id	sid	tid	token	lemma	pos
ejpam-4905	1	1	european	european	PROPN
ejpam-4905	1	2	journal	journal	PROPN
ejpam-4905	1	3	of	of	ADP
ejpam-4905	1	4	pure	pure	ADJ
ejpam-4905	1	5	and	and	CCONJ
ejpam-4905	1	6	applied	apply	VERB
ejpam-4905	1	7	mathematics	mathematic	NOUN
ejpam-4905	1	8	vol	vol	NOUN
ejpam-4905	1	9	.	.	PUNCT
ejpam-4905	2	1	16	16	NUM
ejpam-4905	2	2	,	,	PUNCT
ejpam-4905	2	3	no	no	INTJ
ejpam-4905	2	4	.	.	NOUN
ejpam-4905	2	5	4	4	NUM
ejpam-4905	2	6	,	,	PUNCT
ejpam-4905	2	7	2023	2023	NUM
ejpam-4905	2	8	,	,	PUNCT
ejpam-4905	2	9	2118	2118	NUM
ejpam-4905	2	10	-	-	SYM
ejpam-4905	2	11	2131	2131	NUM
ejpam-4905	2	12	issn	issn	VERB
ejpam-4905	2	13	1307	1307	NUM
ejpam-4905	2	14	-	-	SYM
ejpam-4905	2	15	5543	5543	NUM
ejpam-4905	2	16	–	–	PUNCT
ejpam-4905	2	17	ejpam.com	ejpam.com	X
ejpam-4905	2	18	published	publish	VERB
ejpam-4905	2	19	by	by	ADP
ejpam-4905	2	20	new	new	PROPN
ejpam-4905	2	21	york	york	PROPN
ejpam-4905	2	22	business	business	PROPN
ejpam-4905	2	23	global	global	PROPN
ejpam-4905	2	24	j2hop	j2hop	PROPN
ejpam-4905	2	25	domination	domination	NOUN
ejpam-4905	2	26	in	in	ADP
ejpam-4905	2	27	graphs	graph	NOUN
ejpam-4905	2	28	:	:	PUNCT
ejpam-4905	2	29	properties	property	NOUN
ejpam-4905	2	30	and	and	CCONJ
ejpam-4905	2	31	connections	connection	NOUN
ejpam-4905	2	32	with	with	ADP
ejpam-4905	2	33	other	other	ADJ
ejpam-4905	2	34	parameters	parameter	NOUN
ejpam-4905	2	35	javier	javier	PROPN
ejpam-4905	2	36	a.	a.	PROPN
ejpam-4905	2	37	hassan1,∗	hassan1,∗	PROPN
ejpam-4905	2	38	,	,	PUNCT
ejpam-4905	2	39	alcyn	alcyn	PROPN
ejpam-4905	2	40	r.	r.	PROPN
ejpam-4905	2	41	bakkang2	bakkang2	PROPN
ejpam-4905	2	42	,	,	PUNCT
ejpam-4905	2	43	amil	amil	NOUN
ejpam-4905	2	44	-	-	PUNCT
ejpam-4905	2	45	shab	shab	PROPN
ejpam-4905	2	46	s.	s.	PROPN
ejpam-4905	2	47	sappari1	sappari1	PROPN
ejpam-4905	3	1	1mathematics	1mathematics	NUM
ejpam-4905	3	2	and	and	CCONJ
ejpam-4905	3	3	sciences	sciences	PROPN
ejpam-4905	3	4	department	department	PROPN
ejpam-4905	3	5	,	,	PUNCT
ejpam-4905	3	6	college	college	NOUN
ejpam-4905	3	7	of	of	ADP
ejpam-4905	3	8	arts	art	NOUN
ejpam-4905	3	9	and	and	CCONJ
ejpam-4905	3	10	sciences	science	NOUN
ejpam-4905	3	11	,	,	PUNCT
ejpam-4905	3	12	msu	msu	PROPN
ejpam-4905	3	13	tawi	tawi	PROPN
ejpam-4905	3	14	-	-	PUNCT
ejpam-4905	3	15	tawi	tawi	PROPN
ejpam-4905	3	16	college	college	PROPN
ejpam-4905	3	17	of	of	ADP
ejpam-4905	3	18	technology	technology	NOUN
ejpam-4905	3	19	and	and	CCONJ
ejpam-4905	3	20	oceanography	oceanography	NOUN
ejpam-4905	3	21	,	,	PUNCT
ejpam-4905	3	22	bongao	bongao	NOUN
ejpam-4905	3	23	,	,	PUNCT
ejpam-4905	3	24	tawi	tawi	NOUN
ejpam-4905	3	25	-	-	PUNCT
ejpam-4905	3	26	tawi	tawi	NOUN
ejpam-4905	3	27	,	,	PUNCT
ejpam-4905	3	28	philippines	philippine	NOUN
ejpam-4905	3	29	2	2	NUM
ejpam-4905	3	30	secondary	secondary	ADJ
ejpam-4905	3	31	education	education	NOUN
ejpam-4905	3	32	department	department	NOUN
ejpam-4905	3	33	,	,	PUNCT
ejpam-4905	3	34	college	college	NOUN
ejpam-4905	3	35	of	of	ADP
ejpam-4905	3	36	education	education	NOUN
ejpam-4905	3	37	,	,	PUNCT
ejpam-4905	3	38	msu	msu	PROPN
ejpam-4905	3	39	tawi	tawi	PROPN
ejpam-4905	3	40	-	-	PUNCT
ejpam-4905	3	41	tawi	tawi	PROPN
ejpam-4905	3	42	college	college	PROPN
ejpam-4905	3	43	of	of	ADP
ejpam-4905	3	44	technology	technology	NOUN
ejpam-4905	3	45	and	and	CCONJ
ejpam-4905	3	46	oceanography	oceanography	NOUN
ejpam-4905	3	47	,	,	PUNCT
ejpam-4905	3	48	bongao	bongao	NOUN
ejpam-4905	3	49	,	,	PUNCT
ejpam-4905	3	50	tawi	tawi	NOUN
ejpam-4905	3	51	-	-	PUNCT
ejpam-4905	3	52	tawi	tawi	NOUN
ejpam-4905	3	53	,	,	PUNCT
ejpam-4905	3	54	philippines	philippine	NOUN
ejpam-4905	3	55	abstract	abstract	ADJ
ejpam-4905	3	56	.	.	PUNCT
ejpam-4905	4	1	a	a	DET
ejpam-4905	4	2	subset	subset	NOUN
ejpam-4905	4	3	t	t	NOUN
ejpam-4905	4	4	=	=	SYM
ejpam-4905	4	5	{	{	PUNCT
ejpam-4905	4	6	v1	v1	PROPN
ejpam-4905	4	7	,	,	PUNCT
ejpam-4905	4	8	v2	v2	PROPN
ejpam-4905	4	9	,	,	PUNCT
ejpam-4905	4	10	·	·	PUNCT
ejpam-4905	4	11	·	·	PUNCT
ejpam-4905	4	12	·	·	PUNCT
ejpam-4905	4	13	,	,	PUNCT
ejpam-4905	4	14	vm	vm	NOUN
ejpam-4905	4	15	}	}	PUNCT
ejpam-4905	4	16	of	of	ADP
ejpam-4905	4	17	vertices	vertex	NOUN
ejpam-4905	4	18	of	of	ADP
ejpam-4905	4	19	a	a	DET
ejpam-4905	4	20	graph	graph	NOUN
ejpam-4905	4	21	g	g	NOUN
ejpam-4905	4	22	is	be	AUX
ejpam-4905	4	23	called	call	VERB
ejpam-4905	4	24	a	a	DET
ejpam-4905	4	25	j2	j2	PROPN
ejpam-4905	4	26	-	-	PUNCT
ejpam-4905	4	27	set	set	VERB
ejpam-4905	4	28	if	if	SCONJ
ejpam-4905	4	29	n2	n2	ADJ
ejpam-4905	4	30	g[vi	g[vi	PROPN
ejpam-4905	4	31	]	]	PUNCT
ejpam-4905	4	32	\	\	PROPN
ejpam-4905	4	33	n2	n2	PROPN
ejpam-4905	4	34	g[vj	g[vj	PROPN
ejpam-4905	4	35	]	]	PUNCT
ejpam-4905	4	36	̸=	̸=	PROPN
ejpam-4905	4	37	∅	∅	NOUN
ejpam-4905	4	38	for	for	ADP
ejpam-4905	4	39	every	every	DET
ejpam-4905	4	40	i	i	PROPN
ejpam-4905	4	41	̸=	̸=	PROPN
ejpam-4905	4	42	j	j	PROPN
ejpam-4905	4	43	,	,	PUNCT
ejpam-4905	4	44	where	where	SCONJ
ejpam-4905	4	45	i	i	PRON
ejpam-4905	4	46	,	,	PUNCT
ejpam-4905	4	47	j	j	PROPN
ejpam-4905	4	48	∈	∈	PROPN
ejpam-4905	4	49	{	{	PUNCT
ejpam-4905	4	50	1	1	NUM
ejpam-4905	4	51	,	,	PUNCT
ejpam-4905	4	52	2	2	NUM
ejpam-4905	4	53	,	,	PUNCT
ejpam-4905	4	54	.	.	PUNCT
ejpam-4905	4	55	.	.	PUNCT
ejpam-4905	5	1	.	.	PUNCT
ejpam-4905	6	1	,	,	PUNCT
ejpam-4905	6	2	m	m	VERB
ejpam-4905	6	3	}	}	PUNCT
ejpam-4905	6	4	.	.	PUNCT
ejpam-4905	7	1	a	a	DET
ejpam-4905	7	2	j2	j2	PROPN
ejpam-4905	7	3	-	-	PUNCT
ejpam-4905	7	4	set	set	VERB
ejpam-4905	7	5	t	t	PROPN
ejpam-4905	7	6	is	be	AUX
ejpam-4905	7	7	called	call	VERB
ejpam-4905	7	8	a	a	DET
ejpam-4905	7	9	j2	j2	PROPN
ejpam-4905	7	10	-	-	PUNCT
ejpam-4905	7	11	hop	hop	NOUN
ejpam-4905	7	12	dominating	dominating	NOUN
ejpam-4905	7	13	in	in	ADP
ejpam-4905	7	14	g	g	PROPN
ejpam-4905	7	15	if	if	SCONJ
ejpam-4905	7	16	for	for	ADP
ejpam-4905	7	17	every	every	DET
ejpam-4905	7	18	a	a	DET
ejpam-4905	7	19	∈	∈	PROPN
ejpam-4905	7	20	v	v	NOUN
ejpam-4905	7	21	(	(	PUNCT
ejpam-4905	7	22	g	g	NOUN
ejpam-4905	7	23	)	)	PUNCT
ejpam-4905	7	24	\	\	PROPN
ejpam-4905	7	25	t	t	NOUN
ejpam-4905	7	26	,	,	PUNCT
ejpam-4905	7	27	there	there	PRON
ejpam-4905	7	28	exists	exist	VERB
ejpam-4905	7	29	b	b	PROPN
ejpam-4905	7	30	∈	∈	PROPN
ejpam-4905	7	31	t	t	NOUN
ejpam-4905	7	32	such	such	ADJ
ejpam-4905	7	33	that	that	PRON
ejpam-4905	7	34	dg(a	dg(a	PROPN
ejpam-4905	7	35	,	,	PUNCT
ejpam-4905	7	36	b	b	X
ejpam-4905	7	37	)	)	PUNCT
ejpam-4905	8	1	=	=	SYM
ejpam-4905	8	2	2	2	X
ejpam-4905	8	3	.	.	X
ejpam-4905	9	1	the	the	DET
ejpam-4905	9	2	j2	j2	PROPN
ejpam-4905	9	3	-	-	PUNCT
ejpam-4905	9	4	hop	hop	PROPN
ejpam-4905	9	5	domination	domination	NOUN
ejpam-4905	9	6	number	number	NOUN
ejpam-4905	9	7	of	of	ADP
ejpam-4905	9	8	g	g	NOUN
ejpam-4905	9	9	,	,	PUNCT
ejpam-4905	9	10	denoted	denote	VERB
ejpam-4905	9	11	by	by	ADP
ejpam-4905	9	12	γj2h(g	γj2h(g	NOUN
ejpam-4905	9	13	)	)	PUNCT
ejpam-4905	9	14	,	,	PUNCT
ejpam-4905	9	15	is	be	AUX
ejpam-4905	9	16	the	the	DET
ejpam-4905	9	17	maximum	maximum	ADJ
ejpam-4905	9	18	cardinality	cardinality	NOUN
ejpam-4905	9	19	among	among	ADP
ejpam-4905	9	20	all	all	DET
ejpam-4905	9	21	j2	j2	PROPN
ejpam-4905	9	22	-	-	PUNCT
ejpam-4905	9	23	hop	hop	NOUN
ejpam-4905	9	24	dominating	dominating	NOUN
ejpam-4905	9	25	sets	set	NOUN
ejpam-4905	9	26	in	in	ADP
ejpam-4905	9	27	g.	g.	PROPN
ejpam-4905	9	28	in	in	ADP
ejpam-4905	9	29	this	this	DET
ejpam-4905	9	30	paper	paper	NOUN
ejpam-4905	9	31	,	,	PUNCT
ejpam-4905	9	32	we	we	PRON
ejpam-4905	9	33	initiate	initiate	VERB
ejpam-4905	9	34	the	the	DET
ejpam-4905	9	35	study	study	NOUN
ejpam-4905	9	36	on	on	ADP
ejpam-4905	9	37	j2	j2	PROPN
ejpam-4905	9	38	-	-	PUNCT
ejpam-4905	9	39	hop	hop	NOUN
ejpam-4905	9	40	domination	domination	NOUN
ejpam-4905	9	41	and	and	CCONJ
ejpam-4905	9	42	we	we	PRON
ejpam-4905	9	43	establish	establish	VERB
ejpam-4905	9	44	its	its	PRON
ejpam-4905	9	45	properties	property	NOUN
ejpam-4905	9	46	and	and	CCONJ
ejpam-4905	9	47	connections	connection	NOUN
ejpam-4905	9	48	with	with	ADP
ejpam-4905	9	49	other	other	ADJ
ejpam-4905	9	50	known	know	VERB
ejpam-4905	9	51	parameters	parameter	NOUN
ejpam-4905	9	52	in	in	ADP
ejpam-4905	9	53	graph	graph	NOUN
ejpam-4905	9	54	theory	theory	NOUN
ejpam-4905	9	55	.	.	PUNCT
ejpam-4905	10	1	we	we	PRON
ejpam-4905	10	2	show	show	VERB
ejpam-4905	10	3	that	that	SCONJ
ejpam-4905	10	4	every	every	DET
ejpam-4905	10	5	maximum	maximum	ADJ
ejpam-4905	10	6	hop	hop	NOUN
ejpam-4905	10	7	independent	independent	ADJ
ejpam-4905	10	8	set	set	NOUN
ejpam-4905	10	9	is	be	AUX
ejpam-4905	10	10	a	a	DET
ejpam-4905	10	11	j2	j2	PROPN
ejpam-4905	10	12	-	-	PUNCT
ejpam-4905	10	13	hop	hop	NOUN
ejpam-4905	10	14	dominating	dominating	NOUN
ejpam-4905	10	15	,	,	PUNCT
ejpam-4905	10	16	hence	hence	ADV
ejpam-4905	10	17	,	,	PUNCT
ejpam-4905	10	18	this	this	DET
ejpam-4905	10	19	parameter	parameter	NOUN
ejpam-4905	10	20	is	be	AUX
ejpam-4905	10	21	greater	great	ADJ
ejpam-4905	10	22	than	than	SCONJ
ejpam-4905	10	23	compare	compare	VERB
ejpam-4905	10	24	to	to	ADP
ejpam-4905	10	25	the	the	DET
ejpam-4905	10	26	hop	hop	NOUN
ejpam-4905	10	27	independence	independence	NOUN
ejpam-4905	10	28	parameter	parameter	NOUN
ejpam-4905	10	29	on	on	ADP
ejpam-4905	10	30	any	any	DET
ejpam-4905	10	31	graph	graph	NOUN
ejpam-4905	10	32	.	.	PUNCT
ejpam-4905	11	1	moreover	moreover	ADV
ejpam-4905	11	2	,	,	PUNCT
ejpam-4905	11	3	we	we	PRON
ejpam-4905	11	4	derive	derive	VERB
ejpam-4905	11	5	some	some	DET
ejpam-4905	11	6	lower	low	ADJ
ejpam-4905	11	7	and	and	CCONJ
ejpam-4905	11	8	upper	upper	ADJ
ejpam-4905	11	9	bounds	bound	NOUN
ejpam-4905	11	10	of	of	ADP
ejpam-4905	11	11	the	the	DET
ejpam-4905	11	12	parameter	parameter	NOUN
ejpam-4905	11	13	for	for	ADP
ejpam-4905	11	14	a	a	DET
ejpam-4905	11	15	generalized	generalized	ADJ
ejpam-4905	11	16	graph	graph	NOUN
ejpam-4905	11	17	,	,	PUNCT
ejpam-4905	11	18	join	join	VERB
ejpam-4905	11	19	and	and	CCONJ
ejpam-4905	11	20	corona	corona	NOUN
ejpam-4905	11	21	of	of	ADP
ejpam-4905	11	22	two	two	NUM
ejpam-4905	11	23	graphs	graph	NOUN
ejpam-4905	11	24	,	,	PUNCT
ejpam-4905	11	25	respectively	respectively	ADV
ejpam-4905	11	26	.	.	PUNCT
ejpam-4905	12	1	finally	finally	ADV
ejpam-4905	12	2	,	,	PUNCT
ejpam-4905	12	3	we	we	PRON
ejpam-4905	12	4	obtain	obtain	VERB
ejpam-4905	12	5	exact	exact	ADJ
ejpam-4905	12	6	values	value	NOUN
ejpam-4905	12	7	of	of	ADP
ejpam-4905	12	8	the	the	DET
ejpam-4905	12	9	parameter	parameter	NOUN
ejpam-4905	12	10	for	for	ADP
ejpam-4905	12	11	some	some	DET
ejpam-4905	12	12	special	special	ADJ
ejpam-4905	12	13	graphs	graph	NOUN
ejpam-4905	12	14	and	and	CCONJ
ejpam-4905	12	15	shadow	shadow	NOUN
ejpam-4905	12	16	graph	graph	NOUN
ejpam-4905	12	17	using	use	VERB
ejpam-4905	12	18	the	the	DET
ejpam-4905	12	19	characterization	characterization	NOUN
ejpam-4905	12	20	results	result	NOUN
ejpam-4905	12	21	that	that	PRON
ejpam-4905	12	22	are	be	AUX
ejpam-4905	12	23	formulated	formulate	VERB
ejpam-4905	12	24	in	in	ADP
ejpam-4905	12	25	this	this	DET
ejpam-4905	12	26	study	study	NOUN
ejpam-4905	12	27	.	.	PUNCT
ejpam-4905	13	1	2020	2020	NUM
ejpam-4905	13	2	mathematics	mathematic	NOUN
ejpam-4905	13	3	subject	subject	NOUN
ejpam-4905	13	4	classifications	classification	NOUN
ejpam-4905	13	5	:	:	PUNCT
ejpam-4905	13	6	05c69	05c69	X
ejpam-4905	13	7	key	key	ADJ
ejpam-4905	13	8	words	word	NOUN
ejpam-4905	13	9	and	and	CCONJ
ejpam-4905	13	10	phrases	phrase	NOUN
ejpam-4905	13	11	:	:	PUNCT
ejpam-4905	13	12	j2	j2	PROPN
ejpam-4905	13	13	-	-	PUNCT
ejpam-4905	13	14	set	set	PROPN
ejpam-4905	13	15	,	,	PUNCT
ejpam-4905	13	16	j2	j2	PROPN
ejpam-4905	13	17	-	-	PUNCT
ejpam-4905	13	18	hop	hop	PROPN
ejpam-4905	13	19	dominating	dominating	NOUN
ejpam-4905	13	20	set	set	NOUN
ejpam-4905	13	21	,	,	PUNCT
ejpam-4905	13	22	j2	j2	PROPN
ejpam-4905	13	23	-	-	PUNCT
ejpam-4905	13	24	hop	hop	PROPN
ejpam-4905	13	25	domination	domination	NOUN
ejpam-4905	13	26	number	number	NOUN
ejpam-4905	13	27	1	1	NUM
ejpam-4905	13	28	.	.	PUNCT
ejpam-4905	14	1	introduction	introduction	NOUN
ejpam-4905	14	2	hop	hop	PROPN
ejpam-4905	14	3	domination	domination	NOUN
ejpam-4905	14	4	was	be	AUX
ejpam-4905	14	5	introduced	introduce	VERB
ejpam-4905	14	6	by	by	ADP
ejpam-4905	14	7	natarajan	natarajan	PROPN
ejpam-4905	14	8	et	et	PROPN
ejpam-4905	14	9	al	al	PROPN
ejpam-4905	14	10	.	.	PUNCT
ejpam-4905	15	1	in	in	ADP
ejpam-4905	15	2	[	[	X
ejpam-4905	15	3	9	9	NUM
ejpam-4905	15	4	]	]	PUNCT
ejpam-4905	15	5	.	.	PUNCT
ejpam-4905	16	1	a	a	DET
ejpam-4905	16	2	subset	subset	NOUN
ejpam-4905	16	3	s	s	NOUN
ejpam-4905	16	4	of	of	ADP
ejpam-4905	16	5	a	a	DET
ejpam-4905	16	6	vertices	vertex	NOUN
ejpam-4905	16	7	of	of	ADP
ejpam-4905	16	8	a	a	DET
ejpam-4905	16	9	graph	graph	NOUN
ejpam-4905	16	10	g	g	NOUN
ejpam-4905	16	11	is	be	AUX
ejpam-4905	16	12	called	call	VERB
ejpam-4905	16	13	a	a	DET
ejpam-4905	16	14	hop	hop	NOUN
ejpam-4905	16	15	dominating	dominating	NOUN
ejpam-4905	16	16	if	if	SCONJ
ejpam-4905	16	17	for	for	ADP
ejpam-4905	16	18	every	every	DET
ejpam-4905	16	19	a	a	DET
ejpam-4905	16	20	∈	∈	PROPN
ejpam-4905	16	21	v	v	NOUN
ejpam-4905	16	22	(	(	PUNCT
ejpam-4905	16	23	g)\s	g)\s	NOUN
ejpam-4905	16	24	,	,	PUNCT
ejpam-4905	16	25	there	there	PRON
ejpam-4905	16	26	exists	exist	VERB
ejpam-4905	16	27	b	b	PROPN
ejpam-4905	16	28	∈	∈	PROPN
ejpam-4905	16	29	s	s	VERB
ejpam-4905	16	30	such	such	ADJ
ejpam-4905	16	31	that	that	SCONJ
ejpam-4905	16	32	dg(a	dg(a	PROPN
ejpam-4905	16	33	,	,	PUNCT
ejpam-4905	16	34	b	b	X
ejpam-4905	16	35	)	)	PUNCT
ejpam-4905	16	36	=	=	SYM
ejpam-4905	16	37	2	2	X
ejpam-4905	16	38	.	.	PUNCT
ejpam-4905	16	39	the	the	DET
ejpam-4905	16	40	minimum	minimum	ADJ
ejpam-4905	16	41	cardinality	cardinality	NOUN
ejpam-4905	16	42	among	among	ADP
ejpam-4905	16	43	all	all	DET
ejpam-4905	16	44	hop	hop	NOUN
ejpam-4905	16	45	dominating	dominating	NOUN
ejpam-4905	16	46	sets	set	NOUN
ejpam-4905	16	47	of	of	ADP
ejpam-4905	16	48	g	g	NOUN
ejpam-4905	16	49	,	,	PUNCT
ejpam-4905	16	50	denoted	denote	VERB
ejpam-4905	16	51	by	by	ADP
ejpam-4905	16	52	γh(g	γh(g	NOUN
ejpam-4905	16	53	)	)	PUNCT
ejpam-4905	16	54	,	,	PUNCT
ejpam-4905	16	55	is	be	AUX
ejpam-4905	16	56	called	call	VERB
ejpam-4905	16	57	the	the	DET
ejpam-4905	16	58	hop	hop	NOUN
ejpam-4905	16	59	domination	domination	NOUN
ejpam-4905	16	60	number	number	NOUN
ejpam-4905	16	61	of	of	ADP
ejpam-4905	16	62	g.	g.	PROPN
ejpam-4905	16	63	this	this	DET
ejpam-4905	16	64	parameter	parameter	NOUN
ejpam-4905	16	65	had	have	AUX
ejpam-4905	16	66	studied	study	VERB
ejpam-4905	16	67	on	on	ADP
ejpam-4905	16	68	some	some	DET
ejpam-4905	16	69	families	family	NOUN
ejpam-4905	16	70	of	of	ADP
ejpam-4905	16	71	graphs	graph	NOUN
ejpam-4905	16	72	and	and	CCONJ
ejpam-4905	16	73	graphs	graph	NOUN
ejpam-4905	16	74	obtained	obtain	VERB
ejpam-4905	16	75	from	from	ADP
ejpam-4905	16	76	some	some	DET
ejpam-4905	16	77	operations	operation	NOUN
ejpam-4905	16	78	in	in	ADP
ejpam-4905	16	79	[	[	X
ejpam-4905	16	80	1	1	NUM
ejpam-4905	16	81	,	,	PUNCT
ejpam-4905	16	82	2	2	NUM
ejpam-4905	16	83	,	,	PUNCT
ejpam-4905	16	84	9	9	NUM
ejpam-4905	16	85	]	]	PUNCT
ejpam-4905	16	86	.	.	PUNCT
ejpam-4905	17	1	researchers	researcher	NOUN
ejpam-4905	17	2	in	in	ADP
ejpam-4905	17	3	the	the	DET
ejpam-4905	17	4	field	field	NOUN
ejpam-4905	17	5	had	have	AUX
ejpam-4905	17	6	further	far	ADV
ejpam-4905	17	7	investigated	investigate	VERB
ejpam-4905	17	8	this	this	DET
ejpam-4905	17	9	concept	concept	NOUN
ejpam-4905	17	10	,	,	PUNCT
ejpam-4905	17	11	and	and	CCONJ
ejpam-4905	17	12	introduced	introduce	VERB
ejpam-4905	17	13	new	new	ADJ
ejpam-4905	17	14	variants	variant	NOUN
ejpam-4905	17	15	and	and	CCONJ
ejpam-4905	17	16	obtained	obtain	VERB
ejpam-4905	17	17	some	some	DET
ejpam-4905	17	18	significant	significant	ADJ
ejpam-4905	17	19	results	result	NOUN
ejpam-4905	17	20	that	that	PRON
ejpam-4905	17	21	contributed	contribute	VERB
ejpam-4905	17	22	a	a	DET
ejpam-4905	17	23	lot	lot	NOUN
ejpam-4905	17	24	to	to	ADP
ejpam-4905	17	25	the	the	DET
ejpam-4905	17	26	hop	hop	NOUN
ejpam-4905	17	27	domination	domination	NOUN
ejpam-4905	17	28	theory	theory	NOUN
ejpam-4905	17	29	(	(	PUNCT
ejpam-4905	17	30	see	see	VERB
ejpam-4905	17	31	[	[	X
ejpam-4905	17	32	3–8	3–8	NUM
ejpam-4905	17	33	,	,	PUNCT
ejpam-4905	17	34	10–12	10–12	NUM
ejpam-4905	17	35	]	]	PUNCT
ejpam-4905	17	36	)	)	PUNCT
ejpam-4905	17	37	.	.	PUNCT
ejpam-4905	18	1	in	in	ADP
ejpam-4905	18	2	this	this	DET
ejpam-4905	18	3	paper	paper	NOUN
ejpam-4905	18	4	,	,	PUNCT
ejpam-4905	18	5	new	new	ADJ
ejpam-4905	18	6	parameter	parameter	NOUN
ejpam-4905	18	7	called	call	VERB
ejpam-4905	18	8	j2	j2	PROPN
ejpam-4905	18	9	-	-	PUNCT
ejpam-4905	18	10	hop	hop	NOUN
ejpam-4905	18	11	domination	domination	NOUN
ejpam-4905	18	12	in	in	ADP
ejpam-4905	18	13	a	a	DET
ejpam-4905	18	14	graph	graph	NOUN
ejpam-4905	18	15	will	will	AUX
ejpam-4905	18	16	be	be	AUX
ejpam-4905	18	17	introduced	introduce	VERB
ejpam-4905	18	18	and	and	CCONJ
ejpam-4905	18	19	investigated	investigate	VERB
ejpam-4905	18	20	.	.	PUNCT
ejpam-4905	19	1	we	we	PRON
ejpam-4905	19	2	will	will	AUX
ejpam-4905	19	3	establish	establish	VERB
ejpam-4905	19	4	its	its	PRON
ejpam-4905	19	5	relationships	relationship	NOUN
ejpam-4905	19	6	with	with	ADP
ejpam-4905	19	7	other	other	ADJ
ejpam-4905	19	8	known	know	VERB
ejpam-4905	19	9	parameters	parameter	NOUN
ejpam-4905	19	10	in	in	ADP
ejpam-4905	19	11	graph	graph	NOUN
ejpam-4905	19	12	doi	doi	NOUN
ejpam-4905	19	13	:	:	PUNCT
ejpam-4905	19	14	https://doi.org/10.29020/nybg.ejpam.v16i4.4905	https://doi.org/10.29020/nybg.ejpam.v16i4.4905	ADJ
ejpam-4905	19	15	email	email	NOUN
ejpam-4905	19	16	addresses	address	NOUN
ejpam-4905	19	17	:	:	PUNCT
ejpam-4905	19	18	javierhassan@msutawi-tawi.edu.ph	javierhassan@msutawi-tawi.edu.ph	PROPN
ejpam-4905	19	19	(	(	PUNCT
ejpam-4905	19	20	j.	j.	PROPN
ejpam-4905	19	21	hassan	hassan	PROPN
ejpam-4905	19	22	)	)	PUNCT
ejpam-4905	20	1	alcynbakkang@msutawi-tawi.edu.ph	alcynbakkang@msutawi-tawi.edu.ph	PROPN
ejpam-4905	20	2	(	(	PUNCT
ejpam-4905	20	3	a.	a.	PROPN
ejpam-4905	20	4	bakkang	bakkang	PROPN
ejpam-4905	20	5	)	)	PUNCT
ejpam-4905	20	6	amilshabsappari@msutawi-tawi.edu.ph	amilshabsappari@msutawi-tawi.edu.ph	PROPN
ejpam-4905	20	7	(	(	PUNCT
ejpam-4905	20	8	a.	a.	NOUN
ejpam-4905	20	9	sappari	sappari	PROPN
ejpam-4905	20	10	)	)	PUNCT
ejpam-4905	20	11	https://www.ejpam.com	https://www.ejpam.com	X
ejpam-4905	21	1	2118	2118	NUM
ejpam-4905	21	2	©	©	PROPN
ejpam-4905	21	3	2023	2023	NUM
ejpam-4905	21	4	ejpam	ejpam	NOUN
ejpam-4905	21	5	all	all	DET
ejpam-4905	21	6	rights	right	NOUN
ejpam-4905	21	7	reserved	reserve	VERB
ejpam-4905	21	8	.	.	PUNCT
ejpam-4905	22	1	j.	j.	PROPN
ejpam-4905	22	2	hassan	hassan	PROPN
ejpam-4905	22	3	,	,	PUNCT
ejpam-4905	22	4	a.	a.	PROPN
ejpam-4905	22	5	bakkang	bakkang	PROPN
ejpam-4905	22	6	,	,	PUNCT
ejpam-4905	22	7	a.	a.	NOUN
ejpam-4905	22	8	sappari	sappari	PROPN
ejpam-4905	22	9	/	/	SYM
ejpam-4905	22	10	eur	eur	PROPN
ejpam-4905	22	11	.	.	PUNCT
ejpam-4905	23	1	j.	j.	PROPN
ejpam-4905	23	2	pure	pure	PROPN
ejpam-4905	23	3	appl	appl	PROPN
ejpam-4905	23	4	.	.	PROPN
ejpam-4905	23	5	math	math	PROPN
ejpam-4905	23	6	,	,	PUNCT
ejpam-4905	23	7	16	16	NUM
ejpam-4905	23	8	(	(	PUNCT
ejpam-4905	23	9	4	4	NUM
ejpam-4905	23	10	)	)	PUNCT
ejpam-4905	23	11	(	(	PUNCT
ejpam-4905	23	12	2023	2023	NUM
ejpam-4905	23	13	)	)	PUNCT
ejpam-4905	23	14	,	,	PUNCT
ejpam-4905	23	15	2118	2118	NUM
ejpam-4905	23	16	-	-	SYM
ejpam-4905	23	17	2131	2131	NUM
ejpam-4905	23	18	2119	2119	NUM
ejpam-4905	23	19	theory	theory	NOUN
ejpam-4905	23	20	.	.	PUNCT
ejpam-4905	24	1	moreover	moreover	ADV
ejpam-4905	24	2	,	,	PUNCT
ejpam-4905	24	3	we	we	PRON
ejpam-4905	24	4	will	will	AUX
ejpam-4905	24	5	determine	determine	VERB
ejpam-4905	24	6	its	its	PRON
ejpam-4905	24	7	bounds	bound	NOUN
ejpam-4905	24	8	or	or	CCONJ
ejpam-4905	24	9	exact	exact	ADJ
ejpam-4905	24	10	values	value	NOUN
ejpam-4905	24	11	on	on	ADP
ejpam-4905	24	12	some	some	DET
ejpam-4905	24	13	special	special	ADJ
ejpam-4905	24	14	graphs	graph	NOUN
ejpam-4905	24	15	,	,	PUNCT
ejpam-4905	24	16	shadow	shadow	NOUN
ejpam-4905	24	17	graph	graph	NOUN
ejpam-4905	24	18	and	and	CCONJ
ejpam-4905	24	19	join	join	NOUN
ejpam-4905	24	20	of	of	ADP
ejpam-4905	24	21	two	two	NUM
ejpam-4905	24	22	graphs	graph	NOUN
ejpam-4905	24	23	.	.	PUNCT
ejpam-4905	25	1	we	we	PRON
ejpam-4905	25	2	believe	believe	VERB
ejpam-4905	25	3	that	that	SCONJ
ejpam-4905	25	4	this	this	DET
ejpam-4905	25	5	parameter	parameter	NOUN
ejpam-4905	25	6	and	and	CCONJ
ejpam-4905	25	7	its	its	PRON
ejpam-4905	25	8	results	result	NOUN
ejpam-4905	25	9	would	would	AUX
ejpam-4905	25	10	give	give	VERB
ejpam-4905	25	11	additional	additional	ADJ
ejpam-4905	25	12	insights	insight	NOUN
ejpam-4905	25	13	to	to	ADP
ejpam-4905	25	14	researchers	researcher	NOUN
ejpam-4905	25	15	in	in	ADP
ejpam-4905	25	16	the	the	DET
ejpam-4905	25	17	field	field	NOUN
ejpam-4905	25	18	and	and	CCONJ
ejpam-4905	25	19	would	would	AUX
ejpam-4905	25	20	help	help	VERB
ejpam-4905	25	21	them	they	PRON
ejpam-4905	25	22	for	for	ADP
ejpam-4905	25	23	more	more	ADJ
ejpam-4905	25	24	research	research	NOUN
ejpam-4905	25	25	directions	direction	NOUN
ejpam-4905	25	26	in	in	ADP
ejpam-4905	25	27	the	the	DET
ejpam-4905	25	28	future	future	NOUN
ejpam-4905	25	29	.	.	PUNCT
ejpam-4905	26	1	2	2	X
ejpam-4905	26	2	.	.	X
ejpam-4905	26	3	terminology	terminology	NOUN
ejpam-4905	26	4	and	and	CCONJ
ejpam-4905	26	5	notation	notation	NOUN
ejpam-4905	26	6	a	a	DET
ejpam-4905	26	7	path	path	NOUN
ejpam-4905	26	8	graph	graph	NOUN
ejpam-4905	26	9	is	be	AUX
ejpam-4905	26	10	a	a	DET
ejpam-4905	26	11	non	non	ADJ
ejpam-4905	26	12	-	-	ADJ
ejpam-4905	26	13	empty	empty	ADJ
ejpam-4905	26	14	graph	graph	NOUN
ejpam-4905	26	15	with	with	ADP
ejpam-4905	26	16	vertex	vertex	NOUN
ejpam-4905	26	17	-	-	PUNCT
ejpam-4905	26	18	set	set	VERB
ejpam-4905	26	19	{	{	PUNCT
ejpam-4905	26	20	x1	x1	PROPN
ejpam-4905	26	21	,	,	PUNCT
ejpam-4905	26	22	x2	x2	PROPN
ejpam-4905	26	23	,	,	PUNCT
ejpam-4905	26	24	.	.	PUNCT
ejpam-4905	26	25	.	.	PUNCT
ejpam-4905	27	1	.	.	PUNCT
ejpam-4905	28	1	,	,	PUNCT
ejpam-4905	28	2	xn	xn	X
ejpam-4905	28	3	}	}	PUNCT
ejpam-4905	28	4	and	and	CCONJ
ejpam-4905	28	5	edge	edge	NOUN
ejpam-4905	28	6	-	-	PUNCT
ejpam-4905	28	7	set	set	NOUN
ejpam-4905	28	8	{	{	PUNCT
ejpam-4905	28	9	x1x2	x1x2	NOUN
ejpam-4905	28	10	,	,	PUNCT
ejpam-4905	28	11	x2x3	x2x3	PROPN
ejpam-4905	28	12	,	,	PUNCT
ejpam-4905	28	13	.	.	PUNCT
ejpam-4905	28	14	.	.	PUNCT
ejpam-4905	29	1	.	.	PUNCT
ejpam-4905	30	1	,	,	PUNCT
ejpam-4905	30	2	xn−1xn	xn−1xn	PROPN
ejpam-4905	30	3	}	}	PUNCT
ejpam-4905	30	4	,	,	PUNCT
ejpam-4905	30	5	where	where	SCONJ
ejpam-4905	30	6	the	the	DET
ejpam-4905	30	7	x	x	NOUN
ejpam-4905	30	8	′	′	NOUN
ejpam-4905	30	9	is	be	AUX
ejpam-4905	30	10	are	be	AUX
ejpam-4905	30	11	all	all	ADV
ejpam-4905	30	12	distinct	distinct	ADJ
ejpam-4905	30	13	.	.	PUNCT
ejpam-4905	31	1	the	the	DET
ejpam-4905	31	2	path	path	NOUN
ejpam-4905	31	3	of	of	ADP
ejpam-4905	31	4	order	order	NOUN
ejpam-4905	31	5	n	n	NOUN
ejpam-4905	31	6	is	be	AUX
ejpam-4905	31	7	denoted	denote	VERB
ejpam-4905	31	8	by	by	ADP
ejpam-4905	31	9	pn	pn	PROPN
ejpam-4905	31	10	.	.	PUNCT
ejpam-4905	32	1	if	if	SCONJ
ejpam-4905	32	2	g	g	PROPN
ejpam-4905	32	3	is	be	AUX
ejpam-4905	32	4	a	a	DET
ejpam-4905	32	5	graph	graph	NOUN
ejpam-4905	32	6	and	and	CCONJ
ejpam-4905	32	7	u	u	NOUN
ejpam-4905	32	8	and	and	CCONJ
ejpam-4905	32	9	v	v	NOUN
ejpam-4905	32	10	are	be	AUX
ejpam-4905	32	11	vertices	vertex	NOUN
ejpam-4905	32	12	of	of	ADP
ejpam-4905	32	13	g	g	NOUN
ejpam-4905	32	14	,	,	PUNCT
ejpam-4905	32	15	then	then	ADV
ejpam-4905	32	16	a	a	DET
ejpam-4905	32	17	path	path	NOUN
ejpam-4905	32	18	from	from	ADP
ejpam-4905	32	19	vertex	vertex	NOUN
ejpam-4905	32	20	u	u	NOUN
ejpam-4905	32	21	to	to	PART
ejpam-4905	32	22	vertex	vertex	NOUN
ejpam-4905	32	23	v	v	NOUN
ejpam-4905	32	24	is	be	AUX
ejpam-4905	32	25	sometimes	sometimes	ADV
ejpam-4905	32	26	called	call	VERB
ejpam-4905	32	27	a	a	DET
ejpam-4905	32	28	u	u	NOUN
ejpam-4905	32	29	-	-	NOUN
ejpam-4905	32	30	v	v	ADJ
ejpam-4905	32	31	path	path	NOUN
ejpam-4905	32	32	.	.	PUNCT
ejpam-4905	33	1	the	the	DET
ejpam-4905	33	2	cycle	cycle	NOUN
ejpam-4905	33	3	graph	graph	NOUN
ejpam-4905	33	4	cn	cn	PROPN
ejpam-4905	33	5	is	be	AUX
ejpam-4905	33	6	the	the	DET
ejpam-4905	33	7	graph	graph	NOUN
ejpam-4905	33	8	of	of	ADP
ejpam-4905	33	9	order	order	NOUN
ejpam-4905	33	10	n	n	PRON
ejpam-4905	33	11	≥	≥	NOUN
ejpam-4905	33	12	3	3	NUM
ejpam-4905	33	13	with	with	ADP
ejpam-4905	33	14	vertex	vertex	NOUN
ejpam-4905	33	15	-	-	PUNCT
ejpam-4905	33	16	set	set	VERB
ejpam-4905	33	17	{	{	PUNCT
ejpam-4905	33	18	x1	x1	PROPN
ejpam-4905	33	19	,	,	PUNCT
ejpam-4905	33	20	x2	x2	PROPN
ejpam-4905	33	21	,	,	PUNCT
ejpam-4905	33	22	.	.	PUNCT
ejpam-4905	33	23	.	.	PUNCT
ejpam-4905	33	24	.	.	PUNCT
ejpam-4905	34	1	,	,	PUNCT
ejpam-4905	34	2	xn	xn	X
ejpam-4905	34	3	}	}	PUNCT
ejpam-4905	34	4	and	and	CCONJ
ejpam-4905	34	5	edge	edge	NOUN
ejpam-4905	34	6	-	-	PUNCT
ejpam-4905	34	7	set	set	NOUN
ejpam-4905	34	8	{	{	PUNCT
ejpam-4905	34	9	x1x2	x1x2	NOUN
ejpam-4905	34	10	,	,	PUNCT
ejpam-4905	34	11	x2x3	x2x3	PROPN
ejpam-4905	34	12	,	,	PUNCT
ejpam-4905	34	13	.	.	PUNCT
ejpam-4905	34	14	.	.	PUNCT
ejpam-4905	35	1	.	.	PUNCT
ejpam-4905	36	1	,	,	PUNCT
ejpam-4905	36	2	xn−1xn	xn−1xn	PROPN
ejpam-4905	36	3	,	,	PUNCT
ejpam-4905	36	4	xnx1	xnx1	PROPN
ejpam-4905	36	5	}	}	PUNCT
ejpam-4905	36	6	.	.	PUNCT
ejpam-4905	37	1	let	let	VERB
ejpam-4905	37	2	g	g	PROPN
ejpam-4905	37	3	=	=	SYM
ejpam-4905	37	4	(	(	PUNCT
ejpam-4905	37	5	v	v	NOUN
ejpam-4905	37	6	(	(	PUNCT
ejpam-4905	37	7	g	g	NOUN
ejpam-4905	37	8	)	)	PUNCT
ejpam-4905	37	9	,	,	PUNCT
ejpam-4905	37	10	e(g	e(g	PROPN
ejpam-4905	37	11	)	)	PUNCT
ejpam-4905	37	12	)	)	PUNCT
ejpam-4905	38	1	be	be	AUX
ejpam-4905	38	2	a	a	DET
ejpam-4905	38	3	simple	simple	ADJ
ejpam-4905	38	4	and	and	CCONJ
ejpam-4905	38	5	undirected	undirected	ADJ
ejpam-4905	38	6	graph	graph	NOUN
ejpam-4905	38	7	.	.	PUNCT
ejpam-4905	39	1	the	the	DET
ejpam-4905	39	2	distance	distance	NOUN
ejpam-4905	39	3	dg(u	dg(u	NOUN
ejpam-4905	39	4	,	,	PUNCT
ejpam-4905	39	5	v	v	NOUN
ejpam-4905	39	6	)	)	PUNCT
ejpam-4905	39	7	in	in	ADP
ejpam-4905	39	8	g	g	NOUN
ejpam-4905	39	9	of	of	ADP
ejpam-4905	39	10	two	two	NUM
ejpam-4905	39	11	vertices	vertex	NOUN
ejpam-4905	39	12	u	u	NOUN
ejpam-4905	39	13	,	,	PUNCT
ejpam-4905	39	14	v	v	PROPN
ejpam-4905	39	15	is	be	AUX
ejpam-4905	39	16	the	the	DET
ejpam-4905	39	17	length	length	NOUN
ejpam-4905	39	18	of	of	ADP
ejpam-4905	39	19	a	a	DET
ejpam-4905	39	20	shortest	short	ADJ
ejpam-4905	39	21	u	u	NOUN
ejpam-4905	39	22	-	-	NOUN
ejpam-4905	39	23	v	v	ADJ
ejpam-4905	39	24	path	path	NOUN
ejpam-4905	39	25	in	in	ADP
ejpam-4905	39	26	g.	g.	PROPN
ejpam-4905	39	27	the	the	DET
ejpam-4905	39	28	greatest	great	ADJ
ejpam-4905	39	29	distance	distance	NOUN
ejpam-4905	39	30	between	between	ADP
ejpam-4905	39	31	any	any	DET
ejpam-4905	39	32	two	two	NUM
ejpam-4905	39	33	vertices	vertex	NOUN
ejpam-4905	39	34	in	in	ADP
ejpam-4905	39	35	g	g	NOUN
ejpam-4905	39	36	,	,	PUNCT
ejpam-4905	39	37	denoted	denote	VERB
ejpam-4905	39	38	by	by	ADP
ejpam-4905	39	39	diam(g	diam(g	PROPN
ejpam-4905	39	40	)	)	PUNCT
ejpam-4905	39	41	,	,	PUNCT
ejpam-4905	39	42	is	be	AUX
ejpam-4905	39	43	called	call	VERB
ejpam-4905	39	44	the	the	DET
ejpam-4905	39	45	diameter	diameter	NOUN
ejpam-4905	39	46	of	of	ADP
ejpam-4905	39	47	g.	g.	PROPN
ejpam-4905	39	48	two	two	NUM
ejpam-4905	39	49	vertices	vertice	VERB
ejpam-4905	39	50	x	x	X
ejpam-4905	39	51	,	,	PUNCT
ejpam-4905	39	52	y	y	PROPN
ejpam-4905	39	53	of	of	ADP
ejpam-4905	39	54	g	g	PROPN
ejpam-4905	39	55	are	be	AUX
ejpam-4905	39	56	adjacent	adjacent	ADJ
ejpam-4905	39	57	,	,	PUNCT
ejpam-4905	39	58	or	or	CCONJ
ejpam-4905	39	59	neighbors	neighbor	NOUN
ejpam-4905	39	60	,	,	PUNCT
ejpam-4905	39	61	if	if	SCONJ
ejpam-4905	39	62	xy	xy	PROPN
ejpam-4905	39	63	is	be	AUX
ejpam-4905	39	64	an	an	DET
ejpam-4905	39	65	edge	edge	NOUN
ejpam-4905	39	66	of	of	ADP
ejpam-4905	39	67	g.	g.	PROPN
ejpam-4905	39	68	the	the	DET
ejpam-4905	39	69	open	open	ADJ
ejpam-4905	39	70	neighborhood	neighborhood	NOUN
ejpam-4905	39	71	of	of	ADP
ejpam-4905	39	72	x	x	PUNCT
ejpam-4905	39	73	in	in	ADP
ejpam-4905	39	74	g	g	PROPN
ejpam-4905	39	75	is	be	AUX
ejpam-4905	39	76	the	the	DET
ejpam-4905	39	77	set	set	NOUN
ejpam-4905	39	78	ng(x	ng(x	NUM
ejpam-4905	39	79	)	)	PUNCT
ejpam-4905	39	80	=	=	PRON
ejpam-4905	39	81	{	{	PUNCT
ejpam-4905	39	82	y	y	PROPN
ejpam-4905	39	83	∈	∈	PROPN
ejpam-4905	39	84	v	v	NOUN
ejpam-4905	39	85	(	(	PUNCT
ejpam-4905	39	86	g	g	NOUN
ejpam-4905	39	87	)	)	PUNCT
ejpam-4905	39	88	:	:	PUNCT
ejpam-4905	39	89	xy	xy	PROPN
ejpam-4905	39	90	∈	∈	PROPN
ejpam-4905	39	91	e(g	e(g	PROPN
ejpam-4905	39	92	)	)	PUNCT
ejpam-4905	39	93	}	}	PUNCT
ejpam-4905	39	94	.	.	PUNCT
ejpam-4905	40	1	the	the	DET
ejpam-4905	40	2	closed	closed	ADJ
ejpam-4905	40	3	neighborhood	neighborhood	NOUN
ejpam-4905	40	4	of	of	ADP
ejpam-4905	40	5	x	x	SYM
ejpam-4905	40	6	ing	ing	NOUN
ejpam-4905	40	7	is	be	AUX
ejpam-4905	40	8	the	the	DET
ejpam-4905	40	9	setng[x	setng[x	NOUN
ejpam-4905	40	10	]	]	X
ejpam-4905	40	11	=	=	SYM
ejpam-4905	40	12	ng(x)∪{x	ng(x)∪{x	NOUN
ejpam-4905	40	13	}	}	PUNCT
ejpam-4905	40	14	.	.	PUNCT
ejpam-4905	41	1	ifx	ifx	PROPN
ejpam-4905	41	2	⊆	⊆	NUM
ejpam-4905	41	3	v	v	NOUN
ejpam-4905	41	4	(	(	PUNCT
ejpam-4905	41	5	g	g	NOUN
ejpam-4905	41	6	)	)	PUNCT
ejpam-4905	41	7	,	,	PUNCT
ejpam-4905	41	8	the	the	DET
ejpam-4905	41	9	open	open	ADJ
ejpam-4905	41	10	neighborhood	neighborhood	NOUN
ejpam-4905	41	11	of	of	ADP
ejpam-4905	41	12	x	x	PUNCT
ejpam-4905	41	13	in	in	ADP
ejpam-4905	41	14	g	g	PROPN
ejpam-4905	41	15	is	be	AUX
ejpam-4905	41	16	the	the	DET
ejpam-4905	41	17	set	set	NOUN
ejpam-4905	41	18	ng(x	ng(x	NUM
ejpam-4905	41	19	)	)	PUNCT
ejpam-4905	42	1	=	=	SYM
ejpam-4905	42	2	⋃	⋃	NOUN
ejpam-4905	42	3	x∈x	x∈x	NOUN
ejpam-4905	42	4	ng(x	ng(x	NUM
ejpam-4905	42	5	)	)	PUNCT
ejpam-4905	42	6	.	.	PUNCT
ejpam-4905	43	1	the	the	DET
ejpam-4905	43	2	closed	closed	ADJ
ejpam-4905	43	3	neighborhood	neighborhood	NOUN
ejpam-4905	43	4	of	of	ADP
ejpam-4905	43	5	x	x	PUNCT
ejpam-4905	43	6	in	in	ADP
ejpam-4905	43	7	g	g	PROPN
ejpam-4905	43	8	is	be	AUX
ejpam-4905	43	9	the	the	DET
ejpam-4905	43	10	set	set	NOUN
ejpam-4905	43	11	ng[x	ng[x	PROPN
ejpam-4905	43	12	]	]	X
ejpam-4905	43	13	=	=	PUNCT
ejpam-4905	43	14	ng(x	ng(x	X
ejpam-4905	43	15	)	)	PUNCT
ejpam-4905	44	1	∪x	∪x	ADP
ejpam-4905	44	2	.	.	PUNCT
ejpam-4905	45	1	a	a	DET
ejpam-4905	45	2	vertex	vertex	NOUN
ejpam-4905	45	3	a	a	PRON
ejpam-4905	45	4	in	in	ADP
ejpam-4905	45	5	g	g	PROPN
ejpam-4905	45	6	is	be	AUX
ejpam-4905	45	7	a	a	DET
ejpam-4905	45	8	hop	hop	NOUN
ejpam-4905	45	9	neighbor	neighbor	NOUN
ejpam-4905	45	10	of	of	ADP
ejpam-4905	45	11	a	a	DET
ejpam-4905	45	12	vertex	vertex	NOUN
ejpam-4905	45	13	b	b	NOUN
ejpam-4905	45	14	in	in	ADP
ejpam-4905	45	15	g	g	PROPN
ejpam-4905	45	16	if	if	SCONJ
ejpam-4905	45	17	dg(a	dg(a	X
ejpam-4905	45	18	,	,	PUNCT
ejpam-4905	45	19	b	b	X
ejpam-4905	45	20	)	)	PUNCT
ejpam-4905	45	21	=	=	SYM
ejpam-4905	45	22	2	2	X
ejpam-4905	45	23	.	.	X
ejpam-4905	45	24	the	the	DET
ejpam-4905	45	25	set	set	ADJ
ejpam-4905	45	26	n2	n2	PROPN
ejpam-4905	45	27	g(a	g(a	PROPN
ejpam-4905	45	28	)	)	PUNCT
ejpam-4905	45	29	=	=	PRON
ejpam-4905	46	1	{	{	PUNCT
ejpam-4905	46	2	b	b	PROPN
ejpam-4905	46	3	∈	∈	ADJ
ejpam-4905	46	4	v	v	NOUN
ejpam-4905	46	5	(	(	PUNCT
ejpam-4905	46	6	g	g	NOUN
ejpam-4905	46	7	)	)	PUNCT
ejpam-4905	46	8	:	:	PUNCT
ejpam-4905	47	1	dg(a	dg(a	X
ejpam-4905	47	2	,	,	PUNCT
ejpam-4905	47	3	b	b	X
ejpam-4905	47	4	)	)	PUNCT
ejpam-4905	47	5	=	=	SYM
ejpam-4905	47	6	2	2	X
ejpam-4905	47	7	}	}	PUNCT
ejpam-4905	47	8	is	be	AUX
ejpam-4905	47	9	called	call	VERB
ejpam-4905	47	10	the	the	DET
ejpam-4905	47	11	open	open	ADJ
ejpam-4905	47	12	hop	hop	NOUN
ejpam-4905	47	13	neighborhood	neighborhood	NOUN
ejpam-4905	47	14	of	of	ADP
ejpam-4905	47	15	a.	a.	NOUN
ejpam-4905	47	16	the	the	DET
ejpam-4905	47	17	closed	closed	ADJ
ejpam-4905	47	18	hop	hop	NOUN
ejpam-4905	47	19	neighborhood	neighborhood	NOUN
ejpam-4905	47	20	of	of	ADP
ejpam-4905	47	21	a	a	PRON
ejpam-4905	47	22	in	in	ADP
ejpam-4905	47	23	g	g	PROPN
ejpam-4905	47	24	is	be	AUX
ejpam-4905	47	25	given	give	VERB
ejpam-4905	47	26	by	by	ADP
ejpam-4905	47	27	n2	n2	ADJ
ejpam-4905	47	28	g[a	g[a	PROPN
ejpam-4905	47	29	]	]	X
ejpam-4905	47	30	=	=	SYM
ejpam-4905	47	31	n2	n2	ADJ
ejpam-4905	47	32	g(a)∪{a	g(a)∪{a	NOUN
ejpam-4905	47	33	}	}	PUNCT
ejpam-4905	47	34	.	.	PUNCT
ejpam-4905	48	1	the	the	DET
ejpam-4905	48	2	open	open	ADJ
ejpam-4905	48	3	hop	hop	NOUN
ejpam-4905	48	4	neighborhood	neighborhood	NOUN
ejpam-4905	48	5	of	of	ADP
ejpam-4905	48	6	s	s	NOUN
ejpam-4905	48	7	⊆	⊆	NUM
ejpam-4905	48	8	v	v	NOUN
ejpam-4905	48	9	(	(	PUNCT
ejpam-4905	48	10	g	g	NOUN
ejpam-4905	48	11	)	)	PUNCT
ejpam-4905	48	12	is	be	AUX
ejpam-4905	48	13	the	the	DET
ejpam-4905	48	14	set	set	ADJ
ejpam-4905	48	15	n2	n2	ADJ
ejpam-4905	48	16	g(s	g(s	PROPN
ejpam-4905	48	17	)	)	PUNCT
ejpam-4905	48	18	=	=	SYM
ejpam-4905	49	1	⋃	⋃	ADP
ejpam-4905	49	2	a∈s	a∈s	ADJ
ejpam-4905	49	3	n2	n2	NOUN
ejpam-4905	49	4	g(a	g(a	PROPN
ejpam-4905	49	5	)	)	PUNCT
ejpam-4905	49	6	.	.	PUNCT
ejpam-4905	50	1	the	the	DET
ejpam-4905	50	2	closed	closed	ADJ
ejpam-4905	50	3	hop	hop	NOUN
ejpam-4905	50	4	neighborhood	neighborhood	NOUN
ejpam-4905	50	5	of	of	ADP
ejpam-4905	50	6	s	s	PRON
ejpam-4905	50	7	in	in	ADP
ejpam-4905	50	8	g	g	PROPN
ejpam-4905	50	9	is	be	AUX
ejpam-4905	50	10	the	the	DET
ejpam-4905	50	11	set	set	ADJ
ejpam-4905	50	12	n2	n2	ADJ
ejpam-4905	50	13	g[s	g[s	PROPN
ejpam-4905	50	14	]	]	PUNCT
ejpam-4905	50	15	=	=	SYM
ejpam-4905	50	16	n2	n2	ADJ
ejpam-4905	50	17	g(s	g(s	PROPN
ejpam-4905	50	18	)	)	PUNCT
ejpam-4905	50	19	∪	∪	ADP
ejpam-4905	50	20	s.	s.	PROPN
ejpam-4905	50	21	a	a	DET
ejpam-4905	50	22	subset	subset	NOUN
