id	sid	tid	token	lemma	pos
ejpam-4883	1	1	european	european	PROPN
ejpam-4883	1	2	journal	journal	PROPN
ejpam-4883	1	3	of	of	ADP
ejpam-4883	1	4	pure	pure	ADJ
ejpam-4883	1	5	and	and	CCONJ
ejpam-4883	1	6	applied	apply	VERB
ejpam-4883	1	7	mathematics	mathematic	NOUN
ejpam-4883	1	8	vol	vol	NOUN
ejpam-4883	1	9	.	.	PUNCT
ejpam-4883	2	1	16	16	NUM
ejpam-4883	2	2	,	,	PUNCT
ejpam-4883	2	3	no	no	INTJ
ejpam-4883	2	4	.	.	NOUN
ejpam-4883	2	5	4	4	NUM
ejpam-4883	2	6	,	,	PUNCT
ejpam-4883	2	7	2023	2023	NUM
ejpam-4883	2	8	,	,	PUNCT
ejpam-4883	2	9	2082	2082	NUM
ejpam-4883	2	10	-	-	SYM
ejpam-4883	2	11	2095	2095	NUM
ejpam-4883	2	12	issn	issn	PROPN
ejpam-4883	2	13	1307	1307	NUM
ejpam-4883	2	14	-	-	SYM
ejpam-4883	2	15	5543	5543	NUM
ejpam-4883	2	16	–	–	PUNCT
ejpam-4883	2	17	ejpam.com	ejpam.com	X
ejpam-4883	2	18	published	publish	VERB
ejpam-4883	2	19	by	by	ADP
ejpam-4883	2	20	new	new	PROPN
ejpam-4883	2	21	york	york	PROPN
ejpam-4883	2	22	business	business	PROPN
ejpam-4883	2	23	global	global	PROPN
ejpam-4883	2	24	j	j	PROPN
ejpam-4883	2	25	-	-	PUNCT
ejpam-4883	2	26	domination	domination	NOUN
ejpam-4883	2	27	in	in	ADP
ejpam-4883	2	28	graphs	graph	NOUN
ejpam-4883	2	29	javier	javier	PROPN
ejpam-4883	2	30	a.	a.	PROPN
ejpam-4883	2	31	hassan1,∗	hassan1,∗	PROPN
ejpam-4883	2	32	,	,	PUNCT
ejpam-4883	2	33	jeffrey	jeffrey	PROPN
ejpam-4883	2	34	imer	imer	PROPN
ejpam-4883	2	35	salim1	salim1	PROPN
ejpam-4883	3	1	1mathematics	1mathematics	NUM
ejpam-4883	3	2	and	and	CCONJ
ejpam-4883	3	3	sciences	sciences	PROPN
ejpam-4883	3	4	department	department	PROPN
ejpam-4883	3	5	,	,	PUNCT
ejpam-4883	3	6	college	college	NOUN
ejpam-4883	3	7	of	of	ADP
ejpam-4883	3	8	arts	art	NOUN
ejpam-4883	3	9	and	and	CCONJ
ejpam-4883	3	10	sciences	science	NOUN
ejpam-4883	3	11	,	,	PUNCT
ejpam-4883	3	12	msu	msu	PROPN
ejpam-4883	3	13	-	-	PUNCT
ejpam-4883	3	14	tawi	tawi	NOUN
ejpam-4883	3	15	-	-	PUNCT
ejpam-4883	3	16	tawi	tawi	NOUN
ejpam-4883	3	17	college	college	PROPN
ejpam-4883	3	18	of	of	ADP
ejpam-4883	3	19	technology	technology	NOUN
ejpam-4883	3	20	and	and	CCONJ
ejpam-4883	3	21	oceanography	oceanography	NOUN
ejpam-4883	3	22	,	,	PUNCT
ejpam-4883	3	23	bongao	bongao	NOUN
ejpam-4883	3	24	,	,	PUNCT
ejpam-4883	3	25	tawi	tawi	NOUN
ejpam-4883	3	26	-	-	PUNCT
ejpam-4883	3	27	tawi	tawi	NOUN
ejpam-4883	3	28	,	,	PUNCT
ejpam-4883	3	29	philippines	philippine	NOUN
ejpam-4883	3	30	abstract	abstract	ADJ
ejpam-4883	3	31	.	.	PUNCT
ejpam-4883	4	1	let	let	VERB
ejpam-4883	4	2	g	g	PRON
ejpam-4883	4	3	be	be	AUX
ejpam-4883	4	4	a	a	DET
ejpam-4883	4	5	graph	graph	NOUN
ejpam-4883	4	6	.	.	PUNCT
ejpam-4883	5	1	a	a	DET
ejpam-4883	5	2	subset	subset	NOUN
ejpam-4883	5	3	d	d	NOUN
ejpam-4883	5	4	=	=	PUNCT
ejpam-4883	5	5	{	{	PUNCT
ejpam-4883	5	6	d1	d1	PROPN
ejpam-4883	5	7	,	,	PUNCT
ejpam-4883	5	8	d2	d2	PROPN
ejpam-4883	5	9	,	,	PUNCT
ejpam-4883	5	10	·	·	PUNCT
ejpam-4883	5	11	·	·	PUNCT
ejpam-4883	5	12	·	·	PUNCT
ejpam-4883	5	13	,	,	PUNCT
ejpam-4883	5	14	dm	dm	INTJ
ejpam-4883	5	15	}	}	PUNCT
ejpam-4883	5	16	of	of	ADP
ejpam-4883	5	17	vertices	vertex	NOUN
ejpam-4883	5	18	of	of	ADP
ejpam-4883	5	19	g	g	PROPN
ejpam-4883	5	20	is	be	AUX
ejpam-4883	5	21	called	call	VERB
ejpam-4883	5	22	a	a	DET
ejpam-4883	5	23	j	j	NOUN
ejpam-4883	5	24	-	-	PUNCT
ejpam-4883	5	25	set	set	VERB
ejpam-4883	5	26	if	if	SCONJ
ejpam-4883	5	27	ng[di	ng[di	X
ejpam-4883	5	28	]	]	X
ejpam-4883	5	29	\ng[dj	\ng[dj	X
ejpam-4883	5	30	]	]	PUNCT
ejpam-4883	5	31	̸=	̸=	NOUN
ejpam-4883	5	32	∅	∅	NOUN
ejpam-4883	5	33	for	for	ADP
ejpam-4883	5	34	every	every	DET
ejpam-4883	5	35	i	i	PROPN
ejpam-4883	5	36	̸=	̸=	PROPN
ejpam-4883	5	37	j	j	PROPN
ejpam-4883	5	38	,	,	PUNCT
ejpam-4883	5	39	where	where	SCONJ
ejpam-4883	5	40	i	i	PRON
ejpam-4883	5	41	,	,	PUNCT
ejpam-4883	5	42	j	j	PROPN
ejpam-4883	5	43	∈	∈	PROPN
ejpam-4883	5	44	{	{	PUNCT
ejpam-4883	5	45	1	1	NUM
ejpam-4883	5	46	,	,	PUNCT
ejpam-4883	5	47	2	2	NUM
ejpam-4883	5	48	,	,	PUNCT
ejpam-4883	5	49	.	.	PUNCT
ejpam-4883	5	50	.	.	PUNCT
ejpam-4883	6	1	.	.	PUNCT
ejpam-4883	7	1	,	,	PUNCT
ejpam-4883	7	2	m	m	VERB
ejpam-4883	7	3	}	}	PUNCT
ejpam-4883	7	4	.	.	PUNCT
ejpam-4883	8	1	a	a	DET
ejpam-4883	8	2	j	j	NOUN
ejpam-4883	8	3	-	-	PUNCT
ejpam-4883	8	4	set	set	PROPN
ejpam-4883	8	5	is	be	AUX
ejpam-4883	8	6	called	call	VERB
ejpam-4883	8	7	a	a	DET
ejpam-4883	8	8	j	j	PROPN
ejpam-4883	8	9	-	-	PUNCT
ejpam-4883	8	10	dominating	dominating	ADJ
ejpam-4883	8	11	set	set	NOUN
ejpam-4883	8	12	of	of	ADP
ejpam-4883	8	13	g	g	PROPN
ejpam-4883	8	14	if	if	SCONJ
ejpam-4883	8	15	d	d	PROPN
ejpam-4883	8	16	=	=	PRON
ejpam-4883	8	17	{	{	PUNCT
ejpam-4883	8	18	d1	d1	PROPN
ejpam-4883	8	19	,	,	PUNCT
ejpam-4883	8	20	d2	d2	PROPN
ejpam-4883	8	21	,	,	PUNCT
ejpam-4883	8	22	.	.	PUNCT
ejpam-4883	8	23	.	.	PUNCT
ejpam-4883	9	1	.	.	PUNCT
ejpam-4883	10	1	,	,	PUNCT
ejpam-4883	10	2	dm	dm	PROPN
ejpam-4883	10	3	}	}	PUNCT
ejpam-4883	10	4	is	be	AUX
ejpam-4883	10	5	a	a	DET
ejpam-4883	10	6	dominating	dominating	NOUN
ejpam-4883	10	7	set	set	NOUN
ejpam-4883	10	8	of	of	ADP
ejpam-4883	10	9	g.	g.	PROPN
ejpam-4883	10	10	the	the	DET
ejpam-4883	10	11	j	j	PROPN
ejpam-4883	10	12	-	-	PUNCT
ejpam-4883	10	13	domination	domination	NOUN
ejpam-4883	10	14	number	number	NOUN
ejpam-4883	10	15	of	of	ADP
ejpam-4883	10	16	g	g	NOUN
ejpam-4883	10	17	,	,	PUNCT
ejpam-4883	10	18	denoted	denote	VERB
ejpam-4883	10	19	by	by	ADP
ejpam-4883	10	20	γj(g	γj(g	NOUN
ejpam-4883	10	21	)	)	PUNCT
ejpam-4883	10	22	,	,	PUNCT
ejpam-4883	10	23	is	be	AUX
ejpam-4883	10	24	the	the	DET
ejpam-4883	10	25	maximum	maximum	ADJ
ejpam-4883	10	26	cardinality	cardinality	NOUN
ejpam-4883	10	27	of	of	ADP
ejpam-4883	10	28	a	a	DET
ejpam-4883	10	29	j	j	PROPN
ejpam-4883	10	30	-	-	PUNCT
ejpam-4883	10	31	dominating	dominating	ADJ
ejpam-4883	10	32	set	set	NOUN
ejpam-4883	10	33	of	of	ADP
ejpam-4883	10	34	g.	g.	PROPN
ejpam-4883	10	35	in	in	ADP
ejpam-4883	10	36	this	this	DET
ejpam-4883	10	37	paper	paper	NOUN
ejpam-4883	10	38	,	,	PUNCT
ejpam-4883	10	39	we	we	PRON
ejpam-4883	10	40	introduce	introduce	VERB
ejpam-4883	10	41	this	this	DET
ejpam-4883	10	42	new	new	ADJ
ejpam-4883	10	43	concept	concept	NOUN
ejpam-4883	10	44	and	and	CCONJ
ejpam-4883	10	45	we	we	PRON
ejpam-4883	10	46	establish	establish	VERB
ejpam-4883	10	47	formulas	formula	NOUN
ejpam-4883	10	48	and	and	CCONJ
ejpam-4883	10	49	properties	property	NOUN
ejpam-4883	10	50	on	on	ADP
ejpam-4883	10	51	some	some	DET
ejpam-4883	10	52	classes	class	NOUN
ejpam-4883	10	53	of	of	ADP
ejpam-4883	10	54	graphs	graph	NOUN
ejpam-4883	10	55	and	and	CCONJ
ejpam-4883	10	56	in	in	ADP
ejpam-4883	10	57	join	join	NOUN
ejpam-4883	10	58	of	of	ADP
ejpam-4883	10	59	two	two	NUM
ejpam-4883	10	60	graphs	graph	NOUN
ejpam-4883	10	61	.	.	PUNCT
ejpam-4883	11	1	upper	upper	ADJ
ejpam-4883	11	2	and	and	CCONJ
ejpam-4883	11	3	lower	low	ADJ
ejpam-4883	11	4	bounds	bound	NOUN
ejpam-4883	11	5	of	of	ADP
ejpam-4883	11	6	j	j	NOUN
ejpam-4883	11	7	-	-	PUNCT
ejpam-4883	11	8	domination	domination	NOUN
ejpam-4883	11	9	parameter	parameter	NOUN
ejpam-4883	11	10	with	with	ADP
ejpam-4883	11	11	respect	respect	NOUN
ejpam-4883	11	12	to	to	ADP
ejpam-4883	11	13	the	the	DET
ejpam-4883	11	14	order	order	NOUN
ejpam-4883	11	15	of	of	ADP
ejpam-4883	11	16	a	a	DET
ejpam-4883	11	17	graph	graph	NOUN
ejpam-4883	11	18	and	and	CCONJ
ejpam-4883	11	19	other	other	ADJ
ejpam-4883	11	20	parameters	parameter	NOUN
ejpam-4883	11	21	in	in	ADP
ejpam-4883	11	22	graph	graph	NOUN
ejpam-4883	11	23	theory	theory	NOUN
ejpam-4883	11	24	are	be	AUX
ejpam-4883	11	25	obtained	obtain	VERB
ejpam-4883	11	26	.	.	PUNCT
ejpam-4883	12	1	in	in	ADP
ejpam-4883	12	2	addition	addition	NOUN
ejpam-4883	12	3	,	,	PUNCT
ejpam-4883	12	4	we	we	PRON
ejpam-4883	12	5	present	present	VERB
ejpam-4883	12	6	realization	realization	NOUN
ejpam-4883	12	7	result	result	NOUN
ejpam-4883	12	8	involving	involve	VERB
ejpam-4883	12	9	this	this	DET
ejpam-4883	12	10	parameter	parameter	NOUN
ejpam-4883	12	11	and	and	CCONJ
ejpam-4883	12	12	the	the	DET
ejpam-4883	12	13	standard	standard	ADJ
ejpam-4883	12	14	domination	domination	NOUN
ejpam-4883	12	15	.	.	PUNCT
ejpam-4883	13	1	moreover	moreover	ADV
ejpam-4883	13	2	,	,	PUNCT
ejpam-4883	13	3	we	we	PRON
ejpam-4883	13	4	characterize	characterize	VERB
ejpam-4883	13	5	j	j	PROPN
ejpam-4883	13	6	-	-	PUNCT
ejpam-4883	13	7	dominating	dominating	NOUN
ejpam-4883	13	8	sets	set	NOUN
ejpam-4883	13	9	in	in	ADP
ejpam-4883	13	10	some	some	DET
ejpam-4883	13	11	classes	class	NOUN
ejpam-4883	13	12	of	of	ADP
ejpam-4883	13	13	graphs	graph	NOUN
ejpam-4883	13	14	and	and	CCONJ
ejpam-4883	13	15	join	join	NOUN
ejpam-4883	13	16	of	of	ADP
ejpam-4883	13	17	two	two	NUM
ejpam-4883	13	18	graphs	graph	NOUN
ejpam-4883	13	19	and	and	CCONJ
ejpam-4883	13	20	finally	finally	ADV
ejpam-4883	13	21	determine	determine	VERB
ejpam-4883	13	22	the	the	DET
ejpam-4883	13	23	exact	exact	ADJ
ejpam-4883	13	24	value	value	NOUN
ejpam-4883	13	25	of	of	ADP
ejpam-4883	13	26	the	the	DET
ejpam-4883	13	27	parameter	parameter	NOUN
ejpam-4883	13	28	of	of	ADP
ejpam-4883	13	29	each	each	PRON
ejpam-4883	13	30	of	of	ADP
ejpam-4883	13	31	these	these	DET
ejpam-4883	13	32	graphs	graph	NOUN
ejpam-4883	13	33	.	.	PUNCT
ejpam-4883	14	1	2020	2020	NUM
ejpam-4883	14	2	mathematics	mathematic	NOUN
ejpam-4883	14	3	subject	subject	NOUN
ejpam-4883	14	4	classifications	classification	NOUN
ejpam-4883	14	5	:	:	PUNCT
ejpam-4883	14	6	05c69	05c69	X
ejpam-4883	14	7	key	key	ADJ
ejpam-4883	14	8	words	word	NOUN
ejpam-4883	14	9	and	and	CCONJ
ejpam-4883	14	10	phrases	phrase	NOUN
ejpam-4883	14	11	:	:	PUNCT
ejpam-4883	14	12	j	j	NOUN
ejpam-4883	14	13	-	-	PUNCT
ejpam-4883	14	14	set	set	PROPN
ejpam-4883	14	15	,	,	PUNCT
ejpam-4883	14	16	j	j	PROPN
ejpam-4883	14	17	-	-	PUNCT
ejpam-4883	14	18	dominating	dominating	NOUN
ejpam-4883	14	19	set	set	NOUN
ejpam-4883	14	20	,	,	PUNCT
ejpam-4883	14	21	j	j	NOUN
ejpam-4883	14	22	-	-	PUNCT
ejpam-4883	14	23	domination	domination	NOUN
ejpam-4883	14	24	number	number	NOUN
ejpam-4883	14	25	1	1	NUM
ejpam-4883	14	26	.	.	PUNCT
ejpam-4883	15	1	introduction	introduction	NOUN
ejpam-4883	15	2	domination	domination	NOUN
ejpam-4883	15	3	in	in	ADP
ejpam-4883	15	4	a	a	DET
ejpam-4883	15	5	graph	graph	NOUN
ejpam-4883	15	6	has	have	AUX
ejpam-4883	15	7	been	be	AUX
ejpam-4883	15	8	one	one	NUM
ejpam-4883	15	9	of	of	ADP
ejpam-4883	15	10	the	the	DET
ejpam-4883	15	11	most	most	ADV
ejpam-4883	15	12	interesting	interesting	ADJ
ejpam-4883	15	13	and	and	CCONJ
ejpam-4883	15	14	widely	widely	ADV
ejpam-4883	15	15	studied	study	VERB
ejpam-4883	15	16	topics	topic	NOUN
ejpam-4883	15	17	in	in	ADP
ejpam-4883	15	18	graph	graph	NOUN
ejpam-4883	15	19	theory	theory	NOUN
ejpam-4883	15	20	.	.	PUNCT
ejpam-4883	16	1	it	it	PRON
ejpam-4883	16	2	has	have	VERB
ejpam-4883	16	3	many	many	ADJ
ejpam-4883	16	4	applications	application	NOUN
ejpam-4883	16	5	in	in	ADP
ejpam-4883	16	6	various	various	ADJ
ejpam-4883	16	7	fields	field	NOUN
ejpam-4883	16	8	and	and	CCONJ
ejpam-4883	16	9	in	in	ADP
ejpam-4883	16	10	networks	network	NOUN
ejpam-4883	16	11	.	.	PUNCT
ejpam-4883	17	1	the	the	DET
ejpam-4883	17	2	study	study	NOUN
ejpam-4883	17	3	of	of	ADP
ejpam-4883	17	4	domination	domination	NOUN
ejpam-4883	17	5	in	in	ADP
ejpam-4883	17	6	graphs	graph	NOUN
ejpam-4883	17	7	came	come	VERB
ejpam-4883	17	8	about	about	ADV
ejpam-4883	17	9	partially	partially	ADV
ejpam-4883	17	10	as	as	ADP
ejpam-4883	17	11	a	a	DET
ejpam-4883	17	12	result	result	NOUN
ejpam-4883	17	13	of	of	ADP
ejpam-4883	17	14	the	the	DET
ejpam-4883	17	15	study	study	NOUN
ejpam-4883	17	16	of	of	ADP
ejpam-4883	17	17	games	game	NOUN
ejpam-4883	17	18	and	and	CCONJ
ejpam-4883	17	19	recreational	recreational	ADJ
ejpam-4883	17	20	mathematics	mathematic	NOUN
ejpam-4883	17	21	.	.	PUNCT
ejpam-4883	18	1	in	in	ADP
ejpam-4883	18	2	particular	particular	ADJ
ejpam-4883	18	3	,	,	PUNCT
ejpam-4883	18	4	mathematicians	mathematician	NOUN
ejpam-4883	18	5	studied	study	VERB
ejpam-4883	18	6	how	how	SCONJ
ejpam-4883	18	7	chess	chess	NOUN
ejpam-4883	18	8	pieces	piece	NOUN
ejpam-4883	18	9	of	of	ADP
ejpam-4883	18	10	a	a	DET
ejpam-4883	18	11	particular	particular	ADJ
ejpam-4883	18	12	type	type	NOUN
ejpam-4883	18	13	could	could	AUX
ejpam-4883	18	14	be	be	AUX
ejpam-4883	18	15	placed	place	VERB
ejpam-4883	18	16	on	on	ADP
ejpam-4883	18	17	a	a	DET
ejpam-4883	18	18	chessboard	chessboard	NOUN
ejpam-4883	18	19	in	in	ADP
ejpam-4883	18	20	such	such	DET
ejpam-4883	18	21	a	a	DET
ejpam-4883	18	22	way	way	NOUN
ejpam-4883	18	23	that	that	PRON
ejpam-4883	18	24	they	they	PRON
ejpam-4883	18	25	would	would	AUX
ejpam-4883	18	26	attack	attack	VERB
ejpam-4883	18	27	,	,	PUNCT
ejpam-4883	18	28	or	or	CCONJ
ejpam-4883	18	29	dominate	dominate	VERB
ejpam-4883	18	30	,	,	PUNCT
ejpam-4883	18	31	every	every	DET
ejpam-4883	18	32	square	square	NOUN
ejpam-4883	18	33	on	on	ADP
ejpam-4883	18	34	the	the	DET
ejpam-4883	18	35	board	board	NOUN
ejpam-4883	18	36	.	.	PUNCT
ejpam-4883	19	1	in	in	ADP
ejpam-4883	19	2	1962	1962	NUM
ejpam-4883	19	3	,	,	PUNCT
ejpam-4883	19	4	oystein	oystein	ADJ
ejpam-4883	19	5	ore	ore	NOUN
ejpam-4883	19	6	introduced	introduce	VERB
ejpam-4883	19	7	the	the	DET
ejpam-4883	19	8	concepts	concept	NOUN
ejpam-4883	19	9	of	of	ADP
ejpam-4883	19	10	dominating	dominate	VERB
ejpam-4883	19	11	set	set	NOUN
ejpam-4883	19	12	and	and	CCONJ
ejpam-4883	19	13	domination	domination	NOUN
ejpam-4883	19	14	number	number	NOUN
ejpam-4883	19	15	in	in	ADP
ejpam-4883	19	16	a	a	DET
ejpam-4883	19	17	graph	graph	NOUN
ejpam-4883	19	18	in	in	ADP
ejpam-4883	19	19	his	his	PRON
ejpam-4883	19	20	book	book	NOUN
ejpam-4883	19	21	on	on	ADP
ejpam-4883	19	22	graph	graph	NOUN
ejpam-4883	19	23	theory	theory	NOUN
ejpam-4883	19	24	[	[	X
ejpam-4883	19	25	12	12	NUM
ejpam-4883	19	26	]	]	PUNCT
ejpam-4883	19	27	.	.	PUNCT
ejpam-4883	20	1	a	a	DET
ejpam-4883	20	2	decade	decade	NOUN
ejpam-4883	20	3	later	later	ADV
ejpam-4883	20	4	,	,	PUNCT
ejpam-4883	20	5	cockayne	cockayne	NOUN
ejpam-4883	20	6	and	and	CCONJ
ejpam-4883	20	7	hedetniemi	hedetniemi	ADV
ejpam-4883	20	8	published	publish	VERB
ejpam-4883	20	9	a	a	DET
ejpam-4883	20	10	survey	survey	NOUN
ejpam-4883	20	11	paper	paper	NOUN
ejpam-4883	20	12	,	,	PUNCT
ejpam-4883	20	13	in	in	ADP
ejpam-4883	20	14	which	which	PRON
ejpam-4883	20	15	the	the	DET
ejpam-4883	20	16	notation	notation	PROPN
ejpam-4883	20	17	γ(g	γ(g	PROPN
ejpam-4883	20	18	)	)	PUNCT
ejpam-4883	20	19	was	be	AUX
ejpam-4883	20	20	first	first	ADV
ejpam-4883	20	21	used	use	VERB
ejpam-4883	20	22	for	for	ADP
ejpam-4883	20	23	the	the	DET
ejpam-4883	20	24	domination	domination	NOUN
ejpam-4883	20	25	number	number	NOUN
ejpam-4883	20	26	of	of	ADP
ejpam-4883	20	27	a	a	DET
ejpam-4883	20	28	graph	graph	NOUN
ejpam-4883	20	29	g	g	PROPN
ejpam-4883	20	30	[	[	X
ejpam-4883	20	31	3	3	NUM
ejpam-4883	20	32	]	]	PUNCT
ejpam-4883	20	33	.	.	PUNCT
ejpam-4883	21	1	since	since	SCONJ
ejpam-4883	21	2	then	then	ADV
ejpam-4883	21	3	,	,	PUNCT
ejpam-4883	21	4	several	several	ADJ
ejpam-4883	21	5	mathematicians	mathematician	NOUN
ejpam-4883	21	6	had	have	AUX
ejpam-4883	21	7	studied	study	VERB
ejpam-4883	21	8	and	and	CCONJ
ejpam-4883	21	9	introduced	introduce	VERB
ejpam-4883	21	10	new	new	ADJ
ejpam-4883	21	11	domination	domination	NOUN
ejpam-4883	21	12	parameters	parameter	NOUN
ejpam-4883	21	13	in	in	ADP
ejpam-4883	21	14	graphs	graph	NOUN
ejpam-4883	21	15	.	.	PUNCT
ejpam-4883	22	1	some	some	DET
ejpam-4883	22	2	variants	variant	NOUN
ejpam-4883	22	3	of	of	ADP
ejpam-4883	22	4	domination	domination	NOUN
ejpam-4883	22	5	were	be	AUX
ejpam-4883	22	6	defined	define	VERB
ejpam-4883	22	7	and	and	CCONJ
ejpam-4883	22	8	further	far	ADV
ejpam-4883	22	9	studied	study	VERB
ejpam-4883	22	10	by	by	ADP
ejpam-4883	22	11	researchers	researcher	NOUN
ejpam-4883	22	12	can	can	AUX
ejpam-4883	22	13	be	be	AUX
ejpam-4883	22	14	found	find	VERB
ejpam-4883	22	15	in	in	ADP
ejpam-4883	22	16	[	[	X
ejpam-4883	22	17	1	1	NUM
ejpam-4883	22	18	,	,	PUNCT
ejpam-4883	22	19	2	2	NUM
ejpam-4883	22	20	,	,	PUNCT
ejpam-4883	22	21	4–11	4–11	NOUN
ejpam-4883	22	22	,	,	PUNCT
ejpam-4883	22	23	13	13	NUM
ejpam-4883	22	24	,	,	PUNCT
ejpam-4883	22	25	14	14	NUM
ejpam-4883	22	26	]	]	PUNCT
ejpam-4883	22	27	.	.	PUNCT
ejpam-4883	23	1	∗corresponding	∗corresponde	VERB
ejpam-4883	23	2	author	author	NOUN
ejpam-4883	23	3	.	.	PUNCT
ejpam-4883	24	1	doi	doi	NOUN
ejpam-4883	24	2	:	:	PUNCT
ejpam-4883	24	3	https://doi.org/10.29020/nybg.ejpam.v16i4.4883	https://doi.org/10.29020/nybg.ejpam.v16i4.4883	ADJ
ejpam-4883	24	4	email	email	NOUN
ejpam-4883	24	5	addresses	address	VERB
ejpam-4883	24	6	:	:	PUNCT
ejpam-4883	24	7	javierhassan@msutawi-tawi.edu.ph	javierhassan@msutawi-tawi.edu.ph	PROPN
ejpam-4883	24	8	(	(	PUNCT
ejpam-4883	24	9	j.	j.	PROPN
ejpam-4883	24	10	a.	a.	PROPN
ejpam-4883	24	11	hassan	hassan	PROPN
ejpam-4883	24	12	)	)	PUNCT
ejpam-4883	25	1	jeffreyimersalim@msutawi-tawi.edu.ph	jeffreyimersalim@msutawi-tawi.edu.ph	PROPN
ejpam-4883	25	2	(	(	PUNCT
ejpam-4883	25	3	j.	j.	PROPN
ejpam-4883	25	4	i.	i.	PROPN
ejpam-4883	25	5	salim	salim	PROPN
ejpam-4883	25	6	)	)	PUNCT
ejpam-4883	25	7	https://www.ejpam.com	https://www.ejpam.com	X
ejpam-4883	25	8	2082	2082	NUM
ejpam-4883	25	9	©	©	ADP
ejpam-4883	25	10	2023	2023	NUM
ejpam-4883	25	11	ejpam	ejpam	NOUN
ejpam-4883	25	12	all	all	DET
ejpam-4883	25	13	rights	right	NOUN
ejpam-4883	25	14	reserved	reserve	VERB
ejpam-4883	25	15	.	.	PUNCT
ejpam-4883	26	1	j.	j.	PROPN
ejpam-4883	26	2	hassan	hassan	PROPN
ejpam-4883	26	3	,	,	PUNCT
ejpam-4883	26	4	j.	j.	PROPN
ejpam-4883	26	5	salim	salim	PROPN
ejpam-4883	26	6	/	/	SYM
ejpam-4883	26	7	eur	eur	PROPN
ejpam-4883	26	8	.	.	PUNCT
ejpam-4883	27	1	j.	j.	PROPN
ejpam-4883	27	2	pure	pure	PROPN
ejpam-4883	27	3	appl	appl	PROPN
ejpam-4883	27	4	.	.	PROPN
ejpam-4883	27	5	math	math	PROPN
ejpam-4883	27	6	,	,	PUNCT
ejpam-4883	27	7	16	16	NUM
ejpam-4883	27	8	(	(	PUNCT
ejpam-4883	27	9	4	4	NUM
ejpam-4883	27	10	)	)	PUNCT
ejpam-4883	27	11	(	(	PUNCT
ejpam-4883	27	12	2023	2023	NUM
ejpam-4883	27	13	)	)	PUNCT
ejpam-4883	27	14	,	,	PUNCT
ejpam-4883	27	15	2082	2082	NUM
ejpam-4883	27	16	-	-	SYM
ejpam-4883	27	17	2095	2095	NUM
ejpam-4883	27	18	2083	2083	NUM
ejpam-4883	27	19	in	in	ADP
ejpam-4883	27	20	this	this	DET
ejpam-4883	27	21	paper	paper	NOUN
ejpam-4883	27	22	,	,	PUNCT
ejpam-4883	27	23	new	new	ADJ
ejpam-4883	27	24	variant	variant	NOUN
ejpam-4883	27	25	of	of	ADP
ejpam-4883	27	26	domination	domination	NOUN
ejpam-4883	27	27	called	call	VERB
ejpam-4883	27	28	j	j	NOUN
ejpam-4883	27	29	-	-	PUNCT
ejpam-4883	27	30	domination	domination	NOUN
ejpam-4883	27	31	in	in	ADP
ejpam-4883	27	32	a	a	DET
ejpam-4883	27	33	graph	graph	NOUN
ejpam-4883	27	34	will	will	AUX
ejpam-4883	27	35	be	be	AUX
ejpam-4883	27	36	introduced	introduce	VERB
ejpam-4883	27	37	and	and	CCONJ
ejpam-4883	27	38	investigated	investigate	VERB
ejpam-4883	27	39	.	.	PUNCT
ejpam-4883	28	1	its	its	PRON
ejpam-4883	28	2	relationships	relationship	NOUN
ejpam-4883	28	3	with	with	ADP
ejpam-4883	28	4	other	other	ADJ
ejpam-4883	28	5	variants	variant	NOUN
ejpam-4883	28	6	of	of	ADP
ejpam-4883	28	7	domination	domination	NOUN
ejpam-4883	28	8	and	and	CCONJ
ejpam-4883	28	9	other	other	ADJ
ejpam-4883	28	10	concepts	concept	NOUN
ejpam-4883	28	11	in	in	ADP
ejpam-4883	28	12	graph	graph	NOUN
ejpam-4883	28	13	theory	theory	NOUN
ejpam-4883	28	14	will	will	AUX
ejpam-4883	28	15	be	be	AUX
ejpam-4883	28	16	determined	determine	VERB
ejpam-4883	28	17	.	.	PUNCT
ejpam-4883	29	1	moreover	moreover	ADV
ejpam-4883	29	2	,	,	PUNCT
ejpam-4883	29	3	characterizations	characterization	NOUN
ejpam-4883	29	4	of	of	ADP
ejpam-4883	29	5	j	j	PROPN
ejpam-4883	29	6	-	-	PUNCT
ejpam-4883	29	7	dominating	dominating	NOUN
ejpam-4883	29	8	sets	set	NOUN
ejpam-4883	29	9	in	in	ADP
ejpam-4883	29	10	some	some	DET
ejpam-4883	29	11	classes	class	NOUN
ejpam-4883	29	12	of	of	ADP
ejpam-4883	29	13	graphs	graph	NOUN
ejpam-4883	29	14	and	and	CCONJ
ejpam-4883	29	15	join	join	NOUN
ejpam-4883	29	16	of	of	ADP
ejpam-4883	29	17	two	two	NUM
ejpam-4883	29	18	of	of	ADP
ejpam-4883	29	19	graphs	graph	NOUN
ejpam-4883	29	20	will	will	AUX
ejpam-4883	29	21	be	be	AUX
ejpam-4883	29	22	presented	present	VERB
ejpam-4883	29	23	.	.	PUNCT
ejpam-4883	30	1	these	these	DET
ejpam-4883	30	2	results	result	NOUN
ejpam-4883	30	3	will	will	AUX
ejpam-4883	30	4	be	be	AUX
ejpam-4883	30	5	used	use	VERB
ejpam-4883	30	6	to	to	PART
ejpam-4883	30	7	determine	determine	VERB
ejpam-4883	30	8	exact	exact	ADJ
ejpam-4883	30	9	values	value	NOUN
ejpam-4883	30	10	or	or	CCONJ
ejpam-4883	30	11	bounds	bound	NOUN
ejpam-4883	30	12	of	of	ADP
ejpam-4883	30	13	the	the	DET
ejpam-4883	30	14	parameter	parameter	NOUN
ejpam-4883	30	15	for	for	ADP
ejpam-4883	30	16	these	these	DET
ejpam-4883	30	17	graphs	graph	NOUN
ejpam-4883	30	18	.	.	PUNCT
ejpam-4883	31	1	we	we	PRON
ejpam-4883	31	2	believe	believe	VERB
ejpam-4883	31	3	that	that	SCONJ
ejpam-4883	31	4	this	this	DET
ejpam-4883	31	5	study	study	NOUN
ejpam-4883	31	6	and	and	CCONJ
ejpam-4883	31	7	its	its	PRON
ejpam-4883	31	8	results	result	NOUN
ejpam-4883	31	9	will	will	AUX
ejpam-4883	31	10	help	help	VERB
ejpam-4883	31	11	other	other	ADJ
ejpam-4883	31	12	researchers	researcher	NOUN
ejpam-4883	31	13	in	in	ADP
ejpam-4883	31	14	the	the	DET
ejpam-4883	31	15	field	field	NOUN
ejpam-4883	31	16	for	for	ADP
ejpam-4883	31	17	more	more	ADJ
ejpam-4883	31	18	research	research	NOUN
ejpam-4883	31	19	directions	direction	NOUN
ejpam-4883	31	20	in	in	ADP
ejpam-4883	31	21	the	the	DET
ejpam-4883	31	22	future	future	NOUN
ejpam-4883	31	23	.	.	PUNCT
ejpam-4883	32	1	2	2	X
ejpam-4883	32	2	.	.	X
ejpam-4883	32	3	terminology	terminology	NOUN
ejpam-4883	32	4	and	and	CCONJ
ejpam-4883	32	5	notation	notation	NOUN
ejpam-4883	32	6	let	let	VERB
ejpam-4883	32	7	g	g	NOUN
ejpam-4883	32	8	=	=	SYM
ejpam-4883	32	9	(	(	PUNCT
ejpam-4883	32	10	v	v	NOUN
ejpam-4883	32	11	(	(	PUNCT
ejpam-4883	32	12	g	g	NOUN
ejpam-4883	32	13	)	)	PUNCT
ejpam-4883	32	14	,	,	PUNCT
ejpam-4883	32	15	e(g	e(g	PROPN
ejpam-4883	32	16	)	)	PUNCT
ejpam-4883	32	17	)	)	PUNCT
ejpam-4883	32	18	be	be	AUX
ejpam-4883	32	19	a	a	DET
ejpam-4883	32	20	simple	simple	ADJ
ejpam-4883	32	21	and	and	CCONJ
ejpam-4883	32	22	undirected	undirected	ADJ
ejpam-4883	32	23	graph	graph	NOUN
ejpam-4883	32	24	.	.	PUNCT
ejpam-4883	33	1	two	two	NUM
ejpam-4883	33	2	vertices	vertex	NOUN
ejpam-4883	33	3	x	x	X
ejpam-4883	33	4	,	,	PUNCT
ejpam-4883	33	5	y	y	PROPN
ejpam-4883	33	6	of	of	ADP
ejpam-4883	33	7	g	g	PROPN
ejpam-4883	33	8	are	be	AUX
ejpam-4883	33	9	adjacent	adjacent	ADJ
ejpam-4883	33	10	,	,	PUNCT
ejpam-4883	33	11	or	or	CCONJ
ejpam-4883	33	12	neighbors	neighbor	NOUN
ejpam-4883	33	13	,	,	PUNCT
ejpam-4883	33	14	if	if	SCONJ
ejpam-4883	33	15	xy	xy	PROPN
ejpam-4883	33	16	is	be	AUX
ejpam-4883	33	17	an	an	DET
ejpam-4883	33	18	edge	edge	NOUN
ejpam-4883	33	19	of	of	ADP
ejpam-4883	33	20	g.	g.	PROPN
ejpam-4883	33	21	the	the	DET
ejpam-4883	33	22	open	open	ADJ
ejpam-4883	33	23	neighborhood	neighborhood	NOUN
ejpam-4883	33	24	of	of	ADP
ejpam-4883	33	25	x	x	PUNCT
ejpam-4883	33	26	in	in	ADP
ejpam-4883	33	27	g	g	PROPN
ejpam-4883	33	28	is	be	AUX
ejpam-4883	33	29	the	the	DET
ejpam-4883	33	30	set	set	NOUN
ejpam-4883	33	31	ng(x	ng(x	NUM
ejpam-4883	33	32	)	)	PUNCT
ejpam-4883	34	1	=	=	PRON
ejpam-4883	34	2	{	{	PUNCT
ejpam-4883	34	3	y	y	PROPN
ejpam-4883	34	4	∈	∈	PROPN
ejpam-4883	34	5	v	v	NOUN
ejpam-4883	34	6	(	(	PUNCT
ejpam-4883	34	7	g	g	NOUN
ejpam-4883	34	8	)	)	PUNCT
ejpam-4883	34	9	:	:	PUNCT
ejpam-4883	34	10	xy	xy	PROPN
ejpam-4883	34	11	∈	∈	PROPN
ejpam-4883	34	12	e(g	e(g	PROPN
ejpam-4883	34	13	)	)	PUNCT
ejpam-4883	34	14	}	}	PUNCT
ejpam-4883	34	15	.	.	PUNCT
ejpam-4883	35	1	the	the	DET
ejpam-4883	35	2	closed	closed	ADJ
ejpam-4883	35	3	neighborhood	neighborhood	NOUN
ejpam-4883	35	4	of	of	ADP
ejpam-4883	35	5	x	x	PUNCT
ejpam-4883	35	6	in	in	ADP
ejpam-4883	35	7	g	g	PROPN
ejpam-4883	35	8	is	be	AUX
ejpam-4883	35	9	the	the	DET
ejpam-4883	35	10	set	set	NOUN
ejpam-4883	35	11	ng[x	ng[x	PROPN
ejpam-4883	35	12	]	]	X
ejpam-4883	35	13	=	=	PUNCT
ejpam-4883	35	14	ng(x	ng(x	X
ejpam-4883	35	15	)	)	PUNCT
ejpam-4883	35	16	∪	∪	ADP
ejpam-4883	35	17	{	{	PUNCT
ejpam-4883	35	18	x	x	NOUN
ejpam-4883	35	19	}	}	PUNCT
ejpam-4883	35	20	.	.	PUNCT
ejpam-4883	36	1	if	if	SCONJ
ejpam-4883	36	2	x	x	PROPN
ejpam-4883	36	3	⊆	⊆	NUM
ejpam-4883	36	4	v	v	X
ejpam-4883	36	5	(	(	PUNCT
ejpam-4883	36	6	g	g	NOUN
ejpam-4883	36	7	)	)	PUNCT
ejpam-4883	36	8	,	,	PUNCT
ejpam-4883	36	9	the	the	DET
ejpam-4883	36	10	open	open	ADJ
ejpam-4883	36	11	neighborhood	neighborhood	NOUN
ejpam-4883	36	12	of	of	ADP
ejpam-4883	36	13	x	x	PUNCT
ejpam-4883	36	14	in	in	ADP
ejpam-4883	36	15	g	g	PROPN
ejpam-4883	36	16	is	be	AUX
ejpam-4883	36	17	the	the	DET
ejpam-4883	36	18	set	set	NOUN
ejpam-4883	36	19	ng(x	ng(x	NUM
ejpam-4883	36	20	)	)	PUNCT
ejpam-4883	37	1	=	=	SYM
ejpam-4883	37	2	⋃	⋃	NOUN
ejpam-4883	37	3	x∈x	x∈x	NOUN
ejpam-4883	37	4	ng(x	ng(x	NUM
ejpam-4883	37	5	)	)	PUNCT
ejpam-4883	37	6	.	.	PUNCT
ejpam-4883	38	1	the	the	DET
ejpam-4883	38	2	closed	closed	ADJ
ejpam-4883	38	3	neighborhood	neighborhood	NOUN
ejpam-4883	38	4	of	of	ADP
ejpam-4883	38	5	x	x	PUNCT
ejpam-4883	38	6	in	in	ADP
ejpam-4883	38	7	g	g	PROPN
ejpam-4883	38	8	is	be	AUX
ejpam-4883	38	9	the	the	DET
ejpam-4883	38	10	set	set	NOUN
ejpam-4883	38	11	ng[x	ng[x	PROPN
ejpam-4883	38	12	]	]	X
ejpam-4883	38	13	=	=	SYM
ejpam-4883	38	14	ng(x)∪x	ng(x)∪x	PROPN
ejpam-4883	38	15	.	.	PUNCT
ejpam-4883	39	1	a	a	DET
ejpam-4883	39	2	subset	subset	NOUN
ejpam-4883	39	3	d	d	NOUN
ejpam-4883	39	4	of	of	ADP
ejpam-4883	39	5	v	v	NOUN
ejpam-4883	39	6	(	(	PUNCT
ejpam-4883	39	7	g	g	NOUN
ejpam-4883	39	8	)	)	PUNCT
ejpam-4883	39	9	is	be	AUX
ejpam-4883	39	10	called	call	VERB
ejpam-4883	39	11	a	a	DET
ejpam-4883	39	12	dominating	dominating	NOUN
ejpam-4883	39	13	of	of	ADP
ejpam-4883	39	14	g	g	PROPN
ejpam-4883	39	15	if	if	SCONJ
ejpam-4883	39	16	for	for	ADP
ejpam-4883	39	17	every	every	DET
ejpam-4883	39	18	x	x	SYM
ejpam-4883	39	19	∈	∈	PROPN
ejpam-4883	39	20	v	v	ADP
ejpam-4883	39	21	(	(	PUNCT
ejpam-4883	39	22	g	g	NOUN
ejpam-4883	39	23	)	)	PUNCT
ejpam-4883	39	24	\	\	PUNCT
ejpam-4883	40	1	d	d	X
ejpam-4883	40	2	,	,	PUNCT
ejpam-4883	40	3	there	there	PRON
ejpam-4883	40	4	exists	exist	VERB
ejpam-4883	40	5	y	y	PROPN
ejpam-4883	40	6	∈	∈	PROPN
ejpam-4883	41	1	d	d	X
ejpam-4883	41	2	such	such	ADJ
ejpam-4883	41	3	that	that	SCONJ
ejpam-4883	41	4	xy	xy	PROPN
ejpam-4883	41	5	∈	∈	PROPN
ejpam-4883	41	6	e(g	e(g	PROPN
ejpam-4883	41	7	)	)	PUNCT
ejpam-4883	41	8	.	.	PUNCT
ejpam-4883	42	1	the	the	DET
ejpam-4883	42	2	domination	domination	NOUN
ejpam-4883	42	3	number	number	NOUN
ejpam-4883	42	4	of	of	ADP
ejpam-4883	42	5	g	g	NOUN
ejpam-4883	42	6	,	,	PUNCT
ejpam-4883	42	7	denoted	denote	VERB
ejpam-4883	42	8	by	by	ADP
ejpam-4883	42	9	γ(g	γ(g	PROPN
ejpam-4883	42	10	)	)	PUNCT
ejpam-4883	42	11	,	,	PUNCT
ejpam-4883	42	12	is	be	AUX
ejpam-4883	42	13	the	the	DET
ejpam-4883	42	14	minimum	minimum	ADJ
ejpam-4883	42	15	cardinality	cardinality	NOUN
ejpam-4883	42	16	of	of	ADP
ejpam-4883	42	17	a	a	DET
ejpam-4883	42	18	dominating	dominating	NOUN
ejpam-4883	42	19	set	set	VERB
ejpam-4883	42	20	in	in	ADP
ejpam-4883	42	21	g.	g.	PROPN
ejpam-4883	42	22	a	a	DET
ejpam-4883	42	23	graph	graph	NOUN
ejpam-4883	42	24	g	g	NOUN
ejpam-4883	42	25	is	be	AUX
ejpam-4883	42	26	connected	connect	VERB
ejpam-4883	42	27	if	if	SCONJ
ejpam-4883	42	28	every	every	DET
ejpam-4883	42	29	pair	pair	NOUN
ejpam-4883	42	30	of	of	ADP
ejpam-4883	42	31	its	its	PRON
ejpam-4883	42	32	vertices	vertex	NOUN
ejpam-4883	42	33	can	can	AUX
ejpam-4883	42	34	be	be	AUX
ejpam-4883	42	35	joined	join	VERB
ejpam-4883	42	36	by	by	ADP
ejpam-4883	42	37	a	a	DET
ejpam-4883	42	38	path	path	NOUN
ejpam-4883	42	39	.	.	PUNCT
ejpam-4883	43	1	otherwise	otherwise	ADV
ejpam-4883	43	2	,	,	PUNCT
ejpam-4883	43	3	g	g	PROPN
ejpam-4883	43	4	is	be	AUX
ejpam-4883	43	5	disconnected	disconnect	VERB
ejpam-4883	43	6	.	.	PUNCT
ejpam-4883	44	1	a	a	DET
ejpam-4883	44	2	maximal	maximal	ADJ
ejpam-4883	44	3	connected	connected	ADJ
ejpam-4883	44	4	subgraph	subgraph	NOUN
ejpam-4883	44	5	(	(	PUNCT
ejpam-4883	44	6	not	not	PART
ejpam-4883	44	7	a	a	DET
ejpam-4883	44	8	subgraph	subgraph	NOUN
ejpam-4883	44	9	of	of	ADP
ejpam-4883	44	10	any	any	DET
ejpam-4883	44	11	connected	connected	ADJ
ejpam-4883	44	12	subgraph	subgraph	NOUN
ejpam-4883	44	13	)	)	PUNCT
ejpam-4883	44	14	of	of	ADP
ejpam-4883	44	15	g	g	PROPN
ejpam-4883	44	16	is	be	AUX
ejpam-4883	44	17	called	call	VERB
ejpam-4883	44	18	a	a	DET
ejpam-4883	44	19	component	component	NOUN
ejpam-4883	44	20	of	of	ADP
ejpam-4883	44	21	g.	g.	PROPN
ejpam-4883	44	22	a	a	DET
ejpam-4883	44	23	dominating	dominating	NOUN
ejpam-4883	44	24	set	set	NOUN
ejpam-4883	44	25	d	d	NOUN
ejpam-4883	44	26	of	of	ADP
ejpam-4883	44	27	g	g	PROPN
ejpam-4883	44	28	is	be	AUX
ejpam-4883	44	29	called	call	VERB
ejpam-4883	44	30	a	a	DET
ejpam-4883	44	31	connected	connect	VERB
ejpam-4883	44	32	dominating	dominating	NOUN
ejpam-4883	44	33	if	if	SCONJ
ejpam-4883	44	34	the	the	DET
ejpam-4883	44	35	induced	induced	ADJ
ejpam-4883	44	36	subgraph	subgraph	NOUN
ejpam-4883	44	37	⟨d⟩	⟨d⟩	PROPN
ejpam-4883	44	38	of	of	ADP
ejpam-4883	44	39	d	d	PROPN
ejpam-4883	44	40	is	be	AUX
ejpam-4883	44	41	connected	connect	VERB
ejpam-4883	44	42	.	.	PUNCT
ejpam-4883	45	1	the	the	DET
ejpam-4883	45	2	connected	connected	ADJ
ejpam-4883	45	3	domination	domination	NOUN
ejpam-4883	45	4	number	number	NOUN
ejpam-4883	45	5	of	of	ADP
ejpam-4883	45	6	g	g	NOUN
ejpam-4883	45	7	,	,	PUNCT
ejpam-4883	45	8	denoted	denote	VERB
ejpam-4883	45	9	by	by	ADP
ejpam-4883	45	10	γc(g	γc(g	NOUN
ejpam-4883	45	11	)	)	PUNCT
ejpam-4883	45	12	,	,	PUNCT
ejpam-4883	45	13	is	be	AUX
ejpam-4883	45	14	the	the	DET
ejpam-4883	45	15	minimum	minimum	ADJ
ejpam-4883	45	16	cardinality	cardinality	NOUN
ejpam-4883	45	17	of	of	ADP
ejpam-4883	45	18	a	a	DET
ejpam-4883	45	19	connected	connect	VERB
ejpam-4883	45	20	dominating	dominating	NOUN
ejpam-4883	45	21	set	set	VERB
ejpam-4883	45	22	in	in	ADP
ejpam-4883	45	23	g.	g.	PROPN
ejpam-4883	45	24	any	any	DET
ejpam-4883	45	25	connected	connected	ADJ
ejpam-4883	45	26	dominating	dominating	NOUN
ejpam-4883	45	27	set	set	VERB
ejpam-4883	45	28	d	d	NOUN
ejpam-4883	45	29	with	with	ADP
ejpam-4883	45	30	cardinality	cardinality	NOUN
ejpam-4883	45	31	equal	equal	ADJ
ejpam-4883	45	32	to	to	ADP
ejpam-4883	45	33	γc(g	γc(g	NUM
ejpam-4883	45	34	)	)	PUNCT
ejpam-4883	45	35	is	be	AUX
ejpam-4883	45	36	called	call	VERB
ejpam-4883	45	37	a	a	DET
ejpam-4883	45	38	γc	γc	NOUN
ejpam-4883	45	39	-	-	PUNCT
ejpam-4883	45	40	set	set	NOUN
ejpam-4883	45	41	of	of	ADP
ejpam-4883	45	42	g.	g.	PROPN
ejpam-4883	45	43	the	the	DET
ejpam-4883	45	44	distance	distance	NOUN
ejpam-4883	45	45	dg(u	dg(u	X
ejpam-4883	45	46	,	,	PUNCT
ejpam-4883	45	47	v	v	NOUN
ejpam-4883	45	48	)	)	PUNCT
ejpam-4883	45	49	in	in	ADP
ejpam-4883	45	50	g	g	NOUN
ejpam-4883	45	51	of	of	ADP
ejpam-4883	45	52	two	two	NUM
ejpam-4883	45	53	vertices	vertex	NOUN
ejpam-4883	45	54	u	u	NOUN
ejpam-4883	45	55	,	,	PUNCT
ejpam-4883	45	56	v	v	PROPN
ejpam-4883	45	57	is	be	AUX
ejpam-4883	45	58	the	the	DET
ejpam-4883	45	59	length	length	NOUN
ejpam-4883	45	60	of	of	ADP
ejpam-4883	45	61	a	a	DET
ejpam-4883	45	62	shortest	short	ADJ
ejpam-4883	45	63	u	u	NOUN
ejpam-4883	45	64	-	-	NOUN
ejpam-4883	45	65	v	v	ADJ
ejpam-4883	45	66	path	path	NOUN
ejpam-4883	45	67	in	in	ADP
ejpam-4883	45	68	g.	g.	PROPN
ejpam-4883	45	69	the	the	DET
ejpam-4883	45	70	greatest	great	ADJ
ejpam-4883	45	71	distance	distance	NOUN
ejpam-4883	45	72	between	between	ADP
ejpam-4883	45	73	any	any	DET
ejpam-4883	45	74	two	two	NUM
ejpam-4883	45	75	vertices	vertex	NOUN
ejpam-4883	45	76	in	in	ADP
ejpam-4883	45	77	g	g	NOUN
ejpam-4883	45	78	,	,	PUNCT
ejpam-4883	45	79	denoted	denote	VERB
ejpam-4883	45	80	by	by	ADP
ejpam-4883	45	81	diam(g	diam(g	PROPN
ejpam-4883	45	82	)	)	PUNCT
ejpam-4883	45	83	,	,	PUNCT
ejpam-4883	45	84	is	be	AUX
ejpam-4883	45	85	called	call	VERB
ejpam-4883	45	86	the	the	DET
ejpam-4883	45	87	diameter	diameter	NOUN
ejpam-4883	45	88	of	of	ADP
