id	sid	tid	token	lemma	pos
ejpam-5625	1	1	european	european	PROPN
ejpam-5625	1	2	journal	journal	PROPN
ejpam-5625	1	3	of	of	ADP
ejpam-5625	1	4	pure	pure	ADJ
ejpam-5625	1	5	and	and	CCONJ
ejpam-5625	1	6	applied	applied	ADJ
ejpam-5625	1	7	mathematics	mathematic	NOUN
ejpam-5625	1	8	2025	2025	NUM
ejpam-5625	1	9	,	,	PUNCT
ejpam-5625	1	10	vol	vol	NOUN
ejpam-5625	1	11	.	.	PROPN
ejpam-5625	1	12	18	18	NUM
ejpam-5625	1	13	,	,	PUNCT
ejpam-5625	1	14	issue	issue	NOUN
ejpam-5625	1	15	1	1	NUM
ejpam-5625	1	16	,	,	PUNCT
ejpam-5625	1	17	article	article	NOUN
ejpam-5625	1	18	number	number	NOUN
ejpam-5625	1	19	5625	5625	NUM
ejpam-5625	1	20	issn	issn	PROPN
ejpam-5625	1	21	1307	1307	NUM
ejpam-5625	1	22	-	-	SYM
ejpam-5625	1	23	5543	5543	NUM
ejpam-5625	1	24	–	–	PUNCT
ejpam-5625	1	25	ejpam.com	ejpam.com	X
ejpam-5625	1	26	published	publish	VERB
ejpam-5625	1	27	by	by	ADP
ejpam-5625	1	28	new	new	PROPN
ejpam-5625	1	29	york	york	PROPN
ejpam-5625	1	30	business	business	PROPN
ejpam-5625	1	31	global	global	ADJ
ejpam-5625	1	32	characterizing	characterize	VERB
ejpam-5625	1	33	2	2	NUM
ejpam-5625	1	34	-	-	PUNCT
ejpam-5625	1	35	distance	distance	NOUN
ejpam-5625	1	36	certified	certify	VERB
ejpam-5625	1	37	hop	hop	NOUN
ejpam-5625	1	38	dominating	dominating	NOUN
ejpam-5625	1	39	sets	set	NOUN
ejpam-5625	1	40	using	use	VERB
ejpam-5625	1	41	co	co	ADJ
ejpam-5625	1	42	-	-	ADJ
ejpam-5625	1	43	certified	certify	VERB
ejpam-5625	1	44	pointwise	pointwise	PROPN
ejpam-5625	1	45	non	non	ADJ
ejpam-5625	1	46	-	-	ADJ
ejpam-5625	1	47	domination	domination	ADJ
ejpam-5625	1	48	concept	concept	NOUN
ejpam-5625	1	49	noor	noor	PROPN
ejpam-5625	1	50	-	-	PUNCT
ejpam-5625	1	51	sharief	sharief	PROPN
ejpam-5625	1	52	n.	n.	PROPN
ejpam-5625	1	53	ulal1	ulal1	PROPN
ejpam-5625	1	54	,	,	PUNCT
ejpam-5625	1	55	javier	javier	PROPN
ejpam-5625	1	56	a.	a.	PROPN
ejpam-5625	1	57	hassan1,2,∗	hassan1,2,∗	PROPN
ejpam-5625	1	58	,	,	PUNCT
ejpam-5625	1	59	mercedita	mercedita	PROPN
ejpam-5625	1	60	a.	a.	NOUN
ejpam-5625	1	61	langamin1	langamin1	PROPN
ejpam-5625	1	62	,	,	PUNCT
ejpam-5625	1	63	noor	noor	PROPN
ejpam-5625	1	64	-	-	PUNCT
ejpam-5625	1	65	han	han	PROPN
ejpam-5625	1	66	n.	n.	PROPN
ejpam-5625	1	67	ulal1	ulal1	PROPN
ejpam-5625	2	1	1mathematics	1mathematics	NUM
ejpam-5625	2	2	and	and	CCONJ
ejpam-5625	2	3	sciences	sciences	PROPN
ejpam-5625	2	4	department	department	PROPN
ejpam-5625	2	5	,	,	PUNCT
ejpam-5625	2	6	college	college	NOUN
ejpam-5625	2	7	of	of	ADP
ejpam-5625	2	8	arts	art	NOUN
ejpam-5625	2	9	and	and	CCONJ
ejpam-5625	2	10	sciences	science	NOUN
ejpam-5625	2	11	,	,	PUNCT
ejpam-5625	2	12	msu	msu	PROPN
ejpam-5625	2	13	tawi	tawi	PROPN
ejpam-5625	2	14	-	-	PUNCT
ejpam-5625	2	15	tawi	tawi	PROPN
ejpam-5625	2	16	college	college	PROPN
ejpam-5625	2	17	of	of	ADP
ejpam-5625	2	18	technology	technology	NOUN
ejpam-5625	2	19	and	and	CCONJ
ejpam-5625	2	20	oceanography	oceanography	NOUN
ejpam-5625	2	21	,	,	PUNCT
ejpam-5625	2	22	bongao	bongao	NOUN
ejpam-5625	2	23	,	,	PUNCT
ejpam-5625	2	24	tawi	tawi	NOUN
ejpam-5625	2	25	-	-	PUNCT
ejpam-5625	2	26	tawi	tawi	NOUN
ejpam-5625	2	27	,	,	PUNCT
ejpam-5625	2	28	philippines	philippine	NOUN
ejpam-5625	2	29	2department	2department	NUM
ejpam-5625	2	30	of	of	ADP
ejpam-5625	2	31	mathematics	mathematic	NOUN
ejpam-5625	2	32	,	,	PUNCT
ejpam-5625	2	33	college	college	NOUN
ejpam-5625	2	34	of	of	ADP
ejpam-5625	2	35	science	science	PROPN
ejpam-5625	2	36	,	,	PUNCT
ejpam-5625	2	37	korea	korea	PROPN
ejpam-5625	2	38	university	university	PROPN
ejpam-5625	2	39	,	,	PUNCT
ejpam-5625	2	40	seoul	seoul	PROPN
ejpam-5625	2	41	,	,	PUNCT
ejpam-5625	2	42	south	south	PROPN
ejpam-5625	2	43	korea	korea	PROPN
ejpam-5625	2	44	abstract	abstract	NOUN
ejpam-5625	2	45	.	.	PUNCT
ejpam-5625	3	1	let	let	VERB
ejpam-5625	3	2	g	g	PRON
ejpam-5625	3	3	be	be	AUX
ejpam-5625	3	4	a	a	DET
ejpam-5625	3	5	graph	graph	NOUN
ejpam-5625	3	6	.	.	PUNCT
ejpam-5625	4	1	then	then	ADV
ejpam-5625	4	2	c	c	PROPN
ejpam-5625	4	3	⊆	⊆	NUM
ejpam-5625	4	4	v	v	X
ejpam-5625	4	5	(	(	PUNCT
ejpam-5625	4	6	g	g	NOUN
ejpam-5625	4	7	)	)	PUNCT
ejpam-5625	4	8	is	be	AUX
ejpam-5625	4	9	called	call	VERB
ejpam-5625	4	10	a	a	DET
ejpam-5625	4	11	2	2	NUM
ejpam-5625	4	12	-	-	PUNCT
ejpam-5625	4	13	distance	distance	NOUN
ejpam-5625	4	14	certified	certify	VERB
ejpam-5625	4	15	hop	hop	NOUN
ejpam-5625	4	16	dominating	dominating	NOUN
ejpam-5625	4	17	if	if	SCONJ
ejpam-5625	4	18	∀	∀	NOUN
ejpam-5625	4	19	x	x	SYM
ejpam-5625	4	20	∈	∈	NOUN
ejpam-5625	4	21	v	v	X
ejpam-5625	4	22	(	(	PUNCT
ejpam-5625	4	23	g)\c	g)\c	NOUN
ejpam-5625	4	24	,	,	PUNCT
ejpam-5625	4	25	there	there	PRON
ejpam-5625	4	26	exists	exist	VERB
ejpam-5625	4	27	y	y	PROPN
ejpam-5625	4	28	∈	∈	PROPN
ejpam-5625	4	29	c	c	PROPN
ejpam-5625	4	30	such	such	ADJ
ejpam-5625	4	31	that	that	DET
ejpam-5625	4	32	dg(x	dg(x	PROPN
ejpam-5625	4	33	,	,	PUNCT
ejpam-5625	4	34	y	y	NOUN
ejpam-5625	4	35	)	)	PUNCT
ejpam-5625	4	36	=	=	SYM
ejpam-5625	4	37	2	2	NUM
ejpam-5625	4	38	and	and	CCONJ
ejpam-5625	4	39	∀	∀	NOUN
ejpam-5625	4	40	a	a	DET
ejpam-5625	4	41	∈	∈	PROPN
ejpam-5625	4	42	c	c	X
ejpam-5625	4	43	,	,	PUNCT
ejpam-5625	4	44	|n2	|n2	X
ejpam-5625	4	45	g(a)\c|	g(a)\c|	PUNCT
ejpam-5625	5	1	=	=	PUNCT
ejpam-5625	5	2	0	0	PUNCT
ejpam-5625	5	3	or	or	CCONJ
ejpam-5625	5	4	|n2	|n2	PROPN
ejpam-5625	5	5	g(a)\c|	g(a)\c|	X
ejpam-5625	5	6	≥	≥	NOUN
ejpam-5625	5	7	2	2	NUM
ejpam-5625	5	8	.	.	PUNCT
ejpam-5625	6	1	the	the	DET
ejpam-5625	6	2	2	2	NUM
ejpam-5625	6	3	-	-	PUNCT
ejpam-5625	6	4	distance	distance	NOUN
ejpam-5625	6	5	certified	certify	VERB
ejpam-5625	6	6	hop	hop	NOUN
ejpam-5625	6	7	domination	domination	NOUN
ejpam-5625	6	8	number	number	NOUN
ejpam-5625	6	9	of	of	ADP
ejpam-5625	6	10	g	g	NOUN
ejpam-5625	6	11	,	,	PUNCT
ejpam-5625	6	12	denoted	denote	VERB
ejpam-5625	6	13	by	by	ADP
ejpam-5625	6	14	γ2ch(g	γ2ch(g	NOUN
ejpam-5625	6	15	)	)	PUNCT
ejpam-5625	6	16	,	,	PUNCT
ejpam-5625	6	17	is	be	AUX
ejpam-5625	6	18	the	the	DET
ejpam-5625	6	19	minimum	minimum	ADJ
ejpam-5625	6	20	cardinality	cardinality	NOUN
ejpam-5625	6	21	among	among	ADP
ejpam-5625	6	22	all	all	DET
ejpam-5625	6	23	2	2	NUM
ejpam-5625	6	24	-	-	PUNCT
ejpam-5625	6	25	distance	distance	NOUN
ejpam-5625	6	26	certified	certify	VERB
ejpam-5625	6	27	hop	hop	NOUN
ejpam-5625	6	28	dominating	dominating	NOUN
ejpam-5625	6	29	sets	set	NOUN
ejpam-5625	6	30	of	of	ADP
ejpam-5625	6	31	g.	g.	PROPN
ejpam-5625	6	32	in	in	ADP
ejpam-5625	6	33	this	this	DET
ejpam-5625	6	34	study	study	NOUN
ejpam-5625	6	35	,	,	PUNCT
ejpam-5625	6	36	the	the	DET
ejpam-5625	6	37	researchers	researcher	NOUN
ejpam-5625	6	38	give	give	VERB
ejpam-5625	6	39	some	some	DET
ejpam-5625	6	40	properties	property	NOUN
ejpam-5625	6	41	of	of	ADP
ejpam-5625	6	42	this	this	DET
ejpam-5625	6	43	new	new	ADJ
ejpam-5625	6	44	concept	concept	NOUN
ejpam-5625	6	45	on	on	ADP
ejpam-5625	6	46	some	some	DET
ejpam-5625	6	47	graphs	graph	NOUN
ejpam-5625	6	48	,	,	PUNCT
ejpam-5625	6	49	and	and	CCONJ
ejpam-5625	6	50	present	present	VERB
ejpam-5625	6	51	some	some	DET
ejpam-5625	6	52	its	its	PRON
ejpam-5625	6	53	connections	connection	NOUN
ejpam-5625	6	54	with	with	ADP
ejpam-5625	6	55	others	other	NOUN
ejpam-5625	6	56	parameters	parameter	NOUN
ejpam-5625	6	57	.	.	PUNCT
ejpam-5625	7	1	also	also	ADV
ejpam-5625	7	2	,	,	PUNCT
ejpam-5625	7	3	the	the	DET
ejpam-5625	7	4	researchers	researcher	NOUN
ejpam-5625	7	5	introduce	introduce	VERB
ejpam-5625	7	6	co	co	ADJ
ejpam-5625	7	7	-	-	ADJ
ejpam-5625	7	8	certified	certify	VERB
ejpam-5625	7	9	pointwise	pointwise	NOUN
ejpam-5625	7	10	nondomination	nondomination	NOUN
ejpam-5625	7	11	(	(	PUNCT
ejpam-5625	7	12	co	co	ADJ
ejpam-5625	7	13	-	-	ADJ
ejpam-5625	7	14	certified	certified	ADJ
ejpam-5625	7	15	pnd	pnd	NOUN
ejpam-5625	7	16	)	)	PUNCT
ejpam-5625	7	17	to	to	PART
ejpam-5625	7	18	characterize	characterize	VERB
ejpam-5625	7	19	the	the	DET
ejpam-5625	7	20	2	2	NUM
ejpam-5625	7	21	-	-	PUNCT
ejpam-5625	7	22	distance	distance	NOUN
ejpam-5625	7	23	certified	certify	VERB
ejpam-5625	7	24	hop	hop	NOUN
ejpam-5625	7	25	dominating	dominating	NOUN
ejpam-5625	7	26	sets	set	NOUN
ejpam-5625	7	27	in	in	ADP
ejpam-5625	7	28	the	the	DET
ejpam-5625	7	29	join	join	NOUN
ejpam-5625	7	30	of	of	ADP
ejpam-5625	7	31	two	two	NUM
ejpam-5625	7	32	graphs	graph	NOUN
ejpam-5625	7	33	.	.	PUNCT
ejpam-5625	8	1	they	they	PRON
ejpam-5625	8	2	obtain	obtain	VERB
ejpam-5625	8	3	some	some	DET
ejpam-5625	8	4	simplified	simplified	ADJ
ejpam-5625	8	5	formulas	formula	NOUN
ejpam-5625	8	6	of	of	ADP
ejpam-5625	8	7	the	the	DET
ejpam-5625	8	8	said	say	VERB
ejpam-5625	8	9	parameter	parameter	NOUN
ejpam-5625	8	10	on	on	ADP
ejpam-5625	8	11	this	this	DET
ejpam-5625	8	12	graph	graph	NOUN
ejpam-5625	8	13	using	use	VERB
ejpam-5625	8	14	this	this	DET
ejpam-5625	8	15	newly	newly	ADV
ejpam-5625	8	16	defined	define	VERB
ejpam-5625	8	17	concept	concept	NOUN
ejpam-5625	8	18	and	and	CCONJ
ejpam-5625	8	19	some	some	DET
ejpam-5625	8	20	characterizations	characterization	NOUN
ejpam-5625	8	21	formulated	formulate	VERB
ejpam-5625	8	22	.	.	PUNCT
ejpam-5625	9	1	2020	2020	NUM
ejpam-5625	9	2	mathematics	mathematic	NOUN
ejpam-5625	9	3	subject	subject	NOUN
ejpam-5625	9	4	classifications	classification	NOUN
ejpam-5625	9	5	:	:	PUNCT
ejpam-5625	9	6	05c69	05c69	X
ejpam-5625	9	7	key	key	ADJ
ejpam-5625	9	8	words	word	NOUN
ejpam-5625	9	9	and	and	CCONJ
ejpam-5625	9	10	phrases	phrase	NOUN
ejpam-5625	9	11	:	:	PUNCT
ejpam-5625	9	12	2	2	NUM
ejpam-5625	9	13	-	-	PUNCT
ejpam-5625	9	14	distance	distance	NOUN
ejpam-5625	9	15	certified	certify	VERB
ejpam-5625	9	16	set	set	NOUN
ejpam-5625	9	17	,	,	PUNCT
ejpam-5625	9	18	co	co	ADJ
ejpam-5625	9	19	-	-	ADJ
ejpam-5625	9	20	certified	certify	VERB
ejpam-5625	9	21	pointwise	pointwise	PROPN
ejpam-5625	9	22	non	non	ADJ
ejpam-5625	9	23	-	-	NOUN
ejpam-5625	9	24	domination	domination	ADJ
ejpam-5625	9	25	,	,	PUNCT
ejpam-5625	9	26	2distance	2distance	NUM
ejpam-5625	9	27	certified	certify	VERB
ejpam-5625	9	28	hop	hop	NOUN
ejpam-5625	9	29	dominating	dominating	NOUN
ejpam-5625	9	30	set	set	NOUN
ejpam-5625	9	31	,	,	PUNCT
ejpam-5625	9	32	2	2	NUM
ejpam-5625	9	33	-	-	PUNCT
ejpam-5625	9	34	distance	distance	NOUN
ejpam-5625	9	35	certified	certify	VERB
ejpam-5625	9	36	hop	hop	NOUN
ejpam-5625	9	37	domination	domination	NOUN
ejpam-5625	9	38	number	number	NOUN
ejpam-5625	9	39	1	1	NUM
ejpam-5625	9	40	.	.	PUNCT
ejpam-5625	10	1	introduction	introduction	NOUN
ejpam-5625	10	2	domination	domination	NOUN
ejpam-5625	10	3	in	in	ADP
ejpam-5625	10	4	graph	graph	NOUN
ejpam-5625	10	5	theory	theory	NOUN
ejpam-5625	10	6	is	be	AUX
ejpam-5625	10	7	a	a	DET
ejpam-5625	10	8	fundamental	fundamental	ADJ
ejpam-5625	10	9	concept	concept	NOUN
ejpam-5625	10	10	that	that	PRON
ejpam-5625	10	11	explores	explore	VERB
ejpam-5625	10	12	how	how	SCONJ
ejpam-5625	10	13	subsets	subset	NOUN
ejpam-5625	10	14	of	of	ADP
ejpam-5625	10	15	vertices	vertex	NOUN
ejpam-5625	10	16	can	can	AUX
ejpam-5625	10	17	control	control	VERB
ejpam-5625	10	18	or	or	CCONJ
ejpam-5625	10	19	influence	influence	VERB
ejpam-5625	10	20	the	the	DET
ejpam-5625	10	21	entire	entire	ADJ
ejpam-5625	10	22	graph	graph	NOUN
ejpam-5625	10	23	.	.	PUNCT
ejpam-5625	11	1	a	a	DET
ejpam-5625	11	2	dominating	dominating	NOUN
ejpam-5625	11	3	set	set	NOUN
ejpam-5625	11	4	for	for	ADP
ejpam-5625	11	5	a	a	DET
ejpam-5625	11	6	graph	graph	NOUN
ejpam-5625	11	7	g	g	NOUN
ejpam-5625	11	8	is	be	AUX
ejpam-5625	11	9	defined	define	VERB
ejpam-5625	11	10	as	as	ADP
ejpam-5625	11	11	a	a	DET
ejpam-5625	11	12	subset	subset	NOUN
ejpam-5625	11	13	of	of	ADP
ejpam-5625	11	14	vertices	vertex	NOUN
ejpam-5625	11	15	d	d	X
ejpam-5625	11	16	such	such	ADJ
ejpam-5625	11	17	that	that	SCONJ
ejpam-5625	11	18	every	every	DET
ejpam-5625	11	19	vertex	vertex	NOUN
ejpam-5625	11	20	in	in	ADP
ejpam-5625	11	21	g	g	PROPN
ejpam-5625	11	22	is	be	AUX
ejpam-5625	11	23	either	either	CCONJ
ejpam-5625	11	24	included	include	VERB
ejpam-5625	11	25	in	in	ADP
ejpam-5625	11	26	d	d	NOUN
ejpam-5625	11	27	or	or	CCONJ
ejpam-5625	11	28	is	be	AUX
ejpam-5625	11	29	adjacent	adjacent	ADJ
ejpam-5625	11	30	to	to	ADP
ejpam-5625	11	31	at	at	ADV
ejpam-5625	11	32	least	least	ADV
ejpam-5625	11	33	one	one	NUM
ejpam-5625	11	34	vertex	vertex	NOUN
ejpam-5625	11	35	in	in	ADP
ejpam-5625	11	36	d.	d.	PROPN
ejpam-5625	11	37	this	this	DET
ejpam-5625	11	38	concept	concept	NOUN
ejpam-5625	11	39	is	be	AUX
ejpam-5625	11	40	crucial	crucial	ADJ
ejpam-5625	11	41	in	in	ADP
ejpam-5625	11	42	various	various	ADJ
ejpam-5625	11	43	applications	application	NOUN
ejpam-5625	11	44	such	such	ADJ
ejpam-5625	11	45	as	as	ADP
ejpam-5625	11	46	in	in	ADP
ejpam-5625	11	47	network	network	NOUN
ejpam-5625	11	48	design	design	NOUN
ejpam-5625	11	49	,	,	PUNCT
ejpam-5625	11	50	resource	resource	NOUN
ejpam-5625	11	51	allocation	allocation	NOUN
ejpam-5625	11	52	,	,	PUNCT
ejpam-5625	11	53	social	social	ADJ
ejpam-5625	11	54	network	network	NOUN
ejpam-5625	11	55	analysis	analysis	NOUN
ejpam-5625	11	56	,	,	PUNCT
ejpam-5625	11	57	and	and	CCONJ
ejpam-5625	11	58	in	in	ADP
ejpam-5625	11	59	infrastructure	infrastructure	NOUN
ejpam-5625	11	60	networks	network	NOUN
ejpam-5625	11	61	.	.	PUNCT
ejpam-5625	12	1	∗corresponding	∗corresponde	VERB
ejpam-5625	12	2	author	author	NOUN
ejpam-5625	12	3	.	.	PUNCT
ejpam-5625	13	1	doi	doi	NOUN
ejpam-5625	13	2	:	:	PUNCT
ejpam-5625	13	3	https://doi.org/10.29020/nybg.ejpam.v18i1.5625	https://doi.org/10.29020/nybg.ejpam.v18i1.5625	NOUN
ejpam-5625	13	4	email	email	NOUN
ejpam-5625	13	5	addresses	address	NOUN
ejpam-5625	13	6	:	:	PUNCT
ejpam-5625	13	7	noor-shariefulal@msutawi-tawi.edu.ph	noor-shariefulal@msutawi-tawi.edu.ph	PROPN
ejpam-5625	13	8	(	(	PUNCT
ejpam-5625	13	9	n.s	n.s	PROPN
ejpam-5625	13	10	.	.	PROPN
ejpam-5625	13	11	ulal	ulal	PROPN
ejpam-5625	13	12	)	)	PUNCT
ejpam-5625	14	1	javierhassan@msutawi-tawi.edu.ph	javierhassan@msutawi-tawi.edu.ph	PROPN
ejpam-5625	14	2	(	(	PUNCT
ejpam-5625	14	3	j.	j.	PROPN
ejpam-5625	14	4	hassan	hassan	PROPN
ejpam-5625	14	5	)	)	PUNCT
ejpam-5625	14	6	merceditalangamin@msutawi-tawi.edu.ph	merceditalangamin@msutawi-tawi.edu.ph	PROPN
ejpam-5625	14	7	(	(	PUNCT
ejpam-5625	14	8	m.	m.	NOUN
ejpam-5625	14	9	langamin	langamin	PROPN
ejpam-5625	14	10	)	)	PUNCT
ejpam-5625	15	1	noorhanulal@msutawi-tawi.edu.ph	noorhanulal@msutawi-tawi.edu.ph	PROPN
ejpam-5625	15	2	(	(	PUNCT
ejpam-5625	15	3	n.h	n.h	PROPN
ejpam-5625	15	4	.	.	PUNCT
ejpam-5625	15	5	ulal	ulal	PROPN
ejpam-5625	15	6	)	)	PUNCT
ejpam-5625	16	1	https://www.ejpam.com	https://www.ejpam.com	X
ejpam-5625	17	1	1	1	NUM
ejpam-5625	17	2	copyright	copyright	NOUN
ejpam-5625	17	3	:	:	PUNCT
ejpam-5625	17	4	©	©	PROPN
ejpam-5625	17	5	2025	2025	NUM
ejpam-5625	17	6	the	the	DET
ejpam-5625	17	7	author(s	author(s	NOUN
ejpam-5625	17	8	)	)	PUNCT
ejpam-5625	17	9	.	.	PUNCT
ejpam-5625	18	1	(	(	PUNCT
ejpam-5625	18	2	cc	cc	NOUN
ejpam-5625	18	3	by	by	ADP
ejpam-5625	18	4	-	-	PUNCT
ejpam-5625	18	5	nc	nc	PROPN
ejpam-5625	18	6	4.0	4.0	NUM
ejpam-5625	18	7	)	)	PUNCT
ejpam-5625	18	8	n.	n.	PROPN
ejpam-5625	18	9	s.	s.	PROPN
ejpam-5625	18	10	ulal	ulal	PROPN
ejpam-5625	19	1	et	et	PROPN
ejpam-5625	19	2	al	al	PROPN
ejpam-5625	19	3	.	.	PUNCT
ejpam-5625	19	4	/	/	SYM
ejpam-5625	19	5	eur	eur	PROPN
ejpam-5625	19	6	.	.	PUNCT
ejpam-5625	20	1	j.	j.	PROPN
ejpam-5625	20	2	pure	pure	PROPN
ejpam-5625	20	3	appl	appl	PROPN
ejpam-5625	20	4	.	.	PROPN
ejpam-5625	20	5	math	math	PROPN
ejpam-5625	20	6	,	,	PUNCT
ejpam-5625	20	7	18	18	NUM
ejpam-5625	20	8	(	(	PUNCT
ejpam-5625	20	9	1	1	NUM
ejpam-5625	20	10	)	)	PUNCT
ejpam-5625	20	11	(	(	PUNCT
ejpam-5625	20	12	2025	2025	NUM
ejpam-5625	20	13	)	)	PUNCT
ejpam-5625	20	14	,	,	PUNCT
ejpam-5625	20	15	5625	5625	NUM
ejpam-5625	20	16	2	2	NUM
ejpam-5625	20	17	of	of	ADP
ejpam-5625	20	18	9	9	NUM
ejpam-5625	20	19	the	the	DET
ejpam-5625	20	20	study	study	NOUN
ejpam-5625	20	21	of	of	ADP
ejpam-5625	20	22	domination	domination	NOUN
ejpam-5625	20	23	encompasses	encompass	VERB
ejpam-5625	20	24	various	various	ADJ
ejpam-5625	20	25	parameters	parameter	NOUN
ejpam-5625	20	26	,	,	PUNCT
ejpam-5625	20	27	including	include	VERB
ejpam-5625	20	28	the	the	DET
ejpam-5625	20	29	domination	domination	NOUN
ejpam-5625	20	30	number	number	NOUN
ejpam-5625	20	31	,	,	PUNCT
ejpam-5625	20	32	which	which	PRON
ejpam-5625	20	33	is	be	AUX
ejpam-5625	20	34	the	the	DET
ejpam-5625	20	35	minimum	minimum	ADJ
ejpam-5625	20	36	size	size	NOUN
ejpam-5625	20	37	of	of	ADP
ejpam-5625	20	38	a	a	DET
ejpam-5625	20	39	dominating	dominating	NOUN
ejpam-5625	20	40	set	set	NOUN
ejpam-5625	20	41	,	,	PUNCT
ejpam-5625	20	42	and	and	CCONJ
ejpam-5625	20	43	different	different	ADJ
ejpam-5625	20	44	types	type	NOUN
ejpam-5625	20	45	or	or	CCONJ
ejpam-5625	20	46	variants	variant	NOUN
ejpam-5625	20	47	of	of	ADP
ejpam-5625	20	48	domination	domination	NOUN
ejpam-5625	20	49	such	such	ADJ
ejpam-5625	20	50	as	as	ADP
ejpam-5625	20	51	certified	certify	VERB
ejpam-5625	20	52	domination	domination	NOUN
ejpam-5625	20	53	[	[	X
ejpam-5625	20	54	5	5	NUM
ejpam-5625	20	55	]	]	PUNCT
ejpam-5625	20	56	,	,	PUNCT
ejpam-5625	20	57	grundy	grundy	PROPN
ejpam-5625	20	58	hop	hop	PROPN
ejpam-5625	20	59	domination	domination	PROPN
ejpam-5625	20	60	variants	variant	NOUN
ejpam-5625	21	1	[	[	X
ejpam-5625	21	2	8	8	NUM
ejpam-5625	21	3	,	,	PUNCT
ejpam-5625	21	4	9	9	NUM
ejpam-5625	21	5	]	]	PUNCT
ejpam-5625	21	6	,	,	PUNCT
ejpam-5625	21	7	convex	convex	VERB
ejpam-5625	21	8	hop	hop	NOUN
ejpam-5625	21	9	domination	domination	NOUN
ejpam-5625	21	10	[	[	X
ejpam-5625	21	11	7	7	NUM
ejpam-5625	21	12	]	]	PUNCT
ejpam-5625	21	13	,	,	PUNCT
ejpam-5625	21	14	double	double	ADJ
ejpam-5625	21	15	domination	domination	NOUN
ejpam-5625	21	16	[	[	X
ejpam-5625	21	17	6	6	NUM
ejpam-5625	21	18	]	]	PUNCT
ejpam-5625	21	19	,	,	PUNCT
ejpam-5625	21	20	and	and	CCONJ
ejpam-5625	21	21	many	many	ADJ
ejpam-5625	21	22	more	more	ADJ
ejpam-5625	21	23	.	.	PUNCT
ejpam-5625	22	1	each	each	DET
ejpam-5625	22	2	variation	variation	NOUN
ejpam-5625	22	3	provides	provide	VERB
ejpam-5625	22	4	unique	unique	ADJ
ejpam-5625	22	5	insights	insight	NOUN
ejpam-5625	22	6	and	and	CCONJ
ejpam-5625	22	7	tools	tool	NOUN
ejpam-5625	22	8	for	for	ADP
ejpam-5625	22	9	addressing	address	VERB
ejpam-5625	22	10	specific	specific	ADJ
ejpam-5625	22	11	problems	problem	NOUN
ejpam-5625	22	12	within	within	ADP
ejpam-5625	22	13	graphs	graph	NOUN
ejpam-5625	22	14	.	.	PUNCT
ejpam-5625	23	1	some	some	DET
ejpam-5625	23	2	interesting	interesting	ADJ
ejpam-5625	23	3	studies	study	NOUN
ejpam-5625	23	4	related	relate	VERB
ejpam-5625	23	5	to	to	ADP
ejpam-5625	23	6	domination	domination	NOUN
ejpam-5625	23	7	,	,	PUNCT
ejpam-5625	23	8	certified	certified	ADJ
ejpam-5625	23	9	domination	domination	NOUN
ejpam-5625	23	10	,	,	PUNCT
ejpam-5625	23	11	and	and	CCONJ
