id	sid	tid	token	lemma	pos
ejpam-6076	1	1	european	european	PROPN
ejpam-6076	1	2	journal	journal	PROPN
ejpam-6076	1	3	of	of	ADP
ejpam-6076	1	4	pure	pure	ADJ
ejpam-6076	1	5	and	and	CCONJ
ejpam-6076	1	6	applied	applied	ADJ
ejpam-6076	1	7	mathematics	mathematic	NOUN
ejpam-6076	1	8	2025	2025	NUM
ejpam-6076	1	9	,	,	PUNCT
ejpam-6076	1	10	vol	vol	NOUN
ejpam-6076	1	11	.	.	PROPN
ejpam-6076	1	12	18	18	NUM
ejpam-6076	1	13	,	,	PUNCT
ejpam-6076	1	14	issue	issue	NOUN
ejpam-6076	1	15	3	3	NUM
ejpam-6076	1	16	,	,	PUNCT
ejpam-6076	1	17	article	article	NOUN
ejpam-6076	1	18	number	number	NOUN
ejpam-6076	1	19	6076	6076	NUM
ejpam-6076	1	20	issn	issn	PROPN
ejpam-6076	1	21	1307	1307	NUM
ejpam-6076	1	22	-	-	SYM
ejpam-6076	1	23	5543	5543	NUM
ejpam-6076	1	24	–	–	PUNCT
ejpam-6076	1	25	ejpam.com	ejpam.com	X
ejpam-6076	1	26	published	publish	VERB
ejpam-6076	1	27	by	by	ADP
ejpam-6076	1	28	new	new	PROPN
ejpam-6076	1	29	york	york	PROPN
ejpam-6076	1	30	business	business	PROPN
ejpam-6076	1	31	global	global	PROPN
ejpam-6076	1	32	on	on	ADP
ejpam-6076	1	33	certified	certify	VERB
ejpam-6076	1	34	perfect	perfect	ADJ
ejpam-6076	1	35	domination	domination	NOUN
ejpam-6076	1	36	in	in	ADP
ejpam-6076	1	37	graphs	graph	NOUN
ejpam-6076	1	38	jamil	jamil	PROPN
ejpam-6076	1	39	j.	j.	PROPN
ejpam-6076	1	40	hamja1,∗	hamja1,∗	PROPN
ejpam-6076	1	41	,	,	PUNCT
ejpam-6076	1	42	amy	amy	PROPN
ejpam-6076	1	43	a.	a.	PROPN
ejpam-6076	1	44	laja1	laja1	PROPN
ejpam-6076	1	45	,	,	PUNCT
ejpam-6076	1	46	hounam	hounam	PROPN
ejpam-6076	1	47	b.	b.	PROPN
ejpam-6076	1	48	copel1	copel1	PROPN
ejpam-6076	1	49	,	,	PUNCT
ejpam-6076	1	50	bayah	bayah	PROPN
ejpam-6076	1	51	j.	j.	PROPN
ejpam-6076	1	52	amiruddin	amiruddin	PROPN
ejpam-6076	1	53	-	-	PUNCT
ejpam-6076	1	54	rajik1	rajik1	PROPN
ejpam-6076	1	55	,	,	PUNCT
ejpam-6076	1	56	nurijam	nurijam	PROPN
ejpam-6076	1	57	hanna	hanna	PROPN
ejpam-6076	1	58	m.	m.	PROPN
ejpam-6076	1	59	mohammad1	mohammad1	PROPN
ejpam-6076	1	60	1	1	NUM
ejpam-6076	1	61	department	department	NOUN
ejpam-6076	1	62	of	of	ADP
ejpam-6076	1	63	mathematics	mathematic	NOUN
ejpam-6076	1	64	,	,	PUNCT
ejpam-6076	1	65	college	college	NOUN
ejpam-6076	1	66	of	of	ADP
ejpam-6076	1	67	arts	art	NOUN
ejpam-6076	1	68	and	and	CCONJ
ejpam-6076	1	69	sciences	science	NOUN
ejpam-6076	1	70	,	,	PUNCT
ejpam-6076	1	71	msu	msu	PROPN
ejpam-6076	1	72	-	-	PUNCT
ejpam-6076	1	73	tawi	tawi	NOUN
ejpam-6076	1	74	-	-	PUNCT
ejpam-6076	1	75	tawi	tawi	NOUN
ejpam-6076	1	76	college	college	PROPN
ejpam-6076	1	77	of	of	ADP
ejpam-6076	1	78	technology	technology	NOUN
ejpam-6076	1	79	and	and	CCONJ
ejpam-6076	1	80	oceanography	oceanography	NOUN
ejpam-6076	1	81	,	,	PUNCT
ejpam-6076	1	82	7500	7500	NUM
ejpam-6076	1	83	tawi	tawi	NOUN
ejpam-6076	1	84	-	-	PUNCT
ejpam-6076	1	85	tawi	tawi	NOUN
ejpam-6076	1	86	,	,	PUNCT
ejpam-6076	1	87	philippines	philippine	NOUN
ejpam-6076	1	88	.	.	PUNCT
ejpam-6076	2	1	abstract	abstract	ADJ
ejpam-6076	2	2	.	.	PUNCT
ejpam-6076	3	1	let	let	VERB
ejpam-6076	3	2	g	g	PROPN
ejpam-6076	3	3	=	=	SYM
ejpam-6076	3	4	(	(	PUNCT
ejpam-6076	3	5	v	v	NOUN
ejpam-6076	3	6	(	(	PUNCT
ejpam-6076	3	7	g	g	NOUN
ejpam-6076	3	8	)	)	PUNCT
ejpam-6076	3	9	,	,	PUNCT
ejpam-6076	3	10	e(g	e(g	PROPN
ejpam-6076	3	11	)	)	PUNCT
ejpam-6076	3	12	)	)	PUNCT
ejpam-6076	4	1	be	be	AUX
ejpam-6076	4	2	a	a	DET
ejpam-6076	4	3	simple	simple	ADJ
ejpam-6076	4	4	graph	graph	NOUN
ejpam-6076	4	5	.	.	PUNCT
ejpam-6076	5	1	a	a	DET
ejpam-6076	5	2	perfect	perfect	ADJ
ejpam-6076	5	3	dominating	dominating	NOUN
ejpam-6076	5	4	set	set	VERB
ejpam-6076	5	5	j	j	PROPN
ejpam-6076	5	6	⊆	⊆	NUM
ejpam-6076	5	7	v	v	NOUN
ejpam-6076	5	8	(	(	PUNCT
ejpam-6076	5	9	g	g	NOUN
ejpam-6076	5	10	)	)	PUNCT
ejpam-6076	5	11	is	be	AUX
ejpam-6076	5	12	called	call	VERB
ejpam-6076	5	13	a	a	DET
ejpam-6076	5	14	certified	certify	VERB
ejpam-6076	5	15	perfect	perfect	ADJ
ejpam-6076	5	16	dominating	dominating	NOUN
ejpam-6076	5	17	set	set	NOUN
ejpam-6076	5	18	of	of	ADP
ejpam-6076	5	19	g	g	PROPN
ejpam-6076	5	20	if	if	SCONJ
ejpam-6076	5	21	each	each	DET
ejpam-6076	5	22	vertex	vertex	VERB
ejpam-6076	5	23	a	a	DET
ejpam-6076	5	24	∈	∈	PROPN
ejpam-6076	5	25	j	j	PROPN
ejpam-6076	5	26	has	have	AUX
ejpam-6076	5	27	either	either	CCONJ
ejpam-6076	5	28	no	no	DET
ejpam-6076	5	29	neighbors	neighbor	NOUN
ejpam-6076	5	30	or	or	CCONJ
ejpam-6076	5	31	at	at	ADP
ejpam-6076	5	32	least	least	ADV
ejpam-6076	5	33	two	two	NUM
ejpam-6076	5	34	neighbors	neighbor	NOUN
ejpam-6076	5	35	in	in	ADP
ejpam-6076	5	36	v	v	NOUN
ejpam-6076	5	37	(	(	PUNCT
ejpam-6076	5	38	g	g	NOUN
ejpam-6076	5	39	)	)	PUNCT
ejpam-6076	5	40	\j	\j	NOUN
ejpam-6076	5	41	.	.	PUNCT
ejpam-6076	6	1	the	the	DET
ejpam-6076	6	2	certified	certify	VERB
ejpam-6076	6	3	perfect	perfect	ADJ
ejpam-6076	6	4	domination	domination	NOUN
ejpam-6076	6	5	number	number	NOUN
ejpam-6076	6	6	of	of	ADP
ejpam-6076	6	7	g	g	PROPN
ejpam-6076	6	8	γcerp(g	γcerp(g	NOUN
ejpam-6076	6	9	)	)	PUNCT
ejpam-6076	6	10	represents	represent	VERB
ejpam-6076	6	11	the	the	DET
ejpam-6076	6	12	smallest	small	ADJ
ejpam-6076	6	13	size	size	NOUN
ejpam-6076	6	14	of	of	ADP
ejpam-6076	6	15	a	a	DET
ejpam-6076	6	16	certified	certify	VERB
ejpam-6076	6	17	perfect	perfect	ADJ
ejpam-6076	6	18	dominating	dominating	NOUN
ejpam-6076	6	19	set	set	VERB
ejpam-6076	6	20	in	in	ADP
ejpam-6076	6	21	g.	g.	PROPN
ejpam-6076	6	22	a	a	DET
ejpam-6076	6	23	certified	certify	VERB
ejpam-6076	6	24	perfect	perfect	ADJ
ejpam-6076	6	25	dominating	dominating	NOUN
ejpam-6076	6	26	set	set	NOUN
ejpam-6076	6	27	of	of	ADP
ejpam-6076	6	28	g	g	PROPN
ejpam-6076	6	29	that	that	PRON
ejpam-6076	6	30	attains	attain	VERB
ejpam-6076	6	31	this	this	DET
ejpam-6076	6	32	minimum	minimum	ADJ
ejpam-6076	6	33	size	size	NOUN
ejpam-6076	6	34	,	,	PUNCT
ejpam-6076	6	35	i.e.	i.e.	X
ejpam-6076	6	36	,	,	PUNCT
ejpam-6076	6	37	|j	|j	NOUN
ejpam-6076	6	38	|	|	NOUN
ejpam-6076	6	39	=	=	SYM
ejpam-6076	6	40	γcerp(g	γcerp(g	PROPN
ejpam-6076	6	41	)	)	PUNCT
ejpam-6076	6	42	is	be	AUX
ejpam-6076	6	43	referred	refer	VERB
ejpam-6076	6	44	to	to	ADP
ejpam-6076	6	45	as	as	ADP
ejpam-6076	6	46	a	a	DET
ejpam-6076	6	47	γcerp	γcerp	NOUN
ejpam-6076	6	48	-	-	PUNCT
ejpam-6076	6	49	set	set	NOUN
ejpam-6076	6	50	.	.	PUNCT
ejpam-6076	7	1	in	in	ADP
ejpam-6076	7	2	this	this	DET
ejpam-6076	7	3	paper	paper	NOUN
ejpam-6076	7	4	,	,	PUNCT
ejpam-6076	7	5	we	we	PRON
ejpam-6076	7	6	first	first	ADV
ejpam-6076	7	7	present	present	VERB
ejpam-6076	7	8	some	some	DET
ejpam-6076	7	9	upper	upper	ADJ
ejpam-6076	7	10	bounds	bound	NOUN
ejpam-6076	7	11	for	for	ADP
ejpam-6076	7	12	the	the	DET
ejpam-6076	7	13	certified	certify	VERB
ejpam-6076	7	14	perfect	perfect	ADJ
ejpam-6076	7	15	domination	domination	NOUN
ejpam-6076	7	16	number	number	NOUN
ejpam-6076	7	17	of	of	ADP
ejpam-6076	7	18	g	g	NOUN
ejpam-6076	7	19	,	,	PUNCT
ejpam-6076	7	20	investigate	investigate	VERB
ejpam-6076	7	21	the	the	DET
ejpam-6076	7	22	relationship	relationship	NOUN
ejpam-6076	7	23	between	between	ADP
ejpam-6076	7	24	certified	certified	ADJ
ejpam-6076	7	25	domination	domination	NOUN
ejpam-6076	7	26	and	and	CCONJ
ejpam-6076	7	27	certified	certify	VERB
ejpam-6076	7	28	perfect	perfect	ADJ
ejpam-6076	7	29	domination	domination	NOUN
ejpam-6076	7	30	parameters	parameter	NOUN
ejpam-6076	7	31	,	,	PUNCT
ejpam-6076	7	32	and	and	CCONJ
ejpam-6076	7	33	determine	determine	VERB
ejpam-6076	7	34	graphs	graph	NOUN
ejpam-6076	7	35	with	with	ADP
ejpam-6076	7	36	small	small	ADJ
ejpam-6076	7	37	and	and	CCONJ
ejpam-6076	7	38	large	large	ADJ
ejpam-6076	7	39	values	value	NOUN
ejpam-6076	7	40	of	of	ADP
ejpam-6076	7	41	these	these	DET
ejpam-6076	7	42	parameters	parameter	NOUN
ejpam-6076	7	43	.	.	PUNCT
ejpam-6076	8	1	secondly	secondly	ADV
ejpam-6076	8	2	,	,	PUNCT
ejpam-6076	8	3	we	we	PRON
ejpam-6076	8	4	characterize	characterize	VERB
ejpam-6076	8	5	the	the	DET
ejpam-6076	8	6	graphs	graph	NOUN
ejpam-6076	8	7	with	with	ADP
ejpam-6076	8	8	γcerp(g	γcerp(g	PROPN
ejpam-6076	8	9	)	)	PUNCT
ejpam-6076	8	10	=	=	SYM
ejpam-6076	8	11	n	n	NOUN
ejpam-6076	8	12	and	and	CCONJ
ejpam-6076	8	13	γcerp(g	γcerp(g	ADJ
ejpam-6076	8	14	)	)	PUNCT
ejpam-6076	8	15	=	=	SYM
ejpam-6076	8	16	γcer(g	γcer(g	NOUN
ejpam-6076	8	17	)	)	PUNCT
ejpam-6076	8	18	.	.	PUNCT
ejpam-6076	9	1	finally	finally	ADV
ejpam-6076	9	2	,	,	PUNCT
ejpam-6076	9	3	we	we	PRON
ejpam-6076	9	4	characterize	characterize	VERB
ejpam-6076	9	5	the	the	DET
ejpam-6076	9	6	certified	certify	VERB
ejpam-6076	9	7	perfect	perfect	ADJ
ejpam-6076	9	8	dominating	dominating	NOUN
ejpam-6076	9	9	set	set	VERB
ejpam-6076	9	10	under	under	ADP
ejpam-6076	9	11	the	the	DET
ejpam-6076	9	12	lexicographic	lexicographic	ADJ
ejpam-6076	9	13	and	and	CCONJ
ejpam-6076	9	14	cartesian	cartesian	ADJ
ejpam-6076	9	15	products	product	NOUN
ejpam-6076	9	16	of	of	ADP
ejpam-6076	9	17	two	two	NUM
ejpam-6076	9	18	graphs	graph	NOUN
ejpam-6076	9	19	,	,	PUNCT
ejpam-6076	9	20	determine	determine	VERB
ejpam-6076	9	21	its	its	PRON
ejpam-6076	9	22	certified	certify	VERB
ejpam-6076	9	23	perfect	perfect	ADJ
ejpam-6076	9	24	domination	domination	NOUN
ejpam-6076	9	25	number	number	NOUN
ejpam-6076	9	26	,	,	PUNCT
ejpam-6076	9	27	and	and	CCONJ
ejpam-6076	9	28	identify	identify	VERB
ejpam-6076	9	29	a	a	DET
ejpam-6076	9	30	non	non	ADJ
ejpam-6076	9	31	-	-	ADJ
ejpam-6076	9	32	γcerp	γcerp	ADJ
ejpam-6076	9	33	-	-	PUNCT
ejpam-6076	9	34	graph	graph	NOUN
ejpam-6076	9	35	under	under	ADP
ejpam-6076	9	36	these	these	DET
ejpam-6076	9	37	binary	binary	ADJ
ejpam-6076	9	38	operations	operation	NOUN
ejpam-6076	9	39	.	.	PUNCT
ejpam-6076	10	1	2020	2020	NUM
ejpam-6076	10	2	mathematics	mathematic	NOUN
ejpam-6076	10	3	subject	subject	NOUN
ejpam-6076	10	4	classifications	classification	NOUN
ejpam-6076	10	5	:	:	PUNCT
ejpam-6076	10	6	05c69	05c69	X
ejpam-6076	10	7	key	key	ADJ
ejpam-6076	10	8	words	word	NOUN
ejpam-6076	10	9	and	and	CCONJ
ejpam-6076	10	10	phrases	phrase	NOUN
ejpam-6076	10	11	:	:	PUNCT
ejpam-6076	10	12	certified	certified	ADJ
ejpam-6076	10	13	domination	domination	NOUN
ejpam-6076	10	14	,	,	PUNCT
ejpam-6076	10	15	perfect	perfect	ADJ
ejpam-6076	10	16	domination	domination	NOUN
ejpam-6076	10	17	,	,	PUNCT
ejpam-6076	10	18	certified	certify	VERB
ejpam-6076	10	19	perfect	perfect	ADJ
ejpam-6076	10	20	domination	domination	NOUN
ejpam-6076	10	21	,	,	PUNCT
ejpam-6076	10	22	corona	corona	PROPN
ejpam-6076	10	23	,	,	PUNCT
ejpam-6076	10	24	lexicographic	lexicographic	ADJ
ejpam-6076	10	25	product	product	NOUN
ejpam-6076	10	26	1	1	NUM
ejpam-6076	10	27	.	.	PUNCT
ejpam-6076	10	28	introduction	introduction	NOUN
ejpam-6076	10	29	in	in	ADP
ejpam-6076	10	30	2020	2020	NUM
ejpam-6076	10	31	,	,	PUNCT
ejpam-6076	10	32	dettlaff	dettlaff	VERB
ejpam-6076	10	33	et	et	PROPN
ejpam-6076	10	34	al	al	PROPN
ejpam-6076	10	35	.	.	PUNCT
ejpam-6076	11	1	[	[	X
ejpam-6076	11	2	1	1	NUM
ejpam-6076	11	3	,	,	PUNCT
ejpam-6076	11	4	2	2	NUM
ejpam-6076	11	5	]	]	PUNCT
ejpam-6076	11	6	introduced	introduce	VERB
ejpam-6076	11	7	the	the	DET
ejpam-6076	11	8	concept	concept	NOUN
ejpam-6076	11	9	of	of	ADP
ejpam-6076	11	10	certified	certified	ADJ
ejpam-6076	11	11	domination	domination	NOUN
ejpam-6076	11	12	in	in	ADP
ejpam-6076	11	13	graphs	graph	NOUN
ejpam-6076	11	14	.	.	PUNCT
ejpam-6076	12	1	they	they	PRON
ejpam-6076	12	2	determined	determine	VERB
ejpam-6076	12	3	the	the	DET
ejpam-6076	12	4	exact	exact	ADJ
ejpam-6076	12	5	values	value	NOUN
ejpam-6076	12	6	of	of	ADP
ejpam-6076	12	7	the	the	DET
ejpam-6076	12	8	certified	certify	VERB
ejpam-6076	12	9	domination	domination	NOUN
ejpam-6076	12	10	number	number	NOUN
ejpam-6076	12	11	for	for	ADP
ejpam-6076	12	12	certain	certain	ADJ
ejpam-6076	12	13	classes	class	NOUN
ejpam-6076	12	14	of	of	ADP
ejpam-6076	12	15	graphs	graph	NOUN
ejpam-6076	12	16	and	and	CCONJ
ejpam-6076	12	17	provided	provide	VERB
ejpam-6076	12	18	upper	upper	ADJ
ejpam-6076	12	19	bounds	bound	NOUN
ejpam-6076	12	20	for	for	ADP
ejpam-6076	12	21	this	this	DET
ejpam-6076	12	22	parameter	parameter	NOUN
ejpam-6076	12	23	in	in	ADP
ejpam-6076	12	24	arbitrary	arbitrary	ADJ
ejpam-6076	12	25	graphs	graph	NOUN
ejpam-6076	12	26	.	.	PUNCT
ejpam-6076	13	1	additionally	additionally	ADV
ejpam-6076	13	2	,	,	PUNCT
ejpam-6076	13	3	they	they	PRON
ejpam-6076	13	4	characterized	characterize	VERB
ejpam-6076	13	5	a	a	DET
ejpam-6076	13	6	broad	broad	ADJ
ejpam-6076	13	7	class	class	NOUN
ejpam-6076	13	8	of	of	ADP
ejpam-6076	13	9	graphs	graph	NOUN
ejpam-6076	13	10	in	in	ADP
ejpam-6076	13	11	which	which	PRON
ejpam-6076	13	12	the	the	DET
ejpam-6076	13	13	domination	domination	NOUN
ejpam-6076	13	14	number	number	NOUN
ejpam-6076	13	15	and	and	CCONJ
ejpam-6076	13	16	the	the	DET
ejpam-6076	13	17	certified	certify	VERB
ejpam-6076	13	18	domination	domination	NOUN
ejpam-6076	13	19	number	number	NOUN
ejpam-6076	13	20	are	be	AUX
ejpam-6076	13	21	equal	equal	ADJ
ejpam-6076	13	22	and	and	CCONJ
ejpam-6076	13	23	identified	identify	VERB
ejpam-6076	13	24	graphs	graph	NOUN
ejpam-6076	13	25	with	with	ADP
ejpam-6076	13	26	large	large	ADJ
ejpam-6076	13	27	certified	certified	ADJ
ejpam-6076	13	28	domination	domination	NOUN
ejpam-6076	13	29	numbers	number	NOUN
ejpam-6076	13	30	.	.	PUNCT
ejpam-6076	14	1	they	they	PRON
ejpam-6076	14	2	also	also	ADV
ejpam-6076	14	3	analyzed	analyze	VERB
ejpam-6076	14	4	how	how	SCONJ
ejpam-6076	14	5	the	the	DET
ejpam-6076	14	6	certified	certified	ADJ
ejpam-6076	14	7	domination	domination	NOUN
ejpam-6076	14	8	number	number	NOUN
ejpam-6076	14	9	changes	change	NOUN
ejpam-6076	14	10	when	when	SCONJ
ejpam-6076	14	11	a	a	DET
ejpam-6076	14	12	graph	graph	NOUN
ejpam-6076	14	13	is	be	AUX
ejpam-6076	14	14	modified	modify	VERB
ejpam-6076	14	15	by	by	ADP
ejpam-6076	14	16	adding	add	VERB
ejpam-6076	14	17	or	or	CCONJ
ejpam-6076	14	18	deleting	delete	VERB
ejpam-6076	14	19	an	an	DET
ejpam-6076	14	20	edge	edge	NOUN
ejpam-6076	14	21	or	or	CCONJ
ejpam-6076	14	22	a	a	DET
ejpam-6076	14	23	vertex	vertex	NOUN
ejpam-6076	14	24	.	.	PUNCT
ejpam-6076	15	1	moreover	moreover	ADV
ejpam-6076	15	2	,	,	PUNCT
ejpam-6076	15	3	they	they	PRON
ejpam-6076	15	4	established	establish	VERB
ejpam-6076	15	5	nordhaus	nordhaus	NOUN
ejpam-6076	15	6	–	–	PUNCT
ejpam-6076	15	7	gaddum	gaddum	NOUN
ejpam-6076	15	8	type	type	NOUN
ejpam-6076	15	9	inequalities	inequality	NOUN
ejpam-6076	15	10	for	for	ADP
ejpam-6076	15	11	the	the	DET
ejpam-6076	15	12	certified	certify	VERB
ejpam-6076	15	13	domination	domination	NOUN
ejpam-6076	15	14	number	number	NOUN
ejpam-6076	15	15	and	and	CCONJ
ejpam-6076	15	16	explored	explore	VERB
ejpam-6076	15	17	∗corresponding	∗corresponde	VERB
ejpam-6076	15	18	author	author	NOUN
ejpam-6076	15	19	.	.	PUNCT
ejpam-6076	16	1	doi	doi	NOUN
ejpam-6076	16	2	:	:	PUNCT
ejpam-6076	16	3	https://doi.org/10.29020/nybg.ejpam.v18i3.6076	https://doi.org/10.29020/nybg.ejpam.v18i3.6076	ADJ
ejpam-6076	16	4	email	email	NOUN
ejpam-6076	16	5	addresses	address	NOUN
ejpam-6076	16	6	:	:	PUNCT
ejpam-6076	16	7	jamilhamja@msutawi-tawi.edu.ph	jamilhamja@msutawi-tawi.edu.ph	PROPN
ejpam-6076	16	8	(	(	PUNCT
ejpam-6076	16	9	j.	j.	PROPN
ejpam-6076	16	10	j.	j.	PROPN
ejpam-6076	16	11	hamja	hamja	PROPN
ejpam-6076	16	12	)	)	PUNCT
ejpam-6076	16	13	,	,	PUNCT
ejpam-6076	16	14	amylaja@msutawi-tawi.edu.ph	amylaja@msutawi-tawi.edu.ph	PROPN
ejpam-6076	16	15	(	(	PUNCT
ejpam-6076	16	16	a.	a.	NOUN
ejpam-6076	16	17	a.	a.	NOUN
ejpam-6076	16	18	laja	laja	PROPN
ejpam-6076	16	19	)	)	PUNCT
ejpam-6076	16	20	,	,	PUNCT
ejpam-6076	16	21	hounamcopel@msutawi-tawi.edu.ph	hounamcopel@msutawi-tawi.edu.ph	PROPN
ejpam-6076	16	22	(	(	PUNCT
ejpam-6076	16	23	h.	h.	PROPN
ejpam-6076	16	24	b.	b.	PROPN
ejpam-6076	16	25	copel	copel	PROPN
ejpam-6076	16	26	)	)	PUNCT
ejpam-6076	16	27	,	,	PUNCT
ejpam-6076	16	28	bayahamiruddin@msutawi-tawi.edu.ph	bayahamiruddin@msutawi-tawi.edu.ph	PROPN
ejpam-6076	16	29	(	(	PUNCT
ejpam-6076	16	30	b.	b.	PROPN
ejpam-6076	16	31	j.	j.	PROPN
ejpam-6076	16	32	amiruddin	amiruddin	PROPN
ejpam-6076	16	33	-	-	PUNCT
ejpam-6076	16	34	rajik	rajik	NOUN
ejpam-6076	16	35	)	)	PUNCT
ejpam-6076	16	36	,	,	PUNCT
ejpam-6076	16	37	hannamohammad@msutawi-tawi.edu.ph	hannamohammad@msutawi-tawi.edu.ph	PROPN
ejpam-6076	16	38	(	(	PUNCT
ejpam-6076	16	39	n.	n.	PROPN
ejpam-6076	16	40	h.	h.	PROPN
ejpam-6076	16	41	m.	m.	PROPN
ejpam-6076	16	42	mohammad	mohammad	PROPN
ejpam-6076	16	43	)	)	PUNCT
ejpam-6076	16	44	https://www.ejpam.com	https://www.ejpam.com	X
ejpam-6076	16	45	1	1	NUM
ejpam-6076	16	46	copyright	copyright	NOUN
ejpam-6076	16	47	:	:	PUNCT
ejpam-6076	17	1	©	©	PROPN
ejpam-6076	17	2	2025	2025	NUM
ejpam-6076	17	3	the	the	DET
ejpam-6076	17	4	author(s	author(s	NOUN
ejpam-6076	17	5	)	)	PUNCT
ejpam-6076	17	6	.	.	PUNCT
ejpam-6076	18	1	(	(	PUNCT
ejpam-6076	18	2	cc	cc	NOUN
ejpam-6076	18	3	by	by	ADP
ejpam-6076	18	4	-	-	PUNCT
ejpam-6076	18	5	nc	nc	PROPN
ejpam-6076	18	6	4.0	4.0	NUM
ejpam-6076	18	7	)	)	PUNCT
ejpam-6076	18	8	j.	j.	PROPN
ejpam-6076	18	9	j.	j.	PROPN
ejpam-6076	18	10	hamja	hamja	PROPN
ejpam-6076	19	1	et	et	PROPN
ejpam-6076	19	2	al	al	PROPN
ejpam-6076	19	3	.	.	PUNCT
ejpam-6076	19	4	/	/	SYM
ejpam-6076	19	5	eur	eur	PROPN
ejpam-6076	19	6	.	.	PUNCT
ejpam-6076	20	1	j.	j.	PROPN
ejpam-6076	20	2	pure	pure	PROPN
ejpam-6076	20	3	appl	appl	PROPN
ejpam-6076	20	4	.	.	PROPN
ejpam-6076	20	5	math	math	PROPN
ejpam-6076	20	6	,	,	PUNCT
ejpam-6076	20	7	18	18	NUM
ejpam-6076	20	8	(	(	PUNCT
ejpam-6076	20	9	3	3	NUM
ejpam-6076	20	10	)	)	PUNCT
ejpam-6076	20	11	(	(	PUNCT
ejpam-6076	20	12	2025	2025	NUM
ejpam-6076	20	13	)	)	PUNCT
ejpam-6076	20	14	,	,	PUNCT
ejpam-6076	20	15	6076	6076	NUM
ejpam-6076	20	16	2	2	NUM
ejpam-6076	20	17	of	of	ADP
ejpam-6076	20	18	13	13	NUM
ejpam-6076	20	19	the	the	DET
ejpam-6076	20	20	relationships	relationship	NOUN
ejpam-6076	20	21	among	among	ADP
ejpam-6076	20	22	domination	domination	NOUN
ejpam-6076	20	23	,	,	PUNCT
ejpam-6076	20	24	upper	upper	ADJ
ejpam-6076	20	25	domination	domination	NOUN
ejpam-6076	20	26	,	,	PUNCT
ejpam-6076	20	27	certified	certified	ADJ
ejpam-6076	20	28	domination	domination	NOUN
ejpam-6076	20	29	,	,	PUNCT
ejpam-6076	20	30	and	and	CCONJ
ejpam-6076	20	31	upper	upper	ADJ
ejpam-6076	20	32	certified	certify	VERB
ejpam-6076	20	33	domination	domination	NOUN
ejpam-6076	20	34	numbers	number	NOUN
ejpam-6076	20	35	.	.	PUNCT
ejpam-6076	21	1	in	in	ADP
ejpam-6076	21	2	2023	2023	NUM
ejpam-6076	21	3	,	,	PUNCT
ejpam-6076	21	4	hamja	hamja	VERB
ejpam-6076	21	5	[	[	X
ejpam-6076	21	6	3	3	X
ejpam-6076	21	7	]	]	PUNCT
ejpam-6076	21	8	introduced	introduce	VERB
ejpam-6076	21	9	the	the	DET
ejpam-6076	21	10	concept	concept	NOUN
ejpam-6076	21	11	of	of	ADP
ejpam-6076	21	12	certified	certify	VERB
ejpam-6076	21	13	perfect	perfect	ADJ
ejpam-6076	21	14	domination	domination	NOUN
ejpam-6076	21	15	in	in	ADP
ejpam-6076	21	16	graphs	graph	NOUN
ejpam-6076	21	17	.	.	PUNCT
ejpam-6076	22	1	he	he	PRON
ejpam-6076	22	2	characterized	characterize	VERB
ejpam-6076	22	3	the	the	DET
ejpam-6076	22	4	certified	certify	VERB
ejpam-6076	22	5	perfect	perfect	ADJ
ejpam-6076	22	6	dominating	dominating	NOUN
ejpam-6076	22	7	set	set	NOUN
ejpam-6076	22	8	,	,	PUNCT
ejpam-6076	22	9	determined	determine	VERB
ejpam-6076	22	10	the	the	DET
ejpam-6076	22	11	exact	exact	ADJ
ejpam-6076	22	12	values	value	NOUN
ejpam-6076	22	13	of	of	ADP
ejpam-6076	22	14	the	the	DET
ejpam-6076	22	15	certified	certify	VERB
ejpam-6076	22	16	perfect	perfect	ADJ
ejpam-6076	22	17	domination	domination	NOUN
ejpam-6076	22	18	number	number	NOUN
ejpam-6076	22	19	for	for	ADP
ejpam-6076	22	20	specific	specific	ADJ
ejpam-6076	22	21	graphs	graph	NOUN
ejpam-6076	22	22	,	,	PUNCT
ejpam-6076	22	23	and	and	CCONJ
ejpam-6076	22	24	examined	examine	VERB
ejpam-6076	22	25	this	this	DET
ejpam-6076	22	26	parameter	parameter	NOUN
ejpam-6076	22	27	in	in	ADP
ejpam-6076	22	28	graphs	graph	NOUN
ejpam-6076	22	29	formed	form	VERB
ejpam-6076	22	30	by	by	ADP
ejpam-6076	22	31	the	the	DET
ejpam-6076	22	32	join	join	NOUN
ejpam-6076	22	33	and	and	CCONJ
ejpam-6076	22	34	corona	corona	NOUN
ejpam-6076	22	35	operations	operation	NOUN
ejpam-6076	22	36	.	.	PUNCT
ejpam-6076	23	1	furthermore	furthermore	ADV
ejpam-6076	23	2	,	,	PUNCT
ejpam-6076	23	3	he	he	PRON
ejpam-6076	23	4	investigated	investigate	VERB
ejpam-6076	23	5	the	the	DET
ejpam-6076	23	6	relationship	relationship	NOUN
ejpam-6076	23	7	between	between	ADP
ejpam-6076	23	8	the	the	DET
ejpam-6076	23	9	perfect	perfect	ADJ
ejpam-6076	23	10	dominating	dominating	NOUN
ejpam-6076	23	11	set	set	NOUN
ejpam-6076	23	12	and	and	CCONJ
ejpam-6076	23	13	the	the	DET
ejpam-6076	23	14	certified	certify	VERB
ejpam-6076	23	15	perfect	perfect	ADJ
ejpam-6076	23	16	dominating	dominating	NOUN
ejpam-6076	23	17	set	set	NOUN
ejpam-6076	23	18	of	of	ADP
ejpam-6076	23	19	a	a	DET
ejpam-6076	23	20	graph	graph	NOUN
ejpam-6076	23	21	g	g	NOUN
ejpam-6076	23	22	suppose	suppose	VERB
ejpam-6076	23	23	we	we	PRON
ejpam-6076	23	24	have	have	VERB
ejpam-6076	23	25	a	a	DET
ejpam-6076	23	26	situation	situation	NOUN
ejpam-6076	23	27	modeled	model	VERB
ejpam-6076	23	28	by	by	ADP
ejpam-6076	23	29	a	a	DET
ejpam-6076	23	30	graph	graph	NOUN
ejpam-6076	23	31	g	g	NOUN
ejpam-6076	23	32	,	,	PUNCT
ejpam-6076	23	33	where	where	SCONJ
ejpam-6076	23	34	a	a	DET
ejpam-6076	23	35	subset	subset	NOUN
ejpam-6076	23	36	s	s	VERB
ejpam-6076	23	37	⊆	⊆	NUM
ejpam-6076	23	38	v	v	NOUN
ejpam-6076	23	39	(	(	PUNCT
ejpam-6076	23	40	g	g	NOUN
ejpam-6076	23	41	)	)	PUNCT
ejpam-6076	23	42	represents	represent	VERB
ejpam-6076	23	43	a	a	DET
ejpam-6076	23	44	group	group	NOUN
ejpam-6076	23	45	of	of	ADP
ejpam-6076	23	46	officials	official	NOUN
ejpam-6076	23	47	,	,	PUNCT
ejpam-6076	23	48	and	and	CCONJ
ejpam-6076	23	49	the	the	DET
ejpam-6076	23	50	remaining	remain	VERB
ejpam-6076	23	51	vertices	vertex	NOUN
ejpam-6076	23	52	h	h	NOUN
ejpam-6076	23	53	=	=	SYM
ejpam-6076	23	54	v	v	PROPN
ejpam-6076	23	55	(	(	PUNCT
ejpam-6076	23	56	g	g	NOUN
ejpam-6076	23	57	)	)	PUNCT
ejpam-6076	23	58	\	\	PUNCT
ejpam-6076	24	1	s	s	VERB
ejpam-6076	24	2	represent	represent	VERB
ejpam-6076	24	3	civilians	civilian	NOUN
ejpam-6076	24	4	.	.	PUNCT
ejpam-6076	25	1	each	each	DET
ejpam-6076	25	2	civilian	civilian	ADJ
ejpam-6076	25	3	x	x	SYM
ejpam-6076	25	4	∈	∈	PROPN
ejpam-6076	25	5	h	h	NOUN
ejpam-6076	25	6	must	must	AUX
ejpam-6076	25	7	be	be	AUX
ejpam-6076	25	8	assigned	assign	VERB
ejpam-6076	25	9	exactly	exactly	ADV
ejpam-6076	25	10	one	one	NUM
ejpam-6076	25	11	official	official	ADJ
ejpam-6076	25	12	u	u	NOUN
ejpam-6076	25	13	∈	∈	PROPN
ejpam-6076	25	14	s	s	X
ejpam-6076	25	15	who	who	PRON
ejpam-6076	25	16	is	be	AUX
ejpam-6076	25	17	responsible	responsible	ADJ
ejpam-6076	25	18	for	for	ADP
ejpam-6076	25	19	serving	serve	VERB
ejpam-6076	25	20	them	they	PRON
ejpam-6076	25	21	.	.	PUNCT
ejpam-6076	26	1	additionally	additionally	ADV
ejpam-6076	26	2	,	,	PUNCT
ejpam-6076	26	3	when	when	SCONJ
ejpam-6076	26	4	an	an	DET
ejpam-6076	26	5	official	official	ADJ
ejpam-6076	26	6	u	u	NOUN
ejpam-6076	26	7	serves	serve	VERB
ejpam-6076	26	8	a	a	DET
ejpam-6076	26	9	civilian	civilian	ADJ
ejpam-6076	26	10	x	x	NOUN
ejpam-6076	26	11	,	,	PUNCT
ejpam-6076	26	12	there	there	PRON
ejpam-6076	26	13	must	must	AUX
ejpam-6076	26	14	exist	exist	VERB
ejpam-6076	26	15	another	another	DET
ejpam-6076	26	16	civilian	civilian	ADJ
ejpam-6076	26	17	y	y	PROPN
ejpam-6076	26	18	∈	∈	PROPN
ejpam-6076	26	19	h	h	NOUN
ejpam-6076	26	20	who	who	PRON
ejpam-6076	26	21	only	only	ADV
ejpam-6076	26	22	observes	observe	VERB
ejpam-6076	26	23	the	the	DET
ejpam-6076	26	24	service	service	NOUN
ejpam-6076	26	25	provided	provide	VERB
ejpam-6076	26	26	by	by	ADP
ejpam-6076	26	27	the	the	DET
ejpam-6076	26	28	official	official	ADJ
ejpam-6076	26	29	u	u	NOUN
ejpam-6076	26	30	to	to	ADP
ejpam-6076	26	31	the	the	DET
ejpam-6076	26	32	civilian	civilian	PROPN
ejpam-6076	26	33	x.	x.	NOUN
ejpam-6076	27	1	this	this	DET
ejpam-6076	27	2	observer	observer	NOUN
ejpam-6076	27	3	,	,	PUNCT
ejpam-6076	27	4	or	or	CCONJ
ejpam-6076	27	5	witness	witness	VERB
ejpam-6076	27	6	y	y	PROPN
ejpam-6076	27	7	,	,	PUNCT
ejpam-6076	27	8	ensures	ensure	VERB
ejpam-6076	27	9	that	that	SCONJ
ejpam-6076	27	10	the	the	DET
ejpam-6076	27	11	service	service	NOUN
ejpam-6076	27	12	is	be	AUX
ejpam-6076	27	13	performed	perform	VERB
ejpam-6076	27	14	ethically	ethically	ADV
ejpam-6076	27	15	and	and	CCONJ
ejpam-6076	27	16	without	without	ADP
ejpam-6076	27	17	any	any	DET
ejpam-6076	27	18	misconduct	misconduct	NOUN
ejpam-6076	27	19	from	from	ADP
ejpam-6076	27	20	the	the	DET
ejpam-6076	27	21	official	official	NOUN
ejpam-6076	27	22	.	.	PUNCT
ejpam-6076	28	1	this	this	DET
ejpam-6076	28	2	setup	setup	NOUN
ejpam-6076	28	3	prompts	prompt	VERB
ejpam-6076	28	4	the	the	DET
ejpam-6076	28	5	question	question	NOUN
ejpam-6076	28	6	:	:	PUNCT
ejpam-6076	28	7	what	what	PRON
ejpam-6076	28	8	is	be	AUX
ejpam-6076	28	9	the	the	DET
ejpam-6076	28	10	smallest	small	ADJ
ejpam-6076	28	11	number	number	NOUN
ejpam-6076	28	12	of	of	ADP
ejpam-6076	28	13	officials	official	NOUN
ejpam-6076	28	14	needed	need	VERB
ejpam-6076	28	15	to	to	PART
ejpam-6076	28	16	ensure	ensure	VERB
ejpam-6076	28	17	every	every	DET
ejpam-6076	28	18	civilian	civilian	NOUN
ejpam-6076	28	19	is	be	AUX
ejpam-6076	28	20	served	serve	VERB
ejpam-6076	28	21	in	in	ADP
ejpam-6076	28	22	such	such	DET
ejpam-6076	28	23	a	a	DET
ejpam-6076	28	24	monitored	monitored	ADJ
ejpam-6076	28	25	manner	manner	NOUN
ejpam-6076	28	26	within	within	ADP
ejpam-6076	28	27	a	a	DET
ejpam-6076	28	28	given	give	VERB
ejpam-6076	28	29	social	social	ADJ
ejpam-6076	28	30	structure	structure	NOUN
ejpam-6076	28	31	?	?	PUNCT
ejpam-6076	29	1	this	this	DET
ejpam-6076	29	2	question	question	NOUN
ejpam-6076	29	3	motivated	motivate	VERB
ejpam-6076	29	4	hamja	hamja	VERB
ejpam-6076	30	1	[	[	X
ejpam-6076	30	2	3	3	X
ejpam-6076	30	3	]	]	PUNCT
ejpam-6076	30	4	to	to	PART
ejpam-6076	30	5	define	define	VERB
ejpam-6076	30	6	a	a	DET
ejpam-6076	30	7	new	new	ADJ
ejpam-6076	30	8	concept	concept	NOUN
ejpam-6076	30	9	in	in	ADP
ejpam-6076	30	10	graph	graph	NOUN
ejpam-6076	30	11	theory	theory	NOUN
ejpam-6076	30	12	,	,	PUNCT
ejpam-6076	30	13	known	know	VERB
ejpam-6076	30	14	as	as	ADP
ejpam-6076	30	15	the	the	DET
ejpam-6076	30	16	certified	certify	VERB
ejpam-6076	30	17	perfect	perfect	ADJ
ejpam-6076	30	18	dominating	dominating	NOUN
ejpam-6076	30	19	set	set	NOUN
ejpam-6076	30	20	of	of	ADP
ejpam-6076	30	21	a	a	DET
ejpam-6076	30	22	graph	graph	NOUN
ejpam-6076	30	23	g.	g.	NOUN
ejpam-6076	30	24	in	in	ADP
ejpam-6076	30	25	this	this	DET
ejpam-6076	30	26	paper	paper	NOUN
ejpam-6076	30	27	,	,	PUNCT
ejpam-6076	30	28	we	we	PRON
ejpam-6076	30	29	extend	extend	VERB
ejpam-6076	30	30	the	the	DET
ejpam-6076	30	31	work	work	NOUN
ejpam-6076	30	32	of	of	ADP
ejpam-6076	30	33	hamja	hamja	PROPN
ejpam-6076	30	34	[	[	X
ejpam-6076	30	35	3	3	X
ejpam-6076	30	36	]	]	PUNCT
ejpam-6076	30	37	by	by	ADP
ejpam-6076	30	38	presenting	present	VERB
ejpam-6076	30	39	new	new	ADJ
ejpam-6076	30	40	results	result	NOUN
ejpam-6076	30	41	related	relate	VERB
ejpam-6076	30	42	to	to	ADP
ejpam-6076	30	43	certified	certify	VERB
ejpam-6076	30	44	perfect	perfect	ADJ
ejpam-6076	30	45	domination	domination	NOUN
ejpam-6076	30	46	.	.	PUNCT
ejpam-6076	31	1	we	we	PRON
ejpam-6076	31	2	establish	establish	VERB
ejpam-6076	31	3	some	some	DET
ejpam-6076	31	4	upper	upper	ADJ
ejpam-6076	31	5	bounds	bound	NOUN
ejpam-6076	31	6	for	for	ADP
ejpam-6076	31	7	the	the	DET
ejpam-6076	31	8	certified	certify	VERB
ejpam-6076	31	9	perfect	perfect	ADJ
ejpam-6076	31	10	domination	domination	NOUN
ejpam-6076	31	11	number	number	NOUN
ejpam-6076	31	12	of	of	ADP
ejpam-6076	31	13	a	a	DET
ejpam-6076	31	14	graph	graph	NOUN
ejpam-6076	31	15	g	g	NOUN
ejpam-6076	31	16	and	and	CCONJ
ejpam-6076	31	17	explore	explore	VERB
ejpam-6076	31	18	the	the	DET
ejpam-6076	31	19	relationship	relationship	NOUN
ejpam-6076	31	20	between	between	ADP
ejpam-6076	31	21	certified	certified	ADJ
ejpam-6076	31	22	domination	domination	NOUN
ejpam-6076	31	23	and	and	CCONJ
ejpam-6076	31	24	certified	certify	VERB
ejpam-6076	31	25	perfect	perfect	ADJ
ejpam-6076	31	26	domination	domination	NOUN
ejpam-6076	31	27	parameters	parameter	NOUN
ejpam-6076	31	28	.	.	PUNCT
ejpam-6076	32	1	additionally	additionally	ADV
ejpam-6076	32	2	,	,	PUNCT
ejpam-6076	32	3	we	we	PRON
ejpam-6076	32	4	identify	identify	VERB
ejpam-6076	32	5	classes	class	NOUN
ejpam-6076	32	6	of	of	ADP
ejpam-6076	32	7	graphs	graph	NOUN
ejpam-6076	32	8	with	with	ADP
ejpam-6076	32	9	both	both	CCONJ
ejpam-6076	32	10	small	small	ADJ
ejpam-6076	32	11	and	and	CCONJ
ejpam-6076	32	12	large	large	ADJ
ejpam-6076	32	13	values	value	NOUN
ejpam-6076	32	14	of	of	ADP
ejpam-6076	32	15	these	these	DET
ejpam-6076	32	16	parameters	parameter	NOUN
ejpam-6076	32	17	.	.	PUNCT
ejpam-6076	33	1	furthermore	furthermore	ADV
ejpam-6076	33	2	,	,	PUNCT
ejpam-6076	33	3	we	we	PRON
ejpam-6076	33	4	provide	provide	VERB
ejpam-6076	33	5	a	a	DET
ejpam-6076	33	6	characterization	characterization	NOUN
ejpam-6076	33	7	of	of	ADP
ejpam-6076	33	8	graphs	graph	NOUN
ejpam-6076	33	9	for	for	ADP
ejpam-6076	33	10	which	which	PRON
ejpam-6076	33	11	γcerp(g	γcerp(g	PROPN
ejpam-6076	33	12	)	)	PUNCT
ejpam-6076	33	13	=	=	SYM
ejpam-6076	33	14	n	n	NOUN
ejpam-6076	33	15	and	and	CCONJ
ejpam-6076	33	16	γcerp(g	γcerp(g	ADJ
ejpam-6076	33	17	)	)	PUNCT
ejpam-6076	33	18	=	=	SYM
ejpam-6076	33	19	γcer(g	γcer(g	NOUN
ejpam-6076	33	20	)	)	PUNCT
ejpam-6076	33	21	.	.	PUNCT
ejpam-6076	34	1	we	we	PRON
ejpam-6076	34	2	also	also	ADV
ejpam-6076	34	3	examine	examine	VERB
ejpam-6076	34	4	the	the	DET
ejpam-6076	34	5	behavior	behavior	NOUN
ejpam-6076	34	6	of	of	ADP
ejpam-6076	34	7	certified	certify	VERB
ejpam-6076	34	8	perfect	perfect	ADJ
ejpam-6076	34	9	dominating	dominating	NOUN
ejpam-6076	34	10	sets	set	NOUN
ejpam-6076	34	11	under	under	ADP
ejpam-6076	34	12	graph	graph	NOUN
ejpam-6076	34	13	operations	operation	NOUN
ejpam-6076	34	14	such	such	ADJ
ejpam-6076	34	15	as	as	ADP
ejpam-6076	34	16	the	the	DET
ejpam-6076	34	17	lexicographic	lexicographic	ADJ
ejpam-6076	34	18	and	and	CCONJ
ejpam-6076	34	19	cartesian	cartesian	ADJ
ejpam-6076	34	20	products	product	NOUN
