id	sid	tid	token	lemma	pos
ejpam-4894	1	1	european	european	PROPN
ejpam-4894	1	2	journal	journal	PROPN
ejpam-4894	1	3	of	of	ADP
ejpam-4894	1	4	pure	pure	ADJ
ejpam-4894	1	5	and	and	CCONJ
ejpam-4894	1	6	applied	apply	VERB
ejpam-4894	1	7	mathematics	mathematic	NOUN
ejpam-4894	1	8	vol	vol	NOUN
ejpam-4894	1	9	.	.	PUNCT
ejpam-4894	2	1	16	16	NUM
ejpam-4894	2	2	,	,	PUNCT
ejpam-4894	2	3	no	no	INTJ
ejpam-4894	2	4	.	.	NOUN
ejpam-4894	2	5	4	4	NUM
ejpam-4894	2	6	,	,	PUNCT
ejpam-4894	2	7	2023	2023	NUM
ejpam-4894	2	8	,	,	PUNCT
ejpam-4894	2	9	2763	2763	NUM
ejpam-4894	2	10	-	-	SYM
ejpam-4894	2	11	2774	2774	NUM
ejpam-4894	2	12	issn	issn	PROPN
ejpam-4894	2	13	1307	1307	NUM
ejpam-4894	2	14	-	-	SYM
ejpam-4894	2	15	5543	5543	NUM
ejpam-4894	2	16	–	–	PUNCT
ejpam-4894	3	1	ejpam.com	ejpam.com	X
ejpam-4894	3	2	published	publish	VERB
ejpam-4894	3	3	by	by	ADP
ejpam-4894	3	4	new	new	PROPN
ejpam-4894	3	5	york	york	PROPN
ejpam-4894	3	6	business	business	PROPN
ejpam-4894	3	7	global	global	ADJ
ejpam-4894	3	8	certified	certify	VERB
ejpam-4894	3	9	perfect	perfect	ADJ
ejpam-4894	3	10	domination	domination	NOUN
ejpam-4894	3	11	in	in	ADP
ejpam-4894	3	12	graphs	graph	NOUN
ejpam-4894	3	13	jamil	jamil	PROPN
ejpam-4894	3	14	j.	j.	PROPN
ejpam-4894	3	15	hamja	hamja	PROPN
ejpam-4894	3	16	office	office	PROPN
ejpam-4894	3	17	of	of	ADP
ejpam-4894	3	18	the	the	DET
ejpam-4894	3	19	vice	vice	NOUN
ejpam-4894	3	20	chancellor	chancellor	NOUN
ejpam-4894	3	21	for	for	ADP
ejpam-4894	3	22	academic	academic	ADJ
ejpam-4894	3	23	affairs	affair	NOUN
ejpam-4894	3	24	,	,	PUNCT
ejpam-4894	3	25	msu	msu	PROPN
ejpam-4894	3	26	-	-	PUNCT
ejpam-4894	3	27	tawi	tawi	NOUN
ejpam-4894	3	28	-	-	PUNCT
ejpam-4894	3	29	tawi	tawi	NOUN
ejpam-4894	3	30	college	college	PROPN
ejpam-4894	3	31	of	of	ADP
ejpam-4894	3	32	technology	technology	NOUN
ejpam-4894	3	33	and	and	CCONJ
ejpam-4894	3	34	oceanography	oceanography	NOUN
ejpam-4894	3	35	,	,	PUNCT
ejpam-4894	3	36	7500	7500	NUM
ejpam-4894	3	37	tawi	tawi	NOUN
ejpam-4894	3	38	-	-	PUNCT
ejpam-4894	3	39	tawi	tawi	NOUN
ejpam-4894	3	40	,	,	PUNCT
ejpam-4894	3	41	philippines	philippine	NOUN
ejpam-4894	3	42	abstract	abstract	ADJ
ejpam-4894	3	43	.	.	PUNCT
ejpam-4894	4	1	let	let	VERB
ejpam-4894	4	2	g	g	PROPN
ejpam-4894	4	3	=	=	SYM
ejpam-4894	4	4	(	(	PUNCT
ejpam-4894	4	5	v	v	NOUN
ejpam-4894	4	6	,	,	PUNCT
ejpam-4894	4	7	e	e	NOUN
ejpam-4894	4	8	)	)	PUNCT
ejpam-4894	4	9	be	be	AUX
ejpam-4894	4	10	a	a	DET
ejpam-4894	4	11	simple	simple	ADJ
ejpam-4894	4	12	connected	connected	ADJ
ejpam-4894	4	13	graph	graph	NOUN
ejpam-4894	4	14	.	.	PUNCT
ejpam-4894	5	1	a	a	DET
ejpam-4894	5	2	set	set	NOUN
ejpam-4894	5	3	s	s	NOUN
ejpam-4894	5	4	⊆	⊆	NUM
ejpam-4894	5	5	v	v	NOUN
ejpam-4894	5	6	(	(	PUNCT
ejpam-4894	5	7	g	g	NOUN
ejpam-4894	5	8	)	)	PUNCT
ejpam-4894	5	9	is	be	AUX
ejpam-4894	5	10	called	call	VERB
ejpam-4894	5	11	a	a	DET
ejpam-4894	5	12	certified	certify	VERB
ejpam-4894	5	13	perfect	perfect	ADJ
ejpam-4894	5	14	dominating	dominating	NOUN
ejpam-4894	5	15	set	set	NOUN
ejpam-4894	5	16	of	of	ADP
ejpam-4894	5	17	g	g	PROPN
ejpam-4894	5	18	if	if	SCONJ
ejpam-4894	5	19	every	every	DET
ejpam-4894	5	20	vertex	vertex	NOUN
ejpam-4894	5	21	v	v	ADP
ejpam-4894	5	22	∈	∈	PROPN
ejpam-4894	5	23	v	v	NOUN
ejpam-4894	5	24	(	(	PUNCT
ejpam-4894	5	25	g	g	NOUN
ejpam-4894	5	26	)	)	PUNCT
ejpam-4894	5	27	\	\	PROPN
ejpam-4894	6	1	s	s	PART
ejpam-4894	6	2	is	be	AUX
ejpam-4894	6	3	dominated	dominate	VERB
ejpam-4894	6	4	by	by	ADP
ejpam-4894	6	5	exactly	exactly	ADV
ejpam-4894	6	6	one	one	NUM
ejpam-4894	6	7	element	element	NOUN
ejpam-4894	6	8	u	u	NOUN
ejpam-4894	6	9	∈	∈	PROPN
ejpam-4894	6	10	s	s	PROPN
ejpam-4894	6	11	,	,	PUNCT
ejpam-4894	6	12	such	such	ADJ
ejpam-4894	6	13	that	that	SCONJ
ejpam-4894	6	14	u	u	PROPN
ejpam-4894	6	15	has	have	VERB
ejpam-4894	6	16	either	either	CCONJ
ejpam-4894	6	17	zero	zero	NUM
ejpam-4894	6	18	or	or	CCONJ
ejpam-4894	6	19	at	at	ADP
ejpam-4894	6	20	least	least	ADV
ejpam-4894	6	21	two	two	NUM
ejpam-4894	6	22	neighbors	neighbor	NOUN
ejpam-4894	6	23	in	in	ADP
ejpam-4894	6	24	v	v	NOUN
ejpam-4894	6	25	(	(	PUNCT
ejpam-4894	6	26	g)\s	g)\s	NOUN
ejpam-4894	6	27	.	.	PUNCT
ejpam-4894	7	1	the	the	DET
ejpam-4894	7	2	minimum	minimum	ADJ
ejpam-4894	7	3	cardinality	cardinality	NOUN
ejpam-4894	7	4	of	of	ADP
ejpam-4894	7	5	a	a	DET
ejpam-4894	7	6	certified	certify	VERB
ejpam-4894	7	7	perfect	perfect	ADJ
ejpam-4894	7	8	dominating	dominating	NOUN
ejpam-4894	7	9	set	set	NOUN
ejpam-4894	7	10	of	of	ADP
ejpam-4894	7	11	g	g	PROPN
ejpam-4894	7	12	is	be	AUX
ejpam-4894	7	13	called	call	VERB
ejpam-4894	7	14	the	the	DET
ejpam-4894	7	15	certified	certify	VERB
ejpam-4894	7	16	perfect	perfect	ADJ
ejpam-4894	7	17	domination	domination	NOUN
ejpam-4894	7	18	number	number	NOUN
ejpam-4894	7	19	of	of	ADP
ejpam-4894	7	20	g	g	NOUN
ejpam-4894	7	21	and	and	CCONJ
ejpam-4894	7	22	denoted	denote	VERB
ejpam-4894	7	23	by	by	ADP
ejpam-4894	7	24	γcerp(g	γcerp(g	PROPN
ejpam-4894	7	25	)	)	PUNCT
ejpam-4894	7	26	.	.	PUNCT
ejpam-4894	8	1	a	a	DET
ejpam-4894	8	2	certified	certify	VERB
ejpam-4894	8	3	perfect	perfect	ADJ
ejpam-4894	8	4	dominating	dominating	NOUN
ejpam-4894	8	5	set	set	NOUN
ejpam-4894	8	6	s	s	NOUN
ejpam-4894	8	7	of	of	ADP
ejpam-4894	8	8	g	g	NOUN
ejpam-4894	8	9	with	with	ADP
ejpam-4894	8	10	|s|	|s|	NOUN
ejpam-4894	8	11	=	=	PUNCT
ejpam-4894	8	12	γcerp(g	γcerp(g	PROPN
ejpam-4894	8	13	)	)	PUNCT
ejpam-4894	8	14	is	be	AUX
ejpam-4894	8	15	called	call	VERB
ejpam-4894	8	16	a	a	DET
ejpam-4894	8	17	γcerp	γcerp	NOUN
ejpam-4894	8	18	-	-	PUNCT
ejpam-4894	8	19	set	set	NOUN
ejpam-4894	8	20	.	.	PUNCT
ejpam-4894	9	1	in	in	ADP
ejpam-4894	9	2	this	this	DET
ejpam-4894	9	3	paper	paper	NOUN
ejpam-4894	9	4	,	,	PUNCT
ejpam-4894	9	5	the	the	DET
ejpam-4894	9	6	author	author	NOUN
ejpam-4894	9	7	focuses	focus	VERB
ejpam-4894	9	8	on	on	ADP
ejpam-4894	9	9	several	several	ADJ
ejpam-4894	9	10	key	key	ADJ
ejpam-4894	9	11	aspects	aspect	NOUN
ejpam-4894	9	12	:	:	PUNCT
ejpam-4894	9	13	a	a	DET
ejpam-4894	9	14	characterization	characterization	NOUN
ejpam-4894	9	15	of	of	ADP
ejpam-4894	9	16	the	the	DET
ejpam-4894	9	17	certified	certify	VERB
ejpam-4894	9	18	perfect	perfect	ADJ
ejpam-4894	9	19	dominating	dominating	NOUN
ejpam-4894	9	20	set	set	NOUN
ejpam-4894	9	21	,	,	PUNCT
ejpam-4894	9	22	determining	determine	VERB
ejpam-4894	9	23	the	the	DET
ejpam-4894	9	24	exact	exact	ADJ
ejpam-4894	9	25	values	value	NOUN
ejpam-4894	9	26	of	of	ADP
ejpam-4894	9	27	the	the	DET
ejpam-4894	9	28	certified	certify	VERB
ejpam-4894	9	29	perfect	perfect	ADJ
ejpam-4894	9	30	domination	domination	NOUN
ejpam-4894	9	31	number	number	NOUN
ejpam-4894	9	32	for	for	ADP
ejpam-4894	9	33	specific	specific	ADJ
ejpam-4894	9	34	graphs	graph	NOUN
ejpam-4894	9	35	,	,	PUNCT
ejpam-4894	9	36	and	and	CCONJ
ejpam-4894	9	37	investigating	investigate	VERB
ejpam-4894	9	38	the	the	DET
ejpam-4894	9	39	certified	certify	VERB
ejpam-4894	9	40	perfect	perfect	ADJ
ejpam-4894	9	41	domination	domination	NOUN
ejpam-4894	9	42	number	number	NOUN
ejpam-4894	9	43	of	of	ADP
ejpam-4894	9	44	graphs	graph	NOUN
ejpam-4894	9	45	resulting	result	VERB
ejpam-4894	9	46	from	from	ADP
ejpam-4894	9	47	the	the	DET
ejpam-4894	9	48	join	join	NOUN
ejpam-4894	9	49	and	and	CCONJ
ejpam-4894	9	50	corona	corona	NOUN
ejpam-4894	9	51	of	of	ADP
ejpam-4894	9	52	graphs	graph	NOUN
ejpam-4894	9	53	.	.	PUNCT
ejpam-4894	10	1	furthermore	furthermore	ADV
ejpam-4894	10	2	,	,	PUNCT
ejpam-4894	10	3	the	the	DET
ejpam-4894	10	4	relationship	relationship	NOUN
ejpam-4894	10	5	between	between	ADP
ejpam-4894	10	6	the	the	DET
ejpam-4894	10	7	perfect	perfect	ADJ
ejpam-4894	10	8	dominating	dominating	NOUN
ejpam-4894	10	9	set	set	NOUN
ejpam-4894	10	10	,	,	PUNCT
ejpam-4894	10	11	and	and	CCONJ
ejpam-4894	10	12	the	the	DET
ejpam-4894	10	13	certified	certify	VERB
ejpam-4894	10	14	perfect	perfect	ADJ
ejpam-4894	10	15	dominating	dominating	NOUN
ejpam-4894	10	16	set	set	NOUN
ejpam-4894	10	17	of	of	ADP
ejpam-4894	10	18	a	a	DET
ejpam-4894	10	19	graph	graph	NOUN
ejpam-4894	10	20	g	g	NOUN
ejpam-4894	10	21	are	be	AUX
ejpam-4894	10	22	established	establish	VERB
ejpam-4894	10	23	.	.	PUNCT
ejpam-4894	11	1	2020	2020	NUM
ejpam-4894	11	2	mathematics	mathematics	PROPN
ejpam-4894	11	3	subject	subject	NOUN
ejpam-4894	11	4	classifications	classification	NOUN
ejpam-4894	11	5	:	:	PUNCT
ejpam-4894	11	6	05c69,05c38,05c76	05c69,05c38,05c76	ADJ
ejpam-4894	11	7	key	key	ADJ
ejpam-4894	11	8	words	word	NOUN
ejpam-4894	11	9	and	and	CCONJ
ejpam-4894	11	10	phrases	phrase	NOUN
ejpam-4894	11	11	:	:	PUNCT
ejpam-4894	11	12	certified	certified	ADJ
ejpam-4894	11	13	domination	domination	NOUN
ejpam-4894	11	14	,	,	PUNCT
ejpam-4894	11	15	perfect	perfect	ADJ
ejpam-4894	11	16	domination	domination	NOUN
ejpam-4894	11	17	,	,	PUNCT
ejpam-4894	11	18	certified	certify	VERB
ejpam-4894	11	19	perfect	perfect	ADJ
ejpam-4894	11	20	domination	domination	NOUN
ejpam-4894	11	21	1	1	NUM
ejpam-4894	11	22	.	.	PUNCT
ejpam-4894	11	23	introduction	introduction	NOUN
ejpam-4894	11	24	in	in	ADP
ejpam-4894	11	25	1970	1970	NUM
ejpam-4894	11	26	,	,	PUNCT
ejpam-4894	11	27	dominating	dominating	NOUN
ejpam-4894	11	28	sets	set	NOUN
ejpam-4894	11	29	were	be	AUX
ejpam-4894	11	30	investigated	investigate	VERB
ejpam-4894	11	31	in	in	ADP
ejpam-4894	11	32	the	the	DET
ejpam-4894	11	33	context	context	NOUN
ejpam-4894	11	34	of	of	ADP
ejpam-4894	11	35	social	social	ADJ
ejpam-4894	11	36	networks	network	NOUN
ejpam-4894	11	37	,	,	PUNCT
ejpam-4894	11	38	where	where	SCONJ
ejpam-4894	11	39	vertices	vertex	NOUN
ejpam-4894	11	40	represented	represent	VERB
ejpam-4894	11	41	individuals	individual	NOUN
ejpam-4894	11	42	and	and	CCONJ
ejpam-4894	11	43	edges	edge	NOUN
ejpam-4894	11	44	represented	represent	VERB
ejpam-4894	11	45	relationships	relationship	NOUN
ejpam-4894	11	46	between	between	ADP
ejpam-4894	11	47	them	they	PRON
ejpam-4894	11	48	.	.	PUNCT
ejpam-4894	12	1	the	the	DET
ejpam-4894	12	2	concept	concept	NOUN
ejpam-4894	12	3	of	of	ADP
ejpam-4894	12	4	dominating	dominate	VERB
ejpam-4894	12	5	set	set	NOUN
ejpam-4894	12	6	helps	help	VERB
ejpam-4894	12	7	to	to	PART
ejpam-4894	12	8	identify	identify	VERB
ejpam-4894	12	9	key	key	ADJ
ejpam-4894	12	10	individuals	individual	NOUN
ejpam-4894	12	11	who	who	PRON
ejpam-4894	12	12	could	could	AUX
ejpam-4894	12	13	exert	exert	VERB
ejpam-4894	12	14	influence	influence	NOUN
ejpam-4894	12	15	or	or	CCONJ
ejpam-4894	12	16	control	control	NOUN
ejpam-4894	12	17	over	over	ADP
ejpam-4894	12	18	the	the	DET
ejpam-4894	12	19	entire	entire	ADJ
ejpam-4894	12	20	network	network	NOUN
ejpam-4894	12	21	.	.	PUNCT
ejpam-4894	13	1	however	however	ADV
ejpam-4894	13	2	,	,	PUNCT
ejpam-4894	13	3	the	the	DET
ejpam-4894	13	4	concept	concept	NOUN
ejpam-4894	13	5	quickly	quickly	ADV
ejpam-4894	13	6	found	find	VERB
ejpam-4894	13	7	applications	application	NOUN
ejpam-4894	13	8	in	in	ADP
ejpam-4894	13	9	various	various	ADJ
ejpam-4894	13	10	other	other	ADJ
ejpam-4894	13	11	fields	field	NOUN
ejpam-4894	13	12	,	,	PUNCT
ejpam-4894	13	13	including	include	VERB
ejpam-4894	13	14	computer	computer	NOUN
ejpam-4894	13	15	science	science	NOUN
ejpam-4894	13	16	,	,	PUNCT
ejpam-4894	13	17	operations	operation	NOUN
ejpam-4894	13	18	research	research	NOUN
ejpam-4894	13	19	,	,	PUNCT
ejpam-4894	13	20	and	and	CCONJ
ejpam-4894	13	21	biology	biology	NOUN
ejpam-4894	13	22	.	.	PUNCT
ejpam-4894	14	1	in	in	ADP
ejpam-4894	14	2	2018	2018	NUM
ejpam-4894	14	3	,	,	PUNCT
ejpam-4894	14	4	dettlaff	dettlaff	VERB
ejpam-4894	14	5	et	et	NOUN
ejpam-4894	14	6	.	.	PUNCT
ejpam-4894	15	1	al	al	PROPN
ejpam-4894	15	2	,	,	PUNCT
ejpam-4894	15	3	introduced	introduce	VERB
ejpam-4894	15	4	the	the	DET
ejpam-4894	15	5	concept	concept	NOUN
ejpam-4894	15	6	of	of	ADP
ejpam-4894	15	7	certified	certify	VERB
ejpam-4894	15	8	dominating	dominating	NOUN
ejpam-4894	15	9	set	set	VERB
ejpam-4894	15	10	in	in	ADP
ejpam-4894	15	11	a	a	DET
ejpam-4894	15	12	graphs	graph	NOUN
ejpam-4894	15	13	.	.	PUNCT
ejpam-4894	16	1	therein	therein	ADV
ejpam-4894	16	2	,	,	PUNCT
ejpam-4894	16	3	they	they	PRON
ejpam-4894	16	4	presented	present	VERB
ejpam-4894	16	5	the	the	DET
ejpam-4894	16	6	exact	exact	ADJ
ejpam-4894	16	7	values	value	NOUN
ejpam-4894	16	8	of	of	ADP
ejpam-4894	16	9	the	the	DET
ejpam-4894	16	10	certified	certify	VERB
ejpam-4894	16	11	domination	domination	NOUN
ejpam-4894	16	12	number	number	NOUN
ejpam-4894	16	13	for	for	ADP
ejpam-4894	16	14	some	some	DET
ejpam-4894	16	15	classes	class	NOUN
ejpam-4894	16	16	of	of	ADP
ejpam-4894	16	17	graphs	graph	NOUN
ejpam-4894	16	18	as	as	ADV
ejpam-4894	16	19	well	well	ADV
ejpam-4894	16	20	as	as	ADP
ejpam-4894	16	21	provided	provide	VERB
ejpam-4894	16	22	some	some	DET
ejpam-4894	16	23	upper	upper	ADJ
ejpam-4894	16	24	bounds	bound	NOUN
ejpam-4894	16	25	on	on	ADP
ejpam-4894	16	26	this	this	DET
ejpam-4894	16	27	parameter	parameter	NOUN
ejpam-4894	16	28	for	for	ADP
ejpam-4894	16	29	arbitrary	arbitrary	ADJ
ejpam-4894	16	30	graphs	graph	NOUN
ejpam-4894	16	31	.	.	PUNCT
ejpam-4894	17	1	they	they	PRON
ejpam-4894	17	2	then	then	ADV
ejpam-4894	17	3	characterised	characterise	VERB
ejpam-4894	17	4	a	a	DET
ejpam-4894	17	5	wide	wide	ADJ
ejpam-4894	17	6	class	class	NOUN
ejpam-4894	17	7	of	of	ADP
ejpam-4894	17	8	graphs	graph	NOUN
ejpam-4894	17	9	with	with	ADP
ejpam-4894	17	10	equal	equal	ADJ
ejpam-4894	17	11	domination	domination	NOUN
ejpam-4894	17	12	and	and	CCONJ
ejpam-4894	17	13	certified	certify	VERB
ejpam-4894	17	14	domination	domination	NOUN
ejpam-4894	17	15	numbers	number	NOUN
ejpam-4894	17	16	and	and	CCONJ
ejpam-4894	17	17	characterise	characterise	NOUN
ejpam-4894	17	18	graphs	graph	NOUN
ejpam-4894	17	19	with	with	ADP
ejpam-4894	17	20	large	large	ADJ
ejpam-4894	17	21	values	value	NOUN
ejpam-4894	17	22	of	of	ADP
ejpam-4894	17	23	certified	certified	ADJ
ejpam-4894	17	24	domination	domination	NOUN
ejpam-4894	17	25	numbers	number	NOUN
ejpam-4894	17	26	[	[	X
ejpam-4894	17	27	4	4	NUM
ejpam-4894	17	28	]	]	PUNCT
ejpam-4894	17	29	.	.	PUNCT
ejpam-4894	18	1	moreover	moreover	ADV
ejpam-4894	18	2	,	,	PUNCT
ejpam-4894	18	3	several	several	ADJ
ejpam-4894	18	4	authors	author	NOUN
ejpam-4894	18	5	investigated	investigate	VERB
ejpam-4894	18	6	further	far	ADV
ejpam-4894	18	7	in	in	ADP
ejpam-4894	18	8	this	this	DET
ejpam-4894	18	9	concept	concept	NOUN
ejpam-4894	18	10	.	.	PUNCT
ejpam-4894	19	1	they	they	PRON
ejpam-4894	19	2	obtained	obtain	VERB
ejpam-4894	19	3	the	the	DET
ejpam-4894	19	4	certified	certify	VERB
ejpam-4894	19	5	domination	domination	NOUN
ejpam-4894	19	6	number	number	NOUN
ejpam-4894	19	7	of	of	ADP
ejpam-4894	19	8	cartesian	cartesian	ADJ
ejpam-4894	19	9	product	product	NOUN
ejpam-4894	19	10	,	,	PUNCT
ejpam-4894	19	11	corona	corona	NOUN
ejpam-4894	19	12	product	product	NOUN
ejpam-4894	19	13	of	of	ADP
ejpam-4894	19	14	some	some	DET
ejpam-4894	19	15	standard	standard	ADJ
ejpam-4894	19	16	graphs	graph	NOUN
ejpam-4894	19	17	and	and	CCONJ
ejpam-4894	19	18	special	special	ADJ
ejpam-4894	19	19	subdivision	subdivision	NOUN
ejpam-4894	19	20	graphs	graph	NOUN
ejpam-4894	19	21	of	of	ADP
ejpam-4894	19	22	certain	certain	ADJ
ejpam-4894	19	23	families	family	NOUN
ejpam-4894	19	24	of	of	ADP
ejpam-4894	19	25	graphs	graph	NOUN
ejpam-4894	19	26	(	(	PUNCT
ejpam-4894	19	27	see	see	VERB
ejpam-4894	19	28	[	[	X
ejpam-4894	19	29	9	9	X
ejpam-4894	19	30	]	]	X
ejpam-4894	20	1	[	[	X
ejpam-4894	20	2	1	1	NUM
ejpam-4894	20	3	]	]	PUNCT
ejpam-4894	20	4	,	,	PUNCT
ejpam-4894	20	5	[	[	X
ejpam-4894	20	6	8	8	NUM
ejpam-4894	20	7	]	]	PUNCT
ejpam-4894	20	8	and	and	CCONJ
ejpam-4894	20	9	[	[	X
ejpam-4894	20	10	10	10	NUM
ejpam-4894	20	11	]	]	NUM
ejpam-4894	20	12	)	)	PUNCT
ejpam-4894	20	13	.	.	PUNCT
ejpam-4894	21	1	doi	doi	NOUN
ejpam-4894	21	2	:	:	PUNCT
ejpam-4894	21	3	https://doi.org/10.29020/nybg.ejpam.v16i4.4894	https://doi.org/10.29020/nybg.ejpam.v16i4.4894	ADJ
ejpam-4894	21	4	email	email	NOUN
ejpam-4894	21	5	address	address	NOUN
ejpam-4894	21	6	:	:	PUNCT
ejpam-4894	21	7	jamilhamja@msutawi-tawi.edu.ph	jamilhamja@msutawi-tawi.edu.ph	PROPN
ejpam-4894	21	8	(	(	PUNCT
ejpam-4894	21	9	j.	j.	PROPN
ejpam-4894	21	10	hamja	hamja	PROPN
ejpam-4894	21	11	)	)	PUNCT
ejpam-4894	21	12	https://www.ejpam.com	https://www.ejpam.com	X
ejpam-4894	21	13	2763	2763	NUM
ejpam-4894	21	14	©	©	PROPN
ejpam-4894	21	15	2023	2023	NUM
ejpam-4894	21	16	ejpam	ejpam	NOUN
ejpam-4894	21	17	all	all	DET
ejpam-4894	21	18	rights	right	NOUN
ejpam-4894	21	19	reserved	reserve	VERB
ejpam-4894	21	20	.	.	PUNCT
ejpam-4894	22	1	j.	j.	PROPN
ejpam-4894	22	2	hamja	hamja	PROPN
ejpam-4894	22	3	/	/	SYM
ejpam-4894	22	4	eur	eur	PROPN
ejpam-4894	22	5	.	.	PUNCT
ejpam-4894	23	1	j.	j.	PROPN
ejpam-4894	23	2	pure	pure	PROPN
ejpam-4894	23	3	appl	appl	PROPN
ejpam-4894	23	4	.	.	PROPN
ejpam-4894	23	5	math	math	PROPN
ejpam-4894	23	6	,	,	PUNCT
ejpam-4894	23	7	16	16	NUM
ejpam-4894	23	8	(	(	PUNCT
ejpam-4894	23	9	4	4	NUM
ejpam-4894	23	10	)	)	PUNCT
ejpam-4894	23	11	(	(	PUNCT
ejpam-4894	23	12	2023	2023	NUM
ejpam-4894	23	13	)	)	PUNCT
ejpam-4894	23	14	,	,	PUNCT
ejpam-4894	23	15	2763	2763	NUM
ejpam-4894	23	16	-	-	SYM
ejpam-4894	23	17	2774	2774	NUM
ejpam-4894	23	18	2764	2764	NUM
ejpam-4894	23	19	in	in	ADP
ejpam-4894	23	20	1990	1990	NUM
ejpam-4894	23	21	,	,	PUNCT
ejpam-4894	23	22	livingston	livingston	PROPN
ejpam-4894	23	23	m.	m.	PROPN
ejpam-4894	23	24	et.al	et.al	PROPN
ejpam-4894	23	25	,	,	PUNCT
ejpam-4894	23	26	investigated	investigate	VERB
ejpam-4894	23	27	the	the	DET
ejpam-4894	23	28	concept	concept	NOUN
ejpam-4894	23	29	of	of	ADP
ejpam-4894	23	30	perfect	perfect	ADJ
ejpam-4894	23	31	dominating	dominating	NOUN
ejpam-4894	23	32	set	set	NOUN
ejpam-4894	23	33	of	of	ADP
ejpam-4894	23	34	a	a	DET
ejpam-4894	23	35	graph	graph	NOUN
ejpam-4894	23	36	.	.	PUNCT
ejpam-4894	24	1	they	they	PRON
ejpam-4894	24	2	studied	study	VERB
ejpam-4894	24	3	the	the	DET
ejpam-4894	24	4	existence	existence	NOUN
ejpam-4894	24	5	and	and	CCONJ
ejpam-4894	24	6	construction	construction	NOUN
ejpam-4894	24	7	of	of	ADP
ejpam-4894	24	8	perfect	perfect	ADJ
ejpam-4894	24	9	dominating	dominating	NOUN
ejpam-4894	24	10	sets	set	NOUN
ejpam-4894	24	11	in	in	ADP
ejpam-4894	24	12	families	family	NOUN
ejpam-4894	24	13	of	of	ADP
ejpam-4894	24	14	a	a	DET
ejpam-4894	24	15	graphs	graph	NOUN
ejpam-4894	24	16	arising	arise	VERB
ejpam-4894	24	17	from	from	ADP
ejpam-4894	24	18	the	the	DET