ejpam-5625	23	12	hop	hop	NOUN
ejpam-5625	23	13	domination	domination	NOUN
ejpam-5625	23	14	can	can	AUX
ejpam-5625	23	15	be	be	AUX
ejpam-5625	23	16	found	find	VERB
ejpam-5625	23	17	in	in	ADP
ejpam-5625	23	18	[	[	X
ejpam-5625	23	19	1–4	1–4	NOUN
ejpam-5625	23	20	,	,	PUNCT
ejpam-5625	23	21	10	10	NUM
ejpam-5625	23	22	,	,	PUNCT
ejpam-5625	23	23	11	11	NUM
ejpam-5625	23	24	]	]	PUNCT
ejpam-5625	23	25	.	.	PUNCT
ejpam-5625	24	1	in	in	ADP
ejpam-5625	24	2	this	this	DET
ejpam-5625	24	3	paper	paper	NOUN
ejpam-5625	24	4	,	,	PUNCT
ejpam-5625	24	5	new	new	ADJ
ejpam-5625	24	6	parameter	parameter	NOUN
ejpam-5625	24	7	called	call	VERB
ejpam-5625	24	8	2	2	NUM
ejpam-5625	24	9	-	-	PUNCT
ejpam-5625	24	10	distance	distance	NOUN
ejpam-5625	24	11	certified	certify	VERB
ejpam-5625	24	12	hop	hop	NOUN
ejpam-5625	24	13	domination	domination	NOUN
ejpam-5625	24	14	in	in	ADP
ejpam-5625	24	15	a	a	DET
ejpam-5625	24	16	graph	graph	NOUN
ejpam-5625	24	17	is	be	AUX
ejpam-5625	24	18	introduced	introduce	VERB
ejpam-5625	24	19	and	and	CCONJ
ejpam-5625	24	20	investigated	investigate	VERB
ejpam-5625	24	21	.	.	PUNCT
ejpam-5625	25	1	the	the	DET
ejpam-5625	25	2	researchers	researcher	NOUN
ejpam-5625	25	3	believe	believe	VERB
ejpam-5625	25	4	that	that	SCONJ
ejpam-5625	25	5	his	his	PRON
ejpam-5625	25	6	parameter	parameter	NOUN
ejpam-5625	25	7	and	and	CCONJ
ejpam-5625	25	8	its	its	PRON
ejpam-5625	25	9	results	result	NOUN
ejpam-5625	25	10	would	would	AUX
ejpam-5625	25	11	give	give	VERB
ejpam-5625	25	12	additional	additional	ADJ
ejpam-5625	25	13	insights	insight	NOUN
ejpam-5625	25	14	to	to	ADP
ejpam-5625	25	15	researchers	researcher	NOUN
ejpam-5625	25	16	in	in	ADP
ejpam-5625	25	17	the	the	DET
ejpam-5625	25	18	field	field	NOUN
ejpam-5625	25	19	and	and	CCONJ
ejpam-5625	25	20	would	would	AUX
ejpam-5625	25	21	lead	lead	VERB
ejpam-5625	25	22	to	to	ADP
ejpam-5625	25	23	another	another	DET
ejpam-5625	25	24	interesting	interesting	ADJ
ejpam-5625	25	25	topics	topic	NOUN
ejpam-5625	25	26	or	or	CCONJ
ejpam-5625	25	27	network	network	NOUN
ejpam-5625	25	28	application	application	NOUN
ejpam-5625	25	29	in	in	ADP
ejpam-5625	25	30	the	the	DET
ejpam-5625	25	31	future	future	NOUN
ejpam-5625	25	32	.	.	PUNCT
ejpam-5625	26	1	2	2	X
ejpam-5625	26	2	.	.	X
ejpam-5625	26	3	terminology	terminology	NOUN
ejpam-5625	26	4	and	and	CCONJ
ejpam-5625	26	5	notation	notation	NOUN
ejpam-5625	26	6	let	let	VERB
ejpam-5625	26	7	g	g	NOUN
ejpam-5625	26	8	=	=	SYM
ejpam-5625	26	9	(	(	PUNCT
ejpam-5625	26	10	v	v	NOUN
ejpam-5625	26	11	(	(	PUNCT
ejpam-5625	26	12	g	g	NOUN
ejpam-5625	26	13	)	)	PUNCT
ejpam-5625	26	14	,	,	PUNCT
ejpam-5625	26	15	e(g	e(g	PROPN
ejpam-5625	26	16	)	)	PUNCT
ejpam-5625	26	17	)	)	PUNCT
ejpam-5625	26	18	be	be	AUX
ejpam-5625	26	19	a	a	DET
ejpam-5625	26	20	simple	simple	ADJ
ejpam-5625	26	21	and	and	CCONJ
ejpam-5625	26	22	undirected	undirected	ADJ
ejpam-5625	26	23	graph	graph	NOUN
ejpam-5625	26	24	.	.	PUNCT
ejpam-5625	27	1	the	the	DET
ejpam-5625	27	2	distance	distance	NOUN
ejpam-5625	27	3	dg(u	dg(u	NOUN
ejpam-5625	27	4	,	,	PUNCT
ejpam-5625	27	5	v	v	NOUN
ejpam-5625	27	6	)	)	PUNCT
ejpam-5625	27	7	in	in	ADP
ejpam-5625	27	8	g	g	NOUN
ejpam-5625	27	9	of	of	ADP
ejpam-5625	27	10	two	two	NUM
ejpam-5625	27	11	vertices	vertex	NOUN
ejpam-5625	27	12	u	u	NOUN
ejpam-5625	27	13	,	,	PUNCT
ejpam-5625	27	14	v	v	PROPN
ejpam-5625	27	15	is	be	AUX
ejpam-5625	27	16	the	the	DET
ejpam-5625	27	17	length	length	NOUN
ejpam-5625	27	18	of	of	ADP
ejpam-5625	27	19	a	a	DET
ejpam-5625	27	20	shortest	short	ADJ
ejpam-5625	27	21	u	u	NOUN
ejpam-5625	27	22	-	-	NOUN
ejpam-5625	27	23	v	v	ADJ
ejpam-5625	27	24	path	path	NOUN
ejpam-5625	27	25	in	in	ADP
ejpam-5625	27	26	g.	g.	PROPN
ejpam-5625	27	27	the	the	DET
ejpam-5625	27	28	greatest	great	ADJ
ejpam-5625	27	29	distance	distance	NOUN
ejpam-5625	27	30	between	between	ADP
ejpam-5625	27	31	any	any	DET
ejpam-5625	27	32	two	two	NUM
ejpam-5625	27	33	vertices	vertex	NOUN
ejpam-5625	27	34	in	in	ADP
ejpam-5625	27	35	g	g	NOUN
ejpam-5625	27	36	,	,	PUNCT
ejpam-5625	27	37	denoted	denote	VERB
ejpam-5625	27	38	by	by	ADP
ejpam-5625	27	39	diam(g	diam(g	PROPN
ejpam-5625	27	40	)	)	PUNCT
ejpam-5625	27	41	,	,	PUNCT
ejpam-5625	27	42	is	be	AUX
ejpam-5625	27	43	called	call	VERB
ejpam-5625	27	44	the	the	DET
ejpam-5625	27	45	diameter	diameter	NOUN
ejpam-5625	27	46	of	of	ADP
ejpam-5625	27	47	g.	g.	PROPN
ejpam-5625	27	48	two	two	NUM
ejpam-5625	27	49	vertices	vertice	VERB
ejpam-5625	27	50	x	x	X
ejpam-5625	27	51	,	,	PUNCT
ejpam-5625	27	52	y	y	PROPN
ejpam-5625	27	53	of	of	ADP
ejpam-5625	27	54	g	g	PROPN
ejpam-5625	27	55	are	be	AUX
ejpam-5625	27	56	adjacent	adjacent	ADJ
ejpam-5625	27	57	,	,	PUNCT
ejpam-5625	27	58	or	or	CCONJ
ejpam-5625	27	59	neighbors	neighbor	NOUN
ejpam-5625	27	60	,	,	PUNCT
ejpam-5625	27	61	if	if	SCONJ
ejpam-5625	27	62	xy	xy	PROPN
ejpam-5625	27	63	is	be	AUX
ejpam-5625	27	64	an	an	DET
ejpam-5625	27	65	edge	edge	NOUN
ejpam-5625	27	66	of	of	ADP
ejpam-5625	27	67	g.	g.	PROPN
ejpam-5625	27	68	the	the	DET
ejpam-5625	27	69	open	open	ADJ
ejpam-5625	27	70	neighborhood	neighborhood	NOUN
ejpam-5625	27	71	of	of	ADP
ejpam-5625	27	72	x	x	PUNCT
ejpam-5625	27	73	in	in	ADP
ejpam-5625	27	74	g	g	PROPN
ejpam-5625	27	75	is	be	AUX
ejpam-5625	27	76	the	the	DET
ejpam-5625	27	77	set	set	NOUN
ejpam-5625	27	78	ng(x	ng(x	NUM
ejpam-5625	27	79	)	)	PUNCT
ejpam-5625	27	80	=	=	PRON
ejpam-5625	27	81	{	{	PUNCT
ejpam-5625	27	82	y	y	PROPN
ejpam-5625	27	83	∈	∈	PROPN
ejpam-5625	27	84	v	v	NOUN
ejpam-5625	27	85	(	(	PUNCT
ejpam-5625	27	86	g	g	NOUN
ejpam-5625	27	87	)	)	PUNCT
ejpam-5625	27	88	:	:	PUNCT
ejpam-5625	27	89	xy	xy	PROPN
ejpam-5625	27	90	∈	∈	PROPN
ejpam-5625	27	91	e(g	e(g	PROPN
ejpam-5625	27	92	)	)	PUNCT
ejpam-5625	27	93	}	}	PUNCT
ejpam-5625	27	94	.	.	PUNCT
ejpam-5625	28	1	the	the	DET
ejpam-5625	28	2	closed	closed	ADJ
ejpam-5625	28	3	neighborhood	neighborhood	NOUN
ejpam-5625	28	4	of	of	ADP
ejpam-5625	28	5	x	x	SYM
ejpam-5625	28	6	ing	ing	NOUN
ejpam-5625	28	7	is	be	AUX
ejpam-5625	28	8	the	the	DET
ejpam-5625	28	9	setng[x	setng[x	NOUN
ejpam-5625	28	10	]	]	X
ejpam-5625	28	11	=	=	SYM
ejpam-5625	28	12	ng(x)∪{x	ng(x)∪{x	NOUN
ejpam-5625	28	13	}	}	PUNCT
ejpam-5625	28	14	.	.	PUNCT
ejpam-5625	29	1	ifx	ifx	PROPN
ejpam-5625	29	2	⊆	⊆	NUM
ejpam-5625	29	3	v	v	NOUN
ejpam-5625	29	4	(	(	PUNCT
ejpam-5625	29	5	g	g	NOUN
ejpam-5625	29	6	)	)	PUNCT
ejpam-5625	29	7	,	,	PUNCT
ejpam-5625	29	8	the	the	DET
ejpam-5625	29	9	open	open	ADJ
ejpam-5625	29	10	neighborhood	neighborhood	NOUN
ejpam-5625	29	11	of	of	ADP
ejpam-5625	29	12	x	x	PUNCT
ejpam-5625	29	13	in	in	ADP
ejpam-5625	29	14	g	g	PROPN
ejpam-5625	29	15	is	be	AUX
ejpam-5625	29	16	the	the	DET
ejpam-5625	29	17	set	set	NOUN
ejpam-5625	29	18	ng(x	ng(x	NUM
ejpam-5625	29	19	)	)	PUNCT
ejpam-5625	30	1	=	=	SYM
ejpam-5625	30	2	⋃	⋃	NOUN
ejpam-5625	30	3	x∈x	x∈x	NOUN
ejpam-5625	30	4	ng(x	ng(x	NUM
ejpam-5625	30	5	)	)	PUNCT
ejpam-5625	30	6	.	.	PUNCT
ejpam-5625	31	1	the	the	DET
ejpam-5625	31	2	closed	closed	ADJ
ejpam-5625	31	3	neighborhood	neighborhood	NOUN
ejpam-5625	31	4	of	of	ADP
ejpam-5625	31	5	x	x	PUNCT
ejpam-5625	31	6	in	in	ADP
ejpam-5625	31	7	g	g	PROPN
ejpam-5625	31	8	is	be	AUX
ejpam-5625	31	9	the	the	DET
ejpam-5625	31	10	set	set	NOUN
ejpam-5625	31	11	ng[x	ng[x	PROPN
ejpam-5625	31	12	]	]	X
ejpam-5625	31	13	=	=	PUNCT
ejpam-5625	31	14	ng(x	ng(x	X
ejpam-5625	31	15	)	)	PUNCT
ejpam-5625	31	16	∪	∪	ADP
ejpam-5625	31	17	x.	x.	NOUN
ejpam-5625	31	18	a	a	DET
ejpam-5625	31	19	subset	subset	NOUN
ejpam-5625	31	20	s	s	NOUN
ejpam-5625	31	21	of	of	ADP
ejpam-5625	31	22	v	v	NOUN
ejpam-5625	31	23	(	(	PUNCT
ejpam-5625	31	24	g	g	NOUN
ejpam-5625	31	25	)	)	PUNCT
ejpam-5625	31	26	is	be	AUX
ejpam-5625	31	27	called	call	VERB
ejpam-5625	31	28	a	a	DET
ejpam-5625	31	29	dominating	dominating	NOUN
ejpam-5625	31	30	set	set	NOUN
ejpam-5625	31	31	of	of	ADP
ejpam-5625	31	32	g	g	PROPN
ejpam-5625	31	33	if	if	SCONJ
ejpam-5625	31	34	for	for	ADP
ejpam-5625	31	35	every	every	DET
ejpam-5625	31	36	a	a	DET
ejpam-5625	31	37	∈	∈	PROPN
ejpam-5625	31	38	v	v	NOUN
ejpam-5625	31	39	(	(	PUNCT
ejpam-5625	31	40	g)\s	g)\s	NOUN
ejpam-5625	31	41	,	,	PUNCT
ejpam-5625	31	42	there	there	PRON
ejpam-5625	31	43	exists	exist	VERB
ejpam-5625	31	44	b	b	PROPN
ejpam-5625	31	45	∈	∈	PROPN
ejpam-5625	31	46	s	s	VERB
ejpam-5625	31	47	such	such	ADJ
ejpam-5625	31	48	that	that	SCONJ
ejpam-5625	31	49	dg(a	dg(a	PROPN
ejpam-5625	31	50	,	,	PUNCT
ejpam-5625	31	51	b	b	X
ejpam-5625	31	52	)	)	PUNCT
ejpam-5625	31	53	=	=	SYM
ejpam-5625	31	54	1	1	NUM
ejpam-5625	31	55	,	,	PUNCT
ejpam-5625	31	56	that	that	ADV
ejpam-5625	31	57	is	is	ADV
ejpam-5625	31	58	,	,	PUNCT
ejpam-5625	31	59	s	s	VERB
ejpam-5625	31	60	is	be	AUX
ejpam-5625	31	61	a	a	DET
ejpam-5625	31	62	dominating	dominating	NOUN
ejpam-5625	31	63	set	set	NOUN
ejpam-5625	31	64	of	of	ADP
ejpam-5625	31	65	g	g	PROPN
ejpam-5625	31	66	if	if	SCONJ
ejpam-5625	31	67	ng[s	ng[	NOUN
ejpam-5625	31	68	]	]	PUNCT
ejpam-5625	31	69	=	=	SYM
ejpam-5625	31	70	v	v	NOUN
ejpam-5625	31	71	(	(	PUNCT
ejpam-5625	31	72	g	g	NOUN
ejpam-5625	31	73	)	)	PUNCT
ejpam-5625	31	74	.	.	PUNCT
ejpam-5625	32	1	the	the	DET
ejpam-5625	32	2	minimum	minimum	ADJ
ejpam-5625	32	3	cardinality	cardinality	NOUN
ejpam-5625	32	4	among	among	ADP
ejpam-5625	32	5	all	all	DET
ejpam-5625	32	6	dominating	dominating	NOUN
ejpam-5625	32	7	sets	set	NOUN
ejpam-5625	32	8	of	of	ADP
ejpam-5625	32	9	g	g	NOUN
ejpam-5625	32	10	,	,	PUNCT
ejpam-5625	32	11	denoted	denote	VERB
ejpam-5625	32	12	by	by	ADP
ejpam-5625	32	13	γ(g	γ(g	PROPN
ejpam-5625	32	14	)	)	PUNCT
ejpam-5625	32	15	,	,	PUNCT
ejpam-5625	32	16	is	be	AUX
ejpam-5625	32	17	called	call	VERB
ejpam-5625	32	18	the	the	DET
ejpam-5625	32	19	domination	domination	NOUN
ejpam-5625	32	20	number	number	NOUN
ejpam-5625	32	21	of	of	ADP
ejpam-5625	32	22	g.	g.	PROPN
ejpam-5625	32	23	a	a	DET
ejpam-5625	32	24	dominating	dominating	NOUN
ejpam-5625	32	25	set	set	NOUN
ejpam-5625	32	26	s	s	PROPN
ejpam-5625	32	27	⊆	⊆	NUM
ejpam-5625	32	28	v	v	NOUN
ejpam-5625	32	29	(	(	PUNCT
ejpam-5625	32	30	g	g	NOUN
ejpam-5625	32	31	)	)	PUNCT
ejpam-5625	32	32	is	be	AUX
ejpam-5625	32	33	called	call	VERB
ejpam-5625	32	34	a	a	DET
ejpam-5625	32	35	certified	certify	VERB
ejpam-5625	32	36	dominating	dominating	NOUN
ejpam-5625	32	37	set	set	NOUN
ejpam-5625	32	38	of	of	ADP
ejpam-5625	32	39	g	g	PROPN
ejpam-5625	32	40	if	if	SCONJ
ejpam-5625	32	41	every	every	DET
ejpam-5625	32	42	a	a	DET
ejpam-5625	32	43	∈	∈	PROPN
ejpam-5625	32	44	s	s	NOUN
ejpam-5625	32	45	,	,	PUNCT
ejpam-5625	32	46	a	a	PRON
ejpam-5625	32	47	has	have	VERB
ejpam-5625	32	48	either	either	CCONJ
ejpam-5625	32	49	zero	zero	NUM
ejpam-5625	32	50	or	or	CCONJ
ejpam-5625	32	51	atleast	atleast	ADJ
ejpam-5625	32	52	two	two	NUM
ejpam-5625	32	53	neighbors	neighbor	NOUN
ejpam-5625	32	54	in	in	ADP
ejpam-5625	32	55	v	v	NOUN
ejpam-5625	32	56	(	(	PUNCT
ejpam-5625	32	57	g	g	NOUN
ejpam-5625	32	58	)	)	PUNCT
ejpam-5625	32	59	\	\	PUNCT
ejpam-5625	33	1	s.	s.	PROPN
ejpam-5625	33	2	the	the	DET
ejpam-5625	33	3	minimum	minimum	ADJ
ejpam-5625	33	4	cardinality	cardinality	NOUN
ejpam-5625	33	5	among	among	ADP
ejpam-5625	33	6	all	all	DET
ejpam-5625	33	7	certified	certify	VERB
ejpam-5625	33	8	dominating	dominating	NOUN
ejpam-5625	33	9	sets	set	NOUN
ejpam-5625	33	10	of	of	ADP
ejpam-5625	33	11	g	g	NOUN
ejpam-5625	33	12	,	,	PUNCT
ejpam-5625	33	13	denoted	denote	VERB
ejpam-5625	33	14	by	by	ADP
ejpam-5625	33	15	γcer(g	γcer(g	PROPN
ejpam-5625	33	16	)	)	PUNCT
ejpam-5625	33	17	,	,	PUNCT
ejpam-5625	33	18	is	be	AUX
ejpam-5625	33	19	called	call	VERB
ejpam-5625	33	20	the	the	DET
ejpam-5625	33	21	certified	certify	VERB
ejpam-5625	33	22	domination	domination	NOUN
ejpam-5625	33	23	number	number	NOUN
ejpam-5625	33	24	of	of	ADP
ejpam-5625	33	25	g.	g.	PROPN
ejpam-5625	33	26	let	let	VERB
ejpam-5625	33	27	g	g	NOUN
ejpam-5625	33	28	be	be	AUX
ejpam-5625	33	29	a	a	DET
ejpam-5625	33	30	graph	graph	NOUN
ejpam-5625	33	31	.	.	PUNCT
ejpam-5625	34	1	then	then	ADV
ejpam-5625	34	2	s	s	VERB
ejpam-5625	34	3	⊆	⊆	NUM
ejpam-5625	34	4	v	v	NOUN
ejpam-5625	34	5	(	(	PUNCT
ejpam-5625	34	6	g	g	NOUN
ejpam-5625	34	7	)	)	PUNCT
ejpam-5625	34	8	is	be	AUX
ejpam-5625	34	9	called	call	VERB
ejpam-5625	34	10	a	a	DET
ejpam-5625	34	11	pointwise	pointwise	ADJ
ejpam-5625	34	12	non	non	ADJ
ejpam-5625	34	13	-	-	ADJ
ejpam-5625	34	14	dominating	dominating	ADJ
ejpam-5625	34	15	(	(	PUNCT
ejpam-5625	34	16	pnd	pnd	NOUN
ejpam-5625	34	17	)	)	PUNCT
ejpam-5625	34	18	set	set	NOUN
ejpam-5625	34	19	of	of	ADP
ejpam-5625	34	20	g	g	PROPN
ejpam-5625	34	21	if	if	SCONJ
ejpam-5625	34	22	for	for	ADP
ejpam-5625	34	23	each	each	DET
ejpam-5625	34	24	v	v	NUM
ejpam-5625	34	25	∈	∈	NOUN
ejpam-5625	34	26	v	v	NOUN
ejpam-5625	34	27	(	(	PUNCT
ejpam-5625	34	28	g)\s	g)\s	NOUN
ejpam-5625	34	29	,	,	PUNCT
ejpam-5625	34	30	there	there	PRON
ejpam-5625	34	31	exists	exist	VERB
ejpam-5625	34	32	w	w	PROPN
ejpam-5625	34	33	∈	∈	PROPN
ejpam-5625	34	34	s	s	VERB
ejpam-5625	34	35	such	such	ADJ
ejpam-5625	34	36	that	that	DET
ejpam-5625	34	37	v	v	NOUN
ejpam-5625	34	38	/∈	/∈	PUNCT
ejpam-5625	34	39	ng(w	ng(w	NOUN
ejpam-5625	34	40	)	)	PUNCT
ejpam-5625	34	41	.	.	PUNCT
ejpam-5625	35	1	the	the	DET
ejpam-5625	35	2	minimum	minimum	ADJ
ejpam-5625	35	3	cardinality	cardinality	NOUN
ejpam-5625	35	4	of	of	ADP
ejpam-5625	35	5	a	a	DET
ejpam-5625	35	6	pointwise	pointwise	ADJ
ejpam-5625	35	7	non	non	ADJ
ejpam-5625	35	8	-	-	ADJ
ejpam-5625	35	9	dominating	dominating	ADJ
ejpam-5625	35	10	(	(	PUNCT
ejpam-5625	35	11	pnd	pnd	NOUN
ejpam-5625	35	12	)	)	PUNCT
ejpam-5625	35	13	set	set	NOUN
ejpam-5625	35	14	of	of	ADP
ejpam-5625	35	15	g	g	NOUN
ejpam-5625	35	16	,	,	PUNCT
ejpam-5625	35	17	is	be	AUX
ejpam-5625	35	18	called	call	VERB
ejpam-5625	35	19	the	the	DET
ejpam-5625	35	20	pointwise	pointwise	ADJ
ejpam-5625	35	21	non	non	ADJ
ejpam-5625	35	22	-	-	NOUN
ejpam-5625	35	23	domination	domination	ADJ
ejpam-5625	35	24	(	(	PUNCT
ejpam-5625	35	25	pnd	pnd	NOUN
ejpam-5625	35	26	)	)	PUNCT
ejpam-5625	35	27	number	number	NOUN
ejpam-5625	35	28	of	of	ADP
ejpam-5625	35	29	g.	g.	PROPN
ejpam-5625	35	30	a	a	DET
ejpam-5625	35	31	vertex	vertex	NOUN
ejpam-5625	35	32	a	a	PRON
ejpam-5625	35	33	in	in	ADP
ejpam-5625	35	34	g	g	PROPN
ejpam-5625	35	35	is	be	AUX
ejpam-5625	35	36	a	a	DET
ejpam-5625	35	37	hop	hop	NOUN
ejpam-5625	35	38	neighbor	neighbor	NOUN
ejpam-5625	35	39	of	of	ADP
ejpam-5625	35	40	a	a	DET
ejpam-5625	35	41	vertex	vertex	NOUN
ejpam-5625	35	42	b	b	NOUN
ejpam-5625	35	43	in	in	ADP
ejpam-5625	35	44	g	g	PROPN
ejpam-5625	35	45	if	if	SCONJ
ejpam-5625	35	46	dg(a	dg(a	X
ejpam-5625	35	47	,	,	PUNCT
ejpam-5625	35	48	b	b	X
ejpam-5625	35	49	)	)	PUNCT
ejpam-5625	36	1	=	=	SYM
ejpam-5625	36	2	2	2	X
ejpam-5625	36	3	.	.	X
ejpam-5625	36	4	the	the	DET
ejpam-5625	36	5	set	set	ADJ
ejpam-5625	36	6	n2	n2	PROPN
ejpam-5625	36	7	g(a	g(a	PROPN
ejpam-5625	36	8	)	)	PUNCT
ejpam-5625	36	9	=	=	PRON
ejpam-5625	37	1	{	{	PUNCT
ejpam-5625	37	2	b	b	PROPN
ejpam-5625	37	3	∈	∈	ADJ
ejpam-5625	37	4	v	v	NOUN
ejpam-5625	37	5	(	(	PUNCT
ejpam-5625	37	6	g	g	NOUN
ejpam-5625	37	7	)	)	PUNCT
ejpam-5625	37	8	:	:	PUNCT
ejpam-5625	38	1	dg(a	dg(a	X
ejpam-5625	38	2	,	,	PUNCT
ejpam-5625	38	3	b	b	X
ejpam-5625	38	4	)	)	PUNCT
ejpam-5625	38	5	=	=	SYM
ejpam-5625	38	6	2	2	X
ejpam-5625	38	7	}	}	PUNCT
ejpam-5625	38	8	is	be	AUX
ejpam-5625	38	9	called	call	VERB
ejpam-5625	38	10	the	the	DET
ejpam-5625	38	11	open	open	ADJ
ejpam-5625	38	12	hop	hop	NOUN
ejpam-5625	38	13	neighborhood	neighborhood	NOUN
ejpam-5625	38	14	of	of	ADP
ejpam-5625	38	15	a.	a.	NOUN
ejpam-5625	38	16	the	the	DET
ejpam-5625	38	17	closed	closed	ADJ
ejpam-5625	38	18	hop	hop	NOUN
ejpam-5625	38	19	neighborhood	neighborhood	NOUN
ejpam-5625	38	20	of	of	ADP
ejpam-5625	38	21	a	a	PRON
ejpam-5625	38	22	in	in	ADP
ejpam-5625	38	23	g	g	PROPN
ejpam-5625	38	24	is	be	AUX
ejpam-5625	38	25	given	give	VERB
ejpam-5625	38	26	by	by	ADP
ejpam-5625	38	27	n2	n2	ADJ
ejpam-5625	38	28	g[a	g[a	PROPN
ejpam-5625	38	29	]	]	X
ejpam-5625	38	30	=	=	SYM
ejpam-5625	38	31	n2	n2	ADJ
ejpam-5625	38	32	g(a)∪{a	g(a)∪{a	NOUN
ejpam-5625	38	33	}	}	PUNCT
ejpam-5625	38	34	.	.	PUNCT
ejpam-5625	39	1	the	the	DET
ejpam-5625	39	2	open	open	ADJ
ejpam-5625	39	3	hop	hop	NOUN
ejpam-5625	39	4	neighborhood	neighborhood	NOUN
ejpam-5625	39	5	of	of	ADP
ejpam-5625	39	6	s	s	NOUN
ejpam-5625	39	7	⊆	⊆	NUM
ejpam-5625	39	8	v	v	NOUN
ejpam-5625	39	9	(	(	PUNCT
ejpam-5625	39	10	g	g	NOUN
ejpam-5625	39	11	)	)	PUNCT
ejpam-5625	39	12	is	be	AUX
ejpam-5625	39	13	the	the	DET
ejpam-5625	39	14	set	set	ADJ
ejpam-5625	39	15	n2	n2	ADJ
ejpam-5625	39	16	g(s	g(s	PROPN
ejpam-5625	39	17	)	)	PUNCT
ejpam-5625	39	18	=	=	SYM
ejpam-5625	40	1	⋃	⋃	ADP
ejpam-5625	40	2	a∈s	a∈s	ADJ
ejpam-5625	40	3	n2	n2	NOUN
ejpam-5625	40	4	g(a	g(a	PROPN
ejpam-5625	40	5	)	)	PUNCT
ejpam-5625	40	6	.	.	PUNCT
ejpam-5625	41	1	the	the	DET
ejpam-5625	41	2	closed	closed	ADJ
ejpam-5625	41	3	hop	hop	NOUN
ejpam-5625	41	4	neighborhood	neighborhood	NOUN
ejpam-5625	41	5	of	of	ADP
ejpam-5625	41	6	s	s	PRON
ejpam-5625	41	7	in	in	ADP
ejpam-5625	41	8	g	g	PROPN
ejpam-5625	41	9	is	be	AUX
ejpam-5625	41	10	the	the	DET
ejpam-5625	41	11	set	set	ADJ
ejpam-5625	41	12	n2	n2	ADJ
ejpam-5625	41	13	g[s	g[s	PROPN
ejpam-5625	41	14	]	]	PUNCT
ejpam-5625	41	15	=	=	SYM
ejpam-5625	41	16	n2	n2	ADJ
ejpam-5625	41	17	g(s	g(s	PROPN
ejpam-5625	41	18	)	)	PUNCT
ejpam-5625	41	19	∪	∪	ADP
ejpam-5625	41	20	s.	s.	PROPN
ejpam-5625	41	21	a	a	DET
ejpam-5625	41	22	subset	subset	NOUN
ejpam-5625	41	23	s	s	X
ejpam-5625	41	24	of	of	ADP
ejpam-5625	41	25	v	v	NOUN
ejpam-5625	41	26	(	(	PUNCT
ejpam-5625	41	27	g	g	NOUN
ejpam-5625	41	28	)	)	PUNCT
ejpam-5625	41	29	is	be	AUX
ejpam-5625	41	30	a	a	DET
ejpam-5625	41	31	hop	hop	NOUN
ejpam-5625	41	32	dominating	dominating	NOUN
ejpam-5625	41	33	of	of	ADP
ejpam-5625	41	34	g	g	PROPN
ejpam-5625	41	35	if	if	SCONJ
ejpam-5625	41	36	for	for	ADP
ejpam-5625	41	37	every	every	DET
ejpam-5625	41	38	a	a	DET
ejpam-5625	41	39	∈	∈	PROPN
ejpam-5625	41	40	v	v	NOUN
ejpam-5625	41	41	(	(	PUNCT
ejpam-5625	41	42	g)\s	g)\s	NOUN
ejpam-5625	41	43	,	,	PUNCT
ejpam-5625	41	44	there	there	PRON
ejpam-5625	41	45	exists	exist	VERB
ejpam-5625	41	46	b	b	PROPN
ejpam-5625	41	47	∈	∈	PROPN
ejpam-5625	41	48	s	s	VERB
ejpam-5625	41	49	such	such	ADJ
ejpam-5625	41	50	that	that	SCONJ
ejpam-5625	41	51	dg(a	dg(a	PROPN
ejpam-5625	41	52	,	,	PUNCT
ejpam-5625	41	53	b	b	X
ejpam-5625	41	54	)	)	PUNCT
ejpam-5625	41	55	=	=	SYM
ejpam-5625	41	56	2	2	NUM
ejpam-5625	41	57	,	,	PUNCT
ejpam-5625	41	58	that	that	ADV
ejpam-5625	41	59	is	is	ADV
ejpam-5625	41	60	,	,	PUNCT
ejpam-5625	41	61	s	s	VERB
ejpam-5625	41	62	is	be	AUX
ejpam-5625	41	63	a	a	DET
ejpam-5625	41	64	hop	hop	NOUN
ejpam-5625	41	65	dominating	dominating	NOUN
ejpam-5625	41	66	set	set	NOUN
ejpam-5625	41	67	of	of	ADP
ejpam-5625	41	68	g	g	PROPN