ejpam-6076	34	21	.	.	PUNCT
ejpam-6076	35	1	in	in	ADP
ejpam-6076	35	2	doing	do	VERB
ejpam-6076	35	3	so	so	ADV
ejpam-6076	35	4	,	,	PUNCT
ejpam-6076	35	5	we	we	PRON
ejpam-6076	35	6	determine	determine	VERB
ejpam-6076	35	7	the	the	DET
ejpam-6076	35	8	certified	certify	VERB
ejpam-6076	35	9	perfect	perfect	ADJ
ejpam-6076	35	10	domination	domination	NOUN
ejpam-6076	35	11	number	number	NOUN
ejpam-6076	35	12	for	for	ADP
ejpam-6076	35	13	these	these	DET
ejpam-6076	35	14	products	product	NOUN
ejpam-6076	35	15	and	and	CCONJ
ejpam-6076	35	16	identify	identify	VERB
ejpam-6076	35	17	non	non	ADJ
ejpam-6076	35	18	-	-	ADJ
ejpam-6076	35	19	γcerp	γcerp	ADJ
ejpam-6076	35	20	-	-	PUNCT
ejpam-6076	35	21	graphs	graph	NOUN
ejpam-6076	35	22	arising	arise	VERB
ejpam-6076	35	23	from	from	ADP
ejpam-6076	35	24	these	these	DET
ejpam-6076	35	25	operations	operation	NOUN
ejpam-6076	35	26	.	.	PUNCT
ejpam-6076	36	1	this	this	DET
ejpam-6076	36	2	work	work	NOUN
ejpam-6076	36	3	contributes	contribute	VERB
ejpam-6076	36	4	to	to	ADP
ejpam-6076	36	5	the	the	DET
ejpam-6076	36	6	ongoing	ongoing	ADJ
ejpam-6076	36	7	development	development	NOUN
ejpam-6076	36	8	of	of	ADP
ejpam-6076	36	9	domination	domination	NOUN
ejpam-6076	36	10	parameters	parameter	NOUN
ejpam-6076	36	11	in	in	ADP
ejpam-6076	36	12	graphs	graph	NOUN
ejpam-6076	36	13	and	and	CCONJ
ejpam-6076	36	14	offers	offer	VERB
ejpam-6076	36	15	new	new	ADJ
ejpam-6076	36	16	insights	insight	NOUN
ejpam-6076	36	17	into	into	ADP
ejpam-6076	36	18	their	their	PRON
ejpam-6076	36	19	structural	structural	ADJ
ejpam-6076	36	20	properties	property	NOUN
ejpam-6076	36	21	.	.	PUNCT
ejpam-6076	37	1	2	2	X
ejpam-6076	37	2	.	.	X
ejpam-6076	37	3	terminology	terminology	NOUN
ejpam-6076	37	4	and	and	CCONJ
ejpam-6076	37	5	notation	notation	NOUN
ejpam-6076	37	6	for	for	ADP
ejpam-6076	37	7	standard	standard	ADJ
ejpam-6076	37	8	graph	graph	NOUN
ejpam-6076	37	9	theory	theory	NOUN
ejpam-6076	37	10	terminology	terminology	NOUN
ejpam-6076	37	11	,	,	PUNCT
ejpam-6076	37	12	we	we	PRON
ejpam-6076	37	13	follow	follow	VERB
ejpam-6076	37	14	the	the	DET
ejpam-6076	37	15	book	book	NOUN
ejpam-6076	37	16	graph	graph	NOUN
ejpam-6076	37	17	theory	theory	NOUN
ejpam-6076	37	18	by	by	ADP
ejpam-6076	37	19	diestel	diestel	NOUN
ejpam-6076	37	20	[	[	X
ejpam-6076	37	21	4	4	NUM
ejpam-6076	37	22	]	]	PUNCT
ejpam-6076	37	23	.	.	PUNCT
ejpam-6076	38	1	let	let	VERB
ejpam-6076	38	2	g	g	PROPN
ejpam-6076	38	3	=	=	SYM
ejpam-6076	38	4	(	(	PUNCT
ejpam-6076	38	5	v	v	NOUN
ejpam-6076	38	6	,	,	PUNCT
ejpam-6076	38	7	e	e	NOUN
ejpam-6076	38	8	)	)	PUNCT
ejpam-6076	38	9	be	be	AUX
ejpam-6076	38	10	a	a	DET
ejpam-6076	38	11	simple	simple	ADJ
ejpam-6076	38	12	,	,	PUNCT
ejpam-6076	38	13	connected	connected	ADJ
ejpam-6076	38	14	graph	graph	NOUN
ejpam-6076	38	15	,	,	PUNCT
ejpam-6076	38	16	where	where	SCONJ
ejpam-6076	38	17	v	v	NOUN
ejpam-6076	38	18	=	=	SYM
ejpam-6076	38	19	v	v	NOUN
ejpam-6076	38	20	(	(	PUNCT
ejpam-6076	38	21	g	g	NOUN
ejpam-6076	38	22	)	)	PUNCT
ejpam-6076	38	23	is	be	AUX
ejpam-6076	38	24	the	the	DET
ejpam-6076	38	25	vertex	vertex	NOUN
ejpam-6076	38	26	set	set	NOUN
ejpam-6076	38	27	and	and	CCONJ
ejpam-6076	38	28	e	e	NOUN
ejpam-6076	38	29	=	=	PROPN
ejpam-6076	38	30	e(g	e(g	PROPN
ejpam-6076	38	31	)	)	PUNCT
ejpam-6076	38	32	is	be	AUX
ejpam-6076	38	33	the	the	DET
ejpam-6076	38	34	edge	edge	NOUN
ejpam-6076	38	35	set	set	NOUN
ejpam-6076	38	36	of	of	ADP
ejpam-6076	38	37	g.	g.	PROPN
ejpam-6076	38	38	the	the	DET
ejpam-6076	38	39	degree	degree	NOUN
ejpam-6076	38	40	of	of	ADP
ejpam-6076	38	41	a	a	DET
ejpam-6076	38	42	vertex	vertex	NOUN
ejpam-6076	38	43	v	v	NOUN
ejpam-6076	38	44	,	,	PUNCT
ejpam-6076	38	45	denoted	denote	VERB
ejpam-6076	38	46	by	by	ADP
ejpam-6076	38	47	deg(v	deg(v	NOUN
ejpam-6076	38	48	)	)	PUNCT
ejpam-6076	38	49	,	,	PUNCT
ejpam-6076	38	50	is	be	AUX
ejpam-6076	38	51	the	the	DET
ejpam-6076	38	52	number	number	NOUN
ejpam-6076	38	53	of	of	ADP
ejpam-6076	38	54	edges	edge	NOUN
ejpam-6076	38	55	incident	incident	NOUN
ejpam-6076	38	56	to	to	ADP
ejpam-6076	38	57	v.	v.	ADP
ejpam-6076	38	58	the	the	DET
ejpam-6076	38	59	maximum	maximum	ADJ
ejpam-6076	38	60	degree	degree	NOUN
ejpam-6076	38	61	of	of	ADP
ejpam-6076	38	62	g	g	NOUN
ejpam-6076	38	63	,	,	PUNCT
ejpam-6076	38	64	denoted	denote	VERB
ejpam-6076	38	65	by	by	ADP
ejpam-6076	38	66	∆(g	∆(g	PROPN
ejpam-6076	38	67	)	)	PUNCT
ejpam-6076	38	68	,	,	PUNCT
ejpam-6076	38	69	is	be	AUX
ejpam-6076	38	70	the	the	DET
ejpam-6076	38	71	highest	high	ADJ
ejpam-6076	38	72	degree	degree	NOUN
ejpam-6076	38	73	among	among	ADP
ejpam-6076	38	74	all	all	DET
ejpam-6076	38	75	vertices	vertex	NOUN
ejpam-6076	38	76	of	of	ADP
ejpam-6076	38	77	g	g	NOUN
ejpam-6076	38	78	,	,	PUNCT
ejpam-6076	38	79	defined	define	VERB
ejpam-6076	38	80	as	as	ADP
ejpam-6076	38	81	∆(g	∆(g	NOUN
ejpam-6076	38	82	)	)	PUNCT
ejpam-6076	38	83	=	=	SYM
ejpam-6076	38	84	max{deg(v	max{deg(v	PROPN
ejpam-6076	38	85	)	)	PUNCT
ejpam-6076	38	86	:	:	PUNCT
ejpam-6076	38	87	v	v	X
ejpam-6076	38	88	∈	∈	PROPN
ejpam-6076	38	89	v	v	NOUN
ejpam-6076	38	90	(	(	PUNCT
ejpam-6076	38	91	g	g	NOUN
ejpam-6076	38	92	)	)	PUNCT
ejpam-6076	38	93	}	}	PUNCT
ejpam-6076	38	94	.	.	PUNCT
ejpam-6076	39	1	the	the	DET
ejpam-6076	39	2	open	open	ADJ
ejpam-6076	39	3	neighborhood	neighborhood	NOUN
ejpam-6076	39	4	of	of	ADP
ejpam-6076	39	5	a	a	DET
ejpam-6076	39	6	vertex	vertex	NOUN
ejpam-6076	39	7	u	u	NOUN
ejpam-6076	39	8	,	,	PUNCT
ejpam-6076	39	9	denoted	denote	VERB
ejpam-6076	39	10	by	by	ADP
ejpam-6076	39	11	ng(u	ng(u	NOUN
ejpam-6076	39	12	)	)	PUNCT
ejpam-6076	39	13	,	,	PUNCT
ejpam-6076	39	14	is	be	AUX
ejpam-6076	39	15	the	the	DET
ejpam-6076	39	16	set	set	NOUN
ejpam-6076	39	17	of	of	ADP
ejpam-6076	39	18	all	all	DET
ejpam-6076	39	19	vertices	vertex	NOUN
ejpam-6076	39	20	adjacent	adjacent	ADJ
ejpam-6076	39	21	to	to	ADP
ejpam-6076	39	22	u	u	PRON
ejpam-6076	39	23	,	,	PUNCT
ejpam-6076	39	24	formally	formally	ADV
ejpam-6076	39	25	given	give	VERB
ejpam-6076	39	26	by	by	ADP
ejpam-6076	39	27	ng(u	ng(u	NOUN
ejpam-6076	39	28	)	)	PUNCT
ejpam-6076	39	29	=	=	PRON
ejpam-6076	39	30	{	{	PUNCT
ejpam-6076	39	31	v	v	NUM
ejpam-6076	39	32	∈	∈	NOUN
ejpam-6076	39	33	v	v	NOUN
ejpam-6076	39	34	(	(	PUNCT
ejpam-6076	39	35	g	g	NOUN
ejpam-6076	39	36	)	)	PUNCT
ejpam-6076	39	37	:	:	PUNCT
ejpam-6076	39	38	uv	uv	PROPN
ejpam-6076	39	39	∈	∈	PROPN
ejpam-6076	39	40	e(g	e(g	PROPN
ejpam-6076	39	41	)	)	PUNCT
ejpam-6076	39	42	}	}	PUNCT
ejpam-6076	39	43	.	.	PUNCT
ejpam-6076	40	1	the	the	DET
ejpam-6076	40	2	closed	closed	ADJ
ejpam-6076	40	3	neighborhood	neighborhood	NOUN
ejpam-6076	40	4	of	of	ADP
ejpam-6076	40	5	a	a	DET
ejpam-6076	40	6	vertex	vertex	NOUN
ejpam-6076	40	7	u	u	NOUN
ejpam-6076	40	8	,	,	PUNCT
ejpam-6076	40	9	denoted	denote	VERB
ejpam-6076	40	10	by	by	ADP
ejpam-6076	40	11	ng[u	ng[u	PROPN
ejpam-6076	40	12	]	]	PUNCT
ejpam-6076	40	13	,	,	PUNCT
ejpam-6076	40	14	is	be	AUX
ejpam-6076	40	15	the	the	DET
ejpam-6076	40	16	set	set	NOUN
ejpam-6076	40	17	containing	contain	VERB
ejpam-6076	40	18	u	u	NOUN
ejpam-6076	40	19	along	along	ADP
ejpam-6076	40	20	with	with	ADP
ejpam-6076	40	21	all	all	DET
ejpam-6076	40	22	vertices	vertex	NOUN
ejpam-6076	40	23	adjacent	adjacent	ADJ
ejpam-6076	40	24	to	to	ADP
ejpam-6076	40	25	it	it	PRON
ejpam-6076	40	26	,	,	PUNCT
ejpam-6076	40	27	expressed	express	VERB
ejpam-6076	40	28	as	as	ADP
ejpam-6076	40	29	ng[u	ng[u	PROPN
ejpam-6076	40	30	]	]	X
ejpam-6076	40	31	=	=	SYM
ejpam-6076	40	32	ng(u	ng(u	PROPN
ejpam-6076	40	33	)	)	PUNCT
ejpam-6076	40	34	∪	∪	NOUN
ejpam-6076	40	35	{	{	PUNCT
ejpam-6076	40	36	u	u	NOUN
ejpam-6076	40	37	}	}	PUNCT
ejpam-6076	40	38	.	.	PUNCT
ejpam-6076	41	1	similarly	similarly	ADV
ejpam-6076	41	2	,	,	PUNCT
ejpam-6076	41	3	the	the	DET
ejpam-6076	41	4	closed	closed	ADJ
ejpam-6076	41	5	neighborhood	neighborhood	NOUN
ejpam-6076	41	6	of	of	ADP
ejpam-6076	41	7	a	a	DET
ejpam-6076	41	8	subset	subset	NOUN
ejpam-6076	41	9	s	s	VERB
ejpam-6076	41	10	⊆	⊆	NUM
ejpam-6076	41	11	v	v	NOUN
ejpam-6076	41	12	(	(	PUNCT
ejpam-6076	41	13	g	g	NOUN
ejpam-6076	41	14	)	)	PUNCT
ejpam-6076	41	15	is	be	AUX
ejpam-6076	41	16	dej	dej	PROPN
ejpam-6076	41	17	.	.	PUNCT
ejpam-6076	42	1	j.	j.	PROPN
ejpam-6076	42	2	hamja	hamja	PROPN
ejpam-6076	42	3	et	et	PROPN
ejpam-6076	42	4	al	al	PROPN
ejpam-6076	42	5	.	.	PUNCT
ejpam-6076	42	6	/	/	SYM
ejpam-6076	42	7	eur	eur	PROPN
ejpam-6076	42	8	.	.	PUNCT
ejpam-6076	43	1	j.	j.	PROPN
ejpam-6076	43	2	pure	pure	PROPN
ejpam-6076	43	3	appl	appl	PROPN
ejpam-6076	43	4	.	.	PROPN
ejpam-6076	43	5	math	math	PROPN
ejpam-6076	43	6	,	,	PUNCT
ejpam-6076	43	7	18	18	NUM
ejpam-6076	43	8	(	(	PUNCT
ejpam-6076	43	9	3	3	NUM
ejpam-6076	43	10	)	)	PUNCT
ejpam-6076	43	11	(	(	PUNCT
ejpam-6076	43	12	2025	2025	NUM
ejpam-6076	43	13	)	)	PUNCT
ejpam-6076	43	14	,	,	PUNCT
ejpam-6076	43	15	6076	6076	NUM
ejpam-6076	43	16	3	3	NUM
ejpam-6076	43	17	of	of	ADP
ejpam-6076	43	18	13	13	NUM
ejpam-6076	43	19	fined	fine	VERB
ejpam-6076	43	20	as	as	ADP
ejpam-6076	43	21	the	the	DET
ejpam-6076	43	22	union	union	NOUN
ejpam-6076	43	23	of	of	ADP
ejpam-6076	43	24	the	the	DET
ejpam-6076	43	25	closed	closed	ADJ
ejpam-6076	43	26	neighborhoods	neighborhood	NOUN
ejpam-6076	43	27	of	of	ADP
ejpam-6076	43	28	all	all	DET
ejpam-6076	43	29	vertices	vertex	NOUN
ejpam-6076	43	30	in	in	ADP
ejpam-6076	43	31	s	s	NOUN
ejpam-6076	43	32	,	,	PUNCT
ejpam-6076	43	33	i.e.	i.e.	X
ejpam-6076	43	34	,	,	PUNCT
ejpam-6076	43	35	ng[s	ng[s	PROPN
ejpam-6076	43	36	]	]	X
ejpam-6076	43	37	=	=	PUNCT
ejpam-6076	44	1	⋃	⋃	VERB
ejpam-6076	44	2	v∈s	v∈s	ADJ
ejpam-6076	44	3	ng[v	ng[v	NOUN
ejpam-6076	44	4	]	]	PUNCT
ejpam-6076	44	5	.	.	PUNCT
ejpam-6076	45	1	a	a	DET
ejpam-6076	45	2	vertex	vertex	NOUN
ejpam-6076	45	3	of	of	ADP
ejpam-6076	45	4	degree	degree	NOUN
ejpam-6076	45	5	|v	|v	NOUN
ejpam-6076	45	6	(	(	PUNCT
ejpam-6076	45	7	g)|	g)|	NOUN
ejpam-6076	45	8	−	−	NOUN
ejpam-6076	45	9	1	1	NUM
ejpam-6076	45	10	is	be	AUX
ejpam-6076	45	11	called	call	VERB
ejpam-6076	45	12	a	a	DET
ejpam-6076	45	13	universal	universal	ADJ
ejpam-6076	45	14	vertex	vertex	NOUN
ejpam-6076	45	15	of	of	ADP
ejpam-6076	45	16	g.	g.	PROPN
ejpam-6076	45	17	a	a	DET
ejpam-6076	45	18	vertex	vertex	NOUN
ejpam-6076	45	19	with	with	ADP
ejpam-6076	45	20	degree	degree	NOUN
ejpam-6076	45	21	one	one	PRON
ejpam-6076	45	22	is	be	AUX
ejpam-6076	45	23	referred	refer	VERB
ejpam-6076	45	24	to	to	ADP
ejpam-6076	45	25	as	as	ADP
ejpam-6076	45	26	a	a	DET
ejpam-6076	45	27	leaf	leaf	NOUN
ejpam-6076	45	28	,	,	PUNCT
ejpam-6076	45	29	and	and	CCONJ
ejpam-6076	45	30	its	its	PRON
ejpam-6076	45	31	unique	unique	ADJ
ejpam-6076	45	32	adjacent	adjacent	ADJ
ejpam-6076	45	33	vertex	vertex	NOUN
ejpam-6076	45	34	is	be	AUX
ejpam-6076	45	35	called	call	VERB
ejpam-6076	45	36	its	its	PRON
ejpam-6076	45	37	support	support	NOUN
ejpam-6076	45	38	vertex	vertex	NOUN
ejpam-6076	45	39	(	(	PUNCT
ejpam-6076	45	40	or	or	CCONJ
ejpam-6076	45	41	simply	simply	ADV
ejpam-6076	45	42	,	,	PUNCT
ejpam-6076	45	43	its	its	PRON
ejpam-6076	45	44	support	support	NOUN
ejpam-6076	45	45	)	)	PUNCT
ejpam-6076	45	46	.	.	PUNCT
ejpam-6076	46	1	a	a	DET
ejpam-6076	46	2	support	support	NOUN
ejpam-6076	46	3	vertex	vertex	NOUN
ejpam-6076	46	4	that	that	PRON
ejpam-6076	46	5	is	be	AUX
ejpam-6076	46	6	adjacent	adjacent	ADJ
ejpam-6076	46	7	to	to	ADP
ejpam-6076	46	8	at	at	ADV
ejpam-6076	46	9	least	least	ADV
ejpam-6076	46	10	two	two	NUM
ejpam-6076	46	11	leaves	leave	NOUN
ejpam-6076	46	12	is	be	AUX
ejpam-6076	46	13	known	know	VERB
ejpam-6076	46	14	as	as	ADP
ejpam-6076	46	15	a	a	DET
ejpam-6076	46	16	strong	strong	ADJ
ejpam-6076	46	17	support	support	NOUN
ejpam-6076	46	18	,	,	PUNCT
ejpam-6076	46	19	whereas	whereas	SCONJ
ejpam-6076	46	20	one	one	NUM
ejpam-6076	46	21	that	that	PRON
ejpam-6076	46	22	is	be	AUX
ejpam-6076	46	23	adjacent	adjacent	ADJ
ejpam-6076	46	24	to	to	ADP
ejpam-6076	46	25	only	only	ADV
ejpam-6076	46	26	a	a	DET
ejpam-6076	46	27	single	single	ADJ
ejpam-6076	46	28	leaf	leaf	NOUN
ejpam-6076	46	29	is	be	AUX
ejpam-6076	46	30	called	call	VERB
ejpam-6076	46	31	a	a	DET
ejpam-6076	46	32	weak	weak	ADJ
ejpam-6076	46	33	support	support	NOUN
ejpam-6076	46	34	.	.	PUNCT
ejpam-6076	47	1	the	the	DET
ejpam-6076	47	2	set	set	NOUN
ejpam-6076	47	3	of	of	ADP
ejpam-6076	47	4	all	all	DET
ejpam-6076	47	5	leaves	leave	NOUN
ejpam-6076	47	6	in	in	ADP
ejpam-6076	47	7	g	g	PROPN
ejpam-6076	47	8	is	be	AUX
ejpam-6076	47	9	denoted	denote	VERB
ejpam-6076	47	10	by	by	ADP
ejpam-6076	47	11	l(g	l(g	NOUN
ejpam-6076	47	12	)	)	PUNCT
ejpam-6076	47	13	.	.	PUNCT
ejpam-6076	48	1	for	for	ADP
ejpam-6076	48	2	a	a	DET
ejpam-6076	48	3	leaf	leaf	NOUN
ejpam-6076	48	4	v	v	ADP
ejpam-6076	48	5	∈	∈	PROPN
ejpam-6076	48	6	l(g	l(g	NOUN
ejpam-6076	48	7	)	)	PUNCT
ejpam-6076	48	8	,	,	PUNCT
ejpam-6076	48	9	its	its	PRON
ejpam-6076	48	10	support	support	NOUN
ejpam-6076	48	11	vertex	vertex	NOUN
ejpam-6076	48	12	is	be	AUX
ejpam-6076	48	13	represented	represent	VERB
ejpam-6076	48	14	as	as	ADP
ejpam-6076	48	15	sl	sl	NOUN
ejpam-6076	48	16	,	,	PUNCT
ejpam-6076	48	17	while	while	SCONJ
ejpam-6076	48	18	for	for	ADP
ejpam-6076	48	19	a	a	DET
ejpam-6076	48	20	weak	weak	ADJ
ejpam-6076	48	21	support	support	NOUN
ejpam-6076	48	22	v	v	NOUN
ejpam-6076	48	23	,	,	PUNCT
ejpam-6076	48	24	the	the	DET
ejpam-6076	48	25	unique	unique	ADJ
ejpam-6076	48	26	leaf	leaf	NOUN
ejpam-6076	48	27	adjacent	adjacent	ADJ
ejpam-6076	48	28	to	to	ADP
ejpam-6076	48	29	v	v	NOUN
ejpam-6076	48	30	is	be	AUX
ejpam-6076	48	31	denoted	denote	VERB
ejpam-6076	48	32	by	by	ADP
ejpam-6076	48	33	wl	wl	PROPN
ejpam-6076	48	34	.	.	PUNCT
ejpam-6076	49	1	the	the	DET
ejpam-6076	49	2	set	set	NOUN
ejpam-6076	49	3	of	of	ADP
ejpam-6076	49	4	all	all	DET
ejpam-6076	49	5	strong	strong	ADJ
ejpam-6076	49	6	support	support	NOUN
ejpam-6076	49	7	vertices	vertex	NOUN
ejpam-6076	49	8	in	in	ADP
ejpam-6076	49	9	g	g	PROPN
ejpam-6076	49	10	is	be	AUX
ejpam-6076	49	11	denoted	denote	VERB
ejpam-6076	49	12	by	by	ADP
ejpam-6076	49	13	ss	ss	NOUN
ejpam-6076	49	14	,	,	PUNCT
ejpam-6076	49	15	and	and	CCONJ
ejpam-6076	49	16	the	the	DET
ejpam-6076	49	17	set	set	NOUN
ejpam-6076	49	18	of	of	ADP
ejpam-6076	49	19	all	all	DET
ejpam-6076	49	20	weak	weak	ADJ
ejpam-6076	49	21	support	support	NOUN
ejpam-6076	49	22	vertices	vertex	NOUN
ejpam-6076	49	23	is	be	AUX
ejpam-6076	49	24	denoted	denote	VERB
ejpam-6076	49	25	by	by	ADP
ejpam-6076	49	26	ws	ws	NOUN
ejpam-6076	49	27	,	,	PUNCT
ejpam-6076	49	28	as	as	SCONJ
ejpam-6076	49	29	defined	define	VERB
ejpam-6076	49	30	by	by	ADP
ejpam-6076	49	31	dettlaff	dettlaff	VERB
ejpam-6076	49	32	et	et	PROPN
ejpam-6076	49	33	al	al	PROPN
ejpam-6076	49	34	.	.	PUNCT
ejpam-6076	50	1	(	(	PUNCT
ejpam-6076	50	2	see	see	VERB
ejpam-6076	50	3	[	[	X
ejpam-6076	50	4	1	1	NUM
ejpam-6076	50	5	]	]	NUM
ejpam-6076	50	6	)	)	PUNCT
ejpam-6076	50	7	.	.	PUNCT
ejpam-6076	51	1	a	a	DET
ejpam-6076	51	2	subset	subset	NOUN
ejpam-6076	51	3	j	j	PROPN
ejpam-6076	51	4	of	of	ADP
ejpam-6076	51	5	v	v	PROPN
ejpam-6076	51	6	(	(	PUNCT
ejpam-6076	51	7	g	g	NOUN
ejpam-6076	51	8	)	)	PUNCT
ejpam-6076	51	9	is	be	AUX
ejpam-6076	51	10	called	call	VERB
ejpam-6076	51	11	a	a	DET
ejpam-6076	51	12	dominating	dominating	NOUN
ejpam-6076	51	13	set	set	NOUN
ejpam-6076	51	14	if	if	SCONJ
ejpam-6076	51	15	every	every	DET
ejpam-6076	51	16	vertex	vertex	NOUN
ejpam-6076	51	17	in	in	ADP
ejpam-6076	51	18	g	g	PROPN
ejpam-6076	51	19	is	be	AUX
ejpam-6076	51	20	either	either	CCONJ
ejpam-6076	51	21	in	in	ADP
ejpam-6076	51	22	j	j	PROPN
ejpam-6076	51	23	or	or	CCONJ
ejpam-6076	51	24	adjacent	adjacent	ADJ
ejpam-6076	51	25	to	to	ADP
ejpam-6076	51	26	at	at	ADV
ejpam-6076	51	27	least	least	ADV
ejpam-6076	51	28	one	one	NUM
ejpam-6076	51	29	vertex	vertex	NOUN
ejpam-6076	51	30	in	in	ADP
ejpam-6076	51	31	j	j	PROPN
ejpam-6076	51	32	,	,	PUNCT
ejpam-6076	51	33	i.e.	i.e.	X
ejpam-6076	51	34	,	,	PUNCT
ejpam-6076	51	35	ng[j	ng[j	NOUN
ejpam-6076	51	36	]	]	PUNCT
ejpam-6076	51	37	=	=	SYM
ejpam-6076	51	38	v	v	X
ejpam-6076	51	39	(	(	PUNCT
ejpam-6076	51	40	g	g	NOUN
ejpam-6076	51	41	)	)	PUNCT
ejpam-6076	51	42	.	.	PUNCT
ejpam-6076	52	1	a	a	DET
ejpam-6076	52	2	dominating	dominating	NOUN
ejpam-6076	52	3	set	set	NOUN
ejpam-6076	52	4	j	j	PROPN
ejpam-6076	52	5	is	be	AUX
ejpam-6076	52	6	said	say	VERB
ejpam-6076	52	7	to	to	PART
ejpam-6076	52	8	be	be	AUX
ejpam-6076	52	9	minimal	minimal	ADJ
ejpam-6076	52	10	if	if	SCONJ
ejpam-6076	52	11	no	no	DET
ejpam-6076	52	12	proper	proper	ADJ
ejpam-6076	52	13	subset	subset	NOUN
ejpam-6076	52	14	j	j	PROPN
ejpam-6076	52	15	′	′	NUM
ejpam-6076	53	1	⊂	⊂	PROPN
ejpam-6076	53	2	j	j	PROPN
ejpam-6076	53	3	remains	remain	VERB
ejpam-6076	53	4	a	a	DET
ejpam-6076	53	5	dominating	dominating	NOUN
ejpam-6076	53	6	set	set	NOUN
ejpam-6076	53	7	.	.	PUNCT
ejpam-6076	54	1	the	the	DET
ejpam-6076	54	2	domination	domination	NOUN
ejpam-6076	54	3	number	number	NOUN
ejpam-6076	54	4	γ(g	γ(g	PROPN
ejpam-6076	54	5	)	)	PUNCT
ejpam-6076	54	6	represents	represent	VERB
ejpam-6076	54	7	the	the	DET
ejpam-6076	54	8	smallest	small	ADJ
ejpam-6076	54	9	size	size	NOUN
ejpam-6076	54	10	of	of	ADP
ejpam-6076	54	11	a	a	DET
ejpam-6076	54	12	dominating	dominating	NOUN
ejpam-6076	54	13	set	set	VERB
ejpam-6076	54	14	in	in	ADP
ejpam-6076	54	15	g.	g.	PROPN
ejpam-6076	54	16	a	a	DET
ejpam-6076	54	17	dominating	dominating	NOUN
ejpam-6076	54	18	set	set	VERB
ejpam-6076	54	19	j	j	PROPN
ejpam-6076	54	20	that	that	PRON
ejpam-6076	54	21	attains	attain	VERB
ejpam-6076	54	22	this	this	DET
ejpam-6076	54	23	minimum	minimum	ADJ
ejpam-6076	54	24	size	size	NOUN
ejpam-6076	54	25	,	,	PUNCT
ejpam-6076	54	26	i.e.	i.e.	X
ejpam-6076	54	27	,	,	PUNCT
ejpam-6076	54	28	|j	|j	NOUN
ejpam-6076	54	29	|	|	NOUN
ejpam-6076	55	1	=	=	SYM
ejpam-6076	55	2	γ(g	γ(g	PROPN
ejpam-6076	55	3	)	)	PUNCT
ejpam-6076	55	4	,	,	PUNCT
ejpam-6076	55	5	is	be	AUX
ejpam-6076	55	6	referred	refer	VERB
ejpam-6076	55	7	to	to	ADP
ejpam-6076	55	8	as	as	ADP
ejpam-6076	55	9	a	a	DET
ejpam-6076	55	10	γ	γ	NOUN
ejpam-6076	55	11	-	-	PUNCT
ejpam-6076	55	12	set	set	NOUN
ejpam-6076	55	13	,	,	PUNCT
ejpam-6076	55	14	as	as	SCONJ
ejpam-6076	55	15	defined	define	VERB
ejpam-6076	55	16	by	by	ADP
ejpam-6076	55	17	haynes	hayne	NOUN
ejpam-6076	55	18	et	et	PROPN
ejpam-6076	55	19	al	al	PROPN
ejpam-6076	55	20	.	.	PUNCT
ejpam-6076	56	1	(	(	PUNCT
ejpam-6076	56	2	see	see	VERB
ejpam-6076	56	3	[	[	X
ejpam-6076	56	4	5	5	NUM
ejpam-6076	56	5	]	]	NUM
ejpam-6076	56	6	)	)	PUNCT
ejpam-6076	56	7	.	.	PUNCT
ejpam-6076	57	1	a	a	DET
ejpam-6076	57	2	subset	subset	NOUN
ejpam-6076	57	3	j	j	PROPN
ejpam-6076	57	4	of	of	ADP
ejpam-6076	57	5	v	v	PROPN
ejpam-6076	57	6	(	(	PUNCT
ejpam-6076	57	7	g	g	NOUN
ejpam-6076	57	8	)	)	PUNCT
ejpam-6076	57	9	is	be	AUX
ejpam-6076	57	10	called	call	VERB
ejpam-6076	57	11	a	a	DET
ejpam-6076	57	12	certified	certify	VERB
ejpam-6076	57	13	dominating	dominating	NOUN
ejpam-6076	57	14	set	set	NOUN
ejpam-6076	57	15	if	if	SCONJ
ejpam-6076	57	16	each	each	DET
ejpam-6076	57	17	vertex	vertex	NOUN
ejpam-6076	57	18	a	a	DET
ejpam-6076	57	19	∈	∈	PROPN
ejpam-6076	57	20	j	j	PROPN
ejpam-6076	57	21	has	have	VERB
ejpam-6076	57	22	either	either	CCONJ
ejpam-6076	57	23	no	no	DET
ejpam-6076	57	24	neighbors	neighbor	NOUN
ejpam-6076	57	25	or	or	CCONJ
ejpam-6076	57	26	at	at	ADP
ejpam-6076	57	27	least	least	ADV
ejpam-6076	57	28	two	two	NUM
ejpam-6076	57	29	neighbors	neighbor	NOUN
ejpam-6076	57	30	in	in	ADP
ejpam-6076	57	31	v	v	NOUN
ejpam-6076	57	32	(	(	PUNCT
ejpam-6076	57	33	g	g	NOUN
ejpam-6076	57	34	)	)	PUNCT
ejpam-6076	57	35	\	\	PROPN
ejpam-6076	57	36	j	j	PROPN
ejpam-6076	57	37	.	.	PUNCT
ejpam-6076	58	1	the	the	DET
ejpam-6076	58	2	certified	certify	VERB
ejpam-6076	58	3	domination	domination	NOUN
ejpam-6076	58	4	number	number	NOUN
ejpam-6076	58	5	γcer(g	γcer(g	PROPN
ejpam-6076	58	6	)	)	PUNCT
ejpam-6076	58	7	represents	represent	VERB
ejpam-6076	58	8	the	the	DET
ejpam-6076	58	9	smallest	small	ADJ
ejpam-6076	58	10	size	size	NOUN
ejpam-6076	58	11	of	of	ADP
ejpam-6076	58	12	a	a	DET
ejpam-6076	58	13	certified	certify	VERB
ejpam-6076	58	14	dominating	dominating	NOUN
ejpam-6076	58	15	set	set	VERB
ejpam-6076	58	16	in	in	ADP
ejpam-6076	58	17	g.	g.	PROPN
ejpam-6076	58	18	a	a	DET
ejpam-6076	58	19	certified	certify	VERB
ejpam-6076	58	20	dominating	dominating	NOUN
ejpam-6076	58	21	set	set	NOUN
ejpam-6076	58	22	of	of	ADP
ejpam-6076	58	23	g	g	PROPN
ejpam-6076	58	24	that	that	PRON
ejpam-6076	58	25	attains	attain	VERB
ejpam-6076	58	26	this	this	DET
ejpam-6076	58	27	minimum	minimum	ADJ
ejpam-6076	58	28	size	size	NOUN
ejpam-6076	58	29	,	,	PUNCT
ejpam-6076	58	30	i.e.	i.e.	X
ejpam-6076	58	31	,	,	PUNCT
ejpam-6076	58	32	|j	|j	NOUN
ejpam-6076	58	33	|	|	NOUN
ejpam-6076	58	34	=	=	SYM
ejpam-6076	58	35	γcerp(g	γcerp(g	PROPN
ejpam-6076	58	36	)	)	PUNCT
ejpam-6076	58	37	is	be	AUX
ejpam-6076	58	38	referred	refer	VERB
ejpam-6076	58	39	to	to	ADP
ejpam-6076	58	40	as	as	ADP
ejpam-6076	58	41	a	a	DET
ejpam-6076	58	42	γcer	γcer	NOUN
ejpam-6076	58	43	-	-	PUNCT
ejpam-6076	58	44	set	set	NOUN
ejpam-6076	58	45	,	,	PUNCT
ejpam-6076	58	46	as	as	SCONJ
ejpam-6076	58	47	defined	define	VERB
ejpam-6076	58	48	by	by	ADP
ejpam-6076	58	49	dettlaff	dettlaff	VERB
ejpam-6076	58	50	et	et	PROPN
ejpam-6076	58	51	al	al	PROPN
ejpam-6076	58	52	.	.	PUNCT
ejpam-6076	59	1	(	(	PUNCT
ejpam-6076	59	2	see	see	VERB
ejpam-6076	59	3	[	[	X
ejpam-6076	59	4	1	1	NUM
ejpam-6076	59	5	]	]	NUM
ejpam-6076	59	6	)	)	PUNCT
ejpam-6076	59	7	.	.	PUNCT
ejpam-6076	60	1	a	a	DET
ejpam-6076	60	2	subset	subset	NOUN
ejpam-6076	60	3	j	j	PROPN
ejpam-6076	60	4	of	of	ADP
ejpam-6076	60	5	v	v	PROPN
ejpam-6076	60	6	(	(	PUNCT
ejpam-6076	60	7	g	g	NOUN
ejpam-6076	60	8	)	)	PUNCT
ejpam-6076	60	9	is	be	AUX
ejpam-6076	60	10	called	call	VERB
ejpam-6076	60	11	an	an	DET
ejpam-6076	60	12	independent	independent	ADJ
ejpam-6076	60	13	set	set	NOUN
ejpam-6076	60	14	if	if	SCONJ
ejpam-6076	60	15	for	for	ADP
ejpam-6076	60	16	each	each	DET
ejpam-6076	60	17	pair	pair	NOUN
ejpam-6076	60	18	of	of	ADP
ejpam-6076	60	19	distinct	distinct	ADJ
ejpam-6076	60	20	vertices	vertex	NOUN
ejpam-6076	60	21	a	a	DET
ejpam-6076	60	22	,	,	PUNCT
ejpam-6076	60	23	b	b	PROPN
ejpam-6076	60	24	∈	∈	PROPN
ejpam-6076	60	25	j	j	PROPN
ejpam-6076	60	26	,	,	PUNCT
ejpam-6076	60	27	ab	ab	PROPN
ejpam-6076	60	28	/∈	/∈	PUNCT
ejpam-6076	60	29	e(g	e(g	PROPN
ejpam-6076	60	30	)	)	PUNCT
ejpam-6076	60	31	.	.	PUNCT
ejpam-6076	61	1	the	the	DET
ejpam-6076	61	2	independence	independence	NOUN
ejpam-6076	61	3	number	number	NOUN
ejpam-6076	61	4	α(g	α(g	NUM
ejpam-6076	61	5	)	)	PUNCT
ejpam-6076	61	6	represents	represent	VERB
ejpam-6076	61	7	the	the	DET
ejpam-6076	61	8	maximum	maximum	ADJ
ejpam-6076	61	9	size	size	NOUN
ejpam-6076	61	10	of	of	ADP
ejpam-6076	61	11	an	an	DET
ejpam-6076	61	12	independent	independent	ADJ
ejpam-6076	61	13	set	set	NOUN
ejpam-6076	61	14	of	of	ADP
ejpam-6076	61	15	g.	g.	PROPN
ejpam-6076	61	16	additionally	additionally	ADV
ejpam-6076	61	17	,	,	PUNCT
ejpam-6076	61	18	the	the	DET
ejpam-6076	61	19	independent	independent	ADJ
ejpam-6076	61	20	domination	domination	NOUN
ejpam-6076	61	21	number	number	NOUN
ejpam-6076	61	22	γi(g	γi(g	PUNCT
ejpam-6076	61	23	)	)	PUNCT
ejpam-6076	61	24	represents	represent	VERB
ejpam-6076	61	25	the	the	DET
ejpam-6076	61	26	minimum	minimum	ADJ
ejpam-6076	61	27	size	size	NOUN
ejpam-6076	61	28	of	of	ADP
ejpam-6076	61	29	an	an	DET
ejpam-6076	61	30	independent	independent	ADJ
ejpam-6076	61	31	set	set	NOUN
ejpam-6076	61	32	of	of	ADP
ejpam-6076	61	33	g	g	NOUN
ejpam-6076	61	34	,	,	PUNCT
ejpam-6076	61	35	as	as	SCONJ
ejpam-6076	61	36	defined	define	VERB
ejpam-6076	61	37	by	by	ADP
ejpam-6076	61	38	paraic	paraic	PROPN
ejpam-6076	61	39	et	et	PROPN
ejpam-6076	61	40	al	al	PROPN
ejpam-6076	61	41	.	.	PUNCT
ejpam-6076	62	1	(	(	PUNCT
ejpam-6076	62	2	see[6	see[6	X
ejpam-6076	62	3	]	]	PUNCT
ejpam-6076	62	4	)	)	PUNCT
ejpam-6076	62	5	.	.	PUNCT
ejpam-6076	63	1	a	a	DET
ejpam-6076	63	2	subset	subset	NOUN
ejpam-6076	63	3	j	j	PROPN
ejpam-6076	63	4	of	of	ADP
ejpam-6076	63	5	v	v	PROPN
ejpam-6076	63	6	(	(	PUNCT
ejpam-6076	63	7	g	g	NOUN
ejpam-6076	63	8	)	)	PUNCT
ejpam-6076	63	9	is	be	AUX
ejpam-6076	63	10	called	call	VERB
ejpam-6076	63	11	a	a	DET
ejpam-6076	63	12	perfect	perfect	ADJ
ejpam-6076	63	13	dominating	dominating	NOUN
ejpam-6076	63	14	set	set	NOUN
ejpam-6076	63	15	if	if	SCONJ
ejpam-6076	63	16	each	each	DET
ejpam-6076	63	17	vertex	vertex	NOUN
ejpam-6076	63	18	in	in	ADP
ejpam-6076	63	19	v	v	NOUN
ejpam-6076	63	20	(	(	PUNCT
ejpam-6076	63	21	g	g	NOUN
ejpam-6076	63	22	)	)	PUNCT
ejpam-6076	63	23	\	\	PROPN
ejpam-6076	64	1	j	j	PROPN
ejpam-6076	64	2	is	be	AUX
ejpam-6076	64	3	adjacent	adjacent	ADJ
ejpam-6076	64	4	to	to	ADP
ejpam-6076	64	5	exactly	exactly	ADV
ejpam-6076	64	6	one	one	NUM
ejpam-6076	64	7	vertex	vertex	NOUN
ejpam-6076	64	8	in	in	ADP
ejpam-6076	64	9	j	j	PROPN
ejpam-6076	64	10	.	.	PUNCT
ejpam-6076	65	1	the	the	DET
ejpam-6076	65	2	perfect	perfect	ADJ
ejpam-6076	65	3	domination	domination	NOUN
ejpam-6076	65	4	number	number	NOUN
ejpam-6076	65	5	γp(g	γp(g	PUNCT
ejpam-6076	65	6	)	)	PUNCT
ejpam-6076	65	7	represents	represent	VERB
ejpam-6076	65	8	the	the	DET
ejpam-6076	65	9	smallest	small	ADJ
ejpam-6076	65	10	size	size	NOUN
ejpam-6076	65	11	of	of	ADP
ejpam-6076	65	12	a	a	DET
ejpam-6076	65	13	perfect	perfect	ADJ
ejpam-6076	65	14	dominating	dominating	NOUN
ejpam-6076	65	15	set	set	VERB
ejpam-6076	65	16	in	in	ADP
ejpam-6076	65	17	g.	g.	PROPN
ejpam-6076	65	18	a	a	DET
ejpam-6076	65	19	perfect	perfect	ADJ
ejpam-6076	65	20	dominating	dominating	NOUN
ejpam-6076	65	21	set	set	NOUN
ejpam-6076	65	22	of	of	ADP
ejpam-6076	65	23	g	g	PROPN
ejpam-6076	65	24	that	that	PRON
ejpam-6076	65	25	attains	attain	VERB
ejpam-6076	65	26	this	this	DET
ejpam-6076	65	27	minimum	minimum	ADJ
ejpam-6076	65	28	size	size	NOUN
ejpam-6076	65	29	,	,	PUNCT
ejpam-6076	65	30	i.e	i.e	PRON
ejpam-6076	65	31	,	,	PUNCT
ejpam-6076	65	32	|j	|j	NOUN
ejpam-6076	65	33	|	|	NOUN
ejpam-6076	65	34	=	=	SYM
ejpam-6076	65	35	γcerp(g	γcerp(g	PROPN
ejpam-6076	65	36	)	)	PUNCT
ejpam-6076	65	37	is	be	AUX
ejpam-6076	65	38	referred	refer	VERB
ejpam-6076	65	39	to	to	ADP
ejpam-6076	65	40	as	as	ADP
ejpam-6076	65	41	a	a	DET
ejpam-6076	65	42	γp	γp	NOUN
ejpam-6076	65	43	-	-	PUNCT
ejpam-6076	65	44	set	set	NOUN
ejpam-6076	65	45	,	,	PUNCT
ejpam-6076	65	46	as	as	SCONJ
ejpam-6076	65	47	defined	define	VERB
ejpam-6076	65	48	by	by	ADP
ejpam-6076	65	49	livingston	livingston	PROPN
ejpam-6076	65	50	et	et	PROPN
ejpam-6076	65	51	al	al	PROPN
ejpam-6076	65	52	.	.	PUNCT
ejpam-6076	66	1	(	(	PUNCT
ejpam-6076	66	2	see	see	VERB
ejpam-6076	66	3	[	[	X
ejpam-6076	66	4	7	7	NUM
ejpam-6076	66	5	]	]	NUM
ejpam-6076	66	6	)	)	PUNCT
ejpam-6076	66	7	.	.	PUNCT
ejpam-6076	67	1	a	a	DET
ejpam-6076	67	2	perfect	perfect	ADJ
ejpam-6076	67	3	dominating	dominating	NOUN
ejpam-6076	67	4	set	set	VERB
ejpam-6076	67	5	j	j	PROPN
ejpam-6076	67	6	⊆	⊆	NUM
ejpam-6076	67	7	v	v	NOUN
ejpam-6076	67	8	(	(	PUNCT
ejpam-6076	67	9	g	g	NOUN
ejpam-6076	67	10	)	)	PUNCT
ejpam-6076	67	11	is	be	AUX
ejpam-6076	67	12	called	call	VERB
ejpam-6076	67	13	an	an	DET
ejpam-6076	67	14	independent	independent	ADJ
ejpam-6076	67	15	perfect	perfect	ADJ
ejpam-6076	67	16	dominating	dominating	NOUN
ejpam-6076	67	17	set	set	NOUN
ejpam-6076	67	18	if	if	SCONJ
ejpam-6076	67	19	no	no	DET
ejpam-6076	67	20	two	two	NUM
ejpam-6076	67	21	vertices	vertex	NOUN
ejpam-6076	67	22	in	in	ADP
ejpam-6076	67	23	j	j	PROPN
ejpam-6076	67	24	are	be	AUX
ejpam-6076	67	25	adjacent	adjacent	ADJ
ejpam-6076	67	26	.	.	PUNCT
ejpam-6076	68	1	the	the	DET
ejpam-6076	68	2	independent	independent	ADJ
ejpam-6076	68	3	perfect	perfect	ADJ
ejpam-6076	68	4	domination	domination	NOUN
ejpam-6076	68	5	number	number	NOUN
ejpam-6076	68	6	γip(g	γip(g	PROPN
ejpam-6076	68	7	)	)	PUNCT
ejpam-6076	68	8	represents	represent	VERB
ejpam-6076	68	9	the	the	DET
ejpam-6076	68	10	smallest	small	ADJ
ejpam-6076	68	11	size	size	NOUN
ejpam-6076	68	12	of	of	ADP
ejpam-6076	68	13	an	an	DET
ejpam-6076	68	14	independent	independent	ADJ
ejpam-6076	68	15	perfect	perfect	ADJ
ejpam-6076	68	16	dominating	dominating	NOUN
ejpam-6076	68	17	set	set	NOUN
ejpam-6076	68	18	of	of	ADP
ejpam-6076	68	19	g.	g.	PROPN
ejpam-6076	68	20	an	an	DET
ejpam-6076	68	21	independent	independent	ADJ
ejpam-6076	68	22	perfect	perfect	ADJ
ejpam-6076	68	23	dominating	dominating	NOUN
ejpam-6076	68	24	set	set	NOUN
ejpam-6076	68	25	of	of	ADP
ejpam-6076	68	26	g	g	PROPN
ejpam-6076	68	27	that	that	PRON
ejpam-6076	68	28	attains	attain	VERB
ejpam-6076	68	29	this	this	DET
ejpam-6076	68	30	minimum	minimum	ADJ
ejpam-6076	68	31	size	size	NOUN
ejpam-6076	68	32	,	,	PUNCT
ejpam-6076	68	33	i.e.	i.e.	X
ejpam-6076	68	34	,	,	PUNCT
ejpam-6076	68	35	|j	|j	NOUN
ejpam-6076	69	1	|	|	NOUN
ejpam-6076	69	2	=	=	SYM
ejpam-6076	69	3	γip(g	γip(g	PROPN
ejpam-6076	69	4	)	)	PUNCT
ejpam-6076	69	5	is	be	AUX
ejpam-6076	69	6	referred	refer	VERB
ejpam-6076	69	7	to	to	ADP
ejpam-6076	69	8	as	as	ADP
ejpam-6076	69	9	an	an	DET
ejpam-6076	69	10	γip	γip	NOUN
ejpam-6076	69	11	-	-	PUNCT
ejpam-6076	69	12	set	set	NOUN
ejpam-6076	69	13	,	,	PUNCT
ejpam-6076	69	14	as	as	SCONJ
ejpam-6076	69	15	defined	define	VERB
ejpam-6076	69	16	by	by	ADP
ejpam-6076	69	17	armada	armada	PROPN
ejpam-6076	69	18	et	et	PROPN
ejpam-6076	69	19	al	al	PROPN
ejpam-6076	69	20	(	(	PUNCT
ejpam-6076	69	21	see	see	VERB
ejpam-6076	69	22	[	[	X
ejpam-6076	69	23	8	8	NUM
ejpam-6076	69	24	]	]	NUM
ejpam-6076	69	25	)	)	PUNCT
ejpam-6076	69	26	.	.	PUNCT
ejpam-6076	70	1	an	an	DET
ejpam-6076	70	2	independent	independent	ADJ
ejpam-6076	70	3	perfect	perfect	ADJ
ejpam-6076	70	4	dominating	dominating	NOUN
ejpam-6076	70	5	set	set	VERB
ejpam-6076	70	6	j	j	PROPN
ejpam-6076	70	7	⊆	⊆	NUM
ejpam-6076	70	8	v	v	NOUN
ejpam-6076	70	9	(	(	PUNCT
ejpam-6076	70	10	g	g	NOUN
ejpam-6076	70	11	)	)	PUNCT
ejpam-6076	70	12	is	be	AUX
ejpam-6076	70	13	called	call	VERB
ejpam-6076	70	14	a	a	DET
ejpam-6076	70	15	certified	certify	VERB
ejpam-6076	70	16	independent	independent	ADJ
ejpam-6076	70	17	perfect	perfect	ADJ
ejpam-6076	70	18	dominating	dominating	NOUN
ejpam-6076	70	19	set	set	NOUN
ejpam-6076	70	20	if	if	SCONJ
ejpam-6076	70	21	every	every	DET
