id	sid	tid	token	lemma	pos
ejpam-5157	1	1	european	european	PROPN
ejpam-5157	1	2	journal	journal	PROPN
ejpam-5157	1	3	of	of	ADP
ejpam-5157	1	4	pure	pure	ADJ
ejpam-5157	1	5	and	and	CCONJ
ejpam-5157	1	6	applied	apply	VERB
ejpam-5157	1	7	mathematics	mathematic	NOUN
ejpam-5157	1	8	vol	vol	NOUN
ejpam-5157	1	9	.	.	PROPN
ejpam-5157	2	1	17	17	NUM
ejpam-5157	2	2	,	,	PUNCT
ejpam-5157	2	3	no	no	INTJ
ejpam-5157	2	4	.	.	NOUN
ejpam-5157	2	5	2	2	NUM
ejpam-5157	2	6	,	,	PUNCT
ejpam-5157	2	7	2024	2024	NUM
ejpam-5157	2	8	,	,	PUNCT
ejpam-5157	2	9	1038	1038	NUM
ejpam-5157	2	10	-	-	SYM
ejpam-5157	2	11	1045	1045	NUM
ejpam-5157	2	12	issn	issn	PROPN
ejpam-5157	2	13	1307	1307	NUM
ejpam-5157	2	14	-	-	SYM
ejpam-5157	2	15	5543	5543	NUM
ejpam-5157	2	16	–	–	PUNCT
ejpam-5157	2	17	ejpam.com	ejpam.com	X
ejpam-5157	2	18	published	publish	VERB
ejpam-5157	2	19	by	by	ADP
ejpam-5157	2	20	new	new	PROPN
ejpam-5157	2	21	york	york	PROPN
ejpam-5157	2	22	business	business	PROPN
ejpam-5157	2	23	global	global	ADJ
ejpam-5157	2	24	certified	certify	VERB
ejpam-5157	2	25	vertex	vertex	NOUN
ejpam-5157	2	26	cover	cover	NOUN
ejpam-5157	2	27	of	of	ADP
ejpam-5157	2	28	a	a	DET
ejpam-5157	2	29	graph	graph	NOUN
ejpam-5157	2	30	javier	javier	PROPN
ejpam-5157	2	31	a.	a.	PROPN
ejpam-5157	2	32	hassan1,∗	hassan1,∗	PROPN
ejpam-5157	2	33	,	,	PUNCT
ejpam-5157	2	34	maria	maria	PROPN
ejpam-5157	2	35	andrea	andrea	PROPN
ejpam-5157	2	36	o.	o.	PROPN
ejpam-5157	2	37	bonsocan2	bonsocan2	PROPN
ejpam-5157	2	38	,	,	PUNCT
ejpam-5157	2	39	regimar	regimar	PROPN
ejpam-5157	2	40	a.	a.	PROPN
ejpam-5157	2	41	rasid1	rasid1	ADV
ejpam-5157	2	42	,	,	PUNCT
ejpam-5157	2	43	amil	amil	NOUN
ejpam-5157	2	44	-	-	PUNCT
ejpam-5157	2	45	shab	shab	PROPN
ejpam-5157	2	46	s.	s.	PROPN
ejpam-5157	2	47	sappari1	sappari1	PROPN
ejpam-5157	2	48	1mathematics	1mathematics	NUM
ejpam-5157	2	49	and	and	CCONJ
ejpam-5157	2	50	sciences	sciences	PROPN
ejpam-5157	2	51	department	department	PROPN
ejpam-5157	2	52	,	,	PUNCT
ejpam-5157	2	53	college	college	NOUN
ejpam-5157	2	54	of	of	ADP
ejpam-5157	2	55	arts	art	NOUN
ejpam-5157	2	56	and	and	CCONJ
ejpam-5157	2	57	sciences	science	NOUN
ejpam-5157	2	58	,	,	PUNCT
ejpam-5157	2	59	msu	msu	PROPN
ejpam-5157	2	60	tawi	tawi	PROPN
ejpam-5157	2	61	-	-	PUNCT
ejpam-5157	2	62	tawi	tawi	PROPN
ejpam-5157	2	63	college	college	PROPN
ejpam-5157	2	64	of	of	ADP
ejpam-5157	2	65	technology	technology	NOUN
ejpam-5157	2	66	and	and	CCONJ
ejpam-5157	2	67	oceanography	oceanography	NOUN
ejpam-5157	2	68	,	,	PUNCT
ejpam-5157	2	69	bongao	bongao	NOUN
ejpam-5157	2	70	,	,	PUNCT
ejpam-5157	2	71	tawi	tawi	NOUN
ejpam-5157	2	72	-	-	PUNCT
ejpam-5157	2	73	tawi	tawi	NOUN
ejpam-5157	2	74	,	,	PUNCT
ejpam-5157	2	75	philippines	philippine	NOUN
ejpam-5157	2	76	2department	2department	NUM
ejpam-5157	2	77	of	of	ADP
ejpam-5157	2	78	mathematics	mathematic	NOUN
ejpam-5157	2	79	,	,	PUNCT
ejpam-5157	2	80	ateneo	ateneo	X
ejpam-5157	2	81	de	de	PROPN
ejpam-5157	2	82	davao	davao	PROPN
ejpam-5157	2	83	university	university	PROPN
ejpam-5157	2	84	,	,	PUNCT
ejpam-5157	2	85	e.	e.	PROPN
ejpam-5157	2	86	jacinto	jacinto	PROPN
ejpam-5157	2	87	street	street	PROPN
ejpam-5157	2	88	,	,	PUNCT
ejpam-5157	2	89	8016	8016	NUM
ejpam-5157	2	90	davao	davao	PROPN
ejpam-5157	2	91	city	city	NOUN
ejpam-5157	2	92	,	,	PUNCT
ejpam-5157	3	1	philippines	philippine	NOUN
ejpam-5157	3	2	abstract	abstract	ADJ
ejpam-5157	3	3	.	.	PUNCT
ejpam-5157	4	1	let	let	VERB
ejpam-5157	4	2	g	g	PRON
ejpam-5157	4	3	be	be	AUX
ejpam-5157	4	4	a	a	DET
ejpam-5157	4	5	graph	graph	NOUN
ejpam-5157	4	6	.	.	PUNCT
ejpam-5157	5	1	then	then	ADV
ejpam-5157	5	2	q	q	X
ejpam-5157	5	3	⊆	⊆	NUM
ejpam-5157	5	4	v	v	NOUN
ejpam-5157	5	5	(	(	PUNCT
ejpam-5157	5	6	g	g	NOUN
ejpam-5157	5	7	)	)	PUNCT
ejpam-5157	5	8	is	be	AUX
ejpam-5157	5	9	called	call	VERB
ejpam-5157	5	10	a	a	DET
ejpam-5157	5	11	certified	certify	VERB
ejpam-5157	5	12	vertex	vertex	NOUN
ejpam-5157	5	13	cover	cover	NOUN
ejpam-5157	5	14	of	of	ADP
ejpam-5157	5	15	g	g	PROPN
ejpam-5157	5	16	if	if	SCONJ
ejpam-5157	5	17	q	q	NOUN
ejpam-5157	5	18	is	be	AUX
ejpam-5157	5	19	a	a	DET
ejpam-5157	5	20	vertex	vertex	NOUN
ejpam-5157	5	21	cover	cover	NOUN
ejpam-5157	5	22	of	of	ADP
ejpam-5157	5	23	g	g	NOUN
ejpam-5157	5	24	and	and	CCONJ
ejpam-5157	5	25	every	every	DET
ejpam-5157	5	26	x	x	X
ejpam-5157	5	27	∈	∈	PROPN
ejpam-5157	5	28	q	q	NOUN
ejpam-5157	5	29	,	,	PUNCT
ejpam-5157	5	30	x	x	PUNCT
ejpam-5157	5	31	has	have	VERB
ejpam-5157	5	32	either	either	CCONJ
ejpam-5157	5	33	zero	zero	NUM
ejpam-5157	5	34	or	or	CCONJ
ejpam-5157	5	35	at	at	ADP
ejpam-5157	5	36	least	least	ADV
ejpam-5157	5	37	two	two	NUM
ejpam-5157	5	38	neighbors	neighbor	NOUN
ejpam-5157	5	39	in	in	ADP
ejpam-5157	5	40	v	v	NOUN
ejpam-5157	5	41	(	(	PUNCT
ejpam-5157	5	42	g	g	NOUN
ejpam-5157	5	43	)	)	PUNCT
ejpam-5157	5	44	\q	\q	NOUN
ejpam-5157	5	45	.	.	PUNCT
ejpam-5157	6	1	the	the	DET
ejpam-5157	6	2	certified	certify	VERB
ejpam-5157	6	3	vertex	vertex	NOUN
ejpam-5157	6	4	cover	cover	NOUN
ejpam-5157	6	5	number	number	NOUN
ejpam-5157	6	6	of	of	ADP
ejpam-5157	6	7	g	g	NOUN
ejpam-5157	6	8	,	,	PUNCT
ejpam-5157	6	9	denoted	denote	VERB
ejpam-5157	6	10	by	by	ADP
ejpam-5157	6	11	βcer(g	βcer(g	NOUN
ejpam-5157	6	12	)	)	PUNCT
ejpam-5157	6	13	,	,	PUNCT
ejpam-5157	6	14	is	be	AUX
ejpam-5157	6	15	the	the	DET
ejpam-5157	6	16	minimum	minimum	ADJ
ejpam-5157	6	17	cardinality	cardinality	NOUN
ejpam-5157	6	18	of	of	ADP
ejpam-5157	6	19	a	a	DET
ejpam-5157	6	20	certified	certify	VERB
ejpam-5157	6	21	vertex	vertex	NOUN
ejpam-5157	6	22	cover	cover	NOUN
ejpam-5157	6	23	of	of	ADP
ejpam-5157	6	24	g.	g.	PROPN
ejpam-5157	6	25	in	in	ADP
ejpam-5157	6	26	this	this	DET
ejpam-5157	6	27	paper	paper	NOUN
ejpam-5157	6	28	,	,	PUNCT
ejpam-5157	6	29	we	we	PRON
ejpam-5157	6	30	investigate	investigate	VERB
ejpam-5157	6	31	this	this	DET
ejpam-5157	6	32	newly	newly	ADV
ejpam-5157	6	33	defined	define	VERB
ejpam-5157	6	34	concept	concept	NOUN
ejpam-5157	6	35	on	on	ADP
ejpam-5157	6	36	some	some	DET
ejpam-5157	6	37	special	special	ADJ
ejpam-5157	6	38	graphs	graph	NOUN
ejpam-5157	6	39	and	and	CCONJ
ejpam-5157	6	40	on	on	ADP
ejpam-5157	6	41	the	the	DET
ejpam-5157	6	42	join	join	NOUN
ejpam-5157	6	43	of	of	ADP
ejpam-5157	6	44	two	two	NUM
ejpam-5157	6	45	graphs	graph	NOUN
ejpam-5157	6	46	.	.	PUNCT
ejpam-5157	7	1	we	we	PRON
ejpam-5157	7	2	characterize	characterize	VERB
ejpam-5157	7	3	certified	certify	VERB
ejpam-5157	7	4	vertex	vertex	NOUN
ejpam-5157	7	5	cover	cover	NOUN
ejpam-5157	7	6	in	in	ADP
ejpam-5157	7	7	these	these	DET
ejpam-5157	7	8	graphs	graph	NOUN
ejpam-5157	7	9	and	and	CCONJ
ejpam-5157	7	10	subsequently	subsequently	ADV
ejpam-5157	7	11	derive	derive	VERB
ejpam-5157	7	12	the	the	DET
ejpam-5157	7	13	simplified	simplified	ADJ
ejpam-5157	7	14	formulas	formula	NOUN
ejpam-5157	7	15	for	for	ADP
ejpam-5157	7	16	calculating	calculate	VERB
ejpam-5157	7	17	the	the	DET
ejpam-5157	7	18	certified	certify	VERB
ejpam-5157	7	19	vertex	vertex	NOUN
ejpam-5157	7	20	cover	cover	NOUN
ejpam-5157	7	21	number	number	NOUN
ejpam-5157	7	22	.	.	PUNCT
ejpam-5157	8	1	moreover	moreover	ADV
ejpam-5157	8	2	,	,	PUNCT
ejpam-5157	8	3	we	we	PRON
ejpam-5157	8	4	present	present	VERB
ejpam-5157	8	5	some	some	DET
ejpam-5157	8	6	bounds	bound	NOUN
ejpam-5157	8	7	and	and	CCONJ
ejpam-5157	8	8	properties	property	NOUN
ejpam-5157	8	9	of	of	ADP
ejpam-5157	8	10	certified	certify	VERB
ejpam-5157	8	11	vertex	vertex	NOUN
ejpam-5157	8	12	cover	cover	NOUN
ejpam-5157	8	13	.	.	PUNCT
ejpam-5157	9	1	2020	2020	NUM
ejpam-5157	9	2	mathematics	mathematic	NOUN
ejpam-5157	9	3	subject	subject	NOUN
ejpam-5157	9	4	classifications	classification	NOUN
ejpam-5157	9	5	:	:	PUNCT
ejpam-5157	9	6	05c69	05c69	X
ejpam-5157	9	7	key	key	ADJ
ejpam-5157	9	8	words	word	NOUN
ejpam-5157	9	9	and	and	CCONJ
ejpam-5157	9	10	phrases	phrase	NOUN
ejpam-5157	9	11	:	:	PUNCT
ejpam-5157	9	12	certified	certify	VERB
ejpam-5157	9	13	set	set	NOUN
ejpam-5157	9	14	,	,	PUNCT
ejpam-5157	9	15	vertex	vertex	NOUN
ejpam-5157	9	16	cover	cover	NOUN
ejpam-5157	9	17	,	,	PUNCT
ejpam-5157	9	18	certified	certify	VERB
ejpam-5157	9	19	vertex	vertex	NOUN
ejpam-5157	9	20	covering	covering	NOUN
ejpam-5157	9	21	set	set	NOUN
ejpam-5157	9	22	,	,	PUNCT
ejpam-5157	9	23	certified	certify	VERB
ejpam-5157	9	24	vertex	vertex	NOUN
ejpam-5157	9	25	cover	cover	NOUN
ejpam-5157	9	26	number	number	NOUN
ejpam-5157	9	27	1	1	NUM
ejpam-5157	9	28	.	.	PUNCT
ejpam-5157	9	29	introduction	introduction	NOUN
ejpam-5157	9	30	vertex	vertex	NOUN
ejpam-5157	9	31	cover	cover	NOUN
ejpam-5157	9	32	of	of	ADP
ejpam-5157	9	33	a	a	DET
ejpam-5157	9	34	graph	graph	NOUN
ejpam-5157	9	35	is	be	AUX
ejpam-5157	9	36	an	an	DET
ejpam-5157	9	37	important	important	ADJ
ejpam-5157	9	38	concept	concept	NOUN
ejpam-5157	9	39	in	in	ADP
ejpam-5157	9	40	graph	graph	NOUN
ejpam-5157	9	41	theory	theory	NOUN
ejpam-5157	9	42	.	.	PUNCT
ejpam-5157	10	1	a	a	DET
ejpam-5157	10	2	vertex	vertex	NOUN
ejpam-5157	10	3	cover	cover	NOUN
ejpam-5157	10	4	of	of	ADP
ejpam-5157	10	5	a	a	DET
ejpam-5157	10	6	graph	graph	NOUN
ejpam-5157	10	7	is	be	AUX
ejpam-5157	10	8	a	a	DET
ejpam-5157	10	9	subset	subset	NOUN
ejpam-5157	10	10	of	of	ADP
ejpam-5157	10	11	vertices	vertex	NOUN
ejpam-5157	10	12	that	that	PRON
ejpam-5157	10	13	covers	cover	VERB
ejpam-5157	10	14	all	all	DET
ejpam-5157	10	15	the	the	DET
ejpam-5157	10	16	edges	edge	NOUN
ejpam-5157	10	17	in	in	ADP
ejpam-5157	10	18	the	the	DET
ejpam-5157	10	19	graph	graph	NOUN
ejpam-5157	10	20	.	.	PUNCT
ejpam-5157	11	1	in	in	ADP
ejpam-5157	11	2	other	other	ADJ
ejpam-5157	11	3	words	word	NOUN
ejpam-5157	11	4	,	,	PUNCT
ejpam-5157	11	5	every	every	DET
ejpam-5157	11	6	edge	edge	NOUN
ejpam-5157	11	7	in	in	ADP
ejpam-5157	11	8	the	the	DET
ejpam-5157	11	9	graph	graph	NOUN
ejpam-5157	11	10	is	be	AUX
ejpam-5157	11	11	incident	incident	NOUN
ejpam-5157	11	12	to	to	ADP
ejpam-5157	11	13	at	at	ADV
ejpam-5157	11	14	least	least	ADV
ejpam-5157	11	15	one	one	NUM
ejpam-5157	11	16	vertex	vertex	NOUN
ejpam-5157	11	17	in	in	ADP
ejpam-5157	11	18	the	the	DET
ejpam-5157	11	19	vertex	vertex	NOUN
ejpam-5157	11	20	cover	cover	NOUN
ejpam-5157	11	21	.	.	PUNCT
ejpam-5157	12	1	the	the	DET
ejpam-5157	12	2	concept	concept	NOUN
ejpam-5157	12	3	of	of	ADP
ejpam-5157	12	4	vertex	vertex	NOUN
ejpam-5157	12	5	covering	cover	VERB
ejpam-5157	12	6	set	set	NOUN
ejpam-5157	12	7	has	have	VERB
ejpam-5157	12	8	practical	practical	ADJ
ejpam-5157	12	9	applications	application	NOUN
ejpam-5157	12	10	in	in	ADP
ejpam-5157	12	11	various	various	ADJ
ejpam-5157	12	12	fields	field	NOUN
ejpam-5157	12	13	,	,	PUNCT
ejpam-5157	12	14	such	such	ADJ
ejpam-5157	12	15	as	as	ADP
ejpam-5157	12	16	network	network	NOUN
ejpam-5157	12	17	design	design	NOUN
ejpam-5157	12	18	,	,	PUNCT
ejpam-5157	12	19	optimization	optimization	NOUN
ejpam-5157	12	20	,	,	PUNCT
ejpam-5157	12	21	and	and	CCONJ
ejpam-5157	12	22	resource	resource	NOUN
ejpam-5157	12	23	allocation	allocation	NOUN
ejpam-5157	12	24	.	.	PUNCT
ejpam-5157	13	1	the	the	DET
ejpam-5157	13	2	size	size	NOUN
ejpam-5157	13	3	of	of	ADP
ejpam-5157	13	4	the	the	DET
ejpam-5157	13	5	smallest	small	ADJ
ejpam-5157	13	6	vertex	vertex	NOUN
ejpam-5157	13	7	cover	cover	NOUN
ejpam-5157	13	8	is	be	AUX
ejpam-5157	13	9	called	call	VERB
ejpam-5157	13	10	the	the	DET
ejpam-5157	13	11	vertex	vertex	NOUN
ejpam-5157	13	12	cover	cover	NOUN
ejpam-5157	13	13	number	number	NOUN
ejpam-5157	13	14	of	of	ADP
ejpam-5157	13	15	the	the	DET
ejpam-5157	13	16	graph	graph	NOUN
ejpam-5157	13	17	.	.	PUNCT
ejpam-5157	14	1	finding	find	VERB
ejpam-5157	14	2	the	the	DET
ejpam-5157	14	3	minimum	minimum	ADJ
ejpam-5157	14	4	vertex	vertex	NOUN
ejpam-5157	14	5	cover	cover	NOUN
ejpam-5157	14	6	or	or	CCONJ
ejpam-5157	14	7	the	the	DET
ejpam-5157	14	8	vertex	vertex	NOUN
ejpam-5157	14	9	cover	cover	NOUN
ejpam-5157	14	10	number	number	NOUN
ejpam-5157	14	11	of	of	ADP
ejpam-5157	14	12	a	a	DET
ejpam-5157	14	13	graph	graph	NOUN
ejpam-5157	14	14	is	be	AUX
ejpam-5157	14	15	an	an	DET
ejpam-5157	14	16	important	important	ADJ
ejpam-5157	14	17	problem	problem	NOUN
ejpam-5157	14	18	in	in	ADP
ejpam-5157	14	19	graph	graph	NOUN
ejpam-5157	14	20	theory	theory	NOUN
ejpam-5157	14	21	.	.	PUNCT
ejpam-5157	15	1	in	in	ADP
ejpam-5157	15	2	practical	practical	ADJ
ejpam-5157	15	3	scenarios	scenario	NOUN
ejpam-5157	15	4	,	,	PUNCT
ejpam-5157	15	5	finding	find	VERB
ejpam-5157	15	6	a	a	DET
ejpam-5157	15	7	minimum	minimum	ADJ
ejpam-5157	15	8	vertex	vertex	NOUN
ejpam-5157	15	9	cover	cover	NOUN
ejpam-5157	15	10	helps	help	VERB
ejpam-5157	15	11	in	in	ADP
ejpam-5157	15	12	minimizing	minimize	VERB
ejpam-5157	15	13	costs	cost	NOUN
ejpam-5157	15	14	or	or	CCONJ
ejpam-5157	15	15	maximizing	maximize	VERB
ejpam-5157	15	16	efficiency	efficiency	NOUN
ejpam-5157	15	17	.	.	PUNCT
ejpam-5157	16	1	researchers	researcher	NOUN
ejpam-5157	16	2	had	have	AUX
ejpam-5157	16	3	studied	study	VERB
ejpam-5157	16	4	vertex	vertex	NOUN
ejpam-5157	16	5	cover	cover	NOUN
ejpam-5157	16	6	parameter	parameter	NOUN
ejpam-5157	16	7	and	and	CCONJ
ejpam-5157	16	8	its	its	PRON
ejpam-5157	16	9	variants	variant	NOUN
ejpam-5157	16	10	on	on	ADP
ejpam-5157	16	11	different	different	ADJ
ejpam-5157	16	12	types	type	NOUN
ejpam-5157	16	13	of	of	ADP
ejpam-5157	16	14	graphs	graph	NOUN
ejpam-5157	16	15	(	(	PUNCT
ejpam-5157	16	16	see	see	VERB
ejpam-5157	16	17	[	[	X
ejpam-5157	16	18	1	1	NUM
ejpam-5157	16	19	,	,	PUNCT
ejpam-5157	16	20	12	12	NUM
ejpam-5157	16	21	,	,	PUNCT
ejpam-5157	16	22	13	13	NUM
ejpam-5157	16	23	]	]	NUM
ejpam-5157	16	24	)	)	PUNCT
ejpam-5157	16	25	.	.	PUNCT
ejpam-5157	17	1	∗corresponding	∗corresponde	VERB
ejpam-5157	17	2	author	author	NOUN
ejpam-5157	17	3	.	.	PUNCT
ejpam-5157	18	1	doi	doi	NOUN
ejpam-5157	18	2	:	:	PUNCT
ejpam-5157	18	3	https://doi.org/10.29020/nybg.ejpam.v17i2.5157	https://doi.org/10.29020/nybg.ejpam.v17i2.5157	NOUN
ejpam-5157	18	4	email	email	NOUN
ejpam-5157	18	5	addresses	address	NOUN
ejpam-5157	18	6	:	:	PUNCT
ejpam-5157	18	7	javierhassan@msutawi-tawi.edu.ph	javierhassan@msutawi-tawi.edu.ph	PROPN
ejpam-5157	18	8	(	(	PUNCT
ejpam-5157	18	9	j.	j.	PROPN
ejpam-5157	18	10	a.	a.	PROPN
ejpam-5157	18	11	hassan	hassan	PROPN
ejpam-5157	18	12	)	)	PUNCT
ejpam-5157	19	1	mariaandrea.bonsocan@g.msuiit.edu.ph	mariaandrea.bonsocan@g.msuiit.edu.ph	PROPN
ejpam-5157	19	2	(	(	PUNCT
ejpam-5157	19	3	m.	m.	NOUN
ejpam-5157	19	4	a.	a.	PROPN
ejpam-5157	19	5	bonsocan	bonsocan	PROPN
ejpam-5157	19	6	)	)	PUNCT
ejpam-5157	19	7	regimarrasid@msutawi-tawi.edu.ph	regimarrasid@msutawi-tawi.edu.ph	PROPN
ejpam-5157	19	8	(	(	PUNCT
ejpam-5157	19	9	r.	r.	PROPN
ejpam-5157	19	10	a.	a.	PROPN
ejpam-5157	19	11	rasid	rasid	PROPN
ejpam-5157	19	12	)	)	PUNCT
ejpam-5157	19	13	amilshabsappari@msutawi-tawi.edu.ph	amilshabsappari@msutawi-tawi.edu.ph	PROPN
ejpam-5157	19	14	(	(	PUNCT
ejpam-5157	19	15	a.	a.	PROPN
ejpam-5157	19	16	s.	s.	PROPN
ejpam-5157	19	17	sappari	sappari	PROPN
ejpam-5157	19	18	)	)	PUNCT
ejpam-5157	19	19	https://www.ejpam.com	https://www.ejpam.com	X
ejpam-5157	19	20	1038	1038	NUM
ejpam-5157	20	1	©	©	PROPN
ejpam-5157	20	2	2023	2023	NUM
ejpam-5157	20	3	ejpam	ejpam	NOUN
ejpam-5157	20	4	all	all	DET
ejpam-5157	20	5	rights	right	NOUN
ejpam-5157	20	6	reserved	reserve	VERB
ejpam-5157	20	7	.	.	PUNCT
ejpam-5157	21	1	j.	j.	PROPN
ejpam-5157	21	2	a.	a.	PROPN
ejpam-5157	21	3	hassan	hassan	PROPN
ejpam-5157	21	4	et	et	PROPN
ejpam-5157	21	5	al	al	PROPN
ejpam-5157	21	6	.	.	PUNCT
ejpam-5157	21	7	/	/	SYM
ejpam-5157	21	8	eur	eur	PROPN
ejpam-5157	21	9	.	.	PUNCT
ejpam-5157	22	1	j.	j.	PROPN
ejpam-5157	22	2	pure	pure	PROPN
ejpam-5157	22	3	appl	appl	PROPN
ejpam-5157	22	4	.	.	PROPN
ejpam-5157	22	5	math	math	PROPN
ejpam-5157	22	6	,	,	PUNCT
ejpam-5157	22	7	17	17	NUM
ejpam-5157	22	8	(	(	PUNCT
ejpam-5157	22	9	2	2	NUM
ejpam-5157	22	10	)	)	PUNCT
ejpam-5157	22	11	(	(	PUNCT
ejpam-5157	22	12	2024	2024	NUM
ejpam-5157	22	13	)	)	PUNCT
ejpam-5157	22	14	,	,	PUNCT
ejpam-5157	22	15	1038	1038	NUM
ejpam-5157	22	16	-	-	SYM
ejpam-5157	22	17	1045	1045	NUM
ejpam-5157	22	18	1039	1039	NUM
ejpam-5157	22	19	recently	recently	ADV
ejpam-5157	22	20	,	,	PUNCT
ejpam-5157	22	21	v.	v.	PROPN
ejpam-5157	22	22	bilar	bilar	PROPN
ejpam-5157	22	23	et	et	PROPN
ejpam-5157	22	24	al	al	PROPN
ejpam-5157	22	25	.	.	PUNCT
ejpam-5157	23	1	[	[	X
ejpam-5157	23	2	1	1	X
ejpam-5157	23	3	]	]	PUNCT
ejpam-5157	23	4	introduced	introduce	VERB
ejpam-5157	23	5	and	and	CCONJ
ejpam-5157	23	6	investigated	investigate	VERB
ejpam-5157	23	7	another	another	DET
ejpam-5157	23	8	variant	variant	NOUN
ejpam-5157	23	9	of	of	ADP
ejpam-5157	23	10	a	a	DET
ejpam-5157	23	11	vertex	vertex	NOUN
ejpam-5157	23	12	cover	cover	NOUN
ejpam-5157	23	13	called	call	VERB
ejpam-5157	23	14	vertex	vertex	NOUN
ejpam-5157	23	15	cover	cover	VERB
ejpam-5157	23	16	hop	hop	NOUN
ejpam-5157	23	17	domination	domination	NOUN
ejpam-5157	23	18	.	.	PUNCT
ejpam-5157	24	1	let	let	VERB
ejpam-5157	24	2	g	g	PRON
ejpam-5157	24	3	be	be	AUX
ejpam-5157	24	4	a	a	DET
ejpam-5157	24	5	graph	graph	NOUN
ejpam-5157	24	6	.	.	PUNCT
ejpam-5157	25	1	then	then	ADV
ejpam-5157	25	2	a	a	DET
ejpam-5157	25	3	subset	subset	NOUN
ejpam-5157	25	4	c	c	NOUN
ejpam-5157	25	5	of	of	ADP
ejpam-5157	25	6	a	a	DET
ejpam-5157	25	7	vertex	vertex	NOUN
ejpam-5157	25	8	-	-	PUNCT
ejpam-5157	25	9	set	set	NOUN
ejpam-5157	25	10	of	of	ADP
ejpam-5157	25	11	g	g	PROPN
ejpam-5157	25	12	is	be	AUX
ejpam-5157	25	13	called	call	VERB
ejpam-5157	25	14	a	a	DET
ejpam-5157	25	15	vertex	vertex	NOUN
ejpam-5157	25	16	cover	cover	NOUN
ejpam-5157	25	17	hop	hop	NOUN
ejpam-5157	25	18	dominating	dominating	NOUN
ejpam-5157	25	19	if	if	SCONJ
ejpam-5157	25	20	c	c	PROPN
ejpam-5157	25	21	is	be	AUX
ejpam-5157	25	22	both	both	CCONJ
ejpam-5157	25	23	a	a	DET
ejpam-5157	25	24	vertex	vertex	NOUN
ejpam-5157	25	25	cover	cover	NOUN
ejpam-5157	25	26	and	and	CCONJ
ejpam-5157	25	27	a	a	DET
ejpam-5157	25	28	hop	hop	NOUN
ejpam-5157	25	29	dominating	dominating	NOUN
ejpam-5157	25	30	of	of	ADP