ejpam-5625	41	69	if	if	SCONJ
ejpam-5625	41	70	n2	n2	ADJ
ejpam-5625	41	71	g[s	g[s	PROPN
ejpam-5625	41	72	]	]	X
ejpam-5625	41	73	=	=	SYM
ejpam-5625	41	74	v	v	NOUN
ejpam-5625	41	75	(	(	PUNCT
ejpam-5625	41	76	g	g	NOUN
ejpam-5625	41	77	)	)	PUNCT
ejpam-5625	41	78	.	.	PUNCT
ejpam-5625	42	1	the	the	DET
ejpam-5625	42	2	minimum	minimum	ADJ
ejpam-5625	42	3	cardinality	cardinality	NOUN
ejpam-5625	42	4	among	among	ADP
ejpam-5625	42	5	all	all	DET
ejpam-5625	42	6	hop	hop	NOUN
ejpam-5625	42	7	dominating	dominating	NOUN
ejpam-5625	42	8	sets	set	NOUN
ejpam-5625	42	9	of	of	ADP
ejpam-5625	42	10	g	g	NOUN
ejpam-5625	42	11	,	,	PUNCT
ejpam-5625	42	12	denoted	denote	VERB
ejpam-5625	42	13	by	by	ADP
ejpam-5625	42	14	γh(g	γh(g	NOUN
ejpam-5625	42	15	)	)	PUNCT
ejpam-5625	42	16	,	,	PUNCT
ejpam-5625	42	17	is	be	AUX
ejpam-5625	42	18	called	call	VERB
ejpam-5625	42	19	n.	n.	PROPN
ejpam-5625	42	20	s.	s.	PROPN
ejpam-5625	42	21	ulal	ulal	PROPN
ejpam-5625	42	22	et	et	PROPN
ejpam-5625	42	23	al	al	PROPN
ejpam-5625	42	24	.	.	PUNCT
ejpam-5625	42	25	/	/	SYM
ejpam-5625	42	26	eur	eur	PROPN
ejpam-5625	42	27	.	.	PUNCT
ejpam-5625	43	1	j.	j.	PROPN
ejpam-5625	43	2	pure	pure	PROPN
ejpam-5625	43	3	appl	appl	PROPN
ejpam-5625	43	4	.	.	PROPN
ejpam-5625	43	5	math	math	PROPN
ejpam-5625	43	6	,	,	PUNCT
ejpam-5625	43	7	18	18	NUM
ejpam-5625	43	8	(	(	PUNCT
ejpam-5625	43	9	1	1	NUM
ejpam-5625	43	10	)	)	PUNCT
ejpam-5625	43	11	(	(	PUNCT
ejpam-5625	43	12	2025	2025	NUM
ejpam-5625	43	13	)	)	PUNCT
ejpam-5625	43	14	,	,	PUNCT
ejpam-5625	43	15	5625	5625	NUM
ejpam-5625	43	16	3	3	NUM
ejpam-5625	43	17	of	of	ADP
ejpam-5625	43	18	9	9	NUM
ejpam-5625	43	19	the	the	DET
ejpam-5625	43	20	hop	hop	NOUN
ejpam-5625	43	21	domination	domination	NOUN
ejpam-5625	43	22	number	number	NOUN
ejpam-5625	43	23	of	of	ADP
ejpam-5625	43	24	g.	g.	PROPN
ejpam-5625	43	25	let	let	VERB
ejpam-5625	43	26	g	g	NOUN
ejpam-5625	43	27	and	and	CCONJ
ejpam-5625	43	28	h	h	NOUN
ejpam-5625	43	29	be	be	VERB
ejpam-5625	43	30	any	any	DET
ejpam-5625	43	31	two	two	NUM
ejpam-5625	43	32	graphs	graph	NOUN
ejpam-5625	43	33	.	.	PUNCT
ejpam-5625	44	1	the	the	DET
ejpam-5625	44	2	join	join	NOUN
ejpam-5625	44	3	of	of	ADP
ejpam-5625	44	4	g	g	PROPN
ejpam-5625	44	5	and	and	CCONJ
ejpam-5625	44	6	h	h	NOUN
ejpam-5625	44	7	,	,	PUNCT
ejpam-5625	44	8	denoted	denote	VERB
ejpam-5625	44	9	by	by	ADP
ejpam-5625	44	10	g+h	g+h	PROPN
ejpam-5625	44	11	is	be	AUX
ejpam-5625	44	12	the	the	DET
ejpam-5625	44	13	graph	graph	NOUN
ejpam-5625	44	14	with	with	ADP
ejpam-5625	44	15	vertex	vertex	NOUN
ejpam-5625	44	16	set	set	VERB
ejpam-5625	44	17	v	v	NOUN
ejpam-5625	44	18	(	(	PUNCT
ejpam-5625	44	19	g+h	g+h	NOUN
ejpam-5625	44	20	)	)	PUNCT
ejpam-5625	45	1	=	=	SYM
ejpam-5625	45	2	v	v	X
ejpam-5625	45	3	(	(	PUNCT
ejpam-5625	45	4	g	g	NOUN
ejpam-5625	45	5	)	)	PUNCT
ejpam-5625	45	6	∪	∪	NOUN
ejpam-5625	45	7	v	v	NOUN
ejpam-5625	45	8	(	(	PUNCT
ejpam-5625	45	9	h	h	NOUN
ejpam-5625	45	10	)	)	PUNCT
ejpam-5625	45	11	and	and	CCONJ
ejpam-5625	45	12	edge	edge	NOUN
ejpam-5625	45	13	set	set	VERB
ejpam-5625	45	14	e(g+h	e(g+h	NUM
ejpam-5625	45	15	)	)	PUNCT
ejpam-5625	45	16	=	=	SYM
ejpam-5625	45	17	e(g	e(g	NOUN
ejpam-5625	45	18	)	)	PUNCT
ejpam-5625	45	19	∪	∪	ADP
ejpam-5625	45	20	e(h	e(h	PROPN
ejpam-5625	45	21	)	)	PUNCT
ejpam-5625	45	22	∪	∪	NOUN
ejpam-5625	45	23	{	{	PUNCT
ejpam-5625	45	24	uv	uv	NOUN
ejpam-5625	45	25	:	:	PUNCT
ejpam-5625	45	26	u	u	PROPN
ejpam-5625	45	27	∈	∈	PROPN
ejpam-5625	45	28	v	v	ADP
ejpam-5625	45	29	(	(	PUNCT
ejpam-5625	45	30	g	g	NOUN
ejpam-5625	45	31	)	)	PUNCT
ejpam-5625	45	32	,	,	PUNCT
ejpam-5625	45	33	v	v	X
ejpam-5625	45	34	∈	∈	PROPN
ejpam-5625	45	35	v	v	NOUN
ejpam-5625	45	36	(	(	PUNCT
ejpam-5625	45	37	h	h	NOUN
ejpam-5625	45	38	)	)	PUNCT
ejpam-5625	45	39	}	}	PUNCT
ejpam-5625	45	40	.	.	PUNCT
ejpam-5625	46	1	3	3	X
ejpam-5625	46	2	.	.	NOUN
ejpam-5625	46	3	results	result	VERB
ejpam-5625	46	4	4	4	NUM
ejpam-5625	46	5	.	.	PUNCT
ejpam-5625	46	6	properties	property	NOUN
ejpam-5625	46	7	and	and	CCONJ
ejpam-5625	46	8	bounds	bound	NOUN
ejpam-5625	46	9	of	of	ADP
ejpam-5625	46	10	2	2	NUM
ejpam-5625	46	11	-	-	PUNCT
ejpam-5625	46	12	distance	distance	NOUN
ejpam-5625	46	13	certified	certify	VERB
ejpam-5625	46	14	hop	hop	NOUN
ejpam-5625	46	15	domination	domination	NOUN
ejpam-5625	46	16	in	in	ADP
ejpam-5625	46	17	some	some	DET
ejpam-5625	46	18	graphs	graph	NOUN
ejpam-5625	46	19	definition	definition	NOUN
ejpam-5625	46	20	1	1	NUM
ejpam-5625	46	21	.	.	PUNCT
ejpam-5625	47	1	let	let	VERB
ejpam-5625	47	2	g	g	PRON
ejpam-5625	47	3	be	be	AUX
ejpam-5625	47	4	a	a	DET
ejpam-5625	47	5	graph	graph	NOUN
ejpam-5625	47	6	.	.	PUNCT
ejpam-5625	48	1	then	then	ADV
ejpam-5625	48	2	a	a	DET
ejpam-5625	48	3	set	set	NOUN
ejpam-5625	48	4	c	c	NOUN
ejpam-5625	48	5	⊆	⊆	NUM
ejpam-5625	48	6	v	v	NOUN
ejpam-5625	48	7	(	(	PUNCT
ejpam-5625	48	8	g	g	NOUN
ejpam-5625	48	9	)	)	PUNCT
ejpam-5625	48	10	is	be	AUX
ejpam-5625	48	11	called	call	VERB
ejpam-5625	48	12	a	a	DET
ejpam-5625	48	13	2	2	NUM
ejpam-5625	48	14	-	-	PUNCT
ejpam-5625	48	15	distance	distance	NOUN
ejpam-5625	48	16	certified	certify	VERB
ejpam-5625	48	17	hop	hop	NOUN
ejpam-5625	48	18	dominating	dominating	NOUN
ejpam-5625	48	19	if	if	SCONJ
ejpam-5625	48	20	∀	∀	NOUN
ejpam-5625	48	21	x	x	SYM
ejpam-5625	48	22	∈	∈	NOUN
ejpam-5625	48	23	v	v	X
ejpam-5625	48	24	(	(	PUNCT
ejpam-5625	48	25	g)\c	g)\c	NOUN
ejpam-5625	48	26	,	,	PUNCT
ejpam-5625	48	27	there	there	PRON
ejpam-5625	48	28	exists	exist	VERB
ejpam-5625	48	29	y	y	PROPN
ejpam-5625	48	30	∈	∈	PROPN
ejpam-5625	48	31	c	c	PROPN
ejpam-5625	48	32	such	such	ADJ
ejpam-5625	48	33	that	that	DET
ejpam-5625	48	34	dg(x	dg(x	PROPN
ejpam-5625	48	35	,	,	PUNCT
ejpam-5625	48	36	y	y	NOUN
ejpam-5625	48	37	)	)	PUNCT
ejpam-5625	48	38	=	=	SYM
ejpam-5625	48	39	2	2	NUM
ejpam-5625	48	40	and	and	CCONJ
ejpam-5625	48	41	∀	∀	NOUN
ejpam-5625	48	42	a	a	DET
ejpam-5625	48	43	∈	∈	PROPN
ejpam-5625	48	44	c	c	X
ejpam-5625	48	45	,	,	PUNCT
ejpam-5625	48	46	|n2	|n2	X
ejpam-5625	48	47	g(a)\c|	g(a)\c|	PUNCT
ejpam-5625	49	1	=	=	PUNCT
ejpam-5625	49	2	0	0	PUNCT
ejpam-5625	49	3	or	or	CCONJ
ejpam-5625	49	4	|n2	|n2	PROPN
ejpam-5625	49	5	g(a)\c|	g(a)\c|	X
ejpam-5625	49	6	≥	≥	NOUN
ejpam-5625	49	7	2	2	NUM
ejpam-5625	49	8	.	.	PUNCT
ejpam-5625	50	1	the	the	DET
ejpam-5625	50	2	2	2	NUM
ejpam-5625	50	3	-	-	PUNCT
ejpam-5625	50	4	distance	distance	NOUN
ejpam-5625	50	5	certified	certify	VERB
ejpam-5625	50	6	hop	hop	NOUN
ejpam-5625	50	7	domination	domination	NOUN
ejpam-5625	50	8	number	number	NOUN
ejpam-5625	50	9	of	of	ADP
ejpam-5625	50	10	g	g	NOUN
ejpam-5625	50	11	,	,	PUNCT
ejpam-5625	50	12	denoted	denote	VERB
ejpam-5625	50	13	by	by	ADP
ejpam-5625	50	14	γ2ch(g	γ2ch(g	NOUN
ejpam-5625	50	15	)	)	PUNCT
ejpam-5625	50	16	,	,	PUNCT
ejpam-5625	50	17	is	be	AUX
ejpam-5625	50	18	the	the	DET
ejpam-5625	50	19	minimum	minimum	ADJ
ejpam-5625	50	20	cardinality	cardinality	NOUN
ejpam-5625	50	21	among	among	ADP
ejpam-5625	50	22	all	all	DET
ejpam-5625	50	23	2	2	NUM
ejpam-5625	50	24	-	-	PUNCT
ejpam-5625	50	25	distace	distace	NOUN
ejpam-5625	50	26	certified	certify	VERB
ejpam-5625	50	27	hop	hop	NOUN
ejpam-5625	50	28	dominating	dominating	NOUN
ejpam-5625	50	29	sets	set	NOUN
ejpam-5625	50	30	of	of	ADP
ejpam-5625	50	31	g.	g.	PROPN
ejpam-5625	50	32	example	example	NOUN
ejpam-5625	50	33	1	1	X
ejpam-5625	50	34	.	.	PUNCT
ejpam-5625	50	35	below	below	ADV
ejpam-5625	50	36	is	be	AUX
ejpam-5625	50	37	an	an	DET
ejpam-5625	50	38	example	example	NOUN
ejpam-5625	50	39	of	of	ADP
ejpam-5625	50	40	2	2	NUM
ejpam-5625	50	41	-	-	PUNCT
ejpam-5625	50	42	distance	distance	NOUN
ejpam-5625	50	43	certified	certify	VERB
ejpam-5625	50	44	hop	hop	NOUN
ejpam-5625	50	45	domination	domination	NOUN
ejpam-5625	50	46	.	.	PUNCT
ejpam-5625	51	1	p5	p5	ADJ
ejpam-5625	51	2	:	:	PUNCT
ejpam-5625	51	3	a	a	DET
ejpam-5625	51	4	b	b	X
ejpam-5625	51	5	c	c	NOUN
ejpam-5625	51	6	d	d	X
ejpam-5625	51	7	e	e	X
ejpam-5625	51	8	figure	figure	NOUN
ejpam-5625	51	9	1	1	NUM
ejpam-5625	51	10	:	:	PUNCT
ejpam-5625	51	11	a	a	DET
ejpam-5625	51	12	path	path	NOUN
ejpam-5625	51	13	graph	graph	NOUN
ejpam-5625	51	14	of	of	ADP
ejpam-5625	51	15	p5	p5	ADJ
ejpam-5625	51	16	with	with	ADP
ejpam-5625	51	17	γ2ch(p5	γ2ch(p5	ADJ
ejpam-5625	51	18	)	)	PUNCT
ejpam-5625	51	19	=	=	SYM
ejpam-5625	51	20	3	3	NUM
ejpam-5625	51	21	consider	consider	VERB
ejpam-5625	51	22	the	the	DET
ejpam-5625	51	23	path	path	NOUN
ejpam-5625	51	24	graph	graph	NOUN
ejpam-5625	51	25	given	give	VERB
ejpam-5625	51	26	above	above	ADV
ejpam-5625	51	27	.	.	PUNCT
ejpam-5625	52	1	let	let	VERB
ejpam-5625	52	2	c	c	NOUN
ejpam-5625	52	3	=	=	PUNCT
ejpam-5625	52	4	{	{	PUNCT
ejpam-5625	52	5	b	b	PROPN
ejpam-5625	52	6	,	,	PUNCT
ejpam-5625	52	7	c	c	NOUN
ejpam-5625	52	8	,	,	PUNCT
ejpam-5625	52	9	d	d	NOUN
ejpam-5625	52	10	}	}	PUNCT
ejpam-5625	52	11	.	.	PUNCT
ejpam-5625	53	1	then	then	ADV
ejpam-5625	53	2	n2	n2	PROPN
ejpam-5625	53	3	g[b	g[b	PROPN
ejpam-5625	53	4	]	]	X
ejpam-5625	53	5	=	=	PUNCT
ejpam-5625	53	6	{	{	PUNCT
ejpam-5625	53	7	b	b	PROPN
ejpam-5625	53	8	,	,	PUNCT
ejpam-5625	53	9	d	d	NOUN
ejpam-5625	53	10	}	}	PUNCT
ejpam-5625	53	11	,	,	PUNCT
ejpam-5625	53	12	n2	n2	PROPN
ejpam-5625	53	13	g[c	g[c	PROPN
ejpam-5625	53	14	]	]	X
ejpam-5625	53	15	=	=	X
ejpam-5625	53	16	{	{	PUNCT
ejpam-5625	53	17	a	a	X
ejpam-5625	53	18	,	,	PUNCT
ejpam-5625	53	19	c	c	NOUN
ejpam-5625	53	20	,	,	PUNCT
ejpam-5625	53	21	e	e	NOUN
ejpam-5625	53	22	}	}	PUNCT
ejpam-5625	53	23	and	and	CCONJ
ejpam-5625	53	24	n2	n2	PROPN
ejpam-5625	53	25	g[d	g[d	PROPN
ejpam-5625	53	26	]	]	X
ejpam-5625	53	27	=	=	SYM
ejpam-5625	53	28	{	{	PUNCT
ejpam-5625	53	29	b	b	PROPN
ejpam-5625	53	30	,	,	PUNCT
ejpam-5625	53	31	d	d	NOUN
ejpam-5625	53	32	}	}	PUNCT
ejpam-5625	53	33	.	.	PUNCT
ejpam-5625	54	1	thus	thus	ADV
ejpam-5625	54	2	,	,	PUNCT
ejpam-5625	54	3	n2	n2	PROPN
ejpam-5625	54	4	g[c	g[c	PROPN
ejpam-5625	54	5	]	]	X
ejpam-5625	54	6	=	=	SYM
ejpam-5625	54	7	v	v	X
ejpam-5625	54	8	(	(	PUNCT
ejpam-5625	54	9	g	g	NOUN
ejpam-5625	54	10	)	)	PUNCT
ejpam-5625	54	11	,	,	PUNCT
ejpam-5625	54	12	and	and	CCONJ
ejpam-5625	54	13	so	so	ADV
ejpam-5625	54	14	c	c	PROPN
ejpam-5625	54	15	is	be	AUX
ejpam-5625	54	16	a	a	DET
ejpam-5625	54	17	hop	hop	NOUN
ejpam-5625	54	18	dominating	dominating	NOUN
ejpam-5625	54	19	set	set	NOUN
ejpam-5625	54	20	of	of	ADP
ejpam-5625	54	21	g.	g.	PROPN
ejpam-5625	54	22	observe	observe	VERB
ejpam-5625	54	23	that	that	SCONJ
ejpam-5625	54	24	,	,	PUNCT
ejpam-5625	54	25	vertices	vertice	VERB
ejpam-5625	54	26	b	b	NOUN
ejpam-5625	54	27	and	and	CCONJ
ejpam-5625	54	28	d	d	PROPN
ejpam-5625	54	29	have	have	VERB
ejpam-5625	54	30	zero	zero	NUM
ejpam-5625	54	31	hop	hop	NOUN
ejpam-5625	54	32	neighbor	neighbor	NOUN
ejpam-5625	54	33	in	in	ADP
ejpam-5625	54	34	v	v	NOUN
ejpam-5625	54	35	(	(	PUNCT
ejpam-5625	54	36	g)\c	g)\c	NOUN
ejpam-5625	54	37	and	and	CCONJ
ejpam-5625	54	38	vertex	vertex	NOUN
ejpam-5625	54	39	c	c	PROPN
ejpam-5625	54	40	has	have	VERB
ejpam-5625	54	41	two	two	NUM
ejpam-5625	54	42	hop	hop	NOUN
ejpam-5625	54	43	neighbors	neighbor	NOUN
ejpam-5625	54	44	a	a	PRON
ejpam-5625	54	45	,	,	PUNCT
ejpam-5625	54	46	e	e	X
ejpam-5625	54	47	in	in	ADP
ejpam-5625	54	48	v	v	NUM
ejpam-5625	54	49	(	(	PUNCT
ejpam-5625	54	50	g)\c	g)\c	NOUN
ejpam-5625	54	51	.	.	PUNCT
ejpam-5625	55	1	therefore	therefore	ADV
ejpam-5625	55	2	,	,	PUNCT
ejpam-5625	55	3	c	c	PROPN
ejpam-5625	55	4	is	be	AUX
ejpam-5625	55	5	a	a	DET
ejpam-5625	55	6	2	2	NUM
ejpam-5625	55	7	-	-	PUNCT
ejpam-5625	55	8	distance	distance	NOUN
ejpam-5625	55	9	certified	certify	VERB
ejpam-5625	55	10	hop	hop	NOUN
ejpam-5625	55	11	dominating	dominating	NOUN
ejpam-5625	55	12	set	set	NOUN
ejpam-5625	55	13	of	of	ADP
ejpam-5625	55	14	g.	g.	PROPN
ejpam-5625	55	15	moreover	moreover	ADV
ejpam-5625	55	16	,	,	PUNCT
ejpam-5625	55	17	it	it	PRON
ejpam-5625	55	18	can	can	AUX
ejpam-5625	55	19	be	be	AUX
ejpam-5625	55	20	verified	verify	VERB
ejpam-5625	55	21	that	that	SCONJ
ejpam-5625	55	22	γ2ch(g	γ2ch(g	NOUN
ejpam-5625	55	23	)	)	PUNCT
ejpam-5625	55	24	=	=	SYM
ejpam-5625	55	25	3	3	X
ejpam-5625	55	26	.	.	X
ejpam-5625	55	27	theorem	theorem	NOUN
ejpam-5625	55	28	1	1	NUM
ejpam-5625	55	29	.	.	PUNCT
ejpam-5625	56	1	let	let	VERB
ejpam-5625	56	2	g	g	PRON
ejpam-5625	56	3	be	be	AUX
ejpam-5625	56	4	a	a	DET
ejpam-5625	56	5	graph	graph	NOUN
ejpam-5625	56	6	.	.	PUNCT
ejpam-5625	57	1	then	then	ADV
ejpam-5625	57	2	(	(	PUNCT
ejpam-5625	57	3	i	i	NOUN
ejpam-5625	57	4	)	)	PUNCT
ejpam-5625	57	5	γh(g	γh(g	NOUN
ejpam-5625	57	6	)	)	PUNCT
ejpam-5625	57	7	≤	≤	NUM
ejpam-5625	57	8	γ2ch(g	γ2ch(g	NOUN
ejpam-5625	57	9	)	)	PUNCT
ejpam-5625	57	10	(	(	PUNCT
ejpam-5625	57	11	ii	ii	NOUN
ejpam-5625	57	12	)	)	PUNCT
ejpam-5625	57	13	1	1	NUM
ejpam-5625	57	14	≤	≤	NUM
ejpam-5625	57	15	γ2ch(g	γ2ch(g	NOUN
ejpam-5625	57	16	)	)	PUNCT
ejpam-5625	57	17	≤	≤	PUNCT
ejpam-5625	57	18	|v	|v	X
ejpam-5625	57	19	(	(	PUNCT
ejpam-5625	57	20	g)|	g)|	PROPN
ejpam-5625	57	21	(	(	PUNCT
ejpam-5625	57	22	iii	iii	NOUN
ejpam-5625	57	23	)	)	PUNCT
ejpam-5625	57	24	γ2ch(g	γ2ch(g	NOUN
ejpam-5625	57	25	)	)	PUNCT
ejpam-5625	57	26	=	=	SYM
ejpam-5625	57	27	1	1	NUM
ejpam-5625	57	28	if	if	SCONJ
ejpam-5625	57	29	and	and	CCONJ
ejpam-5625	57	30	only	only	ADV
ejpam-5625	57	31	if	if	SCONJ
ejpam-5625	57	32	g	g	PROPN
ejpam-5625	57	33	=	=	SYM
ejpam-5625	57	34	k1	k1	NOUN
ejpam-5625	57	35	proof	proof	NOUN
ejpam-5625	57	36	.	.	PUNCT
ejpam-5625	58	1	(	(	PUNCT
ejpam-5625	58	2	i	i	NOUN
ejpam-5625	58	3	)	)	PUNCT
ejpam-5625	58	4	let	let	VERB
ejpam-5625	58	5	d	d	PRON
ejpam-5625	58	6	be	be	AUX
ejpam-5625	58	7	a	a	DET
ejpam-5625	58	8	minimum	minimum	ADJ
ejpam-5625	58	9	2	2	NUM
ejpam-5625	58	10	-	-	PUNCT
ejpam-5625	58	11	distance	distance	NOUN
ejpam-5625	58	12	certified	certify	VERB
ejpam-5625	58	13	hop	hop	NOUN
ejpam-5625	58	14	dominating	dominating	NOUN
ejpam-5625	58	15	set	set	NOUN
ejpam-5625	58	16	of	of	ADP
ejpam-5625	58	17	g.	g.	PROPN
ejpam-5625	59	1	then	then	ADV
ejpam-5625	59	2	d	d	PROPN
ejpam-5625	59	3	is	be	AUX
ejpam-5625	59	4	a	a	DET
ejpam-5625	59	5	hop	hop	NOUN
ejpam-5625	59	6	dominating	dominating	NOUN
ejpam-5625	59	7	set	set	NOUN
ejpam-5625	59	8	of	of	ADP
ejpam-5625	59	9	g	g	PROPN
ejpam-5625	59	10	and	and	CCONJ
ejpam-5625	59	11	γ2ch(g	γ2ch(g	NOUN
ejpam-5625	59	12	)	)	PUNCT
ejpam-5625	59	13	=	=	SYM
ejpam-5625	59	14	|d|	|d|	PROPN
ejpam-5625	59	15	.	.	PUNCT
ejpam-5625	60	1	thus	thus	ADV
ejpam-5625	60	2	,	,	PUNCT
ejpam-5625	60	3	γh(g	γh(g	NOUN
ejpam-5625	60	4	)	)	PUNCT
ejpam-5625	60	5	≤	≤	NUM
ejpam-5625	60	6	|d|	|d|	PROPN
ejpam-5625	60	7	=	=	SYM
ejpam-5625	60	8	γ2ch(g	γ2ch(g	PROPN
ejpam-5625	60	9	)	)	PUNCT
ejpam-5625	60	10	by	by	ADP
ejpam-5625	60	11	definition	definition	NOUN
ejpam-5625	60	12	.	.	PUNCT
ejpam-5625	61	1	(	(	PUNCT
ejpam-5625	61	2	ii	ii	NOUN
ejpam-5625	61	3	)	)	PUNCT
ejpam-5625	61	4	since	since	SCONJ
ejpam-5625	61	5	γh(g	γh(g	NOUN
ejpam-5625	61	6	)	)	PUNCT
ejpam-5625	61	7	≥	≥	NOUN
ejpam-5625	61	8	1	1	NUM
ejpam-5625	61	9	for	for	ADP
ejpam-5625	61	10	any	any	DET
ejpam-5625	61	11	graph	graph	NOUN
ejpam-5625	61	12	(	(	PUNCT
ejpam-5625	61	13	g	g	NOUN
ejpam-5625	61	14	)	)	PUNCT
ejpam-5625	61	15	,	,	PUNCT
ejpam-5625	61	16	γ2ch(g	γ2ch(g	PROPN
ejpam-5625	61	17	)	)	PUNCT
ejpam-5625	61	18	≥	≥	NOUN
ejpam-5625	61	19	1	1	NUM
ejpam-5625	61	20	by	by	ADP
ejpam-5625	61	21	(	(	PUNCT
ejpam-5625	61	22	i	i	NOUN
ejpam-5625	61	23	)	)	PUNCT
ejpam-5625	61	24	.	.	PUNCT
ejpam-5625	62	1	the	the	DET
ejpam-5625	62	2	upperbound	upperbound	PROPN
ejpam-5625	62	3	is	be	AUX
ejpam-5625	62	4	clear	clear	ADJ
ejpam-5625	62	5	since	since	SCONJ
ejpam-5625	62	6	any	any	DET
ejpam-5625	62	7	2	2	NUM
ejpam-5625	62	8	-	-	PUNCT
ejpam-5625	62	9	distance	distance	NOUN
ejpam-5625	62	10	certified	certify	VERB
ejpam-5625	62	11	hop	hop	NOUN
ejpam-5625	62	12	dominating	dominating	NOUN
ejpam-5625	62	13	set	set	NOUN
ejpam-5625	62	14	is	be	AUX
ejpam-5625	62	15	always	always	ADV
ejpam-5625	62	16	a	a	DET
ejpam-5625	62	17	subset	subset	NOUN
ejpam-5625	62	18	of	of	ADP
ejpam-5625	62	19	v	v	NOUN
ejpam-5625	62	20	(	(	PUNCT
ejpam-5625	62	21	g	g	NOUN
ejpam-5625	62	22	)	)	PUNCT
ejpam-5625	62	23	.	.	PUNCT
ejpam-5625	63	1	n.	n.	PROPN
ejpam-5625	63	2	s.	s.	PROPN
ejpam-5625	63	3	ulal	ulal	PROPN
ejpam-5625	63	4	et	et	PROPN
ejpam-5625	63	5	al	al	PROPN
ejpam-5625	63	6	.	.	PUNCT
ejpam-5625	63	7	/	/	SYM
ejpam-5625	63	8	eur	eur	PROPN
ejpam-5625	63	9	.	.	PUNCT
ejpam-5625	64	1	j.	j.	PROPN
ejpam-5625	64	2	pure	pure	PROPN
ejpam-5625	64	3	appl	appl	PROPN
ejpam-5625	64	4	.	.	PROPN
ejpam-5625	64	5	math	math	PROPN
ejpam-5625	64	6	,	,	PUNCT
ejpam-5625	64	7	18	18	NUM
ejpam-5625	64	8	(	(	PUNCT
ejpam-5625	64	9	1	1	NUM
ejpam-5625	64	10	)	)	PUNCT
ejpam-5625	64	11	(	(	PUNCT
ejpam-5625	64	12	2025	2025	NUM
ejpam-5625	64	13	)	)	PUNCT
ejpam-5625	64	14	,	,	PUNCT
ejpam-5625	64	15	5625	5625	NUM
ejpam-5625	64	16	4	4	NUM
ejpam-5625	64	17	of	of	ADP
ejpam-5625	64	18	9	9	NUM
ejpam-5625	64	19	(	(	PUNCT
ejpam-5625	64	20	iii	iii	NOUN
ejpam-5625	64	21	)	)	PUNCT
ejpam-5625	64	22	suppose	suppose	VERB
ejpam-5625	64	23	that	that	SCONJ
ejpam-5625	64	24	γ2ch(g	γ2ch(g	NOUN
ejpam-5625	64	25	)	)	PUNCT
ejpam-5625	64	26	=	=	SYM
ejpam-5625	64	27	1	1	X
ejpam-5625	64	28	.	.	PUNCT
ejpam-5625	64	29	then	then	ADV
ejpam-5625	64	30	γh(g	γh(g	NOUN
ejpam-5625	64	31	)	)	PUNCT
ejpam-5625	64	32	=	=	SYM
ejpam-5625	64	33	1	1	X
ejpam-5625	64	34	by	by	ADP
ejpam-5625	64	35	(	(	PUNCT
ejpam-5625	64	36	i	i	NOUN
ejpam-5625	64	37	)	)	PUNCT
ejpam-5625	64	38	.	.	PUNCT
ejpam-5625	65	1	it	it	PRON
ejpam-5625	65	2	follows	follow	VERB
ejpam-5625	65	3	that	that	SCONJ
ejpam-5625	65	4	g	g	PROPN
ejpam-5625	65	5	=	=	PROPN
ejpam-5625	65	6	k1	k1	PROPN
ejpam-5625	65	7	.	.	PUNCT
ejpam-5625	66	1	conversely	conversely	ADV
ejpam-5625	66	2	,	,	PUNCT
ejpam-5625	66	3	suppose	suppose	VERB
ejpam-5625	66	4	that	that	SCONJ
ejpam-5625	66	5	g	g	PROPN
ejpam-5625	66	6	=	=	PROPN
ejpam-5625	66	7	k1	k1	PROPN
ejpam-5625	66	8	.	.	PUNCT
ejpam-5625	67	1	then	then	ADV
ejpam-5625	67	2	by	by	ADP
ejpam-5625	67	3	(	(	PUNCT
ejpam-5625	67	4	ii	ii	NOUN
ejpam-5625	67	5	)	)	PUNCT
ejpam-5625	67	6	γ2ch(g	γ2ch(g	NOUN
ejpam-5625	67	7	)	)	PUNCT
ejpam-5625	67	8	=	=	SYM
ejpam-5625	67	9	1	1	NUM
ejpam-5625	67	10	theorem	theorem	NOUN
ejpam-5625	67	11	2	2	NUM
ejpam-5625	67	12	.	.	PUNCT
ejpam-5625	68	1	let	let	VERB
ejpam-5625	68	2	n	n	PRON
ejpam-5625	68	3	be	be	AUX
ejpam-5625	68	4	a	a	DET
ejpam-5625	68	5	positive	positive	ADJ
ejpam-5625	68	6	integer	integer	NOUN
ejpam-5625	68	7	.	.	PUNCT
ejpam-5625	69	1	then	then	ADV
ejpam-5625	69	2	s	s	VERB
ejpam-5625	69	3	⊆	⊆	NUM
ejpam-5625	69	4	v	v	NOUN
ejpam-5625	69	5	(	(	PUNCT
ejpam-5625	69	6	kn	kn	PROPN
ejpam-5625	69	7	)	)	PUNCT
ejpam-5625	69	8	is	be	AUX
ejpam-5625	69	9	a	a	DET
ejpam-5625	69	10	2	2	NUM
ejpam-5625	69	11	-	-	PUNCT
ejpam-5625	69	12	distance	distance	NOUN