ejpam-4883	45	89	g.	g.	PROPN
ejpam-4883	45	90	a	a	DET
ejpam-4883	45	91	subset	subset	NOUN
ejpam-4883	45	92	i	i	PRON
ejpam-4883	45	93	of	of	ADP
ejpam-4883	45	94	v	v	NOUN
ejpam-4883	45	95	(	(	PUNCT
ejpam-4883	45	96	g	g	NOUN
ejpam-4883	45	97	)	)	PUNCT
ejpam-4883	45	98	is	be	AUX
ejpam-4883	45	99	called	call	VERB
ejpam-4883	45	100	an	an	DET
ejpam-4883	45	101	independent	independent	ADJ
ejpam-4883	45	102	if	if	SCONJ
ejpam-4883	45	103	for	for	ADP
ejpam-4883	45	104	every	every	DET
ejpam-4883	45	105	pair	pair	NOUN
ejpam-4883	45	106	of	of	ADP
ejpam-4883	45	107	distinct	distinct	ADJ
ejpam-4883	45	108	vertices	vertex	NOUN
ejpam-4883	45	109	x	x	X
ejpam-4883	45	110	,	,	PUNCT
ejpam-4883	45	111	y	y	PROPN
ejpam-4883	45	112	∈	∈	PROPN
ejpam-4883	45	113	i	i	PRON
ejpam-4883	45	114	,	,	PUNCT
ejpam-4883	45	115	dg(x	dg(x	X
ejpam-4883	45	116	,	,	PUNCT
ejpam-4883	45	117	y	y	NOUN
ejpam-4883	45	118	)	)	PUNCT
ejpam-4883	45	119	̸=	̸=	PROPN
ejpam-4883	45	120	1	1	NUM
ejpam-4883	45	121	.	.	PUNCT
ejpam-4883	46	1	the	the	DET
ejpam-4883	46	2	maximum	maximum	ADJ
ejpam-4883	46	3	cardinality	cardinality	NOUN
ejpam-4883	46	4	of	of	ADP
ejpam-4883	46	5	an	an	DET
ejpam-4883	46	6	independent	independent	ADJ
ejpam-4883	46	7	set	set	NOUN
ejpam-4883	46	8	in	in	ADP
ejpam-4883	46	9	g	g	NOUN
ejpam-4883	46	10	,	,	PUNCT
ejpam-4883	46	11	denoted	denote	VERB
ejpam-4883	46	12	by	by	ADP
ejpam-4883	46	13	α(g	α(g	NOUN
ejpam-4883	46	14	)	)	PUNCT
ejpam-4883	46	15	,	,	PUNCT
ejpam-4883	46	16	is	be	AUX
ejpam-4883	46	17	called	call	VERB
ejpam-4883	46	18	the	the	DET
ejpam-4883	46	19	independence	independence	NOUN
ejpam-4883	46	20	number	number	NOUN
ejpam-4883	46	21	of	of	ADP
ejpam-4883	46	22	g.	g.	PROPN
ejpam-4883	46	23	any	any	DET
ejpam-4883	46	24	independent	independent	ADJ
ejpam-4883	46	25	set	set	NOUN
ejpam-4883	46	26	i	i	PRON
ejpam-4883	46	27	with	with	ADP
ejpam-4883	46	28	cardinality	cardinality	NOUN
ejpam-4883	46	29	equal	equal	ADJ
ejpam-4883	46	30	to	to	ADP
ejpam-4883	46	31	α(g	α(g	NUM
ejpam-4883	46	32	)	)	PUNCT
ejpam-4883	46	33	is	be	AUX
ejpam-4883	46	34	called	call	VERB
ejpam-4883	46	35	an	an	DET
ejpam-4883	46	36	α	α	NOUN
ejpam-4883	46	37	-	-	PUNCT
ejpam-4883	46	38	set	set	NOUN
ejpam-4883	46	39	of	of	ADP
ejpam-4883	46	40	g.	g.	PROPN
ejpam-4883	46	41	a	a	DET
ejpam-4883	46	42	graph	graph	NOUN
ejpam-4883	46	43	is	be	AUX
ejpam-4883	46	44	complete	complete	ADJ
ejpam-4883	46	45	if	if	SCONJ
ejpam-4883	46	46	every	every	DET
ejpam-4883	46	47	pair	pair	NOUN
ejpam-4883	46	48	of	of	ADP
ejpam-4883	46	49	distinct	distinct	ADJ
ejpam-4883	46	50	vertices	vertex	NOUN
ejpam-4883	46	51	are	be	AUX
ejpam-4883	46	52	adjacent	adjacent	ADJ
ejpam-4883	46	53	.	.	PUNCT
ejpam-4883	47	1	a	a	DET
ejpam-4883	47	2	complete	complete	ADJ
ejpam-4883	47	3	graph	graph	NOUN
ejpam-4883	47	4	of	of	ADP
ejpam-4883	47	5	order	order	NOUN
ejpam-4883	47	6	n	n	NOUN
ejpam-4883	47	7	is	be	AUX
ejpam-4883	47	8	denoted	denote	VERB
ejpam-4883	47	9	by	by	ADP
ejpam-4883	47	10	kn	kn	PROPN
ejpam-4883	47	11	.	.	PUNCT
ejpam-4883	48	1	the	the	DET
ejpam-4883	48	2	complement	complement	NOUN
ejpam-4883	48	3	of	of	ADP
ejpam-4883	48	4	a	a	DET
ejpam-4883	48	5	graph	graph	NOUN
ejpam-4883	48	6	g	g	NOUN
ejpam-4883	48	7	,	,	PUNCT
ejpam-4883	48	8	denoted	denote	VERB
ejpam-4883	48	9	by	by	ADP
ejpam-4883	48	10	g	g	NOUN
ejpam-4883	48	11	,	,	PUNCT
ejpam-4883	48	12	is	be	AUX
ejpam-4883	48	13	the	the	DET
ejpam-4883	48	14	graph	graph	NOUN
ejpam-4883	48	15	with	with	ADP
ejpam-4883	48	16	v	v	NOUN
ejpam-4883	48	17	(	(	PUNCT
ejpam-4883	48	18	g	g	NOUN
ejpam-4883	48	19	)	)	PUNCT
ejpam-4883	48	20	=	=	NOUN
ejpam-4883	48	21	v	v	X
ejpam-4883	48	22	(	(	PUNCT
ejpam-4883	48	23	g	g	NOUN
ejpam-4883	48	24	)	)	PUNCT
ejpam-4883	48	25	and	and	CCONJ
ejpam-4883	48	26	e(g	e(g	PROPN
ejpam-4883	48	27	)	)	PUNCT
ejpam-4883	49	1	=	=	PRON
ejpam-4883	49	2	{	{	PUNCT
ejpam-4883	49	3	uv	uv	NOUN
ejpam-4883	49	4	:	:	PUNCT
ejpam-4883	49	5	u	u	NOUN
ejpam-4883	49	6	,	,	PUNCT
ejpam-4883	49	7	v	v	PROPN
ejpam-4883	49	8	∈	∈	PROPN
ejpam-4883	49	9	v	v	NOUN
ejpam-4883	49	10	(	(	PUNCT
ejpam-4883	49	11	g	g	NOUN
ejpam-4883	49	12	)	)	PUNCT
ejpam-4883	49	13	and	and	CCONJ
ejpam-4883	49	14	uv	uv	NOUN
ejpam-4883	49	15	/∈	/∈	PROPN
ejpam-4883	49	16	e(g	e(g	PROPN
ejpam-4883	49	17	)	)	PUNCT
ejpam-4883	49	18	}	}	PUNCT
ejpam-4883	49	19	.	.	PUNCT
ejpam-4883	50	1	let	let	VERB
ejpam-4883	50	2	g	g	NOUN
ejpam-4883	50	3	and	and	CCONJ
ejpam-4883	50	4	h	h	NOUN
ejpam-4883	50	5	be	be	VERB
ejpam-4883	50	6	any	any	DET
ejpam-4883	50	7	two	two	NUM
ejpam-4883	50	8	graphs	graph	NOUN
ejpam-4883	50	9	.	.	PUNCT
ejpam-4883	51	1	the	the	DET
ejpam-4883	51	2	join	join	NOUN
ejpam-4883	51	3	of	of	ADP
ejpam-4883	51	4	g	g	PROPN
ejpam-4883	51	5	and	and	CCONJ
ejpam-4883	51	6	h	h	NOUN
ejpam-4883	51	7	,	,	PUNCT
ejpam-4883	51	8	denoted	denote	VERB
ejpam-4883	51	9	by	by	ADP
ejpam-4883	51	10	g+h	g+h	PROPN
ejpam-4883	51	11	is	be	AUX
ejpam-4883	51	12	the	the	DET
ejpam-4883	51	13	graph	graph	NOUN
ejpam-4883	51	14	with	with	ADP
ejpam-4883	51	15	vertex	vertex	NOUN
ejpam-4883	51	16	set	set	VERB
ejpam-4883	51	17	v	v	NOUN
ejpam-4883	51	18	(	(	PUNCT
ejpam-4883	51	19	g+h	g+h	NOUN
ejpam-4883	51	20	)	)	PUNCT
ejpam-4883	52	1	=	=	SYM
ejpam-4883	52	2	v	v	X
ejpam-4883	52	3	(	(	PUNCT
ejpam-4883	52	4	g	g	NOUN
ejpam-4883	52	5	)	)	PUNCT
ejpam-4883	52	6	∪	∪	NOUN
ejpam-4883	52	7	v	v	NOUN
ejpam-4883	52	8	(	(	PUNCT
ejpam-4883	52	9	h	h	NOUN
ejpam-4883	52	10	)	)	PUNCT
ejpam-4883	52	11	and	and	CCONJ
ejpam-4883	52	12	edge	edge	NOUN
ejpam-4883	52	13	set	set	VERB
ejpam-4883	52	14	e(g+h	e(g+h	NUM
ejpam-4883	52	15	)	)	PUNCT
ejpam-4883	52	16	=	=	SYM
ejpam-4883	52	17	e(g	e(g	NOUN
ejpam-4883	52	18	)	)	PUNCT
ejpam-4883	52	19	∪	∪	ADP
ejpam-4883	52	20	e(h	e(h	PROPN
ejpam-4883	52	21	)	)	PUNCT
ejpam-4883	52	22	∪	∪	NOUN
ejpam-4883	52	23	{	{	PUNCT
ejpam-4883	52	24	uv	uv	NOUN
ejpam-4883	52	25	:	:	PUNCT
ejpam-4883	52	26	u	u	PROPN
ejpam-4883	52	27	∈	∈	PROPN
ejpam-4883	52	28	v	v	ADP
ejpam-4883	52	29	(	(	PUNCT
ejpam-4883	52	30	g	g	NOUN
ejpam-4883	52	31	)	)	PUNCT
ejpam-4883	52	32	,	,	PUNCT
ejpam-4883	52	33	v	v	X
ejpam-4883	52	34	∈	∈	PROPN
ejpam-4883	52	35	v	v	NOUN
ejpam-4883	52	36	(	(	PUNCT
ejpam-4883	52	37	h	h	NOUN
ejpam-4883	52	38	)	)	PUNCT
ejpam-4883	52	39	}	}	PUNCT
ejpam-4883	52	40	.	.	PUNCT
ejpam-4883	53	1	j.	j.	PROPN
ejpam-4883	53	2	hassan	hassan	PROPN
ejpam-4883	53	3	,	,	PUNCT
ejpam-4883	53	4	j.	j.	PROPN
ejpam-4883	53	5	salim	salim	PROPN
ejpam-4883	53	6	/	/	SYM
ejpam-4883	53	7	eur	eur	PROPN
ejpam-4883	53	8	.	.	PUNCT
ejpam-4883	54	1	j.	j.	PROPN
ejpam-4883	54	2	pure	pure	PROPN
ejpam-4883	54	3	appl	appl	PROPN
ejpam-4883	54	4	.	.	PROPN
ejpam-4883	54	5	math	math	PROPN
ejpam-4883	54	6	,	,	PUNCT
ejpam-4883	54	7	16	16	NUM
ejpam-4883	54	8	(	(	PUNCT
ejpam-4883	54	9	4	4	NUM
ejpam-4883	54	10	)	)	PUNCT
ejpam-4883	54	11	(	(	PUNCT
ejpam-4883	54	12	2023	2023	NUM
ejpam-4883	54	13	)	)	PUNCT
ejpam-4883	54	14	,	,	PUNCT
ejpam-4883	54	15	2082	2082	NUM
ejpam-4883	54	16	-	-	SYM
ejpam-4883	54	17	2095	2095	NUM
ejpam-4883	54	18	2084	2084	NUM
ejpam-4883	54	19	3	3	NUM
ejpam-4883	54	20	.	.	PUNCT
ejpam-4883	54	21	results	result	NOUN
ejpam-4883	54	22	we	we	PRON
ejpam-4883	54	23	begin	begin	VERB
ejpam-4883	54	24	this	this	DET
ejpam-4883	54	25	section	section	NOUN
ejpam-4883	54	26	by	by	ADP
ejpam-4883	54	27	introducing	introduce	VERB
ejpam-4883	54	28	the	the	DET
ejpam-4883	54	29	concept	concept	NOUN
ejpam-4883	54	30	of	of	ADP
ejpam-4883	54	31	j	j	NOUN
ejpam-4883	54	32	-	-	PUNCT
ejpam-4883	54	33	domination	domination	NOUN
ejpam-4883	54	34	in	in	ADP
ejpam-4883	54	35	a	a	DET
ejpam-4883	54	36	graph	graph	NOUN
ejpam-4883	54	37	.	.	PUNCT
ejpam-4883	55	1	definition	definition	NOUN
ejpam-4883	55	2	1	1	NUM
ejpam-4883	55	3	.	.	PUNCT
ejpam-4883	56	1	let	let	VERB
ejpam-4883	56	2	g	g	PRON
ejpam-4883	56	3	be	be	AUX
ejpam-4883	56	4	a	a	DET
ejpam-4883	56	5	simple	simple	ADJ
ejpam-4883	56	6	and	and	CCONJ
ejpam-4883	56	7	undirected	undirected	ADJ
ejpam-4883	56	8	graph	graph	NOUN
ejpam-4883	56	9	and	and	CCONJ
ejpam-4883	56	10	m	m	NOUN
ejpam-4883	56	11	∈	∈	PROPN
ejpam-4883	56	12	n	n	NOUN
ejpam-4883	56	13	.	.	PUNCT
ejpam-4883	57	1	a	a	DET
ejpam-4883	57	2	subset	subset	NOUN
ejpam-4883	57	3	d	d	NOUN
ejpam-4883	57	4	=	=	PUNCT
ejpam-4883	57	5	{	{	PUNCT
ejpam-4883	57	6	d1	d1	PROPN
ejpam-4883	57	7	,	,	PUNCT
ejpam-4883	57	8	d2	d2	PROPN
ejpam-4883	57	9	,	,	PUNCT
ejpam-4883	57	10	·	·	PUNCT
ejpam-4883	57	11	·	·	PUNCT
ejpam-4883	57	12	·	·	PUNCT
ejpam-4883	57	13	,	,	PUNCT
ejpam-4883	57	14	dm	dm	INTJ
ejpam-4883	57	15	}	}	PUNCT
ejpam-4883	57	16	of	of	ADP
ejpam-4883	57	17	vertices	vertex	NOUN
ejpam-4883	57	18	of	of	ADP
ejpam-4883	57	19	g	g	PROPN
ejpam-4883	57	20	is	be	AUX
ejpam-4883	57	21	called	call	VERB
ejpam-4883	57	22	a	a	DET
ejpam-4883	57	23	j	j	NOUN
ejpam-4883	57	24	-	-	PUNCT
ejpam-4883	57	25	set	set	VERB
ejpam-4883	57	26	if	if	SCONJ
ejpam-4883	57	27	ng[di	ng[di	X
ejpam-4883	57	28	]	]	PUNCT
ejpam-4883	57	29	\	\	PUNCT
ejpam-4883	58	1	ng[dj	ng[dj	PROPN
ejpam-4883	58	2	]	]	PUNCT
ejpam-4883	58	3	̸=	̸=	PROPN
ejpam-4883	58	4	∅	∅	NOUN
ejpam-4883	58	5	for	for	ADP
ejpam-4883	58	6	every	every	DET
ejpam-4883	58	7	i	i	PROPN
ejpam-4883	58	8	̸=	̸=	PROPN
ejpam-4883	58	9	j	j	PROPN
ejpam-4883	58	10	,	,	PUNCT
ejpam-4883	58	11	where	where	SCONJ
ejpam-4883	58	12	i	i	PRON
ejpam-4883	58	13	,	,	PUNCT
ejpam-4883	58	14	j	j	PROPN
ejpam-4883	58	15	∈	∈	PROPN
ejpam-4883	58	16	{	{	PUNCT
ejpam-4883	58	17	1	1	NUM
ejpam-4883	58	18	,	,	PUNCT
ejpam-4883	58	19	2	2	NUM
ejpam-4883	58	20	,	,	PUNCT
ejpam-4883	58	21	.	.	PUNCT
ejpam-4883	58	22	.	.	PUNCT
ejpam-4883	58	23	.	.	PUNCT
ejpam-4883	59	1	,	,	PUNCT
ejpam-4883	59	2	m	m	VERB
ejpam-4883	59	3	}	}	PUNCT
ejpam-4883	59	4	.	.	PUNCT
ejpam-4883	60	1	a	a	DET
ejpam-4883	60	2	j	j	NOUN
ejpam-4883	60	3	-	-	PUNCT
ejpam-4883	60	4	set	set	PROPN
ejpam-4883	60	5	is	be	AUX
ejpam-4883	60	6	called	call	VERB
ejpam-4883	60	7	a	a	DET
ejpam-4883	60	8	j	j	PROPN
ejpam-4883	60	9	-	-	PUNCT
ejpam-4883	60	10	dominating	dominating	ADJ
ejpam-4883	60	11	set	set	NOUN
ejpam-4883	60	12	of	of	ADP
ejpam-4883	60	13	g	g	PROPN
ejpam-4883	60	14	if	if	SCONJ
ejpam-4883	60	15	d	d	PROPN
ejpam-4883	60	16	=	=	PRON
ejpam-4883	60	17	{	{	PUNCT
ejpam-4883	60	18	d1	d1	PROPN
ejpam-4883	60	19	,	,	PUNCT
ejpam-4883	60	20	d2	d2	PROPN
ejpam-4883	60	21	,	,	PUNCT
ejpam-4883	60	22	.	.	PUNCT
ejpam-4883	60	23	.	.	PUNCT
ejpam-4883	61	1	.	.	PUNCT
ejpam-4883	62	1	,	,	PUNCT
ejpam-4883	62	2	dm	dm	PROPN
ejpam-4883	62	3	}	}	PUNCT
ejpam-4883	62	4	is	be	AUX
ejpam-4883	62	5	a	a	DET
ejpam-4883	62	6	dominating	dominating	NOUN
ejpam-4883	62	7	set	set	NOUN
ejpam-4883	62	8	of	of	ADP
ejpam-4883	62	9	g.	g.	PROPN
ejpam-4883	62	10	the	the	DET
ejpam-4883	62	11	j	j	PROPN
ejpam-4883	62	12	-	-	PUNCT
ejpam-4883	62	13	domination	domination	NOUN
ejpam-4883	62	14	number	number	NOUN
ejpam-4883	62	15	of	of	ADP
ejpam-4883	62	16	g	g	NOUN
ejpam-4883	62	17	,	,	PUNCT
ejpam-4883	62	18	denoted	denote	VERB
ejpam-4883	62	19	by	by	ADP
ejpam-4883	62	20	γj(g	γj(g	NOUN
ejpam-4883	62	21	)	)	PUNCT
ejpam-4883	62	22	,	,	PUNCT
ejpam-4883	62	23	is	be	AUX
ejpam-4883	62	24	the	the	DET
ejpam-4883	62	25	maximum	maximum	ADJ
ejpam-4883	62	26	cardinality	cardinality	NOUN
ejpam-4883	62	27	of	of	ADP
ejpam-4883	62	28	a	a	DET
ejpam-4883	62	29	j	j	PROPN
ejpam-4883	62	30	-	-	PUNCT
ejpam-4883	62	31	dominating	dominating	ADJ
ejpam-4883	62	32	set	set	NOUN
ejpam-4883	62	33	of	of	ADP
ejpam-4883	62	34	g.	g.	PROPN
ejpam-4883	62	35	any	any	DET
ejpam-4883	62	36	j	j	PROPN
ejpam-4883	62	37	-	-	PUNCT
ejpam-4883	62	38	dominating	dominating	NOUN
ejpam-4883	62	39	set	set	NOUN
ejpam-4883	62	40	d	d	NOUN
ejpam-4883	62	41	with	with	ADP
ejpam-4883	62	42	|d|	|d|	PROPN
ejpam-4883	62	43	=	=	SYM
ejpam-4883	62	44	γj(g	γj(g	PROPN
ejpam-4883	62	45	)	)	PUNCT
ejpam-4883	62	46	(	(	PUNCT
ejpam-4883	62	47	resp	resp	NOUN
ejpam-4883	62	48	.	.	PUNCT
ejpam-4883	63	1	|d|	|d|	PROPN
ejpam-4883	63	2	=	=	SYM
ejpam-4883	63	3	γ(g	γ(g	PROPN
ejpam-4883	63	4	)	)	PUNCT
ejpam-4883	63	5	)	)	PUNCT
ejpam-4883	63	6	,	,	PUNCT
ejpam-4883	63	7	is	be	AUX
ejpam-4883	63	8	called	call	VERB
ejpam-4883	63	9	a	a	DET
ejpam-4883	63	10	γj	γj	NOUN
ejpam-4883	63	11	-set	-set	PUNCT
ejpam-4883	63	12	or	or	CCONJ
ejpam-4883	63	13	the	the	DET
ejpam-4883	63	14	maximum	maximum	ADJ
ejpam-4883	63	15	(	(	PUNCT
ejpam-4883	63	16	resp	resp	NOUN
ejpam-4883	63	17	.	.	PUNCT
ejpam-4883	64	1	minimum	minimum	ADJ
ejpam-4883	64	2	)	)	PUNCT
ejpam-4883	64	3	j	j	NOUN
ejpam-4883	64	4	-	-	PUNCT
ejpam-4883	64	5	dominating	dominate	VERB
ejpam-4883	64	6	set	set	NOUN
ejpam-4883	64	7	of	of	ADP
ejpam-4883	64	8	g.	g.	PROPN
ejpam-4883	64	9	moreover	moreover	ADV
ejpam-4883	64	10	,	,	PUNCT
ejpam-4883	64	11	if	if	SCONJ
ejpam-4883	64	12	x	x	X
ejpam-4883	64	13	∈	∈	PROPN
ejpam-4883	64	14	ng[di	ng[di	NOUN
ejpam-4883	64	15	]	]	X
ejpam-4883	64	16	\ng[dj	\ng[dj	X
ejpam-4883	64	17	]	]	PUNCT
ejpam-4883	64	18	,	,	PUNCT
ejpam-4883	64	19	then	then	ADV
ejpam-4883	64	20	we	we	PRON
ejpam-4883	64	21	say	say	VERB
ejpam-4883	64	22	di	di	PROPN
ejpam-4883	64	23	j	j	PROPN
ejpam-4883	64	24	-	-	VERB
ejpam-4883	64	25	footprinted	footprinte	VERB
ejpam-4883	64	26	a	a	DET
ejpam-4883	64	27	vertex	vertex	NOUN
ejpam-4883	64	28	x	x	PUNCT
ejpam-4883	64	29	in	in	ADP
ejpam-4883	64	30	g.	g.	PROPN
ejpam-4883	64	31	example	example	NOUN
ejpam-4883	65	1	1	1	X
ejpam-4883	65	2	.	.	X
ejpam-4883	66	1	consider	consider	VERB
ejpam-4883	66	2	the	the	DET
ejpam-4883	66	3	graph	graph	NOUN
ejpam-4883	66	4	g	g	NOUN
ejpam-4883	66	5	in	in	ADP
ejpam-4883	66	6	figure	figure	NOUN
ejpam-4883	66	7	1	1	NUM
ejpam-4883	66	8	.	.	PUNCT
ejpam-4883	67	1	let	let	VERB
ejpam-4883	67	2	d	d	NOUN
ejpam-4883	67	3	=	=	PRON
ejpam-4883	67	4	{	{	PUNCT
ejpam-4883	67	5	a	a	PRON
ejpam-4883	67	6	,	,	PUNCT
ejpam-4883	67	7	b	b	NOUN
ejpam-4883	67	8	,	,	PUNCT
ejpam-4883	67	9	c	c	NOUN
ejpam-4883	67	10	,	,	PUNCT
ejpam-4883	67	11	d	d	NOUN
ejpam-4883	67	12	}	}	PUNCT
ejpam-4883	67	13	.	.	PUNCT
ejpam-4883	68	1	observe	observe	VERB
ejpam-4883	68	2	that	that	SCONJ
ejpam-4883	68	3	a	a	DET
ejpam-4883	68	4	∈	∈	PROPN
ejpam-4883	68	5	ng[a	ng[a	NOUN
ejpam-4883	68	6	]	]	PUNCT
ejpam-4883	68	7	\	\	PROPN
ejpam-4883	69	1	ng[u	ng[u	PROPN
ejpam-4883	69	2	]	]	PUNCT
ejpam-4883	69	3	,	,	PUNCT
ejpam-4883	69	4	∀	∀	X
ejpam-4883	69	5	u	u	NOUN
ejpam-4883	69	6	∈	∈	PROPN
ejpam-4883	69	7	{	{	PUNCT
ejpam-4883	69	8	b	b	NOUN
ejpam-4883	69	9	,	,	PUNCT
ejpam-4883	69	10	c	c	NOUN
ejpam-4883	69	11	,	,	PUNCT
ejpam-4883	69	12	d	d	NOUN
ejpam-4883	69	13	}	}	PUNCT
ejpam-4883	69	14	,	,	PUNCT
ejpam-4883	69	15	b	b	X
ejpam-4883	69	16	∈	∈	PROPN
ejpam-4883	69	17	ng[b	ng[b	NOUN
ejpam-4883	69	18	]	]	PUNCT
ejpam-4883	69	19	\	\	PUNCT
ejpam-4883	70	1	ng[v	ng[v	NOUN
ejpam-4883	70	2	]	]	PUNCT
ejpam-4883	70	3	,	,	PUNCT
ejpam-4883	70	4	∀	∀	X
ejpam-4883	70	5	v	v	ADP
ejpam-4883	70	6	∈	∈	PROPN
ejpam-4883	70	7	{	{	PUNCT
ejpam-4883	70	8	a	a	NOUN
ejpam-4883	70	9	,	,	PUNCT
ejpam-4883	70	10	c	c	NOUN
ejpam-4883	70	11	,	,	PUNCT
ejpam-4883	70	12	d	d	NOUN
ejpam-4883	70	13	}	}	PUNCT
ejpam-4883	70	14	,	,	PUNCT
ejpam-4883	70	15	c	c	PROPN
ejpam-4883	70	16	∈	∈	PROPN
ejpam-4883	70	17	ng[c	ng[c	PROPN
ejpam-4883	70	18	]	]	PUNCT
ejpam-4883	70	19	\	\	PUNCT
ejpam-4883	71	1	ng[w	ng[w	PROPN
ejpam-4883	71	2	]	]	PUNCT
ejpam-4883	71	3	,	,	PUNCT
ejpam-4883	71	4	∀	∀	PUNCT
ejpam-4883	71	5	w	w	PROPN
ejpam-4883	71	6	∈	∈	PROPN
ejpam-4883	71	7	{	{	PUNCT
ejpam-4883	71	8	a	a	PROPN
ejpam-4883	71	9	,	,	PUNCT
ejpam-4883	71	10	b	b	NOUN
ejpam-4883	71	11	,	,	PUNCT
ejpam-4883	71	12	d	d	NOUN
ejpam-4883	71	13	}	}	PUNCT
ejpam-4883	71	14	and	and	CCONJ
ejpam-4883	71	15	d	d	PROPN
ejpam-4883	71	16	∈	∈	PROPN
ejpam-4883	71	17	ng[d	ng[d	PROPN
ejpam-4883	71	18	]	]	PUNCT
ejpam-4883	71	19	\	\	PROPN
ejpam-4883	71	20	ng[x	ng[x	PROPN
ejpam-4883	71	21	]	]	PUNCT
ejpam-4883	71	22	,	,	PUNCT
ejpam-4883	71	23	∀	∀	PUNCT
ejpam-4883	71	24	x	x	X
ejpam-4883	71	25	∈	∈	NOUN
ejpam-4883	71	26	{	{	PUNCT
ejpam-4883	71	27	a	a	PROPN
ejpam-4883	71	28	,	,	PUNCT
ejpam-4883	71	29	b	b	NOUN
ejpam-4883	71	30	,	,	PUNCT
ejpam-4883	71	31	c	c	NOUN
ejpam-4883	71	32	}	}	PUNCT
ejpam-4883	71	33	.	.	PUNCT
ejpam-4883	72	1	it	it	PRON
ejpam-4883	72	2	follows	follow	VERB
ejpam-4883	72	3	that	that	SCONJ
ejpam-4883	72	4	ng[s	ng[s	PROPN
ejpam-4883	72	5	]	]	PUNCT
ejpam-4883	72	6	\	\	PROPN
ejpam-4883	72	7	ng[t	ng[t	PROPN
ejpam-4883	72	8	]	]	PUNCT
ejpam-4883	72	9	̸=	̸=	PROPN
ejpam-4883	72	10	∅	∅	NOUN
ejpam-4883	72	11	for	for	ADP
ejpam-4883	72	12	every	every	DET
ejpam-4883	72	13	s	s	PROPN
ejpam-4883	72	14	,	,	PUNCT
ejpam-4883	72	15	t	t	PROPN
ejpam-4883	72	16	∈	∈	PROPN
ejpam-4883	73	1	d	d	PROPN
ejpam-4883	73	2	,	,	PUNCT
ejpam-4883	73	3	s	s	VERB
ejpam-4883	73	4	̸=	̸=	PROPN
ejpam-4883	73	5	t.	t.	PROPN
ejpam-4883	73	6	thus	thus	ADV
ejpam-4883	73	7	,	,	PUNCT
ejpam-4883	73	8	d	d	PROPN
ejpam-4883	73	9	is	be	AUX
ejpam-4883	73	10	a	a	DET
ejpam-4883	73	11	j	j	NOUN
ejpam-4883	73	12	-	-	PUNCT
ejpam-4883	73	13	set	set	NOUN
ejpam-4883	73	14	of	of	ADP
ejpam-4883	73	15	g.	g.	PROPN
ejpam-4883	73	16	since	since	SCONJ
ejpam-4883	73	17	ng[d	ng[d	PROPN
ejpam-4883	73	18	]	]	PUNCT
ejpam-4883	73	19	=	=	SYM
ejpam-4883	73	20	v	v	X
ejpam-4883	73	21	(	(	PUNCT
ejpam-4883	73	22	g	g	NOUN
ejpam-4883	73	23	)	)	PUNCT
ejpam-4883	73	24	,	,	PUNCT
ejpam-4883	73	25	it	it	PRON
ejpam-4883	73	26	follows	follow	VERB
ejpam-4883	73	27	that	that	SCONJ
ejpam-4883	73	28	d	d	NOUN
ejpam-4883	73	29	is	be	AUX
ejpam-4883	73	30	a	a	DET
ejpam-4883	73	31	dominating	dominating	NOUN
ejpam-4883	73	32	set	set	NOUN
ejpam-4883	73	33	of	of	ADP
ejpam-4883	73	34	g.	g.	PROPN
ejpam-4883	73	35	consequently	consequently	ADV
ejpam-4883	73	36	,	,	PUNCT
ejpam-4883	73	37	d	d	PROPN
ejpam-4883	73	38	is	be	AUX
ejpam-4883	73	39	a	a	DET
ejpam-4883	73	40	j	j	PROPN
ejpam-4883	73	41	-	-	PUNCT
ejpam-4883	73	42	dominating	dominating	ADJ
ejpam-4883	73	43	set	set	NOUN
ejpam-4883	73	44	of	of	ADP
ejpam-4883	73	45	g.	g.	PROPN
ejpam-4883	73	46	moreover	moreover	ADV
ejpam-4883	73	47	,	,	PUNCT
ejpam-4883	73	48	it	it	PRON
ejpam-4883	73	49	can	can	AUX
ejpam-4883	73	50	be	be	AUX
ejpam-4883	73	51	verified	verify	VERB
ejpam-4883	73	52	that	that	SCONJ
ejpam-4883	73	53	γj(g	γj(g	PUNCT
ejpam-4883	73	54	)	)	PUNCT
ejpam-4883	73	55	=	=	SYM
ejpam-4883	74	1	4	4	X
ejpam-4883	74	2	.	.	X
ejpam-4883	75	1	g	g	NOUN
ejpam-4883	75	2	:	:	PUNCT
ejpam-4883	76	1	b	b	X
ejpam-4883	76	2	c	c	NOUN
ejpam-4883	76	3	f	f	PROPN
ejpam-4883	77	1	e	e	PROPN
ejpam-4883	77	2	d	d	X
ejpam-4883	77	3	a	a	DET
ejpam-4883	77	4	g	g	NOUN
ejpam-4883	77	5	figure	figure	NOUN
ejpam-4883	77	6	1	1	NUM
ejpam-4883	77	7	:	:	PUNCT
ejpam-4883	77	8	a	a	DET
ejpam-4883	77	9	graph	graph	NOUN
ejpam-4883	77	10	g	g	NOUN
ejpam-4883	77	11	with	with	ADP
ejpam-4883	77	12	γj	γj	ADP
ejpam-4883	77	13	(	(	PUNCT
ejpam-4883	77	14	g	g	NOUN
ejpam-4883	77	15	)	)	PUNCT
ejpam-4883	77	16	=	=	SYM
ejpam-4883	77	17	4	4	NUM
ejpam-4883	77	18	theorem	theorem	NOUN
ejpam-4883	77	19	1	1	NUM
ejpam-4883	77	20	.	.	PUNCT
ejpam-4883	78	1	let	let	VERB
ejpam-4883	78	2	g	g	NOUN
ejpam-4883	78	3	be	be	AUX
ejpam-4883	78	4	any	any	DET
ejpam-4883	78	5	graph	graph	NOUN
ejpam-4883	78	6	and	and	CCONJ
ejpam-4883	78	7	m	m	PROPN
ejpam-4883	78	8	∈	∈	PROPN
ejpam-4883	78	9	n.	n.	NOUN
ejpam-4883	78	10	then	then	ADV
ejpam-4883	78	11	t	t	PROPN
ejpam-4883	78	12	=	=	SYM
ejpam-4883	78	13	{	{	PUNCT
ejpam-4883	78	14	t1	t1	NOUN
ejpam-4883	78	15	,	,	PUNCT
ejpam-4883	78	16	t2	t2	NOUN
ejpam-4883	78	17	,	,	PUNCT
ejpam-4883	78	18	.	.	PUNCT
ejpam-4883	78	19	.	.	PUNCT
ejpam-4883	79	1	.	.	PUNCT
ejpam-4883	80	1	,	,	PUNCT
ejpam-4883	80	2	tm	tm	NOUN
ejpam-4883	80	3	}	}	PUNCT
ejpam-4883	80	4	⊆	⊆	NUM
ejpam-4883	80	5	v	v	NOUN
ejpam-4883	80	6	(	(	PUNCT
ejpam-4883	80	7	g	g	NOUN
ejpam-4883	80	8	)	)	PUNCT
ejpam-4883	80	9	is	be	AUX
ejpam-4883	80	10	a	a	DET
ejpam-4883	80	11	minimum	minimum	ADJ
ejpam-4883	80	12	dominating	dominating	NOUN
ejpam-4883	80	13	set	set	VERB
ejpam-4883	80	14	in	in	ADP
ejpam-4883	80	15	g	g	PROPN
ejpam-4883	80	16	if	if	SCONJ
ejpam-4883	81	1	and	and	CCONJ
ejpam-4883	81	2	only	only	ADV
ejpam-4883	81	3	if	if	SCONJ
ejpam-4883	81	4	t	t	PROPN
ejpam-4883	81	5	is	be	AUX
ejpam-4883	81	6	a	a	DET
ejpam-4883	81	7	minimum	minimum	ADJ
ejpam-4883	81	8	j	j	NOUN
ejpam-4883	81	9	-	-	PUNCT
ejpam-4883	81	10	dominating	dominating	NOUN
ejpam-4883	81	11	set	set	NOUN
ejpam-4883	81	12	in	in	ADP
ejpam-4883	81	13	g.	g.	PROPN
ejpam-4883	81	14	proof	proof	PROPN
ejpam-4883	81	15	.	.	PUNCT
ejpam-4883	82	1	suppose	suppose	VERB
ejpam-4883	82	2	that	that	SCONJ
ejpam-4883	82	3	t	t	NOUN
ejpam-4883	82	4	=	=	SYM
ejpam-4883	82	5	{	{	PUNCT
ejpam-4883	82	6	t1	t1	NOUN
ejpam-4883	82	7	,	,	PUNCT
ejpam-4883	82	8	t2	t2	NOUN
ejpam-4883	82	9	,	,	PUNCT
ejpam-4883	82	10	.	.	PUNCT
ejpam-4883	82	11	.	.	PUNCT
ejpam-4883	82	12	.	.	PUNCT
ejpam-4883	83	1	,	,	PUNCT
ejpam-4883	83	2	tm	tm	NOUN
ejpam-4883	83	3	}	}	PUNCT
ejpam-4883	83	4	⊆	⊆	NUM
ejpam-4883	83	5	v	v	NOUN
ejpam-4883	83	6	(	(	PUNCT
ejpam-4883	83	7	g	g	NOUN
ejpam-4883	83	8	)	)	PUNCT
ejpam-4883	83	9	is	be	AUX
ejpam-4883	83	10	a	a	DET
ejpam-4883	83	11	minimum	minimum	ADJ
ejpam-4883	83	12	dominating	dominating	NOUN
ejpam-4883	83	13	set	set	VERB
ejpam-4883	83	14	in	in	ADP
ejpam-4883	83	15	g.	g.	PROPN
ejpam-4883	83	16	then	then	ADV
ejpam-4883	83	17	it	it	PRON
ejpam-4883	83	18	remains	remain	VERB
ejpam-4883	83	19	to	to	PART
ejpam-4883	83	20	show	show	VERB
ejpam-4883	83	21	that	that	SCONJ
ejpam-4883	83	22	t	t	PROPN
ejpam-4883	83	23	is	be	AUX
ejpam-4883	83	24	a	a	DET
ejpam-4883	83	25	j	j	NOUN
ejpam-4883	83	26	-	-	PUNCT
ejpam-4883	83	27	set	set	NOUN
ejpam-4883	83	28	in	in	ADP
ejpam-4883	83	29	g.	g.	PROPN
ejpam-4883	83	30	suppose	suppose	VERB
ejpam-4883	83	31	on	on	ADP
ejpam-4883	83	32	the	the	DET
ejpam-4883	83	33	contrary	contrary	NOUN
ejpam-4883	83	34	that	that	PRON
ejpam-4883	83	35	t	t	NOUN
ejpam-4883	83	36	=	=	SYM
ejpam-4883	83	37	{	{	PUNCT
ejpam-4883	83	38	t1	t1	NOUN
ejpam-4883	83	39	,	,	PUNCT
ejpam-4883	83	40	t2	t2	NOUN
ejpam-4883	83	41	,	,	PUNCT
ejpam-4883	83	42	·	·	PUNCT
ejpam-4883	83	43	·	·	PUNCT
ejpam-4883	83	44	·	·	PUNCT
ejpam-4883	83	45	,	,	PUNCT
ejpam-4883	83	46	tm	tm	PROPN
ejpam-4883	83	47	}	}	PUNCT
ejpam-4883	83	48	is	be	AUX
ejpam-4883	83	49	not	not	PART
ejpam-4883	83	50	a	a	DET
ejpam-4883	83	51	j	j	NOUN
ejpam-4883	83	52	-	-	PUNCT
ejpam-4883	83	53	set	set	NOUN
ejpam-4883	83	54	in	in	ADP
ejpam-4883	83	55	g.	g.	PROPN
ejpam-4883	83	56	then	then	ADV
ejpam-4883	83	57	there	there	PRON
ejpam-4883	83	58	exist	exist	VERB
ejpam-4883	83	59	a	a	DET
ejpam-4883	83	60	,	,	PUNCT
ejpam-4883	83	61	b	b	PROPN
ejpam-4883	83	62	∈	∈	PROPN
ejpam-4883	83	63	t	t	NOUN
ejpam-4883	84	1	such	such	ADJ
ejpam-4883	84	2	that	that	SCONJ
ejpam-4883	84	3	either	either	CCONJ
ejpam-4883	84	4	ng[a	ng[a	PROPN
ejpam-4883	84	5	]	]	PUNCT
ejpam-4883	84	6	\	\	PROPN
ejpam-4883	84	7	ng[b	ng[b	NOUN
ejpam-4883	84	8	]	]	PUNCT
ejpam-4883	84	9	=	=	SYM
ejpam-4883	84	10	∅	∅	NOUN
ejpam-4883	84	11	or	or	CCONJ
ejpam-4883	84	12	ng[b	ng[b	NOUN
ejpam-4883	84	13	]	]	PUNCT
ejpam-4883	84	14	\	\	PUNCT
ejpam-4883	85	1	ng[a	ng[a	NOUN
ejpam-4883	85	2	]	]	X
ejpam-4883	85	3	=	=	PUNCT
ejpam-4883	85	4	∅.	∅.	ADP
ejpam-4883	85	5	this	this	PRON
ejpam-4883	85	6	means	mean	VERB
ejpam-4883	85	7	that	that	SCONJ
ejpam-4883	85	8	either	either	CCONJ
ejpam-4883	85	9	ng[a	ng[a	PROPN
ejpam-4883	85	10	]	]	PUNCT
ejpam-4883	85	11	⊆	⊆	NUM
ejpam-4883	85	12	ng[b	ng[b	NOUN
ejpam-4883	85	13	]	]	PUNCT
ejpam-4883	85	14	or	or	CCONJ
ejpam-4883	85	15	ng[b	ng[b	NOUN
ejpam-4883	85	16	]	]	PUNCT
ejpam-4883	85	17	⊆	⊆	NUM
ejpam-4883	85	18	ng[a	ng[a	NOUN
ejpam-4883	85	19	]	]	X
ejpam-4883	85	20	.	.	PUNCT
ejpam-4883	86	1	if	if	SCONJ
ejpam-4883	86	2	ng[a	ng[a	PROPN
ejpam-4883	86	3	]	]	X
ejpam-4883	86	4	⊆	⊆	NUM
ejpam-4883	86	5	ng[b	ng[b	NOUN
ejpam-4883	86	6	]	]	PUNCT
ejpam-4883	86	7	,	,	PUNCT
ejpam-4883	86	8	then	then	ADV
ejpam-4883	86	9	d	d	PROPN
ejpam-4883	86	10	=	=	SYM
ejpam-4883	86	11	t	t	PROPN
ejpam-4883	86	12	\	\	PROPN
ejpam-4883	86	13	{	{	PUNCT
ejpam-4883	86	14	a	a	PRON
ejpam-4883	86	15	}	}	PUNCT
ejpam-4883	86	16	is	be	AUX
ejpam-4883	86	17	a	a	DET
ejpam-4883	86	18	dominating	dominating	NOUN
ejpam-4883	86	19	set	set	NOUN
ejpam-4883	86	20	in	in	ADP
ejpam-4883	86	21	g	g	NOUN
ejpam-4883	86	22	,	,	PUNCT
ejpam-4883	86	23	contradicting	contradict	VERB
ejpam-4883	86	24	to	to	ADP
ejpam-4883	86	25	the	the	DET
ejpam-4883	86	26	minimality	minimality	NOUN
ejpam-4883	86	27	of	of	ADP
ejpam-4883	86	28	t	t	PROPN
ejpam-4883	86	29	.	.	PUNCT
ejpam-4883	87	1	similarly	similarly	ADV
ejpam-4883	87	2	,	,	PUNCT
ejpam-4883	87	3	when	when	SCONJ
ejpam-4883	87	4	ng[b	ng[b	NOUN
ejpam-4883	87	5	]	]	X
ejpam-4883	87	6	⊆	⊆	NUM
ejpam-4883	87	7	ng[a	ng[a	NOUN
ejpam-4883	87	8	]	]	X
ejpam-4883	87	9	=	=	PUNCT
ejpam-4883	87	10	∅.	∅.	PRON
ejpam-4883	87	11	hence	hence	ADV
ejpam-4883	87	12	,	,	PUNCT
ejpam-4883	87	13	t	t	PROPN
ejpam-4883	87	14	is	be	AUX
ejpam-4883	87	15	a	a	DET
ejpam-4883	87	16	j.	j.	PROPN
ejpam-4883	87	17	hassan	hassan	PROPN
ejpam-4883	87	18	,	,	PUNCT
ejpam-4883	87	19	j.	j.	PROPN
ejpam-4883	87	20	salim	salim	PROPN
ejpam-4883	87	21	/	/	SYM
ejpam-4883	87	22	eur	eur	PROPN
ejpam-4883	87	23	.	.	PUNCT
ejpam-4883	88	1	j.	j.	PROPN
ejpam-4883	88	2	pure	pure	PROPN
ejpam-4883	88	3	appl	appl	PROPN
ejpam-4883	88	4	.	.	PROPN
ejpam-4883	88	5	math	math	PROPN
ejpam-4883	88	6	,	,	PUNCT
ejpam-4883	88	7	16	16	NUM
ejpam-4883	88	8	(	(	PUNCT
ejpam-4883	88	9	4	4	NUM
ejpam-4883	88	10	)	)	PUNCT
ejpam-4883	88	11	(	(	PUNCT
ejpam-4883	88	12	2023	2023	NUM
ejpam-4883	88	13	)	)	PUNCT
ejpam-4883	88	14	,	,	PUNCT
ejpam-4883	88	15	2082	2082	NUM
ejpam-4883	88	16	-	-	SYM
ejpam-4883	88	17	2095	2095	NUM
ejpam-4883	88	18	2085	2085	NUM
ejpam-4883	88	19	j	j	X
ejpam-4883	88	20	-	-	PUNCT
ejpam-4883	88	21	dominating	dominating	NOUN
ejpam-4883	88	22	set	set	NOUN
ejpam-4883	88	23	in	in	ADP
ejpam-4883	88	24	g.	g.	PROPN
ejpam-4883	88	25	since	since	SCONJ
ejpam-4883	88	26	γ(g	γ(g	PROPN
ejpam-4883	88	27	)	)	PUNCT
ejpam-4883	89	1	=	=	VERB
ejpam-4883	89	2	|t	|t	VERB
ejpam-4883	90	1	|	|	ADV
ejpam-4883	90	2	,	,	PUNCT
ejpam-4883	90	3	the	the	DET
ejpam-4883	90	4	assertion	assertion	NOUN
ejpam-4883	90	5	follows	follow	VERB
ejpam-4883	90	6	.	.	PUNCT
ejpam-4883	91	1	conversely	conversely	ADV
ejpam-4883	91	2	,	,	PUNCT
ejpam-4883	91	3	suppose	suppose	VERB
ejpam-4883	91	4	that	that	SCONJ
ejpam-4883	91	5	t	t	NOUN
ejpam-4883	91	6	=	=	SYM
ejpam-4883	91	7	{	{	PUNCT
ejpam-4883	91	8	t1	t1	NOUN
ejpam-4883	91	9	,	,	PUNCT
ejpam-4883	91	10	t2	t2	NOUN
ejpam-4883	91	11	,	,	PUNCT
ejpam-4883	91	12	.	.	PUNCT
ejpam-4883	91	13	.	.	PUNCT
ejpam-4883	91	14	.	.	PUNCT
ejpam-4883	92	1	,	,	PUNCT
ejpam-4883	92	2	tm	tm	PROPN
ejpam-4883	92	3	}	}	PUNCT
ejpam-4883	92	4	is	be	AUX
ejpam-4883	92	5	a	a	DET
ejpam-4883	92	6	minimum	minimum	ADJ
ejpam-4883	92	7	j	j	NOUN
ejpam-4883	92	8	-	-	PUNCT
ejpam-4883	92	9	dominating	dominating	ADJ
ejpam-4883	92	10	set	set	NOUN
ejpam-4883	92	11	of	of	ADP
ejpam-4883	92	12	g.	g.	PROPN
ejpam-4883	92	13	then	then	ADV
ejpam-4883	92	14	|t	|t	VERB
ejpam-4883	92	15	|	|	ADV
ejpam-4883	92	16	=	=	SYM
ejpam-4883	92	17	γ(g	γ(g	PROPN
ejpam-4883	92	18	)	)	PUNCT
ejpam-4883	92	19	(	(	PUNCT
ejpam-4883	92	20	by	by	ADP
ejpam-4883	92	21	definition	definition	NOUN
ejpam-4883	92	22	)	)	PUNCT
ejpam-4883	92	23	.	.	PUNCT
ejpam-4883	93	1	it	it	PRON
ejpam-4883	93	2	follows	follow	VERB
ejpam-4883	93	3	that	that	SCONJ
ejpam-4883	93	4	t	t	PROPN
ejpam-4883	93	5	is	be	AUX
ejpam-4883	93	6	a	a	DET
ejpam-4883	93	7	minimum	minimum	ADJ
ejpam-4883	93	8	dominating	dominating	NOUN
ejpam-4883	93	9	set	set	VERB
ejpam-4883	93	10	in	in	ADP
ejpam-4883	93	11	g.	g.	PROPN
ejpam-4883	93	12	proposition	proposition	PROPN
ejpam-4883	93	13	1	1	X
ejpam-4883	93	14	.	.	PUNCT
ejpam-4883	94	1	let	let	VERB
ejpam-4883	94	2	g	g	NOUN
ejpam-4883	94	3	be	be	AUX
ejpam-4883	94	4	any	any	DET
ejpam-4883	94	5	graph	graph	NOUN
ejpam-4883	94	6	and	and	CCONJ
ejpam-4883	94	7	r	r	NOUN
ejpam-4883	94	8	∈	∈	PROPN
ejpam-4883	94	9	n.	n.	NOUN
ejpam-4883	94	10	then	then	ADV
ejpam-4883	94	11	each	each	PRON
ejpam-4883	94	12	of	of	ADP
ejpam-4883	94	13	the	the	DET
ejpam-4883	94	14	following	follow	VERB
ejpam-4883	94	15	holds	hold	VERB
ejpam-4883	94	16	:	:	PUNCT
ejpam-4883	94	17	(	(	PUNCT
ejpam-4883	94	18	i	i	NOUN
ejpam-4883	94	19	)	)	PUNCT
ejpam-4883	94	20	a	a	DET
ejpam-4883	94	21	graph	graph	NOUN
ejpam-4883	94	22	g	g	PROPN
ejpam-4883	94	23	admits	admit	VERB
ejpam-4883	94	24	a	a	DET
ejpam-4883	94	25	j	j	NOUN
ejpam-4883	94	26	-	-	PUNCT
ejpam-4883	94	27	domination	domination	NOUN
ejpam-4883	94	28	.	.	PUNCT
ejpam-4883	95	1	(	(	PUNCT
ejpam-4883	95	2	ii	ii	NOUN
ejpam-4883	95	3	)	)	PUNCT
ejpam-4883	95	4	given	give	VERB
ejpam-4883	95	5	any	any	DET
ejpam-4883	95	6	j	j	PROPN
ejpam-4883	95	7	-	-	PUNCT
ejpam-4883	95	8	dominating	dominate	VERB
ejpam-4883	95	9	set	set	NOUN
ejpam-4883	96	1	d	d	NOUN
ejpam-4883	96	2	=	=	PUNCT
ejpam-4883	96	3	{	{	PUNCT
ejpam-4883	96	4	d1	d1	PROPN
ejpam-4883	96	5	,	,	PUNCT
ejpam-4883	96	6	d2	d2	PROPN
ejpam-4883	96	7	,	,	PUNCT
ejpam-4883	96	8	·	·	PUNCT
ejpam-4883	96	9	·	·	PUNCT
ejpam-4883	96	10	·	·	PUNCT
ejpam-4883	96	11	,	,	PUNCT
ejpam-4883	96	12	dr	dr	PROPN
ejpam-4883	96	13	}	}	PUNCT
ejpam-4883	96	14	of	of	ADP
ejpam-4883	96	15	g	g	PROPN
ejpam-4883	96	16	,	,	PUNCT
ejpam-4883	96	17	we	we	PRON
ejpam-4883	96	18	have	have	VERB
ejpam-4883	96	19	|d|	|d|	NOUN
ejpam-4883	96	20	=	=	SYM
ejpam-4883	96	21	r	r	NOUN
ejpam-4883	96	22	≤	≤	NOUN
ejpam-4883	96	23	γj(g	γj(g	PUNCT
ejpam-4883	96	24	)	)	PUNCT
ejpam-4883	96	25	.	.	PUNCT
ejpam-4883	97	1	(	(	PUNCT
ejpam-4883	97	2	iii	iii	X
ejpam-4883	97	3	)	)	PUNCT
ejpam-4883	97	4	γ(g	γ(g	PROPN
ejpam-4883	97	5	)	)	PUNCT
ejpam-4883	97	6	≤	≤	NOUN
ejpam-4883	97	7	γj(g	γj(g	NUM
ejpam-4883	97	8	)	)	PUNCT
ejpam-4883	97	9	,	,	PUNCT
ejpam-4883	97	10	and	and	CCONJ
ejpam-4883	97	11	this	this	DET
ejpam-4883	97	12	bound	bind	VERB
ejpam-4883	97	13	is	be	AUX
ejpam-4883	97	14	sharp	sharp	ADJ
ejpam-4883	97	15	.	.	PUNCT
ejpam-4883	98	1	proof	proof	NOUN
ejpam-4883	98	2	.	.	PUNCT
ejpam-4883	99	1	(	(	PUNCT
ejpam-4883	99	2	i	i	NOUN
ejpam-4883	99	3	)	)	PUNCT
ejpam-4883	99	4	since	since	SCONJ
ejpam-4883	99	5	any	any	DET
ejpam-4883	99	6	graph	graph	NOUN
ejpam-4883	99	7	g	g	PROPN
ejpam-4883	99	8	admits	admit	VERB
ejpam-4883	99	9	domination	domination	NOUN
ejpam-4883	99	10	,	,	PUNCT
ejpam-4883	99	11	the	the	DET
ejpam-4883	99	12	result	result	NOUN
ejpam-4883	99	13	follows	follow	VERB
ejpam-4883	99	14	from	from	ADP
ejpam-4883	99	15	theorem	theorem	ADJ
ejpam-4883	99	16	1	1	NUM
ejpam-4883	99	17	.	.	PUNCT
ejpam-4883	99	18	(	(	PUNCT
ejpam-4883	99	19	ii	ii	NOUN
ejpam-4883	99	20	)	)	PUNCT
ejpam-4883	99	21	let	let	VERB
ejpam-4883	99	22	d	d	PRON
ejpam-4883	99	23	be	be	AUX
ejpam-4883	99	24	any	any	DET
ejpam-4883	99	25	j	j	PROPN
ejpam-4883	99	26	-	-	PUNCT
ejpam-4883	99	27	dominating	dominating	ADJ
ejpam-4883	99	28	set	set	NOUN
ejpam-4883	99	29	of	of	ADP
ejpam-4883	99	30	g.	g.	PROPN
ejpam-4883	99	31	if	if	SCONJ
ejpam-4883	99	32	d	d	PROPN
ejpam-4883	99	33	is	be	AUX
ejpam-4883	99	34	the	the	DET
ejpam-4883	99	35	maximum	maximum	ADJ
ejpam-4883	99	36	,	,	PUNCT
ejpam-4883	99	37	then	then	ADV
ejpam-4883	99	38	γj(g	γj(g	PUNCT
ejpam-4883	99	39	)	)	PUNCT
ejpam-4883	100	1	=	=	SYM
ejpam-4883	100	2	|d|	|d|	PROPN
ejpam-4883	100	3	and	and	CCONJ
ejpam-4883	100	4	we	we	PRON
ejpam-4883	100	5	are	be	AUX
ejpam-4883	100	6	done	do	VERB
ejpam-4883	100	7	.	.	PUNCT
ejpam-4883	101	1	if	if	SCONJ
ejpam-4883	101	2	d	d	NOUN
ejpam-4883	101	3	is	be	AUX
ejpam-4883	101	4	not	not	PART
ejpam-4883	101	5	the	the	DET
ejpam-4883	101	6	maximum	maximum	ADJ
ejpam-4883	101	7	,	,	PUNCT
ejpam-4883	101	8	then	then	ADV
ejpam-4883	101	9	γj(g	γj(g	PUNCT
ejpam-4883	101	10	)	)	PUNCT
ejpam-4883	101	11	>	>	X
ejpam-4883	101	12	|d|	|d|	PROPN
ejpam-4883	101	13	.	.	PUNCT
ejpam-4883	102	1	consequently	consequently	ADV
ejpam-4883	102	2	,	,	PUNCT
ejpam-4883	102	3	|d|	|d|	PROPN
ejpam-4883	102	4	≤	≤	PROPN