ejpam-5157	25	31	g.	g.	PROPN
ejpam-5157	25	32	the	the	DET
ejpam-5157	25	33	vertex	vertex	NOUN
ejpam-5157	25	34	cover	cover	VERB
ejpam-5157	25	35	hop	hop	NOUN
ejpam-5157	25	36	domination	domination	NOUN
ejpam-5157	25	37	number	number	NOUN
ejpam-5157	25	38	of	of	ADP
ejpam-5157	25	39	g	g	NOUN
ejpam-5157	25	40	,	,	PUNCT
ejpam-5157	25	41	denoted	denote	VERB
ejpam-5157	25	42	by	by	ADP
ejpam-5157	25	43	γvch(g	γvch(g	NOUN
ejpam-5157	25	44	)	)	PUNCT
ejpam-5157	25	45	,	,	PUNCT
ejpam-5157	25	46	is	be	AUX
ejpam-5157	25	47	the	the	DET
ejpam-5157	25	48	minimum	minimum	ADJ
ejpam-5157	25	49	cardinality	cardinality	NOUN
ejpam-5157	25	50	among	among	ADP
ejpam-5157	25	51	all	all	DET
ejpam-5157	25	52	vertex	vertex	NOUN
ejpam-5157	25	53	cover	cover	NOUN
ejpam-5157	25	54	hop	hop	NOUN
ejpam-5157	25	55	dominating	dominating	NOUN
ejpam-5157	25	56	sets	set	NOUN
ejpam-5157	25	57	in	in	ADP
ejpam-5157	25	58	g.	g.	PROPN
ejpam-5157	25	59	they	they	PRON
ejpam-5157	25	60	have	have	AUX
ejpam-5157	25	61	determined	determine	VERB
ejpam-5157	25	62	its	its	PRON
ejpam-5157	25	63	relations	relation	NOUN
ejpam-5157	25	64	with	with	ADP
ejpam-5157	25	65	other	other	ADJ
ejpam-5157	25	66	parameters	parameter	NOUN
ejpam-5157	25	67	in	in	ADP
ejpam-5157	25	68	graph	graph	NOUN
ejpam-5157	25	69	theory	theory	NOUN
ejpam-5157	25	70	such	such	ADJ
ejpam-5157	25	71	as	as	ADP
ejpam-5157	25	72	vertex	vertex	NOUN
ejpam-5157	25	73	cover	cover	NOUN
ejpam-5157	25	74	,	,	PUNCT
ejpam-5157	25	75	hop	hop	NOUN
ejpam-5157	25	76	domination	domination	NOUN
ejpam-5157	25	77	parameter	parameter	NOUN
ejpam-5157	25	78	.	.	PUNCT
ejpam-5157	26	1	moreover	moreover	ADV
ejpam-5157	26	2	,	,	PUNCT
ejpam-5157	26	3	they	they	PRON
ejpam-5157	26	4	have	have	AUX
ejpam-5157	26	5	characterized	characterize	VERB
ejpam-5157	26	6	the	the	DET
ejpam-5157	26	7	vertex	vertex	NOUN
ejpam-5157	26	8	cover	cover	NOUN
ejpam-5157	26	9	hop	hop	NOUN
ejpam-5157	26	10	dominating	dominating	NOUN
ejpam-5157	26	11	sets	set	NOUN
ejpam-5157	26	12	in	in	ADP
ejpam-5157	26	13	some	some	DET
ejpam-5157	26	14	special	special	ADJ
ejpam-5157	26	15	graphs	graph	NOUN
ejpam-5157	26	16	,	,	PUNCT
ejpam-5157	26	17	join	join	NOUN
ejpam-5157	26	18	,	,	PUNCT
ejpam-5157	26	19	and	and	CCONJ
ejpam-5157	26	20	corona	corona	NOUN
ejpam-5157	26	21	of	of	ADP
ejpam-5157	26	22	two	two	NUM
ejpam-5157	26	23	graphs	graph	NOUN
ejpam-5157	26	24	,	,	PUNCT
ejpam-5157	26	25	and	and	CCONJ
ejpam-5157	26	26	obtained	obtain	VERB
ejpam-5157	26	27	the	the	DET
ejpam-5157	26	28	exact	exact	ADJ
ejpam-5157	26	29	values	value	NOUN
ejpam-5157	26	30	or	or	CCONJ
ejpam-5157	26	31	bounds	bound	NOUN
ejpam-5157	26	32	of	of	ADP
ejpam-5157	26	33	the	the	DET
ejpam-5157	26	34	parameter	parameter	NOUN
ejpam-5157	26	35	of	of	ADP
ejpam-5157	26	36	these	these	DET
ejpam-5157	26	37	graphs	graph	NOUN
ejpam-5157	26	38	.	.	PUNCT
ejpam-5157	27	1	some	some	DET
ejpam-5157	27	2	studies	study	NOUN
ejpam-5157	27	3	related	relate	VERB
ejpam-5157	27	4	to	to	ADP
ejpam-5157	27	5	vertex	vertex	NOUN
ejpam-5157	27	6	cover	cover	NOUN
ejpam-5157	27	7	hop	hop	NOUN
ejpam-5157	27	8	domination	domination	NOUN
ejpam-5157	27	9	can	can	AUX
ejpam-5157	27	10	be	be	AUX
ejpam-5157	27	11	found	find	VERB
ejpam-5157	27	12	in	in	ADP
ejpam-5157	27	13	[	[	X
ejpam-5157	27	14	2–11	2–11	NOUN
ejpam-5157	27	15	]	]	PUNCT
ejpam-5157	27	16	.	.	PUNCT
ejpam-5157	28	1	in	in	ADP
ejpam-5157	28	2	this	this	DET
ejpam-5157	28	3	study	study	NOUN
ejpam-5157	28	4	,	,	PUNCT
ejpam-5157	28	5	we	we	PRON
ejpam-5157	28	6	initiate	initiate	VERB
ejpam-5157	28	7	the	the	DET
ejpam-5157	28	8	study	study	NOUN
ejpam-5157	28	9	of	of	ADP
ejpam-5157	28	10	certified	certify	VERB
ejpam-5157	28	11	vertex	vertex	NOUN
ejpam-5157	28	12	cover	cover	NOUN
ejpam-5157	28	13	of	of	ADP
ejpam-5157	28	14	a	a	DET
ejpam-5157	28	15	graph	graph	NOUN
ejpam-5157	28	16	.	.	PUNCT
ejpam-5157	29	1	we	we	PRON
ejpam-5157	29	2	do	do	AUX
ejpam-5157	29	3	believe	believe	VERB
ejpam-5157	29	4	,	,	PUNCT
ejpam-5157	29	5	this	this	DET
ejpam-5157	29	6	parameter	parameter	NOUN
ejpam-5157	29	7	and	and	CCONJ
ejpam-5157	29	8	its	its	PRON
ejpam-5157	29	9	results	result	NOUN
ejpam-5157	29	10	would	would	AUX
ejpam-5157	29	11	lead	lead	VERB
ejpam-5157	29	12	to	to	ADP
ejpam-5157	29	13	another	another	DET
ejpam-5157	29	14	interesting	interesting	ADJ
ejpam-5157	29	15	studies	study	NOUN
ejpam-5157	29	16	and	and	CCONJ
ejpam-5157	29	17	applications	application	NOUN
ejpam-5157	29	18	in	in	ADP
ejpam-5157	29	19	the	the	DET
ejpam-5157	29	20	future	future	NOUN
ejpam-5157	29	21	.	.	PUNCT
ejpam-5157	30	1	further	far	ADV
ejpam-5157	30	2	,	,	PUNCT
ejpam-5157	30	3	this	this	DET
ejpam-5157	30	4	study	study	NOUN
ejpam-5157	30	5	would	would	AUX
ejpam-5157	30	6	serve	serve	VERB
ejpam-5157	30	7	as	as	ADP
ejpam-5157	30	8	reference	reference	NOUN
ejpam-5157	30	9	to	to	ADP
ejpam-5157	30	10	future	future	ADJ
ejpam-5157	30	11	researchers	researcher	NOUN
ejpam-5157	30	12	who	who	PRON
ejpam-5157	30	13	will	will	AUX
ejpam-5157	30	14	study	study	VERB
ejpam-5157	30	15	on	on	ADP
ejpam-5157	30	16	concept	concept	NOUN
ejpam-5157	30	17	or	or	CCONJ
ejpam-5157	30	18	problem	problem	NOUN
ejpam-5157	30	19	related	relate	VERB
ejpam-5157	30	20	to	to	ADP
ejpam-5157	30	21	vertex	vertex	NOUN
ejpam-5157	30	22	cover	cover	NOUN
ejpam-5157	30	23	of	of	ADP
ejpam-5157	30	24	a	a	DET
ejpam-5157	30	25	graph	graph	NOUN
ejpam-5157	30	26	.	.	PUNCT
ejpam-5157	31	1	2	2	X
ejpam-5157	31	2	.	.	X
ejpam-5157	31	3	terminology	terminology	NOUN
ejpam-5157	31	4	and	and	CCONJ
ejpam-5157	31	5	notation	notation	NOUN
ejpam-5157	31	6	let	let	VERB
ejpam-5157	31	7	g	g	NOUN
ejpam-5157	31	8	=	=	SYM
ejpam-5157	31	9	(	(	PUNCT
ejpam-5157	31	10	v	v	NOUN
ejpam-5157	31	11	(	(	PUNCT
ejpam-5157	31	12	g	g	NOUN
ejpam-5157	31	13	)	)	PUNCT
ejpam-5157	31	14	,	,	PUNCT
ejpam-5157	31	15	e(g	e(g	PROPN
ejpam-5157	31	16	)	)	PUNCT
ejpam-5157	31	17	)	)	PUNCT
ejpam-5157	31	18	be	be	AUX
ejpam-5157	31	19	a	a	DET
ejpam-5157	31	20	simple	simple	ADJ
ejpam-5157	31	21	and	and	CCONJ
ejpam-5157	31	22	undirected	undirected	ADJ
ejpam-5157	31	23	graph	graph	NOUN
ejpam-5157	31	24	.	.	PUNCT
ejpam-5157	32	1	then	then	ADV
ejpam-5157	32	2	s	s	VERB
ejpam-5157	32	3	⊆	⊆	NUM
ejpam-5157	32	4	v	v	NOUN
ejpam-5157	32	5	(	(	PUNCT
ejpam-5157	32	6	g	g	NOUN
ejpam-5157	32	7	)	)	PUNCT
ejpam-5157	32	8	is	be	AUX
ejpam-5157	32	9	called	call	VERB
ejpam-5157	32	10	a	a	DET
ejpam-5157	32	11	vertex	vertex	NOUN
ejpam-5157	32	12	cover	cover	NOUN
ejpam-5157	32	13	of	of	ADP
ejpam-5157	32	14	g	g	PROPN
ejpam-5157	32	15	if	if	SCONJ
ejpam-5157	32	16	every	every	DET
ejpam-5157	32	17	edge	edge	NOUN
ejpam-5157	32	18	in	in	ADP
ejpam-5157	32	19	g	g	PROPN
ejpam-5157	32	20	is	be	AUX
ejpam-5157	32	21	incident	incident	NOUN
ejpam-5157	32	22	to	to	ADP
ejpam-5157	32	23	some	some	DET
ejpam-5157	32	24	vertex	vertex	NOUN
ejpam-5157	32	25	in	in	ADP
ejpam-5157	32	26	s.	s.	PROPN
ejpam-5157	32	27	the	the	DET
ejpam-5157	32	28	minimum	minimum	ADJ
ejpam-5157	32	29	cardinality	cardinality	NOUN
ejpam-5157	32	30	of	of	ADP
ejpam-5157	32	31	a	a	DET
ejpam-5157	32	32	vertex	vertex	NOUN
ejpam-5157	32	33	cover	cover	NOUN
ejpam-5157	32	34	of	of	ADP
ejpam-5157	32	35	g	g	NOUN
ejpam-5157	32	36	,	,	PUNCT
ejpam-5157	32	37	denoted	denote	VERB
ejpam-5157	32	38	by	by	ADP
ejpam-5157	32	39	β(g	β(g	PROPN
ejpam-5157	32	40	)	)	PUNCT
ejpam-5157	32	41	,	,	PUNCT
ejpam-5157	32	42	is	be	AUX
ejpam-5157	32	43	called	call	VERB
ejpam-5157	32	44	the	the	DET
ejpam-5157	32	45	vertex	vertex	NOUN
ejpam-5157	32	46	cover	cover	NOUN
ejpam-5157	32	47	number	number	NOUN
ejpam-5157	32	48	of	of	ADP
ejpam-5157	32	49	g.	g.	PROPN
ejpam-5157	32	50	a	a	DET
ejpam-5157	32	51	set	set	NOUN
ejpam-5157	32	52	q	q	NOUN
ejpam-5157	32	53	⊆	⊆	NUM
ejpam-5157	32	54	v	v	NOUN
ejpam-5157	32	55	(	(	PUNCT
ejpam-5157	32	56	g	g	NOUN
ejpam-5157	32	57	)	)	PUNCT
ejpam-5157	32	58	is	be	AUX
ejpam-5157	32	59	called	call	VERB
ejpam-5157	32	60	a	a	DET
ejpam-5157	32	61	certified	certify	VERB
ejpam-5157	32	62	set	set	NOUN
ejpam-5157	32	63	of	of	ADP
ejpam-5157	32	64	g	g	PROPN
ejpam-5157	32	65	if	if	SCONJ
ejpam-5157	32	66	every	every	DET
ejpam-5157	32	67	vertex	vertex	NOUN
ejpam-5157	32	68	x	x	X
ejpam-5157	32	69	∈	∈	NOUN
ejpam-5157	32	70	q	q	NOUN
ejpam-5157	32	71	,	,	PUNCT
ejpam-5157	32	72	x	x	PUNCT
ejpam-5157	32	73	has	have	VERB
ejpam-5157	32	74	either	either	CCONJ
ejpam-5157	32	75	zero	zero	NUM
ejpam-5157	32	76	or	or	CCONJ
ejpam-5157	32	77	at	at	ADP
ejpam-5157	32	78	least	least	ADV
ejpam-5157	32	79	two	two	NUM
ejpam-5157	32	80	neighbors	neighbor	NOUN
ejpam-5157	32	81	in	in	ADP
ejpam-5157	32	82	v	v	NOUN
ejpam-5157	32	83	(	(	PUNCT
ejpam-5157	32	84	g	g	NOUN
ejpam-5157	32	85	)	)	PUNCT
ejpam-5157	32	86	\q	\q	NOUN
ejpam-5157	32	87	.	.	PUNCT
ejpam-5157	33	1	let	let	VERB
ejpam-5157	33	2	g	g	NOUN
ejpam-5157	33	3	and	and	CCONJ
ejpam-5157	33	4	h	h	NOUN
ejpam-5157	33	5	be	be	VERB
ejpam-5157	33	6	any	any	DET
ejpam-5157	33	7	two	two	NUM
ejpam-5157	33	8	graphs	graph	NOUN
ejpam-5157	33	9	.	.	PUNCT
ejpam-5157	34	1	the	the	DET
ejpam-5157	34	2	join	join	NOUN
ejpam-5157	34	3	of	of	ADP
ejpam-5157	34	4	g	g	PROPN
ejpam-5157	34	5	and	and	CCONJ
ejpam-5157	34	6	h	h	NOUN
ejpam-5157	34	7	,	,	PUNCT
ejpam-5157	34	8	denoted	denote	VERB
ejpam-5157	34	9	by	by	ADP
ejpam-5157	34	10	g+h	g+h	PROPN
ejpam-5157	34	11	is	be	AUX
ejpam-5157	34	12	the	the	DET
ejpam-5157	34	13	graph	graph	NOUN
ejpam-5157	34	14	with	with	ADP
ejpam-5157	34	15	vertex	vertex	NOUN
ejpam-5157	34	16	set	set	VERB
ejpam-5157	34	17	v	v	NOUN
ejpam-5157	34	18	(	(	PUNCT
ejpam-5157	34	19	g+h	g+h	NOUN
ejpam-5157	34	20	)	)	PUNCT
ejpam-5157	35	1	=	=	SYM
ejpam-5157	35	2	v	v	X
ejpam-5157	35	3	(	(	PUNCT
ejpam-5157	35	4	g	g	NOUN
ejpam-5157	35	5	)	)	PUNCT
ejpam-5157	35	6	∪	∪	NOUN
ejpam-5157	35	7	v	v	NOUN
ejpam-5157	35	8	(	(	PUNCT
ejpam-5157	35	9	h	h	NOUN
ejpam-5157	35	10	)	)	PUNCT
ejpam-5157	35	11	and	and	CCONJ
ejpam-5157	35	12	edge	edge	NOUN
ejpam-5157	35	13	set	set	VERB
ejpam-5157	35	14	e(g+h	e(g+h	NUM
ejpam-5157	35	15	)	)	PUNCT
ejpam-5157	35	16	=	=	SYM
ejpam-5157	35	17	e(g	e(g	NOUN
ejpam-5157	35	18	)	)	PUNCT
ejpam-5157	35	19	∪	∪	ADP
ejpam-5157	35	20	e(h	e(h	PROPN
ejpam-5157	35	21	)	)	PUNCT
ejpam-5157	35	22	∪	∪	NOUN
ejpam-5157	35	23	{	{	PUNCT
ejpam-5157	35	24	uv	uv	NOUN
ejpam-5157	35	25	:	:	PUNCT
ejpam-5157	35	26	u	u	PROPN
ejpam-5157	35	27	∈	∈	PROPN
ejpam-5157	35	28	v	v	ADP
ejpam-5157	35	29	(	(	PUNCT
ejpam-5157	35	30	g	g	NOUN
ejpam-5157	35	31	)	)	PUNCT
ejpam-5157	35	32	,	,	PUNCT
ejpam-5157	35	33	v	v	X
ejpam-5157	35	34	∈	∈	PROPN
ejpam-5157	35	35	v	v	NOUN
ejpam-5157	35	36	(	(	PUNCT
ejpam-5157	35	37	h	h	NOUN
ejpam-5157	35	38	)	)	PUNCT
ejpam-5157	35	39	}	}	PUNCT
ejpam-5157	35	40	.	.	PUNCT
ejpam-5157	36	1	3	3	X
ejpam-5157	36	2	.	.	X
ejpam-5157	36	3	results	result	NOUN
ejpam-5157	36	4	we	we	PRON
ejpam-5157	36	5	begin	begin	VERB
ejpam-5157	36	6	this	this	DET
ejpam-5157	36	7	section	section	NOUN
ejpam-5157	36	8	by	by	ADP
ejpam-5157	36	9	introducing	introduce	VERB
ejpam-5157	36	10	the	the	DET
ejpam-5157	36	11	concept	concept	NOUN
ejpam-5157	36	12	of	of	ADP
ejpam-5157	36	13	certified	certify	VERB
ejpam-5157	36	14	vertex	vertex	NOUN
ejpam-5157	36	15	cover	cover	NOUN
ejpam-5157	36	16	of	of	ADP
ejpam-5157	36	17	a	a	DET
ejpam-5157	36	18	graph	graph	NOUN
ejpam-5157	36	19	.	.	PUNCT
ejpam-5157	37	1	definition	definition	NOUN
ejpam-5157	37	2	1	1	NUM
ejpam-5157	37	3	.	.	PUNCT
ejpam-5157	38	1	let	let	VERB
ejpam-5157	38	2	g	g	PRON
ejpam-5157	38	3	be	be	AUX
ejpam-5157	38	4	a	a	DET
ejpam-5157	38	5	graph	graph	NOUN
ejpam-5157	38	6	.	.	PUNCT
ejpam-5157	39	1	then	then	ADV
ejpam-5157	39	2	q	q	X
ejpam-5157	39	3	⊆	⊆	NUM
ejpam-5157	39	4	v	v	NOUN
ejpam-5157	39	5	(	(	PUNCT
ejpam-5157	39	6	g	g	NOUN
ejpam-5157	39	7	)	)	PUNCT
ejpam-5157	39	8	is	be	AUX
ejpam-5157	39	9	called	call	VERB
ejpam-5157	39	10	a	a	DET
ejpam-5157	39	11	certified	certify	VERB
ejpam-5157	39	12	vertex	vertex	NOUN
ejpam-5157	39	13	cover	cover	NOUN
ejpam-5157	39	14	of	of	ADP
ejpam-5157	39	15	g	g	PROPN
ejpam-5157	39	16	if	if	SCONJ
ejpam-5157	39	17	q	q	NOUN
ejpam-5157	39	18	is	be	AUX
ejpam-5157	39	19	a	a	DET
ejpam-5157	39	20	vertex	vertex	NOUN
ejpam-5157	39	21	cover	cover	NOUN
ejpam-5157	39	22	of	of	ADP
ejpam-5157	39	23	g	g	NOUN
ejpam-5157	39	24	and	and	CCONJ
ejpam-5157	39	25	every	every	DET
ejpam-5157	39	26	x	x	X
ejpam-5157	39	27	∈	∈	PROPN
ejpam-5157	39	28	q	q	NOUN
ejpam-5157	39	29	,	,	PUNCT
ejpam-5157	39	30	x	x	PUNCT
ejpam-5157	39	31	has	have	VERB
ejpam-5157	39	32	either	either	CCONJ
ejpam-5157	39	33	zero	zero	NUM
ejpam-5157	39	34	or	or	CCONJ
ejpam-5157	39	35	at	at	ADP
ejpam-5157	39	36	least	least	ADV
ejpam-5157	39	37	two	two	NUM
ejpam-5157	39	38	neighbors	neighbor	NOUN
ejpam-5157	39	39	in	in	ADP
ejpam-5157	39	40	v	v	NOUN
ejpam-5157	39	41	(	(	PUNCT
ejpam-5157	39	42	g	g	NOUN
ejpam-5157	39	43	)	)	PUNCT
ejpam-5157	39	44	\	\	PUNCT
ejpam-5157	39	45	q.	q.	NOUN
ejpam-5157	39	46	the	the	DET
ejpam-5157	39	47	certified	certify	VERB
ejpam-5157	39	48	vertex	vertex	NOUN
ejpam-5157	39	49	cover	cover	NOUN
ejpam-5157	39	50	number	number	NOUN
ejpam-5157	39	51	of	of	ADP
ejpam-5157	39	52	g	g	NOUN
ejpam-5157	39	53	,	,	PUNCT
ejpam-5157	39	54	denoted	denote	VERB
ejpam-5157	39	55	by	by	ADP
ejpam-5157	39	56	βcer(g	βcer(g	NOUN
ejpam-5157	39	57	)	)	PUNCT
ejpam-5157	39	58	,	,	PUNCT
ejpam-5157	39	59	is	be	AUX
ejpam-5157	39	60	the	the	DET
ejpam-5157	39	61	minimum	minimum	ADJ
ejpam-5157	39	62	cardinality	cardinality	NOUN
ejpam-5157	39	63	of	of	ADP
ejpam-5157	39	64	a	a	DET
ejpam-5157	39	65	certified	certify	VERB
ejpam-5157	39	66	vertex	vertex	NOUN
ejpam-5157	39	67	cover	cover	NOUN
ejpam-5157	39	68	of	of	ADP
ejpam-5157	39	69	g.	g.	PROPN
ejpam-5157	39	70	example	example	NOUN
ejpam-5157	39	71	1	1	X
ejpam-5157	39	72	.	.	X
ejpam-5157	39	73	consider	consider	VERB
ejpam-5157	39	74	the	the	DET
ejpam-5157	39	75	graph	graph	NOUN
ejpam-5157	39	76	g	g	NOUN
ejpam-5157	39	77	below	below	ADV
ejpam-5157	39	78	.	.	PUNCT
ejpam-5157	40	1	let	let	VERB
ejpam-5157	40	2	q	q	NOUN
ejpam-5157	41	1	=	=	PUNCT
ejpam-5157	41	2	{	{	PUNCT
ejpam-5157	41	3	u4	u4	PROPN
ejpam-5157	41	4	,	,	PUNCT
ejpam-5157	41	5	u5	u5	PROPN
ejpam-5157	41	6	,	,	PUNCT
ejpam-5157	41	7	u6	u6	PROPN
ejpam-5157	41	8	,	,	PUNCT
ejpam-5157	41	9	u7	u7	PROPN
ejpam-5157	41	10	}	}	PUNCT
ejpam-5157	41	11	.	.	PUNCT
ejpam-5157	42	1	then	then	ADV
ejpam-5157	42	2	vertex	vertex	PROPN
ejpam-5157	42	3	u4	u4	PROPN
ejpam-5157	42	4	has	have	VERB
ejpam-5157	42	5	three	three	NUM
ejpam-5157	42	6	neighbors	neighbor	NOUN
ejpam-5157	42	7	outside	outside	ADP
ejpam-5157	42	8	q	q	NOUN
ejpam-5157	42	9	and	and	CCONJ
ejpam-5157	42	10	vertices	vertice	VERB
ejpam-5157	42	11	u5	u5	PROPN
ejpam-5157	42	12	,	,	PUNCT
ejpam-5157	42	13	u6	u6	NOUN
ejpam-5157	42	14	and	and	CCONJ
ejpam-5157	42	15	u7	u7	PROPN
ejpam-5157	42	16	have	have	VERB
ejpam-5157	42	17	zero	zero	NUM
ejpam-5157	42	18	neighbor	neighbor	NOUN
ejpam-5157	42	19	outside	outside	ADP
ejpam-5157	42	20	q.	q.	PROPN
ejpam-5157	42	21	it	it	PRON
ejpam-5157	42	22	follows	follow	VERB
ejpam-5157	42	23	that	that	SCONJ
ejpam-5157	42	24	q	q	NOUN
ejpam-5157	42	25	is	be	AUX
ejpam-5157	42	26	a	a	DET
ejpam-5157	42	27	certified	certify	VERB
ejpam-5157	42	28	vertex	vertex	NOUN
ejpam-5157	42	29	cover	cover	NOUN
ejpam-5157	42	30	of	of	ADP
ejpam-5157	42	31	g.	g.	PROPN
ejpam-5157	42	32	observe	observe	VERB
ejpam-5157	42	33	that	that	SCONJ
ejpam-5157	42	34	every	every	DET
ejpam-5157	42	35	edge	edge	NOUN
ejpam-5157	42	36	of	of	ADP
ejpam-5157	42	37	g	g	PROPN
ejpam-5157	42	38	is	be	AUX
ejpam-5157	42	39	incident	incident	NOUN
ejpam-5157	42	40	to	to	ADP
ejpam-5157	42	41	some	some	DET
ejpam-5157	42	42	vertex	vertex	NOUN
ejpam-5157	42	43	in	in	ADP
ejpam-5157	42	44	q.	q.	PROPN
ejpam-5157	42	45	thus	thus	ADV
ejpam-5157	42	46	,	,	PUNCT
ejpam-5157	42	47	q	q	PROPN
ejpam-5157	42	48	is	be	AUX
ejpam-5157	42	49	a	a	DET
ejpam-5157	42	50	vertex	vertex	NOUN
ejpam-5157	42	51	cover	cover	NOUN
ejpam-5157	42	52	of	of	ADP
ejpam-5157	42	53	g	g	NOUN
ejpam-5157	42	54	,	,	PUNCT
ejpam-5157	42	55	showing	show	VERB
ejpam-5157	42	56	that	that	PRON
ejpam-5157	42	57	q	q	NOUN
ejpam-5157	42	58	is	be	AUX
ejpam-5157	42	59	a	a	DET
ejpam-5157	42	60	certified	certify	VERB
ejpam-5157	42	61	vertex	vertex	NOUN
ejpam-5157	42	62	cover	cover	NOUN
ejpam-5157	42	63	of	of	ADP
ejpam-5157	42	64	g.	g.	PROPN
ejpam-5157	42	65	moreover	moreover	ADV
ejpam-5157	42	66	,	,	PUNCT
ejpam-5157	42	67	it	it	PRON
ejpam-5157	42	68	can	can	AUX
ejpam-5157	42	69	be	be	AUX
ejpam-5157	42	70	verified	verify	VERB
ejpam-5157	42	71	that	that	SCONJ
ejpam-5157	42	72	βcer(g	βcer(g	NOUN
ejpam-5157	42	73	)	)	PUNCT
ejpam-5157	43	1	=	=	SYM
ejpam-5157	43	2	4	4	X
ejpam-5157	43	3	.	.	PUNCT
ejpam-5157	43	4	j.	j.	PROPN
ejpam-5157	43	5	a.	a.	PROPN
ejpam-5157	43	6	hassan	hassan	PROPN
ejpam-5157	43	7	et	et	PROPN
ejpam-5157	43	8	al	al	PROPN
ejpam-5157	43	9	.	.	PUNCT
ejpam-5157	43	10	/	/	SYM
ejpam-5157	43	11	eur	eur	PROPN
ejpam-5157	43	12	.	.	PUNCT
ejpam-5157	44	1	j.	j.	PROPN
ejpam-5157	44	2	pure	pure	PROPN
ejpam-5157	44	3	appl	appl	PROPN
ejpam-5157	44	4	.	.	PROPN
ejpam-5157	44	5	math	math	PROPN
ejpam-5157	44	6	,	,	PUNCT
ejpam-5157	44	7	17	17	NUM
ejpam-5157	44	8	(	(	PUNCT
ejpam-5157	44	9	2	2	NUM
ejpam-5157	44	10	)	)	PUNCT
ejpam-5157	44	11	(	(	PUNCT
ejpam-5157	44	12	2024	2024	NUM
ejpam-5157	44	13	)	)	PUNCT
ejpam-5157	44	14	,	,	PUNCT
ejpam-5157	44	15	1038	1038	NUM
ejpam-5157	44	16	-	-	SYM
ejpam-5157	44	17	1045	1045	NUM
ejpam-5157	44	18	1040	1040	NUM
ejpam-5157	44	19	u3	u3	PROPN
ejpam-5157	44	20	u4	u4	PROPN
ejpam-5157	44	21	u2	u2	PROPN
ejpam-5157	44	22	u5	u5	PROPN
ejpam-5157	44	23	u1	u1	PROPN
ejpam-5157	44	24	g	g	NOUN
ejpam-5157	44	25	:	:	PUNCT
ejpam-5157	44	26	u6	u6	PROPN
ejpam-5157	44	27	u7	u7	PROPN
ejpam-5157	44	28	figure	figure	NOUN
ejpam-5157	44	29	1	1	NUM
ejpam-5157	44	30	:	:	PUNCT
ejpam-5157	44	31	graph	graph	VERB
ejpam-5157	44	32	g	g	NOUN