ejpam-5625	69	13	certified	certify	VERB
ejpam-5625	69	14	hop	hop	NOUN
ejpam-5625	69	15	dominating	dominating	NOUN
ejpam-5625	69	16	set	set	VERB
ejpam-5625	69	17	if	if	SCONJ
ejpam-5625	69	18	and	and	CCONJ
ejpam-5625	69	19	only	only	ADV
ejpam-5625	69	20	if	if	SCONJ
ejpam-5625	69	21	s	s	VERB
ejpam-5625	69	22	=	=	SYM
ejpam-5625	69	23	v	v	PROPN
ejpam-5625	69	24	(	(	PUNCT
ejpam-5625	69	25	kn	kn	PROPN
ejpam-5625	69	26	)	)	PUNCT
ejpam-5625	69	27	.	.	PUNCT
ejpam-5625	70	1	proof	proof	NOUN
ejpam-5625	70	2	.	.	PUNCT
ejpam-5625	71	1	suppose	suppose	VERB
ejpam-5625	71	2	that	that	SCONJ
ejpam-5625	71	3	s	s	VERB
ejpam-5625	71	4	is	be	AUX
ejpam-5625	71	5	2	2	NUM
ejpam-5625	71	6	-	-	PUNCT
ejpam-5625	71	7	distance	distance	NOUN
ejpam-5625	71	8	certified	certify	VERB
ejpam-5625	71	9	hop	hop	NOUN
ejpam-5625	71	10	dominating	dominating	NOUN
ejpam-5625	71	11	of	of	ADP
ejpam-5625	71	12	kn	kn	PROPN
ejpam-5625	71	13	.	.	PROPN
ejpam-5625	71	14	assume	assume	VERB
ejpam-5625	71	15	that	that	SCONJ
ejpam-5625	71	16	s	s	VERB
ejpam-5625	71	17	̸=	̸=	PROPN
ejpam-5625	71	18	v	v	NOUN
ejpam-5625	71	19	(	(	PUNCT
ejpam-5625	71	20	kn	kn	PROPN
ejpam-5625	71	21	)	)	PUNCT
ejpam-5625	71	22	.	.	PUNCT
ejpam-5625	72	1	then	then	ADV
ejpam-5625	72	2	there	there	PRON
ejpam-5625	72	3	exists	exist	VERB
ejpam-5625	72	4	at	at	ADV
ejpam-5625	72	5	least	least	ADV
ejpam-5625	72	6	one	one	NUM
ejpam-5625	72	7	vertex	vertex	NOUN
ejpam-5625	72	8	v	v	ADP
ejpam-5625	72	9	∈	∈	NOUN
ejpam-5625	72	10	v	v	NOUN
ejpam-5625	72	11	(	(	PUNCT
ejpam-5625	72	12	kn	kn	PROPN
ejpam-5625	72	13	)	)	PUNCT
ejpam-5625	72	14	such	such	ADJ
ejpam-5625	72	15	that	that	DET
ejpam-5625	72	16	v	v	NOUN
ejpam-5625	72	17	/∈	/∈	PUNCT
ejpam-5625	72	18	s.	s.	PROPN
ejpam-5625	72	19	since	since	SCONJ
ejpam-5625	72	20	the	the	DET
ejpam-5625	72	21	graph	graph	NOUN
ejpam-5625	72	22	is	be	AUX
ejpam-5625	72	23	complete	complete	ADJ
ejpam-5625	72	24	,	,	PUNCT
ejpam-5625	72	25	it	it	PRON
ejpam-5625	72	26	implies	imply	VERB
ejpam-5625	72	27	that	that	SCONJ
ejpam-5625	72	28	every	every	DET
ejpam-5625	72	29	vertex	vertex	NOUN
ejpam-5625	72	30	is	be	AUX
ejpam-5625	72	31	adjacent	adjacent	ADJ
ejpam-5625	72	32	to	to	ADP
ejpam-5625	72	33	every	every	DET
ejpam-5625	72	34	other	other	ADJ
ejpam-5625	72	35	vertex	vertex	NOUN
ejpam-5625	72	36	.	.	PUNCT
ejpam-5625	73	1	then	then	ADV
ejpam-5625	73	2	for	for	SCONJ
ejpam-5625	73	3	s	s	PRON
ejpam-5625	73	4	to	to	PART
ejpam-5625	73	5	be	be	AUX
ejpam-5625	73	6	a	a	DET
ejpam-5625	73	7	2	2	NUM
ejpam-5625	73	8	-	-	PUNCT
ejpam-5625	73	9	distance	distance	NOUN
ejpam-5625	73	10	certified	certify	VERB
ejpam-5625	73	11	hop	hop	NOUN
ejpam-5625	73	12	dominating	dominating	NOUN
ejpam-5625	73	13	,	,	PUNCT
ejpam-5625	73	14	v	v	X
ejpam-5625	73	15	must	must	AUX
ejpam-5625	73	16	be	be	AUX
ejpam-5625	73	17	included	include	VERB
ejpam-5625	73	18	in	in	ADP
ejpam-5625	73	19	s.	s.	PROPN
ejpam-5625	73	20	since	since	SCONJ
ejpam-5625	73	21	v	v	NUM
ejpam-5625	73	22	/∈	/∈	PUNCT
ejpam-5625	73	23	n2	n2	ADJ
ejpam-5625	73	24	g[s	g[s	PROPN
ejpam-5625	73	25	]	]	PUNCT
ejpam-5625	73	26	,	,	PUNCT
ejpam-5625	73	27	a	a	DET
ejpam-5625	73	28	contradiction	contradiction	NOUN
ejpam-5625	73	29	.	.	PUNCT
ejpam-5625	74	1	for	for	ADP
ejpam-5625	74	2	the	the	DET
ejpam-5625	74	3	converse	converse	NOUN
ejpam-5625	74	4	,	,	PUNCT
ejpam-5625	74	5	suppose	suppose	VERB
ejpam-5625	74	6	that	that	SCONJ
ejpam-5625	74	7	s	s	VERB
ejpam-5625	74	8	=	=	SYM
ejpam-5625	74	9	v	v	PROPN
ejpam-5625	74	10	(	(	PUNCT
ejpam-5625	74	11	kn	kn	PROPN
ejpam-5625	74	12	)	)	PUNCT
ejpam-5625	74	13	.	.	PUNCT
ejpam-5625	75	1	then	then	ADV
ejpam-5625	75	2	s	s	VERB
ejpam-5625	75	3	is	be	AUX
ejpam-5625	75	4	a	a	DET
ejpam-5625	75	5	2	2	NUM
ejpam-5625	75	6	-	-	PUNCT
ejpam-5625	75	7	distance	distance	NOUN
ejpam-5625	75	8	certified	certify	VERB
ejpam-5625	75	9	hop	hop	NOUN
ejpam-5625	75	10	dominating	dominating	NOUN
ejpam-5625	75	11	set	set	NOUN
ejpam-5625	75	12	of	of	ADP
ejpam-5625	75	13	kn	kn	PROPN
ejpam-5625	75	14	.	.	PUNCT
ejpam-5625	75	15	corollary	corollary	ADJ
ejpam-5625	76	1	1	1	NUM
ejpam-5625	76	2	.	.	PUNCT
ejpam-5625	77	1	let	let	VERB
ejpam-5625	77	2	m	m	PRON
ejpam-5625	77	3	be	be	AUX
ejpam-5625	77	4	a	a	DET
ejpam-5625	77	5	positive	positive	ADJ
ejpam-5625	77	6	integer	integer	NOUN
ejpam-5625	77	7	.	.	PUNCT
ejpam-5625	78	1	then	then	ADV
ejpam-5625	78	2	γ2ch(km	γ2ch(km	NOUN
ejpam-5625	78	3	)	)	PUNCT
ejpam-5625	78	4	=	=	PRON
ejpam-5625	78	5	m	m	VERB
ejpam-5625	78	6	for	for	ADP
ejpam-5625	78	7	all	all	DET
ejpam-5625	78	8	m	m	PROPN
ejpam-5625	78	9	≥	≥	NOUN
ejpam-5625	78	10	1	1	NUM
ejpam-5625	78	11	.	.	NOUN
ejpam-5625	78	12	5	5	NUM
ejpam-5625	78	13	.	.	X
ejpam-5625	78	14	co	co	VERB
ejpam-5625	78	15	-	-	ADJ
ejpam-5625	78	16	certified	certify	VERB
ejpam-5625	78	17	pointwise	pointwise	PROPN
ejpam-5625	78	18	non	non	ADJ
ejpam-5625	78	19	-	-	NOUN
ejpam-5625	78	20	domination	domination	NOUN
ejpam-5625	78	21	in	in	ADP
ejpam-5625	78	22	graphs	graph	NOUN
ejpam-5625	78	23	the	the	DET
ejpam-5625	78	24	following	follow	VERB
ejpam-5625	78	25	definiton	definiton	PROPN
ejpam-5625	78	26	will	will	AUX
ejpam-5625	78	27	be	be	AUX
ejpam-5625	78	28	used	use	VERB
ejpam-5625	78	29	to	to	PART
ejpam-5625	78	30	characterize	characterize	VERB
ejpam-5625	78	31	the	the	DET
ejpam-5625	78	32	2	2	NUM
ejpam-5625	78	33	-	-	PUNCT
ejpam-5625	78	34	distnace	distnace	NOUN
ejpam-5625	78	35	certified	certify	VERB
ejpam-5625	78	36	hop	hop	NOUN
ejpam-5625	78	37	dominating	dominating	NOUN
ejpam-5625	78	38	sets	set	NOUN
ejpam-5625	78	39	in	in	ADP
ejpam-5625	78	40	the	the	DET
ejpam-5625	78	41	join	join	NOUN
ejpam-5625	78	42	of	of	ADP
ejpam-5625	78	43	two	two	NUM
ejpam-5625	78	44	graphs	graph	NOUN
ejpam-5625	78	45	as	as	ADV
ejpam-5625	78	46	well	well	ADV
ejpam-5625	78	47	as	as	ADP
ejpam-5625	78	48	to	to	PART
ejpam-5625	78	49	solve	solve	VERB
ejpam-5625	78	50	its	its	PRON
ejpam-5625	78	51	2	2	NUM
ejpam-5625	78	52	-	-	PUNCT
ejpam-5625	78	53	distance	distance	NOUN
ejpam-5625	78	54	certified	certify	VERB
ejpam-5625	78	55	hop	hop	NOUN
ejpam-5625	78	56	domination	domination	NOUN
ejpam-5625	78	57	numbers	number	NOUN
ejpam-5625	78	58	.	.	PUNCT
ejpam-5625	79	1	definition	definition	NOUN
ejpam-5625	79	2	2	2	NUM
ejpam-5625	79	3	.	.	PUNCT
ejpam-5625	80	1	let	let	VERB
ejpam-5625	80	2	g	g	PRON
ejpam-5625	80	3	be	be	AUX
ejpam-5625	80	4	a	a	DET
ejpam-5625	80	5	graph	graph	NOUN
ejpam-5625	80	6	.	.	PUNCT
ejpam-5625	81	1	then	then	ADV
ejpam-5625	81	2	s	s	VERB
ejpam-5625	81	3	⊆	⊆	NUM
ejpam-5625	81	4	v	v	NOUN
ejpam-5625	81	5	(	(	PUNCT
ejpam-5625	81	6	g	g	NOUN
ejpam-5625	81	7	)	)	PUNCT
ejpam-5625	81	8	is	be	AUX
ejpam-5625	81	9	called	call	VERB
ejpam-5625	81	10	a	a	DET
ejpam-5625	81	11	co	co	ADJ
ejpam-5625	81	12	-	-	ADJ
ejpam-5625	81	13	certified	certify	VERB
ejpam-5625	81	14	pointwise	pointwise	PROPN
ejpam-5625	81	15	non	non	ADJ
ejpam-5625	81	16	-	-	ADJ
ejpam-5625	81	17	dominating	dominating	ADJ
ejpam-5625	81	18	(	(	PUNCT
ejpam-5625	81	19	co	co	ADJ
ejpam-5625	81	20	-	-	ADJ
ejpam-5625	81	21	certified	certify	VERB
ejpam-5625	81	22	pnd	pnd	NOUN
ejpam-5625	81	23	)	)	PUNCT
ejpam-5625	81	24	set	set	NOUN
ejpam-5625	81	25	of	of	ADP
ejpam-5625	81	26	g	g	PROPN
ejpam-5625	81	27	if	if	SCONJ
ejpam-5625	81	28	s	s	PART
ejpam-5625	81	29	satisfies	satisfy	VERB
ejpam-5625	81	30	the	the	DET
ejpam-5625	81	31	following	follow	VERB
ejpam-5625	81	32	two	two	NUM
ejpam-5625	81	33	conditions	condition	NOUN
ejpam-5625	81	34	:	:	PUNCT
ejpam-5625	81	35	(	(	PUNCT
ejpam-5625	81	36	i	i	NOUN
ejpam-5625	81	37	)	)	PUNCT
ejpam-5625	81	38	for	for	ADP
ejpam-5625	81	39	each	each	DET
ejpam-5625	81	40	u	u	PROPN
ejpam-5625	81	41	∈	∈	PROPN
ejpam-5625	81	42	s	s	NOUN
ejpam-5625	81	43	,	,	PUNCT
ejpam-5625	81	44	there	there	PRON
ejpam-5625	81	45	exist	exist	VERB
ejpam-5625	81	46	either	either	DET
ejpam-5625	81	47	zero	zero	NUM
ejpam-5625	81	48	or	or	CCONJ
ejpam-5625	81	49	at	at	ADP
ejpam-5625	81	50	least	least	ADV
ejpam-5625	81	51	two	two	NUM
ejpam-5625	81	52	vertices	vertex	NOUN
ejpam-5625	81	53	x	x	X
ejpam-5625	81	54	,	,	PUNCT
ejpam-5625	81	55	y	y	PROPN
ejpam-5625	81	56	∈	∈	PROPN
ejpam-5625	81	57	v	v	X
ejpam-5625	81	58	(	(	PUNCT
ejpam-5625	81	59	g)\s	g)\s	VERB
ejpam-5625	81	60	such	such	ADJ
ejpam-5625	81	61	that	that	SCONJ
ejpam-5625	81	62	x	x	NOUN
ejpam-5625	81	63	,	,	PUNCT
ejpam-5625	81	64	y	y	PROPN
ejpam-5625	81	65	/∈	/∈	PUNCT
ejpam-5625	81	66	ng(u	ng(u	PROPN
ejpam-5625	81	67	)	)	PUNCT
ejpam-5625	81	68	.	.	PUNCT
ejpam-5625	82	1	(	(	PUNCT
ejpam-5625	82	2	ii	ii	NOUN
ejpam-5625	82	3	)	)	PUNCT
ejpam-5625	82	4	for	for	ADP
ejpam-5625	82	5	each	each	DET
ejpam-5625	82	6	v	v	NUM
ejpam-5625	82	7	∈	∈	NOUN
ejpam-5625	82	8	v	v	NOUN
ejpam-5625	82	9	(	(	PUNCT
ejpam-5625	82	10	g)\s	g)\s	NOUN
ejpam-5625	82	11	,	,	PUNCT
ejpam-5625	82	12	there	there	PRON
ejpam-5625	82	13	exists	exist	VERB
ejpam-5625	82	14	w	w	PROPN
ejpam-5625	82	15	∈	∈	PROPN
ejpam-5625	82	16	s	s	VERB
ejpam-5625	82	17	such	such	ADJ
ejpam-5625	82	18	that	that	DET
ejpam-5625	82	19	v	v	NOUN
ejpam-5625	82	20	/∈	/∈	PUNCT
ejpam-5625	82	21	ng(w	ng(w	NOUN
ejpam-5625	82	22	)	)	PUNCT
ejpam-5625	82	23	.	.	PUNCT
ejpam-5625	83	1	the	the	DET
ejpam-5625	83	2	minimum	minimum	ADJ
ejpam-5625	83	3	cardinality	cardinality	NOUN
ejpam-5625	83	4	of	of	ADP
ejpam-5625	83	5	a	a	DET
ejpam-5625	83	6	co	co	ADJ
ejpam-5625	83	7	-	-	ADJ
ejpam-5625	83	8	certified	certify	VERB
ejpam-5625	83	9	pointwise	pointwise	PROPN
ejpam-5625	83	10	non	non	ADJ
ejpam-5625	83	11	-	-	ADJ
ejpam-5625	83	12	dominating	dominating	ADJ
ejpam-5625	83	13	(	(	PUNCT
ejpam-5625	83	14	co	co	ADJ
ejpam-5625	83	15	-	-	ADJ
ejpam-5625	83	16	certified	certify	VERB
ejpam-5625	83	17	pnd	pnd	NOUN
ejpam-5625	83	18	)	)	PUNCT
ejpam-5625	83	19	set	set	NOUN
ejpam-5625	83	20	of	of	ADP
ejpam-5625	83	21	g	g	NOUN
ejpam-5625	83	22	,	,	PUNCT
ejpam-5625	83	23	is	be	AUX
ejpam-5625	83	24	called	call	VERB
ejpam-5625	83	25	the	the	DET
ejpam-5625	83	26	co	co	ADJ
ejpam-5625	83	27	-	-	ADJ
ejpam-5625	83	28	certified	certify	VERB
ejpam-5625	83	29	pointwise	pointwise	PROPN
ejpam-5625	83	30	non	non	ADJ
ejpam-5625	83	31	-	-	NOUN
ejpam-5625	83	32	domination	domination	ADJ
ejpam-5625	83	33	(	(	PUNCT
ejpam-5625	83	34	co	co	ADJ
ejpam-5625	83	35	-	-	ADJ
ejpam-5625	83	36	certified	certify	VERB
ejpam-5625	83	37	pnd	pnd	NOUN
ejpam-5625	83	38	)	)	PUNCT
ejpam-5625	83	39	number	number	NOUN
ejpam-5625	83	40	of	of	ADP
ejpam-5625	83	41	g.	g.	PROPN
ejpam-5625	83	42	remark	remark	PROPN
ejpam-5625	83	43	1	1	NUM
ejpam-5625	83	44	.	.	PUNCT
ejpam-5625	84	1	let	let	VERB
ejpam-5625	84	2	g	g	PRON
ejpam-5625	84	3	be	be	AUX
ejpam-5625	84	4	a	a	DET
ejpam-5625	84	5	graph	graph	NOUN
ejpam-5625	84	6	.	.	PUNCT
ejpam-5625	85	1	then	then	ADV
ejpam-5625	85	2	(	(	PUNCT
ejpam-5625	85	3	i	i	NOUN
ejpam-5625	85	4	)	)	PUNCT
ejpam-5625	85	5	pnd(g	pnd(g	ADP
ejpam-5625	85	6	)	)	PUNCT
ejpam-5625	85	7	≤	≤	NUM
ejpam-5625	85	8	ccpnd(g	ccpnd(g	NOUN
ejpam-5625	85	9	)	)	PUNCT
ejpam-5625	85	10	;	;	PUNCT
ejpam-5625	85	11	and	and	CCONJ
ejpam-5625	85	12	(	(	PUNCT
ejpam-5625	85	13	ii	ii	NOUN
ejpam-5625	85	14	)	)	PUNCT
ejpam-5625	85	15	1	1	NUM
ejpam-5625	85	16	≤	≤	NUM
ejpam-5625	85	17	ccpnd(g	ccpnd(g	NOUN
ejpam-5625	85	18	)	)	PUNCT
ejpam-5625	85	19	≤	≤	NOUN
ejpam-5625	86	1	|v	|v	X
ejpam-5625	86	2	(	(	PUNCT
ejpam-5625	86	3	g)|	g)|	PROPN
ejpam-5625	86	4	.	.	PUNCT
ejpam-5625	86	5	theorem	theorem	NOUN
ejpam-5625	86	6	3	3	X
ejpam-5625	86	7	.	.	PUNCT
ejpam-5625	87	1	let	let	VERB
ejpam-5625	87	2	g	g	PRON
ejpam-5625	87	3	be	be	AUX
ejpam-5625	87	4	a	a	DET
ejpam-5625	87	5	graph	graph	NOUN
ejpam-5625	87	6	.	.	PUNCT
ejpam-5625	88	1	then	then	ADV
ejpam-5625	88	2	ccpnd(g	ccpnd(g	PROPN
ejpam-5625	88	3	)	)	PUNCT
ejpam-5625	88	4	̸=	̸=	PROPN
ejpam-5625	88	5	|v	|v	NOUN
ejpam-5625	88	6	(	(	PUNCT
ejpam-5625	88	7	g)|	g)|	VERB
ejpam-5625	88	8	if	if	SCONJ
ejpam-5625	88	9	and	and	CCONJ
ejpam-5625	88	10	only	only	ADV
ejpam-5625	88	11	if	if	SCONJ
ejpam-5625	88	12	ccpnd(g	ccpnd(g	NUM
ejpam-5625	88	13	)	)	PUNCT
ejpam-5625	88	14	≤	≤	NOUN
ejpam-5625	88	15	|v	|v	X
ejpam-5625	88	16	(	(	PUNCT
ejpam-5625	88	17	g)|	g)|	INTJ
ejpam-5625	88	18	−	−	NOUN
ejpam-5625	88	19	2	2	NUM
ejpam-5625	88	20	.	.	PUNCT
ejpam-5625	88	21	proof	proof	NOUN
ejpam-5625	88	22	.	.	PUNCT
ejpam-5625	88	23	suppose	suppose	VERB
ejpam-5625	88	24	that	that	SCONJ
ejpam-5625	88	25	ccpnd(g	ccpnd(g	NOUN
ejpam-5625	88	26	)	)	PUNCT
ejpam-5625	88	27	=	=	SYM
ejpam-5625	88	28	|v	|v	PROPN
ejpam-5625	88	29	(	(	PUNCT
ejpam-5625	88	30	g)|	g)|	INTJ
ejpam-5625	88	31	−	−	NOUN
ejpam-5625	88	32	1	1	NUM
ejpam-5625	88	33	,	,	PUNCT
ejpam-5625	88	34	say	say	VERB
ejpam-5625	88	35	s	s	PRON
ejpam-5625	88	36	is	be	AUX
ejpam-5625	88	37	the	the	DET
ejpam-5625	88	38	minimum	minimum	ADJ
ejpam-5625	88	39	co	co	ADJ
ejpam-5625	88	40	-	-	ADJ
ejpam-5625	88	41	certified	certify	VERB
ejpam-5625	88	42	pnd	pnd	NOUN
ejpam-5625	88	43	set	set	NOUN
ejpam-5625	88	44	.	.	PUNCT
ejpam-5625	89	1	then	then	ADV
ejpam-5625	89	2	there	there	PRON
ejpam-5625	89	3	exists	exist	VERB
ejpam-5625	89	4	x	x	X
ejpam-5625	89	5	∈	∈	PROPN
ejpam-5625	89	6	v	v	X
ejpam-5625	89	7	(	(	PUNCT
ejpam-5625	89	8	g	g	NOUN
ejpam-5625	89	9	)	)	PUNCT
ejpam-5625	89	10	such	such	ADJ
ejpam-5625	89	11	that	that	SCONJ
ejpam-5625	89	12	x	x	PROPN
ejpam-5625	89	13	/∈	/∈	PROPN
ejpam-5625	89	14	s.	s.	PROPN
ejpam-5625	89	15	let	let	VERB
ejpam-5625	89	16	y	y	PROPN
ejpam-5625	89	17	∈	∈	PROPN
ejpam-5625	89	18	s.	s.	PROPN
ejpam-5625	89	19	if	if	SCONJ
ejpam-5625	89	20	x	x	PROPN
ejpam-5625	89	21	and	and	CCONJ
ejpam-5625	89	22	y	y	PROPN
ejpam-5625	89	23	are	be	AUX
ejpam-5625	89	24	non	non	ADJ
ejpam-5625	89	25	-	-	ADJ
ejpam-5625	89	26	adjacent	adjacent	ADJ
ejpam-5625	89	27	.	.	PUNCT
ejpam-5625	90	1	n.	n.	PROPN
ejpam-5625	90	2	s.	s.	PROPN
ejpam-5625	90	3	ulal	ulal	PROPN
ejpam-5625	90	4	et	et	PROPN
ejpam-5625	90	5	al	al	PROPN
ejpam-5625	90	6	.	.	PUNCT
ejpam-5625	90	7	/	/	SYM
ejpam-5625	90	8	eur	eur	PROPN
ejpam-5625	90	9	.	.	PUNCT
ejpam-5625	91	1	j.	j.	PROPN
ejpam-5625	91	2	pure	pure	PROPN
ejpam-5625	91	3	appl	appl	PROPN
ejpam-5625	91	4	.	.	PROPN
ejpam-5625	91	5	math	math	PROPN
ejpam-5625	91	6	,	,	PUNCT
ejpam-5625	91	7	18	18	NUM
ejpam-5625	91	8	(	(	PUNCT
ejpam-5625	91	9	1	1	NUM
ejpam-5625	91	10	)	)	PUNCT
ejpam-5625	91	11	(	(	PUNCT
ejpam-5625	91	12	2025	2025	NUM
ejpam-5625	91	13	)	)	PUNCT
ejpam-5625	91	14	,	,	PUNCT
ejpam-5625	91	15	5625	5625	NUM
ejpam-5625	91	16	5	5	NUM
ejpam-5625	91	17	of	of	ADP
ejpam-5625	91	18	9	9	NUM
ejpam-5625	91	19	then	then	ADV
ejpam-5625	91	20	x	x	X
ejpam-5625	91	21	/∈	/∈	PUNCT
ejpam-5625	91	22	ng(y	ng(y	NOUN
ejpam-5625	91	23	)	)	PUNCT
ejpam-5625	91	24	.	.	PUNCT
ejpam-5625	92	1	that	that	PRON
ejpam-5625	92	2	is	is	ADV
ejpam-5625	92	3	,	,	PUNCT
ejpam-5625	92	4	y	y	PROPN
ejpam-5625	92	5	has	have	VERB
ejpam-5625	92	6	only	only	ADV
ejpam-5625	92	7	one	one	NUM
ejpam-5625	92	8	non	non	ADJ
ejpam-5625	92	9	-	-	NOUN
ejpam-5625	92	10	neighbor	neighbor	NOUN
ejpam-5625	92	11	x	x	INTJ
ejpam-5625	92	12	in	in	ADP
ejpam-5625	92	13	v	v	NOUN
ejpam-5625	92	14	(	(	PUNCT
ejpam-5625	92	15	g)\s	g)\s	NOUN
ejpam-5625	92	16	,	,	PUNCT
ejpam-5625	92	17	a	a	DET
ejpam-5625	92	18	contradiction	contradiction	NOUN
ejpam-5625	92	19	.	.	PUNCT
ejpam-5625	93	1	therefore	therefore	ADV
ejpam-5625	93	2	,	,	PUNCT
ejpam-5625	93	3	the	the	DET
ejpam-5625	93	4	assertion	assertion	NOUN
ejpam-5625	93	5	follows	follow	VERB
ejpam-5625	93	6	.	.	PUNCT
ejpam-5625	94	1	the	the	DET
ejpam-5625	94	2	converse	converse	NOUN
ejpam-5625	94	3	is	be	AUX
ejpam-5625	94	4	clear	clear	ADJ
ejpam-5625	94	5	.	.	PUNCT
ejpam-5625	95	1	proposition	proposition	NOUN
ejpam-5625	95	2	1	1	NUM
ejpam-5625	95	3	.	.	PUNCT
ejpam-5625	96	1	let	let	VERB
ejpam-5625	96	2	k	k	PRON
ejpam-5625	96	3	be	be	AUX
ejpam-5625	96	4	a	a	DET
ejpam-5625	96	5	positive	positive	ADJ
ejpam-5625	96	6	integer	integer	NOUN
ejpam-5625	96	7	.	.	PUNCT
ejpam-5625	97	1	then	then	ADV
ejpam-5625	97	2	,	,	PUNCT
ejpam-5625	97	3	(	(	PUNCT
ejpam-5625	97	4	i	i	NOUN
ejpam-5625	97	5	)	)	PUNCT
ejpam-5625	97	6	ccpnd(pk	ccpnd(pk	PROPN
ejpam-5625	97	7	)	)	PUNCT
ejpam-5625	98	1	=	=	PRON
ejpam-5625	98	2	{	{	PUNCT
ejpam-5625	98	3	k	k	NOUN
ejpam-5625	98	4	,	,	PUNCT
ejpam-5625	98	5	if	if	SCONJ
ejpam-5625	98	6	k	k	PROPN
ejpam-5625	98	7	=	=	SYM
ejpam-5625	98	8	1	1	NUM
ejpam-5625	98	9	,	,	PUNCT
ejpam-5625	98	10	2	2	NUM
ejpam-5625	98	11	,	,	PUNCT
ejpam-5625	98	12	3	3	NUM
ejpam-5625	98	13	,	,	PUNCT
ejpam-5625	98	14	4	4	NUM
ejpam-5625	98	15	2	2	NUM
ejpam-5625	98	16	,	,	PUNCT
ejpam-5625	98	17	if	if	SCONJ
ejpam-5625	98	18	k	k	PROPN
ejpam-5625	98	19	≥	≥	NOUN
ejpam-5625	98	20	5	5	NUM
ejpam-5625	98	21	;	;	PUNCT
ejpam-5625	98	22	(	(	PUNCT
ejpam-5625	98	23	ii	ii	NOUN
ejpam-5625	98	24	)	)	PUNCT
ejpam-5625	98	25	ccpnd(ck	ccpnd(ck	NOUN
ejpam-5625	98	26	)	)	PUNCT
ejpam-5625	98	27	=	=	SYM
ejpam-5625	98	28	{	{	PUNCT
ejpam-5625	98	29	k	k	NOUN
ejpam-5625	98	30	,	,	PUNCT
ejpam-5625	98	31	if	if	SCONJ
ejpam-5625	98	32	k	k	PROPN
ejpam-5625	98	33	=	=	SYM
ejpam-5625	98	34	3	3	NUM
ejpam-5625	98	35	,	,	PUNCT
ejpam-5625	98	36	4	4	NUM
ejpam-5625	98	37	2	2	NUM
ejpam-5625	98	38	,	,	PUNCT
ejpam-5625	98	39	if	if	SCONJ
ejpam-5625	98	40	k	k	PROPN
ejpam-5625	98	41	≥	≥	NOUN
ejpam-5625	98	42	5	5	NUM
ejpam-5625	98	43	;	;	PUNCT
ejpam-5625	98	44	(	(	PUNCT
ejpam-5625	98	45	iii	iii	X
ejpam-5625	98	46	)	)	PUNCT
ejpam-5625	98	47	ccpnd(kk	ccpnd(kk	ADJ
ejpam-5625	98	48	)	)	PUNCT
ejpam-5625	98	49	=	=	SYM
ejpam-5625	98	50	k	k	PROPN
ejpam-5625	98	51	for	for	ADP
ejpam-5625	98	52	all	all	DET
ejpam-5625	98	53	k	k	PROPN
ejpam-5625	98	54	≥	≥	NUM
ejpam-5625	98	55	1	1	NUM
ejpam-5625	98	56	;	;	PUNCT
ejpam-5625	98	57	and	and	CCONJ
ejpam-5625	98	58	(	(	PUNCT
ejpam-5625	98	59	iv	iv	X
ejpam-5625	98	60	)	)	PUNCT
ejpam-5625	98	61	ccpnd	ccpnd	NOUN
ejpam-5625	98	62	¯(kk	¯(kk	PROPN
ejpam-5625	98	63	)	)	PUNCT
ejpam-5625	98	64	=	=	PRON
ejpam-5625	98	65	{	{	PUNCT
ejpam-5625	98	66	1	1	NUM
ejpam-5625	98	67	,	,	PUNCT
ejpam-5625	98	68	if	if	SCONJ
ejpam-5625	98	69	for	for	ADP
ejpam-5625	98	70	all	all	DET
ejpam-5625	98	71	k	k	PROPN
ejpam-5625	98	72	≥	≥	NUM
ejpam-5625	98	73	3	3	NUM
ejpam-5625	98	74	k	k	NOUN
ejpam-5625	98	75	,	,	PUNCT
ejpam-5625	98	76	if	if	SCONJ
ejpam-5625	98	77	k	k	PROPN
ejpam-5625	98	78	=	=	SYM
ejpam-5625	98	79	1	1	NUM
ejpam-5625	98	80	,	,	PUNCT
ejpam-5625	98	81	2	2	NUM
ejpam-5625	98	82	.	.	PUNCT
ejpam-5625	98	83	proof	proof	NOUN
ejpam-5625	98	84	.	.	PUNCT
ejpam-5625	99	1	(	(	PUNCT
ejpam-5625	99	2	i	i	NOUN
ejpam-5625	99	3	)	)	PUNCT
ejpam-5625	99	4	since	since	SCONJ
ejpam-5625	99	5	pnd(pk	pnd(pk	NOUN
ejpam-5625	99	6	)	)	PUNCT
ejpam-5625	99	7	=	=	SYM
ejpam-5625	99	8	k	k	PROPN
ejpam-5625	99	9	for	for	ADP
ejpam-5625	99	10	k	k	PROPN