ejpam-4905	50	23	s	s	X
ejpam-4905	50	24	of	of	ADP
ejpam-4905	50	25	v	v	NOUN
ejpam-4905	50	26	(	(	PUNCT
ejpam-4905	50	27	g	g	NOUN
ejpam-4905	50	28	)	)	PUNCT
ejpam-4905	50	29	is	be	AUX
ejpam-4905	50	30	a	a	DET
ejpam-4905	50	31	hop	hop	NOUN
ejpam-4905	50	32	dominating	dominating	NOUN
ejpam-4905	50	33	of	of	ADP
ejpam-4905	50	34	g	g	PROPN
ejpam-4905	50	35	if	if	SCONJ
ejpam-4905	50	36	for	for	ADP
ejpam-4905	50	37	every	every	DET
ejpam-4905	50	38	a	a	DET
ejpam-4905	50	39	∈	∈	PROPN
ejpam-4905	50	40	v	v	NOUN
ejpam-4905	50	41	(	(	PUNCT
ejpam-4905	50	42	g)\s	g)\s	NOUN
ejpam-4905	50	43	,	,	PUNCT
ejpam-4905	50	44	there	there	PRON
ejpam-4905	50	45	exists	exist	VERB
ejpam-4905	50	46	b	b	PROPN
ejpam-4905	50	47	∈	∈	PROPN
ejpam-4905	50	48	s	s	VERB
ejpam-4905	50	49	such	such	ADJ
ejpam-4905	50	50	that	that	SCONJ
ejpam-4905	50	51	dg(a	dg(a	PROPN
ejpam-4905	50	52	,	,	PUNCT
ejpam-4905	50	53	b	b	X
ejpam-4905	50	54	)	)	PUNCT
ejpam-4905	50	55	=	=	SYM
ejpam-4905	50	56	2	2	X
ejpam-4905	50	57	.	.	PUNCT
ejpam-4905	50	58	the	the	DET
ejpam-4905	50	59	minimum	minimum	ADJ
ejpam-4905	50	60	cardinality	cardinality	NOUN
ejpam-4905	50	61	among	among	ADP
ejpam-4905	50	62	all	all	DET
ejpam-4905	50	63	hop	hop	NOUN
ejpam-4905	50	64	dominating	dominating	NOUN
ejpam-4905	50	65	sets	set	NOUN
ejpam-4905	50	66	of	of	ADP
ejpam-4905	50	67	g	g	NOUN
ejpam-4905	50	68	,	,	PUNCT
ejpam-4905	50	69	denoted	denote	VERB
ejpam-4905	50	70	by	by	ADP
ejpam-4905	50	71	γh(g	γh(g	NOUN
ejpam-4905	50	72	)	)	PUNCT
ejpam-4905	50	73	,	,	PUNCT
ejpam-4905	50	74	is	be	AUX
ejpam-4905	50	75	called	call	VERB
ejpam-4905	50	76	the	the	DET
ejpam-4905	50	77	hop	hop	NOUN
ejpam-4905	50	78	domination	domination	NOUN
ejpam-4905	50	79	number	number	NOUN
ejpam-4905	50	80	of	of	ADP
ejpam-4905	50	81	g.	g.	PROPN
ejpam-4905	50	82	any	any	DET
ejpam-4905	50	83	hop	hop	NOUN
ejpam-4905	50	84	dominating	dominating	NOUN
ejpam-4905	50	85	set	set	VERB
ejpam-4905	50	86	with	with	ADP
ejpam-4905	50	87	cardinality	cardinality	NOUN
ejpam-4905	50	88	equal	equal	ADJ
ejpam-4905	50	89	to	to	ADP
ejpam-4905	50	90	γh(g	γh(g	NOUN
ejpam-4905	50	91	)	)	PUNCT
ejpam-4905	50	92	is	be	AUX
ejpam-4905	50	93	called	call	VERB
ejpam-4905	50	94	a	a	DET
ejpam-4905	50	95	γh	γh	ADV
ejpam-4905	50	96	-	-	PUNCT
ejpam-4905	50	97	set	set	NOUN
ejpam-4905	50	98	of	of	ADP
ejpam-4905	50	99	g.	g.	PROPN
ejpam-4905	50	100	a	a	DET
ejpam-4905	50	101	subset	subset	NOUN
ejpam-4905	50	102	s	s	NOUN
ejpam-4905	50	103	of	of	ADP
ejpam-4905	50	104	v	v	NOUN
ejpam-4905	50	105	(	(	PUNCT
ejpam-4905	50	106	g	g	NOUN
ejpam-4905	50	107	)	)	PUNCT
ejpam-4905	50	108	is	be	AUX
ejpam-4905	50	109	called	call	VERB
ejpam-4905	50	110	a	a	DET
ejpam-4905	50	111	hop	hop	NOUN
ejpam-4905	50	112	independent	independent	ADJ
ejpam-4905	50	113	if	if	SCONJ
ejpam-4905	50	114	for	for	ADP
ejpam-4905	50	115	every	every	DET
ejpam-4905	50	116	pair	pair	NOUN
ejpam-4905	50	117	of	of	ADP
ejpam-4905	50	118	distinct	distinct	ADJ
ejpam-4905	50	119	vertices	vertex	NOUN
ejpam-4905	50	120	x	x	X
ejpam-4905	50	121	,	,	PUNCT
ejpam-4905	50	122	y	y	PROPN
ejpam-4905	50	123	∈	∈	PROPN
ejpam-4905	50	124	s	s	PROPN
ejpam-4905	50	125	,	,	PUNCT
ejpam-4905	50	126	dg(x	dg(x	NUM
ejpam-4905	50	127	,	,	PUNCT
ejpam-4905	50	128	y	y	NOUN
ejpam-4905	50	129	)	)	PUNCT
ejpam-4905	50	130	̸=	̸=	PROPN
ejpam-4905	50	131	2	2	NUM
ejpam-4905	50	132	.	.	PUNCT
ejpam-4905	51	1	the	the	DET
ejpam-4905	51	2	maximum	maximum	ADJ
ejpam-4905	51	3	cardinality	cardinality	NOUN
ejpam-4905	51	4	of	of	ADP
ejpam-4905	51	5	a	a	DET
ejpam-4905	51	6	hop	hop	NOUN
ejpam-4905	51	7	independent	independent	ADJ
ejpam-4905	51	8	set	set	NOUN
ejpam-4905	51	9	in	in	ADP
ejpam-4905	51	10	g	g	NOUN
ejpam-4905	51	11	,	,	PUNCT
ejpam-4905	51	12	denoted	denote	VERB
ejpam-4905	51	13	by	by	ADP
ejpam-4905	51	14	αh(g	αh(g	NOUN
ejpam-4905	51	15	)	)	PUNCT
ejpam-4905	51	16	,	,	PUNCT
ejpam-4905	51	17	is	be	AUX
ejpam-4905	51	18	called	call	VERB
ejpam-4905	51	19	the	the	DET
ejpam-4905	51	20	hop	hop	NOUN
ejpam-4905	51	21	independence	independence	NOUN
ejpam-4905	51	22	number	number	NOUN
ejpam-4905	51	23	of	of	ADP
ejpam-4905	51	24	g.	g.	PROPN
ejpam-4905	51	25	any	any	DET
ejpam-4905	51	26	hop	hop	NOUN
ejpam-4905	51	27	independent	independent	ADJ
ejpam-4905	51	28	set	set	NOUN
ejpam-4905	51	29	s	s	PROPN
ejpam-4905	51	30	with	with	ADP
ejpam-4905	51	31	cardinality	cardinality	NOUN
ejpam-4905	51	32	equal	equal	ADJ
ejpam-4905	51	33	to	to	ADP
ejpam-4905	51	34	αh(g	αh(g	NOUN
ejpam-4905	51	35	)	)	PUNCT
ejpam-4905	51	36	is	be	AUX
ejpam-4905	51	37	called	call	VERB
ejpam-4905	51	38	an	an	DET
ejpam-4905	51	39	αh	αh	NOUN
ejpam-4905	51	40	-	-	PUNCT
ejpam-4905	51	41	set	set	NOUN
ejpam-4905	51	42	of	of	ADP
ejpam-4905	51	43	g.	g.	PROPN
ejpam-4905	51	44	let	let	VERB
ejpam-4905	51	45	g	g	NOUN
ejpam-4905	51	46	and	and	CCONJ
ejpam-4905	51	47	h	h	NOUN
ejpam-4905	51	48	be	be	VERB
ejpam-4905	51	49	any	any	DET
ejpam-4905	51	50	two	two	NUM
ejpam-4905	51	51	graphs	graph	NOUN
ejpam-4905	51	52	.	.	PUNCT
ejpam-4905	52	1	the	the	DET
ejpam-4905	52	2	join	join	NOUN
ejpam-4905	52	3	of	of	ADP
ejpam-4905	52	4	g	g	PROPN
ejpam-4905	52	5	and	and	CCONJ
ejpam-4905	52	6	h	h	NOUN
ejpam-4905	52	7	,	,	PUNCT
ejpam-4905	52	8	denoted	denote	VERB
ejpam-4905	52	9	by	by	ADP
ejpam-4905	52	10	g+h	g+h	PROPN
ejpam-4905	52	11	is	be	AUX
ejpam-4905	52	12	the	the	DET
ejpam-4905	52	13	graph	graph	NOUN
ejpam-4905	52	14	with	with	ADP
ejpam-4905	52	15	vertex	vertex	NOUN
ejpam-4905	52	16	set	set	VERB
ejpam-4905	52	17	v	v	NOUN
ejpam-4905	52	18	(	(	PUNCT
ejpam-4905	52	19	g+h	g+h	NOUN
ejpam-4905	52	20	)	)	PUNCT
ejpam-4905	53	1	=	=	SYM
ejpam-4905	53	2	v	v	X
ejpam-4905	53	3	(	(	PUNCT
ejpam-4905	53	4	g	g	NOUN
ejpam-4905	53	5	)	)	PUNCT
ejpam-4905	53	6	∪	∪	NOUN
ejpam-4905	53	7	v	v	NOUN
ejpam-4905	53	8	(	(	PUNCT
ejpam-4905	53	9	h	h	NOUN
ejpam-4905	53	10	)	)	PUNCT
ejpam-4905	53	11	and	and	CCONJ
ejpam-4905	53	12	edge	edge	NOUN
ejpam-4905	53	13	set	set	VERB
ejpam-4905	53	14	e(g+h	e(g+h	NUM
ejpam-4905	53	15	)	)	PUNCT
ejpam-4905	53	16	=	=	SYM
ejpam-4905	53	17	e(g	e(g	NOUN
ejpam-4905	53	18	)	)	PUNCT
ejpam-4905	53	19	∪	∪	ADP
ejpam-4905	53	20	e(h	e(h	PROPN
ejpam-4905	53	21	)	)	PUNCT
ejpam-4905	53	22	∪	∪	NOUN
ejpam-4905	53	23	{	{	PUNCT
ejpam-4905	53	24	uv	uv	NOUN
ejpam-4905	53	25	:	:	PUNCT
ejpam-4905	53	26	u	u	PROPN
ejpam-4905	53	27	∈	∈	PROPN
ejpam-4905	53	28	v	v	ADP
ejpam-4905	53	29	(	(	PUNCT
ejpam-4905	53	30	g	g	NOUN
ejpam-4905	53	31	)	)	PUNCT
ejpam-4905	53	32	,	,	PUNCT
ejpam-4905	53	33	v	v	X
ejpam-4905	53	34	∈	∈	PROPN
ejpam-4905	53	35	v	v	NOUN
ejpam-4905	53	36	(	(	PUNCT
ejpam-4905	53	37	h	h	NOUN
ejpam-4905	53	38	)	)	PUNCT
ejpam-4905	53	39	}	}	PUNCT
ejpam-4905	53	40	.	.	PUNCT
ejpam-4905	54	1	the	the	DET
ejpam-4905	54	2	corona	corona	NOUN
ejpam-4905	54	3	g	g	PROPN
ejpam-4905	54	4	and	and	CCONJ
ejpam-4905	54	5	h	h	NOUN
ejpam-4905	54	6	,	,	PUNCT
ejpam-4905	54	7	denoted	denote	VERB
ejpam-4905	54	8	by	by	ADP
ejpam-4905	54	9	g	g	PROPN
ejpam-4905	54	10	◦	◦	NOUN
ejpam-4905	54	11	h	h	NOUN
ejpam-4905	54	12	,	,	PUNCT
ejpam-4905	54	13	the	the	DET
ejpam-4905	54	14	graph	graph	NOUN
ejpam-4905	54	15	obtained	obtain	VERB
ejpam-4905	54	16	by	by	ADP
ejpam-4905	54	17	taking	take	VERB
ejpam-4905	54	18	one	one	NUM
ejpam-4905	54	19	copy	copy	NOUN
ejpam-4905	54	20	of	of	ADP
ejpam-4905	54	21	g	g	PROPN
ejpam-4905	54	22	and	and	CCONJ
ejpam-4905	54	23	|v	|v	PROPN
ejpam-4905	54	24	(	(	PUNCT
ejpam-4905	54	25	g)|	g)|	NOUN
ejpam-4905	54	26	copies	copy	NOUN
ejpam-4905	54	27	of	of	ADP
ejpam-4905	54	28	h	h	NOUN
ejpam-4905	54	29	,	,	PUNCT
ejpam-4905	54	30	and	and	CCONJ
ejpam-4905	54	31	then	then	ADV
ejpam-4905	54	32	joining	join	VERB
ejpam-4905	54	33	the	the	DET
ejpam-4905	54	34	ith	ith	PROPN
ejpam-4905	54	35	vertex	vertex	NOUN
ejpam-4905	54	36	of	of	ADP
ejpam-4905	54	37	g	g	NOUN
ejpam-4905	54	38	to	to	ADP
ejpam-4905	54	39	every	every	DET
ejpam-4905	54	40	vertex	vertex	NOUN
ejpam-4905	54	41	of	of	ADP
ejpam-4905	54	42	the	the	DET
ejpam-4905	54	43	ith	ith	PROPN
ejpam-4905	54	44	copy	copy	NOUN
ejpam-4905	54	45	of	of	ADP
ejpam-4905	54	46	h.	h.	PROPN
ejpam-4905	54	47	we	we	PRON
ejpam-4905	54	48	denote	denote	VERB
ejpam-4905	54	49	by	by	ADP
ejpam-4905	54	50	hv	hv	PROPN
ejpam-4905	54	51	the	the	DET
ejpam-4905	54	52	copy	copy	NOUN
ejpam-4905	54	53	of	of	ADP
ejpam-4905	54	54	h	h	NOUN
ejpam-4905	54	55	in	in	ADP
ejpam-4905	54	56	g	g	PROPN
ejpam-4905	54	57	◦	◦	NOUN
ejpam-4905	54	58	h	h	NOUN
ejpam-4905	54	59	corresponding	correspond	VERB
ejpam-4905	54	60	to	to	ADP
ejpam-4905	54	61	the	the	DET
ejpam-4905	54	62	vertex	vertex	NOUN
ejpam-4905	54	63	v	v	ADP
ejpam-4905	54	64	∈	∈	PROPN
ejpam-4905	54	65	g	g	PROPN
ejpam-4905	54	66	j.	j.	PROPN
ejpam-4905	54	67	hassan	hassan	PROPN
ejpam-4905	54	68	,	,	PUNCT
ejpam-4905	54	69	a.	a.	PROPN
ejpam-4905	54	70	bakkang	bakkang	PROPN
ejpam-4905	54	71	,	,	PUNCT
ejpam-4905	54	72	a.	a.	NOUN
ejpam-4905	54	73	sappari	sappari	PROPN
ejpam-4905	54	74	/	/	SYM
ejpam-4905	54	75	eur	eur	PROPN
ejpam-4905	54	76	.	.	PUNCT
ejpam-4905	55	1	j.	j.	PROPN
ejpam-4905	55	2	pure	pure	PROPN
ejpam-4905	55	3	appl	appl	PROPN
ejpam-4905	55	4	.	.	PROPN
ejpam-4905	55	5	math	math	PROPN
ejpam-4905	55	6	,	,	PUNCT
ejpam-4905	55	7	16	16	NUM
ejpam-4905	55	8	(	(	PUNCT
ejpam-4905	55	9	4	4	NUM
ejpam-4905	55	10	)	)	PUNCT
ejpam-4905	55	11	(	(	PUNCT
ejpam-4905	55	12	2023	2023	NUM
ejpam-4905	55	13	)	)	PUNCT
ejpam-4905	55	14	,	,	PUNCT
ejpam-4905	55	15	2118	2118	NUM
ejpam-4905	55	16	-	-	SYM
ejpam-4905	55	17	2131	2131	NUM
ejpam-4905	55	18	2120	2120	NUM
ejpam-4905	55	19	and	and	CCONJ
ejpam-4905	55	20	write	write	VERB
ejpam-4905	55	21	v	v	ADP
ejpam-4905	55	22	+	+	PROPN
ejpam-4905	55	23	hv	hv	NOUN
ejpam-4905	55	24	for	for	ADP
ejpam-4905	55	25	⟨{v}+hv⟩.	⟨{v}+hv⟩.	NOUN
ejpam-4905	55	26	the	the	DET
ejpam-4905	55	27	shadow	shadow	NOUN
ejpam-4905	55	28	graph	graph	NOUN
ejpam-4905	55	29	s(g	s(g	PROPN
ejpam-4905	55	30	)	)	PUNCT
ejpam-4905	55	31	of	of	ADP
ejpam-4905	55	32	graph	graph	NOUN
ejpam-4905	55	33	g	g	PROPN
ejpam-4905	55	34	is	be	AUX
ejpam-4905	55	35	constructed	construct	VERB
ejpam-4905	55	36	by	by	ADP
ejpam-4905	55	37	taking	take	VERB
ejpam-4905	55	38	two	two	NUM
ejpam-4905	55	39	copies	copy	NOUN
ejpam-4905	55	40	of	of	ADP
ejpam-4905	55	41	g	g	NOUN
ejpam-4905	55	42	,	,	PUNCT
ejpam-4905	55	43	say	say	VERB
ejpam-4905	55	44	g1	g1	PROPN
ejpam-4905	55	45	and	and	CCONJ
ejpam-4905	55	46	g2	g2	PROPN
ejpam-4905	55	47	,	,	PUNCT
ejpam-4905	55	48	and	and	CCONJ
ejpam-4905	55	49	then	then	ADV
ejpam-4905	55	50	joining	join	VERB
ejpam-4905	55	51	each	each	DET
ejpam-4905	55	52	vertex	vertex	NOUN
ejpam-4905	55	53	u	u	NOUN
ejpam-4905	55	54	∈	∈	PROPN
ejpam-4905	55	55	v	v	NOUN
ejpam-4905	55	56	(	(	PUNCT
ejpam-4905	55	57	g1	g1	PROPN
ejpam-4905	55	58	)	)	PUNCT
ejpam-4905	55	59	to	to	ADP
ejpam-4905	55	60	the	the	DET
ejpam-4905	55	61	neighbors	neighbor	NOUN
ejpam-4905	55	62	of	of	ADP
ejpam-4905	55	63	its	its	PRON
ejpam-4905	55	64	corresponding	correspond	VERB
ejpam-4905	55	65	vertex	vertex	NOUN
ejpam-4905	55	66	u′	u′	PROPN
ejpam-4905	55	67	∈	∈	PROPN
ejpam-4905	55	68	v	v	NOUN
ejpam-4905	55	69	(	(	PUNCT
ejpam-4905	55	70	g2	g2	PROPN
ejpam-4905	55	71	)	)	PUNCT
ejpam-4905	55	72	.	.	PUNCT
ejpam-4905	56	1	3	3	X
ejpam-4905	56	2	.	.	X
ejpam-4905	56	3	results	result	NOUN
ejpam-4905	56	4	we	we	PRON
ejpam-4905	56	5	begin	begin	VERB
ejpam-4905	56	6	this	this	DET
ejpam-4905	56	7	section	section	NOUN
ejpam-4905	56	8	by	by	ADP
ejpam-4905	56	9	introducing	introduce	VERB
ejpam-4905	56	10	the	the	DET
ejpam-4905	56	11	concept	concept	NOUN
ejpam-4905	56	12	of	of	ADP
ejpam-4905	56	13	j2	j2	PROPN
ejpam-4905	56	14	-	-	PUNCT
ejpam-4905	56	15	hop	hop	PROPN
ejpam-4905	56	16	domination	domination	NOUN
ejpam-4905	56	17	in	in	ADP
ejpam-4905	56	18	a	a	DET
ejpam-4905	56	19	graph	graph	NOUN
ejpam-4905	56	20	.	.	PUNCT
ejpam-4905	57	1	definition	definition	NOUN
ejpam-4905	57	2	1	1	NUM
ejpam-4905	57	3	.	.	PUNCT
ejpam-4905	58	1	let	let	VERB
ejpam-4905	58	2	g	g	NOUN
ejpam-4905	58	3	be	be	AUX
ejpam-4905	58	4	an	an	DET
ejpam-4905	58	5	undirected	undirected	ADJ
ejpam-4905	58	6	graph	graph	NOUN
ejpam-4905	58	7	andm	andm	PROPN
ejpam-4905	58	8	∈	∈	PROPN
ejpam-4905	58	9	n	n	X
ejpam-4905	58	10	.	.	PUNCT
ejpam-4905	59	1	a	a	DET
ejpam-4905	59	2	subset	subset	NOUN
ejpam-4905	59	3	t	t	NOUN
ejpam-4905	59	4	=	=	SYM
ejpam-4905	59	5	{	{	PUNCT
ejpam-4905	59	6	v1	v1	PROPN
ejpam-4905	59	7	,	,	PUNCT
ejpam-4905	59	8	v2	v2	PROPN
ejpam-4905	59	9	,	,	PUNCT
ejpam-4905	59	10	·	·	PUNCT
ejpam-4905	59	11	·	·	PUNCT
ejpam-4905	59	12	·	·	PUNCT
ejpam-4905	59	13	,	,	PUNCT
ejpam-4905	59	14	vm	vm	NOUN
ejpam-4905	59	15	}	}	PUNCT
ejpam-4905	59	16	of	of	ADP
ejpam-4905	59	17	vertices	vertex	NOUN
ejpam-4905	59	18	of	of	ADP
ejpam-4905	59	19	g	g	PROPN
ejpam-4905	59	20	is	be	AUX
ejpam-4905	59	21	called	call	VERB
ejpam-4905	59	22	a	a	DET
ejpam-4905	59	23	j2	j2	PROPN
ejpam-4905	59	24	-	-	PUNCT
ejpam-4905	59	25	set	set	VERB
ejpam-4905	59	26	if	if	SCONJ
ejpam-4905	59	27	n2	n2	ADJ
ejpam-4905	59	28	g[vi	g[vi	PROPN
ejpam-4905	59	29	]	]	PUNCT
ejpam-4905	59	30	\	\	PROPN
ejpam-4905	59	31	n2	n2	PROPN
ejpam-4905	59	32	g[vj	g[vj	PROPN
ejpam-4905	59	33	]	]	PUNCT
ejpam-4905	59	34	̸=	̸=	PROPN
ejpam-4905	59	35	∅	∅	NOUN
ejpam-4905	59	36	for	for	ADP
ejpam-4905	59	37	every	every	DET
ejpam-4905	59	38	i	i	PROPN
ejpam-4905	59	39	̸=	̸=	PROPN
ejpam-4905	59	40	j	j	PROPN
ejpam-4905	59	41	,	,	PUNCT
ejpam-4905	59	42	where	where	SCONJ
ejpam-4905	59	43	i	i	PRON
ejpam-4905	59	44	,	,	PUNCT
ejpam-4905	59	45	j	j	PROPN
ejpam-4905	59	46	∈	∈	PROPN
ejpam-4905	59	47	{	{	PUNCT
ejpam-4905	59	48	1	1	NUM
ejpam-4905	59	49	,	,	PUNCT
ejpam-4905	59	50	2	2	NUM
ejpam-4905	59	51	,	,	PUNCT
ejpam-4905	59	52	.	.	PUNCT
ejpam-4905	59	53	.	.	PUNCT
ejpam-4905	60	1	.	.	PUNCT
ejpam-4905	61	1	,	,	PUNCT
ejpam-4905	61	2	m	m	VERB
ejpam-4905	61	3	}	}	PUNCT
ejpam-4905	61	4	.	.	PUNCT
ejpam-4905	62	1	a	a	DET
ejpam-4905	62	2	j2	j2	PROPN
ejpam-4905	62	3	-	-	PUNCT
ejpam-4905	62	4	set	set	VERB
ejpam-4905	62	5	t	t	PROPN
ejpam-4905	62	6	is	be	AUX
ejpam-4905	62	7	called	call	VERB
ejpam-4905	62	8	a	a	DET
ejpam-4905	62	9	j2	j2	PROPN
ejpam-4905	62	10	-	-	PUNCT
ejpam-4905	62	11	hop	hop	NOUN
ejpam-4905	62	12	dominating	dominating	NOUN
ejpam-4905	62	13	in	in	ADP
ejpam-4905	62	14	g	g	PROPN
ejpam-4905	62	15	,	,	PUNCT
ejpam-4905	62	16	if	if	SCONJ
ejpam-4905	62	17	t	t	PROPN
ejpam-4905	62	18	is	be	AUX
ejpam-4905	62	19	a	a	DET
ejpam-4905	62	20	hop	hop	NOUN
ejpam-4905	62	21	dominating	dominating	NOUN
ejpam-4905	62	22	set	set	VERB
ejpam-4905	62	23	in	in	ADP
ejpam-4905	62	24	g.	g.	PROPN
ejpam-4905	62	25	the	the	DET
ejpam-4905	62	26	j2	j2	PROPN
ejpam-4905	62	27	-	-	PUNCT
ejpam-4905	62	28	hop	hop	PROPN
ejpam-4905	62	29	domination	domination	NOUN
ejpam-4905	62	30	number	number	NOUN
ejpam-4905	62	31	of	of	ADP
ejpam-4905	62	32	g	g	NOUN
ejpam-4905	62	33	,	,	PUNCT
ejpam-4905	62	34	denoted	denote	VERB
ejpam-4905	62	35	by	by	ADP
ejpam-4905	62	36	γj2h(g	γj2h(g	NOUN
ejpam-4905	62	37	)	)	PUNCT
ejpam-4905	62	38	,	,	PUNCT
ejpam-4905	62	39	is	be	AUX
ejpam-4905	62	40	the	the	DET
ejpam-4905	62	41	maximum	maximum	ADJ
ejpam-4905	62	42	cardinality	cardinality	NOUN
ejpam-4905	62	43	among	among	ADP
ejpam-4905	62	44	all	all	DET
ejpam-4905	62	45	j2	j2	PROPN
ejpam-4905	62	46	-	-	PUNCT
ejpam-4905	62	47	hop	hop	NOUN
ejpam-4905	62	48	dominating	dominating	NOUN
ejpam-4905	62	49	sets	set	NOUN
ejpam-4905	62	50	in	in	ADP
ejpam-4905	62	51	g.	g.	PROPN
ejpam-4905	62	52	any	any	DET
ejpam-4905	62	53	j2	j2	PROPN
ejpam-4905	62	54	-	-	PUNCT
ejpam-4905	62	55	hop	hop	NOUN
ejpam-4905	62	56	dominating	dominating	NOUN
ejpam-4905	62	57	set	set	VERB
ejpam-4905	62	58	t	t	PROPN
ejpam-4905	62	59	with	with	ADP
ejpam-4905	62	60	|t	|t	PROPN
ejpam-4905	62	61	|	|	ADV
ejpam-4905	62	62	=	=	SYM
ejpam-4905	62	63	γj2h(g	γj2h(g	NOUN
ejpam-4905	62	64	)	)	PUNCT
ejpam-4905	62	65	(	(	PUNCT
ejpam-4905	62	66	resp	resp	NOUN
ejpam-4905	62	67	.	.	PUNCT
ejpam-4905	63	1	|t	|t	PROPN
ejpam-4905	64	1	|	|	ADV
ejpam-4905	64	2	=	=	PRON
ejpam-4905	64	3	γh(g	γh(g	NOUN
ejpam-4905	64	4	)	)	PUNCT
ejpam-4905	64	5	)	)	PUNCT
ejpam-4905	64	6	,	,	PUNCT
ejpam-4905	64	7	is	be	AUX
ejpam-4905	64	8	called	call	VERB
ejpam-4905	64	9	a	a	DET
ejpam-4905	64	10	γj2h	γj2h	NOUN
ejpam-4905	64	11	-	-	PUNCT
ejpam-4905	64	12	set	set	NOUN
ejpam-4905	64	13	or	or	CCONJ
ejpam-4905	64	14	the	the	DET
ejpam-4905	64	15	maximum	maximum	ADJ
ejpam-4905	64	16	(	(	PUNCT
ejpam-4905	64	17	resp	resp	NOUN
ejpam-4905	64	18	.	.	PUNCT
ejpam-4905	65	1	minimum	minimum	ADJ
ejpam-4905	65	2	)	)	PUNCT
ejpam-4905	65	3	j2	j2	PROPN
ejpam-4905	65	4	-	-	PUNCT
ejpam-4905	65	5	hop	hop	PROPN
ejpam-4905	65	6	dominating	dominating	NOUN
ejpam-4905	65	7	set	set	NOUN
ejpam-4905	65	8	of	of	ADP
ejpam-4905	65	9	g.	g.	PROPN
ejpam-4905	65	10	example	example	NOUN
ejpam-4905	66	1	1	1	X
ejpam-4905	66	2	.	.	X
ejpam-4905	66	3	consider	consider	VERB
ejpam-4905	66	4	the	the	DET
ejpam-4905	66	5	graph	graph	NOUN
ejpam-4905	66	6	g	g	NOUN
ejpam-4905	66	7	in	in	ADP
ejpam-4905	66	8	figure	figure	NOUN
ejpam-4905	66	9	1	1	NUM
ejpam-4905	66	10	and	and	CCONJ
ejpam-4905	66	11	let	let	VERB
ejpam-4905	66	12	t	t	NOUN
ejpam-4905	66	13	=	=	SYM
ejpam-4905	66	14	{	{	PUNCT
ejpam-4905	66	15	u1	u1	NOUN
ejpam-4905	66	16	,	,	PUNCT
ejpam-4905	66	17	u2	u2	NOUN
ejpam-4905	66	18	,	,	PUNCT
ejpam-4905	66	19	.	.	PUNCT
ejpam-4905	66	20	.	.	PUNCT
ejpam-4905	67	1	.	.	PUNCT
ejpam-4905	67	2	,	,	PUNCT
ejpam-4905	67	3	u6	u6	PROPN
ejpam-4905	67	4	}	}	PUNCT
ejpam-4905	67	5	.	.	PUNCT
ejpam-4905	68	1	notice	notice	VERB
ejpam-4905	68	2	that	that	SCONJ
ejpam-4905	68	3	ui	ui	PROPN
ejpam-4905	68	4	∈	∈	PROPN
ejpam-4905	68	5	n2	n2	PROPN
ejpam-4905	68	6	g[ui	g[ui	PROPN
ejpam-4905	68	7	]	]	PUNCT
ejpam-4905	68	8	\	\	PROPN
ejpam-4905	68	9	n2	n2	PROPN
ejpam-4905	68	10	g[uj	g[uj	PROPN
ejpam-4905	68	11	]	]	PUNCT
ejpam-4905	68	12	∀	∀	PUNCT
ejpam-4905	69	1	i	i	NOUN
ejpam-4905	69	2	̸=	̸=	PROPN
ejpam-4905	69	3	j	j	PROPN
ejpam-4905	69	4	where	where	SCONJ
ejpam-4905	69	5	i	i	PRON
ejpam-4905	69	6	,	,	PUNCT
ejpam-4905	69	7	j	j	PROPN
ejpam-4905	69	8	∈	∈	PROPN
ejpam-4905	69	9	{	{	PUNCT
ejpam-4905	69	10	1	1	NUM
ejpam-4905	69	11	,	,	PUNCT
ejpam-4905	69	12	2	2	NUM
ejpam-4905	69	13	,	,	PUNCT
ejpam-4905	69	14	.	.	PUNCT
ejpam-4905	69	15	.	.	PUNCT
ejpam-4905	69	16	.	.	PUNCT
ejpam-4905	70	1	,	,	PUNCT
ejpam-4905	70	2	6	6	NUM
ejpam-4905	70	3	}	}	PUNCT
ejpam-4905	70	4	.	.	PUNCT
ejpam-4905	71	1	thus	thus	ADV
ejpam-4905	71	2	,	,	PUNCT
ejpam-4905	71	3	t	t	PROPN
ejpam-4905	71	4	is	be	AUX
ejpam-4905	71	5	a	a	DET
ejpam-4905	71	6	j2	j2	PROPN
ejpam-4905	71	7	-	-	PUNCT
ejpam-4905	71	8	set	set	NOUN
ejpam-4905	71	9	of	of	ADP
ejpam-4905	71	10	g.	g.	PROPN
ejpam-4905	71	11	since	since	SCONJ
ejpam-4905	71	12	n2	n2	PROPN
ejpam-4905	71	13	g[t	g[t	NOUN
ejpam-4905	71	14	]	]	X
ejpam-4905	71	15	=	=	SYM
ejpam-4905	71	16	v	v	X
ejpam-4905	71	17	(	(	PUNCT
ejpam-4905	71	18	g	g	NOUN
ejpam-4905	71	19	)	)	PUNCT
ejpam-4905	71	20	,	,	PUNCT
ejpam-4905	71	21	it	it	PRON
ejpam-4905	71	22	follows	follow	VERB
ejpam-4905	71	23	that	that	SCONJ
ejpam-4905	71	24	t	t	PROPN
ejpam-4905	71	25	is	be	AUX
ejpam-4905	71	26	a	a	DET
ejpam-4905	71	27	j2	j2	PROPN
ejpam-4905	71	28	-	-	PUNCT
ejpam-4905	71	29	hop	hop	NOUN
ejpam-4905	71	30	dominating	dominating	NOUN
ejpam-4905	71	31	set	set	NOUN
ejpam-4905	71	32	of	of	ADP
ejpam-4905	71	33	g.	g.	PROPN
ejpam-4905	71	34	observe	observe	VERB
ejpam-4905	71	35	that	that	SCONJ
ejpam-4905	71	36	n2	n2	PROPN
ejpam-4905	71	37	g[u7	g[u7	PROPN
ejpam-4905	71	38	]	]	PUNCT
ejpam-4905	71	39	⊆	⊆	NUM
ejpam-4905	71	40	n2	n2	ADJ
ejpam-4905	71	41	g[u3	g[u3	NOUN
ejpam-4905	71	42	]	]	PUNCT
ejpam-4905	71	43	,	,	PUNCT
ejpam-4905	71	44	n	n	PROPN
ejpam-4905	71	45	2	2	NUM
ejpam-4905	71	46	g[u8	g[u8	NOUN
ejpam-4905	71	47	]	]	X
ejpam-4905	71	48	⊆	⊆	NUM
ejpam-4905	71	49	n2	n2	ADJ
ejpam-4905	71	50	g[u3	g[u3	NOUN
ejpam-4905	71	51	]	]	PUNCT
ejpam-4905	71	52	,	,	PUNCT
ejpam-4905	71	53	n	n	PROPN
ejpam-4905	71	54	2	2	NUM
ejpam-4905	71	55	g[u9	g[u9	NOUN
ejpam-4905	71	56	]	]	X
ejpam-4905	71	57	⊆	⊆	NUM
ejpam-4905	71	58	n2	n2	ADJ
ejpam-4905	71	59	g[u1	g[u1	NOUN
ejpam-4905	71	60	]	]	X
ejpam-4905	71	61	,	,	PUNCT
ejpam-4905	71	62	and	and	CCONJ
ejpam-4905	71	63	n2	n2	ADJ
ejpam-4905	71	64	g[u10	g[u10	PROPN
ejpam-4905	71	65	]	]	X
ejpam-4905	71	66	⊆	⊆	NUM
ejpam-4905	71	67	n2	n2	ADJ
ejpam-4905	71	68	g[u3	g[u3	NOUN
ejpam-4905	71	69	]	]	PUNCT
ejpam-4905	71	70	.	.	PUNCT
ejpam-4905	72	1	thus	thus	ADV
ejpam-4905	72	2	,	,	PUNCT
ejpam-4905	72	3	t	t	PROPN
ejpam-4905	72	4	is	be	AUX
ejpam-4905	72	5	a	a	DET
ejpam-4905	72	6	maximum	maximum	ADJ
ejpam-4905	72	7	j2	j2	PROPN
ejpam-4905	72	8	-	-	PUNCT
ejpam-4905	72	9	hop	hop	NOUN
ejpam-4905	72	10	dominating	dominating	NOUN
ejpam-4905	72	11	set	set	NOUN
ejpam-4905	72	12	of	of	ADP
ejpam-4905	72	13	g.	g.	PROPN
ejpam-4905	72	14	hence	hence	ADV
ejpam-4905	72	15	,	,	PUNCT
ejpam-4905	72	16	γj2h(g	γj2h(g	NOUN
ejpam-4905	72	17	)	)	PUNCT
ejpam-4905	72	18	=	=	SYM
ejpam-4905	72	19	6	6	NUM
ejpam-4905	72	20	.	.	PUNCT
ejpam-4905	73	1	g	g	NOUN
ejpam-4905	73	2	:	:	PUNCT
ejpam-4905	73	3	u3	u3	PROPN
ejpam-4905	73	4	u5	u5	PROPN
ejpam-4905	73	5	u1	u1	PROPN
ejpam-4905	73	6	u2	u2	PROPN
ejpam-4905	73	7	u4	u4	PROPN
ejpam-4905	73	8	u6	u6	PROPN
ejpam-4905	73	9	u7	u7	PROPN
ejpam-4905	73	10	u8	u8	PROPN
ejpam-4905	73	11	u9	u9	PROPN
ejpam-4905	73	12	u10	u10	PROPN
ejpam-4905	73	13	figure	figure	NOUN
ejpam-4905	73	14	1	1	NUM
ejpam-4905	73	15	:	:	PUNCT
ejpam-4905	73	16	graph	graph	VERB
ejpam-4905	73	17	g	g	NOUN
ejpam-4905	73	18	with	with	ADP
ejpam-4905	73	19	γj2h(g	γj2h(g	NOUN
ejpam-4905	73	20	)	)	PUNCT
ejpam-4905	73	21	=	=	SYM
ejpam-4905	73	22	6	6	NUM
ejpam-4905	73	23	j.	j.	PROPN
ejpam-4905	73	24	hassan	hassan	PROPN
ejpam-4905	73	25	,	,	PUNCT
ejpam-4905	73	26	a.	a.	PROPN
ejpam-4905	73	27	bakkang	bakkang	PROPN
ejpam-4905	73	28	,	,	PUNCT
ejpam-4905	73	29	a.	a.	NOUN
ejpam-4905	73	30	sappari	sappari	PROPN
ejpam-4905	73	31	/	/	SYM
ejpam-4905	73	32	eur	eur	PROPN
ejpam-4905	73	33	.	.	PUNCT
ejpam-4905	74	1	j.	j.	PROPN
ejpam-4905	74	2	pure	pure	PROPN
ejpam-4905	74	3	appl	appl	PROPN
ejpam-4905	74	4	.	.	PROPN
ejpam-4905	74	5	math	math	PROPN
ejpam-4905	74	6	,	,	PUNCT
ejpam-4905	74	7	16	16	NUM
ejpam-4905	74	8	(	(	PUNCT
ejpam-4905	74	9	4	4	NUM
ejpam-4905	74	10	)	)	PUNCT
ejpam-4905	74	11	(	(	PUNCT
ejpam-4905	74	12	2023	2023	NUM
ejpam-4905	74	13	)	)	PUNCT
ejpam-4905	74	14	,	,	PUNCT
ejpam-4905	74	15	2118	2118	NUM
ejpam-4905	74	16	-	-	SYM
ejpam-4905	74	17	2131	2131	NUM
ejpam-4905	74	18	2121	2121	NUM
ejpam-4905	74	19	theorem	theorem	VERB
ejpam-4905	74	20	1	1	NUM
ejpam-4905	74	21	.	.	PUNCT
ejpam-4905	75	1	let	let	VERB
ejpam-4905	75	2	g	g	NOUN
ejpam-4905	75	3	be	be	AUX
ejpam-4905	75	4	any	any	DET
ejpam-4905	75	5	graph	graph	NOUN
ejpam-4905	75	6	of	of	ADP
ejpam-4905	75	7	order	order	NOUN
ejpam-4905	75	8	m	m	VERB
ejpam-4905	75	9	≥	≥	NOUN
ejpam-4905	75	10	1	1	NUM
ejpam-4905	75	11	.	.	PUNCT
ejpam-4905	76	1	then	then	ADV
ejpam-4905	76	2	each	each	PRON
ejpam-4905	76	3	of	of	ADP
ejpam-4905	76	4	the	the	DET
ejpam-4905	76	5	following	follow	VERB
ejpam-4905	76	6	holds	hold	VERB
ejpam-4905	76	7	:	:	PUNCT
ejpam-4905	76	8	(	(	PUNCT
ejpam-4905	76	9	i	i	NOUN
ejpam-4905	76	10	)	)	PUNCT
ejpam-4905	76	11	n	n	CCONJ
ejpam-4905	76	12	⊆	⊆	NUM
ejpam-4905	76	13	v	v	NOUN
ejpam-4905	76	14	(	(	PUNCT
ejpam-4905	76	15	g	g	NOUN
ejpam-4905	76	16	)	)	PUNCT
ejpam-4905	76	17	is	be	AUX
ejpam-4905	76	18	a	a	DET
ejpam-4905	76	19	γh	γh	ADV
ejpam-4905	76	20	-	-	PUNCT
ejpam-4905	76	21	set	set	NOUN
ejpam-4905	76	22	in	in	ADP
ejpam-4905	76	23	g	g	PROPN
ejpam-4905	76	24	if	if	SCONJ
ejpam-4905	77	1	and	and	CCONJ
ejpam-4905	77	2	only	only	ADV
ejpam-4905	77	3	if	if	SCONJ
ejpam-4905	77	4	n	n	PRON
ejpam-4905	77	5	is	be	AUX
ejpam-4905	77	6	a	a	DET
ejpam-4905	77	7	minimum	minimum	ADJ
ejpam-4905	77	8	j2hop	j2hop	NOUN
ejpam-4905	77	9	dominating	dominating	NOUN
ejpam-4905	77	10	set	set	VERB
ejpam-4905	77	11	in	in	ADP
ejpam-4905	77	12	g.	g.	PROPN
ejpam-4905	77	13	(	(	PUNCT
ejpam-4905	77	14	ii	ii	PROPN
ejpam-4905	77	15	)	)	PUNCT
ejpam-4905	77	16	γh(g	γh(g	NOUN
ejpam-4905	77	17	)	)	PUNCT
ejpam-4905	78	1	≤	≤	NUM
ejpam-4905	78	2	γj2h(g	γj2h(g	NOUN
ejpam-4905	78	3	)	)	PUNCT
ejpam-4905	78	4	.	.	PUNCT
ejpam-4905	79	1	(	(	PUNCT
ejpam-4905	79	2	iii	iii	X
ejpam-4905	79	3	)	)	PUNCT
ejpam-4905	79	4	1	1	NUM
ejpam-4905	79	5	≤	≤	NUM
ejpam-4905	79	6	γj2h(g	γj2h(g	NOUN
ejpam-4905	79	7	)	)	PUNCT
ejpam-4905	79	8	≤	≤	NUM
ejpam-4905	79	9	m.	m.	NOUN
ejpam-4905	79	10	proof	proof	NOUN
ejpam-4905	79	11	.	.	PUNCT
ejpam-4905	80	1	(	(	PUNCT
ejpam-4905	80	2	i	i	NOUN
ejpam-4905	80	3	)	)	PUNCT
ejpam-4905	80	4	suppose	suppose	VERB
ejpam-4905	80	5	that	that	SCONJ
ejpam-4905	80	6	n	n	PROPN
ejpam-4905	80	7	⊆	⊆	NUM
ejpam-4905	80	8	v	v	NOUN
ejpam-4905	80	9	(	(	PUNCT
ejpam-4905	80	10	g	g	NOUN
ejpam-4905	80	11	)	)	PUNCT
ejpam-4905	80	12	is	be	AUX
ejpam-4905	80	13	a	a	DET
ejpam-4905	80	14	γh	γh	ADV
ejpam-4905	80	15	-	-	PUNCT
ejpam-4905	80	16	set	set	NOUN
ejpam-4905	80	17	in	in	ADP
ejpam-4905	80	18	g.	g.	PROPN
ejpam-4905	80	19	then	then	ADV
ejpam-4905	80	20	n	n	PROPN
ejpam-4905	80	21	is	be	AUX
ejpam-4905	80	22	a	a	DET
ejpam-4905	80	23	minimum	minimum	ADJ
ejpam-4905	80	24	hop	hop	NOUN
ejpam-4905	80	25	dominating	dominating	NOUN
ejpam-4905	80	26	set	set	VERB
ejpam-4905	80	27	in	in	ADP
ejpam-4905	80	28	g.	g.	PROPN
ejpam-4905	80	29	it	it	PRON
ejpam-4905	80	30	suffices	suffice	VERB
ejpam-4905	80	31	to	to	PART
ejpam-4905	80	32	show	show	VERB
ejpam-4905	80	33	that	that	SCONJ
ejpam-4905	80	34	n	n	PRON
ejpam-4905	80	35	is	be	AUX
ejpam-4905	80	36	a	a	DET
ejpam-4905	80	37	j2	j2	NOUN
ejpam-4905	80	38	-	-	PUNCT
ejpam-4905	80	39	set	set	NOUN
ejpam-4905	80	40	in	in	ADP
ejpam-4905	80	41	g.	g.	PROPN
ejpam-4905	80	42	suppose	suppose	VERB
ejpam-4905	80	43	on	on	ADP
ejpam-4905	80	44	the	the	DET
ejpam-4905	80	45	contrary	contrary	NOUN
ejpam-4905	80	46	that	that	PRON
ejpam-4905	80	47	n	n	VERB
ejpam-4905	80	48	is	be	AUX
ejpam-4905	80	49	not	not	PART
ejpam-4905	80	50	a	a	DET
ejpam-4905	80	51	j2	j2	NOUN
ejpam-4905	80	52	-	-	PUNCT
ejpam-4905	80	53	set	set	NOUN
ejpam-4905	80	54	in	in	ADP
ejpam-4905	80	55	g.	g.	PROPN
ejpam-4905	80	56	then	then	ADV
ejpam-4905	80	57	there	there	PRON
ejpam-4905	80	58	exist	exist	VERB
ejpam-4905	80	59	x	x	NOUN
ejpam-4905	80	60	,	,	PUNCT
ejpam-4905	80	61	y	y	PROPN
ejpam-4905	80	62	∈	∈	PROPN
ejpam-4905	80	63	n	n	PRON
ejpam-4905	80	64	such	such	ADJ
ejpam-4905	80	65	that	that	SCONJ
ejpam-4905	80	66	either	either	CCONJ
ejpam-4905	80	67	n2	n2	PROPN
ejpam-4905	80	68	g[x]\n2	g[x]\n2	X
ejpam-4905	80	69	g[y	g[y	NOUN
ejpam-4905	80	70	]	]	X
ejpam-4905	80	71	=	=	SYM
ejpam-4905	80	72	∅	∅	NOUN
ejpam-4905	80	73	or	or	CCONJ
ejpam-4905	80	74	n2	n2	ADJ
ejpam-4905	80	75	g[y	g[y	NOUN
ejpam-4905	80	76	]	]	PUNCT
ejpam-4905	80	77	\	\	PROPN
ejpam-4905	80	78	n2	n2	PROPN
ejpam-4905	80	79	g[x	g[x	PROPN
ejpam-4905	80	80	]	]	X
ejpam-4905	80	81	=	=	PUNCT
ejpam-4905	80	82	∅.	∅.	NOUN
ejpam-4905	80	83	this	this	PRON
ejpam-4905	80	84	means	mean	VERB
ejpam-4905	80	85	that	that	SCONJ
ejpam-4905	80	86	either	either	CCONJ
ejpam-4905	80	87	n2	n2	PROPN
ejpam-4905	80	88	g[x	g[x	PROPN
ejpam-4905	80	89	]	]	X
ejpam-4905	80	90	⊆	⊆	NUM
ejpam-4905	80	91	n2	n2	ADJ
ejpam-4905	80	92	g[y	g[y	NOUN
ejpam-4905	80	93	]	]	PUNCT
ejpam-4905	80	94	or	or	CCONJ
ejpam-4905	80	95	n2	n2	ADJ
ejpam-4905	80	96	g[y	g[y	NOUN
ejpam-4905	80	97	]	]	X
ejpam-4905	80	98	⊆	⊆	NUM
ejpam-4905	80	99	n2	n2	NOUN
ejpam-4905	80	100	g[x	g[x	NOUN
ejpam-4905	80	101	]	]	PUNCT
ejpam-4905	80	102	.	.	PUNCT
ejpam-4905	81	1	if	if	SCONJ
ejpam-4905	81	2	n2	n2	PROPN
ejpam-4905	81	3	g[x	g[x	PROPN
ejpam-4905	81	4	]	]	X
ejpam-4905	81	5	⊆	⊆	NUM
ejpam-4905	81	6	n2	n2	ADJ
ejpam-4905	81	7	g[y	g[y	NOUN
ejpam-4905	81	8	]	]	PUNCT
ejpam-4905	81	9	,	,	PUNCT
ejpam-4905	81	10	then	then	ADV
ejpam-4905	81	11	d′	d′	X
ejpam-4905	81	12	=	=	SYM
ejpam-4905	81	13	n	n	CCONJ
ejpam-4905	81	14	\	\	NOUN
ejpam-4905	81	15	{	{	PUNCT
ejpam-4905	81	16	x	x	NOUN
ejpam-4905	81	17	}	}	PUNCT
ejpam-4905	81	18	is	be	AUX
ejpam-4905	81	19	a	a	DET
ejpam-4905	81	20	hop	hop	NOUN
ejpam-4905	81	21	dominating	dominating	NOUN
ejpam-4905	81	22	set	set	NOUN
ejpam-4905	81	23	in	in	ADP
ejpam-4905	81	24	g	g	NOUN
ejpam-4905	81	25	,	,	PUNCT
ejpam-4905	81	26	contradicting	contradict	VERB
ejpam-4905	81	27	the	the	DET
ejpam-4905	81	28	minimality	minimality	NOUN
ejpam-4905	81	29	of	of	ADP
ejpam-4905	81	30	n	n	PROPN
ejpam-4905	81	31	.	.	PUNCT
ejpam-4905	82	1	similarly	similarly	ADV
ejpam-4905	82	2	,	,	PUNCT
ejpam-4905	82	3	when	when	SCONJ
ejpam-4905	82	4	n2	n2	ADJ
ejpam-4905	82	5	g[y	g[y	NOUN
ejpam-4905	82	6	]	]	X
ejpam-4905	82	7	⊆	⊆	NUM
ejpam-4905	82	8	n2	n2	NOUN
ejpam-4905	82	9	g[x	g[x	PROPN
ejpam-4905	82	10	]	]	PUNCT
ejpam-4905	82	11	.	.	PUNCT
ejpam-4905	83	1	consequently	consequently	ADV
ejpam-4905	83	2	,	,	PUNCT
ejpam-4905	83	3	n	n	PRON
ejpam-4905	83	4	is	be	AUX
ejpam-4905	83	5	a	a	DET
ejpam-4905	83	6	minimum	minimum	ADJ
ejpam-4905	83	7	j2	j2	NOUN
ejpam-4905	83	8	-	-	PUNCT
ejpam-4905	83	9	hop	hop	NOUN
ejpam-4905	83	10	dominating	dominating	NOUN
ejpam-4905	83	11	set	set	NOUN
ejpam-4905	83	12	of	of	ADP
ejpam-4905	83	13	g.	g.	PROPN
ejpam-4905	83	14	conversely	conversely	ADV
ejpam-4905	83	15	,	,	PUNCT
ejpam-4905	83	16	suppose	suppose	VERB
ejpam-4905	83	17	that	that	SCONJ
ejpam-4905	83	18	n	n	PRON
ejpam-4905	83	19	is	be	AUX
ejpam-4905	83	20	a	a	DET
ejpam-4905	83	21	minimum	minimum	ADJ
ejpam-4905	83	22	j2	j2	NOUN
ejpam-4905	83	23	-	-	PUNCT
ejpam-4905	83	24	hop	hop	NOUN
ejpam-4905	83	25	dominating	dominating	NOUN
ejpam-4905	83	26	set	set	NOUN
ejpam-4905	83	27	of	of	ADP
ejpam-4905	83	28	g.	g.	PROPN
ejpam-4905	83	29	then	then	ADV
ejpam-4905	83	30	|n	|n	PUNCT
ejpam-4905	83	31	|	|	ADV
ejpam-4905	83	32	=	=	PRON
ejpam-4905	83	33	γh(g	γh(g	NOUN
ejpam-4905	83	34	)	)	PUNCT
ejpam-4905	83	35	(	(	PUNCT
ejpam-4905	83	36	by	by	ADP
ejpam-4905	83	37	definition	definition	NOUN
ejpam-4905	83	38	)	)	PUNCT
ejpam-4905	83	39	.	.	PUNCT
ejpam-4905	84	1	it	it	PRON
ejpam-4905	84	2	follows	follow	VERB
ejpam-4905	84	3	that	that	SCONJ
ejpam-4905	84	4	n	n	PRON
ejpam-4905	84	5	is	be	AUX
ejpam-4905	84	6	a	a	DET
ejpam-4905	84	7	γh	γh	ADV
ejpam-4905	84	8	-	-	PUNCT
ejpam-4905	84	9	set	set	NOUN
ejpam-4905	84	10	in	in	ADP
ejpam-4905	84	11	g.	g.	PROPN
ejpam-4905	84	12	(	(	PUNCT
ejpam-4905	84	13	ii	ii	PROPN
ejpam-4905	84	14	)	)	PUNCT
ejpam-4905	84	15	let	let	VERB
ejpam-4905	84	16	s	s	PRON
ejpam-4905	84	17	be	be	AUX
ejpam-4905	84	18	a	a	DET
ejpam-4905	84	19	γh	γh	ADV
ejpam-4905	84	20	-	-	PUNCT
ejpam-4905	84	21	set	set	NOUN
ejpam-4905	84	22	of	of	ADP
ejpam-4905	84	23	g.	g.	PROPN
ejpam-4905	84	24	then	then	ADV
ejpam-4905	84	25	by	by	ADP
ejpam-4905	84	26	(	(	PUNCT
ejpam-4905	84	27	i	i	NOUN
ejpam-4905	84	28	)	)	PUNCT
ejpam-4905	84	29	,	,	PUNCT
ejpam-4905	84	30	s	s	VERB
ejpam-4905	84	31	is	be	AUX
ejpam-4905	84	32	a	a	DET
ejpam-4905	84	33	minimum	minimum	ADJ
ejpam-4905	84	34	j2	j2	NOUN
ejpam-4905	84	35	-	-	PUNCT
ejpam-4905	84	36	hop	hop	NOUN
ejpam-4905	84	37	dominating	dominating	NOUN
ejpam-4905	84	38	set	set	VERB
ejpam-4905	84	39	in	in	ADP
ejpam-4905	84	40	g.	g.	PROPN
ejpam-4905	84	41	since	since	SCONJ
ejpam-4905	84	42	γj2h(g	γj2h(g	NOUN
ejpam-4905	84	43	)	)	PUNCT
ejpam-4905	84	44	is	be	AUX
ejpam-4905	84	45	the	the	DET
ejpam-4905	84	46	maximum	maximum	ADJ
ejpam-4905	84	47	cardinality	cardinality	NOUN
ejpam-4905	84	48	among	among	ADP
ejpam-4905	84	49	all	all	DET
ejpam-4905	84	50	j2	j2	PROPN
ejpam-4905	84	51	-	-	PUNCT
ejpam-4905	84	52	hop	hop	NOUN
ejpam-4905	84	53	dominating	dominating	NOUN
ejpam-4905	84	54	sets	set	NOUN
ejpam-4905	84	55	in	in	ADP
ejpam-4905	84	56	g	g	NOUN
ejpam-4905	84	57	,	,	PUNCT
ejpam-4905	84	58	it	it	PRON
ejpam-4905	84	59	follows	follow	VERB
ejpam-4905	84	60	that	that	PRON
ejpam-4905	84	61	γh(g	γh(g	NOUN
ejpam-4905	84	62	)	)	PUNCT
ejpam-4905	84	63	=	=	SYM
ejpam-4905	84	64	|s|	|s|	VERB
ejpam-4905	84	65	≤	≤	NUM
ejpam-4905	84	66	γj2h(g	γj2h(g	NOUN
ejpam-4905	84	67	)	)	PUNCT
ejpam-4905	84	68	.	.	PUNCT
ejpam-4905	85	1	(	(	PUNCT
ejpam-4905	85	2	iii	iii	NOUN
ejpam-4905	85	3	)	)	PUNCT
ejpam-4905	85	4	since	since	SCONJ
ejpam-4905	85	5	γh(g	γh(g	NOUN
ejpam-4905	85	6	)	)	PUNCT
ejpam-4905	85	7	≥	≥	NOUN
ejpam-4905	85	8	1	1	NUM
ejpam-4905	85	9	for	for	ADP
ejpam-4905	85	10	any	any	DET
ejpam-4905	85	11	graph	graph	NOUN
ejpam-4905	85	12	g	g	NOUN
ejpam-4905	85	13	of	of	ADP
ejpam-4905	85	14	order	order	NOUN
ejpam-4905	85	15	m	m	VERB
ejpam-4905	85	16	≥	≥	NOUN
ejpam-4905	85	17	1	1	NUM
ejpam-4905	85	18	,	,	PUNCT
ejpam-4905	85	19	we	we	PRON
ejpam-4905	85	20	have	have	VERB
ejpam-4905	85	21	γj2h(g	γj2h(g	NOUN
ejpam-4905	85	22	)	)	PUNCT
ejpam-4905	85	23	≥	≥	NOUN
ejpam-4905	85	24	1	1	NUM
ejpam-4905	85	25	by	by	ADP
ejpam-4905	85	26	(	(	PUNCT
ejpam-4905	85	27	ii	ii	NOUN
ejpam-4905	85	28	)	)	PUNCT
ejpam-4905	85	29	.	.	PUNCT
ejpam-4905	86	1	since	since	SCONJ
ejpam-4905	86	2	any	any	DET
ejpam-4905	86	3	j2	j2	PROPN
ejpam-4905	86	4	-	-	PUNCT
ejpam-4905	86	5	hop	hop	NOUN
ejpam-4905	86	6	dominating	dominating	NOUN
ejpam-4905	86	7	set	set	NOUN
ejpam-4905	86	8	n	n	PROPN
ejpam-4905	86	9	of	of	ADP
ejpam-4905	86	10	g	g	PROPN
ejpam-4905	86	11	is	be	AUX
ejpam-4905	86	12	always	always	ADV
ejpam-4905	86	13	a	a	DET
ejpam-4905	86	14	subset	subset	NOUN
ejpam-4905	86	15	of	of	ADP
ejpam-4905	86	16	v	v	NOUN
ejpam-4905	86	17	(	(	PUNCT
ejpam-4905	86	18	g	g	NOUN
ejpam-4905	86	19	)	)	PUNCT
ejpam-4905	86	20	,	,	PUNCT
ejpam-4905	86	21	it	it	PRON
ejpam-4905	86	22	follows	follow	VERB
ejpam-4905	86	23	that	that	SCONJ
ejpam-4905	86	24	γj2h(g	γj2h(g	NOUN
ejpam-4905	86	25	)	)	PUNCT
ejpam-4905	86	26	≤	≤	NOUN
ejpam-4905	86	27	|v	|v	X