ejpam-5157	44	33	with	with	ADP
ejpam-5157	44	34	βcer(g	βcer(g	NOUN
ejpam-5157	44	35	)	)	PUNCT
ejpam-5157	44	36	=	=	SYM
ejpam-5157	44	37	4	4	NUM
ejpam-5157	44	38	proposition	proposition	NOUN
ejpam-5157	44	39	1	1	NUM
ejpam-5157	44	40	.	.	PUNCT
ejpam-5157	45	1	let	let	VERB
ejpam-5157	45	2	g	g	PRON
ejpam-5157	45	3	be	be	AUX
ejpam-5157	45	4	a	a	DET
ejpam-5157	45	5	graph	graph	NOUN
ejpam-5157	45	6	.	.	PUNCT
ejpam-5157	46	1	then	then	ADV
ejpam-5157	46	2	(	(	PUNCT
ejpam-5157	46	3	i	i	NOUN
ejpam-5157	46	4	)	)	PUNCT
ejpam-5157	46	5	β(g	β(g	PROPN
ejpam-5157	46	6	)	)	PUNCT
ejpam-5157	46	7	≤	≤	PUNCT
ejpam-5157	46	8	βcer(g	βcer(g	NOUN
ejpam-5157	46	9	)	)	PUNCT
ejpam-5157	46	10	;	;	PUNCT
ejpam-5157	46	11	(	(	PUNCT
ejpam-5157	46	12	ii	ii	NOUN
ejpam-5157	46	13	)	)	PUNCT
ejpam-5157	46	14	1	1	NUM
ejpam-5157	46	15	≤	≤	NUM
ejpam-5157	46	16	βcer(g	βcer(g	NOUN
ejpam-5157	46	17	)	)	PUNCT
ejpam-5157	46	18	≤	≤	NUM
ejpam-5157	46	19	|v	|v	X
ejpam-5157	46	20	(	(	PUNCT
ejpam-5157	46	21	g)|	g)|	NOUN
ejpam-5157	46	22	;	;	PUNCT
ejpam-5157	46	23	(	(	PUNCT
ejpam-5157	46	24	iii	iii	X
ejpam-5157	46	25	)	)	PUNCT
ejpam-5157	46	26	a	a	DET
ejpam-5157	46	27	certified	certify	VERB
ejpam-5157	46	28	set	set	NOUN
ejpam-5157	46	29	of	of	ADP
ejpam-5157	46	30	g	g	NOUN
ejpam-5157	46	31	may	may	AUX
ejpam-5157	46	32	not	not	PART
ejpam-5157	46	33	be	be	AUX
ejpam-5157	46	34	a	a	DET
ejpam-5157	46	35	vertex	vertex	NOUN
ejpam-5157	46	36	cover	cover	NOUN
ejpam-5157	46	37	of	of	ADP
ejpam-5157	46	38	g	g	NOUN
ejpam-5157	46	39	;	;	PUNCT
ejpam-5157	46	40	(	(	PUNCT
ejpam-5157	46	41	iv	iv	X
ejpam-5157	46	42	)	)	PUNCT
ejpam-5157	46	43	a	a	DET
ejpam-5157	46	44	vertex	vertex	NOUN
ejpam-5157	46	45	cover	cover	NOUN
ejpam-5157	46	46	of	of	ADP
ejpam-5157	46	47	g	g	NOUN
ejpam-5157	46	48	may	may	AUX
ejpam-5157	46	49	not	not	PART
ejpam-5157	46	50	be	be	AUX
ejpam-5157	46	51	a	a	DET
ejpam-5157	46	52	certified	certified	ADJ
ejpam-5157	46	53	set	set	NOUN
ejpam-5157	46	54	of	of	ADP
ejpam-5157	46	55	g	g	NOUN
ejpam-5157	46	56	;	;	PUNCT
ejpam-5157	46	57	and	and	CCONJ
ejpam-5157	46	58	(	(	PUNCT
ejpam-5157	46	59	v	v	NOUN
ejpam-5157	46	60	)	)	PUNCT
ejpam-5157	46	61	if	if	SCONJ
ejpam-5157	46	62	q	q	PROPN
ejpam-5157	46	63	⊆	⊆	NUM
ejpam-5157	46	64	v	v	NOUN
ejpam-5157	46	65	(	(	PUNCT
ejpam-5157	46	66	g	g	NOUN
ejpam-5157	46	67	)	)	PUNCT
ejpam-5157	46	68	is	be	AUX
ejpam-5157	46	69	both	both	PRON
ejpam-5157	46	70	certified	certified	ADJ
ejpam-5157	46	71	and	and	CCONJ
ejpam-5157	46	72	minimum	minimum	ADJ
ejpam-5157	46	73	vertex	vertex	NOUN
ejpam-5157	46	74	cover	cover	NOUN
ejpam-5157	46	75	of	of	ADP
ejpam-5157	46	76	g	g	NOUN
ejpam-5157	46	77	,	,	PUNCT
ejpam-5157	46	78	then	then	ADV
ejpam-5157	46	79	q	q	X
ejpam-5157	46	80	is	be	AUX
ejpam-5157	46	81	a	a	DET
ejpam-5157	46	82	minimum	minimum	ADJ
ejpam-5157	46	83	certified	certify	VERB
ejpam-5157	46	84	vertex	vertex	NOUN
ejpam-5157	46	85	cover	cover	NOUN
ejpam-5157	46	86	of	of	ADP
ejpam-5157	46	87	g	g	NOUN
ejpam-5157	46	88	and	and	CCONJ
ejpam-5157	46	89	βcer(g	βcer(g	NOUN
ejpam-5157	46	90	)	)	PUNCT
ejpam-5157	47	1	=	=	SYM
ejpam-5157	47	2	|q|	|q|	VERB
ejpam-5157	47	3	.	.	NOUN
ejpam-5157	47	4	proof	proof	NOUN
ejpam-5157	47	5	.	.	PUNCT
ejpam-5157	48	1	(	(	PUNCT
ejpam-5157	48	2	i	i	NOUN
ejpam-5157	48	3	)	)	PUNCT
ejpam-5157	48	4	let	let	VERB
ejpam-5157	48	5	g	g	NOUN
ejpam-5157	48	6	be	be	AUX
ejpam-5157	48	7	a	a	DET
ejpam-5157	48	8	graph	graph	NOUN
ejpam-5157	48	9	and	and	CCONJ
ejpam-5157	48	10	let	let	VERB
ejpam-5157	48	11	q	q	PART
ejpam-5157	48	12	be	be	AUX
ejpam-5157	48	13	a	a	DET
ejpam-5157	48	14	minimum	minimum	ADJ
ejpam-5157	48	15	certified	certify	VERB
ejpam-5157	48	16	vertex	vertex	NOUN
ejpam-5157	48	17	cover	cover	NOUN
ejpam-5157	48	18	of	of	ADP
ejpam-5157	48	19	g.	g.	PROPN
ejpam-5157	49	1	then	then	ADV
ejpam-5157	49	2	q	q	X
ejpam-5157	49	3	is	be	AUX
ejpam-5157	49	4	vertex	vertex	NOUN
ejpam-5157	49	5	cover	cover	NOUN
ejpam-5157	49	6	of	of	ADP
ejpam-5157	49	7	g.	g.	PROPN
ejpam-5157	49	8	since	since	SCONJ
ejpam-5157	49	9	β(g	β(g	PROPN
ejpam-5157	49	10	)	)	PUNCT
ejpam-5157	49	11	is	be	AUX
ejpam-5157	49	12	the	the	DET
ejpam-5157	49	13	smallest	small	ADJ
ejpam-5157	49	14	cardinality	cardinality	NOUN
ejpam-5157	49	15	of	of	ADP
ejpam-5157	49	16	a	a	DET
ejpam-5157	49	17	vertex	vertex	NOUN
ejpam-5157	49	18	cover	cover	NOUN
ejpam-5157	49	19	of	of	ADP
ejpam-5157	49	20	g	g	NOUN
ejpam-5157	49	21	,	,	PUNCT
ejpam-5157	49	22	it	it	PRON
ejpam-5157	49	23	follows	follow	VERB
ejpam-5157	49	24	that	that	SCONJ
ejpam-5157	49	25	β(g	β(g	PROPN
ejpam-5157	49	26	)	)	PUNCT
ejpam-5157	49	27	≤	≤	PUNCT
ejpam-5157	49	28	|q|	|q|	VERB
ejpam-5157	49	29	=	=	SYM
ejpam-5157	49	30	βcer(g	βcer(g	NOUN
ejpam-5157	49	31	)	)	PUNCT
ejpam-5157	49	32	.	.	PUNCT
ejpam-5157	50	1	(	(	PUNCT
ejpam-5157	50	2	ii	ii	NOUN
ejpam-5157	50	3	)	)	PUNCT
ejpam-5157	50	4	since	since	SCONJ
ejpam-5157	50	5	β(g	β(g	PROPN
ejpam-5157	50	6	)	)	PUNCT
ejpam-5157	50	7	≥	≥	NOUN
ejpam-5157	50	8	1	1	NUM
ejpam-5157	50	9	for	for	ADP
ejpam-5157	50	10	any	any	DET
ejpam-5157	50	11	graph	graph	NOUN
ejpam-5157	50	12	g	g	NOUN
ejpam-5157	50	13	,	,	PUNCT
ejpam-5157	50	14	βcer(g	βcer(g	NOUN
ejpam-5157	50	15	)	)	PUNCT
ejpam-5157	50	16	≥	≥	NOUN
ejpam-5157	50	17	1	1	NUM
ejpam-5157	50	18	by	by	ADP
ejpam-5157	50	19	(	(	PUNCT
ejpam-5157	50	20	i	i	NOUN
ejpam-5157	50	21	)	)	PUNCT
ejpam-5157	50	22	.	.	PUNCT
ejpam-5157	51	1	clearly	clearly	ADV
ejpam-5157	51	2	,	,	PUNCT
ejpam-5157	51	3	βcer(g	βcer(g	NOUN
ejpam-5157	51	4	)	)	PUNCT
ejpam-5157	51	5	≤	≤	NOUN
ejpam-5157	51	6	|v	|v	X
ejpam-5157	51	7	(	(	PUNCT
ejpam-5157	51	8	g)|	g)|	PROPN
ejpam-5157	51	9	.	.	PUNCT
ejpam-5157	52	1	therefore	therefore	ADV
ejpam-5157	52	2	,	,	PUNCT
ejpam-5157	52	3	1	1	NUM
ejpam-5157	52	4	≤	≤	NUM
ejpam-5157	52	5	βcer(g	βcer(g	NOUN
ejpam-5157	52	6	)	)	PUNCT
ejpam-5157	52	7	≤	≤	NUM
ejpam-5157	52	8	|v	|v	X
ejpam-5157	52	9	(	(	PUNCT
ejpam-5157	52	10	g)|	g)|	NOUN
ejpam-5157	52	11	.	.	PUNCT
ejpam-5157	53	1	(	(	PUNCT
ejpam-5157	53	2	iii	iii	X
ejpam-5157	53	3	)	)	PUNCT
ejpam-5157	53	4	consider	consider	VERB
ejpam-5157	53	5	again	again	ADV
ejpam-5157	53	6	the	the	DET
ejpam-5157	53	7	graph	graph	NOUN
ejpam-5157	53	8	g	g	NOUN
ejpam-5157	53	9	below	below	ADV
ejpam-5157	53	10	:	:	PUNCT
ejpam-5157	53	11	u3	u3	PROPN
ejpam-5157	53	12	u4	u4	PROPN
ejpam-5157	53	13	u2	u2	PROPN
ejpam-5157	53	14	u5	u5	PROPN
ejpam-5157	53	15	u1	u1	PROPN
ejpam-5157	53	16	g	g	NOUN
ejpam-5157	53	17	:	:	PUNCT
ejpam-5157	53	18	u6	u6	PROPN
ejpam-5157	53	19	u7	u7	PROPN
ejpam-5157	53	20	j.	j.	PROPN
ejpam-5157	53	21	a.	a.	PROPN
ejpam-5157	53	22	hassan	hassan	PROPN
ejpam-5157	53	23	et	et	PROPN
ejpam-5157	53	24	al	al	PROPN
ejpam-5157	53	25	.	.	PUNCT
ejpam-5157	53	26	/	/	SYM
ejpam-5157	53	27	eur	eur	PROPN
ejpam-5157	53	28	.	.	PUNCT
ejpam-5157	54	1	j.	j.	PROPN
ejpam-5157	54	2	pure	pure	PROPN
ejpam-5157	54	3	appl	appl	PROPN
ejpam-5157	54	4	.	.	PROPN
ejpam-5157	54	5	math	math	PROPN
ejpam-5157	54	6	,	,	PUNCT
ejpam-5157	54	7	17	17	NUM
ejpam-5157	54	8	(	(	PUNCT
ejpam-5157	54	9	2	2	NUM
ejpam-5157	54	10	)	)	PUNCT
ejpam-5157	54	11	(	(	PUNCT
ejpam-5157	54	12	2024	2024	NUM
ejpam-5157	54	13	)	)	PUNCT
ejpam-5157	54	14	,	,	PUNCT
ejpam-5157	54	15	1038	1038	NUM
ejpam-5157	54	16	-	-	SYM
ejpam-5157	54	17	1045	1045	NUM
ejpam-5157	54	18	1041	1041	NUM
ejpam-5157	54	19	let	let	VERB
ejpam-5157	54	20	q	q	NOUN
ejpam-5157	55	1	=	=	PUNCT
ejpam-5157	55	2	{	{	PUNCT
ejpam-5157	55	3	u4	u4	PROPN
ejpam-5157	55	4	,	,	PUNCT
ejpam-5157	55	5	u5	u5	PROPN
ejpam-5157	55	6	}	}	PUNCT
ejpam-5157	55	7	.	.	PUNCT
ejpam-5157	56	1	then	then	ADV
ejpam-5157	56	2	u4	u4	PROPN
ejpam-5157	56	3	and	and	CCONJ
ejpam-5157	56	4	u5	u5	PROPN
ejpam-5157	56	5	have	have	VERB
ejpam-5157	56	6	three	three	NUM
ejpam-5157	56	7	and	and	CCONJ
ejpam-5157	56	8	two	two	NUM
ejpam-5157	56	9	neighbors	neighbor	NOUN
ejpam-5157	56	10	in	in	ADP
ejpam-5157	56	11	v	v	NOUN
ejpam-5157	56	12	(	(	PUNCT
ejpam-5157	56	13	g	g	NOUN
ejpam-5157	56	14	)	)	PUNCT
ejpam-5157	56	15	\q	\q	NOUN
ejpam-5157	56	16	,	,	PUNCT
ejpam-5157	56	17	respectively	respectively	ADV
ejpam-5157	56	18	.	.	PUNCT
ejpam-5157	57	1	it	it	PRON
ejpam-5157	57	2	follows	follow	VERB
ejpam-5157	57	3	that	that	SCONJ
ejpam-5157	57	4	q	q	NOUN
ejpam-5157	57	5	is	be	AUX
ejpam-5157	57	6	a	a	DET
ejpam-5157	57	7	certified	certified	ADJ
ejpam-5157	57	8	set	set	NOUN
ejpam-5157	57	9	of	of	ADP
ejpam-5157	57	10	g.	g.	PROPN
ejpam-5157	57	11	however	however	ADV
ejpam-5157	57	12	,	,	PUNCT
ejpam-5157	57	13	q	q	X
ejpam-5157	57	14	is	be	AUX
ejpam-5157	57	15	not	not	PART
ejpam-5157	57	16	a	a	DET
ejpam-5157	57	17	vertex	vertex	NOUN
ejpam-5157	57	18	cover	cover	NOUN
ejpam-5157	57	19	of	of	ADP
ejpam-5157	57	20	g	g	NOUN
ejpam-5157	57	21	since	since	SCONJ
ejpam-5157	57	22	edge	edge	NOUN
ejpam-5157	57	23	u6u7	u6u7	VERB
ejpam-5157	57	24	is	be	AUX
ejpam-5157	57	25	not	not	PART
ejpam-5157	57	26	incident	incident	NOUN
ejpam-5157	57	27	to	to	ADP
ejpam-5157	57	28	either	either	CCONJ
ejpam-5157	57	29	u4	u4	PROPN
ejpam-5157	57	30	or	or	CCONJ
ejpam-5157	57	31	u5	u5	PROPN
ejpam-5157	57	32	.	.	PUNCT
ejpam-5157	58	1	hence	hence	ADV
ejpam-5157	58	2	,	,	PUNCT
ejpam-5157	58	3	the	the	DET
ejpam-5157	58	4	assertion	assertion	NOUN
ejpam-5157	58	5	follows	follow	VERB
ejpam-5157	58	6	.	.	PUNCT
ejpam-5157	59	1	(	(	PUNCT
ejpam-5157	59	2	iv	iv	AUX
ejpam-5157	59	3	)	)	PUNCT
ejpam-5157	59	4	consider	consider	VERB
ejpam-5157	59	5	again	again	ADV
ejpam-5157	59	6	the	the	DET
ejpam-5157	59	7	graph	graph	NOUN
ejpam-5157	59	8	g	g	PROPN
ejpam-5157	59	9	in	in	ADP
ejpam-5157	59	10	(	(	PUNCT
ejpam-5157	59	11	iii	iii	NOUN
ejpam-5157	59	12	)	)	PUNCT
ejpam-5157	59	13	and	and	CCONJ
ejpam-5157	59	14	let	let	VERB
ejpam-5157	59	15	s	s	PRON
ejpam-5157	59	16	=	=	PUNCT
ejpam-5157	59	17	{	{	PUNCT
ejpam-5157	59	18	u4	u4	PROPN
ejpam-5157	59	19	,	,	PUNCT
ejpam-5157	59	20	u5	u5	PROPN
ejpam-5157	59	21	,	,	PUNCT
ejpam-5157	59	22	u6	u6	NOUN
ejpam-5157	59	23	}	}	PUNCT
ejpam-5157	59	24	.	.	PUNCT
ejpam-5157	60	1	then	then	ADV
ejpam-5157	60	2	every	every	DET
ejpam-5157	60	3	edge	edge	NOUN
ejpam-5157	60	4	of	of	ADP
ejpam-5157	60	5	g	g	PROPN
ejpam-5157	60	6	is	be	AUX
ejpam-5157	60	7	incident	incident	NOUN
ejpam-5157	60	8	to	to	ADP
ejpam-5157	60	9	some	some	DET
ejpam-5157	60	10	elements	element	NOUN
ejpam-5157	60	11	of	of	ADP
ejpam-5157	60	12	s	s	PRON
ejpam-5157	60	13	,	,	PUNCT
ejpam-5157	60	14	showing	show	VERB
ejpam-5157	60	15	that	that	SCONJ
ejpam-5157	60	16	s	s	VERB
ejpam-5157	60	17	is	be	AUX
ejpam-5157	60	18	a	a	DET
ejpam-5157	60	19	vertex	vertex	NOUN
ejpam-5157	60	20	cover	cover	NOUN
ejpam-5157	60	21	of	of	ADP
ejpam-5157	60	22	g.	g.	PROPN
ejpam-5157	60	23	now	now	ADV
ejpam-5157	60	24	,	,	PUNCT
ejpam-5157	60	25	observe	observe	VERB
ejpam-5157	60	26	that	that	SCONJ
ejpam-5157	60	27	vertex	vertex	NOUN
ejpam-5157	60	28	u5	u5	PROPN
ejpam-5157	60	29	and	and	CCONJ
ejpam-5157	60	30	u6	u6	NOUN
ejpam-5157	60	31	have	have	VERB
ejpam-5157	60	32	only	only	ADV
ejpam-5157	60	33	one	one	NUM
ejpam-5157	60	34	neighbor	neighbor	NOUN
ejpam-5157	60	35	u7	u7	PROPN
ejpam-5157	60	36	outside	outside	ADP
ejpam-5157	60	37	s.	s.	PROPN
ejpam-5157	60	38	it	it	PRON
ejpam-5157	60	39	follows	follow	VERB
ejpam-5157	60	40	that	that	SCONJ
ejpam-5157	60	41	s	s	VERB
ejpam-5157	60	42	is	be	AUX
ejpam-5157	60	43	not	not	PART
ejpam-5157	60	44	a	a	DET
ejpam-5157	60	45	certified	certified	ADJ
ejpam-5157	60	46	set	set	NOUN
ejpam-5157	60	47	of	of	ADP
ejpam-5157	60	48	g.	g.	PROPN
ejpam-5157	60	49	hence	hence	ADV
ejpam-5157	60	50	,	,	PUNCT
ejpam-5157	60	51	(	(	PUNCT
ejpam-5157	60	52	iv	iv	X
ejpam-5157	60	53	)	)	PUNCT
ejpam-5157	60	54	holds	hold	NOUN
ejpam-5157	60	55	.	.	PUNCT
ejpam-5157	61	1	(	(	PUNCT
ejpam-5157	61	2	v	v	NOUN
ejpam-5157	61	3	)	)	PUNCT
ejpam-5157	61	4	let	let	VERB
ejpam-5157	61	5	q	q	NOUN
ejpam-5157	61	6	be	be	AUX
ejpam-5157	61	7	both	both	PRON
ejpam-5157	61	8	certified	certified	ADJ
ejpam-5157	61	9	and	and	CCONJ
ejpam-5157	61	10	minimum	minimum	ADJ
ejpam-5157	61	11	vertex	vertex	NOUN
ejpam-5157	61	12	cover	cover	NOUN
ejpam-5157	61	13	of	of	ADP
ejpam-5157	61	14	g.	g.	PROPN
ejpam-5157	61	15	suppose	suppose	VERB
ejpam-5157	61	16	that	that	SCONJ
ejpam-5157	61	17	q	q	NOUN
ejpam-5157	61	18	is	be	AUX
ejpam-5157	61	19	not	not	PART
ejpam-5157	61	20	a	a	DET
ejpam-5157	61	21	minimum	minimum	ADJ
ejpam-5157	61	22	certified	certify	VERB
ejpam-5157	61	23	vertex	vertex	NOUN
ejpam-5157	61	24	cover	cover	NOUN
ejpam-5157	61	25	of	of	ADP
ejpam-5157	61	26	g.	g.	PROPN
ejpam-5157	61	27	then	then	ADV
ejpam-5157	61	28	there	there	PRON
ejpam-5157	61	29	exists	exist	VERB
ejpam-5157	61	30	t	t	PROPN
ejpam-5157	61	31	⊂	⊂	PROPN
ejpam-5157	61	32	v	v	X
ejpam-5157	61	33	(	(	PUNCT
ejpam-5157	61	34	g	g	NOUN
ejpam-5157	61	35	)	)	PUNCT
ejpam-5157	61	36	such	such	ADJ
ejpam-5157	61	37	that	that	SCONJ
ejpam-5157	61	38	t	t	PROPN
ejpam-5157	61	39	is	be	AUX
ejpam-5157	61	40	a	a	DET
ejpam-5157	61	41	certified	certify	VERB
ejpam-5157	61	42	vertex	vertex	NOUN
ejpam-5157	61	43	cover	cover	NOUN
ejpam-5157	61	44	and	and	CCONJ
ejpam-5157	61	45	|q|	|q|	VERB
ejpam-5157	61	46	>	>	X
ejpam-5157	61	47	|t	|t	NOUN
ejpam-5157	61	48	|	|	NOUN
ejpam-5157	61	49	.	.	PUNCT
ejpam-5157	62	1	note	note	VERB
ejpam-5157	62	2	that	that	SCONJ
ejpam-5157	62	3	t	t	PROPN
ejpam-5157	62	4	is	be	AUX
ejpam-5157	62	5	a	a	DET
ejpam-5157	62	6	vertex	vertex	NOUN
ejpam-5157	62	7	cover	cover	NOUN
ejpam-5157	62	8	of	of	ADP
ejpam-5157	62	9	g	g	NOUN
ejpam-5157	62	10	,	,	PUNCT
ejpam-5157	62	11	a	a	DET
ejpam-5157	62	12	contradiction	contradiction	NOUN
ejpam-5157	62	13	to	to	ADP
ejpam-5157	62	14	the	the	DET
ejpam-5157	62	15	fact	fact	NOUN
ejpam-5157	62	16	q	q	NOUN
ejpam-5157	62	17	is	be	AUX
ejpam-5157	62	18	a	a	DET
ejpam-5157	62	19	minimum	minimum	ADJ
ejpam-5157	62	20	vertex	vertex	NOUN
ejpam-5157	62	21	cover	cover	NOUN
ejpam-5157	62	22	of	of	ADP
ejpam-5157	62	23	g.	g.	PROPN
ejpam-5157	62	24	therefore	therefore	ADV
ejpam-5157	62	25	,	,	PUNCT
ejpam-5157	62	26	q	q	PROPN
ejpam-5157	62	27	is	be	AUX
ejpam-5157	62	28	a	a	DET
ejpam-5157	62	29	minimum	minimum	ADJ
ejpam-5157	62	30	certified	certify	VERB
ejpam-5157	62	31	vertex	vertex	NOUN
ejpam-5157	62	32	cove	cove	NOUN
ejpam-5157	62	33	of	of	ADP
ejpam-5157	62	34	g	g	PROPN
ejpam-5157	62	35	,	,	PUNCT
ejpam-5157	62	36	and	and	CCONJ
ejpam-5157	62	37	so	so	ADV
ejpam-5157	62	38	βcer(g	βcer(g	ADJ
ejpam-5157	62	39	)	)	PUNCT
ejpam-5157	62	40	=	=	SYM
ejpam-5157	63	1	|q|	|q|	PROPN
ejpam-5157	63	2	.	.	PUNCT
ejpam-5157	63	3	lemma	lemma	PROPN
ejpam-5157	63	4	1	1	NUM
ejpam-5157	63	5	.	.	PUNCT
ejpam-5157	64	1	let	let	VERB
ejpam-5157	64	2	n	n	PRON
ejpam-5157	64	3	be	be	AUX
ejpam-5157	64	4	a	a	DET
ejpam-5157	64	5	positive	positive	ADJ
ejpam-5157	64	6	integer	integer	NOUN
ejpam-5157	64	7	and	and	CCONJ
ejpam-5157	64	8	let	let	VERB
ejpam-5157	64	9	q	q	PUNCT
ejpam-5157	64	10	be	be	AUX
ejpam-5157	64	11	a	a	DET
ejpam-5157	64	12	certified	certify	VERB
ejpam-5157	64	13	vertex	vertex	NOUN
ejpam-5157	64	14	cover	cover	NOUN
ejpam-5157	64	15	of	of	ADP
ejpam-5157	64	16	pn	pn	NOUN
ejpam-5157	64	17	,	,	PUNCT
ejpam-5157	64	18	where	where	SCONJ
ejpam-5157	64	19	v	v	X
ejpam-5157	64	20	(	(	PUNCT
ejpam-5157	64	21	pn	pn	NOUN
ejpam-5157	64	22	)	)	PUNCT
ejpam-5157	64	23	=	=	SYM
ejpam-5157	64	24	{	{	PUNCT
ejpam-5157	64	25	v1	v1	PROPN
ejpam-5157	64	26	,	,	PUNCT
ejpam-5157	64	27	v2	v2	PROPN
ejpam-5157	64	28	,	,	PUNCT
ejpam-5157	64	29	...	...	PUNCT
ejpam-5157	64	30	,	,	PUNCT
ejpam-5157	64	31	vn	vn	PROPN
ejpam-5157	64	32	}	}	PUNCT
ejpam-5157	64	33	.	.	PUNCT
ejpam-5157	65	1	then	then	ADV
ejpam-5157	65	2	q	q	X
ejpam-5157	65	3	=	=	SYM
ejpam-5157	65	4	v	v	X
ejpam-5157	65	5	(	(	PUNCT
ejpam-5157	65	6	pn	pn	NOUN
ejpam-5157	65	7	)	)	PUNCT
ejpam-5157	65	8	if	if	SCONJ
ejpam-5157	65	9	and	and	CCONJ
ejpam-5157	65	10	only	only	ADV
ejpam-5157	65	11	if	if	SCONJ
ejpam-5157	65	12	one	one	NUM
ejpam-5157	65	13	of	of	ADP
ejpam-5157	65	14	the	the	DET
ejpam-5157	65	15	following	follow	VERB
ejpam-5157	65	16	holds	hold	VERB
ejpam-5157	65	17	:	:	PUNCT
ejpam-5157	65	18	(	(	PUNCT
ejpam-5157	65	19	i	i	NOUN
ejpam-5157	65	20	)	)	PUNCT
ejpam-5157	65	21	vi	vi	PROPN
ejpam-5157	65	22	,	,	PUNCT
ejpam-5157	65	23	vj	vj	X
ejpam-5157	65	24	∈	∈	PROPN
ejpam-5157	65	25	q	q	X
ejpam-5157	65	26	where	where	SCONJ
ejpam-5157	65	27	dpn(vi	dpn(vi	NOUN
ejpam-5157	65	28	,	,	PUNCT
ejpam-5157	65	29	vj	vj	ADJ
ejpam-5157	65	30	)	)	PUNCT
ejpam-5157	65	31	=	=	SYM
ejpam-5157	65	32	1	1	NUM
ejpam-5157	65	33	for	for	ADP
ejpam-5157	65	34	some	some	DET
ejpam-5157	65	35	i	i	PROPN
ejpam-5157	65	36	,	,	PUNCT
ejpam-5157	65	37	j	j	PROPN
ejpam-5157	65	38	∈	∈	PROPN
ejpam-5157	65	39	{	{	PUNCT
ejpam-5157	65	40	1	1	NUM
ejpam-5157	65	41	,	,	PUNCT
ejpam-5157	65	42	2	2	NUM
ejpam-5157	65	43	,	,	PUNCT
ejpam-5157	65	44	.	.	PUNCT
ejpam-5157	65	45	.	.	PUNCT
ejpam-5157	65	46	.	.	PUNCT
ejpam-5157	65	47	,	,	PUNCT
ejpam-5157	65	48	n	n	CCONJ
ejpam-5157	65	49	}	}	PUNCT
ejpam-5157	65	50	.	.	PUNCT
ejpam-5157	66	1	(	(	PUNCT
ejpam-5157	66	2	ii	ii	NOUN
ejpam-5157	66	3	)	)	PUNCT
ejpam-5157	66	4	v1	v1	PROPN
ejpam-5157	66	5	∈	∈	PROPN
ejpam-5157	66	6	q.	q.	PROPN
ejpam-5157	66	7	(	(	PUNCT
ejpam-5157	66	8	iii	iii	NOUN
ejpam-5157	66	9	)	)	PUNCT
ejpam-5157	66	10	vn	vn	PROPN
ejpam-5157	66	11	∈	∈	PROPN
ejpam-5157	66	12	q.	q.	PROPN
ejpam-5157	66	13	proof	proof	NOUN
ejpam-5157	66	14	.	.	PUNCT
ejpam-5157	67	1	suppose	suppose	VERB
ejpam-5157	67	2	that	that	PRON
ejpam-5157	67	3	q	q	NOUN
ejpam-5157	67	4	is	be	AUX
ejpam-5157	67	5	a	a	DET
ejpam-5157	67	6	certified	certify	VERB
ejpam-5157	67	7	vertex	vertex	NOUN
ejpam-5157	67	8	cover	cover	NOUN
ejpam-5157	67	9	of	of	ADP
ejpam-5157	67	10	pn	pn	PROPN
ejpam-5157	67	11	.	.	PROPN
ejpam-5157	67	12	ifq	ifq	PROPN
ejpam-5157	67	13	=	=	SYM
ejpam-5157	67	14	v	v	NOUN
ejpam-5157	67	15	(	(	PUNCT
ejpam-5157	67	16	pn	pn	NOUN
ejpam-5157	67	17	)	)	PUNCT
ejpam-5157	67	18	=	=	SYM
ejpam-5157	67	19	{	{	PUNCT
ejpam-5157	67	20	v1	v1	PROPN