ejpam-5625	99	11	=	=	SYM
ejpam-5625	99	12	1	1	NUM
ejpam-5625	99	13	,	,	PUNCT
ejpam-5625	99	14	2	2	NUM
ejpam-5625	99	15	,	,	PUNCT
ejpam-5625	99	16	it	it	PRON
ejpam-5625	99	17	follows	follow	VERB
ejpam-5625	99	18	that	that	SCONJ
ejpam-5625	99	19	ccpnd(pk	ccpnd(pk	NOUN
ejpam-5625	99	20	)	)	PUNCT
ejpam-5625	100	1	=	=	SYM
ejpam-5625	100	2	k	k	PROPN
ejpam-5625	100	3	for	for	ADP
ejpam-5625	100	4	k	k	PROPN
ejpam-5625	100	5	=	=	SYM
ejpam-5625	100	6	1	1	NUM
ejpam-5625	100	7	,	,	PUNCT
ejpam-5625	100	8	2	2	NUM
ejpam-5625	100	9	by	by	ADP
ejpam-5625	100	10	remark	remark	NOUN
ejpam-5625	100	11	1	1	NUM
ejpam-5625	100	12	.	.	PUNCT
ejpam-5625	101	1	for	for	ADP
ejpam-5625	101	2	k	k	PROPN
ejpam-5625	101	3	=	=	SYM
ejpam-5625	101	4	3	3	NUM
ejpam-5625	101	5	,	,	PUNCT
ejpam-5625	101	6	let	let	VERB
ejpam-5625	101	7	v	v	NOUN
ejpam-5625	101	8	(	(	PUNCT
ejpam-5625	101	9	p3	p3	PROPN
ejpam-5625	101	10	)	)	PUNCT
ejpam-5625	102	1	=	=	PRON
ejpam-5625	102	2	{	{	PUNCT
ejpam-5625	102	3	a1	a1	PROPN
ejpam-5625	102	4	,	,	PUNCT
ejpam-5625	102	5	a2	a2	PROPN
ejpam-5625	102	6	,	,	PUNCT
ejpam-5625	102	7	a3	a3	NOUN
ejpam-5625	102	8	}	}	PUNCT
ejpam-5625	102	9	.	.	PUNCT
ejpam-5625	103	1	consider	consider	VERB
ejpam-5625	103	2	r	r	NOUN
ejpam-5625	103	3	=	=	SYM
ejpam-5625	103	4	{	{	PUNCT
ejpam-5625	103	5	a1	a1	PROPN
ejpam-5625	103	6	,	,	PUNCT
ejpam-5625	103	7	a2	a2	PROPN
ejpam-5625	103	8	}	}	PUNCT
ejpam-5625	103	9	.	.	PUNCT
ejpam-5625	104	1	then	then	ADV
ejpam-5625	104	2	r	r	NOUN
ejpam-5625	104	3	is	be	AUX
ejpam-5625	104	4	a	a	DET
ejpam-5625	104	5	minimum	minimum	ADJ
ejpam-5625	104	6	pnd	pnd	NOUN
ejpam-5625	104	7	set	set	NOUN
ejpam-5625	104	8	of	of	ADP
ejpam-5625	104	9	p3	p3	PROPN
ejpam-5625	104	10	,	,	PUNCT
ejpam-5625	104	11	and	and	CCONJ
ejpam-5625	104	12	so	so	ADV
ejpam-5625	104	13	by	by	ADP
ejpam-5625	104	14	remark	remark	NOUN
ejpam-5625	104	15	1	1	NUM
ejpam-5625	104	16	,	,	PUNCT
ejpam-5625	104	17	ccpnd(p3	ccpnd(p3	NOUN
ejpam-5625	104	18	)	)	PUNCT
ejpam-5625	104	19	≥	≥	NOUN
ejpam-5625	104	20	2	2	NUM
ejpam-5625	104	21	.	.	PUNCT
ejpam-5625	104	22	by	by	ADP
ejpam-5625	104	23	theorem	theorem	ADJ
ejpam-5625	104	24	3	3	NUM
ejpam-5625	104	25	,	,	PUNCT
ejpam-5625	104	26	ccpnd(p3	ccpnd(p3	NOUN
ejpam-5625	104	27	)	)	PUNCT
ejpam-5625	104	28	=	=	SYM
ejpam-5625	105	1	3	3	X
ejpam-5625	105	2	.	.	X
ejpam-5625	105	3	for	for	ADP
ejpam-5625	105	4	k	k	PROPN
ejpam-5625	105	5	=	=	SYM
ejpam-5625	105	6	4	4	NUM
ejpam-5625	105	7	,	,	PUNCT
ejpam-5625	105	8	let	let	VERB
ejpam-5625	105	9	v	v	NOUN
ejpam-5625	105	10	(	(	PUNCT
ejpam-5625	105	11	p4	p4	ADJ
ejpam-5625	105	12	)	)	PUNCT
ejpam-5625	105	13	=	=	SYM
ejpam-5625	105	14	{	{	PUNCT
ejpam-5625	105	15	a1	a1	PROPN
ejpam-5625	105	16	,	,	PUNCT
ejpam-5625	105	17	a2	a2	PROPN
ejpam-5625	105	18	,	,	PUNCT
ejpam-5625	105	19	a3	a3	NOUN
ejpam-5625	105	20	,	,	PUNCT
ejpam-5625	105	21	a4	a4	PROPN
ejpam-5625	105	22	}	}	PUNCT
ejpam-5625	105	23	.	.	PUNCT
ejpam-5625	106	1	consider	consider	VERB
ejpam-5625	106	2	q	q	NOUN
ejpam-5625	106	3	=	=	PUNCT
ejpam-5625	106	4	{	{	PUNCT
ejpam-5625	106	5	a1	a1	PROPN
ejpam-5625	106	6	,	,	PUNCT
ejpam-5625	106	7	a2	a2	PROPN
ejpam-5625	106	8	}	}	PUNCT
ejpam-5625	106	9	.	.	PUNCT
ejpam-5625	107	1	then	then	ADV
ejpam-5625	107	2	q	q	X
ejpam-5625	107	3	is	be	AUX
ejpam-5625	107	4	a	a	DET
ejpam-5625	107	5	minimum	minimum	ADJ
ejpam-5625	107	6	pnd	pnd	NOUN
ejpam-5625	107	7	set	set	NOUN
ejpam-5625	107	8	of	of	ADP
ejpam-5625	107	9	p4	p4	NOUN
ejpam-5625	107	10	.	.	PUNCT
ejpam-5625	108	1	thus	thus	ADV
ejpam-5625	108	2	,	,	PUNCT
ejpam-5625	108	3	ccpnd(p4	ccpnd(p4	NOUN
ejpam-5625	108	4	)	)	PUNCT
ejpam-5625	108	5	≥	≥	NOUN
ejpam-5625	108	6	2	2	NUM
ejpam-5625	108	7	by	by	ADP
ejpam-5625	108	8	remark	remark	NOUN
ejpam-5625	108	9	1	1	NUM
ejpam-5625	108	10	.	.	PUNCT
ejpam-5625	108	11	suppose	suppose	VERB
ejpam-5625	108	12	that	that	SCONJ
ejpam-5625	108	13	ccpnd(g	ccpnd(g	NOUN
ejpam-5625	108	14	)	)	PUNCT
ejpam-5625	108	15	=	=	SYM
ejpam-5625	109	1	2	2	NUM
ejpam-5625	109	2	,	,	PUNCT
ejpam-5625	109	3	say	say	VERB
ejpam-5625	109	4	m	m	NOUN
ejpam-5625	109	5	is	be	AUX
ejpam-5625	109	6	a	a	DET
ejpam-5625	109	7	minimum	minimum	ADJ
ejpam-5625	109	8	co	co	VERB
ejpam-5625	109	9	-	-	ADJ
ejpam-5625	109	10	certified	certify	VERB
ejpam-5625	109	11	pnd	pnd	NOUN
ejpam-5625	109	12	set	set	NOUN
ejpam-5625	109	13	of	of	ADP
ejpam-5625	109	14	p4	p4	NOUN
ejpam-5625	109	15	.	.	PUNCT
ejpam-5625	110	1	then	then	ADV
ejpam-5625	110	2	m	m	PROPN
ejpam-5625	110	3	is	be	AUX
ejpam-5625	110	4	either	either	PRON
ejpam-5625	110	5	of	of	ADP
ejpam-5625	110	6	the	the	DET
ejpam-5625	110	7	following	follow	VERB
ejpam-5625	110	8	sets	set	NOUN
ejpam-5625	110	9	:	:	PUNCT
ejpam-5625	110	10	{	{	PUNCT
ejpam-5625	110	11	a1	a1	NOUN
ejpam-5625	110	12	,	,	PUNCT
ejpam-5625	110	13	a2	a2	PROPN
ejpam-5625	110	14	}	}	PUNCT
ejpam-5625	110	15	,	,	PUNCT
ejpam-5625	110	16	{	{	PUNCT
ejpam-5625	110	17	a2	a2	NOUN
ejpam-5625	110	18	,	,	PUNCT
ejpam-5625	110	19	a3	a3	NOUN
ejpam-5625	110	20	}	}	PUNCT
ejpam-5625	110	21	,	,	PUNCT
ejpam-5625	110	22	{	{	PUNCT
ejpam-5625	110	23	a3	a3	NOUN
ejpam-5625	110	24	,	,	PUNCT
ejpam-5625	110	25	a4	a4	PROPN
ejpam-5625	110	26	}	}	PUNCT
ejpam-5625	110	27	or	or	CCONJ
ejpam-5625	110	28	{	{	PUNCT
ejpam-5625	110	29	a1	a1	NOUN
ejpam-5625	110	30	,	,	PUNCT
ejpam-5625	110	31	a4	a4	NOUN
ejpam-5625	110	32	}	}	PUNCT
ejpam-5625	110	33	.	.	PUNCT
ejpam-5625	111	1	however	however	ADV
ejpam-5625	111	2	,	,	PUNCT
ejpam-5625	111	3	either	either	PRON
ejpam-5625	111	4	of	of	ADP
ejpam-5625	111	5	these	these	DET
ejpam-5625	111	6	cases	case	NOUN
ejpam-5625	111	7	contradicts	contradict	VERB
ejpam-5625	111	8	our	our	PRON
ejpam-5625	111	9	assumption	assumption	NOUN
ejpam-5625	111	10	of	of	ADP
ejpam-5625	111	11	being	be	AUX
ejpam-5625	111	12	a	a	DET
ejpam-5625	111	13	co	co	ADJ
ejpam-5625	111	14	-	-	ADJ
ejpam-5625	111	15	certified	certify	VERB
ejpam-5625	111	16	pnd	pnd	NOUN
ejpam-5625	111	17	set	set	NOUN
ejpam-5625	111	18	of	of	ADP
ejpam-5625	111	19	p4	p4	NOUN
ejpam-5625	111	20	.	.	PUNCT
ejpam-5625	112	1	therefore	therefore	ADV
ejpam-5625	112	2	,	,	PUNCT
ejpam-5625	112	3	by	by	ADP
ejpam-5625	112	4	theorem	theorem	ADJ
ejpam-5625	112	5	3	3	NUM
ejpam-5625	112	6	,	,	PUNCT
ejpam-5625	112	7	ccpnd(p4	ccpnd(p4	NOUN
ejpam-5625	112	8	)	)	PUNCT
ejpam-5625	112	9	=	=	SYM
ejpam-5625	113	1	4	4	X
ejpam-5625	113	2	.	.	PUNCT
ejpam-5625	114	1	next	next	ADV
ejpam-5625	114	2	,	,	PUNCT
ejpam-5625	114	3	for	for	ADP
ejpam-5625	114	4	k	k	PROPN
ejpam-5625	114	5	≥	≥	NUM
ejpam-5625	114	6	5	5	NUM
ejpam-5625	114	7	,	,	PUNCT
ejpam-5625	114	8	let	let	VERB
ejpam-5625	114	9	v	v	NOUN
ejpam-5625	114	10	(	(	PUNCT
ejpam-5625	114	11	pk	pk	NOUN
ejpam-5625	114	12	)	)	PUNCT
ejpam-5625	114	13	=	=	NOUN
ejpam-5625	114	14	{	{	PUNCT
ejpam-5625	114	15	v1	v1	PROPN
ejpam-5625	114	16	,	,	PUNCT
ejpam-5625	114	17	v2	v2	PROPN
ejpam-5625	114	18	,	,	PUNCT
ejpam-5625	114	19	...	...	PUNCT
ejpam-5625	114	20	,	,	PUNCT
ejpam-5625	114	21	vk	vk	ADP
ejpam-5625	114	22	}	}	PUNCT
ejpam-5625	114	23	.	.	PUNCT
ejpam-5625	115	1	consider	consider	VERB
ejpam-5625	115	2	n	n	NOUN
ejpam-5625	115	3	=	=	SYM
ejpam-5625	115	4	{	{	PUNCT
ejpam-5625	115	5	v1	v1	PROPN
ejpam-5625	115	6	,	,	PUNCT
ejpam-5625	115	7	v2	v2	PROPN
ejpam-5625	115	8	}	}	PUNCT
ejpam-5625	115	9	.	.	PUNCT
ejpam-5625	116	1	then	then	ADV
ejpam-5625	116	2	n	n	PRON
ejpam-5625	116	3	is	be	AUX
ejpam-5625	116	4	a	a	DET
ejpam-5625	116	5	minimum	minimum	ADJ
ejpam-5625	116	6	pnd	pnd	NOUN
ejpam-5625	116	7	set	set	NOUN
ejpam-5625	116	8	of	of	ADP
ejpam-5625	116	9	pk	pk	PROPN
ejpam-5625	116	10	.	.	PROPN
ejpam-5625	117	1	since	since	SCONJ
ejpam-5625	117	2	k	k	PROPN
ejpam-5625	117	3	≥	≥	NUM
ejpam-5625	117	4	5	5	NUM
ejpam-5625	117	5	,	,	PUNCT
ejpam-5625	117	6	both	both	PRON
ejpam-5625	117	7	v1	v1	VERB
ejpam-5625	117	8	and	and	CCONJ
ejpam-5625	117	9	v2	v2	NOUN
ejpam-5625	117	10	has	have	VERB
ejpam-5625	117	11	at	at	ADV
ejpam-5625	117	12	least	least	ADV
ejpam-5625	117	13	two	two	NUM
ejpam-5625	117	14	non	non	NOUN
ejpam-5625	117	15	-	-	NOUN
ejpam-5625	117	16	neighbors	neighbor	NOUN
ejpam-5625	117	17	in	in	ADP
ejpam-5625	117	18	v	v	NOUN
ejpam-5625	117	19	(	(	PUNCT
ejpam-5625	117	20	pk)\n	pk)\n	NOUN
ejpam-5625	117	21	,	,	PUNCT
ejpam-5625	117	22	respectively	respectively	ADV
ejpam-5625	117	23	.	.	PUNCT
ejpam-5625	118	1	therefore	therefore	ADV
ejpam-5625	118	2	,	,	PUNCT
ejpam-5625	118	3	n	n	PRON
ejpam-5625	118	4	is	be	AUX
ejpam-5625	118	5	a	a	DET
ejpam-5625	118	6	minimum	minimum	ADJ
ejpam-5625	118	7	co	co	VERB
ejpam-5625	118	8	-	-	ADJ
ejpam-5625	118	9	certified	certify	VERB
ejpam-5625	118	10	pnd	pnd	NOUN
ejpam-5625	118	11	set	set	NOUN
ejpam-5625	118	12	of	of	ADP
ejpam-5625	118	13	pk	pk	NOUN
ejpam-5625	118	14	,	,	PUNCT
ejpam-5625	118	15	and	and	CCONJ
ejpam-5625	118	16	so	so	ADV
ejpam-5625	118	17	ccpnd(pk	ccpnd(pk	ADJ
ejpam-5625	118	18	)	)	PUNCT
ejpam-5625	118	19	=	=	SYM
ejpam-5625	118	20	2	2	NUM
ejpam-5625	118	21	for	for	ADP
ejpam-5625	118	22	all	all	DET
ejpam-5625	118	23	k	k	PROPN
ejpam-5625	118	24	≥	≥	NUM
ejpam-5625	118	25	5	5	NUM
ejpam-5625	118	26	.	.	PUNCT
ejpam-5625	118	27	(	(	PUNCT
ejpam-5625	118	28	ii	ii	NOUN
ejpam-5625	118	29	)	)	PUNCT
ejpam-5625	118	30	since	since	SCONJ
ejpam-5625	118	31	pnd(c3	pnd(c3	NOUN
ejpam-5625	118	32	)	)	PUNCT
ejpam-5625	118	33	=	=	SYM
ejpam-5625	118	34	3	3	X
ejpam-5625	118	35	,	,	PUNCT
ejpam-5625	118	36	it	it	PRON
ejpam-5625	118	37	follows	follow	VERB
ejpam-5625	118	38	that	that	SCONJ
ejpam-5625	118	39	ccpnd(c3	ccpnd(c3	NOUN
ejpam-5625	118	40	)	)	PUNCT
ejpam-5625	118	41	=	=	SYM
ejpam-5625	118	42	3	3	NUM
ejpam-5625	118	43	by	by	ADP
ejpam-5625	118	44	remark	remark	NOUN
ejpam-5625	118	45	1	1	NUM
ejpam-5625	118	46	.	.	PUNCT
ejpam-5625	119	1	for	for	ADP
ejpam-5625	119	2	n	n	NOUN
ejpam-5625	119	3	=	=	SYM
ejpam-5625	119	4	4	4	NUM
ejpam-5625	119	5	,	,	PUNCT
ejpam-5625	119	6	let	let	VERB
ejpam-5625	119	7	v	v	X
ejpam-5625	119	8	(	(	PUNCT
ejpam-5625	119	9	c4	c4	NOUN
ejpam-5625	119	10	)	)	PUNCT
ejpam-5625	119	11	=	=	SYM
ejpam-5625	119	12	{	{	PUNCT
ejpam-5625	119	13	u1	u1	NOUN
ejpam-5625	119	14	,	,	PUNCT
ejpam-5625	119	15	u2	u2	NOUN
ejpam-5625	119	16	,	,	PUNCT
ejpam-5625	119	17	u3	u3	NOUN
ejpam-5625	119	18	,	,	PUNCT
ejpam-5625	119	19	u4	u4	PROPN
ejpam-5625	119	20	}	}	PUNCT
ejpam-5625	119	21	.	.	PUNCT
ejpam-5625	120	1	consider	consider	VERB
ejpam-5625	120	2	q	q	NOUN
ejpam-5625	120	3	=	=	NUM
ejpam-5625	120	4	{	{	PUNCT
ejpam-5625	120	5	u1	u1	NOUN
ejpam-5625	120	6	,	,	PUNCT
ejpam-5625	120	7	u2	u2	PROPN
ejpam-5625	120	8	}	}	PUNCT
ejpam-5625	120	9	.	.	PUNCT
ejpam-5625	121	1	then	then	ADV
ejpam-5625	121	2	q	q	X
ejpam-5625	121	3	is	be	AUX
ejpam-5625	121	4	a	a	DET
ejpam-5625	121	5	minimum	minimum	ADJ
ejpam-5625	121	6	pnd	pnd	NOUN
ejpam-5625	121	7	set	set	NOUN
ejpam-5625	121	8	of	of	ADP
ejpam-5625	121	9	c4	c4	NOUN
ejpam-5625	121	10	.	.	PUNCT
ejpam-5625	122	1	by	by	ADP
ejpam-5625	122	2	remark	remark	NOUN
ejpam-5625	122	3	1	1	NUM
ejpam-5625	122	4	,	,	PUNCT
ejpam-5625	122	5	ccpnd(c4	ccpnd(c4	NOUN
ejpam-5625	122	6	)	)	PUNCT
ejpam-5625	122	7	≥	≥	NOUN
ejpam-5625	122	8	2	2	NUM
ejpam-5625	122	9	.	.	PUNCT
ejpam-5625	122	10	suppose	suppose	VERB
ejpam-5625	122	11	that	that	SCONJ
ejpam-5625	122	12	ccpnd(c4	ccpnd(c4	NOUN
ejpam-5625	122	13	)	)	PUNCT
ejpam-5625	122	14	=	=	SYM
ejpam-5625	122	15	2	2	NUM
ejpam-5625	122	16	,	,	PUNCT
ejpam-5625	122	17	say	say	VERB
ejpam-5625	122	18	r	r	NOUN
ejpam-5625	122	19	is	be	AUX
ejpam-5625	122	20	a	a	DET
ejpam-5625	122	21	minimum	minimum	ADJ
ejpam-5625	122	22	co	co	VERB
ejpam-5625	122	23	-	-	ADJ
ejpam-5625	122	24	certified	certify	VERB
ejpam-5625	122	25	pnd	pnd	NOUN
ejpam-5625	122	26	set	set	NOUN
ejpam-5625	122	27	of	of	ADP
ejpam-5625	122	28	c4	c4	NOUN
ejpam-5625	122	29	.	.	PUNCT
ejpam-5625	123	1	then	then	ADV
ejpam-5625	123	2	r	r	NOUN
ejpam-5625	123	3	is	be	AUX
ejpam-5625	123	4	either	either	PRON
ejpam-5625	123	5	of	of	ADP
ejpam-5625	123	6	the	the	DET
ejpam-5625	123	7	following	follow	VERB
ejpam-5625	123	8	sets	set	NOUN
ejpam-5625	123	9	:	:	PUNCT
ejpam-5625	123	10	{	{	PUNCT
ejpam-5625	123	11	u1	u1	NOUN
ejpam-5625	123	12	,	,	PUNCT
ejpam-5625	123	13	u2	u2	PROPN
ejpam-5625	123	14	}	}	PUNCT
ejpam-5625	123	15	,	,	PUNCT
ejpam-5625	123	16	{	{	PUNCT
ejpam-5625	123	17	u2	u2	NOUN
ejpam-5625	123	18	,	,	PUNCT
ejpam-5625	123	19	u3	u3	NOUN
ejpam-5625	123	20	}	}	PUNCT
ejpam-5625	123	21	,	,	PUNCT
ejpam-5625	123	22	{	{	PUNCT
ejpam-5625	123	23	u3	u3	PROPN
ejpam-5625	123	24	,	,	PUNCT
ejpam-5625	123	25	u4	u4	PROPN
ejpam-5625	123	26	}	}	PUNCT
ejpam-5625	123	27	or	or	CCONJ
ejpam-5625	123	28	{	{	PUNCT
ejpam-5625	123	29	u4	u4	PROPN
ejpam-5625	123	30	,	,	PUNCT
ejpam-5625	123	31	u1	u1	NOUN
ejpam-5625	123	32	}	}	PUNCT
ejpam-5625	123	33	.	.	PUNCT
ejpam-5625	124	1	however	however	ADV
ejpam-5625	124	2	,	,	PUNCT
ejpam-5625	124	3	either	either	PRON
ejpam-5625	124	4	of	of	ADP
ejpam-5625	124	5	these	these	DET
ejpam-5625	124	6	cases	case	NOUN
ejpam-5625	124	7	violates	violate	VERB
ejpam-5625	124	8	the	the	DET
ejpam-5625	124	9	properties	property	NOUN
ejpam-5625	124	10	of	of	ADP
ejpam-5625	124	11	a	a	DET
ejpam-5625	124	12	co	co	ADJ
ejpam-5625	124	13	-	-	ADJ
ejpam-5625	124	14	certified	certify	VERB
ejpam-5625	124	15	pnd	pnd	NOUN
ejpam-5625	124	16	set	set	NOUN
ejpam-5625	124	17	.	.	PUNCT
ejpam-5625	125	1	by	by	ADP
ejpam-5625	125	2	theorem	theorem	ADJ
ejpam-5625	125	3	3	3	NUM
ejpam-5625	125	4	,	,	PUNCT
ejpam-5625	125	5	ccpnd(c4	ccpnd(c4	NOUN
ejpam-5625	125	6	)	)	PUNCT
ejpam-5625	125	7	=	=	SYM
ejpam-5625	125	8	4	4	X
ejpam-5625	125	9	.	.	PUNCT
ejpam-5625	125	10	now	now	ADV
ejpam-5625	125	11	,	,	PUNCT
ejpam-5625	125	12	suppose	suppose	VERB
ejpam-5625	125	13	that	that	SCONJ
ejpam-5625	126	1	k	k	PROPN
ejpam-5625	126	2	≥	≥	NUM
ejpam-5625	126	3	5	5	NUM
ejpam-5625	126	4	.	.	PUNCT
ejpam-5625	126	5	let	let	VERB
ejpam-5625	126	6	v	v	NOUN
ejpam-5625	126	7	(	(	PUNCT
ejpam-5625	126	8	ck	ck	PROPN
ejpam-5625	126	9	)	)	PUNCT
ejpam-5625	126	10	=	=	SYM
ejpam-5625	126	11	{	{	PUNCT
ejpam-5625	126	12	u1	u1	NOUN
ejpam-5625	126	13	,	,	PUNCT
ejpam-5625	126	14	u2	u2	NOUN
ejpam-5625	126	15	,	,	PUNCT
ejpam-5625	126	16	...	...	PUNCT
ejpam-5625	126	17	,	,	PUNCT
ejpam-5625	126	18	uk	uk	PROPN
ejpam-5625	126	19	}	}	PUNCT
ejpam-5625	126	20	.	.	PUNCT
ejpam-5625	127	1	consider	consider	VERB
ejpam-5625	127	2	p	p	NOUN
ejpam-5625	127	3	=	=	NOUN
ejpam-5625	127	4	{	{	PUNCT
ejpam-5625	127	5	u1	u1	NOUN
ejpam-5625	127	6	,	,	PUNCT
ejpam-5625	127	7	u2	u2	PROPN
ejpam-5625	127	8	}	}	PUNCT
ejpam-5625	127	9	.	.	PUNCT
ejpam-5625	128	1	then	then	ADV
ejpam-5625	128	2	p	p	NOUN
ejpam-5625	128	3	is	be	AUX
ejpam-5625	128	4	a	a	DET
ejpam-5625	128	5	minimum	minimum	ADJ
ejpam-5625	128	6	pnd	pnd	NOUN
ejpam-5625	128	7	set	set	NOUN
ejpam-5625	128	8	of	of	ADP
ejpam-5625	128	9	ck	ck	PROPN
ejpam-5625	128	10	.	.	PUNCT
ejpam-5625	129	1	since	since	SCONJ
ejpam-5625	129	2	k	k	PROPN
ejpam-5625	129	3	≥	≥	PROPN
ejpam-5625	129	4	5	5	NUM
ejpam-5625	129	5	,	,	PUNCT
ejpam-5625	129	6	u1	u1	NOUN
ejpam-5625	129	7	and	and	CCONJ
ejpam-5625	129	8	u2	u2	NOUN
ejpam-5625	129	9	has	have	VERB
ejpam-5625	129	10	at	at	ADV
ejpam-5625	129	11	least	least	ADV
ejpam-5625	129	12	two	two	NUM
ejpam-5625	129	13	neighbors	neighbor	NOUN
ejpam-5625	129	14	in	in	ADP
ejpam-5625	129	15	v	v	NOUN
ejpam-5625	129	16	(	(	PUNCT
ejpam-5625	129	17	ck)\p	ck)\p	NOUN
ejpam-5625	129	18	.	.	PUNCT
ejpam-5625	130	1	therefore	therefore	ADV
ejpam-5625	130	2	,	,	PUNCT
ejpam-5625	130	3	p	p	PRON
ejpam-5625	130	4	is	be	AUX
ejpam-5625	130	5	a	a	DET
ejpam-5625	130	6	minimum	minimum	ADJ
ejpam-5625	130	7	co	co	VERB
ejpam-5625	130	8	-	-	ADJ
ejpam-5625	130	9	certified	certify	VERB
ejpam-5625	130	10	pnd	pnd	NOUN
ejpam-5625	130	11	set	set	NOUN
ejpam-5625	130	12	of	of	ADP
ejpam-5625	130	13	ck	ck	PROPN
ejpam-5625	130	14	,	,	PUNCT
ejpam-5625	130	15	and	and	CCONJ
ejpam-5625	130	16	so	so	ADV
ejpam-5625	130	17	ccpnd(ck	ccpnd(ck	ADJ
ejpam-5625	130	18	)	)	PUNCT
ejpam-5625	130	19	=	=	SYM
ejpam-5625	130	20	2	2	NUM
ejpam-5625	130	21	for	for	ADP
ejpam-5625	130	22	all	all	DET
ejpam-5625	130	23	k	k	PROPN
ejpam-5625	130	24	≥	≥	NUM
ejpam-5625	130	25	5	5	NUM
ejpam-5625	130	26	.	.	PUNCT
ejpam-5625	130	27	n.	n.	PROPN
ejpam-5625	130	28	s.	s.	PROPN
ejpam-5625	130	29	ulal	ulal	PROPN
ejpam-5625	130	30	et	et	PROPN
ejpam-5625	130	31	al	al	PROPN
ejpam-5625	130	32	.	.	PUNCT
ejpam-5625	130	33	/	/	SYM
ejpam-5625	130	34	eur	eur	PROPN
ejpam-5625	130	35	.	.	PUNCT
ejpam-5625	131	1	j.	j.	PROPN
ejpam-5625	131	2	pure	pure	PROPN
ejpam-5625	131	3	appl	appl	PROPN
ejpam-5625	131	4	.	.	PROPN
ejpam-5625	131	5	math	math	PROPN
ejpam-5625	131	6	,	,	PUNCT
ejpam-5625	131	7	18	18	NUM
ejpam-5625	131	8	(	(	PUNCT
ejpam-5625	131	9	1	1	NUM
ejpam-5625	131	10	)	)	PUNCT
ejpam-5625	131	11	(	(	PUNCT
ejpam-5625	131	12	2025	2025	NUM
ejpam-5625	131	13	)	)	PUNCT
ejpam-5625	131	14	,	,	PUNCT
ejpam-5625	131	15	5625	5625	NUM
ejpam-5625	131	16	6	6	NUM
ejpam-5625	131	17	of	of	ADP
ejpam-5625	131	18	9	9	NUM
ejpam-5625	131	19	(	(	PUNCT
ejpam-5625	131	20	iii	iii	NOUN
ejpam-5625	131	21	)	)	PUNCT
ejpam-5625	131	22	let	let	VERB
ejpam-5625	131	23	s	s	NOUN
ejpam-5625	131	24	=	=	NOUN
ejpam-5625	131	25	v	v	PROPN
ejpam-5625	131	26	(	(	PUNCT
ejpam-5625	131	27	kk	kk	PROPN
ejpam-5625	131	28	)	)	PUNCT
ejpam-5625	131	29	=	=	SYM
ejpam-5625	131	30	{	{	PUNCT
ejpam-5625	131	31	a1	a1	PROPN
ejpam-5625	131	32	,	,	PUNCT
ejpam-5625	131	33	a2	a2	PROPN
ejpam-5625	131	34	,	,	PUNCT
ejpam-5625	131	35	...	...	PUNCT
ejpam-5625	131	36	,	,	PUNCT
ejpam-5625	131	37	ak	ak	PROPN
ejpam-5625	131	38	}	}	PUNCT
ejpam-5625	131	39	.	.	PUNCT
ejpam-5625	132	1	then	then	ADV
ejpam-5625	132	2	s	s	VERB
ejpam-5625	132	3	is	be	AUX
ejpam-5625	132	4	a	a	DET
ejpam-5625	132	5	co	co	ADJ
ejpam-5625	132	6	-	-	ADJ
ejpam-5625	132	7	certified	certify	VERB
ejpam-5625	132	8	pnd	pnd	NOUN
ejpam-5625	132	9	set	set	NOUN
ejpam-5625	132	10	of	of	ADP
ejpam-5625	132	11	kk	kk	PROPN
ejpam-5625	132	12	.	.	PROPN
ejpam-5625	132	13	suppose	suppose	VERB
ejpam-5625	132	14	that	that	SCONJ
ejpam-5625	132	15	s	s	VERB
ejpam-5625	132	16	is	be	AUX
ejpam-5625	132	17	not	not	PART
ejpam-5625	132	18	a	a	DET
ejpam-5625	132	19	minimum	minimum	ADJ
ejpam-5625	132	20	co	co	ADJ
ejpam-5625	132	21	-	-	ADJ
ejpam-5625	132	22	certified	certify	VERB
ejpam-5625	132	23	pnd	pnd	NOUN
ejpam-5625	132	24	set	set	NOUN
ejpam-5625	132	25	of	of	ADP
ejpam-5625	132	26	kk	kk	PROPN
ejpam-5625	132	27	.	.	PUNCT
ejpam-5625	133	1	then	then	ADV
ejpam-5625	133	2	there	there	PRON
ejpam-5625	133	3	exists	exist	VERB
ejpam-5625	133	4	x	x	X
ejpam-5625	133	5	∈	∈	PROPN
ejpam-5625	133	6	v	v	ADP
ejpam-5625	133	7	(	(	PUNCT
ejpam-5625	133	8	kk	kk	PROPN
ejpam-5625	133	9	)	)	PUNCT
ejpam-5625	133	10	such	such	ADJ
