id	sid	tid	token	lemma	pos
ejpam-6063	1	1	european	european	PROPN
ejpam-6063	1	2	journal	journal	PROPN
ejpam-6063	1	3	of	of	ADP
ejpam-6063	1	4	pure	pure	ADJ
ejpam-6063	1	5	and	and	CCONJ
ejpam-6063	1	6	applied	applied	ADJ
ejpam-6063	1	7	mathematics	mathematic	NOUN
ejpam-6063	1	8	2025	2025	NUM
ejpam-6063	1	9	,	,	PUNCT
ejpam-6063	1	10	vol	vol	NOUN
ejpam-6063	1	11	.	.	PROPN
ejpam-6063	1	12	18	18	NUM
ejpam-6063	1	13	,	,	PUNCT
ejpam-6063	1	14	issue	issue	NOUN
ejpam-6063	1	15	2	2	NUM
ejpam-6063	1	16	,	,	PUNCT
ejpam-6063	1	17	article	article	NOUN
ejpam-6063	1	18	number	number	NOUN
ejpam-6063	1	19	6063	6063	NUM
ejpam-6063	1	20	issn	issn	PROPN
ejpam-6063	1	21	1307	1307	NUM
ejpam-6063	1	22	-	-	SYM
ejpam-6063	1	23	5543	5543	NUM
ejpam-6063	1	24	–	–	PUNCT
ejpam-6063	1	25	ejpam.com	ejpam.com	X
ejpam-6063	1	26	published	publish	VERB
ejpam-6063	1	27	by	by	ADP
ejpam-6063	1	28	new	new	PROPN
ejpam-6063	1	29	york	york	PROPN
ejpam-6063	1	30	business	business	PROPN
ejpam-6063	1	31	global	global	ADJ
ejpam-6063	1	32	2	2	NUM
ejpam-6063	1	33	-	-	PUNCT
ejpam-6063	1	34	vertex	vertex	NOUN
ejpam-6063	1	35	covering	covering	NOUN
ejpam-6063	1	36	of	of	ADP
ejpam-6063	1	37	a	a	DET
ejpam-6063	1	38	graph	graph	NOUN
ejpam-6063	1	39	javier	javier	PROPN
ejpam-6063	1	40	a.	a.	PROPN
ejpam-6063	1	41	hassan1,2,∗	hassan1,2,∗	PROPN
ejpam-6063	1	42	,	,	PUNCT
ejpam-6063	1	43	sergio	sergio	PROPN
ejpam-6063	1	44	r.	r.	PROPN
ejpam-6063	1	45	canoy	canoy	PROPN
ejpam-6063	1	46	,	,	PUNCT
ejpam-6063	1	47	jr.3,4	jr.3,4	PROPN
ejpam-6063	1	48	,	,	PUNCT
ejpam-6063	1	49	anabel	anabel	PROPN
ejpam-6063	1	50	gamorez5	gamorez5	PROPN
ejpam-6063	1	51	,	,	PUNCT
ejpam-6063	1	52	eman	eman	PROPN
ejpam-6063	1	53	c.	c.	PROPN
ejpam-6063	1	54	ahmad5	ahmad5	PROPN
ejpam-6063	1	55	,	,	PUNCT
ejpam-6063	1	56	an	an	DET
ejpam-6063	1	57	-	-	PUNCT
ejpam-6063	1	58	nadzwie	nadzwie	NOUN
ejpam-6063	1	59	s.	s.	PROPN
ejpam-6063	1	60	sappari1	sappari1	PROPN
ejpam-6063	1	61	1mathematics	1mathematics	NUM
ejpam-6063	1	62	and	and	CCONJ
ejpam-6063	1	63	sciences	sciences	PROPN
ejpam-6063	1	64	department	department	PROPN
ejpam-6063	1	65	,	,	PUNCT
ejpam-6063	1	66	college	college	NOUN
ejpam-6063	1	67	of	of	ADP
ejpam-6063	1	68	arts	art	NOUN
ejpam-6063	1	69	and	and	CCONJ
ejpam-6063	1	70	sciences	science	NOUN
ejpam-6063	1	71	,	,	PUNCT
ejpam-6063	1	72	msu	msu	PROPN
ejpam-6063	1	73	tawi	tawi	PROPN
ejpam-6063	1	74	-	-	PUNCT
ejpam-6063	1	75	tawi	tawi	PROPN
ejpam-6063	1	76	college	college	PROPN
ejpam-6063	1	77	of	of	ADP
ejpam-6063	1	78	technology	technology	NOUN
ejpam-6063	1	79	and	and	CCONJ
ejpam-6063	1	80	oceanography	oceanography	NOUN
ejpam-6063	1	81	,	,	PUNCT
ejpam-6063	1	82	bongao	bongao	NOUN
ejpam-6063	1	83	,	,	PUNCT
ejpam-6063	1	84	tawi	tawi	NOUN
ejpam-6063	1	85	-	-	PUNCT
ejpam-6063	1	86	tawi	tawi	NOUN
ejpam-6063	1	87	,	,	PUNCT
ejpam-6063	1	88	philippines	philippine	NOUN
ejpam-6063	1	89	2department	2department	NUM
ejpam-6063	1	90	of	of	ADP
ejpam-6063	1	91	mathematics	mathematic	NOUN
ejpam-6063	1	92	,	,	PUNCT
ejpam-6063	1	93	college	college	NOUN
ejpam-6063	1	94	of	of	ADP
ejpam-6063	1	95	science	science	PROPN
ejpam-6063	1	96	,	,	PUNCT
ejpam-6063	1	97	korea	korea	PROPN
ejpam-6063	1	98	university	university	PROPN
ejpam-6063	1	99	,	,	PUNCT
ejpam-6063	1	100	seoul	seoul	PROPN
ejpam-6063	1	101	,	,	PUNCT
ejpam-6063	1	102	south	south	PROPN
ejpam-6063	1	103	korea	korea	PROPN
ejpam-6063	2	1	3department	3department	NUM
ejpam-6063	2	2	of	of	ADP
ejpam-6063	2	3	mathematics	mathematic	NOUN
ejpam-6063	2	4	and	and	CCONJ
ejpam-6063	2	5	statistics	statistic	NOUN
ejpam-6063	2	6	,	,	PUNCT
ejpam-6063	2	7	college	college	NOUN
ejpam-6063	2	8	of	of	ADP
ejpam-6063	2	9	science	science	NOUN
ejpam-6063	2	10	and	and	CCONJ
ejpam-6063	2	11	mathematics	mathematic	NOUN
ejpam-6063	2	12	,	,	PUNCT
ejpam-6063	2	13	msu	msu	PROPN
ejpam-6063	2	14	-	-	PUNCT
ejpam-6063	2	15	iligan	iligan	PROPN
ejpam-6063	2	16	institute	institute	PROPN
ejpam-6063	2	17	of	of	ADP
ejpam-6063	2	18	technology	technology	PROPN
ejpam-6063	2	19	,	,	PUNCT
ejpam-6063	2	20	iligan	iligan	PROPN
ejpam-6063	2	21	city	city	PROPN
ejpam-6063	2	22	,	,	PUNCT
ejpam-6063	2	23	philippines	philippine	NOUN
ejpam-6063	2	24	4center	4center	PROPN
ejpam-6063	2	25	of	of	ADP
ejpam-6063	2	26	mathematical	mathematical	ADJ
ejpam-6063	2	27	and	and	CCONJ
ejpam-6063	2	28	theoretical	theoretical	ADJ
ejpam-6063	2	29	physical	physical	ADJ
ejpam-6063	2	30	sciences	science	NOUN
ejpam-6063	2	31	,	,	PUNCT
ejpam-6063	2	32	prism	prism	NOUN
ejpam-6063	2	33	,	,	PUNCT
ejpam-6063	2	34	msu	msu	PROPN
ejpam-6063	2	35	-	-	PUNCT
ejpam-6063	2	36	iligan	iligan	PROPN
ejpam-6063	2	37	institute	institute	PROPN
ejpam-6063	2	38	of	of	ADP
ejpam-6063	2	39	technology	technology	PROPN
ejpam-6063	2	40	,	,	PUNCT
ejpam-6063	2	41	iligan	iligan	PROPN
ejpam-6063	2	42	city	city	PROPN
ejpam-6063	2	43	,	,	PUNCT
ejpam-6063	2	44	philippines	philippine	VERB
ejpam-6063	2	45	5department	5department	NUM
ejpam-6063	2	46	of	of	ADP
ejpam-6063	2	47	mathematics	mathematic	NOUN
ejpam-6063	2	48	and	and	CCONJ
ejpam-6063	2	49	statistics	statistic	NOUN
ejpam-6063	2	50	,	,	PUNCT
ejpam-6063	2	51	college	college	NOUN
ejpam-6063	2	52	of	of	ADP
ejpam-6063	2	53	science	science	NOUN
ejpam-6063	2	54	and	and	CCONJ
ejpam-6063	2	55	mathematics	mathematic	NOUN
ejpam-6063	2	56	,	,	PUNCT
ejpam-6063	2	57	western	western	ADJ
ejpam-6063	2	58	mindanao	mindanao	PROPN
ejpam-6063	2	59	state	state	PROPN
ejpam-6063	2	60	university	university	PROPN
ejpam-6063	2	61	,	,	PUNCT
ejpam-6063	2	62	zamboanga	zamboanga	PROPN
ejpam-6063	2	63	city	city	PROPN
ejpam-6063	2	64	,	,	PUNCT
ejpam-6063	2	65	philippines	philippine	NOUN
ejpam-6063	2	66	abstract	abstract	ADJ
ejpam-6063	2	67	.	.	PUNCT
ejpam-6063	3	1	in	in	ADP
ejpam-6063	3	2	this	this	DET
ejpam-6063	3	3	paper	paper	NOUN
ejpam-6063	3	4	,	,	PUNCT
ejpam-6063	3	5	we	we	PRON
ejpam-6063	3	6	initiate	initiate	VERB
ejpam-6063	3	7	the	the	DET
ejpam-6063	3	8	study	study	NOUN
ejpam-6063	3	9	on	on	ADP
ejpam-6063	3	10	2	2	NUM
ejpam-6063	3	11	-	-	PUNCT
ejpam-6063	3	12	vertex	vertex	NOUN
ejpam-6063	3	13	covering	covering	NOUN
ejpam-6063	3	14	of	of	ADP
ejpam-6063	3	15	a	a	DET
ejpam-6063	3	16	graph	graph	NOUN
ejpam-6063	3	17	.	.	PUNCT
ejpam-6063	4	1	we	we	PRON
ejpam-6063	4	2	characterize	characterize	VERB
ejpam-6063	4	3	the	the	DET
ejpam-6063	4	4	2	2	NUM
ejpam-6063	4	5	-	-	PUNCT
ejpam-6063	4	6	vertex	vertex	NOUN
ejpam-6063	4	7	covering	covering	NOUN
ejpam-6063	4	8	sets	set	NOUN
ejpam-6063	4	9	in	in	ADP
ejpam-6063	4	10	some	some	DET
ejpam-6063	4	11	special	special	ADJ
ejpam-6063	4	12	graphs	graph	NOUN
ejpam-6063	4	13	,	,	PUNCT
ejpam-6063	4	14	join	join	VERB
ejpam-6063	4	15	and	and	CCONJ
ejpam-6063	4	16	corona	corona	NOUN
ejpam-6063	4	17	of	of	ADP
ejpam-6063	4	18	two	two	NUM
ejpam-6063	4	19	graphs	graph	NOUN
ejpam-6063	4	20	,	,	PUNCT
ejpam-6063	4	21	and	and	CCONJ
ejpam-6063	4	22	we	we	PRON
ejpam-6063	4	23	derive	derive	VERB
ejpam-6063	4	24	some	some	DET
ejpam-6063	4	25	bounds	bound	NOUN
ejpam-6063	4	26	or	or	CCONJ
ejpam-6063	4	27	formulas	formula	NOUN
ejpam-6063	4	28	of	of	ADP
ejpam-6063	4	29	the	the	DET
ejpam-6063	4	30	said	say	VERB
ejpam-6063	4	31	parameter	parameter	NOUN
ejpam-6063	4	32	of	of	ADP
ejpam-6063	4	33	each	each	PRON
ejpam-6063	4	34	of	of	ADP
ejpam-6063	4	35	these	these	DET
ejpam-6063	4	36	graphs	graph	NOUN
ejpam-6063	4	37	.	.	PUNCT
ejpam-6063	5	1	2020	2020	NUM
ejpam-6063	5	2	mathematics	mathematic	NOUN
ejpam-6063	5	3	subject	subject	NOUN
ejpam-6063	5	4	classifications	classification	NOUN
ejpam-6063	5	5	:	:	PUNCT
ejpam-6063	5	6	05c69	05c69	X
ejpam-6063	5	7	key	key	ADJ
ejpam-6063	5	8	words	word	NOUN
ejpam-6063	5	9	and	and	CCONJ
ejpam-6063	5	10	phrases	phrase	NOUN
ejpam-6063	5	11	:	:	PUNCT
ejpam-6063	5	12	vertex	vertex	NOUN
ejpam-6063	5	13	cover	cover	NOUN
ejpam-6063	5	14	,	,	PUNCT
ejpam-6063	5	15	2	2	NUM
ejpam-6063	5	16	-	-	PUNCT
ejpam-6063	5	17	domination	domination	NOUN
ejpam-6063	5	18	,	,	PUNCT
ejpam-6063	5	19	2	2	NUM
ejpam-6063	5	20	-	-	PUNCT
ejpam-6063	5	21	vertex	vertex	NOUN
ejpam-6063	5	22	covering	covering	NOUN
ejpam-6063	5	23	set	set	NOUN
ejpam-6063	5	24	,	,	PUNCT
ejpam-6063	5	25	2	2	NUM
ejpam-6063	5	26	-	-	PUNCT
ejpam-6063	5	27	vertex	vertex	NOUN
ejpam-6063	5	28	cover	cover	NOUN
ejpam-6063	5	29	number	number	NOUN
ejpam-6063	5	30	1	1	NUM
ejpam-6063	5	31	.	.	PUNCT
ejpam-6063	5	32	introduction	introduction	NOUN
ejpam-6063	5	33	vertex	vertex	NOUN
ejpam-6063	5	34	cover	cover	NOUN
ejpam-6063	5	35	of	of	ADP
ejpam-6063	5	36	a	a	DET
ejpam-6063	5	37	graph	graph	NOUN
ejpam-6063	5	38	is	be	AUX
ejpam-6063	5	39	one	one	NUM
ejpam-6063	5	40	the	the	DET
ejpam-6063	5	41	well	well	ADV
ejpam-6063	5	42	-	-	PUNCT
ejpam-6063	5	43	studied	study	VERB
ejpam-6063	5	44	parameters	parameter	NOUN
ejpam-6063	5	45	in	in	ADP
ejpam-6063	5	46	the	the	DET
ejpam-6063	5	47	theory	theory	NOUN
ejpam-6063	5	48	of	of	ADP
ejpam-6063	5	49	graphs	graph	NOUN
ejpam-6063	5	50	.	.	PUNCT
ejpam-6063	6	1	in	in	ADP
ejpam-6063	6	2	optimization	optimization	NOUN
ejpam-6063	6	3	,	,	PUNCT
ejpam-6063	6	4	the	the	DET
ejpam-6063	6	5	parameter	parameter	NOUN
ejpam-6063	6	6	can	can	AUX
ejpam-6063	6	7	be	be	AUX
ejpam-6063	6	8	used	use	VERB
ejpam-6063	6	9	to	to	PART
ejpam-6063	6	10	model	model	VERB
ejpam-6063	6	11	some	some	DET
ejpam-6063	6	12	real	real	ADJ
ejpam-6063	6	13	-	-	PUNCT
ejpam-6063	6	14	world	world	NOUN
ejpam-6063	6	15	problems	problem	NOUN
ejpam-6063	6	16	and	and	CCONJ
ejpam-6063	6	17	in	in	ADP
ejpam-6063	6	18	the	the	DET
ejpam-6063	6	19	elimination	elimination	NOUN
ejpam-6063	6	20	of	of	ADP
ejpam-6063	6	21	repetitive	repetitive	ADJ
ejpam-6063	6	22	dna	dna	NOUN
ejpam-6063	6	23	sequences	sequence	NOUN
ejpam-6063	6	24	for	for	ADP
ejpam-6063	6	25	synthetic	synthetic	ADJ
ejpam-6063	6	26	biology	biology	NOUN
ejpam-6063	7	1	[	[	X
ejpam-6063	7	2	1	1	NUM
ejpam-6063	7	3	]	]	PUNCT
ejpam-6063	7	4	.	.	PUNCT
ejpam-6063	8	1	the	the	DET
ejpam-6063	8	2	parameter	parameter	NOUN
ejpam-6063	8	3	,	,	PUNCT
ejpam-6063	8	4	as	as	SCONJ
ejpam-6063	8	5	pointed	point	VERB
ejpam-6063	8	6	out	out	ADP
ejpam-6063	8	7	by	by	ADP
ejpam-6063	8	8	angel	angel	NOUN
ejpam-6063	8	9	and	and	CCONJ
ejpam-6063	8	10	toregas	torega	NOUN
ejpam-6063	8	11	et	et	PROPN
ejpam-6063	8	12	al	al	PROPN
ejpam-6063	8	13	.	.	PUNCT
ejpam-6063	9	1	in	in	ADP
ejpam-6063	9	2	[	[	X
ejpam-6063	9	3	2	2	NUM
ejpam-6063	9	4	]	]	PUNCT
ejpam-6063	9	5	and	and	CCONJ
ejpam-6063	9	6	[	[	X
ejpam-6063	9	7	3	3	NUM
ejpam-6063	9	8	]	]	PUNCT
ejpam-6063	9	9	,	,	PUNCT
ejpam-6063	9	10	respectively	respectively	ADV
ejpam-6063	9	11	,	,	PUNCT
ejpam-6063	9	12	can	can	AUX
ejpam-6063	9	13	also	also	ADV
ejpam-6063	9	14	serve	serve	VERB
ejpam-6063	9	15	to	to	PART
ejpam-6063	9	16	model	model	NOUN
ejpam-6063	9	17	safety	safety	NOUN
ejpam-6063	9	18	,	,	PUNCT
ejpam-6063	9	19	defense	defense	NOUN
ejpam-6063	9	20	strategy	strategy	NOUN
ejpam-6063	9	21	,	,	PUNCT
ejpam-6063	9	22	and	and	CCONJ
ejpam-6063	9	23	emergency	emergency	NOUN
ejpam-6063	9	24	facility	facility	NOUN
ejpam-6063	9	25	location	location	NOUN
ejpam-6063	9	26	problems	problem	NOUN
ejpam-6063	9	27	.	.	PUNCT
ejpam-6063	10	1	it	it	PRON
ejpam-6063	10	2	is	be	AUX
ejpam-6063	10	3	well	well	ADV
ejpam-6063	10	4	-	-	PUNCT
ejpam-6063	10	5	known	know	VERB
ejpam-6063	10	6	that	that	SCONJ
ejpam-6063	10	7	the	the	DET
ejpam-6063	10	8	vertex	vertex	NOUN
ejpam-6063	10	9	cover	cover	NOUN
ejpam-6063	10	10	problem	problem	NOUN
ejpam-6063	10	11	is	be	AUX
ejpam-6063	10	12	an	an	DET
ejpam-6063	10	13	np	np	ADJ
ejpam-6063	10	14	-	-	PUNCT
ejpam-6063	10	15	hard	hard	ADJ
ejpam-6063	10	16	optimization	optimization	NOUN
ejpam-6063	10	17	problem	problem	NOUN
ejpam-6063	10	18	.	.	PUNCT
ejpam-6063	11	1	karp	karp	NOUN
ejpam-6063	11	2	in	in	ADP
ejpam-6063	11	3	[	[	X
ejpam-6063	11	4	4	4	NUM
ejpam-6063	11	5	]	]	PUNCT
ejpam-6063	11	6	used	use	VERB
ejpam-6063	11	7	the	the	DET
ejpam-6063	11	8	np	np	ADJ
ejpam-6063	11	9	-completeness	-completeness	NOUN
ejpam-6063	11	10	of	of	ADP
ejpam-6063	11	11	the	the	DET
ejpam-6063	11	12	clique	clique	ADJ
ejpam-6063	11	13	problem	problem	NOUN
ejpam-6063	11	14	to	to	PART
ejpam-6063	11	15	show	show	VERB
ejpam-6063	11	16	that	that	SCONJ
ejpam-6063	11	17	the	the	DET
ejpam-6063	11	18	vertex	vertex	NOUN
ejpam-6063	11	19	cover	cover	NOUN
ejpam-6063	11	20	problem	problem	NOUN
ejpam-6063	11	21	is	be	AUX
ejpam-6063	11	22	np	np	INTJ
ejpam-6063	11	23	complete	complete	ADJ
ejpam-6063	11	24	.	.	PUNCT
ejpam-6063	12	1	np	np	PRON
ejpam-6063	12	2	-completeness	-completeness	NOUN
ejpam-6063	12	3	of	of	ADP
ejpam-6063	12	4	the	the	DET
ejpam-6063	12	5	vertex	vertex	NOUN
ejpam-6063	12	6	problem	problem	NOUN
ejpam-6063	12	7	was	be	AUX
ejpam-6063	12	8	also	also	ADV
ejpam-6063	12	9	investigated	investigate	VERB
ejpam-6063	12	10	by	by	ADP
ejpam-6063	12	11	garey	garey	PROPN
ejpam-6063	12	12	et	et	PROPN
ejpam-6063	12	13	al	al	PROPN
ejpam-6063	12	14	.	.	PUNCT
ejpam-6063	13	1	∗corresponding	∗corresponde	VERB
ejpam-6063	13	2	author	author	NOUN
ejpam-6063	13	3	.	.	PUNCT
ejpam-6063	14	1	doi	doi	NOUN
ejpam-6063	14	2	:	:	PUNCT
ejpam-6063	14	3	https://doi.org/10.29020/nybg.ejpam.v18i2.6063	https://doi.org/10.29020/nybg.ejpam.v18i2.6063	ADJ
ejpam-6063	14	4	email	email	NOUN
ejpam-6063	14	5	addresses	address	NOUN
ejpam-6063	14	6	:	:	PUNCT
ejpam-6063	14	7	javierhassan@msutawi-tawi.edu.ph	javierhassan@msutawi-tawi.edu.ph	PROPN
ejpam-6063	14	8	(	(	PUNCT
ejpam-6063	14	9	j.	j.	PROPN
ejpam-6063	14	10	hassan	hassan	PROPN
ejpam-6063	14	11	)	)	PUNCT
ejpam-6063	15	1	sergio.canoy@g.msuiit.edu.ph	sergio.canoy@g.msuiit.edu.ph	PROPN
ejpam-6063	15	2	(	(	PUNCT
ejpam-6063	15	3	s.	s.	PROPN
ejpam-6063	15	4	canoy	canoy	PROPN
ejpam-6063	15	5	,	,	PUNCT
ejpam-6063	15	6	jr	jr	PROPN
ejpam-6063	15	7	.	.	PUNCT
ejpam-6063	15	8	)	)	PUNCT
ejpam-6063	16	1	anabel.gamorez@wmsu.edu.ph	anabel.gamorez@wmsu.edu.ph	PROPN
ejpam-6063	16	2	(	(	PUNCT
ejpam-6063	16	3	a.	a.	NOUN
ejpam-6063	16	4	gamorez	gamorez	PROPN
ejpam-6063	16	5	)	)	PUNCT
ejpam-6063	17	1	ahmad.eman@wmsu.edu.ph	ahmad.eman@wmsu.edu.ph	PROPN
ejpam-6063	17	2	(	(	PUNCT
ejpam-6063	17	3	e.	e.	PROPN
ejpam-6063	17	4	ahmad	ahmad	PROPN
ejpam-6063	17	5	)	)	PUNCT
ejpam-6063	17	6	annadzwiesappari@msutawi-tawi.edu.ph	annadzwiesappari@msutawi-tawi.edu.ph	PROPN
ejpam-6063	17	7	(	(	PUNCT
ejpam-6063	17	8	a.	a.	NOUN
ejpam-6063	17	9	sappari	sappari	PROPN
ejpam-6063	17	10	)	)	PUNCT
ejpam-6063	17	11	https://www.ejpam.com	https://www.ejpam.com	X
ejpam-6063	18	1	1	1	NUM
ejpam-6063	18	2	copyright	copyright	NOUN
ejpam-6063	18	3	:	:	PUNCT
ejpam-6063	18	4	©	©	PROPN
ejpam-6063	18	5	2025	2025	NUM
ejpam-6063	18	6	the	the	DET
ejpam-6063	18	7	author(s	author(s	NOUN
ejpam-6063	18	8	)	)	PUNCT
ejpam-6063	18	9	.	.	PUNCT
ejpam-6063	19	1	(	(	PUNCT
ejpam-6063	19	2	cc	cc	NOUN
ejpam-6063	19	3	by	by	ADP
ejpam-6063	19	4	-	-	PUNCT
ejpam-6063	19	5	nc	nc	PROPN
ejpam-6063	19	6	4.0	4.0	NUM
ejpam-6063	19	7	)	)	PUNCT
ejpam-6063	19	8	j.	j.	PROPN
ejpam-6063	19	9	hassan	hassan	PROPN
ejpam-6063	19	10	et	et	PROPN
ejpam-6063	19	11	.	.	PUNCT
ejpam-6063	20	1	al	al	PROPN
ejpam-6063	20	2	/	/	PUNCT
ejpam-6063	20	3	eur	eur	PROPN
ejpam-6063	20	4	.	.	PUNCT
ejpam-6063	21	1	j.	j.	PROPN
ejpam-6063	21	2	pure	pure	PROPN
ejpam-6063	21	3	appl	appl	PROPN
ejpam-6063	21	4	.	.	PROPN
ejpam-6063	21	5	math	math	PROPN
ejpam-6063	21	6	,	,	PUNCT
ejpam-6063	21	7	18	18	NUM
ejpam-6063	21	8	(	(	PUNCT
ejpam-6063	21	9	2	2	NUM
ejpam-6063	21	10	)	)	PUNCT
ejpam-6063	21	11	(	(	PUNCT
ejpam-6063	21	12	2025	2025	NUM
ejpam-6063	21	13	)	)	PUNCT
ejpam-6063	21	14	,	,	PUNCT
ejpam-6063	21	15	6063	6063	NUM
ejpam-6063	21	16	2	2	NUM
ejpam-6063	21	17	of	of	ADP
ejpam-6063	21	18	11	11	NUM
ejpam-6063	21	19	in	in	ADP
ejpam-6063	21	20	[	[	X
ejpam-6063	21	21	5	5	NUM
ejpam-6063	21	22	]	]	PUNCT
ejpam-6063	21	23	and	and	CCONJ
ejpam-6063	21	24	[	[	X
ejpam-6063	21	25	6	6	NUM
ejpam-6063	21	26	]	]	PUNCT
ejpam-6063	21	27	.	.	PUNCT
ejpam-6063	22	1	other	other	ADJ
ejpam-6063	22	2	studies	study	NOUN
ejpam-6063	22	3	on	on	ADP
ejpam-6063	22	4	vertex	vertex	NOUN
ejpam-6063	22	5	cover	cover	NOUN
ejpam-6063	22	6	and	and	CCONJ
ejpam-6063	22	7	its	its	PRON
ejpam-6063	22	8	variations	variation	NOUN
ejpam-6063	22	9	can	can	AUX
ejpam-6063	22	10	be	be	AUX
ejpam-6063	22	11	found	find	VERB
ejpam-6063	22	12	in	in	ADP
ejpam-6063	22	13	[	[	X
ejpam-6063	22	14	7	7	NUM
ejpam-6063	22	15	]	]	PUNCT
ejpam-6063	22	16	,	,	PUNCT
ejpam-6063	22	17	[	[	X
ejpam-6063	22	18	8	8	NUM
ejpam-6063	22	19	]	]	PUNCT
ejpam-6063	22	20	,	,	PUNCT
ejpam-6063	22	21	[	[	X
ejpam-6063	22	22	9	9	NUM
ejpam-6063	22	23	]	]	PUNCT
ejpam-6063	22	24	,	,	PUNCT
ejpam-6063	22	25	[	[	X
ejpam-6063	22	26	10	10	NUM
ejpam-6063	22	27	]	]	PUNCT
ejpam-6063	22	28	,	,	PUNCT
ejpam-6063	22	29	[	[	X
ejpam-6063	22	30	11	11	NUM
ejpam-6063	22	31	]	]	PUNCT
ejpam-6063	22	32	,	,	PUNCT
ejpam-6063	22	33	[	[	X
ejpam-6063	22	34	12	12	NUM
ejpam-6063	22	35	]	]	PUNCT
ejpam-6063	22	36	,	,	PUNCT
ejpam-6063	22	37	and	and	CCONJ
ejpam-6063	22	38	[	[	X
ejpam-6063	22	39	13	13	NUM
ejpam-6063	22	40	]	]	PUNCT
ejpam-6063	22	41	.	.	PUNCT
ejpam-6063	23	1	vertex	vertex	NOUN
ejpam-6063	23	2	cover	cover	NOUN
ejpam-6063	23	3	is	be	AUX
ejpam-6063	23	4	closely	closely	ADV
ejpam-6063	23	5	related	relate	VERB
ejpam-6063	23	6	to	to	ADP
ejpam-6063	23	7	the	the	DET
ejpam-6063	23	8	concept	concept	NOUN
ejpam-6063	23	9	of	of	ADP
ejpam-6063	23	10	domination	domination	NOUN
ejpam-6063	23	11	.	.	PUNCT
ejpam-6063	24	1	in	in	ADP
ejpam-6063	24	2	fact	fact	NOUN
ejpam-6063	24	3	,	,	PUNCT
ejpam-6063	24	4	for	for	ADP
ejpam-6063	24	5	a	a	DET
ejpam-6063	24	6	non	non	ADJ
ejpam-6063	24	7	-	-	ADJ
ejpam-6063	24	8	trivial	trivial	ADJ
ejpam-6063	24	9	connected	connected	ADJ
ejpam-6063	24	10	graph	graph	NOUN
ejpam-6063	24	11	,	,	PUNCT
ejpam-6063	24	12	a	a	DET
ejpam-6063	24	13	vertex	vertex	NOUN
ejpam-6063	24	14	cover	cover	NOUN
ejpam-6063	24	15	is	be	AUX
ejpam-6063	24	16	a	a	DET
ejpam-6063	24	17	dominating	dominating	NOUN
ejpam-6063	24	18	set	set	NOUN
ejpam-6063	24	19	.	.	PUNCT
ejpam-6063	25	1	undoubtedly	undoubtedly	ADV
ejpam-6063	25	2	,	,	PUNCT
ejpam-6063	25	3	a	a	DET
ejpam-6063	25	4	vertex	vertex	NOUN
ejpam-6063	25	5	cover	cover	NOUN
ejpam-6063	25	6	can	can	AUX
ejpam-6063	25	7	be	be	AUX
ejpam-6063	25	8	made	make	VERB
ejpam-6063	25	9	a	a	DET
ejpam-6063	25	10	dominating	dominating	NOUN
ejpam-6063	25	11	set	set	VERB
ejpam-6063	25	12	in	in	ADP
ejpam-6063	25	13	any	any	DET
ejpam-6063	25	14	graph	graph	NOUN
ejpam-6063	25	15	by	by	ADP
ejpam-6063	25	16	incorporating	incorporate	VERB
ejpam-6063	25	17	a	a	DET
ejpam-6063	25	18	domination	domination	NOUN
ejpam-6063	25	19	-	-	PUNCT
ejpam-6063	25	20	related	relate	VERB
ejpam-6063	25	21	concept	concept	NOUN
ejpam-6063	25	22	in	in	ADP
ejpam-6063	25	23	its	its	PRON
ejpam-6063	25	24	definition	definition	NOUN
ejpam-6063	25	25	.	.	PUNCT
ejpam-6063	26	1	in	in	ADP
ejpam-6063	26	2	this	this	DET
ejpam-6063	26	3	way	way	NOUN
ejpam-6063	26	4	,	,	PUNCT
ejpam-6063	26	5	a	a	DET
ejpam-6063	26	6	variant	variant	NOUN
ejpam-6063	26	7	of	of	ADP
ejpam-6063	26	8	vertex	vertex	NOUN
ejpam-6063	26	9	cover	cover	NOUN
ejpam-6063	26	10	emerges	emerge	NOUN
ejpam-6063	26	11	(	(	PUNCT
ejpam-6063	26	12	see	see	VERB
ejpam-6063	26	13	,	,	PUNCT
ejpam-6063	26	14	for	for	ADP
ejpam-6063	26	15	example	example	NOUN
ejpam-6063	26	16	,	,	PUNCT
ejpam-6063	26	17	[	[	X
ejpam-6063	26	18	8	8	NUM
ejpam-6063	26	19	]	]	PUNCT
ejpam-6063	26	20	,	,	PUNCT
ejpam-6063	26	21	[	[	X
ejpam-6063	26	22	9	9	NUM
ejpam-6063	26	23	]	]	PUNCT
ejpam-6063	26	24	,	,	PUNCT
ejpam-6063	26	25	[	[	X
ejpam-6063	26	26	10	10	NUM
ejpam-6063	26	27	]	]	PUNCT
ejpam-6063	26	28	,	,	PUNCT
ejpam-6063	26	29	[	[	X
ejpam-6063	26	30	11	11	NUM
ejpam-6063	26	31	]	]	PUNCT
ejpam-6063	26	32	,	,	PUNCT
ejpam-6063	26	33	and	and	CCONJ
ejpam-6063	26	34	[	[	X
ejpam-6063	26	35	12	12	NUM
ejpam-6063	26	36	]	]	PUNCT
ejpam-6063	26	37	)	)	PUNCT
ejpam-6063	26	38	.	.	PUNCT
ejpam-6063	27	1	using	use	VERB
ejpam-6063	27	2	the	the	DET
ejpam-6063	27	3	concept	concept	NOUN
ejpam-6063	27	4	of	of	ADP
ejpam-6063	27	5	2	2	NUM
ejpam-6063	27	6	-	-	PUNCT
ejpam-6063	27	7	domination	domination	NOUN
ejpam-6063	27	8	,	,	PUNCT
ejpam-6063	27	9	we	we	PRON
ejpam-6063	27	10	introduce	introduce	VERB
ejpam-6063	27	11	the	the	DET
ejpam-6063	27	12	parameter	parameter	NOUN
ejpam-6063	27	13	called	call	VERB
ejpam-6063	27	14	2	2	NUM
ejpam-6063	27	15	-	-	PUNCT
ejpam-6063	27	16	vertex	vertex	NOUN
ejpam-6063	27	17	cover	cover	NOUN
ejpam-6063	27	18	of	of	ADP
ejpam-6063	27	19	a	a	DET
ejpam-6063	27	20	graph	graph	NOUN
ejpam-6063	27	21	.	.	PUNCT
ejpam-6063	28	1	as	as	SCONJ
ejpam-6063	28	2	used	use	VERB
ejpam-6063	28	3	to	to	PART
ejpam-6063	28	4	model	model	VERB
ejpam-6063	28	5	a	a	DET
ejpam-6063	28	6	protection	protection	NOUN
ejpam-6063	28	7	strategy	strategy	NOUN
ejpam-6063	28	8	in	in	ADP
ejpam-6063	28	9	a	a	DET
ejpam-6063	28	10	network	network	NOUN
ejpam-6063	28	11	,	,	PUNCT
ejpam-6063	28	12	the	the	DET
ejpam-6063	28	13	2	2	NUM
ejpam-6063	28	14	-	-	PUNCT
ejpam-6063	28	15	vertex	vertex	NOUN
ejpam-6063	28	16	covering	covering	NOUN
ejpam-6063	28	17	ensures	ensure	VERB
ejpam-6063	28	18	that	that	SCONJ
ejpam-6063	28	19	every	every	DET
ejpam-6063	28	20	node	node	NOUN
ejpam-6063	28	21	or	or	CCONJ
ejpam-6063	28	22	vertex	vertex	NOUN
ejpam-6063	28	23	outside	outside	ADP
ejpam-6063	28	24	the	the	DET
ejpam-6063	28	25	cover	cover	NOUN
ejpam-6063	28	26	has	have	VERB
ejpam-6063	28	27	at	at	ADV
ejpam-6063	28	28	least	least	ADV
ejpam-6063	28	29	two	two	NUM
ejpam-6063	28	30	neighbors	neighbor	NOUN
ejpam-6063	28	31	coming	come	VERB
ejpam-6063	28	32	from	from	ADP
ejpam-6063	28	33	the	the	DET
ejpam-6063	28	34	covering	covering	NOUN
ejpam-6063	28	35	.	.	PUNCT
ejpam-6063	29	1	for	for	ADP
ejpam-6063	29	2	studies	study	NOUN
ejpam-6063	29	3	that	that	PRON
ejpam-6063	29	4	deal	deal	VERB
ejpam-6063	29	5	with	with	ADP
ejpam-6063	29	6	2	2	NUM
ejpam-6063	29	7	-	-	PUNCT
ejpam-6063	29	8	domination	domination	NOUN
ejpam-6063	29	9	and	and	CCONJ
ejpam-6063	29	10	its	its	PRON
ejpam-6063	29	11	variants	variant	NOUN
ejpam-6063	29	12	,	,	PUNCT
ejpam-6063	29	13	readers	reader	NOUN
ejpam-6063	29	14	may	may	AUX
ejpam-6063	29	15	consider	consider	VERB
ejpam-6063	29	16	[	[	X
ejpam-6063	29	17	14	14	NUM
ejpam-6063	29	18	]	]	PUNCT
ejpam-6063	29	19	,	,	PUNCT
ejpam-6063	30	1	[	[	X
ejpam-6063	30	2	15],[16	15],[16	X
ejpam-6063	30	3	]	]	X
ejpam-6063	30	4	,	,	PUNCT
ejpam-6063	30	5	[	[	X
ejpam-6063	30	6	17	17	NUM
ejpam-6063	30	7	]	]	PUNCT
ejpam-6063	30	8	,	,	PUNCT
ejpam-6063	30	9	and	and	CCONJ
ejpam-6063	30	10	[	[	X
ejpam-6063	30	11	18	18	NUM
ejpam-6063	30	12	]	]	SYM
ejpam-6063	30	13	.	.	PUNCT
ejpam-6063	30	14	2	2	X
ejpam-6063	30	15	.	.	X
ejpam-6063	30	16	terminologies	terminology	NOUN
ejpam-6063	30	17	and	and	CCONJ
ejpam-6063	30	18	notations	notation	NOUN
ejpam-6063	30	19	the	the	DET
ejpam-6063	30	20	open	open	ADJ
ejpam-6063	30	21	neighborhood	neighborhood	NOUN
ejpam-6063	30	22	of	of	ADP
ejpam-6063	30	23	a	a	DET
ejpam-6063	30	24	vertex	vertex	NOUN
ejpam-6063	30	25	v	v	NOUN
ejpam-6063	30	26	of	of	ADP
ejpam-6063	30	27	a	a	DET
ejpam-6063	30	28	simple	simple	ADJ
ejpam-6063	30	29	undirected	undirected	ADJ
ejpam-6063	30	30	graph	graph	NOUN
ejpam-6063	30	31	g	g	PROPN
ejpam-6063	30	32	is	be	AUX
ejpam-6063	30	33	the	the	DET
ejpam-6063	30	34	set	set	NOUN
ejpam-6063	30	35	ng(v	ng(v	PUNCT
ejpam-6063	30	36	)	)	PUNCT
ejpam-6063	30	37	=	=	SYM
ejpam-6063	31	1	{	{	PUNCT
ejpam-6063	31	2	u	u	NOUN
ejpam-6063	31	3	∈	∈	PROPN
ejpam-6063	31	4	v	v	NOUN
ejpam-6063	31	5	(	(	PUNCT
ejpam-6063	31	6	g	g	NOUN
ejpam-6063	31	7	)	)	PUNCT
ejpam-6063	31	8	:	:	PUNCT
ejpam-6063	31	9	uv	uv	PROPN
ejpam-6063	31	10	∈	∈	PROPN
ejpam-6063	31	11	e(g	e(g	PROPN
ejpam-6063	31	12	)	)	PUNCT
ejpam-6063	31	13	}	}	PUNCT
ejpam-6063	31	14	and	and	CCONJ
ejpam-6063	31	15	its	its	PRON
ejpam-6063	31	16	closed	closed	ADJ
ejpam-6063	31	17	neighborhood	neighborhood	NOUN
ejpam-6063	31	18	is	be	AUX
ejpam-6063	31	19	the	the	DET
ejpam-6063	31	20	set	set	NOUN
ejpam-6063	31	21	ng[v	ng[v	NOUN
ejpam-6063	31	22	]	]	X
ejpam-6063	31	23	=	=	SYM
ejpam-6063	31	24	ng(v	ng(v	X
ejpam-6063	31	25	)	)	PUNCT
ejpam-6063	31	26	∪	∪	ADP
ejpam-6063	31	27	{	{	PUNCT
ejpam-6063	31	28	v	v	NOUN
ejpam-6063	31	29	}	}	PUNCT
ejpam-6063	31	30	.	.	PUNCT
ejpam-6063	32	1	the	the	DET
ejpam-6063	32	2	open	open	ADJ
ejpam-6063	32	3	neighborhood	neighborhood	NOUN
ejpam-6063	32	4	of	of	ADP
ejpam-6063	32	5	a	a	DET
ejpam-6063	32	6	set	set	NOUN
ejpam-6063	32	7	s	s	NOUN
ejpam-6063	32	8	⊆	⊆	NUM
ejpam-6063	32	9	v	v	NOUN
ejpam-6063	32	10	(	(	PUNCT
ejpam-6063	32	11	g	g	NOUN
ejpam-6063	32	12	)	)	PUNCT
ejpam-6063	32	13	is	be	AUX
ejpam-6063	32	14	the	the	DET
ejpam-6063	32	15	set	set	NOUN
ejpam-6063	32	16	ng(s	ng(s	NOUN
ejpam-6063	32	17	)	)	PUNCT
ejpam-6063	32	18	=	=	SYM
ejpam-6063	32	19	∪v∈sng(v	∪v∈sng(v	PROPN
ejpam-6063	32	20	)	)	PUNCT
ejpam-6063	32	21	,	,	PUNCT
ejpam-6063	32	22	and	and	CCONJ
ejpam-6063	32	23	its	its	PRON
ejpam-6063	32	24	closed	closed	ADJ
ejpam-6063	32	25	neighborhood	neighborhood	NOUN
ejpam-6063	32	26	is	be	AUX
ejpam-6063	32	27	the	the	DET
ejpam-6063	32	28	set	set	VERB
ejpam-6063	32	29	ng[s	ng[	NOUN
ejpam-6063	32	30	]	]	PUNCT
ejpam-6063	32	31	=	=	SYM
ejpam-6063	32	32	s	s	X
ejpam-6063	32	33	∪	∪	NOUN
ejpam-6063	32	34	ng(s	ng(s	NUM
ejpam-6063	32	35	)	)	PUNCT
ejpam-6063	32	36	.	.	PUNCT
ejpam-6063	33	1	a	a	DET
ejpam-6063	33	2	vertex	vertex	NOUN
ejpam-6063	33	3	v	v	ADP
ejpam-6063	33	4	∈	∈	PROPN
ejpam-6063	33	5	v	v	NOUN
ejpam-6063	33	6	(	(	PUNCT
ejpam-6063	33	7	g	g	NOUN
ejpam-6063	33	8	)	)	PUNCT
ejpam-6063	33	9	is	be	AUX
ejpam-6063	33	10	an	an	DET
ejpam-6063	33	11	isolated	isolated	ADJ
ejpam-6063	33	12	vertex	vertex	NOUN
ejpam-6063	33	13	if	if	SCONJ
ejpam-6063	33	14	|ng(v)|	|ng(v)|	NOUN
ejpam-6063	33	15	=	=	SYM
ejpam-6063	33	16	0	0	NUM
ejpam-6063	33	17	.	.	PUNCT
ejpam-6063	34	1	the	the	DET
ejpam-6063	34	2	set	set	NOUN
ejpam-6063	34	3	containing	contain	VERB
ejpam-6063	34	4	all	all	DET
ejpam-6063	34	5	the	the	DET
ejpam-6063	34	6	isolated	isolated	ADJ
ejpam-6063	34	7	vertices	vertex	NOUN
ejpam-6063	34	8	in	in	ADP
ejpam-6063	34	9	g	g	PROPN
ejpam-6063	34	10	will	will	AUX
ejpam-6063	34	11	be	be	AUX
ejpam-6063	34	12	denoted	denote	VERB
ejpam-6063	34	13	by	by	ADP
ejpam-6063	34	14	i(g	i(g	NOUN
ejpam-6063	34	15	)	)	PUNCT
ejpam-6063	34	16	.	.	PUNCT
ejpam-6063	35	1	a	a	DET
ejpam-6063	35	2	vertex	vertex	NOUN
ejpam-6063	35	3	v	v	NOUN
ejpam-6063	35	4	is	be	AUX
ejpam-6063	35	5	a	a	DET
ejpam-6063	35	6	leaf	leaf	NOUN
ejpam-6063	35	7	or	or	CCONJ
ejpam-6063	35	8	an	an	DET
ejpam-6063	35	9	endvertex	endvertex	NOUN
ejpam-6063	35	10	if	if	SCONJ
ejpam-6063	35	11	|ng(v)|	|ng(v)|	VERB
ejpam-6063	35	12	=	=	SYM
ejpam-6063	35	13	1	1	X
ejpam-6063	35	14	.	.	PUNCT
ejpam-6063	36	1	the	the	DET
ejpam-6063	36	2	set	set	NOUN
ejpam-6063	36	3	l(g	l(g	NOUN
ejpam-6063	36	4	)	)	PUNCT
ejpam-6063	36	5	will	will	AUX
ejpam-6063	36	6	denote	denote	VERB
ejpam-6063	36	7	the	the	DET
ejpam-6063	36	8	set	set	NOUN
ejpam-6063	36	9	consisting	consist	VERB
ejpam-6063	36	10	of	of	ADP
ejpam-6063	36	11	all	all	DET
ejpam-6063	36	12	the	the	DET
ejpam-6063	36	13	leaves	leave	NOUN
ejpam-6063	36	14	in	in	ADP
ejpam-6063	36	15	g.	g.	PROPN
ejpam-6063	36	16	let	let	VERB
ejpam-6063	36	17	g	g	NOUN
ejpam-6063	36	18	and	and	CCONJ
ejpam-6063	36	19	h	h	NOUN
ejpam-6063	36	20	be	be	VERB
ejpam-6063	36	21	any	any	DET
ejpam-6063	36	22	two	two	NUM
ejpam-6063	36	23	graphs	graph	NOUN
ejpam-6063	36	24	.	.	PUNCT
ejpam-6063	37	1	the	the	DET
ejpam-6063	37	2	join	join	NOUN
ejpam-6063	37	3	g	g	PROPN
ejpam-6063	37	4	+	+	CCONJ
ejpam-6063	37	5	h	h	NOUN
ejpam-6063	37	6	is	be	AUX
ejpam-6063	37	7	the	the	DET
ejpam-6063	37	8	graph	graph	NOUN
ejpam-6063	37	9	with	with	ADP
ejpam-6063	37	10	vertex	vertex	NOUN
ejpam-6063	37	11	set	set	VERB
ejpam-6063	37	12	v	v	NOUN
ejpam-6063	37	13	(	(	PUNCT
ejpam-6063	37	14	g+h	g+h	NOUN
ejpam-6063	37	15	)	)	PUNCT
ejpam-6063	37	16	=	=	SYM
ejpam-6063	37	17	v	v	NOUN
ejpam-6063	37	18	(	(	PUNCT
ejpam-6063	37	19	g)∪	g)∪	VERB
ejpam-6063	37	20	v	v	NUM
ejpam-6063	37	21	(	(	PUNCT
ejpam-6063	37	22	h	h	NOUN
ejpam-6063	37	23	)	)	PUNCT
ejpam-6063	37	24	and	and	CCONJ
ejpam-6063	37	25	edge	edge	NOUN
ejpam-6063	37	26	set	set	VERB
ejpam-6063	37	27	e(g+h	e(g+h	NUM
ejpam-6063	37	28	)	)	PUNCT
ejpam-6063	38	1	=	=	SYM
ejpam-6063	38	2	e(g)∪e(h)∪	e(g)∪e(h)∪	NOUN
ejpam-6063	38	3	{	{	PUNCT
ejpam-6063	38	4	uv	uv	NOUN
ejpam-6063	38	5	:	:	PUNCT
ejpam-6063	38	6	u	u	PROPN
ejpam-6063	38	7	∈	∈	PROPN
ejpam-6063	38	8	v	v	ADP
ejpam-6063	38	9	(	(	PUNCT
ejpam-6063	38	10	g	g	NOUN
ejpam-6063	38	11	)	)	PUNCT
ejpam-6063	38	12	,	,	PUNCT
ejpam-6063	38	13	v	v	X