ejpam-4894	24	19	interconnection	interconnection	NOUN
ejpam-4894	24	20	networks	network	NOUN
ejpam-4894	24	21	of	of	ADP
ejpam-4894	24	22	parallel	parallel	ADJ
ejpam-4894	24	23	computers	computer	NOUN
ejpam-4894	24	24	.	.	PUNCT
ejpam-4894	25	1	these	these	PRON
ejpam-4894	25	2	include	include	VERB
ejpam-4894	25	3	trees	tree	NOUN
ejpam-4894	25	4	,	,	PUNCT
ejpam-4894	25	5	dags	dag	NOUN
ejpam-4894	25	6	,	,	PUNCT
ejpam-4894	25	7	series	series	NOUN
ejpam-4894	25	8	-	-	PUNCT
ejpam-4894	25	9	parallel	parallel	ADJ
ejpam-4894	25	10	graphs	graph	NOUN
ejpam-4894	25	11	,	,	PUNCT
ejpam-4894	25	12	meshes	mesh	NOUN
ejpam-4894	25	13	,	,	PUNCT
ejpam-4894	25	14	tori	tori	PROPN
ejpam-4894	25	15	,	,	PUNCT
ejpam-4894	25	16	hypercubes	hypercube	NOUN
ejpam-4894	25	17	,	,	PUNCT
ejpam-4894	25	18	cube	cube	NOUN
ejpam-4894	25	19	-	-	PUNCT
ejpam-4894	25	20	connected	connect	VERB
ejpam-4894	25	21	cycles	cycle	NOUN
ejpam-4894	25	22	,	,	PUNCT
ejpam-4894	25	23	cubeconnected	cubeconnected	ADJ
ejpam-4894	25	24	paths	path	NOUN
ejpam-4894	25	25	,	,	PUNCT
ejpam-4894	25	26	and	and	CCONJ
ejpam-4894	25	27	de	de	ADP
ejpam-4894	25	28	bruijn	bruijn	NOUN
ejpam-4894	25	29	graphs	graph	NOUN
ejpam-4894	25	30	[	[	X
ejpam-4894	25	31	7	7	NUM
ejpam-4894	25	32	]	]	PUNCT
ejpam-4894	25	33	.	.	PUNCT
ejpam-4894	26	1	in	in	ADP
ejpam-4894	26	2	2014	2014	NUM
ejpam-4894	26	3	,	,	PUNCT
ejpam-4894	26	4	kwon	kwon	VERB
ejpam-4894	26	5	y.s	y.s	PROPN
ejpam-4894	26	6	.	.	PROPN
ejpam-4894	26	7	et	et	PROPN
ejpam-4894	26	8	.	.	PUNCT
ejpam-4894	27	1	al	al	PROPN
ejpam-4894	27	2	.	.	PROPN
ejpam-4894	27	3	,	,	PUNCT
ejpam-4894	27	4	got	get	VERB
ejpam-4894	27	5	some	some	DET
ejpam-4894	27	6	results	result	NOUN
ejpam-4894	27	7	related	relate	VERB
ejpam-4894	27	8	to	to	ADP
ejpam-4894	27	9	perfect	perfect	ADJ
ejpam-4894	27	10	domination	domination	NOUN
ejpam-4894	27	11	sets	set	NOUN
ejpam-4894	27	12	of	of	ADP
ejpam-4894	27	13	cayley	cayley	ADJ
ejpam-4894	27	14	graphs	graph	NOUN
ejpam-4894	27	15	.	.	PUNCT
ejpam-4894	28	1	they	they	PRON
ejpam-4894	28	2	showed	show	VERB
ejpam-4894	28	3	that	that	SCONJ
ejpam-4894	28	4	if	if	SCONJ
ejpam-4894	28	5	a	a	DET
ejpam-4894	28	6	cayley	cayley	ADJ
ejpam-4894	28	7	graph	graph	NOUN
ejpam-4894	28	8	c(a	c(a	PROPN
ejpam-4894	28	9	,	,	PUNCT
ejpam-4894	28	10	x	x	PRON
ejpam-4894	28	11	)	)	PUNCT
ejpam-4894	28	12	has	have	VERB
ejpam-4894	28	13	a	a	DET
ejpam-4894	28	14	perfect	perfect	ADJ
ejpam-4894	28	15	dominating	dominating	NOUN
ejpam-4894	28	16	set	set	NOUN
ejpam-4894	28	17	s	s	PRON
ejpam-4894	28	18	which	which	PRON
ejpam-4894	28	19	is	be	AUX
ejpam-4894	28	20	a	a	DET
ejpam-4894	28	21	normal	normal	ADJ
ejpam-4894	28	22	subgroup	subgroup	NOUN
ejpam-4894	28	23	of	of	ADP
ejpam-4894	28	24	a	a	PRON
ejpam-4894	28	25	and	and	CCONJ
ejpam-4894	28	26	whose	whose	DET
ejpam-4894	28	27	induced	induced	ADJ
ejpam-4894	28	28	subgraph	subgraph	NOUN
ejpam-4894	28	29	is	be	AUX
ejpam-4894	28	30	f	f	PROPN
ejpam-4894	28	31	,	,	PUNCT
ejpam-4894	28	32	then	then	ADV
ejpam-4894	28	33	there	there	PRON
ejpam-4894	28	34	exists	exist	VERB
ejpam-4894	28	35	an	an	DET
ejpam-4894	28	36	f	f	PROPN
ejpam-4894	28	37	-bundle	-bundle	PROPN
ejpam-4894	28	38	projection	projection	NOUN
ejpam-4894	28	39	p	p	NOUN
ejpam-4894	28	40	:	:	PUNCT
ejpam-4894	28	41	c(a	c(a	PROPN
ejpam-4894	28	42	,	,	PUNCT
ejpam-4894	28	43	x	x	X
ejpam-4894	28	44	)	)	PUNCT
ejpam-4894	28	45	→	→	SYM
ejpam-4894	28	46	km	km	NOUN
ejpam-4894	28	47	for	for	ADP
ejpam-4894	28	48	some	some	DET
ejpam-4894	28	49	positive	positive	ADJ
ejpam-4894	28	50	integer	integer	NOUN
ejpam-4894	28	51	m	m	VERB
ejpam-4894	29	1	[	[	X
ejpam-4894	29	2	6	6	NUM
ejpam-4894	29	3	]	]	PUNCT
ejpam-4894	29	4	.	.	PUNCT
ejpam-4894	30	1	moreover	moreover	ADV
ejpam-4894	30	2	,	,	PUNCT
ejpam-4894	30	3	numerous	numerous	ADJ
ejpam-4894	30	4	classes	class	NOUN
ejpam-4894	30	5	of	of	ADP
ejpam-4894	30	6	graphs	graph	NOUN
ejpam-4894	30	7	have	have	AUX
ejpam-4894	30	8	been	be	AUX
ejpam-4894	30	9	investigated	investigate	VERB
ejpam-4894	30	10	to	to	PART
ejpam-4894	30	11	explore	explore	VERB
ejpam-4894	30	12	variations	variation	NOUN
ejpam-4894	30	13	and	and	CCONJ
ejpam-4894	30	14	parameters	parameter	NOUN
ejpam-4894	30	15	of	of	ADP
ejpam-4894	30	16	the	the	DET
ejpam-4894	30	17	perfect	perfect	ADJ
ejpam-4894	30	18	dominating	dominating	NOUN
ejpam-4894	30	19	set	set	NOUN
ejpam-4894	30	20	(	(	PUNCT
ejpam-4894	30	21	see	see	VERB
ejpam-4894	30	22	[	[	X
ejpam-4894	30	23	3	3	NUM
ejpam-4894	30	24	]	]	PUNCT
ejpam-4894	30	25	,	,	PUNCT
ejpam-4894	30	26	[	[	X
ejpam-4894	30	27	11	11	NUM
ejpam-4894	30	28	]	]	PUNCT
ejpam-4894	30	29	and	and	CCONJ
ejpam-4894	30	30	[	[	X
ejpam-4894	30	31	2	2	NUM
ejpam-4894	30	32	]	]	PUNCT
ejpam-4894	30	33	)	)	PUNCT
ejpam-4894	30	34	.	.	PUNCT
ejpam-4894	31	1	now	now	ADV
ejpam-4894	31	2	,	,	PUNCT
ejpam-4894	31	3	let	let	VERB
ejpam-4894	31	4	us	we	PRON
ejpam-4894	31	5	consider	consider	VERB
ejpam-4894	31	6	a	a	DET
ejpam-4894	31	7	scenario	scenario	NOUN
ejpam-4894	31	8	,	,	PUNCT
ejpam-4894	31	9	denoted	denote	VERB
ejpam-4894	31	10	as	as	ADP
ejpam-4894	31	11	g	g	NOUN
ejpam-4894	31	12	,	,	PUNCT
ejpam-4894	31	13	where	where	SCONJ
ejpam-4894	31	14	we	we	PRON
ejpam-4894	31	15	are	be	AUX
ejpam-4894	31	16	given	give	VERB
ejpam-4894	31	17	a	a	DET
ejpam-4894	31	18	set	set	NOUN
ejpam-4894	31	19	of	of	ADP
ejpam-4894	31	20	officials	official	NOUN
ejpam-4894	31	21	,	,	PUNCT
ejpam-4894	31	22	denoted	denote	VERB
ejpam-4894	31	23	as	as	ADP
ejpam-4894	31	24	s	s	NOUN
ejpam-4894	31	25	⊆	⊆	NUM
ejpam-4894	31	26	v	v	NOUN
ejpam-4894	31	27	(	(	PUNCT
ejpam-4894	31	28	g	g	NOUN
ejpam-4894	31	29	)	)	PUNCT
ejpam-4894	31	30	,	,	PUNCT
ejpam-4894	31	31	and	and	CCONJ
ejpam-4894	31	32	a	a	DET
ejpam-4894	31	33	set	set	NOUN
ejpam-4894	31	34	of	of	ADP
ejpam-4894	31	35	civilians	civilian	NOUN
ejpam-4894	31	36	,	,	PUNCT
ejpam-4894	31	37	denoted	denote	VERB
ejpam-4894	31	38	as	as	ADP
ejpam-4894	31	39	h	h	NOUN
ejpam-4894	31	40	=	=	PROPN
ejpam-4894	31	41	v	v	X
ejpam-4894	31	42	(	(	PUNCT
ejpam-4894	31	43	g	g	NOUN
ejpam-4894	31	44	)	)	PUNCT
ejpam-4894	31	45	\	\	NOUN
ejpam-4894	32	1	s.	s.	PROPN
ejpam-4894	32	2	for	for	ADP
ejpam-4894	32	3	each	each	DET
ejpam-4894	32	4	civilian	civilian	NOUN
ejpam-4894	32	5	,	,	PUNCT
ejpam-4894	32	6	represented	represent	VERB
ejpam-4894	32	7	by	by	ADP
ejpam-4894	32	8	x	x	PROPN
ejpam-4894	32	9	∈	∈	PROPN
ejpam-4894	32	10	h	h	NOUN
ejpam-4894	32	11	,	,	PUNCT
ejpam-4894	32	12	it	it	PRON
ejpam-4894	32	13	is	be	AUX
ejpam-4894	32	14	necessary	necessary	ADJ
ejpam-4894	32	15	to	to	PART
ejpam-4894	32	16	have	have	VERB
ejpam-4894	32	17	precisely	precisely	ADV
ejpam-4894	32	18	one	one	NUM
ejpam-4894	32	19	official	official	NOUN
ejpam-4894	32	20	,	,	PUNCT
ejpam-4894	32	21	denoted	denote	VERB
ejpam-4894	32	22	as	as	ADP
ejpam-4894	32	23	u	u	NOUN
ejpam-4894	32	24	∈	∈	PROPN
ejpam-4894	32	25	s	s	PROPN
ejpam-4894	32	26	,	,	PUNCT
ejpam-4894	32	27	who	who	PRON
ejpam-4894	32	28	can	can	AUX
ejpam-4894	32	29	serve	serve	VERB
ejpam-4894	32	30	that	that	DET
ejpam-4894	32	31	civilian	civilian	NOUN
ejpam-4894	32	32	.	.	PUNCT
ejpam-4894	33	1	furthermore	furthermore	ADV
ejpam-4894	33	2	,	,	PUNCT
ejpam-4894	33	3	whenever	whenever	SCONJ
ejpam-4894	33	4	such	such	DET
ejpam-4894	33	5	an	an	DET
ejpam-4894	33	6	official	official	ADJ
ejpam-4894	33	7	u	u	NOUN
ejpam-4894	33	8	serves	serve	VERB
ejpam-4894	33	9	a	a	DET
ejpam-4894	33	10	civilian	civilian	ADJ
ejpam-4894	33	11	x	x	NOUN
ejpam-4894	33	12	,	,	PUNCT
ejpam-4894	33	13	there	there	PRON
ejpam-4894	33	14	must	must	AUX
ejpam-4894	33	15	exist	exist	VERB
ejpam-4894	33	16	another	another	DET
ejpam-4894	33	17	civilian	civilian	NOUN
ejpam-4894	33	18	,	,	PUNCT
ejpam-4894	33	19	denoted	denote	VERB
ejpam-4894	33	20	as	as	ADP
ejpam-4894	33	21	y	y	PROPN
ejpam-4894	33	22	∈	∈	PROPN
ejpam-4894	33	23	h	h	NOUN
ejpam-4894	33	24	,	,	PUNCT
ejpam-4894	33	25	who	who	PRON
ejpam-4894	33	26	only	only	ADV
ejpam-4894	33	27	observes	observe	VERB
ejpam-4894	33	28	the	the	DET
ejpam-4894	33	29	service	service	NOUN
ejpam-4894	33	30	provided	provide	VERB
ejpam-4894	33	31	by	by	ADP
ejpam-4894	33	32	the	the	DET
ejpam-4894	33	33	official	official	ADJ
ejpam-4894	33	34	u	u	NOUN
ejpam-4894	33	35	to	to	ADP
ejpam-4894	33	36	civilian	civilian	VERB
ejpam-4894	33	37	x.	x.	NOUN
ejpam-4894	33	38	in	in	ADP
ejpam-4894	33	39	other	other	ADJ
ejpam-4894	33	40	words	word	NOUN
ejpam-4894	33	41	,	,	PUNCT
ejpam-4894	33	42	y	y	PROPN
ejpam-4894	33	43	acts	act	VERB
ejpam-4894	33	44	as	as	ADP
ejpam-4894	33	45	a	a	DET
ejpam-4894	33	46	witness	witness	NOUN
ejpam-4894	33	47	,	,	PUNCT
ejpam-4894	33	48	ensuring	ensure	VERB
ejpam-4894	33	49	that	that	SCONJ
ejpam-4894	33	50	there	there	PRON
ejpam-4894	33	51	is	be	VERB
ejpam-4894	33	52	no	no	DET
ejpam-4894	33	53	abuse	abuse	NOUN
ejpam-4894	33	54	or	or	CCONJ
ejpam-4894	33	55	misconduct	misconduct	NOUN
ejpam-4894	33	56	from	from	ADP
ejpam-4894	33	57	official	official	ADJ
ejpam-4894	33	58	u.	u.	NOUN
ejpam-4894	34	1	the	the	DET
ejpam-4894	34	2	question	question	NOUN
ejpam-4894	34	3	arises	arise	VERB
ejpam-4894	34	4	:	:	PUNCT
ejpam-4894	34	5	what	what	PRON
ejpam-4894	34	6	is	be	AUX
ejpam-4894	34	7	the	the	DET
ejpam-4894	34	8	minimum	minimum	ADJ
ejpam-4894	34	9	number	number	NOUN
ejpam-4894	34	10	of	of	ADP
ejpam-4894	34	11	officials	official	NOUN
ejpam-4894	34	12	required	require	VERB
ejpam-4894	34	13	to	to	PART
ejpam-4894	34	14	guarantee	guarantee	VERB
ejpam-4894	34	15	such	such	DET
ejpam-4894	34	16	a	a	DET
ejpam-4894	34	17	service	service	NOUN
ejpam-4894	34	18	,	,	PUNCT
ejpam-4894	34	19	considering	consider	VERB
ejpam-4894	34	20	a	a	DET
ejpam-4894	34	21	given	give	VERB
ejpam-4894	34	22	social	social	ADJ
ejpam-4894	34	23	network	network	NOUN
ejpam-4894	34	24	?	?	PUNCT
ejpam-4894	35	1	this	this	DET
ejpam-4894	35	2	problem	problem	NOUN
ejpam-4894	35	3	leads	lead	VERB
ejpam-4894	35	4	us	we	PRON
ejpam-4894	35	5	to	to	PART
ejpam-4894	35	6	introduce	introduce	VERB
ejpam-4894	35	7	the	the	DET
ejpam-4894	35	8	concept	concept	NOUN
ejpam-4894	35	9	of	of	ADP
ejpam-4894	35	10	a	a	DET
ejpam-4894	35	11	certified	certify	VERB
ejpam-4894	35	12	perfect	perfect	ADJ
ejpam-4894	35	13	dominating	dominating	NOUN
ejpam-4894	35	14	set	set	NOUN
ejpam-4894	35	15	of	of	ADP
ejpam-4894	35	16	a	a	DET
ejpam-4894	35	17	graph	graph	NOUN
ejpam-4894	35	18	g.	g.	NOUN
ejpam-4894	35	19	2	2	NUM
ejpam-4894	35	20	.	.	PUNCT
ejpam-4894	35	21	terminology	terminology	NOUN
ejpam-4894	35	22	and	and	CCONJ
ejpam-4894	35	23	notation	notation	NOUN
ejpam-4894	35	24	this	this	DET
ejpam-4894	35	25	section	section	NOUN
ejpam-4894	35	26	comprises	comprise	VERB
ejpam-4894	35	27	essential	essential	ADJ
ejpam-4894	35	28	definitions	definition	NOUN
ejpam-4894	35	29	required	require	VERB
ejpam-4894	35	30	for	for	ADP
ejpam-4894	35	31	the	the	DET
ejpam-4894	35	32	study	study	NOUN
ejpam-4894	35	33	.	.	PUNCT
ejpam-4894	36	1	let	let	VERB
ejpam-4894	36	2	g	g	PROPN
ejpam-4894	36	3	=	=	SYM
ejpam-4894	36	4	(	(	PUNCT
ejpam-4894	36	5	v	v	NOUN
ejpam-4894	36	6	,	,	PUNCT
ejpam-4894	36	7	e	e	NOUN
ejpam-4894	36	8	)	)	PUNCT
ejpam-4894	36	9	,	,	PUNCT
ejpam-4894	36	10	where	where	SCONJ
ejpam-4894	36	11	v	v	NOUN
ejpam-4894	36	12	represents	represent	VERB
ejpam-4894	36	13	the	the	DET
ejpam-4894	36	14	vertex	vertex	NOUN
ejpam-4894	36	15	set	set	NOUN
ejpam-4894	36	16	of	of	ADP
ejpam-4894	36	17	g	g	PROPN
ejpam-4894	36	18	and	and	CCONJ
ejpam-4894	36	19	e	e	NOUN
ejpam-4894	36	20	represents	represent	VERB
ejpam-4894	36	21	the	the	DET
ejpam-4894	36	22	edge	edge	NOUN
ejpam-4894	36	23	set	set	NOUN
ejpam-4894	36	24	of	of	ADP
ejpam-4894	36	25	g.	g.	PROPN
ejpam-4894	36	26	the	the	DET
ejpam-4894	36	27	elements	element	NOUN
ejpam-4894	36	28	of	of	ADP
ejpam-4894	36	29	v	v	NOUN
ejpam-4894	36	30	(	(	PUNCT
ejpam-4894	36	31	g	g	NOUN
ejpam-4894	36	32	)	)	PUNCT
ejpam-4894	36	33	are	be	AUX
ejpam-4894	36	34	called	call	VERB
ejpam-4894	36	35	vertices	vertex	NOUN
ejpam-4894	36	36	and	and	CCONJ
ejpam-4894	36	37	the	the	DET
ejpam-4894	36	38	cardinality	cardinality	NOUN
ejpam-4894	36	39	|v	|v	PROPN
ejpam-4894	36	40	(	(	PUNCT
ejpam-4894	36	41	g)|	g)|	NOUN
ejpam-4894	36	42	of	of	ADP
ejpam-4894	36	43	v	v	NOUN
ejpam-4894	36	44	is	be	AUX
ejpam-4894	36	45	the	the	DET
ejpam-4894	36	46	order	order	NOUN
ejpam-4894	36	47	of	of	ADP
ejpam-4894	36	48	g.	g.	PROPN
ejpam-4894	36	49	the	the	DET
ejpam-4894	36	50	elements	element	NOUN
ejpam-4894	36	51	of	of	ADP
ejpam-4894	36	52	e(g	e(g	PROPN
ejpam-4894	36	53	)	)	PUNCT
ejpam-4894	36	54	are	be	AUX
ejpam-4894	36	55	called	call	VERB
ejpam-4894	36	56	edges	edge	NOUN
ejpam-4894	36	57	and	and	CCONJ
ejpam-4894	36	58	the	the	DET
ejpam-4894	36	59	cardinality	cardinality	NOUN
ejpam-4894	36	60	|e(g)|	|e(g)|	PROPN
ejpam-4894	36	61	of	of	ADP
ejpam-4894	36	62	e	e	PROPN
ejpam-4894	36	63	is	be	AUX
ejpam-4894	36	64	the	the	DET
ejpam-4894	36	65	size	size	NOUN
ejpam-4894	36	66	of	of	ADP
ejpam-4894	36	67	g.	g.	PROPN
ejpam-4894	36	68	the	the	DET
ejpam-4894	36	69	degree	degree	NOUN
ejpam-4894	36	70	of	of	ADP
ejpam-4894	36	71	a	a	DET
ejpam-4894	36	72	vertex	vertex	NOUN
ejpam-4894	36	73	v	v	NOUN
ejpam-4894	36	74	,	,	PUNCT
ejpam-4894	36	75	denoted	denote	VERB
ejpam-4894	36	76	as	as	ADP
ejpam-4894	36	77	deg(v	deg(v	PROPN
ejpam-4894	36	78	)	)	PUNCT
ejpam-4894	36	79	,	,	PUNCT
ejpam-4894	36	80	refers	refer	VERB
ejpam-4894	36	81	to	to	ADP
ejpam-4894	36	82	the	the	DET
ejpam-4894	36	83	number	number	NOUN
ejpam-4894	36	84	of	of	ADP
ejpam-4894	36	85	edges	edge	NOUN
ejpam-4894	36	86	incident	incident	NOUN
ejpam-4894	36	87	with	with	ADP
ejpam-4894	36	88	v.	v.	ADP
ejpam-4894	36	89	the	the	DET
ejpam-4894	36	90	maximum	maximum	ADJ
ejpam-4894	36	91	degree	degree	NOUN
ejpam-4894	36	92	among	among	ADP
ejpam-4894	36	93	all	all	DET
ejpam-4894	36	94	vertices	vertex	NOUN
ejpam-4894	36	95	in	in	ADP
ejpam-4894	36	96	g	g	PROPN
ejpam-4894	36	97	is	be	AUX
ejpam-4894	36	98	denoted	denote	VERB
ejpam-4894	36	99	as	as	ADP
ejpam-4894	36	100	∆(g	∆(g	NOUN
ejpam-4894	36	101	)	)	PUNCT
ejpam-4894	36	102	.	.	PUNCT
ejpam-4894	37	1	the	the	DET
ejpam-4894	37	2	open	open	ADJ
ejpam-4894	37	3	neighborhood	neighborhood	NOUN
ejpam-4894	37	4	of	of	ADP
ejpam-4894	37	5	a	a	DET
ejpam-4894	37	6	vertex	vertex	NOUN
ejpam-4894	37	7	u	u	NOUN
ejpam-4894	37	8	in	in	ADP
ejpam-4894	37	9	g	g	PROPN
ejpam-4894	37	10	is	be	AUX
ejpam-4894	37	11	the	the	DET
ejpam-4894	37	12	set	set	NOUN
ejpam-4894	37	13	of	of	ADP
ejpam-4894	37	14	its	its	PRON
ejpam-4894	37	15	neighboring	neighboring	NOUN
ejpam-4894	37	16	vertices	vertex	NOUN
ejpam-4894	37	17	and	and	CCONJ
ejpam-4894	37	18	is	be	AUX
ejpam-4894	37	19	denoted	denote	VERB
ejpam-4894	37	20	as	as	ADP
ejpam-4894	37	21	ng(u	ng(u	NOUN
ejpam-4894	37	22	)	)	PUNCT
ejpam-4894	37	23	=	=	PRON
ejpam-4894	37	24	{	{	PUNCT
ejpam-4894	37	25	v	v	NUM
ejpam-4894	37	26	∈	∈	NOUN
ejpam-4894	37	27	v	v	NOUN
ejpam-4894	37	28	(	(	PUNCT
ejpam-4894	37	29	g	g	NOUN
ejpam-4894	37	30	)	)	PUNCT
ejpam-4894	37	31	:	:	PUNCT
ejpam-4894	37	32	uv	uv	PROPN
ejpam-4894	37	33	∈	∈	PROPN
ejpam-4894	37	34	e(g	e(g	PROPN
ejpam-4894	37	35	)	)	PUNCT
ejpam-4894	37	36	}	}	PUNCT
ejpam-4894	37	37	.	.	PUNCT
ejpam-4894	38	1	the	the	DET
ejpam-4894	38	2	closed	closed	ADJ
ejpam-4894	38	3	neighborhood	neighborhood	NOUN
ejpam-4894	38	4	of	of	ADP
ejpam-4894	38	5	u	u	NOUN
ejpam-4894	38	6	in	in	ADP
ejpam-4894	38	7	g	g	PROPN
ejpam-4894	38	8	is	be	AUX
ejpam-4894	38	9	the	the	DET
ejpam-4894	38	10	open	open	ADJ
ejpam-4894	38	11	neighborhood	neighborhood	NOUN
ejpam-4894	38	12	of	of	ADP
ejpam-4894	38	13	u	u	NOUN
ejpam-4894	38	14	along	along	ADP
ejpam-4894	38	15	with	with	ADP
ejpam-4894	38	16	the	the	DET
ejpam-4894	38	17	vertex	vertex	NOUN
ejpam-4894	38	18	u	u	NOUN
ejpam-4894	38	19	itself	itself	PRON
ejpam-4894	38	20	,	,	PUNCT
ejpam-4894	38	21	expressed	express	VERB
ejpam-4894	38	22	as	as	ADP
ejpam-4894	38	23	ng[u	ng[u	PROPN
ejpam-4894	38	24	]	]	X
ejpam-4894	38	25	=	=	SYM
ejpam-4894	38	26	ng(u	ng(u	PROPN
ejpam-4894	38	27	)	)	PUNCT
ejpam-4894	38	28	∪	∪	NOUN
ejpam-4894	38	29	{	{	PUNCT
ejpam-4894	38	30	u	u	NOUN
ejpam-4894	38	31	}	}	PUNCT
ejpam-4894	38	32	.	.	PUNCT
ejpam-4894	39	1	similarly	similarly	ADV
ejpam-4894	39	2	,	,	PUNCT
ejpam-4894	39	3	the	the	DET
ejpam-4894	39	4	closed	closed	ADJ
ejpam-4894	39	5	neighborhood	neighborhood	NOUN
ejpam-4894	39	6	of	of	ADP
ejpam-4894	39	7	a	a	DET
ejpam-4894	39	8	subset	subset	NOUN
ejpam-4894	39	9	s	s	NOUN
ejpam-4894	39	10	of	of	ADP
ejpam-4894	39	11	v	v	NOUN
ejpam-4894	39	12	(	(	PUNCT
ejpam-4894	39	13	g	g	NOUN
ejpam-4894	39	14	)	)	PUNCT
ejpam-4894	39	15	,	,	PUNCT
ejpam-4894	39	16	denoted	denote	VERB
ejpam-4894	39	17	as	as	ADP
ejpam-4894	39	18	ng[s	ng[	NOUN
ejpam-4894	39	19	]	]	PUNCT
ejpam-4894	39	20	=	=	SYM
ejpam-4894	39	21	∪v∈sng[v	∪v∈sng[v	NOUN
ejpam-4894	39	22	]	]	PUNCT
ejpam-4894	39	23	,	,	PUNCT
ejpam-4894	39	24	represents	represent	VERB
ejpam-4894	39	25	the	the	DET
ejpam-4894	39	26	set	set	NOUN
ejpam-4894	39	27	of	of	ADP
ejpam-4894	39	28	vertices	vertex	NOUN
ejpam-4894	39	29	in	in	ADP
ejpam-4894	39	30	g	g	PROPN
ejpam-4894	39	31	that	that	PRON
ejpam-4894	39	32	are	be	AUX
ejpam-4894	39	33	either	either	CCONJ
ejpam-4894	39	34	in	in	ADP
ejpam-4894	39	35	s	s	PRON
ejpam-4894	39	36	or	or	CCONJ
ejpam-4894	39	37	are	be	AUX
ejpam-4894	39	38	adjacent	adjacent	ADJ
ejpam-4894	39	39	to	to	ADP
ejpam-4894	39	40	a	a	DET
ejpam-4894	39	41	vertex	vertex	NOUN
ejpam-4894	39	42	in	in	ADP
ejpam-4894	39	43	s.	s.	PROPN
ejpam-4894	39	44	the	the	DET
ejpam-4894	39	45	join	join	NOUN
ejpam-4894	39	46	of	of	ADP
ejpam-4894	39	47	two	two	NUM
ejpam-4894	39	48	graphs	graph	NOUN
ejpam-4894	39	49	g	g	NOUN
ejpam-4894	39	50	and	and	CCONJ
ejpam-4894	39	51	h	h	NOUN
ejpam-4894	39	52	,	,	PUNCT
ejpam-4894	39	53	denoted	denote	VERB
ejpam-4894	39	54	by	by	ADP
ejpam-4894	39	55	g	g	PROPN
ejpam-4894	39	56	+	+	PROPN
ejpam-4894	39	57	h	h	NOUN
ejpam-4894	39	58	,	,	PUNCT
ejpam-4894	39	59	is	be	AUX
ejpam-4894	39	60	the	the	DET
ejpam-4894	39	61	graph	graph	NOUN
ejpam-4894	39	62	with	with	ADP
ejpam-4894	39	63	v	v	NOUN
ejpam-4894	39	64	(	(	PUNCT
ejpam-4894	39	65	g+h	g+h	NOUN
ejpam-4894	39	66	)	)	PUNCT
ejpam-4894	40	1	=	=	SYM
ejpam-4894	40	2	v	v	X
ejpam-4894	40	3	(	(	PUNCT
ejpam-4894	40	4	g	g	NOUN
ejpam-4894	40	5	)	)	PUNCT
ejpam-4894	40	6	∪	∪	NOUN
ejpam-4894	40	7	v	v	NOUN
ejpam-4894	40	8	(	(	PUNCT