ejpam-5625	133	11	that	that	SCONJ
ejpam-5625	133	12	x	x	PROPN
ejpam-5625	133	13	/∈	/∈	PROPN
ejpam-5625	133	14	s.	s.	PROPN
ejpam-5625	133	15	however	however	ADV
ejpam-5625	133	16	,	,	PUNCT
ejpam-5625	133	17	x	x	PUNCT
ejpam-5625	133	18	is	be	AUX
ejpam-5625	133	19	adjacent	adjacent	ADJ
ejpam-5625	133	20	to	to	ADP
ejpam-5625	133	21	every	every	DET
ejpam-5625	133	22	other	other	ADJ
ejpam-5625	133	23	vertex	vertex	NOUN
ejpam-5625	133	24	in	in	ADP
ejpam-5625	133	25	v	v	NOUN
ejpam-5625	133	26	(	(	PUNCT
ejpam-5625	133	27	kk)\{x	kk)\{x	PROPN
ejpam-5625	133	28	}	}	PUNCT
ejpam-5625	133	29	,	,	PUNCT
ejpam-5625	133	30	a	a	DET
ejpam-5625	133	31	contradiction	contradiction	NOUN
ejpam-5625	133	32	to	to	ADP
ejpam-5625	133	33	the	the	DET
ejpam-5625	133	34	fact	fact	NOUN
ejpam-5625	133	35	that	that	SCONJ
ejpam-5625	133	36	s	s	VERB
ejpam-5625	133	37	is	be	AUX
ejpam-5625	133	38	a	a	DET
ejpam-5625	133	39	pnd	pnd	NOUN
ejpam-5625	133	40	set	set	NOUN
ejpam-5625	133	41	of	of	ADP
ejpam-5625	133	42	kk	kk	PROPN
ejpam-5625	133	43	.	.	PUNCT
ejpam-5625	134	1	therefore	therefore	ADV
ejpam-5625	134	2	,	,	PUNCT
ejpam-5625	134	3	s	s	NOUN
ejpam-5625	134	4	=	=	SYM
ejpam-5625	134	5	v	v	PROPN
ejpam-5625	134	6	(	(	PUNCT
ejpam-5625	134	7	kk	kk	PROPN
ejpam-5625	134	8	)	)	PUNCT
ejpam-5625	134	9	is	be	AUX
ejpam-5625	134	10	a	a	DET
ejpam-5625	134	11	minimum	minimum	ADJ
ejpam-5625	134	12	co	co	VERB
ejpam-5625	134	13	-	-	ADJ
ejpam-5625	134	14	certified	certify	VERB
ejpam-5625	134	15	pnd	pnd	NOUN
ejpam-5625	134	16	set	set	NOUN
ejpam-5625	134	17	of	of	ADP
ejpam-5625	134	18	kk	kk	PROPN
ejpam-5625	134	19	,	,	PUNCT
ejpam-5625	134	20	and	and	CCONJ
ejpam-5625	134	21	so	so	ADV
ejpam-5625	134	22	ccpnd(kk	ccpnd(kk	ADJ
ejpam-5625	134	23	)	)	PUNCT
ejpam-5625	135	1	=	=	SYM
ejpam-5625	135	2	k	k	NOUN
ejpam-5625	135	3	∀	∀	X
ejpam-5625	136	1	k	k	X
ejpam-5625	136	2	≥	≥	NUM
ejpam-5625	136	3	1	1	NUM
ejpam-5625	136	4	(	(	PUNCT
ejpam-5625	136	5	iv	iv	X
ejpam-5625	136	6	)	)	PUNCT
ejpam-5625	136	7	clearly	clearly	ADV
ejpam-5625	136	8	,	,	PUNCT
ejpam-5625	136	9	ccpnd	ccpnd	NOUN
ejpam-5625	136	10	¯(kk	¯(kk	PROPN
ejpam-5625	136	11	)	)	PUNCT
ejpam-5625	136	12	=	=	SYM
ejpam-5625	136	13	1	1	X
ejpam-5625	136	14	.	.	X
ejpam-5625	137	1	for	for	ADP
ejpam-5625	137	2	k	k	PROPN
ejpam-5625	137	3	=	=	SYM
ejpam-5625	137	4	2	2	NUM
ejpam-5625	137	5	,	,	PUNCT
ejpam-5625	137	6	let	let	VERB
ejpam-5625	137	7	v	v	NOUN
ejpam-5625	137	8	¯(kk	¯(kk	VERB
ejpam-5625	137	9	)	)	PUNCT
ejpam-5625	138	1	=	=	PRON
ejpam-5625	138	2	{	{	PUNCT
ejpam-5625	138	3	v1	v1	NOUN
ejpam-5625	138	4	,	,	PUNCT
ejpam-5625	138	5	v2	v2	PROPN
ejpam-5625	138	6	}	}	PUNCT
ejpam-5625	138	7	.	.	PUNCT
ejpam-5625	139	1	consider	consider	VERB
ejpam-5625	139	2	r	r	NOUN
ejpam-5625	139	3	=	=	SYM
ejpam-5625	139	4	{	{	PUNCT
ejpam-5625	139	5	v1	v1	NOUN
ejpam-5625	139	6	}	}	PUNCT
ejpam-5625	139	7	.	.	PUNCT
ejpam-5625	140	1	then	then	ADV
ejpam-5625	140	2	r	r	NOUN
ejpam-5625	140	3	is	be	AUX
ejpam-5625	140	4	a	a	DET
ejpam-5625	140	5	minimum	minimum	ADJ
ejpam-5625	140	6	pnd	pnd	NOUN
ejpam-5625	140	7	set	set	NOUN
ejpam-5625	140	8	of	of	ADP
ejpam-5625	140	9	¯(k2	¯(k2	NOUN
ejpam-5625	140	10	)	)	PUNCT
ejpam-5625	140	11	,	,	PUNCT
ejpam-5625	140	12	and	and	CCONJ
ejpam-5625	140	13	so	so	ADV
ejpam-5625	140	14	pnd	pnd	NOUN
ejpam-5625	140	15	¯(k2	¯(k2	NOUN
ejpam-5625	140	16	)	)	PUNCT
ejpam-5625	140	17	=	=	SYM
ejpam-5625	140	18	1	1	X
ejpam-5625	140	19	.	.	PUNCT
ejpam-5625	141	1	thus	thus	ADV
ejpam-5625	141	2	,	,	PUNCT
ejpam-5625	141	3	ccpnd	ccpnd	NOUN
ejpam-5625	141	4	¯(k2	¯(k2	NOUN
ejpam-5625	141	5	)	)	PUNCT
ejpam-5625	141	6	≥	≥	NOUN
ejpam-5625	141	7	1	1	X
ejpam-5625	141	8	.	.	PUNCT
ejpam-5625	141	9	assume	assume	VERB
ejpam-5625	141	10	that	that	SCONJ
ejpam-5625	141	11	ccpnd	ccpnd	NOUN
ejpam-5625	141	12	¯(k2	¯(k2	NOUN
ejpam-5625	141	13	)	)	PUNCT
ejpam-5625	141	14	=	=	SYM
ejpam-5625	142	1	1	1	X
ejpam-5625	142	2	.	.	PUNCT
ejpam-5625	142	3	then	then	ADV
ejpam-5625	142	4	either	either	CCONJ
ejpam-5625	142	5	{	{	PUNCT
ejpam-5625	142	6	v1	v1	NOUN
ejpam-5625	142	7	}	}	PUNCT
ejpam-5625	142	8	or	or	CCONJ
ejpam-5625	142	9	{	{	PUNCT
ejpam-5625	142	10	v2	v2	NOUN
ejpam-5625	142	11	}	}	PUNCT
ejpam-5625	142	12	is	be	AUX
ejpam-5625	142	13	a	a	DET
ejpam-5625	142	14	minimum	minimum	ADJ
ejpam-5625	142	15	co	co	VERB
ejpam-5625	142	16	-	-	ADJ
ejpam-5625	142	17	certified	certify	VERB
ejpam-5625	142	18	pnd	pnd	NOUN
ejpam-5625	142	19	set	set	NOUN
ejpam-5625	142	20	of	of	ADP
ejpam-5625	142	21	k̄2	k̄2	PROPN
ejpam-5625	142	22	.	.	PUNCT
ejpam-5625	143	1	suppose	suppose	VERB
ejpam-5625	143	2	that	that	SCONJ
ejpam-5625	143	3	m	m	VERB
ejpam-5625	143	4	=	=	SYM
ejpam-5625	143	5	{	{	PUNCT
ejpam-5625	143	6	v2	v2	NOUN
ejpam-5625	143	7	}	}	PUNCT
ejpam-5625	143	8	is	be	AUX
ejpam-5625	143	9	a	a	DET
ejpam-5625	143	10	minimum	minimum	ADJ
ejpam-5625	143	11	co	co	ADJ
ejpam-5625	143	12	-	-	ADJ
ejpam-5625	143	13	certifed	certifed	ADJ
ejpam-5625	143	14	pnd	pnd	NOUN
ejpam-5625	143	15	set	set	NOUN
ejpam-5625	143	16	of	of	ADP
ejpam-5625	143	17	¯(k2	¯(k2	NOUN
ejpam-5625	143	18	)	)	PUNCT
ejpam-5625	143	19	.	.	PUNCT
ejpam-5625	144	1	however	however	ADV
ejpam-5625	144	2	,	,	PUNCT
ejpam-5625	144	3	v2	v2	PROPN
ejpam-5625	144	4	has	have	VERB
ejpam-5625	144	5	only	only	ADV
ejpam-5625	144	6	one	one	NUM
ejpam-5625	144	7	non	non	ADJ
ejpam-5625	144	8	-	-	ADJ
ejpam-5625	144	9	neighbor	neighbor	ADJ
ejpam-5625	144	10	v1	v1	PROPN
ejpam-5625	144	11	∈	∈	PROPN
ejpam-5625	144	12	v	v	ADP
ejpam-5625	144	13	¯(k2)\m	¯(k2)\m	PRON
ejpam-5625	144	14	,	,	PUNCT
ejpam-5625	144	15	a	a	DET
ejpam-5625	144	16	contradiction	contradiction	NOUN
ejpam-5625	144	17	.	.	PUNCT
ejpam-5625	145	1	similarly	similarly	ADV
ejpam-5625	145	2	the	the	DET
ejpam-5625	145	3	assertion	assertion	NOUN
ejpam-5625	145	4	follows	follow	VERB
ejpam-5625	145	5	when	when	SCONJ
ejpam-5625	145	6	m	m	VERB
ejpam-5625	145	7	=	=	SYM
ejpam-5625	145	8	{	{	PUNCT
ejpam-5625	145	9	v1	v1	NOUN
ejpam-5625	145	10	}	}	PUNCT
ejpam-5625	145	11	.	.	PUNCT
ejpam-5625	146	1	thus	thus	ADV
ejpam-5625	146	2	,	,	PUNCT
ejpam-5625	146	3	ccpnd	ccpnd	NOUN
ejpam-5625	146	4	¯(k2	¯(k2	NOUN
ejpam-5625	146	5	)	)	PUNCT
ejpam-5625	146	6	=	=	SYM
ejpam-5625	146	7	2	2	X
ejpam-5625	146	8	.	.	PUNCT
ejpam-5625	147	1	next	next	ADV
ejpam-5625	147	2	,	,	PUNCT
ejpam-5625	147	3	suppose	suppose	VERB
ejpam-5625	147	4	that	that	SCONJ
ejpam-5625	147	5	k	k	PROPN
ejpam-5625	147	6	≥	≥	NUM
ejpam-5625	147	7	3	3	X
ejpam-5625	147	8	.	.	PUNCT
ejpam-5625	147	9	let	let	VERB
ejpam-5625	147	10	v	v	PART
ejpam-5625	147	11	¯(kk	¯(kk	VERB
ejpam-5625	147	12	)	)	PUNCT
ejpam-5625	148	1	=	=	PRON
ejpam-5625	148	2	{	{	PUNCT
ejpam-5625	148	3	a1	a1	PROPN
ejpam-5625	148	4	,	,	PUNCT
ejpam-5625	148	5	a2	a2	PROPN
ejpam-5625	148	6	,	,	PUNCT
ejpam-5625	148	7	...	...	PUNCT
ejpam-5625	148	8	,	,	PUNCT
ejpam-5625	148	9	ak	ak	PROPN
ejpam-5625	148	10	}	}	PUNCT
ejpam-5625	148	11	.	.	PUNCT
ejpam-5625	149	1	consider	consider	VERB
ejpam-5625	149	2	b	b	NOUN
ejpam-5625	149	3	=	=	SYM
ejpam-5625	149	4	{	{	PUNCT
ejpam-5625	149	5	a1	a1	PROPN
ejpam-5625	149	6	}	}	PUNCT
ejpam-5625	149	7	.	.	PUNCT
ejpam-5625	150	1	then	then	ADV
ejpam-5625	150	2	b	b	X
ejpam-5625	150	3	is	be	AUX
ejpam-5625	150	4	a	a	DET
ejpam-5625	150	5	minimum	minimum	ADJ
ejpam-5625	150	6	pnd	pnd	NOUN
ejpam-5625	150	7	set	set	NOUN
ejpam-5625	150	8	of	of	ADP
ejpam-5625	150	9	¯(kk	¯(kk	PROPN
ejpam-5625	150	10	)	)	PUNCT
ejpam-5625	150	11	.	.	PUNCT
ejpam-5625	151	1	since	since	SCONJ
ejpam-5625	151	2	k	k	PROPN
ejpam-5625	151	3	≥	≥	PROPN
ejpam-5625	151	4	3	3	NUM
ejpam-5625	151	5	,	,	PUNCT
ejpam-5625	151	6	a1	a1	NOUN
ejpam-5625	151	7	has	have	VERB
ejpam-5625	151	8	at	at	ADV
ejpam-5625	151	9	least	least	ADV
ejpam-5625	151	10	two	two	NUM
ejpam-5625	151	11	non	non	NOUN
ejpam-5625	151	12	-	-	NOUN
ejpam-5625	151	13	neighbors	neighbor	NOUN
ejpam-5625	151	14	in	in	ADP
ejpam-5625	151	15	v	v	NOUN
ejpam-5625	151	16	(	(	PUNCT
ejpam-5625	151	17	k̄	k̄	NOUN
ejpam-5625	151	18	)	)	PUNCT
ejpam-5625	151	19	\b	\b	NOUN
ejpam-5625	151	20	.	.	PUNCT
ejpam-5625	152	1	therefore	therefore	ADV
ejpam-5625	152	2	,	,	PUNCT
ejpam-5625	152	3	ccpnd	ccpnd	NOUN
ejpam-5625	152	4	¯(kk	¯(kk	PROPN
ejpam-5625	152	5	)	)	PUNCT
ejpam-5625	152	6	=	=	SYM
ejpam-5625	152	7	1	1	NUM
ejpam-5625	152	8	for	for	ADP
ejpam-5625	152	9	all	all	DET
ejpam-5625	152	10	k	k	PROPN
ejpam-5625	152	11	≥	≥	NUM
ejpam-5625	152	12	3	3	NUM
ejpam-5625	152	13	.	.	NOUN
ejpam-5625	153	1	6	6	NUM
ejpam-5625	153	2	.	.	NOUN
ejpam-5625	153	3	2	2	NUM
ejpam-5625	153	4	-	-	PUNCT
ejpam-5625	153	5	distance	distance	NOUN
ejpam-5625	153	6	certified	certify	VERB
ejpam-5625	153	7	hop	hop	NOUN
ejpam-5625	153	8	domination	domination	NOUN
ejpam-5625	153	9	in	in	ADP
ejpam-5625	153	10	the	the	DET
ejpam-5625	153	11	join	join	NOUN
ejpam-5625	153	12	of	of	ADP
ejpam-5625	153	13	two	two	NUM
ejpam-5625	153	14	graphs	graph	NOUN
ejpam-5625	153	15	theorem	theorem	VERB
ejpam-5625	153	16	4	4	NUM
ejpam-5625	153	17	.	.	PUNCT
ejpam-5625	154	1	let	let	VERB
ejpam-5625	154	2	g	g	NOUN
ejpam-5625	154	3	and	and	CCONJ
ejpam-5625	154	4	h	h	PROPN
ejpam-5625	154	5	be	be	AUX
ejpam-5625	154	6	graphs	graph	NOUN
ejpam-5625	154	7	.	.	PUNCT
ejpam-5625	155	1	then	then	ADV
ejpam-5625	155	2	o	o	X
ejpam-5625	155	3	⊆	⊆	NUM
ejpam-5625	155	4	(	(	PUNCT
ejpam-5625	155	5	g	g	NOUN
ejpam-5625	155	6	+	+	NOUN
ejpam-5625	155	7	h	h	NOUN
ejpam-5625	155	8	)	)	PUNCT
ejpam-5625	155	9	is	be	AUX
ejpam-5625	155	10	a	a	DET
ejpam-5625	155	11	2	2	NUM
ejpam-5625	155	12	-	-	PUNCT
ejpam-5625	155	13	distance	distance	NOUN
ejpam-5625	155	14	certified	certify	VERB
ejpam-5625	155	15	hop	hop	NOUN
ejpam-5625	155	16	dominating	dominating	NOUN
ejpam-5625	155	17	set	set	NOUN
ejpam-5625	155	18	of	of	ADP
ejpam-5625	155	19	g+h	g+h	PROPN
ejpam-5625	156	1	if	if	SCONJ
ejpam-5625	156	2	and	and	CCONJ
ejpam-5625	156	3	only	only	ADV
ejpam-5625	156	4	if	if	SCONJ
ejpam-5625	156	5	o	o	NOUN
ejpam-5625	156	6	=	=	PUNCT
ejpam-5625	156	7	og	og	PROPN
ejpam-5625	156	8	∪oh	∪oh	PROPN
ejpam-5625	156	9	,	,	PUNCT
ejpam-5625	156	10	where	where	SCONJ
ejpam-5625	156	11	og	og	PROPN
ejpam-5625	156	12	and	and	CCONJ
ejpam-5625	156	13	oh	oh	INTJ
ejpam-5625	156	14	are	be	AUX
ejpam-5625	156	15	co	co	ADJ
ejpam-5625	156	16	-	-	ADJ
ejpam-5625	156	17	certified	certify	VERB
ejpam-5625	156	18	pnd	pnd	NOUN
ejpam-5625	156	19	sets	set	NOUN
ejpam-5625	156	20	in	in	ADP
ejpam-5625	156	21	g	g	PROPN
ejpam-5625	156	22	and	and	CCONJ
ejpam-5625	156	23	h	h	NOUN
ejpam-5625	156	24	,	,	PUNCT
ejpam-5625	156	25	respectively	respectively	ADV
ejpam-5625	156	26	.	.	PUNCT
ejpam-5625	157	1	proof	proof	NOUN
ejpam-5625	157	2	.	.	PUNCT
ejpam-5625	158	1	suppose	suppose	VERB
ejpam-5625	158	2	that	that	SCONJ
ejpam-5625	158	3	o	o	PROPN
ejpam-5625	158	4	⊆	⊆	NUM
ejpam-5625	158	5	v	v	NOUN
ejpam-5625	158	6	(	(	PUNCT
ejpam-5625	158	7	g	g	PROPN
ejpam-5625	158	8	+	+	NOUN
ejpam-5625	158	9	h	h	NOUN
ejpam-5625	158	10	)	)	PUNCT
ejpam-5625	158	11	is	be	AUX
ejpam-5625	158	12	a	a	DET
ejpam-5625	158	13	2	2	NUM
ejpam-5625	158	14	-	-	PUNCT
ejpam-5625	158	15	distance	distance	NOUN
ejpam-5625	158	16	certified	certify	VERB
ejpam-5625	158	17	hop	hop	NOUN
ejpam-5625	158	18	dominating	dominating	NOUN
ejpam-5625	158	19	set	set	NOUN
ejpam-5625	158	20	of	of	ADP
ejpam-5625	158	21	g	g	PROPN
ejpam-5625	158	22	+	+	PROPN
ejpam-5625	158	23	h.	h.	PROPN
ejpam-5625	158	24	then	then	ADV
ejpam-5625	158	25	o	o	PROPN
ejpam-5625	158	26	is	be	AUX
ejpam-5625	158	27	a	a	DET
ejpam-5625	158	28	hop	hop	NOUN
ejpam-5625	158	29	dominating	dominating	NOUN
ejpam-5625	158	30	set	set	NOUN
ejpam-5625	158	31	of	of	ADP
ejpam-5625	158	32	g	g	PROPN
ejpam-5625	158	33	+	+	PROPN
ejpam-5625	158	34	h.	h.	PROPN
ejpam-5625	158	35	if	if	SCONJ
ejpam-5625	158	36	o	o	PROPN
ejpam-5625	158	37	⊆	⊆	NUM
ejpam-5625	158	38	v	v	X
ejpam-5625	158	39	(	(	PUNCT
ejpam-5625	158	40	g	g	NOUN
ejpam-5625	158	41	)	)	PUNCT
ejpam-5625	158	42	,	,	PUNCT
ejpam-5625	158	43	then	then	ADV
ejpam-5625	158	44	n2	n2	PROPN
ejpam-5625	158	45	g[o	g[o	PROPN
ejpam-5625	158	46	]	]	X
ejpam-5625	158	47	⊆	⊆	NUM
ejpam-5625	158	48	v	v	NOUN
ejpam-5625	158	49	(	(	PUNCT
ejpam-5625	158	50	g	g	NOUN
ejpam-5625	158	51	)	)	PUNCT
ejpam-5625	158	52	,	,	PUNCT
ejpam-5625	158	53	a	a	DET
ejpam-5625	158	54	contradiction	contradiction	NOUN
ejpam-5625	158	55	.	.	PUNCT
ejpam-5625	159	1	hence	hence	ADV
ejpam-5625	159	2	,	,	PUNCT
ejpam-5625	159	3	o	o	NOUN
ejpam-5625	159	4	⊈	⊈	PROPN
ejpam-5625	159	5	v	v	NOUN
ejpam-5625	159	6	(	(	PUNCT
ejpam-5625	159	7	g	g	NOUN
ejpam-5625	159	8	)	)	PUNCT
ejpam-5625	159	9	.	.	PUNCT
ejpam-5625	160	1	similarly	similarly	ADV
ejpam-5625	160	2	,	,	PUNCT
ejpam-5625	160	3	o	o	NOUN
ejpam-5625	160	4	⊈	⊈	PROPN
ejpam-5625	160	5	v	v	NOUN
ejpam-5625	160	6	(	(	PUNCT
ejpam-5625	160	7	h	h	NOUN
ejpam-5625	160	8	)	)	PUNCT
ejpam-5625	160	9	.	.	PUNCT
ejpam-5625	161	1	thus	thus	ADV
ejpam-5625	161	2	,	,	PUNCT
ejpam-5625	161	3	o	o	PROPN
ejpam-5625	161	4	=	=	PUNCT
ejpam-5625	161	5	og	og	PROPN
ejpam-5625	161	6	∪	∪	PROPN
ejpam-5625	162	1	oh	oh	INTJ
ejpam-5625	162	2	,	,	PUNCT
ejpam-5625	162	3	where	where	SCONJ
ejpam-5625	162	4	og	og	PROPN
ejpam-5625	162	5	⊆	⊆	NUM
ejpam-5625	162	6	v	v	NOUN
ejpam-5625	162	7	(	(	PUNCT
ejpam-5625	162	8	g	g	NOUN
ejpam-5625	162	9	)	)	PUNCT
ejpam-5625	162	10	and	and	CCONJ
ejpam-5625	162	11	oh	oh	NUM
ejpam-5625	162	12	⊆	⊆	NUM
ejpam-5625	162	13	v	v	X
ejpam-5625	162	14	(	(	PUNCT
ejpam-5625	162	15	h	h	NOUN
ejpam-5625	162	16	)	)	PUNCT
ejpam-5625	162	17	.	.	PUNCT
ejpam-5625	163	1	let	let	VERB
ejpam-5625	163	2	x	x	SYM
ejpam-5625	163	3	∈	∈	PROPN
ejpam-5625	163	4	v	v	X
ejpam-5625	163	5	(	(	PUNCT
ejpam-5625	163	6	g	g	NOUN
ejpam-5625	163	7	+	+	X
ejpam-5625	163	8	h)\o	h)\o	NOUN
ejpam-5625	163	9	.	.	PUNCT
ejpam-5625	164	1	assume	assume	VERB
ejpam-5625	164	2	that	that	SCONJ
ejpam-5625	164	3	x	x	SYM
ejpam-5625	164	4	∈	∈	NOUN
ejpam-5625	164	5	v	v	NOUN
ejpam-5625	164	6	(	(	PUNCT
ejpam-5625	164	7	g)\og	g)\og	PROPN
ejpam-5625	164	8	.	.	PROPN
ejpam-5625	164	9	since	since	SCONJ
ejpam-5625	164	10	o	o	PROPN
ejpam-5625	164	11	is	be	AUX
ejpam-5625	164	12	a	a	DET
ejpam-5625	164	13	hop	hop	NOUN
ejpam-5625	164	14	dominating	dominating	NOUN
ejpam-5625	164	15	,	,	PUNCT
ejpam-5625	164	16	there	there	PRON
ejpam-5625	164	17	exists	exist	VERB
ejpam-5625	164	18	y	y	PROPN
ejpam-5625	164	19	∈	∈	PROPN
ejpam-5625	164	20	o	o	NOUN
ejpam-5625	165	1	such	such	ADJ
ejpam-5625	165	2	that	that	SCONJ
ejpam-5625	165	3	dg+h(x	dg+h(x	PROPN
ejpam-5625	165	4	,	,	PUNCT
ejpam-5625	165	5	y	y	NOUN
ejpam-5625	165	6	)	)	PUNCT
ejpam-5625	165	7	=	=	SYM
ejpam-5625	165	8	2	2	X
ejpam-5625	165	9	.	.	PUNCT
ejpam-5625	165	10	it	it	PRON
ejpam-5625	165	11	follows	follow	VERB
ejpam-5625	165	12	that	that	SCONJ
ejpam-5625	165	13	x	x	SYM
ejpam-5625	165	14	/∈	/∈	PUNCT
ejpam-5625	165	15	ng(y	ng(y	NOUN
ejpam-5625	165	16	)	)	PUNCT
ejpam-5625	165	17	,	,	PUNCT
ejpam-5625	165	18	and	and	CCONJ
ejpam-5625	165	19	so	so	ADV
ejpam-5625	165	20	og	og	PROPN
ejpam-5625	165	21	is	be	AUX
ejpam-5625	165	22	a	a	DET
ejpam-5625	165	23	pnd	pnd	NOUN
ejpam-5625	165	24	set	set	NOUN
ejpam-5625	165	25	of	of	ADP
ejpam-5625	165	26	g.	g.	PROPN
ejpam-5625	165	27	since	since	SCONJ
ejpam-5625	165	28	o	o	PROPN
ejpam-5625	165	29	is	be	AUX
ejpam-5625	165	30	a	a	DET
ejpam-5625	165	31	2	2	NUM
ejpam-5625	165	32	-	-	PUNCT
ejpam-5625	165	33	distance	distance	NOUN
ejpam-5625	165	34	certified	certify	VERB
ejpam-5625	165	35	set	set	NOUN
ejpam-5625	165	36	,	,	PUNCT
ejpam-5625	165	37	for	for	ADP
ejpam-5625	165	38	every	every	DET
ejpam-5625	165	39	w	w	PROPN
ejpam-5625	165	40	∈	∈	PROPN
ejpam-5625	165	41	og	og	PROPN
ejpam-5625	165	42	,	,	PUNCT
ejpam-5625	165	43	there	there	PRON
ejpam-5625	165	44	exist	exist	VERB
ejpam-5625	165	45	either	either	DET
ejpam-5625	165	46	zero	zero	NUM
ejpam-5625	165	47	or	or	CCONJ
ejpam-5625	165	48	at	at	ADP
ejpam-5625	165	49	least	least	ADV
ejpam-5625	165	50	two	two	NUM
ejpam-5625	165	51	vertices	vertex	NOUN
ejpam-5625	165	52	u	u	NOUN
ejpam-5625	165	53	,	,	PUNCT
ejpam-5625	165	54	v	v	NOUN
ejpam-5625	165	55	∈	∈	NOUN
ejpam-5625	165	56	v	v	NOUN
ejpam-5625	165	57	(	(	PUNCT
ejpam-5625	165	58	g)\og	g)\og	PROPN
ejpam-5625	165	59	such	such	ADJ
ejpam-5625	165	60	that	that	SCONJ
ejpam-5625	165	61	dg+h(w	dg+h(w	PROPN
ejpam-5625	165	62	,	,	PUNCT
ejpam-5625	165	63	u	u	NOUN
ejpam-5625	165	64	)	)	PUNCT
ejpam-5625	165	65	=	=	SYM
ejpam-5625	165	66	(	(	PUNCT
ejpam-5625	165	67	w	w	PROPN
ejpam-5625	165	68	,	,	PUNCT
ejpam-5625	165	69	v	v	NOUN
ejpam-5625	165	70	)	)	PUNCT
ejpam-5625	165	71	=	=	SYM
ejpam-5625	166	1	2	2	X
ejpam-5625	166	2	.	.	X
ejpam-5625	166	3	hence	hence	ADV
ejpam-5625	166	4	,	,	PUNCT
ejpam-5625	166	5	u	u	NOUN
ejpam-5625	166	6	,	,	PUNCT
ejpam-5625	166	7	v	v	NOUN
ejpam-5625	166	8	/∈	/∈	PUNCT
ejpam-5625	166	9	ng(w	ng(w	NOUN
ejpam-5625	166	10	)	)	PUNCT
ejpam-5625	166	11	.	.	PUNCT
ejpam-5625	167	1	consequently	consequently	ADV
ejpam-5625	167	2	,	,	PUNCT
ejpam-5625	167	3	og	og	PROPN
ejpam-5625	167	4	is	be	AUX
ejpam-5625	167	5	a	a	DET
ejpam-5625	167	6	co	co	ADJ
ejpam-5625	167	7	-	-	ADJ
ejpam-5625	167	8	certified	certify	VERB
ejpam-5625	167	9	pnd	pnd	NOUN
ejpam-5625	167	10	set	set	NOUN
ejpam-5625	167	11	of	of	ADP
ejpam-5625	167	12	g.	g.	PROPN
ejpam-5625	167	13	similarly	similarly	ADV
ejpam-5625	167	14	,	,	PUNCT
ejpam-5625	167	15	oh	oh	INTJ
ejpam-5625	167	16	is	be	AUX
ejpam-5625	167	17	a	a	DET
ejpam-5625	167	18	co	co	ADJ
ejpam-5625	167	19	-	-	ADJ
ejpam-5625	167	20	certified	certify	VERB
ejpam-5625	167	21	pnd	pnd	NOUN
ejpam-5625	167	22	set	set	NOUN
ejpam-5625	167	23	of	of	ADP
ejpam-5625	167	24	h.	h.	NOUN
ejpam-5625	167	25	conversely	conversely	ADV
ejpam-5625	167	26	,	,	PUNCT
ejpam-5625	167	27	suppose	suppose	VERB
ejpam-5625	167	28	that	that	SCONJ
ejpam-5625	167	29	o	o	NOUN
ejpam-5625	168	1	=	=	PUNCT
ejpam-5625	168	2	og	og	PROPN
ejpam-5625	168	3	∪	∪	PROPN
ejpam-5625	169	1	oh	oh	INTJ
ejpam-5625	169	2	,	,	PUNCT
ejpam-5625	169	3	where	where	SCONJ
ejpam-5625	169	4	og	og	PROPN
ejpam-5625	170	1	and	and	CCONJ
ejpam-5625	170	2	oh	oh	INTJ
ejpam-5625	170	3	are	be	VERB
ejpam-5625	170	4	co	co	ADJ
ejpam-5625	170	5	-	-	ADJ
ejpam-5625	170	6	certified	certify	VERB
ejpam-5625	170	7	pnd	pnd	NOUN
ejpam-5625	170	8	sets	set	NOUN
ejpam-5625	170	9	in	in	ADP
ejpam-5625	170	10	g	g	PROPN
ejpam-5625	170	11	and	and	CCONJ
ejpam-5625	170	12	h	h	NOUN
ejpam-5625	170	13	,	,	PUNCT
ejpam-5625	170	14	respectively	respectively	ADV
ejpam-5625	170	15	.	.	PUNCT
ejpam-5625	171	1	let	let	VERB
ejpam-5625	171	2	a	a	DET
ejpam-5625	171	3	∈	∈	PROPN
ejpam-5625	171	4	v	v	NOUN
ejpam-5625	171	5	(	(	PUNCT
ejpam-5625	171	6	g+h)\o	g+h)\o	PROPN
ejpam-5625	171	7	.	.	PUNCT
ejpam-5625	171	8	assume	assume	VERB
ejpam-5625	171	9	that	that	SCONJ
ejpam-5625	171	10	a	a	DET