ejpam-4905	86	28	(	(	PUNCT
ejpam-4905	86	29	g)|	g)|	NOUN
ejpam-4905	86	30	=	=	NOUN
ejpam-4905	86	31	m.	m.	NOUN
ejpam-4905	86	32	therefore	therefore	ADV
ejpam-4905	86	33	,	,	PUNCT
ejpam-4905	86	34	1	1	NUM
ejpam-4905	86	35	≤	≤	NUM
ejpam-4905	86	36	γj2h(g	γj2h(g	NOUN
ejpam-4905	86	37	)	)	PUNCT
ejpam-4905	86	38	≤	≤	NOUN
ejpam-4905	86	39	m.	m.	NOUN
ejpam-4905	86	40	theorem	theorem	NOUN
ejpam-4905	86	41	2	2	X
ejpam-4905	86	42	.	.	PUNCT
ejpam-4905	87	1	let	let	VERB
ejpam-4905	87	2	g	g	NOUN
ejpam-4905	87	3	be	be	AUX
ejpam-4905	87	4	any	any	DET
ejpam-4905	87	5	graph	graph	NOUN
ejpam-4905	87	6	.	.	PUNCT
ejpam-4905	88	1	then	then	ADV
ejpam-4905	88	2	n	n	PROPN
ejpam-4905	88	3	⊆	⊆	NUM
ejpam-4905	88	4	v	v	NOUN
ejpam-4905	88	5	(	(	PUNCT
ejpam-4905	88	6	g	g	NOUN
ejpam-4905	88	7	)	)	PUNCT
ejpam-4905	88	8	is	be	AUX
ejpam-4905	88	9	a	a	DET
ejpam-4905	88	10	maximum	maximum	ADJ
ejpam-4905	88	11	j2	j2	NOUN
ejpam-4905	88	12	-	-	PUNCT
ejpam-4905	88	13	set	set	VERB
ejpam-4905	88	14	if	if	SCONJ
ejpam-4905	88	15	and	and	CCONJ
ejpam-4905	88	16	only	only	ADV
ejpam-4905	88	17	if	if	SCONJ
ejpam-4905	88	18	n	n	PRON
ejpam-4905	88	19	is	be	AUX
ejpam-4905	88	20	a	a	DET
ejpam-4905	88	21	γj2h	γj2h	NOUN
ejpam-4905	88	22	-	-	PUNCT
ejpam-4905	88	23	set	set	NOUN
ejpam-4905	88	24	of	of	ADP
ejpam-4905	88	25	g.	g.	PROPN
ejpam-4905	88	26	proof	proof	PROPN
ejpam-4905	88	27	.	.	PUNCT
ejpam-4905	89	1	let	let	VERB
ejpam-4905	89	2	n	n	PRON
ejpam-4905	89	3	be	be	AUX
ejpam-4905	89	4	a	a	DET
ejpam-4905	89	5	maximum	maximum	ADJ
ejpam-4905	89	6	j2	j2	NOUN
ejpam-4905	89	7	-	-	PUNCT
ejpam-4905	89	8	set	set	NOUN
ejpam-4905	89	9	of	of	ADP
ejpam-4905	89	10	g.	g.	PROPN
ejpam-4905	89	11	assume	assume	VERB
ejpam-4905	89	12	that	that	SCONJ
ejpam-4905	89	13	n	n	PRON
ejpam-4905	89	14	is	be	AUX
ejpam-4905	89	15	not	not	PART
ejpam-4905	89	16	a	a	DET
ejpam-4905	89	17	hop	hop	NOUN
ejpam-4905	89	18	dominating	dominating	NOUN
ejpam-4905	89	19	set	set	VERB
ejpam-4905	89	20	in	in	ADP
ejpam-4905	89	21	g.	g.	PROPN
ejpam-4905	89	22	then	then	ADV
ejpam-4905	89	23	there	there	PRON
ejpam-4905	89	24	exists	exist	VERB
ejpam-4905	89	25	u	u	PROPN
ejpam-4905	89	26	∈	∈	PROPN
ejpam-4905	89	27	v	v	ADP
ejpam-4905	89	28	(	(	PUNCT
ejpam-4905	89	29	g	g	NOUN
ejpam-4905	89	30	)	)	PUNCT
ejpam-4905	89	31	\n	\n	PUNCT
ejpam-4905	90	1	such	such	ADJ
ejpam-4905	90	2	that	that	DET
ejpam-4905	90	3	u	u	PROPN
ejpam-4905	90	4	/∈	/∈	PROPN
ejpam-4905	90	5	n2	n2	NOUN
ejpam-4905	90	6	g[n	g[n	X
ejpam-4905	90	7	]	]	X
ejpam-4905	90	8	.	.	PUNCT
ejpam-4905	91	1	this	this	PRON
ejpam-4905	91	2	implies	imply	VERB
ejpam-4905	91	3	that	that	SCONJ
ejpam-4905	91	4	u	u	PROPN
ejpam-4905	91	5	/∈	/∈	PROPN
ejpam-4905	91	6	n2	n2	PROPN
ejpam-4905	91	7	g[v	g[v	PROPN
ejpam-4905	91	8	]	]	PUNCT
ejpam-4905	91	9	for	for	ADP
ejpam-4905	91	10	every	every	DET
ejpam-4905	91	11	v	v	NOUN
ejpam-4905	91	12	∈	∈	NOUN
ejpam-4905	91	13	n	n	NOUN
ejpam-4905	91	14	.	.	PUNCT
ejpam-4905	92	1	let	let	VERB
ejpam-4905	92	2	n	n	PRON
ejpam-4905	92	3	′	′	NOUN
ejpam-4905	92	4	=	=	PUNCT
ejpam-4905	92	5	{	{	PUNCT
ejpam-4905	92	6	u}∪n	u}∪n	NOUN
ejpam-4905	92	7	.	.	PUNCT
ejpam-4905	93	1	since	since	SCONJ
ejpam-4905	93	2	n	n	NUM
ejpam-4905	93	3	is	be	AUX
ejpam-4905	93	4	a	a	DET
ejpam-4905	93	5	j2	j2	NOUN
ejpam-4905	93	6	-	-	PUNCT
ejpam-4905	93	7	set	set	NOUN
ejpam-4905	93	8	in	in	ADP
ejpam-4905	93	9	g	g	PROPN
ejpam-4905	93	10	and	and	CCONJ
ejpam-4905	93	11	u	u	NOUN
ejpam-4905	93	12	∈	∈	PROPN
ejpam-4905	93	13	n2	n2	PROPN
ejpam-4905	93	14	g[u	g[u	PROPN
ejpam-4905	93	15	]	]	PUNCT
ejpam-4905	93	16	,	,	PUNCT
ejpam-4905	93	17	it	it	PRON
ejpam-4905	93	18	follows	follow	VERB
ejpam-4905	93	19	that	that	PRON
ejpam-4905	93	20	n2	n2	ADJ
ejpam-4905	93	21	g[a	g[a	PROPN
ejpam-4905	93	22	]	]	PUNCT
ejpam-4905	93	23	\	\	PROPN
ejpam-4905	93	24	n2	n2	PROPN
ejpam-4905	93	25	g[b	g[b	PROPN
ejpam-4905	93	26	]	]	PUNCT
ejpam-4905	93	27	̸=	̸=	PROPN
ejpam-4905	93	28	∅	∅	NOUN
ejpam-4905	93	29	and	and	CCONJ
ejpam-4905	93	30	n2	n2	PROPN
ejpam-4905	93	31	g[b	g[b	PROPN
ejpam-4905	93	32	]	]	PUNCT
ejpam-4905	93	33	\	\	PROPN
ejpam-4905	93	34	n2	n2	PROPN
ejpam-4905	93	35	g[a	g[a	PROPN
ejpam-4905	93	36	]	]	X
ejpam-4905	93	37	̸=	̸=	PROPN
ejpam-4905	93	38	∅	∅	NOUN
ejpam-4905	93	39	for	for	ADP
ejpam-4905	93	40	every	every	DET
ejpam-4905	93	41	a	a	DET
ejpam-4905	93	42	̸=	̸=	PROPN
ejpam-4905	93	43	b	b	PROPN
ejpam-4905	93	44	,	,	PUNCT
ejpam-4905	93	45	where	where	SCONJ
ejpam-4905	93	46	a	a	PRON
ejpam-4905	93	47	,	,	PUNCT
ejpam-4905	93	48	b	b	X
ejpam-4905	93	49	∈	∈	PROPN
ejpam-4905	93	50	n	n	PRON
ejpam-4905	93	51	′.	′.	NOUN
ejpam-4905	93	52	this	this	PRON
ejpam-4905	93	53	means	mean	VERB
ejpam-4905	93	54	that	that	SCONJ
ejpam-4905	93	55	n	n	ADV
ejpam-4905	93	56	′	′	NOUN
ejpam-4905	93	57	is	be	AUX
ejpam-4905	93	58	a	a	DET
ejpam-4905	93	59	j2	j2	NOUN
ejpam-4905	93	60	-	-	PUNCT
ejpam-4905	93	61	set	set	NOUN
ejpam-4905	93	62	in	in	ADP
ejpam-4905	93	63	g	g	NOUN
ejpam-4905	93	64	,	,	PUNCT
ejpam-4905	93	65	contradicting	contradict	VERB
ejpam-4905	93	66	the	the	DET
ejpam-4905	93	67	maximality	maximality	NOUN
ejpam-4905	93	68	of	of	ADP
ejpam-4905	93	69	n	n	PROPN
ejpam-4905	93	70	.	.	PUNCT
ejpam-4905	94	1	hence	hence	ADV
ejpam-4905	94	2	,	,	PUNCT
ejpam-4905	94	3	n	n	PRON
ejpam-4905	94	4	is	be	AUX
ejpam-4905	94	5	a	a	DET
ejpam-4905	94	6	hop	hop	NOUN
ejpam-4905	94	7	dominating	dominating	NOUN
ejpam-4905	94	8	set	set	NOUN
ejpam-4905	94	9	of	of	ADP
ejpam-4905	94	10	g.	g.	PROPN
ejpam-4905	94	11	since	since	SCONJ
ejpam-4905	94	12	n	n	NUM
ejpam-4905	94	13	is	be	AUX
ejpam-4905	94	14	a	a	DET
ejpam-4905	94	15	maximum	maximum	ADJ
ejpam-4905	94	16	j2	j2	NOUN
ejpam-4905	94	17	-	-	PUNCT
ejpam-4905	94	18	set	set	NOUN
ejpam-4905	94	19	of	of	ADP
ejpam-4905	94	20	g	g	NOUN
ejpam-4905	94	21	,	,	PUNCT
ejpam-4905	94	22	n	n	X
ejpam-4905	94	23	is	be	AUX
ejpam-4905	94	24	a	a	DET
ejpam-4905	94	25	maximum	maximum	ADJ
ejpam-4905	94	26	j2	j2	PROPN
ejpam-4905	94	27	-	-	PUNCT
ejpam-4905	94	28	hop	hop	NOUN
ejpam-4905	94	29	dominating	dominating	NOUN
ejpam-4905	94	30	set	set	NOUN
ejpam-4905	94	31	of	of	ADP
ejpam-4905	94	32	g	g	PROPN
ejpam-4905	94	33	,	,	PUNCT
ejpam-4905	94	34	that	that	ADV
ejpam-4905	94	35	is	is	ADV
ejpam-4905	94	36	,	,	PUNCT
ejpam-4905	94	37	n	n	PRON
ejpam-4905	94	38	is	be	AUX
ejpam-4905	94	39	a	a	DET
ejpam-4905	94	40	γj2h	γj2h	NOUN
ejpam-4905	94	41	-	-	PUNCT
ejpam-4905	94	42	set	set	NOUN
ejpam-4905	94	43	of	of	ADP
ejpam-4905	94	44	g.	g.	NOUN
ejpam-4905	94	45	conversely	conversely	ADV
ejpam-4905	94	46	,	,	PUNCT
ejpam-4905	94	47	suppose	suppose	VERB
ejpam-4905	94	48	that	that	SCONJ
ejpam-4905	94	49	n	n	PRON
ejpam-4905	94	50	is	be	AUX
ejpam-4905	94	51	a	a	DET
ejpam-4905	94	52	γj2h	γj2h	NOUN
ejpam-4905	94	53	-	-	PUNCT
ejpam-4905	94	54	set	set	NOUN
ejpam-4905	94	55	of	of	ADP
ejpam-4905	94	56	g.	g.	PROPN
ejpam-4905	94	57	then	then	ADV
ejpam-4905	94	58	n	n	PROPN
ejpam-4905	94	59	is	be	AUX
ejpam-4905	94	60	a	a	DET
ejpam-4905	94	61	maximum	maximum	ADJ
ejpam-4905	94	62	j2	j2	PROPN
ejpam-4905	94	63	-	-	PUNCT
ejpam-4905	94	64	hop	hop	NOUN
ejpam-4905	94	65	dominating	dominating	NOUN
ejpam-4905	94	66	set	set	NOUN
ejpam-4905	94	67	of	of	ADP
ejpam-4905	94	68	g.	g.	PROPN
ejpam-4905	94	69	hence	hence	ADV
ejpam-4905	94	70	,	,	PUNCT
ejpam-4905	94	71	the	the	DET
ejpam-4905	94	72	assertion	assertion	NOUN
ejpam-4905	94	73	follows	follow	VERB
ejpam-4905	94	74	.	.	PUNCT
ejpam-4905	95	1	the	the	DET
ejpam-4905	95	2	following	following	ADJ
ejpam-4905	95	3	result	result	NOUN
ejpam-4905	95	4	follows	follow	VERB
ejpam-4905	95	5	from	from	ADP
ejpam-4905	95	6	theorem	theorem	ADJ
ejpam-4905	95	7	2	2	NUM
ejpam-4905	95	8	.	.	PUNCT
ejpam-4905	95	9	corollary	corollary	ADJ
ejpam-4905	95	10	1	1	NUM
ejpam-4905	95	11	.	.	PUNCT
ejpam-4905	96	1	let	let	VERB
ejpam-4905	96	2	g	g	PRON
ejpam-4905	96	3	be	be	AUX
ejpam-4905	96	4	a	a	DET
ejpam-4905	96	5	graph	graph	NOUN
ejpam-4905	96	6	and	and	CCONJ
ejpam-4905	96	7	let	let	VERB
ejpam-4905	96	8	n	n	X
ejpam-4905	96	9	=	=	PRON
ejpam-4905	96	10	{	{	PUNCT
ejpam-4905	96	11	x1	x1	PROPN
ejpam-4905	96	12	,	,	PUNCT
ejpam-4905	96	13	x2	x2	PROPN
ejpam-4905	96	14	,	,	PUNCT
ejpam-4905	96	15	.	.	PUNCT
ejpam-4905	96	16	.	.	PUNCT
ejpam-4905	96	17	.	.	PUNCT
ejpam-4905	97	1	,	,	PUNCT
ejpam-4905	97	2	xk	xk	AUX
ejpam-4905	97	3	}	}	PUNCT
ejpam-4905	97	4	be	be	AUX
ejpam-4905	97	5	a	a	DET
ejpam-4905	97	6	j2	j2	PROPN
ejpam-4905	97	7	-	-	PUNCT
ejpam-4905	97	8	set	set	NOUN
ejpam-4905	97	9	of	of	ADP
ejpam-4905	97	10	g.	g.	PROPN
ejpam-4905	97	11	then	then	ADV
ejpam-4905	97	12	|n	|n	ADV
ejpam-4905	98	1	|	|	PROPN
ejpam-4905	98	2	=	=	SYM
ejpam-4905	98	3	k	k	PROPN
ejpam-4905	98	4	≤	≤	NUM
ejpam-4905	98	5	γj2h(g	γj2h(g	NOUN
ejpam-4905	98	6	)	)	PUNCT
ejpam-4905	98	7	.	.	PUNCT
ejpam-4905	99	1	j.	j.	PROPN
ejpam-4905	99	2	hassan	hassan	PROPN
ejpam-4905	99	3	,	,	PUNCT
ejpam-4905	99	4	a.	a.	PROPN
ejpam-4905	99	5	bakkang	bakkang	PROPN
ejpam-4905	99	6	,	,	PUNCT
ejpam-4905	99	7	a.	a.	NOUN
ejpam-4905	99	8	sappari	sappari	PROPN
ejpam-4905	99	9	/	/	SYM
ejpam-4905	99	10	eur	eur	PROPN
ejpam-4905	99	11	.	.	PUNCT
ejpam-4905	100	1	j.	j.	PROPN
ejpam-4905	100	2	pure	pure	PROPN
ejpam-4905	100	3	appl	appl	PROPN
ejpam-4905	100	4	.	.	PROPN
ejpam-4905	100	5	math	math	PROPN
ejpam-4905	100	6	,	,	PUNCT
ejpam-4905	100	7	16	16	NUM
ejpam-4905	100	8	(	(	PUNCT
ejpam-4905	100	9	4	4	NUM
ejpam-4905	100	10	)	)	PUNCT
ejpam-4905	100	11	(	(	PUNCT
ejpam-4905	100	12	2023	2023	NUM
ejpam-4905	100	13	)	)	PUNCT
ejpam-4905	100	14	,	,	PUNCT
ejpam-4905	100	15	2118	2118	NUM
ejpam-4905	100	16	-	-	SYM
ejpam-4905	100	17	2131	2131	NUM
ejpam-4905	100	18	2122	2122	NUM
ejpam-4905	100	19	proposition	proposition	NOUN
ejpam-4905	100	20	1	1	NUM
ejpam-4905	100	21	.	.	PUNCT
ejpam-4905	100	22	given	give	VERB
ejpam-4905	100	23	any	any	DET
ejpam-4905	100	24	positive	positive	ADJ
ejpam-4905	100	25	integer	integer	NOUN
ejpam-4905	100	26	k	k	PROPN
ejpam-4905	100	27	≥	≥	NUM
ejpam-4905	100	28	1	1	NUM
ejpam-4905	100	29	,	,	PUNCT
ejpam-4905	100	30	we	we	PRON
ejpam-4905	100	31	have	have	AUX
ejpam-4905	100	32	γj2h(pk	γj2h(pk	VERB
ejpam-4905	100	33	)	)	PUNCT
ejpam-4905	100	34	=	=	PUNCT
ejpam-4905	101	1			NUM
ejpam-4905	101	2	1	1	NUM
ejpam-4905	102	1	if	if	SCONJ
ejpam-4905	102	2	k	k	NOUN
ejpam-4905	102	3	=	=	NOUN
ejpam-4905	102	4	1	1	NUM
ejpam-4905	102	5	2	2	NUM
ejpam-4905	102	6	if	if	SCONJ
ejpam-4905	102	7	k	k	NOUN
ejpam-4905	102	8	=	=	SYM
ejpam-4905	102	9	2	2	NUM
ejpam-4905	102	10	,	,	PUNCT
ejpam-4905	102	11	3	3	NUM
ejpam-4905	102	12	,	,	PUNCT
ejpam-4905	102	13	4	4	NUM
ejpam-4905	102	14	3	3	NUM
ejpam-4905	102	15	if	if	SCONJ
ejpam-4905	102	16	k	k	NOUN
ejpam-4905	102	17	=	=	NOUN
ejpam-4905	102	18	5	5	NUM
ejpam-4905	102	19	4	4	NUM
ejpam-4905	102	20	if	if	SCONJ
ejpam-4905	102	21	k	k	NOUN
ejpam-4905	102	22	=	=	SYM
ejpam-4905	102	23	6	6	NUM
ejpam-4905	102	24	,	,	PUNCT
ejpam-4905	102	25	7	7	NUM
ejpam-4905	102	26	k	k	NOUN
ejpam-4905	102	27	−	−	PROPN
ejpam-4905	102	28	4	4	NUM
ejpam-4905	102	29	if	if	SCONJ
ejpam-4905	102	30	k	k	PROPN
ejpam-4905	102	31	≥	≥	VERB
ejpam-4905	102	32	8	8	NUM
ejpam-4905	102	33	proof	proof	NOUN
ejpam-4905	102	34	.	.	PUNCT
ejpam-4905	103	1	clearly	clearly	ADV
ejpam-4905	103	2	,	,	PUNCT
ejpam-4905	103	3	γj2h(p1	γj2h(p1	NOUN
ejpam-4905	103	4	)	)	PUNCT
ejpam-4905	103	5	=	=	SYM
ejpam-4905	103	6	1	1	NUM
ejpam-4905	103	7	,	,	PUNCT
ejpam-4905	103	8	γj2h(pk	γj2h(pk	ADJ
ejpam-4905	103	9	)	)	PUNCT
ejpam-4905	103	10	=	=	SYM
ejpam-4905	103	11	2	2	NUM
ejpam-4905	103	12	for	for	ADP
ejpam-4905	103	13	k	k	X
ejpam-4905	103	14	=	=	SYM
ejpam-4905	103	15	2	2	NUM
ejpam-4905	103	16	,	,	PUNCT
ejpam-4905	103	17	3	3	NUM
ejpam-4905	103	18	,	,	PUNCT
ejpam-4905	103	19	4	4	NUM
ejpam-4905	103	20	,	,	PUNCT
ejpam-4905	103	21	γj2h(p5	γj2h(p5	X
ejpam-4905	103	22	)	)	PUNCT
ejpam-4905	103	23	=	=	SYM
ejpam-4905	103	24	3	3	NUM
ejpam-4905	103	25	and	and	CCONJ
ejpam-4905	103	26	γj2h(pk	γj2h(pk	ADJ
ejpam-4905	103	27	)	)	PUNCT
ejpam-4905	103	28	=	=	SYM
ejpam-4905	103	29	4	4	NUM
ejpam-4905	103	30	for	for	ADP
ejpam-4905	103	31	k	k	NOUN
ejpam-4905	103	32	=	=	SYM
ejpam-4905	103	33	6	6	NUM
ejpam-4905	103	34	,	,	PUNCT
ejpam-4905	103	35	7	7	NUM
ejpam-4905	103	36	.	.	PUNCT
ejpam-4905	103	37	suppose	suppose	VERB
ejpam-4905	104	1	that	that	SCONJ
ejpam-4905	104	2	k	k	PROPN
ejpam-4905	104	3	≥	≥	NUM
ejpam-4905	104	4	8	8	NUM
ejpam-4905	104	5	.	.	PUNCT
ejpam-4905	105	1	let	let	VERB
ejpam-4905	105	2	pk	pk	NOUN
ejpam-4905	105	3	=	=	PUNCT
ejpam-4905	106	1	[	[	X
ejpam-4905	106	2	a1	a1	NOUN
ejpam-4905	106	3	,	,	PUNCT
ejpam-4905	106	4	a2	a2	PROPN
ejpam-4905	106	5	,	,	PUNCT
ejpam-4905	106	6	.	.	PUNCT
ejpam-4905	106	7	.	.	PUNCT
ejpam-4905	107	1	.	.	PUNCT
ejpam-4905	108	1	,	,	PUNCT
ejpam-4905	108	2	ak	ak	PROPN
ejpam-4905	108	3	]	]	PUNCT
ejpam-4905	108	4	and	and	CCONJ
ejpam-4905	108	5	let	let	VERB
ejpam-4905	108	6	s	s	PRON
ejpam-4905	108	7	=	=	NOUN
ejpam-4905	108	8	{	{	PUNCT
ejpam-4905	108	9	a3	a3	NOUN
ejpam-4905	108	10	,	,	PUNCT
ejpam-4905	108	11	a4	a4	PROPN
ejpam-4905	108	12	,	,	PUNCT
ejpam-4905	108	13	.	.	PUNCT
ejpam-4905	108	14	.	.	PUNCT
ejpam-4905	109	1	.	.	PUNCT
ejpam-4905	110	1	,	,	PUNCT
ejpam-4905	110	2	ak−3	ak−3	PROPN
ejpam-4905	110	3	,	,	PUNCT
ejpam-4905	110	4	ak−2	ak−2	PROPN
ejpam-4905	110	5	}	}	PUNCT
ejpam-4905	110	6	.	.	PUNCT
ejpam-4905	111	1	then	then	ADV
ejpam-4905	111	2	n2	n2	PROPN
ejpam-4905	111	3	pk	pk	PROPN
ejpam-4905	111	4	[	[	X
ejpam-4905	111	5	s	s	X
ejpam-4905	111	6	]	]	X
ejpam-4905	111	7	=	=	SYM
ejpam-4905	111	8	v	v	X
ejpam-4905	111	9	(	(	PUNCT
ejpam-4905	111	10	pk	pk	NOUN
ejpam-4905	111	11	)	)	PUNCT
ejpam-4905	111	12	,	,	PUNCT
ejpam-4905	111	13	showing	show	VERB
ejpam-4905	111	14	that	that	SCONJ
ejpam-4905	111	15	s	s	VERB
ejpam-4905	111	16	is	be	AUX
ejpam-4905	111	17	a	a	DET
ejpam-4905	111	18	hop	hop	NOUN
ejpam-4905	111	19	dominating	dominating	NOUN
ejpam-4905	111	20	set	set	VERB
ejpam-4905	111	21	in	in	ADP
ejpam-4905	111	22	pk	pk	PROPN
ejpam-4905	111	23	.	.	PROPN
ejpam-4905	111	24	observe	observe	VERB
ejpam-4905	111	25	that	that	SCONJ
ejpam-4905	111	26	ai−2	ai−2	PROPN
ejpam-4905	111	27	∈	∈	PROPN
ejpam-4905	111	28	n2	n2	NOUN
ejpam-4905	111	29	pk	pk	NOUN
ejpam-4905	111	30	[	[	X
ejpam-4905	111	31	ai	ai	NOUN
ejpam-4905	111	32	]	]	PUNCT
ejpam-4905	111	33	\	\	PROPN
ejpam-4905	111	34	n2	n2	ADJ
ejpam-4905	111	35	pk	pk	PROPN
ejpam-4905	112	1	[	[	X
ejpam-4905	112	2	aj	aj	X
ejpam-4905	112	3	]	]	PUNCT
ejpam-4905	112	4	and	and	CCONJ
ejpam-4905	112	5	aj+2	aj+2	NUM
ejpam-4905	112	6	∈	∈	PROPN
ejpam-4905	112	7	n2	n2	ADJ
ejpam-4905	112	8	pk	pk	NOUN
ejpam-4905	112	9	[	[	X
ejpam-4905	112	10	aj	aj	PROPN
ejpam-4905	112	11	]	]	PUNCT
ejpam-4905	112	12	\	\	PROPN
ejpam-4905	112	13	n2	n2	ADJ
ejpam-4905	112	14	pk	pk	NOUN
ejpam-4905	112	15	[	[	X
ejpam-4905	112	16	ai	ai	NOUN
ejpam-4905	112	17	]	]	X
ejpam-4905	112	18	for	for	ADP
ejpam-4905	112	19	all	all	PRON
ejpam-4905	112	20	j	j	PROPN
ejpam-4905	112	21	>	>	X
ejpam-4905	112	22	i	i	PROPN
ejpam-4905	112	23	,	,	PUNCT
ejpam-4905	112	24	where	where	SCONJ
ejpam-4905	112	25	i	i	PRON
ejpam-4905	112	26	,	,	PUNCT
ejpam-4905	112	27	j	j	PROPN
ejpam-4905	112	28	∈	∈	PROPN
ejpam-4905	112	29	{	{	PUNCT
ejpam-4905	112	30	3	3	NUM
ejpam-4905	112	31	,	,	PUNCT
ejpam-4905	112	32	4	4	NUM
ejpam-4905	112	33	,	,	PUNCT
ejpam-4905	112	34	.	.	PUNCT
ejpam-4905	112	35	.	.	PUNCT
ejpam-4905	113	1	.	.	PUNCT
ejpam-4905	114	1	,	,	PUNCT
ejpam-4905	115	1	k	k	PROPN
ejpam-4905	116	1	−	−	PROPN
ejpam-4905	116	2	3	3	NUM
ejpam-4905	116	3	,	,	PUNCT
ejpam-4905	116	4	k	k	PROPN
ejpam-4905	116	5	−	−	PROPN
ejpam-4905	116	6	2	2	NUM
ejpam-4905	116	7	}	}	PUNCT
ejpam-4905	116	8	.	.	PUNCT
ejpam-4905	117	1	thus	thus	ADV
ejpam-4905	117	2	,	,	PUNCT
ejpam-4905	117	3	n2	n2	ADJ
ejpam-4905	117	4	pk	pk	NOUN
ejpam-4905	117	5	[	[	X
ejpam-4905	117	6	ai	ai	NOUN
ejpam-4905	117	7	]	]	PUNCT
ejpam-4905	117	8	\	\	PROPN
ejpam-4905	117	9	n2	n2	ADJ
ejpam-4905	117	10	pk	pk	PROPN
ejpam-4905	117	11	[	[	X
ejpam-4905	117	12	aj	aj	X
ejpam-4905	117	13	]	]	X
ejpam-4905	117	14	̸=	̸=	PROPN
ejpam-4905	117	15	∅	∅	NOUN
ejpam-4905	117	16	for	for	ADP
ejpam-4905	117	17	all	all	PRON
ejpam-4905	117	18	i	i	PRON
ejpam-4905	117	19	̸=	̸=	PROPN
ejpam-4905	117	20	j	j	PROPN
ejpam-4905	117	21	,	,	PUNCT
ejpam-4905	117	22	where	where	SCONJ
ejpam-4905	117	23	i	i	PRON
ejpam-4905	117	24	,	,	PUNCT
ejpam-4905	117	25	j	j	PROPN
ejpam-4905	117	26	∈	∈	PROPN
ejpam-4905	117	27	{	{	PUNCT
ejpam-4905	117	28	3	3	NUM
ejpam-4905	117	29	,	,	PUNCT
ejpam-4905	117	30	4	4	NUM
ejpam-4905	117	31	,	,	PUNCT
ejpam-4905	117	32	.	.	PUNCT
ejpam-4905	117	33	.	.	PUNCT
ejpam-4905	117	34	.	.	PUNCT
ejpam-4905	118	1	,	,	PUNCT
ejpam-4905	119	1	k	k	PROPN
ejpam-4905	120	1	−	−	PROPN
ejpam-4905	120	2	3	3	NUM
ejpam-4905	120	3	,	,	PUNCT
ejpam-4905	120	4	k	k	PROPN
ejpam-4905	120	5	−	−	PROPN
ejpam-4905	120	6	2	2	NUM
ejpam-4905	120	7	}	}	PUNCT
ejpam-4905	120	8	,	,	PUNCT
ejpam-4905	120	9	that	that	ADV
ejpam-4905	120	10	is	is	ADV
ejpam-4905	120	11	,	,	PUNCT
ejpam-4905	120	12	s	s	VERB
ejpam-4905	120	13	is	be	AUX
ejpam-4905	120	14	a	a	DET
ejpam-4905	120	15	j2	j2	NOUN
ejpam-4905	120	16	-	-	PUNCT
ejpam-4905	120	17	set	set	NOUN
ejpam-4905	120	18	in	in	ADP
ejpam-4905	120	19	pk	pk	NOUN
ejpam-4905	120	20	.	.	PROPN
ejpam-4905	120	21	therefore	therefore	ADV
ejpam-4905	120	22	,	,	PUNCT
ejpam-4905	120	23	s	s	VERB
ejpam-4905	120	24	is	be	AUX
ejpam-4905	120	25	a	a	DET
ejpam-4905	120	26	j2	j2	PROPN
ejpam-4905	120	27	-	-	PUNCT
ejpam-4905	120	28	hop	hop	NOUN
ejpam-4905	120	29	dominating	dominating	NOUN
ejpam-4905	120	30	set	set	VERB
ejpam-4905	120	31	in	in	ADP
ejpam-4905	120	32	pk	pk	PROPN
ejpam-4905	120	33	.	.	PROPN
ejpam-4905	121	1	since	since	SCONJ
ejpam-4905	121	2	n2	n2	PROPN
ejpam-4905	121	3	pk	pk	PROPN
ejpam-4905	121	4	[	[	X
ejpam-4905	121	5	a1	a1	NOUN
ejpam-4905	121	6	]	]	PUNCT
ejpam-4905	121	7	⊆	⊆	NUM
ejpam-4905	121	8	n2	n2	ADJ
ejpam-4905	121	9	pk	pk	NOUN
ejpam-4905	121	10	[	[	X
ejpam-4905	121	11	a3	a3	NOUN
ejpam-4905	121	12	]	]	PUNCT
ejpam-4905	121	13	,	,	PUNCT
ejpam-4905	121	14	n	n	PROPN
ejpam-4905	122	1	2	2	NUM
ejpam-4905	122	2	pk	pk	NOUN
ejpam-4905	122	3	[	[	X
ejpam-4905	122	4	a2	a2	X
ejpam-4905	122	5	]	]	PUNCT
ejpam-4905	122	6	⊆	⊆	NUM
ejpam-4905	122	7	n2	n2	ADJ
ejpam-4905	122	8	pk	pk	NOUN
ejpam-4905	122	9	[	[	X
ejpam-4905	122	10	a4	a4	NOUN
ejpam-4905	122	11	]	]	PUNCT
ejpam-4905	122	12	,	,	PUNCT
ejpam-4905	122	13	n	n	PROPN
ejpam-4905	122	14	2	2	NUM
ejpam-4905	122	15	pk	pk	NOUN
ejpam-4905	122	16	[	[	X
ejpam-4905	122	17	ak	ak	X
ejpam-4905	122	18	]	]	X
ejpam-4905	122	19	⊆	⊆	NUM
ejpam-4905	122	20	n2	n2	ADJ
ejpam-4905	122	21	pk	pk	NOUN
ejpam-4905	122	22	[	[	X
ejpam-4905	122	23	ak−2	ak−2	X
ejpam-4905	122	24	]	]	X
ejpam-4905	122	25	,	,	PUNCT
ejpam-4905	122	26	and	and	CCONJ
ejpam-4905	122	27	n2	n2	ADJ
ejpam-4905	122	28	pk	pk	NOUN
ejpam-4905	122	29	[	[	X
ejpam-4905	122	30	ak−1	ak−1	NOUN
ejpam-4905	122	31	]	]	X
ejpam-4905	122	32	⊆	⊆	NUM
ejpam-4905	122	33	n2	n2	ADJ
ejpam-4905	122	34	pk	pk	NOUN
ejpam-4905	122	35	[	[	X
ejpam-4905	122	36	ak−3	ak−3	NOUN
ejpam-4905	122	37	]	]	X
ejpam-4905	122	38	,	,	PUNCT
ejpam-4905	122	39	it	it	PRON
ejpam-4905	122	40	follows	follow	VERB
ejpam-4905	122	41	that	that	SCONJ
ejpam-4905	122	42	s	s	VERB
ejpam-4905	122	43	is	be	AUX
ejpam-4905	122	44	a	a	DET
ejpam-4905	122	45	maximum	maximum	ADJ
ejpam-4905	122	46	j2	j2	PROPN
ejpam-4905	122	47	-	-	PUNCT
ejpam-4905	122	48	hop	hop	NOUN
ejpam-4905	122	49	dominating	dominating	NOUN
ejpam-4905	122	50	set	set	NOUN
ejpam-4905	122	51	of	of	ADP
ejpam-4905	122	52	pk	pk	PROPN
ejpam-4905	122	53	.	.	PROPN
ejpam-4905	122	54	hence	hence	ADV
ejpam-4905	122	55	,	,	PUNCT
ejpam-4905	122	56	γj2h(pk	γj2h(pk	ADJ
ejpam-4905	122	57	)	)	PUNCT
ejpam-4905	123	1	=	=	SYM
ejpam-4905	123	2	k	k	NOUN
ejpam-4905	124	1	−	−	NOUN
ejpam-4905	124	2	4	4	NUM
ejpam-4905	124	3	for	for	ADP
ejpam-4905	124	4	all	all	DET
ejpam-4905	124	5	k	k	PROPN
ejpam-4905	124	6	≥	≥	NUM
ejpam-4905	124	7	8	8	NUM
ejpam-4905	124	8	.	.	PUNCT
ejpam-4905	124	9	theorem	theorem	NOUN
ejpam-4905	124	10	3	3	X
ejpam-4905	124	11	.	.	PUNCT
ejpam-4905	125	1	let	let	VERB
ejpam-4905	125	2	g	g	NOUN
ejpam-4905	125	3	be	be	AUX
ejpam-4905	125	4	any	any	DET
ejpam-4905	125	5	graph	graph	NOUN
ejpam-4905	125	6	of	of	ADP
ejpam-4905	125	7	order	order	NOUN
ejpam-4905	125	8	n	n	NOUN
ejpam-4905	125	9	and	and	CCONJ
ejpam-4905	125	10	n	n	CCONJ
ejpam-4905	125	11	be	be	VERB
ejpam-4905	125	12	any	any	DET
ejpam-4905	125	13	j2	j2	PROPN
ejpam-4905	125	14	-	-	PUNCT
ejpam-4905	125	15	hop	hop	NOUN
ejpam-4905	125	16	dominating	dominating	NOUN
ejpam-4905	125	17	set	set	NOUN
ejpam-4905	125	18	of	of	ADP
ejpam-4905	125	19	g.	g.	PROPN
ejpam-4905	126	1	then	then	ADV
ejpam-4905	126	2	each	each	PRON
ejpam-4905	126	3	of	of	ADP
ejpam-4905	126	4	the	the	DET
ejpam-4905	126	5	following	follow	VERB
ejpam-4905	126	6	holds	hold	VERB
ejpam-4905	126	7	:	:	PUNCT
ejpam-4905	126	8	(	(	PUNCT
ejpam-4905	126	9	i	i	NOUN
ejpam-4905	126	10	)	)	PUNCT
ejpam-4905	126	11	a	a	DET
ejpam-4905	126	12	∈	∈	ADJ
ejpam-4905	126	13	n	n	CCONJ
ejpam-4905	126	14	if	if	SCONJ
ejpam-4905	127	1	and	and	CCONJ
ejpam-4905	127	2	only	only	ADV
ejpam-4905	127	3	if	if	SCONJ
ejpam-4905	127	4	n2	n2	ADJ
ejpam-4905	127	5	g[a	g[a	NOUN
ejpam-4905	127	6	]	]	PUNCT
ejpam-4905	127	7	⊈	⊈	PROPN
ejpam-4905	127	8	n2	n2	PROPN
ejpam-4905	127	9	g[b	g[b	PROPN
ejpam-4905	127	10	]	]	PUNCT
ejpam-4905	127	11	and	and	CCONJ
ejpam-4905	127	12	n2	n2	PROPN
ejpam-4905	127	13	g[b	g[b	PROPN
ejpam-4905	127	14	]	]	PUNCT
ejpam-4905	127	15	⊈	⊈	PROPN
ejpam-4905	127	16	n2	n2	ADJ
ejpam-4905	127	17	g[a	g[a	PROPN
ejpam-4905	127	18	]	]	X
ejpam-4905	127	19	∀	∀	PUNCT
ejpam-4905	127	20	b	b	NOUN
ejpam-4905	127	21	∈	∈	PROPN
ejpam-4905	127	22	n	n	PRON
ejpam-4905	127	23	\	\	NOUN
ejpam-4905	127	24	{	{	PUNCT
ejpam-4905	127	25	a	a	NOUN
ejpam-4905	127	26	}	}	PUNCT
ejpam-4905	127	27	.	.	PUNCT
ejpam-4905	128	1	(	(	PUNCT
ejpam-4905	128	2	ii	ii	NOUN
ejpam-4905	128	3	)	)	PUNCT
ejpam-4905	128	4	γj2h(g	γj2h(g	NOUN
ejpam-4905	128	5	)	)	PUNCT
ejpam-4905	129	1	=	=	SYM
ejpam-4905	129	2	|v	|v	X
ejpam-4905	129	3	(	(	PUNCT
ejpam-4905	129	4	g)|	g)|	NOUN
ejpam-4905	129	5	=	=	PUNCT
ejpam-4905	129	6	n	n	NOUN
ejpam-4905	129	7	if	if	SCONJ
ejpam-4905	129	8	and	and	CCONJ
ejpam-4905	129	9	only	only	ADV
ejpam-4905	129	10	if	if	SCONJ
ejpam-4905	129	11	n2	n2	ADJ
ejpam-4905	129	12	g[vi	g[vi	PROPN
ejpam-4905	129	13	]	]	X
ejpam-4905	129	14	⊈	⊈	PROPN
ejpam-4905	129	15	n2	n2	PROPN
ejpam-4905	129	16	g[vj	g[vj	PROPN
ejpam-4905	129	17	]	]	PUNCT
ejpam-4905	129	18	∀	∀	PUNCT
ejpam-4905	130	1	i	i	NOUN
ejpam-4905	130	2	̸=	̸=	PROPN
ejpam-4905	130	3	j	j	PROPN
ejpam-4905	130	4	where	where	SCONJ
ejpam-4905	130	5	i	i	PRON
ejpam-4905	130	6	,	,	PUNCT
ejpam-4905	130	7	j	j	PROPN
ejpam-4905	130	8	∈	∈	PROPN
ejpam-4905	130	9	{	{	PUNCT
ejpam-4905	130	10	1	1	NUM
ejpam-4905	130	11	,	,	PUNCT
ejpam-4905	130	12	2	2	NUM
ejpam-4905	130	13	,	,	PUNCT
ejpam-4905	130	14	.	.	PUNCT
ejpam-4905	130	15	.	.	PUNCT
ejpam-4905	131	1	.	.	PUNCT
ejpam-4905	131	2	,	,	PUNCT
ejpam-4905	131	3	n	n	CCONJ
ejpam-4905	131	4	}	}	PUNCT
ejpam-4905	131	5	.	.	PUNCT
ejpam-4905	132	1	(	(	PUNCT
ejpam-4905	132	2	iii	iii	X
ejpam-4905	132	3	)	)	PUNCT
ejpam-4905	132	4	if	if	SCONJ
ejpam-4905	132	5	g	g	PROPN
ejpam-4905	132	6	is	be	AUX
ejpam-4905	132	7	kn	kn	PROPN
ejpam-4905	132	8	or	or	CCONJ
ejpam-4905	132	9	kn	kn	PROPN
ejpam-4905	132	10	,	,	PUNCT
ejpam-4905	132	11	then	then	ADV
ejpam-4905	132	12	γj2h(g	γj2h(g	NOUN
ejpam-4905	132	13	)	)	PUNCT
ejpam-4905	132	14	=	=	SYM
ejpam-4905	133	1	n	n	CCONJ
ejpam-4905	133	2	for	for	ADP
ejpam-4905	133	3	all	all	DET
ejpam-4905	133	4	n	n	PRON
ejpam-4905	133	5	≥	≥	NOUN
ejpam-4905	133	6	1	1	NUM
ejpam-4905	133	7	.	.	PUNCT
ejpam-4905	134	1	proof	proof	NOUN
ejpam-4905	134	2	.	.	PUNCT
ejpam-4905	135	1	(	(	PUNCT
ejpam-4905	135	2	i	i	NOUN
ejpam-4905	135	3	)	)	PUNCT
ejpam-4905	135	4	let	let	VERB
ejpam-4905	135	5	g	g	NOUN
ejpam-4905	135	6	be	be	AUX
ejpam-4905	135	7	a	a	DET
ejpam-4905	135	8	graph	graph	NOUN
ejpam-4905	135	9	and	and	CCONJ
ejpam-4905	135	10	n	n	PRON
ejpam-4905	135	11	be	be	AUX
ejpam-4905	135	12	a	a	DET
ejpam-4905	135	13	j2	j2	PROPN
ejpam-4905	135	14	-	-	PUNCT
ejpam-4905	135	15	hop	hop	NOUN
ejpam-4905	135	16	dominating	dominating	NOUN
ejpam-4905	135	17	set	set	NOUN
ejpam-4905	135	18	of	of	ADP
ejpam-4905	135	19	g.	g.	PROPN
ejpam-4905	135	20	suppose	suppose	VERB
ejpam-4905	135	21	that	that	SCONJ
ejpam-4905	135	22	a	a	DET
ejpam-4905	135	23	∈	∈	PROPN
ejpam-4905	135	24	n	n	NOUN
ejpam-4905	135	25	.	.	PUNCT
ejpam-4905	136	1	then	then	ADV
ejpam-4905	136	2	n2	n2	PROPN
ejpam-4905	136	3	g[a	g[a	PROPN
ejpam-4905	136	4	]	]	PUNCT
ejpam-4905	136	5	\	\	PROPN
ejpam-4905	136	6	n2	n2	PROPN
ejpam-4905	136	7	g[b	g[b	PROPN
ejpam-4905	136	8	]	]	PUNCT
ejpam-4905	136	9	̸=	̸=	PROPN
ejpam-4905	136	10	∅	∅	NOUN
ejpam-4905	136	11	and	and	CCONJ
ejpam-4905	136	12	n2	n2	PROPN
ejpam-4905	136	13	g[b	g[b	PROPN
ejpam-4905	136	14	]	]	PUNCT
ejpam-4905	136	15	\	\	PROPN
ejpam-4905	136	16	n2	n2	PROPN
ejpam-4905	136	17	g[a	g[a	PROPN
ejpam-4905	136	18	]	]	X
ejpam-4905	136	19	̸=	̸=	PROPN
ejpam-4905	136	20	∅	∅	NOUN
ejpam-4905	136	21	∀	∀	NOUN
ejpam-4905	136	22	b	b	X
ejpam-4905	136	23	∈	∈	PROPN
ejpam-4905	136	24	n	n	PRON
ejpam-4905	136	25	\	\	NOUN
ejpam-4905	136	26	{	{	PUNCT
ejpam-4905	136	27	a	a	NOUN
ejpam-4905	136	28	}	}	PUNCT
ejpam-4905	136	29	.	.	PUNCT
ejpam-4905	137	1	it	it	PRON
ejpam-4905	137	2	follows	follow	VERB
ejpam-4905	137	3	that	that	DET
ejpam-4905	137	4	n2	n2	ADJ
ejpam-4905	137	5	g[a	g[a	PROPN
ejpam-4905	137	6	]	]	PUNCT
ejpam-4905	137	7	⊈	⊈	PROPN
ejpam-4905	137	8	n2	n2	PROPN
ejpam-4905	137	9	g[b	g[b	PROPN
ejpam-4905	137	10	]	]	PUNCT
ejpam-4905	137	11	and	and	CCONJ
ejpam-4905	137	12	n2	n2	PROPN
ejpam-4905	137	13	g[b	g[b	PROPN
ejpam-4905	137	14	]	]	PUNCT
ejpam-4905	137	15	⊈	⊈	PROPN
ejpam-4905	137	16	n2	n2	ADJ
ejpam-4905	137	17	g[a	g[a	PROPN
ejpam-4905	137	18	]	]	X
ejpam-4905	137	19	∀	∀	PUNCT
ejpam-4905	138	1	b	b	NOUN
ejpam-4905	138	2	∈	∈	PROPN
ejpam-4905	138	3	n	n	PRON
ejpam-4905	138	4	\	\	NOUN
ejpam-4905	138	5	{	{	PUNCT
ejpam-4905	138	6	a	a	NOUN
ejpam-4905	138	7	}	}	PUNCT
ejpam-4905	138	8	.	.	PUNCT
ejpam-4905	139	1	conversely	conversely	ADV
ejpam-4905	139	2	,	,	PUNCT
ejpam-4905	139	3	suppose	suppose	VERB
ejpam-4905	139	4	that	that	SCONJ
ejpam-4905	139	5	n2	n2	PROPN
ejpam-4905	139	6	g[a	g[a	PROPN
ejpam-4905	139	7	]	]	PUNCT
ejpam-4905	139	8	⊈	⊈	PROPN
ejpam-4905	139	9	n2	n2	PROPN
ejpam-4905	139	10	g[b	g[b	PROPN
ejpam-4905	139	11	]	]	PUNCT
ejpam-4905	139	12	and	and	CCONJ
ejpam-4905	139	13	n2	n2	PROPN
ejpam-4905	139	14	g[b	g[b	PROPN
ejpam-4905	139	15	]	]	PUNCT
ejpam-4905	139	16	⊈	⊈	PROPN
ejpam-4905	139	17	n2	n2	ADJ
ejpam-4905	139	18	g[a	g[a	PROPN
ejpam-4905	139	19	]	]	X
ejpam-4905	139	20	∀	∀	PUNCT
ejpam-4905	139	21	b	b	X
ejpam-4905	139	22	∈	∈	PROPN
ejpam-4905	139	23	n	n	PRON
ejpam-4905	139	24	\{a	\{a	NOUN
ejpam-4905	139	25	}	}	PUNCT
ejpam-4905	139	26	.	.	PUNCT
ejpam-4905	140	1	this	this	PRON
ejpam-4905	140	2	means	mean	VERB
ejpam-4905	140	3	that	that	SCONJ
ejpam-4905	140	4	n2	n2	ADJ
ejpam-4905	140	5	g[a	g[a	PROPN
ejpam-4905	140	6	]	]	PUNCT
ejpam-4905	140	7	\n2	\n2	PROPN
ejpam-4905	140	8	g[b	g[b	NOUN
ejpam-4905	140	9	]	]	PUNCT
ejpam-4905	140	10	̸=	̸=	PROPN
ejpam-4905	140	11	∅	∅	NOUN
ejpam-4905	140	12	and	and	CCONJ
ejpam-4905	140	13	n2	n2	PROPN
ejpam-4905	140	14	g[b	g[b	PROPN
ejpam-4905	140	15	]	]	PUNCT
ejpam-4905	140	16	\n2	\n2	ADP
ejpam-4905	140	17	g[a	g[a	PROPN
ejpam-4905	140	18	]	]	X
ejpam-4905	140	19	̸=	̸=	PROPN
ejpam-4905	140	20	∅	∅	NOUN
ejpam-4905	140	21	∀	∀	NOUN
ejpam-4905	140	22	b	b	X
ejpam-4905	140	23	∈	∈	PROPN
ejpam-4905	140	24	n	n	PRON
ejpam-4905	140	25	\	\	NOUN
ejpam-4905	140	26	{	{	PUNCT
ejpam-4905	140	27	a	a	NOUN
ejpam-4905	140	28	}	}	PUNCT
ejpam-4905	140	29	.	.	PUNCT
ejpam-4905	141	1	hence	hence	ADV
ejpam-4905	141	2	,	,	PUNCT
ejpam-4905	141	3	a	a	DET
ejpam-4905	141	4	∈	∈	PROPN
ejpam-4905	141	5	n	n	X
ejpam-4905	141	6	.	.	PUNCT
ejpam-4905	142	1	(	(	PUNCT
ejpam-4905	142	2	ii	ii	NOUN
ejpam-4905	142	3	)	)	PUNCT
ejpam-4905	142	4	suppose	suppose	VERB
ejpam-4905	142	5	that	that	SCONJ
ejpam-4905	142	6	γj2h(g	γj2h(g	NOUN
ejpam-4905	142	7	)	)	PUNCT
ejpam-4905	142	8	=	=	SYM
ejpam-4905	142	9	|v	|v	X
ejpam-4905	142	10	(	(	PUNCT
ejpam-4905	142	11	g)|	g)|	NOUN
ejpam-4905	142	12	=	=	NOUN
ejpam-4905	142	13	n.	n.	NOUN
ejpam-4905	142	14	then	then	ADV
ejpam-4905	142	15	n	n	PROPN
ejpam-4905	142	16	=	=	SYM
ejpam-4905	142	17	v	v	PROPN
ejpam-4905	142	18	(	(	PUNCT
ejpam-4905	142	19	g	g	NOUN
ejpam-4905	142	20	)	)	PUNCT
ejpam-4905	142	21	=	=	SYM
ejpam-4905	142	22	{	{	PUNCT
ejpam-4905	142	23	v1	v1	PROPN
ejpam-4905	142	24	,	,	PUNCT
ejpam-4905	142	25	v2	v2	PROPN
ejpam-4905	142	26	,	,	PUNCT
ejpam-4905	142	27	.	.	PUNCT
ejpam-4905	142	28	.	.	PUNCT
ejpam-4905	142	29	.	.	PUNCT
ejpam-4905	143	1	,	,	PUNCT
ejpam-4905	143	2	vn	vn	PROPN
ejpam-4905	143	3	}	}	PUNCT
ejpam-4905	143	4	is	be	AUX
ejpam-4905	143	5	the	the	DET
ejpam-4905	143	6	γj2h	γj2h	NOUN
ejpam-4905	143	7	-	-	PUNCT
ejpam-4905	143	8	set	set	NOUN
ejpam-4905	143	9	of	of	ADP
ejpam-4905	143	10	g.	g.	PROPN
ejpam-4905	143	11	thus	thus	ADV
ejpam-4905	143	12	,	,	PUNCT
ejpam-4905	143	13	n2	n2	PROPN
ejpam-4905	143	14	g[vi]\n2	g[vi]\n2	PROPN
ejpam-4905	143	15	g[vj	g[vj	PROPN
ejpam-4905	143	16	]	]	PUNCT
ejpam-4905	143	17	̸=	̸=	PROPN
ejpam-4905	143	18	∅	∅	NOUN
ejpam-4905	143	19	∀	∀	NOUN
ejpam-4905	144	1	i	i	PRON
ejpam-4905	144	2	̸=	̸=	PROPN
ejpam-4905	144	3	j	j	PROPN
ejpam-4905	144	4	,	,	PUNCT
ejpam-4905	144	5	where	where	SCONJ
ejpam-4905	144	6	i	i	PRON
ejpam-4905	144	7	,	,	PUNCT
ejpam-4905	144	8	j	j	PROPN
ejpam-4905	144	9	∈	∈	PROPN
ejpam-4905	144	10	{	{	PUNCT
ejpam-4905	144	11	1	1	NUM
ejpam-4905	144	12	,	,	PUNCT
ejpam-4905	144	13	2	2	NUM
ejpam-4905	144	14	,	,	PUNCT
ejpam-4905	144	15	.	.	PUNCT
ejpam-4905	144	16	.	.	PUNCT
ejpam-4905	145	1	.	.	PUNCT
ejpam-4905	145	2	,	,	PUNCT
ejpam-4905	146	1	n	n	CCONJ
ejpam-4905	146	2	}	}	PUNCT
ejpam-4905	146	3	.	.	PUNCT
ejpam-4905	147	1	it	it	PRON
ejpam-4905	147	2	follows	follow	VERB
ejpam-4905	147	3	that	that	SCONJ
ejpam-4905	147	4	n2	n2	PROPN
ejpam-4905	147	5	g[vi	g[vi	PROPN
ejpam-4905	147	6	]	]	PUNCT
ejpam-4905	147	7	⊈	⊈	PROPN
ejpam-4905	147	8	n2	n2	PROPN
ejpam-4905	147	9	g[vj	g[vj	PROPN
ejpam-4905	147	10	]	]	PUNCT
ejpam-4905	147	11	∀	∀	PUNCT
ejpam-4905	148	1	i	i	NOUN
ejpam-4905	148	2	̸=	̸=	PROPN
ejpam-4905	148	3	j	j	PROPN
ejpam-4905	148	4	,	,	PUNCT
ejpam-4905	148	5	where	where	SCONJ
ejpam-4905	148	6	i	i	PRON
ejpam-4905	148	7	,	,	PUNCT
ejpam-4905	148	8	j	j	PROPN
ejpam-4905	148	9	∈	∈	PROPN
ejpam-4905	148	10	{	{	PUNCT
ejpam-4905	148	11	1	1	NUM
ejpam-4905	148	12	,	,	PUNCT
ejpam-4905	148	13	2	2	NUM
ejpam-4905	148	14	,	,	PUNCT
ejpam-4905	148	15	.	.	PUNCT
ejpam-4905	148	16	.	.	PUNCT
ejpam-4905	149	1	.	.	PUNCT
ejpam-4905	149	2	,	,	PUNCT
ejpam-4905	149	3	n	n	CCONJ
ejpam-4905	149	4	}	}	PUNCT
ejpam-4905	149	5	.	.	PUNCT
ejpam-4905	150	1	conversely	conversely	ADV
ejpam-4905	150	2	,	,	PUNCT
ejpam-4905	150	3	suppose	suppose	VERB
ejpam-4905	150	4	that	that	SCONJ
ejpam-4905	150	5	n2	n2	PROPN
ejpam-4905	150	6	g[vi	g[vi	PROPN
ejpam-4905	150	7	]	]	PUNCT
ejpam-4905	150	8	⊈	⊈	PROPN
ejpam-4905	150	9	n2	n2	PROPN
ejpam-4905	150	10	g[vj	g[vj	PROPN
ejpam-4905	150	11	]	]	PUNCT
ejpam-4905	150	12	∀	∀	PUNCT
ejpam-4905	150	13	i	i	NOUN
ejpam-4905	150	14	̸=	̸=	PROPN
ejpam-4905	150	15	j	j	PROPN
ejpam-4905	150	16	,	,	PUNCT
ejpam-4905	150	17	where	where	SCONJ
ejpam-4905	150	18	i	i	PRON
ejpam-4905	150	19	,	,	PUNCT
ejpam-4905	150	20	j	j	PROPN
ejpam-4905	150	21	∈	∈	PROPN
ejpam-4905	150	22	{	{	PUNCT
ejpam-4905	150	23	1	1	NUM
ejpam-4905	150	24	,	,	PUNCT
ejpam-4905	150	25	2	2	NUM
ejpam-4905	150	26	,	,	PUNCT
ejpam-4905	150	27	.	.	PUNCT
ejpam-4905	150	28	.	.	PUNCT
ejpam-4905	151	1	.	.	PUNCT
ejpam-4905	151	2	,	,	PUNCT
ejpam-4905	151	3	n	n	CCONJ
ejpam-4905	151	4	}	}	PUNCT
ejpam-4905	151	5	.	.	PUNCT
ejpam-4905	152	1	then	then	ADV
ejpam-4905	152	2	n2	n2	PROPN
ejpam-4905	152	3	g[vi	g[vi	PROPN
ejpam-4905	152	4	]	]	PUNCT
ejpam-4905	152	5	\n2	\n2	PROPN
ejpam-4905	152	6	g[vj	g[vj	PROPN
ejpam-4905	152	7	]	]	PUNCT
ejpam-4905	152	8	̸=	̸=	PROPN
ejpam-4905	152	9	∅	∅	NOUN
ejpam-4905	152	10	∀	∀	NOUN
ejpam-4905	153	1	i	i	PRON
ejpam-4905	153	2	̸=	̸=	PROPN
ejpam-4905	153	3	j	j	PROPN
ejpam-4905	153	4	,	,	PUNCT
ejpam-4905	153	5	i	i	PRON
ejpam-4905	153	6	,	,	PUNCT
ejpam-4905	153	7	j	j	PROPN
ejpam-4905	153	8	∈	∈	PROPN
ejpam-4905	153	9	{	{	PUNCT
ejpam-4905	153	10	1	1	NUM
ejpam-4905	153	11	,	,	PUNCT
ejpam-4905	153	12	2	2	NUM
ejpam-4905	153	13	,	,	PUNCT
ejpam-4905	153	14	.	.	PUNCT
ejpam-4905	153	15	.	.	PUNCT
ejpam-4905	154	1	.	.	PUNCT
ejpam-4905	154	2	,	,	PUNCT
ejpam-4905	155	1	n	n	CCONJ
ejpam-4905	155	2	}	}	PUNCT
ejpam-4905	155	3	.	.	PUNCT
ejpam-4905	156	1	it	it	PRON
ejpam-4905	156	2	follows	follow	VERB
ejpam-4905	156	3	that	that	SCONJ
ejpam-4905	156	4	vi	vi	NOUN
ejpam-4905	156	5	,	,	PUNCT
ejpam-4905	156	6	vj	vj	NOUN
ejpam-4905	156	7	are	be	AUX
ejpam-4905	156	8	in	in	ADP
ejpam-4905	156	9	j2	j2	PROPN
ejpam-4905	156	10	-	-	PUNCT
ejpam-4905	156	11	set	set	PROPN
ejpam-4905	156	12	s	s	NOUN
ejpam-4905	156	13	of	of	ADP
ejpam-4905	156	14	g	g	NOUN
ejpam-4905	156	15	∀	∀	NOUN
ejpam-4905	157	1	i	i	NOUN
ejpam-4905	157	2	̸=	̸=	PROPN
ejpam-4905	157	3	j	j	PROPN
ejpam-4905	157	4	,	,	PUNCT
ejpam-4905	157	5	where	where	SCONJ
ejpam-4905	157	6	i	i	PRON
ejpam-4905	157	7	,	,	PUNCT
ejpam-4905	157	8	j	j	PROPN
ejpam-4905	157	9	∈	∈	PROPN
ejpam-4905	157	10	{	{	PUNCT
ejpam-4905	157	11	1	1	NUM
ejpam-4905	157	12	,	,	PUNCT
ejpam-4905	157	13	2	2	NUM
ejpam-4905	157	14	,	,	PUNCT
ejpam-4905	157	15	.	.	PUNCT
ejpam-4905	157	16	.	.	PUNCT
ejpam-4905	158	1	.	.	PUNCT
ejpam-4905	158	2	,	,	PUNCT
ejpam-4905	158	3	n	n	CCONJ
