4914	239	7	integer	integer	NOUN
ejpam-4914	239	8	k	k	NOUN
ejpam-4914	239	9	,	,	PUNCT
ejpam-4914	239	10	there	there	PRON
ejpam-4914	239	11	exists	exist	VERB
ejpam-4914	239	12	a	a	DET
ejpam-4914	239	13	connected	connected	ADJ
ejpam-4914	239	14	graph	graph	NOUN
ejpam-4914	239	15	g	g	NOUN
ejpam-4914	239	16	for	for	ADP
ejpam-4914	239	17	which	which	PRON
ejpam-4914	239	18	γhr(g	γhr(g	NUM
ejpam-4914	239	19	)	)	PUNCT
ejpam-4914	239	20	=	=	SYM
ejpam-4914	239	21	γhi(g	γhi(g	PROPN
ejpam-4914	239	22	)	)	PUNCT
ejpam-4914	240	1	+	+	CCONJ
ejpam-4914	240	2	k.	k.	PROPN
ejpam-4914	240	3	(	(	PUNCT
ejpam-4914	240	4	ii	ii	PROPN
ejpam-4914	240	5	)	)	PUNCT
ejpam-4914	240	6	for	for	ADP
ejpam-4914	240	7	every	every	DET
ejpam-4914	240	8	pair	pair	NOUN
ejpam-4914	240	9	of	of	ADP
ejpam-4914	240	10	positive	positive	ADJ
ejpam-4914	240	11	integers	integer	NOUN
ejpam-4914	240	12	a	a	PRON
ejpam-4914	240	13	and	and	CCONJ
ejpam-4914	240	14	b	b	NOUN
ejpam-4914	240	15	with	with	ADP
ejpam-4914	240	16	4	4	NUM
ejpam-4914	240	17	≤	≤	NOUN
ejpam-4914	240	18	a	a	DET
ejpam-4914	240	19	≤	≤	NUM
ejpam-4914	240	20	b	b	NOUN
ejpam-4914	240	21	,	,	PUNCT
ejpam-4914	240	22	there	there	PRON
ejpam-4914	240	23	exists	exist	VERB
ejpam-4914	240	24	a	a	DET
ejpam-4914	240	25	connected	connected	ADJ
ejpam-4914	240	26	graph	graph	NOUN
ejpam-4914	240	27	g	g	NOUN
ejpam-4914	240	28	for	for	ADP
ejpam-4914	240	29	which	which	PRON
ejpam-4914	240	30	γhi(g	γhi(g	ADP
ejpam-4914	240	31	)	)	PUNCT
ejpam-4914	241	1	=	=	SYM
ejpam-4914	241	2	a	a	PRON
ejpam-4914	241	3	and	and	CCONJ
ejpam-4914	241	4	γ2h(g	γ2h(g	PROPN
ejpam-4914	241	5	)	)	PUNCT
ejpam-4914	242	1	=	=	SYM
ejpam-4914	242	2	b.	b.	PROPN
ejpam-4914	242	3	consequently	consequently	ADV
ejpam-4914	242	4	,	,	PUNCT
ejpam-4914	242	5	for	for	ADP
ejpam-4914	242	6	each	each	DET
ejpam-4914	242	7	nonnegative	nonnegative	ADJ
ejpam-4914	242	8	integer	integer	NOUN
ejpam-4914	242	9	k	k	NOUN
ejpam-4914	242	10	,	,	PUNCT
ejpam-4914	242	11	there	there	PRON
ejpam-4914	242	12	exists	exist	VERB
ejpam-4914	242	13	a	a	DET
ejpam-4914	242	14	connected	connected	ADJ
ejpam-4914	242	15	graph	graph	NOUN
ejpam-4914	242	16	g	g	NOUN
ejpam-4914	242	17	with	with	ADP
ejpam-4914	242	18	γ2h(g	γ2h(g	NOUN
ejpam-4914	242	19	)	)	PUNCT
ejpam-4914	242	20	=	=	SYM
ejpam-4914	242	21	γhi(g	γhi(g	PROPN
ejpam-4914	242	22	)	)	PUNCT
ejpam-4914	243	1	+	+	CCONJ
ejpam-4914	243	2	k.	k.	PROPN
ejpam-4914	243	3	s.r	s.r	PROPN
ejpam-4914	243	4	.	.	PROPN
ejpam-4914	243	5	jr	jr	PROPN
ejpam-4914	243	6	.	.	PROPN
ejpam-4914	243	7	canoy	canoy	PROPN
ejpam-4914	243	8	,	,	PUNCT
ejpam-4914	243	9	f.p	f.p	PROPN
ejpam-4914	243	10	.	.	PROPN
ejpam-4914	243	11	jamil	jamil	PROPN
ejpam-4914	243	12	and	and	CCONJ
ejpam-4914	243	13	s.m	s.m	PROPN
ejpam-4914	243	14	.	.	PROPN
ejpam-4914	243	15	menchavez	menchavez	PROPN
ejpam-4914	243	16	/	/	PUNCT
ejpam-4914	243	17	eur	eur	PROPN
ejpam-4914	243	18	.	.	PUNCT
ejpam-4914	244	1	j.	j.	PROPN
ejpam-4914	244	2	pure	pure	PROPN
ejpam-4914	244	3	appl	appl	PROPN
ejpam-4914	244	4	.	.	PROPN
ejpam-4914	244	5	math	math	PROPN
ejpam-4914	244	6	,	,	PUNCT
ejpam-4914	244	7	16	16	NUM
ejpam-4914	244	8	(	(	PUNCT
ejpam-4914	244	9	4	4	NUM
ejpam-4914	244	10	)	)	PUNCT
ejpam-4914	244	11	(	(	PUNCT
ejpam-4914	244	12	2023	2023	NUM
ejpam-4914	244	13	)	)	PUNCT
ejpam-4914	244	14	,	,	PUNCT
ejpam-4914	244	15	2431	2431	NUM
ejpam-4914	244	16	-	-	SYM
ejpam-4914	244	17	2449	2449	NUM
ejpam-4914	244	18	2438	2438	NUM
ejpam-4914	244	19	proof	proof	NOUN
ejpam-4914	244	20	:	:	PUNCT
ejpam-4914	244	21	for	for	ADP
ejpam-4914	244	22	(	(	PUNCT
ejpam-4914	244	23	i	i	NOUN
ejpam-4914	244	24	):	):	PUNCT
ejpam-4914	244	25	if	if	SCONJ
ejpam-4914	244	26	k	k	PROPN
ejpam-4914	244	27	=	=	SYM
ejpam-4914	244	28	0	0	PROPN
ejpam-4914	244	29	,	,	PUNCT
ejpam-4914	244	30	then	then	ADV
ejpam-4914	244	31	we	we	PRON
ejpam-4914	244	32	take	take	VERB
ejpam-4914	244	33	a	a	DET
ejpam-4914	244	34	complete	complete	ADJ
ejpam-4914	244	35	graph	graph	NOUN
ejpam-4914	244	36	g.	g.	PROPN
ejpam-4914	244	37	suppose	suppose	VERB
ejpam-4914	244	38	that	that	SCONJ
ejpam-4914	244	39	k	k	PROPN
ejpam-4914	244	40	≥	≥	NUM
ejpam-4914	244	41	1	1	NUM
ejpam-4914	244	42	.	.	PUNCT
ejpam-4914	245	1	first	first	ADV
ejpam-4914	245	2	,	,	PUNCT
ejpam-4914	245	3	suppose	suppose	VERB
ejpam-4914	245	4	that	that	SCONJ
ejpam-4914	245	5	k	k	PROPN
ejpam-4914	245	6	is	be	AUX
ejpam-4914	245	7	even	even	ADV
ejpam-4914	245	8	,	,	PUNCT
ejpam-4914	245	9	say	say	VERB
ejpam-4914	245	10	k	k	X
ejpam-4914	245	11	=	=	PUNCT
ejpam-4914	245	12	2j	2j	NUM
ejpam-4914	245	13	for	for	ADP
ejpam-4914	245	14	some	some	DET
ejpam-4914	245	15	integer	integer	PROPN
ejpam-4914	245	16	j	j	PROPN
ejpam-4914	245	17	≥	≥	NUM
ejpam-4914	245	18	1	1	NUM
ejpam-4914	245	19	.	.	PUNCT
ejpam-4914	246	1	choose	choose	VERB
ejpam-4914	246	2	g	g	NOUN
ejpam-4914	246	3	to	to	PART
ejpam-4914	246	4	be	be	AUX
ejpam-4914	246	5	the	the	DET
ejpam-4914	246	6	path	path	NOUN
ejpam-4914	246	7	pn	pn	NOUN
ejpam-4914	246	8	,	,	PUNCT
ejpam-4914	246	9	where	where	SCONJ
ejpam-4914	246	10	n	n	NOUN
ejpam-4914	246	11	=	=	NOUN
ejpam-4914	246	12	12j	12j	NOUN
ejpam-4914	246	13	+	+	X
ejpam-4914	246	14	3	3	X
ejpam-4914	246	15	.	.	X
ejpam-4914	246	16	writing	write	VERB
ejpam-4914	246	17	n	n	NOUN
ejpam-4914	246	18	=	=	SYM
ejpam-4914	246	19	6(2j	6(2j	NUM
ejpam-4914	246	20	)	)	PUNCT
ejpam-4914	246	21	+	+	SYM
ejpam-4914	246	22	3	3	NUM
ejpam-4914	246	23	,	,	PUNCT
ejpam-4914	246	24	observation	observation	NOUN
ejpam-4914	246	25	1.2	1.2	NUM
ejpam-4914	246	26	yields	yield	NOUN
ejpam-4914	246	27	γhr(g	γhr(g	NUM
ejpam-4914	246	28	)	)	PUNCT
ejpam-4914	246	29	=	=	SYM
ejpam-4914	246	30	γhr(pn	γhr(pn	NOUN
ejpam-4914	246	31	)	)	PUNCT
ejpam-4914	246	32	=	=	SYM
ejpam-4914	246	33	4(2j	4(2j	X
ejpam-4914	246	34	)	)	PUNCT
ejpam-4914	246	35	+	+	CCONJ
ejpam-4914	246	36	3	3	NUM
ejpam-4914	246	37	=	=	SYM
ejpam-4914	246	38	8j	8j	NUM
ejpam-4914	246	39	+	+	CCONJ
ejpam-4914	246	40	3	3	X
ejpam-4914	246	41	.	.	PUNCT
ejpam-4914	246	42	similarly	similarly	ADV
ejpam-4914	246	43	,	,	PUNCT
ejpam-4914	246	44	by	by	ADP
ejpam-4914	246	45	observation	observation	NOUN
ejpam-4914	246	46	2.2	2.2	NUM
ejpam-4914	246	47	,	,	PUNCT
ejpam-4914	246	48	γhi(g	γhi(g	PROPN
ejpam-4914	246	49	)	)	PUNCT
ejpam-4914	247	1	=	=	SYM
ejpam-4914	247	2	2(3j	2(3j	NUM
ejpam-4914	247	3	)	)	PUNCT
ejpam-4914	248	1	+	+	CCONJ
ejpam-4914	248	2	3	3	X
ejpam-4914	248	3	.	.	PUNCT
ejpam-4914	248	4	thus	thus	ADV
ejpam-4914	248	5	,	,	PUNCT
ejpam-4914	248	6	γhr(g	γhr(g	PROPN
ejpam-4914	248	7	)	)	PUNCT
ejpam-4914	248	8	=	=	NOUN
ejpam-4914	249	1	(	(	PUNCT
ejpam-4914	249	2	6j	6j	NOUN
ejpam-4914	249	3	+	+	CCONJ
ejpam-4914	249	4	3	3	X
ejpam-4914	249	5	)	)	PUNCT
ejpam-4914	249	6	+	+	NUM
ejpam-4914	249	7	2j	2j	NUM
ejpam-4914	249	8	=	=	SYM
ejpam-4914	249	9	γhi(g	γhi(g	PROPN
ejpam-4914	249	10	)	)	PUNCT
ejpam-4914	250	1	+	+	CCONJ
ejpam-4914	250	2	k.	k.	PROPN
ejpam-4914	250	3	next	next	ADV
ejpam-4914	250	4	,	,	PUNCT
ejpam-4914	250	5	suppose	suppose	VERB
ejpam-4914	250	6	that	that	SCONJ
ejpam-4914	250	7	k	k	PROPN
ejpam-4914	250	8	=	=	PUNCT
ejpam-4914	250	9	2j	2j	NOUN
ejpam-4914	250	10	+	+	CCONJ
ejpam-4914	250	11	1	1	NUM
ejpam-4914	250	12	for	for	ADP
ejpam-4914	250	13	some	some	DET
ejpam-4914	250	14	integer	integer	PROPN
ejpam-4914	250	15	j	j	PROPN
ejpam-4914	250	16	≥	≥	NUM
ejpam-4914	250	17	0	0	NUM
ejpam-4914	250	18	.	.	PUNCT
ejpam-4914	251	1	if	if	SCONJ
ejpam-4914	251	2	j	j	PROPN
ejpam-4914	251	3	=	=	SYM
ejpam-4914	251	4	0	0	PROPN
ejpam-4914	251	5	,	,	PUNCT
ejpam-4914	251	6	then	then	ADV
ejpam-4914	251	7	we	we	PRON
ejpam-4914	251	8	take	take	VERB
ejpam-4914	251	9	g	g	PROPN
ejpam-4914	251	10	=	=	PROPN
ejpam-4914	251	11	c7	c7	PROPN
ejpam-4914	251	12	.	.	PROPN
ejpam-4914	252	1	assume	assume	VERB
ejpam-4914	252	2	that	that	SCONJ
ejpam-4914	252	3	j	j	PROPN
ejpam-4914	252	4	≥	≥	NUM
ejpam-4914	252	5	1	1	NUM
ejpam-4914	252	6	.	.	PUNCT
ejpam-4914	252	7	consider	consider	VERB
ejpam-4914	252	8	the	the	DET
ejpam-4914	252	9	graph	graph	NOUN
ejpam-4914	252	10	g	g	PROPN
ejpam-4914	252	11	=	=	SYM
ejpam-4914	252	12	cn	cn	PROPN
ejpam-4914	252	13	,	,	PUNCT
ejpam-4914	252	14	a	a	DET
ejpam-4914	252	15	cycle	cycle	NOUN
ejpam-4914	252	16	on	on	ADP
ejpam-4914	252	17	n	n	DET
ejpam-4914	252	18	vertices	vertex	NOUN
ejpam-4914	252	19	,	,	PUNCT
ejpam-4914	252	20	where	where	SCONJ
ejpam-4914	252	21	n	n	PROPN
ejpam-4914	252	22	=	=	SYM
ejpam-4914	252	23	12j+3	12j+3	NUM
ejpam-4914	252	24	.	.	PUNCT
ejpam-4914	253	1	by	by	ADP
ejpam-4914	253	2	observation	observation	NOUN
ejpam-4914	253	3	1.2	1.2	NUM
ejpam-4914	253	4	and	and	CCONJ
ejpam-4914	253	5	observation	observation	NOUN
ejpam-4914	253	6	2.2	2.2	NUM
ejpam-4914	253	7	,	,	PUNCT
ejpam-4914	253	8	γhr(g	γhr(g	PROPN
ejpam-4914	253	9	)	)	PUNCT
ejpam-4914	253	10	=	=	SYM
ejpam-4914	253	11	4(2j	4(2j	X
ejpam-4914	253	12	)	)	PUNCT
ejpam-4914	253	13	+	+	CCONJ
ejpam-4914	253	14	3	3	NUM
ejpam-4914	253	15	=	=	SYM
ejpam-4914	253	16	(	(	PUNCT
ejpam-4914	253	17	6j	6j	NOUN
ejpam-4914	253	18	+	+	CCONJ
ejpam-4914	253	19	2	2	X
ejpam-4914	253	20	)	)	PUNCT
ejpam-4914	253	21	+	+	CCONJ
ejpam-4914	253	22	(	(	PUNCT
ejpam-4914	253	23	2j	2j	NUM
ejpam-4914	253	24	+	+	CCONJ
ejpam-4914	253	25	1	1	X
ejpam-4914	253	26	)	)	PUNCT
ejpam-4914	253	27	=	=	NOUN
ejpam-4914	254	1	[	[	X
ejpam-4914	254	2	(	(	PUNCT
ejpam-4914	254	3	2(3j	2(3j	NUM
ejpam-4914	254	4	)	)	PUNCT
ejpam-4914	254	5	+	+	CCONJ
ejpam-4914	254	6	2	2	X
ejpam-4914	254	7	]	]	PUNCT
ejpam-4914	254	8	+	+	CCONJ
ejpam-4914	254	9	(	(	PUNCT
ejpam-4914	254	10	2j	2j	NUM
ejpam-4914	254	11	+	+	CCONJ
ejpam-4914	254	12	1	1	X
ejpam-4914	254	13	)	)	PUNCT
ejpam-4914	254	14	=	=	SYM
ejpam-4914	254	15	γhi(g	γhi(g	PROPN
ejpam-4914	254	16	)	)	PUNCT
ejpam-4914	255	1	+	+	CCONJ
ejpam-4914	255	2	k.	k.	NOUN
ejpam-4914	255	3	for	for	ADP
ejpam-4914	255	4	(	(	PUNCT
ejpam-4914	255	5	ii	ii	PROPN
ejpam-4914	255	6	):	):	PUNCT
ejpam-4914	255	7	if	if	SCONJ
ejpam-4914	255	8	a	a	DET
ejpam-4914	255	9	=	=	SYM
ejpam-4914	255	10	b	b	NOUN
ejpam-4914	255	11	,	,	PUNCT
ejpam-4914	255	12	then	then	ADV
ejpam-4914	255	13	we	we	PRON
ejpam-4914	255	14	take	take	VERB
ejpam-4914	255	15	g	g	PROPN
ejpam-4914	255	16	=	=	SYM
ejpam-4914	255	17	ka	ka	PROPN
ejpam-4914	255	18	,	,	PUNCT
ejpam-4914	255	19	the	the	DET
ejpam-4914	255	20	complete	complete	ADJ
ejpam-4914	255	21	graph	graph	NOUN
ejpam-4914	255	22	on	on	ADP
ejpam-4914	255	23	a	a	DET
ejpam-4914	255	24	vertices	vertex	NOUN
ejpam-4914	255	25	.	.	PUNCT
ejpam-4914	256	1	suppose	suppose	VERB
ejpam-4914	256	2	that	that	SCONJ
ejpam-4914	256	3	b	b	X
ejpam-4914	256	4	=	=	PRON
ejpam-4914	256	5	a+	a+	PUNCT
ejpam-4914	256	6	k	k	PROPN
ejpam-4914	256	7	with	with	ADP
ejpam-4914	256	8	k	k	PROPN
ejpam-4914	256	9	≥	≥	NUM
ejpam-4914	256	10	1	1	NUM
ejpam-4914	256	11	.	.	PUNCT
ejpam-4914	257	1	we	we	PRON
ejpam-4914	257	2	consider	consider	VERB
ejpam-4914	257	3	the	the	DET
ejpam-4914	257	4	following	follow	VERB
ejpam-4914	257	5	cases	case	NOUN
ejpam-4914	257	6	:	:	PUNCT
ejpam-4914	257	7	case	case	NOUN
ejpam-4914	257	8	1	1	NUM
ejpam-4914	257	9	:	:	PUNCT
ejpam-4914	257	10	suppose	suppose	VERB
ejpam-4914	257	11	that	that	SCONJ
ejpam-4914	257	12	a	a	DET
ejpam-4914	257	13	=	=	SYM
ejpam-4914	257	14	2n+2	2n+2	NUM
ejpam-4914	257	15	for	for	ADP
ejpam-4914	257	16	some	some	DET
ejpam-4914	257	17	n	n	PRON
ejpam-4914	257	18	≥	≥	NOUN
ejpam-4914	257	19	1	1	NUM
ejpam-4914	257	20	.	.	PUNCT
ejpam-4914	258	1	let	let	VERB
ejpam-4914	258	2	t	t	NOUN
ejpam-4914	258	3	=	=	PUNCT
ejpam-4914	258	4	4n	4n	NOUN
ejpam-4914	258	5	,	,	PUNCT
ejpam-4914	258	6	and	and	CCONJ
ejpam-4914	258	7	put	put	VERB
ejpam-4914	258	8	pt	pt	NOUN
ejpam-4914	259	1	=	=	SYM
ejpam-4914	260	1	[	[	X
ejpam-4914	260	2	x1	x1	PROPN
ejpam-4914	260	3	,	,	PUNCT
ejpam-4914	260	4	x2	x2	PROPN
ejpam-4914	260	5	,	,	PUNCT
ejpam-4914	260	6	.	.	PUNCT
ejpam-4914	260	7	.	.	PUNCT
ejpam-4914	261	1	.	.	PUNCT
ejpam-4914	262	1	,	,	PUNCT
ejpam-4914	262	2	xt	xt	ADP
ejpam-4914	262	3	]	]	PUNCT
ejpam-4914	262	4	,	,	PUNCT
ejpam-4914	262	5	a	a	DET
ejpam-4914	262	6	path	path	NOUN
ejpam-4914	262	7	on	on	ADP
ejpam-4914	262	8	t	t	PROPN
ejpam-4914	262	9	vertices	vertex	NOUN
ejpam-4914	262	10	.	.	PUNCT
ejpam-4914	263	1	if	if	SCONJ
ejpam-4914	263	2	n	n	NOUN
ejpam-4914	263	3	=	=	SYM
ejpam-4914	263	4	1	1	NUM
ejpam-4914	263	5	,	,	PUNCT
ejpam-4914	263	6	then	then	ADV
ejpam-4914	263	7	we	we	PRON
ejpam-4914	263	8	take	take	VERB
ejpam-4914	263	9	g	g	NOUN
ejpam-4914	263	10	=	=	SYM
ejpam-4914	263	11	g1	g1	PROPN
ejpam-4914	263	12	,	,	PUNCT
ejpam-4914	263	13	where	where	SCONJ
ejpam-4914	263	14	g1	g1	PROPN
ejpam-4914	263	15	is	be	AUX
ejpam-4914	263	16	the	the	DET
ejpam-4914	263	17	graph	graph	NOUN
ejpam-4914	263	18	in	in	ADP
ejpam-4914	263	19	figure	figure	NOUN
ejpam-4914	263	20	3	3	NUM
ejpam-4914	263	21	obtained	obtain	VERB
ejpam-4914	263	22	from	from	ADP
ejpam-4914	263	23	p4	p4	NOUN
ejpam-4914	263	24	by	by	ADP
ejpam-4914	263	25	adding	add	VERB
ejpam-4914	263	26	k	k	PROPN
ejpam-4914	264	1	+	+	CCONJ
ejpam-4914	264	2	1	1	NUM
ejpam-4914	264	3	distinct	distinct	ADJ
ejpam-4914	264	4	paths	path	NOUN
ejpam-4914	264	5	[	[	X
ejpam-4914	264	6	x3	x3	ADJ
ejpam-4914	264	7	,	,	PUNCT
ejpam-4914	264	8	yj	yj	PROPN
ejpam-4914	264	9	,	,	PUNCT
ejpam-4914	264	10	zj	zj	PROPN
ejpam-4914	264	11	]	]	PUNCT
ejpam-4914	264	12	,	,	PUNCT
ejpam-4914	264	13	j	j	PROPN
ejpam-4914	264	14	=	=	SYM
ejpam-4914	264	15	1	1	NUM
ejpam-4914	264	16	,	,	PUNCT
ejpam-4914	264	17	2	2	NUM
ejpam-4914	264	18	,	,	PUNCT
ejpam-4914	264	19	.	.	PUNCT
ejpam-4914	264	20	.	.	PUNCT
ejpam-4914	264	21	.	.	PUNCT
ejpam-4914	265	1	,	,	PUNCT
ejpam-4914	266	1	k	k	PROPN
ejpam-4914	266	2	+	+	NOUN
ejpam-4914	266	3	1	1	X
ejpam-4914	266	4	.	.	X
ejpam-4914	266	5	define	define	VERB
ejpam-4914	266	6	...........................................................................................................................	...........................................................................................................................	PUNCT
ejpam-4914	266	7	...........................................................................................................................	...........................................................................................................................	PUNCT
ejpam-4914	266	8	...........................................................................................................................	...........................................................................................................................	PUNCT
ejpam-4914	266	9	........................................................................	........................................................................	PUNCT
ejpam-4914	266	10	....................................	....................................	PUNCT
ejpam-4914	266	11	....................................	....................................	PUNCT
ejpam-4914	267	1	....................................	....................................	PUNCT
ejpam-4914	267	2	....................................	....................................	PUNCT
ejpam-4914	268	1	....................................	....................................	PUNCT
ejpam-4914	268	2	....................................	....................................	PUNCT
ejpam-4914	269	1	....................................	....................................	PUNCT
ejpam-4914	269	2	....................................	....................................	PUNCT
ejpam-4914	270	1	......................................................................	......................................................................	PUNCT
ejpam-4914	270	2	....................................	....................................	PUNCT
ejpam-4914	271	1	.....................................................................................................	.....................................................................................................	PUNCT
ejpam-4914	271	2	....................................	....................................	PUNCT
ejpam-4914	272	1	....................................	....................................	PUNCT
ejpam-4914	272	2	......................................................................	......................................................................	PUNCT
ejpam-4914	273	1	....................................	....................................	PUNCT
ejpam-4914	273	2	.....................................................................	.....................................................................	PUNCT
ejpam-4914	274	1	....................................	....................................	PUNCT
ejpam-4914	274	2	....................................	....................................	PUNCT
ejpam-4914	275	1	..........	..........	PUNCT
ejpam-4914	275	2	.........	.........	PUNCT
ejpam-4914	276	1	.........	.........	PUNCT
ejpam-4914	276	2	.........	.........	PUNCT
ejpam-4914	277	1	.........	.........	PUNCT
ejpam-4914	277	2	.........	.........	PUNCT
ejpam-4914	278	1	.........	.........	PUNCT
ejpam-4914	278	2	......	......	PUNCT
ejpam-4914	278	3	....................................	....................................	PUNCT
ejpam-4914	279	1	..........	..........	PUNCT
ejpam-4914	279	2	.........	.........	PUNCT
ejpam-4914	280	1	.........	.........	PUNCT
ejpam-4914	280	2	.........	.........	PUNCT
ejpam-4914	281	1	.........	.........	PUNCT
ejpam-4914	281	2	.........	.........	PUNCT
ejpam-4914	282	1	.........	.........	PUNCT
ejpam-4914	282	2	.....	.....	PUNCT
ejpam-4914	282	3	....................................	....................................	PUNCT
ejpam-4914	283	1	....................................	....................................	PUNCT
ejpam-4914	283	2	....................	....................	PUNCT
ejpam-4914	284	1	...................	...................	PUNCT
ejpam-4914	284	2	...................	...................	PUNCT
ejpam-4914	284	3	............	............	PUNCT
ejpam-4914	284	4	....................................	....................................	PUNCT
ejpam-4914	284	5	..................	..................	PUNCT
ejpam-4914	285	1	.................	.................	PUNCT
ejpam-4914	285	2	.................	.................	PUNCT
ejpam-4914	286	1	.................	.................	PUNCT
ejpam-4914	286	2	.................	.................	PUNCT
ejpam-4914	286	3	...............	...............	PUNCT
ejpam-4914	287	1	....................................	....................................	PUNCT
ejpam-4914	288	1	....................................	....................................	PUNCT
ejpam-4914	289	1	•	•	NUM
ejpam-4914	289	2	•	•	NUM
ejpam-4914	289	3	•	•	NUM
ejpam-4914	289	4	•	•	NUM
ejpam-4914	289	5	•	•	NUM
ejpam-4914	289	6	•	•	NUM
ejpam-4914	289	7	•	•	NUM
ejpam-4914	289	8	•	•	NUM
ejpam-4914	289	9	•	•	NUM
ejpam-4914	289	10	•	•	NUM
ejpam-4914	289	11	•	•	NOUN
ejpam-4914	289	12	•	•	NUM
ejpam-4914	289	13	·	·	PUNCT
ejpam-4914	289	14	·	·	PUNCT
ejpam-4914	289	15	·	·	PUNCT
ejpam-4914	289	16	·	·	PUNCT
ejpam-4914	289	17	·	·	PUNCT
ejpam-4914	289	18	·	·	PUNCT
ejpam-4914	290	1	x1	x1	PUNCT
ejpam-4914	291	1	x2	x2	NOUN
ejpam-4914	291	2	x3	x3	PROPN
ejpam-4914	291	3	x4	x4	PROPN
ejpam-4914	292	1	y1	y1	NOUN
ejpam-4914	293	1	y2	y2	INTJ
ejpam-4914	293	2	yk	yk	NOUN
ejpam-4914	293	3	yk+1	yk+1	PRON
ejpam-4914	293	4	z1	z1	PROPN
ejpam-4914	293	5	z2	z2	PROPN
ejpam-4914	293	6	zk	zk	PROPN
ejpam-4914	293	7	zk+1	zk+1	PROPN
ejpam-4914	293	8	g1	g1	PROPN
ejpam-4914	293	9	...........................................................................................................................	...........................................................................................................................	PUNCT
ejpam-4914	293	10	...........................................................................................................................	...........................................................................................................................	PROPN
ejpam-4914	293	11	...........................................................................................................................	...........................................................................................................................	PUNCT
ejpam-4914	293	12	...........................................................................................................................	...........................................................................................................................	PUNCT
ejpam-4914	293	13	...........................................................................................................................	...........................................................................................................................	PUNCT
ejpam-4914	293	14	....................................	....................................	PUNCT
ejpam-4914	293	15	...........................................................................................................................	...........................................................................................................................	PUNCT
ejpam-4914	293	16	...........................................................................................................................	...........................................................................................................................	PUNCT
ejpam-4914	293	17	...........................................................................................................................	...........................................................................................................................	PUNCT
ejpam-4914	293	18	........................................................................	........................................................................	PUNCT
ejpam-4914	293	19	....................................	....................................	PUNCT
ejpam-4914	293	20	....................................	....................................	PUNCT
ejpam-4914	293	21	....................................	....................................	PUNCT
ejpam-4914	293	22	....................................	....................................	PUNCT
ejpam-4914	293	23	....................................	....................................	PUNCT
ejpam-4914	293	24	....................................	....................................	PUNCT
ejpam-4914	293	25	....................................	....................................	PUNCT
ejpam-4914	293	26	....................................	....................................	PUNCT
ejpam-4914	293	27	......................................................................	......................................................................	PUNCT
ejpam-4914	293	28	....................................	....................................	PUNCT
ejpam-4914	293	29	.....................................................................................................	.....................................................................................................	PUNCT
ejpam-4914	293	30	....................................	....................................	PUNCT
ejpam-4914	293	31	....................................	....................................	PUNCT
ejpam-4914	293	32	......................................................................	......................................................................	PUNCT
ejpam-4914	293	33	....................................	....................................	PUNCT
ejpam-4914	293	34	.....................................................................	.....................................................................	PUNCT
ejpam-4914	293	35	....................................	....................................	PUNCT
ejpam-4914	293	36	....................................	....................................	PUNCT
ejpam-4914	294	1	..........	..........	PUNCT
ejpam-4914	294	2	.........	.........	PUNCT
ejpam-4914	295	1	.........	.........	PUNCT
ejpam-4914	295	2	.........	.........	PUNCT
ejpam-4914	296	1	.........	.........	PUNCT
ejpam-4914	296	2	.........	.........	PUNCT
ejpam-4914	297	1	.........	.........	PUNCT
ejpam-4914	297	2	......	......	PUNCT
ejpam-4914	297	3	....................................	....................................	PUNCT
ejpam-4914	298	1	..........	..........	PUNCT
ejpam-4914	298	2	.........	.........	PUNCT
ejpam-4914	299	1	.........	.........	PUNCT
ejpam-4914	299	2	.........	.........	PUNCT
ejpam-4914	300	1	.........	.........	PUNCT
ejpam-4914	300	2	.........	.........	PUNCT
ejpam-4914	301	1	.........	.........	PUNCT
ejpam-4914	301	2	.....	.....	PUNCT
ejpam-4914	301	3	....................................	....................................	PUNCT
ejpam-4914	302	1	....................................	....................................	PUNCT
ejpam-4914	302	2	....................	....................	PUNCT
ejpam-4914	303	1	...................	...................	PUNCT
ejpam-4914	303	2	...................	...................	PUNCT
ejpam-4914	303	3	............	............	PUNCT
ejpam-4914	303	4	....................................	....................................	PUNCT
ejpam-4914	303	5	..................	..................	PUNCT
ejpam-4914	304	1	.................	.................	PUNCT
ejpam-4914	304	2	.................	.................	PUNCT
ejpam-4914	305	1	.................	.................	PUNCT
ejpam-4914	305	2	.................	.................	PUNCT
ejpam-4914	305	3	...............	...............	PUNCT
ejpam-4914	306	1	....................................	....................................	PUNCT
ejpam-4914	307	1	....................................	....................................	PUNCT
ejpam-4914	308	1	•	•	NUM
ejpam-4914	308	2	•	•	NUM
ejpam-4914	308	3	•	•	NUM
ejpam-4914	308	4	•	•	NUM
ejpam-4914	308	5	•	•	NUM
ejpam-4914	308	6	•	•	NUM
ejpam-4914	308	7	•	•	NUM
ejpam-4914	308	8	•	•	NUM
ejpam-4914	308	9	•	•	NUM
ejpam-4914	308	10	•	•	NUM
ejpam-4914	308	11	•	•	NUM
ejpam-4914	308	12	•	•	NUM
ejpam-4914	308	13	•	•	NUM
ejpam-4914	308	14	•	•	NUM
ejpam-4914	308	15	•	•	NUM
ejpam-4914	308	16	•	•	NUM
ejpam-4914	308	17	•	•	NOUN
ejpam-4914	308	18	•	•	NUM
ejpam-4914	308	19	·	·	PUNCT
ejpam-4914	308	20	·	·	PUNCT
ejpam-4914	308	21	·	·	PUNCT
ejpam-4914	308	22	·	·	PUNCT
ejpam-4914	308	23	·	·	PUNCT
ejpam-4914	308	24	·	·	PUNCT
ejpam-4914	308	25	·	·	PUNCT
ejpam-4914	308	26	·	·	PUNCT
ejpam-4914	308	27	·	·	PUNCT
ejpam-4914	309	1	x1	x1	PUNCT
ejpam-4914	310	1	x2	x2	NOUN
ejpam-4914	310	2	x3	x3	PROPN
ejpam-4914	310	3	x4	x4	PROPN
ejpam-4914	310	4	x5	x5	PROPN
ejpam-4914	310	5	x6	x6	PROPN
ejpam-4914	310	6	xt−3	xt−3	PROPN
ejpam-4914	310	7	xt−2	xt−2	PROPN
ejpam-4914	310	8	xt−1	xt−1	PROPN
ejpam-4914	310	9	xt	xt	PROPN
ejpam-4914	311	1	y1	y1	INTJ
ejpam-4914	311	2	y2	y2	NOUN
ejpam-4914	311	3	yk−1	yk−1	PROPN
ejpam-4914	311	4	ykz1	ykz1	PROPN
ejpam-4914	311	5	z2	z2	PROPN
ejpam-4914	311	6	zk−1	zk−1	PROPN
ejpam-4914	311	7	zk	zk	PROPN
ejpam-4914	311	8	.......................................	.......................................	PUNCT
ejpam-4914	311	9	.......................................	.......................................	PUNCT
ejpam-4914	312	1	g2	g2	PROPN
ejpam-4914	312	2	figure	figure	VERB
ejpam-4914	312	3	3	3	NUM
ejpam-4914	312	4	:	:	PUNCT
ejpam-4914	312	5	examples	example	NOUN
ejpam-4914	312	6	of	of	ADP
ejpam-4914	312	7	graphs	graph	NOUN
ejpam-4914	312	8	g	g	NOUN
ejpam-4914	312	9	with	with	ADP
ejpam-4914	312	10	γhi(g	γhi(g	PROPN
ejpam-4914	312	11	)	)	PUNCT
ejpam-4914	313	1	=	=	SYM
ejpam-4914	313	2	a	a	DET
ejpam-4914	313	3	and	and	CCONJ
ejpam-4914	313	4	γ2h(g	γ2h(g	PROPN
ejpam-4914	313	5	)	)	PUNCT
ejpam-4914	314	1	=	=	SYM
ejpam-4914	314	2	b	b	X
ejpam-4914	314	3	v2	v2	NOUN
ejpam-4914	314	4	=	=	SYM
ejpam-4914	314	5	{	{	PUNCT
ejpam-4914	314	6	x3	x3	PROPN
ejpam-4914	314	7	,	,	PUNCT
ejpam-4914	314	8	x4	x4	PROPN
ejpam-4914	314	9	}	}	PUNCT
ejpam-4914	314	10	,	,	PUNCT
ejpam-4914	314	11	v1	v1	NOUN
ejpam-4914	314	12	=	=	SYM
ejpam-4914	314	13	∅	∅	NOUN
ejpam-4914	314	14	and	and	CCONJ
ejpam-4914	314	15	v0	v0	NOUN
ejpam-4914	314	16	=	=	SYM
ejpam-4914	314	17	v	v	PROPN
ejpam-4914	314	18	(	(	PUNCT
ejpam-4914	314	19	g)\{x3	g)\{x3	PROPN
ejpam-4914	314	20	,	,	PUNCT
ejpam-4914	314	21	x4	x4	ADJ
ejpam-4914	314	22	}	}	PUNCT
ejpam-4914	314	23	.	.	PUNCT
ejpam-4914	315	1	then	then	ADV
ejpam-4914	315	2	f	f	X
ejpam-4914	315	3	=	=	SYM
ejpam-4914	315	4	(	(	PUNCT
ejpam-4914	315	5	v0	v0	PROPN
ejpam-4914	315	6	,	,	PUNCT
ejpam-4914	315	7	v1	v1	NOUN
ejpam-4914	315	8	,	,	PUNCT
ejpam-4914	315	9	v2	v2	PROPN
ejpam-4914	315	10	)	)	PUNCT
ejpam-4914	315	11	is	be	AUX
ejpam-4914	315	12	a	a	DET
ejpam-4914	315	13	γhi	γhi	ADJ
ejpam-4914	315	14	-function	-function	NOUN
ejpam-4914	315	15	of	of	ADP
ejpam-4914	315	16	g.	g.	NOUN
ejpam-4914	315	17	thus	thus	ADV
ejpam-4914	315	18	,	,	PUNCT
ejpam-4914	315	19	γhi(g	γhi(g	PROPN
ejpam-4914	315	20	)	)	PUNCT
ejpam-4914	315	21	=	=	SYM
ejpam-4914	316	1	4	4	NUM
ejpam-4914	316	2	=	=	SYM
ejpam-4914	316	3	a.	a.	NOUN
ejpam-4914	316	4	on	on	ADP
ejpam-4914	316	5	the	the	DET
ejpam-4914	316	6	other	other	ADJ
ejpam-4914	316	7	hand	hand	NOUN
ejpam-4914	316	8	,	,	PUNCT
ejpam-4914	316	9	the	the	DET
ejpam-4914	316	10	set	set	NOUN
ejpam-4914	316	11	{	{	PUNCT
ejpam-4914	316	12	x1	x1	PROPN
ejpam-4914	316	13	,	,	PUNCT
ejpam-4914	316	14	x2	x2	PROPN
ejpam-4914	316	15	,	,	PUNCT
ejpam-4914	316	16	x4}∪{zj	x4}∪{zj	PROPN
ejpam-4914	316	17	:	:	PUNCT
ejpam-4914	317	1	j	j	X
ejpam-4914	317	2	=	=	SYM
ejpam-4914	317	3	1	1	NUM
ejpam-4914	317	4	,	,	PUNCT
ejpam-4914	317	5	2	2	NUM
ejpam-4914	317	6	,	,	PUNCT
ejpam-4914	317	7	.	.	PUNCT
ejpam-4914	317	8	.	.	PUNCT
ejpam-4914	317	9	.	.	PUNCT
ejpam-4914	318	1	,	,	PUNCT
ejpam-4914	318	2	k+1	k+1	X
ejpam-4914	318	3	}	}	PUNCT
ejpam-4914	318	4	is	be	AUX
ejpam-4914	318	5	a	a	DET
ejpam-4914	318	6	γ2h	γ2h	NOUN
ejpam-4914	318	7	-	-	PUNCT
ejpam-4914	318	8	set	set	NOUN
ejpam-4914	318	9	of	of	ADP
ejpam-4914	318	10	g	g	NOUN
ejpam-4914	318	11	,	,	PUNCT
ejpam-4914	318	12	implying	imply	VERB
ejpam-4914	318	13	that	that	SCONJ
ejpam-4914	318	14	γ2h(g	γ2h(g	NOUN
ejpam-4914	318	15	)	)	PUNCT
ejpam-4914	319	1	=	=	SYM
ejpam-4914	319	2	3+k+1	3+k+1	NUM
ejpam-4914	319	3	=	=	SYM
ejpam-4914	319	4	4+k	4+k	PROPN
ejpam-4914	319	5	=	=	SYM
ejpam-4914	319	6	b.	b.	PROPN
ejpam-4914	319	7	suppose	suppose	VERB
ejpam-4914	319	8	that	that	SCONJ
ejpam-4914	319	9	n	n	PROPN
ejpam-4914	319	10	≥	≥	NUM
ejpam-4914	319	11	2	2	NUM
ejpam-4914	319	12	.	.	PUNCT
ejpam-4914	319	13	obtain	obtain	VERB
ejpam-4914	319	14	g	g	NOUN
ejpam-4914	319	15	as	as	ADP
ejpam-4914	319	16	the	the	DET
ejpam-4914	319	17	graph	graph	NOUN
ejpam-4914	319	18	g2	g2	PROPN
ejpam-4914	319	19	in	in	ADP
ejpam-4914	319	20	figure	figure	NOUN
ejpam-4914	319	21	3	3	NUM
ejpam-4914	319	22	from	from	ADP
ejpam-4914	319	23	pt	pt	NOUN
ejpam-4914	319	24	by	by	ADP
ejpam-4914	319	25	adding	add	VERB
ejpam-4914	319	26	k	k	PROPN
ejpam-4914	319	27	distinct	distinct	ADJ
ejpam-4914	319	28	paths	path	NOUN
ejpam-4914	319	29	[	[	X
ejpam-4914	319	30	x3	x3	ADJ
ejpam-4914	319	31	,	,	PUNCT
ejpam-4914	319	32	yj	yj	PROPN
ejpam-4914	319	33	,	,	PUNCT
ejpam-4914	319	34	zj	zj	PROPN
ejpam-4914	319	35	]	]	PUNCT
ejpam-4914	319	36	,	,	PUNCT
ejpam-4914	319	37	j	j	PROPN
ejpam-4914	319	38	=	=	SYM
ejpam-4914	319	39	1	1	NUM
ejpam-4914	319	40	,	,	PUNCT
ejpam-4914	319	41	2	2	NUM
ejpam-4914	319	42	,	,	PUNCT
ejpam-4914	319	43	.	.	PUNCT
ejpam-4914	319	44	.	.	PUNCT
ejpam-4914	320	1	.	.	PUNCT
ejpam-4914	321	1	,	,	PUNCT
ejpam-4914	321	2	k.	k.	PROPN
ejpam-4914	321	3	define	define	VERB
ejpam-4914	321	4	v2	v2	PROPN
ejpam-4914	321	5	=	=	PUNCT
ejpam-4914	321	6	{	{	PUNCT
ejpam-4914	321	7	x3	x3	PROPN
ejpam-4914	321	8	,	,	PUNCT
ejpam-4914	321	9	x4	x4	PROPN
ejpam-4914	321	10	}	}	PUNCT
ejpam-4914	321	11	,	,	PUNCT
ejpam-4914	321	12	v1	v1	NOUN
ejpam-4914	321	13	=	=	SYM
ejpam-4914	321	14	{	{	PUNCT
ejpam-4914	321	15	x7	x7	NOUN
ejpam-4914	321	16	,	,	PUNCT
ejpam-4914	321	17	x8	x8	PROPN
ejpam-4914	321	18	,	,	PUNCT
ejpam-4914	321	19	x11	x11	PROPN
ejpam-4914	321	20	,	,	PUNCT
ejpam-4914	321	21	x12	x12	NUM
ejpam-4914	321	22	,	,	PUNCT
ejpam-4914	321	23	.	.	PUNCT
ejpam-4914	321	24	.	.	PUNCT
ejpam-4914	322	1	.	.	PUNCT
ejpam-4914	323	1	,	,	PUNCT
ejpam-4914	323	2	xt−1	xt−1	PROPN
ejpam-4914	323	3	,	,	PUNCT
ejpam-4914	323	4	xt	xt	ADP
ejpam-4914	323	5	}	}	PUNCT
ejpam-4914	323	6	and	and	CCONJ
ejpam-4914	323	7	v0	v0	PROPN
ejpam-4914	323	8	=	=	SYM
ejpam-4914	323	9	v	v	PROPN
ejpam-4914	323	10	(	(	PUNCT
ejpam-4914	323	11	g)\	g)\	NOUN
ejpam-4914	323	12	(	(	PUNCT
ejpam-4914	323	13	v1	v1	VERB
ejpam-4914	323	14	∪	∪	NOUN
ejpam-4914	323	15	v2	v2	NOUN
ejpam-4914	323	16	)	)	PUNCT
ejpam-4914	323	17	.	.	PUNCT
ejpam-4914	324	1	then	then	ADV
ejpam-4914	324	2	f	f	PROPN
ejpam-4914	324	3	=	=	SYM
ejpam-4914	324	4	(	(	PUNCT
ejpam-4914	324	5	v0	v0	PROPN
ejpam-4914	324	6	,	,	PUNCT
ejpam-4914	324	7	v1	v1	NOUN
ejpam-4914	324	8	,	,	PUNCT
ejpam-4914	324	9	v2	v2	PROPN
ejpam-4914	324	10	)	)	PUNCT
ejpam-4914	324	11	is	be	AUX
ejpam-4914	324	12	a	a	DET
ejpam-4914	324	13	γhi	γhi	ADJ
ejpam-4914	324	14	-function	-function	NOUN
ejpam-4914	324	15	of	of	ADP
ejpam-4914	324	16	g	g	NOUN
ejpam-4914	324	17	,	,	PUNCT
ejpam-4914	324	18	implying	imply	VERB
ejpam-4914	324	19	that	that	SCONJ
ejpam-4914	324	20	γhi(g	γhi(g	PROPN
ejpam-4914	324	21	)	)	PUNCT
ejpam-4914	325	1	=	=	SYM
ejpam-4914	325	2	γhi(pt	γhi(pt	NOUN
ejpam-4914	325	3	)	)	PUNCT
ejpam-4914	325	4	=	=	SYM
ejpam-4914	325	5	2n+	2n+	NUM
ejpam-4914	325	6	2	2	NUM
ejpam-4914	325	7	=	=	SYM
ejpam-4914	325	8	a.	a.	NOUN
ejpam-4914	325	9	on	on	ADP
ejpam-4914	325	10	the	the	DET
ejpam-4914	325	11	other	other	ADJ
ejpam-4914	325	12	hand	hand	NOUN
ejpam-4914	325	13	,	,	PUNCT
ejpam-4914	325	14	necessarily	necessarily	ADV
ejpam-4914	325	15	,	,	PUNCT
ejpam-4914	325	16	s	s	PART
ejpam-4914	325	17	=	=	SYM
ejpam-4914	325	18	{	{	PUNCT
ejpam-4914	325	19	zj	zj	NOUN
ejpam-4914	325	20	:	:	PUNCT
ejpam-4914	325	21	j	j	PROPN
ejpam-4914	326	1	=	=	SYM
ejpam-4914	326	2	1	1	NUM
ejpam-4914	326	3	,	,	PUNCT
ejpam-4914	326	4	2	2	NUM
ejpam-4914	326	5	,	,	PUNCT
ejpam-4914	326	6	.	.	PUNCT
ejpam-4914	326	7	.	.	PUNCT
ejpam-4914	327	1	.	.	PUNCT
ejpam-4914	328	1	,	,	PUNCT
ejpam-4914	328	2	k	k	X
ejpam-4914	328	3	}	}	PUNCT
ejpam-4914	328	4	is	be	AUX
ejpam-4914	328	5	contained	contain	VERB
ejpam-4914	328	6	in	in	ADP
ejpam-4914	328	7	any	any	DET
ejpam-4914	328	8	2	2	NUM
ejpam-4914	328	9	-	-	PUNCT
ejpam-4914	328	10	hop	hop	NOUN
ejpam-4914	328	11	dominating	dominating	NOUN
ejpam-4914	328	12	set	set	NOUN
ejpam-4914	328	13	of	of	ADP
ejpam-4914	328	14	g.	g.	PROPN
ejpam-4914	328	15	observe	observe	VERB
ejpam-4914	328	16	that	that	SCONJ
ejpam-4914	328	17	s	s	VERB
ejpam-4914	328	18	∪	∪	X
ejpam-4914	328	19	{	{	PUNCT
ejpam-4914	328	20	x1	x1	PROPN