ejpam-4894	40	9	h	h	NOUN
ejpam-4894	40	10	)	)	PUNCT
ejpam-4894	40	11	and	and	CCONJ
ejpam-4894	40	12	e(g+h	e(g+h	NUM
ejpam-4894	40	13	)	)	PUNCT
ejpam-4894	40	14	=	=	SYM
ejpam-4894	40	15	e(g	e(g	PROPN
ejpam-4894	40	16	)	)	PUNCT
ejpam-4894	40	17	∪e(h	∪e(h	PROPN
ejpam-4894	40	18	)	)	PUNCT
ejpam-4894	40	19	∪	∪	NOUN
ejpam-4894	40	20	{	{	PUNCT
ejpam-4894	40	21	uv	uv	NOUN
ejpam-4894	40	22	:	:	PUNCT
ejpam-4894	40	23	u	u	PROPN
ejpam-4894	40	24	∈	∈	PROPN
ejpam-4894	40	25	v	v	ADP
ejpam-4894	40	26	(	(	PUNCT
ejpam-4894	40	27	g	g	NOUN
ejpam-4894	40	28	)	)	PUNCT
ejpam-4894	40	29	,	,	PUNCT
ejpam-4894	40	30	v	v	X
ejpam-4894	40	31	∈	∈	PROPN
ejpam-4894	40	32	v	v	NOUN
ejpam-4894	40	33	(	(	PUNCT
ejpam-4894	40	34	h	h	NOUN
ejpam-4894	40	35	)	)	PUNCT
ejpam-4894	40	36	}	}	PUNCT
ejpam-4894	40	37	.	.	PUNCT
ejpam-4894	41	1	the	the	DET
ejpam-4894	41	2	corona	corona	NOUN
ejpam-4894	41	3	of	of	ADP
ejpam-4894	41	4	graphs	graph	NOUN
ejpam-4894	41	5	g	g	PROPN
ejpam-4894	41	6	and	and	CCONJ
ejpam-4894	41	7	h	h	NOUN
ejpam-4894	41	8	,	,	PUNCT
ejpam-4894	41	9	g	g	PROPN
ejpam-4894	41	10	◦	◦	NOUN
ejpam-4894	41	11	h	h	NOUN
ejpam-4894	41	12	,	,	PUNCT
ejpam-4894	41	13	is	be	AUX
ejpam-4894	41	14	the	the	DET
ejpam-4894	41	15	graph	graph	NOUN
ejpam-4894	41	16	obtained	obtain	VERB
ejpam-4894	41	17	by	by	ADP
ejpam-4894	41	18	taking	take	VERB
ejpam-4894	41	19	one	one	NUM
ejpam-4894	41	20	copy	copy	NOUN
ejpam-4894	41	21	of	of	ADP
ejpam-4894	41	22	g	g	PROPN
ejpam-4894	41	23	and	and	CCONJ
ejpam-4894	41	24	|v	|v	PROPN
ejpam-4894	41	25	(	(	PUNCT
ejpam-4894	41	26	g)|	g)|	NOUN
ejpam-4894	41	27	copies	copy	NOUN
ejpam-4894	41	28	of	of	ADP
ejpam-4894	41	29	h	h	NOUN
ejpam-4894	41	30	,	,	PUNCT
ejpam-4894	41	31	and	and	CCONJ
ejpam-4894	41	32	then	then	ADV
ejpam-4894	41	33	joining	join	VERB
ejpam-4894	41	34	the	the	DET
ejpam-4894	41	35	ith	ith	PROPN
ejpam-4894	41	36	vertex	vertex	NOUN
ejpam-4894	41	37	of	of	ADP
ejpam-4894	41	38	g	g	NOUN
ejpam-4894	41	39	to	to	ADP
ejpam-4894	41	40	every	every	DET
ejpam-4894	41	41	vertex	vertex	NOUN
ejpam-4894	41	42	of	of	ADP
ejpam-4894	41	43	the	the	DET
ejpam-4894	41	44	ith	ith	PROPN
ejpam-4894	41	45	copy	copy	NOUN
ejpam-4894	41	46	of	of	ADP
ejpam-4894	41	47	h.	h.	PROPN
ejpam-4894	41	48	for	for	ADP
ejpam-4894	41	49	every	every	DET
ejpam-4894	41	50	v	v	NUM
ejpam-4894	41	51	∈	∈	PROPN
ejpam-4894	41	52	v	v	NOUN
ejpam-4894	41	53	(	(	PUNCT
ejpam-4894	41	54	g	g	NOUN
ejpam-4894	41	55	)	)	PUNCT
ejpam-4894	41	56	,	,	PUNCT
ejpam-4894	41	57	denote	denote	VERB
ejpam-4894	41	58	by	by	ADP
ejpam-4894	41	59	hv	hv	PROPN
ejpam-4894	41	60	the	the	DET
ejpam-4894	41	61	copy	copy	NOUN
ejpam-4894	41	62	of	of	ADP
ejpam-4894	41	63	h	h	NOUN
ejpam-4894	41	64	whose	whose	DET
ejpam-4894	41	65	vertices	vertex	NOUN
ejpam-4894	41	66	are	be	AUX
ejpam-4894	41	67	attached	attach	VERB
ejpam-4894	41	68	one	one	NUM
ejpam-4894	41	69	by	by	ADP
ejpam-4894	41	70	one	one	NUM
ejpam-4894	41	71	to	to	ADP
ejpam-4894	41	72	the	the	DET
ejpam-4894	41	73	vertex	vertex	NOUN
ejpam-4894	41	74	v.	v.	ADP
ejpam-4894	41	75	subsequently	subsequently	ADV
ejpam-4894	41	76	,	,	PUNCT
ejpam-4894	41	77	denote	denote	VERB
ejpam-4894	41	78	by	by	ADP
ejpam-4894	41	79	v+hv	v+hv	NOUN
ejpam-4894	41	80	the	the	DET
ejpam-4894	41	81	subgraph	subgraph	NOUN
ejpam-4894	41	82	of	of	ADP
ejpam-4894	41	83	the	the	DET
ejpam-4894	41	84	corona	corona	NOUN
ejpam-4894	41	85	g	g	PROPN
ejpam-4894	41	86	◦	◦	PROPN
ejpam-4894	41	87	h	h	PROPN
ejpam-4894	41	88	j.	j.	PROPN
ejpam-4894	41	89	hamja	hamja	PROPN
ejpam-4894	41	90	/	/	SYM
ejpam-4894	41	91	eur	eur	PROPN
ejpam-4894	41	92	.	.	PUNCT
ejpam-4894	42	1	j.	j.	PROPN
ejpam-4894	42	2	pure	pure	PROPN
ejpam-4894	42	3	appl	appl	PROPN
ejpam-4894	42	4	.	.	PROPN
ejpam-4894	42	5	math	math	PROPN
ejpam-4894	42	6	,	,	PUNCT
ejpam-4894	42	7	16	16	NUM
ejpam-4894	42	8	(	(	PUNCT
ejpam-4894	42	9	4	4	NUM
ejpam-4894	42	10	)	)	PUNCT
ejpam-4894	42	11	(	(	PUNCT
ejpam-4894	42	12	2023	2023	NUM
ejpam-4894	42	13	)	)	PUNCT
ejpam-4894	42	14	,	,	PUNCT
ejpam-4894	42	15	2763	2763	NUM
ejpam-4894	42	16	-	-	SYM
ejpam-4894	42	17	2774	2774	NUM
ejpam-4894	42	18	2765	2765	NUM
ejpam-4894	42	19	corresponding	correspond	VERB
ejpam-4894	42	20	to	to	ADP
ejpam-4894	42	21	the	the	DET
ejpam-4894	42	22	join	join	NOUN
ejpam-4894	42	23	⟨{v}⟩+hv	⟨{v}⟩+hv	PROPN
ejpam-4894	42	24	,	,	PUNCT
ejpam-4894	42	25	v	v	PROPN
ejpam-4894	42	26	∈	∈	PROPN
ejpam-4894	42	27	v	v	NOUN
ejpam-4894	42	28	(	(	PUNCT
ejpam-4894	42	29	g	g	NOUN
ejpam-4894	42	30	)	)	PUNCT
ejpam-4894	43	1	[	[	X
ejpam-4894	43	2	5	5	NUM
ejpam-4894	43	3	]	]	PUNCT
ejpam-4894	43	4	.	.	PUNCT
ejpam-4894	44	1	a	a	DET
ejpam-4894	44	2	set	set	NOUN
ejpam-4894	44	3	s	s	NOUN
ejpam-4894	44	4	⊆	⊆	NUM
ejpam-4894	44	5	v	v	NOUN
ejpam-4894	44	6	(	(	PUNCT
ejpam-4894	44	7	g	g	NOUN
ejpam-4894	44	8	)	)	PUNCT
ejpam-4894	44	9	is	be	AUX
ejpam-4894	44	10	called	call	VERB
ejpam-4894	44	11	dominating	dominating	NOUN
ejpam-4894	44	12	set	set	NOUN
ejpam-4894	44	13	if	if	SCONJ
ejpam-4894	44	14	ng[s	ng[	NOUN
ejpam-4894	44	15	]	]	PUNCT
ejpam-4894	44	16	=	=	SYM
ejpam-4894	44	17	v	v	NOUN
ejpam-4894	44	18	(	(	PUNCT
ejpam-4894	44	19	g	g	NOUN
ejpam-4894	44	20	)	)	PUNCT
ejpam-4894	44	21	.	.	PUNCT
ejpam-4894	45	1	a	a	DET
ejpam-4894	45	2	dominating	dominating	NOUN
ejpam-4894	45	3	set	set	NOUN
ejpam-4894	45	4	s	s	VERB
ejpam-4894	45	5	is	be	AUX
ejpam-4894	45	6	a	a	DET
ejpam-4894	45	7	minimal	minimal	ADJ
ejpam-4894	45	8	dominating	dominating	NOUN
ejpam-4894	45	9	set	set	NOUN
ejpam-4894	45	10	if	if	SCONJ
ejpam-4894	45	11	no	no	DET
ejpam-4894	45	12	proper	proper	ADJ
ejpam-4894	45	13	subset	subset	NOUN
ejpam-4894	45	14	s′	s′	VERB
ejpam-4894	45	15	⊂	⊂	ADJ
ejpam-4894	45	16	s	s	X
ejpam-4894	45	17	is	be	AUX
ejpam-4894	45	18	a	a	DET
ejpam-4894	45	19	dominating	dominating	NOUN
ejpam-4894	45	20	set	set	NOUN
ejpam-4894	45	21	.	.	PUNCT
ejpam-4894	46	1	a	a	DET
ejpam-4894	46	2	minimum	minimum	ADJ
ejpam-4894	46	3	cardinality	cardinality	NOUN
ejpam-4894	46	4	of	of	ADP
ejpam-4894	46	5	a	a	DET
ejpam-4894	46	6	dominating	dominating	NOUN
ejpam-4894	46	7	set	set	NOUN
ejpam-4894	46	8	of	of	ADP
ejpam-4894	46	9	g	g	PROPN
ejpam-4894	46	10	is	be	AUX
ejpam-4894	46	11	called	call	VERB
ejpam-4894	46	12	domination	domination	NOUN
ejpam-4894	46	13	number	number	NOUN
ejpam-4894	46	14	of	of	ADP
ejpam-4894	46	15	g	g	NOUN
ejpam-4894	46	16	,	,	PUNCT
ejpam-4894	46	17	and	and	CCONJ
ejpam-4894	46	18	is	be	AUX
ejpam-4894	46	19	denoted	denote	VERB
ejpam-4894	46	20	by	by	ADP
ejpam-4894	46	21	γ(g	γ(g	PROPN
ejpam-4894	46	22	)	)	PUNCT
ejpam-4894	46	23	.	.	PUNCT
ejpam-4894	47	1	a	a	DET
ejpam-4894	47	2	dominating	dominating	NOUN
ejpam-4894	47	3	set	set	NOUN
ejpam-4894	47	4	s	s	NOUN
ejpam-4894	47	5	with	with	ADP
ejpam-4894	47	6	|s|	|s|	PROPN
ejpam-4894	47	7	=	=	SYM
ejpam-4894	47	8	γ(g	γ(g	PROPN
ejpam-4894	47	9	)	)	PUNCT
ejpam-4894	47	10	is	be	AUX
ejpam-4894	47	11	called	call	VERB
ejpam-4894	47	12	a	a	DET
ejpam-4894	47	13	γ	γ	NOUN
ejpam-4894	47	14	-	-	PUNCT
ejpam-4894	47	15	set	set	NOUN
ejpam-4894	47	16	.	.	PUNCT
ejpam-4894	48	1	a	a	DET
ejpam-4894	48	2	dominating	dominating	NOUN
ejpam-4894	48	3	set	set	NOUN
ejpam-4894	48	4	s	s	PROPN
ejpam-4894	48	5	⊆	⊆	NUM
ejpam-4894	48	6	v	v	NOUN
ejpam-4894	48	7	(	(	PUNCT
ejpam-4894	48	8	g	g	NOUN
ejpam-4894	48	9	)	)	PUNCT
ejpam-4894	48	10	is	be	AUX
ejpam-4894	48	11	called	call	VERB
ejpam-4894	48	12	certified	certified	ADJ
ejpam-4894	48	13	dominating	dominating	NOUN
ejpam-4894	48	14	set	set	NOUN
ejpam-4894	48	15	of	of	ADP
ejpam-4894	48	16	g	g	PROPN
ejpam-4894	49	1	if	if	SCONJ
ejpam-4894	49	2	every	every	DET
ejpam-4894	49	3	vertex	vertex	NOUN
ejpam-4894	49	4	v	v	ADP
ejpam-4894	49	5	∈	∈	NOUN
ejpam-4894	49	6	s	s	PART
ejpam-4894	49	7	has	have	VERB
ejpam-4894	49	8	either	either	CCONJ
ejpam-4894	49	9	zero	zero	NUM
ejpam-4894	49	10	or	or	CCONJ
ejpam-4894	49	11	at	at	ADP
ejpam-4894	49	12	least	least	ADV
ejpam-4894	49	13	two	two	NUM
ejpam-4894	49	14	neighbors	neighbor	NOUN
ejpam-4894	49	15	in	in	ADP
ejpam-4894	49	16	v	v	NOUN
ejpam-4894	49	17	(	(	PUNCT
ejpam-4894	49	18	g	g	NOUN
ejpam-4894	49	19	)	)	PUNCT
ejpam-4894	49	20	\s	\s	NOUN
ejpam-4894	49	21	.	.	PUNCT
ejpam-4894	50	1	a	a	DET
ejpam-4894	50	2	minimum	minimum	ADJ
ejpam-4894	50	3	cardinality	cardinality	NOUN
ejpam-4894	50	4	of	of	ADP
ejpam-4894	50	5	a	a	DET
ejpam-4894	50	6	certified	certify	VERB
ejpam-4894	50	7	dominating	dominating	NOUN
ejpam-4894	50	8	set	set	NOUN
ejpam-4894	50	9	of	of	ADP
ejpam-4894	50	10	g	g	PROPN
ejpam-4894	50	11	is	be	AUX
ejpam-4894	50	12	called	call	VERB
ejpam-4894	50	13	certified	certified	ADJ
ejpam-4894	50	14	domination	domination	NOUN
ejpam-4894	50	15	number	number	NOUN
ejpam-4894	50	16	of	of	ADP
ejpam-4894	50	17	g	g	NOUN
ejpam-4894	50	18	and	and	CCONJ
ejpam-4894	50	19	denoted	denote	VERB
ejpam-4894	50	20	by	by	ADP
ejpam-4894	50	21	γcer(g	γcer(g	PROPN
ejpam-4894	50	22	)	)	PUNCT
ejpam-4894	50	23	.	.	PUNCT
ejpam-4894	51	1	a	a	DET
ejpam-4894	51	2	certified	certify	VERB
ejpam-4894	51	3	dominating	dominating	NOUN
ejpam-4894	51	4	set	set	NOUN
ejpam-4894	51	5	of	of	ADP
ejpam-4894	51	6	s	s	PRON
ejpam-4894	51	7	with	with	ADP
ejpam-4894	51	8	|s|	|s|	PROPN
ejpam-4894	51	9	=	=	PUNCT
ejpam-4894	51	10	γcer(g	γcer(g	PROPN
ejpam-4894	51	11	)	)	PUNCT
ejpam-4894	51	12	is	be	AUX
ejpam-4894	51	13	called	call	VERB
ejpam-4894	51	14	a	a	DET
ejpam-4894	51	15	γcer	γcer	NOUN
ejpam-4894	51	16	-	-	PUNCT
ejpam-4894	51	17	set	set	NOUN
ejpam-4894	51	18	[	[	X
ejpam-4894	51	19	4	4	NUM
ejpam-4894	51	20	]	]	PUNCT
ejpam-4894	51	21	.	.	PUNCT
ejpam-4894	52	1	a	a	DET
ejpam-4894	52	2	set	set	NOUN
ejpam-4894	52	3	s	s	NOUN
ejpam-4894	52	4	⊆	⊆	NUM
ejpam-4894	52	5	v	v	NOUN
ejpam-4894	52	6	(	(	PUNCT
ejpam-4894	52	7	g	g	NOUN
ejpam-4894	52	8	)	)	PUNCT
ejpam-4894	52	9	is	be	AUX
ejpam-4894	52	10	called	call	VERB
ejpam-4894	52	11	perfect	perfect	ADJ
ejpam-4894	52	12	dominating	dominating	NOUN
ejpam-4894	52	13	set	set	NOUN
ejpam-4894	52	14	if	if	SCONJ
ejpam-4894	52	15	every	every	DET
ejpam-4894	52	16	vertex	vertex	NOUN
ejpam-4894	52	17	v	v	ADP
ejpam-4894	52	18	∈	∈	PROPN
ejpam-4894	52	19	v	v	NOUN
ejpam-4894	52	20	(	(	PUNCT
ejpam-4894	52	21	g	g	NOUN
ejpam-4894	52	22	)	)	PUNCT
ejpam-4894	52	23	\	\	PROPN
ejpam-4894	53	1	s	s	PART
ejpam-4894	53	2	is	be	AUX
ejpam-4894	53	3	dominated	dominate	VERB
ejpam-4894	53	4	by	by	ADP
ejpam-4894	53	5	exactly	exactly	ADV
ejpam-4894	53	6	one	one	NUM
ejpam-4894	53	7	element	element	NOUN
ejpam-4894	53	8	in	in	ADP
ejpam-4894	53	9	s.	s.	PROPN
ejpam-4894	53	10	the	the	DET
ejpam-4894	53	11	minimum	minimum	ADJ
ejpam-4894	53	12	cardinality	cardinality	NOUN
ejpam-4894	53	13	of	of	ADP
ejpam-4894	53	14	a	a	DET
ejpam-4894	53	15	perfect	perfect	ADJ
ejpam-4894	53	16	dominating	dominating	NOUN
ejpam-4894	53	17	set	set	NOUN
ejpam-4894	53	18	of	of	ADP
ejpam-4894	53	19	g	g	PROPN
ejpam-4894	53	20	is	be	AUX
ejpam-4894	53	21	called	call	VERB
ejpam-4894	53	22	perfect	perfect	ADJ
ejpam-4894	53	23	domination	domination	NOUN
ejpam-4894	53	24	number	number	NOUN
ejpam-4894	53	25	,	,	PUNCT
ejpam-4894	53	26	and	and	CCONJ
ejpam-4894	53	27	is	be	AUX
ejpam-4894	53	28	denoted	denote	VERB
ejpam-4894	53	29	by	by	ADP
ejpam-4894	53	30	γp(g	γp(g	NOUN
ejpam-4894	53	31	)	)	PUNCT
ejpam-4894	53	32	.	.	PUNCT
ejpam-4894	54	1	a	a	DET
ejpam-4894	54	2	perfect	perfect	ADJ
ejpam-4894	54	3	dominating	dominating	NOUN
ejpam-4894	54	4	set	set	NOUN
ejpam-4894	54	5	s	s	NOUN
ejpam-4894	54	6	with	with	ADP
ejpam-4894	54	7	|s|	|s|	PROPN
ejpam-4894	54	8	=	=	SYM
ejpam-4894	54	9	γp(g	γp(g	X
ejpam-4894	54	10	)	)	PUNCT
ejpam-4894	54	11	is	be	AUX
ejpam-4894	54	12	said	say	VERB
ejpam-4894	54	13	to	to	PART
ejpam-4894	54	14	be	be	AUX
ejpam-4894	54	15	a	a	DET
ejpam-4894	54	16	γp	γp	NOUN
ejpam-4894	54	17	-	-	PUNCT
ejpam-4894	54	18	set	set	NOUN
ejpam-4894	54	19	[	[	X
ejpam-4894	54	20	7	7	NUM
ejpam-4894	54	21	]	]	PUNCT
ejpam-4894	54	22	.	.	PUNCT
ejpam-4894	55	1	a	a	DET
ejpam-4894	55	2	perfect	perfect	ADJ
ejpam-4894	55	3	dominating	dominating	NOUN
ejpam-4894	55	4	set	set	NOUN
ejpam-4894	55	5	s	s	PROPN
ejpam-4894	55	6	⊆	⊆	NUM
ejpam-4894	55	7	v	v	NOUN
ejpam-4894	55	8	(	(	PUNCT
ejpam-4894	55	9	g	g	NOUN
ejpam-4894	55	10	)	)	PUNCT
ejpam-4894	55	11	is	be	AUX
ejpam-4894	55	12	called	call	VERB
ejpam-4894	55	13	certified	certify	VERB
ejpam-4894	55	14	perfect	perfect	ADJ
ejpam-4894	55	15	dominating	dominating	NOUN
ejpam-4894	55	16	set	set	NOUN
ejpam-4894	55	17	of	of	ADP
ejpam-4894	55	18	g	g	PROPN
ejpam-4894	55	19	if	if	SCONJ
ejpam-4894	55	20	every	every	DET
ejpam-4894	55	21	u	u	NOUN
ejpam-4894	55	22	∈	∈	X
ejpam-4894	55	23	s	s	AUX
ejpam-4894	55	24	has	have	VERB
ejpam-4894	55	25	either	either	CCONJ
ejpam-4894	55	26	zero	zero	NUM
ejpam-4894	55	27	or	or	CCONJ
ejpam-4894	55	28	at	at	ADP
ejpam-4894	55	29	least	least	ADV
ejpam-4894	55	30	two	two	NUM
ejpam-4894	55	31	neighbors	neighbor	NOUN
ejpam-4894	55	32	in	in	ADP
ejpam-4894	55	33	v	v	NOUN
ejpam-4894	55	34	(	(	PUNCT
ejpam-4894	55	35	g	g	NOUN
ejpam-4894	55	36	)	)	PUNCT
ejpam-4894	55	37	\	\	PUNCT
ejpam-4894	56	1	s.	s.	PROPN
ejpam-4894	56	2	a	a	DET
ejpam-4894	56	3	minimum	minimum	ADJ
ejpam-4894	56	4	cardinality	cardinality	NOUN
ejpam-4894	56	5	of	of	ADP
ejpam-4894	56	6	a	a	DET
ejpam-4894	56	7	certified	certify	VERB
ejpam-4894	56	8	perfect	perfect	ADJ
ejpam-4894	56	9	dominating	dominating	NOUN
ejpam-4894	56	10	set	set	NOUN
ejpam-4894	56	11	of	of	ADP
ejpam-4894	56	12	g	g	PROPN
ejpam-4894	56	13	is	be	AUX
ejpam-4894	56	14	called	call	VERB
ejpam-4894	56	15	certified	certify	VERB
ejpam-4894	56	16	perfect	perfect	ADJ
ejpam-4894	56	17	domination	domination	NOUN
ejpam-4894	56	18	number	number	NOUN
ejpam-4894	56	19	of	of	ADP
ejpam-4894	56	20	g	g	NOUN
ejpam-4894	56	21	and	and	CCONJ
ejpam-4894	56	22	denoted	denote	VERB
ejpam-4894	56	23	by	by	ADP
ejpam-4894	56	24	γcerp(g	γcerp(g	PROPN
ejpam-4894	56	25	)	)	PUNCT
ejpam-4894	56	26	.	.	PUNCT
ejpam-4894	57	1	a	a	DET
ejpam-4894	57	2	certified	certify	VERB
ejpam-4894	57	3	perfect	perfect	ADJ
ejpam-4894	57	4	dominating	dominating	NOUN
ejpam-4894	57	5	set	set	NOUN
ejpam-4894	57	6	s	s	NOUN
ejpam-4894	57	7	of	of	ADP
ejpam-4894	57	8	g	g	NOUN
ejpam-4894	57	9	with	with	ADP
ejpam-4894	57	10	|s|	|s|	NOUN
ejpam-4894	57	11	=	=	PUNCT
ejpam-4894	57	12	γcerp(g	γcerp(g	PROPN
ejpam-4894	57	13	)	)	PUNCT
ejpam-4894	57	14	is	be	AUX
ejpam-4894	57	15	called	call	VERB
ejpam-4894	57	16	a	a	DET
ejpam-4894	57	17	γcerp	γcerp	NOUN
ejpam-4894	57	18	-	-	PUNCT
ejpam-4894	57	19	set	set	NOUN
ejpam-4894	57	20	.	.	PUNCT
ejpam-4894	57	21	example	example	NOUN
ejpam-4894	58	1	1	1	NUM
ejpam-4894	58	2	.	.	X
ejpam-4894	58	3	consider	consider	VERB
ejpam-4894	58	4	the	the	DET
ejpam-4894	58	5	three	three	NUM
ejpam-4894	58	6	subsets	subset	NOUN
ejpam-4894	58	7	s1	s1	NOUN
ejpam-4894	58	8	,	,	PUNCT
ejpam-4894	58	9	s2	s2	NOUN
ejpam-4894	58	10	,	,	PUNCT
ejpam-4894	58	11	and	and	CCONJ
ejpam-4894	58	12	s3	s3	PROPN
ejpam-4894	58	13	of	of	ADP
ejpam-4894	58	14	the	the	DET
ejpam-4894	58	15	graph	graph	NOUN
ejpam-4894	58	16	g	g	NOUN
ejpam-4894	58	17	shown	show	VERB
ejpam-4894	58	18	in	in	ADP
ejpam-4894	58	19	figure	figure	NOUN
ejpam-4894	58	20	1	1	NUM
ejpam-4894	58	21	.	.	PUNCT
ejpam-4894	59	1	first	first	ADV
ejpam-4894	59	2	,	,	PUNCT
ejpam-4894	59	3	let	let	VERB
ejpam-4894	59	4	s1	s1	PROPN
ejpam-4894	59	5	=	=	PUNCT
ejpam-4894	59	6	{	{	PUNCT
ejpam-4894	59	7	a	a	X
ejpam-4894	59	8	,	,	PUNCT
ejpam-4894	59	9	e	e	NOUN
ejpam-4894	59	10	,	,	PUNCT
ejpam-4894	59	11	g	g	PROPN
ejpam-4894	59	12	,	,	PUNCT
ejpam-4894	59	13	j	j	PROPN
ejpam-4894	59	14	,	,	PUNCT
ejpam-4894	59	15	n	n	CCONJ
ejpam-4894	59	16	,	,	PUNCT
ejpam-4894	59	17	r	r	NOUN
ejpam-4894	59	18	}	}	PUNCT
ejpam-4894	59	19	⊆	⊆	NUM
ejpam-4894	59	20	v	v	NOUN
ejpam-4894	59	21	(	(	PUNCT
ejpam-4894	59	22	g	g	NOUN
ejpam-4894	59	23	)	)	PUNCT
ejpam-4894	59	24	be	be	AUX
ejpam-4894	59	25	a	a	DET
ejpam-4894	59	26	first	first	ADJ
ejpam-4894	59	27	dominating	dominating	NOUN
ejpam-4894	59	28	set	set	NOUN
ejpam-4894	59	29	of	of	ADP
ejpam-4894	59	30	g.	g.	PROPN
ejpam-4894	59	31	observe	observe	VERB
ejpam-4894	59	32	that	that	SCONJ
ejpam-4894	59	33	all	all	PRON
ejpam-4894	59	34	vertices	vertice	VERB
ejpam-4894	59	35	a	a	DET
ejpam-4894	59	36	,	,	PUNCT
ejpam-4894	59	37	e	e	NOUN
ejpam-4894	59	38	,	,	PUNCT
ejpam-4894	59	39	g	g	PROPN
ejpam-4894	59	40	,	,	PUNCT
ejpam-4894	59	41	j	j	PROPN
ejpam-4894	59	42	,	,	PUNCT
ejpam-4894	59	43	n	n	CCONJ
ejpam-4894	59	44	,	,	PUNCT
ejpam-4894	59	45	r	r	NOUN
ejpam-4894	59	46	∈	∈	PROPN
ejpam-4894	59	47	s1	s1	NOUN
ejpam-4894	59	48	have	have	VERB
ejpam-4894	59	49	at	at	ADV
ejpam-4894	59	50	least	least	ADV
ejpam-4894	59	51	two	two	NUM
ejpam-4894	59	52	neighbors	neighbor	NOUN
ejpam-4894	59	53	in	in	ADP
ejpam-4894	59	54	v	v	NOUN
ejpam-4894	59	55	(	(	PUNCT
ejpam-4894	59	56	g	g	NOUN
ejpam-4894	59	57	)	)	PUNCT
ejpam-4894	59	58	\	\	NOUN
ejpam-4894	59	59	s1	s1	NOUN
ejpam-4894	59	60	.	.	PUNCT
ejpam-4894	60	1	thus	thus	ADV
ejpam-4894	60	2	,	,	PUNCT
ejpam-4894	60	3	s1	s1	PROPN
ejpam-4894	60	4	is	be	AUX
ejpam-4894	60	5	a	a	DET
ejpam-4894	60	6	certified	certify	VERB
ejpam-4894	60	7	dominating	dominating	NOUN
ejpam-4894	60	8	set	set	NOUN
ejpam-4894	60	9	of	of	ADP
ejpam-4894	60	10	g	g	NOUN
ejpam-4894	60	11	,	,	PUNCT
ejpam-4894	60	12	and	and	CCONJ
ejpam-4894	60	13	γcer(g	γcer(g	NUM
ejpam-4894	60	14	)	)	PUNCT
ejpam-4894	60	15	=	=	SYM
ejpam-4894	60	16	|s1|	|s1|	NOUN
ejpam-4894	60	17	=	=	SYM
ejpam-4894	60	18	6	6	X
ejpam-4894	60	19	.	.	PUNCT
ejpam-4894	61	1	however	however	ADV
ejpam-4894	61	2	,	,	PUNCT
ejpam-4894	61	3	s1	s1	PROPN
ejpam-4894	61	4	is	be	AUX
ejpam-4894	61	5	not	not	PART
ejpam-4894	61	6	a	a	DET
ejpam-4894	61	7	perfect	perfect	ADJ
ejpam-4894	61	8	dominating	dominating	NOUN
ejpam-4894	61	9	set	set	NOUN
ejpam-4894	61	10	since	since	SCONJ
ejpam-4894	61	11	there	there	PRON
ejpam-4894	61	12	exist	exist	VERB
ejpam-4894	61	13	vertices	vertex	NOUN
ejpam-4894	61	14	b	b	NUM
ejpam-4894	61	15	,	,	PUNCT
ejpam-4894	61	16	c	c	NOUN
ejpam-4894	61	17	,	,	PUNCT
ejpam-4894	61	18	d	d	PROPN
ejpam-4894	61	19	∈	∈	PROPN
ejpam-4894	61	20	v	v	ADP
ejpam-4894	61	21	(	(	PUNCT
ejpam-4894	61	22	g	g	NOUN
ejpam-4894	61	23	)	)	PUNCT
ejpam-4894	61	24	\	\	NOUN
ejpam-4894	61	25	s1	s1	NOUN
ejpam-4894	61	26	dominated	dominate	VERB
ejpam-4894	61	27	by	by	ADP
ejpam-4894	61	28	two	two	NUM
ejpam-4894	61	29	vertices	vertex	NOUN
ejpam-4894	61	30	a	a	PRON