ejpam-6063	38	14	∈	∈	PROPN
ejpam-6063	38	15	v	v	NOUN
ejpam-6063	38	16	(	(	PUNCT
ejpam-6063	38	17	h	h	NOUN
ejpam-6063	38	18	)	)	PUNCT
ejpam-6063	38	19	}	}	PUNCT
ejpam-6063	38	20	.	.	PUNCT
ejpam-6063	39	1	the	the	DET
ejpam-6063	39	2	corona	corona	NOUN
ejpam-6063	39	3	g	g	PROPN
ejpam-6063	39	4	◦	◦	NOUN
ejpam-6063	39	5	h	h	NOUN
ejpam-6063	39	6	is	be	AUX
ejpam-6063	39	7	the	the	DET
ejpam-6063	39	8	graph	graph	NOUN
ejpam-6063	39	9	obtained	obtain	VERB
ejpam-6063	39	10	by	by	ADP
ejpam-6063	39	11	taking	take	VERB
ejpam-6063	39	12	one	one	NUM
ejpam-6063	39	13	copy	copy	NOUN
ejpam-6063	39	14	of	of	ADP
ejpam-6063	39	15	g	g	PROPN
ejpam-6063	39	16	and	and	CCONJ
ejpam-6063	39	17	|v	|v	PROPN
ejpam-6063	39	18	(	(	PUNCT
ejpam-6063	39	19	g)|	g)|	NOUN
ejpam-6063	39	20	copies	copy	NOUN
ejpam-6063	39	21	of	of	ADP
ejpam-6063	39	22	h	h	NOUN
ejpam-6063	39	23	,	,	PUNCT
ejpam-6063	39	24	and	and	CCONJ
ejpam-6063	39	25	then	then	ADV
ejpam-6063	39	26	joining	join	VERB
ejpam-6063	39	27	the	the	DET
ejpam-6063	39	28	ith	ith	PROPN
ejpam-6063	39	29	vertex	vertex	NOUN
ejpam-6063	39	30	of	of	ADP
ejpam-6063	39	31	g	g	NOUN
ejpam-6063	39	32	to	to	ADP
ejpam-6063	39	33	every	every	DET
ejpam-6063	39	34	vertex	vertex	NOUN
ejpam-6063	39	35	of	of	ADP
ejpam-6063	39	36	the	the	DET
ejpam-6063	39	37	ith	ith	PROPN
ejpam-6063	39	38	copy	copy	NOUN
ejpam-6063	39	39	of	of	ADP
ejpam-6063	39	40	h.	h.	PROPN
ejpam-6063	39	41	we	we	PRON
ejpam-6063	39	42	denote	denote	VERB
ejpam-6063	39	43	by	by	ADP
ejpam-6063	39	44	hv	hv	PROPN
ejpam-6063	40	1	the	the	DET
ejpam-6063	40	2	copy	copy	NOUN
ejpam-6063	40	3	of	of	ADP
ejpam-6063	40	4	h	h	NOUN
ejpam-6063	40	5	in	in	ADP
ejpam-6063	40	6	g	g	PROPN
ejpam-6063	40	7	◦	◦	NOUN
ejpam-6063	40	8	h	h	NOUN
ejpam-6063	40	9	corresponding	correspond	VERB
ejpam-6063	40	10	to	to	ADP
ejpam-6063	40	11	the	the	DET
ejpam-6063	40	12	vertex	vertex	NOUN
ejpam-6063	40	13	v	v	ADP
ejpam-6063	40	14	∈	∈	PROPN
ejpam-6063	40	15	g	g	NOUN
ejpam-6063	40	16	and	and	CCONJ
ejpam-6063	40	17	write	write	VERB
ejpam-6063	40	18	v+hv	v+hv	PROPN
ejpam-6063	40	19	for	for	ADP
ejpam-6063	40	20	⟨{v}⟩+hv	⟨{v}⟩+hv	PROPN
ejpam-6063	40	21	.	.	PUNCT
ejpam-6063	41	1	a	a	DET
ejpam-6063	41	2	subset	subset	NOUN
ejpam-6063	41	3	a	a	PRON
ejpam-6063	41	4	of	of	ADP
ejpam-6063	41	5	v	v	NOUN
ejpam-6063	41	6	(	(	PUNCT
ejpam-6063	41	7	g	g	NOUN
ejpam-6063	41	8	)	)	PUNCT
ejpam-6063	41	9	is	be	AUX
ejpam-6063	41	10	independent	independent	ADJ
ejpam-6063	41	11	if	if	SCONJ
ejpam-6063	41	12	for	for	SCONJ
ejpam-6063	41	13	every	every	DET
ejpam-6063	41	14	pair	pair	NOUN
ejpam-6063	41	15	of	of	ADP
ejpam-6063	41	16	distinct	distinct	ADJ
ejpam-6063	41	17	vertices	vertex	NOUN
ejpam-6063	41	18	in	in	ADP
ejpam-6063	41	19	a	a	DET
ejpam-6063	41	20	do	do	AUX
ejpam-6063	41	21	not	not	PART
ejpam-6063	41	22	form	form	VERB
ejpam-6063	41	23	an	an	DET
ejpam-6063	41	24	edge	edge	NOUN
ejpam-6063	41	25	.	.	PUNCT
ejpam-6063	42	1	the	the	DET
ejpam-6063	42	2	maximum	maximum	ADJ
ejpam-6063	42	3	cardinality	cardinality	NOUN
ejpam-6063	42	4	of	of	ADP
ejpam-6063	42	5	an	an	DET
ejpam-6063	42	6	independent	independent	ADJ
ejpam-6063	42	7	set	set	NOUN
ejpam-6063	42	8	in	in	ADP
ejpam-6063	42	9	g	g	NOUN
ejpam-6063	42	10	,	,	PUNCT
ejpam-6063	42	11	denoted	denote	VERB
ejpam-6063	42	12	by	by	ADP
ejpam-6063	42	13	α(g	α(g	NOUN
ejpam-6063	42	14	)	)	PUNCT
ejpam-6063	42	15	,	,	PUNCT
ejpam-6063	42	16	is	be	AUX
ejpam-6063	42	17	called	call	VERB
ejpam-6063	42	18	the	the	DET
ejpam-6063	42	19	independence	independence	NOUN
ejpam-6063	42	20	number	number	NOUN
ejpam-6063	42	21	of	of	ADP
ejpam-6063	42	22	g.	g.	PROPN
ejpam-6063	42	23	any	any	DET
ejpam-6063	42	24	independent	independent	ADJ
ejpam-6063	42	25	set	set	NOUN
ejpam-6063	42	26	with	with	ADP
ejpam-6063	42	27	cardinality	cardinality	NOUN
ejpam-6063	42	28	equal	equal	ADJ
ejpam-6063	42	29	to	to	ADP
ejpam-6063	42	30	α(g	α(g	NUM
ejpam-6063	42	31	)	)	PUNCT
ejpam-6063	42	32	is	be	AUX
ejpam-6063	42	33	called	call	VERB
ejpam-6063	42	34	an	an	DET
ejpam-6063	42	35	α	α	NOUN
ejpam-6063	42	36	-	-	PUNCT
ejpam-6063	42	37	set	set	VERB
ejpam-6063	42	38	in	in	ADP
ejpam-6063	42	39	g.	g.	PROPN
ejpam-6063	42	40	a	a	DET
ejpam-6063	42	41	set	set	NOUN
ejpam-6063	42	42	s	s	PROPN
ejpam-6063	42	43	⊆	⊆	NUM
ejpam-6063	42	44	v	v	NOUN
ejpam-6063	42	45	(	(	PUNCT
ejpam-6063	42	46	g	g	NOUN
ejpam-6063	42	47	)	)	PUNCT
ejpam-6063	42	48	is	be	AUX
ejpam-6063	42	49	a	a	DET
ejpam-6063	42	50	dominating	dominating	NOUN
ejpam-6063	42	51	set	set	VERB
ejpam-6063	42	52	in	in	ADP
ejpam-6063	42	53	g	g	PROPN
ejpam-6063	42	54	if	if	SCONJ
ejpam-6063	42	55	ng[s	ng[	NOUN
ejpam-6063	42	56	]	]	PUNCT
ejpam-6063	42	57	=	=	SYM
ejpam-6063	42	58	v	v	NOUN
ejpam-6063	42	59	(	(	PUNCT
ejpam-6063	42	60	g	g	NOUN
ejpam-6063	42	61	)	)	PUNCT
ejpam-6063	42	62	.	.	PUNCT
ejpam-6063	43	1	it	it	PRON
ejpam-6063	43	2	is	be	AUX
ejpam-6063	43	3	a	a	DET
ejpam-6063	43	4	2	2	NUM
ejpam-6063	43	5	-	-	PUNCT
ejpam-6063	43	6	dominating	dominating	NOUN
ejpam-6063	43	7	set	set	NOUN
ejpam-6063	43	8	if	if	SCONJ
ejpam-6063	43	9	for	for	ADP
ejpam-6063	43	10	every	every	PRON
ejpam-6063	43	11	v	v	NUM
ejpam-6063	43	12	∈	∈	NOUN
ejpam-6063	43	13	v	v	NOUN
ejpam-6063	43	14	(	(	PUNCT
ejpam-6063	43	15	g	g	NOUN
ejpam-6063	43	16	)	)	PUNCT
ejpam-6063	43	17	\	\	PROPN
ejpam-6063	44	1	s	s	X
ejpam-6063	44	2	,	,	PUNCT
ejpam-6063	44	3	|ng(v	|ng(v	ADJ
ejpam-6063	44	4	)	)	PUNCT
ejpam-6063	44	5	∩	∩	NOUN
ejpam-6063	44	6	s|	s|	VERB
ejpam-6063	44	7	≥	≥	NOUN
ejpam-6063	44	8	2	2	NUM
ejpam-6063	44	9	,	,	PUNCT
ejpam-6063	44	10	i.e.	i.e.	X
ejpam-6063	44	11	,	,	PUNCT
ejpam-6063	44	12	v	v	NOUN
ejpam-6063	44	13	has	have	VERB
ejpam-6063	44	14	at	at	ADV
ejpam-6063	44	15	least	least	ADV
ejpam-6063	44	16	two	two	NUM
ejpam-6063	44	17	neighbors	neighbor	NOUN
ejpam-6063	44	18	in	in	ADP
ejpam-6063	44	19	s.	s.	PROPN
ejpam-6063	44	20	the	the	DET
ejpam-6063	44	21	domination	domination	NOUN
ejpam-6063	44	22	number	number	NOUN
ejpam-6063	44	23	(	(	PUNCT
ejpam-6063	44	24	2	2	NUM
ejpam-6063	44	25	-	-	PUNCT
ejpam-6063	44	26	domination	domination	NOUN
ejpam-6063	44	27	number	number	NOUN
ejpam-6063	44	28	)	)	PUNCT
ejpam-6063	44	29	of	of	ADP
ejpam-6063	44	30	g	g	NOUN
ejpam-6063	44	31	,	,	PUNCT
ejpam-6063	44	32	denoted	denote	VERB
ejpam-6063	44	33	γ(g	γ(g	PROPN
ejpam-6063	44	34	)	)	PUNCT
ejpam-6063	44	35	(	(	PUNCT
ejpam-6063	44	36	resp	resp	NOUN
ejpam-6063	44	37	.	.	PUNCT
ejpam-6063	45	1	γ2(g	γ2(g	VERB
ejpam-6063	45	2	)	)	PUNCT
ejpam-6063	45	3	)	)	PUNCT
ejpam-6063	45	4	,	,	PUNCT
ejpam-6063	45	5	is	be	AUX
ejpam-6063	45	6	the	the	DET
ejpam-6063	45	7	minimum	minimum	ADJ
ejpam-6063	45	8	cardinality	cardinality	NOUN
ejpam-6063	45	9	of	of	ADP
ejpam-6063	45	10	a	a	DET
ejpam-6063	45	11	dominating	dominating	NOUN
ejpam-6063	45	12	(	(	PUNCT
ejpam-6063	45	13	resp	resp	NOUN
ejpam-6063	45	14	.	.	PUNCT
ejpam-6063	46	1	2	2	NUM
ejpam-6063	46	2	-	-	PUNCT
ejpam-6063	46	3	dominating	dominating	NOUN
ejpam-6063	46	4	)	)	PUNCT
ejpam-6063	46	5	set	set	VERB
ejpam-6063	46	6	in	in	ADP
ejpam-6063	46	7	g.	g.	PROPN
ejpam-6063	46	8	any	any	DET
ejpam-6063	46	9	dominating	dominating	NOUN
ejpam-6063	46	10	set	set	NOUN
ejpam-6063	46	11	(	(	PUNCT
ejpam-6063	46	12	2	2	NUM
ejpam-6063	46	13	-	-	PUNCT
ejpam-6063	46	14	dominating	dominating	NOUN
ejpam-6063	46	15	set	set	NOUN
ejpam-6063	46	16	)	)	PUNCT
ejpam-6063	46	17	with	with	ADP
ejpam-6063	46	18	cardinality	cardinality	PROPN
ejpam-6063	46	19	γ(g	γ(g	PROPN
ejpam-6063	46	20	)	)	PUNCT
ejpam-6063	46	21	(	(	PUNCT
ejpam-6063	46	22	resp	resp	NOUN
ejpam-6063	46	23	.	.	PUNCT
ejpam-6063	47	1	γ2(g	γ2(g	VERB
ejpam-6063	47	2	)	)	PUNCT
ejpam-6063	47	3	)	)	PUNCT
ejpam-6063	47	4	is	be	AUX
ejpam-6063	47	5	called	call	VERB
ejpam-6063	47	6	a	a	DET
ejpam-6063	47	7	γ	γ	NOUN
ejpam-6063	47	8	-	-	PUNCT
ejpam-6063	47	9	set	set	ADJ
ejpam-6063	47	10	(	(	PUNCT
ejpam-6063	47	11	resp	resp	NOUN
ejpam-6063	47	12	.	.	PUNCT
ejpam-6063	48	1	γ2	γ2	NOUN
ejpam-6063	48	2	-	-	PUNCT
ejpam-6063	48	3	set	set	NOUN
ejpam-6063	48	4	)	)	PUNCT
ejpam-6063	48	5	.	.	PUNCT
ejpam-6063	49	1	a	a	DET
ejpam-6063	49	2	subset	subset	NOUN
ejpam-6063	49	3	s	s	NOUN
ejpam-6063	49	4	of	of	ADP
ejpam-6063	49	5	vertices	vertex	NOUN
ejpam-6063	49	6	of	of	ADP
ejpam-6063	49	7	a	a	DET
ejpam-6063	49	8	graph	graph	NOUN
ejpam-6063	49	9	g	g	NOUN
ejpam-6063	49	10	is	be	AUX
ejpam-6063	49	11	called	call	VERB
ejpam-6063	49	12	a	a	DET
ejpam-6063	49	13	vertex	vertex	NOUN
ejpam-6063	49	14	cover	cover	NOUN
ejpam-6063	49	15	of	of	ADP
ejpam-6063	49	16	g	g	PROPN
ejpam-6063	49	17	if	if	SCONJ
ejpam-6063	49	18	for	for	ADP
ejpam-6063	49	19	every	every	DET
ejpam-6063	49	20	edge	edge	NOUN
ejpam-6063	49	21	e	e	NOUN
ejpam-6063	49	22	=	=	PUNCT
ejpam-6063	49	23	uv	uv	PROPN
ejpam-6063	49	24	∈	∈	PROPN
ejpam-6063	49	25	e(g	e(g	PROPN
ejpam-6063	49	26	)	)	PUNCT
ejpam-6063	49	27	,	,	PUNCT
ejpam-6063	49	28	either	either	CCONJ
ejpam-6063	49	29	u	u	PROPN
ejpam-6063	49	30	∈	∈	PROPN
ejpam-6063	49	31	s	s	X
ejpam-6063	49	32	or	or	CCONJ
ejpam-6063	49	33	v	v	ADP
ejpam-6063	49	34	∈	∈	PROPN
ejpam-6063	49	35	s.	s.	PROPN
ejpam-6063	49	36	the	the	DET
ejpam-6063	49	37	minimum	minimum	ADJ
ejpam-6063	49	38	cardinality	cardinality	NOUN
ejpam-6063	49	39	of	of	ADP
ejpam-6063	49	40	a	a	DET
ejpam-6063	49	41	vertex	vertex	NOUN
ejpam-6063	49	42	cover	cover	NOUN
ejpam-6063	49	43	of	of	ADP
ejpam-6063	49	44	g	g	PROPN
ejpam-6063	49	45	is	be	AUX
ejpam-6063	49	46	the	the	DET
ejpam-6063	49	47	vertex	vertex	NOUN
ejpam-6063	49	48	cover	cover	NOUN
ejpam-6063	49	49	number	number	NOUN
ejpam-6063	49	50	of	of	ADP
ejpam-6063	49	51	g	g	NOUN
ejpam-6063	49	52	and	and	CCONJ
ejpam-6063	49	53	is	be	AUX
ejpam-6063	49	54	denoted	denote	VERB
ejpam-6063	49	55	by	by	ADP
ejpam-6063	49	56	β(g	β(g	PROPN
ejpam-6063	49	57	)	)	PUNCT
ejpam-6063	49	58	.	.	PUNCT
ejpam-6063	50	1	any	any	DET
ejpam-6063	50	2	vertex	vertex	NOUN
ejpam-6063	50	3	cover	cover	NOUN
ejpam-6063	50	4	of	of	ADP
ejpam-6063	50	5	g	g	NOUN
ejpam-6063	50	6	with	with	ADP
ejpam-6063	50	7	cardinality	cardinality	PROPN
ejpam-6063	50	8	β(g	β(g	PROPN
ejpam-6063	50	9	)	)	PUNCT
ejpam-6063	50	10	is	be	AUX
ejpam-6063	50	11	called	call	VERB
ejpam-6063	50	12	a	a	DET
ejpam-6063	50	13	β	β	NOUN
ejpam-6063	50	14	-	-	NOUN
ejpam-6063	50	15	set	set	NOUN
ejpam-6063	50	16	.	.	PUNCT
ejpam-6063	51	1	a	a	DET
ejpam-6063	51	2	set	set	NOUN
ejpam-6063	51	3	s	s	NOUN
ejpam-6063	51	4	⊆	⊆	NUM
ejpam-6063	51	5	v	v	NOUN
ejpam-6063	51	6	(	(	PUNCT
ejpam-6063	51	7	g	g	NOUN
ejpam-6063	51	8	)	)	PUNCT
ejpam-6063	51	9	is	be	AUX
ejpam-6063	51	10	a	a	DET
ejpam-6063	51	11	2	2	NUM
ejpam-6063	51	12	-	-	PUNCT
ejpam-6063	51	13	vertex	vertex	NOUN
ejpam-6063	51	14	cover	cover	NOUN
ejpam-6063	51	15	(	(	PUNCT
ejpam-6063	51	16	or	or	CCONJ
ejpam-6063	51	17	covering	cover	VERB
ejpam-6063	51	18	)	)	PUNCT
ejpam-6063	51	19	of	of	ADP
ejpam-6063	51	20	g	g	PROPN
ejpam-6063	51	21	if	if	SCONJ
ejpam-6063	51	22	s	s	VERB
ejpam-6063	51	23	is	be	AUX
ejpam-6063	51	24	both	both	PRON
ejpam-6063	51	25	a	a	DET
ejpam-6063	51	26	vertex	vertex	NOUN
ejpam-6063	51	27	cover	cover	NOUN
ejpam-6063	51	28	and	and	CCONJ
ejpam-6063	51	29	a	a	DET
ejpam-6063	51	30	2	2	NUM
ejpam-6063	51	31	-	-	PUNCT
ejpam-6063	51	32	dominating	dominating	NOUN
ejpam-6063	51	33	set	set	NOUN
ejpam-6063	51	34	in	in	ADP
ejpam-6063	51	35	g.	g.	PROPN
ejpam-6063	51	36	the	the	DET
ejpam-6063	51	37	2	2	NUM
ejpam-6063	51	38	-	-	PUNCT
ejpam-6063	51	39	vertex	vertex	NOUN
ejpam-6063	51	40	covering	covering	NOUN
ejpam-6063	51	41	number	number	NOUN
ejpam-6063	51	42	of	of	ADP
ejpam-6063	51	43	g	g	NOUN
ejpam-6063	51	44	,	,	PUNCT
ejpam-6063	51	45	denoted	denote	VERB
ejpam-6063	51	46	by	by	ADP
ejpam-6063	51	47	β2(g	β2(g	NOUN
ejpam-6063	51	48	)	)	PUNCT
ejpam-6063	51	49	,	,	PUNCT
ejpam-6063	51	50	is	be	AUX
ejpam-6063	51	51	the	the	DET
ejpam-6063	51	52	minimum	minimum	ADJ
ejpam-6063	51	53	cardinality	cardinality	NOUN
ejpam-6063	51	54	of	of	ADP
ejpam-6063	51	55	a	a	DET
ejpam-6063	51	56	2	2	NUM
ejpam-6063	51	57	-	-	PUNCT
ejpam-6063	51	58	vertex	vertex	NOUN
ejpam-6063	51	59	covering	covering	NOUN
ejpam-6063	51	60	of	of	ADP
ejpam-6063	51	61	g.	g.	PROPN
ejpam-6063	51	62	any	any	DET
ejpam-6063	51	63	2	2	NUM
ejpam-6063	51	64	-	-	PUNCT
ejpam-6063	51	65	vertex	vertex	NOUN
ejpam-6063	51	66	covering	covering	NOUN
ejpam-6063	51	67	of	of	ADP
ejpam-6063	51	68	g	g	NOUN
ejpam-6063	51	69	with	with	ADP
ejpam-6063	51	70	cardinality	cardinality	NOUN
ejpam-6063	51	71	β2(g	β2(g	NUM
ejpam-6063	51	72	)	)	PUNCT
ejpam-6063	51	73	is	be	AUX
ejpam-6063	51	74	called	call	VERB
ejpam-6063	51	75	a	a	DET
ejpam-6063	51	76	β2	β2	NOUN
ejpam-6063	51	77	-	-	PUNCT
ejpam-6063	51	78	set	set	NOUN
ejpam-6063	51	79	.	.	PUNCT
ejpam-6063	52	1	j.	j.	PROPN
ejpam-6063	52	2	hassan	hassan	PROPN
ejpam-6063	52	3	et	et	PROPN
ejpam-6063	52	4	.	.	PUNCT
ejpam-6063	53	1	al	al	PROPN
ejpam-6063	53	2	/	/	PUNCT
ejpam-6063	53	3	eur	eur	PROPN
ejpam-6063	53	4	.	.	PUNCT
ejpam-6063	54	1	j.	j.	PROPN
ejpam-6063	54	2	pure	pure	PROPN
ejpam-6063	54	3	appl	appl	PROPN
ejpam-6063	54	4	.	.	PROPN
ejpam-6063	54	5	math	math	PROPN
ejpam-6063	54	6	,	,	PUNCT
ejpam-6063	54	7	18	18	NUM
ejpam-6063	54	8	(	(	PUNCT
ejpam-6063	54	9	2	2	NUM
ejpam-6063	54	10	)	)	PUNCT
ejpam-6063	54	11	(	(	PUNCT
ejpam-6063	54	12	2025	2025	NUM
ejpam-6063	54	13	)	)	PUNCT
ejpam-6063	54	14	,	,	PUNCT
ejpam-6063	54	15	6063	6063	NUM
ejpam-6063	54	16	3	3	NUM
ejpam-6063	54	17	of	of	ADP
ejpam-6063	54	18	11	11	NUM
ejpam-6063	54	19	readers	reader	NOUN
ejpam-6063	54	20	are	be	AUX
ejpam-6063	54	21	referred	refer	VERB
ejpam-6063	54	22	to	to	ADP
ejpam-6063	54	23	[	[	X
ejpam-6063	54	24	19	19	NUM
ejpam-6063	54	25	]	]	PUNCT
ejpam-6063	54	26	for	for	ADP
ejpam-6063	54	27	other	other	ADJ
ejpam-6063	54	28	basic	basic	ADJ
ejpam-6063	54	29	definitions	definition	NOUN
ejpam-6063	54	30	that	that	PRON
ejpam-6063	54	31	are	be	AUX
ejpam-6063	54	32	not	not	PART
ejpam-6063	54	33	given	give	VERB
ejpam-6063	54	34	here	here	ADV
ejpam-6063	54	35	.	.	PUNCT
ejpam-6063	55	1	3	3	X
ejpam-6063	55	2	.	.	X
ejpam-6063	55	3	main	main	ADJ
ejpam-6063	55	4	results	result	NOUN
ejpam-6063	55	5	theorem	theorem	VERB
ejpam-6063	55	6	1	1	NUM
ejpam-6063	55	7	.	.	PUNCT
ejpam-6063	56	1	let	let	VERB
ejpam-6063	56	2	g1	g1	PROPN
ejpam-6063	56	3	,	,	PUNCT
ejpam-6063	56	4	g2	g2	PROPN
ejpam-6063	56	5	,	,	PUNCT
ejpam-6063	56	6	.	.	PUNCT
ejpam-6063	56	7	.	.	PUNCT
ejpam-6063	57	1	.	.	PUNCT
ejpam-6063	58	1	,	,	PUNCT
ejpam-6063	58	2	gk	gk	PROPN
ejpam-6063	58	3	be	be	AUX
ejpam-6063	58	4	the	the	DET
ejpam-6063	58	5	components	component	NOUN
ejpam-6063	58	6	of	of	ADP
ejpam-6063	58	7	g.	g.	PROPN
ejpam-6063	58	8	then	then	ADV
ejpam-6063	58	9	β2(g	β2(g	NUM
ejpam-6063	58	10	)	)	PUNCT
ejpam-6063	58	11	=	=	SYM
ejpam-6063	59	1	∑k	∑k	PROPN
ejpam-6063	59	2	j=1	j=1	NOUN
ejpam-6063	59	3	β2(gj	β2(gj	NUM
ejpam-6063	59	4	)	)	PUNCT
ejpam-6063	59	5	.	.	PUNCT
ejpam-6063	60	1	proof	proof	NOUN
ejpam-6063	60	2	.	.	PUNCT
ejpam-6063	61	1	let	let	VERB
ejpam-6063	61	2	s	s	PRON
ejpam-6063	61	3	be	be	AUX
ejpam-6063	61	4	a	a	DET
ejpam-6063	61	5	β2	β2	NOUN
ejpam-6063	61	6	-	-	PUNCT
ejpam-6063	61	7	set	set	NOUN
ejpam-6063	61	8	in	in	ADP
ejpam-6063	61	9	g.	g.	PROPN
ejpam-6063	61	10	for	for	ADP
ejpam-6063	61	11	each	each	DET
ejpam-6063	61	12	j	j	PROPN
ejpam-6063	61	13	∈	∈	PROPN
ejpam-6063	62	1	[	[	X
ejpam-6063	62	2	k	k	X
ejpam-6063	62	3	]	]	X
ejpam-6063	62	4	=	=	X
ejpam-6063	62	5	{	{	PUNCT
ejpam-6063	62	6	1	1	NUM
ejpam-6063	62	7	,	,	PUNCT
ejpam-6063	62	8	2	2	NUM
ejpam-6063	62	9	,	,	PUNCT
ejpam-6063	62	10	·	·	PUNCT
ejpam-6063	62	11	·	·	PUNCT
ejpam-6063	62	12	·	·	PUNCT
ejpam-6063	62	13	,	,	PUNCT
ejpam-6063	62	14	k	k	X
ejpam-6063	62	15	}	}	PUNCT
ejpam-6063	62	16	,	,	PUNCT
ejpam-6063	62	17	let	let	VERB
ejpam-6063	62	18	sj	sj	INTJ
ejpam-6063	62	19	=	=	NOUN
ejpam-6063	62	20	s	s	PART
ejpam-6063	62	21	∩	∩	ADJ
ejpam-6063	62	22	v	v	NOUN
ejpam-6063	62	23	(	(	PUNCT
ejpam-6063	62	24	gj	gj	NOUN
ejpam-6063	62	25	)	)	PUNCT
ejpam-6063	62	26	.	.	PUNCT
ejpam-6063	63	1	then	then	ADV
ejpam-6063	63	2	s	s	VERB
ejpam-6063	63	3	=	=	SYM
ejpam-6063	63	4	∪k	∪k	PROPN
ejpam-6063	63	5	j=1sj	j=1sj	PROPN
ejpam-6063	63	6	.	.	PUNCT
ejpam-6063	64	1	since	since	SCONJ
ejpam-6063	64	2	s	s	PROPN
ejpam-6063	64	3	is	be	AUX
ejpam-6063	64	4	a	a	DET
ejpam-6063	64	5	vertex	vertex	NOUN
ejpam-6063	64	6	cover	cover	NOUN
ejpam-6063	64	7	of	of	ADP
ejpam-6063	64	8	g	g	NOUN
ejpam-6063	64	9	,	,	PUNCT
ejpam-6063	64	10	it	it	PRON
ejpam-6063	64	11	follows	follow	VERB
ejpam-6063	64	12	that	that	SCONJ
ejpam-6063	64	13	sj	sj	PROPN
ejpam-6063	64	14	is	be	AUX
ejpam-6063	64	15	a	a	DET
ejpam-6063	64	16	vertex	vertex	NOUN
ejpam-6063	64	17	cover	cover	NOUN
ejpam-6063	64	18	of	of	ADP
ejpam-6063	64	19	gj	gj	NOUN
ejpam-6063	64	20	for	for	ADP
ejpam-6063	64	21	each	each	DET
ejpam-6063	64	22	j	j	PROPN
ejpam-6063	64	23	∈	∈	PROPN
ejpam-6063	65	1	[	[	X
ejpam-6063	65	2	k	k	X
ejpam-6063	65	3	]	]	X
ejpam-6063	65	4	.	.	PUNCT
ejpam-6063	66	1	now	now	ADV
ejpam-6063	66	2	,	,	PUNCT
ejpam-6063	66	3	let	let	VERB
ejpam-6063	66	4	j	j	PROPN
ejpam-6063	66	5	∈	∈	PROPN
ejpam-6063	67	1	[	[	X
ejpam-6063	67	2	k	k	X
ejpam-6063	67	3	]	]	PUNCT
ejpam-6063	67	4	and	and	CCONJ
ejpam-6063	67	5	let	let	VERB
ejpam-6063	67	6	v	v	NUM
ejpam-6063	67	7	∈	∈	PROPN
ejpam-6063	67	8	v	v	NOUN
ejpam-6063	67	9	(	(	PUNCT
ejpam-6063	67	10	gj	gj	NOUN
ejpam-6063	67	11	)	)	PUNCT
ejpam-6063	67	12	\	\	PUNCT
ejpam-6063	67	13	sj	sj	INTJ
ejpam-6063	67	14	.	.	PUNCT
ejpam-6063	68	1	since	since	SCONJ
ejpam-6063	68	2	s	s	PROPN
ejpam-6063	68	3	is	be	AUX
ejpam-6063	68	4	a	a	DET
ejpam-6063	68	5	2	2	NUM
ejpam-6063	68	6	-	-	PUNCT
ejpam-6063	68	7	dominating	dominating	NOUN
ejpam-6063	68	8	set	set	NOUN
ejpam-6063	68	9	in	in	ADP
ejpam-6063	68	10	g	g	PROPN
ejpam-6063	68	11	,	,	PUNCT
ejpam-6063	68	12	we	we	PRON
ejpam-6063	68	13	have	have	VERB
ejpam-6063	68	14	|ng(v	|ng(v	VERB
ejpam-6063	68	15	)	)	PUNCT
ejpam-6063	68	16	∩	∩	NOUN
ejpam-6063	68	17	s|	s|	VERB
ejpam-6063	68	18	≥	≥	NOUN
ejpam-6063	68	19	2	2	X
ejpam-6063	68	20	.	.	PUNCT
ejpam-6063	69	1	it	it	PRON
ejpam-6063	69	2	follows	follow	VERB
ejpam-6063	69	3	that	that	SCONJ
ejpam-6063	69	4	|ngj	|ngj	NOUN
ejpam-6063	69	5	(	(	PUNCT
ejpam-6063	69	6	v	v	NOUN
ejpam-6063	69	7	)	)	PUNCT
ejpam-6063	69	8	∩	∩	NOUN
ejpam-6063	69	9	sj	sj	X
ejpam-6063	69	10	|	|	ADV
ejpam-6063	69	11	≥	≥	NOUN
ejpam-6063	69	12	2	2	NUM
ejpam-6063	69	13	.	.	PUNCT
ejpam-6063	70	1	therefore	therefore	ADV
ejpam-6063	70	2	,	,	PUNCT
ejpam-6063	70	3	sj	sj	PROPN
ejpam-6063	70	4	is	be	AUX
ejpam-6063	70	5	a	a	DET
ejpam-6063	70	6	2	2	NUM
ejpam-6063	70	7	-	-	PUNCT
ejpam-6063	70	8	vertex	vertex	NOUN
ejpam-6063	70	9	cover	cover	NOUN
ejpam-6063	70	10	of	of	ADP
ejpam-6063	70	11	gj	gj	NOUN
ejpam-6063	70	12	for	for	ADP
ejpam-6063	70	13	each	each	DET
ejpam-6063	70	14	j	j	PROPN
ejpam-6063	70	15	∈	∈	PROPN
ejpam-6063	71	1	[	[	X
ejpam-6063	71	2	k	k	X
ejpam-6063	71	3	]	]	X
ejpam-6063	71	4	.	.	PUNCT
ejpam-6063	72	1	thus	thus	ADV
ejpam-6063	72	2	,	,	PUNCT
ejpam-6063	72	3	β2(g	β2(g	NUM
ejpam-6063	72	4	)	)	PUNCT
ejpam-6063	72	5	=	=	SYM
ejpam-6063	72	6	|s|	|s|	PROPN
ejpam-6063	72	7	=	=	SYM
ejpam-6063	73	1	|	|	NOUN
ejpam-6063	73	2	∪k	∪k	NUM
ejpam-6063	73	3	j=1	j=1	NOUN
ejpam-6063	73	4	sj	sj	INTJ
ejpam-6063	73	5	|	|	ADV
ejpam-6063	73	6	=	=	SYM
ejpam-6063	73	7	k∑	k∑	PROPN
ejpam-6063	74	1	j=1	j=1	NOUN
ejpam-6063	74	2	|sj	|sj	X
ejpam-6063	74	3	|	|	ADV
ejpam-6063	74	4	≥	≥	NOUN
ejpam-6063	74	5	k∑	k∑	VERB
ejpam-6063	74	6	j=1	j=1	PROPN
ejpam-6063	74	7	β2(gj	β2(gj	NUM
ejpam-6063	74	8	)	)	PUNCT
ejpam-6063	74	9	.	.	PUNCT
ejpam-6063	75	1	for	for	ADP
ejpam-6063	75	2	each	each	DET
ejpam-6063	75	3	j	j	PROPN
ejpam-6063	75	4	∈	∈	PROPN
ejpam-6063	75	5	[	[	X
ejpam-6063	75	6	k	k	X
ejpam-6063	75	7	]	]	X
ejpam-6063	75	8	,	,	PUNCT
ejpam-6063	75	9	let	let	VERB
ejpam-6063	75	10	dj	dj	PRON
ejpam-6063	75	11	be	be	AUX
ejpam-6063	75	12	a	a	DET
ejpam-6063	75	13	β2	β2	NOUN
ejpam-6063	75	14	-	-	PUNCT
ejpam-6063	75	15	set	set	NOUN
ejpam-6063	75	16	in	in	ADP
ejpam-6063	75	17	gj	gj	NOUN
ejpam-6063	75	18	.	.	PUNCT
ejpam-6063	76	1	clearly	clearly	ADV
ejpam-6063	76	2	,	,	PUNCT
ejpam-6063	76	3	d	d	PROPN
ejpam-6063	76	4	=	=	PUNCT
ejpam-6063	76	5	∪k	∪k	NUM
ejpam-6063	76	6	j=1dj	j=1dj	X
ejpam-6063	76	7	is	be	AUX
ejpam-6063	76	8	a	a	DET
ejpam-6063	76	9	2	2	NUM
ejpam-6063	76	10	-	-	PUNCT
ejpam-6063	76	11	vertex	vertex	NOUN
ejpam-6063	76	12	cover	cover	NOUN
ejpam-6063	76	13	of	of	ADP
ejpam-6063	76	14	g.	g.	PROPN
ejpam-6063	76	15	hence	hence	ADV
ejpam-6063	76	16	,	,	PUNCT
ejpam-6063	76	17	β2(g	β2(g	SYM
ejpam-6063	76	18	)	)	PUNCT
ejpam-6063	76	19	≤	≤	PUNCT
ejpam-6063	76	20	|d|	|d|	PROPN
ejpam-6063	76	21	=	=	PUNCT
ejpam-6063	77	1	|	|	ADV
ejpam-6063	77	2	∪k	∪k	NUM
ejpam-6063	77	3	j=1	j=1	NOUN
ejpam-6063	77	4	dj	dj	NOUN
ejpam-6063	77	5	|	|	NOUN
ejpam-6063	77	6	=	=	SYM
ejpam-6063	77	7	k∑	k∑	PROPN
ejpam-6063	78	1	j=1	j=1	NOUN
ejpam-6063	78	2	|dj	|dj	PUNCT
ejpam-6063	78	3	|	|	NOUN
ejpam-6063	78	4	=	=	SYM
ejpam-6063	78	5	k∑	k∑	NOUN
ejpam-6063	78	6	j=1	j=1	PROPN
ejpam-6063	78	7	β2(gj	β2(gj	NUM
ejpam-6063	78	8	)	)	PUNCT
ejpam-6063	78	9	.	.	PUNCT
ejpam-6063	79	1	this	this	PRON
ejpam-6063	79	2	proves	prove	VERB
ejpam-6063	79	3	the	the	DET
ejpam-6063	79	4	assertion	assertion	NOUN
ejpam-6063	79	5	.	.	PUNCT
ejpam-6063	80	1	theorem	theorem	NOUN
ejpam-6063	80	2	2	2	NUM
ejpam-6063	80	3	.	.	PUNCT
ejpam-6063	81	1	let	let	VERB
ejpam-6063	81	2	g	g	PRON
ejpam-6063	81	3	be	be	AUX
ejpam-6063	81	4	a	a	DET
ejpam-6063	81	5	graph	graph	NOUN
ejpam-6063	81	6	on	on	ADP
ejpam-6063	81	7	n	n	DET
ejpam-6063	81	8	vertices	vertex	NOUN
ejpam-6063	81	9	.	.	PUNCT
ejpam-6063	82	1	then	then	ADV
ejpam-6063	82	2	max{γ2(g	max{γ2(g	PROPN
ejpam-6063	82	3	)	)	PUNCT
ejpam-6063	82	4	,	,	PUNCT
ejpam-6063	82	5	β(g	β(g	PROPN
ejpam-6063	82	6	)	)	PUNCT
ejpam-6063	82	7	}	}	PUNCT
ejpam-6063	82	8	≤	≤	NOUN
ejpam-6063	82	9	β2(g	β2(g	NUM
ejpam-6063	82	10	)	)	PUNCT
ejpam-6063	82	11	≤	≤	NOUN
ejpam-6063	82	12	n.	n.	NOUN
ejpam-6063	82	13	moreover	moreover	ADV
ejpam-6063	82	14	,	,	PUNCT
ejpam-6063	82	15	each	each	PRON
ejpam-6063	82	16	of	of	ADP
ejpam-6063	82	17	the	the	DET
ejpam-6063	82	18	following	follow	VERB
ejpam-6063	82	19	holds	hold	VERB
ejpam-6063	82	20	:	:	PUNCT
ejpam-6063	82	21	(	(	PUNCT
ejpam-6063	82	22	i	i	NOUN
ejpam-6063	82	23	)	)	PUNCT
ejpam-6063	82	24	if	if	SCONJ
ejpam-6063	82	25	g	g	PROPN
ejpam-6063	82	26	has	have	VERB
ejpam-6063	82	27	vertex	vertex	NOUN
ejpam-6063	82	28	v	v	NOUN
ejpam-6063	82	29	with	with	ADP
ejpam-6063	82	30	|ng(v)|	|ng(v)|	NOUN
ejpam-6063	82	31	≥	≥	NOUN
ejpam-6063	82	32	2	2	NUM
ejpam-6063	82	33	,	,	PUNCT
ejpam-6063	82	34	then	then	ADV
ejpam-6063	82	35	β2(g	β2(g	NUM
ejpam-6063	82	36	)	)	PUNCT
ejpam-6063	82	37	≤	≤	NOUN
ejpam-6063	83	1	n	n	CCONJ
ejpam-6063	83	2	−	−	PROPN
ejpam-6063	83	3	1	1	NUM
ejpam-6063	83	4	.	.	PUNCT
ejpam-6063	84	1	in	in	ADP
ejpam-6063	84	2	particular	particular	ADJ
ejpam-6063	84	3	,	,	PUNCT
ejpam-6063	84	4	if	if	SCONJ
ejpam-6063	84	5	g	g	PROPN
ejpam-6063	84	6	is	be	AUX
ejpam-6063	84	7	a	a	DET
ejpam-6063	84	8	connected	connected	ADJ
ejpam-6063	84	9	graph	graph	NOUN
ejpam-6063	84	10	and	and	CCONJ
ejpam-6063	84	11	n	n	PRON
ejpam-6063	84	12	≥	≥	NOUN
ejpam-6063	84	13	3	3	NUM
ejpam-6063	84	14	,	,	PUNCT
ejpam-6063	84	15	then	then	ADV
ejpam-6063	84	16	β2(g	β2(g	NUM
ejpam-6063	84	17	)	)	PUNCT
ejpam-6063	84	18	≤	≤	NUM
ejpam-6063	84	19	n−	n−	NOUN
ejpam-6063	84	20	1	1	NUM
ejpam-6063	84	21	.	.	PUNCT
ejpam-6063	84	22	(	(	PUNCT
ejpam-6063	84	23	ii	ii	NOUN
ejpam-6063	84	24	)	)	PUNCT
ejpam-6063	84	25	β2(g	β2(g	NUM
ejpam-6063	84	26	)	)	PUNCT
ejpam-6063	84	27	=	=	SYM
ejpam-6063	84	28	1	1	NUM
ejpam-6063	84	29	if	if	SCONJ
ejpam-6063	84	30	and	and	CCONJ
ejpam-6063	84	31	only	only	ADV
ejpam-6063	84	32	if	if	SCONJ
ejpam-6063	84	33	g	g	PROPN
ejpam-6063	84	34	=	=	PROPN
ejpam-6063	84	35	k1	k1	PROPN
ejpam-6063	84	36	.	.	PUNCT
ejpam-6063	85	1	(	(	PUNCT
ejpam-6063	85	2	iii	iii	NOUN
ejpam-6063	85	3	)	)	PUNCT
ejpam-6063	85	4	β2(g	β2(g	NUM
ejpam-6063	85	5	)	)	PUNCT
ejpam-6063	85	6	=	=	SYM
ejpam-6063	85	7	2	2	NUM
ejpam-6063	85	8	if	if	SCONJ
ejpam-6063	85	9	and	and	CCONJ
ejpam-6063	85	10	only	only	ADV
ejpam-6063	85	11	if	if	SCONJ
ejpam-6063	85	12	g	g	PROPN
ejpam-6063	85	13	∈	∈	PROPN
ejpam-6063	85	14	{	{	PUNCT
ejpam-6063	85	15	k2,k2,k2	k2,k2,k2	VERB
ejpam-6063	85	16	+	+	PROPN
ejpam-6063	85	17	h	h	NOUN
ejpam-6063	85	18	,	,	PUNCT
ejpam-6063	85	19	k2	k2	ADJ
ejpam-6063	85	20	+	+	NOUN
ejpam-6063	85	21	h	h	NOUN
ejpam-6063	85	22	}	}	PUNCT
ejpam-6063	85	23	for	for	ADP
ejpam-6063	85	24	some	some	DET
ejpam-6063	85	25	graph	graph	NOUN
ejpam-6063	85	26	h	h	NOUN
ejpam-6063	85	27	of	of	ADP
ejpam-6063	85	28	order	order	NOUN
ejpam-6063	85	29	n−	n−	NOUN
ejpam-6063	85	30	2	2	NUM
ejpam-6063	85	31	.	.	PUNCT
ejpam-6063	85	32	(	(	PUNCT
ejpam-6063	85	33	iv	iv	X
ejpam-6063	85	34	)	)	PUNCT
ejpam-6063	85	35	β2(g	β2(g	NUM
ejpam-6063	85	36	)	)	PUNCT
ejpam-6063	85	37	=	=	SYM
ejpam-6063	86	1	n	n	NOUN
ejpam-6063	86	2	if	if	SCONJ
ejpam-6063	86	3	and	and	CCONJ
ejpam-6063	86	4	only	only	ADV
ejpam-6063	86	5	if	if	SCONJ
ejpam-6063	86	6	g′	g′	PROPN
ejpam-6063	86	7	∈	∈	PROPN
ejpam-6063	86	8	{	{	PUNCT
ejpam-6063	86	9	k1,k2	k1,k2	PROPN
ejpam-6063	86	10	}	}	PUNCT
ejpam-6063	86	11	for	for	ADP
ejpam-6063	86	12	every	every	DET
ejpam-6063	86	13	component	component	NOUN
ejpam-6063	86	14	g′	g′	NOUN
ejpam-6063	86	15	of	of	ADP
ejpam-6063	86	16	g.	g.	PROPN
ejpam-6063	86	17	proof	proof	NOUN
ejpam-6063	86	18	.	.	PUNCT
ejpam-6063	87	1	since	since	SCONJ
ejpam-6063	87	2	every	every	DET
ejpam-6063	87	3	2	2	NUM
ejpam-6063	87	4	-	-	PUNCT
ejpam-6063	87	5	vertex	vertex	NOUN
ejpam-6063	87	6	covering	covering	NOUN
ejpam-6063	87	7	of	of	ADP
ejpam-6063	87	8	g	g	PROPN
ejpam-6063	87	9	is	be	AUX
ejpam-6063	87	10	both	both	CCONJ
ejpam-6063	87	11	a	a	DET
ejpam-6063	87	12	vertex	vertex	NOUN
ejpam-6063	87	13	cover	cover	NOUN
ejpam-6063	87	14	and	and	CCONJ
ejpam-6063	87	15	a	a	DET
ejpam-6063	87	16	2	2	NUM
ejpam-6063	87	17	-	-	PUNCT
ejpam-6063	87	18	dominating	dominating	NOUN
ejpam-6063	87	19	set	set	NOUN
ejpam-6063	87	20	in	in	ADP
ejpam-6063	87	21	g	g	PROPN
ejpam-6063	87	22	,	,	PUNCT
ejpam-6063	87	23	it	it	PRON
ejpam-6063	87	24	follows	follow	VERB
ejpam-6063	87	25	that	that	SCONJ
ejpam-6063	87	26	max{γ2(g	max{γ2(g	PROPN
ejpam-6063	87	27	)	)	PUNCT
ejpam-6063	87	28	,	,	PUNCT
ejpam-6063	87	29	β(g	β(g	PROPN
ejpam-6063	87	30	)	)	PUNCT
ejpam-6063	87	31	}	}	PUNCT
ejpam-6063	87	32	≤	≤	NOUN
ejpam-6063	87	33	β2(g	β2(g	NUM
ejpam-6063	87	34	)	)	PUNCT
ejpam-6063	87	35	.	.	PUNCT
ejpam-6063	88	1	clealy	clealy	PROPN
ejpam-6063	88	2	,	,	PUNCT
ejpam-6063	88	3	β2(g	β2(g	NUM
ejpam-6063	88	4	)	)	PUNCT
ejpam-6063	88	5	≤	≤	NOUN
ejpam-6063	88	6	n.	n.	NOUN
ejpam-6063	88	7	(	(	PUNCT
ejpam-6063	88	8	i	i	NOUN
ejpam-6063	88	9	)	)	PUNCT
ejpam-6063	88	10	suppose	suppose	VERB
ejpam-6063	88	11	g	g	PROPN
ejpam-6063	88	12	has	have	VERB
ejpam-6063	88	13	vertex	vertex	NOUN
ejpam-6063	88	14	v	v	NOUN
ejpam-6063	88	15	with	with	ADP
ejpam-6063	88	16	|ng(v)|	|ng(v)|	NOUN
ejpam-6063	88	17	≥	≥	NOUN
ejpam-6063	88	18	2	2	NUM
ejpam-6063	88	19	.	.	PUNCT
ejpam-6063	89	1	then	then	ADV
ejpam-6063	89	2	clealy	clealy	PROPN
ejpam-6063	89	3	,	,	PUNCT
ejpam-6063	89	4	s	s	PART
ejpam-6063	89	5	=	=	SYM
ejpam-6063	89	6	v	v	X
ejpam-6063	89	7	(	(	PUNCT
ejpam-6063	89	8	g	g	NOUN
ejpam-6063	89	9	)	)	PUNCT
ejpam-6063	89	10	\	\	NOUN
ejpam-6063	89	11	{	{	PUNCT
ejpam-6063	89	12	v	v	NOUN
ejpam-6063	89	13	}	}	PUNCT
ejpam-6063	89	14	is	be	AUX
ejpam-6063	89	15	a	a	DET
ejpam-6063	89	16	2	2	NUM
ejpam-6063	89	17	-	-	PUNCT
ejpam-6063	89	18	vertex	vertex	NOUN
ejpam-6063	89	19	cover	cover	NOUN
ejpam-6063	89	20	of	of	ADP
ejpam-6063	89	21	g.	g.	PROPN
ejpam-6063	89	22	hence	hence	ADV
ejpam-6063	89	23	,	,	PUNCT
ejpam-6063	89	24	β2(g	β2(g	NUM
ejpam-6063	89	25	)	)	PUNCT
ejpam-6063	89	26	≤	≤	NUM
ejpam-6063	89	27	|s|	|s|	PROPN
ejpam-6063	89	28	=	=	PUNCT
ejpam-6063	89	29	n	n	CCONJ
ejpam-6063	89	30	−	−	PROPN
ejpam-6063	89	31	1	1	X
ejpam-6063	89	32	.	.	PUNCT
ejpam-6063	90	1	if	if	SCONJ
ejpam-6063	90	2	g	g	PROPN
ejpam-6063	90	3	is	be	AUX
ejpam-6063	90	4	connected	connect	VERB
ejpam-6063	90	5	and	and	CCONJ
ejpam-6063	90	6	n	n	PRON
ejpam-6063	90	7	≥	≥	NOUN
ejpam-6063	90	8	3	3	NUM
ejpam-6063	90	9	,	,	PUNCT
ejpam-6063	90	10	then	then	ADV
ejpam-6063	90	11	there	there	PRON
ejpam-6063	90	12	exists	exist	VERB
ejpam-6063	90	13	w	w	PROPN
ejpam-6063	90	14	∈	∈	PROPN
ejpam-6063	90	15	v	v	ADP
ejpam-6063	90	16	(	(	PUNCT
ejpam-6063	90	17	g	g	NOUN
ejpam-6063	90	18	)	)	PUNCT
ejpam-6063	90	19	with	with	ADP
ejpam-6063	90	20	|ng(w)|	|ng(w)|	PROPN
ejpam-6063	90	21	≥	≥	NOUN
ejpam-6063	90	22	2	2	NUM
ejpam-6063	90	23	.	.	PUNCT
ejpam-6063	90	24	therefore	therefore	ADV
ejpam-6063	90	25	,	,	PUNCT