ejpam-4914	328	21	,	,	PUNCT
ejpam-4914	328	22	x2	x2	PROPN
ejpam-4914	328	23	,	,	PUNCT
ejpam-4914	328	24	x4	x4	PROPN
ejpam-4914	328	25	,	,	PUNCT
ejpam-4914	328	26	x5	x5	PROPN
ejpam-4914	328	27	,	,	PUNCT
ejpam-4914	328	28	x8	x8	PROPN
ejpam-4914	328	29	,	,	PUNCT
ejpam-4914	328	30	x9	x9	PROPN
ejpam-4914	328	31	,	,	PUNCT
ejpam-4914	328	32	.	.	PUNCT
ejpam-4914	328	33	.	.	PUNCT
ejpam-4914	329	1	.	.	PUNCT
ejpam-4914	330	1	,	,	PUNCT
ejpam-4914	330	2	xt−4	xt−4	PROPN
ejpam-4914	330	3	,	,	PUNCT
ejpam-4914	330	4	xt−3	xt−3	PROPN
ejpam-4914	330	5	,	,	PUNCT
ejpam-4914	330	6	xt−1	xt−1	PROPN
ejpam-4914	330	7	,	,	PUNCT
ejpam-4914	330	8	xt	xt	ADP
ejpam-4914	330	9	}	}	PUNCT
ejpam-4914	330	10	is	be	AUX
ejpam-4914	330	11	a	a	DET
ejpam-4914	330	12	γ2h	γ2h	NOUN
ejpam-4914	330	13	-	-	PUNCT
ejpam-4914	330	14	set	set	NOUN
ejpam-4914	330	15	of	of	ADP
ejpam-4914	330	16	g.	g.	PROPN
ejpam-4914	330	17	thus	thus	ADV
ejpam-4914	330	18	,	,	PUNCT
ejpam-4914	330	19	γ2h(g	γ2h(g	PROPN
ejpam-4914	330	20	)	)	PUNCT
ejpam-4914	330	21	=	=	SYM
ejpam-4914	331	1	(	(	PUNCT
ejpam-4914	331	2	2n+	2n+	NUM
ejpam-4914	331	3	2	2	NUM
ejpam-4914	331	4	)	)	PUNCT
ejpam-4914	332	1	+	+	CCONJ
ejpam-4914	332	2	k	k	NOUN
ejpam-4914	332	3	=	=	SYM
ejpam-4914	332	4	a+	a+	PUNCT
ejpam-4914	332	5	k	k	PROPN
ejpam-4914	332	6	=	=	PROPN
ejpam-4914	332	7	b.	b.	PROPN
ejpam-4914	332	8	s.r	s.r	PROPN
ejpam-4914	332	9	.	.	PROPN
ejpam-4914	332	10	jr	jr	PROPN
ejpam-4914	332	11	.	.	PROPN
ejpam-4914	332	12	canoy	canoy	PROPN
ejpam-4914	332	13	,	,	PUNCT
ejpam-4914	332	14	f.p	f.p	PROPN
ejpam-4914	332	15	.	.	PROPN
ejpam-4914	332	16	jamil	jamil	PROPN
ejpam-4914	332	17	and	and	CCONJ
ejpam-4914	332	18	s.m	s.m	PROPN
ejpam-4914	332	19	.	.	PROPN
ejpam-4914	332	20	menchavez	menchavez	PROPN
ejpam-4914	332	21	/	/	PUNCT
ejpam-4914	332	22	eur	eur	PROPN
ejpam-4914	332	23	.	.	PUNCT
ejpam-4914	333	1	j.	j.	PROPN
ejpam-4914	333	2	pure	pure	PROPN
ejpam-4914	333	3	appl	appl	PROPN
ejpam-4914	333	4	.	.	PROPN
ejpam-4914	333	5	math	math	PROPN
ejpam-4914	333	6	,	,	PUNCT
ejpam-4914	333	7	16	16	NUM
ejpam-4914	333	8	(	(	PUNCT
ejpam-4914	333	9	4	4	NUM
ejpam-4914	333	10	)	)	PUNCT
ejpam-4914	333	11	(	(	PUNCT
ejpam-4914	333	12	2023	2023	NUM
ejpam-4914	333	13	)	)	PUNCT
ejpam-4914	333	14	,	,	PUNCT
ejpam-4914	333	15	2431	2431	NUM
ejpam-4914	333	16	-	-	SYM
ejpam-4914	333	17	2449	2449	NUM
ejpam-4914	333	18	2439	2439	NUM
ejpam-4914	333	19	case	case	NOUN
ejpam-4914	333	20	2	2	NUM
ejpam-4914	333	21	:	:	PUNCT
ejpam-4914	333	22	suppose	suppose	VERB
ejpam-4914	333	23	that	that	SCONJ
ejpam-4914	333	24	a	a	DET
ejpam-4914	333	25	=	=	X
ejpam-4914	333	26	2n+3	2n+3	NOUN
ejpam-4914	333	27	for	for	ADP
ejpam-4914	333	28	some	some	DET
ejpam-4914	333	29	n	n	PRON
ejpam-4914	333	30	≥	≥	NOUN
ejpam-4914	333	31	1	1	NUM
ejpam-4914	333	32	.	.	PUNCT
ejpam-4914	333	33	put	put	VERB
ejpam-4914	333	34	t	t	NOUN
ejpam-4914	333	35	=	=	PUNCT
ejpam-4914	333	36	4n	4n	NOUN
ejpam-4914	333	37	and	and	CCONJ
ejpam-4914	333	38	let	let	VERB
ejpam-4914	333	39	pt	pt	X
ejpam-4914	333	40	=	=	PUNCT
ejpam-4914	334	1	[	[	X
ejpam-4914	334	2	x1	x1	PROPN
ejpam-4914	334	3	,	,	PUNCT
ejpam-4914	334	4	x2	x2	PROPN
ejpam-4914	334	5	,	,	PUNCT
ejpam-4914	334	6	.	.	PUNCT
ejpam-4914	334	7	.	.	PUNCT
ejpam-4914	335	1	.	.	PUNCT
ejpam-4914	336	1	,	,	PUNCT
ejpam-4914	336	2	xt	xt	ADP
ejpam-4914	336	3	]	]	PUNCT
ejpam-4914	336	4	.	.	PUNCT
ejpam-4914	337	1	if	if	SCONJ
ejpam-4914	337	2	n	n	NOUN
ejpam-4914	337	3	=	=	SYM
ejpam-4914	337	4	1	1	NUM
ejpam-4914	337	5	,	,	PUNCT
ejpam-4914	337	6	then	then	ADV
ejpam-4914	337	7	obtain	obtain	VERB
ejpam-4914	337	8	g	g	NOUN
ejpam-4914	337	9	as	as	ADP
ejpam-4914	337	10	the	the	DET
ejpam-4914	337	11	graph	graph	NOUN
ejpam-4914	337	12	g1	g1	NOUN
ejpam-4914	337	13	in	in	ADP
ejpam-4914	337	14	figure	figure	NOUN
ejpam-4914	337	15	4by	4by	NOUN
ejpam-4914	337	16	adding	add	VERB
ejpam-4914	337	17	to	to	AUX
ejpam-4914	337	18	p6	p6	VERB
ejpam-4914	337	19	k	k	PROPN
ejpam-4914	337	20	geodesics	geodesic	NOUN
ejpam-4914	337	21	,	,	PUNCT
ejpam-4914	337	22	namely	namely	ADV
ejpam-4914	337	23	...........................................................................................................................	...........................................................................................................................	PUNCT
ejpam-4914	337	24	...........................................................................................................................	...........................................................................................................................	PUNCT
ejpam-4914	337	25	...........................................................................................................................	...........................................................................................................................	PUNCT
ejpam-4914	337	26	...........................................................................................................................	...........................................................................................................................	PUNCT
ejpam-4914	337	27	...........................................................................................................................	...........................................................................................................................	PUNCT
ejpam-4914	337	28	........................................................................	........................................................................	PUNCT
ejpam-4914	337	29	....................................	....................................	PUNCT
ejpam-4914	337	30	....................................	....................................	PUNCT
ejpam-4914	338	1	....................................	....................................	PUNCT
ejpam-4914	338	2	....................................	....................................	PUNCT
ejpam-4914	339	1	....................................	....................................	PUNCT
ejpam-4914	339	2	....................................	....................................	PUNCT
ejpam-4914	340	1	....................................	....................................	PUNCT
ejpam-4914	340	2	....................................	....................................	PUNCT
ejpam-4914	341	1	......................................................................	......................................................................	PUNCT
ejpam-4914	341	2	....................................	....................................	PUNCT
ejpam-4914	342	1	.....................................................................................................	.....................................................................................................	PUNCT
ejpam-4914	342	2	....................................	....................................	PUNCT
ejpam-4914	343	1	....................................	....................................	PUNCT
ejpam-4914	343	2	......................................................................	......................................................................	PUNCT
ejpam-4914	344	1	....................................	....................................	PUNCT
ejpam-4914	344	2	.....................................................................	.....................................................................	PUNCT
ejpam-4914	345	1	....................................	....................................	PUNCT
ejpam-4914	345	2	....................................	....................................	PUNCT
ejpam-4914	346	1	..........	..........	PUNCT
ejpam-4914	346	2	.........	.........	PUNCT
ejpam-4914	347	1	.........	.........	PUNCT
ejpam-4914	347	2	.........	.........	PUNCT
ejpam-4914	348	1	.........	.........	PUNCT
ejpam-4914	348	2	.........	.........	PUNCT
ejpam-4914	349	1	.........	.........	PUNCT
ejpam-4914	349	2	......	......	PUNCT
ejpam-4914	349	3	....................................	....................................	PUNCT
ejpam-4914	350	1	..........	..........	PUNCT
ejpam-4914	350	2	.........	.........	PUNCT
ejpam-4914	351	1	.........	.........	PUNCT
ejpam-4914	351	2	.........	.........	PUNCT
ejpam-4914	352	1	.........	.........	PUNCT
ejpam-4914	352	2	.........	.........	PUNCT
ejpam-4914	353	1	.........	.........	PUNCT
ejpam-4914	353	2	.....	.....	PUNCT
ejpam-4914	353	3	....................................	....................................	PUNCT
ejpam-4914	354	1	....................................	....................................	PUNCT
ejpam-4914	354	2	....................	....................	PUNCT
ejpam-4914	355	1	...................	...................	PUNCT
ejpam-4914	355	2	...................	...................	PUNCT
ejpam-4914	355	3	............	............	PUNCT
ejpam-4914	355	4	....................................	....................................	PUNCT
ejpam-4914	355	5	..................	..................	PUNCT
ejpam-4914	356	1	.................	.................	PUNCT
ejpam-4914	356	2	.................	.................	PUNCT
ejpam-4914	357	1	.................	.................	PUNCT
ejpam-4914	357	2	.................	.................	PUNCT
ejpam-4914	357	3	...............	...............	PUNCT
ejpam-4914	358	1	....................................	....................................	PUNCT
ejpam-4914	359	1	....................................	....................................	PUNCT
ejpam-4914	360	1	•	•	NUM
ejpam-4914	360	2	•	•	NUM
ejpam-4914	360	3	•	•	NUM
ejpam-4914	360	4	•	•	NUM
ejpam-4914	360	5	•	•	NUM
ejpam-4914	360	6	•	•	NUM
ejpam-4914	360	7	•	•	NUM
ejpam-4914	360	8	•	•	NUM
ejpam-4914	360	9	•	•	NUM
ejpam-4914	360	10	•	•	NUM
ejpam-4914	360	11	•	•	NUM
ejpam-4914	360	12	•	•	NOUN
ejpam-4914	360	13	•	•	NOUN
ejpam-4914	360	14	••	••	NOUN
ejpam-4914	360	15	•	•	NUM
ejpam-4914	360	16	•	•	NOUN
ejpam-4914	360	17	•	•	NUM
ejpam-4914	360	18	·	·	PUNCT
ejpam-4914	360	19	·	·	PUNCT
ejpam-4914	360	20	·	·	PUNCT
ejpam-4914	360	21	·	·	PUNCT
ejpam-4914	360	22	·	·	PUNCT
ejpam-4914	360	23	·	·	PUNCT
ejpam-4914	361	1	x1	x1	PUNCT
ejpam-4914	362	1	x2	x2	NOUN
ejpam-4914	362	2	x3	x3	PROPN
ejpam-4914	362	3	x4	x4	PROPN
ejpam-4914	362	4	x5	x5	PROPN
ejpam-4914	362	5	x6	x6	PROPN
ejpam-4914	362	6	y1	y1	NOUN
ejpam-4914	362	7	y2	y2	NOUN
ejpam-4914	362	8	yk−1	yk−1	PROPN
ejpam-4914	362	9	ykz1	ykz1	PROPN
ejpam-4914	362	10	z2	z2	PROPN
ejpam-4914	362	11	zk−1	zk−1	PROPN
ejpam-4914	362	12	zk	zk	PROPN
ejpam-4914	362	13	g1	g1	PROPN
ejpam-4914	362	14	...........................................................................................................................	...........................................................................................................................	PUNCT
ejpam-4914	362	15	...........................................................................................................................	...........................................................................................................................	PROPN
ejpam-4914	362	16	...........................................................................................................................	...........................................................................................................................	PUNCT
ejpam-4914	362	17	...........................................................................................................................	...........................................................................................................................	PUNCT
ejpam-4914	362	18	....................................	....................................	PUNCT
ejpam-4914	362	19	...........................................................................................................................	...........................................................................................................................	PUNCT
ejpam-4914	362	20	...........................................................................................................................	...........................................................................................................................	PUNCT
ejpam-4914	362	21	...........................................................................................................................	...........................................................................................................................	PUNCT
ejpam-4914	362	22	........................................................................	........................................................................	PUNCT
ejpam-4914	362	23	....................................	....................................	PUNCT
ejpam-4914	362	24	....................................	....................................	PUNCT
ejpam-4914	363	1	....................................	....................................	PUNCT
ejpam-4914	363	2	....................................	....................................	PUNCT
ejpam-4914	364	1	....................................	....................................	PUNCT
ejpam-4914	364	2	....................................	....................................	PUNCT
ejpam-4914	365	1	....................................	....................................	PUNCT
ejpam-4914	365	2	....................................	....................................	PUNCT
ejpam-4914	366	1	......................................................................	......................................................................	PUNCT
ejpam-4914	366	2	....................................	....................................	PUNCT
ejpam-4914	367	1	.....................................................................................................	.....................................................................................................	PUNCT
ejpam-4914	367	2	....................................	....................................	PUNCT
ejpam-4914	368	1	....................................	....................................	PUNCT
ejpam-4914	368	2	......................................................................	......................................................................	PUNCT
ejpam-4914	369	1	....................................	....................................	PUNCT
ejpam-4914	369	2	.....................................................................	.....................................................................	PUNCT
ejpam-4914	370	1	....................................	....................................	PUNCT
ejpam-4914	370	2	....................................	....................................	PUNCT
ejpam-4914	371	1	..........	..........	PUNCT
ejpam-4914	371	2	.........	.........	PUNCT
ejpam-4914	372	1	.........	.........	PUNCT
ejpam-4914	372	2	.........	.........	PUNCT
ejpam-4914	373	1	.........	.........	PUNCT
ejpam-4914	373	2	.........	.........	PUNCT
ejpam-4914	374	1	.........	.........	PUNCT
ejpam-4914	374	2	......	......	PUNCT
ejpam-4914	374	3	....................................	....................................	PUNCT
ejpam-4914	375	1	..........	..........	PUNCT
ejpam-4914	375	2	.........	.........	PUNCT
ejpam-4914	376	1	.........	.........	PUNCT
ejpam-4914	376	2	.........	.........	PUNCT
ejpam-4914	377	1	.........	.........	PUNCT
ejpam-4914	377	2	.........	.........	PUNCT
ejpam-4914	378	1	.........	.........	PUNCT
ejpam-4914	378	2	.....	.....	PUNCT
ejpam-4914	378	3	....................................	....................................	PUNCT
ejpam-4914	379	1	....................................	....................................	PUNCT
ejpam-4914	379	2	....................	....................	PUNCT
ejpam-4914	380	1	...................	...................	PUNCT
ejpam-4914	380	2	...................	...................	PUNCT
ejpam-4914	380	3	............	............	PUNCT
ejpam-4914	380	4	....................................	....................................	PUNCT
ejpam-4914	380	5	..................	..................	PUNCT
ejpam-4914	381	1	.................	.................	PUNCT
ejpam-4914	381	2	.................	.................	PUNCT
ejpam-4914	382	1	.................	.................	PUNCT
ejpam-4914	382	2	.................	.................	PUNCT
ejpam-4914	382	3	...............	...............	PUNCT
ejpam-4914	383	1	....................................	....................................	PUNCT
ejpam-4914	384	1	....................................	....................................	PUNCT
ejpam-4914	385	1	•	•	NUM
ejpam-4914	385	2	•	•	NUM
ejpam-4914	385	3	•	•	NUM
ejpam-4914	385	4	•	•	NUM
ejpam-4914	385	5	•	•	NUM
ejpam-4914	385	6	•	•	NUM
ejpam-4914	385	7	•	•	NUM
ejpam-4914	385	8	•	•	NUM
ejpam-4914	385	9	•	•	NUM
ejpam-4914	385	10	•	•	NUM
ejpam-4914	385	11	•	•	NOUN
ejpam-4914	385	12	•	•	NOUN
ejpam-4914	385	13	••	••	NOUN
ejpam-4914	385	14	•	•	NUM
ejpam-4914	385	15	•	•	NUM
ejpam-4914	385	16	•	•	NUM
ejpam-4914	385	17	•	•	NUM
ejpam-4914	385	18	•	•	NUM
ejpam-4914	385	19	•	•	NOUN
ejpam-4914	385	20	•	•	NUM
ejpam-4914	385	21	·	·	PUNCT
ejpam-4914	385	22	·	·	PUNCT
ejpam-4914	385	23	·	·	PUNCT
ejpam-4914	385	24	·	·	PUNCT
ejpam-4914	385	25	·	·	PUNCT
ejpam-4914	385	26	·	·	PUNCT
ejpam-4914	385	27	·	·	PUNCT
ejpam-4914	385	28	·	·	PUNCT
ejpam-4914	385	29	·	·	PUNCT
ejpam-4914	386	1	x1	x1	PUNCT
ejpam-4914	387	1	x2	x2	NOUN
ejpam-4914	387	2	x3	x3	PROPN
ejpam-4914	387	3	x4	x4	PROPN
ejpam-4914	387	4	x5	x5	PROPN
ejpam-4914	387	5	xt−3	xt−3	PROPN
ejpam-4914	387	6	xt−2	xt−2	PROPN
ejpam-4914	387	7	xt−1	xt−1	PROPN
ejpam-4914	388	1	xt	xt	PROPN
ejpam-4914	389	1	y1	y1	INTJ
ejpam-4914	390	1	y2	y2	NOUN
ejpam-4914	390	2	yk+1	yk+1	NOUN
ejpam-4914	390	3	yk+2z1	yk+2z1	PROPN
ejpam-4914	390	4	z2	z2	PROPN
ejpam-4914	390	5	zk+1	zk+1	NUM
ejpam-4914	390	6	zk+2	zk+2	NUM
ejpam-4914	390	7	.......................................	.......................................	PUNCT
ejpam-4914	390	8	.......................................	.......................................	PUNCT
ejpam-4914	391	1	g2	g2	PROPN
ejpam-4914	391	2	figure	figure	VERB
ejpam-4914	391	3	4	4	NUM
ejpam-4914	391	4	:	:	PUNCT
ejpam-4914	391	5	examples	example	NOUN
ejpam-4914	391	6	of	of	ADP
ejpam-4914	391	7	graphs	graph	NOUN
ejpam-4914	391	8	g	g	NOUN
ejpam-4914	391	9	with	with	ADP
ejpam-4914	391	10	γhi(g	γhi(g	PROPN
ejpam-4914	391	11	)	)	PUNCT
ejpam-4914	392	1	=	=	SYM
ejpam-4914	392	2	a	a	DET
ejpam-4914	392	3	and	and	CCONJ
ejpam-4914	392	4	γ2h(g	γ2h(g	PROPN
ejpam-4914	392	5	)	)	PUNCT
ejpam-4914	393	1	=	=	SYM
ejpam-4914	393	2	b	b	X
ejpam-4914	394	1	[	[	X
ejpam-4914	394	2	x2	x2	PROPN
ejpam-4914	394	3	,	,	PUNCT
ejpam-4914	394	4	yj	yj	PROPN
ejpam-4914	394	5	,	,	PUNCT
ejpam-4914	394	6	zj	zj	PROPN
ejpam-4914	394	7	]	]	PUNCT
ejpam-4914	394	8	,	,	PUNCT
ejpam-4914	394	9	j	j	PROPN
ejpam-4914	394	10	=	=	SYM
ejpam-4914	394	11	1	1	NUM
ejpam-4914	394	12	,	,	PUNCT
ejpam-4914	394	13	2	2	NUM
ejpam-4914	394	14	,	,	PUNCT
ejpam-4914	394	15	.	.	PUNCT
ejpam-4914	394	16	.	.	PUNCT
ejpam-4914	395	1	.	.	PUNCT
ejpam-4914	396	1	,	,	PUNCT
ejpam-4914	396	2	k.	k.	PROPN
ejpam-4914	396	3	then	then	ADV
ejpam-4914	396	4	γhi(g	γhi(g	PROPN
ejpam-4914	396	5	)	)	PUNCT
ejpam-4914	397	1	=	=	SYM
ejpam-4914	397	2	5	5	NUM
ejpam-4914	397	3	=	=	SYM
ejpam-4914	397	4	a	a	NOUN
ejpam-4914	397	5	,	,	PUNCT
ejpam-4914	397	6	which	which	PRON
ejpam-4914	397	7	is	be	AUX
ejpam-4914	397	8	determined	determine	VERB
ejpam-4914	397	9	by	by	ADP
ejpam-4914	397	10	the	the	DET
ejpam-4914	397	11	γhi	γhi	ADV
ejpam-4914	397	12	-function	-function	NOUN
ejpam-4914	397	13	f	f	NOUN
ejpam-4914	397	14	=	=	SYM
ejpam-4914	397	15	(	(	PUNCT
ejpam-4914	397	16	v0	v0	PROPN
ejpam-4914	397	17	,	,	PUNCT
ejpam-4914	397	18	v1	v1	NOUN
ejpam-4914	397	19	,	,	PUNCT
ejpam-4914	397	20	v2	v2	PROPN
ejpam-4914	397	21	)	)	PUNCT
ejpam-4914	397	22	with	with	ADP
ejpam-4914	397	23	v1	v1	NOUN
ejpam-4914	397	24	=	=	SYM
ejpam-4914	397	25	{	{	PUNCT
ejpam-4914	397	26	x6	x6	PROPN
ejpam-4914	397	27	}	}	PUNCT
ejpam-4914	397	28	,	,	PUNCT
ejpam-4914	397	29	v2	v2	PROPN
ejpam-4914	397	30	=	=	SYM
ejpam-4914	397	31	{	{	PUNCT
ejpam-4914	397	32	x2	x2	PROPN
ejpam-4914	397	33	,	,	PUNCT
ejpam-4914	397	34	x3	x3	ADJ
ejpam-4914	397	35	}	}	PUNCT
ejpam-4914	397	36	and	and	CCONJ
ejpam-4914	397	37	v0	v0	PROPN
ejpam-4914	397	38	=	=	SYM
ejpam-4914	397	39	v	v	PROPN
ejpam-4914	397	40	(	(	PUNCT
ejpam-4914	397	41	g	g	NOUN
ejpam-4914	397	42	)	)	PUNCT
ejpam-4914	397	43	\	\	NOUN
ejpam-4914	397	44	{	{	PUNCT
ejpam-4914	397	45	x2	x2	PROPN
ejpam-4914	397	46	,	,	PUNCT
ejpam-4914	397	47	x3	x3	ADJ
ejpam-4914	397	48	,	,	PUNCT
ejpam-4914	397	49	x6	x6	PROPN
ejpam-4914	397	50	}	}	PUNCT
ejpam-4914	397	51	.	.	PUNCT
ejpam-4914	398	1	on	on	ADP
ejpam-4914	398	2	the	the	DET
ejpam-4914	398	3	other	other	ADJ
ejpam-4914	398	4	hand	hand	NOUN
ejpam-4914	398	5	,	,	PUNCT
ejpam-4914	398	6	s	s	PART
ejpam-4914	398	7	=	=	SYM
ejpam-4914	398	8	{	{	PUNCT
ejpam-4914	398	9	zj	zj	NOUN
ejpam-4914	398	10	:	:	PUNCT
ejpam-4914	399	1	j	j	PROPN
ejpam-4914	400	1	=	=	SYM
ejpam-4914	401	1	1	1	NUM
ejpam-4914	401	2	,	,	PUNCT
ejpam-4914	401	3	2	2	NUM
ejpam-4914	401	4	,	,	PUNCT
ejpam-4914	401	5	.	.	PUNCT
ejpam-4914	401	6	.	.	PUNCT
ejpam-4914	402	1	.	.	PUNCT
ejpam-4914	403	1	,	,	PUNCT
ejpam-4914	403	2	k	k	X
ejpam-4914	403	3	}	}	PUNCT
ejpam-4914	403	4	is	be	AUX
ejpam-4914	403	5	always	always	ADV
ejpam-4914	403	6	contained	contain	VERB
ejpam-4914	403	7	in	in	ADP
ejpam-4914	403	8	a	a	DET
ejpam-4914	403	9	2	2	NUM
ejpam-4914	403	10	-	-	PUNCT
ejpam-4914	403	11	hop	hop	NOUN
ejpam-4914	403	12	dominating	dominating	NOUN
ejpam-4914	403	13	set	set	NOUN
ejpam-4914	403	14	of	of	ADP
ejpam-4914	403	15	g	g	PROPN
ejpam-4914	403	16	so	so	SCONJ
ejpam-4914	403	17	that	that	PRON
ejpam-4914	403	18	s	s	VERB
ejpam-4914	403	19	∪	∪	X
ejpam-4914	403	20	(	(	PUNCT
ejpam-4914	403	21	v	v	NOUN
ejpam-4914	403	22	(	(	PUNCT
ejpam-4914	403	23	p6	p6	PROPN
ejpam-4914	403	24	)	)	PUNCT
ejpam-4914	403	25	\	\	NOUN
ejpam-4914	404	1	{	{	PUNCT
ejpam-4914	404	2	x2	x2	NOUN
ejpam-4914	404	3	}	}	PUNCT
ejpam-4914	404	4	)	)	PUNCT
ejpam-4914	404	5	is	be	AUX
ejpam-4914	404	6	a	a	DET
ejpam-4914	404	7	γ2h	γ2h	NOUN
ejpam-4914	404	8	-	-	PUNCT
ejpam-4914	404	9	set	set	NOUN
ejpam-4914	404	10	of	of	ADP
ejpam-4914	404	11	g.	g.	PROPN
ejpam-4914	404	12	thus	thus	ADV
ejpam-4914	404	13	,	,	PUNCT
ejpam-4914	404	14	γ2h(g	γ2h(g	PROPN
ejpam-4914	404	15	)	)	PUNCT
ejpam-4914	404	16	=	=	SYM
ejpam-4914	405	1	5	5	NUM
ejpam-4914	405	2	+	+	CCONJ
ejpam-4914	405	3	k	k	PROPN
ejpam-4914	405	4	=	=	SYM
ejpam-4914	405	5	b.	b.	PROPN
ejpam-4914	405	6	now	now	ADV
ejpam-4914	405	7	,	,	PUNCT
ejpam-4914	405	8	suppose	suppose	VERB
ejpam-4914	405	9	that	that	SCONJ
ejpam-4914	405	10	n	n	PROPN
ejpam-4914	405	11	≥	≥	NUM
ejpam-4914	405	12	2	2	NUM
ejpam-4914	405	13	.	.	PUNCT
ejpam-4914	406	1	obtain	obtain	VERB
ejpam-4914	406	2	g	g	NOUN
ejpam-4914	406	3	as	as	ADP
ejpam-4914	406	4	the	the	DET
ejpam-4914	406	5	graph	graph	NOUN
ejpam-4914	406	6	g2	g2	PROPN
ejpam-4914	406	7	in	in	ADP
ejpam-4914	406	8	figure	figure	NOUN
ejpam-4914	406	9	4	4	NUM
ejpam-4914	406	10	from	from	ADP
ejpam-4914	406	11	pt	pt	NOUN
ejpam-4914	406	12	by	by	ADP
ejpam-4914	406	13	adding	add	VERB
ejpam-4914	406	14	k	k	PROPN
ejpam-4914	406	15	distinct	distinct	ADJ
ejpam-4914	406	16	paths	path	NOUN
ejpam-4914	406	17	[	[	X
ejpam-4914	406	18	x3	x3	ADJ
ejpam-4914	406	19	,	,	PUNCT
ejpam-4914	406	20	yj	yj	PROPN
ejpam-4914	406	21	,	,	PUNCT
ejpam-4914	406	22	zj	zj	PROPN
ejpam-4914	406	23	]	]	PUNCT
ejpam-4914	406	24	,	,	PUNCT
ejpam-4914	406	25	j	j	PROPN
ejpam-4914	406	26	=	=	SYM
ejpam-4914	406	27	1	1	NUM
ejpam-4914	406	28	,	,	PUNCT
ejpam-4914	406	29	2	2	NUM
ejpam-4914	406	30	,	,	PUNCT
ejpam-4914	406	31	.	.	PUNCT
ejpam-4914	406	32	.	.	PUNCT
ejpam-4914	407	1	.	.	PUNCT
ejpam-4914	408	1	,	,	PUNCT
ejpam-4914	409	1	k	k	PROPN
ejpam-4914	409	2	+	+	PROPN
ejpam-4914	409	3	2	2	X
ejpam-4914	409	4	.	.	PUNCT
ejpam-4914	409	5	define	define	VERB
ejpam-4914	409	6	v2	v2	PROPN
ejpam-4914	409	7	=	=	SYM
ejpam-4914	409	8	{	{	PUNCT
ejpam-4914	409	9	x2	x2	NOUN
ejpam-4914	409	10	}	}	PUNCT
ejpam-4914	409	11	,	,	PUNCT
ejpam-4914	409	12	v1	v1	NOUN
ejpam-4914	409	13	=	=	SYM
ejpam-4914	409	14	{	{	PUNCT
ejpam-4914	409	15	x1	x1	PROPN
ejpam-4914	409	16	,	,	PUNCT
ejpam-4914	409	17	x3	x3	ADJ
ejpam-4914	409	18	,	,	PUNCT
ejpam-4914	409	19	x4	x4	PROPN
ejpam-4914	409	20	,	,	PUNCT
ejpam-4914	409	21	x7	x7	NOUN
ejpam-4914	409	22	,	,	PUNCT
ejpam-4914	409	23	x8	x8	PROPN
ejpam-4914	409	24	,	,	PUNCT
ejpam-4914	409	25	.	.	PUNCT
ejpam-4914	409	26	.	.	PUNCT
ejpam-4914	410	1	.	.	PUNCT
ejpam-4914	411	1	,	,	PUNCT
ejpam-4914	411	2	xt−1	xt−1	PROPN
ejpam-4914	411	3	,	,	PUNCT
ejpam-4914	411	4	xt	xt	ADP
ejpam-4914	411	5	}	}	PUNCT
ejpam-4914	411	6	and	and	CCONJ
ejpam-4914	411	7	v0	v0	PROPN
ejpam-4914	411	8	=	=	SYM
ejpam-4914	411	9	v	v	PROPN
ejpam-4914	411	10	(	(	PUNCT
ejpam-4914	411	11	g	g	NOUN
ejpam-4914	411	12	)	)	PUNCT
ejpam-4914	411	13	\	\	PUNCT
ejpam-4914	412	1	(	(	PUNCT
ejpam-4914	412	2	v1	v1	VERB
ejpam-4914	412	3	∪	∪	NOUN
ejpam-4914	412	4	v2	v2	NOUN
ejpam-4914	412	5	)	)	PUNCT
ejpam-4914	412	6	.	.	PUNCT
ejpam-4914	413	1	then	then	ADV
ejpam-4914	413	2	f	f	PROPN
ejpam-4914	413	3	=	=	SYM
ejpam-4914	413	4	(	(	PUNCT
ejpam-4914	413	5	v0	v0	PROPN
ejpam-4914	413	6	,	,	PUNCT
ejpam-4914	413	7	v1	v1	NOUN
ejpam-4914	413	8	,	,	PUNCT
ejpam-4914	413	9	v2	v2	PROPN
ejpam-4914	413	10	)	)	PUNCT
ejpam-4914	413	11	is	be	AUX
ejpam-4914	413	12	a	a	DET
ejpam-4914	413	13	γhi	γhi	ADJ
ejpam-4914	413	14	-function	-function	NOUN
ejpam-4914	413	15	of	of	ADP
ejpam-4914	413	16	g	g	NOUN
ejpam-4914	413	17	,	,	PUNCT
ejpam-4914	413	18	implying	imply	VERB
ejpam-4914	413	19	that	that	SCONJ
ejpam-4914	413	20	γhi(g	γhi(g	PROPN
ejpam-4914	413	21	)	)	PUNCT
ejpam-4914	414	1	=	=	SYM
ejpam-4914	414	2	γhi(pt	γhi(pt	NOUN
ejpam-4914	414	3	)	)	PUNCT
ejpam-4914	414	4	=	=	SYM
ejpam-4914	414	5	2n	2n	NUM
ejpam-4914	415	1	+	+	CCONJ
ejpam-4914	415	2	3	3	NUM
ejpam-4914	415	3	=	=	SYM
ejpam-4914	415	4	a.	a.	NOUN
ejpam-4914	415	5	on	on	ADP
ejpam-4914	415	6	the	the	DET
ejpam-4914	415	7	other	other	ADJ
ejpam-4914	415	8	hand	hand	NOUN
ejpam-4914	415	9	,	,	PUNCT
ejpam-4914	415	10	if	if	SCONJ
ejpam-4914	415	11	s	s	VERB
ejpam-4914	415	12	=	=	SYM
ejpam-4914	415	13	{	{	PUNCT
ejpam-4914	415	14	zj	zj	NOUN
ejpam-4914	415	15	:	:	PUNCT
ejpam-4914	415	16	j	j	PROPN
ejpam-4914	416	1	=	=	SYM
ejpam-4914	416	2	1	1	NUM
ejpam-4914	416	3	,	,	PUNCT
ejpam-4914	416	4	2	2	NUM
ejpam-4914	416	5	,	,	PUNCT
ejpam-4914	416	6	.	.	PUNCT
ejpam-4914	416	7	.	.	PUNCT
ejpam-4914	417	1	.	.	PUNCT
ejpam-4914	418	1	,	,	PUNCT
ejpam-4914	418	2	k	k	X
ejpam-4914	418	3	}	}	PUNCT
ejpam-4914	418	4	,	,	PUNCT
ejpam-4914	418	5	then	then	ADV
ejpam-4914	418	6	s	s	VERB
ejpam-4914	418	7	∪	∪	X
ejpam-4914	418	8	{	{	PUNCT
ejpam-4914	418	9	x1	x1	PROPN
ejpam-4914	418	10	,	,	PUNCT
ejpam-4914	418	11	x3	x3	ADJ
ejpam-4914	418	12	,	,	PUNCT
ejpam-4914	418	13	x4	x4	PROPN
ejpam-4914	418	14	,	,	PUNCT
ejpam-4914	418	15	x7	x7	NOUN
ejpam-4914	418	16	,	,	PUNCT
ejpam-4914	418	17	x8	x8	PROPN
ejpam-4914	418	18	,	,	PUNCT
ejpam-4914	418	19	x11	x11	PROPN
ejpam-4914	418	20	,	,	PUNCT
ejpam-4914	418	21	x12	x12	NUM
ejpam-4914	418	22	,	,	PUNCT
ejpam-4914	418	23	.	.	PUNCT
ejpam-4914	418	24	.	.	PUNCT
ejpam-4914	418	25	.	.	PUNCT
ejpam-4914	419	1	,	,	PUNCT
ejpam-4914	419	2	xt−1	xt−1	PROPN
ejpam-4914	419	3	,	,	PUNCT
ejpam-4914	419	4	xt	xt	ADP
ejpam-4914	419	5	}	}	PUNCT
ejpam-4914	419	6	is	be	AUX
ejpam-4914	419	7	a	a	DET
ejpam-4914	419	8	γ2h	γ2h	NOUN
ejpam-4914	419	9	-	-	PUNCT
ejpam-4914	419	10	set	set	NOUN
ejpam-4914	419	11	of	of	ADP
ejpam-4914	419	12	g.	g.	PROPN
ejpam-4914	419	13	thus	thus	ADV
ejpam-4914	419	14	,	,	PUNCT
ejpam-4914	419	15	γ2h(g	γ2h(g	PROPN
ejpam-4914	419	16	)	)	PUNCT
ejpam-4914	419	17	=	=	SYM
ejpam-4914	419	18	(	(	PUNCT
ejpam-4914	419	19	2n	2n	NUM
ejpam-4914	419	20	+	+	CCONJ
ejpam-4914	419	21	1	1	NUM
ejpam-4914	419	22	)	)	PUNCT
ejpam-4914	420	1	+	+	CCONJ
ejpam-4914	420	2	(	(	PUNCT
ejpam-4914	420	3	k	k	X
ejpam-4914	420	4	+	+	PROPN
ejpam-4914	420	5	2	2	X
ejpam-4914	420	6	)	)	PUNCT
ejpam-4914	420	7	=	=	PRON
ejpam-4914	420	8	a+	a+	PUNCT
ejpam-4914	420	9	k	k	PROPN
ejpam-4914	420	10	=	=	PROPN
ejpam-4914	420	11	b.	b.	PROPN
ejpam-4914	421	1	■	■	PUNCT
ejpam-4914	421	2	2.2	2.2	NUM
ejpam-4914	421	3	.	.	PUNCT
ejpam-4914	422	1	pndi	pndi	NOUN
ejpam-4914	422	2	-	-	PUNCT
ejpam-4914	422	3	functions	function	NOUN
ejpam-4914	422	4	a	a	DET
ejpam-4914	422	5	function	function	NOUN
ejpam-4914	422	6	f	f	PROPN
ejpam-4914	422	7	=	=	SYM
ejpam-4914	422	8	(	(	PUNCT
ejpam-4914	422	9	v0	v0	PROPN
ejpam-4914	422	10	,	,	PUNCT
ejpam-4914	422	11	v1	v1	NOUN
ejpam-4914	422	12	,	,	PUNCT
ejpam-4914	422	13	v2	v2	PROPN
ejpam-4914	422	14	)	)	PUNCT
ejpam-4914	422	15	on	on	ADP
ejpam-4914	422	16	v	v	ADP
ejpam-4914	422	17	(	(	PUNCT
ejpam-4914	422	18	g	g	NOUN
ejpam-4914	422	19	)	)	PUNCT
ejpam-4914	422	20	is	be	AUX
ejpam-4914	422	21	a	a	DET
ejpam-4914	422	22	pndi	pndi	ADJ
ejpam-4914	422	23	-	-	PUNCT
ejpam-4914	422	24	function	function	NOUN
ejpam-4914	422	25	of	of	ADP
ejpam-4914	422	26	g	g	PROPN
ejpam-4914	422	27	if	if	SCONJ
ejpam-4914	422	28	for	for	ADP
ejpam-4914	422	29	each	each	DET
ejpam-4914	422	30	v	v	NOUN
ejpam-4914	422	31	∈	∈	PROPN
ejpam-4914	422	32	v0	v0	NOUN
ejpam-4914	422	33	one	one	NUM
ejpam-4914	422	34	of	of	ADP
ejpam-4914	422	35	the	the	DET
ejpam-4914	422	36	following	follow	VERB
ejpam-4914	422	37	holds	hold	VERB
ejpam-4914	422	38	:	:	PUNCT
ejpam-4914	422	39	(	(	PUNCT
ejpam-4914	422	40	i	i	NOUN
ejpam-4914	422	41	)	)	PUNCT
ejpam-4914	422	42	there	there	PRON
ejpam-4914	422	43	exists	exist	VERB
ejpam-4914	422	44	u	u	PROPN
ejpam-4914	422	45	∈	∈	PROPN
ejpam-4914	422	46	v2	v2	NOUN
ejpam-4914	422	47	for	for	ADP
ejpam-4914	422	48	which	which	PRON
ejpam-4914	422	49	v	v	NOUN
ejpam-4914	422	50	/∈	/∈	PUNCT
ejpam-4914	422	51	ng(u	ng(u	NOUN
ejpam-4914	422	52	)	)	PUNCT
ejpam-4914	422	53	;	;	PUNCT
ejpam-4914	422	54	(	(	PUNCT
ejpam-4914	422	55	ii	ii	NOUN
ejpam-4914	422	56	)	)	PUNCT
ejpam-4914	422	57	there	there	PRON
ejpam-4914	422	58	exist	exist	VERB
ejpam-4914	422	59	vertices	vertex	NOUN
ejpam-4914	422	60	u	u	NOUN
ejpam-4914	422	61	and	and	CCONJ
ejpam-4914	422	62	w	w	NOUN
ejpam-4914	422	63	in	in	ADP
ejpam-4914	422	64	v1	v1	NOUN
ejpam-4914	422	65	for	for	ADP
ejpam-4914	422	66	which	which	PRON
ejpam-4914	422	67	v	v	NOUN
ejpam-4914	422	68	/∈	/∈	PUNCT
ejpam-4914	422	69	ng(u	ng(u	NOUN
ejpam-4914	422	70	)	)	PUNCT
ejpam-4914	422	71	∪ng(w	∪ng(w	NOUN
ejpam-4914	422	72	)	)	PUNCT
ejpam-4914	422	73	.	.	PUNCT
ejpam-4914	423	1	the	the	DET
ejpam-4914	423	2	minimum	minimum	ADJ
ejpam-4914	423	3	weight	weight	NOUN
ejpam-4914	423	4	of	of	ADP
ejpam-4914	423	5	an	an	DET
ejpam-4914	423	6	pndi	pndi	ADJ
ejpam-4914	423	7	-function	-function	NOUN
ejpam-4914	423	8	of	of	ADP
ejpam-4914	423	9	g	g	NOUN
ejpam-4914	423	10	is	be	AUX
ejpam-4914	423	11	the	the	DET
ejpam-4914	423	12	pndi	pndi	ADJ
ejpam-4914	423	13	number	number	NOUN
ejpam-4914	423	14	of	of	ADP
ejpam-4914	423	15	g	g	NOUN
ejpam-4914	423	16	,	,	PUNCT
ejpam-4914	423	17	denoted	denote	VERB
ejpam-4914	423	18	by	by	ADP
ejpam-4914	423	19	pndi(g	pndi(g	NOUN
ejpam-4914	423	20	)	)	PUNCT
ejpam-4914	423	21	.	.	PUNCT
ejpam-4914	424	1	any	any	DET
ejpam-4914	424	2	pndi	pndi	ADJ
ejpam-4914	424	3	-function	-function	NOUN
ejpam-4914	424	4	of	of	ADP
ejpam-4914	424	5	g	g	NOUN
ejpam-4914	424	6	with	with	ADP
ejpam-4914	424	7	weight	weight	NOUN
ejpam-4914	424	8	equal	equal	ADJ
ejpam-4914	424	9	to	to	ADP
ejpam-4914	424	10	pndi(g	pndi(g	NUM
ejpam-4914	424	11	)	)	PUNCT
ejpam-4914	424	12	is	be	AUX
ejpam-4914	424	13	a	a	DET
ejpam-4914	424	14	pndi	pndi	ADJ
ejpam-4914	424	15	-	-	PUNCT
ejpam-4914	424	16	function	function	NOUN
ejpam-4914	424	17	.	.	PUNCT
ejpam-4914	425	1	s.r	s.r	PROPN
ejpam-4914	425	2	.	.	PROPN
ejpam-4914	425	3	jr	jr	PROPN
ejpam-4914	425	4	.	.	PROPN
ejpam-4914	425	5	canoy	canoy	PROPN
ejpam-4914	425	6	,	,	PUNCT
ejpam-4914	425	7	f.p	f.p	PROPN
ejpam-4914	425	8	.	.	PROPN
ejpam-4914	425	9	jamil	jamil	PROPN
ejpam-4914	425	10	and	and	CCONJ
ejpam-4914	425	11	s.m	s.m	PROPN
ejpam-4914	425	12	.	.	PROPN
ejpam-4914	425	13	menchavez	menchavez	PROPN
ejpam-4914	425	14	/	/	PUNCT
ejpam-4914	425	15	eur	eur	PROPN
ejpam-4914	425	16	.	.	PUNCT
ejpam-4914	426	1	j.	j.	PROPN
ejpam-4914	426	2	pure	pure	PROPN
ejpam-4914	426	3	appl	appl	PROPN
ejpam-4914	426	4	.	.	PROPN
ejpam-4914	426	5	math	math	PROPN
ejpam-4914	426	6	,	,	PUNCT
ejpam-4914	426	7	16	16	NUM
ejpam-4914	426	8	(	(	PUNCT
ejpam-4914	426	9	4	4	NUM
ejpam-4914	426	10	)	)	PUNCT
ejpam-4914	426	11	(	(	PUNCT
ejpam-4914	426	12	2023	2023	NUM
ejpam-4914	426	13	)	)	PUNCT
ejpam-4914	426	14	,	,	PUNCT
ejpam-4914	426	15	2431	2431	NUM
ejpam-4914	426	16	-	-	SYM
ejpam-4914	426	17	2449	2449	NUM
ejpam-4914	426	18	2440	2440	NUM
ejpam-4914	426	19	example	example	NOUN
ejpam-4914	426	20	2.6	2.6	NUM
ejpam-4914	426	21	.	.	PUNCT
ejpam-4914	427	1	(	(	PUNCT
ejpam-4914	427	2	1	1	X
ejpam-4914	427	3	)	)	PUNCT
ejpam-4914	427	4	pndi(pn	pndi(pn	NOUN
ejpam-4914	427	5	)	)	PUNCT
ejpam-4914	427	6	=	=	PUNCT
ejpam-4914	428	1			NOUN
ejpam-4914	428	2	1	1	NUM
ejpam-4914	428	3	,	,	PUNCT
ejpam-4914	428	4	if	if	SCONJ
ejpam-4914	428	5	n	n	NOUN
ejpam-4914	428	6	=	=	SYM
ejpam-4914	428	7	1	1	NUM
ejpam-4914	428	8	;	;	PUNCT
ejpam-4914	428	9	2	2	NUM
ejpam-4914	428	10	,	,	PUNCT
ejpam-4914	428	11	if	if	SCONJ
ejpam-4914	428	12	n	n	NOUN
ejpam-4914	428	13	=	=	SYM
ejpam-4914	428	14	2	2	NUM
ejpam-4914	428	15	;	;	PUNCT
ejpam-4914	428	16	3	3	NUM
ejpam-4914	428	17	,	,	PUNCT
ejpam-4914	428	18	if	if	SCONJ
ejpam-4914	428	19	n	n	PRON
ejpam-4914	428	20	≥	≥	NOUN
ejpam-4914	428	21	3	3	NUM
ejpam-4914	428	22	.	.	PUNCT
ejpam-4914	428	23	(	(	PUNCT
ejpam-4914	428	24	2	2	X
ejpam-4914	428	25	)	)	PUNCT
ejpam-4914	428	26	pndi(cn	pndi(cn	NOUN
ejpam-4914	428	27	)	)	PUNCT
ejpam-4914	428	28	=	=	PUNCT
ejpam-4914	428	29	{	{	PUNCT
ejpam-4914	428	30	4	4	NUM
ejpam-4914	428	31	,	,	PUNCT
ejpam-4914	428	32	if	if	SCONJ
ejpam-4914	428	33	n	n	NOUN
ejpam-4914	428	34	=	=	SYM
ejpam-4914	428	35	4	4	NUM
ejpam-4914	428	36	;	;	PUNCT
ejpam-4914	428	37	3	3	NUM
ejpam-4914	428	38	,	,	PUNCT
ejpam-4914	428	39	otherwise	otherwise	ADV
ejpam-4914	428	40	.	.	PUNCT
ejpam-4914	429	1	(	(	PUNCT
ejpam-4914	429	2	3	3	X
ejpam-4914	429	3	)	)	PUNCT
ejpam-4914	429	4	pndi(kp	pndi(kp	NOUN
ejpam-4914	429	5	)	)	PUNCT
ejpam-4914	429	6	=	=	PUNCT
ejpam-4914	430	1	p	p	NOUN
ejpam-4914	430	2	for	for	ADP
ejpam-4914	430	3	p	p	PRON
ejpam-4914	430	4	≥	≥	NUM
ejpam-4914	430	5	1	1	NUM
ejpam-4914	430	6	and	and	CCONJ
ejpam-4914	430	7	pndi(km	pndi(km	NOUN
ejpam-4914	430	8	,	,	PUNCT
ejpam-4914	430	9	n	n	CCONJ
ejpam-4914	430	10	)	)	PUNCT
ejpam-4914	430	11	=	=	SYM
ejpam-4914	430	12	4	4	NUM
ejpam-4914	430	13	for	for	ADP
ejpam-4914	430	14	m	m	PROPN
ejpam-4914	430	15	,	,	PUNCT
ejpam-4914	430	16	n	n	PRON