ejpam-4905	158	4	}	}	PUNCT
ejpam-4905	158	5	.	.	PUNCT
ejpam-4905	159	1	thus	thus	ADV
ejpam-4905	159	2	,	,	PUNCT
ejpam-4905	159	3	s	s	VERB
ejpam-4905	159	4	=	=	SYM
ejpam-4905	159	5	v	v	X
ejpam-4905	159	6	(	(	PUNCT
ejpam-4905	159	7	g	g	NOUN
ejpam-4905	159	8	)	)	PUNCT
ejpam-4905	159	9	.	.	PUNCT
ejpam-4905	160	1	consequently	consequently	ADV
ejpam-4905	160	2	,	,	PUNCT
ejpam-4905	160	3	γj2h(g	γj2h(g	NOUN
ejpam-4905	160	4	)	)	PUNCT
ejpam-4905	160	5	=	=	SYM
ejpam-4905	160	6	|s|	|s|	PROPN
ejpam-4905	160	7	=	=	PUNCT
ejpam-4905	160	8	|v	|v	PROPN
ejpam-4905	160	9	(	(	PUNCT
ejpam-4905	160	10	g)|	g)|	NOUN
ejpam-4905	160	11	=	=	PROPN
ejpam-4905	160	12	n.	n.	PROPN
ejpam-4905	160	13	j.	j.	PROPN
ejpam-4905	160	14	hassan	hassan	PROPN
ejpam-4905	160	15	,	,	PUNCT
ejpam-4905	160	16	a.	a.	PROPN
ejpam-4905	160	17	bakkang	bakkang	PROPN
ejpam-4905	160	18	,	,	PUNCT
ejpam-4905	160	19	a.	a.	NOUN
ejpam-4905	160	20	sappari	sappari	PROPN
ejpam-4905	160	21	/	/	SYM
ejpam-4905	160	22	eur	eur	PROPN
ejpam-4905	160	23	.	.	PUNCT
ejpam-4905	161	1	j.	j.	PROPN
ejpam-4905	161	2	pure	pure	PROPN
ejpam-4905	161	3	appl	appl	PROPN
ejpam-4905	161	4	.	.	PROPN
ejpam-4905	161	5	math	math	PROPN
ejpam-4905	161	6	,	,	PUNCT
ejpam-4905	161	7	16	16	NUM
ejpam-4905	161	8	(	(	PUNCT
ejpam-4905	161	9	4	4	NUM
ejpam-4905	161	10	)	)	PUNCT
ejpam-4905	161	11	(	(	PUNCT
ejpam-4905	161	12	2023	2023	NUM
ejpam-4905	161	13	)	)	PUNCT
ejpam-4905	161	14	,	,	PUNCT
ejpam-4905	161	15	2118	2118	NUM
ejpam-4905	161	16	-	-	SYM
ejpam-4905	161	17	2131	2131	NUM
ejpam-4905	161	18	2123	2123	NUM
ejpam-4905	161	19	(	(	PUNCT
ejpam-4905	161	20	iii	iii	NOUN
ejpam-4905	161	21	)	)	PUNCT
ejpam-4905	161	22	let	let	VERB
ejpam-4905	161	23	g	g	PROPN
ejpam-4905	161	24	=	=	PROPN
ejpam-4905	161	25	kn	kn	PROPN
ejpam-4905	161	26	and	and	CCONJ
ejpam-4905	161	27	v	v	PROPN
ejpam-4905	161	28	(	(	PUNCT
ejpam-4905	161	29	g	g	NOUN
ejpam-4905	161	30	)	)	PUNCT
ejpam-4905	161	31	=	=	SYM
ejpam-4905	161	32	{	{	PUNCT
ejpam-4905	161	33	a1	a1	PROPN
ejpam-4905	161	34	,	,	PUNCT
ejpam-4905	161	35	a2	a2	PROPN
ejpam-4905	161	36	,	,	PUNCT
ejpam-4905	161	37	.	.	PUNCT
ejpam-4905	161	38	.	.	PUNCT
ejpam-4905	162	1	.	.	PUNCT
ejpam-4905	163	1	,	,	PUNCT
ejpam-4905	163	2	an	an	PRON
ejpam-4905	163	3	}	}	PUNCT
ejpam-4905	163	4	.	.	PUNCT
ejpam-4905	164	1	then	then	ADV
ejpam-4905	164	2	{	{	PUNCT
ejpam-4905	164	3	ai	ai	VERB
ejpam-4905	164	4	}	}	PUNCT
ejpam-4905	164	5	=	=	SYM
ejpam-4905	164	6	n2	n2	PROPN
ejpam-4905	164	7	g[ai	g[ai	PROPN
ejpam-4905	164	8	]	]	PUNCT
ejpam-4905	164	9	⊈	⊈	PROPN
ejpam-4905	164	10	n2	n2	NOUN
ejpam-4905	164	11	g[aj	g[aj	PROPN
ejpam-4905	164	12	]	]	X
ejpam-4905	164	13	=	=	PUNCT
ejpam-4905	164	14	{	{	PUNCT
ejpam-4905	164	15	aj	aj	PROPN
ejpam-4905	164	16	}	}	PUNCT
ejpam-4905	164	17	∀	∀	NOUN
ejpam-4905	165	1	i	i	NOUN
ejpam-4905	165	2	̸=	̸=	PROPN
ejpam-4905	165	3	j	j	PROPN
ejpam-4905	165	4	,	,	PUNCT
ejpam-4905	165	5	where	where	SCONJ
ejpam-4905	165	6	i	i	PRON
ejpam-4905	165	7	,	,	PUNCT
ejpam-4905	165	8	j	j	PROPN
ejpam-4905	165	9	∈	∈	PROPN
ejpam-4905	165	10	{	{	PUNCT
ejpam-4905	165	11	1	1	NUM
ejpam-4905	165	12	,	,	PUNCT
ejpam-4905	165	13	2	2	NUM
ejpam-4905	165	14	,	,	PUNCT
ejpam-4905	165	15	.	.	PUNCT
ejpam-4905	165	16	.	.	PUNCT
ejpam-4905	166	1	.	.	PUNCT
ejpam-4905	166	2	,	,	PUNCT
ejpam-4905	166	3	n	n	CCONJ
ejpam-4905	166	4	}	}	PUNCT
ejpam-4905	166	5	.	.	PUNCT
ejpam-4905	167	1	thus	thus	ADV
ejpam-4905	167	2	,	,	PUNCT
ejpam-4905	167	3	by	by	ADP
ejpam-4905	167	4	(	(	PUNCT
ejpam-4905	167	5	ii	ii	NOUN
ejpam-4905	167	6	)	)	PUNCT
ejpam-4905	167	7	,	,	PUNCT
ejpam-4905	167	8	γj2h(g	γj2h(g	NOUN
ejpam-4905	167	9	)	)	PUNCT
ejpam-4905	167	10	=	=	SYM
ejpam-4905	167	11	|v	|v	X
ejpam-4905	167	12	(	(	PUNCT
ejpam-4905	167	13	g)|	g)|	NOUN
ejpam-4905	167	14	=	=	PUNCT
ejpam-4905	167	15	n.	n.	NOUN
ejpam-4905	167	16	similarly	similarly	ADV
ejpam-4905	167	17	,	,	PUNCT
ejpam-4905	167	18	if	if	SCONJ
ejpam-4905	167	19	g	g	PROPN
ejpam-4905	167	20	=	=	SYM
ejpam-4905	167	21	kn	kn	PROPN
ejpam-4905	167	22	,	,	PUNCT
ejpam-4905	167	23	then	then	ADV
ejpam-4905	167	24	γj2h(g	γj2h(g	NOUN
ejpam-4905	167	25	)	)	PUNCT
ejpam-4905	167	26	=	=	SYM
ejpam-4905	167	27	|v	|v	X
ejpam-4905	167	28	(	(	PUNCT
ejpam-4905	167	29	g)|	g)|	NOUN
ejpam-4905	167	30	=	=	PUNCT
ejpam-4905	167	31	n.	n.	NOUN
ejpam-4905	167	32	theorem	theorem	VERB
ejpam-4905	167	33	4	4	NUM
ejpam-4905	167	34	.	.	PUNCT
ejpam-4905	167	35	let	let	VERB
ejpam-4905	167	36	a	a	PRON
ejpam-4905	167	37	,	,	PUNCT
ejpam-4905	167	38	b	b	NOUN
ejpam-4905	167	39	be	be	AUX
ejpam-4905	167	40	positive	positive	ADJ
ejpam-4905	167	41	integers	integer	NOUN
ejpam-4905	167	42	with	with	ADP
ejpam-4905	167	43	2	2	NUM
ejpam-4905	167	44	≤	≤	NOUN
ejpam-4905	167	45	a	a	DET
ejpam-4905	167	46	≤	≤	PROPN
ejpam-4905	167	47	b.	b.	NOUN
ejpam-4905	167	48	then	then	ADV
ejpam-4905	167	49	there	there	PRON
ejpam-4905	167	50	exists	exist	VERB
ejpam-4905	167	51	a	a	DET
ejpam-4905	167	52	connected	connected	ADJ
ejpam-4905	167	53	graph	graph	NOUN
ejpam-4905	167	54	g	g	ADP
ejpam-4905	167	55	such	such	ADJ
ejpam-4905	167	56	that	that	PRON
ejpam-4905	167	57	γh(g	γh(g	NOUN
ejpam-4905	167	58	)	)	PUNCT
ejpam-4905	167	59	=	=	SYM
ejpam-4905	167	60	a	a	PRON
ejpam-4905	167	61	and	and	CCONJ
ejpam-4905	167	62	γj2h(g	γj2h(g	NOUN
ejpam-4905	167	63	)	)	PUNCT
ejpam-4905	167	64	=	=	SYM
ejpam-4905	167	65	b.	b.	PROPN
ejpam-4905	168	1	in	in	ADP
ejpam-4905	168	2	other	other	ADJ
ejpam-4905	168	3	words	word	NOUN
ejpam-4905	168	4	,	,	PUNCT
ejpam-4905	168	5	γj2h(g	γj2h(g	NOUN
ejpam-4905	168	6	)	)	PUNCT
ejpam-4905	168	7	−	−	NOUN
ejpam-4905	168	8	γh(g	γh(g	NOUN
ejpam-4905	168	9	)	)	PUNCT
ejpam-4905	168	10	can	can	AUX
ejpam-4905	168	11	be	be	AUX
ejpam-4905	168	12	made	make	VERB
ejpam-4905	168	13	arbitrarily	arbitrarily	ADV
ejpam-4905	168	14	large	large	ADJ
ejpam-4905	168	15	.	.	PUNCT
ejpam-4905	169	1	proof	proof	NOUN
ejpam-4905	169	2	.	.	PUNCT
ejpam-4905	170	1	for	for	ADP
ejpam-4905	170	2	a	a	DET
ejpam-4905	170	3	=	=	SYM
ejpam-4905	170	4	b	b	NOUN
ejpam-4905	170	5	,	,	PUNCT
ejpam-4905	170	6	consider	consider	VERB
ejpam-4905	170	7	ka	ka	PROPN
ejpam-4905	170	8	.	.	PUNCT
ejpam-4905	171	1	then	then	ADV
ejpam-4905	171	2	by	by	ADP
ejpam-4905	171	3	theorem	theorem	ADJ
ejpam-4905	171	4	3	3	NUM
ejpam-4905	171	5	,	,	PUNCT
ejpam-4905	171	6	γj2h(ka	γj2h(ka	NOUN
ejpam-4905	171	7	)	)	PUNCT
ejpam-4905	171	8	=	=	SYM
ejpam-4905	171	9	a	a	DET
ejpam-4905	171	10	=	=	SYM
ejpam-4905	171	11	γh(ka	γh(ka	NOUN
ejpam-4905	171	12	)	)	PUNCT
ejpam-4905	171	13	.	.	PUNCT
ejpam-4905	172	1	suppose	suppose	VERB
ejpam-4905	172	2	that	that	SCONJ
ejpam-4905	172	3	a	a	DET
ejpam-4905	172	4	<	<	X
ejpam-4905	172	5	b.	b.	NOUN
ejpam-4905	172	6	consider	consider	VERB
ejpam-4905	172	7	the	the	DET
ejpam-4905	172	8	following	follow	VERB
ejpam-4905	172	9	two	two	NUM
ejpam-4905	172	10	cases	case	NOUN
ejpam-4905	172	11	:	:	PUNCT
ejpam-4905	172	12	case	case	NOUN
ejpam-4905	172	13	1	1	NUM
ejpam-4905	172	14	:	:	PUNCT
ejpam-4905	172	15	a	a	PRON
ejpam-4905	172	16	is	be	AUX
ejpam-4905	172	17	odd	odd	ADJ
ejpam-4905	172	18	.	.	PUNCT
ejpam-4905	173	1	consider	consider	VERB
ejpam-4905	173	2	the	the	DET
ejpam-4905	173	3	graph	graph	NOUN
ejpam-4905	173	4	g	g	NOUN
ejpam-4905	173	5	in	in	ADP
ejpam-4905	173	6	figure	figure	NOUN
ejpam-4905	173	7	2	2	NUM
ejpam-4905	173	8	.	.	PUNCT
ejpam-4905	174	1	let	let	VERB
ejpam-4905	174	2	m	m	VERB
ejpam-4905	174	3	=	=	VERB
ejpam-4905	175	1	b	b	X
ejpam-4905	175	2	−	−	PROPN
ejpam-4905	175	3	a	a	PRON
ejpam-4905	175	4	and	and	CCONJ
ejpam-4905	175	5	let	let	VERB
ejpam-4905	175	6	s	s	PRON
ejpam-4905	175	7	=	=	PUNCT
ejpam-4905	175	8	{	{	PUNCT
ejpam-4905	175	9	x1	x1	PROPN
ejpam-4905	175	10	,	,	PUNCT
ejpam-4905	175	11	x2	x2	PROPN
ejpam-4905	175	12	,	,	PUNCT
ejpam-4905	175	13	.	.	PUNCT
ejpam-4905	175	14	.	.	PUNCT
ejpam-4905	176	1	.	.	PUNCT
ejpam-4905	177	1	,	,	PUNCT
ejpam-4905	177	2	xa	xa	PROPN
ejpam-4905	177	3	}	}	PUNCT
ejpam-4905	177	4	and	and	CCONJ
ejpam-4905	177	5	s′	s′	ADJ
ejpam-4905	177	6	=	=	PUNCT
ejpam-4905	177	7	{	{	PUNCT
ejpam-4905	177	8	x1	x1	PROPN
ejpam-4905	177	9	,	,	PUNCT
ejpam-4905	177	10	x2	x2	PROPN
ejpam-4905	177	11	,	,	PUNCT
ejpam-4905	177	12	.	.	PUNCT
ejpam-4905	177	13	.	.	PUNCT
ejpam-4905	178	1	.	.	PUNCT
ejpam-4905	179	1	,	,	PUNCT
ejpam-4905	179	2	xa−3	xa−3	PROPN
ejpam-4905	179	3	,	,	PUNCT
ejpam-4905	179	4	u	u	NOUN
ejpam-4905	179	5	,	,	PUNCT
ejpam-4905	179	6	v	v	PROPN
ejpam-4905	179	7	,	,	PUNCT
ejpam-4905	179	8	xa	xa	PROPN
ejpam-4905	179	9	,	,	PUNCT
ejpam-4905	179	10	c1	c1	PROPN
ejpam-4905	179	11	,	,	PUNCT
ejpam-4905	179	12	c2	c2	PROPN
ejpam-4905	179	13	,	,	PUNCT
ejpam-4905	179	14	.	.	PUNCT
ejpam-4905	179	15	.	.	PUNCT
ejpam-4905	179	16	.	.	PUNCT
ejpam-4905	180	1	,	,	PUNCT
ejpam-4905	180	2	cm	cm	NOUN
ejpam-4905	180	3	}	}	PUNCT
ejpam-4905	180	4	.	.	PUNCT
ejpam-4905	181	1	then	then	ADV
ejpam-4905	181	2	s	s	PRON
ejpam-4905	181	3	and	and	CCONJ
ejpam-4905	181	4	s′	s′	NOUN
ejpam-4905	181	5	are	be	AUX
ejpam-4905	181	6	γh	γh	ADV
ejpam-4905	181	7	-	-	PUNCT
ejpam-4905	181	8	set	set	VERB
ejpam-4905	181	9	and	and	CCONJ
ejpam-4905	181	10	γj2h	γj2h	NOUN
ejpam-4905	181	11	-	-	PUNCT
ejpam-4905	181	12	set	set	NOUN
ejpam-4905	181	13	in	in	ADP
ejpam-4905	181	14	g	g	NOUN
ejpam-4905	181	15	,	,	PUNCT
ejpam-4905	181	16	respectively	respectively	ADV
ejpam-4905	181	17	.	.	PUNCT
ejpam-4905	182	1	hence	hence	ADV
ejpam-4905	182	2	,	,	PUNCT
ejpam-4905	182	3	γh(g	γh(g	NOUN
ejpam-4905	182	4	)	)	PUNCT
ejpam-4905	182	5	=	=	SYM
ejpam-4905	182	6	a	a	PRON
ejpam-4905	182	7	and	and	CCONJ
ejpam-4905	182	8	γj2h(g	γj2h(g	NOUN
ejpam-4905	182	9	)	)	PUNCT
ejpam-4905	182	10	=	=	PUNCT
ejpam-4905	182	11	a+m	a+m	NUM
ejpam-4905	182	12	=	=	SYM
ejpam-4905	182	13	b.	b.	PROPN
ejpam-4905	182	14	consequently	consequently	ADV
ejpam-4905	182	15	,	,	PUNCT
ejpam-4905	182	16	γh(g	γh(g	NOUN
ejpam-4905	182	17	)	)	PUNCT
ejpam-4905	182	18	<	<	X
ejpam-4905	182	19	γj2h(g	γj2h(g	NOUN
ejpam-4905	182	20	)	)	PUNCT
ejpam-4905	182	21	.	.	PUNCT
ejpam-4905	182	22	.	.	PUNCT
ejpam-4905	182	23	.	.	PUNCT
ejpam-4905	183	1	.	.	PUNCT
ejpam-4905	184	1	g	g	NOUN
ejpam-4905	184	2	:	:	PUNCT
ejpam-4905	185	1	x2x1	x2x1	PROPN
ejpam-4905	185	2	v	v	X
ejpam-4905	185	3	c1	c1	PROPN
ejpam-4905	185	4	xa−1x3	xa−1x3	PROPN
ejpam-4905	186	1	x4	x4	PROPN
ejpam-4905	186	2	.	.	PUNCT
ejpam-4905	186	3	.	.	PUNCT
ejpam-4905	186	4	.	.	PUNCT
ejpam-4905	187	1	xa	xa	PROPN
ejpam-4905	188	1	xa−2	xa−2	PROPN
ejpam-4905	188	2	c2	c2	PROPN
ejpam-4905	188	3	u	u	NOUN
ejpam-4905	188	4	cm	cm	NOUN
ejpam-4905	188	5	figure	figure	NOUN
ejpam-4905	188	6	2	2	NUM
ejpam-4905	188	7	:	:	PUNCT
ejpam-4905	188	8	graph	graph	VERB
ejpam-4905	188	9	g	g	NOUN
ejpam-4905	188	10	with	with	ADP
ejpam-4905	188	11	γh(g	γh(g	NOUN
ejpam-4905	188	12	)	)	PUNCT
ejpam-4905	188	13	<	<	X
ejpam-4905	188	14	γj2h(g	γj2h(g	NOUN
ejpam-4905	188	15	)	)	PUNCT
ejpam-4905	188	16	case	case	NOUN
ejpam-4905	188	17	2	2	NUM
ejpam-4905	188	18	:	:	PUNCT
ejpam-4905	188	19	a	a	PRON
ejpam-4905	188	20	is	be	AUX
ejpam-4905	188	21	even	even	ADV
ejpam-4905	188	22	.	.	PUNCT
ejpam-4905	189	1	consider	consider	VERB
ejpam-4905	189	2	the	the	DET
ejpam-4905	189	3	graph	graph	NOUN
ejpam-4905	189	4	h	h	NOUN
ejpam-4905	189	5	in	in	ADP
ejpam-4905	189	6	figure	figure	NOUN
ejpam-4905	189	7	3	3	NUM
ejpam-4905	189	8	.	.	PUNCT
ejpam-4905	190	1	let	let	VERB
ejpam-4905	190	2	t	t	NOUN
ejpam-4905	190	3	=	=	SYM
ejpam-4905	190	4	b	b	PROPN
ejpam-4905	190	5	−	−	PROPN
ejpam-4905	190	6	a	a	PRON
ejpam-4905	190	7	and	and	CCONJ
ejpam-4905	190	8	let	let	VERB
ejpam-4905	190	9	c	c	NOUN
ejpam-4905	190	10	=	=	PUNCT
ejpam-4905	190	11	{	{	PUNCT
ejpam-4905	190	12	x1	x1	PROPN
ejpam-4905	190	13	,	,	PUNCT
ejpam-4905	190	14	x2	x2	PROPN
ejpam-4905	190	15	,	,	PUNCT
ejpam-4905	190	16	.	.	PUNCT
ejpam-4905	190	17	.	.	PUNCT
ejpam-4905	191	1	.	.	PUNCT
ejpam-4905	192	1	,	,	PUNCT
ejpam-4905	192	2	xa	xa	PROPN
ejpam-4905	192	3	}	}	PUNCT
ejpam-4905	192	4	and	and	CCONJ
ejpam-4905	192	5	c	c	NOUN
ejpam-4905	192	6	′	′	NUM
ejpam-4905	192	7	=	=	PUNCT
ejpam-4905	192	8	{	{	PUNCT
ejpam-4905	193	1	x1	x1	PROPN
ejpam-4905	193	2	,	,	PUNCT
ejpam-4905	193	3	x2	x2	PROPN
ejpam-4905	193	4	,	,	PUNCT
ejpam-4905	193	5	.	.	PUNCT
ejpam-4905	193	6	.	.	PUNCT
ejpam-4905	193	7	.	.	PUNCT
ejpam-4905	194	1	,	,	PUNCT
ejpam-4905	194	2	xa−2	xa−2	PROPN
ejpam-4905	194	3	,	,	PUNCT
ejpam-4905	194	4	v	v	NOUN
ejpam-4905	194	5	,	,	PUNCT
ejpam-4905	194	6	w	w	PROPN
ejpam-4905	194	7	,	,	PUNCT
ejpam-4905	194	8	c1	c1	PROPN
ejpam-4905	194	9	,	,	PUNCT
ejpam-4905	194	10	c2	c2	PROPN
ejpam-4905	194	11	,	,	PUNCT
ejpam-4905	194	12	.	.	PUNCT
ejpam-4905	194	13	.	.	PUNCT
ejpam-4905	194	14	.	.	PUNCT
ejpam-4905	195	1	,	,	PUNCT
ejpam-4905	195	2	ct	ct	NOUN
ejpam-4905	195	3	}	}	PUNCT
ejpam-4905	195	4	.	.	PUNCT
ejpam-4905	196	1	then	then	ADV
ejpam-4905	196	2	c	c	PROPN
ejpam-4905	196	3	and	and	CCONJ
ejpam-4905	196	4	c	c	PROPN
ejpam-4905	196	5	′	′	NOUN
ejpam-4905	196	6	are	be	AUX
ejpam-4905	196	7	γh	γh	ADV
ejpam-4905	196	8	-	-	PUNCT
ejpam-4905	196	9	set	set	VERB
ejpam-4905	196	10	and	and	CCONJ
ejpam-4905	196	11	γj2h	γj2h	NOUN
ejpam-4905	196	12	-	-	PUNCT
ejpam-4905	196	13	set	set	VERB
ejpam-4905	196	14	in	in	ADP
ejpam-4905	196	15	h	h	NOUN
ejpam-4905	196	16	,	,	PUNCT
ejpam-4905	196	17	respectively	respectively	ADV
ejpam-4905	196	18	.	.	PUNCT
ejpam-4905	197	1	therefore	therefore	ADV
ejpam-4905	197	2	,	,	PUNCT
ejpam-4905	197	3	γh(h	γh(h	PUNCT
ejpam-4905	197	4	)	)	PUNCT
ejpam-4905	197	5	=	=	SYM
ejpam-4905	197	6	a	a	DET
ejpam-4905	197	7	and	and	CCONJ
ejpam-4905	197	8	γj2h(h	γj2h(h	NUM
ejpam-4905	197	9	)	)	PUNCT
ejpam-4905	197	10	=	=	PUNCT
ejpam-4905	197	11	a	a	PRON
ejpam-4905	197	12	+	+	NUM
ejpam-4905	197	13	t	t	NOUN
ejpam-4905	197	14	=	=	SYM
ejpam-4905	197	15	b	b	PROPN
ejpam-4905	197	16	,	,	PUNCT
ejpam-4905	197	17	showing	show	VERB
ejpam-4905	197	18	that	that	PRON
ejpam-4905	197	19	γh(h	γh(h	PUNCT
ejpam-4905	197	20	)	)	PUNCT
ejpam-4905	197	21	<	<	X
ejpam-4905	197	22	γj2h(h	γj2h(h	NOUN
ejpam-4905	197	23	)	)	PUNCT
ejpam-4905	197	24	.	.	PUNCT
ejpam-4905	198	1	j.	j.	PROPN
ejpam-4905	198	2	hassan	hassan	PROPN
ejpam-4905	198	3	,	,	PUNCT
ejpam-4905	198	4	a.	a.	PROPN
ejpam-4905	198	5	bakkang	bakkang	PROPN
ejpam-4905	198	6	,	,	PUNCT
ejpam-4905	198	7	a.	a.	NOUN
ejpam-4905	198	8	sappari	sappari	PROPN
ejpam-4905	198	9	/	/	SYM
ejpam-4905	198	10	eur	eur	PROPN
ejpam-4905	198	11	.	.	PUNCT
ejpam-4905	199	1	j.	j.	PROPN
ejpam-4905	199	2	pure	pure	PROPN
ejpam-4905	199	3	appl	appl	PROPN
ejpam-4905	199	4	.	.	PROPN
ejpam-4905	199	5	math	math	PROPN
ejpam-4905	199	6	,	,	PUNCT
ejpam-4905	199	7	16	16	NUM
ejpam-4905	199	8	(	(	PUNCT
ejpam-4905	199	9	4	4	NUM
ejpam-4905	199	10	)	)	PUNCT
ejpam-4905	199	11	(	(	PUNCT
ejpam-4905	199	12	2023	2023	NUM
ejpam-4905	199	13	)	)	PUNCT
ejpam-4905	199	14	,	,	PUNCT
ejpam-4905	199	15	2118	2118	NUM
ejpam-4905	199	16	-	-	SYM
ejpam-4905	199	17	2131	2131	NUM
ejpam-4905	199	18	2124	2124	NUM
ejpam-4905	199	19	.	.	PUNCT
ejpam-4905	199	20	.	.	PUNCT
ejpam-4905	199	21	.	.	PUNCT
ejpam-4905	200	1	h	h	NOUN
ejpam-4905	200	2	:	:	PUNCT
ejpam-4905	201	1	x2x1	x2x1	PUNCT
ejpam-4905	201	2	v	v	X
ejpam-4905	201	3	c1	c1	PROPN
ejpam-4905	201	4	xax3	xax3	PROPN
ejpam-4905	201	5	x4	x4	PROPN
ejpam-4905	201	6	.	.	PUNCT
ejpam-4905	201	7	.	.	PUNCT
ejpam-4905	201	8	.	.	PUNCT
ejpam-4905	202	1	w	w	PROPN
ejpam-4905	202	2	xa−1	xa−1	PROPN
ejpam-4905	202	3	c2	c2	PROPN
ejpam-4905	202	4	ct	ct	PROPN
ejpam-4905	202	5	figure	figure	NOUN
ejpam-4905	202	6	3	3	NUM
ejpam-4905	202	7	:	:	PUNCT
ejpam-4905	202	8	graph	graph	NOUN
ejpam-4905	202	9	h	h	NOUN
ejpam-4905	202	10	with	with	ADP
ejpam-4905	202	11	γh(h	γh(h	NOUN
ejpam-4905	202	12	)	)	PUNCT
ejpam-4905	202	13	<	<	X
ejpam-4905	202	14	γj2h(h	γj2h(h	NOUN
ejpam-4905	202	15	)	)	PUNCT
ejpam-4905	202	16	theorem	theorem	NOUN
ejpam-4905	202	17	5	5	NUM
ejpam-4905	202	18	.	.	PUNCT
ejpam-4905	203	1	let	let	VERB
ejpam-4905	203	2	g	g	NOUN
ejpam-4905	203	3	be	be	AUX
ejpam-4905	203	4	any	any	DET
ejpam-4905	203	5	graph	graph	NOUN
ejpam-4905	203	6	and	and	CCONJ
ejpam-4905	203	7	let	let	VERB
ejpam-4905	203	8	s	s	PRON
ejpam-4905	203	9	⊆	⊆	NUM
ejpam-4905	203	10	v	v	NOUN
ejpam-4905	203	11	(	(	PUNCT
ejpam-4905	203	12	g	g	NOUN
ejpam-4905	203	13	)	)	PUNCT
ejpam-4905	203	14	.	.	PUNCT
ejpam-4905	204	1	then	then	ADV
ejpam-4905	204	2	every	every	DET
ejpam-4905	204	3	hop	hop	NOUN
ejpam-4905	204	4	independent	independent	ADJ
ejpam-4905	204	5	set	set	PROPN
ejpam-4905	204	6	s	s	PART
ejpam-4905	204	7	is	be	AUX
ejpam-4905	204	8	a	a	DET
ejpam-4905	204	9	j2	j2	NOUN
ejpam-4905	204	10	-	-	PUNCT
ejpam-4905	204	11	set	set	NOUN
ejpam-4905	204	12	in	in	ADP
ejpam-4905	204	13	g.	g.	PROPN
ejpam-4905	204	14	in	in	ADP
ejpam-4905	204	15	particular	particular	ADJ
ejpam-4905	204	16	,	,	PUNCT
ejpam-4905	204	17	every	every	DET
ejpam-4905	204	18	αh	αh	NOUN
ejpam-4905	204	19	-	-	PUNCT
ejpam-4905	204	20	set	set	NOUN
ejpam-4905	204	21	is	be	AUX
ejpam-4905	204	22	a	a	DET
ejpam-4905	204	23	j2	j2	PROPN
ejpam-4905	204	24	-	-	PUNCT
ejpam-4905	204	25	hop	hop	NOUN
ejpam-4905	204	26	dominating	dominating	NOUN
ejpam-4905	204	27	set	set	NOUN
ejpam-4905	204	28	.	.	PUNCT
ejpam-4905	205	1	moreover	moreover	ADV
ejpam-4905	205	2	,	,	PUNCT
ejpam-4905	205	3	αh(g	αh(g	NOUN
ejpam-4905	205	4	)	)	PUNCT
ejpam-4905	205	5	≤	≤	NUM
ejpam-4905	205	6	γj2h(g	γj2h(g	NOUN
ejpam-4905	205	7	)	)	PUNCT
ejpam-4905	205	8	.	.	PUNCT
ejpam-4905	206	1	proof	proof	NOUN
ejpam-4905	206	2	.	.	PUNCT
ejpam-4905	207	1	let	let	VERB
ejpam-4905	207	2	s	s	PRON
ejpam-4905	207	3	be	be	AUX
ejpam-4905	207	4	a	a	DET
ejpam-4905	207	5	hop	hop	NOUN
ejpam-4905	207	6	independent	independent	ADJ
ejpam-4905	207	7	set	set	NOUN
ejpam-4905	207	8	in	in	ADP
ejpam-4905	207	9	g.	g.	PROPN
ejpam-4905	207	10	then	then	ADV
ejpam-4905	207	11	dg(a	dg(a	PROPN
ejpam-4905	207	12	,	,	PUNCT
ejpam-4905	207	13	b	b	X
ejpam-4905	207	14	)	)	PUNCT
ejpam-4905	207	15	̸=	̸=	PROPN
ejpam-4905	207	16	2	2	NUM
ejpam-4905	207	17	for	for	ADP
ejpam-4905	207	18	every	every	DET
ejpam-4905	207	19	a	a	PROPN
ejpam-4905	207	20	,	,	PUNCT
ejpam-4905	207	21	b	b	X
ejpam-4905	207	22	∈	∈	PROPN
ejpam-4905	207	23	s.	s.	PROPN
ejpam-4905	207	24	suppose	suppose	VERB
ejpam-4905	207	25	on	on	ADP
ejpam-4905	207	26	the	the	DET
ejpam-4905	207	27	contrary	contrary	NOUN
ejpam-4905	207	28	that	that	PRON
ejpam-4905	207	29	s	s	VERB
ejpam-4905	207	30	is	be	AUX
ejpam-4905	207	31	not	not	PART
ejpam-4905	207	32	a	a	DET
ejpam-4905	207	33	j2	j2	NOUN
ejpam-4905	207	34	-	-	PUNCT
ejpam-4905	207	35	set	set	NOUN
ejpam-4905	207	36	in	in	ADP
ejpam-4905	207	37	g.	g.	PROPN
ejpam-4905	207	38	then	then	ADV
ejpam-4905	207	39	there	there	PRON
ejpam-4905	207	40	exist	exist	VERB
ejpam-4905	207	41	x	x	NOUN
ejpam-4905	207	42	,	,	PUNCT
ejpam-4905	207	43	y	y	PROPN
ejpam-4905	207	44	∈	∈	PROPN
ejpam-4905	207	45	s	s	VERB
ejpam-4905	207	46	such	such	ADJ
ejpam-4905	207	47	that	that	DET
ejpam-4905	207	48	n2	n2	PROPN
ejpam-4905	207	49	g[x]\n2	g[x]\n2	X
ejpam-4905	207	50	g[y	g[y	NOUN
ejpam-4905	207	51	]	]	X
ejpam-4905	207	52	=	=	SYM
ejpam-4905	207	53	∅	∅	NOUN
ejpam-4905	207	54	or	or	CCONJ
ejpam-4905	207	55	n2	n2	ADJ
ejpam-4905	207	56	g[y]\n2	g[y]\n2	ADJ
ejpam-4905	207	57	g[x	g[x	ADP
ejpam-4905	207	58	]	]	X
ejpam-4905	208	1	=	=	PUNCT
ejpam-4905	208	2	∅.	∅.	NOUN
ejpam-4905	208	3	it	it	PRON
ejpam-4905	208	4	follows	follow	VERB
ejpam-4905	208	5	that	that	DET
ejpam-4905	208	6	n2	n2	PROPN
ejpam-4905	208	7	g[x	g[x	PROPN
ejpam-4905	208	8	]	]	X
ejpam-4905	208	9	⊆	⊆	NUM
ejpam-4905	208	10	n2	n2	ADJ
ejpam-4905	208	11	g[y	g[y	NOUN
ejpam-4905	208	12	]	]	PUNCT
ejpam-4905	208	13	or	or	CCONJ
ejpam-4905	208	14	n	n	PRON
ejpam-4905	208	15	2	2	NUM
ejpam-4905	208	16	g[y	g[y	NOUN
ejpam-4905	208	17	]	]	PUNCT
ejpam-4905	208	18	⊆	⊆	NUM
ejpam-4905	208	19	n2	n2	NOUN
ejpam-4905	208	20	g[x	g[x	NOUN
ejpam-4905	208	21	]	]	PUNCT
ejpam-4905	208	22	.	.	PUNCT
ejpam-4905	209	1	in	in	ADP
ejpam-4905	209	2	either	either	DET
ejpam-4905	209	3	case	case	NOUN
ejpam-4905	209	4	,	,	PUNCT
ejpam-4905	209	5	we	we	PRON
ejpam-4905	209	6	have	have	VERB
ejpam-4905	209	7	dg(x	dg(x	NUM
ejpam-4905	209	8	,	,	PUNCT
ejpam-4905	209	9	y	y	NOUN
ejpam-4905	209	10	)	)	PUNCT
ejpam-4905	209	11	=	=	SYM
ejpam-4905	209	12	2	2	NUM
ejpam-4905	209	13	,	,	PUNCT
ejpam-4905	209	14	a	a	DET
ejpam-4905	209	15	contradiction	contradiction	NOUN
ejpam-4905	209	16	to	to	ADP
ejpam-4905	209	17	the	the	DET
ejpam-4905	209	18	fact	fact	NOUN
ejpam-4905	209	19	that	that	SCONJ
ejpam-4905	209	20	s	s	VERB
ejpam-4905	209	21	is	be	AUX
ejpam-4905	209	22	a	a	DET
ejpam-4905	209	23	hop	hop	NOUN
ejpam-4905	209	24	independent	independent	ADJ
ejpam-4905	209	25	set	set	NOUN
ejpam-4905	209	26	in	in	ADP
ejpam-4905	209	27	g.	g.	PROPN
ejpam-4905	209	28	therefore	therefore	ADV
ejpam-4905	209	29	,	,	PUNCT
ejpam-4905	209	30	s	s	VERB
ejpam-4905	209	31	is	be	AUX
ejpam-4905	209	32	a	a	DET
ejpam-4905	209	33	j2	j2	NOUN
ejpam-4905	209	34	-	-	PUNCT
ejpam-4905	209	35	set	set	NOUN
ejpam-4905	209	36	in	in	ADP
ejpam-4905	209	37	g.	g.	PROPN
ejpam-4905	209	38	next	next	ADV
ejpam-4905	209	39	,	,	PUNCT
ejpam-4905	209	40	let	let	VERB
ejpam-4905	209	41	s′	s′	NOUN
ejpam-4905	209	42	be	be	AUX
ejpam-4905	209	43	an	an	DET
ejpam-4905	209	44	αh	αh	NOUN
ejpam-4905	209	45	-	-	PUNCT
ejpam-4905	209	46	set	set	NOUN
ejpam-4905	209	47	of	of	ADP
ejpam-4905	209	48	g.	g.	PROPN
ejpam-4905	209	49	then	then	ADV
ejpam-4905	209	50	s′	s′	PROPN
ejpam-4905	209	51	is	be	AUX
ejpam-4905	209	52	a	a	DET
ejpam-4905	209	53	maximum	maximum	ADJ
ejpam-4905	209	54	hop	hop	NOUN
ejpam-4905	209	55	independent	independent	ADJ
ejpam-4905	209	56	set	set	NOUN
ejpam-4905	209	57	of	of	ADP
ejpam-4905	209	58	g	g	PROPN
ejpam-4905	209	59	(	(	PUNCT
ejpam-4905	209	60	by	by	ADP
ejpam-4905	209	61	definition	definition	NOUN
ejpam-4905	209	62	)	)	PUNCT
ejpam-4905	209	63	.	.	PUNCT
ejpam-4905	210	1	thus	thus	ADV
ejpam-4905	210	2	,	,	PUNCT
ejpam-4905	210	3	s′	s′	PROPN
ejpam-4905	210	4	is	be	AUX
ejpam-4905	210	5	a	a	DET
ejpam-4905	210	6	j2	j2	NOUN
ejpam-4905	210	7	-	-	PUNCT
ejpam-4905	210	8	set	set	NOUN
ejpam-4905	210	9	in	in	ADP
ejpam-4905	210	10	g	g	NOUN
ejpam-4905	210	11	by	by	ADP
ejpam-4905	210	12	the	the	DET
ejpam-4905	210	13	first	first	ADJ
ejpam-4905	210	14	part	part	NOUN
ejpam-4905	210	15	.	.	PUNCT
ejpam-4905	211	1	now	now	ADV
ejpam-4905	211	2	,	,	PUNCT
ejpam-4905	211	3	suppose	suppose	VERB
ejpam-4905	211	4	on	on	ADP
ejpam-4905	211	5	the	the	DET
ejpam-4905	211	6	contrary	contrary	NOUN
ejpam-4905	211	7	that	that	SCONJ
ejpam-4905	211	8	s′	s′	ADJ
ejpam-4905	211	9	is	be	AUX
ejpam-4905	211	10	not	not	PART
ejpam-4905	211	11	a	a	DET
ejpam-4905	211	12	hop	hop	NOUN
ejpam-4905	211	13	dominating	dominating	NOUN
ejpam-4905	211	14	set	set	NOUN
ejpam-4905	211	15	of	of	ADP
ejpam-4905	211	16	g.	g.	PROPN
ejpam-4905	211	17	then	then	ADV
ejpam-4905	211	18	there	there	PRON
ejpam-4905	211	19	exists	exist	VERB
ejpam-4905	211	20	x	x	X
ejpam-4905	211	21	∈	∈	PROPN
ejpam-4905	211	22	v	v	X
ejpam-4905	211	23	(	(	PUNCT
ejpam-4905	211	24	g	g	NOUN
ejpam-4905	211	25	)	)	PUNCT
ejpam-4905	211	26	\	\	NOUN
ejpam-4905	212	1	s′	s′	VERB
ejpam-4905	212	2	such	such	ADJ
ejpam-4905	212	3	that	that	SCONJ
ejpam-4905	212	4	x	x	PROPN
ejpam-4905	212	5	/∈	/∈	PUNCT
ejpam-4905	212	6	n2	n2	PROPN
ejpam-4905	212	7	g[y	g[y	NOUN
ejpam-4905	212	8	]	]	X
ejpam-4905	212	9	∀	∀	PUNCT
ejpam-4905	212	10	y	y	PROPN
ejpam-4905	212	11	∈	∈	PROPN
ejpam-4905	212	12	s′.	s′.	PROPN
ejpam-4905	213	1	this	this	PRON
ejpam-4905	213	2	means	mean	VERB
ejpam-4905	213	3	that	that	SCONJ
ejpam-4905	213	4	dg(x	dg(x	ADV
ejpam-4905	213	5	,	,	PUNCT
ejpam-4905	213	6	y	y	NOUN
ejpam-4905	213	7	)	)	PUNCT
ejpam-4905	213	8	̸=	̸=	PROPN
ejpam-4905	213	9	2	2	NUM
ejpam-4905	213	10	for	for	ADP
ejpam-4905	213	11	all	all	DET
ejpam-4905	213	12	y	y	PROPN
ejpam-4905	213	13	∈	∈	PROPN
ejpam-4905	213	14	s′.	s′.	PROPN
ejpam-4905	213	15	thus	thus	ADV
ejpam-4905	213	16	,	,	PUNCT
ejpam-4905	213	17	s∗	s∗	PROPN
ejpam-4905	213	18	=	=	SYM
ejpam-4905	213	19	{	{	PUNCT
ejpam-4905	213	20	x	x	NOUN
ejpam-4905	213	21	}	}	PUNCT
ejpam-4905	213	22	∪	∪	ADJ
ejpam-4905	213	23	s′	s′	NUM
ejpam-4905	213	24	is	be	AUX
ejpam-4905	213	25	a	a	DET
ejpam-4905	213	26	hop	hop	NOUN
ejpam-4905	213	27	independent	independent	ADJ
ejpam-4905	213	28	set	set	NOUN
ejpam-4905	213	29	in	in	ADP
ejpam-4905	213	30	g	g	NOUN
ejpam-4905	213	31	,	,	PUNCT
ejpam-4905	213	32	contradicting	contradict	VERB
ejpam-4905	213	33	the	the	DET
ejpam-4905	213	34	maximality	maximality	NOUN
ejpam-4905	213	35	of	of	ADP
ejpam-4905	213	36	s′.	s′.	PROPN
ejpam-4905	213	37	hence	hence	ADV
ejpam-4905	213	38	,	,	PUNCT
ejpam-4905	213	39	s′	s′	PROPN
ejpam-4905	213	40	is	be	AUX
ejpam-4905	213	41	a	a	DET
ejpam-4905	213	42	hop	hop	NOUN
ejpam-4905	213	43	dominating	dominating	NOUN
ejpam-4905	213	44	set	set	NOUN
ejpam-4905	213	45	of	of	ADP
ejpam-4905	213	46	g	g	NOUN
ejpam-4905	213	47	,	,	PUNCT
ejpam-4905	213	48	showing	show	VERB
ejpam-4905	213	49	that	that	SCONJ
ejpam-4905	213	50	s′	s′	ADJ
ejpam-4905	213	51	is	be	AUX
ejpam-4905	213	52	a	a	DET
ejpam-4905	213	53	j2	j2	PROPN
ejpam-4905	213	54	-	-	PUNCT
ejpam-4905	213	55	hop	hop	NOUN
ejpam-4905	213	56	dominating	dominating	NOUN
ejpam-4905	213	57	in	in	ADP
ejpam-4905	213	58	g.	g.	PROPN
ejpam-4905	213	59	consequently	consequently	ADV
ejpam-4905	213	60	,	,	PUNCT
ejpam-4905	213	61	αh(g	αh(g	NOUN
ejpam-4905	213	62	)	)	PUNCT
ejpam-4905	213	63	≤	≤	NUM
ejpam-4905	213	64	γj2h(g	γj2h(g	NOUN
ejpam-4905	213	65	)	)	PUNCT
ejpam-4905	213	66	.	.	PUNCT
ejpam-4905	214	1	theorem	theorem	NOUN
ejpam-4905	214	2	6	6	NUM
ejpam-4905	214	3	.	.	PUNCT
ejpam-4905	215	1	let	let	VERB
ejpam-4905	215	2	g	g	NOUN
ejpam-4905	215	3	and	and	CCONJ
ejpam-4905	215	4	h	h	NOUN
ejpam-4905	215	5	be	be	VERB
ejpam-4905	215	6	two	two	NUM
ejpam-4905	215	7	connected	connected	ADJ
ejpam-4905	215	8	graphs	graph	NOUN
ejpam-4905	215	9	.	.	PUNCT
ejpam-4905	216	1	if	if	SCONJ
ejpam-4905	216	2	n	n	NOUN
ejpam-4905	216	3	=	=	SYM
ejpam-4905	216	4	ng∪nh	ng∪nh	NUM
ejpam-4905	216	5	⊆	⊆	NUM
ejpam-4905	216	6	v	v	NOUN
ejpam-4905	216	7	(	(	PUNCT
ejpam-4905	216	8	g+h	g+h	PROPN
ejpam-4905	216	9	)	)	PUNCT
ejpam-4905	216	10	,	,	PUNCT
ejpam-4905	216	11	where	where	SCONJ
ejpam-4905	216	12	ng	ng	PROPN
ejpam-4905	216	13	and	and	CCONJ
ejpam-4905	216	14	nh	nh	PROPN
ejpam-4905	216	15	are	be	AUX
ejpam-4905	216	16	j2	j2	NOUN
ejpam-4905	216	17	-	-	PUNCT
ejpam-4905	216	18	sets	set	NOUN
ejpam-4905	216	19	in	in	ADP
ejpam-4905	216	20	g	g	PROPN
ejpam-4905	216	21	and	and	CCONJ
ejpam-4905	216	22	h	h	NOUN
ejpam-4905	216	23	,	,	PUNCT
ejpam-4905	216	24	respectively	respectively	ADV
ejpam-4905	216	25	,	,	PUNCT
ejpam-4905	216	26	then	then	ADV
ejpam-4905	216	27	n	n	PRON
ejpam-4905	216	28	is	be	AUX
ejpam-4905	216	29	a	a	DET
ejpam-4905	216	30	j2	j2	NOUN
ejpam-4905	216	31	-	-	PUNCT
ejpam-4905	216	32	set	set	NOUN
ejpam-4905	216	33	in	in	ADP
ejpam-4905	216	34	g+h	g+h	PROPN
ejpam-4905	216	35	.	.	PUNCT
ejpam-4905	217	1	proof	proof	NOUN
ejpam-4905	217	2	.	.	PUNCT
ejpam-4905	218	1	let	let	VERB
ejpam-4905	218	2	a	a	DET
ejpam-4905	218	3	,	,	PUNCT
ejpam-4905	218	4	b	b	PROPN
ejpam-4905	218	5	∈	∈	PROPN
ejpam-4905	218	6	n	n	ADV
ejpam-4905	218	7	.	.	PUNCT
ejpam-4905	219	1	suppose	suppose	VERB
ejpam-4905	219	2	that	that	SCONJ
ejpam-4905	219	3	a	a	PRON
ejpam-4905	219	4	,	,	PUNCT
ejpam-4905	219	5	b	b	PROPN
ejpam-4905	219	6	∈	∈	PROPN
ejpam-4905	219	7	ng	ng	PROPN
ejpam-4905	219	8	.	.	PUNCT
ejpam-4905	220	1	if	if	SCONJ
ejpam-4905	220	2	dg(a	dg(a	NUM
ejpam-4905	220	3	,	,	PUNCT
ejpam-4905	220	4	b	b	X
ejpam-4905	220	5	)	)	PUNCT
ejpam-4905	220	6	=	=	SYM
ejpam-4905	220	7	1	1	NUM
ejpam-4905	220	8	,	,	PUNCT
ejpam-4905	220	9	then	then	ADV
ejpam-4905	220	10	a	a	DET
ejpam-4905	220	11	∈	∈	PROPN
ejpam-4905	220	12	n2	n2	NOUN
ejpam-4905	220	13	g+h	g+h	PROPN
ejpam-4905	221	1	[	[	X
ejpam-4905	221	2	a	a	X
ejpam-4905	221	3	]	]	PUNCT
ejpam-4905	221	4	\	\	PROPN
ejpam-4905	221	5	n2	n2	NOUN
ejpam-4905	221	6	g+h	g+h	PUNCT
ejpam-4905	222	1	[	[	X
ejpam-4905	222	2	b	b	X
ejpam-4905	222	3	]	]	X
ejpam-4905	222	4	and	and	CCONJ
ejpam-4905	222	5	b	b	PROPN
ejpam-4905	222	6	∈	∈	PROPN
ejpam-4905	222	7	n2	n2	NOUN
ejpam-4905	222	8	g+h	g+h	PUNCT
ejpam-4905	223	1	[	[	X
ejpam-4905	223	2	b	b	X
ejpam-4905	223	3	]	]	PUNCT
ejpam-4905	223	4	\	\	PROPN
ejpam-4905	223	5	n2	n2	NOUN
ejpam-4905	223	6	g+h	g+h	PROPN
ejpam-4905	224	1	[	[	X
ejpam-4905	224	2	a	a	X
ejpam-4905	224	3	]	]	X
ejpam-4905	224	4	.	.	PUNCT
ejpam-4905	225	1	since	since	SCONJ
ejpam-4905	225	2	a	a	DET
ejpam-4905	225	3	,	,	PUNCT
ejpam-4905	225	4	b	b	NOUN
ejpam-4905	225	5	are	be	AUX
ejpam-4905	225	6	arbitrary	arbitrary	ADJ
ejpam-4905	225	7	,	,	PUNCT
ejpam-4905	225	8	the	the	DET
ejpam-4905	225	9	assertion	assertion	NOUN
ejpam-4905	225	10	follows	follow	VERB
ejpam-4905	225	11	.	.	PUNCT
ejpam-4905	226	1	assume	assume	VERB
ejpam-4905	226	2	that	that	SCONJ
ejpam-4905	226	3	dg(a	dg(a	PROPN
ejpam-4905	226	4	,	,	PUNCT
ejpam-4905	226	5	b	b	X
ejpam-4905	226	6	)	)	PUNCT
ejpam-4905	226	7	=	=	SYM
ejpam-4905	226	8	2	2	X
ejpam-4905	226	9	.	.	PUNCT
ejpam-4905	226	10	since	since	SCONJ
ejpam-4905	226	11	ng	ng	PROPN
ejpam-4905	226	12	is	be	AUX
ejpam-4905	226	13	a	a	DET
ejpam-4905	226	14	j2set	j2set	NOUN
ejpam-4905	226	15	in	in	ADP
ejpam-4905	226	16	g	g	NOUN
ejpam-4905	226	17	,	,	PUNCT
ejpam-4905	226	18	there	there	PRON
ejpam-4905	226	19	exist	exist	VERB
ejpam-4905	226	20	w	w	NOUN
ejpam-4905	226	21	,	,	PUNCT
ejpam-4905	226	22	z	z	PROPN
ejpam-4905	226	23	∈	∈	PROPN
ejpam-4905	226	24	v	v	ADP
ejpam-4905	226	25	(	(	PUNCT
ejpam-4905	226	26	g	g	NOUN
ejpam-4905	226	27	)	)	PUNCT
ejpam-4905	226	28	such	such	ADJ
ejpam-4905	226	29	that	that	SCONJ
ejpam-4905	226	30	w	w	PROPN
ejpam-4905	226	31	∈	∈	PROPN
ejpam-4905	226	32	n2	n2	NOUN
ejpam-4905	226	33	g[a]\n2	g[a]\n2	NOUN
ejpam-4905	226	34	g[b	g[b	NOUN
ejpam-4905	226	35	]	]	PUNCT
ejpam-4905	226	36	and	and	CCONJ
ejpam-4905	226	37	z	z	NOUN
ejpam-4905	226	38	∈	∈	PROPN
ejpam-4905	226	39	n2	n2	NOUN
ejpam-4905	226	40	g[b]\n2	g[b]\n2	VERB
ejpam-4905	226	41	g[a	g[a	PROPN
ejpam-4905	226	42	]	]	PUNCT
ejpam-4905	226	43	.	.	PUNCT
ejpam-4905	227	1	let	let	VERB
ejpam-4905	227	2	s	s	PRON
ejpam-4905	227	3	∈	∈	PROPN
ejpam-4905	227	4	ng(w)∩ng(a	ng(w)∩ng(a	PROPN
ejpam-4905	227	5	)	)	PUNCT
ejpam-4905	227	6	and	and	CCONJ
ejpam-4905	227	7	t	t	PROPN
ejpam-4905	227	8	∈	∈	PROPN
ejpam-4905	227	9	ng(z)∩ng(b	ng(z)∩ng(b	PROPN
ejpam-4905	227	10	)	)	PUNCT
ejpam-4905	227	11	.	.	PUNCT
ejpam-4905	228	1	then	then	ADV
ejpam-4905	228	2	s	s	VERB
ejpam-4905	228	3	∈	∈	PROPN
ejpam-4905	228	4	n2	n2	NOUN
ejpam-4905	228	5	g+h	g+h	PROPN
ejpam-4905	229	1	[	[	X
ejpam-4905	229	2	b]\n2	b]\n2	PROPN
ejpam-4905	229	3	g+h	g+h	PROPN
ejpam-4905	230	1	[	[	X
ejpam-4905	230	2	a	a	X
ejpam-4905	230	3	]	]	X
ejpam-4905	230	4	and	and	CCONJ
ejpam-4905	230	5	t	t	PROPN
ejpam-4905	230	6	∈	∈	PROPN
ejpam-4905	230	7	n2	n2	NOUN
ejpam-4905	230	8	g+h	g+h	PUNCT
ejpam-4905	231	1	[	[	X
ejpam-4905	231	2	a]\n2	a]\n2	PROPN
ejpam-4905	231	3	g+h	g+h	PUNCT
ejpam-4905	232	1	[	[	X
ejpam-4905	232	2	b	b	X
ejpam-4905	232	3	]	]	X
ejpam-4905	232	4	.	.	PUNCT
ejpam-4905	233	1	since	since	SCONJ
ejpam-4905	233	2	a	a	DET
ejpam-4905	233	3	,	,	PUNCT
ejpam-4905	233	4	b	b	NOUN
ejpam-4905	233	5	are	be	AUX
ejpam-4905	233	6	arbitrary	arbitrary	ADJ
ejpam-4905	233	7	,	,	PUNCT
ejpam-4905	233	8	n	n	PRON
ejpam-4905	233	9	is	be	AUX
ejpam-4905	233	10	a	a	DET
ejpam-4905	233	11	j2	j2	PROPN
ejpam-4905	233	12	-	-	PUNCT
ejpam-4905	233	13	set	set	NOUN
ejpam-4905	233	14	of	of	ADP
ejpam-4905	233	15	g	g	PROPN
ejpam-4905	233	16	+	+	CCONJ
ejpam-4905	233	17	h.	h.	PROPN
ejpam-4905	233	18	next	next	ADV
ejpam-4905	233	19	,	,	PUNCT
ejpam-4905	233	20	suppose	suppose	VERB
ejpam-4905	233	21	that	that	SCONJ
ejpam-4905	233	22	dg(a	dg(a	PROPN
ejpam-4905	233	23	,	,	PUNCT
ejpam-4905	233	24	b	b	X
ejpam-4905	233	25	)	)	PUNCT
ejpam-4905	233	26	≥	≥	NOUN
ejpam-4905	233	27	3	3	NUM
ejpam-4905	233	28	.	.	PUNCT
ejpam-4905	233	29	let	let	VERB
ejpam-4905	233	30	u	u	PRON
ejpam-4905	233	31	∈	∈	PROPN
ejpam-4905	233	32	ng(a	ng(a	NOUN
ejpam-4905	233	33	)	)	PUNCT
ejpam-4905	233	34	and	and	CCONJ
ejpam-4905	233	35	v	v	ADP
ejpam-4905	233	36	∈	∈	PROPN
ejpam-4905	233	37	ng(b	ng(b	NOUN
ejpam-4905	233	38	)	)	PUNCT
ejpam-4905	233	39	,	,	PUNCT
ejpam-4905	233	40	then	then	ADV
ejpam-4905	233	41	u	u	PROPN
ejpam-4905	233	42	∈	∈	PROPN
ejpam-4905	233	43	n2	n2	NOUN
ejpam-4905	233	44	g+h	g+h	PROPN
ejpam-4905	234	1	[	[	X
ejpam-4905	234	2	b]\n2	b]\n2	PROPN
ejpam-4905	234	3	g+h	g+h	PROPN
ejpam-4905	235	1	[	[	X
ejpam-4905	235	2	a	a	X
ejpam-4905	235	3	]	]	X
ejpam-4905	235	4	and	and	CCONJ
ejpam-4905	235	5	v	v	ADP
ejpam-4905	235	6	∈	∈	PROPN
ejpam-4905	235	7	n2	n2	NOUN
ejpam-4905	235	8	g+h	g+h	PUNCT
ejpam-4905	236	1	[	[	X
ejpam-4905	236	2	a]\n2	a]\n2	PROPN
ejpam-4905	236	3	g+h	g+h	PUNCT
ejpam-4905	237	1	[	[	X
ejpam-4905	237	2	b	b	X
ejpam-4905	237	3	]	]	X
ejpam-4905	237	4	.	.	PUNCT
ejpam-4905	238	1	since	since	SCONJ
ejpam-4905	238	2	a	a	DET
ejpam-4905	238	3	,	,	PUNCT
ejpam-4905	238	4	b	b	NOUN
ejpam-4905	238	5	are	be	AUX
ejpam-4905	238	6	arbitrary	arbitrary	ADJ
ejpam-4905	238	7	,	,	PUNCT
ejpam-4905	238	8	n	n	PRON
ejpam-4905	238	9	is	be	AUX
ejpam-4905	238	10	a	a	DET
ejpam-4905	238	11	j2	j2	PROPN
ejpam-4905	238	12	-	-	PUNCT
ejpam-4905	238	13	set	set	NOUN
ejpam-4905	238	14	of	of	ADP
ejpam-4905	238	15	g+h	g+h	PROPN
ejpam-4905	238	16	.	.	PUNCT
ejpam-4905	239	1	similarly	similarly	ADV
ejpam-4905	239	2	,	,	PUNCT
ejpam-4905	239	3	if	if	SCONJ
ejpam-4905	239	4	a	a	DET
ejpam-4905	239	5	,	,	PUNCT
ejpam-4905	239	6	b	b	PROPN
ejpam-4905	239	7	∈	∈	PROPN
ejpam-4905	239	8	nh	nh	PROPN
ejpam-4905	239	9	,	,	PUNCT
ejpam-4905	239	10	thenn	thenn	PROPN
ejpam-4905	239	11	is	be	AUX