ejpam-6063	90	26	β2(g	β2(g	NUM
ejpam-6063	90	27	)	)	PUNCT
ejpam-6063	90	28	≤	≤	NUM
ejpam-6063	90	29	n−	n−	NOUN
ejpam-6063	90	30	1	1	NUM
ejpam-6063	90	31	.	.	PUNCT
ejpam-6063	91	1	(	(	PUNCT
ejpam-6063	91	2	ii	ii	NOUN
ejpam-6063	91	3	)	)	PUNCT
ejpam-6063	91	4	suppose	suppose	VERB
ejpam-6063	91	5	β2(g	β2(g	SYM
ejpam-6063	91	6	)	)	PUNCT
ejpam-6063	91	7	=	=	SYM
ejpam-6063	91	8	1	1	NUM
ejpam-6063	91	9	,	,	PUNCT
ejpam-6063	91	10	and	and	CCONJ
ejpam-6063	91	11	let	let	VERB
ejpam-6063	91	12	s	s	AUX
ejpam-6063	91	13	=	=	NOUN
ejpam-6063	91	14	{	{	PUNCT
ejpam-6063	91	15	v	v	NOUN
ejpam-6063	91	16	}	}	PUNCT
ejpam-6063	91	17	be	be	AUX
ejpam-6063	91	18	a	a	DET
ejpam-6063	91	19	β2	β2	NOUN
ejpam-6063	91	20	-	-	PUNCT
ejpam-6063	91	21	set	set	NOUN
ejpam-6063	91	22	of	of	ADP
ejpam-6063	91	23	g.	g.	PROPN
ejpam-6063	91	24	since	since	SCONJ
ejpam-6063	91	25	s	s	PROPN
ejpam-6063	91	26	is	be	AUX
ejpam-6063	91	27	a	a	DET
ejpam-6063	91	28	2	2	NUM
ejpam-6063	91	29	-	-	PUNCT
ejpam-6063	91	30	dominating	dominating	NOUN
ejpam-6063	91	31	set	set	NOUN
ejpam-6063	91	32	,	,	PUNCT
ejpam-6063	91	33	there	there	PRON
ejpam-6063	91	34	can	can	AUX
ejpam-6063	91	35	be	be	AUX
ejpam-6063	91	36	no	no	DET
ejpam-6063	91	37	vertex	vertex	NOUN
ejpam-6063	91	38	outside	outside	ADP
ejpam-6063	91	39	s.	s.	PROPN
ejpam-6063	91	40	hence	hence	PROPN
ejpam-6063	91	41	,	,	PUNCT
ejpam-6063	91	42	g	g	PROPN
ejpam-6063	91	43	=	=	PROPN
ejpam-6063	91	44	k1	k1	PROPN
ejpam-6063	91	45	.	.	PUNCT
ejpam-6063	92	1	j.	j.	PROPN
ejpam-6063	92	2	hassan	hassan	PROPN
ejpam-6063	92	3	et	et	PROPN
ejpam-6063	92	4	.	.	PUNCT
ejpam-6063	93	1	al	al	PROPN
ejpam-6063	93	2	/	/	PUNCT
ejpam-6063	93	3	eur	eur	PROPN
ejpam-6063	93	4	.	.	PUNCT
ejpam-6063	94	1	j.	j.	PROPN
ejpam-6063	94	2	pure	pure	PROPN
ejpam-6063	94	3	appl	appl	PROPN
ejpam-6063	94	4	.	.	PROPN
ejpam-6063	94	5	math	math	PROPN
ejpam-6063	94	6	,	,	PUNCT
ejpam-6063	94	7	18	18	NUM
ejpam-6063	94	8	(	(	PUNCT
ejpam-6063	94	9	2	2	NUM
ejpam-6063	94	10	)	)	PUNCT
ejpam-6063	94	11	(	(	PUNCT
ejpam-6063	94	12	2025	2025	NUM
ejpam-6063	94	13	)	)	PUNCT
ejpam-6063	94	14	,	,	PUNCT
ejpam-6063	94	15	6063	6063	NUM
ejpam-6063	94	16	4	4	NUM
ejpam-6063	94	17	of	of	ADP
ejpam-6063	94	18	11	11	NUM
ejpam-6063	94	19	the	the	DET
ejpam-6063	94	20	converse	converse	NOUN
ejpam-6063	94	21	is	be	AUX
ejpam-6063	94	22	clear	clear	ADJ
ejpam-6063	94	23	.	.	PUNCT
ejpam-6063	95	1	(	(	PUNCT
ejpam-6063	95	2	iii	iii	NOUN
ejpam-6063	95	3	)	)	PUNCT
ejpam-6063	95	4	suppose	suppose	VERB
ejpam-6063	95	5	β2(g	β2(g	SYM
ejpam-6063	95	6	)	)	PUNCT
ejpam-6063	95	7	=	=	SYM
ejpam-6063	95	8	2	2	NUM
ejpam-6063	95	9	,	,	PUNCT
ejpam-6063	95	10	say	say	VERB
ejpam-6063	95	11	s	s	X
ejpam-6063	95	12	=	=	PUNCT
ejpam-6063	95	13	{	{	PUNCT
ejpam-6063	95	14	x	x	PROPN
ejpam-6063	95	15	,	,	PUNCT
ejpam-6063	95	16	y	y	PRON
ejpam-6063	95	17	}	}	PUNCT
ejpam-6063	95	18	is	be	AUX
ejpam-6063	95	19	a	a	DET
ejpam-6063	95	20	β2	β2	NOUN
ejpam-6063	95	21	-	-	PUNCT
ejpam-6063	95	22	set	set	NOUN
ejpam-6063	95	23	of	of	ADP
ejpam-6063	95	24	g.	g.	PROPN
ejpam-6063	95	25	suppose	suppose	VERB
ejpam-6063	95	26	first	first	ADV
ejpam-6063	95	27	that	that	SCONJ
ejpam-6063	95	28	xy	xy	PROPN
ejpam-6063	95	29	∈	∈	PROPN
ejpam-6063	95	30	e(g	e(g	PROPN
ejpam-6063	95	31	)	)	PUNCT
ejpam-6063	95	32	.	.	PUNCT
ejpam-6063	96	1	if	if	SCONJ
ejpam-6063	96	2	n	n	NOUN
ejpam-6063	96	3	=	=	SYM
ejpam-6063	96	4	2	2	NUM
ejpam-6063	96	5	,	,	PUNCT
ejpam-6063	96	6	then	then	ADV
ejpam-6063	96	7	g	g	PROPN
ejpam-6063	96	8	=	=	PROPN
ejpam-6063	96	9	k2	k2	PROPN
ejpam-6063	96	10	.	.	PUNCT
ejpam-6063	97	1	suppose	suppose	VERB
ejpam-6063	97	2	n	n	PRON
ejpam-6063	97	3	≥	≥	X
ejpam-6063	97	4	2	2	NUM
ejpam-6063	97	5	and	and	CCONJ
ejpam-6063	97	6	z	z	NOUN
ejpam-6063	97	7	∈	∈	PROPN
ejpam-6063	97	8	v	v	ADP
ejpam-6063	97	9	(	(	PUNCT
ejpam-6063	97	10	g	g	NOUN
ejpam-6063	97	11	)	)	PUNCT
ejpam-6063	97	12	\	\	PUNCT
ejpam-6063	98	1	s.	s.	PROPN
ejpam-6063	98	2	since	since	SCONJ
ejpam-6063	98	3	s	s	PROPN
ejpam-6063	98	4	is	be	AUX
ejpam-6063	98	5	a	a	DET
ejpam-6063	98	6	2	2	NUM
ejpam-6063	98	7	-	-	PUNCT
ejpam-6063	98	8	dominating	dominating	NOUN
ejpam-6063	98	9	set	set	NOUN
ejpam-6063	98	10	in	in	ADP
ejpam-6063	98	11	g	g	PROPN
ejpam-6063	98	12	,	,	PUNCT
ejpam-6063	98	13	we	we	PRON
ejpam-6063	98	14	have	have	VERB
ejpam-6063	98	15	z	z	PROPN
ejpam-6063	98	16	∈	∈	PROPN
ejpam-6063	98	17	ng(x	ng(x	NUM
ejpam-6063	98	18	)	)	PUNCT
ejpam-6063	98	19	∩ng(y	∩ng(y	PROPN
ejpam-6063	98	20	)	)	PUNCT
ejpam-6063	98	21	.	.	PUNCT
ejpam-6063	99	1	this	this	PRON
ejpam-6063	99	2	implies	imply	VERB
ejpam-6063	99	3	that	that	SCONJ
ejpam-6063	99	4	g	g	PROPN
ejpam-6063	99	5	=	=	SYM
ejpam-6063	99	6	⟨{x	⟨{x	PROPN
ejpam-6063	99	7	,	,	PUNCT
ejpam-6063	99	8	y}⟩+h	y}⟩+h	NOUN
ejpam-6063	99	9	=	=	SYM
ejpam-6063	99	10	k2	k2	PROPN
ejpam-6063	100	1	+	+	PROPN
ejpam-6063	100	2	h	h	NOUN
ejpam-6063	100	3	,	,	PUNCT
ejpam-6063	100	4	where	where	SCONJ
ejpam-6063	100	5	h	h	NOUN
ejpam-6063	100	6	=	=	SYM
ejpam-6063	100	7	⟨v	⟨v	X
ejpam-6063	100	8	(	(	PUNCT
ejpam-6063	100	9	g	g	NOUN
ejpam-6063	100	10	)	)	PUNCT
ejpam-6063	100	11	\	\	PROPN
ejpam-6063	100	12	s⟩	s⟩	NOUN
ejpam-6063	100	13	is	be	AUX
ejpam-6063	100	14	a	a	DET
ejpam-6063	100	15	graph	graph	NOUN
ejpam-6063	100	16	of	of	ADP
ejpam-6063	100	17	order	order	NOUN
ejpam-6063	100	18	n	n	CCONJ
ejpam-6063	100	19	−	−	PROPN
ejpam-6063	100	20	2	2	NUM
ejpam-6063	100	21	.	.	PUNCT
ejpam-6063	101	1	next	next	ADV
ejpam-6063	101	2	,	,	PUNCT
ejpam-6063	101	3	suppose	suppose	VERB
ejpam-6063	101	4	xy	xy	PROPN
ejpam-6063	101	5	/∈	/∈	PUNCT
ejpam-6063	101	6	e(g	e(g	PROPN
ejpam-6063	101	7	)	)	PUNCT
ejpam-6063	101	8	.	.	PUNCT
ejpam-6063	102	1	if	if	SCONJ
ejpam-6063	102	2	n	n	NOUN
ejpam-6063	102	3	=	=	SYM
ejpam-6063	102	4	2	2	NUM
ejpam-6063	102	5	,	,	PUNCT
ejpam-6063	102	6	then	then	ADV
ejpam-6063	102	7	g	g	PROPN
ejpam-6063	102	8	=	=	PROPN
ejpam-6063	102	9	k2	k2	PROPN
ejpam-6063	102	10	by	by	ADP
ejpam-6063	102	11	(	(	PUNCT
ejpam-6063	102	12	ii	ii	NOUN
ejpam-6063	102	13	)	)	PUNCT
ejpam-6063	102	14	and	and	CCONJ
ejpam-6063	102	15	theorem	theorem	VERB
ejpam-6063	102	16	1	1	NUM
ejpam-6063	102	17	.	.	PUNCT
ejpam-6063	102	18	suppose	suppose	VERB
ejpam-6063	102	19	n	n	PRON
ejpam-6063	102	20	≥	≥	NUM
ejpam-6063	102	21	3	3	NUM
ejpam-6063	102	22	.	.	PUNCT
ejpam-6063	103	1	following	follow	VERB
ejpam-6063	103	2	an	an	DET
ejpam-6063	103	3	earlier	early	ADJ
ejpam-6063	103	4	argument	argument	NOUN
ejpam-6063	103	5	,	,	PUNCT
ejpam-6063	103	6	we	we	PRON
ejpam-6063	103	7	have	have	VERB
ejpam-6063	103	8	v	v	NOUN
ejpam-6063	103	9	(	(	PUNCT
ejpam-6063	103	10	g)\s	g)\s	VERB
ejpam-6063	103	11	⊆	⊆	NUM
ejpam-6063	103	12	ng(x)∩ng(y	ng(x)∩ng(y	NOUN
ejpam-6063	103	13	)	)	PUNCT
ejpam-6063	103	14	.	.	PUNCT
ejpam-6063	104	1	therefore	therefore	ADV
ejpam-6063	104	2	,	,	PUNCT
ejpam-6063	104	3	g	g	PROPN
ejpam-6063	104	4	=	=	SYM
ejpam-6063	104	5	⟨{x	⟨{x	PROPN
ejpam-6063	104	6	,	,	PUNCT
ejpam-6063	104	7	y}⟩+h	y}⟩+h	NOUN
ejpam-6063	104	8	=	=	PUNCT
ejpam-6063	104	9	k2+h	k2+h	PROPN
ejpam-6063	104	10	,	,	PUNCT
ejpam-6063	104	11	where	where	SCONJ
ejpam-6063	104	12	h	h	NOUN
ejpam-6063	104	13	=	=	SYM
ejpam-6063	104	14	⟨v	⟨v	PROPN
ejpam-6063	104	15	(	(	PUNCT
ejpam-6063	104	16	g)\s⟩	g)\s⟩	PROPN
ejpam-6063	104	17	is	be	AUX
ejpam-6063	104	18	a	a	DET
ejpam-6063	104	19	graph	graph	NOUN
ejpam-6063	104	20	of	of	ADP
ejpam-6063	104	21	order	order	NOUN
ejpam-6063	104	22	n−	n−	NOUN
ejpam-6063	104	23	2	2	NUM
ejpam-6063	104	24	.	.	PUNCT
ejpam-6063	105	1	accordingly	accordingly	ADV
ejpam-6063	105	2	,	,	PUNCT
ejpam-6063	105	3	g	g	PROPN
ejpam-6063	105	4	∈	∈	PROPN
ejpam-6063	105	5	{	{	PUNCT
ejpam-6063	105	6	k2,k2,k2	k2,k2,k2	VERB
ejpam-6063	105	7	+	+	PROPN
ejpam-6063	105	8	h	h	NOUN
ejpam-6063	105	9	,	,	PUNCT
ejpam-6063	105	10	k2	k2	ADJ
ejpam-6063	105	11	+	+	NOUN
ejpam-6063	105	12	h	h	NOUN
ejpam-6063	105	13	}	}	PUNCT
ejpam-6063	105	14	for	for	ADP
ejpam-6063	105	15	some	some	DET
ejpam-6063	105	16	graph	graph	NOUN
ejpam-6063	105	17	h	h	NOUN
ejpam-6063	105	18	of	of	ADP
ejpam-6063	105	19	order	order	NOUN
ejpam-6063	105	20	n−	n−	NOUN
ejpam-6063	105	21	2	2	NUM
ejpam-6063	105	22	.	.	PUNCT
ejpam-6063	106	1	the	the	DET
ejpam-6063	106	2	converse	converse	NOUN
ejpam-6063	106	3	is	be	AUX
ejpam-6063	106	4	clear	clear	ADJ
ejpam-6063	106	5	.	.	PUNCT
ejpam-6063	107	1	(	(	PUNCT
ejpam-6063	107	2	iv	iv	X
ejpam-6063	107	3	)	)	PUNCT
ejpam-6063	107	4	suppose	suppose	VERB
ejpam-6063	107	5	β2(g	β2(g	SYM
ejpam-6063	107	6	)	)	PUNCT
ejpam-6063	107	7	=	=	VERB
ejpam-6063	108	1	n.	n.	NOUN
ejpam-6063	108	2	from	from	ADP
ejpam-6063	108	3	(	(	PUNCT
ejpam-6063	108	4	i	i	NOUN
ejpam-6063	108	5	)	)	PUNCT
ejpam-6063	108	6	,	,	PUNCT
ejpam-6063	108	7	it	it	PRON
ejpam-6063	108	8	follows	follow	VERB
ejpam-6063	108	9	that	that	SCONJ
ejpam-6063	108	10	|ng(v)|	|ng(v)|	NOUN
ejpam-6063	108	11	≤	≤	NOUN
ejpam-6063	108	12	1	1	NUM
ejpam-6063	108	13	for	for	ADP
ejpam-6063	108	14	every	every	DET
ejpam-6063	108	15	v	v	NUM
ejpam-6063	108	16	∈	∈	NOUN
ejpam-6063	108	17	v	v	NOUN
ejpam-6063	108	18	(	(	PUNCT
ejpam-6063	108	19	g	g	NOUN
ejpam-6063	108	20	)	)	PUNCT
ejpam-6063	108	21	.	.	PUNCT
ejpam-6063	109	1	this	this	PRON
ejpam-6063	109	2	implies	imply	VERB
ejpam-6063	109	3	that	that	SCONJ
ejpam-6063	109	4	g′	g′	NOUN
ejpam-6063	109	5	∈	∈	PROPN
ejpam-6063	109	6	{	{	PUNCT
ejpam-6063	109	7	k1,k2	k1,k2	PROPN
ejpam-6063	109	8	}	}	PUNCT
ejpam-6063	109	9	for	for	ADP
ejpam-6063	109	10	every	every	DET
ejpam-6063	109	11	component	component	NOUN
ejpam-6063	109	12	g′	g′	NOUN
ejpam-6063	109	13	of	of	ADP
ejpam-6063	109	14	g.	g.	PROPN
ejpam-6063	109	15	for	for	ADP
ejpam-6063	109	16	the	the	DET
ejpam-6063	109	17	converse	converse	NOUN
ejpam-6063	109	18	,	,	PUNCT
ejpam-6063	109	19	suppose	suppose	VERB
ejpam-6063	109	20	that	that	SCONJ
ejpam-6063	109	21	g′	g′	PROPN
ejpam-6063	109	22	∈	∈	PROPN
ejpam-6063	109	23	{	{	PUNCT
ejpam-6063	109	24	k1,k2	k1,k2	PROPN
ejpam-6063	109	25	}	}	PUNCT
ejpam-6063	109	26	for	for	ADP
ejpam-6063	109	27	every	every	DET
ejpam-6063	109	28	component	component	NOUN
ejpam-6063	109	29	g′	g′	NOUN
ejpam-6063	109	30	of	of	ADP
ejpam-6063	109	31	g.	g.	PROPN
ejpam-6063	109	32	from	from	ADP
ejpam-6063	109	33	(	(	PUNCT
ejpam-6063	109	34	ii	ii	NOUN
ejpam-6063	109	35	)	)	PUNCT
ejpam-6063	109	36	and	and	CCONJ
ejpam-6063	109	37	(	(	PUNCT
ejpam-6063	109	38	iii	iii	NOUN
ejpam-6063	109	39	)	)	PUNCT
ejpam-6063	109	40	,	,	PUNCT
ejpam-6063	109	41	and	and	CCONJ
ejpam-6063	109	42	by	by	ADP
ejpam-6063	109	43	theorem	theorem	NOUN
ejpam-6063	109	44	1	1	NUM
ejpam-6063	109	45	,	,	PUNCT
ejpam-6063	109	46	it	it	PRON
ejpam-6063	109	47	follows	follow	VERB
ejpam-6063	109	48	that	that	SCONJ
ejpam-6063	109	49	β2(g	β2(g	X
ejpam-6063	109	50	)	)	PUNCT
ejpam-6063	109	51	=	=	SYM
ejpam-6063	109	52	n.	n.	NOUN
ejpam-6063	109	53	theorem	theorem	NOUN
ejpam-6063	109	54	3	3	X
ejpam-6063	109	55	.	.	PUNCT
ejpam-6063	110	1	let	let	VERB
ejpam-6063	110	2	g	g	PRON
ejpam-6063	110	3	be	be	AUX
ejpam-6063	110	4	a	a	DET
ejpam-6063	110	5	graph	graph	NOUN
ejpam-6063	110	6	on	on	ADP
ejpam-6063	110	7	n	n	CCONJ
ejpam-6063	110	8	vertices	vertex	NOUN
ejpam-6063	110	9	such	such	ADJ
ejpam-6063	110	10	that	that	SCONJ
ejpam-6063	110	11	|ng(v)|	|ng(v)|	NOUN
ejpam-6063	110	12	≥	≥	NOUN
ejpam-6063	110	13	2	2	NUM
ejpam-6063	110	14	for	for	ADP
ejpam-6063	110	15	some	some	DET
ejpam-6063	110	16	vertex	vertex	NOUN
ejpam-6063	110	17	v	v	ADP
ejpam-6063	110	18	∈	∈	NOUN
ejpam-6063	110	19	v	v	NOUN
ejpam-6063	110	20	(	(	PUNCT
ejpam-6063	110	21	g	g	NOUN
ejpam-6063	110	22	)	)	PUNCT
ejpam-6063	110	23	.	.	PUNCT
ejpam-6063	111	1	then	then	ADV
ejpam-6063	111	2	β2(g	β2(g	NUM
ejpam-6063	111	3	)	)	PUNCT
ejpam-6063	112	1	=	=	SYM
ejpam-6063	112	2	n−1	n−1	PROPN
ejpam-6063	112	3	if	if	SCONJ
ejpam-6063	112	4	and	and	CCONJ
ejpam-6063	112	5	only	only	ADV
ejpam-6063	112	6	if	if	SCONJ
ejpam-6063	112	7	for	for	ADP
ejpam-6063	112	8	every	every	DET
ejpam-6063	112	9	pair	pair	NOUN
ejpam-6063	112	10	of	of	ADP
ejpam-6063	112	11	non	non	ADJ
ejpam-6063	112	12	-	-	ADJ
ejpam-6063	112	13	adjacent	adjacent	ADJ
ejpam-6063	112	14	vertices	vertex	NOUN
ejpam-6063	112	15	p	p	NOUN
ejpam-6063	112	16	and	and	CCONJ
ejpam-6063	112	17	q	q	NOUN
ejpam-6063	112	18	of	of	ADP
ejpam-6063	112	19	g	g	NOUN
ejpam-6063	112	20	,	,	PUNCT
ejpam-6063	112	21	it	it	PRON
ejpam-6063	112	22	holds	hold	VERB
ejpam-6063	112	23	that	that	SCONJ
ejpam-6063	112	24	p	p	X
ejpam-6063	112	25	,	,	PUNCT
ejpam-6063	112	26	q	q	PROPN
ejpam-6063	112	27	∈	∈	PROPN
ejpam-6063	112	28	l(g	l(g	PROPN
ejpam-6063	112	29	)	)	PUNCT
ejpam-6063	112	30	∪	∪	ADP
ejpam-6063	112	31	i(g	i(g	NOUN
ejpam-6063	112	32	)	)	PUNCT
ejpam-6063	112	33	.	.	PUNCT
ejpam-6063	113	1	proof	proof	NOUN
ejpam-6063	113	2	.	.	PUNCT
ejpam-6063	114	1	suppose	suppose	VERB
ejpam-6063	114	2	β2(g	β2(g	SYM
ejpam-6063	114	3	)	)	PUNCT
ejpam-6063	114	4	=	=	SYM
ejpam-6063	115	1	n	n	CCONJ
ejpam-6063	115	2	−	−	PROPN
ejpam-6063	115	3	1	1	X
ejpam-6063	115	4	.	.	PUNCT
ejpam-6063	116	1	suppose	suppose	VERB
ejpam-6063	116	2	,	,	PUNCT
ejpam-6063	116	3	for	for	ADP
ejpam-6063	116	4	a	a	DET
ejpam-6063	116	5	contradiction	contradiction	NOUN
ejpam-6063	116	6	,	,	PUNCT
ejpam-6063	116	7	that	that	SCONJ
ejpam-6063	116	8	there	there	PRON
ejpam-6063	116	9	exist	exist	VERB
ejpam-6063	116	10	nonadjacent	nonadjacent	ADJ
ejpam-6063	116	11	vertices	vertex	NOUN
ejpam-6063	116	12	p	p	NOUN
ejpam-6063	116	13	and	and	CCONJ
ejpam-6063	116	14	q	q	NOUN
ejpam-6063	116	15	such	such	ADJ
ejpam-6063	116	16	that	that	SCONJ
ejpam-6063	116	17	p	p	X
ejpam-6063	116	18	,	,	PUNCT
ejpam-6063	116	19	q	q	X
ejpam-6063	116	20	/∈	/∈	PUNCT
ejpam-6063	116	21	l(g)∪i(g	l(g)∪i(g	PROPN
ejpam-6063	116	22	)	)	PUNCT
ejpam-6063	116	23	.	.	PUNCT
ejpam-6063	117	1	then	then	ADV
ejpam-6063	117	2	|ng(p)|	|ng(p)|	VERB
ejpam-6063	117	3	≥	≥	NOUN
ejpam-6063	117	4	2	2	NUM
ejpam-6063	117	5	and	and	CCONJ
ejpam-6063	117	6	|ng(q)|	|ng(q)|	PROPN
ejpam-6063	117	7	≥	≥	NUM
ejpam-6063	117	8	2	2	NUM
ejpam-6063	117	9	.	.	PUNCT
ejpam-6063	118	1	it	it	PRON
ejpam-6063	118	2	follows	follow	VERB
ejpam-6063	118	3	that	that	PRON
ejpam-6063	118	4	s	s	VERB
ejpam-6063	118	5	=	=	SYM
ejpam-6063	118	6	v	v	X
ejpam-6063	118	7	(	(	PUNCT
ejpam-6063	118	8	g	g	NOUN
ejpam-6063	118	9	)	)	PUNCT
ejpam-6063	118	10	\	\	NOUN
ejpam-6063	119	1	{	{	PUNCT
ejpam-6063	119	2	p	p	X
ejpam-6063	119	3	,	,	PUNCT
ejpam-6063	119	4	q	q	X
ejpam-6063	119	5	}	}	PUNCT
ejpam-6063	119	6	is	be	AUX
ejpam-6063	119	7	a	a	DET
ejpam-6063	119	8	2	2	NUM
ejpam-6063	119	9	-	-	PUNCT
ejpam-6063	119	10	vertex	vertex	NOUN
ejpam-6063	119	11	covering	covering	NOUN
ejpam-6063	119	12	of	of	ADP
ejpam-6063	119	13	g	g	NOUN
ejpam-6063	119	14	,	,	PUNCT
ejpam-6063	119	15	implying	imply	VERB
ejpam-6063	119	16	that	that	SCONJ
ejpam-6063	119	17	β(g	β(g	PROPN
ejpam-6063	119	18	)	)	PUNCT
ejpam-6063	119	19	≤	≤	NUM
ejpam-6063	119	20	|s|	|s|	PROPN
ejpam-6063	119	21	=	=	SYM
ejpam-6063	119	22	n−	n−	NOUN
ejpam-6063	119	23	2	2	NUM
ejpam-6063	119	24	,	,	PUNCT
ejpam-6063	119	25	a	a	DET
ejpam-6063	119	26	contradiction	contradiction	NOUN
ejpam-6063	119	27	to	to	ADP
ejpam-6063	119	28	our	our	PRON
ejpam-6063	119	29	assumption	assumption	NOUN
ejpam-6063	119	30	.	.	PUNCT
ejpam-6063	120	1	therefore	therefore	ADV
ejpam-6063	120	2	,	,	PUNCT
ejpam-6063	120	3	g	g	NOUN
ejpam-6063	120	4	satisfies	satisfy	VERB
ejpam-6063	120	5	the	the	DET
ejpam-6063	120	6	given	give	VERB
ejpam-6063	120	7	property	property	NOUN
ejpam-6063	120	8	.	.	PUNCT
ejpam-6063	121	1	for	for	ADP
ejpam-6063	121	2	the	the	DET
ejpam-6063	121	3	converse	converse	NOUN
ejpam-6063	121	4	,	,	PUNCT
ejpam-6063	121	5	suppose	suppose	VERB
ejpam-6063	121	6	that	that	SCONJ
ejpam-6063	121	7	g	g	PROPN
ejpam-6063	121	8	satisfies	satisfy	VERB
ejpam-6063	121	9	the	the	DET
ejpam-6063	121	10	property	property	NOUN
ejpam-6063	121	11	and	and	CCONJ
ejpam-6063	121	12	let	let	VERB
ejpam-6063	121	13	s′	s′	PROPN
ejpam-6063	121	14	be	be	AUX
ejpam-6063	121	15	a	a	DET
ejpam-6063	121	16	β2	β2	NOUN
ejpam-6063	121	17	-	-	PUNCT
ejpam-6063	121	18	set	set	NOUN
ejpam-6063	121	19	in	in	ADP
ejpam-6063	121	20	g.	g.	PROPN
ejpam-6063	121	21	since	since	SCONJ
ejpam-6063	121	22	|ng(v)|	|ng(v)|	NOUN
ejpam-6063	121	23	≥	≥	NOUN
ejpam-6063	121	24	2	2	NUM
ejpam-6063	121	25	for	for	ADP
ejpam-6063	121	26	some	some	DET
ejpam-6063	121	27	vertex	vertex	NOUN
ejpam-6063	121	28	v	v	ADP
ejpam-6063	121	29	∈	∈	NOUN
ejpam-6063	121	30	v	v	NOUN
ejpam-6063	121	31	(	(	PUNCT
ejpam-6063	121	32	g	g	NOUN
ejpam-6063	121	33	)	)	PUNCT
ejpam-6063	121	34	,	,	PUNCT
ejpam-6063	121	35	it	it	PRON
ejpam-6063	121	36	follows	follow	VERB
ejpam-6063	121	37	from	from	ADP
ejpam-6063	121	38	theorem	theorem	ADJ
ejpam-6063	121	39	2	2	NUM
ejpam-6063	121	40	that	that	SCONJ
ejpam-6063	121	41	β2(g	β2(g	NUM
ejpam-6063	121	42	)	)	PUNCT
ejpam-6063	121	43	=	=	PRON
ejpam-6063	121	44	|s′|	|s′|	VERB
ejpam-6063	121	45	≤	≤	NOUN
ejpam-6063	121	46	n	n	CCONJ
ejpam-6063	121	47	−	−	PROPN
ejpam-6063	121	48	1	1	NUM
ejpam-6063	121	49	.	.	PUNCT
ejpam-6063	122	1	suppose	suppose	VERB
ejpam-6063	122	2	for	for	ADP
ejpam-6063	122	3	a	a	DET
ejpam-6063	122	4	contradiction	contradiction	NOUN
ejpam-6063	122	5	that	that	SCONJ
ejpam-6063	122	6	β2(g	β2(g	NUM
ejpam-6063	122	7	)	)	PUNCT
ejpam-6063	122	8	≤	≤	NOUN
ejpam-6063	123	1	n	n	CCONJ
ejpam-6063	123	2	−	−	PROPN
ejpam-6063	123	3	2	2	NUM
ejpam-6063	123	4	.	.	PUNCT
ejpam-6063	123	5	then	then	ADV
ejpam-6063	123	6	there	there	PRON
ejpam-6063	123	7	exist	exist	VERB
ejpam-6063	123	8	distinct	distinct	ADJ
ejpam-6063	123	9	vertices	vertex	NOUN
ejpam-6063	123	10	x	x	X
ejpam-6063	123	11	,	,	PUNCT
ejpam-6063	123	12	y	y	PROPN
ejpam-6063	123	13	∈	∈	PROPN
ejpam-6063	123	14	v	v	X
ejpam-6063	123	15	(	(	PUNCT
ejpam-6063	123	16	g)\s′.	g)\s′.	PROPN
ejpam-6063	123	17	since	since	SCONJ
ejpam-6063	123	18	s′	s′	ADJ
ejpam-6063	123	19	is	be	AUX
ejpam-6063	123	20	a	a	DET
ejpam-6063	123	21	vertex	vertex	NOUN
ejpam-6063	123	22	cover	cover	NOUN
ejpam-6063	123	23	of	of	ADP
ejpam-6063	123	24	g	g	NOUN
ejpam-6063	123	25	,	,	PUNCT
ejpam-6063	123	26	we	we	PRON
ejpam-6063	123	27	have	have	VERB
ejpam-6063	123	28	xy	xy	PROPN
ejpam-6063	123	29	/∈	/∈	PUNCT
ejpam-6063	123	30	e(g	e(g	PROPN
ejpam-6063	123	31	)	)	PUNCT
ejpam-6063	123	32	.	.	PUNCT
ejpam-6063	124	1	this	this	PRON
ejpam-6063	124	2	implies	imply	VERB
ejpam-6063	124	3	that	that	SCONJ
ejpam-6063	124	4	x	x	X
ejpam-6063	124	5	,	,	PUNCT
ejpam-6063	124	6	y	y	PROPN
ejpam-6063	124	7	∈	∈	PROPN
ejpam-6063	124	8	l(g	l(g	PROPN
ejpam-6063	124	9	)	)	PUNCT
ejpam-6063	124	10	∪	∪	ADP
ejpam-6063	124	11	i(g	i(g	NOUN
ejpam-6063	124	12	)	)	PUNCT
ejpam-6063	124	13	by	by	ADP
ejpam-6063	124	14	the	the	DET
ejpam-6063	124	15	assumption	assumption	NOUN
ejpam-6063	124	16	.	.	PUNCT
ejpam-6063	125	1	therefore	therefore	ADV
ejpam-6063	125	2	,	,	PUNCT
ejpam-6063	125	3	s	s	VERB
ejpam-6063	125	4	is	be	AUX
ejpam-6063	125	5	not	not	PART
ejpam-6063	125	6	a	a	DET
ejpam-6063	125	7	2	2	NUM
ejpam-6063	125	8	-	-	PUNCT
ejpam-6063	125	9	dominating	dominating	NOUN
ejpam-6063	125	10	set	set	NOUN
ejpam-6063	125	11	in	in	ADP
ejpam-6063	125	12	g	g	PROPN
ejpam-6063	125	13	,	,	PUNCT
ejpam-6063	125	14	a	a	DET
ejpam-6063	125	15	contradiction	contradiction	NOUN
ejpam-6063	125	16	.	.	PUNCT
ejpam-6063	126	1	accordingly	accordingly	ADV
ejpam-6063	126	2	,	,	PUNCT
ejpam-6063	126	3	β2(g	β2(g	NUM
ejpam-6063	126	4	)	)	PUNCT
ejpam-6063	126	5	=	=	PUNCT
ejpam-6063	126	6	|s′|	|s′|	NOUN
ejpam-6063	126	7	=	=	PUNCT
ejpam-6063	126	8	n−	n−	NOUN
ejpam-6063	126	9	1	1	NUM
ejpam-6063	126	10	.	.	PUNCT
ejpam-6063	127	1	the	the	DET
ejpam-6063	127	2	next	next	ADJ
ejpam-6063	127	3	result	result	NOUN
ejpam-6063	127	4	is	be	AUX
ejpam-6063	127	5	immediate	immediate	ADJ
ejpam-6063	127	6	from	from	ADP
ejpam-6063	127	7	theorem	theorem	ADJ
ejpam-6063	127	8	3	3	NUM
ejpam-6063	127	9	.	.	PUNCT
ejpam-6063	127	10	corollary	corollary	ADJ
ejpam-6063	127	11	1	1	NUM
ejpam-6063	127	12	.	.	PUNCT
ejpam-6063	128	1	let	let	VERB
ejpam-6063	128	2	n	n	PRON
ejpam-6063	128	3	be	be	AUX
ejpam-6063	128	4	a	a	DET
ejpam-6063	128	5	positive	positive	ADJ
ejpam-6063	128	6	integer	integer	NOUN
ejpam-6063	128	7	such	such	ADJ
ejpam-6063	128	8	that	that	SCONJ
ejpam-6063	128	9	n	n	NUM
ejpam-6063	128	10	≥	≥	NOUN
ejpam-6063	128	11	3	3	NUM
ejpam-6063	128	12	.	.	PUNCT
ejpam-6063	129	1	then	then	ADV
ejpam-6063	129	2	(	(	PUNCT
ejpam-6063	129	3	i	i	NOUN
ejpam-6063	129	4	)	)	PUNCT
ejpam-6063	129	5	β2(kn	β2(kn	NUM
ejpam-6063	129	6	)	)	PUNCT
ejpam-6063	130	1	=	=	PUNCT
ejpam-6063	130	2	n−	n−	NOUN
ejpam-6063	130	3	1	1	NUM
ejpam-6063	130	4	,	,	PUNCT
ejpam-6063	130	5	and	and	CCONJ
ejpam-6063	130	6	(	(	PUNCT
ejpam-6063	130	7	ii	ii	NOUN
ejpam-6063	130	8	)	)	PUNCT
ejpam-6063	130	9	β2(k1,n−1	β2(k1,n−1	PROPN
ejpam-6063	130	10	)	)	PUNCT
ejpam-6063	131	1	=	=	PUNCT
ejpam-6063	131	2	n−	n−	NOUN
ejpam-6063	131	3	1	1	NUM
ejpam-6063	131	4	.	.	PUNCT
ejpam-6063	131	5	theorem	theorem	VERB
ejpam-6063	131	6	4	4	NUM
ejpam-6063	131	7	.	.	PUNCT
ejpam-6063	132	1	let	let	VERB
ejpam-6063	132	2	n	n	PRON
ejpam-6063	132	3	be	be	AUX
ejpam-6063	132	4	a	a	DET
ejpam-6063	132	5	positive	positive	ADJ
ejpam-6063	132	6	integer	integer	NOUN
ejpam-6063	132	7	.	.	PUNCT
ejpam-6063	133	1	then	then	ADV
ejpam-6063	133	2	β2(pn	β2(pn	PRON
ejpam-6063	133	3	)	)	PUNCT
ejpam-6063	133	4	=	=	SYM
ejpam-6063	133	5	{	{	PUNCT
ejpam-6063	133	6	⌈	⌈	NOUN
ejpam-6063	133	7	n	n	CCONJ
ejpam-6063	133	8	2	2	NUM
ejpam-6063	133	9	⌉	⌉	NOUN
ejpam-6063	133	10	,	,	PUNCT
ejpam-6063	133	11	if	if	SCONJ
ejpam-6063	133	12	n	n	PRON
ejpam-6063	133	13	is	be	AUX
ejpam-6063	133	14	odd	odd	ADJ
ejpam-6063	133	15	n	n	PRON
ejpam-6063	133	16	2	2	NUM
ejpam-6063	133	17	+	+	NUM
ejpam-6063	133	18	1	1	NUM
ejpam-6063	133	19	,	,	PUNCT
ejpam-6063	133	20	if	if	SCONJ
ejpam-6063	133	21	n	n	PRON
ejpam-6063	133	22	is	be	AUX
ejpam-6063	133	23	even	even	ADV
ejpam-6063	133	24	j.	j.	PROPN
ejpam-6063	133	25	hassan	hassan	PROPN
ejpam-6063	133	26	et	et	PROPN
ejpam-6063	133	27	.	.	PUNCT
ejpam-6063	134	1	al	al	PROPN
ejpam-6063	134	2	/	/	PUNCT
ejpam-6063	134	3	eur	eur	PROPN
ejpam-6063	134	4	.	.	PUNCT
ejpam-6063	135	1	j.	j.	PROPN
ejpam-6063	135	2	pure	pure	PROPN
ejpam-6063	135	3	appl	appl	PROPN
ejpam-6063	135	4	.	.	PROPN
ejpam-6063	135	5	math	math	PROPN
ejpam-6063	135	6	,	,	PUNCT
ejpam-6063	135	7	18	18	NUM
ejpam-6063	135	8	(	(	PUNCT
ejpam-6063	135	9	2	2	NUM
ejpam-6063	135	10	)	)	PUNCT
ejpam-6063	135	11	(	(	PUNCT
ejpam-6063	135	12	2025	2025	NUM
ejpam-6063	135	13	)	)	PUNCT
ejpam-6063	135	14	,	,	PUNCT
ejpam-6063	135	15	6063	6063	NUM
ejpam-6063	135	16	5	5	NUM
ejpam-6063	135	17	of	of	ADP
ejpam-6063	135	18	11	11	NUM
ejpam-6063	135	19	proof	proof	NOUN
ejpam-6063	135	20	.	.	PUNCT
ejpam-6063	136	1	from	from	ADP
ejpam-6063	136	2	(	(	PUNCT
ejpam-6063	136	3	ii	ii	NOUN
ejpam-6063	136	4	)	)	PUNCT
ejpam-6063	136	5	and	and	CCONJ
ejpam-6063	136	6	(	(	PUNCT
ejpam-6063	136	7	iii	iii	NOUN
ejpam-6063	136	8	)	)	PUNCT
ejpam-6063	136	9	of	of	ADP
ejpam-6063	136	10	theorem	theorem	NOUN
ejpam-6063	136	11	2	2	NUM
ejpam-6063	136	12	,	,	PUNCT
ejpam-6063	136	13	we	we	PRON
ejpam-6063	136	14	have	have	VERB
ejpam-6063	136	15	β2(p1	β2(p1	NUM
ejpam-6063	136	16	)	)	PUNCT
ejpam-6063	136	17	=	=	SYM
ejpam-6063	137	1	1	1	NUM
ejpam-6063	137	2	=	=	SYM
ejpam-6063	137	3	1	1	NUM
ejpam-6063	137	4	+	+	NOUN
ejpam-6063	137	5	1	1	NUM
ejpam-6063	137	6	2	2	NUM
ejpam-6063	137	7	and	and	CCONJ
ejpam-6063	137	8	β2(p2	β2(p2	PRON
ejpam-6063	137	9	)	)	PUNCT
ejpam-6063	137	10	=	=	SYM
ejpam-6063	137	11	2	2	NUM
ejpam-6063	137	12	=	=	SYM
ejpam-6063	137	13	2	2	NUM
ejpam-6063	137	14	2	2	NUM
ejpam-6063	137	15	+	+	NUM
ejpam-6063	137	16	1	1	X
ejpam-6063	137	17	.	.	PUNCT
ejpam-6063	137	18	let	let	VERB
ejpam-6063	137	19	s	s	PRON
ejpam-6063	137	20	be	be	AUX
ejpam-6063	137	21	a	a	DET
ejpam-6063	137	22	β2	β2	NOUN
ejpam-6063	137	23	-	-	PUNCT
ejpam-6063	137	24	set	set	NOUN
ejpam-6063	137	25	on	on	ADP
ejpam-6063	137	26	pn	pn	PROPN
ejpam-6063	137	27	.	.	PROPN
ejpam-6063	137	28	suppose	suppose	VERB
ejpam-6063	137	29	first	first	ADV
ejpam-6063	137	30	that	that	SCONJ
ejpam-6063	137	31	n	n	PROPN
ejpam-6063	137	32	is	be	AUX
ejpam-6063	137	33	odd	odd	ADJ
ejpam-6063	137	34	and	and	CCONJ
ejpam-6063	137	35	n	n	PRON
ejpam-6063	137	36	≥	≥	NOUN
ejpam-6063	137	37	3	3	NUM
ejpam-6063	137	38	,	,	PUNCT
ejpam-6063	137	39	say	say	VERB
ejpam-6063	137	40	n	n	NOUN
ejpam-6063	137	41	=	=	SYM
ejpam-6063	137	42	2r	2r	NUM
ejpam-6063	138	1	+	+	CCONJ
ejpam-6063	138	2	1	1	NUM
ejpam-6063	139	1	where	where	SCONJ
ejpam-6063	139	2	r	r	NOUN
ejpam-6063	139	3	≥	≥	NUM
ejpam-6063	139	4	1	1	NUM
ejpam-6063	139	5	.	.	PUNCT
ejpam-6063	139	6	let	let	VERB
ejpam-6063	139	7	pn	pn	VERB
ejpam-6063	139	8	=	=	PUNCT
ejpam-6063	140	1	[	[	X
ejpam-6063	140	2	a1	a1	NOUN
ejpam-6063	140	3	,	,	PUNCT
ejpam-6063	140	4	a2	a2	PROPN
ejpam-6063	140	5	,	,	PUNCT
ejpam-6063	140	6	.	.	PUNCT
ejpam-6063	140	7	.	.	PUNCT
ejpam-6063	140	8	.	.	PUNCT
ejpam-6063	141	1	,	,	PUNCT
ejpam-6063	141	2	a2r	a2r	NOUN
ejpam-6063	141	3	,	,	PUNCT
ejpam-6063	141	4	a2r+1	a2r+1	ADJ
ejpam-6063	141	5	]	]	PUNCT
ejpam-6063	141	6	.	.	PUNCT
ejpam-6063	142	1	since	since	SCONJ
ejpam-6063	142	2	s	s	PROPN
ejpam-6063	142	3	is	be	AUX
ejpam-6063	142	4	a	a	DET
ejpam-6063	142	5	2	2	NUM
ejpam-6063	142	6	-	-	PUNCT
ejpam-6063	142	7	dominating	dominating	NOUN
ejpam-6063	142	8	set	set	NOUN
ejpam-6063	142	9	,	,	PUNCT
ejpam-6063	142	10	we	we	PRON
ejpam-6063	142	11	have	have	VERB
ejpam-6063	142	12	a1	a1	NOUN
ejpam-6063	142	13	,	,	PUNCT
ejpam-6063	142	14	a2r+1	a2r+1	PROPN
ejpam-6063	142	15	∈	∈	PROPN
ejpam-6063	142	16	s.	s.	PROPN
ejpam-6063	142	17	again	again	ADV
ejpam-6063	142	18	,	,	PUNCT
ejpam-6063	142	19	since	since	SCONJ
ejpam-6063	142	20	s	s	NOUN
ejpam-6063	142	21	is	be	AUX
ejpam-6063	142	22	a	a	DET
ejpam-6063	142	23	β2	β2	NOUN
ejpam-6063	142	24	-	-	PUNCT
ejpam-6063	142	25	set	set	NOUN
ejpam-6063	142	26	of	of	ADP
ejpam-6063	142	27	pn	pn	PROPN
ejpam-6063	142	28	,	,	PUNCT
ejpam-6063	142	29	it	it	PRON
ejpam-6063	142	30	follows	follow	VERB
ejpam-6063	142	31	that	that	DET
ejpam-6063	142	32	s	s	VERB
ejpam-6063	142	33	=	=	PUNCT
ejpam-6063	142	34	{	{	PUNCT
ejpam-6063	142	35	a1	a1	NOUN
ejpam-6063	142	36	,	,	PUNCT
ejpam-6063	142	37	a3	a3	NOUN
ejpam-6063	142	38	,	,	PUNCT
ejpam-6063	142	39	.	.	PUNCT
ejpam-6063	142	40	.	.	PUNCT
ejpam-6063	143	1	.	.	PUNCT
ejpam-6063	144	1	,	,	PUNCT
ejpam-6063	144	2	a2r−1	a2r−1	PROPN
ejpam-6063	144	3	,	,	PUNCT
ejpam-6063	144	4	a2r+1	a2r+1	ADJ
ejpam-6063	144	5	}	}	PUNCT
ejpam-6063	144	6	.	.	PUNCT
ejpam-6063	145	1	hence	hence	ADV
ejpam-6063	145	2	,	,	PUNCT
ejpam-6063	145	3	β2(pn	β2(pn	PRON
ejpam-6063	145	4	)	)	PUNCT
ejpam-6063	145	5	=	=	PUNCT
ejpam-6063	145	6	|s|	|s|	NOUN
ejpam-6063	145	7	=	=	NOUN
ejpam-6063	145	8	n+	n+	NUM
ejpam-6063	145	9	1	1	NUM
ejpam-6063	145	10	2	2	NUM
ejpam-6063	145	11	=	=	NOUN
ejpam-6063	145	12	⌈n	⌈n	NOUN
ejpam-6063	145	13	2	2	NUM
ejpam-6063	145	14	⌉	⌉	X
ejpam-6063	145	15	.	.	PUNCT
ejpam-6063	146	1	next	next	ADV
ejpam-6063	146	2	,	,	PUNCT
ejpam-6063	146	3	suppose	suppose	VERB
ejpam-6063	146	4	n	n	PRON
ejpam-6063	146	5	is	be	AUX
ejpam-6063	146	6	even	even	ADV
ejpam-6063	146	7	and	and	CCONJ
ejpam-6063	146	8	n	n	PRON
ejpam-6063	146	9	≥	≥	NOUN
ejpam-6063	146	10	4	4	NUM
ejpam-6063	146	11	,	,	PUNCT
ejpam-6063	146	12	say	say	VERB
ejpam-6063	146	13	n	n	NOUN
ejpam-6063	146	14	=	=	SYM
ejpam-6063	146	15	2	2	NUM
ejpam-6063	146	16	t	t	NUM
ejpam-6063	146	17	where	where	SCONJ
ejpam-6063	146	18	t	t	PROPN
ejpam-6063	146	19	≥	≥	PROPN
ejpam-6063	146	20	2	2	X
ejpam-6063	146	21	.	.	PUNCT
ejpam-6063	147	1	let	let	VERB
ejpam-6063	147	2	pn	pn	VERB
ejpam-6063	147	3	=	=	PUNCT
ejpam-6063	148	1	[	[	X
ejpam-6063	148	2	a1	a1	NOUN
ejpam-6063	148	3	,	,	PUNCT
ejpam-6063	148	4	a2	a2	PROPN
ejpam-6063	148	5	,	,	PUNCT
ejpam-6063	148	6	.	.	PUNCT
ejpam-6063	148	7	.	.	PUNCT
ejpam-6063	149	1	.	.	PUNCT
ejpam-6063	150	1	,	,	PUNCT
ejpam-6063	150	2	a2r−1	a2r−1	PROPN
ejpam-6063	150	3	,	,	PUNCT
ejpam-6063	150	4	a2r	a2r	NOUN
ejpam-6063	150	5	]	]	PUNCT
ejpam-6063	150	6	.	.	PUNCT
ejpam-6063	151	1	since	since	SCONJ
ejpam-6063	151	2	s	s	PROPN
ejpam-6063	151	3	is	be	AUX
ejpam-6063	151	4	2	2	NUM
ejpam-6063	151	5	-	-	PUNCT
ejpam-6063	151	6	dominating	dominating	NOUN
ejpam-6063	151	7	,	,	PUNCT
ejpam-6063	151	8	a1	a1	NOUN
ejpam-6063	151	9	,	,	PUNCT
ejpam-6063	151	10	a2r	a2r	PROPN
ejpam-6063	151	11	∈	∈	PROPN
ejpam-6063	151	12	s.	s.	PROPN
ejpam-6063	151	13	again	again	ADV
ejpam-6063	151	14	,	,	PUNCT
ejpam-6063	151	15	since	since	SCONJ
ejpam-6063	151	16	s	s	NOUN
ejpam-6063	151	17	is	be	AUX
ejpam-6063	151	18	a	a	DET
ejpam-6063	151	19	β2	β2	NOUN
ejpam-6063	151	20	-	-	PUNCT
ejpam-6063	151	21	set	set	NOUN
ejpam-6063	151	22	of	of	ADP