ejpam-4914	430	17	≥	≥	NOUN
ejpam-4914	430	18	2	2	NUM
ejpam-4914	430	19	.	.	PUNCT
ejpam-4914	431	1	if	if	SCONJ
ejpam-4914	431	2	f	f	PROPN
ejpam-4914	431	3	=	=	SYM
ejpam-4914	431	4	(	(	PUNCT
ejpam-4914	431	5	v0	v0	PROPN
ejpam-4914	431	6	,	,	PUNCT
ejpam-4914	431	7	v1	v1	NOUN
ejpam-4914	431	8	,	,	PUNCT
ejpam-4914	431	9	v2	v2	PROPN
ejpam-4914	431	10	)	)	PUNCT
ejpam-4914	431	11	is	be	AUX
ejpam-4914	431	12	a	a	DET
ejpam-4914	431	13	pndi	pndi	ADJ
ejpam-4914	431	14	-function	-function	NOUN
ejpam-4914	431	15	of	of	ADP
ejpam-4914	431	16	g	g	NOUN
ejpam-4914	431	17	,	,	PUNCT
ejpam-4914	431	18	then	then	ADV
ejpam-4914	431	19	v1	v1	VERB
ejpam-4914	431	20	∪	∪	ADJ
ejpam-4914	431	21	v2	v2	PROPN
ejpam-4914	431	22	is	be	AUX
ejpam-4914	431	23	a	a	DET
ejpam-4914	431	24	pnd	pnd	NOUN
ejpam-4914	431	25	-	-	PUNCT
ejpam-4914	431	26	set	set	NOUN
ejpam-4914	431	27	of	of	ADP
ejpam-4914	431	28	g.	g.	PROPN
ejpam-4914	431	29	thus	thus	ADV
ejpam-4914	431	30	,	,	PUNCT
ejpam-4914	431	31	pnd(g	pnd(g	ADP
ejpam-4914	431	32	)	)	PUNCT
ejpam-4914	431	33	≤	≤	NOUN
ejpam-4914	431	34	|v1|	|v1|	NOUN
ejpam-4914	431	35	+	+	CCONJ
ejpam-4914	431	36	|v2|	|v2|	ADV
ejpam-4914	431	37	≤	≤	NUM
ejpam-4914	431	38	ωg(f	ωg(f	NUM
ejpam-4914	431	39	)	)	PUNCT
ejpam-4914	431	40	for	for	ADP
ejpam-4914	431	41	all	all	DET
ejpam-4914	431	42	pndi	pndi	NOUN
ejpam-4914	431	43	-functions	-function	NOUN
ejpam-4914	431	44	f	f	X
ejpam-4914	431	45	=	=	SYM
ejpam-4914	431	46	(	(	PUNCT
ejpam-4914	431	47	v0	v0	PROPN
ejpam-4914	431	48	,	,	PUNCT
ejpam-4914	431	49	v1	v1	NOUN
ejpam-4914	431	50	,	,	PUNCT
ejpam-4914	431	51	v2	v2	PROPN
ejpam-4914	431	52	)	)	PUNCT
ejpam-4914	431	53	of	of	ADP
ejpam-4914	431	54	g.	g.	PROPN
ejpam-4914	431	55	on	on	ADP
ejpam-4914	431	56	the	the	DET
ejpam-4914	431	57	other	other	ADJ
ejpam-4914	431	58	hand	hand	NOUN
ejpam-4914	431	59	,	,	PUNCT
ejpam-4914	431	60	if	if	SCONJ
ejpam-4914	431	61	s	s	VERB
ejpam-4914	431	62	⊆	⊆	NUM
ejpam-4914	431	63	v	v	NOUN
ejpam-4914	431	64	(	(	PUNCT
ejpam-4914	431	65	g	g	NOUN
ejpam-4914	431	66	)	)	PUNCT
ejpam-4914	431	67	is	be	AUX
ejpam-4914	431	68	a	a	DET
ejpam-4914	431	69	pnd	pnd	NOUN
ejpam-4914	431	70	-	-	PUNCT
ejpam-4914	431	71	set	set	NOUN
ejpam-4914	431	72	of	of	ADP
ejpam-4914	431	73	g	g	NOUN
ejpam-4914	431	74	,	,	PUNCT
ejpam-4914	431	75	then	then	ADV
ejpam-4914	431	76	f	f	PROPN
ejpam-4914	431	77	=	=	PUNCT
ejpam-4914	431	78	(	(	PUNCT
ejpam-4914	431	79	v	v	NOUN
ejpam-4914	431	80	(	(	PUNCT
ejpam-4914	431	81	g	g	NOUN
ejpam-4914	431	82	)	)	PUNCT
ejpam-4914	431	83	\	\	NOUN
ejpam-4914	431	84	s,∅	s,∅	NOUN
ejpam-4914	431	85	,	,	PUNCT
ejpam-4914	431	86	s	s	PART
ejpam-4914	431	87	)	)	PUNCT
ejpam-4914	431	88	is	be	AUX
ejpam-4914	431	89	a	a	DET
ejpam-4914	431	90	pndi	pndi	ADJ
ejpam-4914	431	91	function	function	NOUN
ejpam-4914	431	92	of	of	ADP
ejpam-4914	431	93	g.	g.	PROPN
ejpam-4914	431	94	also	also	ADV
ejpam-4914	431	95	,	,	PUNCT
ejpam-4914	431	96	every	every	DET
ejpam-4914	431	97	hop	hop	NOUN
ejpam-4914	431	98	italian	italian	ADJ
ejpam-4914	431	99	dominating	dominating	NOUN
ejpam-4914	431	100	function	function	NOUN
ejpam-4914	431	101	is	be	AUX
ejpam-4914	431	102	a	a	DET
ejpam-4914	431	103	pndi	pndi	ADJ
ejpam-4914	431	104	-function	-function	NOUN
ejpam-4914	431	105	.	.	PUNCT
ejpam-4914	432	1	thus	thus	ADV
ejpam-4914	432	2	,	,	PUNCT
ejpam-4914	432	3	pnd(g	pnd(g	ADP
ejpam-4914	432	4	)	)	PUNCT
ejpam-4914	432	5	≤	≤	NUM
ejpam-4914	432	6	pndi(g	pndi(g	NOUN
ejpam-4914	432	7	)	)	PUNCT
ejpam-4914	432	8	≤	≤	NUM
ejpam-4914	432	9	min{2	min{2	NUM
ejpam-4914	432	10	pnd(g	pnd(g	NOUN
ejpam-4914	432	11	)	)	PUNCT
ejpam-4914	432	12	,	,	PUNCT
ejpam-4914	432	13	γhi(g	γhi(g	PROPN
ejpam-4914	432	14	)	)	PUNCT
ejpam-4914	432	15	}	}	PUNCT
ejpam-4914	432	16	.	.	PUNCT
ejpam-4914	433	1	observation	observation	NOUN
ejpam-4914	433	2	2.7	2.7	NUM
ejpam-4914	433	3	.	.	PUNCT
ejpam-4914	434	1	let	let	VERB
ejpam-4914	434	2	g	g	NOUN
ejpam-4914	434	3	be	be	AUX
ejpam-4914	434	4	any	any	DET
ejpam-4914	434	5	graph	graph	NOUN
ejpam-4914	434	6	.	.	PUNCT
ejpam-4914	435	1	then	then	ADV
ejpam-4914	435	2	(	(	PUNCT
ejpam-4914	435	3	i	i	NOUN
ejpam-4914	435	4	)	)	PUNCT
ejpam-4914	435	5	pndi(g	pndi(g	NOUN
ejpam-4914	435	6	)	)	PUNCT
ejpam-4914	435	7	=	=	SYM
ejpam-4914	436	1	1	1	NUM
ejpam-4914	436	2	if	if	SCONJ
ejpam-4914	436	3	and	and	CCONJ
ejpam-4914	436	4	only	only	ADV
ejpam-4914	436	5	if	if	SCONJ
ejpam-4914	436	6	g	g	PROPN
ejpam-4914	436	7	=	=	SYM
ejpam-4914	436	8	k1	k1	PROPN
ejpam-4914	436	9	;	;	PUNCT
ejpam-4914	436	10	(	(	PUNCT
ejpam-4914	436	11	ii	ii	NOUN
ejpam-4914	436	12	)	)	PUNCT
ejpam-4914	436	13	pndi(g	pndi(g	NOUN
ejpam-4914	436	14	)	)	PUNCT
ejpam-4914	436	15	=	=	SYM
ejpam-4914	436	16	2	2	NUM
ejpam-4914	436	17	if	if	SCONJ
ejpam-4914	436	18	and	and	CCONJ
ejpam-4914	436	19	only	only	ADV
ejpam-4914	436	20	if	if	SCONJ
ejpam-4914	436	21	either	either	CCONJ
ejpam-4914	436	22	g	g	PROPN
ejpam-4914	436	23	=	=	SYM
ejpam-4914	436	24	k2	k2	PROPN
ejpam-4914	436	25	or	or	CCONJ
ejpam-4914	436	26	g	g	PROPN
ejpam-4914	436	27	is	be	AUX
ejpam-4914	436	28	a	a	DET
ejpam-4914	436	29	nontrivial	nontrivial	ADJ
ejpam-4914	436	30	graph	graph	NOUN
ejpam-4914	436	31	with	with	ADP
ejpam-4914	436	32	an	an	DET
ejpam-4914	436	33	isolated	isolated	ADJ
ejpam-4914	436	34	vertex	vertex	NOUN
ejpam-4914	436	35	;	;	PUNCT
ejpam-4914	436	36	(	(	PUNCT
ejpam-4914	436	37	iii	iii	X
ejpam-4914	436	38	)	)	PUNCT
ejpam-4914	436	39	pndi(g	pndi(g	NOUN
ejpam-4914	436	40	)	)	PUNCT
ejpam-4914	436	41	=	=	SYM
ejpam-4914	436	42	3	3	NUM
ejpam-4914	436	43	if	if	SCONJ
ejpam-4914	436	44	and	and	CCONJ
ejpam-4914	436	45	only	only	ADV
ejpam-4914	436	46	if	if	SCONJ
ejpam-4914	436	47	one	one	NUM
ejpam-4914	436	48	of	of	ADP
ejpam-4914	436	49	the	the	DET
ejpam-4914	436	50	following	following	NOUN
ejpam-4914	436	51	holds	hold	VERB
ejpam-4914	436	52	:	:	PUNCT
ejpam-4914	436	53	(	(	PUNCT
ejpam-4914	436	54	a	a	X
ejpam-4914	436	55	)	)	PUNCT
ejpam-4914	436	56	g	g	NOUN
ejpam-4914	436	57	has	have	VERB
ejpam-4914	436	58	an	an	DET
ejpam-4914	436	59	endvertex	endvertex	NOUN
ejpam-4914	436	60	;	;	PUNCT
ejpam-4914	436	61	(	(	PUNCT
ejpam-4914	436	62	b	b	X
ejpam-4914	436	63	)	)	PUNCT
ejpam-4914	436	64	g	g	NOUN
ejpam-4914	436	65	has	have	VERB
ejpam-4914	436	66	a	a	DET
ejpam-4914	436	67	set	set	NOUN
ejpam-4914	436	68	of	of	ADP
ejpam-4914	436	69	vertices	vertex	NOUN
ejpam-4914	436	70	s	s	PART
ejpam-4914	436	71	=	=	PUNCT
ejpam-4914	436	72	{	{	PUNCT
ejpam-4914	436	73	x	x	PROPN
ejpam-4914	436	74	,	,	PUNCT
ejpam-4914	436	75	y	y	PROPN
ejpam-4914	436	76	,	,	PUNCT
ejpam-4914	436	77	z	z	NOUN
ejpam-4914	436	78	}	}	PUNCT
ejpam-4914	436	79	for	for	ADP
ejpam-4914	436	80	which	which	PRON
ejpam-4914	436	81	every	every	DET
ejpam-4914	436	82	v	v	NOUN
ejpam-4914	436	83	∈	∈	NOUN
ejpam-4914	436	84	v	v	NOUN
ejpam-4914	436	85	(	(	PUNCT
ejpam-4914	436	86	g	g	NOUN
ejpam-4914	436	87	)	)	PUNCT
ejpam-4914	436	88	\	\	PROPN
ejpam-4914	437	1	s	s	PART
ejpam-4914	437	2	is	be	AUX
ejpam-4914	437	3	adjacent	adjacent	ADJ
ejpam-4914	437	4	to	to	ADP
ejpam-4914	437	5	at	at	ADP
ejpam-4914	437	6	most	most	ADV
ejpam-4914	437	7	one	one	NUM
ejpam-4914	437	8	vertex	vertex	NOUN
ejpam-4914	437	9	in	in	ADP
ejpam-4914	437	10	s.	s.	PROPN
ejpam-4914	437	11	lemma	lemma	PROPN
ejpam-4914	437	12	2.8	2.8	NUM
ejpam-4914	437	13	.	.	PUNCT
ejpam-4914	438	1	let	let	VERB
ejpam-4914	438	2	g	g	PRON
ejpam-4914	438	3	be	be	AUX
ejpam-4914	438	4	a	a	DET
ejpam-4914	438	5	noncomplete	noncomplete	ADJ
ejpam-4914	438	6	graph	graph	NOUN
ejpam-4914	438	7	.	.	PUNCT
ejpam-4914	439	1	then	then	ADV
ejpam-4914	439	2	g	g	PROPN
ejpam-4914	439	3	admits	admit	VERB
ejpam-4914	439	4	a	a	DET
ejpam-4914	439	5	pndi	pndi	ADJ
ejpam-4914	439	6	-	-	PUNCT
ejpam-4914	439	7	function	function	NOUN
ejpam-4914	439	8	f	f	NOUN
ejpam-4914	439	9	=	=	SYM
ejpam-4914	439	10	(	(	PUNCT
ejpam-4914	439	11	v0	v0	PROPN
ejpam-4914	439	12	,	,	PUNCT
ejpam-4914	439	13	v1	v1	NOUN
ejpam-4914	439	14	,	,	PUNCT
ejpam-4914	439	15	v2	v2	PROPN
ejpam-4914	439	16	)	)	PUNCT
ejpam-4914	439	17	for	for	ADP
ejpam-4914	439	18	which	which	PRON
ejpam-4914	439	19	v2	v2	VERB
ejpam-4914	439	20	̸=	̸=	PROPN
ejpam-4914	439	21	∅.	∅.	PRON
ejpam-4914	439	22	proof	proof	NOUN
ejpam-4914	439	23	:	:	PUNCT
ejpam-4914	439	24	let	let	VERB
ejpam-4914	439	25	f	f	PROPN
ejpam-4914	439	26	=	=	SYM
ejpam-4914	439	27	(	(	PUNCT
ejpam-4914	439	28	v0	v0	PROPN
ejpam-4914	439	29	,	,	PUNCT
ejpam-4914	439	30	v1	v1	NOUN
ejpam-4914	439	31	,	,	PUNCT
ejpam-4914	439	32	v2	v2	PROPN
ejpam-4914	439	33	)	)	PUNCT
ejpam-4914	439	34	be	be	AUX
ejpam-4914	439	35	a	a	DET
ejpam-4914	439	36	pndi	pndi	ADJ
ejpam-4914	439	37	-function	-function	NOUN
ejpam-4914	439	38	of	of	ADP
ejpam-4914	439	39	g	g	NOUN
ejpam-4914	439	40	with	with	ADP
ejpam-4914	439	41	v2	v2	NOUN
ejpam-4914	439	42	=	=	PUNCT
ejpam-4914	439	43	∅.	∅.	NOUN
ejpam-4914	439	44	then	then	ADV
ejpam-4914	439	45	v	v	NOUN
ejpam-4914	439	46	(	(	PUNCT
ejpam-4914	439	47	g	g	NOUN
ejpam-4914	439	48	)	)	PUNCT
ejpam-4914	439	49	=	=	SYM
ejpam-4914	439	50	v1	v1	NOUN
ejpam-4914	439	51	.	.	PUNCT
ejpam-4914	440	1	since	since	SCONJ
ejpam-4914	440	2	g	g	PROPN
ejpam-4914	440	3	is	be	AUX
ejpam-4914	440	4	noncomplete	noncomplete	ADJ
ejpam-4914	440	5	,	,	PUNCT
ejpam-4914	440	6	there	there	PRON
ejpam-4914	440	7	exist	exist	VERB
ejpam-4914	440	8	u	u	NOUN
ejpam-4914	440	9	,	,	PUNCT
ejpam-4914	440	10	v	v	NOUN
ejpam-4914	440	11	∈	∈	PROPN
ejpam-4914	440	12	v	v	NOUN
ejpam-4914	440	13	(	(	PUNCT
ejpam-4914	440	14	g	g	NOUN
ejpam-4914	440	15	)	)	PUNCT
ejpam-4914	440	16	such	such	ADJ
ejpam-4914	440	17	thatdg(u	thatdg(u	PROPN
ejpam-4914	440	18	,	,	PUNCT
ejpam-4914	440	19	v	v	NOUN
ejpam-4914	440	20	)	)	PUNCT
ejpam-4914	440	21	=	=	SYM
ejpam-4914	440	22	2	2	X
ejpam-4914	440	23	.	.	X
ejpam-4914	440	24	observe	observe	VERB
ejpam-4914	440	25	that	that	SCONJ
ejpam-4914	440	26	g	g	NOUN
ejpam-4914	440	27	=	=	PUNCT
ejpam-4914	440	28	(	(	PUNCT
ejpam-4914	440	29	{	{	PUNCT
ejpam-4914	440	30	u	u	NOUN
ejpam-4914	440	31	}	}	PUNCT
ejpam-4914	440	32	,	,	PUNCT
ejpam-4914	440	33	v1	v1	VERB
ejpam-4914	440	34	\	\	PROPN
ejpam-4914	440	35	{	{	PUNCT
ejpam-4914	440	36	u	u	NOUN
ejpam-4914	440	37	,	,	PUNCT
ejpam-4914	440	38	v	v	NOUN
ejpam-4914	440	39	}	}	PUNCT
ejpam-4914	440	40	,	,	PUNCT
ejpam-4914	440	41	{	{	PUNCT
ejpam-4914	440	42	v	v	NOUN
ejpam-4914	440	43	}	}	PUNCT
ejpam-4914	440	44	)	)	PUNCT
ejpam-4914	440	45	is	be	AUX
ejpam-4914	440	46	a	a	DET
ejpam-4914	440	47	pndi	pndi	ADJ
ejpam-4914	440	48	-function	-function	NOUN
ejpam-4914	440	49	of	of	ADP
ejpam-4914	440	50	g	g	NOUN
ejpam-4914	440	51	with	with	ADP
ejpam-4914	440	52	ωg(g	ωg(g	NOUN
ejpam-4914	440	53	)	)	PUNCT
ejpam-4914	440	54	=	=	NOUN
ejpam-4914	440	55	ωg(f	ωg(f	NOUN
ejpam-4914	440	56	)	)	PUNCT
ejpam-4914	440	57	.	.	PUNCT
ejpam-4914	441	1	■	■	PUNCT
ejpam-4914	441	2	3	3	X
ejpam-4914	441	3	.	.	X
ejpam-4914	441	4	graphs	graph	NOUN
ejpam-4914	441	5	under	under	ADP
ejpam-4914	441	6	binary	binary	ADJ
ejpam-4914	441	7	operations	operation	NOUN
ejpam-4914	441	8	in	in	ADP
ejpam-4914	441	9	view	view	NOUN
ejpam-4914	441	10	of	of	ADP
ejpam-4914	441	11	proposition	proposition	NOUN
ejpam-4914	441	12	2.4	2.4	NUM
ejpam-4914	441	13	,	,	PUNCT
ejpam-4914	441	14	γhi(gg	γhi(gg	NUM
ejpam-4914	441	15	)	)	PUNCT
ejpam-4914	441	16	≥	≥	NOUN
ejpam-4914	441	17	2	2	NUM
ejpam-4914	441	18	for	for	ADP
ejpam-4914	441	19	any	any	DET
ejpam-4914	441	20	graph	graph	NOUN
ejpam-4914	441	21	g.	g.	NOUN
ejpam-4914	441	22	proposition	proposition	NOUN
ejpam-4914	441	23	3.1	3.1	NUM
ejpam-4914	441	24	.	.	PUNCT
ejpam-4914	442	1	(	(	PUNCT
ejpam-4914	442	2	complementary	complementary	ADJ
ejpam-4914	442	3	prism	prism	NOUN
ejpam-4914	442	4	of	of	ADP
ejpam-4914	442	5	graphs	graph	NOUN
ejpam-4914	442	6	)	)	PUNCT
ejpam-4914	442	7	let	let	VERB
ejpam-4914	442	8	g	g	NOUN
ejpam-4914	442	9	be	be	AUX
ejpam-4914	442	10	any	any	DET
ejpam-4914	442	11	graph	graph	NOUN
ejpam-4914	442	12	.	.	PUNCT
ejpam-4914	443	1	then	then	ADV
ejpam-4914	443	2	(	(	PUNCT
ejpam-4914	443	3	i	i	NOUN
ejpam-4914	443	4	)	)	PUNCT
ejpam-4914	443	5	γhi(gg	γhi(gg	PROPN
ejpam-4914	443	6	)	)	PUNCT
ejpam-4914	443	7	=	=	SYM
ejpam-4914	443	8	2	2	NUM
ejpam-4914	443	9	if	if	SCONJ
ejpam-4914	443	10	and	and	CCONJ
ejpam-4914	443	11	only	only	ADV
ejpam-4914	443	12	if	if	SCONJ
ejpam-4914	443	13	g	g	PROPN
ejpam-4914	443	14	=	=	PROPN
ejpam-4914	443	15	k1	k1	PROPN
ejpam-4914	443	16	.	.	PUNCT
ejpam-4914	443	17	(	(	PUNCT
ejpam-4914	443	18	ii	ii	NOUN
ejpam-4914	443	19	)	)	PUNCT
ejpam-4914	443	20	γhi(gg	γhi(gg	PROPN
ejpam-4914	443	21	)	)	PUNCT
ejpam-4914	443	22	=	=	SYM
ejpam-4914	443	23	4	4	NUM
ejpam-4914	443	24	for	for	ADP
ejpam-4914	443	25	all	all	DET
ejpam-4914	443	26	nontrivial	nontrivial	ADJ
ejpam-4914	443	27	graphs	graph	NOUN
ejpam-4914	443	28	g.	g.	PROPN
ejpam-4914	443	29	s.r	s.r	PROPN
ejpam-4914	443	30	.	.	PROPN
ejpam-4914	443	31	jr	jr	PROPN
ejpam-4914	443	32	.	.	PROPN
ejpam-4914	443	33	canoy	canoy	PROPN
ejpam-4914	443	34	,	,	PUNCT
ejpam-4914	443	35	f.p	f.p	PROPN
ejpam-4914	443	36	.	.	PROPN
ejpam-4914	443	37	jamil	jamil	PROPN
ejpam-4914	443	38	and	and	CCONJ
ejpam-4914	443	39	s.m	s.m	PROPN
ejpam-4914	443	40	.	.	PROPN
ejpam-4914	443	41	menchavez	menchavez	PROPN
ejpam-4914	443	42	/	/	PUNCT
ejpam-4914	443	43	eur	eur	PROPN
ejpam-4914	443	44	.	.	PUNCT
ejpam-4914	444	1	j.	j.	PROPN
ejpam-4914	444	2	pure	pure	PROPN
ejpam-4914	444	3	appl	appl	PROPN
ejpam-4914	444	4	.	.	PROPN
ejpam-4914	444	5	math	math	PROPN
ejpam-4914	444	6	,	,	PUNCT
ejpam-4914	444	7	16	16	NUM
ejpam-4914	444	8	(	(	PUNCT
ejpam-4914	444	9	4	4	NUM
ejpam-4914	444	10	)	)	PUNCT
ejpam-4914	444	11	(	(	PUNCT
ejpam-4914	444	12	2023	2023	NUM
ejpam-4914	444	13	)	)	PUNCT
ejpam-4914	444	14	,	,	PUNCT
ejpam-4914	444	15	2431	2431	NUM
ejpam-4914	444	16	-	-	SYM
ejpam-4914	444	17	2449	2449	NUM
ejpam-4914	444	18	2441	2441	NUM
ejpam-4914	444	19	proof	proof	NOUN
ejpam-4914	444	20	:	:	PUNCT
ejpam-4914	444	21	if	if	SCONJ
ejpam-4914	444	22	g	g	PROPN
ejpam-4914	444	23	=	=	SYM
ejpam-4914	444	24	k1	k1	PROPN
ejpam-4914	444	25	,	,	PUNCT
ejpam-4914	444	26	then	then	ADV
ejpam-4914	444	27	γhi(gg	γhi(gg	NUM
ejpam-4914	444	28	)	)	PUNCT
ejpam-4914	444	29	=	=	SYM
ejpam-4914	445	1	γhi(p2	γhi(p2	ADJ
ejpam-4914	445	2	)	)	PUNCT
ejpam-4914	445	3	=	=	SYM
ejpam-4914	445	4	2	2	X
ejpam-4914	445	5	.	.	X
ejpam-4914	446	1	conversely	conversely	ADV
ejpam-4914	446	2	,	,	PUNCT
ejpam-4914	446	3	if	if	SCONJ
ejpam-4914	446	4	γhi(gg	γhi(gg	NOUN
ejpam-4914	446	5	)	)	PUNCT
ejpam-4914	446	6	=	=	SYM
ejpam-4914	446	7	2	2	NUM
ejpam-4914	446	8	,	,	PUNCT
ejpam-4914	446	9	then	then	ADV
ejpam-4914	446	10	gg	gg	PROPN
ejpam-4914	446	11	=	=	SYM
ejpam-4914	446	12	k2	k2	PROPN
ejpam-4914	446	13	by	by	ADP
ejpam-4914	446	14	proposition	proposition	NOUN
ejpam-4914	446	15	2.4(ii	2.4(ii	NUM
ejpam-4914	446	16	)	)	PUNCT
ejpam-4914	446	17	.	.	PUNCT
ejpam-4914	447	1	this	this	PRON
ejpam-4914	447	2	means	mean	VERB
ejpam-4914	447	3	that	that	SCONJ
ejpam-4914	447	4	g	g	PROPN
ejpam-4914	447	5	=	=	PROPN
ejpam-4914	447	6	k1	k1	PROPN
ejpam-4914	447	7	.	.	PUNCT
ejpam-4914	447	8	suppose	suppose	VERB
ejpam-4914	447	9	that	that	SCONJ
ejpam-4914	447	10	g	g	PROPN
ejpam-4914	447	11	̸=	̸=	PROPN
ejpam-4914	447	12	k1	k1	NOUN
ejpam-4914	447	13	.	.	PUNCT
ejpam-4914	448	1	first	first	ADV
ejpam-4914	448	2	,	,	PUNCT
ejpam-4914	448	3	we	we	PRON
ejpam-4914	448	4	claim	claim	VERB
ejpam-4914	448	5	that	that	SCONJ
ejpam-4914	448	6	γhi(gg	γhi(gg	NOUN
ejpam-4914	448	7	)	)	PUNCT
ejpam-4914	448	8	≥	≥	NOUN
ejpam-4914	448	9	4	4	NUM
ejpam-4914	448	10	.	.	PUNCT
ejpam-4914	449	1	by	by	ADP
ejpam-4914	449	2	(	(	PUNCT
ejpam-4914	449	3	i	i	NOUN
ejpam-4914	449	4	)	)	PUNCT
ejpam-4914	449	5	,	,	PUNCT
ejpam-4914	449	6	γhi(gg	γhi(gg	PROPN
ejpam-4914	449	7	)	)	PUNCT
ejpam-4914	449	8	≥	≥	NOUN
ejpam-4914	449	9	3	3	NUM
ejpam-4914	449	10	.	.	PUNCT
ejpam-4914	449	11	suppose	suppose	VERB
ejpam-4914	449	12	that	that	SCONJ
ejpam-4914	449	13	γhi(gg	γhi(gg	PROPN
ejpam-4914	449	14	)	)	PUNCT
ejpam-4914	449	15	=	=	SYM
ejpam-4914	450	1	3	3	X
ejpam-4914	450	2	.	.	X
ejpam-4914	451	1	in	in	ADP
ejpam-4914	451	2	view	view	NOUN
ejpam-4914	451	3	of	of	ADP
ejpam-4914	451	4	proposition	proposition	NOUN
ejpam-4914	451	5	2.4(iii	2.4(iii	NUM
ejpam-4914	451	6	)	)	PUNCT
ejpam-4914	451	7	,	,	PUNCT
ejpam-4914	451	8	γ(gg	γ(gg	NUM
ejpam-4914	451	9	)	)	PUNCT
ejpam-4914	451	10	=	=	SYM
ejpam-4914	452	1	1	1	X
ejpam-4914	452	2	.	.	PUNCT
ejpam-4914	453	1	this	this	PRON
ejpam-4914	453	2	is	be	AUX
ejpam-4914	453	3	possible	possible	ADJ
ejpam-4914	453	4	only	only	ADV
ejpam-4914	453	5	when	when	SCONJ
ejpam-4914	453	6	gg	gg	PROPN
ejpam-4914	453	7	=	=	SYM
ejpam-4914	453	8	k2	k2	PROPN
ejpam-4914	453	9	so	so	SCONJ
ejpam-4914	453	10	that	that	SCONJ
ejpam-4914	453	11	γhi(gg	γhi(gg	NOUN
ejpam-4914	453	12	)	)	PUNCT
ejpam-4914	453	13	=	=	SYM
ejpam-4914	453	14	2	2	NUM
ejpam-4914	453	15	,	,	PUNCT
ejpam-4914	453	16	a	a	DET
ejpam-4914	453	17	contradiction	contradiction	NOUN
ejpam-4914	453	18	.	.	PUNCT
ejpam-4914	454	1	thus	thus	ADV
ejpam-4914	454	2	,	,	PUNCT
ejpam-4914	454	3	γhi(gg	γhi(gg	PROPN
ejpam-4914	454	4	)	)	PUNCT
ejpam-4914	454	5	≥	≥	NOUN
ejpam-4914	455	1	4	4	NUM
ejpam-4914	455	2	.	.	PUNCT
ejpam-4914	456	1	let	let	VERB
ejpam-4914	456	2	v	v	NUM
ejpam-4914	456	3	∈	∈	PROPN
ejpam-4914	456	4	v	v	NOUN
ejpam-4914	456	5	(	(	PUNCT
ejpam-4914	456	6	g	g	NOUN
ejpam-4914	456	7	)	)	PUNCT
ejpam-4914	456	8	and	and	CCONJ
ejpam-4914	456	9	define	define	VERB
ejpam-4914	456	10	v2	v2	NOUN
ejpam-4914	456	11	=	=	SYM
ejpam-4914	456	12	{	{	PUNCT
ejpam-4914	456	13	v	v	NOUN
ejpam-4914	456	14	,	,	PUNCT
ejpam-4914	456	15	v	v	NOUN
ejpam-4914	456	16	}	}	PUNCT
ejpam-4914	456	17	,	,	PUNCT
ejpam-4914	456	18	v1	v1	NOUN
ejpam-4914	456	19	=	=	SYM
ejpam-4914	456	20	∅	∅	NOUN
ejpam-4914	456	21	and	and	CCONJ
ejpam-4914	456	22	v0	v0	NOUN
ejpam-4914	456	23	=	=	SYM
ejpam-4914	456	24	v	v	PROPN
ejpam-4914	456	25	(	(	PUNCT
ejpam-4914	456	26	gg	gg	NOUN
ejpam-4914	456	27	)	)	PUNCT
ejpam-4914	456	28	\	\	PROPN
ejpam-4914	456	29	{	{	PUNCT
ejpam-4914	456	30	v	v	NOUN
ejpam-4914	456	31	,	,	PUNCT
ejpam-4914	456	32	v	v	NOUN
ejpam-4914	456	33	}	}	PUNCT
ejpam-4914	456	34	.	.	PUNCT
ejpam-4914	457	1	let	let	VERB
ejpam-4914	457	2	z	z	NOUN
ejpam-4914	457	3	∈	∈	PROPN
ejpam-4914	457	4	v0	v0	PROPN
ejpam-4914	457	5	.	.	PUNCT
ejpam-4914	458	1	assume	assume	VERB
ejpam-4914	458	2	that	that	SCONJ
ejpam-4914	458	3	z	z	PROPN
ejpam-4914	458	4	∈	∈	PROPN
ejpam-4914	458	5	v	v	ADP
ejpam-4914	458	6	(	(	PUNCT
ejpam-4914	458	7	g	g	NOUN
ejpam-4914	458	8	)	)	PUNCT
ejpam-4914	458	9	(	(	PUNCT
ejpam-4914	458	10	the	the	DET
ejpam-4914	458	11	case	case	NOUN
ejpam-4914	458	12	where	where	SCONJ
ejpam-4914	458	13	z	z	PROPN
ejpam-4914	458	14	∈	∈	PROPN
ejpam-4914	458	15	v	v	ADP
ejpam-4914	458	16	(	(	PUNCT
ejpam-4914	458	17	g	g	NOUN
ejpam-4914	458	18	)	)	PUNCT
ejpam-4914	458	19	is	be	AUX
ejpam-4914	458	20	done	do	VERB
ejpam-4914	458	21	similarly	similarly	ADV
ejpam-4914	458	22	)	)	PUNCT
ejpam-4914	458	23	.	.	PUNCT
ejpam-4914	459	1	if	if	SCONJ
ejpam-4914	459	2	zv	zv	PROPN
ejpam-4914	459	3	∈	∈	PROPN
ejpam-4914	459	4	e(g	e(g	PROPN
ejpam-4914	459	5	)	)	PUNCT
ejpam-4914	459	6	,	,	PUNCT
ejpam-4914	459	7	then	then	ADV
ejpam-4914	459	8	[	[	X
ejpam-4914	459	9	z	z	NOUN
ejpam-4914	459	10	,	,	PUNCT
ejpam-4914	459	11	v	v	NOUN
ejpam-4914	459	12	,	,	PUNCT
ejpam-4914	459	13	v	v	NOUN
ejpam-4914	459	14	]	]	PUNCT
ejpam-4914	459	15	is	be	AUX
ejpam-4914	459	16	a	a	DET
ejpam-4914	459	17	geodesic	geodesic	NOUN
ejpam-4914	459	18	in	in	ADP
ejpam-4914	459	19	gg	gg	NOUN
ejpam-4914	459	20	so	so	SCONJ
ejpam-4914	459	21	that	that	SCONJ
ejpam-4914	459	22	v	v	ADP
ejpam-4914	459	23	∈	∈	PROPN
ejpam-4914	459	24	v2	v2	PROPN
ejpam-4914	459	25	∩ngg(z	∩ngg(z	PROPN
ejpam-4914	459	26	,	,	PUNCT
ejpam-4914	459	27	2	2	NUM
ejpam-4914	459	28	)	)	PUNCT
ejpam-4914	459	29	.	.	PUNCT
ejpam-4914	460	1	on	on	ADP
ejpam-4914	460	2	the	the	DET
ejpam-4914	460	3	other	other	ADJ
ejpam-4914	460	4	hand	hand	NOUN
ejpam-4914	460	5	,	,	PUNCT
ejpam-4914	460	6	if	if	SCONJ
ejpam-4914	460	7	zv	zv	PROPN
ejpam-4914	460	8	/∈	/∈	PUNCT
ejpam-4914	460	9	e(g	e(g	PROPN
ejpam-4914	460	10	)	)	PUNCT
ejpam-4914	460	11	,	,	PUNCT
ejpam-4914	460	12	then	then	ADV
ejpam-4914	460	13	[	[	X
ejpam-4914	460	14	z	z	X
ejpam-4914	460	15	,	,	PUNCT
ejpam-4914	460	16	z	z	PROPN
ejpam-4914	460	17	,	,	PUNCT
ejpam-4914	460	18	v	v	NOUN
ejpam-4914	460	19	]	]	PUNCT
ejpam-4914	460	20	is	be	AUX
ejpam-4914	460	21	a	a	DET
ejpam-4914	460	22	geodesic	geodesic	NOUN
ejpam-4914	460	23	in	in	ADP
ejpam-4914	460	24	gg	gg	NOUN
ejpam-4914	460	25	so	so	SCONJ
ejpam-4914	460	26	that	that	SCONJ
ejpam-4914	460	27	v	v	ADP
ejpam-4914	460	28	∈	∈	PROPN
ejpam-4914	460	29	v2∩ngg(z	v2∩ngg(z	NOUN
ejpam-4914	460	30	,	,	PUNCT
ejpam-4914	460	31	2	2	NUM
ejpam-4914	460	32	)	)	PUNCT
ejpam-4914	460	33	.	.	PUNCT
ejpam-4914	461	1	this	this	PRON
ejpam-4914	461	2	shows	show	VERB
ejpam-4914	461	3	that	that	SCONJ
ejpam-4914	461	4	f	f	PROPN
ejpam-4914	461	5	=	=	SYM
ejpam-4914	461	6	(	(	PUNCT
ejpam-4914	461	7	v0	v0	PROPN
ejpam-4914	461	8	,	,	PUNCT
ejpam-4914	461	9	v1	v1	NOUN
ejpam-4914	461	10	,	,	PUNCT
ejpam-4914	461	11	v2	v2	NOUN
ejpam-4914	461	12	)	)	PUNCT
ejpam-4914	461	13	∈	∈	PROPN
ejpam-4914	461	14	hi(gg	hi(gg	PROPN
ejpam-4914	461	15	)	)	PUNCT
ejpam-4914	461	16	,	,	PUNCT
ejpam-4914	461	17	and	and	CCONJ
ejpam-4914	461	18	consequently	consequently	ADV
ejpam-4914	461	19	,	,	PUNCT
ejpam-4914	461	20	γhi(gg	γhi(gg	PROPN
ejpam-4914	461	21	)	)	PUNCT
ejpam-4914	461	22	≤	≤	NOUN
ejpam-4914	461	23	ωgg(f	ωgg(f	NUM
ejpam-4914	461	24	)	)	PUNCT
ejpam-4914	461	25	=	=	PUNCT
ejpam-4914	461	26	4	4	X
ejpam-4914	461	27	.	.	X
ejpam-4914	461	28	■	■	PUNCT
ejpam-4914	461	29	from	from	ADP
ejpam-4914	461	30	proposition	proposition	NOUN
ejpam-4914	461	31	3.1(ii	3.1(ii	NUM
ejpam-4914	461	32	)	)	PUNCT
ejpam-4914	461	33	,	,	PUNCT
ejpam-4914	461	34	for	for	ADP
ejpam-4914	461	35	n	n	X
ejpam-4914	461	36	≥	≥	NOUN
ejpam-4914	461	37	5	5	NUM
ejpam-4914	461	38	,	,	PUNCT
ejpam-4914	461	39	γhi(knkn	γhi(knkn	PROPN
ejpam-4914	461	40	)	)	PUNCT
ejpam-4914	461	41	=	=	SYM
ejpam-4914	461	42	4	4	NUM
ejpam-4914	461	43	while	while	SCONJ
ejpam-4914	461	44	max{γhi(kn	max{γhi(kn	NOUN
ejpam-4914	461	45	)	)	PUNCT
ejpam-4914	461	46	,	,	PUNCT
ejpam-4914	461	47	γhi(kn	γhi(kn	NOUN
ejpam-4914	461	48	)	)	PUNCT
ejpam-4914	461	49	}	}	PUNCT
ejpam-4914	461	50	=	=	PUNCT
ejpam-4914	461	51	n.	n.	PROPN
ejpam-4914	461	52	thus	thus	ADV
ejpam-4914	461	53	,	,	PUNCT
ejpam-4914	461	54	contrary	contrary	ADV
ejpam-4914	461	55	to	to	ADP
ejpam-4914	461	56	the	the	DET
ejpam-4914	461	57	case	case	NOUN
ejpam-4914	461	58	of	of	ADP
ejpam-4914	461	59	italian	italian	ADJ
ejpam-4914	461	60	domination	domination	NOUN
ejpam-4914	461	61	in	in	ADP
ejpam-4914	461	62	complementary	complementary	ADJ
ejpam-4914	461	63	prisms	prism	NOUN
ejpam-4914	461	64	,	,	PUNCT
ejpam-4914	461	65	it	it	PRON
ejpam-4914	461	66	is	be	AUX
ejpam-4914	461	67	not	not	PART
ejpam-4914	461	68	always	always	ADV
ejpam-4914	461	69	true	true	ADJ
ejpam-4914	461	70	that	that	SCONJ
ejpam-4914	461	71	γhi(gg	γhi(gg	NUM
ejpam-4914	461	72	)	)	PUNCT
ejpam-4914	461	73	≥	≥	PROPN
ejpam-4914	461	74	max{γhi(g	max{γhi(g	PROPN
ejpam-4914	461	75	)	)	PUNCT
ejpam-4914	461	76	,	,	PUNCT
ejpam-4914	461	77	γhi(g	γhi(g	PROPN
ejpam-4914	461	78	)	)	PUNCT
ejpam-4914	461	79	}	}	PUNCT
ejpam-4914	461	80	.	.	PUNCT
ejpam-4914	462	1	corollary	corollary	ADJ
ejpam-4914	462	2	3.2	3.2	NUM
ejpam-4914	462	3	.	.	PUNCT
ejpam-4914	463	1	for	for	ADP
ejpam-4914	463	2	all	all	DET
ejpam-4914	463	3	graphs	graph	NOUN
ejpam-4914	463	4	g	g	ADP
ejpam-4914	463	5	,	,	PUNCT
ejpam-4914	463	6	γhi(gg	γhi(gg	PROPN
ejpam-4914	463	7	)	)	PUNCT
ejpam-4914	463	8	≤	≤	NOUN
ejpam-4914	463	9	γhi(g	γhi(g	PROPN
ejpam-4914	463	10	)	)	PUNCT
ejpam-4914	464	1	+	+	CCONJ
ejpam-4914	464	2	γhi(g	γhi(g	NUM
ejpam-4914	464	3	)	)	PUNCT
ejpam-4914	464	4	,	,	PUNCT
ejpam-4914	464	5	and	and	CCONJ
ejpam-4914	464	6	this	this	DET
ejpam-4914	464	7	bound	bind	VERB
ejpam-4914	464	8	is	be	AUX
ejpam-4914	464	9	sharp	sharp	ADJ
ejpam-4914	464	10	.	.	PUNCT
ejpam-4914	465	1	proof	proof	NOUN
ejpam-4914	465	2	:	:	PUNCT
ejpam-4914	465	3	if	if	SCONJ
ejpam-4914	465	4	g	g	PROPN
ejpam-4914	465	5	=	=	SYM
ejpam-4914	465	6	k1	k1	PROPN
ejpam-4914	465	7	,	,	PUNCT
ejpam-4914	465	8	then	then	ADV
ejpam-4914	465	9	by	by	ADP
ejpam-4914	465	10	proposition	proposition	NOUN
ejpam-4914	465	11	3.1(i	3.1(i	NUM
ejpam-4914	465	12	)	)	PUNCT
ejpam-4914	465	13	and	and	CCONJ
ejpam-4914	465	14	proposition	proposition	NOUN
ejpam-4914	465	15	2.4(i	2.4(i	NUM
ejpam-4914	465	16	)	)	PUNCT
ejpam-4914	465	17	,	,	PUNCT
ejpam-4914	465	18	γhi(gg	γhi(gg	NUM
ejpam-4914	465	19	)	)	PUNCT
ejpam-4914	465	20	=	=	SYM
ejpam-4914	465	21	2	2	NUM
ejpam-4914	465	22	=	=	SYM
ejpam-4914	465	23	γhi(g	γhi(g	PROPN
ejpam-4914	465	24	)	)	PUNCT
ejpam-4914	466	1	+	+	CCONJ
ejpam-4914	466	2	γhi(g	γhi(g	NUM
ejpam-4914	466	3	)	)	PUNCT
ejpam-4914	466	4	.	.	PUNCT
ejpam-4914	467	1	suppose	suppose	VERB
ejpam-4914	467	2	that	that	SCONJ
ejpam-4914	467	3	g	g	PROPN
ejpam-4914	467	4	̸=	̸=	PROPN
ejpam-4914	467	5	k1	k1	NOUN
ejpam-4914	467	6	.	.	PUNCT
ejpam-4914	468	1	since	since	SCONJ
ejpam-4914	468	2	2	2	NUM
ejpam-4914	468	3	≤	≤	NUM
ejpam-4914	468	4	γhi(g	γhi(g	PROPN
ejpam-4914	468	5	)	)	PUNCT
ejpam-4914	468	6	and	and	CCONJ
ejpam-4914	468	7	2	2	NUM
ejpam-4914	468	8	≤	≤	NUM
ejpam-4914	468	9	γhi(g	γhi(g	PROPN
ejpam-4914	468	10	)	)	PUNCT
ejpam-4914	468	11	,	,	PUNCT
ejpam-4914	468	12	4	4	NUM
ejpam-4914	468	13	≤	≤	NUM
ejpam-4914	468	14	γhi(g	γhi(g	PROPN
ejpam-4914	468	15	)	)	PUNCT
ejpam-4914	469	1	+	+	CCONJ
ejpam-4914	469	2	γhi(g	γhi(g	NUM
ejpam-4914	469	3	)	)	PUNCT
ejpam-4914	469	4	.	.	PUNCT
ejpam-4914	470	1	the	the	DET
ejpam-4914	470	2	conclusion	conclusion	NOUN
ejpam-4914	470	3	follows	follow	VERB
ejpam-4914	470	4	immediately	immediately	ADV
ejpam-4914	470	5	from	from	ADP
ejpam-4914	470	6	proposition	proposition	NOUN
ejpam-4914	470	7	3.1	3.1	NUM
ejpam-4914	470	8	.	.	PUNCT
ejpam-4914	470	9	to	to	PART
ejpam-4914	470	10	show	show	VERB
ejpam-4914	470	11	sharpness	sharpness	NOUN
ejpam-4914	470	12	of	of	ADP
ejpam-4914	470	13	the	the	DET
ejpam-4914	470	14	bound	bind	VERB
ejpam-4914	470	15	,	,	PUNCT
ejpam-4914	470	16	consider	consider	VERB
ejpam-4914	470	17	g	g	NOUN
ejpam-4914	470	18	=	=	NOUN
ejpam-4914	470	19	p2	p2	X
ejpam-4914	470	20	.	.	PUNCT
ejpam-4914	471	1	by	by	ADP
ejpam-4914	471	2	observation	observation	NOUN
ejpam-4914	471	3	2.1(ii	2.1(ii	NUM
ejpam-4914	471	4	)	)	PUNCT
ejpam-4914	471	5	,	,	PUNCT
ejpam-4914	471	6	γhi(gg	γhi(gg	NUM
ejpam-4914	471	7	)	)	PUNCT
ejpam-4914	471	8	=	=	PUNCT
ejpam-4914	471	9	γhi(p4	γhi(p4	ADJ
ejpam-4914	471	10	)	)	PUNCT
ejpam-4914	471	11	=	=	SYM
ejpam-4914	471	12	4	4	NUM
ejpam-4914	471	13	=	=	SYM
ejpam-4914	471	14	γhi(g	γhi(g	PROPN
ejpam-4914	471	15	)	)	PUNCT
ejpam-4914	471	16	+	+	CCONJ
ejpam-4914	471	17	γhi(g	γhi(g	NUM
ejpam-4914	471	18	)	)	PUNCT
ejpam-4914	471	19	.	.	PUNCT
ejpam-4914	472	1	■	■	PUNCT
ejpam-4914	472	2	strict	strict	ADJ
ejpam-4914	472	3	inequality	inequality	NOUN
ejpam-4914	472	4	can	can	AUX
ejpam-4914	472	5	be	be	AUX
ejpam-4914	472	6	obtained	obtain	VERB
ejpam-4914	472	7	in	in	ADP
ejpam-4914	472	8	theorem	theorem	ADJ
ejpam-4914	472	9	3.2	3.2	NUM
ejpam-4914	472	10	.	.	PUNCT
ejpam-4914	473	1	consider	consider	VERB
ejpam-4914	473	2	g	g	NOUN
ejpam-4914	473	3	=	=	NOUN
ejpam-4914	473	4	k1	k1	NOUN
ejpam-4914	473	5	∪k3	∪k3	NOUN
ejpam-4914	473	6	.	.	PUNCT
ejpam-4914	474	1	the	the	DET
ejpam-4914	474	2	graph	graph	NOUN
ejpam-4914	474	3	gg	gg	NOUN
ejpam-4914	474	4	is	be	AUX
ejpam-4914	474	5	as	as	SCONJ
ejpam-4914	474	6	shown	show	VERB
ejpam-4914	474	7	in	in	ADP
ejpam-4914	474	8	figure	figure	NOUN
ejpam-4914	474	9	5	5	NUM
ejpam-4914	474	10	.	.	PUNCT
ejpam-4914	475	1	for	for	ADP
ejpam-4914	475	2	this	this	DET
ejpam-4914	475	3	graph	graph	NOUN
ejpam-4914	475	4	,	,	PUNCT
ejpam-4914	475	5	γhi(gg	γhi(gg	PROPN
ejpam-4914	475	6	)	)	PUNCT
ejpam-4914	475	7	=	=	SYM
ejpam-4914	475	8	4	4	NUM
ejpam-4914	475	9	,	,	PUNCT
ejpam-4914	475	10	γhi(g	γhi(g	PROPN
ejpam-4914	475	11	)	)	PUNCT
ejpam-4914	475	12	=	=	SYM
ejpam-4914	475	13	4	4	NUM
ejpam-4914	475	14	and	and	CCONJ
ejpam-4914	475	15	γhi(g	γhi(g	NUM
ejpam-4914	475	16	)	)	PUNCT
ejpam-4914	475	17	=	=	SYM
ejpam-4914	475	18	3	3	X
ejpam-4914	475	19	.	.	NUM
ejpam-4914	475	20	............................................................................................................................................................................	............................................................................................................................................................................	PUNCT
ejpam-4914	475	21	............	............	PUNCT
ejpam-4914	475	22	...........	...........	PUNCT
ejpam-4914	475	23	...........	...........	PUNCT
ejpam-4914	475	24	...........	...........	PUNCT
ejpam-4914	475	25	...........	...........	PUNCT
ejpam-4914	475	26	...........	...........	PUNCT
ejpam-4914	475	27	...........	...........	PUNCT
ejpam-4914	475	28	...........	...........	PUNCT
ejpam-4914	475	29	....	....	PUNCT
ejpam-4914	475	30	..	..	PUNCT
ejpam-4914	475	31	...............................................................................................................................	...............................................................................................................................	PUNCT
ejpam-4914	475	32	....................................	....................................	PUNCT
ejpam-4914	475	33	........................................................................	........................................................................	PUNCT
ejpam-4914	475	34	....................................	....................................	PUNCT
ejpam-4914	475	35	....................................	....................................	PUNCT
ejpam-4914	475	36	....................................	....................................	PUNCT
ejpam-4914	475	37	....................................	....................................	PUNCT
ejpam-4914	475	38	.........	.........	PUNCT
ejpam-4914	475	39	........	........	PUNCT
ejpam-4914	475	40	........	........	PUNCT
ejpam-4914	475	41	........	........	PUNCT
ejpam-4914	475	42	........	........	PUNCT
ejpam-4914	475	43	........	........	PUNCT
ejpam-4914	475	44	........	........	PUNCT
ejpam-4914	475	45	........	........	PUNCT
ejpam-4914	475	46	........	........	PUNCT
ejpam-4914	475	47	........	........	PUNCT
ejpam-4914	475	48	........	........	PUNCT
ejpam-4914	475	49	........	........	PUNCT
ejpam-4914	475	50	........	........	PUNCT
ejpam-4914	475	51	........	........	PUNCT
ejpam-4914	475	52	........	........	PUNCT
ejpam-4914	475	53	........	........	PUNCT
ejpam-4914	475	54	.......	.......	PUNCT
ejpam-4914	475	55	.........	.........	PUNCT
ejpam-4914	475	56	........	........	PUNCT
ejpam-4914	475	57	........	........	PUNCT
ejpam-4914	475	58	........	........	PUNCT
ejpam-4914	475	59	........	........	PUNCT
ejpam-4914	475	60	........	........	PUNCT
ejpam-4914	475	61	........	........	PUNCT
ejpam-4914	475	62	........	........	PUNCT
ejpam-4914	475	63	........	........	PUNCT
ejpam-4914	475	64	........	........	PUNCT
ejpam-4914	475	65	........	........	PUNCT
ejpam-4914	475	66	........	........	PUNCT
ejpam-4914	475	67	........	........	PUNCT
ejpam-4914	475	68	........	........	PUNCT
ejpam-4914	475	69	........	........	PUNCT
ejpam-4914	475	70	........	........	PUNCT
ejpam-4914	475	71	........	........	PUNCT
ejpam-4914	475	72	........	........	PUNCT
ejpam-4914	475	73	........	........	PUNCT
ejpam-4914	475	74	........	........	PUNCT
ejpam-4914	475	75	........	........	PUNCT
ejpam-4914	475	76	........	........	PUNCT
ejpam-4914	475	77	........	........	PUNCT
ejpam-4914	475	78	.........	.........	PUNCT
ejpam-4914	475	79	........	........	PUNCT
ejpam-4914	475	80	........	........	PUNCT
ejpam-4914	475	81	........	........	PUNCT
ejpam-4914	475	82	........	........	PUNCT
ejpam-4914	475	83	........	........	PUNCT
ejpam-4914	475	84	........	........	PUNCT
ejpam-4914	475	85	........	........	PUNCT
ejpam-4914	475	86	........	........	PUNCT
ejpam-4914	475	87	........	........	PUNCT
ejpam-4914	475	88	........	........	PUNCT
ejpam-4914	475	89	........	........	PUNCT
ejpam-4914	475	90	........	........	PUNCT
ejpam-4914	475	91	........	........	PUNCT
ejpam-4914	475	92	........	........	PUNCT
ejpam-4914	475	93	........	........	PUNCT
ejpam-4914	475	94	.......	.......	PUNCT
ejpam-4914	475	95	.........	.........	PUNCT
ejpam-4914	475	96	........	........	PUNCT
ejpam-4914	475	97	........	........	PUNCT
ejpam-4914	475	98	........	........	PUNCT
ejpam-4914	475	99	........	........	PUNCT
ejpam-4914	475	100	........	........	PUNCT
ejpam-4914	475	101	........	........	PUNCT
ejpam-4914	475	102	........	........	PUNCT
ejpam-4914	475	103	........	........	PUNCT
ejpam-4914	475	104	........	........	PUNCT
ejpam-4914	475	105	........	........	PUNCT
ejpam-4914	475	106	........	........	PUNCT
ejpam-4914	475	107	........	........	PUNCT
ejpam-4914	475	108	........	........	PUNCT
ejpam-4914	475	109	........	........	PUNCT
ejpam-4914	475	110	........	........	PUNCT
ejpam-4914	475	111	.......	.......	PUNCT
ejpam-4914	475	112	..........................	..........................	PUNCT
ejpam-4914	475	113	.........................	.........................	PUNCT
ejpam-4914	475	114	.........................	.........................	PUNCT
ejpam-4914	475	115	.........................	.........................	PUNCT
ejpam-4914	475	116	.........................	.........................	PUNCT
ejpam-4914	475	117	..................	..................	PUNCT
ejpam-4914	475	118	.............................................................................................................................................................................................................................	.............................................................................................................................................................................................................................	X
ejpam-4914	475	119	..........................................................................................................................................................................................................................................................................................	..........................................................................................................................................................................................................................................................................................	PUNCT
ejpam-4914	476	1	figure	figure	NOUN
ejpam-4914	476	2	5	5	NUM
ejpam-4914	476	3	:	:	PUNCT
ejpam-4914	476	4	the	the	DET
ejpam-4914	476	5	graph	graph	NOUN
ejpam-4914	476	6	of	of	ADP
ejpam-4914	476	7	gg	gg	NOUN
ejpam-4914	476	8	where	where	SCONJ
ejpam-4914	476	9	g	g	NOUN
ejpam-4914	476	10	=	=	PROPN
ejpam-4914	476	11	k1	k1	PROPN
ejpam-4914	476	12	∪k3	∪k3	NOUN
ejpam-4914	476	13	theorem	theorem	VERB