ejpam-4894	61	31	,	,	PUNCT
ejpam-4894	61	32	e	e	PROPN
ejpam-4894	61	33	∈	∈	PROPN
ejpam-4894	61	34	s	s	NOUN
ejpam-4894	61	35	,	,	PUNCT
ejpam-4894	61	36	and	and	CCONJ
ejpam-4894	62	1	vertex	vertex	NOUN
ejpam-4894	62	2	f	f	PROPN
ejpam-4894	62	3	∈	∈	PROPN
ejpam-4894	62	4	v	v	PROPN
ejpam-4894	62	5	(	(	PUNCT
ejpam-4894	62	6	g	g	NOUN
ejpam-4894	62	7	)	)	PUNCT
ejpam-4894	62	8	\	\	NOUN
ejpam-4894	62	9	s1	s1	NOUN
ejpam-4894	62	10	is	be	AUX
ejpam-4894	62	11	dominated	dominate	VERB
ejpam-4894	62	12	by	by	ADP
ejpam-4894	62	13	e	e	PROPN
ejpam-4894	62	14	,	,	PUNCT
ejpam-4894	62	15	f	f	PROPN
ejpam-4894	62	16	∈	∈	PROPN
ejpam-4894	62	17	s1	s1	PROPN
ejpam-4894	62	18	.	.	PUNCT
ejpam-4894	63	1	therefore	therefore	ADV
ejpam-4894	63	2	,	,	PUNCT
ejpam-4894	63	3	s1	s1	NOUN
ejpam-4894	63	4	is	be	AUX
ejpam-4894	63	5	not	not	PART
ejpam-4894	63	6	a	a	DET
ejpam-4894	63	7	certified	certify	VERB
ejpam-4894	63	8	perfect	perfect	ADJ
ejpam-4894	63	9	dominating	dominating	NOUN
ejpam-4894	63	10	set	set	NOUN
ejpam-4894	63	11	of	of	ADP
ejpam-4894	63	12	g.	g.	PROPN
ejpam-4894	63	13	second	second	PROPN
ejpam-4894	63	14	,	,	PUNCT
ejpam-4894	63	15	let	let	VERB
ejpam-4894	63	16	s2	s2	VERB
ejpam-4894	63	17	=	=	PRON
ejpam-4894	63	18	{	{	PUNCT
ejpam-4894	63	19	a	a	DET
ejpam-4894	63	20	,	,	PUNCT
ejpam-4894	63	21	b	b	NOUN
ejpam-4894	63	22	,	,	PUNCT
ejpam-4894	63	23	g	g	PROPN
ejpam-4894	63	24	,	,	PUNCT
ejpam-4894	63	25	j	j	PROPN
ejpam-4894	63	26	,	,	PUNCT
ejpam-4894	63	27	n	n	CCONJ
ejpam-4894	63	28	,	,	PUNCT
ejpam-4894	63	29	r	r	NOUN
ejpam-4894	63	30	}	}	PUNCT
ejpam-4894	63	31	⊆	⊆	NUM
ejpam-4894	63	32	v	v	NOUN
ejpam-4894	63	33	(	(	PUNCT
ejpam-4894	63	34	g	g	NOUN
ejpam-4894	63	35	)	)	PUNCT
ejpam-4894	63	36	be	be	AUX
ejpam-4894	63	37	a	a	DET
ejpam-4894	63	38	second	second	ADJ
ejpam-4894	63	39	dominanting	dominanting	NOUN
ejpam-4894	63	40	set	set	NOUN
ejpam-4894	63	41	of	of	ADP
ejpam-4894	63	42	g.	g.	PROPN
ejpam-4894	63	43	observe	observe	VERB
ejpam-4894	63	44	that	that	SCONJ
ejpam-4894	63	45	all	all	DET
ejpam-4894	63	46	vertices	vertex	NOUN
ejpam-4894	63	47	in	in	ADP
ejpam-4894	63	48	v	v	ADP
ejpam-4894	63	49	(	(	PUNCT
ejpam-4894	63	50	g	g	NOUN
ejpam-4894	63	51	)	)	PUNCT
ejpam-4894	63	52	\	\	NOUN
ejpam-4894	63	53	s2	s2	PROPN
ejpam-4894	63	54	are	be	AUX
ejpam-4894	63	55	dominated	dominate	VERB
ejpam-4894	63	56	by	by	ADP
ejpam-4894	63	57	exactly	exactly	ADV
ejpam-4894	63	58	one	one	NUM
ejpam-4894	63	59	vertex	vertex	NOUN
ejpam-4894	63	60	in	in	ADP
ejpam-4894	63	61	s2	s2	PROPN
ejpam-4894	63	62	.	.	PUNCT
ejpam-4894	64	1	thus	thus	ADV
ejpam-4894	64	2	,	,	PUNCT
ejpam-4894	64	3	s2	s2	PROPN
ejpam-4894	64	4	is	be	AUX
ejpam-4894	64	5	a	a	DET
ejpam-4894	64	6	perfect	perfect	ADJ
ejpam-4894	64	7	dominating	dominating	NOUN
ejpam-4894	64	8	set	set	NOUN
ejpam-4894	64	9	of	of	ADP
ejpam-4894	64	10	g	g	NOUN
ejpam-4894	64	11	,	,	PUNCT
ejpam-4894	64	12	and	and	CCONJ
ejpam-4894	64	13	γp(g	γp(g	PUNCT
ejpam-4894	64	14	)	)	PUNCT
ejpam-4894	64	15	=	=	SYM
ejpam-4894	64	16	|s2|	|s2|	NOUN
ejpam-4894	64	17	=	=	NOUN
ejpam-4894	64	18	6	6	NUM
ejpam-4894	64	19	.	.	PUNCT
ejpam-4894	65	1	however	however	ADV
ejpam-4894	65	2	,	,	PUNCT
ejpam-4894	65	3	s2	s2	PROPN
ejpam-4894	65	4	is	be	AUX
ejpam-4894	65	5	not	not	PART
ejpam-4894	65	6	a	a	DET
ejpam-4894	65	7	certified	certify	VERB
ejpam-4894	65	8	dominating	dominating	NOUN
ejpam-4894	65	9	set	set	NOUN
ejpam-4894	65	10	of	of	ADP
ejpam-4894	65	11	g	g	PROPN
ejpam-4894	65	12	since	since	SCONJ
ejpam-4894	65	13	there	there	PRON
ejpam-4894	65	14	exists	exist	VERB
ejpam-4894	65	15	vertex	vertex	NOUN
ejpam-4894	65	16	b	b	PROPN
ejpam-4894	65	17	∈	∈	PROPN
ejpam-4894	65	18	s2	s2	NOUN
ejpam-4894	65	19	that	that	PRON
ejpam-4894	65	20	has	have	VERB
ejpam-4894	65	21	only	only	ADV
ejpam-4894	65	22	one	one	NUM
ejpam-4894	65	23	neighbor	neighbor	NOUN
ejpam-4894	65	24	in	in	ADP
ejpam-4894	65	25	v	v	PROPN
ejpam-4894	65	26	(	(	PUNCT
ejpam-4894	65	27	g	g	NOUN
ejpam-4894	65	28	)	)	PUNCT
ejpam-4894	65	29	\	\	NOUN
ejpam-4894	65	30	s2	s2	PROPN
ejpam-4894	65	31	.	.	PUNCT
ejpam-4894	66	1	therefore	therefore	ADV
ejpam-4894	66	2	,	,	PUNCT
ejpam-4894	66	3	s2	s2	PROPN
ejpam-4894	66	4	is	be	AUX
ejpam-4894	66	5	not	not	PART
ejpam-4894	66	6	a	a	DET
ejpam-4894	66	7	perfect	perfect	ADJ
ejpam-4894	66	8	certified	certify	VERB
ejpam-4894	66	9	dominating	dominating	NOUN
ejpam-4894	66	10	set	set	NOUN
ejpam-4894	66	11	of	of	ADP
ejpam-4894	66	12	g.	g.	PROPN
ejpam-4894	66	13	lastly	lastly	ADV
ejpam-4894	66	14	,	,	PUNCT
ejpam-4894	66	15	let	let	VERB
ejpam-4894	66	16	s3	s3	PROPN
ejpam-4894	66	17	=	=	SYM
ejpam-4894	66	18	{	{	PUNCT
ejpam-4894	66	19	a	a	PRON
ejpam-4894	66	20	,	,	PUNCT
ejpam-4894	66	21	b	b	NOUN
ejpam-4894	66	22	,	,	PUNCT
ejpam-4894	66	23	c	c	NOUN
ejpam-4894	66	24	,	,	PUNCT
ejpam-4894	66	25	d	d	NOUN
ejpam-4894	66	26	,	,	PUNCT
ejpam-4894	66	27	e	e	NOUN
ejpam-4894	66	28	,	,	PUNCT
ejpam-4894	66	29	f	f	PROPN
ejpam-4894	66	30	,	,	PUNCT
ejpam-4894	66	31	g	g	PROPN
ejpam-4894	66	32	,	,	PUNCT
ejpam-4894	66	33	j	j	PROPN
ejpam-4894	66	34	,	,	PUNCT
ejpam-4894	66	35	n	n	CCONJ
ejpam-4894	66	36	,	,	PUNCT
ejpam-4894	66	37	r	r	NOUN
ejpam-4894	66	38	}	}	PUNCT
ejpam-4894	66	39	⊆	⊆	NUM
ejpam-4894	66	40	v	v	NOUN
ejpam-4894	66	41	(	(	PUNCT
ejpam-4894	66	42	g	g	NOUN
ejpam-4894	66	43	)	)	PUNCT
ejpam-4894	66	44	be	be	VERB
ejpam-4894	66	45	a	a	DET
ejpam-4894	66	46	third	third	ADJ
ejpam-4894	66	47	dominating	dominating	NOUN
ejpam-4894	66	48	set	set	NOUN
ejpam-4894	66	49	of	of	ADP
ejpam-4894	66	50	g.	g.	PROPN
ejpam-4894	66	51	observe	observe	VERB
ejpam-4894	66	52	that	that	SCONJ
ejpam-4894	66	53	all	all	DET
ejpam-4894	66	54	vertices	vertex	NOUN
ejpam-4894	66	55	in	in	ADP
ejpam-4894	66	56	v	v	ADP
ejpam-4894	66	57	(	(	PUNCT
ejpam-4894	66	58	g	g	NOUN
ejpam-4894	66	59	)	)	PUNCT
ejpam-4894	66	60	\	\	NOUN
ejpam-4894	66	61	s3	s3	PROPN
ejpam-4894	66	62	are	be	AUX
ejpam-4894	66	63	dominated	dominate	VERB
ejpam-4894	66	64	by	by	ADP
ejpam-4894	66	65	exactly	exactly	ADV
ejpam-4894	66	66	one	one	NUM
ejpam-4894	66	67	vertex	vertex	NOUN
ejpam-4894	66	68	in	in	ADP
ejpam-4894	66	69	s3	s3	PROPN
ejpam-4894	66	70	,	,	PUNCT
ejpam-4894	66	71	and	and	CCONJ
ejpam-4894	66	72	all	all	DET
ejpam-4894	66	73	vertices	vertex	NOUN
ejpam-4894	66	74	in	in	ADP
ejpam-4894	66	75	s3	s3	PROPN
ejpam-4894	66	76	have	have	VERB
ejpam-4894	66	77	either	either	CCONJ
ejpam-4894	66	78	zero	zero	NUM
ejpam-4894	66	79	or	or	CCONJ
ejpam-4894	66	80	at	at	ADP
ejpam-4894	66	81	least	least	ADV
ejpam-4894	66	82	two	two	NUM
ejpam-4894	66	83	neighbors	neighbor	NOUN
ejpam-4894	66	84	in	in	ADP
ejpam-4894	66	85	v	v	NOUN
ejpam-4894	66	86	(	(	PUNCT
ejpam-4894	66	87	g	g	NOUN
ejpam-4894	66	88	)	)	PUNCT
ejpam-4894	66	89	\	\	PROPN
ejpam-4894	66	90	s3	s3	PROPN
ejpam-4894	66	91	.	.	PUNCT
ejpam-4894	67	1	therefore	therefore	ADV
ejpam-4894	67	2	,	,	PUNCT
ejpam-4894	67	3	s3	s3	PROPN
ejpam-4894	67	4	is	be	AUX
ejpam-4894	67	5	a	a	DET
ejpam-4894	67	6	certified	certify	VERB
ejpam-4894	67	7	perfect	perfect	ADJ
ejpam-4894	67	8	dominating	dominating	NOUN
ejpam-4894	67	9	set	set	NOUN
ejpam-4894	67	10	of	of	ADP
ejpam-4894	67	11	g	g	NOUN
ejpam-4894	67	12	,	,	PUNCT
ejpam-4894	67	13	and	and	CCONJ
ejpam-4894	67	14	γcerp(g	γcerp(g	ADJ
ejpam-4894	67	15	)	)	PUNCT
ejpam-4894	67	16	=	=	SYM
ejpam-4894	67	17	|s3|	|s3|	NOUN
ejpam-4894	67	18	=	=	SYM
ejpam-4894	67	19	10	10	NUM
ejpam-4894	67	20	.	.	PUNCT
ejpam-4894	68	1	j.	j.	PROPN
ejpam-4894	68	2	hamja	hamja	PROPN
ejpam-4894	68	3	/	/	SYM
ejpam-4894	68	4	eur	eur	PROPN
ejpam-4894	68	5	.	.	PUNCT
ejpam-4894	69	1	j.	j.	PROPN
ejpam-4894	69	2	pure	pure	PROPN
ejpam-4894	69	3	appl	appl	PROPN
ejpam-4894	69	4	.	.	PROPN
ejpam-4894	69	5	math	math	PROPN
ejpam-4894	69	6	,	,	PUNCT
ejpam-4894	69	7	16	16	NUM
ejpam-4894	69	8	(	(	PUNCT
ejpam-4894	69	9	4	4	NUM
ejpam-4894	69	10	)	)	PUNCT
ejpam-4894	69	11	(	(	PUNCT
ejpam-4894	69	12	2023	2023	NUM
ejpam-4894	69	13	)	)	PUNCT
ejpam-4894	69	14	,	,	PUNCT
ejpam-4894	69	15	2763	2763	NUM
ejpam-4894	69	16	-	-	SYM
ejpam-4894	69	17	2774	2774	NUM
ejpam-4894	69	18	2766	2766	NUM
ejpam-4894	69	19	a	a	DET
ejpam-4894	69	20	b	b	PROPN
ejpam-4894	69	21	rc	rc	PROPN
ejpam-4894	69	22	e	e	PROPN
ejpam-4894	70	1	i	i	PRON
ejpam-4894	70	2	jh	jh	VERB
ejpam-4894	70	3	m	m	PROPN
ejpam-4894	70	4	s	s	PROPN
ejpam-4894	70	5	l	l	NOUN
ejpam-4894	70	6	n	n	ADJ
ejpam-4894	70	7	qp	qp	ADP
ejpam-4894	70	8	f	f	PROPN
ejpam-4894	70	9	g	g	PROPN
ejpam-4894	70	10	:	:	PUNCT
ejpam-4894	70	11	k	k	PROPN
ejpam-4894	70	12	od	od	NUM
ejpam-4894	70	13	g	g	PROPN
ejpam-4894	70	14	figure	figure	NOUN
ejpam-4894	70	15	1	1	NUM
ejpam-4894	70	16	:	:	PUNCT
ejpam-4894	70	17	graph	graph	VERB
ejpam-4894	70	18	g	g	NOUN
ejpam-4894	70	19	with	with	ADP
ejpam-4894	70	20	γcerp(g	γcerp(g	PROPN
ejpam-4894	70	21	)	)	PUNCT
ejpam-4894	70	22	=	=	SYM
ejpam-4894	70	23	10	10	NUM
ejpam-4894	70	24	3	3	NUM
ejpam-4894	70	25	.	.	PUNCT
ejpam-4894	70	26	main	main	ADJ
ejpam-4894	70	27	results	result	NOUN
ejpam-4894	70	28	proposition	proposition	NOUN
ejpam-4894	70	29	1	1	NUM
ejpam-4894	70	30	.	.	PUNCT
ejpam-4894	71	1	let	let	VERB
ejpam-4894	71	2	g	g	PRON
ejpam-4894	71	3	be	be	AUX
ejpam-4894	71	4	a	a	DET
ejpam-4894	71	5	connected	connected	ADJ
ejpam-4894	71	6	graph	graph	NOUN
ejpam-4894	71	7	of	of	ADP
ejpam-4894	71	8	order	order	NOUN
ejpam-4894	71	9	n.	n.	NOUN
ejpam-4894	71	10	then	then	ADV
ejpam-4894	71	11	every	every	DET
ejpam-4894	71	12	support	support	NOUN
ejpam-4894	71	13	vertex	vertex	NOUN
ejpam-4894	71	14	of	of	ADP
ejpam-4894	71	15	g	g	PROPN
ejpam-4894	71	16	belongs	belong	VERB
ejpam-4894	71	17	to	to	ADP
ejpam-4894	71	18	every	every	DET
ejpam-4894	71	19	certified	certify	VERB
ejpam-4894	71	20	perfect	perfect	ADJ
ejpam-4894	71	21	dominating	dominating	NOUN
ejpam-4894	71	22	set	set	NOUN
ejpam-4894	71	23	of	of	ADP
ejpam-4894	71	24	g.	g.	PROPN
ejpam-4894	71	25	proof	proof	PROPN
ejpam-4894	71	26	.	.	PUNCT
ejpam-4894	72	1	assume	assume	VERB
ejpam-4894	72	2	that	that	SCONJ
ejpam-4894	72	3	s	s	VERB
ejpam-4894	72	4	is	be	AUX
ejpam-4894	72	5	a	a	DET
ejpam-4894	72	6	certified	certify	VERB
ejpam-4894	72	7	perfect	perfect	ADJ
ejpam-4894	72	8	dominating	dominating	NOUN
ejpam-4894	72	9	set	set	NOUN
ejpam-4894	72	10	of	of	ADP
ejpam-4894	72	11	g.	g.	PROPN
ejpam-4894	72	12	let	let	VERB
ejpam-4894	72	13	u	u	PRON
ejpam-4894	72	14	be	be	AUX
ejpam-4894	72	15	a	a	DET
ejpam-4894	72	16	support	support	NOUN
ejpam-4894	72	17	vertex	vertex	NOUN
ejpam-4894	72	18	of	of	ADP
ejpam-4894	72	19	g	g	NOUN
ejpam-4894	72	20	,	,	PUNCT
ejpam-4894	72	21	and	and	CCONJ
ejpam-4894	72	22	v	v	X
ejpam-4894	72	23	be	be	AUX
ejpam-4894	72	24	a	a	DET
ejpam-4894	72	25	leaf	leaf	NOUN
ejpam-4894	72	26	adjacent	adjacent	ADJ
ejpam-4894	72	27	to	to	PART
ejpam-4894	72	28	u.	u.	VERB
ejpam-4894	72	29	if	if	SCONJ
ejpam-4894	72	30	u	u	PROPN
ejpam-4894	72	31	/∈	/∈	PUNCT
ejpam-4894	73	1	s	s	PART
ejpam-4894	73	2	,	,	PUNCT
ejpam-4894	73	3	then	then	ADV
ejpam-4894	73	4	v	v	X
ejpam-4894	73	5	∈	∈	NOUN
ejpam-4894	73	6	s.	s.	PROPN
ejpam-4894	73	7	however	however	ADV
ejpam-4894	73	8	,	,	PUNCT
ejpam-4894	73	9	since	since	SCONJ
ejpam-4894	73	10	v	v	NOUN
ejpam-4894	73	11	would	would	AUX
ejpam-4894	73	12	have	have	VERB
ejpam-4894	73	13	only	only	ADV
ejpam-4894	73	14	one	one	NUM
ejpam-4894	73	15	neighbor	neighbor	NOUN
ejpam-4894	73	16	in	in	ADP
ejpam-4894	73	17	v	v	PROPN
ejpam-4894	73	18	(	(	PUNCT
ejpam-4894	73	19	g	g	NOUN
ejpam-4894	73	20	)	)	PUNCT
ejpam-4894	73	21	\	\	PROPN
ejpam-4894	74	1	s	s	X
ejpam-4894	74	2	,	,	PUNCT
ejpam-4894	74	3	s	s	VERB
ejpam-4894	74	4	is	be	AUX
ejpam-4894	74	5	not	not	PART
ejpam-4894	74	6	a	a	DET
ejpam-4894	74	7	certified	certify	VERB
ejpam-4894	74	8	dominating	dominating	NOUN
ejpam-4894	74	9	set	set	NOUN
ejpam-4894	74	10	of	of	ADP
ejpam-4894	74	11	g.	g.	PROPN
ejpam-4894	74	12	therefore	therefore	ADV
ejpam-4894	74	13	,	,	PUNCT
ejpam-4894	74	14	we	we	PRON
ejpam-4894	74	15	can	can	AUX
ejpam-4894	74	16	conclude	conclude	VERB
ejpam-4894	74	17	that	that	PRON
ejpam-4894	74	18	s	s	VERB
ejpam-4894	74	19	can	can	AUX
ejpam-4894	74	20	not	not	PART
ejpam-4894	74	21	be	be	AUX
ejpam-4894	74	22	a	a	DET
ejpam-4894	74	23	certified	certify	VERB
ejpam-4894	74	24	perfect	perfect	ADJ
ejpam-4894	74	25	dominating	dominating	NOUN
ejpam-4894	74	26	set	set	NOUN
ejpam-4894	74	27	.	.	PUNCT
ejpam-4894	75	1	this	this	PRON
ejpam-4894	75	2	contradicts	contradict	VERB
ejpam-4894	75	3	the	the	DET
ejpam-4894	75	4	initial	initial	ADJ
ejpam-4894	75	5	assumption	assumption	NOUN
ejpam-4894	75	6	that	that	SCONJ
ejpam-4894	75	7	s	s	VERB
ejpam-4894	75	8	is	be	AUX
ejpam-4894	75	9	a	a	DET
ejpam-4894	75	10	certified	certify	VERB
ejpam-4894	75	11	perfect	perfect	ADJ
ejpam-4894	75	12	dominating	dominating	NOUN
ejpam-4894	75	13	set	set	NOUN
ejpam-4894	75	14	.	.	PUNCT
ejpam-4894	76	1	theorem	theorem	NOUN
ejpam-4894	76	2	1	1	NUM
ejpam-4894	76	3	.	.	X
ejpam-4894	76	4	for	for	ADP
ejpam-4894	76	5	any	any	DET
ejpam-4894	76	6	graph	graph	NOUN
ejpam-4894	76	7	g	g	NOUN
ejpam-4894	76	8	of	of	ADP
ejpam-4894	76	9	order	order	NOUN
ejpam-4894	76	10	n	n	CCONJ
ejpam-4894	76	11	,	,	PUNCT
ejpam-4894	76	12	γp(g	γp(g	NUM
ejpam-4894	76	13	)	)	PUNCT
ejpam-4894	76	14	≤	≤	NUM
ejpam-4894	76	15	γcp(g	γcp(g	NOUN
ejpam-4894	76	16	)	)	PUNCT
ejpam-4894	76	17	≤	≤	NOUN
ejpam-4894	76	18	n.	n.	NOUN
ejpam-4894	76	19	proof	proof	NOUN
ejpam-4894	76	20	.	.	PUNCT
ejpam-4894	77	1	let	let	VERB
ejpam-4894	77	2	g	g	PRON
ejpam-4894	77	3	be	be	AUX
ejpam-4894	77	4	a	a	DET
ejpam-4894	77	5	graph	graph	NOUN
ejpam-4894	77	6	of	of	ADP
ejpam-4894	77	7	order	order	NOUN
ejpam-4894	77	8	n.	n.	NOUN
ejpam-4894	77	9	let	let	VERB
ejpam-4894	77	10	us	we	PRON
ejpam-4894	77	11	first	first	ADV
ejpam-4894	77	12	show	show	VERB
ejpam-4894	77	13	its	its	PRON
ejpam-4894	77	14	upper	upper	ADJ
ejpam-4894	77	15	bounds	bound	NOUN
ejpam-4894	77	16	.	.	PUNCT
ejpam-4894	78	1	let	let	VERB
ejpam-4894	78	2	s1	s1	NOUN
ejpam-4894	78	3	be	be	AUX
ejpam-4894	78	4	a	a	DET
ejpam-4894	78	5	perfect	perfect	ADJ
ejpam-4894	78	6	dominasting	dominaste	VERB
ejpam-4894	78	7	set	set	NOUN
ejpam-4894	78	8	,	,	PUNCT
ejpam-4894	78	9	and	and	CCONJ
ejpam-4894	78	10	s2	s2	NOUN
ejpam-4894	78	11	be	be	VERB
ejpam-4894	78	12	a	a	DET
ejpam-4894	78	13	certified	certify	VERB
ejpam-4894	78	14	perfect	perfect	ADJ
ejpam-4894	78	15	dominating	dominating	NOUN
ejpam-4894	78	16	set	set	NOUN
ejpam-4894	78	17	of	of	ADP
ejpam-4894	78	18	g.	g.	PROPN
ejpam-4894	78	19	since	since	SCONJ
ejpam-4894	78	20	every	every	DET
ejpam-4894	78	21	certified	certify	VERB
ejpam-4894	78	22	perfect	perfect	ADJ
ejpam-4894	78	23	dominating	dominating	NOUN
ejpam-4894	78	24	set	set	VERB
ejpam-4894	78	25	s1	s1	PROPN
ejpam-4894	78	26	⊆	⊆	NUM
ejpam-4894	78	27	s2	s2	PROPN
ejpam-4894	78	28	,	,	PUNCT
ejpam-4894	78	29	γp(g	γp(g	NUM
ejpam-4894	78	30	)	)	PUNCT
ejpam-4894	78	31	≤	≤	NUM
ejpam-4894	78	32	γcerp(g	γcerp(g	PROPN
ejpam-4894	78	33	)	)	PUNCT
ejpam-4894	78	34	.	.	PUNCT
ejpam-4894	79	1	so	so	ADV
ejpam-4894	79	2	,	,	PUNCT
ejpam-4894	79	3	we	we	PRON
ejpam-4894	79	4	are	be	AUX
ejpam-4894	79	5	left	leave	VERB
ejpam-4894	79	6	to	to	PART
ejpam-4894	79	7	show	show	VERB
ejpam-4894	79	8	its	its	PRON
ejpam-4894	79	9	lower	low	ADJ
ejpam-4894	79	10	bounds	bound	NOUN
ejpam-4894	79	11	.	.	PUNCT
ejpam-4894	80	1	since	since	SCONJ
ejpam-4894	80	2	every	every	DET
ejpam-4894	80	3	certified	certify	VERB
ejpam-4894	80	4	dominating	dominating	NOUN
ejpam-4894	80	5	set	set	NOUN
ejpam-4894	80	6	s2	s2	NOUN
ejpam-4894	80	7	⊆	⊆	NUM
ejpam-4894	80	8	v	v	NOUN
ejpam-4894	80	9	(	(	PUNCT
ejpam-4894	80	10	g	g	NOUN
ejpam-4894	80	11	)	)	PUNCT
ejpam-4894	80	12	,	,	PUNCT
ejpam-4894	80	13	γcerp(g	γcerp(g	PROPN
ejpam-4894	80	14	)	)	PUNCT
ejpam-4894	80	15	≤	≤	NOUN
ejpam-4894	80	16	|v	|v	X
ejpam-4894	80	17	(	(	PUNCT
ejpam-4894	80	18	g)|	g)|	PROPN
ejpam-4894	80	19	=	=	PROPN
ejpam-4894	80	20	n.	n.	PROPN
ejpam-4894	80	21	therefore	therefore	ADV
ejpam-4894	80	22	,	,	PUNCT
ejpam-4894	80	23	the	the	DET
ejpam-4894	80	24	assertion	assertion	NOUN
ejpam-4894	80	25	holds	hold	VERB
ejpam-4894	80	26	.	.	PUNCT
ejpam-4894	81	1	theorem	theorem	NOUN
ejpam-4894	81	2	2	2	NUM
ejpam-4894	81	3	.	.	PUNCT
ejpam-4894	81	4	let	let	VERB
ejpam-4894	81	5	a	a	DET
ejpam-4894	81	6	and	and	CCONJ
ejpam-4894	81	7	b	b	NOUN
ejpam-4894	81	8	positive	positive	ADJ
ejpam-4894	81	9	integers	integer	NOUN
ejpam-4894	81	10	with	with	ADP
ejpam-4894	81	11	1	1	NUM
ejpam-4894	81	12	≤	≤	NOUN
ejpam-4894	81	13	a	a	DET
ejpam-4894	81	14	≤	≤	PROPN
ejpam-4894	81	15	b.	b.	NOUN
ejpam-4894	82	1	then	then	ADV
ejpam-4894	82	2	there	there	PRON
ejpam-4894	82	3	exists	exist	VERB
ejpam-4894	82	4	a	a	DET
ejpam-4894	82	5	connected	connected	ADJ
ejpam-4894	82	6	graph	graph	NOUN
ejpam-4894	82	7	g	g	ADP
ejpam-4894	82	8	such	such	ADJ
ejpam-4894	82	9	that	that	PRON
ejpam-4894	82	10	γp(g	γp(g	PUNCT
ejpam-4894	82	11	)	)	PUNCT
ejpam-4894	82	12	=	=	SYM
ejpam-4894	82	13	a	a	PRON
ejpam-4894	82	14	and	and	CCONJ
ejpam-4894	82	15	γcerp(g	γcerp(g	ADJ
ejpam-4894	82	16	)	)	PUNCT
ejpam-4894	82	17	=	=	SYM
ejpam-4894	82	18	b.	b.	PROPN
ejpam-4894	82	19	proof	proof	NOUN
ejpam-4894	82	20	.	.	PUNCT
ejpam-4894	83	1	consider	consider	VERB
ejpam-4894	83	2	the	the	DET
ejpam-4894	83	3	following	follow	VERB
ejpam-4894	83	4	cases	case	NOUN
ejpam-4894	83	5	:	:	PUNCT
ejpam-4894	83	6	case	case	NOUN
ejpam-4894	83	7	1	1	NUM
ejpam-4894	83	8	:	:	PUNCT
ejpam-4894	83	9	a	a	DET
ejpam-4894	83	10	=	=	SYM
ejpam-4894	83	11	b	b	NOUN
ejpam-4894	83	12	let	let	VERB
ejpam-4894	83	13	g1	g1	PROPN
ejpam-4894	83	14	be	be	AUX
ejpam-4894	83	15	the	the	DET
ejpam-4894	83	16	graph	graph	NOUN
ejpam-4894	83	17	shown	show	VERB
ejpam-4894	83	18	in	in	ADP
ejpam-4894	83	19	figure	figure	NOUN
ejpam-4894	83	20	2	2	NUM
ejpam-4894	83	21	.	.	PUNCT
ejpam-4894	84	1	let	let	VERB
ejpam-4894	84	2	s	s	VERB
ejpam-4894	84	3	=	=	PUNCT
ejpam-4894	84	4	{	{	PUNCT
ejpam-4894	84	5	x1	x1	PROPN
ejpam-4894	84	6	,	,	PUNCT
ejpam-4894	84	7	x2	x2	PROPN
ejpam-4894	84	8	,	,	PUNCT
ejpam-4894	84	9	.	.	PUNCT
ejpam-4894	84	10	.	.	PUNCT
ejpam-4894	85	1	.	.	PUNCT
ejpam-4894	86	1	,	,	PUNCT
ejpam-4894	86	2	xc−1	xc−1	PROPN
ejpam-4894	86	3	,	,	PUNCT
ejpam-4894	86	4	xc	xc	PROPN
ejpam-4894	86	5	,	,	PUNCT
ejpam-4894	86	6	}	}	PUNCT
ejpam-4894	86	7	⊆	⊆	NUM
ejpam-4894	86	8	v	v	NOUN
ejpam-4894	86	9	(	(	PUNCT
ejpam-4894	86	10	g1	g1	PROPN
ejpam-4894	86	11	)	)	PUNCT