ejpam-6063	151	23	pn	pn	PROPN
ejpam-6063	151	24	,	,	PUNCT
ejpam-6063	151	25	s	s	PART
ejpam-6063	151	26	=	=	NOUN
ejpam-6063	151	27	{	{	PUNCT
ejpam-6063	151	28	a1	a1	NOUN
ejpam-6063	151	29	,	,	PUNCT
ejpam-6063	151	30	a3	a3	NOUN
ejpam-6063	151	31	,	,	PUNCT
ejpam-6063	151	32	.	.	PUNCT
ejpam-6063	151	33	.	.	PUNCT
ejpam-6063	152	1	.	.	PUNCT
ejpam-6063	153	1	,	,	PUNCT
ejpam-6063	153	2	a2r−1}∪	a2r−1}∪	NOUN
ejpam-6063	153	3	{	{	PUNCT
ejpam-6063	153	4	a2r+1	a2r+1	ADJ
ejpam-6063	153	5	}	}	PUNCT
ejpam-6063	153	6	.	.	PUNCT
ejpam-6063	154	1	hence	hence	ADV
ejpam-6063	154	2	,	,	PUNCT
ejpam-6063	154	3	β2(pn	β2(pn	PRON
ejpam-6063	154	4	)	)	PUNCT
ejpam-6063	154	5	=	=	PUNCT
ejpam-6063	154	6	|s|	|s|	NOUN
ejpam-6063	154	7	=	=	SYM
ejpam-6063	154	8	n	n	PRON
ejpam-6063	154	9	2	2	NUM
ejpam-6063	154	10	+	+	NUM
ejpam-6063	154	11	1	1	NUM
ejpam-6063	154	12	.	.	X
ejpam-6063	154	13	theorem	theorem	NOUN
ejpam-6063	154	14	5	5	NUM
ejpam-6063	154	15	.	.	PUNCT
ejpam-6063	155	1	let	let	VERB
ejpam-6063	155	2	n	n	PRON
ejpam-6063	155	3	be	be	AUX
ejpam-6063	155	4	a	a	DET
ejpam-6063	155	5	positive	positive	ADJ
ejpam-6063	155	6	integer	integer	NOUN
ejpam-6063	155	7	where	where	SCONJ
ejpam-6063	155	8	n	n	NUM
ejpam-6063	155	9	≥	≥	NOUN
ejpam-6063	155	10	3	3	NUM
ejpam-6063	155	11	.	.	PUNCT
ejpam-6063	155	12	then	then	ADV
ejpam-6063	155	13	β2(cn	β2(cn	NUM
ejpam-6063	155	14	)	)	PUNCT
ejpam-6063	156	1	=	=	PRON
ejpam-6063	156	2	{	{	PUNCT
ejpam-6063	156	3	⌈	⌈	NOUN
ejpam-6063	156	4	n	n	CCONJ
ejpam-6063	156	5	2	2	NUM
ejpam-6063	156	6	⌉	⌉	NOUN
ejpam-6063	156	7	,	,	PUNCT
ejpam-6063	156	8	if	if	SCONJ
ejpam-6063	156	9	n	n	PRON
ejpam-6063	156	10	is	be	AUX
ejpam-6063	156	11	odd	odd	ADJ
ejpam-6063	156	12	n	n	PRON
ejpam-6063	156	13	2	2	NUM
ejpam-6063	156	14	,	,	PUNCT
ejpam-6063	156	15	if	if	SCONJ
ejpam-6063	156	16	n	n	PRON
ejpam-6063	156	17	is	be	AUX
ejpam-6063	156	18	even	even	ADV
ejpam-6063	156	19	proof	proof	ADJ
ejpam-6063	156	20	.	.	PUNCT
ejpam-6063	157	1	let	let	VERB
ejpam-6063	157	2	d	d	PRON
ejpam-6063	157	3	be	be	AUX
ejpam-6063	157	4	a	a	DET
ejpam-6063	157	5	β2	β2	NOUN
ejpam-6063	157	6	-	-	PUNCT
ejpam-6063	157	7	set	set	NOUN
ejpam-6063	157	8	on	on	ADP
ejpam-6063	157	9	pn	pn	PROPN
ejpam-6063	157	10	.	.	PROPN
ejpam-6063	157	11	suppose	suppose	VERB
ejpam-6063	157	12	n	n	PRON
ejpam-6063	157	13	is	be	AUX
ejpam-6063	157	14	odd	odd	ADJ
ejpam-6063	157	15	.	.	PUNCT
ejpam-6063	158	1	clearly	clearly	ADV
ejpam-6063	158	2	,	,	PUNCT
ejpam-6063	158	3	β2(c3	β2(c3	NOUN
ejpam-6063	158	4	)	)	PUNCT
ejpam-6063	158	5	=	=	SYM
ejpam-6063	158	6	2	2	NUM
ejpam-6063	158	7	=	=	SYM
ejpam-6063	158	8	1	1	NUM
ejpam-6063	158	9	+	+	NOUN
ejpam-6063	158	10	1	1	NUM
ejpam-6063	158	11	2	2	NUM
ejpam-6063	158	12	.	.	PUNCT
ejpam-6063	159	1	so	so	ADV
ejpam-6063	159	2	suppose	suppose	VERB
ejpam-6063	159	3	that	that	SCONJ
ejpam-6063	159	4	n	n	NUM
ejpam-6063	159	5	≥	≥	NUM
ejpam-6063	159	6	5	5	NUM
ejpam-6063	159	7	,	,	PUNCT
ejpam-6063	159	8	say	say	VERB
ejpam-6063	159	9	n	n	NOUN
ejpam-6063	159	10	=	=	SYM
ejpam-6063	159	11	2r	2r	NUM
ejpam-6063	160	1	+	+	CCONJ
ejpam-6063	160	2	1	1	NUM
ejpam-6063	161	1	where	where	SCONJ
ejpam-6063	161	2	r	r	NOUN
ejpam-6063	161	3	≥	≥	NOUN
ejpam-6063	161	4	2	2	NUM
ejpam-6063	161	5	.	.	PUNCT
ejpam-6063	161	6	let	let	VERB
ejpam-6063	161	7	cn	cn	PROPN
ejpam-6063	162	1	=	=	PUNCT
ejpam-6063	162	2	[	[	X
ejpam-6063	162	3	b1	b1	NOUN
ejpam-6063	162	4	,	,	PUNCT
ejpam-6063	162	5	b2	b2	NOUN
ejpam-6063	162	6	,	,	PUNCT
ejpam-6063	162	7	.	.	PUNCT
ejpam-6063	162	8	.	.	PUNCT
ejpam-6063	163	1	.	.	PUNCT
ejpam-6063	164	1	,	,	PUNCT
ejpam-6063	164	2	b2r	b2r	PROPN
ejpam-6063	164	3	,	,	PUNCT
ejpam-6063	164	4	b2r+1	b2r+1	PROPN
ejpam-6063	164	5	,	,	PUNCT
ejpam-6063	164	6	b1	b1	NOUN
ejpam-6063	164	7	]	]	PUNCT
ejpam-6063	164	8	.	.	PUNCT
ejpam-6063	165	1	we	we	PRON
ejpam-6063	165	2	may	may	AUX
ejpam-6063	165	3	assume	assume	VERB
ejpam-6063	165	4	that	that	SCONJ
ejpam-6063	165	5	b1	b1	PROPN
ejpam-6063	165	6	∈	∈	PROPN
ejpam-6063	165	7	d.	d.	PROPN
ejpam-6063	165	8	since	since	SCONJ
ejpam-6063	165	9	d	d	PROPN
ejpam-6063	165	10	is	be	AUX
ejpam-6063	165	11	a	a	DET
ejpam-6063	165	12	β2	β2	NOUN
ejpam-6063	165	13	-	-	PUNCT
ejpam-6063	165	14	set	set	NOUN
ejpam-6063	165	15	of	of	ADP
ejpam-6063	165	16	cn	cn	PROPN
ejpam-6063	165	17	,	,	PUNCT
ejpam-6063	165	18	it	it	PRON
ejpam-6063	165	19	follows	follow	VERB
ejpam-6063	165	20	that	that	SCONJ
ejpam-6063	165	21	d	d	PROPN
ejpam-6063	165	22	=	=	PRON
ejpam-6063	165	23	{	{	PUNCT
ejpam-6063	165	24	b1	b1	NOUN
ejpam-6063	165	25	,	,	PUNCT
ejpam-6063	165	26	b3	b3	PROPN
ejpam-6063	165	27	,	,	PUNCT
ejpam-6063	165	28	.	.	PUNCT
ejpam-6063	165	29	.	.	PUNCT
ejpam-6063	166	1	.	.	PUNCT
ejpam-6063	167	1	,	,	PUNCT
ejpam-6063	167	2	b2r−1	b2r−1	PROPN
ejpam-6063	167	3	,	,	PUNCT
ejpam-6063	167	4	b2r+1	b2r+1	NOUN
ejpam-6063	167	5	}	}	PUNCT
ejpam-6063	167	6	.	.	PUNCT
ejpam-6063	168	1	hence	hence	ADV
ejpam-6063	168	2	,	,	PUNCT
ejpam-6063	168	3	β2(cn	β2(cn	PROPN
ejpam-6063	168	4	)	)	PUNCT
ejpam-6063	168	5	=	=	SYM
ejpam-6063	168	6	|d|	|d|	PROPN
ejpam-6063	168	7	=	=	PUNCT
ejpam-6063	168	8	n+	n+	PUNCT
ejpam-6063	168	9	1	1	NUM
ejpam-6063	168	10	2	2	NUM
ejpam-6063	168	11	=	=	NOUN
ejpam-6063	168	12	⌈n	⌈n	NOUN
ejpam-6063	168	13	2	2	NUM
ejpam-6063	168	14	⌉	⌉	X
ejpam-6063	168	15	.	.	PUNCT
ejpam-6063	169	1	now	now	ADV
ejpam-6063	169	2	,	,	PUNCT
ejpam-6063	169	3	suppose	suppose	VERB
ejpam-6063	169	4	that	that	SCONJ
ejpam-6063	169	5	n	n	PRON
ejpam-6063	169	6	is	be	AUX
ejpam-6063	169	7	even	even	ADV
ejpam-6063	169	8	and	and	CCONJ
ejpam-6063	169	9	n	n	PRON
ejpam-6063	169	10	≥	≥	NOUN
ejpam-6063	169	11	4	4	NUM
ejpam-6063	169	12	,	,	PUNCT
ejpam-6063	169	13	say	say	VERB
ejpam-6063	169	14	n	n	NOUN
ejpam-6063	170	1	=	=	SYM
ejpam-6063	170	2	2	2	NUM
ejpam-6063	170	3	m	m	NOUN
ejpam-6063	170	4	form	form	NOUN
ejpam-6063	170	5	≥	≥	NOUN
ejpam-6063	170	6	2	2	NUM
ejpam-6063	170	7	.	.	PUNCT
ejpam-6063	170	8	let	let	VERB
ejpam-6063	170	9	cn	cn	PROPN
ejpam-6063	170	10	=	=	PUNCT
ejpam-6063	171	1	[	[	X
ejpam-6063	171	2	b1	b1	NOUN
ejpam-6063	171	3	,	,	PUNCT
ejpam-6063	171	4	b2	b2	NOUN
ejpam-6063	171	5	,	,	PUNCT
ejpam-6063	171	6	.	.	PUNCT
ejpam-6063	171	7	.	.	PUNCT
ejpam-6063	172	1	.	.	PUNCT
ejpam-6063	173	1	,	,	PUNCT
ejpam-6063	173	2	b2m−1	b2m−1	PROPN
ejpam-6063	173	3	,	,	PUNCT
ejpam-6063	173	4	b2	b2	NOUN
ejpam-6063	173	5	m	m	PROPN
ejpam-6063	173	6	,	,	PUNCT
ejpam-6063	173	7	b1	b1	NOUN
ejpam-6063	173	8	]	]	PUNCT
ejpam-6063	173	9	.	.	PUNCT
ejpam-6063	174	1	again	again	ADV
ejpam-6063	174	2	,	,	PUNCT
ejpam-6063	174	3	we	we	PRON
ejpam-6063	174	4	may	may	AUX
ejpam-6063	174	5	assume	assume	VERB
ejpam-6063	174	6	that	that	SCONJ
ejpam-6063	174	7	b1	b1	PROPN
ejpam-6063	174	8	∈	∈	PROPN
ejpam-6063	174	9	d.	d.	PROPN
ejpam-6063	174	10	sinced	since	VERB
ejpam-6063	174	11	is	be	AUX
ejpam-6063	174	12	a	a	DET
ejpam-6063	174	13	β2	β2	NOUN
ejpam-6063	174	14	-	-	PUNCT
ejpam-6063	174	15	set	set	NOUN
ejpam-6063	174	16	of	of	ADP
ejpam-6063	174	17	cn	cn	PROPN
ejpam-6063	174	18	,	,	PUNCT
ejpam-6063	174	19	we	we	PRON
ejpam-6063	174	20	haved	have	VERB
ejpam-6063	174	21	=	=	PUNCT
ejpam-6063	174	22	{	{	PUNCT
ejpam-6063	174	23	b1	b1	NOUN
ejpam-6063	174	24	,	,	PUNCT
ejpam-6063	174	25	b3	b3	PROPN
ejpam-6063	174	26	,	,	PUNCT
ejpam-6063	174	27	.	.	PUNCT
ejpam-6063	174	28	.	.	PUNCT
ejpam-6063	175	1	.	.	PUNCT
ejpam-6063	176	1	,	,	PUNCT
ejpam-6063	176	2	b2m−1	b2m−1	PROPN
ejpam-6063	176	3	}	}	PUNCT
ejpam-6063	176	4	.	.	PUNCT
ejpam-6063	177	1	hence	hence	ADV
ejpam-6063	177	2	,	,	PUNCT
ejpam-6063	177	3	β2(pn	β2(pn	PROPN
ejpam-6063	177	4	)	)	PUNCT
ejpam-6063	177	5	=	=	SYM
ejpam-6063	177	6	|d|	|d|	PROPN
ejpam-6063	177	7	=	=	SYM
ejpam-6063	177	8	n	n	PRON
ejpam-6063	177	9	2	2	NUM
ejpam-6063	177	10	.	.	PUNCT
ejpam-6063	177	11	theorem	theorem	VERB
ejpam-6063	177	12	6	6	NUM
ejpam-6063	177	13	.	.	PUNCT
ejpam-6063	178	1	let	let	VERB
ejpam-6063	178	2	g	g	PROPN
ejpam-6063	178	3	=	=	PUNCT
ejpam-6063	178	4	km1,m2	km1,m2	PROPN
ejpam-6063	178	5	,	,	PUNCT
ejpam-6063	178	6	·	·	PUNCT
ejpam-6063	178	7	·	·	PUNCT
ejpam-6063	178	8	·	·	PUNCT
ejpam-6063	178	9	,	,	PUNCT
ejpam-6063	178	10	mk	mk	PROPN
ejpam-6063	178	11	be	be	AUX
ejpam-6063	178	12	a	a	DET
ejpam-6063	178	13	complete	complete	ADJ
ejpam-6063	178	14	k	k	ADJ
ejpam-6063	178	15	-	-	ADJ
ejpam-6063	178	16	partite	partite	ADJ
ejpam-6063	178	17	graph	graph	NOUN
ejpam-6063	178	18	with	with	ADP
ejpam-6063	178	19	2	2	NUM
ejpam-6063	178	20	≤	≤	NOUN
ejpam-6063	178	21	m1	m1	NOUN
ejpam-6063	178	22	≤	≤	NUM
ejpam-6063	178	23	m2	m2	PROPN
ejpam-6063	178	24	≤	≤	NOUN
ejpam-6063	178	25	·	·	PUNCT
ejpam-6063	178	26	·	·	PUNCT
ejpam-6063	178	27	·	·	PUNCT
ejpam-6063	179	1	≤	≤	NUM
ejpam-6063	179	2	mk	mk	PROPN
ejpam-6063	179	3	.	.	PUNCT
ejpam-6063	180	1	then	then	ADV
ejpam-6063	180	2	β2(g	β2(g	NUM
ejpam-6063	180	3	)	)	PUNCT
ejpam-6063	180	4	=	=	SYM
ejpam-6063	180	5	∑	∑	PUNCT
ejpam-6063	180	6	i∈[k]\{k	i∈[k]\{k	PROPN
ejpam-6063	180	7	}	}	PUNCT
ejpam-6063	180	8	mi	mi	NOUN
ejpam-6063	180	9	where	where	SCONJ
ejpam-6063	180	10	[	[	X
ejpam-6063	180	11	k	k	X
ejpam-6063	180	12	]	]	X
ejpam-6063	180	13	=	=	X
ejpam-6063	180	14	{	{	PUNCT
ejpam-6063	180	15	1	1	NUM
ejpam-6063	180	16	,	,	PUNCT
ejpam-6063	180	17	2	2	NUM
ejpam-6063	180	18	,	,	PUNCT
ejpam-6063	180	19	·	·	PUNCT
ejpam-6063	180	20	·	·	PUNCT
ejpam-6063	180	21	·	·	PUNCT
ejpam-6063	180	22	,	,	PUNCT
ejpam-6063	180	23	k	k	NOUN
ejpam-6063	180	24	}	}	PUNCT
ejpam-6063	180	25	.	.	PUNCT
ejpam-6063	181	1	proof	proof	NOUN
ejpam-6063	181	2	.	.	PUNCT
ejpam-6063	182	1	let	let	VERB
ejpam-6063	182	2	s1	s1	NOUN
ejpam-6063	182	3	,	,	PUNCT
ejpam-6063	182	4	s2	s2	PROPN
ejpam-6063	182	5	,	,	PUNCT
ejpam-6063	182	6	·	·	PUNCT
ejpam-6063	182	7	·	·	PUNCT
ejpam-6063	182	8	·	·	PUNCT
ejpam-6063	182	9	,	,	PUNCT
ejpam-6063	182	10	sk	sk	NOUN
ejpam-6063	182	11	be	be	AUX
ejpam-6063	182	12	the	the	DET
ejpam-6063	182	13	partite	partite	ADJ
ejpam-6063	182	14	sets	set	NOUN
ejpam-6063	182	15	of	of	ADP
ejpam-6063	182	16	g	g	NOUN
ejpam-6063	182	17	and	and	CCONJ
ejpam-6063	182	18	let	let	VERB
ejpam-6063	182	19	s	s	PRON
ejpam-6063	182	20	be	be	AUX
ejpam-6063	182	21	a	a	DET
ejpam-6063	182	22	β2	β2	NOUN
ejpam-6063	182	23	-	-	PUNCT
ejpam-6063	182	24	set	set	NOUN
ejpam-6063	182	25	of	of	ADP
ejpam-6063	182	26	g.	g.	PROPN
ejpam-6063	182	27	suppose	suppose	VERB
ejpam-6063	182	28	v	v	ADP
ejpam-6063	182	29	∈	∈	PROPN
ejpam-6063	182	30	v	v	NOUN
ejpam-6063	182	31	(	(	PUNCT
ejpam-6063	182	32	g	g	NOUN
ejpam-6063	182	33	)	)	PUNCT
ejpam-6063	182	34	\	\	PUNCT
ejpam-6063	183	1	s.	s.	PROPN
ejpam-6063	183	2	then	then	ADV
ejpam-6063	183	3	there	there	PRON
ejpam-6063	183	4	exists	exist	VERB
ejpam-6063	183	5	j	j	PROPN
ejpam-6063	183	6	∈	∈	PROPN
ejpam-6063	184	1	[	[	X
ejpam-6063	184	2	k	k	X
ejpam-6063	184	3	]	]	X
ejpam-6063	184	4	=	=	X
ejpam-6063	184	5	{	{	PUNCT
ejpam-6063	184	6	1	1	NUM
ejpam-6063	184	7	,	,	PUNCT
ejpam-6063	184	8	2	2	NUM
ejpam-6063	184	9	,	,	PUNCT
ejpam-6063	184	10	·	·	PUNCT
ejpam-6063	184	11	·	·	PUNCT
ejpam-6063	184	12	·	·	PUNCT
ejpam-6063	184	13	,	,	PUNCT
ejpam-6063	184	14	k	k	X
ejpam-6063	184	15	}	}	PUNCT
ejpam-6063	184	16	such	such	ADJ
ejpam-6063	184	17	that	that	SCONJ
ejpam-6063	184	18	v	v	NOUN
ejpam-6063	184	19	∈	∈	NOUN
ejpam-6063	184	20	sj	sj	INTJ
ejpam-6063	184	21	.	.	PUNCT
ejpam-6063	185	1	since	since	SCONJ
ejpam-6063	185	2	s	s	PROPN
ejpam-6063	185	3	is	be	AUX
ejpam-6063	185	4	a	a	DET
ejpam-6063	185	5	vertex	vertex	NOUN
ejpam-6063	185	6	cover	cover	NOUN
ejpam-6063	185	7	of	of	ADP
ejpam-6063	185	8	g	g	NOUN
ejpam-6063	185	9	,	,	PUNCT
ejpam-6063	185	10	it	it	PRON
ejpam-6063	185	11	follows	follow	VERB
ejpam-6063	185	12	that	that	SCONJ
ejpam-6063	185	13	∪i∈[k]\{j}si	∪i∈[k]\{j}si	NOUN
ejpam-6063	185	14	⊆	⊆	NUM
ejpam-6063	185	15	s.	s.	PROPN
ejpam-6063	185	16	moreover	moreover	ADV
ejpam-6063	185	17	,	,	PUNCT
ejpam-6063	185	18	since	since	SCONJ
ejpam-6063	185	19	v	v	NOUN
ejpam-6063	185	20	(	(	PUNCT
ejpam-6063	185	21	g	g	NOUN
ejpam-6063	185	22	)	)	PUNCT
ejpam-6063	185	23	\	\	PUNCT
ejpam-6063	186	1	sj	sj	PROPN
ejpam-6063	186	2	is	be	AUX
ejpam-6063	186	3	a	a	DET
ejpam-6063	186	4	2	2	NUM
ejpam-6063	186	5	-	-	PUNCT
ejpam-6063	186	6	vertex	vertex	NOUN
ejpam-6063	186	7	cover	cover	NOUN
ejpam-6063	186	8	of	of	ADP
ejpam-6063	186	9	g	g	PROPN
ejpam-6063	186	10	and	and	CCONJ
ejpam-6063	186	11	s	s	NOUN
ejpam-6063	186	12	is	be	AUX
ejpam-6063	186	13	a	a	DET
ejpam-6063	186	14	β2	β2	NOUN
ejpam-6063	186	15	-	-	PUNCT
ejpam-6063	186	16	set	set	NOUN
ejpam-6063	186	17	of	of	ADP
ejpam-6063	186	18	g	g	NOUN
ejpam-6063	186	19	,	,	PUNCT
ejpam-6063	186	20	s	s	PART
ejpam-6063	186	21	=	=	SYM
ejpam-6063	186	22	v	v	X
ejpam-6063	186	23	(	(	PUNCT
ejpam-6063	186	24	g	g	NOUN
ejpam-6063	186	25	)	)	PUNCT
ejpam-6063	186	26	\	\	PUNCT
ejpam-6063	187	1	sj	sj	INTJ
ejpam-6063	187	2	.	.	PUNCT
ejpam-6063	188	1	again	again	ADV
ejpam-6063	188	2	,	,	PUNCT
ejpam-6063	188	3	because	because	SCONJ
ejpam-6063	188	4	s	s	NOUN
ejpam-6063	188	5	is	be	AUX
ejpam-6063	188	6	a	a	DET
ejpam-6063	188	7	β2	β2	NOUN
ejpam-6063	188	8	-	-	PUNCT
ejpam-6063	188	9	set	set	NOUN
ejpam-6063	188	10	of	of	ADP
ejpam-6063	188	11	g	g	NOUN
ejpam-6063	188	12	,	,	PUNCT
ejpam-6063	188	13	we	we	PRON
ejpam-6063	188	14	must	must	AUX
ejpam-6063	188	15	have	have	VERB
ejpam-6063	188	16	j	j	PROPN
ejpam-6063	188	17	=	=	PROPN
ejpam-6063	188	18	k.	k.	PROPN
ejpam-6063	188	19	therefore	therefore	ADV
ejpam-6063	188	20	,	,	PUNCT
ejpam-6063	188	21	β2(g	β2(g	NUM
ejpam-6063	188	22	)	)	PUNCT
ejpam-6063	188	23	=	=	PUNCT
ejpam-6063	188	24	|s|	|s|	NOUN
ejpam-6063	188	25	=	=	SYM
ejpam-6063	188	26	∑	∑	PUNCT
ejpam-6063	188	27	i∈[k]\{k}mi	i∈[k]\{k}mi	X
ejpam-6063	188	28	.	.	PUNCT
ejpam-6063	189	1	the	the	DET
ejpam-6063	189	2	next	next	ADJ
ejpam-6063	189	3	result	result	NOUN
ejpam-6063	189	4	follows	follow	VERB
ejpam-6063	189	5	from	from	ADP
ejpam-6063	189	6	theorem	theorem	ADJ
ejpam-6063	189	7	6	6	NUM
ejpam-6063	189	8	.	.	PUNCT
ejpam-6063	190	1	j.	j.	PROPN
ejpam-6063	190	2	hassan	hassan	PROPN
ejpam-6063	190	3	et	et	PROPN
ejpam-6063	190	4	.	.	PUNCT
ejpam-6063	191	1	al	al	PROPN
ejpam-6063	191	2	/	/	PUNCT
ejpam-6063	191	3	eur	eur	PROPN
ejpam-6063	191	4	.	.	PUNCT
ejpam-6063	192	1	j.	j.	PROPN
ejpam-6063	192	2	pure	pure	PROPN
ejpam-6063	192	3	appl	appl	PROPN
ejpam-6063	192	4	.	.	PROPN
ejpam-6063	192	5	math	math	PROPN
ejpam-6063	192	6	,	,	PUNCT
ejpam-6063	192	7	18	18	NUM
ejpam-6063	192	8	(	(	PUNCT
ejpam-6063	192	9	2	2	NUM
ejpam-6063	192	10	)	)	PUNCT
ejpam-6063	192	11	(	(	PUNCT
ejpam-6063	192	12	2025	2025	NUM
ejpam-6063	192	13	)	)	PUNCT
ejpam-6063	192	14	,	,	PUNCT
ejpam-6063	192	15	6063	6063	NUM
ejpam-6063	192	16	6	6	NUM
ejpam-6063	192	17	of	of	ADP
ejpam-6063	192	18	11	11	NUM
ejpam-6063	192	19	corollary	corollary	ADJ
ejpam-6063	192	20	2	2	NUM
ejpam-6063	192	21	.	.	PUNCT
ejpam-6063	193	1	let	let	VERB
ejpam-6063	193	2	m	m	PRON
ejpam-6063	193	3	and	and	CCONJ
ejpam-6063	193	4	n	n	ADV
ejpam-6063	193	5	be	be	VERB
ejpam-6063	193	6	positive	positive	ADJ
ejpam-6063	193	7	integers	integer	NOUN
ejpam-6063	193	8	such	such	ADJ
ejpam-6063	193	9	that	that	SCONJ
ejpam-6063	193	10	2	2	NUM
ejpam-6063	193	11	≤	≤	NUM
ejpam-6063	193	12	m	m	VERB
ejpam-6063	193	13	≤	≤	NOUN
ejpam-6063	193	14	n.	n.	NOUN
ejpam-6063	193	15	for	for	ADP
ejpam-6063	193	16	the	the	DET
ejpam-6063	193	17	complete	complete	ADJ
ejpam-6063	193	18	bipartite	bipartite	PROPN
ejpam-6063	193	19	km	km	PROPN
ejpam-6063	193	20	,	,	PUNCT
ejpam-6063	193	21	n	n	CCONJ
ejpam-6063	193	22	,	,	PUNCT
ejpam-6063	193	23	we	we	PRON
ejpam-6063	193	24	have	have	VERB
ejpam-6063	193	25	β2(km	β2(km	NUM
ejpam-6063	193	26	,	,	PUNCT
ejpam-6063	193	27	n	n	CCONJ
ejpam-6063	193	28	)	)	PUNCT
ejpam-6063	193	29	=	=	VERB
ejpam-6063	193	30	m.	m.	NOUN
ejpam-6063	193	31	remark	remark	NOUN
ejpam-6063	193	32	1	1	NUM
ejpam-6063	193	33	.	.	PUNCT
ejpam-6063	194	1	let	let	VERB
ejpam-6063	194	2	g	g	PRON
ejpam-6063	194	3	be	be	AUX
ejpam-6063	194	4	a	a	DET
ejpam-6063	194	5	graph	graph	NOUN
ejpam-6063	194	6	and	and	CCONJ
ejpam-6063	194	7	let	let	VERB
ejpam-6063	194	8	s	s	PRON
ejpam-6063	194	9	be	be	AUX
ejpam-6063	194	10	a	a	DET
ejpam-6063	194	11	2	2	NUM
ejpam-6063	194	12	-	-	PUNCT
ejpam-6063	194	13	vertex	vertex	NOUN
ejpam-6063	194	14	cover	cover	NOUN
ejpam-6063	194	15	of	of	ADP
ejpam-6063	194	16	g.	g.	PROPN
ejpam-6063	194	17	then	then	ADV
ejpam-6063	194	18	l(g)∪	l(g)∪	PROPN
ejpam-6063	194	19	i(g	i(g	NOUN
ejpam-6063	194	20	)	)	PUNCT
ejpam-6063	195	1	⊆	⊆	NUM
ejpam-6063	195	2	s.	s.	PROPN
ejpam-6063	195	3	theorem	theorem	VERB
ejpam-6063	195	4	7	7	NUM
ejpam-6063	195	5	.	.	PUNCT
ejpam-6063	196	1	let	let	VERB
ejpam-6063	196	2	a	a	PRON
ejpam-6063	196	3	and	and	CCONJ
ejpam-6063	196	4	b	b	NOUN
ejpam-6063	196	5	be	be	AUX
ejpam-6063	196	6	positive	positive	ADJ
ejpam-6063	196	7	integers	integer	NOUN
ejpam-6063	196	8	such	such	ADJ
ejpam-6063	196	9	that	that	SCONJ
ejpam-6063	196	10	3	3	NUM
ejpam-6063	196	11	≤	≤	NOUN
ejpam-6063	196	12	a	a	DET
ejpam-6063	196	13	≤	≤	PROPN
ejpam-6063	196	14	b.	b.	NOUN
ejpam-6063	197	1	then	then	ADV
ejpam-6063	197	2	there	there	PRON
ejpam-6063	197	3	exists	exist	VERB
ejpam-6063	197	4	a	a	DET
ejpam-6063	197	5	connected	connected	ADJ
ejpam-6063	197	6	graph	graph	NOUN
ejpam-6063	197	7	g	g	ADP
ejpam-6063	197	8	such	such	ADJ
ejpam-6063	197	9	that	that	PRON
ejpam-6063	197	10	β(g	β(g	PROPN
ejpam-6063	197	11	)	)	PUNCT
ejpam-6063	197	12	=	=	SYM
ejpam-6063	197	13	a	a	PRON
ejpam-6063	197	14	and	and	CCONJ
ejpam-6063	197	15	β2(g	β2(g	NUM
ejpam-6063	197	16	)	)	PUNCT
ejpam-6063	197	17	=	=	SYM
ejpam-6063	197	18	b.	b.	NOUN
ejpam-6063	197	19	proof	proof	NOUN
ejpam-6063	197	20	.	.	PUNCT
ejpam-6063	198	1	if	if	SCONJ
ejpam-6063	198	2	a	a	DET
ejpam-6063	198	3	=	=	SYM
ejpam-6063	198	4	b	b	NOUN
ejpam-6063	198	5	,	,	PUNCT
ejpam-6063	198	6	then	then	ADV
ejpam-6063	198	7	consider	consider	VERB
ejpam-6063	198	8	g	g	NOUN
ejpam-6063	198	9	=	=	SYM
ejpam-6063	198	10	ka+1	ka+1	PROPN
ejpam-6063	198	11	.	.	PUNCT
ejpam-6063	199	1	then	then	ADV
ejpam-6063	199	2	β(g	β(g	NUM
ejpam-6063	199	3	)	)	PUNCT
ejpam-6063	200	1	=	=	SYM
ejpam-6063	200	2	a	a	PROPN
ejpam-6063	200	3	and	and	CCONJ
ejpam-6063	200	4	,	,	PUNCT
ejpam-6063	200	5	by	by	ADP
ejpam-6063	200	6	corollary	corollary	ADJ
ejpam-6063	200	7	1(i	1(i	NUM
ejpam-6063	200	8	)	)	PUNCT
ejpam-6063	200	9	,	,	PUNCT
ejpam-6063	200	10	β2(g	β2(g	NUM
ejpam-6063	200	11	)	)	PUNCT
ejpam-6063	200	12	=	=	VERB
ejpam-6063	200	13	a.	a.	NOUN
ejpam-6063	200	14	next	next	ADV
ejpam-6063	200	15	,	,	PUNCT
ejpam-6063	200	16	suppose	suppose	VERB
ejpam-6063	200	17	a	a	DET
ejpam-6063	200	18	<	<	X
ejpam-6063	200	19	b	b	NOUN
ejpam-6063	200	20	and	and	CCONJ
ejpam-6063	200	21	letm	letm	NOUN
ejpam-6063	200	22	=	=	SYM
ejpam-6063	200	23	b−a	b−a	PROPN
ejpam-6063	200	24	.	.	PUNCT
ejpam-6063	201	1	let	let	VERB
ejpam-6063	201	2	g	g	NOUN
ejpam-6063	201	3	be	be	AUX
ejpam-6063	201	4	the	the	DET
ejpam-6063	201	5	graph	graph	NOUN
ejpam-6063	201	6	obtained	obtain	VERB
ejpam-6063	201	7	fromka+1	fromka+1	PROPN
ejpam-6063	201	8	by	by	ADP
ejpam-6063	201	9	adding	add	VERB
ejpam-6063	201	10	m	m	PRON
ejpam-6063	201	11	pendant	pendant	ADJ
ejpam-6063	201	12	edges	edge	NOUN
ejpam-6063	201	13	v1x1	v1x1	NOUN
ejpam-6063	201	14	,	,	PUNCT
ejpam-6063	201	15	v1x2	v1x2	NOUN
ejpam-6063	201	16	,	,	PUNCT
ejpam-6063	201	17	.	.	PUNCT
ejpam-6063	201	18	.	.	PUNCT
ejpam-6063	202	1	.	.	PUNCT
ejpam-6063	203	1	,	,	PUNCT
ejpam-6063	203	2	v1xm	v1xm	PROPN
ejpam-6063	203	3	,	,	PUNCT
ejpam-6063	203	4	where	where	SCONJ
ejpam-6063	203	5	v	v	X
ejpam-6063	203	6	(	(	PUNCT
ejpam-6063	203	7	ka+1	ka+1	PROPN
ejpam-6063	203	8	)	)	PUNCT
ejpam-6063	203	9	=	=	NOUN
ejpam-6063	203	10	{	{	PUNCT
ejpam-6063	203	11	v1	v1	PROPN
ejpam-6063	203	12	,	,	PUNCT
ejpam-6063	203	13	v2	v2	PROPN
ejpam-6063	203	14	,	,	PUNCT
ejpam-6063	203	15	·	·	PUNCT
ejpam-6063	203	16	·	·	PUNCT
ejpam-6063	203	17	·	·	PUNCT
ejpam-6063	203	18	,	,	PUNCT
ejpam-6063	203	19	va	va	NOUN
ejpam-6063	203	20	,	,	PUNCT
ejpam-6063	203	21	va+1	va+1	ADJ
ejpam-6063	203	22	}	}	PUNCT
ejpam-6063	203	23	(	(	PUNCT
ejpam-6063	203	24	see	see	VERB
ejpam-6063	203	25	figure	figure	NOUN
ejpam-6063	203	26	2	2	NUM
ejpam-6063	203	27	)	)	PUNCT
ejpam-6063	203	28	.	.	PUNCT
ejpam-6063	204	1	clearly	clearly	ADV
ejpam-6063	204	2	,	,	PUNCT
ejpam-6063	204	3	s1	s1	PROPN
ejpam-6063	204	4	=	=	PUNCT
ejpam-6063	204	5	{	{	PUNCT
ejpam-6063	204	6	v1	v1	PROPN
ejpam-6063	204	7	,	,	PUNCT
ejpam-6063	204	8	v2	v2	PROPN
ejpam-6063	204	9	,	,	PUNCT
ejpam-6063	204	10	·	·	PUNCT
ejpam-6063	204	11	·	·	PUNCT
ejpam-6063	204	12	·	·	PUNCT
ejpam-6063	204	13	,	,	PUNCT
ejpam-6063	204	14	va	va	NOUN
ejpam-6063	204	15	}	}	PUNCT
ejpam-6063	204	16	is	be	AUX
ejpam-6063	204	17	a	a	DET
ejpam-6063	204	18	vertex	vertex	NOUN
ejpam-6063	204	19	cover	cover	NOUN
ejpam-6063	204	20	of	of	ADP
ejpam-6063	204	21	g.	g.	PROPN
ejpam-6063	204	22	hence	hence	ADV
ejpam-6063	204	23	,	,	PUNCT
ejpam-6063	204	24	β(g	β(g	PROPN
ejpam-6063	204	25	)	)	PUNCT
ejpam-6063	204	26	≤	≤	NUM
ejpam-6063	204	27	|s1|	|s1|	NOUN
ejpam-6063	204	28	=	=	PUNCT
ejpam-6063	204	29	a.	a.	NOUN
ejpam-6063	204	30	let	let	VERB
ejpam-6063	204	31	s	s	PRON
ejpam-6063	204	32	be	be	AUX
ejpam-6063	204	33	a	a	DET
ejpam-6063	204	34	β	β	NOUN
ejpam-6063	204	35	-	-	VERB
ejpam-6063	204	36	set	set	VERB
ejpam-6063	204	37	in	in	ADP
ejpam-6063	204	38	g.	g.	PROPN
ejpam-6063	204	39	if	if	SCONJ
ejpam-6063	204	40	v1	v1	PROPN
ejpam-6063	204	41	/∈	/∈	PUNCT
ejpam-6063	205	1	s	s	X
ejpam-6063	205	2	,	,	PUNCT
ejpam-6063	205	3	then	then	ADV
ejpam-6063	205	4	{	{	PUNCT
ejpam-6063	205	5	v2	v2	PROPN
ejpam-6063	205	6	,	,	PUNCT
ejpam-6063	205	7	v3	v3	PROPN
ejpam-6063	205	8	,	,	PUNCT
ejpam-6063	205	9	·	·	PUNCT
ejpam-6063	205	10	·	·	PUNCT
ejpam-6063	205	11	·	·	PUNCT
ejpam-6063	205	12	,	,	PUNCT
ejpam-6063	205	13	va+1	va+1	X
ejpam-6063	205	14	}	}	PUNCT
ejpam-6063	205	15	⊆	⊆	NUM
ejpam-6063	205	16	s	s	NOUN
ejpam-6063	205	17	since	since	SCONJ
ejpam-6063	205	18	s	s	NOUN
ejpam-6063	205	19	is	be	AUX
ejpam-6063	205	20	a	a	DET
ejpam-6063	205	21	vertex	vertex	NOUN
ejpam-6063	205	22	cover	cover	NOUN
ejpam-6063	205	23	of	of	ADP
ejpam-6063	205	24	g.	g.	PROPN
ejpam-6063	205	25	again	again	ADV
ejpam-6063	205	26	,	,	PUNCT
ejpam-6063	205	27	since	since	SCONJ
ejpam-6063	205	28	s	s	NOUN
ejpam-6063	205	29	is	be	AUX
ejpam-6063	205	30	a	a	DET
ejpam-6063	205	31	vertex	vertex	NOUN
ejpam-6063	205	32	cover	cover	NOUN
ejpam-6063	205	33	of	of	ADP
ejpam-6063	205	34	g	g	NOUN
ejpam-6063	205	35	,	,	PUNCT
ejpam-6063	205	36	it	it	PRON
ejpam-6063	205	37	follows	follow	VERB
ejpam-6063	205	38	that	that	SCONJ
ejpam-6063	205	39	{	{	PUNCT
ejpam-6063	205	40	x1	x1	ADJ
ejpam-6063	205	41	,	,	PUNCT
ejpam-6063	205	42	x2	x2	PROPN
ejpam-6063	205	43	,	,	PUNCT
ejpam-6063	205	44	·	·	PUNCT
ejpam-6063	205	45	·	·	PUNCT
ejpam-6063	205	46	·	·	PUNCT
ejpam-6063	205	47	,	,	PUNCT
ejpam-6063	205	48	xm	xm	X
ejpam-6063	205	49	}	}	PUNCT
ejpam-6063	205	50	⊆	⊆	NUM
ejpam-6063	205	51	s.	s.	PROPN
ejpam-6063	205	52	thus	thus	ADV
ejpam-6063	205	53	,	,	PUNCT
ejpam-6063	205	54	s	s	VERB
ejpam-6063	205	55	=	=	PUNCT
ejpam-6063	205	56	{	{	PUNCT
ejpam-6063	205	57	x1	x1	PROPN
ejpam-6063	205	58	,	,	PUNCT
ejpam-6063	205	59	x2	x2	PROPN
ejpam-6063	205	60	,	,	PUNCT
ejpam-6063	205	61	·	·	PUNCT
ejpam-6063	205	62	·	·	PUNCT
ejpam-6063	205	63	·	·	PUNCT
ejpam-6063	205	64	,	,	PUNCT
ejpam-6063	205	65	xm	xm	PROPN
ejpam-6063	205	66	,	,	PUNCT
ejpam-6063	205	67	v2	v2	PROPN
ejpam-6063	205	68	,	,	PUNCT
ejpam-6063	205	69	v3	v3	PROPN
ejpam-6063	205	70	,	,	PUNCT
ejpam-6063	205	71	·	·	PUNCT
ejpam-6063	205	72	·	·	PUNCT
ejpam-6063	205	73	·	·	PUNCT
ejpam-6063	205	74	,	,	PUNCT
ejpam-6063	205	75	va+1	va+1	ADJ
ejpam-6063	205	76	}	}	PUNCT
ejpam-6063	205	77	.	.	PUNCT
ejpam-6063	206	1	consequently	consequently	ADV
ejpam-6063	206	2	,	,	PUNCT
ejpam-6063	206	3	β(g	β(g	PROPN
ejpam-6063	206	4	)	)	PUNCT
ejpam-6063	206	5	=	=	SYM
ejpam-6063	206	6	|s|	|s|	PROPN
ejpam-6063	206	7	=	=	SYM
ejpam-6063	206	8	m+	m+	NUM
ejpam-6063	206	9	a	a	DET
ejpam-6063	206	10	=	=	NOUN
ejpam-6063	206	11	b−	b−	PROPN
ejpam-6063	206	12	a+	a+	PUNCT
ejpam-6063	206	13	a	a	PRON
ejpam-6063	206	14	=	=	SYM
ejpam-6063	206	15	b	b	PROPN
ejpam-6063	206	16	,	,	PUNCT
ejpam-6063	206	17	which	which	PRON
ejpam-6063	206	18	is	be	AUX
ejpam-6063	206	19	not	not	PART
ejpam-6063	206	20	possible	possible	ADJ
ejpam-6063	206	21	.	.	PUNCT
ejpam-6063	207	1	thus	thus	ADV
ejpam-6063	207	2	,	,	PUNCT
ejpam-6063	207	3	v1	v1	PROPN
ejpam-6063	207	4	∈	∈	PROPN
ejpam-6063	207	5	s.	s.	PROPN
ejpam-6063	207	6	suppose	suppose	VERB
ejpam-6063	207	7	|(v	|(v	PROPN
ejpam-6063	207	8	(	(	PUNCT
ejpam-6063	207	9	ka+1	ka+1	PROPN
ejpam-6063	207	10	)	)	PUNCT
ejpam-6063	207	11	\	\	NOUN
ejpam-6063	207	12	{	{	PUNCT
ejpam-6063	207	13	v1	v1	NOUN
ejpam-6063	207	14	}	}	PUNCT
ejpam-6063	207	15	)	)	PUNCT
ejpam-6063	207	16	∩	∩	NOUN
ejpam-6063	207	17	s|	s|	VERB
ejpam-6063	207	18	<	<	X
ejpam-6063	207	19	a−	a−	PROPN
ejpam-6063	207	20	1	1	NUM
ejpam-6063	207	21	.	.	PUNCT
ejpam-6063	208	1	then	then	ADV
ejpam-6063	208	2	there	there	PRON
ejpam-6063	208	3	exist	exist	VERB
ejpam-6063	208	4	r	r	NOUN
ejpam-6063	208	5	,	,	PUNCT
ejpam-6063	208	6	t	t	PROPN
ejpam-6063	208	7	∈	∈	PROPN
ejpam-6063	208	8	{	{	PUNCT
ejpam-6063	208	9	2	2	NUM
ejpam-6063	208	10	,	,	PUNCT
ejpam-6063	208	11	3	3	NUM
ejpam-6063	208	12	,	,	PUNCT
ejpam-6063	208	13	·	·	PUNCT
ejpam-6063	208	14	·	·	PUNCT
ejpam-6063	208	15	·	·	PUNCT
ejpam-6063	208	16	,	,	PUNCT
ejpam-6063	208	17	a	a	DET
ejpam-6063	208	18	+	+	NOUN
ejpam-6063	208	19	1	1	NUM
ejpam-6063	208	20	}	}	PUNCT
ejpam-6063	208	21	such	such	ADJ
ejpam-6063	208	22	that	that	SCONJ
ejpam-6063	208	23	vr	vr	PROPN
ejpam-6063	208	24	,	,	PUNCT
ejpam-6063	208	25	vt	vt	PROPN
ejpam-6063	208	26	/∈	/∈	PUNCT
ejpam-6063	208	27	s.	s.	PROPN
ejpam-6063	209	1	this	this	PRON
ejpam-6063	209	2	,	,	PUNCT
ejpam-6063	209	3	however	however	ADV
ejpam-6063	209	4	,	,	PUNCT
ejpam-6063	209	5	is	be	AUX
ejpam-6063	209	6	not	not	PART
ejpam-6063	209	7	possible	possible	ADJ
ejpam-6063	209	8	because	because	SCONJ
ejpam-6063	209	9	vrvt	vrvt	PROPN
ejpam-6063	209	10	∈	∈	PROPN
ejpam-6063	209	11	e(g	e(g	PROPN
ejpam-6063	209	12	)	)	PUNCT
ejpam-6063	209	13	and	and	CCONJ
ejpam-6063	209	14	s	s	VERB
ejpam-6063	209	15	is	be	AUX
ejpam-6063	209	16	a	a	DET
ejpam-6063	209	17	vertex	vertex	NOUN
ejpam-6063	209	18	cover	cover	NOUN
ejpam-6063	209	19	.	.	PUNCT
ejpam-6063	210	1	therefore	therefore	ADV
ejpam-6063	210	2	,	,	PUNCT
ejpam-6063	210	3	|(v	|(v	PROPN
ejpam-6063	210	4	(	(	PUNCT
ejpam-6063	210	5	ka+1)\{v1})∩s|	ka+1)\{v1})∩s|	PROPN
ejpam-6063	210	6	=	=	SYM
ejpam-6063	210	7	a−1	a−1	PROPN
ejpam-6063	210	8	.	.	PUNCT
ejpam-6063	211	1	therefore	therefore	ADV
ejpam-6063	211	2	,	,	PUNCT
ejpam-6063	211	3	since	since	SCONJ
ejpam-6063	211	4	s	s	NOUN
ejpam-6063	211	5	is	be	AUX
ejpam-6063	211	6	a	a	DET
ejpam-6063	211	7	β	β	NOUN
ejpam-6063	211	8	-	-	VERB
ejpam-6063	211	9	set	set	VERB
ejpam-6063	211	10	in	in	ADP
ejpam-6063	211	11	g	g	NOUN
ejpam-6063	211	12	,	,	PUNCT
ejpam-6063	211	13	β(g	β(g	PROPN
ejpam-6063	211	14	)	)	PUNCT
ejpam-6063	212	1	=	=	SYM
ejpam-6063	212	2	|s|	|s|	NOUN
ejpam-6063	212	3	=	=	NOUN
ejpam-6063	212	4	a.	a.	NOUN
ejpam-6063	212	5	let	let	VERB
ejpam-6063	212	6	d	d	PRON
ejpam-6063	212	7	be	be	AUX
ejpam-6063	212	8	a	a	DET
ejpam-6063	212	9	β2	β2	NOUN
ejpam-6063	212	10	-	-	PUNCT
ejpam-6063	212	11	set	set	NOUN
ejpam-6063	212	12	in	in	ADP
ejpam-6063	212	13	g.	g.	NOUN
ejpam-6063	212	14	by	by	ADP
ejpam-6063	212	15	remark	remark	NOUN
ejpam-6063	212	16	1	1	NUM
ejpam-6063	212	17	,	,	PUNCT
ejpam-6063	212	18	{	{	PUNCT
ejpam-6063	212	19	x1	x1	PROPN
ejpam-6063	212	20	,	,	PUNCT
ejpam-6063	212	21	x2	x2	PROPN
ejpam-6063	212	22	,	,	PUNCT
ejpam-6063	212	23	·	·	PUNCT
ejpam-6063	212	24	·	·	PUNCT
ejpam-6063	212	25	·	·	PUNCT
ejpam-6063	212	26	,	,	PUNCT
ejpam-6063	212	27	xm	xm	X
ejpam-6063	212	28	}	}	PUNCT
ejpam-6063	212	29	⊆	⊆	NUM
ejpam-6063	212	30	s.	s.	PROPN
ejpam-6063	212	31	now	now	ADV
ejpam-6063	212	32	suppose	suppose	VERB
ejpam-6063	212	33	v1	v1	PROPN
ejpam-6063	212	34	/∈	/∈	PUNCT
ejpam-6063	213	1	s.	s.	PROPN
ejpam-6063	213	2	since	since	SCONJ
ejpam-6063	213	3	d	d	PROPN
ejpam-6063	213	4	is	be	AUX
ejpam-6063	213	5	a	a	DET
ejpam-6063	213	6	vertex	vertex	NOUN
ejpam-6063	213	7	cover	cover	NOUN
ejpam-6063	213	8	of	of	ADP
ejpam-6063	213	9	g	g	NOUN
ejpam-6063	213	10	,	,	PUNCT
ejpam-6063	213	11	it	it	PRON
ejpam-6063	213	12	follows	follow	VERB
ejpam-6063	213	13	that	that	SCONJ
ejpam-6063	213	14	{	{	PUNCT
ejpam-6063	213	15	v2	v2	PROPN
ejpam-6063	213	16	,	,	PUNCT
ejpam-6063	213	17	v3	v3	PROPN
ejpam-6063	213	18	,	,	PUNCT
ejpam-6063	213	19	·	·	PUNCT
ejpam-6063	213	20	·	·	PUNCT
ejpam-6063	213	21	·	·	PUNCT
ejpam-6063	213	22	,	,	PUNCT
ejpam-6063	213	23	va+1	va+1	ADJ