ejpam-4914	476	14	3.3	3.3	NUM
ejpam-4914	476	15	.	.	PUNCT
ejpam-4914	477	1	(	(	PUNCT
ejpam-4914	477	2	join	join	VERB
ejpam-4914	477	3	of	of	ADP
ejpam-4914	477	4	graphs	graph	NOUN
ejpam-4914	477	5	)	)	PUNCT
ejpam-4914	477	6	let	let	VERB
ejpam-4914	477	7	g	g	NOUN
ejpam-4914	477	8	and	and	CCONJ
ejpam-4914	477	9	h	h	NOUN
ejpam-4914	477	10	be	be	VERB
ejpam-4914	477	11	any	any	DET
ejpam-4914	477	12	graphs	graph	NOUN
ejpam-4914	477	13	,	,	PUNCT
ejpam-4914	477	14	and	and	CCONJ
ejpam-4914	477	15	f	f	X
ejpam-4914	477	16	=	=	SYM
ejpam-4914	477	17	(	(	PUNCT
ejpam-4914	477	18	v0	v0	PROPN
ejpam-4914	477	19	,	,	PUNCT
ejpam-4914	477	20	v1	v1	NOUN
ejpam-4914	477	21	,	,	PUNCT
ejpam-4914	477	22	v0	v0	NOUN
ejpam-4914	477	23	)	)	PUNCT
ejpam-4914	477	24	be	be	VERB
ejpam-4914	477	25	a	a	DET
ejpam-4914	477	26	function	function	NOUN
ejpam-4914	477	27	on	on	ADP
ejpam-4914	477	28	v	v	NOUN
ejpam-4914	477	29	(	(	PUNCT
ejpam-4914	477	30	g	g	PROPN
ejpam-4914	477	31	+	+	NOUN
ejpam-4914	477	32	h	h	NOUN
ejpam-4914	477	33	)	)	PUNCT
ejpam-4914	477	34	.	.	PUNCT
ejpam-4914	478	1	then	then	ADV
ejpam-4914	478	2	f	f	PROPN
ejpam-4914	478	3	∈	∈	PROPN
ejpam-4914	478	4	hid(g	hid(g	PROPN
ejpam-4914	478	5	+	+	CCONJ
ejpam-4914	478	6	h	h	NOUN
ejpam-4914	478	7	)	)	PUNCT
ejpam-4914	478	8	if	if	SCONJ
ejpam-4914	478	9	and	and	CCONJ
ejpam-4914	478	10	only	only	ADV
ejpam-4914	478	11	if	if	SCONJ
ejpam-4914	478	12	f	f	PROPN
ejpam-4914	478	13	|g	|g	VERB
ejpam-4914	478	14	∈	∈	PROPN
ejpam-4914	478	15	pndi(g	pndi(g	PROPN
ejpam-4914	478	16	)	)	PUNCT
ejpam-4914	478	17	and	and	CCONJ
ejpam-4914	478	18	f	f	PROPN
ejpam-4914	478	19	|h	|h	X
ejpam-4914	478	20	∈	∈	PROPN
ejpam-4914	478	21	pndi(h	pndi(h	PROPN
ejpam-4914	478	22	)	)	PUNCT
ejpam-4914	478	23	,	,	PUNCT
ejpam-4914	478	24	where	where	SCONJ
ejpam-4914	478	25	f	f	PROPN
ejpam-4914	478	26	|g	|g	VERB
ejpam-4914	478	27	and	and	CCONJ
ejpam-4914	478	28	f	f	PROPN
ejpam-4914	478	29	|h	|h	NOUN
ejpam-4914	478	30	are	be	AUX
ejpam-4914	478	31	the	the	DET
ejpam-4914	478	32	restrictions	restriction	NOUN
ejpam-4914	478	33	of	of	ADP
ejpam-4914	478	34	f	f	PROPN
ejpam-4914	478	35	to	to	ADP
ejpam-4914	478	36	g	g	PROPN
ejpam-4914	478	37	and	and	CCONJ
ejpam-4914	478	38	h	h	NOUN
ejpam-4914	478	39	,	,	PUNCT
ejpam-4914	478	40	respectively	respectively	ADV
ejpam-4914	478	41	.	.	PUNCT
ejpam-4914	479	1	s.r	s.r	PROPN
ejpam-4914	479	2	.	.	PROPN
ejpam-4914	479	3	jr	jr	PROPN
ejpam-4914	479	4	.	.	PROPN
ejpam-4914	479	5	canoy	canoy	PROPN
ejpam-4914	479	6	,	,	PUNCT
ejpam-4914	479	7	f.p	f.p	PROPN
ejpam-4914	479	8	.	.	PROPN
ejpam-4914	479	9	jamil	jamil	PROPN
ejpam-4914	479	10	and	and	CCONJ
ejpam-4914	479	11	s.m	s.m	PROPN
ejpam-4914	479	12	.	.	PROPN
ejpam-4914	479	13	menchavez	menchavez	PROPN
ejpam-4914	479	14	/	/	PUNCT
ejpam-4914	479	15	eur	eur	PROPN
ejpam-4914	479	16	.	.	PUNCT
ejpam-4914	480	1	j.	j.	PROPN
ejpam-4914	480	2	pure	pure	PROPN
ejpam-4914	480	3	appl	appl	PROPN
ejpam-4914	480	4	.	.	PROPN
ejpam-4914	480	5	math	math	PROPN
ejpam-4914	480	6	,	,	PUNCT
ejpam-4914	480	7	16	16	NUM
ejpam-4914	480	8	(	(	PUNCT
ejpam-4914	480	9	4	4	NUM
ejpam-4914	480	10	)	)	PUNCT
ejpam-4914	480	11	(	(	PUNCT
ejpam-4914	480	12	2023	2023	NUM
ejpam-4914	480	13	)	)	PUNCT
ejpam-4914	480	14	,	,	PUNCT
ejpam-4914	480	15	2431	2431	NUM
ejpam-4914	480	16	-	-	SYM
ejpam-4914	480	17	2449	2449	NUM
ejpam-4914	480	18	2442	2442	NUM
ejpam-4914	480	19	proof	proof	NOUN
ejpam-4914	480	20	:	:	PUNCT
ejpam-4914	480	21	let	let	VERB
ejpam-4914	480	22	f	f	PROPN
ejpam-4914	480	23	=	=	SYM
ejpam-4914	480	24	(	(	PUNCT
ejpam-4914	480	25	v0	v0	PROPN
ejpam-4914	480	26	,	,	PUNCT
ejpam-4914	480	27	v1	v1	NOUN
ejpam-4914	480	28	,	,	PUNCT
ejpam-4914	480	29	v2	v2	NOUN
ejpam-4914	480	30	)	)	PUNCT
ejpam-4914	480	31	∈	∈	PROPN
ejpam-4914	480	32	hid(g+h	hid(g+h	PROPN
ejpam-4914	480	33	)	)	PUNCT
ejpam-4914	480	34	.	.	PUNCT
ejpam-4914	481	1	let	let	VERB
ejpam-4914	481	2	v	v	X
ejpam-4914	481	3	∈	∈	NOUN
ejpam-4914	481	4	v0∩v	v0∩v	ADP
ejpam-4914	481	5	(	(	PUNCT
ejpam-4914	481	6	g	g	NOUN
ejpam-4914	481	7	)	)	PUNCT
ejpam-4914	481	8	.	.	PUNCT
ejpam-4914	482	1	then	then	ADV
ejpam-4914	482	2	|v2∩ng+h(v	|v2∩ng+h(v	ADV
ejpam-4914	482	3	,	,	PUNCT
ejpam-4914	482	4	2)|	2)|	NUM
ejpam-4914	482	5	≥	≥	NUM
ejpam-4914	482	6	1	1	NUM
ejpam-4914	482	7	or	or	CCONJ
ejpam-4914	482	8	|v1∩ng+h(v	|v1∩ng+h(v	NOUN
ejpam-4914	482	9	,	,	PUNCT
ejpam-4914	482	10	2)|	2)|	NUM
ejpam-4914	482	11	≥	≥	NUM
ejpam-4914	482	12	2	2	NUM
ejpam-4914	482	13	.	.	PUNCT
ejpam-4914	482	14	suppose	suppose	VERB
ejpam-4914	482	15	that	that	SCONJ
ejpam-4914	482	16	|v2∩ng+h(v	|v2∩ng+h(v	NOUN
ejpam-4914	482	17	,	,	PUNCT
ejpam-4914	482	18	2)|	2)|	NUM
ejpam-4914	482	19	≥	≥	NOUN
ejpam-4914	482	20	1	1	NUM
ejpam-4914	482	21	,	,	PUNCT
ejpam-4914	482	22	and	and	CCONJ
ejpam-4914	482	23	let	let	VERB
ejpam-4914	482	24	u	u	PRON
ejpam-4914	482	25	∈	∈	PROPN
ejpam-4914	482	26	v2∩ng+h(v	v2∩ng+h(v	NOUN
ejpam-4914	482	27	,	,	PUNCT
ejpam-4914	482	28	2	2	NUM
ejpam-4914	482	29	)	)	PUNCT
ejpam-4914	482	30	.	.	PUNCT
ejpam-4914	483	1	since	since	SCONJ
ejpam-4914	483	2	dg+h(u	dg+h(u	PROPN
ejpam-4914	483	3	,	,	PUNCT
ejpam-4914	483	4	v	v	NOUN
ejpam-4914	483	5	)	)	PUNCT
ejpam-4914	483	6	=	=	SYM
ejpam-4914	483	7	2	2	NUM
ejpam-4914	483	8	,	,	PUNCT
ejpam-4914	483	9	u	u	NOUN
ejpam-4914	483	10	∈	∈	PROPN
ejpam-4914	483	11	v2	v2	PROPN
ejpam-4914	483	12	∩	∩	ADJ
ejpam-4914	483	13	v	v	NOUN
ejpam-4914	483	14	(	(	PUNCT
ejpam-4914	483	15	g	g	NOUN
ejpam-4914	483	16	)	)	PUNCT
ejpam-4914	483	17	and	and	CCONJ
ejpam-4914	483	18	u	u	NOUN
ejpam-4914	483	19	/∈	/∈	PUNCT
ejpam-4914	483	20	ng(v	ng(v	NUM
ejpam-4914	483	21	)	)	PUNCT
ejpam-4914	483	22	.	.	PUNCT
ejpam-4914	484	1	suppose	suppose	VERB
ejpam-4914	484	2	,	,	PUNCT
ejpam-4914	484	3	on	on	ADP
ejpam-4914	484	4	the	the	DET
ejpam-4914	484	5	other	other	ADJ
ejpam-4914	484	6	hand	hand	NOUN
ejpam-4914	484	7	,	,	PUNCT
ejpam-4914	484	8	that	that	SCONJ
ejpam-4914	484	9	|v1	|v1	ADJ
ejpam-4914	484	10	∩	∩	NOUN
ejpam-4914	484	11	ng+h(v	ng+h(v	PROPN
ejpam-4914	484	12	,	,	PUNCT
ejpam-4914	484	13	2)|	2)|	NUM
ejpam-4914	484	14	≥	≥	NOUN
ejpam-4914	484	15	2	2	NUM
ejpam-4914	484	16	,	,	PUNCT
ejpam-4914	484	17	say	say	VERB
ejpam-4914	484	18	u	u	NOUN
ejpam-4914	484	19	,	,	PUNCT
ejpam-4914	484	20	w	w	PROPN
ejpam-4914	484	21	∈	∈	PROPN
ejpam-4914	484	22	v1	v1	NOUN
ejpam-4914	484	23	∩	∩	X
ejpam-4914	484	24	ng+h(v	ng+h(v	NOUN
ejpam-4914	484	25	,	,	PUNCT
ejpam-4914	484	26	2	2	NUM
ejpam-4914	484	27	)	)	PUNCT
ejpam-4914	484	28	.	.	PUNCT
ejpam-4914	485	1	then	then	ADV
ejpam-4914	485	2	u	u	NOUN
ejpam-4914	485	3	,	,	PUNCT
ejpam-4914	485	4	w	w	PROPN
ejpam-4914	485	5	∈	∈	PROPN
ejpam-4914	485	6	v1	v1	NOUN
ejpam-4914	485	7	∩	∩	ADJ
ejpam-4914	485	8	v	v	NOUN
ejpam-4914	485	9	(	(	PUNCT
ejpam-4914	485	10	g	g	NOUN
ejpam-4914	485	11	)	)	PUNCT
ejpam-4914	485	12	and	and	CCONJ
ejpam-4914	485	13	v	v	NOUN
ejpam-4914	485	14	/∈	/∈	PUNCT
ejpam-4914	485	15	ng(u)∪ng(w	ng(u)∪ng(w	PROPN
ejpam-4914	485	16	)	)	PUNCT
ejpam-4914	485	17	.	.	PUNCT
ejpam-4914	486	1	this	this	PRON
ejpam-4914	486	2	shows	show	VERB
ejpam-4914	486	3	that	that	SCONJ
ejpam-4914	486	4	f	f	PROPN
ejpam-4914	486	5	|g	|g	PROPN
ejpam-4914	486	6	=	=	SYM
ejpam-4914	486	7	(	(	PUNCT
ejpam-4914	486	8	v0∩v	v0∩v	X
ejpam-4914	486	9	(	(	PUNCT
ejpam-4914	486	10	g	g	NOUN
ejpam-4914	486	11	)	)	PUNCT
ejpam-4914	486	12	,	,	PUNCT
ejpam-4914	486	13	v1∩v	v1∩v	NOUN
ejpam-4914	486	14	(	(	PUNCT
ejpam-4914	486	15	g	g	NOUN
ejpam-4914	486	16	)	)	PUNCT
ejpam-4914	486	17	,	,	PUNCT
ejpam-4914	486	18	v2∩v	v2∩v	PROPN
ejpam-4914	486	19	(	(	PUNCT
ejpam-4914	486	20	g	g	NOUN
ejpam-4914	486	21	)	)	PUNCT
ejpam-4914	486	22	)	)	PUNCT
ejpam-4914	487	1	∈	∈	PROPN
ejpam-4914	487	2	pndi(g	pndi(g	NOUN
ejpam-4914	487	3	)	)	PUNCT
ejpam-4914	487	4	.	.	PUNCT
ejpam-4914	488	1	similarly	similarly	ADV
ejpam-4914	488	2	,	,	PUNCT
ejpam-4914	488	3	f	f	PROPN
ejpam-4914	488	4	|h	|h	X
ejpam-4914	488	5	∈	∈	PROPN
ejpam-4914	488	6	pndi(h	pndi(h	PROPN
ejpam-4914	488	7	)	)	PUNCT
ejpam-4914	488	8	.	.	PUNCT
ejpam-4914	489	1	conversely	conversely	ADV
ejpam-4914	489	2	,	,	PUNCT
ejpam-4914	489	3	let	let	VERB
ejpam-4914	489	4	v	v	PRON
ejpam-4914	489	5	∈	∈	PROPN
ejpam-4914	489	6	v0	v0	NOUN
ejpam-4914	489	7	.	.	PUNCT
ejpam-4914	489	8	suppose	suppose	VERB
ejpam-4914	489	9	that	that	SCONJ
ejpam-4914	489	10	v	v	NUM
ejpam-4914	489	11	∈	∈	PROPN
ejpam-4914	489	12	v	v	NOUN
ejpam-4914	489	13	(	(	PUNCT
ejpam-4914	489	14	g	g	NOUN
ejpam-4914	489	15	)	)	PUNCT
ejpam-4914	489	16	.	.	PUNCT
ejpam-4914	490	1	if	if	SCONJ
ejpam-4914	490	2	f	f	PROPN
ejpam-4914	490	3	|g	|g	VERB
ejpam-4914	490	4	∈	∈	PROPN
ejpam-4914	490	5	pndi(g	pndi(g	PROPN
ejpam-4914	490	6	)	)	PUNCT
ejpam-4914	490	7	,	,	PUNCT
ejpam-4914	490	8	then	then	ADV
ejpam-4914	490	9	there	there	PRON
ejpam-4914	490	10	exists	exist	VERB
ejpam-4914	490	11	u	u	PROPN
ejpam-4914	490	12	∈	∈	PROPN
ejpam-4914	490	13	v2	v2	PROPN
ejpam-4914	490	14	∩	∩	ADJ
ejpam-4914	490	15	v	v	NOUN
ejpam-4914	490	16	(	(	PUNCT
ejpam-4914	490	17	g	g	NOUN
ejpam-4914	490	18	)	)	PUNCT
ejpam-4914	490	19	for	for	ADP
ejpam-4914	490	20	which	which	PRON
ejpam-4914	490	21	v	v	NOUN
ejpam-4914	490	22	/∈	/∈	PUNCT
ejpam-4914	490	23	ng(u	ng(u	NOUN
ejpam-4914	490	24	)	)	PUNCT
ejpam-4914	490	25	or	or	CCONJ
ejpam-4914	490	26	there	there	PRON
ejpam-4914	490	27	exist	exist	VERB
ejpam-4914	490	28	w	w	PROPN
ejpam-4914	490	29	,	,	PUNCT
ejpam-4914	490	30	z	z	NOUN
ejpam-4914	490	31	∈	∈	PROPN
ejpam-4914	490	32	v1	v1	NOUN
ejpam-4914	490	33	∩	∩	ADJ
ejpam-4914	490	34	v	v	NOUN
ejpam-4914	490	35	(	(	PUNCT
ejpam-4914	490	36	g	g	NOUN
ejpam-4914	490	37	)	)	PUNCT
ejpam-4914	490	38	for	for	ADP
ejpam-4914	490	39	which	which	PRON
ejpam-4914	490	40	v	v	NOUN
ejpam-4914	490	41	/∈	/∈	PUNCT
ejpam-4914	490	42	ng(w	ng(w	NOUN
ejpam-4914	490	43	)	)	PUNCT
ejpam-4914	490	44	∪	∪	ADP
ejpam-4914	490	45	ng(z	ng(z	PROPN
ejpam-4914	490	46	)	)	PUNCT
ejpam-4914	490	47	.	.	PUNCT
ejpam-4914	491	1	the	the	DET
ejpam-4914	491	2	former	former	ADJ
ejpam-4914	491	3	implies	imply	VERB
ejpam-4914	491	4	that	that	SCONJ
ejpam-4914	491	5	u	u	PROPN
ejpam-4914	491	6	∈	∈	PROPN
ejpam-4914	491	7	v2	v2	PROPN
ejpam-4914	491	8	∩	∩	NOUN
ejpam-4914	491	9	ng+h(v	ng+h(v	NOUN
ejpam-4914	491	10	,	,	PUNCT
ejpam-4914	491	11	2	2	NUM
ejpam-4914	491	12	)	)	PUNCT
ejpam-4914	491	13	,	,	PUNCT
ejpam-4914	491	14	while	while	SCONJ
ejpam-4914	491	15	the	the	DET
ejpam-4914	491	16	latter	latter	ADJ
ejpam-4914	491	17	implies	imply	VERB
ejpam-4914	491	18	that	that	SCONJ
ejpam-4914	491	19	w	w	X
ejpam-4914	491	20	,	,	PUNCT
ejpam-4914	491	21	z	z	NOUN
ejpam-4914	491	22	∈	∈	NOUN
ejpam-4914	491	23	v1	v1	NOUN
ejpam-4914	491	24	∩	∩	X
ejpam-4914	491	25	ng+h(v	ng+h(v	NOUN
ejpam-4914	491	26	,	,	PUNCT
ejpam-4914	491	27	2	2	NUM
ejpam-4914	491	28	)	)	PUNCT
ejpam-4914	491	29	.	.	PUNCT
ejpam-4914	492	1	similarly	similarly	ADV
ejpam-4914	492	2	,	,	PUNCT
ejpam-4914	492	3	if	if	SCONJ
ejpam-4914	492	4	v	v	NUM
ejpam-4914	492	5	∈	∈	PROPN
ejpam-4914	492	6	v	v	NOUN
ejpam-4914	492	7	(	(	PUNCT
ejpam-4914	492	8	h	h	NOUN
ejpam-4914	492	9	)	)	PUNCT
ejpam-4914	492	10	and	and	CCONJ
ejpam-4914	492	11	f	f	PROPN
ejpam-4914	492	12	|h	|h	X
ejpam-4914	492	13	∈	∈	PROPN
ejpam-4914	492	14	pndi(h	pndi(h	PROPN
ejpam-4914	492	15	)	)	PUNCT
ejpam-4914	492	16	,	,	PUNCT
ejpam-4914	492	17	then	then	ADV
ejpam-4914	492	18	|v2	|v2	X
ejpam-4914	492	19	∩ng+h(v	∩ng+h(v	PROPN
ejpam-4914	492	20	,	,	PUNCT
ejpam-4914	492	21	2)|	2)|	NUM
ejpam-4914	492	22	≥	≥	NUM
ejpam-4914	492	23	1	1	NUM
ejpam-4914	492	24	or	or	CCONJ
ejpam-4914	492	25	|v1	|v1	NUM
ejpam-4914	492	26	∩ng+h(v	∩ng+h(v	PROPN
ejpam-4914	492	27	,	,	PUNCT
ejpam-4914	492	28	2)|	2)|	NUM
ejpam-4914	492	29	≥	≥	NOUN
ejpam-4914	492	30	2	2	NUM
ejpam-4914	492	31	.	.	PUNCT
ejpam-4914	493	1	therefore	therefore	ADV
ejpam-4914	493	2	,	,	PUNCT
ejpam-4914	493	3	f	f	PROPN
ejpam-4914	493	4	∈	∈	PROPN
ejpam-4914	493	5	hid(g+h	hid(g+h	PROPN
ejpam-4914	493	6	)	)	PUNCT
ejpam-4914	493	7	.	.	PUNCT
ejpam-4914	494	1	■	■	PUNCT
ejpam-4914	494	2	corollary	corollary	ADJ
ejpam-4914	494	3	3.4	3.4	NUM
ejpam-4914	494	4	.	.	PUNCT
ejpam-4914	495	1	let	let	VERB
ejpam-4914	495	2	g	g	NOUN
ejpam-4914	495	3	and	and	CCONJ
ejpam-4914	495	4	h	h	NOUN
ejpam-4914	495	5	be	be	VERB
ejpam-4914	495	6	any	any	DET
ejpam-4914	495	7	graphs	graph	NOUN
ejpam-4914	495	8	of	of	ADP
ejpam-4914	495	9	orders	order	NOUN
ejpam-4914	495	10	m	m	VERB
ejpam-4914	495	11	and	and	CCONJ
ejpam-4914	495	12	n	n	CCONJ
ejpam-4914	495	13	,	,	PUNCT
ejpam-4914	495	14	respectively	respectively	ADV
ejpam-4914	495	15	.	.	PUNCT
ejpam-4914	496	1	then	then	ADV
ejpam-4914	496	2	γhi(g+h	γhi(g+h	VERB
ejpam-4914	496	3	)	)	PUNCT
ejpam-4914	497	1	=	=	SYM
ejpam-4914	497	2	pndi(g	pndi(g	PROPN
ejpam-4914	497	3	)	)	PUNCT
ejpam-4914	498	1	+	+	CCONJ
ejpam-4914	498	2	pndi(h	pndi(h	ADP
ejpam-4914	498	3	)	)	PUNCT
ejpam-4914	498	4	.	.	PUNCT
ejpam-4914	499	1	in	in	ADP
ejpam-4914	499	2	particular	particular	ADJ
ejpam-4914	499	3	,	,	PUNCT
ejpam-4914	499	4	(	(	PUNCT
ejpam-4914	499	5	i	i	NOUN
ejpam-4914	499	6	)	)	PUNCT
ejpam-4914	499	7	γhi(g+h	γhi(g+h	NOUN
ejpam-4914	499	8	)	)	PUNCT
ejpam-4914	500	1	=	=	PUNCT
ejpam-4914	501	1	m+	m+	NUM
ejpam-4914	501	2	n	n	NOUN
ejpam-4914	501	3	if	if	SCONJ
ejpam-4914	501	4	g	g	PROPN
ejpam-4914	501	5	and	and	CCONJ
ejpam-4914	501	6	h	h	NOUN
ejpam-4914	501	7	are	be	AUX
ejpam-4914	501	8	complete	complete	ADJ
ejpam-4914	501	9	graphs	graph	NOUN
ejpam-4914	501	10	;	;	PUNCT
ejpam-4914	501	11	(	(	PUNCT
ejpam-4914	501	12	ii	ii	NOUN
ejpam-4914	501	13	)	)	PUNCT
ejpam-4914	501	14	γhi(g+h	γhi(g+h	NOUN
ejpam-4914	501	15	)	)	PUNCT
ejpam-4914	502	1	=	=	PUNCT
ejpam-4914	502	2	4	4	NUM
ejpam-4914	502	3	if	if	SCONJ
ejpam-4914	502	4	both	both	PRON
ejpam-4914	502	5	g	g	PROPN
ejpam-4914	502	6	and	and	CCONJ
ejpam-4914	502	7	h	h	NOUN
ejpam-4914	502	8	have	have	VERB
ejpam-4914	502	9	isolated	isolate	VERB
ejpam-4914	502	10	vertices	vertex	NOUN
ejpam-4914	502	11	;	;	PUNCT
ejpam-4914	502	12	(	(	PUNCT
ejpam-4914	502	13	iii	iii	NOUN
ejpam-4914	502	14	)	)	PUNCT
ejpam-4914	502	15	γhi(g+h	γhi(g+h	NOUN
ejpam-4914	502	16	)	)	PUNCT
ejpam-4914	502	17	=	=	SYM
ejpam-4914	503	1	1	1	NUM
ejpam-4914	503	2	+	+	CCONJ
ejpam-4914	503	3	pndi(h	pndi(h	ADP
ejpam-4914	503	4	)	)	PUNCT
ejpam-4914	503	5	if	if	SCONJ
ejpam-4914	503	6	g	g	PROPN
ejpam-4914	503	7	=	=	SYM
ejpam-4914	503	8	k1	k1	PROPN
ejpam-4914	503	9	;	;	PUNCT
ejpam-4914	503	10	proposition	proposition	NOUN
ejpam-4914	503	11	3.5	3.5	NUM
ejpam-4914	503	12	.	.	PUNCT
ejpam-4914	504	1	let	let	VERB
ejpam-4914	504	2	g	g	PRON
ejpam-4914	504	3	be	be	AUX
ejpam-4914	504	4	a	a	DET
ejpam-4914	504	5	graph	graph	NOUN
ejpam-4914	504	6	with	with	ADP
ejpam-4914	504	7	no	no	DET
ejpam-4914	504	8	isolated	isolated	ADJ
ejpam-4914	504	9	vertices	vertex	NOUN
ejpam-4914	504	10	.	.	PUNCT
ejpam-4914	505	1	then	then	ADV
ejpam-4914	505	2	γhi(g	γhi(g	PROPN
ejpam-4914	505	3	◦	◦	NOUN
ejpam-4914	505	4	h	h	NOUN
ejpam-4914	505	5	)	)	PUNCT
ejpam-4914	505	6	≤	≤	NOUN
ejpam-4914	505	7	γ∗t1,2(g	γ∗t1,2(g	PROPN
ejpam-4914	505	8	)	)	PUNCT
ejpam-4914	505	9	.	.	PUNCT
ejpam-4914	506	1	proof	proof	NOUN
ejpam-4914	506	2	:	:	PUNCT
ejpam-4914	506	3	let	let	VERB
ejpam-4914	506	4	s	s	PRON
ejpam-4914	506	5	⊆	⊆	NUM
ejpam-4914	506	6	v	v	NOUN
ejpam-4914	506	7	(	(	PUNCT
ejpam-4914	506	8	g	g	NOUN
ejpam-4914	506	9	)	)	PUNCT
ejpam-4914	506	10	be	be	AUX
ejpam-4914	506	11	a	a	DET
ejpam-4914	506	12	γ∗t1,2	γ∗t1,2	NOUN
ejpam-4914	506	13	-	-	PUNCT
ejpam-4914	506	14	set	set	NOUN
ejpam-4914	506	15	of	of	ADP
ejpam-4914	506	16	g	g	NOUN
ejpam-4914	506	17	,	,	PUNCT
ejpam-4914	506	18	and	and	CCONJ
ejpam-4914	506	19	define	define	VERB
ejpam-4914	506	20	f	f	PROPN
ejpam-4914	506	21	=	=	SYM
ejpam-4914	506	22	(	(	PUNCT
ejpam-4914	506	23	v0	v0	PROPN
ejpam-4914	506	24	,	,	PUNCT
ejpam-4914	506	25	v1	v1	NOUN
ejpam-4914	506	26	,	,	PUNCT
ejpam-4914	506	27	v2	v2	PROPN
ejpam-4914	506	28	)	)	PUNCT
ejpam-4914	506	29	,	,	PUNCT
ejpam-4914	506	30	where	where	SCONJ
ejpam-4914	506	31	v0	v0	NOUN
ejpam-4914	506	32	=	=	SYM
ejpam-4914	506	33	v	v	PROPN
ejpam-4914	506	34	(	(	PUNCT
ejpam-4914	506	35	g	g	PROPN
ejpam-4914	506	36	◦	◦	NOUN
ejpam-4914	506	37	h	h	NOUN
ejpam-4914	506	38	)	)	PUNCT
ejpam-4914	506	39	\	\	PROPN
ejpam-4914	506	40	s	s	PROPN
ejpam-4914	506	41	,	,	PUNCT
ejpam-4914	506	42	v1	v1	NOUN
ejpam-4914	506	43	=	=	SYM
ejpam-4914	506	44	∅	∅	NOUN
ejpam-4914	506	45	and	and	CCONJ
ejpam-4914	506	46	v2	v2	PROPN
ejpam-4914	506	47	=	=	SYM
ejpam-4914	506	48	s.	s.	PROPN
ejpam-4914	506	49	let	let	VERB
ejpam-4914	506	50	v	v	NUM
ejpam-4914	506	51	∈	∈	PROPN
ejpam-4914	506	52	v0	v0	NOUN
ejpam-4914	506	53	∩	∩	X
ejpam-4914	506	54	v	v	X
ejpam-4914	506	55	(	(	PUNCT
ejpam-4914	506	56	g	g	NOUN
ejpam-4914	506	57	)	)	PUNCT
ejpam-4914	506	58	.	.	PUNCT
ejpam-4914	507	1	since	since	SCONJ
ejpam-4914	507	2	v2	v2	PROPN
ejpam-4914	507	3	is	be	AUX
ejpam-4914	507	4	a	a	DET
ejpam-4914	507	5	hop	hop	NOUN
ejpam-4914	507	6	dominating	dominating	NOUN
ejpam-4914	507	7	set	set	NOUN
ejpam-4914	507	8	of	of	ADP
ejpam-4914	507	9	g	g	NOUN
ejpam-4914	507	10	,	,	PUNCT
ejpam-4914	507	11	there	there	PRON
ejpam-4914	507	12	exists	exist	VERB
ejpam-4914	507	13	u	u	PROPN
ejpam-4914	507	14	∈	∈	PROPN
ejpam-4914	507	15	v2	v2	NOUN
ejpam-4914	507	16	for	for	ADP
ejpam-4914	507	17	which	which	PRON
ejpam-4914	507	18	dg(u	dg(u	X
ejpam-4914	507	19	,	,	PUNCT
ejpam-4914	507	20	v	v	NOUN
ejpam-4914	507	21	)	)	PUNCT
ejpam-4914	507	22	=	=	SYM
ejpam-4914	507	23	2	2	X
ejpam-4914	507	24	.	.	X
ejpam-4914	507	25	let	let	VERB
ejpam-4914	507	26	v	v	NUM
ejpam-4914	507	27	∈	∈	PROPN
ejpam-4914	507	28	v0∩v	v0∩v	X
ejpam-4914	507	29	(	(	PUNCT
ejpam-4914	507	30	hu	hu	PROPN
ejpam-4914	507	31	)	)	PUNCT
ejpam-4914	507	32	,	,	PUNCT
ejpam-4914	507	33	where	where	SCONJ
ejpam-4914	507	34	u	u	PROPN
ejpam-4914	507	35	∈	∈	PROPN
ejpam-4914	507	36	v	v	X
ejpam-4914	507	37	(	(	PUNCT
ejpam-4914	507	38	g	g	NOUN
ejpam-4914	507	39	)	)	PUNCT
ejpam-4914	507	40	.	.	PUNCT
ejpam-4914	508	1	since	since	SCONJ
ejpam-4914	508	2	v2	v2	PROPN
ejpam-4914	508	3	is	be	AUX
ejpam-4914	508	4	a	a	DET
ejpam-4914	508	5	total	total	ADJ
ejpam-4914	508	6	dominating	dominating	NOUN
ejpam-4914	508	7	set	set	NOUN
ejpam-4914	508	8	of	of	ADP
ejpam-4914	508	9	g	g	NOUN
ejpam-4914	508	10	,	,	PUNCT
ejpam-4914	508	11	there	there	PRON
ejpam-4914	508	12	exists	exist	VERB
ejpam-4914	508	13	w	w	PROPN
ejpam-4914	508	14	∈	∈	PROPN
ejpam-4914	508	15	v2	v2	PROPN
ejpam-4914	508	16	∩ng(u	∩ng(u	PROPN
ejpam-4914	508	17	)	)	PUNCT
ejpam-4914	508	18	.	.	PUNCT
ejpam-4914	509	1	then	then	ADV
ejpam-4914	509	2	dg	dg	VERB
ejpam-4914	509	3	◦	◦	PROPN
ejpam-4914	509	4	h(u	h(u	PROPN
ejpam-4914	509	5	,	,	PUNCT
ejpam-4914	509	6	w	w	NOUN
ejpam-4914	509	7	)	)	PUNCT
ejpam-4914	509	8	=	=	SYM
ejpam-4914	509	9	2	2	X
ejpam-4914	509	10	.	.	PUNCT
ejpam-4914	510	1	thus	thus	ADV
ejpam-4914	510	2	,	,	PUNCT
ejpam-4914	510	3	f	f	PROPN
ejpam-4914	510	4	∈	∈	PROPN
ejpam-4914	510	5	hid(g	hid(g	PROPN
ejpam-4914	510	6	◦	◦	NOUN
ejpam-4914	510	7	h	h	NOUN
ejpam-4914	510	8	)	)	PUNCT
ejpam-4914	510	9	.	.	PUNCT
ejpam-4914	511	1	consequently	consequently	ADV
ejpam-4914	511	2	,	,	PUNCT
ejpam-4914	511	3	γhi(g	γhi(g	PROPN
ejpam-4914	511	4	◦	◦	NOUN
ejpam-4914	511	5	h	h	NOUN
ejpam-4914	511	6	)	)	PUNCT
ejpam-4914	511	7	≤	≤	NOUN
ejpam-4914	511	8	ωg	ωg	NOUN
ejpam-4914	511	9	◦	◦	NOUN
ejpam-4914	511	10	h(f	h(f	X
ejpam-4914	511	11	)	)	PUNCT
ejpam-4914	511	12	=	=	SYM
ejpam-4914	511	13	2|s|	2|s|	NUM
ejpam-4914	511	14	=	=	SYM
ejpam-4914	511	15	2γ∗t1,2(g	2γ∗t1,2(g	NUM
ejpam-4914	511	16	)	)	PUNCT
ejpam-4914	511	17	.	.	PUNCT
ejpam-4914	512	1	■	■	PUNCT
ejpam-4914	512	2	theorem	theorem	VERB
ejpam-4914	512	3	3.6	3.6	NUM
ejpam-4914	512	4	.	.	PUNCT
ejpam-4914	513	1	(	(	PUNCT
ejpam-4914	513	2	corona	corona	NOUN
ejpam-4914	513	3	of	of	ADP
ejpam-4914	513	4	graphs	graph	NOUN
ejpam-4914	513	5	)	)	PUNCT
ejpam-4914	513	6	let	let	VERB
ejpam-4914	513	7	g	g	NOUN
ejpam-4914	513	8	be	be	AUX
ejpam-4914	513	9	a	a	DET
ejpam-4914	513	10	nontrivial	nontrivial	ADJ
ejpam-4914	513	11	connected	connect	VERB
ejpam-4914	513	12	graph	graph	NOUN
ejpam-4914	513	13	and	and	CCONJ
ejpam-4914	513	14	h	h	NOUN
ejpam-4914	513	15	any	any	DET
ejpam-4914	513	16	graph	graph	NOUN
ejpam-4914	513	17	,	,	PUNCT
ejpam-4914	513	18	and	and	CCONJ
ejpam-4914	513	19	let	let	VERB
ejpam-4914	513	20	f	f	PROPN
ejpam-4914	513	21	=	=	SYM
ejpam-4914	513	22	(	(	PUNCT
ejpam-4914	513	23	v0	v0	PROPN
ejpam-4914	513	24	,	,	PUNCT
ejpam-4914	513	25	v1	v1	NOUN
ejpam-4914	513	26	,	,	PUNCT
ejpam-4914	513	27	v2	v2	PROPN
ejpam-4914	513	28	)	)	PUNCT
ejpam-4914	513	29	be	be	AUX
ejpam-4914	513	30	a	a	DET
ejpam-4914	513	31	function	function	NOUN
ejpam-4914	513	32	on	on	ADP
ejpam-4914	513	33	v	v	NOUN
ejpam-4914	513	34	(	(	PUNCT
ejpam-4914	513	35	g	g	PROPN
ejpam-4914	513	36	◦	◦	NOUN
ejpam-4914	513	37	h	h	NOUN
ejpam-4914	513	38	)	)	PUNCT
ejpam-4914	513	39	.	.	PUNCT
ejpam-4914	514	1	then	then	ADV
ejpam-4914	514	2	f	f	PROPN
ejpam-4914	514	3	∈	∈	PROPN
ejpam-4914	514	4	hid(g	hid(g	PROPN
ejpam-4914	514	5	◦	◦	NOUN
ejpam-4914	514	6	h	h	NOUN
ejpam-4914	514	7	)	)	PUNCT
ejpam-4914	514	8	if	if	SCONJ
ejpam-4914	514	9	and	and	CCONJ
ejpam-4914	514	10	only	only	ADV
ejpam-4914	514	11	if	if	SCONJ
ejpam-4914	514	12	each	each	PRON
ejpam-4914	514	13	of	of	ADP
ejpam-4914	514	14	the	the	DET
ejpam-4914	514	15	following	follow	VERB
ejpam-4914	514	16	holds	hold	VERB
ejpam-4914	514	17	:	:	PUNCT
ejpam-4914	514	18	(	(	PUNCT
ejpam-4914	514	19	i	i	NOUN
ejpam-4914	514	20	)	)	PUNCT
ejpam-4914	514	21	one	one	NUM
ejpam-4914	514	22	of	of	ADP
ejpam-4914	514	23	the	the	DET
ejpam-4914	514	24	following	following	NOUN
ejpam-4914	514	25	holds	hold	VERB
ejpam-4914	514	26	for	for	ADP
ejpam-4914	514	27	each	each	DET
ejpam-4914	514	28	v	v	NOUN
ejpam-4914	514	29	∈	∈	PROPN
ejpam-4914	514	30	v0	v0	NOUN
ejpam-4914	514	31	∩	∩	X
ejpam-4914	514	32	v	v	X
ejpam-4914	514	33	(	(	PUNCT
ejpam-4914	514	34	g	g	NOUN
ejpam-4914	514	35	):	):	PUNCT
ejpam-4914	514	36	(	(	PUNCT
ejpam-4914	514	37	a	a	X
ejpam-4914	514	38	)	)	PUNCT
ejpam-4914	514	39	|v2	|v2	PROPN
ejpam-4914	514	40	∩ng(v	∩ng(v	PROPN
ejpam-4914	514	41	,	,	PUNCT
ejpam-4914	514	42	2)|	2)|	NUM
ejpam-4914	514	43	≥	≥	NUM
ejpam-4914	514	44	1	1	NUM
ejpam-4914	514	45	or	or	CCONJ
ejpam-4914	514	46	|v1	|v1	PROPN
ejpam-4914	514	47	∩ng(v	∩ng(v	PROPN
ejpam-4914	514	48	,	,	PUNCT
ejpam-4914	514	49	2)|	2)|	NUM
ejpam-4914	514	50	≥	≥	NOUN
ejpam-4914	514	51	2	2	NUM
ejpam-4914	514	52	;	;	PUNCT
ejpam-4914	514	53	(	(	PUNCT
ejpam-4914	514	54	b	b	X
ejpam-4914	514	55	)	)	PUNCT
ejpam-4914	514	56	there	there	PRON
ejpam-4914	514	57	exists	exist	VERB
ejpam-4914	514	58	w	w	PROPN
ejpam-4914	514	59	∈	∈	PROPN
ejpam-4914	514	60	ng(v	ng(v	PUNCT
ejpam-4914	514	61	)	)	PUNCT
ejpam-4914	514	62	for	for	ADP
ejpam-4914	514	63	which	which	PRON
ejpam-4914	514	64	|v2	|v2	NOUN
ejpam-4914	514	65	∩	∩	NOUN
ejpam-4914	514	66	v	v	X
ejpam-4914	514	67	(	(	PUNCT
ejpam-4914	514	68	hw)|	hw)|	ADJ
ejpam-4914	514	69	≥	≥	NOUN
ejpam-4914	514	70	1	1	NUM
ejpam-4914	514	71	;	;	PUNCT
ejpam-4914	514	72	c	c	X
ejpam-4914	514	73	)	)	PUNCT
ejpam-4914	514	74	there	there	PRON
ejpam-4914	514	75	exists	exist	VERB
ejpam-4914	514	76	w	w	PROPN
ejpam-4914	514	77	∈	∈	PROPN
ejpam-4914	514	78	ng(v	ng(v	PUNCT
ejpam-4914	514	79	)	)	PUNCT
ejpam-4914	514	80	for	for	ADP
ejpam-4914	514	81	which	which	PRON
ejpam-4914	514	82	|v1	|v1	PROPN
ejpam-4914	514	83	∩	∩	ADJ
ejpam-4914	514	84	v	v	X
ejpam-4914	514	85	(	(	PUNCT
ejpam-4914	514	86	hw)|	hw)|	PRON
ejpam-4914	514	87	≥	≥	NOUN
ejpam-4914	514	88	2	2	NUM
ejpam-4914	514	89	;	;	PUNCT
ejpam-4914	514	90	(	(	PUNCT
ejpam-4914	514	91	d	d	X
ejpam-4914	514	92	)	)	PUNCT
ejpam-4914	514	93	there	there	PRON
ejpam-4914	514	94	exist	exist	VERB
ejpam-4914	514	95	u	u	NOUN
ejpam-4914	514	96	,	,	PUNCT
ejpam-4914	514	97	w	w	PROPN
ejpam-4914	514	98	∈	∈	PROPN
ejpam-4914	514	99	ng(v	ng(v	PUNCT
ejpam-4914	514	100	)	)	PUNCT
ejpam-4914	514	101	for	for	ADP
ejpam-4914	514	102	which	which	PRON
ejpam-4914	514	103	|v1	|v1	PROPN
ejpam-4914	514	104	∩	∩	ADJ
ejpam-4914	514	105	v	v	X
ejpam-4914	514	106	(	(	PUNCT
ejpam-4914	514	107	hw)|	hw)|	PROPN
ejpam-4914	514	108	=	=	SYM
ejpam-4914	514	109	1	1	NUM
ejpam-4914	514	110	=	=	SYM
ejpam-4914	514	111	|v1	|v1	PROPN
ejpam-4914	514	112	∩	∩	ADJ
ejpam-4914	514	113	v	v	NOUN
ejpam-4914	514	114	(	(	PUNCT
ejpam-4914	514	115	hu)|	hu)|	NOUN
ejpam-4914	514	116	;	;	PUNCT
ejpam-4914	514	117	(	(	PUNCT
ejpam-4914	514	118	e	e	NOUN
ejpam-4914	514	119	)	)	PUNCT
ejpam-4914	514	120	|v1	|v1	PROPN
ejpam-4914	514	121	∩ng(v	∩ng(v	PROPN
ejpam-4914	514	122	,	,	PUNCT
ejpam-4914	514	123	2)|	2)|	NUM
ejpam-4914	515	1	=	=	SYM
ejpam-4914	515	2	1	1	NUM
ejpam-4914	516	1	and	and	CCONJ
ejpam-4914	516	2	there	there	PRON
ejpam-4914	516	3	exists	exist	VERB
ejpam-4914	516	4	w	w	PROPN
ejpam-4914	516	5	∈	∈	PROPN
ejpam-4914	516	6	ng(v	ng(v	PUNCT
ejpam-4914	516	7	)	)	PUNCT
ejpam-4914	516	8	for	for	ADP
ejpam-4914	516	9	which	which	PRON
ejpam-4914	516	10	|v1	|v1	PROPN
ejpam-4914	516	11	∩	∩	ADJ
ejpam-4914	516	12	v	v	X
ejpam-4914	516	13	(	(	PUNCT
ejpam-4914	516	14	hw)|	hw)|	PROPN
ejpam-4914	516	15	=	=	SYM
ejpam-4914	516	16	1	1	NUM
ejpam-4914	516	17	.	.	PUNCT
ejpam-4914	516	18	(	(	PUNCT
ejpam-4914	516	19	ii	ii	NOUN
ejpam-4914	516	20	)	)	PUNCT
ejpam-4914	516	21	each	each	PRON
ejpam-4914	516	22	of	of	ADP
ejpam-4914	516	23	the	the	DET
ejpam-4914	516	24	following	following	NOUN
ejpam-4914	516	25	holds	hold	VERB
ejpam-4914	516	26	for	for	ADP
ejpam-4914	516	27	every	every	PRON
ejpam-4914	516	28	v	v	NUM
ejpam-4914	516	29	∈	∈	NOUN
ejpam-4914	516	30	v	v	NOUN
ejpam-4914	516	31	(	(	PUNCT
ejpam-4914	516	32	g	g	NOUN
ejpam-4914	516	33	)	)	PUNCT
ejpam-4914	516	34	with	with	ADP
ejpam-4914	516	35	v2	v2	PROPN
ejpam-4914	516	36	∩ng(v	∩ng(v	PROPN
ejpam-4914	516	37	)	)	PUNCT
ejpam-4914	516	38	=	=	SYM
ejpam-4914	516	39	∅	∅	NOUN
ejpam-4914	516	40	:	:	PUNCT
ejpam-4914	516	41	(	(	PUNCT
ejpam-4914	516	42	a	a	X
ejpam-4914	516	43	)	)	PUNCT
ejpam-4914	516	44	f	f	NOUN
ejpam-4914	516	45	|hv	|hv	NOUN
ejpam-4914	516	46	is	be	AUX
ejpam-4914	516	47	a	a	DET
ejpam-4914	516	48	pndi	pndi	ADJ
ejpam-4914	516	49	-	-	PUNCT
ejpam-4914	516	50	function	function	NOUN
ejpam-4914	516	51	of	of	ADP
ejpam-4914	516	52	hv	hv	PROPN
ejpam-4914	516	53	if	if	SCONJ
ejpam-4914	516	54	ng(v	ng(v	NOUN
ejpam-4914	516	55	)	)	PUNCT
ejpam-4914	516	56	⊆	⊆	NUM
ejpam-4914	516	57	v0	v0	NOUN
ejpam-4914	516	58	;	;	PUNCT
ejpam-4914	516	59	s.r	s.r	PROPN
ejpam-4914	516	60	.	.	PROPN
ejpam-4914	516	61	jr	jr	PROPN
ejpam-4914	516	62	.	.	PROPN
ejpam-4914	516	63	canoy	canoy	PROPN
ejpam-4914	516	64	,	,	PUNCT
ejpam-4914	516	65	f.p	f.p	PROPN
ejpam-4914	516	66	.	.	PROPN
ejpam-4914	516	67	jamil	jamil	PROPN
ejpam-4914	516	68	and	and	CCONJ
ejpam-4914	516	69	s.m	s.m	PROPN
ejpam-4914	516	70	.	.	PROPN
ejpam-4914	516	71	menchavez	menchavez	PROPN
ejpam-4914	516	72	/	/	PUNCT
ejpam-4914	516	73	eur	eur	PROPN
ejpam-4914	516	74	.	.	PUNCT
ejpam-4914	517	1	j.	j.	PROPN
ejpam-4914	517	2	pure	pure	PROPN
ejpam-4914	517	3	appl	appl	PROPN
ejpam-4914	517	4	.	.	PROPN
ejpam-4914	517	5	math	math	PROPN
ejpam-4914	517	6	,	,	PUNCT
ejpam-4914	517	7	16	16	NUM
ejpam-4914	517	8	(	(	PUNCT
ejpam-4914	517	9	4	4	NUM
ejpam-4914	517	10	)	)	PUNCT
ejpam-4914	517	11	(	(	PUNCT
ejpam-4914	517	12	2023	2023	NUM
ejpam-4914	517	13	)	)	PUNCT
ejpam-4914	517	14	,	,	PUNCT
ejpam-4914	517	15	2431	2431	NUM
ejpam-4914	517	16	-	-	SYM
ejpam-4914	517	17	2449	2449	NUM
ejpam-4914	517	18	2443	2443	NUM
ejpam-4914	517	19	(	(	PUNCT
ejpam-4914	517	20	b	b	NOUN
ejpam-4914	517	21	)	)	PUNCT
ejpam-4914	517	22	v	v	NOUN
ejpam-4914	517	23	(	(	PUNCT
ejpam-4914	517	24	hv	hv	NOUN
ejpam-4914	517	25	)	)	PUNCT
ejpam-4914	517	26	\	\	PROPN
ejpam-4914	517	27	v0	v0	NOUN
ejpam-4914	517	28	is	be	AUX
ejpam-4914	517	29	a	a	DET
ejpam-4914	517	30	pnd	pnd	NOUN
ejpam-4914	517	31	-	-	PUNCT
ejpam-4914	517	32	set	set	NOUN
ejpam-4914	517	33	of	of	ADP
ejpam-4914	517	34	hv	hv	PRON
ejpam-4914	517	35	if	if	SCONJ
ejpam-4914	517	36	|v1	|v1	PROPN
ejpam-4914	517	37	∩ng(v)|	∩ng(v)|	PROPN
ejpam-4914	517	38	=	=	SYM
ejpam-4914	517	39	1	1	X
ejpam-4914	517	40	.	.	PUNCT
ejpam-4914	517	41	proof	proof	NOUN
ejpam-4914	517	42	:	:	PUNCT
ejpam-4914	517	43	suppose	suppose	VERB
ejpam-4914	517	44	that	that	SCONJ
ejpam-4914	517	45	f	f	PROPN
ejpam-4914	517	46	∈	∈	PROPN
ejpam-4914	517	47	hid(g	hid(g	PROPN
ejpam-4914	517	48	◦	◦	NOUN
ejpam-4914	517	49	h	h	NOUN
ejpam-4914	517	50	)	)	PUNCT
ejpam-4914	517	51	.	.	PUNCT
ejpam-4914	518	1	then	then	ADV
ejpam-4914	518	2	(	(	PUNCT
ejpam-4914	518	3	i	i	NOUN
ejpam-4914	518	4	)	)	PUNCT
ejpam-4914	518	5	is	be	AUX
ejpam-4914	518	6	clear	clear	ADJ
ejpam-4914	518	7	.	.	PUNCT
ejpam-4914	519	1	let	let	VERB
ejpam-4914	519	2	v	v	NUM
ejpam-4914	519	3	∈	∈	PROPN
ejpam-4914	519	4	v	v	NOUN
ejpam-4914	519	5	(	(	PUNCT
ejpam-4914	519	6	g	g	NOUN
ejpam-4914	519	7	)	)	PUNCT
ejpam-4914	519	8	with	with	ADP
ejpam-4914	519	9	v2	v2	PROPN
ejpam-4914	519	10	∩ng(v	∩ng(v	PROPN
ejpam-4914	519	11	)	)	PUNCT
ejpam-4914	519	12	=	=	PUNCT
ejpam-4914	519	13	∅.	∅.	PROPN
ejpam-4914	519	14	suppose	suppose	VERB
ejpam-4914	519	15	that	that	SCONJ
ejpam-4914	519	16	ng(v	ng(v	NOUN
ejpam-4914	519	17	)	)	PUNCT
ejpam-4914	519	18	⊆	⊆	NUM
ejpam-4914	519	19	v0	v0	NOUN
ejpam-4914	519	20	,	,	PUNCT
ejpam-4914	519	21	and	and	CCONJ
ejpam-4914	519	22	let	let	VERB
ejpam-4914	519	23	u	u	PRON
ejpam-4914	519	24	∈	∈	PROPN
ejpam-4914	519	25	v0	v0	NOUN
ejpam-4914	519	26	∩	∩	X
ejpam-4914	519	27	v	v	X
ejpam-4914	519	28	(	(	PUNCT
ejpam-4914	519	29	hv	hv	PROPN
ejpam-4914	519	30	)	)	PUNCT
ejpam-4914	519	31	.	.	PUNCT
ejpam-4914	520	1	then	then	ADV
ejpam-4914	520	2	|v2	|v2	PROPN
ejpam-4914	520	3	∩	∩	PROPN
ejpam-4914	520	4	ng	ng	PROPN
ejpam-4914	520	5	◦	◦	PROPN
ejpam-4914	520	6	h(u	h(u	PROPN
ejpam-4914	520	7	,	,	PUNCT
ejpam-4914	520	8	2)|	2)|	NUM
ejpam-4914	520	9	≥	≥	NOUN
ejpam-4914	520	10	1	1	NUM
ejpam-4914	520	11	of	of	ADP
ejpam-4914	520	12	|v1	|v1	PROPN
ejpam-4914	520	13	∩	∩	PROPN
ejpam-4914	520	14	ng	ng	PROPN
ejpam-4914	520	15	◦	◦	PROPN
ejpam-4914	520	16	h(u	h(u	PROPN
ejpam-4914	520	17	,	,	PUNCT
ejpam-4914	520	18	2)|	2)|	NUM
ejpam-4914	520	19	≥	≥	NOUN
ejpam-4914	520	20	2	2	NUM
ejpam-4914	520	21	.	.	PUNCT
ejpam-4914	520	22	since	since	SCONJ
ejpam-4914	520	23	ng(v	ng(v	NOUN
ejpam-4914	520	24	)	)	PUNCT
ejpam-4914	520	25	⊆	⊆	NUM
ejpam-4914	520	26	v0	v0	NOUN
ejpam-4914	520	27	,	,	PUNCT
ejpam-4914	520	28	the	the	DET
ejpam-4914	520	29	preceding	precede	VERB
ejpam-4914	520	30	statement	statement	NOUN
ejpam-4914	520	31	implies	imply	VERB
ejpam-4914	520	32	that	that	SCONJ
ejpam-4914	520	33	|v2∩nhv(u	|v2∩nhv(u	NOUN
ejpam-4914	520	34	,	,	PUNCT
ejpam-4914	520	35	2)|	2)|	NUM
ejpam-4914	520	36	≥	≥	NUM
ejpam-4914	520	37	1	1	NUM
ejpam-4914	520	38	or	or	CCONJ
ejpam-4914	520	39	|v1∩nhv(u	|v1∩nhv(u	ADJ
ejpam-4914	520	40	,	,	PUNCT
ejpam-4914	520	41	2)|	2)|	NUM
ejpam-4914	520	42	≥	≥	NOUN
ejpam-4914	520	43	2	2	NUM
ejpam-4914	520	44	.	.	PUNCT
ejpam-4914	521	1	it	it	PRON
ejpam-4914	521	2	means	mean	VERB
ejpam-4914	521	3	that	that	SCONJ
ejpam-4914	521	4	there	there	PRON
ejpam-4914	521	5	exists	exist	VERB
ejpam-4914	521	6	w	w	PROPN
ejpam-4914	521	7	∈	∈	PROPN
ejpam-4914	521	8	v2∩v	v2∩v	NOUN
ejpam-4914	521	9	(	(	PUNCT
ejpam-4914	521	10	hv	hv	PROPN
ejpam-4914	521	11	)	)	PUNCT
ejpam-4914	521	12	for	for	ADP
ejpam-4914	521	13	which	which	PRON
ejpam-4914	521	14	u	u	X
ejpam-4914	521	15	/∈	/∈	PUNCT
ejpam-4914	521	16	nhv(w	nhv(w	PROPN
ejpam-4914	521	17	)	)	PUNCT
ejpam-4914	521	18	or	or	CCONJ
ejpam-4914	521	19	there	there	PRON
ejpam-4914	521	20	exist	exist	VERB
ejpam-4914	521	21	w	w	ADP
ejpam-4914	521	22	and	and	CCONJ
ejpam-4914	521	23	z	z	NOUN
ejpam-4914	521	24	in	in	ADP
ejpam-4914	521	25	v1	v1	PROPN
ejpam-4914	521	26	∩	∩	ADJ
ejpam-4914	521	27	v	v	X
ejpam-4914	521	28	(	(	PUNCT
ejpam-4914	521	29	hv	hv	PROPN
ejpam-4914	521	30	)	)	PUNCT
ejpam-4914	521	31	for	for	ADP
ejpam-4914	521	32	which	which	PRON
ejpam-4914	521	33	u	u	X
ejpam-4914	521	34	/∈	/∈	PROPN
ejpam-4914	521	35	nhv(w)∪nhv(z	nhv(w)∪nhv(z	PROPN
ejpam-4914	521	36	)	)	PUNCT
ejpam-4914	521	37	.	.	PUNCT
ejpam-4914	522	1	thus	thus	ADV
ejpam-4914	522	2	,	,	PUNCT
ejpam-4914	522	3	f	f	PROPN
ejpam-4914	522	4	|hv	|hv	NUM
ejpam-4914	522	5	=	=	SYM
ejpam-4914	522	6	(	(	PUNCT
ejpam-4914	522	7	v0∩v	v0∩v	X
ejpam-4914	522	8	(	(	PUNCT
ejpam-4914	522	9	hv	hv	NOUN
ejpam-4914	522	10	)	)	PUNCT
ejpam-4914	522	11	,	,	PUNCT
ejpam-4914	522	12	v1∩v	v1∩v	PROPN
ejpam-4914	522	13	(	(	PUNCT
ejpam-4914	522	14	hv	hv	PROPN
ejpam-4914	522	15	)	)	PUNCT
ejpam-4914	522	16	,	,	PUNCT
ejpam-4914	522	17	v2∩v	v2∩v	PROPN
ejpam-4914	522	18	(	(	PUNCT
ejpam-4914	522	19	hv	hv	NOUN
ejpam-4914	522	20	)	)	PUNCT
ejpam-4914	522	21	)	)	PUNCT
ejpam-4914	522	22	is	be	AUX
ejpam-4914	522	23	a	a	DET
ejpam-4914	522	24	pndi	pndi	ADJ
ejpam-4914	522	25	-function	-function	NOUN
ejpam-4914	522	26	of	of	ADP
ejpam-4914	522	27	hv	hv	PROPN
ejpam-4914	522	28	and	and	CCONJ
ejpam-4914	522	29	(	(	PUNCT
ejpam-4914	522	30	ii)(a	ii)(a	PROPN
ejpam-4914	522	31	)	)	PUNCT
ejpam-4914	522	32	holds	hold	VERB
ejpam-4914	522	33	.	.	PUNCT
ejpam-4914	523	1	suppose	suppose	VERB
ejpam-4914	523	2	that	that	SCONJ
ejpam-4914	523	3	|v1	|v1	PROPN
ejpam-4914	523	4	∩ng(v)|	∩ng(v)|	PROPN
ejpam-4914	523	5	=	=	SYM
ejpam-4914	523	6	1	1	X
ejpam-4914	523	7	.	.	PUNCT
ejpam-4914	523	8	let	let	VERB
ejpam-4914	523	9	u	u	PRON
ejpam-4914	523	10	∈	∈	PROPN
ejpam-4914	523	11	v0	v0	NOUN
ejpam-4914	523	12	∩	∩	X
ejpam-4914	523	13	v	v	X
ejpam-4914	523	14	(	(	PUNCT
ejpam-4914	523	15	hv	hv	PROPN
ejpam-4914	523	16	)	)	PUNCT
ejpam-4914	523	17	.	.	PUNCT
ejpam-4914	524	1	following	follow	VERB
ejpam-4914	524	2	similar	similar	ADJ
ejpam-4914	524	3	argument	argument	NOUN
ejpam-4914	524	4	,	,	PUNCT
ejpam-4914	524	5	since	since	SCONJ
ejpam-4914	524	6	|v1	|v1	PROPN
ejpam-4914	524	7	∩ng(v)|	∩ng(v)|	PROPN
ejpam-4914	524	8	=	=	SYM
ejpam-4914	524	9	1	1	NUM
ejpam-4914	524	10	,	,	PUNCT
ejpam-4914	524	11	we	we	PRON
ejpam-4914	524	12	have	have	VERB
ejpam-4914	524	13	|v2	|v2	NOUN
ejpam-4914	524	14	∩nhv(u	∩nhv(u	PRON
ejpam-4914	524	15	,	,	PUNCT
ejpam-4914	524	16	2)|	2)|	NUM
ejpam-4914	524	17	≥	≥	NOUN
ejpam-4914	524	18	1	1	NUM
ejpam-4914	524	19	or	or	CCONJ
ejpam-4914	524	20	|v1	|v1	NUM
ejpam-4914	524	21	∩nhv(u	∩nhv(u	NUM
ejpam-4914	524	22	,	,	PUNCT
ejpam-4914	524	23	2)|	2)|	NUM
ejpam-4914	524	24	≥	≥	NOUN
ejpam-4914	524	25	1	1	NUM
ejpam-4914	524	26	.	.	PUNCT
ejpam-4914	525	1	in	in	ADP
ejpam-4914	525	2	any	any	DET
ejpam-4914	525	3	case	case	NOUN
ejpam-4914	525	4	,	,	PUNCT
ejpam-4914	525	5	there	there	PRON
ejpam-4914	525	6	exists	exist	VERB
ejpam-4914	525	7	w	w	PROPN
ejpam-4914	525	8	∈	∈	PROPN
ejpam-4914	525	9	v	v	NOUN
ejpam-4914	525	10	(	(	PUNCT
ejpam-4914	525	11	hv)\v0	hv)\v0	INTJ
ejpam-4914	525	12	such	such	ADJ
ejpam-4914	525	13	that	that	DET
ejpam-4914	525	14	u	u	NOUN
ejpam-4914	525	15	/∈	/∈	PUNCT
ejpam-4914	525	16	nhv(w	nhv(w	PROPN
ejpam-4914	525	17	)	)	PUNCT
ejpam-4914	525	18	,	,	PUNCT
ejpam-4914	525	19	showing	show	VERB
ejpam-4914	525	20	that	that	DET
ejpam-4914	525	21	v	v	NOUN
ejpam-4914	525	22	(	(	PUNCT
ejpam-4914	525	23	hv)\v0	hv)\v0	PROPN
ejpam-4914	525	24	is	be	AUX
ejpam-4914	525	25	a	a	DET
ejpam-4914	525	26	pnd	pnd	NOUN
ejpam-4914	525	27	-	-	PUNCT
ejpam-4914	525	28	set	set	NOUN