ejpam-6076	70	22	a	a	DET
ejpam-6076	70	23	∈	∈	PROPN
ejpam-6076	70	24	j	j	PROPN
ejpam-6076	70	25	has	have	VERB
ejpam-6076	70	26	either	either	CCONJ
ejpam-6076	70	27	zero	zero	NUM
ejpam-6076	70	28	or	or	CCONJ
ejpam-6076	70	29	at	at	ADP
ejpam-6076	70	30	least	least	ADV
ejpam-6076	70	31	two	two	NUM
ejpam-6076	70	32	neighbors	neighbor	NOUN
ejpam-6076	70	33	in	in	ADP
ejpam-6076	70	34	v	v	NOUN
ejpam-6076	70	35	(	(	PUNCT
ejpam-6076	70	36	g	g	NOUN
ejpam-6076	70	37	)	)	PUNCT
ejpam-6076	70	38	\	\	PROPN
ejpam-6076	70	39	j	j	PROPN
ejpam-6076	70	40	.	.	PUNCT
ejpam-6076	71	1	the	the	DET
ejpam-6076	71	2	certified	certify	VERB
ejpam-6076	71	3	independence	independence	NOUN
ejpam-6076	71	4	perfect	perfect	ADJ
ejpam-6076	71	5	number	number	NOUN
ejpam-6076	71	6	the	the	DET
ejpam-6076	71	7	maximum	maximum	ADJ
ejpam-6076	71	8	size	size	NOUN
ejpam-6076	71	9	of	of	ADP
ejpam-6076	71	10	a	a	DET
ejpam-6076	71	11	certified	certify	VERB
ejpam-6076	71	12	independent	independent	ADJ
ejpam-6076	71	13	perfect	perfect	ADJ
ejpam-6076	71	14	set	set	NOUN
ejpam-6076	71	15	of	of	ADP
ejpam-6076	71	16	g.	g.	PROPN
ejpam-6076	71	17	additionally	additionally	ADV
ejpam-6076	71	18	,	,	PUNCT
ejpam-6076	71	19	the	the	DET
ejpam-6076	71	20	certified	certify	VERB
ejpam-6076	71	21	independent	independent	ADJ
ejpam-6076	71	22	perfect	perfect	ADJ
ejpam-6076	71	23	domination	domination	NOUN
ejpam-6076	71	24	number	number	NOUN
ejpam-6076	71	25	γcerip(g	γcerip(g	PROPN
ejpam-6076	71	26	)	)	PUNCT
ejpam-6076	71	27	represents	represent	VERB
ejpam-6076	71	28	the	the	DET
ejpam-6076	71	29	minimum	minimum	ADJ
ejpam-6076	71	30	size	size	NOUN
ejpam-6076	71	31	of	of	ADP
ejpam-6076	71	32	a	a	DET
ejpam-6076	71	33	certified	certify	VERB
ejpam-6076	71	34	independent	independent	ADJ
ejpam-6076	71	35	perfect	perfect	ADJ
ejpam-6076	71	36	dominating	dominating	NOUN
ejpam-6076	71	37	set	set	NOUN
ejpam-6076	71	38	of	of	ADP
ejpam-6076	71	39	g.	g.	PROPN
ejpam-6076	71	40	a	a	DET
ejpam-6076	71	41	certified	certify	VERB
ejpam-6076	71	42	independent	independent	ADJ
ejpam-6076	71	43	perfect	perfect	ADJ
ejpam-6076	71	44	dominating	dominating	NOUN
ejpam-6076	71	45	set	set	NOUN
ejpam-6076	71	46	of	of	ADP
ejpam-6076	71	47	g	g	PROPN
ejpam-6076	71	48	that	that	PRON
ejpam-6076	71	49	attains	attain	VERB
ejpam-6076	71	50	this	this	DET
ejpam-6076	71	51	minimum	minimum	ADJ
ejpam-6076	71	52	size	size	NOUN
ejpam-6076	71	53	,	,	PUNCT
ejpam-6076	71	54	i.e.	i.e.	X
ejpam-6076	71	55	,	,	PUNCT
ejpam-6076	71	56	|j	|j	NOUN
ejpam-6076	71	57	|	|	NOUN
ejpam-6076	71	58	=	=	SYM
ejpam-6076	71	59	γcerip(g	γcerip(g	PROPN
ejpam-6076	71	60	)	)	PUNCT
ejpam-6076	71	61	is	be	AUX
ejpam-6076	71	62	referred	refer	VERB
ejpam-6076	71	63	to	to	ADP
ejpam-6076	71	64	as	as	ADP
ejpam-6076	71	65	a	a	DET
ejpam-6076	71	66	γcerip	γcerip	NOUN
ejpam-6076	71	67	-	-	PUNCT
ejpam-6076	71	68	set	set	NOUN
ejpam-6076	71	69	.	.	PUNCT
ejpam-6076	72	1	a	a	DET
ejpam-6076	72	2	perfect	perfect	ADJ
ejpam-6076	72	3	dominating	dominating	NOUN
ejpam-6076	72	4	set	set	VERB
ejpam-6076	72	5	j	j	PROPN
ejpam-6076	72	6	⊆	⊆	NUM
ejpam-6076	72	7	v	v	NOUN
ejpam-6076	72	8	(	(	PUNCT
ejpam-6076	72	9	g	g	NOUN
ejpam-6076	72	10	)	)	PUNCT
ejpam-6076	72	11	is	be	AUX
ejpam-6076	72	12	called	call	VERB
ejpam-6076	72	13	a	a	DET
ejpam-6076	72	14	certified	certify	VERB
ejpam-6076	72	15	perfect	perfect	ADJ
ejpam-6076	72	16	dominating	dominating	NOUN
ejpam-6076	72	17	set	set	NOUN
ejpam-6076	72	18	of	of	ADP
ejpam-6076	72	19	g	g	PROPN
ejpam-6076	72	20	j.	j.	PROPN
ejpam-6076	72	21	j.	j.	PROPN
ejpam-6076	72	22	hamja	hamja	PROPN
ejpam-6076	72	23	et	et	PROPN
ejpam-6076	72	24	al	al	PROPN
ejpam-6076	72	25	.	.	PUNCT
ejpam-6076	72	26	/	/	SYM
ejpam-6076	72	27	eur	eur	PROPN
ejpam-6076	72	28	.	.	PUNCT
ejpam-6076	73	1	j.	j.	PROPN
ejpam-6076	73	2	pure	pure	PROPN
ejpam-6076	73	3	appl	appl	PROPN
ejpam-6076	73	4	.	.	PROPN
ejpam-6076	73	5	math	math	PROPN
ejpam-6076	73	6	,	,	PUNCT
ejpam-6076	73	7	18	18	NUM
ejpam-6076	73	8	(	(	PUNCT
ejpam-6076	73	9	3	3	NUM
ejpam-6076	73	10	)	)	PUNCT
ejpam-6076	73	11	(	(	PUNCT
ejpam-6076	73	12	2025	2025	NUM
ejpam-6076	73	13	)	)	PUNCT
ejpam-6076	73	14	,	,	PUNCT
ejpam-6076	73	15	6076	6076	NUM
ejpam-6076	73	16	4	4	NUM
ejpam-6076	73	17	of	of	ADP
ejpam-6076	73	18	13	13	NUM
ejpam-6076	73	19	if	if	SCONJ
ejpam-6076	73	20	each	each	DET
ejpam-6076	73	21	vertex	vertex	VERB
ejpam-6076	73	22	a	a	DET
ejpam-6076	73	23	∈	∈	PROPN
ejpam-6076	73	24	j	j	PROPN
ejpam-6076	73	25	has	have	VERB
ejpam-6076	73	26	either	either	CCONJ
ejpam-6076	73	27	no	no	DET
ejpam-6076	73	28	neighbors	neighbor	NOUN
ejpam-6076	73	29	or	or	CCONJ
ejpam-6076	73	30	at	at	ADP
ejpam-6076	73	31	least	least	ADV
ejpam-6076	73	32	two	two	NUM
ejpam-6076	73	33	neighbors	neighbor	NOUN
ejpam-6076	73	34	in	in	ADP
ejpam-6076	73	35	v	v	NOUN
ejpam-6076	73	36	(	(	PUNCT
ejpam-6076	73	37	g	g	NOUN
ejpam-6076	73	38	)	)	PUNCT
ejpam-6076	73	39	\	\	PROPN
ejpam-6076	73	40	j	j	PROPN
ejpam-6076	73	41	.	.	PUNCT
ejpam-6076	74	1	the	the	DET
ejpam-6076	74	2	certified	certify	VERB
ejpam-6076	74	3	perfect	perfect	ADJ
ejpam-6076	74	4	domination	domination	NOUN
ejpam-6076	74	5	number	number	NOUN
ejpam-6076	74	6	of	of	ADP
ejpam-6076	74	7	g	g	PROPN
ejpam-6076	74	8	γcerp(g	γcerp(g	NOUN
ejpam-6076	74	9	)	)	PUNCT
ejpam-6076	74	10	represents	represent	VERB
ejpam-6076	74	11	the	the	DET
ejpam-6076	74	12	smallest	small	ADJ
ejpam-6076	74	13	size	size	NOUN
ejpam-6076	74	14	of	of	ADP
ejpam-6076	74	15	a	a	DET
ejpam-6076	74	16	certified	certify	VERB
ejpam-6076	74	17	perfect	perfect	ADJ
ejpam-6076	74	18	dominating	dominating	NOUN
ejpam-6076	74	19	set	set	VERB
ejpam-6076	74	20	in	in	ADP
ejpam-6076	74	21	g.	g.	PROPN
ejpam-6076	74	22	a	a	DET
ejpam-6076	74	23	certified	certify	VERB
ejpam-6076	74	24	perfect	perfect	ADJ
ejpam-6076	74	25	dominating	dominating	NOUN
ejpam-6076	74	26	set	set	NOUN
ejpam-6076	74	27	of	of	ADP
ejpam-6076	74	28	g	g	PROPN
ejpam-6076	74	29	that	that	PRON
ejpam-6076	74	30	attains	attain	VERB
ejpam-6076	74	31	this	this	DET
ejpam-6076	74	32	minimum	minimum	ADJ
ejpam-6076	74	33	size	size	NOUN
ejpam-6076	74	34	,	,	PUNCT
ejpam-6076	74	35	i.e.	i.e.	X
ejpam-6076	74	36	,	,	PUNCT
ejpam-6076	74	37	|j	|j	NOUN
ejpam-6076	74	38	|	|	NOUN
ejpam-6076	75	1	=	=	SYM
ejpam-6076	75	2	γcep(g	γcep(g	NOUN
ejpam-6076	75	3	)	)	PUNCT
ejpam-6076	75	4	is	be	AUX
ejpam-6076	75	5	referred	refer	VERB
ejpam-6076	75	6	to	to	ADP
ejpam-6076	75	7	as	as	ADP
ejpam-6076	75	8	a	a	DET
ejpam-6076	75	9	γcerp	γcerp	NOUN
ejpam-6076	75	10	-	-	PUNCT
ejpam-6076	75	11	set	set	NOUN
ejpam-6076	75	12	,	,	PUNCT
ejpam-6076	75	13	as	as	SCONJ
ejpam-6076	75	14	defined	define	VERB
ejpam-6076	75	15	by	by	ADP
ejpam-6076	75	16	hamja	hamja	PROPN
ejpam-6076	75	17	(	(	PUNCT
ejpam-6076	75	18	see	see	VERB
ejpam-6076	75	19	[	[	X
ejpam-6076	75	20	3	3	NUM
ejpam-6076	75	21	]	]	NUM
ejpam-6076	75	22	)	)	PUNCT
ejpam-6076	75	23	.	.	PUNCT
ejpam-6076	76	1	it	it	PRON
ejpam-6076	76	2	is	be	AUX
ejpam-6076	76	3	worth	worth	ADJ
ejpam-6076	76	4	noting	note	VERB
ejpam-6076	76	5	that	that	SCONJ
ejpam-6076	76	6	if	if	SCONJ
ejpam-6076	76	7	a	a	DET
ejpam-6076	76	8	graph	graph	NOUN
ejpam-6076	76	9	g	g	NOUN
ejpam-6076	76	10	has	have	VERB
ejpam-6076	76	11	no	no	DET
ejpam-6076	76	12	certified	certify	VERB
ejpam-6076	76	13	perfect	perfect	ADJ
ejpam-6076	76	14	dominating	dominating	NOUN
ejpam-6076	76	15	set	set	NOUN
ejpam-6076	76	16	,	,	PUNCT
ejpam-6076	76	17	then	then	ADV
ejpam-6076	76	18	we	we	PRON
ejpam-6076	76	19	say	say	VERB
ejpam-6076	76	20	that	that	SCONJ
ejpam-6076	76	21	g	g	PROPN
ejpam-6076	76	22	is	be	AUX
ejpam-6076	76	23	a	a	DET
ejpam-6076	76	24	non	non	ADJ
ejpam-6076	76	25	-	-	ADJ
ejpam-6076	76	26	γcerp	γcerp	ADJ
ejpam-6076	76	27	-	-	PUNCT
ejpam-6076	76	28	graph	graph	NOUN
ejpam-6076	76	29	.	.	PUNCT
ejpam-6076	77	1	3	3	X
ejpam-6076	77	2	.	.	X
ejpam-6076	77	3	preliminary	preliminary	ADJ
ejpam-6076	77	4	results	result	NOUN
ejpam-6076	77	5	this	this	DET
ejpam-6076	77	6	section	section	NOUN
ejpam-6076	77	7	presents	present	VERB
ejpam-6076	77	8	the	the	DET
ejpam-6076	77	9	bounds	bound	NOUN
ejpam-6076	77	10	,	,	PUNCT
ejpam-6076	77	11	properties	property	NOUN
ejpam-6076	77	12	,	,	PUNCT
ejpam-6076	77	13	and	and	CCONJ
ejpam-6076	77	14	exact	exact	ADJ
ejpam-6076	77	15	values	value	NOUN
ejpam-6076	77	16	for	for	ADP
ejpam-6076	77	17	various	various	ADJ
ejpam-6076	77	18	graphs	graph	NOUN
ejpam-6076	77	19	.	.	PUNCT
ejpam-6076	78	1	we	we	PRON
ejpam-6076	78	2	begin	begin	VERB
ejpam-6076	78	3	by	by	ADP
ejpam-6076	78	4	providing	provide	VERB
ejpam-6076	78	5	some	some	DET
ejpam-6076	78	6	upper	upper	ADJ
ejpam-6076	78	7	bounds	bound	NOUN
ejpam-6076	78	8	for	for	ADP
ejpam-6076	78	9	the	the	DET
ejpam-6076	78	10	certified	certify	VERB
ejpam-6076	78	11	perfect	perfect	ADJ
ejpam-6076	78	12	domination	domination	NOUN
ejpam-6076	78	13	in	in	ADP
ejpam-6076	78	14	graphs	graph	NOUN
ejpam-6076	78	15	.	.	PUNCT
ejpam-6076	79	1	proposition	proposition	NOUN
ejpam-6076	79	2	1	1	NUM
ejpam-6076	79	3	.	.	PUNCT
ejpam-6076	80	1	[	[	X
ejpam-6076	80	2	3	3	X
ejpam-6076	80	3	]	]	PUNCT
ejpam-6076	80	4	for	for	ADP
ejpam-6076	80	5	a	a	DET
ejpam-6076	80	6	path	path	NOUN
ejpam-6076	80	7	pn	pn	NOUN
ejpam-6076	80	8	of	of	ADP
ejpam-6076	80	9	order	order	NOUN
ejpam-6076	80	10	n	n	PRON
ejpam-6076	80	11	≥	≥	NOUN
ejpam-6076	80	12	1	1	NUM
ejpam-6076	80	13	,	,	PUNCT
ejpam-6076	80	14	γcerp(pn	γcerp(pn	NOUN
ejpam-6076	80	15	)	)	PUNCT
ejpam-6076	80	16	=	=	NOUN
ejpam-6076	80	17	{	{	PUNCT
ejpam-6076	80	18	n	n	NOUN
ejpam-6076	80	19	3	3	NUM
ejpam-6076	80	20	,	,	PUNCT
ejpam-6076	80	21	if	if	SCONJ
ejpam-6076	80	22	n	n	PRON
ejpam-6076	80	23	≡	≡	PROPN
ejpam-6076	80	24	0	0	PUNCT
ejpam-6076	81	1	(	(	PUNCT
ejpam-6076	81	2	mod	mod	NOUN
ejpam-6076	81	3	3	3	NUM
ejpam-6076	81	4	)	)	PUNCT
ejpam-6076	81	5	,	,	PUNCT
ejpam-6076	81	6	n	n	CCONJ
ejpam-6076	81	7	,	,	PUNCT
ejpam-6076	81	8	otherwise	otherwise	ADV
ejpam-6076	81	9	.	.	PUNCT
ejpam-6076	82	1	proposition	proposition	NOUN
ejpam-6076	82	2	2	2	NUM
ejpam-6076	82	3	.	.	PUNCT
ejpam-6076	83	1	[	[	X
ejpam-6076	83	2	3	3	X
ejpam-6076	83	3	]	]	PUNCT
ejpam-6076	83	4	for	for	ADP
ejpam-6076	83	5	a	a	DET
ejpam-6076	83	6	cycle	cycle	NOUN
ejpam-6076	83	7	cn	cn	NOUN
ejpam-6076	83	8	of	of	ADP
ejpam-6076	83	9	order	order	NOUN
ejpam-6076	83	10	n	n	PRON
ejpam-6076	83	11	≥	≥	NOUN
ejpam-6076	83	12	3	3	NUM
ejpam-6076	83	13	,	,	PUNCT
ejpam-6076	83	14	γcerp(cn	γcerp(cn	NOUN
ejpam-6076	83	15	)	)	PUNCT
ejpam-6076	83	16	=	=	PUNCT
ejpam-6076	83	17	{	{	PUNCT
ejpam-6076	83	18	n	n	NOUN
ejpam-6076	83	19	3	3	NUM
ejpam-6076	83	20	,	,	PUNCT
ejpam-6076	83	21	if	if	SCONJ
ejpam-6076	83	22	n	n	PRON
ejpam-6076	83	23	≡	≡	PROPN
ejpam-6076	83	24	0	0	PUNCT
ejpam-6076	83	25	(	(	PUNCT
ejpam-6076	83	26	mod	mod	NOUN
ejpam-6076	83	27	3	3	NUM
ejpam-6076	83	28	)	)	PUNCT
ejpam-6076	83	29	,	,	PUNCT
ejpam-6076	83	30	n	n	CCONJ
ejpam-6076	83	31	,	,	PUNCT
ejpam-6076	83	32	otherwise	otherwise	ADV
ejpam-6076	83	33	.	.	PUNCT
ejpam-6076	84	1	proposition	proposition	NOUN
ejpam-6076	84	2	3	3	NUM
ejpam-6076	84	3	.	.	PUNCT
ejpam-6076	85	1	[	[	X
ejpam-6076	85	2	3	3	X
ejpam-6076	85	3	]	]	PUNCT
ejpam-6076	85	4	for	for	ADP
ejpam-6076	85	5	a	a	DET
ejpam-6076	85	6	complete	complete	ADJ
ejpam-6076	85	7	graph	graph	NOUN
ejpam-6076	85	8	kn	kn	NOUN
ejpam-6076	85	9	of	of	ADP
ejpam-6076	85	10	order	order	NOUN
ejpam-6076	85	11	n	n	CCONJ
ejpam-6076	85	12	,	,	PUNCT
ejpam-6076	85	13	γcerp(kn	γcerp(kn	NOUN
ejpam-6076	85	14	)	)	PUNCT
ejpam-6076	85	15	=	=	PUNCT
ejpam-6076	85	16	{	{	PUNCT
ejpam-6076	85	17	1	1	NUM
ejpam-6076	85	18	,	,	PUNCT
ejpam-6076	85	19	if	if	SCONJ
ejpam-6076	85	20	n	n	CCONJ
ejpam-6076	85	21	=	=	SYM
ejpam-6076	85	22	1	1	NUM
ejpam-6076	85	23	or	or	CCONJ
ejpam-6076	85	24	n	n	PRON
ejpam-6076	85	25	≥	≥	NOUN
ejpam-6076	85	26	3	3	NUM
ejpam-6076	85	27	,	,	PUNCT
ejpam-6076	85	28	2	2	NUM
ejpam-6076	85	29	,	,	PUNCT
ejpam-6076	85	30	if	if	SCONJ
ejpam-6076	85	31	n	n	NOUN
ejpam-6076	85	32	=	=	SYM
ejpam-6076	85	33	2	2	X
ejpam-6076	85	34	.	.	X
ejpam-6076	85	35	proposition	proposition	NOUN
ejpam-6076	85	36	4	4	NUM
ejpam-6076	85	37	.	.	PUNCT
ejpam-6076	86	1	[	[	X
ejpam-6076	86	2	3	3	X
ejpam-6076	86	3	]	]	PUNCT
ejpam-6076	86	4	for	for	ADP
ejpam-6076	86	5	a	a	DET
ejpam-6076	86	6	complete	complete	ADJ
ejpam-6076	86	7	bipartite	bipartite	NOUN
ejpam-6076	86	8	graph	graph	NOUN
ejpam-6076	86	9	km	km	PROPN
ejpam-6076	86	10	,	,	PUNCT
ejpam-6076	86	11	n	n	NUM
ejpam-6076	86	12	of	of	ADP
ejpam-6076	86	13	orders	order	NOUN
ejpam-6076	86	14	m+	m+	NUM
ejpam-6076	86	15	n	n	CCONJ
ejpam-6076	86	16	,	,	PUNCT
ejpam-6076	86	17	γcerp(km	γcerp(km	PROPN
ejpam-6076	86	18	,	,	PUNCT
ejpam-6076	86	19	n	n	CCONJ
ejpam-6076	86	20	)	)	PUNCT
ejpam-6076	86	21	=	=	SYM
ejpam-6076	87	1			NOUN
ejpam-6076	87	2	1	1	NUM
ejpam-6076	87	3	,	,	PUNCT
ejpam-6076	87	4	if	if	SCONJ
ejpam-6076	87	5	m	m	ADV
ejpam-6076	87	6	=	=	SYM
ejpam-6076	87	7	1	1	NUM
ejpam-6076	87	8	or	or	CCONJ
ejpam-6076	87	9	n	n	NOUN
ejpam-6076	87	10	=	=	SYM
ejpam-6076	87	11	1	1	NUM
ejpam-6076	87	12	,	,	PUNCT
ejpam-6076	87	13	4	4	NUM
ejpam-6076	87	14	,	,	PUNCT
ejpam-6076	87	15	if	if	SCONJ
ejpam-6076	87	16	m	m	VERB
ejpam-6076	87	17	=	=	SYM
ejpam-6076	87	18	n	n	NOUN
ejpam-6076	87	19	=	=	SYM
ejpam-6076	87	20	2	2	NUM
ejpam-6076	87	21	,	,	PUNCT
ejpam-6076	87	22	2	2	NUM
ejpam-6076	87	23	,	,	PUNCT
ejpam-6076	87	24	otherwise	otherwise	ADV
ejpam-6076	87	25	.	.	PUNCT
ejpam-6076	88	1	proposition	proposition	NOUN
ejpam-6076	88	2	5	5	NUM
ejpam-6076	88	3	.	.	PUNCT
ejpam-6076	89	1	if	if	SCONJ
ejpam-6076	89	2	g	g	PROPN
ejpam-6076	89	3	is	be	AUX
ejpam-6076	89	4	a	a	DET
ejpam-6076	89	5	connected	connected	ADJ
ejpam-6076	89	6	graph	graph	NOUN
ejpam-6076	89	7	,	,	PUNCT
ejpam-6076	89	8	then	then	ADV
ejpam-6076	89	9	each	each	DET
ejpam-6076	89	10	support	support	NOUN
ejpam-6076	89	11	vertex	vertex	NOUN
ejpam-6076	89	12	of	of	ADP
ejpam-6076	89	13	g	g	PROPN
ejpam-6076	89	14	is	be	AUX
ejpam-6076	89	15	contained	contain	VERB
ejpam-6076	89	16	in	in	ADP
ejpam-6076	89	17	every	every	DET
ejpam-6076	89	18	certified	certify	VERB
ejpam-6076	89	19	perfect	perfect	ADJ
ejpam-6076	89	20	dominating	dominating	NOUN
ejpam-6076	89	21	set	set	NOUN
ejpam-6076	89	22	of	of	ADP
ejpam-6076	89	23	g.	g.	PROPN
ejpam-6076	89	24	proof	proof	PROPN
ejpam-6076	89	25	.	.	PUNCT
ejpam-6076	90	1	let	let	VERB
ejpam-6076	90	2	j	j	PROPN
ejpam-6076	90	3	be	be	AUX
ejpam-6076	90	4	a	a	DET
ejpam-6076	90	5	certified	certify	VERB
ejpam-6076	90	6	perfect	perfect	ADJ
ejpam-6076	90	7	dominating	dominating	NOUN
ejpam-6076	90	8	set	set	NOUN
ejpam-6076	90	9	of	of	ADP
ejpam-6076	90	10	g.	g.	PROPN
ejpam-6076	90	11	assume	assume	PROPN
ejpam-6076	90	12	,	,	PUNCT
ejpam-6076	90	13	for	for	ADP
ejpam-6076	90	14	the	the	DET
ejpam-6076	90	15	sake	sake	NOUN
ejpam-6076	90	16	of	of	ADP
ejpam-6076	90	17	contradiction	contradiction	NOUN
ejpam-6076	90	18	,	,	PUNCT
ejpam-6076	90	19	that	that	SCONJ
ejpam-6076	90	20	there	there	PRON
ejpam-6076	90	21	exists	exist	VERB
ejpam-6076	90	22	a	a	DET
ejpam-6076	90	23	support	support	NOUN
ejpam-6076	90	24	vertex	vertex	NOUN
ejpam-6076	90	25	a	a	DET
ejpam-6076	90	26	∈	∈	NOUN
ejpam-6076	90	27	v	v	ADP
ejpam-6076	90	28	(	(	PUNCT
ejpam-6076	90	29	g	g	NOUN
ejpam-6076	90	30	)	)	PUNCT
ejpam-6076	90	31	such	such	ADJ
ejpam-6076	90	32	that	that	SCONJ
ejpam-6076	90	33	a	a	DET
ejpam-6076	90	34	/∈	/∈	NOUN
ejpam-6076	90	35	j	j	PROPN
ejpam-6076	90	36	.	.	PUNCT
ejpam-6076	91	1	since	since	SCONJ
ejpam-6076	91	2	a	a	PRON
ejpam-6076	91	3	is	be	AUX
ejpam-6076	91	4	a	a	DET
ejpam-6076	91	5	support	support	NOUN
ejpam-6076	91	6	vertex	vertex	NOUN
ejpam-6076	91	7	,	,	PUNCT
ejpam-6076	91	8	there	there	PRON
ejpam-6076	91	9	must	must	AUX
ejpam-6076	91	10	be	be	AUX
ejpam-6076	91	11	at	at	ADV
ejpam-6076	91	12	least	least	ADV
ejpam-6076	91	13	one	one	NUM
ejpam-6076	91	14	leaf	leaf	NOUN
ejpam-6076	91	15	b	b	NOUN
ejpam-6076	91	16	∈	∈	NOUN
ejpam-6076	91	17	v	v	NOUN
ejpam-6076	91	18	(	(	PUNCT
ejpam-6076	91	19	g	g	NOUN
ejpam-6076	91	20	)	)	PUNCT
ejpam-6076	91	21	with	with	ADP
ejpam-6076	91	22	ng(b	ng(b	NOUN
ejpam-6076	91	23	)	)	PUNCT
ejpam-6076	91	24	=	=	PRON
ejpam-6076	91	25	{	{	PUNCT
ejpam-6076	91	26	a	a	X
ejpam-6076	91	27	}	}	PUNCT
ejpam-6076	91	28	.	.	PUNCT
ejpam-6076	92	1	as	as	ADP
ejpam-6076	92	2	a	a	DET
ejpam-6076	92	3	/∈	/∈	NOUN
ejpam-6076	92	4	j	j	NOUN
ejpam-6076	92	5	,	,	PUNCT
ejpam-6076	92	6	it	it	PRON
ejpam-6076	92	7	follows	follow	VERB
ejpam-6076	92	8	that	that	PRON
ejpam-6076	92	9	b	b	PROPN
ejpam-6076	92	10	∈	∈	PROPN
ejpam-6076	92	11	j	j	PROPN
ejpam-6076	92	12	.	.	PUNCT
ejpam-6076	93	1	however	however	ADV
ejpam-6076	93	2	,	,	PUNCT
ejpam-6076	93	3	this	this	PRON
ejpam-6076	93	4	implies	imply	VERB
ejpam-6076	93	5	that	that	SCONJ
ejpam-6076	93	6	b	b	NOUN
ejpam-6076	93	7	has	have	VERB
ejpam-6076	93	8	exactly	exactly	ADV
ejpam-6076	93	9	one	one	NUM
ejpam-6076	93	10	neighbor	neighbor	NOUN
ejpam-6076	93	11	in	in	ADP
ejpam-6076	93	12	v	v	PROPN
ejpam-6076	93	13	(	(	PUNCT
ejpam-6076	93	14	g	g	NOUN
ejpam-6076	93	15	)	)	PUNCT
ejpam-6076	93	16	\	\	PROPN
ejpam-6076	94	1	j	j	PROPN
ejpam-6076	94	2	,	,	PUNCT
ejpam-6076	94	3	say	say	VERB
ejpam-6076	94	4	a	a	PRON
ejpam-6076	94	5	,	,	PUNCT
ejpam-6076	94	6	contradicting	contradict	VERB
ejpam-6076	94	7	the	the	DET
ejpam-6076	94	8	definition	definition	NOUN
ejpam-6076	94	9	of	of	ADP
ejpam-6076	94	10	a	a	DET
ejpam-6076	94	11	certified	certify	VERB
ejpam-6076	94	12	perfect	perfect	ADJ
ejpam-6076	94	13	dominating	dominating	NOUN
ejpam-6076	94	14	set	set	NOUN
ejpam-6076	94	15	.	.	PUNCT
ejpam-6076	95	1	thus	thus	ADV
ejpam-6076	95	2	,	,	PUNCT
ejpam-6076	95	3	every	every	DET
ejpam-6076	95	4	support	support	NOUN
ejpam-6076	95	5	vertex	vertex	NOUN
ejpam-6076	95	6	of	of	ADP
ejpam-6076	95	7	g	g	PROPN
ejpam-6076	95	8	is	be	AUX
ejpam-6076	95	9	included	include	VERB
ejpam-6076	95	10	in	in	ADP
ejpam-6076	95	11	every	every	DET
ejpam-6076	95	12	certified	certify	VERB
ejpam-6076	95	13	perfect	perfect	ADJ
ejpam-6076	95	14	dominating	dominating	NOUN
ejpam-6076	95	15	set	set	NOUN
ejpam-6076	95	16	of	of	ADP
ejpam-6076	95	17	g.	g.	PROPN
ejpam-6076	95	18	j.	j.	PROPN
ejpam-6076	95	19	j.	j.	PROPN
ejpam-6076	95	20	hamja	hamja	PROPN
ejpam-6076	95	21	et	et	PROPN
ejpam-6076	95	22	al	al	PROPN
ejpam-6076	95	23	.	.	PUNCT
ejpam-6076	95	24	/	/	SYM
ejpam-6076	95	25	eur	eur	PROPN
ejpam-6076	95	26	.	.	PUNCT
ejpam-6076	96	1	j.	j.	PROPN
ejpam-6076	96	2	pure	pure	PROPN
ejpam-6076	96	3	appl	appl	PROPN
ejpam-6076	96	4	.	.	PROPN
ejpam-6076	96	5	math	math	PROPN
ejpam-6076	96	6	,	,	PUNCT
ejpam-6076	96	7	18	18	NUM
ejpam-6076	96	8	(	(	PUNCT
ejpam-6076	96	9	3	3	NUM
ejpam-6076	96	10	)	)	PUNCT
ejpam-6076	96	11	(	(	PUNCT
ejpam-6076	96	12	2025	2025	NUM
ejpam-6076	96	13	)	)	PUNCT
ejpam-6076	96	14	,	,	PUNCT
ejpam-6076	96	15	6076	6076	NUM
ejpam-6076	96	16	5	5	NUM
ejpam-6076	96	17	of	of	ADP
ejpam-6076	96	18	13	13	NUM
ejpam-6076	96	19	proposition	proposition	NOUN
ejpam-6076	96	20	6	6	NUM
ejpam-6076	96	21	.	.	PUNCT
ejpam-6076	97	1	let	let	VERB
ejpam-6076	97	2	k	k	PRON
ejpam-6076	97	3	be	be	AUX
ejpam-6076	97	4	a	a	DET
ejpam-6076	97	5	positive	positive	ADJ
ejpam-6076	97	6	integer	integer	NOUN
ejpam-6076	97	7	and	and	CCONJ
ejpam-6076	97	8	g	g	PROPN
ejpam-6076	97	9	be	be	AUX
ejpam-6076	97	10	a	a	DET
ejpam-6076	97	11	graph	graph	NOUN
ejpam-6076	97	12	of	of	ADP
ejpam-6076	97	13	order	order	NOUN
ejpam-6076	97	14	n.	n.	NOUN
ejpam-6076	97	15	if	if	SCONJ
ejpam-6076	97	16	the	the	DET
ejpam-6076	97	17	strong	strong	ADJ
ejpam-6076	97	18	support	support	NOUN
ejpam-6076	97	19	vertices	vertex	NOUN
ejpam-6076	97	20	of	of	ADP
ejpam-6076	97	21	g	g	NOUN
ejpam-6076	97	22	are	be	AUX
ejpam-6076	97	23	collectively	collectively	ADV
ejpam-6076	97	24	adjacent	adjacent	ADJ
ejpam-6076	97	25	to	to	ADP
ejpam-6076	97	26	k	k	PROPN
ejpam-6076	97	27	leaves	leave	NOUN
ejpam-6076	97	28	,	,	PUNCT
ejpam-6076	97	29	then	then	ADV
ejpam-6076	97	30	γcerp(g	γcerp(g	PROPN
ejpam-6076	97	31	)	)	PUNCT
ejpam-6076	97	32	≤	≤	NUM
ejpam-6076	97	33	n−	n−	PROPN
ejpam-6076	97	34	k.	k.	PROPN
ejpam-6076	97	35	moreover	moreover	ADV
ejpam-6076	97	36	,	,	PUNCT
ejpam-6076	97	37	γcerp(g	γcerp(g	PROPN
ejpam-6076	97	38	)	)	PUNCT
ejpam-6076	97	39	≤	≤	NUM
ejpam-6076	97	40	n−	n−	NOUN
ejpam-6076	97	41	2|ss	2|ss	NUM
ejpam-6076	97	42	|	|	ADV
ejpam-6076	97	43	.	.	PUNCT
ejpam-6076	98	1	proof	proof	NOUN
ejpam-6076	98	2	.	.	PUNCT
ejpam-6076	99	1	let	let	VERB
ejpam-6076	99	2	lg	lg	NOUN
ejpam-6076	99	3	be	be	AUX
ejpam-6076	99	4	the	the	DET
ejpam-6076	99	5	set	set	NOUN
ejpam-6076	99	6	of	of	ADP
ejpam-6076	99	7	all	all	DET
ejpam-6076	99	8	leaves	leave	NOUN
ejpam-6076	99	9	adjacent	adjacent	ADJ
ejpam-6076	99	10	to	to	ADP
ejpam-6076	99	11	strong	strong	ADJ
ejpam-6076	99	12	support	support	NOUN
ejpam-6076	99	13	vertices	vertex	NOUN
ejpam-6076	99	14	in	in	ADP
ejpam-6076	99	15	g	g	PROPN
ejpam-6076	99	16	,	,	PUNCT
ejpam-6076	99	17	where	where	SCONJ
ejpam-6076	99	18	|lg|	|lg|	NOUN
ejpam-6076	99	19	=	=	SYM
ejpam-6076	99	20	k.	k.	NOUN
ejpam-6076	99	21	consider	consider	VERB
ejpam-6076	99	22	the	the	DET
ejpam-6076	99	23	set	set	NOUN
ejpam-6076	99	24	v	v	NOUN
ejpam-6076	99	25	(	(	PUNCT
ejpam-6076	99	26	g	g	NOUN
ejpam-6076	99	27	)	)	PUNCT
ejpam-6076	99	28	\	\	PROPN
ejpam-6076	100	1	lg	lg	PROPN
ejpam-6076	100	2	.	.	PROPN
ejpam-6076	101	1	since	since	SCONJ
ejpam-6076	101	2	each	each	DET
ejpam-6076	101	3	leaf	leaf	NOUN
ejpam-6076	101	4	in	in	ADP
ejpam-6076	101	5	lg	lg	NOUN
ejpam-6076	101	6	is	be	AUX
ejpam-6076	101	7	adjacent	adjacent	ADJ
ejpam-6076	101	8	to	to	ADP
ejpam-6076	101	9	exactly	exactly	ADV
ejpam-6076	101	10	one	one	NUM
ejpam-6076	101	11	strong	strong	ADJ
ejpam-6076	101	12	support	support	NOUN
ejpam-6076	101	13	vertex	vertex	NOUN
ejpam-6076	101	14	,	,	PUNCT
ejpam-6076	101	15	the	the	DET
ejpam-6076	101	16	set	set	NOUN
ejpam-6076	101	17	v	v	NOUN
ejpam-6076	101	18	(	(	PUNCT
ejpam-6076	101	19	g	g	NOUN
ejpam-6076	101	20	)	)	PUNCT
ejpam-6076	101	21	\	\	NOUN
ejpam-6076	102	1	lg	lg	NOUN
ejpam-6076	102	2	forms	form	VERB
ejpam-6076	102	3	a	a	DET
ejpam-6076	102	4	certified	certify	VERB
ejpam-6076	102	5	perfect	perfect	ADJ
ejpam-6076	102	6	dominating	dominating	NOUN
ejpam-6076	102	7	set	set	NOUN
ejpam-6076	102	8	of	of	ADP
ejpam-6076	102	9	g.	g.	PROPN
ejpam-6076	102	10	therefore	therefore	ADV
ejpam-6076	102	11	,	,	PUNCT
ejpam-6076	102	12	γcerp(g	γcerp(g	PROPN
ejpam-6076	102	13	)	)	PUNCT
ejpam-6076	102	14	≤	≤	NOUN
ejpam-6076	102	15	|v	|v	X
ejpam-6076	102	16	(	(	PUNCT
ejpam-6076	102	17	g	g	NOUN
ejpam-6076	102	18	)	)	PUNCT
ejpam-6076	102	19	\	\	PUNCT
ejpam-6076	103	1	lg|	lg|	NOUN
ejpam-6076	103	2	=	=	SYM
ejpam-6076	103	3	n−	n−	PROPN
ejpam-6076	103	4	k.	k.	PROPN
ejpam-6076	103	5	moreover	moreover	ADV
ejpam-6076	103	6	,	,	PUNCT
ejpam-6076	103	7	each	each	DET
ejpam-6076	103	8	strong	strong	ADJ
ejpam-6076	103	9	support	support	NOUN
ejpam-6076	103	10	vertex	vertex	NOUN
ejpam-6076	103	11	in	in	ADP
ejpam-6076	103	12	ss	ss	PROPN
ejpam-6076	103	13	is	be	AUX
ejpam-6076	103	14	adjacent	adjacent	ADJ
ejpam-6076	103	15	to	to	ADP
ejpam-6076	103	16	at	at	ADV
ejpam-6076	103	17	least	least	ADV
ejpam-6076	103	18	two	two	NUM
ejpam-6076	103	19	leaves	leave	NOUN
ejpam-6076	103	20	,	,	PUNCT
ejpam-6076	103	21	implying	imply	VERB
ejpam-6076	103	22	k	k	PROPN
ejpam-6076	103	23	≥	≥	X
ejpam-6076	103	24	2|ss	2|ss	NUM
ejpam-6076	103	25	|	|	NOUN
ejpam-6076	103	26	.	.	PUNCT
ejpam-6076	104	1	hence	hence	ADV
ejpam-6076	104	2	,	,	PUNCT
ejpam-6076	104	3	it	it	PRON
ejpam-6076	104	4	follows	follow	VERB
ejpam-6076	104	5	that	that	SCONJ
ejpam-6076	104	6	γcerp(g	γcerp(g	PROPN
ejpam-6076	104	7	)	)	PUNCT
ejpam-6076	104	8	≤	≤	NUM
ejpam-6076	104	9	n−	n−	NOUN
ejpam-6076	104	10	2|ss	2|ss	NUM
ejpam-6076	104	11	|	|	ADV
ejpam-6076	104	12	.	.	PUNCT
ejpam-6076	105	1	theorem	theorem	NOUN
ejpam-6076	105	2	1	1	NUM
ejpam-6076	105	3	.	.	PUNCT
ejpam-6076	106	1	let	let	VERB
ejpam-6076	106	2	g	g	PRON
ejpam-6076	106	3	be	be	AUX
ejpam-6076	106	4	a	a	DET
ejpam-6076	106	5	graph	graph	NOUN
ejpam-6076	106	6	.	.	PUNCT
ejpam-6076	107	1	then	then	ADV
ejpam-6076	107	2	for	for	ADP
ejpam-6076	107	3	every	every	DET
ejpam-6076	107	4	maximum	maximum	ADJ
ejpam-6076	107	5	certified	certify	VERB
ejpam-6076	107	6	independent	independent	ADJ
ejpam-6076	107	7	perfect	perfect	ADJ
ejpam-6076	107	8	set	set	NOUN
ejpam-6076	107	9	j	j	PROPN
ejpam-6076	107	10	of	of	ADP
ejpam-6076	107	11	g	g	PROPN
ejpam-6076	107	12	is	be	AUX
ejpam-6076	107	13	a	a	DET
ejpam-6076	107	14	certified	certify	VERB
ejpam-6076	107	15	perfect	perfect	ADJ
ejpam-6076	107	16	dominating	dominating	NOUN
ejpam-6076	107	17	set	set	NOUN
ejpam-6076	107	18	of	of	ADP
ejpam-6076	107	19	g.	g.	PROPN
ejpam-6076	107	20	moreover	moreover	ADV
ejpam-6076	107	21	,	,	PUNCT
ejpam-6076	107	22	γcerp(g	γcerp(g	PROPN
ejpam-6076	107	23	)	)	PUNCT
ejpam-6076	107	24	≤	≤	NOUN
ejpam-6076	107	25	αcerp(g	αcerp(g	NOUN
ejpam-6076	107	26	)	)	PUNCT
ejpam-6076	107	27	.	.	PUNCT
ejpam-6076	108	1	proof	proof	NOUN
ejpam-6076	108	2	.	.	PUNCT
ejpam-6076	109	1	let	let	VERB
ejpam-6076	109	2	g	g	PRON
ejpam-6076	109	3	be	be	AUX
ejpam-6076	109	4	a	a	DET
ejpam-6076	109	5	graph	graph	NOUN
ejpam-6076	109	6	and	and	CCONJ
ejpam-6076	109	7	j	j	PROPN
ejpam-6076	109	8	a	a	DET
ejpam-6076	109	9	maximum	maximum	ADV
ejpam-6076	109	10	certified	certify	VERB
ejpam-6076	109	11	independent	independent	ADJ
ejpam-6076	109	12	perfect	perfect	ADJ
ejpam-6076	109	13	set	set	NOUN
ejpam-6076	109	14	of	of	ADP
ejpam-6076	109	15	g.	g.	PROPN
ejpam-6076	109	16	since	since	SCONJ
ejpam-6076	109	17	j	j	PROPN
ejpam-6076	109	18	is	be	AUX
ejpam-6076	109	19	an	an	DET
ejpam-6076	109	20	independent	independent	ADJ
ejpam-6076	109	21	set	set	NOUN
ejpam-6076	109	22	,	,	PUNCT
ejpam-6076	109	23	for	for	ADP
ejpam-6076	109	24	any	any	DET
ejpam-6076	109	25	a	a	NOUN
ejpam-6076	109	26	,	,	PUNCT
ejpam-6076	109	27	b	b	PROPN
ejpam-6076	109	28	∈	∈	PROPN
ejpam-6076	109	29	j	j	NOUN
ejpam-6076	109	30	,	,	PUNCT
ejpam-6076	109	31	we	we	PRON
ejpam-6076	109	32	have	have	VERB
ejpam-6076	109	33	dg(a	dg(a	NOUN
ejpam-6076	109	34	,	,	PUNCT
ejpam-6076	109	35	b	b	X
ejpam-6076	109	36	)	)	PUNCT
ejpam-6076	109	37	̸=	̸=	PROPN
ejpam-6076	109	38	1	1	NUM
ejpam-6076	109	39	.	.	PUNCT
ejpam-6076	110	1	furthermore	furthermore	ADV
ejpam-6076	110	2	,	,	PUNCT
ejpam-6076	110	3	because	because	SCONJ
ejpam-6076	110	4	j	j	PROPN
ejpam-6076	110	5	is	be	AUX
ejpam-6076	110	6	a	a	DET
ejpam-6076	110	7	certified	certify	VERB
ejpam-6076	110	8	independent	independent	ADJ
ejpam-6076	110	9	perfect	perfect	ADJ
ejpam-6076	110	10	set	set	NOUN
ejpam-6076	110	11	,	,	PUNCT
ejpam-6076	110	12	it	it	PRON
ejpam-6076	110	13	follows	follow	VERB
ejpam-6076	110	14	that	that	SCONJ
ejpam-6076	110	15	dg(a	dg(a	ADP
ejpam-6076	110	16	,	,	PUNCT
ejpam-6076	110	17	b	b	X
ejpam-6076	110	18	)	)	PUNCT
ejpam-6076	110	19	̸=	̸=	PROPN
ejpam-6076	110	20	2	2	NUM
ejpam-6076	110	21	for	for	ADP
ejpam-6076	110	22	all	all	DET
ejpam-6076	110	23	a	a	DET
ejpam-6076	110	24	,	,	PUNCT
ejpam-6076	110	25	b	b	X
ejpam-6076	110	26	∈	∈	PROPN
ejpam-6076	110	27	d.	d.	PROPN
ejpam-6076	110	28	now	now	ADV
ejpam-6076	110	29	,	,	PUNCT
ejpam-6076	110	30	let	let	VERB
ejpam-6076	110	31	y	y	PROPN
ejpam-6076	110	32	∈	∈	PROPN
ejpam-6076	110	33	v	v	ADP
ejpam-6076	110	34	(	(	PUNCT
ejpam-6076	110	35	g	g	NOUN
ejpam-6076	110	36	)	)	PUNCT
ejpam-6076	110	37	\	\	PROPN
ejpam-6076	111	1	j	j	PROPN
ejpam-6076	111	2	.	.	PUNCT
ejpam-6076	112	1	since	since	SCONJ
ejpam-6076	112	2	j	j	PROPN
ejpam-6076	112	3	is	be	AUX
ejpam-6076	112	4	a	a	DET
ejpam-6076	112	5	maximum	maximum	ADJ
ejpam-6076	112	6	-	-	PUNCT
ejpam-6076	112	7	size	size	NOUN
ejpam-6076	112	8	certified	certify	VERB
ejpam-6076	112	9	independent	independent	ADJ
ejpam-6076	112	10	perfect	perfect	ADJ
ejpam-6076	112	11	set	set	NOUN
ejpam-6076	112	12	,	,	PUNCT
ejpam-6076	112	13	y	y	PROPN
ejpam-6076	112	14	must	must	AUX
ejpam-6076	112	15	be	be	AUX
ejpam-6076	112	16	dominated	dominate	VERB
ejpam-6076	112	17	by	by	ADP
ejpam-6076	112	18	at	at	ADV
ejpam-6076	112	19	least	least	ADV
ejpam-6076	112	20	one	one	NUM
ejpam-6076	112	21	vertex	vertex	NOUN
ejpam-6076	112	22	in	in	ADP
ejpam-6076	112	23	j	j	PROPN
ejpam-6076	112	24	.	.	PUNCT
ejpam-6076	113	1	hence	hence	ADV
ejpam-6076	113	2	,	,	PUNCT
ejpam-6076	113	3	there	there	PRON
ejpam-6076	113	4	exists	exist	VERB
ejpam-6076	113	5	some	some	DET
ejpam-6076	113	6	z	z	NOUN
ejpam-6076	113	7	∈	∈	PROPN
ejpam-6076	113	8	j	j	NOUN
ejpam-6076	113	9	such	such	ADJ
ejpam-6076	113	10	that	that	SCONJ
ejpam-6076	113	11	y	y	PROPN
ejpam-6076	113	12	∈	∈	PROPN
ejpam-6076	113	13	ng(z	ng(z	PROPN
ejpam-6076	113	14	)	)	PUNCT
ejpam-6076	113	15	.	.	PUNCT
ejpam-6076	114	1	therefore	therefore	ADV
ejpam-6076	114	2	,	,	PUNCT
ejpam-6076	114	3	ng[j	ng[j	PROPN
ejpam-6076	114	4	]	]	PUNCT
ejpam-6076	114	5	=	=	SYM
ejpam-6076	114	6	v	v	X
ejpam-6076	114	7	(	(	PUNCT
ejpam-6076	114	8	g	g	NOUN
ejpam-6076	114	9	)	)	PUNCT
ejpam-6076	114	10	,	,	PUNCT
ejpam-6076	114	11	which	which	PRON
ejpam-6076	114	12	implies	imply	VERB
ejpam-6076	114	13	that	that	SCONJ
ejpam-6076	114	14	j	j	PROPN
ejpam-6076	114	15	is	be	AUX
ejpam-6076	114	16	a	a	DET
ejpam-6076	114	17	dominating	dominating	NOUN
ejpam-6076	114	18	set	set	NOUN
ejpam-6076	114	19	of	of	ADP
ejpam-6076	114	20	g.	g.	PROPN
ejpam-6076	114	21	additionally	additionally	ADV
ejpam-6076	114	22	,	,	PUNCT
ejpam-6076	114	23	since	since	SCONJ
ejpam-6076	114	24	j	j	PROPN
ejpam-6076	114	25	is	be	AUX
ejpam-6076	114	26	an	an	DET
ejpam-6076	114	27	independent	independent	ADJ
ejpam-6076	114	28	perfect	perfect	ADJ
ejpam-6076	114	29	set	set	NOUN
ejpam-6076	114	30	,	,	PUNCT
ejpam-6076	114	31	each	each	DET
ejpam-6076	114	32	vertex	vertex	NOUN
ejpam-6076	114	33	in	in	ADP
ejpam-6076	114	34	v	v	NOUN
ejpam-6076	114	35	(	(	PUNCT
ejpam-6076	114	36	g	g	NOUN
ejpam-6076	114	37	)	)	PUNCT
ejpam-6076	114	38	\	\	PROPN
ejpam-6076	115	1	j	j	PROPN
ejpam-6076	115	2	is	be	AUX
ejpam-6076	115	3	dominated	dominate	VERB
ejpam-6076	115	4	by	by	ADP
ejpam-6076	115	5	exactly	exactly	ADV
ejpam-6076	115	6	one	one	NUM