ejpam-6063	213	24	}	}	PUNCT
ejpam-6063	213	25	⊆	⊆	NUM
ejpam-6063	213	26	d.	d.	NOUN
ejpam-6063	213	27	hence	hence	ADV
ejpam-6063	213	28	,	,	PUNCT
ejpam-6063	213	29	d	d	PROPN
ejpam-6063	213	30	=	=	PRON
ejpam-6063	213	31	{	{	PUNCT
ejpam-6063	213	32	x1	x1	PROPN
ejpam-6063	213	33	,	,	PUNCT
ejpam-6063	213	34	x2	x2	PROPN
ejpam-6063	213	35	,	,	PUNCT
ejpam-6063	213	36	·	·	PUNCT
ejpam-6063	213	37	·	·	PUNCT
ejpam-6063	213	38	·	·	PUNCT
ejpam-6063	213	39	,	,	PUNCT
ejpam-6063	213	40	xm	xm	PROPN
ejpam-6063	213	41	,	,	PUNCT
ejpam-6063	213	42	v2	v2	PROPN
ejpam-6063	213	43	,	,	PUNCT
ejpam-6063	213	44	v3	v3	PROPN
ejpam-6063	213	45	,	,	PUNCT
ejpam-6063	213	46	·	·	PUNCT
ejpam-6063	213	47	·	·	PUNCT
ejpam-6063	213	48	·	·	PUNCT
ejpam-6063	213	49	,	,	PUNCT
ejpam-6063	213	50	va+1	va+1	ADJ
ejpam-6063	213	51	}	}	PUNCT
ejpam-6063	213	52	.	.	PUNCT
ejpam-6063	214	1	this	this	PRON
ejpam-6063	214	2	implies	imply	VERB
ejpam-6063	214	3	that	that	SCONJ
ejpam-6063	214	4	β2(g	β2(g	NUM
ejpam-6063	214	5	)	)	PUNCT
ejpam-6063	214	6	=	=	SYM
ejpam-6063	214	7	|d|	|d|	PROPN
ejpam-6063	214	8	=	=	SYM
ejpam-6063	215	1	m	m	VERB
ejpam-6063	215	2	+	+	NUM
ejpam-6063	215	3	a	a	DET
ejpam-6063	215	4	=	=	X
ejpam-6063	215	5	b.	b.	PROPN
ejpam-6063	215	6	suppose	suppose	VERB
ejpam-6063	215	7	v1	v1	PROPN
ejpam-6063	215	8	∈	∈	PROPN
ejpam-6063	215	9	d.	d.	NOUN
ejpam-6063	215	10	since	since	SCONJ
ejpam-6063	215	11	d	d	PROPN
ejpam-6063	215	12	is	be	AUX
ejpam-6063	215	13	a	a	DET
ejpam-6063	215	14	vertex	vertex	NOUN
ejpam-6063	215	15	cover	cover	NOUN
ejpam-6063	215	16	of	of	ADP
ejpam-6063	215	17	g	g	NOUN
ejpam-6063	215	18	,	,	PUNCT
ejpam-6063	215	19	|v	|v	PROPN
ejpam-6063	215	20	(	(	PUNCT
ejpam-6063	215	21	ka+1	ka+1	PROPN
ejpam-6063	215	22	)	)	PUNCT
ejpam-6063	215	23	\d|	\d|	NOUN
ejpam-6063	215	24	≤	≤	NUM
ejpam-6063	215	25	1	1	NUM
ejpam-6063	215	26	.	.	PUNCT
ejpam-6063	216	1	the	the	DET
ejpam-6063	216	2	assumption	assumption	NOUN
ejpam-6063	216	3	that	that	SCONJ
ejpam-6063	216	4	d	d	NOUN
ejpam-6063	216	5	is	be	AUX
ejpam-6063	216	6	a	a	DET
ejpam-6063	216	7	β2	β2	NOUN
ejpam-6063	216	8	-	-	PUNCT
ejpam-6063	216	9	set	set	NOUN
ejpam-6063	216	10	in	in	ADP
ejpam-6063	216	11	g	g	PROPN
ejpam-6063	216	12	forces	force	NOUN
ejpam-6063	216	13	|v	|v	X
ejpam-6063	216	14	(	(	PUNCT
ejpam-6063	216	15	ka+1	ka+1	PROPN
ejpam-6063	216	16	)	)	PUNCT
ejpam-6063	216	17	\d|	\d|	NOUN
ejpam-6063	216	18	=	=	SYM
ejpam-6063	216	19	1	1	X
ejpam-6063	216	20	.	.	PUNCT
ejpam-6063	216	21	therefore	therefore	ADV
ejpam-6063	216	22	,	,	PUNCT
ejpam-6063	216	23	β2(g	β2(g	NUM
ejpam-6063	216	24	)	)	PUNCT
ejpam-6063	216	25	=	=	SYM
ejpam-6063	216	26	|d|	|d|	PROPN
ejpam-6063	216	27	=	=	PUNCT
ejpam-6063	216	28	m+	m+	PRON
ejpam-6063	216	29	a	a	DET
ejpam-6063	216	30	=	=	X
ejpam-6063	216	31	b.	b.	PROPN
ejpam-6063	216	32	v1	v1	PROPN
ejpam-6063	216	33	va+1vav8v7	va+1vav8v7	PROPN
ejpam-6063	216	34	v6	v6	NOUN
ejpam-6063	216	35	v5	v5	PROPN
ejpam-6063	216	36	v4	v4	PROPN
ejpam-6063	216	37	v3	v3	PROPN
ejpam-6063	216	38	v2	v2	PROPN
ejpam-6063	217	1	x1	x1	PROPN
ejpam-6063	217	2	xm	xm	PROPN
ejpam-6063	218	1	xm−1	xm−1	PROPN
ejpam-6063	218	2	x3	x3	PROPN
ejpam-6063	218	3	x2	x2	PROPN
ejpam-6063	218	4	...	...	PUNCT
ejpam-6063	218	5	...	...	PUNCT
ejpam-6063	219	1	therefore	therefore	ADV
ejpam-6063	219	2	,	,	PUNCT
ejpam-6063	219	3	the	the	DET
ejpam-6063	219	4	assertion	assertion	NOUN
ejpam-6063	219	5	holds	hold	VERB
ejpam-6063	219	6	.	.	PUNCT
ejpam-6063	220	1	the	the	DET
ejpam-6063	220	2	next	next	ADJ
ejpam-6063	220	3	result	result	NOUN
ejpam-6063	220	4	is	be	AUX
ejpam-6063	220	5	a	a	DET
ejpam-6063	220	6	consequence	consequence	NOUN
ejpam-6063	220	7	of	of	ADP
ejpam-6063	220	8	theorem	theorem	ADJ
ejpam-6063	220	9	7	7	NUM
ejpam-6063	220	10	.	.	PUNCT
ejpam-6063	220	11	corollary	corollary	ADJ
ejpam-6063	220	12	3	3	X
ejpam-6063	220	13	.	.	PUNCT
ejpam-6063	221	1	let	let	VERB
ejpam-6063	221	2	n	n	PRON
ejpam-6063	221	3	be	be	AUX
ejpam-6063	221	4	a	a	DET
ejpam-6063	221	5	positive	positive	ADJ
ejpam-6063	221	6	integer	integer	NOUN
ejpam-6063	221	7	.	.	PUNCT
ejpam-6063	222	1	then	then	ADV
ejpam-6063	222	2	there	there	PRON
ejpam-6063	222	3	exists	exist	VERB
ejpam-6063	222	4	a	a	DET
ejpam-6063	222	5	connected	connected	ADJ
ejpam-6063	222	6	graph	graph	NOUN
ejpam-6063	222	7	g	g	ADP
ejpam-6063	222	8	such	such	ADJ
ejpam-6063	222	9	that	that	SCONJ
ejpam-6063	222	10	β2(g	β2(g	NUM
ejpam-6063	222	11	)	)	PUNCT
ejpam-6063	222	12	−	−	ADP
ejpam-6063	222	13	β(g	β(g	PROPN
ejpam-6063	222	14	)	)	PUNCT
ejpam-6063	222	15	=	=	VERB
ejpam-6063	223	1	n.	n.	NOUN
ejpam-6063	223	2	in	in	ADP
ejpam-6063	223	3	other	other	ADJ
ejpam-6063	223	4	words	word	NOUN
ejpam-6063	223	5	,	,	PUNCT
ejpam-6063	223	6	the	the	DET
ejpam-6063	223	7	difference	difference	NOUN
ejpam-6063	223	8	β2(g	β2(g	NUM
ejpam-6063	223	9	)	)	PUNCT
ejpam-6063	223	10	−	−	ADP
ejpam-6063	223	11	β(g	β(g	PROPN
ejpam-6063	223	12	)	)	PUNCT
ejpam-6063	223	13	can	can	AUX
ejpam-6063	223	14	be	be	AUX
ejpam-6063	223	15	made	make	VERB
ejpam-6063	223	16	arbitrarily	arbitrarily	ADV
ejpam-6063	223	17	large	large	ADJ
ejpam-6063	223	18	.	.	PUNCT
ejpam-6063	224	1	theorem	theorem	ADJ
ejpam-6063	224	2	8	8	NUM
ejpam-6063	224	3	.	.	PUNCT
ejpam-6063	225	1	let	let	VERB
ejpam-6063	225	2	h	h	PRON
ejpam-6063	225	3	be	be	AUX
ejpam-6063	225	4	a	a	DET
ejpam-6063	225	5	non	non	ADJ
ejpam-6063	225	6	-	-	ADJ
ejpam-6063	225	7	trivial	trivial	ADJ
ejpam-6063	225	8	graph	graph	NOUN
ejpam-6063	225	9	and	and	CCONJ
ejpam-6063	225	10	let	let	VERB
ejpam-6063	225	11	g	g	PROPN
ejpam-6063	225	12	=	=	PROPN
ejpam-6063	225	13	k1+h	k1+h	PROPN
ejpam-6063	225	14	,	,	PUNCT
ejpam-6063	226	1	where	where	SCONJ
ejpam-6063	226	2	k1	k1	NOUN
ejpam-6063	226	3	=	=	PUNCT
ejpam-6063	226	4	⟨{v}⟩.	⟨{v}⟩.	NOUN
ejpam-6063	226	5	then	then	ADV
ejpam-6063	226	6	a	a	DET
ejpam-6063	226	7	set	set	NOUN
ejpam-6063	226	8	s	s	NOUN
ejpam-6063	226	9	⊆	⊆	NUM
ejpam-6063	226	10	v	v	NOUN
ejpam-6063	226	11	(	(	PUNCT
ejpam-6063	226	12	g	g	NOUN
ejpam-6063	226	13	)	)	PUNCT
ejpam-6063	226	14	is	be	AUX
ejpam-6063	226	15	a	a	DET
ejpam-6063	226	16	2	2	NUM
ejpam-6063	226	17	-	-	PUNCT
ejpam-6063	226	18	vertex	vertex	NOUN
ejpam-6063	226	19	cover	cover	NOUN
ejpam-6063	226	20	of	of	ADP
ejpam-6063	226	21	g+h	g+h	PROPN
ejpam-6063	227	1	if	if	SCONJ
ejpam-6063	227	2	and	and	CCONJ
ejpam-6063	227	3	only	only	ADV
ejpam-6063	227	4	if	if	SCONJ
ejpam-6063	227	5	s	s	VERB
ejpam-6063	227	6	=	=	SYM
ejpam-6063	227	7	v	v	PROPN
ejpam-6063	227	8	(	(	PUNCT
ejpam-6063	227	9	h	h	NOUN
ejpam-6063	227	10	)	)	PUNCT
ejpam-6063	227	11	or	or	CCONJ
ejpam-6063	227	12	s	s	NOUN
ejpam-6063	227	13	=	=	SYM
ejpam-6063	227	14	{	{	PUNCT
ejpam-6063	227	15	v}∪dh∪i(h	v}∪dh∪i(h	PROPN
ejpam-6063	227	16	)	)	PUNCT
ejpam-6063	227	17	where	where	SCONJ
ejpam-6063	227	18	dh	dh	NOUN
ejpam-6063	227	19	is	be	AUX
ejpam-6063	227	20	a	a	DET
ejpam-6063	227	21	vertex	vertex	NOUN
ejpam-6063	227	22	cover	cover	NOUN
ejpam-6063	227	23	of	of	ADP
ejpam-6063	227	24	h	h	NOUN
ejpam-6063	227	25	such	such	ADJ
ejpam-6063	227	26	that	that	DET
ejpam-6063	227	27	dh	dh	PROPN
ejpam-6063	227	28	∩	∩	NOUN
ejpam-6063	227	29	i(g	i(g	ADV
ejpam-6063	227	30	)	)	PUNCT
ejpam-6063	227	31	=	=	PUNCT
ejpam-6063	227	32	∅.	∅.	PROPN
ejpam-6063	227	33	j.	j.	PROPN
ejpam-6063	227	34	hassan	hassan	PROPN
ejpam-6063	227	35	et	et	PROPN
ejpam-6063	227	36	.	.	PUNCT
ejpam-6063	228	1	al	al	PROPN
ejpam-6063	228	2	/	/	PUNCT
ejpam-6063	228	3	eur	eur	PROPN
ejpam-6063	228	4	.	.	PUNCT
ejpam-6063	229	1	j.	j.	PROPN
ejpam-6063	229	2	pure	pure	PROPN
ejpam-6063	229	3	appl	appl	PROPN
ejpam-6063	229	4	.	.	PROPN
ejpam-6063	229	5	math	math	PROPN
ejpam-6063	229	6	,	,	PUNCT
ejpam-6063	229	7	18	18	NUM
ejpam-6063	229	8	(	(	PUNCT
ejpam-6063	229	9	2	2	NUM
ejpam-6063	229	10	)	)	PUNCT
ejpam-6063	229	11	(	(	PUNCT
ejpam-6063	229	12	2025	2025	NUM
ejpam-6063	229	13	)	)	PUNCT
ejpam-6063	229	14	,	,	PUNCT
ejpam-6063	229	15	6063	6063	NUM
ejpam-6063	229	16	7	7	NUM
ejpam-6063	229	17	of	of	ADP
ejpam-6063	229	18	11	11	NUM
ejpam-6063	229	19	proof	proof	NOUN
ejpam-6063	229	20	.	.	PUNCT
ejpam-6063	230	1	suppose	suppose	VERB
ejpam-6063	230	2	s	s	NOUN
ejpam-6063	230	3	is	be	AUX
ejpam-6063	230	4	a	a	DET
ejpam-6063	230	5	2	2	NUM
ejpam-6063	230	6	-	-	PUNCT
ejpam-6063	230	7	vertex	vertex	NOUN
ejpam-6063	230	8	cover	cover	NOUN
ejpam-6063	230	9	of	of	ADP
ejpam-6063	230	10	g.	g.	PROPN
ejpam-6063	230	11	suppose	suppose	VERB
ejpam-6063	230	12	v	v	ADP
ejpam-6063	230	13	∈	∈	PROPN
ejpam-6063	230	14	s	s	PART
ejpam-6063	230	15	and	and	CCONJ
ejpam-6063	230	16	let	let	VERB
ejpam-6063	230	17	w	w	PROPN
ejpam-6063	230	18	∈	∈	PROPN
ejpam-6063	230	19	i(h	i(h	NOUN
ejpam-6063	230	20	)	)	PUNCT
ejpam-6063	230	21	.	.	PUNCT
ejpam-6063	231	1	since	since	SCONJ
ejpam-6063	231	2	ng(w	ng(w	NOUN
ejpam-6063	231	3	)	)	PUNCT
ejpam-6063	231	4	∩	∩	NOUN
ejpam-6063	231	5	s	s	PART
ejpam-6063	231	6	=	=	PUNCT
ejpam-6063	231	7	{	{	PUNCT
ejpam-6063	231	8	v	v	NOUN
ejpam-6063	231	9	}	}	PUNCT
ejpam-6063	231	10	and	and	CCONJ
ejpam-6063	231	11	s	s	VERB
ejpam-6063	231	12	is	be	AUX
ejpam-6063	231	13	2	2	NUM
ejpam-6063	231	14	-	-	PUNCT
ejpam-6063	231	15	dominating	dominating	NOUN
ejpam-6063	231	16	set	set	NOUN
ejpam-6063	231	17	in	in	ADP
ejpam-6063	231	18	g	g	PROPN
ejpam-6063	231	19	,	,	PUNCT
ejpam-6063	231	20	w	w	PROPN
ejpam-6063	231	21	∈	∈	PROPN
ejpam-6063	231	22	s.	s.	PROPN
ejpam-6063	231	23	thus	thus	ADV
ejpam-6063	231	24	,	,	PUNCT
ejpam-6063	231	25	i(h	i(h	NOUN
ejpam-6063	231	26	)	)	PUNCT
ejpam-6063	231	27	⊆	⊆	NUM
ejpam-6063	231	28	s.	s.	PROPN
ejpam-6063	231	29	let	let	VERB
ejpam-6063	231	30	dh	dh	NOUN
ejpam-6063	231	31	=	=	PUNCT
ejpam-6063	232	1	[	[	X
ejpam-6063	232	2	v	v	X
ejpam-6063	232	3	(	(	PUNCT
ejpam-6063	232	4	h)\	h)\	NOUN
ejpam-6063	232	5	i(h)]∩s	i(h)]∩	NOUN
ejpam-6063	232	6	.	.	PUNCT
ejpam-6063	233	1	since	since	SCONJ
ejpam-6063	233	2	s	s	PROPN
ejpam-6063	233	3	is	be	AUX
ejpam-6063	233	4	a	a	DET
ejpam-6063	233	5	vertex	vertex	NOUN
ejpam-6063	233	6	cover	cover	NOUN
ejpam-6063	233	7	of	of	ADP
ejpam-6063	233	8	g	g	NOUN
ejpam-6063	233	9	,	,	PUNCT
ejpam-6063	233	10	dh	dh	PROPN
ejpam-6063	233	11	is	be	AUX
ejpam-6063	233	12	a	a	DET
ejpam-6063	233	13	vertex	vertex	NOUN
ejpam-6063	233	14	cover	cover	NOUN
ejpam-6063	233	15	of	of	ADP
ejpam-6063	233	16	h.	h.	PROPN
ejpam-6063	233	17	thus	thus	ADV
ejpam-6063	233	18	,	,	PUNCT
ejpam-6063	233	19	(	(	PUNCT
ejpam-6063	233	20	i	i	NOUN
ejpam-6063	233	21	)	)	PUNCT
ejpam-6063	233	22	holds	hold	VERB
ejpam-6063	233	23	.	.	PUNCT
ejpam-6063	234	1	if	if	SCONJ
ejpam-6063	234	2	v	v	NUM
ejpam-6063	234	3	/∈	/∈	SYM
ejpam-6063	234	4	s	s	X
ejpam-6063	234	5	,	,	PUNCT
ejpam-6063	234	6	then	then	ADV
ejpam-6063	234	7	s	s	VERB
ejpam-6063	234	8	=	=	SYM
ejpam-6063	234	9	v	v	PROPN
ejpam-6063	234	10	(	(	PUNCT
ejpam-6063	234	11	h	h	NOUN
ejpam-6063	234	12	)	)	PUNCT
ejpam-6063	234	13	because	because	SCONJ
ejpam-6063	234	14	vp	vp	PROPN
ejpam-6063	234	15	∈	∈	PROPN
ejpam-6063	234	16	e(g	e(g	PROPN
ejpam-6063	234	17	)	)	PUNCT
ejpam-6063	234	18	for	for	ADP
ejpam-6063	234	19	all	all	DET
ejpam-6063	234	20	p	p	PROPN
ejpam-6063	234	21	∈	∈	PROPN
ejpam-6063	234	22	v	v	NOUN
ejpam-6063	234	23	(	(	PUNCT
ejpam-6063	234	24	h	h	NOUN
ejpam-6063	234	25	)	)	PUNCT
ejpam-6063	234	26	and	and	CCONJ
ejpam-6063	234	27	s	s	VERB
ejpam-6063	234	28	is	be	AUX
ejpam-6063	234	29	a	a	DET
ejpam-6063	234	30	vertex	vertex	NOUN
ejpam-6063	234	31	cover	cover	NOUN
ejpam-6063	234	32	of	of	ADP
ejpam-6063	234	33	g.	g.	PROPN
ejpam-6063	234	34	this	this	PRON
ejpam-6063	234	35	implies	imply	VERB
ejpam-6063	234	36	that	that	SCONJ
ejpam-6063	234	37	(	(	PUNCT
ejpam-6063	234	38	ii	ii	NOUN
ejpam-6063	234	39	)	)	PUNCT
ejpam-6063	234	40	holds	hold	VERB
ejpam-6063	234	41	.	.	PUNCT
ejpam-6063	235	1	for	for	ADP
ejpam-6063	235	2	the	the	DET
ejpam-6063	235	3	converse	converse	NOUN
ejpam-6063	235	4	,	,	PUNCT
ejpam-6063	235	5	suppose	suppose	VERB
ejpam-6063	235	6	that	that	SCONJ
ejpam-6063	235	7	(	(	PUNCT
ejpam-6063	235	8	i	i	NOUN
ejpam-6063	235	9	)	)	PUNCT
ejpam-6063	235	10	holds	hold	VERB
ejpam-6063	235	11	.	.	PUNCT
ejpam-6063	236	1	clearly	clearly	ADV
ejpam-6063	236	2	,	,	PUNCT
ejpam-6063	236	3	s	s	VERB
ejpam-6063	236	4	is	be	AUX
ejpam-6063	236	5	a	a	DET
ejpam-6063	236	6	vertex	vertex	NOUN
ejpam-6063	236	7	cover	cover	NOUN
ejpam-6063	236	8	of	of	ADP
ejpam-6063	236	9	g.	g.	PROPN
ejpam-6063	236	10	let	let	VERB
ejpam-6063	236	11	x	x	SYM
ejpam-6063	236	12	∈	∈	PROPN
ejpam-6063	236	13	v	v	X
ejpam-6063	236	14	(	(	PUNCT
ejpam-6063	236	15	g	g	NOUN
ejpam-6063	236	16	)	)	PUNCT
ejpam-6063	236	17	\	\	PUNCT
ejpam-6063	237	1	s.	s.	PROPN
ejpam-6063	237	2	then	then	ADV
ejpam-6063	237	3	x	x	SYM
ejpam-6063	237	4	∈	∈	PROPN
ejpam-6063	237	5	v	v	ADP
ejpam-6063	237	6	(	(	PUNCT
ejpam-6063	237	7	h	h	NOUN
ejpam-6063	237	8	)	)	PUNCT
ejpam-6063	237	9	\	\	PUNCT
ejpam-6063	238	1	(	(	PUNCT
ejpam-6063	238	2	dh	dh	NOUN
ejpam-6063	238	3	∪	∪	ADP
ejpam-6063	238	4	i(g	i(g	NOUN
ejpam-6063	238	5	)	)	PUNCT
ejpam-6063	238	6	)	)	PUNCT
ejpam-6063	238	7	.	.	PUNCT
ejpam-6063	239	1	since	since	SCONJ
ejpam-6063	239	2	x	x	PROPN
ejpam-6063	239	3	/∈	/∈	PUNCT
ejpam-6063	239	4	i(g	i(g	NOUN
ejpam-6063	239	5	)	)	PUNCT
ejpam-6063	239	6	,	,	PUNCT
ejpam-6063	239	7	xq	xq	PROPN
ejpam-6063	239	8	∈	∈	PROPN
ejpam-6063	239	9	e(h	e(h	PROPN
ejpam-6063	239	10	)	)	PUNCT
ejpam-6063	239	11	for	for	ADP
ejpam-6063	239	12	some	some	DET
ejpam-6063	239	13	q	q	PROPN
ejpam-6063	239	14	∈	∈	PROPN
ejpam-6063	239	15	v	v	NOUN
ejpam-6063	239	16	(	(	PUNCT
ejpam-6063	239	17	h	h	NOUN
ejpam-6063	239	18	)	)	PUNCT
ejpam-6063	239	19	.	.	PUNCT
ejpam-6063	240	1	since	since	SCONJ
ejpam-6063	240	2	dh	dh	PROPN
ejpam-6063	240	3	is	be	AUX
ejpam-6063	240	4	a	a	DET
ejpam-6063	240	5	vertex	vertex	NOUN
ejpam-6063	240	6	cover	cover	NOUN
ejpam-6063	240	7	of	of	ADP
ejpam-6063	240	8	h	h	NOUN
ejpam-6063	240	9	,	,	PUNCT
ejpam-6063	240	10	q	q	PROPN
ejpam-6063	241	1	∈	∈	PROPN
ejpam-6063	241	2	dh	dh	NOUN
ejpam-6063	241	3	.	.	PUNCT
ejpam-6063	242	1	it	it	PRON
ejpam-6063	242	2	follows	follow	VERB
ejpam-6063	242	3	that	that	SCONJ
ejpam-6063	242	4	v	v	NOUN
ejpam-6063	242	5	,	,	PUNCT
ejpam-6063	242	6	q	q	PROPN
ejpam-6063	242	7	∈	∈	PROPN
ejpam-6063	242	8	ng(x	ng(x	NUM
ejpam-6063	242	9	)	)	PUNCT
ejpam-6063	242	10	∩	∩	PROPN
ejpam-6063	242	11	s.	s.	PROPN
ejpam-6063	242	12	hence	hence	ADV
ejpam-6063	242	13	,	,	PUNCT
ejpam-6063	242	14	s	s	VERB
ejpam-6063	242	15	is	be	AUX
ejpam-6063	242	16	a	a	DET
ejpam-6063	242	17	2	2	NUM
ejpam-6063	242	18	-	-	PUNCT
ejpam-6063	242	19	dominating	dominating	NOUN
ejpam-6063	242	20	set	set	NOUN
ejpam-6063	242	21	in	in	ADP
ejpam-6063	242	22	g.	g.	PROPN
ejpam-6063	242	23	therefore	therefore	ADV
ejpam-6063	242	24	,	,	PUNCT
ejpam-6063	242	25	s	s	VERB
ejpam-6063	242	26	is	be	AUX
ejpam-6063	242	27	a	a	DET
ejpam-6063	242	28	2	2	NUM
ejpam-6063	242	29	-	-	PUNCT
ejpam-6063	242	30	vertex	vertex	NOUN
ejpam-6063	242	31	cover	cover	NOUN
ejpam-6063	242	32	of	of	ADP
ejpam-6063	242	33	g.	g.	PROPN
ejpam-6063	242	34	if	if	SCONJ
ejpam-6063	242	35	(	(	PUNCT
ejpam-6063	242	36	ii	ii	NOUN
ejpam-6063	242	37	)	)	PUNCT
ejpam-6063	242	38	holds	hold	VERB
ejpam-6063	242	39	,	,	PUNCT
ejpam-6063	242	40	then	then	ADV
ejpam-6063	242	41	s	s	VERB
ejpam-6063	242	42	=	=	SYM
ejpam-6063	242	43	v	v	PROPN
ejpam-6063	242	44	(	(	PUNCT
ejpam-6063	242	45	h	h	NOUN
ejpam-6063	242	46	)	)	PUNCT
ejpam-6063	242	47	is	be	AUX
ejpam-6063	242	48	a	a	DET
ejpam-6063	242	49	2	2	NUM
ejpam-6063	242	50	-	-	PUNCT
ejpam-6063	242	51	vertex	vertex	NOUN
ejpam-6063	242	52	cover	cover	NOUN
ejpam-6063	242	53	of	of	ADP
ejpam-6063	242	54	g	g	NOUN
ejpam-6063	242	55	because	because	SCONJ
ejpam-6063	242	56	h	h	NOUN
ejpam-6063	242	57	is	be	AUX
ejpam-6063	242	58	non	non	ADJ
ejpam-6063	242	59	-	-	ADJ
ejpam-6063	242	60	trivial	trivial	ADJ
ejpam-6063	242	61	.	.	PUNCT
ejpam-6063	243	1	lemma	lemma	PROPN
ejpam-6063	243	2	1	1	X
ejpam-6063	243	3	.	.	PUNCT
ejpam-6063	244	1	let	let	VERB
ejpam-6063	244	2	g	g	PRON
ejpam-6063	244	3	be	be	AUX
ejpam-6063	244	4	a	a	DET
ejpam-6063	244	5	graph	graph	NOUN
ejpam-6063	244	6	of	of	ADP
ejpam-6063	244	7	order	order	NOUN
ejpam-6063	244	8	n	n	NOUN
ejpam-6063	244	9	and	and	CCONJ
ejpam-6063	244	10	let	let	VERB
ejpam-6063	244	11	s	s	PRON
ejpam-6063	244	12	be	be	AUX
ejpam-6063	244	13	a	a	DET
ejpam-6063	244	14	β	β	NOUN
ejpam-6063	244	15	-	-	NOUN
ejpam-6063	244	16	set	set	NOUN
ejpam-6063	244	17	of	of	ADP
ejpam-6063	244	18	g.	g.	PROPN
ejpam-6063	244	19	then	then	ADV
ejpam-6063	244	20	each	each	PRON
ejpam-6063	244	21	of	of	ADP
ejpam-6063	244	22	the	the	DET
ejpam-6063	244	23	following	follow	VERB
ejpam-6063	244	24	holds	hold	VERB
ejpam-6063	244	25	:	:	PUNCT
ejpam-6063	244	26	(	(	PUNCT
ejpam-6063	244	27	i	i	NOUN
ejpam-6063	244	28	)	)	PUNCT
ejpam-6063	244	29	s	s	PART
ejpam-6063	244	30	∩	∩	ADJ
ejpam-6063	244	31	i(g	i(g	NOUN
ejpam-6063	244	32	)	)	PUNCT
ejpam-6063	245	1	=	=	PUNCT
ejpam-6063	245	2	∅.	∅.	PRON
ejpam-6063	245	3	(	(	PUNCT
ejpam-6063	245	4	ii	ii	NOUN
ejpam-6063	245	5	)	)	PUNCT
ejpam-6063	245	6	n	n	PROPN
ejpam-6063	245	7	=	=	SYM
ejpam-6063	245	8	β(g	β(g	PROPN
ejpam-6063	245	9	)	)	PUNCT
ejpam-6063	246	1	+	+	CCONJ
ejpam-6063	246	2	|i(g)|+	|i(g)|+	ADV
ejpam-6063	246	3	|(v	|(v	ADJ
ejpam-6063	246	4	(	(	PUNCT
ejpam-6063	246	5	g	g	NOUN
ejpam-6063	246	6	)	)	PUNCT
ejpam-6063	246	7	\	\	PUNCT
ejpam-6063	246	8	i(g	i(g	NOUN
ejpam-6063	246	9	)	)	PUNCT
ejpam-6063	246	10	)	)	PUNCT
ejpam-6063	247	1	\	\	PROPN
ejpam-6063	247	2	s|	s|	PROPN
ejpam-6063	247	3	.	.	PUNCT
ejpam-6063	248	1	(	(	PUNCT
ejpam-6063	248	2	iii	iii	X
ejpam-6063	248	3	)	)	PUNCT
ejpam-6063	248	4	if	if	SCONJ
ejpam-6063	248	5	g	g	PROPN
ejpam-6063	248	6	is	be	AUX
ejpam-6063	248	7	not	not	PART
ejpam-6063	248	8	the	the	DET
ejpam-6063	248	9	empty	empty	ADJ
ejpam-6063	248	10	graph	graph	NOUN
ejpam-6063	248	11	,	,	PUNCT
ejpam-6063	248	12	then	then	ADV
ejpam-6063	248	13	|(v	|(v	PROPN
ejpam-6063	248	14	(	(	PUNCT
ejpam-6063	248	15	g	g	NOUN
ejpam-6063	248	16	)	)	PUNCT
ejpam-6063	248	17	\	\	PUNCT
ejpam-6063	248	18	i(g	i(g	NOUN
ejpam-6063	248	19	)	)	PUNCT
ejpam-6063	248	20	)	)	PUNCT
ejpam-6063	249	1	\	\	PROPN
ejpam-6063	249	2	s|	s|	VERB
ejpam-6063	249	3	≥	≥	NOUN
ejpam-6063	249	4	1	1	NUM
ejpam-6063	249	5	.	.	PUNCT
ejpam-6063	250	1	hence	hence	ADV
ejpam-6063	250	2	,	,	PUNCT
ejpam-6063	250	3	n	n	PROPN
ejpam-6063	250	4	=	=	SYM
ejpam-6063	250	5	β(g	β(g	PROPN
ejpam-6063	250	6	)	)	PUNCT
ejpam-6063	251	1	+	+	CCONJ
ejpam-6063	251	2	|i(g)|+	|i(g)|+	ADV
ejpam-6063	251	3	|(v	|(v	ADJ
ejpam-6063	251	4	(	(	PUNCT
ejpam-6063	251	5	g	g	NOUN
ejpam-6063	251	6	)	)	PUNCT
ejpam-6063	251	7	\	\	PUNCT
ejpam-6063	251	8	i(g	i(g	NOUN
ejpam-6063	251	9	)	)	PUNCT
ejpam-6063	251	10	)	)	PUNCT
ejpam-6063	252	1	\	\	PROPN
ejpam-6063	252	2	s|	s|	VERB
ejpam-6063	252	3	≥	≥	NUM
ejpam-6063	252	4	β(g	β(g	NUM
ejpam-6063	252	5	)	)	PUNCT
ejpam-6063	253	1	+	+	CCONJ
ejpam-6063	253	2	|i(g)|+	|i(g)|+	NOUN
ejpam-6063	253	3	1	1	X
ejpam-6063	253	4	.	.	PUNCT
ejpam-6063	253	5	proof	proof	NOUN
ejpam-6063	253	6	.	.	PUNCT
ejpam-6063	254	1	(	(	PUNCT
ejpam-6063	254	2	i	i	NOUN
ejpam-6063	254	3	)	)	PUNCT
ejpam-6063	254	4	since	since	SCONJ
ejpam-6063	254	5	a	a	DET
ejpam-6063	254	6	vertex	vertex	NOUN
ejpam-6063	254	7	cover	cover	NOUN
ejpam-6063	254	8	only	only	ADV
ejpam-6063	254	9	ensures	ensure	VERB
ejpam-6063	254	10	that	that	SCONJ
ejpam-6063	254	11	every	every	DET
ejpam-6063	254	12	edge	edge	NOUN
ejpam-6063	254	13	is	be	AUX
ejpam-6063	254	14	incident	incident	NOUN
ejpam-6063	254	15	to	to	ADP
ejpam-6063	254	16	a	a	DET
ejpam-6063	254	17	vertex	vertex	NOUN
ejpam-6063	254	18	inside	inside	ADP
ejpam-6063	254	19	the	the	DET
ejpam-6063	254	20	cover	cover	NOUN
ejpam-6063	254	21	,	,	PUNCT
ejpam-6063	254	22	s	s	AUX
ejpam-6063	254	23	being	be	AUX
ejpam-6063	254	24	a	a	DET
ejpam-6063	254	25	β	β	NOUN
ejpam-6063	254	26	-	-	NOUN
ejpam-6063	254	27	set	set	NOUN
ejpam-6063	254	28	of	of	ADP
ejpam-6063	254	29	g	g	PROPN
ejpam-6063	254	30	implies	imply	VERB
ejpam-6063	254	31	that	that	SCONJ
ejpam-6063	254	32	s	s	VERB
ejpam-6063	254	33	∩	∩	NOUN
ejpam-6063	254	34	i(g	i(g	NOUN
ejpam-6063	254	35	)	)	PUNCT
ejpam-6063	255	1	=	=	PUNCT
ejpam-6063	255	2	∅.	∅.	VERB
ejpam-6063	255	3	hence	hence	ADV
ejpam-6063	255	4	,	,	PUNCT
ejpam-6063	255	5	(	(	PUNCT
ejpam-6063	255	6	i	i	NOUN
ejpam-6063	255	7	)	)	PUNCT
ejpam-6063	255	8	holds	hold	VERB
ejpam-6063	255	9	.	.	PUNCT
ejpam-6063	256	1	(	(	PUNCT
ejpam-6063	256	2	ii	ii	NOUN
ejpam-6063	256	3	)	)	PUNCT
ejpam-6063	256	4	since	since	SCONJ
ejpam-6063	256	5	v	v	NOUN
ejpam-6063	256	6	(	(	PUNCT
ejpam-6063	256	7	g	g	NOUN
ejpam-6063	256	8	)	)	PUNCT
ejpam-6063	256	9	=	=	SYM
ejpam-6063	256	10	s	s	X
ejpam-6063	256	11	∪	∪	ADP
ejpam-6063	256	12	i(g	i(g	NOUN
ejpam-6063	256	13	)	)	PUNCT
ejpam-6063	256	14	∪	∪	ADP
ejpam-6063	256	15	[	[	X
ejpam-6063	256	16	v	v	X
ejpam-6063	256	17	(	(	PUNCT
ejpam-6063	256	18	g	g	NOUN
ejpam-6063	256	19	)	)	PUNCT
ejpam-6063	256	20	\	\	PUNCT
ejpam-6063	256	21	i(g	i(g	NOUN
ejpam-6063	256	22	)	)	PUNCT
ejpam-6063	256	23	)	)	PUNCT
ejpam-6063	256	24	\	\	PUNCT
ejpam-6063	257	1	s	s	X
ejpam-6063	257	2	]	]	X
ejpam-6063	257	3	,	,	PUNCT
ejpam-6063	257	4	(	(	PUNCT
ejpam-6063	257	5	i	i	NOUN
ejpam-6063	257	6	)	)	PUNCT
ejpam-6063	257	7	implies	imply	VERB
ejpam-6063	257	8	that	that	SCONJ
ejpam-6063	257	9	n	n	PROPN
ejpam-6063	257	10	=	=	SYM
ejpam-6063	257	11	β(g	β(g	PROPN
ejpam-6063	257	12	)	)	PUNCT
ejpam-6063	258	1	+	+	CCONJ
ejpam-6063	259	1	|i(g)|	|i(g)|	NOUN
ejpam-6063	259	2	+	+	CCONJ
ejpam-6063	259	3	|(v	|(v	PROPN
ejpam-6063	259	4	(	(	PUNCT
ejpam-6063	259	5	g	g	NOUN
ejpam-6063	259	6	)	)	PUNCT
ejpam-6063	259	7	\	\	PUNCT
ejpam-6063	259	8	i(g	i(g	NOUN
ejpam-6063	259	9	)	)	PUNCT
ejpam-6063	259	10	)	)	PUNCT
ejpam-6063	259	11	\	\	PROPN
ejpam-6063	259	12	s|	s|	PROPN
ejpam-6063	259	13	.	.	PUNCT
ejpam-6063	260	1	(	(	PUNCT
ejpam-6063	260	2	iii	iii	X
ejpam-6063	260	3	)	)	PUNCT
ejpam-6063	260	4	suppose	suppose	VERB
ejpam-6063	260	5	g	g	PROPN
ejpam-6063	260	6	̸=	̸=	PROPN
ejpam-6063	260	7	kn	kn	PROPN
ejpam-6063	260	8	.	.	PUNCT
ejpam-6063	261	1	then	then	ADV
ejpam-6063	261	2	s	s	VERB
ejpam-6063	261	3	̸=	̸=	PROPN
ejpam-6063	261	4	∅	∅	NOUN
ejpam-6063	261	5	and	and	CCONJ
ejpam-6063	261	6	β(g	β(g	NUM
ejpam-6063	261	7	)	)	PUNCT
ejpam-6063	261	8	=	=	SYM
ejpam-6063	261	9	|s|	|s|	PROPN
ejpam-6063	261	10	≤	≤	NOUN
ejpam-6063	261	11	n	n	CCONJ
ejpam-6063	261	12	−	−	PROPN
ejpam-6063	261	13	1	1	NUM
ejpam-6063	261	14	.	.	PUNCT
ejpam-6063	262	1	it	it	PRON
ejpam-6063	262	2	follows	follow	VERB
ejpam-6063	262	3	that	that	SCONJ
ejpam-6063	262	4	|(v	|(v	PROPN
ejpam-6063	262	5	(	(	PUNCT
ejpam-6063	262	6	g	g	NOUN
ejpam-6063	262	7	)	)	PUNCT
ejpam-6063	262	8	\	\	PUNCT
ejpam-6063	262	9	i(g	i(g	NOUN
ejpam-6063	262	10	)	)	PUNCT
ejpam-6063	262	11	)	)	PUNCT
ejpam-6063	262	12	\	\	PROPN
ejpam-6063	263	1	s|	s|	VERB
ejpam-6063	263	2	≥	≥	NOUN
ejpam-6063	263	3	1	1	NUM
ejpam-6063	263	4	.	.	PUNCT
ejpam-6063	264	1	therefore	therefore	ADV
ejpam-6063	264	2	,	,	PUNCT
ejpam-6063	264	3	n	n	PROPN
ejpam-6063	264	4	=	=	SYM
ejpam-6063	264	5	β(g	β(g	PROPN
ejpam-6063	264	6	)	)	PUNCT
ejpam-6063	265	1	+	+	CCONJ
ejpam-6063	265	2	|i(g)|+	|i(g)|+	ADV
ejpam-6063	265	3	|(v	|(v	ADJ
ejpam-6063	265	4	(	(	PUNCT
ejpam-6063	265	5	g	g	NOUN
ejpam-6063	265	6	)	)	PUNCT
ejpam-6063	265	7	\	\	PUNCT
ejpam-6063	265	8	i(g	i(g	NOUN
ejpam-6063	265	9	)	)	PUNCT
ejpam-6063	265	10	)	)	PUNCT
ejpam-6063	266	1	\	\	PROPN
ejpam-6063	266	2	s|	s|	VERB
ejpam-6063	266	3	≥	≥	NUM
ejpam-6063	266	4	β(g	β(g	NUM
ejpam-6063	266	5	)	)	PUNCT
ejpam-6063	267	1	+	+	CCONJ
ejpam-6063	267	2	|i(g)|+	|i(g)|+	ADJ
ejpam-6063	267	3	1	1	NUM
ejpam-6063	267	4	.	.	PUNCT
ejpam-6063	267	5	corollary	corollary	ADJ
ejpam-6063	267	6	4	4	NUM
ejpam-6063	267	7	.	.	PUNCT
ejpam-6063	268	1	let	let	VERB
ejpam-6063	268	2	h	h	PRON
ejpam-6063	268	3	be	be	AUX
ejpam-6063	268	4	a	a	DET
ejpam-6063	268	5	non	non	ADJ
ejpam-6063	268	6	-	-	ADJ
ejpam-6063	268	7	trivial	trivial	ADJ
ejpam-6063	268	8	graph	graph	NOUN
ejpam-6063	268	9	of	of	ADP
ejpam-6063	268	10	order	order	NOUN
ejpam-6063	268	11	n	n	NOUN
ejpam-6063	268	12	and	and	CCONJ
ejpam-6063	268	13	g	g	PROPN
ejpam-6063	268	14	=	=	PROPN
ejpam-6063	268	15	k1+h	k1+h	PROPN
ejpam-6063	268	16	.	.	PUNCT
ejpam-6063	269	1	then	then	ADV
ejpam-6063	269	2	each	each	PRON
ejpam-6063	269	3	of	of	ADP
ejpam-6063	269	4	the	the	DET
ejpam-6063	269	5	following	follow	VERB
ejpam-6063	269	6	holds	hold	VERB
ejpam-6063	269	7	:	:	PUNCT
ejpam-6063	269	8	(	(	PUNCT
ejpam-6063	269	9	i	i	NOUN
ejpam-6063	269	10	)	)	PUNCT
ejpam-6063	269	11	if	if	SCONJ
ejpam-6063	269	12	h	h	NOUN
ejpam-6063	269	13	̸=	̸=	PROPN
ejpam-6063	269	14	kn	kn	PROPN
ejpam-6063	269	15	,	,	PUNCT
ejpam-6063	269	16	then	then	ADV
ejpam-6063	269	17	β2(g	β2(g	NUM
ejpam-6063	269	18	)	)	PUNCT
ejpam-6063	269	19	=	=	SYM
ejpam-6063	269	20	β(h	β(h	PUNCT
ejpam-6063	269	21	)	)	PUNCT
ejpam-6063	270	1	+	+	CCONJ
ejpam-6063	270	2	|i(h)|	|i(h)|	PROPN
ejpam-6063	270	3	+	+	CCONJ
ejpam-6063	270	4	1	1	X
ejpam-6063	270	5	.	.	PUNCT
ejpam-6063	271	1	moreover	moreover	ADV
ejpam-6063	271	2	,	,	PUNCT
ejpam-6063	271	3	if	if	SCONJ
ejpam-6063	271	4	h	h	NOUN
ejpam-6063	271	5	is	be	AUX
ejpam-6063	271	6	connected	connect	VERB
ejpam-6063	271	7	,	,	PUNCT
ejpam-6063	271	8	then	then	ADV
ejpam-6063	271	9	β2(g	β2(g	NUM
ejpam-6063	271	10	)	)	PUNCT
ejpam-6063	271	11	=	=	SYM
ejpam-6063	271	12	β(h	β(h	PUNCT
ejpam-6063	271	13	)	)	PUNCT
ejpam-6063	271	14	+	+	NUM
ejpam-6063	271	15	1	1	X
ejpam-6063	271	16	.	.	X
ejpam-6063	271	17	(	(	PUNCT
ejpam-6063	271	18	ii	ii	NOUN
ejpam-6063	271	19	)	)	PUNCT
ejpam-6063	271	20	if	if	SCONJ
ejpam-6063	271	21	h	h	NOUN
ejpam-6063	271	22	=	=	SYM
ejpam-6063	271	23	kn	kn	PROPN
ejpam-6063	271	24	,	,	PUNCT
ejpam-6063	271	25	then	then	ADV
ejpam-6063	271	26	β2(g	β2(g	NUM
ejpam-6063	271	27	)	)	PUNCT
ejpam-6063	271	28	=	=	SYM
ejpam-6063	271	29	n.	n.	NOUN
ejpam-6063	271	30	proof	proof	NOUN
ejpam-6063	271	31	.	.	PUNCT
ejpam-6063	272	1	(	(	PUNCT
ejpam-6063	272	2	i	i	NOUN
ejpam-6063	272	3	)	)	PUNCT
ejpam-6063	272	4	suppose	suppose	VERB
ejpam-6063	272	5	h	h	PROPN
ejpam-6063	272	6	̸=	̸=	PROPN
ejpam-6063	272	7	kn	kn	PROPN
ejpam-6063	272	8	.	.	PUNCT
ejpam-6063	273	1	then	then	ADV
ejpam-6063	273	2	v	v	X
ejpam-6063	273	3	(	(	PUNCT
ejpam-6063	273	4	h	h	NOUN
ejpam-6063	273	5	)	)	PUNCT
ejpam-6063	273	6	\	\	NOUN
ejpam-6063	273	7	i(h	i(h	NOUN
ejpam-6063	273	8	)	)	PUNCT
ejpam-6063	273	9	̸=	̸=	PROPN
ejpam-6063	273	10	∅.	∅.	ADV
ejpam-6063	273	11	let	let	VERB
ejpam-6063	273	12	dh	dh	NOUN
ejpam-6063	273	13	be	be	AUX
ejpam-6063	273	14	a	a	DET
ejpam-6063	273	15	β	β	NOUN
ejpam-6063	273	16	-	-	VERB
ejpam-6063	273	17	set	set	NOUN
ejpam-6063	273	18	of	of	ADP
ejpam-6063	273	19	h.	h.	PROPN
ejpam-6063	273	20	then	then	ADV
ejpam-6063	273	21	dh	dh	PROPN
ejpam-6063	273	22	∩	∩	NOUN
ejpam-6063	273	23	i(h	i(h	NOUN
ejpam-6063	273	24	)	)	PUNCT
ejpam-6063	273	25	=	=	PUNCT
ejpam-6063	273	26	∅.	∅.	AUX
ejpam-6063	273	27	let	let	VERB
ejpam-6063	273	28	s	s	NOUN
ejpam-6063	273	29	=	=	NOUN
ejpam-6063	273	30	{	{	PUNCT
ejpam-6063	273	31	v	v	NOUN
ejpam-6063	273	32	}	}	PUNCT
ejpam-6063	273	33	∪dh	∪dh	PROPN
ejpam-6063	273	34	∪	∪	ADP
ejpam-6063	273	35	i(h	i(h	NOUN
ejpam-6063	273	36	)	)	PUNCT
ejpam-6063	273	37	.	.	PUNCT
ejpam-6063	274	1	then	then	ADV
ejpam-6063	274	2	s	s	VERB
ejpam-6063	274	3	is	be	AUX
ejpam-6063	274	4	a	a	DET
ejpam-6063	274	5	2	2	NUM
ejpam-6063	274	6	-	-	PUNCT
ejpam-6063	274	7	vertex	vertex	NOUN
ejpam-6063	274	8	cover	cover	NOUN
ejpam-6063	274	9	of	of	ADP
ejpam-6063	274	10	g	g	NOUN
ejpam-6063	274	11	by	by	ADP
ejpam-6063	274	12	theorem	theorem	NOUN
ejpam-6063	274	13	8	8	NUM
ejpam-6063	274	14	.	.	PUNCT
ejpam-6063	275	1	it	it	PRON
ejpam-6063	275	2	follows	follow	VERB
ejpam-6063	275	3	that	that	SCONJ
ejpam-6063	275	4	β2(g	β2(g	NUM
ejpam-6063	275	5	)	)	PUNCT
ejpam-6063	275	6	≤	≤	NUM
ejpam-6063	275	7	|s|	|s|	PROPN
ejpam-6063	275	8	=	=	SYM
ejpam-6063	275	9	β(h	β(h	PROPN
ejpam-6063	275	10	)	)	PUNCT
ejpam-6063	275	11	+	+	NUM
ejpam-6063	275	12	|i(h)|+	|i(h)|+	NOUN
ejpam-6063	275	13	1	1	NUM
ejpam-6063	275	14	.	.	PUNCT
ejpam-6063	276	1	next	next	ADV
ejpam-6063	276	2	,	,	PUNCT
ejpam-6063	276	3	suppose	suppose	VERB
ejpam-6063	276	4	s0	s0	PROPN
ejpam-6063	276	5	is	be	AUX
ejpam-6063	276	6	a	a	DET
ejpam-6063	276	7	β2	β2	NOUN
ejpam-6063	276	8	-	-	PUNCT
ejpam-6063	276	9	set	set	NOUN
ejpam-6063	276	10	of	of	ADP
ejpam-6063	276	11	g.	g.	PROPN
ejpam-6063	277	1	if	if	SCONJ
ejpam-6063	277	2	s0	s0	PROPN
ejpam-6063	277	3	satisfies	satisfie	NOUN
ejpam-6063	277	4	(	(	PUNCT
ejpam-6063	277	5	ii	ii	NOUN
ejpam-6063	277	6	)	)	PUNCT
ejpam-6063	277	7	of	of	ADP
ejpam-6063	277	8	theorem	theorem	ADJ
ejpam-6063	277	9	8	8	NUM
ejpam-6063	277	10	,	,	PUNCT
ejpam-6063	277	11	then	then	ADV