ejpam-4894	86	12	.	.	PUNCT
ejpam-4894	87	1	then	then	ADV
ejpam-4894	87	2	s	s	VERB
ejpam-4894	87	3	is	be	AUX
ejpam-4894	87	4	both	both	PRON
ejpam-4894	87	5	γp	γp	NOUN
ejpam-4894	87	6	-	-	PUNCT
ejpam-4894	87	7	set	set	VERB
ejpam-4894	87	8	and	and	CCONJ
ejpam-4894	87	9	γcerp	γcerp	NOUN
ejpam-4894	87	10	-	-	PUNCT
ejpam-4894	87	11	set	set	NOUN
ejpam-4894	87	12	of	of	ADP
ejpam-4894	87	13	g.	g.	PROPN
ejpam-4894	87	14	therefore	therefore	ADV
ejpam-4894	87	15	,	,	PUNCT
ejpam-4894	87	16	a	a	DET
ejpam-4894	87	17	=	=	PUNCT
ejpam-4894	87	18	γp(g3	γp(g3	NOUN
ejpam-4894	87	19	)	)	PUNCT
ejpam-4894	87	20	=	=	SYM
ejpam-4894	87	21	γcerp(g3	γcerp(g3	NOUN
ejpam-4894	87	22	)	)	PUNCT
ejpam-4894	87	23	=	=	SYM
ejpam-4894	87	24	b.	b.	NOUN
ejpam-4894	87	25	case	case	NOUN
ejpam-4894	87	26	2	2	NUM
ejpam-4894	87	27	:	:	PUNCT
ejpam-4894	87	28	a	a	DET
ejpam-4894	87	29	<	<	X
ejpam-4894	87	30	b.	b.	PROPN
ejpam-4894	87	31	let	let	VERB
ejpam-4894	87	32	g2	g2	PROPN
ejpam-4894	87	33	be	be	AUX
ejpam-4894	87	34	the	the	DET
ejpam-4894	87	35	graph	graph	NOUN
ejpam-4894	87	36	shown	show	VERB
ejpam-4894	87	37	in	in	ADP
ejpam-4894	87	38	figure	figure	NOUN
ejpam-4894	87	39	3	3	NUM
ejpam-4894	87	40	and	and	CCONJ
ejpam-4894	87	41	figure	figure	VERB
ejpam-4894	87	42	4	4	NUM
ejpam-4894	87	43	for	for	ADP
ejpam-4894	87	44	γp(g2	γp(g2	NOUN
ejpam-4894	87	45	)	)	PUNCT
ejpam-4894	87	46	and	and	CCONJ
ejpam-4894	87	47	γcerp(g2	γcerp(g2	NOUN
ejpam-4894	87	48	)	)	PUNCT
ejpam-4894	87	49	,	,	PUNCT
ejpam-4894	87	50	respectively	respectively	ADV
ejpam-4894	87	51	.	.	PUNCT
ejpam-4894	88	1	let	let	VERB
ejpam-4894	88	2	n	n	NOUN
ejpam-4894	88	3	=	=	PUNCT
ejpam-4894	88	4	b−a	b−a	X
ejpam-4894	88	5	and	and	CCONJ
ejpam-4894	88	6	a	a	DET
ejpam-4894	88	7	=	=	SYM
ejpam-4894	88	8	c+n	c+n	PROPN
ejpam-4894	88	9	with	with	ADP
ejpam-4894	88	10	c	c	PROPN
ejpam-4894	88	11	≥	≥	NUM
ejpam-4894	88	12	2	2	NUM
ejpam-4894	88	13	and	and	CCONJ
ejpam-4894	88	14	n	n	PRON
ejpam-4894	88	15	≥	≥	NOUN
ejpam-4894	88	16	2	2	NUM
ejpam-4894	88	17	.	.	PUNCT
ejpam-4894	89	1	let	let	VERB
ejpam-4894	89	2	s	s	VERB
ejpam-4894	89	3	=	=	PUNCT
ejpam-4894	89	4	{	{	PUNCT
ejpam-4894	89	5	x1	x1	PROPN
ejpam-4894	89	6	,	,	PUNCT
ejpam-4894	89	7	x2	x2	PROPN
ejpam-4894	89	8	,	,	PUNCT
ejpam-4894	89	9	.	.	PUNCT
ejpam-4894	89	10	.	.	PUNCT
ejpam-4894	90	1	.	.	PUNCT
ejpam-4894	91	1	,	,	PUNCT
ejpam-4894	91	2	xc−1	xc−1	PROPN
ejpam-4894	91	3	,	,	PUNCT
ejpam-4894	91	4	xc	xc	PROPN
ejpam-4894	91	5	,	,	PUNCT
ejpam-4894	91	6	y1	y1	PROPN
ejpam-4894	91	7	,	,	PUNCT
ejpam-4894	91	8	y2	y2	PROPN
ejpam-4894	91	9	,	,	PUNCT
ejpam-4894	91	10	.	.	PUNCT
ejpam-4894	91	11	.	.	PUNCT
ejpam-4894	92	1	.	.	PUNCT
ejpam-4894	93	1	,	,	PUNCT
ejpam-4894	93	2	yn	yn	PROPN
ejpam-4894	93	3	}	}	PUNCT
ejpam-4894	93	4	and	and	CCONJ
ejpam-4894	93	5	s∗	s∗	PROPN
ejpam-4894	94	1	=	=	SYM
ejpam-4894	94	2	s	s	PART
ejpam-4894	94	3	∪	∪	X
ejpam-4894	94	4	{	{	PUNCT
ejpam-4894	94	5	z1	z1	ADJ
ejpam-4894	94	6	,	,	PUNCT
ejpam-4894	94	7	z2	z2	PROPN
ejpam-4894	94	8	,	,	PUNCT
ejpam-4894	94	9	.	.	PUNCT
ejpam-4894	94	10	.	.	PUNCT
ejpam-4894	95	1	.	.	PUNCT
ejpam-4894	96	1	,	,	PUNCT
ejpam-4894	96	2	zn	zn	X
ejpam-4894	96	3	}	}	PUNCT
ejpam-4894	96	4	.	.	PUNCT
ejpam-4894	97	1	hence	hence	ADV
ejpam-4894	97	2	,	,	PUNCT
ejpam-4894	97	3	γp(g2	γp(g2	NOUN
ejpam-4894	97	4	)	)	PUNCT
ejpam-4894	97	5	=	=	PUNCT
ejpam-4894	97	6	|s|	|s|	NOUN
ejpam-4894	97	7	=	=	SYM
ejpam-4894	97	8	c	c	PROPN
ejpam-4894	97	9	+	+	CCONJ
ejpam-4894	97	10	n	n	PROPN
ejpam-4894	97	11	=	=	SYM
ejpam-4894	97	12	a	a	PRON
ejpam-4894	97	13	and	and	CCONJ
ejpam-4894	97	14	γcerp(g2	γcerp(g2	VERB
ejpam-4894	97	15	)	)	PUNCT
ejpam-4894	97	16	=	=	NOUN
ejpam-4894	97	17	|s∗|	|s∗|	NOUN
ejpam-4894	97	18	=	=	SYM
ejpam-4894	97	19	c+	c+	X
ejpam-4894	97	20	n+	n+	X
ejpam-4894	97	21	n	n	NOUN
ejpam-4894	97	22	=	=	SYM
ejpam-4894	97	23	a+	a+	PUNCT
ejpam-4894	97	24	n	n	PROPN
ejpam-4894	97	25	=	=	PROPN
ejpam-4894	97	26	b.	b.	PROPN
ejpam-4894	98	1	j.	j.	PROPN
ejpam-4894	98	2	hamja	hamja	PROPN
ejpam-4894	98	3	/	/	SYM
ejpam-4894	98	4	eur	eur	PROPN
ejpam-4894	98	5	.	.	PUNCT
ejpam-4894	99	1	j.	j.	PROPN
ejpam-4894	99	2	pure	pure	PROPN
ejpam-4894	99	3	appl	appl	PROPN
ejpam-4894	99	4	.	.	PROPN
ejpam-4894	99	5	math	math	PROPN
ejpam-4894	99	6	,	,	PUNCT
ejpam-4894	99	7	16	16	NUM
ejpam-4894	99	8	(	(	PUNCT
ejpam-4894	99	9	4	4	NUM
ejpam-4894	99	10	)	)	PUNCT
ejpam-4894	99	11	(	(	PUNCT
ejpam-4894	99	12	2023	2023	NUM
ejpam-4894	99	13	)	)	PUNCT
ejpam-4894	99	14	,	,	PUNCT
ejpam-4894	99	15	2763	2763	NUM
ejpam-4894	99	16	-	-	SYM
ejpam-4894	99	17	2774	2774	NUM
ejpam-4894	99	18	2767	2767	NUM
ejpam-4894	99	19	...	...	PUNCT
ejpam-4894	100	1	x1	x1	PROPN
ejpam-4894	100	2	x2	x2	NOUN
ejpam-4894	101	1	x3	x3	PROPN
ejpam-4894	101	2	xc−1	xc−1	PROPN
ejpam-4894	101	3	xc	xc	PROPN
ejpam-4894	101	4	g1	g1	PROPN
ejpam-4894	101	5	:	:	PUNCT
ejpam-4894	101	6	figure	figure	NOUN
ejpam-4894	101	7	2	2	NUM
ejpam-4894	101	8	:	:	PUNCT
ejpam-4894	101	9	graph	graph	NOUN
ejpam-4894	101	10	g1	g1	NOUN
ejpam-4894	101	11	with	with	ADP
ejpam-4894	101	12	γp(g1	γp(g1	NOUN
ejpam-4894	101	13	)	)	PUNCT
ejpam-4894	101	14	=	=	SYM
ejpam-4894	101	15	γcerp(g1	γcerp(g1	NOUN
ejpam-4894	101	16	)	)	PUNCT
ejpam-4894	101	17	...	...	PUNCT
ejpam-4894	102	1	x1	x1	NUM
ejpam-4894	102	2	x2	x2	NOUN
ejpam-4894	103	1	x3	x3	PROPN
ejpam-4894	103	2	xc−1	xc−1	VERB
ejpam-4894	103	3	xc	xc	PROPN
ejpam-4894	103	4	g2	g2	PROPN
ejpam-4894	103	5	:	:	PUNCT
ejpam-4894	103	6	...	...	PUNCT
ejpam-4894	104	1	y1	y1	INTJ
ejpam-4894	104	2	y2	y2	INTJ
ejpam-4894	104	3	yn	yn	INTJ
ejpam-4894	104	4	...	...	PUNCT
ejpam-4894	105	1	z1	z1	PROPN
ejpam-4894	105	2	z2	z2	PROPN
ejpam-4894	105	3	zn	zn	PROPN
ejpam-4894	105	4	figure	figure	NOUN
ejpam-4894	105	5	3	3	NUM
ejpam-4894	105	6	:	:	PUNCT
ejpam-4894	105	7	graph	graph	NOUN
ejpam-4894	105	8	g2	g2	PROPN
ejpam-4894	105	9	with	with	ADP
ejpam-4894	105	10	γp(g2	γp(g2	NOUN
ejpam-4894	105	11	)	)	PUNCT
ejpam-4894	105	12	=	=	PUNCT
ejpam-4894	106	1	a	a	PRON
ejpam-4894	106	2	...	...	PUNCT
ejpam-4894	107	1	x1	x1	NUM
ejpam-4894	107	2	x2	x2	PROPN
ejpam-4894	108	1	x3	x3	PROPN
ejpam-4894	108	2	xc−1	xc−1	VERB
ejpam-4894	108	3	xc	xc	PROPN
ejpam-4894	108	4	g2	g2	PROPN
ejpam-4894	108	5	:	:	PUNCT
ejpam-4894	108	6	...	...	PUNCT
ejpam-4894	109	1	y1	y1	INTJ
ejpam-4894	109	2	y2	y2	INTJ
ejpam-4894	109	3	yn	yn	INTJ
ejpam-4894	109	4	...	...	PUNCT
ejpam-4894	110	1	z1	z1	PROPN
ejpam-4894	110	2	z2	z2	PROPN
ejpam-4894	110	3	zn	zn	PROPN
ejpam-4894	110	4	figure	figure	NOUN
ejpam-4894	110	5	4	4	NUM
ejpam-4894	110	6	:	:	PUNCT
ejpam-4894	110	7	graph	graph	NOUN
ejpam-4894	110	8	g2	g2	PROPN
ejpam-4894	110	9	with	with	ADP
ejpam-4894	110	10	γcerp(g2	γcerp(g2	NOUN
ejpam-4894	110	11	)	)	PUNCT
ejpam-4894	110	12	=	=	SYM
ejpam-4894	110	13	b	b	NOUN
ejpam-4894	110	14	consequently	consequently	ADV
ejpam-4894	110	15	,	,	PUNCT
ejpam-4894	110	16	the	the	DET
ejpam-4894	110	17	statement	statement	NOUN
ejpam-4894	110	18	is	be	AUX
ejpam-4894	110	19	substantiated	substantiate	VERB
ejpam-4894	110	20	by	by	ADP
ejpam-4894	110	21	this	this	DET
ejpam-4894	110	22	evidence	evidence	NOUN
ejpam-4894	110	23	.	.	PUNCT
ejpam-4894	111	1	therefore	therefore	ADV
ejpam-4894	111	2	,	,	PUNCT
ejpam-4894	111	3	this	this	PRON
ejpam-4894	111	4	completes	complete	VERB
ejpam-4894	111	5	the	the	DET
ejpam-4894	111	6	proof	proof	NOUN
ejpam-4894	111	7	.	.	PUNCT
ejpam-4894	112	1	j.	j.	PROPN
ejpam-4894	112	2	hamja	hamja	PROPN
ejpam-4894	112	3	/	/	SYM
ejpam-4894	112	4	eur	eur	PROPN
ejpam-4894	112	5	.	.	PUNCT
ejpam-4894	113	1	j.	j.	PROPN
ejpam-4894	113	2	pure	pure	PROPN
ejpam-4894	113	3	appl	appl	PROPN
ejpam-4894	113	4	.	.	PROPN
ejpam-4894	113	5	math	math	PROPN
ejpam-4894	113	6	,	,	PUNCT
ejpam-4894	113	7	16	16	NUM
ejpam-4894	113	8	(	(	PUNCT
ejpam-4894	113	9	4	4	NUM
ejpam-4894	113	10	)	)	PUNCT
ejpam-4894	113	11	(	(	PUNCT
ejpam-4894	113	12	2023	2023	NUM
ejpam-4894	113	13	)	)	PUNCT
ejpam-4894	113	14	,	,	PUNCT
ejpam-4894	113	15	2763	2763	NUM
ejpam-4894	113	16	-	-	SYM
ejpam-4894	113	17	2774	2774	NUM
ejpam-4894	113	18	2768	2768	NUM
ejpam-4894	113	19	theorem	theorem	NOUN
ejpam-4894	113	20	3	3	X
ejpam-4894	113	21	.	.	PUNCT
ejpam-4894	114	1	let	let	VERB
ejpam-4894	114	2	g	g	PRON
ejpam-4894	114	3	be	be	AUX
ejpam-4894	114	4	a	a	DET
ejpam-4894	114	5	graph	graph	NOUN
ejpam-4894	114	6	with	with	ADP
ejpam-4894	114	7	components	component	NOUN
ejpam-4894	114	8	g1	g1	PROPN
ejpam-4894	114	9	,	,	PUNCT
ejpam-4894	114	10	g2	g2	PROPN
ejpam-4894	114	11	,	,	PUNCT
ejpam-4894	114	12	.	.	PUNCT
ejpam-4894	114	13	.	.	PUNCT
ejpam-4894	115	1	.	.	PUNCT
ejpam-4894	116	1	,	,	PUNCT
ejpam-4894	116	2	gk	gk	PROPN
ejpam-4894	116	3	,	,	PUNCT
ejpam-4894	116	4	where	where	SCONJ
ejpam-4894	116	5	k	k	PROPN
ejpam-4894	116	6	≥	≥	NUM
ejpam-4894	116	7	2	2	NUM
ejpam-4894	116	8	.	.	PUNCT
ejpam-4894	117	1	then	then	ADV
ejpam-4894	117	2	γcp(g	γcp(g	X
ejpam-4894	117	3	)	)	PUNCT
ejpam-4894	117	4	=	=	SYM
ejpam-4894	117	5	k∑	k∑	PROPN
ejpam-4894	117	6	i=1	i=1	PROPN
ejpam-4894	117	7	γcp(gi	γcp(gi	PROPN
ejpam-4894	117	8	)	)	PUNCT
ejpam-4894	117	9	.	.	PUNCT
ejpam-4894	118	1	proof	proof	NOUN
ejpam-4894	118	2	.	.	PUNCT
ejpam-4894	119	1	let	let	VERB
ejpam-4894	119	2	si	si	PRON
ejpam-4894	119	3	be	be	AUX
ejpam-4894	119	4	a	a	DET
ejpam-4894	119	5	γcerp	γcerp	NOUN
ejpam-4894	119	6	-	-	PUNCT
ejpam-4894	119	7	set	set	NOUN
ejpam-4894	119	8	of	of	ADP
ejpam-4894	119	9	gi	gi	NOUN
ejpam-4894	119	10	for	for	ADP
ejpam-4894	119	11	each	each	DET
ejpam-4894	119	12	i	i	PRON
ejpam-4894	119	13	∈	∈	PROPN
ejpam-4894	119	14	{	{	PUNCT
ejpam-4894	119	15	1	1	NUM
ejpam-4894	119	16	,	,	PUNCT
ejpam-4894	119	17	2	2	NUM
ejpam-4894	119	18	,	,	PUNCT
ejpam-4894	119	19	.	.	PUNCT
ejpam-4894	119	20	.	.	PUNCT
ejpam-4894	120	1	.	.	PUNCT
ejpam-4894	121	1	,	,	PUNCT
ejpam-4894	121	2	k	k	X
ejpam-4894	121	3	}	}	PUNCT
ejpam-4894	121	4	.	.	PUNCT
ejpam-4894	122	1	then	then	ADV
ejpam-4894	122	2	s	s	VERB
ejpam-4894	122	3	=	=	PROPN
ejpam-4894	122	4	⋃k	⋃k	PROPN
ejpam-4894	122	5	i=1	i=1	PROPN
ejpam-4894	122	6	si	si	PROPN
ejpam-4894	122	7	forms	form	VERB
ejpam-4894	122	8	a	a	DET
ejpam-4894	122	9	γcerp	γcerp	NOUN
ejpam-4894	122	10	-	-	PUNCT
ejpam-4894	122	11	set	set	NOUN
ejpam-4894	122	12	of	of	ADP
ejpam-4894	122	13	g.	g.	PROPN
ejpam-4894	122	14	therefore	therefore	ADV
ejpam-4894	122	15	,	,	PUNCT
ejpam-4894	122	16	we	we	PRON
ejpam-4894	122	17	have	have	AUX
ejpam-4894	122	18	γcerp(g	γcerp(g	VERB
ejpam-4894	122	19	)	)	PUNCT
ejpam-4894	122	20	≤	≤	NUM
ejpam-4894	122	21	|s|	|s|	PROPN
ejpam-4894	122	22	=	=	PUNCT
ejpam-4894	122	23	k∑	k∑	PROPN
ejpam-4894	122	24	i=1	i=1	PROPN
ejpam-4894	123	1	|si|	|si|	PROPN
ejpam-4894	124	1	=	=	SYM
ejpam-4894	125	1	k∑	k∑	PROPN
ejpam-4894	126	1	i=1	i=1	PROPN
ejpam-4894	127	1	γcerp(gi	γcerp(gi	PROPN
ejpam-4894	127	2	)	)	PUNCT
ejpam-4894	127	3	.	.	PUNCT
ejpam-4894	128	1	conversely	conversely	ADV
ejpam-4894	128	2	,	,	PUNCT
ejpam-4894	128	3	suppose	suppose	VERB
ejpam-4894	128	4	s∗	s∗	PROPN
ejpam-4894	128	5	is	be	AUX
ejpam-4894	128	6	a	a	DET
ejpam-4894	128	7	γcerp	γcerp	NOUN
ejpam-4894	128	8	-	-	PUNCT
ejpam-4894	128	9	set	set	NOUN
ejpam-4894	128	10	of	of	ADP
ejpam-4894	128	11	g.	g.	PROPN
ejpam-4894	128	12	for	for	ADP
ejpam-4894	128	13	each	each	DET
ejpam-4894	128	14	i	i	PRON
ejpam-4894	128	15	∈	∈	PROPN
ejpam-4894	128	16	{	{	PUNCT
ejpam-4894	128	17	1	1	NUM
ejpam-4894	128	18	,	,	PUNCT
ejpam-4894	128	19	2	2	NUM
ejpam-4894	128	20	,	,	PUNCT
ejpam-4894	128	21	.	.	PUNCT
ejpam-4894	128	22	.	.	PUNCT
ejpam-4894	128	23	.	.	PUNCT
ejpam-4894	129	1	,	,	PUNCT
ejpam-4894	129	2	k	k	X
ejpam-4894	129	3	}	}	PUNCT
ejpam-4894	129	4	,	,	PUNCT
ejpam-4894	129	5	let	let	VERB
ejpam-4894	129	6	s∗	s∗	PROPN
ejpam-4894	130	1	i	i	PRON
ejpam-4894	130	2	=	=	PUNCT
ejpam-4894	130	3	s∗	s∗	PROPN
ejpam-4894	130	4	∩	∩	PROPN
ejpam-4894	130	5	v	v	X
ejpam-4894	130	6	(	(	PUNCT
ejpam-4894	130	7	gi	gi	NOUN
ejpam-4894	130	8	)	)	PUNCT
ejpam-4894	130	9	.	.	PUNCT
ejpam-4894	131	1	since	since	SCONJ
ejpam-4894	131	2	s∗	s∗	PROPN
ejpam-4894	131	3	is	be	AUX
ejpam-4894	131	4	a	a	DET
ejpam-4894	131	5	γcerp	γcerp	NOUN
ejpam-4894	131	6	-	-	PUNCT
ejpam-4894	131	7	set	set	NOUN
ejpam-4894	131	8	of	of	ADP
ejpam-4894	131	9	g	g	NOUN
ejpam-4894	131	10	,	,	PUNCT
ejpam-4894	131	11	s∗	s∗	PROPN
ejpam-4894	131	12	i	i	PRON
ejpam-4894	131	13	is	be	AUX
ejpam-4894	131	14	a	a	DET
ejpam-4894	131	15	γcerp	γcerp	NOUN
ejpam-4894	131	16	-	-	PUNCT
ejpam-4894	131	17	set	set	NOUN
ejpam-4894	131	18	of	of	ADP
ejpam-4894	131	19	gi	gi	NOUN
ejpam-4894	131	20	for	for	ADP
ejpam-4894	131	21	each	each	DET
ejpam-4894	131	22	i	i	PRON
ejpam-4894	131	23	∈	∈	PROPN
ejpam-4894	131	24	{	{	PUNCT
ejpam-4894	131	25	1	1	NUM
ejpam-4894	131	26	,	,	PUNCT
ejpam-4894	131	27	2	2	NUM
ejpam-4894	131	28	,	,	PUNCT
ejpam-4894	131	29	.	.	PUNCT
ejpam-4894	131	30	.	.	PUNCT
ejpam-4894	132	1	.	.	PUNCT
ejpam-4894	133	1	,	,	PUNCT
ejpam-4894	133	2	k	k	X
ejpam-4894	133	3	}	}	PUNCT
ejpam-4894	133	4	.	.	PUNCT
ejpam-4894	134	1	this	this	PRON
ejpam-4894	134	2	implies	imply	VERB
ejpam-4894	134	3	that	that	SCONJ
ejpam-4894	134	4	γcerp(g	γcerp(g	PROPN
ejpam-4894	134	5	)	)	PUNCT
ejpam-4894	134	6	=	=	NOUN
ejpam-4894	134	7	|s∗|	|s∗|	NUM
ejpam-4894	135	1	=	=	SYM
ejpam-4894	135	2	k∑	k∑	VERB
ejpam-4894	135	3	i=1	i=1	X
ejpam-4894	136	1	|s∗	|s∗	PROPN
ejpam-4894	136	2	i	i	PRON
ejpam-4894	136	3	|	|	ADV
ejpam-4894	136	4	≥	≥	NOUN
ejpam-4894	136	5	k∑	k∑	VERB
ejpam-4894	136	6	i=1	i=1	PROPN
ejpam-4894	136	7	γcerp(gi	γcerp(gi	PROPN
ejpam-4894	136	8	)	)	PUNCT
ejpam-4894	136	9	.	.	PUNCT
ejpam-4894	137	1	therefore	therefore	ADV
ejpam-4894	137	2	,	,	PUNCT
ejpam-4894	137	3	γcerp(g	γcerp(g	PROPN
ejpam-4894	137	4	)	)	PUNCT
ejpam-4894	137	5	=	=	PUNCT
ejpam-4894	138	1	∑k	∑k	PROPN
ejpam-4894	138	2	i=1	i=1	PROPN
ejpam-4894	138	3	γcerp(gi	γcerp(gi	PROPN
ejpam-4894	138	4	)	)	PUNCT
ejpam-4894	138	5	.	.	PUNCT
ejpam-4894	139	1	theorem	theorem	ADJ
ejpam-4894	139	2	4	4	NUM
ejpam-4894	139	3	.	.	X
ejpam-4894	140	1	for	for	ADP
ejpam-4894	140	2	a	a	DET
ejpam-4894	140	3	path	path	NOUN
ejpam-4894	140	4	pn	pn	NOUN
ejpam-4894	140	5	of	of	ADP
ejpam-4894	140	6	order	order	NOUN
ejpam-4894	140	7	n	n	PRON
ejpam-4894	140	8	≥	≥	NOUN
ejpam-4894	140	9	1	1	NUM
ejpam-4894	140	10	,	,	PUNCT
ejpam-4894	140	11	γcerp(pn	γcerp(pn	NOUN
ejpam-4894	140	12	)	)	PUNCT
ejpam-4894	140	13	=	=	NOUN
ejpam-4894	140	14	{	{	PUNCT
ejpam-4894	140	15	n	n	NOUN
ejpam-4894	140	16	3	3	NUM
ejpam-4894	140	17	if	if	SCONJ
ejpam-4894	140	18	n	n	PRON
ejpam-4894	140	19	≡	≡	PROPN
ejpam-4894	140	20	0	0	PUNCT
ejpam-4894	140	21	(	(	PUNCT
ejpam-4894	140	22	mod	mod	NOUN
ejpam-4894	140	23	3	3	NUM
ejpam-4894	140	24	)	)	PUNCT
ejpam-4894	140	25	;	;	PUNCT
ejpam-4894	140	26	n	n	X
ejpam-4894	140	27	otherwise	otherwise	ADV
ejpam-4894	140	28	.	.	PUNCT
ejpam-4894	141	1	proof	proof	NOUN
ejpam-4894	141	2	.	.	PUNCT
ejpam-4894	142	1	suppose	suppose	VERB
ejpam-4894	142	2	that	that	SCONJ
ejpam-4894	142	3	v	v	INTJ
ejpam-4894	142	4	(	(	PUNCT
ejpam-4894	142	5	pn	pn	NOUN
ejpam-4894	142	6	)	)	PUNCT
ejpam-4894	142	7	=	=	SYM
ejpam-4894	142	8	{	{	PUNCT
ejpam-4894	142	9	v1	v1	PROPN
ejpam-4894	142	10	,	,	PUNCT
ejpam-4894	142	11	v2	v2	PROPN
ejpam-4894	142	12	,	,	PUNCT
ejpam-4894	142	13	.	.	PUNCT
ejpam-4894	142	14	.	.	PUNCT
ejpam-4894	143	1	.	.	PUNCT
ejpam-4894	144	1	,	,	PUNCT
ejpam-4894	144	2	vn−1	vn−1	PROPN
ejpam-4894	144	3	,	,	PUNCT
ejpam-4894	144	4	vn	vn	VERB
ejpam-4894	144	5	}	}	PUNCT
ejpam-4894	144	6	such	such	ADJ
ejpam-4894	144	7	that	that	SCONJ
ejpam-4894	144	8	deg(v1	deg(v1	NOUN
ejpam-4894	144	9	)	)	PUNCT
ejpam-4894	144	10	=	=	SYM
ejpam-4894	144	11	deg(vn	deg(vn	X
ejpam-4894	144	12	)	)	PUNCT
ejpam-4894	144	13	=	=	SYM
ejpam-4894	144	14	1	1	NUM
ejpam-4894	144	15	and	and	CCONJ
ejpam-4894	144	16	deg(vi	deg(vi	NOUN
ejpam-4894	144	17	)	)	PUNCT
ejpam-4894	144	18	=	=	SYM
ejpam-4894	144	19	2	2	NUM
ejpam-4894	144	20	for	for	ADP
ejpam-4894	144	21	each	each	DET
ejpam-4894	144	22	i	i	PRON
ejpam-4894	144	23	∈	∈	PROPN
ejpam-4894	144	24	{	{	PUNCT
ejpam-4894	144	25	2	2	NUM
ejpam-4894	144	26	,	,	PUNCT
ejpam-4894	144	27	3	3	NUM
ejpam-4894	144	28	,	,	PUNCT
ejpam-4894	144	29	.	.	PUNCT
ejpam-4894	144	30	.	.	PUNCT
ejpam-4894	145	1	.	.	PUNCT
ejpam-4894	146	1	,	,	PUNCT
ejpam-4894	146	2	n−	n−	NOUN
ejpam-4894	146	3	1	1	NUM
ejpam-4894	146	4	}	}	PUNCT
ejpam-4894	146	5	.	.	PUNCT
ejpam-4894	147	1	consider	consider	VERB
ejpam-4894	147	2	the	the	DET
ejpam-4894	147	3	following	follow	VERB
ejpam-4894	147	4	cases	case	NOUN
ejpam-4894	147	5	:	:	PUNCT
ejpam-4894	147	6	case	case	NOUN
ejpam-4894	147	7	1	1	NUM
ejpam-4894	147	8	.	.	PUNCT
ejpam-4894	147	9	suppose	suppose	VERB
ejpam-4894	147	10	that	that	SCONJ
ejpam-4894	147	11	n	n	PROPN
ejpam-4894	147	12	≡	≡	PROPN
ejpam-4894	147	13	0	0	PUNCT
ejpam-4894	148	1	(	(	PUNCT
ejpam-4894	148	2	mod	mod	NOUN
ejpam-4894	148	3	3	3	NUM
ejpam-4894	148	4	)	)	PUNCT
ejpam-4894	148	5	.	.	PUNCT
ejpam-4894	149	1	suppose	suppose	VERB
ejpam-4894	149	2	that	that	SCONJ
ejpam-4894	149	3	n	n	PROPN
ejpam-4894	149	4	=	=	SYM
ejpam-4894	149	5	3	3	X
ejpam-4894	149	6	.	.	PUNCT
ejpam-4894	149	7	then	then	ADV
ejpam-4894	149	8	γcerp(p3	γcerp(p3	VERB
ejpam-4894	149	9	)	)	PUNCT
ejpam-4894	150	1	=	=	SYM
ejpam-4894	150	2	1	1	NUM
ejpam-4894	150	3	=	=	SYM
ejpam-4894	150	4	3	3	NUM
ejpam-4894	150	5	3	3	NUM
ejpam-4894	150	6	.	.	PUNCT
ejpam-4894	150	7	suppose	suppose	VERB
ejpam-4894	150	8	that	that	SCONJ
ejpam-4894	150	9	n	n	PROPN
ejpam-4894	150	10	>	>	X
ejpam-4894	150	11	3	3	X
ejpam-4894	150	12	.	.	PUNCT
ejpam-4894	150	13	let	let	VERB
ejpam-4894	150	14	q	q	NOUN
ejpam-4894	150	15	=	=	PUNCT
ejpam-4894	150	16	n	n	NUM
ejpam-4894	150	17	3	3	NUM
ejpam-4894	150	18	and	and	CCONJ
ejpam-4894	150	19	r	r	PROPN
ejpam-4894	150	20	∈	∈	PROPN
ejpam-4894	150	21	{	{	PUNCT
ejpam-4894	150	22	1	1	NUM
ejpam-4894	150	23	,	,	PUNCT
ejpam-4894	150	24	2	2	NUM
ejpam-4894	150	25	,	,	PUNCT
ejpam-4894	150	26	.	.	PUNCT
ejpam-4894	150	27	.	.	PUNCT
ejpam-4894	151	1	.	.	PUNCT
ejpam-4894	152	1	,	,	PUNCT
ejpam-4894	152	2	q	q	X
ejpam-4894	153	1	−	−	PROPN
ejpam-4894	153	2	1	1	NUM
ejpam-4894	153	3	,	,	PUNCT
ejpam-4894	153	4	q	q	NOUN
ejpam-4894	153	5	}	}	PUNCT
ejpam-4894	153	6	.	.	PUNCT
ejpam-4894	154	1	then	then	ADV
ejpam-4894	154	2	let	let	VERB
ejpam-4894	154	3	us	we	PRON
ejpam-4894	154	4	denote	denote	VERB
ejpam-4894	154	5	a	a	DET
ejpam-4894	154	6	group	group	NOUN
ejpam-4894	154	7	of	of	ADP
ejpam-4894	154	8	vertices	vertex	NOUN