ejpam-4883	102	5	γj(g	γj(g	PUNCT
ejpam-4883	102	6	)	)	PUNCT
ejpam-4883	102	7	for	for	ADP
ejpam-4883	102	8	any	any	DET
ejpam-4883	102	9	j	j	PROPN
ejpam-4883	102	10	-	-	PUNCT
ejpam-4883	102	11	dominating	dominate	VERB
ejpam-4883	102	12	set	set	NOUN
ejpam-4883	102	13	d	d	PROPN
ejpam-4883	102	14	of	of	ADP
ejpam-4883	102	15	g.	g.	PROPN
ejpam-4883	102	16	(	(	PUNCT
ejpam-4883	102	17	iii	iii	NOUN
ejpam-4883	102	18	)	)	PUNCT
ejpam-4883	102	19	let	let	VERB
ejpam-4883	102	20	d′	d′	PRON
ejpam-4883	102	21	be	be	AUX
ejpam-4883	102	22	a	a	DET
ejpam-4883	102	23	minimum	minimum	ADJ
ejpam-4883	102	24	dominating	dominating	NOUN
ejpam-4883	102	25	set	set	NOUN
ejpam-4883	102	26	of	of	ADP
ejpam-4883	102	27	g.	g.	PROPN
ejpam-4883	102	28	then	then	ADV
ejpam-4883	102	29	d′	d′	PRON
ejpam-4883	102	30	is	be	AUX
ejpam-4883	102	31	a	a	DET
ejpam-4883	102	32	minimum	minimum	ADJ
ejpam-4883	102	33	j	j	NOUN
ejpam-4883	102	34	-	-	PUNCT
ejpam-4883	102	35	dominating	dominating	NOUN
ejpam-4883	102	36	set	set	NOUN
ejpam-4883	102	37	in	in	ADP
ejpam-4883	102	38	g	g	NOUN
ejpam-4883	102	39	by	by	ADP
ejpam-4883	102	40	theorem	theorem	NOUN
ejpam-4883	102	41	1	1	NUM
ejpam-4883	102	42	.	.	PUNCT
ejpam-4883	103	1	thus	thus	ADV
ejpam-4883	103	2	,	,	PUNCT
ejpam-4883	103	3	by	by	ADP
ejpam-4883	103	4	(	(	PUNCT
ejpam-4883	103	5	ii	ii	NOUN
ejpam-4883	103	6	)	)	PUNCT
ejpam-4883	103	7	,	,	PUNCT
ejpam-4883	103	8	γ(g	γ(g	PROPN
ejpam-4883	103	9	)	)	PUNCT
ejpam-4883	103	10	=	=	PRON
ejpam-4883	104	1	|d′|	|d′|	PROPN
ejpam-4883	104	2	≤	≤	NOUN
ejpam-4883	104	3	γj(g	γj(g	PUNCT
ejpam-4883	104	4	)	)	PUNCT
ejpam-4883	104	5	.	.	PUNCT
ejpam-4883	105	1	to	to	PART
ejpam-4883	105	2	see	see	VERB
ejpam-4883	105	3	the	the	DET
ejpam-4883	105	4	bound	bind	VERB
ejpam-4883	105	5	is	be	AUX
ejpam-4883	105	6	sharp	sharp	ADJ
ejpam-4883	105	7	,	,	PUNCT
ejpam-4883	105	8	consider	consider	VERB
ejpam-4883	105	9	p4	p4	ADJ
ejpam-4883	105	10	.	.	PUNCT
ejpam-4883	106	1	then	then	ADV
ejpam-4883	106	2	γj(p4	γj(p4	PROPN
ejpam-4883	106	3	)	)	PUNCT
ejpam-4883	106	4	=	=	SYM
ejpam-4883	106	5	2	2	NUM
ejpam-4883	106	6	=	=	SYM
ejpam-4883	106	7	γ(p4	γ(p4	NOUN
ejpam-4883	106	8	)	)	PUNCT
ejpam-4883	106	9	.	.	PUNCT
ejpam-4883	107	1	theorem	theorem	NOUN
ejpam-4883	107	2	2	2	NUM
ejpam-4883	107	3	.	.	PUNCT
ejpam-4883	108	1	let	let	VERB
ejpam-4883	108	2	g	g	NOUN
ejpam-4883	108	3	be	be	AUX
ejpam-4883	108	4	any	any	DET
ejpam-4883	108	5	graph	graph	NOUN
ejpam-4883	108	6	.	.	PUNCT
ejpam-4883	109	1	then	then	ADV
ejpam-4883	109	2	each	each	PRON
ejpam-4883	109	3	of	of	ADP
ejpam-4883	109	4	the	the	DET
ejpam-4883	109	5	following	following	NOUN
ejpam-4883	109	6	is	be	AUX
ejpam-4883	109	7	true	true	ADJ
ejpam-4883	109	8	.	.	PUNCT
ejpam-4883	110	1	(	(	PUNCT
ejpam-4883	110	2	i	i	NOUN
ejpam-4883	110	3	)	)	PUNCT
ejpam-4883	110	4	a	a	DET
ejpam-4883	110	5	j	j	NOUN
ejpam-4883	110	6	-	-	PUNCT
ejpam-4883	110	7	set	set	NOUN
ejpam-4883	110	8	may	may	AUX
ejpam-4883	110	9	not	not	PART
ejpam-4883	110	10	be	be	AUX
ejpam-4883	110	11	a	a	DET
ejpam-4883	110	12	j	j	PROPN
ejpam-4883	110	13	-	-	PUNCT
ejpam-4883	110	14	dominating	dominating	NOUN
ejpam-4883	110	15	set	set	NOUN
ejpam-4883	110	16	.	.	PUNCT
ejpam-4883	111	1	in	in	ADP
ejpam-4883	111	2	particular	particular	ADJ
ejpam-4883	111	3	,	,	PUNCT
ejpam-4883	111	4	if	if	SCONJ
ejpam-4883	111	5	the	the	DET
ejpam-4883	111	6	cardinality	cardinality	NOUN
ejpam-4883	111	7	of	of	ADP
ejpam-4883	111	8	a	a	DET
ejpam-4883	111	9	j	j	NOUN
ejpam-4883	111	10	-	-	PUNCT
ejpam-4883	111	11	set	set	VERB
ejpam-4883	111	12	d	d	NOUN
ejpam-4883	111	13	is	be	AUX
ejpam-4883	111	14	strictly	strictly	ADV
ejpam-4883	111	15	less	less	ADJ
ejpam-4883	111	16	than	than	ADP
ejpam-4883	111	17	the	the	DET
ejpam-4883	111	18	domination	domination	NOUN
ejpam-4883	111	19	number	number	NOUN
ejpam-4883	111	20	of	of	ADP
ejpam-4883	111	21	g	g	NOUN
ejpam-4883	111	22	,	,	PUNCT
ejpam-4883	111	23	then	then	ADV
ejpam-4883	111	24	d	d	X
ejpam-4883	111	25	can	can	AUX
ejpam-4883	111	26	not	not	PART
ejpam-4883	111	27	be	be	AUX
ejpam-4883	111	28	a	a	DET
ejpam-4883	111	29	j	j	PROPN
ejpam-4883	111	30	-	-	PUNCT
ejpam-4883	111	31	dominating	dominating	ADJ
ejpam-4883	111	32	set	set	NOUN
ejpam-4883	111	33	of	of	ADP
ejpam-4883	111	34	g.	g.	PROPN
ejpam-4883	111	35	(	(	PUNCT
ejpam-4883	111	36	ii	ii	PROPN
ejpam-4883	111	37	)	)	PUNCT
ejpam-4883	111	38	a	a	DET
ejpam-4883	111	39	vertex	vertex	NOUN
ejpam-4883	111	40	set	set	VERB
ejpam-4883	111	41	v	v	NOUN
ejpam-4883	111	42	(	(	PUNCT
ejpam-4883	111	43	g	g	NOUN
ejpam-4883	111	44	)	)	PUNCT
ejpam-4883	111	45	of	of	ADP
ejpam-4883	111	46	g	g	NOUN
ejpam-4883	111	47	may	may	AUX
ejpam-4883	111	48	not	not	PART
ejpam-4883	111	49	be	be	AUX
ejpam-4883	111	50	a	a	DET
ejpam-4883	111	51	j	j	NOUN
ejpam-4883	111	52	-	-	PUNCT
ejpam-4883	111	53	set	set	NOUN
ejpam-4883	111	54	in	in	ADP
ejpam-4883	111	55	g.	g.	PROPN
ejpam-4883	111	56	(	(	PUNCT
ejpam-4883	111	57	iii	iii	PROPN
ejpam-4883	111	58	)	)	PUNCT
ejpam-4883	111	59	every	every	DET
ejpam-4883	111	60	j	j	PROPN
ejpam-4883	111	61	-	-	PUNCT
ejpam-4883	111	62	dominating	dominating	NOUN
ejpam-4883	111	63	set	set	NOUN
ejpam-4883	111	64	in	in	ADP
ejpam-4883	111	65	g	g	PROPN
ejpam-4883	111	66	is	be	AUX
ejpam-4883	111	67	a	a	DET
ejpam-4883	111	68	dominating	dominating	NOUN
ejpam-4883	111	69	set	set	NOUN
ejpam-4883	111	70	but	but	CCONJ
ejpam-4883	111	71	the	the	DET
ejpam-4883	111	72	converse	converse	NOUN
ejpam-4883	111	73	is	be	AUX
ejpam-4883	111	74	not	not	PART
ejpam-4883	111	75	always	always	ADV
ejpam-4883	111	76	true	true	ADJ
ejpam-4883	111	77	.	.	PUNCT
ejpam-4883	112	1	proof	proof	NOUN
ejpam-4883	112	2	.	.	PUNCT
ejpam-4883	113	1	(	(	PUNCT
ejpam-4883	113	2	i	i	NOUN
ejpam-4883	113	3	)	)	PUNCT
ejpam-4883	113	4	consider	consider	VERB
ejpam-4883	113	5	the	the	DET
ejpam-4883	113	6	example	example	NOUN
ejpam-4883	113	7	1	1	NUM
ejpam-4883	113	8	and	and	CCONJ
ejpam-4883	113	9	let	let	VERB
ejpam-4883	113	10	d	d	NOUN
ejpam-4883	113	11	=	=	SYM
ejpam-4883	113	12	(	(	PUNCT
ejpam-4883	113	13	b	b	NOUN
ejpam-4883	113	14	,	,	PUNCT
ejpam-4883	113	15	e	e	NOUN
ejpam-4883	113	16	)	)	PUNCT
ejpam-4883	113	17	.	.	PUNCT
ejpam-4883	114	1	then	then	ADV
ejpam-4883	114	2	ng[b	ng[b	NOUN
ejpam-4883	114	3	]	]	PUNCT
ejpam-4883	114	4	\ng[e	\ng[e	X
ejpam-4883	114	5	]	]	X
ejpam-4883	114	6	=	=	SYM
ejpam-4883	114	7	{	{	PUNCT
ejpam-4883	114	8	b	b	NOUN
ejpam-4883	114	9	}	}	PUNCT
ejpam-4883	114	10	̸=	̸=	PROPN
ejpam-4883	114	11	∅	∅	NOUN
ejpam-4883	114	12	and	and	CCONJ
ejpam-4883	114	13	ng[e	ng[e	PROPN
ejpam-4883	114	14	]	]	PUNCT
ejpam-4883	114	15	\ng[b	\ng[b	NOUN
ejpam-4883	114	16	]	]	X
ejpam-4883	115	1	=	=	X
ejpam-4883	115	2	{	{	PUNCT
ejpam-4883	115	3	a	a	X
ejpam-4883	115	4	,	,	PUNCT
ejpam-4883	115	5	d	d	NOUN
ejpam-4883	115	6	,	,	PUNCT
ejpam-4883	115	7	e	e	NOUN
ejpam-4883	115	8	,	,	PUNCT
ejpam-4883	115	9	g	g	NOUN
ejpam-4883	115	10	}	}	PUNCT
ejpam-4883	115	11	=	=	NOUN
ejpam-4883	115	12	̸	̸	ADV
ejpam-4883	115	13	∅.	∅.	NOUN
ejpam-4883	115	14	it	it	PRON
ejpam-4883	115	15	follows	follow	VERB
ejpam-4883	115	16	that	that	SCONJ
ejpam-4883	115	17	d	d	NOUN
ejpam-4883	115	18	is	be	AUX
ejpam-4883	115	19	a	a	DET
ejpam-4883	115	20	j	j	NOUN
ejpam-4883	115	21	-	-	PUNCT
ejpam-4883	115	22	set	set	NOUN
ejpam-4883	115	23	in	in	ADP
ejpam-4883	115	24	g.	g.	PROPN
ejpam-4883	115	25	however	however	ADV
ejpam-4883	115	26	,	,	PUNCT
ejpam-4883	115	27	d	d	PROPN
ejpam-4883	115	28	is	be	AUX
ejpam-4883	115	29	not	not	PART
ejpam-4883	115	30	a	a	DET
ejpam-4883	115	31	j	j	PROPN
ejpam-4883	115	32	-	-	PUNCT
ejpam-4883	115	33	dominating	dominating	NOUN
ejpam-4883	115	34	set	set	NOUN
ejpam-4883	115	35	in	in	ADP
ejpam-4883	115	36	g	g	PROPN
ejpam-4883	115	37	since	since	SCONJ
ejpam-4883	115	38	c	c	PROPN
ejpam-4883	115	39	/∈	/∈	PUNCT
ejpam-4883	116	1	ng[d	ng[d	PROPN
ejpam-4883	116	2	]	]	PUNCT
ejpam-4883	116	3	,	,	PUNCT
ejpam-4883	116	4	that	that	ADV
ejpam-4883	116	5	is	is	ADV
ejpam-4883	116	6	,	,	PUNCT
ejpam-4883	116	7	d	d	X
ejpam-4883	116	8	not	not	PART
ejpam-4883	116	9	a	a	DET
ejpam-4883	116	10	dominating	dominating	NOUN
ejpam-4883	116	11	set	set	VERB
ejpam-4883	116	12	in	in	ADP
ejpam-4883	116	13	g.	g.	PROPN
ejpam-4883	116	14	for	for	ADP
ejpam-4883	116	15	the	the	DET
ejpam-4883	116	16	particular	particular	ADJ
ejpam-4883	116	17	case	case	NOUN
ejpam-4883	116	18	,	,	PUNCT
ejpam-4883	116	19	suppose	suppose	VERB
ejpam-4883	116	20	on	on	ADP
ejpam-4883	116	21	the	the	DET
ejpam-4883	116	22	contrary	contrary	NOUN
ejpam-4883	116	23	that	that	SCONJ
ejpam-4883	116	24	a	a	DET
ejpam-4883	116	25	j	j	NOUN
ejpam-4883	116	26	-	-	PUNCT
ejpam-4883	116	27	set	set	VERB
ejpam-4883	116	28	d	d	NOUN
ejpam-4883	116	29	can	can	AUX
ejpam-4883	116	30	be	be	AUX
ejpam-4883	116	31	a	a	DET
ejpam-4883	116	32	j	j	PROPN
ejpam-4883	116	33	-	-	PUNCT
ejpam-4883	116	34	dominating	dominating	ADJ
ejpam-4883	116	35	set	set	NOUN
ejpam-4883	116	36	of	of	ADP
ejpam-4883	116	37	g.	g.	PROPN
ejpam-4883	117	1	then	then	ADV
ejpam-4883	117	2	d	d	PROPN
ejpam-4883	117	3	is	be	AUX
ejpam-4883	117	4	a	a	DET
ejpam-4883	117	5	dominating	dominating	NOUN
ejpam-4883	117	6	set	set	NOUN
ejpam-4883	117	7	(	(	PUNCT
ejpam-4883	117	8	by	by	ADP
ejpam-4883	117	9	definition	definition	NOUN
ejpam-4883	117	10	)	)	PUNCT
ejpam-4883	117	11	.	.	PUNCT
ejpam-4883	118	1	thus	thus	ADV
ejpam-4883	118	2	,	,	PUNCT
ejpam-4883	118	3	γ(g	γ(g	PROPN
ejpam-4883	118	4	)	)	PUNCT
ejpam-4883	118	5	≤	≤	NOUN
ejpam-4883	118	6	|d|	|d|	PROPN
ejpam-4883	118	7	,	,	PUNCT
ejpam-4883	118	8	a	a	DET
ejpam-4883	118	9	contradiction	contradiction	NOUN
ejpam-4883	118	10	.	.	PUNCT
ejpam-4883	119	1	therefore	therefore	ADV
ejpam-4883	119	2	,	,	PUNCT
ejpam-4883	119	3	d	d	PROPN
ejpam-4883	119	4	can	can	AUX
ejpam-4883	119	5	not	not	PART
ejpam-4883	119	6	be	be	AUX
ejpam-4883	119	7	a	a	DET
ejpam-4883	119	8	j	j	PROPN
ejpam-4883	119	9	-	-	PUNCT
ejpam-4883	119	10	dominating	dominating	ADJ
ejpam-4883	119	11	set	set	NOUN
ejpam-4883	119	12	of	of	ADP
ejpam-4883	119	13	g.	g.	PROPN
ejpam-4883	119	14	(	(	PUNCT
ejpam-4883	119	15	ii	ii	NOUN
ejpam-4883	119	16	)	)	PUNCT
ejpam-4883	119	17	consider	consider	VERB
ejpam-4883	119	18	again	again	ADV
ejpam-4883	119	19	the	the	DET
ejpam-4883	119	20	graph	graph	NOUN
ejpam-4883	119	21	in	in	ADP
ejpam-4883	119	22	example	example	NOUN
ejpam-4883	119	23	1	1	X
ejpam-4883	119	24	.	.	X
ejpam-4883	119	25	observe	observe	VERB
ejpam-4883	119	26	that	that	SCONJ
ejpam-4883	119	27	ng[c	ng[c	NOUN
ejpam-4883	119	28	]	]	PUNCT
ejpam-4883	119	29	\ng[g	\ng[g	ADV
ejpam-4883	119	30	]	]	X
ejpam-4883	119	31	=	=	SYM
ejpam-4883	119	32	{	{	PUNCT
ejpam-4883	119	33	c	c	NOUN
ejpam-4883	119	34	,	,	PUNCT
ejpam-4883	119	35	g	g	NOUN
ejpam-4883	119	36	}	}	PUNCT
ejpam-4883	119	37	\	\	NOUN
ejpam-4883	119	38	{	{	PUNCT
ejpam-4883	119	39	c	c	NOUN
ejpam-4883	119	40	,	,	PUNCT
ejpam-4883	119	41	e	e	NOUN
ejpam-4883	119	42	,	,	PUNCT
ejpam-4883	119	43	g	g	NOUN
ejpam-4883	119	44	}	}	PUNCT
ejpam-4883	119	45	=	=	PUNCT
ejpam-4883	119	46	∅.	∅.	ADP
ejpam-4883	119	47	this	this	PRON
ejpam-4883	119	48	shows	show	VERB
ejpam-4883	119	49	that	that	SCONJ
ejpam-4883	119	50	v	v	X
ejpam-4883	119	51	(	(	PUNCT
ejpam-4883	119	52	g	g	NOUN
ejpam-4883	119	53	)	)	PUNCT
ejpam-4883	119	54	is	be	AUX
ejpam-4883	119	55	not	not	PART
ejpam-4883	119	56	a	a	DET
ejpam-4883	119	57	j	j	NOUN
ejpam-4883	119	58	-	-	PUNCT
ejpam-4883	119	59	set	set	NOUN
ejpam-4883	119	60	of	of	ADP
ejpam-4883	119	61	g.	g.	PROPN
ejpam-4883	119	62	(	(	PUNCT
ejpam-4883	119	63	iii	iii	X
ejpam-4883	119	64	)	)	PUNCT
ejpam-4883	119	65	let	let	VERB
ejpam-4883	119	66	g	g	NOUN
ejpam-4883	119	67	be	be	AUX
ejpam-4883	119	68	a	a	DET
ejpam-4883	119	69	graph	graph	NOUN
ejpam-4883	119	70	and	and	CCONJ
ejpam-4883	119	71	let	let	VERB
ejpam-4883	119	72	d	d	PRON
ejpam-4883	119	73	be	be	AUX
ejpam-4883	119	74	a	a	DET
ejpam-4883	119	75	j	j	PROPN
ejpam-4883	119	76	-	-	PUNCT
ejpam-4883	119	77	dominating	dominating	NOUN
ejpam-4883	119	78	set	set	NOUN
ejpam-4883	119	79	.	.	PUNCT
ejpam-4883	120	1	then	then	ADV
ejpam-4883	120	2	d	d	PROPN
ejpam-4883	120	3	is	be	AUX
ejpam-4883	120	4	a	a	DET
ejpam-4883	120	5	dominating	dominating	NOUN
ejpam-4883	120	6	set	set	NOUN
ejpam-4883	120	7	in	in	ADP
ejpam-4883	120	8	g	g	PROPN
ejpam-4883	120	9	(	(	PUNCT
ejpam-4883	120	10	by	by	ADP
ejpam-4883	120	11	definition	definition	NOUN
ejpam-4883	120	12	)	)	PUNCT
ejpam-4883	120	13	.	.	PUNCT
ejpam-4883	121	1	to	to	PART
ejpam-4883	121	2	see	see	VERB
ejpam-4883	121	3	that	that	SCONJ
ejpam-4883	121	4	the	the	DET
ejpam-4883	121	5	converse	converse	NOUN
ejpam-4883	121	6	is	be	AUX
ejpam-4883	121	7	not	not	PART
ejpam-4883	121	8	true	true	ADJ
ejpam-4883	121	9	,	,	PUNCT
ejpam-4883	121	10	consider	consider	VERB
ejpam-4883	121	11	the	the	DET
ejpam-4883	121	12	graph	graph	NOUN
ejpam-4883	121	13	in	in	ADP
ejpam-4883	121	14	example	example	NOUN
ejpam-4883	121	15	1	1	NUM
ejpam-4883	121	16	and	and	CCONJ
ejpam-4883	121	17	let	let	VERB
ejpam-4883	121	18	ã	ã	PROPN
ejpam-4883	121	19	=	=	PRON
ejpam-4883	121	20	{	{	PUNCT
ejpam-4883	121	21	d	d	PROPN
ejpam-4883	121	22	,	,	PUNCT
ejpam-4883	121	23	e	e	NOUN
ejpam-4883	121	24	,	,	PUNCT
ejpam-4883	121	25	f	f	X
ejpam-4883	121	26	,	,	PUNCT
ejpam-4883	121	27	g	g	PROPN
ejpam-4883	121	28	}	}	PUNCT
ejpam-4883	121	29	.	.	PUNCT
ejpam-4883	122	1	then	then	ADV
ejpam-4883	122	2	ng[ã	ng[ã	VERB
ejpam-4883	122	3	]	]	X
ejpam-4883	122	4	=	=	SYM
ejpam-4883	122	5	v	v	X
ejpam-4883	122	6	(	(	PUNCT
ejpam-4883	122	7	g	g	NOUN
ejpam-4883	122	8	)	)	PUNCT
ejpam-4883	122	9	,	,	PUNCT
ejpam-4883	122	10	and	and	CCONJ
ejpam-4883	122	11	so	so	ADV
ejpam-4883	122	12	ã	ã	PROPN
ejpam-4883	122	13	is	be	AUX
ejpam-4883	122	14	a	a	DET
ejpam-4883	122	15	dominating	dominating	NOUN
ejpam-4883	122	16	set	set	NOUN
ejpam-4883	122	17	of	of	ADP
ejpam-4883	122	18	g.	g.	PROPN
ejpam-4883	122	19	however	however	ADV
ejpam-4883	122	20	,	,	PUNCT
ejpam-4883	122	21	j.	j.	PROPN
ejpam-4883	122	22	hassan	hassan	PROPN
ejpam-4883	122	23	,	,	PUNCT
ejpam-4883	122	24	j.	j.	PROPN
ejpam-4883	122	25	salim	salim	PROPN
ejpam-4883	122	26	/	/	SYM
ejpam-4883	122	27	eur	eur	PROPN
ejpam-4883	122	28	.	.	PUNCT
ejpam-4883	123	1	j.	j.	PROPN
ejpam-4883	123	2	pure	pure	PROPN
ejpam-4883	123	3	appl	appl	PROPN
ejpam-4883	123	4	.	.	PROPN
ejpam-4883	123	5	math	math	PROPN
ejpam-4883	123	6	,	,	PUNCT
ejpam-4883	123	7	16	16	NUM
ejpam-4883	123	8	(	(	PUNCT
ejpam-4883	123	9	4	4	NUM
ejpam-4883	123	10	)	)	PUNCT
ejpam-4883	123	11	(	(	PUNCT
ejpam-4883	123	12	2023	2023	NUM
ejpam-4883	123	13	)	)	PUNCT
ejpam-4883	123	14	,	,	PUNCT
ejpam-4883	123	15	2082	2082	NUM
ejpam-4883	123	16	-	-	SYM
ejpam-4883	123	17	2095	2095	NUM
ejpam-4883	123	18	2086	2086	NUM
ejpam-4883	123	19	ng[d	ng[d	PROPN
ejpam-4883	123	20	]	]	PUNCT
ejpam-4883	123	21	\	\	PROPN
ejpam-4883	124	1	ng[e	ng[e	PROPN
ejpam-4883	124	2	]	]	X
ejpam-4883	124	3	=	=	PUNCT
ejpam-4883	124	4	{	{	PUNCT
ejpam-4883	124	5	d	d	NOUN
ejpam-4883	124	6	,	,	PUNCT
ejpam-4883	124	7	e	e	NOUN
ejpam-4883	124	8	}	}	PUNCT
ejpam-4883	124	9	\	\	NOUN
ejpam-4883	124	10	{	{	PUNCT
ejpam-4883	124	11	a	a	PRON
ejpam-4883	124	12	,	,	PUNCT
ejpam-4883	124	13	d	d	NOUN
ejpam-4883	124	14	,	,	PUNCT
ejpam-4883	124	15	e	e	NOUN
ejpam-4883	124	16	,	,	PUNCT
ejpam-4883	124	17	f	f	X
ejpam-4883	124	18	,	,	PUNCT
ejpam-4883	124	19	g	g	NOUN
ejpam-4883	124	20	}	}	PUNCT
ejpam-4883	124	21	=	=	NOUN
ejpam-4883	124	22	∅	∅	NOUN
ejpam-4883	124	23	,	,	PUNCT
ejpam-4883	124	24	showing	show	VERB
ejpam-4883	124	25	that	that	SCONJ
ejpam-4883	124	26	ã	ã	PROPN
ejpam-4883	124	27	is	be	AUX
ejpam-4883	124	28	not	not	PART
ejpam-4883	124	29	a	a	DET
ejpam-4883	124	30	j	j	NOUN
ejpam-4883	124	31	-	-	PUNCT
ejpam-4883	124	32	set	set	NOUN
ejpam-4883	124	33	in	in	ADP
ejpam-4883	124	34	g.	g.	PROPN
ejpam-4883	124	35	thus	thus	ADV
ejpam-4883	124	36	,	,	PUNCT
ejpam-4883	124	37	ã	ã	PROPN
ejpam-4883	124	38	can	can	AUX
ejpam-4883	124	39	not	not	PART
ejpam-4883	124	40	be	be	AUX
ejpam-4883	124	41	a	a	DET
ejpam-4883	124	42	j	j	PROPN
ejpam-4883	124	43	-	-	PUNCT
ejpam-4883	124	44	dominating	dominating	NOUN
ejpam-4883	124	45	set	set	NOUN
ejpam-4883	124	46	in	in	ADP
ejpam-4883	124	47	g.	g.	PROPN
ejpam-4883	124	48	theorem	theorem	PROPN
ejpam-4883	124	49	3	3	X
ejpam-4883	124	50	.	.	PUNCT
ejpam-4883	125	1	let	let	VERB
ejpam-4883	125	2	g	g	NOUN
ejpam-4883	125	3	be	be	AUX
ejpam-4883	125	4	any	any	DET
ejpam-4883	125	5	graph	graph	NOUN
ejpam-4883	125	6	and	and	CCONJ
ejpam-4883	126	1	k	k	PROPN
ejpam-4883	126	2	∈	∈	PROPN
ejpam-4883	126	3	n.	n.	NOUN
ejpam-4883	126	4	then	then	ADV
ejpam-4883	126	5	t	t	PROPN
ejpam-4883	126	6	=	=	SYM
ejpam-4883	126	7	{	{	PUNCT
ejpam-4883	126	8	v1	v1	PROPN
ejpam-4883	126	9	,	,	PUNCT
ejpam-4883	126	10	v2	v2	PROPN
ejpam-4883	126	11	,	,	PUNCT
ejpam-4883	126	12	·	·	PUNCT
ejpam-4883	126	13	·	·	PUNCT
ejpam-4883	126	14	·	·	PUNCT
ejpam-4883	126	15	,	,	PUNCT
ejpam-4883	126	16	vk	vk	CCONJ
ejpam-4883	126	17	}	}	PUNCT
ejpam-4883	126	18	is	be	AUX
ejpam-4883	126	19	a	a	DET
ejpam-4883	126	20	maximum	maximum	ADJ
ejpam-4883	126	21	j	j	NOUN
ejpam-4883	126	22	-	-	NOUN
ejpam-4883	126	23	set	set	NOUN
ejpam-4883	126	24	of	of	ADP
ejpam-4883	126	25	g	g	PROPN
ejpam-4883	126	26	if	if	SCONJ
ejpam-4883	127	1	and	and	CCONJ
ejpam-4883	127	2	only	only	ADV
ejpam-4883	127	3	if	if	SCONJ
ejpam-4883	127	4	t	t	PROPN
ejpam-4883	127	5	is	be	AUX
ejpam-4883	127	6	a	a	DET
ejpam-4883	127	7	maximum	maximum	ADJ
ejpam-4883	127	8	j	j	NOUN
ejpam-4883	127	9	-	-	PUNCT
ejpam-4883	127	10	dominating	dominating	ADJ
ejpam-4883	127	11	set	set	NOUN
ejpam-4883	127	12	of	of	ADP
ejpam-4883	127	13	g.	g.	PROPN
ejpam-4883	127	14	in	in	ADP
ejpam-4883	127	15	particular	particular	ADJ
ejpam-4883	127	16	,	,	PUNCT
ejpam-4883	127	17	γj(g	γj(g	PUNCT
ejpam-4883	127	18	)	)	PUNCT
ejpam-4883	128	1	=	=	SYM
ejpam-4883	128	2	k.	k.	NOUN
ejpam-4883	128	3	proof	proof	NOUN
ejpam-4883	128	4	.	.	PUNCT
ejpam-4883	129	1	let	let	VERB
ejpam-4883	129	2	t	t	NOUN
ejpam-4883	129	3	=	=	SYM
ejpam-4883	129	4	{	{	PUNCT
ejpam-4883	129	5	v1	v1	PROPN
ejpam-4883	129	6	,	,	PUNCT
ejpam-4883	129	7	·	·	PUNCT
ejpam-4883	129	8	·	·	PUNCT
ejpam-4883	129	9	·	·	PUNCT
ejpam-4883	129	10	,	,	PUNCT
ejpam-4883	129	11	vk	vk	PART
ejpam-4883	129	12	}	}	PUNCT
ejpam-4883	129	13	be	be	AUX
ejpam-4883	129	14	a	a	DET
ejpam-4883	129	15	maximum	maximum	ADJ
ejpam-4883	129	16	j	j	NOUN
ejpam-4883	129	17	-	-	NOUN
ejpam-4883	129	18	set	set	NOUN
ejpam-4883	129	19	of	of	ADP
ejpam-4883	129	20	g.	g.	PROPN
ejpam-4883	129	21	suppose	suppose	VERB
ejpam-4883	129	22	t	t	PROPN
ejpam-4883	129	23	is	be	AUX
ejpam-4883	129	24	not	not	PART
ejpam-4883	129	25	a	a	DET
ejpam-4883	129	26	dominating	dominating	NOUN
ejpam-4883	129	27	set	set	NOUN
ejpam-4883	129	28	of	of	ADP
ejpam-4883	129	29	g.	g.	PROPN
ejpam-4883	129	30	then	then	ADV
ejpam-4883	129	31	there	there	PRON
ejpam-4883	129	32	exists	exist	VERB
ejpam-4883	129	33	a	a	DET
ejpam-4883	129	34	∈	∈	PROPN
ejpam-4883	129	35	v	v	NOUN
ejpam-4883	129	36	(	(	PUNCT
ejpam-4883	129	37	g	g	NOUN
ejpam-4883	129	38	)	)	PUNCT
ejpam-4883	129	39	\	\	PROPN
ejpam-4883	129	40	t	t	NOUN
ejpam-4883	129	41	such	such	ADJ
ejpam-4883	129	42	that	that	SCONJ
ejpam-4883	129	43	a	a	DET
ejpam-4883	129	44	/∈	/∈	PUNCT
ejpam-4883	129	45	ng[t	ng[t	NOUN
ejpam-4883	129	46	]	]	X
ejpam-4883	129	47	.	.	PUNCT
ejpam-4883	130	1	this	this	PRON
ejpam-4883	130	2	implies	imply	VERB
ejpam-4883	130	3	that	that	SCONJ
ejpam-4883	130	4	a	a	DET
ejpam-4883	130	5	/∈	/∈	PUNCT
ejpam-4883	130	6	ng[b	ng[b	NOUN
ejpam-4883	130	7	]	]	PUNCT
ejpam-4883	130	8	for	for	ADP
ejpam-4883	130	9	every	every	DET
ejpam-4883	130	10	b	b	PROPN
ejpam-4883	130	11	∈	∈	PROPN
ejpam-4883	130	12	t	t	PROPN
ejpam-4883	130	13	.	.	PUNCT
ejpam-4883	131	1	let	let	VERB
ejpam-4883	131	2	t0	t0	PROPN
ejpam-4883	131	3	=	=	SYM
ejpam-4883	131	4	{	{	PUNCT
ejpam-4883	131	5	v0	v0	NOUN
ejpam-4883	131	6	,	,	PUNCT
ejpam-4883	131	7	v1	v1	NOUN
ejpam-4883	131	8	,	,	PUNCT
ejpam-4883	131	9	·	·	PUNCT
ejpam-4883	131	10	·	·	PUNCT
ejpam-4883	131	11	·	·	PUNCT
ejpam-4883	131	12	,	,	PUNCT
ejpam-4883	131	13	vk	vk	ADP
ejpam-4883	131	14	}	}	PUNCT
ejpam-4883	131	15	,	,	PUNCT
ejpam-4883	131	16	where	where	SCONJ
ejpam-4883	131	17	v0	v0	NOUN
ejpam-4883	131	18	=	=	SYM
ejpam-4883	131	19	a.	a.	NOUN
ejpam-4883	131	20	since	since	SCONJ
ejpam-4883	131	21	t	t	PROPN
ejpam-4883	131	22	is	be	AUX
ejpam-4883	131	23	a	a	DET
ejpam-4883	131	24	j	j	NOUN
ejpam-4883	131	25	-	-	PUNCT
ejpam-4883	131	26	set	set	VERB
ejpam-4883	131	27	and	and	CCONJ
ejpam-4883	131	28	v0	v0	NOUN
ejpam-4883	131	29	∈	∈	NOUN
ejpam-4883	131	30	ng[v0	ng[v0	PROPN
ejpam-4883	131	31	]	]	PUNCT
ejpam-4883	131	32	,	,	PUNCT
ejpam-4883	131	33	it	it	PRON
ejpam-4883	131	34	follows	follow	VERB
ejpam-4883	131	35	that	that	SCONJ
ejpam-4883	131	36	ng[vi	ng[vi	PROPN
ejpam-4883	131	37	]	]	X
ejpam-4883	131	38	\ng[vj	\ng[vj	NOUN
ejpam-4883	131	39	]	]	PUNCT
ejpam-4883	131	40	̸=	̸=	NOUN
ejpam-4883	131	41	∅	∅	NOUN
ejpam-4883	131	42	for	for	ADP
ejpam-4883	131	43	every	every	DET
ejpam-4883	131	44	i	i	PROPN
ejpam-4883	131	45	̸=	̸=	PROPN
ejpam-4883	131	46	j	j	PROPN
ejpam-4883	131	47	,	,	PUNCT
ejpam-4883	131	48	where	where	SCONJ
ejpam-4883	131	49	i	i	PRON
ejpam-4883	131	50	,	,	PUNCT
ejpam-4883	131	51	j	j	PROPN
ejpam-4883	131	52	∈	∈	PROPN
ejpam-4883	131	53	{	{	PUNCT
ejpam-4883	131	54	0	0	NUM
ejpam-4883	131	55	,	,	PUNCT
ejpam-4883	131	56	1	1	NUM
ejpam-4883	131	57	,	,	PUNCT
ejpam-4883	131	58	.	.	PUNCT
ejpam-4883	131	59	.	.	PUNCT
ejpam-4883	132	1	.	.	PUNCT
ejpam-4883	133	1	,	,	PUNCT
ejpam-4883	133	2	k	k	X
ejpam-4883	133	3	}	}	PUNCT
ejpam-4883	133	4	.	.	PUNCT
ejpam-4883	134	1	hence	hence	ADV
ejpam-4883	134	2	,	,	PUNCT
ejpam-4883	134	3	t0	t0	PROPN
ejpam-4883	134	4	is	be	AUX
ejpam-4883	134	5	a	a	DET
ejpam-4883	134	6	j	j	NOUN
ejpam-4883	134	7	-	-	NOUN
ejpam-4883	134	8	set	set	NOUN
ejpam-4883	134	9	in	in	ADP
ejpam-4883	134	10	g	g	NOUN
ejpam-4883	134	11	,	,	PUNCT
ejpam-4883	134	12	contradicting	contradict	VERB
ejpam-4883	134	13	the	the	DET
ejpam-4883	134	14	maximality	maximality	NOUN
ejpam-4883	134	15	of	of	ADP
ejpam-4883	134	16	t	t	PROPN
ejpam-4883	134	17	.	.	PUNCT
ejpam-4883	135	1	therefore	therefore	ADV
ejpam-4883	135	2	,	,	PUNCT
ejpam-4883	135	3	t	t	PROPN
ejpam-4883	135	4	is	be	AUX
ejpam-4883	135	5	a	a	DET
ejpam-4883	135	6	dominating	dominating	NOUN
ejpam-4883	135	7	set	set	NOUN
ejpam-4883	135	8	of	of	ADP
ejpam-4883	135	9	g.	g.	PROPN
ejpam-4883	135	10	since	since	SCONJ
ejpam-4883	135	11	t	t	PROPN
ejpam-4883	135	12	is	be	AUX
ejpam-4883	135	13	a	a	DET
ejpam-4883	135	14	maximum	maximum	ADJ
ejpam-4883	135	15	j	j	NOUN
ejpam-4883	135	16	-	-	NOUN
ejpam-4883	135	17	set	set	NOUN
ejpam-4883	135	18	in	in	ADP
ejpam-4883	135	19	g	g	PROPN
ejpam-4883	135	20	,	,	PUNCT
ejpam-4883	135	21	it	it	PRON
ejpam-4883	135	22	follows	follow	VERB
ejpam-4883	135	23	that	that	SCONJ
ejpam-4883	135	24	t	t	PROPN
ejpam-4883	135	25	is	be	AUX
ejpam-4883	135	26	a	a	DET
ejpam-4883	135	27	maximum	maximum	ADJ
ejpam-4883	135	28	j	j	NOUN
ejpam-4883	135	29	-	-	PUNCT
ejpam-4883	135	30	dominating	dominating	NOUN
ejpam-4883	135	31	set	set	NOUN
ejpam-4883	135	32	in	in	ADP
ejpam-4883	135	33	g.	g.	PROPN
ejpam-4883	135	34	consequently	consequently	ADV
ejpam-4883	135	35	,	,	PUNCT
ejpam-4883	135	36	γj(g	γj(g	PUNCT
ejpam-4883	135	37	)	)	PUNCT
ejpam-4883	135	38	=	=	VERB
ejpam-4883	136	1	k.	k.	NOUN
ejpam-4883	137	1	the	the	DET
ejpam-4883	137	2	converse	converse	NOUN
ejpam-4883	137	3	is	be	AUX
ejpam-4883	137	4	clear	clear	ADJ
ejpam-4883	137	5	.	.	PUNCT
ejpam-4883	138	1	the	the	DET
ejpam-4883	138	2	following	following	ADJ
ejpam-4883	138	3	result	result	NOUN
ejpam-4883	138	4	follows	follow	VERB
ejpam-4883	138	5	from	from	ADP
ejpam-4883	138	6	theorem	theorem	ADJ
ejpam-4883	138	7	3	3	NUM
ejpam-4883	138	8	.	.	PUNCT
ejpam-4883	138	9	corollary	corollary	ADJ
ejpam-4883	138	10	1	1	NUM
ejpam-4883	138	11	.	.	PUNCT
ejpam-4883	139	1	let	let	VERB
ejpam-4883	139	2	g	g	PRON
ejpam-4883	139	3	be	be	AUX
ejpam-4883	139	4	a	a	DET
ejpam-4883	139	5	graph	graph	NOUN
ejpam-4883	139	6	and	and	CCONJ
ejpam-4883	139	7	let	let	VERB
ejpam-4883	139	8	d	d	NOUN
ejpam-4883	139	9	=	=	PRON
ejpam-4883	139	10	{	{	PUNCT
ejpam-4883	139	11	x1	x1	PROPN
ejpam-4883	139	12	,	,	PUNCT
ejpam-4883	139	13	x2	x2	PROPN
ejpam-4883	139	14	,	,	PUNCT
ejpam-4883	139	15	·	·	PUNCT
ejpam-4883	139	16	·	·	PUNCT
ejpam-4883	139	17	·	·	PUNCT
ejpam-4883	139	18	,	,	PUNCT
ejpam-4883	139	19	xs	xs	PROPN
ejpam-4883	139	20	}	}	PUNCT
ejpam-4883	139	21	be	be	VERB
ejpam-4883	139	22	a	a	DET
ejpam-4883	139	23	j	j	NOUN
ejpam-4883	139	24	-	-	PUNCT
ejpam-4883	139	25	set	set	NOUN
ejpam-4883	139	26	of	of	ADP
ejpam-4883	139	27	g.	g.	PROPN
ejpam-4883	139	28	then	then	ADV
ejpam-4883	139	29	|d|	|d|	PROPN
ejpam-4883	139	30	=	=	SYM
ejpam-4883	139	31	s	s	PART
ejpam-4883	139	32	≤	≤	NOUN
ejpam-4883	139	33	γj(g	γj(g	NUM
ejpam-4883	139	34	)	)	PUNCT
ejpam-4883	139	35	.	.	PUNCT
ejpam-4883	140	1	theorem	theorem	ADJ
ejpam-4883	140	2	4	4	NUM
ejpam-4883	140	3	.	.	PUNCT
ejpam-4883	141	1	let	let	VERB
ejpam-4883	141	2	g	g	NOUN
ejpam-4883	141	3	be	be	AUX
ejpam-4883	141	4	any	any	DET
ejpam-4883	141	5	graph	graph	NOUN
ejpam-4883	141	6	and	and	CCONJ
ejpam-4883	141	7	d	d	NOUN
ejpam-4883	141	8	be	be	AUX
ejpam-4883	141	9	any	any	DET
ejpam-4883	141	10	j	j	PROPN
ejpam-4883	141	11	-	-	PUNCT
ejpam-4883	141	12	dominating	dominating	ADJ
ejpam-4883	141	13	set	set	NOUN
ejpam-4883	141	14	of	of	ADP
ejpam-4883	141	15	g.	g.	PROPN
ejpam-4883	142	1	then	then	ADV
ejpam-4883	142	2	(	(	PUNCT
ejpam-4883	142	3	i	i	NOUN
ejpam-4883	142	4	)	)	PUNCT
ejpam-4883	142	5	u	u	NOUN
ejpam-4883	142	6	∈	∈	PROPN
ejpam-4883	143	1	d	d	NOUN
ejpam-4883	143	2	if	if	SCONJ
ejpam-4883	143	3	and	and	CCONJ
ejpam-4883	143	4	only	only	ADV
ejpam-4883	143	5	if	if	SCONJ
ejpam-4883	143	6	ng[u	ng[u	PROPN
ejpam-4883	143	7	]	]	PUNCT
ejpam-4883	143	8	⊈	⊈	X
ejpam-4883	144	1	ng[v	ng[v	X
ejpam-4883	144	2	]	]	PUNCT
ejpam-4883	144	3	and	and	CCONJ
ejpam-4883	144	4	ng[v	ng[v	PROPN
ejpam-4883	144	5	]	]	X
ejpam-4883	144	6	⊈	⊈	PROPN
ejpam-4883	144	7	ng[u	ng[u	PROPN
ejpam-4883	144	8	]	]	X
ejpam-4883	144	9	∀	∀	X
ejpam-4883	144	10	v	v	ADP
ejpam-4883	144	11	∈	∈	PROPN
ejpam-4883	144	12	d	d	X
ejpam-4883	144	13	\	\	X
ejpam-4883	144	14	{	{	PUNCT
ejpam-4883	144	15	u	u	NOUN
ejpam-4883	144	16	}	}	PUNCT
ejpam-4883	144	17	;	;	PUNCT
ejpam-4883	144	18	and	and	CCONJ
ejpam-4883	144	19	(	(	PUNCT
ejpam-4883	144	20	ii	ii	NOUN
ejpam-4883	144	21	)	)	PUNCT
ejpam-4883	144	22	a	a	DET
ejpam-4883	144	23	pendant	pendant	ADJ
ejpam-4883	144	24	vertex	vertex	NOUN
ejpam-4883	144	25	x	x	PUNCT
ejpam-4883	144	26	is	be	AUX
ejpam-4883	144	27	in	in	ADP
ejpam-4883	144	28	d	d	PROPN
ejpam-4883	144	29	if	if	SCONJ
ejpam-4883	144	30	and	and	CCONJ
ejpam-4883	144	31	only	only	ADV
ejpam-4883	144	32	if	if	SCONJ
ejpam-4883	144	33	its	its	PRON
ejpam-4883	144	34	neighbor	neighbor	NOUN
ejpam-4883	144	35	y	y	PROPN
ejpam-4883	144	36	is	be	AUX
ejpam-4883	144	37	not	not	PART
ejpam-4883	144	38	in	in	ADP
ejpam-4883	144	39	d.	d.	PROPN
ejpam-4883	144	40	proof	proof	NOUN
ejpam-4883	144	41	.	.	PUNCT
ejpam-4883	145	1	(	(	PUNCT
ejpam-4883	145	2	i	i	NOUN
ejpam-4883	145	3	)	)	PUNCT
ejpam-4883	145	4	let	let	VERB
ejpam-4883	145	5	d	d	PRON
ejpam-4883	145	6	be	be	AUX
ejpam-4883	145	7	a	a	DET
ejpam-4883	145	8	j	j	PROPN
ejpam-4883	145	9	-	-	PUNCT
ejpam-4883	145	10	dominating	dominating	ADJ
ejpam-4883	145	11	set	set	NOUN
ejpam-4883	145	12	of	of	ADP
ejpam-4883	145	13	g	g	PROPN
ejpam-4883	145	14	and	and	CCONJ
ejpam-4883	145	15	u	u	PROPN
ejpam-4883	145	16	∈	∈	PROPN
ejpam-4883	145	17	d.	d.	PROPN
ejpam-4883	145	18	since	since	SCONJ
ejpam-4883	145	19	d	d	PROPN
ejpam-4883	145	20	is	be	AUX
ejpam-4883	145	21	a	a	DET
ejpam-4883	145	22	j	j	NOUN
ejpam-4883	145	23	-	-	PUNCT
ejpam-4883	145	24	set	set	NOUN
ejpam-4883	145	25	,	,	PUNCT
ejpam-4883	145	26	we	we	PRON
ejpam-4883	145	27	have	have	AUX
ejpam-4883	145	28	ng[u	ng[u	VERB
ejpam-4883	145	29	]	]	PUNCT
ejpam-4883	145	30	\ng[v	\ng[v	NUM
ejpam-4883	145	31	]	]	PUNCT
ejpam-4883	145	32	̸=	̸=	PROPN
ejpam-4883	145	33	∅	∅	NOUN
ejpam-4883	145	34	and	and	CCONJ
ejpam-4883	145	35	ng[v	ng[v	NOUN
ejpam-4883	145	36	]	]	X
ejpam-4883	146	1	\ng[u	\ng[u	NOUN
ejpam-4883	146	2	]	]	PUNCT
ejpam-4883	146	3	̸=	̸=	NOUN
ejpam-4883	146	4	∅	∅	NOUN
ejpam-4883	146	5	∀	∀	NOUN
ejpam-4883	146	6	v	v	ADP
ejpam-4883	146	7	∈	∈	PROPN
ejpam-4883	146	8	d	d	X
ejpam-4883	146	9	\	\	X
ejpam-4883	146	10	{	{	PUNCT
ejpam-4883	146	11	u	u	NOUN
ejpam-4883	146	12	}	}	PUNCT
ejpam-4883	146	13	.	.	PUNCT
ejpam-4883	147	1	it	it	PRON
ejpam-4883	147	2	follows	follow	VERB
ejpam-4883	147	3	that	that	SCONJ
ejpam-4883	147	4	ng[u	ng[u	PROPN
ejpam-4883	147	5	]	]	PUNCT
ejpam-4883	147	6	⊈	⊈	X
ejpam-4883	148	1	ng[v	ng[v	X
ejpam-4883	148	2	]	]	PUNCT
ejpam-4883	148	3	and	and	CCONJ
ejpam-4883	148	4	ng[v	ng[v	PROPN
ejpam-4883	148	5	]	]	X
ejpam-4883	148	6	⊈	⊈	PROPN
ejpam-4883	148	7	ng[u	ng[u	PROPN
ejpam-4883	148	8	]	]	X
ejpam-4883	148	9	∀	∀	X
ejpam-4883	148	10	v	v	ADP
ejpam-4883	148	11	∈	∈	PROPN
ejpam-4883	148	12	d	d	X
ejpam-4883	148	13	\	\	X
ejpam-4883	148	14	{	{	PUNCT
ejpam-4883	148	15	u	u	NOUN
ejpam-4883	148	16	}	}	PUNCT
ejpam-4883	148	17	.	.	PUNCT
ejpam-4883	149	1	conversely	conversely	ADV
ejpam-4883	149	2	,	,	PUNCT
ejpam-4883	149	3	suppose	suppose	VERB
ejpam-4883	149	4	that	that	SCONJ
ejpam-4883	149	5	ng[u	ng[u	PROPN
ejpam-4883	149	6	]	]	PUNCT
ejpam-4883	149	7	⊈	⊈	X
ejpam-4883	149	8	ng[v	ng[v	X
ejpam-4883	149	9	]	]	PUNCT
ejpam-4883	149	10	and	and	CCONJ
ejpam-4883	150	1	ng[v	ng[v	PROPN
ejpam-4883	150	2	]	]	X
ejpam-4883	150	3	⊈	⊈	PROPN
ejpam-4883	150	4	ng[u	ng[u	PROPN
ejpam-4883	150	5	]	]	X
ejpam-4883	150	6	∀	∀	X
ejpam-4883	150	7	v	v	ADP
ejpam-4883	150	8	∈	∈	PROPN
ejpam-4883	150	9	d	d	X
ejpam-4883	150	10	\	\	X
ejpam-4883	150	11	{	{	PUNCT
ejpam-4883	150	12	u	u	NOUN
ejpam-4883	150	13	}	}	PUNCT
ejpam-4883	150	14	.	.	PUNCT
ejpam-4883	151	1	then	then	ADV
ejpam-4883	151	2	ng[u	ng[u	PROPN
ejpam-4883	151	3	]	]	PUNCT
ejpam-4883	151	4	\ng[v	\ng[v	NUM
ejpam-4883	151	5	]	]	PUNCT
ejpam-4883	151	6	̸=	̸=	PROPN
ejpam-4883	151	7	∅	∅	NOUN
ejpam-4883	151	8	and	and	CCONJ
ejpam-4883	151	9	ng[v	ng[v	NOUN
ejpam-4883	151	10	]	]	X
ejpam-4883	151	11	\ng[u	\ng[u	NOUN
ejpam-4883	151	12	]	]	PUNCT
ejpam-4883	151	13	̸=	̸=	NOUN
ejpam-4883	151	14	∅	∅	NOUN
ejpam-4883	151	15	∀	∀	NOUN
ejpam-4883	151	16	v	v	ADP
ejpam-4883	151	17	∈	∈	PROPN
ejpam-4883	151	18	d	d	X
ejpam-4883	151	19	\	\	X
ejpam-4883	151	20	{	{	PUNCT
ejpam-4883	151	21	u	u	NOUN
ejpam-4883	151	22	}	}	PUNCT
ejpam-4883	151	23	.	.	PUNCT
ejpam-4883	152	1	this	this	PRON
ejpam-4883	152	2	means	mean	VERB
ejpam-4883	152	3	that	that	SCONJ
ejpam-4883	152	4	u	u	PROPN
ejpam-4883	152	5	∈	∈	PROPN
ejpam-4883	152	6	d.	d.	PROPN
ejpam-4883	152	7	(	(	PUNCT
ejpam-4883	152	8	ii	ii	NOUN
ejpam-4883	152	9	)	)	PUNCT
ejpam-4883	152	10	let	let	VERB
ejpam-4883	152	11	x	x	PUNCT
ejpam-4883	152	12	∈	∈	PROPN
ejpam-4883	152	13	d	d	AUX
ejpam-4883	152	14	be	be	AUX
ejpam-4883	152	15	a	a	DET
ejpam-4883	152	16	pendant	pendant	ADJ
ejpam-4883	152	17	vertex	vertex	NOUN
ejpam-4883	152	18	of	of	ADP
ejpam-4883	152	19	g	g	NOUN
ejpam-4883	152	20	and	and	CCONJ
ejpam-4883	152	21	let	let	VERB
ejpam-4883	152	22	y	y	PRON
ejpam-4883	152	23	be	be	AUX
ejpam-4883	152	24	a	a	DET
ejpam-4883	152	25	neighbor	neighbor	NOUN
ejpam-4883	152	26	of	of	ADP
ejpam-4883	152	27	x.	x.	PROPN
ejpam-4883	152	28	then	then	ADV
ejpam-4883	152	29	ng[x	ng[x	PROPN
ejpam-4883	152	30	]	]	PUNCT
ejpam-4883	152	31	⊆	⊆	NUM
ejpam-4883	152	32	ng[y	ng[y	PROPN
ejpam-4883	152	33	]	]	PUNCT
ejpam-4883	152	34	.	.	PUNCT
ejpam-4883	153	1	suppose	suppose	VERB
ejpam-4883	153	2	on	on	ADP
ejpam-4883	153	3	the	the	DET
ejpam-4883	153	4	contrary	contrary	NOUN
ejpam-4883	153	5	that	that	PRON
ejpam-4883	153	6	y	y	PROPN
ejpam-4883	153	7	∈	∈	PROPN
ejpam-4883	153	8	d.	d.	PROPN
ejpam-4883	153	9	then	then	ADV
ejpam-4883	153	10	by	by	ADP
ejpam-4883	153	11	(	(	PUNCT
ejpam-4883	153	12	i	i	NOUN
ejpam-4883	153	13	)	)	PUNCT
ejpam-4883	153	14	,	,	PUNCT