ejpam-4914	525	29	of	of	ADP
ejpam-4914	525	30	hv	hv	PROPN
ejpam-4914	525	31	.	.	PUNCT
ejpam-4914	526	1	this	this	PRON
ejpam-4914	526	2	proves	prove	VERB
ejpam-4914	526	3	(	(	PUNCT
ejpam-4914	526	4	ii)(b	ii)(b	ADJ
ejpam-4914	526	5	)	)	PUNCT
ejpam-4914	526	6	.	.	PUNCT
ejpam-4914	527	1	conversely	conversely	ADV
ejpam-4914	527	2	,	,	PUNCT
ejpam-4914	527	3	suppose	suppose	VERB
ejpam-4914	527	4	that	that	SCONJ
ejpam-4914	527	5	(	(	PUNCT
ejpam-4914	527	6	i	i	NOUN
ejpam-4914	527	7	)	)	PUNCT
ejpam-4914	527	8	and	and	CCONJ
ejpam-4914	527	9	(	(	PUNCT
ejpam-4914	527	10	ii	ii	NOUN
ejpam-4914	527	11	)	)	PUNCT
ejpam-4914	527	12	hold	hold	VERB
ejpam-4914	527	13	for	for	ADP
ejpam-4914	527	14	f	f	PROPN
ejpam-4914	527	15	.	.	PUNCT
ejpam-4914	528	1	let	let	VERB
ejpam-4914	528	2	v	v	NUM
ejpam-4914	528	3	∈	∈	PROPN
ejpam-4914	528	4	v0	v0	NOUN
ejpam-4914	528	5	.	.	PUNCT
ejpam-4914	529	1	if	if	SCONJ
ejpam-4914	529	2	v	v	NUM
ejpam-4914	529	3	∈	∈	PROPN
ejpam-4914	529	4	v	v	NOUN
ejpam-4914	529	5	(	(	PUNCT
ejpam-4914	529	6	g	g	NOUN
ejpam-4914	529	7	)	)	PUNCT
ejpam-4914	529	8	,	,	PUNCT
ejpam-4914	529	9	then	then	ADV
ejpam-4914	529	10	(	(	PUNCT
ejpam-4914	529	11	i	i	NOUN
ejpam-4914	529	12	)	)	PUNCT
ejpam-4914	529	13	implies	imply	VERB
ejpam-4914	529	14	the	the	DET
ejpam-4914	529	15	existence	existence	NOUN
ejpam-4914	529	16	of	of	ADP
ejpam-4914	529	17	u	u	PROPN
ejpam-4914	529	18	∈	∈	PROPN
ejpam-4914	529	19	v2	v2	NOUN
ejpam-4914	529	20	such	such	ADJ
ejpam-4914	529	21	that	that	SCONJ
ejpam-4914	529	22	dg	dg	VERB
ejpam-4914	529	23	◦	◦	PROPN
ejpam-4914	529	24	h(u	h(u	PROPN
ejpam-4914	529	25	,	,	PUNCT
ejpam-4914	529	26	v	v	NOUN
ejpam-4914	529	27	)	)	PUNCT
ejpam-4914	529	28	=	=	SYM
ejpam-4914	529	29	2	2	NUM
ejpam-4914	529	30	or	or	CCONJ
ejpam-4914	529	31	of	of	ADP
ejpam-4914	529	32	vertices	vertex	NOUN
ejpam-4914	529	33	u	u	NOUN
ejpam-4914	529	34	and	and	CCONJ
ejpam-4914	529	35	w	w	NOUN
ejpam-4914	529	36	in	in	ADP
ejpam-4914	529	37	v1	v1	NOUN
ejpam-4914	529	38	such	such	ADJ
ejpam-4914	529	39	that	that	SCONJ
ejpam-4914	529	40	dg	dg	PROPN
ejpam-4914	529	41	◦	◦	PROPN
ejpam-4914	529	42	h(u	h(u	PROPN
ejpam-4914	529	43	,	,	PUNCT
ejpam-4914	529	44	v	v	NOUN
ejpam-4914	529	45	)	)	PUNCT
ejpam-4914	529	46	=	=	SYM
ejpam-4914	529	47	2	2	NUM
ejpam-4914	529	48	=	=	SYM
ejpam-4914	529	49	dg	dg	NOUN
ejpam-4914	529	50	◦	◦	NOUN
ejpam-4914	529	51	h(w	h(w	PROPN
ejpam-4914	529	52	,	,	PUNCT
ejpam-4914	529	53	v	v	NOUN
ejpam-4914	529	54	)	)	PUNCT
ejpam-4914	529	55	.	.	PUNCT
ejpam-4914	530	1	now	now	ADV
ejpam-4914	530	2	,	,	PUNCT
ejpam-4914	530	3	suppose	suppose	VERB
ejpam-4914	530	4	that	that	SCONJ
ejpam-4914	530	5	v	v	ADP
ejpam-4914	530	6	∈	∈	PROPN
ejpam-4914	530	7	v	v	NOUN
ejpam-4914	530	8	(	(	PUNCT
ejpam-4914	530	9	hu	hu	PROPN
ejpam-4914	530	10	)	)	PUNCT
ejpam-4914	530	11	for	for	ADP
ejpam-4914	530	12	some	some	DET
ejpam-4914	530	13	u	u	NOUN
ejpam-4914	530	14	∈	∈	PROPN
ejpam-4914	530	15	v	v	NOUN
ejpam-4914	530	16	(	(	PUNCT
ejpam-4914	530	17	g	g	NOUN
ejpam-4914	530	18	)	)	PUNCT
ejpam-4914	530	19	.	.	PUNCT
ejpam-4914	531	1	if	if	SCONJ
ejpam-4914	531	2	v2	v2	PROPN
ejpam-4914	531	3	∩ng(u	∩ng(u	PROPN
ejpam-4914	531	4	)	)	PUNCT
ejpam-4914	531	5	̸=	̸=	PROPN
ejpam-4914	531	6	∅	∅	NOUN
ejpam-4914	531	7	,	,	PUNCT
ejpam-4914	531	8	and	and	CCONJ
ejpam-4914	531	9	w	w	PROPN
ejpam-4914	531	10	∈	∈	PROPN
ejpam-4914	531	11	v2	v2	PROPN
ejpam-4914	531	12	∩ng(u	∩ng(u	PROPN
ejpam-4914	531	13	)	)	PUNCT
ejpam-4914	531	14	,	,	PUNCT
ejpam-4914	531	15	then	then	ADV
ejpam-4914	531	16	w	w	PROPN
ejpam-4914	531	17	is	be	AUX
ejpam-4914	531	18	the	the	DET
ejpam-4914	531	19	desired	desire	VERB
ejpam-4914	531	20	vertex	vertex	NOUN
ejpam-4914	531	21	for	for	ADP
ejpam-4914	531	22	which	which	PRON
ejpam-4914	531	23	w	w	PROPN
ejpam-4914	531	24	∈	∈	PROPN
ejpam-4914	531	25	v2	v2	PROPN
ejpam-4914	531	26	and	and	CCONJ
ejpam-4914	531	27	dg	dg	NOUN
ejpam-4914	531	28	◦	◦	NOUN
ejpam-4914	531	29	h(v	h(v	PROPN
ejpam-4914	531	30	,	,	PUNCT
ejpam-4914	531	31	w	w	NOUN
ejpam-4914	531	32	)	)	PUNCT
ejpam-4914	531	33	=	=	SYM
ejpam-4914	531	34	2	2	X
ejpam-4914	531	35	.	.	PUNCT
ejpam-4914	531	36	suppose	suppose	VERB
ejpam-4914	531	37	that	that	SCONJ
ejpam-4914	531	38	v2	v2	PROPN
ejpam-4914	531	39	∩ng(u	∩ng(u	PROPN
ejpam-4914	531	40	)	)	PUNCT
ejpam-4914	531	41	=	=	PUNCT
ejpam-4914	531	42	∅.	∅.	NOUN
ejpam-4914	531	43	we	we	PRON
ejpam-4914	531	44	consider	consider	VERB
ejpam-4914	531	45	two	two	NUM
ejpam-4914	531	46	cases	case	NOUN
ejpam-4914	531	47	:	:	PUNCT
ejpam-4914	531	48	case	case	NOUN
ejpam-4914	531	49	1	1	NUM
ejpam-4914	531	50	:	:	PUNCT
ejpam-4914	531	51	if	if	SCONJ
ejpam-4914	531	52	ng(u	ng(u	NOUN
ejpam-4914	531	53	)	)	PUNCT
ejpam-4914	531	54	⊆	⊆	NUM
ejpam-4914	531	55	v0	v0	NOUN
ejpam-4914	531	56	,	,	PUNCT
ejpam-4914	531	57	then	then	ADV
ejpam-4914	531	58	by	by	ADP
ejpam-4914	531	59	condition	condition	NOUN
ejpam-4914	531	60	(	(	PUNCT
ejpam-4914	531	61	ii)(a	ii)(a	PROPN
ejpam-4914	531	62	)	)	PUNCT
ejpam-4914	531	63	,	,	PUNCT
ejpam-4914	531	64	there	there	PRON
ejpam-4914	531	65	exists	exist	VERB
ejpam-4914	531	66	there	there	PRON
ejpam-4914	531	67	exists	exist	VERB
ejpam-4914	531	68	w	w	PROPN
ejpam-4914	531	69	∈	∈	PROPN
ejpam-4914	531	70	v2∩v	v2∩v	NOUN
ejpam-4914	531	71	(	(	PUNCT
ejpam-4914	531	72	hu	hu	PROPN
ejpam-4914	531	73	)	)	PUNCT
ejpam-4914	531	74	for	for	ADP
ejpam-4914	531	75	which	which	PRON
ejpam-4914	531	76	v	v	X
ejpam-4914	531	77	/∈	/∈	PUNCT
ejpam-4914	531	78	nhu(w	nhu(w	PROPN
ejpam-4914	531	79	)	)	PUNCT
ejpam-4914	531	80	or	or	CCONJ
ejpam-4914	531	81	there	there	PRON
ejpam-4914	531	82	exist	exist	VERB
ejpam-4914	531	83	vertices	vertex	NOUN
ejpam-4914	531	84	z	z	NOUN
ejpam-4914	531	85	and	and	CCONJ
ejpam-4914	531	86	w	w	PROPN
ejpam-4914	531	87	in	in	ADP
ejpam-4914	531	88	v1	v1	NOUN
ejpam-4914	531	89	∩	∩	ADJ
ejpam-4914	531	90	v	v	X
ejpam-4914	531	91	(	(	PUNCT
ejpam-4914	531	92	hu	hu	PROPN
ejpam-4914	531	93	)	)	PUNCT
ejpam-4914	531	94	for	for	ADP
ejpam-4914	531	95	which	which	PRON
ejpam-4914	531	96	v	v	X
ejpam-4914	531	97	/∈	/∈	PUNCT
ejpam-4914	531	98	nhu(w	nhu(w	PROPN
ejpam-4914	531	99	)	)	PUNCT
ejpam-4914	531	100	∪	∪	ADP
ejpam-4914	531	101	nhv(z	nhv(z	PROPN
ejpam-4914	531	102	)	)	PUNCT
ejpam-4914	531	103	.	.	PUNCT
ejpam-4914	532	1	the	the	DET
ejpam-4914	532	2	former	former	ADJ
ejpam-4914	532	3	implies	imply	VERB
ejpam-4914	532	4	that	that	SCONJ
ejpam-4914	532	5	dg	dg	VERB
ejpam-4914	532	6	◦	◦	NOUN
ejpam-4914	532	7	h(w	h(w	PROPN
ejpam-4914	532	8	,	,	PUNCT
ejpam-4914	532	9	v	v	NOUN
ejpam-4914	532	10	)	)	PUNCT
ejpam-4914	532	11	=	=	SYM
ejpam-4914	532	12	2	2	NUM
ejpam-4914	532	13	,	,	PUNCT
ejpam-4914	532	14	while	while	SCONJ
ejpam-4914	532	15	latter	latter	ADJ
ejpam-4914	532	16	implies	imply	VERB
ejpam-4914	532	17	that	that	SCONJ
ejpam-4914	532	18	dg	dg	VERB
ejpam-4914	532	19	◦	◦	NOUN
ejpam-4914	532	20	h(w	h(w	PROPN
ejpam-4914	532	21	,	,	PUNCT
ejpam-4914	532	22	v	v	NOUN
ejpam-4914	532	23	)	)	PUNCT
ejpam-4914	532	24	=	=	SYM
ejpam-4914	532	25	2	2	NUM
ejpam-4914	532	26	=	=	SYM
ejpam-4914	532	27	dg	dg	NOUN
ejpam-4914	532	28	◦	◦	NOUN
ejpam-4914	532	29	h(z	h(z	NOUN
ejpam-4914	532	30	,	,	PUNCT
ejpam-4914	532	31	v	v	NOUN
ejpam-4914	532	32	)	)	PUNCT
ejpam-4914	532	33	.	.	PUNCT
ejpam-4914	533	1	case	case	NOUN
ejpam-4914	533	2	2	2	NUM
ejpam-4914	533	3	:	:	PUNCT
ejpam-4914	533	4	suppose	suppose	VERB
ejpam-4914	533	5	that	that	SCONJ
ejpam-4914	533	6	ng(u	ng(u	NOUN
ejpam-4914	533	7	)	)	PUNCT
ejpam-4914	533	8	∩	∩	NOUN
ejpam-4914	533	9	v1	v1	NOUN
ejpam-4914	533	10	̸=	̸=	PROPN
ejpam-4914	533	11	∅.	∅.	ADV
ejpam-4914	533	12	if	if	SCONJ
ejpam-4914	533	13	|ng(u	|ng(u	NUM
ejpam-4914	533	14	)	)	PUNCT
ejpam-4914	533	15	∩	∩	NOUN
ejpam-4914	533	16	v1|	v1|	NOUN
ejpam-4914	533	17	≥	≥	NUM
ejpam-4914	533	18	2	2	NUM
ejpam-4914	533	19	,	,	PUNCT
ejpam-4914	533	20	say	say	VERB
ejpam-4914	533	21	w	w	NOUN
ejpam-4914	533	22	,	,	PUNCT
ejpam-4914	533	23	z	z	PROPN
ejpam-4914	533	24	∈	∈	PROPN
ejpam-4914	533	25	ng(u	ng(u	NOUN
ejpam-4914	533	26	)	)	PUNCT
ejpam-4914	533	27	∩	∩	NOUN
ejpam-4914	533	28	v1	v1	NOUN
ejpam-4914	533	29	,	,	PUNCT
ejpam-4914	533	30	then	then	ADV
ejpam-4914	533	31	dg	dg	VERB
ejpam-4914	533	32	◦	◦	NOUN
ejpam-4914	533	33	h(w	h(w	PROPN
ejpam-4914	533	34	,	,	PUNCT
ejpam-4914	533	35	v	v	NOUN
ejpam-4914	533	36	)	)	PUNCT
ejpam-4914	533	37	=	=	SYM
ejpam-4914	533	38	2	2	NUM
ejpam-4914	533	39	=	=	SYM
ejpam-4914	533	40	dg	dg	NOUN
ejpam-4914	533	41	◦	◦	NOUN
ejpam-4914	533	42	h(z	h(z	NOUN
ejpam-4914	533	43	,	,	PUNCT
ejpam-4914	533	44	v	v	NOUN
ejpam-4914	533	45	)	)	PUNCT
ejpam-4914	533	46	.	.	PUNCT
ejpam-4914	534	1	suppose	suppose	VERB
ejpam-4914	534	2	that	that	SCONJ
ejpam-4914	534	3	|ng(u	|ng(u	X
ejpam-4914	534	4	)	)	PUNCT
ejpam-4914	534	5	∩	∩	ADJ
ejpam-4914	534	6	v1|	v1|	NOUN
ejpam-4914	534	7	=	=	SYM
ejpam-4914	534	8	1	1	NUM
ejpam-4914	534	9	,	,	PUNCT
ejpam-4914	534	10	say	say	VERB
ejpam-4914	534	11	x	x	SYM
ejpam-4914	534	12	∈	∈	PROPN
ejpam-4914	534	13	ng(u	ng(u	NOUN
ejpam-4914	534	14	)	)	PUNCT
ejpam-4914	534	15	∩	∩	NOUN
ejpam-4914	534	16	v1	v1	NOUN
ejpam-4914	534	17	.	.	PUNCT
ejpam-4914	535	1	by	by	ADP
ejpam-4914	535	2	(	(	PUNCT
ejpam-4914	535	3	ii)(b	ii)(b	PROPN
ejpam-4914	535	4	)	)	PUNCT
ejpam-4914	535	5	,	,	PUNCT
ejpam-4914	535	6	v	v	X
ejpam-4914	535	7	(	(	PUNCT
ejpam-4914	535	8	hu	hu	PROPN
ejpam-4914	535	9	)	)	PUNCT
ejpam-4914	535	10	\	\	PROPN
ejpam-4914	535	11	v0	v0	NOUN
ejpam-4914	535	12	is	be	AUX
ejpam-4914	535	13	a	a	DET
ejpam-4914	535	14	pnd	pnd	NOUN
ejpam-4914	535	15	-	-	PUNCT
ejpam-4914	535	16	set	set	NOUN
ejpam-4914	535	17	of	of	ADP
ejpam-4914	535	18	hu	hu	PROPN
ejpam-4914	535	19	so	so	SCONJ
ejpam-4914	535	20	that	that	SCONJ
ejpam-4914	535	21	there	there	PRON
ejpam-4914	535	22	exists	exist	VERB
ejpam-4914	535	23	w	w	PROPN
ejpam-4914	535	24	∈	∈	PROPN
ejpam-4914	535	25	v	v	ADP
ejpam-4914	535	26	(	(	PUNCT
ejpam-4914	535	27	hu	hu	PROPN
ejpam-4914	535	28	)	)	PUNCT
ejpam-4914	535	29	\	\	PROPN
ejpam-4914	535	30	v0	v0	NOUN
ejpam-4914	535	31	such	such	ADJ
ejpam-4914	535	32	that	that	DET
ejpam-4914	535	33	v	v	NOUN
ejpam-4914	535	34	/∈	/∈	PUNCT
ejpam-4914	535	35	nhu(w	nhu(w	PROPN
ejpam-4914	535	36	)	)	PUNCT
ejpam-4914	535	37	.	.	PUNCT
ejpam-4914	536	1	we	we	PRON
ejpam-4914	536	2	either	either	CCONJ
ejpam-4914	536	3	have	have	VERB
ejpam-4914	536	4	w	w	PROPN
ejpam-4914	536	5	∈	∈	PROPN
ejpam-4914	536	6	v2	v2	PROPN
ejpam-4914	536	7	and	and	CCONJ
ejpam-4914	536	8	dg	dg	NOUN
ejpam-4914	536	9	◦	◦	NOUN
ejpam-4914	536	10	h(w	h(w	PROPN
ejpam-4914	536	11	,	,	PUNCT
ejpam-4914	536	12	v	v	NOUN
ejpam-4914	536	13	)	)	PUNCT
ejpam-4914	536	14	=	=	SYM
ejpam-4914	536	15	2	2	NUM
ejpam-4914	536	16	or	or	CCONJ
ejpam-4914	536	17	w	w	PROPN
ejpam-4914	536	18	∈	∈	PROPN
ejpam-4914	536	19	v1	v1	NOUN
ejpam-4914	536	20	and	and	CCONJ
ejpam-4914	536	21	dg	dg	NOUN
ejpam-4914	536	22	◦	◦	NOUN
ejpam-4914	536	23	h(w	h(w	PROPN
ejpam-4914	536	24	,	,	PUNCT
ejpam-4914	536	25	v	v	NOUN
ejpam-4914	536	26	)	)	PUNCT
ejpam-4914	536	27	=	=	SYM
ejpam-4914	536	28	2	2	NUM
ejpam-4914	536	29	=	=	SYM
ejpam-4914	536	30	dg	dg	NOUN
ejpam-4914	536	31	◦	◦	NOUN
ejpam-4914	536	32	h(x	h(x	PROPN
ejpam-4914	536	33	,	,	PUNCT
ejpam-4914	536	34	v	v	NOUN
ejpam-4914	536	35	)	)	PUNCT
ejpam-4914	536	36	.	.	PUNCT
ejpam-4914	537	1	accordingly	accordingly	ADV
ejpam-4914	537	2	,	,	PUNCT
ejpam-4914	537	3	f	f	PROPN
ejpam-4914	537	4	∈	∈	PROPN
ejpam-4914	537	5	hid(g	hid(g	PROPN
ejpam-4914	537	6	◦	◦	NOUN
ejpam-4914	537	7	h	h	NOUN
ejpam-4914	537	8	)	)	PUNCT
ejpam-4914	537	9	.	.	PUNCT
ejpam-4914	538	1	■	■	PUNCT
ejpam-4914	538	2	corollary	corollary	ADJ
ejpam-4914	538	3	3.7	3.7	NUM
ejpam-4914	538	4	.	.	PUNCT
ejpam-4914	539	1	let	let	VERB
ejpam-4914	539	2	g	g	PRON
ejpam-4914	539	3	be	be	AUX
ejpam-4914	539	4	a	a	DET
ejpam-4914	539	5	connected	connected	ADJ
ejpam-4914	539	6	graph	graph	NOUN
ejpam-4914	539	7	of	of	ADP
ejpam-4914	539	8	order	order	NOUN
ejpam-4914	539	9	n	n	NOUN
ejpam-4914	540	1	and	and	CCONJ
ejpam-4914	540	2	h	h	NOUN
ejpam-4914	540	3	be	be	AUX
ejpam-4914	540	4	any	any	DET
ejpam-4914	540	5	graph	graph	NOUN
ejpam-4914	540	6	.	.	PUNCT
ejpam-4914	541	1	(	(	PUNCT
ejpam-4914	541	2	i	i	NOUN
ejpam-4914	541	3	)	)	PUNCT
ejpam-4914	541	4	if	if	SCONJ
ejpam-4914	541	5	γ(g	γ(g	PROPN
ejpam-4914	541	6	)	)	PUNCT
ejpam-4914	541	7	=	=	SYM
ejpam-4914	542	1	1	1	NUM
ejpam-4914	542	2	,	,	PUNCT
ejpam-4914	542	3	then	then	ADV
ejpam-4914	542	4	4	4	NUM
ejpam-4914	542	5	≤	≤	NOUN
ejpam-4914	542	6	γhi(g	γhi(g	ADP
ejpam-4914	542	7	◦	◦	NOUN
ejpam-4914	542	8	h	h	NOUN
ejpam-4914	542	9	)	)	PUNCT
ejpam-4914	542	10	≤	≤	NOUN
ejpam-4914	542	11	6	6	NUM
ejpam-4914	542	12	.	.	PUNCT
ejpam-4914	543	1	more	more	ADV
ejpam-4914	543	2	precisely	precisely	ADV
ejpam-4914	543	3	,	,	PUNCT
ejpam-4914	543	4	(	(	PUNCT
ejpam-4914	543	5	a	a	X
ejpam-4914	543	6	)	)	PUNCT
ejpam-4914	543	7	γhi(g	γhi(g	PROPN
ejpam-4914	543	8	◦	◦	NOUN
ejpam-4914	543	9	h	h	NOUN
ejpam-4914	543	10	)	)	PUNCT
ejpam-4914	543	11	=	=	SYM
ejpam-4914	543	12	4	4	NUM
ejpam-4914	543	13	if	if	SCONJ
ejpam-4914	543	14	γh(g	γh(g	NOUN
ejpam-4914	543	15	)	)	PUNCT
ejpam-4914	543	16	=	=	SYM
ejpam-4914	543	17	2	2	NUM
ejpam-4914	543	18	or	or	CCONJ
ejpam-4914	543	19	h	h	NOUN
ejpam-4914	543	20	=	=	SYM
ejpam-4914	543	21	k2	k2	PROPN
ejpam-4914	543	22	or	or	CCONJ
ejpam-4914	543	23	h	h	NOUN
ejpam-4914	543	24	has	have	VERB
ejpam-4914	543	25	an	an	DET
ejpam-4914	543	26	isolated	isolated	ADJ
ejpam-4914	543	27	vertex	vertex	NOUN
ejpam-4914	543	28	;	;	PUNCT
ejpam-4914	543	29	(	(	PUNCT
ejpam-4914	543	30	b	b	X
ejpam-4914	543	31	)	)	PUNCT
ejpam-4914	543	32	γhi(g	γhi(g	PROPN
ejpam-4914	544	1	◦	◦	NOUN
ejpam-4914	544	2	h	h	NOUN
ejpam-4914	544	3	)	)	PUNCT
ejpam-4914	544	4	=	=	SYM
ejpam-4914	544	5	5	5	NUM
ejpam-4914	544	6	if	if	SCONJ
ejpam-4914	544	7	pndi(h	pndi(h	ADJ
ejpam-4914	544	8	)	)	PUNCT
ejpam-4914	544	9	=	=	SYM
ejpam-4914	544	10	3	3	NUM
ejpam-4914	544	11	;	;	PUNCT
ejpam-4914	544	12	and	and	CCONJ
ejpam-4914	544	13	(	(	PUNCT
ejpam-4914	544	14	c	c	X
ejpam-4914	544	15	)	)	PUNCT
ejpam-4914	544	16	γhi(g	γhi(g	PROPN
ejpam-4914	545	1	◦	◦	NOUN
ejpam-4914	545	2	h	h	NOUN
ejpam-4914	545	3	)	)	PUNCT
ejpam-4914	545	4	=	=	NOUN
ejpam-4914	545	5	6	6	NUM
ejpam-4914	545	6	if	if	SCONJ
ejpam-4914	545	7	pndi(h	pndi(h	ADJ
ejpam-4914	545	8	)	)	PUNCT
ejpam-4914	545	9	≥	≥	NOUN
ejpam-4914	545	10	4	4	NUM
ejpam-4914	545	11	.	.	PUNCT
ejpam-4914	545	12	(	(	PUNCT
ejpam-4914	545	13	ii	ii	NOUN
ejpam-4914	545	14	)	)	PUNCT
ejpam-4914	545	15	in	in	ADP
ejpam-4914	545	16	general	general	ADJ
ejpam-4914	545	17	,	,	PUNCT
ejpam-4914	545	18	4	4	NUM
ejpam-4914	545	19	≤	≤	NOUN
ejpam-4914	545	20	γhi(g	γhi(g	ADP
ejpam-4914	545	21	◦	◦	NOUN
ejpam-4914	545	22	h	h	NOUN
ejpam-4914	545	23	)	)	PUNCT
ejpam-4914	545	24	≤	≤	NUM
ejpam-4914	545	25	ρh(g	ρh(g	NOUN
ejpam-4914	545	26	)	)	PUNCT
ejpam-4914	545	27	,	,	PUNCT
ejpam-4914	545	28	where	where	SCONJ
ejpam-4914	545	29	ρh(g	ρh(g	NOUN
ejpam-4914	545	30	)	)	PUNCT
ejpam-4914	545	31	=	=	SYM
ejpam-4914	545	32	min{2|s|+(n−	min{2|s|+(n−	NOUN
ejpam-4914	545	33	|ng(s)|	|ng(s)|	NOUN
ejpam-4914	545	34	)	)	PUNCT
ejpam-4914	545	35	pndi(h	pndi(h	ADP
ejpam-4914	545	36	)	)	PUNCT
ejpam-4914	545	37	:	:	PUNCT
ejpam-4914	545	38	s	s	VERB
ejpam-4914	545	39	∈	∈	NOUN
ejpam-4914	545	40	hd(g	hd(g	NOUN
ejpam-4914	545	41	)	)	PUNCT
ejpam-4914	545	42	}	}	PUNCT
ejpam-4914	545	43	,	,	PUNCT
ejpam-4914	545	44	and	and	CCONJ
ejpam-4914	545	45	this	this	DET
ejpam-4914	545	46	bound	bind	VERB
ejpam-4914	545	47	is	be	AUX
ejpam-4914	545	48	tight	tight	ADJ
ejpam-4914	545	49	.	.	PUNCT
ejpam-4914	546	1	s.r	s.r	PROPN
ejpam-4914	546	2	.	.	PROPN
ejpam-4914	546	3	jr	jr	PROPN
ejpam-4914	546	4	.	.	PROPN
ejpam-4914	546	5	canoy	canoy	PROPN
ejpam-4914	546	6	,	,	PUNCT
ejpam-4914	546	7	f.p	f.p	PROPN
ejpam-4914	546	8	.	.	PROPN
ejpam-4914	546	9	jamil	jamil	PROPN
ejpam-4914	546	10	and	and	CCONJ
ejpam-4914	546	11	s.m	s.m	PROPN
ejpam-4914	546	12	.	.	PROPN
ejpam-4914	546	13	menchavez	menchavez	PROPN
ejpam-4914	546	14	/	/	PUNCT
ejpam-4914	546	15	eur	eur	PROPN
ejpam-4914	546	16	.	.	PUNCT
ejpam-4914	547	1	j.	j.	PROPN
ejpam-4914	547	2	pure	pure	PROPN
ejpam-4914	547	3	appl	appl	PROPN
ejpam-4914	547	4	.	.	PROPN
ejpam-4914	547	5	math	math	PROPN
ejpam-4914	547	6	,	,	PUNCT
ejpam-4914	547	7	16	16	NUM
ejpam-4914	547	8	(	(	PUNCT
ejpam-4914	547	9	4	4	NUM
ejpam-4914	547	10	)	)	PUNCT
ejpam-4914	547	11	(	(	PUNCT
ejpam-4914	547	12	2023	2023	NUM
ejpam-4914	547	13	)	)	PUNCT
ejpam-4914	547	14	,	,	PUNCT
ejpam-4914	547	15	2431	2431	NUM
ejpam-4914	547	16	-	-	SYM
ejpam-4914	547	17	2449	2449	NUM
ejpam-4914	547	18	2444	2444	NUM
ejpam-4914	547	19	proof	proof	NOUN
ejpam-4914	547	20	:	:	PUNCT
ejpam-4914	547	21	in	in	ADP
ejpam-4914	547	22	any	any	DET
ejpam-4914	547	23	case	case	NOUN
ejpam-4914	547	24	γhi(g	γhi(g	ADP
ejpam-4914	547	25	◦	◦	NOUN
ejpam-4914	547	26	h	h	NOUN
ejpam-4914	547	27	)	)	PUNCT
ejpam-4914	547	28	≥	≥	NOUN
ejpam-4914	547	29	4	4	NUM
ejpam-4914	547	30	by	by	ADP
ejpam-4914	547	31	proposition	proposition	NOUN
ejpam-4914	547	32	2.4	2.4	NUM
ejpam-4914	547	33	.	.	PUNCT
ejpam-4914	547	34	suppose	suppose	VERB
ejpam-4914	547	35	that	that	SCONJ
ejpam-4914	547	36	γ(g	γ(g	PROPN
ejpam-4914	547	37	)	)	PUNCT
ejpam-4914	547	38	=	=	SYM
ejpam-4914	547	39	1	1	NUM
ejpam-4914	547	40	,	,	PUNCT
ejpam-4914	547	41	and	and	CCONJ
ejpam-4914	547	42	let	let	VERB
ejpam-4914	547	43	u	u	PRON
ejpam-4914	547	44	∈	∈	PROPN
ejpam-4914	547	45	v	v	ADP
ejpam-4914	547	46	(	(	PUNCT
ejpam-4914	547	47	g	g	NOUN
ejpam-4914	547	48	)	)	PUNCT
ejpam-4914	547	49	for	for	ADP
ejpam-4914	547	50	which	which	PRON
ejpam-4914	547	51	ng[u	ng[u	ADP
ejpam-4914	547	52	]	]	X
ejpam-4914	547	53	=	=	SYM
ejpam-4914	547	54	v	v	NOUN
ejpam-4914	547	55	(	(	PUNCT
ejpam-4914	547	56	g	g	NOUN
ejpam-4914	547	57	)	)	PUNCT
ejpam-4914	547	58	.	.	PUNCT
ejpam-4914	548	1	pick	pick	VERB
ejpam-4914	548	2	vu	vu	NOUN
ejpam-4914	548	3	∈	∈	PROPN
ejpam-4914	548	4	v	v	PROPN
ejpam-4914	548	5	(	(	PUNCT
ejpam-4914	548	6	hu	hu	PROPN
ejpam-4914	548	7	)	)	PUNCT
ejpam-4914	548	8	and	and	CCONJ
ejpam-4914	548	9	w	w	PROPN
ejpam-4914	548	10	∈	∈	PROPN
ejpam-4914	548	11	v	v	ADP
ejpam-4914	548	12	(	(	PUNCT
ejpam-4914	548	13	g	g	NOUN
ejpam-4914	548	14	)	)	PUNCT
ejpam-4914	548	15	\	\	NOUN
ejpam-4914	548	16	{	{	PUNCT
ejpam-4914	548	17	u	u	NOUN
ejpam-4914	548	18	}	}	PUNCT
ejpam-4914	548	19	.	.	PUNCT
ejpam-4914	549	1	put	put	VERB
ejpam-4914	549	2	s	s	PART
ejpam-4914	549	3	=	=	PUNCT
ejpam-4914	549	4	{	{	PUNCT
ejpam-4914	549	5	u	u	NOUN
ejpam-4914	549	6	,	,	PUNCT
ejpam-4914	549	7	vu	vu	X
ejpam-4914	549	8	,	,	PUNCT
ejpam-4914	549	9	w	w	NOUN
ejpam-4914	549	10	}	}	PUNCT
ejpam-4914	549	11	,	,	PUNCT
ejpam-4914	549	12	and	and	CCONJ
ejpam-4914	549	13	define	define	VERB
ejpam-4914	549	14	v2	v2	PROPN
ejpam-4914	549	15	=	=	SYM
ejpam-4914	549	16	s	s	NOUN
ejpam-4914	549	17	,	,	PUNCT
ejpam-4914	549	18	v1	v1	NOUN
ejpam-4914	549	19	=	=	SYM
ejpam-4914	549	20	∅	∅	NOUN
ejpam-4914	549	21	and	and	CCONJ
ejpam-4914	549	22	v0	v0	NOUN
ejpam-4914	549	23	=	=	SYM
ejpam-4914	549	24	v	v	PROPN
ejpam-4914	549	25	(	(	PUNCT
ejpam-4914	549	26	g	g	PROPN
ejpam-4914	549	27	◦	◦	NOUN
ejpam-4914	549	28	h	h	NOUN
ejpam-4914	549	29	)	)	PUNCT
ejpam-4914	549	30	\	\	PUNCT
ejpam-4914	549	31	s.	s.	PROPN
ejpam-4914	549	32	by	by	ADP
ejpam-4914	549	33	theorem	theorem	NOUN
ejpam-4914	549	34	3.6	3.6	NUM
ejpam-4914	549	35	,	,	PUNCT
ejpam-4914	549	36	f	f	X
ejpam-4914	549	37	=	=	SYM
ejpam-4914	549	38	(	(	PUNCT
ejpam-4914	549	39	v0	v0	PROPN
ejpam-4914	549	40	,	,	PUNCT
ejpam-4914	549	41	v1	v1	NOUN
ejpam-4914	549	42	,	,	PUNCT
ejpam-4914	549	43	v2	v2	NOUN
ejpam-4914	549	44	)	)	PUNCT
ejpam-4914	549	45	∈	∈	PROPN
ejpam-4914	549	46	hid(g	hid(g	PROPN
ejpam-4914	549	47	◦	◦	NOUN
ejpam-4914	549	48	h	h	NOUN
ejpam-4914	549	49	)	)	PUNCT
ejpam-4914	549	50	.	.	PUNCT
ejpam-4914	550	1	thus	thus	ADV
ejpam-4914	550	2	,	,	PUNCT
ejpam-4914	550	3	γhi(g	γhi(g	PROPN
ejpam-4914	550	4	◦	◦	NOUN
ejpam-4914	550	5	h	h	NOUN
ejpam-4914	550	6	)	)	PUNCT
ejpam-4914	550	7	≤	≤	NOUN
ejpam-4914	550	8	ωg	ωg	NOUN
ejpam-4914	550	9	◦	◦	NOUN
ejpam-4914	550	10	h(f	h(f	NOUN
ejpam-4914	550	11	)	)	PUNCT
ejpam-4914	550	12	=	=	SYM
ejpam-4914	550	13	6	6	X
ejpam-4914	550	14	.	.	PUNCT
ejpam-4914	550	15	suppose	suppose	VERB
ejpam-4914	550	16	that	that	SCONJ
ejpam-4914	550	17	γh(g	γh(g	NOUN
ejpam-4914	550	18	)	)	PUNCT
ejpam-4914	550	19	=	=	SYM
ejpam-4914	550	20	2	2	NUM
ejpam-4914	550	21	,	,	PUNCT
ejpam-4914	550	22	and	and	CCONJ
ejpam-4914	550	23	let	let	VERB
ejpam-4914	550	24	s	s	PRON
ejpam-4914	550	25	be	be	AUX
ejpam-4914	550	26	a	a	DET
ejpam-4914	550	27	γh	γh	ADV
ejpam-4914	550	28	-	-	PUNCT
ejpam-4914	550	29	set	set	NOUN
ejpam-4914	550	30	of	of	ADP
ejpam-4914	550	31	g.	g.	PROPN
ejpam-4914	550	32	necessarily	necessarily	ADV
ejpam-4914	550	33	,	,	PUNCT
ejpam-4914	550	34	u	u	PROPN
ejpam-4914	550	35	∈	∈	PROPN
ejpam-4914	550	36	s.	s.	PROPN
ejpam-4914	550	37	put	put	VERB
ejpam-4914	550	38	s	s	VERB
ejpam-4914	550	39	=	=	PUNCT
ejpam-4914	550	40	{	{	PUNCT
ejpam-4914	550	41	u	u	NOUN
ejpam-4914	550	42	,	,	PUNCT
ejpam-4914	550	43	v	v	NOUN
ejpam-4914	550	44	}	}	PUNCT
ejpam-4914	550	45	,	,	PUNCT
ejpam-4914	550	46	where	where	SCONJ
ejpam-4914	550	47	dg(x	dg(x	NUM
ejpam-4914	550	48	,	,	PUNCT
ejpam-4914	550	49	v	v	NOUN
ejpam-4914	550	50	)	)	PUNCT
ejpam-4914	550	51	=	=	SYM
ejpam-4914	550	52	2	2	NUM
ejpam-4914	550	53	for	for	ADP
ejpam-4914	550	54	all	all	PRON
ejpam-4914	550	55	x	x	SYM
ejpam-4914	550	56	∈	∈	NOUN
ejpam-4914	550	57	v	v	NOUN
ejpam-4914	550	58	(	(	PUNCT
ejpam-4914	550	59	g	g	NOUN
ejpam-4914	550	60	)	)	PUNCT
ejpam-4914	550	61	\	\	NOUN
ejpam-4914	550	62	{	{	PUNCT
ejpam-4914	550	63	u	u	NOUN
ejpam-4914	550	64	}	}	PUNCT
ejpam-4914	550	65	.	.	PUNCT
ejpam-4914	551	1	define	define	VERB
ejpam-4914	551	2	v2	v2	PROPN
ejpam-4914	551	3	=	=	SYM
ejpam-4914	551	4	s	s	NOUN
ejpam-4914	551	5	,	,	PUNCT
ejpam-4914	551	6	v1	v1	NOUN
ejpam-4914	551	7	=	=	SYM
ejpam-4914	551	8	∅	∅	NOUN
ejpam-4914	551	9	and	and	CCONJ
ejpam-4914	551	10	v0	v0	NOUN
ejpam-4914	551	11	=	=	SYM
ejpam-4914	551	12	(	(	PUNCT
ejpam-4914	551	13	v	v	NOUN
ejpam-4914	551	14	(	(	PUNCT
ejpam-4914	551	15	g	g	NOUN
ejpam-4914	551	16	)	)	PUNCT
ejpam-4914	551	17	\	\	PROPN
ejpam-4914	552	1	s	s	X
ejpam-4914	552	2	)	)	PUNCT
ejpam-4914	552	3	∪	∪	PROPN
ejpam-4914	552	4	(	(	PUNCT
ejpam-4914	552	5	∪x∈v	∪x∈v	X
ejpam-4914	552	6	(	(	PUNCT
ejpam-4914	552	7	g)v	g)v	X
ejpam-4914	552	8	(	(	PUNCT
ejpam-4914	552	9	hx	hx	PROPN
ejpam-4914	552	10	)	)	PUNCT
ejpam-4914	552	11	)	)	PUNCT
ejpam-4914	552	12	.	.	PUNCT
ejpam-4914	553	1	since	since	SCONJ
ejpam-4914	553	2	v	v	NUM
ejpam-4914	553	3	(	(	PUNCT
ejpam-4914	553	4	hu	hu	PROPN
ejpam-4914	553	5	)	)	PUNCT
ejpam-4914	553	6	∪	∪	NOUN
ejpam-4914	553	7	(	(	PUNCT
ejpam-4914	553	8	v	v	NOUN
ejpam-4914	553	9	(	(	PUNCT
ejpam-4914	553	10	g	g	NOUN
ejpam-4914	553	11	)	)	PUNCT
ejpam-4914	553	12	\	\	PROPN
ejpam-4914	554	1	s	s	X
ejpam-4914	554	2	)	)	PUNCT
ejpam-4914	554	3	⊆	⊆	NUM
ejpam-4914	554	4	ng	ng	PROPN
ejpam-4914	554	5	◦	◦	NOUN
ejpam-4914	554	6	h(v	h(v	PROPN
ejpam-4914	554	7	,	,	PUNCT
ejpam-4914	554	8	2	2	NUM
ejpam-4914	554	9	)	)	PUNCT
ejpam-4914	554	10	and	and	CCONJ
ejpam-4914	554	11	∪x∈v	∪x∈v	PROPN
ejpam-4914	554	12	(	(	PUNCT
ejpam-4914	554	13	g)\{u}v	g)\{u}v	PROPN
ejpam-4914	554	14	(	(	PUNCT
ejpam-4914	554	15	hx	hx	PROPN
ejpam-4914	554	16	)	)	PUNCT
ejpam-4914	554	17	⊆	⊆	NUM
ejpam-4914	554	18	ng	ng	PROPN
ejpam-4914	554	19	◦	◦	PROPN
ejpam-4914	554	20	h(u	h(u	PROPN
ejpam-4914	554	21	,	,	PUNCT
ejpam-4914	554	22	2	2	NUM
ejpam-4914	554	23	)	)	PUNCT
ejpam-4914	554	24	,	,	PUNCT
ejpam-4914	554	25	f	f	PROPN
ejpam-4914	554	26	∈	∈	PROPN
ejpam-4914	554	27	hid(g	hid(g	PROPN
ejpam-4914	554	28	◦	◦	NOUN
ejpam-4914	554	29	h	h	NOUN
ejpam-4914	554	30	)	)	PUNCT
ejpam-4914	554	31	.	.	PUNCT
ejpam-4914	555	1	thus	thus	ADV
ejpam-4914	555	2	,	,	PUNCT
ejpam-4914	555	3	γhi(g	γhi(g	PROPN
ejpam-4914	555	4	◦	◦	NOUN
ejpam-4914	555	5	h	h	NOUN
ejpam-4914	555	6	)	)	PUNCT
ejpam-4914	555	7	≤	≤	NOUN
ejpam-4914	555	8	ωg	ωg	NOUN
ejpam-4914	555	9	◦	◦	NOUN
ejpam-4914	555	10	h(f	h(f	X
ejpam-4914	555	11	)	)	PUNCT
ejpam-4914	555	12	=	=	SYM
ejpam-4914	555	13	4	4	X
ejpam-4914	555	14	.	.	PUNCT
ejpam-4914	555	15	suppose	suppose	VERB
ejpam-4914	555	16	that	that	SCONJ
ejpam-4914	555	17	h	h	NOUN
ejpam-4914	555	18	has	have	VERB
ejpam-4914	555	19	an	an	DET
ejpam-4914	555	20	isolated	isolated	ADJ
ejpam-4914	555	21	vertex	vertex	NOUN
ejpam-4914	555	22	v.	v.	ADP
ejpam-4914	555	23	put	put	NOUN
ejpam-4914	555	24	s	s	PART
ejpam-4914	555	25	=	=	PUNCT
ejpam-4914	555	26	{	{	PUNCT
ejpam-4914	555	27	u	u	NOUN
ejpam-4914	555	28	,	,	PUNCT
ejpam-4914	555	29	vu	vu	PROPN
ejpam-4914	555	30	}	}	PUNCT
ejpam-4914	555	31	,	,	PUNCT
ejpam-4914	555	32	where	where	SCONJ
ejpam-4914	555	33	vu	vu	PROPN
ejpam-4914	555	34	is	be	AUX
ejpam-4914	555	35	the	the	DET
ejpam-4914	555	36	copy	copy	NOUN
ejpam-4914	555	37	of	of	ADP
ejpam-4914	555	38	vertex	vertex	NOUN
ejpam-4914	555	39	v	v	ADP
ejpam-4914	555	40	inhu	inhu	NOUN
ejpam-4914	555	41	.	.	PUNCT
ejpam-4914	556	1	define	define	VERB
ejpam-4914	556	2	v2	v2	PROPN
ejpam-4914	556	3	=	=	SYM
ejpam-4914	556	4	s	s	NOUN
ejpam-4914	556	5	,	,	PUNCT
ejpam-4914	556	6	v1	v1	NOUN
ejpam-4914	556	7	=	=	SYM
ejpam-4914	556	8	∅	∅	NOUN
ejpam-4914	556	9	and	and	CCONJ
ejpam-4914	556	10	v0	v0	NOUN
ejpam-4914	556	11	=	=	SYM
ejpam-4914	556	12	(	(	PUNCT
ejpam-4914	556	13	v	v	NOUN
ejpam-4914	556	14	(	(	PUNCT
ejpam-4914	556	15	g	g	NOUN
ejpam-4914	556	16	)	)	PUNCT
ejpam-4914	556	17	\	\	NOUN
ejpam-4914	556	18	{	{	PUNCT
ejpam-4914	556	19	u})∪	u})∪	PROPN
ejpam-4914	556	20	(	(	PUNCT
ejpam-4914	556	21	(	(	PUNCT
ejpam-4914	556	22	∪x∈v	∪x∈v	X
ejpam-4914	556	23	(	(	PUNCT
ejpam-4914	556	24	g)v	g)v	X
ejpam-4914	556	25	(	(	PUNCT
ejpam-4914	556	26	hx	hx	PROPN
ejpam-4914	556	27	)	)	PUNCT
ejpam-4914	556	28	)	)	PUNCT
ejpam-4914	556	29	\	\	NOUN
ejpam-4914	556	30	{	{	PUNCT
ejpam-4914	556	31	vu	vu	NOUN
ejpam-4914	556	32	}	}	PUNCT
ejpam-4914	556	33	)	)	PUNCT
ejpam-4914	556	34	.	.	PUNCT
ejpam-4914	557	1	then	then	ADV
ejpam-4914	557	2	(	(	PUNCT
ejpam-4914	557	3	v	v	X
ejpam-4914	557	4	(	(	PUNCT
ejpam-4914	557	5	g	g	NOUN
ejpam-4914	557	6	)	)	PUNCT
ejpam-4914	557	7	\	\	NOUN
ejpam-4914	557	8	{	{	PUNCT
ejpam-4914	557	9	u})∪(v	u})∪(v	PROPN
ejpam-4914	557	10	(	(	PUNCT
ejpam-4914	557	11	hu	hu	PROPN
ejpam-4914	557	12	)	)	PUNCT
ejpam-4914	557	13	\	\	NOUN
ejpam-4914	557	14	{	{	PUNCT
ejpam-4914	557	15	vu	vu	NOUN
ejpam-4914	557	16	}	}	PUNCT
ejpam-4914	557	17	)	)	PUNCT
ejpam-4914	558	1	⊆	⊆	NUM
ejpam-4914	558	2	ng	ng	PROPN
ejpam-4914	558	3	◦	◦	PROPN
ejpam-4914	558	4	h(vu	h(vu	PROPN
ejpam-4914	558	5	,	,	PUNCT
ejpam-4914	558	6	2	2	NUM
ejpam-4914	558	7	)	)	PUNCT
ejpam-4914	558	8	and	and	CCONJ
ejpam-4914	558	9	∪x∈v	∪x∈v	PROPN
ejpam-4914	558	10	(	(	PUNCT
ejpam-4914	558	11	g)\{u}v	g)\{u}v	PROPN
ejpam-4914	558	12	(	(	PUNCT
ejpam-4914	558	13	hx	hx	PROPN
ejpam-4914	558	14	)	)	PUNCT
ejpam-4914	558	15	⊆	⊆	NUM
ejpam-4914	558	16	ng	ng	PROPN
ejpam-4914	558	17	◦	◦	PROPN
ejpam-4914	558	18	h(u	h(u	PROPN
ejpam-4914	558	19	,	,	PUNCT
ejpam-4914	558	20	2	2	NUM
ejpam-4914	558	21	)	)	PUNCT
ejpam-4914	558	22	.	.	PUNCT
ejpam-4914	559	1	thus	thus	ADV
ejpam-4914	559	2	,	,	PUNCT
ejpam-4914	559	3	f	f	PROPN
ejpam-4914	559	4	∈	∈	PROPN
ejpam-4914	559	5	hid(g	hid(g	PROPN
ejpam-4914	559	6	◦	◦	NOUN
ejpam-4914	559	7	h	h	NOUN
ejpam-4914	559	8	)	)	PUNCT
ejpam-4914	560	1	so	so	SCONJ
ejpam-4914	560	2	that	that	SCONJ
ejpam-4914	560	3	γhi(g	γhi(g	ADP
ejpam-4914	560	4	◦	◦	NOUN
ejpam-4914	560	5	h	h	NOUN
ejpam-4914	560	6	)	)	PUNCT
ejpam-4914	560	7	≤	≤	NOUN
ejpam-4914	560	8	ωg	ωg	NOUN
ejpam-4914	560	9	◦	◦	NOUN
ejpam-4914	560	10	h(f	h(f	X
ejpam-4914	560	11	)	)	PUNCT
ejpam-4914	561	1	=	=	SYM
ejpam-4914	562	1	4	4	X
ejpam-4914	562	2	.	.	PUNCT
ejpam-4914	562	3	suppose	suppose	VERB
ejpam-4914	562	4	that	that	SCONJ
ejpam-4914	562	5	γh(g	γh(g	NOUN
ejpam-4914	562	6	)	)	PUNCT
ejpam-4914	562	7	̸=	̸=	PROPN
ejpam-4914	562	8	2	2	NUM
ejpam-4914	562	9	and	and	CCONJ
ejpam-4914	562	10	pndi(h	pndi(h	PROPN
ejpam-4914	562	11	)	)	PUNCT
ejpam-4914	562	12	≥	≥	NOUN
ejpam-4914	562	13	2	2	NUM
ejpam-4914	562	14	.	.	PUNCT
ejpam-4914	563	1	let	let	VERB
ejpam-4914	563	2	fu	fu	NOUN
ejpam-4914	563	3	=	=	PUNCT
ejpam-4914	563	4	(	(	PUNCT
ejpam-4914	563	5	v	v	NUM
ejpam-4914	563	6	u	u	NOUN
ejpam-4914	563	7	0	0	NUM
ejpam-4914	563	8	,	,	PUNCT
ejpam-4914	563	9	v	v	PART
ejpam-4914	563	10	u	u	NOUN
ejpam-4914	563	11	1	1	NUM
ejpam-4914	563	12	,	,	PUNCT
ejpam-4914	563	13	v	v	NOUN
ejpam-4914	563	14	u	u	NOUN
ejpam-4914	563	15	2	2	NUM
ejpam-4914	563	16	)	)	PUNCT
ejpam-4914	563	17	be	be	AUX
ejpam-4914	563	18	a	a	DET
ejpam-4914	563	19	pndi	pndi	ADJ
ejpam-4914	563	20	-function	-function	NOUN
ejpam-4914	563	21	of	of	ADP
ejpam-4914	563	22	hu	hu	PROPN
ejpam-4914	563	23	.	.	PROPN
ejpam-4914	564	1	define	define	VERB
ejpam-4914	564	2	v2	v2	PROPN
ejpam-4914	564	3	=	=	PUNCT
ejpam-4914	564	4	{	{	PUNCT
ejpam-4914	564	5	u}∪v	u}∪v	NUM
ejpam-4914	564	6	u	u	NOUN
ejpam-4914	564	7	2	2	NUM
ejpam-4914	564	8	,	,	PUNCT
ejpam-4914	564	9	v1	v1	NOUN
ejpam-4914	564	10	=	=	SYM
ejpam-4914	564	11	v	v	PART
ejpam-4914	564	12	u	u	NOUN
ejpam-4914	564	13	1	1	NUM
ejpam-4914	564	14	and	and	CCONJ
ejpam-4914	564	15	v0	v0	NOUN
ejpam-4914	564	16	=	=	SYM
ejpam-4914	564	17	(	(	PUNCT
ejpam-4914	564	18	v	v	NOUN
ejpam-4914	564	19	(	(	PUNCT
ejpam-4914	564	20	g	g	NOUN
ejpam-4914	564	21	)	)	PUNCT
ejpam-4914	564	22	\	\	NOUN
ejpam-4914	564	23	{	{	PUNCT
ejpam-4914	564	24	u})∪	u})∪	PROPN
ejpam-4914	564	25	(	(	PUNCT
ejpam-4914	564	26	∪x∈v	∪x∈v	X
ejpam-4914	564	27	(	(	PUNCT
ejpam-4914	564	28	g)\{u}v	g)\{u}v	PROPN
ejpam-4914	564	29	(	(	PUNCT
ejpam-4914	564	30	hx	hx	PROPN
ejpam-4914	564	31	)	)	PUNCT
ejpam-4914	564	32	)	)	PUNCT
ejpam-4914	564	33	∪v	∪v	ADP
ejpam-4914	565	1	u	u	NOUN
ejpam-4914	565	2	0	0	NUM
ejpam-4914	565	3	.	.	PUNCT
ejpam-4914	566	1	then	then	ADV
ejpam-4914	566	2	f	f	PROPN
ejpam-4914	566	3	=	=	SYM
ejpam-4914	566	4	(	(	PUNCT
ejpam-4914	566	5	v0	v0	PROPN
ejpam-4914	566	6	,	,	PUNCT
ejpam-4914	566	7	v1	v1	NOUN
ejpam-4914	566	8	,	,	PUNCT
ejpam-4914	566	9	v2	v2	NOUN
ejpam-4914	566	10	)	)	PUNCT
ejpam-4914	566	11	∈	∈	PROPN
ejpam-4914	566	12	hid(g	hid(g	PROPN
ejpam-4914	566	13	)	)	PUNCT
ejpam-4914	566	14	with	with	ADP
ejpam-4914	566	15	ωg	ωg	NOUN
ejpam-4914	566	16	◦	◦	VERB
ejpam-4914	566	17	h(f	h(f	X
ejpam-4914	566	18	)	)	PUNCT
ejpam-4914	566	19	=	=	SYM
ejpam-4914	567	1	2	2	NUM
ejpam-4914	567	2	+	+	CCONJ
ejpam-4914	567	3	(	(	PUNCT
ejpam-4914	567	4	|v	|v	PROPN
ejpam-4914	567	5	u	u	NOUN
ejpam-4914	567	6	1	1	NUM
ejpam-4914	567	7	|+	|+	NOUN
ejpam-4914	567	8	2|v	2|v	NUM
ejpam-4914	568	1	u	u	NOUN
ejpam-4914	568	2	2	2	NUM
ejpam-4914	568	3	|	|	NOUN
ejpam-4914	568	4	)	)	PUNCT
ejpam-4914	568	5	=	=	SYM
ejpam-4914	568	6	2	2	NUM
ejpam-4914	568	7	+	+	CCONJ
ejpam-4914	568	8	pndi(h	pndi(h	PROPN
ejpam-4914	568	9	)	)	PUNCT
ejpam-4914	568	10	.	.	PUNCT
ejpam-4914	569	1	if	if	SCONJ
ejpam-4914	569	2	pndi(h	pndi(h	ADV
ejpam-4914	569	3	)	)	PUNCT
ejpam-4914	569	4	=	=	SYM
ejpam-4914	569	5	2	2	NUM
ejpam-4914	569	6	(	(	PUNCT
ejpam-4914	569	7	i.e.	i.e.	X
ejpam-4914	569	8	,	,	PUNCT
ejpam-4914	569	9	h	h	NOUN
ejpam-4914	569	10	=	=	SYM
ejpam-4914	569	11	k2	k2	PROPN
ejpam-4914	569	12	)	)	PUNCT
ejpam-4914	569	13	,	,	PUNCT
ejpam-4914	569	14	then	then	ADV
ejpam-4914	569	15	γhi(g	γhi(g	ADP
ejpam-4914	569	16	◦	◦	NOUN
ejpam-4914	569	17	h	h	NOUN
ejpam-4914	569	18	)	)	PUNCT
ejpam-4914	569	19	=	=	PUNCT
ejpam-4914	570	1	4	4	X
ejpam-4914	570	2	.	.	X
ejpam-4914	571	1	if	if	SCONJ
ejpam-4914	571	2	pndi(h	pndi(h	ADV
ejpam-4914	571	3	)	)	PUNCT
ejpam-4914	571	4	=	=	SYM
ejpam-4914	571	5	3	3	NUM
ejpam-4914	571	6	,	,	PUNCT
ejpam-4914	571	7	then	then	ADV
ejpam-4914	571	8	the	the	DET
ejpam-4914	571	9	preceding	precede	VERB
ejpam-4914	571	10	result	result	NOUN
ejpam-4914	571	11	implies	imply	VERB
ejpam-4914	571	12	that	that	SCONJ
ejpam-4914	571	13	γhi(g	γhi(g	PROPN
ejpam-4914	571	14	◦	◦	NOUN
ejpam-4914	571	15	h	h	NOUN
ejpam-4914	571	16	)	)	PUNCT
ejpam-4914	571	17	=	=	SYM
ejpam-4914	571	18	5	5	NUM
ejpam-4914	571	19	;	;	PUNCT
ejpam-4914	571	20	and	and	CCONJ
ejpam-4914	571	21	by	by	ADP
ejpam-4914	571	22	a	a	DET
ejpam-4914	571	23	similar	similar	ADJ
ejpam-4914	571	24	reason	reason	NOUN
ejpam-4914	571	25	,	,	PUNCT
ejpam-4914	571	26	if	if	SCONJ
ejpam-4914	571	27	pndi(h	pndi(h	ADP
ejpam-4914	571	28	)	)	PUNCT
ejpam-4914	571	29	≥	≥	NOUN
ejpam-4914	571	30	4	4	NUM
ejpam-4914	571	31	,	,	PUNCT
ejpam-4914	571	32	then	then	ADV
ejpam-4914	571	33	γhi(g	γhi(g	ADP
ejpam-4914	571	34	◦	◦	NOUN
ejpam-4914	571	35	h	h	NOUN
ejpam-4914	571	36	)	)	PUNCT
ejpam-4914	571	37	=	=	SYM
ejpam-4914	571	38	6	6	X
ejpam-4914	571	39	.	.	PUNCT
ejpam-4914	571	40	to	to	PART
ejpam-4914	571	41	prove	prove	VERB
ejpam-4914	571	42	(	(	PUNCT
ejpam-4914	571	43	ii	ii	NOUN
ejpam-4914	571	44	)	)	PUNCT
ejpam-4914	571	45	,	,	PUNCT
ejpam-4914	571	46	let	let	VERB
ejpam-4914	571	47	s	s	PRON
ejpam-4914	571	48	⊆	⊆	NUM
ejpam-4914	571	49	v	v	NOUN
ejpam-4914	571	50	(	(	PUNCT
ejpam-4914	571	51	g	g	NOUN
ejpam-4914	571	52	)	)	PUNCT
ejpam-4914	571	53	be	be	AUX
ejpam-4914	571	54	a	a	DET
ejpam-4914	571	55	hop	hop	NOUN
ejpam-4914	571	56	dominating	dominating	NOUN
ejpam-4914	571	57	set	set	NOUN
ejpam-4914	571	58	of	of	ADP
ejpam-4914	571	59	g.	g.	PROPN
ejpam-4914	571	60	for	for	ADP
ejpam-4914	571	61	each	each	DET
ejpam-4914	571	62	v	v	NUM
ejpam-4914	571	63	∈	∈	PROPN
ejpam-4914	571	64	v	v	NOUN
ejpam-4914	571	65	(	(	PUNCT
ejpam-4914	571	66	g)\ng(s	g)\ng(s	NOUN
ejpam-4914	571	67	)	)	PUNCT
ejpam-4914	571	68	,	,	PUNCT
ejpam-4914	571	69	let	let	VERB
ejpam-4914	571	70	fv	fv	X
ejpam-4914	571	71	=	=	PUNCT
ejpam-4914	571	72	(	(	PUNCT
ejpam-4914	571	73	v	v	NOUN
ejpam-4914	571	74	v	v	NOUN
ejpam-4914	571	75	0	0	NUM
ejpam-4914	571	76	,	,	PUNCT
ejpam-4914	571	77	v	v	NOUN
ejpam-4914	571	78	v	v	NUM
ejpam-4914	571	79	1	1	NUM
ejpam-4914	571	80	,	,	PUNCT
ejpam-4914	571	81	v	v	NOUN
ejpam-4914	571	82	v	v	NOUN
ejpam-4914	571	83	2	2	NUM
ejpam-4914	571	84	)	)	PUNCT
ejpam-4914	571	85	be	be	AUX
ejpam-4914	571	86	a	a	DET
ejpam-4914	571	87	pndi	pndi	ADJ
ejpam-4914	571	88	-function	-function	NOUN
ejpam-4914	571	89	of	of	ADP
ejpam-4914	571	90	h	h	NOUN
ejpam-4914	571	91	=	=	SYM
ejpam-4914	571	92	hv	hv	PROPN
ejpam-4914	571	93	.	.	PROPN
ejpam-4914	571	94	define	define	VERB
ejpam-4914	571	95	the	the	DET
ejpam-4914	571	96	following	following	ADJ
ejpam-4914	571	97	•	•	NUM
ejpam-4914	571	98	v0	v0	NOUN
ejpam-4914	571	99	=	=	PUNCT
ejpam-4914	572	1	[	[	X
ejpam-4914	572	2	v	v	X
ejpam-4914	572	3	(	(	PUNCT
ejpam-4914	572	4	g	g	NOUN
ejpam-4914	572	5	)	)	PUNCT
ejpam-4914	572	6	\	\	PUNCT
ejpam-4914	573	1	s	s	X
ejpam-4914	573	2	]	]	X
ejpam-4914	573	3	∪	∪	X
ejpam-4914	573	4	[	[	PUNCT
ejpam-4914	573	5	∪v∈v	∪v∈v	X
ejpam-4914	573	6	(	(	PUNCT
ejpam-4914	573	7	g)∩ng(s)v	g)∩ng(s)v	NUM
ejpam-4914	573	8	(	(	PUNCT
ejpam-4914	573	9	hv	hv	PROPN
ejpam-4914	573	10	)	)	PUNCT
ejpam-4914	573	11	]	]	PUNCT
ejpam-4914	573	12	∪	∪	ADP
ejpam-4914	573	13	[	[	PUNCT
ejpam-4914	573	14	∪v∈v	∪v∈v	NOUN
ejpam-4914	573	15	(	(	PUNCT
ejpam-4914	573	16	g)\ng(s)v	g)\ng(s)v	PROPN
ejpam-4914	573	17	v	v	NOUN
ejpam-4914	573	18	0	0	NUM
ejpam-4914	573	19	]	]	PUNCT
ejpam-4914	573	20	;	;	PUNCT
ejpam-4914	573	21	•	•	NUM
ejpam-4914	573	22	v1	v1	NOUN
ejpam-4914	573	23	=	=	SYM