ejpam-4905	239	12	a	a	DET
ejpam-4905	239	13	j2	j2	PROPN
ejpam-4905	239	14	-	-	PUNCT
ejpam-4905	239	15	set	set	NOUN
ejpam-4905	239	16	of	of	ADP
ejpam-4905	239	17	g+h	g+h	PROPN
ejpam-4905	239	18	.	.	PUNCT
ejpam-4905	240	1	next	next	ADV
ejpam-4905	240	2	,	,	PUNCT
ejpam-4905	240	3	suppose	suppose	VERB
ejpam-4905	240	4	that	that	SCONJ
ejpam-4905	240	5	a	a	DET
ejpam-4905	240	6	∈	∈	PROPN
ejpam-4905	240	7	ng	ng	PROPN
ejpam-4905	240	8	and	and	CCONJ
ejpam-4905	240	9	b	b	PROPN
ejpam-4905	240	10	∈	∈	PROPN
ejpam-4905	240	11	nh	nh	PROPN
ejpam-4905	240	12	.	.	PUNCT
ejpam-4905	241	1	then	then	ADV
ejpam-4905	241	2	a	a	DET
ejpam-4905	241	3	∈	∈	PROPN
ejpam-4905	241	4	n2	n2	NOUN
ejpam-4905	241	5	g+h	g+h	PROPN
ejpam-4905	242	1	[	[	X
ejpam-4905	242	2	a	a	X
ejpam-4905	242	3	]	]	X
ejpam-4905	242	4	\n2	\n2	ADJ
ejpam-4905	242	5	g+h	g+h	X
ejpam-4905	243	1	[	[	X
ejpam-4905	243	2	b	b	X
ejpam-4905	243	3	]	]	X
ejpam-4905	243	4	and	and	CCONJ
ejpam-4905	243	5	b	b	PROPN
ejpam-4905	243	6	∈	∈	PROPN
ejpam-4905	243	7	n2	n2	NOUN
ejpam-4905	243	8	g+h	g+h	PUNCT
ejpam-4905	244	1	[	[	X
ejpam-4905	244	2	b	b	X
ejpam-4905	244	3	]	]	X
ejpam-4905	244	4	\n2	\n2	ADJ
ejpam-4905	244	5	g+h	g+h	X
ejpam-4905	245	1	[	[	X
ejpam-4905	245	2	a	a	X
ejpam-4905	245	3	]	]	X
ejpam-4905	245	4	.	.	PUNCT
ejpam-4905	246	1	since	since	SCONJ
ejpam-4905	246	2	a	a	DET
ejpam-4905	246	3	,	,	PUNCT
ejpam-4905	246	4	b	b	NOUN
ejpam-4905	246	5	are	be	AUX
ejpam-4905	246	6	arbitrary	arbitrary	ADJ
ejpam-4905	246	7	,	,	PUNCT
ejpam-4905	246	8	it	it	PRON
ejpam-4905	246	9	follows	follow	VERB
ejpam-4905	246	10	that	that	SCONJ
ejpam-4905	246	11	n	n	PRON
ejpam-4905	246	12	is	be	AUX
ejpam-4905	246	13	a	a	DET
ejpam-4905	246	14	j2	j2	PROPN
ejpam-4905	246	15	-	-	PUNCT
ejpam-4905	246	16	set	set	NOUN
ejpam-4905	246	17	of	of	ADP
ejpam-4905	246	18	g+h	g+h	PROPN
ejpam-4905	246	19	.	.	PUNCT
ejpam-4905	247	1	j.	j.	PROPN
ejpam-4905	247	2	hassan	hassan	PROPN
ejpam-4905	247	3	,	,	PUNCT
ejpam-4905	247	4	a.	a.	PROPN
ejpam-4905	247	5	bakkang	bakkang	PROPN
ejpam-4905	247	6	,	,	PUNCT
ejpam-4905	247	7	a.	a.	NOUN
ejpam-4905	247	8	sappari	sappari	PROPN
ejpam-4905	247	9	/	/	SYM
ejpam-4905	247	10	eur	eur	PROPN
ejpam-4905	247	11	.	.	PUNCT
ejpam-4905	248	1	j.	j.	PROPN
ejpam-4905	248	2	pure	pure	PROPN
ejpam-4905	248	3	appl	appl	PROPN
ejpam-4905	248	4	.	.	PROPN
ejpam-4905	248	5	math	math	PROPN
ejpam-4905	248	6	,	,	PUNCT
ejpam-4905	248	7	16	16	NUM
ejpam-4905	248	8	(	(	PUNCT
ejpam-4905	248	9	4	4	NUM
ejpam-4905	248	10	)	)	PUNCT
ejpam-4905	248	11	(	(	PUNCT
ejpam-4905	248	12	2023	2023	NUM
ejpam-4905	248	13	)	)	PUNCT
ejpam-4905	248	14	,	,	PUNCT
ejpam-4905	248	15	2118	2118	NUM
ejpam-4905	248	16	-	-	SYM
ejpam-4905	248	17	2131	2131	NUM
ejpam-4905	248	18	2125	2125	NUM
ejpam-4905	248	19	theorem	theorem	VERB
ejpam-4905	248	20	7	7	NUM
ejpam-4905	248	21	.	.	PUNCT
ejpam-4905	249	1	let	let	VERB
ejpam-4905	249	2	g	g	NOUN
ejpam-4905	249	3	and	and	CCONJ
ejpam-4905	249	4	h	h	NOUN
ejpam-4905	249	5	be	be	VERB
ejpam-4905	249	6	two	two	NUM
ejpam-4905	249	7	connected	connected	ADJ
ejpam-4905	249	8	graphs	graph	NOUN
ejpam-4905	249	9	.	.	PUNCT
ejpam-4905	250	1	if	if	SCONJ
ejpam-4905	250	2	n	n	NOUN
ejpam-4905	250	3	=	=	SYM
ejpam-4905	250	4	ng∪nh	ng∪nh	NUM
ejpam-4905	250	5	⊆	⊆	NUM
ejpam-4905	250	6	v	v	NOUN
ejpam-4905	250	7	(	(	PUNCT
ejpam-4905	250	8	g+h	g+h	PROPN
ejpam-4905	250	9	)	)	PUNCT
ejpam-4905	250	10	,	,	PUNCT
ejpam-4905	250	11	where	where	SCONJ
ejpam-4905	250	12	ng	ng	PROPN
ejpam-4905	250	13	and	and	CCONJ
ejpam-4905	250	14	nh	nh	PROPN
ejpam-4905	250	15	are	be	AUX
ejpam-4905	250	16	j2	j2	PROPN
ejpam-4905	250	17	-	-	PUNCT
ejpam-4905	250	18	hop	hop	NOUN
ejpam-4905	250	19	dominating	dominating	NOUN
ejpam-4905	250	20	sets	set	NOUN
ejpam-4905	250	21	in	in	ADP
ejpam-4905	250	22	g	g	PROPN
ejpam-4905	250	23	and	and	CCONJ
ejpam-4905	250	24	h	h	NOUN
ejpam-4905	250	25	,	,	PUNCT
ejpam-4905	250	26	respectively	respectively	ADV
ejpam-4905	250	27	,	,	PUNCT
ejpam-4905	250	28	then	then	ADV
ejpam-4905	250	29	n	n	PRON
ejpam-4905	250	30	is	be	AUX
ejpam-4905	250	31	a	a	DET
ejpam-4905	250	32	j2	j2	PROPN
ejpam-4905	250	33	-	-	PUNCT
ejpam-4905	250	34	hop	hop	NOUN
ejpam-4905	250	35	dominating	dominating	NOUN
ejpam-4905	250	36	set	set	VERB
ejpam-4905	250	37	in	in	ADP
ejpam-4905	250	38	g+h	g+h	PROPN
ejpam-4905	250	39	.	.	PUNCT
ejpam-4905	251	1	moreover	moreover	ADV
ejpam-4905	251	2	,	,	PUNCT
ejpam-4905	251	3	γj2h(g+h	γj2h(g+h	PROPN
ejpam-4905	251	4	)	)	PUNCT
ejpam-4905	251	5	≥	≥	NOUN
ejpam-4905	251	6	γj2h(g	γj2h(g	NOUN
ejpam-4905	251	7	)	)	PUNCT
ejpam-4905	251	8	+	+	CCONJ
ejpam-4905	251	9	γj2h(h	γj2h(h	NUM
ejpam-4905	251	10	)	)	PUNCT
ejpam-4905	251	11	.	.	PUNCT
ejpam-4905	252	1	proof	proof	NOUN
ejpam-4905	252	2	.	.	PUNCT
ejpam-4905	253	1	let	let	VERB
ejpam-4905	253	2	n	n	PROPN
ejpam-4905	253	3	=	=	SYM
ejpam-4905	253	4	ng	ng	PROPN
ejpam-4905	253	5	∪	∪	X
ejpam-4905	253	6	nh	nh	PROPN
ejpam-4905	253	7	,	,	PUNCT
ejpam-4905	253	8	where	where	SCONJ
ejpam-4905	253	9	ng	ng	PROPN
ejpam-4905	253	10	and	and	CCONJ
ejpam-4905	253	11	nh	nh	PROPN
ejpam-4905	253	12	are	be	AUX
ejpam-4905	253	13	j2	j2	PROPN
ejpam-4905	253	14	-	-	PUNCT
ejpam-4905	253	15	hop	hop	NOUN
ejpam-4905	253	16	dominating	dominating	NOUN
ejpam-4905	253	17	sets	set	NOUN
ejpam-4905	253	18	in	in	ADP
ejpam-4905	253	19	g	g	PROPN
ejpam-4905	253	20	and	and	CCONJ
ejpam-4905	253	21	h	h	NOUN
ejpam-4905	253	22	,	,	PUNCT
ejpam-4905	253	23	respectively	respectively	ADV
ejpam-4905	253	24	.	.	PUNCT
ejpam-4905	254	1	since	since	SCONJ
ejpam-4905	254	2	ng	ng	PROPN
ejpam-4905	254	3	and	and	CCONJ
ejpam-4905	254	4	nh	nh	PROPN
ejpam-4905	254	5	are	be	AUX
ejpam-4905	254	6	j2	j2	NOUN
ejpam-4905	254	7	-	-	PUNCT
ejpam-4905	254	8	sets	set	NOUN
ejpam-4905	254	9	in	in	ADP
ejpam-4905	254	10	g	g	PROPN
ejpam-4905	254	11	and	and	CCONJ
ejpam-4905	254	12	h	h	NOUN
ejpam-4905	254	13	,	,	PUNCT
ejpam-4905	254	14	respectively	respectively	ADV
ejpam-4905	254	15	,	,	PUNCT
ejpam-4905	254	16	it	it	PRON
ejpam-4905	254	17	follows	follow	VERB
ejpam-4905	254	18	that	that	SCONJ
ejpam-4905	254	19	n	n	PRON
ejpam-4905	254	20	is	be	AUX
ejpam-4905	254	21	a	a	DET
ejpam-4905	254	22	j2	j2	NOUN
ejpam-4905	254	23	-	-	PUNCT
ejpam-4905	254	24	set	set	NOUN
ejpam-4905	254	25	in	in	ADP
ejpam-4905	254	26	g	g	PROPN
ejpam-4905	254	27	+	+	CCONJ
ejpam-4905	254	28	h	h	NOUN
ejpam-4905	254	29	by	by	ADP
ejpam-4905	254	30	theorem	theorem	NOUN
ejpam-4905	254	31	6	6	NUM
ejpam-4905	254	32	.	.	PUNCT
ejpam-4905	255	1	since	since	SCONJ
ejpam-4905	255	2	ng	ng	PROPN
ejpam-4905	255	3	and	and	CCONJ
ejpam-4905	255	4	nh	nh	PROPN
ejpam-4905	255	5	are	be	AUX
ejpam-4905	255	6	hop	hop	NOUN
ejpam-4905	255	7	dominating	dominating	NOUN
ejpam-4905	255	8	sets	set	NOUN
ejpam-4905	255	9	in	in	ADP
ejpam-4905	255	10	g	g	PROPN
ejpam-4905	255	11	and	and	CCONJ
ejpam-4905	255	12	h	h	NOUN
ejpam-4905	255	13	,	,	PUNCT
ejpam-4905	255	14	respectively	respectively	ADV
ejpam-4905	255	15	,	,	PUNCT
ejpam-4905	255	16	we	we	PRON
ejpam-4905	255	17	have	have	VERB
ejpam-4905	255	18	n2	n2	ADJ
ejpam-4905	255	19	g[ng	g[ng	NOUN
ejpam-4905	255	20	]	]	X
ejpam-4905	255	21	=	=	SYM
ejpam-4905	255	22	v	v	X
ejpam-4905	255	23	(	(	PUNCT
ejpam-4905	255	24	g	g	NOUN
ejpam-4905	255	25	)	)	PUNCT
ejpam-4905	255	26	and	and	CCONJ
ejpam-4905	255	27	n2	n2	ADJ
ejpam-4905	255	28	g[nh	g[nh	NOUN
ejpam-4905	255	29	]	]	PUNCT
ejpam-4905	255	30	=	=	SYM
ejpam-4905	255	31	v	v	X
ejpam-4905	255	32	(	(	PUNCT
ejpam-4905	255	33	h	h	NOUN
ejpam-4905	255	34	)	)	PUNCT
ejpam-4905	255	35	.	.	PUNCT
ejpam-4905	256	1	observe	observe	VERB
ejpam-4905	256	2	that	that	DET
ejpam-4905	256	3	n2	n2	PROPN
ejpam-4905	256	4	g[ng	g[ng	PROPN
ejpam-4905	256	5	]	]	X
ejpam-4905	256	6	⊆	⊆	NUM
ejpam-4905	256	7	n2	n2	NOUN
ejpam-4905	256	8	g+h	g+h	PROPN
ejpam-4905	257	1	[	[	X
ejpam-4905	257	2	ng	ng	X
ejpam-4905	257	3	]	]	X
ejpam-4905	257	4	and	and	CCONJ
ejpam-4905	257	5	n2	n2	ADJ
ejpam-4905	257	6	h	h	PROPN
ejpam-4905	258	1	[	[	X
ejpam-4905	258	2	nh	nh	X
ejpam-4905	258	3	]	]	PUNCT
ejpam-4905	258	4	⊆	⊆	NUM
ejpam-4905	258	5	n2	n2	NOUN
ejpam-4905	258	6	g+h	g+h	PROPN
ejpam-4905	259	1	[	[	X
ejpam-4905	259	2	nh	nh	X
ejpam-4905	259	3	]	]	PUNCT
ejpam-4905	259	4	.	.	PUNCT
ejpam-4905	260	1	thus	thus	ADV
ejpam-4905	260	2	,	,	PUNCT
ejpam-4905	260	3	n2	n2	PROPN
ejpam-4905	260	4	g+h	g+h	PROPN
ejpam-4905	261	1	[	[	X
ejpam-4905	261	2	n	n	X
ejpam-4905	261	3	]	]	X
ejpam-4905	261	4	=	=	SYM
ejpam-4905	261	5	n2	n2	PROPN
ejpam-4905	261	6	g+h	g+h	PROPN
ejpam-4905	262	1	[	[	X
ejpam-4905	262	2	ng	ng	X
ejpam-4905	262	3	∪nh	∪nh	PROPN
ejpam-4905	262	4	]	]	PUNCT
ejpam-4905	263	1	=	=	SYM
ejpam-4905	263	2	v	v	X
ejpam-4905	263	3	(	(	PUNCT
ejpam-4905	263	4	g+h	g+h	PROPN
ejpam-4905	263	5	)	)	PUNCT
ejpam-4905	263	6	,	,	PUNCT
ejpam-4905	263	7	showing	show	VERB
ejpam-4905	263	8	that	that	SCONJ
ejpam-4905	263	9	n	n	PRON
ejpam-4905	263	10	is	be	AUX
ejpam-4905	263	11	a	a	DET
ejpam-4905	263	12	hop	hop	NOUN
ejpam-4905	263	13	dominating	dominating	NOUN
ejpam-4905	263	14	set	set	VERB
ejpam-4905	263	15	in	in	ADP
ejpam-4905	263	16	g+h	g+h	PROPN
ejpam-4905	263	17	.	.	PUNCT
ejpam-4905	264	1	therefore	therefore	ADV
ejpam-4905	264	2	,	,	PUNCT
ejpam-4905	264	3	n	n	PRON
ejpam-4905	264	4	is	be	AUX
ejpam-4905	264	5	a	a	DET
ejpam-4905	264	6	j2	j2	PROPN
ejpam-4905	264	7	-	-	PUNCT
ejpam-4905	264	8	hop	hop	NOUN
ejpam-4905	264	9	dominating	dominating	NOUN
ejpam-4905	264	10	set	set	VERB
ejpam-4905	264	11	in	in	ADP
ejpam-4905	264	12	g+h	g+h	PROPN
ejpam-4905	264	13	.	.	PUNCT
ejpam-4905	265	1	next	next	ADV
ejpam-4905	265	2	,	,	PUNCT
ejpam-4905	265	3	let	let	VERB
ejpam-4905	265	4	n	n	PRON
ejpam-4905	265	5	′	′	NOUN
ejpam-4905	266	1	=	=	PUNCT
ejpam-4905	266	2	n	n	CCONJ
ejpam-4905	266	3	′	′	NUM
ejpam-4905	266	4	g	g	NOUN
ejpam-4905	266	5	∪	∪	ADJ
ejpam-4905	266	6	n	n	CCONJ
ejpam-4905	266	7	′	′	NUM
ejpam-4905	266	8	h	h	NOUN
ejpam-4905	266	9	,	,	PUNCT
ejpam-4905	266	10	where	where	SCONJ
ejpam-4905	266	11	n	n	PRON
ejpam-4905	266	12	′	′	VERB
ejpam-4905	266	13	g	g	NOUN
ejpam-4905	266	14	and	and	CCONJ
ejpam-4905	266	15	n	n	PROPN
ejpam-4905	266	16	′	′	NOUN
ejpam-4905	266	17	h	h	NOUN
ejpam-4905	266	18	are	be	AUX
ejpam-4905	266	19	γj2h	γj2h	NOUN
ejpam-4905	266	20	-	-	PUNCT
ejpam-4905	266	21	sets	set	NOUN
ejpam-4905	266	22	in	in	ADP
ejpam-4905	266	23	g	g	PROPN
ejpam-4905	266	24	and	and	CCONJ
ejpam-4905	266	25	h	h	NOUN
ejpam-4905	266	26	,	,	PUNCT
ejpam-4905	266	27	respectively	respectively	ADV
ejpam-4905	266	28	.	.	PUNCT
ejpam-4905	267	1	then	then	ADV
ejpam-4905	267	2	by	by	ADP
ejpam-4905	267	3	the	the	DET
ejpam-4905	267	4	first	first	ADJ
ejpam-4905	267	5	part	part	NOUN
ejpam-4905	267	6	,	,	PUNCT
ejpam-4905	267	7	n	n	CCONJ
ejpam-4905	267	8	′	′	NUM
ejpam-4905	267	9	is	be	AUX
ejpam-4905	267	10	a	a	DET
ejpam-4905	267	11	j2	j2	PROPN
ejpam-4905	267	12	-	-	PUNCT
ejpam-4905	267	13	hop	hop	NOUN
ejpam-4905	267	14	dominating	dominating	NOUN
ejpam-4905	267	15	set	set	VERB
ejpam-4905	267	16	in	in	ADP
ejpam-4905	267	17	g+h	g+h	PROPN
ejpam-4905	267	18	.	.	PUNCT
ejpam-4905	268	1	consequently	consequently	ADV
ejpam-4905	268	2	,	,	PUNCT
ejpam-4905	268	3	γj2h(g+h	γj2h(g+h	PROPN
ejpam-4905	268	4	)	)	PUNCT
ejpam-4905	268	5	≥	≥	NOUN
ejpam-4905	268	6	|n	|n	X
ejpam-4905	268	7	′|	′|	NUM
ejpam-4905	268	8	=	=	SYM
ejpam-4905	268	9	|n	|n	NOUN
ejpam-4905	268	10	′	′	NUM
ejpam-4905	269	1	g|+	g|+	PROPN
ejpam-4905	269	2	|n	|n	AUX
ejpam-4905	269	3	′	′	NUM
ejpam-4905	269	4	h	h	NOUN
ejpam-4905	270	1	|	|	ADV
ejpam-4905	270	2	=	=	SYM
ejpam-4905	270	3	γj2h(g	γj2h(g	NOUN
ejpam-4905	270	4	)	)	PUNCT
ejpam-4905	270	5	+	+	CCONJ
ejpam-4905	270	6	γj2h(h	γj2h(h	NUM
ejpam-4905	270	7	)	)	PUNCT
ejpam-4905	270	8	.	.	PUNCT
ejpam-4905	271	1	remark	remark	PROPN
ejpam-4905	271	2	1	1	NUM
ejpam-4905	271	3	.	.	PUNCT
ejpam-4905	272	1	the	the	DET
ejpam-4905	272	2	bound	bind	VERB
ejpam-4905	272	3	given	give	VERB
ejpam-4905	272	4	in	in	ADP
ejpam-4905	272	5	theorem	theorem	ADJ
ejpam-4905	272	6	7	7	NUM
ejpam-4905	272	7	is	be	AUX
ejpam-4905	272	8	sharp	sharp	ADJ
ejpam-4905	272	9	.	.	PUNCT
ejpam-4905	273	1	moreover	moreover	ADV
ejpam-4905	273	2	,	,	PUNCT
ejpam-4905	273	3	strict	strict	ADJ
ejpam-4905	273	4	inequality	inequality	NOUN
ejpam-4905	273	5	is	be	AUX
ejpam-4905	273	6	attainable	attainable	ADJ
ejpam-4905	273	7	.	.	PUNCT
ejpam-4905	274	1	for	for	ADP
ejpam-4905	274	2	the	the	DET
ejpam-4905	274	3	sharpness	sharpness	NOUN
ejpam-4905	274	4	,	,	PUNCT
ejpam-4905	274	5	consider	consider	VERB
ejpam-4905	274	6	the	the	DET
ejpam-4905	274	7	join	join	NOUN
ejpam-4905	274	8	graph	graph	NOUN
ejpam-4905	274	9	p3	p3	NOUN
ejpam-4905	274	10	+	+	CCONJ
ejpam-4905	274	11	p4	p4	ADJ
ejpam-4905	274	12	in	in	ADP
ejpam-4905	274	13	figure	figure	NOUN
ejpam-4905	274	14	4	4	NUM
ejpam-4905	274	15	.	.	PUNCT
ejpam-4905	275	1	let	let	VERB
ejpam-4905	275	2	s	s	VERB
ejpam-4905	275	3	=	=	X
ejpam-4905	275	4	{	{	PUNCT
ejpam-4905	275	5	a	a	DET
ejpam-4905	275	6	,	,	PUNCT
ejpam-4905	275	7	b	b	NOUN
ejpam-4905	275	8	,	,	PUNCT
ejpam-4905	275	9	e	e	NOUN
ejpam-4905	275	10	,	,	PUNCT
ejpam-4905	275	11	f	f	NOUN
ejpam-4905	275	12	}	}	PUNCT
ejpam-4905	275	13	.	.	PUNCT
ejpam-4905	276	1	then	then	ADV
ejpam-4905	276	2	n2	n2	PROPN
ejpam-4905	276	3	p3+p4	p3+p4	PROPN
ejpam-4905	276	4	[	[	X
ejpam-4905	276	5	s	s	X
ejpam-4905	276	6	]	]	X
ejpam-4905	276	7	=	=	SYM
ejpam-4905	276	8	v	v	X
ejpam-4905	276	9	(	(	PUNCT
ejpam-4905	276	10	p3	p3	NOUN
ejpam-4905	276	11	+	+	CCONJ
ejpam-4905	276	12	p4	p4	ADJ
ejpam-4905	276	13	)	)	PUNCT
ejpam-4905	276	14	,	,	PUNCT
ejpam-4905	276	15	showing	show	VERB
ejpam-4905	276	16	that	that	SCONJ
ejpam-4905	276	17	s	s	VERB
ejpam-4905	276	18	is	be	AUX
ejpam-4905	276	19	a	a	DET
ejpam-4905	276	20	hop	hop	NOUN
ejpam-4905	276	21	dominating	dominating	NOUN
ejpam-4905	276	22	set	set	VERB
ejpam-4905	276	23	in	in	ADP
ejpam-4905	276	24	p3	p3	PROPN
ejpam-4905	276	25	+	+	CCONJ
ejpam-4905	276	26	p4	p4	ADJ
ejpam-4905	276	27	.	.	PUNCT
ejpam-4905	277	1	observe	observe	VERB
ejpam-4905	277	2	that	that	SCONJ
ejpam-4905	277	3	x	x	SYM
ejpam-4905	277	4	∈	∈	PROPN
ejpam-4905	277	5	n2	n2	NOUN
ejpam-4905	277	6	p3+p4	p3+p4	PROPN
ejpam-4905	278	1	[	[	X
ejpam-4905	278	2	x	x	X
ejpam-4905	278	3	]	]	PUNCT
ejpam-4905	278	4	\	\	PROPN
ejpam-4905	278	5	n2	n2	PROPN
ejpam-4905	278	6	p3+p4	p3+p4	PROPN
ejpam-4905	278	7	[	[	X
ejpam-4905	278	8	y	y	X
ejpam-4905	278	9	]	]	X
ejpam-4905	278	10	and	and	CCONJ
ejpam-4905	278	11	y	y	PROPN
ejpam-4905	278	12	∈	∈	PROPN
ejpam-4905	278	13	n2	n2	PROPN
ejpam-4905	278	14	p3+p4	p3+p4	PROPN
ejpam-4905	279	1	[	[	X
ejpam-4905	279	2	y	y	X
ejpam-4905	279	3	]	]	PUNCT
ejpam-4905	279	4	\	\	PROPN
ejpam-4905	279	5	n2	n2	PROPN
ejpam-4905	279	6	p3+p4	p3+p4	PROPN
ejpam-4905	280	1	[	[	X
ejpam-4905	280	2	x	x	X
ejpam-4905	280	3	]	]	X
ejpam-4905	280	4	for	for	ADP
ejpam-4905	280	5	every	every	DET
ejpam-4905	280	6	x	x	SYM
ejpam-4905	280	7	̸=	̸=	PROPN
ejpam-4905	280	8	y	y	PROPN
ejpam-4905	280	9	where	where	SCONJ
ejpam-4905	280	10	x	x	X
ejpam-4905	280	11	,	,	PUNCT
ejpam-4905	280	12	y	y	PROPN
ejpam-4905	280	13	∈	∈	PROPN
ejpam-4905	280	14	s.	s.	PROPN
ejpam-4905	280	15	this	this	PRON
ejpam-4905	280	16	means	mean	VERB
ejpam-4905	280	17	that	that	SCONJ
ejpam-4905	280	18	n2	n2	PROPN
ejpam-4905	280	19	p3+p4	p3+p4	PROPN
ejpam-4905	280	20	[	[	X
ejpam-4905	280	21	x]\n2	x]\n2	PROPN
ejpam-4905	280	22	p3+p4	p3+p4	PROPN
ejpam-4905	280	23	[	[	X
ejpam-4905	280	24	y	y	X
ejpam-4905	280	25	]	]	X
ejpam-4905	280	26	̸=	̸=	PROPN
ejpam-4905	280	27	∅	∅	NOUN
ejpam-4905	280	28	and	and	CCONJ
ejpam-4905	280	29	n2	n2	PROPN
ejpam-4905	280	30	p3+p4	p3+p4	PROPN
ejpam-4905	281	1	[	[	X
ejpam-4905	281	2	y]\n2	y]\n2	PROPN
ejpam-4905	281	3	p3+p4	p3+p4	PROPN
ejpam-4905	281	4	[	[	X
ejpam-4905	281	5	x	x	X
ejpam-4905	281	6	]	]	X
ejpam-4905	281	7	̸=	̸=	NOUN
ejpam-4905	281	8	∅	∅	NOUN
ejpam-4905	281	9	for	for	ADP
ejpam-4905	281	10	every	every	DET
ejpam-4905	281	11	x	x	SYM
ejpam-4905	281	12	̸=	̸=	PROPN
ejpam-4905	281	13	y	y	PROPN
ejpam-4905	281	14	where	where	SCONJ
ejpam-4905	281	15	x	x	X
ejpam-4905	281	16	,	,	PUNCT
ejpam-4905	281	17	y	y	PROPN
ejpam-4905	281	18	∈	∈	PROPN
ejpam-4905	281	19	s.	s.	PROPN
ejpam-4905	281	20	thus	thus	ADV
ejpam-4905	281	21	,	,	PUNCT
ejpam-4905	281	22	s	s	VERB
ejpam-4905	281	23	is	be	AUX
ejpam-4905	281	24	a	a	DET
ejpam-4905	281	25	j2	j2	PROPN
ejpam-4905	281	26	-	-	PUNCT
ejpam-4905	281	27	hop	hop	NOUN
ejpam-4905	281	28	dominating	dominating	NOUN
ejpam-4905	281	29	set	set	VERB
ejpam-4905	281	30	in	in	ADP
ejpam-4905	281	31	p3	p3	PROPN
ejpam-4905	281	32	+	+	CCONJ
ejpam-4905	281	33	p4	p4	ADJ
ejpam-4905	281	34	.	.	PUNCT
ejpam-4905	282	1	since	since	SCONJ
ejpam-4905	282	2	n2	n2	PROPN
ejpam-4905	282	3	p3+p4	p3+p4	PROPN
ejpam-4905	282	4	[	[	X
ejpam-4905	282	5	c	c	X
ejpam-4905	282	6	]	]	X
ejpam-4905	282	7	⊆	⊆	NUM
ejpam-4905	282	8	n2	n2	NOUN
ejpam-4905	282	9	p3+p4	p3+p4	PROPN
ejpam-4905	283	1	[	[	X
ejpam-4905	283	2	a	a	X
ejpam-4905	283	3	]	]	X
ejpam-4905	283	4	,	,	PUNCT
ejpam-4905	283	5	n2	n2	PROPN
ejpam-4905	283	6	p3+p4	p3+p4	PROPN
ejpam-4905	284	1	[	[	X
ejpam-4905	284	2	f	f	X
ejpam-4905	284	3	]	]	PUNCT
ejpam-4905	284	4	⊆	⊆	NUM
ejpam-4905	284	5	n2	n2	NOUN
ejpam-4905	284	6	p3+p4	p3+p4	PROPN
ejpam-4905	285	1	[	[	X
ejpam-4905	285	2	d	d	X
ejpam-4905	285	3	]	]	X
ejpam-4905	285	4	,	,	PUNCT
ejpam-4905	285	5	and	and	CCONJ
ejpam-4905	285	6	n2	n2	PROPN
ejpam-4905	285	7	p3+p4	p3+p4	PROPN
ejpam-4905	286	1	[	[	X
ejpam-4905	286	2	e	e	X
ejpam-4905	286	3	]	]	X
ejpam-4905	286	4	⊆	⊆	NUM
ejpam-4905	286	5	n2	n2	NOUN
ejpam-4905	286	6	p3+p4	p3+p4	PROPN
ejpam-4905	287	1	[	[	X
ejpam-4905	287	2	g	g	X
ejpam-4905	287	3	]	]	X
ejpam-4905	287	4	,	,	PUNCT
ejpam-4905	287	5	it	it	PRON
ejpam-4905	287	6	follows	follow	VERB
ejpam-4905	287	7	that	that	SCONJ
ejpam-4905	287	8	s	s	VERB
ejpam-4905	287	9	is	be	AUX
ejpam-4905	287	10	a	a	DET
ejpam-4905	287	11	maximum	maximum	ADJ
ejpam-4905	287	12	j2	j2	PROPN
ejpam-4905	287	13	-	-	PUNCT
ejpam-4905	287	14	hop	hop	NOUN
ejpam-4905	287	15	dominating	dominating	NOUN
ejpam-4905	287	16	set	set	NOUN
ejpam-4905	287	17	of	of	ADP
ejpam-4905	287	18	p3	p3	PROPN
ejpam-4905	287	19	+	+	CCONJ
ejpam-4905	287	20	p4	p4	ADJ
ejpam-4905	287	21	.	.	PUNCT
ejpam-4905	288	1	hence	hence	ADV
ejpam-4905	288	2	,	,	PUNCT
ejpam-4905	288	3	γj2h(p3	γj2h(p3	ADJ
ejpam-4905	288	4	+	+	CCONJ
ejpam-4905	288	5	p5	p5	ADJ
ejpam-4905	288	6	)	)	PUNCT
ejpam-4905	288	7	=	=	SYM
ejpam-4905	289	1	4	4	X
ejpam-4905	289	2	.	.	PUNCT
ejpam-4905	290	1	by	by	ADP
ejpam-4905	290	2	proposition	proposition	NOUN
ejpam-4905	290	3	1	1	NUM
ejpam-4905	290	4	,	,	PUNCT
ejpam-4905	290	5	γj2h(p3	γj2h(p3	ADJ
ejpam-4905	290	6	)	)	PUNCT
ejpam-4905	290	7	=	=	SYM
ejpam-4905	290	8	2	2	NUM
ejpam-4905	290	9	and	and	CCONJ
ejpam-4905	290	10	γj2h(p4	γj2h(p4	NOUN
ejpam-4905	290	11	)	)	PUNCT
ejpam-4905	290	12	=	=	SYM
ejpam-4905	290	13	2	2	X
ejpam-4905	290	14	.	.	PUNCT
ejpam-4905	290	15	consequently	consequently	ADV
ejpam-4905	290	16	,	,	PUNCT
ejpam-4905	290	17	γj2h(p3	γj2h(p3	ADJ
ejpam-4905	290	18	+	+	CCONJ
ejpam-4905	290	19	p4	p4	ADJ
ejpam-4905	290	20	)	)	PUNCT
ejpam-4905	290	21	=	=	SYM
ejpam-4905	290	22	4	4	NUM
ejpam-4905	290	23	=	=	SYM
ejpam-4905	290	24	γj2h(p3	γj2h(p3	ADJ
ejpam-4905	290	25	)	)	PUNCT
ejpam-4905	291	1	+	+	CCONJ
ejpam-4905	291	2	γj2h(p4	γj2h(p4	NOUN
ejpam-4905	291	3	)	)	PUNCT
ejpam-4905	291	4	.	.	PUNCT
ejpam-4905	292	1	j.	j.	PROPN
ejpam-4905	292	2	hassan	hassan	PROPN
ejpam-4905	292	3	,	,	PUNCT
ejpam-4905	292	4	a.	a.	PROPN
ejpam-4905	292	5	bakkang	bakkang	PROPN
ejpam-4905	292	6	,	,	PUNCT
ejpam-4905	292	7	a.	a.	NOUN
ejpam-4905	292	8	sappari	sappari	PROPN
ejpam-4905	292	9	/	/	SYM
ejpam-4905	292	10	eur	eur	PROPN
ejpam-4905	292	11	.	.	PUNCT
ejpam-4905	293	1	j.	j.	PROPN
ejpam-4905	293	2	pure	pure	PROPN
ejpam-4905	293	3	appl	appl	PROPN
ejpam-4905	293	4	.	.	PROPN
ejpam-4905	293	5	math	math	PROPN
ejpam-4905	293	6	,	,	PUNCT
ejpam-4905	293	7	16	16	NUM
ejpam-4905	293	8	(	(	PUNCT
ejpam-4905	293	9	4	4	NUM
ejpam-4905	293	10	)	)	PUNCT
ejpam-4905	293	11	(	(	PUNCT
ejpam-4905	293	12	2023	2023	NUM
ejpam-4905	293	13	)	)	PUNCT
ejpam-4905	293	14	,	,	PUNCT
ejpam-4905	293	15	2118	2118	NUM
ejpam-4905	293	16	-	-	SYM
ejpam-4905	293	17	2131	2131	NUM
ejpam-4905	293	18	2126	2126	NUM
ejpam-4905	293	19	p3	p3	PROPN
ejpam-4905	293	20	+	+	CCONJ
ejpam-4905	293	21	p4	p4	ADJ
ejpam-4905	293	22	:	:	PUNCT
ejpam-4905	293	23	ba	ba	PROPN
ejpam-4905	293	24	e	e	NOUN
ejpam-4905	293	25	fd	fd	AUX
ejpam-4905	293	26	g	g	PROPN
ejpam-4905	293	27	c	c	PROPN
ejpam-4905	293	28	figure	figure	NOUN
ejpam-4905	293	29	4	4	NUM
ejpam-4905	293	30	:	:	PUNCT
ejpam-4905	293	31	graph	graph	NOUN
ejpam-4905	293	32	p3	p3	NOUN
ejpam-4905	293	33	+	+	CCONJ
ejpam-4905	293	34	p4	p4	ADJ
ejpam-4905	293	35	with	with	ADP
ejpam-4905	293	36	γj2h(p3	γj2h(p3	NOUN
ejpam-4905	293	37	+	+	CCONJ
ejpam-4905	293	38	p4	p4	ADJ
ejpam-4905	293	39	)	)	PUNCT
ejpam-4905	293	40	=	=	SYM
ejpam-4905	293	41	4	4	NUM
ejpam-4905	293	42	=	=	SYM
ejpam-4905	293	43	γj2h(p3	γj2h(p3	ADJ
ejpam-4905	293	44	)	)	PUNCT
ejpam-4905	294	1	+	+	CCONJ
ejpam-4905	294	2	γj2h(p4	γj2h(p4	NOUN
ejpam-4905	294	3	)	)	PUNCT
ejpam-4905	294	4	for	for	ADP
ejpam-4905	294	5	strict	strict	ADJ
ejpam-4905	294	6	inequality	inequality	NOUN
ejpam-4905	294	7	,	,	PUNCT
ejpam-4905	294	8	consider	consider	VERB
ejpam-4905	294	9	the	the	DET
ejpam-4905	294	10	graph	graph	NOUN
ejpam-4905	294	11	p2+p8	p2+p8	PROPN
ejpam-4905	294	12	in	in	ADP
ejpam-4905	294	13	figure	figure	NOUN
ejpam-4905	294	14	5	5	NUM
ejpam-4905	294	15	.	.	PUNCT
ejpam-4905	295	1	let	let	VERB
ejpam-4905	295	2	s′	s′	ADJ
ejpam-4905	295	3	=	=	PUNCT
ejpam-4905	295	4	{	{	PUNCT
ejpam-4905	295	5	a	a	PRON
ejpam-4905	295	6	,	,	PUNCT
ejpam-4905	295	7	b	b	NOUN
ejpam-4905	295	8	,	,	PUNCT
ejpam-4905	295	9	d	d	NOUN
ejpam-4905	295	10	,	,	PUNCT
ejpam-4905	295	11	e	e	NOUN
ejpam-4905	295	12	,	,	PUNCT
ejpam-4905	295	13	f	f	PROPN
ejpam-4905	295	14	,	,	PUNCT
ejpam-4905	295	15	g	g	PROPN
ejpam-4905	295	16	,	,	PUNCT
ejpam-4905	295	17	h	h	NOUN
ejpam-4905	295	18	,	,	PUNCT
ejpam-4905	295	19	i	i	PROPN
ejpam-4905	295	20	}	}	PUNCT
ejpam-4905	295	21	.	.	PUNCT
ejpam-4905	296	1	then	then	ADV
ejpam-4905	296	2	s′	s′	PROPN
ejpam-4905	296	3	is	be	AUX
ejpam-4905	296	4	a	a	DET
ejpam-4905	296	5	γj2h	γj2h	NOUN
ejpam-4905	296	6	-	-	PUNCT
ejpam-4905	296	7	set	set	NOUN
ejpam-4905	296	8	in	in	ADP
ejpam-4905	296	9	p2+p8	p2+p8	PROPN
ejpam-4905	296	10	.	.	PUNCT
ejpam-4905	297	1	thus	thus	ADV
ejpam-4905	297	2	,	,	PUNCT
ejpam-4905	297	3	γj2h(p2+p8	γj2h(p2+p8	PROPN
ejpam-4905	297	4	)	)	PUNCT
ejpam-4905	297	5	=	=	SYM
ejpam-4905	298	1	8	8	X
ejpam-4905	298	2	.	.	PUNCT
ejpam-4905	298	3	by	by	ADP
ejpam-4905	298	4	proposition	proposition	NOUN
ejpam-4905	298	5	1	1	NUM
ejpam-4905	298	6	,	,	PUNCT
ejpam-4905	298	7	γj2h(p2	γj2h(p2	PROPN
ejpam-4905	298	8	)	)	PUNCT
ejpam-4905	298	9	=	=	SYM
ejpam-4905	298	10	2	2	NUM
ejpam-4905	298	11	and	and	CCONJ
ejpam-4905	298	12	γj2h(p8	γj2h(p8	NOUN
ejpam-4905	298	13	)	)	PUNCT
ejpam-4905	298	14	=	=	SYM
ejpam-4905	299	1	4	4	X
ejpam-4905	299	2	.	.	PUNCT
ejpam-4905	300	1	hence	hence	ADV
ejpam-4905	300	2	,	,	PUNCT
ejpam-4905	300	3	γj2h(p2	γj2h(p2	PROPN
ejpam-4905	300	4	+	+	SYM
ejpam-4905	300	5	p8	p8	PROPN
ejpam-4905	300	6	)	)	PUNCT
ejpam-4905	300	7	=	=	SYM
ejpam-4905	300	8	8	8	NUM
ejpam-4905	300	9	>	>	SYM
ejpam-4905	300	10	6	6	NUM
ejpam-4905	300	11	=	=	SYM
ejpam-4905	300	12	γj2h(p2	γj2h(p2	PROPN
ejpam-4905	300	13	)	)	PUNCT
ejpam-4905	300	14	+	+	NUM
ejpam-4905	300	15	γj2h(p8	γj2h(p8	NOUN
ejpam-4905	300	16	)	)	PUNCT
ejpam-4905	300	17	.	.	PUNCT
ejpam-4905	301	1	p2	p2	PROPN
ejpam-4905	301	2	+	+	CCONJ
ejpam-4905	301	3	p8	p8	ADJ
ejpam-4905	301	4	:	:	PUNCT
ejpam-4905	301	5	b	b	X
ejpam-4905	301	6	e	e	NOUN
ejpam-4905	301	7	a	a	PROPN
ejpam-4905	301	8	c	c	X
ejpam-4905	301	9	gd	gd	NOUN
ejpam-4905	302	1	hf	hf	PROPN
ejpam-4905	302	2	i	i	PRON
ejpam-4905	303	1	j	j	PROPN
ejpam-4905	303	2	figure	figure	VERB
ejpam-4905	303	3	5	5	NUM
ejpam-4905	303	4	:	:	PUNCT
ejpam-4905	303	5	graph	graph	NOUN
ejpam-4905	303	6	p2	p2	PROPN
ejpam-4905	303	7	+	+	CCONJ
ejpam-4905	303	8	p8	p8	ADJ
ejpam-4905	303	9	with	with	ADP
ejpam-4905	303	10	γj2h(p2	γj2h(p2	PROPN
ejpam-4905	303	11	+	+	SYM
ejpam-4905	303	12	p8	p8	PROPN
ejpam-4905	303	13	)	)	PUNCT
ejpam-4905	303	14	>	>	PUNCT
ejpam-4905	304	1	γj2h(p2	γj2h(p2	PROPN
ejpam-4905	304	2	)	)	PUNCT
ejpam-4905	305	1	+	+	CCONJ
ejpam-4905	305	2	γj2h(p8	γj2h(p8	NOUN
ejpam-4905	305	3	)	)	PUNCT
ejpam-4905	305	4	theorem	theorem	VERB
ejpam-4905	305	5	8	8	NUM
ejpam-4905	305	6	.	.	PUNCT
ejpam-4905	306	1	let	let	VERB
ejpam-4905	306	2	g	g	NOUN
ejpam-4905	306	3	be	be	AUX
ejpam-4905	306	4	any	any	DET
ejpam-4905	306	5	non	non	ADJ
ejpam-4905	306	6	-	-	ADJ
ejpam-4905	306	7	trivial	trivial	ADJ
ejpam-4905	306	8	connected	connected	ADJ
ejpam-4905	306	9	graph	graph	NOUN
ejpam-4905	306	10	and	and	CCONJ
ejpam-4905	306	11	h	h	NOUN
ejpam-4905	306	12	be	be	AUX
ejpam-4905	306	13	any	any	DET
ejpam-4905	306	14	connected	connected	ADJ
ejpam-4905	306	15	graph	graph	NOUN
ejpam-4905	306	16	.	.	PUNCT
ejpam-4905	307	1	if	if	SCONJ
ejpam-4905	307	2	t	t	NOUN
ejpam-4905	307	3	=	=	SYM
ejpam-4905	307	4	⋃	⋃	NOUN
ejpam-4905	307	5	v∈v	v∈v	NOUN
ejpam-4905	307	6	(	(	PUNCT
ejpam-4905	307	7	g	g	NOUN
ejpam-4905	307	8	)	)	PUNCT
ejpam-4905	307	9	tv	tv	NOUN
ejpam-4905	307	10	,	,	PUNCT
ejpam-4905	307	11	where	where	SCONJ
ejpam-4905	307	12	tv	tv	NOUN
ejpam-4905	307	13	is	be	AUX
ejpam-4905	307	14	a	a	DET
ejpam-4905	307	15	maximum	maximum	ADJ
ejpam-4905	307	16	j2	j2	NOUN
ejpam-4905	307	17	-	-	PUNCT
ejpam-4905	307	18	set	set	NOUN
ejpam-4905	307	19	in	in	ADP
ejpam-4905	307	20	hv	hv	PROPN
ejpam-4905	307	21	for	for	ADP
ejpam-4905	307	22	each	each	DET
ejpam-4905	307	23	v	v	NUM
ejpam-4905	307	24	∈	∈	PROPN
ejpam-4905	307	25	v	v	NOUN
ejpam-4905	307	26	(	(	PUNCT
ejpam-4905	307	27	g	g	NOUN
ejpam-4905	307	28	)	)	PUNCT
ejpam-4905	307	29	,	,	PUNCT
ejpam-4905	307	30	then	then	ADV
ejpam-4905	307	31	t	t	PROPN
ejpam-4905	307	32	is	be	AUX
ejpam-4905	307	33	a	a	DET
ejpam-4905	307	34	j2	j2	PROPN
ejpam-4905	307	35	-	-	PUNCT
ejpam-4905	307	36	hop	hop	NOUN
ejpam-4905	307	37	dominating	dominating	NOUN
ejpam-4905	307	38	set	set	VERB
ejpam-4905	307	39	in	in	ADP
ejpam-4905	307	40	g	g	PROPN
ejpam-4905	307	41	◦	◦	NOUN
ejpam-4905	307	42	h.	h.	NOUN
ejpam-4905	307	43	moreover	moreover	ADV
ejpam-4905	307	44	,	,	PUNCT
ejpam-4905	307	45	γj2h(g	γj2h(g	VERB
ejpam-4905	307	46	◦	◦	NOUN
ejpam-4905	307	47	h	h	NOUN
ejpam-4905	307	48	)	)	PUNCT
ejpam-4905	307	49	≥	≥	NOUN
ejpam-4905	307	50	|v	|v	PROPN
ejpam-4905	307	51	(	(	PUNCT
ejpam-4905	307	52	g)|	g)|	NOUN
ejpam-4905	307	53	·	·	PUNCT
ejpam-4905	307	54	γj2h(h	γj2h(h	NUM
ejpam-4905	307	55	)	)	PUNCT
ejpam-4905	307	56	.	.	PUNCT
ejpam-4905	308	1	proof	proof	NOUN
ejpam-4905	308	2	.	.	PUNCT
ejpam-4905	309	1	suppose	suppose	VERB
ejpam-4905	309	2	that	that	SCONJ
ejpam-4905	309	3	t	t	NOUN
ejpam-4905	309	4	=	=	PUNCT
ejpam-4905	309	5	⋃	⋃	NOUN
ejpam-4905	309	6	v∈v	v∈v	NOUN
ejpam-4905	309	7	(	(	PUNCT
ejpam-4905	309	8	g	g	NOUN
ejpam-4905	309	9	)	)	PUNCT
ejpam-4905	309	10	tv	tv	NOUN
ejpam-4905	309	11	,	,	PUNCT
ejpam-4905	309	12	where	where	SCONJ
ejpam-4905	309	13	tv	tv	NOUN
ejpam-4905	309	14	is	be	AUX
ejpam-4905	309	15	a	a	DET
ejpam-4905	309	16	maximum	maximum	ADJ
ejpam-4905	309	17	j2	j2	NOUN
ejpam-4905	309	18	-	-	PUNCT
ejpam-4905	309	19	set	set	NOUN
ejpam-4905	309	20	in	in	ADP
ejpam-4905	309	21	hv	hv	PROPN
ejpam-4905	309	22	for	for	ADP
ejpam-4905	309	23	each	each	DET
ejpam-4905	309	24	v	v	NUM
ejpam-4905	309	25	∈	∈	PROPN
ejpam-4905	309	26	v	v	NOUN
ejpam-4905	309	27	(	(	PUNCT
ejpam-4905	309	28	g	g	NOUN
ejpam-4905	309	29	)	)	PUNCT
ejpam-4905	309	30	.	.	PUNCT
ejpam-4905	310	1	let	let	VERB
ejpam-4905	310	2	a	a	DET
ejpam-4905	310	3	,	,	PUNCT
ejpam-4905	310	4	b	b	PROPN
ejpam-4905	310	5	∈	∈	PROPN
ejpam-4905	310	6	t	t	PROPN
ejpam-4905	310	7	.	.	PUNCT
ejpam-4905	311	1	suppose	suppose	VERB
ejpam-4905	311	2	that	that	SCONJ
ejpam-4905	311	3	a	a	PRON
ejpam-4905	311	4	,	,	PUNCT
ejpam-4905	311	5	b	b	PROPN
ejpam-4905	311	6	∈	∈	PROPN
ejpam-4905	311	7	tu	tu	PROPN
ejpam-4905	311	8	for	for	ADP
ejpam-4905	311	9	some	some	DET
ejpam-4905	311	10	u	u	NOUN
ejpam-4905	311	11	∈	∈	PROPN
ejpam-4905	311	12	v	v	NOUN
ejpam-4905	311	13	(	(	PUNCT
ejpam-4905	311	14	g	g	NOUN
ejpam-4905	311	15	)	)	PUNCT
ejpam-4905	311	16	.	.	PUNCT
ejpam-4905	312	1	if	if	SCONJ
ejpam-4905	312	2	dh(a	dh(a	ADJ
ejpam-4905	312	3	,	,	PUNCT
ejpam-4905	312	4	b	b	NOUN
ejpam-4905	312	5	)	)	PUNCT
ejpam-4905	312	6	=	=	SYM
ejpam-4905	312	7	1	1	NUM
ejpam-4905	312	8	,	,	PUNCT
ejpam-4905	312	9	then	then	ADV
ejpam-4905	312	10	a	a	DET
ejpam-4905	312	11	∈	∈	PROPN
ejpam-4905	312	12	n2	n2	NOUN
ejpam-4905	312	13	g	g	PROPN
ejpam-4905	312	14	◦	◦	NOUN
ejpam-4905	312	15	h	h	NOUN
ejpam-4905	313	1	[	[	X
ejpam-4905	313	2	a	a	X
ejpam-4905	313	3	]	]	PUNCT
ejpam-4905	313	4	\	\	PROPN
ejpam-4905	313	5	n2	n2	PROPN
ejpam-4905	313	6	g	g	PROPN
ejpam-4905	313	7	◦	◦	NOUN
ejpam-4905	313	8	h	h	NOUN
ejpam-4905	314	1	[	[	X
ejpam-4905	314	2	b	b	X
ejpam-4905	314	3	]	]	X
ejpam-4905	314	4	and	and	CCONJ
ejpam-4905	314	5	b	b	PROPN
ejpam-4905	314	6	∈	∈	PROPN
ejpam-4905	314	7	n2	n2	NOUN
ejpam-4905	314	8	g	g	PROPN
ejpam-4905	314	9	◦	◦	NOUN
ejpam-4905	314	10	h	h	NOUN
ejpam-4905	315	1	[	[	X
ejpam-4905	315	2	b	b	X
ejpam-4905	315	3	]	]	PUNCT
ejpam-4905	315	4	\	\	PROPN
ejpam-4905	315	5	n2	n2	PROPN
ejpam-4905	315	6	g	g	PROPN
ejpam-4905	315	7	◦	◦	NOUN
ejpam-4905	315	8	h	h	NOUN
ejpam-4905	316	1	[	[	X
ejpam-4905	316	2	a	a	X
ejpam-4905	316	3	]	]	X
ejpam-4905	316	4	.	.	PUNCT
ejpam-4905	317	1	it	it	PRON
ejpam-4905	317	2	follows	follow	VERB
ejpam-4905	317	3	that	that	SCONJ
ejpam-4905	317	4	t	t	PROPN
ejpam-4905	317	5	is	be	AUX
ejpam-4905	317	6	a	a	DET
ejpam-4905	317	7	j2	j2	NOUN
ejpam-4905	317	8	-	-	PUNCT
ejpam-4905	317	9	set	set	NOUN
ejpam-4905	317	10	in	in	ADP
ejpam-4905	317	11	g	g	PROPN
ejpam-4905	317	12	◦	◦	PROPN
ejpam-4905	317	13	h.	h.	NOUN
ejpam-4905	317	14	assume	assume	VERB
ejpam-4905	317	15	that	that	SCONJ
ejpam-4905	317	16	dh(a	dh(a	ADJ
ejpam-4905	317	17	,	,	PUNCT
ejpam-4905	317	18	b	b	NOUN
ejpam-4905	318	1	)	)	PUNCT
ejpam-4905	318	2	=	=	SYM
ejpam-4905	318	3	2	2	X
ejpam-4905	318	4	.	.	PUNCT
ejpam-4905	318	5	since	since	SCONJ
ejpam-4905	318	6	tu	tu	PROPN
ejpam-4905	318	7	is	be	AUX
ejpam-4905	318	8	a	a	DET
ejpam-4905	318	9	j2set	j2set	NOUN
ejpam-4905	318	10	in	in	ADP
ejpam-4905	318	11	hu	hu	PROPN
ejpam-4905	318	12	,	,	PUNCT
ejpam-4905	318	13	there	there	PRON
ejpam-4905	318	14	exist	exist	VERB
ejpam-4905	318	15	w	w	ADP
ejpam-4905	318	16	,	,	PUNCT
ejpam-4905	318	17	z	z	PROPN
ejpam-4905	318	18	∈	∈	PROPN
ejpam-4905	318	19	v	v	ADP
ejpam-4905	318	20	(	(	PUNCT
ejpam-4905	318	21	hu	hu	PROPN
ejpam-4905	318	22	)	)	PUNCT
ejpam-4905	318	23	such	such	ADJ
ejpam-4905	318	24	that	that	SCONJ
ejpam-4905	318	25	w	w	PROPN
ejpam-4905	318	26	∈	∈	PROPN
ejpam-4905	318	27	n2	n2	PROPN
ejpam-4905	318	28	hu	hu	PROPN
ejpam-4905	319	1	[	[	X
ejpam-4905	319	2	a]\n2	a]\n2	PROPN
ejpam-4905	319	3	hu	hu	PROPN
ejpam-4905	320	1	[	[	X
ejpam-4905	320	2	b	b	X
ejpam-4905	320	3	]	]	X
ejpam-4905	320	4	and	and	CCONJ
ejpam-4905	320	5	z	z	NOUN
ejpam-4905	320	6	∈	∈	PROPN
ejpam-4905	320	7	n2	n2	PROPN
ejpam-4905	320	8	hu	hu	PROPN
ejpam-4905	321	1	[	[	X
ejpam-4905	321	2	b]\n2	b]\n2	NOUN
ejpam-4905	321	3	hu	hu	PROPN
ejpam-4905	322	1	[	[	X
ejpam-4905	322	2	a	a	X
ejpam-4905	322	3	]	]	X
ejpam-4905	322	4	.	.	PUNCT
ejpam-4905	323	1	let	let	VERB
ejpam-4905	323	2	s	s	PRON
ejpam-4905	323	3	∈	∈	PROPN
ejpam-4905	323	4	nhu(w	nhu(w	PROPN
ejpam-4905	323	5	)	)	PUNCT
ejpam-4905	323	6	∩	∩	NOUN
ejpam-4905	323	7	nhu(a	nhu(a	PROPN
ejpam-4905	323	8	)	)	PUNCT
ejpam-4905	323	9	and	and	CCONJ
ejpam-4905	323	10	t	t	PROPN
ejpam-4905	323	11	∈	∈	PROPN
ejpam-4905	323	12	nhu(z	nhu(z	PROPN
ejpam-4905	323	13	)	)	PUNCT
ejpam-4905	323	14	∩	∩	NOUN
ejpam-4905	323	15	nhu(b	nhu(b	NUM
ejpam-4905	323	16	)	)	PUNCT
ejpam-4905	323	17	.	.	PUNCT
ejpam-4905	324	1	then	then	ADV
ejpam-4905	324	2	s	s	VERB
ejpam-4905	324	3	∈	∈	PROPN
ejpam-4905	324	4	n2	n2	NOUN
ejpam-4905	324	5	g	g	PROPN
ejpam-4905	324	6	◦	◦	NOUN
ejpam-4905	324	7	h	h	NOUN
ejpam-4905	325	1	[	[	X
ejpam-4905	325	2	b]\n2	b]\n2	NOUN
ejpam-4905	325	3	g	g	NOUN
ejpam-4905	325	4	◦	◦	NOUN
ejpam-4905	325	5	h	h	NOUN
ejpam-4905	326	1	[	[	X
ejpam-4905	326	2	a	a	X
ejpam-4905	326	3	]	]	X
ejpam-4905	326	4	and	and	CCONJ
ejpam-4905	326	5	t	t	PROPN
ejpam-4905	326	6	∈	∈	PROPN
ejpam-4905	326	7	n2	n2	PROPN
ejpam-4905	326	8	g	g	PROPN
ejpam-4905	326	9	◦	◦	NOUN
ejpam-4905	326	10	h	h	NOUN
ejpam-4905	327	1	[	[	X
ejpam-4905	327	2	a]\n2	a]\n2	ADP
ejpam-4905	327	3	g	g	NOUN
ejpam-4905	327	4	◦	◦	NOUN
ejpam-4905	327	5	h	h	NOUN
ejpam-4905	328	1	[	[	X
ejpam-4905	328	2	b	b	X
ejpam-4905	328	3	]	]	X
ejpam-4905	328	4	.	.	PUNCT
ejpam-4905	329	1	since	since	SCONJ
ejpam-4905	329	2	a	a	DET
ejpam-4905	329	3	,	,	PUNCT
ejpam-4905	329	4	b	b	NOUN
ejpam-4905	329	5	are	be	AUX
ejpam-4905	329	6	arbitrary	arbitrary	ADJ
ejpam-4905	329	7	,	,	PUNCT
ejpam-4905	329	8	t	t	PROPN
ejpam-4905	329	9	is	be	AUX
ejpam-4905	329	10	a	a	DET
ejpam-4905	329	11	j2	j2	PROPN
ejpam-4905	329	12	-	-	PUNCT
ejpam-4905	329	13	set	set	NOUN
ejpam-4905	329	14	of	of	ADP
ejpam-4905	329	15	g	g	PROPN
ejpam-4905	329	16	◦	◦	NOUN
ejpam-4905	329	17	h.	h.	PROPN
ejpam-4905	329	18	next	next	ADV
ejpam-4905	329	19	,	,	PUNCT
ejpam-4905	329	20	suppose	suppose	VERB
ejpam-4905	329	21	that	that	SCONJ
ejpam-4905	329	22	dh(a	dh(a	NOUN
ejpam-4905	329	23	,	,	PUNCT
ejpam-4905	329	24	b	b	NOUN
ejpam-4905	329	25	)	)	PUNCT
ejpam-4905	329	26	≥	≥	NOUN
ejpam-4905	329	27	3	3	NUM
ejpam-4905	329	28	.	.	PUNCT
ejpam-4905	330	1	let	let	VERB
ejpam-4905	330	2	u	u	PRON
ejpam-4905	330	3	∈	∈	PROPN
ejpam-4905	330	4	nh(a	nh(a	NUM
ejpam-4905	330	5	)	)	PUNCT