ejpam-4883	153	15	ng[x	ng[x	PROPN
ejpam-4883	153	16	]	]	PUNCT
ejpam-4883	153	17	⊈	⊈	PROPN
ejpam-4883	154	1	ng[y	ng[y	PROPN
ejpam-4883	154	2	]	]	PUNCT
ejpam-4883	154	3	and	and	CCONJ
ejpam-4883	154	4	ng[y	ng[y	PROPN
ejpam-4883	154	5	]	]	PUNCT
ejpam-4883	154	6	⊈	⊈	PROPN
ejpam-4883	154	7	ng[x	ng[x	PROPN
ejpam-4883	154	8	]	]	X
ejpam-4883	154	9	∀	∀	PUNCT
ejpam-4883	154	10	y	y	PROPN
ejpam-4883	154	11	∈	∈	PROPN
ejpam-4883	155	1	d	d	X
ejpam-4883	155	2	\	\	X
ejpam-4883	155	3	{	{	PUNCT
ejpam-4883	155	4	x	x	NOUN
ejpam-4883	155	5	}	}	PUNCT
ejpam-4883	155	6	,	,	PUNCT
ejpam-4883	155	7	a	a	DET
ejpam-4883	155	8	contradiction	contradiction	NOUN
ejpam-4883	155	9	.	.	PUNCT
ejpam-4883	156	1	thus	thus	ADV
ejpam-4883	156	2	,	,	PUNCT
ejpam-4883	156	3	y	y	PROPN
ejpam-4883	156	4	/∈	/∈	PUNCT
ejpam-4883	156	5	d.	d.	PROPN
ejpam-4883	156	6	conversely	conversely	ADV
ejpam-4883	156	7	,	,	PUNCT
ejpam-4883	156	8	suppose	suppose	VERB
ejpam-4883	156	9	on	on	ADP
ejpam-4883	156	10	the	the	DET
ejpam-4883	156	11	contrary	contrary	NOUN
ejpam-4883	156	12	that	that	SCONJ
ejpam-4883	156	13	x	x	X
ejpam-4883	156	14	/∈	/∈	PROPN
ejpam-4883	156	15	d.	d.	PROPN
ejpam-4883	156	16	then	then	ADV
ejpam-4883	156	17	by	by	ADP
ejpam-4883	156	18	(	(	PUNCT
ejpam-4883	156	19	i	i	NOUN
ejpam-4883	156	20	)	)	PUNCT
ejpam-4883	156	21	,	,	PUNCT
ejpam-4883	156	22	ng[x	ng[x	PROPN
ejpam-4883	156	23	]	]	PUNCT
ejpam-4883	156	24	⊆	⊆	NUM
ejpam-4883	156	25	ng[a	ng[a	NOUN
ejpam-4883	156	26	]	]	PUNCT
ejpam-4883	156	27	or	or	CCONJ
ejpam-4883	156	28	ng[a	ng[a	PROPN
ejpam-4883	156	29	]	]	X
ejpam-4883	156	30	⊆	⊆	NUM
ejpam-4883	156	31	ng[x	ng[x	PROPN
ejpam-4883	156	32	]	]	PUNCT
ejpam-4883	156	33	for	for	ADP
ejpam-4883	156	34	some	some	DET
ejpam-4883	156	35	a	a	DET
ejpam-4883	156	36	∈	∈	PROPN
ejpam-4883	156	37	d.	d.	NOUN
ejpam-4883	156	38	since	since	SCONJ
ejpam-4883	156	39	its	its	PRON
ejpam-4883	156	40	neighbor	neighbor	NOUN
ejpam-4883	156	41	y	y	PROPN
ejpam-4883	156	42	is	be	AUX
ejpam-4883	156	43	not	not	PART
ejpam-4883	156	44	in	in	ADP
ejpam-4883	156	45	d	d	PROPN
ejpam-4883	156	46	,	,	PUNCT
ejpam-4883	156	47	we	we	PRON
ejpam-4883	156	48	have	have	VERB
ejpam-4883	156	49	x	x	X
ejpam-4883	156	50	/∈	/∈	PUNCT
ejpam-4883	157	1	ng[w	ng[w	PROPN
ejpam-4883	157	2	]	]	PUNCT
ejpam-4883	157	3	for	for	ADP
ejpam-4883	157	4	every	every	DET
ejpam-4883	157	5	w	w	PROPN
ejpam-4883	157	6	∈	∈	PROPN
ejpam-4883	157	7	d.	d.	PROPN
ejpam-4883	157	8	thus	thus	ADV
ejpam-4883	157	9	,	,	PUNCT
ejpam-4883	157	10	ng[d	ng[d	PROPN
ejpam-4883	157	11	]	]	PUNCT
ejpam-4883	157	12	̸=	̸=	PROPN
ejpam-4883	157	13	v	v	NOUN
ejpam-4883	157	14	(	(	PUNCT
ejpam-4883	157	15	g	g	NOUN
ejpam-4883	157	16	)	)	PUNCT
ejpam-4883	157	17	,	,	PUNCT
ejpam-4883	157	18	showing	show	VERB
ejpam-4883	157	19	that	that	SCONJ
ejpam-4883	157	20	d	d	NOUN
ejpam-4883	157	21	is	be	AUX
ejpam-4883	157	22	not	not	PART
ejpam-4883	157	23	a	a	DET
ejpam-4883	157	24	dominating	dominating	NOUN
ejpam-4883	157	25	set	set	VERB
ejpam-4883	157	26	in	in	ADP
ejpam-4883	157	27	g.	g.	PROPN
ejpam-4883	157	28	however	however	ADV
ejpam-4883	157	29	,	,	PUNCT
ejpam-4883	157	30	this	this	PRON
ejpam-4883	157	31	is	be	AUX
ejpam-4883	157	32	a	a	DET
ejpam-4883	157	33	contradiction	contradiction	NOUN
ejpam-4883	157	34	to	to	ADP
ejpam-4883	157	35	the	the	DET
ejpam-4883	157	36	fact	fact	NOUN
ejpam-4883	157	37	that	that	SCONJ
ejpam-4883	157	38	d	d	NOUN
ejpam-4883	157	39	is	be	AUX
ejpam-4883	157	40	a	a	DET
ejpam-4883	157	41	j	j	PROPN
ejpam-4883	157	42	-	-	PUNCT
ejpam-4883	157	43	dominating	dominating	NOUN
ejpam-4883	157	44	set	set	NOUN
ejpam-4883	157	45	in	in	ADP
ejpam-4883	157	46	g.	g.	PROPN
ejpam-4883	157	47	therefore	therefore	ADV
ejpam-4883	157	48	,	,	PUNCT
ejpam-4883	157	49	x	x	PROPN
ejpam-4883	157	50	∈	∈	PROPN
ejpam-4883	157	51	d.	d.	PROPN
ejpam-4883	157	52	j.	j.	PROPN
ejpam-4883	157	53	hassan	hassan	PROPN
ejpam-4883	157	54	,	,	PUNCT
ejpam-4883	157	55	j.	j.	PROPN
ejpam-4883	157	56	salim	salim	PROPN
ejpam-4883	157	57	/	/	SYM
ejpam-4883	157	58	eur	eur	PROPN
ejpam-4883	157	59	.	.	PUNCT
ejpam-4883	158	1	j.	j.	PROPN
ejpam-4883	158	2	pure	pure	PROPN
ejpam-4883	158	3	appl	appl	PROPN
ejpam-4883	158	4	.	.	PROPN
ejpam-4883	158	5	math	math	PROPN
ejpam-4883	158	6	,	,	PUNCT
ejpam-4883	158	7	16	16	NUM
ejpam-4883	158	8	(	(	PUNCT
ejpam-4883	158	9	4	4	NUM
ejpam-4883	158	10	)	)	PUNCT
ejpam-4883	158	11	(	(	PUNCT
ejpam-4883	158	12	2023	2023	NUM
ejpam-4883	158	13	)	)	PUNCT
ejpam-4883	158	14	,	,	PUNCT
ejpam-4883	158	15	2082	2082	NUM
ejpam-4883	158	16	-	-	SYM
ejpam-4883	158	17	2095	2095	NUM
ejpam-4883	158	18	2087	2087	NUM
ejpam-4883	158	19	theorem	theorem	VERB
ejpam-4883	158	20	5	5	NUM
ejpam-4883	158	21	.	.	PUNCT
ejpam-4883	159	1	let	let	VERB
ejpam-4883	159	2	g	g	NOUN
ejpam-4883	159	3	be	be	AUX
ejpam-4883	159	4	any	any	DET
ejpam-4883	159	5	graph	graph	NOUN
ejpam-4883	159	6	and	and	CCONJ
ejpam-4883	159	7	let	let	VERB
ejpam-4883	159	8	s	s	PRON
ejpam-4883	159	9	⊆	⊆	NUM
ejpam-4883	159	10	v	v	NOUN
ejpam-4883	159	11	(	(	PUNCT
ejpam-4883	159	12	g	g	NOUN
ejpam-4883	159	13	)	)	PUNCT
ejpam-4883	159	14	.	.	PUNCT
ejpam-4883	160	1	then	then	ADV
ejpam-4883	160	2	(	(	PUNCT
ejpam-4883	160	3	i	i	NOUN
ejpam-4883	160	4	)	)	PUNCT
ejpam-4883	160	5	every	every	DET
ejpam-4883	160	6	independent	independent	ADJ
ejpam-4883	160	7	set	set	NOUN
ejpam-4883	160	8	s	s	PART
ejpam-4883	160	9	is	be	AUX
ejpam-4883	160	10	a	a	DET
ejpam-4883	160	11	j	j	NOUN
ejpam-4883	160	12	-	-	PUNCT
ejpam-4883	160	13	set	set	NOUN
ejpam-4883	160	14	;	;	PUNCT
ejpam-4883	160	15	and	and	CCONJ
ejpam-4883	160	16	(	(	PUNCT
ejpam-4883	160	17	ii	ii	NOUN
ejpam-4883	160	18	)	)	PUNCT
ejpam-4883	160	19	every	every	DET
ejpam-4883	160	20	maximum	maximum	ADJ
ejpam-4883	160	21	independent	independent	ADJ
ejpam-4883	160	22	set	set	NOUN
ejpam-4883	160	23	s	s	PART
ejpam-4883	160	24	is	be	AUX
ejpam-4883	160	25	a	a	DET
ejpam-4883	160	26	j	j	PROPN
ejpam-4883	160	27	-	-	PUNCT
ejpam-4883	160	28	dominating	dominating	NOUN
ejpam-4883	160	29	set	set	NOUN
ejpam-4883	160	30	.	.	PUNCT
ejpam-4883	161	1	moreover	moreover	ADV
ejpam-4883	161	2	,	,	PUNCT
ejpam-4883	161	3	α(g	α(g	NUM
ejpam-4883	161	4	)	)	PUNCT
ejpam-4883	161	5	≤	≤	NOUN
ejpam-4883	161	6	γj(g	γj(g	NUM
ejpam-4883	161	7	)	)	PUNCT
ejpam-4883	161	8	.	.	PUNCT
ejpam-4883	162	1	proof	proof	NOUN
ejpam-4883	162	2	.	.	PUNCT
ejpam-4883	163	1	(	(	PUNCT
ejpam-4883	163	2	i	i	NOUN
ejpam-4883	163	3	)	)	PUNCT
ejpam-4883	163	4	let	let	VERB
ejpam-4883	163	5	s	s	PRON
ejpam-4883	163	6	=	=	NOUN
ejpam-4883	163	7	{	{	PUNCT
ejpam-4883	163	8	s1	s1	NOUN
ejpam-4883	163	9	,	,	PUNCT
ejpam-4883	163	10	s2	s2	NOUN
ejpam-4883	163	11	,	,	PUNCT
ejpam-4883	163	12	.	.	PUNCT
ejpam-4883	163	13	.	.	PUNCT
ejpam-4883	164	1	.	.	PUNCT
ejpam-4883	165	1	,	,	PUNCT
ejpam-4883	165	2	sk	sk	PART
ejpam-4883	165	3	}	}	PUNCT
ejpam-4883	165	4	be	be	AUX
ejpam-4883	165	5	an	an	DET
ejpam-4883	165	6	independent	independent	ADJ
ejpam-4883	165	7	set	set	NOUN
ejpam-4883	165	8	in	in	ADP
ejpam-4883	165	9	g.	g.	PROPN
ejpam-4883	165	10	then	then	ADV
ejpam-4883	165	11	dg(si	dg(si	PROPN
ejpam-4883	165	12	,	,	PUNCT
ejpam-4883	165	13	sj	sj	PROPN
ejpam-4883	165	14	)	)	PUNCT
ejpam-4883	165	15	≥	≥	NOUN
ejpam-4883	165	16	2	2	NUM
ejpam-4883	165	17	for	for	ADP
ejpam-4883	165	18	every	every	DET
ejpam-4883	165	19	i	i	PROPN
ejpam-4883	165	20	̸=	̸=	PROPN
ejpam-4883	165	21	j	j	PROPN
ejpam-4883	165	22	,	,	PUNCT
ejpam-4883	165	23	where	where	SCONJ
ejpam-4883	165	24	i	i	PRON
ejpam-4883	165	25	,	,	PUNCT
ejpam-4883	165	26	j	j	PROPN
ejpam-4883	165	27	∈	∈	PROPN
ejpam-4883	165	28	{	{	PUNCT
ejpam-4883	165	29	1	1	NUM
ejpam-4883	165	30	,	,	PUNCT
ejpam-4883	165	31	2	2	NUM
ejpam-4883	165	32	,	,	PUNCT
ejpam-4883	165	33	.	.	PUNCT
ejpam-4883	165	34	.	.	PUNCT
ejpam-4883	166	1	.	.	PUNCT
ejpam-4883	167	1	,	,	PUNCT
ejpam-4883	167	2	k	k	X
ejpam-4883	167	3	}	}	PUNCT
ejpam-4883	167	4	.	.	PUNCT
ejpam-4883	168	1	it	it	PRON
ejpam-4883	168	2	follows	follow	VERB
ejpam-4883	168	3	that	that	SCONJ
ejpam-4883	168	4	si	si	PROPN
ejpam-4883	168	5	∈	∈	PROPN
ejpam-4883	168	6	ng[si	ng[si	PROPN
ejpam-4883	168	7	]	]	PUNCT
ejpam-4883	168	8	\ng[sj	\ng[sj	NOUN
ejpam-4883	168	9	]	]	PUNCT
ejpam-4883	168	10	for	for	ADP
ejpam-4883	168	11	every	every	DET
ejpam-4883	168	12	i	i	PROPN
ejpam-4883	168	13	̸=	̸=	PROPN
ejpam-4883	168	14	j.	j.	PROPN
ejpam-4883	168	15	thus	thus	ADV
ejpam-4883	168	16	,	,	PUNCT
ejpam-4883	168	17	ng[si	ng[si	PROPN
ejpam-4883	168	18	]	]	PUNCT
ejpam-4883	168	19	\ng[sj	\ng[sj	NOUN
ejpam-4883	168	20	]	]	PUNCT
ejpam-4883	168	21	̸=	̸=	NOUN
ejpam-4883	168	22	∅	∅	NOUN
ejpam-4883	168	23	for	for	ADP
ejpam-4883	168	24	every	every	DET
ejpam-4883	168	25	i	i	PROPN
ejpam-4883	168	26	̸=	̸=	PROPN
ejpam-4883	168	27	j.	j.	PROPN
ejpam-4883	168	28	this	this	PRON
ejpam-4883	168	29	shows	show	VERB
ejpam-4883	168	30	that	that	SCONJ
ejpam-4883	168	31	s	s	VERB
ejpam-4883	168	32	is	be	AUX
ejpam-4883	168	33	a	a	DET
ejpam-4883	168	34	j	j	NOUN
ejpam-4883	168	35	-	-	PUNCT
ejpam-4883	168	36	set	set	NOUN
ejpam-4883	168	37	of	of	ADP
ejpam-4883	168	38	g.	g.	PROPN
ejpam-4883	168	39	(	(	PUNCT
ejpam-4883	168	40	ii	ii	PROPN
ejpam-4883	168	41	)	)	PUNCT
ejpam-4883	168	42	next	next	ADV
ejpam-4883	168	43	,	,	PUNCT
ejpam-4883	168	44	let	let	VERB
ejpam-4883	168	45	s′	s′	ADJ
ejpam-4883	168	46	=	=	PUNCT
ejpam-4883	168	47	{	{	PUNCT
ejpam-4883	168	48	a1	a1	PROPN
ejpam-4883	168	49	,	,	PUNCT
ejpam-4883	168	50	a2	a2	PROPN
ejpam-4883	168	51	,	,	PUNCT
ejpam-4883	168	52	.	.	PUNCT
ejpam-4883	168	53	.	.	PUNCT
ejpam-4883	169	1	.	.	PUNCT
ejpam-4883	170	1	,	,	PUNCT
ejpam-4883	170	2	am	be	AUX
ejpam-4883	170	3	}	}	PUNCT
ejpam-4883	170	4	be	be	AUX
ejpam-4883	170	5	a	a	DET
ejpam-4883	170	6	maximum	maximum	ADJ
ejpam-4883	170	7	independent	independent	ADJ
ejpam-4883	170	8	set	set	NOUN
ejpam-4883	170	9	of	of	ADP
ejpam-4883	170	10	g.	g.	PROPN
ejpam-4883	170	11	then	then	ADV
ejpam-4883	170	12	s′	s′	PROPN
ejpam-4883	170	13	is	be	AUX
ejpam-4883	170	14	a	a	DET
ejpam-4883	170	15	j	j	NOUN
ejpam-4883	170	16	-	-	PUNCT
ejpam-4883	170	17	set	set	NOUN
ejpam-4883	170	18	in	in	ADP
ejpam-4883	170	19	g	g	NOUN
ejpam-4883	170	20	by	by	ADP
ejpam-4883	170	21	(	(	PUNCT
ejpam-4883	170	22	i	i	NOUN
ejpam-4883	170	23	)	)	PUNCT
ejpam-4883	170	24	.	.	PUNCT
ejpam-4883	171	1	now	now	ADV
ejpam-4883	171	2	,	,	PUNCT
ejpam-4883	171	3	suppose	suppose	VERB
ejpam-4883	171	4	on	on	ADP
ejpam-4883	171	5	the	the	DET
ejpam-4883	171	6	contrary	contrary	NOUN
ejpam-4883	171	7	that	that	SCONJ
ejpam-4883	171	8	s′	s′	ADJ
ejpam-4883	171	9	is	be	AUX
ejpam-4883	171	10	not	not	PART
ejpam-4883	171	11	a	a	DET
ejpam-4883	171	12	dominating	dominating	NOUN
ejpam-4883	171	13	set	set	NOUN
ejpam-4883	171	14	of	of	ADP
ejpam-4883	171	15	g.	g.	PROPN
ejpam-4883	171	16	then	then	ADV
ejpam-4883	171	17	there	there	PRON
ejpam-4883	171	18	exists	exist	VERB
ejpam-4883	171	19	x	x	X
ejpam-4883	171	20	∈	∈	PROPN
ejpam-4883	171	21	v	v	X
ejpam-4883	171	22	(	(	PUNCT
ejpam-4883	171	23	g	g	NOUN
ejpam-4883	171	24	)	)	PUNCT
ejpam-4883	171	25	\	\	NOUN
ejpam-4883	172	1	s′	s′	VERB
ejpam-4883	172	2	such	such	ADJ
ejpam-4883	172	3	that	that	SCONJ
ejpam-4883	172	4	x	x	SYM
ejpam-4883	172	5	/∈	/∈	PUNCT
ejpam-4883	172	6	ng[y	ng[y	PROPN
ejpam-4883	172	7	]	]	X
ejpam-4883	172	8	∀	∀	PUNCT
ejpam-4883	172	9	y	y	PROPN
ejpam-4883	172	10	∈	∈	PROPN
ejpam-4883	172	11	s′.	s′.	PROPN
ejpam-4883	172	12	this	this	PRON
ejpam-4883	172	13	means	mean	VERB
ejpam-4883	172	14	that	that	SCONJ
ejpam-4883	172	15	dg(x	dg(x	ADV
ejpam-4883	172	16	,	,	PUNCT
ejpam-4883	172	17	y	y	PROPN
ejpam-4883	172	18	)	)	PUNCT
ejpam-4883	172	19	≥	≥	NOUN
ejpam-4883	172	20	2	2	NUM
ejpam-4883	172	21	for	for	ADP
ejpam-4883	172	22	all	all	DET
ejpam-4883	172	23	y	y	PROPN
ejpam-4883	172	24	∈	∈	PROPN
ejpam-4883	172	25	s′.	s′.	PROPN
ejpam-4883	172	26	thus	thus	ADV
ejpam-4883	172	27	,	,	PUNCT
ejpam-4883	172	28	s∗	s∗	PROPN
ejpam-4883	172	29	=	=	SYM
ejpam-4883	172	30	{	{	PUNCT
ejpam-4883	172	31	x	x	NOUN
ejpam-4883	172	32	}	}	PUNCT
ejpam-4883	172	33	∪	∪	ADJ
ejpam-4883	172	34	s′	s′	NUM
ejpam-4883	172	35	is	be	AUX
ejpam-4883	172	36	an	an	DET
ejpam-4883	172	37	independent	independent	ADJ
ejpam-4883	172	38	set	set	NOUN
ejpam-4883	172	39	in	in	ADP
ejpam-4883	172	40	g	g	NOUN
ejpam-4883	172	41	,	,	PUNCT
ejpam-4883	172	42	contradicting	contradict	VERB
ejpam-4883	172	43	the	the	DET
ejpam-4883	172	44	maximality	maximality	NOUN
ejpam-4883	172	45	of	of	ADP
ejpam-4883	172	46	s′.	s′.	PROPN
ejpam-4883	172	47	therefore	therefore	ADV
ejpam-4883	172	48	,	,	PUNCT
ejpam-4883	172	49	s′	s′	PROPN
ejpam-4883	172	50	is	be	AUX
ejpam-4883	172	51	a	a	DET
ejpam-4883	172	52	dominating	dominating	NOUN
ejpam-4883	172	53	set	set	NOUN
ejpam-4883	172	54	of	of	ADP
ejpam-4883	172	55	g	g	NOUN
ejpam-4883	172	56	,	,	PUNCT
ejpam-4883	172	57	and	and	CCONJ
ejpam-4883	172	58	so	so	ADV
ejpam-4883	172	59	s′	s′	ADJ
ejpam-4883	172	60	is	be	AUX
ejpam-4883	172	61	a	a	DET
ejpam-4883	172	62	j	j	NOUN
ejpam-4883	172	63	-	-	PUNCT
ejpam-4883	172	64	dominating	dominating	NOUN
ejpam-4883	172	65	in	in	ADP
ejpam-4883	172	66	g.	g.	PROPN
ejpam-4883	172	67	consequently	consequently	ADV
ejpam-4883	172	68	,	,	PUNCT
ejpam-4883	172	69	α(g	α(g	NUM
ejpam-4883	172	70	)	)	PUNCT
ejpam-4883	172	71	≤	≤	NOUN
ejpam-4883	172	72	γj(g	γj(g	PUNCT
ejpam-4883	172	73	)	)	PUNCT
ejpam-4883	172	74	by	by	ADP
ejpam-4883	172	75	proposition	proposition	NOUN
ejpam-4883	172	76	1	1	NUM
ejpam-4883	172	77	.	.	PUNCT
ejpam-4883	172	78	remark	remark	PROPN
ejpam-4883	172	79	1	1	NUM
ejpam-4883	172	80	.	.	PUNCT
ejpam-4883	173	1	the	the	DET
ejpam-4883	173	2	converse	converse	NOUN
ejpam-4883	173	3	of	of	ADP
ejpam-4883	173	4	the	the	DET
ejpam-4883	173	5	theorem	theorem	NOUN
ejpam-4883	173	6	5	5	NUM
ejpam-4883	173	7	is	be	AUX
ejpam-4883	173	8	not	not	PART
ejpam-4883	173	9	true	true	ADJ
ejpam-4883	173	10	.	.	PUNCT
ejpam-4883	174	1	moreover	moreover	ADV
ejpam-4883	174	2	,	,	PUNCT
ejpam-4883	174	3	the	the	DET
ejpam-4883	174	4	sharpness	sharpness	NOUN
ejpam-4883	174	5	of	of	ADP
ejpam-4883	174	6	the	the	DET
ejpam-4883	174	7	bound	bind	VERB
ejpam-4883	174	8	is	be	AUX
ejpam-4883	174	9	attainable	attainable	ADJ
ejpam-4883	174	10	.	.	PUNCT
ejpam-4883	175	1	to	to	PART
ejpam-4883	175	2	see	see	VERB
ejpam-4883	175	3	this	this	PRON
ejpam-4883	175	4	,	,	PUNCT
ejpam-4883	175	5	consider	consider	VERB
ejpam-4883	175	6	the	the	DET
ejpam-4883	175	7	graph	graph	NOUN
ejpam-4883	175	8	g	g	NOUN
ejpam-4883	175	9	in	in	ADP
ejpam-4883	175	10	figure	figure	NOUN
ejpam-4883	175	11	2	2	NUM
ejpam-4883	175	12	.	.	PUNCT
ejpam-4883	176	1	let	let	VERB
ejpam-4883	176	2	c	c	NOUN
ejpam-4883	176	3	=	=	PUNCT
ejpam-4883	176	4	{	{	PUNCT
ejpam-4883	176	5	a	a	X
ejpam-4883	176	6	,	,	PUNCT
ejpam-4883	176	7	e	e	NOUN
ejpam-4883	176	8	,	,	PUNCT
ejpam-4883	176	9	h	h	NOUN
ejpam-4883	176	10	}	}	PUNCT
ejpam-4883	176	11	.	.	PUNCT
ejpam-4883	177	1	then	then	ADV
ejpam-4883	177	2	ng[a	ng[a	X
ejpam-4883	177	3	]	]	X
ejpam-4883	177	4	=	=	X
ejpam-4883	177	5	{	{	PUNCT
ejpam-4883	177	6	a	a	DET
ejpam-4883	177	7	,	,	PUNCT
ejpam-4883	177	8	b	b	NOUN
ejpam-4883	177	9	}	}	PUNCT
ejpam-4883	177	10	,	,	PUNCT
ejpam-4883	177	11	ng[e	ng[e	ADV
ejpam-4883	177	12	]	]	PUNCT
ejpam-4883	177	13	=	=	PUNCT
ejpam-4883	177	14	{	{	PUNCT
ejpam-4883	177	15	b	b	PROPN
ejpam-4883	177	16	,	,	PUNCT
ejpam-4883	177	17	c	c	NOUN
ejpam-4883	177	18	,	,	PUNCT
ejpam-4883	177	19	d	d	NOUN
ejpam-4883	177	20	,	,	PUNCT
ejpam-4883	177	21	e	e	NOUN
ejpam-4883	177	22	,	,	PUNCT
ejpam-4883	177	23	f	f	PROPN
ejpam-4883	177	24	,	,	PUNCT
ejpam-4883	177	25	g	g	PROPN
ejpam-4883	177	26	,	,	PUNCT
ejpam-4883	177	27	h	h	NOUN
ejpam-4883	177	28	}	}	PUNCT
ejpam-4883	177	29	and	and	CCONJ
ejpam-4883	177	30	ng[h	ng[h	PROPN
ejpam-4883	177	31	]	]	X
ejpam-4883	177	32	=	=	X
ejpam-4883	177	33	{	{	PUNCT
ejpam-4883	177	34	e	e	NOUN
ejpam-4883	177	35	,	,	PUNCT
ejpam-4883	177	36	g	g	PROPN
ejpam-4883	177	37	,	,	PUNCT
ejpam-4883	177	38	h	h	NOUN
ejpam-4883	177	39	,	,	PUNCT
ejpam-4883	177	40	i	i	NOUN
ejpam-4883	177	41	}	}	PUNCT
ejpam-4883	177	42	.	.	PUNCT
ejpam-4883	178	1	thus	thus	ADV
ejpam-4883	178	2	,	,	PUNCT
ejpam-4883	178	3	ng[a	ng[a	PROPN
ejpam-4883	178	4	]	]	PUNCT
ejpam-4883	178	5	\	\	PUNCT
ejpam-4883	178	6	ng[e	ng[e	PROPN
ejpam-4883	178	7	]	]	X
ejpam-4883	178	8	=	=	X
ejpam-4883	178	9	{	{	PUNCT
ejpam-4883	178	10	a	a	NOUN
ejpam-4883	178	11	}	}	PUNCT
ejpam-4883	178	12	=	=	NOUN
ejpam-4883	178	13	̸	̸	ADJ
ejpam-4883	178	14	∅	∅	NOUN
ejpam-4883	178	15	,	,	PUNCT
ejpam-4883	178	16	ng[a	ng[a	PROPN
ejpam-4883	178	17	]	]	PUNCT
ejpam-4883	178	18	\	\	PUNCT
ejpam-4883	179	1	ng[h	ng[h	PROPN
ejpam-4883	179	2	]	]	X
ejpam-4883	179	3	=	=	PUNCT
ejpam-4883	179	4	ng[a	ng[a	NOUN
ejpam-4883	179	5	]	]	X
ejpam-4883	179	6	̸=	̸=	PROPN
ejpam-4883	179	7	∅	∅	NOUN
ejpam-4883	179	8	,	,	PUNCT
ejpam-4883	179	9	ng[e	ng[e	PROPN
ejpam-4883	179	10	]	]	PUNCT
ejpam-4883	179	11	\	\	PUNCT
ejpam-4883	179	12	ng[a	ng[a	NOUN
ejpam-4883	179	13	]	]	X
ejpam-4883	179	14	=	=	X
ejpam-4883	179	15	{	{	PUNCT
ejpam-4883	179	16	c	c	NOUN
ejpam-4883	179	17	,	,	PUNCT
ejpam-4883	179	18	d	d	NOUN
ejpam-4883	179	19	,	,	PUNCT
ejpam-4883	179	20	e	e	NOUN
ejpam-4883	179	21	,	,	PUNCT
ejpam-4883	179	22	f	f	PROPN
ejpam-4883	179	23	,	,	PUNCT
ejpam-4883	179	24	g	g	PROPN
ejpam-4883	179	25	,	,	PUNCT
ejpam-4883	179	26	h	h	NOUN
ejpam-4883	179	27	}	}	PUNCT
ejpam-4883	179	28	=	=	NOUN
ejpam-4883	179	29	̸	̸	ADJ
ejpam-4883	179	30	∅	∅	NOUN
ejpam-4883	179	31	,	,	PUNCT
ejpam-4883	179	32	ng[e	ng[e	PROPN
ejpam-4883	179	33	]	]	PUNCT
ejpam-4883	179	34	\	\	PROPN
ejpam-4883	179	35	ng[h	ng[h	PROPN
ejpam-4883	179	36	]	]	X
ejpam-4883	179	37	=	=	PUNCT
ejpam-4883	179	38	{	{	PUNCT
ejpam-4883	179	39	b	b	PROPN
ejpam-4883	179	40	,	,	PUNCT
ejpam-4883	179	41	c	c	NOUN
ejpam-4883	179	42	,	,	PUNCT
ejpam-4883	179	43	d	d	NOUN
ejpam-4883	179	44	,	,	PUNCT
ejpam-4883	179	45	f	f	NOUN
ejpam-4883	179	46	}	}	PUNCT
ejpam-4883	179	47	=	=	NOUN
ejpam-4883	179	48	̸	̸	ADJ
ejpam-4883	179	49	∅	∅	NOUN
ejpam-4883	179	50	,	,	PUNCT
ejpam-4883	179	51	ng[h	ng[h	PROPN
ejpam-4883	179	52	]	]	PUNCT
ejpam-4883	179	53	\ng[a	\ng[a	NOUN
ejpam-4883	179	54	]	]	X
ejpam-4883	179	55	=	=	SYM
ejpam-4883	179	56	ng[h	ng[h	PROPN
ejpam-4883	179	57	]	]	X
ejpam-4883	179	58	̸=	̸=	PROPN
ejpam-4883	179	59	∅	∅	NOUN
ejpam-4883	179	60	and	and	CCONJ
ejpam-4883	179	61	ng[h	ng[h	PROPN
ejpam-4883	179	62	]	]	X
ejpam-4883	179	63	\ng[e	\ng[e	X
ejpam-4883	179	64	]	]	X
ejpam-4883	179	65	=	=	SYM
ejpam-4883	179	66	{	{	PUNCT
ejpam-4883	179	67	i	i	NOUN
ejpam-4883	179	68	}	}	PUNCT
ejpam-4883	179	69	=	=	NOUN
ejpam-4883	179	70	̸	̸	X
ejpam-4883	179	71	∅.	∅.	ADV
ejpam-4883	179	72	therefore	therefore	ADV
ejpam-4883	179	73	,	,	PUNCT
ejpam-4883	179	74	c	c	PROPN
ejpam-4883	179	75	is	be	AUX
ejpam-4883	179	76	a	a	DET
ejpam-4883	179	77	j	j	NOUN
ejpam-4883	179	78	-	-	PUNCT
ejpam-4883	179	79	set	set	NOUN
ejpam-4883	179	80	in	in	ADP
ejpam-4883	179	81	g.	g.	PROPN
ejpam-4883	179	82	however	however	ADV
ejpam-4883	179	83	,	,	PUNCT
ejpam-4883	179	84	c	c	PROPN
ejpam-4883	179	85	is	be	AUX
ejpam-4883	179	86	not	not	PART
ejpam-4883	179	87	an	an	DET
ejpam-4883	179	88	independent	independent	ADJ
ejpam-4883	179	89	set	set	NOUN
ejpam-4883	179	90	in	in	ADP
ejpam-4883	179	91	g	g	PROPN
ejpam-4883	179	92	since	since	SCONJ
ejpam-4883	179	93	dg(e	dg(e	ADV
ejpam-4883	179	94	,	,	PUNCT
ejpam-4883	179	95	h	h	NOUN
ejpam-4883	179	96	)	)	PUNCT
ejpam-4883	179	97	=	=	SYM
ejpam-4883	179	98	1	1	X
ejpam-4883	179	99	.	.	X
ejpam-4883	179	100	for	for	ADP
ejpam-4883	179	101	the	the	DET
ejpam-4883	179	102	sharpness	sharpness	NOUN
ejpam-4883	179	103	,	,	PUNCT
ejpam-4883	179	104	consider	consider	VERB
ejpam-4883	179	105	again	again	ADV
ejpam-4883	179	106	the	the	DET
ejpam-4883	179	107	graph	graph	NOUN
ejpam-4883	179	108	g	g	PROPN
ejpam-4883	179	109	in	in	ADP
ejpam-4883	179	110	figure	figure	NOUN
ejpam-4883	179	111	2	2	NUM
ejpam-4883	179	112	and	and	CCONJ
ejpam-4883	179	113	let	let	VERB
ejpam-4883	179	114	t	t	NOUN
ejpam-4883	179	115	=	=	PUNCT
ejpam-4883	179	116	{	{	PUNCT
ejpam-4883	179	117	a	a	X
ejpam-4883	179	118	,	,	PUNCT
ejpam-4883	179	119	c	c	NOUN
ejpam-4883	179	120	,	,	PUNCT
ejpam-4883	179	121	d	d	NOUN
ejpam-4883	179	122	,	,	PUNCT
ejpam-4883	179	123	f	f	X
ejpam-4883	179	124	,	,	PUNCT
ejpam-4883	179	125	g	g	PROPN
ejpam-4883	179	126	,	,	PUNCT
ejpam-4883	179	127	i	i	PROPN
ejpam-4883	179	128	}	}	PUNCT
ejpam-4883	179	129	.	.	PUNCT
ejpam-4883	180	1	then	then	ADV
ejpam-4883	180	2	t	t	PROPN
ejpam-4883	180	3	is	be	AUX
ejpam-4883	180	4	an	an	DET
ejpam-4883	180	5	α	α	NOUN
ejpam-4883	180	6	-	-	PUNCT
ejpam-4883	180	7	set	set	NOUN
ejpam-4883	180	8	of	of	ADP
ejpam-4883	180	9	g	g	NOUN
ejpam-4883	180	10	,	,	PUNCT
ejpam-4883	180	11	and	and	CCONJ
ejpam-4883	180	12	so	so	ADV
ejpam-4883	180	13	α(g	α(g	NUM
ejpam-4883	180	14	)	)	PUNCT
ejpam-4883	180	15	=	=	PUNCT
ejpam-4883	180	16	6	6	X
ejpam-4883	180	17	.	.	PUNCT
ejpam-4883	181	1	now	now	ADV
ejpam-4883	181	2	,	,	PUNCT
ejpam-4883	181	3	since	since	SCONJ
ejpam-4883	181	4	x	x	PROPN
ejpam-4883	181	5	∈	∈	PROPN
ejpam-4883	181	6	ng[x	ng[x	PROPN
ejpam-4883	181	7	]	]	PUNCT
ejpam-4883	181	8	\	\	PROPN
ejpam-4883	181	9	ng[y	ng[y	PROPN
ejpam-4883	181	10	]	]	X
ejpam-4883	181	11	∀	∀	PUNCT
ejpam-4883	181	12	x	x	NOUN
ejpam-4883	181	13	,	,	PUNCT
ejpam-4883	181	14	y	y	PROPN
ejpam-4883	181	15	∈	∈	PROPN
ejpam-4883	181	16	t	t	PROPN
ejpam-4883	181	17	,	,	PUNCT
ejpam-4883	181	18	it	it	PRON
ejpam-4883	181	19	follows	follow	VERB
ejpam-4883	181	20	that	that	SCONJ
ejpam-4883	181	21	t	t	PROPN
ejpam-4883	181	22	is	be	AUX
ejpam-4883	181	23	a	a	DET
ejpam-4883	181	24	j	j	NOUN
ejpam-4883	181	25	-	-	PUNCT
ejpam-4883	181	26	set	set	NOUN
ejpam-4883	181	27	in	in	ADP
ejpam-4883	181	28	g.	g.	PROPN
ejpam-4883	181	29	moreover	moreover	ADV
ejpam-4883	181	30	,	,	PUNCT
ejpam-4883	181	31	observe	observe	VERB
ejpam-4883	181	32	that	that	SCONJ
ejpam-4883	181	33	ng[t	ng[t	NOUN
ejpam-4883	181	34	]	]	PUNCT
ejpam-4883	182	1	=	=	SYM
ejpam-4883	182	2	v	v	X
ejpam-4883	182	3	(	(	PUNCT
ejpam-4883	182	4	g	g	NOUN
ejpam-4883	182	5	)	)	PUNCT
ejpam-4883	182	6	.	.	PUNCT
ejpam-4883	183	1	hence	hence	ADV
ejpam-4883	183	2	,	,	PUNCT
ejpam-4883	183	3	t	t	PROPN
ejpam-4883	183	4	is	be	AUX
ejpam-4883	183	5	j	j	PROPN
ejpam-4883	183	6	-	-	PUNCT
ejpam-4883	183	7	dominating	dominating	NOUN
ejpam-4883	183	8	set	set	NOUN
ejpam-4883	183	9	in	in	ADP
ejpam-4883	183	10	g.	g.	PROPN
ejpam-4883	183	11	since	since	SCONJ
ejpam-4883	183	12	a	a	DET
ejpam-4883	183	13	,	,	PUNCT
ejpam-4883	183	14	d	d	PROPN
ejpam-4883	183	15	,	,	PUNCT
ejpam-4883	183	16	f	f	PROPN
ejpam-4883	184	1	and	and	CCONJ
ejpam-4883	184	2	i	i	PRON
ejpam-4883	184	3	are	be	AUX
ejpam-4883	184	4	pendant	pendant	ADJ
ejpam-4883	184	5	vertices	vertex	NOUN
ejpam-4883	184	6	of	of	ADP
ejpam-4883	184	7	g	g	NOUN
ejpam-4883	184	8	,	,	PUNCT
ejpam-4883	184	9	it	it	PRON
ejpam-4883	184	10	follows	follow	VERB
ejpam-4883	184	11	that	that	SCONJ
ejpam-4883	184	12	t	t	PROPN
ejpam-4883	184	13	is	be	AUX
ejpam-4883	184	14	the	the	DET
ejpam-4883	184	15	maximum	maximum	ADJ
ejpam-4883	184	16	j	j	PROPN
ejpam-4883	184	17	-	-	PUNCT
ejpam-4883	184	18	dominating	dominating	NOUN
ejpam-4883	184	19	of	of	ADP
ejpam-4883	184	20	g	g	NOUN
ejpam-4883	184	21	by	by	ADP
ejpam-4883	184	22	theorem	theorem	NOUN
ejpam-4883	184	23	4	4	NUM
ejpam-4883	184	24	.	.	PUNCT
ejpam-4883	184	25	consequently	consequently	ADV
ejpam-4883	184	26	,	,	PUNCT
ejpam-4883	184	27	α(g	α(g	NUM
ejpam-4883	184	28	)	)	PUNCT
ejpam-4883	184	29	=	=	SYM
ejpam-4883	184	30	6	6	NUM
ejpam-4883	184	31	=	=	SYM
ejpam-4883	184	32	γj(g	γj(g	NUM
ejpam-4883	184	33	)	)	PUNCT
ejpam-4883	184	34	.	.	PUNCT
ejpam-4883	185	1	g	g	NOUN
ejpam-4883	185	2	:	:	PUNCT
ejpam-4883	185	3	b	b	X
ejpam-4883	185	4	c	c	NOUN
ejpam-4883	185	5	fed	feed	VERB
ejpam-4883	185	6	a	a	DET
ejpam-4883	185	7	g	g	NOUN
ejpam-4883	185	8	h	h	NOUN
ejpam-4883	186	1	i	i	PRON
ejpam-4883	186	2	figure	figure	VERB
ejpam-4883	186	3	2	2	NUM
ejpam-4883	186	4	:	:	PUNCT
ejpam-4883	186	5	a	a	DET
ejpam-4883	186	6	graph	graph	NOUN
ejpam-4883	186	7	g	g	NOUN
ejpam-4883	186	8	with	with	ADP
ejpam-4883	186	9	a	a	DET
ejpam-4883	186	10	j	j	NOUN
ejpam-4883	186	11	-	-	PUNCT
ejpam-4883	186	12	set	set	VERB
ejpam-4883	186	13	c	c	NOUN
ejpam-4883	186	14	which	which	PRON
ejpam-4883	186	15	is	be	AUX
ejpam-4883	186	16	not	not	PART
ejpam-4883	186	17	an	an	DET
ejpam-4883	186	18	independent	independent	ADJ
ejpam-4883	186	19	set	set	NOUN
ejpam-4883	186	20	in	in	ADP
ejpam-4883	186	21	g	g	PROPN
ejpam-4883	186	22	j.	j.	PROPN
ejpam-4883	186	23	hassan	hassan	PROPN
ejpam-4883	186	24	,	,	PUNCT
ejpam-4883	186	25	j.	j.	PROPN
ejpam-4883	186	26	salim	salim	PROPN
ejpam-4883	186	27	/	/	SYM
ejpam-4883	186	28	eur	eur	PROPN
ejpam-4883	186	29	.	.	PUNCT
ejpam-4883	187	1	j.	j.	PROPN
ejpam-4883	187	2	pure	pure	PROPN
ejpam-4883	187	3	appl	appl	PROPN
ejpam-4883	187	4	.	.	PROPN
ejpam-4883	187	5	math	math	PROPN
ejpam-4883	187	6	,	,	PUNCT
ejpam-4883	187	7	16	16	NUM
ejpam-4883	187	8	(	(	PUNCT
ejpam-4883	187	9	4	4	NUM
ejpam-4883	187	10	)	)	PUNCT
ejpam-4883	187	11	(	(	PUNCT
ejpam-4883	187	12	2023	2023	NUM
ejpam-4883	187	13	)	)	PUNCT
ejpam-4883	187	14	,	,	PUNCT
ejpam-4883	187	15	2082	2082	NUM
ejpam-4883	187	16	-	-	SYM
ejpam-4883	187	17	2095	2095	NUM
ejpam-4883	187	18	2088	2088	NUM
ejpam-4883	187	19	theorem	theorem	VERB
ejpam-4883	187	20	6	6	NUM
ejpam-4883	187	21	.	.	PUNCT
ejpam-4883	188	1	let	let	VERB
ejpam-4883	188	2	g	g	NOUN
ejpam-4883	188	3	be	be	AUX
ejpam-4883	188	4	any	any	DET
ejpam-4883	188	5	graph	graph	NOUN
ejpam-4883	188	6	on	on	ADP
ejpam-4883	188	7	n	n	PRON
ejpam-4883	188	8	≥	≥	NUM
ejpam-4883	188	9	1	1	NUM
ejpam-4883	188	10	vertices	vertex	NOUN
ejpam-4883	188	11	.	.	PUNCT
ejpam-4883	189	1	then	then	ADV
ejpam-4883	189	2	each	each	PRON
ejpam-4883	189	3	of	of	ADP
ejpam-4883	189	4	the	the	DET
ejpam-4883	189	5	following	following	ADJ
ejpam-4883	189	6	statements	statement	NOUN
ejpam-4883	189	7	holds	hold	VERB
ejpam-4883	189	8	.	.	PUNCT
ejpam-4883	190	1	(	(	PUNCT
ejpam-4883	190	2	i	i	NOUN
ejpam-4883	190	3	)	)	PUNCT
ejpam-4883	190	4	1	1	NUM
ejpam-4883	190	5	≤	≤	NOUN
ejpam-4883	190	6	γj(g	γj(g	PROPN
ejpam-4883	190	7	)	)	PUNCT
ejpam-4883	190	8	≤	≤	NOUN
ejpam-4883	190	9	n.	n.	NOUN
ejpam-4883	190	10	(	(	PUNCT
ejpam-4883	190	11	ii	ii	PROPN
ejpam-4883	190	12	)	)	PUNCT
ejpam-4883	190	13	γj(g	γj(g	PUNCT
ejpam-4883	190	14	)	)	PUNCT
ejpam-4883	190	15	=	=	SYM
ejpam-4883	190	16	1	1	NUM
ejpam-4883	190	17	if	if	SCONJ
ejpam-4883	190	18	and	and	CCONJ
ejpam-4883	190	19	only	only	ADV
ejpam-4883	190	20	if	if	SCONJ
ejpam-4883	190	21	g	g	PROPN
ejpam-4883	190	22	is	be	AUX
ejpam-4883	190	23	complete	complete	ADJ
ejpam-4883	190	24	.	.	PUNCT
ejpam-4883	191	1	(	(	PUNCT
ejpam-4883	191	2	iii	iii	X
ejpam-4883	191	3	)	)	PUNCT
ejpam-4883	191	4	if	if	SCONJ
ejpam-4883	191	5	γj(g	γj(g	NOUN
ejpam-4883	191	6	)	)	PUNCT
ejpam-4883	191	7	≤	≤	NOUN
ejpam-4883	191	8	n	n	CCONJ
ejpam-4883	191	9	−	−	PROPN
ejpam-4883	191	10	1	1	NUM
ejpam-4883	191	11	,	,	PUNCT
ejpam-4883	191	12	then	then	ADV
ejpam-4883	191	13	|ng[v]|	|ng[v]|	PROPN
ejpam-4883	191	14	≥	≥	NUM
ejpam-4883	191	15	2	2	NUM
ejpam-4883	191	16	for	for	ADP
ejpam-4883	191	17	some	some	DET
ejpam-4883	191	18	v	v	ADP
ejpam-4883	191	19	∈	∈	NOUN
ejpam-4883	191	20	v	v	NOUN
ejpam-4883	191	21	(	(	PUNCT
ejpam-4883	191	22	g	g	NOUN
ejpam-4883	191	23	)	)	PUNCT
ejpam-4883	191	24	.	.	PUNCT
ejpam-4883	192	1	however	however	ADV
ejpam-4883	192	2	,	,	PUNCT
ejpam-4883	192	3	the	the	DET
ejpam-4883	192	4	converse	converse	NOUN
ejpam-4883	192	5	is	be	AUX
ejpam-4883	192	6	not	not	PART
ejpam-4883	192	7	true	true	ADJ
ejpam-4883	192	8	.	.	PUNCT
ejpam-4883	193	1	(	(	PUNCT
ejpam-4883	193	2	iv	iv	X
ejpam-4883	193	3	)	)	PUNCT
ejpam-4883	193	4	if	if	SCONJ
ejpam-4883	193	5	g	g	PROPN
ejpam-4883	193	6	=	=	SYM
ejpam-4883	193	7	kn	kn	PROPN
ejpam-4883	193	8	,	,	PUNCT
ejpam-4883	193	9	then	then	ADV
ejpam-4883	193	10	γj(g	γj(g	PUNCT
ejpam-4883	193	11	)	)	PUNCT
ejpam-4883	194	1	=	=	SYM
ejpam-4883	194	2	|v	|v	X
ejpam-4883	194	3	(	(	PUNCT
ejpam-4883	194	4	g)|	g)|	PROPN
ejpam-4883	194	5	=	=	NOUN
ejpam-4883	194	6	n.	n.	PROPN
ejpam-4883	194	7	however	however	ADV
ejpam-4883	194	8	,	,	PUNCT
ejpam-4883	194	9	the	the	DET
ejpam-4883	194	10	converse	converse	NOUN
ejpam-4883	194	11	is	be	AUX
ejpam-4883	194	12	not	not	PART
ejpam-4883	194	13	true	true	ADJ
ejpam-4883	194	14	.	.	PUNCT
ejpam-4883	195	1	proof	proof	NOUN
ejpam-4883	195	2	.	.	PUNCT
ejpam-4883	196	1	(	(	PUNCT
ejpam-4883	196	2	i	i	NOUN
ejpam-4883	196	3	)	)	PUNCT
ejpam-4883	196	4	since	since	SCONJ
ejpam-4883	196	5	any	any	DET
ejpam-4883	196	6	singleton	singleton	NOUN
ejpam-4883	196	7	set	set	NOUN
ejpam-4883	196	8	{	{	PUNCT
ejpam-4883	196	9	a	a	PRON
ejpam-4883	196	10	}	}	PUNCT
ejpam-4883	196	11	is	be	AUX
ejpam-4883	196	12	a	a	DET
ejpam-4883	196	13	j	j	NOUN
ejpam-4883	196	14	-	-	PUNCT
ejpam-4883	196	15	set	set	NOUN
ejpam-4883	196	16	for	for	ADP
ejpam-4883	196	17	any	any	DET
ejpam-4883	196	18	a	a	DET
ejpam-4883	196	19	∈	∈	PROPN
ejpam-4883	196	20	v	v	NOUN
ejpam-4883	196	21	(	(	PUNCT
ejpam-4883	196	22	g	g	NOUN
ejpam-4883	196	23	)	)	PUNCT
ejpam-4883	196	24	,	,	PUNCT
ejpam-4883	196	25	the	the	DET
ejpam-4883	196	26	lower	low	ADJ
ejpam-4883	196	27	bound	bind	VERB
ejpam-4883	196	28	follows	follow	VERB
ejpam-4883	196	29	by	by	ADP
ejpam-4883	196	30	corollary	corollary	ADJ
ejpam-4883	196	31	1	1	NUM
ejpam-4883	196	32	.	.	PUNCT
ejpam-4883	197	1	moreover	moreover	ADV
ejpam-4883	197	2	,	,	PUNCT
ejpam-4883	197	3	since	since	SCONJ
ejpam-4883	197	4	any	any	DET
ejpam-4883	197	5	γj	γj	NOUN
ejpam-4883	197	6	-set	-set	PUNCT
ejpam-4883	197	7	d	d	NOUN
ejpam-4883	197	8	is	be	AUX
ejpam-4883	197	9	always	always	ADV
ejpam-4883	197	10	a	a	DET
ejpam-4883	197	11	subset	subset	NOUN
ejpam-4883	197	12	of	of	ADP
ejpam-4883	197	13	v	v	NOUN
ejpam-4883	197	14	(	(	PUNCT
ejpam-4883	197	15	g	g	NOUN
ejpam-4883	197	16	)	)	PUNCT
ejpam-4883	197	17	,	,	PUNCT
ejpam-4883	197	18	the	the	DET
ejpam-4883	197	19	upper	upper	ADJ
ejpam-4883	197	20	bound	bind	VERB
ejpam-4883	197	21	follows	follow	VERB
ejpam-4883	197	22	.	.	PUNCT
ejpam-4883	198	1	consequently	consequently	ADV
ejpam-4883	198	2	,	,	PUNCT
ejpam-4883	198	3	1	1	NUM
ejpam-4883	198	4	≤	≤	NOUN
ejpam-4883	198	5	γj(g	γj(g	PROPN
ejpam-4883	198	6	)	)	PUNCT
ejpam-4883	198	7	≤	≤	NOUN
ejpam-4883	198	8	n.	n.	NOUN
ejpam-4883	198	9	(	(	PUNCT
ejpam-4883	198	10	ii	ii	NOUN
ejpam-4883	198	11	)	)	PUNCT
ejpam-4883	198	12	assume	assume	VERB
ejpam-4883	198	13	that	that	SCONJ
ejpam-4883	198	14	γj(g	γj(g	PUNCT
ejpam-4883	198	15	)	)	PUNCT
ejpam-4883	198	16	=	=	SYM
ejpam-4883	198	17	1	1	X
ejpam-4883	198	18	.	.	PUNCT
ejpam-4883	198	19	suppose	suppose	VERB
ejpam-4883	198	20	g	g	PROPN
ejpam-4883	198	21	is	be	AUX
ejpam-4883	198	22	non	non	ADJ
ejpam-4883	198	23	-	-	ADJ
ejpam-4883	198	24	complete	complete	ADJ
ejpam-4883	198	25	graph	graph	NOUN
ejpam-4883	198	26	.	.	PUNCT
ejpam-4883	199	1	then	then	ADV
ejpam-4883	199	2	there	there	PRON
ejpam-4883	199	3	exist	exist	VERB
ejpam-4883	199	4	a	a	DET
ejpam-4883	199	5	,	,	PUNCT
ejpam-4883	199	6	b	b	PROPN
ejpam-4883	199	7	∈	∈	PROPN
ejpam-4883	199	8	v	v	NOUN
ejpam-4883	199	9	(	(	PUNCT
ejpam-4883	199	10	g	g	NOUN
ejpam-4883	199	11	)	)	PUNCT
ejpam-4883	199	12	such	such	ADJ
ejpam-4883	199	13	that	that	SCONJ
ejpam-4883	199	14	dg(a	dg(a	PROPN
ejpam-4883	199	15	,	,	PUNCT
ejpam-4883	199	16	b	b	NOUN
ejpam-4883	199	17	)	)	PUNCT
ejpam-4883	199	18	≥	≥	NOUN
ejpam-4883	199	19	2	2	NUM
ejpam-4883	199	20	.	.	PUNCT
ejpam-4883	200	1	let	let	VERB
ejpam-4883	200	2	c	c	NOUN
ejpam-4883	200	3	=	=	PUNCT
ejpam-4883	200	4	{	{	PUNCT
ejpam-4883	200	5	a	a	DET
ejpam-4883	200	6	,	,	PUNCT
ejpam-4883	200	7	b	b	NOUN
ejpam-4883	200	8	}	}	PUNCT
ejpam-4883	200	9	.	.	PUNCT
ejpam-4883	201	1	since	since	SCONJ
ejpam-4883	201	2	dg(a	dg(a	NUM
ejpam-4883	201	3	,	,	PUNCT
ejpam-4883	201	4	b	b	X
ejpam-4883	201	5	)	)	PUNCT
ejpam-4883	201	6	≥	≥	NOUN
ejpam-4883	201	7	2	2	NUM
ejpam-4883	201	8	,	,	PUNCT