ejpam-4894	154	9	of	of	ADP
ejpam-4894	154	10	pn	pn	NOUN
ejpam-4894	154	11	into	into	ADP
ejpam-4894	154	12	q	q	PROPN
ejpam-4894	154	13	disjoint	disjoint	PROPN
ejpam-4894	154	14	subsets	subset	NOUN
ejpam-4894	154	15	gr	gr	ADP
ejpam-4894	154	16	,	,	PUNCT
ejpam-4894	154	17	these	these	PRON
ejpam-4894	154	18	are	be	AUX
ejpam-4894	154	19	,	,	PUNCT
ejpam-4894	154	20	g1	g1	PROPN
ejpam-4894	154	21	=	=	SYM
ejpam-4894	154	22	{	{	PUNCT
ejpam-4894	154	23	v1	v1	PROPN
ejpam-4894	154	24	,	,	PUNCT
ejpam-4894	154	25	v2	v2	PROPN
ejpam-4894	154	26	,	,	PUNCT
ejpam-4894	154	27	v3	v3	PROPN
ejpam-4894	154	28	}	}	PUNCT
ejpam-4894	154	29	g2	g2	PROPN
ejpam-4894	154	30	=	=	SYM
ejpam-4894	154	31	{	{	PUNCT
ejpam-4894	154	32	v4	v4	PROPN
ejpam-4894	154	33	,	,	PUNCT
ejpam-4894	154	34	v5	v5	PROPN
ejpam-4894	154	35	,	,	PUNCT
ejpam-4894	154	36	v6	v6	NOUN
ejpam-4894	154	37	}	}	PUNCT
ejpam-4894	154	38	...	...	PUNCT
ejpam-4894	155	1	gq	gq	PROPN
ejpam-4894	155	2	=	=	PRON
ejpam-4894	155	3	{	{	PUNCT
ejpam-4894	155	4	vn−2	vn−2	PROPN
ejpam-4894	155	5	,	,	PUNCT
ejpam-4894	155	6	vn−1	vn−1	PROPN
ejpam-4894	155	7	,	,	PUNCT
ejpam-4894	155	8	vn	vn	NOUN
ejpam-4894	155	9	}	}	PUNCT
ejpam-4894	155	10	clearly	clearly	ADV
ejpam-4894	155	11	,	,	PUNCT
ejpam-4894	155	12	the	the	DET
ejpam-4894	155	13	set	set	NOUN
ejpam-4894	155	14	s	s	PART
ejpam-4894	155	15	=	=	PUNCT
ejpam-4894	155	16	{	{	PUNCT
ejpam-4894	155	17	v2	v2	PROPN
ejpam-4894	155	18	,	,	PUNCT
ejpam-4894	155	19	v5	v5	NOUN
ejpam-4894	155	20	,	,	PUNCT
ejpam-4894	155	21	.	.	PUNCT
ejpam-4894	155	22	.	.	PUNCT
ejpam-4894	156	1	.	.	PUNCT
ejpam-4894	157	1	,	,	PUNCT
ejpam-4894	157	2	vn−4	vn−4	NOUN
ejpam-4894	157	3	,	,	PUNCT
ejpam-4894	157	4	vn−1	vn−1	ADJ
ejpam-4894	157	5	}	}	PUNCT
ejpam-4894	157	6	⊆	⊆	NUM
ejpam-4894	157	7	v	v	NOUN
ejpam-4894	157	8	(	(	PUNCT
ejpam-4894	157	9	pn	pn	NOUN
ejpam-4894	157	10	)	)	PUNCT
ejpam-4894	157	11	is	be	AUX
ejpam-4894	157	12	a	a	DET
ejpam-4894	157	13	γcerp	γcerp	NOUN
ejpam-4894	157	14	-	-	PUNCT
ejpam-4894	157	15	set	set	NOUN
ejpam-4894	157	16	of	of	ADP
ejpam-4894	157	17	pn	pn	PROPN
ejpam-4894	157	18	since	since	SCONJ
ejpam-4894	157	19	ng[s	ng[s	PROPN
ejpam-4894	157	20	]	]	PUNCT
ejpam-4894	157	21	=	=	SYM
ejpam-4894	157	22	v	v	X
ejpam-4894	157	23	(	(	PUNCT
ejpam-4894	157	24	pn	pn	NOUN
ejpam-4894	157	25	)	)	PUNCT
ejpam-4894	157	26	and	and	CCONJ
ejpam-4894	157	27	every	every	DET
ejpam-4894	157	28	vertex	vertex	NOUN
ejpam-4894	157	29	vj	vj	X
ejpam-4894	157	30	∈	∈	PROPN
ejpam-4894	157	31	v	v	PROPN
ejpam-4894	157	32	(	(	PUNCT
ejpam-4894	157	33	pn	pn	PROPN
ejpam-4894	157	34	)	)	PUNCT
ejpam-4894	157	35	,	,	PUNCT
ejpam-4894	157	36	j	j	PROPN
ejpam-4894	157	37	∈	∈	PROPN
ejpam-4894	157	38	{	{	PUNCT
ejpam-4894	157	39	2	2	NUM
ejpam-4894	157	40	,	,	PUNCT
ejpam-4894	157	41	5	5	NUM
ejpam-4894	157	42	,	,	PUNCT
ejpam-4894	157	43	.	.	PUNCT
ejpam-4894	157	44	.	.	PUNCT
ejpam-4894	158	1	.	.	PUNCT
ejpam-4894	159	1	,	,	PUNCT
ejpam-4894	160	1	n	n	CCONJ
ejpam-4894	160	2	−	−	PROPN
ejpam-4894	160	3	4	4	NUM
ejpam-4894	160	4	,	,	PUNCT
ejpam-4894	160	5	n	n	CCONJ
ejpam-4894	160	6	−	−	PROPN
ejpam-4894	160	7	1	1	NUM
ejpam-4894	160	8	}	}	PUNCT
ejpam-4894	160	9	has	have	VERB
ejpam-4894	160	10	two	two	NUM
ejpam-4894	160	11	neighbors	neighbor	NOUN
ejpam-4894	160	12	in	in	ADP
ejpam-4894	160	13	v	v	PROPN
ejpam-4894	160	14	(	(	PUNCT
ejpam-4894	160	15	pn	pn	NOUN
ejpam-4894	160	16	)	)	PUNCT
ejpam-4894	160	17	\	\	PROPN
ejpam-4894	160	18	s.	s.	PROPN
ejpam-4894	161	1	it	it	PRON
ejpam-4894	161	2	follows	follow	VERB
ejpam-4894	161	3	that	that	SCONJ
ejpam-4894	161	4	all	all	DET
ejpam-4894	161	5	other	other	ADJ
ejpam-4894	161	6	vertices	vertex	NOUN
ejpam-4894	161	7	vi	vi	PROPN
ejpam-4894	161	8	∈	∈	PROPN
ejpam-4894	161	9	v	v	NOUN
ejpam-4894	161	10	(	(	PUNCT
ejpam-4894	161	11	pn	pn	NOUN
ejpam-4894	161	12	)	)	PUNCT
ejpam-4894	161	13	\	\	PROPN
ejpam-4894	162	1	s	s	X
ejpam-4894	162	2	,	,	PUNCT
ejpam-4894	162	3	i	i	PRON
ejpam-4894	162	4	∈	∈	PROPN
ejpam-4894	162	5	{	{	PUNCT
ejpam-4894	162	6	1	1	NUM
ejpam-4894	162	7	,	,	PUNCT
ejpam-4894	162	8	3	3	NUM
ejpam-4894	162	9	,	,	PUNCT
ejpam-4894	162	10	4	4	NUM
ejpam-4894	162	11	,	,	PUNCT
ejpam-4894	162	12	.	.	PUNCT
ejpam-4894	162	13	.	.	PUNCT
ejpam-4894	162	14	.	.	PUNCT
ejpam-4894	163	1	,	,	PUNCT
ejpam-4894	164	1	n	n	CCONJ
ejpam-4894	164	2	−	−	PROPN
ejpam-4894	164	3	3	3	NUM
ejpam-4894	164	4	,	,	PUNCT
ejpam-4894	164	5	n	n	CCONJ
ejpam-4894	164	6	−	−	PROPN
ejpam-4894	164	7	2	2	NUM
ejpam-4894	164	8	,	,	PUNCT
ejpam-4894	164	9	n	n	CCONJ
ejpam-4894	164	10	}	}	PUNCT
ejpam-4894	164	11	are	be	AUX
ejpam-4894	164	12	dominated	dominate	VERB
ejpam-4894	164	13	by	by	ADP
ejpam-4894	164	14	exactly	exactly	ADV
ejpam-4894	164	15	one	one	NUM
ejpam-4894	164	16	vertex	vertex	NOUN
ejpam-4894	164	17	in	in	ADP
ejpam-4894	164	18	s.	s.	PROPN
ejpam-4894	164	19	therefore	therefore	ADV
ejpam-4894	164	20	,	,	PUNCT
ejpam-4894	164	21	γcerp(pn	γcerp(pn	NOUN
ejpam-4894	164	22	)	)	PUNCT
ejpam-4894	164	23	=	=	PUNCT
ejpam-4894	164	24	|s|	|s|	NOUN
ejpam-4894	164	25	=	=	PROPN
ejpam-4894	164	26	n	n	PRON
ejpam-4894	164	27	3	3	NUM
ejpam-4894	164	28	.	.	PUNCT
ejpam-4894	165	1	j.	j.	PROPN
ejpam-4894	165	2	hamja	hamja	PROPN
ejpam-4894	165	3	/	/	SYM
ejpam-4894	165	4	eur	eur	PROPN
ejpam-4894	165	5	.	.	PUNCT
ejpam-4894	166	1	j.	j.	PROPN
ejpam-4894	166	2	pure	pure	PROPN
ejpam-4894	166	3	appl	appl	PROPN
ejpam-4894	166	4	.	.	PROPN
ejpam-4894	166	5	math	math	PROPN
ejpam-4894	166	6	,	,	PUNCT
ejpam-4894	166	7	16	16	NUM
ejpam-4894	166	8	(	(	PUNCT
ejpam-4894	166	9	4	4	NUM
ejpam-4894	166	10	)	)	PUNCT
ejpam-4894	166	11	(	(	PUNCT
ejpam-4894	166	12	2023	2023	NUM
ejpam-4894	166	13	)	)	PUNCT
ejpam-4894	166	14	,	,	PUNCT
ejpam-4894	166	15	2763	2763	NUM
ejpam-4894	166	16	-	-	SYM
ejpam-4894	166	17	2774	2774	NUM
ejpam-4894	166	18	2769	2769	NUM
ejpam-4894	166	19	case	case	NOUN
ejpam-4894	166	20	2	2	NUM
ejpam-4894	166	21	.	.	PUNCT
ejpam-4894	166	22	suppose	suppose	VERB
ejpam-4894	166	23	that	that	SCONJ
ejpam-4894	166	24	n	n	PROPN
ejpam-4894	166	25	≡	≡	PROPN
ejpam-4894	166	26	1(mod3	1(mod3	NUM
ejpam-4894	166	27	)	)	PUNCT
ejpam-4894	166	28	.	.	PUNCT
ejpam-4894	167	1	clearly	clearly	ADV
ejpam-4894	167	2	if	if	SCONJ
ejpam-4894	167	3	n	n	NOUN
ejpam-4894	167	4	=	=	SYM
ejpam-4894	167	5	1	1	NUM
ejpam-4894	167	6	,	,	PUNCT
ejpam-4894	167	7	γcerp(p1	γcerp(p1	NOUN
ejpam-4894	167	8	)	)	PUNCT
ejpam-4894	167	9	=	=	SYM
ejpam-4894	167	10	1	1	X
ejpam-4894	167	11	.	.	PUNCT
ejpam-4894	167	12	suppose	suppose	VERB
ejpam-4894	167	13	that	that	SCONJ
ejpam-4894	167	14	n	n	PROPN
ejpam-4894	167	15	=	=	SYM
ejpam-4894	167	16	4	4	X
ejpam-4894	167	17	.	.	PUNCT
ejpam-4894	167	18	let	let	VERB
ejpam-4894	167	19	s1	s1	PROPN
ejpam-4894	167	20	=	=	SYM
ejpam-4894	167	21	{	{	PUNCT
ejpam-4894	167	22	v1	v1	PROPN
ejpam-4894	167	23	,	,	PUNCT
ejpam-4894	167	24	v2	v2	PROPN
ejpam-4894	167	25	,	,	PUNCT
ejpam-4894	167	26	v3	v3	PROPN
ejpam-4894	167	27	,	,	PUNCT
ejpam-4894	167	28	v4	v4	PROPN
ejpam-4894	167	29	}	}	PUNCT
ejpam-4894	167	30	.	.	PUNCT
ejpam-4894	168	1	observe	observe	VERB
ejpam-4894	168	2	that	that	SCONJ
ejpam-4894	168	3	every	every	DET
ejpam-4894	168	4	vertex	vertex	NOUN
ejpam-4894	168	5	in	in	ADP
ejpam-4894	168	6	s1	s1	PROPN
ejpam-4894	168	7	has	have	VERB
ejpam-4894	168	8	zero	zero	NUM
ejpam-4894	168	9	neighbor	neighbor	NOUN
ejpam-4894	168	10	in	in	ADP
ejpam-4894	168	11	v	v	NOUN
ejpam-4894	168	12	(	(	PUNCT
ejpam-4894	168	13	p4	p4	ADJ
ejpam-4894	168	14	)	)	PUNCT
ejpam-4894	168	15	\	\	NOUN
ejpam-4894	168	16	s1	s1	NOUN
ejpam-4894	168	17	,	,	PUNCT
ejpam-4894	168	18	this	this	PRON
ejpam-4894	168	19	means	mean	VERB
ejpam-4894	168	20	that	that	SCONJ
ejpam-4894	168	21	γcerp(s1	γcerp(s1	NOUN
ejpam-4894	168	22	)	)	PUNCT
ejpam-4894	169	1	=	=	SYM
ejpam-4894	169	2	|s1|	|s1|	NOUN
ejpam-4894	169	3	=	=	SYM
ejpam-4894	169	4	4	4	X
ejpam-4894	169	5	.	.	PUNCT
ejpam-4894	169	6	suppose	suppose	VERB
ejpam-4894	169	7	that	that	SCONJ
ejpam-4894	169	8	n	n	PROPN
ejpam-4894	169	9	≥	≥	NUM
ejpam-4894	169	10	4	4	NUM
ejpam-4894	169	11	.	.	PUNCT
ejpam-4894	170	1	let	let	VERB
ejpam-4894	170	2	q	q	NOUN
ejpam-4894	170	3	=	=	PUNCT
ejpam-4894	170	4	n	n	DET
ejpam-4894	170	5	4	4	NUM
ejpam-4894	170	6	and	and	CCONJ
ejpam-4894	170	7	r	r	NOUN
ejpam-4894	170	8	∈	∈	PROPN
ejpam-4894	170	9	{	{	PUNCT
ejpam-4894	170	10	1	1	NUM
ejpam-4894	170	11	,	,	PUNCT
ejpam-4894	170	12	2	2	NUM
ejpam-4894	170	13	,	,	PUNCT
ejpam-4894	170	14	.	.	PUNCT
ejpam-4894	170	15	.	.	PUNCT
ejpam-4894	171	1	.	.	PUNCT
ejpam-4894	172	1	,	,	PUNCT
ejpam-4894	172	2	q−	q−	PROPN
ejpam-4894	172	3	1	1	NUM
ejpam-4894	172	4	,	,	PUNCT
ejpam-4894	172	5	q	q	NOUN
ejpam-4894	172	6	}	}	PUNCT
ejpam-4894	172	7	.	.	PUNCT
ejpam-4894	173	1	then	then	ADV
ejpam-4894	173	2	let	let	VERB
ejpam-4894	173	3	us	we	PRON
ejpam-4894	173	4	denote	denote	VERB
ejpam-4894	173	5	a	a	DET
ejpam-4894	173	6	group	group	NOUN
ejpam-4894	173	7	of	of	ADP
ejpam-4894	173	8	vertices	vertex	NOUN
ejpam-4894	173	9	of	of	ADP
ejpam-4894	173	10	pn	pn	NOUN
ejpam-4894	173	11	into	into	ADP
ejpam-4894	173	12	q	q	PROPN
ejpam-4894	173	13	disjoint	disjoint	NOUN
ejpam-4894	173	14	subsets	subset	NOUN
ejpam-4894	173	15	as	as	ADP
ejpam-4894	173	16	gr	gr	PROPN
ejpam-4894	173	17	,	,	PUNCT
ejpam-4894	173	18	these	these	PRON
ejpam-4894	173	19	are	be	AUX
ejpam-4894	173	20	,	,	PUNCT
ejpam-4894	173	21	g1	g1	PROPN
ejpam-4894	173	22	=	=	SYM
ejpam-4894	173	23	{	{	PUNCT
ejpam-4894	173	24	v1	v1	PROPN
ejpam-4894	173	25	,	,	PUNCT
ejpam-4894	173	26	v2	v2	PROPN
ejpam-4894	173	27	,	,	PUNCT
ejpam-4894	173	28	v3	v3	PROPN
ejpam-4894	173	29	,	,	PUNCT
ejpam-4894	173	30	v4	v4	PROPN
ejpam-4894	173	31	}	}	PUNCT
ejpam-4894	173	32	g2	g2	PROPN
ejpam-4894	173	33	=	=	PRON
ejpam-4894	173	34	{	{	PUNCT
ejpam-4894	173	35	v5	v5	PROPN
ejpam-4894	173	36	,	,	PUNCT
ejpam-4894	173	37	v6	v6	NOUN
ejpam-4894	173	38	,	,	PUNCT
ejpam-4894	173	39	v7	v7	NUM
ejpam-4894	173	40	,	,	PUNCT
ejpam-4894	173	41	v8	v8	PROPN
ejpam-4894	173	42	}	}	PUNCT
ejpam-4894	173	43	...	...	PUNCT
ejpam-4894	174	1	gq	gq	PROPN
ejpam-4894	174	2	=	=	PRON
ejpam-4894	174	3	{	{	PUNCT
ejpam-4894	174	4	vn−3	vn−3	PROPN
ejpam-4894	174	5	,	,	PUNCT
ejpam-4894	174	6	vn−2	vn−2	PROPN
ejpam-4894	174	7	,	,	PUNCT
ejpam-4894	174	8	vn−1	vn−1	PROPN
ejpam-4894	174	9	,	,	PUNCT
ejpam-4894	174	10	vn	vn	NOUN
ejpam-4894	174	11	}	}	PUNCT
ejpam-4894	174	12	.	.	PUNCT
ejpam-4894	175	1	clearly	clearly	ADV
ejpam-4894	175	2	,	,	PUNCT
ejpam-4894	175	3	the	the	DET
ejpam-4894	175	4	set	set	NOUN
ejpam-4894	175	5	s	s	PART
ejpam-4894	175	6	′	′	NOUN
ejpam-4894	175	7	=	=	SYM
ejpam-4894	175	8	⋃n	⋃n	NOUN
ejpam-4894	175	9	r=1	r=1	NOUN
ejpam-4894	175	10	sr	sr	PROPN
ejpam-4894	175	11	is	be	AUX
ejpam-4894	175	12	a	a	DET
ejpam-4894	175	13	γcerp	γcerp	NOUN
ejpam-4894	175	14	-	-	PUNCT
ejpam-4894	175	15	set	set	NOUN
ejpam-4894	175	16	of	of	ADP
ejpam-4894	175	17	pn	pn	PROPN
ejpam-4894	175	18	since	since	SCONJ
ejpam-4894	175	19	ng[s	ng[s	PROPN
ejpam-4894	175	20	′	′	VERB
ejpam-4894	175	21	]	]	PUNCT
ejpam-4894	176	1	=	=	SYM
ejpam-4894	176	2	v	v	X
ejpam-4894	176	3	(	(	PUNCT
ejpam-4894	176	4	pn	pn	NOUN
ejpam-4894	176	5	)	)	PUNCT
ejpam-4894	176	6	and	and	CCONJ
ejpam-4894	176	7	every	every	DET
ejpam-4894	176	8	vertex	vertex	NOUN
ejpam-4894	176	9	vj	vj	X
ejpam-4894	176	10	∈	∈	PROPN
ejpam-4894	176	11	v	v	PROPN
ejpam-4894	176	12	(	(	PUNCT
ejpam-4894	176	13	pn	pn	PROPN
ejpam-4894	176	14	)	)	PUNCT
ejpam-4894	176	15	,	,	PUNCT
ejpam-4894	176	16	j	j	PROPN
ejpam-4894	176	17	∈	∈	PROPN
ejpam-4894	176	18	{	{	PUNCT
ejpam-4894	176	19	1	1	NUM
ejpam-4894	176	20	,	,	PUNCT
ejpam-4894	176	21	2	2	NUM
ejpam-4894	176	22	,	,	PUNCT
ejpam-4894	176	23	.	.	PUNCT
ejpam-4894	176	24	.	.	PUNCT
ejpam-4894	176	25	.	.	PUNCT
ejpam-4894	177	1	,	,	PUNCT
ejpam-4894	177	2	n−1	n−1	PROPN
ejpam-4894	177	3	,	,	PUNCT
ejpam-4894	177	4	n	n	CCONJ
ejpam-4894	177	5	}	}	PUNCT
ejpam-4894	177	6	has	have	VERB
ejpam-4894	177	7	zero	zero	NUM
ejpam-4894	177	8	neighbor	neighbor	NOUN
ejpam-4894	177	9	in	in	ADP
ejpam-4894	177	10	v	v	NOUN
ejpam-4894	177	11	(	(	PUNCT
ejpam-4894	177	12	pn)\s	pn)\s	NOUN
ejpam-4894	177	13	′	′	NUM
ejpam-4894	177	14	.	.	PUNCT
ejpam-4894	178	1	consequently	consequently	ADV
ejpam-4894	178	2	,	,	PUNCT
ejpam-4894	178	3	γcerp(pn	γcerp(pn	NOUN
ejpam-4894	178	4	)	)	PUNCT
ejpam-4894	178	5	=	=	SYM
ejpam-4894	178	6	|s′	|s′	NOUN
ejpam-4894	178	7	|=	|=	PUNCT
ejpam-4894	178	8	|v	|v	NOUN
ejpam-4894	178	9	(	(	PUNCT
ejpam-4894	178	10	pn)|=	pn)|=	X
ejpam-4894	178	11	n.	n.	NOUN
ejpam-4894	178	12	case	case	NOUN
ejpam-4894	178	13	3	3	X
ejpam-4894	178	14	.	.	PUNCT
ejpam-4894	178	15	suppose	suppose	VERB
ejpam-4894	178	16	that	that	SCONJ
ejpam-4894	178	17	n	n	NUM
ejpam-4894	178	18	≡	≡	PROPN
ejpam-4894	178	19	2	2	NUM
ejpam-4894	178	20	(	(	PUNCT
ejpam-4894	178	21	mod	mod	NOUN
ejpam-4894	178	22	3	3	NUM
ejpam-4894	178	23	)	)	PUNCT
ejpam-4894	178	24	.	.	PUNCT
ejpam-4894	179	1	clearly	clearly	ADV
ejpam-4894	179	2	,	,	PUNCT
ejpam-4894	179	3	if	if	SCONJ
ejpam-4894	179	4	n	n	NOUN
ejpam-4894	179	5	=	=	SYM
ejpam-4894	179	6	2	2	NUM
ejpam-4894	179	7	,	,	PUNCT
ejpam-4894	179	8	then	then	ADV
ejpam-4894	179	9	γcerp(p2	γcerp(p2	ADJ
ejpam-4894	179	10	)	)	PUNCT
ejpam-4894	179	11	=	=	SYM
ejpam-4894	180	1	2	2	X
ejpam-4894	180	2	.	.	PUNCT
ejpam-4894	180	3	suppose	suppose	VERB
ejpam-4894	180	4	that	that	SCONJ
ejpam-4894	180	5	n	n	NOUN
ejpam-4894	180	6	=	=	SYM
ejpam-4894	180	7	5	5	X
ejpam-4894	180	8	.	.	PUNCT
ejpam-4894	180	9	let	let	VERB
ejpam-4894	180	10	s1	s1	PROPN
ejpam-4894	180	11	=	=	SYM
ejpam-4894	180	12	{	{	PUNCT
ejpam-4894	180	13	v1	v1	PROPN
ejpam-4894	180	14	,	,	PUNCT
ejpam-4894	180	15	v2	v2	PROPN
ejpam-4894	180	16	,	,	PUNCT
ejpam-4894	180	17	v3	v3	PROPN
ejpam-4894	180	18	,	,	PUNCT
ejpam-4894	180	19	v4	v4	PROPN
ejpam-4894	180	20	,	,	PUNCT
ejpam-4894	180	21	v5	v5	PROPN
ejpam-4894	180	22	}	}	PUNCT
ejpam-4894	180	23	.	.	PUNCT
ejpam-4894	181	1	observe	observe	VERB
ejpam-4894	181	2	that	that	SCONJ
ejpam-4894	181	3	every	every	DET
ejpam-4894	181	4	vertex	vertex	NOUN
ejpam-4894	181	5	in	in	ADP
ejpam-4894	181	6	s1	s1	PROPN
ejpam-4894	181	7	has	have	VERB
ejpam-4894	181	8	zero	zero	NUM
ejpam-4894	181	9	neighbor	neighbor	NOUN
ejpam-4894	181	10	in	in	ADP
ejpam-4894	181	11	v	v	NOUN
ejpam-4894	181	12	(	(	PUNCT
ejpam-4894	181	13	p5	p5	ADJ
ejpam-4894	181	14	)	)	PUNCT
ejpam-4894	181	15	\	\	NOUN
ejpam-4894	181	16	s1	s1	NOUN
ejpam-4894	181	17	.	.	PUNCT
ejpam-4894	182	1	thus	thus	ADV
ejpam-4894	182	2	,	,	PUNCT
ejpam-4894	182	3	γcerp(s1	γcerp(s1	ADJ
ejpam-4894	182	4	)	)	PUNCT
ejpam-4894	183	1	=	=	SYM
ejpam-4894	183	2	|s1|	|s1|	NOUN
ejpam-4894	183	3	=	=	SYM
ejpam-4894	183	4	5	5	X
ejpam-4894	183	5	.	.	PUNCT
ejpam-4894	183	6	let	let	VERB
ejpam-4894	183	7	q	q	NOUN
ejpam-4894	183	8	=	=	PUNCT
ejpam-4894	183	9	n	n	NUM
ejpam-4894	183	10	5	5	NUM
ejpam-4894	183	11	and	and	CCONJ
ejpam-4894	183	12	r	r	NOUN
ejpam-4894	183	13	∈	∈	PROPN
ejpam-4894	183	14	{	{	PUNCT
ejpam-4894	183	15	1	1	NUM
ejpam-4894	183	16	,	,	PUNCT
ejpam-4894	183	17	2	2	NUM
ejpam-4894	183	18	,	,	PUNCT
ejpam-4894	183	19	.	.	PUNCT
ejpam-4894	183	20	.	.	PUNCT
ejpam-4894	184	1	.	.	PUNCT
ejpam-4894	185	1	,	,	PUNCT
ejpam-4894	185	2	q	q	X
ejpam-4894	186	1	−	−	PROPN
ejpam-4894	186	2	1	1	NUM
ejpam-4894	186	3	,	,	PUNCT
ejpam-4894	186	4	q	q	NOUN
ejpam-4894	186	5	}	}	PUNCT
ejpam-4894	186	6	.	.	PUNCT
ejpam-4894	187	1	then	then	ADV
ejpam-4894	187	2	let	let	VERB
ejpam-4894	187	3	us	we	PRON
ejpam-4894	187	4	denote	denote	VERB
ejpam-4894	187	5	a	a	DET
ejpam-4894	187	6	group	group	NOUN
ejpam-4894	187	7	of	of	ADP
ejpam-4894	187	8	vertices	vertex	NOUN
ejpam-4894	187	9	of	of	ADP
ejpam-4894	187	10	pn	pn	NOUN
ejpam-4894	187	11	into	into	ADP
ejpam-4894	187	12	q	q	PROPN
ejpam-4894	187	13	disjoint	disjoint	NOUN
ejpam-4894	187	14	subsets	subset	NOUN
ejpam-4894	187	15	as	as	ADP
ejpam-4894	187	16	gr	gr	PROPN
ejpam-4894	187	17	,	,	PUNCT
ejpam-4894	187	18	these	these	PRON
ejpam-4894	187	19	are	be	AUX
ejpam-4894	187	20	,	,	PUNCT
ejpam-4894	187	21	g1	g1	PROPN
ejpam-4894	187	22	=	=	SYM
ejpam-4894	187	23	{	{	PUNCT
ejpam-4894	187	24	v1	v1	PROPN
ejpam-4894	187	25	,	,	PUNCT
ejpam-4894	187	26	v2	v2	PROPN
ejpam-4894	187	27	,	,	PUNCT
ejpam-4894	187	28	v3	v3	PROPN
ejpam-4894	187	29	,	,	PUNCT
ejpam-4894	187	30	v4	v4	PROPN
ejpam-4894	187	31	,	,	PUNCT
ejpam-4894	187	32	v5	v5	PROPN
ejpam-4894	187	33	}	}	PUNCT
ejpam-4894	187	34	g2	g2	PROPN
ejpam-4894	187	35	=	=	SYM
ejpam-4894	187	36	{	{	PUNCT
ejpam-4894	187	37	v6	v6	NOUN
ejpam-4894	187	38	,	,	PUNCT
ejpam-4894	187	39	v7	v7	NUM
ejpam-4894	187	40	,	,	PUNCT
ejpam-4894	187	41	v8	v8	PROPN
ejpam-4894	187	42	,	,	PUNCT
ejpam-4894	187	43	v9	v9	PROPN
ejpam-4894	187	44	,	,	PUNCT
ejpam-4894	187	45	v10	v10	PROPN
ejpam-4894	187	46	}	}	PUNCT
ejpam-4894	187	47	...	...	PUNCT
ejpam-4894	188	1	gq	gq	PROPN
ejpam-4894	188	2	=	=	PRON
ejpam-4894	188	3	{	{	PUNCT
ejpam-4894	188	4	vn−5	vn−5	NOUN
ejpam-4894	188	5	,	,	PUNCT
ejpam-4894	188	6	vn−4	vn−4	NOUN
ejpam-4894	188	7	,	,	PUNCT
ejpam-4894	188	8	vn−3	vn−3	PROPN
ejpam-4894	188	9	,	,	PUNCT
ejpam-4894	188	10	vn−2	vn−2	PROPN
ejpam-4894	188	11	,	,	PUNCT
ejpam-4894	188	12	vn−1	vn−1	PROPN
ejpam-4894	188	13	,	,	PUNCT
ejpam-4894	188	14	vn	vn	NOUN
ejpam-4894	188	15	}	}	PUNCT
ejpam-4894	188	16	.	.	PUNCT
ejpam-4894	189	1	clearly	clearly	ADV
ejpam-4894	189	2	,	,	PUNCT
ejpam-4894	189	3	the	the	DET
ejpam-4894	189	4	set	set	NOUN
ejpam-4894	189	5	s	s	VERB
ejpam-4894	189	6	′′	′′	PROPN
ejpam-4894	189	7	=	=	SYM
ejpam-4894	189	8	⋃n	⋃n	PROPN
ejpam-4894	189	9	r=1	r=1	PROPN
ejpam-4894	189	10	sr	sr	PROPN
ejpam-4894	189	11	,	,	PUNCT
ejpam-4894	189	12	r	r	NOUN
ejpam-4894	189	13	∈	∈	PROPN
ejpam-4894	189	14	{	{	PUNCT
ejpam-4894	189	15	1	1	NUM
ejpam-4894	189	16	,	,	PUNCT
ejpam-4894	189	17	2	2	NUM
ejpam-4894	189	18	,	,	PUNCT
ejpam-4894	189	19	.	.	PUNCT
ejpam-4894	189	20	.	.	PUNCT
ejpam-4894	189	21	.	.	PUNCT
ejpam-4894	190	1	,	,	PUNCT
ejpam-4894	190	2	q	q	X
ejpam-4894	191	1	−	−	PROPN
ejpam-4894	191	2	1	1	NUM
ejpam-4894	191	3	,	,	PUNCT
ejpam-4894	191	4	q	q	X
ejpam-4894	191	5	}	}	PUNCT
ejpam-4894	191	6	is	be	AUX
ejpam-4894	191	7	a	a	DET
ejpam-4894	191	8	γcerp	γcerp	NOUN
ejpam-4894	191	9	-	-	PUNCT
ejpam-4894	191	10	set	set	NOUN
ejpam-4894	191	11	of	of	ADP
ejpam-4894	191	12	pn	pn	PROPN
ejpam-4894	191	13	since	since	SCONJ
ejpam-4894	191	14	ng[s	ng[s	PROPN
ejpam-4894	191	15	′′	′′	PROPN
ejpam-4894	191	16	]	]	PUNCT
ejpam-4894	192	1	=	=	SYM