ejpam-4905	330	6	and	and	CCONJ
ejpam-4905	330	7	j.	j.	PROPN
ejpam-4905	330	8	hassan	hassan	PROPN
ejpam-4905	330	9	,	,	PUNCT
ejpam-4905	330	10	a.	a.	PROPN
ejpam-4905	330	11	bakkang	bakkang	PROPN
ejpam-4905	330	12	,	,	PUNCT
ejpam-4905	330	13	a.	a.	NOUN
ejpam-4905	330	14	sappari	sappari	PROPN
ejpam-4905	330	15	/	/	SYM
ejpam-4905	330	16	eur	eur	PROPN
ejpam-4905	330	17	.	.	PUNCT
ejpam-4905	331	1	j.	j.	PROPN
ejpam-4905	331	2	pure	pure	PROPN
ejpam-4905	331	3	appl	appl	PROPN
ejpam-4905	331	4	.	.	PROPN
ejpam-4905	331	5	math	math	PROPN
ejpam-4905	331	6	,	,	PUNCT
ejpam-4905	331	7	16	16	NUM
ejpam-4905	331	8	(	(	PUNCT
ejpam-4905	331	9	4	4	NUM
ejpam-4905	331	10	)	)	PUNCT
ejpam-4905	331	11	(	(	PUNCT
ejpam-4905	331	12	2023	2023	NUM
ejpam-4905	331	13	)	)	PUNCT
ejpam-4905	331	14	,	,	PUNCT
ejpam-4905	331	15	2118	2118	NUM
ejpam-4905	331	16	-	-	SYM
ejpam-4905	331	17	2131	2131	NUM
ejpam-4905	331	18	2127	2127	NUM
ejpam-4905	331	19	v	v	ADP
ejpam-4905	331	20	∈	∈	PROPN
ejpam-4905	331	21	nh(b	nh(b	NOUN
ejpam-4905	331	22	)	)	PUNCT
ejpam-4905	331	23	,	,	PUNCT
ejpam-4905	331	24	then	then	ADV
ejpam-4905	331	25	u	u	PROPN
ejpam-4905	331	26	∈	∈	PROPN
ejpam-4905	331	27	n2	n2	NOUN
ejpam-4905	331	28	g	g	PROPN
ejpam-4905	331	29	◦	◦	NOUN
ejpam-4905	331	30	h	h	NOUN
ejpam-4905	332	1	[	[	X
ejpam-4905	332	2	b]\n2	b]\n2	NOUN
ejpam-4905	332	3	g	g	NOUN
ejpam-4905	332	4	◦	◦	NOUN
ejpam-4905	332	5	h	h	NOUN
ejpam-4905	333	1	[	[	X
ejpam-4905	333	2	a	a	X
ejpam-4905	333	3	]	]	X
ejpam-4905	333	4	and	and	CCONJ
ejpam-4905	333	5	v	v	ADP
ejpam-4905	333	6	∈	∈	PROPN
ejpam-4905	333	7	n2	n2	NOUN
ejpam-4905	333	8	g	g	PROPN
ejpam-4905	333	9	◦	◦	NOUN
ejpam-4905	333	10	h	h	NOUN
ejpam-4905	334	1	[	[	X
ejpam-4905	334	2	a]\n2	a]\n2	ADP
ejpam-4905	334	3	g	g	NOUN
ejpam-4905	334	4	◦	◦	NOUN
ejpam-4905	334	5	h	h	NOUN
ejpam-4905	335	1	[	[	X
ejpam-4905	335	2	b	b	X
ejpam-4905	335	3	]	]	X
ejpam-4905	335	4	.	.	PUNCT
ejpam-4905	336	1	since	since	SCONJ
ejpam-4905	336	2	a	a	DET
ejpam-4905	336	3	,	,	PUNCT
ejpam-4905	336	4	b	b	NOUN
ejpam-4905	336	5	are	be	AUX
ejpam-4905	336	6	arbitrary	arbitrary	ADJ
ejpam-4905	336	7	,	,	PUNCT
ejpam-4905	336	8	t	t	PROPN
ejpam-4905	336	9	is	be	AUX
ejpam-4905	336	10	a	a	DET
ejpam-4905	336	11	j2	j2	PROPN
ejpam-4905	336	12	-	-	PUNCT
ejpam-4905	336	13	set	set	NOUN
ejpam-4905	336	14	of	of	ADP
ejpam-4905	336	15	g	g	PROPN
ejpam-4905	336	16	◦	◦	NOUN
ejpam-4905	336	17	h.	h.	PROPN
ejpam-4905	336	18	next	next	ADV
ejpam-4905	336	19	,	,	PUNCT
ejpam-4905	336	20	assume	assume	VERB
ejpam-4905	336	21	that	that	SCONJ
ejpam-4905	336	22	a	a	DET
ejpam-4905	336	23	∈	∈	PROPN
ejpam-4905	336	24	tx	tx	NOUN
ejpam-4905	336	25	and	and	CCONJ
ejpam-4905	336	26	b	b	X
ejpam-4905	336	27	∈	∈	NOUN
ejpam-4905	336	28	ty	ty	INTJ
ejpam-4905	336	29	for	for	ADP
ejpam-4905	336	30	some	some	DET
ejpam-4905	336	31	x	x	NOUN
ejpam-4905	336	32	,	,	PUNCT
ejpam-4905	336	33	y	y	PROPN
ejpam-4905	336	34	∈	∈	PROPN
ejpam-4905	336	35	v	v	NOUN
ejpam-4905	336	36	(	(	PUNCT
ejpam-4905	336	37	g	g	NOUN
ejpam-4905	336	38	)	)	PUNCT
ejpam-4905	336	39	,	,	PUNCT
ejpam-4905	336	40	x	x	PROPN
ejpam-4905	336	41	̸=	̸=	PROPN
ejpam-4905	336	42	y.	y.	NOUN
ejpam-4905	336	43	then	then	ADV
ejpam-4905	336	44	a	a	DET
ejpam-4905	336	45	∈	∈	PROPN
ejpam-4905	336	46	n2	n2	NOUN
ejpam-4905	336	47	g	g	PROPN
ejpam-4905	336	48	◦	◦	NOUN
ejpam-4905	336	49	h	h	NOUN
ejpam-4905	337	1	[	[	X
ejpam-4905	337	2	a	a	X
ejpam-4905	337	3	]	]	PUNCT
ejpam-4905	337	4	\	\	PROPN
ejpam-4905	337	5	n2	n2	PROPN
ejpam-4905	337	6	g	g	PROPN
ejpam-4905	337	7	◦	◦	NOUN
ejpam-4905	337	8	h	h	NOUN
ejpam-4905	338	1	[	[	X
ejpam-4905	338	2	b	b	X
ejpam-4905	338	3	]	]	X
ejpam-4905	338	4	and	and	CCONJ
ejpam-4905	338	5	b	b	PROPN
ejpam-4905	338	6	∈	∈	PROPN
ejpam-4905	338	7	n2	n2	NOUN
ejpam-4905	338	8	g	g	PROPN
ejpam-4905	338	9	◦	◦	NOUN
ejpam-4905	338	10	h	h	NOUN
ejpam-4905	339	1	[	[	X
ejpam-4905	339	2	b	b	X
ejpam-4905	339	3	]	]	PUNCT
ejpam-4905	339	4	\	\	PROPN
ejpam-4905	339	5	n2	n2	PROPN
ejpam-4905	339	6	g	g	PROPN
ejpam-4905	339	7	◦	◦	NOUN
ejpam-4905	339	8	h	h	NOUN
ejpam-4905	340	1	[	[	X
ejpam-4905	340	2	a	a	X
ejpam-4905	340	3	]	]	X
ejpam-4905	340	4	.	.	PUNCT
ejpam-4905	341	1	thus	thus	ADV
ejpam-4905	341	2	,	,	PUNCT
ejpam-4905	341	3	n2	n2	ADJ
ejpam-4905	341	4	g	g	PROPN
ejpam-4905	341	5	◦	◦	NOUN
ejpam-4905	341	6	h	h	NOUN
ejpam-4905	342	1	[	[	X
ejpam-4905	342	2	a	a	X
ejpam-4905	342	3	]	]	PUNCT
ejpam-4905	342	4	\	\	PROPN
ejpam-4905	342	5	n2	n2	PROPN
ejpam-4905	342	6	g	g	PROPN
ejpam-4905	342	7	◦	◦	NOUN
ejpam-4905	342	8	h	h	NOUN
ejpam-4905	343	1	[	[	X
ejpam-4905	343	2	b	b	X
ejpam-4905	343	3	]	]	X
ejpam-4905	343	4	̸=	̸=	PROPN
ejpam-4905	343	5	∅	∅	NOUN
ejpam-4905	343	6	and	and	CCONJ
ejpam-4905	343	7	n2	n2	ADJ
ejpam-4905	343	8	g	g	PROPN
ejpam-4905	343	9	◦	◦	NOUN
ejpam-4905	343	10	h	h	NOUN
ejpam-4905	344	1	[	[	X
ejpam-4905	344	2	b	b	X
ejpam-4905	344	3	]	]	PUNCT
ejpam-4905	344	4	\	\	PROPN
ejpam-4905	344	5	n2	n2	PROPN
ejpam-4905	344	6	g	g	PROPN
ejpam-4905	344	7	◦	◦	NOUN
ejpam-4905	344	8	h	h	NOUN
ejpam-4905	345	1	[	[	X
ejpam-4905	345	2	a	a	X
ejpam-4905	345	3	]	]	X
ejpam-4905	345	4	̸=	̸=	PROPN
ejpam-4905	345	5	∅.	∅.	NOUN
ejpam-4905	345	6	since	since	SCONJ
ejpam-4905	345	7	a	a	PRON
ejpam-4905	345	8	and	and	CCONJ
ejpam-4905	345	9	b	b	NOUN
ejpam-4905	345	10	are	be	AUX
ejpam-4905	345	11	arbitrary	arbitrary	ADJ
ejpam-4905	345	12	,	,	PUNCT
ejpam-4905	345	13	t	t	PROPN
ejpam-4905	345	14	is	be	AUX
ejpam-4905	345	15	a	a	DET
ejpam-4905	345	16	j2	j2	NOUN
ejpam-4905	345	17	-	-	PUNCT
ejpam-4905	345	18	set	set	NOUN
ejpam-4905	345	19	in	in	ADP
ejpam-4905	345	20	g	g	PROPN
ejpam-4905	345	21	◦	◦	NOUN
ejpam-4905	345	22	h.	h.	NOUN
ejpam-4905	346	1	now	now	ADV
ejpam-4905	346	2	,	,	PUNCT
ejpam-4905	346	3	since	since	SCONJ
ejpam-4905	346	4	tv	tv	NOUN
ejpam-4905	346	5	is	be	AUX
ejpam-4905	346	6	a	a	DET
ejpam-4905	346	7	maximum	maximum	ADJ
ejpam-4905	346	8	j2	j2	NOUN
ejpam-4905	346	9	-	-	PUNCT
ejpam-4905	346	10	set	set	NOUN
ejpam-4905	346	11	in	in	ADP
ejpam-4905	346	12	hv	hv	PROPN
ejpam-4905	346	13	for	for	ADP
ejpam-4905	346	14	each	each	DET
ejpam-4905	346	15	v	v	NUM
ejpam-4905	346	16	∈	∈	PROPN
ejpam-4905	346	17	v	v	NOUN
ejpam-4905	346	18	(	(	PUNCT
ejpam-4905	346	19	g	g	NOUN
ejpam-4905	346	20	)	)	PUNCT
ejpam-4905	346	21	,	,	PUNCT
ejpam-4905	346	22	it	it	PRON
ejpam-4905	346	23	follows	follow	VERB
ejpam-4905	346	24	that	that	SCONJ
ejpam-4905	346	25	tv	tv	NOUN
ejpam-4905	346	26	is	be	AUX
ejpam-4905	346	27	a	a	DET
ejpam-4905	346	28	maximum	maximum	ADJ
ejpam-4905	346	29	j2	j2	PROPN
ejpam-4905	346	30	-	-	PUNCT
ejpam-4905	346	31	hop	hop	NOUN
ejpam-4905	346	32	dominating	dominating	NOUN
ejpam-4905	346	33	set	set	VERB
ejpam-4905	346	34	in	in	ADP
ejpam-4905	346	35	hv	hv	PROPN
ejpam-4905	346	36	for	for	ADP
ejpam-4905	346	37	every	every	DET
ejpam-4905	346	38	v	v	NUM
ejpam-4905	346	39	∈	∈	PROPN
ejpam-4905	346	40	v	v	NOUN
ejpam-4905	346	41	(	(	PUNCT
ejpam-4905	346	42	g	g	NOUN
ejpam-4905	346	43	)	)	PUNCT
ejpam-4905	346	44	by	by	ADP
ejpam-4905	346	45	theorem	theorem	NOUN
ejpam-4905	346	46	2	2	NUM
ejpam-4905	346	47	.	.	X
ejpam-4905	346	48	thus,⋃	thus,⋃	NOUN
ejpam-4905	346	49	v∈v	v∈v	NOUN
ejpam-4905	346	50	(	(	PUNCT
ejpam-4905	346	51	g	g	NOUN
ejpam-4905	346	52	)	)	PUNCT
ejpam-4905	346	53	v	v	NOUN
ejpam-4905	346	54	(	(	PUNCT
ejpam-4905	346	55	hv	hv	PROPN
ejpam-4905	346	56	)	)	PUNCT
ejpam-4905	346	57	⊆	⊆	NUM
ejpam-4905	346	58	n2	n2	NOUN
ejpam-4905	346	59	g	g	PROPN
ejpam-4905	346	60	◦	◦	NOUN
ejpam-4905	346	61	h	h	NOUN
ejpam-4905	347	1	[	[	X
ejpam-4905	347	2	t	t	X
ejpam-4905	347	3	]	]	PUNCT
ejpam-4905	347	4	.	.	PUNCT
ejpam-4905	348	1	now	now	ADV
ejpam-4905	348	2	,	,	PUNCT
ejpam-4905	348	3	let	let	VERB
ejpam-4905	348	4	r	r	NOUN
ejpam-4905	348	5	∈	∈	PROPN
ejpam-4905	348	6	v	v	NOUN
ejpam-4905	348	7	(	(	PUNCT
ejpam-4905	348	8	g	g	PROPN
ejpam-4905	348	9	◦	◦	NOUN
ejpam-4905	348	10	h	h	NOUN
ejpam-4905	348	11	)	)	PUNCT
ejpam-4905	348	12	\	\	NOUN
ejpam-4905	349	1	⋃	⋃	NOUN
ejpam-4905	349	2	v∈v	v∈v	NOUN
ejpam-4905	349	3	(	(	PUNCT
ejpam-4905	349	4	g	g	NOUN
ejpam-4905	349	5	)	)	PUNCT
ejpam-4905	349	6	v	v	NOUN
ejpam-4905	349	7	(	(	PUNCT
ejpam-4905	349	8	hv	hv	PROPN
ejpam-4905	349	9	)	)	PUNCT
ejpam-4905	349	10	.	.	PUNCT
ejpam-4905	350	1	then	then	ADV
ejpam-4905	350	2	r	r	NOUN
ejpam-4905	350	3	∈	∈	PROPN
ejpam-4905	350	4	v	v	NOUN
ejpam-4905	350	5	(	(	PUNCT
ejpam-4905	350	6	g	g	NOUN
ejpam-4905	350	7	)	)	PUNCT
ejpam-4905	350	8	.	.	PUNCT
ejpam-4905	351	1	since	since	SCONJ
ejpam-4905	351	2	g	g	PROPN
ejpam-4905	351	3	is	be	AUX
ejpam-4905	351	4	a	a	DET
ejpam-4905	351	5	non	non	ADJ
ejpam-4905	351	6	-	-	ADJ
ejpam-4905	351	7	trivial	trivial	ADJ
ejpam-4905	351	8	connected	connected	ADJ
ejpam-4905	351	9	graph	graph	NOUN
ejpam-4905	351	10	,	,	PUNCT
ejpam-4905	351	11	there	there	PRON
ejpam-4905	351	12	exists	exist	VERB
ejpam-4905	351	13	q	q	PROPN
ejpam-4905	351	14	∈	∈	NOUN
ejpam-4905	351	15	ts	ts	ADP
ejpam-4905	351	16	such	such	ADJ
ejpam-4905	351	17	that	that	SCONJ
ejpam-4905	351	18	dg	dg	PROPN
ejpam-4905	351	19	◦	◦	PROPN
ejpam-4905	351	20	h(r	h(r	NOUN
ejpam-4905	351	21	,	,	PUNCT
ejpam-4905	351	22	q	q	NOUN
ejpam-4905	351	23	)	)	PUNCT
ejpam-4905	351	24	=	=	SYM
ejpam-4905	351	25	2	2	NUM
ejpam-4905	351	26	for	for	ADP
ejpam-4905	351	27	some	some	PRON
ejpam-4905	351	28	s	s	NOUN
ejpam-4905	351	29	∈	∈	PROPN
ejpam-4905	351	30	v	v	NOUN
ejpam-4905	351	31	(	(	PUNCT
ejpam-4905	351	32	g	g	NOUN
ejpam-4905	351	33	)	)	PUNCT
ejpam-4905	351	34	.	.	PUNCT
ejpam-4905	352	1	hence	hence	ADV
ejpam-4905	352	2	,	,	PUNCT
ejpam-4905	352	3	n2	n2	ADJ
ejpam-4905	352	4	g	g	PROPN
ejpam-4905	352	5	◦	◦	NOUN
ejpam-4905	352	6	h	h	NOUN
ejpam-4905	353	1	[	[	X
ejpam-4905	353	2	t	t	X
ejpam-4905	353	3	]	]	X
ejpam-4905	353	4	=	=	SYM
ejpam-4905	353	5	v	v	X
ejpam-4905	353	6	(	(	PUNCT
ejpam-4905	353	7	g	g	PROPN
ejpam-4905	353	8	◦	◦	NOUN
ejpam-4905	353	9	h	h	NOUN
ejpam-4905	353	10	)	)	PUNCT
ejpam-4905	353	11	,	,	PUNCT
ejpam-4905	353	12	and	and	CCONJ
ejpam-4905	353	13	so	so	ADV
ejpam-4905	353	14	t	t	PROPN
ejpam-4905	353	15	is	be	AUX
ejpam-4905	353	16	a	a	DET
ejpam-4905	353	17	j2	j2	PROPN
ejpam-4905	353	18	-	-	PUNCT
ejpam-4905	353	19	hop	hop	NOUN
ejpam-4905	353	20	dominating	dominating	NOUN
ejpam-4905	353	21	set	set	VERB
ejpam-4905	353	22	in	in	ADP
ejpam-4905	353	23	g	g	PROPN
ejpam-4905	353	24	◦	◦	NOUN
ejpam-4905	353	25	h.	h.	PROPN
ejpam-4905	353	26	consequently	consequently	ADV
ejpam-4905	353	27	,	,	PUNCT
ejpam-4905	353	28	γj2h(g	γj2h(g	NOUN
ejpam-4905	353	29	◦	◦	NOUN
ejpam-4905	353	30	h	h	NOUN
ejpam-4905	353	31	)	)	PUNCT
ejpam-4905	353	32	≥	≥	NOUN
ejpam-4905	353	33	|v	|v	PROPN
ejpam-4905	353	34	(	(	PUNCT
ejpam-4905	353	35	g)|	g)|	NOUN
ejpam-4905	353	36	·	·	PUNCT
ejpam-4905	353	37	γj2h(h	γj2h(h	NUM
ejpam-4905	353	38	)	)	PUNCT
ejpam-4905	353	39	.	.	PUNCT
ejpam-4905	354	1	lemma	lemma	PROPN
ejpam-4905	354	2	1	1	NUM
ejpam-4905	354	3	.	.	PUNCT
ejpam-4905	355	1	[	[	X
ejpam-4905	355	2	7	7	X
ejpam-4905	355	3	]	]	PUNCT
ejpam-4905	355	4	let	let	VERB
ejpam-4905	355	5	g	g	PRON
ejpam-4905	355	6	be	be	AUX
ejpam-4905	355	7	a	a	DET
ejpam-4905	355	8	non	non	ADJ
ejpam-4905	355	9	-	-	ADJ
ejpam-4905	355	10	trivial	trivial	ADJ
ejpam-4905	355	11	connected	connected	ADJ
ejpam-4905	355	12	graph	graph	NOUN
ejpam-4905	355	13	and	and	CCONJ
ejpam-4905	355	14	let	let	VERB
ejpam-4905	355	15	g1	g1	PROPN
ejpam-4905	355	16	and	and	CCONJ
ejpam-4905	355	17	g2	g2	PROPN
ejpam-4905	355	18	be	be	VERB
ejpam-4905	355	19	two	two	NUM
ejpam-4905	355	20	copies	copy	NOUN
ejpam-4905	355	21	of	of	ADP
ejpam-4905	355	22	g	g	NOUN
ejpam-4905	355	23	in	in	ADP
ejpam-4905	355	24	the	the	DET
ejpam-4905	355	25	graph	graph	NOUN
ejpam-4905	355	26	s(g	s(g	PROPN
ejpam-4905	355	27	)	)	PUNCT
ejpam-4905	355	28	.	.	PUNCT
ejpam-4905	356	1	if	if	SCONJ
ejpam-4905	356	2	w	w	PROPN
ejpam-4905	356	3	∈	∈	PROPN
ejpam-4905	356	4	v	v	X
ejpam-4905	356	5	(	(	PUNCT
ejpam-4905	356	6	g1	g1	PROPN
ejpam-4905	356	7	)	)	PUNCT
ejpam-4905	356	8	and	and	CCONJ
ejpam-4905	356	9	w′	w′	PROPN
ejpam-4905	356	10	∈	∈	PROPN
ejpam-4905	356	11	v	v	X
ejpam-4905	356	12	(	(	PUNCT
ejpam-4905	356	13	g2	g2	PROPN
ejpam-4905	356	14	)	)	PUNCT
ejpam-4905	356	15	is	be	AUX
ejpam-4905	356	16	the	the	DET
ejpam-4905	356	17	corresponding	corresponding	ADJ
ejpam-4905	356	18	vertex	vertex	NOUN
ejpam-4905	356	19	of	of	ADP
ejpam-4905	356	20	w	w	PROPN
ejpam-4905	356	21	,	,	PUNCT
ejpam-4905	356	22	then	then	ADV
ejpam-4905	356	23	n2	n2	PROPN
ejpam-4905	356	24	s(g)[w	s(g)[w	PROPN
ejpam-4905	356	25	]	]	PUNCT
ejpam-4905	356	26	=	=	SYM
ejpam-4905	356	27	n2	n2	PROPN
ejpam-4905	356	28	g1	g1	PROPN
ejpam-4905	357	1	[	[	X
ejpam-4905	357	2	w	w	X
ejpam-4905	357	3	]	]	X
ejpam-4905	357	4	∪n2	∪n2	PROPN
ejpam-4905	357	5	g2	g2	PROPN
ejpam-4905	358	1	[	[	X
ejpam-4905	358	2	w′	w′	ADP
ejpam-4905	358	3	]	]	X
ejpam-4905	358	4	=	=	SYM
ejpam-4905	358	5	n2	n2	ADJ
ejpam-4905	358	6	s(g)[w	s(g)[w	NOUN
ejpam-4905	358	7	′	′	NOUN
ejpam-4905	358	8	]	]	PUNCT
ejpam-4905	358	9	.	.	PUNCT
ejpam-4905	359	1	lemma	lemma	PROPN
ejpam-4905	359	2	2	2	X
ejpam-4905	359	3	.	.	PUNCT
ejpam-4905	360	1	let	let	VERB
ejpam-4905	360	2	g	g	PRON
ejpam-4905	360	3	be	be	AUX
ejpam-4905	360	4	a	a	DET
ejpam-4905	360	5	non	non	ADJ
ejpam-4905	360	6	-	-	ADJ
ejpam-4905	360	7	trivial	trivial	ADJ
ejpam-4905	360	8	connected	connected	ADJ
ejpam-4905	360	9	graph	graph	NOUN
ejpam-4905	360	10	and	and	CCONJ
ejpam-4905	360	11	let	let	VERB
ejpam-4905	360	12	g1	g1	PROPN
ejpam-4905	360	13	and	and	CCONJ
ejpam-4905	360	14	g2	g2	PROPN
ejpam-4905	360	15	be	be	VERB
ejpam-4905	360	16	two	two	NUM
ejpam-4905	360	17	copies	copy	NOUN
ejpam-4905	360	18	of	of	ADP
ejpam-4905	360	19	g	g	NOUN
ejpam-4905	360	20	in	in	ADP
ejpam-4905	360	21	the	the	DET
ejpam-4905	360	22	graph	graph	NOUN
ejpam-4905	360	23	s(g	s(g	PROPN
ejpam-4905	360	24	)	)	PUNCT
ejpam-4905	360	25	.	.	PUNCT
ejpam-4905	361	1	if	if	SCONJ
ejpam-4905	361	2	n2	n2	ADJ
ejpam-4905	361	3	g1	g1	PROPN
ejpam-4905	361	4	[	[	X
ejpam-4905	361	5	a	a	X
ejpam-4905	361	6	]	]	X
ejpam-4905	361	7	⊆	⊆	NUM
ejpam-4905	361	8	n2	n2	ADJ
ejpam-4905	361	9	g1	g1	NOUN
ejpam-4905	361	10	[	[	X
ejpam-4905	361	11	b	b	X
ejpam-4905	361	12	]	]	X
ejpam-4905	361	13	or	or	CCONJ
ejpam-4905	361	14	n2	n2	ADJ
ejpam-4905	361	15	g2	g2	PROPN
ejpam-4905	361	16	[	[	X
ejpam-4905	361	17	a	a	X
ejpam-4905	361	18	]	]	X
ejpam-4905	361	19	⊆	⊆	NUM
ejpam-4905	361	20	n2	n2	NOUN
ejpam-4905	361	21	g2	g2	PROPN
ejpam-4905	362	1	[	[	X
ejpam-4905	362	2	b	b	X
ejpam-4905	362	3	]	]	X
ejpam-4905	362	4	,	,	PUNCT
ejpam-4905	362	5	then	then	ADV
ejpam-4905	362	6	n2	n2	ADJ
ejpam-4905	362	7	s(g)[a	s(g)[a	PROPN
ejpam-4905	362	8	]	]	PUNCT
ejpam-4905	362	9	⊆	⊆	NUM
ejpam-4905	362	10	n2	n2	PROPN
ejpam-4905	362	11	s(g)[b	s(g)[b	PROPN
ejpam-4905	362	12	]	]	PUNCT
ejpam-4905	362	13	.	.	PUNCT
ejpam-4905	363	1	proof	proof	NOUN
ejpam-4905	363	2	.	.	PUNCT
ejpam-4905	364	1	let	let	VERB
ejpam-4905	364	2	a	a	DET
ejpam-4905	364	3	,	,	PUNCT
ejpam-4905	364	4	b	b	PROPN
ejpam-4905	364	5	∈	∈	PROPN
ejpam-4905	364	6	v	v	NOUN
ejpam-4905	364	7	(	(	PUNCT
ejpam-4905	364	8	g1	g1	PROPN
ejpam-4905	364	9	)	)	PUNCT
ejpam-4905	364	10	and	and	CCONJ
ejpam-4905	364	11	suppose	suppose	VERB
ejpam-4905	364	12	that	that	SCONJ
ejpam-4905	364	13	n2	n2	PROPN
ejpam-4905	364	14	g1	g1	PROPN
ejpam-4905	364	15	[	[	X
ejpam-4905	364	16	a	a	X
ejpam-4905	364	17	]	]	X
ejpam-4905	364	18	⊆	⊆	NUM
ejpam-4905	364	19	n2	n2	ADJ
ejpam-4905	364	20	g1	g1	NOUN
ejpam-4905	364	21	[	[	X
ejpam-4905	364	22	b	b	X
ejpam-4905	364	23	]	]	X
ejpam-4905	364	24	.	.	PUNCT
ejpam-4905	365	1	let	let	VERB
ejpam-4905	365	2	x	x	SYM
ejpam-4905	365	3	∈	∈	PROPN
ejpam-4905	365	4	n2	n2	ADJ
ejpam-4905	365	5	s(g)[a	s(g)[a	NOUN
ejpam-4905	365	6	]	]	PUNCT
ejpam-4905	365	7	.	.	PUNCT
ejpam-4905	366	1	then	then	ADV
ejpam-4905	366	2	ds(g)(a	ds(g)(a	PROPN
ejpam-4905	366	3	,	,	PUNCT
ejpam-4905	366	4	x	x	NOUN
ejpam-4905	366	5	)	)	PUNCT
ejpam-4905	366	6	=	=	SYM
ejpam-4905	366	7	2	2	X
ejpam-4905	366	8	.	.	PUNCT
ejpam-4905	367	1	if	if	SCONJ
ejpam-4905	367	2	x	x	SYM
ejpam-4905	367	3	∈	∈	PROPN
ejpam-4905	367	4	v	v	NOUN
ejpam-4905	367	5	(	(	PUNCT
ejpam-4905	367	6	g1	g1	PROPN
ejpam-4905	367	7	)	)	PUNCT
ejpam-4905	367	8	,	,	PUNCT
ejpam-4905	367	9	then	then	ADV
ejpam-4905	367	10	dg1(a	dg1(a	PROPN
ejpam-4905	367	11	,	,	PUNCT
ejpam-4905	367	12	x	x	NOUN
ejpam-4905	367	13	)	)	PUNCT
ejpam-4905	367	14	=	=	SYM
ejpam-4905	367	15	2	2	X
ejpam-4905	367	16	.	.	PUNCT
ejpam-4905	368	1	so	so	ADV
ejpam-4905	368	2	,	,	PUNCT
ejpam-4905	368	3	x	x	PROPN
ejpam-4905	368	4	∈	∈	PROPN
ejpam-4905	368	5	n2	n2	NOUN
ejpam-4905	368	6	g1	g1	PROPN
ejpam-4905	368	7	[	[	X
ejpam-4905	368	8	a	a	X
ejpam-4905	368	9	]	]	X
ejpam-4905	368	10	.	.	PUNCT
ejpam-4905	369	1	thus	thus	ADV
ejpam-4905	369	2	,	,	PUNCT
ejpam-4905	369	3	by	by	ADP
ejpam-4905	369	4	assumption	assumption	NOUN
ejpam-4905	369	5	,	,	PUNCT
ejpam-4905	369	6	x	x	SYM
ejpam-4905	369	7	∈	∈	PROPN
ejpam-4905	369	8	n2	n2	NOUN
ejpam-4905	369	9	g1	g1	PROPN
ejpam-4905	370	1	[	[	X
ejpam-4905	370	2	b	b	X
ejpam-4905	370	3	]	]	X
ejpam-4905	370	4	.	.	PUNCT
ejpam-4905	371	1	by	by	ADP
ejpam-4905	371	2	lemma	lemma	PROPN
ejpam-4905	371	3	1	1	NUM
ejpam-4905	371	4	,	,	PUNCT
ejpam-4905	371	5	x	x	SYM
ejpam-4905	371	6	∈	∈	PROPN
ejpam-4905	371	7	n2	n2	PROPN
ejpam-4905	371	8	s(g)[b	s(g)[b	PROPN
ejpam-4905	371	9	]	]	PUNCT
ejpam-4905	371	10	,	,	PUNCT
ejpam-4905	371	11	and	and	CCONJ
ejpam-4905	371	12	we	we	PRON
ejpam-4905	371	13	are	be	AUX
ejpam-4905	371	14	done	do	VERB
ejpam-4905	371	15	.	.	PUNCT
ejpam-4905	371	16	suppose	suppose	VERB
ejpam-4905	371	17	that	that	SCONJ
ejpam-4905	371	18	x	x	SYM
ejpam-4905	371	19	∈	∈	NOUN
ejpam-4905	371	20	v	v	X
ejpam-4905	371	21	(	(	PUNCT
ejpam-4905	371	22	g2	g2	PROPN
ejpam-4905	371	23	)	)	PUNCT
ejpam-4905	371	24	.	.	PUNCT
ejpam-4905	372	1	then	then	ADV
ejpam-4905	372	2	x	x	SYM
ejpam-4905	372	3	∈	∈	PROPN
ejpam-4905	372	4	n2	n2	NOUN
ejpam-4905	372	5	g2	g2	PROPN
ejpam-4905	373	1	[	[	X
ejpam-4905	373	2	a′	a′	X
ejpam-4905	373	3	]	]	X
ejpam-4905	373	4	for	for	ADP
ejpam-4905	373	5	some	some	DET
ejpam-4905	373	6	a′	a′	NOUN
ejpam-4905	373	7	∈	∈	PROPN
ejpam-4905	373	8	v	v	NOUN
ejpam-4905	373	9	(	(	PUNCT
ejpam-4905	373	10	g2	g2	PROPN
ejpam-4905	373	11	)	)	PUNCT
ejpam-4905	373	12	.	.	PUNCT
ejpam-4905	374	1	since	since	SCONJ
ejpam-4905	374	2	n2	n2	PROPN
ejpam-4905	374	3	g2	g2	PROPN
ejpam-4905	374	4	[	[	X
ejpam-4905	374	5	a′	a′	X
ejpam-4905	374	6	]	]	X
ejpam-4905	374	7	⊆	⊆	NUM
ejpam-4905	374	8	n2	n2	NOUN
ejpam-4905	374	9	g2	g2	PROPN
ejpam-4905	374	10	[	[	X
ejpam-4905	374	11	b′	b′	X
ejpam-4905	374	12	]	]	X
ejpam-4905	374	13	⊆	⊆	NUM
ejpam-4905	374	14	n2	n2	PROPN
ejpam-4905	374	15	s(g)[b	s(g)[b	PROPN
ejpam-4905	374	16	]	]	PUNCT
ejpam-4905	374	17	,	,	PUNCT
ejpam-4905	374	18	it	it	PRON
ejpam-4905	374	19	follows	follow	VERB
ejpam-4905	374	20	that	that	SCONJ
ejpam-4905	374	21	x	x	SYM
ejpam-4905	374	22	∈	∈	PROPN
ejpam-4905	374	23	n2	n2	PROPN
ejpam-4905	374	24	s(g)[b	s(g)[b	PROPN
ejpam-4905	374	25	]	]	PUNCT
ejpam-4905	374	26	,	,	PUNCT
ejpam-4905	374	27	and	and	CCONJ
ejpam-4905	374	28	so	so	ADV
ejpam-4905	374	29	n2	n2	ADJ
ejpam-4905	374	30	s(g)[a	s(g)[a	NOUN
ejpam-4905	374	31	]	]	PUNCT
ejpam-4905	374	32	⊆	⊆	NUM
ejpam-4905	374	33	n2	n2	PROPN
ejpam-4905	374	34	s(g)[b	s(g)[b	PROPN
ejpam-4905	374	35	]	]	PUNCT
ejpam-4905	374	36	.	.	PUNCT
ejpam-4905	375	1	similarly	similarly	ADV
ejpam-4905	375	2	,	,	PUNCT
ejpam-4905	375	3	if	if	SCONJ
ejpam-4905	375	4	n2	n2	PROPN
ejpam-4905	375	5	g2	g2	PROPN
ejpam-4905	375	6	[	[	X
ejpam-4905	375	7	a	a	X
ejpam-4905	375	8	]	]	X
ejpam-4905	375	9	⊆	⊆	NUM
ejpam-4905	375	10	n2	n2	NOUN
ejpam-4905	375	11	g2	g2	PROPN
ejpam-4905	375	12	[	[	X
ejpam-4905	375	13	b	b	X
ejpam-4905	375	14	]	]	X
ejpam-4905	375	15	,	,	PUNCT
ejpam-4905	375	16	then	then	ADV
ejpam-4905	375	17	n2	n2	ADJ
ejpam-4905	375	18	s(g)[a	s(g)[a	PROPN
ejpam-4905	375	19	]	]	PUNCT
ejpam-4905	375	20	⊆	⊆	NUM
ejpam-4905	375	21	n2	n2	PROPN
ejpam-4905	375	22	s(g)[b	s(g)[b	PROPN
ejpam-4905	375	23	]	]	PUNCT
ejpam-4905	375	24	.	.	PUNCT
ejpam-4905	376	1	theorem	theorem	NOUN
ejpam-4905	376	2	9	9	NUM
ejpam-4905	376	3	.	.	PUNCT
ejpam-4905	377	1	let	let	VERB
ejpam-4905	377	2	g	g	PRON
ejpam-4905	377	3	be	be	AUX
ejpam-4905	377	4	a	a	DET
ejpam-4905	377	5	connected	connected	ADJ
ejpam-4905	377	6	non	non	ADJ
ejpam-4905	377	7	-	-	ADJ
ejpam-4905	377	8	trivial	trivial	ADJ
ejpam-4905	377	9	graph	graph	NOUN
ejpam-4905	377	10	.	.	PUNCT
ejpam-4905	378	1	then	then	ADV
ejpam-4905	378	2	t	t	PROPN
ejpam-4905	378	3	⊆	⊆	NUM
ejpam-4905	378	4	v	v	NOUN
ejpam-4905	378	5	(	(	PUNCT
ejpam-4905	378	6	s(g	s(g	PROPN
ejpam-4905	378	7	)	)	PUNCT
ejpam-4905	378	8	)	)	PUNCT
ejpam-4905	378	9	is	be	AUX
ejpam-4905	378	10	a	a	DET
ejpam-4905	378	11	j2	j2	NOUN
ejpam-4905	378	12	-	-	PUNCT
ejpam-4905	378	13	set	set	NOUN
ejpam-4905	378	14	in	in	ADP
ejpam-4905	378	15	s(g	s(g	PROPN
ejpam-4905	378	16	)	)	PUNCT
ejpam-4905	378	17	if	if	SCONJ
ejpam-4905	378	18	and	and	CCONJ
ejpam-4905	378	19	only	only	ADV
ejpam-4905	378	20	if	if	SCONJ
ejpam-4905	378	21	t	t	PROPN
ejpam-4905	378	22	satisfies	satisfy	VERB
ejpam-4905	378	23	one	one	NUM
ejpam-4905	378	24	of	of	ADP
ejpam-4905	378	25	the	the	DET
ejpam-4905	378	26	following	following	ADJ
ejpam-4905	378	27	conditions	condition	NOUN
ejpam-4905	378	28	:	:	PUNCT
ejpam-4905	378	29	(	(	PUNCT
ejpam-4905	378	30	i	i	NOUN
ejpam-4905	378	31	)	)	PUNCT
ejpam-4905	378	32	t	t	PROPN
ejpam-4905	378	33	is	be	AUX
ejpam-4905	378	34	a	a	DET
ejpam-4905	378	35	j2	j2	NOUN
ejpam-4905	378	36	-	-	PUNCT
ejpam-4905	378	37	set	set	NOUN
ejpam-4905	378	38	in	in	ADP
ejpam-4905	378	39	g1	g1	PROPN
ejpam-4905	378	40	.	.	PUNCT
ejpam-4905	379	1	(	(	PUNCT
ejpam-4905	379	2	ii	ii	NOUN
ejpam-4905	379	3	)	)	PUNCT
ejpam-4905	379	4	t	t	PROPN
ejpam-4905	379	5	is	be	AUX
ejpam-4905	379	6	a	a	DET
ejpam-4905	379	7	j2	j2	NOUN
ejpam-4905	379	8	-	-	PUNCT
ejpam-4905	379	9	set	set	NOUN
ejpam-4905	379	10	in	in	ADP
ejpam-4905	379	11	g2	g2	PROPN
ejpam-4905	379	12	.	.	PUNCT
ejpam-4905	380	1	(	(	PUNCT
ejpam-4905	380	2	iii	iii	X
ejpam-4905	380	3	)	)	PUNCT
ejpam-4905	380	4	t	t	NOUN
ejpam-4905	380	5	=	=	SYM
ejpam-4905	380	6	tg1∪tg2	tg1∪tg2	PROPN
ejpam-4905	380	7	,	,	PUNCT
ejpam-4905	380	8	where	where	SCONJ
ejpam-4905	380	9	tg1∪t	tg1∪t	PROPN
ejpam-4905	380	10	′	′	NUM
ejpam-4905	381	1	g2	g2	PROPN
ejpam-4905	381	2	and	and	CCONJ
ejpam-4905	381	3	t	t	PROPN
ejpam-4905	381	4	′	′	PROPN
ejpam-4905	381	5	g1	g1	PROPN
ejpam-4905	381	6	∪tg2	∪tg2	PROPN
ejpam-4905	381	7	are	be	AUX
ejpam-4905	381	8	j2	j2	NOUN
ejpam-4905	381	9	-	-	PUNCT
ejpam-4905	381	10	sets	set	NOUN
ejpam-4905	381	11	in	in	ADP
ejpam-4905	381	12	g1	g1	PROPN
ejpam-4905	381	13	and	and	CCONJ
ejpam-4905	381	14	g2	g2	PROPN
ejpam-4905	381	15	,	,	PUNCT
ejpam-4905	381	16	respectively	respectively	ADV
ejpam-4905	381	17	,	,	PUNCT
ejpam-4905	382	1	where	where	SCONJ
ejpam-4905	382	2	t	t	PROPN
ejpam-4905	382	3	′	′	NUM
ejpam-4905	383	1	g2	g2	PROPN
ejpam-4905	384	1	=	=	PRON
ejpam-4905	385	1	{	{	PUNCT
ejpam-4905	385	2	x	x	PROPN
ejpam-4905	385	3	∈	∈	PROPN
ejpam-4905	385	4	v	v	NOUN
ejpam-4905	385	5	(	(	PUNCT
ejpam-4905	385	6	g1	g1	PROPN
ejpam-4905	385	7	)	)	PUNCT
ejpam-4905	385	8	:	:	PUNCT
ejpam-4905	385	9	x	x	X
ejpam-4905	385	10	′	′	NUM
ejpam-4905	385	11	∈	∈	ADJ
ejpam-4905	385	12	tg2	tg2	NOUN
ejpam-4905	385	13	}	}	PUNCT
ejpam-4905	385	14	and	and	CCONJ
ejpam-4905	385	15	t	t	PROPN
ejpam-4905	385	16	′	′	NUM
ejpam-4905	385	17	g1	g1	PROPN
ejpam-4905	385	18	=	=	PRON
ejpam-4905	385	19	{	{	PUNCT
ejpam-4905	385	20	y	y	PROPN
ejpam-4905	385	21	∈	∈	PROPN
ejpam-4905	385	22	v	v	PROPN
ejpam-4905	385	23	(	(	PUNCT
ejpam-4905	385	24	g2	g2	PROPN
ejpam-4905	385	25	)	)	PUNCT
ejpam-4905	385	26	:	:	PUNCT
ejpam-4905	386	1	y	y	X
ejpam-4905	386	2	′	′	NUM
ejpam-4905	386	3	∈	∈	PROPN
ejpam-4905	386	4	tg1	tg1	NOUN
ejpam-4905	386	5	}	}	PUNCT
ejpam-4905	386	6	.	.	PUNCT
ejpam-4905	387	1	proof	proof	NOUN
ejpam-4905	387	2	.	.	PUNCT
ejpam-4905	388	1	suppose	suppose	VERB
ejpam-4905	388	2	that	that	SCONJ
ejpam-4905	388	3	t	t	PROPN
ejpam-4905	388	4	is	be	AUX
ejpam-4905	388	5	a	a	DET
ejpam-4905	388	6	j2	j2	NOUN
ejpam-4905	388	7	-	-	PUNCT
ejpam-4905	388	8	set	set	NOUN
ejpam-4905	388	9	in	in	ADP
ejpam-4905	388	10	s(g	s(g	PROPN
ejpam-4905	388	11	)	)	PUNCT
ejpam-4905	388	12	.	.	PUNCT
ejpam-4905	389	1	let	let	VERB
ejpam-4905	389	2	tg1	tg1	NOUN
ejpam-4905	389	3	=	=	SYM
ejpam-4905	389	4	t	t	NOUN
ejpam-4905	389	5	∩v	∩v	NOUN
ejpam-4905	389	6	(	(	PUNCT
ejpam-4905	389	7	g1	g1	PROPN
ejpam-4905	389	8	)	)	PUNCT
ejpam-4905	389	9	and	and	CCONJ
ejpam-4905	390	1	tg2	tg2	NOUN
ejpam-4905	390	2	=	=	PUNCT
ejpam-4905	390	3	t	t	NOUN
ejpam-4905	390	4	∩v	∩v	NOUN
ejpam-4905	390	5	(	(	PUNCT
ejpam-4905	390	6	g2	g2	PROPN
ejpam-4905	390	7	)	)	PUNCT
ejpam-4905	390	8	.	.	PUNCT
ejpam-4905	391	1	if	if	SCONJ
ejpam-4905	391	2	tg2	tg2	NOUN
ejpam-4905	391	3	=	=	SYM
ejpam-4905	391	4	∅	∅	NOUN
ejpam-4905	391	5	,	,	PUNCT
ejpam-4905	391	6	then	then	ADV
ejpam-4905	391	7	t	t	NOUN
ejpam-4905	391	8	=	=	PUNCT
ejpam-4905	391	9	tg1	tg1	NOUN
ejpam-4905	391	10	is	be	AUX
ejpam-4905	391	11	a	a	DET
ejpam-4905	391	12	j2	j2	NOUN
ejpam-4905	391	13	-	-	PUNCT
ejpam-4905	391	14	set	set	NOUN
ejpam-4905	391	15	in	in	ADP
ejpam-4905	391	16	g1	g1	NOUN
ejpam-4905	391	17	.	.	PUNCT
ejpam-4905	392	1	if	if	SCONJ
ejpam-4905	392	2	tg1	tg1	NOUN
ejpam-4905	392	3	=	=	SYM
ejpam-4905	392	4	∅	∅	NOUN
ejpam-4905	392	5	,	,	PUNCT
ejpam-4905	392	6	then	then	ADV
ejpam-4905	392	7	t	t	PROPN
ejpam-4905	392	8	=	=	PUNCT
ejpam-4905	392	9	tg2	tg2	PROPN
ejpam-4905	392	10	is	be	AUX
ejpam-4905	392	11	a	a	DET
ejpam-4905	392	12	j2	j2	NOUN
ejpam-4905	392	13	-	-	PUNCT
ejpam-4905	392	14	set	set	NOUN
ejpam-4905	392	15	in	in	ADP
ejpam-4905	392	16	g2	g2	PROPN
ejpam-4905	392	17	,	,	PUNCT
ejpam-4905	392	18	showing	show	VERB
ejpam-4905	392	19	that	that	SCONJ
ejpam-4905	392	20	(	(	PUNCT
ejpam-4905	392	21	i	i	NOUN
ejpam-4905	392	22	)	)	PUNCT
ejpam-4905	392	23	or	or	CCONJ
ejpam-4905	392	24	(	(	PUNCT
ejpam-4905	392	25	ii	ii	NOUN
ejpam-4905	392	26	)	)	PUNCT
ejpam-4905	392	27	holds	hold	VERB
ejpam-4905	392	28	.	.	PUNCT
ejpam-4905	393	1	assume	assume	VERB
ejpam-4905	393	2	that	that	SCONJ
ejpam-4905	393	3	tg1	tg1	VERB
ejpam-4905	393	4	̸=	̸=	PROPN
ejpam-4905	393	5	∅	∅	NOUN
ejpam-4905	393	6	and	and	CCONJ
ejpam-4905	393	7	tg2	tg2	PROPN
ejpam-4905	393	8	̸=	̸=	PROPN
ejpam-4905	393	9	∅.	∅.	ADV
ejpam-4905	393	10	suppose	suppose	VERB
ejpam-4905	393	11	on	on	ADP
ejpam-4905	393	12	the	the	DET
ejpam-4905	393	13	contrary	contrary	NOUN
ejpam-4905	393	14	that	that	PRON
ejpam-4905	393	15	s	s	VERB
ejpam-4905	393	16	=	=	NOUN
ejpam-4905	393	17	tg1	tg1	X
ejpam-4905	393	18	∪	∪	PROPN
ejpam-4905	393	19	t	t	PROPN
ejpam-4905	393	20	′	′	NUM
ejpam-4905	393	21	g2	g2	PROPN
ejpam-4905	393	22	is	be	AUX
ejpam-4905	393	23	not	not	PART
ejpam-4905	393	24	a	a	DET
ejpam-4905	393	25	j2	j2	NOUN
ejpam-4905	393	26	-	-	PUNCT
ejpam-4905	393	27	set	set	NOUN
ejpam-4905	393	28	in	in	ADP
ejpam-4905	393	29	g1	g1	PROPN
ejpam-4905	393	30	.	.	PUNCT
ejpam-4905	394	1	then	then	ADV
ejpam-4905	394	2	there	there	PRON
ejpam-4905	394	3	exist	exist	VERB
ejpam-4905	394	4	a	a	DET
ejpam-4905	394	5	,	,	PUNCT
ejpam-4905	394	6	b	b	X
ejpam-4905	394	7	∈	∈	PROPN
ejpam-4905	394	8	s	s	VERB
ejpam-4905	394	9	such	such	ADJ
ejpam-4905	394	10	j.	j.	PROPN
ejpam-4905	394	11	hassan	hassan	PROPN
ejpam-4905	394	12	,	,	PUNCT
ejpam-4905	394	13	a.	a.	PROPN
ejpam-4905	394	14	bakkang	bakkang	PROPN
ejpam-4905	394	15	,	,	PUNCT
ejpam-4905	394	16	a.	a.	NOUN
ejpam-4905	394	17	sappari	sappari	PROPN
ejpam-4905	394	18	/	/	SYM
ejpam-4905	394	19	eur	eur	PROPN
ejpam-4905	394	20	.	.	PUNCT
ejpam-4905	395	1	j.	j.	PROPN
ejpam-4905	395	2	pure	pure	PROPN
ejpam-4905	395	3	appl	appl	PROPN
ejpam-4905	395	4	.	.	PROPN
ejpam-4905	395	5	math	math	PROPN
ejpam-4905	395	6	,	,	PUNCT
ejpam-4905	395	7	16	16	NUM
ejpam-4905	395	8	(	(	PUNCT
ejpam-4905	395	9	4	4	NUM
ejpam-4905	395	10	)	)	PUNCT
ejpam-4905	395	11	(	(	PUNCT
ejpam-4905	395	12	2023	2023	NUM
ejpam-4905	395	13	)	)	PUNCT
ejpam-4905	395	14	,	,	PUNCT
ejpam-4905	395	15	2118	2118	NUM
ejpam-4905	395	16	-	-	SYM
ejpam-4905	395	17	2131	2131	NUM
ejpam-4905	395	18	2128	2128	NUM
ejpam-4905	395	19	that	that	SCONJ
ejpam-4905	395	20	n2	n2	ADJ
ejpam-4905	395	21	g1	g1	PROPN
ejpam-4905	395	22	[	[	X
ejpam-4905	395	23	a	a	X
ejpam-4905	395	24	]	]	PUNCT
ejpam-4905	395	25	\	\	PROPN
ejpam-4905	395	26	n2	n2	ADJ
ejpam-4905	395	27	g1	g1	PROPN
ejpam-4905	396	1	[	[	X
ejpam-4905	396	2	b	b	X
ejpam-4905	396	3	]	]	X
ejpam-4905	396	4	=	=	SYM
ejpam-4905	396	5	∅	∅	NOUN
ejpam-4905	396	6	or	or	CCONJ
ejpam-4905	396	7	n2	n2	ADJ
ejpam-4905	396	8	g1	g1	PROPN
ejpam-4905	396	9	[	[	X
ejpam-4905	396	10	b	b	X
ejpam-4905	396	11	]	]	PUNCT
ejpam-4905	396	12	\	\	PROPN
ejpam-4905	396	13	n2	n2	ADJ
ejpam-4905	396	14	g1	g1	PROPN
ejpam-4905	397	1	[	[	X
ejpam-4905	397	2	a	a	X
ejpam-4905	397	3	]	]	X
ejpam-4905	397	4	=	=	PUNCT
ejpam-4905	397	5	∅.	∅.	NOUN
ejpam-4905	397	6	it	it	PRON
ejpam-4905	397	7	follows	follow	VERB
ejpam-4905	397	8	that	that	SCONJ
ejpam-4905	397	9	n2	n2	ADJ
ejpam-4905	397	10	g1	g1	PROPN
ejpam-4905	397	11	[	[	X
ejpam-4905	397	12	a	a	X
ejpam-4905	397	13	]	]	X
ejpam-4905	397	14	⊆	⊆	NUM
ejpam-4905	397	15	n2	n2	ADJ
ejpam-4905	397	16	g1	g1	NOUN
ejpam-4905	397	17	[	[	X
ejpam-4905	397	18	b	b	X
ejpam-4905	397	19	]	]	X
ejpam-4905	397	20	or	or	CCONJ
ejpam-4905	397	21	n2	n2	ADJ
ejpam-4905	397	22	g1	g1	PROPN
ejpam-4905	397	23	[	[	X
ejpam-4905	397	24	b	b	X
ejpam-4905	397	25	]	]	X
ejpam-4905	397	26	⊆	⊆	NUM
ejpam-4905	397	27	n2	n2	ADJ
ejpam-4905	397	28	g1	g1	NOUN
ejpam-4905	397	29	[	[	X
ejpam-4905	397	30	a	a	X
ejpam-4905	397	31	]	]	X
ejpam-4905	397	32	.	.	PUNCT
ejpam-4905	398	1	if	if	SCONJ
ejpam-4905	398	2	a	a	DET
ejpam-4905	398	3	,	,	PUNCT
ejpam-4905	398	4	b	b	X
ejpam-4905	398	5	∈	∈	PROPN
ejpam-4905	398	6	tg1	tg1	NOUN
ejpam-4905	398	7	,	,	PUNCT
ejpam-4905	398	8	then	then	ADV
ejpam-4905	398	9	a	a	PRON
ejpam-4905	398	10	,	,	PUNCT
ejpam-4905	398	11	b	b	PROPN
ejpam-4905	398	12	∈	∈	PROPN
ejpam-4905	398	13	t	t	NOUN
ejpam-4905	398	14	.	.	PUNCT
ejpam-4905	399	1	since	since	SCONJ
ejpam-4905	399	2	n2	n2	PROPN
ejpam-4905	399	3	g1	g1	PROPN
ejpam-4905	399	4	[	[	X
ejpam-4905	399	5	a	a	X
ejpam-4905	399	6	]	]	X
ejpam-4905	399	7	⊆	⊆	NUM
ejpam-4905	399	8	n2	n2	ADJ
ejpam-4905	399	9	g1	g1	NOUN
ejpam-4905	399	10	[	[	X
ejpam-4905	399	11	b	b	X
ejpam-4905	399	12	]	]	X
ejpam-4905	399	13	or	or	CCONJ
ejpam-4905	399	14	n2	n2	ADJ
ejpam-4905	399	15	g1	g1	PROPN
ejpam-4905	399	16	[	[	X
ejpam-4905	399	17	b	b	X
ejpam-4905	399	18	]	]	X
ejpam-4905	399	19	⊆	⊆	NUM
ejpam-4905	399	20	n2	n2	ADJ
ejpam-4905	399	21	g1	g1	NOUN
ejpam-4905	399	22	[	[	X
ejpam-4905	399	23	a	a	X
ejpam-4905	399	24	]	]	X
ejpam-4905	399	25	,	,	PUNCT
ejpam-4905	399	26	we	we	PRON
ejpam-4905	399	27	have	have	VERB
ejpam-4905	399	28	n2	n2	ADJ
ejpam-4905	399	29	s(g)[a	s(g)[a	NOUN
ejpam-4905	399	30	]	]	PUNCT
ejpam-4905	399	31	⊆	⊆	NUM
ejpam-4905	399	32	n2	n2	PROPN
ejpam-4905	399	33	s(g)[b	s(g)[b	PROPN
ejpam-4905	399	34	]	]	PUNCT
ejpam-4905	399	35	or	or	CCONJ
ejpam-4905	399	36	n2	n2	PROPN
ejpam-4905	399	37	s(g)[b	s(g)[b	PROPN
ejpam-4905	399	38	]	]	PUNCT
ejpam-4905	399	39	⊆	⊆	NUM
ejpam-4905	399	40	n2	n2	ADJ
ejpam-4905	399	41	s(g)[a	s(g)[a	NOUN
ejpam-4905	399	42	]	]	PUNCT
ejpam-4905	399	43	by	by	ADP
ejpam-4905	399	44	lemma	lemma	PROPN
ejpam-4905	399	45	2	2	NUM
ejpam-4905	399	46	.	.	PUNCT
ejpam-4905	400	1	thus	thus	ADV
ejpam-4905	400	2	,	,	PUNCT
ejpam-4905	400	3	n2	n2	ADJ
ejpam-4905	400	4	s(g)[a	s(g)[a	NOUN
ejpam-4905	400	5	]	]	PUNCT
ejpam-4905	400	6	\n	\n	PUNCT
ejpam-4905	400	7	2	2	NUM
ejpam-4905	400	8	s(g)[b	s(g)[b	NOUN
ejpam-4905	400	9	]	]	PUNCT
ejpam-4905	400	10	=	=	SYM
ejpam-4905	400	11	∅	∅	NOUN
ejpam-4905	400	12	or	or	CCONJ
ejpam-4905	400	13	n2	n2	PROPN
ejpam-4905	400	14	s(g)[b	s(g)[b	PROPN
ejpam-4905	400	15	]	]	PUNCT
ejpam-4905	400	16	\n	\n	PUNCT
ejpam-4905	400	17	2	2	NUM
ejpam-4905	400	18	s(g)[a	s(g)[a	NOUN
ejpam-4905	400	19	]	]	X
ejpam-4905	400	20	=	=	SYM
ejpam-4905	400	21	∅	∅	NOUN
ejpam-4905	400	22	,	,	PUNCT
ejpam-4905	400	23	a	a	DET
ejpam-4905	400	24	contradiction	contradiction	NOUN
ejpam-4905	400	25	to	to	ADP
ejpam-4905	400	26	the	the	DET
ejpam-4905	400	27	fact	fact	NOUN
ejpam-4905	400	28	that	that	SCONJ
ejpam-4905	400	29	t	t	PROPN
ejpam-4905	400	30	is	be	AUX
ejpam-4905	400	31	a	a	DET
ejpam-4905	400	32	j2	j2	NOUN
ejpam-4905	400	33	-	-	PUNCT
ejpam-4905	400	34	set	set	NOUN
ejpam-4905	400	35	in	in	ADP
ejpam-4905	400	36	s(g	s(g	PROPN
ejpam-4905	400	37	)	)	PUNCT
ejpam-4905	400	38	.	.	PUNCT
ejpam-4905	400	39	suppose	suppose	VERB
ejpam-4905	400	40	that	that	SCONJ
ejpam-4905	400	41	a	a	PRON
ejpam-4905	400	42	,	,	PUNCT
ejpam-4905	400	43	b	b	PROPN
ejpam-4905	400	44	∈	∈	PROPN
ejpam-4905	400	45	t	t	NOUN
ejpam-4905	400	46	′	′	NUM
ejpam-4905	400	47	g2	g2	PROPN
ejpam-4905	400	48	.	.	PUNCT
ejpam-4905	401	1	then	then	ADV
ejpam-4905	401	2	a′	a′	PROPN
ejpam-4905	401	3	,	,	PUNCT
ejpam-4905	401	4	b′	b′	NUM
ejpam-4905	401	5	∈	∈	NOUN
ejpam-4905	401	6	tg2	tg2	NOUN
ejpam-4905	401	7	⊆	⊆	NUM
ejpam-4905	401	8	t	t	NOUN
ejpam-4905	401	9	.	.	PUNCT
ejpam-4905	402	1	since	since	SCONJ
ejpam-4905	402	2	n2	n2	ADJ
ejpam-4905	402	3	g1	g1	PROPN
ejpam-4905	402	4	[	[	X
ejpam-4905	402	5	a]\n2	a]\n2	NOUN
ejpam-4905	402	6	g1	g1	NOUN
ejpam-4905	402	7	[	[	X
ejpam-4905	402	8	b	b	X
ejpam-4905	402	9	]	]	X
ejpam-4905	402	10	=	=	SYM
ejpam-4905	402	11	∅	∅	NOUN
ejpam-4905	402	12	or	or	CCONJ
ejpam-4905	402	13	n2	n2	ADJ
ejpam-4905	402	14	g1	g1	PROPN
ejpam-4905	402	15	[	[	X
ejpam-4905	402	16	b	b	X
ejpam-4905	402	17	]	]	PUNCT
ejpam-4905	402	18	\	\	PROPN
ejpam-4905	402	19	n2	n2	ADJ
ejpam-4905	402	20	g1	g1	PROPN
ejpam-4905	402	21	[	[	X
ejpam-4905	402	22	a	a	X
ejpam-4905	402	23	]	]	X