ejpam-4883	201	9	it	it	PRON
ejpam-4883	201	10	follows	follow	VERB
ejpam-4883	201	11	that	that	SCONJ
ejpam-4883	201	12	a	a	DET
ejpam-4883	201	13	∈	∈	PROPN
ejpam-4883	201	14	ng[a	ng[a	NOUN
ejpam-4883	201	15	]	]	PUNCT
ejpam-4883	201	16	\ng[b	\ng[b	PROPN
ejpam-4883	201	17	]	]	PUNCT
ejpam-4883	201	18	and	and	CCONJ
ejpam-4883	201	19	b	b	X
ejpam-4883	201	20	∈	∈	PROPN
ejpam-4883	201	21	ng[b	ng[b	NOUN
ejpam-4883	201	22	]	]	PUNCT
ejpam-4883	201	23	\ng[a	\ng[a	NOUN
ejpam-4883	201	24	]	]	PUNCT
ejpam-4883	201	25	.	.	PUNCT
ejpam-4883	202	1	thus	thus	ADV
ejpam-4883	202	2	,	,	PUNCT
ejpam-4883	202	3	ng[a	ng[a	PROPN
ejpam-4883	202	4	]	]	PUNCT
ejpam-4883	202	5	\ng[b	\ng[b	PROPN
ejpam-4883	202	6	]	]	X
ejpam-4883	202	7	̸=	̸=	PROPN
ejpam-4883	202	8	∅	∅	NOUN
ejpam-4883	202	9	and	and	CCONJ
ejpam-4883	202	10	ng[b	ng[b	NOUN
ejpam-4883	202	11	]	]	PUNCT
ejpam-4883	202	12	\ng[a	\ng[a	NOUN
ejpam-4883	202	13	]	]	X
ejpam-4883	202	14	̸=	̸=	PROPN
ejpam-4883	202	15	∅	∅	NOUN
ejpam-4883	202	16	c	c	NOUN
ejpam-4883	202	17	,	,	PUNCT
ejpam-4883	202	18	showing	show	VERB
ejpam-4883	202	19	that	that	SCONJ
ejpam-4883	202	20	c	c	PROPN
ejpam-4883	202	21	is	be	AUX
ejpam-4883	202	22	a	a	DET
ejpam-4883	202	23	j	j	NOUN
ejpam-4883	202	24	-	-	PUNCT
ejpam-4883	202	25	set	set	NOUN
ejpam-4883	202	26	in	in	ADP
ejpam-4883	202	27	g.	g.	NOUN
ejpam-4883	202	28	by	by	ADP
ejpam-4883	202	29	corollary	corollary	ADJ
ejpam-4883	202	30	1	1	NUM
ejpam-4883	202	31	,	,	PUNCT
ejpam-4883	202	32	γj(g	γj(g	NUM
ejpam-4883	202	33	)	)	PUNCT
ejpam-4883	202	34	≥	≥	NOUN
ejpam-4883	202	35	2	2	NUM
ejpam-4883	202	36	which	which	PRON
ejpam-4883	202	37	is	be	AUX
ejpam-4883	202	38	a	a	DET
ejpam-4883	202	39	contradiction	contradiction	NOUN
ejpam-4883	202	40	.	.	PUNCT
ejpam-4883	203	1	hence	hence	ADV
ejpam-4883	203	2	,	,	PUNCT
ejpam-4883	203	3	g	g	PROPN
ejpam-4883	203	4	is	be	AUX
ejpam-4883	203	5	complete	complete	ADJ
ejpam-4883	203	6	.	.	PUNCT
ejpam-4883	204	1	conversely	conversely	ADV
ejpam-4883	204	2	,	,	PUNCT
ejpam-4883	204	3	suppose	suppose	VERB
ejpam-4883	204	4	that	that	SCONJ
ejpam-4883	204	5	g	g	PROPN
ejpam-4883	204	6	is	be	AUX
ejpam-4883	204	7	complete	complete	ADJ
ejpam-4883	204	8	.	.	PUNCT
ejpam-4883	205	1	then	then	ADV
ejpam-4883	205	2	ng[a	ng[a	X
ejpam-4883	205	3	]	]	X
ejpam-4883	205	4	=	=	SYM
ejpam-4883	205	5	v	v	X
ejpam-4883	205	6	(	(	PUNCT
ejpam-4883	205	7	g	g	NOUN
ejpam-4883	205	8	)	)	PUNCT
ejpam-4883	205	9	for	for	ADP
ejpam-4883	205	10	any	any	DET
ejpam-4883	205	11	a	a	DET
ejpam-4883	205	12	∈	∈	PROPN
ejpam-4883	205	13	v	v	NOUN
ejpam-4883	205	14	(	(	PUNCT
ejpam-4883	205	15	g	g	NOUN
ejpam-4883	205	16	)	)	PUNCT
ejpam-4883	205	17	.	.	PUNCT
ejpam-4883	206	1	it	it	PRON
ejpam-4883	206	2	follows	follow	VERB
ejpam-4883	206	3	that	that	SCONJ
ejpam-4883	206	4	for	for	ADP
ejpam-4883	206	5	every	every	DET
ejpam-4883	206	6	two	two	NUM
ejpam-4883	206	7	distinct	distinct	ADJ
ejpam-4883	206	8	vertices	vertex	NOUN
ejpam-4883	206	9	x	x	X
ejpam-4883	206	10	,	,	PUNCT
ejpam-4883	206	11	y	y	PROPN
ejpam-4883	206	12	∈	∈	PROPN
ejpam-4883	206	13	v	v	NOUN
ejpam-4883	206	14	(	(	PUNCT
ejpam-4883	206	15	g	g	NOUN
ejpam-4883	206	16	)	)	PUNCT
ejpam-4883	206	17	,	,	PUNCT
ejpam-4883	206	18	ng[x]\ng[y	ng[x]\ng[y	PROPN
ejpam-4883	206	19	]	]	X
ejpam-4883	206	20	=	=	SYM
ejpam-4883	206	21	v	v	X
ejpam-4883	206	22	(	(	PUNCT
ejpam-4883	206	23	g)\v	g)\v	NOUN
ejpam-4883	206	24	(	(	PUNCT
ejpam-4883	206	25	g	g	NOUN
ejpam-4883	206	26	)	)	PUNCT
ejpam-4883	206	27	=	=	NOUN
ejpam-4883	206	28	∅.	∅.	ADP
ejpam-4883	206	29	thus	thus	ADV
ejpam-4883	206	30	,	,	PUNCT
ejpam-4883	206	31	γj(g	γj(g	NUM
ejpam-4883	206	32	)	)	PUNCT
ejpam-4883	206	33	≥	≥	PRON
ejpam-4883	206	34	2	2	NUM
ejpam-4883	206	35	is	be	AUX
ejpam-4883	206	36	impossible	impossible	ADJ
ejpam-4883	206	37	.	.	PUNCT
ejpam-4883	207	1	hence	hence	ADV
ejpam-4883	207	2	,	,	PUNCT
ejpam-4883	207	3	γj(g	γj(g	PUNCT
ejpam-4883	207	4	)	)	PUNCT
ejpam-4883	207	5	=	=	SYM
ejpam-4883	207	6	1	1	X
ejpam-4883	207	7	by	by	ADP
ejpam-4883	207	8	(	(	PUNCT
ejpam-4883	207	9	i	i	NOUN
ejpam-4883	207	10	)	)	PUNCT
ejpam-4883	207	11	.	.	PUNCT
ejpam-4883	208	1	(	(	PUNCT
ejpam-4883	208	2	iii	iii	X
ejpam-4883	208	3	)	)	PUNCT
ejpam-4883	208	4	assume	assume	VERB
ejpam-4883	208	5	that	that	SCONJ
ejpam-4883	208	6	γj(g	γj(g	NOUN
ejpam-4883	208	7	)	)	PUNCT
ejpam-4883	208	8	≤	≤	NOUN
ejpam-4883	208	9	n−1	n−1	PROPN
ejpam-4883	208	10	.	.	PROPN
ejpam-4883	208	11	suppose	suppose	VERB
ejpam-4883	208	12	on	on	ADP
ejpam-4883	208	13	the	the	DET
ejpam-4883	208	14	contrary	contrary	NOUN
ejpam-4883	208	15	that	that	SCONJ
ejpam-4883	208	16	|ng[v]|	|ng[v]|	PROPN
ejpam-4883	208	17	<	<	X
ejpam-4883	208	18	2	2	NUM
ejpam-4883	208	19	for	for	ADP
ejpam-4883	208	20	every	every	DET
ejpam-4883	208	21	v	v	NUM
ejpam-4883	208	22	∈	∈	NOUN
ejpam-4883	208	23	v	v	NOUN
ejpam-4883	208	24	(	(	PUNCT
ejpam-4883	208	25	g	g	NOUN
ejpam-4883	208	26	)	)	PUNCT
ejpam-4883	208	27	.	.	PUNCT
ejpam-4883	209	1	this	this	PRON
ejpam-4883	209	2	means	mean	VERB
ejpam-4883	209	3	that	that	SCONJ
ejpam-4883	209	4	|ng[v]|	|ng[v]|	PROPN
ejpam-4883	209	5	=	=	SYM
ejpam-4883	209	6	1	1	NUM
ejpam-4883	209	7	for	for	ADP
ejpam-4883	209	8	every	every	DET
ejpam-4883	209	9	v	v	NUM
ejpam-4883	209	10	∈	∈	NOUN
ejpam-4883	209	11	v	v	NOUN
ejpam-4883	209	12	(	(	PUNCT
ejpam-4883	209	13	g	g	NOUN
ejpam-4883	209	14	)	)	PUNCT
ejpam-4883	209	15	.	.	PUNCT
ejpam-4883	210	1	it	it	PRON
ejpam-4883	210	2	follows	follow	VERB
ejpam-4883	210	3	that	that	SCONJ
ejpam-4883	210	4	g	g	PROPN
ejpam-4883	210	5	=	=	SYM
ejpam-4883	210	6	kn	kn	PROPN
ejpam-4883	210	7	.	.	PROPN
ejpam-4883	210	8	observe	observe	VERB
ejpam-4883	210	9	that	that	SCONJ
ejpam-4883	210	10	γ(kn	γ(kn	VERB
ejpam-4883	210	11	)	)	PUNCT
ejpam-4883	210	12	=	=	SYM
ejpam-4883	210	13	n.	n.	PROPN
ejpam-4883	210	14	thus	thus	ADV
ejpam-4883	210	15	,	,	PUNCT
ejpam-4883	210	16	γ(kn	γ(kn	NUM
ejpam-4883	210	17	)	)	PUNCT
ejpam-4883	210	18	=	=	SYM
ejpam-4883	210	19	n	n	PRON
ejpam-4883	210	20	≤	≤	NOUN
ejpam-4883	210	21	γj(g	γj(g	PUNCT
ejpam-4883	210	22	)	)	PUNCT
ejpam-4883	210	23	by	by	ADP
ejpam-4883	210	24	proposition	proposition	NOUN
ejpam-4883	210	25	1	1	NUM
ejpam-4883	210	26	,	,	PUNCT
ejpam-4883	210	27	a	a	DET
ejpam-4883	210	28	contradiction	contradiction	NOUN
ejpam-4883	210	29	.	.	PUNCT
ejpam-4883	211	1	to	to	PART
ejpam-4883	211	2	see	see	VERB
ejpam-4883	211	3	that	that	SCONJ
ejpam-4883	211	4	the	the	DET
ejpam-4883	211	5	converse	converse	NOUN
ejpam-4883	211	6	is	be	AUX
ejpam-4883	211	7	not	not	PART
ejpam-4883	211	8	true	true	ADJ
ejpam-4883	211	9	,	,	PUNCT
ejpam-4883	211	10	consider	consider	VERB
ejpam-4883	211	11	the	the	DET
ejpam-4883	211	12	graphg	graphg	NOUN
ejpam-4883	211	13	in	in	ADP
ejpam-4883	211	14	figure	figure	NOUN
ejpam-4883	211	15	3	3	NUM
ejpam-4883	211	16	.	.	PUNCT
ejpam-4883	211	17	letk	letk	PROPN
ejpam-4883	211	18	=	=	SYM
ejpam-4883	211	19	v	v	NOUN
ejpam-4883	211	20	(	(	PUNCT
ejpam-4883	211	21	g	g	NOUN
ejpam-4883	211	22	)	)	PUNCT
ejpam-4883	211	23	=	=	NOUN
ejpam-4883	211	24	{	{	PUNCT
ejpam-4883	211	25	a	a	PRON
ejpam-4883	211	26	,	,	PUNCT
ejpam-4883	211	27	b	b	NOUN
ejpam-4883	211	28	,	,	PUNCT
ejpam-4883	211	29	c	c	NOUN
ejpam-4883	211	30	,	,	PUNCT
ejpam-4883	211	31	d	d	NOUN
ejpam-4883	211	32	,	,	PUNCT
ejpam-4883	211	33	e	e	NOUN
ejpam-4883	211	34	,	,	PUNCT
ejpam-4883	211	35	f	f	NOUN
ejpam-4883	211	36	}	}	PUNCT
ejpam-4883	211	37	.	.	PUNCT
ejpam-4883	212	1	then	then	ADV
ejpam-4883	212	2	ng[a	ng[a	PROPN
ejpam-4883	212	3	]	]	PUNCT
ejpam-4883	212	4	\	\	PROPN
ejpam-4883	212	5	ng[b	ng[b	NOUN
ejpam-4883	212	6	]	]	X
ejpam-4883	212	7	=	=	PUNCT
ejpam-4883	212	8	{	{	PUNCT
ejpam-4883	212	9	d	d	NOUN
ejpam-4883	212	10	}	}	PUNCT
ejpam-4883	212	11	,	,	PUNCT
ejpam-4883	212	12	ng[a	ng[a	PROPN
ejpam-4883	212	13	]	]	PUNCT
ejpam-4883	212	14	\	\	PROPN
ejpam-4883	212	15	ng[c	ng[c	PROPN
ejpam-4883	212	16	]	]	X
ejpam-4883	212	17	=	=	X
ejpam-4883	212	18	{	{	PUNCT
ejpam-4883	212	19	a	a	NOUN
ejpam-4883	212	20	}	}	PUNCT
ejpam-4883	212	21	,	,	PUNCT
ejpam-4883	212	22	ng[a	ng[a	PROPN
ejpam-4883	212	23	]	]	PUNCT
ejpam-4883	212	24	\	\	PROPN
ejpam-4883	212	25	ng[d	ng[d	PROPN
ejpam-4883	212	26	]	]	X
ejpam-4883	212	27	=	=	PUNCT
ejpam-4883	212	28	{	{	PUNCT
ejpam-4883	212	29	b	b	NOUN
ejpam-4883	212	30	}	}	PUNCT
ejpam-4883	212	31	,	,	PUNCT
ejpam-4883	212	32	ng[a	ng[a	PROPN
ejpam-4883	212	33	]	]	PUNCT
ejpam-4883	212	34	\	\	PUNCT
ejpam-4883	212	35	ng[e	ng[e	PROPN
ejpam-4883	212	36	]	]	X
ejpam-4883	212	37	=	=	X
ejpam-4883	212	38	{	{	PUNCT
ejpam-4883	212	39	a	a	PRON
ejpam-4883	212	40	,	,	PUNCT
ejpam-4883	212	41	b	b	NOUN
ejpam-4883	212	42	}	}	PUNCT
ejpam-4883	212	43	,	,	PUNCT
ejpam-4883	212	44	ng[a	ng[a	PROPN
ejpam-4883	212	45	]	]	PUNCT
ejpam-4883	212	46	\	\	PUNCT
ejpam-4883	213	1	ng[f	ng[f	PROPN
ejpam-4883	213	2	]	]	PUNCT
ejpam-4883	213	3	=	=	PUNCT
ejpam-4883	213	4	{	{	PUNCT
ejpam-4883	213	5	a	a	X
ejpam-4883	213	6	,	,	PUNCT
ejpam-4883	213	7	d	d	NOUN
ejpam-4883	213	8	}	}	PUNCT
ejpam-4883	213	9	,	,	PUNCT
ejpam-4883	213	10	ng[b	ng[b	PROPN
ejpam-4883	213	11	]	]	PUNCT
ejpam-4883	213	12	\	\	PROPN
ejpam-4883	214	1	ng[a	ng[a	NOUN
ejpam-4883	214	2	]	]	X
ejpam-4883	214	3	=	=	X
ejpam-4883	214	4	{	{	PUNCT
ejpam-4883	214	5	c	c	NOUN
ejpam-4883	214	6	,	,	PUNCT
ejpam-4883	214	7	f	f	NOUN
ejpam-4883	214	8	}	}	PUNCT
ejpam-4883	214	9	,	,	PUNCT
ejpam-4883	214	10	ng[b	ng[b	PROPN
ejpam-4883	214	11	]	]	PUNCT
ejpam-4883	214	12	\	\	PROPN
ejpam-4883	214	13	ng[c	ng[c	PROPN
ejpam-4883	214	14	]	]	X
ejpam-4883	214	15	=	=	X
ejpam-4883	214	16	{	{	PUNCT
ejpam-4883	214	17	a	a	X
ejpam-4883	214	18	,	,	PUNCT
ejpam-4883	214	19	f	f	NOUN
ejpam-4883	214	20	}	}	PUNCT
ejpam-4883	214	21	,	,	PUNCT
ejpam-4883	214	22	ng[b	ng[b	PROPN
ejpam-4883	214	23	]	]	PUNCT
ejpam-4883	214	24	\	\	PROPN
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ejpam-4883	216	9	]	]	PUNCT
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ejpam-4883	226	5	:	:	PUNCT
ejpam-4883	226	6	a	a	DET
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ejpam-4883	226	10	|ng[v]|	|ng[v]|	PROPN
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ejpam-4883	226	12	2	2	NUM
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ejpam-4883	226	15	v	v	ADP
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ejpam-4883	226	19	g	g	NOUN
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ejpam-4883	226	25	(	(	PUNCT
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ejpam-4883	226	28	=	=	SYM
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ejpam-4883	227	4	(	(	PUNCT
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ejpam-4883	227	6	)	)	PUNCT
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ejpam-4883	230	1	,	,	PUNCT
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ejpam-4883	230	11	ng[vi	ng[vi	X
ejpam-4883	230	12	]	]	X
ejpam-4883	230	13	=	=	SYM
ejpam-4883	230	14	{	{	PUNCT
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ejpam-4883	230	20	∈	∈	PROPN
ejpam-4883	230	21	{	{	PUNCT
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ejpam-4883	230	25	,	,	PUNCT
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ejpam-4883	232	5	ng[vi	ng[vi	PROPN
ejpam-4883	232	6	]	]	X
ejpam-4883	232	7	\ng[vj	\ng[vj	NOUN
ejpam-4883	232	8	]	]	PUNCT
ejpam-4883	232	9	for	for	ADP
ejpam-4883	232	10	each	each	DET
ejpam-4883	232	11	i	i	PRON
ejpam-4883	232	12	̸=	̸=	PROPN
ejpam-4883	232	13	j	j	PROPN
ejpam-4883	232	14	,	,	PUNCT
ejpam-4883	232	15	where	where	SCONJ
ejpam-4883	232	16	i	i	PRON
ejpam-4883	232	17	,	,	PUNCT
ejpam-4883	232	18	j	j	PROPN
ejpam-4883	232	19	∈	∈	PROPN
ejpam-4883	232	20	{	{	PUNCT
ejpam-4883	232	21	1	1	NUM
ejpam-4883	232	22	,	,	PUNCT
ejpam-4883	232	23	2	2	NUM
ejpam-4883	232	24	,	,	PUNCT
ejpam-4883	232	25	.	.	PUNCT
ejpam-4883	232	26	.	.	PUNCT
ejpam-4883	233	1	.	.	PUNCT
ejpam-4883	233	2	,	,	PUNCT
ejpam-4883	233	3	n	n	CCONJ
ejpam-4883	233	4	}	}	PUNCT
ejpam-4883	233	5	.	.	PUNCT
ejpam-4883	234	1	this	this	PRON
ejpam-4883	234	2	means	mean	VERB
ejpam-4883	234	3	thatng[vi]\ng[vj	thatng[vi]\ng[vj	NOUN
ejpam-4883	234	4	]	]	PUNCT
ejpam-4883	234	5	̸=	̸=	PROPN
ejpam-4883	234	6	∅	∅	NOUN
ejpam-4883	234	7	for	for	ADP
ejpam-4883	234	8	each	each	DET
ejpam-4883	234	9	i	i	PRON
ejpam-4883	234	10	̸=	̸=	PROPN
ejpam-4883	234	11	j.	j.	PROPN
ejpam-4883	234	12	thus	thus	ADV
ejpam-4883	234	13	,	,	PUNCT
ejpam-4883	234	14	v	v	X
ejpam-4883	234	15	(	(	PUNCT
ejpam-4883	234	16	g	g	NOUN
ejpam-4883	234	17	)	)	PUNCT
ejpam-4883	234	18	is	be	AUX
ejpam-4883	234	19	a	a	DET
ejpam-4883	234	20	j	j	NOUN
ejpam-4883	234	21	-	-	PUNCT
ejpam-4883	234	22	set	set	VERB
ejpam-4883	234	23	ofg	ofg	PROPN
ejpam-4883	234	24	.	.	PUNCT
ejpam-4883	235	1	since	since	SCONJ
ejpam-4883	235	2	v	v	NOUN
ejpam-4883	235	3	(	(	PUNCT
ejpam-4883	235	4	g	g	NOUN
ejpam-4883	235	5	)	)	PUNCT
ejpam-4883	235	6	is	be	AUX
ejpam-4883	235	7	a	a	DET
ejpam-4883	235	8	dominating	dominating	NOUN
ejpam-4883	235	9	set	set	NOUN
ejpam-4883	235	10	of	of	ADP
ejpam-4883	235	11	g	g	PROPN
ejpam-4883	235	12	,	,	PUNCT
ejpam-4883	235	13	v	v	X
ejpam-4883	235	14	(	(	PUNCT
ejpam-4883	235	15	g	g	NOUN
ejpam-4883	235	16	)	)	PUNCT
ejpam-4883	235	17	is	be	AUX
ejpam-4883	235	18	a	a	DET
ejpam-4883	235	19	j	j	PROPN
ejpam-4883	235	20	-	-	PUNCT
ejpam-4883	235	21	dominating	dominating	NOUN
ejpam-4883	235	22	set	set	NOUN
ejpam-4883	235	23	in	in	ADP
ejpam-4883	235	24	g.	g.	PROPN
ejpam-4883	235	25	hence	hence	ADV
ejpam-4883	235	26	,	,	PUNCT
ejpam-4883	235	27	γj(g	γj(g	PUNCT
ejpam-4883	235	28	)	)	PUNCT
ejpam-4883	235	29	=	=	SYM
ejpam-4883	235	30	|v	|v	X
ejpam-4883	235	31	(	(	PUNCT
ejpam-4883	235	32	g)|	g)|	NOUN
ejpam-4883	235	33	=	=	PUNCT
ejpam-4883	235	34	n	n	X
ejpam-4883	235	35	by	by	X
ejpam-4883	235	36	(	(	PUNCT
ejpam-4883	235	37	i	i	NOUN
ejpam-4883	235	38	)	)	PUNCT
ejpam-4883	235	39	.	.	PUNCT
ejpam-4883	236	1	the	the	DET
ejpam-4883	236	2	converse	converse	NOUN
ejpam-4883	236	3	follows	follow	VERB
ejpam-4883	236	4	by	by	ADP
ejpam-4883	236	5	considering	consider	VERB
ejpam-4883	236	6	the	the	DET
ejpam-4883	236	7	graph	graph	NOUN
ejpam-4883	236	8	in	in	ADP
ejpam-4883	236	9	figure	figure	NOUN
ejpam-4883	236	10	3	3	NUM
ejpam-4883	236	11	.	.	PUNCT
ejpam-4883	236	12	theorem	theorem	NOUN
ejpam-4883	236	13	7	7	NUM
ejpam-4883	236	14	.	.	PUNCT
ejpam-4883	237	1	let	let	VERB
ejpam-4883	237	2	g	g	NOUN
ejpam-4883	237	3	be	be	AUX
ejpam-4883	237	4	any	any	DET
ejpam-4883	237	5	non	non	ADJ
ejpam-4883	237	6	-	-	ADJ
ejpam-4883	237	7	complete	complete	ADJ
ejpam-4883	237	8	graph	graph	NOUN
ejpam-4883	237	9	.	.	PUNCT
ejpam-4883	238	1	then	then	ADV
ejpam-4883	238	2	every	every	DET
ejpam-4883	238	3	dominating	dominating	NOUN
ejpam-4883	238	4	vertex	vertex	NOUN
ejpam-4883	238	5	v	v	NOUN
ejpam-4883	238	6	of	of	ADP
ejpam-4883	238	7	g	g	PROPN
ejpam-4883	238	8	is	be	AUX
ejpam-4883	238	9	not	not	PART
ejpam-4883	238	10	in	in	ADP
ejpam-4883	238	11	γj	γj	PROPN
ejpam-4883	238	12	-set	-set	PUNCT
ejpam-4883	238	13	d	d	PROPN
ejpam-4883	238	14	of	of	ADP
ejpam-4883	238	15	g.	g.	PROPN
ejpam-4883	238	16	proof	proof	NOUN
ejpam-4883	238	17	.	.	PUNCT
ejpam-4883	239	1	suppose	suppose	VERB
ejpam-4883	239	2	g	g	PROPN
ejpam-4883	239	3	is	be	AUX
ejpam-4883	239	4	non	non	ADJ
ejpam-4883	239	5	-	-	ADJ
ejpam-4883	239	6	complete	complete	ADJ
ejpam-4883	239	7	graph	graph	NOUN
ejpam-4883	239	8	.	.	PUNCT
ejpam-4883	240	1	then	then	ADV
ejpam-4883	240	2	γj(g	γj(g	PUNCT
ejpam-4883	240	3	)	)	PUNCT
ejpam-4883	240	4	≥	≥	NOUN
ejpam-4883	240	5	2	2	NUM
ejpam-4883	240	6	by	by	ADP
ejpam-4883	240	7	theorem	theorem	ADJ
ejpam-4883	240	8	6(ii	6(ii	NOUN
ejpam-4883	240	9	)	)	PUNCT
ejpam-4883	240	10	.	.	PUNCT
ejpam-4883	241	1	now	now	ADV
ejpam-4883	241	2	,	,	PUNCT
ejpam-4883	241	3	let	let	VERB
ejpam-4883	241	4	v	v	NUM
ejpam-4883	241	5	∈	∈	PROPN
ejpam-4883	241	6	v	v	NOUN
ejpam-4883	241	7	(	(	PUNCT
ejpam-4883	241	8	g	g	NOUN
ejpam-4883	241	9	)	)	PUNCT
ejpam-4883	241	10	be	be	AUX
ejpam-4883	241	11	a	a	DET
ejpam-4883	241	12	dominating	dominating	NOUN
ejpam-4883	241	13	vertex	vertex	NOUN
ejpam-4883	241	14	of	of	ADP
ejpam-4883	241	15	g.	g.	PROPN
ejpam-4883	241	16	suppose	suppose	VERB
ejpam-4883	241	17	on	on	ADP
ejpam-4883	241	18	the	the	DET
ejpam-4883	241	19	contrary	contrary	NOUN
ejpam-4883	241	20	that	that	SCONJ
ejpam-4883	241	21	v	v	ADP
ejpam-4883	241	22	∈	∈	PROPN
ejpam-4883	241	23	d.	d.	NOUN
ejpam-4883	241	24	since	since	SCONJ
ejpam-4883	241	25	v	v	NUM
ejpam-4883	241	26	is	be	AUX
ejpam-4883	241	27	a	a	DET
ejpam-4883	241	28	dominating	dominating	NOUN
ejpam-4883	241	29	vertex	vertex	NOUN
ejpam-4883	241	30	of	of	ADP
ejpam-4883	241	31	g	g	PROPN
ejpam-4883	241	32	,	,	PUNCT
ejpam-4883	241	33	we	we	PRON
ejpam-4883	241	34	have	have	VERB
ejpam-4883	241	35	ng[v	ng[v	ADV
ejpam-4883	241	36	]	]	X
ejpam-4883	242	1	=	=	SYM
ejpam-4883	242	2	v	v	X
ejpam-4883	242	3	(	(	PUNCT
ejpam-4883	242	4	g	g	NOUN
ejpam-4883	242	5	)	)	PUNCT
ejpam-4883	242	6	.	.	PUNCT
ejpam-4883	243	1	thus	thus	ADV
ejpam-4883	243	2	,	,	PUNCT
ejpam-4883	243	3	ng[u	ng[u	PROPN
ejpam-4883	243	4	]	]	PUNCT
ejpam-4883	243	5	\ng[v	\ng[v	NUM
ejpam-4883	243	6	]	]	X
ejpam-4883	243	7	=	=	SYM
ejpam-4883	243	8	ng[u	ng[u	PROPN
ejpam-4883	243	9	]	]	PUNCT
ejpam-4883	243	10	\	\	PROPN
ejpam-4883	243	11	v	v	X
ejpam-4883	243	12	(	(	PUNCT
ejpam-4883	243	13	g	g	NOUN
ejpam-4883	243	14	)	)	PUNCT
ejpam-4883	243	15	=	=	NOUN
ejpam-4883	243	16	∅	∅	NOUN
ejpam-4883	243	17	∀	∀	X
ejpam-4883	243	18	u	u	NOUN
ejpam-4883	243	19	∈	∈	PROPN
ejpam-4883	243	20	d	d	X
ejpam-4883	243	21	\	\	X
ejpam-4883	243	22	{	{	PUNCT
ejpam-4883	243	23	v	v	NOUN
ejpam-4883	243	24	}	}	PUNCT
ejpam-4883	243	25	.	.	PUNCT
ejpam-4883	244	1	it	it	PRON
ejpam-4883	244	2	follows	follow	VERB
ejpam-4883	244	3	that	that	SCONJ
ejpam-4883	244	4	{	{	PUNCT
ejpam-4883	244	5	v	v	NOUN
ejpam-4883	244	6	}	}	PUNCT
ejpam-4883	244	7	is	be	AUX
ejpam-4883	244	8	a	a	DET
ejpam-4883	244	9	maximum	maximum	ADJ
ejpam-4883	244	10	j	j	NOUN
ejpam-4883	244	11	-	-	NOUN
ejpam-4883	244	12	set	set	NOUN
ejpam-4883	244	13	of	of	ADP
ejpam-4883	244	14	g.	g.	PROPN
ejpam-4883	244	15	by	by	ADP
ejpam-4883	244	16	theorem	theorem	NOUN
ejpam-4883	244	17	3	3	NUM
ejpam-4883	244	18	,	,	PUNCT
ejpam-4883	244	19	γj(g	γj(g	PUNCT
ejpam-4883	244	20	)	)	PUNCT
ejpam-4883	244	21	=	=	SYM
ejpam-4883	244	22	1	1	NUM
ejpam-4883	244	23	,	,	PUNCT
ejpam-4883	244	24	a	a	DET
ejpam-4883	244	25	contradiction	contradiction	NOUN
ejpam-4883	244	26	.	.	PUNCT
ejpam-4883	245	1	therefore	therefore	ADV
ejpam-4883	245	2	,	,	PUNCT
ejpam-4883	245	3	v	v	X
ejpam-4883	245	4	/∈	/∈	PUNCT
ejpam-4883	245	5	d.	d.	PROPN
ejpam-4883	246	1	the	the	DET
ejpam-4883	246	2	next	next	ADJ
ejpam-4883	246	3	result	result	NOUN
ejpam-4883	246	4	is	be	AUX
ejpam-4883	246	5	a	a	DET
ejpam-4883	246	6	realization	realization	NOUN
ejpam-4883	246	7	result	result	NOUN
ejpam-4883	246	8	involving	involve	VERB
ejpam-4883	246	9	j	j	PROPN
ejpam-4883	246	10	-	-	PUNCT
ejpam-4883	246	11	domination	domination	NOUN
ejpam-4883	246	12	number	number	NOUN
ejpam-4883	246	13	and	and	CCONJ
ejpam-4883	246	14	domination	domination	NOUN
ejpam-4883	246	15	number	number	NOUN
ejpam-4883	246	16	of	of	ADP
ejpam-4883	246	17	a	a	DET
ejpam-4883	246	18	graph	graph	NOUN
ejpam-4883	246	19	.	.	PUNCT
ejpam-4883	247	1	theorem	theorem	NOUN
ejpam-4883	247	2	8	8	NUM
ejpam-4883	247	3	.	.	PUNCT
ejpam-4883	248	1	let	let	VERB
ejpam-4883	248	2	m	m	PRON
ejpam-4883	248	3	and	and	CCONJ
ejpam-4883	248	4	n	n	ADV
ejpam-4883	248	5	be	be	VERB
ejpam-4883	248	6	positive	positive	ADJ
ejpam-4883	248	7	integers	integer	NOUN
ejpam-4883	248	8	such	such	ADJ
ejpam-4883	248	9	that	that	SCONJ
ejpam-4883	248	10	2	2	NUM
ejpam-4883	248	11	≤	≤	NUM
ejpam-4883	248	12	a	a	DET
ejpam-4883	248	13	≤	≤	PROPN
ejpam-4883	248	14	b.	b.	NOUN
ejpam-4883	249	1	then	then	ADV
ejpam-4883	249	2	there	there	PRON
ejpam-4883	249	3	exists	exist	VERB
ejpam-4883	249	4	a	a	DET
ejpam-4883	249	5	connected	connected	ADJ
ejpam-4883	249	6	graph	graph	NOUN
ejpam-4883	249	7	g	g	ADP
ejpam-4883	249	8	such	such	ADJ
ejpam-4883	249	9	that	that	DET
ejpam-4883	249	10	γ(g	γ(g	PROPN
ejpam-4883	249	11	)	)	PUNCT
ejpam-4883	250	1	=	=	SYM
ejpam-4883	250	2	a	a	PRON
ejpam-4883	250	3	and	and	CCONJ
ejpam-4883	250	4	γj(g	γj(g	NUM
ejpam-4883	250	5	)	)	PUNCT
ejpam-4883	250	6	=	=	SYM
ejpam-4883	251	1	b.	b.	PROPN
ejpam-4883	251	2	that	that	ADV
ejpam-4883	251	3	is	be	AUX
ejpam-4883	251	4	,	,	PUNCT
ejpam-4883	251	5	γj(g)−γ(g	γj(g)−γ(g	PROPN
ejpam-4883	251	6	)	)	PUNCT
ejpam-4883	251	7	can	can	AUX
ejpam-4883	251	8	be	be	AUX
ejpam-4883	251	9	made	make	VERB
ejpam-4883	251	10	arbitrarily	arbitrarily	ADV
ejpam-4883	251	11	large	large	ADJ
ejpam-4883	251	12	.	.	PUNCT
ejpam-4883	252	1	proof	proof	NOUN
ejpam-4883	252	2	.	.	PUNCT
ejpam-4883	253	1	for	for	ADP
ejpam-4883	253	2	equality	equality	NOUN
ejpam-4883	253	3	,	,	PUNCT
ejpam-4883	253	4	consider	consider	VERB
ejpam-4883	253	5	the	the	DET
ejpam-4883	253	6	graph	graph	NOUN
ejpam-4883	253	7	ka	ka	PROPN
ejpam-4883	253	8	.	.	PROPN
ejpam-4883	254	1	since	since	SCONJ
ejpam-4883	254	2	γ(ks	γ(ks	PROPN
ejpam-4883	254	3	)	)	PUNCT
ejpam-4883	255	1	=	=	SYM
ejpam-4883	255	2	s	s	X
ejpam-4883	255	3	,	,	PUNCT
ejpam-4883	255	4	it	it	PRON
ejpam-4883	255	5	follows	follow	VERB
ejpam-4883	255	6	that	that	SCONJ
ejpam-4883	255	7	γj(ka	γj(ka	NOUN
ejpam-4883	255	8	)	)	PUNCT
ejpam-4883	255	9	=	=	SYM
ejpam-4883	256	1	a	a	DET
ejpam-4883	256	2	=	=	SYM
ejpam-4883	256	3	γ(ka	γ(ka	NOUN
ejpam-4883	256	4	)	)	PUNCT
ejpam-4883	256	5	by	by	ADP
ejpam-4883	256	6	theorem	theorem	ADJ
ejpam-4883	256	7	6(iv	6(iv	NOUN
ejpam-4883	256	8	)	)	PUNCT
ejpam-4883	256	9	.	.	PUNCT
ejpam-4883	257	1	for	for	ADP
ejpam-4883	257	2	the	the	DET
ejpam-4883	257	3	inequality(a	inequality(a	PROPN
ejpam-4883	257	4	<	<	X
ejpam-4883	257	5	b	b	PROPN
ejpam-4883	257	6	)	)	PUNCT
ejpam-4883	257	7	,	,	PUNCT
ejpam-4883	257	8	consider	consider	VERB
ejpam-4883	257	9	the	the	DET
ejpam-4883	257	10	following	follow	VERB
ejpam-4883	257	11	cases	case	NOUN
ejpam-4883	257	12	:	:	PUNCT
ejpam-4883	257	13	case	case	NOUN
ejpam-4883	257	14	1	1	NUM
ejpam-4883	257	15	:	:	PUNCT
ejpam-4883	257	16	a=2	a=2	VERB
ejpam-4883	257	17	let	let	VERB
ejpam-4883	257	18	m	m	NOUN
ejpam-4883	257	19	=	=	VERB
ejpam-4883	257	20	b	b	NOUN
ejpam-4883	257	21	−	−	PROPN
ejpam-4883	257	22	2	2	NUM
ejpam-4883	257	23	and	and	CCONJ
ejpam-4883	257	24	consider	consider	VERB
ejpam-4883	257	25	consider	consider	VERB
ejpam-4883	257	26	the	the	DET
ejpam-4883	257	27	graph	graph	NOUN
ejpam-4883	257	28	g	g	NOUN
ejpam-4883	257	29	in	in	ADP
ejpam-4883	257	30	figure	figure	NOUN
ejpam-4883	257	31	4	4	NUM
ejpam-4883	257	32	.	.	PUNCT
ejpam-4883	258	1	let	let	VERB
ejpam-4883	258	2	d1	d1	PROPN
ejpam-4883	258	3	=	=	SYM
ejpam-4883	258	4	{	{	PUNCT
ejpam-4883	258	5	y1	y1	NOUN
ejpam-4883	258	6	,	,	PUNCT
ejpam-4883	258	7	y2	y2	NOUN
ejpam-4883	258	8	}	}	PUNCT
ejpam-4883	258	9	and	and	CCONJ
ejpam-4883	258	10	j.	j.	PROPN
ejpam-4883	258	11	hassan	hassan	PROPN
ejpam-4883	258	12	,	,	PUNCT
ejpam-4883	258	13	j.	j.	PROPN
ejpam-4883	258	14	salim	salim	PROPN
ejpam-4883	258	15	/	/	SYM
ejpam-4883	258	16	eur	eur	PROPN
ejpam-4883	258	17	.	.	PUNCT
ejpam-4883	259	1	j.	j.	PROPN
ejpam-4883	259	2	pure	pure	PROPN
ejpam-4883	259	3	appl	appl	PROPN
ejpam-4883	259	4	.	.	PROPN
ejpam-4883	259	5	math	math	PROPN
ejpam-4883	259	6	,	,	PUNCT
ejpam-4883	259	7	16	16	NUM
ejpam-4883	259	8	(	(	PUNCT
ejpam-4883	259	9	4	4	NUM
ejpam-4883	259	10	)	)	PUNCT
ejpam-4883	259	11	(	(	PUNCT
ejpam-4883	259	12	2023	2023	NUM
ejpam-4883	259	13	)	)	PUNCT
ejpam-4883	259	14	,	,	PUNCT
ejpam-4883	259	15	2082	2082	NUM
ejpam-4883	259	16	-	-	SYM
ejpam-4883	259	17	2095	2095	NUM
ejpam-4883	259	18	2090	2090	NUM
ejpam-4883	259	19	d2	d2	PROPN
ejpam-4883	259	20	=	=	SYM
ejpam-4883	259	21	{	{	PUNCT
ejpam-4883	259	22	x1	x1	PROPN
ejpam-4883	259	23	,	,	PUNCT
ejpam-4883	259	24	x2	x2	PROPN
ejpam-4883	259	25	,	,	PUNCT
ejpam-4883	259	26	z1	z1	PROPN
ejpam-4883	259	27	,	,	PUNCT
ejpam-4883	259	28	z2	z2	PROPN
ejpam-4883	259	29	,	,	PUNCT
ejpam-4883	259	30	.	.	PUNCT
ejpam-4883	259	31	.	.	PUNCT
ejpam-4883	260	1	.	.	PUNCT
ejpam-4883	261	1	,	,	PUNCT
ejpam-4883	261	2	zm	zm	PROPN
ejpam-4883	261	3	}	}	PUNCT
ejpam-4883	261	4	.	.	PUNCT
ejpam-4883	262	1	since	since	SCONJ
ejpam-4883	262	2	for	for	ADP
ejpam-4883	262	3	any	any	DET
ejpam-4883	262	4	x	x	SYM
ejpam-4883	262	5	∈	∈	PROPN
ejpam-4883	262	6	v	v	NOUN
ejpam-4883	262	7	(	(	PUNCT
ejpam-4883	262	8	g	g	NOUN
ejpam-4883	262	9	)	)	PUNCT
ejpam-4883	262	10	is	be	AUX
ejpam-4883	262	11	not	not	PART
ejpam-4883	262	12	a	a	DET
ejpam-4883	262	13	dominating	dominating	NOUN
ejpam-4883	262	14	vertex	vertex	NOUN
ejpam-4883	262	15	,	,	PUNCT
ejpam-4883	262	16	it	it	PRON
ejpam-4883	262	17	follows	follow	VERB
ejpam-4883	262	18	that	that	SCONJ
ejpam-4883	262	19	d1	d1	PROPN
ejpam-4883	262	20	is	be	AUX
ejpam-4883	262	21	a	a	DET
ejpam-4883	262	22	γ	γ	NOUN
ejpam-4883	262	23	-	-	PUNCT
ejpam-4883	262	24	set	set	NOUN
ejpam-4883	262	25	of	of	ADP
ejpam-4883	262	26	g	g	NOUN
ejpam-4883	262	27	,	,	PUNCT
ejpam-4883	262	28	that	that	ADV
ejpam-4883	262	29	is	is	ADV
ejpam-4883	262	30	,	,	PUNCT
ejpam-4883	262	31	γ(g	γ(g	PROPN
ejpam-4883	262	32	)	)	PUNCT
ejpam-4883	262	33	=	=	SYM
ejpam-4883	263	1	2	2	X
ejpam-4883	263	2	.	.	PUNCT
ejpam-4883	263	3	now	now	ADV
ejpam-4883	263	4	,	,	PUNCT
ejpam-4883	263	5	observe	observe	VERB
ejpam-4883	263	6	that	that	SCONJ
ejpam-4883	263	7	x1	x1	PROPN
ejpam-4883	263	8	,	,	PUNCT
ejpam-4883	263	9	y1	y1	PROPN
ejpam-4883	263	10	∈	∈	PROPN
ejpam-4883	263	11	ng[x1]\ng[u	ng[x1]\ng[u	NOUN
ejpam-4883	263	12	]	]	X
ejpam-4883	263	13	∀	∀	PUNCT
ejpam-4883	263	14	u	u	NOUN
ejpam-4883	263	15	∈	∈	PROPN
ejpam-4883	263	16	d2	d2	PROPN
ejpam-4883	263	17	\	\	PROPN
ejpam-4883	263	18	{	{	PUNCT
ejpam-4883	263	19	x1	x1	PROPN
ejpam-4883	263	20	}	}	PUNCT
ejpam-4883	263	21	,	,	PUNCT
ejpam-4883	263	22	x2	x2	PROPN
ejpam-4883	263	23	∈	∈	PROPN
ejpam-4883	263	24	ng[x2	ng[x2	PROPN
ejpam-4883	263	25	]	]	PUNCT
ejpam-4883	263	26	\ng[v	\ng[v	NUM
ejpam-4883	263	27	]	]	PUNCT
ejpam-4883	263	28	∀	∀	X
ejpam-4883	263	29	v	v	ADP
ejpam-4883	263	30	∈	∈	PROPN
ejpam-4883	263	31	d2	d2	PROPN
ejpam-4883	263	32	\	\	PROPN
ejpam-4883	263	33	{	{	PUNCT
ejpam-4883	263	34	x2	x2	NOUN
ejpam-4883	263	35	}	}	PUNCT
ejpam-4883	263	36	,	,	PUNCT
ejpam-4883	263	37	and	and	CCONJ
ejpam-4883	263	38	zi	zi	NOUN
ejpam-4883	263	39	∈	∈	PROPN
ejpam-4883	263	40	ng[zi	ng[zi	NOUN
ejpam-4883	263	41	]	]	X
ejpam-4883	263	42	\ng[w	\ng[w	PROPN
ejpam-4883	263	43	]	]	PUNCT
ejpam-4883	263	44	∀	∀	PUNCT
ejpam-4883	263	45	w	w	PROPN
ejpam-4883	263	46	∈	∈	PROPN
ejpam-4883	263	47	d2	d2	PROPN
ejpam-4883	263	48	\	\	PROPN
ejpam-4883	263	49	{	{	PUNCT
ejpam-4883	263	50	zi	zi	NOUN
ejpam-4883	263	51	}	}	PUNCT
ejpam-4883	263	52	,	,	PUNCT
ejpam-4883	263	53	i	i	PRON
ejpam-4883	263	54	∈	∈	PROPN
ejpam-4883	263	55	{	{	PUNCT
ejpam-4883	263	56	1	1	NUM
ejpam-4883	263	57	,	,	PUNCT
ejpam-4883	263	58	2	2	NUM
ejpam-4883	263	59	,	,	PUNCT
ejpam-4883	263	60	.	.	PUNCT
ejpam-4883	263	61	.	.	PUNCT
ejpam-4883	264	1	.	.	PUNCT
ejpam-4883	265	1	,	,	PUNCT
ejpam-4883	265	2	m	m	VERB
ejpam-4883	265	3	}	}	PUNCT
ejpam-4883	265	4	.	.	PUNCT
ejpam-4883	266	1	thus	thus	ADV
ejpam-4883	266	2	,	,	PUNCT
ejpam-4883	266	3	d2	d2	PROPN
ejpam-4883	266	4	is	be	AUX
ejpam-4883	266	5	a	a	DET
ejpam-4883	266	6	j	j	NOUN
ejpam-4883	266	7	-	-	PUNCT
ejpam-4883	266	8	set	set	NOUN
ejpam-4883	266	9	of	of	ADP
ejpam-4883	266	10	g.	g.	PROPN
ejpam-4883	266	11	since	since	SCONJ
ejpam-4883	266	12	d2	d2	PROPN
ejpam-4883	266	13	is	be	AUX
ejpam-4883	266	14	a	a	DET
ejpam-4883	266	15	dominating	dominating	NOUN
ejpam-4883	266	16	set	set	NOUN
ejpam-4883	266	17	,	,	PUNCT
ejpam-4883	266	18	d2	d2	PROPN
ejpam-4883	266	19	is	be	AUX
ejpam-4883	266	20	a	a	DET
ejpam-4883	266	21	j	j	PROPN
ejpam-4883	266	22	-	-	PUNCT
ejpam-4883	266	23	dominating	dominating	NOUN
ejpam-4883	266	24	of	of	ADP
ejpam-4883	266	25	g.	g.	PROPN
ejpam-4883	266	26	since	since	SCONJ
ejpam-4883	266	27	xi	xi	PROPN
ejpam-4883	266	28	and	and	CCONJ
ejpam-4883	266	29	zj	zj	PROPN
ejpam-4883	266	30	are	be	AUX
ejpam-4883	266	31	pendant	pendant	ADJ
ejpam-4883	266	32	vertices	vertex	NOUN
ejpam-4883	266	33	for	for	ADP
ejpam-4883	266	34	each	each	DET
ejpam-4883	266	35	i	i	PRON
ejpam-4883	266	36	∈	∈	PROPN
ejpam-4883	266	37	{	{	PUNCT
ejpam-4883	266	38	1	1	NUM
ejpam-4883	266	39	,	,	PUNCT
ejpam-4883	266	40	2	2	NUM
ejpam-4883	266	41	}	}	PUNCT
ejpam-4883	266	42	and	and	CCONJ
ejpam-4883	266	43	j	j	PROPN
ejpam-4883	266	44	∈	∈	PROPN
ejpam-4883	266	45	{	{	PUNCT
ejpam-4883	266	46	1	1	NUM
ejpam-4883	266	47	,	,	PUNCT
ejpam-4883	266	48	2	2	NUM
ejpam-4883	266	49	,	,	PUNCT
ejpam-4883	266	50	.	.	PUNCT
ejpam-4883	266	51	.	.	PUNCT
ejpam-4883	267	1	.	.	PUNCT
ejpam-4883	268	1	,	,	PUNCT
ejpam-4883	268	2	m	m	VERB
ejpam-4883	268	3	}	}	PUNCT
ejpam-4883	268	4	,	,	PUNCT
ejpam-4883	268	5	it	it	PRON
ejpam-4883	268	6	follows	follow	VERB
ejpam-4883	268	7	that	that	SCONJ
ejpam-4883	268	8	d2	d2	PROPN
ejpam-4883	268	9	is	be	AUX
ejpam-4883	268	10	a	a	DET
ejpam-4883	268	11	maximum	maximum	ADJ
ejpam-4883	268	12	j	j	NOUN
ejpam-4883	268	13	-	-	PUNCT
ejpam-4883	268	14	dominating	dominating	ADJ
ejpam-4883	268	15	set	set	NOUN
ejpam-4883	268	16	of	of	ADP
ejpam-4883	268	17	g	g	NOUN
ejpam-4883	268	18	by	by	ADP
ejpam-4883	268	19	theorem	theorem	NOUN
ejpam-4883	268	20	4	4	NUM
ejpam-4883	268	21	.	.	PUNCT
ejpam-4883	268	22	hence	hence	ADV
ejpam-4883	268	23	,	,	PUNCT
ejpam-4883	268	24	γj(g	γj(g	PUNCT
ejpam-4883	268	25	)	)	PUNCT
ejpam-4883	268	26	=	=	PUNCT
ejpam-4883	269	1	m+	m+	NUM
ejpam-4883	269	2	2	2	NUM
ejpam-4883	269	3	=	=	SYM
ejpam-4883	269	4	b	b	NOUN
ejpam-4883	269	5	by	by	ADP
ejpam-4883	269	6	theorem	theorem	NOUN
ejpam-4883	269	7	3	3	NUM
ejpam-4883	269	8	.	.	PUNCT
ejpam-4883	269	9	consequently	consequently	ADV
ejpam-4883	269	10	,	,	PUNCT
ejpam-4883	269	11	γ(g	γ(g	PROPN
ejpam-4883	269	12	)	)	PUNCT
ejpam-4883	269	13	=	=	PUNCT
ejpam-4883	270	1	a	a	DET
ejpam-4883	270	2	<	<	X
ejpam-4883	270	3	b	b	NOUN
ejpam-4883	270	4	=	=	PUNCT
ejpam-4883	270	5	γj(g	γj(g	NUM
ejpam-4883	270	6	)	)	PUNCT
ejpam-4883	270	7	.	.	PUNCT
ejpam-4883	271	1	g	g	NOUN
ejpam-4883	271	2	:	:	PUNCT
ejpam-4883	271	3	z1x2x1	z1x2x1	X
ejpam-4883	271	4	.	.	PUNCT
ejpam-4883	271	5	.	.	PUNCT
ejpam-4883	271	6	.	.	PUNCT
ejpam-4883	272	1	zm	zm	PROPN
ejpam-4883	272	2	z2	z2	PROPN
ejpam-4883	272	3	y1	y1	PROPN
ejpam-4883	272	4	y2	y2	NOUN
ejpam-4883	272	5	figure	figure	NOUN
ejpam-4883	272	6	4	4	NUM
ejpam-4883	272	7	:	:	PUNCT
ejpam-4883	272	8	a	a	DET
ejpam-4883	272	9	graph	graph	NOUN
ejpam-4883	272	10	g	g	NOUN
ejpam-4883	272	11	with	with	ADP
ejpam-4883	272	12	γ(g	γ(g	PROPN
ejpam-4883	272	13	)	)	PUNCT
ejpam-4883	272	14	<	<	X
ejpam-4883	272	15	γj	γj	PROPN
ejpam-4883	272	16	(	(	PUNCT
ejpam-4883	272	17	g	g	NOUN
ejpam-4883	272	18	)	)	PUNCT
ejpam-4883	272	19	case	case	NOUN
ejpam-4883	272	20	2	2	NUM
ejpam-4883	272	21	:	:	PUNCT
ejpam-4883	272	22	a	a	DET
ejpam-4883	272	23	≥	≥	NOUN
ejpam-4883	272	24	3	3	NUM
ejpam-4883	272	25	let	let	VERB
ejpam-4883	272	26	m	m	VERB
ejpam-4883	272	27	=	=	VERB
ejpam-4883	273	1	b	b	X
ejpam-4883	273	2	−	−	PROPN
ejpam-4883	273	3	a	a	PRON
ejpam-4883	274	1	and	and	CCONJ
ejpam-4883	274	2	consider	consider	VERB
ejpam-4883	274	3	the	the	DET
ejpam-4883	274	4	graph	graph	NOUN
ejpam-4883	274	5	g′	g′	NOUN
ejpam-4883	274	6	in	in	ADP
ejpam-4883	274	7	figure	figure	NOUN
ejpam-4883	274	8	5	5	NUM
ejpam-4883	274	9	.	.	PUNCT
ejpam-4883	275	1	let	let	VERB
ejpam-4883	275	2	d′	d′	X
ejpam-4883	275	3	=	=	SYM
ejpam-4883	275	4	{	{	PUNCT
ejpam-4883	275	5	y1	y1	PROPN
ejpam-4883	275	6	,	,	PUNCT
ejpam-4883	275	7	y2	y2	PROPN
ejpam-4883	275	8	,	,	PUNCT
ejpam-4883	275	9	.	.	PUNCT
ejpam-4883	275	10	.	.	PUNCT
ejpam-4883	276	1	.	.	PUNCT
ejpam-4883	277	1	,	,	PUNCT
ejpam-4883	277	2	ya	ya	NOUN
ejpam-4883	277	3	}	}	PUNCT
ejpam-4883	277	4	and	and	CCONJ
ejpam-4883	277	5	d∗	d∗	PROPN
ejpam-4883	277	6	=	=	SYM
ejpam-4883	277	7	(	(	PUNCT
ejpam-4883	277	8	x1	x1	PROPN
ejpam-4883	277	9	,	,	PUNCT