ejpam-4894	192	2	v	v	X
ejpam-4894	192	3	(	(	PUNCT
ejpam-4894	192	4	pn	pn	NOUN
ejpam-4894	192	5	)	)	PUNCT
ejpam-4894	192	6	and	and	CCONJ
ejpam-4894	192	7	every	every	DET
ejpam-4894	192	8	vertex	vertex	NOUN
ejpam-4894	192	9	vj	vj	X
ejpam-4894	192	10	∈	∈	PROPN
ejpam-4894	192	11	v	v	PROPN
ejpam-4894	192	12	(	(	PUNCT
ejpam-4894	192	13	pn	pn	PROPN
ejpam-4894	192	14	)	)	PUNCT
ejpam-4894	192	15	,	,	PUNCT
ejpam-4894	192	16	j	j	PROPN
ejpam-4894	192	17	∈	∈	PROPN
ejpam-4894	192	18	{	{	PUNCT
ejpam-4894	192	19	1	1	NUM
ejpam-4894	192	20	,	,	PUNCT
ejpam-4894	192	21	2	2	NUM
ejpam-4894	192	22	,	,	PUNCT
ejpam-4894	192	23	.	.	PUNCT
ejpam-4894	192	24	.	.	PUNCT
ejpam-4894	192	25	.	.	PUNCT
ejpam-4894	193	1	,	,	PUNCT
ejpam-4894	193	2	n−1	n−1	PROPN
ejpam-4894	193	3	,	,	PUNCT
ejpam-4894	193	4	n	n	CCONJ
ejpam-4894	193	5	}	}	PUNCT
ejpam-4894	193	6	has	have	AUX
ejpam-4894	193	7	zero	zero	NUM
ejpam-4894	193	8	neighbor	neighbor	NOUN
ejpam-4894	193	9	in	in	ADP
ejpam-4894	193	10	v	v	NOUN
ejpam-4894	193	11	(	(	PUNCT
ejpam-4894	193	12	pn)\s	pn)\s	VERB
ejpam-4894	193	13	′′	′′	PROPN
ejpam-4894	193	14	.	.	PUNCT
ejpam-4894	194	1	consequently	consequently	ADV
ejpam-4894	194	2	,	,	PUNCT
ejpam-4894	194	3	γcerp(pn	γcerp(pn	NOUN
ejpam-4894	194	4	)	)	PUNCT
ejpam-4894	194	5	=	=	PUNCT
ejpam-4894	195	1	|s′′	|s′′	PRON
ejpam-4894	195	2	|=	|=	X
ejpam-4894	195	3	|v	|v	X
ejpam-4894	195	4	(	(	PUNCT
ejpam-4894	195	5	pn)|=	pn)|=	X
ejpam-4894	195	6	n.	n.	NOUN
ejpam-4894	195	7	theorem	theorem	VERB
ejpam-4894	195	8	5	5	NUM
ejpam-4894	195	9	.	.	X
ejpam-4894	195	10	for	for	ADP
ejpam-4894	195	11	a	a	DET
ejpam-4894	195	12	cycle	cycle	NOUN
ejpam-4894	195	13	cn	cn	NOUN
ejpam-4894	195	14	of	of	ADP
ejpam-4894	195	15	order	order	NOUN
ejpam-4894	195	16	n	n	PRON
ejpam-4894	195	17	≥	≥	NOUN
ejpam-4894	195	18	3	3	NUM
ejpam-4894	195	19	,	,	PUNCT
ejpam-4894	195	20	γcerp(cn	γcerp(cn	NOUN
ejpam-4894	195	21	)	)	PUNCT
ejpam-4894	195	22	=	=	PUNCT
ejpam-4894	195	23	{	{	PUNCT
ejpam-4894	195	24	n	n	NOUN
ejpam-4894	195	25	3	3	NUM
ejpam-4894	195	26	if	if	SCONJ
ejpam-4894	195	27	n	n	PRON
ejpam-4894	195	28	≡	≡	PROPN
ejpam-4894	195	29	0	0	PUNCT
ejpam-4894	195	30	(	(	PUNCT
ejpam-4894	195	31	mod	mod	NOUN
ejpam-4894	195	32	3	3	NUM
ejpam-4894	195	33	)	)	PUNCT
ejpam-4894	195	34	;	;	PUNCT
ejpam-4894	195	35	n	n	X
ejpam-4894	195	36	otherwise	otherwise	ADV
ejpam-4894	195	37	.	.	PUNCT
ejpam-4894	196	1	proof	proof	NOUN
ejpam-4894	196	2	.	.	PUNCT
ejpam-4894	197	1	suppose	suppose	VERB
ejpam-4894	197	2	that	that	SCONJ
ejpam-4894	197	3	v	v	INTJ
ejpam-4894	197	4	(	(	PUNCT
ejpam-4894	197	5	cn	cn	PROPN
ejpam-4894	197	6	)	)	PUNCT
ejpam-4894	197	7	=	=	SYM
ejpam-4894	197	8	{	{	PUNCT
ejpam-4894	197	9	v1	v1	PROPN
ejpam-4894	197	10	,	,	PUNCT
ejpam-4894	197	11	v2	v2	PROPN
ejpam-4894	197	12	,	,	PUNCT
ejpam-4894	197	13	.	.	PUNCT
ejpam-4894	197	14	.	.	PUNCT
ejpam-4894	197	15	.	.	PUNCT
ejpam-4894	198	1	,	,	PUNCT
ejpam-4894	198	2	vn−1	vn−1	PROPN
ejpam-4894	198	3	,	,	PUNCT
ejpam-4894	198	4	vn	vn	VERB
ejpam-4894	198	5	}	}	PUNCT
ejpam-4894	198	6	such	such	ADJ
ejpam-4894	198	7	that	that	SCONJ
ejpam-4894	198	8	deg(vi	deg(vi	NOUN
ejpam-4894	198	9	)	)	PUNCT
ejpam-4894	198	10	=	=	SYM
ejpam-4894	198	11	2	2	NUM
ejpam-4894	198	12	,	,	PUNCT
ejpam-4894	198	13	i	i	PRON
ejpam-4894	198	14	∈	∈	PROPN
ejpam-4894	198	15	{	{	PUNCT
ejpam-4894	198	16	1	1	NUM
ejpam-4894	198	17	,	,	PUNCT
ejpam-4894	198	18	2	2	NUM
ejpam-4894	198	19	,	,	PUNCT
ejpam-4894	198	20	.	.	PUNCT
ejpam-4894	198	21	.	.	PUNCT
ejpam-4894	199	1	.	.	PUNCT
ejpam-4894	200	1	,	,	PUNCT
ejpam-4894	200	2	n−	n−	NOUN
ejpam-4894	200	3	1	1	NUM
ejpam-4894	200	4	,	,	PUNCT
ejpam-4894	200	5	n	n	CCONJ
ejpam-4894	200	6	}	}	PUNCT
ejpam-4894	200	7	.	.	PUNCT
ejpam-4894	201	1	consider	consider	VERB
ejpam-4894	201	2	the	the	DET
ejpam-4894	201	3	following	follow	VERB
ejpam-4894	201	4	cases	case	NOUN
ejpam-4894	201	5	:	:	PUNCT
ejpam-4894	201	6	case	case	NOUN
ejpam-4894	201	7	1	1	NUM
ejpam-4894	201	8	:	:	PUNCT
ejpam-4894	201	9	suppose	suppose	VERB
ejpam-4894	201	10	that	that	SCONJ
ejpam-4894	201	11	n	n	PROPN
ejpam-4894	201	12	≡	≡	PROPN
ejpam-4894	201	13	0	0	PUNCT
ejpam-4894	202	1	(	(	PUNCT
ejpam-4894	202	2	mod	mod	NOUN
ejpam-4894	202	3	3	3	NUM
ejpam-4894	202	4	)	)	PUNCT
ejpam-4894	202	5	.	.	PUNCT
ejpam-4894	203	1	if	if	SCONJ
ejpam-4894	203	2	n	n	NUM
ejpam-4894	203	3	=	=	SYM
ejpam-4894	203	4	3	3	X
ejpam-4894	203	5	.	.	PUNCT
ejpam-4894	203	6	then	then	ADV
ejpam-4894	203	7	γcerp(c3	γcerp(c3	NUM
ejpam-4894	203	8	)	)	PUNCT
ejpam-4894	203	9	=	=	SYM
ejpam-4894	203	10	1	1	NUM
ejpam-4894	203	11	=	=	SYM
ejpam-4894	203	12	3	3	NUM
ejpam-4894	203	13	3	3	NUM
ejpam-4894	203	14	.	.	PUNCT
ejpam-4894	203	15	suppose	suppose	VERB
ejpam-4894	203	16	that	that	SCONJ
ejpam-4894	203	17	n	n	PROPN
ejpam-4894	203	18	>	>	X
ejpam-4894	203	19	3	3	X
ejpam-4894	203	20	.	.	PUNCT
ejpam-4894	203	21	let	let	VERB
ejpam-4894	203	22	q	q	NOUN
ejpam-4894	203	23	=	=	PUNCT
ejpam-4894	203	24	n	n	NUM
ejpam-4894	203	25	3	3	NUM
ejpam-4894	203	26	and	and	CCONJ
ejpam-4894	203	27	r	r	PROPN
ejpam-4894	203	28	∈	∈	PROPN
ejpam-4894	203	29	{	{	PUNCT
ejpam-4894	203	30	1	1	NUM
ejpam-4894	203	31	,	,	PUNCT
ejpam-4894	203	32	2	2	NUM
ejpam-4894	203	33	,	,	PUNCT
ejpam-4894	203	34	.	.	PUNCT
ejpam-4894	203	35	.	.	PUNCT
ejpam-4894	204	1	.	.	PUNCT
ejpam-4894	205	1	,	,	PUNCT
ejpam-4894	205	2	q	q	X
ejpam-4894	206	1	−	−	PROPN
ejpam-4894	206	2	1	1	NUM
ejpam-4894	206	3	,	,	PUNCT
ejpam-4894	206	4	q	q	NOUN
ejpam-4894	206	5	}	}	PUNCT
ejpam-4894	206	6	.	.	PUNCT
ejpam-4894	207	1	then	then	ADV
ejpam-4894	207	2	let	let	VERB
ejpam-4894	207	3	us	we	PRON
ejpam-4894	207	4	denote	denote	VERB
ejpam-4894	207	5	a	a	DET
ejpam-4894	207	6	group	group	NOUN
ejpam-4894	207	7	of	of	ADP
ejpam-4894	207	8	vertices	vertex	NOUN
ejpam-4894	207	9	of	of	ADP
ejpam-4894	207	10	cn	cn	PROPN
ejpam-4894	207	11	into	into	ADP
ejpam-4894	207	12	q	q	PROPN
ejpam-4894	207	13	disjoint	disjoint	PROPN
ejpam-4894	207	14	subsets	subsets	PROPN
ejpam-4894	207	15	hr	hr	NOUN
ejpam-4894	207	16	,	,	PUNCT
ejpam-4894	207	17	these	these	PRON
ejpam-4894	207	18	are	be	AUX
ejpam-4894	207	19	,	,	PUNCT
ejpam-4894	207	20	h1	h1	PROPN
ejpam-4894	207	21	=	=	SYM
ejpam-4894	207	22	{	{	PUNCT
ejpam-4894	207	23	v1	v1	PROPN
ejpam-4894	207	24	,	,	PUNCT
ejpam-4894	207	25	v2	v2	PROPN
ejpam-4894	207	26	,	,	PUNCT
ejpam-4894	207	27	v3	v3	PROPN
ejpam-4894	207	28	}	}	PUNCT
ejpam-4894	208	1	j.	j.	PROPN
ejpam-4894	208	2	hamja	hamja	PROPN
ejpam-4894	208	3	/	/	SYM
ejpam-4894	208	4	eur	eur	PROPN
ejpam-4894	208	5	.	.	PUNCT
ejpam-4894	209	1	j.	j.	PROPN
ejpam-4894	209	2	pure	pure	PROPN
ejpam-4894	209	3	appl	appl	PROPN
ejpam-4894	209	4	.	.	PROPN
ejpam-4894	209	5	math	math	PROPN
ejpam-4894	209	6	,	,	PUNCT
ejpam-4894	209	7	16	16	NUM
ejpam-4894	209	8	(	(	PUNCT
ejpam-4894	209	9	4	4	NUM
ejpam-4894	209	10	)	)	PUNCT
ejpam-4894	209	11	(	(	PUNCT
ejpam-4894	209	12	2023	2023	NUM
ejpam-4894	209	13	)	)	PUNCT
ejpam-4894	209	14	,	,	PUNCT
ejpam-4894	209	15	2763	2763	NUM
ejpam-4894	209	16	-	-	SYM
ejpam-4894	209	17	2774	2774	NUM
ejpam-4894	209	18	2770	2770	NUM
ejpam-4894	209	19	h2	h2	NOUN
ejpam-4894	209	20	=	=	PUNCT
ejpam-4894	209	21	{	{	PUNCT
ejpam-4894	209	22	v4	v4	PROPN
ejpam-4894	209	23	,	,	PUNCT
ejpam-4894	209	24	v5	v5	PROPN
ejpam-4894	209	25	,	,	PUNCT
ejpam-4894	209	26	v6	v6	NOUN
ejpam-4894	209	27	}	}	PUNCT
ejpam-4894	209	28	...	...	PUNCT
ejpam-4894	210	1	hq	hq	INTJ
ejpam-4894	210	2	=	=	PUNCT
ejpam-4894	210	3	{	{	PUNCT
ejpam-4894	210	4	vn−2	vn−2	PROPN
ejpam-4894	210	5	,	,	PUNCT
ejpam-4894	210	6	vn−1	vn−1	PROPN
ejpam-4894	210	7	,	,	PUNCT
ejpam-4894	210	8	vn	vn	NOUN
ejpam-4894	210	9	}	}	PUNCT
ejpam-4894	210	10	clearly	clearly	ADV
ejpam-4894	210	11	,	,	PUNCT
ejpam-4894	210	12	the	the	DET
ejpam-4894	210	13	set	set	NOUN
ejpam-4894	210	14	s	s	PART
ejpam-4894	210	15	=	=	PUNCT
ejpam-4894	210	16	{	{	PUNCT
ejpam-4894	210	17	v2	v2	PROPN
ejpam-4894	210	18	,	,	PUNCT
ejpam-4894	210	19	v5	v5	NOUN
ejpam-4894	210	20	,	,	PUNCT
ejpam-4894	210	21	.	.	PUNCT
ejpam-4894	210	22	.	.	PUNCT
ejpam-4894	211	1	.	.	PUNCT
ejpam-4894	212	1	,	,	PUNCT
ejpam-4894	212	2	vn−4	vn−4	NOUN
ejpam-4894	212	3	,	,	PUNCT
ejpam-4894	212	4	vn−1	vn−1	ADJ
ejpam-4894	212	5	}	}	PUNCT
ejpam-4894	212	6	⊆	⊆	NUM
ejpam-4894	212	7	v	v	NOUN
ejpam-4894	212	8	(	(	PUNCT
ejpam-4894	212	9	cn	cn	PROPN
ejpam-4894	212	10	)	)	PUNCT
ejpam-4894	212	11	is	be	AUX
ejpam-4894	212	12	a	a	DET
ejpam-4894	212	13	γcerp	γcerp	NOUN
ejpam-4894	212	14	-	-	PUNCT
ejpam-4894	212	15	set	set	NOUN
ejpam-4894	212	16	of	of	ADP
ejpam-4894	212	17	cn	cn	PROPN
ejpam-4894	212	18	since	since	SCONJ
ejpam-4894	212	19	ng[s	ng[	NOUN
ejpam-4894	212	20	]	]	PUNCT
ejpam-4894	212	21	=	=	SYM
ejpam-4894	212	22	v	v	X
ejpam-4894	212	23	(	(	PUNCT
ejpam-4894	212	24	cn	cn	PROPN
ejpam-4894	212	25	)	)	PUNCT
ejpam-4894	212	26	and	and	CCONJ
ejpam-4894	212	27	every	every	DET
ejpam-4894	212	28	vertex	vertex	NOUN
ejpam-4894	212	29	vj	vj	X
ejpam-4894	212	30	∈	∈	PROPN
ejpam-4894	212	31	v	v	PROPN
ejpam-4894	212	32	(	(	PUNCT
ejpam-4894	212	33	cn	cn	PROPN
ejpam-4894	212	34	)	)	PUNCT
ejpam-4894	212	35	,	,	PUNCT
ejpam-4894	212	36	j	j	PROPN
ejpam-4894	212	37	∈	∈	PROPN
ejpam-4894	212	38	{	{	PUNCT
ejpam-4894	212	39	2	2	NUM
ejpam-4894	212	40	,	,	PUNCT
ejpam-4894	212	41	5	5	NUM
ejpam-4894	212	42	,	,	PUNCT
ejpam-4894	212	43	.	.	PUNCT
ejpam-4894	212	44	.	.	PUNCT
ejpam-4894	213	1	.	.	PUNCT
ejpam-4894	214	1	,	,	PUNCT
ejpam-4894	215	1	n	n	CCONJ
ejpam-4894	215	2	−	−	PROPN
ejpam-4894	215	3	4	4	NUM
ejpam-4894	215	4	,	,	PUNCT
ejpam-4894	215	5	n	n	CCONJ
ejpam-4894	215	6	−	−	PROPN
ejpam-4894	215	7	1	1	NUM
ejpam-4894	215	8	}	}	PUNCT
ejpam-4894	215	9	has	have	VERB
ejpam-4894	215	10	two	two	NUM
ejpam-4894	215	11	neighbors	neighbor	NOUN
ejpam-4894	215	12	in	in	ADP
ejpam-4894	215	13	v	v	NOUN
ejpam-4894	215	14	(	(	PUNCT
ejpam-4894	215	15	cn)\s	cn)\s	NOUN
ejpam-4894	215	16	.	.	PUNCT
ejpam-4894	216	1	furthermore	furthermore	ADV
ejpam-4894	216	2	,	,	PUNCT
ejpam-4894	216	3	every	every	DET
ejpam-4894	216	4	vertex	vertex	NOUN
ejpam-4894	216	5	vi	vi	NOUN
ejpam-4894	216	6	∈	∈	PROPN
ejpam-4894	216	7	v	v	NOUN
ejpam-4894	216	8	(	(	PUNCT
ejpam-4894	216	9	cn)\s	cn)\s	PROPN
ejpam-4894	216	10	,	,	PUNCT
ejpam-4894	216	11	i	i	PRON
ejpam-4894	216	12	∈	∈	PROPN
ejpam-4894	216	13	{	{	PUNCT
ejpam-4894	216	14	1	1	NUM
ejpam-4894	216	15	,	,	PUNCT
ejpam-4894	216	16	3	3	NUM
ejpam-4894	216	17	,	,	PUNCT
ejpam-4894	216	18	4	4	NUM
ejpam-4894	216	19	,	,	PUNCT
ejpam-4894	216	20	6	6	NUM
ejpam-4894	216	21	,	,	PUNCT
ejpam-4894	216	22	.	.	PUNCT
ejpam-4894	216	23	.	.	PUNCT
ejpam-4894	217	1	.	.	PUNCT
ejpam-4894	217	2	,	,	PUNCT
ejpam-4894	218	1	n−5	n−5	PROPN
ejpam-4894	218	2	,	,	PUNCT
ejpam-4894	218	3	n−3	n−3	PROPN
ejpam-4894	218	4	,	,	PUNCT
ejpam-4894	218	5	n−2	n−2	PROPN
ejpam-4894	218	6	,	,	PUNCT
ejpam-4894	218	7	n	n	CCONJ
ejpam-4894	218	8	}	}	PUNCT
ejpam-4894	218	9	is	be	AUX
ejpam-4894	218	10	dominated	dominate	VERB
ejpam-4894	218	11	by	by	ADP
ejpam-4894	218	12	exactly	exactly	ADV
ejpam-4894	218	13	one	one	NUM
ejpam-4894	218	14	vertex	vertex	NOUN
ejpam-4894	218	15	in	in	ADP
ejpam-4894	218	16	s.	s.	PROPN
ejpam-4894	218	17	therefore	therefore	ADV
ejpam-4894	218	18	,	,	PUNCT
ejpam-4894	218	19	γcerp(cn	γcerp(cn	NOUN
ejpam-4894	218	20	)	)	PUNCT
ejpam-4894	218	21	=	=	PUNCT
ejpam-4894	218	22	|s|	|s|	NOUN
ejpam-4894	218	23	=	=	PROPN
ejpam-4894	218	24	n	n	PRON
ejpam-4894	218	25	3	3	NUM
ejpam-4894	218	26	.	.	PUNCT
ejpam-4894	219	1	case	case	NOUN
ejpam-4894	219	2	2	2	X
ejpam-4894	219	3	.	.	PUNCT
ejpam-4894	219	4	suppose	suppose	VERB
ejpam-4894	219	5	that	that	SCONJ
ejpam-4894	219	6	n	n	PROPN
ejpam-4894	219	7	≡	≡	PROPN
ejpam-4894	219	8	1(mod3	1(mod3	NUM
ejpam-4894	219	9	)	)	PUNCT
ejpam-4894	219	10	.	.	PUNCT
ejpam-4894	220	1	suppose	suppose	VERB
ejpam-4894	220	2	that	that	SCONJ
ejpam-4894	220	3	n	n	PROPN
ejpam-4894	220	4	=	=	SYM
ejpam-4894	220	5	4	4	X
ejpam-4894	220	6	.	.	PUNCT
ejpam-4894	220	7	let	let	VERB
ejpam-4894	220	8	s1	s1	PROPN
ejpam-4894	220	9	=	=	SYM
ejpam-4894	220	10	{	{	PUNCT
ejpam-4894	220	11	v1	v1	PROPN
ejpam-4894	220	12	,	,	PUNCT
ejpam-4894	220	13	v2	v2	PROPN
ejpam-4894	220	14	,	,	PUNCT
ejpam-4894	220	15	v3	v3	PROPN
ejpam-4894	220	16	,	,	PUNCT
ejpam-4894	220	17	v4	v4	PROPN
ejpam-4894	220	18	}	}	PUNCT
ejpam-4894	220	19	.	.	PUNCT
ejpam-4894	221	1	observe	observe	VERB
ejpam-4894	221	2	that	that	SCONJ
ejpam-4894	221	3	every	every	DET
ejpam-4894	221	4	vertex	vertex	NOUN
ejpam-4894	221	5	in	in	ADP
ejpam-4894	221	6	∈	∈	PROPN
ejpam-4894	221	7	s1	s1	NOUN
ejpam-4894	221	8	has	have	VERB
ejpam-4894	221	9	zero	zero	NUM
ejpam-4894	221	10	neighbor	neighbor	NOUN
ejpam-4894	221	11	in	in	ADP
ejpam-4894	221	12	v	v	PROPN
ejpam-4894	221	13	(	(	PUNCT
ejpam-4894	221	14	c4	c4	NOUN
ejpam-4894	221	15	)	)	PUNCT
ejpam-4894	221	16	\	\	NOUN
ejpam-4894	221	17	s1	s1	NOUN
ejpam-4894	221	18	.	.	PUNCT
ejpam-4894	222	1	this	this	PRON
ejpam-4894	222	2	means	mean	VERB
ejpam-4894	222	3	that	that	SCONJ
ejpam-4894	222	4	γcerp(s1	γcerp(s1	NOUN
ejpam-4894	222	5	)	)	PUNCT
ejpam-4894	223	1	=	=	SYM
ejpam-4894	223	2	|s1|	|s1|	NOUN
ejpam-4894	223	3	=	=	SYM
ejpam-4894	223	4	4	4	X
ejpam-4894	223	5	.	.	PUNCT
ejpam-4894	223	6	suppose	suppose	VERB
ejpam-4894	223	7	that	that	SCONJ
ejpam-4894	223	8	n	n	PROPN
ejpam-4894	223	9	>	>	X
ejpam-4894	223	10	4	4	X
ejpam-4894	223	11	.	.	PUNCT
ejpam-4894	223	12	let	let	VERB
ejpam-4894	223	13	q	q	NOUN
ejpam-4894	223	14	=	=	PUNCT
ejpam-4894	223	15	n	n	PRON
ejpam-4894	223	16	4	4	NUM
ejpam-4894	223	17	and	and	CCONJ
ejpam-4894	223	18	r	r	NOUN
ejpam-4894	223	19	∈	∈	PROPN
ejpam-4894	223	20	{	{	PUNCT
ejpam-4894	223	21	1	1	NUM
ejpam-4894	223	22	,	,	PUNCT
ejpam-4894	223	23	2	2	NUM
ejpam-4894	223	24	,	,	PUNCT
ejpam-4894	223	25	.	.	PUNCT
ejpam-4894	223	26	.	.	PUNCT
ejpam-4894	224	1	.	.	PUNCT
ejpam-4894	225	1	,	,	PUNCT
ejpam-4894	225	2	q	q	X
ejpam-4894	226	1	−	−	PROPN
ejpam-4894	226	2	1	1	NUM
ejpam-4894	226	3	,	,	PUNCT
ejpam-4894	226	4	q	q	NOUN
ejpam-4894	226	5	}	}	PUNCT
ejpam-4894	226	6	.	.	PUNCT
ejpam-4894	227	1	then	then	ADV
ejpam-4894	227	2	let	let	VERB
ejpam-4894	227	3	us	we	PRON
ejpam-4894	227	4	denote	denote	VERB
ejpam-4894	227	5	a	a	DET
ejpam-4894	227	6	group	group	NOUN
ejpam-4894	227	7	of	of	ADP
ejpam-4894	227	8	vertices	vertex	NOUN
ejpam-4894	227	9	of	of	ADP
ejpam-4894	227	10	cn	cn	PROPN
ejpam-4894	227	11	into	into	ADP
ejpam-4894	227	12	q	q	PROPN
ejpam-4894	227	13	disjoint	disjoint	NOUN
ejpam-4894	227	14	subsets	subset	NOUN
ejpam-4894	227	15	as	as	ADP
ejpam-4894	227	16	hr	hr	NOUN
ejpam-4894	227	17	,	,	PUNCT
ejpam-4894	227	18	these	these	PRON
ejpam-4894	227	19	are	be	AUX
ejpam-4894	227	20	,	,	PUNCT
ejpam-4894	227	21	h1	h1	PROPN
ejpam-4894	227	22	=	=	SYM
ejpam-4894	227	23	{	{	PUNCT
ejpam-4894	227	24	v1	v1	PROPN
ejpam-4894	227	25	,	,	PUNCT
ejpam-4894	227	26	v2	v2	PROPN
ejpam-4894	227	27	,	,	PUNCT
ejpam-4894	227	28	v3	v3	PROPN
ejpam-4894	227	29	,	,	PUNCT
ejpam-4894	227	30	v4	v4	PROPN
ejpam-4894	227	31	}	}	PUNCT
ejpam-4894	227	32	h2	h2	NOUN
ejpam-4894	227	33	=	=	SYM
ejpam-4894	227	34	{	{	PUNCT
ejpam-4894	227	35	v5	v5	PROPN
ejpam-4894	227	36	,	,	PUNCT
ejpam-4894	227	37	v6	v6	NOUN
ejpam-4894	227	38	,	,	PUNCT
ejpam-4894	227	39	v7	v7	NUM
ejpam-4894	227	40	,	,	PUNCT
ejpam-4894	227	41	v8	v8	PROPN
ejpam-4894	227	42	}	}	PUNCT
ejpam-4894	227	43	...	...	PUNCT
ejpam-4894	228	1	hq	hq	NOUN
ejpam-4894	228	2	=	=	PUNCT
ejpam-4894	228	3	{	{	PUNCT
ejpam-4894	228	4	vn−3	vn−3	PROPN
ejpam-4894	228	5	,	,	PUNCT
ejpam-4894	228	6	vn−2	vn−2	PROPN
ejpam-4894	228	7	,	,	PUNCT
ejpam-4894	228	8	vn−1	vn−1	PROPN
ejpam-4894	228	9	,	,	PUNCT
ejpam-4894	228	10	vn	vn	NOUN
ejpam-4894	228	11	}	}	PUNCT
ejpam-4894	228	12	.	.	PUNCT
ejpam-4894	229	1	clearly	clearly	ADV
ejpam-4894	229	2	,	,	PUNCT
ejpam-4894	229	3	the	the	DET
ejpam-4894	229	4	set	set	NOUN
ejpam-4894	229	5	s∗	s∗	PROPN
ejpam-4894	229	6	=	=	SYM
ejpam-4894	229	7	⋃n	⋃n	NOUN
ejpam-4894	229	8	r=1	r=1	PROPN
ejpam-4894	229	9	sr	sr	PROPN
ejpam-4894	229	10	is	be	AUX
ejpam-4894	229	11	a	a	DET
ejpam-4894	229	12	γcerp	γcerp	NOUN
ejpam-4894	229	13	-	-	PUNCT
ejpam-4894	229	14	set	set	NOUN
ejpam-4894	229	15	of	of	ADP
ejpam-4894	229	16	cn	cn	PROPN
ejpam-4894	229	17	since	since	SCONJ
ejpam-4894	229	18	ng[s	ng[s	PROPN
ejpam-4894	229	19	∗	∗	NOUN
ejpam-4894	229	20	]	]	PUNCT
ejpam-4894	229	21	=	=	SYM
ejpam-4894	229	22	v	v	X
ejpam-4894	229	23	(	(	PUNCT
ejpam-4894	229	24	cn	cn	PROPN
ejpam-4894	229	25	)	)	PUNCT
ejpam-4894	229	26	and	and	CCONJ
ejpam-4894	229	27	every	every	DET
ejpam-4894	229	28	vertex	vertex	NOUN
ejpam-4894	229	29	vj	vj	X
ejpam-4894	229	30	∈	∈	PROPN
ejpam-4894	229	31	v	v	PROPN
ejpam-4894	229	32	(	(	PUNCT
ejpam-4894	229	33	cn	cn	PROPN
ejpam-4894	229	34	)	)	PUNCT
ejpam-4894	229	35	,	,	PUNCT
ejpam-4894	229	36	j	j	PROPN
ejpam-4894	229	37	∈	∈	PROPN
ejpam-4894	229	38	{	{	PUNCT
ejpam-4894	229	39	1	1	NUM
ejpam-4894	229	40	,	,	PUNCT
ejpam-4894	229	41	2	2	NUM
ejpam-4894	229	42	,	,	PUNCT
ejpam-4894	229	43	.	.	PUNCT
ejpam-4894	229	44	.	.	PUNCT
ejpam-4894	230	1	.	.	PUNCT
ejpam-4894	231	1	,	,	PUNCT
ejpam-4894	231	2	n	n	CCONJ
ejpam-4894	231	3	−	−	PROPN
ejpam-4894	231	4	1	1	NUM
ejpam-4894	231	5	,	,	PUNCT
ejpam-4894	231	6	n	n	CCONJ
ejpam-4894	231	7	}	}	PUNCT
ejpam-4894	231	8	has	have	VERB
ejpam-4894	231	9	zero	zero	NUM
ejpam-4894	231	10	neighbor	neighbor	NOUN
ejpam-4894	231	11	in	in	ADP
ejpam-4894	231	12	v	v	PROPN
ejpam-4894	231	13	(	(	PUNCT
ejpam-4894	231	14	cn	cn	PROPN
ejpam-4894	231	15	)	)	PUNCT
ejpam-4894	231	16	\	\	PROPN
ejpam-4894	232	1	s∗.	s∗.	ADJ
ejpam-4894	232	2	consequently	consequently	ADV
ejpam-4894	232	3	,	,	PUNCT
ejpam-4894	232	4	γcerp(cn	γcerp(cn	NOUN
ejpam-4894	232	5	)	)	PUNCT
ejpam-4894	232	6	=	=	NOUN
ejpam-4894	232	7	|s∗|	|s∗|	NUM
ejpam-4894	232	8	=	=	SYM
ejpam-4894	232	9	|v	|v	X
ejpam-4894	232	10	(	(	PUNCT
ejpam-4894	232	11	cn)|	cn)|	X
ejpam-4894	232	12	=	=	SYM
ejpam-4894	232	13	n.	n.	NOUN
ejpam-4894	232	14	case	case	NOUN
ejpam-4894	232	15	3	3	X
ejpam-4894	232	16	.	.	PUNCT
ejpam-4894	232	17	suppose	suppose	VERB
ejpam-4894	232	18	that	that	SCONJ
ejpam-4894	232	19	n	n	PROPN
ejpam-4894	232	20	≡	≡	PROPN
ejpam-4894	232	21	2(mod3	2(mod3	NUM
ejpam-4894	232	22	)	)	PUNCT
ejpam-4894	232	23	.	.	PUNCT
ejpam-4894	233	1	suppose	suppose	VERB
ejpam-4894	233	2	that	that	SCONJ
ejpam-4894	233	3	n	n	NOUN
ejpam-4894	233	4	=	=	SYM
ejpam-4894	233	5	5	5	X
ejpam-4894	233	6	.	.	PUNCT