ejpam-5157	67	21	,	,	PUNCT
ejpam-5157	67	22	v2	v2	PROPN
ejpam-5157	67	23	,	,	PUNCT
ejpam-5157	67	24	.	.	PUNCT
ejpam-5157	67	25	.	.	PUNCT
ejpam-5157	68	1	.	.	PUNCT
ejpam-5157	69	1	,	,	PUNCT
ejpam-5157	69	2	vn	vn	PROPN
ejpam-5157	69	3	}	}	PUNCT
ejpam-5157	69	4	,	,	PUNCT
ejpam-5157	69	5	then	then	ADV
ejpam-5157	69	6	(	(	PUNCT
ejpam-5157	69	7	i),(ii	i),(ii	PROPN
ejpam-5157	69	8	)	)	PUNCT
ejpam-5157	69	9	and	and	CCONJ
ejpam-5157	69	10	(	(	PUNCT
ejpam-5157	69	11	iii	iii	NOUN
ejpam-5157	69	12	)	)	PUNCT
ejpam-5157	69	13	follow	follow	NOUN
ejpam-5157	69	14	.	.	PUNCT
ejpam-5157	70	1	conversely	conversely	ADV
ejpam-5157	70	2	,	,	PUNCT
ejpam-5157	70	3	suppose	suppose	VERB
ejpam-5157	70	4	that	that	SCONJ
ejpam-5157	70	5	(	(	PUNCT
ejpam-5157	70	6	i	i	NOUN
ejpam-5157	70	7	)	)	PUNCT
ejpam-5157	70	8	holds	hold	VERB
ejpam-5157	70	9	.	.	PUNCT
ejpam-5157	71	1	that	that	PRON
ejpam-5157	71	2	is	be	AUX
ejpam-5157	71	3	,	,	PUNCT
ejpam-5157	71	4	vi	vi	PROPN
ejpam-5157	71	5	,	,	PUNCT
ejpam-5157	71	6	vj	vj	X
ejpam-5157	71	7	∈	∈	PROPN
ejpam-5157	71	8	q	q	NOUN
ejpam-5157	72	1	such	such	ADJ
ejpam-5157	72	2	that	that	SCONJ
ejpam-5157	72	3	dpn(vi	dpn(vi	NOUN
ejpam-5157	72	4	,	,	PUNCT
ejpam-5157	72	5	vj	vj	ADJ
ejpam-5157	72	6	)	)	PUNCT
ejpam-5157	72	7	=	=	SYM
ejpam-5157	72	8	1	1	NUM
ejpam-5157	72	9	for	for	ADP
ejpam-5157	72	10	some	some	DET
ejpam-5157	72	11	i	i	PROPN
ejpam-5157	72	12	,	,	PUNCT
ejpam-5157	72	13	j	j	PROPN
ejpam-5157	72	14	∈	∈	PROPN
ejpam-5157	72	15	{	{	PUNCT
ejpam-5157	72	16	1	1	NUM
ejpam-5157	72	17	,	,	PUNCT
ejpam-5157	72	18	2	2	NUM
ejpam-5157	72	19	,	,	PUNCT
ejpam-5157	72	20	.	.	PUNCT
ejpam-5157	72	21	.	.	PUNCT
ejpam-5157	72	22	.	.	PUNCT
ejpam-5157	72	23	,	,	PUNCT
ejpam-5157	72	24	n	n	CCONJ
ejpam-5157	72	25	}	}	PUNCT
ejpam-5157	72	26	.	.	PUNCT
ejpam-5157	73	1	assume	assume	VERB
ejpam-5157	73	2	that	that	SCONJ
ejpam-5157	74	1	i	i	PRON
ejpam-5157	74	2	<	<	X
ejpam-5157	74	3	j.	j.	PROPN
ejpam-5157	75	1	if	if	SCONJ
ejpam-5157	75	2	j	j	PROPN
ejpam-5157	75	3	=	=	SYM
ejpam-5157	75	4	n	n	CCONJ
ejpam-5157	75	5	,	,	PUNCT
ejpam-5157	75	6	then	then	ADV
ejpam-5157	75	7	vi	vi	PROPN
ejpam-5157	75	8	has	have	VERB
ejpam-5157	75	9	only	only	ADV
ejpam-5157	75	10	one	one	NUM
ejpam-5157	75	11	neighbor	neighbor	NOUN
ejpam-5157	75	12	vi−1	vi−1	PROPN
ejpam-5157	75	13	in	in	ADP
ejpam-5157	75	14	v	v	NOUN
ejpam-5157	75	15	(	(	PUNCT
ejpam-5157	75	16	pn)\q	pn)\q	PROPN
ejpam-5157	75	17	.	.	PUNCT
ejpam-5157	76	1	since	since	SCONJ
ejpam-5157	76	2	q	q	PROPN
ejpam-5157	76	3	is	be	AUX
ejpam-5157	76	4	a	a	DET
ejpam-5157	76	5	certified	certified	ADJ
ejpam-5157	76	6	set	set	NOUN
ejpam-5157	76	7	,	,	PUNCT
ejpam-5157	76	8	vi−1	vi−1	PROPN
ejpam-5157	76	9	must	must	AUX
ejpam-5157	76	10	be	be	AUX
ejpam-5157	76	11	in	in	ADP
ejpam-5157	76	12	q.	q.	PROPN
ejpam-5157	76	13	if	if	SCONJ
ejpam-5157	76	14	vi−1	vi−1	PROPN
ejpam-5157	76	15	is	be	AUX
ejpam-5157	76	16	in	in	ADP
ejpam-5157	76	17	q	q	NOUN
ejpam-5157	76	18	,	,	PUNCT
ejpam-5157	76	19	then	then	ADV
ejpam-5157	76	20	applying	apply	VERB
ejpam-5157	76	21	the	the	DET
ejpam-5157	76	22	same	same	ADJ
ejpam-5157	76	23	argument	argument	NOUN
ejpam-5157	76	24	,	,	PUNCT
ejpam-5157	76	25	vi−2	vi−2	PROPN
ejpam-5157	76	26	must	must	AUX
ejpam-5157	76	27	also	also	ADV
ejpam-5157	76	28	be	be	AUX
ejpam-5157	76	29	in	in	ADP
ejpam-5157	76	30	q.	q.	NOUN
ejpam-5157	76	31	continuing	continue	VERB
ejpam-5157	76	32	this	this	DET
ejpam-5157	76	33	process	process	NOUN
ejpam-5157	76	34	,	,	PUNCT
ejpam-5157	76	35	all	all	DET
ejpam-5157	76	36	other	other	ADJ
ejpam-5157	76	37	vertices	vertex	NOUN
ejpam-5157	76	38	in	in	ADP
ejpam-5157	76	39	v	v	NOUN
ejpam-5157	76	40	(	(	PUNCT
ejpam-5157	76	41	pn)\q	pn)\q	PROPN
ejpam-5157	76	42	must	must	AUX
ejpam-5157	76	43	be	be	AUX
ejpam-5157	76	44	in	in	ADP
ejpam-5157	76	45	q.	q.	NOUN
ejpam-5157	76	46	hence	hence	ADV
ejpam-5157	76	47	,	,	PUNCT
ejpam-5157	76	48	q	q	NOUN
ejpam-5157	76	49	=	=	SYM
ejpam-5157	76	50	v	v	NOUN
ejpam-5157	76	51	(	(	PUNCT
ejpam-5157	76	52	pn	pn	NOUN
ejpam-5157	76	53	)	)	PUNCT
ejpam-5157	76	54	.	.	PUNCT
ejpam-5157	77	1	suppose	suppose	VERB
ejpam-5157	77	2	that	that	SCONJ
ejpam-5157	77	3	j	j	PROPN
ejpam-5157	77	4	̸=	̸=	PROPN
ejpam-5157	77	5	n.	n.	NOUN
ejpam-5157	77	6	since	since	SCONJ
ejpam-5157	77	7	dpn(vi	dpn(vi	NOUN
ejpam-5157	77	8	,	,	PUNCT
ejpam-5157	77	9	vj	vj	ADJ
ejpam-5157	77	10	)	)	PUNCT
ejpam-5157	77	11	=	=	SYM
ejpam-5157	77	12	1	1	NUM
ejpam-5157	77	13	,	,	PUNCT
ejpam-5157	77	14	both	both	DET
ejpam-5157	77	15	vi	vi	NOUN
ejpam-5157	77	16	and	and	CCONJ
ejpam-5157	77	17	vj	vj	INTJ
ejpam-5157	77	18	have	have	VERB
ejpam-5157	77	19	only	only	ADV
ejpam-5157	77	20	one	one	NUM
ejpam-5157	77	21	neighbor	neighbor	NOUN
ejpam-5157	77	22	vi−1	vi−1	PROPN
ejpam-5157	77	23	and	and	CCONJ
ejpam-5157	77	24	vj+1	vj+1	PROPN
ejpam-5157	77	25	in	in	ADP
ejpam-5157	77	26	v	v	NUM
ejpam-5157	77	27	(	(	PUNCT
ejpam-5157	77	28	pn)\q	pn)\q	PROPN
ejpam-5157	77	29	,	,	PUNCT
ejpam-5157	77	30	respectively	respectively	ADV
ejpam-5157	77	31	.	.	PUNCT
ejpam-5157	78	1	applying	apply	VERB
ejpam-5157	78	2	the	the	DET
ejpam-5157	78	3	same	same	ADJ
ejpam-5157	78	4	argument	argument	NOUN
ejpam-5157	78	5	,	,	PUNCT
ejpam-5157	78	6	all	all	DET
ejpam-5157	78	7	other	other	ADJ
ejpam-5157	78	8	vertices	vertex	NOUN
ejpam-5157	78	9	in	in	ADP
ejpam-5157	78	10	v	v	NOUN
ejpam-5157	78	11	(	(	PUNCT
ejpam-5157	78	12	pn)\q	pn)\q	PROPN
ejpam-5157	78	13	must	must	AUX
ejpam-5157	78	14	also	also	ADV
ejpam-5157	78	15	be	be	AUX
ejpam-5157	78	16	in	in	ADP
ejpam-5157	78	17	q.	q.	PROPN
ejpam-5157	78	18	thus	thus	ADV
ejpam-5157	78	19	,	,	PUNCT
ejpam-5157	78	20	q	q	PROPN
ejpam-5157	78	21	=	=	SYM
ejpam-5157	78	22	v	v	NOUN
ejpam-5157	78	23	(	(	PUNCT
ejpam-5157	78	24	pn	pn	NOUN
ejpam-5157	78	25	)	)	PUNCT
ejpam-5157	78	26	.	.	PUNCT
ejpam-5157	79	1	similarly	similarly	ADV
ejpam-5157	79	2	,	,	PUNCT
ejpam-5157	79	3	the	the	DET
ejpam-5157	79	4	same	same	ADJ
ejpam-5157	79	5	result	result	NOUN
ejpam-5157	79	6	follows	follow	VERB
ejpam-5157	79	7	when	when	SCONJ
ejpam-5157	79	8	i	i	PRON
ejpam-5157	79	9	>	>	X
ejpam-5157	79	10	j.	j.	PROPN
ejpam-5157	79	11	now	now	ADV
ejpam-5157	79	12	,	,	PUNCT
ejpam-5157	79	13	suppose	suppose	VERB
ejpam-5157	79	14	that	that	SCONJ
ejpam-5157	79	15	(	(	PUNCT
ejpam-5157	79	16	ii	ii	NOUN
ejpam-5157	79	17	)	)	PUNCT
ejpam-5157	79	18	holds	hold	VERB
ejpam-5157	79	19	,	,	PUNCT
ejpam-5157	79	20	that	that	ADV
ejpam-5157	79	21	is	is	ADV
ejpam-5157	79	22	,	,	PUNCT
ejpam-5157	79	23	v1	v1	PROPN
ejpam-5157	79	24	∈	∈	PROPN
ejpam-5157	79	25	q.	q.	NOUN
ejpam-5157	79	26	observe	observe	VERB
ejpam-5157	79	27	that	that	SCONJ
ejpam-5157	79	28	v1	v1	NOUN
ejpam-5157	79	29	has	have	VERB
ejpam-5157	79	30	only	only	ADV
ejpam-5157	79	31	one	one	NUM
ejpam-5157	79	32	neighbor	neighbor	NOUN
ejpam-5157	79	33	in	in	ADP
ejpam-5157	79	34	v	v	PROPN
ejpam-5157	79	35	(	(	PUNCT
ejpam-5157	79	36	pn	pn	NOUN
ejpam-5157	79	37	)	)	PUNCT
ejpam-5157	79	38	\q	\q	NOUN
ejpam-5157	79	39	,	,	PUNCT
ejpam-5157	79	40	which	which	PRON
ejpam-5157	79	41	is	be	AUX
ejpam-5157	79	42	v2	v2	PROPN
ejpam-5157	79	43	.	.	PUNCT
ejpam-5157	80	1	since	since	SCONJ
ejpam-5157	80	2	q	q	PROPN
ejpam-5157	80	3	is	be	AUX
ejpam-5157	80	4	a	a	DET
ejpam-5157	80	5	certified	certify	VERB
ejpam-5157	80	6	set	set	NOUN
ejpam-5157	80	7	in	in	ADP
ejpam-5157	80	8	pn	pn	PROPN
ejpam-5157	80	9	,	,	PUNCT
ejpam-5157	80	10	v2	v2	PROPN
ejpam-5157	80	11	must	must	AUX
ejpam-5157	80	12	be	be	AUX
ejpam-5157	80	13	in	in	ADP
ejpam-5157	80	14	q.	q.	PROPN
ejpam-5157	80	15	however	however	ADV
ejpam-5157	80	16	,	,	PUNCT
ejpam-5157	80	17	v2	v2	PROPN
ejpam-5157	80	18	has	have	VERB
ejpam-5157	80	19	only	only	ADV
ejpam-5157	80	20	one	one	NUM
ejpam-5157	80	21	neighbor	neighbor	NOUN
ejpam-5157	80	22	in	in	ADP
ejpam-5157	80	23	v	v	PROPN
ejpam-5157	80	24	(	(	PUNCT
ejpam-5157	80	25	pn	pn	NOUN
ejpam-5157	80	26	)	)	PUNCT
ejpam-5157	80	27	\q	\q	NOUN
ejpam-5157	80	28	,	,	PUNCT
ejpam-5157	80	29	which	which	PRON
ejpam-5157	80	30	is	be	AUX
ejpam-5157	80	31	v3	v3	PROPN
ejpam-5157	80	32	.	.	PUNCT
ejpam-5157	81	1	since	since	SCONJ
ejpam-5157	81	2	q	q	PROPN
ejpam-5157	81	3	is	be	AUX
ejpam-5157	81	4	a	a	DET
ejpam-5157	81	5	certified	certify	VERB
ejpam-5157	81	6	set	set	NOUN
ejpam-5157	81	7	in	in	ADP
ejpam-5157	81	8	pn	pn	PROPN
ejpam-5157	81	9	,	,	PUNCT
ejpam-5157	81	10	v3	v3	PROPN
ejpam-5157	81	11	must	must	AUX
ejpam-5157	81	12	also	also	ADV
ejpam-5157	81	13	be	be	AUX
ejpam-5157	81	14	in	in	ADP
ejpam-5157	81	15	q.	q.	NOUN
ejpam-5157	81	16	continuing	continue	VERB
ejpam-5157	81	17	this	this	DET
ejpam-5157	81	18	process	process	NOUN
ejpam-5157	81	19	,	,	PUNCT
ejpam-5157	81	20	all	all	DET
ejpam-5157	81	21	other	other	ADJ
ejpam-5157	81	22	vertices	vertex	NOUN
ejpam-5157	81	23	in	in	ADP
ejpam-5157	81	24	v	v	NOUN
ejpam-5157	81	25	(	(	PUNCT
ejpam-5157	81	26	pn)\q	pn)\q	PROPN
ejpam-5157	81	27	must	must	AUX
ejpam-5157	81	28	be	be	AUX
ejpam-5157	81	29	included	include	VERB
ejpam-5157	81	30	as	as	ADP
ejpam-5157	81	31	elements	element	NOUN
ejpam-5157	81	32	of	of	ADP
ejpam-5157	81	33	q.	q.	PROPN
ejpam-5157	81	34	therefore	therefore	ADV
ejpam-5157	81	35	,	,	PUNCT
ejpam-5157	81	36	q	q	PROPN
ejpam-5157	81	37	=	=	SYM
ejpam-5157	81	38	v	v	NOUN
ejpam-5157	81	39	(	(	PUNCT
ejpam-5157	81	40	pn	pn	NOUN
ejpam-5157	81	41	)	)	PUNCT
ejpam-5157	81	42	.	.	PUNCT
ejpam-5157	82	1	similarly	similarly	ADV
ejpam-5157	82	2	,	,	PUNCT
ejpam-5157	82	3	the	the	DET
ejpam-5157	82	4	assertion	assertion	NOUN
ejpam-5157	82	5	follows	follow	VERB
ejpam-5157	82	6	when	when	SCONJ
ejpam-5157	82	7	(	(	PUNCT
ejpam-5157	82	8	iii	iii	NOUN
ejpam-5157	82	9	)	)	PUNCT
ejpam-5157	82	10	holds	hold	VERB
ejpam-5157	82	11	.	.	PUNCT
ejpam-5157	83	1	the	the	DET
ejpam-5157	83	2	following	follow	VERB
ejpam-5157	83	3	lemma	lemma	PROPN
ejpam-5157	83	4	can	can	AUX
ejpam-5157	83	5	be	be	AUX
ejpam-5157	83	6	proved	prove	VERB
ejpam-5157	83	7	similarly	similarly	ADV
ejpam-5157	83	8	.	.	PUNCT
ejpam-5157	84	1	j.	j.	PROPN
ejpam-5157	84	2	a.	a.	PROPN
ejpam-5157	84	3	hassan	hassan	PROPN
ejpam-5157	84	4	et	et	PROPN
ejpam-5157	84	5	al	al	PROPN
ejpam-5157	84	6	.	.	PUNCT
ejpam-5157	84	7	/	/	SYM
ejpam-5157	84	8	eur	eur	PROPN
ejpam-5157	84	9	.	.	PUNCT
ejpam-5157	85	1	j.	j.	PROPN
ejpam-5157	85	2	pure	pure	PROPN
ejpam-5157	85	3	appl	appl	PROPN
ejpam-5157	85	4	.	.	PROPN
ejpam-5157	85	5	math	math	PROPN
ejpam-5157	85	6	,	,	PUNCT
ejpam-5157	85	7	17	17	NUM
ejpam-5157	85	8	(	(	PUNCT
ejpam-5157	85	9	2	2	NUM
ejpam-5157	85	10	)	)	PUNCT
ejpam-5157	85	11	(	(	PUNCT
ejpam-5157	85	12	2024	2024	NUM
ejpam-5157	85	13	)	)	PUNCT
ejpam-5157	85	14	,	,	PUNCT
ejpam-5157	85	15	1038	1038	NUM
ejpam-5157	85	16	-	-	SYM
ejpam-5157	85	17	1045	1045	NUM
ejpam-5157	85	18	1042	1042	NUM
ejpam-5157	85	19	lemma	lemma	PROPN
ejpam-5157	85	20	2	2	NUM
ejpam-5157	85	21	.	.	PUNCT
ejpam-5157	86	1	let	let	VERB
ejpam-5157	86	2	n	n	PRON
ejpam-5157	86	3	be	be	AUX
ejpam-5157	86	4	a	a	DET
ejpam-5157	86	5	positive	positive	ADJ
ejpam-5157	86	6	integer	integer	NOUN
ejpam-5157	86	7	and	and	CCONJ
ejpam-5157	86	8	let	let	VERB
ejpam-5157	86	9	q	q	PUNCT
ejpam-5157	86	10	be	be	AUX
ejpam-5157	86	11	a	a	DET
ejpam-5157	86	12	certified	certify	VERB
ejpam-5157	86	13	vertex	vertex	NOUN
ejpam-5157	86	14	cover	cover	NOUN
ejpam-5157	86	15	of	of	ADP
ejpam-5157	86	16	cn	cn	PROPN
ejpam-5157	86	17	,	,	PUNCT
ejpam-5157	86	18	where	where	SCONJ
ejpam-5157	86	19	v	v	X
ejpam-5157	86	20	(	(	PUNCT
ejpam-5157	86	21	cn	cn	PROPN
ejpam-5157	86	22	)	)	PUNCT
ejpam-5157	86	23	=	=	SYM
ejpam-5157	86	24	{	{	PUNCT
ejpam-5157	86	25	v1	v1	PROPN
ejpam-5157	86	26	,	,	PUNCT
ejpam-5157	86	27	v2	v2	PROPN
ejpam-5157	86	28	,	,	PUNCT
ejpam-5157	86	29	.	.	PUNCT
ejpam-5157	86	30	.	.	PUNCT
ejpam-5157	87	1	.	.	PUNCT
ejpam-5157	88	1	,	,	PUNCT
ejpam-5157	88	2	vn	vn	PROPN
ejpam-5157	88	3	}	}	PUNCT
ejpam-5157	88	4	.	.	PUNCT
ejpam-5157	89	1	then	then	ADV
ejpam-5157	89	2	q	q	X
ejpam-5157	89	3	=	=	SYM
ejpam-5157	89	4	v	v	X
ejpam-5157	89	5	(	(	PUNCT
ejpam-5157	89	6	cn	cn	PROPN
ejpam-5157	89	7	)	)	PUNCT
ejpam-5157	89	8	if	if	SCONJ
ejpam-5157	89	9	and	and	CCONJ
ejpam-5157	89	10	only	only	ADV
ejpam-5157	89	11	if	if	SCONJ
ejpam-5157	89	12	vi	vi	PROPN
ejpam-5157	89	13	,	,	PUNCT
ejpam-5157	89	14	vj	vj	X
ejpam-5157	89	15	∈	∈	PROPN
ejpam-5157	89	16	q	q	NOUN
ejpam-5157	89	17	such	such	ADJ
ejpam-5157	89	18	that	that	SCONJ
ejpam-5157	89	19	dcn(vi	dcn(vi	NOUN
ejpam-5157	89	20	,	,	PUNCT
ejpam-5157	89	21	vj	vj	INTJ
ejpam-5157	89	22	)	)	PUNCT
ejpam-5157	89	23	=	=	SYM
ejpam-5157	89	24	1	1	X
ejpam-5157	89	25	.	.	X
ejpam-5157	89	26	theorem	theorem	NOUN
ejpam-5157	89	27	1	1	NUM
ejpam-5157	89	28	.	.	PUNCT
ejpam-5157	90	1	let	let	VERB
ejpam-5157	90	2	n	n	PRON
ejpam-5157	90	3	be	be	AUX
ejpam-5157	90	4	a	a	DET
ejpam-5157	90	5	positive	positive	ADJ
ejpam-5157	90	6	integer	integer	NOUN
ejpam-5157	90	7	.	.	PUNCT
ejpam-5157	91	1	then	then	ADV
ejpam-5157	91	2	βcer(pn	βcer(pn	NOUN
ejpam-5157	91	3	)	)	PUNCT
ejpam-5157	91	4	=	=	SYM
ejpam-5157	91	5	{	{	PUNCT
ejpam-5157	92	1	n	n	NOUN
ejpam-5157	92	2	,	,	PUNCT
ejpam-5157	92	3	n	n	NOUN
ejpam-5157	92	4	=	=	SYM
ejpam-5157	92	5	1	1	NUM
ejpam-5157	92	6	or	or	CCONJ
ejpam-5157	92	7	even⌊	even⌊	VERB
ejpam-5157	92	8	n	n	DET
ejpam-5157	92	9	2	2	NUM
ejpam-5157	92	10	⌋	⌋	NOUN
ejpam-5157	92	11	,	,	PUNCT
ejpam-5157	92	12	otherwise	otherwise	ADV
ejpam-5157	92	13	.	.	PUNCT
ejpam-5157	93	1	proof	proof	NOUN
ejpam-5157	93	2	.	.	PUNCT
ejpam-5157	94	1	let	let	VERB
ejpam-5157	94	2	v	v	X
ejpam-5157	94	3	(	(	PUNCT
ejpam-5157	94	4	pn	pn	NOUN
ejpam-5157	94	5	)	)	PUNCT
ejpam-5157	94	6	=	=	SYM
ejpam-5157	94	7	{	{	PUNCT
ejpam-5157	94	8	v1	v1	PROPN
ejpam-5157	94	9	,	,	PUNCT
ejpam-5157	94	10	v2	v2	PROPN
ejpam-5157	94	11	,	,	PUNCT
ejpam-5157	94	12	.	.	PUNCT
ejpam-5157	94	13	.	.	PUNCT
ejpam-5157	95	1	.	.	PUNCT
ejpam-5157	96	1	,	,	PUNCT
ejpam-5157	96	2	vn	vn	PROPN
ejpam-5157	96	3	}	}	PUNCT
ejpam-5157	96	4	.	.	PUNCT
ejpam-5157	97	1	clearly	clearly	ADV
ejpam-5157	97	2	,	,	PUNCT
ejpam-5157	97	3	βcer(p1	βcer(p1	NOUN
ejpam-5157	97	4	)	)	PUNCT
ejpam-5157	97	5	=	=	SYM
ejpam-5157	97	6	1	1	NUM
ejpam-5157	97	7	,	,	PUNCT
ejpam-5157	97	8	βcer(p2	βcer(p2	ADJ
ejpam-5157	97	9	)	)	PUNCT
ejpam-5157	97	10	=	=	SYM
ejpam-5157	97	11	2	2	NUM
ejpam-5157	97	12	and	and	CCONJ
ejpam-5157	97	13	βcer(p4	βcer(p4	NOUN
ejpam-5157	97	14	)	)	PUNCT
ejpam-5157	97	15	=	=	SYM
ejpam-5157	98	1	4	4	X
ejpam-5157	98	2	.	.	PUNCT
ejpam-5157	98	3	suppose	suppose	VERB
ejpam-5157	98	4	that	that	SCONJ
ejpam-5157	98	5	n	n	PROPN
ejpam-5157	98	6	≥	≥	NOUN
ejpam-5157	98	7	6	6	NUM
ejpam-5157	98	8	and	and	CCONJ
ejpam-5157	98	9	even	even	ADV
ejpam-5157	98	10	.	.	PUNCT
ejpam-5157	99	1	let	let	VERB
ejpam-5157	99	2	q	q	PRON
ejpam-5157	99	3	be	be	AUX
ejpam-5157	99	4	a	a	DET
ejpam-5157	99	5	certified	certify	VERB
ejpam-5157	99	6	vertex	vertex	NOUN
ejpam-5157	99	7	cover	cover	NOUN
ejpam-5157	99	8	of	of	ADP
ejpam-5157	99	9	pn	pn	PROPN
ejpam-5157	99	10	.	.	PUNCT
ejpam-5157	100	1	if	if	SCONJ
ejpam-5157	100	2	v1	v1	PROPN
ejpam-5157	100	3	,	,	PUNCT
ejpam-5157	100	4	vn	vn	PROPN
ejpam-5157	100	5	∈	∈	PROPN
ejpam-5157	100	6	q	q	PROPN
ejpam-5157	100	7	or	or	CCONJ
ejpam-5157	100	8	vi	vi	NOUN
ejpam-5157	100	9	,	,	PUNCT
ejpam-5157	100	10	vj	vj	X
ejpam-5157	100	11	∈	∈	PROPN
ejpam-5157	101	1	q	q	NOUN
ejpam-5157	101	2	,	,	PUNCT
ejpam-5157	101	3	where	where	SCONJ
ejpam-5157	101	4	dpn(vi	dpn(vi	NOUN
ejpam-5157	101	5	,	,	PUNCT
ejpam-5157	101	6	vj	vj	ADJ
ejpam-5157	101	7	)	)	PUNCT
ejpam-5157	101	8	=	=	SYM
ejpam-5157	101	9	1	1	NUM
ejpam-5157	101	10	,	,	PUNCT
ejpam-5157	101	11	then	then	ADV
ejpam-5157	101	12	q	q	NOUN
ejpam-5157	101	13	=	=	SYM
ejpam-5157	101	14	v	v	X
ejpam-5157	101	15	(	(	PUNCT
ejpam-5157	101	16	pn	pn	NOUN
ejpam-5157	101	17	)	)	PUNCT
ejpam-5157	101	18	by	by	ADP
ejpam-5157	101	19	lemma	lemma	PROPN
ejpam-5157	101	20	1	1	NUM
ejpam-5157	101	21	,	,	PUNCT
ejpam-5157	101	22	and	and	CCONJ
ejpam-5157	101	23	the	the	DET
ejpam-5157	101	24	proof	proof	NOUN
ejpam-5157	101	25	is	be	AUX
ejpam-5157	101	26	complete	complete	ADJ
ejpam-5157	101	27	.	.	PUNCT
ejpam-5157	102	1	now	now	ADV
ejpam-5157	102	2	,	,	PUNCT
ejpam-5157	102	3	suppose	suppose	VERB
ejpam-5157	102	4	that	that	SCONJ
ejpam-5157	102	5	v1	v1	NOUN
ejpam-5157	102	6	,	,	PUNCT
ejpam-5157	102	7	vn	vn	PROPN
ejpam-5157	102	8	/∈	/∈	PUNCT
ejpam-5157	102	9	q	q	PROPN
ejpam-5157	102	10	and	and	CCONJ
ejpam-5157	102	11	dpn(vs	dpn(vs	PROPN
ejpam-5157	102	12	,	,	PUNCT
ejpam-5157	102	13	vt	vt	PROPN
ejpam-5157	102	14	)	)	PUNCT
ejpam-5157	102	15	≥	≥	NOUN
ejpam-5157	102	16	2	2	NUM
ejpam-5157	102	17	for	for	ADP
ejpam-5157	102	18	every	every	DET
ejpam-5157	102	19	s	s	PROPN
ejpam-5157	102	20	,	,	PUNCT
ejpam-5157	102	21	t	t	PROPN
ejpam-5157	102	22	∈	∈	PROPN
ejpam-5157	102	23	{	{	PUNCT
ejpam-5157	102	24	2	2	NUM
ejpam-5157	102	25	,	,	PUNCT
ejpam-5157	102	26	.	.	PUNCT
ejpam-5157	102	27	.	.	PUNCT
ejpam-5157	103	1	.	.	PUNCT
ejpam-5157	104	1	,	,	PUNCT
ejpam-5157	104	2	n–1	n–1	NOUN
ejpam-5157	104	3	}	}	PUNCT
ejpam-5157	104	4	.	.	PUNCT
ejpam-5157	105	1	since	since	SCONJ
ejpam-5157	105	2	q	q	PROPN
ejpam-5157	105	3	is	be	AUX
ejpam-5157	105	4	a	a	DET
ejpam-5157	105	5	vertex	vertex	NOUN
ejpam-5157	105	6	cover	cover	NOUN
ejpam-5157	105	7	of	of	ADP
ejpam-5157	105	8	pn	pn	PROPN
ejpam-5157	105	9	,	,	PUNCT
ejpam-5157	105	10	it	it	PRON
ejpam-5157	105	11	follows	follow	VERB
ejpam-5157	105	12	that	that	SCONJ
ejpam-5157	105	13	either	either	CCONJ
ejpam-5157	105	14	{	{	PUNCT
ejpam-5157	105	15	v2	v2	PROPN
ejpam-5157	105	16	,	,	PUNCT
ejpam-5157	105	17	v4	v4	NOUN
ejpam-5157	105	18	,	,	PUNCT
ejpam-5157	105	19	.	.	PUNCT
ejpam-5157	105	20	.	.	PUNCT
ejpam-5157	106	1	.	.	PUNCT
ejpam-5157	107	1	,	,	PUNCT
ejpam-5157	107	2	vn−2	vn−2	PROPN
ejpam-5157	107	3	,	,	PUNCT
ejpam-5157	107	4	vn−1	vn−1	ADJ
ejpam-5157	107	5	}	}	PUNCT
ejpam-5157	107	6	⊆	⊆	NUM
ejpam-5157	107	7	q	q	NOUN
ejpam-5157	107	8	or	or	CCONJ
ejpam-5157	107	9	{	{	PUNCT