ejpam-6076	115	7	vertex	vertex	NOUN
ejpam-6076	115	8	in	in	ADP
ejpam-6076	115	9	j	j	PROPN
ejpam-6076	115	10	.	.	PUNCT
ejpam-6076	116	1	thus	thus	ADV
ejpam-6076	116	2	,	,	PUNCT
ejpam-6076	116	3	j	j	PROPN
ejpam-6076	116	4	is	be	AUX
ejpam-6076	116	5	an	an	DET
ejpam-6076	116	6	independent	independent	ADJ
ejpam-6076	116	7	perfect	perfect	ADJ
ejpam-6076	116	8	dominating	dominating	NOUN
ejpam-6076	116	9	set	set	NOUN
ejpam-6076	116	10	of	of	ADP
ejpam-6076	116	11	g.	g.	PROPN
ejpam-6076	116	12	given	give	VERB
ejpam-6076	116	13	that	that	SCONJ
ejpam-6076	116	14	j	j	PROPN
ejpam-6076	116	15	is	be	AUX
ejpam-6076	116	16	also	also	ADV
ejpam-6076	116	17	a	a	DET
ejpam-6076	116	18	certified	certify	VERB
ejpam-6076	116	19	independent	independent	ADJ
ejpam-6076	116	20	perfect	perfect	ADJ
ejpam-6076	116	21	set	set	NOUN
ejpam-6076	116	22	,	,	PUNCT
ejpam-6076	116	23	it	it	PRON
ejpam-6076	116	24	follows	follow	VERB
ejpam-6076	116	25	that	that	SCONJ
ejpam-6076	116	26	j	j	PROPN
ejpam-6076	116	27	is	be	AUX
ejpam-6076	116	28	a	a	DET
ejpam-6076	116	29	certified	certify	VERB
ejpam-6076	116	30	perfect	perfect	ADJ
ejpam-6076	116	31	dominating	dominating	NOUN
ejpam-6076	116	32	set	set	NOUN
ejpam-6076	116	33	of	of	ADP
ejpam-6076	116	34	g.	g.	PROPN
ejpam-6076	116	35	let	let	VERB
ejpam-6076	116	36	j	j	PROPN
ejpam-6076	116	37	be	be	AUX
ejpam-6076	116	38	a	a	DET
ejpam-6076	116	39	maximum	maximum	ADV
ejpam-6076	116	40	certified	certify	VERB
ejpam-6076	116	41	perfect	perfect	ADJ
ejpam-6076	116	42	independent	independent	ADJ
ejpam-6076	116	43	set	set	NOUN
ejpam-6076	116	44	of	of	ADP
ejpam-6076	116	45	g.	g.	PROPN
ejpam-6076	116	46	therefore	therefore	ADV
ejpam-6076	116	47	,	,	PUNCT
ejpam-6076	116	48	|j	|j	VERB
ejpam-6076	116	49	|	|	NOUN
ejpam-6076	116	50	=	=	SYM
ejpam-6076	116	51	αcerp(g	αcerp(g	NOUN
ejpam-6076	116	52	)	)	PUNCT
ejpam-6076	116	53	.	.	PUNCT
ejpam-6076	117	1	since	since	SCONJ
ejpam-6076	117	2	j	j	PROPN
ejpam-6076	117	3	is	be	AUX
ejpam-6076	117	4	also	also	ADV
ejpam-6076	117	5	a	a	DET
ejpam-6076	117	6	certified	certify	VERB
ejpam-6076	117	7	perfect	perfect	ADJ
ejpam-6076	117	8	dominating	dominating	NOUN
ejpam-6076	117	9	set	set	NOUN
ejpam-6076	117	10	,	,	PUNCT
ejpam-6076	117	11	we	we	PRON
ejpam-6076	117	12	have	have	AUX
ejpam-6076	117	13	γcerp(g	γcerp(g	VERB
ejpam-6076	117	14	)	)	PUNCT
ejpam-6076	117	15	≤	≤	NUM
ejpam-6076	117	16	αcerp(g	αcerp(g	NOUN
ejpam-6076	117	17	)	)	PUNCT
ejpam-6076	117	18	.	.	PUNCT
ejpam-6076	118	1	this	this	PRON
ejpam-6076	118	2	completes	complete	VERB
ejpam-6076	118	3	the	the	DET
ejpam-6076	118	4	proof	proof	NOUN
ejpam-6076	118	5	.	.	PUNCT
ejpam-6076	119	1	the	the	DET
ejpam-6076	119	2	next	next	ADJ
ejpam-6076	119	3	result	result	NOUN
ejpam-6076	119	4	shows	show	VERB
ejpam-6076	119	5	the	the	DET
ejpam-6076	119	6	relationship	relationship	NOUN
ejpam-6076	119	7	between	between	ADP
ejpam-6076	119	8	certified	certified	ADJ
ejpam-6076	119	9	domination	domination	NOUN
ejpam-6076	119	10	and	and	CCONJ
ejpam-6076	119	11	certified	certify	VERB
ejpam-6076	119	12	perfect	perfect	ADJ
ejpam-6076	119	13	domination	domination	NOUN
ejpam-6076	119	14	parameters	parameter	NOUN
ejpam-6076	119	15	.	.	PUNCT
ejpam-6076	120	1	remark	remark	PROPN
ejpam-6076	120	2	1	1	NUM
ejpam-6076	120	3	.	.	PUNCT
ejpam-6076	121	1	every	every	DET
ejpam-6076	121	2	certified	certify	VERB
ejpam-6076	121	3	perfect	perfect	ADJ
ejpam-6076	121	4	dominating	dominating	NOUN
ejpam-6076	121	5	set	set	NOUN
ejpam-6076	121	6	of	of	ADP
ejpam-6076	121	7	g	g	PROPN
ejpam-6076	121	8	is	be	AUX
ejpam-6076	121	9	also	also	ADV
ejpam-6076	121	10	a	a	DET
ejpam-6076	121	11	certified	certify	VERB
ejpam-6076	121	12	dominating	dominating	NOUN
ejpam-6076	121	13	set	set	NOUN
ejpam-6076	121	14	of	of	ADP
ejpam-6076	121	15	g.	g.	PROPN
ejpam-6076	121	16	hence	hence	ADV
ejpam-6076	121	17	,	,	PUNCT
ejpam-6076	121	18	γcer(g	γcer(g	PROPN
ejpam-6076	121	19	)	)	PUNCT
ejpam-6076	121	20	≤	≤	NOUN
ejpam-6076	121	21	γcerp(g	γcerp(g	PROPN
ejpam-6076	121	22	)	)	PUNCT
ejpam-6076	121	23	.	.	PUNCT
ejpam-6076	122	1	theorem	theorem	NOUN
ejpam-6076	122	2	2	2	NUM
ejpam-6076	122	3	.	.	PUNCT
ejpam-6076	122	4	let	let	VERB
ejpam-6076	122	5	a	a	PRON
ejpam-6076	122	6	and	and	CCONJ
ejpam-6076	122	7	b	b	NOUN
ejpam-6076	122	8	be	be	AUX
ejpam-6076	122	9	positive	positive	ADJ
ejpam-6076	122	10	integers	integer	NOUN
ejpam-6076	122	11	with	with	ADP
ejpam-6076	122	12	3	3	NUM
ejpam-6076	122	13	≤	≤	NOUN
ejpam-6076	122	14	a	a	DET
ejpam-6076	122	15	≤	≤	PROPN
ejpam-6076	122	16	b.	b.	NOUN
ejpam-6076	123	1	then	then	ADV
ejpam-6076	123	2	there	there	PRON
ejpam-6076	123	3	exists	exist	VERB
ejpam-6076	123	4	a	a	DET
ejpam-6076	123	5	connected	connected	ADJ
ejpam-6076	123	6	graph	graph	NOUN
ejpam-6076	123	7	g	g	ADP
ejpam-6076	123	8	such	such	ADJ
ejpam-6076	123	9	that	that	DET
ejpam-6076	123	10	γcer(g	γcer(g	NOUN
ejpam-6076	123	11	)	)	PUNCT
ejpam-6076	123	12	=	=	PUNCT
ejpam-6076	123	13	a	a	PRON
ejpam-6076	123	14	and	and	CCONJ
ejpam-6076	123	15	γcerp(g	γcerp(g	ADJ
ejpam-6076	123	16	)	)	PUNCT
ejpam-6076	123	17	=	=	SYM
ejpam-6076	123	18	b.	b.	PROPN
ejpam-6076	123	19	j.	j.	PROPN
ejpam-6076	123	20	j.	j.	PROPN
ejpam-6076	124	1	hamja	hamja	PROPN
ejpam-6076	124	2	et	et	PROPN
ejpam-6076	124	3	al	al	PROPN
ejpam-6076	124	4	.	.	PUNCT
ejpam-6076	124	5	/	/	SYM
ejpam-6076	124	6	eur	eur	PROPN
ejpam-6076	124	7	.	.	PUNCT
ejpam-6076	125	1	j.	j.	PROPN
ejpam-6076	125	2	pure	pure	PROPN
ejpam-6076	125	3	appl	appl	PROPN
ejpam-6076	125	4	.	.	PROPN
ejpam-6076	125	5	math	math	PROPN
ejpam-6076	125	6	,	,	PUNCT
ejpam-6076	125	7	18	18	NUM
ejpam-6076	125	8	(	(	PUNCT
ejpam-6076	125	9	3	3	NUM
ejpam-6076	125	10	)	)	PUNCT
ejpam-6076	125	11	(	(	PUNCT
ejpam-6076	125	12	2025	2025	NUM
ejpam-6076	125	13	)	)	PUNCT
ejpam-6076	125	14	,	,	PUNCT
ejpam-6076	125	15	6076	6076	NUM
ejpam-6076	125	16	6	6	NUM
ejpam-6076	125	17	of	of	ADP
ejpam-6076	125	18	13	13	NUM
ejpam-6076	125	19	proof	proof	NOUN
ejpam-6076	125	20	.	.	PUNCT
ejpam-6076	126	1	for	for	ADP
ejpam-6076	126	2	a	a	DET
ejpam-6076	126	3	=	=	SYM
ejpam-6076	126	4	b	b	NOUN
ejpam-6076	126	5	,	,	PUNCT
ejpam-6076	126	6	consider	consider	VERB
ejpam-6076	126	7	g	g	PROPN
ejpam-6076	126	8	=	=	SYM
ejpam-6076	126	9	ka	ka	PROPN
ejpam-6076	126	10	.	.	PROPN
ejpam-6076	127	1	in	in	ADP
ejpam-6076	127	2	this	this	DET
ejpam-6076	127	3	case	case	NOUN
ejpam-6076	127	4	,	,	PUNCT
ejpam-6076	127	5	γcer(g	γcer(g	NOUN
ejpam-6076	127	6	)	)	PUNCT
ejpam-6076	127	7	=	=	PUNCT
ejpam-6076	127	8	a	a	PRON
ejpam-6076	127	9	and	and	CCONJ
ejpam-6076	127	10	γcerp(g	γcerp(g	ADJ
ejpam-6076	127	11	)	)	PUNCT
ejpam-6076	127	12	=	=	SYM
ejpam-6076	127	13	a	a	DET
ejpam-6076	127	14	,	,	PUNCT
ejpam-6076	127	15	so	so	ADV
ejpam-6076	127	16	γcer(g	γcer(g	ADJ
ejpam-6076	127	17	)	)	PUNCT
ejpam-6076	128	1	=	=	SYM
ejpam-6076	128	2	γcerp(g	γcerp(g	PROPN
ejpam-6076	128	3	)	)	PUNCT
ejpam-6076	128	4	.	.	PUNCT
ejpam-6076	129	1	for	for	ADP
ejpam-6076	129	2	a	a	DET
ejpam-6076	129	3	<	<	X
ejpam-6076	129	4	b	b	NOUN
ejpam-6076	129	5	,	,	PUNCT
ejpam-6076	129	6	we	we	PRON
ejpam-6076	129	7	examine	examine	VERB
ejpam-6076	129	8	the	the	DET
ejpam-6076	129	9	following	follow	VERB
ejpam-6076	129	10	cases	case	NOUN
ejpam-6076	129	11	:	:	PUNCT
ejpam-6076	129	12	case	case	NOUN
ejpam-6076	129	13	1	1	NUM
ejpam-6076	129	14	.	.	PUNCT
ejpam-6076	130	1	a	a	PRON
ejpam-6076	130	2	is	be	AUX
ejpam-6076	130	3	odd	odd	ADJ
ejpam-6076	130	4	.	.	PUNCT
ejpam-6076	131	1	let	let	VERB
ejpam-6076	131	2	m	m	VERB
ejpam-6076	131	3	=	=	VERB
ejpam-6076	132	1	b	b	X
ejpam-6076	132	2	−	−	NOUN
ejpam-6076	132	3	a	a	NOUN
ejpam-6076	132	4	,	,	PUNCT
ejpam-6076	132	5	and	and	CCONJ
ejpam-6076	132	6	consider	consider	VERB
ejpam-6076	132	7	the	the	DET
ejpam-6076	132	8	graph	graph	NOUN
ejpam-6076	132	9	g	g	NOUN
ejpam-6076	132	10	as	as	SCONJ
ejpam-6076	132	11	illustrated	illustrate	VERB
ejpam-6076	132	12	in	in	ADP
ejpam-6076	132	13	figure	figure	NOUN
ejpam-6076	132	14	1	1	NUM
ejpam-6076	132	15	.	.	PUNCT
ejpam-6076	132	16	define	define	VERB
ejpam-6076	132	17	j1	j1	PROPN
ejpam-6076	132	18	=	=	PUNCT
ejpam-6076	132	19	{	{	PUNCT
ejpam-6076	132	20	x1	x1	PROPN
ejpam-6076	132	21	,	,	PUNCT
ejpam-6076	132	22	x2	x2	PROPN
ejpam-6076	132	23	,	,	PUNCT
ejpam-6076	132	24	.	.	PUNCT
ejpam-6076	132	25	.	.	PUNCT
ejpam-6076	133	1	.	.	PUNCT
ejpam-6076	134	1	,	,	PUNCT
ejpam-6076	134	2	xa	xa	PROPN
ejpam-6076	134	3	}	}	PUNCT
ejpam-6076	134	4	and	and	CCONJ
ejpam-6076	134	5	j2	j2	PROPN
ejpam-6076	134	6	=	=	SYM
ejpam-6076	134	7	{	{	PUNCT
ejpam-6076	134	8	x1	x1	PROPN
ejpam-6076	134	9	,	,	PUNCT
ejpam-6076	134	10	x2	x2	PROPN
ejpam-6076	134	11	,	,	PUNCT
ejpam-6076	134	12	.	.	PUNCT
ejpam-6076	134	13	.	.	PUNCT
ejpam-6076	135	1	.	.	PUNCT
ejpam-6076	136	1	,	,	PUNCT
ejpam-6076	136	2	xa	xa	PROPN
ejpam-6076	136	3	,	,	PUNCT
ejpam-6076	136	4	y1	y1	PROPN
ejpam-6076	136	5	,	,	PUNCT
ejpam-6076	136	6	y2	y2	PROPN
ejpam-6076	136	7	,	,	PUNCT
ejpam-6076	136	8	.	.	PUNCT
ejpam-6076	136	9	.	.	PUNCT
ejpam-6076	136	10	.	.	PUNCT
ejpam-6076	137	1	,	,	PUNCT
ejpam-6076	137	2	ym	ym	PROPN
ejpam-6076	137	3	}	}	PUNCT
ejpam-6076	137	4	.	.	PUNCT
ejpam-6076	138	1	then	then	ADV
ejpam-6076	138	2	,	,	PUNCT
ejpam-6076	138	3	j1	j1	PROPN
ejpam-6076	138	4	and	and	CCONJ
ejpam-6076	138	5	j2	j2	PROPN
ejpam-6076	138	6	are	be	AUX
ejpam-6076	138	7	the	the	DET
ejpam-6076	138	8	γcer	γcer	NOUN
ejpam-6076	138	9	-	-	PUNCT
ejpam-6076	138	10	set	set	VERB
ejpam-6076	138	11	and	and	CCONJ
ejpam-6076	138	12	γcerp	γcerp	NOUN
ejpam-6076	138	13	-	-	PUNCT
ejpam-6076	138	14	set	set	NOUN
ejpam-6076	138	15	of	of	ADP
ejpam-6076	138	16	g	g	NOUN
ejpam-6076	138	17	,	,	PUNCT
ejpam-6076	138	18	respectively	respectively	ADV
ejpam-6076	138	19	.	.	PUNCT
ejpam-6076	139	1	therefore	therefore	ADV
ejpam-6076	139	2	,	,	PUNCT
ejpam-6076	139	3	we	we	PRON
ejpam-6076	139	4	have	have	VERB
ejpam-6076	139	5	γcer(g	γcer(g	NUM
ejpam-6076	139	6	)	)	PUNCT
ejpam-6076	140	1	=	=	SYM
ejpam-6076	140	2	|j1|	|j1|	NOUN
ejpam-6076	141	1	=	=	PUNCT
ejpam-6076	141	2	a	a	PRON
ejpam-6076	141	3	and	and	CCONJ
ejpam-6076	141	4	γcerp(g	γcerp(g	ADJ
ejpam-6076	141	5	)	)	PUNCT
ejpam-6076	141	6	=	=	SYM
ejpam-6076	141	7	|j2|	|j2|	NOUN
ejpam-6076	141	8	=	=	SYM
ejpam-6076	141	9	a+m	a+m	NUM
ejpam-6076	142	1	=	=	SYM
ejpam-6076	142	2	b.	b.	PROPN
ejpam-6076	143	1	x1	x1	PROPN
ejpam-6076	144	1	x2	x2	NOUN
ejpam-6076	144	2	x3	x3	PROPN
ejpam-6076	144	3	x4	x4	PROPN
ejpam-6076	144	4	x5	x5	PROPN
ejpam-6076	144	5	x6	x6	PROPN
ejpam-6076	144	6	xa−4	xa−4	PROPN
ejpam-6076	144	7	xa−3	xa−3	PROPN
ejpam-6076	144	8	xa−2	xa−2	PROPN
ejpam-6076	144	9	xa−1	xa−1	PROPN
ejpam-6076	145	1	xa	xa	PROPN
ejpam-6076	146	1	y1	y1	NOUN
ejpam-6076	146	2	y2	y2	PROPN
ejpam-6076	146	3	y3	y3	NOUN
ejpam-6076	146	4	y4	y4	NOUN
ejpam-6076	146	5	y5	y5	PROPN
ejpam-6076	146	6	y1	y1	PROPN
ejpam-6076	146	7	ym−5	ym−5	PROPN
ejpam-6076	146	8	ym−4	ym−4	PROPN
ejpam-6076	146	9	ym−2	ym−2	PROPN
ejpam-6076	146	10	ym−1	ym−1	PROPN
ejpam-6076	146	11	ym	ym	INTJ
ejpam-6076	146	12	g	g	PROPN
ejpam-6076	146	13	:	:	PUNCT
ejpam-6076	146	14	.	.	PUNCT
ejpam-6076	146	15	.	.	PUNCT
ejpam-6076	146	16	.	.	PUNCT
ejpam-6076	147	1	ya−3	ya−3	ADV
ejpam-6076	147	2	figure	figure	VERB
ejpam-6076	147	3	1	1	NUM
ejpam-6076	147	4	:	:	PUNCT
ejpam-6076	147	5	a	a	DET
ejpam-6076	147	6	graph	graph	NOUN
ejpam-6076	147	7	g	g	NOUN
ejpam-6076	147	8	with	with	ADP
ejpam-6076	147	9	γcer(g	γcer(g	NOUN
ejpam-6076	147	10	)	)	PUNCT
ejpam-6076	147	11	<	<	X
ejpam-6076	147	12	γcerp(g	γcerp(g	PROPN
ejpam-6076	147	13	)	)	PUNCT
ejpam-6076	147	14	.	.	PUNCT
ejpam-6076	148	1	case	case	NOUN
ejpam-6076	148	2	2	2	NUM
ejpam-6076	148	3	.	.	X
ejpam-6076	149	1	a	a	PRON
ejpam-6076	149	2	is	be	AUX
ejpam-6076	149	3	even	even	ADV
ejpam-6076	149	4	.	.	PUNCT
ejpam-6076	150	1	let	let	VERB
ejpam-6076	150	2	m	m	VERB
ejpam-6076	150	3	=	=	VERB
ejpam-6076	150	4	b	b	X
ejpam-6076	150	5	−	−	NOUN
ejpam-6076	150	6	a	a	NOUN
ejpam-6076	150	7	,	,	PUNCT
ejpam-6076	150	8	and	and	CCONJ
ejpam-6076	150	9	consider	consider	VERB
ejpam-6076	150	10	the	the	DET
ejpam-6076	150	11	graph	graph	NOUN
ejpam-6076	150	12	g′	g′	NOUN
ejpam-6076	150	13	as	as	SCONJ
ejpam-6076	150	14	shown	show	VERB
ejpam-6076	150	15	in	in	ADP
ejpam-6076	150	16	figure	figure	NOUN
ejpam-6076	150	17	2	2	NUM
ejpam-6076	150	18	.	.	PUNCT
ejpam-6076	150	19	define	define	VERB
ejpam-6076	150	20	j	j	PROPN
ejpam-6076	150	21	′	′	NOUN
ejpam-6076	150	22	1	1	NUM
ejpam-6076	150	23	=	=	SYM
ejpam-6076	150	24	{	{	PUNCT
ejpam-6076	150	25	x1	x1	PROPN
ejpam-6076	150	26	,	,	PUNCT
ejpam-6076	150	27	x2	x2	PROPN
ejpam-6076	150	28	,	,	PUNCT
ejpam-6076	150	29	.	.	PUNCT
ejpam-6076	150	30	.	.	PUNCT
ejpam-6076	151	1	.	.	PUNCT
ejpam-6076	152	1	,	,	PUNCT
ejpam-6076	152	2	xa	xa	PROPN
ejpam-6076	152	3	}	}	PUNCT
ejpam-6076	152	4	and	and	CCONJ
ejpam-6076	152	5	j	j	PROPN
ejpam-6076	152	6	′	′	NOUN
ejpam-6076	152	7	2	2	NUM
ejpam-6076	152	8	=	=	SYM
ejpam-6076	152	9	{	{	PUNCT
ejpam-6076	152	10	x1	x1	PROPN
ejpam-6076	152	11	,	,	PUNCT
ejpam-6076	152	12	x2	x2	PROPN
ejpam-6076	152	13	,	,	PUNCT
ejpam-6076	152	14	.	.	PUNCT
ejpam-6076	152	15	.	.	PUNCT
ejpam-6076	153	1	.	.	PUNCT
ejpam-6076	154	1	,	,	PUNCT
ejpam-6076	154	2	xa	xa	PROPN
ejpam-6076	154	3	,	,	PUNCT
ejpam-6076	154	4	y1	y1	PROPN
ejpam-6076	154	5	,	,	PUNCT
ejpam-6076	154	6	y2	y2	PROPN
ejpam-6076	154	7	,	,	PUNCT
ejpam-6076	154	8	.	.	PUNCT
ejpam-6076	154	9	.	.	PUNCT
ejpam-6076	154	10	.	.	PUNCT
ejpam-6076	155	1	,	,	PUNCT
ejpam-6076	155	2	ym	ym	PROPN
ejpam-6076	155	3	}	}	PUNCT
ejpam-6076	155	4	.	.	PUNCT
ejpam-6076	156	1	then	then	ADV
ejpam-6076	156	2	,	,	PUNCT
ejpam-6076	156	3	j	j	PROPN
ejpam-6076	156	4	′	′	NOUN
ejpam-6076	156	5	1	1	NUM
ejpam-6076	156	6	and	and	CCONJ
ejpam-6076	156	7	j	j	PROPN
ejpam-6076	156	8	′	′	NOUN
ejpam-6076	156	9	2	2	NUM
ejpam-6076	156	10	are	be	AUX
ejpam-6076	156	11	the	the	DET
ejpam-6076	156	12	γcer	γcer	NOUN
ejpam-6076	156	13	-	-	PUNCT
ejpam-6076	156	14	set	set	VERB
ejpam-6076	156	15	and	and	CCONJ
ejpam-6076	156	16	γcerp	γcerp	NOUN
ejpam-6076	156	17	-	-	PUNCT
ejpam-6076	156	18	set	set	NOUN
ejpam-6076	156	19	of	of	ADP
ejpam-6076	156	20	g	g	NOUN
ejpam-6076	156	21	,	,	PUNCT
ejpam-6076	156	22	respectively	respectively	ADV
ejpam-6076	156	23	.	.	PUNCT
ejpam-6076	157	1	consequently	consequently	ADV
ejpam-6076	157	2	,	,	PUNCT
ejpam-6076	157	3	γcer(g	γcer(g	NOUN
ejpam-6076	157	4	)	)	PUNCT
ejpam-6076	157	5	=	=	PUNCT
ejpam-6076	158	1	|j	|j	NOUN
ejpam-6076	158	2	′	′	NUM
ejpam-6076	159	1	1|	1|	NUM
ejpam-6076	160	1	=	=	NOUN
ejpam-6076	160	2	a	a	DET
ejpam-6076	160	3	and	and	CCONJ
ejpam-6076	160	4	γcerp(g	γcerp(g	ADJ
ejpam-6076	160	5	)	)	PUNCT
ejpam-6076	160	6	=	=	PUNCT
ejpam-6076	161	1	|j	|j	NOUN
ejpam-6076	162	1	′	′	NUM
ejpam-6076	162	2	2|	2|	NUM
ejpam-6076	163	1	=	=	PUNCT
ejpam-6076	163	2	a+m	a+m	NUM
ejpam-6076	164	1	=	=	SYM
ejpam-6076	164	2	b.	b.	PROPN
ejpam-6076	165	1	x1	x1	PROPN
ejpam-6076	166	1	x2	x2	NOUN
ejpam-6076	166	2	x3	x3	PROPN
ejpam-6076	166	3	x4	x4	PROPN
ejpam-6076	166	4	x5	x5	PROPN
ejpam-6076	166	5	x6	x6	PROPN
ejpam-6076	166	6	xa−4	xa−4	PROPN
ejpam-6076	166	7	xa−5	xa−5	PROPN
ejpam-6076	166	8	xa−2	xa−2	PROPN
ejpam-6076	166	9	xa−1	xa−1	PROPN
ejpam-6076	166	10	xa	xa	PROPN
ejpam-6076	167	1	y1	y1	NOUN
ejpam-6076	167	2	y2	y2	PROPN
ejpam-6076	167	3	y3	y3	NOUN
ejpam-6076	167	4	y4	y4	NOUN
ejpam-6076	167	5	y5	y5	PROPN
ejpam-6076	167	6	y1	y1	PROPN
ejpam-6076	167	7	ym−5	ym−5	PROPN
ejpam-6076	167	8	ym−4	ym−4	PROPN
ejpam-6076	167	9	ym−2	ym−2	PROPN
ejpam-6076	167	10	ym−1	ym−1	PROPN
ejpam-6076	167	11	ym	ym	PROPN
ejpam-6076	167	12	g′	g′	NOUN
ejpam-6076	167	13	:	:	PUNCT
ejpam-6076	167	14	.	.	PUNCT
ejpam-6076	167	15	.	.	PUNCT
ejpam-6076	167	16	.	.	PUNCT
ejpam-6076	168	1	ya−3	ya−3	INTJ
ejpam-6076	168	2	.	.	PUNCT
ejpam-6076	168	3	.	.	PUNCT
ejpam-6076	168	4	.	.	PUNCT
ejpam-6076	169	1	xa−3	xa−3	PROPN
ejpam-6076	170	1	z1	z1	PROPN
ejpam-6076	170	2	z2	z2	PROPN
ejpam-6076	170	3	zm	zm	PROPN
ejpam-6076	170	4	figure	figure	NOUN
ejpam-6076	170	5	2	2	NUM
ejpam-6076	170	6	:	:	PUNCT
ejpam-6076	170	7	a	a	DET
ejpam-6076	170	8	graph	graph	NOUN
ejpam-6076	170	9	g′	g′	NOUN
ejpam-6076	170	10	with	with	ADP
ejpam-6076	170	11	γcer(g	γcer(g	PROPN
ejpam-6076	170	12	′	′	NUM
ejpam-6076	170	13	)	)	PUNCT
ejpam-6076	170	14	<	<	X
ejpam-6076	170	15	γcerp(g	γcerp(g	PROPN
ejpam-6076	170	16	′	′	NUM
ejpam-6076	170	17	)	)	PUNCT
ejpam-6076	170	18	.	.	PUNCT
ejpam-6076	171	1	this	this	PRON
ejpam-6076	171	2	completes	complete	VERB
ejpam-6076	171	3	the	the	DET
ejpam-6076	171	4	proof	proof	NOUN
ejpam-6076	171	5	.	.	PUNCT
ejpam-6076	172	1	j.	j.	PROPN
ejpam-6076	172	2	j.	j.	PROPN
ejpam-6076	172	3	hamja	hamja	PROPN
ejpam-6076	172	4	et	et	PROPN
ejpam-6076	172	5	al	al	PROPN
ejpam-6076	172	6	.	.	PUNCT
ejpam-6076	172	7	/	/	SYM
ejpam-6076	172	8	eur	eur	PROPN
ejpam-6076	172	9	.	.	PUNCT
ejpam-6076	173	1	j.	j.	PROPN
ejpam-6076	173	2	pure	pure	PROPN
ejpam-6076	173	3	appl	appl	PROPN
ejpam-6076	173	4	.	.	PROPN
ejpam-6076	173	5	math	math	PROPN
ejpam-6076	173	6	,	,	PUNCT
ejpam-6076	173	7	18	18	NUM
ejpam-6076	173	8	(	(	PUNCT
ejpam-6076	173	9	3	3	NUM
ejpam-6076	173	10	)	)	PUNCT
ejpam-6076	173	11	(	(	PUNCT
ejpam-6076	173	12	2025	2025	NUM
ejpam-6076	173	13	)	)	PUNCT
ejpam-6076	173	14	,	,	PUNCT
ejpam-6076	173	15	6076	6076	NUM
ejpam-6076	173	16	7	7	NUM
ejpam-6076	173	17	of	of	ADP
ejpam-6076	173	18	13	13	NUM
ejpam-6076	173	19	the	the	DET
ejpam-6076	173	20	following	following	ADJ
ejpam-6076	173	21	result	result	NOUN
ejpam-6076	173	22	is	be	AUX
ejpam-6076	173	23	a	a	DET
ejpam-6076	173	24	direct	direct	ADJ
ejpam-6076	173	25	consequence	consequence	NOUN
ejpam-6076	173	26	of	of	ADP
ejpam-6076	173	27	theorem	theorem	ADJ
ejpam-6076	173	28	2	2	NUM
ejpam-6076	173	29	.	.	PUNCT
ejpam-6076	173	30	corollary	corollary	ADJ
ejpam-6076	173	31	1	1	NUM
ejpam-6076	173	32	.	.	PUNCT
ejpam-6076	174	1	for	for	ADP
ejpam-6076	174	2	any	any	DET
ejpam-6076	174	3	non	non	ADJ
ejpam-6076	174	4	-	-	ADJ
ejpam-6076	174	5	negative	negative	ADJ
ejpam-6076	174	6	integer	integer	NOUN
ejpam-6076	174	7	c	c	NOUN
ejpam-6076	174	8	,	,	PUNCT
ejpam-6076	174	9	there	there	PRON
ejpam-6076	174	10	exists	exist	VERB
ejpam-6076	174	11	a	a	DET
ejpam-6076	174	12	connected	connected	ADJ
ejpam-6076	174	13	graph	graph	NOUN
ejpam-6076	174	14	g	g	ADP
ejpam-6076	174	15	such	such	ADJ
ejpam-6076	174	16	that	that	SCONJ
ejpam-6076	174	17	γcerp(g)−	γcerp(g)−	PROPN
ejpam-6076	174	18	γcer(g	γcer(g	PROPN
ejpam-6076	174	19	)	)	PUNCT
ejpam-6076	174	20	=	=	SYM
ejpam-6076	174	21	c.	c.	NOUN
ejpam-6076	174	22	in	in	ADP
ejpam-6076	174	23	other	other	ADJ
ejpam-6076	174	24	words	word	NOUN
ejpam-6076	174	25	,	,	PUNCT
ejpam-6076	174	26	the	the	DET
ejpam-6076	174	27	difference	difference	NOUN
ejpam-6076	174	28	between	between	ADP
ejpam-6076	174	29	γcerp(g	γcerp(g	PROPN
ejpam-6076	174	30	)	)	PUNCT
ejpam-6076	174	31	and	and	CCONJ
ejpam-6076	174	32	γcer(g	γcer(g	NOUN
ejpam-6076	174	33	)	)	PUNCT
ejpam-6076	174	34	can	can	AUX
ejpam-6076	174	35	be	be	AUX
ejpam-6076	174	36	made	make	VERB
ejpam-6076	174	37	arbitrarily	arbitrarily	ADV
ejpam-6076	174	38	large	large	ADJ
ejpam-6076	174	39	.	.	PUNCT
ejpam-6076	175	1	we	we	PRON
ejpam-6076	175	2	will	will	AUX
ejpam-6076	175	3	now	now	ADV
ejpam-6076	175	4	determine	determine	VERB
ejpam-6076	175	5	the	the	DET
ejpam-6076	175	6	small	small	ADJ
ejpam-6076	175	7	and	and	CCONJ
ejpam-6076	175	8	large	large	ADJ
ejpam-6076	175	9	values	value	NOUN
ejpam-6076	175	10	of	of	ADP
ejpam-6076	175	11	graph	graph	NOUN
ejpam-6076	175	12	g.	g.	PROPN
ejpam-6076	175	13	theorem	theorem	NOUN
ejpam-6076	175	14	3	3	X
ejpam-6076	175	15	.	.	PUNCT
ejpam-6076	176	1	let	let	VERB
ejpam-6076	176	2	g	g	PRON
ejpam-6076	176	3	be	be	AUX
ejpam-6076	176	4	a	a	DET
ejpam-6076	176	5	graph	graph	NOUN
ejpam-6076	176	6	with	with	ADP
ejpam-6076	176	7	order	order	NOUN
ejpam-6076	176	8	n	n	PRON
ejpam-6076	176	9	≥	≥	NOUN
ejpam-6076	176	10	1	1	NUM
ejpam-6076	176	11	.	.	PUNCT
ejpam-6076	177	1	then	then	ADV
ejpam-6076	177	2	the	the	DET
ejpam-6076	177	3	following	follow	VERB
ejpam-6076	177	4	hold	hold	NOUN
ejpam-6076	177	5	:	:	PUNCT
ejpam-6076	177	6	(	(	PUNCT
ejpam-6076	177	7	i	i	NOUN
ejpam-6076	177	8	)	)	PUNCT
ejpam-6076	177	9	1	1	NUM
ejpam-6076	177	10	≤	≤	NUM
ejpam-6076	177	11	γcerp(g	γcerp(g	PROPN
ejpam-6076	177	12	)	)	PUNCT
ejpam-6076	177	13	≤	≤	NOUN
ejpam-6076	177	14	n	n	CCONJ
ejpam-6076	177	15	;	;	PUNCT
ejpam-6076	177	16	(	(	PUNCT
ejpam-6076	177	17	ii	ii	NOUN
ejpam-6076	177	18	)	)	PUNCT
ejpam-6076	177	19	γcerp(g	γcerp(g	PROPN
ejpam-6076	177	20	)	)	PUNCT
ejpam-6076	177	21	=	=	SYM
ejpam-6076	177	22	1	1	NUM
ejpam-6076	178	1	if	if	SCONJ
ejpam-6076	178	2	and	and	CCONJ
ejpam-6076	178	3	only	only	ADV
ejpam-6076	178	4	if	if	SCONJ
ejpam-6076	178	5	γcer(g	γcer(g	NUM
ejpam-6076	178	6	)	)	PUNCT
ejpam-6076	178	7	=	=	SYM
ejpam-6076	178	8	1	1	NUM
ejpam-6076	178	9	;	;	PUNCT
ejpam-6076	178	10	(	(	PUNCT
ejpam-6076	178	11	iii	iii	X
ejpam-6076	178	12	)	)	PUNCT
ejpam-6076	178	13	if	if	SCONJ
ejpam-6076	178	14	γcerp(g	γcerp(g	VERB
ejpam-6076	178	15	)	)	PUNCT
ejpam-6076	178	16	=	=	SYM
ejpam-6076	178	17	2	2	NUM
ejpam-6076	178	18	,	,	PUNCT
ejpam-6076	178	19	then	then	ADV
ejpam-6076	178	20	γcer(g	γcer(g	NUM
ejpam-6076	178	21	)	)	PUNCT
ejpam-6076	178	22	=	=	SYM
ejpam-6076	179	1	2	2	X
ejpam-6076	179	2	.	.	PUNCT
ejpam-6076	180	1	however	however	ADV
ejpam-6076	180	2	,	,	PUNCT
ejpam-6076	180	3	the	the	DET
ejpam-6076	180	4	reverse	reverse	NOUN
ejpam-6076	180	5	is	be	AUX
ejpam-6076	180	6	not	not	PART
ejpam-6076	180	7	necessarily	necessarily	ADV
ejpam-6076	180	8	true	true	ADJ
ejpam-6076	180	9	.	.	PUNCT
ejpam-6076	181	1	(	(	PUNCT
ejpam-6076	181	2	iv	iv	X
ejpam-6076	181	3	)	)	PUNCT
ejpam-6076	181	4	if	if	SCONJ
ejpam-6076	181	5	γcer(g	γcer(g	NOUN
ejpam-6076	181	6	)	)	PUNCT
ejpam-6076	181	7	=	=	SYM
ejpam-6076	182	1	n	n	CCONJ
ejpam-6076	182	2	,	,	PUNCT
ejpam-6076	182	3	then	then	ADV
ejpam-6076	182	4	γcerp(g	γcerp(g	PROPN
ejpam-6076	182	5	)	)	PUNCT
ejpam-6076	182	6	=	=	VERB
ejpam-6076	182	7	n.	n.	NOUN
ejpam-6076	182	8	however	however	ADV
ejpam-6076	182	9	,	,	PUNCT
ejpam-6076	182	10	the	the	DET
ejpam-6076	182	11	reverse	reverse	NOUN
ejpam-6076	182	12	is	be	AUX
ejpam-6076	182	13	not	not	PART
ejpam-6076	182	14	necessarily	necessarily	ADV
ejpam-6076	182	15	true	true	ADJ
ejpam-6076	182	16	.	.	PUNCT
ejpam-6076	183	1	proof	proof	NOUN
ejpam-6076	183	2	.	.	PUNCT
ejpam-6076	184	1	(	(	PUNCT
ejpam-6076	184	2	i	i	NOUN
ejpam-6076	184	3	)	)	PUNCT
ejpam-6076	184	4	let	let	VERB
ejpam-6076	184	5	g	g	NOUN
ejpam-6076	184	6	be	be	AUX
ejpam-6076	184	7	a	a	DET
ejpam-6076	184	8	graph	graph	NOUN
ejpam-6076	184	9	with	with	ADP
ejpam-6076	184	10	|v	|v	PROPN
ejpam-6076	184	11	(	(	PUNCT
ejpam-6076	184	12	g)|	g)|	NOUN
ejpam-6076	184	13	=	=	SYM
ejpam-6076	184	14	n.	n.	NOUN
ejpam-6076	184	15	since	since	SCONJ
ejpam-6076	184	16	the	the	DET
ejpam-6076	184	17	empty	empty	ADJ
ejpam-6076	184	18	set	set	NOUN
ejpam-6076	184	19	is	be	AUX
ejpam-6076	184	20	not	not	PART
ejpam-6076	184	21	a	a	DET
ejpam-6076	184	22	certified	certify	VERB
ejpam-6076	184	23	dominating	dominating	NOUN
ejpam-6076	184	24	set	set	NOUN
ejpam-6076	184	25	of	of	ADP
ejpam-6076	184	26	g	g	PROPN
ejpam-6076	184	27	,	,	PUNCT
ejpam-6076	184	28	we	we	PRON
ejpam-6076	184	29	conclude	conclude	VERB
ejpam-6076	184	30	that	that	PRON
ejpam-6076	184	31	γcer(g	γcer(g	NOUN
ejpam-6076	184	32	)	)	PUNCT
ejpam-6076	184	33	≥	≥	NOUN
ejpam-6076	184	34	1	1	NUM
ejpam-6076	184	35	.	.	PUNCT
ejpam-6076	184	36	additionally	additionally	ADV
ejpam-6076	184	37	,	,	PUNCT
ejpam-6076	184	38	note	note	VERB
ejpam-6076	184	39	that	that	SCONJ
ejpam-6076	184	40	any	any	DET
ejpam-6076	184	41	certified	certify	VERB
ejpam-6076	184	42	perfect	perfect	ADJ
ejpam-6076	184	43	dominating	dominating	NOUN
ejpam-6076	184	44	set	set	NOUN
ejpam-6076	184	45	j	j	PROPN
ejpam-6076	184	46	is	be	AUX
ejpam-6076	184	47	a	a	DET
ejpam-6076	184	48	subset	subset	NOUN
ejpam-6076	184	49	of	of	ADP
ejpam-6076	184	50	the	the	DET
ejpam-6076	184	51	vertex	vertex	NOUN
ejpam-6076	184	52	set	set	NOUN
ejpam-6076	184	53	of	of	ADP
ejpam-6076	184	54	g	g	NOUN
ejpam-6076	184	55	,	,	PUNCT
ejpam-6076	184	56	i.e.	i.e.	X
ejpam-6076	184	57	,	,	PUNCT
ejpam-6076	184	58	j	j	PROPN
ejpam-6076	184	59	⊆	⊆	NUM
ejpam-6076	184	60	v	v	NOUN
ejpam-6076	184	61	(	(	PUNCT
ejpam-6076	184	62	g	g	NOUN
ejpam-6076	184	63	)	)	PUNCT
ejpam-6076	184	64	.	.	PUNCT
ejpam-6076	185	1	thus	thus	ADV
ejpam-6076	185	2	,	,	PUNCT
ejpam-6076	185	3	we	we	PRON
ejpam-6076	185	4	have	have	AUX
ejpam-6076	185	5	γcerp(g	γcerp(g	VERB
ejpam-6076	185	6	)	)	PUNCT
ejpam-6076	185	7	≤	≤	NOUN
ejpam-6076	185	8	|v	|v	X
ejpam-6076	185	9	(	(	PUNCT
ejpam-6076	185	10	g)|	g)|	PROPN
ejpam-6076	185	11	=	=	SYM
ejpam-6076	185	12	n.	n.	PROPN
ejpam-6076	185	13	therefore	therefore	ADV
ejpam-6076	185	14	,	,	PUNCT
ejpam-6076	185	15	1	1	NUM
ejpam-6076	185	16	≤	≤	NUM
ejpam-6076	185	17	γcerp(g	γcerp(g	PROPN
ejpam-6076	185	18	)	)	PUNCT
ejpam-6076	185	19	≤	≤	NOUN
ejpam-6076	185	20	n.	n.	NOUN
ejpam-6076	185	21	(	(	PUNCT
ejpam-6076	185	22	ii	ii	NOUN
ejpam-6076	185	23	)	)	PUNCT
ejpam-6076	185	24	assume	assume	VERB
ejpam-6076	185	25	that	that	SCONJ
ejpam-6076	185	26	γcerp(g	γcerp(g	VERB
ejpam-6076	185	27	)	)	PUNCT
ejpam-6076	185	28	=	=	SYM
ejpam-6076	186	1	1	1	X
ejpam-6076	186	2	.	.	PUNCT
ejpam-6076	187	1	then	then	ADV
ejpam-6076	187	2	j	j	X
ejpam-6076	187	3	=	=	PRON
ejpam-6076	187	4	{	{	PUNCT
ejpam-6076	187	5	x	x	NOUN
ejpam-6076	187	6	}	}	PUNCT
ejpam-6076	187	7	is	be	AUX
ejpam-6076	187	8	the	the	DET
ejpam-6076	187	9	minimum	minimum	ADJ
ejpam-6076	187	10	certified	certify	VERB
ejpam-6076	187	11	perfect	perfect	ADJ
ejpam-6076	187	12	dominating	dominating	NOUN
ejpam-6076	187	13	set	set	NOUN
ejpam-6076	187	14	of	of	ADP
ejpam-6076	187	15	g.	g.	PROPN
ejpam-6076	187	16	this	this	PRON
ejpam-6076	187	17	implies	imply	VERB
ejpam-6076	187	18	that	that	SCONJ
ejpam-6076	187	19	j	j	PROPN
ejpam-6076	187	20	is	be	AUX
ejpam-6076	187	21	also	also	ADV
ejpam-6076	187	22	a	a	DET
ejpam-6076	187	23	minimum	minimum	ADJ
ejpam-6076	187	24	certified	certify	VERB
ejpam-6076	187	25	dominating	dominating	NOUN
ejpam-6076	187	26	set	set	NOUN
ejpam-6076	187	27	of	of	ADP
ejpam-6076	187	28	g	g	PROPN
ejpam-6076	187	29	,	,	PUNCT
ejpam-6076	187	30	so	so	ADV
ejpam-6076	187	31	γcer(g	γcer(g	ADJ
ejpam-6076	187	32	)	)	PUNCT
ejpam-6076	187	33	=	=	PUNCT
ejpam-6076	188	1	|j	|j	NOUN
ejpam-6076	189	1	|	|	NOUN
ejpam-6076	189	2	=	=	NOUN
ejpam-6076	189	3	1	1	X
ejpam-6076	189	4	.	.	PUNCT
ejpam-6076	190	1	conversely	conversely	ADV
ejpam-6076	190	2	,	,	PUNCT
ejpam-6076	190	3	suppose	suppose	VERB
ejpam-6076	190	4	γcer(g	γcer(g	X
ejpam-6076	190	5	)	)	PUNCT
ejpam-6076	190	6	=	=	SYM
ejpam-6076	191	1	1	1	X
ejpam-6076	191	2	.	.	PUNCT
ejpam-6076	192	1	then	then	ADV
ejpam-6076	192	2	j	j	PROPN
ejpam-6076	192	3	′	′	NUM
ejpam-6076	192	4	=	=	PUNCT
ejpam-6076	192	5	{	{	PUNCT
ejpam-6076	192	6	x′	x′	NUM
ejpam-6076	192	7	}	}	PUNCT
ejpam-6076	192	8	.	.	PUNCT
ejpam-6076	193	1	for	for	ADP
ejpam-6076	193	2	every	every	DET
ejpam-6076	193	3	y′	y′	NOUN
ejpam-6076	193	4	∈	∈	PROPN
ejpam-6076	193	5	v	v	NOUN
ejpam-6076	193	6	(	(	PUNCT
ejpam-6076	193	7	g	g	NOUN
ejpam-6076	193	8	)	)	PUNCT
ejpam-6076	193	9	\	\	PROPN
ejpam-6076	194	1	j	j	PROPN
ejpam-6076	194	2	′	′	PROPN
ejpam-6076	194	3	,	,	PUNCT
ejpam-6076	194	4	the	the	DET
ejpam-6076	194	5	only	only	ADJ
ejpam-6076	194	6	vertex	vertex	NOUN
ejpam-6076	194	7	dominating	dominating	NOUN
ejpam-6076	194	8	y′	y′	NOUN
ejpam-6076	194	9	is	be	AUX
ejpam-6076	194	10	x′	x′	PROPN
ejpam-6076	194	11	∈	∈	PROPN
ejpam-6076	194	12	j	j	PROPN
ejpam-6076	194	13	′.	′.	PROPN
ejpam-6076	194	14	hence	hence	ADV
ejpam-6076	194	15	,	,	PUNCT
ejpam-6076	194	16	j	j	PROPN
ejpam-6076	194	17	′	′	PROPN
ejpam-6076	194	18	is	be	AUX
ejpam-6076	194	19	a	a	DET
ejpam-6076	194	20	perfect	perfect	ADJ
ejpam-6076	194	21	dominating	dominating	NOUN
ejpam-6076	194	22	set	set	NOUN
ejpam-6076	194	23	of	of	ADP
ejpam-6076	194	24	g	g	NOUN
ejpam-6076	194	25	,	,	PUNCT
ejpam-6076	194	26	which	which	PRON
ejpam-6076	194	27	implies	imply	VERB
ejpam-6076	194	28	γcerp(g	γcerp(g	PROPN
ejpam-6076	194	29	)	)	PUNCT
ejpam-6076	194	30	=	=	PUNCT
ejpam-6076	195	1	|j	|j	NOUN
ejpam-6076	195	2	′|	′|	NUM
ejpam-6076	195	3	=	=	SYM
ejpam-6076	195	4	1	1	X
ejpam-6076	195	5	.	.	PUNCT
ejpam-6076	195	6	(	(	PUNCT
ejpam-6076	195	7	iii	iii	NOUN
ejpam-6076	195	8	)	)	PUNCT
ejpam-6076	195	9	assume	assume	VERB
ejpam-6076	195	10	that	that	SCONJ
ejpam-6076	195	11	γcerp(g	γcerp(g	VERB
ejpam-6076	195	12	)	)	PUNCT
ejpam-6076	195	13	=	=	SYM
ejpam-6076	196	1	2	2	X
ejpam-6076	196	2	.	.	X
ejpam-6076	197	1	then	then	ADV
ejpam-6076	197	2	j	j	X
ejpam-6076	197	3	=	=	PUNCT
ejpam-6076	197	4	{	{	PUNCT
ejpam-6076	197	5	x	x	PROPN
ejpam-6076	197	6	,	,	PUNCT
ejpam-6076	197	7	y	y	PRON
ejpam-6076	197	8	}	}	PUNCT
ejpam-6076	197	9	is	be	AUX
ejpam-6076	197	10	the	the	DET
ejpam-6076	197	11	minimum	minimum	ADJ
ejpam-6076	197	12	certified	certify	VERB
ejpam-6076	197	13	perfect	perfect	ADJ
ejpam-6076	197	14	dominating	dominating	NOUN
ejpam-6076	197	15	set	set	NOUN
ejpam-6076	197	16	of	of	ADP
ejpam-6076	197	17	g.	g.	PROPN
ejpam-6076	197	18	since	since	SCONJ
ejpam-6076	197	19	every	every	DET
ejpam-6076	197	20	certified	certify	VERB
ejpam-6076	197	21	perfect	perfect	ADJ
ejpam-6076	197	22	dominating	dominating	NOUN
ejpam-6076	197	23	set	set	NOUN
ejpam-6076	197	24	is	be	AUX
ejpam-6076	197	25	also	also	ADV
ejpam-6076	197	26	a	a	DET
ejpam-6076	197	27	certified	certify	VERB
ejpam-6076	197	28	dominating	dominating	NOUN
ejpam-6076	197	29	set	set	NOUN
ejpam-6076	197	30	,	,	PUNCT
ejpam-6076	197	31	we	we	PRON
ejpam-6076	197	32	conclude	conclude	VERB
ejpam-6076	197	33	that	that	PRON
ejpam-6076	197	34	γcer(g	γcer(g	NOUN
ejpam-6076	197	35	)	)	PUNCT
ejpam-6076	197	36	=	=	SYM
ejpam-6076	198	1	2	2	X
ejpam-6076	198	2	.	.	PUNCT
ejpam-6076	198	3	however	however	ADV
ejpam-6076	198	4	,	,	PUNCT
ejpam-6076	198	5	the	the	DET
ejpam-6076	198	6	reverse	reverse	NOUN
ejpam-6076	198	7	does	do	AUX
ejpam-6076	198	8	not	not	PART
ejpam-6076	198	9	always	always	ADV