ejpam-6063	277	12	s0	s0	PROPN
ejpam-6063	277	13	=	=	PUNCT
ejpam-6063	277	14	{	{	PUNCT
ejpam-6063	277	15	v	v	NOUN
ejpam-6063	277	16	}	}	PUNCT
ejpam-6063	277	17	∪	∪	ADP
ejpam-6063	277	18	dh	dh	PROPN
ejpam-6063	277	19	∪	∪	ADJ
ejpam-6063	277	20	i(h	i(h	NOUN
ejpam-6063	277	21	)	)	PUNCT
ejpam-6063	277	22	where	where	SCONJ
ejpam-6063	277	23	dh	dh	NOUN
ejpam-6063	277	24	is	be	AUX
ejpam-6063	277	25	a	a	DET
ejpam-6063	277	26	vertex	vertex	NOUN
ejpam-6063	277	27	cover	cover	NOUN
ejpam-6063	277	28	of	of	ADP
ejpam-6063	277	29	h	h	NOUN
ejpam-6063	277	30	such	such	ADJ
ejpam-6063	277	31	that	that	DET
ejpam-6063	277	32	dh	dh	PROPN
ejpam-6063	277	33	∩	∩	NOUN
ejpam-6063	277	34	i(g	i(g	ADV
ejpam-6063	277	35	)	)	PUNCT
ejpam-6063	277	36	=	=	PUNCT
ejpam-6063	277	37	∅.	∅.	VERB
ejpam-6063	277	38	hence	hence	ADV
ejpam-6063	277	39	,	,	PUNCT
ejpam-6063	277	40	β2(g	β2(g	NUM
ejpam-6063	277	41	)	)	PUNCT
ejpam-6063	277	42	=	=	PUNCT
ejpam-6063	277	43	j.	j.	PROPN
ejpam-6063	277	44	hassan	hassan	PROPN
ejpam-6063	277	45	et	et	PROPN
ejpam-6063	277	46	.	.	PUNCT
ejpam-6063	278	1	al	al	PROPN
ejpam-6063	278	2	/	/	PUNCT
ejpam-6063	278	3	eur	eur	PROPN
ejpam-6063	278	4	.	.	PUNCT
ejpam-6063	279	1	j.	j.	PROPN
ejpam-6063	279	2	pure	pure	PROPN
ejpam-6063	279	3	appl	appl	PROPN
ejpam-6063	279	4	.	.	PROPN
ejpam-6063	279	5	math	math	PROPN
ejpam-6063	279	6	,	,	PUNCT
ejpam-6063	279	7	18	18	NUM
ejpam-6063	279	8	(	(	PUNCT
ejpam-6063	279	9	2	2	NUM
ejpam-6063	279	10	)	)	PUNCT
ejpam-6063	279	11	(	(	PUNCT
ejpam-6063	279	12	2025	2025	NUM
ejpam-6063	279	13	)	)	PUNCT
ejpam-6063	279	14	,	,	PUNCT
ejpam-6063	279	15	6063	6063	NUM
ejpam-6063	279	16	8	8	NUM
ejpam-6063	279	17	of	of	ADP
ejpam-6063	279	18	11	11	NUM
ejpam-6063	279	19	|s0|	|s0|	NOUN
ejpam-6063	279	20	≥	≥	NOUN
ejpam-6063	279	21	β(h	β(h	PUNCT
ejpam-6063	279	22	)	)	PUNCT
ejpam-6063	280	1	+	+	CCONJ
ejpam-6063	280	2	|i(h)|	|i(h)|	PROPN
ejpam-6063	280	3	+	+	CCONJ
ejpam-6063	280	4	1	1	X
ejpam-6063	280	5	.	.	PUNCT
ejpam-6063	280	6	suppose	suppose	VERB
ejpam-6063	280	7	s0	s0	PROPN
ejpam-6063	280	8	=	=	SYM
ejpam-6063	280	9	v	v	PROPN
ejpam-6063	280	10	(	(	PUNCT
ejpam-6063	280	11	h	h	NOUN
ejpam-6063	280	12	)	)	PUNCT
ejpam-6063	280	13	.	.	PUNCT
ejpam-6063	281	1	then	then	ADV
ejpam-6063	281	2	β2(g	β2(g	NUM
ejpam-6063	281	3	)	)	PUNCT
ejpam-6063	281	4	=	=	SYM
ejpam-6063	281	5	|s0|	|s0|	NOUN
ejpam-6063	281	6	=	=	SYM
ejpam-6063	281	7	n.	n.	NOUN
ejpam-6063	281	8	by	by	ADP
ejpam-6063	281	9	lemma	lemma	PROPN
ejpam-6063	281	10	1	1	NUM
ejpam-6063	281	11	,	,	PUNCT
ejpam-6063	281	12	β2(g	β2(g	NUM
ejpam-6063	281	13	)	)	PUNCT
ejpam-6063	281	14	≥	≥	NOUN
ejpam-6063	281	15	β(h	β(h	NUM
ejpam-6063	281	16	)	)	PUNCT
ejpam-6063	282	1	+	+	NUM
ejpam-6063	282	2	|i(h)|+	|i(h)|+	NOUN
ejpam-6063	282	3	1	1	NUM
ejpam-6063	282	4	.	.	PUNCT
ejpam-6063	282	5	therefore	therefore	ADV
ejpam-6063	282	6	,	,	PUNCT
ejpam-6063	282	7	β2(g	β2(g	NUM
ejpam-6063	282	8	)	)	PUNCT
ejpam-6063	282	9	=	=	SYM
ejpam-6063	282	10	β(h	β(h	PUNCT
ejpam-6063	282	11	)	)	PUNCT
ejpam-6063	282	12	+	+	NUM
ejpam-6063	282	13	|i(h)|+	|i(h)|+	NOUN
ejpam-6063	282	14	1	1	NUM
ejpam-6063	282	15	.	.	PUNCT
ejpam-6063	282	16	(	(	PUNCT
ejpam-6063	282	17	ii	ii	NOUN
ejpam-6063	282	18	)	)	PUNCT
ejpam-6063	282	19	suppose	suppose	VERB
ejpam-6063	282	20	h	h	PROPN
ejpam-6063	282	21	=	=	PROPN
ejpam-6063	282	22	kn	kn	PROPN
ejpam-6063	282	23	.	.	PUNCT
ejpam-6063	283	1	then	then	ADV
ejpam-6063	283	2	|i(h)|	|i(h)|	PROPN
ejpam-6063	283	3	=	=	SYM
ejpam-6063	283	4	n	n	PROPN
ejpam-6063	283	5	and	and	CCONJ
ejpam-6063	283	6	dh	dh	NOUN
ejpam-6063	283	7	=	=	NOUN
ejpam-6063	283	8	∅	∅	NOUN
ejpam-6063	283	9	is	be	AUX
ejpam-6063	283	10	the	the	DET
ejpam-6063	283	11	only	only	ADJ
ejpam-6063	283	12	vertex	vertex	NOUN
ejpam-6063	283	13	cover	cover	NOUN
ejpam-6063	283	14	of	of	ADP
ejpam-6063	283	15	h.	h.	PROPN
ejpam-6063	283	16	hence	hence	ADV
ejpam-6063	283	17	,	,	PUNCT
ejpam-6063	283	18	by	by	ADP
ejpam-6063	283	19	theorem	theorem	NOUN
ejpam-6063	283	20	8	8	NUM
ejpam-6063	283	21	,	,	PUNCT
ejpam-6063	283	22	s	s	PART
ejpam-6063	283	23	=	=	SYM
ejpam-6063	283	24	v	v	PROPN
ejpam-6063	283	25	(	(	PUNCT
ejpam-6063	283	26	h	h	NOUN
ejpam-6063	283	27	)	)	PUNCT
ejpam-6063	283	28	is	be	AUX
ejpam-6063	283	29	a	a	DET
ejpam-6063	283	30	β2	β2	NOUN
ejpam-6063	283	31	-	-	PUNCT
ejpam-6063	283	32	set	set	NOUN
ejpam-6063	283	33	of	of	ADP
ejpam-6063	283	34	g.	g.	PROPN
ejpam-6063	283	35	thus	thus	ADV
ejpam-6063	283	36	,	,	PUNCT
ejpam-6063	283	37	β2(g	β2(g	NUM
ejpam-6063	283	38	)	)	PUNCT
ejpam-6063	283	39	=	=	SYM
ejpam-6063	283	40	n.	n.	NOUN
ejpam-6063	283	41	theorem	theorem	VERB
ejpam-6063	283	42	9	9	NUM
ejpam-6063	283	43	.	.	PUNCT
ejpam-6063	284	1	let	let	VERB
ejpam-6063	284	2	g	g	NOUN
ejpam-6063	284	3	and	and	CCONJ
ejpam-6063	284	4	h	h	PROPN
ejpam-6063	284	5	be	be	VERB
ejpam-6063	284	6	non	non	ADJ
ejpam-6063	284	7	-	-	ADJ
ejpam-6063	284	8	trivial	trivial	ADJ
ejpam-6063	284	9	graphs	graph	NOUN
ejpam-6063	284	10	.	.	PUNCT
ejpam-6063	285	1	then	then	ADV
ejpam-6063	285	2	s	s	VERB
ejpam-6063	285	3	⊆	⊆	NUM
ejpam-6063	285	4	v	v	NOUN
ejpam-6063	285	5	(	(	PUNCT
ejpam-6063	285	6	g	g	PROPN
ejpam-6063	285	7	+	+	NOUN
ejpam-6063	285	8	h	h	NOUN
ejpam-6063	285	9	)	)	PUNCT
ejpam-6063	285	10	is	be	AUX
ejpam-6063	285	11	a	a	DET
ejpam-6063	285	12	2	2	NUM
ejpam-6063	285	13	-	-	PUNCT
ejpam-6063	285	14	vertex	vertex	NOUN
ejpam-6063	285	15	covering	covering	NOUN
ejpam-6063	285	16	of	of	ADP
ejpam-6063	285	17	g+h	g+h	PROPN
ejpam-6063	286	1	if	if	SCONJ
ejpam-6063	286	2	and	and	CCONJ
ejpam-6063	286	3	only	only	ADV
ejpam-6063	286	4	if	if	SCONJ
ejpam-6063	286	5	s	s	X
ejpam-6063	286	6	=	=	PUNCT
ejpam-6063	286	7	sg∪sh	sg∪sh	PROPN
ejpam-6063	286	8	and	and	CCONJ
ejpam-6063	286	9	satisfies	satisfy	VERB
ejpam-6063	286	10	one	one	NUM
ejpam-6063	286	11	of	of	ADP
ejpam-6063	286	12	the	the	DET
ejpam-6063	286	13	following	following	ADJ
ejpam-6063	286	14	conditions	condition	NOUN
ejpam-6063	286	15	:	:	PUNCT
ejpam-6063	286	16	(	(	PUNCT
ejpam-6063	286	17	i	i	NOUN
ejpam-6063	286	18	)	)	PUNCT
ejpam-6063	286	19	sg	sg	PROPN
ejpam-6063	286	20	=	=	SYM
ejpam-6063	286	21	v	v	PROPN
ejpam-6063	286	22	(	(	PUNCT
ejpam-6063	286	23	g	g	NOUN
ejpam-6063	286	24	)	)	PUNCT
ejpam-6063	286	25	and	and	CCONJ
ejpam-6063	286	26	sh	sh	PROPN
ejpam-6063	286	27	is	be	AUX
ejpam-6063	286	28	a	a	DET
ejpam-6063	286	29	vertex	vertex	NOUN
ejpam-6063	286	30	cover	cover	NOUN
ejpam-6063	286	31	of	of	ADP
ejpam-6063	286	32	h.	h.	PROPN
ejpam-6063	286	33	(	(	PUNCT
ejpam-6063	286	34	ii	ii	PROPN
ejpam-6063	286	35	)	)	PUNCT
ejpam-6063	286	36	sh	sh	PROPN
ejpam-6063	286	37	=	=	SYM
ejpam-6063	286	38	v	v	PROPN
ejpam-6063	286	39	(	(	PUNCT
ejpam-6063	286	40	h	h	NOUN
ejpam-6063	286	41	)	)	PUNCT
ejpam-6063	286	42	and	and	CCONJ
ejpam-6063	286	43	sg	sg	PROPN
ejpam-6063	286	44	is	be	AUX
ejpam-6063	286	45	a	a	DET
ejpam-6063	286	46	vertex	vertex	NOUN
ejpam-6063	286	47	cover	cover	NOUN
ejpam-6063	286	48	of	of	ADP
ejpam-6063	286	49	g.	g.	PROPN
ejpam-6063	286	50	proof	proof	NOUN
ejpam-6063	286	51	.	.	PUNCT
ejpam-6063	287	1	suppose	suppose	VERB
ejpam-6063	287	2	s	s	NOUN
ejpam-6063	287	3	is	be	AUX
ejpam-6063	287	4	a	a	DET
ejpam-6063	287	5	2	2	NUM
ejpam-6063	287	6	-	-	PUNCT
ejpam-6063	287	7	vertex	vertex	NOUN
ejpam-6063	287	8	cover	cover	NOUN
ejpam-6063	287	9	of	of	ADP
ejpam-6063	287	10	g+h	g+h	PROPN
ejpam-6063	287	11	.	.	PUNCT
ejpam-6063	288	1	let	let	VERB
ejpam-6063	288	2	sg	sg	VERB
ejpam-6063	288	3	=	=	PUNCT
ejpam-6063	288	4	s∩v	s∩v	PROPN
ejpam-6063	288	5	(	(	PUNCT
ejpam-6063	288	6	g	g	NOUN
ejpam-6063	288	7	)	)	PUNCT
ejpam-6063	288	8	and	and	CCONJ
ejpam-6063	288	9	sh	sh	INTJ
ejpam-6063	288	10	=	=	SYM
ejpam-6063	288	11	s∩v	s∩v	PROPN
ejpam-6063	288	12	(	(	PUNCT
ejpam-6063	288	13	h	h	NOUN
ejpam-6063	288	14	)	)	PUNCT
ejpam-6063	288	15	.	.	PUNCT
ejpam-6063	289	1	then	then	ADV
ejpam-6063	289	2	s	s	VERB
ejpam-6063	289	3	=	=	PUNCT
ejpam-6063	289	4	sg	sg	X
ejpam-6063	289	5	∪	∪	VERB
ejpam-6063	289	6	sh	sh	PROPN
ejpam-6063	289	7	.	.	PUNCT
ejpam-6063	290	1	suppose	suppose	VERB
ejpam-6063	290	2	sg	sg	ADP
ejpam-6063	290	3	̸=	̸=	PROPN
ejpam-6063	290	4	v	v	NOUN
ejpam-6063	290	5	(	(	PUNCT
ejpam-6063	290	6	g	g	NOUN
ejpam-6063	290	7	)	)	PUNCT
ejpam-6063	290	8	and	and	CCONJ
ejpam-6063	290	9	sh	sh	INTJ
ejpam-6063	290	10	̸=	̸=	PROPN
ejpam-6063	290	11	v	v	NOUN
ejpam-6063	290	12	(	(	PUNCT
ejpam-6063	290	13	h	h	NOUN
ejpam-6063	290	14	)	)	PUNCT
ejpam-6063	290	15	.	.	PUNCT
ejpam-6063	291	1	pick	pick	VERB
ejpam-6063	291	2	any	any	DET
ejpam-6063	291	3	x	x	SYM
ejpam-6063	291	4	∈	∈	PROPN
ejpam-6063	291	5	v	v	ADP
ejpam-6063	291	6	(	(	PUNCT
ejpam-6063	291	7	g	g	NOUN
ejpam-6063	291	8	)	)	PUNCT
ejpam-6063	291	9	\	\	PROPN
ejpam-6063	291	10	sg	sg	NOUN
ejpam-6063	291	11	and	and	CCONJ
ejpam-6063	291	12	p	p	PROPN
ejpam-6063	291	13	∈	∈	PROPN
ejpam-6063	291	14	v	v	ADP
ejpam-6063	291	15	(	(	PUNCT
ejpam-6063	291	16	h	h	NOUN
ejpam-6063	291	17	)	)	PUNCT
ejpam-6063	291	18	\	\	PUNCT
ejpam-6063	292	1	sh	sh	INTJ
ejpam-6063	292	2	.	.	PUNCT
ejpam-6063	293	1	since	since	SCONJ
ejpam-6063	293	2	xp	xp	PROPN
ejpam-6063	293	3	∈	∈	PROPN
ejpam-6063	293	4	e(g+h	e(g+h	NUM
ejpam-6063	293	5	)	)	PUNCT
ejpam-6063	293	6	,	,	PUNCT
ejpam-6063	293	7	it	it	PRON
ejpam-6063	293	8	follows	follow	VERB
ejpam-6063	293	9	that	that	SCONJ
ejpam-6063	293	10	s	s	VERB
ejpam-6063	293	11	is	be	AUX
ejpam-6063	293	12	not	not	PART
ejpam-6063	293	13	a	a	DET
ejpam-6063	293	14	vertex	vertex	NOUN
ejpam-6063	293	15	cover	cover	NOUN
ejpam-6063	293	16	of	of	ADP
ejpam-6063	293	17	g+h	g+h	PROPN
ejpam-6063	293	18	,	,	PUNCT
ejpam-6063	293	19	a	a	DET
ejpam-6063	293	20	contradiction	contradiction	NOUN
ejpam-6063	293	21	.	.	PUNCT
ejpam-6063	294	1	thus	thus	ADV
ejpam-6063	294	2	,	,	PUNCT
ejpam-6063	294	3	sg	sg	PROPN
ejpam-6063	294	4	=	=	SYM
ejpam-6063	294	5	v	v	PROPN
ejpam-6063	294	6	(	(	PUNCT
ejpam-6063	294	7	g	g	NOUN
ejpam-6063	294	8	)	)	PUNCT
ejpam-6063	294	9	or	or	CCONJ
ejpam-6063	294	10	sh	sh	INTJ
ejpam-6063	294	11	=	=	SYM
ejpam-6063	294	12	v	v	PROPN
ejpam-6063	294	13	(	(	PUNCT
ejpam-6063	294	14	h	h	NOUN
ejpam-6063	294	15	)	)	PUNCT
ejpam-6063	294	16	.	.	PUNCT
ejpam-6063	295	1	suppose	suppose	VERB
ejpam-6063	295	2	sg	sg	PROPN
ejpam-6063	295	3	=	=	SYM
ejpam-6063	295	4	v	v	PROPN
ejpam-6063	295	5	(	(	PUNCT
ejpam-6063	295	6	g	g	NOUN
ejpam-6063	295	7	)	)	PUNCT
ejpam-6063	295	8	and	and	CCONJ
ejpam-6063	295	9	let	let	VERB
ejpam-6063	295	10	st	st	PROPN
ejpam-6063	295	11	∈	∈	PROPN
ejpam-6063	295	12	e(h	e(h	PROPN
ejpam-6063	295	13	)	)	PUNCT
ejpam-6063	295	14	.	.	PUNCT
ejpam-6063	296	1	since	since	SCONJ
ejpam-6063	296	2	s	s	PROPN
ejpam-6063	296	3	is	be	AUX
ejpam-6063	296	4	a	a	DET
ejpam-6063	296	5	vertex	vertex	NOUN
ejpam-6063	296	6	cover	cover	NOUN
ejpam-6063	296	7	of	of	ADP
ejpam-6063	296	8	g	g	PROPN
ejpam-6063	296	9	+	+	CCONJ
ejpam-6063	296	10	h	h	NOUN
ejpam-6063	296	11	,	,	PUNCT
ejpam-6063	296	12	s	s	VERB
ejpam-6063	296	13	∈	∈	X
ejpam-6063	296	14	sh	sh	INTJ
ejpam-6063	296	15	or	or	CCONJ
ejpam-6063	296	16	t	t	PROPN
ejpam-6063	296	17	∈	∈	PROPN
ejpam-6063	297	1	sh	sh	INTJ
ejpam-6063	297	2	.	.	PUNCT
ejpam-6063	298	1	hence	hence	ADV
ejpam-6063	298	2	,	,	PUNCT
ejpam-6063	298	3	sh	sh	PROPN
ejpam-6063	298	4	is	be	AUX
ejpam-6063	298	5	a	a	DET
ejpam-6063	298	6	vertex	vertex	NOUN
ejpam-6063	298	7	cover	cover	NOUN
ejpam-6063	298	8	of	of	ADP
ejpam-6063	298	9	h	h	NOUN
ejpam-6063	298	10	,	,	PUNCT
ejpam-6063	298	11	showing	show	VERB
ejpam-6063	298	12	that	that	SCONJ
ejpam-6063	298	13	(	(	PUNCT
ejpam-6063	298	14	i	i	NOUN
ejpam-6063	298	15	)	)	PUNCT
ejpam-6063	298	16	holds	hold	VERB
ejpam-6063	298	17	.	.	PUNCT
ejpam-6063	299	1	similarly	similarly	ADV
ejpam-6063	299	2	,	,	PUNCT
ejpam-6063	299	3	sg	sg	PROPN
ejpam-6063	299	4	is	be	AUX
ejpam-6063	299	5	a	a	DET
ejpam-6063	299	6	vertex	vertex	NOUN
ejpam-6063	299	7	cover	cover	NOUN
ejpam-6063	299	8	of	of	ADP
ejpam-6063	299	9	g	g	NOUN
ejpam-6063	299	10	whenever	whenever	SCONJ
ejpam-6063	299	11	sh	sh	PROPN
ejpam-6063	299	12	=	=	SYM
ejpam-6063	299	13	v	v	PROPN
ejpam-6063	299	14	(	(	PUNCT
ejpam-6063	299	15	h	h	NOUN
ejpam-6063	299	16	)	)	PUNCT
ejpam-6063	299	17	,	,	PUNCT
ejpam-6063	299	18	showing	show	VERB
ejpam-6063	299	19	that	that	SCONJ
ejpam-6063	299	20	(	(	PUNCT
ejpam-6063	299	21	ii	ii	NOUN
ejpam-6063	299	22	)	)	PUNCT
ejpam-6063	299	23	holds	hold	VERB
ejpam-6063	299	24	.	.	PUNCT
ejpam-6063	300	1	for	for	ADP
ejpam-6063	300	2	the	the	DET
ejpam-6063	300	3	converse	converse	NOUN
ejpam-6063	300	4	,	,	PUNCT
ejpam-6063	300	5	suppose	suppose	VERB
ejpam-6063	300	6	that	that	SCONJ
ejpam-6063	300	7	s	s	VERB
ejpam-6063	300	8	=	=	PUNCT
ejpam-6063	300	9	sg	sg	PART
ejpam-6063	300	10	∪	∪	NOUN
ejpam-6063	300	11	sh	sh	PROPN
ejpam-6063	300	12	and	and	CCONJ
ejpam-6063	300	13	(	(	PUNCT
ejpam-6063	300	14	i	i	NOUN
ejpam-6063	300	15	)	)	PUNCT
ejpam-6063	300	16	holds	hold	VERB
ejpam-6063	300	17	.	.	PUNCT
ejpam-6063	301	1	let	let	VERB
ejpam-6063	301	2	pq	pq	INTJ
ejpam-6063	301	3	∈	∈	PROPN
ejpam-6063	301	4	e(g	e(g	PROPN
ejpam-6063	302	1	+	+	CCONJ
ejpam-6063	302	2	h	h	NOUN
ejpam-6063	302	3	)	)	PUNCT
ejpam-6063	302	4	.	.	PUNCT
ejpam-6063	303	1	if	if	SCONJ
ejpam-6063	303	2	p	p	PROPN
ejpam-6063	303	3	∈	∈	PROPN
ejpam-6063	303	4	v	v	ADP
ejpam-6063	303	5	(	(	PUNCT
ejpam-6063	303	6	g	g	NOUN
ejpam-6063	303	7	)	)	PUNCT
ejpam-6063	303	8	or	or	CCONJ
ejpam-6063	303	9	q	q	ADJ
ejpam-6063	303	10	∈	∈	PROPN
ejpam-6063	303	11	v	v	NOUN
ejpam-6063	303	12	(	(	PUNCT
ejpam-6063	303	13	g	g	NOUN
ejpam-6063	303	14	)	)	PUNCT
ejpam-6063	303	15	,	,	PUNCT
ejpam-6063	303	16	then	then	ADV
ejpam-6063	303	17	p	p	PROPN
ejpam-6063	303	18	∈	∈	PROPN
ejpam-6063	303	19	s	s	PART
ejpam-6063	303	20	or	or	CCONJ
ejpam-6063	303	21	q	q	PROPN
ejpam-6063	303	22	∈	∈	PROPN
ejpam-6063	303	23	s.	s.	PROPN
ejpam-6063	303	24	suppose	suppose	VERB
ejpam-6063	303	25	pq	pq	PROPN
ejpam-6063	303	26	∈	∈	PROPN
ejpam-6063	303	27	e(h	e(h	PROPN
ejpam-6063	303	28	)	)	PUNCT
ejpam-6063	303	29	.	.	PUNCT
ejpam-6063	304	1	since	since	SCONJ
ejpam-6063	304	2	sh	sh	PROPN
ejpam-6063	304	3	is	be	AUX
ejpam-6063	304	4	a	a	DET
ejpam-6063	304	5	vertex	vertex	NOUN
ejpam-6063	304	6	cover	cover	NOUN
ejpam-6063	304	7	of	of	ADP
ejpam-6063	304	8	h	h	NOUN
ejpam-6063	304	9	,	,	PUNCT
ejpam-6063	304	10	p	p	PROPN
ejpam-6063	304	11	∈	∈	PROPN
ejpam-6063	304	12	sh	sh	INTJ
ejpam-6063	304	13	⊂	⊂	PROPN
ejpam-6063	304	14	s	s	PART
ejpam-6063	304	15	or	or	CCONJ
ejpam-6063	304	16	q	q	ADJ
ejpam-6063	304	17	∈	∈	PROPN
ejpam-6063	305	1	sh	sh	INTJ
ejpam-6063	305	2	⊂	⊂	PROPN
ejpam-6063	305	3	s.	s.	PROPN
ejpam-6063	305	4	this	this	PRON
ejpam-6063	305	5	implies	imply	VERB
ejpam-6063	305	6	that	that	SCONJ
ejpam-6063	305	7	s	s	VERB
ejpam-6063	305	8	is	be	AUX
ejpam-6063	305	9	a	a	DET
ejpam-6063	305	10	vertex	vertex	NOUN
ejpam-6063	305	11	cover	cover	NOUN
ejpam-6063	305	12	of	of	ADP
ejpam-6063	305	13	g	g	PROPN
ejpam-6063	305	14	+	+	PROPN
ejpam-6063	305	15	h.	h.	PROPN
ejpam-6063	305	16	now	now	ADV
ejpam-6063	305	17	let	let	VERB
ejpam-6063	305	18	z	z	NOUN
ejpam-6063	305	19	∈	∈	PROPN
ejpam-6063	305	20	v	v	X
ejpam-6063	305	21	(	(	PUNCT
ejpam-6063	305	22	g+h	g+h	NOUN
ejpam-6063	305	23	)	)	PUNCT
ejpam-6063	305	24	\	\	PUNCT
ejpam-6063	306	1	s.	s.	PROPN
ejpam-6063	306	2	since	since	SCONJ
ejpam-6063	306	3	sg	sg	PROPN
ejpam-6063	306	4	=	=	SYM
ejpam-6063	306	5	v	v	PROPN
ejpam-6063	306	6	(	(	PUNCT
ejpam-6063	306	7	g	g	NOUN
ejpam-6063	306	8	)	)	PUNCT
ejpam-6063	306	9	,	,	PUNCT
ejpam-6063	306	10	z	z	PROPN
ejpam-6063	306	11	∈	∈	PROPN
ejpam-6063	306	12	v	v	ADP
ejpam-6063	306	13	(	(	PUNCT
ejpam-6063	306	14	h	h	NOUN
ejpam-6063	306	15	)	)	PUNCT
ejpam-6063	306	16	\	\	PUNCT
ejpam-6063	307	1	sh	sh	INTJ
ejpam-6063	307	2	.	.	PUNCT
ejpam-6063	308	1	the	the	DET
ejpam-6063	308	2	assumption	assumption	NOUN
ejpam-6063	308	3	that	that	SCONJ
ejpam-6063	308	4	g	g	PROPN
ejpam-6063	308	5	is	be	AUX
ejpam-6063	308	6	non	non	ADJ
ejpam-6063	308	7	-	-	ADJ
ejpam-6063	308	8	trivial	trivial	ADJ
ejpam-6063	308	9	assures	assure	NOUN
ejpam-6063	308	10	that	that	SCONJ
ejpam-6063	308	11	|ng+h(z	|ng+h(z	PRON
ejpam-6063	308	12	)	)	PUNCT
ejpam-6063	308	13	∩	∩	NOUN
ejpam-6063	308	14	s|	s|	VERB
ejpam-6063	308	15	≥	≥	NUM
ejpam-6063	308	16	|ng+h(z	|ng+h(z	NOUN
ejpam-6063	308	17	)	)	PUNCT
ejpam-6063	308	18	∩	∩	NOUN
ejpam-6063	308	19	sg|	sg|	NOUN
ejpam-6063	308	20	=	=	SYM
ejpam-6063	308	21	|sg|	|sg|	NOUN
ejpam-6063	308	22	≥	≥	NOUN
ejpam-6063	308	23	2	2	NUM
ejpam-6063	308	24	.	.	PUNCT
ejpam-6063	309	1	therefore	therefore	ADV
ejpam-6063	309	2	,	,	PUNCT
ejpam-6063	309	3	s	s	VERB
ejpam-6063	309	4	is	be	AUX
ejpam-6063	309	5	a	a	DET
ejpam-6063	309	6	2	2	NUM
ejpam-6063	309	7	-	-	PUNCT
ejpam-6063	309	8	vertex	vertex	NOUN
ejpam-6063	309	9	covering	covering	NOUN
ejpam-6063	309	10	of	of	ADP
ejpam-6063	309	11	g.	g.	PROPN
ejpam-6063	309	12	the	the	DET
ejpam-6063	309	13	same	same	ADJ
ejpam-6063	309	14	conclusion	conclusion	NOUN
ejpam-6063	309	15	is	be	AUX
ejpam-6063	309	16	true	true	ADJ
ejpam-6063	309	17	for	for	SCONJ
ejpam-6063	309	18	s	s	PRON
ejpam-6063	309	19	if	if	SCONJ
ejpam-6063	309	20	(	(	PUNCT
ejpam-6063	309	21	ii	ii	NOUN
ejpam-6063	309	22	)	)	PUNCT
ejpam-6063	309	23	holds	hold	VERB
ejpam-6063	309	24	.	.	PUNCT
ejpam-6063	310	1	the	the	DET
ejpam-6063	310	2	next	next	ADJ
ejpam-6063	310	3	result	result	NOUN
ejpam-6063	310	4	is	be	AUX
ejpam-6063	310	5	a	a	DET
ejpam-6063	310	6	consequence	consequence	NOUN
ejpam-6063	310	7	of	of	ADP
ejpam-6063	310	8	theorem	theorem	ADJ
ejpam-6063	310	9	9	9	NUM
ejpam-6063	310	10	corollary	corollary	NOUN
ejpam-6063	310	11	5	5	NUM
ejpam-6063	310	12	.	.	PUNCT
ejpam-6063	311	1	let	let	VERB
ejpam-6063	311	2	g	g	NOUN
ejpam-6063	311	3	and	and	CCONJ
ejpam-6063	311	4	h	h	PROPN
ejpam-6063	311	5	be	be	VERB
ejpam-6063	311	6	non	non	ADJ
ejpam-6063	311	7	-	-	ADJ
ejpam-6063	311	8	trivial	trivial	ADJ
ejpam-6063	311	9	graphs	graph	NOUN
ejpam-6063	311	10	of	of	ADP
ejpam-6063	311	11	orders	order	NOUN
ejpam-6063	311	12	m	m	VERB
ejpam-6063	311	13	and	and	CCONJ
ejpam-6063	311	14	n	n	CCONJ
ejpam-6063	311	15	,	,	PUNCT
ejpam-6063	311	16	respectively	respectively	ADV
ejpam-6063	311	17	.	.	PUNCT
ejpam-6063	312	1	then	then	ADV
ejpam-6063	312	2	β2(g+h	β2(g+h	NOUN
ejpam-6063	312	3	)	)	PUNCT
ejpam-6063	313	1	=	=	PUNCT
ejpam-6063	313	2	min{m+	min{m+	PROPN
ejpam-6063	313	3	β(h	β(h	PROPN
ejpam-6063	313	4	)	)	PUNCT
ejpam-6063	313	5	,	,	PUNCT
ejpam-6063	313	6	n+	n+	NUM
ejpam-6063	313	7	β(g	β(g	PROPN
ejpam-6063	313	8	)	)	PUNCT
ejpam-6063	313	9	}	}	PUNCT
ejpam-6063	313	10	.	.	PUNCT
ejpam-6063	314	1	theorem	theorem	ADJ
ejpam-6063	314	2	10	10	NUM
ejpam-6063	314	3	.	.	PUNCT
ejpam-6063	315	1	let	let	VERB
ejpam-6063	315	2	g	g	PRON
ejpam-6063	315	3	be	be	AUX
ejpam-6063	315	4	a	a	DET
ejpam-6063	315	5	non	non	ADJ
ejpam-6063	315	6	-	-	ADJ
ejpam-6063	315	7	trivial	trivial	ADJ
ejpam-6063	315	8	connected	connected	ADJ
ejpam-6063	315	9	graph	graph	NOUN
ejpam-6063	315	10	and	and	CCONJ
ejpam-6063	315	11	let	let	VERB
ejpam-6063	315	12	h	h	NOUN
ejpam-6063	315	13	be	be	AUX
ejpam-6063	315	14	any	any	DET
ejpam-6063	315	15	graph	graph	NOUN
ejpam-6063	315	16	.	.	PUNCT
ejpam-6063	316	1	then	then	ADV
ejpam-6063	316	2	s	s	VERB
ejpam-6063	316	3	⊆	⊆	NUM
ejpam-6063	316	4	v	v	NOUN
ejpam-6063	316	5	(	(	PUNCT
ejpam-6063	316	6	g	g	PROPN
ejpam-6063	316	7	◦	◦	NOUN
ejpam-6063	316	8	h	h	NOUN
ejpam-6063	316	9	)	)	PUNCT
ejpam-6063	316	10	is	be	AUX
ejpam-6063	316	11	a	a	DET
ejpam-6063	316	12	2	2	NUM
ejpam-6063	316	13	-	-	PUNCT
ejpam-6063	316	14	vertex	vertex	NOUN
ejpam-6063	316	15	cover	cover	NOUN
ejpam-6063	316	16	of	of	ADP
ejpam-6063	316	17	g	g	PROPN
ejpam-6063	316	18	◦	◦	NOUN
ejpam-6063	316	19	h	h	NOUN
ejpam-6063	316	20	if	if	SCONJ
ejpam-6063	317	1	and	and	CCONJ
ejpam-6063	317	2	only	only	ADV
ejpam-6063	317	3	if	if	SCONJ
ejpam-6063	317	4	d	d	PROPN
ejpam-6063	317	5	=	=	SYM
ejpam-6063	317	6	q	q	NOUN
ejpam-6063	317	7	∪	∪	X
ejpam-6063	317	8	(	(	PUNCT
ejpam-6063	317	9	∪v∈v	∪v∈v	PROPN
ejpam-6063	317	10	(	(	PUNCT
ejpam-6063	317	11	g)rv	g)rv	PROPN
ejpam-6063	317	12	)	)	PUNCT
ejpam-6063	317	13	and	and	CCONJ
ejpam-6063	317	14	satisfies	satisfy	VERB
ejpam-6063	317	15	the	the	DET
ejpam-6063	317	16	following	follow	VERB
ejpam-6063	317	17	conditions	condition	NOUN
ejpam-6063	317	18	:	:	PUNCT
ejpam-6063	317	19	(	(	PUNCT
ejpam-6063	317	20	i	i	NOUN
ejpam-6063	317	21	)	)	PUNCT
ejpam-6063	317	22	q	q	X
ejpam-6063	317	23	is	be	AUX
ejpam-6063	317	24	a	a	DET
ejpam-6063	317	25	vertex	vertex	NOUN
ejpam-6063	317	26	cover	cover	NOUN
ejpam-6063	317	27	of	of	ADP
ejpam-6063	317	28	g.	g.	PROPN
ejpam-6063	317	29	(	(	PUNCT
ejpam-6063	317	30	ii	ii	PROPN
ejpam-6063	317	31	)	)	PUNCT
ejpam-6063	317	32	i(hw	i(hw	PROPN
ejpam-6063	317	33	)	)	PUNCT
ejpam-6063	317	34	⊆	⊆	NUM
ejpam-6063	317	35	rw	rw	NOUN
ejpam-6063	317	36	and	and	CCONJ
ejpam-6063	317	37	rw	rw	PROPN
ejpam-6063	317	38	\	\	PROPN
ejpam-6063	317	39	i(hw	i(hw	PROPN
ejpam-6063	317	40	)	)	PUNCT
ejpam-6063	317	41	is	be	AUX
ejpam-6063	317	42	a	a	DET
ejpam-6063	317	43	vertex	vertex	NOUN
ejpam-6063	317	44	cover	cover	NOUN
ejpam-6063	317	45	of	of	ADP
ejpam-6063	317	46	hv	hv	PROPN
ejpam-6063	317	47	for	for	ADP
ejpam-6063	317	48	each	each	DET
ejpam-6063	317	49	w	w	PROPN
ejpam-6063	317	50	∈	∈	PROPN
ejpam-6063	317	51	q.	q.	NOUN
ejpam-6063	317	52	(	(	PUNCT
ejpam-6063	317	53	iii	iii	NOUN
ejpam-6063	317	54	)	)	PUNCT
ejpam-6063	317	55	sv	sv	NOUN
ejpam-6063	317	56	=	=	SYM
ejpam-6063	317	57	v	v	PROPN
ejpam-6063	317	58	(	(	PUNCT
ejpam-6063	317	59	hv	hv	PROPN
ejpam-6063	317	60	)	)	PUNCT
ejpam-6063	317	61	for	for	ADP
ejpam-6063	317	62	each	each	DET
ejpam-6063	317	63	v	v	NUM
ejpam-6063	317	64	∈	∈	PROPN
ejpam-6063	317	65	v	v	NOUN
ejpam-6063	317	66	(	(	PUNCT
ejpam-6063	317	67	g	g	NOUN
ejpam-6063	317	68	)	)	PUNCT
ejpam-6063	317	69	\q	\q	NOUN
ejpam-6063	317	70	.	.	PUNCT
ejpam-6063	318	1	proof	proof	NOUN
ejpam-6063	318	2	.	.	PUNCT
ejpam-6063	319	1	assume	assume	VERB
ejpam-6063	319	2	that	that	SCONJ
ejpam-6063	319	3	s	s	VERB
ejpam-6063	319	4	is	be	AUX
ejpam-6063	319	5	a	a	DET
ejpam-6063	319	6	2	2	NUM
ejpam-6063	319	7	-	-	PUNCT
ejpam-6063	319	8	vertex	vertex	NOUN
ejpam-6063	319	9	cover	cover	NOUN
ejpam-6063	319	10	of	of	ADP
ejpam-6063	319	11	g	g	PROPN
ejpam-6063	319	12	◦	◦	NOUN
ejpam-6063	319	13	h.	h.	NOUN
ejpam-6063	319	14	let	let	VERB
ejpam-6063	319	15	q	q	NOUN
ejpam-6063	319	16	=	=	SYM
ejpam-6063	319	17	d	d	PROPN
ejpam-6063	319	18	∩	∩	ADJ
ejpam-6063	319	19	v	v	X
ejpam-6063	319	20	(	(	PUNCT
ejpam-6063	319	21	g	g	NOUN
ejpam-6063	319	22	)	)	PUNCT
ejpam-6063	319	23	and	and	CCONJ
ejpam-6063	319	24	let	let	VERB
ejpam-6063	319	25	rv	rv	NOUN
ejpam-6063	319	26	=	=	SYM
ejpam-6063	319	27	d∩v	d∩v	PROPN
ejpam-6063	319	28	(	(	PUNCT
ejpam-6063	319	29	hv	hv	NOUN
ejpam-6063	319	30	)	)	PUNCT
ejpam-6063	319	31	for	for	ADP
ejpam-6063	319	32	each	each	DET
ejpam-6063	319	33	v	v	NUM
ejpam-6063	319	34	∈	∈	PROPN
ejpam-6063	319	35	v	v	NOUN
ejpam-6063	319	36	(	(	PUNCT
ejpam-6063	319	37	g	g	NOUN
ejpam-6063	319	38	)	)	PUNCT
ejpam-6063	319	39	.	.	PUNCT
ejpam-6063	320	1	clearly	clearly	ADV
ejpam-6063	320	2	,	,	PUNCT
ejpam-6063	320	3	d	d	PROPN
ejpam-6063	320	4	=	=	PUNCT
ejpam-6063	320	5	q∪(∪v∈v	q∪(∪v∈v	PROPN
ejpam-6063	320	6	(	(	PUNCT
ejpam-6063	320	7	g)rv	g)rv	PROPN
ejpam-6063	320	8	)	)	PUNCT
ejpam-6063	320	9	.	.	PUNCT
ejpam-6063	321	1	let	let	VERB
ejpam-6063	321	2	pq	pq	INTJ
ejpam-6063	321	3	∈	∈	PROPN
ejpam-6063	321	4	e(g	e(g	PROPN
ejpam-6063	321	5	)	)	PUNCT
ejpam-6063	322	1	⊂	⊂	PROPN
ejpam-6063	322	2	e(g	e(g	PROPN
ejpam-6063	322	3	◦	◦	NOUN
ejpam-6063	322	4	h	h	NOUN
ejpam-6063	322	5	)	)	PUNCT
ejpam-6063	322	6	.	.	PUNCT
ejpam-6063	323	1	the	the	DET
ejpam-6063	323	2	assumption	assumption	NOUN
ejpam-6063	323	3	that	that	SCONJ
ejpam-6063	323	4	d	d	NOUN
ejpam-6063	323	5	is	be	AUX
ejpam-6063	323	6	a	a	DET
ejpam-6063	323	7	vertex	vertex	NOUN
ejpam-6063	323	8	cover	cover	NOUN
ejpam-6063	323	9	of	of	ADP
ejpam-6063	323	10	g	g	PROPN
ejpam-6063	323	11	◦	◦	NOUN
ejpam-6063	323	12	h	h	NOUN
ejpam-6063	323	13	implies	imply	VERB
ejpam-6063	323	14	that	that	SCONJ
ejpam-6063	323	15	p	p	PROPN
ejpam-6063	323	16	∈	∈	PROPN
ejpam-6063	323	17	q	q	NOUN
ejpam-6063	323	18	or	or	CCONJ
ejpam-6063	323	19	b	b	PROPN
ejpam-6063	323	20	∈	∈	PROPN
ejpam-6063	323	21	q.	q.	NOUN
ejpam-6063	323	22	this	this	PRON
ejpam-6063	323	23	shows	show	VERB
ejpam-6063	323	24	that	that	SCONJ
ejpam-6063	323	25	q	q	NOUN
ejpam-6063	323	26	is	be	AUX
ejpam-6063	323	27	a	a	DET
ejpam-6063	323	28	vertex	vertex	NOUN
ejpam-6063	323	29	cover	cover	NOUN
ejpam-6063	323	30	of	of	ADP
ejpam-6063	323	31	g.	g.	PROPN
ejpam-6063	323	32	this	this	PRON
ejpam-6063	323	33	,	,	PUNCT
ejpam-6063	323	34	in	in	ADP
ejpam-6063	323	35	turn	turn	NOUN
ejpam-6063	323	36	,	,	PUNCT
ejpam-6063	323	37	shows	show	VERB
ejpam-6063	323	38	that	that	SCONJ
ejpam-6063	323	39	(	(	PUNCT
ejpam-6063	323	40	i	i	NOUN
ejpam-6063	323	41	)	)	PUNCT
ejpam-6063	323	42	holds	hold	VERB
ejpam-6063	323	43	.	.	PUNCT
ejpam-6063	324	1	let	let	VERB
ejpam-6063	324	2	w	w	PROPN
ejpam-6063	324	3	∈	∈	PROPN
ejpam-6063	324	4	q.	q.	PROPN
ejpam-6063	324	5	j.	j.	PROPN
ejpam-6063	324	6	hassan	hassan	PROPN
ejpam-6063	324	7	et	et	PROPN
ejpam-6063	324	8	.	.	PUNCT
ejpam-6063	325	1	al	al	PROPN
ejpam-6063	325	2	/	/	PUNCT
ejpam-6063	325	3	eur	eur	PROPN
ejpam-6063	325	4	.	.	PUNCT
ejpam-6063	326	1	j.	j.	PROPN
ejpam-6063	326	2	pure	pure	PROPN
ejpam-6063	326	3	appl	appl	PROPN
ejpam-6063	326	4	.	.	PROPN
ejpam-6063	326	5	math	math	PROPN
ejpam-6063	326	6	,	,	PUNCT
ejpam-6063	326	7	18	18	NUM
ejpam-6063	326	8	(	(	PUNCT
ejpam-6063	326	9	2	2	NUM
ejpam-6063	326	10	)	)	PUNCT
ejpam-6063	326	11	(	(	PUNCT
ejpam-6063	326	12	2025	2025	NUM
ejpam-6063	326	13	)	)	PUNCT
ejpam-6063	326	14	,	,	PUNCT
ejpam-6063	326	15	6063	6063	NUM
ejpam-6063	326	16	9	9	NUM
ejpam-6063	326	17	of	of	ADP
ejpam-6063	326	18	11	11	NUM
ejpam-6063	326	19	since	since	SCONJ
ejpam-6063	326	20	d	d	PROPN
ejpam-6063	326	21	is	be	AUX
ejpam-6063	326	22	a	a	DET
ejpam-6063	326	23	2	2	NUM
ejpam-6063	326	24	-	-	PUNCT
ejpam-6063	326	25	dominating	dominating	NOUN
ejpam-6063	326	26	set	set	NOUN
ejpam-6063	327	1	,	,	PUNCT
ejpam-6063	327	2	it	it	PRON
ejpam-6063	327	3	follows	follow	VERB
ejpam-6063	327	4	that	that	SCONJ
ejpam-6063	327	5	i(hw	i(hw	NUM
ejpam-6063	327	6	)	)	PUNCT
ejpam-6063	327	7	⊆	⊆	NUM
ejpam-6063	327	8	rw	rw	NOUN
ejpam-6063	327	9	.	.	PUNCT
ejpam-6063	328	1	let	let	VERB
ejpam-6063	328	2	st	st	PROPN
ejpam-6063	328	3	∈	∈	PROPN
ejpam-6063	328	4	e(hw	e(hw	PROPN
ejpam-6063	328	5	)	)	PUNCT
ejpam-6063	328	6	.	.	PUNCT
ejpam-6063	329	1	then	then	ADV
ejpam-6063	329	2	s	s	PROPN
ejpam-6063	329	3	,	,	PUNCT
ejpam-6063	329	4	t	t	PROPN
ejpam-6063	329	5	∈	∈	PROPN
ejpam-6063	329	6	v	v	PROPN
ejpam-6063	329	7	(	(	PUNCT
ejpam-6063	329	8	hw	hw	NOUN
ejpam-6063	329	9	)	)	PUNCT
ejpam-6063	329	10	\	\	NOUN
ejpam-6063	329	11	i(hw	i(hw	NUM
ejpam-6063	329	12	)	)	PUNCT
ejpam-6063	329	13	.	.	PUNCT
ejpam-6063	330	1	since	since	SCONJ
ejpam-6063	330	2	d	d	PROPN
ejpam-6063	330	3	is	be	AUX
ejpam-6063	330	4	a	a	DET
ejpam-6063	330	5	vertex	vertex	NOUN
ejpam-6063	330	6	cover	cover	NOUN
ejpam-6063	330	7	of	of	ADP
ejpam-6063	330	8	g	g	PROPN
ejpam-6063	330	9	◦	◦	NOUN
ejpam-6063	330	10	h	h	NOUN
ejpam-6063	330	11	)	)	PUNCT
ejpam-6063	330	12	,	,	PUNCT
ejpam-6063	330	13	s	s	PROPN
ejpam-6063	330	14	∈	∈	PROPN
ejpam-6063	330	15	rw	rw	NOUN
ejpam-6063	330	16	or	or	CCONJ
ejpam-6063	330	17	t	t	PROPN
ejpam-6063	330	18	∈	∈	PROPN
ejpam-6063	330	19	rw	rw	NOUN
ejpam-6063	330	20	.	.	PUNCT
ejpam-6063	331	1	hence	hence	ADV
ejpam-6063	331	2	,	,	PUNCT
ejpam-6063	331	3	s	s	PROPN
ejpam-6063	331	4	∈	∈	PROPN
ejpam-6063	331	5	rw	rw	NOUN
ejpam-6063	331	6	\	\	PROPN
ejpam-6063	331	7	i(hw	i(hw	PROPN
ejpam-6063	331	8	)	)	PUNCT
ejpam-6063	331	9	or	or	CCONJ
ejpam-6063	331	10	t	t	PROPN
ejpam-6063	331	11	∈	∈	PROPN
ejpam-6063	331	12	rw	rw	PROPN
ejpam-6063	331	13	\	\	PROPN
ejpam-6063	331	14	i(hw	i(hw	PROPN
ejpam-6063	331	15	)	)	PUNCT
ejpam-6063	331	16	.	.	PUNCT
ejpam-6063	332	1	this	this	PRON
ejpam-6063	332	2	implies	imply	VERB
ejpam-6063	332	3	that	that	SCONJ
ejpam-6063	332	4	rw	rw	PROPN
ejpam-6063	332	5	\	\	PROPN
ejpam-6063	332	6	i(hw	i(hw	PROPN
ejpam-6063	332	7	)	)	PUNCT
ejpam-6063	332	8	is	be	AUX
ejpam-6063	332	9	a	a	DET
ejpam-6063	332	10	vertex	vertex	NOUN
ejpam-6063	332	11	cover	cover	NOUN
ejpam-6063	332	12	of	of	ADP
ejpam-6063	332	13	hw	hw	PRON
ejpam-6063	332	14	.	.	PUNCT
ejpam-6063	333	1	thus	thus	ADV
ejpam-6063	333	2	,	,	PUNCT
ejpam-6063	333	3	(	(	PUNCT
ejpam-6063	333	4	ii	ii	NOUN
ejpam-6063	333	5	)	)	PUNCT
ejpam-6063	333	6	holds	hold	VERB
ejpam-6063	333	7	.	.	PUNCT
ejpam-6063	334	1	next	next	ADV
ejpam-6063	334	2	,	,	PUNCT
ejpam-6063	334	3	let	let	VERB
ejpam-6063	334	4	v	v	NOUN