ejpam-4883	277	10	x2	x2	PROPN
ejpam-4883	277	11	,	,	PUNCT
ejpam-4883	277	12	·	·	PUNCT
ejpam-4883	277	13	·	·	PUNCT
ejpam-4883	277	14	·	·	PUNCT
ejpam-4883	277	15	,	,	PUNCT
ejpam-4883	277	16	xa	xa	PROPN
ejpam-4883	277	17	,	,	PUNCT
ejpam-4883	277	18	z1	z1	PROPN
ejpam-4883	277	19	,	,	PUNCT
ejpam-4883	277	20	z2	z2	PROPN
ejpam-4883	277	21	,	,	PUNCT
ejpam-4883	277	22	·	·	PUNCT
ejpam-4883	277	23	·	·	PUNCT
ejpam-4883	277	24	·	·	PUNCT
ejpam-4883	277	25	,	,	PUNCT
ejpam-4883	277	26	zm	zm	PROPN
ejpam-4883	277	27	)	)	PUNCT
ejpam-4883	277	28	.	.	PUNCT
ejpam-4883	278	1	then	then	ADV
ejpam-4883	278	2	d′	d′	PRON
ejpam-4883	278	3	is	be	AUX
ejpam-4883	278	4	a	a	DET
ejpam-4883	278	5	γ	γ	NOUN
ejpam-4883	278	6	-	-	PUNCT
ejpam-4883	278	7	set	set	NOUN
ejpam-4883	278	8	of	of	ADP
ejpam-4883	278	9	g.	g.	PROPN
ejpam-4883	278	10	thus	thus	ADV
ejpam-4883	278	11	,	,	PUNCT
ejpam-4883	278	12	γ(g	γ(g	PROPN
ejpam-4883	278	13	)	)	PUNCT
ejpam-4883	279	1	=	=	SYM
ejpam-4883	279	2	a.	a.	NOUN
ejpam-4883	279	3	observe	observe	VERB
ejpam-4883	279	4	that	that	SCONJ
ejpam-4883	279	5	xi	xi	PROPN
ejpam-4883	279	6	∈	∈	PROPN
ejpam-4883	279	7	ng[xi	ng[xi	PROPN
ejpam-4883	279	8	]	]	PUNCT
ejpam-4883	279	9	\	\	PROPN
ejpam-4883	280	1	ng[u	ng[u	PROPN
ejpam-4883	280	2	]	]	PUNCT
ejpam-4883	280	3	∀	∀	PUNCT
ejpam-4883	280	4	u	u	NOUN
ejpam-4883	280	5	∈	∈	PROPN
ejpam-4883	280	6	d∗	d∗	PROPN
ejpam-4883	280	7	\	\	PROPN
ejpam-4883	280	8	{	{	PUNCT
ejpam-4883	280	9	xi	xi	NOUN
ejpam-4883	280	10	}	}	PUNCT
ejpam-4883	280	11	,	,	PUNCT
ejpam-4883	280	12	zk−1	zk−1	PROPN
ejpam-4883	280	13	∈	∈	PROPN
ejpam-4883	280	14	ng[zk	ng[zk	SYM
ejpam-4883	280	15	]	]	PUNCT
ejpam-4883	280	16	\	\	PUNCT
ejpam-4883	280	17	ng[zl	ng[zl	X
ejpam-4883	280	18	]	]	X
ejpam-4883	280	19	∀	∀	X
ejpam-4883	280	20	l	l	NOUN
ejpam-4883	280	21	>	>	X
ejpam-4883	281	1	k	k	X
ejpam-4883	281	2	,	,	PUNCT
ejpam-4883	281	3	zs+1	zs+1	PROPN
ejpam-4883	281	4	∈	∈	PROPN
ejpam-4883	281	5	ng[zs	ng[zs	PART
ejpam-4883	281	6	]	]	PUNCT
ejpam-4883	281	7	\	\	PROPN
ejpam-4883	281	8	ng[zt	ng[zt	PROPN
ejpam-4883	281	9	]	]	X
ejpam-4883	281	10	∀	∀	X
ejpam-4883	281	11	s	s	PART
ejpam-4883	281	12	>	>	X
ejpam-4883	281	13	t	t	PROPN
ejpam-4883	281	14	,	,	PUNCT
ejpam-4883	281	15	and	and	CCONJ
ejpam-4883	281	16	zq	zq	PROPN
ejpam-4883	281	17	∈	∈	PROPN
ejpam-4883	281	18	ng[zq	ng[zq	PROPN
ejpam-4883	281	19	]	]	PUNCT
ejpam-4883	281	20	\	\	PROPN
ejpam-4883	281	21	ng[xr	ng[xr	PROPN
ejpam-4883	281	22	]	]	X
ejpam-4883	281	23	,	,	PUNCT
ejpam-4883	281	24	where	where	SCONJ
ejpam-4883	281	25	i	i	PRON
ejpam-4883	281	26	,	,	PUNCT
ejpam-4883	281	27	r	r	NOUN
ejpam-4883	281	28	∈	∈	PROPN
ejpam-4883	281	29	{	{	PUNCT
ejpam-4883	281	30	1	1	NUM
ejpam-4883	281	31	,	,	PUNCT
ejpam-4883	281	32	2	2	NUM
ejpam-4883	281	33	,	,	PUNCT
ejpam-4883	281	34	.	.	PUNCT
ejpam-4883	281	35	.	.	PUNCT
ejpam-4883	282	1	.	.	PUNCT
ejpam-4883	283	1	,	,	PUNCT
ejpam-4883	283	2	a	a	PRON
ejpam-4883	283	3	}	}	PUNCT
ejpam-4883	283	4	,	,	PUNCT
ejpam-4883	283	5	and	and	CCONJ
ejpam-4883	283	6	i	i	PRON
ejpam-4883	283	7	,	,	PUNCT
ejpam-4883	283	8	k	k	PROPN
ejpam-4883	283	9	,	,	PUNCT
ejpam-4883	283	10	l	l	PROPN
ejpam-4883	283	11	,	,	PUNCT
ejpam-4883	283	12	s	s	PROPN
ejpam-4883	283	13	,	,	PUNCT
ejpam-4883	283	14	t	t	PROPN
ejpam-4883	283	15	,	,	PUNCT
ejpam-4883	283	16	q	q	PROPN
ejpam-4883	283	17	∈	∈	PROPN
ejpam-4883	283	18	{	{	PUNCT
ejpam-4883	283	19	1	1	NUM
ejpam-4883	283	20	,	,	PUNCT
ejpam-4883	283	21	2	2	NUM
ejpam-4883	283	22	,	,	PUNCT
ejpam-4883	283	23	.	.	PUNCT
ejpam-4883	283	24	.	.	PUNCT
ejpam-4883	284	1	.	.	PUNCT
ejpam-4883	285	1	,	,	PUNCT
ejpam-4883	285	2	m	m	VERB
ejpam-4883	285	3	}	}	PUNCT
ejpam-4883	285	4	.	.	PUNCT
ejpam-4883	286	1	thus	thus	ADV
ejpam-4883	286	2	,	,	PUNCT
ejpam-4883	286	3	d∗	d∗	PROPN
ejpam-4883	286	4	is	be	AUX
ejpam-4883	286	5	a	a	DET
ejpam-4883	286	6	j	j	NOUN
ejpam-4883	286	7	-	-	PUNCT
ejpam-4883	286	8	set	set	NOUN
ejpam-4883	286	9	of	of	ADP
ejpam-4883	286	10	g.	g.	PROPN
ejpam-4883	286	11	since	since	SCONJ
ejpam-4883	286	12	d∗	d∗	PROPN
ejpam-4883	286	13	is	be	AUX
ejpam-4883	286	14	a	a	DET
ejpam-4883	286	15	dominating	dominating	NOUN
ejpam-4883	286	16	set	set	NOUN
ejpam-4883	286	17	,	,	PUNCT
ejpam-4883	286	18	d∗	d∗	PROPN
ejpam-4883	286	19	is	be	AUX
ejpam-4883	286	20	a	a	DET
ejpam-4883	286	21	j	j	PROPN
ejpam-4883	286	22	-	-	PUNCT
ejpam-4883	286	23	dominating	dominating	NOUN
ejpam-4883	286	24	of	of	ADP
ejpam-4883	286	25	g.	g.	PROPN
ejpam-4883	286	26	since	since	SCONJ
ejpam-4883	286	27	xi	xi	PROPN
ejpam-4883	286	28	is	be	AUX
ejpam-4883	286	29	a	a	DET
ejpam-4883	286	30	pendant	pendant	ADJ
ejpam-4883	286	31	vertex	vertex	NOUN
ejpam-4883	286	32	for	for	ADP
ejpam-4883	286	33	each	each	DET
ejpam-4883	286	34	i	i	PRON
ejpam-4883	286	35	∈	∈	PROPN
ejpam-4883	286	36	{	{	PUNCT
ejpam-4883	286	37	1	1	NUM
ejpam-4883	286	38	,	,	PUNCT
ejpam-4883	286	39	2	2	NUM
ejpam-4883	286	40	,	,	PUNCT
ejpam-4883	286	41	.	.	PUNCT
ejpam-4883	286	42	.	.	PUNCT
ejpam-4883	287	1	.	.	PUNCT
ejpam-4883	288	1	,	,	PUNCT
ejpam-4883	288	2	a	a	X
ejpam-4883	288	3	}	}	PUNCT
ejpam-4883	288	4	,	,	PUNCT
ejpam-4883	288	5	it	it	PRON
ejpam-4883	288	6	follows	follow	VERB
ejpam-4883	288	7	that	that	SCONJ
ejpam-4883	288	8	d∗	d∗	PROPN
ejpam-4883	288	9	is	be	AUX
ejpam-4883	288	10	a	a	DET
ejpam-4883	288	11	maximum	maximum	ADJ
ejpam-4883	288	12	j	j	NOUN
ejpam-4883	288	13	-	-	PUNCT
ejpam-4883	288	14	dominating	dominating	ADJ
ejpam-4883	288	15	set	set	NOUN
ejpam-4883	288	16	of	of	ADP
ejpam-4883	288	17	g	g	NOUN
ejpam-4883	288	18	by	by	ADP
ejpam-4883	288	19	theorem	theorem	NOUN
ejpam-4883	288	20	4	4	NUM
ejpam-4883	288	21	.	.	PUNCT
ejpam-4883	288	22	hence	hence	ADV
ejpam-4883	288	23	,	,	PUNCT
ejpam-4883	288	24	γj(g	γj(g	PUNCT
ejpam-4883	288	25	)	)	PUNCT
ejpam-4883	288	26	=	=	VERB
ejpam-4883	289	1	m+	m+	NUM
ejpam-4883	289	2	a	a	DET
ejpam-4883	289	3	=	=	X
ejpam-4883	289	4	b	b	NOUN
ejpam-4883	289	5	by	by	ADP
ejpam-4883	289	6	theorem	theorem	NOUN
ejpam-4883	289	7	3	3	NUM
ejpam-4883	289	8	.	.	PUNCT
ejpam-4883	289	9	therefore	therefore	ADV
ejpam-4883	289	10	,	,	PUNCT
ejpam-4883	289	11	γ(g	γ(g	PROPN
ejpam-4883	289	12	)	)	PUNCT
ejpam-4883	290	1	=	=	PUNCT
ejpam-4883	290	2	a	a	DET
ejpam-4883	290	3	<	<	X
ejpam-4883	290	4	b	b	NOUN
ejpam-4883	290	5	=	=	PUNCT
ejpam-4883	290	6	γj(g	γj(g	NUM
ejpam-4883	290	7	)	)	PUNCT
ejpam-4883	290	8	.	.	PUNCT
ejpam-4883	291	1	j.	j.	PROPN
ejpam-4883	291	2	hassan	hassan	PROPN
ejpam-4883	291	3	,	,	PUNCT
ejpam-4883	291	4	j.	j.	PROPN
ejpam-4883	291	5	salim	salim	PROPN
ejpam-4883	291	6	/	/	SYM
ejpam-4883	291	7	eur	eur	PROPN
ejpam-4883	291	8	.	.	PUNCT
ejpam-4883	292	1	j.	j.	PROPN
ejpam-4883	292	2	pure	pure	PROPN
ejpam-4883	292	3	appl	appl	PROPN
ejpam-4883	292	4	.	.	PROPN
ejpam-4883	292	5	math	math	PROPN
ejpam-4883	292	6	,	,	PUNCT
ejpam-4883	292	7	16	16	NUM
ejpam-4883	292	8	(	(	PUNCT
ejpam-4883	292	9	4	4	NUM
ejpam-4883	292	10	)	)	PUNCT
ejpam-4883	292	11	(	(	PUNCT
ejpam-4883	292	12	2023	2023	NUM
ejpam-4883	292	13	)	)	PUNCT
ejpam-4883	292	14	,	,	PUNCT
ejpam-4883	292	15	2082	2082	NUM
ejpam-4883	292	16	-	-	SYM
ejpam-4883	292	17	2095	2095	NUM
ejpam-4883	292	18	2091	2091	NUM
ejpam-4883	292	19	g′	g′	NOUN
ejpam-4883	292	20	:	:	PUNCT
ejpam-4883	293	1	x2	x2	PROPN
ejpam-4883	293	2	z1	z1	PROPN
ejpam-4883	293	3	.	.	PUNCT
ejpam-4883	293	4	.	.	PUNCT
ejpam-4883	293	5	.	.	PUNCT
ejpam-4883	294	1	xaxa−1x3	xaxa−1x3	X
ejpam-4883	294	2	x1	x1	PROPN
ejpam-4883	294	3	.	.	PUNCT
ejpam-4883	294	4	.	.	PUNCT
ejpam-4883	294	5	.	.	PUNCT
ejpam-4883	295	1	x4	x4	PROPN
ejpam-4883	295	2	zm	zm	PROPN
ejpam-4883	295	3	z2	z2	PROPN
ejpam-4883	295	4	y1	y1	NOUN
ejpam-4883	296	1	y2	y2	NOUN
ejpam-4883	296	2	y3	y3	NOUN
ejpam-4883	296	3	ya−1	ya−1	PROPN
ejpam-4883	296	4	yaya−2	yaya−2	NUM
ejpam-4883	296	5	figure	figure	NOUN
ejpam-4883	296	6	5	5	NUM
ejpam-4883	296	7	:	:	PUNCT
ejpam-4883	296	8	a	a	DET
ejpam-4883	296	9	graph	graph	NOUN
ejpam-4883	296	10	g′	g′	NOUN
ejpam-4883	296	11	with	with	ADP
ejpam-4883	296	12	γ(g′	γ(g′	NUM
ejpam-4883	296	13	)	)	PUNCT
ejpam-4883	296	14	<	<	X
ejpam-4883	296	15	γj	γj	PROPN
ejpam-4883	296	16	(	(	PUNCT
ejpam-4883	296	17	g	g	PROPN
ejpam-4883	296	18	′	′	NUM
ejpam-4883	296	19	)	)	PUNCT
ejpam-4883	296	20	proposition	proposition	NOUN
ejpam-4883	296	21	2	2	NUM
ejpam-4883	296	22	.	.	PUNCT
ejpam-4883	296	23	given	give	VERB
ejpam-4883	296	24	any	any	DET
ejpam-4883	296	25	positive	positive	ADJ
ejpam-4883	296	26	integer	integer	NOUN
ejpam-4883	296	27	n	n	PRON
ejpam-4883	296	28	≥	≥	NOUN
ejpam-4883	296	29	1	1	NUM
ejpam-4883	296	30	,	,	PUNCT
ejpam-4883	296	31	we	we	PRON
ejpam-4883	296	32	have	have	AUX
ejpam-4883	296	33	(	(	PUNCT
ejpam-4883	296	34	i	i	NOUN
ejpam-4883	296	35	)	)	PUNCT
ejpam-4883	297	1	γj(pn	γj(pn	PROPN
ejpam-4883	297	2	)	)	PUNCT
ejpam-4883	297	3	=	=	SYM
ejpam-4883	298	1			NOUN
ejpam-4883	298	2	1	1	NUM
ejpam-4883	298	3	if	if	SCONJ
ejpam-4883	298	4	n	n	NOUN
ejpam-4883	298	5	=	=	SYM
ejpam-4883	298	6	1	1	NUM
ejpam-4883	298	7	,	,	PUNCT
ejpam-4883	298	8	2	2	NUM
ejpam-4883	298	9	2	2	NUM
ejpam-4883	298	10	if	if	SCONJ
ejpam-4883	298	11	n	n	NOUN
ejpam-4883	298	12	=	=	SYM
ejpam-4883	298	13	3	3	NUM
ejpam-4883	298	14	,	,	PUNCT
ejpam-4883	298	15	4	4	NUM
ejpam-4883	298	16	n−	n−	NOUN
ejpam-4883	298	17	2	2	NUM
ejpam-4883	298	18	if	if	SCONJ
ejpam-4883	298	19	n	n	PRON
ejpam-4883	298	20	≥	≥	NOUN
ejpam-4883	298	21	5	5	NUM
ejpam-4883	298	22	,	,	PUNCT
ejpam-4883	298	23	(	(	PUNCT
ejpam-4883	298	24	ii	ii	NOUN
ejpam-4883	298	25	)	)	PUNCT
ejpam-4883	298	26	γj(cn	γj(cn	PROPN
ejpam-4883	298	27	)	)	PUNCT
ejpam-4883	299	1	=	=	PRON
ejpam-4883	299	2	{	{	PUNCT
ejpam-4883	299	3	1	1	NUM
ejpam-4883	299	4	if	if	SCONJ
ejpam-4883	299	5	n	n	ADV
ejpam-4883	299	6	=	=	SYM
ejpam-4883	299	7	3	3	NUM
ejpam-4883	299	8	n	n	NOUN
ejpam-4883	299	9	if	if	SCONJ
ejpam-4883	299	10	n	n	PRON
ejpam-4883	299	11	≥	≥	NOUN
ejpam-4883	299	12	4	4	NUM
ejpam-4883	299	13	.	.	PUNCT
ejpam-4883	300	1	proof	proof	NOUN
ejpam-4883	300	2	.	.	PUNCT
ejpam-4883	301	1	(	(	PUNCT
ejpam-4883	301	2	i	i	NOUN
ejpam-4883	301	3	)	)	PUNCT
ejpam-4883	301	4	clearly	clearly	ADV
ejpam-4883	301	5	,	,	PUNCT
ejpam-4883	301	6	γj(pn	γj(pn	ADJ
ejpam-4883	301	7	)	)	PUNCT
ejpam-4883	301	8	=	=	SYM
ejpam-4883	301	9	1	1	NUM
ejpam-4883	301	10	for	for	ADP
ejpam-4883	301	11	n	n	NOUN
ejpam-4883	301	12	=	=	SYM
ejpam-4883	301	13	1	1	NUM
ejpam-4883	301	14	,	,	PUNCT
ejpam-4883	301	15	2	2	NUM
ejpam-4883	301	16	and	and	CCONJ
ejpam-4883	301	17	γj(pn	γj(pn	ADJ
ejpam-4883	301	18	)	)	PUNCT
ejpam-4883	301	19	=	=	SYM
ejpam-4883	301	20	2	2	NUM
ejpam-4883	301	21	for	for	ADP
ejpam-4883	301	22	n	n	NOUN
ejpam-4883	301	23	=	=	SYM
ejpam-4883	301	24	3	3	NUM
ejpam-4883	301	25	,	,	PUNCT
ejpam-4883	301	26	4	4	NUM
ejpam-4883	301	27	.	.	PUNCT
ejpam-4883	301	28	suppose	suppose	VERB
ejpam-4883	301	29	n	n	PRON
ejpam-4883	301	30	≥	≥	NUM
ejpam-4883	301	31	5	5	NUM
ejpam-4883	301	32	.	.	PUNCT
ejpam-4883	302	1	let	let	VERB
ejpam-4883	302	2	pn	pn	VERB
ejpam-4883	302	3	=	=	VERB
ejpam-4883	302	4	g	g	PROPN
ejpam-4883	302	5	=	=	PUNCT
ejpam-4883	303	1	[	[	X
ejpam-4883	303	2	v1	v1	NOUN
ejpam-4883	303	3	,	,	PUNCT
ejpam-4883	303	4	v2	v2	NOUN
ejpam-4883	303	5	,	,	PUNCT
ejpam-4883	303	6	.	.	PUNCT
ejpam-4883	303	7	.	.	PUNCT
ejpam-4883	303	8	.	.	PUNCT
ejpam-4883	304	1	,	,	PUNCT
ejpam-4883	304	2	vn	vn	X
ejpam-4883	304	3	]	]	X
ejpam-4883	304	4	,	,	PUNCT
ejpam-4883	304	5	d	d	X
ejpam-4883	304	6	=	=	PRON
ejpam-4883	304	7	{	{	PUNCT
ejpam-4883	304	8	v2	v2	PROPN
ejpam-4883	304	9	,	,	PUNCT
ejpam-4883	304	10	v3	v3	PROPN
ejpam-4883	304	11	·	·	PUNCT
ejpam-4883	304	12	·	·	PUNCT
ejpam-4883	304	13	·	·	PUNCT
ejpam-4883	304	14	,	,	PUNCT
ejpam-4883	304	15	vn−2	vn−2	PROPN
ejpam-4883	304	16	,	,	PUNCT
ejpam-4883	304	17	vn−1	vn−1	ADJ
ejpam-4883	304	18	}	}	PUNCT
ejpam-4883	304	19	and	and	CCONJ
ejpam-4883	304	20	let	let	VERB
ejpam-4883	304	21	i	i	PRON
ejpam-4883	304	22	,	,	PUNCT
ejpam-4883	304	23	j	j	PROPN
ejpam-4883	304	24	∈	∈	PROPN
ejpam-4883	304	25	{	{	PUNCT
ejpam-4883	304	26	2	2	NUM
ejpam-4883	304	27	,	,	PUNCT
ejpam-4883	304	28	3	3	NUM
ejpam-4883	304	29	,	,	PUNCT
ejpam-4883	304	30	.	.	PUNCT
ejpam-4883	304	31	.	.	PUNCT
ejpam-4883	305	1	.	.	PUNCT
ejpam-4883	306	1	,	,	PUNCT
ejpam-4883	306	2	n	n	CCONJ
ejpam-4883	306	3	−	−	PROPN
ejpam-4883	306	4	1	1	NUM
ejpam-4883	306	5	}	}	PUNCT
ejpam-4883	306	6	.	.	PUNCT
ejpam-4883	307	1	then	then	ADV
ejpam-4883	307	2	vi−1	vi−1	PROPN
ejpam-4883	307	3	∈	∈	PROPN
ejpam-4883	307	4	ng[vi]\ng[vj	ng[vi]\ng[vj	VERB
ejpam-4883	307	5	]	]	PUNCT
ejpam-4883	307	6	and	and	CCONJ
ejpam-4883	307	7	vj+1	vj+1	NUM
ejpam-4883	307	8	∈	∈	NOUN
ejpam-4883	307	9	ng[vj	ng[vj	NOUN
ejpam-4883	307	10	]	]	X
ejpam-4883	307	11	\ng[vi	\ng[vi	X
ejpam-4883	307	12	]	]	X
ejpam-4883	307	13	for	for	ADP
ejpam-4883	307	14	all	all	PRON
ejpam-4883	307	15	j	j	PROPN
ejpam-4883	307	16	>	>	X
ejpam-4883	308	1	i	i	PROPN
ejpam-4883	308	2	,	,	PUNCT
ejpam-4883	308	3	i	i	PRON
ejpam-4883	308	4	,	,	PUNCT
ejpam-4883	308	5	j	j	PROPN
ejpam-4883	308	6	∈	∈	PROPN
ejpam-4883	308	7	{	{	PUNCT
ejpam-4883	308	8	2	2	NUM
ejpam-4883	308	9	,	,	PUNCT
ejpam-4883	308	10	3	3	NUM
ejpam-4883	308	11	,	,	PUNCT
ejpam-4883	308	12	.	.	PUNCT
ejpam-4883	308	13	.	.	PUNCT
ejpam-4883	308	14	.	.	PUNCT
ejpam-4883	309	1	,	,	PUNCT
ejpam-4883	309	2	n−1	n−1	PROPN
ejpam-4883	309	3	}	}	PUNCT
ejpam-4883	309	4	.	.	PUNCT
ejpam-4883	310	1	thus	thus	ADV
ejpam-4883	310	2	,	,	PUNCT
ejpam-4883	310	3	ng[vi]\ng[vj	ng[vi]\ng[vj	VERB
ejpam-4883	310	4	]	]	PUNCT
ejpam-4883	310	5	̸=	̸=	NOUN
ejpam-4883	310	6	∅	∅	NOUN
ejpam-4883	310	7	for	for	ADP
ejpam-4883	310	8	all	all	PRON
ejpam-4883	310	9	i	i	PRON
ejpam-4883	310	10	̸=	̸=	PROPN
ejpam-4883	310	11	j	j	PROPN
ejpam-4883	310	12	,	,	PUNCT
ejpam-4883	310	13	i	i	PRON
ejpam-4883	310	14	,	,	PUNCT
ejpam-4883	310	15	j	j	PROPN
ejpam-4883	310	16	∈	∈	PROPN
ejpam-4883	310	17	{	{	PUNCT
ejpam-4883	310	18	2	2	NUM
ejpam-4883	310	19	,	,	PUNCT
ejpam-4883	310	20	3	3	NUM
ejpam-4883	310	21	,	,	PUNCT
ejpam-4883	310	22	.	.	PUNCT
ejpam-4883	310	23	.	.	PUNCT
ejpam-4883	311	1	.	.	PUNCT
ejpam-4883	312	1	,	,	PUNCT
ejpam-4883	312	2	n−1	n−1	PROPN
ejpam-4883	312	3	}	}	PUNCT
ejpam-4883	312	4	,	,	PUNCT
ejpam-4883	312	5	showing	show	VERB
ejpam-4883	312	6	that	that	SCONJ
ejpam-4883	312	7	d	d	NOUN
ejpam-4883	312	8	is	be	AUX
ejpam-4883	312	9	a	a	DET
ejpam-4883	312	10	j	j	NOUN
ejpam-4883	312	11	-	-	PUNCT
ejpam-4883	312	12	set	set	NOUN
ejpam-4883	312	13	in	in	ADP
ejpam-4883	312	14	g.	g.	PROPN
ejpam-4883	312	15	therefore	therefore	ADV
ejpam-4883	312	16	,	,	PUNCT
ejpam-4883	312	17	γj(pn	γj(pn	PROPN
ejpam-4883	312	18	)	)	PUNCT
ejpam-4883	312	19	≥	≥	NOUN
ejpam-4883	312	20	n−	n−	NOUN
ejpam-4883	312	21	2	2	NUM
ejpam-4883	312	22	by	by	ADP
ejpam-4883	312	23	corollary	corollary	ADJ
ejpam-4883	312	24	1	1	NUM
ejpam-4883	312	25	.	.	PUNCT
ejpam-4883	313	1	if	if	SCONJ
ejpam-4883	313	2	γj(pn	γj(pn	NOUN
ejpam-4883	313	3	)	)	PUNCT
ejpam-4883	313	4	=	=	SYM
ejpam-4883	314	1	n	n	CCONJ
ejpam-4883	314	2	,	,	PUNCT
ejpam-4883	314	3	then	then	ADV
ejpam-4883	314	4	v	v	X
ejpam-4883	314	5	(	(	PUNCT
ejpam-4883	314	6	pn	pn	NOUN
ejpam-4883	314	7	)	)	PUNCT
ejpam-4883	314	8	is	be	AUX
ejpam-4883	314	9	the	the	DET
ejpam-4883	314	10	maximum	maximum	ADJ
ejpam-4883	314	11	j	j	PROPN
ejpam-4883	314	12	-	-	PUNCT
ejpam-4883	314	13	dominating	dominating	NOUN
ejpam-4883	314	14	set	set	NOUN
ejpam-4883	314	15	in	in	ADP
ejpam-4883	314	16	pn	pn	PROPN
ejpam-4883	314	17	.	.	PUNCT
ejpam-4883	315	1	however	however	ADV
ejpam-4883	315	2	,	,	PUNCT
ejpam-4883	315	3	ng[vn	ng[vn	PROPN
ejpam-4883	315	4	]	]	X
ejpam-4883	315	5	⊆	⊆	NUM
ejpam-4883	315	6	ng[vn−1	ng[vn−1	PROPN
ejpam-4883	315	7	]	]	PUNCT
ejpam-4883	315	8	.	.	PUNCT
ejpam-4883	316	1	thus	thus	ADV
ejpam-4883	316	2	,	,	PUNCT
ejpam-4883	316	3	ng[vn	ng[vn	PROPN
ejpam-4883	316	4	]	]	PUNCT
ejpam-4883	316	5	\	\	PROPN
ejpam-4883	316	6	ng[vn−1	ng[vn−1	PROPN
ejpam-4883	316	7	]	]	X
ejpam-4883	316	8	=	=	SYM
ejpam-4883	316	9	∅	∅	NOUN
ejpam-4883	316	10	,	,	PUNCT
ejpam-4883	316	11	a	a	DET
ejpam-4883	316	12	contradiction	contradiction	NOUN
ejpam-4883	316	13	.	.	PUNCT
ejpam-4883	317	1	next	next	ADV
ejpam-4883	317	2	,	,	PUNCT
ejpam-4883	317	3	suppose	suppose	VERB
ejpam-4883	317	4	that	that	SCONJ
ejpam-4883	317	5	γj(g	γj(g	PUNCT
ejpam-4883	317	6	)	)	PUNCT
ejpam-4883	317	7	=	=	SYM
ejpam-4883	317	8	n	n	CCONJ
ejpam-4883	317	9	−	−	PROPN
ejpam-4883	317	10	1	1	NUM
ejpam-4883	317	11	,	,	PUNCT
ejpam-4883	317	12	say	say	VERB
ejpam-4883	317	13	a	a	DET
ejpam-4883	317	14	=	=	PUNCT
ejpam-4883	317	15	{	{	PUNCT
ejpam-4883	317	16	a1	a1	PROPN
ejpam-4883	317	17	,	,	PUNCT
ejpam-4883	317	18	a2	a2	PROPN
ejpam-4883	317	19	,	,	PUNCT
ejpam-4883	317	20	·	·	PUNCT
ejpam-4883	317	21	·	·	PUNCT
ejpam-4883	317	22	·	·	PUNCT
ejpam-4883	317	23	,	,	PUNCT
ejpam-4883	317	24	an−1	an−1	ADJ
ejpam-4883	317	25	}	}	PUNCT
ejpam-4883	317	26	is	be	AUX
ejpam-4883	317	27	a	a	DET
ejpam-4883	317	28	j	j	PROPN
ejpam-4883	317	29	-	-	PUNCT
ejpam-4883	317	30	dominating	dominating	ADJ
ejpam-4883	317	31	set	set	NOUN
ejpam-4883	317	32	of	of	ADP
ejpam-4883	317	33	g.	g.	PROPN
ejpam-4883	317	34	then	then	ADV
ejpam-4883	317	35	either	either	CCONJ
ejpam-4883	317	36	⟨a⟩	⟨a⟩	PROPN
ejpam-4883	317	37	is	be	AUX
ejpam-4883	317	38	connected	connect	VERB
ejpam-4883	317	39	or	or	CCONJ
ejpam-4883	317	40	disconnected	disconnected	ADJ
ejpam-4883	317	41	.	.	PUNCT
ejpam-4883	318	1	assume	assume	VERB
ejpam-4883	318	2	that	that	SCONJ
ejpam-4883	318	3	⟨a⟩	⟨a⟩	PROPN
ejpam-4883	318	4	is	be	AUX
ejpam-4883	318	5	connected	connect	VERB
ejpam-4883	318	6	.	.	PUNCT
ejpam-4883	319	1	then	then	ADV
ejpam-4883	319	2	either	either	CCONJ
ejpam-4883	319	3	v1	v1	VERB
ejpam-4883	319	4	or	or	CCONJ
ejpam-4883	319	5	vn	vn	PROPN
ejpam-4883	319	6	is	be	AUX
ejpam-4883	319	7	not	not	PART
ejpam-4883	319	8	in	in	ADP
ejpam-4883	319	9	a.	a.	NOUN
ejpam-4883	319	10	if	if	SCONJ
ejpam-4883	319	11	v1	v1	PROPN
ejpam-4883	319	12	/∈	/∈	PUNCT
ejpam-4883	320	1	a	a	PRON
ejpam-4883	320	2	,	,	PUNCT
ejpam-4883	320	3	then	then	ADV
ejpam-4883	320	4	vn−1	vn−1	ADJ
ejpam-4883	320	5	,	,	PUNCT
ejpam-4883	320	6	vn	vn	PROPN
ejpam-4883	320	7	∈	∈	PROPN
ejpam-4883	320	8	a.	a.	NOUN
ejpam-4883	320	9	however	however	ADV
ejpam-4883	320	10	,	,	PUNCT
ejpam-4883	320	11	ng[vn	ng[vn	PROPN
ejpam-4883	320	12	]	]	PUNCT
ejpam-4883	320	13	\	\	PROPN
ejpam-4883	321	1	ng[vn−1	ng[vn−1	PROPN
ejpam-4883	321	2	]	]	X
ejpam-4883	321	3	=	=	SYM
ejpam-4883	321	4	{	{	PUNCT
ejpam-4883	321	5	vn	vn	NOUN
ejpam-4883	321	6	,	,	PUNCT
ejpam-4883	321	7	vn−1	vn−1	ADJ
ejpam-4883	321	8	}	}	PUNCT
ejpam-4883	321	9	\	\	NOUN
ejpam-4883	321	10	{	{	PUNCT
ejpam-4883	321	11	vn	vn	NOUN
ejpam-4883	321	12	,	,	PUNCT
ejpam-4883	321	13	vn−1	vn−1	ADJ
ejpam-4883	321	14	,	,	PUNCT
ejpam-4883	321	15	vn−2	vn−2	NOUN
ejpam-4883	321	16	}	}	PUNCT
ejpam-4883	321	17	=	=	SYM
ejpam-4883	321	18	∅	∅	NOUN
ejpam-4883	321	19	,	,	PUNCT
ejpam-4883	321	20	a	a	DET
ejpam-4883	321	21	contradiction	contradiction	NOUN
ejpam-4883	321	22	.	.	PUNCT
ejpam-4883	322	1	if	if	SCONJ
ejpam-4883	322	2	vn	vn	PROPN
ejpam-4883	322	3	/∈	/∈	PROPN
ejpam-4883	323	1	a	a	DET
ejpam-4883	323	2	,	,	PUNCT
ejpam-4883	323	3	then	then	ADV
ejpam-4883	323	4	v1	v1	VERB
ejpam-4883	323	5	,	,	PUNCT
ejpam-4883	323	6	v2	v2	PROPN
ejpam-4883	323	7	∈	∈	PROPN
ejpam-4883	323	8	a.	a.	NOUN
ejpam-4883	323	9	however	however	ADV
ejpam-4883	323	10	,	,	PUNCT
ejpam-4883	323	11	ng[v1	ng[v1	PROPN
ejpam-4883	323	12	]	]	PUNCT
ejpam-4883	323	13	\	\	PROPN
ejpam-4883	323	14	ng[v2	ng[v2	PROPN
ejpam-4883	323	15	]	]	X
ejpam-4883	323	16	=	=	PRON
ejpam-4883	323	17	{	{	PUNCT
ejpam-4883	323	18	v1	v1	NOUN
ejpam-4883	323	19	,	,	PUNCT
ejpam-4883	323	20	v2	v2	PROPN
ejpam-4883	323	21	}	}	PUNCT
ejpam-4883	323	22	\	\	NOUN
ejpam-4883	323	23	{	{	PUNCT
ejpam-4883	323	24	v1	v1	NOUN
ejpam-4883	323	25	,	,	PUNCT
ejpam-4883	323	26	v2	v2	PROPN
ejpam-4883	323	27	,	,	PUNCT
ejpam-4883	323	28	v3	v3	PROPN
ejpam-4883	323	29	}	}	PUNCT
ejpam-4883	323	30	=	=	SYM
ejpam-4883	323	31	∅	∅	NOUN
ejpam-4883	323	32	,	,	PUNCT
ejpam-4883	323	33	a	a	DET
ejpam-4883	323	34	contradiction	contradiction	NOUN
ejpam-4883	323	35	.	.	PUNCT
ejpam-4883	324	1	next	next	ADV
ejpam-4883	324	2	,	,	PUNCT
ejpam-4883	324	3	suppose	suppose	VERB
ejpam-4883	324	4	that	that	SCONJ
ejpam-4883	324	5	⟨a⟩	⟨a⟩	PROPN
ejpam-4883	324	6	is	be	AUX
ejpam-4883	324	7	disconnected	disconnect	VERB
ejpam-4883	324	8	.	.	PUNCT
ejpam-4883	325	1	then	then	ADV
ejpam-4883	325	2	there	there	PRON
ejpam-4883	325	3	exists	exist	VERB
ejpam-4883	325	4	vj	vj	PROPN
ejpam-4883	325	5	∈	∈	PROPN
ejpam-4883	325	6	v	v	PROPN
ejpam-4883	325	7	(	(	PUNCT
ejpam-4883	325	8	g	g	NOUN
ejpam-4883	325	9	)	)	PUNCT
ejpam-4883	325	10	such	such	ADJ
ejpam-4883	325	11	that	that	SCONJ
ejpam-4883	325	12	vj	vj	PROPN
ejpam-4883	325	13	/∈	/∈	PROPN
ejpam-4883	326	1	a	a	PRON
ejpam-4883	327	1	for	for	ADP
ejpam-4883	327	2	some	some	DET
ejpam-4883	327	3	j	j	PROPN
ejpam-4883	327	4	∈	∈	PROPN
ejpam-4883	327	5	{	{	PUNCT
ejpam-4883	327	6	2	2	NUM
ejpam-4883	327	7	,	,	PUNCT
ejpam-4883	327	8	3	3	NUM
ejpam-4883	327	9	,	,	PUNCT
ejpam-4883	327	10	.	.	PUNCT
ejpam-4883	327	11	.	.	PUNCT
ejpam-4883	327	12	.	.	PUNCT
ejpam-4883	328	1	,	,	PUNCT
ejpam-4883	329	1	n	n	CCONJ
ejpam-4883	329	2	−	−	PROPN
ejpam-4883	329	3	1	1	NUM
ejpam-4883	329	4	}	}	PUNCT
ejpam-4883	329	5	.	.	PUNCT
ejpam-4883	330	1	if	if	SCONJ
ejpam-4883	330	2	v2	v2	PROPN
ejpam-4883	330	3	/∈	/∈	VERB
ejpam-4883	331	1	a	a	PRON
ejpam-4883	331	2	,	,	PUNCT
ejpam-4883	331	3	then	then	ADV
ejpam-4883	331	4	vn−1	vn−1	ADJ
ejpam-4883	331	5	,	,	PUNCT
ejpam-4883	331	6	vn	vn	PROPN
ejpam-4883	331	7	∈	∈	PROPN
ejpam-4883	331	8	a	a	PRON
ejpam-4883	331	9	and	and	CCONJ
ejpam-4883	331	10	ng[vn	ng[vn	PROPN
ejpam-4883	331	11	]	]	PUNCT
ejpam-4883	331	12	\	\	PROPN
ejpam-4883	332	1	ng[vn−1	ng[vn−1	PROPN
ejpam-4883	332	2	]	]	X
ejpam-4883	332	3	=	=	SYM
ejpam-4883	332	4	{	{	PUNCT
ejpam-4883	332	5	vn	vn	NOUN
ejpam-4883	332	6	,	,	PUNCT
ejpam-4883	332	7	vn−1	vn−1	ADJ
ejpam-4883	332	8	}	}	PUNCT
ejpam-4883	332	9	\	\	NOUN
ejpam-4883	332	10	{	{	PUNCT
ejpam-4883	332	11	vn	vn	NOUN
ejpam-4883	332	12	,	,	PUNCT
ejpam-4883	332	13	vn−1	vn−1	ADJ
ejpam-4883	332	14	,	,	PUNCT
ejpam-4883	332	15	vn−2	vn−2	NOUN
ejpam-4883	332	16	}	}	PUNCT
ejpam-4883	332	17	=	=	SYM
ejpam-4883	332	18	∅	∅	NOUN
ejpam-4883	332	19	,	,	PUNCT
ejpam-4883	332	20	a	a	DET
ejpam-4883	332	21	contradiction	contradiction	NOUN
ejpam-4883	332	22	.	.	PUNCT
ejpam-4883	333	1	if	if	SCONJ
ejpam-4883	333	2	vn−1	vn−1	PROPN
ejpam-4883	333	3	/∈	/∈	PUNCT
ejpam-4883	334	1	a	a	DET
ejpam-4883	334	2	,	,	PUNCT
ejpam-4883	334	3	then	then	ADV
ejpam-4883	334	4	v1	v1	VERB
ejpam-4883	334	5	,	,	PUNCT
ejpam-4883	334	6	v2	v2	PROPN
ejpam-4883	334	7	∈	∈	PROPN
ejpam-4883	334	8	a	a	PRON
ejpam-4883	334	9	and	and	CCONJ
ejpam-4883	334	10	ng[v1	ng[v1	PROPN
ejpam-4883	334	11	]	]	X
ejpam-4883	334	12	\ng[v2	\ng[v2	NOUN
ejpam-4883	334	13	]	]	X
ejpam-4883	334	14	=	=	SYM
ejpam-4883	334	15	{	{	PUNCT
ejpam-4883	334	16	v1	v1	PROPN
ejpam-4883	334	17	,	,	PUNCT
ejpam-4883	334	18	v2	v2	PROPN
ejpam-4883	334	19	}	}	PUNCT
ejpam-4883	334	20	\	\	NOUN
ejpam-4883	334	21	{	{	PUNCT
ejpam-4883	334	22	v1	v1	NOUN
ejpam-4883	334	23	,	,	PUNCT
ejpam-4883	334	24	v2	v2	PROPN
ejpam-4883	334	25	,	,	PUNCT
ejpam-4883	334	26	v3	v3	PROPN
ejpam-4883	334	27	}	}	PUNCT
ejpam-4883	334	28	=	=	SYM
ejpam-4883	334	29	∅	∅	NOUN
ejpam-4883	334	30	,	,	PUNCT
ejpam-4883	334	31	a	a	DET
ejpam-4883	334	32	contradiction	contradiction	NOUN
ejpam-4883	334	33	.	.	PUNCT
ejpam-4883	335	1	lastly	lastly	ADV
ejpam-4883	335	2	,	,	PUNCT
ejpam-4883	335	3	suppose	suppose	VERB
ejpam-4883	335	4	that	that	SCONJ
ejpam-4883	335	5	vj	vj	PROPN
ejpam-4883	335	6	/∈	/∈	PROPN
ejpam-4883	336	1	a	a	INTJ
ejpam-4883	336	2	,	,	PUNCT
ejpam-4883	336	3	where	where	SCONJ
ejpam-4883	336	4	2	2	NUM
ejpam-4883	336	5	<	<	X
ejpam-4883	336	6	j	j	X
ejpam-4883	336	7	<	<	X
ejpam-4883	336	8	n−	n−	PROPN
ejpam-4883	336	9	1	1	NUM
ejpam-4883	336	10	.	.	PUNCT
ejpam-4883	337	1	then	then	ADV
ejpam-4883	337	2	vn−1	vn−1	PROPN
ejpam-4883	337	3	,	,	PUNCT
ejpam-4883	337	4	vn	vn	PROPN
ejpam-4883	337	5	∈	∈	PROPN
ejpam-4883	337	6	a	a	PRON
ejpam-4883	337	7	and	and	CCONJ
ejpam-4883	337	8	v1	v1	NOUN
ejpam-4883	337	9	,	,	PUNCT
ejpam-4883	337	10	v2	v2	PROPN
ejpam-4883	337	11	∈	∈	PROPN
ejpam-4883	337	12	a	a	PRON
ejpam-4883	337	13	,	,	PUNCT
ejpam-4883	337	14	a	a	DET
ejpam-4883	337	15	contradiction	contradiction	NOUN
ejpam-4883	337	16	by	by	ADP
ejpam-4883	337	17	the	the	DET
ejpam-4883	337	18	preceding	precede	VERB
ejpam-4883	337	19	arguments	argument	NOUN
ejpam-4883	337	20	.	.	PUNCT
ejpam-4883	338	1	therefore	therefore	ADV
ejpam-4883	338	2	,	,	PUNCT
ejpam-4883	338	3	γj(g	γj(g	PUNCT
ejpam-4883	338	4	)	)	PUNCT
ejpam-4883	338	5	=	=	SYM
ejpam-4883	339	1	n	n	CCONJ
ejpam-4883	339	2	−	−	NOUN
ejpam-4883	339	3	1	1	NUM
ejpam-4883	339	4	is	be	AUX
ejpam-4883	339	5	impossible	impossible	ADJ
ejpam-4883	339	6	.	.	PUNCT
ejpam-4883	340	1	consequently	consequently	ADV
ejpam-4883	340	2	,	,	PUNCT
ejpam-4883	340	3	j.	j.	PROPN
ejpam-4883	340	4	hassan	hassan	PROPN
ejpam-4883	340	5	,	,	PUNCT
ejpam-4883	340	6	j.	j.	PROPN
ejpam-4883	340	7	salim	salim	PROPN
ejpam-4883	340	8	/	/	SYM
ejpam-4883	340	9	eur	eur	PROPN
ejpam-4883	340	10	.	.	PUNCT
ejpam-4883	341	1	j.	j.	PROPN
ejpam-4883	341	2	pure	pure	PROPN
ejpam-4883	341	3	appl	appl	PROPN
ejpam-4883	341	4	.	.	PROPN
ejpam-4883	341	5	math	math	PROPN
ejpam-4883	341	6	,	,	PUNCT
ejpam-4883	341	7	16	16	NUM
ejpam-4883	341	8	(	(	PUNCT
ejpam-4883	341	9	4	4	NUM
ejpam-4883	341	10	)	)	PUNCT
ejpam-4883	341	11	(	(	PUNCT
ejpam-4883	341	12	2023	2023	NUM
ejpam-4883	341	13	)	)	PUNCT
ejpam-4883	341	14	,	,	PUNCT
ejpam-4883	341	15	2082	2082	NUM
ejpam-4883	341	16	-	-	SYM
ejpam-4883	341	17	2095	2095	NUM
ejpam-4883	341	18	2092	2092	NUM
ejpam-4883	341	19	γj(pn	γj(pn	NUM
ejpam-4883	341	20	)	)	PUNCT
ejpam-4883	341	21	=	=	NUM
ejpam-4883	341	22	n−	n−	NOUN
ejpam-4883	341	23	2	2	NUM
ejpam-4883	341	24	for	for	ADP
ejpam-4883	341	25	all	all	DET
ejpam-4883	341	26	n	n	PRON
ejpam-4883	341	27	≥	≥	NOUN
ejpam-4883	341	28	5	5	NUM
ejpam-4883	341	29	.	.	PUNCT
ejpam-4883	341	30	(	(	PUNCT
ejpam-4883	341	31	ii	ii	NOUN
ejpam-4883	341	32	)	)	PUNCT
ejpam-4883	341	33	clearly	clearly	ADV
ejpam-4883	341	34	,	,	PUNCT
ejpam-4883	341	35	γj(c3	γj(c3	NOUN
ejpam-4883	341	36	)	)	PUNCT
ejpam-4883	341	37	=	=	SYM
ejpam-4883	342	1	1	1	X
ejpam-4883	342	2	.	.	PUNCT
ejpam-4883	342	3	suppose	suppose	VERB
ejpam-4883	342	4	n	n	PRON
ejpam-4883	342	5	≥	≥	NUM
ejpam-4883	342	6	4	4	NUM
ejpam-4883	342	7	.	.	PUNCT
ejpam-4883	343	1	let	let	VERB
ejpam-4883	343	2	cn	cn	PROPN
ejpam-4883	344	1	=	=	NOUN
ejpam-4883	344	2	g	g	PROPN
ejpam-4883	344	3	=	=	PUNCT
ejpam-4883	345	1	[	[	X
ejpam-4883	345	2	v1	v1	NOUN
ejpam-4883	345	3	,	,	PUNCT
ejpam-4883	345	4	v2	v2	NOUN
ejpam-4883	345	5	,	,	PUNCT
ejpam-4883	345	6	.	.	PUNCT
ejpam-4883	345	7	.	.	PUNCT
ejpam-4883	345	8	.	.	PUNCT
ejpam-4883	346	1	,	,	PUNCT
ejpam-4883	346	2	vn	vn	X
ejpam-4883	346	3	,	,	PUNCT
ejpam-4883	346	4	v1	v1	PROPN
ejpam-4883	346	5	]	]	PUNCT
ejpam-4883	346	6	and	and	CCONJ
ejpam-4883	346	7	let	let	VERB
ejpam-4883	346	8	d∗	d∗	NOUN
ejpam-4883	346	9	=	=	SYM
ejpam-4883	346	10	v	v	NOUN
ejpam-4883	346	11	(	(	PUNCT
ejpam-4883	346	12	g	g	NOUN
ejpam-4883	346	13	)	)	PUNCT
ejpam-4883	346	14	=	=	SYM
ejpam-4883	346	15	{	{	PUNCT
ejpam-4883	346	16	v1	v1	PROPN
ejpam-4883	346	17	,	,	PUNCT
ejpam-4883	346	18	v2	v2	PROPN
ejpam-4883	346	19	,	,	PUNCT
ejpam-4883	346	20	.	.	PUNCT
ejpam-4883	346	21	.	.	PUNCT
ejpam-4883	346	22	.	.	PUNCT
ejpam-4883	347	1	,	,	PUNCT
ejpam-4883	347	2	vn	vn	PROPN
ejpam-4883	347	3	}	}	PUNCT
ejpam-4883	347	4	.	.	PUNCT
ejpam-4883	348	1	then	then	ADV
ejpam-4883	348	2	vn	vn	PROPN
ejpam-4883	348	3	∈	∈	PROPN
ejpam-4883	348	4	ng[v1	ng[v1	PROPN
ejpam-4883	348	5	]	]	X
ejpam-4883	348	6	\ng[vj	\ng[vj	NOUN
ejpam-4883	348	7	]	]	PUNCT
ejpam-4883	348	8	for	for	ADP
ejpam-4883	348	9	all	all	DET
ejpam-4883	348	10	j	j	PROPN
ejpam-4883	348	11	∈	∈	PROPN
ejpam-4883	348	12	{	{	PUNCT
ejpam-4883	348	13	2	2	NUM
ejpam-4883	348	14	,	,	PUNCT
ejpam-4883	348	15	3	3	NUM
ejpam-4883	348	16	,	,	PUNCT
ejpam-4883	348	17	.	.	PUNCT
ejpam-4883	348	18	.	.	PUNCT
ejpam-4883	348	19	.	.	PUNCT
ejpam-4883	349	1	,	,	PUNCT
ejpam-4883	349	2	n−	n−	NOUN
ejpam-4883	349	3	2	2	NUM
ejpam-4883	349	4	}	}	PUNCT
ejpam-4883	349	5	,	,	PUNCT
ejpam-4883	349	6	v1	v1	PROPN
ejpam-4883	349	7	∈	∈	PROPN
ejpam-4883	349	8	ng[v1	ng[v1	PROPN
ejpam-4883	349	9	]	]	PUNCT
ejpam-4883	349	10	\	\	PROPN
ejpam-4883	349	11	ng[vn−1	ng[vn−1	PROPN
ejpam-4883	349	12	]	]	PUNCT
ejpam-4883	349	13	,	,	PUNCT
ejpam-4883	349	14	v2	v2	PROPN
ejpam-4883	349	15	∈	∈	PROPN
ejpam-4883	349	16	ng[v1	ng[v1	PROPN
ejpam-4883	349	17	]	]	PUNCT
ejpam-4883	349	18	\	\	PUNCT
ejpam-4883	350	1	ng[vn	ng[vn	PROPN
ejpam-4883	350	2	]	]	X
ejpam-4883	350	3	,	,	PUNCT
ejpam-4883	350	4	v1	v1	PROPN
ejpam-4883	350	5	∈	∈	PROPN
ejpam-4883	350	6	ng[vn	ng[vn	PROPN
ejpam-4883	350	7	]	]	PUNCT
ejpam-4883	350	8	\	\	PUNCT
ejpam-4883	350	9	ng[vs	ng[vs	PROPN
ejpam-4883	350	10	]	]	PUNCT
ejpam-4883	350	11	for	for	ADP
ejpam-4883	350	12	all	all	DET
ejpam-4883	350	13	s	s	PROPN
ejpam-4883	350	14	∈	∈	NOUN
ejpam-4883	350	15	{	{	PUNCT
ejpam-4883	350	16	3	3	NUM
ejpam-4883	350	17	,	,	PUNCT
ejpam-4883	350	18	4	4	NUM
ejpam-4883	350	19	,	,	PUNCT
ejpam-4883	350	20	.	.	PUNCT
ejpam-4883	350	21	.	.	PUNCT
ejpam-4883	351	1	.	.	PUNCT
ejpam-4883	352	1	,	,	PUNCT
ejpam-4883	352	2	n	n	CCONJ
ejpam-4883	352	3	−	−	PROPN
ejpam-4883	352	4	1	1	NUM
ejpam-4883	352	5	}	}	PUNCT
ejpam-4883	352	6	,	,	PUNCT
ejpam-4883	352	7	vn	vn	PROPN
ejpam-4883	352	8	∈	∈	PROPN
ejpam-4883	352	9	ng[vn	ng[vn	PROPN
ejpam-4883	352	10	]	]	PUNCT
ejpam-4883	352	11	\	\	PROPN
ejpam-4883	352	12	ng[v2	ng[v2	PROPN
ejpam-4883	352	13	]	]	PUNCT
ejpam-4883	352	14	,	,	PUNCT
ejpam-4883	352	15	vn−1	vn−1	PROPN
ejpam-4883	352	16	∈	∈	PROPN
ejpam-4883	352	17	ng[vn	ng[vn	PROPN
ejpam-4883	352	18	]	]	PUNCT
ejpam-4883	352	19	\	\	PUNCT
ejpam-4883	353	1	ng[v1	ng[v1	PROPN
ejpam-4883	353	2	]	]	PUNCT
ejpam-4883	353	3	,	,	PUNCT
ejpam-4883	353	4	vi+1	vi+1	NOUN
ejpam-4883	353	5	∈	∈	PROPN
ejpam-4883	353	6	ng[vi	ng[vi	PROPN
ejpam-4883	353	7	]	]	PUNCT
ejpam-4883	353	8	\	\	PROPN
ejpam-4883	353	9	ng[vj	ng[vj	PROPN
ejpam-4883	353	10	]	]	PUNCT
ejpam-4883	353	11	∀	∀	PUNCT
ejpam-4883	354	1	i	i	NOUN
ejpam-4883	354	2	,	,	PUNCT
ejpam-4883	354	3	j	j	PROPN
ejpam-4883	354	4	∈	∈	PROPN
ejpam-4883	354	5	{	{	PUNCT
ejpam-4883	354	6	1	1	NUM
ejpam-4883	354	7	,	,	PUNCT
ejpam-4883	354	8	2	2	NUM
ejpam-4883	354	9	,	,	PUNCT
ejpam-4883	354	10	.	.	PUNCT
ejpam-4883	354	11	.	.	PUNCT
ejpam-4883	354	12	.	.	PUNCT
ejpam-4883	355	1	,	,	PUNCT
ejpam-4883	356	1	n	n	CCONJ
ejpam-4883	356	2	−	−	PROPN
ejpam-4883	356	3	1	1	NUM
ejpam-4883	356	4	}	}	PUNCT
ejpam-4883	356	5	,	,	PUNCT
ejpam-4883	356	6	i	i	PRON
ejpam-4883	356	7	>	>	X
ejpam-4883	356	8	j	j	PROPN
ejpam-4883	356	9	,	,	PUNCT
ejpam-4883	356	10	and	and	CCONJ
ejpam-4883	356	11	vk−1	vk−1	VERB
ejpam-4883	356	12	∈	∈	PROPN
ejpam-4883	356	13	ng[vk	ng[vk	ADP
ejpam-4883	356	14	]	]	X
ejpam-4883	356	15	\	\	PUNCT
ejpam-4883	357	1	ng[vl	ng[vl	X
ejpam-4883	357	2	]	]	X
ejpam-4883	357	3	∀	∀	X
ejpam-4883	358	1	k	k	X
ejpam-4883	358	2	,	,	PUNCT
ejpam-4883	358	3	l	l	PROPN
ejpam-4883	358	4	∈	∈	PROPN
ejpam-4883	358	5	{	{	PUNCT
ejpam-4883	358	6	2	2	NUM
ejpam-4883	358	7	,	,	PUNCT
ejpam-4883	358	8	3	3	NUM
ejpam-4883	358	9	,	,	PUNCT
ejpam-4883	358	10	.	.	PUNCT
ejpam-4883	358	11	.	.	PUNCT
ejpam-4883	358	12	.	.	PUNCT
ejpam-4883	358	13	,	,	PUNCT
ejpam-4883	358	14	n	n	CCONJ
ejpam-4883	358	15	}	}	PUNCT
ejpam-4883	358	16	,	,	PUNCT
ejpam-4883	358	17	k	k	PROPN
ejpam-4883	358	18	<	<	X
ejpam-4883	358	19	l.	l.	PROPN
ejpam-4883	358	20	it	it	PRON
ejpam-4883	358	21	follows	follow	VERB
ejpam-4883	358	22	that	that	SCONJ
ejpam-4883	358	23	ng[vq	ng[vq	NOUN
ejpam-4883	358	24	]	]	SYM
ejpam-4883	358	25	\ng[vr	\ng[vr	NOUN
ejpam-4883	358	26	]	]	PUNCT
ejpam-4883	358	27	̸=	̸=	PROPN
ejpam-4883	358	28	∅	∅	NOUN
ejpam-4883	358	29	for	for	ADP
ejpam-4883	358	30	every	every	DET