ejpam-6076	198	10	hold	hold	VERB
ejpam-6076	198	11	.	.	PUNCT
ejpam-6076	199	1	for	for	ADP
ejpam-6076	199	2	example	example	NOUN
ejpam-6076	199	3	,	,	PUNCT
ejpam-6076	199	4	consider	consider	VERB
ejpam-6076	199	5	g	g	PROPN
ejpam-6076	199	6	=	=	SYM
ejpam-6076	199	7	c5	c5	PROPN
ejpam-6076	199	8	.	.	PUNCT
ejpam-6076	200	1	in	in	ADP
ejpam-6076	200	2	this	this	DET
ejpam-6076	200	3	case	case	NOUN
ejpam-6076	200	4	,	,	PUNCT
ejpam-6076	200	5	γcer(c5	γcer(c5	ADJ
ejpam-6076	200	6	)	)	PUNCT
ejpam-6076	200	7	=	=	SYM
ejpam-6076	201	1	2	2	NUM
ejpam-6076	201	2	,	,	PUNCT
ejpam-6076	201	3	but	but	CCONJ
ejpam-6076	201	4	γcerp(c5	γcerp(c5	NOUN
ejpam-6076	201	5	)	)	PUNCT
ejpam-6076	201	6	=	=	SYM
ejpam-6076	202	1	5	5	X
ejpam-6076	202	2	.	.	PUNCT
ejpam-6076	202	3	(	(	PUNCT
ejpam-6076	202	4	iv	iv	X
ejpam-6076	202	5	)	)	PUNCT
ejpam-6076	202	6	suppose	suppose	VERB
ejpam-6076	202	7	γcer(g	γcer(g	X
ejpam-6076	202	8	)	)	PUNCT
ejpam-6076	202	9	=	=	VERB
ejpam-6076	203	1	n.	n.	NOUN
ejpam-6076	203	2	since	since	SCONJ
ejpam-6076	203	3	γcerp(g	γcerp(g	PROPN
ejpam-6076	203	4	)	)	PUNCT
ejpam-6076	203	5	≥	≥	NOUN
ejpam-6076	203	6	γcer(g	γcer(g	NUM
ejpam-6076	203	7	)	)	PUNCT
ejpam-6076	203	8	and	and	CCONJ
ejpam-6076	203	9	γcerp(g	γcerp(g	NOUN
ejpam-6076	203	10	)	)	PUNCT
ejpam-6076	203	11	≤	≤	NOUN
ejpam-6076	203	12	n	n	CCONJ
ejpam-6076	203	13	,	,	PUNCT
ejpam-6076	203	14	it	it	PRON
ejpam-6076	203	15	follows	follow	VERB
ejpam-6076	203	16	that	that	SCONJ
ejpam-6076	203	17	γcerp(g	γcerp(g	VERB
ejpam-6076	203	18	)	)	PUNCT
ejpam-6076	203	19	=	=	VERB
ejpam-6076	203	20	n.	n.	NOUN
ejpam-6076	203	21	to	to	PART
ejpam-6076	203	22	show	show	VERB
ejpam-6076	203	23	that	that	SCONJ
ejpam-6076	203	24	the	the	DET
ejpam-6076	203	25	reverse	reverse	NOUN
ejpam-6076	203	26	is	be	AUX
ejpam-6076	203	27	not	not	PART
ejpam-6076	203	28	necessarily	necessarily	ADV
ejpam-6076	203	29	true	true	ADJ
ejpam-6076	203	30	,	,	PUNCT
ejpam-6076	203	31	consider	consider	VERB
ejpam-6076	203	32	g	g	NOUN
ejpam-6076	203	33	=	=	SYM
ejpam-6076	203	34	p5	p5	PROPN
ejpam-6076	203	35	.	.	PUNCT
ejpam-6076	204	1	in	in	ADP
ejpam-6076	204	2	this	this	DET
ejpam-6076	204	3	case	case	NOUN
ejpam-6076	204	4	,	,	PUNCT
ejpam-6076	204	5	γcerp(p5	γcerp(p5	ADJ
ejpam-6076	204	6	)	)	PUNCT
ejpam-6076	204	7	=	=	SYM
ejpam-6076	204	8	5	5	NUM
ejpam-6076	204	9	,	,	PUNCT
ejpam-6076	204	10	but	but	CCONJ
ejpam-6076	204	11	γcer(p5	γcer(p5	NOUN
ejpam-6076	204	12	)	)	PUNCT
ejpam-6076	204	13	=	=	SYM
ejpam-6076	204	14	2	2	X
ejpam-6076	204	15	.	.	PUNCT
ejpam-6076	205	1	in	in	ADP
ejpam-6076	205	2	what	what	PRON
ejpam-6076	205	3	follows	follow	VERB
ejpam-6076	205	4	the	the	DET
ejpam-6076	205	5	following	follow	VERB
ejpam-6076	205	6	result	result	NOUN
ejpam-6076	205	7	will	will	AUX
ejpam-6076	205	8	be	be	AUX
ejpam-6076	205	9	used	use	VERB
ejpam-6076	205	10	.	.	PUNCT
ejpam-6076	206	1	theorem	theorem	ADJ
ejpam-6076	206	2	4	4	NUM
ejpam-6076	206	3	.	.	PUNCT
ejpam-6076	207	1	[	[	X
ejpam-6076	207	2	3	3	X
ejpam-6076	207	3	]	]	PUNCT
ejpam-6076	207	4	let	let	VERB
ejpam-6076	207	5	g	g	PRON
ejpam-6076	207	6	be	be	AUX
ejpam-6076	207	7	a	a	DET
ejpam-6076	207	8	graph	graph	NOUN
ejpam-6076	207	9	consisting	consist	VERB
ejpam-6076	207	10	of	of	ADP
ejpam-6076	207	11	components	component	NOUN
ejpam-6076	207	12	g1	g1	NOUN
ejpam-6076	207	13	,	,	PUNCT
ejpam-6076	207	14	g2	g2	PROPN
ejpam-6076	207	15	,	,	PUNCT
ejpam-6076	207	16	.	.	PUNCT
ejpam-6076	207	17	.	.	PUNCT
ejpam-6076	208	1	.	.	PUNCT
ejpam-6076	209	1	,	,	PUNCT
ejpam-6076	209	2	gk	gk	PROPN
ejpam-6076	209	3	,	,	PUNCT
ejpam-6076	209	4	where	where	SCONJ
ejpam-6076	209	5	k	k	PROPN
ejpam-6076	209	6	≥	≥	NUM
ejpam-6076	209	7	2	2	NUM
ejpam-6076	209	8	.	.	PUNCT
ejpam-6076	209	9	then	then	ADV
ejpam-6076	209	10	γcerp(g	γcerp(g	PROPN
ejpam-6076	209	11	)	)	PUNCT
ejpam-6076	209	12	=	=	SYM
ejpam-6076	210	1	k∑	k∑	VERB
ejpam-6076	210	2	i=1	i=1	PROPN
ejpam-6076	210	3	γcerp(gi	γcerp(gi	PROPN
ejpam-6076	210	4	)	)	PUNCT
ejpam-6076	210	5	.	.	PUNCT
ejpam-6076	211	1	proposition	proposition	NOUN
ejpam-6076	211	2	7	7	NUM
ejpam-6076	211	3	.	.	PUNCT
ejpam-6076	212	1	let	let	VERB
ejpam-6076	212	2	g	g	PROPN
ejpam-6076	212	3	=	=	PROPN
ejpam-6076	212	4	kn	kn	PROPN
ejpam-6076	212	5	be	be	AUX
ejpam-6076	212	6	the	the	DET
ejpam-6076	212	7	complement	complement	NOUN
ejpam-6076	212	8	of	of	ADP
ejpam-6076	212	9	a	a	DET
ejpam-6076	212	10	complete	complete	ADJ
ejpam-6076	212	11	graph	graph	NOUN
ejpam-6076	212	12	of	of	ADP
ejpam-6076	212	13	order	order	NOUN
ejpam-6076	212	14	n.	n.	NOUN
ejpam-6076	212	15	then	then	ADV
ejpam-6076	212	16	γcerp(kn	γcerp(kn	VERB
ejpam-6076	212	17	)	)	PUNCT
ejpam-6076	213	1	=	=	SYM
ejpam-6076	213	2	n.	n.	PROPN
ejpam-6076	213	3	j.	j.	PROPN
ejpam-6076	213	4	j.	j.	PROPN
ejpam-6076	213	5	hamja	hamja	PROPN
ejpam-6076	213	6	et	et	PROPN
ejpam-6076	213	7	al	al	PROPN
ejpam-6076	213	8	.	.	PUNCT
ejpam-6076	213	9	/	/	SYM
ejpam-6076	213	10	eur	eur	PROPN
ejpam-6076	213	11	.	.	PUNCT
ejpam-6076	214	1	j.	j.	PROPN
ejpam-6076	214	2	pure	pure	PROPN
ejpam-6076	214	3	appl	appl	PROPN
ejpam-6076	214	4	.	.	PROPN
ejpam-6076	214	5	math	math	PROPN
ejpam-6076	214	6	,	,	PUNCT
ejpam-6076	214	7	18	18	NUM
ejpam-6076	214	8	(	(	PUNCT
ejpam-6076	214	9	3	3	NUM
ejpam-6076	214	10	)	)	PUNCT
ejpam-6076	214	11	(	(	PUNCT
ejpam-6076	214	12	2025	2025	NUM
ejpam-6076	214	13	)	)	PUNCT
ejpam-6076	214	14	,	,	PUNCT
ejpam-6076	214	15	6076	6076	NUM
ejpam-6076	214	16	8	8	NUM
ejpam-6076	214	17	of	of	ADP
ejpam-6076	214	18	13	13	NUM
ejpam-6076	214	19	proof	proof	NOUN
ejpam-6076	214	20	.	.	PUNCT
ejpam-6076	215	1	let	let	VERB
ejpam-6076	215	2	g	g	PROPN
ejpam-6076	215	3	=	=	PROPN
ejpam-6076	215	4	kn	kn	PROPN
ejpam-6076	215	5	be	be	AUX
ejpam-6076	215	6	the	the	DET
ejpam-6076	215	7	complement	complement	NOUN
ejpam-6076	215	8	of	of	ADP
ejpam-6076	215	9	a	a	DET
ejpam-6076	215	10	complete	complete	ADJ
ejpam-6076	215	11	graph	graph	NOUN
ejpam-6076	215	12	of	of	ADP
ejpam-6076	215	13	order	order	NOUN
ejpam-6076	215	14	n.	n.	NOUN
ejpam-6076	215	15	then	then	ADV
ejpam-6076	215	16	g	g	PROPN
ejpam-6076	215	17	consists	consist	VERB
ejpam-6076	215	18	of	of	ADP
ejpam-6076	215	19	n	n	DET
ejpam-6076	215	20	isolated	isolated	ADJ
ejpam-6076	215	21	vertices	vertex	NOUN
ejpam-6076	215	22	.	.	PUNCT
ejpam-6076	216	1	since	since	SCONJ
ejpam-6076	216	2	every	every	DET
ejpam-6076	216	3	vertex	vertex	NOUN
ejpam-6076	216	4	is	be	AUX
ejpam-6076	216	5	an	an	DET
ejpam-6076	216	6	isolated	isolate	VERB
ejpam-6076	216	7	,	,	PUNCT
ejpam-6076	216	8	each	each	DET
ejpam-6076	216	9	component	component	NOUN
ejpam-6076	216	10	is	be	AUX
ejpam-6076	216	11	a	a	DET
ejpam-6076	216	12	trivial	trivial	ADJ
ejpam-6076	216	13	graph	graph	NOUN
ejpam-6076	216	14	k1	k1	NOUN
ejpam-6076	216	15	.	.	PUNCT
ejpam-6076	217	1	from	from	ADP
ejpam-6076	217	2	[	[	X
ejpam-6076	217	3	3	3	NUM
ejpam-6076	217	4	]	]	PUNCT
ejpam-6076	217	5	,	,	PUNCT
ejpam-6076	217	6	we	we	PRON
ejpam-6076	217	7	know	know	VERB
ejpam-6076	217	8	that	that	SCONJ
ejpam-6076	217	9	γcerp(k1	γcerp(k1	ADV
ejpam-6076	217	10	)	)	PUNCT
ejpam-6076	217	11	=	=	SYM
ejpam-6076	217	12	1	1	NUM
ejpam-6076	217	13	for	for	ADP
ejpam-6076	217	14	each	each	DET
ejpam-6076	217	15	component	component	NOUN
ejpam-6076	217	16	.	.	PUNCT
ejpam-6076	218	1	since	since	SCONJ
ejpam-6076	218	2	g	g	PROPN
ejpam-6076	218	3	has	have	VERB
ejpam-6076	218	4	exactly	exactly	ADV
ejpam-6076	218	5	n	n	PRON
ejpam-6076	218	6	such	such	ADJ
ejpam-6076	218	7	components	component	NOUN
ejpam-6076	218	8	,	,	PUNCT
ejpam-6076	218	9	theorem	theorem	VERB
ejpam-6076	218	10	4	4	NUM
ejpam-6076	218	11	implies	imply	VERB
ejpam-6076	218	12	that	that	SCONJ
ejpam-6076	218	13	γcerp(g	γcerp(g	PROPN
ejpam-6076	218	14	)	)	PUNCT
ejpam-6076	218	15	=	=	SYM
ejpam-6076	219	1	n∑	n∑	PROPN
ejpam-6076	219	2	i=1	i=1	PROPN
ejpam-6076	219	3	γcerp(k1	γcerp(k1	ADV
ejpam-6076	219	4	)	)	PUNCT
ejpam-6076	220	1	=	=	VERB
ejpam-6076	220	2	n.	n.	PROPN
ejpam-6076	220	3	thus	thus	ADV
ejpam-6076	220	4	,	,	PUNCT
ejpam-6076	220	5	we	we	PRON
ejpam-6076	220	6	conclude	conclude	VERB
ejpam-6076	220	7	that	that	PRON
ejpam-6076	220	8	γcerp(g	γcerp(g	PROPN
ejpam-6076	220	9	)	)	PUNCT
ejpam-6076	220	10	=	=	SYM
ejpam-6076	220	11	γcerp(kn	γcerp(kn	NOUN
ejpam-6076	220	12	)	)	PUNCT
ejpam-6076	220	13	=	=	VERB
ejpam-6076	220	14	n.	n.	NOUN
ejpam-6076	220	15	as	as	SCONJ
ejpam-6076	220	16	previously	previously	ADV
ejpam-6076	220	17	noted	note	VERB
ejpam-6076	220	18	,	,	PUNCT
ejpam-6076	220	19	for	for	ADP
ejpam-6076	220	20	any	any	DET
ejpam-6076	220	21	graph	graph	NOUN
ejpam-6076	220	22	g	g	NOUN
ejpam-6076	220	23	with	with	ADP
ejpam-6076	220	24	order	order	NOUN
ejpam-6076	220	25	n	n	CCONJ
ejpam-6076	220	26	,	,	PUNCT
ejpam-6076	220	27	we	we	PRON
ejpam-6076	220	28	have	have	AUX
ejpam-6076	220	29	γcerp(g	γcerp(g	VERB
ejpam-6076	220	30	)	)	PUNCT
ejpam-6076	220	31	≤	≤	NOUN
ejpam-6076	220	32	n	n	ADP
ejpam-6076	220	33	and	and	CCONJ
ejpam-6076	220	34	γcerp(g	γcerp(g	ADJ
ejpam-6076	220	35	)	)	PUNCT
ejpam-6076	220	36	̸=	̸=	PROPN
ejpam-6076	220	37	n	n	CCONJ
ejpam-6076	220	38	−	−	PROPN
ejpam-6076	220	39	1	1	NUM
ejpam-6076	220	40	.	.	PUNCT
ejpam-6076	221	1	furthermore	furthermore	ADV
ejpam-6076	221	2	,	,	PUNCT
ejpam-6076	221	3	there	there	PRON
ejpam-6076	221	4	exist	exist	VERB
ejpam-6076	221	5	graphs	graph	NOUN
ejpam-6076	221	6	such	such	ADJ
ejpam-6076	221	7	as	as	ADP
ejpam-6076	221	8	a	a	DET
ejpam-6076	221	9	path	path	NOUN
ejpam-6076	221	10	pn	pn	PROPN
ejpam-6076	221	11	,	,	PUNCT
ejpam-6076	221	12	a	a	DET
ejpam-6076	221	13	cycle	cycle	NOUN
ejpam-6076	221	14	cn	cn	NOUN
ejpam-6076	221	15	for	for	ADP
ejpam-6076	221	16	n	n	X
ejpam-6076	221	17	̸≡	̸≡	X
ejpam-6076	221	18	0	0	PUNCT
ejpam-6076	221	19	(	(	PUNCT
ejpam-6076	221	20	mod	mod	PROPN
ejpam-6076	221	21	3	3	NUM
ejpam-6076	221	22	)	)	PUNCT
ejpam-6076	221	23	(	(	PUNCT
ejpam-6076	221	24	see	see	VERB
ejpam-6076	221	25	[	[	X
ejpam-6076	221	26	3	3	NUM
ejpam-6076	221	27	]	]	NUM
ejpam-6076	221	28	)	)	PUNCT
ejpam-6076	221	29	,	,	PUNCT
ejpam-6076	221	30	the	the	DET
ejpam-6076	221	31	complement	complement	NOUN
ejpam-6076	221	32	of	of	ADP
ejpam-6076	221	33	a	a	DET
ejpam-6076	221	34	complete	complete	ADJ
ejpam-6076	221	35	graph	graph	NOUN
ejpam-6076	221	36	on	on	ADP
ejpam-6076	221	37	n	n	DET
ejpam-6076	221	38	vertices	vertex	NOUN
ejpam-6076	221	39	,	,	PUNCT
ejpam-6076	221	40	where	where	SCONJ
ejpam-6076	221	41	γcerp(g	γcerp(g	NOUN
ejpam-6076	221	42	)	)	PUNCT
ejpam-6076	221	43	=	=	VERB
ejpam-6076	222	1	n.	n.	NOUN
ejpam-6076	222	2	this	this	PRON
ejpam-6076	222	3	motivates	motivate	VERB
ejpam-6076	222	4	the	the	DET
ejpam-6076	222	5	investigation	investigation	NOUN
ejpam-6076	222	6	into	into	ADP
ejpam-6076	222	7	the	the	DET
ejpam-6076	222	8	characterization	characterization	NOUN
ejpam-6076	222	9	of	of	ADP
ejpam-6076	222	10	all	all	DET
ejpam-6076	222	11	graphs	graph	NOUN
ejpam-6076	222	12	for	for	ADP
ejpam-6076	222	13	which	which	PRON
ejpam-6076	222	14	γcerp(g	γcerp(g	PROPN
ejpam-6076	222	15	)	)	PUNCT
ejpam-6076	222	16	=	=	SYM
ejpam-6076	223	1	n	n	CCONJ
ejpam-6076	223	2	,	,	PUNCT
ejpam-6076	223	3	which	which	PRON
ejpam-6076	223	4	is	be	AUX
ejpam-6076	223	5	addressed	address	VERB
ejpam-6076	223	6	in	in	ADP
ejpam-6076	223	7	this	this	DET
ejpam-6076	223	8	section	section	NOUN
ejpam-6076	223	9	.	.	PUNCT
ejpam-6076	224	1	recall	recall	VERB
ejpam-6076	224	2	that	that	SCONJ
ejpam-6076	224	3	f.	f.	PROPN
ejpam-6076	224	4	harary	harary	PROPN
ejpam-6076	224	5	(	(	PUNCT
ejpam-6076	224	6	see	see	VERB
ejpam-6076	224	7	[	[	X
ejpam-6076	224	8	9	9	NUM
ejpam-6076	224	9	]	]	PUNCT
ejpam-6076	224	10	)	)	PUNCT
ejpam-6076	224	11	defined	define	VERB
ejpam-6076	224	12	the	the	DET
ejpam-6076	224	13	corona	corona	NOUN
ejpam-6076	224	14	of	of	ADP
ejpam-6076	224	15	two	two	NUM
ejpam-6076	224	16	graphs	graph	NOUN
ejpam-6076	224	17	g	g	NOUN
ejpam-6076	224	18	and	and	CCONJ
ejpam-6076	224	19	h	h	NOUN
ejpam-6076	224	20	,	,	PUNCT
ejpam-6076	224	21	denoted	denote	VERB
ejpam-6076	224	22	by	by	ADP
ejpam-6076	224	23	g	g	PROPN
ejpam-6076	224	24	◦	◦	NOUN
ejpam-6076	224	25	h	h	NOUN
ejpam-6076	224	26	,	,	PUNCT
ejpam-6076	224	27	is	be	AUX
ejpam-6076	224	28	constructed	construct	VERB
ejpam-6076	224	29	by	by	ADP
ejpam-6076	224	30	taking	take	VERB
ejpam-6076	224	31	one	one	NUM
ejpam-6076	224	32	copy	copy	NOUN
ejpam-6076	224	33	of	of	ADP
ejpam-6076	224	34	g	g	PROPN
ejpam-6076	224	35	and	and	CCONJ
ejpam-6076	224	36	|v	|v	PROPN
ejpam-6076	224	37	(	(	PUNCT
ejpam-6076	224	38	g)|	g)|	NOUN
ejpam-6076	224	39	copies	copy	NOUN
ejpam-6076	224	40	of	of	ADP
ejpam-6076	224	41	h	h	NOUN
ejpam-6076	224	42	,	,	PUNCT
ejpam-6076	224	43	then	then	ADV
ejpam-6076	224	44	connecting	connect	VERB
ejpam-6076	224	45	each	each	DET
ejpam-6076	224	46	vertex	vertex	NOUN
ejpam-6076	224	47	i	i	PRON
ejpam-6076	224	48	of	of	ADP
ejpam-6076	224	49	g	g	NOUN
ejpam-6076	224	50	to	to	ADP
ejpam-6076	224	51	every	every	DET
ejpam-6076	224	52	vertex	vertex	NOUN
ejpam-6076	224	53	in	in	ADP
ejpam-6076	224	54	the	the	DET
ejpam-6076	224	55	corresponding	correspond	VERB
ejpam-6076	224	56	ith	ith	PROPN
ejpam-6076	224	57	copy	copy	NOUN
ejpam-6076	224	58	of	of	ADP
ejpam-6076	224	59	h.	h.	PROPN
ejpam-6076	224	60	for	for	ADP
ejpam-6076	224	61	each	each	DET
ejpam-6076	224	62	v	v	NUM
ejpam-6076	224	63	∈	∈	PROPN
ejpam-6076	224	64	v	v	NOUN
ejpam-6076	224	65	(	(	PUNCT
ejpam-6076	224	66	g	g	NOUN
ejpam-6076	224	67	)	)	PUNCT
ejpam-6076	224	68	,	,	PUNCT
ejpam-6076	224	69	let	let	VERB
ejpam-6076	224	70	hv	hv	PROPN
ejpam-6076	224	71	represent	represent	VERB
ejpam-6076	224	72	the	the	DET
ejpam-6076	224	73	copy	copy	NOUN
ejpam-6076	224	74	of	of	ADP
ejpam-6076	224	75	h	h	NOUN
ejpam-6076	224	76	whose	whose	DET
ejpam-6076	224	77	vertices	vertex	NOUN
ejpam-6076	224	78	are	be	AUX
ejpam-6076	224	79	individually	individually	ADV
ejpam-6076	224	80	linked	link	VERB
ejpam-6076	224	81	to	to	ADP
ejpam-6076	224	82	v.	v.	ADP
ejpam-6076	224	83	the	the	DET
ejpam-6076	224	84	subgraph	subgraph	NOUN
ejpam-6076	224	85	of	of	ADP
ejpam-6076	224	86	the	the	DET
ejpam-6076	224	87	corona	corona	NOUN
ejpam-6076	224	88	g	g	PROPN
ejpam-6076	224	89	◦	◦	NOUN
ejpam-6076	224	90	h	h	NOUN
ejpam-6076	224	91	associated	associate	VERB
ejpam-6076	224	92	with	with	ADP
ejpam-6076	224	93	the	the	DET
ejpam-6076	224	94	join	join	NOUN
ejpam-6076	224	95	⟨{v}⟩	⟨{v}⟩	NOUN
ejpam-6076	225	1	+	+	CCONJ
ejpam-6076	225	2	hv	hv	PROPN
ejpam-6076	225	3	is	be	AUX
ejpam-6076	225	4	denoted	denote	VERB
ejpam-6076	225	5	by	by	ADP
ejpam-6076	225	6	v	v	PRON
ejpam-6076	225	7	+	+	CCONJ
ejpam-6076	225	8	hv	hv	PROPN
ejpam-6076	225	9	,	,	PUNCT
ejpam-6076	225	10	where	where	SCONJ
ejpam-6076	225	11	v	v	X
ejpam-6076	225	12	∈	∈	PROPN
ejpam-6076	225	13	v	v	NOUN
ejpam-6076	225	14	(	(	PUNCT
ejpam-6076	225	15	g	g	NOUN
ejpam-6076	225	16	)	)	PUNCT
ejpam-6076	226	1	[	[	X
ejpam-6076	226	2	9	9	NUM
ejpam-6076	226	3	]	]	PUNCT
ejpam-6076	226	4	.	.	PUNCT
ejpam-6076	227	1	in	in	ADP
ejpam-6076	227	2	particular	particular	ADJ
ejpam-6076	227	3	,	,	PUNCT
ejpam-6076	227	4	when	when	SCONJ
ejpam-6076	227	5	h	h	NOUN
ejpam-6076	227	6	is	be	AUX
ejpam-6076	227	7	a	a	DET
ejpam-6076	227	8	single	single	ADJ
ejpam-6076	227	9	-	-	PUNCT
ejpam-6076	227	10	vertex	vertex	NOUN
ejpam-6076	227	11	graph	graph	NOUN
ejpam-6076	227	12	,	,	PUNCT
ejpam-6076	227	13	h	h	NOUN
ejpam-6076	227	14	=	=	SYM
ejpam-6076	227	15	k1	k1	PROPN
ejpam-6076	227	16	,	,	PUNCT
ejpam-6076	227	17	the	the	DET
ejpam-6076	227	18	corona	corona	NOUN
ejpam-6076	227	19	g	g	PROPN
ejpam-6076	227	20	◦	◦	NOUN
ejpam-6076	227	21	k1	k1	NOUN
ejpam-6076	227	22	is	be	AUX
ejpam-6076	227	23	referred	refer	VERB
ejpam-6076	227	24	to	to	ADP
ejpam-6076	227	25	as	as	SCONJ
ejpam-6076	227	26	the	the	DET
ejpam-6076	227	27	corona	corona	NOUN
ejpam-6076	227	28	of	of	ADP
ejpam-6076	227	29	g.	g.	PROPN
ejpam-6076	227	30	theorem	theorem	VERB
ejpam-6076	227	31	5	5	NUM
ejpam-6076	227	32	.	.	PUNCT
ejpam-6076	228	1	[	[	X
ejpam-6076	228	2	3	3	X
ejpam-6076	228	3	]	]	X
ejpam-6076	228	4	if	if	SCONJ
ejpam-6076	228	5	g	g	PROPN
ejpam-6076	228	6	is	be	AUX
ejpam-6076	228	7	a	a	DET
ejpam-6076	228	8	connected	connected	ADJ
ejpam-6076	228	9	graph	graph	NOUN
ejpam-6076	228	10	with	with	ADP
ejpam-6076	228	11	|v	|v	PROPN
ejpam-6076	228	12	(	(	PUNCT
ejpam-6076	228	13	g)|	g)|	NOUN
ejpam-6076	228	14	=	=	NOUN
ejpam-6076	228	15	m	m	PROPN
ejpam-6076	228	16	,	,	PUNCT
ejpam-6076	228	17	and	and	CCONJ
ejpam-6076	228	18	h	h	NOUN
ejpam-6076	228	19	is	be	AUX
ejpam-6076	228	20	a	a	DET
ejpam-6076	228	21	trivial	trivial	ADJ
ejpam-6076	228	22	graph	graph	NOUN
ejpam-6076	228	23	,	,	PUNCT
ejpam-6076	228	24	then	then	ADV
ejpam-6076	228	25	γcerp(g	γcerp(g	VERB
ejpam-6076	228	26	◦	◦	NOUN
ejpam-6076	228	27	h	h	NOUN
ejpam-6076	228	28	)	)	PUNCT
ejpam-6076	228	29	=	=	SYM
ejpam-6076	228	30	2	2	NUM
ejpam-6076	228	31	m.	m.	NOUN
ejpam-6076	228	32	corollary	corollary	NOUN
ejpam-6076	228	33	2	2	NUM
ejpam-6076	228	34	.	.	PUNCT
ejpam-6076	229	1	if	if	SCONJ
ejpam-6076	229	2	g	g	PROPN
ejpam-6076	229	3	is	be	AUX
ejpam-6076	229	4	the	the	DET
ejpam-6076	229	5	corona	corona	NOUN
ejpam-6076	229	6	of	of	ADP
ejpam-6076	229	7	a	a	DET
ejpam-6076	229	8	graph	graph	NOUN
ejpam-6076	229	9	with	with	ADP
ejpam-6076	229	10	|v	|v	PROPN
ejpam-6076	229	11	(	(	PUNCT
ejpam-6076	229	12	g)|	g)|	NOUN
ejpam-6076	229	13	=	=	PUNCT
ejpam-6076	229	14	n	n	CCONJ
ejpam-6076	229	15	,	,	PUNCT
ejpam-6076	229	16	then	then	ADV
ejpam-6076	229	17	γcerp(g	γcerp(g	PROPN
ejpam-6076	229	18	)	)	PUNCT
ejpam-6076	229	19	=	=	VERB
ejpam-6076	229	20	n.	n.	NOUN
ejpam-6076	229	21	proof	proof	NOUN
ejpam-6076	229	22	.	.	PUNCT
ejpam-6076	230	1	let	let	VERB
ejpam-6076	230	2	d	d	PRON
ejpam-6076	230	3	be	be	AUX
ejpam-6076	230	4	a	a	DET
ejpam-6076	230	5	γcerp	γcerp	NOUN
ejpam-6076	230	6	-	-	PUNCT
ejpam-6076	230	7	set	set	NOUN
ejpam-6076	230	8	of	of	ADP
ejpam-6076	230	9	g.	g.	PROPN
ejpam-6076	230	10	let	let	VERB
ejpam-6076	230	11	g	g	NOUN
ejpam-6076	230	12	be	be	AUX
ejpam-6076	230	13	the	the	DET
ejpam-6076	230	14	corona	corona	NOUN
ejpam-6076	230	15	of	of	ADP
ejpam-6076	230	16	some	some	DET
ejpam-6076	230	17	graph	graph	NOUN
ejpam-6076	230	18	.	.	PUNCT
ejpam-6076	231	1	then	then	ADV
ejpam-6076	231	2	g	g	PROPN
ejpam-6076	231	3	=	=	PROPN
ejpam-6076	231	4	h	h	PROPN
ejpam-6076	231	5	◦	◦	NOUN
ejpam-6076	231	6	k1	k1	NOUN
ejpam-6076	231	7	for	for	ADP
ejpam-6076	231	8	some	some	DET
ejpam-6076	231	9	graph	graph	NOUN
ejpam-6076	231	10	h.	h.	PROPN
ejpam-6076	231	11	by	by	ADP
ejpam-6076	231	12	theorem	theorem	NOUN
ejpam-6076	231	13	5	5	NUM
ejpam-6076	231	14	,	,	PUNCT
ejpam-6076	231	15	|d|	|d|	PROPN
ejpam-6076	231	16	=	=	SYM
ejpam-6076	231	17	γcerp(g	γcerp(g	PROPN
ejpam-6076	231	18	)	)	PUNCT
ejpam-6076	231	19	=	=	SYM
ejpam-6076	231	20	γcerp(h	γcerp(h	PROPN
ejpam-6076	231	21	◦	◦	NOUN
ejpam-6076	231	22	k1	k1	PROPN
ejpam-6076	231	23	)	)	PUNCT
ejpam-6076	231	24	=	=	SYM
ejpam-6076	232	1	2	2	NUM
ejpam-6076	232	2	m	m	NOUN
ejpam-6076	232	3	=	=	NOUN
ejpam-6076	232	4	|v	|v	X
ejpam-6076	232	5	(	(	PUNCT
ejpam-6076	232	6	g)|	g)|	PROPN
ejpam-6076	232	7	=	=	PROPN
ejpam-6076	232	8	n.	n.	PROPN
ejpam-6076	232	9	therefore	therefore	ADV
ejpam-6076	232	10	,	,	PUNCT
ejpam-6076	232	11	γcerp(g	γcerp(g	PROPN
ejpam-6076	232	12	)	)	PUNCT
ejpam-6076	232	13	=	=	SYM
ejpam-6076	232	14	n.	n.	NOUN
ejpam-6076	232	15	remark	remark	NOUN
ejpam-6076	232	16	2	2	NUM
ejpam-6076	232	17	.	.	PUNCT
ejpam-6076	233	1	let	let	VERB
ejpam-6076	233	2	g	g	PRON
ejpam-6076	233	3	be	be	AUX
ejpam-6076	233	4	a	a	DET
ejpam-6076	233	5	graph	graph	NOUN
ejpam-6076	233	6	with	with	ADP
ejpam-6076	233	7	|v	|v	PROPN
ejpam-6076	233	8	(	(	PUNCT
ejpam-6076	233	9	g)|	g)|	NOUN
ejpam-6076	233	10	=	=	SYM
ejpam-6076	233	11	n.	n.	NOUN
ejpam-6076	233	12	if	if	SCONJ
ejpam-6076	233	13	g	g	PROPN
ejpam-6076	233	14	is	be	AUX
ejpam-6076	233	15	either	either	CCONJ
ejpam-6076	233	16	the	the	DET
ejpam-6076	233	17	corona	corona	NOUN
ejpam-6076	233	18	of	of	ADP
ejpam-6076	233	19	a	a	DET
ejpam-6076	233	20	graph	graph	NOUN
ejpam-6076	233	21	,	,	PUNCT
ejpam-6076	233	22	the	the	DET
ejpam-6076	233	23	complement	complement	NOUN
ejpam-6076	233	24	of	of	ADP
ejpam-6076	233	25	a	a	DET
ejpam-6076	233	26	complete	complete	ADJ
ejpam-6076	233	27	graph	graph	NOUN
ejpam-6076	233	28	,	,	PUNCT
ejpam-6076	233	29	or	or	CCONJ
ejpam-6076	233	30	the	the	DET
ejpam-6076	233	31	union	union	NOUN
ejpam-6076	233	32	of	of	ADP
ejpam-6076	233	33	these	these	DET
ejpam-6076	233	34	two	two	NUM
ejpam-6076	233	35	types	type	NOUN
ejpam-6076	233	36	,	,	PUNCT
ejpam-6076	233	37	then	then	ADV
ejpam-6076	233	38	γcerp(g	γcerp(g	PROPN
ejpam-6076	233	39	)	)	PUNCT
ejpam-6076	233	40	=	=	SYM
ejpam-6076	233	41	n.	n.	NOUN
ejpam-6076	233	42	remark	remark	NOUN
ejpam-6076	233	43	3	3	NUM
ejpam-6076	233	44	.	.	PUNCT
ejpam-6076	234	1	it	it	PRON
ejpam-6076	234	2	is	be	AUX
ejpam-6076	234	3	important	important	ADJ
ejpam-6076	234	4	to	to	PART
ejpam-6076	234	5	observe	observe	VERB
ejpam-6076	234	6	that	that	SCONJ
ejpam-6076	234	7	the	the	DET
ejpam-6076	234	8	result	result	NOUN
ejpam-6076	234	9	above	above	ADV
ejpam-6076	234	10	indicates	indicate	VERB
ejpam-6076	234	11	the	the	DET
ejpam-6076	234	12	sharpness	sharpness	NOUN
ejpam-6076	234	13	of	of	ADP
ejpam-6076	234	14	the	the	DET
ejpam-6076	234	15	upper	upper	ADJ
ejpam-6076	234	16	bound	bind	VERB
ejpam-6076	234	17	in	in	ADP
ejpam-6076	234	18	the	the	DET
ejpam-6076	234	19	inequality	inequality	NOUN
ejpam-6076	234	20	γcerp(g	γcerp(g	NOUN
ejpam-6076	234	21	)	)	PUNCT
ejpam-6076	234	22	≤	≤	NOUN
ejpam-6076	234	23	2γ(g	2γ(g	NUM
ejpam-6076	234	24	)	)	PUNCT
ejpam-6076	234	25	,	,	PUNCT
ejpam-6076	234	26	because	because	SCONJ
ejpam-6076	234	27	for	for	ADP
ejpam-6076	234	28	the	the	DET
ejpam-6076	234	29	corona	corona	NOUN
ejpam-6076	234	30	g	g	NOUN
ejpam-6076	234	31	of	of	ADP
ejpam-6076	234	32	any	any	DET
ejpam-6076	234	33	graph	graph	NOUN
ejpam-6076	234	34	that	that	PRON
ejpam-6076	234	35	does	do	AUX
ejpam-6076	234	36	not	not	PART
ejpam-6076	234	37	contain	contain	VERB
ejpam-6076	234	38	an	an	DET
ejpam-6076	234	39	isolated	isolated	ADJ
ejpam-6076	234	40	vertex	vertex	NOUN
ejpam-6076	234	41	,	,	PUNCT
ejpam-6076	234	42	we	we	PRON
ejpam-6076	234	43	have	have	AUX
ejpam-6076	234	44	γcerp(g	γcerp(g	VERB
ejpam-6076	234	45	)	)	PUNCT
ejpam-6076	234	46	=	=	SYM
ejpam-6076	234	47	2γ(g	2γ(g	NUM
ejpam-6076	234	48	)	)	PUNCT
ejpam-6076	234	49	.	.	PUNCT
ejpam-6076	235	1	4	4	X
ejpam-6076	235	2	.	.	X
ejpam-6076	235	3	graphs	graph	NOUN
ejpam-6076	235	4	with	with	ADP
ejpam-6076	235	5	γcerp(g	γcerp(g	PROPN
ejpam-6076	235	6	)	)	PUNCT
ejpam-6076	235	7	=	=	SYM
ejpam-6076	235	8	γcer(g	γcer(g	NOUN
ejpam-6076	235	9	)	)	PUNCT
ejpam-6076	235	10	we	we	PRON
ejpam-6076	235	11	proceed	proceed	VERB
ejpam-6076	235	12	with	with	ADP
ejpam-6076	235	13	our	our	PRON
ejpam-6076	235	14	investigation	investigation	NOUN
ejpam-6076	235	15	of	of	ADP
ejpam-6076	235	16	the	the	DET
ejpam-6076	235	17	certified	certify	VERB
ejpam-6076	235	18	perfect	perfect	ADJ
ejpam-6076	235	19	domination	domination	NOUN
ejpam-6076	235	20	number	number	NOUN
ejpam-6076	235	21	by	by	ADP
ejpam-6076	235	22	examining	examine	VERB
ejpam-6076	235	23	the	the	DET
ejpam-6076	235	24	class	class	NOUN
ejpam-6076	235	25	of	of	ADP
ejpam-6076	235	26	graphs	graph	NOUN
ejpam-6076	235	27	where	where	SCONJ
ejpam-6076	235	28	γcerp(g	γcerp(g	NOUN
ejpam-6076	235	29	)	)	PUNCT
ejpam-6076	235	30	=	=	SYM
ejpam-6076	235	31	γcer(g	γcer(g	PROPN
ejpam-6076	235	32	)	)	PUNCT
ejpam-6076	235	33	.	.	PUNCT
ejpam-6076	236	1	j.	j.	PROPN
ejpam-6076	236	2	j.	j.	PROPN
ejpam-6076	236	3	hamja	hamja	PROPN
ejpam-6076	236	4	et	et	PROPN
ejpam-6076	236	5	al	al	PROPN
ejpam-6076	236	6	.	.	PUNCT
ejpam-6076	236	7	/	/	SYM
ejpam-6076	236	8	eur	eur	PROPN
ejpam-6076	236	9	.	.	PUNCT
ejpam-6076	237	1	j.	j.	PROPN
ejpam-6076	237	2	pure	pure	PROPN
ejpam-6076	237	3	appl	appl	PROPN
ejpam-6076	237	4	.	.	PROPN
ejpam-6076	237	5	math	math	PROPN
ejpam-6076	237	6	,	,	PUNCT
ejpam-6076	237	7	18	18	NUM
ejpam-6076	237	8	(	(	PUNCT
ejpam-6076	237	9	3	3	NUM
ejpam-6076	237	10	)	)	PUNCT
ejpam-6076	237	11	(	(	PUNCT
ejpam-6076	237	12	2025	2025	NUM
ejpam-6076	237	13	)	)	PUNCT
ejpam-6076	237	14	,	,	PUNCT
ejpam-6076	237	15	6076	6076	NUM
ejpam-6076	237	16	9	9	NUM
ejpam-6076	237	17	of	of	ADP
ejpam-6076	237	18	13	13	NUM
ejpam-6076	237	19	theorem	theorem	NOUN
ejpam-6076	237	20	6	6	NUM
ejpam-6076	237	21	.	.	PUNCT
ejpam-6076	238	1	let	let	VERB
ejpam-6076	238	2	g	g	PRON
ejpam-6076	238	3	be	be	AUX
ejpam-6076	238	4	a	a	DET
ejpam-6076	238	5	connected	connected	ADJ
ejpam-6076	238	6	graph	graph	NOUN
ejpam-6076	238	7	with	with	ADP
ejpam-6076	238	8	|v	|v	PROPN
ejpam-6076	238	9	(	(	PUNCT
ejpam-6076	238	10	g)|	g)|	X
ejpam-6076	238	11	≥	≥	NOUN
ejpam-6076	238	12	3	3	NUM
ejpam-6076	238	13	.	.	PUNCT
ejpam-6076	239	1	then	then	ADV
ejpam-6076	239	2	γcer(g	γcer(g	NUM
ejpam-6076	239	3	)	)	PUNCT
ejpam-6076	240	1	=	=	SYM
ejpam-6076	240	2	γcerp(g	γcerp(g	PROPN
ejpam-6076	240	3	)	)	PUNCT
ejpam-6076	240	4	if	if	SCONJ
ejpam-6076	240	5	and	and	CCONJ
ejpam-6076	240	6	only	only	ADV
ejpam-6076	240	7	if	if	SCONJ
ejpam-6076	240	8	there	there	PRON
ejpam-6076	240	9	exists	exist	VERB
ejpam-6076	240	10	a	a	DET
ejpam-6076	240	11	γcer	γcer	NOUN
ejpam-6076	240	12	-	-	PUNCT
ejpam-6076	240	13	set	set	VERB
ejpam-6076	240	14	j	j	NOUN
ejpam-6076	240	15	of	of	ADP
ejpam-6076	240	16	g	g	PROPN
ejpam-6076	240	17	such	such	ADJ
ejpam-6076	240	18	that	that	SCONJ
ejpam-6076	240	19	each	each	DET
ejpam-6076	240	20	vertex	vertex	NOUN
ejpam-6076	240	21	u	u	NOUN
ejpam-6076	240	22	∈	∈	PROPN
ejpam-6076	240	23	v	v	ADP
ejpam-6076	240	24	(	(	PUNCT
ejpam-6076	240	25	g	g	NOUN
ejpam-6076	240	26	)	)	PUNCT
ejpam-6076	240	27	\	\	PROPN
ejpam-6076	241	1	j	j	PROPN
ejpam-6076	241	2	is	be	AUX
ejpam-6076	241	3	dominated	dominate	VERB
ejpam-6076	241	4	by	by	ADP
ejpam-6076	241	5	exactly	exactly	ADV
ejpam-6076	241	6	one	one	NUM
ejpam-6076	241	7	vertex	vertex	NOUN
ejpam-6076	241	8	v	v	ADP
ejpam-6076	241	9	∈	∈	PROPN
ejpam-6076	241	10	j	j	PROPN
ejpam-6076	241	11	.	.	PUNCT
ejpam-6076	242	1	proof	proof	NOUN
ejpam-6076	242	2	.	.	PUNCT
ejpam-6076	243	1	assume	assume	VERB
ejpam-6076	243	2	that	that	SCONJ
ejpam-6076	243	3	γcer(g	γcer(g	NOUN
ejpam-6076	243	4	)	)	PUNCT
ejpam-6076	243	5	=	=	SYM
ejpam-6076	243	6	γcerp(g	γcerp(g	PROPN
ejpam-6076	243	7	)	)	PUNCT
ejpam-6076	243	8	.	.	PUNCT
ejpam-6076	244	1	let	let	VERB
ejpam-6076	244	2	j	j	PROPN
ejpam-6076	244	3	be	be	AUX
ejpam-6076	244	4	a	a	DET
ejpam-6076	244	5	γcerp	γcerp	NOUN
ejpam-6076	244	6	-	-	PUNCT
ejpam-6076	244	7	set	set	NOUN
ejpam-6076	244	8	of	of	ADP
ejpam-6076	244	9	g.	g.	PROPN
ejpam-6076	244	10	since	since	SCONJ
ejpam-6076	244	11	j	j	PROPN
ejpam-6076	244	12	is	be	AUX
ejpam-6076	244	13	a	a	DET
ejpam-6076	244	14	γcerp	γcerp	NOUN
ejpam-6076	244	15	-	-	PUNCT
ejpam-6076	244	16	set	set	VERB
ejpam-6076	244	17	and	and	CCONJ
ejpam-6076	244	18	γ(g	γ(g	PROPN
ejpam-6076	244	19	)	)	PUNCT
ejpam-6076	245	1	=	=	SYM
ejpam-6076	245	2	γcer(g	γcer(g	PROPN
ejpam-6076	245	3	)	)	PUNCT
ejpam-6076	245	4	,	,	PUNCT
ejpam-6076	245	5	it	it	PRON
ejpam-6076	245	6	follows	follow	VERB
ejpam-6076	245	7	that	that	SCONJ
ejpam-6076	245	8	j	j	PROPN
ejpam-6076	245	9	is	be	AUX
ejpam-6076	245	10	also	also	ADV
ejpam-6076	245	11	a	a	DET
ejpam-6076	245	12	γcer	γcer	NOUN
ejpam-6076	245	13	-	-	PUNCT
ejpam-6076	245	14	set	set	NOUN
ejpam-6076	245	15	of	of	ADP
ejpam-6076	245	16	g.	g.	PROPN
ejpam-6076	245	17	now	now	ADV
ejpam-6076	245	18	,	,	PUNCT
ejpam-6076	245	19	for	for	ADP
ejpam-6076	245	20	the	the	DET
ejpam-6076	245	21	sake	sake	NOUN
ejpam-6076	245	22	of	of	ADP
ejpam-6076	245	23	contradiction	contradiction	NOUN
ejpam-6076	245	24	,	,	PUNCT
ejpam-6076	245	25	assume	assume	VERB
ejpam-6076	245	26	that	that	SCONJ
ejpam-6076	245	27	there	there	PRON
ejpam-6076	245	28	exists	exist	VERB
ejpam-6076	245	29	a	a	DET
ejpam-6076	245	30	vertex	vertex	NOUN
ejpam-6076	245	31	u	u	NOUN
ejpam-6076	245	32	∈	∈	PROPN
ejpam-6076	245	33	v	v	ADP
ejpam-6076	245	34	(	(	PUNCT
ejpam-6076	245	35	g	g	NOUN
ejpam-6076	245	36	)	)	PUNCT
ejpam-6076	245	37	\	\	PROPN
ejpam-6076	245	38	j	j	PROPN
ejpam-6076	245	39	that	that	PRON
ejpam-6076	245	40	is	be	AUX
ejpam-6076	245	41	dominated	dominate	VERB
ejpam-6076	245	42	by	by	ADP
ejpam-6076	245	43	at	at	ADV
ejpam-6076	245	44	least	least	ADV
ejpam-6076	245	45	two	two	NUM
ejpam-6076	245	46	distinct	distinct	ADJ
ejpam-6076	245	47	vertices	vertex	NOUN
ejpam-6076	245	48	v1	v1	NOUN
ejpam-6076	245	49	,	,	PUNCT
ejpam-6076	245	50	v2	v2	PROPN
ejpam-6076	245	51	∈	∈	PROPN
ejpam-6076	245	52	j	j	PROPN
ejpam-6076	245	53	.	.	PUNCT
ejpam-6076	246	1	in	in	ADP
ejpam-6076	246	2	this	this	DET
ejpam-6076	246	3	case	case	NOUN
ejpam-6076	246	4	,	,	PUNCT
ejpam-6076	246	5	j	j	PROPN
ejpam-6076	246	6	would	would	AUX
ejpam-6076	246	7	not	not	PART
ejpam-6076	246	8	be	be	AUX
ejpam-6076	246	9	a	a	DET
ejpam-6076	246	10	certified	certify	VERB
ejpam-6076	246	11	perfect	perfect	ADJ
ejpam-6076	246	12	dominating	dominating	NOUN
ejpam-6076	246	13	set	set	NOUN
ejpam-6076	246	14	,	,	PUNCT
ejpam-6076	246	15	which	which	PRON
ejpam-6076	246	16	contradicts	contradict	VERB
ejpam-6076	246	17	our	our	PRON
ejpam-6076	246	18	assumption	assumption	NOUN
ejpam-6076	246	19	that	that	SCONJ
ejpam-6076	246	20	j	j	PROPN
ejpam-6076	246	21	is	be	AUX
ejpam-6076	246	22	a	a	DET
ejpam-6076	246	23	γcerp	γcerp	NOUN
ejpam-6076	246	24	-	-	PUNCT
ejpam-6076	246	25	set	set	NOUN
ejpam-6076	246	26	.	.	PUNCT
ejpam-6076	247	1	therefore	therefore	ADV
ejpam-6076	247	2	,	,	PUNCT
ejpam-6076	247	3	every	every	DET
ejpam-6076	247	4	vertex	vertex	NOUN
ejpam-6076	247	5	u	u	NOUN
ejpam-6076	247	6	∈	∈	PROPN
ejpam-6076	247	7	v	v	ADP
ejpam-6076	247	8	(	(	PUNCT
ejpam-6076	247	9	g	g	NOUN
ejpam-6076	247	10	)	)	PUNCT
ejpam-6076	247	11	\	\	PROPN
ejpam-6076	247	12	j	j	PROPN
ejpam-6076	247	13	is	be	AUX
ejpam-6076	247	14	dominated	dominate	VERB
ejpam-6076	247	15	by	by	ADP
ejpam-6076	247	16	exactly	exactly	ADV
ejpam-6076	247	17	one	one	NUM
ejpam-6076	247	18	vertex	vertex	NOUN
ejpam-6076	247	19	in	in	ADP