ejpam-6063	334	5	/∈	/∈	PUNCT
ejpam-6063	334	6	q	q	PUNCT
ejpam-6063	334	7	and	and	CCONJ
ejpam-6063	334	8	let	let	VERB
ejpam-6063	334	9	q	q	PROPN
ejpam-6063	334	10	∈	∈	PROPN
ejpam-6063	334	11	v	v	X
ejpam-6063	334	12	(	(	PUNCT
ejpam-6063	334	13	hv	hv	PROPN
ejpam-6063	334	14	)	)	PUNCT
ejpam-6063	334	15	.	.	PUNCT
ejpam-6063	335	1	since	since	SCONJ
ejpam-6063	335	2	d	d	PROPN
ejpam-6063	335	3	is	be	AUX
ejpam-6063	335	4	a	a	DET
ejpam-6063	335	5	vertex	vertex	NOUN
ejpam-6063	335	6	cover	cover	NOUN
ejpam-6063	335	7	of	of	ADP
ejpam-6063	335	8	g	g	PROPN
ejpam-6063	335	9	◦	◦	NOUN
ejpam-6063	335	10	h	h	NOUN
ejpam-6063	335	11	and	and	CCONJ
ejpam-6063	335	12	vq	vq	PROPN
ejpam-6063	335	13	∈	∈	PROPN
ejpam-6063	335	14	e(g	e(g	PROPN
ejpam-6063	335	15	◦	◦	PROPN
ejpam-6063	335	16	h	h	NOUN
ejpam-6063	335	17	)	)	PUNCT
ejpam-6063	335	18	,	,	PUNCT
ejpam-6063	335	19	q	q	PROPN
ejpam-6063	335	20	∈	∈	PROPN
ejpam-6063	336	1	rv	rv	X
ejpam-6063	336	2	.	.	PUNCT
ejpam-6063	337	1	since	since	SCONJ
ejpam-6063	337	2	v	v	NUM
ejpam-6063	337	3	was	be	AUX
ejpam-6063	337	4	an	an	DET
ejpam-6063	337	5	arbitrary	arbitrary	ADJ
ejpam-6063	337	6	vertex	vertex	NOUN
ejpam-6063	337	7	of	of	ADP
ejpam-6063	337	8	hv	hv	PROPN
ejpam-6063	337	9	,	,	PUNCT
ejpam-6063	337	10	it	it	PRON
ejpam-6063	337	11	follows	follow	VERB
ejpam-6063	337	12	that	that	PRON
ejpam-6063	337	13	rv	rv	PROPN
ejpam-6063	337	14	=	=	SYM
ejpam-6063	337	15	v	v	PROPN
ejpam-6063	337	16	(	(	PUNCT
ejpam-6063	337	17	hv	hv	PROPN
ejpam-6063	337	18	)	)	PUNCT
ejpam-6063	337	19	.	.	PUNCT
ejpam-6063	338	1	this	this	PRON
ejpam-6063	338	2	shows	show	VERB
ejpam-6063	338	3	that	that	SCONJ
ejpam-6063	338	4	(	(	PUNCT
ejpam-6063	338	5	iii	iii	NOUN
ejpam-6063	338	6	)	)	PUNCT
ejpam-6063	338	7	also	also	ADV
ejpam-6063	338	8	holds	hold	VERB
ejpam-6063	338	9	.	.	PUNCT
ejpam-6063	339	1	for	for	ADP
ejpam-6063	339	2	the	the	DET
ejpam-6063	339	3	converse	converse	NOUN
ejpam-6063	339	4	,	,	PUNCT
ejpam-6063	339	5	suppose	suppose	VERB
ejpam-6063	339	6	that	that	SCONJ
ejpam-6063	339	7	d	d	NOUN
ejpam-6063	339	8	is	be	AUX
ejpam-6063	339	9	as	as	SCONJ
ejpam-6063	339	10	described	describe	VERB
ejpam-6063	339	11	and	and	CCONJ
ejpam-6063	339	12	satisfies	satisfie	NOUN
ejpam-6063	339	13	(	(	PUNCT
ejpam-6063	339	14	i	i	NOUN
ejpam-6063	339	15	)	)	PUNCT
ejpam-6063	339	16	,	,	PUNCT
ejpam-6063	339	17	(	(	PUNCT
ejpam-6063	339	18	ii	ii	NOUN
ejpam-6063	339	19	)	)	PUNCT
ejpam-6063	339	20	,	,	PUNCT
ejpam-6063	339	21	and	and	CCONJ
ejpam-6063	339	22	(	(	PUNCT
ejpam-6063	339	23	iii	iii	NOUN
ejpam-6063	339	24	)	)	PUNCT
ejpam-6063	339	25	.	.	PUNCT
ejpam-6063	340	1	let	let	VERB
ejpam-6063	340	2	vw	vw	PRON
ejpam-6063	340	3	∈	∈	PROPN
ejpam-6063	340	4	e(g	e(g	PROPN
ejpam-6063	340	5	◦	◦	NOUN
ejpam-6063	340	6	h	h	NOUN
ejpam-6063	340	7	)	)	PUNCT
ejpam-6063	340	8	.	.	PUNCT
ejpam-6063	341	1	if	if	SCONJ
ejpam-6063	341	2	v	v	X
ejpam-6063	341	3	,	,	PUNCT
ejpam-6063	341	4	w	w	PROPN
ejpam-6063	341	5	∈	∈	PROPN
ejpam-6063	341	6	v	v	ADP
ejpam-6063	341	7	(	(	PUNCT
ejpam-6063	341	8	g	g	NOUN
ejpam-6063	341	9	)	)	PUNCT
ejpam-6063	341	10	,	,	PUNCT
ejpam-6063	341	11	then	then	ADV
ejpam-6063	341	12	v	v	X
ejpam-6063	341	13	∈	∈	PROPN
ejpam-6063	341	14	q	q	NOUN
ejpam-6063	341	15	or	or	CCONJ
ejpam-6063	341	16	v	v	ADP
ejpam-6063	341	17	∈	∈	NOUN
ejpam-6063	341	18	q	q	PUNCT
ejpam-6063	341	19	by	by	ADP
ejpam-6063	341	20	(	(	PUNCT
ejpam-6063	341	21	i	i	NOUN
ejpam-6063	341	22	)	)	PUNCT
ejpam-6063	341	23	.	.	PUNCT
ejpam-6063	342	1	suppose	suppose	VERB
ejpam-6063	342	2	v	v	ADP
ejpam-6063	342	3	∈	∈	PROPN
ejpam-6063	342	4	v	v	NOUN
ejpam-6063	342	5	(	(	PUNCT
ejpam-6063	342	6	g	g	NOUN
ejpam-6063	342	7	)	)	PUNCT
ejpam-6063	342	8	and	and	CCONJ
ejpam-6063	342	9	w	w	PROPN
ejpam-6063	342	10	∈	∈	PROPN
ejpam-6063	342	11	v	v	ADP
ejpam-6063	342	12	(	(	PUNCT
ejpam-6063	342	13	hv	hv	PROPN
ejpam-6063	342	14	)	)	PUNCT
ejpam-6063	342	15	.	.	PUNCT
ejpam-6063	343	1	if	if	SCONJ
ejpam-6063	343	2	v	v	NUM
ejpam-6063	343	3	∈	∈	PROPN
ejpam-6063	343	4	q	q	NOUN
ejpam-6063	343	5	,	,	PUNCT
ejpam-6063	343	6	then	then	ADV
ejpam-6063	343	7	vw	vw	PROPN
ejpam-6063	343	8	is	be	AUX
ejpam-6063	343	9	incident	incident	NOUN
ejpam-6063	343	10	to	to	ADP
ejpam-6063	343	11	v	v	PROPN
ejpam-6063	343	12	∈	∈	PROPN
ejpam-6063	343	13	d.	d.	NOUN
ejpam-6063	343	14	suppose	suppose	VERB
ejpam-6063	343	15	v	v	X
ejpam-6063	343	16	/∈	/∈	PUNCT
ejpam-6063	343	17	q.	q.	PROPN
ejpam-6063	344	1	then	then	ADV
ejpam-6063	344	2	rv	rv	PROPN
ejpam-6063	345	1	=	=	SYM
ejpam-6063	345	2	v	v	PROPN
ejpam-6063	345	3	(	(	PUNCT
ejpam-6063	345	4	hv	hv	NOUN
ejpam-6063	345	5	)	)	PUNCT
ejpam-6063	345	6	by	by	ADP
ejpam-6063	345	7	(	(	PUNCT
ejpam-6063	345	8	iii	iii	NOUN
ejpam-6063	345	9	)	)	PUNCT
ejpam-6063	345	10	.	.	PUNCT
ejpam-6063	346	1	hence	hence	ADV
ejpam-6063	346	2	,	,	PUNCT
ejpam-6063	346	3	w	w	PROPN
ejpam-6063	346	4	∈	∈	PROPN
ejpam-6063	346	5	sv	sv	NOUN
ejpam-6063	346	6	and	and	CCONJ
ejpam-6063	346	7	vw	vw	PROPN
ejpam-6063	346	8	is	be	AUX
ejpam-6063	346	9	incident	incident	NOUN
ejpam-6063	346	10	to	to	ADP
ejpam-6063	346	11	w	w	PROPN
ejpam-6063	346	12	∈	∈	PROPN
ejpam-6063	346	13	d.	d.	PROPN
ejpam-6063	346	14	next	next	ADV
ejpam-6063	346	15	,	,	PUNCT
ejpam-6063	346	16	suppose	suppose	VERB
ejpam-6063	346	17	that	that	SCONJ
ejpam-6063	346	18	that	that	PRON
ejpam-6063	346	19	v	v	NOUN
ejpam-6063	346	20	,	,	PUNCT
ejpam-6063	346	21	w	w	PROPN
ejpam-6063	346	22	∈	∈	PROPN
ejpam-6063	346	23	v	v	X
ejpam-6063	346	24	(	(	PUNCT
ejpam-6063	346	25	hz	hz	NOUN
ejpam-6063	346	26	)	)	PUNCT
ejpam-6063	346	27	for	for	ADP
ejpam-6063	346	28	some	some	DET
ejpam-6063	346	29	z	z	NOUN
ejpam-6063	346	30	∈	∈	PROPN
ejpam-6063	346	31	v	v	ADP
ejpam-6063	346	32	(	(	PUNCT
ejpam-6063	346	33	g	g	NOUN
ejpam-6063	346	34	)	)	PUNCT
ejpam-6063	346	35	.	.	PUNCT
ejpam-6063	347	1	if	if	SCONJ
ejpam-6063	347	2	z	z	NOUN
ejpam-6063	347	3	/∈	/∈	PUNCT
ejpam-6063	348	1	q	q	ADJ
ejpam-6063	348	2	,	,	PUNCT
ejpam-6063	348	3	then	then	ADV
ejpam-6063	348	4	rz	rz	PROPN
ejpam-6063	348	5	=	=	SYM
ejpam-6063	348	6	v	v	PROPN
ejpam-6063	348	7	(	(	PUNCT
ejpam-6063	348	8	hz	hz	PROPN
ejpam-6063	348	9	)	)	PUNCT
ejpam-6063	348	10	.	.	PUNCT
ejpam-6063	349	1	this	this	PRON
ejpam-6063	349	2	implies	imply	VERB
ejpam-6063	349	3	that	that	SCONJ
ejpam-6063	349	4	v	v	NOUN
ejpam-6063	349	5	,	,	PUNCT
ejpam-6063	349	6	w	w	PROPN
ejpam-6063	349	7	∈	∈	PROPN
ejpam-6063	349	8	rz	rz	NOUN
ejpam-6063	349	9	⊂	⊂	PROPN
ejpam-6063	349	10	d.	d.	PROPN
ejpam-6063	349	11	suppose	suppose	VERB
ejpam-6063	349	12	that	that	SCONJ
ejpam-6063	349	13	z	z	PROPN
ejpam-6063	349	14	∈	∈	PROPN
ejpam-6063	349	15	q.	q.	PROPN
ejpam-6063	349	16	since	since	SCONJ
ejpam-6063	349	17	vw	vw	PROPN
ejpam-6063	349	18	∈	∈	PROPN
ejpam-6063	349	19	e(g	e(g	PROPN
ejpam-6063	349	20	◦	◦	PROPN
ejpam-6063	349	21	h	h	NOUN
ejpam-6063	349	22	)	)	PUNCT
ejpam-6063	349	23	,	,	PUNCT
ejpam-6063	349	24	v	v	NOUN
ejpam-6063	349	25	,	,	PUNCT
ejpam-6063	349	26	w	w	PROPN
ejpam-6063	349	27	∈	∈	PROPN
ejpam-6063	349	28	v	v	X
ejpam-6063	349	29	(	(	PUNCT
ejpam-6063	349	30	hz	hz	NOUN
ejpam-6063	349	31	)	)	PUNCT
ejpam-6063	349	32	\	\	NOUN
ejpam-6063	349	33	i(hz	i(hz	PROPN
ejpam-6063	349	34	)	)	PUNCT
ejpam-6063	349	35	.	.	PUNCT
ejpam-6063	350	1	by	by	ADP
ejpam-6063	350	2	(	(	PUNCT
ejpam-6063	350	3	ii	ii	NOUN
ejpam-6063	350	4	)	)	PUNCT
ejpam-6063	350	5	,	,	PUNCT
ejpam-6063	350	6	rz	rz	NOUN
ejpam-6063	350	7	\	\	PROPN
ejpam-6063	350	8	i(hz	i(hz	X
ejpam-6063	350	9	)	)	PUNCT
ejpam-6063	350	10	is	be	AUX
ejpam-6063	350	11	a	a	DET
ejpam-6063	350	12	vertex	vertex	NOUN
ejpam-6063	350	13	cover	cover	NOUN
ejpam-6063	350	14	of	of	ADP
ejpam-6063	350	15	hz	hz	PROPN
ejpam-6063	350	16	.	.	PUNCT
ejpam-6063	351	1	it	it	PRON
ejpam-6063	351	2	follows	follow	VERB
ejpam-6063	351	3	that	that	SCONJ
ejpam-6063	351	4	v	v	NUM
ejpam-6063	351	5	∈	∈	PROPN
ejpam-6063	351	6	rz	rz	NOUN
ejpam-6063	351	7	\	\	PROPN
ejpam-6063	351	8	i(hz	i(hz	PROPN
ejpam-6063	351	9	)	)	PUNCT
ejpam-6063	351	10	or	or	CCONJ
ejpam-6063	351	11	w	w	PROPN
ejpam-6063	351	12	∈	∈	PROPN
ejpam-6063	351	13	rz	rz	NOUN
ejpam-6063	351	14	\	\	PROPN
ejpam-6063	351	15	i(hz	i(hz	PROPN
ejpam-6063	351	16	)	)	PUNCT
ejpam-6063	351	17	.	.	PUNCT
ejpam-6063	352	1	therefore	therefore	ADV
ejpam-6063	352	2	,	,	PUNCT
ejpam-6063	352	3	d	d	X
ejpam-6063	352	4	is	be	AUX
ejpam-6063	352	5	a	a	DET
ejpam-6063	352	6	vertex	vertex	NOUN
ejpam-6063	352	7	cover	cover	NOUN
ejpam-6063	352	8	of	of	ADP
ejpam-6063	352	9	g	g	PROPN
ejpam-6063	352	10	◦	◦	NOUN
ejpam-6063	352	11	h.	h.	PROPN
ejpam-6063	352	12	finally	finally	ADV
ejpam-6063	352	13	,	,	PUNCT
ejpam-6063	352	14	let	let	VERB
ejpam-6063	352	15	x	x	PUNCT
ejpam-6063	352	16	∈	∈	PROPN
ejpam-6063	352	17	v	v	X
ejpam-6063	352	18	(	(	PUNCT
ejpam-6063	352	19	g	g	PROPN
ejpam-6063	352	20	◦	◦	NOUN
ejpam-6063	352	21	h	h	NOUN
ejpam-6063	352	22	)	)	PUNCT
ejpam-6063	352	23	\d	\d	NOUN
ejpam-6063	352	24	and	and	CCONJ
ejpam-6063	352	25	let	let	VERB
ejpam-6063	352	26	v	v	NUM
ejpam-6063	352	27	∈	∈	PROPN
ejpam-6063	352	28	v	v	NOUN
ejpam-6063	352	29	(	(	PUNCT
ejpam-6063	352	30	g	g	NOUN
ejpam-6063	352	31	)	)	PUNCT
ejpam-6063	352	32	such	such	ADJ
ejpam-6063	352	33	that	that	SCONJ
ejpam-6063	352	34	x	x	SYM
ejpam-6063	352	35	∈	∈	NOUN
ejpam-6063	352	36	v	v	NOUN
ejpam-6063	352	37	(	(	PUNCT
ejpam-6063	352	38	v	v	NOUN
ejpam-6063	352	39	+	+	CCONJ
ejpam-6063	352	40	hv	hv	NOUN
ejpam-6063	352	41	)	)	PUNCT
ejpam-6063	352	42	.	.	PUNCT
ejpam-6063	353	1	if	if	SCONJ
ejpam-6063	353	2	x	x	X
ejpam-6063	353	3	=	=	SYM
ejpam-6063	353	4	v	v	NOUN
ejpam-6063	353	5	,	,	PUNCT
ejpam-6063	353	6	then	then	ADV
ejpam-6063	353	7	v	v	ADP
ejpam-6063	353	8	/∈	/∈	PUNCT
ejpam-6063	353	9	q.	q.	PROPN
ejpam-6063	353	10	by	by	ADP
ejpam-6063	353	11	(	(	PUNCT
ejpam-6063	353	12	iii	iii	NOUN
ejpam-6063	353	13	)	)	PUNCT
ejpam-6063	353	14	,	,	PUNCT
ejpam-6063	353	15	rv	rv	PROPN
ejpam-6063	354	1	=	=	SYM
ejpam-6063	354	2	v	v	PROPN
ejpam-6063	354	3	(	(	PUNCT
ejpam-6063	354	4	hv	hv	PROPN
ejpam-6063	354	5	)	)	PUNCT
ejpam-6063	354	6	.	.	PUNCT
ejpam-6063	355	1	since	since	SCONJ
ejpam-6063	355	2	g	g	PROPN
ejpam-6063	355	3	is	be	AUX
ejpam-6063	355	4	a	a	DET
ejpam-6063	355	5	non	non	ADJ
ejpam-6063	355	6	-	-	ADJ
ejpam-6063	355	7	trivial	trivial	ADJ
ejpam-6063	355	8	connected	connected	ADJ
ejpam-6063	355	9	graph	graph	NOUN
ejpam-6063	355	10	and	and	CCONJ
ejpam-6063	355	11	q	q	NOUN
ejpam-6063	355	12	is	be	AUX
ejpam-6063	355	13	a	a	DET
ejpam-6063	355	14	vertex	vertex	NOUN
ejpam-6063	355	15	cover	cover	NOUN
ejpam-6063	355	16	of	of	ADP
ejpam-6063	355	17	g	g	NOUN
ejpam-6063	355	18	,	,	PUNCT
ejpam-6063	355	19	it	it	PRON
ejpam-6063	355	20	follows	follow	VERB
ejpam-6063	355	21	that	that	SCONJ
ejpam-6063	355	22	ng(v)∩q	ng(v)∩q	PROPN
ejpam-6063	355	23	̸=	̸=	PROPN
ejpam-6063	355	24	∅.	∅.	ADV
ejpam-6063	355	25	choose	choose	VERB
ejpam-6063	355	26	any	any	DET
ejpam-6063	355	27	u	u	NOUN
ejpam-6063	355	28	∈	∈	PROPN
ejpam-6063	355	29	ng(v	ng(v	PUNCT
ejpam-6063	355	30	)	)	PUNCT
ejpam-6063	355	31	∩	∩	NOUN
ejpam-6063	355	32	q	q	X
ejpam-6063	355	33	and	and	CCONJ
ejpam-6063	355	34	s	s	PROPN
ejpam-6063	355	35	∈	∈	PROPN
ejpam-6063	355	36	rv	rv	PROPN
ejpam-6063	355	37	.	.	PUNCT
ejpam-6063	356	1	then	then	ADV
ejpam-6063	356	2	u	u	SYM
ejpam-6063	356	3	,	,	PUNCT
ejpam-6063	356	4	s	s	PROPN
ejpam-6063	356	5	∈	∈	PROPN
ejpam-6063	356	6	ng	ng	PROPN
ejpam-6063	356	7	◦	◦	NOUN
ejpam-6063	356	8	h(v	h(v	NOUN
ejpam-6063	356	9	)	)	PUNCT
ejpam-6063	356	10	∩d	∩d	NOUN
ejpam-6063	356	11	.	.	PUNCT
ejpam-6063	356	12	suppose	suppose	VERB
ejpam-6063	356	13	x	x	SYM
ejpam-6063	356	14	∈	∈	PROPN
ejpam-6063	356	15	v	v	ADP
ejpam-6063	356	16	(	(	PUNCT
ejpam-6063	356	17	hv	hv	PROPN
ejpam-6063	356	18	)	)	PUNCT
ejpam-6063	356	19	\	\	PROPN
ejpam-6063	357	1	rv	rv	PROPN
ejpam-6063	357	2	.	.	PUNCT
ejpam-6063	358	1	then	then	ADV
ejpam-6063	358	2	x	x	X
ejpam-6063	358	3	/∈	/∈	PUNCT
ejpam-6063	358	4	i(hv	i(hv	NUM
ejpam-6063	358	5	)	)	PUNCT
ejpam-6063	358	6	because	because	SCONJ
ejpam-6063	358	7	x	x	PROPN
ejpam-6063	358	8	/∈	/∈	PROPN
ejpam-6063	358	9	d.	d.	PROPN
ejpam-6063	358	10	also	also	ADV
ejpam-6063	358	11	,	,	PUNCT
ejpam-6063	358	12	from	from	ADP
ejpam-6063	358	13	(	(	PUNCT
ejpam-6063	358	14	iii	iii	NOUN
ejpam-6063	358	15	)	)	PUNCT
ejpam-6063	358	16	,	,	PUNCT
ejpam-6063	358	17	it	it	PRON
ejpam-6063	358	18	follows	follow	VERB
ejpam-6063	358	19	that	that	SCONJ
ejpam-6063	358	20	v	v	ADP
ejpam-6063	358	21	∈	∈	PROPN
ejpam-6063	358	22	q	q	NOUN
ejpam-6063	359	1	(	(	PUNCT
ejpam-6063	359	2	otherwise	otherwise	ADV
ejpam-6063	359	3	,	,	PUNCT
ejpam-6063	359	4	rv	rv	PROPN
ejpam-6063	359	5	=	=	SYM
ejpam-6063	359	6	v	v	PROPN
ejpam-6063	359	7	(	(	PUNCT
ejpam-6063	359	8	hv	hv	PROPN
ejpam-6063	359	9	)	)	PUNCT
ejpam-6063	359	10	contrary	contrary	ADV
ejpam-6063	359	11	to	to	ADP
ejpam-6063	359	12	the	the	DET
ejpam-6063	359	13	fact	fact	NOUN
ejpam-6063	359	14	that	that	SCONJ
ejpam-6063	359	15	x	x	SYM
ejpam-6063	359	16	∈	∈	NOUN
ejpam-6063	359	17	v	v	ADP
ejpam-6063	359	18	(	(	PUNCT
ejpam-6063	359	19	hv	hv	PROPN
ejpam-6063	359	20	)	)	PUNCT
ejpam-6063	359	21	\	\	PROPN
ejpam-6063	359	22	rv	rv	PROPN
ejpam-6063	359	23	)	)	PUNCT
ejpam-6063	359	24	.	.	PUNCT
ejpam-6063	360	1	hence	hence	ADV
ejpam-6063	360	2	,	,	PUNCT
ejpam-6063	360	3	from	from	ADP
ejpam-6063	360	4	(	(	PUNCT
ejpam-6063	360	5	ii	ii	NOUN
ejpam-6063	360	6	)	)	PUNCT
ejpam-6063	360	7	,	,	PUNCT
ejpam-6063	360	8	rv	rv	PROPN
ejpam-6063	360	9	\	\	PROPN
ejpam-6063	360	10	i(g	i(g	ADV
ejpam-6063	360	11	)	)	PUNCT
ejpam-6063	360	12	is	be	AUX
ejpam-6063	360	13	a	a	DET
ejpam-6063	360	14	vertex	vertex	NOUN
ejpam-6063	360	15	cover	cover	NOUN
ejpam-6063	360	16	of	of	ADP
ejpam-6063	360	17	hv	hv	PROPN
ejpam-6063	360	18	.	.	PUNCT
ejpam-6063	361	1	this	this	PRON
ejpam-6063	361	2	implies	imply	VERB
ejpam-6063	361	3	that	that	SCONJ
ejpam-6063	361	4	nhv(x	nhv(x	PROPN
ejpam-6063	361	5	)	)	PUNCT
ejpam-6063	361	6	∩	∩	NOUN
ejpam-6063	361	7	(	(	PUNCT
ejpam-6063	361	8	rv	rv	NOUN
ejpam-6063	361	9	\	\	PROPN
ejpam-6063	361	10	i(g	i(g	NOUN
ejpam-6063	361	11	)	)	PUNCT
ejpam-6063	361	12	)	)	PUNCT
ejpam-6063	362	1	̸=	̸=	PROPN
ejpam-6063	362	2	∅.	∅.	ADV
ejpam-6063	362	3	since	since	SCONJ
ejpam-6063	362	4	v	v	PROPN
ejpam-6063	362	5	∈	∈	PROPN
ejpam-6063	362	6	ng	ng	PROPN
ejpam-6063	362	7	◦	◦	NOUN
ejpam-6063	362	8	h(x	h(x	PROPN
ejpam-6063	362	9	)	)	PUNCT
ejpam-6063	362	10	,	,	PUNCT
ejpam-6063	362	11	it	it	PRON
ejpam-6063	362	12	follows	follow	VERB
ejpam-6063	362	13	that	that	SCONJ
ejpam-6063	362	14	|ng	|ng	AUX
ejpam-6063	362	15	◦	◦	NOUN
ejpam-6063	362	16	h(x	h(x	PROPN
ejpam-6063	362	17	)	)	PUNCT
ejpam-6063	362	18	∩	∩	PROPN
ejpam-6063	362	19	d|	d|	PROPN
ejpam-6063	362	20	≥	≥	NUM
ejpam-6063	362	21	|nhv(x	|nhv(x	NUM
ejpam-6063	362	22	)	)	PUNCT
ejpam-6063	362	23	∩	∩	NOUN
ejpam-6063	362	24	(	(	PUNCT
ejpam-6063	362	25	rv	rv	NOUN
ejpam-6063	362	26	\	\	PROPN
ejpam-6063	362	27	i(g))|	i(g))|	NOUN
ejpam-6063	362	28	+	+	CCONJ
ejpam-6063	362	29	1	1	NUM
ejpam-6063	362	30	≥	≥	NOUN
ejpam-6063	362	31	2	2	NUM
ejpam-6063	362	32	.	.	PUNCT
ejpam-6063	363	1	this	this	PRON
ejpam-6063	363	2	shows	show	VERB
ejpam-6063	363	3	that	that	SCONJ
ejpam-6063	363	4	d	d	NOUN
ejpam-6063	363	5	is	be	AUX
ejpam-6063	363	6	a	a	DET
ejpam-6063	363	7	2	2	NUM
ejpam-6063	363	8	-	-	PUNCT
ejpam-6063	363	9	dominating	dominating	NOUN
ejpam-6063	363	10	set	set	NOUN
ejpam-6063	363	11	in	in	ADP
ejpam-6063	363	12	g	g	PROPN
ejpam-6063	363	13	◦	◦	NOUN
ejpam-6063	363	14	h.	h.	PROPN
ejpam-6063	363	15	therefore	therefore	ADV
ejpam-6063	363	16	,	,	PUNCT
ejpam-6063	363	17	d	d	X
ejpam-6063	363	18	is	be	AUX
ejpam-6063	363	19	a	a	DET
ejpam-6063	363	20	2	2	NUM
ejpam-6063	363	21	-	-	PUNCT
ejpam-6063	363	22	vertex	vertex	NOUN
ejpam-6063	363	23	cover	cover	NOUN
ejpam-6063	363	24	of	of	ADP
ejpam-6063	363	25	g	g	PROPN
ejpam-6063	363	26	◦	◦	NOUN
ejpam-6063	363	27	h.	h.	PROPN
ejpam-6063	363	28	corollary	corollary	ADJ
ejpam-6063	363	29	6	6	NUM
ejpam-6063	363	30	.	.	PUNCT
ejpam-6063	364	1	let	let	VERB
ejpam-6063	364	2	g	g	PRON
ejpam-6063	364	3	be	be	AUX
ejpam-6063	364	4	a	a	DET
ejpam-6063	364	5	non	non	ADJ
ejpam-6063	364	6	-	-	ADJ
ejpam-6063	364	7	trivial	trivial	ADJ
ejpam-6063	364	8	connected	connected	ADJ
ejpam-6063	364	9	graph	graph	NOUN
ejpam-6063	364	10	of	of	ADP
ejpam-6063	364	11	order	order	NOUN
ejpam-6063	364	12	m	m	VERB
ejpam-6063	364	13	and	and	CCONJ
ejpam-6063	364	14	let	let	VERB
ejpam-6063	364	15	h	h	NOUN
ejpam-6063	364	16	be	be	AUX
ejpam-6063	364	17	any	any	DET
ejpam-6063	364	18	graph	graph	NOUN
ejpam-6063	364	19	of	of	ADP
ejpam-6063	364	20	order	order	NOUN
ejpam-6063	364	21	n.	n.	NOUN
ejpam-6063	364	22	then	then	ADV
ejpam-6063	364	23	β2(g	β2(g	PUNCT
ejpam-6063	364	24	◦	◦	NOUN
ejpam-6063	364	25	h	h	NOUN
ejpam-6063	364	26	)	)	PUNCT
ejpam-6063	365	1	=	=	VERB
ejpam-6063	365	2	mn+	mn+	NOUN
ejpam-6063	365	3	(	(	PUNCT
ejpam-6063	365	4	β(h)−	β(h)−	NOUN
ejpam-6063	365	5	n+	n+	ADJ
ejpam-6063	365	6	|i(h)|+	|i(h)|+	NOUN
ejpam-6063	365	7	1)β(g	1)β(g	NUM
ejpam-6063	365	8	)	)	PUNCT
ejpam-6063	365	9	.	.	PUNCT
ejpam-6063	366	1	in	in	ADP
ejpam-6063	366	2	particular	particular	ADJ
ejpam-6063	366	3	,	,	PUNCT
ejpam-6063	366	4	if	if	SCONJ
ejpam-6063	366	5	h	h	NOUN
ejpam-6063	366	6	is	be	AUX
ejpam-6063	366	7	a	a	DET
ejpam-6063	366	8	non	non	ADJ
ejpam-6063	366	9	-	-	ADJ
ejpam-6063	366	10	trivial	trivial	ADJ
ejpam-6063	366	11	connected	connected	ADJ
ejpam-6063	366	12	graph	graph	NOUN
ejpam-6063	366	13	,	,	PUNCT
ejpam-6063	366	14	then	then	ADV
ejpam-6063	366	15	β2(g	β2(g	PUNCT
ejpam-6063	366	16	◦	◦	NOUN
ejpam-6063	366	17	h	h	NOUN
ejpam-6063	366	18	)	)	PUNCT
ejpam-6063	366	19	=	=	VERB
ejpam-6063	366	20	mn+	mn+	NOUN
ejpam-6063	366	21	(	(	PUNCT
ejpam-6063	366	22	β(h)−	β(h)−	NOUN
ejpam-6063	366	23	n+	n+	NUM
ejpam-6063	366	24	1)β(g	1)β(g	NUM
ejpam-6063	366	25	)	)	PUNCT
ejpam-6063	366	26	.	.	PUNCT
ejpam-6063	367	1	proof	proof	NOUN
ejpam-6063	367	2	.	.	PUNCT
ejpam-6063	368	1	let	let	VERB
ejpam-6063	368	2	q	q	PART
ejpam-6063	368	3	be	be	AUX
ejpam-6063	368	4	a	a	DET
ejpam-6063	368	5	β	β	NOUN
ejpam-6063	368	6	-	-	VERB
ejpam-6063	368	7	set	set	VERB
ejpam-6063	368	8	in	in	ADP
ejpam-6063	368	9	g	g	PROPN
ejpam-6063	368	10	,	,	PUNCT
ejpam-6063	368	11	dv	dv	PROPN
ejpam-6063	368	12	a	a	DET
ejpam-6063	368	13	β	β	X
ejpam-6063	368	14	-	-	PUNCT
ejpam-6063	368	15	set	set	VERB
ejpam-6063	368	16	in	in	ADP
ejpam-6063	368	17	hv	hv	PROPN
ejpam-6063	368	18	and	and	CCONJ
ejpam-6063	368	19	rv	rv	PROPN
ejpam-6063	368	20	=	=	PROPN
ejpam-6063	368	21	dv	dv	PROPN
ejpam-6063	368	22	∪	∪	ADP
ejpam-6063	368	23	i(hv	i(hv	NOUN
ejpam-6063	368	24	)	)	PUNCT
ejpam-6063	368	25	for	for	ADP
ejpam-6063	368	26	each	each	DET
ejpam-6063	368	27	v	v	NOUN
ejpam-6063	368	28	∈	∈	PROPN
ejpam-6063	368	29	q	q	NOUN
ejpam-6063	368	30	,	,	PUNCT
ejpam-6063	368	31	and	and	CCONJ
ejpam-6063	368	32	let	let	VERB
ejpam-6063	368	33	sv	sv	INTJ
ejpam-6063	368	34	=	=	SYM
ejpam-6063	368	35	v	v	PROPN
ejpam-6063	368	36	(	(	PUNCT
ejpam-6063	368	37	hv	hv	PROPN
ejpam-6063	368	38	)	)	PUNCT
ejpam-6063	368	39	for	for	ADP
ejpam-6063	368	40	each	each	PRON
ejpam-6063	368	41	v	v	NUM
ejpam-6063	368	42	∈	∈	PROPN
ejpam-6063	368	43	v	v	NOUN
ejpam-6063	368	44	(	(	PUNCT
ejpam-6063	368	45	g	g	NOUN
ejpam-6063	368	46	)	)	PUNCT
ejpam-6063	368	47	\	\	NOUN
ejpam-6063	369	1	q.	q.	NOUN
ejpam-6063	370	1	then	then	ADV
ejpam-6063	370	2	d	d	PROPN
ejpam-6063	370	3	=	=	SYM
ejpam-6063	370	4	q	q	NOUN
ejpam-6063	370	5	∪	∪	X
ejpam-6063	370	6	(	(	PUNCT
ejpam-6063	370	7	∪v∈v	∪v∈v	PROPN
ejpam-6063	370	8	(	(	PUNCT
ejpam-6063	370	9	g)rv	g)rv	PROPN
ejpam-6063	370	10	)	)	PUNCT
ejpam-6063	370	11	is	be	AUX
ejpam-6063	370	12	a	a	DET
ejpam-6063	370	13	2	2	NUM
ejpam-6063	370	14	-	-	PUNCT
ejpam-6063	370	15	vertex	vertex	NOUN
ejpam-6063	370	16	cover	cover	NOUN
ejpam-6063	370	17	of	of	ADP
ejpam-6063	370	18	g	g	PROPN
ejpam-6063	370	19	◦	◦	NOUN
ejpam-6063	370	20	h	h	NOUN
ejpam-6063	370	21	by	by	ADP
ejpam-6063	370	22	theorem	theorem	NOUN
ejpam-6063	370	23	10	10	NUM
ejpam-6063	370	24	.	.	PUNCT
ejpam-6063	371	1	it	it	PRON
ejpam-6063	371	2	follows	follow	VERB
ejpam-6063	371	3	that	that	SCONJ
ejpam-6063	371	4	β2(g	β2(g	PUNCT
ejpam-6063	371	5	◦	◦	NOUN
ejpam-6063	371	6	h	h	NOUN
ejpam-6063	371	7	)	)	PUNCT
ejpam-6063	371	8	≤	≤	NOUN
ejpam-6063	371	9	|d|	|d|	PROPN
ejpam-6063	371	10	=	=	SYM
ejpam-6063	371	11	|q|+	|q|+	NOUN
ejpam-6063	371	12	∑	∑	NOUN
ejpam-6063	371	13	v∈q	v∈q	NOUN
ejpam-6063	371	14	|rv|+	|rv|+	PROPN
ejpam-6063	371	15	∑	∑	PUNCT
ejpam-6063	371	16	v∈v	v∈v	PROPN
ejpam-6063	371	17	(	(	PUNCT
ejpam-6063	371	18	g)\q	g)\q	PROPN
ejpam-6063	371	19	|rv|	|rv|	PROPN
ejpam-6063	371	20	=	=	SYM
ejpam-6063	371	21	β(g	β(g	PROPN
ejpam-6063	371	22	)	)	PUNCT
ejpam-6063	371	23	+	+	CCONJ
ejpam-6063	371	24	β(g)[β(h	β(g)[β(h	NUM
ejpam-6063	371	25	)	)	PUNCT
ejpam-6063	371	26	+	+	SYM
ejpam-6063	371	27	|i(h)|+	|i(h)|+	NOUN
ejpam-6063	371	28	n(m−	n(m−	PROPN
ejpam-6063	371	29	β(g	β(g	PROPN
ejpam-6063	371	30	)	)	PUNCT
ejpam-6063	371	31	)	)	PUNCT
ejpam-6063	372	1	=	=	SYM
ejpam-6063	372	2	mn+	mn+	NOUN
ejpam-6063	372	3	(	(	PUNCT
ejpam-6063	372	4	β(h)−	β(h)−	NOUN
ejpam-6063	372	5	n+	n+	ADJ
ejpam-6063	372	6	|i(h)|+	|i(h)|+	NOUN
ejpam-6063	372	7	1)β(g	1)β(g	NUM
ejpam-6063	372	8	)	)	PUNCT
ejpam-6063	372	9	.	.	PUNCT
ejpam-6063	373	1	on	on	ADP
ejpam-6063	373	2	the	the	DET
ejpam-6063	373	3	other	other	ADJ
ejpam-6063	373	4	hand	hand	NOUN
ejpam-6063	373	5	,	,	PUNCT
ejpam-6063	373	6	let	let	VERB
ejpam-6063	373	7	d0	d0	NOUN
ejpam-6063	373	8	be	be	AUX
ejpam-6063	373	9	a	a	DET
ejpam-6063	373	10	β2	β2	NOUN
ejpam-6063	373	11	-	-	PUNCT
ejpam-6063	373	12	set	set	NOUN
ejpam-6063	373	13	in	in	ADP
ejpam-6063	373	14	g	g	PROPN
ejpam-6063	373	15	◦	◦	PROPN
ejpam-6063	373	16	h.	h.	NOUN
ejpam-6063	373	17	then	then	ADV
ejpam-6063	373	18	d0	d0	PROPN
ejpam-6063	373	19	=	=	PUNCT
ejpam-6063	373	20	x	x	SYM
ejpam-6063	373	21	∪	∪	X
ejpam-6063	373	22	(	(	PUNCT
ejpam-6063	373	23	∪v∈v	∪v∈v	NOUN
ejpam-6063	373	24	(	(	PUNCT
ejpam-6063	373	25	g)tv	g)tv	NOUN
ejpam-6063	373	26	)	)	PUNCT
ejpam-6063	373	27	and	and	CCONJ
ejpam-6063	373	28	satisfies	satisfy	VERB
ejpam-6063	373	29	properties	property	NOUN
ejpam-6063	373	30	(	(	PUNCT
ejpam-6063	373	31	i	i	NOUN
ejpam-6063	373	32	)	)	PUNCT
ejpam-6063	373	33	,	,	PUNCT
ejpam-6063	373	34	(	(	PUNCT
ejpam-6063	373	35	ii	ii	NOUN
ejpam-6063	373	36	)	)	PUNCT
ejpam-6063	373	37	,	,	PUNCT
ejpam-6063	373	38	and	and	CCONJ
ejpam-6063	373	39	(	(	PUNCT
ejpam-6063	373	40	iii	iii	NOUN
ejpam-6063	373	41	)	)	PUNCT
ejpam-6063	373	42	of	of	ADP
ejpam-6063	373	43	theorem	theorem	ADJ
ejpam-6063	373	44	10	10	NUM
ejpam-6063	373	45	.	.	PUNCT
ejpam-6063	374	1	hence	hence	ADV
ejpam-6063	374	2	,	,	PUNCT
ejpam-6063	374	3	x	x	X
ejpam-6063	374	4	is	be	AUX
ejpam-6063	374	5	a	a	DET
ejpam-6063	374	6	vertex	vertex	NOUN
ejpam-6063	374	7	cover	cover	NOUN
ejpam-6063	374	8	of	of	ADP
ejpam-6063	374	9	g	g	PROPN
ejpam-6063	374	10	j.	j.	PROPN
ejpam-6063	374	11	hassan	hassan	PROPN
ejpam-6063	374	12	et	et	PROPN
ejpam-6063	374	13	.	.	PUNCT
ejpam-6063	375	1	al	al	PROPN
ejpam-6063	375	2	/	/	PUNCT
ejpam-6063	375	3	eur	eur	PROPN
ejpam-6063	375	4	.	.	PUNCT
ejpam-6063	376	1	j.	j.	PROPN
ejpam-6063	376	2	pure	pure	PROPN
ejpam-6063	376	3	appl	appl	PROPN
ejpam-6063	376	4	.	.	PROPN
ejpam-6063	376	5	math	math	PROPN
ejpam-6063	376	6	,	,	PUNCT
ejpam-6063	376	7	18	18	NUM
ejpam-6063	376	8	(	(	PUNCT
ejpam-6063	376	9	2	2	NUM
ejpam-6063	376	10	)	)	PUNCT
ejpam-6063	376	11	(	(	PUNCT
ejpam-6063	376	12	2025	2025	NUM
ejpam-6063	376	13	)	)	PUNCT
ejpam-6063	376	14	,	,	PUNCT
ejpam-6063	376	15	6063	6063	NUM
ejpam-6063	376	16	10	10	NUM
ejpam-6063	376	17	of	of	ADP
ejpam-6063	376	18	11	11	NUM
ejpam-6063	376	19	by	by	ADP
ejpam-6063	376	20	(	(	PUNCT
ejpam-6063	376	21	i	i	NOUN
ejpam-6063	376	22	)	)	PUNCT
ejpam-6063	376	23	,	,	PUNCT
ejpam-6063	376	24	tv	tv	NOUN
ejpam-6063	376	25	=	=	PUNCT
ejpam-6063	376	26	sv	sv	NOUN
ejpam-6063	376	27	∪	∪	ADP
ejpam-6063	376	28	i(hv	i(hv	NOUN
ejpam-6063	376	29	)	)	PUNCT
ejpam-6063	376	30	,	,	PUNCT
ejpam-6063	376	31	where	where	SCONJ
ejpam-6063	376	32	sv	sv	PROPN
ejpam-6063	376	33	is	be	AUX
ejpam-6063	376	34	a	a	DET
ejpam-6063	376	35	vertex	vertex	NOUN
ejpam-6063	376	36	cover	cover	NOUN
ejpam-6063	376	37	of	of	ADP
ejpam-6063	376	38	hv	hv	PROPN
ejpam-6063	376	39	,	,	PUNCT
ejpam-6063	376	40	for	for	ADP
ejpam-6063	376	41	each	each	DET
ejpam-6063	376	42	v	v	NOUN
ejpam-6063	376	43	∈	∈	NOUN
ejpam-6063	376	44	x	x	PUNCT
ejpam-6063	376	45	by	by	ADP
ejpam-6063	376	46	(	(	PUNCT
ejpam-6063	376	47	ii	ii	NOUN
ejpam-6063	376	48	)	)	PUNCT
ejpam-6063	376	49	,	,	PUNCT
ejpam-6063	376	50	and	and	CCONJ
ejpam-6063	376	51	tv	tv	NOUN
ejpam-6063	376	52	=	=	SYM
ejpam-6063	376	53	v	v	PROPN
ejpam-6063	376	54	(	(	PUNCT
ejpam-6063	376	55	hv	hv	PROPN
ejpam-6063	376	56	)	)	PUNCT
ejpam-6063	376	57	for	for	ADP
ejpam-6063	376	58	each	each	DET
ejpam-6063	376	59	v	v	NUM
ejpam-6063	376	60	∈	∈	PROPN
ejpam-6063	376	61	v	v	NOUN
ejpam-6063	376	62	(	(	PUNCT
ejpam-6063	376	63	g	g	NOUN
ejpam-6063	376	64	)	)	PUNCT
ejpam-6063	376	65	\x	\x	NOUN
ejpam-6063	376	66	by	by	ADP
ejpam-6063	376	67	(	(	PUNCT
ejpam-6063	376	68	iii	iii	NOUN
ejpam-6063	376	69	)	)	PUNCT
ejpam-6063	376	70	.	.	PUNCT
ejpam-6063	377	1	thus	thus	ADV
ejpam-6063	377	2	,	,	PUNCT
ejpam-6063	377	3	β2(g	β2(g	PUNCT
ejpam-6063	377	4	◦	◦	NOUN
ejpam-6063	377	5	h	h	NOUN
ejpam-6063	377	6	)	)	PUNCT
ejpam-6063	377	7	=	=	NOUN
ejpam-6063	377	8	|d0|	|d0|	NOUN
ejpam-6063	377	9	=	=	SYM
ejpam-6063	377	10	|x|+	|x|+	NOUN
ejpam-6063	377	11	∑	∑	ADP
ejpam-6063	377	12	v∈x	v∈x	PROPN
ejpam-6063	377	13	|tv|+	|tv|+	X
ejpam-6063	377	14	∑	∑	PUNCT
ejpam-6063	377	15	v∈v	v∈v	PROPN
ejpam-6063	377	16	(	(	PUNCT
ejpam-6063	377	17	g)\x	g)\x	PROPN
ejpam-6063	377	18	|tv|	|tv|	PROPN
ejpam-6063	377	19	≥	≥	NOUN
ejpam-6063	377	20	|x|+	|x|+	NOUN
ejpam-6063	377	21	∑	∑	PROPN
ejpam-6063	377	22	v∈x	v∈x	PROPN
ejpam-6063	377	23	(	(	PUNCT
ejpam-6063	377	24	β(h	β(h	PROPN
ejpam-6063	377	25	)	)	PUNCT
ejpam-6063	377	26	+	+	CCONJ
ejpam-6063	377	27	|i(h)|	|i(h)|	NOUN
ejpam-6063	377	28	)	)	PUNCT
ejpam-6063	377	29	+	+	CCONJ
ejpam-6063	377	30	∑	∑	PUNCT
ejpam-6063	377	31	v∈v	v∈v	NOUN
ejpam-6063	377	32	(	(	PUNCT
ejpam-6063	377	33	g)\x	g)\x	PROPN
ejpam-6063	377	34	n	n	NOUN
ejpam-6063	377	35	=	=	SYM
ejpam-6063	377	36	|x|+	|x|+	NOUN
ejpam-6063	377	37	|x|(β(h	|x|(β(h	NUM
ejpam-6063	377	38	)	)	PUNCT
ejpam-6063	377	39	+	+	CCONJ
ejpam-6063	377	40	|i(h)|	|i(h)|	PROPN
ejpam-6063	377	41	)	)	PUNCT
ejpam-6063	377	42	+	+	CCONJ
ejpam-6063	377	43	(	(	PUNCT
ejpam-6063	377	44	m−	m−	PROPN
ejpam-6063	377	45	|x|)n	|x|)n	NUM
ejpam-6063	377	46	=	=	PUNCT
ejpam-6063	377	47	mn+	mn+	NOUN
ejpam-6063	377	48	(	(	PUNCT
ejpam-6063	377	49	β(h	β(h	NOUN
ejpam-6063	377	50	)	)	PUNCT
ejpam-6063	377	51	+	+	CCONJ
ejpam-6063	377	52	|i(h)|	|i(h)|	PROPN
ejpam-6063	377	53	−	−	PROPN
ejpam-6063	377	54	n+	n+	X
ejpam-6063	377	55	1)|x|	1)|x|	NUM
ejpam-6063	377	56	≥	≥	NOUN
ejpam-6063	377	57	mn+	mn+	NOUN
ejpam-6063	377	58	(	(	PUNCT
ejpam-6063	377	59	β(h	β(h	PROPN
ejpam-6063	377	60	)	)	PUNCT
ejpam-6063	377	61	+	+	CCONJ
ejpam-6063	377	62	|i(h)|	|i(h)|	PROPN
ejpam-6063	377	63	−	−	NUM
ejpam-6063	377	64	n+	n+	NUM
ejpam-6063	377	65	1)β(g	1)β(g	NUM
ejpam-6063	377	66	)	)	PUNCT
ejpam-6063	377	67	.	.	PUNCT
ejpam-6063	378	1	this	this	PRON
ejpam-6063	378	2	establishes	establish	VERB
ejpam-6063	378	3	the	the	DET
ejpam-6063	378	4	desired	desire	VERB
ejpam-6063	378	5	equality	equality	NOUN
ejpam-6063	378	6	.	.	PUNCT
ejpam-6063	379	1	if	if	SCONJ
ejpam-6063	379	2	h	h	NOUN
ejpam-6063	379	3	is	be	AUX
ejpam-6063	379	4	a	a	DET
ejpam-6063	379	5	non	non	ADJ
ejpam-6063	379	6	-	-	ADJ
ejpam-6063	379	7	trivial	trivial	ADJ
ejpam-6063	379	8	connected	connected	ADJ
ejpam-6063	379	9	graph	graph	NOUN
ejpam-6063	379	10	,	,	PUNCT
ejpam-6063	379	11	then	then	ADV
ejpam-6063	379	12	|i(h)|	|i(h)|	PROPN
ejpam-6063	379	13	=	=	SYM
ejpam-6063	379	14	0	0	NUM
ejpam-6063	379	15	.	.	PUNCT
ejpam-6063	380	1	hence	hence	ADV
ejpam-6063	380	2	,	,	PUNCT
ejpam-6063	380	3	the	the	DET
ejpam-6063	380	4	additional	additional	ADJ
ejpam-6063	380	5	assertion	assertion	NOUN
ejpam-6063	380	6	holds	hold	VERB
ejpam-6063	380	7	.	.	PUNCT
ejpam-6063	381	1	4	4	X
ejpam-6063	381	2	.	.	X
ejpam-6063	381	3	conclusion	conclusion	NOUN
ejpam-6063	381	4	the	the	DET
ejpam-6063	381	5	parameter	parameter	NOUN
ejpam-6063	381	6	2	2	NUM
ejpam-6063	381	7	-	-	PUNCT
ejpam-6063	381	8	vertex	vertex	NOUN
ejpam-6063	381	9	cover	cover	NOUN
ejpam-6063	381	10	,	,	PUNCT
ejpam-6063	381	11	a	a	DET
ejpam-6063	381	12	variant	variant	NOUN
ejpam-6063	381	13	of	of	ADP
ejpam-6063	381	14	vertex	vertex	NOUN
ejpam-6063	381	15	cover	cover	NOUN
ejpam-6063	381	16	,	,	PUNCT
ejpam-6063	381	17	had	have	AUX
ejpam-6063	381	18	been	be	AUX