ejpam-4914	573	24	∪v∈v	∪v∈v	X
ejpam-4914	573	25	(	(	PUNCT
ejpam-4914	573	26	g)\ng(s)v	g)\ng(s)v	PROPN
ejpam-4914	573	27	v	v	NUM
ejpam-4914	573	28	1	1	NUM
ejpam-4914	573	29	;	;	PUNCT
ejpam-4914	573	30	•	•	NUM
ejpam-4914	573	31	v2	v2	NOUN
ejpam-4914	573	32	=	=	SYM
ejpam-4914	573	33	s	s	NOUN
ejpam-4914	573	34	∪	∪	X
ejpam-4914	573	35	[	[	PUNCT
ejpam-4914	573	36	∪v∈v	∪v∈v	NOUN
ejpam-4914	573	37	(	(	PUNCT
ejpam-4914	573	38	g)\ng(s)v	g)\ng(s)v	PROPN
ejpam-4914	573	39	v	v	NUM
ejpam-4914	573	40	2	2	NUM
ejpam-4914	573	41	]	]	PUNCT
ejpam-4914	573	42	.	.	PUNCT
ejpam-4914	574	1	put	put	VERB
ejpam-4914	574	2	f	f	PROPN
ejpam-4914	574	3	=	=	SYM
ejpam-4914	574	4	(	(	PUNCT
ejpam-4914	574	5	v0	v0	PROPN
ejpam-4914	574	6	,	,	PUNCT
ejpam-4914	574	7	v1	v1	NOUN
ejpam-4914	574	8	,	,	PUNCT
ejpam-4914	574	9	v2	v2	PROPN
ejpam-4914	574	10	)	)	PUNCT
ejpam-4914	574	11	.	.	PUNCT
ejpam-4914	575	1	let	let	VERB
ejpam-4914	575	2	v	v	NUM
ejpam-4914	575	3	∈	∈	PROPN
ejpam-4914	575	4	v0	v0	NOUN
ejpam-4914	575	5	∩	∩	X
ejpam-4914	575	6	v	v	X
ejpam-4914	575	7	(	(	PUNCT
ejpam-4914	575	8	g	g	NOUN
ejpam-4914	575	9	)	)	PUNCT
ejpam-4914	575	10	.	.	PUNCT
ejpam-4914	576	1	since	since	SCONJ
ejpam-4914	576	2	s	s	PROPN
ejpam-4914	576	3	is	be	AUX
ejpam-4914	576	4	a	a	DET
ejpam-4914	576	5	hop	hop	NOUN
ejpam-4914	576	6	dominating	dominating	NOUN
ejpam-4914	576	7	set	set	NOUN
ejpam-4914	576	8	of	of	ADP
ejpam-4914	576	9	g	g	PROPN
ejpam-4914	576	10	and	and	CCONJ
ejpam-4914	576	11	v	v	ADP
ejpam-4914	576	12	∈	∈	PROPN
ejpam-4914	576	13	v	v	NOUN
ejpam-4914	576	14	(	(	PUNCT
ejpam-4914	576	15	g	g	NOUN
ejpam-4914	576	16	)	)	PUNCT
ejpam-4914	576	17	\	\	PROPN
ejpam-4914	577	1	s	s	X
ejpam-4914	577	2	,	,	PUNCT
ejpam-4914	577	3	there	there	PRON
ejpam-4914	577	4	exists	exist	VERB
ejpam-4914	577	5	u	u	PROPN
ejpam-4914	577	6	∈	∈	PROPN
ejpam-4914	577	7	s	s	PART
ejpam-4914	577	8	⊆	⊆	NUM
ejpam-4914	577	9	v2	v2	PROPN
ejpam-4914	577	10	∩	∩	ADJ
ejpam-4914	577	11	v	v	NOUN
ejpam-4914	577	12	(	(	PUNCT
ejpam-4914	577	13	g	g	NOUN
ejpam-4914	577	14	)	)	PUNCT
ejpam-4914	577	15	for	for	ADP
ejpam-4914	577	16	which	which	PRON
ejpam-4914	577	17	dg(u	dg(u	X
ejpam-4914	577	18	,	,	PUNCT
ejpam-4914	577	19	v	v	NOUN
ejpam-4914	577	20	)	)	PUNCT
ejpam-4914	577	21	=	=	SYM
ejpam-4914	577	22	2	2	NUM
ejpam-4914	577	23	,	,	PUNCT
ejpam-4914	577	24	showing	show	VERB
ejpam-4914	577	25	that	that	DET
ejpam-4914	577	26	condition	condition	NOUN
ejpam-4914	577	27	(	(	PUNCT
ejpam-4914	577	28	i)(b	i)(b	NUM
ejpam-4914	577	29	)	)	PUNCT
ejpam-4914	577	30	of	of	ADP
ejpam-4914	577	31	theorem	theorem	ADJ
ejpam-4914	577	32	3.6	3.6	NUM
ejpam-4914	577	33	is	be	AUX
ejpam-4914	577	34	satisfied	satisfied	ADJ
ejpam-4914	577	35	.	.	PUNCT
ejpam-4914	578	1	let	let	VERB
ejpam-4914	578	2	v	v	NUM
ejpam-4914	578	3	∈	∈	PROPN
ejpam-4914	578	4	v	v	NOUN
ejpam-4914	578	5	(	(	PUNCT
ejpam-4914	578	6	g	g	NOUN
ejpam-4914	578	7	)	)	PUNCT
ejpam-4914	578	8	for	for	ADP
ejpam-4914	578	9	which	which	PRON
ejpam-4914	578	10	v2	v2	NOUN
ejpam-4914	578	11	∩	∩	NOUN
ejpam-4914	578	12	ng(v	ng(v	NUM
ejpam-4914	578	13	)	)	PUNCT
ejpam-4914	578	14	=	=	PUNCT
ejpam-4914	578	15	∅.	∅.	NOUN
ejpam-4914	578	16	since	since	SCONJ
ejpam-4914	578	17	v1	v1	NOUN
ejpam-4914	578	18	∩	∩	ADJ
ejpam-4914	578	19	v	v	NOUN
ejpam-4914	578	20	(	(	PUNCT
ejpam-4914	578	21	g	g	NOUN
ejpam-4914	578	22	)	)	PUNCT
ejpam-4914	578	23	=	=	NOUN
ejpam-4914	578	24	∅	∅	NOUN
ejpam-4914	578	25	,	,	PUNCT
ejpam-4914	578	26	v	v	NOUN
ejpam-4914	578	27	∈	∈	PROPN
ejpam-4914	578	28	v	v	NOUN
ejpam-4914	578	29	(	(	PUNCT
ejpam-4914	578	30	g	g	NOUN
ejpam-4914	578	31	)	)	PUNCT
ejpam-4914	578	32	\	\	NOUN
ejpam-4914	578	33	ng(s	ng(s	NUM
ejpam-4914	578	34	)	)	PUNCT
ejpam-4914	578	35	.	.	PUNCT
ejpam-4914	579	1	then	then	ADV
ejpam-4914	579	2	f	f	PROPN
ejpam-4914	579	3	|hv	|hv	PROPN
ejpam-4914	579	4	=	=	SYM
ejpam-4914	579	5	fv	fv	X
ejpam-4914	579	6	,	,	PUNCT
ejpam-4914	579	7	and	and	CCONJ
ejpam-4914	579	8	therefore	therefore	ADV
ejpam-4914	579	9	f	f	PROPN
ejpam-4914	579	10	|hv	|hv	PROPN
ejpam-4914	579	11	is	be	AUX
ejpam-4914	579	12	a	a	DET
ejpam-4914	579	13	pndi	pndi	ADJ
ejpam-4914	579	14	-function	-function	NOUN
ejpam-4914	579	15	of	of	ADP
ejpam-4914	579	16	hv	hv	PROPN
ejpam-4914	579	17	.	.	PUNCT
ejpam-4914	580	1	by	by	ADP
ejpam-4914	580	2	theorem	theorem	NOUN
ejpam-4914	580	3	3.6	3.6	NUM
ejpam-4914	580	4	,	,	PUNCT
ejpam-4914	580	5	f	f	PROPN
ejpam-4914	580	6	∈	∈	PROPN
ejpam-4914	580	7	hid(g	hid(g	PROPN
ejpam-4914	580	8	◦	◦	NOUN
ejpam-4914	580	9	h	h	NOUN
ejpam-4914	580	10	)	)	PUNCT
ejpam-4914	580	11	.	.	PUNCT
ejpam-4914	581	1	moreover	moreover	ADV
ejpam-4914	581	2	,	,	PUNCT
ejpam-4914	581	3	γhi(g	γhi(g	PROPN
ejpam-4914	581	4	◦	◦	NOUN
ejpam-4914	581	5	h	h	NOUN
ejpam-4914	581	6	)	)	PUNCT
ejpam-4914	581	7	≤	≤	NOUN
ejpam-4914	581	8	ωg	ωg	NOUN
ejpam-4914	581	9	◦	◦	NOUN
ejpam-4914	581	10	h(f	h(f	NOUN
ejpam-4914	581	11	)	)	PUNCT
ejpam-4914	582	1	=	=	PUNCT
ejpam-4914	582	2	|v1|+	|v1|+	PRON
ejpam-4914	582	3	2|v2|	2|v2|	NUM
ejpam-4914	582	4	=	=	SYM
ejpam-4914	582	5	∑	∑	PUNCT
ejpam-4914	582	6	v∈v	v∈v	PROPN
ejpam-4914	582	7	(	(	PUNCT
ejpam-4914	582	8	g)\ng(s	g)\ng(s	NOUN
ejpam-4914	582	9	)	)	PUNCT
ejpam-4914	582	10	|v	|v	PROPN
ejpam-4914	582	11	v	v	ADP
ejpam-4914	582	12	1	1	NUM
ejpam-4914	582	13	|+	|+	NOUN
ejpam-4914	582	14	2|s|+	2|s|+	NUM
ejpam-4914	582	15	∑	∑	SYM
ejpam-4914	582	16	v∈v	v∈v	PROPN
ejpam-4914	582	17	(	(	PUNCT
ejpam-4914	582	18	g)\ng(s	g)\ng(s	NOUN
ejpam-4914	582	19	)	)	PUNCT
ejpam-4914	582	20	|v	|v	PROPN
ejpam-4914	582	21	v	v	ADP
ejpam-4914	582	22	2	2	NUM
ejpam-4914	582	23	|	|	NOUN
ejpam-4914	582	24	=	=	SYM
ejpam-4914	582	25	2|s|+	2|s|+	NUM
ejpam-4914	582	26	∑	∑	PUNCT
ejpam-4914	582	27	v∈v	v∈v	PROPN
ejpam-4914	582	28	(	(	PUNCT
ejpam-4914	582	29	g)\ng(s	g)\ng(s	NOUN
ejpam-4914	582	30	)	)	PUNCT
ejpam-4914	582	31	(	(	PUNCT
ejpam-4914	582	32	|v	|v	PROPN
ejpam-4914	582	33	v	v	NUM
ejpam-4914	582	34	1	1	NUM
ejpam-4914	582	35	|+	|+	NOUN
ejpam-4914	582	36	2|v	2|v	NUM
ejpam-4914	582	37	v	v	ADP
ejpam-4914	582	38	2	2	NUM
ejpam-4914	582	39	|	|	NOUN
ejpam-4914	582	40	)	)	PUNCT
ejpam-4914	582	41	=	=	PUNCT
ejpam-4914	583	1	2|s|+	2|s|+	NUM
ejpam-4914	583	2	[	[	X
ejpam-4914	583	3	n−	n−	NOUN
ejpam-4914	583	4	|ng(s)|	|ng(s)|	VERB
ejpam-4914	583	5	]	]	X
ejpam-4914	583	6	pndi(h	pndi(h	ADP
ejpam-4914	583	7	)	)	PUNCT
ejpam-4914	583	8	.	.	PUNCT
ejpam-4914	584	1	since	since	SCONJ
ejpam-4914	584	2	s	s	NOUN
ejpam-4914	584	3	is	be	AUX
ejpam-4914	584	4	arbitrary	arbitrary	ADJ
ejpam-4914	584	5	,	,	PUNCT
ejpam-4914	584	6	γhi(g	γhi(g	PROPN
ejpam-4914	584	7	◦	◦	NOUN
ejpam-4914	584	8	h	h	NOUN
ejpam-4914	584	9	)	)	PUNCT
ejpam-4914	584	10	≤	≤	NOUN
ejpam-4914	584	11	ρh(g	ρh(g	NOUN
ejpam-4914	584	12	)	)	PUNCT
ejpam-4914	584	13	.	.	PUNCT
ejpam-4914	585	1	s.r	s.r	PROPN
ejpam-4914	585	2	.	.	PROPN
ejpam-4914	585	3	jr	jr	PROPN
ejpam-4914	585	4	.	.	PROPN
ejpam-4914	585	5	canoy	canoy	PROPN
ejpam-4914	585	6	,	,	PUNCT
ejpam-4914	585	7	f.p	f.p	PROPN
ejpam-4914	585	8	.	.	PROPN
ejpam-4914	585	9	jamil	jamil	PROPN
ejpam-4914	585	10	and	and	CCONJ
ejpam-4914	585	11	s.m	s.m	PROPN
ejpam-4914	585	12	.	.	PROPN
ejpam-4914	585	13	menchavez	menchavez	PROPN
ejpam-4914	585	14	/	/	PUNCT
ejpam-4914	585	15	eur	eur	PROPN
ejpam-4914	585	16	.	.	PUNCT
ejpam-4914	586	1	j.	j.	PROPN
ejpam-4914	586	2	pure	pure	PROPN
ejpam-4914	586	3	appl	appl	PROPN
ejpam-4914	586	4	.	.	PROPN
ejpam-4914	586	5	math	math	PROPN
ejpam-4914	586	6	,	,	PUNCT
ejpam-4914	586	7	16	16	NUM
ejpam-4914	586	8	(	(	PUNCT
ejpam-4914	586	9	4	4	NUM
ejpam-4914	586	10	)	)	PUNCT
ejpam-4914	586	11	(	(	PUNCT
ejpam-4914	586	12	2023	2023	NUM
ejpam-4914	586	13	)	)	PUNCT
ejpam-4914	586	14	,	,	PUNCT
ejpam-4914	586	15	2431	2431	NUM
ejpam-4914	586	16	-	-	SYM
ejpam-4914	586	17	2449	2449	NUM
ejpam-4914	586	18	2445	2445	NUM
ejpam-4914	586	19	consider	consider	VERB
ejpam-4914	586	20	g	g	NOUN
ejpam-4914	586	21	=	=	SYM
ejpam-4914	586	22	p4	p4	ADJ
ejpam-4914	586	23	.	.	PUNCT
ejpam-4914	587	1	for	for	ADP
ejpam-4914	587	2	any	any	DET
ejpam-4914	587	3	graph	graph	NOUN
ejpam-4914	587	4	h	h	NOUN
ejpam-4914	587	5	,	,	PUNCT
ejpam-4914	587	6	γhi(g	γhi(g	ADP
ejpam-4914	587	7	◦	◦	NOUN
ejpam-4914	587	8	h	h	NOUN
ejpam-4914	587	9	)	)	PUNCT
ejpam-4914	587	10	=	=	SYM
ejpam-4914	587	11	4	4	NUM
ejpam-4914	587	12	=	=	NOUN
ejpam-4914	587	13	ρh(g	ρh(g	NOUN
ejpam-4914	587	14	)	)	PUNCT
ejpam-4914	587	15	.	.	PUNCT
ejpam-4914	588	1	this	this	PRON
ejpam-4914	588	2	proves	prove	VERB
ejpam-4914	588	3	the	the	DET
ejpam-4914	588	4	tightness	tightness	NOUN
ejpam-4914	588	5	of	of	ADP
ejpam-4914	588	6	the	the	DET
ejpam-4914	588	7	bound	bind	VERB
ejpam-4914	588	8	.	.	PUNCT
ejpam-4914	589	1	■	■	PUNCT
ejpam-4914	589	2	it	it	PRON
ejpam-4914	589	3	is	be	AUX
ejpam-4914	589	4	also	also	ADV
ejpam-4914	589	5	worth	worth	ADJ
ejpam-4914	589	6	noting	note	VERB
ejpam-4914	589	7	that	that	SCONJ
ejpam-4914	589	8	for	for	ADP
ejpam-4914	589	9	graphs	graph	NOUN
ejpam-4914	589	10	g	g	NOUN
ejpam-4914	589	11	with	with	ADP
ejpam-4914	589	12	no	no	DET
ejpam-4914	589	13	isolated	isolated	ADJ
ejpam-4914	589	14	vertices	vertex	NOUN
ejpam-4914	589	15	,	,	PUNCT
ejpam-4914	589	16	corollary	corollary	ADJ
ejpam-4914	589	17	3.7	3.7	NUM
ejpam-4914	589	18	is	be	AUX
ejpam-4914	589	19	an	an	DET
ejpam-4914	589	20	improvement	improvement	NOUN
ejpam-4914	589	21	of	of	ADP
ejpam-4914	589	22	proposition	proposition	NOUN
ejpam-4914	589	23	3.5	3.5	NUM
ejpam-4914	589	24	as	as	ADP
ejpam-4914	589	25	ρh(g	ρh(g	NOUN
ejpam-4914	589	26	)	)	PUNCT
ejpam-4914	589	27	≤	≤	NOUN
ejpam-4914	589	28	2γ∗t1,2(g	2γ∗t1,2(g	NUM
ejpam-4914	589	29	)	)	PUNCT
ejpam-4914	589	30	.	.	PUNCT
ejpam-4914	590	1	theorem	theorem	VERB
ejpam-4914	590	2	3.8	3.8	NUM
ejpam-4914	590	3	.	.	PUNCT
ejpam-4914	591	1	(	(	PUNCT
ejpam-4914	591	2	lexicographic	lexicographic	ADJ
ejpam-4914	591	3	product	product	NOUN
ejpam-4914	591	4	of	of	ADP
ejpam-4914	591	5	graphs	graph	NOUN
ejpam-4914	591	6	)	)	PUNCT
ejpam-4914	591	7	let	let	VERB
ejpam-4914	591	8	g	g	NOUN
ejpam-4914	591	9	and	and	CCONJ
ejpam-4914	591	10	h	h	NOUN
ejpam-4914	591	11	be	be	AUX
ejpam-4914	591	12	connected	connect	VERB
ejpam-4914	591	13	graphs	graph	NOUN
ejpam-4914	591	14	and	and	CCONJ
ejpam-4914	591	15	f	f	NOUN
ejpam-4914	591	16	=	=	SYM
ejpam-4914	591	17	(	(	PUNCT
ejpam-4914	591	18	v0	v0	PROPN
ejpam-4914	591	19	,	,	PUNCT
ejpam-4914	591	20	v1	v1	NOUN
ejpam-4914	591	21	,	,	PUNCT
ejpam-4914	591	22	v2	v2	PROPN
ejpam-4914	591	23	)	)	PUNCT
ejpam-4914	591	24	a	a	DET
ejpam-4914	591	25	function	function	NOUN
ejpam-4914	591	26	on	on	ADP
ejpam-4914	591	27	v	v	NOUN
ejpam-4914	591	28	(	(	PUNCT
ejpam-4914	591	29	g[h	g[h	PROPN
ejpam-4914	591	30	]	]	PUNCT
ejpam-4914	591	31	)	)	PUNCT
ejpam-4914	591	32	.	.	PUNCT
ejpam-4914	592	1	let	let	VERB
ejpam-4914	592	2	a	a	DET
ejpam-4914	592	3	,	,	PUNCT
ejpam-4914	592	4	b	b	NOUN
ejpam-4914	592	5	and	and	CCONJ
ejpam-4914	592	6	c	c	PROPN
ejpam-4914	592	7	be	be	AUX
ejpam-4914	592	8	subsets	subset	NOUN
ejpam-4914	592	9	of	of	ADP
ejpam-4914	592	10	v	v	NOUN
ejpam-4914	592	11	(	(	PUNCT
ejpam-4914	592	12	g	g	NOUN
ejpam-4914	592	13	)	)	PUNCT
ejpam-4914	592	14	and	and	CCONJ
ejpam-4914	592	15	let	let	VERB
ejpam-4914	592	16	ax	ax	NOUN
ejpam-4914	592	17	,	,	PUNCT
ejpam-4914	592	18	bx	bx	PROPN
ejpam-4914	592	19	and	and	CCONJ
ejpam-4914	592	20	cx	cx	PROPN
ejpam-4914	592	21	be	be	VERB
ejpam-4914	592	22	subsets	subset	NOUN
ejpam-4914	592	23	of	of	ADP
ejpam-4914	592	24	v	v	NOUN
ejpam-4914	592	25	(	(	PUNCT
ejpam-4914	592	26	h	h	NOUN
ejpam-4914	592	27	)	)	PUNCT
ejpam-4914	592	28	such	such	ADJ
ejpam-4914	592	29	that	that	DET
ejpam-4914	592	30	v0	v0	NOUN
ejpam-4914	592	31	=	=	SYM
ejpam-4914	592	32	∪x∈a	∪x∈a	PROPN
ejpam-4914	592	33	(	(	PUNCT
ejpam-4914	592	34	{	{	PUNCT
ejpam-4914	592	35	x	x	NOUN
ejpam-4914	592	36	}	}	PUNCT
ejpam-4914	592	37	×ax	×ax	ADJ
ejpam-4914	592	38	)	)	PUNCT
ejpam-4914	592	39	,	,	PUNCT
ejpam-4914	592	40	v1	v1	NOUN
ejpam-4914	592	41	=	=	SYM
ejpam-4914	592	42	∪x∈b	∪x∈b	NOUN
ejpam-4914	592	43	(	(	PUNCT
ejpam-4914	592	44	{	{	PUNCT
ejpam-4914	592	45	x	x	NOUN
ejpam-4914	592	46	}	}	PUNCT
ejpam-4914	592	47	×bx	×bx	NOUN
ejpam-4914	592	48	)	)	PUNCT
ejpam-4914	592	49	and	and	CCONJ
ejpam-4914	592	50	v2	v2	NOUN
ejpam-4914	592	51	=	=	SYM
ejpam-4914	592	52	∪x∈c	∪x∈c	NOUN
ejpam-4914	592	53	(	(	PUNCT
ejpam-4914	592	54	{	{	PUNCT
ejpam-4914	592	55	x	x	NOUN
ejpam-4914	592	56	}	}	PUNCT
ejpam-4914	592	57	×	×	PROPN
ejpam-4914	592	58	cx	cx	PROPN
ejpam-4914	592	59	)	)	PUNCT
ejpam-4914	592	60	.	.	PUNCT
ejpam-4914	593	1	then	then	ADV
ejpam-4914	593	2	f	f	PROPN
ejpam-4914	593	3	∈	∈	PROPN
ejpam-4914	593	4	hid(g[h	hid(g[h	NOUN
ejpam-4914	593	5	]	]	PUNCT
ejpam-4914	593	6	)	)	PUNCT
ejpam-4914	593	7	if	if	SCONJ
ejpam-4914	593	8	and	and	CCONJ
ejpam-4914	593	9	only	only	ADV
ejpam-4914	593	10	if	if	SCONJ
ejpam-4914	593	11	each	each	PRON
ejpam-4914	593	12	of	of	ADP
ejpam-4914	593	13	the	the	DET
ejpam-4914	593	14	following	follow	VERB
ejpam-4914	593	15	holds	hold	VERB
ejpam-4914	593	16	:	:	PUNCT
ejpam-4914	593	17	(	(	PUNCT
ejpam-4914	593	18	i	i	NOUN
ejpam-4914	593	19	)	)	PUNCT
ejpam-4914	593	20	b	b	PROPN
ejpam-4914	593	21	∪	∪	NOUN
ejpam-4914	593	22	c	c	X
ejpam-4914	593	23	is	be	AUX
ejpam-4914	593	24	a	a	DET
ejpam-4914	593	25	hop	hop	NOUN
ejpam-4914	593	26	dominating	dominating	NOUN
ejpam-4914	593	27	set	set	NOUN
ejpam-4914	593	28	of	of	ADP
ejpam-4914	593	29	g	g	NOUN
ejpam-4914	593	30	;	;	PUNCT
ejpam-4914	593	31	(	(	PUNCT
ejpam-4914	593	32	ii	ii	NOUN
ejpam-4914	593	33	)	)	PUNCT
ejpam-4914	593	34	for	for	ADP
ejpam-4914	593	35	each	each	DET
ejpam-4914	593	36	x	x	SYM
ejpam-4914	593	37	∈	∈	PROPN
ejpam-4914	593	38	a	a	PRON
ejpam-4914	593	39	for	for	ADP
ejpam-4914	593	40	which	which	PRON
ejpam-4914	593	41	c	c	PROPN
ejpam-4914	593	42	∩ng(x	∩ng(x	X
ejpam-4914	593	43	,	,	PUNCT
ejpam-4914	593	44	2	2	X
ejpam-4914	593	45	)	)	PUNCT
ejpam-4914	593	46	=	=	NOUN
ejpam-4914	593	47	∅	∅	NOUN
ejpam-4914	593	48	,	,	PUNCT
ejpam-4914	593	49	one	one	NUM
ejpam-4914	593	50	of	of	ADP
ejpam-4914	593	51	the	the	DET
ejpam-4914	593	52	following	following	NOUN
ejpam-4914	593	53	holds	hold	VERB
ejpam-4914	593	54	:	:	PUNCT
ejpam-4914	593	55	(	(	PUNCT
ejpam-4914	593	56	a	a	X
ejpam-4914	593	57	)	)	PUNCT
ejpam-4914	593	58	|b	|b	ADJ
ejpam-4914	593	59	∩ng(x	∩ng(x	NOUN
ejpam-4914	593	60	,	,	PUNCT
ejpam-4914	593	61	2)|	2)|	NUM
ejpam-4914	593	62	≥	≥	NOUN
ejpam-4914	593	63	2	2	NUM
ejpam-4914	593	64	;	;	PUNCT
ejpam-4914	593	65	(	(	PUNCT
ejpam-4914	593	66	b	b	X
ejpam-4914	593	67	)	)	PUNCT
ejpam-4914	593	68	b	b	NOUN
ejpam-4914	593	69	∩ng(x	∩ng(x	NOUN
ejpam-4914	593	70	,	,	PUNCT
ejpam-4914	593	71	2	2	X
ejpam-4914	593	72	)	)	PUNCT
ejpam-4914	593	73	=	=	PRON
ejpam-4914	593	74	{	{	PUNCT
ejpam-4914	593	75	w	w	NOUN
ejpam-4914	593	76	}	}	PUNCT
ejpam-4914	593	77	such	such	ADJ
ejpam-4914	593	78	that	that	SCONJ
ejpam-4914	593	79	|bw|	|bw|	PROPN
ejpam-4914	593	80	≥	≥	NOUN
ejpam-4914	593	81	2	2	NUM
ejpam-4914	593	82	;	;	PUNCT
ejpam-4914	593	83	(	(	PUNCT
ejpam-4914	593	84	c	c	X
ejpam-4914	593	85	)	)	PUNCT
ejpam-4914	593	86	x	x	SYM
ejpam-4914	594	1	∈	∈	NOUN
ejpam-4914	594	2	b	b	X
ejpam-4914	594	3	∪	∪	VERB
ejpam-4914	594	4	c	c	PROPN
ejpam-4914	594	5	,	,	PUNCT
ejpam-4914	594	6	b	b	X
ejpam-4914	594	7	∩ng(x	∩ng(x	X
ejpam-4914	594	8	,	,	PUNCT
ejpam-4914	594	9	2	2	X
ejpam-4914	594	10	)	)	PUNCT
ejpam-4914	594	11	=	=	PRON
ejpam-4914	594	12	{	{	PUNCT
ejpam-4914	594	13	w	w	NOUN
ejpam-4914	594	14	}	}	PUNCT
ejpam-4914	594	15	with	with	ADP
ejpam-4914	594	16	|bw|	|bw|	PROPN
ejpam-4914	594	17	=	=	SYM
ejpam-4914	594	18	1	1	NUM
ejpam-4914	594	19	and	and	CCONJ
ejpam-4914	594	20	bx	bx	PROPN
ejpam-4914	594	21	∪	∪	PROPN
ejpam-4914	594	22	cx	cx	PROPN
ejpam-4914	594	23	is	be	AUX
ejpam-4914	594	24	a	a	DET
ejpam-4914	594	25	pnd	pnd	NOUN
ejpam-4914	594	26	-	-	PUNCT
ejpam-4914	594	27	set	set	NOUN
ejpam-4914	594	28	of	of	ADP
ejpam-4914	594	29	h	h	NOUN
ejpam-4914	594	30	;	;	PUNCT
ejpam-4914	594	31	(	(	PUNCT
ejpam-4914	594	32	d	d	X
ejpam-4914	594	33	)	)	PUNCT
ejpam-4914	594	34	x	x	SYM
ejpam-4914	594	35	∈	∈	NOUN
ejpam-4914	594	36	b	b	X
ejpam-4914	594	37	∪	∪	VERB
ejpam-4914	594	38	c	c	PROPN
ejpam-4914	594	39	,	,	PUNCT
ejpam-4914	594	40	b	b	X
ejpam-4914	594	41	∩ng(x	∩ng(x	X
ejpam-4914	594	42	,	,	PUNCT
ejpam-4914	594	43	2	2	X
ejpam-4914	594	44	)	)	PUNCT
ejpam-4914	594	45	=	=	NOUN
ejpam-4914	594	46	∅	∅	NOUN
ejpam-4914	594	47	and	and	CCONJ
ejpam-4914	594	48	the	the	DET
ejpam-4914	594	49	restriction	restriction	NOUN
ejpam-4914	595	1	f	f	PROPN
ejpam-4914	595	2	|⟨{x}×v	|⟨{x}×v	PROPN
ejpam-4914	595	3	(	(	PUNCT
ejpam-4914	595	4	h)⟩	h)⟩	NOUN
ejpam-4914	595	5	of	of	ADP
ejpam-4914	595	6	f	f	PROPN
ejpam-4914	595	7	on	on	ADP
ejpam-4914	595	8	⟨{x	⟨{x	PROPN
ejpam-4914	595	9	}	}	PUNCT
ejpam-4914	595	10	×	×	NOUN
ejpam-4914	595	11	v	v	NOUN
ejpam-4914	595	12	(	(	PUNCT
ejpam-4914	595	13	h)⟩	h)⟩	PROPN
ejpam-4914	595	14	is	be	AUX
ejpam-4914	595	15	a	a	DET
ejpam-4914	595	16	pndi	pndi	ADJ
ejpam-4914	595	17	-	-	PUNCT
ejpam-4914	595	18	function	function	NOUN
ejpam-4914	595	19	of	of	ADP
ejpam-4914	595	20	⟨{x	⟨{x	NOUN
ejpam-4914	595	21	}	}	PUNCT
ejpam-4914	595	22	×	×	NOUN
ejpam-4914	595	23	v	v	NOUN
ejpam-4914	595	24	(	(	PUNCT
ejpam-4914	595	25	h)⟩.	h)⟩.	NOUN
ejpam-4914	595	26	proof	proof	NOUN
ejpam-4914	595	27	:	:	PUNCT
ejpam-4914	595	28	assume	assume	VERB
ejpam-4914	595	29	that	that	SCONJ
ejpam-4914	595	30	f	f	PROPN
ejpam-4914	595	31	∈	∈	PROPN
ejpam-4914	595	32	hid(g[h	hid(g[h	NOUN
ejpam-4914	595	33	]	]	PUNCT
ejpam-4914	595	34	)	)	PUNCT
ejpam-4914	595	35	.	.	PUNCT
ejpam-4914	596	1	then	then	ADV
ejpam-4914	596	2	v1	v1	VERB
ejpam-4914	596	3	∪	∪	ADJ
ejpam-4914	596	4	v2	v2	NOUN
ejpam-4914	596	5	is	be	AUX
ejpam-4914	596	6	a	a	DET
ejpam-4914	596	7	hop	hop	NOUN
ejpam-4914	596	8	dominating	dominating	NOUN
ejpam-4914	596	9	set	set	NOUN
ejpam-4914	596	10	of	of	ADP
ejpam-4914	596	11	g[h	g[h	PROPN
ejpam-4914	596	12	]	]	PUNCT
ejpam-4914	596	13	.	.	PUNCT
ejpam-4914	597	1	this	this	PRON
ejpam-4914	597	2	implies	imply	VERB
ejpam-4914	597	3	that	that	SCONJ
ejpam-4914	597	4	b	b	NOUN
ejpam-4914	597	5	∪	∪	NOUN
ejpam-4914	597	6	c	c	NOUN
ejpam-4914	597	7	is	be	AUX
ejpam-4914	597	8	a	a	DET
ejpam-4914	597	9	hop	hop	NOUN
ejpam-4914	597	10	dominating	dominating	NOUN
ejpam-4914	597	11	set	set	NOUN
ejpam-4914	597	12	of	of	ADP
ejpam-4914	597	13	g	g	NOUN
ejpam-4914	597	14	,	,	PUNCT
ejpam-4914	597	15	and	and	CCONJ
ejpam-4914	597	16	(	(	PUNCT
ejpam-4914	597	17	i	i	NOUN
ejpam-4914	597	18	)	)	PUNCT
ejpam-4914	597	19	holds	hold	VERB
ejpam-4914	597	20	.	.	PUNCT
ejpam-4914	598	1	next	next	ADV
ejpam-4914	598	2	,	,	PUNCT
ejpam-4914	598	3	to	to	PART
ejpam-4914	598	4	prove	prove	VERB
ejpam-4914	598	5	(	(	PUNCT
ejpam-4914	598	6	ii	ii	NOUN
ejpam-4914	598	7	)	)	PUNCT
ejpam-4914	598	8	,	,	PUNCT
ejpam-4914	598	9	let	let	VERB
ejpam-4914	598	10	x	x	SYM
ejpam-4914	598	11	∈	∈	VERB
ejpam-4914	598	12	a	a	PRON
ejpam-4914	598	13	for	for	ADP
ejpam-4914	598	14	which	which	PRON
ejpam-4914	598	15	c	c	PROPN
ejpam-4914	598	16	∩ng(x	∩ng(x	X
ejpam-4914	598	17	,	,	PUNCT
ejpam-4914	598	18	2	2	X
ejpam-4914	598	19	)	)	PUNCT
ejpam-4914	598	20	=	=	VERB
ejpam-4914	598	21	∅.	∅.	NOUN
ejpam-4914	598	22	we	we	PRON
ejpam-4914	598	23	consider	consider	VERB
ejpam-4914	598	24	the	the	DET
ejpam-4914	598	25	following	follow	VERB
ejpam-4914	598	26	cases	case	NOUN
ejpam-4914	598	27	:	:	PUNCT
ejpam-4914	598	28	case	case	NOUN
ejpam-4914	598	29	1	1	NUM
ejpam-4914	598	30	:	:	PUNCT
ejpam-4914	598	31	suppose	suppose	VERB
ejpam-4914	598	32	that	that	SCONJ
ejpam-4914	598	33	b	b	PROPN
ejpam-4914	598	34	∩ng(x	∩ng(x	NOUN
ejpam-4914	598	35	,	,	PUNCT
ejpam-4914	598	36	2	2	NUM
ejpam-4914	598	37	)	)	PUNCT
ejpam-4914	598	38	=	=	NOUN
ejpam-4914	598	39	∅.	∅.	NOUN
ejpam-4914	598	40	since	since	SCONJ
ejpam-4914	598	41	b	b	PROPN
ejpam-4914	598	42	∪c	∪c	VERB
ejpam-4914	598	43	hop	hop	NOUN
ejpam-4914	598	44	-	-	PUNCT
ejpam-4914	598	45	dominates	dominate	VERB
ejpam-4914	598	46	a	a	DET
ejpam-4914	598	47	,	,	PUNCT
ejpam-4914	598	48	x	x	SYM
ejpam-4914	598	49	∈	∈	PROPN
ejpam-4914	598	50	b	b	PROPN
ejpam-4914	598	51	∪c	∪c	PROPN
ejpam-4914	598	52	.	.	PUNCT
ejpam-4914	599	1	put	put	VERB
ejpam-4914	599	2	tx	tx	PROPN
ejpam-4914	600	1	=	=	PUNCT
ejpam-4914	600	2	⟨{x}×v	⟨{x}×v	X
ejpam-4914	600	3	(	(	PUNCT
ejpam-4914	600	4	h)⟩.	h)⟩.	VERB
ejpam-4914	600	5	let	let	VERB
ejpam-4914	600	6	y	y	PRON
ejpam-4914	600	7	∈	∈	PROPN
ejpam-4914	600	8	ax	ax	NOUN
ejpam-4914	600	9	.	.	PUNCT
ejpam-4914	601	1	then	then	ADV
ejpam-4914	601	2	|v2∩ng[h]((x	|v2∩ng[h]((x	PROPN
ejpam-4914	601	3	,	,	PUNCT
ejpam-4914	601	4	y	y	NOUN
ejpam-4914	601	5	)	)	PUNCT
ejpam-4914	601	6	,	,	PUNCT
ejpam-4914	601	7	2)|	2)|	NUM
ejpam-4914	601	8	≥	≥	NUM
ejpam-4914	601	9	1	1	NUM
ejpam-4914	601	10	or	or	CCONJ
ejpam-4914	601	11	|v1∩ng[h]((x	|v1∩ng[h]((x	PROPN
ejpam-4914	601	12	,	,	PUNCT
ejpam-4914	601	13	y	y	PROPN
ejpam-4914	601	14	)	)	PUNCT
ejpam-4914	601	15	,	,	PUNCT
ejpam-4914	601	16	2)|	2)|	NUM
ejpam-4914	601	17	≥	≥	NOUN
ejpam-4914	601	18	2	2	NUM
ejpam-4914	601	19	.	.	PUNCT
ejpam-4914	602	1	if	if	SCONJ
ejpam-4914	602	2	(	(	PUNCT
ejpam-4914	602	3	u	u	NOUN
ejpam-4914	602	4	,	,	PUNCT
ejpam-4914	602	5	v	v	NOUN
ejpam-4914	602	6	)	)	PUNCT
ejpam-4914	602	7	∈	∈	PROPN
ejpam-4914	602	8	v2	v2	PROPN
ejpam-4914	602	9	∩	∩	NOUN
ejpam-4914	602	10	ng[h]((x	ng[h]((x	NOUN
ejpam-4914	602	11	,	,	PUNCT
ejpam-4914	602	12	y	y	NOUN
ejpam-4914	602	13	)	)	PUNCT
ejpam-4914	602	14	,	,	PUNCT
ejpam-4914	602	15	2	2	NUM
ejpam-4914	602	16	)	)	PUNCT
ejpam-4914	602	17	,	,	PUNCT
ejpam-4914	602	18	then	then	ADV
ejpam-4914	602	19	x	x	X
ejpam-4914	602	20	=	=	PUNCT
ejpam-4914	602	21	u	u	NOUN
ejpam-4914	602	22	so	so	SCONJ
ejpam-4914	602	23	that	that	SCONJ
ejpam-4914	602	24	(	(	PUNCT
ejpam-4914	602	25	u	u	NOUN
ejpam-4914	602	26	,	,	PUNCT
ejpam-4914	602	27	v	v	NOUN
ejpam-4914	602	28	)	)	PUNCT
ejpam-4914	602	29	∈	∈	NOUN
ejpam-4914	602	30	v	v	NOUN
ejpam-4914	602	31	x	x	SYM
ejpam-4914	602	32	2	2	NUM
ejpam-4914	602	33	=	=	SYM
ejpam-4914	602	34	v2	v2	PROPN
ejpam-4914	602	35	∩	∩	ADJ
ejpam-4914	602	36	v	v	NOUN
ejpam-4914	602	37	(	(	PUNCT
ejpam-4914	602	38	tx	tx	PROPN
ejpam-4914	602	39	)	)	PUNCT
ejpam-4914	602	40	and	and	CCONJ
ejpam-4914	602	41	(	(	PUNCT
ejpam-4914	602	42	x	x	X
ejpam-4914	602	43	,	,	PUNCT
ejpam-4914	602	44	y)(u	y)(u	ADJ
ejpam-4914	602	45	,	,	PUNCT
ejpam-4914	602	46	v	v	NOUN
ejpam-4914	602	47	)	)	PUNCT
ejpam-4914	602	48	/∈	/∈	PUNCT
ejpam-4914	603	1	e(tx	e(tx	NUM
ejpam-4914	603	2	)	)	PUNCT
ejpam-4914	603	3	.	.	PUNCT
ejpam-4914	604	1	on	on	ADP
ejpam-4914	604	2	the	the	DET
ejpam-4914	604	3	other	other	ADJ
ejpam-4914	604	4	hand	hand	NOUN
ejpam-4914	604	5	,	,	PUNCT
ejpam-4914	604	6	if	if	SCONJ
ejpam-4914	604	7	(	(	PUNCT
ejpam-4914	604	8	u	u	NOUN
ejpam-4914	604	9	,	,	PUNCT
ejpam-4914	604	10	v	v	NOUN
ejpam-4914	604	11	)	)	PUNCT
ejpam-4914	604	12	,	,	PUNCT
ejpam-4914	604	13	(	(	PUNCT
ejpam-4914	604	14	w	w	PROPN
ejpam-4914	604	15	,	,	PUNCT
ejpam-4914	604	16	z	z	NOUN
ejpam-4914	604	17	)	)	PUNCT
ejpam-4914	604	18	∈	∈	NOUN
ejpam-4914	604	19	v1	v1	NOUN
ejpam-4914	604	20	∩	∩	ADJ
ejpam-4914	604	21	ng[h]((x	ng[h]((x	NOUN
ejpam-4914	604	22	,	,	PUNCT
ejpam-4914	604	23	y	y	NOUN
ejpam-4914	604	24	)	)	PUNCT
ejpam-4914	604	25	,	,	PUNCT
ejpam-4914	604	26	2	2	NUM
ejpam-4914	604	27	)	)	PUNCT
ejpam-4914	604	28	,	,	PUNCT
ejpam-4914	604	29	then	then	ADV
ejpam-4914	604	30	u	u	X
ejpam-4914	604	31	=	=	PROPN
ejpam-4914	604	32	w	w	PROPN
ejpam-4914	604	33	=	=	PUNCT
ejpam-4914	604	34	x	x	PUNCT
ejpam-4914	604	35	so	so	ADV
ejpam-4914	604	36	that	that	SCONJ
ejpam-4914	604	37	(	(	PUNCT
ejpam-4914	604	38	u	u	NOUN
ejpam-4914	604	39	,	,	PUNCT
ejpam-4914	604	40	v	v	NOUN
ejpam-4914	604	41	)	)	PUNCT
ejpam-4914	604	42	,	,	PUNCT
ejpam-4914	604	43	(	(	PUNCT
ejpam-4914	604	44	w	w	PROPN
ejpam-4914	604	45	,	,	PUNCT
ejpam-4914	604	46	z	z	NOUN
ejpam-4914	604	47	)	)	PUNCT
ejpam-4914	604	48	∈	∈	NOUN
ejpam-4914	604	49	v	v	NOUN
ejpam-4914	604	50	x	x	SYM
ejpam-4914	604	51	1	1	NUM
ejpam-4914	604	52	=	=	SYM
ejpam-4914	604	53	v1	v1	NOUN
ejpam-4914	604	54	∩	∩	ADJ
ejpam-4914	604	55	v	v	NOUN
ejpam-4914	604	56	(	(	PUNCT
ejpam-4914	604	57	tx	tx	PROPN
ejpam-4914	604	58	)	)	PUNCT
ejpam-4914	604	59	and	and	CCONJ
ejpam-4914	604	60	(	(	PUNCT
ejpam-4914	604	61	x	x	X
ejpam-4914	604	62	,	,	PUNCT
ejpam-4914	604	63	y)(u	y)(u	ADJ
ejpam-4914	604	64	,	,	PUNCT
ejpam-4914	604	65	v	v	NOUN
ejpam-4914	604	66	)	)	PUNCT
ejpam-4914	604	67	,	,	PUNCT
ejpam-4914	604	68	(	(	PUNCT
ejpam-4914	604	69	x	x	X
ejpam-4914	604	70	,	,	PUNCT
ejpam-4914	604	71	y)(w	y)(w	PROPN
ejpam-4914	604	72	,	,	PUNCT
ejpam-4914	604	73	z	z	NOUN
ejpam-4914	604	74	)	)	PUNCT
ejpam-4914	604	75	/∈	/∈	PUNCT
ejpam-4914	605	1	e(tx	e(tx	NUM
ejpam-4914	605	2	)	)	PUNCT
ejpam-4914	605	3	.	.	PUNCT
ejpam-4914	606	1	since	since	SCONJ
ejpam-4914	606	2	y	y	PROPN
ejpam-4914	606	3	is	be	AUX
ejpam-4914	606	4	arbitrary	arbitrary	ADJ
ejpam-4914	606	5	,	,	PUNCT
ejpam-4914	606	6	f	f	PROPN
ejpam-4914	606	7	|tx	|tx	X
ejpam-4914	606	8	=	=	SYM
ejpam-4914	606	9	(	(	PUNCT
ejpam-4914	606	10	v	v	NOUN
ejpam-4914	606	11	x	x	SYM
ejpam-4914	606	12	0	0	NUM
ejpam-4914	606	13	,	,	PUNCT
ejpam-4914	606	14	v	v	NOUN
ejpam-4914	606	15	x	x	SYM
ejpam-4914	606	16	1	1	NUM
ejpam-4914	606	17	,	,	PUNCT
ejpam-4914	606	18	v	v	NOUN
ejpam-4914	606	19	x	x	SYM
ejpam-4914	606	20	2	2	NUM
ejpam-4914	606	21	)	)	PUNCT
ejpam-4914	606	22	is	be	AUX
ejpam-4914	606	23	a	a	DET
ejpam-4914	606	24	pndi	pndi	ADJ
ejpam-4914	606	25	-function	-function	NOUN
ejpam-4914	606	26	of	of	ADP
ejpam-4914	606	27	tx	tx	PROPN
ejpam-4914	606	28	,	,	PUNCT
ejpam-4914	606	29	where	where	SCONJ
ejpam-4914	606	30	v	v	NOUN
ejpam-4914	606	31	x	x	SYM
ejpam-4914	606	32	0	0	PUNCT
ejpam-4914	607	1	=	=	SYM
ejpam-4914	607	2	v0∩v	v0∩v	PROPN
ejpam-4914	607	3	(	(	PUNCT
ejpam-4914	607	4	tx	tx	PROPN
ejpam-4914	607	5	)	)	PUNCT
ejpam-4914	607	6	.	.	PUNCT
ejpam-4914	608	1	this	this	PRON
ejpam-4914	608	2	proves	prove	VERB
ejpam-4914	608	3	(	(	PUNCT
ejpam-4914	608	4	ii)(d	ii)(d	PROPN
ejpam-4914	608	5	)	)	PUNCT
ejpam-4914	608	6	.	.	PUNCT
ejpam-4914	609	1	case	case	NOUN
ejpam-4914	609	2	2	2	NUM
ejpam-4914	609	3	:	:	PUNCT
ejpam-4914	609	4	suppose	suppose	VERB
ejpam-4914	609	5	that	that	SCONJ
ejpam-4914	609	6	b	b	PROPN
ejpam-4914	609	7	∩	∩	NOUN
ejpam-4914	609	8	ng(x	ng(x	NUM
ejpam-4914	609	9	,	,	PUNCT
ejpam-4914	609	10	2	2	X
ejpam-4914	609	11	)	)	PUNCT
ejpam-4914	609	12	̸=	̸=	PROPN
ejpam-4914	609	13	∅.	∅.	ADV
ejpam-4914	609	14	if	if	SCONJ
ejpam-4914	609	15	|b	|b	ADJ
ejpam-4914	609	16	∩	∩	NOUN
ejpam-4914	609	17	ng(x	ng(x	NUM
ejpam-4914	609	18	,	,	PUNCT
ejpam-4914	609	19	2)|	2)|	NUM
ejpam-4914	609	20	≥	≥	NOUN
ejpam-4914	609	21	2	2	NUM
ejpam-4914	609	22	,	,	PUNCT
ejpam-4914	609	23	then	then	ADV
ejpam-4914	609	24	(	(	PUNCT
ejpam-4914	609	25	ii)(a	ii)(a	PROPN
ejpam-4914	609	26	)	)	PUNCT
ejpam-4914	609	27	is	be	AUX
ejpam-4914	609	28	done	do	VERB
ejpam-4914	609	29	.	.	PUNCT
ejpam-4914	610	1	note	note	VERB
ejpam-4914	610	2	that	that	SCONJ
ejpam-4914	610	3	such	such	ADJ
ejpam-4914	610	4	holds	hold	NOUN
ejpam-4914	610	5	particularly	particularly	ADV
ejpam-4914	610	6	when	when	SCONJ
ejpam-4914	610	7	x	x	PROPN
ejpam-4914	610	8	/∈	/∈	SYM
ejpam-4914	610	9	b	b	X
ejpam-4914	610	10	∪	∪	VERB
ejpam-4914	610	11	c	c	NOUN
ejpam-4914	610	12	,	,	PUNCT
ejpam-4914	610	13	y	y	PROPN
ejpam-4914	610	14	∈	∈	PROPN
ejpam-4914	610	15	ax	ax	NOUN
ejpam-4914	610	16	and	and	CCONJ
ejpam-4914	610	17	we	we	PRON
ejpam-4914	610	18	have	have	VERB
ejpam-4914	610	19	(	(	PUNCT
ejpam-4914	610	20	u	u	NOUN
ejpam-4914	610	21	,	,	PUNCT
ejpam-4914	610	22	v	v	NOUN
ejpam-4914	610	23	)	)	PUNCT
ejpam-4914	610	24	,	,	PUNCT
ejpam-4914	610	25	(	(	PUNCT
ejpam-4914	610	26	w	w	PROPN
ejpam-4914	610	27	,	,	PUNCT
ejpam-4914	610	28	z	z	NOUN
ejpam-4914	610	29	)	)	PUNCT
ejpam-4914	610	30	∈	∈	PROPN
ejpam-4914	610	31	v1	v1	NOUN
ejpam-4914	610	32	∩ng[h]((x	∩ng[h]((x	NOUN
ejpam-4914	610	33	,	,	PUNCT
ejpam-4914	610	34	y	y	PROPN
ejpam-4914	610	35	)	)	PUNCT
ejpam-4914	610	36	,	,	PUNCT
ejpam-4914	610	37	2	2	X
ejpam-4914	610	38	)	)	PUNCT
ejpam-4914	610	39	with	with	ADP
ejpam-4914	610	40	u	u	NOUN
ejpam-4914	610	41	̸=	̸=	PROPN
ejpam-4914	610	42	w.	w.	NOUN
ejpam-4914	610	43	assume	assume	VERB
ejpam-4914	610	44	that	that	SCONJ
ejpam-4914	610	45	|b	|b	ADJ
ejpam-4914	610	46	∩	∩	NOUN
ejpam-4914	610	47	ng(x	ng(x	NUM
ejpam-4914	610	48	,	,	PUNCT
ejpam-4914	610	49	2)|	2)|	NUM
ejpam-4914	610	50	=	=	SYM
ejpam-4914	610	51	1	1	NUM
ejpam-4914	610	52	,	,	PUNCT
ejpam-4914	610	53	say	say	VERB
ejpam-4914	610	54	b	b	NOUN
ejpam-4914	610	55	∩	∩	NOUN
ejpam-4914	610	56	ng(x	ng(x	NUM
ejpam-4914	610	57	,	,	PUNCT
ejpam-4914	610	58	2	2	X
ejpam-4914	610	59	)	)	PUNCT
ejpam-4914	610	60	=	=	PRON
ejpam-4914	610	61	{	{	PUNCT
ejpam-4914	610	62	w	w	NOUN
ejpam-4914	610	63	}	}	PUNCT
ejpam-4914	610	64	.	.	PUNCT
ejpam-4914	611	1	if	if	SCONJ
ejpam-4914	611	2	|bw|	|bw|	PROPN
ejpam-4914	611	3	≥	≥	NOUN
ejpam-4914	611	4	2	2	NUM
ejpam-4914	611	5	,	,	PUNCT
ejpam-4914	611	6	then	then	ADV
ejpam-4914	611	7	(	(	PUNCT
ejpam-4914	611	8	ii)(b	ii)(b	ADJ
ejpam-4914	611	9	)	)	PUNCT
ejpam-4914	611	10	holds	hold	VERB
ejpam-4914	611	11	.	.	PUNCT
ejpam-4914	612	1	note	note	VERB
ejpam-4914	612	2	that	that	SCONJ
ejpam-4914	612	3	this	this	PRON
ejpam-4914	612	4	readily	readily	ADV
ejpam-4914	612	5	follows	follow	VERB
ejpam-4914	612	6	if	if	SCONJ
ejpam-4914	612	7	x	x	PROPN
ejpam-4914	612	8	/∈	/∈	SYM
ejpam-4914	612	9	b	b	NOUN
ejpam-4914	612	10	∪	∪	PROPN
ejpam-4914	612	11	c.	c.	PROPN
ejpam-4914	612	12	now	now	ADV
ejpam-4914	612	13	suppose	suppose	VERB
ejpam-4914	612	14	that	that	SCONJ
ejpam-4914	612	15	|bw|	|bw|	PRON
ejpam-4914	612	16	=	=	SYM
ejpam-4914	612	17	1	1	X
ejpam-4914	612	18	.	.	PUNCT
ejpam-4914	613	1	since	since	SCONJ
ejpam-4914	613	2	c	c	PROPN
ejpam-4914	613	3	∩	∩	PROPN
ejpam-4914	613	4	ng(x	ng(x	NUM
ejpam-4914	613	5	,	,	PUNCT
ejpam-4914	613	6	2	2	X
ejpam-4914	613	7	)	)	PUNCT
ejpam-4914	613	8	=	=	NOUN
ejpam-4914	613	9	∅	∅	NOUN
ejpam-4914	613	10	,	,	PUNCT
ejpam-4914	613	11	necessarily	necessarily	ADV
ejpam-4914	613	12	x	x	ADP
ejpam-4914	613	13	∈	∈	PROPN
ejpam-4914	613	14	b	b	PROPN
ejpam-4914	613	15	∪	∪	VERB
ejpam-4914	613	16	c.	c.	NOUN
ejpam-4914	613	17	we	we	PRON
ejpam-4914	613	18	claim	claim	VERB
ejpam-4914	613	19	that	that	SCONJ
ejpam-4914	613	20	bx	bx	PROPN
ejpam-4914	613	21	∪	∪	PROPN
ejpam-4914	613	22	cx	cx	PROPN
ejpam-4914	613	23	is	be	AUX
ejpam-4914	613	24	a	a	DET
ejpam-4914	613	25	pnd	pnd	NOUN
ejpam-4914	613	26	-	-	PUNCT
ejpam-4914	613	27	set	set	NOUN
ejpam-4914	613	28	of	of	ADP
ejpam-4914	613	29	h.	h.	NOUN
ejpam-4914	613	30	let	let	VERB
ejpam-4914	613	31	y	y	PROPN
ejpam-4914	613	32	∈	∈	PROPN
ejpam-4914	613	33	v	v	ADP
ejpam-4914	613	34	(	(	PUNCT
ejpam-4914	613	35	h	h	NOUN
ejpam-4914	613	36	)	)	PUNCT
ejpam-4914	613	37	\	\	PUNCT
ejpam-4914	614	1	(	(	PUNCT
ejpam-4914	614	2	bx	bx	PROPN
ejpam-4914	614	3	∪	∪	PROPN
ejpam-4914	614	4	cx	cx	PROPN
ejpam-4914	614	5	)	)	PUNCT
ejpam-4914	615	1	=	=	NOUN
ejpam-4914	615	2	ax	ax	NOUN
ejpam-4914	615	3	\	\	PROPN
ejpam-4914	616	1	(	(	PUNCT
ejpam-4914	616	2	bx	bx	PROPN
ejpam-4914	616	3	∪	∪	PROPN
ejpam-4914	616	4	cx	cx	PROPN
ejpam-4914	616	5	)	)	PUNCT
ejpam-4914	616	6	.	.	PUNCT
ejpam-4914	617	1	then	then	ADV
ejpam-4914	617	2	(	(	PUNCT
ejpam-4914	617	3	x	x	X
ejpam-4914	617	4	,	,	PUNCT
ejpam-4914	617	5	y	y	NOUN
ejpam-4914	617	6	)	)	PUNCT
ejpam-4914	617	7	∈	∈	PROPN
ejpam-4914	617	8	v0	v0	NOUN
ejpam-4914	617	9	\	\	PUNCT
ejpam-4914	617	10	(	(	PUNCT
ejpam-4914	617	11	v1	v1	VERB
ejpam-4914	617	12	∪	∪	NOUN
ejpam-4914	617	13	v2	v2	NOUN
ejpam-4914	617	14	)	)	PUNCT
ejpam-4914	617	15	.	.	PUNCT
ejpam-4914	618	1	there	there	PRON
ejpam-4914	618	2	exists	exist	VERB
ejpam-4914	618	3	(	(	PUNCT
ejpam-4914	618	4	a	a	PRON
ejpam-4914	618	5	,	,	PUNCT
ejpam-4914	618	6	b	b	NOUN
ejpam-4914	618	7	)	)	PUNCT
ejpam-4914	618	8	∈	∈	PROPN
ejpam-4914	618	9	v2	v2	PROPN
ejpam-4914	618	10	∩ng[h]((x	∩ng[h]((x	NOUN
ejpam-4914	618	11	,	,	PUNCT
ejpam-4914	618	12	y	y	PROPN
ejpam-4914	618	13	)	)	PUNCT
ejpam-4914	618	14	,	,	PUNCT
ejpam-4914	618	15	2	2	NUM
ejpam-4914	618	16	)	)	PUNCT
ejpam-4914	618	17	or	or	CCONJ
ejpam-4914	618	18	there	there	PRON
ejpam-4914	618	19	exist	exist	VERB
ejpam-4914	618	20	distinct	distinct	ADJ
ejpam-4914	618	21	(	(	PUNCT
ejpam-4914	618	22	a	a	DET
ejpam-4914	618	23	,	,	PUNCT
ejpam-4914	618	24	b	b	NOUN
ejpam-4914	618	25	)	)	PUNCT
ejpam-4914	618	26	,	,	PUNCT
ejpam-4914	618	27	(	(	PUNCT
ejpam-4914	618	28	s	s	X
ejpam-4914	618	29	,	,	PUNCT
ejpam-4914	618	30	t	t	NOUN
ejpam-4914	618	31	)	)	PUNCT
ejpam-4914	618	32	∈	∈	PROPN
ejpam-4914	618	33	v1	v1	PROPN
ejpam-4914	618	34	∩ng[h]((x	∩ng[h]((x	NOUN
ejpam-4914	618	35	,	,	PUNCT
ejpam-4914	618	36	y	y	PROPN
ejpam-4914	618	37	)	)	PUNCT
ejpam-4914	618	38	,	,	PUNCT
ejpam-4914	618	39	2	2	NUM
ejpam-4914	618	40	)	)	PUNCT
ejpam-4914	618	41	.	.	PUNCT
ejpam-4914	619	1	the	the	DET
ejpam-4914	619	2	former	former	ADJ
ejpam-4914	619	3	implies	imply	VERB
ejpam-4914	619	4	that	that	SCONJ
ejpam-4914	619	5	b	b	X
ejpam-4914	619	6	∈	∈	PROPN
ejpam-4914	619	7	cx	cx	NOUN
ejpam-4914	619	8	and	and	CCONJ
ejpam-4914	619	9	by	by	ADP
ejpam-4914	619	10	/∈	/∈	PUNCT
ejpam-4914	619	11	e(h	e(h	PROPN
ejpam-4914	619	12	)	)	PUNCT
ejpam-4914	619	13	.	.	PUNCT
ejpam-4914	620	1	since	since	SCONJ
ejpam-4914	620	2	|bw|	|bw|	PROPN
ejpam-4914	620	3	=	=	SYM
ejpam-4914	620	4	1	1	NUM
ejpam-4914	620	5	,	,	PUNCT
ejpam-4914	620	6	the	the	DET
ejpam-4914	620	7	latter	latter	ADJ
ejpam-4914	620	8	implies	imply	VERB
ejpam-4914	620	9	that	that	SCONJ
ejpam-4914	620	10	a	a	DET
ejpam-4914	620	11	∈	∈	PROPN
ejpam-4914	620	12	bx	bx	NOUN
ejpam-4914	620	13	and	and	CCONJ
ejpam-4914	620	14	by	by	ADP
ejpam-4914	620	15	/∈	/∈	PUNCT
ejpam-4914	620	16	e(h	e(h	PROPN
ejpam-4914	620	17	)	)	PUNCT
ejpam-4914	620	18	or	or	CCONJ
ejpam-4914	620	19	s	s	PROPN
ejpam-4914	620	20	∈	∈	NOUN
ejpam-4914	620	21	bx	bx	NOUN
ejpam-4914	620	22	and	and	CCONJ
ejpam-4914	620	23	ty	ty	INTJ
ejpam-4914	620	24	/∈	/∈	PUNCT
ejpam-4914	620	25	e(h	e(h	PROPN
ejpam-4914	620	26	)	)	PUNCT
ejpam-4914	620	27	.	.	PUNCT
ejpam-4914	621	1	accordingly	accordingly	ADV
ejpam-4914	621	2	,	,	PUNCT
ejpam-4914	621	3	bx	bx	PROPN
ejpam-4914	621	4	∪cx	∪cx	PROPN
ejpam-4914	621	5	is	be	AUX
ejpam-4914	621	6	a	a	DET
ejpam-4914	621	7	pnd	pnd	NOUN
ejpam-4914	621	8	-	-	PUNCT
ejpam-4914	621	9	set	set	NOUN
ejpam-4914	621	10	of	of	ADP
ejpam-4914	621	11	h.	h.	PROPN
ejpam-4914	621	12	this	this	PRON
ejpam-4914	621	13	proves	prove	VERB
ejpam-4914	621	14	(	(	PUNCT
ejpam-4914	621	15	ii)(d	ii)(d	PROPN
ejpam-4914	621	16	)	)	PUNCT
ejpam-4914	621	17	.	.	PUNCT
ejpam-4914	622	1	s.r	s.r	PROPN
ejpam-4914	622	2	.	.	PROPN
ejpam-4914	622	3	jr	jr	PROPN
ejpam-4914	622	4	.	.	PROPN
ejpam-4914	622	5	canoy	canoy	PROPN
ejpam-4914	622	6	,	,	PUNCT
ejpam-4914	622	7	f.p	f.p	PROPN
ejpam-4914	622	8	.	.	PROPN
ejpam-4914	622	9	jamil	jamil	PROPN
ejpam-4914	622	10	and	and	CCONJ
ejpam-4914	622	11	s.m	s.m	PROPN
ejpam-4914	622	12	.	.	PROPN
ejpam-4914	622	13	menchavez	menchavez	PROPN
ejpam-4914	622	14	/	/	PUNCT
ejpam-4914	622	15	eur	eur	PROPN
ejpam-4914	622	16	.	.	PUNCT
ejpam-4914	623	1	j.	j.	PROPN
ejpam-4914	623	2	pure	pure	PROPN
ejpam-4914	623	3	appl	appl	PROPN
ejpam-4914	623	4	.	.	PROPN
ejpam-4914	623	5	math	math	PROPN
ejpam-4914	623	6	,	,	PUNCT