ejpam-4883	358	31	q	q	X
ejpam-4883	358	32	̸=	̸=	PROPN
ejpam-4883	358	33	r	r	NOUN
ejpam-4883	358	34	,	,	PUNCT
ejpam-4883	358	35	q	q	NOUN
ejpam-4883	358	36	,	,	PUNCT
ejpam-4883	358	37	r	r	NOUN
ejpam-4883	358	38	∈	∈	PROPN
ejpam-4883	358	39	{	{	PUNCT
ejpam-4883	358	40	1	1	NUM
ejpam-4883	358	41	,	,	PUNCT
ejpam-4883	358	42	2	2	NUM
ejpam-4883	358	43	,	,	PUNCT
ejpam-4883	358	44	.	.	PUNCT
ejpam-4883	358	45	.	.	PUNCT
ejpam-4883	359	1	.	.	PUNCT
ejpam-4883	359	2	,	,	PUNCT
ejpam-4883	359	3	n	n	CCONJ
ejpam-4883	359	4	}	}	PUNCT
ejpam-4883	359	5	.	.	PUNCT
ejpam-4883	360	1	therefore	therefore	ADV
ejpam-4883	360	2	,	,	PUNCT
ejpam-4883	360	3	d∗	d∗	PROPN
ejpam-4883	360	4	=	=	SYM
ejpam-4883	360	5	v	v	NOUN
ejpam-4883	360	6	(	(	PUNCT
ejpam-4883	360	7	g	g	NOUN
ejpam-4883	360	8	)	)	PUNCT
ejpam-4883	360	9	is	be	AUX
ejpam-4883	360	10	a	a	DET
ejpam-4883	360	11	j	j	PROPN
ejpam-4883	360	12	-	-	PUNCT
ejpam-4883	360	13	dominating	dominating	NOUN
ejpam-4883	360	14	set	set	NOUN
ejpam-4883	360	15	in	in	ADP
ejpam-4883	360	16	g	g	NOUN
ejpam-4883	360	17	,	,	PUNCT
ejpam-4883	360	18	showing	show	VERB
ejpam-4883	360	19	that	that	PRON
ejpam-4883	360	20	γj(cn	γj(cn	PROPN
ejpam-4883	360	21	)	)	PUNCT
ejpam-4883	361	1	=	=	PROPN
ejpam-4883	361	2	n	n	PROPN
ejpam-4883	361	3	for	for	ADP
ejpam-4883	361	4	all	all	DET
ejpam-4883	361	5	n	n	PRON
ejpam-4883	361	6	≥	≥	NUM
ejpam-4883	361	7	4	4	NUM
ejpam-4883	361	8	.	.	PUNCT
ejpam-4883	361	9	theorem	theorem	NOUN
ejpam-4883	361	10	9	9	NUM
ejpam-4883	361	11	.	.	PUNCT
ejpam-4883	362	1	let	let	VERB
ejpam-4883	362	2	g	g	NOUN
ejpam-4883	362	3	and	and	CCONJ
ejpam-4883	362	4	h	h	NOUN
ejpam-4883	362	5	be	be	VERB
ejpam-4883	362	6	two	two	NUM
ejpam-4883	362	7	non	non	ADJ
ejpam-4883	362	8	-	-	ADJ
ejpam-4883	362	9	complete	complete	ADJ
ejpam-4883	362	10	graphs	graph	NOUN
ejpam-4883	362	11	.	.	PUNCT
ejpam-4883	363	1	a	a	DET
ejpam-4883	363	2	subset	subset	NOUN
ejpam-4883	363	3	d	d	NOUN
ejpam-4883	363	4	of	of	ADP
ejpam-4883	363	5	distinct	distinct	ADJ
ejpam-4883	363	6	vertices	vertex	NOUN
ejpam-4883	363	7	of	of	ADP
ejpam-4883	363	8	g+h	g+h	PROPN
ejpam-4883	363	9	is	be	AUX
ejpam-4883	363	10	a	a	DET
ejpam-4883	363	11	j	j	PROPN
ejpam-4883	363	12	-	-	PUNCT
ejpam-4883	363	13	dominating	dominating	NOUN
ejpam-4883	363	14	set	set	NOUN
ejpam-4883	363	15	in	in	ADP
ejpam-4883	363	16	g+h	g+h	PROPN
ejpam-4883	364	1	if	if	SCONJ
ejpam-4883	364	2	and	and	CCONJ
ejpam-4883	364	3	only	only	ADV
ejpam-4883	364	4	if	if	SCONJ
ejpam-4883	364	5	one	one	NUM
ejpam-4883	364	6	of	of	ADP
ejpam-4883	364	7	the	the	DET
ejpam-4883	364	8	following	follow	VERB
ejpam-4883	364	9	condition	condition	NOUN
ejpam-4883	364	10	holds	hold	VERB
ejpam-4883	364	11	:	:	PUNCT
ejpam-4883	364	12	(	(	PUNCT
ejpam-4883	364	13	i	i	NOUN
ejpam-4883	364	14	)	)	PUNCT
ejpam-4883	364	15	d	d	PRON
ejpam-4883	364	16	is	be	AUX
ejpam-4883	364	17	a	a	DET
ejpam-4883	364	18	j	j	PROPN
ejpam-4883	364	19	-	-	PUNCT
ejpam-4883	364	20	dominating	dominating	ADJ
ejpam-4883	364	21	set	set	NOUN
ejpam-4883	364	22	of	of	ADP
ejpam-4883	364	23	g.	g.	PROPN
ejpam-4883	364	24	(	(	PUNCT
ejpam-4883	364	25	ii	ii	PROPN
ejpam-4883	364	26	)	)	PUNCT
ejpam-4883	364	27	d	d	NOUN
ejpam-4883	364	28	is	be	AUX
ejpam-4883	364	29	a	a	DET
ejpam-4883	364	30	j	j	PROPN
ejpam-4883	364	31	-	-	PUNCT
ejpam-4883	364	32	dominating	dominating	ADJ
ejpam-4883	364	33	set	set	NOUN
ejpam-4883	364	34	of	of	ADP
ejpam-4883	364	35	h.	h.	PROPN
ejpam-4883	364	36	(	(	PUNCT
ejpam-4883	364	37	iii	iii	NOUN
ejpam-4883	364	38	)	)	PUNCT
ejpam-4883	364	39	d	d	NOUN
ejpam-4883	364	40	=	=	SYM
ejpam-4883	364	41	dg	dg	X
ejpam-4883	364	42	∪dh	∪dh	NOUN
ejpam-4883	364	43	,	,	PUNCT
ejpam-4883	364	44	where	where	SCONJ
ejpam-4883	364	45	dg	dg	NOUN
ejpam-4883	364	46	and	and	CCONJ
ejpam-4883	364	47	dh	dh	PROPN
ejpam-4883	364	48	are	be	AUX
ejpam-4883	364	49	j	j	NOUN
ejpam-4883	364	50	-	-	PUNCT
ejpam-4883	364	51	sets	set	NOUN
ejpam-4883	364	52	in	in	ADP
ejpam-4883	364	53	g	g	PROPN
ejpam-4883	364	54	and	and	CCONJ
ejpam-4883	364	55	h	h	NOUN
ejpam-4883	364	56	,	,	PUNCT
ejpam-4883	364	57	respectively	respectively	ADV
ejpam-4883	364	58	.	.	PUNCT
ejpam-4883	365	1	proof	proof	NOUN
ejpam-4883	365	2	.	.	PUNCT
ejpam-4883	366	1	suppose	suppose	VERB
ejpam-4883	366	2	that	that	SCONJ
ejpam-4883	366	3	d	d	PROPN
ejpam-4883	366	4	is	be	AUX
ejpam-4883	366	5	j	j	PROPN
ejpam-4883	366	6	-	-	PUNCT
ejpam-4883	366	7	dominating	dominating	NOUN
ejpam-4883	366	8	set	set	NOUN
ejpam-4883	366	9	in	in	ADP
ejpam-4883	366	10	g	g	PROPN
ejpam-4883	367	1	+	+	CCONJ
ejpam-4883	367	2	h.	h.	NOUN
ejpam-4883	368	1	if	if	SCONJ
ejpam-4883	368	2	dh	dh	NOUN
ejpam-4883	368	3	=	=	NOUN
ejpam-4883	368	4	∅	∅	NOUN
ejpam-4883	368	5	,	,	PUNCT
ejpam-4883	368	6	then	then	ADV
ejpam-4883	368	7	d	d	PROPN
ejpam-4883	368	8	=	=	PUNCT
ejpam-4883	368	9	dg	dg	PROPN
ejpam-4883	368	10	is	be	AUX
ejpam-4883	368	11	a	a	DET
ejpam-4883	368	12	j	j	PROPN
ejpam-4883	368	13	-	-	PUNCT
ejpam-4883	368	14	dominating	dominating	NOUN
ejpam-4883	368	15	set	set	NOUN
ejpam-4883	368	16	in	in	ADP
ejpam-4883	368	17	g	g	NOUN
ejpam-4883	368	18	,	,	PUNCT
ejpam-4883	368	19	and	and	CCONJ
ejpam-4883	368	20	so	so	ADV
ejpam-4883	368	21	(	(	PUNCT
ejpam-4883	368	22	i	i	NOUN
ejpam-4883	368	23	)	)	PUNCT
ejpam-4883	368	24	holds	hold	VERB
ejpam-4883	368	25	.	.	PUNCT
ejpam-4883	369	1	if	if	SCONJ
ejpam-4883	369	2	dg	dg	NOUN
ejpam-4883	369	3	=	=	SYM
ejpam-4883	369	4	∅	∅	NOUN
ejpam-4883	369	5	,	,	PUNCT
ejpam-4883	369	6	then	then	ADV
ejpam-4883	369	7	d	d	PROPN
ejpam-4883	369	8	=	=	SYM
ejpam-4883	369	9	dh	dh	PROPN
ejpam-4883	369	10	is	be	AUX
ejpam-4883	369	11	a	a	DET
ejpam-4883	369	12	j	j	PROPN
ejpam-4883	369	13	-	-	PUNCT
ejpam-4883	369	14	dominating	dominating	NOUN
ejpam-4883	369	15	set	set	NOUN
ejpam-4883	369	16	in	in	ADP
ejpam-4883	369	17	h	h	NOUN
ejpam-4883	369	18	,	,	PUNCT
ejpam-4883	369	19	showing	show	VERB
ejpam-4883	369	20	that	that	SCONJ
ejpam-4883	369	21	(	(	PUNCT
ejpam-4883	369	22	ii	ii	NOUN
ejpam-4883	369	23	)	)	PUNCT
ejpam-4883	369	24	holds	hold	VERB
ejpam-4883	369	25	.	.	PUNCT
ejpam-4883	370	1	now	now	ADV
ejpam-4883	370	2	,	,	PUNCT
ejpam-4883	370	3	assume	assume	VERB
ejpam-4883	370	4	that	that	SCONJ
ejpam-4883	370	5	dg	dg	PROPN
ejpam-4883	370	6	and	and	CCONJ
ejpam-4883	370	7	dh	dh	NOUN
ejpam-4883	370	8	are	be	AUX
ejpam-4883	370	9	both	both	PRON
ejpam-4883	370	10	non	non	ADJ
ejpam-4883	370	11	-	-	ADJ
ejpam-4883	370	12	empty	empty	ADJ
ejpam-4883	370	13	.	.	PUNCT
ejpam-4883	371	1	suppose	suppose	VERB
ejpam-4883	371	2	on	on	ADP
ejpam-4883	371	3	the	the	DET
ejpam-4883	371	4	contrary	contrary	NOUN
ejpam-4883	371	5	that	that	PRON
ejpam-4883	371	6	dg	dg	PROPN
ejpam-4883	371	7	is	be	AUX
ejpam-4883	371	8	not	not	PART
ejpam-4883	371	9	a	a	DET
ejpam-4883	371	10	j	j	NOUN
ejpam-4883	371	11	-	-	PUNCT
ejpam-4883	371	12	set	set	NOUN
ejpam-4883	371	13	in	in	ADP
ejpam-4883	371	14	g.	g.	PROPN
ejpam-4883	371	15	then	then	ADV
ejpam-4883	371	16	there	there	PRON
ejpam-4883	371	17	exist	exist	VERB
ejpam-4883	371	18	a	a	PRON
ejpam-4883	371	19	,	,	PUNCT
ejpam-4883	371	20	b	b	X
ejpam-4883	371	21	∈	∈	PROPN
ejpam-4883	371	22	dg	dg	VERB
ejpam-4883	371	23	⊆	⊆	NUM
ejpam-4883	371	24	d	d	NOUN
ejpam-4883	371	25	such	such	ADJ
ejpam-4883	371	26	that	that	SCONJ
ejpam-4883	371	27	either	either	CCONJ
ejpam-4883	371	28	ng[a	ng[a	PROPN
ejpam-4883	371	29	]	]	PUNCT
ejpam-4883	371	30	\ng[b	\ng[b	PROPN
ejpam-4883	371	31	]	]	X
ejpam-4883	372	1	=	=	SYM
ejpam-4883	372	2	∅	∅	NOUN
ejpam-4883	372	3	or	or	CCONJ
ejpam-4883	372	4	ng[b	ng[b	NOUN
ejpam-4883	372	5	]	]	PUNCT
ejpam-4883	372	6	\ng[a	\ng[a	NOUN
ejpam-4883	372	7	]	]	X
ejpam-4883	372	8	=	=	SYM
ejpam-4883	372	9	∅	∅	NOUN
ejpam-4883	372	10	,	,	PUNCT
ejpam-4883	372	11	a	a	DET
ejpam-4883	372	12	contradiction	contradiction	NOUN
ejpam-4883	372	13	.	.	PUNCT
ejpam-4883	373	1	therefore	therefore	ADV
ejpam-4883	373	2	,	,	PUNCT
ejpam-4883	373	3	dg	dg	PROPN
ejpam-4883	373	4	is	be	AUX
ejpam-4883	373	5	a	a	DET
ejpam-4883	373	6	j	j	NOUN
ejpam-4883	373	7	-	-	PUNCT
ejpam-4883	373	8	set	set	NOUN
ejpam-4883	373	9	in	in	ADP
ejpam-4883	373	10	g.	g.	PROPN
ejpam-4883	373	11	similarly	similarly	ADV
ejpam-4883	373	12	,	,	PUNCT
ejpam-4883	373	13	dh	dh	PROPN
ejpam-4883	373	14	is	be	AUX
ejpam-4883	373	15	a	a	DET
ejpam-4883	373	16	j	j	NOUN
ejpam-4883	373	17	-	-	PUNCT
ejpam-4883	373	18	set	set	NOUN
ejpam-4883	373	19	in	in	ADP
ejpam-4883	373	20	h.	h.	PROPN
ejpam-4883	373	21	consequently	consequently	ADV
ejpam-4883	373	22	,	,	PUNCT
ejpam-4883	373	23	(	(	PUNCT
ejpam-4883	373	24	iii	iii	NOUN
ejpam-4883	373	25	)	)	PUNCT
ejpam-4883	373	26	holds	hold	VERB
ejpam-4883	373	27	.	.	PUNCT
ejpam-4883	374	1	conversely	conversely	ADV
ejpam-4883	374	2	,	,	PUNCT
ejpam-4883	374	3	suppose	suppose	VERB
ejpam-4883	374	4	that	that	SCONJ
ejpam-4883	374	5	(	(	PUNCT
ejpam-4883	374	6	i	i	NOUN
ejpam-4883	374	7	)	)	PUNCT
ejpam-4883	374	8	holds	hold	VERB
ejpam-4883	374	9	.	.	PUNCT
ejpam-4883	375	1	then	then	ADV
ejpam-4883	375	2	d	d	X
ejpam-4883	375	3	is	be	AUX
ejpam-4883	375	4	a	a	DET
ejpam-4883	375	5	j	j	PROPN
ejpam-4883	375	6	-	-	PUNCT
ejpam-4883	375	7	dominating	dominating	NOUN
ejpam-4883	375	8	set	set	NOUN
ejpam-4883	375	9	in	in	ADP
ejpam-4883	375	10	g+h	g+h	PROPN
ejpam-4883	375	11	.	.	PUNCT
ejpam-4883	376	1	similarly	similarly	ADV
ejpam-4883	376	2	,	,	PUNCT
ejpam-4883	376	3	if	if	SCONJ
ejpam-4883	376	4	(	(	PUNCT
ejpam-4883	376	5	ii	ii	NOUN
ejpam-4883	376	6	)	)	PUNCT
ejpam-4883	376	7	holds	hold	VERB
ejpam-4883	376	8	,	,	PUNCT
ejpam-4883	376	9	then	then	ADV
ejpam-4883	376	10	d	d	PROPN
ejpam-4883	376	11	is	be	AUX
ejpam-4883	376	12	a	a	DET
ejpam-4883	376	13	j	j	PROPN
ejpam-4883	376	14	-	-	PUNCT
ejpam-4883	376	15	dominating	dominating	NOUN
ejpam-4883	376	16	set	set	NOUN
ejpam-4883	376	17	in	in	ADP
ejpam-4883	376	18	g+h	g+h	PROPN
ejpam-4883	376	19	.	.	PUNCT
ejpam-4883	377	1	now	now	ADV
ejpam-4883	377	2	,	,	PUNCT
ejpam-4883	377	3	assume	assume	VERB
ejpam-4883	377	4	that	that	SCONJ
ejpam-4883	377	5	(	(	PUNCT
ejpam-4883	377	6	iii	iii	NOUN
ejpam-4883	377	7	)	)	PUNCT
ejpam-4883	377	8	holds	hold	VERB
ejpam-4883	377	9	.	.	PUNCT
ejpam-4883	378	1	since	since	SCONJ
ejpam-4883	378	2	dg	dg	PROPN
ejpam-4883	378	3	and	and	CCONJ
ejpam-4883	378	4	dh	dh	NOUN
ejpam-4883	378	5	are	be	AUX
ejpam-4883	378	6	both	both	PRON
ejpam-4883	378	7	non	non	ADJ
ejpam-4883	378	8	-	-	ADJ
ejpam-4883	378	9	empty	empty	ADJ
ejpam-4883	378	10	,	,	PUNCT
ejpam-4883	378	11	it	it	PRON
ejpam-4883	378	12	follows	follow	VERB
ejpam-4883	378	13	that	that	SCONJ
ejpam-4883	378	14	d	d	NOUN
ejpam-4883	378	15	is	be	AUX
ejpam-4883	378	16	a	a	DET
ejpam-4883	378	17	dominating	dominating	NOUN
ejpam-4883	378	18	set	set	VERB
ejpam-4883	378	19	in	in	ADP
ejpam-4883	378	20	g+h	g+h	PROPN
ejpam-4883	378	21	.	.	PUNCT
ejpam-4883	379	1	it	it	PRON
ejpam-4883	379	2	remains	remain	VERB
ejpam-4883	379	3	to	to	PART
ejpam-4883	379	4	show	show	VERB
ejpam-4883	379	5	that	that	SCONJ
ejpam-4883	379	6	d	d	NOUN
ejpam-4883	379	7	is	be	AUX
ejpam-4883	379	8	a	a	DET
ejpam-4883	379	9	j	j	NOUN
ejpam-4883	379	10	-	-	PUNCT
ejpam-4883	379	11	set	set	NOUN
ejpam-4883	379	12	in	in	ADP
ejpam-4883	379	13	g+h	g+h	PROPN
ejpam-4883	379	14	.	.	PUNCT
ejpam-4883	380	1	let	let	VERB
ejpam-4883	380	2	a	a	DET
ejpam-4883	380	3	,	,	PUNCT
ejpam-4883	380	4	b	b	X
ejpam-4883	380	5	∈	∈	PROPN
ejpam-4883	380	6	d.	d.	NOUN
ejpam-4883	380	7	if	if	SCONJ
ejpam-4883	380	8	a	a	PRON
ejpam-4883	380	9	,	,	PUNCT
ejpam-4883	380	10	b	b	X
ejpam-4883	380	11	∈	∈	PROPN
ejpam-4883	380	12	dg	dg	VERB
ejpam-4883	380	13	⊆	⊆	NUM
ejpam-4883	380	14	d	d	PROPN
ejpam-4883	380	15	,	,	PUNCT
ejpam-4883	380	16	then	then	ADV
ejpam-4883	380	17	ng[a]\ng[b	ng[a]\ng[b	NOUN
ejpam-4883	380	18	]	]	X
ejpam-4883	380	19	̸=	̸=	PROPN
ejpam-4883	380	20	∅	∅	NOUN
ejpam-4883	380	21	and	and	CCONJ
ejpam-4883	380	22	ng[b	ng[b	NOUN
ejpam-4883	380	23	]	]	PUNCT
ejpam-4883	380	24	\	\	PROPN
ejpam-4883	381	1	ng[a	ng[a	NOUN
ejpam-4883	381	2	]	]	X
ejpam-4883	381	3	̸=	̸=	PROPN
ejpam-4883	381	4	∅	∅	NOUN
ejpam-4883	381	5	by	by	ADP
ejpam-4883	381	6	assumption	assumption	NOUN
ejpam-4883	381	7	.	.	PUNCT
ejpam-4883	382	1	this	this	PRON
ejpam-4883	382	2	implies	imply	VERB
ejpam-4883	382	3	that	that	SCONJ
ejpam-4883	382	4	ng+h	ng+h	PROPN
ejpam-4883	383	1	[	[	X
ejpam-4883	383	2	a	a	X
ejpam-4883	383	3	]	]	PUNCT
ejpam-4883	383	4	\	\	PROPN
ejpam-4883	383	5	ng+h	ng+h	PROPN
ejpam-4883	384	1	[	[	X
ejpam-4883	384	2	b	b	X
ejpam-4883	384	3	]	]	X
ejpam-4883	384	4	̸=	̸=	PROPN
ejpam-4883	384	5	∅	∅	NOUN
ejpam-4883	384	6	and	and	CCONJ
ejpam-4883	384	7	ng+h	ng+h	NOUN
ejpam-4883	385	1	[	[	X
ejpam-4883	385	2	b	b	X
ejpam-4883	385	3	]	]	X
ejpam-4883	385	4	\ng+h	\ng+h	X
ejpam-4883	385	5	[	[	X
ejpam-4883	385	6	a	a	X
ejpam-4883	385	7	]	]	X
ejpam-4883	385	8	̸=	̸=	PROPN
ejpam-4883	385	9	∅	∅	NOUN
ejpam-4883	385	10	,	,	PUNCT
ejpam-4883	385	11	and	and	CCONJ
ejpam-4883	385	12	we	we	PRON
ejpam-4883	385	13	are	be	AUX
ejpam-4883	385	14	done	do	VERB
ejpam-4883	385	15	.	.	PUNCT
ejpam-4883	386	1	similarly	similarly	ADV
ejpam-4883	386	2	,	,	PUNCT
ejpam-4883	386	3	if	if	SCONJ
ejpam-4883	386	4	a	a	PRON
ejpam-4883	386	5	,	,	PUNCT
ejpam-4883	386	6	b	b	X
ejpam-4883	386	7	∈	∈	PROPN
ejpam-4883	386	8	dh	dh	NOUN
ejpam-4883	387	1	⊆	⊆	NUM
ejpam-4883	387	2	d	d	PROPN
ejpam-4883	387	3	,	,	PUNCT
ejpam-4883	387	4	then	then	ADV
ejpam-4883	387	5	d	d	PROPN
ejpam-4883	387	6	is	be	AUX
ejpam-4883	387	7	a	a	DET
ejpam-4883	387	8	j	j	NOUN
ejpam-4883	387	9	-	-	PUNCT
ejpam-4883	387	10	set	set	NOUN
ejpam-4883	387	11	in	in	ADP
ejpam-4883	387	12	g+h	g+h	PROPN
ejpam-4883	387	13	.	.	PUNCT
ejpam-4883	388	1	now	now	ADV
ejpam-4883	388	2	,	,	PUNCT
ejpam-4883	388	3	assume	assume	VERB
ejpam-4883	388	4	that	that	SCONJ
ejpam-4883	388	5	a	a	DET
ejpam-4883	388	6	∈	∈	NOUN
ejpam-4883	388	7	dg	dg	NOUN
ejpam-4883	388	8	and	and	CCONJ
ejpam-4883	388	9	b	b	X
ejpam-4883	388	10	∈	∈	PROPN
ejpam-4883	388	11	dh	dh	NOUN
ejpam-4883	388	12	.	.	PUNCT
ejpam-4883	389	1	if	if	SCONJ
ejpam-4883	389	2	a	a	PRON
ejpam-4883	389	3	is	be	AUX
ejpam-4883	389	4	a	a	DET
ejpam-4883	389	5	dominating	dominating	NOUN
ejpam-4883	389	6	vertex	vertex	NOUN
ejpam-4883	389	7	of	of	ADP
ejpam-4883	389	8	g	g	NOUN
ejpam-4883	389	9	,	,	PUNCT
ejpam-4883	389	10	then	then	ADV
ejpam-4883	389	11	dh	dh	NOUN
ejpam-4883	389	12	=	=	NOUN
ejpam-4883	389	13	∅	∅	NOUN
ejpam-4883	389	14	,	,	PUNCT
ejpam-4883	389	15	a	a	DET
ejpam-4883	389	16	contradiction	contradiction	NOUN
ejpam-4883	389	17	.	.	PUNCT
ejpam-4883	390	1	thus	thus	ADV
ejpam-4883	390	2	,	,	PUNCT
ejpam-4883	390	3	a	a	PRON
ejpam-4883	390	4	is	be	AUX
ejpam-4883	390	5	not	not	PART
ejpam-4883	390	6	a	a	DET
ejpam-4883	390	7	dominating	dominating	NOUN
ejpam-4883	390	8	vertex	vertex	NOUN
ejpam-4883	390	9	of	of	ADP
ejpam-4883	390	10	g.	g.	PROPN
ejpam-4883	390	11	similarly	similarly	ADV
ejpam-4883	390	12	,	,	PUNCT
ejpam-4883	390	13	b	b	PROPN
ejpam-4883	390	14	is	be	AUX
ejpam-4883	390	15	not	not	PART
ejpam-4883	390	16	a	a	DET
ejpam-4883	390	17	dominating	dominating	NOUN
ejpam-4883	390	18	vertex	vertex	NOUN
ejpam-4883	390	19	of	of	ADP
ejpam-4883	390	20	h.	h.	PROPN
ejpam-4883	390	21	let	let	VERB
ejpam-4883	390	22	x	x	SYM
ejpam-4883	390	23	∈	∈	PROPN
ejpam-4883	390	24	v	v	X
ejpam-4883	390	25	(	(	PUNCT
ejpam-4883	390	26	g	g	NOUN
ejpam-4883	390	27	)	)	PUNCT
ejpam-4883	390	28	and	and	CCONJ
ejpam-4883	390	29	y	y	PROPN
ejpam-4883	390	30	∈	∈	PROPN
ejpam-4883	390	31	v	v	ADP
ejpam-4883	390	32	(	(	PUNCT
ejpam-4883	390	33	h	h	NOUN
ejpam-4883	390	34	)	)	PUNCT
ejpam-4883	390	35	,	,	PUNCT
ejpam-4883	390	36	where	where	SCONJ
ejpam-4883	390	37	x	x	X
ejpam-4883	390	38	/∈	/∈	PUNCT
ejpam-4883	390	39	ng[a	ng[a	NOUN
ejpam-4883	390	40	]	]	PUNCT
ejpam-4883	390	41	and	and	CCONJ
ejpam-4883	390	42	y	y	PROPN
ejpam-4883	390	43	/∈	/∈	PUNCT
ejpam-4883	390	44	nh	nh	PROPN
ejpam-4883	391	1	[	[	X
ejpam-4883	391	2	b	b	X
ejpam-4883	391	3	]	]	X
ejpam-4883	391	4	.	.	PUNCT
ejpam-4883	392	1	then	then	ADV
ejpam-4883	392	2	y	y	PROPN
ejpam-4883	392	3	∈	∈	PROPN
ejpam-4883	392	4	ng+h	ng+h	VERB
ejpam-4883	393	1	[	[	X
ejpam-4883	393	2	a	a	X
ejpam-4883	393	3	]	]	PUNCT
ejpam-4883	393	4	\	\	PROPN
ejpam-4883	393	5	ng+h	ng+h	PROPN
ejpam-4883	394	1	[	[	X
ejpam-4883	394	2	b	b	X
ejpam-4883	394	3	]	]	X
ejpam-4883	394	4	̸=	̸=	PROPN
ejpam-4883	394	5	∅	∅	NOUN
ejpam-4883	394	6	and	and	CCONJ
ejpam-4883	394	7	x	x	PUNCT
ejpam-4883	394	8	∈	∈	PROPN
ejpam-4883	394	9	ng+h	ng+h	VERB
ejpam-4883	395	1	[	[	X
ejpam-4883	395	2	b	b	X
ejpam-4883	395	3	]	]	PUNCT
ejpam-4883	395	4	\	\	VERB
ejpam-4883	396	1	ng+h	ng+h	PROPN
ejpam-4883	397	1	[	[	X
ejpam-4883	397	2	a	a	X
ejpam-4883	397	3	]	]	X
ejpam-4883	397	4	.	.	PUNCT
ejpam-4883	398	1	thus	thus	ADV
ejpam-4883	398	2	,	,	PUNCT
ejpam-4883	398	3	d	d	PRON
ejpam-4883	398	4	is	be	AUX
ejpam-4883	398	5	a	a	DET
ejpam-4883	398	6	j	j	NOUN
ejpam-4883	398	7	-	-	PUNCT
ejpam-4883	398	8	set	set	NOUN
ejpam-4883	398	9	in	in	ADP
ejpam-4883	398	10	g+h	g+h	PROPN
ejpam-4883	398	11	.	.	PUNCT
ejpam-4883	399	1	consequently	consequently	ADV
ejpam-4883	399	2	,	,	PUNCT
ejpam-4883	399	3	d	d	PROPN
ejpam-4883	399	4	is	be	AUX
ejpam-4883	399	5	a	a	DET
ejpam-4883	399	6	j	j	PROPN
ejpam-4883	399	7	-	-	PUNCT
ejpam-4883	399	8	dominating	dominating	NOUN
ejpam-4883	399	9	set	set	NOUN
ejpam-4883	399	10	in	in	ADP
ejpam-4883	399	11	g+h	g+h	PROPN
ejpam-4883	399	12	.	.	PUNCT
ejpam-4883	400	1	corollary	corollary	ADJ
ejpam-4883	400	2	2	2	NUM
ejpam-4883	400	3	.	.	PUNCT
ejpam-4883	401	1	let	let	VERB
ejpam-4883	401	2	g	g	NOUN
ejpam-4883	401	3	and	and	CCONJ
ejpam-4883	401	4	h	h	NOUN
ejpam-4883	401	5	be	be	VERB
ejpam-4883	401	6	two	two	NUM
ejpam-4883	401	7	non	non	ADJ
ejpam-4883	401	8	-	-	ADJ
ejpam-4883	401	9	complete	complete	ADJ
ejpam-4883	401	10	graphs	graph	NOUN
ejpam-4883	401	11	.	.	PUNCT
ejpam-4883	402	1	then	then	ADV
ejpam-4883	402	2	γj(g+h	γj(g+h	PROPN
ejpam-4883	402	3	)	)	PUNCT
ejpam-4883	402	4	=	=	SYM
ejpam-4883	403	1	γj(g	γj(g	X
ejpam-4883	403	2	)	)	PUNCT
ejpam-4883	404	1	+	+	CCONJ
ejpam-4883	404	2	γj(h	γj(h	NUM
ejpam-4883	404	3	)	)	PUNCT
ejpam-4883	404	4	.	.	PUNCT
ejpam-4883	405	1	in	in	ADP
ejpam-4883	405	2	particular	particular	ADJ
ejpam-4883	405	3	,	,	PUNCT
ejpam-4883	405	4	each	each	PRON
ejpam-4883	405	5	of	of	ADP
ejpam-4883	405	6	the	the	DET
ejpam-4883	405	7	following	follow	VERB
ejpam-4883	405	8	holds	hold	VERB
ejpam-4883	405	9	:	:	PUNCT
ejpam-4883	405	10	(	(	PUNCT
ejpam-4883	405	11	i	i	NOUN
ejpam-4883	405	12	)	)	PUNCT
ejpam-4883	405	13	γj(km	γj(km	PROPN
ejpam-4883	405	14	,	,	PUNCT
ejpam-4883	405	15	n	n	CCONJ
ejpam-4883	405	16	)	)	PUNCT
ejpam-4883	405	17	=	=	SYM
ejpam-4883	405	18	γj(km	γj(km	ADJ
ejpam-4883	405	19	)	)	PUNCT
ejpam-4883	406	1	+	+	CCONJ
ejpam-4883	406	2	γj(kn	γj(kn	NOUN
ejpam-4883	406	3	)	)	PUNCT
ejpam-4883	407	1	=	=	PUNCT
ejpam-4883	408	1	m+	m+	NUM
ejpam-4883	408	2	n	n	PROPN
ejpam-4883	408	3	for	for	ADP
ejpam-4883	408	4	all	all	DET
ejpam-4883	408	5	m	m	PROPN
ejpam-4883	408	6	,	,	PUNCT
ejpam-4883	408	7	n	n	PRON
ejpam-4883	408	8	≥	≥	NOUN
ejpam-4883	408	9	2	2	NUM
ejpam-4883	408	10	.	.	PUNCT
ejpam-4883	409	1	j.	j.	PROPN
ejpam-4883	409	2	hassan	hassan	PROPN
ejpam-4883	409	3	,	,	PUNCT
ejpam-4883	409	4	j.	j.	PROPN
ejpam-4883	409	5	salim	salim	PROPN
ejpam-4883	409	6	/	/	SYM
ejpam-4883	409	7	eur	eur	PROPN
ejpam-4883	409	8	.	.	PUNCT
ejpam-4883	410	1	j.	j.	PROPN
ejpam-4883	410	2	pure	pure	PROPN
ejpam-4883	410	3	appl	appl	PROPN
ejpam-4883	410	4	.	.	PROPN
ejpam-4883	410	5	math	math	PROPN
ejpam-4883	410	6	,	,	PUNCT
ejpam-4883	410	7	16	16	NUM
ejpam-4883	410	8	(	(	PUNCT
ejpam-4883	410	9	4	4	NUM
ejpam-4883	410	10	)	)	PUNCT
ejpam-4883	410	11	(	(	PUNCT
ejpam-4883	410	12	2023	2023	NUM
ejpam-4883	410	13	)	)	PUNCT
ejpam-4883	410	14	,	,	PUNCT
ejpam-4883	410	15	2082	2082	NUM
ejpam-4883	410	16	-	-	SYM
ejpam-4883	410	17	2095	2095	NUM
ejpam-4883	410	18	2093	2093	NUM
ejpam-4883	410	19	(	(	PUNCT
ejpam-4883	410	20	ii	ii	NOUN
ejpam-4883	410	21	)	)	PUNCT
ejpam-4883	411	1	γj(pn	γj(pn	PROPN
ejpam-4883	411	2	+	+	NUM
ejpam-4883	411	3	pm	pm	NOUN
ejpam-4883	411	4	)	)	PUNCT
ejpam-4883	411	5	=	=	PUNCT
ejpam-4883	411	6			NOUN
ejpam-4883	411	7	4	4	NUM
ejpam-4883	411	8	if	if	SCONJ
ejpam-4883	411	9	n	n	PRON
ejpam-4883	411	10	=	=	SYM
ejpam-4883	411	11	3,m	3,m	NUM
ejpam-4883	411	12	=	=	NOUN
ejpam-4883	411	13	3	3	NUM
ejpam-4883	411	14	m	m	NOUN
ejpam-4883	411	15	if	if	SCONJ
ejpam-4883	411	16	n	n	ADJ
ejpam-4883	411	17	=	=	SYM
ejpam-4883	411	18	3,m	3,m	NUM
ejpam-4883	411	19	≥	≥	NOUN
ejpam-4883	411	20	4	4	NUM
ejpam-4883	411	21	n	n	NOUN
ejpam-4883	411	22	if	if	SCONJ
ejpam-4883	411	23	n	n	NUM
ejpam-4883	411	24	≥	≥	VERB
ejpam-4883	411	25	4	4	NUM
ejpam-4883	411	26	≥	≥	NOUN
ejpam-4883	411	27	m	m	NOUN
ejpam-4883	411	28	=	=	SYM
ejpam-4883	411	29	3	3	NUM
ejpam-4883	411	30	n+m−	n+m−	NOUN
ejpam-4883	411	31	4	4	NUM
ejpam-4883	411	32	if	if	SCONJ
ejpam-4883	411	33	n	n	NUM
ejpam-4883	411	34	≥	≥	VERB
ejpam-4883	411	35	4,m	4,m	NUM
ejpam-4883	411	36	≥	≥	NOUN
ejpam-4883	411	37	4	4	NUM
ejpam-4883	411	38	.	.	PUNCT
ejpam-4883	412	1	(	(	PUNCT
ejpam-4883	412	2	iii	iii	NOUN
ejpam-4883	412	3	)	)	PUNCT
ejpam-4883	412	4	γj(pn	γj(pn	NOUN
ejpam-4883	412	5	+	+	NOUN
ejpam-4883	412	6	cm	cm	NOUN
ejpam-4883	412	7	)	)	PUNCT
ejpam-4883	412	8	=	=	PRON
ejpam-4883	412	9	{	{	PUNCT
ejpam-4883	412	10	m+	m+	NUM
ejpam-4883	412	11	2	2	NUM
ejpam-4883	412	12	if	if	SCONJ
ejpam-4883	412	13	n	n	NOUN
ejpam-4883	412	14	=	=	SYM
ejpam-4883	412	15	3	3	NUM
ejpam-4883	412	16	≥	≥	NOUN
ejpam-4883	412	17	m	m	PROPN
ejpam-4883	412	18	≥	≥	NUM
ejpam-4883	412	19	4	4	NUM
ejpam-4883	412	20	n+m−	n+m−	NOUN
ejpam-4883	412	21	2	2	NUM
ejpam-4883	412	22	if	if	SCONJ
ejpam-4883	412	23	n	n	NUM
ejpam-4883	412	24	≥	≥	VERB
ejpam-4883	412	25	4,m	4,m	NUM
ejpam-4883	412	26	≥	≥	NOUN
ejpam-4883	412	27	4	4	NUM
ejpam-4883	412	28	.	.	PUNCT
ejpam-4883	412	29	(	(	PUNCT
ejpam-4883	412	30	iv	iv	X
ejpam-4883	412	31	)	)	PUNCT
ejpam-4883	412	32	γj(cn	γj(cn	PROPN
ejpam-4883	413	1	+	+	PUNCT
ejpam-4883	413	2	cm	cm	NOUN
ejpam-4883	413	3	)	)	PUNCT
ejpam-4883	413	4	=	=	SYM
ejpam-4883	414	1	n+m	n+m	PROPN
ejpam-4883	414	2	for	for	ADP
ejpam-4883	414	3	all	all	DET
ejpam-4883	414	4	n	n	CCONJ
ejpam-4883	414	5	,	,	PUNCT
ejpam-4883	414	6	m	m	VERB
ejpam-4883	414	7	≥	≥	NOUN
ejpam-4883	414	8	4	4	NUM
ejpam-4883	414	9	.	.	PUNCT
ejpam-4883	415	1	proof	proof	NOUN
ejpam-4883	415	2	.	.	PUNCT
ejpam-4883	416	1	let	let	VERB
ejpam-4883	416	2	d	d	NOUN
ejpam-4883	416	3	=	=	PRON
ejpam-4883	416	4	dg	dg	PROPN
ejpam-4883	416	5	∪dh	∪dh	PROPN
ejpam-4883	416	6	be	be	AUX
ejpam-4883	416	7	a	a	DET
ejpam-4883	416	8	γj	γj	NOUN
ejpam-4883	416	9	-set	-set	PUNCT
ejpam-4883	416	10	of	of	ADP
ejpam-4883	416	11	g+h	g+h	PROPN
ejpam-4883	416	12	.	.	PUNCT
ejpam-4883	417	1	then	then	ADV
ejpam-4883	417	2	by	by	ADP
ejpam-4883	417	3	theorem	theorem	NOUN
ejpam-4883	417	4	9	9	NUM
ejpam-4883	417	5	,	,	PUNCT
ejpam-4883	417	6	dg	dg	NOUN
ejpam-4883	417	7	and	and	CCONJ
ejpam-4883	417	8	dh	dh	PROPN
ejpam-4883	417	9	are	be	AUX
ejpam-4883	417	10	j	j	NOUN
ejpam-4883	417	11	-	-	PUNCT
ejpam-4883	417	12	sets	set	NOUN
ejpam-4883	417	13	in	in	ADP
ejpam-4883	417	14	g	g	PROPN
ejpam-4883	417	15	and	and	CCONJ
ejpam-4883	417	16	h	h	NOUN
ejpam-4883	417	17	,	,	PUNCT
ejpam-4883	417	18	respectively	respectively	ADV
ejpam-4883	417	19	.	.	PUNCT
ejpam-4883	418	1	it	it	PRON
ejpam-4883	418	2	follows	follow	VERB
ejpam-4883	418	3	that	that	SCONJ
ejpam-4883	418	4	γj(g+h	γj(g+h	VERB
ejpam-4883	418	5	)	)	PUNCT
ejpam-4883	418	6	=	=	SYM
ejpam-4883	418	7	|d|	|d|	PROPN
ejpam-4883	418	8	=	=	PUNCT
ejpam-4883	418	9	|dg|+	|dg|+	ADP
ejpam-4883	418	10	|dh	|dh	NUM
ejpam-4883	418	11	|	|	ADV
ejpam-4883	418	12	≤	≤	NUM
ejpam-4883	418	13	γj(dg	γj(dg	NOUN
ejpam-4883	418	14	)	)	PUNCT
ejpam-4883	418	15	+	+	SYM
ejpam-4883	418	16	γj(dh	γj(dh	PROPN
ejpam-4883	418	17	)	)	PUNCT
ejpam-4883	418	18	by	by	ADP
ejpam-4883	418	19	corollary	corollary	ADJ
ejpam-4883	418	20	1	1	NUM
ejpam-4883	418	21	.	.	PUNCT
ejpam-4883	419	1	on	on	ADP
ejpam-4883	419	2	the	the	DET
ejpam-4883	419	3	other	other	ADJ
ejpam-4883	419	4	hand	hand	NOUN
ejpam-4883	419	5	,	,	PUNCT
ejpam-4883	419	6	suppose	suppose	VERB
ejpam-4883	419	7	that	that	SCONJ
ejpam-4883	419	8	d	d	PROPN
ejpam-4883	419	9	=	=	X
ejpam-4883	419	10	dg	dg	X
ejpam-4883	419	11	∪	∪	PROPN
ejpam-4883	419	12	dh	dh	NOUN
ejpam-4883	419	13	,	,	PUNCT
ejpam-4883	419	14	where	where	SCONJ
ejpam-4883	419	15	dg	dg	NOUN
ejpam-4883	419	16	and	and	CCONJ
ejpam-4883	419	17	dh	dh	NOUN
ejpam-4883	419	18	are	be	AUX
ejpam-4883	419	19	maximum	maximum	ADJ
ejpam-4883	419	20	j	j	NOUN
ejpam-4883	419	21	-	-	PUNCT
ejpam-4883	419	22	sets	set	NOUN
ejpam-4883	419	23	in	in	ADP
ejpam-4883	419	24	g	g	PROPN
ejpam-4883	419	25	and	and	CCONJ
ejpam-4883	419	26	h	h	NOUN
ejpam-4883	419	27	,	,	PUNCT
ejpam-4883	419	28	respectively	respectively	ADV
ejpam-4883	419	29	.	.	PUNCT
ejpam-4883	419	30	then	then	ADV
ejpam-4883	419	31	by	by	ADP
ejpam-4883	419	32	theorem	theorem	NOUN
ejpam-4883	419	33	9	9	NUM
ejpam-4883	419	34	,	,	PUNCT
ejpam-4883	419	35	d	d	PRON
ejpam-4883	419	36	is	be	AUX
ejpam-4883	419	37	a	a	DET
ejpam-4883	419	38	j	j	PROPN
ejpam-4883	419	39	-	-	PUNCT
ejpam-4883	419	40	dominating	dominating	NOUN
ejpam-4883	419	41	set	set	NOUN
ejpam-4883	419	42	in	in	ADP
ejpam-4883	419	43	g+h	g+h	PROPN
ejpam-4883	419	44	.	.	PUNCT
ejpam-4883	420	1	thus	thus	ADV
ejpam-4883	420	2	,	,	PUNCT
ejpam-4883	420	3	γj(g	γj(g	PUNCT
ejpam-4883	420	4	)	)	PUNCT
ejpam-4883	421	1	+	+	CCONJ
ejpam-4883	422	1	γj(h	γj(h	X
ejpam-4883	422	2	)	)	PUNCT
ejpam-4883	422	3	=	=	PUNCT
ejpam-4883	422	4	|dg|+	|dg|+	ADP
ejpam-4883	422	5	|dh	|dh	NUM
ejpam-4883	422	6	|	|	ADV
ejpam-4883	422	7	=	=	SYM
ejpam-4883	422	8	|d|	|d|	PROPN
ejpam-4883	422	9	≤	≤	NUM
ejpam-4883	422	10	γj(g+h	γj(g+h	NOUN
ejpam-4883	422	11	)	)	PUNCT
ejpam-4883	422	12	by	by	ADP
ejpam-4883	422	13	proposition	proposition	NOUN
ejpam-4883	422	14	1	1	NUM
ejpam-4883	422	15	.	.	PUNCT
ejpam-4883	422	16	consequently	consequently	ADV
ejpam-4883	422	17	,	,	PUNCT
ejpam-4883	422	18	γj(g	γj(g	PROPN
ejpam-4883	423	1	+	+	CCONJ
ejpam-4883	423	2	h	h	X
ejpam-4883	423	3	)	)	PUNCT
ejpam-4883	423	4	=	=	PRON
ejpam-4883	423	5	γj(g	γj(g	X
ejpam-4883	423	6	)	)	PUNCT
ejpam-4883	424	1	+	+	CCONJ
ejpam-4883	424	2	γj(h	γj(h	NUM
ejpam-4883	424	3	)	)	PUNCT
ejpam-4883	424	4	.	.	PUNCT
ejpam-4883	425	1	moreover	moreover	ADV
ejpam-4883	425	2	,	,	PUNCT
ejpam-4883	425	3	particular	particular	ADJ
ejpam-4883	425	4	cases	case	NOUN
ejpam-4883	425	5	follow	follow	VERB
ejpam-4883	425	6	from	from	ADP
ejpam-4883	425	7	theorem	theorem	ADJ
ejpam-4883	425	8	6	6	NUM
ejpam-4883	425	9	and	and	CCONJ
ejpam-4883	425	10	proposition	proposition	NOUN
ejpam-4883	425	11	2	2	NUM
ejpam-4883	425	12	.	.	PUNCT
ejpam-4883	425	13	theorem	theorem	NOUN
ejpam-4883	425	14	10	10	NUM
ejpam-4883	425	15	.	.	PUNCT
ejpam-4883	426	1	let	let	VERB
ejpam-4883	426	2	g	g	NOUN
ejpam-4883	426	3	and	and	CCONJ
ejpam-4883	426	4	h	h	NOUN
ejpam-4883	426	5	be	be	VERB
ejpam-4883	426	6	any	any	DET
ejpam-4883	426	7	complete	complete	ADJ
ejpam-4883	426	8	and	and	CCONJ
ejpam-4883	426	9	any	any	DET
ejpam-4883	426	10	non	non	ADJ
ejpam-4883	426	11	-	-	ADJ
ejpam-4883	426	12	complete	complete	ADJ
ejpam-4883	426	13	graphs	graph	NOUN
ejpam-4883	426	14	,	,	PUNCT
ejpam-4883	426	15	respectively	respectively	ADV
ejpam-4883	426	16	.	.	PUNCT
ejpam-4883	427	1	a	a	DET
ejpam-4883	427	2	subset	subset	NOUN
ejpam-4883	427	3	d	d	NOUN
ejpam-4883	427	4	=	=	SYM
ejpam-4883	427	5	dg	dg	PROPN
ejpam-4883	427	6	∪dh	∪dh	NOUN
ejpam-4883	427	7	of	of	ADP
ejpam-4883	427	8	distinct	distinct	ADJ
ejpam-4883	427	9	vertices	vertex	NOUN
ejpam-4883	427	10	of	of	ADP
ejpam-4883	427	11	g	g	PROPN
ejpam-4883	427	12	+	+	NOUN
ejpam-4883	427	13	h	h	NOUN
ejpam-4883	427	14	is	be	AUX
ejpam-4883	427	15	a	a	DET
ejpam-4883	427	16	j	j	PROPN
ejpam-4883	427	17	-	-	PUNCT
ejpam-4883	427	18	dominating	dominating	NOUN
ejpam-4883	427	19	set	set	NOUN
ejpam-4883	427	20	in	in	ADP
ejpam-4883	427	21	g	g	PROPN
ejpam-4883	428	1	+	+	NOUN
ejpam-4883	428	2	h	h	NOUN
ejpam-4883	428	3	if	if	SCONJ
ejpam-4883	428	4	and	and	CCONJ
ejpam-4883	428	5	only	only	ADV
ejpam-4883	428	6	if	if	SCONJ
ejpam-4883	428	7	one	one	NUM
ejpam-4883	428	8	of	of	ADP
ejpam-4883	428	9	the	the	DET
ejpam-4883	428	10	following	follow	VERB
ejpam-4883	428	11	conditions	condition	NOUN
ejpam-4883	428	12	holds	hold	VERB
ejpam-4883	428	13	:	:	PUNCT
ejpam-4883	428	14	(	(	PUNCT
ejpam-4883	428	15	i	i	NOUN
ejpam-4883	428	16	)	)	PUNCT
ejpam-4883	428	17	dh	dh	NOUN
ejpam-4883	428	18	=	=	NOUN
ejpam-4883	428	19	∅	∅	NOUN
ejpam-4883	428	20	and	and	CCONJ
ejpam-4883	428	21	dg	dg	PROPN
ejpam-4883	428	22	is	be	AUX
ejpam-4883	428	23	a	a	DET
ejpam-4883	428	24	j	j	PROPN
ejpam-4883	428	25	-	-	PUNCT
ejpam-4883	428	26	dominating	dominating	ADJ
ejpam-4883	428	27	set	set	NOUN
ejpam-4883	428	28	of	of	ADP
ejpam-4883	428	29	g.	g.	PROPN
ejpam-4883	428	30	(	(	PUNCT
ejpam-4883	428	31	ii	ii	PROPN
ejpam-4883	428	32	)	)	PUNCT
ejpam-4883	428	33	dg	dg	NOUN
ejpam-4883	428	34	=	=	NOUN
ejpam-4883	428	35	∅	∅	NOUN
ejpam-4883	428	36	and	and	CCONJ
ejpam-4883	428	37	dh	dh	NOUN
ejpam-4883	428	38	is	be	AUX
ejpam-4883	428	39	a	a	DET
ejpam-4883	428	40	j	j	PROPN
ejpam-4883	428	41	-	-	PUNCT
ejpam-4883	428	42	dominating	dominating	ADJ
ejpam-4883	428	43	set	set	NOUN
ejpam-4883	428	44	of	of	ADP
ejpam-4883	428	45	h.	h.	PROPN
ejpam-4883	428	46	proof	proof	NOUN
ejpam-4883	428	47	.	.	PUNCT
ejpam-4883	429	1	suppose	suppose	VERB
ejpam-4883	429	2	that	that	SCONJ
ejpam-4883	429	3	d	d	PROPN
ejpam-4883	429	4	is	be	AUX
ejpam-4883	429	5	j	j	PROPN
ejpam-4883	429	6	-	-	PUNCT
ejpam-4883	429	7	dominating	dominating	NOUN
ejpam-4883	429	8	set	set	NOUN
ejpam-4883	429	9	in	in	ADP
ejpam-4883	429	10	g	g	PROPN
ejpam-4883	429	11	+	+	CCONJ
ejpam-4883	429	12	h.	h.	PROPN
ejpam-4883	429	13	since	since	SCONJ
ejpam-4883	429	14	g	g	PROPN
ejpam-4883	429	15	is	be	AUX
ejpam-4883	429	16	complete	complete	ADJ
ejpam-4883	429	17	,	,	PUNCT
ejpam-4883	429	18	both	both	PRON
ejpam-4883	429	19	dg	dg	PROPN
ejpam-4883	429	20	̸=	̸=	PROPN
ejpam-4883	429	21	∅	∅	NOUN
ejpam-4883	429	22	and	and	CCONJ
ejpam-4883	429	23	dh	dh	PROPN
ejpam-4883	429	24	̸=	̸=	PROPN
ejpam-4883	429	25	∅	∅	NOUN
ejpam-4883	429	26	are	be	AUX
ejpam-4883	429	27	impossible	impossible	ADJ
ejpam-4883	429	28	.	.	PUNCT
ejpam-4883	430	1	it	it	PRON
ejpam-4883	430	2	follows	follow	VERB
ejpam-4883	430	3	that	that	SCONJ
ejpam-4883	430	4	either	either	CCONJ
ejpam-4883	430	5	dg	dg	PROPN
ejpam-4883	430	6	̸=	̸=	PROPN
ejpam-4883	430	7	∅	∅	NOUN
ejpam-4883	430	8	and	and	CCONJ
ejpam-4883	430	9	dh	dh	NOUN
ejpam-4883	430	10	=	=	NOUN
ejpam-4883	430	11	∅	∅	NOUN
ejpam-4883	430	12	or	or	CCONJ
ejpam-4883	430	13	dh	dh	NOUN
ejpam-4883	430	14	̸=	̸=	PROPN
ejpam-4883	430	15	∅	∅	NOUN
ejpam-4883	430	16	and	and	CCONJ
ejpam-4883	430	17	dg	dg	X
ejpam-4883	430	18	=	=	PUNCT
ejpam-4883	430	19	∅.	∅.	VERB
ejpam-4883	430	20	if	if	SCONJ
ejpam-4883	430	21	dh	dh	NOUN
ejpam-4883	430	22	=	=	NOUN
ejpam-4883	430	23	∅	∅	NOUN
ejpam-4883	430	24	,	,	PUNCT
ejpam-4883	430	25	then	then	ADV
ejpam-4883	430	26	d	d	PROPN
ejpam-4883	430	27	=	=	PUNCT
ejpam-4883	430	28	dg	dg	PROPN
ejpam-4883	430	29	is	be	AUX
ejpam-4883	430	30	a	a	DET
ejpam-4883	430	31	j	j	PROPN
ejpam-4883	430	32	-	-	PUNCT
ejpam-4883	430	33	dominating	dominating	NOUN
ejpam-4883	430	34	set	set	NOUN
ejpam-4883	430	35	in	in	ADP
ejpam-4883	430	36	g	g	NOUN
ejpam-4883	430	37	,	,	PUNCT
ejpam-4883	430	38	and	and	CCONJ
ejpam-4883	430	39	so	so	ADV
ejpam-4883	430	40	(	(	PUNCT
ejpam-4883	430	41	i	i	NOUN
ejpam-4883	430	42	)	)	PUNCT
ejpam-4883	430	43	holds	hold	VERB
ejpam-4883	430	44	.	.	PUNCT
ejpam-4883	431	1	if	if	SCONJ
ejpam-4883	431	2	dg	dg	NOUN
ejpam-4883	431	3	=	=	SYM
ejpam-4883	431	4	∅	∅	NOUN
ejpam-4883	431	5	,	,	PUNCT
ejpam-4883	431	6	then	then	ADV
ejpam-4883	431	7	d	d	PROPN
ejpam-4883	431	8	=	=	SYM
ejpam-4883	431	9	dh	dh	PROPN
ejpam-4883	431	10	is	be	AUX
ejpam-4883	431	11	a	a	DET
ejpam-4883	431	12	j	j	PROPN