ejpam-5157	107	10	v2	v2	PROPN
ejpam-5157	107	11	,	,	PUNCT
ejpam-5157	107	12	v4	v4	NOUN
ejpam-5157	107	13	,	,	PUNCT
ejpam-5157	107	14	.	.	PUNCT
ejpam-5157	107	15	.	.	PUNCT
ejpam-5157	108	1	.	.	PUNCT
ejpam-5157	109	1	,	,	PUNCT
ejpam-5157	109	2	vn−2	vn−2	PROPN
ejpam-5157	109	3	,	,	PUNCT
ejpam-5157	109	4	vn	vn	VERB
ejpam-5157	109	5	}	}	PUNCT
ejpam-5157	109	6	⊆	⊆	NUM
ejpam-5157	109	7	q	q	NOUN
ejpam-5157	109	8	.	.	PUNCT
ejpam-5157	110	1	by	by	ADP
ejpam-5157	110	2	lemma	lemma	PROPN
ejpam-5157	110	3	1	1	NUM
ejpam-5157	110	4	,	,	PUNCT
ejpam-5157	110	5	either	either	PRON
ejpam-5157	110	6	of	of	ADP
ejpam-5157	110	7	this	this	DET
ejpam-5157	110	8	case	case	NOUN
ejpam-5157	110	9	,	,	PUNCT
ejpam-5157	110	10	we	we	PRON
ejpam-5157	110	11	have	have	VERB
ejpam-5157	110	12	q	q	NOUN
ejpam-5157	110	13	=	=	SYM
ejpam-5157	110	14	v	v	NOUN
ejpam-5157	110	15	(	(	PUNCT
ejpam-5157	110	16	pn	pn	NOUN
ejpam-5157	110	17	)	)	PUNCT
ejpam-5157	110	18	.	.	PUNCT
ejpam-5157	111	1	therefore	therefore	ADV
ejpam-5157	111	2	,	,	PUNCT
ejpam-5157	111	3	βcer(pn	βcer(pn	NOUN
ejpam-5157	111	4	)	)	PUNCT
ejpam-5157	111	5	=	=	SYM
ejpam-5157	111	6	n	n	PROPN
ejpam-5157	111	7	for	for	ADP
ejpam-5157	111	8	all	all	DET
ejpam-5157	111	9	n	n	PRON
ejpam-5157	111	10	≥	≥	NOUN
ejpam-5157	111	11	6	6	NUM
ejpam-5157	111	12	and	and	CCONJ
ejpam-5157	111	13	even	even	ADV
ejpam-5157	111	14	.	.	PUNCT
ejpam-5157	112	1	next	next	ADV
ejpam-5157	112	2	,	,	PUNCT
ejpam-5157	112	3	suppose	suppose	VERB
ejpam-5157	112	4	that	that	SCONJ
ejpam-5157	112	5	n	n	PROPN
ejpam-5157	112	6	≥	≥	NUM
ejpam-5157	112	7	5	5	NUM
ejpam-5157	112	8	and	and	CCONJ
ejpam-5157	112	9	odd	odd	ADJ
ejpam-5157	112	10	.	.	PUNCT
ejpam-5157	113	1	let	let	VERB
ejpam-5157	113	2	s	s	VERB
ejpam-5157	113	3	=	=	PUNCT
ejpam-5157	113	4	{	{	PUNCT
ejpam-5157	113	5	v2	v2	PROPN
ejpam-5157	113	6	,	,	PUNCT
ejpam-5157	113	7	v4	v4	NOUN
ejpam-5157	113	8	,	,	PUNCT
ejpam-5157	113	9	.	.	PUNCT
ejpam-5157	113	10	.	.	PUNCT
ejpam-5157	114	1	.	.	PUNCT
ejpam-5157	115	1	,	,	PUNCT
ejpam-5157	115	2	vn−1	vn−1	ADJ
ejpam-5157	115	3	}	}	PUNCT
ejpam-5157	115	4	.	.	PUNCT
ejpam-5157	116	1	then	then	ADV
ejpam-5157	116	2	s	s	VERB
ejpam-5157	116	3	is	be	AUX
ejpam-5157	116	4	a	a	DET
ejpam-5157	116	5	minimum	minimum	ADJ
ejpam-5157	116	6	vertex	vertex	NOUN
ejpam-5157	116	7	cover	cover	NOUN
ejpam-5157	116	8	of	of	ADP
ejpam-5157	116	9	pn	pn	PROPN
ejpam-5157	116	10	.	.	PUNCT
ejpam-5157	116	11	clearly	clearly	ADV
ejpam-5157	116	12	,	,	PUNCT
ejpam-5157	116	13	s	s	VERB
ejpam-5157	116	14	is	be	AUX
ejpam-5157	116	15	a	a	DET
ejpam-5157	116	16	certified	certified	ADJ
ejpam-5157	116	17	set	set	NOUN
ejpam-5157	116	18	of	of	ADP
ejpam-5157	116	19	pn	pn	PROPN
ejpam-5157	116	20	.	.	PROPN
ejpam-5157	117	1	therefore	therefore	ADV
ejpam-5157	117	2	,	,	PUNCT
ejpam-5157	117	3	βcer(pn	βcer(pn	NOUN
ejpam-5157	117	4	)	)	PUNCT
ejpam-5157	117	5	=	=	PUNCT
ejpam-5157	117	6	|s|	|s|	NOUN
ejpam-5157	117	7	=	=	SYM
ejpam-5157	117	8	⌊	⌊	PROPN
ejpam-5157	117	9	n	n	ADV
ejpam-5157	117	10	2	2	NUM
ejpam-5157	117	11	⌋	⌋	NOUN
ejpam-5157	117	12	since	since	SCONJ
ejpam-5157	117	13	n	n	NUM
ejpam-5157	117	14	is	be	AUX
ejpam-5157	117	15	odd	odd	ADJ
ejpam-5157	117	16	.	.	PUNCT
ejpam-5157	118	1	theorem	theorem	NOUN
ejpam-5157	118	2	2	2	NUM
ejpam-5157	118	3	.	.	PUNCT
ejpam-5157	119	1	let	let	VERB
ejpam-5157	119	2	n	n	PRON
ejpam-5157	119	3	be	be	AUX
ejpam-5157	119	4	a	a	DET
ejpam-5157	119	5	positive	positive	ADJ
ejpam-5157	119	6	integer	integer	NOUN
ejpam-5157	119	7	.	.	PUNCT
ejpam-5157	120	1	then	then	ADV
ejpam-5157	120	2	βcer(cn	βcer(cn	ADJ
ejpam-5157	120	3	)	)	PUNCT
ejpam-5157	120	4	=	=	SYM
ejpam-5157	120	5	{	{	PUNCT
ejpam-5157	121	1	n	n	CCONJ
ejpam-5157	121	2	,	,	PUNCT
ejpam-5157	121	3	if	if	SCONJ
ejpam-5157	121	4	n	n	PRON
ejpam-5157	121	5	is	be	AUX
ejpam-5157	121	6	odd	odd	ADJ
ejpam-5157	121	7	n	n	PRON
ejpam-5157	121	8	2	2	NUM
ejpam-5157	121	9	,	,	PUNCT
ejpam-5157	121	10	if	if	SCONJ
ejpam-5157	121	11	n	n	PRON
ejpam-5157	121	12	is	be	AUX
ejpam-5157	121	13	even	even	ADV
ejpam-5157	121	14	.	.	PUNCT
ejpam-5157	122	1	proof	proof	NOUN
ejpam-5157	122	2	.	.	PUNCT
ejpam-5157	123	1	let	let	VERB
ejpam-5157	123	2	n	n	PRON
ejpam-5157	123	3	be	be	AUX
ejpam-5157	123	4	a	a	DET
ejpam-5157	123	5	positive	positive	ADJ
ejpam-5157	123	6	integer	integer	NOUN
ejpam-5157	123	7	and	and	CCONJ
ejpam-5157	123	8	v(cn	v(cn	NUM
ejpam-5157	123	9	)	)	PUNCT
ejpam-5157	123	10	=	=	PRON
ejpam-5157	123	11	{	{	PUNCT
ejpam-5157	123	12	v1	v1	PROPN
ejpam-5157	123	13	,	,	PUNCT
ejpam-5157	123	14	v2	v2	PROPN
ejpam-5157	123	15	,	,	PUNCT
ejpam-5157	123	16	.	.	PUNCT
ejpam-5157	123	17	.	.	PUNCT
ejpam-5157	124	1	.	.	PUNCT
ejpam-5157	125	1	,	,	PUNCT
ejpam-5157	125	2	vn	vn	PROPN
ejpam-5157	125	3	}	}	PUNCT
ejpam-5157	125	4	.	.	PUNCT
ejpam-5157	126	1	let	let	AUX
ejpam-5157	126	2	n	n	NOUN
ejpam-5157	126	3	=	=	SYM
ejpam-5157	126	4	3	3	X
ejpam-5157	126	5	.	.	PUNCT
ejpam-5157	126	6	then	then	ADV
ejpam-5157	126	7	β(c3	β(c3	NOUN
ejpam-5157	126	8	)	)	PUNCT
ejpam-5157	126	9	=	=	SYM
ejpam-5157	126	10	2	2	NUM
ejpam-5157	126	11	,	,	PUNCT
ejpam-5157	126	12	and	and	CCONJ
ejpam-5157	126	13	so	so	ADV
ejpam-5157	126	14	βcer(c3	βcer(c3	NOUN
ejpam-5157	126	15	)	)	PUNCT
ejpam-5157	126	16	≥	≥	NOUN
ejpam-5157	126	17	2	2	NUM
ejpam-5157	126	18	by	by	ADP
ejpam-5157	126	19	proposition	proposition	NOUN
ejpam-5157	126	20	1	1	NUM
ejpam-5157	126	21	.	.	PUNCT
ejpam-5157	126	22	suppose	suppose	VERB
ejpam-5157	126	23	that	that	SCONJ
ejpam-5157	126	24	βcer(c3	βcer(c3	NOUN
ejpam-5157	126	25	)	)	PUNCT
ejpam-5157	126	26	=	=	SYM
ejpam-5157	127	1	2	2	X
ejpam-5157	127	2	.	.	PUNCT
ejpam-5157	127	3	then	then	ADV
ejpam-5157	127	4	there	there	PRON
ejpam-5157	127	5	exists	exist	VERB
ejpam-5157	127	6	vi	vi	PROPN
ejpam-5157	127	7	∈	∈	PROPN
ejpam-5157	127	8	v	v	NOUN
ejpam-5157	127	9	(	(	PUNCT
ejpam-5157	127	10	c3)\n	c3)\n	PROPN
ejpam-5157	127	11	for	for	ADP
ejpam-5157	127	12	some	some	DET
ejpam-5157	127	13	i	i	PRON
ejpam-5157	127	14	∈	∈	PROPN
ejpam-5157	127	15	{	{	PUNCT
ejpam-5157	127	16	1	1	NUM
ejpam-5157	127	17	,	,	PUNCT
ejpam-5157	127	18	2	2	NUM
ejpam-5157	127	19	,	,	PUNCT
ejpam-5157	127	20	3	3	NUM
ejpam-5157	127	21	}	}	PUNCT
ejpam-5157	127	22	,	,	PUNCT
ejpam-5157	127	23	where	where	SCONJ
ejpam-5157	127	24	n	n	PRON
ejpam-5157	127	25	is	be	AUX
ejpam-5157	127	26	a	a	DET
ejpam-5157	127	27	certified	certify	VERB
ejpam-5157	127	28	vertex	vertex	NOUN
ejpam-5157	127	29	cover	cover	NOUN
ejpam-5157	127	30	of	of	ADP
ejpam-5157	127	31	c3	c3	PROPN
ejpam-5157	127	32	.	.	PUNCT
ejpam-5157	128	1	assume	assume	VERB
ejpam-5157	128	2	that	that	SCONJ
ejpam-5157	128	3	i	i	PRON
ejpam-5157	128	4	=	=	NOUN
ejpam-5157	129	1	1	1	X
ejpam-5157	129	2	.	.	PUNCT
ejpam-5157	129	3	then	then	ADV
ejpam-5157	129	4	both	both	DET
ejpam-5157	129	5	v2	v2	PROPN
ejpam-5157	129	6	,	,	PUNCT
ejpam-5157	129	7	v3	v3	PROPN
ejpam-5157	129	8	∈	∈	PROPN
ejpam-5157	129	9	n	n	AUX
ejpam-5157	129	10	have	have	VERB
ejpam-5157	129	11	only	only	ADV
ejpam-5157	129	12	one	one	NUM
ejpam-5157	129	13	neighbor	neighbor	NOUN
ejpam-5157	129	14	v1	v1	NOUN
ejpam-5157	129	15	outside	outside	ADP
ejpam-5157	129	16	n	n	NOUN
ejpam-5157	129	17	,	,	PUNCT
ejpam-5157	129	18	a	a	DET
ejpam-5157	129	19	contradiction	contradiction	NOUN
ejpam-5157	129	20	.	.	PUNCT
ejpam-5157	130	1	similarly	similarly	ADV
ejpam-5157	130	2	,	,	PUNCT
ejpam-5157	130	3	when	when	SCONJ
ejpam-5157	130	4	i	i	PRON
ejpam-5157	130	5	=	=	SYM
ejpam-5157	130	6	2	2	NUM
ejpam-5157	130	7	or	or	CCONJ
ejpam-5157	130	8	i	i	PRON
ejpam-5157	130	9	=	=	NOUN
ejpam-5157	130	10	3	3	X
ejpam-5157	130	11	.	.	PUNCT
ejpam-5157	130	12	therefore	therefore	ADV
ejpam-5157	130	13	,	,	PUNCT
ejpam-5157	130	14	βcer(c3	βcer(c3	NOUN
ejpam-5157	130	15	)	)	PUNCT
ejpam-5157	130	16	=	=	SYM
ejpam-5157	131	1	3	3	X
ejpam-5157	131	2	.	.	PUNCT
ejpam-5157	131	3	now	now	ADV
ejpam-5157	131	4	,	,	PUNCT
ejpam-5157	131	5	suppose	suppose	VERB
ejpam-5157	131	6	that	that	SCONJ
ejpam-5157	131	7	n	n	PROPN
ejpam-5157	131	8	≥	≥	NUM
ejpam-5157	131	9	5	5	NUM
ejpam-5157	131	10	and	and	CCONJ
ejpam-5157	131	11	odd	odd	ADJ
ejpam-5157	131	12	.	.	PUNCT
ejpam-5157	132	1	let	let	VERB
ejpam-5157	132	2	q	q	PRON
ejpam-5157	132	3	be	be	AUX
ejpam-5157	132	4	a	a	DET
ejpam-5157	132	5	certified	certify	VERB
ejpam-5157	132	6	vertex	vertex	NOUN
ejpam-5157	132	7	cover	cover	NOUN
ejpam-5157	132	8	of	of	ADP
ejpam-5157	132	9	cn	cn	PROPN
ejpam-5157	132	10	and	and	CCONJ
ejpam-5157	132	11	let	let	VERB
ejpam-5157	132	12	vi	vi	NOUN
ejpam-5157	132	13	,	,	PUNCT
ejpam-5157	132	14	vj	vj	X
ejpam-5157	132	15	∈	∈	PROPN
ejpam-5157	132	16	q	q	PROPN
ejpam-5157	132	17	for	for	ADP
ejpam-5157	132	18	some	some	DET
ejpam-5157	132	19	i	i	PROPN
ejpam-5157	132	20	,	,	PUNCT
ejpam-5157	132	21	j	j	PROPN
ejpam-5157	132	22	∈	∈	PROPN
ejpam-5157	132	23	{	{	PUNCT
ejpam-5157	132	24	1	1	NUM
ejpam-5157	132	25	,	,	PUNCT
ejpam-5157	132	26	2	2	NUM
ejpam-5157	132	27	,	,	PUNCT
ejpam-5157	132	28	.	.	PUNCT
ejpam-5157	132	29	.	.	PUNCT
ejpam-5157	133	1	.	.	PUNCT
ejpam-5157	133	2	,	,	PUNCT
ejpam-5157	133	3	n	n	CCONJ
ejpam-5157	133	4	}	}	PUNCT
ejpam-5157	133	5	.	.	PUNCT
ejpam-5157	134	1	if	if	SCONJ
ejpam-5157	134	2	dcn(vi	dcn(vi	NOUN
ejpam-5157	134	3	,	,	PUNCT
ejpam-5157	134	4	vj	vj	INTJ
ejpam-5157	134	5	)	)	PUNCT
ejpam-5157	134	6	=	=	SYM
ejpam-5157	134	7	1	1	NUM
ejpam-5157	134	8	,	,	PUNCT
ejpam-5157	134	9	then	then	ADV
ejpam-5157	134	10	by	by	ADP
ejpam-5157	134	11	lemma	lemma	PROPN
ejpam-5157	134	12	2	2	NUM
ejpam-5157	134	13	,	,	PUNCT
ejpam-5157	134	14	q	q	NOUN
ejpam-5157	134	15	=	=	SYM
ejpam-5157	134	16	v	v	X
ejpam-5157	134	17	(	(	PUNCT
ejpam-5157	134	18	cn	cn	PROPN
ejpam-5157	134	19	)	)	PUNCT
ejpam-5157	134	20	and	and	CCONJ
ejpam-5157	134	21	we	we	PRON
ejpam-5157	134	22	are	be	AUX
ejpam-5157	134	23	done	do	VERB
ejpam-5157	134	24	.	.	PUNCT
ejpam-5157	135	1	now	now	ADV
ejpam-5157	135	2	,	,	PUNCT
ejpam-5157	135	3	since	since	SCONJ
ejpam-5157	135	4	q	q	NOUN
ejpam-5157	135	5	is	be	AUX
ejpam-5157	135	6	a	a	DET
ejpam-5157	135	7	vertex	vertex	NOUN
ejpam-5157	135	8	cover	cover	NOUN
ejpam-5157	135	9	,	,	PUNCT
ejpam-5157	135	10	dcn(vi	dcn(vi	NOUN
ejpam-5157	135	11	,	,	PUNCT
ejpam-5157	135	12	vj	vj	INTJ
ejpam-5157	135	13	)	)	PUNCT
ejpam-5157	135	14	<	<	X
ejpam-5157	135	15	3	3	NUM
ejpam-5157	135	16	.	.	PUNCT
ejpam-5157	136	1	thus	thus	ADV
ejpam-5157	136	2	,	,	PUNCT
ejpam-5157	136	3	dcn(vi	dcn(vi	NOUN
ejpam-5157	136	4	,	,	PUNCT
ejpam-5157	136	5	vj	vj	INTJ
ejpam-5157	136	6	)	)	PUNCT
ejpam-5157	136	7	=	=	SYM
ejpam-5157	136	8	2	2	NUM
ejpam-5157	136	9	.	.	NOUN
ejpam-5157	136	10	wlog	wlog	NOUN
ejpam-5157	136	11	,	,	PUNCT
ejpam-5157	136	12	assume	assume	VERB
ejpam-5157	136	13	that	that	SCONJ
ejpam-5157	136	14	v1	v1	PROPN
ejpam-5157	136	15	∈	∈	PROPN
ejpam-5157	136	16	q.	q.	NOUN
ejpam-5157	136	17	then	then	ADV
ejpam-5157	136	18	v1	v1	PROPN
ejpam-5157	136	19	,	,	PUNCT
ejpam-5157	136	20	v3	v3	PROPN
ejpam-5157	136	21	,	,	PUNCT
ejpam-5157	136	22	.	.	PUNCT
ejpam-5157	136	23	.	.	PUNCT
ejpam-5157	137	1	.	.	PUNCT
ejpam-5157	138	1	,	,	PUNCT
ejpam-5157	138	2	vn−2	vn−2	PROPN
ejpam-5157	138	3	∈	∈	PROPN
ejpam-5157	138	4	q.	q.	PROPN
ejpam-5157	138	5	since	since	SCONJ
ejpam-5157	138	6	q	q	PROPN
ejpam-5157	138	7	is	be	AUX
ejpam-5157	138	8	vertex	vertex	NOUN
ejpam-5157	138	9	cover	cover	NOUN
ejpam-5157	138	10	,	,	PUNCT
ejpam-5157	138	11	either	either	CCONJ
ejpam-5157	138	12	vn−1	vn−1	ADJ
ejpam-5157	138	13	or	or	CCONJ
ejpam-5157	138	14	vn	vn	PROPN
ejpam-5157	138	15	must	must	AUX
ejpam-5157	138	16	be	be	AUX
ejpam-5157	138	17	in	in	ADP
ejpam-5157	138	18	q.	q.	PROPN
ejpam-5157	138	19	if	if	SCONJ
ejpam-5157	138	20	vn−1	vn−1	PROPN
ejpam-5157	138	21	∈	∈	PROPN
ejpam-5157	138	22	q	q	NOUN
ejpam-5157	138	23	,	,	PUNCT
ejpam-5157	138	24	then	then	ADV
ejpam-5157	138	25	v1	v1	PROPN
ejpam-5157	138	26	,	,	PUNCT
ejpam-5157	138	27	v3	v3	PROPN
ejpam-5157	138	28	,	,	PUNCT
ejpam-5157	138	29	.	.	PUNCT
ejpam-5157	138	30	.	.	PUNCT
ejpam-5157	139	1	.	.	PUNCT
ejpam-5157	140	1	,	,	PUNCT
ejpam-5157	140	2	vn−2	vn−2	PROPN
ejpam-5157	140	3	,	,	PUNCT
ejpam-5157	140	4	vn−1	vn−1	PROPN
ejpam-5157	140	5	∈	∈	PROPN
ejpam-5157	140	6	q.	q.	NOUN
ejpam-5157	140	7	since	since	SCONJ
ejpam-5157	140	8	dcn(vn−1	dcn(vn−1	PROPN
ejpam-5157	140	9	,	,	PUNCT
ejpam-5157	140	10	vn−2	vn−2	PROPN
ejpam-5157	140	11	)	)	PUNCT
ejpam-5157	140	12	=	=	SYM
ejpam-5157	141	1	1	1	X
ejpam-5157	141	2	,	,	PUNCT
ejpam-5157	141	3	it	it	PRON
ejpam-5157	141	4	follows	follow	VERB
ejpam-5157	141	5	that	that	PRON
ejpam-5157	141	6	q	q	PROPN
ejpam-5157	141	7	=	=	SYM
ejpam-5157	141	8	v	v	X
ejpam-5157	141	9	(	(	PUNCT
ejpam-5157	141	10	cn	cn	PROPN
ejpam-5157	141	11	)	)	PUNCT
ejpam-5157	141	12	by	by	ADP
ejpam-5157	141	13	lemma	lemma	PROPN
ejpam-5157	141	14	2	2	NUM
ejpam-5157	141	15	.	.	PUNCT
ejpam-5157	141	16	similarly	similarly	ADV
ejpam-5157	141	17	,	,	PUNCT
ejpam-5157	141	18	when	when	SCONJ
ejpam-5157	141	19	vn	vn	PROPN
ejpam-5157	141	20	∈	∈	PROPN
ejpam-5157	141	21	q	q	NOUN
ejpam-5157	141	22	,	,	PUNCT
ejpam-5157	141	23	then	then	ADV
ejpam-5157	141	24	q	q	NOUN
ejpam-5157	141	25	=	=	SYM
ejpam-5157	141	26	v	v	X
ejpam-5157	141	27	(	(	PUNCT
ejpam-5157	141	28	cn	cn	PROPN
ejpam-5157	141	29	)	)	PUNCT
ejpam-5157	141	30	.	.	PUNCT
ejpam-5157	142	1	therefore	therefore	ADV
ejpam-5157	142	2	,	,	PUNCT
ejpam-5157	142	3	βcer(cn	βcer(cn	NOUN
ejpam-5157	142	4	)	)	PUNCT
ejpam-5157	142	5	=	=	SYM
ejpam-5157	143	1	n	n	PROPN
ejpam-5157	143	2	for	for	ADP
ejpam-5157	143	3	all	all	DET
ejpam-5157	143	4	n	n	PRON
ejpam-5157	143	5	≥	≥	NOUN
ejpam-5157	143	6	3	3	NUM
ejpam-5157	143	7	and	and	CCONJ
ejpam-5157	143	8	odd	odd	ADJ
ejpam-5157	143	9	.	.	PUNCT
ejpam-5157	144	1	next	next	ADV
ejpam-5157	144	2	,	,	PUNCT
ejpam-5157	144	3	suppose	suppose	VERB
ejpam-5157	144	4	that	that	SCONJ
ejpam-5157	144	5	n	n	PROPN
ejpam-5157	144	6	≥	≥	NOUN
ejpam-5157	144	7	4	4	NUM
ejpam-5157	144	8	and	and	CCONJ
ejpam-5157	144	9	even	even	ADV
ejpam-5157	144	10	.	.	PUNCT
ejpam-5157	145	1	let	let	VERB
ejpam-5157	145	2	q1	q1	PROPN
ejpam-5157	145	3	=	=	SYM
ejpam-5157	145	4	{	{	PUNCT
ejpam-5157	145	5	v1	v1	PROPN
ejpam-5157	145	6	,	,	PUNCT
ejpam-5157	145	7	v3	v3	PROPN
ejpam-5157	145	8	,	,	PUNCT
ejpam-5157	145	9	.	.	PUNCT
ejpam-5157	145	10	.	.	PUNCT
ejpam-5157	146	1	.	.	PUNCT
ejpam-5157	147	1	,	,	PUNCT
ejpam-5157	147	2	vn−1	vn−1	ADJ
ejpam-5157	147	3	}	}	PUNCT
ejpam-5157	147	4	.	.	PUNCT
ejpam-5157	148	1	then	then	ADV
ejpam-5157	148	2	q1	q1	PROPN
ejpam-5157	148	3	is	be	AUX
ejpam-5157	148	4	a	a	DET
ejpam-5157	148	5	minimum	minimum	ADJ
ejpam-5157	148	6	vertex	vertex	NOUN
ejpam-5157	148	7	cover	cover	NOUN
ejpam-5157	148	8	of	of	ADP
ejpam-5157	148	9	cn	cn	PROPN
ejpam-5157	148	10	.	.	PUNCT
ejpam-5157	148	11	clearly	clearly	ADV
ejpam-5157	148	12	,	,	PUNCT
ejpam-5157	148	13	q1	q1	PROPN
ejpam-5157	148	14	is	be	AUX
ejpam-5157	148	15	a	a	DET
ejpam-5157	148	16	certified	certified	ADJ
ejpam-5157	148	17	set	set	NOUN
ejpam-5157	148	18	of	of	ADP
ejpam-5157	148	19	cn	cn	PROPN
ejpam-5157	148	20	.	.	PUNCT
ejpam-5157	149	1	since	since	SCONJ
ejpam-5157	149	2	n	n	NUM
ejpam-5157	149	3	is	be	AUX
ejpam-5157	149	4	even	even	ADV
ejpam-5157	149	5	,	,	PUNCT
ejpam-5157	149	6	it	it	PRON
ejpam-5157	149	7	follows	follow	VERB
ejpam-5157	149	8	that	that	SCONJ
ejpam-5157	149	9	βcer(cn	βcer(cn	NOUN
ejpam-5157	149	10	)	)	PUNCT
ejpam-5157	149	11	=	=	SYM
ejpam-5157	149	12	|q1|	|q1|	ADV
ejpam-5157	149	13	=	=	SYM
ejpam-5157	149	14	n	n	PRON
ejpam-5157	149	15	2	2	NUM
ejpam-5157	149	16	for	for	ADP
ejpam-5157	149	17	all	all	DET
ejpam-5157	149	18	n	n	PRON
ejpam-5157	149	19	≥	≥	NOUN
ejpam-5157	149	20	4	4	NUM
ejpam-5157	149	21	and	and	CCONJ
ejpam-5157	149	22	even	even	ADV
ejpam-5157	149	23	.	.	PUNCT
ejpam-5157	150	1	j.	j.	PROPN
ejpam-5157	150	2	a.	a.	PROPN
ejpam-5157	150	3	hassan	hassan	PROPN
ejpam-5157	150	4	et	et	PROPN
ejpam-5157	150	5	al	al	PROPN
ejpam-5157	150	6	.	.	PUNCT
ejpam-5157	150	7	/	/	SYM
ejpam-5157	150	8	eur	eur	PROPN
ejpam-5157	150	9	.	.	PUNCT
ejpam-5157	151	1	j.	j.	PROPN
ejpam-5157	151	2	pure	pure	PROPN
ejpam-5157	151	3	appl	appl	PROPN
ejpam-5157	151	4	.	.	PROPN
ejpam-5157	151	5	math	math	PROPN
ejpam-5157	151	6	,	,	PUNCT
ejpam-5157	151	7	17	17	NUM
ejpam-5157	151	8	(	(	PUNCT
ejpam-5157	151	9	2	2	NUM
ejpam-5157	151	10	)	)	PUNCT
ejpam-5157	151	11	(	(	PUNCT
ejpam-5157	151	12	2024	2024	NUM
ejpam-5157	151	13	)	)	PUNCT
ejpam-5157	151	14	,	,	PUNCT
ejpam-5157	151	15	1038	1038	NUM
ejpam-5157	151	16	-	-	SYM
ejpam-5157	151	17	1045	1045	NUM
ejpam-5157	151	18	1043	1043	NUM
ejpam-5157	151	19	theorem	theorem	NOUN
ejpam-5157	151	20	3	3	NUM
ejpam-5157	151	21	.	.	PUNCT
ejpam-5157	152	1	let	let	VERB
ejpam-5157	152	2	m	m	PRON
ejpam-5157	152	3	be	be	AUX
ejpam-5157	152	4	a	a	DET
ejpam-5157	152	5	positive	positive	ADJ
ejpam-5157	152	6	integer	integer	NOUN
ejpam-5157	152	7	.	.	PUNCT
ejpam-5157	153	1	then	then	ADV
ejpam-5157	153	2	n	n	PRON
ejpam-5157	153	3	is	be	AUX
ejpam-5157	153	4	a	a	DET
ejpam-5157	153	5	certified	certify	VERB
ejpam-5157	153	6	vertex	vertex	NOUN
ejpam-5157	153	7	cover	cover	NOUN
ejpam-5157	153	8	of	of	ADP
ejpam-5157	153	9	km	km	PROPN
ejpam-5157	153	10	if	if	SCONJ
ejpam-5157	153	11	and	and	CCONJ
ejpam-5157	153	12	only	only	ADV
ejpam-5157	153	13	if	if	SCONJ
ejpam-5157	153	14	n	n	PROPN
ejpam-5157	153	15	=	=	SYM
ejpam-5157	153	16	v	v	NOUN
ejpam-5157	153	17	(	(	PUNCT
ejpam-5157	153	18	km	km	PROPN
ejpam-5157	153	19	)	)	PUNCT
ejpam-5157	153	20	.	.	PUNCT
ejpam-5157	154	1	proof	proof	NOUN
ejpam-5157	154	2	.	.	PUNCT
ejpam-5157	155	1	let	let	VERB
ejpam-5157	155	2	n	n	PRON
ejpam-5157	155	3	be	be	AUX
ejpam-5157	155	4	a	a	DET
ejpam-5157	155	5	certified	certify	VERB
ejpam-5157	155	6	vertex	vertex	NOUN
ejpam-5157	155	7	cover	cover	NOUN
ejpam-5157	155	8	of	of	ADP
ejpam-5157	155	9	km	km	PROPN
ejpam-5157	155	10	.	.	PUNCT
ejpam-5157	156	1	then	then	ADV
ejpam-5157	156	2	n	n	PRON
ejpam-5157	156	3	is	be	AUX
ejpam-5157	156	4	a	a	DET
ejpam-5157	156	5	vertex	vertex	NOUN