ejpam-4905	402	24	=	=	SYM
ejpam-4905	402	25	∅	∅	NOUN
ejpam-4905	402	26	,	,	PUNCT
ejpam-4905	402	27	it	it	PRON
ejpam-4905	402	28	follows	follow	VERB
ejpam-4905	402	29	that	that	SCONJ
ejpam-4905	402	30	n2	n2	PROPN
ejpam-4905	402	31	g2	g2	PROPN
ejpam-4905	402	32	[	[	X
ejpam-4905	402	33	a′	a′	X
ejpam-4905	402	34	]	]	PUNCT
ejpam-4905	402	35	\	\	PROPN
ejpam-4905	402	36	n2	n2	PROPN
ejpam-4905	402	37	g2	g2	PROPN
ejpam-4905	403	1	[	[	X
ejpam-4905	403	2	b′	b′	X
ejpam-4905	403	3	]	]	X
ejpam-4905	403	4	=	=	SYM
ejpam-4905	403	5	∅	∅	NOUN
ejpam-4905	403	6	or	or	CCONJ
ejpam-4905	403	7	n2	n2	ADJ
ejpam-4905	403	8	g2	g2	PROPN
ejpam-4905	404	1	[	[	X
ejpam-4905	404	2	b′	b′	X
ejpam-4905	404	3	]	]	PUNCT
ejpam-4905	404	4	\	\	PROPN
ejpam-4905	404	5	n2	n2	PROPN
ejpam-4905	404	6	g2	g2	PROPN
ejpam-4905	405	1	[	[	X
ejpam-4905	405	2	a′	a′	X
ejpam-4905	405	3	]	]	X
ejpam-4905	405	4	=	=	PUNCT
ejpam-4905	405	5	∅.	∅.	VERB
ejpam-4905	405	6	thus	thus	ADV
ejpam-4905	405	7	,	,	PUNCT
ejpam-4905	405	8	n2	n2	PROPN
ejpam-4905	405	9	g2	g2	PROPN
ejpam-4905	405	10	[	[	X
ejpam-4905	405	11	a′	a′	X
ejpam-4905	405	12	]	]	X
ejpam-4905	405	13	⊆	⊆	NUM
ejpam-4905	405	14	n2	n2	NOUN
ejpam-4905	405	15	g2	g2	PROPN
ejpam-4905	406	1	[	[	X
ejpam-4905	406	2	b′	b′	X
ejpam-4905	406	3	]	]	X
ejpam-4905	406	4	or	or	CCONJ
ejpam-4905	406	5	n2	n2	PROPN
ejpam-4905	406	6	g2	g2	PROPN
ejpam-4905	407	1	[	[	X
ejpam-4905	407	2	b′	b′	X
ejpam-4905	407	3	]	]	X
ejpam-4905	407	4	⊆	⊆	NUM
ejpam-4905	407	5	n2	n2	NOUN
ejpam-4905	407	6	g2	g2	PROPN
ejpam-4905	408	1	[	[	X
ejpam-4905	408	2	a′	a′	NOUN
ejpam-4905	408	3	]	]	X
ejpam-4905	408	4	,	,	PUNCT
ejpam-4905	408	5	and	and	CCONJ
ejpam-4905	408	6	so	so	ADV
ejpam-4905	408	7	n2	n2	ADJ
ejpam-4905	408	8	s(g)[a	s(g)[a	PROPN
ejpam-4905	408	9	′	′	NOUN
ejpam-4905	408	10	]	]	X
ejpam-4905	409	1	⊆	⊆	NUM
ejpam-4905	409	2	n2	n2	PROPN
ejpam-4905	409	3	s(g)[b	s(g)[b	PROPN
ejpam-4905	409	4	′	′	PROPN
ejpam-4905	409	5	]	]	PUNCT
ejpam-4905	409	6	or	or	CCONJ
ejpam-4905	409	7	n2	n2	PROPN
ejpam-4905	409	8	s(g)[b	s(g)[b	PROPN
ejpam-4905	409	9	′	′	NOUN
ejpam-4905	409	10	]	]	PUNCT
ejpam-4905	409	11	⊆	⊆	NUM
ejpam-4905	409	12	n2	n2	ADJ
ejpam-4905	409	13	s(g)[a	s(g)[a	PROPN
ejpam-4905	409	14	′	′	NOUN
ejpam-4905	409	15	]	]	PUNCT
ejpam-4905	409	16	by	by	ADP
ejpam-4905	409	17	lemma	lemma	PROPN
ejpam-4905	409	18	2	2	NUM
ejpam-4905	409	19	,	,	PUNCT
ejpam-4905	409	20	which	which	PRON
ejpam-4905	409	21	is	be	AUX
ejpam-4905	409	22	a	a	DET
ejpam-4905	409	23	contradiction	contradiction	NOUN
ejpam-4905	409	24	.	.	PUNCT
ejpam-4905	410	1	now	now	ADV
ejpam-4905	410	2	,	,	PUNCT
ejpam-4905	410	3	suppose	suppose	VERB
ejpam-4905	410	4	that	that	SCONJ
ejpam-4905	410	5	a	a	DET
ejpam-4905	410	6	∈	∈	PROPN
ejpam-4905	410	7	tg1	tg1	NOUN
ejpam-4905	410	8	and	and	CCONJ
ejpam-4905	410	9	b	b	PROPN
ejpam-4905	410	10	∈	∈	PROPN
ejpam-4905	410	11	t	t	NOUN
ejpam-4905	410	12	′	′	NUM
ejpam-4905	410	13	g2	g2	PROPN
ejpam-4905	410	14	.	.	PUNCT
ejpam-4905	411	1	then	then	ADV
ejpam-4905	411	2	b′	b′	NUM
ejpam-4905	411	3	∈	∈	NOUN
ejpam-4905	411	4	tg2	tg2	ADJ
ejpam-4905	411	5	.	.	PUNCT
ejpam-4905	412	1	since	since	SCONJ
ejpam-4905	412	2	n2	n2	ADJ
ejpam-4905	412	3	g1	g1	PROPN
ejpam-4905	412	4	[	[	X
ejpam-4905	412	5	a	a	X
ejpam-4905	412	6	]	]	X
ejpam-4905	412	7	⊆	⊆	NUM
ejpam-4905	412	8	n2	n2	ADJ
ejpam-4905	412	9	g1	g1	NOUN
ejpam-4905	413	1	[	[	X
ejpam-4905	413	2	b	b	X
ejpam-4905	413	3	]	]	X
ejpam-4905	413	4	or	or	CCONJ
ejpam-4905	413	5	n2	n2	ADJ
ejpam-4905	413	6	g1	g1	PROPN
ejpam-4905	413	7	[	[	X
ejpam-4905	413	8	b	b	X
ejpam-4905	413	9	]	]	X
ejpam-4905	413	10	⊆	⊆	NUM
ejpam-4905	413	11	n2	n2	ADJ
ejpam-4905	413	12	g1	g1	NOUN
ejpam-4905	413	13	[	[	X
ejpam-4905	413	14	a	a	X
ejpam-4905	413	15	]	]	X
ejpam-4905	413	16	,	,	PUNCT
ejpam-4905	413	17	it	it	PRON
ejpam-4905	413	18	follows	follow	VERB
ejpam-4905	413	19	that	that	DET
ejpam-4905	413	20	n2	n2	ADJ
ejpam-4905	413	21	s(g)[a	s(g)[a	NOUN
ejpam-4905	413	22	]	]	PUNCT
ejpam-4905	413	23	⊆	⊆	NUM
ejpam-4905	413	24	n2	n2	PROPN
ejpam-4905	413	25	s(g)[b	s(g)[b	PROPN
ejpam-4905	413	26	′	′	PROPN
ejpam-4905	413	27	]	]	PUNCT
ejpam-4905	413	28	or	or	CCONJ
ejpam-4905	413	29	n2	n2	PROPN
ejpam-4905	413	30	s(g)[b	s(g)[b	PROPN
ejpam-4905	413	31	′	′	PROPN
ejpam-4905	413	32	]	]	PUNCT
ejpam-4905	413	33	⊆	⊆	NUM
ejpam-4905	413	34	n2	n2	ADJ
ejpam-4905	413	35	s(g)[a	s(g)[a	NOUN
ejpam-4905	413	36	]	]	PUNCT
ejpam-4905	413	37	by	by	ADP
ejpam-4905	413	38	lemma	lemma	PROPN
ejpam-4905	413	39	1	1	NUM
ejpam-4905	413	40	and	and	CCONJ
ejpam-4905	413	41	lemma	lemma	PROPN
ejpam-4905	413	42	2	2	NUM
ejpam-4905	413	43	,	,	PUNCT
ejpam-4905	413	44	a	a	DET
ejpam-4905	413	45	contradiction	contradiction	NOUN
ejpam-4905	413	46	.	.	PUNCT
ejpam-4905	414	1	thus	thus	ADV
ejpam-4905	414	2	,	,	PUNCT
ejpam-4905	414	3	s	s	PART
ejpam-4905	414	4	=	=	NOUN
ejpam-4905	414	5	tg1	tg1	X
ejpam-4905	414	6	∪	∪	PROPN
ejpam-4905	414	7	t	t	PROPN
ejpam-4905	414	8	′	′	NUM
ejpam-4905	415	1	g2	g2	PROPN
ejpam-4905	415	2	is	be	AUX
ejpam-4905	415	3	a	a	DET
ejpam-4905	415	4	j2	j2	NOUN
ejpam-4905	415	5	-	-	PUNCT
ejpam-4905	415	6	set	set	NOUN
ejpam-4905	415	7	in	in	ADP
ejpam-4905	415	8	g1	g1	PROPN
ejpam-4905	415	9	.	.	PUNCT
ejpam-4905	416	1	similarly	similarly	ADV
ejpam-4905	416	2	,	,	PUNCT
ejpam-4905	416	3	t	t	PROPN
ejpam-4905	416	4	′	′	NUM
ejpam-4905	416	5	g1	g1	PROPN
ejpam-4905	416	6	∪	∪	VERB
ejpam-4905	416	7	tg2	tg2	PROPN
ejpam-4905	416	8	is	be	AUX
ejpam-4905	416	9	a	a	DET
ejpam-4905	416	10	j2	j2	NOUN
ejpam-4905	416	11	-	-	PUNCT
ejpam-4905	416	12	set	set	NOUN
ejpam-4905	416	13	in	in	ADP
ejpam-4905	416	14	g2	g2	PROPN
ejpam-4905	416	15	.	.	PUNCT
ejpam-4905	417	1	thus	thus	ADV
ejpam-4905	417	2	,	,	PUNCT
ejpam-4905	417	3	(	(	PUNCT
ejpam-4905	417	4	iii	iii	NOUN
ejpam-4905	417	5	)	)	PUNCT
ejpam-4905	417	6	holds	hold	VERB
ejpam-4905	417	7	.	.	PUNCT
ejpam-4905	418	1	conversely	conversely	ADV
ejpam-4905	418	2	,	,	PUNCT
ejpam-4905	418	3	if	if	SCONJ
ejpam-4905	418	4	(	(	PUNCT
ejpam-4905	418	5	i	i	NOUN
ejpam-4905	418	6	)	)	PUNCT
ejpam-4905	418	7	or	or	CCONJ
ejpam-4905	418	8	(	(	PUNCT
ejpam-4905	418	9	ii	ii	NOUN
ejpam-4905	418	10	)	)	PUNCT
ejpam-4905	418	11	holds	hold	VERB
ejpam-4905	418	12	,	,	PUNCT
ejpam-4905	418	13	then	then	ADV
ejpam-4905	418	14	the	the	DET
ejpam-4905	418	15	assertion	assertion	NOUN
ejpam-4905	418	16	follows	follow	VERB
ejpam-4905	418	17	.	.	PUNCT
ejpam-4905	419	1	assume	assume	VERB
ejpam-4905	419	2	that	that	SCONJ
ejpam-4905	419	3	(	(	PUNCT
ejpam-4905	419	4	iii	iii	NOUN
ejpam-4905	419	5	)	)	PUNCT
ejpam-4905	419	6	holds	hold	VERB
ejpam-4905	419	7	.	.	PUNCT
ejpam-4905	420	1	let	let	VERB
ejpam-4905	420	2	x	x	PRON
ejpam-4905	420	3	,	,	PUNCT
ejpam-4905	420	4	y	y	PROPN
ejpam-4905	420	5	∈	∈	PROPN
ejpam-4905	420	6	t	t	NOUN
ejpam-4905	420	7	=	=	PUNCT
ejpam-4905	420	8	tg1	tg1	X
ejpam-4905	420	9	∪	∪	X
ejpam-4905	420	10	tg2	tg2	ADJ
ejpam-4905	420	11	.	.	PUNCT
ejpam-4905	421	1	if	if	SCONJ
ejpam-4905	421	2	x	x	X
ejpam-4905	421	3	,	,	PUNCT
ejpam-4905	421	4	y	y	PROPN
ejpam-4905	421	5	∈	∈	PROPN
ejpam-4905	421	6	tg1	tg1	VERB
ejpam-4905	421	7	⊆	⊆	NUM
ejpam-4905	421	8	tg1	tg1	X
ejpam-4905	421	9	∪	∪	ADP
ejpam-4905	421	10	t	t	PROPN
ejpam-4905	421	11	′	′	NUM
ejpam-4905	421	12	g2	g2	PROPN
ejpam-4905	421	13	,	,	PUNCT
ejpam-4905	421	14	then	then	ADV
ejpam-4905	421	15	n2	n2	PROPN
ejpam-4905	421	16	g1	g1	PROPN
ejpam-4905	421	17	[	[	X
ejpam-4905	421	18	x	x	X
ejpam-4905	421	19	]	]	X
ejpam-4905	421	20	\	\	PROPN
ejpam-4905	421	21	n2	n2	ADJ
ejpam-4905	421	22	g1	g1	PROPN
ejpam-4905	422	1	[	[	X
ejpam-4905	422	2	y	y	X
ejpam-4905	422	3	]	]	X
ejpam-4905	422	4	̸=	̸=	PROPN
ejpam-4905	422	5	∅	∅	NOUN
ejpam-4905	422	6	and	and	CCONJ
ejpam-4905	422	7	n2	n2	ADJ
ejpam-4905	422	8	g1	g1	PROPN
ejpam-4905	422	9	[	[	X
ejpam-4905	422	10	y	y	X
ejpam-4905	422	11	]	]	PUNCT
ejpam-4905	422	12	\	\	PROPN
ejpam-4905	422	13	n2	n2	ADJ
ejpam-4905	422	14	g1	g1	PROPN
ejpam-4905	422	15	[	[	X
ejpam-4905	422	16	x	x	X
ejpam-4905	422	17	]	]	X
ejpam-4905	422	18	̸=	̸=	PROPN
ejpam-4905	422	19	∅	∅	NOUN
ejpam-4905	422	20	by	by	ADP
ejpam-4905	422	21	assumption	assumption	NOUN
ejpam-4905	422	22	.	.	PUNCT
ejpam-4905	423	1	this	this	PRON
ejpam-4905	423	2	means	mean	VERB
ejpam-4905	423	3	that	that	SCONJ
ejpam-4905	423	4	n2	n2	ADJ
ejpam-4905	423	5	g1	g1	PROPN
ejpam-4905	423	6	[	[	X
ejpam-4905	423	7	x	x	X
ejpam-4905	423	8	]	]	X
ejpam-4905	423	9	⊈	⊈	PROPN
ejpam-4905	423	10	n2	n2	ADJ
ejpam-4905	423	11	g1	g1	PROPN
ejpam-4905	424	1	[	[	X
ejpam-4905	424	2	y	y	X
ejpam-4905	424	3	]	]	X
ejpam-4905	424	4	and	and	CCONJ
ejpam-4905	424	5	n2	n2	PROPN
ejpam-4905	424	6	g1	g1	PROPN
ejpam-4905	424	7	[	[	X
ejpam-4905	424	8	y	y	X
ejpam-4905	424	9	]	]	X
ejpam-4905	424	10	⊈	⊈	PROPN
ejpam-4905	424	11	n2	n2	ADJ
ejpam-4905	424	12	g1	g1	PROPN
ejpam-4905	424	13	[	[	X
ejpam-4905	424	14	x	x	X
ejpam-4905	424	15	]	]	X
ejpam-4905	424	16	.	.	PUNCT
ejpam-4905	425	1	thus	thus	ADV
ejpam-4905	425	2	,	,	PUNCT
ejpam-4905	425	3	n2	n2	ADJ
ejpam-4905	425	4	s(g)[x	s(g)[x	NOUN
ejpam-4905	425	5	]	]	PUNCT
ejpam-4905	425	6	⊈	⊈	PROPN
ejpam-4905	425	7	n2	n2	NOUN
ejpam-4905	425	8	s(g)[y	s(g)[y	NOUN
ejpam-4905	425	9	]	]	PUNCT
ejpam-4905	425	10	and	and	CCONJ
ejpam-4905	425	11	n2	n2	PROPN
ejpam-4905	425	12	s(g)[y	s(g)[y	PROPN
ejpam-4905	425	13	]	]	PUNCT
ejpam-4905	425	14	⊈	⊈	PROPN
ejpam-4905	425	15	n2	n2	ADJ
ejpam-4905	425	16	s(g)[x	s(g)[x	NOUN
ejpam-4905	425	17	]	]	PUNCT
ejpam-4905	425	18	,	,	PUNCT
ejpam-4905	425	19	and	and	CCONJ
ejpam-4905	425	20	we	we	PRON
ejpam-4905	425	21	are	be	AUX
ejpam-4905	425	22	done	do	VERB
ejpam-4905	425	23	.	.	PUNCT
ejpam-4905	426	1	if	if	SCONJ
ejpam-4905	426	2	x	x	X
ejpam-4905	426	3	,	,	PUNCT
ejpam-4905	426	4	y	y	PROPN
ejpam-4905	426	5	∈	∈	PROPN
ejpam-4905	426	6	tg2	tg2	ADV
ejpam-4905	426	7	,	,	PUNCT
ejpam-4905	426	8	then	then	ADV
ejpam-4905	426	9	x′	x′	NUM
ejpam-4905	426	10	,	,	PUNCT
ejpam-4905	426	11	y′	y′	NOUN
ejpam-4905	426	12	∈	∈	PROPN
ejpam-4905	426	13	t	t	NOUN
ejpam-4905	426	14	′	′	NUM
ejpam-4905	426	15	g2	g2	PROPN
ejpam-4905	426	16	⊆	⊆	NUM
ejpam-4905	426	17	tg1	tg1	X
ejpam-4905	426	18	∪	∪	ADP
ejpam-4905	426	19	t	t	PROPN
ejpam-4905	426	20	′	′	NUM
ejpam-4905	426	21	g2	g2	PROPN
ejpam-4905	426	22	.	.	PUNCT
ejpam-4905	427	1	since	since	SCONJ
ejpam-4905	427	2	tg1	tg1	NOUN
ejpam-4905	427	3	∪	∪	ADP
ejpam-4905	427	4	t	t	PROPN
ejpam-4905	427	5	′	′	NUM
ejpam-4905	427	6	g2	g2	PROPN
ejpam-4905	427	7	is	be	AUX
ejpam-4905	427	8	a	a	DET
ejpam-4905	427	9	j2	j2	NOUN
ejpam-4905	427	10	-	-	PUNCT
ejpam-4905	427	11	set	set	NOUN
ejpam-4905	427	12	in	in	ADP
ejpam-4905	427	13	g1	g1	PROPN
ejpam-4905	427	14	,	,	PUNCT
ejpam-4905	427	15	we	we	PRON
ejpam-4905	427	16	have	have	VERB
ejpam-4905	427	17	n2	n2	ADJ
ejpam-4905	427	18	g1	g1	PROPN
ejpam-4905	427	19	[	[	X
ejpam-4905	427	20	x′	x′	X
ejpam-4905	427	21	]	]	X
ejpam-4905	427	22	⊈	⊈	PROPN
ejpam-4905	427	23	n2	n2	ADJ
ejpam-4905	427	24	g1	g1	PROPN
ejpam-4905	427	25	[	[	X
ejpam-4905	427	26	y′	y′	X
ejpam-4905	427	27	]	]	PUNCT
ejpam-4905	427	28	and	and	CCONJ
ejpam-4905	427	29	n2	n2	PROPN
ejpam-4905	427	30	g1	g1	PROPN
ejpam-4905	427	31	[	[	X
ejpam-4905	427	32	y′	y′	NUM
ejpam-4905	427	33	]	]	PUNCT
ejpam-4905	427	34	⊈	⊈	PROPN
ejpam-4905	427	35	n2	n2	ADJ
ejpam-4905	427	36	g1	g1	PROPN
ejpam-4905	427	37	[	[	X
ejpam-4905	427	38	x′	x′	X
ejpam-4905	427	39	]	]	X
ejpam-4905	427	40	.	.	PUNCT
ejpam-4905	428	1	thus	thus	ADV
ejpam-4905	428	2	,	,	PUNCT
ejpam-4905	428	3	by	by	ADP
ejpam-4905	428	4	lemma	lemma	PROPN
ejpam-4905	428	5	1	1	NUM
ejpam-4905	428	6	,	,	PUNCT
ejpam-4905	428	7	n2	n2	ADJ
ejpam-4905	428	8	s(g)[x	s(g)[x	NOUN
ejpam-4905	428	9	]	]	PUNCT
ejpam-4905	428	10	⊈	⊈	PROPN
ejpam-4905	428	11	n2	n2	NOUN
ejpam-4905	428	12	s(g)[y	s(g)[y	NOUN
ejpam-4905	428	13	]	]	PUNCT
ejpam-4905	428	14	and	and	CCONJ
ejpam-4905	428	15	n2	n2	PROPN
ejpam-4905	428	16	s(g)[y	s(g)[y	PROPN
ejpam-4905	428	17	]	]	PUNCT
ejpam-4905	428	18	⊈	⊈	PROPN
ejpam-4905	428	19	n2	n2	ADJ
ejpam-4905	428	20	s(g)[x	s(g)[x	NOUN
ejpam-4905	428	21	]	]	PUNCT
ejpam-4905	428	22	.	.	PUNCT
ejpam-4905	429	1	now	now	ADV
ejpam-4905	429	2	,	,	PUNCT
ejpam-4905	429	3	assume	assume	VERB
ejpam-4905	429	4	that	that	SCONJ
ejpam-4905	429	5	x	x	PUNCT
ejpam-4905	429	6	∈	∈	PROPN
ejpam-4905	429	7	tg1	tg1	NOUN
ejpam-4905	429	8	and	and	CCONJ
ejpam-4905	429	9	y	y	PROPN
ejpam-4905	429	10	∈	∈	PROPN
ejpam-4905	429	11	tg2	tg2	ADJ
ejpam-4905	429	12	.	.	PUNCT
ejpam-4905	430	1	then	then	ADV
ejpam-4905	430	2	y′	y′	NUM
ejpam-4905	430	3	∈	∈	PROPN
ejpam-4905	430	4	t	t	NOUN
ejpam-4905	430	5	′	′	NUM
ejpam-4905	430	6	g2	g2	PROPN
ejpam-4905	430	7	,	,	PUNCT
ejpam-4905	430	8	and	and	CCONJ
ejpam-4905	430	9	so	so	ADV
ejpam-4905	430	10	x	x	NOUN
ejpam-4905	430	11	,	,	PUNCT
ejpam-4905	430	12	y′	y′	NOUN
ejpam-4905	430	13	∈	∈	PROPN
ejpam-4905	430	14	tg1	tg1	X
ejpam-4905	430	15	∪	∪	PROPN
ejpam-4905	430	16	t	t	PROPN
ejpam-4905	430	17	′	′	NUM
ejpam-4905	430	18	g2	g2	PROPN
ejpam-4905	430	19	.	.	PUNCT
ejpam-4905	431	1	since	since	SCONJ
ejpam-4905	431	2	tg1	tg1	NOUN
ejpam-4905	431	3	∪	∪	ADP
ejpam-4905	431	4	t	t	PROPN
ejpam-4905	431	5	′	′	NUM
ejpam-4905	431	6	g2	g2	PROPN
ejpam-4905	431	7	is	be	AUX
ejpam-4905	431	8	a	a	DET
ejpam-4905	431	9	j2	j2	NOUN
ejpam-4905	431	10	-	-	PUNCT
ejpam-4905	431	11	set	set	NOUN
ejpam-4905	431	12	in	in	ADP
ejpam-4905	431	13	g1	g1	PROPN
ejpam-4905	431	14	,	,	PUNCT
ejpam-4905	431	15	we	we	PRON
ejpam-4905	431	16	have	have	VERB
ejpam-4905	431	17	n2	n2	ADJ
ejpam-4905	431	18	g1	g1	NOUN
ejpam-4905	431	19	[	[	X
ejpam-4905	431	20	x	x	X
ejpam-4905	431	21	]	]	X
ejpam-4905	431	22	⊈	⊈	PROPN
ejpam-4905	431	23	n2	n2	ADJ
ejpam-4905	431	24	g1	g1	PROPN
ejpam-4905	431	25	[	[	X
ejpam-4905	431	26	y′	y′	X
ejpam-4905	431	27	]	]	PUNCT
ejpam-4905	431	28	and	and	CCONJ
ejpam-4905	431	29	n2	n2	PROPN
ejpam-4905	431	30	g1	g1	PROPN
ejpam-4905	431	31	[	[	X
ejpam-4905	431	32	y′	y′	NUM
ejpam-4905	431	33	]	]	PUNCT
ejpam-4905	431	34	⊈	⊈	PROPN
ejpam-4905	431	35	n2	n2	ADJ
ejpam-4905	431	36	g1	g1	PROPN
ejpam-4905	432	1	[	[	X
ejpam-4905	432	2	x	x	X
ejpam-4905	432	3	]	]	X
ejpam-4905	432	4	.	.	PUNCT
ejpam-4905	433	1	thus	thus	ADV
ejpam-4905	433	2	,	,	PUNCT
ejpam-4905	433	3	by	by	ADP
ejpam-4905	433	4	lemma	lemma	PROPN
ejpam-4905	433	5	1	1	NUM
ejpam-4905	433	6	,	,	PUNCT
ejpam-4905	433	7	n2	n2	ADJ
ejpam-4905	433	8	s(g)[x	s(g)[x	NOUN
ejpam-4905	433	9	]	]	PUNCT
ejpam-4905	433	10	⊈	⊈	PROPN
ejpam-4905	433	11	n2	n2	NOUN
ejpam-4905	433	12	s(g)[y	s(g)[y	NOUN
ejpam-4905	433	13	]	]	PUNCT
ejpam-4905	433	14	and	and	CCONJ
ejpam-4905	433	15	n2	n2	PROPN
ejpam-4905	433	16	s(g)[y	s(g)[y	PROPN
ejpam-4905	433	17	]	]	PUNCT
ejpam-4905	433	18	⊈	⊈	PROPN
ejpam-4905	433	19	n2	n2	ADJ
ejpam-4905	433	20	s(g)[x	s(g)[x	NOUN
ejpam-4905	433	21	]	]	PUNCT
ejpam-4905	433	22	.	.	PUNCT
ejpam-4905	434	1	since	since	SCONJ
ejpam-4905	434	2	x	x	X
ejpam-4905	434	3	,	,	PUNCT
ejpam-4905	434	4	y	y	PROPN
ejpam-4905	434	5	are	be	AUX
ejpam-4905	434	6	arbitrary	arbitrary	ADJ
ejpam-4905	434	7	,	,	PUNCT
ejpam-4905	434	8	it	it	PRON
ejpam-4905	434	9	follows	follow	VERB
ejpam-4905	434	10	that	that	SCONJ
ejpam-4905	434	11	t	t	PROPN
ejpam-4905	434	12	is	be	AUX
ejpam-4905	434	13	a	a	DET
ejpam-4905	434	14	j2	j2	NOUN
ejpam-4905	434	15	-	-	PUNCT
ejpam-4905	434	16	set	set	NOUN
ejpam-4905	434	17	in	in	ADP
ejpam-4905	434	18	s(g	s(g	PROPN
ejpam-4905	434	19	)	)	PUNCT
ejpam-4905	434	20	.	.	PUNCT
ejpam-4905	435	1	theorem	theorem	VERB
ejpam-4905	435	2	10	10	NUM
ejpam-4905	435	3	.	.	PUNCT
ejpam-4905	436	1	[	[	X
ejpam-4905	436	2	5	5	X
ejpam-4905	436	3	]	]	PUNCT
ejpam-4905	436	4	let	let	VERB
ejpam-4905	436	5	g	g	PRON
ejpam-4905	436	6	be	be	AUX
ejpam-4905	436	7	a	a	DET
ejpam-4905	436	8	non	non	ADJ
ejpam-4905	436	9	-	-	ADJ
ejpam-4905	436	10	trivial	trivial	ADJ
ejpam-4905	436	11	connected	connected	ADJ
ejpam-4905	436	12	graph	graph	NOUN
ejpam-4905	436	13	.	.	PUNCT
ejpam-4905	437	1	then	then	ADV
ejpam-4905	437	2	s	s	VERB
ejpam-4905	437	3	is	be	AUX
ejpam-4905	437	4	a	a	DET
ejpam-4905	437	5	hop	hop	NOUN
ejpam-4905	437	6	dominating	dominating	NOUN
ejpam-4905	437	7	set	set	NOUN
ejpam-4905	437	8	in	in	ADP
ejpam-4905	437	9	s(g	s(g	PROPN
ejpam-4905	437	10	)	)	PUNCT
ejpam-4905	437	11	if	if	SCONJ
ejpam-4905	437	12	and	and	CCONJ
ejpam-4905	437	13	only	only	ADV
ejpam-4905	437	14	if	if	SCONJ
ejpam-4905	437	15	one	one	NUM
ejpam-4905	437	16	of	of	ADP
ejpam-4905	437	17	the	the	DET
ejpam-4905	437	18	following	follow	VERB
ejpam-4905	437	19	conditions	condition	NOUN
ejpam-4905	437	20	holds	hold	VERB
ejpam-4905	437	21	:	:	PUNCT
ejpam-4905	437	22	(	(	PUNCT
ejpam-4905	437	23	i	i	NOUN
ejpam-4905	437	24	)	)	PUNCT
ejpam-4905	437	25	s	s	VERB
ejpam-4905	437	26	is	be	AUX
ejpam-4905	437	27	a	a	DET
ejpam-4905	437	28	hop	hop	NOUN
ejpam-4905	437	29	dominating	dominating	NOUN
ejpam-4905	437	30	set	set	VERB
ejpam-4905	437	31	in	in	ADP
ejpam-4905	437	32	g1	g1	PROPN
ejpam-4905	437	33	.	.	PUNCT
ejpam-4905	438	1	(	(	PUNCT
ejpam-4905	438	2	ii	ii	X
ejpam-4905	438	3	)	)	PUNCT
ejpam-4905	438	4	s	s	VERB
ejpam-4905	438	5	is	be	AUX
ejpam-4905	438	6	a	a	DET
ejpam-4905	438	7	hop	hop	NOUN
ejpam-4905	438	8	dominating	dominating	NOUN
ejpam-4905	438	9	set	set	VERB
ejpam-4905	438	10	in	in	ADP
ejpam-4905	438	11	g2	g2	PROPN
ejpam-4905	438	12	.	.	PUNCT
ejpam-4905	439	1	(	(	PUNCT
ejpam-4905	439	2	iii	iii	X
ejpam-4905	439	3	)	)	PUNCT
ejpam-4905	439	4	s	s	PART
ejpam-4905	439	5	=	=	NOUN
ejpam-4905	439	6	sg1	sg1	NOUN
ejpam-4905	439	7	∪	∪	VERB
ejpam-4905	439	8	sg2	sg2	PROPN
ejpam-4905	439	9	such	such	ADJ
ejpam-4905	439	10	that	that	SCONJ
ejpam-4905	439	11	sg1	sg1	NOUN
ejpam-4905	439	12	∪	∪	ADP
ejpam-4905	439	13	s′	s′	ADJ
ejpam-4905	439	14	g2	g2	PROPN
ejpam-4905	439	15	and	and	CCONJ
ejpam-4905	439	16	s′	s′	ADJ
ejpam-4905	439	17	g1	g1	PROPN
ejpam-4905	439	18	∪	∪	ADP
ejpam-4905	439	19	sg2	sg2	PROPN
ejpam-4905	439	20	are	be	AUX
ejpam-4905	439	21	hop	hop	NOUN
ejpam-4905	439	22	dominating	dominating	NOUN
ejpam-4905	439	23	sets	set	NOUN
ejpam-4905	439	24	in	in	ADP
ejpam-4905	439	25	g1	g1	PROPN
ejpam-4905	439	26	and	and	CCONJ
ejpam-4905	439	27	g2	g2	PROPN
ejpam-4905	439	28	,	,	PUNCT
ejpam-4905	439	29	respectively	respectively	ADV
ejpam-4905	439	30	,	,	PUNCT
ejpam-4905	439	31	where	where	SCONJ
ejpam-4905	439	32	s′	s′	ADJ
ejpam-4905	439	33	g2	g2	PROPN
ejpam-4905	439	34	=	=	PRON
ejpam-4905	439	35	{	{	PUNCT
ejpam-4905	439	36	a	a	PRON
ejpam-4905	439	37	∈	∈	PROPN
ejpam-4905	439	38	v	v	NOUN
ejpam-4905	439	39	(	(	PUNCT
ejpam-4905	439	40	g1	g1	PROPN
ejpam-4905	439	41	)	)	PUNCT
ejpam-4905	439	42	:	:	PUNCT
ejpam-4905	439	43	a	a	DET
ejpam-4905	439	44	′	′	NUM
ejpam-4905	439	45	∈	∈	PROPN
ejpam-4905	439	46	sg2	sg2	PROPN
ejpam-4905	439	47	}	}	PUNCT
ejpam-4905	439	48	and	and	CCONJ
ejpam-4905	439	49	s′	s′	ADJ
ejpam-4905	439	50	g1	g1	PROPN
ejpam-4905	439	51	=	=	SYM
ejpam-4905	439	52	{	{	PUNCT
ejpam-4905	439	53	b	b	PROPN
ejpam-4905	439	54	∈	∈	PROPN
ejpam-4905	439	55	v	v	NOUN
ejpam-4905	439	56	(	(	PUNCT
ejpam-4905	439	57	g2	g2	PROPN
ejpam-4905	439	58	)	)	PUNCT
ejpam-4905	439	59	:	:	PUNCT
ejpam-4905	440	1	b	b	X
ejpam-4905	440	2	′	′	NUM
ejpam-4905	440	3	∈	∈	PROPN
ejpam-4905	440	4	sg1	sg1	NOUN
ejpam-4905	440	5	}	}	PUNCT
ejpam-4905	440	6	.	.	PUNCT
ejpam-4905	441	1	theorem	theorem	NOUN
ejpam-4905	441	2	11	11	NUM
ejpam-4905	441	3	.	.	PUNCT
ejpam-4905	442	1	let	let	VERB
ejpam-4905	442	2	g	g	PRON
ejpam-4905	442	3	be	be	AUX
ejpam-4905	442	4	a	a	DET
ejpam-4905	442	5	connected	connected	ADJ
ejpam-4905	442	6	non	non	ADJ
ejpam-4905	442	7	-	-	ADJ
ejpam-4905	442	8	trivial	trivial	ADJ
ejpam-4905	442	9	graph	graph	NOUN
ejpam-4905	442	10	.	.	PUNCT
ejpam-4905	443	1	then	then	ADV
ejpam-4905	443	2	t	t	PROPN
ejpam-4905	443	3	⊆	⊆	NUM
ejpam-4905	443	4	v	v	NOUN
ejpam-4905	443	5	(	(	PUNCT
ejpam-4905	443	6	s(g	s(g	PROPN
ejpam-4905	443	7	)	)	PUNCT
ejpam-4905	443	8	)	)	PUNCT
ejpam-4905	443	9	is	be	AUX
ejpam-4905	443	10	a	a	DET
ejpam-4905	443	11	j2	j2	PROPN
ejpam-4905	443	12	-	-	PUNCT
ejpam-4905	443	13	hop	hop	NOUN
ejpam-4905	443	14	dominating	dominating	NOUN
ejpam-4905	443	15	set	set	NOUN
ejpam-4905	443	16	in	in	ADP
ejpam-4905	443	17	s(g	s(g	PROPN
ejpam-4905	443	18	)	)	PUNCT
ejpam-4905	443	19	if	if	SCONJ
ejpam-4905	443	20	and	and	CCONJ
ejpam-4905	443	21	only	only	ADV
ejpam-4905	443	22	if	if	SCONJ
ejpam-4905	443	23	t	t	PROPN
ejpam-4905	443	24	satisfies	satisfy	VERB
ejpam-4905	443	25	one	one	NUM
ejpam-4905	443	26	of	of	ADP
ejpam-4905	443	27	the	the	DET
ejpam-4905	443	28	following	following	ADJ
ejpam-4905	443	29	conditions	condition	NOUN
ejpam-4905	443	30	:	:	PUNCT
ejpam-4905	443	31	(	(	PUNCT
ejpam-4905	443	32	i	i	NOUN
ejpam-4905	443	33	)	)	PUNCT
ejpam-4905	443	34	t	t	PROPN
ejpam-4905	443	35	is	be	AUX
ejpam-4905	443	36	a	a	DET
ejpam-4905	443	37	j2	j2	PROPN
ejpam-4905	443	38	-	-	PUNCT
ejpam-4905	443	39	hop	hop	NOUN
ejpam-4905	443	40	dominating	dominating	NOUN
ejpam-4905	443	41	set	set	VERB
ejpam-4905	443	42	in	in	ADP
ejpam-4905	443	43	g1	g1	PROPN
ejpam-4905	443	44	.	.	PUNCT
ejpam-4905	444	1	(	(	PUNCT
ejpam-4905	444	2	ii	ii	NOUN
ejpam-4905	444	3	)	)	PUNCT
ejpam-4905	444	4	t	t	PROPN
ejpam-4905	444	5	is	be	AUX
ejpam-4905	444	6	a	a	DET
ejpam-4905	444	7	j2	j2	PROPN
ejpam-4905	444	8	-	-	PUNCT
ejpam-4905	444	9	hop	hop	NOUN
ejpam-4905	444	10	dominating	dominating	NOUN
ejpam-4905	444	11	set	set	NOUN
ejpam-4905	444	12	in	in	ADP
ejpam-4905	444	13	g2	g2	PROPN
ejpam-4905	444	14	.	.	PUNCT
ejpam-4905	445	1	j.	j.	PROPN
ejpam-4905	445	2	hassan	hassan	PROPN
ejpam-4905	445	3	,	,	PUNCT
ejpam-4905	445	4	a.	a.	PROPN
ejpam-4905	445	5	bakkang	bakkang	PROPN
ejpam-4905	445	6	,	,	PUNCT
ejpam-4905	445	7	a.	a.	NOUN
ejpam-4905	445	8	sappari	sappari	PROPN
ejpam-4905	445	9	/	/	SYM
ejpam-4905	445	10	eur	eur	PROPN
ejpam-4905	445	11	.	.	PUNCT
ejpam-4905	446	1	j.	j.	PROPN
ejpam-4905	446	2	pure	pure	PROPN
ejpam-4905	446	3	appl	appl	PROPN
ejpam-4905	446	4	.	.	PROPN
ejpam-4905	446	5	math	math	PROPN
ejpam-4905	446	6	,	,	PUNCT
ejpam-4905	446	7	16	16	NUM
ejpam-4905	446	8	(	(	PUNCT
ejpam-4905	446	9	4	4	NUM
ejpam-4905	446	10	)	)	PUNCT
ejpam-4905	446	11	(	(	PUNCT
ejpam-4905	446	12	2023	2023	NUM
ejpam-4905	446	13	)	)	PUNCT
ejpam-4905	446	14	,	,	PUNCT
ejpam-4905	446	15	2118	2118	NUM
ejpam-4905	446	16	-	-	SYM
ejpam-4905	446	17	2131	2131	NUM
ejpam-4905	446	18	2129	2129	NUM
ejpam-4905	446	19	(	(	PUNCT
ejpam-4905	446	20	iii	iii	NOUN
ejpam-4905	446	21	)	)	PUNCT
ejpam-4905	446	22	t	t	NOUN
ejpam-4905	446	23	=	=	SYM
ejpam-4905	446	24	tg	tg	PROPN
ejpam-4905	446	25	∪th	∪th	PROPN
ejpam-4905	446	26	,	,	PUNCT
ejpam-4905	446	27	where	where	SCONJ
ejpam-4905	446	28	tg1	tg1	VERB
ejpam-4905	446	29	∪t	∪t	NUM
ejpam-4905	446	30	′	′	NUM
ejpam-4905	446	31	g2	g2	PROPN
ejpam-4905	446	32	and	and	CCONJ
ejpam-4905	446	33	t	t	PROPN
ejpam-4905	446	34	′	′	PROPN
ejpam-4905	446	35	g1	g1	PROPN
ejpam-4905	446	36	∪tg2	∪tg2	PROPN
ejpam-4905	446	37	are	be	AUX
ejpam-4905	446	38	j2	j2	NOUN
ejpam-4905	446	39	-	-	PUNCT
ejpam-4905	446	40	hop	hop	NOUN
ejpam-4905	446	41	dominating	dominating	NOUN
ejpam-4905	446	42	sets	set	NOUN
ejpam-4905	446	43	in	in	ADP
ejpam-4905	446	44	g1	g1	PROPN
ejpam-4905	446	45	and	and	CCONJ
ejpam-4905	446	46	g2	g2	PROPN
ejpam-4905	446	47	,	,	PUNCT
ejpam-4905	446	48	respectively	respectively	ADV
ejpam-4905	446	49	,	,	PUNCT
ejpam-4905	446	50	where	where	SCONJ
ejpam-4905	446	51	t	t	PROPN
ejpam-4905	446	52	′	′	NUM
ejpam-4905	447	1	g2	g2	PROPN
ejpam-4905	448	1	=	=	PRON
ejpam-4905	449	1	{	{	PUNCT
ejpam-4905	449	2	x	x	PROPN
ejpam-4905	449	3	∈	∈	PROPN
ejpam-4905	449	4	v	v	NOUN
ejpam-4905	449	5	(	(	PUNCT
ejpam-4905	449	6	g1	g1	PROPN
ejpam-4905	449	7	)	)	PUNCT
ejpam-4905	449	8	:	:	PUNCT
ejpam-4905	449	9	x	x	X
ejpam-4905	449	10	′	′	NUM
ejpam-4905	449	11	∈	∈	ADJ
ejpam-4905	449	12	tg2	tg2	NOUN
ejpam-4905	449	13	}	}	PUNCT
ejpam-4905	449	14	and	and	CCONJ
ejpam-4905	449	15	t	t	PROPN
ejpam-4905	449	16	′	′	NUM
ejpam-4905	449	17	g1	g1	PROPN
ejpam-4905	449	18	=	=	PRON
ejpam-4905	449	19	{	{	PUNCT
ejpam-4905	449	20	y	y	PROPN
ejpam-4905	449	21	∈	∈	PROPN
ejpam-4905	449	22	v	v	PROPN
ejpam-4905	449	23	(	(	PUNCT
ejpam-4905	449	24	g2	g2	PROPN
ejpam-4905	449	25	)	)	PUNCT
ejpam-4905	449	26	:	:	PUNCT
ejpam-4905	450	1	y	y	X
ejpam-4905	450	2	′	′	NUM
ejpam-4905	450	3	∈	∈	PROPN
ejpam-4905	450	4	tg1	tg1	NOUN
ejpam-4905	450	5	}	}	PUNCT
ejpam-4905	450	6	.	.	PUNCT
ejpam-4905	451	1	proof	proof	NOUN
ejpam-4905	451	2	.	.	PUNCT
ejpam-4905	452	1	suppose	suppose	VERB
ejpam-4905	452	2	that	that	SCONJ
ejpam-4905	452	3	t	t	PROPN
ejpam-4905	452	4	is	be	AUX
ejpam-4905	452	5	a	a	DET
ejpam-4905	452	6	j2	j2	PROPN
ejpam-4905	452	7	-	-	PUNCT
ejpam-4905	452	8	hop	hop	NOUN
ejpam-4905	452	9	dominating	dominating	NOUN
ejpam-4905	452	10	set	set	NOUN
ejpam-4905	452	11	in	in	ADP
ejpam-4905	452	12	s(g	s(g	PROPN
ejpam-4905	452	13	)	)	PUNCT
ejpam-4905	452	14	.	.	PUNCT
ejpam-4905	453	1	let	let	VERB
ejpam-4905	453	2	tg1	tg1	NOUN
ejpam-4905	453	3	=	=	SYM
ejpam-4905	453	4	t	t	PROPN
ejpam-4905	453	5	∩	∩	ADJ
ejpam-4905	453	6	v	v	X
ejpam-4905	453	7	(	(	PUNCT
ejpam-4905	453	8	g1	g1	PROPN
ejpam-4905	453	9	)	)	PUNCT
ejpam-4905	453	10	and	and	CCONJ
ejpam-4905	453	11	tg2	tg2	NOUN
ejpam-4905	453	12	=	=	PUNCT
ejpam-4905	454	1	t	t	NOUN
ejpam-4905	454	2	∩v	∩v	NOUN
ejpam-4905	454	3	(	(	PUNCT
ejpam-4905	454	4	g2	g2	PROPN
ejpam-4905	454	5	)	)	PUNCT
ejpam-4905	454	6	.	.	PUNCT
ejpam-4905	455	1	if	if	SCONJ
ejpam-4905	455	2	tg2	tg2	NOUN
ejpam-4905	455	3	=	=	SYM
ejpam-4905	455	4	∅	∅	NOUN
ejpam-4905	455	5	,	,	PUNCT
ejpam-4905	455	6	then	then	ADV
ejpam-4905	455	7	t	t	PROPN
ejpam-4905	455	8	=	=	PUNCT
ejpam-4905	455	9	sg1	sg1	NOUN
ejpam-4905	455	10	is	be	AUX
ejpam-4905	455	11	a	a	DET
ejpam-4905	455	12	j2	j2	PROPN
ejpam-4905	455	13	-	-	PUNCT
ejpam-4905	455	14	hop	hop	NOUN
ejpam-4905	455	15	dominating	dominating	NOUN
ejpam-4905	455	16	set	set	NOUN
ejpam-4905	455	17	of	of	ADP
ejpam-4905	455	18	g1	g1	PROPN
ejpam-4905	455	19	.	.	PUNCT
ejpam-4905	456	1	if	if	SCONJ
ejpam-4905	456	2	tg1	tg1	NOUN
ejpam-4905	456	3	=	=	SYM
ejpam-4905	456	4	∅	∅	NOUN
ejpam-4905	456	5	,	,	PUNCT
ejpam-4905	456	6	then	then	ADV
ejpam-4905	456	7	t	t	PROPN
ejpam-4905	456	8	=	=	PUNCT
ejpam-4905	456	9	tg2	tg2	PROPN
ejpam-4905	456	10	is	be	AUX
ejpam-4905	456	11	a	a	DET
ejpam-4905	456	12	j2	j2	PROPN
ejpam-4905	456	13	-	-	PUNCT
ejpam-4905	456	14	hop	hop	NOUN
ejpam-4905	456	15	dominating	dominating	NOUN
ejpam-4905	456	16	set	set	NOUN
ejpam-4905	456	17	of	of	ADP
ejpam-4905	456	18	g2	g2	PROPN
ejpam-4905	456	19	,	,	PUNCT
ejpam-4905	456	20	showing	show	VERB
ejpam-4905	456	21	that	that	SCONJ
ejpam-4905	456	22	(	(	PUNCT
ejpam-4905	456	23	i	i	NOUN
ejpam-4905	456	24	)	)	PUNCT
ejpam-4905	456	25	or	or	CCONJ
ejpam-4905	456	26	(	(	PUNCT
ejpam-4905	456	27	ii	ii	NOUN
ejpam-4905	456	28	)	)	PUNCT
ejpam-4905	456	29	holds	hold	VERB
ejpam-4905	456	30	.	.	PUNCT
ejpam-4905	457	1	now	now	ADV
ejpam-4905	457	2	,	,	PUNCT
ejpam-4905	457	3	since	since	SCONJ
ejpam-4905	457	4	t	t	PROPN
ejpam-4905	457	5	is	be	AUX
ejpam-4905	457	6	a	a	DET
ejpam-4905	457	7	j2	j2	NOUN
ejpam-4905	457	8	-	-	PUNCT
ejpam-4905	457	9	set	set	NOUN
ejpam-4905	457	10	in	in	ADP
ejpam-4905	457	11	s(g	s(g	PROPN
ejpam-4905	457	12	)	)	PUNCT
ejpam-4905	457	13	,	,	PUNCT
ejpam-4905	457	14	tg1	tg1	VERB
ejpam-4905	457	15	∪	∪	ADP
ejpam-4905	457	16	t	t	PROPN
ejpam-4905	457	17	′	′	NUM
ejpam-4905	457	18	g2	g2	PROPN
ejpam-4905	457	19	and	and	CCONJ
ejpam-4905	457	20	t	t	PROPN
ejpam-4905	457	21	′	′	PROPN
ejpam-4905	457	22	g1	g1	PROPN
ejpam-4905	457	23	∪	∪	ADP
ejpam-4905	457	24	tg2	tg2	PROPN
ejpam-4905	457	25	are	be	AUX
ejpam-4905	457	26	j2	j2	NOUN
ejpam-4905	457	27	-	-	PUNCT
ejpam-4905	457	28	sets	set	NOUN
ejpam-4905	457	29	in	in	ADP
ejpam-4905	457	30	g1	g1	PROPN
ejpam-4905	457	31	and	and	CCONJ
ejpam-4905	457	32	g2	g2	PROPN
ejpam-4905	457	33	,	,	PUNCT
ejpam-4905	457	34	respectively	respectively	ADV
ejpam-4905	457	35	,	,	PUNCT
ejpam-4905	457	36	by	by	ADP
ejpam-4905	457	37	theorem	theorem	NOUN
ejpam-4905	457	38	9	9	NUM
ejpam-4905	457	39	.	.	PUNCT
ejpam-4905	458	1	also	also	ADV
ejpam-4905	458	2	,	,	PUNCT
ejpam-4905	458	3	since	since	SCONJ
ejpam-4905	458	4	t	t	PROPN
ejpam-4905	458	5	is	be	AUX
ejpam-4905	458	6	a	a	DET
ejpam-4905	458	7	hop	hop	NOUN
ejpam-4905	458	8	dominating	dominating	NOUN
ejpam-4905	458	9	set	set	NOUN
ejpam-4905	458	10	in	in	ADP
ejpam-4905	458	11	s(g	s(g	PROPN
ejpam-4905	458	12	)	)	PUNCT
ejpam-4905	458	13	,	,	PUNCT
ejpam-4905	458	14	tg1	tg1	VERB
ejpam-4905	458	15	∪	∪	ADP
ejpam-4905	458	16	t	t	PROPN
ejpam-4905	458	17	′	′	NUM
ejpam-4905	458	18	g2	g2	PROPN
ejpam-4905	458	19	and	and	CCONJ
ejpam-4905	458	20	t	t	PROPN
ejpam-4905	458	21	′	′	PROPN
ejpam-4905	458	22	g1	g1	PROPN
ejpam-4905	458	23	∪	∪	ADP
ejpam-4905	458	24	tg2	tg2	PROPN
ejpam-4905	458	25	are	be	AUX
ejpam-4905	458	26	hop	hop	NOUN
ejpam-4905	458	27	dominating	dominating	NOUN
ejpam-4905	458	28	sets	set	NOUN
ejpam-4905	458	29	in	in	ADP
ejpam-4905	458	30	g1	g1	PROPN
ejpam-4905	458	31	and	and	CCONJ
ejpam-4905	458	32	g2	g2	PROPN
ejpam-4905	458	33	,	,	PUNCT
ejpam-4905	458	34	respectively	respectively	ADV
ejpam-4905	458	35	,	,	PUNCT
ejpam-4905	458	36	by	by	ADP
ejpam-4905	458	37	theorem	theorem	NOUN
ejpam-4905	458	38	10	10	NUM
ejpam-4905	458	39	.	.	PUNCT
ejpam-4905	459	1	consequently	consequently	ADV
ejpam-4905	459	2	,	,	PUNCT
ejpam-4905	459	3	tg1	tg1	VERB
ejpam-4905	459	4	∪t	∪t	NUM
ejpam-4905	459	5	′	′	NUM
ejpam-4905	459	6	g2	g2	PROPN
ejpam-4905	459	7	and	and	CCONJ
ejpam-4905	459	8	t	t	PROPN
ejpam-4905	459	9	′	′	PROPN
ejpam-4905	459	10	g1	g1	PROPN
ejpam-4905	459	11	∪	∪	ADP
ejpam-4905	459	12	tg2	tg2	PROPN
ejpam-4905	459	13	are	be	AUX
ejpam-4905	459	14	j2	j2	NOUN
ejpam-4905	459	15	-	-	PUNCT
ejpam-4905	459	16	hop	hop	NOUN
ejpam-4905	459	17	dominating	dominating	NOUN
ejpam-4905	459	18	sets	set	NOUN
ejpam-4905	459	19	in	in	ADP
ejpam-4905	459	20	g1	g1	PROPN
ejpam-4905	459	21	and	and	CCONJ
ejpam-4905	459	22	g2	g2	PROPN
ejpam-4905	459	23	,	,	PUNCT
ejpam-4905	459	24	respectively	respectively	ADV
ejpam-4905	459	25	.	.	PUNCT
ejpam-4905	460	1	for	for	ADP
ejpam-4905	460	2	the	the	DET
ejpam-4905	460	3	converse	converse	NOUN
ejpam-4905	460	4	,	,	PUNCT
ejpam-4905	460	5	suppose	suppose	VERB
ejpam-4905	460	6	(	(	PUNCT
ejpam-4905	460	7	i	i	NOUN
ejpam-4905	460	8	)	)	PUNCT
ejpam-4905	460	9	holds	hold	VERB
ejpam-4905	460	10	.	.	PUNCT
ejpam-4905	461	1	then	then	ADV
ejpam-4905	461	2	t	t	PROPN
ejpam-4905	461	3	is	be	AUX
ejpam-4905	461	4	both	both	PRON
ejpam-4905	461	5	a	a	DET
ejpam-4905	461	6	j2	j2	PROPN
ejpam-4905	461	7	-	-	PUNCT
ejpam-4905	461	8	set	set	NOUN
ejpam-4905	461	9	and	and	CCONJ
ejpam-4905	461	10	a	a	DET
ejpam-4905	461	11	hop	hop	NOUN
ejpam-4905	461	12	dominating	dominating	NOUN
ejpam-4905	461	13	in	in	ADP
ejpam-4905	461	14	g1	g1	PROPN
ejpam-4905	461	15	.	.	PUNCT
ejpam-4905	462	1	thus	thus	ADV
ejpam-4905	462	2	,	,	PUNCT
ejpam-4905	462	3	by	by	ADP
ejpam-4905	462	4	theorem	theorem	NOUN
ejpam-4905	462	5	9	9	NUM
ejpam-4905	462	6	and	and	CCONJ
ejpam-4905	462	7	by	by	ADP
ejpam-4905	462	8	theorem	theorem	NOUN
ejpam-4905	462	9	10	10	NUM
ejpam-4905	462	10	,	,	PUNCT
ejpam-4905	462	11	t	t	PROPN
ejpam-4905	462	12	is	be	AUX
ejpam-4905	462	13	a	a	DET
ejpam-4905	462	14	j2	j2	PROPN
ejpam-4905	462	15	-	-	PUNCT
ejpam-4905	462	16	hop	hop	NOUN
ejpam-4905	462	17	dominating	dominating	NOUN
ejpam-4905	462	18	set	set	NOUN
ejpam-4905	462	19	in	in	ADP
ejpam-4905	462	20	s(g	s(g	PROPN
ejpam-4905	462	21	)	)	PUNCT
ejpam-4905	462	22	.	.	PUNCT
ejpam-4905	463	1	similarly	similarly	ADV
ejpam-4905	463	2	,	,	PUNCT
ejpam-4905	463	3	if	if	SCONJ
ejpam-4905	463	4	(	(	PUNCT
ejpam-4905	463	5	ii	ii	NOUN
ejpam-4905	463	6	)	)	PUNCT
ejpam-4905	463	7	holds	hold	VERB
ejpam-4905	463	8	,	,	PUNCT
ejpam-4905	463	9	then	then	ADV
ejpam-4905	463	10	t	t	PROPN
ejpam-4905	463	11	is	be	AUX
ejpam-4905	463	12	a	a	DET
ejpam-4905	463	13	j2	j2	PROPN
ejpam-4905	463	14	-	-	PUNCT
ejpam-4905	463	15	hop	hop	NOUN
ejpam-4905	463	16	dominating	dominating	NOUN
ejpam-4905	463	17	set	set	NOUN
ejpam-4905	463	18	of	of	ADP
ejpam-4905	463	19	s(g	s(g	PROPN
ejpam-4905	463	20	)	)	PUNCT
ejpam-4905	463	21	.	.	PUNCT
ejpam-4905	464	1	suppose	suppose	VERB
ejpam-4905	464	2	that	that	SCONJ
ejpam-4905	464	3	(	(	PUNCT
ejpam-4905	464	4	iii	iii	NOUN
ejpam-4905	464	5	)	)	PUNCT
ejpam-4905	464	6	holds	hold	VERB
ejpam-4905	464	7	.	.	PUNCT
ejpam-4905	465	1	then	then	ADV
ejpam-4905	465	2	by	by	ADP
ejpam-4905	465	3	theorem	theorem	ADJ
ejpam-4905	465	4	9	9	NUM
ejpam-4905	465	5	and	and	CCONJ
ejpam-4905	465	6	theorem	theorem	VERB
ejpam-4905	465	7	10	10	NUM
ejpam-4905	465	8	,	,	PUNCT
ejpam-4905	465	9	t	t	PROPN
ejpam-4905	465	10	is	be	AUX
ejpam-4905	465	11	a	a	DET
ejpam-4905	465	12	j2	j2	PROPN
ejpam-4905	465	13	-	-	PUNCT
ejpam-4905	465	14	hop	hop	NOUN
ejpam-4905	465	15	dominating	dominating	NOUN
ejpam-4905	465	16	set	set	NOUN
ejpam-4905	465	17	of	of	ADP
ejpam-4905	465	18	s(g	s(g	PROPN
ejpam-4905	465	19	)	)	PUNCT
ejpam-4905	465	20	.	.	PUNCT
ejpam-4905	466	1	corollary	corollary	ADJ
ejpam-4905	466	2	2	2	NUM
ejpam-4905	466	3	.	.	PUNCT
ejpam-4905	467	1	let	let	VERB
ejpam-4905	467	2	g	g	PRON
ejpam-4905	467	3	be	be	AUX
ejpam-4905	467	4	a	a	DET
ejpam-4905	467	5	connected	connected	ADJ
ejpam-4905	467	6	non	non	ADJ
ejpam-4905	467	7	-	-	ADJ
ejpam-4905	467	8	trivial	trivial	ADJ
ejpam-4905	467	9	graph	graph	NOUN
ejpam-4905	467	10	.	.	PUNCT
ejpam-4905	468	1	then	then	ADV
ejpam-4905	468	2	γj2h(s(g	γj2h(s(g	NOUN
ejpam-4905	468	3	)	)	PUNCT
ejpam-4905	468	4	)	)	PUNCT
ejpam-4905	469	1	=	=	SYM
ejpam-4905	469	2	γj2h(g	γj2h(g	NOUN
ejpam-4905	469	3	)	)	PUNCT
ejpam-4905	469	4	.	.	PUNCT
ejpam-4905	470	1	proof	proof	NOUN
ejpam-4905	470	2	.	.	PUNCT
ejpam-4905	471	1	let	let	VERB
ejpam-4905	471	2	t	t	NOUN
ejpam-4905	471	3	be	be	AUX
ejpam-4905	471	4	a	a	DET
ejpam-4905	471	5	γj2h	γj2h	NOUN