ejpam-4914	623	7	16	16	NUM
ejpam-4914	623	8	(	(	PUNCT
ejpam-4914	623	9	4	4	NUM
ejpam-4914	623	10	)	)	PUNCT
ejpam-4914	623	11	(	(	PUNCT
ejpam-4914	623	12	2023	2023	NUM
ejpam-4914	623	13	)	)	PUNCT
ejpam-4914	623	14	,	,	PUNCT
ejpam-4914	623	15	2431	2431	NUM
ejpam-4914	623	16	-	-	SYM
ejpam-4914	623	17	2449	2449	NUM
ejpam-4914	623	18	2446	2446	NUM
ejpam-4914	623	19	conversely	conversely	ADV
ejpam-4914	623	20	,	,	PUNCT
ejpam-4914	623	21	suppose	suppose	VERB
ejpam-4914	623	22	that	that	SCONJ
ejpam-4914	623	23	conditions	condition	NOUN
ejpam-4914	623	24	(	(	PUNCT
ejpam-4914	623	25	i	i	NOUN
ejpam-4914	623	26	)	)	PUNCT
ejpam-4914	623	27	and	and	CCONJ
ejpam-4914	623	28	(	(	PUNCT
ejpam-4914	623	29	ii	ii	NOUN
ejpam-4914	623	30	)	)	PUNCT
ejpam-4914	623	31	all	all	PRON
ejpam-4914	623	32	hold	hold	VERB
ejpam-4914	623	33	for	for	ADP
ejpam-4914	623	34	f	f	PROPN
ejpam-4914	623	35	.	.	PUNCT
ejpam-4914	624	1	let	let	VERB
ejpam-4914	624	2	(	(	PUNCT
ejpam-4914	624	3	x	x	NOUN
ejpam-4914	624	4	,	,	PUNCT
ejpam-4914	624	5	y	y	NOUN
ejpam-4914	624	6	)	)	PUNCT
ejpam-4914	624	7	∈	∈	PROPN
ejpam-4914	624	8	v0	v0	NOUN
ejpam-4914	624	9	.	.	PUNCT
ejpam-4914	625	1	then	then	ADV
ejpam-4914	625	2	x	x	SYM
ejpam-4914	625	3	∈	∈	NOUN
ejpam-4914	625	4	a.	a.	NOUN
ejpam-4914	625	5	if	if	SCONJ
ejpam-4914	625	6	u	u	PROPN
ejpam-4914	625	7	∈	∈	PROPN
ejpam-4914	625	8	c∩ng(x	c∩ng(x	NOUN
ejpam-4914	625	9	,	,	PUNCT
ejpam-4914	625	10	2	2	NUM
ejpam-4914	625	11	)	)	PUNCT
ejpam-4914	625	12	,	,	PUNCT
ejpam-4914	625	13	then	then	ADV
ejpam-4914	625	14	for	for	ADP
ejpam-4914	625	15	any	any	DET
ejpam-4914	625	16	v	v	PROPN
ejpam-4914	625	17	∈	∈	PROPN
ejpam-4914	625	18	cu	cu	PROPN
ejpam-4914	625	19	,	,	PUNCT
ejpam-4914	625	20	(	(	PUNCT
ejpam-4914	625	21	u	u	NOUN
ejpam-4914	625	22	,	,	PUNCT
ejpam-4914	625	23	v	v	NOUN
ejpam-4914	625	24	)	)	PUNCT
ejpam-4914	625	25	∈	∈	PROPN
ejpam-4914	625	26	v2∩ng[h]((x	v2∩ng[h]((x	NOUN
ejpam-4914	625	27	,	,	PUNCT
ejpam-4914	625	28	y	y	PROPN
ejpam-4914	625	29	)	)	PUNCT
ejpam-4914	625	30	,	,	PUNCT
ejpam-4914	625	31	2	2	NUM
ejpam-4914	625	32	)	)	PUNCT
ejpam-4914	625	33	.	.	PUNCT
ejpam-4914	626	1	now	now	ADV
ejpam-4914	626	2	assume	assume	VERB
ejpam-4914	626	3	that	that	SCONJ
ejpam-4914	626	4	c	c	PROPN
ejpam-4914	626	5	∩ng(x	∩ng(x	PROPN
ejpam-4914	626	6	,	,	PUNCT
ejpam-4914	626	7	2	2	X
ejpam-4914	626	8	)	)	PUNCT
ejpam-4914	626	9	=	=	VERB
ejpam-4914	626	10	∅.	∅.	NOUN
ejpam-4914	626	11	it	it	PRON
ejpam-4914	626	12	is	be	AUX
ejpam-4914	626	13	straightforward	straightforward	ADJ
ejpam-4914	626	14	to	to	PART
ejpam-4914	626	15	show	show	VERB
ejpam-4914	626	16	that	that	SCONJ
ejpam-4914	626	17	if	if	SCONJ
ejpam-4914	626	18	(	(	PUNCT
ejpam-4914	626	19	ii)(a	ii)(a	NOUN
ejpam-4914	626	20	)	)	PUNCT
ejpam-4914	626	21	or	or	CCONJ
ejpam-4914	626	22	(	(	PUNCT
ejpam-4914	626	23	ii)(b	ii)(b	ADJ
ejpam-4914	626	24	)	)	PUNCT
ejpam-4914	626	25	holds	hold	VERB
ejpam-4914	626	26	for	for	ADP
ejpam-4914	626	27	x	x	X
ejpam-4914	626	28	,	,	PUNCT
ejpam-4914	626	29	then	then	ADV
ejpam-4914	626	30	|v1	|v1	PROPN
ejpam-4914	626	31	∩ng[h]((x	∩ng[h]((x	NOUN
ejpam-4914	626	32	,	,	PUNCT
ejpam-4914	626	33	y	y	PROPN
ejpam-4914	626	34	)	)	PUNCT
ejpam-4914	626	35	,	,	PUNCT
ejpam-4914	626	36	2)|	2)|	NUM
ejpam-4914	626	37	≥	≥	NUM
ejpam-4914	626	38	2	2	NUM
ejpam-4914	626	39	.	.	PUNCT
ejpam-4914	626	40	suppose	suppose	VERB
ejpam-4914	626	41	that	that	SCONJ
ejpam-4914	626	42	(	(	PUNCT
ejpam-4914	626	43	ii)(c	ii)(c	PROPN
ejpam-4914	626	44	)	)	PUNCT
ejpam-4914	626	45	holds	hold	VERB
ejpam-4914	626	46	for	for	SCONJ
ejpam-4914	626	47	x.	x.	NOUN
ejpam-4914	626	48	let	let	VERB
ejpam-4914	626	49	b	b	X
ejpam-4914	626	50	∩ng(x	∩ng(x	VERB
ejpam-4914	626	51	,	,	PUNCT
ejpam-4914	626	52	2	2	NUM
ejpam-4914	626	53	)	)	PUNCT
ejpam-4914	626	54	=	=	PRON
ejpam-4914	626	55	{	{	PUNCT
ejpam-4914	626	56	w	w	NOUN
ejpam-4914	626	57	}	}	PUNCT
ejpam-4914	626	58	and	and	CCONJ
ejpam-4914	626	59	let	let	VERB
ejpam-4914	626	60	z	z	NOUN
ejpam-4914	626	61	∈	∈	PROPN
ejpam-4914	626	62	bw	bw	NOUN
ejpam-4914	626	63	.	.	PUNCT
ejpam-4914	627	1	if	if	SCONJ
ejpam-4914	627	2	t	t	PROPN
ejpam-4914	627	3	∈	∈	PROPN
ejpam-4914	627	4	cx	cx	PROPN
ejpam-4914	627	5	for	for	ADP
ejpam-4914	627	6	which	which	PRON
ejpam-4914	627	7	ty	ty	NUM
ejpam-4914	627	8	/∈	/∈	PUNCT
ejpam-4914	627	9	e(h	e(h	PROPN
ejpam-4914	627	10	)	)	PUNCT
ejpam-4914	627	11	,	,	PUNCT
ejpam-4914	627	12	then	then	ADV
ejpam-4914	627	13	(	(	PUNCT
ejpam-4914	627	14	x	x	X
ejpam-4914	627	15	,	,	PUNCT
ejpam-4914	627	16	t	t	PROPN
ejpam-4914	627	17	)	)	PUNCT
ejpam-4914	627	18	∈	∈	PROPN
ejpam-4914	627	19	v2	v2	PROPN
ejpam-4914	627	20	∩ng[h]((x	∩ng[h]((x	NOUN
ejpam-4914	627	21	,	,	PUNCT
ejpam-4914	627	22	y	y	PROPN
ejpam-4914	627	23	)	)	PUNCT
ejpam-4914	627	24	,	,	PUNCT
ejpam-4914	627	25	2	2	NUM
ejpam-4914	627	26	)	)	PUNCT
ejpam-4914	627	27	.	.	PUNCT
ejpam-4914	628	1	on	on	ADP
ejpam-4914	628	2	the	the	DET
ejpam-4914	628	3	other	other	ADJ
ejpam-4914	628	4	hand	hand	NOUN
ejpam-4914	628	5	,	,	PUNCT
ejpam-4914	628	6	if	if	SCONJ
ejpam-4914	628	7	t	t	PROPN
ejpam-4914	628	8	∈	∈	PROPN
ejpam-4914	628	9	bx	bx	PROPN
ejpam-4914	628	10	for	for	ADP
ejpam-4914	628	11	which	which	PRON
ejpam-4914	628	12	ty	ty	NUM
ejpam-4914	628	13	/∈	/∈	PUNCT
ejpam-4914	628	14	e(h	e(h	PROPN
ejpam-4914	628	15	)	)	PUNCT
ejpam-4914	628	16	,	,	PUNCT
ejpam-4914	628	17	then	then	ADV
ejpam-4914	628	18	(	(	PUNCT
ejpam-4914	628	19	x	x	X
ejpam-4914	628	20	,	,	PUNCT
ejpam-4914	628	21	t	t	PROPN
ejpam-4914	628	22	)	)	PUNCT
ejpam-4914	628	23	and	and	CCONJ
ejpam-4914	628	24	(	(	PUNCT
ejpam-4914	628	25	w	w	PROPN
ejpam-4914	628	26	,	,	PUNCT
ejpam-4914	628	27	z	z	NOUN
ejpam-4914	628	28	)	)	PUNCT
ejpam-4914	628	29	are	be	AUX
ejpam-4914	628	30	distinct	distinct	ADJ
ejpam-4914	628	31	vertices	vertex	NOUN
ejpam-4914	628	32	in	in	ADP
ejpam-4914	628	33	v1∩ng[h]((x	v1∩ng[h]((x	ADP
ejpam-4914	628	34	,	,	PUNCT
ejpam-4914	628	35	y	y	PROPN
ejpam-4914	628	36	)	)	PUNCT
ejpam-4914	628	37	,	,	PUNCT
ejpam-4914	628	38	2	2	NUM
ejpam-4914	628	39	)	)	PUNCT
ejpam-4914	628	40	.	.	PUNCT
ejpam-4914	629	1	finally	finally	ADV
ejpam-4914	629	2	,	,	PUNCT
ejpam-4914	629	3	suppose	suppose	VERB
ejpam-4914	629	4	that	that	SCONJ
ejpam-4914	629	5	(	(	PUNCT
ejpam-4914	629	6	ii)(d	ii)(d	PROPN
ejpam-4914	629	7	)	)	PUNCT
ejpam-4914	629	8	holds	hold	VERB
ejpam-4914	629	9	for	for	ADP
ejpam-4914	629	10	x.	x.	NOUN
ejpam-4914	629	11	put	put	VERB
ejpam-4914	629	12	tx	tx	PROPN
ejpam-4914	629	13	=	=	PUNCT
ejpam-4914	629	14	⟨{x}×v	⟨{x}×v	PROPN
ejpam-4914	629	15	(	(	PUNCT
ejpam-4914	629	16	h)⟩	h)⟩	NOUN
ejpam-4914	629	17	and	and	CCONJ
ejpam-4914	629	18	define	define	VERB
ejpam-4914	629	19	v	v	NUM
ejpam-4914	629	20	x	x	PUNCT
ejpam-4914	629	21	i	i	PROPN
ejpam-4914	629	22	=	=	SYM
ejpam-4914	629	23	vi	vi	PROPN
ejpam-4914	629	24	∩	∩	ADJ
ejpam-4914	629	25	v	v	NOUN
ejpam-4914	629	26	(	(	PUNCT
ejpam-4914	629	27	tx	tx	PROPN
ejpam-4914	629	28	)	)	PUNCT
ejpam-4914	629	29	for	for	ADP
ejpam-4914	629	30	i	i	PROPN
ejpam-4914	629	31	=	=	SYM
ejpam-4914	629	32	0	0	NUM
ejpam-4914	629	33	,	,	PUNCT
ejpam-4914	629	34	1	1	NUM
ejpam-4914	629	35	,	,	PUNCT
ejpam-4914	629	36	2	2	NUM
ejpam-4914	629	37	.	.	PUNCT
ejpam-4914	630	1	then	then	ADV
ejpam-4914	630	2	f	f	X
ejpam-4914	630	3	|tx	|tx	PROPN
ejpam-4914	631	1	=	=	SYM
ejpam-4914	631	2	(	(	PUNCT
ejpam-4914	631	3	v	v	NOUN
ejpam-4914	631	4	x	x	SYM
ejpam-4914	631	5	0	0	NUM
ejpam-4914	631	6	,	,	PUNCT
ejpam-4914	631	7	v	v	NOUN
ejpam-4914	631	8	x	x	SYM
ejpam-4914	631	9	1	1	NUM
ejpam-4914	631	10	,	,	PUNCT
ejpam-4914	631	11	v	v	NOUN
ejpam-4914	631	12	x	x	SYM
ejpam-4914	631	13	2	2	NUM
ejpam-4914	631	14	)	)	PUNCT
ejpam-4914	631	15	.	.	PUNCT
ejpam-4914	632	1	since	since	SCONJ
ejpam-4914	632	2	(	(	PUNCT
ejpam-4914	632	3	x	x	NOUN
ejpam-4914	632	4	,	,	PUNCT
ejpam-4914	632	5	y	y	NOUN
ejpam-4914	632	6	)	)	PUNCT
ejpam-4914	632	7	∈	∈	PROPN
ejpam-4914	632	8	v	v	NOUN
ejpam-4914	632	9	x	x	SYM
ejpam-4914	632	10	0	0	NUM
ejpam-4914	632	11	and	and	CCONJ
ejpam-4914	632	12	f	f	PROPN
ejpam-4914	632	13	|tx	|tx	X
ejpam-4914	632	14	is	be	AUX
ejpam-4914	632	15	a	a	DET
ejpam-4914	632	16	pndi	pndi	ADJ
ejpam-4914	632	17	-function	-function	NOUN
ejpam-4914	632	18	of	of	ADP
ejpam-4914	632	19	tx	tx	PROPN
ejpam-4914	632	20	,	,	PUNCT
ejpam-4914	632	21	there	there	PRON
ejpam-4914	632	22	exists	exist	VERB
ejpam-4914	632	23	(	(	PUNCT
ejpam-4914	632	24	x	x	X
ejpam-4914	632	25	,	,	PUNCT
ejpam-4914	632	26	v	v	NOUN
ejpam-4914	632	27	)	)	PUNCT
ejpam-4914	632	28	∈	∈	NOUN
ejpam-4914	632	29	v	v	NOUN
ejpam-4914	632	30	x	x	SYM
ejpam-4914	632	31	2	2	NUM
ejpam-4914	632	32	with	with	ADP
ejpam-4914	632	33	(	(	PUNCT
ejpam-4914	632	34	x	x	NOUN
ejpam-4914	632	35	,	,	PUNCT
ejpam-4914	632	36	y)(x	y)(x	PROPN
ejpam-4914	632	37	,	,	PUNCT
ejpam-4914	632	38	v	v	NOUN
ejpam-4914	632	39	)	)	PUNCT
ejpam-4914	632	40	/∈	/∈	PUNCT
ejpam-4914	633	1	e(tx	e(tx	NUM
ejpam-4914	633	2	)	)	PUNCT
ejpam-4914	633	3	or	or	CCONJ
ejpam-4914	633	4	there	there	PRON
ejpam-4914	633	5	exist	exist	VERB
ejpam-4914	633	6	distinct	distinct	ADJ
ejpam-4914	633	7	(	(	PUNCT
ejpam-4914	633	8	x	x	NOUN
ejpam-4914	633	9	,	,	PUNCT
ejpam-4914	633	10	v	v	NOUN
ejpam-4914	633	11	)	)	PUNCT
ejpam-4914	633	12	,	,	PUNCT
ejpam-4914	634	1	(	(	PUNCT
ejpam-4914	634	2	x	x	X
ejpam-4914	634	3	,	,	PUNCT
ejpam-4914	634	4	z	z	NOUN
ejpam-4914	634	5	)	)	PUNCT
ejpam-4914	634	6	∈	∈	NOUN
ejpam-4914	634	7	v	v	NOUN
ejpam-4914	634	8	x	x	SYM
ejpam-4914	634	9	1	1	NUM
ejpam-4914	634	10	such	such	ADJ
ejpam-4914	634	11	that	that	SCONJ
ejpam-4914	634	12	(	(	PUNCT
ejpam-4914	634	13	x	x	NOUN
ejpam-4914	634	14	,	,	PUNCT
ejpam-4914	634	15	y)(x	y)(x	PROPN
ejpam-4914	634	16	,	,	PUNCT
ejpam-4914	634	17	v	v	NOUN
ejpam-4914	634	18	)	)	PUNCT
ejpam-4914	634	19	,	,	PUNCT
ejpam-4914	634	20	(	(	PUNCT
ejpam-4914	634	21	x	x	NOUN
ejpam-4914	634	22	,	,	PUNCT
ejpam-4914	634	23	y)(x	y)(x	PROPN
ejpam-4914	634	24	,	,	PUNCT
ejpam-4914	634	25	z	z	NOUN
ejpam-4914	634	26	)	)	PUNCT
ejpam-4914	634	27	/∈	/∈	PUNCT
ejpam-4914	635	1	e(tx	e(tx	NUM
ejpam-4914	635	2	)	)	PUNCT
ejpam-4914	635	3	.	.	PUNCT
ejpam-4914	636	1	the	the	DET
ejpam-4914	636	2	former	former	ADJ
ejpam-4914	636	3	implies	imply	VERB
ejpam-4914	636	4	that	that	SCONJ
ejpam-4914	636	5	(	(	PUNCT
ejpam-4914	636	6	x	x	NOUN
ejpam-4914	636	7	,	,	PUNCT
ejpam-4914	636	8	v	v	NOUN
ejpam-4914	636	9	)	)	PUNCT
ejpam-4914	636	10	∈	∈	PROPN
ejpam-4914	636	11	v2	v2	PROPN
ejpam-4914	636	12	∩	∩	NOUN
ejpam-4914	636	13	ng[h]((x	ng[h]((x	NOUN
ejpam-4914	636	14	,	,	PUNCT
ejpam-4914	636	15	y	y	NOUN
ejpam-4914	636	16	)	)	PUNCT
ejpam-4914	636	17	,	,	PUNCT
ejpam-4914	636	18	2	2	NUM
ejpam-4914	636	19	)	)	PUNCT
ejpam-4914	636	20	,	,	PUNCT
ejpam-4914	636	21	while	while	SCONJ
ejpam-4914	636	22	the	the	DET
ejpam-4914	636	23	latter	latter	ADJ
ejpam-4914	636	24	implies	imply	VERB
ejpam-4914	636	25	that	that	SCONJ
ejpam-4914	636	26	(	(	PUNCT
ejpam-4914	636	27	x	x	NOUN
ejpam-4914	636	28	,	,	PUNCT
ejpam-4914	636	29	v	v	NOUN
ejpam-4914	636	30	)	)	PUNCT
ejpam-4914	636	31	,	,	PUNCT
ejpam-4914	636	32	(	(	PUNCT
ejpam-4914	636	33	x	x	X
ejpam-4914	636	34	,	,	PUNCT
ejpam-4914	636	35	z	z	NOUN
ejpam-4914	636	36	)	)	PUNCT
ejpam-4914	636	37	∈	∈	PROPN
ejpam-4914	636	38	v1	v1	NOUN
ejpam-4914	636	39	∩ng[h]((x	∩ng[h]((x	NOUN
ejpam-4914	636	40	,	,	PUNCT
ejpam-4914	636	41	y	y	PROPN
ejpam-4914	636	42	)	)	PUNCT
ejpam-4914	636	43	,	,	PUNCT
ejpam-4914	636	44	2	2	NUM
ejpam-4914	636	45	)	)	PUNCT
ejpam-4914	636	46	.	.	PUNCT
ejpam-4914	637	1	therefore	therefore	ADV
ejpam-4914	637	2	,	,	PUNCT
ejpam-4914	637	3	f	f	PROPN
ejpam-4914	637	4	∈	∈	PROPN
ejpam-4914	637	5	hid(g[h	hid(g[h	NOUN
ejpam-4914	637	6	]	]	PUNCT
ejpam-4914	637	7	)	)	PUNCT
ejpam-4914	637	8	.	.	PUNCT
ejpam-4914	638	1	■	■	PUNCT
ejpam-4914	638	2	corollary	corollary	ADJ
ejpam-4914	638	3	3.9	3.9	NUM
ejpam-4914	638	4	.	.	PUNCT
ejpam-4914	639	1	let	let	VERB
ejpam-4914	639	2	g	g	NOUN
ejpam-4914	639	3	and	and	CCONJ
ejpam-4914	639	4	h	h	NOUN
ejpam-4914	639	5	be	be	AUX
ejpam-4914	639	6	nontrivial	nontrivial	ADJ
ejpam-4914	639	7	connected	connect	VERB
ejpam-4914	639	8	graphs	graph	NOUN
ejpam-4914	639	9	where	where	SCONJ
ejpam-4914	639	10	h	h	NOUN
ejpam-4914	639	11	is	be	AUX
ejpam-4914	639	12	noncomplete	noncomplete	ADJ
ejpam-4914	639	13	.	.	PUNCT
ejpam-4914	640	1	then	then	ADV
ejpam-4914	640	2	γhi(g[h	γhi(g[h	NUM
ejpam-4914	640	3	]	]	NOUN
ejpam-4914	640	4	)	)	PUNCT
ejpam-4914	640	5	≤	≤	NOUN
ejpam-4914	640	6	min{2|s	min{2|s	PUNCT
ejpam-4914	640	7	∩ng(s	∩ng(s	VERB
ejpam-4914	640	8	,	,	PUNCT
ejpam-4914	640	9	2)|+	2)|+	NUM
ejpam-4914	640	10	pndi(h)|s	pndi(h)|s	NOUN
ejpam-4914	640	11	\ng(s	\ng(s	NOUN
ejpam-4914	640	12	,	,	PUNCT
ejpam-4914	640	13	2)|	2)|	NUM
ejpam-4914	640	14	:	:	PUNCT
ejpam-4914	641	1	s	s	X
ejpam-4914	641	2	∈	∈	NOUN
ejpam-4914	641	3	hd(g	hd(g	NOUN
ejpam-4914	641	4	)	)	PUNCT
ejpam-4914	641	5	}	}	PUNCT
ejpam-4914	641	6	,	,	PUNCT
ejpam-4914	641	7	and	and	CCONJ
ejpam-4914	641	8	this	this	DET
ejpam-4914	641	9	bound	bind	VERB
ejpam-4914	641	10	is	be	AUX
ejpam-4914	641	11	sharp	sharp	ADJ
ejpam-4914	641	12	.	.	PUNCT
ejpam-4914	642	1	proof	proof	NOUN
ejpam-4914	642	2	:	:	PUNCT
ejpam-4914	642	3	put	put	VERB
ejpam-4914	642	4	αh(g	αh(g	NOUN
ejpam-4914	642	5	)	)	PUNCT
ejpam-4914	642	6	=	=	SYM
ejpam-4914	642	7	min{2|s	min{2|	NOUN
ejpam-4914	642	8	∩	∩	NOUN
ejpam-4914	642	9	ng(s	ng(s	NUM
ejpam-4914	642	10	,	,	PUNCT
ejpam-4914	642	11	2)|	2)|	NUM
ejpam-4914	642	12	+	+	CCONJ
ejpam-4914	642	13	pndi(h)|s	pndi(h)|s	NOUN
ejpam-4914	642	14	\	\	NOUN
ejpam-4914	642	15	ng(s	ng(s	PUNCT
ejpam-4914	642	16	,	,	PUNCT
ejpam-4914	642	17	2)|	2)|	NUM
ejpam-4914	642	18	:	:	PUNCT
ejpam-4914	642	19	s	s	VERB
ejpam-4914	642	20	∈	∈	NOUN
ejpam-4914	642	21	hd(g	hd(g	NOUN
ejpam-4914	642	22	)	)	PUNCT
ejpam-4914	642	23	}	}	PUNCT
ejpam-4914	642	24	.	.	PUNCT
ejpam-4914	643	1	let	let	VERB
ejpam-4914	643	2	s	s	PRON
ejpam-4914	643	3	⊆	⊆	NUM
ejpam-4914	643	4	v	v	NOUN
ejpam-4914	643	5	(	(	PUNCT
ejpam-4914	643	6	g	g	NOUN
ejpam-4914	643	7	)	)	PUNCT
ejpam-4914	643	8	be	be	AUX
ejpam-4914	643	9	a	a	DET
ejpam-4914	643	10	hop	hop	NOUN
ejpam-4914	643	11	dominating	dominating	NOUN
ejpam-4914	643	12	set	set	NOUN
ejpam-4914	643	13	of	of	ADP
ejpam-4914	643	14	g.	g.	PROPN
ejpam-4914	643	15	for	for	ADP
ejpam-4914	643	16	each	each	DET
ejpam-4914	643	17	x	x	SYM
ejpam-4914	643	18	∈	∈	PROPN
ejpam-4914	643	19	s	s	PART
ejpam-4914	643	20	\ng(s	\ng(s	NOUN
ejpam-4914	643	21	,	,	PUNCT
ejpam-4914	643	22	2	2	NUM
ejpam-4914	643	23	)	)	PUNCT
ejpam-4914	643	24	,	,	PUNCT
ejpam-4914	643	25	let	let	VERB
ejpam-4914	643	26	fx	fx	NOUN
ejpam-4914	643	27	=	=	PUNCT
ejpam-4914	643	28	(	(	PUNCT
ejpam-4914	643	29	v	v	NOUN
ejpam-4914	643	30	x	x	SYM
ejpam-4914	643	31	0	0	NUM
ejpam-4914	643	32	,	,	PUNCT
ejpam-4914	643	33	v	v	NOUN
ejpam-4914	643	34	x	x	SYM
ejpam-4914	643	35	1	1	NUM
ejpam-4914	643	36	,	,	PUNCT
ejpam-4914	643	37	v	v	NOUN
ejpam-4914	643	38	x	x	SYM
ejpam-4914	643	39	2	2	NUM
ejpam-4914	643	40	)	)	PUNCT
ejpam-4914	643	41	be	be	AUX
ejpam-4914	643	42	a	a	DET
ejpam-4914	643	43	pndi	pndi	ADJ
ejpam-4914	643	44	-function	-function	NOUN
ejpam-4914	643	45	of	of	ADP
ejpam-4914	643	46	⟨{x	⟨{x	NOUN
ejpam-4914	643	47	}	}	PUNCT
ejpam-4914	643	48	×	×	NOUN
ejpam-4914	643	49	v	v	NOUN
ejpam-4914	643	50	(	(	PUNCT
ejpam-4914	643	51	h)⟩.	h)⟩.	NOUN
ejpam-4914	643	52	by	by	ADP
ejpam-4914	643	53	lemma	lemma	PROPN
ejpam-4914	643	54	2.8	2.8	NUM
ejpam-4914	643	55	,	,	PUNCT
ejpam-4914	643	56	since	since	SCONJ
ejpam-4914	643	57	⟨{x	⟨{x	PROPN
ejpam-4914	643	58	}	}	PUNCT
ejpam-4914	643	59	×	×	NOUN
ejpam-4914	643	60	v	v	NOUN
ejpam-4914	644	1	(	(	PUNCT
ejpam-4914	644	2	h)⟩	h)⟩	PROPN
ejpam-4914	644	3	is	be	AUX
ejpam-4914	644	4	noncomplete	noncomplete	ADJ
ejpam-4914	644	5	,	,	PUNCT
ejpam-4914	644	6	we	we	PRON
ejpam-4914	644	7	assume	assume	VERB
ejpam-4914	644	8	that	that	SCONJ
ejpam-4914	644	9	v	v	X
ejpam-4914	644	10	x	x	SYM
ejpam-4914	644	11	2	2	NUM
ejpam-4914	644	12	̸=	̸=	PROPN
ejpam-4914	644	13	∅	∅	NOUN
ejpam-4914	644	14	for	for	ADP
ejpam-4914	644	15	each	each	DET
ejpam-4914	644	16	x	x	PUNCT
ejpam-4914	644	17	∈	∈	PROPN
ejpam-4914	644	18	s	s	PART
ejpam-4914	644	19	\ng(s	\ng(s	NOUN
ejpam-4914	644	20	,	,	PUNCT
ejpam-4914	644	21	2	2	NUM
ejpam-4914	644	22	)	)	PUNCT
ejpam-4914	644	23	.	.	PUNCT
ejpam-4914	645	1	pick	pick	VERB
ejpam-4914	645	2	y	y	PROPN
ejpam-4914	645	3	∈	∈	PROPN
ejpam-4914	645	4	v	v	PROPN
ejpam-4914	645	5	(	(	PUNCT
ejpam-4914	645	6	h	h	NOUN
ejpam-4914	645	7	)	)	PUNCT
ejpam-4914	645	8	.	.	PUNCT
ejpam-4914	646	1	define	define	VERB
ejpam-4914	646	2	the	the	DET
ejpam-4914	646	3	following	follow	VERB
ejpam-4914	646	4	sets	set	NOUN
ejpam-4914	646	5	:	:	PUNCT
ejpam-4914	646	6	•	•	NUM
ejpam-4914	646	7	v2	v2	NOUN
ejpam-4914	646	8	=	=	SYM
ejpam-4914	646	9	[	[	PUNCT
ejpam-4914	646	10	∪x∈s∩ng(s,2){(x	∪x∈s∩ng(s,2){(x	PROPN
ejpam-4914	646	11	,	,	PUNCT
ejpam-4914	646	12	y	y	NOUN
ejpam-4914	646	13	)	)	PUNCT
ejpam-4914	646	14	}	}	PUNCT
ejpam-4914	646	15	]	]	PUNCT
ejpam-4914	646	16	∪	∪	X
ejpam-4914	646	17	[	[	PUNCT
ejpam-4914	646	18	∪x∈s\ng(s,2)v	∪x∈s\ng(s,2)v	PUNCT
ejpam-4914	646	19	x	x	SYM
ejpam-4914	646	20	2	2	NUM
ejpam-4914	646	21	]	]	PUNCT
ejpam-4914	646	22	;	;	PUNCT
ejpam-4914	646	23	•	•	NUM
ejpam-4914	646	24	v1	v1	NOUN
ejpam-4914	646	25	=	=	SYM
ejpam-4914	646	26	∪x∈s\ng(s,2)v	∪x∈s\ng(s,2)v	SYM
ejpam-4914	646	27	x	x	SYM
ejpam-4914	646	28	1	1	NUM
ejpam-4914	646	29	;	;	PUNCT
ejpam-4914	646	30	and	and	CCONJ
ejpam-4914	646	31	•	•	NUM
ejpam-4914	646	32	v0	v0	NOUN
ejpam-4914	646	33	=	=	SYM
ejpam-4914	646	34	v	v	NOUN
ejpam-4914	646	35	(	(	PUNCT
ejpam-4914	646	36	g[h	g[h	PROPN
ejpam-4914	646	37	]	]	PUNCT
ejpam-4914	646	38	)	)	PUNCT
ejpam-4914	646	39	\	\	PUNCT
ejpam-4914	647	1	(	(	PUNCT
ejpam-4914	647	2	v1	v1	VERB
ejpam-4914	647	3	∪	∪	NOUN
ejpam-4914	647	4	v2	v2	NOUN
ejpam-4914	647	5	)	)	PUNCT
ejpam-4914	647	6	.	.	PUNCT
ejpam-4914	648	1	let	let	VERB
ejpam-4914	648	2	f	f	PROPN
ejpam-4914	648	3	=	=	SYM
ejpam-4914	648	4	(	(	PUNCT
ejpam-4914	648	5	v0	v0	PROPN
ejpam-4914	648	6	,	,	PUNCT
ejpam-4914	648	7	v1	v1	NOUN
ejpam-4914	648	8	,	,	PUNCT
ejpam-4914	648	9	v2	v2	PROPN
ejpam-4914	648	10	)	)	PUNCT
ejpam-4914	648	11	.	.	PUNCT
ejpam-4914	649	1	as	as	ADP
ejpam-4914	649	2	in	in	ADP
ejpam-4914	649	3	theorem	theorem	NOUN
ejpam-4914	649	4	3.8	3.8	NUM
ejpam-4914	649	5	,	,	PUNCT
ejpam-4914	649	6	write	write	VERB
ejpam-4914	649	7	v0	v0	NOUN
ejpam-4914	649	8	=	=	SYM
ejpam-4914	649	9	∪x∈a	∪x∈a	PROPN
ejpam-4914	649	10	(	(	PUNCT
ejpam-4914	649	11	{	{	PUNCT
ejpam-4914	649	12	x	x	NOUN
ejpam-4914	649	13	}	}	PUNCT
ejpam-4914	649	14	×ax	×ax	ADJ
ejpam-4914	649	15	)	)	PUNCT
ejpam-4914	649	16	,	,	PUNCT
ejpam-4914	649	17	v1	v1	NOUN
ejpam-4914	649	18	=	=	SYM
ejpam-4914	649	19	∪x∈b	∪x∈b	NOUN
ejpam-4914	649	20	(	(	PUNCT
ejpam-4914	649	21	{	{	PUNCT
ejpam-4914	649	22	x	x	NOUN
ejpam-4914	649	23	}	}	PUNCT
ejpam-4914	649	24	×bx	×bx	NOUN
ejpam-4914	649	25	)	)	PUNCT
ejpam-4914	649	26	and	and	CCONJ
ejpam-4914	649	27	v2	v2	NOUN
ejpam-4914	649	28	=	=	SYM
ejpam-4914	649	29	∪x∈c	∪x∈c	NOUN
ejpam-4914	649	30	(	(	PUNCT
ejpam-4914	649	31	{	{	PUNCT
ejpam-4914	649	32	x	x	NOUN
ejpam-4914	649	33	}	}	PUNCT
ejpam-4914	649	34	×	×	PROPN
ejpam-4914	649	35	cx	cx	NOUN
ejpam-4914	649	36	)	)	PUNCT
ejpam-4914	649	37	.	.	PUNCT
ejpam-4914	650	1	since	since	SCONJ
ejpam-4914	650	2	v	v	NUM
ejpam-4914	650	3	x	x	SYM
ejpam-4914	650	4	2	2	NUM
ejpam-4914	650	5	̸=	̸=	PROPN
ejpam-4914	650	6	∅	∅	NOUN
ejpam-4914	650	7	for	for	ADP
ejpam-4914	650	8	each	each	DET
ejpam-4914	650	9	x	x	PUNCT
ejpam-4914	650	10	∈	∈	PROPN
ejpam-4914	650	11	s	s	PART
ejpam-4914	650	12	\	\	NOUN
ejpam-4914	650	13	ng(s	ng(s	NUM
ejpam-4914	650	14	,	,	PUNCT
ejpam-4914	650	15	2	2	NUM
ejpam-4914	650	16	)	)	PUNCT
ejpam-4914	650	17	,	,	PUNCT
ejpam-4914	650	18	c	c	X
ejpam-4914	650	19	=	=	SYM
ejpam-4914	650	20	s	s	PROPN
ejpam-4914	650	21	and	and	CCONJ
ejpam-4914	650	22	b	b	X
ejpam-4914	650	23	=	=	SYM
ejpam-4914	650	24	s	s	PART
ejpam-4914	650	25	\	\	NOUN
ejpam-4914	650	26	ng(s	ng(s	PUNCT
ejpam-4914	650	27	,	,	PUNCT
ejpam-4914	650	28	2	2	NUM
ejpam-4914	650	29	)	)	PUNCT
ejpam-4914	650	30	.	.	PUNCT
ejpam-4914	651	1	thus	thus	ADV
ejpam-4914	651	2	b	b	X
ejpam-4914	651	3	∪	∪	NOUN
ejpam-4914	651	4	c	c	X
ejpam-4914	651	5	is	be	AUX
ejpam-4914	651	6	a	a	DET
ejpam-4914	651	7	hop	hop	NOUN
ejpam-4914	651	8	dominating	dominating	NOUN
ejpam-4914	651	9	set	set	NOUN
ejpam-4914	651	10	of	of	ADP
ejpam-4914	651	11	g.	g.	PROPN
ejpam-4914	651	12	let	let	VERB
ejpam-4914	651	13	x	x	SYM
ejpam-4914	651	14	∈	∈	VERB
ejpam-4914	651	15	a	a	PRON
ejpam-4914	651	16	with	with	ADP
ejpam-4914	651	17	c	c	NOUN
ejpam-4914	651	18	∩	∩	NOUN
ejpam-4914	651	19	ng(x	ng(x	NUM
ejpam-4914	651	20	,	,	PUNCT
ejpam-4914	651	21	2	2	X
ejpam-4914	651	22	)	)	PUNCT
ejpam-4914	651	23	=	=	NOUN
ejpam-4914	651	24	∅.	∅.	NOUN
ejpam-4914	651	25	since	since	SCONJ
ejpam-4914	651	26	c	c	PROPN
ejpam-4914	651	27	is	be	AUX
ejpam-4914	651	28	a	a	DET
ejpam-4914	651	29	hop	hop	NOUN
ejpam-4914	651	30	dominating	dominating	NOUN
ejpam-4914	651	31	set	set	NOUN
ejpam-4914	651	32	of	of	ADP
ejpam-4914	651	33	g	g	NOUN
ejpam-4914	651	34	,	,	PUNCT
ejpam-4914	651	35	x	x	SYM
ejpam-4914	651	36	∈	∈	PROPN
ejpam-4914	651	37	c	c	NOUN
ejpam-4914	651	38	\	\	PROPN
ejpam-4914	651	39	ng(c	ng(c	SYM
ejpam-4914	651	40	,	,	PUNCT
ejpam-4914	651	41	2	2	X
ejpam-4914	651	42	)	)	PUNCT
ejpam-4914	651	43	=	=	SYM
ejpam-4914	651	44	b.	b.	PROPN
ejpam-4914	651	45	note	note	VERB
ejpam-4914	651	46	that	that	SCONJ
ejpam-4914	651	47	if	if	SCONJ
ejpam-4914	651	48	b	b	PROPN
ejpam-4914	651	49	∩	∩	NOUN
ejpam-4914	651	50	ng(x	ng(x	NUM
ejpam-4914	651	51	,	,	PUNCT
ejpam-4914	651	52	2	2	X
ejpam-4914	651	53	)	)	PUNCT
ejpam-4914	651	54	̸=	̸=	PROPN
ejpam-4914	651	55	∅	∅	NOUN
ejpam-4914	651	56	and	and	CCONJ
ejpam-4914	651	57	u	u	NOUN
ejpam-4914	651	58	∈	∈	PROPN
ejpam-4914	651	59	b	b	PROPN
ejpam-4914	651	60	∩	∩	NOUN
ejpam-4914	651	61	ng(x	ng(x	NUM
ejpam-4914	651	62	,	,	PUNCT
ejpam-4914	651	63	2	2	NUM
ejpam-4914	651	64	)	)	PUNCT
ejpam-4914	651	65	,	,	PUNCT
ejpam-4914	651	66	then	then	ADV
ejpam-4914	651	67	u	u	PROPN
ejpam-4914	651	68	∈	∈	PROPN
ejpam-4914	651	69	c	c	NOUN
ejpam-4914	651	70	∩	∩	NOUN
ejpam-4914	651	71	ng(x	ng(x	NUM
ejpam-4914	651	72	,	,	PUNCT
ejpam-4914	651	73	2	2	NUM
ejpam-4914	651	74	)	)	PUNCT
ejpam-4914	651	75	,	,	PUNCT
ejpam-4914	651	76	a	a	DET
ejpam-4914	651	77	contradiction	contradiction	NOUN
ejpam-4914	651	78	.	.	PUNCT
ejpam-4914	652	1	thus	thus	ADV
ejpam-4914	652	2	,	,	PUNCT
ejpam-4914	652	3	b	b	PROPN
ejpam-4914	652	4	∩	∩	NOUN
ejpam-4914	652	5	ng(x	ng(x	NUM
ejpam-4914	652	6	,	,	PUNCT
ejpam-4914	652	7	2	2	X
ejpam-4914	652	8	)	)	PUNCT
ejpam-4914	652	9	=	=	NOUN
ejpam-4914	652	10	∅.	∅.	NOUN
ejpam-4914	652	11	since	since	SCONJ
ejpam-4914	652	12	f	f	PROPN
ejpam-4914	652	13	|⟨{x}×v	|⟨{x}×v	PROPN
ejpam-4914	652	14	(	(	PUNCT
ejpam-4914	652	15	h)⟩	h)⟩	NOUN
ejpam-4914	652	16	=	=	PUNCT
ejpam-4914	652	17	fx	fx	PROPN
ejpam-4914	652	18	for	for	ADP
ejpam-4914	652	19	each	each	DET
ejpam-4914	652	20	x	x	SYM
ejpam-4914	652	21	∈	∈	PROPN
ejpam-4914	652	22	c	c	NOUN
ejpam-4914	652	23	\	\	PROPN
ejpam-4914	652	24	ng(c	ng(c	SYM
ejpam-4914	652	25	,	,	PUNCT
ejpam-4914	652	26	2	2	NUM
ejpam-4914	652	27	)	)	PUNCT
ejpam-4914	652	28	,	,	PUNCT
ejpam-4914	652	29	f	f	PROPN
ejpam-4914	652	30	∈	∈	PROPN
ejpam-4914	652	31	hid(g[h	hid(g[h	NOUN
ejpam-4914	652	32	]	]	PUNCT
ejpam-4914	652	33	)	)	PUNCT
ejpam-4914	652	34	by	by	ADP
ejpam-4914	652	35	theorem	theorem	ADJ
ejpam-4914	652	36	3.8	3.8	NUM
ejpam-4914	652	37	.	.	PUNCT
ejpam-4914	653	1	therefore	therefore	ADV
ejpam-4914	653	2	,	,	PUNCT
ejpam-4914	653	3	γhi(g[h	γhi(g[h	ADV
ejpam-4914	653	4	]	]	PUNCT
ejpam-4914	653	5	)	)	PUNCT
ejpam-4914	653	6	≤	≤	NUM
ejpam-4914	653	7	2|v2|+	2|v2|+	NUM
ejpam-4914	653	8	|v1|	|v1|	NOUN
ejpam-4914	653	9	=	=	SYM
ejpam-4914	653	10	2|s	2|s	NUM
ejpam-4914	653	11	∩ng(s	∩ng(s	VERB
ejpam-4914	653	12	,	,	PUNCT
ejpam-4914	653	13	2)|+	2)|+	NUM
ejpam-4914	653	14	∑	∑	PUNCT
ejpam-4914	653	15	u∈s\ng(s,2	u∈s\ng(s,2	VERB
ejpam-4914	653	16	)	)	PUNCT
ejpam-4914	654	1	[	[	X
ejpam-4914	654	2	2|v	2|v	NUM
ejpam-4914	654	3	u	u	NOUN
ejpam-4914	654	4	2	2	NUM
ejpam-4914	654	5	|+	|+	NOUN
ejpam-4914	654	6	|v	|v	NOUN
ejpam-4914	654	7	u	u	NOUN
ejpam-4914	654	8	1	1	NUM
ejpam-4914	654	9	|	|	NOUN
ejpam-4914	654	10	]	]	X
ejpam-4914	654	11	=	=	SYM
ejpam-4914	654	12	2|s	2|s	NUM
ejpam-4914	654	13	∩ng(s	∩ng(s	PROPN
ejpam-4914	654	14	,	,	PUNCT
ejpam-4914	654	15	2)|+	2)|+	NUM
ejpam-4914	654	16	pndi(h)|s	pndi(h)|s	NOUN
ejpam-4914	654	17	\ng(s	\ng(s	NOUN
ejpam-4914	654	18	,	,	PUNCT
ejpam-4914	654	19	2)|	2)|	NUM
ejpam-4914	654	20	.	.	PUNCT
ejpam-4914	655	1	since	since	SCONJ
ejpam-4914	655	2	s	s	NOUN
ejpam-4914	655	3	is	be	AUX
ejpam-4914	655	4	arbitrary	arbitrary	ADJ
ejpam-4914	655	5	,	,	PUNCT
ejpam-4914	655	6	γhi(g[h	γhi(g[h	ADJ
ejpam-4914	655	7	]	]	NOUN
ejpam-4914	655	8	)	)	PUNCT
ejpam-4914	655	9	≤	≤	NOUN
ejpam-4914	655	10	αh(g	αh(g	NOUN
ejpam-4914	655	11	)	)	PUNCT
ejpam-4914	655	12	.	.	PUNCT
ejpam-4914	656	1	to	to	PART
ejpam-4914	656	2	show	show	VERB
ejpam-4914	656	3	the	the	DET
ejpam-4914	656	4	sharpness	sharpness	NOUN
ejpam-4914	656	5	of	of	ADP
ejpam-4914	656	6	the	the	DET
ejpam-4914	656	7	upperbound	upperbound	NOUN
ejpam-4914	656	8	,	,	PUNCT
ejpam-4914	656	9	consider	consider	VERB
ejpam-4914	656	10	the	the	DET
ejpam-4914	656	11	graph	graph	NOUN
ejpam-4914	656	12	g	g	NOUN
ejpam-4914	656	13	in	in	ADP
ejpam-4914	656	14	figure	figure	NOUN
ejpam-4914	656	15	6	6	NUM
ejpam-4914	656	16	.	.	PUNCT
ejpam-4914	656	17	verify	verify	VERB
ejpam-4914	656	18	references	reference	NOUN
ejpam-4914	656	19	2447	2447	NUM
ejpam-4914	656	20	•	•	NUM
ejpam-4914	656	21	•	•	NUM
ejpam-4914	656	22	•	•	NUM
ejpam-4914	656	23	•	•	NUM
ejpam-4914	656	24	•	•	NUM
ejpam-4914	656	25	•	•	NUM
ejpam-4914	656	26	•	•	NUM
ejpam-4914	656	27	•	•	NUM
ejpam-4914	656	28	•	•	NUM
ejpam-4914	656	29	•	•	NUM
ejpam-4914	656	30	•	•	NUM
ejpam-4914	656	31	•	•	NUM
ejpam-4914	656	32	•	•	NUM
ejpam-4914	656	33	•	•	NOUN
ejpam-4914	656	34	•	•	NOUN
ejpam-4914	656	35	...........................................................................................................................	...........................................................................................................................	PUNCT
ejpam-4914	656	36	...........................................................................................................................	...........................................................................................................................	PUNCT
ejpam-4914	656	37	...........................................................................................................................	...........................................................................................................................	PUNCT
ejpam-4914	657	1	...........................................................................................................................	...........................................................................................................................	PUNCT
ejpam-4914	657	2	...........................................................................................................................	...........................................................................................................................	PUNCT
ejpam-4914	658	1	...........................................................................................................................	...........................................................................................................................	PUNCT
ejpam-4914	658	2	....................................	....................................	PUNCT
ejpam-4914	659	1	...........................................................................................................................	...........................................................................................................................	PUNCT
ejpam-4914	659	2	................................................................................................................................	................................................................................................................................	PUNCT
ejpam-4914	660	1	....................................	....................................	PUNCT
ejpam-4914	660	2	....................................	....................................	PUNCT
ejpam-4914	660	3	............	............	PUNCT
ejpam-4914	660	4	...........	...........	PUNCT
ejpam-4914	660	5	...........	...........	PUNCT
ejpam-4914	660	6	...........	...........	PUNCT
ejpam-4914	660	7	...........	...........	PUNCT
ejpam-4914	660	8	...........	...........	PUNCT
ejpam-4914	660	9	...........	...........	PUNCT
ejpam-4914	660	10	...........	...........	PUNCT
ejpam-4914	660	11	...........	...........	PUNCT
ejpam-4914	660	12	...........	...........	PUNCT
ejpam-4914	660	13	...........	...........	PUNCT
ejpam-4914	660	14	......	......	PUNCT
ejpam-4914	660	15	....................................	....................................	PUNCT
ejpam-4914	661	1	...........................................................................................................................	...........................................................................................................................	PUNCT
ejpam-4914	661	2	....................................	....................................	PUNCT
ejpam-4914	662	1	...........................................................................................................................	...........................................................................................................................	PUNCT
ejpam-4914	662	2	............	............	PUNCT
ejpam-4914	662	3	...........	...........	PUNCT
ejpam-4914	662	4	...........	...........	PUNCT
ejpam-4914	662	5	...........	...........	PUNCT
ejpam-4914	662	6	...........	...........	PUNCT
ejpam-4914	662	7	...........	...........	PUNCT
ejpam-4914	662	8	...........	...........	PUNCT
ejpam-4914	662	9	...........	...........	PUNCT
ejpam-4914	662	10	...........	...........	PUNCT
ejpam-4914	662	11	...........	...........	PUNCT
ejpam-4914	662	12	...........	...........	PUNCT
ejpam-4914	662	13	......	......	PUNCT
ejpam-4914	663	1	....................................	....................................	PUNCT
ejpam-4914	663	2	....................................	....................................	PUNCT
ejpam-4914	664	1	................................................................................................................................	................................................................................................................................	PUNCT
ejpam-4914	664	2	....................................	....................................	PUNCT
ejpam-4914	664	3	...........................................................................................................................	...........................................................................................................................	PUNCT
ejpam-4914	665	1	....................................	....................................	PUNCT
ejpam-4914	666	1	x	x	PUNCT
ejpam-4914	666	2	y	y	NOUN
ejpam-4914	666	3	z	z	NOUN
ejpam-4914	666	4	g	g	PROPN
ejpam-4914	666	5	figure	figure	NOUN
ejpam-4914	666	6	6	6	NUM
ejpam-4914	666	7	:	:	PUNCT
ejpam-4914	666	8	graph	graph	NOUN
ejpam-4914	666	9	g	g	NOUN
ejpam-4914	666	10	showing	show	VERB
ejpam-4914	666	11	sharpness	sharpness	NOUN
ejpam-4914	666	12	of	of	ADP
ejpam-4914	666	13	the	the	DET
ejpam-4914	666	14	bound	bind	VERB
ejpam-4914	666	15	in	in	ADP
ejpam-4914	666	16	corollary	corollary	ADJ
ejpam-4914	666	17	3.9	3.9	NUM
ejpam-4914	666	18	that	that	PRON
ejpam-4914	666	19	for	for	ADP
ejpam-4914	666	20	n	n	PRON
ejpam-4914	666	21	≥	≥	NOUN
ejpam-4914	666	22	3	3	NUM
ejpam-4914	666	23	,	,	PUNCT
ejpam-4914	666	24	γhi(g[pn	γhi(g[pn	PROPN
ejpam-4914	666	25	]	]	PUNCT
ejpam-4914	666	26	)	)	PUNCT
ejpam-4914	666	27	=	=	SYM
ejpam-4914	667	1	7	7	X
ejpam-4914	667	2	.	.	X
ejpam-4914	667	3	note	note	VERB
ejpam-4914	667	4	that	that	SCONJ
ejpam-4914	667	5	pndi(pn	pndi(pn	NOUN
ejpam-4914	667	6	)	)	PUNCT
ejpam-4914	667	7	=	=	SYM
ejpam-4914	667	8	3	3	NUM
ejpam-4914	667	9	while	while	SCONJ
ejpam-4914	667	10	the	the	DET
ejpam-4914	667	11	set	set	NOUN
ejpam-4914	667	12	s	s	X
ejpam-4914	667	13	=	=	X
ejpam-4914	667	14	{	{	PUNCT
ejpam-4914	667	15	x	x	PROPN
ejpam-4914	667	16	,	,	PUNCT
ejpam-4914	667	17	y	y	PROPN
ejpam-4914	667	18	,	,	PUNCT
ejpam-4914	667	19	z	z	NOUN
ejpam-4914	667	20	}	}	PUNCT
ejpam-4914	667	21	is	be	AUX
ejpam-4914	667	22	a	a	DET
ejpam-4914	667	23	hop	hop	NOUN
ejpam-4914	667	24	dominating	dominating	NOUN
ejpam-4914	667	25	set	set	NOUN
ejpam-4914	667	26	of	of	ADP
ejpam-4914	667	27	g	g	PROPN
ejpam-4914	667	28	with	with	ADP
ejpam-4914	667	29	|s	|s	PROPN
ejpam-4914	667	30	∩	∩	NOUN
ejpam-4914	667	31	ng(s	ng(s	CCONJ
ejpam-4914	667	32	,	,	PUNCT
ejpam-4914	667	33	2)|	2)|	NUM
ejpam-4914	667	34	=	=	SYM
ejpam-4914	667	35	2	2	NUM
ejpam-4914	667	36	and	and	CCONJ
ejpam-4914	667	37	|s	|s	PROPN
ejpam-4914	667	38	\	\	PROPN
ejpam-4914	667	39	ng(s	ng(s	PUNCT
ejpam-4914	667	40	,	,	PUNCT
ejpam-4914	667	41	2)|	2)|	NUM
ejpam-4914	667	42	=	=	SYM
ejpam-4914	667	43	1	1	X
ejpam-4914	667	44	.	.	PUNCT
ejpam-4914	668	1	in	in	ADP
ejpam-4914	668	2	this	this	DET
ejpam-4914	668	3	case	case	NOUN
ejpam-4914	668	4	,	,	PUNCT
ejpam-4914	668	5	αh(g	αh(g	NOUN
ejpam-4914	668	6	)	)	PUNCT
ejpam-4914	668	7	=	=	SYM
ejpam-4914	668	8	2(2	2(2	NUM
ejpam-4914	668	9	)	)	PUNCT
ejpam-4914	668	10	+	+	CCONJ
ejpam-4914	668	11	pndi(pn)(1	pndi(pn)(1	NOUN
ejpam-4914	668	12	)	)	PUNCT
ejpam-4914	668	13	=	=	SYM
ejpam-4914	668	14	7	7	X
ejpam-4914	668	15	.	.	X
ejpam-4914	668	16	■	■	PUNCT
ejpam-4914	668	17	strict	strict	ADJ
ejpam-4914	668	18	inequality	inequality	NOUN
ejpam-4914	668	19	in	in	ADP
ejpam-4914	668	20	corollary	corollary	ADJ
ejpam-4914	668	21	3.9	3.9	NUM
ejpam-4914	668	22	can	can	AUX
ejpam-4914	668	23	also	also	ADV
ejpam-4914	668	24	be	be	AUX
ejpam-4914	668	25	attained	attain	VERB
ejpam-4914	668	26	.	.	PUNCT
ejpam-4914	669	1	note	note	VERB
ejpam-4914	669	2	that	that	SCONJ
ejpam-4914	669	3	for	for	ADP
ejpam-4914	669	4	n	n	PRON
ejpam-4914	669	5	≥	≥	NOUN
ejpam-4914	669	6	3	3	NUM
ejpam-4914	669	7	,	,	PUNCT
ejpam-4914	669	8	γhi(c5[pn	γhi(c5[pn	PROPN
ejpam-4914	669	9	]	]	X
ejpam-4914	669	10	)	)	PUNCT
ejpam-4914	669	11	=	=	SYM
ejpam-4914	669	12	5	5	NUM
ejpam-4914	669	13	while	while	SCONJ
ejpam-4914	669	14	αpn(c5	αpn(c5	VERB
ejpam-4914	669	15	)	)	PUNCT
ejpam-4914	669	16	=	=	SYM
ejpam-4914	670	1	6	6	X
ejpam-4914	670	2	.	.	PUNCT
ejpam-4914	671	1	the	the	DET
ejpam-4914	671	2	same	same	ADJ
ejpam-4914	671	3	example	example	NOUN
ejpam-4914	671	4	also	also	ADV
ejpam-4914	671	5	shows	show	VERB
ejpam-4914	671	6	that	that	SCONJ
ejpam-4914	671	7	αh(g	αh(g	NOUN
ejpam-4914	671	8	)	)	PUNCT
ejpam-4914	671	9	need	need	AUX
ejpam-4914	671	10	not	not	PART
ejpam-4914	671	11	be	be	AUX