ejpam-6076	247	20	j	j	PROPN
ejpam-6076	247	21	.	.	PUNCT
ejpam-6076	248	1	conversely	conversely	ADV
ejpam-6076	248	2	,	,	PUNCT
ejpam-6076	248	3	suppose	suppose	VERB
ejpam-6076	248	4	that	that	SCONJ
ejpam-6076	248	5	j	j	PROPN
ejpam-6076	248	6	is	be	AUX
ejpam-6076	248	7	a	a	DET
ejpam-6076	248	8	γcer	γcer	NOUN
ejpam-6076	248	9	-	-	PUNCT
ejpam-6076	248	10	set	set	NOUN
ejpam-6076	248	11	of	of	ADP
ejpam-6076	248	12	g	g	NOUN
ejpam-6076	248	13	such	such	ADJ
ejpam-6076	248	14	that	that	SCONJ
ejpam-6076	248	15	each	each	DET
ejpam-6076	248	16	vertex	vertex	NOUN
ejpam-6076	248	17	u	u	NOUN
ejpam-6076	248	18	∈	∈	PROPN
ejpam-6076	248	19	v	v	ADP
ejpam-6076	248	20	(	(	PUNCT
ejpam-6076	248	21	g	g	NOUN
ejpam-6076	248	22	)	)	PUNCT
ejpam-6076	248	23	\	\	PROPN
ejpam-6076	249	1	j	j	PROPN
ejpam-6076	249	2	is	be	AUX
ejpam-6076	249	3	dominated	dominate	VERB
ejpam-6076	249	4	by	by	ADP
ejpam-6076	249	5	exactly	exactly	ADV
ejpam-6076	249	6	one	one	NUM
ejpam-6076	249	7	vertex	vertex	NOUN
ejpam-6076	249	8	in	in	ADP
ejpam-6076	249	9	j	j	PROPN
ejpam-6076	249	10	.	.	PUNCT
ejpam-6076	250	1	for	for	ADP
ejpam-6076	250	2	contradiction	contradiction	NOUN
ejpam-6076	250	3	,	,	PUNCT
ejpam-6076	250	4	assume	assume	VERB
ejpam-6076	250	5	that	that	SCONJ
ejpam-6076	250	6	there	there	PRON
ejpam-6076	250	7	exists	exist	VERB
ejpam-6076	250	8	a	a	DET
ejpam-6076	250	9	vertex	vertex	NOUN
ejpam-6076	250	10	v	v	ADP
ejpam-6076	250	11	∈	∈	PROPN
ejpam-6076	250	12	j	j	NOUN
ejpam-6076	250	13	such	such	ADJ
ejpam-6076	250	14	that	that	DET
ejpam-6076	250	15	|ng(v	|ng(v	NOUN
ejpam-6076	250	16	)	)	PUNCT
ejpam-6076	250	17	∩	∩	NOUN
ejpam-6076	250	18	(	(	PUNCT
ejpam-6076	250	19	v	v	NOUN
ejpam-6076	250	20	(	(	PUNCT
ejpam-6076	250	21	g	g	NOUN
ejpam-6076	250	22	)	)	PUNCT
ejpam-6076	250	23	\	\	NOUN
ejpam-6076	250	24	j)|	j)|	NOUN
ejpam-6076	250	25	=	=	SYM
ejpam-6076	250	26	1	1	X
ejpam-6076	250	27	.	.	PUNCT
ejpam-6076	251	1	this	this	PRON
ejpam-6076	251	2	implies	imply	VERB
ejpam-6076	251	3	that	that	SCONJ
ejpam-6076	251	4	the	the	DET
ejpam-6076	251	5	unique	unique	ADJ
ejpam-6076	251	6	neighbor	neighbor	NOUN
ejpam-6076	251	7	of	of	ADP
ejpam-6076	251	8	v	v	NOUN
ejpam-6076	251	9	in	in	ADP
ejpam-6076	251	10	v	v	NOUN
ejpam-6076	251	11	(	(	PUNCT
ejpam-6076	251	12	g)\j	g)\j	PROPN
ejpam-6076	251	13	contradicts	contradict	VERB
ejpam-6076	251	14	our	our	PRON
ejpam-6076	251	15	assumption	assumption	NOUN
ejpam-6076	251	16	that	that	SCONJ
ejpam-6076	251	17	every	every	DET
ejpam-6076	251	18	vertex	vertex	NOUN
ejpam-6076	251	19	in	in	ADP
ejpam-6076	251	20	v	v	NOUN
ejpam-6076	251	21	(	(	PUNCT
ejpam-6076	251	22	g)\j	g)\j	PROPN
ejpam-6076	251	23	is	be	AUX
ejpam-6076	251	24	dominated	dominate	VERB
ejpam-6076	251	25	by	by	ADP
ejpam-6076	251	26	exactly	exactly	ADV
ejpam-6076	251	27	one	one	NUM
ejpam-6076	251	28	vertex	vertex	NOUN
ejpam-6076	251	29	in	in	ADP
ejpam-6076	251	30	j	j	PROPN
ejpam-6076	251	31	.	.	PUNCT
ejpam-6076	252	1	hence	hence	ADV
ejpam-6076	252	2	,	,	PUNCT
ejpam-6076	252	3	we	we	PRON
ejpam-6076	252	4	conclude	conclude	VERB
ejpam-6076	252	5	that	that	SCONJ
ejpam-6076	252	6	|ng(v	|ng(v	ADP
ejpam-6076	252	7	)	)	PUNCT
ejpam-6076	252	8	∩	∩	NOUN
ejpam-6076	252	9	(	(	PUNCT
ejpam-6076	252	10	v	v	NOUN
ejpam-6076	252	11	(	(	PUNCT
ejpam-6076	252	12	g	g	NOUN
ejpam-6076	252	13	)	)	PUNCT
ejpam-6076	252	14	\	\	PROPN
ejpam-6076	252	15	j)|	j)|	NOUN
ejpam-6076	252	16	≥	≥	NOUN
ejpam-6076	252	17	2	2	NUM
ejpam-6076	252	18	for	for	ADP
ejpam-6076	252	19	all	all	DET
ejpam-6076	252	20	v	v	ADP
ejpam-6076	252	21	∈	∈	PROPN
ejpam-6076	252	22	j	j	NOUN
ejpam-6076	252	23	,	,	PUNCT
ejpam-6076	252	24	meaning	mean	VERB
ejpam-6076	252	25	that	that	SCONJ
ejpam-6076	252	26	j	j	PROPN
ejpam-6076	252	27	is	be	AUX
ejpam-6076	252	28	a	a	DET
ejpam-6076	252	29	certified	certify	VERB
ejpam-6076	252	30	perfect	perfect	ADJ
ejpam-6076	252	31	dominating	dominating	NOUN
ejpam-6076	252	32	set	set	NOUN
ejpam-6076	252	33	.	.	PUNCT
ejpam-6076	253	1	therefore	therefore	ADV
ejpam-6076	253	2	,	,	PUNCT
ejpam-6076	253	3	γcerp(g	γcerp(g	PROPN
ejpam-6076	253	4	)	)	PUNCT
ejpam-6076	253	5	≤	≤	NOUN
ejpam-6076	253	6	|j	|j	PUNCT
ejpam-6076	254	1	|	|	NOUN
ejpam-6076	254	2	=	=	SYM
ejpam-6076	254	3	γcer(g	γcer(g	NOUN
ejpam-6076	254	4	)	)	PUNCT
ejpam-6076	254	5	.	.	PUNCT
ejpam-6076	255	1	by	by	ADP
ejpam-6076	255	2	remark	remark	NOUN
ejpam-6076	255	3	1	1	NUM
ejpam-6076	255	4	,	,	PUNCT
ejpam-6076	255	5	we	we	PRON
ejpam-6076	255	6	also	also	ADV
ejpam-6076	255	7	have	have	VERB
ejpam-6076	255	8	γcer(g	γcer(g	NUM
ejpam-6076	255	9	)	)	PUNCT
ejpam-6076	255	10	≤	≤	NOUN
ejpam-6076	255	11	γcerp(g	γcerp(g	PROPN
ejpam-6076	255	12	)	)	PUNCT
ejpam-6076	255	13	.	.	PUNCT
ejpam-6076	256	1	thus	thus	ADV
ejpam-6076	256	2	,	,	PUNCT
ejpam-6076	256	3	γcer(g	γcer(g	NOUN
ejpam-6076	256	4	)	)	PUNCT
ejpam-6076	256	5	=	=	SYM
ejpam-6076	256	6	γcerp(g	γcerp(g	PROPN
ejpam-6076	256	7	)	)	PUNCT
ejpam-6076	256	8	.	.	PUNCT
ejpam-6076	257	1	corollary	corollary	ADJ
ejpam-6076	257	2	3	3	X
ejpam-6076	257	3	.	.	PUNCT
ejpam-6076	258	1	let	let	VERB
ejpam-6076	258	2	g	g	PRON
ejpam-6076	258	3	be	be	AUX
ejpam-6076	258	4	a	a	DET
ejpam-6076	258	5	connected	connected	ADJ
ejpam-6076	258	6	graph	graph	NOUN
ejpam-6076	258	7	with	with	ADP
ejpam-6076	258	8	|v	|v	PROPN
ejpam-6076	258	9	(	(	PUNCT
ejpam-6076	258	10	g)|	g)|	X
ejpam-6076	258	11	≥	≥	NOUN
ejpam-6076	258	12	3	3	NUM
ejpam-6076	258	13	.	.	PUNCT
ejpam-6076	259	1	if	if	SCONJ
ejpam-6076	259	2	g	g	PROPN
ejpam-6076	259	3	has	have	VERB
ejpam-6076	259	4	a	a	DET
ejpam-6076	259	5	γip	γip	ADV
ejpam-6076	259	6	-	-	PUNCT
ejpam-6076	259	7	set	set	NOUN
ejpam-6076	259	8	that	that	PRON
ejpam-6076	259	9	does	do	AUX
ejpam-6076	259	10	not	not	PART
ejpam-6076	259	11	include	include	VERB
ejpam-6076	259	12	any	any	DET
ejpam-6076	259	13	leaf	leaf	NOUN
ejpam-6076	259	14	of	of	ADP
ejpam-6076	259	15	g	g	NOUN
ejpam-6076	259	16	,	,	PUNCT
ejpam-6076	259	17	then	then	ADV
ejpam-6076	259	18	γcer(g	γcer(g	NUM
ejpam-6076	259	19	)	)	PUNCT
ejpam-6076	260	1	=	=	SYM
ejpam-6076	260	2	γcerp(g	γcerp(g	PROPN
ejpam-6076	260	3	)	)	PUNCT
ejpam-6076	260	4	.	.	PUNCT
ejpam-6076	261	1	proof	proof	NOUN
ejpam-6076	261	2	.	.	PUNCT
ejpam-6076	262	1	let	let	VERB
ejpam-6076	262	2	j	j	PROPN
ejpam-6076	262	3	be	be	AUX
ejpam-6076	262	4	a	a	DET
ejpam-6076	262	5	γip	γip	NOUN
ejpam-6076	262	6	-	-	PUNCT
ejpam-6076	262	7	set	set	NOUN
ejpam-6076	262	8	of	of	ADP
ejpam-6076	262	9	g	g	NOUN
ejpam-6076	262	10	that	that	PRON
ejpam-6076	262	11	does	do	AUX
ejpam-6076	262	12	not	not	PART
ejpam-6076	262	13	contain	contain	VERB
ejpam-6076	262	14	any	any	DET
ejpam-6076	262	15	leaf	leaf	NOUN
ejpam-6076	262	16	of	of	ADP
ejpam-6076	262	17	g.	g.	PROPN
ejpam-6076	262	18	since	since	SCONJ
ejpam-6076	262	19	j	j	PROPN
ejpam-6076	262	20	is	be	AUX
ejpam-6076	262	21	independent	independent	ADJ
ejpam-6076	262	22	,	,	PUNCT
ejpam-6076	262	23	every	every	DET
ejpam-6076	262	24	vertex	vertex	NOUN
ejpam-6076	262	25	v	v	ADP
ejpam-6076	262	26	∈	∈	PROPN
ejpam-6076	262	27	j	j	NOUN
ejpam-6076	262	28	has	have	VERB
ejpam-6076	262	29	all	all	PRON
ejpam-6076	262	30	of	of	ADP
ejpam-6076	262	31	its	its	PRON
ejpam-6076	262	32	neighbors	neighbor	NOUN
ejpam-6076	262	33	in	in	ADP
ejpam-6076	262	34	v	v	NOUN
ejpam-6076	262	35	(	(	PUNCT
ejpam-6076	262	36	g	g	NOUN
ejpam-6076	262	37	)	)	PUNCT
ejpam-6076	262	38	\	\	PROPN
ejpam-6076	262	39	j	j	PROPN
ejpam-6076	262	40	.	.	PUNCT
ejpam-6076	263	1	this	this	PRON
ejpam-6076	263	2	implies	imply	VERB
ejpam-6076	263	3	that	that	SCONJ
ejpam-6076	263	4	ng(v	ng(v	NOUN
ejpam-6076	263	5	)	)	PUNCT
ejpam-6076	263	6	⊆	⊆	NUM
ejpam-6076	263	7	v	v	NOUN
ejpam-6076	263	8	(	(	PUNCT
ejpam-6076	263	9	g	g	NOUN
ejpam-6076	263	10	)	)	PUNCT
ejpam-6076	263	11	\	\	PROPN
ejpam-6076	263	12	j	j	PROPN
ejpam-6076	263	13	.	.	PUNCT
ejpam-6076	264	1	furthermore	furthermore	ADV
ejpam-6076	264	2	,	,	PUNCT
ejpam-6076	264	3	since	since	SCONJ
ejpam-6076	264	4	v	v	NOUN
ejpam-6076	264	5	is	be	AUX
ejpam-6076	264	6	neither	neither	CCONJ
ejpam-6076	264	7	a	a	DET
ejpam-6076	264	8	leaf	leaf	NOUN
ejpam-6076	264	9	nor	nor	CCONJ
ejpam-6076	264	10	an	an	DET
ejpam-6076	264	11	isolated	isolated	ADJ
ejpam-6076	264	12	vertex	vertex	NOUN
ejpam-6076	264	13	,	,	PUNCT
ejpam-6076	264	14	we	we	PRON
ejpam-6076	264	15	have	have	VERB
ejpam-6076	264	16	|ng(v)|	|ng(v)|	NOUN
ejpam-6076	264	17	≥	≥	NOUN
ejpam-6076	264	18	2	2	NUM
ejpam-6076	264	19	.	.	PUNCT
ejpam-6076	265	1	thus	thus	ADV
ejpam-6076	265	2	,	,	PUNCT
ejpam-6076	265	3	|ng(v)∩	|ng(v)∩	PROPN
ejpam-6076	265	4	(	(	PUNCT
ejpam-6076	265	5	v	v	NOUN
ejpam-6076	265	6	(	(	PUNCT
ejpam-6076	265	7	g	g	NOUN
ejpam-6076	265	8	)	)	PUNCT
ejpam-6076	265	9	\	\	NOUN
ejpam-6076	265	10	j)|	j)|	NOUN
ejpam-6076	265	11	=	=	SYM
ejpam-6076	265	12	|ng(v)|	|ng(v)|	NOUN
ejpam-6076	265	13	≥	≥	NOUN
ejpam-6076	265	14	2	2	NUM
ejpam-6076	265	15	.	.	PUNCT
ejpam-6076	266	1	therefore	therefore	ADV
ejpam-6076	266	2	,	,	PUNCT
ejpam-6076	266	3	j	j	PROPN
ejpam-6076	266	4	is	be	AUX
ejpam-6076	266	5	a	a	DET
ejpam-6076	266	6	certified	certify	VERB
ejpam-6076	266	7	perfect	perfect	ADJ
ejpam-6076	266	8	dominating	dominating	NOUN
ejpam-6076	266	9	set	set	NOUN
ejpam-6076	266	10	.	.	PUNCT
ejpam-6076	267	1	by	by	ADP
ejpam-6076	267	2	theorem	theorem	NOUN
ejpam-6076	267	3	6	6	NUM
ejpam-6076	267	4	,	,	PUNCT
ejpam-6076	267	5	we	we	PRON
ejpam-6076	267	6	conclude	conclude	VERB
ejpam-6076	267	7	that	that	PRON
ejpam-6076	267	8	γcer(g	γcer(g	NOUN
ejpam-6076	267	9	)	)	PUNCT
ejpam-6076	267	10	=	=	SYM
ejpam-6076	267	11	γcerp(g	γcerp(g	PROPN
ejpam-6076	267	12	)	)	PUNCT
ejpam-6076	267	13	.	.	PUNCT
ejpam-6076	268	1	the	the	DET
ejpam-6076	268	2	following	following	ADJ
ejpam-6076	268	3	result	result	NOUN
ejpam-6076	268	4	immediately	immediately	ADV
ejpam-6076	268	5	follows	follow	VERB
ejpam-6076	268	6	from	from	ADP
ejpam-6076	268	7	theorem	theorem	ADJ
ejpam-6076	268	8	6	6	NUM
ejpam-6076	268	9	and	and	CCONJ
ejpam-6076	268	10	corollary	corollary	ADJ
ejpam-6076	268	11	3	3	NUM
ejpam-6076	268	12	.	.	PUNCT
ejpam-6076	268	13	corollary	corollary	ADJ
ejpam-6076	268	14	4	4	NUM
ejpam-6076	268	15	.	.	PUNCT
ejpam-6076	269	1	let	let	VERB
ejpam-6076	269	2	g	g	PRON
ejpam-6076	269	3	be	be	AUX
ejpam-6076	269	4	a	a	DET
ejpam-6076	269	5	graph	graph	NOUN
ejpam-6076	269	6	where	where	SCONJ
ejpam-6076	269	7	δ(g	δ(g	PROPN
ejpam-6076	269	8	)	)	PUNCT
ejpam-6076	269	9	≥	≥	NOUN
ejpam-6076	270	1	2	2	NUM
ejpam-6076	270	2	.	.	PUNCT
ejpam-6076	271	1	if	if	SCONJ
ejpam-6076	271	2	g	g	PROPN
ejpam-6076	271	3	contains	contain	VERB
ejpam-6076	271	4	a	a	DET
ejpam-6076	271	5	γp	γp	NOUN
ejpam-6076	271	6	-	-	PUNCT
ejpam-6076	271	7	set	set	VERB
ejpam-6076	271	8	j	j	NOUN
ejpam-6076	271	9	such	such	ADJ
ejpam-6076	271	10	that	that	SCONJ
ejpam-6076	271	11	each	each	DET
ejpam-6076	271	12	vertex	vertex	NOUN
ejpam-6076	271	13	in	in	ADP
ejpam-6076	271	14	j	j	PROPN
ejpam-6076	271	15	has	have	VERB
ejpam-6076	271	16	at	at	ADV
ejpam-6076	271	17	least	least	ADV
ejpam-6076	271	18	two	two	NUM
ejpam-6076	271	19	neighbors	neighbor	NOUN
ejpam-6076	271	20	in	in	ADP
ejpam-6076	271	21	v	v	NOUN
ejpam-6076	271	22	(	(	PUNCT
ejpam-6076	271	23	g	g	NOUN
ejpam-6076	271	24	)	)	PUNCT
ejpam-6076	271	25	\	\	PROPN
ejpam-6076	272	1	j	j	PROPN
ejpam-6076	272	2	,	,	PUNCT
ejpam-6076	272	3	then	then	ADV
ejpam-6076	272	4	γcerp(g	γcerp(g	PROPN
ejpam-6076	272	5	)	)	PUNCT
ejpam-6076	272	6	=	=	SYM
ejpam-6076	272	7	γcer(g	γcer(g	NOUN
ejpam-6076	272	8	)	)	PUNCT
ejpam-6076	272	9	.	.	PUNCT
ejpam-6076	273	1	5	5	X
ejpam-6076	273	2	.	.	NOUN
ejpam-6076	273	3	lexicographic	lexicographic	ADJ
ejpam-6076	273	4	product	product	NOUN
ejpam-6076	273	5	of	of	ADP
ejpam-6076	273	6	two	two	NUM
ejpam-6076	273	7	graphs	graph	NOUN
ejpam-6076	273	8	in	in	ADP
ejpam-6076	273	9	this	this	DET
ejpam-6076	273	10	section	section	NOUN
ejpam-6076	273	11	,	,	PUNCT
ejpam-6076	273	12	we	we	PRON
ejpam-6076	273	13	characterize	characterize	VERB
ejpam-6076	273	14	the	the	DET
ejpam-6076	273	15	lexicographic	lexicographic	ADJ
ejpam-6076	273	16	product	product	NOUN
ejpam-6076	273	17	of	of	ADP
ejpam-6076	273	18	two	two	NUM
ejpam-6076	273	19	graphs	graph	NOUN
ejpam-6076	273	20	and	and	CCONJ
ejpam-6076	273	21	determine	determine	VERB
ejpam-6076	273	22	its	its	PRON
ejpam-6076	273	23	certified	certify	VERB
ejpam-6076	273	24	perfect	perfect	ADJ
ejpam-6076	273	25	domination	domination	NOUN
ejpam-6076	273	26	number	number	NOUN
ejpam-6076	273	27	.	.	PUNCT
ejpam-6076	274	1	recall	recall	VERB
ejpam-6076	274	2	that	that	SCONJ
ejpam-6076	274	3	f.	f.	PROPN
ejpam-6076	274	4	harary	harary	PROPN
ejpam-6076	274	5	(	(	PUNCT
ejpam-6076	274	6	see	see	VERB
ejpam-6076	274	7	[	[	X
ejpam-6076	274	8	9	9	NUM
ejpam-6076	274	9	]	]	PUNCT
ejpam-6076	274	10	)	)	PUNCT
ejpam-6076	274	11	defined	define	VERB
ejpam-6076	274	12	the	the	DET
ejpam-6076	274	13	lexicographic	lexicographic	ADJ
ejpam-6076	274	14	product	product	NOUN
ejpam-6076	274	15	or	or	CCONJ
ejpam-6076	274	16	composition	composition	NOUN
ejpam-6076	274	17	of	of	ADP
ejpam-6076	274	18	two	two	NUM
ejpam-6076	274	19	graphs	graph	NOUN
ejpam-6076	274	20	g	g	NOUN
ejpam-6076	274	21	and	and	CCONJ
ejpam-6076	274	22	h	h	NOUN
ejpam-6076	274	23	is	be	AUX
ejpam-6076	274	24	the	the	DET
ejpam-6076	274	25	graph	graph	NOUN
ejpam-6076	274	26	g[h	g[h	PROPN
ejpam-6076	274	27	]	]	PUNCT
ejpam-6076	274	28	with	with	ADP
ejpam-6076	274	29	vertex	vertex	NOUN
ejpam-6076	274	30	set	set	VERB
ejpam-6076	274	31	v	v	NOUN
ejpam-6076	274	32	(	(	PUNCT
ejpam-6076	274	33	g[h	g[h	PROPN
ejpam-6076	274	34	]	]	PUNCT
ejpam-6076	274	35	)	)	PUNCT
ejpam-6076	275	1	=	=	SYM
ejpam-6076	275	2	v	v	X
ejpam-6076	275	3	(	(	PUNCT
ejpam-6076	275	4	g)×v	g)×v	PROPN
ejpam-6076	275	5	(	(	PUNCT
ejpam-6076	275	6	h	h	NOUN
ejpam-6076	275	7	)	)	PUNCT
ejpam-6076	275	8	and	and	CCONJ
ejpam-6076	275	9	edge	edge	VERB
ejpam-6076	275	10	set	set	VERB
ejpam-6076	275	11	e(g[h	e(g[h	NOUN
ejpam-6076	275	12	]	]	PUNCT
ejpam-6076	275	13	)	)	PUNCT
ejpam-6076	275	14	satisfying	satisfy	VERB
ejpam-6076	275	15	the	the	DET
ejpam-6076	275	16	following	follow	VERB
ejpam-6076	275	17	condition	condition	NOUN
ejpam-6076	275	18	:	:	PUNCT
ejpam-6076	275	19	(	(	PUNCT
ejpam-6076	275	20	x	x	X
ejpam-6076	275	21	,	,	PUNCT
ejpam-6076	275	22	u)(y	u)(y	PROPN
ejpam-6076	275	23	,	,	PUNCT
ejpam-6076	275	24	v	v	NOUN
ejpam-6076	275	25	)	)	PUNCT
ejpam-6076	275	26	∈	∈	NOUN
ejpam-6076	275	27	e(g[h	e(g[h	NOUN
ejpam-6076	275	28	]	]	PUNCT
ejpam-6076	275	29	)	)	PUNCT
ejpam-6076	275	30	if	if	SCONJ
ejpam-6076	275	31	and	and	CCONJ
ejpam-6076	275	32	only	only	ADV
ejpam-6076	275	33	if	if	SCONJ
ejpam-6076	275	34	xy	xy	PROPN
ejpam-6076	275	35	∈	∈	PROPN
ejpam-6076	275	36	e(g	e(g	PROPN
ejpam-6076	275	37	)	)	PUNCT
ejpam-6076	275	38	or	or	CCONJ
ejpam-6076	275	39	x	x	X
ejpam-6076	275	40	=	=	SYM
ejpam-6076	275	41	y	y	PROPN
ejpam-6076	275	42	and	and	CCONJ
ejpam-6076	275	43	uv	uv	PROPN
ejpam-6076	275	44	∈	∈	PROPN
ejpam-6076	275	45	e(h	e(h	PROPN
ejpam-6076	275	46	)	)	PUNCT
ejpam-6076	275	47	.	.	PUNCT
ejpam-6076	276	1	any	any	DET
ejpam-6076	276	2	subset	subset	NOUN
ejpam-6076	276	3	c	c	NOUN
ejpam-6076	276	4	of	of	ADP
ejpam-6076	276	5	v	v	PROPN
ejpam-6076	276	6	(	(	PUNCT
ejpam-6076	276	7	g[h	g[h	PROPN
ejpam-6076	276	8	]	]	PUNCT
ejpam-6076	276	9	)	)	PUNCT
ejpam-6076	276	10	can	can	AUX
ejpam-6076	276	11	be	be	AUX
ejpam-6076	276	12	expressed	express	VERB
ejpam-6076	276	13	as	as	ADP
ejpam-6076	276	14	q	q	NOUN
ejpam-6076	276	15	=	=	NOUN
ejpam-6076	276	16	⋃	⋃	PROPN
ejpam-6076	276	17	x∈s{x	x∈s{x	PROPN
ejpam-6076	276	18	}	}	PUNCT
ejpam-6076	276	19	×	×	PROPN
ejpam-6076	276	20	tx	tx	PROPN
ejpam-6076	276	21	,	,	PUNCT
ejpam-6076	276	22	where	where	SCONJ
ejpam-6076	276	23	s	s	VERB
ejpam-6076	276	24	⊆	⊆	NUM
ejpam-6076	276	25	v	v	NOUN
ejpam-6076	276	26	(	(	PUNCT
ejpam-6076	276	27	g	g	NOUN
ejpam-6076	276	28	)	)	PUNCT
ejpam-6076	276	29	and	and	CCONJ
ejpam-6076	276	30	tx	tx	VERB
ejpam-6076	276	31	⊆	⊆	NUM
ejpam-6076	276	32	v	v	NOUN
ejpam-6076	276	33	(	(	PUNCT
ejpam-6076	276	34	h	h	NOUN
ejpam-6076	276	35	)	)	PUNCT
ejpam-6076	276	36	for	for	ADP
ejpam-6076	276	37	each	each	DET
ejpam-6076	276	38	x	x	PROPN
ejpam-6076	276	39	∈	∈	PROPN
ejpam-6076	276	40	s.	s.	PROPN
ejpam-6076	276	41	j.	j.	PROPN
ejpam-6076	276	42	j.	j.	PROPN
ejpam-6076	276	43	hamja	hamja	PROPN
ejpam-6076	276	44	et	et	PROPN
ejpam-6076	276	45	al	al	PROPN
ejpam-6076	276	46	.	.	PUNCT
ejpam-6076	276	47	/	/	SYM
ejpam-6076	276	48	eur	eur	PROPN
ejpam-6076	276	49	.	.	PUNCT
ejpam-6076	277	1	j.	j.	PROPN
ejpam-6076	277	2	pure	pure	PROPN
ejpam-6076	277	3	appl	appl	PROPN
ejpam-6076	277	4	.	.	PROPN
ejpam-6076	277	5	math	math	PROPN
ejpam-6076	277	6	,	,	PUNCT
ejpam-6076	277	7	18	18	NUM
ejpam-6076	277	8	(	(	PUNCT
ejpam-6076	277	9	3	3	NUM
ejpam-6076	277	10	)	)	PUNCT
ejpam-6076	277	11	(	(	PUNCT
ejpam-6076	277	12	2025	2025	NUM
ejpam-6076	277	13	)	)	PUNCT
ejpam-6076	277	14	,	,	PUNCT
ejpam-6076	277	15	6076	6076	NUM
ejpam-6076	277	16	10	10	NUM
ejpam-6076	277	17	of	of	ADP
ejpam-6076	277	18	13	13	NUM
ejpam-6076	277	19	theorem	theorem	NOUN
ejpam-6076	277	20	7	7	NUM
ejpam-6076	277	21	.	.	PUNCT
ejpam-6076	278	1	let	let	VERB
ejpam-6076	278	2	g	g	NOUN
ejpam-6076	278	3	and	and	CCONJ
ejpam-6076	278	4	h	h	NOUN
ejpam-6076	278	5	be	be	AUX
ejpam-6076	278	6	connected	connect	VERB
ejpam-6076	278	7	nontrivial	nontrivial	ADJ
ejpam-6076	278	8	graphs	graph	NOUN
ejpam-6076	278	9	.	.	PUNCT
ejpam-6076	279	1	a	a	DET
ejpam-6076	279	2	subset	subset	NOUN
ejpam-6076	279	3	q	q	NOUN
ejpam-6076	280	1	=	=	PUNCT
ejpam-6076	280	2	⋃	⋃	PROPN
ejpam-6076	280	3	u∈j	u∈j	NOUN
ejpam-6076	280	4	[	[	PUNCT
ejpam-6076	280	5	{	{	PUNCT
ejpam-6076	280	6	u}×tu	u}×tu	X
ejpam-6076	280	7	]	]	PUNCT
ejpam-6076	280	8	of	of	ADP
ejpam-6076	280	9	v	v	NOUN
ejpam-6076	280	10	(	(	PUNCT
ejpam-6076	280	11	g[h	g[h	PROPN
ejpam-6076	280	12	]	]	PUNCT
ejpam-6076	280	13	)	)	PUNCT
ejpam-6076	280	14	,	,	PUNCT
ejpam-6076	280	15	where	where	SCONJ
ejpam-6076	280	16	j	j	PROPN
ejpam-6076	280	17	⊆	⊆	NUM
ejpam-6076	280	18	v	v	NOUN
ejpam-6076	280	19	(	(	PUNCT
ejpam-6076	280	20	g	g	NOUN
ejpam-6076	280	21	)	)	PUNCT
ejpam-6076	280	22	and	and	CCONJ
ejpam-6076	280	23	tu	tu	PROPN
ejpam-6076	280	24	⊆	⊆	NUM
ejpam-6076	280	25	v	v	ADP
ejpam-6076	280	26	(	(	PUNCT
ejpam-6076	280	27	h	h	NOUN
ejpam-6076	280	28	)	)	PUNCT
ejpam-6076	280	29	for	for	ADP
ejpam-6076	280	30	each	each	DET
ejpam-6076	280	31	u	u	PROPN
ejpam-6076	280	32	∈	∈	PROPN
ejpam-6076	280	33	j	j	PROPN
ejpam-6076	280	34	,	,	PUNCT
ejpam-6076	280	35	is	be	AUX
ejpam-6076	280	36	a	a	DET
ejpam-6076	280	37	certified	certify	VERB
ejpam-6076	280	38	perfect	perfect	ADJ
ejpam-6076	280	39	dominating	dominating	NOUN
ejpam-6076	280	40	set	set	NOUN
ejpam-6076	280	41	of	of	ADP
ejpam-6076	280	42	g[h	g[h	PROPN
ejpam-6076	280	43	]	]	PUNCT
ejpam-6076	281	1	if	if	SCONJ
ejpam-6076	282	1	and	and	CCONJ
ejpam-6076	282	2	only	only	ADV
ejpam-6076	282	3	if	if	SCONJ
ejpam-6076	282	4	j	j	PROPN
ejpam-6076	282	5	is	be	AUX
ejpam-6076	282	6	a	a	DET
ejpam-6076	282	7	certified	certify	VERB
ejpam-6076	282	8	independent	independent	ADJ
ejpam-6076	282	9	perfect	perfect	ADJ
ejpam-6076	282	10	dominating	dominating	NOUN
ejpam-6076	282	11	set	set	NOUN
ejpam-6076	282	12	of	of	ADP
ejpam-6076	282	13	g	g	PROPN
ejpam-6076	282	14	and	and	CCONJ
ejpam-6076	282	15	tu	tu	PROPN
ejpam-6076	282	16	is	be	AUX
ejpam-6076	282	17	a	a	DET
ejpam-6076	282	18	dominating	dominating	NOUN
ejpam-6076	282	19	set	set	NOUN
ejpam-6076	282	20	of	of	ADP
ejpam-6076	282	21	h	h	NOUN
ejpam-6076	282	22	satisfying	satisfy	VERB
ejpam-6076	282	23	|tx|	|tx|	NOUN
ejpam-6076	282	24	=	=	SYM
ejpam-6076	282	25	1	1	NUM
ejpam-6076	282	26	for	for	ADP
ejpam-6076	282	27	all	all	PRON
ejpam-6076	282	28	u	u	PROPN
ejpam-6076	282	29	∈	∈	PROPN
ejpam-6076	282	30	j	j	PROPN
ejpam-6076	282	31	.	.	PUNCT
ejpam-6076	283	1	proof	proof	NOUN
ejpam-6076	283	2	.	.	PUNCT
ejpam-6076	284	1	suppose	suppose	VERB
ejpam-6076	284	2	that	that	PRON
ejpam-6076	284	3	q	q	NOUN
ejpam-6076	284	4	is	be	AUX
ejpam-6076	284	5	a	a	DET
ejpam-6076	284	6	certified	certify	VERB
ejpam-6076	284	7	perfect	perfect	ADJ
ejpam-6076	284	8	dominating	dominating	NOUN
ejpam-6076	284	9	set	set	NOUN
ejpam-6076	284	10	of	of	ADP
ejpam-6076	284	11	g[h	g[h	PROPN
ejpam-6076	284	12	]	]	PUNCT
ejpam-6076	284	13	.	.	PUNCT
ejpam-6076	285	1	let	let	VERB
ejpam-6076	285	2	a	a	DET
ejpam-6076	285	3	∈	∈	PROPN
ejpam-6076	285	4	v	v	NOUN
ejpam-6076	285	5	(	(	PUNCT
ejpam-6076	285	6	g	g	NOUN
ejpam-6076	285	7	)	)	PUNCT
ejpam-6076	285	8	\	\	PROPN
ejpam-6076	285	9	j	j	PROPN
ejpam-6076	285	10	and	and	CCONJ
ejpam-6076	285	11	select	select	VERB
ejpam-6076	285	12	any	any	DET
ejpam-6076	285	13	b	b	PROPN
ejpam-6076	285	14	∈	∈	ADP
ejpam-6076	285	15	v	v	NOUN
ejpam-6076	285	16	(	(	PUNCT
ejpam-6076	285	17	h	h	NOUN
ejpam-6076	285	18	)	)	PUNCT
ejpam-6076	285	19	.	.	PUNCT
ejpam-6076	286	1	since	since	SCONJ
ejpam-6076	286	2	q	q	PROPN
ejpam-6076	286	3	is	be	AUX
ejpam-6076	286	4	a	a	DET
ejpam-6076	286	5	perfect	perfect	ADJ
ejpam-6076	286	6	dominating	dominating	NOUN
ejpam-6076	286	7	set	set	NOUN
ejpam-6076	286	8	,	,	PUNCT
ejpam-6076	286	9	there	there	PRON
ejpam-6076	286	10	exists	exist	VERB
ejpam-6076	286	11	a	a	DET
ejpam-6076	286	12	vertex	vertex	NOUN
ejpam-6076	286	13	(	(	PUNCT
ejpam-6076	286	14	y	y	NOUN
ejpam-6076	286	15	,	,	PUNCT
ejpam-6076	286	16	z	z	NOUN
ejpam-6076	286	17	)	)	PUNCT
ejpam-6076	286	18	∈	∈	PROPN
ejpam-6076	286	19	q	q	NOUN
ejpam-6076	286	20	such	such	ADJ
ejpam-6076	286	21	that	that	DET
ejpam-6076	286	22	ng[h](a	ng[h](a	NOUN
ejpam-6076	286	23	,	,	PUNCT
ejpam-6076	286	24	b	b	NOUN
ejpam-6076	286	25	)	)	PUNCT
ejpam-6076	286	26	∩	∩	NOUN
ejpam-6076	286	27	q	q	X
ejpam-6076	286	28	=	=	SYM
ejpam-6076	286	29	{	{	PUNCT
ejpam-6076	286	30	(	(	PUNCT
ejpam-6076	286	31	y	y	PROPN
ejpam-6076	286	32	,	,	PUNCT
ejpam-6076	286	33	z	z	NOUN
ejpam-6076	286	34	)	)	PUNCT
ejpam-6076	286	35	}	}	PUNCT
ejpam-6076	286	36	.	.	PUNCT
ejpam-6076	287	1	this	this	PRON
ejpam-6076	287	2	implies	imply	VERB
ejpam-6076	287	3	that	that	PRON
ejpam-6076	287	4	ng(a	ng(a	NOUN
ejpam-6076	287	5	)	)	PUNCT
ejpam-6076	287	6	∩	∩	NOUN
ejpam-6076	287	7	j	j	PROPN
ejpam-6076	287	8	=	=	PUNCT
ejpam-6076	287	9	{	{	PUNCT
ejpam-6076	287	10	y	y	NOUN
ejpam-6076	287	11	}	}	PUNCT
ejpam-6076	287	12	,	,	PUNCT
ejpam-6076	287	13	meaning	mean	VERB
ejpam-6076	287	14	that	that	SCONJ
ejpam-6076	287	15	each	each	DET
ejpam-6076	287	16	vertex	vertex	NOUN
ejpam-6076	287	17	a	a	DET
ejpam-6076	287	18	∈	∈	NOUN
ejpam-6076	287	19	v	v	NOUN
ejpam-6076	287	20	(	(	PUNCT
ejpam-6076	287	21	g)\j	g)\j	PROPN
ejpam-6076	287	22	is	be	AUX
ejpam-6076	287	23	dominated	dominate	VERB
ejpam-6076	287	24	by	by	ADP
ejpam-6076	287	25	exactly	exactly	ADV
ejpam-6076	287	26	one	one	NUM
ejpam-6076	287	27	vertex	vertex	NOUN
ejpam-6076	287	28	in	in	ADP
ejpam-6076	287	29	j	j	PROPN
ejpam-6076	287	30	.	.	PUNCT
ejpam-6076	288	1	therefore	therefore	ADV
ejpam-6076	288	2	,	,	PUNCT
ejpam-6076	288	3	j	j	PROPN
ejpam-6076	288	4	is	be	AUX
ejpam-6076	288	5	a	a	DET
ejpam-6076	288	6	perfect	perfect	ADJ
ejpam-6076	288	7	dominating	dominating	NOUN
ejpam-6076	288	8	set	set	VERB
ejpam-6076	288	9	in	in	ADP
ejpam-6076	288	10	g.	g.	PROPN
ejpam-6076	288	11	next	next	ADV
ejpam-6076	288	12	,	,	PUNCT
ejpam-6076	288	13	assume	assume	VERB
ejpam-6076	288	14	that	that	SCONJ
ejpam-6076	288	15	there	there	PRON
ejpam-6076	288	16	exist	exist	VERB
ejpam-6076	288	17	vertices	vertex	NOUN
ejpam-6076	288	18	r	r	NOUN
ejpam-6076	288	19	,	,	PUNCT
ejpam-6076	288	20	s	s	PART
ejpam-6076	288	21	∈	∈	PROPN
ejpam-6076	288	22	j	j	NOUN
ejpam-6076	288	23	such	such	ADJ
ejpam-6076	288	24	that	that	SCONJ
ejpam-6076	288	25	rs	rs	PROPN
ejpam-6076	288	26	∈	∈	PROPN
ejpam-6076	288	27	e(g	e(g	PROPN
ejpam-6076	288	28	)	)	PUNCT
ejpam-6076	288	29	.	.	PUNCT
ejpam-6076	289	1	since	since	SCONJ
ejpam-6076	289	2	h	h	NOUN
ejpam-6076	289	3	is	be	AUX
ejpam-6076	289	4	connected	connect	VERB
ejpam-6076	289	5	,	,	PUNCT
ejpam-6076	289	6	there	there	PRON
ejpam-6076	289	7	exist	exist	VERB
ejpam-6076	289	8	vertices	vertex	NOUN
ejpam-6076	289	9	k	k	X
ejpam-6076	289	10	,	,	PUNCT
ejpam-6076	290	1	l	l	PROPN
ejpam-6076	290	2	∈	∈	PROPN
ejpam-6076	290	3	v	v	ADP
ejpam-6076	290	4	(	(	PUNCT
ejpam-6076	290	5	h	h	NOUN
ejpam-6076	290	6	)	)	PUNCT
ejpam-6076	290	7	such	such	ADJ
ejpam-6076	290	8	that	that	SCONJ
ejpam-6076	290	9	kl	kl	PROPN
ejpam-6076	290	10	∈	∈	PROPN
ejpam-6076	290	11	e(h	e(h	PROPN
ejpam-6076	290	12	)	)	PUNCT
ejpam-6076	290	13	,	,	PUNCT
ejpam-6076	290	14	and	and	CCONJ
ejpam-6076	290	15	consequently	consequently	ADV
ejpam-6076	290	16	,	,	PUNCT
ejpam-6076	290	17	(	(	PUNCT
ejpam-6076	290	18	r	r	NOUN
ejpam-6076	290	19	,	,	PUNCT
ejpam-6076	290	20	k)(s	k)(s	PROPN
ejpam-6076	290	21	,	,	PUNCT
ejpam-6076	290	22	k	k	X
ejpam-6076	290	23	)	)	PUNCT
ejpam-6076	290	24	∈	∈	NOUN
ejpam-6076	290	25	e(g[h	e(g[h	NOUN
ejpam-6076	290	26	]	]	PUNCT
ejpam-6076	290	27	)	)	PUNCT
ejpam-6076	290	28	with	with	ADP
ejpam-6076	290	29	both	both	DET
ejpam-6076	290	30	(	(	PUNCT
ejpam-6076	290	31	r	r	NOUN
ejpam-6076	290	32	,	,	PUNCT
ejpam-6076	290	33	k	k	NOUN
ejpam-6076	290	34	)	)	PUNCT
ejpam-6076	290	35	and	and	CCONJ
ejpam-6076	290	36	(	(	PUNCT
ejpam-6076	290	37	s	s	X
ejpam-6076	290	38	,	,	PUNCT
ejpam-6076	290	39	k	k	NOUN
ejpam-6076	290	40	)	)	PUNCT
ejpam-6076	290	41	belonging	belong	VERB
ejpam-6076	290	42	to	to	ADP
ejpam-6076	290	43	q.	q.	NOUN
ejpam-6076	290	44	this	this	PRON
ejpam-6076	290	45	leads	lead	VERB
ejpam-6076	290	46	to	to	ADP
ejpam-6076	290	47	a	a	DET
ejpam-6076	290	48	contradiction	contradiction	NOUN
ejpam-6076	290	49	,	,	PUNCT
ejpam-6076	290	50	as	as	SCONJ
ejpam-6076	290	51	both	both	PRON
ejpam-6076	290	52	(	(	PUNCT
ejpam-6076	290	53	r	r	NOUN
ejpam-6076	290	54	,	,	PUNCT
ejpam-6076	290	55	l	l	NOUN
ejpam-6076	290	56	)	)	PUNCT
ejpam-6076	290	57	and	and	CCONJ
ejpam-6076	290	58	(	(	PUNCT
ejpam-6076	290	59	s	s	X
ejpam-6076	290	60	,	,	PUNCT
ejpam-6076	290	61	l	l	NOUN
ejpam-6076	290	62	)	)	PUNCT
ejpam-6076	290	63	are	be	AUX
ejpam-6076	290	64	dominated	dominate	VERB
ejpam-6076	290	65	by	by	ADP
ejpam-6076	290	66	(	(	PUNCT
ejpam-6076	290	67	r	r	NOUN
ejpam-6076	290	68	,	,	PUNCT
ejpam-6076	290	69	k	k	NOUN
ejpam-6076	290	70	)	)	PUNCT
ejpam-6076	290	71	and	and	CCONJ
ejpam-6076	290	72	(	(	PUNCT
ejpam-6076	290	73	s	s	X
ejpam-6076	290	74	,	,	PUNCT
ejpam-6076	290	75	k	k	NOUN
ejpam-6076	290	76	)	)	PUNCT
ejpam-6076	290	77	in	in	ADP
ejpam-6076	290	78	q.	q.	PROPN
ejpam-6076	290	79	thus	thus	ADV
ejpam-6076	290	80	,	,	PUNCT
ejpam-6076	290	81	rs	rs	PROPN
ejpam-6076	290	82	/∈	/∈	PUNCT
ejpam-6076	290	83	e(g	e(g	PROPN
ejpam-6076	290	84	)	)	PUNCT
ejpam-6076	290	85	for	for	ADP
ejpam-6076	290	86	any	any	DET
ejpam-6076	290	87	r	r	NOUN
ejpam-6076	290	88	,	,	PUNCT
ejpam-6076	290	89	s	s	NOUN
ejpam-6076	290	90	∈	∈	PROPN
ejpam-6076	290	91	j	j	PROPN
ejpam-6076	290	92	,	,	PUNCT
ejpam-6076	290	93	meaning	mean	VERB
ejpam-6076	290	94	that	that	SCONJ
ejpam-6076	290	95	j	j	PROPN
ejpam-6076	290	96	is	be	AUX
ejpam-6076	290	97	an	an	DET
ejpam-6076	290	98	independent	independent	ADJ
ejpam-6076	290	99	set	set	NOUN
ejpam-6076	290	100	.	.	PUNCT
ejpam-6076	291	1	therefore	therefore	ADV
ejpam-6076	291	2	,	,	PUNCT
ejpam-6076	291	3	j	j	PROPN
ejpam-6076	291	4	is	be	AUX
ejpam-6076	291	5	an	an	DET
ejpam-6076	291	6	independent	independent	ADJ
ejpam-6076	291	7	perfect	perfect	ADJ
ejpam-6076	291	8	dominating	dominating	NOUN
ejpam-6076	291	9	set	set	VERB
ejpam-6076	291	10	in	in	ADP
ejpam-6076	291	11	g	g	NOUN
ejpam-6076	291	12	,	,	PUNCT
ejpam-6076	291	13	and	and	CCONJ
ejpam-6076	291	14	hence	hence	ADV
ejpam-6076	291	15	a	a	DET
ejpam-6076	291	16	certified	certify	VERB
ejpam-6076	291	17	independent	independent	ADJ
ejpam-6076	291	18	perfect	perfect	ADJ
ejpam-6076	291	19	dominating	dominating	NOUN
ejpam-6076	291	20	set	set	NOUN
ejpam-6076	291	21	of	of	ADP
ejpam-6076	291	22	g	g	PROPN
ejpam-6076	291	23	since	since	SCONJ
ejpam-6076	291	24	ng(j	ng(j	NUM
ejpam-6076	291	25	)	)	PUNCT
ejpam-6076	291	26	≥	≥	NOUN
ejpam-6076	292	1	2	2	NUM
ejpam-6076	292	2	.	.	PUNCT
ejpam-6076	292	3	now	now	ADV
ejpam-6076	292	4	,	,	PUNCT
ejpam-6076	292	5	let	let	VERB
ejpam-6076	292	6	u	u	PRON
ejpam-6076	292	7	∈	∈	PROPN
ejpam-6076	292	8	j	j	PROPN
ejpam-6076	292	9	,	,	PUNCT
ejpam-6076	292	10	and	and	CCONJ
ejpam-6076	292	11	suppose	suppose	VERB
ejpam-6076	292	12	that	that	SCONJ
ejpam-6076	292	13	|tu|	|tu|	PROPN
ejpam-6076	292	14	≥	≥	NOUN
ejpam-6076	292	15	2	2	NUM
ejpam-6076	292	16	.	.	PUNCT
ejpam-6076	292	17	choose	choose	VERB
ejpam-6076	292	18	distinct	distinct	ADJ
ejpam-6076	292	19	c	c	NOUN
ejpam-6076	292	20	,	,	PUNCT
ejpam-6076	292	21	d	d	PROPN
ejpam-6076	292	22	∈	∈	PROPN
ejpam-6076	292	23	tu	tu	PROPN
ejpam-6076	292	24	,	,	PUNCT
ejpam-6076	292	25	so	so	SCONJ
ejpam-6076	292	26	that	that	SCONJ
ejpam-6076	292	27	both	both	DET
ejpam-6076	292	28	(	(	PUNCT
ejpam-6076	292	29	u	u	NOUN
ejpam-6076	292	30	,	,	PUNCT
ejpam-6076	292	31	f	f	PROPN
ejpam-6076	292	32	)	)	PUNCT
ejpam-6076	292	33	and	and	CCONJ
ejpam-6076	292	34	(	(	PUNCT
ejpam-6076	292	35	u	u	NOUN
ejpam-6076	292	36	,	,	PUNCT
ejpam-6076	292	37	g	g	NOUN
ejpam-6076	292	38	)	)	PUNCT
ejpam-6076	292	39	belong	belong	VERB
ejpam-6076	292	40	to	to	ADP
ejpam-6076	292	41	q.	q.	NOUN
ejpam-6076	292	42	consider	consider	VERB
ejpam-6076	292	43	an	an	DET
ejpam-6076	292	44	adjacent	adjacent	ADJ
ejpam-6076	292	45	vertex	vertex	NOUN
ejpam-6076	292	46	h	h	NOUN
ejpam-6076	292	47	∈	∈	PROPN
ejpam-6076	292	48	v	v	ADP
ejpam-6076	292	49	(	(	PUNCT
ejpam-6076	292	50	g	g	NOUN
ejpam-6076	292	51	)	)	PUNCT
ejpam-6076	292	52	such	such	ADJ
ejpam-6076	292	53	that	that	SCONJ
ejpam-6076	292	54	uh	uh	INTJ
ejpam-6076	292	55	∈	∈	PROPN
ejpam-6076	292	56	e(g	e(g	PROPN
ejpam-6076	292	57	)	)	PUNCT
ejpam-6076	292	58	.	.	PUNCT
ejpam-6076	293	1	since	since	SCONJ
ejpam-6076	293	2	j	j	PROPN
ejpam-6076	293	3	is	be	AUX
ejpam-6076	293	4	independent	independent	ADJ
ejpam-6076	293	5	,	,	PUNCT
ejpam-6076	293	6	we	we	PRON
ejpam-6076	293	7	have	have	VERB
ejpam-6076	293	8	h	h	NOUN
ejpam-6076	293	9	/∈	/∈	PUNCT