ejpam-4905	471	6	-	-	PUNCT
ejpam-4905	471	7	set	set	NOUN
ejpam-4905	471	8	of	of	ADP
ejpam-4905	471	9	g.	g.	PROPN
ejpam-4905	471	10	then	then	ADV
ejpam-4905	471	11	by	by	ADP
ejpam-4905	471	12	theorem	theorem	NOUN
ejpam-4905	471	13	11	11	NUM
ejpam-4905	471	14	,	,	PUNCT
ejpam-4905	471	15	t	t	PROPN
ejpam-4905	471	16	is	be	AUX
ejpam-4905	471	17	a	a	DET
ejpam-4905	471	18	j2	j2	PROPN
ejpam-4905	471	19	-	-	PUNCT
ejpam-4905	471	20	hop	hop	NOUN
ejpam-4905	471	21	dominating	dominating	NOUN
ejpam-4905	471	22	set	set	NOUN
ejpam-4905	471	23	of	of	ADP
ejpam-4905	471	24	s(g	s(g	PROPN
ejpam-4905	471	25	)	)	PUNCT
ejpam-4905	471	26	.	.	PUNCT
ejpam-4905	472	1	thus	thus	ADV
ejpam-4905	472	2	,	,	PUNCT
ejpam-4905	472	3	γj2h(s(g	γj2h(s(g	NOUN
ejpam-4905	472	4	)	)	PUNCT
ejpam-4905	472	5	)	)	PUNCT
ejpam-4905	473	1	≥	≥	PRON
ejpam-4905	473	2	|t	|t	VERB
ejpam-4905	473	3	|	|	ADV
ejpam-4905	473	4	=	=	PUNCT
ejpam-4905	473	5	γj2h(g	γj2h(g	NOUN
ejpam-4905	473	6	)	)	PUNCT
ejpam-4905	473	7	.	.	PUNCT
ejpam-4905	474	1	on	on	ADP
ejpam-4905	474	2	the	the	DET
ejpam-4905	474	3	other	other	ADJ
ejpam-4905	474	4	hand	hand	NOUN
ejpam-4905	474	5	,	,	PUNCT
ejpam-4905	474	6	suppose	suppose	VERB
ejpam-4905	474	7	t	t	PROPN
ejpam-4905	474	8	∗	∗	NOUN
ejpam-4905	474	9	is	be	AUX
ejpam-4905	474	10	a	a	DET
ejpam-4905	474	11	γj2h	γj2h	NOUN
ejpam-4905	474	12	-	-	PUNCT
ejpam-4905	474	13	set	set	NOUN
ejpam-4905	474	14	of	of	ADP
ejpam-4905	474	15	s(g	s(g	PROPN
ejpam-4905	474	16	)	)	PUNCT
ejpam-4905	474	17	.	.	PUNCT
ejpam-4905	475	1	if	if	SCONJ
ejpam-4905	475	2	t	t	PROPN
ejpam-4905	475	3	∗	∗	NOUN
ejpam-4905	475	4	is	be	AUX
ejpam-4905	475	5	of	of	ADP
ejpam-4905	475	6	type	type	NOUN
ejpam-4905	475	7	(	(	PUNCT
ejpam-4905	475	8	i	i	NOUN
ejpam-4905	475	9	)	)	PUNCT
ejpam-4905	475	10	or	or	CCONJ
ejpam-4905	475	11	(	(	PUNCT
ejpam-4905	475	12	ii	ii	NOUN
ejpam-4905	475	13	)	)	PUNCT
ejpam-4905	475	14	,	,	PUNCT
ejpam-4905	475	15	then	then	ADV
ejpam-4905	475	16	t	t	PROPN
ejpam-4905	475	17	∗	∗	NOUN
ejpam-4905	475	18	is	be	AUX
ejpam-4905	475	19	a	a	DET
ejpam-4905	475	20	j2	j2	PROPN
ejpam-4905	475	21	-	-	PUNCT
ejpam-4905	475	22	hop	hop	NOUN
ejpam-4905	475	23	dominating	dominating	NOUN
ejpam-4905	475	24	set	set	NOUN
ejpam-4905	475	25	of	of	ADP
ejpam-4905	475	26	g	g	NOUN
ejpam-4905	475	27	by	by	ADP
ejpam-4905	475	28	theorem	theorem	NOUN
ejpam-4905	475	29	11(i	11(i	NUM
ejpam-4905	475	30	)	)	PUNCT
ejpam-4905	475	31	and	and	CCONJ
ejpam-4905	475	32	(	(	PUNCT
ejpam-4905	475	33	ii	ii	NOUN
ejpam-4905	475	34	)	)	PUNCT
ejpam-4905	475	35	.	.	PUNCT
ejpam-4905	476	1	hence	hence	ADV
ejpam-4905	476	2	,	,	PUNCT
ejpam-4905	476	3	γj2h(s(g	γj2h(s(g	NOUN
ejpam-4905	476	4	)	)	PUNCT
ejpam-4905	476	5	)	)	PUNCT
ejpam-4905	477	1	=	=	PUNCT
ejpam-4905	477	2	|t	|t	VERB
ejpam-4905	478	1	∗|	∗|	NOUN
ejpam-4905	478	2	≤	≤	NUM
ejpam-4905	478	3	γj2h(g	γj2h(g	NOUN
ejpam-4905	478	4	)	)	PUNCT
ejpam-4905	478	5	.	.	PUNCT
ejpam-4905	479	1	next	next	ADV
ejpam-4905	479	2	,	,	PUNCT
ejpam-4905	479	3	suppose	suppose	VERB
ejpam-4905	479	4	t	t	PROPN
ejpam-4905	479	5	∗	∗	NOUN
ejpam-4905	479	6	is	be	AUX
ejpam-4905	479	7	of	of	ADP
ejpam-4905	479	8	type	type	NOUN
ejpam-4905	479	9	(	(	PUNCT
ejpam-4905	479	10	iii	iii	NOUN
ejpam-4905	479	11	)	)	PUNCT
ejpam-4905	479	12	,	,	PUNCT
ejpam-4905	479	13	say	say	VERB
ejpam-4905	479	14	t	t	PROPN
ejpam-4905	479	15	∗	∗	NOUN
ejpam-4905	479	16	=	=	PUNCT
ejpam-4905	479	17	tg1	tg1	PROPN
ejpam-4905	479	18	∪tg2	∪tg2	PROPN
ejpam-4905	479	19	.	.	PUNCT
ejpam-4905	480	1	then	then	ADV
ejpam-4905	480	2	t	t	PROPN
ejpam-4905	480	3	∗	∗	NOUN
ejpam-4905	480	4	g	g	NOUN
ejpam-4905	480	5	=	=	PUNCT
ejpam-4905	480	6	tg1	tg1	X
ejpam-4905	480	7	∪t	∪t	NUM
ejpam-4905	480	8	′	′	NUM
ejpam-4905	480	9	g2	g2	PROPN
ejpam-4905	480	10	is	be	AUX
ejpam-4905	480	11	a	a	DET
ejpam-4905	480	12	j2	j2	PROPN
ejpam-4905	480	13	-	-	PUNCT
ejpam-4905	480	14	hop	hop	NOUN
ejpam-4905	480	15	dominating	dominating	NOUN
ejpam-4905	480	16	set	set	NOUN
ejpam-4905	480	17	of	of	ADP
ejpam-4905	480	18	g1	g1	NOUN
ejpam-4905	480	19	by	by	ADP
ejpam-4905	480	20	theorem	theorem	NOUN
ejpam-4905	480	21	11(iii	11(iii	NUM
ejpam-4905	480	22	)	)	PUNCT
ejpam-4905	480	23	.	.	PUNCT
ejpam-4905	481	1	this	this	PRON
ejpam-4905	481	2	implies	imply	VERB
ejpam-4905	481	3	that	that	SCONJ
ejpam-4905	481	4	γj2h(s(g	γj2h(s(g	NOUN
ejpam-4905	481	5	)	)	PUNCT
ejpam-4905	481	6	)	)	PUNCT
ejpam-4905	482	1	=	=	PUNCT
ejpam-4905	482	2	|t	|t	VERB
ejpam-4905	483	1	∗|	∗|	NOUN
ejpam-4905	483	2	=	=	SYM
ejpam-4905	483	3	|t	|t	PROPN
ejpam-4905	483	4	∗	∗	NOUN
ejpam-4905	483	5	g|	g|	PROPN
ejpam-4905	483	6	≤	≤	ADJ
ejpam-4905	483	7	γj2h(g	γj2h(g	NOUN
ejpam-4905	483	8	)	)	PUNCT
ejpam-4905	483	9	.	.	PUNCT
ejpam-4905	484	1	consequently	consequently	ADV
ejpam-4905	484	2	,	,	PUNCT
ejpam-4905	484	3	γj2h(s(g	γj2h(s(g	NOUN
ejpam-4905	484	4	)	)	PUNCT
ejpam-4905	484	5	)	)	PUNCT
ejpam-4905	485	1	=	=	SYM
ejpam-4905	485	2	γj2h(g	γj2h(g	NOUN
ejpam-4905	485	3	)	)	PUNCT
ejpam-4905	485	4	.	.	PUNCT
ejpam-4905	486	1	example	example	NOUN
ejpam-4905	487	1	2	2	NUM
ejpam-4905	487	2	.	.	X
ejpam-4905	487	3	consider	consider	VERB
ejpam-4905	487	4	the	the	DET
ejpam-4905	487	5	shadow	shadow	NOUN
ejpam-4905	487	6	graph	graph	VERB
ejpam-4905	487	7	s(c4	s(c4	NOUN
ejpam-4905	487	8	)	)	PUNCT
ejpam-4905	487	9	of	of	ADP
ejpam-4905	487	10	c4	c4	NOUN
ejpam-4905	487	11	in	in	ADP
ejpam-4905	487	12	figure	figure	NOUN
ejpam-4905	487	13	6	6	NUM
ejpam-4905	487	14	.	.	PUNCT
ejpam-4905	488	1	let	let	VERB
ejpam-4905	488	2	v	v	NOUN
ejpam-4905	488	3	(	(	PUNCT
ejpam-4905	488	4	c4	c4	NOUN
ejpam-4905	488	5	)	)	PUNCT
ejpam-4905	488	6	=	=	PUNCT
ejpam-4905	488	7	{	{	PUNCT
ejpam-4905	488	8	a	a	PRON
ejpam-4905	488	9	,	,	PUNCT
ejpam-4905	488	10	b	b	NOUN
ejpam-4905	488	11	,	,	PUNCT
ejpam-4905	488	12	c	c	NOUN
ejpam-4905	488	13	,	,	PUNCT
ejpam-4905	488	14	d	d	NOUN
ejpam-4905	488	15	}	}	PUNCT
ejpam-4905	488	16	and	and	CCONJ
ejpam-4905	488	17	let	let	VERB
ejpam-4905	488	18	n	n	X
ejpam-4905	488	19	=	=	PRON
ejpam-4905	488	20	{	{	PUNCT
ejpam-4905	488	21	a	a	PRON
ejpam-4905	488	22	,	,	PUNCT
ejpam-4905	488	23	b	b	NOUN
ejpam-4905	488	24	}	}	PUNCT
ejpam-4905	488	25	.	.	PUNCT
ejpam-4905	489	1	then	then	ADV
ejpam-4905	489	2	n2	n2	PROPN
ejpam-4905	489	3	c4	c4	NOUN
ejpam-4905	489	4	[	[	X
ejpam-4905	489	5	n	n	X
ejpam-4905	489	6	]	]	X
ejpam-4905	489	7	=	=	SYM
ejpam-4905	489	8	v	v	X
ejpam-4905	489	9	(	(	PUNCT
ejpam-4905	489	10	c4	c4	NOUN
ejpam-4905	489	11	)	)	PUNCT
ejpam-4905	489	12	,	,	PUNCT
ejpam-4905	489	13	a	a	DET
ejpam-4905	489	14	∈	∈	PROPN
ejpam-4905	489	15	n2	n2	NOUN
ejpam-4905	489	16	c4	c4	NOUN
ejpam-4905	489	17	[	[	X
ejpam-4905	489	18	a	a	X
ejpam-4905	489	19	]	]	X
ejpam-4905	489	20	\n2	\n2	ADJ
ejpam-4905	489	21	c4	c4	NOUN
ejpam-4905	490	1	[	[	X
ejpam-4905	490	2	b	b	X
ejpam-4905	490	3	]	]	X
ejpam-4905	490	4	and	and	CCONJ
ejpam-4905	490	5	b	b	PROPN
ejpam-4905	490	6	∈	∈	PROPN
ejpam-4905	490	7	n2	n2	NOUN
ejpam-4905	490	8	c4	c4	NOUN
ejpam-4905	490	9	[	[	X
ejpam-4905	490	10	b	b	X
ejpam-4905	490	11	]	]	X
ejpam-4905	490	12	\n2	\n2	ADJ
ejpam-4905	490	13	c4	c4	NOUN
ejpam-4905	490	14	[	[	X
ejpam-4905	490	15	a	a	X
ejpam-4905	490	16	]	]	X
ejpam-4905	490	17	.	.	PUNCT
ejpam-4905	491	1	thus	thus	ADV
ejpam-4905	491	2	,	,	PUNCT
ejpam-4905	491	3	n	n	PRON
ejpam-4905	491	4	is	be	AUX
ejpam-4905	491	5	a	a	DET
ejpam-4905	491	6	j2	j2	PROPN
ejpam-4905	491	7	-	-	PUNCT
ejpam-4905	491	8	hop	hop	NOUN
ejpam-4905	491	9	dominating	dominating	NOUN
ejpam-4905	491	10	set	set	NOUN
ejpam-4905	491	11	of	of	ADP
ejpam-4905	491	12	c4	c4	NOUN
ejpam-4905	491	13	.	.	PUNCT
ejpam-4905	492	1	since	since	SCONJ
ejpam-4905	492	2	n2	n2	ADJ
ejpam-4905	492	3	c4	c4	NOUN
ejpam-4905	492	4	[	[	X
ejpam-4905	492	5	d	d	X
ejpam-4905	492	6	]	]	X
ejpam-4905	492	7	=	=	SYM
ejpam-4905	492	8	n2	n2	ADJ
ejpam-4905	492	9	c4	c4	NOUN
ejpam-4905	492	10	[	[	X
ejpam-4905	492	11	a	a	X
ejpam-4905	492	12	]	]	X
ejpam-4905	492	13	and	and	CCONJ
ejpam-4905	492	14	n2	n2	ADJ
ejpam-4905	492	15	c4	c4	NOUN
ejpam-4905	492	16	[	[	X
ejpam-4905	492	17	c	c	X
ejpam-4905	492	18	]	]	X
ejpam-4905	492	19	=	=	SYM
ejpam-4905	492	20	n2	n2	ADJ
ejpam-4905	492	21	c4	c4	NOUN
ejpam-4905	492	22	[	[	X
ejpam-4905	492	23	b	b	X
ejpam-4905	492	24	]	]	X
ejpam-4905	492	25	,	,	PUNCT
ejpam-4905	492	26	it	it	PRON
ejpam-4905	492	27	follows	follow	VERB
ejpam-4905	492	28	that	that	SCONJ
ejpam-4905	492	29	n	n	PRON
ejpam-4905	492	30	is	be	AUX
ejpam-4905	492	31	a	a	DET
ejpam-4905	492	32	maximum	maximum	ADJ
ejpam-4905	492	33	j2	j2	PROPN
ejpam-4905	492	34	-	-	PUNCT
ejpam-4905	492	35	hop	hop	NOUN
ejpam-4905	492	36	dominating	dominating	NOUN
ejpam-4905	492	37	set	set	NOUN
ejpam-4905	492	38	of	of	ADP
ejpam-4905	492	39	c4	c4	NOUN
ejpam-4905	492	40	.	.	PUNCT
ejpam-4905	493	1	hence	hence	ADV
ejpam-4905	493	2	,	,	PUNCT
ejpam-4905	493	3	γj2h(c4	γj2h(c4	NOUN
ejpam-4905	493	4	)	)	PUNCT
ejpam-4905	493	5	=	=	SYM
ejpam-4905	494	1	2	2	X
ejpam-4905	494	2	.	.	X
ejpam-4905	494	3	observe	observe	VERB
ejpam-4905	494	4	that	that	SCONJ
ejpam-4905	494	5	a	a	DET
ejpam-4905	494	6	∈	∈	PROPN
ejpam-4905	494	7	n2	n2	NOUN
ejpam-4905	494	8	s(c4	s(c4	NOUN
ejpam-4905	494	9	)	)	PUNCT
ejpam-4905	495	1	[	[	X
ejpam-4905	495	2	a	a	X
ejpam-4905	495	3	]	]	X
ejpam-4905	495	4	\n2	\n2	ADJ
ejpam-4905	495	5	s(c4	s(c4	NOUN
ejpam-4905	495	6	)	)	PUNCT
ejpam-4905	496	1	[	[	X
ejpam-4905	496	2	b	b	X
ejpam-4905	496	3	]	]	X
ejpam-4905	496	4	and	and	CCONJ
ejpam-4905	496	5	b	b	PROPN
ejpam-4905	496	6	∈	∈	PROPN
ejpam-4905	496	7	n2	n2	NOUN
ejpam-4905	496	8	s(c4	s(c4	NOUN
ejpam-4905	496	9	)	)	PUNCT
ejpam-4905	497	1	[	[	X
ejpam-4905	497	2	b	b	X
ejpam-4905	497	3	]	]	X
ejpam-4905	497	4	\n2	\n2	ADJ
ejpam-4905	497	5	s(c4	s(c4	NOUN
ejpam-4905	497	6	)	)	PUNCT
ejpam-4905	498	1	[	[	X
ejpam-4905	498	2	a	a	X
ejpam-4905	498	3	]	]	X
ejpam-4905	498	4	,	,	PUNCT
ejpam-4905	498	5	showing	show	VERB
ejpam-4905	498	6	that	that	SCONJ
ejpam-4905	498	7	n	n	X
ejpam-4905	498	8	is	be	AUX
ejpam-4905	498	9	a	a	DET
ejpam-4905	498	10	j2	j2	NOUN
ejpam-4905	498	11	-	-	PUNCT
ejpam-4905	498	12	set	set	NOUN
ejpam-4905	498	13	in	in	ADP
ejpam-4905	498	14	s(c4	s(c4	NOUN
ejpam-4905	498	15	)	)	PUNCT
ejpam-4905	498	16	.	.	PUNCT
ejpam-4905	499	1	sincen	sincen	NOUN
ejpam-4905	499	2	2	2	NUM
ejpam-4905	499	3	s(c4	s(c4	NOUN
ejpam-4905	499	4	)	)	PUNCT
ejpam-4905	500	1	[	[	X
ejpam-4905	500	2	n	n	X
ejpam-4905	500	3	]	]	X
ejpam-4905	500	4	=	=	SYM
ejpam-4905	500	5	v	v	X
ejpam-4905	500	6	(	(	PUNCT
ejpam-4905	500	7	s(c4	s(c4	NOUN
ejpam-4905	500	8	)	)	PUNCT
ejpam-4905	500	9	)	)	PUNCT
ejpam-4905	500	10	,	,	PUNCT
ejpam-4905	500	11	it	it	PRON
ejpam-4905	500	12	follows	follow	VERB
ejpam-4905	500	13	thatn	thatn	ADV
ejpam-4905	500	14	is	be	AUX
ejpam-4905	500	15	a	a	DET
ejpam-4905	500	16	j2	j2	PROPN
ejpam-4905	500	17	-	-	PUNCT
ejpam-4905	500	18	hop	hop	NOUN
ejpam-4905	500	19	dominating	dominating	NOUN
ejpam-4905	500	20	set	set	VERB
ejpam-4905	500	21	in	in	ADP
ejpam-4905	500	22	s(c4	s(c4	NOUN
ejpam-4905	500	23	)	)	PUNCT
ejpam-4905	500	24	.	.	PUNCT
ejpam-4905	501	1	by	by	ADP
ejpam-4905	501	2	lemma	lemma	PROPN
ejpam-4905	501	3	1	1	NUM
ejpam-4905	501	4	,	,	PUNCT
ejpam-4905	501	5	n2	n2	ADJ
ejpam-4905	501	6	s(c4	s(c4	NOUN
ejpam-4905	501	7	)	)	PUNCT
ejpam-4905	502	1	[	[	X
ejpam-4905	502	2	u	u	X
ejpam-4905	502	3	]	]	X
ejpam-4905	502	4	=	=	PUNCT
ejpam-4905	502	5	n2	n2	ADJ
ejpam-4905	502	6	s(c4	s(c4	NOUN
ejpam-4905	502	7	)	)	PUNCT
ejpam-4905	503	1	[	[	X
ejpam-4905	503	2	u′	u′	X
ejpam-4905	503	3	]	]	PUNCT
ejpam-4905	503	4	for	for	ADP
ejpam-4905	503	5	every	every	DET
ejpam-4905	503	6	u	u	PROPN
ejpam-4905	503	7	∈	∈	PROPN
ejpam-4905	503	8	v	v	NOUN
ejpam-4905	503	9	(	(	PUNCT
ejpam-4905	503	10	c4	c4	NOUN
ejpam-4905	503	11	)	)	PUNCT
ejpam-4905	503	12	.	.	PUNCT
ejpam-4905	504	1	since	since	SCONJ
ejpam-4905	504	2	n	n	ADV
ejpam-4905	504	3	2	2	NUM
ejpam-4905	504	4	s(c4	s(c4	NOUN
ejpam-4905	504	5	)	)	PUNCT
ejpam-4905	505	1	[	[	X
ejpam-4905	505	2	a	a	X
ejpam-4905	505	3	]	]	X
ejpam-4905	505	4	=	=	PUNCT
ejpam-4905	505	5	n2	n2	ADJ
ejpam-4905	505	6	s(c4	s(c4	NOUN
ejpam-4905	505	7	)	)	PUNCT
ejpam-4905	506	1	[	[	X
ejpam-4905	506	2	d	d	X
ejpam-4905	506	3	]	]	X
ejpam-4905	506	4	and	and	CCONJ
ejpam-4905	506	5	n2	n2	ADJ
ejpam-4905	506	6	s(c4	s(c4	NOUN
ejpam-4905	506	7	)	)	PUNCT
ejpam-4905	507	1	[	[	X
ejpam-4905	507	2	b	b	X
ejpam-4905	507	3	]	]	X
ejpam-4905	507	4	=	=	PUNCT
ejpam-4905	507	5	n2	n2	ADJ
ejpam-4905	507	6	s(c4	s(c4	NOUN
ejpam-4905	507	7	)	)	PUNCT
ejpam-4905	508	1	[	[	X
ejpam-4905	508	2	c	c	X
ejpam-4905	508	3	]	]	X
ejpam-4905	508	4	,	,	PUNCT
ejpam-4905	508	5	it	it	PRON
ejpam-4905	508	6	follows	follow	VERB
ejpam-4905	508	7	that	that	SCONJ
ejpam-4905	508	8	n	n	PRON
ejpam-4905	508	9	is	be	AUX
ejpam-4905	508	10	a	a	DET
ejpam-4905	508	11	maximum	maximum	ADJ
ejpam-4905	508	12	j2	j2	PROPN
ejpam-4905	508	13	-	-	PUNCT
ejpam-4905	508	14	hop	hop	NOUN
ejpam-4905	508	15	dominating	dominating	NOUN
ejpam-4905	508	16	set	set	NOUN
ejpam-4905	508	17	of	of	ADP
ejpam-4905	508	18	s(c4	s(c4	NOUN
ejpam-4905	508	19	)	)	PUNCT
ejpam-4905	508	20	.	.	PUNCT
ejpam-4905	509	1	thus	thus	ADV
ejpam-4905	509	2	,	,	PUNCT
ejpam-4905	509	3	γj2h(c4	γj2h(c4	NOUN
ejpam-4905	509	4	)	)	PUNCT
ejpam-4905	509	5	=	=	SYM
ejpam-4905	509	6	2	2	NUM
ejpam-4905	509	7	=	=	SYM
ejpam-4905	509	8	γj2h(s(c4	γj2h(s(c4	NUM
ejpam-4905	509	9	)	)	PUNCT
ejpam-4905	509	10	)	)	PUNCT
ejpam-4905	509	11	.	.	PUNCT
ejpam-4905	510	1	references	reference	NOUN
ejpam-4905	510	2	2130	2130	NUM
ejpam-4905	510	3	a′	a′	PROPN
ejpam-4905	510	4	b′	b′	NUM
ejpam-4905	510	5	ba	ba	NOUN
ejpam-4905	510	6	c	c	PROPN
ejpam-4905	510	7	d	d	PROPN
ejpam-4905	510	8	d′c′	d′c′	NOUN
ejpam-4905	510	9	s(c4	s(c4	NOUN
ejpam-4905	510	10	)	)	PUNCT
ejpam-4905	510	11	:	:	PUNCT
ejpam-4905	510	12	figure	figure	VERB
ejpam-4905	510	13	6	6	NUM
ejpam-4905	510	14	:	:	PUNCT
ejpam-4905	510	15	graph	graph	NOUN
ejpam-4905	510	16	c4	c4	NOUN
ejpam-4905	510	17	with	with	ADP
ejpam-4905	510	18	γj2h(c4	γj2h(c4	NOUN
ejpam-4905	510	19	)	)	PUNCT
ejpam-4905	510	20	=	=	SYM
ejpam-4905	510	21	2	2	NUM
ejpam-4905	510	22	=	=	SYM
ejpam-4905	510	23	γj2h(s(c4	γj2h(s(c4	NUM
ejpam-4905	510	24	)	)	PUNCT
ejpam-4905	510	25	)	)	PUNCT
ejpam-4905	511	1	4	4	X
ejpam-4905	511	2	.	.	X
ejpam-4905	511	3	conclusion	conclusion	VERB
ejpam-4905	511	4	the	the	DET
ejpam-4905	511	5	concept	concept	NOUN
ejpam-4905	511	6	of	of	ADP
ejpam-4905	511	7	j2	j2	PROPN
ejpam-4905	511	8	-	-	PUNCT
ejpam-4905	511	9	hop	hop	NOUN
ejpam-4905	511	10	domination	domination	NOUN
ejpam-4905	511	11	has	have	AUX
ejpam-4905	511	12	been	be	AUX
ejpam-4905	511	13	introduced	introduce	VERB
ejpam-4905	511	14	and	and	CCONJ
ejpam-4905	511	15	initially	initially	ADV
ejpam-4905	511	16	investigated	investigate	VERB
ejpam-4905	511	17	in	in	ADP
ejpam-4905	511	18	this	this	DET
ejpam-4905	511	19	study	study	NOUN
ejpam-4905	511	20	.	.	PUNCT
ejpam-4905	512	1	its	its	PRON
ejpam-4905	512	2	bounds	bound	NOUN
ejpam-4905	512	3	with	with	ADP
ejpam-4905	512	4	respect	respect	NOUN
ejpam-4905	512	5	to	to	ADP
ejpam-4905	512	6	other	other	ADJ
ejpam-4905	512	7	known	know	VERB
ejpam-4905	512	8	parameters	parameter	NOUN
ejpam-4905	512	9	in	in	ADP
ejpam-4905	512	10	graph	graph	NOUN
ejpam-4905	512	11	theory	theory	NOUN
ejpam-4905	512	12	have	have	AUX
ejpam-4905	512	13	been	be	AUX
ejpam-4905	512	14	determined	determine	VERB
ejpam-4905	512	15	.	.	PUNCT
ejpam-4905	513	1	in	in	ADP
ejpam-4905	513	2	addition	addition	NOUN
ejpam-4905	513	3	,	,	PUNCT
ejpam-4905	513	4	characterizations	characterization	NOUN
ejpam-4905	513	5	of	of	ADP
ejpam-4905	513	6	j2	j2	PROPN
ejpam-4905	513	7	-	-	PUNCT
ejpam-4905	513	8	hop	hop	NOUN
ejpam-4905	513	9	dominating	dominating	NOUN
ejpam-4905	513	10	sets	set	NOUN
ejpam-4905	513	11	in	in	ADP
ejpam-4905	513	12	some	some	DET
ejpam-4905	513	13	graphs	graph	NOUN
ejpam-4905	513	14	and	and	CCONJ
ejpam-4905	513	15	shadow	shadow	NOUN
ejpam-4905	513	16	graph	graph	NOUN
ejpam-4905	513	17	have	have	AUX
ejpam-4905	513	18	been	be	AUX
ejpam-4905	513	19	formulated	formulate	VERB
ejpam-4905	513	20	and	and	CCONJ
ejpam-4905	513	21	were	be	AUX
ejpam-4905	513	22	used	use	VERB
ejpam-4905	513	23	to	to	PART
ejpam-4905	513	24	solve	solve	VERB
ejpam-4905	513	25	exact	exact	ADJ
ejpam-4905	513	26	value	value	NOUN
ejpam-4905	513	27	of	of	ADP
ejpam-4905	513	28	the	the	DET
ejpam-4905	513	29	parameter	parameter	NOUN
ejpam-4905	513	30	of	of	ADP
ejpam-4905	513	31	each	each	PRON
ejpam-4905	513	32	of	of	ADP
ejpam-4905	513	33	these	these	DET
ejpam-4905	513	34	graphs	graph	NOUN
ejpam-4905	513	35	.	.	PUNCT
ejpam-4905	514	1	interested	interested	ADJ
ejpam-4905	514	2	researchers	researcher	NOUN
ejpam-4905	514	3	may	may	AUX
ejpam-4905	514	4	study	study	VERB
ejpam-4905	514	5	further	far	ADV
ejpam-4905	514	6	this	this	DET
ejpam-4905	514	7	parameter	parameter	NOUN
ejpam-4905	514	8	on	on	ADP
ejpam-4905	514	9	graphs	graph	NOUN
ejpam-4905	514	10	that	that	PRON
ejpam-4905	514	11	were	be	AUX
ejpam-4905	514	12	not	not	PART
ejpam-4905	514	13	considered	consider	VERB
ejpam-4905	514	14	in	in	ADP
ejpam-4905	514	15	this	this	DET
ejpam-4905	514	16	study	study	NOUN
ejpam-4905	514	17	.	.	PUNCT
ejpam-4905	515	1	further	far	ADV
ejpam-4905	515	2	,	,	PUNCT
ejpam-4905	515	3	researchers	researcher	NOUN
ejpam-4905	515	4	may	may	AUX
ejpam-4905	515	5	consider	consider	VERB
ejpam-4905	515	6	the	the	DET
ejpam-4905	515	7	investigation	investigation	NOUN
ejpam-4905	515	8	on	on	ADP
ejpam-4905	515	9	the	the	DET
ejpam-4905	515	10	complexity	complexity	NOUN
ejpam-4905	515	11	of	of	ADP
ejpam-4905	515	12	solving	solve	VERB
ejpam-4905	515	13	this	this	DET
ejpam-4905	515	14	parameter	parameter	NOUN
ejpam-4905	515	15	and	and	CCONJ
ejpam-4905	515	16	provide	provide	VERB
ejpam-4905	515	17	application	application	NOUN
ejpam-4905	515	18	especially	especially	ADV
ejpam-4905	515	19	in	in	ADP
ejpam-4905	515	20	real	real	ADJ
ejpam-4905	515	21	-	-	PUNCT
ejpam-4905	515	22	life	life	NOUN
ejpam-4905	515	23	situation	situation	NOUN
ejpam-4905	515	24	,	,	PUNCT
ejpam-4905	515	25	network	network	NOUN
ejpam-4905	515	26	and	and	CCONJ
ejpam-4905	515	27	other	other	ADJ
ejpam-4905	515	28	fields	field	NOUN
ejpam-4905	515	29	.	.	PUNCT
ejpam-4905	516	1	acknowledgements	acknowledgement	NOUN
ejpam-4905	516	2	the	the	DET
ejpam-4905	516	3	authors	author	NOUN
ejpam-4905	516	4	would	would	AUX
ejpam-4905	516	5	like	like	VERB
ejpam-4905	516	6	to	to	PART
ejpam-4905	516	7	thank	thank	VERB
ejpam-4905	516	8	mindanao	mindanao	PROPN
ejpam-4905	516	9	state	state	PROPN
ejpam-4905	516	10	university	university	PROPN
ejpam-4905	516	11	tawi	tawi	PROPN
ejpam-4905	516	12	-	-	PUNCT
ejpam-4905	516	13	tawi	tawi	PROPN
ejpam-4905	516	14	college	college	PROPN
ejpam-4905	516	15	of	of	ADP
ejpam-4905	516	16	technology	technology	NOUN
ejpam-4905	516	17	and	and	CCONJ
ejpam-4905	516	18	oceanography	oceanography	NOUN
ejpam-4905	516	19	for	for	ADP
ejpam-4905	516	20	funding	fund	VERB
ejpam-4905	516	21	this	this	DET
ejpam-4905	516	22	research	research	NOUN
ejpam-4905	516	23	.	.	PUNCT
ejpam-4905	517	1	also	also	ADV
ejpam-4905	517	2	,	,	PUNCT
ejpam-4905	517	3	the	the	DET
ejpam-4905	517	4	authors	author	NOUN
ejpam-4905	517	5	would	would	AUX
ejpam-4905	517	6	like	like	VERB
ejpam-4905	517	7	to	to	PART
ejpam-4905	517	8	thank	thank	VERB
ejpam-4905	517	9	the	the	DET
ejpam-4905	517	10	referees	referee	NOUN
ejpam-4905	517	11	for	for	ADP
ejpam-4905	517	12	their	their	PRON
ejpam-4905	517	13	invaluable	invaluable	ADJ
ejpam-4905	517	14	comments	comment	NOUN
ejpam-4905	517	15	and	and	CCONJ
ejpam-4905	517	16	suggestions	suggestion	NOUN
ejpam-4905	517	17	that	that	PRON
ejpam-4905	517	18	led	lead	VERB
ejpam-4905	517	19	to	to	ADP
ejpam-4905	517	20	the	the	DET
ejpam-4905	517	21	improvement	improvement	NOUN
ejpam-4905	517	22	of	of	ADP
ejpam-4905	517	23	the	the	DET
ejpam-4905	517	24	paper	paper	NOUN
ejpam-4905	517	25	.	.	PUNCT
ejpam-4905	518	1	references	reference	NOUN
ejpam-4905	518	2	[	[	X
ejpam-4905	518	3	1	1	NUM
ejpam-4905	518	4	]	]	PUNCT
ejpam-4905	518	5	s.	s.	PROPN
ejpam-4905	518	6	ayyaswamy	ayyaswamy	PROPN
ejpam-4905	518	7	,	,	PUNCT
ejpam-4905	518	8	b.	b.	PROPN
ejpam-4905	518	9	krishnakumari	krishnakumari	PROPN
ejpam-4905	518	10	,	,	PUNCT
ejpam-4905	518	11	b.	b.	PROPN
ejpam-4905	518	12	natarjan	natarjan	PROPN
ejpam-4905	518	13	,	,	PUNCT
ejpam-4905	518	14	and	and	CCONJ
ejpam-4905	518	15	y.	y.	PROPN
ejpam-4905	518	16	venkatakrishnan	venkatakrishnan	PROPN
ejpam-4905	518	17	.	.	PUNCT
ejpam-4905	519	1	bounds	bound	NOUN
ejpam-4905	519	2	on	on	ADP
ejpam-4905	519	3	the	the	DET
ejpam-4905	519	4	hop	hop	NOUN
ejpam-4905	519	5	domination	domination	NOUN
ejpam-4905	519	6	number	number	NOUN
ejpam-4905	519	7	of	of	ADP
ejpam-4905	519	8	a	a	DET
ejpam-4905	519	9	tree	tree	NOUN
ejpam-4905	519	10	.	.	PUNCT
ejpam-4905	520	1	proceedings	proceeding	NOUN
ejpam-4905	520	2	-	-	PUNCT
ejpam-4905	520	3	mathematical	mathematical	ADJ
ejpam-4905	520	4	sciences	science	NOUN
ejpam-4905	520	5	.	.	PUNCT
ejpam-4905	520	6	,	,	PUNCT
ejpam-4905	520	7	125(4):449–455	125(4):449–455	ADP
ejpam-4905	520	8	,	,	PUNCT
ejpam-4905	520	9	2015	2015	NUM
ejpam-4905	520	10	.	.	PUNCT
ejpam-4905	521	1	[	[	X
ejpam-4905	521	2	2	2	NUM
ejpam-4905	521	3	]	]	PUNCT
ejpam-4905	521	4	s.	s.	PROPN
ejpam-4905	521	5	ayyaswamy	ayyaswamy	PROPN
ejpam-4905	521	6	,	,	PUNCT
ejpam-4905	521	7	c.	c.	PROPN
ejpam-4905	521	8	natarajan	natarajan	PROPN
ejpam-4905	521	9	,	,	PUNCT
ejpam-4905	521	10	and	and	CCONJ
ejpam-4905	521	11	g.	g.	PROPN
ejpam-4905	521	12	sathiamoorphy	sathiamoorphy	PROPN
ejpam-4905	521	13	.	.	PUNCT
ejpam-4905	522	1	a	a	DET
ejpam-4905	522	2	note	note	NOUN
ejpam-4905	522	3	on	on	ADP
ejpam-4905	522	4	hop	hop	NOUN
ejpam-4905	522	5	domination	domination	NOUN
ejpam-4905	522	6	references	reference	NOUN
ejpam-4905	522	7	2131	2131	NUM
ejpam-4905	522	8	number	number	NOUN
ejpam-4905	522	9	of	of	ADP
ejpam-4905	522	10	some	some	DET
ejpam-4905	522	11	special	special	ADJ
ejpam-4905	522	12	families	family	NOUN
ejpam-4905	522	13	of	of	ADP
ejpam-4905	522	14	graphs	graph	NOUN
ejpam-4905	522	15	.	.	PUNCT
ejpam-4905	523	1	international	international	ADJ
ejpam-4905	523	2	journal	journal	NOUN
ejpam-4905	523	3	of	of	ADP
ejpam-4905	523	4	pure	pure	ADJ
ejpam-4905	523	5	and	and	CCONJ
ejpam-4905	523	6	applied	applied	ADJ
ejpam-4905	523	7	mathematics	mathematic	NOUN
ejpam-4905	523	8	.	.	PUNCT
ejpam-4905	523	9	,	,	PUNCT
ejpam-4905	523	10	119(12):11465–14171	119(12):11465–14171	NUM
ejpam-4905	523	11	,	,	PUNCT
ejpam-4905	523	12	2018	2018	NUM
ejpam-4905	523	13	.	.	PUNCT
ejpam-4905	524	1	[	[	X
ejpam-4905	524	2	3	3	X
ejpam-4905	524	3	]	]	X
ejpam-4905	524	4	j.	j.	PROPN
ejpam-4905	524	5	hassan	hassan	PROPN
ejpam-4905	524	6	and	and	CCONJ
ejpam-4905	524	7	s.	s.	PROPN
ejpam-4905	524	8	canoy	canoy	PROPN
ejpam-4905	524	9	jr	jr	PROPN
ejpam-4905	524	10	.	.	PROPN
ejpam-4905	524	11	connected	connect	VERB
ejpam-4905	524	12	grundy	grundy	PROPN
ejpam-4905	524	13	hop	hop	NOUN
ejpam-4905	524	14	dominating	dominate	VERB
ejpam-4905	524	15	sequences	sequence	NOUN
ejpam-4905	524	16	in	in	ADP
ejpam-4905	524	17	graphs	graph	NOUN
ejpam-4905	524	18	.	.	PUNCT
ejpam-4905	525	1	eur	eur	PROPN
ejpam-4905	525	2	.	.	PUNCT
ejpam-4905	526	1	j.	j.	PROPN
ejpam-4905	526	2	pure	pure	PROPN
ejpam-4905	526	3	appl	appl	PROPN
ejpam-4905	526	4	.	.	PUNCT
ejpam-4905	526	5	math	math	PROPN
ejpam-4905	526	6	.	.	PUNCT
ejpam-4905	526	7	,	,	PUNCT
ejpam-4905	527	1	16(2):1212–1227	16(2):1212–1227	NUM
ejpam-4905	527	2	,	,	PUNCT
ejpam-4905	527	3	2023	2023	NUM
ejpam-4905	527	4	.	.	PUNCT
ejpam-4905	528	1	[	[	X
ejpam-4905	528	2	4	4	X
ejpam-4905	528	3	]	]	PUNCT
ejpam-4905	528	4	j.	j.	PROPN
ejpam-4905	528	5	hassan	hassan	PROPN
ejpam-4905	528	6	and	and	CCONJ
ejpam-4905	528	7	s.	s.	PROPN
ejpam-4905	528	8	canoy	canoy	PROPN
ejpam-4905	528	9	jr	jr	PROPN
ejpam-4905	528	10	.	.	PUNCT
ejpam-4905	529	1	grundy	grundy	PROPN
ejpam-4905	529	2	dominating	dominating	PROPN
ejpam-4905	529	3	and	and	CCONJ
ejpam-4905	529	4	grundy	grundy	PROPN
ejpam-4905	529	5	hop	hop	NOUN
ejpam-4905	529	6	dominating	dominate	VERB
ejpam-4905	529	7	sequences	sequence	NOUN
ejpam-4905	529	8	in	in	ADP
ejpam-4905	529	9	graphs	graph	NOUN
ejpam-4905	529	10	:	:	PUNCT
ejpam-4905	529	11	relationships	relationship	NOUN
ejpam-4905	529	12	and	and	CCONJ
ejpam-4905	529	13	some	some	DET
ejpam-4905	529	14	structural	structural	ADJ
ejpam-4905	529	15	properties	property	NOUN
ejpam-4905	529	16	.	.	PUNCT
ejpam-4905	530	1	eur	eur	PROPN
ejpam-4905	530	2	.	.	PUNCT
ejpam-4905	531	1	j.	j.	PROPN
ejpam-4905	531	2	pure	pure	PROPN
ejpam-4905	531	3	appl	appl	PROPN
ejpam-4905	531	4	.	.	PUNCT
ejpam-4905	531	5	math	math	PROPN
ejpam-4905	531	6	.	.	PUNCT
ejpam-4905	531	7	,	,	PUNCT
ejpam-4905	532	1	16(2):1154–1166	16(2):1154–1166	NUM
ejpam-4905	532	2	,	,	PUNCT
ejpam-4905	532	3	2023	2023	NUM
ejpam-4905	532	4	.	.	PUNCT
ejpam-4905	533	1	[	[	X
ejpam-4905	533	2	5	5	X
ejpam-4905	533	3	]	]	PUNCT
ejpam-4905	533	4	j.	j.	PROPN
ejpam-4905	533	5	hassan	hassan	PROPN
ejpam-4905	533	6	,	,	PUNCT
ejpam-4905	533	7	s.	s.	PROPN
ejpam-4905	533	8	canoy	canoy	PROPN
ejpam-4905	533	9	jr	jr	PROPN
ejpam-4905	533	10	.	.	PROPN
ejpam-4905	533	11	,	,	PUNCT
ejpam-4905	533	12	and	and	CCONJ
ejpam-4905	533	13	chrisley	chrisley	PROPN
ejpam-4905	533	14	jade	jade	NOUN
ejpam-4905	533	15	saromines	saromine	NOUN
ejpam-4905	533	16	.	.	PUNCT
ejpam-4905	534	1	convex	convex	VERB
ejpam-4905	534	2	hop	hop	NOUN
ejpam-4905	534	3	domination	domination	NOUN
ejpam-4905	534	4	in	in	ADP
ejpam-4905	534	5	graphs	graph	NOUN
ejpam-4905	534	6	.	.	PUNCT
ejpam-4905	535	1	eur	eur	PROPN
ejpam-4905	535	2	.	.	PUNCT
ejpam-4905	536	1	j.	j.	PROPN
ejpam-4905	536	2	pure	pure	PROPN
ejpam-4905	536	3	appl	appl	PROPN
ejpam-4905	536	4	.	.	PUNCT
ejpam-4905	536	5	math	math	PROPN
ejpam-4905	536	6	.	.	PUNCT
ejpam-4905	536	7	,	,	PUNCT
ejpam-4905	536	8	16(1):319–335	16(1):319–335	NOUN
ejpam-4905	536	9	,	,	PUNCT
ejpam-4905	536	10	2023	2023	NUM
ejpam-4905	536	11	.	.	PUNCT
ejpam-4905	537	1	[	[	X
ejpam-4905	537	2	6	6	NUM
ejpam-4905	537	3	]	]	PUNCT
ejpam-4905	537	4	j.	j.	PROPN
ejpam-4905	537	5	hassan	hassan	PROPN
ejpam-4905	537	6	,	,	PUNCT
ejpam-4905	537	7	a.	a.	PROPN
ejpam-4905	537	8	lintasan	lintasan	PROPN
ejpam-4905	537	9	,	,	PUNCT
ejpam-4905	537	10	and	and	CCONJ
ejpam-4905	537	11	n.h	n.h	PROPN
ejpam-4905	537	12	.	.	PUNCT
ejpam-4905	538	1	mohammad	mohammad	PROPN
ejpam-4905	538	2	.	.	PUNCT
ejpam-4905	539	1	some	some	DET
ejpam-4905	539	2	properties	property	NOUN
ejpam-4905	539	3	and	and	CCONJ
ejpam-4905	539	4	realization	realization	NOUN
ejpam-4905	539	5	problems	problem	NOUN
ejpam-4905	539	6	involving	involve	VERB
ejpam-4905	539	7	connected	connected	ADJ
ejpam-4905	539	8	outer	outer	ADJ
ejpam-4905	539	9	-	-	PUNCT
ejpam-4905	539	10	hop	hop	NOUN
ejpam-4905	539	11	independent	independent	ADJ
ejpam-4905	539	12	hop	hop	NOUN
ejpam-4905	539	13	domination	domination	NOUN
ejpam-4905	539	14	in	in	ADP
ejpam-4905	539	15	graphs	graph	NOUN
ejpam-4905	539	16	.	.	PUNCT
ejpam-4905	540	1	eur	eur	PROPN
ejpam-4905	540	2	.	.	PUNCT
ejpam-4905	541	1	j.	j.	PROPN
ejpam-4905	541	2	pure	pure	PROPN
ejpam-4905	541	3	appl	appl	PROPN
ejpam-4905	541	4	.	.	PUNCT
ejpam-4905	541	5	math	math	PROPN
ejpam-4905	541	6	.	.	PUNCT
ejpam-4905	541	7	,	,	PUNCT
ejpam-4905	541	8	16(3):1848–1861	16(3):1848–1861	NUM
ejpam-4905	541	9	,	,	PUNCT
ejpam-4905	541	10	2023	2023	NUM
ejpam-4905	541	11	.	.	PUNCT
ejpam-4905	542	1	[	[	X
ejpam-4905	542	2	7	7	X
ejpam-4905	542	3	]	]	PUNCT
ejpam-4905	542	4	canoy	canoy	ADJ
ejpam-4905	542	5	s.	s.	PROPN
ejpam-4905	542	6	jr	jr	PROPN
ejpam-4905	542	7	.	.	PROPN
ejpam-4905	542	8	and	and	CCONJ
ejpam-4905	542	9	g.	g.	PROPN
ejpam-4905	542	10	salasalan	salasalan	NOUN
ejpam-4905	542	11	.	.	PUNCT
ejpam-4905	543	1	revisiting	revisit	VERB
ejpam-4905	543	2	domination	domination	NOUN
ejpam-4905	543	3	,	,	PUNCT
ejpam-4905	543	4	hop	hop	NOUN
ejpam-4905	543	5	domination	domination	NOUN
ejpam-4905	543	6	,	,	PUNCT
ejpam-4905	543	7	and	and	CCONJ
ejpam-4905	543	8	global	global	ADJ
ejpam-4905	543	9	hop	hop	NOUN
ejpam-4905	543	10	domination	domination	NOUN
ejpam-4905	543	11	in	in	ADP
ejpam-4905	543	12	graphs	graph	NOUN
ejpam-4905	543	13	.	.	PUNCT
ejpam-4905	544	1	eur	eur	PROPN
ejpam-4905	544	2	.	.	PUNCT
ejpam-4905	545	1	j.	j.	PROPN
ejpam-4905	545	2	pure	pure	PROPN
ejpam-4905	545	3	appl	appl	PROPN
ejpam-4905	545	4	.	.	PUNCT
ejpam-4905	545	5	math	math	PROPN
ejpam-4905	545	6	.	.	PUNCT
ejpam-4905	545	7	,	,	PUNCT
ejpam-4905	545	8	14:1415–1428	14:1415–1428	NUM
ejpam-4905	545	9	,	,	PUNCT
ejpam-4905	545	10	2021	2021	NUM
ejpam-4905	545	11	.	.	PUNCT
ejpam-4905	546	1	[	[	X
ejpam-4905	546	2	8	8	NUM
ejpam-4905	546	3	]	]	X
ejpam-4905	546	4	s.	s.	PROPN
ejpam-4905	546	5	canoy	canoy	PROPN
ejpam-4905	546	6	jr	jr	PROPN
ejpam-4905	546	7	and	and	CCONJ
ejpam-4905	546	8	j.	j.	PROPN
ejpam-4905	546	9	hassan	hassan	PROPN
ejpam-4905	546	10	.	.	PUNCT
ejpam-4905	547	1	weakly	weakly	ADJ
ejpam-4905	547	2	convex	convex	VERB
ejpam-4905	547	3	hop	hop	NOUN
ejpam-4905	547	4	dominating	dominating	NOUN
ejpam-4905	547	5	sets	set	NOUN
ejpam-4905	547	6	in	in	ADP
ejpam-4905	547	7	graphs	graph	NOUN
ejpam-4905	547	8	.	.	PUNCT
ejpam-4905	548	1	eur	eur	PROPN
ejpam-4905	548	2	.	.	PUNCT
ejpam-4905	549	1	j.	j.	PROPN
ejpam-4905	549	2	pure	pure	PROPN
ejpam-4905	549	3	appl	appl	PROPN
ejpam-4905	549	4	.	.	PUNCT
ejpam-4905	549	5	math	math	PROPN
ejpam-4905	549	6	.	.	PUNCT
ejpam-4905	549	7	,	,	PUNCT
ejpam-4905	549	8	16(2):1196–1211	16(2):1196–1211	NUM
ejpam-4905	549	9	,	,	PUNCT
ejpam-4905	549	10	2023	2023	NUM
ejpam-4905	549	11	.	.	PUNCT
ejpam-4905	550	1	[	[	X
ejpam-4905	550	2	9	9	NUM
ejpam-4905	550	3	]	]	PUNCT
ejpam-4905	550	4	c.	c.	PROPN
ejpam-4905	550	5	natarajan	natarajan	PROPN
ejpam-4905	550	6	and	and	CCONJ
ejpam-4905	550	7	s.	s.	PROPN
ejpam-4905	550	8	ayyaswamy	ayyaswamy	PROPN
ejpam-4905	550	9	.	.	PUNCT
ejpam-4905	551	1	hop	hop	PROPN
ejpam-4905	551	2	domination	domination	NOUN
ejpam-4905	551	3	in	in	ADP
ejpam-4905	551	4	graphs	graphs	PROPN
ejpam-4905	551	5	ii	ii	PROPN
ejpam-4905	551	6	.	.	PUNCT
ejpam-4905	551	7	versita	versita	PROPN
ejpam-4905	551	8	,	,	PUNCT
ejpam-4905	551	9	,	,	PUNCT
ejpam-4905	551	10	23(2):187	23(2):187	NUM
ejpam-4905	551	11	–	–	PUNCT
ejpam-4905	551	12	199	199	NUM
ejpam-4905	551	13	,	,	PUNCT
ejpam-4905	551	14	2015	2015	NUM
ejpam-4905	551	15	.	.	PUNCT
ejpam-4905	552	1	[	[	X
ejpam-4905	552	2	10	10	NUM
ejpam-4905	552	3	]	]	X
ejpam-4905	552	4	y.	y.	PROPN
ejpam-4905	552	5	pabilona	pabilona	PROPN
ejpam-4905	552	6	and	and	CCONJ
ejpam-4905	552	7	h.	h.	PROPN
ejpam-4905	552	8	rara	rara	PROPN
ejpam-4905	552	9	.	.	PUNCT
ejpam-4905	553	1	total	total	ADJ
ejpam-4905	553	2	hop	hop	NOUN
ejpam-4905	553	3	dominating	dominating	NOUN
ejpam-4905	553	4	sets	set	NOUN
ejpam-4905	553	5	in	in	ADP
ejpam-4905	553	6	the	the	DET
ejpam-4905	553	7	join	join	NOUN
ejpam-4905	553	8	,	,	PUNCT
ejpam-4905	553	9	corona	corona	NOUN
ejpam-4905	553	10	and	and	CCONJ
ejpam-4905	553	11	lexicographic	lexicographic	ADJ
ejpam-4905	553	12	product	product	NOUN
ejpam-4905	553	13	of	of	ADP
ejpam-4905	553	14	graphs	graph	NOUN
ejpam-4905	553	15	.	.	PUNCT
ejpam-4905	554	1	jour	jour	X
ejpam-4905	554	2	.	.	PROPN
ejpam-4905	554	3	of	of	ADP
ejpam-4905	554	4	algebra	algebra	NOUN
ejpam-4905	554	5	and	and	CCONJ
ejpam-4905	554	6	appl	appl	PROPN
ejpam-4905	554	7	.	.	PROPN
ejpam-4905	554	8	math	math	PROPN
ejpam-4905	554	9	.	.	PUNCT
ejpam-4905	554	10	,	,	PUNCT
ejpam-4905	554	11	2:105–115	2:105–115	NUM
ejpam-4905	554	12	,	,	PUNCT
ejpam-4905	554	13	2017	2017	NUM
ejpam-4905	554	14	.	.	PUNCT
ejpam-4905	555	1	[	[	X
ejpam-4905	555	2	11	11	NUM
ejpam-4905	555	3	]	]	X
ejpam-4905	555	4	y.	y.	NOUN
ejpam-4905	555	5	pabilona	pabilona	PROPN
ejpam-4905	555	6	and	and	CCONJ
ejpam-4905	555	7	h.	h.	PROPN
ejpam-4905	555	8	rara	rara	PROPN
ejpam-4905	555	9	.	.	PUNCT
ejpam-4905	556	1	connected	connect	VERB
ejpam-4905	556	2	hop	hop	NOUN
ejpam-4905	556	3	domination	domination	NOUN
ejpam-4905	556	4	in	in	ADP
ejpam-4905	556	5	graphs	graph	NOUN
ejpam-4905	556	6	under	under	ADP
ejpam-4905	556	7	some	some	DET
ejpam-4905	556	8	binary	binary	ADJ
ejpam-4905	556	9	operations	operation	NOUN
ejpam-4905	556	10	.	.	PUNCT
ejpam-4905	557	1	asian	asian	ADJ
ejpam-4905	557	2	-	-	PUNCT
ejpam-4905	557	3	eur	eur	NOUN
ejpam-4905	557	4	.	.	PUNCT
ejpam-4905	558	1	j.	j.	PROPN
ejpam-4905	558	2	math	math	PROPN
ejpam-4905	558	3	.	.	PROPN
ejpam-4905	558	4	,	,	PUNCT
ejpam-4905	558	5	11(5):1850075–1–1850075–11	11(5):1850075–1–1850075–11	NUM
ejpam-4905	558	6	,	,	PUNCT
ejpam-4905	558	7	2018	2018	NUM
ejpam-4905	558	8	.	.	PUNCT
ejpam-4905	559	1	[	[	X
ejpam-4905	559	2	12	12	NUM
ejpam-4905	559	3	]	]	X
ejpam-4905	559	4	g.	g.	PROPN
ejpam-4905	559	5	salasalan	salasalan	NOUN
ejpam-4905	559	6	,	,	PUNCT
ejpam-4905	559	7	s.	s.	PROPN
ejpam-4905	559	8	canoy	canoy	PROPN
ejpam-4905	559	9	jr	jr	PROPN
ejpam-4905	559	10	.	.	PROPN
ejpam-4905	559	11	,	,	PUNCT
ejpam-4905	559	12	and	and	CCONJ
ejpam-4905	559	13	a.	a.	PROPN
ejpam-4905	559	14	aradais	aradais	PROPN
ejpam-4905	559	15	.	.	PROPN
ejpam-4905	560	1	global	global	PROPN
ejpam-4905	560	2	hop	hop	PROPN
ejpam-4905	560	3	domination	domination	PROPN
ejpam-4905	560	4	numbers	number	NOUN
ejpam-4905	560	5	of	of	ADP
ejpam-4905	560	6	graphs	graph	NOUN
ejpam-4905	560	7	.	.	PUNCT
ejpam-4905	561	1	eur	eur	PROPN
ejpam-4905	561	2	.	.	PUNCT
ejpam-4905	562	1	j.	j.	PROPN
ejpam-4905	562	2	pure	pure	PROPN
ejpam-4905	562	3	appl	appl	PROPN
ejpam-4905	562	4	.	.	PUNCT
ejpam-4905	562	5	math	math	PROPN
ejpam-4905	562	6	.	.	PUNCT
ejpam-4905	562	7	,	,	PUNCT
ejpam-4905	562	8	14(1):112–125	14(1):112–125	NUM
ejpam-4905	562	9	,	,	PUNCT
ejpam-4905	562	10	2021	2021	NUM
ejpam-4905	562	11	.	.	PUNCT