ejpam-5625	171	11	∈	∈	PROPN
ejpam-5625	171	12	v	v	NOUN
ejpam-5625	171	13	(	(	PUNCT
ejpam-5625	171	14	g)\og	g)\og	PROPN
ejpam-5625	171	15	.	.	PROPN
ejpam-5625	171	16	since	since	SCONJ
ejpam-5625	171	17	og	og	PROPN
ejpam-5625	171	18	is	be	AUX
ejpam-5625	171	19	a	a	DET
ejpam-5625	171	20	co	co	ADJ
ejpam-5625	171	21	-	-	ADJ
ejpam-5625	171	22	certified	certify	VERB
ejpam-5625	171	23	pnd	pnd	NOUN
ejpam-5625	171	24	set	set	NOUN
ejpam-5625	171	25	of	of	ADP
ejpam-5625	171	26	g	g	NOUN
ejpam-5625	171	27	,	,	PUNCT
ejpam-5625	171	28	there	there	PRON
ejpam-5625	171	29	exists	exist	VERB
ejpam-5625	171	30	b	b	PROPN
ejpam-5625	171	31	∈	∈	PROPN
ejpam-5625	171	32	og	og	PROPN
ejpam-5625	171	33	such	such	ADJ
ejpam-5625	171	34	that	that	SCONJ
ejpam-5625	171	35	a	a	DET
ejpam-5625	171	36	/∈	/∈	NOUN
ejpam-5625	171	37	ng(b	ng(b	X
ejpam-5625	171	38	)	)	PUNCT
ejpam-5625	171	39	and	and	CCONJ
ejpam-5625	171	40	for	for	ADP
ejpam-5625	171	41	each	each	DET
ejpam-5625	171	42	q	q	PROPN
ejpam-5625	171	43	∈	∈	PROPN
ejpam-5625	171	44	og	og	PROPN
ejpam-5625	171	45	,	,	PUNCT
ejpam-5625	171	46	there	there	PRON
ejpam-5625	171	47	exist	exist	VERB
ejpam-5625	171	48	either	either	DET
ejpam-5625	171	49	zero	zero	NUM
ejpam-5625	171	50	or	or	CCONJ
ejpam-5625	171	51	at	at	ADP
ejpam-5625	171	52	least	least	ADV
ejpam-5625	171	53	two	two	NUM
ejpam-5625	171	54	neighbor	neighbor	NOUN
ejpam-5625	171	55	r	r	NOUN
ejpam-5625	171	56	,	,	PUNCT
ejpam-5625	171	57	t	t	PROPN
ejpam-5625	171	58	∈	∈	PROPN
ejpam-5625	171	59	v	v	X
ejpam-5625	171	60	(	(	PUNCT
ejpam-5625	171	61	g)\og	g)\og	PROPN
ejpam-5625	171	62	such	such	ADJ
ejpam-5625	171	63	that	that	DET
ejpam-5625	171	64	r	r	NOUN
ejpam-5625	171	65	,	,	PUNCT
ejpam-5625	171	66	t	t	NOUN
ejpam-5625	171	67	/∈	/∈	PUNCT
ejpam-5625	171	68	ng(q	ng(q	PROPN
ejpam-5625	171	69	)	)	PUNCT
ejpam-5625	171	70	.	.	PUNCT
ejpam-5625	172	1	this	this	PRON
ejpam-5625	172	2	means	mean	VERB
ejpam-5625	172	3	that	that	SCONJ
ejpam-5625	172	4	dg+h(a	dg+h(a	NOUN
ejpam-5625	172	5	,	,	PUNCT
ejpam-5625	172	6	b	b	NOUN
ejpam-5625	172	7	)	)	PUNCT
ejpam-5625	172	8	=	=	SYM
ejpam-5625	172	9	2	2	NUM
ejpam-5625	172	10	and	and	CCONJ
ejpam-5625	172	11	q	q	NOUN
ejpam-5625	172	12	has	have	VERB
ejpam-5625	172	13	either	either	CCONJ
ejpam-5625	172	14	zero	zero	NUM
ejpam-5625	172	15	or	or	CCONJ
ejpam-5625	172	16	at	at	ADP
ejpam-5625	172	17	least	least	ADV
ejpam-5625	172	18	two	two	NUM
ejpam-5625	172	19	hop	hop	NOUN
ejpam-5625	172	20	neighbors	neighbor	NOUN
ejpam-5625	172	21	r	r	PROPN
ejpam-5625	172	22	,	,	PUNCT
ejpam-5625	172	23	t	t	PROPN
ejpam-5625	172	24	∈	∈	PROPN
ejpam-5625	172	25	g	g	PROPN
ejpam-5625	172	26	+	+	PROPN
ejpam-5625	172	27	h.	h.	PROPN
ejpam-5625	172	28	thus	thus	ADV
ejpam-5625	172	29	,	,	PUNCT
ejpam-5625	172	30	o	o	PROPN
ejpam-5625	172	31	is	be	AUX
ejpam-5625	172	32	a	a	DET
ejpam-5625	172	33	2	2	NUM
ejpam-5625	172	34	-	-	PUNCT
ejpam-5625	172	35	distance	distance	NOUN
ejpam-5625	172	36	certified	certify	VERB
ejpam-5625	172	37	hop	hop	NOUN
ejpam-5625	172	38	dominating	dominating	NOUN
ejpam-5625	172	39	set	set	NOUN
ejpam-5625	172	40	of	of	ADP
ejpam-5625	172	41	g	g	PROPN
ejpam-5625	172	42	+	+	PROPN
ejpam-5625	172	43	h.	h.	PROPN
ejpam-5625	172	44	similarly	similarly	ADV
ejpam-5625	172	45	,	,	PUNCT
ejpam-5625	172	46	when	when	SCONJ
ejpam-5625	172	47	a	a	DET
ejpam-5625	172	48	∈	∈	PROPN
ejpam-5625	172	49	v	v	NOUN
ejpam-5625	172	50	(	(	PUNCT
ejpam-5625	172	51	h)\oh	h)\oh	INTJ
ejpam-5625	172	52	,	,	PUNCT
ejpam-5625	172	53	then	then	ADV
ejpam-5625	172	54	o	o	NOUN
ejpam-5625	172	55	is	be	AUX
ejpam-5625	172	56	a	a	DET
ejpam-5625	172	57	2	2	NUM
ejpam-5625	172	58	-	-	PUNCT
ejpam-5625	172	59	distance	distance	NOUN
ejpam-5625	172	60	certified	certify	VERB
ejpam-5625	172	61	hop	hop	NOUN
ejpam-5625	172	62	dominating	dominating	NOUN
ejpam-5625	172	63	set	set	NOUN
ejpam-5625	172	64	of	of	ADP
ejpam-5625	172	65	g+h	g+h	PROPN
ejpam-5625	172	66	.	.	PUNCT
ejpam-5625	173	1	theorem	theorem	NOUN
ejpam-5625	173	2	5	5	NUM
ejpam-5625	173	3	.	.	PUNCT
ejpam-5625	174	1	let	let	VERB
ejpam-5625	174	2	g	g	NOUN
ejpam-5625	175	1	and	and	CCONJ
ejpam-5625	175	2	h	h	NOUN
ejpam-5625	175	3	be	be	VERB
ejpam-5625	175	4	a	a	DET
ejpam-5625	175	5	graphs	graph	NOUN
ejpam-5625	175	6	.	.	PUNCT
ejpam-5625	176	1	then	then	ADV
ejpam-5625	176	2	γ2ch(g+h	γ2ch(g+h	ADJ
ejpam-5625	176	3	)	)	PUNCT
ejpam-5625	176	4	=	=	SYM
ejpam-5625	176	5	ccpnd(g	ccpnd(g	PROPN
ejpam-5625	176	6	)	)	PUNCT
ejpam-5625	177	1	+	+	NUM
ejpam-5625	177	2	ccpnd(h	ccpnd(h	NOUN
ejpam-5625	177	3	)	)	PUNCT
ejpam-5625	177	4	.	.	PUNCT
ejpam-5625	178	1	proof	proof	NOUN
ejpam-5625	178	2	.	.	PUNCT
ejpam-5625	179	1	suppose	suppose	VERB
ejpam-5625	179	2	that	that	SCONJ
ejpam-5625	179	3	o	o	NOUN
ejpam-5625	179	4	=	=	PUNCT
ejpam-5625	179	5	og	og	PROPN
ejpam-5625	179	6	∪oh	∪oh	PROPN
ejpam-5625	179	7	is	be	AUX
ejpam-5625	179	8	a	a	DET
ejpam-5625	179	9	minimum	minimum	ADJ
ejpam-5625	179	10	2	2	NUM
ejpam-5625	179	11	-	-	PUNCT
ejpam-5625	179	12	distance	distance	NOUN
ejpam-5625	179	13	certified	certify	VERB
ejpam-5625	179	14	hop	hop	NOUN
ejpam-5625	179	15	dominating	dominating	NOUN
ejpam-5625	179	16	set	set	NOUN
ejpam-5625	179	17	of	of	ADP
ejpam-5625	179	18	g	g	PROPN
ejpam-5625	179	19	+	+	CCONJ
ejpam-5625	179	20	h.	h.	PROPN
ejpam-5625	179	21	then	then	ADV
ejpam-5625	179	22	by	by	ADP
ejpam-5625	179	23	theorem	theorem	NOUN
ejpam-5625	179	24	4	4	NUM
ejpam-5625	179	25	,	,	PUNCT
ejpam-5625	179	26	og	og	PROPN
ejpam-5625	179	27	and	and	CCONJ
ejpam-5625	179	28	oh	oh	INTJ
ejpam-5625	179	29	are	be	AUX
ejpam-5625	179	30	co	co	ADJ
ejpam-5625	179	31	-	-	ADJ
ejpam-5625	179	32	certified	certify	VERB
ejpam-5625	179	33	pnd	pnd	NOUN
ejpam-5625	179	34	sets	set	NOUN
ejpam-5625	179	35	of	of	ADP
ejpam-5625	179	36	g	g	PROPN
ejpam-5625	179	37	and	and	CCONJ
ejpam-5625	179	38	h	h	NOUN
ejpam-5625	179	39	,	,	PUNCT
ejpam-5625	180	1	n.	n.	PROPN
ejpam-5625	180	2	s.	s.	PROPN
ejpam-5625	180	3	ulal	ulal	PROPN
ejpam-5625	180	4	et	et	PROPN
ejpam-5625	180	5	al	al	PROPN
ejpam-5625	180	6	.	.	PUNCT
ejpam-5625	180	7	/	/	SYM
ejpam-5625	180	8	eur	eur	PROPN
ejpam-5625	180	9	.	.	PUNCT
ejpam-5625	181	1	j.	j.	PROPN
ejpam-5625	181	2	pure	pure	PROPN
ejpam-5625	181	3	appl	appl	PROPN
ejpam-5625	181	4	.	.	PROPN
ejpam-5625	181	5	math	math	PROPN
ejpam-5625	181	6	,	,	PUNCT
ejpam-5625	181	7	18	18	NUM
ejpam-5625	181	8	(	(	PUNCT
ejpam-5625	181	9	1	1	NUM
ejpam-5625	181	10	)	)	PUNCT
ejpam-5625	181	11	(	(	PUNCT
ejpam-5625	181	12	2025	2025	NUM
ejpam-5625	181	13	)	)	PUNCT
ejpam-5625	181	14	,	,	PUNCT
ejpam-5625	181	15	5625	5625	NUM
ejpam-5625	181	16	7	7	NUM
ejpam-5625	181	17	of	of	ADP
ejpam-5625	181	18	9	9	NUM
ejpam-5625	181	19	respectively	respectively	ADV
ejpam-5625	181	20	.	.	PUNCT
ejpam-5625	182	1	thus	thus	ADV
ejpam-5625	182	2	,	,	PUNCT
ejpam-5625	182	3	γ2ch(g+h	γ2ch(g+h	NOUN
ejpam-5625	182	4	)	)	PUNCT
ejpam-5625	182	5	=	=	SYM
ejpam-5625	183	1	|o|	|o|	NOUN
ejpam-5625	183	2	=	=	PUNCT
ejpam-5625	183	3	|og|+	|og|+	PROPN
ejpam-5625	183	4	|oh	|oh	X
ejpam-5625	183	5	|	|	ADV
ejpam-5625	183	6	≥	≥	NOUN
ejpam-5625	183	7	ccpnd(g	ccpnd(g	PROPN
ejpam-5625	183	8	)	)	PUNCT
ejpam-5625	184	1	+	+	NUM
ejpam-5625	184	2	ccpnd(h	ccpnd(h	NOUN
ejpam-5625	184	3	)	)	PUNCT
ejpam-5625	184	4	(	(	PUNCT
ejpam-5625	184	5	i	i	NOUN
ejpam-5625	184	6	)	)	PUNCT
ejpam-5625	184	7	on	on	ADP
ejpam-5625	184	8	the	the	DET
ejpam-5625	184	9	other	other	ADJ
ejpam-5625	184	10	hand	hand	NOUN
ejpam-5625	184	11	,	,	PUNCT
ejpam-5625	184	12	suppose	suppose	VERB
ejpam-5625	184	13	that	that	SCONJ
ejpam-5625	184	14	o	o	PROPN
ejpam-5625	184	15	=	=	SYM
ejpam-5625	184	16	og∪oh	og∪oh	PROPN
ejpam-5625	184	17	,	,	PUNCT
ejpam-5625	184	18	where	where	SCONJ
ejpam-5625	184	19	og	og	PROPN
ejpam-5625	184	20	and	and	CCONJ
ejpam-5625	184	21	oh	oh	INTJ
ejpam-5625	184	22	are	be	AUX
ejpam-5625	184	23	both	both	PRON
ejpam-5625	184	24	minimum	minimum	ADJ
ejpam-5625	184	25	co	co	ADJ
ejpam-5625	184	26	-	-	ADJ
ejpam-5625	184	27	certified	certify	VERB
ejpam-5625	184	28	pnd	pnd	NOUN
ejpam-5625	184	29	sets	set	NOUN
ejpam-5625	184	30	of	of	ADP
ejpam-5625	184	31	g	g	PROPN
ejpam-5625	184	32	and	and	CCONJ
ejpam-5625	184	33	h	h	NOUN
ejpam-5625	184	34	,	,	PUNCT
ejpam-5625	184	35	respectively	respectively	ADV
ejpam-5625	184	36	.	.	PUNCT
ejpam-5625	185	1	then	then	ADV
ejpam-5625	185	2	by	by	ADP
ejpam-5625	185	3	theorem	theorem	NOUN
ejpam-5625	185	4	4	4	NUM
ejpam-5625	185	5	,	,	PUNCT
ejpam-5625	185	6	o	o	PROPN
ejpam-5625	185	7	is	be	AUX
ejpam-5625	185	8	a	a	DET
ejpam-5625	185	9	2	2	NUM
ejpam-5625	185	10	-	-	PUNCT
ejpam-5625	185	11	distance	distance	NOUN
ejpam-5625	185	12	certified	certify	VERB
ejpam-5625	185	13	hop	hop	NOUN
ejpam-5625	185	14	domknating	domknate	VERB
ejpam-5625	185	15	set	set	NOUN
ejpam-5625	185	16	of	of	ADP
ejpam-5625	185	17	g+h	g+h	PROPN
ejpam-5625	185	18	.	.	PUNCT
ejpam-5625	186	1	therefore	therefore	ADV
ejpam-5625	186	2	,	,	PUNCT
ejpam-5625	186	3	ccpnd(g	ccpnd(g	PROPN
ejpam-5625	186	4	)	)	PUNCT
ejpam-5625	187	1	+	+	NUM
ejpam-5625	187	2	ccpnd(h	ccpnd(h	NOUN
ejpam-5625	187	3	)	)	PUNCT
ejpam-5625	188	1	=	=	SYM
ejpam-5625	188	2	|og|+	|og|+	PROPN
ejpam-5625	188	3	|oh	|oh	X
ejpam-5625	188	4	|	|	NOUN
ejpam-5625	188	5	=	=	SYM
ejpam-5625	188	6	|o|	|o|	PROPN
ejpam-5625	188	7	≥	≥	NOUN
ejpam-5625	188	8	γ2ch(g+h	γ2ch(g+h	NOUN
ejpam-5625	188	9	)	)	PUNCT
ejpam-5625	188	10	(	(	PUNCT
ejpam-5625	188	11	ii	ii	NOUN
ejpam-5625	188	12	)	)	PUNCT
ejpam-5625	188	13	combining	combine	VERB
ejpam-5625	188	14	(	(	PUNCT
ejpam-5625	188	15	i	i	NOUN
ejpam-5625	188	16	)	)	PUNCT
ejpam-5625	188	17	and	and	CCONJ
ejpam-5625	188	18	(	(	PUNCT
ejpam-5625	188	19	ii	ii	NOUN
ejpam-5625	188	20	)	)	PUNCT
ejpam-5625	188	21	,	,	PUNCT
ejpam-5625	188	22	we	we	PRON
ejpam-5625	188	23	have	have	VERB
ejpam-5625	188	24	γ2ch(g+h	γ2ch(g+h	NOUN
ejpam-5625	188	25	)	)	PUNCT
ejpam-5625	189	1	=	=	SYM
ejpam-5625	189	2	ccpnd(g	ccpnd(g	PROPN
ejpam-5625	189	3	)	)	PUNCT
ejpam-5625	190	1	+	+	NUM
ejpam-5625	190	2	ccpnd(h	ccpnd(h	NOUN
ejpam-5625	190	3	)	)	PUNCT
ejpam-5625	190	4	.	.	PUNCT
ejpam-5625	191	1	the	the	DET
ejpam-5625	191	2	following	following	ADJ
ejpam-5625	191	3	result	result	NOUN
ejpam-5625	191	4	follows	follow	VERB
ejpam-5625	191	5	from	from	ADP
ejpam-5625	191	6	proposition	proposition	NOUN
ejpam-5625	191	7	1	1	NUM
ejpam-5625	191	8	and	and	CCONJ
ejpam-5625	191	9	theorem	theorem	VERB
ejpam-5625	191	10	5	5	NUM
ejpam-5625	191	11	.	.	PUNCT
ejpam-5625	191	12	corollary	corollary	ADJ
ejpam-5625	191	13	2	2	NUM
ejpam-5625	191	14	.	.	PUNCT
ejpam-5625	192	1	let	let	VERB
ejpam-5625	192	2	n	n	PRON
ejpam-5625	192	3	be	be	AUX
ejpam-5625	192	4	a	a	DET
ejpam-5625	192	5	positive	positive	ADJ
ejpam-5625	192	6	integer	integer	NOUN
ejpam-5625	192	7	.	.	PUNCT
ejpam-5625	193	1	then	then	ADV
ejpam-5625	193	2	(	(	PUNCT
ejpam-5625	193	3	i	i	NOUN
ejpam-5625	193	4	)	)	PUNCT
ejpam-5625	193	5	γ2ch(pn	γ2ch(pn	PROPN
ejpam-5625	193	6	+	+	CCONJ
ejpam-5625	193	7	pn	pn	NOUN
ejpam-5625	193	8	)	)	PUNCT
ejpam-5625	193	9	=	=	SYM
ejpam-5625	193	10	{	{	PUNCT
ejpam-5625	193	11	2n	2n	NUM
ejpam-5625	193	12	,	,	PUNCT
ejpam-5625	193	13	if	if	SCONJ
ejpam-5625	193	14	n	n	NOUN
ejpam-5625	193	15	=	=	SYM
ejpam-5625	193	16	1	1	NUM
ejpam-5625	193	17	,	,	PUNCT
ejpam-5625	193	18	2	2	NUM
ejpam-5625	193	19	,	,	PUNCT
ejpam-5625	193	20	3	3	NUM
ejpam-5625	193	21	,	,	PUNCT
ejpam-5625	193	22	4	4	NUM
ejpam-5625	193	23	4	4	NUM
ejpam-5625	193	24	,	,	PUNCT
ejpam-5625	193	25	if	if	SCONJ
ejpam-5625	193	26	n	n	PRON
ejpam-5625	193	27	≥	≥	NOUN
ejpam-5625	193	28	5	5	NUM
ejpam-5625	193	29	;	;	PUNCT
ejpam-5625	193	30	(	(	PUNCT
ejpam-5625	193	31	ii	ii	NOUN
ejpam-5625	193	32	)	)	PUNCT
ejpam-5625	193	33	γ2ch(cn	γ2ch(cn	VERB
ejpam-5625	194	1	+	+	CCONJ
ejpam-5625	194	2	cn	cn	ADJ
ejpam-5625	194	3	)	)	PUNCT
ejpam-5625	194	4	=	=	SYM
ejpam-5625	194	5	{	{	PUNCT
ejpam-5625	194	6	2n	2n	NUM
ejpam-5625	194	7	,	,	PUNCT
ejpam-5625	194	8	if	if	SCONJ
ejpam-5625	194	9	n	n	CCONJ
ejpam-5625	194	10	=	=	SYM
ejpam-5625	194	11	3	3	NUM
ejpam-5625	194	12	,	,	PUNCT
ejpam-5625	194	13	4	4	NUM
ejpam-5625	194	14	4	4	NUM
ejpam-5625	194	15	,	,	PUNCT
ejpam-5625	194	16	if	if	SCONJ
ejpam-5625	194	17	n	n	PRON
ejpam-5625	194	18	≥	≥	NOUN
ejpam-5625	194	19	5	5	NUM
ejpam-5625	194	20	;	;	PUNCT
ejpam-5625	194	21	(	(	PUNCT
ejpam-5625	194	22	iii	iii	NOUN
ejpam-5625	194	23	)	)	PUNCT
ejpam-5625	194	24	γ2ch(kn	γ2ch(kn	PUNCT
ejpam-5625	195	1	+	+	PROPN
ejpam-5625	195	2	kn	kn	PROPN
ejpam-5625	195	3	)	)	PUNCT
ejpam-5625	195	4	=	=	PUNCT
ejpam-5625	195	5	2n	2n	NUM
ejpam-5625	195	6	for	for	ADP
ejpam-5625	195	7	all	all	DET
ejpam-5625	195	8	n	n	PRON
ejpam-5625	195	9	≥	≥	NOUN
ejpam-5625	195	10	1	1	NUM
ejpam-5625	195	11	;	;	PUNCT
ejpam-5625	195	12	(	(	PUNCT
ejpam-5625	195	13	iv	iv	X
ejpam-5625	195	14	)	)	PUNCT
ejpam-5625	195	15	γ2ch(sn	γ2ch(sn	ADJ
ejpam-5625	195	16	)	)	PUNCT
ejpam-5625	195	17	=	=	PRON
ejpam-5625	195	18	{	{	PUNCT
ejpam-5625	195	19	n	n	CCONJ
ejpam-5625	195	20	,	,	PUNCT
ejpam-5625	195	21	if	if	SCONJ
ejpam-5625	195	22	n	n	CCONJ
ejpam-5625	195	23	=	=	SYM
ejpam-5625	195	24	1	1	NUM
ejpam-5625	195	25	,	,	PUNCT
ejpam-5625	195	26	2	2	NUM
ejpam-5625	195	27	2	2	NUM
ejpam-5625	195	28	,	,	PUNCT
ejpam-5625	195	29	if	if	SCONJ
ejpam-5625	195	30	n	n	PRON
ejpam-5625	195	31	≥	≥	NOUN
ejpam-5625	195	32	3	3	NUM
ejpam-5625	195	33	;	;	PUNCT
ejpam-5625	195	34	(	(	PUNCT
ejpam-5625	195	35	v	v	NOUN
ejpam-5625	195	36	)	)	PUNCT
ejpam-5625	195	37	γ2ch(km	γ2ch(km	NOUN
ejpam-5625	195	38	,	,	PUNCT
ejpam-5625	195	39	n	n	CCONJ
ejpam-5625	195	40	)	)	PUNCT
ejpam-5625	195	41	=	=	SYM
ejpam-5625	196	1			NOUN
ejpam-5625	196	2	4	4	NUM
ejpam-5625	196	3	,	,	PUNCT
ejpam-5625	196	4	if	if	SCONJ
ejpam-5625	196	5	m	m	PROPN
ejpam-5625	196	6	,	,	PUNCT
ejpam-5625	196	7	n	n	NOUN
ejpam-5625	196	8	=	=	SYM
ejpam-5625	196	9	2	2	NUM
ejpam-5625	196	10	3	3	NUM
ejpam-5625	196	11	,	,	PUNCT
ejpam-5625	196	12	if	if	SCONJ
ejpam-5625	196	13	m	m	ADV
ejpam-5625	196	14	=	=	SYM
ejpam-5625	196	15	2	2	NUM
ejpam-5625	196	16	and	and	CCONJ
ejpam-5625	196	17	n	n	PRON
ejpam-5625	196	18	≥	≥	NOUN
ejpam-5625	196	19	3	3	NUM
ejpam-5625	196	20	or	or	CCONJ
ejpam-5625	196	21	m	m	PROPN
ejpam-5625	196	22	≥	≥	NOUN
ejpam-5625	196	23	3	3	NUM
ejpam-5625	196	24	and	and	CCONJ
ejpam-5625	196	25	n	n	NOUN
ejpam-5625	196	26	=	=	SYM
ejpam-5625	196	27	2	2	NUM
ejpam-5625	196	28	2	2	NUM
ejpam-5625	196	29	,	,	PUNCT
ejpam-5625	196	30	if	if	SCONJ
ejpam-5625	196	31	m	m	PROPN
ejpam-5625	196	32	,	,	PUNCT
ejpam-5625	196	33	n	n	PRON
ejpam-5625	196	34	≥	≥	NOUN
ejpam-5625	196	35	3	3	NUM
ejpam-5625	196	36	or	or	CCONJ
ejpam-5625	196	37	m	m	PROPN
ejpam-5625	196	38	,	,	PUNCT
ejpam-5625	196	39	n	n	PROPN
ejpam-5625	196	40	=	=	SYM
ejpam-5625	196	41	1	1	NUM
ejpam-5625	196	42	;	;	PUNCT
ejpam-5625	196	43	(	(	PUNCT
ejpam-5625	196	44	vi	vi	NOUN
ejpam-5625	196	45	)	)	PUNCT
ejpam-5625	196	46	γ2ch(f1,n	γ2ch(f1,n	NOUN
ejpam-5625	196	47	)	)	PUNCT
ejpam-5625	196	48	=	=	PRON
ejpam-5625	196	49	{	{	PUNCT
ejpam-5625	196	50	n+	n+	NOUN
ejpam-5625	196	51	1	1	NUM
ejpam-5625	196	52	,	,	PUNCT
ejpam-5625	196	53	if	if	SCONJ
ejpam-5625	196	54	n	n	NOUN
ejpam-5625	196	55	=	=	SYM
ejpam-5625	196	56	1	1	NUM
ejpam-5625	196	57	,	,	PUNCT
ejpam-5625	196	58	2	2	NUM
ejpam-5625	196	59	,	,	PUNCT
ejpam-5625	196	60	3	3	NUM
ejpam-5625	196	61	,	,	PUNCT
ejpam-5625	196	62	4	4	NUM
ejpam-5625	196	63	3	3	NUM
ejpam-5625	196	64	,	,	PUNCT
ejpam-5625	196	65	if	if	SCONJ
ejpam-5625	196	66	n	n	PRON
ejpam-5625	196	67	≥	≥	NOUN
ejpam-5625	196	68	5	5	NUM
ejpam-5625	196	69	;	;	PUNCT
ejpam-5625	196	70	and	and	CCONJ
ejpam-5625	196	71	(	(	PUNCT
ejpam-5625	196	72	vii	vii	PROPN
ejpam-5625	196	73	)	)	PUNCT
ejpam-5625	196	74	γ2ch(wn	γ2ch(wn	PROPN
ejpam-5625	196	75	)	)	PUNCT
ejpam-5625	196	76	=	=	PRON
ejpam-5625	196	77	{	{	PUNCT
ejpam-5625	196	78	n+	n+	NOUN
ejpam-5625	196	79	1	1	NUM
ejpam-5625	196	80	,	,	PUNCT
ejpam-5625	196	81	if	if	SCONJ
ejpam-5625	196	82	n	n	NOUN
ejpam-5625	196	83	=	=	SYM
ejpam-5625	196	84	3	3	NUM
ejpam-5625	196	85	,	,	PUNCT
ejpam-5625	196	86	4	4	NUM
ejpam-5625	196	87	3	3	NUM
ejpam-5625	196	88	,	,	PUNCT
ejpam-5625	196	89	if	if	SCONJ
ejpam-5625	196	90	n	n	PRON
ejpam-5625	196	91	≥	≥	NOUN
ejpam-5625	196	92	5	5	NUM
ejpam-5625	196	93	.	.	NOUN
ejpam-5625	196	94	7	7	NUM
ejpam-5625	196	95	.	.	X
ejpam-5625	196	96	incomparability	incomparability	NOUN
ejpam-5625	196	97	of	of	ADP
ejpam-5625	196	98	2	2	NUM
ejpam-5625	196	99	-	-	PUNCT
ejpam-5625	196	100	distance	distance	NOUN
ejpam-5625	196	101	certified	certify	VERB
ejpam-5625	196	102	hop	hop	NOUN
ejpam-5625	196	103	domination	domination	NOUN
ejpam-5625	196	104	with	with	ADP
ejpam-5625	196	105	certified	certified	ADJ
ejpam-5625	196	106	domination	domination	NOUN
ejpam-5625	196	107	remark	remark	NOUN
ejpam-5625	196	108	2	2	NUM
ejpam-5625	196	109	.	.	PUNCT
ejpam-5625	197	1	let	let	VERB
ejpam-5625	197	2	h	h	NOUN
ejpam-5625	197	3	be	be	AUX
ejpam-5625	197	4	any	any	DET
ejpam-5625	197	5	graph	graph	NOUN
ejpam-5625	197	6	.	.	PUNCT
ejpam-5625	198	1	then	then	ADV
ejpam-5625	198	2	certified	certify	VERB
ejpam-5625	198	3	domination	domination	NOUN
ejpam-5625	198	4	and	and	CCONJ
ejpam-5625	198	5	2	2	NUM
ejpam-5625	198	6	-	-	PUNCT
ejpam-5625	198	7	distance	distance	NOUN
ejpam-5625	198	8	certified	certify	VERB
ejpam-5625	198	9	hop	hop	NOUN
ejpam-5625	198	10	domination	domination	NOUN
ejpam-5625	198	11	parameters	parameter	NOUN
ejpam-5625	198	12	are	be	AUX
ejpam-5625	198	13	incomparable	incomparable	ADJ
ejpam-5625	198	14	.	.	PUNCT
ejpam-5625	199	1	consider	consider	VERB
ejpam-5625	199	2	the	the	DET
ejpam-5625	199	3	graph	graph	NOUN
ejpam-5625	199	4	g	g	NOUN
ejpam-5625	199	5	below	below	ADV
ejpam-5625	199	6	.	.	PUNCT
ejpam-5625	200	1	n.	n.	PROPN
ejpam-5625	200	2	s.	s.	PROPN
ejpam-5625	200	3	ulal	ulal	PROPN
ejpam-5625	200	4	et	et	PROPN
ejpam-5625	200	5	al	al	PROPN
ejpam-5625	200	6	.	.	PUNCT
ejpam-5625	200	7	/	/	SYM
ejpam-5625	200	8	eur	eur	PROPN
ejpam-5625	200	9	.	.	PUNCT
ejpam-5625	201	1	j.	j.	PROPN
ejpam-5625	201	2	pure	pure	PROPN
ejpam-5625	201	3	appl	appl	PROPN
ejpam-5625	201	4	.	.	PROPN
ejpam-5625	201	5	math	math	PROPN
ejpam-5625	201	6	,	,	PUNCT
ejpam-5625	201	7	18	18	NUM
ejpam-5625	201	8	(	(	PUNCT
ejpam-5625	201	9	1	1	NUM
ejpam-5625	201	10	)	)	PUNCT
ejpam-5625	201	11	(	(	PUNCT
ejpam-5625	201	12	2025	2025	NUM
ejpam-5625	201	13	)	)	PUNCT
ejpam-5625	201	14	,	,	PUNCT
ejpam-5625	201	15	5625	5625	NUM
ejpam-5625	201	16	8	8	NUM
ejpam-5625	201	17	of	of	ADP
ejpam-5625	201	18	9	9	NUM
ejpam-5625	201	19	a	a	DET
ejpam-5625	201	20	b	b	NOUN
ejpam-5625	201	21	c	c	NOUN
ejpam-5625	201	22	d	d	PROPN
ejpam-5625	201	23	e	e	X
ejpam-5625	201	24	f	f	PROPN
ejpam-5625	202	1	g	g	PROPN
ejpam-5625	203	1	h	h	NOUN
ejpam-5625	204	1	i	i	PRON
ejpam-5625	204	2	k	k	PROPN