ejpam-6076	294	1	j	j	PROPN
ejpam-6076	294	2	.	.	PUNCT
ejpam-6076	295	1	clearly	clearly	ADV
ejpam-6076	295	2	,	,	PUNCT
ejpam-6076	295	3	(	(	PUNCT
ejpam-6076	295	4	h	h	NOUN
ejpam-6076	295	5	,	,	PUNCT
ejpam-6076	295	6	f	f	PROPN
ejpam-6076	295	7	)	)	PUNCT
ejpam-6076	295	8	/∈	/∈	PUNCT
ejpam-6076	296	1	q	q	INTJ
ejpam-6076	296	2	,	,	PUNCT
ejpam-6076	296	3	and	and	CCONJ
ejpam-6076	296	4	it	it	PRON
ejpam-6076	296	5	must	must	AUX
ejpam-6076	296	6	be	be	AUX
ejpam-6076	296	7	dominated	dominate	VERB
ejpam-6076	296	8	by	by	ADP
ejpam-6076	296	9	both	both	DET
ejpam-6076	296	10	(	(	PUNCT
ejpam-6076	296	11	u	u	NOUN
ejpam-6076	296	12	,	,	PUNCT
ejpam-6076	296	13	f	f	PROPN
ejpam-6076	296	14	)	)	PUNCT
ejpam-6076	296	15	and	and	CCONJ
ejpam-6076	296	16	(	(	PUNCT
ejpam-6076	296	17	u	u	NOUN
ejpam-6076	296	18	,	,	PUNCT
ejpam-6076	296	19	g	g	NOUN
ejpam-6076	296	20	)	)	PUNCT
ejpam-6076	296	21	in	in	ADP
ejpam-6076	296	22	q	q	NOUN
ejpam-6076	296	23	,	,	PUNCT
ejpam-6076	296	24	which	which	PRON
ejpam-6076	296	25	contradicts	contradict	VERB
ejpam-6076	296	26	the	the	DET
ejpam-6076	296	27	assumption	assumption	NOUN
ejpam-6076	296	28	that	that	SCONJ
ejpam-6076	296	29	q	q	NOUN
ejpam-6076	296	30	is	be	AUX
ejpam-6076	296	31	a	a	DET
ejpam-6076	296	32	perfect	perfect	ADJ
ejpam-6076	296	33	dominating	dominating	NOUN
ejpam-6076	296	34	set	set	NOUN
ejpam-6076	296	35	.	.	PUNCT
ejpam-6076	297	1	therefore	therefore	ADV
ejpam-6076	297	2	,	,	PUNCT
ejpam-6076	297	3	|tu|	|tu|	NOUN
ejpam-6076	297	4	=	=	SYM
ejpam-6076	297	5	1	1	NUM
ejpam-6076	297	6	for	for	ADP
ejpam-6076	297	7	each	each	DET
ejpam-6076	297	8	u	u	PROPN
ejpam-6076	297	9	∈	∈	PROPN
ejpam-6076	297	10	j	j	PROPN
ejpam-6076	297	11	.	.	PUNCT
ejpam-6076	298	1	furthermore	furthermore	ADV
ejpam-6076	298	2	,	,	PUNCT
ejpam-6076	298	3	since	since	SCONJ
ejpam-6076	298	4	j	j	PROPN
ejpam-6076	298	5	is	be	AUX
ejpam-6076	298	6	an	an	DET
ejpam-6076	298	7	independent	independent	ADJ
ejpam-6076	298	8	perfect	perfect	ADJ
ejpam-6076	298	9	dominating	dominating	NOUN
ejpam-6076	298	10	set	set	NOUN
ejpam-6076	298	11	of	of	ADP
ejpam-6076	298	12	g	g	PROPN
ejpam-6076	298	13	and	and	CCONJ
ejpam-6076	298	14	q	q	NOUN
ejpam-6076	298	15	is	be	AUX
ejpam-6076	298	16	a	a	DET
ejpam-6076	298	17	dominating	dominating	NOUN
ejpam-6076	298	18	set	set	VERB
ejpam-6076	298	19	in	in	ADP
ejpam-6076	298	20	g[h	g[h	PROPN
ejpam-6076	298	21	]	]	PUNCT
ejpam-6076	298	22	,	,	PUNCT
ejpam-6076	298	23	it	it	PRON
ejpam-6076	298	24	follows	follow	VERB
ejpam-6076	298	25	that	that	SCONJ
ejpam-6076	298	26	each	each	DET
ejpam-6076	298	27	tu	tu	PROPN
ejpam-6076	298	28	is	be	AUX
ejpam-6076	298	29	a	a	DET
ejpam-6076	298	30	dominating	dominating	NOUN
ejpam-6076	298	31	set	set	NOUN
ejpam-6076	298	32	in	in	ADP
ejpam-6076	298	33	h	h	NOUN
ejpam-6076	298	34	,	,	PUNCT
ejpam-6076	298	35	with	with	ADP
ejpam-6076	298	36	|tu|	|tu|	NOUN
ejpam-6076	298	37	=	=	SYM
ejpam-6076	298	38	1	1	NUM
ejpam-6076	298	39	for	for	ADP
ejpam-6076	298	40	each	each	DET
ejpam-6076	298	41	u	u	PROPN
ejpam-6076	298	42	∈	∈	PROPN
ejpam-6076	298	43	j	j	PROPN
ejpam-6076	298	44	.	.	PUNCT
ejpam-6076	299	1	conversely	conversely	ADV
ejpam-6076	299	2	,	,	PUNCT
ejpam-6076	299	3	suppose	suppose	VERB
ejpam-6076	299	4	that	that	SCONJ
ejpam-6076	299	5	j	j	PROPN
ejpam-6076	299	6	is	be	AUX
ejpam-6076	299	7	a	a	DET
ejpam-6076	299	8	certified	certify	VERB
ejpam-6076	299	9	independent	independent	ADJ
ejpam-6076	299	10	perfect	perfect	ADJ
ejpam-6076	299	11	dominating	dominating	NOUN
ejpam-6076	299	12	set	set	VERB
ejpam-6076	299	13	in	in	ADP
ejpam-6076	299	14	g	g	PROPN
ejpam-6076	299	15	and	and	CCONJ
ejpam-6076	299	16	that	that	SCONJ
ejpam-6076	299	17	tu	tu	PROPN
ejpam-6076	299	18	is	be	AUX
ejpam-6076	299	19	a	a	DET
ejpam-6076	299	20	dominating	dominating	NOUN
ejpam-6076	299	21	set	set	NOUN
ejpam-6076	299	22	of	of	ADP
ejpam-6076	299	23	h	h	NOUN
ejpam-6076	299	24	with	with	ADP
ejpam-6076	299	25	|tu|	|tu|	NOUN
ejpam-6076	299	26	=	=	SYM
ejpam-6076	299	27	1	1	NUM
ejpam-6076	299	28	for	for	ADP
ejpam-6076	299	29	all	all	PRON
ejpam-6076	299	30	u	u	PROPN
ejpam-6076	299	31	∈	∈	PROPN
ejpam-6076	299	32	j	j	PROPN
ejpam-6076	299	33	.	.	PUNCT
ejpam-6076	300	1	let	let	VERB
ejpam-6076	300	2	tu	tu	PROPN
ejpam-6076	300	3	=	=	PUNCT
ejpam-6076	300	4	{	{	PUNCT
ejpam-6076	300	5	w	w	NOUN
ejpam-6076	300	6	}	}	PUNCT
ejpam-6076	300	7	for	for	ADP
ejpam-6076	300	8	each	each	DET
ejpam-6076	300	9	u	u	PROPN
ejpam-6076	300	10	∈	∈	PROPN
ejpam-6076	300	11	j	j	PROPN
ejpam-6076	300	12	,	,	PUNCT
ejpam-6076	300	13	where	where	SCONJ
ejpam-6076	300	14	w	w	NOUN
ejpam-6076	300	15	is	be	AUX
ejpam-6076	300	16	the	the	DET
ejpam-6076	300	17	vertex	vertex	NOUN
ejpam-6076	300	18	in	in	ADP
ejpam-6076	300	19	h	h	NOUN
ejpam-6076	300	20	that	that	PRON
ejpam-6076	300	21	dominates	dominate	VERB
ejpam-6076	300	22	all	all	DET
ejpam-6076	300	23	other	other	ADJ
ejpam-6076	300	24	vertices	vertex	NOUN
ejpam-6076	300	25	of	of	ADP
ejpam-6076	300	26	h.	h.	NOUN
ejpam-6076	300	27	consider	consider	VERB
ejpam-6076	300	28	an	an	DET
ejpam-6076	300	29	arbitrary	arbitrary	ADJ
ejpam-6076	300	30	vertex	vertex	NOUN
ejpam-6076	300	31	(	(	PUNCT
ejpam-6076	300	32	r	r	NOUN
ejpam-6076	300	33	,	,	PUNCT
ejpam-6076	300	34	s	s	PART
ejpam-6076	300	35	)	)	PUNCT
ejpam-6076	300	36	∈	∈	NOUN
ejpam-6076	300	37	v	v	NOUN
ejpam-6076	300	38	(	(	PUNCT
ejpam-6076	300	39	g[h	g[h	PROPN
ejpam-6076	300	40	]	]	PUNCT
ejpam-6076	300	41	)	)	PUNCT
ejpam-6076	300	42	\q	\q	NOUN
ejpam-6076	300	43	,	,	PUNCT
ejpam-6076	300	44	and	and	CCONJ
ejpam-6076	300	45	consider	consider	VERB
ejpam-6076	300	46	the	the	DET
ejpam-6076	300	47	following	follow	VERB
ejpam-6076	300	48	cases	case	NOUN
ejpam-6076	300	49	:	:	PUNCT
ejpam-6076	300	50	case	case	NOUN
ejpam-6076	300	51	1	1	NUM
ejpam-6076	300	52	:	:	PUNCT
ejpam-6076	300	53	r	r	NOUN
ejpam-6076	300	54	/∈	/∈	PROPN
ejpam-6076	300	55	j	j	PROPN
ejpam-6076	300	56	.	.	PUNCT
ejpam-6076	301	1	since	since	SCONJ
ejpam-6076	301	2	j	j	PROPN
ejpam-6076	301	3	is	be	AUX
ejpam-6076	301	4	a	a	DET
ejpam-6076	301	5	perfect	perfect	ADJ
ejpam-6076	301	6	dominating	dominating	NOUN
ejpam-6076	301	7	set	set	NOUN
ejpam-6076	301	8	,	,	PUNCT
ejpam-6076	301	9	there	there	PRON
ejpam-6076	301	10	exists	exist	VERB
ejpam-6076	301	11	u	u	PROPN
ejpam-6076	301	12	∈	∈	PROPN
ejpam-6076	301	13	j	j	NOUN
ejpam-6076	301	14	such	such	ADJ
ejpam-6076	301	15	that	that	PRON
ejpam-6076	301	16	ng(r	ng(r	PUNCT
ejpam-6076	301	17	)	)	PUNCT
ejpam-6076	301	18	∩	∩	NOUN
ejpam-6076	301	19	j	j	PROPN
ejpam-6076	302	1	=	=	PUNCT
ejpam-6076	302	2	{	{	PUNCT
ejpam-6076	302	3	u	u	NOUN
ejpam-6076	302	4	}	}	PUNCT
ejpam-6076	302	5	.	.	PUNCT
ejpam-6076	303	1	hence	hence	ADV
ejpam-6076	303	2	,	,	PUNCT
ejpam-6076	303	3	ng[h](r	ng[h](r	PROPN
ejpam-6076	303	4	,	,	PUNCT
ejpam-6076	303	5	s	s	NOUN
ejpam-6076	303	6	)	)	PUNCT
ejpam-6076	303	7	∩	∩	NOUN
ejpam-6076	303	8	q	q	X
ejpam-6076	303	9	=	=	SYM
ejpam-6076	303	10	{	{	PUNCT
ejpam-6076	303	11	(	(	PUNCT
ejpam-6076	303	12	u	u	NOUN
ejpam-6076	303	13	,	,	PUNCT
ejpam-6076	303	14	w	w	NOUN
ejpam-6076	303	15	)	)	PUNCT
ejpam-6076	303	16	}	}	PUNCT
ejpam-6076	303	17	.	.	PUNCT
ejpam-6076	304	1	therefore	therefore	ADV
ejpam-6076	304	2	,	,	PUNCT
ejpam-6076	304	3	(	(	PUNCT
ejpam-6076	304	4	r	r	NOUN
ejpam-6076	304	5	,	,	PUNCT
ejpam-6076	304	6	s	s	PART
ejpam-6076	304	7	)	)	PUNCT
ejpam-6076	304	8	is	be	AUX
ejpam-6076	304	9	dominated	dominate	VERB
ejpam-6076	304	10	by	by	ADP
ejpam-6076	304	11	exactly	exactly	ADV
ejpam-6076	304	12	one	one	NUM
ejpam-6076	304	13	vertex	vertex	NOUN
ejpam-6076	304	14	in	in	ADP
ejpam-6076	304	15	q	q	NOUN
ejpam-6076	304	16	,	,	PUNCT
ejpam-6076	304	17	and	and	CCONJ
ejpam-6076	304	18	thus	thus	ADV
ejpam-6076	304	19	q	q	X
ejpam-6076	304	20	is	be	AUX
ejpam-6076	304	21	a	a	DET
ejpam-6076	304	22	perfect	perfect	ADJ
ejpam-6076	304	23	dominating	dominating	NOUN
ejpam-6076	304	24	set	set	NOUN
ejpam-6076	304	25	.	.	PUNCT
ejpam-6076	305	1	case	case	NOUN
ejpam-6076	305	2	2	2	NUM
ejpam-6076	305	3	:	:	PUNCT
ejpam-6076	305	4	r	r	NOUN
ejpam-6076	305	5	∈	∈	PROPN
ejpam-6076	305	6	j	j	PROPN
ejpam-6076	305	7	.	.	PUNCT
ejpam-6076	306	1	since	since	SCONJ
ejpam-6076	306	2	tu	tu	PROPN
ejpam-6076	306	3	=	=	PUNCT
ejpam-6076	306	4	{	{	PUNCT
ejpam-6076	306	5	w	w	NOUN
ejpam-6076	306	6	}	}	PUNCT
ejpam-6076	306	7	for	for	ADP
ejpam-6076	306	8	each	each	DET
ejpam-6076	306	9	u	u	PROPN
ejpam-6076	306	10	∈	∈	PROPN
ejpam-6076	306	11	j	j	PROPN
ejpam-6076	306	12	and	and	CCONJ
ejpam-6076	306	13	j	j	PROPN
ejpam-6076	306	14	is	be	AUX
ejpam-6076	306	15	a	a	DET
ejpam-6076	306	16	perfect	perfect	ADJ
ejpam-6076	306	17	dominating	dominating	NOUN
ejpam-6076	306	18	set	set	NOUN
ejpam-6076	306	19	,	,	PUNCT
ejpam-6076	306	20	we	we	PRON
ejpam-6076	306	21	have	have	VERB
ejpam-6076	306	22	ng[h](r	ng[h](r	PROPN
ejpam-6076	306	23	,	,	PUNCT
ejpam-6076	306	24	s	s	PART
ejpam-6076	306	25	)	)	PUNCT
ejpam-6076	306	26	∩	∩	NOUN
ejpam-6076	306	27	q	q	X
ejpam-6076	306	28	=	=	SYM
ejpam-6076	306	29	{	{	PUNCT
ejpam-6076	306	30	(	(	PUNCT
ejpam-6076	306	31	r	r	NOUN
ejpam-6076	306	32	,	,	PUNCT
ejpam-6076	306	33	w	w	NOUN
ejpam-6076	306	34	)	)	PUNCT
ejpam-6076	306	35	}	}	PUNCT
ejpam-6076	306	36	,	,	PUNCT
ejpam-6076	306	37	meaning	mean	VERB
ejpam-6076	306	38	that	that	SCONJ
ejpam-6076	306	39	(	(	PUNCT
ejpam-6076	306	40	r	r	NOUN
ejpam-6076	306	41	,	,	PUNCT
ejpam-6076	306	42	s	s	PART
ejpam-6076	306	43	)	)	PUNCT
ejpam-6076	306	44	is	be	AUX
ejpam-6076	306	45	dominated	dominate	VERB
ejpam-6076	306	46	by	by	ADP
ejpam-6076	306	47	exactly	exactly	ADV
ejpam-6076	306	48	one	one	NUM
ejpam-6076	306	49	vertex	vertex	NOUN
ejpam-6076	306	50	(	(	PUNCT
ejpam-6076	306	51	r	r	NOUN
ejpam-6076	306	52	,	,	PUNCT
ejpam-6076	306	53	w	w	NOUN
ejpam-6076	306	54	)	)	PUNCT
ejpam-6076	306	55	in	in	ADP
ejpam-6076	306	56	q.	q.	PROPN
ejpam-6076	306	57	therefore	therefore	ADV
ejpam-6076	306	58	,	,	PUNCT
ejpam-6076	306	59	q	q	X
ejpam-6076	306	60	is	be	AUX
ejpam-6076	306	61	a	a	DET
ejpam-6076	306	62	perfect	perfect	ADJ
ejpam-6076	306	63	dominating	dominating	NOUN
ejpam-6076	306	64	set	set	NOUN
ejpam-6076	306	65	.	.	PUNCT
ejpam-6076	307	1	finally	finally	ADV
ejpam-6076	307	2	,	,	PUNCT
ejpam-6076	307	3	since	since	SCONJ
ejpam-6076	307	4	j	j	PROPN
ejpam-6076	307	5	is	be	AUX
ejpam-6076	307	6	a	a	DET
ejpam-6076	307	7	certified	certify	VERB
ejpam-6076	307	8	independent	independent	ADJ
ejpam-6076	307	9	perfect	perfect	ADJ
ejpam-6076	307	10	dominating	dominating	NOUN
ejpam-6076	307	11	set	set	VERB
ejpam-6076	307	12	in	in	ADP
ejpam-6076	307	13	g	g	PROPN
ejpam-6076	307	14	and	and	CCONJ
ejpam-6076	307	15	tu	tu	PROPN
ejpam-6076	307	16	is	be	AUX
ejpam-6076	307	17	a	a	DET
ejpam-6076	307	18	dominating	dominating	NOUN
ejpam-6076	307	19	set	set	NOUN
ejpam-6076	307	20	of	of	ADP
ejpam-6076	307	21	h	h	NOUN
ejpam-6076	307	22	with	with	ADP
ejpam-6076	307	23	|tu|	|tu|	NOUN
ejpam-6076	307	24	=	=	SYM
ejpam-6076	307	25	1	1	NUM
ejpam-6076	307	26	for	for	ADP
ejpam-6076	307	27	all	all	PRON
ejpam-6076	307	28	u	u	PROPN
ejpam-6076	307	29	∈	∈	PROPN
ejpam-6076	307	30	j	j	NOUN
ejpam-6076	307	31	,	,	PUNCT
ejpam-6076	307	32	it	it	PRON
ejpam-6076	307	33	follows	follow	VERB
ejpam-6076	307	34	that	that	SCONJ
ejpam-6076	307	35	q	q	NOUN
ejpam-6076	307	36	is	be	AUX
ejpam-6076	307	37	a	a	DET
ejpam-6076	307	38	certified	certify	VERB
ejpam-6076	307	39	perfect	perfect	ADJ
ejpam-6076	307	40	dominating	dominating	NOUN
ejpam-6076	307	41	set	set	NOUN
ejpam-6076	307	42	of	of	ADP
ejpam-6076	307	43	g[h	g[h	PROPN
ejpam-6076	307	44	]	]	PUNCT
ejpam-6076	307	45	.	.	PUNCT
ejpam-6076	308	1	this	this	PRON
ejpam-6076	308	2	completes	complete	VERB
ejpam-6076	308	3	the	the	DET
ejpam-6076	308	4	proof	proof	NOUN
ejpam-6076	308	5	.	.	PUNCT
ejpam-6076	309	1	the	the	DET
ejpam-6076	309	2	following	follow	VERB
ejpam-6076	309	3	results	result	NOUN
ejpam-6076	309	4	immediately	immediately	ADV
ejpam-6076	309	5	follow	follow	VERB
ejpam-6076	309	6	from	from	ADP
ejpam-6076	309	7	theorem	theorem	ADJ
ejpam-6076	309	8	7	7	NUM
ejpam-6076	309	9	.	.	PUNCT
ejpam-6076	309	10	corollary	corollary	ADJ
ejpam-6076	309	11	5	5	NUM
ejpam-6076	309	12	.	.	PUNCT
ejpam-6076	310	1	if	if	SCONJ
ejpam-6076	310	2	g	g	PROPN
ejpam-6076	310	3	and	and	CCONJ
ejpam-6076	310	4	h	h	NOUN
ejpam-6076	310	5	are	be	AUX
ejpam-6076	310	6	connected	connected	ADJ
ejpam-6076	310	7	graphs	graph	NOUN
ejpam-6076	310	8	.	.	PUNCT
ejpam-6076	311	1	then	then	ADV
ejpam-6076	311	2	γcerp(g[h	γcerp(g[h	ADV
ejpam-6076	311	3	]	]	X
ejpam-6076	311	4	)	)	PUNCT
ejpam-6076	312	1	=	=	SYM
ejpam-6076	313	1			PRON
ejpam-6076	313	2	γcerp(g	γcerp(g	PROPN
ejpam-6076	313	3	)	)	PUNCT
ejpam-6076	313	4	,	,	PUNCT
ejpam-6076	313	5	if	if	SCONJ
ejpam-6076	313	6	h	h	NOUN
ejpam-6076	313	7	=	=	SYM
ejpam-6076	313	8	k1	k1	PROPN
ejpam-6076	313	9	,	,	PUNCT
ejpam-6076	313	10	γcerp(h	γcerp(h	NOUN
ejpam-6076	313	11	)	)	PUNCT
ejpam-6076	313	12	,	,	PUNCT
ejpam-6076	313	13	if	if	SCONJ
ejpam-6076	313	14	g	g	PROPN
ejpam-6076	313	15	=	=	SYM
ejpam-6076	313	16	k1	k1	PROPN
ejpam-6076	313	17	,	,	PUNCT
ejpam-6076	313	18	γcerp(g	γcerp(g	PROPN
ejpam-6076	313	19	)	)	PUNCT
ejpam-6076	313	20	,	,	PUNCT
ejpam-6076	313	21	if	if	SCONJ
ejpam-6076	313	22	g	g	PROPN
ejpam-6076	313	23	contains	contain	VERB
ejpam-6076	313	24	γcerip	γcerip	NOUN
ejpam-6076	313	25	−	−	NOUN
ejpam-6076	313	26	set	set	NOUN
ejpam-6076	313	27	and	and	CCONJ
ejpam-6076	313	28	γ(h	γ(h	NOUN
ejpam-6076	313	29	)	)	PUNCT
ejpam-6076	313	30	=	=	SYM
ejpam-6076	314	1	1	1	X
ejpam-6076	314	2	.	.	PUNCT
ejpam-6076	314	3	j.	j.	PROPN
ejpam-6076	314	4	j.	j.	PROPN
ejpam-6076	314	5	hamja	hamja	PROPN
ejpam-6076	314	6	et	et	PROPN
ejpam-6076	314	7	al	al	PROPN
ejpam-6076	314	8	.	.	PUNCT
ejpam-6076	314	9	/	/	SYM
ejpam-6076	314	10	eur	eur	PROPN
ejpam-6076	314	11	.	.	PUNCT
ejpam-6076	315	1	j.	j.	PROPN
ejpam-6076	315	2	pure	pure	PROPN
ejpam-6076	315	3	appl	appl	PROPN
ejpam-6076	315	4	.	.	PROPN
ejpam-6076	315	5	math	math	PROPN
ejpam-6076	315	6	,	,	PUNCT
ejpam-6076	315	7	18	18	NUM
ejpam-6076	315	8	(	(	PUNCT
ejpam-6076	315	9	3	3	NUM
ejpam-6076	315	10	)	)	PUNCT
ejpam-6076	315	11	(	(	PUNCT
ejpam-6076	315	12	2025	2025	NUM
ejpam-6076	315	13	)	)	PUNCT
ejpam-6076	315	14	,	,	PUNCT
ejpam-6076	315	15	6076	6076	NUM
ejpam-6076	315	16	11	11	NUM
ejpam-6076	315	17	of	of	ADP
ejpam-6076	315	18	13	13	NUM
ejpam-6076	315	19	corollary	corollary	ADJ
ejpam-6076	315	20	6	6	NUM
ejpam-6076	315	21	.	.	PUNCT
ejpam-6076	316	1	let	let	VERB
ejpam-6076	316	2	g	g	NOUN
ejpam-6076	316	3	and	and	CCONJ
ejpam-6076	316	4	h	h	NOUN
ejpam-6076	316	5	be	be	AUX
ejpam-6076	316	6	connected	connect	VERB
ejpam-6076	316	7	graphs	graph	NOUN
ejpam-6076	316	8	.	.	PUNCT
ejpam-6076	317	1	then	then	ADV
ejpam-6076	317	2	g[h	g[h	VERB
ejpam-6076	317	3	]	]	PUNCT
ejpam-6076	317	4	is	be	AUX
ejpam-6076	317	5	a	a	DET
ejpam-6076	317	6	non	non	ADJ
ejpam-6076	317	7	-	-	ADJ
ejpam-6076	317	8	γcerp	γcerp	ADJ
ejpam-6076	317	9	-	-	PUNCT
ejpam-6076	317	10	graph	graph	NOUN
ejpam-6076	317	11	if	if	SCONJ
ejpam-6076	318	1	and	and	CCONJ
ejpam-6076	318	2	only	only	ADV
ejpam-6076	318	3	if	if	SCONJ
ejpam-6076	318	4	one	one	NUM
ejpam-6076	318	5	of	of	ADP
ejpam-6076	318	6	the	the	DET
ejpam-6076	318	7	following	follow	VERB
ejpam-6076	318	8	conditions	condition	NOUN
ejpam-6076	318	9	holds	hold	VERB
ejpam-6076	318	10	:	:	PUNCT
ejpam-6076	318	11	(	(	PUNCT
ejpam-6076	318	12	i	i	NOUN
ejpam-6076	318	13	)	)	PUNCT
ejpam-6076	318	14	γ(h	γ(h	PROPN
ejpam-6076	318	15	)	)	PUNCT
ejpam-6076	318	16	≥	≥	NOUN
ejpam-6076	318	17	2	2	NUM
ejpam-6076	318	18	(	(	PUNCT
ejpam-6076	318	19	ii	ii	NOUN
ejpam-6076	318	20	)	)	PUNCT
ejpam-6076	318	21	γ(h	γ(h	NOUN
ejpam-6076	318	22	)	)	PUNCT
ejpam-6076	318	23	=	=	SYM
ejpam-6076	319	1	1	1	NUM
ejpam-6076	319	2	and	and	CCONJ
ejpam-6076	319	3	g	g	PROPN
ejpam-6076	319	4	does	do	AUX
ejpam-6076	319	5	not	not	PART
ejpam-6076	319	6	possess	possess	VERB
ejpam-6076	319	7	a	a	DET
ejpam-6076	319	8	certified	certify	VERB
ejpam-6076	319	9	independent	independent	ADJ
ejpam-6076	319	10	perfect	perfect	ADJ
ejpam-6076	319	11	dominating	dominating	NOUN
ejpam-6076	319	12	set	set	NOUN
ejpam-6076	319	13	.	.	PUNCT
ejpam-6076	320	1	the	the	DET
ejpam-6076	320	2	following	following	ADJ
ejpam-6076	320	3	result	result	NOUN
ejpam-6076	320	4	is	be	AUX
ejpam-6076	320	5	a	a	DET
ejpam-6076	320	6	direct	direct	ADJ
ejpam-6076	320	7	consequence	consequence	NOUN
ejpam-6076	320	8	of	of	ADP
ejpam-6076	320	9	corollary	corollary	ADJ
ejpam-6076	320	10	6	6	NUM
ejpam-6076	320	11	(	(	PUNCT
ejpam-6076	320	12	ii	ii	NOUN
ejpam-6076	320	13	)	)	PUNCT
ejpam-6076	320	14	.	.	PUNCT
ejpam-6076	321	1	corollary	corollary	ADJ
ejpam-6076	321	2	7	7	NUM
ejpam-6076	321	3	.	.	PUNCT
ejpam-6076	322	1	let	let	VERB
ejpam-6076	322	2	g	g	PRON
ejpam-6076	322	3	be	be	AUX
ejpam-6076	322	4	a	a	DET
ejpam-6076	322	5	connected	connected	ADJ
ejpam-6076	322	6	graph	graph	NOUN
ejpam-6076	322	7	that	that	PRON
ejpam-6076	322	8	does	do	AUX
ejpam-6076	322	9	not	not	PART
ejpam-6076	322	10	have	have	VERB
ejpam-6076	322	11	a	a	DET
ejpam-6076	322	12	γcerip	γcerip	NOUN
ejpam-6076	322	13	-	-	PUNCT
ejpam-6076	322	14	set	set	NOUN
ejpam-6076	322	15	,	,	PUNCT
ejpam-6076	322	16	and	and	CCONJ
ejpam-6076	322	17	let	let	VERB
ejpam-6076	322	18	h	h	NOUN
ejpam-6076	322	19	be	be	AUX
ejpam-6076	322	20	any	any	DET
ejpam-6076	322	21	graph	graph	NOUN
ejpam-6076	322	22	.	.	PUNCT
ejpam-6076	323	1	then	then	ADV
ejpam-6076	323	2	g[h	g[h	VERB
ejpam-6076	323	3	]	]	PUNCT
ejpam-6076	323	4	is	be	AUX
ejpam-6076	323	5	a	a	DET
ejpam-6076	323	6	non	non	ADJ
ejpam-6076	323	7	-	-	ADJ
ejpam-6076	323	8	γcerp	γcerp	ADJ
ejpam-6076	323	9	-	-	PUNCT
ejpam-6076	323	10	graph	graph	NOUN
ejpam-6076	323	11	.	.	PUNCT
ejpam-6076	324	1	6	6	X
ejpam-6076	324	2	.	.	X
ejpam-6076	324	3	cartesian	cartesian	ADJ
ejpam-6076	324	4	product	product	NOUN
ejpam-6076	324	5	of	of	ADP
ejpam-6076	324	6	two	two	NUM
ejpam-6076	324	7	graphs	graph	NOUN
ejpam-6076	324	8	recall	recall	VERB
ejpam-6076	324	9	that	that	SCONJ
ejpam-6076	324	10	f.	f.	PROPN
ejpam-6076	324	11	harary	harary	PROPN
ejpam-6076	324	12	(	(	PUNCT
ejpam-6076	324	13	see	see	VERB
ejpam-6076	324	14	[	[	X
ejpam-6076	324	15	9	9	NUM
ejpam-6076	324	16	]	]	PUNCT
ejpam-6076	324	17	)	)	PUNCT
ejpam-6076	324	18	defined	define	VERB
ejpam-6076	324	19	the	the	DET
ejpam-6076	324	20	cartesian	cartesian	ADJ
ejpam-6076	324	21	product	product	NOUN
ejpam-6076	324	22	g	g	NOUN
ejpam-6076	324	23	□	□	PROPN
ejpam-6076	324	24	h	h	NOUN
ejpam-6076	324	25	of	of	ADP
ejpam-6076	324	26	two	two	NUM
ejpam-6076	324	27	graphs	graph	NOUN
ejpam-6076	324	28	g	g	NOUN
ejpam-6076	324	29	and	and	CCONJ
ejpam-6076	324	30	h	h	NOUN
ejpam-6076	324	31	is	be	AUX
ejpam-6076	324	32	a	a	DET
ejpam-6076	324	33	graph	graph	NOUN
ejpam-6076	324	34	with	with	ADP
ejpam-6076	324	35	the	the	DET
ejpam-6076	324	36	vertex	vertex	NOUN
ejpam-6076	324	37	set	set	VERB
ejpam-6076	324	38	v	v	NOUN
ejpam-6076	324	39	(	(	PUNCT
ejpam-6076	324	40	g	g	NOUN
ejpam-6076	324	41	□	□	NOUN
ejpam-6076	324	42	h	h	NOUN
ejpam-6076	324	43	)	)	PUNCT
ejpam-6076	324	44	=	=	NOUN
ejpam-6076	324	45	v	v	X
ejpam-6076	324	46	(	(	PUNCT
ejpam-6076	324	47	g)×	g)×	NOUN
ejpam-6076	324	48	v	v	NOUN
ejpam-6076	324	49	(	(	PUNCT
ejpam-6076	324	50	h	h	NOUN
ejpam-6076	324	51	)	)	PUNCT
ejpam-6076	324	52	.	.	PUNCT
ejpam-6076	325	1	two	two	NUM
ejpam-6076	325	2	vertices	vertex	NOUN
ejpam-6076	325	3	(	(	PUNCT
ejpam-6076	325	4	ui	ui	NOUN
ejpam-6076	325	5	,	,	PUNCT
ejpam-6076	325	6	vj	vj	PROPN
ejpam-6076	325	7	)	)	PUNCT
ejpam-6076	325	8	and	and	CCONJ
ejpam-6076	325	9	(	(	PUNCT
ejpam-6076	325	10	uk	uk	PROPN
ejpam-6076	325	11	,	,	PUNCT
ejpam-6076	325	12	vl	vl	NOUN
ejpam-6076	325	13	)	)	PUNCT
ejpam-6076	325	14	are	be	AUX
ejpam-6076	325	15	adjacent	adjacent	ADJ
ejpam-6076	325	16	in	in	ADP
ejpam-6076	325	17	g	g	PROPN
ejpam-6076	325	18	□	□	PROPN
ejpam-6076	325	19	h	h	NOUN
ejpam-6076	325	20	if	if	SCONJ
ejpam-6076	326	1	and	and	CCONJ
ejpam-6076	326	2	only	only	ADV
ejpam-6076	326	3	if	if	SCONJ
ejpam-6076	326	4	one	one	NUM
ejpam-6076	326	5	of	of	ADP
ejpam-6076	326	6	the	the	DET
ejpam-6076	326	7	following	follow	VERB
ejpam-6076	326	8	conditions	condition	NOUN
ejpam-6076	326	9	holds	hold	VERB
ejpam-6076	326	10	:	:	PUNCT
ejpam-6076	326	11	(	(	PUNCT
ejpam-6076	326	12	i	i	NOUN
ejpam-6076	326	13	)	)	PUNCT
ejpam-6076	326	14	ui	ui	PROPN
ejpam-6076	327	1	=	=	PUNCT
ejpam-6076	327	2	uk	uk	PROPN
ejpam-6076	327	3	and	and	CCONJ
ejpam-6076	327	4	vjvl	vjvl	NOUN
ejpam-6076	327	5	∈	∈	PROPN
ejpam-6076	327	6	e(h	e(h	PROPN
ejpam-6076	327	7	)	)	PUNCT
ejpam-6076	327	8	,	,	PUNCT
ejpam-6076	327	9	(	(	PUNCT
ejpam-6076	327	10	ii	ii	NOUN
ejpam-6076	327	11	)	)	PUNCT
ejpam-6076	327	12	vj	vj	PROPN
ejpam-6076	328	1	=	=	PUNCT
ejpam-6076	328	2	vl	vl	PROPN
ejpam-6076	328	3	and	and	CCONJ
ejpam-6076	328	4	uiuk	uiuk	PROPN
ejpam-6076	328	5	∈	∈	PROPN
ejpam-6076	328	6	e(g	e(g	PROPN
ejpam-6076	328	7	)	)	PUNCT
ejpam-6076	328	8	.	.	PUNCT
ejpam-6076	329	1	proposition	proposition	NOUN
ejpam-6076	329	2	8	8	NUM
ejpam-6076	329	3	.	.	PUNCT
ejpam-6076	330	1	let	let	VERB
ejpam-6076	330	2	g	g	PRON
ejpam-6076	330	3	be	be	AUX
ejpam-6076	330	4	a	a	DET
ejpam-6076	330	5	graph	graph	NOUN
ejpam-6076	330	6	with	with	ADP
ejpam-6076	330	7	γcerp(g	γcerp(g	PROPN
ejpam-6076	330	8	)	)	PUNCT
ejpam-6076	330	9	=	=	SYM
ejpam-6076	331	1	1	1	X
ejpam-6076	331	2	.	.	PUNCT
ejpam-6076	331	3	then	then	ADV
ejpam-6076	331	4	γcerp(g	γcerp(g	PROPN
ejpam-6076	331	5	□	□	SYM
ejpam-6076	331	6	k1	k1	NOUN
ejpam-6076	331	7	)	)	PUNCT
ejpam-6076	331	8	=	=	SYM
ejpam-6076	331	9	1	1	X
ejpam-6076	331	10	.	.	PUNCT
ejpam-6076	332	1	proof	proof	NOUN
ejpam-6076	332	2	.	.	PUNCT
ejpam-6076	333	1	let	let	VERB
ejpam-6076	333	2	g	g	PRON
ejpam-6076	333	3	be	be	AUX
ejpam-6076	333	4	a	a	DET
ejpam-6076	333	5	graph	graph	NOUN
ejpam-6076	333	6	with	with	ADP
ejpam-6076	333	7	γcerp(g	γcerp(g	PROPN
ejpam-6076	333	8	)	)	PUNCT
ejpam-6076	333	9	=	=	SYM
ejpam-6076	334	1	1	1	X
ejpam-6076	334	2	.	.	PUNCT
ejpam-6076	334	3	since	since	SCONJ
ejpam-6076	334	4	g	g	NOUN
ejpam-6076	334	5	□	□	PROPN
ejpam-6076	334	6	k1	k1	NOUN
ejpam-6076	334	7	=	=	SYM
ejpam-6076	334	8	g	g	NOUN
ejpam-6076	334	9	,	,	PUNCT
ejpam-6076	334	10	it	it	PRON
ejpam-6076	334	11	follows	follow	VERB
ejpam-6076	334	12	that	that	SCONJ
ejpam-6076	334	13	γcerp(g	γcerp(g	PROPN
ejpam-6076	334	14	□	□	SYM
ejpam-6076	334	15	k1	k1	NOUN
ejpam-6076	334	16	)	)	PUNCT
ejpam-6076	334	17	=	=	SYM
ejpam-6076	335	1	1	1	X
ejpam-6076	335	2	.	.	X
ejpam-6076	335	3	theorem	theorem	VERB
ejpam-6076	335	4	8	8	NUM
ejpam-6076	335	5	.	.	PUNCT
ejpam-6076	336	1	let	let	VERB
ejpam-6076	336	2	p	p	PRON
ejpam-6076	336	3	≥	≥	NUM
ejpam-6076	336	4	2	2	NUM
ejpam-6076	336	5	be	be	AUX
ejpam-6076	336	6	an	an	DET
ejpam-6076	336	7	integer	integer	NOUN
ejpam-6076	336	8	.	.	PUNCT
ejpam-6076	337	1	then	then	ADV
ejpam-6076	337	2	kp	kp	PROPN
ejpam-6076	337	3	□	□	PROPN
ejpam-6076	337	4	kp	kp	PROPN
ejpam-6076	337	5	is	be	AUX
ejpam-6076	337	6	a	a	DET
ejpam-6076	337	7	non	non	ADJ
ejpam-6076	337	8	-	-	ADJ
ejpam-6076	337	9	γcerp	γcerp	ADJ
ejpam-6076	337	10	-	-	PUNCT
ejpam-6076	337	11	graph	graph	NOUN
ejpam-6076	337	12	.	.	PUNCT
ejpam-6076	338	1	proof	proof	NOUN
ejpam-6076	338	2	.	.	PUNCT
ejpam-6076	339	1	consider	consider	VERB
ejpam-6076	339	2	the	the	DET
ejpam-6076	339	3	complete	complete	ADJ
ejpam-6076	339	4	graph	graph	NOUN
ejpam-6076	339	5	kp	kp	PROPN
ejpam-6076	339	6	with	with	ADP
ejpam-6076	339	7	p	p	NOUN
ejpam-6076	339	8	vertices	vertex	NOUN
ejpam-6076	339	9	.	.	PUNCT
ejpam-6076	340	1	the	the	DET
ejpam-6076	340	2	graph	graph	NOUN
ejpam-6076	340	3	kp	kp	PROPN
ejpam-6076	340	4	□	□	PROPN
ejpam-6076	340	5	kp	kp	PROPN
ejpam-6076	340	6	has	have	VERB
ejpam-6076	340	7	p2	p2	PROPN
ejpam-6076	340	8	vertices	vertex	NOUN
ejpam-6076	340	9	.	.	PUNCT
ejpam-6076	341	1	for	for	ADP
ejpam-6076	341	2	the	the	DET
ejpam-6076	341	3	sake	sake	NOUN
ejpam-6076	341	4	of	of	ADP
ejpam-6076	341	5	contradiction	contradiction	NOUN
ejpam-6076	341	6	,	,	PUNCT
ejpam-6076	341	7	assume	assume	VERB
ejpam-6076	341	8	that	that	SCONJ
ejpam-6076	341	9	kp	kp	PROPN
ejpam-6076	341	10	□	□	PROPN
ejpam-6076	341	11	kp	kp	PROPN
ejpam-6076	341	12	is	be	AUX
ejpam-6076	341	13	a	a	DET
ejpam-6076	341	14	γcerp	γcerp	NOUN
ejpam-6076	341	15	-	-	PUNCT
ejpam-6076	341	16	graph	graph	NOUN
ejpam-6076	341	17	.	.	PUNCT
ejpam-6076	342	1	let	let	VERB
ejpam-6076	342	2	(	(	PUNCT
ejpam-6076	342	3	u	u	NOUN
ejpam-6076	342	4	,	,	PUNCT
ejpam-6076	342	5	v	v	NOUN
ejpam-6076	342	6	)	)	PUNCT
ejpam-6076	342	7	be	be	AUX
ejpam-6076	342	8	any	any	DET
ejpam-6076	342	9	vertex	vertex	NOUN
ejpam-6076	342	10	in	in	ADP
ejpam-6076	342	11	v	v	NUM
ejpam-6076	342	12	(	(	PUNCT
ejpam-6076	342	13	kp	kp	PROPN
ejpam-6076	342	14	□	□	PROPN
ejpam-6076	342	15	kp	kp	PROPN
ejpam-6076	342	16	)	)	PUNCT
ejpam-6076	342	17	,	,	PUNCT
ejpam-6076	342	18	where	where	SCONJ
ejpam-6076	342	19	u	u	NOUN
ejpam-6076	342	20	is	be	AUX
ejpam-6076	342	21	a	a	DET
ejpam-6076	342	22	vertex	vertex	NOUN
ejpam-6076	342	23	in	in	ADP
ejpam-6076	342	24	the	the	DET
ejpam-6076	342	25	first	first	ADJ
ejpam-6076	342	26	kp	kp	PROPN
ejpam-6076	342	27	and	and	CCONJ
ejpam-6076	342	28	v	v	NOUN
ejpam-6076	342	29	is	be	AUX
ejpam-6076	342	30	a	a	DET
ejpam-6076	342	31	vertex	vertex	NOUN
ejpam-6076	342	32	in	in	ADP
ejpam-6076	342	33	the	the	DET
ejpam-6076	342	34	second	second	ADJ
ejpam-6076	342	35	kp	kp	NOUN
ejpam-6076	342	36	.	.	PUNCT
ejpam-6076	343	1	the	the	DET
ejpam-6076	343	2	vertex	vertex	NOUN
ejpam-6076	343	3	(	(	PUNCT
ejpam-6076	343	4	u	u	NOUN
ejpam-6076	343	5	,	,	PUNCT
ejpam-6076	343	6	v	v	NOUN
ejpam-6076	343	7	)	)	PUNCT
ejpam-6076	343	8	is	be	AUX
ejpam-6076	343	9	adjacent	adjacent	ADJ
ejpam-6076	343	10	to	to	ADP
ejpam-6076	343	11	p	p	NOUN
ejpam-6076	343	12	−	−	NUM
ejpam-6076	343	13	1	1	NUM
ejpam-6076	343	14	vertices	vertex	NOUN
ejpam-6076	343	15	of	of	ADP
ejpam-6076	343	16	the	the	DET
ejpam-6076	343	17	form	form	NOUN
ejpam-6076	343	18	(	(	PUNCT
ejpam-6076	343	19	u	u	NOUN
ejpam-6076	343	20	,	,	PUNCT
ejpam-6076	343	21	vi	vi	PROPN
ejpam-6076	343	22	)	)	PUNCT
ejpam-6076	343	23	,	,	PUNCT
ejpam-6076	343	24	where	where	SCONJ
ejpam-6076	343	25	vi	vi	PROPN
ejpam-6076	343	26	is	be	AUX
ejpam-6076	343	27	one	one	NUM
ejpam-6076	343	28	of	of	ADP
ejpam-6076	343	29	the	the	DET
ejpam-6076	343	30	p−	p−	NOUN
ejpam-6076	343	31	1	1	NUM
ejpam-6076	343	32	vertices	vertex	NOUN
ejpam-6076	343	33	of	of	ADP
ejpam-6076	343	34	the	the	DET
ejpam-6076	343	35	second	second	ADJ
ejpam-6076	343	36	kp	kp	NOUN
ejpam-6076	343	37	.	.	PUNCT
ejpam-6076	344	1	therefore	therefore	ADV
ejpam-6076	344	2	,	,	PUNCT
ejpam-6076	344	3	(	(	PUNCT
ejpam-6076	344	4	u	u	NOUN
ejpam-6076	344	5	,	,	PUNCT
ejpam-6076	344	6	v	v	NOUN
ejpam-6076	344	7	)	)	PUNCT
ejpam-6076	344	8	dominates	dominate	VERB
ejpam-6076	344	9	2(p−	2(p−	NUM
ejpam-6076	344	10	1	1	NUM
ejpam-6076	344	11	)	)	PUNCT
ejpam-6076	344	12	vertices	vertex	NOUN
ejpam-6076	344	13	in	in	ADP
ejpam-6076	344	14	kp	kp	PROPN
ejpam-6076	344	15	□	□	PROPN
ejpam-6076	344	16	kp	kp	PROPN
ejpam-6076	344	17	,	,	PUNCT
ejpam-6076	344	18	and	and	CCONJ
ejpam-6076	344	19	so	so	ADV
ejpam-6076	344	20	(	(	PUNCT
ejpam-6076	344	21	u	u	NOUN
ejpam-6076	344	22	,	,	PUNCT
ejpam-6076	344	23	v	v	NOUN
ejpam-6076	344	24	)	)	PUNCT
ejpam-6076	344	25	must	must	AUX
ejpam-6076	344	26	be	be	AUX
ejpam-6076	344	27	included	include	VERB
ejpam-6076	344	28	in	in	ADP
ejpam-6076	344	29	the	the	DET
ejpam-6076	344	30	dominating	dominating	NOUN
ejpam-6076	344	31	set	set	VERB
ejpam-6076	344	32	j	j	PROPN
ejpam-6076	344	33	.	.	PUNCT
ejpam-6076	345	1	now	now	ADV
ejpam-6076	345	2	,	,	PUNCT
ejpam-6076	345	3	we	we	PRON
ejpam-6076	345	4	need	need	VERB
ejpam-6076	345	5	to	to	PART
ejpam-6076	345	6	select	select	VERB
ejpam-6076	345	7	additional	additional	ADJ
ejpam-6076	345	8	vertices	vertex	NOUN
ejpam-6076	345	9	for	for	ADP
ejpam-6076	345	10	j	j	PROPN
ejpam-6076	345	11	from	from	ADP
ejpam-6076	345	12	the	the	DET
ejpam-6076	345	13	remaining	remain	VERB
ejpam-6076	345	14	p2	p2	NOUN
ejpam-6076	345	15	−	−	NOUN
ejpam-6076	345	16	2(p	2(p	NUM
ejpam-6076	345	17	−	−	PROPN
ejpam-6076	345	18	1	1	NUM
ejpam-6076	345	19	)	)	PUNCT
ejpam-6076	345	20	vertices	vertex	NOUN
ejpam-6076	345	21	.	.	PUNCT
ejpam-6076	346	1	without	without	ADP
ejpam-6076	346	2	loss	loss	NOUN
ejpam-6076	346	3	of	of	ADP
ejpam-6076	346	4	generality	generality	NOUN
ejpam-6076	346	5	,	,	PUNCT
ejpam-6076	346	6	assume	assume	VERB
ejpam-6076	346	7	that	that	SCONJ
ejpam-6076	346	8	(	(	PUNCT
ejpam-6076	346	9	u	u	NOUN
ejpam-6076	346	10	,	,	PUNCT
ejpam-6076	346	11	v	v	NOUN
ejpam-6076	346	12	)	)	PUNCT
ejpam-6076	346	13	is	be	AUX
ejpam-6076	346	14	among	among	ADP
ejpam-6076	346	15	these	these	DET
ejpam-6076	346	16	remaining	remain	VERB
ejpam-6076	346	17	vertices	vertex	NOUN
ejpam-6076	346	18	.	.	PUNCT
ejpam-6076	347	1	since	since	SCONJ
ejpam-6076	347	2	(	(	PUNCT
ejpam-6076	347	3	u	u	NOUN
ejpam-6076	347	4	,	,	PUNCT
ejpam-6076	347	5	v	v	NOUN
ejpam-6076	347	6	)	)	PUNCT
ejpam-6076	347	7	is	be	AUX
ejpam-6076	347	8	adjacent	adjacent	ADJ
ejpam-6076	347	9	to	to	ADP
ejpam-6076	347	10	at	at	ADV
ejpam-6076	347	11	least	least	ADV
ejpam-6076	347	12	one	one	NUM
ejpam-6076	347	13	vertex	vertex	NOUN
ejpam-6076	347	14	that	that	PRON
ejpam-6076	347	15	is	be	AUX
ejpam-6076	347	16	already	already	ADV
ejpam-6076	347	17	dominated	dominate	VERB
ejpam-6076	347	18	by	by	ADP
ejpam-6076	347	19	another	another	DET
ejpam-6076	347	20	vertex	vertex	NOUN
ejpam-6076	347	21	in	in	ADP
ejpam-6076	347	22	j	j	PROPN
ejpam-6076	347	23	,	,	PUNCT
ejpam-6076	347	24	this	this	PRON
ejpam-6076	347	25	contradicts	contradict	VERB
ejpam-6076	347	26	the	the	DET
ejpam-6076	347	27	assumption	assumption	NOUN
ejpam-6076	347	28	that	that	SCONJ
ejpam-6076	347	29	j	j	PROPN
ejpam-6076	347	30	is	be	AUX
ejpam-6076	347	31	a	a	DET
ejpam-6076	347	32	perfect	perfect	ADJ
ejpam-6076	347	33	dominating	dominating	NOUN
ejpam-6076	347	34	set	set	NOUN
ejpam-6076	347	35	.	.	PUNCT
ejpam-6076	348	1	thus	thus	ADV
ejpam-6076	348	2	,	,	PUNCT
ejpam-6076	348	3	no	no	DET