ejpam-4883	431	13	-	-	PUNCT
ejpam-4883	431	14	dominating	dominating	NOUN
ejpam-4883	431	15	set	set	NOUN
ejpam-4883	431	16	in	in	ADP
ejpam-4883	431	17	h	h	NOUN
ejpam-4883	431	18	,	,	PUNCT
ejpam-4883	431	19	and	and	CCONJ
ejpam-4883	431	20	so	so	ADV
ejpam-4883	431	21	(	(	PUNCT
ejpam-4883	431	22	ii	ii	NOUN
ejpam-4883	431	23	)	)	PUNCT
ejpam-4883	431	24	holds	hold	VERB
ejpam-4883	431	25	.	.	PUNCT
ejpam-4883	432	1	conversely	conversely	ADV
ejpam-4883	432	2	,	,	PUNCT
ejpam-4883	432	3	suppose	suppose	VERB
ejpam-4883	432	4	that	that	SCONJ
ejpam-4883	432	5	dh	dh	NOUN
ejpam-4883	432	6	=	=	NOUN
ejpam-4883	432	7	∅	∅	NOUN
ejpam-4883	432	8	and	and	CCONJ
ejpam-4883	432	9	dg	dg	PROPN
ejpam-4883	432	10	is	be	AUX
ejpam-4883	432	11	a	a	DET
ejpam-4883	432	12	j	j	PROPN
ejpam-4883	432	13	-	-	PUNCT
ejpam-4883	432	14	dominating	dominating	ADJ
ejpam-4883	432	15	set	set	NOUN
ejpam-4883	432	16	of	of	ADP
ejpam-4883	432	17	g.	g.	PROPN
ejpam-4883	433	1	then	then	ADV
ejpam-4883	433	2	d	d	PROPN
ejpam-4883	433	3	=	=	PUNCT
ejpam-4883	433	4	dg	dg	PROPN
ejpam-4883	433	5	is	be	AUX
ejpam-4883	433	6	a	a	DET
ejpam-4883	433	7	j	j	NOUN
ejpam-4883	433	8	-	-	NOUN
ejpam-4883	433	9	set	set	NOUN
ejpam-4883	433	10	in	in	ADP
ejpam-4883	433	11	g	g	PROPN
ejpam-4883	433	12	+	+	PROPN
ejpam-4883	433	13	h.	h.	PROPN
ejpam-4883	433	14	since	since	SCONJ
ejpam-4883	433	15	dg	dg	PROPN
ejpam-4883	433	16	is	be	AUX
ejpam-4883	433	17	dominating	dominate	VERB
ejpam-4883	433	18	set	set	VERB
ejpam-4883	433	19	in	in	ADP
ejpam-4883	433	20	g	g	PROPN
ejpam-4883	433	21	,	,	PUNCT
ejpam-4883	433	22	ng+h	ng+h	PROPN
ejpam-4883	434	1	[	[	X
ejpam-4883	434	2	d	d	X
ejpam-4883	434	3	]	]	X
ejpam-4883	434	4	=	=	SYM
ejpam-4883	434	5	v	v	NOUN
ejpam-4883	434	6	(	(	PUNCT
ejpam-4883	434	7	g	g	PROPN
ejpam-4883	434	8	+	+	PROPN
ejpam-4883	434	9	h	h	NOUN
ejpam-4883	434	10	)	)	PUNCT
ejpam-4883	434	11	.	.	PUNCT
ejpam-4883	435	1	thus	thus	ADV
ejpam-4883	435	2	,	,	PUNCT
ejpam-4883	435	3	d	d	PRON
ejpam-4883	435	4	is	be	AUX
ejpam-4883	435	5	a	a	DET
ejpam-4883	435	6	dominating	dominating	NOUN
ejpam-4883	435	7	set	set	VERB
ejpam-4883	435	8	in	in	ADP
ejpam-4883	435	9	g+h	g+h	PROPN
ejpam-4883	435	10	,	,	PUNCT
ejpam-4883	435	11	showing	show	VERB
ejpam-4883	435	12	that	that	SCONJ
ejpam-4883	435	13	d	d	NOUN
ejpam-4883	435	14	is	be	AUX
ejpam-4883	435	15	a	a	DET
ejpam-4883	435	16	j	j	PROPN
ejpam-4883	435	17	-	-	PUNCT
ejpam-4883	435	18	dominating	dominating	NOUN
ejpam-4883	435	19	set	set	NOUN
ejpam-4883	435	20	in	in	ADP
ejpam-4883	435	21	g+h	g+h	PROPN
ejpam-4883	435	22	.	.	PUNCT
ejpam-4883	436	1	similarly	similarly	ADV
ejpam-4883	436	2	,	,	PUNCT
ejpam-4883	436	3	if	if	SCONJ
ejpam-4883	436	4	(	(	PUNCT
ejpam-4883	436	5	ii	ii	NOUN
ejpam-4883	436	6	)	)	PUNCT
ejpam-4883	436	7	holds	hold	VERB
ejpam-4883	436	8	,	,	PUNCT
ejpam-4883	436	9	then	then	ADV
ejpam-4883	436	10	d	d	PROPN
ejpam-4883	436	11	is	be	AUX
ejpam-4883	436	12	a	a	DET
ejpam-4883	436	13	j	j	PROPN
ejpam-4883	436	14	-	-	PUNCT
ejpam-4883	436	15	dominating	dominating	NOUN
ejpam-4883	436	16	set	set	NOUN
ejpam-4883	436	17	in	in	ADP
ejpam-4883	436	18	g+h	g+h	PROPN
ejpam-4883	436	19	.	.	PUNCT
ejpam-4883	437	1	corollary	corollary	ADJ
ejpam-4883	437	2	3	3	X
ejpam-4883	437	3	.	.	PUNCT
ejpam-4883	438	1	let	let	VERB
ejpam-4883	438	2	g	g	NOUN
ejpam-4883	438	3	and	and	CCONJ
ejpam-4883	438	4	h	h	NOUN
ejpam-4883	438	5	be	be	AUX
ejpam-4883	438	6	complete	complete	ADJ
ejpam-4883	438	7	and	and	CCONJ
ejpam-4883	438	8	non	non	ADJ
ejpam-4883	438	9	-	-	ADJ
ejpam-4883	438	10	complete	complete	ADJ
ejpam-4883	438	11	graphs	graph	NOUN
ejpam-4883	438	12	,	,	PUNCT
ejpam-4883	438	13	respectively	respectively	ADV
ejpam-4883	438	14	.	.	PUNCT
ejpam-4883	439	1	then	then	ADV
ejpam-4883	439	2	γj(g+h	γj(g+h	PROPN
ejpam-4883	439	3	)	)	PUNCT
ejpam-4883	439	4	=	=	SYM
ejpam-4883	439	5	γj(h	γj(h	NUM
ejpam-4883	439	6	)	)	PUNCT
ejpam-4883	439	7	.	.	PUNCT
ejpam-4883	440	1	in	in	ADP
ejpam-4883	440	2	particular	particular	ADJ
ejpam-4883	440	3	,	,	PUNCT
ejpam-4883	440	4	each	each	PRON
ejpam-4883	440	5	of	of	ADP
ejpam-4883	440	6	the	the	DET
ejpam-4883	440	7	following	follow	VERB
ejpam-4883	440	8	holds	hold	VERB
ejpam-4883	440	9	:	:	PUNCT
ejpam-4883	440	10	(	(	PUNCT
ejpam-4883	440	11	i	i	NOUN
ejpam-4883	440	12	)	)	PUNCT
ejpam-4883	440	13	γj(k1,n	γj(k1,n	PROPN
ejpam-4883	440	14	)	)	PUNCT
ejpam-4883	441	1	=	=	SYM
ejpam-4883	441	2	n	n	PROPN
ejpam-4883	441	3	for	for	ADP
ejpam-4883	441	4	all	all	DET
ejpam-4883	441	5	n	n	PRON
ejpam-4883	441	6	≥	≥	NUM
ejpam-4883	441	7	1	1	NUM
ejpam-4883	441	8	.	.	PUNCT
ejpam-4883	442	1	references	reference	NOUN
ejpam-4883	442	2	2094	2094	NUM
ejpam-4883	442	3	(	(	PUNCT
ejpam-4883	442	4	ii	ii	NOUN
ejpam-4883	442	5	)	)	PUNCT
ejpam-4883	442	6	γj(wn	γj(wn	PROPN
ejpam-4883	442	7	)	)	PUNCT
ejpam-4883	442	8	=	=	SYM
ejpam-4883	442	9	γj(k1	γj(k1	X
ejpam-4883	443	1	+	+	CCONJ
ejpam-4883	443	2	cn	cn	PROPN
ejpam-4883	443	3	)	)	PUNCT
ejpam-4883	443	4	=	=	SYM
ejpam-4883	443	5	n	n	PROPN
ejpam-4883	443	6	for	for	ADP
ejpam-4883	443	7	all	all	DET
ejpam-4883	443	8	n	n	PRON
ejpam-4883	443	9	≥	≥	NOUN
ejpam-4883	443	10	4	4	NUM
ejpam-4883	443	11	.	.	PUNCT
ejpam-4883	443	12	(	(	PUNCT
ejpam-4883	443	13	iii	iii	NOUN
ejpam-4883	443	14	)	)	PUNCT
ejpam-4883	443	15	γj(fn	γj(fn	PROPN
ejpam-4883	443	16	)	)	PUNCT
ejpam-4883	443	17	=	=	SYM
ejpam-4883	443	18	γj(k1	γj(k1	PROPN
ejpam-4883	444	1	+	+	CCONJ
ejpam-4883	444	2	pn	pn	NOUN
ejpam-4883	444	3	)	)	PUNCT
ejpam-4883	445	1	=	=	PUNCT
ejpam-4883	445	2	n−	n−	NOUN
ejpam-4883	445	3	2	2	NUM
ejpam-4883	445	4	for	for	ADP
ejpam-4883	445	5	all	all	DET
ejpam-4883	445	6	n	n	PRON
ejpam-4883	445	7	≥	≥	NOUN
ejpam-4883	445	8	3	3	NUM
ejpam-4883	445	9	.	.	PUNCT
ejpam-4883	446	1	proof	proof	NOUN
ejpam-4883	446	2	.	.	PUNCT
ejpam-4883	447	1	let	let	VERB
ejpam-4883	447	2	d	d	NOUN
ejpam-4883	447	3	=	=	PRON
ejpam-4883	447	4	dg	dg	PROPN
ejpam-4883	447	5	∪dh	∪dh	PROPN
ejpam-4883	447	6	be	be	AUX
ejpam-4883	447	7	a	a	DET
ejpam-4883	447	8	γj	γj	NOUN
ejpam-4883	447	9	-set	-set	PUNCT
ejpam-4883	447	10	of	of	ADP
ejpam-4883	447	11	g+h	g+h	PROPN
ejpam-4883	447	12	.	.	PUNCT
ejpam-4883	448	1	then	then	ADV
ejpam-4883	448	2	by	by	ADP
ejpam-4883	448	3	theorem	theorem	NOUN
ejpam-4883	448	4	10	10	NUM
ejpam-4883	448	5	,	,	PUNCT
ejpam-4883	448	6	either	either	CCONJ
ejpam-4883	448	7	dh	dh	NOUN
ejpam-4883	448	8	=	=	NOUN
ejpam-4883	448	9	∅	∅	NOUN
ejpam-4883	448	10	and	and	CCONJ
ejpam-4883	448	11	dg	dg	PROPN
ejpam-4883	448	12	is	be	AUX
ejpam-4883	448	13	j	j	PROPN
ejpam-4883	448	14	-	-	PUNCT
ejpam-4883	448	15	dominating	dominating	NOUN
ejpam-4883	448	16	set	set	NOUN
ejpam-4883	448	17	in	in	ADP
ejpam-4883	448	18	g	g	PROPN
ejpam-4883	448	19	or	or	CCONJ
ejpam-4883	448	20	dg	dg	NOUN
ejpam-4883	448	21	=	=	NOUN
ejpam-4883	448	22	∅	∅	NOUN
ejpam-4883	448	23	and	and	CCONJ
ejpam-4883	448	24	dh	dh	NOUN
ejpam-4883	448	25	is	be	AUX
ejpam-4883	448	26	j	j	PROPN
ejpam-4883	448	27	-	-	PUNCT
ejpam-4883	448	28	dominating	dominating	NOUN
ejpam-4883	448	29	set	set	NOUN
ejpam-4883	448	30	in	in	ADP
ejpam-4883	448	31	h.	h.	PROPN
ejpam-4883	449	1	it	it	PRON
ejpam-4883	449	2	follows	follow	VERB
ejpam-4883	449	3	that	that	SCONJ
ejpam-4883	449	4	γj(g+h	γj(g+h	VERB
ejpam-4883	449	5	)	)	PUNCT
ejpam-4883	449	6	=	=	SYM
ejpam-4883	449	7	|d|	|d|	PROPN
ejpam-4883	449	8	=	=	PUNCT
ejpam-4883	449	9	|dg|+	|dg|+	ADP
ejpam-4883	449	10	|dh	|dh	NUM
ejpam-4883	449	11	|	|	ADV
ejpam-4883	449	12	=	=	PUNCT
ejpam-4883	449	13	|dg|	|dg|	PROPN
ejpam-4883	449	14	≤	≤	NUM
ejpam-4883	449	15	γj(g	γj(g	PUNCT
ejpam-4883	449	16	)	)	PUNCT
ejpam-4883	449	17	or	or	CCONJ
ejpam-4883	449	18	γj(g+h	γj(g+h	NUM
ejpam-4883	449	19	)	)	PUNCT
ejpam-4883	449	20	=	=	SYM
ejpam-4883	449	21	|d|	|d|	PROPN
ejpam-4883	449	22	=	=	PUNCT
ejpam-4883	449	23	|dg|+	|dg|+	ADP
ejpam-4883	449	24	|dh	|dh	NUM
ejpam-4883	449	25	|	|	ADV
ejpam-4883	449	26	=	=	PUNCT
ejpam-4883	449	27	|dh	|dh	NUM
ejpam-4883	449	28	|	|	ADV
ejpam-4883	449	29	≤	≤	NUM
ejpam-4883	449	30	γj(dh	γj(dh	PROPN
ejpam-4883	449	31	)	)	PUNCT
ejpam-4883	449	32	by	by	ADP
ejpam-4883	449	33	proposition	proposition	NOUN
ejpam-4883	449	34	1	1	NUM
ejpam-4883	449	35	.	.	PUNCT
ejpam-4883	450	1	thus	thus	ADV
ejpam-4883	450	2	,	,	PUNCT
ejpam-4883	450	3	γj(g+h	γj(g+h	NOUN
ejpam-4883	450	4	)	)	PUNCT
ejpam-4883	450	5	≤	≤	NUM
ejpam-4883	450	6	max{γj(g	max{γj(g	NOUN
ejpam-4883	450	7	)	)	PUNCT
ejpam-4883	450	8	,	,	PUNCT
ejpam-4883	450	9	γj(dh	γj(dh	PROPN
ejpam-4883	450	10	)	)	PUNCT
ejpam-4883	450	11	}	}	PUNCT
ejpam-4883	450	12	.	.	PUNCT
ejpam-4883	451	1	since	since	SCONJ
ejpam-4883	451	2	g	g	PROPN
ejpam-4883	451	3	is	be	AUX
ejpam-4883	451	4	complete	complete	ADJ
ejpam-4883	451	5	,	,	PUNCT
ejpam-4883	451	6	it	it	PRON
ejpam-4883	451	7	follows	follow	VERB
ejpam-4883	451	8	that	that	SCONJ
ejpam-4883	451	9	γj(g+h	γj(g+h	NOUN
ejpam-4883	451	10	)	)	PUNCT
ejpam-4883	451	11	≤	≤	NUM
ejpam-4883	451	12	γj(dh	γj(dh	PROPN
ejpam-4883	451	13	)	)	PUNCT
ejpam-4883	451	14	}	}	PUNCT
ejpam-4883	451	15	.	.	PUNCT
ejpam-4883	452	1	on	on	ADP
ejpam-4883	452	2	the	the	DET
ejpam-4883	452	3	other	other	ADJ
ejpam-4883	452	4	hand	hand	NOUN
ejpam-4883	452	5	,	,	PUNCT
ejpam-4883	452	6	suppose	suppose	VERB
ejpam-4883	452	7	that	that	SCONJ
ejpam-4883	452	8	d	d	PROPN
ejpam-4883	452	9	=	=	X
ejpam-4883	452	10	dg	dg	X
ejpam-4883	452	11	∪	∪	PROPN
ejpam-4883	452	12	dh	dh	NOUN
ejpam-4883	452	13	,	,	PUNCT
ejpam-4883	452	14	where	where	SCONJ
ejpam-4883	452	15	either	either	CCONJ
ejpam-4883	452	16	dh	dh	NOUN
ejpam-4883	452	17	=	=	NOUN
ejpam-4883	452	18	∅	∅	NOUN
ejpam-4883	452	19	and	and	CCONJ
ejpam-4883	452	20	dg	dg	X
ejpam-4883	452	21	=	=	SYM
ejpam-4883	452	22	d	d	NOUN
ejpam-4883	452	23	is	be	AUX
ejpam-4883	452	24	γj	γj	SCONJ
ejpam-4883	452	25	set	set	VERB
ejpam-4883	452	26	in	in	ADP
ejpam-4883	452	27	g	g	PROPN
ejpam-4883	452	28	or	or	CCONJ
ejpam-4883	452	29	dg	dg	NOUN
ejpam-4883	452	30	=	=	NOUN
ejpam-4883	452	31	∅	∅	NOUN
ejpam-4883	452	32	and	and	CCONJ
ejpam-4883	452	33	dh	dh	NOUN
ejpam-4883	453	1	=	=	PUNCT
ejpam-4883	453	2	d	d	NOUN
ejpam-4883	453	3	is	be	AUX
ejpam-4883	453	4	γj	γj	ADP
ejpam-4883	453	5	-set	-set	ADJ
ejpam-4883	453	6	in	in	ADP
ejpam-4883	453	7	h.	h.	PROPN
ejpam-4883	453	8	then	then	ADV
ejpam-4883	453	9	by	by	ADP
ejpam-4883	453	10	theorem	theorem	NOUN
ejpam-4883	453	11	10	10	NUM
ejpam-4883	453	12	,	,	PUNCT
ejpam-4883	453	13	d	d	PRON
ejpam-4883	453	14	is	be	AUX
ejpam-4883	453	15	a	a	DET
ejpam-4883	453	16	j	j	PROPN
ejpam-4883	453	17	-	-	PUNCT
ejpam-4883	453	18	dominating	dominating	NOUN
ejpam-4883	453	19	set	set	NOUN
ejpam-4883	453	20	in	in	ADP
ejpam-4883	453	21	g	g	PROPN
ejpam-4883	453	22	+	+	CCONJ
ejpam-4883	453	23	h.	h.	NOUN
ejpam-4883	454	1	it	it	PRON
ejpam-4883	454	2	follows	follow	VERB
ejpam-4883	454	3	that	that	SCONJ
ejpam-4883	454	4	γj(g	γj(g	PUNCT
ejpam-4883	454	5	)	)	PUNCT
ejpam-4883	454	6	=	=	SYM
ejpam-4883	454	7	|dg|	|dg|	PROPN
ejpam-4883	454	8	≤	≤	NUM
ejpam-4883	454	9	γj(g	γj(g	PUNCT
ejpam-4883	455	1	+	+	NUM
ejpam-4883	455	2	h	h	NOUN
ejpam-4883	455	3	)	)	PUNCT
ejpam-4883	455	4	or	or	CCONJ
ejpam-4883	455	5	γj(h	γj(h	NUM
ejpam-4883	455	6	)	)	PUNCT
ejpam-4883	455	7	=	=	SYM
ejpam-4883	456	1	|dh	|dh	NUM
ejpam-4883	456	2	|	|	ADV
ejpam-4883	456	3	≤	≤	NOUN
ejpam-4883	456	4	γj(g	γj(g	PUNCT
ejpam-4883	457	1	+	+	ADJ
ejpam-4883	457	2	h	h	NOUN
ejpam-4883	457	3	)	)	PUNCT
ejpam-4883	457	4	by	by	ADP
ejpam-4883	457	5	proposition	proposition	NOUN
ejpam-4883	457	6	1	1	NUM
ejpam-4883	457	7	.	.	PUNCT
ejpam-4883	457	8	thus	thus	ADV
ejpam-4883	457	9	,	,	PUNCT
ejpam-4883	457	10	max{γj(g	max{γj(g	NOUN
ejpam-4883	457	11	)	)	PUNCT
ejpam-4883	457	12	,	,	PUNCT
ejpam-4883	457	13	γj(h	γj(h	X
ejpam-4883	457	14	)	)	PUNCT
ejpam-4883	457	15	}	}	PUNCT
ejpam-4883	457	16	≤	≤	NOUN
ejpam-4883	457	17	γj(g	γj(g	PUNCT
ejpam-4883	458	1	+	+	NOUN
ejpam-4883	458	2	h	h	NOUN
ejpam-4883	458	3	)	)	PUNCT
ejpam-4883	458	4	.	.	PUNCT
ejpam-4883	459	1	since	since	SCONJ
ejpam-4883	459	2	g	g	PROPN
ejpam-4883	459	3	is	be	AUX
ejpam-4883	459	4	complete	complete	ADJ
ejpam-4883	459	5	,	,	PUNCT
ejpam-4883	459	6	we	we	PRON
ejpam-4883	459	7	have	have	VERB
ejpam-4883	459	8	γj(h	γj(h	NOUN
ejpam-4883	459	9	)	)	PUNCT
ejpam-4883	459	10	≤	≤	NOUN
ejpam-4883	459	11	γj(g	γj(g	X
ejpam-4883	460	1	+	+	NUM
ejpam-4883	460	2	h	h	NOUN
ejpam-4883	460	3	)	)	PUNCT
ejpam-4883	460	4	.	.	PUNCT
ejpam-4883	461	1	consequently	consequently	ADV
ejpam-4883	461	2	,	,	PUNCT
ejpam-4883	461	3	γj(g	γj(g	PROPN
ejpam-4883	462	1	+	+	NUM
ejpam-4883	462	2	h	h	X
ejpam-4883	462	3	)	)	PUNCT
ejpam-4883	462	4	=	=	SYM
ejpam-4883	462	5	γj(h	γj(h	ADJ
ejpam-4883	462	6	)	)	PUNCT
ejpam-4883	462	7	.	.	PUNCT
ejpam-4883	463	1	moreover	moreover	ADV
ejpam-4883	463	2	,	,	PUNCT
ejpam-4883	463	3	particular	particular	ADJ
ejpam-4883	463	4	cases	case	NOUN
ejpam-4883	463	5	follow	follow	VERB
ejpam-4883	463	6	from	from	ADP
ejpam-4883	463	7	theorem	theorem	ADJ
ejpam-4883	463	8	6	6	NUM
ejpam-4883	463	9	and	and	CCONJ
ejpam-4883	463	10	proposition	proposition	NOUN
ejpam-4883	463	11	2	2	NUM
ejpam-4883	463	12	.	.	NOUN
ejpam-4883	463	13	4	4	NUM
ejpam-4883	463	14	.	.	X
ejpam-4883	463	15	conclusion	conclusion	VERB
ejpam-4883	463	16	the	the	DET
ejpam-4883	463	17	concept	concept	NOUN
ejpam-4883	463	18	of	of	ADP
ejpam-4883	463	19	j	j	PROPN
ejpam-4883	463	20	-	-	PUNCT
ejpam-4883	463	21	domination	domination	NOUN
ejpam-4883	463	22	has	have	AUX
ejpam-4883	463	23	been	be	AUX
ejpam-4883	463	24	introduced	introduce	VERB
ejpam-4883	463	25	and	and	CCONJ
ejpam-4883	463	26	investigated	investigate	VERB
ejpam-4883	463	27	in	in	ADP
ejpam-4883	463	28	this	this	DET
ejpam-4883	463	29	study	study	NOUN
ejpam-4883	463	30	.	.	PUNCT
ejpam-4883	464	1	its	its	PRON
ejpam-4883	464	2	bounds	bound	NOUN
ejpam-4883	464	3	with	with	ADP
ejpam-4883	464	4	respect	respect	NOUN
ejpam-4883	464	5	to	to	ADP
ejpam-4883	464	6	the	the	DET
ejpam-4883	464	7	order	order	NOUN
ejpam-4883	464	8	of	of	ADP
ejpam-4883	464	9	a	a	DET
ejpam-4883	464	10	graph	graph	NOUN
ejpam-4883	464	11	and	and	CCONJ
ejpam-4883	464	12	other	other	ADJ
ejpam-4883	464	13	parameters	parameter	NOUN
ejpam-4883	464	14	have	have	AUX
ejpam-4883	464	15	been	be	AUX
ejpam-4883	464	16	determined	determine	VERB
ejpam-4883	464	17	.	.	PUNCT
ejpam-4883	465	1	it	it	PRON
ejpam-4883	465	2	was	be	AUX
ejpam-4883	465	3	shown	show	VERB
ejpam-4883	465	4	that	that	SCONJ
ejpam-4883	465	5	any	any	DET
ejpam-4883	465	6	graph	graph	NOUN
ejpam-4883	465	7	g	g	PROPN
ejpam-4883	465	8	admits	admit	VERB
ejpam-4883	465	9	a	a	DET
ejpam-4883	465	10	j	j	NOUN
ejpam-4883	465	11	-	-	PUNCT
ejpam-4883	465	12	domination	domination	NOUN
ejpam-4883	465	13	and	and	CCONJ
ejpam-4883	465	14	was	be	AUX
ejpam-4883	465	15	found	find	VERB
ejpam-4883	465	16	out	out	ADP
ejpam-4883	465	17	that	that	SCONJ
ejpam-4883	465	18	a	a	DET
ejpam-4883	465	19	graph	graph	NOUN
ejpam-4883	465	20	g	g	NOUN
ejpam-4883	465	21	satisfying	satisfy	VERB
ejpam-4883	465	22	γj(g	γj(g	PUNCT
ejpam-4883	465	23	)	)	PUNCT
ejpam-4883	466	1	=	=	SYM
ejpam-4883	466	2	|v	|v	PROPN
ejpam-4883	466	3	(	(	PUNCT
ejpam-4883	466	4	g)|	g)|	NOUN
ejpam-4883	466	5	is	be	AUX
ejpam-4883	466	6	not	not	PART
ejpam-4883	466	7	unique	unique	ADJ
ejpam-4883	466	8	.	.	PUNCT
ejpam-4883	467	1	moreover	moreover	ADV
ejpam-4883	467	2	,	,	PUNCT
ejpam-4883	467	3	characterizations	characterization	NOUN
ejpam-4883	467	4	of	of	ADP
ejpam-4883	467	5	j	j	PROPN
ejpam-4883	467	6	-	-	PUNCT
ejpam-4883	467	7	dominating	dominating	NOUN
ejpam-4883	467	8	sets	set	NOUN
ejpam-4883	467	9	in	in	ADP
ejpam-4883	467	10	some	some	DET
ejpam-4883	467	11	graphs	graph	NOUN
ejpam-4883	467	12	have	have	AUX
ejpam-4883	467	13	been	be	AUX
ejpam-4883	467	14	used	use	VERB
ejpam-4883	467	15	to	to	PART
ejpam-4883	467	16	determine	determine	VERB
ejpam-4883	467	17	exact	exact	ADJ
ejpam-4883	467	18	values	value	NOUN
ejpam-4883	467	19	of	of	ADP
ejpam-4883	467	20	parameter	parameter	NOUN
ejpam-4883	467	21	of	of	ADP
ejpam-4883	467	22	some	some	DET
ejpam-4883	467	23	graphs	graph	NOUN
ejpam-4883	467	24	.	.	PUNCT
ejpam-4883	468	1	some	some	DET
ejpam-4883	468	2	graphs	graph	NOUN
ejpam-4883	468	3	that	that	PRON
ejpam-4883	468	4	were	be	AUX
ejpam-4883	468	5	not	not	PART
ejpam-4883	468	6	considered	consider	VERB
ejpam-4883	468	7	in	in	ADP
ejpam-4883	468	8	this	this	DET
ejpam-4883	468	9	study	study	NOUN
ejpam-4883	468	10	could	could	AUX
ejpam-4883	468	11	be	be	AUX
ejpam-4883	468	12	an	an	DET
ejpam-4883	468	13	interesting	interesting	ADJ
ejpam-4883	468	14	to	to	PART
ejpam-4883	468	15	investigate	investigate	VERB
ejpam-4883	468	16	further	far	ADV
ejpam-4883	468	17	for	for	ADP
ejpam-4883	468	18	the	the	DET
ejpam-4883	468	19	concept	concept	NOUN
ejpam-4883	468	20	.	.	PUNCT
ejpam-4883	469	1	in	in	ADP
ejpam-4883	469	2	addition	addition	NOUN
ejpam-4883	469	3	,	,	PUNCT
ejpam-4883	469	4	researchers	researcher	NOUN
ejpam-4883	469	5	may	may	AUX
ejpam-4883	469	6	consider	consider	VERB
ejpam-4883	469	7	the	the	DET
ejpam-4883	469	8	complexity	complexity	NOUN
ejpam-4883	469	9	and	and	CCONJ
ejpam-4883	469	10	algorithm	algorithm	NOUN
ejpam-4883	469	11	,	,	PUNCT
ejpam-4883	469	12	and	and	CCONJ
ejpam-4883	469	13	real	real	ADJ
ejpam-4883	469	14	life	life	NOUN
ejpam-4883	469	15	application	application	NOUN
ejpam-4883	469	16	of	of	ADP
ejpam-4883	469	17	the	the	DET
ejpam-4883	469	18	concept	concept	NOUN
ejpam-4883	469	19	.	.	PUNCT
ejpam-4883	470	1	acknowledgements	acknowledgement	NOUN
ejpam-4883	470	2	the	the	DET
ejpam-4883	470	3	authors	author	NOUN
ejpam-4883	470	4	would	would	AUX
ejpam-4883	470	5	like	like	VERB
ejpam-4883	470	6	to	to	PART
ejpam-4883	470	7	thank	thank	VERB
ejpam-4883	470	8	mindanao	mindanao	PROPN
ejpam-4883	470	9	state	state	PROPN
ejpam-4883	470	10	university	university	PROPN
ejpam-4883	470	11	tawi	tawi	PROPN
ejpam-4883	470	12	-	-	PUNCT
ejpam-4883	470	13	tawi	tawi	PROPN
ejpam-4883	470	14	college	college	PROPN
ejpam-4883	470	15	of	of	ADP
ejpam-4883	470	16	technology	technology	NOUN
ejpam-4883	470	17	and	and	CCONJ
ejpam-4883	470	18	oceanography	oceanography	NOUN
ejpam-4883	470	19	for	for	ADP
ejpam-4883	470	20	funding	fund	VERB
ejpam-4883	470	21	this	this	DET
ejpam-4883	470	22	research	research	NOUN
ejpam-4883	470	23	.	.	PUNCT
ejpam-4883	471	1	references	reference	NOUN
ejpam-4883	471	2	[	[	X
ejpam-4883	471	3	1	1	NUM
ejpam-4883	471	4	]	]	X
ejpam-4883	471	5	c.e	c.e	PROPN
ejpam-4883	471	6	.	.	PROPN
ejpam-4883	471	7	adame	adame	PROPN
ejpam-4883	471	8	and	and	CCONJ
ejpam-4883	471	9	c.l	c.l	PROPN
ejpam-4883	471	10	.	.	PROPN
ejpam-4883	471	11	garita	garita	PROPN
ejpam-4883	471	12	.	.	PUNCT
ejpam-4883	472	1	total	total	ADJ
ejpam-4883	472	2	domination	domination	NOUN
ejpam-4883	472	3	on	on	ADP
ejpam-4883	472	4	some	some	DET
ejpam-4883	472	5	graph	graph	NOUN
ejpam-4883	472	6	operators	operator	NOUN
ejpam-4883	472	7	.	.	PUNCT
ejpam-4883	473	1	mathematics	mathematic	NOUN
ejpam-4883	473	2	,	,	PUNCT
ejpam-4883	473	3	9:1–9	9:1–9	NUM
ejpam-4883	473	4	,	,	PUNCT
ejpam-4883	473	5	2021	2021	NUM
ejpam-4883	473	6	.	.	PUNCT
ejpam-4883	474	1	references	reference	NOUN
ejpam-4883	474	2	2095	2095	NUM
ejpam-4883	474	3	[	[	X
ejpam-4883	474	4	2	2	NUM
ejpam-4883	474	5	]	]	X
ejpam-4883	474	6	r.c	r.c	PROPN
ejpam-4883	474	7	.	.	PROPN
ejpam-4883	474	8	brigham	brigham	PROPN
ejpam-4883	474	9	,	,	PUNCT
ejpam-4883	474	10	g.	g.	PROPN
ejpam-4883	474	11	chartrand	chartrand	PROPN
ejpam-4883	474	12	,	,	PUNCT
ejpam-4883	474	13	r.d	r.d	PROPN
ejpam-4883	474	14	.	.	PROPN
ejpam-4883	474	15	dutton	dutton	PROPN
ejpam-4883	474	16	,	,	PUNCT
ejpam-4883	474	17	and	and	CCONJ
ejpam-4883	474	18	p.	p.	PROPN
ejpam-4883	474	19	chang	chang	PROPN
ejpam-4883	474	20	.	.	PUNCT
ejpam-4883	475	1	resolving	resolve	VERB
ejpam-4883	475	2	domination	domination	NOUN
ejpam-4883	475	3	in	in	ADP
ejpam-4883	475	4	graphs	graph	NOUN
ejpam-4883	475	5	.	.	PUNCT
ejpam-4883	476	1	mathematica	mathematica	PROPN
ejpam-4883	476	2	bohemica	bohemica	PROPN
ejpam-4883	476	3	.	.	PUNCT
ejpam-4883	476	4	,	,	PUNCT
ejpam-4883	476	5	25(1):25–36	25(1):25–36	NUM
ejpam-4883	476	6	,	,	PUNCT
ejpam-4883	476	7	2003	2003	NUM
ejpam-4883	476	8	.	.	PUNCT
ejpam-4883	477	1	[	[	X
ejpam-4883	477	2	3	3	NUM
ejpam-4883	477	3	]	]	X
ejpam-4883	477	4	e.j	e.j	PROPN
ejpam-4883	477	5	.	.	PROPN
ejpam-4883	477	6	cockayne	cockayne	PROPN
ejpam-4883	477	7	and	and	CCONJ
ejpam-4883	477	8	s.t	s.t	PROPN
ejpam-4883	477	9	.	.	PROPN
ejpam-4883	477	10	hedetniemi	hedetniemi	PROPN
ejpam-4883	477	11	.	.	PUNCT
ejpam-4883	478	1	towards	towards	ADP
ejpam-4883	478	2	a	a	DET
ejpam-4883	478	3	theory	theory	NOUN
ejpam-4883	478	4	of	of	ADP
ejpam-4883	478	5	domination	domination	NOUN
ejpam-4883	478	6	in	in	ADP
ejpam-4883	478	7	graphs	graph	NOUN
ejpam-4883	478	8	.	.	PUNCT
ejpam-4883	479	1	networks	network	NOUN
ejpam-4883	479	2	,	,	PUNCT
ejpam-4883	479	3	7(3):247–261	7(3):247–261	NUM
ejpam-4883	479	4	,	,	PUNCT
ejpam-4883	479	5	1977	1977	NUM
ejpam-4883	479	6	.	.	PUNCT
ejpam-4883	480	1	[	[	X
ejpam-4883	480	2	4	4	X
ejpam-4883	480	3	]	]	PUNCT
ejpam-4883	480	4	b.	b.	PROPN
ejpam-4883	480	5	gayathri	gayathri	PROPN
ejpam-4883	480	6	and	and	CCONJ
ejpam-4883	480	7	s.	s.	PROPN
ejpam-4883	480	8	kaspar	kaspar	PROPN
ejpam-4883	480	9	.	.	PUNCT
ejpam-4883	481	1	connected	connect	VERB
ejpam-4883	481	2	co	co	ADJ
ejpam-4883	481	3	-	-	ADJ
ejpam-4883	481	4	independent	independent	ADJ
ejpam-4883	481	5	domination	domination	NOUN
ejpam-4883	481	6	of	of	ADP
ejpam-4883	481	7	a	a	DET
ejpam-4883	481	8	graph	graph	NOUN
ejpam-4883	481	9	.	.	PUNCT
ejpam-4883	482	1	int	int	NOUN
ejpam-4883	482	2	.	.	PUNCT
ejpam-4883	483	1	j.	j.	PROPN
ejpam-4883	483	2	contemp	contemp	PROPN
ejpam-4883	483	3	.	.	PUNCT
ejpam-4883	484	1	math	math	NOUN
ejpam-4883	484	2	.	.	PUNCT
ejpam-4883	485	1	sciences	science	NOUN
ejpam-4883	485	2	,	,	PUNCT
ejpam-4883	485	3	,	,	PUNCT
ejpam-4883	485	4	9(6):423–429	9(6):423–429	NUM
ejpam-4883	485	5	,	,	PUNCT
ejpam-4883	485	6	2011	2011	NUM
ejpam-4883	485	7	.	.	PUNCT
ejpam-4883	486	1	[	[	X
ejpam-4883	486	2	5	5	X
ejpam-4883	486	3	]	]	PUNCT
ejpam-4883	486	4	j.	j.	PROPN
ejpam-4883	486	5	hassan	hassan	PROPN
ejpam-4883	486	6	and	and	CCONJ
ejpam-4883	486	7	s.	s.	PROPN
ejpam-4883	486	8	canoy	canoy	PROPN
ejpam-4883	486	9	jr	jr	PROPN
ejpam-4883	486	10	.	.	PUNCT
ejpam-4883	487	1	grundy	grundy	PROPN
ejpam-4883	487	2	hop	hop	PROPN
ejpam-4883	487	3	domination	domination	PROPN
ejpam-4883	487	4	in	in	ADP
ejpam-4883	487	5	graphs	graph	NOUN
ejpam-4883	487	6	.	.	PUNCT
ejpam-4883	488	1	eur	eur	PROPN
ejpam-4883	488	2	.	.	PUNCT
ejpam-4883	489	1	j.	j.	PROPN
ejpam-4883	489	2	pure	pure	PROPN
ejpam-4883	489	3	appl	appl	PROPN
ejpam-4883	489	4	.	.	PUNCT
ejpam-4883	489	5	math	math	PROPN
ejpam-4883	489	6	.	.	PUNCT
ejpam-4883	489	7	,	,	PUNCT
ejpam-4883	489	8	15(4):1623–1636	15(4):1623–1636	NUM
ejpam-4883	489	9	,	,	PUNCT
ejpam-4883	489	10	2022	2022	NUM
ejpam-4883	489	11	.	.	PUNCT
ejpam-4883	490	1	[	[	X
ejpam-4883	490	2	6	6	NUM
ejpam-4883	490	3	]	]	PUNCT
ejpam-4883	490	4	j.	j.	PROPN
ejpam-4883	490	5	hassan	hassan	PROPN
ejpam-4883	490	6	and	and	CCONJ
ejpam-4883	490	7	s.	s.	PROPN
ejpam-4883	490	8	canoy	canoy	PROPN
ejpam-4883	490	9	jr	jr	PROPN
ejpam-4883	490	10	.	.	PROPN
ejpam-4883	490	11	connected	connect	VERB
ejpam-4883	490	12	grundy	grundy	PROPN
ejpam-4883	490	13	hop	hop	NOUN
ejpam-4883	490	14	dominating	dominate	VERB
ejpam-4883	490	15	sequences	sequence	NOUN
ejpam-4883	490	16	in	in	ADP
ejpam-4883	490	17	graphs	graph	NOUN
ejpam-4883	490	18	.	.	PUNCT
ejpam-4883	491	1	eur	eur	PROPN
ejpam-4883	491	2	.	.	PUNCT
ejpam-4883	492	1	j.	j.	PROPN
ejpam-4883	492	2	pure	pure	PROPN
ejpam-4883	492	3	appl	appl	PROPN
ejpam-4883	492	4	.	.	PUNCT
ejpam-4883	492	5	math	math	PROPN
ejpam-4883	492	6	.	.	PUNCT
ejpam-4883	492	7	,	,	PUNCT
ejpam-4883	493	1	16(2):1212–1227	16(2):1212–1227	NUM
ejpam-4883	493	2	,	,	PUNCT
ejpam-4883	493	3	2023	2023	NUM
ejpam-4883	493	4	.	.	PUNCT
ejpam-4883	494	1	[	[	X
ejpam-4883	494	2	7	7	X
ejpam-4883	494	3	]	]	PUNCT
ejpam-4883	494	4	j.	j.	PROPN
ejpam-4883	494	5	hassan	hassan	PROPN
ejpam-4883	494	6	,	,	PUNCT
ejpam-4883	494	7	s.	s.	PROPN
ejpam-4883	494	8	canoy	canoy	PROPN
ejpam-4883	494	9	jr	jr	PROPN
ejpam-4883	494	10	.	.	PROPN
ejpam-4883	494	11	,	,	PUNCT
ejpam-4883	494	12	and	and	CCONJ
ejpam-4883	494	13	chrisley	chrisley	PROPN
ejpam-4883	494	14	jade	jade	NOUN
ejpam-4883	494	15	saromines	saromine	NOUN
ejpam-4883	494	16	.	.	PUNCT
ejpam-4883	495	1	convex	convex	VERB
ejpam-4883	495	2	hop	hop	NOUN
ejpam-4883	495	3	domination	domination	NOUN
ejpam-4883	495	4	in	in	ADP
ejpam-4883	495	5	graphs	graph	NOUN
ejpam-4883	495	6	.	.	PUNCT
ejpam-4883	496	1	eur	eur	PROPN
ejpam-4883	496	2	.	.	PUNCT
ejpam-4883	497	1	j.	j.	PROPN
ejpam-4883	497	2	pure	pure	PROPN
ejpam-4883	497	3	appl	appl	PROPN
ejpam-4883	497	4	.	.	PUNCT
ejpam-4883	497	5	math	math	PROPN
ejpam-4883	497	6	.	.	PUNCT
ejpam-4883	497	7	,	,	PUNCT
ejpam-4883	497	8	16(1):319–335	16(1):319–335	NOUN
ejpam-4883	497	9	,	,	PUNCT
ejpam-4883	497	10	2023	2023	NUM
ejpam-4883	497	11	.	.	PUNCT
ejpam-4883	498	1	[	[	X
ejpam-4883	498	2	8	8	NUM
ejpam-4883	498	3	]	]	X
ejpam-4883	498	4	s.	s.	PROPN
ejpam-4883	498	5	canoy	canoy	PROPN
ejpam-4883	498	6	jr	jr	PROPN
ejpam-4883	498	7	and	and	CCONJ
ejpam-4883	498	8	j.	j.	PROPN
ejpam-4883	498	9	hassan	hassan	PROPN
ejpam-4883	498	10	.	.	PUNCT
ejpam-4883	499	1	weakly	weakly	ADJ
ejpam-4883	499	2	convex	convex	VERB
ejpam-4883	499	3	hop	hop	NOUN
ejpam-4883	499	4	dominating	dominating	NOUN
ejpam-4883	499	5	sets	set	NOUN
ejpam-4883	499	6	in	in	ADP
ejpam-4883	499	7	graphs	graph	NOUN
ejpam-4883	499	8	.	.	PUNCT
ejpam-4883	500	1	eur	eur	PROPN
ejpam-4883	500	2	.	.	PUNCT
ejpam-4883	501	1	j.	j.	PROPN
ejpam-4883	501	2	pure	pure	PROPN
ejpam-4883	501	3	appl	appl	PROPN
ejpam-4883	501	4	.	.	PUNCT
ejpam-4883	501	5	math	math	PROPN
ejpam-4883	501	6	.	.	PUNCT
ejpam-4883	501	7	,	,	PUNCT
ejpam-4883	501	8	16(2):1196–1211	16(2):1196–1211	NUM
ejpam-4883	501	9	,	,	PUNCT
ejpam-4883	501	10	2023	2023	NUM
ejpam-4883	501	11	.	.	PUNCT
ejpam-4883	502	1	[	[	X
ejpam-4883	502	2	9	9	NUM
ejpam-4883	502	3	]	]	PUNCT
ejpam-4883	502	4	m.	m.	NOUN
ejpam-4883	502	5	livingston	livingston	PROPN
ejpam-4883	502	6	and	and	CCONJ
ejpam-4883	502	7	q.f	q.f	PROPN
ejpam-4883	502	8	.	.	PROPN
ejpam-4883	502	9	stout	stout	PROPN
ejpam-4883	502	10	.	.	PUNCT
ejpam-4883	503	1	perfect	perfect	ADJ
ejpam-4883	503	2	dominating	dominating	NOUN
ejpam-4883	503	3	sets	set	NOUN
ejpam-4883	503	4	,	,	PUNCT
ejpam-4883	503	5	.	.	PUNCT
ejpam-4883	504	1	in	in	ADP
ejpam-4883	504	2	congressus	congressus	PROPN
ejpam-4883	504	3	numerantum	numerantum	PROPN
ejpam-4883	504	4	.	.	PUNCT
ejpam-4883	504	5	,	,	PUNCT
ejpam-4883	504	6	79:187–203	79:187–203	NUM
ejpam-4883	504	7	,	,	PUNCT
ejpam-4883	504	8	1990	1990	NUM
ejpam-4883	504	9	.	.	PUNCT
ejpam-4883	505	1	[	[	X
ejpam-4883	505	2	10	10	NUM
ejpam-4883	505	3	]	]	X
ejpam-4883	505	4	j.	j.	PROPN
ejpam-4883	505	5	manditong	manditong	PROPN
ejpam-4883	505	6	,	,	PUNCT
ejpam-4883	505	7	j.	j.	PROPN
ejpam-4883	505	8	hassan	hassan	PROPN
ejpam-4883	505	9	,	,	PUNCT
ejpam-4883	505	10	l.	l.	PROPN
ejpam-4883	505	11	laja	laja	PROPN
ejpam-4883	505	12	,	,	PUNCT
ejpam-4883	505	13	a.	a.	NOUN
ejpam-4883	505	14	laja	laja	PROPN
ejpam-4883	505	15	,	,	PUNCT
ejpam-4883	505	16	n.h	n.h	PROPN
ejpam-4883	505	17	.	.	PUNCT
ejpam-4883	505	18	mohammad	mohammad	PROPN
ejpam-4883	505	19	,	,	PUNCT
ejpam-4883	505	20	and	and	CCONJ
ejpam-4883	505	21	s.	s.	PROPN
ejpam-4883	505	22	kamdon	kamdon	PROPN
ejpam-4883	505	23	.	.	PUNCT
ejpam-4883	506	1	connected	connected	ADJ
ejpam-4883	506	2	outer	outer	ADJ
ejpam-4883	506	3	-	-	PUNCT
ejpam-4883	506	4	hop	hop	NOUN
ejpam-4883	506	5	independent	independent	ADJ
ejpam-4883	506	6	dominating	dominating	NOUN
ejpam-4883	506	7	sets	set	NOUN
ejpam-4883	506	8	in	in	ADP
ejpam-4883	506	9	graphs	graph	NOUN
ejpam-4883	506	10	under	under	ADP
ejpam-4883	506	11	some	some	DET
ejpam-4883	506	12	binary	binary	ADJ
ejpam-4883	506	13	operations	operation	NOUN
ejpam-4883	506	14	.	.	PUNCT
ejpam-4883	507	1	eur	eur	PROPN
ejpam-4883	507	2	.	.	PUNCT
ejpam-4883	508	1	j.	j.	PROPN
ejpam-4883	508	2	pure	pure	PROPN
ejpam-4883	508	3	appl	appl	PROPN
ejpam-4883	508	4	.	.	PUNCT
ejpam-4883	508	5	math	math	PROPN
ejpam-4883	508	6	.	.	PUNCT
ejpam-4883	508	7	,	,	PUNCT
ejpam-4883	509	1	16(3):1817–1829	16(3):1817–1829	NUM
ejpam-4883	509	2	,	,	PUNCT
ejpam-4883	509	3	2023	2023	NUM
ejpam-4883	509	4	.	.	PUNCT
ejpam-4883	510	1	[	[	X
ejpam-4883	510	2	11	11	NUM
ejpam-4883	510	3	]	]	X
ejpam-4883	510	4	c.	c.	PROPN
ejpam-4883	510	5	natarajan	natarajan	PROPN
ejpam-4883	510	6	and	and	CCONJ
ejpam-4883	510	7	s.	s.	PROPN
ejpam-4883	510	8	ayyaswamy	ayyaswamy	PROPN
ejpam-4883	510	9	.	.	PUNCT
ejpam-4883	511	1	hop	hop	PROPN
ejpam-4883	511	2	domination	domination	NOUN
ejpam-4883	511	3	in	in	ADP
ejpam-4883	511	4	graphs	graphs	PROPN
ejpam-4883	511	5	ii	ii	PROPN
ejpam-4883	511	6	.	.	PUNCT
ejpam-4883	511	7	versita	versita	PROPN
ejpam-4883	511	8	,	,	PUNCT
ejpam-4883	511	9	,	,	PUNCT
ejpam-4883	511	10	23(2):187	23(2):187	NUM
ejpam-4883	511	11	–	–	PUNCT
ejpam-4883	511	12	199	199	NUM
ejpam-4883	511	13	,	,	PUNCT
ejpam-4883	511	14	2015	2015	NUM
ejpam-4883	511	15	.	.	PUNCT
ejpam-4883	512	1	[	[	X
ejpam-4883	512	2	12	12	NUM
ejpam-4883	512	3	]	]	SYM
ejpam-4883	512	4	ore	ore	NOUN
ejpam-4883	512	5	o.	o.	NOUN
ejpam-4883	512	6	theory	theory	NOUN
ejpam-4883	512	7	of	of	ADP
ejpam-4883	512	8	graphs	graph	NOUN
ejpam-4883	512	9	,	,	PUNCT
ejpam-4883	512	10	.	.	PUNCT
ejpam-4883	513	1	amer	amer	PROPN
ejpam-4883	513	2	math	math	PROPN
ejpam-4883	513	3	.	.	PUNCT
ejpam-4883	514	1	soc	soc	PROPN
ejpam-4883	514	2	.	.	PUNCT
ejpam-4883	515	1	colloq	colloq	PROPN
ejpam-4883	515	2	.	.	PUNCT
ejpam-4883	516	1	publ	publ	PROPN
ejpam-4883	516	2	.	.	PUNCT
ejpam-4883	517	1	,	,	PUNCT
ejpam-4883	517	2	,	,	PUNCT
ejpam-4883	517	3	38(6	38(6	NOUN
ejpam-4883	517	4	)	)	PUNCT
ejpam-4883	517	5	,	,	PUNCT
ejpam-4883	517	6	1962	1962	NUM
ejpam-4883	517	7	.	.	PUNCT
ejpam-4883	518	1	[	[	X
ejpam-4883	518	2	13	13	NUM
ejpam-4883	518	3	]	]	X
ejpam-4883	518	4	e.	e.	PROPN
ejpam-4883	518	5	sampathkumar	sampathkumar	PROPN
ejpam-4883	518	6	and	and	CCONJ
ejpam-4883	518	7	h.b	h.b	PROPN
ejpam-4883	518	8	.	.	PROPN
ejpam-4883	518	9	walikar	walikar	PROPN
ejpam-4883	518	10	.	.	PUNCT
ejpam-4883	519	1	the	the	DET
ejpam-4883	519	2	connected	connected	ADJ
ejpam-4883	519	3	domination	domination	NOUN
ejpam-4883	519	4	number	number	NOUN
ejpam-4883	519	5	of	of	ADP
ejpam-4883	519	6	a	a	DET
ejpam-4883	519	7	graph	graph	NOUN
ejpam-4883	519	8	.	.	PUNCT
ejpam-4883	519	9	jour	jour	PROPN
ejpam-4883	519	10	.	.	PUNCT
ejpam-4883	519	11	math	math	PROPN
ejpam-4883	519	12	.	.	PUNCT
ejpam-4883	520	1	phy	phy	PROPN
ejpam-4883	520	2	.	.	PUNCT
ejpam-4883	521	1	sci	sci	PROPN
ejpam-4883	521	2	.	.	PROPN
ejpam-4883	521	3	,	,	PUNCT
ejpam-4883	521	4	13(6	13(6	PROPN
ejpam-4883	521	5	)	)	PUNCT
ejpam-4883	521	6	,	,	PUNCT
ejpam-4883	521	7	1979	1979	NUM
ejpam-4883	521	8	.	.	PUNCT
ejpam-4883	522	1	[	[	X
ejpam-4883	522	2	14	14	NUM
ejpam-4883	522	3	]	]	PUNCT
ejpam-4883	522	4	a.	a.	NOUN
ejpam-4883	522	5	sugumaran	sugumaran	NOUN
ejpam-4883	522	6	and	and	CCONJ
ejpam-4883	522	7	e.	e.	PROPN
ejpam-4883	522	8	jayachandran	jayachandran	PROPN
ejpam-4883	522	9	.	.	PUNCT
ejpam-4883	523	1	domination	domination	NOUN
ejpam-4883	523	2	number	number	NOUN
ejpam-4883	523	3	of	of	ADP
ejpam-4883	523	4	some	some	DET
ejpam-4883	523	5	graphs	graph	NOUN
ejpam-4883	523	6	.	.	PUNCT
ejpam-4883	524	1	int’l	int’l	NUM
ejpam-4883	524	2	.	.	PUNCT
ejpam-4883	524	3	jour	jour	AUX
ejpam-4883	524	4	.	.	PUNCT
ejpam-4883	524	5	scientific	scientific	ADJ
ejpam-4883	524	6	.	.	PUNCT
ejpam-4883	525	1	dev’t	dev’t	NOUN
ejpam-4883	525	2	and	and	CCONJ
ejpam-4883	525	3	research	research	NOUN
ejpam-4883	525	4	.	.	PUNCT
ejpam-4883	525	5	,	,	PUNCT
ejpam-4883	525	6	11(3):386–391	11(3):386–391	NUM
ejpam-4883	525	7	,	,	PUNCT
ejpam-4883	525	8	2018	2018	NUM
ejpam-4883	525	9	.	.	PUNCT