ejpam-5157	156	6	cover	cover	NOUN
ejpam-5157	156	7	of	of	ADP
ejpam-5157	156	8	km	km	NOUN
ejpam-5157	156	9	by	by	ADP
ejpam-5157	156	10	definition	definition	NOUN
ejpam-5157	156	11	.	.	PUNCT
ejpam-5157	157	1	thus	thus	ADV
ejpam-5157	157	2	,	,	PUNCT
ejpam-5157	157	3	βcer(km	βcer(km	ADJ
ejpam-5157	157	4	)	)	PUNCT
ejpam-5157	157	5	≥	≥	NOUN
ejpam-5157	157	6	m−1	m−1	PROPN
ejpam-5157	157	7	since	since	SCONJ
ejpam-5157	157	8	β(g	β(g	PROPN
ejpam-5157	157	9	)	)	PUNCT
ejpam-5157	157	10	≤	≤	PUNCT
ejpam-5157	158	1	βcer(g	βcer(g	NOUN
ejpam-5157	158	2	)	)	PUNCT
ejpam-5157	158	3	for	for	ADP
ejpam-5157	158	4	any	any	DET
ejpam-5157	158	5	graph	graph	NOUN
ejpam-5157	158	6	g.	g.	NOUN
ejpam-5157	158	7	if	if	SCONJ
ejpam-5157	158	8	βcer(g	βcer(g	NOUN
ejpam-5157	158	9	)	)	PUNCT
ejpam-5157	159	1	=	=	SYM
ejpam-5157	159	2	m	m	PROPN
ejpam-5157	159	3	,	,	PUNCT
ejpam-5157	159	4	then	then	ADV
ejpam-5157	159	5	n	n	PROPN
ejpam-5157	159	6	=	=	SYM
ejpam-5157	159	7	v	v	PROPN
ejpam-5157	159	8	(	(	PUNCT
ejpam-5157	159	9	km	km	PROPN
ejpam-5157	159	10	)	)	PUNCT
ejpam-5157	159	11	,	,	PUNCT
ejpam-5157	159	12	and	and	CCONJ
ejpam-5157	159	13	we	we	PRON
ejpam-5157	159	14	are	be	AUX
ejpam-5157	159	15	done	do	VERB
ejpam-5157	159	16	.	.	PUNCT
ejpam-5157	160	1	suppose	suppose	VERB
ejpam-5157	160	2	that	that	SCONJ
ejpam-5157	160	3	βcer(km	βcer(km	NOUN
ejpam-5157	160	4	)	)	PUNCT
ejpam-5157	160	5	=	=	SYM
ejpam-5157	161	1	m−	m−	PROPN
ejpam-5157	161	2	1	1	NUM
ejpam-5157	161	3	.	.	PUNCT
ejpam-5157	162	1	then	then	ADV
ejpam-5157	162	2	there	there	PRON
ejpam-5157	162	3	exists	exist	VERB
ejpam-5157	162	4	a	a	DET
ejpam-5157	162	5	unique	unique	ADJ
ejpam-5157	162	6	x	x	SYM
ejpam-5157	162	7	∈	∈	PROPN
ejpam-5157	162	8	v	v	NOUN
ejpam-5157	162	9	(	(	PUNCT
ejpam-5157	162	10	km	km	PROPN
ejpam-5157	162	11	)	)	PUNCT
ejpam-5157	162	12	such	such	ADJ
ejpam-5157	162	13	that	that	SCONJ
ejpam-5157	162	14	x	x	SYM
ejpam-5157	162	15	/∈	/∈	PROPN
ejpam-5157	162	16	n	n	INTJ
ejpam-5157	162	17	.	.	PUNCT
ejpam-5157	163	1	however	however	ADV
ejpam-5157	163	2	,	,	PUNCT
ejpam-5157	163	3	each	each	DET
ejpam-5157	163	4	vertex	vertex	NOUN
ejpam-5157	163	5	in	in	ADP
ejpam-5157	163	6	n	n	NUM
ejpam-5157	163	7	has	have	VERB
ejpam-5157	163	8	only	only	ADV
ejpam-5157	163	9	one	one	NUM
ejpam-5157	163	10	neighbor	neighbor	NOUN
ejpam-5157	163	11	x	x	PUNCT
ejpam-5157	163	12	outside	outside	ADP
ejpam-5157	163	13	n	n	PROPN
ejpam-5157	163	14	,	,	PUNCT
ejpam-5157	163	15	which	which	PRON
ejpam-5157	163	16	is	be	AUX
ejpam-5157	163	17	a	a	DET
ejpam-5157	163	18	contradiction	contradiction	NOUN
ejpam-5157	163	19	to	to	ADP
ejpam-5157	163	20	the	the	DET
ejpam-5157	163	21	fact	fact	NOUN
ejpam-5157	163	22	that	that	SCONJ
ejpam-5157	163	23	n	n	PRON
ejpam-5157	163	24	is	be	AUX
ejpam-5157	163	25	a	a	DET
ejpam-5157	163	26	certified	certified	ADJ
ejpam-5157	163	27	set	set	NOUN
ejpam-5157	163	28	of	of	ADP
ejpam-5157	163	29	km	km	PROPN
ejpam-5157	163	30	.	.	PUNCT
ejpam-5157	164	1	therefore	therefore	ADV
ejpam-5157	164	2	,	,	PUNCT
ejpam-5157	164	3	n	n	PROPN
ejpam-5157	164	4	=	=	SYM
ejpam-5157	164	5	v	v	PROPN
ejpam-5157	164	6	(	(	PUNCT
ejpam-5157	164	7	km	km	PROPN
ejpam-5157	164	8	)	)	PUNCT
ejpam-5157	164	9	for	for	ADP
ejpam-5157	164	10	all	all	DET
ejpam-5157	164	11	m	m	PROPN
ejpam-5157	164	12	≥	≥	NOUN
ejpam-5157	164	13	1	1	NUM
ejpam-5157	164	14	.	.	PUNCT
ejpam-5157	165	1	the	the	DET
ejpam-5157	165	2	converse	converse	NOUN
ejpam-5157	165	3	is	be	AUX
ejpam-5157	165	4	clear	clear	ADJ
ejpam-5157	165	5	.	.	PUNCT
ejpam-5157	166	1	corollary	corollary	ADJ
ejpam-5157	166	2	1	1	NUM
ejpam-5157	166	3	.	.	PUNCT
ejpam-5157	167	1	let	let	VERB
ejpam-5157	167	2	m	m	PRON
ejpam-5157	167	3	be	be	AUX
ejpam-5157	167	4	a	a	DET
ejpam-5157	167	5	positive	positive	ADJ
ejpam-5157	167	6	integer	integer	NOUN
ejpam-5157	167	7	.	.	PUNCT
ejpam-5157	168	1	then	then	ADV
ejpam-5157	168	2	βcer(km	βcer(km	ADJ
ejpam-5157	168	3	)	)	PUNCT
ejpam-5157	168	4	=	=	SYM
ejpam-5157	168	5	m.	m.	NOUN
ejpam-5157	168	6	theorem	theorem	VERB
ejpam-5157	168	7	4	4	NUM
ejpam-5157	168	8	.	.	PUNCT
ejpam-5157	169	1	let	let	VERB
ejpam-5157	169	2	g	g	NOUN
ejpam-5157	169	3	and	and	CCONJ
ejpam-5157	169	4	h	h	NOUN
ejpam-5157	169	5	be	be	VERB
ejpam-5157	169	6	two	two	NUM
ejpam-5157	169	7	graphs	graph	NOUN
ejpam-5157	169	8	with	with	ADP
ejpam-5157	169	9	no	no	DET
ejpam-5157	169	10	trivial	trivial	ADJ
ejpam-5157	169	11	components	component	NOUN
ejpam-5157	169	12	.	.	PUNCT
ejpam-5157	170	1	then	then	ADV
ejpam-5157	170	2	q	q	X
ejpam-5157	170	3	⊆	⊆	NUM
ejpam-5157	170	4	v	v	NOUN
ejpam-5157	170	5	(	(	PUNCT
ejpam-5157	170	6	g+h	g+h	PROPN
ejpam-5157	170	7	)	)	PUNCT
ejpam-5157	170	8	is	be	AUX
ejpam-5157	170	9	a	a	DET
ejpam-5157	170	10	certified	certify	VERB
ejpam-5157	170	11	vertex	vertex	NOUN
ejpam-5157	170	12	cover	cover	NOUN
ejpam-5157	170	13	of	of	ADP
ejpam-5157	170	14	g	g	PROPN
ejpam-5157	170	15	+	+	NOUN
ejpam-5157	170	16	h	h	NOUN
ejpam-5157	170	17	if	if	SCONJ
ejpam-5157	171	1	and	and	CCONJ
ejpam-5157	171	2	only	only	ADV
ejpam-5157	171	3	if	if	SCONJ
ejpam-5157	171	4	q	q	PROPN
ejpam-5157	171	5	=	=	SYM
ejpam-5157	171	6	qg	qg	PROPN
ejpam-5157	171	7	∪	∪	PROPN
ejpam-5157	171	8	qh	qh	PROPN
ejpam-5157	171	9	and	and	CCONJ
ejpam-5157	171	10	satisfies	satisfy	VERB
ejpam-5157	171	11	one	one	NUM
ejpam-5157	171	12	of	of	ADP
ejpam-5157	171	13	the	the	DET
ejpam-5157	171	14	following	following	ADJ
ejpam-5157	171	15	conditions	condition	NOUN
ejpam-5157	171	16	:	:	PUNCT
ejpam-5157	171	17	(	(	PUNCT
ejpam-5157	171	18	i	i	NOUN
ejpam-5157	171	19	)	)	PUNCT
ejpam-5157	171	20	qg	qg	PROPN
ejpam-5157	171	21	=	=	SYM
ejpam-5157	171	22	v	v	PROPN
ejpam-5157	171	23	(	(	PUNCT
ejpam-5157	171	24	g	g	NOUN
ejpam-5157	171	25	)	)	PUNCT
ejpam-5157	171	26	and	and	CCONJ
ejpam-5157	171	27	qh	qh	NOUN
ejpam-5157	171	28	is	be	AUX
ejpam-5157	171	29	a	a	DET
ejpam-5157	171	30	certified	certify	VERB
ejpam-5157	171	31	vertex	vertex	NOUN
ejpam-5157	171	32	cover	cover	NOUN
ejpam-5157	171	33	of	of	ADP
ejpam-5157	171	34	h.	h.	PROPN
ejpam-5157	171	35	(	(	PUNCT
ejpam-5157	171	36	ii	ii	PROPN
ejpam-5157	171	37	)	)	PUNCT
ejpam-5157	171	38	qh	qh	NOUN
ejpam-5157	171	39	=	=	NOUN
ejpam-5157	171	40	v	v	PROPN
ejpam-5157	171	41	(	(	PUNCT
ejpam-5157	171	42	h	h	NOUN
ejpam-5157	171	43	)	)	PUNCT
ejpam-5157	171	44	and	and	CCONJ
ejpam-5157	171	45	qg	qg	PROPN
ejpam-5157	171	46	is	be	AUX
ejpam-5157	171	47	a	a	DET
ejpam-5157	171	48	certified	certify	VERB
ejpam-5157	171	49	vertex	vertex	NOUN
ejpam-5157	171	50	cover	cover	NOUN
ejpam-5157	171	51	of	of	ADP
ejpam-5157	171	52	g.	g.	PROPN
ejpam-5157	171	53	proof	proof	NOUN
ejpam-5157	171	54	.	.	PUNCT
ejpam-5157	172	1	suppose	suppose	VERB
ejpam-5157	172	2	that	that	PRON
ejpam-5157	172	3	q	q	NOUN
ejpam-5157	172	4	is	be	AUX
ejpam-5157	172	5	a	a	DET
ejpam-5157	172	6	certified	certify	VERB
ejpam-5157	172	7	vertex	vertex	NOUN
ejpam-5157	172	8	cover	cover	NOUN
ejpam-5157	172	9	of	of	ADP
ejpam-5157	172	10	g	g	PROPN
ejpam-5157	172	11	+	+	PROPN
ejpam-5157	172	12	h.	h.	NOUN
ejpam-5157	172	13	if	if	SCONJ
ejpam-5157	172	14	q	q	PROPN
ejpam-5157	172	15	=	=	SYM
ejpam-5157	172	16	v	v	NOUN
ejpam-5157	172	17	(	(	PUNCT
ejpam-5157	172	18	g	g	PROPN
ejpam-5157	172	19	+	+	PROPN
ejpam-5157	172	20	h	h	NOUN
ejpam-5157	172	21	)	)	PUNCT
ejpam-5157	172	22	,	,	PUNCT
ejpam-5157	172	23	then	then	ADV
ejpam-5157	172	24	we	we	PRON
ejpam-5157	172	25	are	be	AUX
ejpam-5157	172	26	done	do	VERB
ejpam-5157	172	27	.	.	PUNCT
ejpam-5157	173	1	assume	assume	VERB
ejpam-5157	173	2	that	that	SCONJ
ejpam-5157	173	3	q	q	PROPN
ejpam-5157	173	4	̸=	̸=	PROPN
ejpam-5157	173	5	v	v	NOUN
ejpam-5157	173	6	(	(	PUNCT
ejpam-5157	173	7	g	g	PROPN
ejpam-5157	173	8	+	+	NOUN
ejpam-5157	173	9	h	h	NOUN
ejpam-5157	173	10	)	)	PUNCT
ejpam-5157	173	11	.	.	PUNCT
ejpam-5157	174	1	since	since	SCONJ
ejpam-5157	174	2	q	q	PROPN
ejpam-5157	174	3	is	be	AUX
ejpam-5157	174	4	a	a	DET
ejpam-5157	174	5	vertex	vertex	NOUN
ejpam-5157	174	6	cover	cover	NOUN
ejpam-5157	174	7	of	of	ADP
ejpam-5157	174	8	g	g	PROPN
ejpam-5157	174	9	+	+	CCONJ
ejpam-5157	174	10	h	h	NOUN
ejpam-5157	174	11	,	,	PUNCT
ejpam-5157	174	12	either	either	CCONJ
ejpam-5157	174	13	qg	qg	PROPN
ejpam-5157	174	14	=	=	PROPN
ejpam-5157	174	15	v	v	PROPN
ejpam-5157	174	16	(	(	PUNCT
ejpam-5157	174	17	g	g	NOUN
ejpam-5157	174	18	)	)	PUNCT
ejpam-5157	174	19	and	and	CCONJ
ejpam-5157	174	20	qh	qh	NOUN
ejpam-5157	174	21	̸=	̸=	PROPN
ejpam-5157	174	22	v	v	NOUN
ejpam-5157	174	23	(	(	PUNCT
ejpam-5157	174	24	h	h	NOUN
ejpam-5157	174	25	)	)	PUNCT
ejpam-5157	174	26	or	or	CCONJ
ejpam-5157	174	27	qg	qg	PROPN
ejpam-5157	174	28	̸=	̸=	PROPN
ejpam-5157	174	29	v	v	NOUN
ejpam-5157	174	30	(	(	PUNCT
ejpam-5157	174	31	g	g	NOUN
ejpam-5157	174	32	)	)	PUNCT
ejpam-5157	174	33	and	and	CCONJ
ejpam-5157	174	34	qh	qh	NOUN
ejpam-5157	174	35	=	=	NOUN
ejpam-5157	174	36	v	v	PROPN
ejpam-5157	174	37	(	(	PUNCT
ejpam-5157	174	38	h	h	NOUN
ejpam-5157	174	39	)	)	PUNCT
ejpam-5157	174	40	.	.	PUNCT
ejpam-5157	175	1	suppose	suppose	VERB
ejpam-5157	175	2	that	that	SCONJ
ejpam-5157	175	3	qg	qg	PROPN
ejpam-5157	175	4	=	=	PROPN
ejpam-5157	175	5	v	v	PROPN
ejpam-5157	175	6	(	(	PUNCT
ejpam-5157	175	7	g	g	NOUN
ejpam-5157	175	8	)	)	PUNCT
ejpam-5157	175	9	and	and	CCONJ
ejpam-5157	175	10	qh	qh	NOUN
ejpam-5157	175	11	̸=	̸=	PROPN
ejpam-5157	175	12	v	v	NOUN
ejpam-5157	175	13	(	(	PUNCT
ejpam-5157	175	14	h	h	NOUN
ejpam-5157	175	15	)	)	PUNCT
ejpam-5157	175	16	.	.	PUNCT
ejpam-5157	176	1	since	since	SCONJ
ejpam-5157	176	2	q	q	PROPN
ejpam-5157	176	3	is	be	AUX
ejpam-5157	176	4	a	a	DET
ejpam-5157	176	5	certified	certify	VERB
ejpam-5157	176	6	vertex	vertex	NOUN
ejpam-5157	176	7	cover	cover	NOUN
ejpam-5157	176	8	of	of	ADP
ejpam-5157	176	9	g	g	PROPN
ejpam-5157	176	10	+	+	CCONJ
ejpam-5157	176	11	h	h	NOUN
ejpam-5157	176	12	,	,	PUNCT
ejpam-5157	176	13	qh	qh	NOUN
ejpam-5157	176	14	must	must	AUX
ejpam-5157	176	15	be	be	AUX
ejpam-5157	176	16	a	a	DET
ejpam-5157	176	17	certified	certify	VERB
ejpam-5157	176	18	vertex	vertex	NOUN
ejpam-5157	176	19	cover	cover	NOUN
ejpam-5157	176	20	of	of	ADP
ejpam-5157	176	21	h.	h.	PROPN
ejpam-5157	176	22	thus	thus	ADV
ejpam-5157	176	23	,	,	PUNCT
ejpam-5157	176	24	(	(	PUNCT
ejpam-5157	176	25	i	i	NOUN
ejpam-5157	176	26	)	)	PUNCT
ejpam-5157	176	27	holds	hold	VERB
ejpam-5157	176	28	.	.	PUNCT
ejpam-5157	177	1	similarly	similarly	ADV
ejpam-5157	177	2	,	,	PUNCT
ejpam-5157	177	3	when	when	SCONJ
ejpam-5157	177	4	qg	qg	PROPN
ejpam-5157	177	5	̸=	̸=	PROPN
ejpam-5157	177	6	v	v	ADP
ejpam-5157	177	7	(	(	PUNCT
ejpam-5157	177	8	g	g	NOUN
ejpam-5157	177	9	)	)	PUNCT
ejpam-5157	177	10	and	and	CCONJ
ejpam-5157	177	11	qh	qh	NOUN
ejpam-5157	177	12	=	=	NOUN
ejpam-5157	177	13	v	v	PROPN
ejpam-5157	177	14	(	(	PUNCT
ejpam-5157	177	15	h	h	NOUN
ejpam-5157	177	16	)	)	PUNCT
ejpam-5157	177	17	,	,	PUNCT
ejpam-5157	177	18	then	then	ADV
ejpam-5157	177	19	(	(	PUNCT
ejpam-5157	177	20	ii	ii	NOUN
ejpam-5157	177	21	)	)	PUNCT
ejpam-5157	177	22	holds	hold	VERB
ejpam-5157	177	23	.	.	PUNCT
ejpam-5157	178	1	conversely	conversely	ADV
ejpam-5157	178	2	,	,	PUNCT
ejpam-5157	178	3	suppose	suppose	VERB
ejpam-5157	178	4	that	that	SCONJ
ejpam-5157	178	5	(	(	PUNCT
ejpam-5157	178	6	i	i	NOUN
ejpam-5157	178	7	)	)	PUNCT
ejpam-5157	178	8	holds	hold	VERB
ejpam-5157	178	9	.	.	PUNCT
ejpam-5157	179	1	let	let	VERB
ejpam-5157	179	2	x	x	SYM
ejpam-5157	179	3	∈	∈	PROPN
ejpam-5157	179	4	q.	q.	NOUN
ejpam-5157	179	5	then	then	ADV
ejpam-5157	179	6	either	either	CCONJ
ejpam-5157	179	7	x	x	PROPN
ejpam-5157	179	8	∈	∈	PROPN
ejpam-5157	179	9	qg	qg	PROPN
ejpam-5157	179	10	=	=	SYM
ejpam-5157	179	11	v	v	PROPN
ejpam-5157	179	12	(	(	PUNCT
ejpam-5157	179	13	g	g	NOUN
ejpam-5157	179	14	)	)	PUNCT
ejpam-5157	179	15	or	or	CCONJ
ejpam-5157	179	16	x	x	SYM
ejpam-5157	179	17	∈	∈	PROPN
ejpam-5157	179	18	qh	qh	NOUN
ejpam-5157	179	19	⊆	⊆	PROPN
ejpam-5157	179	20	v	v	NOUN
ejpam-5157	179	21	(	(	PUNCT
ejpam-5157	179	22	h	h	NOUN
ejpam-5157	179	23	)	)	PUNCT
ejpam-5157	179	24	.	.	PUNCT
ejpam-5157	180	1	assume	assume	VERB
ejpam-5157	180	2	that	that	SCONJ
ejpam-5157	180	3	x	x	PUNCT
ejpam-5157	180	4	∈	∈	PROPN
ejpam-5157	180	5	qg	qg	PROPN
ejpam-5157	180	6	=	=	SYM
ejpam-5157	180	7	v	v	PROPN
ejpam-5157	180	8	(	(	PUNCT
ejpam-5157	180	9	g	g	NOUN
ejpam-5157	180	10	)	)	PUNCT
ejpam-5157	180	11	.	.	PUNCT
ejpam-5157	181	1	since	since	SCONJ
ejpam-5157	181	2	h	h	NOUN
ejpam-5157	181	3	has	have	VERB
ejpam-5157	181	4	no	no	DET
ejpam-5157	181	5	trivial	trivial	ADJ
ejpam-5157	181	6	components	component	NOUN
ejpam-5157	181	7	and	and	CCONJ
ejpam-5157	181	8	qh	qh	NOUN
ejpam-5157	181	9	⊆	⊆	NUM
ejpam-5157	181	10	v	v	NOUN
ejpam-5157	181	11	(	(	PUNCT
ejpam-5157	181	12	h	h	NOUN
ejpam-5157	181	13	)	)	PUNCT
ejpam-5157	181	14	is	be	AUX
ejpam-5157	181	15	a	a	DET
ejpam-5157	181	16	certified	certify	VERB
ejpam-5157	181	17	set	set	NOUN
ejpam-5157	181	18	in	in	ADP
ejpam-5157	181	19	h	h	NOUN
ejpam-5157	181	20	,	,	PUNCT
ejpam-5157	181	21	it	it	PRON
ejpam-5157	181	22	follows	follow	VERB
ejpam-5157	181	23	that	that	SCONJ
ejpam-5157	181	24	x	x	PRON
ejpam-5157	181	25	has	have	VERB
ejpam-5157	181	26	either	either	CCONJ
ejpam-5157	181	27	zero	zero	NUM
ejpam-5157	181	28	or	or	CCONJ
ejpam-5157	181	29	at	at	ADP
ejpam-5157	181	30	least	least	ADV
ejpam-5157	181	31	two	two	NUM
ejpam-5157	181	32	neighbors	neighbor	NOUN
ejpam-5157	181	33	in	in	ADP
ejpam-5157	181	34	v	v	PROPN
ejpam-5157	181	35	(	(	PUNCT
ejpam-5157	181	36	h	h	NOUN
ejpam-5157	181	37	)	)	PUNCT
ejpam-5157	181	38	\	\	PROPN
ejpam-5157	181	39	qh	qh	PROPN
ejpam-5157	181	40	.	.	PUNCT
ejpam-5157	182	1	since	since	SCONJ
ejpam-5157	182	2	x	x	PRON
ejpam-5157	182	3	is	be	AUX
ejpam-5157	182	4	arbitrary	arbitrary	ADJ
ejpam-5157	182	5	,	,	PUNCT
ejpam-5157	182	6	q	q	X
ejpam-5157	182	7	is	be	AUX
ejpam-5157	182	8	a	a	DET
ejpam-5157	182	9	certified	certify	VERB
ejpam-5157	182	10	set	set	NOUN
ejpam-5157	182	11	in	in	ADP
ejpam-5157	182	12	g	g	PROPN
ejpam-5157	182	13	+	+	CCONJ
ejpam-5157	182	14	h.	h.	PROPN
ejpam-5157	182	15	since	since	SCONJ
ejpam-5157	182	16	qh	qh	PROPN
ejpam-5157	182	17	is	be	AUX
ejpam-5157	182	18	a	a	DET
ejpam-5157	182	19	vertex	vertex	NOUN
ejpam-5157	182	20	cover	cover	NOUN
ejpam-5157	182	21	of	of	ADP
ejpam-5157	182	22	h	h	NOUN
ejpam-5157	182	23	,	,	PUNCT
ejpam-5157	182	24	q	q	NOUN
ejpam-5157	182	25	=	=	X
ejpam-5157	182	26	v	v	X
ejpam-5157	182	27	(	(	PUNCT
ejpam-5157	182	28	g	g	NOUN
ejpam-5157	182	29	)	)	PUNCT
ejpam-5157	182	30	∪	∪	ADP
ejpam-5157	182	31	qh	qh	PROPN
ejpam-5157	182	32	is	be	AUX
ejpam-5157	182	33	a	a	DET
ejpam-5157	182	34	vertex	vertex	NOUN
ejpam-5157	182	35	cover	cover	NOUN
ejpam-5157	182	36	of	of	ADP
ejpam-5157	182	37	g	g	PROPN
ejpam-5157	182	38	+	+	CCONJ
ejpam-5157	182	39	h.	h.	PROPN
ejpam-5157	182	40	therefore	therefore	ADV
ejpam-5157	182	41	,	,	PUNCT
ejpam-5157	182	42	q	q	X
ejpam-5157	182	43	is	be	AUX
ejpam-5157	182	44	a	a	DET
ejpam-5157	182	45	certified	certify	VERB
ejpam-5157	182	46	vertex	vertex	NOUN
ejpam-5157	182	47	cover	cover	NOUN
ejpam-5157	182	48	of	of	ADP
ejpam-5157	182	49	g	g	PROPN
ejpam-5157	182	50	+	+	CCONJ
ejpam-5157	182	51	h.	h.	PROPN
ejpam-5157	182	52	similarly	similarly	ADV
ejpam-5157	182	53	,	,	PUNCT
ejpam-5157	182	54	the	the	DET
ejpam-5157	182	55	assertion	assertion	NOUN
ejpam-5157	182	56	follows	follow	VERB
ejpam-5157	182	57	when	when	SCONJ
ejpam-5157	182	58	x	x	PROPN
ejpam-5157	182	59	∈	∈	PROPN
ejpam-5157	182	60	qh	qh	NOUN
ejpam-5157	182	61	⊆	⊆	PROPN
ejpam-5157	182	62	v	v	NOUN
ejpam-5157	182	63	(	(	PUNCT
ejpam-5157	182	64	h	h	NOUN
ejpam-5157	182	65	)	)	PUNCT
ejpam-5157	182	66	.	.	PUNCT
ejpam-5157	183	1	the	the	DET
ejpam-5157	183	2	assertion	assertion	NOUN
ejpam-5157	183	3	also	also	ADV
ejpam-5157	183	4	follows	follow	VERB
ejpam-5157	183	5	,	,	PUNCT
ejpam-5157	183	6	when	when	SCONJ
ejpam-5157	183	7	(	(	PUNCT
ejpam-5157	183	8	ii	ii	NOUN
ejpam-5157	183	9	)	)	PUNCT
ejpam-5157	183	10	holds	hold	VERB
ejpam-5157	183	11	.	.	PUNCT
ejpam-5157	184	1	corollary	corollary	ADJ
ejpam-5157	184	2	2	2	NUM
ejpam-5157	184	3	.	.	PUNCT
ejpam-5157	185	1	let	let	VERB
ejpam-5157	185	2	g	g	NOUN
ejpam-5157	185	3	and	and	CCONJ
ejpam-5157	185	4	h	h	NOUN
ejpam-5157	185	5	be	be	AUX
ejpam-5157	185	6	graphs	graph	NOUN
ejpam-5157	185	7	with	with	ADP
ejpam-5157	185	8	no	no	DET
ejpam-5157	185	9	trivial	trivial	ADJ
ejpam-5157	185	10	components	component	NOUN
ejpam-5157	185	11	.	.	PUNCT
ejpam-5157	186	1	then	then	ADV
ejpam-5157	186	2	βcer(g+h	βcer(g+h	PROPN
ejpam-5157	186	3	)	)	PUNCT
ejpam-5157	187	1	=	=	SYM
ejpam-5157	187	2	min{|v	min{|v	PROPN
ejpam-5157	187	3	(	(	PUNCT
ejpam-5157	187	4	g)|+	g)|+	NOUN
ejpam-5157	187	5	βcer(h	βcer(h	PROPN
ejpam-5157	187	6	)	)	PUNCT
ejpam-5157	187	7	,	,	PUNCT
ejpam-5157	187	8	|v	|v	PROPN
ejpam-5157	187	9	(	(	PUNCT
ejpam-5157	187	10	h)|+	h)|+	X
ejpam-5157	187	11	βcer(g	βcer(g	NOUN
ejpam-5157	187	12	)	)	PUNCT
ejpam-5157	187	13	}	}	PUNCT
ejpam-5157	187	14	.	.	PUNCT
ejpam-5157	188	1	theorem	theorem	NOUN
ejpam-5157	188	2	5	5	NUM
ejpam-5157	188	3	.	.	PUNCT
ejpam-5157	189	1	let	let	VERB
ejpam-5157	189	2	g	g	PRON
ejpam-5157	189	3	be	be	AUX
ejpam-5157	189	4	a	a	DET
ejpam-5157	189	5	trivial	trivial	ADJ
ejpam-5157	189	6	graph	graph	NOUN
ejpam-5157	189	7	and	and	CCONJ
ejpam-5157	189	8	h	h	NOUN
ejpam-5157	189	9	be	be	AUX
ejpam-5157	189	10	a	a	DET
ejpam-5157	189	11	graph	graph	NOUN
ejpam-5157	189	12	with	with	ADP
ejpam-5157	189	13	no	no	DET
ejpam-5157	189	14	trivial	trivial	ADJ
ejpam-5157	189	15	components	component	NOUN
ejpam-5157	189	16	.	.	PUNCT
ejpam-5157	190	1	then	then	ADV
ejpam-5157	190	2	q	q	X
ejpam-5157	190	3	⊆	⊆	NUM
ejpam-5157	190	4	v	v	NOUN
ejpam-5157	190	5	(	(	PUNCT
ejpam-5157	190	6	g	g	PROPN
ejpam-5157	190	7	+	+	NOUN
ejpam-5157	190	8	h	h	NOUN
ejpam-5157	190	9	)	)	PUNCT
ejpam-5157	190	10	is	be	AUX
ejpam-5157	190	11	a	a	DET
ejpam-5157	190	12	certified	certify	VERB
ejpam-5157	190	13	vertex	vertex	NOUN
ejpam-5157	190	14	cover	cover	NOUN
ejpam-5157	190	15	of	of	ADP