ejpam-6076	348	4	certified	certify	VERB
ejpam-6076	348	5	perfect	perfect	ADJ
ejpam-6076	348	6	dominating	dominating	NOUN
ejpam-6076	348	7	set	set	NOUN
ejpam-6076	348	8	exists	exist	VERB
ejpam-6076	348	9	in	in	ADP
ejpam-6076	348	10	kp	kp	PROPN
ejpam-6076	348	11	□	□	PROPN
ejpam-6076	348	12	kp	kp	PROPN
ejpam-6076	348	13	,	,	PUNCT
ejpam-6076	348	14	which	which	PRON
ejpam-6076	348	15	implies	imply	VERB
ejpam-6076	348	16	that	that	SCONJ
ejpam-6076	348	17	kp	kp	PROPN
ejpam-6076	348	18	□	□	PROPN
ejpam-6076	348	19	kp	kp	PROPN
ejpam-6076	348	20	is	be	AUX
ejpam-6076	348	21	a	a	DET
ejpam-6076	348	22	non	non	ADJ
ejpam-6076	348	23	-	-	ADJ
ejpam-6076	348	24	γcerp	γcerp	ADJ
ejpam-6076	348	25	-	-	PUNCT
ejpam-6076	348	26	graph	graph	NOUN
ejpam-6076	348	27	for	for	ADP
ejpam-6076	348	28	all	all	DET
ejpam-6076	348	29	p	p	PRON
ejpam-6076	348	30	≥	≥	NUM
ejpam-6076	348	31	2	2	NUM
ejpam-6076	348	32	.	.	PUNCT
ejpam-6076	348	33	theorem	theorem	NOUN
ejpam-6076	348	34	9	9	NUM
ejpam-6076	348	35	.	.	PUNCT
ejpam-6076	349	1	let	let	VERB
ejpam-6076	349	2	m	m	PRON
ejpam-6076	349	3	≥	≥	VERB
ejpam-6076	349	4	3	3	NUM
ejpam-6076	349	5	,	,	PUNCT
ejpam-6076	349	6	n	n	PRON
ejpam-6076	349	7	≥	≥	NOUN
ejpam-6076	349	8	2	2	NUM
ejpam-6076	349	9	be	be	AUX
ejpam-6076	349	10	integers	integer	NOUN
ejpam-6076	349	11	.	.	PUNCT
ejpam-6076	350	1	then	then	ADV
ejpam-6076	350	2	km	km	PROPN
ejpam-6076	350	3	□	□	PROPN
ejpam-6076	350	4	pn	pn	PROPN
ejpam-6076	350	5	is	be	AUX
ejpam-6076	350	6	a	a	DET
ejpam-6076	350	7	non	non	ADJ
ejpam-6076	350	8	-	-	ADJ
ejpam-6076	350	9	γcerp	γcerp	ADJ
ejpam-6076	350	10	-	-	PUNCT
ejpam-6076	350	11	graph	graph	NOUN
ejpam-6076	350	12	.	.	PUNCT
ejpam-6076	351	1	j.	j.	PROPN
ejpam-6076	351	2	j.	j.	PROPN
ejpam-6076	351	3	hamja	hamja	PROPN
ejpam-6076	351	4	et	et	PROPN
ejpam-6076	351	5	al	al	PROPN
ejpam-6076	351	6	.	.	PUNCT
ejpam-6076	351	7	/	/	SYM
ejpam-6076	351	8	eur	eur	PROPN
ejpam-6076	351	9	.	.	PUNCT
ejpam-6076	352	1	j.	j.	PROPN
ejpam-6076	352	2	pure	pure	PROPN
ejpam-6076	352	3	appl	appl	PROPN
ejpam-6076	352	4	.	.	PROPN
ejpam-6076	352	5	math	math	PROPN
ejpam-6076	352	6	,	,	PUNCT
ejpam-6076	352	7	18	18	NUM
ejpam-6076	352	8	(	(	PUNCT
ejpam-6076	352	9	3	3	NUM
ejpam-6076	352	10	)	)	PUNCT
ejpam-6076	352	11	(	(	PUNCT
ejpam-6076	352	12	2025	2025	NUM
ejpam-6076	352	13	)	)	PUNCT
ejpam-6076	352	14	,	,	PUNCT
ejpam-6076	352	15	6076	6076	NUM
ejpam-6076	352	16	12	12	NUM
ejpam-6076	352	17	of	of	ADP
ejpam-6076	352	18	13	13	NUM
ejpam-6076	352	19	proof	proof	NOUN
ejpam-6076	352	20	.	.	PUNCT
ejpam-6076	353	1	let	let	VERB
ejpam-6076	353	2	v	v	NOUN
ejpam-6076	353	3	(	(	PUNCT
ejpam-6076	353	4	km	km	NOUN
ejpam-6076	353	5	)	)	PUNCT
ejpam-6076	353	6	=	=	PRON
ejpam-6076	354	1	{	{	PUNCT
ejpam-6076	354	2	x1	x1	PROPN
ejpam-6076	354	3	,	,	PUNCT
ejpam-6076	354	4	x2	x2	PROPN
ejpam-6076	354	5	,	,	PUNCT
ejpam-6076	354	6	.	.	PUNCT
ejpam-6076	354	7	.	.	PUNCT
ejpam-6076	354	8	.	.	PUNCT
ejpam-6076	355	1	,	,	PUNCT
ejpam-6076	355	2	xm	xm	PROPN
ejpam-6076	355	3	}	}	PUNCT
ejpam-6076	355	4	and	and	CCONJ
ejpam-6076	355	5	v	v	X
ejpam-6076	355	6	(	(	PUNCT
ejpam-6076	355	7	pn	pn	NOUN
ejpam-6076	355	8	)	)	PUNCT
ejpam-6076	355	9	=	=	SYM
ejpam-6076	355	10	{	{	PUNCT
ejpam-6076	355	11	y1	y1	PROPN
ejpam-6076	355	12	,	,	PUNCT
ejpam-6076	355	13	y2	y2	PROPN
ejpam-6076	355	14	,	,	PUNCT
ejpam-6076	355	15	.	.	PUNCT
ejpam-6076	355	16	.	.	PUNCT
ejpam-6076	356	1	.	.	PUNCT
ejpam-6076	357	1	,	,	PUNCT
ejpam-6076	357	2	yn	yn	PROPN
ejpam-6076	357	3	}	}	PUNCT
ejpam-6076	357	4	,	,	PUNCT
ejpam-6076	357	5	so	so	SCONJ
ejpam-6076	357	6	that	that	SCONJ
ejpam-6076	357	7	km	km	PROPN
ejpam-6076	357	8	□	□	PROPN
ejpam-6076	357	9	pn	pn	PROPN
ejpam-6076	357	10	consists	consist	VERB
ejpam-6076	357	11	of	of	ADP
ejpam-6076	357	12	mn	mn	PROPN
ejpam-6076	357	13	vertices	vertex	NOUN
ejpam-6076	357	14	.	.	PUNCT
ejpam-6076	358	1	assume	assume	VERB
ejpam-6076	358	2	,	,	PUNCT
ejpam-6076	358	3	for	for	ADP
ejpam-6076	358	4	the	the	DET
ejpam-6076	358	5	sake	sake	NOUN
ejpam-6076	358	6	of	of	ADP
ejpam-6076	358	7	contradiction	contradiction	NOUN
ejpam-6076	358	8	,	,	PUNCT
ejpam-6076	358	9	that	that	SCONJ
ejpam-6076	358	10	km	km	NOUN
ejpam-6076	358	11	□	□	PROPN
ejpam-6076	358	12	pn	pn	PROPN
ejpam-6076	358	13	is	be	AUX
ejpam-6076	358	14	a	a	DET
ejpam-6076	358	15	γcerpgraph	γcerpgraph	NOUN
ejpam-6076	358	16	.	.	PUNCT
ejpam-6076	359	1	without	without	ADP
ejpam-6076	359	2	loss	loss	NOUN
ejpam-6076	359	3	of	of	ADP
ejpam-6076	359	4	generality	generality	NOUN
ejpam-6076	359	5	,	,	PUNCT
ejpam-6076	359	6	suppose	suppose	VERB
ejpam-6076	359	7	(	(	PUNCT
ejpam-6076	359	8	k1	k1	NOUN
ejpam-6076	359	9	,	,	PUNCT
ejpam-6076	359	10	p1	p1	NOUN
ejpam-6076	359	11	)	)	PUNCT
ejpam-6076	359	12	∈	∈	PROPN
ejpam-6076	359	13	j	j	PROPN
ejpam-6076	359	14	.	.	PUNCT
ejpam-6076	360	1	then	then	ADV
ejpam-6076	360	2	(	(	PUNCT
ejpam-6076	360	3	k1	k1	X
ejpam-6076	360	4	,	,	PUNCT
ejpam-6076	360	5	p1	p1	PROPN
ejpam-6076	360	6	)	)	PUNCT
ejpam-6076	360	7	dominates	dominate	VERB
ejpam-6076	360	8	both	both	PRON
ejpam-6076	360	9	(	(	PUNCT
ejpam-6076	360	10	k1	k1	NOUN
ejpam-6076	360	11	,	,	PUNCT
ejpam-6076	360	12	p2	p2	PROPN
ejpam-6076	360	13	)	)	PUNCT
ejpam-6076	360	14	and	and	CCONJ
ejpam-6076	360	15	(	(	PUNCT
ejpam-6076	360	16	kj	kj	PROPN
ejpam-6076	360	17	,	,	PUNCT
ejpam-6076	360	18	p1	p1	PROPN
ejpam-6076	360	19	)	)	PUNCT
ejpam-6076	360	20	for	for	ADP
ejpam-6076	360	21	some	some	DET
ejpam-6076	360	22	j	j	PROPN
ejpam-6076	360	23	=	=	SYM
ejpam-6076	360	24	2	2	NUM
ejpam-6076	360	25	,	,	PUNCT
ejpam-6076	360	26	3	3	NUM
ejpam-6076	360	27	,	,	PUNCT
ejpam-6076	360	28	.	.	PUNCT
ejpam-6076	360	29	.	.	PUNCT
ejpam-6076	361	1	.	.	PUNCT
ejpam-6076	362	1	,	,	PUNCT
ejpam-6076	362	2	m.	m.	NOUN
ejpam-6076	362	3	note	note	VERB
ejpam-6076	362	4	that	that	SCONJ
ejpam-6076	362	5	(	(	PUNCT
ejpam-6076	362	6	ki	ki	INTJ
ejpam-6076	362	7	,	,	PUNCT
ejpam-6076	362	8	p2	p2	PROPN
ejpam-6076	362	9	)	)	PUNCT
ejpam-6076	362	10	/∈	/∈	PUNCT
ejpam-6076	363	1	j	j	NOUN
ejpam-6076	363	2	,	,	PUNCT
ejpam-6076	363	3	as	as	SCONJ
ejpam-6076	363	4	(	(	PUNCT
ejpam-6076	363	5	ki	ki	PROPN
ejpam-6076	363	6	,	,	PUNCT
ejpam-6076	363	7	p1	p1	PROPN
ejpam-6076	363	8	)	)	PUNCT
ejpam-6076	363	9	is	be	AUX
ejpam-6076	363	10	adjacent	adjacent	ADJ
ejpam-6076	363	11	to	to	ADP
ejpam-6076	363	12	(	(	PUNCT
ejpam-6076	363	13	ki	ki	INTJ
ejpam-6076	363	14	,	,	PUNCT
ejpam-6076	363	15	p2	p2	PROPN
ejpam-6076	363	16	)	)	PUNCT
ejpam-6076	363	17	for	for	ADP
ejpam-6076	363	18	some	some	DET
ejpam-6076	363	19	i	i	NOUN
ejpam-6076	363	20	=	=	NOUN
ejpam-6076	363	21	1	1	NUM
ejpam-6076	363	22	,	,	PUNCT
ejpam-6076	363	23	2	2	NUM
ejpam-6076	363	24	,	,	PUNCT
ejpam-6076	363	25	.	.	PUNCT
ejpam-6076	363	26	.	.	PUNCT
ejpam-6076	364	1	.	.	PUNCT
ejpam-6076	365	1	,	,	PUNCT
ejpam-6076	365	2	m.	m.	NOUN
ejpam-6076	365	3	in	in	ADP
ejpam-6076	365	4	this	this	DET
ejpam-6076	365	5	scenario	scenario	NOUN
ejpam-6076	365	6	,	,	PUNCT
ejpam-6076	365	7	(	(	PUNCT
ejpam-6076	365	8	ki	ki	PROPN
ejpam-6076	365	9	,	,	PUNCT
ejpam-6076	365	10	p3	p3	PROPN
ejpam-6076	365	11	)	)	PUNCT
ejpam-6076	365	12	could	could	AUX
ejpam-6076	365	13	dominate	dominate	VERB
ejpam-6076	365	14	(	(	PUNCT
ejpam-6076	365	15	ki	ki	PROPN
ejpam-6076	365	16	,	,	PUNCT
ejpam-6076	365	17	p2	p2	PROPN
ejpam-6076	365	18	)	)	PUNCT
ejpam-6076	365	19	,	,	PUNCT
ejpam-6076	365	20	but	but	CCONJ
ejpam-6076	365	21	(	(	PUNCT
ejpam-6076	365	22	k1	k1	NOUN
ejpam-6076	365	23	,	,	PUNCT
ejpam-6076	365	24	p2	p2	PROPN
ejpam-6076	365	25	)	)	PUNCT
ejpam-6076	365	26	is	be	AUX
ejpam-6076	365	27	already	already	ADV
ejpam-6076	365	28	dominated	dominate	VERB
ejpam-6076	365	29	by	by	ADP
ejpam-6076	365	30	(	(	PUNCT
ejpam-6076	365	31	k1	k1	PROPN
ejpam-6076	365	32	,	,	PUNCT
ejpam-6076	365	33	p1	p1	PROPN
ejpam-6076	365	34	)	)	PUNCT
ejpam-6076	365	35	,	,	PUNCT
ejpam-6076	365	36	violating	violate	VERB
ejpam-6076	365	37	the	the	DET
ejpam-6076	365	38	condition	condition	NOUN
ejpam-6076	365	39	for	for	ADP
ejpam-6076	365	40	perfect	perfect	ADJ
ejpam-6076	365	41	domination	domination	NOUN
ejpam-6076	365	42	.	.	PUNCT
ejpam-6076	366	1	this	this	DET
ejpam-6076	366	2	contradiction	contradiction	NOUN
ejpam-6076	366	3	implies	imply	VERB
ejpam-6076	366	4	that	that	SCONJ
ejpam-6076	366	5	no	no	DET
ejpam-6076	366	6	certified	certify	VERB
ejpam-6076	366	7	perfect	perfect	ADJ
ejpam-6076	366	8	dominating	dominating	NOUN
ejpam-6076	366	9	set	set	NOUN
ejpam-6076	366	10	exists	exist	VERB
ejpam-6076	366	11	in	in	ADP
ejpam-6076	366	12	km	km	PROPN
ejpam-6076	366	13	□	□	SYM
ejpam-6076	366	14	pn	pn	PROPN
ejpam-6076	366	15	,	,	PUNCT
ejpam-6076	366	16	concluding	conclude	VERB
ejpam-6076	366	17	that	that	SCONJ
ejpam-6076	366	18	km	km	PROPN
ejpam-6076	366	19	□	□	PROPN
ejpam-6076	366	20	pn	pn	PROPN
ejpam-6076	366	21	is	be	AUX
ejpam-6076	366	22	not	not	PART
ejpam-6076	366	23	a	a	DET
ejpam-6076	366	24	γcerp	γcerp	NOUN
ejpam-6076	366	25	-	-	PUNCT
ejpam-6076	366	26	graph	graph	NOUN
ejpam-6076	366	27	.	.	PUNCT
ejpam-6076	367	1	the	the	DET
ejpam-6076	367	2	next	next	ADJ
ejpam-6076	367	3	result	result	NOUN
ejpam-6076	367	4	is	be	AUX
ejpam-6076	367	5	a	a	DET
ejpam-6076	367	6	direct	direct	ADJ
ejpam-6076	367	7	consequence	consequence	NOUN
ejpam-6076	367	8	of	of	ADP
ejpam-6076	367	9	theorems	theorem	NOUN
ejpam-6076	367	10	8	8	NUM
ejpam-6076	367	11	and	and	CCONJ
ejpam-6076	367	12	9	9	NUM
ejpam-6076	367	13	.	.	PUNCT
ejpam-6076	367	14	corollary	corollary	ADJ
ejpam-6076	367	15	8	8	NUM
ejpam-6076	367	16	.	.	PUNCT
ejpam-6076	368	1	for	for	ADP
ejpam-6076	368	2	every	every	DET
ejpam-6076	368	3	even	even	ADV
ejpam-6076	368	4	integer	integer	NOUN
ejpam-6076	368	5	n	n	CCONJ
ejpam-6076	368	6	,	,	PUNCT
ejpam-6076	368	7	k2	k2	PROPN
ejpam-6076	368	8	□	□	SYM
ejpam-6076	368	9	pn	pn	PROPN
ejpam-6076	368	10	is	be	AUX
ejpam-6076	368	11	a	a	DET
ejpam-6076	368	12	non	non	ADJ
ejpam-6076	368	13	-	-	ADJ
ejpam-6076	368	14	γcerp	γcerp	ADJ
ejpam-6076	368	15	-	-	PUNCT
ejpam-6076	368	16	graph	graph	NOUN
ejpam-6076	368	17	.	.	PUNCT
ejpam-6076	369	1	proposition	proposition	NOUN
ejpam-6076	369	2	9	9	NUM
ejpam-6076	369	3	.	.	PUNCT
ejpam-6076	370	1	let	let	VERB
ejpam-6076	370	2	n	n	PRON
ejpam-6076	370	3	be	be	AUX
ejpam-6076	370	4	a	a	DET
ejpam-6076	370	5	positive	positive	ADJ
ejpam-6076	370	6	odd	odd	ADJ
ejpam-6076	370	7	integer	integer	NOUN
ejpam-6076	370	8	.	.	PUNCT
ejpam-6076	371	1	then	then	ADV
ejpam-6076	371	2	γcerp(k2	γcerp(k2	NOUN
ejpam-6076	371	3	□	□	PROPN
ejpam-6076	371	4	pn	pn	ADJ
ejpam-6076	371	5	)	)	PUNCT
ejpam-6076	371	6	=	=	PUNCT
ejpam-6076	371	7	⌈n	⌈n	NOUN
ejpam-6076	371	8	2	2	NUM
ejpam-6076	371	9	⌉	⌉	X
ejpam-6076	371	10	.	.	PUNCT
ejpam-6076	372	1	open	open	ADJ
ejpam-6076	372	2	questions	question	NOUN
ejpam-6076	372	3	and	and	CCONJ
ejpam-6076	372	4	problems	problem	NOUN
ejpam-6076	372	5	:	:	PUNCT
ejpam-6076	372	6	problem	problem	NOUN
ejpam-6076	372	7	1	1	NUM
ejpam-6076	372	8	.	.	PUNCT
ejpam-6076	372	9	characterize	characterize	VERB
ejpam-6076	372	10	the	the	DET
ejpam-6076	372	11	trees	tree	NOUN
ejpam-6076	372	12	t	t	NOUN
ejpam-6076	372	13	,	,	PUNCT
ejpam-6076	372	14	certain	certain	ADJ
ejpam-6076	372	15	classes	class	NOUN
ejpam-6076	372	16	of	of	ADP
ejpam-6076	372	17	graphs	graph	NOUN
ejpam-6076	372	18	,	,	PUNCT
ejpam-6076	372	19	and	and	CCONJ
ejpam-6076	372	20	other	other	ADJ
ejpam-6076	372	21	binary	binary	ADJ
ejpam-6076	372	22	operations	operation	NOUN
ejpam-6076	372	23	that	that	PRON
ejpam-6076	372	24	have	have	AUX
ejpam-6076	372	25	not	not	PART
ejpam-6076	372	26	yet	yet	ADV
ejpam-6076	372	27	been	be	AUX
ejpam-6076	372	28	studied	study	VERB
ejpam-6076	372	29	,	,	PUNCT
ejpam-6076	372	30	and	and	CCONJ
ejpam-6076	372	31	determine	determine	VERB
ejpam-6076	372	32	their	their	PRON
ejpam-6076	372	33	certified	certify	VERB
ejpam-6076	372	34	perfect	perfect	ADJ
ejpam-6076	372	35	domination	domination	NOUN
ejpam-6076	372	36	numbers	number	NOUN
ejpam-6076	372	37	.	.	PUNCT
ejpam-6076	373	1	problem	problem	NOUN
ejpam-6076	373	2	2	2	NUM
ejpam-6076	373	3	.	.	AUX
ejpam-6076	373	4	establish	establish	VERB
ejpam-6076	373	5	nordhaus	nordhaus	NOUN
ejpam-6076	373	6	-	-	PUNCT
ejpam-6076	373	7	gaddum	gaddum	NOUN
ejpam-6076	373	8	type	type	NOUN
ejpam-6076	373	9	results	result	NOUN
ejpam-6076	373	10	for	for	ADP
ejpam-6076	373	11	γcerp(g	γcerp(g	NOUN
ejpam-6076	373	12	)	)	PUNCT
ejpam-6076	373	13	.	.	PUNCT
ejpam-6076	374	1	problem	problem	NOUN
ejpam-6076	374	2	3	3	X
ejpam-6076	374	3	.	.	PUNCT
ejpam-6076	374	4	investigate	investigate	VERB
ejpam-6076	374	5	how	how	SCONJ
ejpam-6076	374	6	the	the	DET
ejpam-6076	374	7	certified	certify	VERB
ejpam-6076	374	8	perfect	perfect	ADJ
ejpam-6076	374	9	domination	domination	NOUN
ejpam-6076	374	10	number	number	NOUN
ejpam-6076	374	11	is	be	AUX
ejpam-6076	374	12	affected	affect	VERB
ejpam-6076	374	13	when	when	SCONJ
ejpam-6076	374	14	a	a	DET
ejpam-6076	374	15	graph	graph	NOUN
ejpam-6076	374	16	is	be	AUX
ejpam-6076	374	17	modified	modify	VERB
ejpam-6076	374	18	by	by	ADP
ejpam-6076	374	19	adding	add	VERB
ejpam-6076	374	20	or	or	CCONJ
ejpam-6076	374	21	removing	remove	VERB
ejpam-6076	374	22	an	an	DET
ejpam-6076	374	23	edge	edge	NOUN
ejpam-6076	374	24	or	or	CCONJ
ejpam-6076	374	25	vertex	vertex	NOUN
ejpam-6076	374	26	.	.	PUNCT
ejpam-6076	375	1	conclusion	conclusion	NOUN
ejpam-6076	375	2	:	:	PUNCT
ejpam-6076	375	3	in	in	ADP
ejpam-6076	375	4	this	this	DET
ejpam-6076	375	5	paper	paper	NOUN
ejpam-6076	375	6	,	,	PUNCT
ejpam-6076	375	7	we	we	PRON
ejpam-6076	375	8	extended	extend	VERB
ejpam-6076	375	9	the	the	DET
ejpam-6076	375	10	study	study	NOUN
ejpam-6076	375	11	of	of	ADP
ejpam-6076	375	12	certified	certify	VERB
ejpam-6076	375	13	perfect	perfect	ADJ
ejpam-6076	375	14	domination	domination	NOUN
ejpam-6076	375	15	in	in	ADP
ejpam-6076	375	16	graphs	graph	NOUN
ejpam-6076	375	17	by	by	ADP
ejpam-6076	375	18	presenting	present	VERB
ejpam-6076	375	19	new	new	ADJ
ejpam-6076	375	20	upper	upper	ADJ
ejpam-6076	375	21	bounds	bound	NOUN
ejpam-6076	375	22	for	for	ADP
ejpam-6076	375	23	the	the	DET
ejpam-6076	375	24	certified	certify	VERB
ejpam-6076	375	25	domination	domination	NOUN
ejpam-6076	375	26	number	number	NOUN
ejpam-6076	375	27	γcerp(g	γcerp(g	PROPN
ejpam-6076	375	28	)	)	PUNCT
ejpam-6076	375	29	.	.	PUNCT
ejpam-6076	376	1	we	we	PRON
ejpam-6076	376	2	explored	explore	VERB
ejpam-6076	376	3	the	the	DET
ejpam-6076	376	4	relationship	relationship	NOUN
ejpam-6076	376	5	between	between	ADP
ejpam-6076	376	6	certified	certified	ADJ
ejpam-6076	376	7	domination	domination	NOUN
ejpam-6076	376	8	and	and	CCONJ
ejpam-6076	376	9	certified	certify	VERB
ejpam-6076	376	10	perfect	perfect	ADJ
ejpam-6076	376	11	domination	domination	NOUN
ejpam-6076	376	12	parameters	parameter	NOUN
ejpam-6076	376	13	,	,	PUNCT
ejpam-6076	376	14	identifying	identify	VERB
ejpam-6076	376	15	graphs	graph	NOUN
ejpam-6076	376	16	with	with	ADP
ejpam-6076	376	17	both	both	CCONJ
ejpam-6076	376	18	small	small	ADJ
ejpam-6076	376	19	and	and	CCONJ
ejpam-6076	376	20	large	large	ADJ
ejpam-6076	376	21	values	value	NOUN
ejpam-6076	376	22	of	of	ADP
ejpam-6076	376	23	these	these	DET
ejpam-6076	376	24	parameters	parameter	NOUN
ejpam-6076	376	25	.	.	PUNCT
ejpam-6076	377	1	our	our	PRON
ejpam-6076	377	2	results	result	NOUN
ejpam-6076	377	3	provide	provide	VERB
ejpam-6076	377	4	a	a	DET
ejpam-6076	377	5	deeper	deep	ADJ
ejpam-6076	377	6	understanding	understanding	NOUN
ejpam-6076	377	7	of	of	ADP
ejpam-6076	377	8	the	the	DET
ejpam-6076	377	9	structural	structural	ADJ
ejpam-6076	377	10	properties	property	NOUN
ejpam-6076	377	11	of	of	ADP
ejpam-6076	377	12	graphs	graph	NOUN
ejpam-6076	377	13	in	in	ADP
ejpam-6076	377	14	the	the	DET
ejpam-6076	377	15	context	context	NOUN
ejpam-6076	377	16	of	of	ADP
ejpam-6076	377	17	certified	certify	VERB
ejpam-6076	377	18	perfect	perfect	ADJ
ejpam-6076	377	19	domination	domination	NOUN
ejpam-6076	377	20	.	.	PUNCT
ejpam-6076	378	1	we	we	PRON
ejpam-6076	378	2	also	also	ADV
ejpam-6076	378	3	characterize	characterize	VERB
ejpam-6076	378	4	graphs	graph	NOUN
ejpam-6076	378	5	for	for	ADP
ejpam-6076	378	6	which	which	PRON
ejpam-6076	378	7	γcerp(g	γcerp(g	PROPN
ejpam-6076	378	8	)	)	PUNCT
ejpam-6076	378	9	=	=	SYM
ejpam-6076	378	10	n	n	NOUN
ejpam-6076	378	11	and	and	CCONJ
ejpam-6076	378	12	γcerp(g	γcerp(g	ADJ
ejpam-6076	378	13	)	)	PUNCT
ejpam-6076	378	14	=	=	SYM
ejpam-6076	378	15	γcer(g	γcer(g	NOUN
ejpam-6076	378	16	)	)	PUNCT
ejpam-6076	378	17	,	,	PUNCT
ejpam-6076	378	18	offering	offer	VERB
ejpam-6076	378	19	insight	insight	NOUN
ejpam-6076	378	20	into	into	ADP
ejpam-6076	378	21	special	special	ADJ
ejpam-6076	378	22	classes	class	NOUN
ejpam-6076	378	23	of	of	ADP
ejpam-6076	378	24	graphs	graph	NOUN
ejpam-6076	378	25	where	where	SCONJ
ejpam-6076	378	26	the	the	DET
ejpam-6076	378	27	certified	certify	VERB
ejpam-6076	378	28	perfect	perfect	ADJ
ejpam-6076	378	29	domination	domination	NOUN
ejpam-6076	378	30	number	number	NOUN
ejpam-6076	378	31	attains	attain	VERB
ejpam-6076	378	32	these	these	DET
ejpam-6076	378	33	specific	specific	ADJ
ejpam-6076	378	34	values	value	NOUN
ejpam-6076	378	35	.	.	PUNCT
ejpam-6076	379	1	furthermore	furthermore	ADV
ejpam-6076	379	2	,	,	PUNCT
ejpam-6076	379	3	our	our	PRON
ejpam-6076	379	4	investigation	investigation	NOUN
ejpam-6076	379	5	into	into	ADP
ejpam-6076	379	6	the	the	DET
ejpam-6076	379	7	lexicographic	lexicographic	ADJ
ejpam-6076	379	8	and	and	CCONJ
ejpam-6076	379	9	cartesian	cartesian	ADJ
ejpam-6076	379	10	products	product	NOUN
ejpam-6076	379	11	of	of	ADP
ejpam-6076	379	12	graphs	graph	NOUN
ejpam-6076	379	13	revealed	reveal	VERB
ejpam-6076	379	14	the	the	DET
ejpam-6076	379	15	behavior	behavior	NOUN
ejpam-6076	379	16	of	of	ADP
ejpam-6076	379	17	certified	certify	VERB
ejpam-6076	379	18	perfect	perfect	ADJ
ejpam-6076	379	19	dominating	dominating	NOUN
ejpam-6076	379	20	sets	set	NOUN
ejpam-6076	379	21	under	under	ADP
ejpam-6076	379	22	these	these	DET
ejpam-6076	379	23	operations	operation	NOUN
ejpam-6076	379	24	.	.	PUNCT
ejpam-6076	380	1	we	we	PRON
ejpam-6076	380	2	determined	determine	VERB
ejpam-6076	380	3	the	the	DET
ejpam-6076	380	4	certified	certify	VERB
ejpam-6076	380	5	perfect	perfect	ADJ
ejpam-6076	380	6	domination	domination	NOUN
ejpam-6076	380	7	number	number	NOUN
ejpam-6076	380	8	for	for	ADP
ejpam-6076	380	9	such	such	ADJ
ejpam-6076	380	10	products	product	NOUN
ejpam-6076	380	11	and	and	CCONJ
ejpam-6076	380	12	identified	identify	VERB
ejpam-6076	380	13	non	non	ADJ
ejpam-6076	380	14	-	-	ADJ
ejpam-6076	380	15	γcerp	γcerp	ADJ
ejpam-6076	380	16	-	-	PUNCT
ejpam-6076	380	17	graphs	graph	NOUN
ejpam-6076	380	18	arising	arise	VERB
ejpam-6076	380	19	from	from	ADP
ejpam-6076	380	20	these	these	DET
ejpam-6076	380	21	binary	binary	ADJ
ejpam-6076	380	22	operations	operation	NOUN
ejpam-6076	380	23	.	.	PUNCT
ejpam-6076	381	1	acknowledgements	acknowledgement	NOUN
ejpam-6076	381	2	the	the	DET
ejpam-6076	381	3	authors	author	NOUN
ejpam-6076	381	4	sincerely	sincerely	ADV
ejpam-6076	381	5	thank	thank	VERB
ejpam-6076	381	6	the	the	DET
ejpam-6076	381	7	reviewers	reviewer	NOUN
ejpam-6076	381	8	for	for	ADP
ejpam-6076	381	9	their	their	PRON
ejpam-6076	381	10	valuable	valuable	ADJ
ejpam-6076	381	11	comments	comment	NOUN
ejpam-6076	381	12	and	and	CCONJ
ejpam-6076	381	13	suggestions	suggestion	NOUN
ejpam-6076	381	14	that	that	PRON
ejpam-6076	381	15	significantly	significantly	ADV
ejpam-6076	381	16	improved	improve	VERB
ejpam-6076	381	17	the	the	DET
ejpam-6076	381	18	quality	quality	NOUN
ejpam-6076	381	19	of	of	ADP
ejpam-6076	381	20	this	this	DET
ejpam-6076	381	21	paper	paper	NOUN
ejpam-6076	381	22	.	.	PUNCT
ejpam-6076	382	1	gratitude	gratitude	NOUN
ejpam-6076	382	2	is	be	AUX
ejpam-6076	382	3	also	also	ADV
ejpam-6076	382	4	extended	extend	VERB
ejpam-6076	382	5	to	to	ADP
ejpam-6076	382	6	mindanao	mindanao	PROPN
ejpam-6076	382	7	state	state	PROPN
ejpam-6076	382	8	university	university	PROPN
ejpam-6076	382	9	–	–	PUNCT
ejpam-6076	382	10	tawi	tawi	NOUN
ejpam-6076	382	11	-	-	PUNCT
ejpam-6076	382	12	tawi	tawi	NOUN
ejpam-6076	382	13	college	college	PROPN
ejpam-6076	382	14	of	of	ADP
ejpam-6076	382	15	technology	technology	NOUN
ejpam-6076	382	16	and	and	CCONJ
ejpam-6076	382	17	oceanography	oceanography	NOUN
ejpam-6076	382	18	(	(	PUNCT
ejpam-6076	382	19	msutcto	msutcto	X
ejpam-6076	382	20	)	)	PUNCT
ejpam-6076	382	21	for	for	ADP
ejpam-6076	382	22	the	the	DET
ejpam-6076	382	23	financial	financial	ADJ
ejpam-6076	382	24	support	support	NOUN
ejpam-6076	382	25	that	that	PRON
ejpam-6076	382	26	made	make	VERB
ejpam-6076	382	27	the	the	DET
ejpam-6076	382	28	publication	publication	NOUN
ejpam-6076	382	29	of	of	ADP
ejpam-6076	382	30	this	this	DET
ejpam-6076	382	31	work	work	NOUN
ejpam-6076	382	32	possible	possible	ADJ
ejpam-6076	382	33	.	.	PUNCT
ejpam-6076	383	1	j.	j.	PROPN
ejpam-6076	383	2	j.	j.	PROPN
ejpam-6076	383	3	hamja	hamja	PROPN
ejpam-6076	383	4	et	et	PROPN
ejpam-6076	383	5	al	al	PROPN
ejpam-6076	383	6	.	.	PUNCT
ejpam-6076	383	7	/	/	SYM
ejpam-6076	383	8	eur	eur	PROPN
ejpam-6076	383	9	.	.	PUNCT
ejpam-6076	384	1	j.	j.	PROPN
ejpam-6076	384	2	pure	pure	PROPN
ejpam-6076	384	3	appl	appl	PROPN
ejpam-6076	384	4	.	.	PROPN
ejpam-6076	384	5	math	math	PROPN
ejpam-6076	384	6	,	,	PUNCT
ejpam-6076	384	7	18	18	NUM
ejpam-6076	384	8	(	(	PUNCT
ejpam-6076	384	9	3	3	NUM
ejpam-6076	384	10	)	)	PUNCT
ejpam-6076	384	11	(	(	PUNCT
ejpam-6076	384	12	2025	2025	NUM
ejpam-6076	384	13	)	)	PUNCT
ejpam-6076	384	14	,	,	PUNCT
ejpam-6076	384	15	6076	6076	NUM
ejpam-6076	384	16	13	13	NUM
ejpam-6076	384	17	of	of	ADP
ejpam-6076	384	18	13	13	NUM
ejpam-6076	384	19	references	reference	NOUN
ejpam-6076	384	20	[	[	X
ejpam-6076	384	21	1	1	NUM
ejpam-6076	384	22	]	]	PUNCT
ejpam-6076	384	23	m.	m.	NOUN
ejpam-6076	384	24	dettlaff	dettlaff	NOUN
ejpam-6076	384	25	,	,	PUNCT
ejpam-6076	384	26	m.	m.	NOUN
ejpam-6076	384	27	lemańska	lemańska	PROPN
ejpam-6076	384	28	,	,	PUNCT
ejpam-6076	384	29	j.	j.	PROPN
ejpam-6076	384	30	topp	topp	PROPN
ejpam-6076	384	31	,	,	PUNCT
ejpam-6076	384	32	r.	r.	PROPN
ejpam-6076	384	33	ziemann	ziemann	PROPN
ejpam-6076	384	34	,	,	PUNCT
ejpam-6076	384	35	and	and	CCONJ
ejpam-6076	384	36	p.	p.	PROPN
ejpam-6076	384	37	żyliński	żyliński	PROPN
ejpam-6076	384	38	.	.	PUNCT
ejpam-6076	385	1	certified	certified	ADJ
ejpam-6076	385	2	domination	domination	NOUN
ejpam-6076	385	3	.	.	PUNCT
ejpam-6076	386	1	akce	akce	PROPN
ejpam-6076	386	2	international	international	PROPN
ejpam-6076	386	3	journal	journal	NOUN
ejpam-6076	386	4	of	of	ADP
ejpam-6076	386	5	graphs	graph	NOUN
ejpam-6076	386	6	and	and	CCONJ
ejpam-6076	386	7	combinatorics	combinatoric	NOUN
ejpam-6076	386	8	,	,	PUNCT
ejpam-6076	386	9	pages	page	NOUN
ejpam-6076	386	10	1–12	1–12	PROPN
ejpam-6076	386	11	,	,	PUNCT
ejpam-6076	386	12	2020	2020	NUM
ejpam-6076	386	13	.	.	PUNCT
ejpam-6076	387	1	[	[	X
ejpam-6076	387	2	2	2	NUM
ejpam-6076	387	3	]	]	PUNCT
ejpam-6076	387	4	m.	m.	NOUN
ejpam-6076	387	5	dettlaff	dettlaff	NOUN
ejpam-6076	387	6	,	,	PUNCT
ejpam-6076	387	7	m.	m.	NOUN
ejpam-6076	387	8	lemańska	lemańska	PROPN
ejpam-6076	387	9	,	,	PUNCT
ejpam-6076	387	10	m.	m.	NOUN
ejpam-6076	387	11	miotk	miotk	NOUN
ejpam-6076	387	12	,	,	PUNCT
ejpam-6076	387	13	j.	j.	PROPN
ejpam-6076	387	14	topp	topp	PROPN
ejpam-6076	387	15	,	,	PUNCT
ejpam-6076	387	16	r.	r.	PROPN
ejpam-6076	387	17	ziemann	ziemann	PROPN
ejpam-6076	387	18	,	,	PUNCT
ejpam-6076	387	19	and	and	CCONJ
ejpam-6076	387	20	p.	p.	PROPN
ejpam-6076	387	21	żyliński	żyliński	PROPN
ejpam-6076	387	22	.	.	PUNCT
ejpam-6076	388	1	graphs	graph	NOUN
ejpam-6076	388	2	with	with	ADP
ejpam-6076	388	3	equal	equal	ADJ
ejpam-6076	388	4	domination	domination	NOUN
ejpam-6076	388	5	and	and	CCONJ
ejpam-6076	388	6	certified	certified	ADJ
ejpam-6076	388	7	domination	domination	NOUN
ejpam-6076	388	8	numbers	number	NOUN
ejpam-6076	388	9	.	.	PUNCT
ejpam-6076	389	1	arxiv	arxiv	PROPN
ejpam-6076	389	2	,	,	PUNCT
ejpam-6076	389	3	pages	page	NOUN
ejpam-6076	389	4	1–15	1–15	PROPN
ejpam-6076	389	5	,	,	PUNCT
ejpam-6076	389	6	2017	2017	NUM
ejpam-6076	389	7	.	.	PUNCT
ejpam-6076	389	8	retrieved	retrieve	VERB
ejpam-6076	389	9	from	from	ADP
ejpam-6076	389	10	http://arxiv.org/abs/1710.02059v2	http://arxiv.org/abs/1710.02059v2	X
ejpam-6076	389	11	.	.	PUNCT
ejpam-6076	390	1	[	[	X
ejpam-6076	390	2	3	3	X
ejpam-6076	390	3	]	]	X
ejpam-6076	390	4	j.	j.	PROPN
ejpam-6076	390	5	j.	j.	PROPN
ejpam-6076	390	6	hamja	hamja	PROPN
ejpam-6076	390	7	.	.	PUNCT
ejpam-6076	391	1	certified	certify	VERB
ejpam-6076	391	2	perfect	perfect	ADJ
ejpam-6076	391	3	domination	domination	NOUN
ejpam-6076	391	4	in	in	ADP
ejpam-6076	391	5	graphs	graph	NOUN
ejpam-6076	391	6	.	.	PUNCT
ejpam-6076	392	1	european	european	ADJ
ejpam-6076	392	2	journal	journal	PROPN
ejpam-6076	392	3	of	of	ADP
ejpam-6076	392	4	pure	pure	ADJ
ejpam-6076	392	5	and	and	CCONJ
ejpam-6076	392	6	applied	applied	ADJ
ejpam-6076	392	7	mathematics	mathematic	NOUN
ejpam-6076	392	8	,	,	PUNCT
ejpam-6076	392	9	16:2763–2774	16:2763–2774	NUM
ejpam-6076	392	10	,	,	PUNCT
ejpam-6076	392	11	2023	2023	NUM
ejpam-6076	392	12	.	.	PUNCT
ejpam-6076	393	1	[	[	X
ejpam-6076	393	2	4	4	NUM
ejpam-6076	393	3	]	]	X
ejpam-6076	393	4	r.	r.	PROPN
ejpam-6076	393	5	diestel	diestel	PROPN
ejpam-6076	393	6	.	.	PUNCT
ejpam-6076	394	1	graph	graph	NOUN
ejpam-6076	394	2	theory	theory	NOUN
ejpam-6076	394	3	.	.	PUNCT
ejpam-6076	395	1	springer	springer	PROPN
ejpam-6076	395	2	,	,	PUNCT
ejpam-6076	395	3	heidelberg	heidelberg	PROPN
ejpam-6076	395	4	,	,	PUNCT
ejpam-6076	395	5	2012	2012	NUM
ejpam-6076	395	6	.	.	PUNCT
ejpam-6076	396	1	[	[	X
ejpam-6076	396	2	5	5	X
ejpam-6076	396	3	]	]	PUNCT
ejpam-6076	396	4	t.	t.	PROPN
ejpam-6076	396	5	w.	w.	PROPN
ejpam-6076	396	6	haynes	haynes	PROPN
ejpam-6076	396	7	,	,	PUNCT
ejpam-6076	396	8	s.	s.	PROPN
ejpam-6076	396	9	t.	t.	PROPN
ejpam-6076	396	10	hedetniemi	hedetniemi	PROPN
ejpam-6076	396	11	,	,	PUNCT
ejpam-6076	396	12	and	and	CCONJ
ejpam-6076	396	13	m.	m.	PROPN
ejpam-6076	396	14	a.	a.	PROPN
ejpam-6076	396	15	henning	henning	PROPN
ejpam-6076	396	16	.	.	PUNCT
ejpam-6076	397	1	topics	topic	NOUN
ejpam-6076	397	2	in	in	ADP
ejpam-6076	397	3	domination	domination	NOUN
ejpam-6076	397	4	in	in	ADP
ejpam-6076	397	5	graphs	graph	NOUN
ejpam-6076	397	6	,	,	PUNCT
ejpam-6076	397	7	volume	volume	NOUN
ejpam-6076	397	8	64	64	NUM
ejpam-6076	397	9	of	of	ADP
ejpam-6076	397	10	developments	development	NOUN
ejpam-6076	397	11	in	in	ADP
ejpam-6076	397	12	mathematics	mathematic	NOUN
ejpam-6076	397	13	.	.	PUNCT
ejpam-6076	398	1	springer	springer	NOUN
ejpam-6076	398	2	,	,	PUNCT
ejpam-6076	398	3	2020	2020	NUM
ejpam-6076	398	4	.	.	PUNCT
ejpam-6076	399	1	[	[	X
ejpam-6076	399	2	6	6	NUM
ejpam-6076	399	3	]	]	PUNCT
ejpam-6076	399	4	s.	s.	PROPN
ejpam-6076	399	5	paraico	paraico	PROPN
ejpam-6076	399	6	and	and	CCONJ
ejpam-6076	399	7	s.	s.	PROPN
ejpam-6076	399	8	canoy	canoy	PROPN
ejpam-6076	399	9	jr	jr	PROPN
ejpam-6076	399	10	.	.	PUNCT
ejpam-6076	399	11	super	super	ADJ
ejpam-6076	399	12	dominating	dominating	NOUN
ejpam-6076	399	13	sets	set	NOUN
ejpam-6076	399	14	in	in	ADP
ejpam-6076	399	15	some	some	DET
ejpam-6076	399	16	products	product	NOUN
ejpam-6076	399	17	of	of	ADP
ejpam-6076	399	18	graphs	graph	NOUN
ejpam-6076	399	19	.	.	PUNCT
ejpam-6076	400	1	far	far	PROPN
ejpam-6076	400	2	east	east	PROPN
ejpam-6076	400	3	journal	journal	PROPN
ejpam-6076	400	4	of	of	ADP
ejpam-6076	400	5	mathematical	mathematical	ADJ
ejpam-6076	400	6	sciences	science	NOUN
ejpam-6076	400	7	,	,	PUNCT
ejpam-6076	400	8	102(10):2393–2402	102(10):2393–2402	NUM
ejpam-6076	400	9	,	,	PUNCT
ejpam-6076	400	10	2017	2017	NUM
ejpam-6076	400	11	.	.	PUNCT
ejpam-6076	401	1	[	[	X
ejpam-6076	401	2	7	7	X
ejpam-6076	401	3	]	]	X
ejpam-6076	401	4	m.	m.	NOUN
ejpam-6076	401	5	livingston	livingston	PROPN
ejpam-6076	401	6	and	and	CCONJ
ejpam-6076	401	7	q.	q.	PROPN
ejpam-6076	401	8	f.	f.	PROPN
ejpam-6076	401	9	stout	stout	PROPN
ejpam-6076	401	10	.	.	PUNCT
ejpam-6076	402	1	perfect	perfect	ADJ
ejpam-6076	402	2	dominating	dominating	NOUN
ejpam-6076	402	3	sets	set	NOUN
ejpam-6076	402	4	.	.	PUNCT
ejpam-6076	403	1	congressus	congressus	PROPN
ejpam-6076	403	2	numerantium	numerantium	PROPN
ejpam-6076	403	3	,	,	PUNCT
ejpam-6076	403	4	pages	page	NOUN
ejpam-6076	403	5	187–203	187–203	NUM
ejpam-6076	403	6	,	,	PUNCT
ejpam-6076	403	7	1990	1990	NUM
ejpam-6076	403	8	.	.	PUNCT
ejpam-6076	404	1	[	[	X
ejpam-6076	404	2	8	8	NUM
ejpam-6076	404	3	]	]	X
ejpam-6076	404	4	c.	c.	PROPN
ejpam-6076	404	5	l.	l.	PROPN
ejpam-6076	404	6	armada	armada	PROPN
ejpam-6076	404	7	and	and	CCONJ
ejpam-6076	404	8	j.	j.	PROPN
ejpam-6076	404	9	j.	j.	PROPN
ejpam-6076	404	10	hamja	hamja	PROPN
ejpam-6076	404	11	.	.	PUNCT
ejpam-6076	405	1	perfect	perfect	PROPN
ejpam-6076	405	2	isolate	isolate	NOUN
ejpam-6076	405	3	domination	domination	NOUN
ejpam-6076	405	4	in	in	ADP
ejpam-6076	405	5	graphs	graph	NOUN
ejpam-6076	405	6	.	.	PUNCT
ejpam-6076	406	1	european	european	ADJ
ejpam-6076	406	2	journal	journal	PROPN
ejpam-6076	406	3	of	of	ADP
ejpam-6076	406	4	pure	pure	ADJ
ejpam-6076	406	5	and	and	CCONJ
ejpam-6076	406	6	applied	applied	ADJ
ejpam-6076	406	7	mathematics	mathematic	NOUN
ejpam-6076	406	8	,	,	PUNCT
ejpam-6076	406	9	16:1326–1341	16:1326–1341	NUM
ejpam-6076	406	10	,	,	PUNCT
ejpam-6076	406	11	2023	2023	NUM
ejpam-6076	406	12	.	.	PUNCT
ejpam-6076	407	1	[	[	X
ejpam-6076	407	2	9	9	NUM
ejpam-6076	407	3	]	]	PUNCT
ejpam-6076	407	4	f.	f.	PROPN
ejpam-6076	407	5	harary	harary	PROPN
ejpam-6076	407	6	.	.	PUNCT
ejpam-6076	408	1	graph	graph	NOUN
ejpam-6076	408	2	theory	theory	NOUN
ejpam-6076	408	3	.	.	PUNCT
ejpam-6076	409	1	addison	addison	PROPN
ejpam-6076	409	2	-	-	PUNCT
ejpam-6076	409	3	wesley	wesley	PROPN
ejpam-6076	409	4	publishing	publishing	PROPN
ejpam-6076	409	5	company	company	PROPN
ejpam-6076	409	6	,	,	PUNCT
ejpam-6076	409	7	inc	inc	PROPN
ejpam-6076	409	8	.	.	PROPN
ejpam-6076	409	9	,	,	PUNCT
ejpam-6076	409	10	massachusetts	massachusetts	PROPN
ejpam-6076	409	11	,	,	PUNCT
ejpam-6076	409	12	1969	1969	NUM
ejpam-6076	409	13	.	.	PUNCT