ejpam-6063	381	19	introduced	introduce	VERB
ejpam-6063	381	20	and	and	CCONJ
ejpam-6063	381	21	initially	initially	ADV
ejpam-6063	381	22	studied	study	VERB
ejpam-6063	381	23	.	.	PUNCT
ejpam-6063	382	1	this	this	DET
ejpam-6063	382	2	newly	newly	ADV
ejpam-6063	382	3	defined	define	VERB
ejpam-6063	382	4	concept	concept	NOUN
ejpam-6063	382	5	incorporates	incorporate	VERB
ejpam-6063	382	6	the	the	DET
ejpam-6063	382	7	concept	concept	NOUN
ejpam-6063	382	8	of	of	ADP
ejpam-6063	382	9	2	2	NUM
ejpam-6063	382	10	-	-	PUNCT
ejpam-6063	382	11	domination	domination	NOUN
ejpam-6063	382	12	in	in	ADP
ejpam-6063	382	13	a	a	DET
ejpam-6063	382	14	graph	graph	NOUN
ejpam-6063	382	15	.	.	PUNCT
ejpam-6063	383	1	we	we	PRON
ejpam-6063	383	2	gave	give	VERB
ejpam-6063	383	3	bounds	bound	NOUN
ejpam-6063	383	4	on	on	ADP
ejpam-6063	383	5	the	the	DET
ejpam-6063	383	6	parameter	parameter	NOUN
ejpam-6063	383	7	,	,	PUNCT
ejpam-6063	383	8	obtained	obtain	VERB
ejpam-6063	383	9	the	the	DET
ejpam-6063	383	10	value	value	NOUN
ejpam-6063	383	11	of	of	ADP
ejpam-6063	383	12	the	the	DET
ejpam-6063	383	13	parameter	parameter	NOUN
ejpam-6063	383	14	for	for	ADP
ejpam-6063	383	15	some	some	DET
ejpam-6063	383	16	well	well	ADV
ejpam-6063	383	17	-	-	PUNCT
ejpam-6063	383	18	known	know	VERB
ejpam-6063	383	19	classes	class	NOUN
ejpam-6063	383	20	of	of	ADP
ejpam-6063	383	21	graphs	graph	NOUN
ejpam-6063	383	22	,	,	PUNCT
ejpam-6063	383	23	and	and	CCONJ
ejpam-6063	383	24	characterized	characterize	VERB
ejpam-6063	383	25	graphs	graph	NOUN
ejpam-6063	383	26	that	that	PRON
ejpam-6063	383	27	attain	attain	VERB
ejpam-6063	383	28	specific	specific	ADJ
ejpam-6063	383	29	values	value	NOUN
ejpam-6063	383	30	such	such	ADJ
ejpam-6063	383	31	as	as	ADP
ejpam-6063	383	32	1	1	NUM
ejpam-6063	383	33	,	,	PUNCT
ejpam-6063	383	34	2	2	NUM
ejpam-6063	383	35	,	,	PUNCT
ejpam-6063	383	36	n	n	CCONJ
ejpam-6063	383	37	−	−	PROPN
ejpam-6063	383	38	1	1	NUM
ejpam-6063	383	39	,	,	PUNCT
ejpam-6063	383	40	and	and	CCONJ
ejpam-6063	383	41	n	n	CCONJ
ejpam-6063	383	42	,	,	PUNCT
ejpam-6063	383	43	where	where	SCONJ
ejpam-6063	383	44	n	n	X
ejpam-6063	383	45	is	be	AUX
ejpam-6063	383	46	the	the	DET
ejpam-6063	383	47	order	order	NOUN
ejpam-6063	383	48	of	of	ADP
ejpam-6063	383	49	the	the	DET
ejpam-6063	383	50	graph	graph	NOUN
ejpam-6063	383	51	.	.	PUNCT
ejpam-6063	384	1	we	we	PRON
ejpam-6063	384	2	also	also	ADV
ejpam-6063	384	3	characterized	characterize	VERB
ejpam-6063	384	4	the	the	DET
ejpam-6063	384	5	2	2	NUM
ejpam-6063	384	6	-	-	PUNCT
ejpam-6063	384	7	vertex	vertex	NOUN
ejpam-6063	384	8	covering	covering	NOUN
ejpam-6063	384	9	in	in	ADP
ejpam-6063	384	10	the	the	DET
ejpam-6063	384	11	join	join	NOUN
ejpam-6063	384	12	and	and	CCONJ
ejpam-6063	384	13	corona	corona	NOUN
ejpam-6063	384	14	of	of	ADP
ejpam-6063	384	15	two	two	NUM
ejpam-6063	384	16	graphs	graph	NOUN
ejpam-6063	384	17	and	and	CCONJ
ejpam-6063	384	18	determined	determine	VERB
ejpam-6063	384	19	their	their	PRON
ejpam-6063	384	20	2	2	NUM
ejpam-6063	384	21	-	-	PUNCT
ejpam-6063	384	22	vertex	vertex	NOUN
ejpam-6063	384	23	cover	cover	NOUN
ejpam-6063	384	24	numbers	number	NOUN
ejpam-6063	384	25	.	.	PUNCT
ejpam-6063	385	1	we	we	PRON
ejpam-6063	385	2	also	also	ADV
ejpam-6063	385	3	showed	show	VERB
ejpam-6063	385	4	that	that	SCONJ
ejpam-6063	385	5	the	the	DET
ejpam-6063	385	6	difference	difference	NOUN
ejpam-6063	385	7	between	between	ADP
ejpam-6063	385	8	2	2	NUM
ejpam-6063	385	9	-	-	PUNCT
ejpam-6063	385	10	vertex	vertex	NOUN
ejpam-6063	385	11	cover	cover	NOUN
ejpam-6063	385	12	number	number	NOUN
ejpam-6063	385	13	and	and	CCONJ
ejpam-6063	385	14	vertex	vertex	NOUN
ejpam-6063	385	15	cover	cover	NOUN
ejpam-6063	385	16	number	number	NOUN
ejpam-6063	385	17	can	can	AUX
ejpam-6063	385	18	be	be	AUX
ejpam-6063	385	19	made	make	VERB
ejpam-6063	385	20	arbitrarily	arbitrarily	ADV
ejpam-6063	385	21	large	large	ADJ
ejpam-6063	385	22	.	.	PUNCT
ejpam-6063	386	1	the	the	DET
ejpam-6063	386	2	new	new	ADJ
ejpam-6063	386	3	parameter	parameter	NOUN
ejpam-6063	386	4	can	can	AUX
ejpam-6063	386	5	be	be	AUX
ejpam-6063	386	6	studied	study	VERB
ejpam-6063	386	7	for	for	ADP
ejpam-6063	386	8	other	other	ADJ
ejpam-6063	386	9	classes	class	NOUN
ejpam-6063	386	10	of	of	ADP
ejpam-6063	386	11	graphs	graph	NOUN
ejpam-6063	386	12	,	,	PUNCT
ejpam-6063	386	13	say	say	VERB
ejpam-6063	386	14	trees	tree	NOUN
ejpam-6063	386	15	and	and	CCONJ
ejpam-6063	386	16	other	other	ADJ
ejpam-6063	386	17	graphs	graph	NOUN
ejpam-6063	386	18	resulting	result	VERB
ejpam-6063	386	19	from	from	ADP
ejpam-6063	386	20	some	some	DET
ejpam-6063	386	21	unary	unary	ADJ
ejpam-6063	386	22	and	and	CCONJ
ejpam-6063	386	23	binary	binary	ADJ
ejpam-6063	386	24	operations	operation	NOUN
ejpam-6063	386	25	,	,	PUNCT
ejpam-6063	386	26	and	and	CCONJ
ejpam-6063	386	27	bounds	bound	NOUN
ejpam-6063	386	28	in	in	ADP
ejpam-6063	386	29	terms	term	NOUN
ejpam-6063	386	30	of	of	ADP
ejpam-6063	386	31	other	other	ADJ
ejpam-6063	386	32	parameters	parameter	NOUN
ejpam-6063	386	33	may	may	AUX
ejpam-6063	386	34	be	be	AUX
ejpam-6063	386	35	determined	determine	VERB
ejpam-6063	386	36	.	.	PUNCT
ejpam-6063	387	1	furthermore	furthermore	ADV
ejpam-6063	387	2	,	,	PUNCT
ejpam-6063	387	3	since	since	SCONJ
ejpam-6063	387	4	the	the	DET
ejpam-6063	387	5	vertex	vertex	NOUN
ejpam-6063	387	6	cover	cover	NOUN
ejpam-6063	387	7	problem	problem	NOUN
ejpam-6063	387	8	is	be	AUX
ejpam-6063	387	9	np	np	ADP
ejpam-6063	387	10	-complete	-complete	ADJ
ejpam-6063	387	11	,	,	PUNCT
ejpam-6063	387	12	the	the	DET
ejpam-6063	387	13	question	question	NOUN
ejpam-6063	387	14	as	as	ADP
ejpam-6063	387	15	to	to	ADP
ejpam-6063	387	16	whether	whether	SCONJ
ejpam-6063	387	17	the	the	DET
ejpam-6063	387	18	2	2	NUM
ejpam-6063	387	19	-	-	PUNCT
ejpam-6063	387	20	vertex	vertex	NOUN
ejpam-6063	387	21	cover	cover	NOUN
ejpam-6063	387	22	problem	problem	NOUN
ejpam-6063	387	23	is	be	AUX
ejpam-6063	387	24	also	also	ADV
ejpam-6063	387	25	np	np	ADP
ejpam-6063	387	26	-complete	-complete	NOUN
ejpam-6063	387	27	remains	remain	VERB
ejpam-6063	387	28	unanswered	unanswered	ADJ
ejpam-6063	387	29	.	.	PUNCT
ejpam-6063	388	1	acknowledgements	acknowledgement	NOUN
ejpam-6063	388	2	the	the	DET
ejpam-6063	388	3	authors	author	NOUN
ejpam-6063	388	4	would	would	AUX
ejpam-6063	388	5	like	like	VERB
ejpam-6063	388	6	to	to	PART
ejpam-6063	388	7	thank	thank	VERB
ejpam-6063	388	8	the	the	DET
ejpam-6063	388	9	referees	referee	NOUN
ejpam-6063	388	10	for	for	ADP
ejpam-6063	388	11	reading	read	VERB
ejpam-6063	388	12	the	the	DET
ejpam-6063	388	13	initial	initial	ADJ
ejpam-6063	388	14	manuscript	manuscript	NOUN
ejpam-6063	388	15	and	and	CCONJ
ejpam-6063	388	16	for	for	ADP
ejpam-6063	388	17	giving	give	VERB
ejpam-6063	388	18	their	their	PRON
ejpam-6063	388	19	comments	comment	NOUN
ejpam-6063	388	20	and	and	CCONJ
ejpam-6063	388	21	suggestions	suggestion	NOUN
ejpam-6063	388	22	.	.	PUNCT
ejpam-6063	389	1	special	special	ADJ
ejpam-6063	389	2	thanks	thank	NOUN
ejpam-6063	389	3	must	must	AUX
ejpam-6063	389	4	go	go	VERB
ejpam-6063	389	5	tomsu	tomsu	NOUN
ejpam-6063	389	6	-	-	PUNCT
ejpam-6063	389	7	iligan	iligan	PROPN
ejpam-6063	389	8	institute	institute	PROPN
ejpam-6063	389	9	of	of	ADP
ejpam-6063	389	10	technology	technology	PROPN
ejpam-6063	389	11	,	,	PUNCT
ejpam-6063	389	12	iligan	iligan	PROPN
ejpam-6063	389	13	city	city	PROPN
ejpam-6063	389	14	,	,	PUNCT
ejpam-6063	389	15	mindanao	mindanao	PROPN
ejpam-6063	389	16	state	state	PROPN
ejpam-6063	389	17	university	university	PROPN
ejpam-6063	389	18	tawi	tawi	PROPN
ejpam-6063	389	19	-	-	PUNCT
ejpam-6063	389	20	tawi	tawi	PROPN
ejpam-6063	389	21	college	college	PROPN
ejpam-6063	389	22	of	of	ADP
ejpam-6063	389	23	technology	technology	NOUN
ejpam-6063	389	24	and	and	CCONJ
ejpam-6063	389	25	oceanography	oceanography	NOUN
ejpam-6063	389	26	,	,	PUNCT
ejpam-6063	389	27	korea	korea	PROPN
ejpam-6063	389	28	university	university	PROPN
ejpam-6063	389	29	,	,	PUNCT
ejpam-6063	389	30	andwestern	andwestern	PROPN
ejpam-6063	389	31	mindanao	mindanao	PROPN
ejpam-6063	389	32	state	state	PROPN
ejpam-6063	389	33	university	university	PROPN
ejpam-6063	389	34	for	for	ADP
ejpam-6063	389	35	funding	fund	VERB
ejpam-6063	389	36	this	this	DET
ejpam-6063	389	37	research	research	NOUN
ejpam-6063	389	38	.	.	PUNCT
ejpam-6063	390	1	references	reference	NOUN
ejpam-6063	390	2	[	[	X
ejpam-6063	390	3	1	1	NUM
ejpam-6063	390	4	]	]	PUNCT
ejpam-6063	390	5	a.	a.	NOUN
ejpam-6063	390	6	reis	reis	NOUN
ejpam-6063	390	7	,	,	PUNCT
ejpam-6063	390	8	s.	s.	PROPN
ejpam-6063	390	9	halper	halper	PROPN
ejpam-6063	390	10	,	,	PUNCT
ejpam-6063	390	11	g.	g.	PROPN
ejpam-6063	390	12	vezeau	vezeau	PROPN
ejpam-6063	390	13	,	,	PUNCT
ejpam-6063	390	14	d.	d.	PROPN
ejpam-6063	390	15	cetnar	cetnar	PROPN
ejpam-6063	390	16	,	,	PUNCT
ejpam-6063	390	17	a.	a.	PROPN
ejpam-6063	390	18	hossain	hossain	PROPN
ejpam-6063	390	19	,	,	PUNCT
ejpam-6063	390	20	p.	p.	PROPN
ejpam-6063	390	21	clauer	clauer	PROPN
ejpam-6063	390	22	,	,	PUNCT
ejpam-6063	390	23	and	and	CCONJ
ejpam-6063	390	24	h.	h.	PROPN
ejpam-6063	390	25	salis	salis	PROPN
ejpam-6063	390	26	.	.	PUNCT
ejpam-6063	391	1	simultaneous	simultaneous	ADJ
ejpam-6063	391	2	repression	repression	NOUN
ejpam-6063	391	3	of	of	ADP
ejpam-6063	391	4	multiple	multiple	ADJ
ejpam-6063	391	5	bacterial	bacterial	ADJ
ejpam-6063	391	6	genes	gene	NOUN
ejpam-6063	391	7	using	use	VERB
ejpam-6063	391	8	nonrepetitive	nonrepetitive	ADJ
ejpam-6063	391	9	extra	extra	ADJ
ejpam-6063	391	10	-	-	ADJ
ejpam-6063	391	11	long	long	ADJ
ejpam-6063	391	12	sgrna	sgrna	NOUN
ejpam-6063	391	13	j.	j.	PROPN
ejpam-6063	391	14	hassan	hassan	PROPN
ejpam-6063	391	15	et	et	PROPN
ejpam-6063	391	16	.	.	PUNCT
ejpam-6063	392	1	al	al	PROPN
ejpam-6063	392	2	/	/	PUNCT
ejpam-6063	392	3	eur	eur	PROPN
ejpam-6063	392	4	.	.	PUNCT
ejpam-6063	393	1	j.	j.	PROPN
ejpam-6063	393	2	pure	pure	PROPN
ejpam-6063	393	3	appl	appl	PROPN
ejpam-6063	393	4	.	.	PROPN
ejpam-6063	393	5	math	math	PROPN
ejpam-6063	393	6	,	,	PUNCT
ejpam-6063	393	7	18	18	NUM
ejpam-6063	393	8	(	(	PUNCT
ejpam-6063	393	9	2	2	NUM
ejpam-6063	393	10	)	)	PUNCT
ejpam-6063	393	11	(	(	PUNCT
ejpam-6063	393	12	2025	2025	NUM
ejpam-6063	393	13	)	)	PUNCT
ejpam-6063	393	14	,	,	PUNCT
ejpam-6063	393	15	6063	6063	NUM
ejpam-6063	393	16	11	11	NUM
ejpam-6063	393	17	of	of	ADP
ejpam-6063	393	18	11	11	NUM
ejpam-6063	393	19	arrays	array	NOUN
ejpam-6063	393	20	.	.	PUNCT
ejpam-6063	394	1	nature	nature	NOUN
ejpam-6063	394	2	biotechnology	biotechnology	NOUN
ejpam-6063	394	3	,	,	PUNCT
ejpam-6063	394	4	37(11):1294–1301	37(11):1294–1301	NUM
ejpam-6063	394	5	,	,	PUNCT
ejpam-6063	394	6	2019	2019	NUM
ejpam-6063	394	7	.	.	PUNCT
ejpam-6063	395	1	[	[	X
ejpam-6063	395	2	2	2	X
ejpam-6063	395	3	]	]	PUNCT
ejpam-6063	395	4	d.	d.	PROPN
ejpam-6063	395	5	angel	angel	NOUN
ejpam-6063	395	6	and	and	CCONJ
ejpam-6063	395	7	a.	a.	NOUN
ejpam-6063	395	8	amutha	amutha	PROPN
ejpam-6063	395	9	.	.	PUNCT
ejpam-6063	396	1	vertex	vertex	NOUN
ejpam-6063	396	2	covering	covering	NOUN
ejpam-6063	396	3	and	and	CCONJ
ejpam-6063	396	4	strong	strong	ADJ
ejpam-6063	396	5	covering	covering	NOUN
ejpam-6063	396	6	of	of	ADP
ejpam-6063	396	7	flower	flower	NOUN
ejpam-6063	396	8	like	like	ADP
ejpam-6063	396	9	network	network	NOUN
ejpam-6063	396	10	structures	structure	NOUN
ejpam-6063	396	11	.	.	PUNCT
ejpam-6063	397	1	procedia	procedia	NOUN
ejpam-6063	397	2	computer	computer	NOUN
ejpam-6063	397	3	science	science	NOUN
ejpam-6063	397	4	,	,	PUNCT
ejpam-6063	397	5	87:164–171	87:164–171	PROPN
ejpam-6063	397	6	,	,	PUNCT
ejpam-6063	397	7	2016	2016	NUM
ejpam-6063	397	8	.	.	PUNCT
ejpam-6063	398	1	[	[	X
ejpam-6063	398	2	3	3	X
ejpam-6063	398	3	]	]	X
ejpam-6063	398	4	c.	c.	PROPN
ejpam-6063	398	5	toregas	toregas	PROPN
ejpam-6063	398	6	,	,	PUNCT
ejpam-6063	398	7	,	,	PUNCT
ejpam-6063	398	8	r.	r.	PROPN
ejpam-6063	398	9	swain	swain	PROPN
ejpam-6063	398	10	,	,	PUNCT
ejpam-6063	398	11	c.	c.	PROPN
ejpam-6063	398	12	revelle	revelle	PROPN
ejpam-6063	398	13	,	,	PUNCT
ejpam-6063	398	14	and	and	CCONJ
ejpam-6063	398	15	l.	l.	PROPN
ejpam-6063	398	16	bercman	bercman	PROPN
ejpam-6063	398	17	.	.	PUNCT
ejpam-6063	399	1	the	the	DET
ejpam-6063	399	2	location	location	NOUN
ejpam-6063	399	3	of	of	ADP
ejpam-6063	399	4	emergency	emergency	NOUN
ejpam-6063	399	5	service	service	NOUN
ejpam-6063	399	6	facilities	facility	NOUN
ejpam-6063	399	7	.	.	PUNCT
ejpam-6063	400	1	journal	journal	NOUN
ejpam-6063	400	2	of	of	ADP
ejpam-6063	400	3	the	the	DET
ejpam-6063	400	4	operations	operation	NOUN
ejpam-6063	400	5	research	research	NOUN
ejpam-6063	400	6	society	society	NOUN
ejpam-6063	400	7	of	of	ADP
ejpam-6063	400	8	america	america	PROPN
ejpam-6063	400	9	,	,	PUNCT
ejpam-6063	400	10	19(6	19(6	NUM
ejpam-6063	400	11	)	)	PUNCT
ejpam-6063	400	12	,	,	PUNCT
ejpam-6063	400	13	1971	1971	NUM
ejpam-6063	400	14	.	.	PUNCT
ejpam-6063	401	1	[	[	X
ejpam-6063	401	2	4	4	NUM
ejpam-6063	401	3	]	]	X
ejpam-6063	401	4	r.m	r.m	PROPN
ejpam-6063	401	5	.	.	PROPN
ejpam-6063	401	6	karp	karp	PROPN
ejpam-6063	401	7	.	.	PUNCT
ejpam-6063	402	1	reducibility	reducibility	PROPN
ejpam-6063	402	2	among	among	ADP
ejpam-6063	402	3	combinatorial	combinatorial	ADJ
ejpam-6063	402	4	problems	problem	NOUN
ejpam-6063	402	5	,	,	PUNCT
ejpam-6063	402	6	complexity	complexity	NOUN
ejpam-6063	402	7	of	of	ADP
ejpam-6063	402	8	computer	computer	NOUN
ejpam-6063	402	9	computations	computation	NOUN
ejpam-6063	402	10	.	.	PUNCT
ejpam-6063	403	1	plenum	plenum	PROPN
ejpam-6063	403	2	press	press	PROPN
ejpam-6063	403	3	,	,	PUNCT
ejpam-6063	403	4	new	new	PROPN
ejpam-6063	403	5	york	york	PROPN
ejpam-6063	403	6	,	,	PUNCT
ejpam-6063	403	7	pages	page	NOUN
ejpam-6063	403	8	85–103	85–103	NUM
ejpam-6063	403	9	,	,	PUNCT
ejpam-6063	403	10	1972	1972	NUM
ejpam-6063	403	11	.	.	PUNCT
ejpam-6063	404	1	[	[	X
ejpam-6063	404	2	5	5	NUM
ejpam-6063	404	3	]	]	X
ejpam-6063	404	4	m.r	m.r	PROPN
ejpam-6063	404	5	.	.	PROPN
ejpam-6063	404	6	garey	garey	PROPN
ejpam-6063	404	7	and	and	CCONJ
ejpam-6063	404	8	d.s	d.s	PROPN
ejpam-6063	404	9	.	.	PROPN
ejpam-6063	404	10	johnson	johnson	PROPN
ejpam-6063	404	11	.	.	PUNCT
ejpam-6063	405	1	the	the	DET
ejpam-6063	405	2	rectilinear	rectilinear	PROPN
ejpam-6063	405	3	steiner	steiner	PROPN
ejpam-6063	405	4	tree	tree	NOUN
ejpam-6063	405	5	problem	problem	NOUN
ejpam-6063	405	6	is	be	AUX
ejpam-6063	405	7	np	np	NOUN
ejpam-6063	405	8	-	-	PUNCT
ejpam-6063	405	9	complete	complete	ADJ
ejpam-6063	405	10	.	.	PUNCT
ejpam-6063	406	1	siam	siam	PROPN
ejpam-6063	406	2	journal	journal	PROPN
ejpam-6063	406	3	on	on	ADP
ejpam-6063	406	4	applied	apply	VERB
ejpam-6063	406	5	mathematics	mathematic	NOUN
ejpam-6063	406	6	,	,	PUNCT
ejpam-6063	406	7	32:826–834	32:826–834	NUM
ejpam-6063	406	8	,	,	PUNCT
ejpam-6063	406	9	1977	1977	NUM
ejpam-6063	406	10	.	.	PUNCT
ejpam-6063	407	1	[	[	X
ejpam-6063	407	2	6	6	NUM
ejpam-6063	407	3	]	]	X
ejpam-6063	407	4	m.r	m.r	PROPN
ejpam-6063	407	5	.	.	PROPN
ejpam-6063	407	6	garey	garey	PROPN
ejpam-6063	407	7	,	,	PUNCT
ejpam-6063	407	8	d.s	d.s	PROPN
ejpam-6063	407	9	.	.	PROPN
ejpam-6063	407	10	johnson	johnson	PROPN
ejpam-6063	407	11	,	,	PUNCT
ejpam-6063	407	12	and	and	CCONJ
ejpam-6063	407	13	l.	l.	PROPN
ejpam-6063	407	14	stockmeyer	stockmeyer	PROPN
ejpam-6063	407	15	.	.	PUNCT
ejpam-6063	408	1	some	some	PRON
ejpam-6063	408	2	simplified	simplified	ADJ
ejpam-6063	408	3	npcomplete	npcomplete	ADJ
ejpam-6063	408	4	problems	problem	NOUN
ejpam-6063	408	5	.	.	PUNCT
ejpam-6063	409	1	proceedings	proceeding	NOUN
ejpam-6063	409	2	of	of	ADP
ejpam-6063	409	3	the	the	DET
ejpam-6063	409	4	sixth	sixth	ADJ
ejpam-6063	409	5	annual	annual	ADJ
ejpam-6063	409	6	acm	acm	NOUN
ejpam-6063	409	7	symposium	symposium	NOUN
ejpam-6063	409	8	on	on	ADP
ejpam-6063	409	9	theory	theory	NOUN
ejpam-6063	409	10	of	of	ADP
ejpam-6063	409	11	computing	computing	NOUN
ejpam-6063	409	12	,	,	PUNCT
ejpam-6063	409	13	pages	page	NOUN
ejpam-6063	409	14	47	47	NUM
ejpam-6063	409	15	–	–	PUNCT
ejpam-6063	409	16	63	63	NUM
ejpam-6063	409	17	,	,	PUNCT
ejpam-6063	409	18	1974	1974	NUM
ejpam-6063	409	19	.	.	PUNCT
ejpam-6063	410	1	[	[	X
ejpam-6063	410	2	7	7	X
ejpam-6063	410	3	]	]	X
ejpam-6063	410	4	b.	b.	PROPN
ejpam-6063	410	5	behsaz	behsaz	PROPN
ejpam-6063	410	6	,	,	PUNCT
ejpam-6063	410	7	p.	p.	PROPN
ejpam-6063	410	8	hatami	hatami	PROPN
ejpam-6063	410	9	,	,	PUNCT
ejpam-6063	410	10	and	and	CCONJ
ejpam-6063	410	11	e.s	e.s	PROPN
ejpam-6063	410	12	.	.	PROPN
ejpam-6063	410	13	mahmoodian	mahmoodian	PROPN
ejpam-6063	410	14	.	.	PUNCT
ejpam-6063	411	1	on	on	ADP
ejpam-6063	411	2	minimum	minimum	ADJ
ejpam-6063	411	3	vertex	vertex	NOUN
ejpam-6063	411	4	cover	cover	NOUN
ejpam-6063	411	5	of	of	ADP
ejpam-6063	411	6	generalized	generalized	ADJ
ejpam-6063	411	7	petersen	petersen	NOUN
ejpam-6063	411	8	graphs	graph	NOUN
ejpam-6063	411	9	.	.	PUNCT
ejpam-6063	412	1	australian	australian	ADJ
ejpam-6063	412	2	journal	journal	NOUN
ejpam-6063	412	3	of	of	ADP
ejpam-6063	412	4	combinatorics	combinatoric	NOUN
ejpam-6063	412	5	,	,	PUNCT
ejpam-6063	412	6	40:253–264	40:253–264	NUM
ejpam-6063	412	7	,	,	PUNCT
ejpam-6063	412	8	2008	2008	NUM
ejpam-6063	412	9	.	.	PUNCT
ejpam-6063	413	1	[	[	X
ejpam-6063	413	2	8	8	X
ejpam-6063	413	3	]	]	X
ejpam-6063	413	4	j.	j.	PROPN
ejpam-6063	413	5	hassan	hassan	PROPN
ejpam-6063	413	6	,	,	PUNCT
ejpam-6063	413	7	m.	m.	NOUN
ejpam-6063	413	8	a.	a.	PROPN
ejpam-6063	413	9	bonsocan	bonsocan	PROPN
ejpam-6063	413	10	,	,	PUNCT
ejpam-6063	413	11	r.	r.	PROPN
ejpam-6063	413	12	rasid	rasid	PROPN
ejpam-6063	413	13	,	,	PUNCT
ejpam-6063	413	14	and	and	CCONJ
ejpam-6063	413	15	a.	a.	NOUN
ejpam-6063	413	16	sappari	sappari	PROPN
ejpam-6063	413	17	.	.	PUNCT
ejpam-6063	414	1	certified	certify	VERB
ejpam-6063	414	2	vertex	vertex	NOUN
ejpam-6063	414	3	cover	cover	NOUN
ejpam-6063	414	4	of	of	ADP
ejpam-6063	414	5	a	a	DET
ejpam-6063	414	6	graph	graph	NOUN
ejpam-6063	414	7	.	.	PUNCT
ejpam-6063	415	1	ur	ur	INTJ
ejpam-6063	415	2	.	.	PUNCT
ejpam-6063	416	1	j.	j.	PROPN
ejpam-6063	416	2	pure	pure	PROPN
ejpam-6063	416	3	appl	appl	PROPN
ejpam-6063	416	4	.	.	PUNCT
ejpam-6063	416	5	math	math	PROPN
ejpam-6063	416	6	.	.	PUNCT
ejpam-6063	416	7	,	,	PUNCT
ejpam-6063	416	8	17(2):1038–1045	17(2):1038–1045	NUM
ejpam-6063	416	9	,	,	PUNCT
ejpam-6063	416	10	2024	2024	NUM
ejpam-6063	416	11	.	.	PUNCT
ejpam-6063	417	1	[	[	X
ejpam-6063	417	2	9	9	NUM
ejpam-6063	417	3	]	]	X
ejpam-6063	417	4	m.	m.	NOUN
ejpam-6063	417	5	henning	henning	PROPN
ejpam-6063	417	6	and	and	CCONJ
ejpam-6063	417	7	a.	a.	PROPN
ejpam-6063	417	8	yeo	yeo	PROPN
ejpam-6063	417	9	.	.	PROPN
ejpam-6063	418	1	identifying	identify	VERB
ejpam-6063	418	2	vertex	vertex	NOUN
ejpam-6063	418	3	covers	cover	VERB
ejpam-6063	418	4	in	in	ADP
ejpam-6063	418	5	graphs	graph	NOUN
ejpam-6063	418	6	.	.	PUNCT
ejpam-6063	419	1	the	the	DET
ejpam-6063	419	2	electric	electric	ADJ
ejpam-6063	419	3	journal	journal	PROPN
ejpam-6063	419	4	in	in	ADP
ejpam-6063	419	5	mathematics	mathematic	NOUN
ejpam-6063	419	6	,	,	PUNCT
ejpam-6063	419	7	,	,	PUNCT
ejpam-6063	419	8	19(4):1038–1045	19(4):1038–1045	NUM
ejpam-6063	419	9	.	.	NOUN
ejpam-6063	419	10	,	,	PUNCT
ejpam-6063	419	11	2012	2012	NUM
ejpam-6063	419	12	.	.	PUNCT
ejpam-6063	420	1	[	[	X
ejpam-6063	420	2	10	10	NUM
ejpam-6063	420	3	]	]	PUNCT
ejpam-6063	420	4	m.	m.	NOUN
ejpam-6063	420	5	marathe	marathe	PROPN
ejpam-6063	420	6	,	,	PUNCT
ejpam-6063	420	7	r.	r.	PROPN
ejpam-6063	420	8	ravi	ravi	PROPN
ejpam-6063	420	9	,	,	PUNCT
ejpam-6063	420	10	and	and	CCONJ
ejpam-6063	420	11	c.	c.	PROPN
ejpam-6063	420	12	p.	p.	PROPN
ejpam-6063	420	13	rangan	rangan	PROPN
ejpam-6063	420	14	.	.	PUNCT
ejpam-6063	421	1	generalized	generalize	VERB
ejpam-6063	421	2	vertex	vertex	NOUN
ejpam-6063	421	3	covering	cover	VERB
ejpam-6063	421	4	in	in	ADP
ejpam-6063	421	5	interval	interval	NOUN
ejpam-6063	421	6	graphs	graph	NOUN
ejpam-6063	421	7	.	.	PUNCT
ejpam-6063	422	1	discrete	discrete	ADJ
ejpam-6063	422	2	applied	applied	ADJ
ejpam-6063	422	3	mathematics	mathematic	NOUN
ejpam-6063	422	4	,	,	PUNCT
ejpam-6063	422	5	39:87–93	39:87–93	PROPN
ejpam-6063	422	6	,	,	PUNCT
ejpam-6063	422	7	1992	1992	NUM
ejpam-6063	422	8	.	.	PUNCT
ejpam-6063	423	1	[	[	X
ejpam-6063	423	2	11	11	NUM
ejpam-6063	423	3	]	]	PUNCT
ejpam-6063	423	4	p.	p.	NOUN
ejpam-6063	423	5	pushpam	pushpam	NOUN
ejpam-6063	423	6	and	and	CCONJ
ejpam-6063	423	7	c.	c.	PROPN
ejpam-6063	423	8	suseendran	suseendran	PROPN
ejpam-6063	423	9	.	.	PUNCT
ejpam-6063	424	1	secure	secure	ADJ
ejpam-6063	424	2	vertex	vertex	NOUN
ejpam-6063	424	3	cover	cover	NOUN
ejpam-6063	424	4	of	of	ADP
ejpam-6063	424	5	a	a	DET
ejpam-6063	424	6	graph	graph	NOUN
ejpam-6063	424	7	.	.	PUNCT
ejpam-6063	425	1	discrete	discrete	ADJ
ejpam-6063	425	2	mathematics	mathematic	NOUN
ejpam-6063	425	3	,	,	PUNCT
ejpam-6063	425	4	algorithms	algorithm	NOUN
ejpam-6063	425	5	and	and	CCONJ
ejpam-6063	425	6	applications	application	NOUN
ejpam-6063	425	7	,	,	PUNCT
ejpam-6063	425	8	9(2	9(2	NUM
ejpam-6063	425	9	)	)	PUNCT
ejpam-6063	425	10	,	,	PUNCT
ejpam-6063	425	11	2017	2017	NUM
ejpam-6063	425	12	.	.	PUNCT
ejpam-6063	426	1	[	[	X
ejpam-6063	426	2	12	12	NUM
ejpam-6063	426	3	]	]	X
ejpam-6063	426	4	l.	l.	PROPN
ejpam-6063	426	5	sathikala	sathikala	PROPN
ejpam-6063	426	6	,	,	PUNCT
ejpam-6063	426	7	k.	k.	PROPN
ejpam-6063	426	8	k.	k.	PROPN
ejpam-6063	426	9	basari	basari	PROPN
ejpam-6063	426	10	,	,	PUNCT
ejpam-6063	426	11	and	and	CCONJ
ejpam-6063	426	12	k.	k.	PROPN
ejpam-6063	426	13	subramanian	subramanian	PROPN
ejpam-6063	426	14	.	.	PROPN
ejpam-6063	427	1	connected	connect	VERB
ejpam-6063	427	2	and	and	CCONJ
ejpam-6063	427	3	total	total	ADJ
ejpam-6063	427	4	vertex	vertex	NOUN
ejpam-6063	427	5	covering	cover	VERB
ejpam-6063	427	6	in	in	ADP
ejpam-6063	427	7	graphs	graph	NOUN
ejpam-6063	427	8	.	.	PUNCT
ejpam-6063	428	1	turkish	turkish	ADJ
ejpam-6063	428	2	journal	journal	NOUN
ejpam-6063	428	3	of	of	ADP
ejpam-6063	428	4	computer	computer	NOUN
ejpam-6063	428	5	and	and	CCONJ
ejpam-6063	428	6	mathematics	mathematic	NOUN
ejpam-6063	428	7	education	education	NOUN
ejpam-6063	428	8	,	,	PUNCT
ejpam-6063	428	9	12(2):2180–2185	12(2):2180–2185	NUM
ejpam-6063	428	10	,	,	PUNCT
ejpam-6063	428	11	2021	2021	NUM
ejpam-6063	428	12	.	.	PUNCT
ejpam-6063	429	1	[	[	X
ejpam-6063	429	2	13	13	NUM
ejpam-6063	429	3	]	]	X
ejpam-6063	429	4	j.	j.	PROPN
ejpam-6063	429	5	uy	uy	PROPN
ejpam-6063	429	6	.	.	PUNCT
ejpam-6063	430	1	vertex	vertex	PROPN
ejpam-6063	430	2	cover	cover	NOUN
ejpam-6063	430	3	of	of	ADP
ejpam-6063	430	4	graphs	graph	NOUN
ejpam-6063	430	5	.	.	PUNCT
ejpam-6063	431	1	journal	journal	NOUN
ejpam-6063	431	2	of	of	ADP
ejpam-6063	431	3	research	research	NOUN
ejpam-6063	431	4	in	in	ADP
ejpam-6063	431	5	science	science	NOUN
ejpam-6063	431	6	and	and	CCONJ
ejpam-6063	431	7	engineering	engineering	NOUN
ejpam-6063	431	8	,	,	PUNCT
ejpam-6063	431	9	1:49–53	1:49–53	NUM
ejpam-6063	431	10	,	,	PUNCT
ejpam-6063	431	11	2003	2003	NUM
ejpam-6063	431	12	.	.	PUNCT
ejpam-6063	432	1	[	[	X
ejpam-6063	432	2	14	14	NUM
ejpam-6063	432	3	]	]	PUNCT
ejpam-6063	432	4	m.	m.	NOUN
ejpam-6063	432	5	blidia	blidia	PROPN
ejpam-6063	432	6	,	,	PUNCT
ejpam-6063	432	7	m.	m.	NOUN
ejpam-6063	432	8	chellali	chellali	PROPN
ejpam-6063	432	9	,	,	PUNCT
ejpam-6063	432	10	and	and	CCONJ
ejpam-6063	432	11	o.	o.	PROPN
ejpam-6063	432	12	favaron	favaron	PROPN
ejpam-6063	432	13	.	.	PUNCT
ejpam-6063	433	1	independence	independence	NOUN
ejpam-6063	433	2	and	and	CCONJ
ejpam-6063	433	3	2	2	NUM
ejpam-6063	433	4	-	-	PUNCT
ejpam-6063	433	5	domination	domination	NOUN
ejpam-6063	433	6	in	in	ADP
ejpam-6063	433	7	trees	tree	NOUN
ejpam-6063	433	8	.	.	PUNCT
ejpam-6063	434	1	australas	australas	PROPN
ejpam-6063	434	2	.	.	PUNCT
ejpam-6063	435	1	j.	j.	PROPN
ejpam-6063	435	2	combin	combin	PROPN
ejpam-6063	435	3	.	.	PROPN
ejpam-6063	435	4	,	,	PUNCT
ejpam-6063	435	5	33:317–327	33:317–327	PROPN
ejpam-6063	435	6	,	,	PUNCT
ejpam-6063	435	7	2005	2005	NUM
ejpam-6063	435	8	.	.	PUNCT
ejpam-6063	436	1	[	[	X
ejpam-6063	436	2	15	15	NUM
ejpam-6063	436	3	]	]	X
ejpam-6063	436	4	m.	m.	NOUN
ejpam-6063	436	5	chellali	chellali	PROPN
ejpam-6063	436	6	.	.	PUNCT
ejpam-6063	437	1	bounds	bound	VERB
ejpam-6063	437	2	on	on	ADP
ejpam-6063	437	3	the	the	DET
ejpam-6063	437	4	2	2	NUM
ejpam-6063	437	5	-	-	PUNCT
ejpam-6063	437	6	domination	domination	NOUN
ejpam-6063	437	7	number	number	NOUN
ejpam-6063	437	8	in	in	ADP
ejpam-6063	437	9	cactus	cactus	NOUN
ejpam-6063	437	10	graphs	graph	NOUN
ejpam-6063	437	11	.	.	PUNCT
ejpam-6063	438	1	theoretical	theoretical	ADJ
ejpam-6063	438	2	computer	computer	NOUN
ejpam-6063	438	3	science	science	NOUN
ejpam-6063	438	4	,	,	PUNCT
ejpam-6063	438	5	26:5–12	26:5–12	NUM
ejpam-6063	438	6	,	,	PUNCT
ejpam-6063	438	7	2006	2006	NUM
ejpam-6063	438	8	.	.	PUNCT
ejpam-6063	439	1	[	[	X
ejpam-6063	439	2	16	16	NUM
ejpam-6063	439	3	]	]	PUNCT
ejpam-6063	439	4	b.	b.	PROPN
ejpam-6063	439	5	domoloan	domoloan	PROPN
ejpam-6063	439	6	and	and	CCONJ
ejpam-6063	439	7	s.	s.	PROPN
ejpam-6063	439	8	canoy	canoy	PROPN
ejpam-6063	439	9	jr	jr	PROPN
ejpam-6063	439	10	.	.	PROPN
ejpam-6063	439	11	2	2	NUM
ejpam-6063	439	12	-	-	PUNCT
ejpam-6063	439	13	domination	domination	NOUN
ejpam-6063	439	14	and	and	CCONJ
ejpam-6063	439	15	restrained	restrain	VERB
ejpam-6063	439	16	2	2	NUM
ejpam-6063	439	17	-	-	PUNCT
ejpam-6063	439	18	domination	domination	NOUN
ejpam-6063	439	19	in	in	ADP
ejpam-6063	439	20	graphs	graph	NOUN
ejpam-6063	439	21	.	.	PUNCT
ejpam-6063	440	1	applied	apply	VERB
ejpam-6063	440	2	mathematical	mathematical	ADJ
ejpam-6063	440	3	sciences	science	NOUN
ejpam-6063	440	4	,	,	PUNCT
ejpam-6063	440	5	9(114):5651–5659	9(114):5651–5659	NUM
ejpam-6063	440	6	,	,	PUNCT
ejpam-6063	440	7	2015	2015	NUM
ejpam-6063	440	8	.	.	PUNCT
ejpam-6063	441	1	[	[	X
ejpam-6063	441	2	17	17	NUM
ejpam-6063	441	3	]	]	X
ejpam-6063	441	4	s.	s.	PROPN
ejpam-6063	441	5	canoy	canoy	PROPN
ejpam-6063	441	6	jr	jr	PROPN
ejpam-6063	441	7	and	and	CCONJ
ejpam-6063	441	8	b.	b.	PROPN
ejpam-6063	441	9	domoloan	domoloan	PROPN
ejpam-6063	441	10	.	.	PUNCT
ejpam-6063	442	1	outer	outer	ADV
ejpam-6063	442	2	-	-	PUNCT
ejpam-6063	442	3	connected	connect	VERB
ejpam-6063	442	4	2	2	NUM
ejpam-6063	442	5	-	-	PUNCT
ejpam-6063	442	6	dominating	dominating	NOUN
ejpam-6063	442	7	sets	set	NOUN
ejpam-6063	442	8	of	of	ADP
ejpam-6063	442	9	graphs	graph	NOUN
ejpam-6063	442	10	.	.	PUNCT
ejpam-6063	443	1	advances	advance	NOUN
ejpam-6063	443	2	and	and	CCONJ
ejpam-6063	443	3	applications	application	NOUN
ejpam-6063	443	4	in	in	ADP
ejpam-6063	443	5	discrete	discrete	ADJ
ejpam-6063	443	6	mathematics	mathematic	NOUN
ejpam-6063	443	7	,	,	PUNCT
ejpam-6063	443	8	20(1):25	20(1):25	PROPN
ejpam-6063	443	9	,	,	PUNCT
ejpam-6063	443	10	2019	2019	NUM
ejpam-6063	443	11	.	.	PUNCT
ejpam-6063	444	1	[	[	X
ejpam-6063	444	2	18	18	NUM
ejpam-6063	444	3	]	]	X
ejpam-6063	444	4	s.	s.	PROPN
ejpam-6063	444	5	canoy	canoy	PROPN
ejpam-6063	444	6	jr	jr	PROPN
ejpam-6063	444	7	,	,	PUNCT
ejpam-6063	444	8	f.	f.	PROPN
ejpam-6063	444	9	jamil	jamil	PROPN
ejpam-6063	444	10	,	,	PUNCT
ejpam-6063	444	11	r.j	r.j	PROPN
ejpam-6063	444	12	.	.	PROPN
ejpam-6063	444	13	fortosa	fortosa	PROPN
ejpam-6063	444	14	,	,	PUNCT
ejpam-6063	444	15	and	and	CCONJ
ejpam-6063	444	16	j.	j.	PROPN
ejpam-6063	444	17	macalisang	macalisang	PROPN
ejpam-6063	444	18	.	.	PUNCT
ejpam-6063	445	1	convex	convex	PROPN
ejpam-6063	445	2	2	2	NUM
ejpam-6063	445	3	-	-	PUNCT
ejpam-6063	445	4	domination	domination	NOUN
ejpam-6063	445	5	in	in	ADP
ejpam-6063	445	6	graphs	graph	NOUN
ejpam-6063	445	7	.	.	PUNCT
ejpam-6063	446	1	european	european	ADJ
ejpam-6063	446	2	journal	journal	PROPN
ejpam-6063	446	3	of	of	ADP
ejpam-6063	446	4	pure	pure	ADJ
ejpam-6063	446	5	and	and	CCONJ
ejpam-6063	446	6	applied	applied	ADJ
ejpam-6063	446	7	mathematics	mathematic	NOUN
ejpam-6063	446	8	,	,	PUNCT
ejpam-6063	446	9	17(3):1539–1552	17(3):1539–1552	NUM
ejpam-6063	446	10	,	,	PUNCT
ejpam-6063	446	11	2024	2024	NUM
ejpam-6063	446	12	.	.	PUNCT
ejpam-6063	447	1	[	[	X
ejpam-6063	447	2	19	19	NUM
ejpam-6063	447	3	]	]	X
ejpam-6063	447	4	f.	f.	PROPN
ejpam-6063	447	5	buckley	buckley	PROPN
ejpam-6063	447	6	and	and	CCONJ
ejpam-6063	447	7	f.	f.	PROPN
ejpam-6063	447	8	harary	harary	PROPN
ejpam-6063	447	9	.	.	PUNCT
ejpam-6063	448	1	distance	distance	NOUN
ejpam-6063	448	2	in	in	ADP
ejpam-6063	448	3	graphs	graph	NOUN
ejpam-6063	448	4	.	.	PUNCT
ejpam-6063	449	1	addison	addison	PROPN
ejpam-6063	449	2	-	-	PUNCT
ejpam-6063	449	3	wesley	wesley	PROPN
ejpam-6063	449	4	,	,	PUNCT
ejpam-6063	449	5	redwood	redwood	NOUN
ejpam-6063	449	6	city	city	NOUN
ejpam-6063	449	7	,	,	PUNCT
ejpam-6063	449	8	1990	1990	NUM
ejpam-6063	449	9	.	.	PUNCT