ejpam-5157	190	16	g	g	PROPN
ejpam-5157	190	17	+	+	NOUN
ejpam-5157	190	18	h	h	NOUN
ejpam-5157	190	19	if	if	SCONJ
ejpam-5157	191	1	and	and	CCONJ
ejpam-5157	191	2	only	only	ADV
ejpam-5157	191	3	if	if	SCONJ
ejpam-5157	191	4	q	q	PROPN
ejpam-5157	191	5	=	=	SYM
ejpam-5157	191	6	qg	qg	PROPN
ejpam-5157	191	7	∪	∪	PROPN
ejpam-5157	191	8	qh	qh	PROPN
ejpam-5157	191	9	,	,	PUNCT
ejpam-5157	191	10	where	where	SCONJ
ejpam-5157	191	11	qg	qg	PROPN
ejpam-5157	191	12	=	=	PROPN
ejpam-5157	191	13	v	v	PROPN
ejpam-5157	191	14	(	(	PUNCT
ejpam-5157	191	15	g	g	NOUN
ejpam-5157	191	16	)	)	PUNCT
ejpam-5157	191	17	and	and	CCONJ
ejpam-5157	191	18	qh	qh	NOUN
ejpam-5157	191	19	is	be	AUX
ejpam-5157	191	20	a	a	DET
ejpam-5157	191	21	certified	certify	VERB
ejpam-5157	191	22	vertex	vertex	NOUN
ejpam-5157	191	23	cover	cover	NOUN
ejpam-5157	191	24	of	of	ADP
ejpam-5157	191	25	h.	h.	PROPN
ejpam-5157	191	26	j.	j.	PROPN
ejpam-5157	191	27	a.	a.	PROPN
ejpam-5157	191	28	hassan	hassan	PROPN
ejpam-5157	191	29	et	et	PROPN
ejpam-5157	191	30	al	al	PROPN
ejpam-5157	191	31	.	.	PUNCT
ejpam-5157	191	32	/	/	SYM
ejpam-5157	191	33	eur	eur	PROPN
ejpam-5157	191	34	.	.	PUNCT
ejpam-5157	192	1	j.	j.	PROPN
ejpam-5157	192	2	pure	pure	PROPN
ejpam-5157	192	3	appl	appl	PROPN
ejpam-5157	192	4	.	.	PROPN
ejpam-5157	192	5	math	math	PROPN
ejpam-5157	192	6	,	,	PUNCT
ejpam-5157	192	7	17	17	NUM
ejpam-5157	192	8	(	(	PUNCT
ejpam-5157	192	9	2	2	NUM
ejpam-5157	192	10	)	)	PUNCT
ejpam-5157	192	11	(	(	PUNCT
ejpam-5157	192	12	2024	2024	NUM
ejpam-5157	192	13	)	)	PUNCT
ejpam-5157	192	14	,	,	PUNCT
ejpam-5157	192	15	1038	1038	NUM
ejpam-5157	192	16	-	-	SYM
ejpam-5157	192	17	1045	1045	NUM
ejpam-5157	192	18	1044	1044	NUM
ejpam-5157	192	19	proof	proof	NOUN
ejpam-5157	192	20	.	.	PUNCT
ejpam-5157	192	21	suppose	suppose	VERB
ejpam-5157	192	22	that	that	PRON
ejpam-5157	192	23	q	q	NOUN
ejpam-5157	192	24	is	be	AUX
ejpam-5157	192	25	a	a	DET
ejpam-5157	192	26	certified	certify	VERB
ejpam-5157	192	27	vertex	vertex	NOUN
ejpam-5157	192	28	cover	cover	NOUN
ejpam-5157	192	29	of	of	ADP
ejpam-5157	192	30	g	g	PROPN
ejpam-5157	192	31	+	+	PROPN
ejpam-5157	192	32	h.	h.	NOUN
ejpam-5157	192	33	if	if	SCONJ
ejpam-5157	192	34	q	q	PROPN
ejpam-5157	192	35	=	=	SYM
ejpam-5157	192	36	v	v	NOUN
ejpam-5157	192	37	(	(	PUNCT
ejpam-5157	192	38	g	g	PROPN
ejpam-5157	192	39	+	+	PROPN
ejpam-5157	192	40	h	h	NOUN
ejpam-5157	192	41	)	)	PUNCT
ejpam-5157	192	42	,	,	PUNCT
ejpam-5157	192	43	then	then	ADV
ejpam-5157	192	44	we	we	PRON
ejpam-5157	192	45	are	be	AUX
ejpam-5157	192	46	done	do	VERB
ejpam-5157	192	47	.	.	PUNCT
ejpam-5157	193	1	assume	assume	VERB
ejpam-5157	193	2	that	that	SCONJ
ejpam-5157	193	3	q	q	PROPN
ejpam-5157	193	4	̸=	̸=	PROPN
ejpam-5157	193	5	v	v	NOUN
ejpam-5157	193	6	(	(	PUNCT
ejpam-5157	193	7	g	g	PROPN
ejpam-5157	193	8	+	+	NOUN
ejpam-5157	193	9	h	h	NOUN
ejpam-5157	193	10	)	)	PUNCT
ejpam-5157	193	11	.	.	PUNCT
ejpam-5157	194	1	since	since	SCONJ
ejpam-5157	194	2	q	q	PROPN
ejpam-5157	194	3	is	be	AUX
ejpam-5157	194	4	a	a	DET
ejpam-5157	194	5	vertex	vertex	NOUN
ejpam-5157	194	6	cover	cover	NOUN
ejpam-5157	194	7	of	of	ADP
ejpam-5157	194	8	g	g	PROPN
ejpam-5157	194	9	+	+	CCONJ
ejpam-5157	194	10	h	h	NOUN
ejpam-5157	194	11	,	,	PUNCT
ejpam-5157	194	12	either	either	CCONJ
ejpam-5157	194	13	qg	qg	PROPN
ejpam-5157	194	14	=	=	PROPN
ejpam-5157	194	15	v	v	PROPN
ejpam-5157	194	16	(	(	PUNCT
ejpam-5157	194	17	g	g	NOUN
ejpam-5157	194	18	)	)	PUNCT
ejpam-5157	194	19	and	and	CCONJ
ejpam-5157	194	20	qh	qh	NOUN
ejpam-5157	194	21	̸=	̸=	PROPN
ejpam-5157	194	22	v	v	NOUN
ejpam-5157	194	23	(	(	PUNCT
ejpam-5157	194	24	h	h	NOUN
ejpam-5157	194	25	)	)	PUNCT
ejpam-5157	194	26	or	or	CCONJ
ejpam-5157	194	27	qg	qg	PROPN
ejpam-5157	194	28	̸=	̸=	PROPN
ejpam-5157	194	29	v	v	NOUN
ejpam-5157	194	30	(	(	PUNCT
ejpam-5157	194	31	g	g	NOUN
ejpam-5157	194	32	)	)	PUNCT
ejpam-5157	194	33	and	and	CCONJ
ejpam-5157	194	34	qh	qh	NOUN
ejpam-5157	194	35	=	=	NOUN
ejpam-5157	194	36	v	v	PROPN
ejpam-5157	194	37	(	(	PUNCT
ejpam-5157	194	38	h	h	NOUN
ejpam-5157	194	39	)	)	PUNCT
ejpam-5157	194	40	.	.	PUNCT
ejpam-5157	195	1	since	since	SCONJ
ejpam-5157	195	2	g	g	PROPN
ejpam-5157	195	3	is	be	AUX
ejpam-5157	195	4	trivial	trivial	ADJ
ejpam-5157	195	5	and	and	CCONJ
ejpam-5157	195	6	q	q	NOUN
ejpam-5157	195	7	is	be	AUX
ejpam-5157	195	8	a	a	DET
ejpam-5157	195	9	certified	certify	VERB
ejpam-5157	195	10	set	set	NOUN
ejpam-5157	195	11	in	in	ADP
ejpam-5157	195	12	g+h	g+h	PROPN
ejpam-5157	195	13	,	,	PUNCT
ejpam-5157	195	14	qg	qg	PROPN
ejpam-5157	195	15	̸=	̸=	PROPN
ejpam-5157	195	16	v	v	PROPN
ejpam-5157	195	17	(	(	PUNCT
ejpam-5157	195	18	g	g	NOUN
ejpam-5157	195	19	)	)	PUNCT
ejpam-5157	195	20	and	and	CCONJ
ejpam-5157	195	21	qh	qh	NOUN
ejpam-5157	195	22	=	=	NOUN
ejpam-5157	195	23	v	v	PROPN
ejpam-5157	195	24	(	(	PUNCT
ejpam-5157	195	25	h	h	NOUN
ejpam-5157	195	26	)	)	PUNCT
ejpam-5157	195	27	is	be	AUX
ejpam-5157	195	28	not	not	PART
ejpam-5157	195	29	possible	possible	ADJ
ejpam-5157	195	30	.	.	PUNCT
ejpam-5157	196	1	hence	hence	ADV
ejpam-5157	196	2	,	,	PUNCT
ejpam-5157	196	3	qg	qg	PROPN
ejpam-5157	196	4	=	=	PROPN
ejpam-5157	196	5	v	v	PROPN
ejpam-5157	196	6	(	(	PUNCT
ejpam-5157	196	7	g	g	NOUN
ejpam-5157	196	8	)	)	PUNCT
ejpam-5157	196	9	and	and	CCONJ
ejpam-5157	196	10	q	q	PROPN
ejpam-5157	196	11	̸=	̸=	PROPN
ejpam-5157	196	12	v	v	NOUN
ejpam-5157	196	13	(	(	PUNCT
ejpam-5157	196	14	h	h	NOUN
ejpam-5157	196	15	)	)	PUNCT
ejpam-5157	196	16	.	.	PUNCT
ejpam-5157	197	1	since	since	SCONJ
ejpam-5157	197	2	q	q	PROPN
ejpam-5157	197	3	is	be	AUX
ejpam-5157	197	4	a	a	DET
ejpam-5157	197	5	certified	certify	VERB
ejpam-5157	197	6	vertex	vertex	NOUN
ejpam-5157	197	7	cover	cover	NOUN
ejpam-5157	197	8	of	of	ADP
ejpam-5157	197	9	g+h	g+h	PROPN
ejpam-5157	197	10	,	,	PUNCT
ejpam-5157	197	11	qh	qh	NOUN
ejpam-5157	197	12	must	must	AUX
ejpam-5157	197	13	be	be	AUX
ejpam-5157	197	14	a	a	DET
ejpam-5157	197	15	certified	certify	VERB
ejpam-5157	197	16	vertex	vertex	NOUN
ejpam-5157	197	17	cover	cover	NOUN
ejpam-5157	197	18	of	of	ADP
ejpam-5157	197	19	h.	h.	NOUN
ejpam-5157	197	20	conversely	conversely	ADV
ejpam-5157	197	21	,	,	PUNCT
ejpam-5157	197	22	suppose	suppose	VERB
ejpam-5157	197	23	that	that	SCONJ
ejpam-5157	197	24	q	q	PROPN
ejpam-5157	197	25	=	=	SYM
ejpam-5157	197	26	qg	qg	PROPN
ejpam-5157	197	27	∪	∪	PROPN
ejpam-5157	197	28	qh	qh	PROPN
ejpam-5157	197	29	,	,	PUNCT
ejpam-5157	197	30	where	where	SCONJ
ejpam-5157	197	31	qg	qg	PROPN
ejpam-5157	197	32	=	=	PROPN
ejpam-5157	197	33	v	v	PROPN
ejpam-5157	197	34	(	(	PUNCT
ejpam-5157	197	35	g	g	NOUN
ejpam-5157	197	36	)	)	PUNCT
ejpam-5157	197	37	and	and	CCONJ
ejpam-5157	197	38	qh	qh	NOUN
ejpam-5157	197	39	is	be	AUX
ejpam-5157	197	40	a	a	DET
ejpam-5157	197	41	certified	certify	VERB
ejpam-5157	197	42	vertex	vertex	NOUN
ejpam-5157	197	43	cover	cover	NOUN
ejpam-5157	197	44	of	of	ADP
ejpam-5157	197	45	h.	h.	NOUN
ejpam-5157	197	46	let	let	VERB
ejpam-5157	197	47	y	y	PROPN
ejpam-5157	197	48	∈	∈	PROPN
ejpam-5157	197	49	q.	q.	NOUN
ejpam-5157	197	50	then	then	ADV
ejpam-5157	197	51	either	either	CCONJ
ejpam-5157	197	52	y	y	PROPN
ejpam-5157	197	53	∈	∈	PROPN
ejpam-5157	197	54	qg	qg	PROPN
ejpam-5157	197	55	=	=	SYM
ejpam-5157	197	56	v	v	PROPN
ejpam-5157	197	57	(	(	PUNCT
ejpam-5157	197	58	g	g	NOUN
ejpam-5157	197	59	)	)	PUNCT
ejpam-5157	197	60	or	or	CCONJ
ejpam-5157	197	61	y	y	PROPN
ejpam-5157	197	62	∈	∈	PROPN
ejpam-5157	197	63	qh	qh	PROPN
ejpam-5157	197	64	⊆	⊆	PROPN
ejpam-5157	197	65	v	v	NOUN
ejpam-5157	197	66	(	(	PUNCT
ejpam-5157	197	67	h	h	NOUN
ejpam-5157	197	68	)	)	PUNCT
ejpam-5157	197	69	.	.	PUNCT
ejpam-5157	198	1	assume	assume	VERB
ejpam-5157	198	2	that	that	SCONJ
ejpam-5157	198	3	y	y	PROPN
ejpam-5157	198	4	∈	∈	PROPN
ejpam-5157	198	5	qg	qg	PROPN
ejpam-5157	198	6	=	=	SYM
ejpam-5157	198	7	v	v	PROPN
ejpam-5157	198	8	(	(	PUNCT
ejpam-5157	198	9	g	g	NOUN
ejpam-5157	198	10	)	)	PUNCT
ejpam-5157	198	11	.	.	PUNCT
ejpam-5157	199	1	since	since	SCONJ
ejpam-5157	199	2	h	h	NOUN
ejpam-5157	199	3	has	have	VERB
ejpam-5157	199	4	no	no	DET
ejpam-5157	199	5	trivial	trivial	ADJ
ejpam-5157	199	6	components	component	NOUN
ejpam-5157	199	7	and	and	CCONJ
ejpam-5157	199	8	qh	qh	NOUN
ejpam-5157	199	9	⊆	⊆	NUM
ejpam-5157	199	10	v	v	NOUN
ejpam-5157	199	11	(	(	PUNCT
ejpam-5157	199	12	h	h	NOUN
ejpam-5157	199	13	)	)	PUNCT
ejpam-5157	199	14	is	be	AUX
ejpam-5157	199	15	a	a	DET
ejpam-5157	199	16	certified	certify	VERB
ejpam-5157	199	17	set	set	NOUN
ejpam-5157	199	18	in	in	ADP
ejpam-5157	199	19	h	h	NOUN
ejpam-5157	199	20	,	,	PUNCT
ejpam-5157	199	21	it	it	PRON
ejpam-5157	199	22	follows	follow	VERB
ejpam-5157	199	23	that	that	SCONJ
ejpam-5157	199	24	y	y	PROPN
ejpam-5157	199	25	has	have	VERB
ejpam-5157	199	26	either	either	CCONJ
ejpam-5157	199	27	zero	zero	NUM
ejpam-5157	199	28	or	or	CCONJ
ejpam-5157	199	29	at	at	ADP
ejpam-5157	199	30	least	least	ADV
ejpam-5157	199	31	two	two	NUM
ejpam-5157	199	32	neighbors	neighbor	NOUN
ejpam-5157	199	33	in	in	ADP
ejpam-5157	199	34	v	v	PROPN
ejpam-5157	199	35	(	(	PUNCT
ejpam-5157	199	36	h	h	NOUN
ejpam-5157	199	37	)	)	PUNCT
ejpam-5157	199	38	\qh	\qh	PROPN
ejpam-5157	199	39	.	.	PUNCT
ejpam-5157	200	1	since	since	SCONJ
ejpam-5157	200	2	y	y	PROPN
ejpam-5157	200	3	is	be	AUX
ejpam-5157	200	4	arbitrary	arbitrary	ADJ
ejpam-5157	200	5	,	,	PUNCT
ejpam-5157	200	6	q	q	X
ejpam-5157	200	7	is	be	AUX
ejpam-5157	200	8	a	a	DET
ejpam-5157	200	9	certified	certify	VERB
ejpam-5157	200	10	set	set	NOUN
ejpam-5157	200	11	in	in	ADP
ejpam-5157	200	12	g	g	PROPN
ejpam-5157	200	13	+	+	CCONJ
ejpam-5157	200	14	h.	h.	PROPN
ejpam-5157	200	15	since	since	SCONJ
ejpam-5157	200	16	qh	qh	PROPN
ejpam-5157	200	17	is	be	AUX
ejpam-5157	200	18	a	a	DET
ejpam-5157	200	19	vertex	vertex	NOUN
ejpam-5157	200	20	cover	cover	NOUN
ejpam-5157	200	21	of	of	ADP
ejpam-5157	200	22	h	h	NOUN
ejpam-5157	200	23	,	,	PUNCT
ejpam-5157	200	24	it	it	PRON
ejpam-5157	200	25	follows	follow	VERB
ejpam-5157	200	26	that	that	DET
ejpam-5157	200	27	q	q	PROPN
ejpam-5157	200	28	=	=	SYM
ejpam-5157	200	29	v	v	X
ejpam-5157	200	30	(	(	PUNCT
ejpam-5157	200	31	g	g	NOUN
ejpam-5157	200	32	)	)	PUNCT
ejpam-5157	200	33	∪qh	∪qh	NOUN
ejpam-5157	200	34	is	be	AUX
ejpam-5157	200	35	a	a	DET
ejpam-5157	200	36	certified	certify	VERB
ejpam-5157	200	37	vertex	vertex	NOUN
ejpam-5157	200	38	cover	cover	NOUN
ejpam-5157	200	39	of	of	ADP
ejpam-5157	200	40	g+h	g+h	PROPN
ejpam-5157	200	41	.	.	PUNCT
ejpam-5157	201	1	similarly	similarly	ADV
ejpam-5157	201	2	,	,	PUNCT
ejpam-5157	201	3	the	the	DET
ejpam-5157	201	4	assertion	assertion	NOUN
ejpam-5157	201	5	follows	follow	VERB
ejpam-5157	201	6	when	when	SCONJ
ejpam-5157	201	7	y	y	PROPN
ejpam-5157	201	8	∈	∈	PROPN
ejpam-5157	201	9	qh	qh	PROPN
ejpam-5157	201	10	⊆	⊆	PROPN
ejpam-5157	201	11	v	v	NOUN
ejpam-5157	201	12	(	(	PUNCT
ejpam-5157	201	13	h	h	NOUN
ejpam-5157	201	14	)	)	PUNCT
ejpam-5157	201	15	.	.	PUNCT
ejpam-5157	202	1	corollary	corollary	ADJ
ejpam-5157	202	2	3	3	X
ejpam-5157	202	3	.	.	PUNCT
ejpam-5157	203	1	let	let	VERB
ejpam-5157	203	2	g	g	PRON
ejpam-5157	203	3	be	be	AUX
ejpam-5157	203	4	a	a	DET
ejpam-5157	203	5	trivial	trivial	ADJ
ejpam-5157	203	6	graph	graph	NOUN
ejpam-5157	203	7	and	and	CCONJ
ejpam-5157	203	8	h	h	NOUN
ejpam-5157	203	9	be	be	AUX
ejpam-5157	203	10	a	a	DET
ejpam-5157	203	11	graph	graph	NOUN
ejpam-5157	203	12	with	with	ADP
ejpam-5157	203	13	no	no	DET
ejpam-5157	203	14	trivial	trivial	ADJ
ejpam-5157	203	15	components	component	NOUN
ejpam-5157	203	16	.	.	PUNCT
ejpam-5157	204	1	then	then	ADV
ejpam-5157	204	2	βcer(g+h	βcer(g+h	PROPN
ejpam-5157	204	3	)	)	PUNCT
ejpam-5157	205	1	=	=	SYM
ejpam-5157	205	2	βcer(h	βcer(h	PROPN
ejpam-5157	205	3	)	)	PUNCT
ejpam-5157	206	1	+	+	CCONJ
ejpam-5157	206	2	1	1	NUM
ejpam-5157	206	3	in	in	ADP
ejpam-5157	206	4	particular	particular	ADJ
ejpam-5157	206	5	,	,	PUNCT
ejpam-5157	206	6	each	each	PRON
ejpam-5157	206	7	of	of	ADP
ejpam-5157	206	8	the	the	DET
ejpam-5157	206	9	following	follow	VERB
ejpam-5157	206	10	holds	hold	VERB
ejpam-5157	206	11	:	:	PUNCT
ejpam-5157	206	12	(	(	PUNCT
ejpam-5157	206	13	i	i	NOUN
ejpam-5157	206	14	)	)	PUNCT
ejpam-5157	206	15	βcer(fn	βcer(fn	NOUN
ejpam-5157	206	16	)	)	PUNCT
ejpam-5157	206	17	=	=	SYM
ejpam-5157	206	18	{	{	PUNCT
ejpam-5157	206	19	n+	n+	NOUN
ejpam-5157	206	20	1	1	NUM
ejpam-5157	206	21	,	,	PUNCT
ejpam-5157	206	22	if	if	SCONJ
ejpam-5157	206	23	n	n	PRON
ejpam-5157	206	24	≥	≥	NOUN
ejpam-5157	206	25	2	2	NUM
ejpam-5157	206	26	and	and	CCONJ
ejpam-5157	206	27	even⌊	even⌊	VERB
ejpam-5157	206	28	n	n	DET
ejpam-5157	206	29	2	2	NUM
ejpam-5157	206	30	⌋	⌋	NOUN
ejpam-5157	206	31	+	+	CCONJ
ejpam-5157	206	32	1	1	NUM
ejpam-5157	206	33	,	,	PUNCT
ejpam-5157	206	34	if	if	SCONJ
ejpam-5157	206	35	n	n	PRON
ejpam-5157	206	36	≥	≥	NOUN
ejpam-5157	206	37	3	3	NUM
ejpam-5157	206	38	and	and	CCONJ
ejpam-5157	206	39	odd	odd	ADJ
ejpam-5157	206	40	.	.	PUNCT
ejpam-5157	207	1	(	(	PUNCT
ejpam-5157	207	2	ii	ii	NOUN
ejpam-5157	207	3	)	)	PUNCT
ejpam-5157	207	4	βcer(wn	βcer(wn	NOUN
ejpam-5157	207	5	)	)	PUNCT
ejpam-5157	208	1	=	=	PRON
ejpam-5157	208	2	{	{	PUNCT
ejpam-5157	208	3	n+	n+	NOUN
ejpam-5157	208	4	1	1	NUM
ejpam-5157	208	5	,	,	PUNCT
ejpam-5157	208	6	if	if	SCONJ
ejpam-5157	208	7	n	n	PRON
ejpam-5157	208	8	is	be	AUX
ejpam-5157	208	9	odd	odd	ADJ
ejpam-5157	208	10	n	n	PRON
ejpam-5157	208	11	2	2	NUM
ejpam-5157	208	12	+	+	NUM
ejpam-5157	208	13	1	1	NUM
ejpam-5157	208	14	,	,	PUNCT
ejpam-5157	208	15	if	if	SCONJ
ejpam-5157	208	16	n	n	PRON
ejpam-5157	208	17	is	be	AUX
ejpam-5157	208	18	even	even	ADV
ejpam-5157	208	19	.	.	PUNCT
ejpam-5157	209	1	4	4	X
ejpam-5157	209	2	.	.	X
ejpam-5157	209	3	conclusion	conclusion	VERB
ejpam-5157	209	4	the	the	DET
ejpam-5157	209	5	concept	concept	NOUN
ejpam-5157	209	6	of	of	ADP
ejpam-5157	209	7	certified	certify	VERB
ejpam-5157	209	8	vertex	vertex	NOUN
ejpam-5157	209	9	cover	cover	NOUN
ejpam-5157	209	10	has	have	AUX
ejpam-5157	209	11	been	be	AUX
ejpam-5157	209	12	introduced	introduce	VERB
ejpam-5157	209	13	and	and	CCONJ
ejpam-5157	209	14	initially	initially	ADV
ejpam-5157	209	15	investigated	investigate	VERB
ejpam-5157	209	16	in	in	ADP
ejpam-5157	209	17	this	this	DET
ejpam-5157	209	18	study	study	NOUN
ejpam-5157	209	19	.	.	PUNCT
ejpam-5157	210	1	defining	define	VERB
ejpam-5157	210	2	the	the	DET
ejpam-5157	210	3	concept	concept	NOUN
ejpam-5157	210	4	introduces	introduce	VERB
ejpam-5157	210	5	a	a	DET
ejpam-5157	210	6	new	new	ADJ
ejpam-5157	210	7	idea	idea	NOUN
ejpam-5157	210	8	in	in	ADP
ejpam-5157	210	9	graph	graph	NOUN
ejpam-5157	210	10	theory	theory	NOUN
ejpam-5157	210	11	.	.	PUNCT
ejpam-5157	211	1	the	the	DET
ejpam-5157	211	2	minimum	minimum	ADJ
ejpam-5157	211	3	cardinality	cardinality	NOUN
ejpam-5157	211	4	of	of	ADP
ejpam-5157	211	5	a	a	DET
ejpam-5157	211	6	certified	certify	VERB
ejpam-5157	211	7	vertex	vertex	NOUN
ejpam-5157	211	8	cover	cover	NOUN
ejpam-5157	211	9	of	of	ADP
ejpam-5157	211	10	some	some	DET
ejpam-5157	211	11	graphs	graph	NOUN
ejpam-5157	211	12	has	have	AUX
ejpam-5157	211	13	been	be	AUX
ejpam-5157	211	14	determined	determine	VERB
ejpam-5157	211	15	in	in	ADP
ejpam-5157	211	16	this	this	DET
ejpam-5157	211	17	study	study	NOUN
ejpam-5157	211	18	.	.	PUNCT
ejpam-5157	212	1	additionally	additionally	ADV
ejpam-5157	212	2	,	,	PUNCT
ejpam-5157	212	3	characterizations	characterization	NOUN
ejpam-5157	212	4	of	of	ADP
ejpam-5157	212	5	vertex	vertex	NOUN
ejpam-5157	212	6	covering	cover	VERB
ejpam-5157	212	7	sets	set	NOUN
ejpam-5157	212	8	of	of	ADP
ejpam-5157	212	9	certain	certain	ADJ
ejpam-5157	212	10	graphs	graph	NOUN
ejpam-5157	212	11	have	have	AUX
ejpam-5157	212	12	aided	aid	VERB
ejpam-5157	212	13	in	in	ADP
ejpam-5157	212	14	determining	determine	VERB
ejpam-5157	212	15	the	the	DET
ejpam-5157	212	16	exact	exact	ADJ
ejpam-5157	212	17	values	value	NOUN
ejpam-5157	212	18	of	of	ADP
ejpam-5157	212	19	parameters	parameter	NOUN
ejpam-5157	212	20	of	of	ADP
ejpam-5157	212	21	some	some	DET
ejpam-5157	212	22	graphs	graph	NOUN
ejpam-5157	212	23	.	.	PUNCT
ejpam-5157	213	1	exploring	explore	VERB
ejpam-5157	213	2	graphs	graph	NOUN
ejpam-5157	213	3	that	that	PRON
ejpam-5157	213	4	have	have	AUX
ejpam-5157	213	5	n’t	not	PART
ejpam-5157	213	6	been	be	AUX
ejpam-5157	213	7	addressed	address	VERB
ejpam-5157	213	8	in	in	ADP
ejpam-5157	213	9	this	this	DET
ejpam-5157	213	10	study	study	NOUN
ejpam-5157	213	11	could	could	AUX
ejpam-5157	213	12	prove	prove	VERB
ejpam-5157	213	13	to	to	PART
ejpam-5157	213	14	be	be	AUX
ejpam-5157	213	15	interesting	interesting	ADJ
ejpam-5157	213	16	,	,	PUNCT
ejpam-5157	213	17	offering	offer	VERB
ejpam-5157	213	18	a	a	DET
ejpam-5157	213	19	fresh	fresh	ADJ
ejpam-5157	213	20	perspective	perspective	NOUN
ejpam-5157	213	21	on	on	ADP
ejpam-5157	213	22	the	the	DET
ejpam-5157	213	23	concept	concept	NOUN
ejpam-5157	213	24	.	.	PUNCT
ejpam-5157	214	1	investigating	investigate	VERB
ejpam-5157	214	2	these	these	DET
ejpam-5157	214	3	unexplored	unexplored	ADJ
ejpam-5157	214	4	graphs	graph	NOUN
ejpam-5157	214	5	might	might	AUX
ejpam-5157	214	6	reveal	reveal	VERB
ejpam-5157	214	7	new	new	ADJ
ejpam-5157	214	8	insights	insight	NOUN
ejpam-5157	214	9	and	and	CCONJ
ejpam-5157	214	10	provide	provide	VERB
ejpam-5157	214	11	a	a	DET
ejpam-5157	214	12	deeper	deep	ADJ
ejpam-5157	214	13	understanding	understanding	NOUN
ejpam-5157	214	14	of	of	ADP
ejpam-5157	214	15	the	the	DET
ejpam-5157	214	16	concept	concept	NOUN
ejpam-5157	214	17	.	.	PUNCT
ejpam-5157	215	1	moreover	moreover	ADV
ejpam-5157	215	2	,	,	PUNCT
ejpam-5157	215	3	interested	interested	ADJ
ejpam-5157	215	4	researchers	researcher	NOUN
ejpam-5157	215	5	may	may	AUX
ejpam-5157	215	6	study	study	VERB
ejpam-5157	215	7	the	the	DET
ejpam-5157	215	8	complexity	complexity	NOUN
ejpam-5157	215	9	and	and	CCONJ
ejpam-5157	215	10	algorithms	algorithm	NOUN
ejpam-5157	215	11	of	of	ADP
ejpam-5157	215	12	solving	solve	VERB
ejpam-5157	215	13	the	the	DET
ejpam-5157	215	14	certified	certify	VERB