ejpam-4914	671	12	determined	determine	VERB
ejpam-4914	671	13	by	by	ADP
ejpam-4914	671	14	a	a	DET
ejpam-4914	671	15	γh	γh	ADV
ejpam-4914	671	16	-	-	PUNCT
ejpam-4914	671	17	set	set	VERB
ejpam-4914	671	18	s	s	PROPN
ejpam-4914	671	19	of	of	ADP
ejpam-4914	671	20	g.	g.	PROPN
ejpam-4914	671	21	if	if	SCONJ
ejpam-4914	671	22	c5	c5	PROPN
ejpam-4914	671	23	=	=	PUNCT
ejpam-4914	672	1	[	[	X
ejpam-4914	672	2	x1	x1	PROPN
ejpam-4914	672	3	,	,	PUNCT
ejpam-4914	672	4	x2	x2	PROPN
ejpam-4914	672	5	,	,	PUNCT
ejpam-4914	672	6	x3	x3	PROPN
ejpam-4914	672	7	,	,	PUNCT
ejpam-4914	672	8	x4	x4	PROPN
ejpam-4914	672	9	,	,	PUNCT
ejpam-4914	672	10	x5	x5	PROPN
ejpam-4914	672	11	,	,	PUNCT
ejpam-4914	672	12	x1	x1	PROPN
ejpam-4914	672	13	]	]	X
ejpam-4914	672	14	,	,	PUNCT
ejpam-4914	672	15	then	then	ADV
ejpam-4914	672	16	s	s	VERB
ejpam-4914	672	17	=	=	PUNCT
ejpam-4914	672	18	{	{	PUNCT
ejpam-4914	672	19	x1	x1	PROPN
ejpam-4914	672	20	,	,	PUNCT
ejpam-4914	672	21	x2	x2	PROPN
ejpam-4914	672	22	,	,	PUNCT
ejpam-4914	672	23	x4	x4	PROPN
ejpam-4914	672	24	}	}	PUNCT
ejpam-4914	672	25	is	be	AUX
ejpam-4914	672	26	a	a	DET
ejpam-4914	672	27	hop	hop	NOUN
ejpam-4914	672	28	dominating	dominating	NOUN
ejpam-4914	672	29	set	set	NOUN
ejpam-4914	672	30	but	but	CCONJ
ejpam-4914	672	31	not	not	PART
ejpam-4914	672	32	a	a	DET
ejpam-4914	672	33	γh	γh	ADV
ejpam-4914	672	34	-	-	PUNCT
ejpam-4914	672	35	set	set	NOUN
ejpam-4914	672	36	of	of	ADP
ejpam-4914	672	37	c5	c5	PROPN
ejpam-4914	672	38	.	.	PUNCT
ejpam-4914	673	1	however	however	ADV
ejpam-4914	673	2	,	,	PUNCT
ejpam-4914	673	3	αpn(c5	αpn(c5	PROPN
ejpam-4914	673	4	)	)	PUNCT
ejpam-4914	673	5	=	=	SYM
ejpam-4914	673	6	2|s∩nc5(s	2|s∩nc5(s	NUM
ejpam-4914	673	7	,	,	PUNCT
ejpam-4914	673	8	2)|+pndi(pn)|s\nc5(s	2)|+pndi(pn)|s\nc5(s	NUM
ejpam-4914	673	9	,	,	PUNCT
ejpam-4914	673	10	2)|	2)|	NUM
ejpam-4914	673	11	=	=	SYM
ejpam-4914	673	12	6	6	NUM
ejpam-4914	673	13	.	.	NOUN
ejpam-4914	673	14	4	4	NUM
ejpam-4914	673	15	.	.	X
ejpam-4914	673	16	conclusion	conclusion	NOUN
ejpam-4914	673	17	it	it	PRON
ejpam-4914	673	18	turned	turn	VERB
ejpam-4914	673	19	out	out	ADP
ejpam-4914	673	20	that	that	SCONJ
ejpam-4914	673	21	the	the	DET
ejpam-4914	673	22	hop	hop	NOUN
ejpam-4914	673	23	italian	italian	ADJ
ejpam-4914	673	24	domination	domination	NOUN
ejpam-4914	673	25	is	be	AUX
ejpam-4914	673	26	directly	directly	ADV
ejpam-4914	673	27	related	relate	VERB
ejpam-4914	673	28	to	to	ADP
ejpam-4914	673	29	both	both	CCONJ
ejpam-4914	673	30	the	the	DET
ejpam-4914	673	31	hop	hop	NOUN
ejpam-4914	673	32	roman	roman	ADJ
ejpam-4914	673	33	domination	domination	NOUN
ejpam-4914	673	34	and	and	CCONJ
ejpam-4914	673	35	the	the	DET
ejpam-4914	673	36	2	2	NUM
ejpam-4914	673	37	-	-	PUNCT
ejpam-4914	673	38	hop	hop	NOUN
ejpam-4914	673	39	domination	domination	NOUN
ejpam-4914	673	40	.	.	PUNCT
ejpam-4914	674	1	more	more	ADV
ejpam-4914	674	2	precisely	precisely	ADV
ejpam-4914	674	3	,	,	PUNCT
ejpam-4914	674	4	γhi(g	γhi(g	PROPN
ejpam-4914	674	5	)	)	PUNCT
ejpam-4914	674	6	≤	≤	NOUN
ejpam-4914	674	7	min{γhr(g	min{γhr(g	PROPN
ejpam-4914	674	8	)	)	PUNCT
ejpam-4914	674	9	,	,	PUNCT
ejpam-4914	674	10	γ2h(g	γ2h(g	PROPN
ejpam-4914	674	11	)	)	PUNCT
ejpam-4914	674	12	}	}	PUNCT
ejpam-4914	674	13	for	for	ADP
ejpam-4914	674	14	all	all	DET
ejpam-4914	674	15	graphs	graph	NOUN
ejpam-4914	674	16	g.	g.	PROPN
ejpam-4914	674	17	more	more	ADV
ejpam-4914	674	18	interestingly	interestingly	ADV
ejpam-4914	674	19	,	,	PUNCT
ejpam-4914	674	20	it	it	PRON
ejpam-4914	674	21	is	be	AUX
ejpam-4914	674	22	shown	show	VERB
ejpam-4914	674	23	that	that	SCONJ
ejpam-4914	674	24	,	,	PUNCT
ejpam-4914	674	25	in	in	ADP
ejpam-4914	674	26	fact	fact	NOUN
ejpam-4914	674	27	,	,	PUNCT
ejpam-4914	674	28	the	the	DET
ejpam-4914	674	29	difference	difference	NOUN
ejpam-4914	674	30	γhr(g)−	γhr(g)−	PROPN
ejpam-4914	674	31	γhi(g	γhi(g	PROPN
ejpam-4914	674	32	)	)	PUNCT
ejpam-4914	674	33	can	can	AUX
ejpam-4914	674	34	be	be	AUX
ejpam-4914	674	35	made	make	VERB
ejpam-4914	674	36	arbitrary	arbitrary	ADJ
ejpam-4914	674	37	large	large	ADJ
ejpam-4914	674	38	,	,	PUNCT
ejpam-4914	674	39	and	and	CCONJ
ejpam-4914	674	40	that	that	SCONJ
ejpam-4914	674	41	any	any	DET
ejpam-4914	674	42	pair	pair	NOUN
ejpam-4914	674	43	of	of	ADP
ejpam-4914	674	44	positive	positive	ADJ
ejpam-4914	674	45	integers	integer	NOUN
ejpam-4914	674	46	a	a	PRON
ejpam-4914	674	47	and	and	CCONJ
ejpam-4914	674	48	b	b	NOUN
ejpam-4914	674	49	with	with	ADP
ejpam-4914	674	50	4	4	NUM
ejpam-4914	674	51	≤	≤	NOUN
ejpam-4914	674	52	a	a	DET
ejpam-4914	674	53	≤	≤	NUM
ejpam-4914	674	54	b	b	NOUN
ejpam-4914	674	55	are	be	AUX
ejpam-4914	674	56	realizable	realizable	ADJ
ejpam-4914	674	57	as	as	ADP
ejpam-4914	674	58	the	the	DET
ejpam-4914	674	59	hop	hop	NOUN
ejpam-4914	674	60	italian	italian	ADJ
ejpam-4914	674	61	domination	domination	NOUN
ejpam-4914	674	62	number	number	NOUN
ejpam-4914	674	63	and	and	CCONJ
ejpam-4914	674	64	the	the	DET
ejpam-4914	674	65	2	2	NUM
ejpam-4914	674	66	-	-	PUNCT
ejpam-4914	674	67	hop	hop	NOUN
ejpam-4914	674	68	domination	domination	NOUN
ejpam-4914	674	69	number	number	NOUN
ejpam-4914	674	70	,	,	PUNCT
ejpam-4914	674	71	respectively	respectively	ADV
ejpam-4914	674	72	,	,	PUNCT
ejpam-4914	674	73	of	of	ADP
ejpam-4914	674	74	some	some	DET
ejpam-4914	674	75	connected	connected	ADJ
ejpam-4914	674	76	graph	graph	NOUN
ejpam-4914	674	77	.	.	PUNCT
ejpam-4914	675	1	finally	finally	ADV
ejpam-4914	675	2	,	,	PUNCT
ejpam-4914	675	3	for	for	ADP
ejpam-4914	675	4	graphs	graph	NOUN
ejpam-4914	675	5	under	under	ADP
ejpam-4914	675	6	the	the	DET
ejpam-4914	675	7	complementary	complementary	ADJ
ejpam-4914	675	8	prism	prism	NOUN
ejpam-4914	675	9	,	,	PUNCT
ejpam-4914	675	10	join	join	NOUN
ejpam-4914	675	11	,	,	PUNCT
ejpam-4914	675	12	corona	corona	NOUN
ejpam-4914	675	13	and	and	CCONJ
ejpam-4914	675	14	lexicographic	lexicographic	ADJ
ejpam-4914	675	15	product	product	NOUN
ejpam-4914	675	16	of	of	ADP
ejpam-4914	675	17	graphs	graph	NOUN
ejpam-4914	675	18	,	,	PUNCT
ejpam-4914	675	19	the	the	DET
ejpam-4914	675	20	hop	hop	NOUN
ejpam-4914	675	21	italian	italian	ADJ
ejpam-4914	675	22	domination	domination	NOUN
ejpam-4914	675	23	number	number	NOUN
ejpam-4914	675	24	is	be	AUX
ejpam-4914	675	25	expressible	expressible	ADJ
ejpam-4914	675	26	in	in	ADP
ejpam-4914	675	27	terms	term	NOUN
ejpam-4914	675	28	of	of	ADP
ejpam-4914	675	29	the	the	DET
ejpam-4914	675	30	hop	hop	NOUN
ejpam-4914	675	31	italian	italian	ADJ
ejpam-4914	675	32	domination	domination	NOUN
ejpam-4914	675	33	numbers	number	NOUN
ejpam-4914	675	34	or	or	CCONJ
ejpam-4914	675	35	of	of	ADP
ejpam-4914	675	36	the	the	DET
ejpam-4914	675	37	pndi	pndi	ADJ
ejpam-4914	675	38	numbers	number	NOUN
ejpam-4914	675	39	of	of	ADP
ejpam-4914	675	40	its	its	PRON
ejpam-4914	675	41	factors	factor	NOUN
ejpam-4914	675	42	.	.	PUNCT
ejpam-4914	676	1	acknowledgements	acknowledgement	NOUN
ejpam-4914	676	2	this	this	DET
ejpam-4914	676	3	project	project	NOUN
ejpam-4914	676	4	is	be	AUX
ejpam-4914	676	5	fully	fully	ADV
ejpam-4914	676	6	supported	support	VERB
ejpam-4914	676	7	by	by	ADP
ejpam-4914	676	8	the	the	DET
ejpam-4914	676	9	msu	msu	PROPN
ejpam-4914	676	10	-	-	PUNCT
ejpam-4914	676	11	iligan	iligan	PROPN
ejpam-4914	676	12	institute	institute	PROPN
ejpam-4914	676	13	of	of	ADP
ejpam-4914	676	14	technology	technology	NOUN
ejpam-4914	676	15	through	through	ADP
ejpam-4914	676	16	the	the	DET
ejpam-4914	676	17	office	office	NOUN
ejpam-4914	676	18	of	of	ADP
ejpam-4914	676	19	the	the	DET
ejpam-4914	676	20	vice	vice	NOUN
ejpam-4914	676	21	chancellor	chancellor	NOUN
ejpam-4914	676	22	for	for	ADP
ejpam-4914	676	23	research	research	NOUN
ejpam-4914	676	24	and	and	CCONJ
ejpam-4914	676	25	extension	extension	NOUN
ejpam-4914	676	26	(	(	PUNCT
ejpam-4914	676	27	ovcre	ovcre	NOUN
ejpam-4914	676	28	)	)	PUNCT
ejpam-4914	676	29	.	.	PUNCT
ejpam-4914	677	1	references	reference	NOUN
ejpam-4914	677	2	[	[	X
ejpam-4914	677	3	1	1	NUM
ejpam-4914	677	4	]	]	X
ejpam-4914	677	5	r.a	r.a	PROPN
ejpam-4914	677	6	.	.	PROPN
ejpam-4914	677	7	beeler	beeler	PROPN
ejpam-4914	677	8	,	,	PUNCT
ejpam-4914	677	9	t.w	t.w	PROPN
ejpam-4914	677	10	.	.	PROPN
ejpam-4914	677	11	haynes	haynes	PROPN
ejpam-4914	677	12	and	and	CCONJ
ejpam-4914	677	13	s.t	s.t	PROPN
ejpam-4914	677	14	.	.	PROPN
ejpam-4914	677	15	hedetniemi	hedetniemi	PROPN
ejpam-4914	677	16	,	,	PUNCT
ejpam-4914	677	17	double	double	ADJ
ejpam-4914	677	18	roman	roman	ADJ
ejpam-4914	677	19	domination	domination	NOUN
ejpam-4914	677	20	.	.	PUNCT
ejpam-4914	678	1	discrete	discrete	ADJ
ejpam-4914	678	2	applied	apply	VERB
ejpam-4914	678	3	mathematics	mathematic	NOUN
ejpam-4914	678	4	,	,	PUNCT
ejpam-4914	678	5	211:23	211:23	PROPN
ejpam-4914	678	6	-	-	SYM
ejpam-4914	678	7	29	29	NUM
ejpam-4914	678	8	,	,	PUNCT
ejpam-4914	678	9	2016	2016	NUM
ejpam-4914	678	10	.	.	PUNCT
ejpam-4914	679	1	[	[	X
ejpam-4914	679	2	2	2	NUM
ejpam-4914	679	3	]	]	PUNCT
ejpam-4914	679	4	c.	c.	PROPN
ejpam-4914	679	5	berge	berge	PROPN
ejpam-4914	679	6	,	,	PUNCT
ejpam-4914	679	7	theorie	theorie	PROPN
ejpam-4914	679	8	des	des	PROPN
ejpam-4914	679	9	graphes	graphes	PROPN
ejpam-4914	679	10	et	et	PROPN
ejpam-4914	679	11	ses	ses	PROPN
ejpam-4914	679	12	applications	application	NOUN
ejpam-4914	679	13	,	,	PUNCT
ejpam-4914	679	14	dunod	dunod	PROPN
ejpam-4914	679	15	,	,	PUNCT
ejpam-4914	679	16	paris	paris	PROPN
ejpam-4914	679	17	,	,	PUNCT
ejpam-4914	679	18	1958	1958	NUM
ejpam-4914	679	19	.	.	PUNCT
ejpam-4914	680	1	translation	translation	NOUN
ejpam-4914	680	2	:	:	PUNCT
ejpam-4914	680	3	the	the	DET
ejpam-4914	680	4	theory	theory	NOUN
ejpam-4914	680	5	of	of	ADP
ejpam-4914	680	6	graphs	graph	NOUN
ejpam-4914	680	7	and	and	CCONJ
ejpam-4914	680	8	its	its	PRON
ejpam-4914	680	9	applications	application	NOUN
ejpam-4914	680	10	,	,	PUNCT
ejpam-4914	680	11	methuen	methuen	PROPN
ejpam-4914	680	12	,	,	PUNCT
ejpam-4914	680	13	london	london	PROPN
ejpam-4914	680	14	and	and	CCONJ
ejpam-4914	680	15	wiley	wiley	PROPN
ejpam-4914	680	16	,	,	PUNCT
ejpam-4914	680	17	new	new	PROPN
ejpam-4914	680	18	york	york	PROPN
ejpam-4914	680	19	,	,	PUNCT
ejpam-4914	680	20	1962	1962	NUM
ejpam-4914	680	21	.	.	PUNCT
ejpam-4914	681	1	references	reference	NOUN
ejpam-4914	681	2	2448	2448	NUM
ejpam-4914	681	3	[	[	X
ejpam-4914	681	4	3	3	NUM
ejpam-4914	681	5	]	]	X
ejpam-4914	681	6	f.	f.	PROPN
ejpam-4914	681	7	buckley	buckley	PROPN
ejpam-4914	681	8	and	and	CCONJ
ejpam-4914	681	9	f.	f.	PROPN
ejpam-4914	681	10	harary	harary	PROPN
ejpam-4914	681	11	.	.	PUNCT
ejpam-4914	682	1	distance	distance	NOUN
ejpam-4914	682	2	in	in	ADP
ejpam-4914	682	3	graphs	graph	NOUN
ejpam-4914	682	4	.	.	PUNCT
ejpam-4914	683	1	addison	addison	PROPN
ejpam-4914	683	2	-	-	PUNCT
ejpam-4914	683	3	wesley	wesley	PROPN
ejpam-4914	683	4	,	,	PUNCT
ejpam-4914	683	5	redwood	redwood	NOUN
ejpam-4914	683	6	city	city	NOUN
ejpam-4914	683	7	,	,	PUNCT
ejpam-4914	683	8	ca	ca	NOUN
ejpam-4914	683	9	,	,	PUNCT
ejpam-4914	683	10	1990	1990	NUM
ejpam-4914	683	11	.	.	PUNCT
ejpam-4914	684	1	[	[	X
ejpam-4914	684	2	4	4	NUM
ejpam-4914	684	3	]	]	X
ejpam-4914	684	4	s.r	s.r	PROPN
ejpam-4914	684	5	.	.	PROPN
ejpam-4914	684	6	canoy	canoy	PROPN
ejpam-4914	684	7	jr	jr	PROPN
ejpam-4914	684	8	.	.	PROPN
ejpam-4914	684	9	and	and	CCONJ
ejpam-4914	684	10	s.	s.	PROPN
ejpam-4914	684	11	arriola	arriola	PROPN
ejpam-4914	684	12	,	,	PUNCT
ejpam-4914	684	13	(	(	PUNCT
ejpam-4914	684	14	1	1	NUM
ejpam-4914	684	15	,	,	PUNCT
ejpam-4914	684	16	2)∗-domination	2)∗-domination	NOUN
ejpam-4914	684	17	in	in	ADP
ejpam-4914	684	18	graphs	graph	NOUN
ejpam-4914	684	19	,	,	PUNCT
ejpam-4914	684	20	advances	advance	NOUN
ejpam-4914	684	21	and	and	CCONJ
ejpam-4914	684	22	applications	application	NOUN
ejpam-4914	684	23	in	in	ADP
ejpam-4914	684	24	discrete	discrete	ADJ
ejpam-4914	684	25	mathematics	mathematic	NOUN
ejpam-4914	684	26	,	,	PUNCT
ejpam-4914	684	27	18(2):179	18(2):179	NUM
ejpam-4914	684	28	-	-	SYM
ejpam-4914	684	29	190	190	NUM
ejpam-4914	684	30	,	,	PUNCT
ejpam-4914	684	31	2017	2017	NUM
ejpam-4914	684	32	.	.	PUNCT
ejpam-4914	685	1	[	[	X
ejpam-4914	685	2	5	5	NUM
ejpam-4914	685	3	]	]	X
ejpam-4914	685	4	s.r	s.r	PROPN
ejpam-4914	685	5	.	.	PROPN
ejpam-4914	685	6	canoy	canoy	PROPN
ejpam-4914	685	7	jr	jr	PROPN
ejpam-4914	685	8	.	.	PROPN
ejpam-4914	685	9	,	,	PUNCT
ejpam-4914	685	10	r.v	r.v	PROPN
ejpam-4914	685	11	.	.	NOUN
ejpam-4914	685	12	mollejon	mollejon	NOUN
ejpam-4914	685	13	and	and	CCONJ
ejpam-4914	685	14	j.g	j.g	PROPN
ejpam-4914	685	15	.	.	PROPN
ejpam-4914	685	16	canoy	canoy	PROPN
ejpam-4914	685	17	,	,	PUNCT
ejpam-4914	685	18	hop	hop	NOUN
ejpam-4914	685	19	dominating	dominating	NOUN
ejpam-4914	685	20	sets	set	NOUN
ejpam-4914	685	21	in	in	ADP
ejpam-4914	685	22	graphs	graph	NOUN
ejpam-4914	685	23	under	under	ADP
ejpam-4914	685	24	binary	binary	ADJ
ejpam-4914	685	25	operations	operation	NOUN
ejpam-4914	685	26	,	,	PUNCT
ejpam-4914	685	27	european	european	PROPN
ejpam-4914	685	28	journal	journal	PROPN
ejpam-4914	685	29	of	of	ADP
ejpam-4914	685	30	pure	pure	ADJ
ejpam-4914	685	31	and	and	CCONJ
ejpam-4914	685	32	applied	applied	ADJ
ejpam-4914	685	33	mathematics	mathematic	NOUN
ejpam-4914	685	34	,	,	PUNCT
ejpam-4914	685	35	12(4):14551463	12(4):14551463	NUM
ejpam-4914	685	36	,	,	PUNCT
ejpam-4914	685	37	2019	2019	NUM
ejpam-4914	685	38	.	.	PUNCT
ejpam-4914	686	1	[	[	X
ejpam-4914	686	2	6	6	NUM
ejpam-4914	686	3	]	]	X
ejpam-4914	686	4	j.b	j.b	PROPN
ejpam-4914	686	5	.	.	PROPN
ejpam-4914	686	6	cariaga	cariaga	PROPN
ejpam-4914	686	7	and	and	CCONJ
ejpam-4914	686	8	f.	f.	PROPN
ejpam-4914	686	9	jamil	jamil	PROPN
ejpam-4914	686	10	,	,	PUNCT
ejpam-4914	686	11	on	on	ADP
ejpam-4914	686	12	double	double	ADJ
ejpam-4914	686	13	roman	roman	ADJ
ejpam-4914	686	14	dominating	dominating	NOUN
ejpam-4914	686	15	functions	function	NOUN
ejpam-4914	686	16	in	in	ADP
ejpam-4914	686	17	graphs	graph	NOUN
ejpam-4914	686	18	,	,	PUNCT
ejpam-4914	686	19	european	european	ADJ
ejpam-4914	686	20	journal	journal	NOUN
ejpam-4914	686	21	of	of	ADP
ejpam-4914	686	22	pure	pure	ADJ
ejpam-4914	686	23	and	and	CCONJ
ejpam-4914	686	24	applied	applied	ADJ
ejpam-4914	686	25	mathematics	mathematic	NOUN
ejpam-4914	686	26	,	,	PUNCT
ejpam-4914	686	27	16(2):847	16(2):847	PROPN
ejpam-4914	686	28	-	-	SYM
ejpam-4914	686	29	863	863	NUM
ejpam-4914	686	30	,	,	PUNCT
ejpam-4914	686	31	2023	2023	NUM
ejpam-4914	686	32	.	.	PUNCT
ejpam-4914	687	1	[	[	X
ejpam-4914	687	2	7	7	NUM
ejpam-4914	687	3	]	]	X
ejpam-4914	687	4	e.w	e.w	PROPN
ejpam-4914	687	5	.	.	PROPN
ejpam-4914	687	6	chambers	chambers	PROPN
ejpam-4914	687	7	,	,	PUNCT
ejpam-4914	687	8	b.	b.	PROPN
ejpam-4914	687	9	kinsley	kinsley	PROPN
ejpam-4914	687	10	,	,	PUNCT
ejpam-4914	687	11	n.	n.	PROPN
ejpam-4914	687	12	prince	prince	PROPN
ejpam-4914	687	13	and	and	CCONJ
ejpam-4914	687	14	d.b	d.b	PROPN
ejpam-4914	687	15	.	.	PROPN
ejpam-4914	687	16	west	west	PROPN
ejpam-4914	687	17	,	,	PUNCT
ejpam-4914	687	18	extremal	extremal	ADJ
ejpam-4914	687	19	problems	problem	NOUN
ejpam-4914	687	20	for	for	ADP
ejpam-4914	687	21	roman	roman	ADJ
ejpam-4914	687	22	domination	domination	NOUN
ejpam-4914	687	23	,	,	PUNCT
ejpam-4914	687	24	siam	siam	ADJ
ejpam-4914	687	25	journal	journal	NOUN
ejpam-4914	687	26	on	on	ADP
ejpam-4914	687	27	discrete	discrete	ADJ
ejpam-4914	687	28	mathematics	mathematic	NOUN
ejpam-4914	687	29	,	,	PUNCT
ejpam-4914	687	30	23(3):1575	23(3):1575	NUM
ejpam-4914	687	31	-	-	SYM
ejpam-4914	687	32	1586	1586	NUM
ejpam-4914	687	33	,	,	PUNCT
ejpam-4914	687	34	2009	2009	NUM
ejpam-4914	687	35	.	.	PUNCT
ejpam-4914	688	1	[	[	X
ejpam-4914	688	2	8	8	NUM
ejpam-4914	688	3	]	]	X
ejpam-4914	688	4	m.	m.	NOUN
ejpam-4914	688	5	chellali	chellali	PROPN
ejpam-4914	688	6	,	,	PUNCT
ejpam-4914	688	7	t.w	t.w	PROPN
ejpam-4914	688	8	.	.	PROPN
ejpam-4914	688	9	haynes	haynes	PROPN
ejpam-4914	688	10	,	,	PUNCT
ejpam-4914	688	11	s.t	s.t	PROPN
ejpam-4914	688	12	.	.	PROPN
ejpam-4914	688	13	hedetniemi	hedetniemi	PROPN
ejpam-4914	688	14	and	and	CCONJ
ejpam-4914	688	15	a.a	a.a	PROPN
ejpam-4914	688	16	.	.	PROPN
ejpam-4914	688	17	mcrae	mcrae	PROPN
ejpam-4914	688	18	,	,	PUNCT
ejpam-4914	688	19	roman	roman	NOUN
ejpam-4914	688	20	{	{	PUNCT
ejpam-4914	688	21	2}-domination	2}-domination	NUM
ejpam-4914	688	22	.	.	PUNCT
ejpam-4914	688	23	discrete	discrete	VERB
ejpam-4914	688	24	applied	apply	VERB
ejpam-4914	688	25	mathematics	mathematic	NOUN
ejpam-4914	688	26	,	,	PUNCT
ejpam-4914	688	27	204:22	204:22	PROPN
ejpam-4914	688	28	-	-	SYM
ejpam-4914	688	29	28	28	NUM
ejpam-4914	688	30	,	,	PUNCT
ejpam-4914	688	31	2016	2016	NUM
ejpam-4914	688	32	.	.	PUNCT
ejpam-4914	689	1	[	[	X
ejpam-4914	689	2	9	9	NUM
ejpam-4914	689	3	]	]	SYM
ejpam-4914	689	4	e.j	e.j	PROPN
ejpam-4914	689	5	.	.	PROPN
ejpam-4914	689	6	cockayne	cockayne	PROPN
ejpam-4914	689	7	,	,	PUNCT
ejpam-4914	689	8	p.a	p.a	PROPN
ejpam-4914	689	9	.	.	PUNCT
ejpam-4914	689	10	dreyer	dreyer	PROPN
ejpam-4914	689	11	,	,	PUNCT
ejpam-4914	689	12	s.m	s.m	PROPN
ejpam-4914	689	13	.	.	PROPN
ejpam-4914	689	14	hedetniemi	hedetniemi	PROPN
ejpam-4914	689	15	and	and	CCONJ
ejpam-4914	689	16	s.t	s.t	PROPN
ejpam-4914	689	17	.	.	PROPN
ejpam-4914	689	18	hedetniemi	hedetniemi	PROPN
ejpam-4914	689	19	,	,	PUNCT
ejpam-4914	689	20	roman	roman	ADJ
ejpam-4914	689	21	domination	domination	NOUN
ejpam-4914	689	22	in	in	ADP
ejpam-4914	689	23	graphs	graph	NOUN
ejpam-4914	689	24	,	,	PUNCT
ejpam-4914	689	25	discrete	discrete	ADJ
ejpam-4914	689	26	mathematics	mathematic	NOUN
ejpam-4914	689	27	,	,	PUNCT
ejpam-4914	689	28	278:11	278:11	NUM
ejpam-4914	689	29	-	-	SYM
ejpam-4914	689	30	22	22	NUM
ejpam-4914	689	31	,	,	PUNCT
ejpam-4914	689	32	2004	2004	NUM
ejpam-4914	689	33	.	.	PUNCT
ejpam-4914	690	1	[	[	X
ejpam-4914	690	2	10	10	NUM
ejpam-4914	690	3	]	]	X
ejpam-4914	690	4	e.	e.	PROPN
ejpam-4914	690	5	cockayne	cockayne	PROPN
ejpam-4914	690	6	and	and	CCONJ
ejpam-4914	690	7	s.	s.	PROPN
ejpam-4914	690	8	hedetniemi	hedetniemi	PROPN
ejpam-4914	690	9	,	,	PUNCT
ejpam-4914	690	10	towards	towards	ADP
ejpam-4914	690	11	a	a	DET
ejpam-4914	690	12	theory	theory	NOUN
ejpam-4914	690	13	of	of	ADP
ejpam-4914	690	14	domination	domination	NOUN
ejpam-4914	690	15	in	in	ADP
ejpam-4914	690	16	graphs	graph	NOUN
ejpam-4914	690	17	,	,	PUNCT
ejpam-4914	690	18	networks	network	NOUN
ejpam-4914	690	19	,	,	PUNCT
ejpam-4914	690	20	7(3):247	7(3):247	PROPN
ejpam-4914	690	21	-	-	SYM
ejpam-4914	690	22	261	261	NUM
ejpam-4914	690	23	,	,	PUNCT
ejpam-4914	690	24	1977	1977	NUM
ejpam-4914	690	25	.	.	PUNCT
ejpam-4914	691	1	[	[	X
ejpam-4914	691	2	11	11	NUM
ejpam-4914	691	3	]	]	X
ejpam-4914	691	4	t.w	t.w	PROPN
ejpam-4914	691	5	.	.	PROPN
ejpam-4914	691	6	haynes	haynes	PROPN
ejpam-4914	691	7	,	,	PUNCT
ejpam-4914	691	8	s.t	s.t	PROPN
ejpam-4914	691	9	.	.	PROPN
ejpam-4914	691	10	hedetniemi	hedetniemi	PROPN
ejpam-4914	691	11	and	and	CCONJ
ejpam-4914	691	12	p.j	p.j	PROPN
ejpam-4914	691	13	.	.	PROPN
ejpam-4914	691	14	slater	slater	PROPN
ejpam-4914	691	15	.	.	PUNCT
ejpam-4914	692	1	fundamentals	fundamental	NOUN
ejpam-4914	692	2	of	of	ADP
ejpam-4914	692	3	domination	domination	NOUN
ejpam-4914	692	4	in	in	ADP
ejpam-4914	692	5	graphs	graph	NOUN
ejpam-4914	692	6	.	.	PUNCT
ejpam-4914	693	1	marcel	marcel	PROPN
ejpam-4914	693	2	dekker	dekker	PROPN
ejpam-4914	693	3	,	,	PUNCT
ejpam-4914	693	4	inc	inc	PROPN
ejpam-4914	693	5	.	.	PROPN
ejpam-4914	693	6	new	new	PROPN
ejpam-4914	693	7	york	york	PROPN
ejpam-4914	693	8	,	,	PUNCT
ejpam-4914	693	9	1998	1998	NUM
ejpam-4914	693	10	.	.	PUNCT
ejpam-4914	694	1	[	[	X
ejpam-4914	694	2	12	12	NUM
ejpam-4914	694	3	]	]	X
ejpam-4914	694	4	c.h	c.h	PROPN
ejpam-4914	694	5	.	.	PROPN
ejpam-4914	695	1	liu	liu	PROPN
ejpam-4914	695	2	and	and	CCONJ
ejpam-4914	695	3	g.j	g.j	PROPN
ejpam-4914	695	4	.	.	PROPN
ejpam-4914	695	5	chang	chang	PROPN
ejpam-4914	695	6	,	,	PUNCT
ejpam-4914	695	7	roman	roman	ADJ
ejpam-4914	695	8	domination	domination	NOUN
ejpam-4914	695	9	on	on	ADP
ejpam-4914	695	10	strongly	strongly	ADV
ejpam-4914	695	11	chordal	chordal	ADJ
ejpam-4914	695	12	graphs	graph	NOUN
ejpam-4914	695	13	,	,	PUNCT
ejpam-4914	695	14	journal	journal	NOUN
ejpam-4914	695	15	of	of	ADP
ejpam-4914	695	16	combinatorial	combinatorial	ADJ
ejpam-4914	695	17	optimization	optimization	NOUN
ejpam-4914	695	18	,	,	PUNCT
ejpam-4914	695	19	26:608	26:608	NUM
ejpam-4914	695	20	-	-	SYM
ejpam-4914	695	21	619	619	NUM
ejpam-4914	695	22	,	,	PUNCT
ejpam-4914	695	23	2013	2013	NUM
ejpam-4914	695	24	.	.	PUNCT
ejpam-4914	696	1	[	[	X
ejpam-4914	696	2	13	13	NUM
ejpam-4914	696	3	]	]	X
ejpam-4914	696	4	r.	r.	PROPN
ejpam-4914	696	5	malalay	malalay	PROPN
ejpam-4914	696	6	and	and	CCONJ
ejpam-4914	696	7	f.	f.	PROPN
ejpam-4914	696	8	jamil	jamil	PROPN
ejpam-4914	696	9	,	,	PUNCT
ejpam-4914	696	10	on	on	ADP
ejpam-4914	696	11	disjunctive	disjunctive	ADJ
ejpam-4914	696	12	domination	domination	NOUN
ejpam-4914	696	13	in	in	ADP
ejpam-4914	696	14	graphs	graph	NOUN
ejpam-4914	696	15	,	,	PUNCT
ejpam-4914	696	16	quaestiones	quaestione	NOUN
ejpam-4914	696	17	mathematicae	mathematicae	PROPN
ejpam-4914	696	18	,	,	PUNCT
ejpam-4914	696	19	43(2):149	43(2):149	PROPN
ejpam-4914	696	20	-	-	PUNCT
ejpam-4914	696	21	168	168	NUM
ejpam-4914	696	22	,	,	PUNCT
ejpam-4914	696	23	2020	2020	NUM
ejpam-4914	696	24	.	.	PUNCT
ejpam-4914	697	1	[	[	X
ejpam-4914	697	2	14	14	NUM
ejpam-4914	697	3	]	]	X
ejpam-4914	697	4	r.v	r.v	PROPN
ejpam-4914	697	5	.	.	NOUN
ejpam-4914	697	6	mollejon	mollejon	NOUN
ejpam-4914	697	7	and	and	CCONJ
ejpam-4914	697	8	s.r	s.r	PROPN
ejpam-4914	697	9	.	.	PROPN
ejpam-4914	697	10	canoy	canoy	PROPN
ejpam-4914	697	11	,	,	PUNCT
ejpam-4914	697	12	jr	jr	PROPN
ejpam-4914	697	13	.	.	PROPN
ejpam-4914	697	14	,	,	PUNCT
ejpam-4914	697	15	double	double	ADJ
ejpam-4914	697	16	hop	hop	NOUN
ejpam-4914	697	17	dominating	dominating	NOUN
ejpam-4914	697	18	sets	set	NOUN
ejpam-4914	697	19	in	in	ADP
ejpam-4914	697	20	graphs	graph	NOUN
ejpam-4914	697	21	,	,	PUNCT
ejpam-4914	697	22	discrete	discrete	ADJ
ejpam-4914	697	23	mathematics	mathematic	NOUN
ejpam-4914	697	24	,	,	PUNCT
ejpam-4914	697	25	algorithm	algorithm	NOUN
ejpam-4914	697	26	and	and	CCONJ
ejpam-4914	697	27	applications	application	NOUN
ejpam-4914	697	28	,	,	PUNCT
ejpam-4914	697	29	13(5):2150057	13(5):2150057	NUM
ejpam-4914	697	30	,	,	PUNCT
ejpam-4914	697	31	2021	2021	NUM
ejpam-4914	697	32	.	.	PUNCT
ejpam-4914	698	1	[	[	X
ejpam-4914	698	2	15	15	NUM
ejpam-4914	698	3	]	]	X
ejpam-4914	698	4	c.	c.	PROPN
ejpam-4914	698	5	natarajan	natarajan	PROPN
ejpam-4914	698	6	and	and	CCONJ
ejpam-4914	698	7	s.k	s.k	PROPN
ejpam-4914	698	8	.	.	PROPN
ejpam-4914	698	9	ayyaswamy	ayyaswamy	PROPN
ejpam-4914	698	10	,	,	PUNCT
ejpam-4914	698	11	hop	hop	NOUN
ejpam-4914	698	12	domination	domination	NOUN
ejpam-4914	698	13	in	in	ADP
ejpam-4914	698	14	graphs	graph	NOUN
ejpam-4914	698	15	-	-	PUNCT
ejpam-4914	698	16	ii	ii	NOUN
ejpam-4914	698	17	,	,	PUNCT
ejpam-4914	698	18	analele	analele	ADP
ejpam-4914	698	19	stiintifice	stiintifice	NOUN
ejpam-4914	698	20	ale	ale	NOUN
ejpam-4914	698	21	universitatii	universitatii	PROPN
ejpam-4914	698	22	”	"	PUNCT
ejpam-4914	698	23	ovidius	ovidius	PROPN
ejpam-4914	698	24	”	"	PUNCT
ejpam-4914	698	25	constanta	constanta	PROPN
ejpam-4914	698	26	.	.	PUNCT
ejpam-4914	699	1	seria	seria	PROPN
ejpam-4914	699	2	matematica	matematica	PROPN
ejpam-4914	699	3	,	,	PUNCT
ejpam-4914	699	4	23(2):187	23(2):187	PROPN
ejpam-4914	699	5	-	-	SYM
ejpam-4914	699	6	199	199	NUM
ejpam-4914	699	7	,	,	PUNCT
ejpam-4914	699	8	2015	2015	NUM
ejpam-4914	699	9	.	.	PUNCT
ejpam-4914	700	1	[	[	X
ejpam-4914	700	2	16	16	NUM
ejpam-4914	700	3	]	]	X
ejpam-4914	700	4	h.	h.	PROPN
ejpam-4914	700	5	nuenay	nuenay	PROPN
ejpam-4914	700	6	-	-	PUNCT
ejpam-4914	700	7	manlanque	manlanque	ADJ
ejpam-4914	700	8	and	and	CCONJ
ejpam-4914	700	9	f.	f.	PROPN
ejpam-4914	700	10	jamil	jamil	PROPN
ejpam-4914	700	11	,	,	PUNCT
ejpam-4914	700	12	cost	cost	VERB
ejpam-4914	700	13	effective	effective	ADJ
ejpam-4914	700	14	domination	domination	NOUN
ejpam-4914	700	15	in	in	ADP
ejpam-4914	700	16	the	the	DET
ejpam-4914	700	17	join	join	NOUN
ejpam-4914	700	18	,	,	PUNCT
ejpam-4914	700	19	corona	corona	NOUN
ejpam-4914	700	20	and	and	CCONJ
ejpam-4914	700	21	composition	composition	NOUN
ejpam-4914	700	22	of	of	ADP
ejpam-4914	700	23	graphs	graph	NOUN
ejpam-4914	700	24	,	,	PUNCT
ejpam-4914	700	25	european	european	ADJ
ejpam-4914	700	26	journal	journal	NOUN
ejpam-4914	700	27	of	of	ADP
ejpam-4914	700	28	pure	pure	ADJ
ejpam-4914	700	29	and	and	CCONJ
ejpam-4914	700	30	applied	applied	ADJ
ejpam-4914	700	31	mathematics	mathematic	NOUN
ejpam-4914	700	32	,	,	PUNCT
ejpam-4914	700	33	12(3):978	12(3):978	NOUN
ejpam-4914	700	34	-	-	SYM
ejpam-4914	700	35	998	998	NUM
ejpam-4914	700	36	,	,	PUNCT
ejpam-4914	700	37	2019	2019	NUM
ejpam-4914	700	38	.	.	PUNCT
ejpam-4914	701	1	[	[	X
ejpam-4914	701	2	17	17	NUM
ejpam-4914	701	3	]	]	X
ejpam-4914	701	4	o.	o.	PROPN
ejpam-4914	701	5	ore	ore	PROPN
ejpam-4914	701	6	,	,	PUNCT
ejpam-4914	701	7	theory	theory	NOUN
ejpam-4914	701	8	of	of	ADP
ejpam-4914	701	9	graphs	graph	NOUN
ejpam-4914	701	10	,	,	PUNCT
ejpam-4914	701	11	american	american	ADJ
ejpam-4914	701	12	mathematics	mathematics	PROPN
ejpam-4914	701	13	society	society	NOUN
ejpam-4914	701	14	,	,	PUNCT
ejpam-4914	701	15	colloquium	colloquium	NOUN
ejpam-4914	701	16	publication	publication	NOUN
ejpam-4914	701	17	38	38	NUM
ejpam-4914	701	18	,	,	PUNCT
ejpam-4914	701	19	1962	1962	NUM
ejpam-4914	701	20	.	.	PUNCT
ejpam-4914	702	1	[	[	X
ejpam-4914	702	2	18	18	NUM
ejpam-4914	702	3	]	]	X
ejpam-4914	702	4	l.	l.	PROPN
ejpam-4914	702	5	paleta	paleta	PROPN
ejpam-4914	702	6	and	and	CCONJ
ejpam-4914	702	7	f.	f.	PROPN
ejpam-4914	702	8	jamil	jamil	PROPN
ejpam-4914	702	9	,	,	PUNCT
ejpam-4914	702	10	more	more	ADV
ejpam-4914	702	11	on	on	ADP
ejpam-4914	702	12	perfect	perfect	ADJ
ejpam-4914	702	13	roman	roman	ADJ
ejpam-4914	702	14	domination	domination	NOUN
ejpam-4914	702	15	in	in	ADP
ejpam-4914	702	16	graphs	graph	NOUN
ejpam-4914	702	17	,	,	PUNCT
ejpam-4914	702	18	european	european	ADJ
ejpam-4914	702	19	journal	journal	NOUN
ejpam-4914	702	20	of	of	ADP
ejpam-4914	702	21	pure	pure	ADJ
ejpam-4914	702	22	and	and	CCONJ
ejpam-4914	702	23	applied	applied	ADJ
ejpam-4914	702	24	mathematics	mathematic	NOUN
ejpam-4914	702	25	,	,	PUNCT
ejpam-4914	702	26	13(3):529	13(3):529	PROPN
ejpam-4914	702	27	-	-	PUNCT
ejpam-4914	702	28	548	548	NUM
ejpam-4914	702	29	,	,	PUNCT
ejpam-4914	702	30	2020	2020	NUM
ejpam-4914	702	31	.	.	PUNCT
ejpam-4914	703	1	references	reference	NOUN
ejpam-4914	703	2	2449	2449	NUM
ejpam-4914	703	3	[	[	X
ejpam-4914	703	4	19	19	NUM
ejpam-4914	703	5	]	]	X
ejpam-4914	703	6	p.	p.	NOUN
ejpam-4914	703	7	pavloc	pavloc	PROPN
ejpam-4914	703	8	and	and	CCONJ
ejpam-4914	703	9	j.	j.	PROPN
ejpam-4914	703	10	zerovnik	zerovnik	PROPN
ejpam-4914	703	11	,	,	PUNCT
ejpam-4914	703	12	roman	roman	ADJ
ejpam-4914	703	13	domination	domination	NOUN
ejpam-4914	703	14	number	number	NOUN
ejpam-4914	703	15	of	of	ADP
ejpam-4914	703	16	the	the	DET
ejpam-4914	703	17	cartesian	cartesian	ADJ
ejpam-4914	703	18	products	product	NOUN
ejpam-4914	703	19	of	of	ADP
ejpam-4914	703	20	paths	path	NOUN
ejpam-4914	703	21	and	and	CCONJ
ejpam-4914	703	22	cycles	cycle	NOUN
ejpam-4914	703	23	,	,	PUNCT
ejpam-4914	703	24	the	the	DET
ejpam-4914	703	25	electronic	electronic	ADJ
ejpam-4914	703	26	journal	journal	NOUN
ejpam-4914	703	27	of	of	ADP
ejpam-4914	703	28	combinatorics	combinatoric	NOUN
ejpam-4914	703	29	19(3):p19	19(3):p19	NUM
ejpam-4914	703	30	,	,	PUNCT
ejpam-4914	703	31	2012	2012	NUM
ejpam-4914	703	32	.	.	PUNCT
ejpam-4914	704	1	[	[	X
ejpam-4914	704	2	20	20	NUM
ejpam-4914	704	3	]	]	X
ejpam-4914	704	4	d.	d.	PROPN
ejpam-4914	704	5	pradhan	pradhan	PROPN
ejpam-4914	704	6	,	,	PUNCT
ejpam-4914	704	7	s.	s.	PROPN
ejpam-4914	704	8	banerjee	banerjee	PROPN
ejpam-4914	704	9	and	and	CCONJ
ejpam-4914	704	10	jia	jia	PROPN
ejpam-4914	704	11	-	-	PROPN
ejpam-4914	704	12	bao	bao	PROPN
ejpam-4914	704	13	liu	liu	PROPN
ejpam-4914	704	14	,	,	PUNCT
ejpam-4914	704	15	perfect	perfect	ADJ
ejpam-4914	704	16	italian	italian	ADJ
ejpam-4914	704	17	domination	domination	NOUN
ejpam-4914	704	18	in	in	ADP
ejpam-4914	704	19	graphs	graph	NOUN
ejpam-4914	704	20	:	:	PUNCT
ejpam-4914	704	21	complexity	complexity	NOUN
ejpam-4914	704	22	and	and	CCONJ
ejpam-4914	704	23	algorithms	algorithm	NOUN
ejpam-4914	704	24	,	,	PUNCT
ejpam-4914	704	25	discrete	discrete	ADJ
ejpam-4914	704	26	applied	apply	VERB
ejpam-4914	704	27	mathematics	mathematic	NOUN
ejpam-4914	704	28	,	,	PUNCT
ejpam-4914	704	29	319:217	319:217	NUM
ejpam-4914	704	30	-	-	SYM
ejpam-4914	704	31	295	295	NUM
ejpam-4914	704	32	,	,	PUNCT
ejpam-4914	704	33	2022	2022	NUM
ejpam-4914	704	34	.	.	PUNCT
ejpam-4914	705	1	[	[	X
ejpam-4914	705	2	21	21	NUM
ejpam-4914	705	3	]	]	X
ejpam-4914	705	4	n.	n.	PROPN
ejpam-4914	705	5	jafari	jafari	PROPN
ejpam-4914	705	6	rad	rad	PROPN
ejpam-4914	705	7	and	and	CCONJ
ejpam-4914	705	8	e.	e.	PROPN
ejpam-4914	705	9	shabanib	shabanib	PROPN
ejpam-4914	705	10	,	,	PUNCT
ejpam-4914	705	11	on	on	ADP
ejpam-4914	705	12	the	the	DET
ejpam-4914	705	13	complexity	complexity	NOUN
ejpam-4914	705	14	of	of	ADP
ejpam-4914	705	15	some	some	DET
ejpam-4914	705	16	hop	hop	NOUN
ejpam-4914	705	17	domination	domination	NOUN
ejpam-4914	705	18	parameters	parameter	NOUN
ejpam-4914	705	19	,	,	PUNCT
ejpam-4914	705	20	electronic	electronic	ADJ
ejpam-4914	705	21	journal	journal	NOUN
ejpam-4914	705	22	of	of	ADP
ejpam-4914	705	23	graph	graph	NOUN
ejpam-4914	705	24	theory	theory	NOUN
ejpam-4914	705	25	and	and	CCONJ
ejpam-4914	705	26	applications	application	NOUN
ejpam-4914	705	27	7(1):74	7(1):74	NUM
ejpam-4914	705	28	-	-	SYM
ejpam-4914	705	29	86	86	NUM
ejpam-4914	705	30	,	,	PUNCT
ejpam-4914	705	31	2019	2019	NUM
ejpam-4914	705	32	.	.	PUNCT
ejpam-4914	706	1	[	[	X
ejpam-4914	706	2	22	22	NUM
ejpam-4914	706	3	]	]	X
ejpam-4914	706	4	n.	n.	PROPN
ejpam-4914	706	5	jafari	jafari	PROPN
ejpam-4914	706	6	rad	rad	PROPN
ejpam-4914	706	7	and	and	CCONJ
ejpam-4914	706	8	a.	a.	NOUN
ejpam-4914	706	9	poureidi	poureidi	PROPN
ejpam-4914	706	10	,	,	PUNCT
ejpam-4914	706	11	on	on	ADP
ejpam-4914	706	12	hop	hop	NOUN
ejpam-4914	706	13	roman	roman	ADJ
ejpam-4914	706	14	domination	domination	NOUN
ejpam-4914	706	15	in	in	ADP
ejpam-4914	706	16	trees	tree	NOUN
ejpam-4914	706	17	,	,	PUNCT
ejpam-4914	706	18	communications	communication	NOUN
ejpam-4914	706	19	in	in	ADP
ejpam-4914	706	20	combinatorics	combinatoric	NOUN
ejpam-4914	706	21	and	and	CCONJ
ejpam-4914	706	22	optimization	optimization	NOUN
ejpam-4914	706	23	,	,	PUNCT
ejpam-4914	706	24	4(2):201	4(2):201	NUM
ejpam-4914	706	25	-	-	SYM
ejpam-4914	706	26	208	208	NUM
ejpam-4914	706	27	,	,	PUNCT
ejpam-4914	706	28	2019	2019	NUM
ejpam-4914	706	29	.	.	PUNCT
ejpam-4914	707	1	[	[	X
ejpam-4914	707	2	23	23	NUM
ejpam-4914	707	3	]	]	X
ejpam-4914	707	4	c.s	c.s	PROPN
ejpam-4914	707	5	.	.	PROPN
ejpam-4914	707	6	revelle	revelle	PROPN
ejpam-4914	707	7	and	and	CCONJ
ejpam-4914	707	8	k.e	k.e	PROPN
ejpam-4914	707	9	.	.	PROPN
ejpam-4914	708	1	rosing	rosing	PROPN
ejpam-4914	708	2	,	,	PUNCT
ejpam-4914	708	3	defendens	defenden	VERB
ejpam-4914	708	4	imperium	imperium	NOUN
ejpam-4914	708	5	romanum	romanum	NOUN
ejpam-4914	708	6	:	:	PUNCT
ejpam-4914	708	7	a	a	DET
ejpam-4914	708	8	classical	classical	ADJ
ejpam-4914	708	9	problem	problem	NOUN
ejpam-4914	708	10	in	in	ADP
ejpam-4914	708	11	military	military	ADJ
ejpam-4914	708	12	strategy	strategy	NOUN
ejpam-4914	708	13	,	,	PUNCT
ejpam-4914	708	14	the	the	DET
ejpam-4914	708	15	american	american	PROPN
ejpam-4914	708	16	mathematical	mathematical	PROPN
ejpam-4914	708	17	monthly	monthly	ADV
ejpam-4914	708	18	.	.	PUNCT
ejpam-4914	709	1	107(7):585	107(7):585	NUM
ejpam-4914	709	2	-	-	SYM
ejpam-4914	709	3	594	594	NUM
ejpam-4914	709	4	,	,	PUNCT
ejpam-4914	709	5	2000	2000	NUM
ejpam-4914	709	6	.	.	PUNCT
ejpam-4914	710	1	[	[	X
ejpam-4914	710	2	24	24	NUM
ejpam-4914	710	3	]	]	X
ejpam-4914	710	4	e.	e.	PROPN
ejpam-4914	710	5	shabani	shabani	PROPN
ejpam-4914	710	6	,	,	PUNCT
ejpam-4914	710	7	hop	hop	PROPN
ejpam-4914	710	8	roman	roman	ADJ
ejpam-4914	710	9	domination	domination	NOUN
ejpam-4914	710	10	in	in	ADP
ejpam-4914	710	11	graphs	graph	NOUN
ejpam-4914	710	12	,	,	PUNCT
ejpam-4914	710	13	manuscript	manuscript	NOUN
ejpam-4914	710	14	(	(	PUNCT
ejpam-4914	710	15	2017	2017	NUM
ejpam-4914	710	16	)	)	PUNCT
ejpam-4914	711	1	[	[	X
ejpam-4914	711	2	25	25	NUM
ejpam-4914	711	3	]	]	X
ejpam-4914	711	4	e.	e.	PROPN
ejpam-4914	711	5	shabani	shabani	PROPN
ejpam-4914	711	6	,	,	PUNCT
ejpam-4914	711	7	n.	n.	PROPN
ejpam-4914	711	8	jafari	jafari	PROPN
ejpam-4914	711	9	rad	rad	PROPN
ejpam-4914	711	10	,	,	PUNCT
ejpam-4914	711	11	and	and	CCONJ
ejpam-4914	711	12	a.	a.	NOUN
ejpam-4914	711	13	poureidi	poureidi	NOUN
ejpam-4914	711	14	,	,	PUNCT
ejpam-4914	711	15	graphs	graph	NOUN
ejpam-4914	711	16	with	with	ADP
ejpam-4914	711	17	large	large	ADJ
ejpam-4914	711	18	hop	hop	NOUN
ejpam-4914	711	19	roman	roman	ADJ
ejpam-4914	711	20	domination	domination	NOUN
ejpam-4914	711	21	numbers	number	NOUN
ejpam-4914	711	22	,	,	PUNCT
ejpam-4914	711	23	computer	computer	NOUN
ejpam-4914	711	24	science	science	NOUN
ejpam-4914	711	25	journal	journal	PROPN
ejpam-4914	711	26	of	of	ADP
ejpam-4914	711	27	moldova	moldova	PROPN
ejpam-4914	711	28	,	,	PUNCT
ejpam-4914	711	29	27(1):1	27(1):1	PROPN
ejpam-4914	711	30	-	-	SYM
ejpam-4914	711	31	20	20	NUM
ejpam-4914	711	32	,	,	PUNCT
ejpam-4914	711	33	2019	2019	NUM
ejpam-4914	711	34	.	.	PUNCT
ejpam-4914	712	1	[	[	X
ejpam-4914	712	2	26	26	NUM
ejpam-4914	712	3	]	]	X
ejpam-4914	712	4	i.	i.	PROPN
ejpam-4914	712	5	stewart	stewart	PROPN
ejpam-4914	712	6	,	,	PUNCT
ejpam-4914	712	7	defend	defend	VERB
ejpam-4914	712	8	the	the	DET
ejpam-4914	712	9	roman	roman	ADJ
ejpam-4914	712	10	empire	empire	NOUN
ejpam-4914	712	11	!	!	PUNCT
ejpam-4914	713	1	,	,	PUNCT
ejpam-4914	713	2	scientific	scientific	ADJ
ejpam-4914	713	3	american	american	ADJ
ejpam-4914	713	4	,	,	PUNCT
ejpam-4914	713	5	281(6):136	281(6):136	NOUN
ejpam-4914	713	6	-	-	SYM
ejpam-4914	713	7	139	139	NUM
ejpam-4914	713	8	,	,	PUNCT
ejpam-4914	713	9	1999	1999	NUM
ejpam-4914	713	10	.	.	PUNCT
ejpam-4914	714	1	[	[	X
ejpam-4914	714	2	27	27	NUM
ejpam-4914	714	3	]	]	X
ejpam-4914	714	4	t.k	t.k	PROPN
ejpam-4914	714	5	.	.	PROPN
ejpam-4914	714	6	sumenjak	sumenjak	PROPN
ejpam-4914	714	7	,	,	PUNCT
ejpam-4914	714	8	p.	p.	NOUN
ejpam-4914	714	9	pavlic	pavlic	NOUN
ejpam-4914	714	10	and	and	CCONJ
ejpam-4914	714	11	a.	a.	NOUN
ejpam-4914	714	12	tepeh	tepeh	NOUN
ejpam-4914	714	13	,	,	PUNCT
ejpam-4914	714	14	on	on	ADP
ejpam-4914	714	15	the	the	DET
ejpam-4914	714	16	roman	roman	ADJ
ejpam-4914	714	17	domination	domination	NOUN
ejpam-4914	714	18	in	in	ADP
ejpam-4914	714	19	the	the	DET
ejpam-4914	714	20	lexicographic	lexicographic	ADJ
ejpam-4914	714	21	product	product	NOUN
ejpam-4914	714	22	of	of	ADP
ejpam-4914	714	23	graphs	graph	NOUN
ejpam-4914	714	24	.	.	PUNCT
ejpam-4914	715	1	discrete	discrete	ADJ
ejpam-4914	715	2	applied	applied	ADJ
ejpam-4914	715	3	mathematics	mathematic	NOUN
ejpam-4914	715	4	,	,	PUNCT
ejpam-4914	715	5	160:2030	160:2030	NUM
ejpam-4914	715	6	-	-	SYM
ejpam-4914	715	7	2036	2036	NUM
ejpam-4914	715	8	,	,	PUNCT
ejpam-4914	715	9	2012	2012	NUM
ejpam-4914	715	10	.	.	PUNCT