ejpam-5625	205	1	j	j	PROPN
ejpam-5625	205	2	l	l	NOUN
ejpam-5625	205	3	m	m	VERB
ejpam-5625	205	4	g	g	NOUN
ejpam-5625	205	5	:	:	PUNCT
ejpam-5625	205	6	let	let	VERB
ejpam-5625	205	7	s1	s1	PROPN
ejpam-5625	205	8	=	=	PUNCT
ejpam-5625	205	9	{	{	PUNCT
ejpam-5625	205	10	a	a	X
ejpam-5625	205	11	,	,	PUNCT
ejpam-5625	205	12	e	e	NOUN
ejpam-5625	205	13	,	,	PUNCT
ejpam-5625	205	14	h	h	NOUN
ejpam-5625	205	15	,	,	PUNCT
ejpam-5625	205	16	k	k	PROPN
ejpam-5625	205	17	,	,	PUNCT
ejpam-5625	205	18	m	m	VERB
ejpam-5625	205	19	}	}	PUNCT
ejpam-5625	205	20	and	and	CCONJ
ejpam-5625	205	21	s2	s2	VERB
ejpam-5625	205	22	=	=	SYM
ejpam-5625	205	23	{	{	PUNCT
ejpam-5625	205	24	c	c	X
ejpam-5625	205	25	,	,	PUNCT
ejpam-5625	205	26	m	m	NOUN
ejpam-5625	205	27	}	}	PUNCT
ejpam-5625	205	28	.	.	PUNCT
ejpam-5625	206	1	then	then	ADV
ejpam-5625	206	2	s1	s1	PROPN
ejpam-5625	206	3	is	be	AUX
ejpam-5625	206	4	minimum	minimum	NOUN
ejpam-5625	206	5	certified	certify	VERB
ejpam-5625	206	6	dominating	dominating	NOUN
ejpam-5625	206	7	set	set	NOUN
ejpam-5625	206	8	of	of	ADP
ejpam-5625	206	9	(	(	PUNCT
ejpam-5625	206	10	g	g	NOUN
ejpam-5625	206	11	)	)	PUNCT
ejpam-5625	206	12	.	.	PUNCT
ejpam-5625	207	1	thus	thus	ADV
ejpam-5625	207	2	γcer(g	γcer(g	NUM
ejpam-5625	207	3	)	)	PUNCT
ejpam-5625	207	4	=	=	SYM
ejpam-5625	208	1	5	5	X
ejpam-5625	208	2	.	.	PUNCT
ejpam-5625	208	3	moreover	moreover	ADV
ejpam-5625	208	4	,	,	PUNCT
ejpam-5625	208	5	s2	s2	PROPN
ejpam-5625	208	6	is	be	AUX
ejpam-5625	208	7	minimum	minimum	ADJ
ejpam-5625	208	8	2	2	NUM
ejpam-5625	208	9	-	-	PUNCT
ejpam-5625	208	10	distance	distance	NOUN
ejpam-5625	208	11	certified	certify	VERB
ejpam-5625	208	12	hop	hop	NOUN
ejpam-5625	208	13	dominating	dominating	NOUN
ejpam-5625	208	14	set	set	NOUN
ejpam-5625	208	15	of	of	ADP
ejpam-5625	208	16	g.	g.	PROPN
ejpam-5625	208	17	hence	hence	ADV
ejpam-5625	208	18	,	,	PUNCT
ejpam-5625	208	19	γ2ch(g	γ2ch(g	NOUN
ejpam-5625	208	20	)	)	PUNCT
ejpam-5625	208	21	=	=	SYM
ejpam-5625	209	1	2	2	X
ejpam-5625	209	2	.	.	X
ejpam-5625	209	3	therefore	therefore	ADV
ejpam-5625	209	4	,	,	PUNCT
ejpam-5625	209	5	γcer(g	γcer(g	PROPN
ejpam-5625	209	6	)	)	PUNCT
ejpam-5625	209	7	>	>	PUNCT
ejpam-5625	209	8	γ2ch(g	γ2ch(g	NOUN
ejpam-5625	209	9	)	)	PUNCT
ejpam-5625	209	10	.	.	PUNCT
ejpam-5625	210	1	next	next	ADV
ejpam-5625	210	2	,	,	PUNCT
ejpam-5625	210	3	consider	consider	VERB
ejpam-5625	210	4	the	the	DET
ejpam-5625	210	5	graph	graph	NOUN
ejpam-5625	210	6	g′	g′	NOUN
ejpam-5625	210	7	below	below	ADV
ejpam-5625	210	8	.	.	PUNCT
ejpam-5625	211	1	a	a	DET
ejpam-5625	211	2	b	b	PROPN
ejpam-5625	211	3	c	c	NOUN
ejpam-5625	211	4	g′	g′	NOUN
ejpam-5625	211	5	:	:	PUNCT
ejpam-5625	212	1	d	d	X
ejpam-5625	212	2	e	e	NOUN
ejpam-5625	212	3	let	let	VERB
ejpam-5625	212	4	u1	u1	NOUN
ejpam-5625	212	5	=	=	PRON
ejpam-5625	212	6	{	{	PUNCT
ejpam-5625	212	7	a	a	DET
ejpam-5625	212	8	,	,	PUNCT
ejpam-5625	212	9	b	b	NOUN
ejpam-5625	212	10	}	}	PUNCT
ejpam-5625	212	11	and	and	CCONJ
ejpam-5625	212	12	u2	u2	PROPN
ejpam-5625	212	13	=	=	PUNCT
ejpam-5625	212	14	{	{	PUNCT
ejpam-5625	212	15	a	a	PRON
ejpam-5625	212	16	,	,	PUNCT
ejpam-5625	212	17	b	b	NOUN
ejpam-5625	212	18	,	,	PUNCT
ejpam-5625	212	19	d	d	NOUN
ejpam-5625	212	20	}	}	PUNCT
ejpam-5625	212	21	.	.	PUNCT
ejpam-5625	213	1	then	then	ADV
ejpam-5625	213	2	u1	u1	NOUN
ejpam-5625	213	3	is	be	AUX
ejpam-5625	213	4	minimum	minimum	NOUN
ejpam-5625	213	5	certified	certify	VERB
ejpam-5625	213	6	dominating	dominating	NOUN
ejpam-5625	213	7	set	set	NOUN
ejpam-5625	213	8	of	of	ADP
ejpam-5625	213	9	g′.	g′.	X
ejpam-5625	213	10	thus	thus	ADV
ejpam-5625	213	11	,	,	PUNCT
ejpam-5625	213	12	γcer	γcer	NOUN
ejpam-5625	213	13	=	=	SYM
ejpam-5625	213	14	2	2	X
ejpam-5625	213	15	.	.	PUNCT
ejpam-5625	213	16	additionally	additionally	ADV
ejpam-5625	213	17	,	,	PUNCT
ejpam-5625	213	18	u2	u2	PROPN
ejpam-5625	213	19	is	be	AUX
ejpam-5625	213	20	minimum	minimum	ADJ
ejpam-5625	213	21	2	2	NUM
ejpam-5625	213	22	-	-	PUNCT
ejpam-5625	213	23	distance	distance	NOUN
ejpam-5625	213	24	certified	certify	VERB
ejpam-5625	213	25	hop	hop	NOUN
ejpam-5625	213	26	dominating	dominating	NOUN
ejpam-5625	213	27	set	set	NOUN
ejpam-5625	213	28	of	of	ADP
ejpam-5625	213	29	g′.	g′.	NOUN
ejpam-5625	213	30	hence	hence	ADV
ejpam-5625	213	31	,	,	PUNCT
ejpam-5625	213	32	γ2ch	γ2ch	PUNCT
ejpam-5625	213	33	=	=	SYM
ejpam-5625	213	34	3	3	X
ejpam-5625	213	35	.	.	X
ejpam-5625	214	1	therefore	therefore	ADV
ejpam-5625	214	2	,	,	PUNCT
ejpam-5625	214	3	γcer	γcer	VERB
ejpam-5625	214	4	<	<	X
ejpam-5625	214	5	γ2ch	γ2ch	PUNCT
ejpam-5625	214	6	.	.	PUNCT
ejpam-5625	215	1	acknowledgements	acknowledgement	VERB
ejpam-5625	215	2	the	the	DET
ejpam-5625	215	3	authors	author	NOUN
ejpam-5625	215	4	would	would	AUX
ejpam-5625	215	5	like	like	VERB
ejpam-5625	215	6	to	to	PART
ejpam-5625	215	7	thank	thank	VERB
ejpam-5625	215	8	mindanao	mindanao	PROPN
ejpam-5625	215	9	state	state	PROPN
ejpam-5625	215	10	university	university	PROPN
ejpam-5625	215	11	tawi	tawi	PROPN
ejpam-5625	215	12	-	-	PUNCT
ejpam-5625	215	13	tawi	tawi	PROPN
ejpam-5625	215	14	college	college	PROPN
ejpam-5625	215	15	of	of	ADP
ejpam-5625	215	16	technology	technology	NOUN
ejpam-5625	215	17	and	and	CCONJ
ejpam-5625	215	18	oceanography	oceanography	NOUN
ejpam-5625	215	19	,	,	PUNCT
ejpam-5625	215	20	and	and	CCONJ
ejpam-5625	215	21	korea	korea	PROPN
ejpam-5625	215	22	university	university	PROPN
ejpam-5625	215	23	for	for	ADP
ejpam-5625	215	24	funding	fund	VERB
ejpam-5625	215	25	this	this	DET
ejpam-5625	215	26	research	research	NOUN
ejpam-5625	215	27	.	.	PUNCT
ejpam-5625	216	1	also	also	ADV
ejpam-5625	216	2	,	,	PUNCT
ejpam-5625	216	3	the	the	DET
ejpam-5625	216	4	authors	author	NOUN
ejpam-5625	216	5	would	would	AUX
ejpam-5625	216	6	like	like	VERB
ejpam-5625	216	7	to	to	PART
ejpam-5625	216	8	thank	thank	VERB
ejpam-5625	216	9	the	the	DET
ejpam-5625	216	10	referees	referee	NOUN
ejpam-5625	216	11	for	for	ADP
ejpam-5625	216	12	their	their	PRON
ejpam-5625	216	13	invaluable	invaluable	ADJ
ejpam-5625	216	14	comments	comment	NOUN
ejpam-5625	216	15	and	and	CCONJ
ejpam-5625	216	16	suggestions	suggestion	NOUN
ejpam-5625	216	17	that	that	PRON
ejpam-5625	216	18	led	lead	VERB
ejpam-5625	216	19	to	to	ADP
ejpam-5625	216	20	the	the	DET
ejpam-5625	216	21	improvement	improvement	NOUN
ejpam-5625	216	22	of	of	ADP
ejpam-5625	216	23	the	the	DET
ejpam-5625	216	24	paper	paper	NOUN
ejpam-5625	216	25	.	.	PUNCT
ejpam-5625	217	1	references	reference	NOUN
ejpam-5625	217	2	[	[	X
ejpam-5625	217	3	1	1	X
ejpam-5625	217	4	]	]	PUNCT
ejpam-5625	217	5	v.	v.	PROPN
ejpam-5625	217	6	g.	g.	PROPN
ejpam-5625	217	7	bhagavathi	bhagavathi	PROPN
ejpam-5625	217	8	ammal	ammal	PROPN
ejpam-5625	217	9	and	and	CCONJ
ejpam-5625	217	10	r.	r.	PROPN
ejpam-5625	217	11	louisa	louisa	PROPN
ejpam-5625	217	12	dickfania	dickfania	PROPN
ejpam-5625	217	13	.	.	PUNCT
ejpam-5625	218	1	accurate	accurate	ADJ
ejpam-5625	218	2	certified	certify	VERB
ejpam-5625	218	3	domination	domination	NOUN
ejpam-5625	218	4	number	number	NOUN
ejpam-5625	218	5	of	of	ADP
ejpam-5625	218	6	graphs	graph	NOUN
ejpam-5625	218	7	.	.	PUNCT
ejpam-5625	219	1	international	international	ADJ
ejpam-5625	219	2	journal	journal	PROPN
ejpam-5625	219	3	of	of	ADP
ejpam-5625	219	4	mathematics	mathematics	NOUN
ejpam-5625	219	5	trends	trend	NOUN
ejpam-5625	219	6	and	and	CCONJ
ejpam-5625	219	7	technology	technology	NOUN
ejpam-5625	219	8	,	,	PUNCT
ejpam-5625	219	9	n.	n.	PROPN
ejpam-5625	219	10	s.	s.	PROPN
ejpam-5625	219	11	ulal	ulal	PROPN
ejpam-5625	219	12	et	et	PROPN
ejpam-5625	219	13	al	al	PROPN
ejpam-5625	219	14	.	.	PUNCT
ejpam-5625	219	15	/	/	SYM
ejpam-5625	219	16	eur	eur	PROPN
ejpam-5625	219	17	.	.	PUNCT
ejpam-5625	220	1	j.	j.	PROPN
ejpam-5625	220	2	pure	pure	PROPN
ejpam-5625	220	3	appl	appl	PROPN
ejpam-5625	220	4	.	.	PROPN
ejpam-5625	220	5	math	math	PROPN
ejpam-5625	220	6	,	,	PUNCT
ejpam-5625	220	7	18	18	NUM
ejpam-5625	220	8	(	(	PUNCT
ejpam-5625	220	9	1	1	NUM
ejpam-5625	220	10	)	)	PUNCT
ejpam-5625	220	11	(	(	PUNCT
ejpam-5625	220	12	2025	2025	NUM
ejpam-5625	220	13	)	)	PUNCT
ejpam-5625	220	14	,	,	PUNCT
ejpam-5625	220	15	5625	5625	NUM
ejpam-5625	220	16	9	9	NUM
ejpam-5625	220	17	of	of	ADP
ejpam-5625	220	18	9	9	NUM
ejpam-5625	220	19	66(05):90–98	66(05):90–98	NUM
ejpam-5625	220	20	,	,	PUNCT
ejpam-5625	220	21	2020	2020	NUM
ejpam-5625	220	22	.	.	PUNCT
ejpam-5625	221	1	[	[	X
ejpam-5625	221	2	2	2	NUM
ejpam-5625	221	3	]	]	PUNCT
ejpam-5625	221	4	s.	s.	PROPN
ejpam-5625	221	5	ayyaswamy	ayyaswamy	PROPN
ejpam-5625	221	6	,	,	PUNCT
ejpam-5625	221	7	b.	b.	PROPN
ejpam-5625	221	8	krishnakumari	krishnakumari	PROPN
ejpam-5625	221	9	,	,	PUNCT
ejpam-5625	221	10	b.	b.	PROPN
ejpam-5625	221	11	natarjan	natarjan	PROPN
ejpam-5625	221	12	,	,	PUNCT
ejpam-5625	221	13	and	and	CCONJ
ejpam-5625	221	14	y.	y.	PROPN
ejpam-5625	221	15	venkatakrishnan	venkatakrishnan	PROPN
ejpam-5625	221	16	.	.	PUNCT
ejpam-5625	222	1	bounds	bound	NOUN
ejpam-5625	222	2	on	on	ADP
ejpam-5625	222	3	the	the	DET
ejpam-5625	222	4	hop	hop	NOUN
ejpam-5625	222	5	domination	domination	NOUN
ejpam-5625	222	6	number	number	NOUN
ejpam-5625	222	7	of	of	ADP
ejpam-5625	222	8	a	a	DET
ejpam-5625	222	9	tree	tree	NOUN
ejpam-5625	222	10	.	.	PUNCT
ejpam-5625	223	1	proceedings	proceeding	NOUN
ejpam-5625	223	2	-	-	PUNCT
ejpam-5625	223	3	mathematical	mathematical	ADJ
ejpam-5625	223	4	sciences	science	NOUN
ejpam-5625	223	5	,	,	PUNCT
ejpam-5625	223	6	125(4):449–455	125(4):449–455	ADP
ejpam-5625	223	7	,	,	PUNCT
ejpam-5625	223	8	2015	2015	NUM
ejpam-5625	223	9	.	.	PUNCT
ejpam-5625	224	1	[	[	X
ejpam-5625	224	2	3	3	X
ejpam-5625	224	3	]	]	X
ejpam-5625	224	4	v.	v.	X
ejpam-5625	224	5	bilar	bilar	PROPN
ejpam-5625	224	6	,	,	PUNCT
ejpam-5625	224	7	m.	m.	NOUN
ejpam-5625	224	8	a.	a.	PROPN
ejpam-5625	224	9	bonsocan	bonsocan	PROPN
ejpam-5625	224	10	,	,	PUNCT
ejpam-5625	224	11	j.	j.	PROPN
ejpam-5625	224	12	hassan	hassan	PROPN
ejpam-5625	224	13	,	,	PUNCT
ejpam-5625	224	14	and	and	CCONJ
ejpam-5625	224	15	s.	s.	PROPN
ejpam-5625	224	16	dagondon	dagondon	PROPN
ejpam-5625	224	17	.	.	PUNCT
ejpam-5625	225	1	vertex	vertex	NOUN
ejpam-5625	225	2	cover	cover	VERB
ejpam-5625	225	3	hop	hop	NOUN
ejpam-5625	225	4	dominating	dominating	NOUN
ejpam-5625	225	5	sets	set	NOUN
ejpam-5625	225	6	in	in	ADP
ejpam-5625	225	7	graphs	graph	NOUN
ejpam-5625	225	8	.	.	PUNCT
ejpam-5625	226	1	european	european	ADJ
ejpam-5625	226	2	journal	journal	PROPN
ejpam-5625	226	3	of	of	ADP
ejpam-5625	226	4	pure	pure	ADJ
ejpam-5625	226	5	and	and	CCONJ
ejpam-5625	226	6	applied	applied	ADJ
ejpam-5625	226	7	mathematics	mathematic	NOUN
ejpam-5625	226	8	,	,	PUNCT
ejpam-5625	226	9	17(1):93–104	17(1):93–104	NUM
ejpam-5625	226	10	,	,	PUNCT
ejpam-5625	226	11	2024	2024	NUM
ejpam-5625	226	12	.	.	PUNCT
ejpam-5625	227	1	[	[	X
ejpam-5625	227	2	4	4	X
ejpam-5625	227	3	]	]	PUNCT
ejpam-5625	227	4	e.	e.	PROPN
ejpam-5625	227	5	j.	j.	PROPN
ejpam-5625	227	6	cockayne	cockayne	PROPN
ejpam-5625	227	7	and	and	CCONJ
ejpam-5625	227	8	s.	s.	PROPN
ejpam-5625	227	9	t.	t.	PROPN
ejpam-5625	227	10	hedetniemi	hedetniemi	PROPN
ejpam-5625	227	11	.	.	PUNCT
ejpam-5625	228	1	towards	towards	ADP
ejpam-5625	228	2	a	a	DET
ejpam-5625	228	3	theory	theory	NOUN
ejpam-5625	228	4	of	of	ADP
ejpam-5625	228	5	domination	domination	NOUN
ejpam-5625	228	6	in	in	ADP
ejpam-5625	228	7	graphs	graph	NOUN
ejpam-5625	228	8	.	.	PUNCT
ejpam-5625	229	1	networks	network	NOUN
ejpam-5625	229	2	,	,	PUNCT
ejpam-5625	229	3	7(3):247–261	7(3):247–261	NUM
ejpam-5625	229	4	,	,	PUNCT
ejpam-5625	229	5	1977	1977	NUM
ejpam-5625	229	6	.	.	PUNCT
ejpam-5625	230	1	[	[	X
ejpam-5625	230	2	5	5	NUM
ejpam-5625	230	3	]	]	PUNCT
ejpam-5625	230	4	m.	m.	NOUN
ejpam-5625	230	5	dettlaff	dettlaff	NOUN
ejpam-5625	230	6	,	,	PUNCT
ejpam-5625	230	7	m.	m.	NOUN
ejpam-5625	230	8	lemanska	lemanska	PROPN
ejpam-5625	230	9	,	,	PUNCT
ejpam-5625	230	10	r.	r.	PROPN
ejpam-5625	230	11	ziemann	ziemann	PROPN
ejpam-5625	230	12	,	,	PUNCT
ejpam-5625	230	13	j.	j.	PROPN
ejpam-5625	230	14	topp	topp	PROPN
ejpam-5625	230	15	,	,	PUNCT
ejpam-5625	230	16	and	and	CCONJ
ejpam-5625	230	17	p.	p.	PROPN
ejpam-5625	230	18	zylinski	zylinski	PROPN
ejpam-5625	230	19	.	.	PUNCT
ejpam-5625	231	1	certified	certified	ADJ
ejpam-5625	231	2	domination	domination	NOUN
ejpam-5625	231	3	.	.	PUNCT
ejpam-5625	232	1	akce	akce	PROPN
ejpam-5625	232	2	international	international	PROPN
ejpam-5625	232	3	journal	journal	NOUN
ejpam-5625	232	4	of	of	ADP
ejpam-5625	232	5	graphs	graph	NOUN
ejpam-5625	232	6	and	and	CCONJ
ejpam-5625	232	7	combinatorics	combinatoric	NOUN
ejpam-5625	232	8	,	,	PUNCT
ejpam-5625	232	9	09(004):1–12	09(004):1–12	PROPN
ejpam-5625	232	10	,	,	PUNCT
ejpam-5625	232	11	2018	2018	NUM
ejpam-5625	232	12	.	.	PUNCT
ejpam-5625	233	1	[	[	X
ejpam-5625	233	2	6	6	NUM
ejpam-5625	233	3	]	]	PUNCT
ejpam-5625	233	4	f.	f.	PROPN
ejpam-5625	233	5	harary	harary	PROPN
ejpam-5625	233	6	and	and	CCONJ
ejpam-5625	233	7	t.	t.	PROPN
ejpam-5625	233	8	w.	w.	PROPN
ejpam-5625	233	9	haynes	haynes	PROPN
ejpam-5625	233	10	.	.	PUNCT
ejpam-5625	234	1	double	double	ADJ
ejpam-5625	234	2	domination	domination	NOUN
ejpam-5625	234	3	in	in	ADP
ejpam-5625	234	4	graphs	graph	NOUN
ejpam-5625	234	5	.	.	PUNCT
ejpam-5625	235	1	ars	ar	NOUN
ejpam-5625	235	2	combinatoria	combinatoria	NOUN
ejpam-5625	235	3	,	,	PUNCT
ejpam-5625	235	4	55:201–213	55:201–213	NUM
ejpam-5625	235	5	,	,	PUNCT
ejpam-5625	235	6	2000	2000	NUM
ejpam-5625	235	7	.	.	PUNCT
ejpam-5625	236	1	[	[	X
ejpam-5625	236	2	7	7	X
ejpam-5625	236	3	]	]	PUNCT
ejpam-5625	236	4	j.	j.	PROPN
ejpam-5625	236	5	hassan	hassan	PROPN
ejpam-5625	236	6	,	,	PUNCT
ejpam-5625	236	7	s.	s.	PROPN
ejpam-5625	236	8	canoy	canoy	PROPN
ejpam-5625	236	9	,	,	PUNCT
ejpam-5625	236	10	and	and	CCONJ
ejpam-5625	236	11	c.	c.	PROPN
ejpam-5625	236	12	j.	j.	PROPN
ejpam-5625	236	13	saromines	saromines	PROPN
ejpam-5625	236	14	.	.	PUNCT
ejpam-5625	237	1	convex	convex	VERB
ejpam-5625	237	2	hop	hop	NOUN
ejpam-5625	237	3	domination	domination	NOUN
ejpam-5625	237	4	in	in	ADP
ejpam-5625	237	5	graphs	graph	NOUN
ejpam-5625	237	6	.	.	PUNCT
ejpam-5625	238	1	european	european	ADJ
ejpam-5625	238	2	journal	journal	PROPN
ejpam-5625	238	3	of	of	ADP
ejpam-5625	238	4	pure	pure	ADJ
ejpam-5625	238	5	and	and	CCONJ
ejpam-5625	238	6	applied	applied	ADJ
ejpam-5625	238	7	mathematics	mathematic	NOUN
ejpam-5625	238	8	,	,	PUNCT
ejpam-5625	238	9	16(1):319–335	16(1):319–335	NUM
ejpam-5625	238	10	,	,	PUNCT
ejpam-5625	238	11	2023	2023	NUM
ejpam-5625	238	12	.	.	PUNCT
ejpam-5625	239	1	[	[	X
ejpam-5625	239	2	8	8	NUM
ejpam-5625	239	3	]	]	PUNCT
ejpam-5625	239	4	j.	j.	PROPN
ejpam-5625	239	5	hassan	hassan	PROPN
ejpam-5625	239	6	and	and	CCONJ
ejpam-5625	239	7	s.	s.	PROPN
ejpam-5625	239	8	canoy	canoy	PROPN
ejpam-5625	239	9	jr	jr	PROPN
ejpam-5625	239	10	.	.	PUNCT
ejpam-5625	240	1	grundy	grundy	PROPN
ejpam-5625	240	2	dominating	dominating	PROPN
ejpam-5625	240	3	and	and	CCONJ
ejpam-5625	240	4	grundy	grundy	PROPN
ejpam-5625	240	5	hop	hop	NOUN
ejpam-5625	240	6	dominating	dominate	VERB
ejpam-5625	240	7	sequences	sequence	NOUN
ejpam-5625	240	8	in	in	ADP
ejpam-5625	240	9	graphs	graph	NOUN
ejpam-5625	240	10	:	:	PUNCT
ejpam-5625	240	11	relationships	relationship	NOUN
ejpam-5625	240	12	and	and	CCONJ
ejpam-5625	240	13	some	some	DET
ejpam-5625	240	14	structural	structural	ADJ
ejpam-5625	240	15	properties	property	NOUN
ejpam-5625	240	16	.	.	PUNCT
ejpam-5625	241	1	european	european	ADJ
ejpam-5625	241	2	journal	journal	PROPN
ejpam-5625	241	3	of	of	ADP
ejpam-5625	241	4	pure	pure	ADJ
ejpam-5625	241	5	and	and	CCONJ
ejpam-5625	241	6	applied	applied	ADJ
ejpam-5625	241	7	mathematics	mathematic	NOUN
ejpam-5625	241	8	,	,	PUNCT
ejpam-5625	241	9	16(2):1212–1227	16(2):1212–1227	NUM
ejpam-5625	241	10	,	,	PUNCT
ejpam-5625	241	11	2023	2023	NUM
ejpam-5625	241	12	.	.	PUNCT
ejpam-5625	242	1	[	[	X
ejpam-5625	242	2	9	9	NUM
ejpam-5625	242	3	]	]	PUNCT
ejpam-5625	242	4	j.	j.	PROPN
ejpam-5625	242	5	hassan	hassan	PROPN
ejpam-5625	242	6	and	and	CCONJ
ejpam-5625	242	7	s.	s.	PROPN
ejpam-5625	242	8	canoy	canoy	PROPN
ejpam-5625	242	9	jr	jr	PROPN
ejpam-5625	242	10	.	.	PUNCT
ejpam-5625	242	11	grundy	grundy	PROPN
ejpam-5625	242	12	total	total	PROPN
ejpam-5625	242	13	hop	hop	PROPN
ejpam-5625	242	14	dominating	dominate	VERB
ejpam-5625	242	15	sequences	sequence	NOUN
ejpam-5625	242	16	in	in	ADP
ejpam-5625	242	17	graphs	graph	NOUN
ejpam-5625	242	18	.	.	PUNCT
ejpam-5625	243	1	european	european	ADJ
ejpam-5625	243	2	journal	journal	PROPN
ejpam-5625	243	3	of	of	ADP
ejpam-5625	243	4	pure	pure	ADJ
ejpam-5625	243	5	and	and	CCONJ
ejpam-5625	243	6	applied	applied	ADJ
ejpam-5625	243	7	mathematics	mathematic	NOUN
ejpam-5625	243	8	,	,	PUNCT
ejpam-5625	243	9	16(4):2597–2612	16(4):2597–2612	NUM
ejpam-5625	243	10	,	,	PUNCT
ejpam-5625	243	11	2023	2023	NUM
ejpam-5625	243	12	.	.	PUNCT
ejpam-5625	244	1	[	[	X
ejpam-5625	244	2	10	10	NUM
ejpam-5625	244	3	]	]	X
ejpam-5625	244	4	c.	c.	PROPN
ejpam-5625	244	5	natarajan	natarajan	PROPN
ejpam-5625	244	6	and	and	CCONJ
ejpam-5625	244	7	s.	s.	PROPN
ejpam-5625	244	8	ayyaswamy	ayyaswamy	PROPN
ejpam-5625	244	9	.	.	PUNCT
ejpam-5625	245	1	hop	hop	PROPN
ejpam-5625	245	2	domination	domination	NOUN
ejpam-5625	245	3	in	in	ADP
ejpam-5625	245	4	graphs	graphs	PROPN
ejpam-5625	245	5	ii	ii	PROPN
ejpam-5625	245	6	.	.	PUNCT
ejpam-5625	245	7	versita	versita	PROPN
ejpam-5625	245	8	,	,	PUNCT
ejpam-5625	245	9	23(2):187	23(2):187	NUM
ejpam-5625	245	10	–	–	PUNCT
ejpam-5625	245	11	199	199	NUM
ejpam-5625	245	12	,	,	PUNCT
ejpam-5625	245	13	2015	2015	NUM
ejpam-5625	245	14	.	.	PUNCT
ejpam-5625	246	1	[	[	X
ejpam-5625	246	2	11	11	NUM
ejpam-5625	246	3	]	]	PUNCT
ejpam-5625	246	4	s.	s.	PROPN
ejpam-5625	246	5	durai	durai	PROPN
ejpam-5625	246	6	raj	raj	PROPN
ejpam-5625	246	7	,	,	PUNCT
ejpam-5625	246	8	s.	s.	PROPN
ejpam-5625	246	9	g.	g.	PROPN
ejpam-5625	246	10	shiji	shiji	PROPN
ejpam-5625	246	11	kumari	kumari	PROPN
ejpam-5625	246	12	,	,	PUNCT
ejpam-5625	246	13	and	and	CCONJ
ejpam-5625	246	14	a.	a.	NOUN
ejpam-5625	246	15	m.	m.	NOUN
ejpam-5625	246	16	anto	anto	PROPN
ejpam-5625	246	17	.	.	PUNCT
ejpam-5625	247	1	certified	certify	VERB
ejpam-5625	247	2	domination	domination	NOUN
ejpam-5625	247	3	number	number	NOUN
ejpam-5625	247	4	in	in	ADP
ejpam-5625	247	5	corona	corona	NOUN
ejpam-5625	247	6	product	product	NOUN
ejpam-5625	247	7	of	of	ADP
ejpam-5625	247	8	graphs	graph	NOUN
ejpam-5625	247	9	.	.	PUNCT
ejpam-5625	248	1	malaya	malaya	PROPN
ejpam-5625	248	2	journal	journal	PROPN
ejpam-5625	248	3	of	of	ADP
ejpam-5625	248	4	matematik	matematik	PROPN
ejpam-5625	248	5	,	,	PUNCT
ejpam-5625	248	6	9(1):1080–1082	9(1):1080–1082	PROPN
ejpam-5625	248	7	,	,	PUNCT
ejpam-5625	248	8	2021	2021	NUM
ejpam-5625	248	9	.	.	PUNCT