ejpam-5157	215	15	vertex	vertex	NOUN
ejpam-5157	215	16	cover	cover	NOUN
ejpam-5157	215	17	number	number	NOUN
ejpam-5157	215	18	of	of	ADP
ejpam-5157	215	19	a	a	DET
ejpam-5157	215	20	graph	graph	NOUN
ejpam-5157	215	21	.	.	PUNCT
ejpam-5157	216	1	acknowledgements	acknowledgement	NOUN
ejpam-5157	216	2	the	the	DET
ejpam-5157	216	3	authors	author	NOUN
ejpam-5157	216	4	would	would	AUX
ejpam-5157	216	5	like	like	VERB
ejpam-5157	216	6	to	to	PART
ejpam-5157	216	7	thank	thank	VERB
ejpam-5157	216	8	mindanao	mindanao	PROPN
ejpam-5157	216	9	state	state	PROPN
ejpam-5157	216	10	university	university	PROPN
ejpam-5157	216	11	tawi	tawi	PROPN
ejpam-5157	216	12	-	-	PUNCT
ejpam-5157	216	13	tawi	tawi	PROPN
ejpam-5157	216	14	college	college	PROPN
ejpam-5157	216	15	of	of	ADP
ejpam-5157	216	16	technology	technology	NOUN
ejpam-5157	216	17	and	and	CCONJ
ejpam-5157	216	18	oceanography	oceanography	NOUN
ejpam-5157	216	19	,	,	PUNCT
ejpam-5157	216	20	and	and	CCONJ
ejpam-5157	216	21	ateneo	ateneo	PROPN
ejpam-5157	216	22	de	de	PROPN
ejpam-5157	216	23	davao	davao	PROPN
ejpam-5157	216	24	university	university	PROPN
ejpam-5157	216	25	for	for	ADP
ejpam-5157	216	26	funding	fund	VERB
ejpam-5157	216	27	this	this	DET
ejpam-5157	216	28	research	research	NOUN
ejpam-5157	216	29	.	.	PUNCT
ejpam-5157	217	1	references	reference	NOUN
ejpam-5157	217	2	1045	1045	NUM
ejpam-5157	217	3	also	also	ADV
ejpam-5157	217	4	,	,	PUNCT
ejpam-5157	217	5	the	the	DET
ejpam-5157	217	6	authors	author	NOUN
ejpam-5157	217	7	would	would	AUX
ejpam-5157	217	8	like	like	VERB
ejpam-5157	217	9	to	to	PART
ejpam-5157	217	10	thank	thank	VERB
ejpam-5157	217	11	the	the	DET
ejpam-5157	217	12	referees	referee	NOUN
ejpam-5157	217	13	for	for	ADP
ejpam-5157	217	14	their	their	PRON
ejpam-5157	217	15	invaluable	invaluable	ADJ
ejpam-5157	217	16	comments	comment	NOUN
ejpam-5157	217	17	and	and	CCONJ
ejpam-5157	217	18	suggestions	suggestion	NOUN
ejpam-5157	217	19	that	that	PRON
ejpam-5157	217	20	led	lead	VERB
ejpam-5157	217	21	to	to	ADP
ejpam-5157	217	22	the	the	DET
ejpam-5157	217	23	improvement	improvement	NOUN
ejpam-5157	217	24	of	of	ADP
ejpam-5157	217	25	the	the	DET
ejpam-5157	217	26	paper	paper	NOUN
ejpam-5157	217	27	.	.	PUNCT
ejpam-5157	218	1	references	reference	NOUN
ejpam-5157	218	2	[	[	X
ejpam-5157	218	3	1	1	NUM
ejpam-5157	218	4	]	]	PUNCT
ejpam-5157	218	5	v.	v.	X
ejpam-5157	218	6	bilar	bilar	PROPN
ejpam-5157	218	7	,	,	PUNCT
ejpam-5157	218	8	m.a	m.a	PROPN
ejpam-5157	218	9	.	.	PROPN
ejpam-5157	218	10	bonsocan	bonsocan	PROPN
ejpam-5157	218	11	,	,	PUNCT
ejpam-5157	218	12	j.	j.	PROPN
ejpam-5157	218	13	hassan	hassan	PROPN
ejpam-5157	218	14	,	,	PUNCT
ejpam-5157	218	15	and	and	CCONJ
ejpam-5157	218	16	s.	s.	PROPN
ejpam-5157	218	17	dagondon	dagondon	PROPN
ejpam-5157	218	18	.	.	PUNCT
ejpam-5157	219	1	vertex	vertex	NOUN
ejpam-5157	219	2	cover	cover	VERB
ejpam-5157	219	3	hop	hop	NOUN
ejpam-5157	219	4	dominating	dominating	NOUN
ejpam-5157	219	5	sets	set	NOUN
ejpam-5157	219	6	in	in	ADP
ejpam-5157	219	7	graphs	graph	NOUN
ejpam-5157	219	8	.	.	PUNCT
ejpam-5157	220	1	eur	eur	PROPN
ejpam-5157	220	2	.	.	PUNCT
ejpam-5157	221	1	j.	j.	PROPN
ejpam-5157	221	2	pure	pure	PROPN
ejpam-5157	221	3	appl	appl	PROPN
ejpam-5157	221	4	.	.	PUNCT
ejpam-5157	221	5	math	math	PROPN
ejpam-5157	221	6	.	.	PUNCT
ejpam-5157	221	7	,	,	PUNCT
ejpam-5157	221	8	17(1):93–104	17(1):93–104	NUM
ejpam-5157	221	9	,	,	PUNCT
ejpam-5157	221	10	2024	2024	NUM
ejpam-5157	221	11	.	.	PUNCT
ejpam-5157	222	1	[	[	X
ejpam-5157	222	2	2	2	X
ejpam-5157	222	3	]	]	PUNCT
ejpam-5157	222	4	j.	j.	PROPN
ejpam-5157	222	5	hassan	hassan	PROPN
ejpam-5157	222	6	,	,	PUNCT
ejpam-5157	222	7	ar	ar	PROPN
ejpam-5157	222	8	.	.	PROPN
ejpam-5157	222	9	bakkang	bakkang	PROPN
ejpam-5157	222	10	,	,	PUNCT
ejpam-5157	222	11	and	and	CCONJ
ejpam-5157	222	12	ass	ass	PROPN
ejpam-5157	222	13	.	.	PROPN
ejpam-5157	222	14	sappari	sappari	PROPN
ejpam-5157	222	15	.	.	PUNCT
ejpam-5157	223	1	j2	j2	PROPN
ejpam-5157	223	2	-	-	PUNCT
ejpam-5157	223	3	hop	hop	PROPN
ejpam-5157	223	4	domination	domination	NOUN
ejpam-5157	223	5	in	in	ADP
ejpam-5157	223	6	graphs	graph	NOUN
ejpam-5157	223	7	:	:	PUNCT
ejpam-5157	223	8	properties	property	NOUN
ejpam-5157	223	9	and	and	CCONJ
ejpam-5157	223	10	connections	connection	NOUN
ejpam-5157	223	11	with	with	ADP
ejpam-5157	223	12	other	other	ADJ
ejpam-5157	223	13	parameters	parameter	NOUN
ejpam-5157	223	14	.	.	PUNCT
ejpam-5157	224	1	eur	eur	PROPN
ejpam-5157	224	2	.	.	PUNCT
ejpam-5157	225	1	j.	j.	PROPN
ejpam-5157	225	2	pure	pure	PROPN
ejpam-5157	225	3	appl	appl	PROPN
ejpam-5157	225	4	.	.	PUNCT
ejpam-5157	225	5	math	math	PROPN
ejpam-5157	225	6	.	.	PUNCT
ejpam-5157	225	7	,	,	PUNCT
ejpam-5157	225	8	16(4):2118–2131	16(4):2118–2131	NUM
ejpam-5157	225	9	,	,	PUNCT
ejpam-5157	225	10	2023	2023	NUM
ejpam-5157	225	11	.	.	PUNCT
ejpam-5157	226	1	[	[	X
ejpam-5157	226	2	3	3	X
ejpam-5157	226	3	]	]	X
ejpam-5157	226	4	j.	j.	PROPN
ejpam-5157	226	5	hassan	hassan	PROPN
ejpam-5157	226	6	and	and	CCONJ
ejpam-5157	226	7	s.	s.	PROPN
ejpam-5157	226	8	canoy	canoy	PROPN
ejpam-5157	226	9	.	.	PUNCT
ejpam-5157	227	1	connected	connect	VERB
ejpam-5157	227	2	grundy	grundy	PROPN
ejpam-5157	227	3	hop	hop	NOUN
ejpam-5157	227	4	dominating	dominate	VERB
ejpam-5157	227	5	sequences	sequence	NOUN
ejpam-5157	227	6	in	in	ADP
ejpam-5157	227	7	graphs	graph	NOUN
ejpam-5157	227	8	.	.	PUNCT
ejpam-5157	228	1	,	,	PUNCT
ejpam-5157	228	2	.	.	PUNCT
ejpam-5157	229	1	eur	eur	PROPN
ejpam-5157	229	2	.	.	PUNCT
ejpam-5157	230	1	j.	j.	PROPN
ejpam-5157	230	2	pure	pure	PROPN
ejpam-5157	230	3	appl	appl	PROPN
ejpam-5157	230	4	.	.	PUNCT
ejpam-5157	230	5	math	math	PROPN
ejpam-5157	230	6	.	.	PUNCT
ejpam-5157	230	7	,	,	PUNCT
ejpam-5157	231	1	16(2):1212–1227	16(2):1212–1227	NUM
ejpam-5157	231	2	,	,	PUNCT
ejpam-5157	231	3	2023	2023	NUM
ejpam-5157	231	4	.	.	PUNCT
ejpam-5157	232	1	[	[	X
ejpam-5157	232	2	4	4	X
ejpam-5157	232	3	]	]	PUNCT
ejpam-5157	232	4	j.	j.	PROPN
ejpam-5157	232	5	hassan	hassan	PROPN
ejpam-5157	232	6	and	and	CCONJ
ejpam-5157	232	7	s.	s.	PROPN
ejpam-5157	232	8	canoy	canoy	PROPN
ejpam-5157	232	9	jr	jr	PROPN
ejpam-5157	232	10	.	.	PUNCT
ejpam-5157	233	1	grundy	grundy	PROPN
ejpam-5157	233	2	dominating	dominating	PROPN
ejpam-5157	233	3	and	and	CCONJ
ejpam-5157	233	4	grundy	grundy	PROPN
ejpam-5157	233	5	hop	hop	NOUN
ejpam-5157	233	6	dominating	dominate	VERB
ejpam-5157	233	7	sequences	sequence	NOUN
ejpam-5157	233	8	in	in	ADP
ejpam-5157	233	9	graphs	graph	NOUN
ejpam-5157	233	10	:	:	PUNCT
ejpam-5157	233	11	relationships	relationship	NOUN
ejpam-5157	233	12	and	and	CCONJ
ejpam-5157	233	13	some	some	DET
ejpam-5157	233	14	structural	structural	ADJ
ejpam-5157	233	15	properties	property	NOUN
ejpam-5157	233	16	.	.	PUNCT
ejpam-5157	234	1	eur	eur	PROPN
ejpam-5157	234	2	.	.	PUNCT
ejpam-5157	235	1	j.	j.	PROPN
ejpam-5157	235	2	pure	pure	PROPN
ejpam-5157	235	3	appl	appl	PROPN
ejpam-5157	235	4	.	.	PUNCT
ejpam-5157	235	5	math	math	PROPN
ejpam-5157	235	6	.	.	PUNCT
ejpam-5157	235	7	,	,	PUNCT
ejpam-5157	236	1	16(2):1154–1166	16(2):1154–1166	NUM
ejpam-5157	236	2	,	,	PUNCT
ejpam-5157	236	3	2023	2023	NUM
ejpam-5157	236	4	.	.	PUNCT
ejpam-5157	237	1	[	[	X
ejpam-5157	237	2	5	5	X
ejpam-5157	237	3	]	]	PUNCT
ejpam-5157	237	4	j.	j.	PROPN
ejpam-5157	237	5	hassan	hassan	PROPN
ejpam-5157	237	6	and	and	CCONJ
ejpam-5157	237	7	s.	s.	PROPN
ejpam-5157	237	8	canoy	canoy	PROPN
ejpam-5157	237	9	jr	jr	PROPN
ejpam-5157	237	10	.	.	PUNCT
ejpam-5157	238	1	grundy	grundy	PROPN
ejpam-5157	238	2	total	total	PROPN
ejpam-5157	238	3	hop	hop	PROPN
ejpam-5157	238	4	dominating	dominate	VERB
ejpam-5157	238	5	sequences	sequence	NOUN
ejpam-5157	238	6	in	in	ADP
ejpam-5157	238	7	graphs	graph	NOUN
ejpam-5157	238	8	.	.	PUNCT
ejpam-5157	239	1	eur	eur	PROPN
ejpam-5157	239	2	.	.	PUNCT
ejpam-5157	240	1	j.	j.	PROPN
ejpam-5157	240	2	pure	pure	PROPN
ejpam-5157	240	3	appl	appl	PROPN
ejpam-5157	240	4	.	.	PUNCT
ejpam-5157	240	5	math	math	PROPN
ejpam-5157	240	6	.	.	PUNCT
ejpam-5157	240	7	,	,	PUNCT
ejpam-5157	240	8	16(4):2597–2612	16(4):2597–2612	NUM
ejpam-5157	240	9	,	,	PUNCT
ejpam-5157	240	10	2023	2023	NUM
ejpam-5157	240	11	.	.	PUNCT
ejpam-5157	241	1	[	[	X
ejpam-5157	241	2	6	6	NUM
ejpam-5157	241	3	]	]	PUNCT
ejpam-5157	241	4	j.	j.	PROPN
ejpam-5157	241	5	hassan	hassan	PROPN
ejpam-5157	241	6	,	,	PUNCT
ejpam-5157	241	7	a.	a.	PROPN
ejpam-5157	241	8	lintasan	lintasan	PROPN
ejpam-5157	241	9	,	,	PUNCT
ejpam-5157	241	10	and	and	CCONJ
ejpam-5157	241	11	n.h	n.h	PROPN
ejpam-5157	241	12	.	.	PUNCT
ejpam-5157	242	1	mohammad	mohammad	PROPN
ejpam-5157	242	2	.	.	PUNCT
ejpam-5157	243	1	some	some	DET
ejpam-5157	243	2	properties	property	NOUN
ejpam-5157	243	3	and	and	CCONJ
ejpam-5157	243	4	realization	realization	NOUN
ejpam-5157	243	5	problems	problem	NOUN
ejpam-5157	243	6	involving	involve	VERB
ejpam-5157	243	7	connected	connected	ADJ
ejpam-5157	243	8	outer	outer	ADJ
ejpam-5157	243	9	-	-	PUNCT
ejpam-5157	243	10	hop	hop	NOUN
ejpam-5157	243	11	independent	independent	ADJ
ejpam-5157	243	12	hop	hop	NOUN
ejpam-5157	243	13	domination	domination	NOUN
ejpam-5157	243	14	in	in	ADP
ejpam-5157	243	15	graphs	graph	NOUN
ejpam-5157	243	16	.	.	PUNCT
ejpam-5157	244	1	eur	eur	PROPN
ejpam-5157	244	2	.	.	PUNCT
ejpam-5157	245	1	j.	j.	PROPN
ejpam-5157	245	2	pure	pure	PROPN
ejpam-5157	245	3	appl	appl	PROPN
ejpam-5157	245	4	.	.	PUNCT
ejpam-5157	245	5	math	math	PROPN
ejpam-5157	245	6	.	.	PUNCT
ejpam-5157	245	7	,	,	PUNCT
ejpam-5157	245	8	16(3):1848–1861	16(3):1848–1861	NUM
ejpam-5157	245	9	,	,	PUNCT
ejpam-5157	245	10	2023	2023	NUM
ejpam-5157	245	11	.	.	PUNCT
ejpam-5157	246	1	[	[	X
ejpam-5157	246	2	7	7	NUM
ejpam-5157	246	3	]	]	X
ejpam-5157	246	4	a.y	a.y	PROPN
ejpam-5157	246	5	.	.	PROPN
ejpam-5157	246	6	isahac	isahac	PROPN
ejpam-5157	246	7	,	,	PUNCT
ejpam-5157	246	8	j.	j.	PROPN
ejpam-5157	246	9	hassan	hassan	PROPN
ejpam-5157	246	10	,	,	PUNCT
ejpam-5157	246	11	ls	ls	PROPN
ejpam-5157	246	12	.	.	PROPN
ejpam-5157	246	13	laja	laja	PROPN
ejpam-5157	246	14	,	,	PUNCT
ejpam-5157	246	15	and	and	CCONJ
ejpam-5157	246	16	hb	hb	PROPN
ejpam-5157	246	17	.	.	PUNCT
ejpam-5157	247	1	copel	copel	ADJ
ejpam-5157	247	2	.	.	PUNCT
ejpam-5157	248	1	outer	outer	ADJ
ejpam-5157	248	2	-	-	PUNCT
ejpam-5157	248	3	convex	convex	ADJ
ejpam-5157	248	4	hop	hop	NOUN
ejpam-5157	248	5	domination	domination	NOUN
ejpam-5157	248	6	in	in	ADP
ejpam-5157	248	7	graphs	graph	NOUN
ejpam-5157	248	8	under	under	ADP
ejpam-5157	248	9	some	some	DET
ejpam-5157	248	10	binary	binary	ADJ
ejpam-5157	248	11	operations	operation	NOUN
ejpam-5157	248	12	.	.	PUNCT
ejpam-5157	249	1	eur	eur	PROPN
ejpam-5157	249	2	.	.	PUNCT
ejpam-5157	250	1	j.	j.	PROPN
ejpam-5157	250	2	pure	pure	PROPN
ejpam-5157	250	3	appl	appl	PROPN
ejpam-5157	250	4	.	.	PUNCT
ejpam-5157	250	5	math	math	PROPN
ejpam-5157	250	6	.	.	PUNCT
ejpam-5157	250	7	,	,	PUNCT
ejpam-5157	250	8	16(4):2035–2048	16(4):2035–2048	NUM
ejpam-5157	250	9	,	,	PUNCT
ejpam-5157	250	10	2023	2023	NUM
ejpam-5157	250	11	.	.	PUNCT
ejpam-5157	251	1	[	[	X
ejpam-5157	251	2	8	8	NUM
ejpam-5157	251	3	]	]	X
ejpam-5157	251	4	s.	s.	PROPN
ejpam-5157	251	5	canoy	canoy	PROPN
ejpam-5157	251	6	jr	jr	PROPN
ejpam-5157	251	7	.	.	PROPN
ejpam-5157	251	8	and	and	CCONJ
ejpam-5157	251	9	j.	j.	PROPN
ejpam-5157	251	10	hassan	hassan	PROPN
ejpam-5157	251	11	.	.	PUNCT
ejpam-5157	252	1	weakly	weakly	ADJ
ejpam-5157	252	2	convex	convex	VERB
ejpam-5157	252	3	hop	hop	NOUN
ejpam-5157	252	4	dominating	dominating	NOUN
ejpam-5157	252	5	sets	set	NOUN
ejpam-5157	252	6	in	in	ADP
ejpam-5157	252	7	graphs	graph	NOUN
ejpam-5157	252	8	.	.	PUNCT
ejpam-5157	253	1	eur	eur	PROPN
ejpam-5157	253	2	.	.	PUNCT
ejpam-5157	254	1	j.	j.	PROPN
ejpam-5157	254	2	pure	pure	PROPN
ejpam-5157	254	3	appl	appl	PROPN
ejpam-5157	254	4	.	.	PUNCT
ejpam-5157	254	5	math	math	PROPN
ejpam-5157	254	6	.	.	PUNCT
ejpam-5157	254	7	,	,	PUNCT
ejpam-5157	254	8	15(4):1783–1796	15(4):1783–1796	NUM
ejpam-5157	254	9	,	,	PUNCT
ejpam-5157	254	10	2022	2022	NUM
ejpam-5157	254	11	.	.	PUNCT
ejpam-5157	255	1	[	[	X
ejpam-5157	255	2	9	9	X
ejpam-5157	255	3	]	]	PUNCT
ejpam-5157	255	4	j.	j.	PROPN
ejpam-5157	255	5	manditong	manditong	PROPN
ejpam-5157	255	6	,	,	PUNCT
ejpam-5157	255	7	j.	j.	PROPN
ejpam-5157	255	8	hassan	hassan	PROPN
ejpam-5157	255	9	,	,	PUNCT
ejpam-5157	255	10	ls	ls	PROPN
ejpam-5157	255	11	laja	laja	PROPN
ejpam-5157	255	12	,	,	PUNCT
ejpam-5157	255	13	aa	aa	INTJ
ejpam-5157	255	14	.	.	PUNCT
ejpam-5157	255	15	laja	laja	PROPN
ejpam-5157	255	16	,	,	PUNCT
ejpam-5157	255	17	nhm	nhm	PROPN
ejpam-5157	255	18	.	.	PUNCT
ejpam-5157	255	19	mohammad	mohammad	PROPN
ejpam-5157	255	20	,	,	PUNCT
ejpam-5157	255	21	and	and	CCONJ
ejpam-5157	255	22	su	su	PROPN
ejpam-5157	255	23	.	.	PROPN
ejpam-5157	255	24	kamdon	kamdon	PROPN
ejpam-5157	255	25	.	.	PUNCT
ejpam-5157	256	1	connected	connected	ADJ
ejpam-5157	256	2	outer	outer	ADJ
ejpam-5157	256	3	-	-	PUNCT
ejpam-5157	256	4	hop	hop	NOUN
ejpam-5157	256	5	independent	independent	ADJ
ejpam-5157	256	6	dominating	dominating	NOUN
ejpam-5157	256	7	sets	set	NOUN
ejpam-5157	256	8	in	in	ADP
ejpam-5157	256	9	graphs	graph	NOUN
ejpam-5157	256	10	under	under	ADP
ejpam-5157	256	11	some	some	DET
ejpam-5157	256	12	binary	binary	ADJ
ejpam-5157	256	13	operations	operation	NOUN
ejpam-5157	256	14	.	.	PUNCT
ejpam-5157	257	1	eur	eur	PROPN
ejpam-5157	257	2	.	.	PUNCT
ejpam-5157	258	1	j.	j.	PROPN
ejpam-5157	258	2	pure	pure	PROPN
ejpam-5157	258	3	appl	appl	PROPN
ejpam-5157	258	4	.	.	PUNCT
ejpam-5157	258	5	math	math	PROPN
ejpam-5157	258	6	.	.	PUNCT
ejpam-5157	258	7	,	,	PUNCT
ejpam-5157	259	1	16(3):1817–1829	16(3):1817–1829	NUM
ejpam-5157	259	2	,	,	PUNCT
ejpam-5157	259	3	2023	2023	NUM
ejpam-5157	259	4	.	.	PUNCT
ejpam-5157	260	1	[	[	X
ejpam-5157	260	2	10	10	NUM
ejpam-5157	260	3	]	]	X
ejpam-5157	260	4	j.	j.	PROPN
ejpam-5157	260	5	manditong	manditong	PROPN
ejpam-5157	260	6	,	,	PUNCT
ejpam-5157	260	7	a.	a.	NOUN
ejpam-5157	260	8	tapeing	tapeing	NOUN
ejpam-5157	260	9	,	,	PUNCT
ejpam-5157	260	10	j.	j.	PROPN
ejpam-5157	260	11	hassan	hassan	PROPN
ejpam-5157	260	12	,	,	PUNCT
ejpam-5157	260	13	a.r	a.r	PROPN
ejpam-5157	260	14	.	.	PROPN
ejpam-5157	260	15	bakkang	bakkang	PROPN
ejpam-5157	260	16	,	,	PUNCT
ejpam-5157	260	17	n.h	n.h	PROPN
ejpam-5157	260	18	.	.	PUNCT
ejpam-5157	260	19	mohammad	mohammad	PROPN
ejpam-5157	260	20	,	,	PUNCT
ejpam-5157	260	21	and	and	CCONJ
ejpam-5157	260	22	s.u	s.u	PROPN
ejpam-5157	260	23	.	.	PROPN
ejpam-5157	260	24	kamdon	kamdon	PROPN
ejpam-5157	260	25	.	.	PUNCT
ejpam-5157	261	1	some	some	DET
ejpam-5157	261	2	properties	property	NOUN
ejpam-5157	261	3	of	of	ADP
ejpam-5157	261	4	zero	zero	NUM
ejpam-5157	261	5	forcing	force	VERB
ejpam-5157	261	6	hop	hop	NOUN
ejpam-5157	261	7	dominating	dominating	NOUN
ejpam-5157	261	8	sets	set	NOUN
ejpam-5157	261	9	in	in	ADP
ejpam-5157	261	10	a	a	DET
ejpam-5157	261	11	graph	graph	NOUN
ejpam-5157	261	12	.	.	PUNCT
ejpam-5157	262	1	eur	eur	PROPN
ejpam-5157	262	2	.	.	PUNCT
ejpam-5157	263	1	j.	j.	PROPN
ejpam-5157	263	2	pure	pure	PROPN
ejpam-5157	263	3	appl	appl	PROPN
ejpam-5157	263	4	.	.	PUNCT
ejpam-5157	263	5	math	math	PROPN
ejpam-5157	263	6	.	.	PUNCT
ejpam-5157	264	1	,	,	PUNCT
ejpam-5157	264	2	17(1):324–337	17(1):324–337	PROPN
ejpam-5157	264	3	,	,	PUNCT
ejpam-5157	264	4	2024	2024	NUM
ejpam-5157	264	5	.	.	PUNCT
ejpam-5157	265	1	[	[	X
ejpam-5157	265	2	11	11	NUM
ejpam-5157	265	3	]	]	PUNCT
ejpam-5157	265	4	j.	j.	PROPN
ejpam-5157	265	5	mohamad	mohamad	PROPN
ejpam-5157	265	6	and	and	CCONJ
ejpam-5157	265	7	h.	h.	PROPN
ejpam-5157	265	8	rara	rara	PROPN
ejpam-5157	265	9	.	.	PUNCT
ejpam-5157	266	1	on	on	ADP
ejpam-5157	266	2	resolving	resolve	VERB
ejpam-5157	266	3	hop	hop	NOUN
ejpam-5157	266	4	domination	domination	NOUN
ejpam-5157	266	5	in	in	ADP
ejpam-5157	266	6	graphs	graph	NOUN
ejpam-5157	266	7	.	.	PUNCT
ejpam-5157	267	1	eur	eur	PROPN
ejpam-5157	267	2	.	.	PUNCT
ejpam-5157	268	1	j.	j.	PROPN
ejpam-5157	268	2	pure	pure	PROPN
ejpam-5157	268	3	appl	appl	PROPN
ejpam-5157	268	4	.	.	PUNCT
ejpam-5157	268	5	math	math	PROPN
ejpam-5157	268	6	.	.	PUNCT
ejpam-5157	268	7	,	,	PUNCT
ejpam-5157	268	8	14(1):324–337	14(1):324–337	PROPN
ejpam-5157	268	9	,	,	PUNCT
ejpam-5157	268	10	2021	2021	NUM
ejpam-5157	268	11	.	.	PUNCT
ejpam-5157	269	1	[	[	X
ejpam-5157	269	2	12	12	NUM
ejpam-5157	269	3	]	]	PUNCT
ejpam-5157	269	4	t.	t.	NOUN
ejpam-5157	269	5	sitthiwirattham	sitthiwirattham	NOUN
ejpam-5157	269	6	.	.	PUNCT
ejpam-5157	270	1	vertex	vertex	NOUN
ejpam-5157	270	2	covering	covering	NOUN
ejpam-5157	270	3	and	and	CCONJ
ejpam-5157	270	4	independent	independent	ADJ
ejpam-5157	270	5	number	number	NOUN
ejpam-5157	270	6	on	on	ADP
ejpam-5157	270	7	difference	difference	NOUN
ejpam-5157	270	8	graphs	graph	NOUN
ejpam-5157	270	9	.	.	PUNCT
ejpam-5157	271	1	international	international	ADJ
ejpam-5157	271	2	journal	journal	NOUN
ejpam-5157	271	3	of	of	ADP
ejpam-5157	271	4	pure	pure	ADJ
ejpam-5157	271	5	and	and	CCONJ
ejpam-5157	271	6	applied	applied	ADJ
ejpam-5157	271	7	mathematics	mathematic	NOUN
ejpam-5157	271	8	.	.	PUNCT
ejpam-5157	271	9	,	,	PUNCT
ejpam-5157	272	1	77(4):543–547	77(4):543–547	NOUN
ejpam-5157	272	2	,	,	PUNCT
ejpam-5157	272	3	2012	2012	NUM
ejpam-5157	272	4	.	.	PUNCT
ejpam-5157	273	1	[	[	X
ejpam-5157	273	2	13	13	NUM
ejpam-5157	273	3	]	]	X
ejpam-5157	273	4	j.	j.	PROPN
ejpam-5157	273	5	uy	uy	PROPN
ejpam-5157	273	6	.	.	PUNCT
ejpam-5157	274	1	revisiting	revisit	VERB
ejpam-5157	274	2	the	the	DET
ejpam-5157	274	3	vertex	vertex	NOUN
ejpam-5157	274	4	cover	cover	NOUN
ejpam-5157	274	5	of	of	ADP
ejpam-5157	274	6	graphs	graph	NOUN
ejpam-5157	274	7	.	.	PUNCT
ejpam-5157	275	1	applied	apply	VERB
ejpam-5157	275	2	mathematical	mathematical	ADJ
ejpam-5157	275	3	sciences	science	NOUN
ejpam-5157	275	4	.	.	PUNCT
ejpam-5157	275	5	,	,	PUNCT
ejpam-5157	275	6	115(9):5707–5714	115(9):5707–5714	PROPN
ejpam-5157	275	7	,	,	PUNCT
ejpam-5157	275	8	2015	2015	NUM
ejpam-5157	275	9	.	.	PUNCT