ejpam-4894	233	7	let	let	VERB
ejpam-4894	233	8	s1	s1	PROPN
ejpam-4894	233	9	=	=	SYM
ejpam-4894	233	10	{	{	PUNCT
ejpam-4894	233	11	v1	v1	PROPN
ejpam-4894	233	12	,	,	PUNCT
ejpam-4894	233	13	v2	v2	PROPN
ejpam-4894	233	14	,	,	PUNCT
ejpam-4894	233	15	v3	v3	PROPN
ejpam-4894	233	16	,	,	PUNCT
ejpam-4894	233	17	v4	v4	PROPN
ejpam-4894	233	18	,	,	PUNCT
ejpam-4894	233	19	v5	v5	PROPN
ejpam-4894	233	20	}	}	PUNCT
ejpam-4894	233	21	.	.	PUNCT
ejpam-4894	234	1	observe	observe	VERB
ejpam-4894	234	2	that	that	SCONJ
ejpam-4894	234	3	every	every	DET
ejpam-4894	234	4	vertex	vertex	NOUN
ejpam-4894	234	5	in	in	ADP
ejpam-4894	234	6	∈	∈	PROPN
ejpam-4894	234	7	s1	s1	NOUN
ejpam-4894	234	8	has	have	VERB
ejpam-4894	234	9	zero	zero	NUM
ejpam-4894	234	10	neighbor	neighbor	NOUN
ejpam-4894	234	11	in	in	ADP
ejpam-4894	234	12	v	v	PROPN
ejpam-4894	234	13	(	(	PUNCT
ejpam-4894	234	14	c5	c5	PROPN
ejpam-4894	234	15	)	)	PUNCT
ejpam-4894	234	16	\	\	PROPN
ejpam-4894	234	17	s1	s1	NOUN
ejpam-4894	234	18	.	.	PUNCT
ejpam-4894	235	1	thus	thus	ADV
ejpam-4894	235	2	,	,	PUNCT
ejpam-4894	235	3	γcerp(s1	γcerp(s1	ADJ
ejpam-4894	235	4	)	)	PUNCT
ejpam-4894	236	1	=	=	SYM
ejpam-4894	236	2	|s1|	|s1|	NOUN
ejpam-4894	236	3	=	=	SYM
ejpam-4894	236	4	5	5	X
ejpam-4894	236	5	.	.	PUNCT
ejpam-4894	236	6	suppose	suppose	VERB
ejpam-4894	236	7	that	that	SCONJ
ejpam-4894	236	8	n	n	PROPN
ejpam-4894	236	9	>	>	X
ejpam-4894	236	10	5	5	X
ejpam-4894	236	11	.	.	PUNCT
ejpam-4894	236	12	let	let	VERB
ejpam-4894	236	13	q	q	NOUN
ejpam-4894	236	14	=	=	PUNCT
ejpam-4894	236	15	n	n	NUM
ejpam-4894	236	16	5	5	NUM
ejpam-4894	236	17	and	and	CCONJ
ejpam-4894	236	18	r	r	NOUN
ejpam-4894	236	19	∈	∈	PROPN
ejpam-4894	236	20	{	{	PUNCT
ejpam-4894	236	21	1	1	NUM
ejpam-4894	236	22	,	,	PUNCT
ejpam-4894	236	23	2	2	NUM
ejpam-4894	236	24	,	,	PUNCT
ejpam-4894	236	25	.	.	PUNCT
ejpam-4894	236	26	.	.	PUNCT
ejpam-4894	237	1	.	.	PUNCT
ejpam-4894	238	1	,	,	PUNCT
ejpam-4894	238	2	q−1	q−1	PROPN
ejpam-4894	238	3	,	,	PUNCT
ejpam-4894	238	4	q	q	NOUN
ejpam-4894	238	5	}	}	PUNCT
ejpam-4894	238	6	.	.	PUNCT
ejpam-4894	239	1	further	far	ADV
ejpam-4894	239	2	,	,	PUNCT
ejpam-4894	239	3	let	let	VERB
ejpam-4894	239	4	us	we	PRON
ejpam-4894	239	5	denote	denote	VERB
ejpam-4894	239	6	a	a	DET
ejpam-4894	239	7	group	group	NOUN
ejpam-4894	239	8	of	of	ADP
ejpam-4894	239	9	vertices	vertex	NOUN
ejpam-4894	239	10	of	of	ADP
ejpam-4894	239	11	cn	cn	PROPN
ejpam-4894	239	12	into	into	ADP
ejpam-4894	239	13	q	q	PROPN
ejpam-4894	239	14	disjoint	disjoint	NOUN
ejpam-4894	239	15	subsets	subset	NOUN
ejpam-4894	239	16	as	as	ADP
ejpam-4894	239	17	hr	hr	NOUN
ejpam-4894	239	18	,	,	PUNCT
ejpam-4894	239	19	these	these	PRON
ejpam-4894	239	20	are	be	AUX
ejpam-4894	239	21	,	,	PUNCT
ejpam-4894	239	22	h1	h1	PROPN
ejpam-4894	239	23	=	=	SYM
ejpam-4894	239	24	{	{	PUNCT
ejpam-4894	239	25	v1	v1	PROPN
ejpam-4894	239	26	,	,	PUNCT
ejpam-4894	239	27	v2	v2	PROPN
ejpam-4894	239	28	,	,	PUNCT
ejpam-4894	239	29	v3	v3	PROPN
ejpam-4894	239	30	,	,	PUNCT
ejpam-4894	239	31	v4	v4	PROPN
ejpam-4894	239	32	,	,	PUNCT
ejpam-4894	239	33	v5	v5	NOUN
ejpam-4894	239	34	}	}	PUNCT
ejpam-4894	239	35	h2	h2	NOUN
ejpam-4894	239	36	=	=	SYM
ejpam-4894	239	37	{	{	PUNCT
ejpam-4894	239	38	v6	v6	NOUN
ejpam-4894	239	39	,	,	PUNCT
ejpam-4894	239	40	v7	v7	NUM
ejpam-4894	239	41	,	,	PUNCT
ejpam-4894	239	42	v8	v8	PROPN
ejpam-4894	239	43	,	,	PUNCT
ejpam-4894	239	44	v9	v9	PROPN
ejpam-4894	239	45	,	,	PUNCT
ejpam-4894	239	46	v10	v10	PROPN
ejpam-4894	239	47	}	}	PUNCT
ejpam-4894	239	48	...	...	PUNCT
ejpam-4894	240	1	hq	hq	NOUN
ejpam-4894	240	2	=	=	PUNCT
ejpam-4894	240	3	{	{	PUNCT
ejpam-4894	240	4	vn−5	vn−5	NOUN
ejpam-4894	240	5	,	,	PUNCT
ejpam-4894	240	6	vn−4	vn−4	NOUN
ejpam-4894	240	7	,	,	PUNCT
ejpam-4894	240	8	vn−3	vn−3	PROPN
ejpam-4894	240	9	,	,	PUNCT
ejpam-4894	240	10	vn−2	vn−2	PROPN
ejpam-4894	240	11	,	,	PUNCT
ejpam-4894	240	12	vn−1	vn−1	PROPN
ejpam-4894	240	13	,	,	PUNCT
ejpam-4894	240	14	vn	vn	NOUN
ejpam-4894	240	15	}	}	PUNCT
ejpam-4894	240	16	.	.	PUNCT
ejpam-4894	241	1	clearly	clearly	ADV
ejpam-4894	241	2	,	,	PUNCT
ejpam-4894	241	3	the	the	DET
ejpam-4894	241	4	set	set	NOUN
ejpam-4894	241	5	s∗∗	s∗∗	NOUN
ejpam-4894	241	6	=	=	SYM
ejpam-4894	241	7	⋃n	⋃n	PROPN
ejpam-4894	241	8	r=1	r=1	PROPN
ejpam-4894	241	9	sr	sr	PROPN
ejpam-4894	241	10	is	be	AUX
ejpam-4894	241	11	a	a	DET
ejpam-4894	241	12	γcerp	γcerp	NOUN
ejpam-4894	241	13	-	-	PUNCT
ejpam-4894	241	14	set	set	NOUN
ejpam-4894	241	15	of	of	ADP
ejpam-4894	241	16	cn	cn	PROPN
ejpam-4894	241	17	since	since	SCONJ
ejpam-4894	241	18	ng[s	ng[s	PROPN
ejpam-4894	241	19	∗∗	∗∗	PROPN
ejpam-4894	241	20	]	]	X
ejpam-4894	241	21	=	=	SYM
ejpam-4894	241	22	v	v	X
ejpam-4894	241	23	(	(	PUNCT
ejpam-4894	241	24	cn	cn	PROPN
ejpam-4894	241	25	)	)	PUNCT
ejpam-4894	241	26	and	and	CCONJ
ejpam-4894	241	27	every	every	DET
ejpam-4894	241	28	vertex	vertex	NOUN
ejpam-4894	241	29	vj	vj	X
ejpam-4894	241	30	∈	∈	PROPN
ejpam-4894	241	31	v	v	PROPN
ejpam-4894	241	32	(	(	PUNCT
ejpam-4894	241	33	cn	cn	PROPN
ejpam-4894	241	34	)	)	PUNCT
ejpam-4894	241	35	,	,	PUNCT
ejpam-4894	241	36	j	j	PROPN
ejpam-4894	241	37	∈	∈	PROPN
ejpam-4894	241	38	{	{	PUNCT
ejpam-4894	241	39	1	1	NUM
ejpam-4894	241	40	,	,	PUNCT
ejpam-4894	241	41	2	2	NUM
ejpam-4894	241	42	,	,	PUNCT
ejpam-4894	241	43	.	.	PUNCT
ejpam-4894	241	44	.	.	PUNCT
ejpam-4894	242	1	.	.	PUNCT
ejpam-4894	243	1	,	,	PUNCT
ejpam-4894	243	2	n	n	CCONJ
ejpam-4894	243	3	−	−	PROPN
ejpam-4894	243	4	1	1	NUM
ejpam-4894	243	5	,	,	PUNCT
ejpam-4894	243	6	n	n	CCONJ
ejpam-4894	243	7	}	}	PUNCT
ejpam-4894	243	8	has	have	VERB
ejpam-4894	243	9	zero	zero	NUM
ejpam-4894	243	10	neighbor	neighbor	NOUN
ejpam-4894	243	11	in	in	ADP
ejpam-4894	243	12	v	v	PROPN
ejpam-4894	243	13	(	(	PUNCT
ejpam-4894	243	14	cn	cn	PROPN
ejpam-4894	243	15	)	)	PUNCT
ejpam-4894	243	16	\	\	PROPN
ejpam-4894	243	17	s∗∗.	s∗∗.	VERB
ejpam-4894	243	18	consequently	consequently	ADV
ejpam-4894	243	19	,	,	PUNCT
ejpam-4894	243	20	γcerp(cn	γcerp(cn	NOUN
ejpam-4894	243	21	)	)	PUNCT
ejpam-4894	243	22	=	=	PUNCT
ejpam-4894	244	1	|s∗∗|	|s∗∗|	PROPN
ejpam-4894	244	2	=	=	X
ejpam-4894	244	3	|v	|v	PROPN
ejpam-4894	244	4	(	(	PUNCT
ejpam-4894	244	5	cn)|	cn)|	X
ejpam-4894	244	6	=	=	SYM
ejpam-4894	244	7	n.	n.	NOUN
ejpam-4894	244	8	theorem	theorem	VERB
ejpam-4894	244	9	6	6	NUM
ejpam-4894	244	10	.	.	PUNCT
ejpam-4894	245	1	for	for	ADP
ejpam-4894	245	2	a	a	DET
ejpam-4894	245	3	complete	complete	ADJ
ejpam-4894	245	4	kn	kn	NOUN
ejpam-4894	245	5	of	of	ADP
ejpam-4894	245	6	order	order	NOUN
ejpam-4894	245	7	n	n	CCONJ
ejpam-4894	245	8	,	,	PUNCT
ejpam-4894	245	9	γcerp(kn	γcerp(kn	NOUN
ejpam-4894	245	10	)	)	PUNCT
ejpam-4894	245	11	=	=	PUNCT
ejpam-4894	245	12	{	{	PUNCT
ejpam-4894	245	13	1	1	NUM
ejpam-4894	245	14	if	if	SCONJ
ejpam-4894	245	15	n	n	ADV
ejpam-4894	245	16	=	=	SYM
ejpam-4894	245	17	1or	1or	NOUN
ejpam-4894	245	18	n	n	CCONJ
ejpam-4894	245	19	≥	≥	NOUN
ejpam-4894	245	20	3	3	NUM
ejpam-4894	245	21	;	;	PUNCT
ejpam-4894	245	22	2	2	NUM
ejpam-4894	245	23	if	if	SCONJ
ejpam-4894	245	24	n	n	X
ejpam-4894	245	25	=	=	SYM
ejpam-4894	245	26	2	2	X
ejpam-4894	245	27	.	.	PUNCT
ejpam-4894	246	1	j.	j.	PROPN
ejpam-4894	246	2	hamja	hamja	PROPN
ejpam-4894	246	3	/	/	SYM
ejpam-4894	246	4	eur	eur	PROPN
ejpam-4894	246	5	.	.	PUNCT
ejpam-4894	247	1	j.	j.	PROPN
ejpam-4894	247	2	pure	pure	PROPN
ejpam-4894	247	3	appl	appl	PROPN
ejpam-4894	247	4	.	.	PROPN
ejpam-4894	247	5	math	math	PROPN
ejpam-4894	247	6	,	,	PUNCT
ejpam-4894	247	7	16	16	NUM
ejpam-4894	247	8	(	(	PUNCT
ejpam-4894	247	9	4	4	NUM
ejpam-4894	247	10	)	)	PUNCT
ejpam-4894	247	11	(	(	PUNCT
ejpam-4894	247	12	2023	2023	NUM
ejpam-4894	247	13	)	)	PUNCT
ejpam-4894	247	14	,	,	PUNCT
ejpam-4894	247	15	2763	2763	NUM
ejpam-4894	247	16	-	-	SYM
ejpam-4894	247	17	2774	2774	NUM
ejpam-4894	247	18	2771	2771	NUM
ejpam-4894	247	19	proof	proof	NOUN
ejpam-4894	247	20	.	.	PUNCT
ejpam-4894	247	21	suppose	suppose	VERB
ejpam-4894	247	22	that	that	SCONJ
ejpam-4894	247	23	kn	kn	PROPN
ejpam-4894	247	24	be	be	AUX
ejpam-4894	247	25	a	a	DET
ejpam-4894	247	26	complete	complete	ADJ
ejpam-4894	247	27	graph	graph	NOUN
ejpam-4894	247	28	of	of	ADP
ejpam-4894	247	29	order	order	NOUN
ejpam-4894	247	30	n	n	PRON
ejpam-4894	247	31	≥	≥	NOUN
ejpam-4894	247	32	1	1	NUM
ejpam-4894	247	33	such	such	ADJ
ejpam-4894	247	34	that	that	SCONJ
ejpam-4894	247	35	every	every	DET
ejpam-4894	247	36	pair	pair	NOUN
ejpam-4894	247	37	of	of	ADP
ejpam-4894	247	38	distinct	distinct	ADJ
ejpam-4894	247	39	vertices	vertex	NOUN
ejpam-4894	247	40	are	be	AUX
ejpam-4894	247	41	adjacent	adjacent	ADJ
ejpam-4894	247	42	.	.	PUNCT
ejpam-4894	248	1	consider	consider	VERB
ejpam-4894	248	2	the	the	DET
ejpam-4894	248	3	following	follow	VERB
ejpam-4894	248	4	cases	case	NOUN
ejpam-4894	248	5	:	:	PUNCT
ejpam-4894	248	6	case	case	NOUN
ejpam-4894	248	7	1	1	NUM
ejpam-4894	248	8	.	.	PUNCT
ejpam-4894	248	9	suppose	suppose	VERB
ejpam-4894	248	10	that	that	SCONJ
ejpam-4894	248	11	n	n	PROPN
ejpam-4894	248	12	=	=	SYM
ejpam-4894	248	13	1	1	NUM
ejpam-4894	248	14	or	or	CCONJ
ejpam-4894	248	15	n	n	PRON
ejpam-4894	248	16	≥	≥	NOUN
ejpam-4894	248	17	3	3	NUM
ejpam-4894	248	18	.	.	PUNCT
ejpam-4894	249	1	if	if	SCONJ
ejpam-4894	249	2	n	n	NOUN
ejpam-4894	249	3	=	=	SYM
ejpam-4894	249	4	1	1	NUM
ejpam-4894	249	5	,	,	PUNCT
ejpam-4894	249	6	then	then	ADV
ejpam-4894	249	7	γcerp(k1	γcerp(k1	ADV
ejpam-4894	249	8	)	)	PUNCT
ejpam-4894	249	9	=	=	SYM
ejpam-4894	250	1	1	1	X
ejpam-4894	250	2	.	.	PUNCT
ejpam-4894	250	3	suppose	suppose	VERB
ejpam-4894	250	4	that	that	SCONJ
ejpam-4894	250	5	n	n	PROPN
ejpam-4894	250	6	≥	≥	NUM
ejpam-4894	250	7	3	3	X
ejpam-4894	250	8	.	.	PUNCT
ejpam-4894	251	1	let	let	VERB
ejpam-4894	251	2	s	s	VERB
ejpam-4894	251	3	=	=	NOUN
ejpam-4894	251	4	{	{	PUNCT
ejpam-4894	251	5	v1	v1	NOUN
ejpam-4894	251	6	}	}	PUNCT
ejpam-4894	251	7	⊆	⊆	NUM
ejpam-4894	251	8	v	v	NOUN
ejpam-4894	251	9	(	(	PUNCT
ejpam-4894	251	10	kn	kn	PROPN
ejpam-4894	251	11	)	)	PUNCT
ejpam-4894	251	12	.	.	PUNCT
ejpam-4894	252	1	since	since	SCONJ
ejpam-4894	252	2	nkn	nkn	PROPN
ejpam-4894	252	3	[	[	X
ejpam-4894	252	4	s	s	X
ejpam-4894	252	5	]	]	X
ejpam-4894	252	6	=	=	SYM
ejpam-4894	252	7	v	v	X
ejpam-4894	252	8	(	(	PUNCT
ejpam-4894	252	9	kn	kn	PROPN
ejpam-4894	252	10	)	)	PUNCT
ejpam-4894	252	11	,	,	PUNCT
ejpam-4894	252	12	all	all	PRON
ejpam-4894	252	13	vertices	vertice	VERB
ejpam-4894	252	14	in	in	ADP
ejpam-4894	252	15	v	v	PROPN
ejpam-4894	252	16	(	(	PUNCT
ejpam-4894	252	17	kn	kn	PROPN
ejpam-4894	252	18	)	)	PUNCT
ejpam-4894	252	19	\	\	PROPN
ejpam-4894	253	1	s	s	PART
ejpam-4894	253	2	are	be	AUX
ejpam-4894	253	3	dominated	dominate	VERB
ejpam-4894	253	4	by	by	ADP
ejpam-4894	253	5	exactly	exactly	ADV
ejpam-4894	253	6	one	one	NUM
ejpam-4894	253	7	vertex	vertex	NOUN
ejpam-4894	253	8	v	v	ADP
ejpam-4894	253	9	∈	∈	PROPN
ejpam-4894	253	10	s	s	PART
ejpam-4894	253	11	and	and	CCONJ
ejpam-4894	253	12	vertex	vertex	NOUN
ejpam-4894	253	13	v	v	ADP
ejpam-4894	253	14	∈	∈	NOUN
ejpam-4894	253	15	s	s	PART
ejpam-4894	253	16	has	have	VERB
ejpam-4894	253	17	at	at	ADV
ejpam-4894	253	18	least	least	ADV
ejpam-4894	253	19	two	two	NUM
ejpam-4894	253	20	neighbors	neighbor	NOUN
ejpam-4894	253	21	in	in	ADP
ejpam-4894	253	22	v	v	PROPN
ejpam-4894	253	23	(	(	PUNCT
ejpam-4894	253	24	kn	kn	PROPN
ejpam-4894	253	25	)	)	PUNCT
ejpam-4894	253	26	\	\	PROPN
ejpam-4894	254	1	s.	s.	PROPN
ejpam-4894	254	2	therefore	therefore	ADV
ejpam-4894	254	3	,	,	PUNCT
ejpam-4894	254	4	γcerp(kn	γcerp(kn	NOUN
ejpam-4894	254	5	)	)	PUNCT
ejpam-4894	254	6	=	=	PUNCT
ejpam-4894	255	1	|s|=	|s|=	PRON
ejpam-4894	255	2	1	1	NUM
ejpam-4894	255	3	.	.	PUNCT
ejpam-4894	255	4	case	case	NOUN
ejpam-4894	255	5	2	2	NUM
ejpam-4894	255	6	.	.	PUNCT
ejpam-4894	255	7	suppose	suppose	VERB
ejpam-4894	255	8	that	that	SCONJ
ejpam-4894	255	9	n	n	PROPN
ejpam-4894	255	10	=	=	SYM
ejpam-4894	255	11	2	2	NUM
ejpam-4894	255	12	.	.	PUNCT
ejpam-4894	255	13	then	then	ADV
ejpam-4894	255	14	γcerp(k2	γcerp(k2	NOUN
ejpam-4894	255	15	)	)	PUNCT
ejpam-4894	255	16	=	=	SYM
ejpam-4894	255	17	2	2	X
ejpam-4894	255	18	.	.	X
ejpam-4894	255	19	theorem	theorem	VERB
ejpam-4894	255	20	7	7	NUM
ejpam-4894	255	21	.	.	X
ejpam-4894	255	22	for	for	ADP
ejpam-4894	255	23	a	a	DET
ejpam-4894	255	24	complete	complete	ADJ
ejpam-4894	255	25	bipartite	bipartite	PROPN
ejpam-4894	255	26	km	km	PROPN
ejpam-4894	255	27	,	,	PUNCT
ejpam-4894	255	28	n	n	CCONJ
ejpam-4894	255	29	with	with	ADP
ejpam-4894	255	30	m	m	PROPN
ejpam-4894	255	31	,	,	PUNCT
ejpam-4894	255	32	n	n	PRON
ejpam-4894	255	33	vertices	vertex	NOUN
ejpam-4894	255	34	,	,	PUNCT
ejpam-4894	255	35	γcerp(km	γcerp(km	NOUN
ejpam-4894	255	36	,	,	PUNCT
ejpam-4894	255	37	n	n	CCONJ
ejpam-4894	255	38	)	)	PUNCT
ejpam-4894	255	39	=	=	SYM
ejpam-4894	256	1			NOUN
ejpam-4894	256	2	1	1	NUM
ejpam-4894	256	3	,	,	PUNCT
ejpam-4894	256	4	if	if	SCONJ
ejpam-4894	256	5	m	m	ADJ
ejpam-4894	256	6	=	=	VERB
ejpam-4894	256	7	1or	1or	PROPN
ejpam-4894	256	8	n	n	PROPN
ejpam-4894	256	9	=	=	SYM
ejpam-4894	256	10	1	1	NUM
ejpam-4894	256	11	;	;	PUNCT
ejpam-4894	256	12	4	4	NUM
ejpam-4894	256	13	,	,	PUNCT
ejpam-4894	256	14	if	if	SCONJ
ejpam-4894	256	15	m	m	VERB
ejpam-4894	256	16	=	=	SYM
ejpam-4894	256	17	n	n	NOUN
ejpam-4894	256	18	=	=	SYM
ejpam-4894	256	19	2	2	NUM
ejpam-4894	256	20	.	.	NOUN
ejpam-4894	256	21	2	2	NUM
ejpam-4894	256	22	,	,	PUNCT
ejpam-4894	256	23	otherwise	otherwise	ADV
ejpam-4894	256	24	.	.	PUNCT
ejpam-4894	257	1	proof	proof	NOUN
ejpam-4894	257	2	.	.	PUNCT
ejpam-4894	258	1	let	let	VERB
ejpam-4894	258	2	m	m	PRON
ejpam-4894	258	3	and	and	CCONJ
ejpam-4894	258	4	n	n	CCONJ
ejpam-4894	258	5	be	be	AUX
ejpam-4894	258	6	a	a	DET
ejpam-4894	258	7	positive	positive	ADJ
ejpam-4894	258	8	integers	integer	NOUN
ejpam-4894	258	9	.	.	PUNCT
ejpam-4894	259	1	suppose	suppose	VERB
ejpam-4894	259	2	that	that	SCONJ
ejpam-4894	259	3	km	km	PROPN
ejpam-4894	259	4	,	,	PUNCT
ejpam-4894	259	5	n	n	PRON
ejpam-4894	259	6	be	be	VERB
ejpam-4894	259	7	a	a	DET
ejpam-4894	259	8	complete	complete	ADJ
ejpam-4894	259	9	bipartite	bipartite	NOUN
ejpam-4894	259	10	graph	graph	NOUN
ejpam-4894	259	11	whose	whose	DET
ejpam-4894	259	12	vertices	vertex	NOUN
ejpam-4894	259	13	can	can	AUX
ejpam-4894	259	14	be	be	AUX
ejpam-4894	259	15	partitioned	partition	VERB
ejpam-4894	259	16	into	into	ADP
ejpam-4894	259	17	two	two	NUM
ejpam-4894	259	18	disjoint	disjoint	NOUN
ejpam-4894	259	19	sets	set	NOUN
ejpam-4894	259	20	such	such	ADJ
ejpam-4894	259	21	that	that	SCONJ
ejpam-4894	259	22	every	every	DET
ejpam-4894	259	23	vertex	vertex	NOUN
ejpam-4894	259	24	in	in	ADP
ejpam-4894	259	25	one	one	NUM
ejpam-4894	259	26	set	set	VERB
ejpam-4894	259	27	u	u	NOUN
ejpam-4894	259	28	of	of	ADP
ejpam-4894	259	29	order	order	NOUN
ejpam-4894	259	30	m	m	VERB
ejpam-4894	259	31	is	be	AUX
ejpam-4894	259	32	connected	connect	VERB
ejpam-4894	259	33	to	to	ADP
ejpam-4894	259	34	every	every	DET
ejpam-4894	259	35	vertex	vertex	NOUN
ejpam-4894	259	36	in	in	ADP
ejpam-4894	259	37	the	the	DET
ejpam-4894	259	38	other	other	ADJ
ejpam-4894	259	39	set	set	VERB
ejpam-4894	259	40	v	v	NOUN
ejpam-4894	259	41	of	of	ADP
ejpam-4894	259	42	order	order	NOUN
ejpam-4894	259	43	n.	n.	NOUN
ejpam-4894	259	44	consider	consider	VERB
ejpam-4894	259	45	the	the	DET
ejpam-4894	259	46	following	follow	VERB
ejpam-4894	259	47	cases	case	NOUN
ejpam-4894	259	48	:	:	PUNCT
ejpam-4894	259	49	case	case	NOUN
ejpam-4894	259	50	1	1	NUM
ejpam-4894	259	51	.	.	PUNCT
ejpam-4894	259	52	suppose	suppose	VERB
ejpam-4894	259	53	that	that	SCONJ
ejpam-4894	259	54	m	m	PROPN
ejpam-4894	259	55	=	=	SYM
ejpam-4894	259	56	1	1	NUM
ejpam-4894	259	57	or	or	CCONJ
ejpam-4894	259	58	n	n	NOUN
ejpam-4894	259	59	=	=	SYM
ejpam-4894	259	60	1	1	X
ejpam-4894	259	61	.	.	PUNCT
ejpam-4894	260	1	clearly	clearly	ADV
ejpam-4894	260	2	,	,	PUNCT
ejpam-4894	260	3	if	if	SCONJ
ejpam-4894	260	4	m	m	ADV
ejpam-4894	260	5	=	=	NOUN
ejpam-4894	260	6	1	1	NUM
ejpam-4894	260	7	,	,	PUNCT
ejpam-4894	260	8	then	then	ADV
ejpam-4894	260	9	γcerp(k1,m	γcerp(k1,m	PROPN
ejpam-4894	260	10	)	)	PUNCT
ejpam-4894	261	1	=	=	SYM
ejpam-4894	261	2	1	1	X
ejpam-4894	261	3	.	.	X
ejpam-4894	261	4	similarly	similarly	ADV
ejpam-4894	261	5	,	,	PUNCT
ejpam-4894	261	6	if	if	SCONJ
ejpam-4894	261	7	n	n	NOUN
ejpam-4894	261	8	=	=	SYM
ejpam-4894	261	9	1	1	NUM
ejpam-4894	261	10	,	,	PUNCT
ejpam-4894	261	11	then	then	ADV
ejpam-4894	261	12	γcerp(km,1	γcerp(km,1	PROPN
ejpam-4894	261	13	)	)	PUNCT
ejpam-4894	261	14	=	=	SYM
ejpam-4894	262	1	1	1	X
ejpam-4894	262	2	.	.	X
ejpam-4894	262	3	case	case	NOUN
ejpam-4894	262	4	2	2	NUM
ejpam-4894	262	5	.	.	PUNCT
ejpam-4894	262	6	suppose	suppose	VERB
ejpam-4894	262	7	that	that	SCONJ
ejpam-4894	262	8	m	m	VERB
ejpam-4894	262	9	=	=	SYM
ejpam-4894	262	10	n	n	PROPN
ejpam-4894	262	11	=	=	SYM
ejpam-4894	262	12	2	2	NUM
ejpam-4894	262	13	.	.	PUNCT
ejpam-4894	262	14	then	then	ADV
ejpam-4894	262	15	km	km	PROPN
ejpam-4894	262	16	,	,	PUNCT
ejpam-4894	262	17	n	n	PROPN
ejpam-4894	262	18	=	=	SYM
ejpam-4894	262	19	k2,2	k2,2	PROPN
ejpam-4894	262	20	.	.	PUNCT
ejpam-4894	263	1	then	then	ADV
ejpam-4894	263	2	k2,2	k2,2	PROPN
ejpam-4894	263	3	∼=	∼=	PROPN
ejpam-4894	263	4	c4	c4	NOUN
ejpam-4894	263	5	.	.	PUNCT
ejpam-4894	264	1	therefore	therefore	ADV
ejpam-4894	264	2	,	,	PUNCT
ejpam-4894	264	3	by	by	ADP
ejpam-4894	264	4	theorem	theorem	NOUN
ejpam-4894	264	5	5	5	NUM
ejpam-4894	264	6	,	,	PUNCT
ejpam-4894	264	7	γcerp(k2,2	γcerp(k2,2	NOUN
ejpam-4894	264	8	)	)	PUNCT
ejpam-4894	264	9	=	=	SYM
ejpam-4894	264	10	4	4	X
ejpam-4894	264	11	.	.	NOUN
ejpam-4894	264	12	case	case	NOUN
ejpam-4894	264	13	3	3	X
ejpam-4894	264	14	.	.	PUNCT
ejpam-4894	264	15	suppose	suppose	VERB
ejpam-4894	264	16	that	that	SCONJ
ejpam-4894	264	17	m	m	PROPN
ejpam-4894	264	18	,	,	PUNCT
ejpam-4894	264	19	n	n	PRON
ejpam-4894	264	20	≥	≥	NOUN
ejpam-4894	264	21	3	3	NUM
ejpam-4894	264	22	.	.	X
ejpam-4894	264	23	write	write	PROPN
ejpam-4894	264	24	km	km	PROPN
ejpam-4894	264	25	,	,	PUNCT
ejpam-4894	264	26	n	n	NOUN
ejpam-4894	264	27	=	=	SYM
ejpam-4894	264	28	km	km	PROPN
ejpam-4894	264	29	+	+	CCONJ
ejpam-4894	264	30	kn	kn	NOUN
ejpam-4894	264	31	=	=	SYM
ejpam-4894	264	32	u	u	PROPN
ejpam-4894	264	33	+	+	X
ejpam-4894	264	34	v	v	NOUN
ejpam-4894	264	35	.	.	PUNCT
ejpam-4894	265	1	let	let	VERB
ejpam-4894	265	2	s	s	PRON
ejpam-4894	265	3	=	=	NOUN
ejpam-4894	265	4	{	{	PUNCT
ejpam-4894	265	5	u1	u1	NOUN
ejpam-4894	265	6	,	,	PUNCT
ejpam-4894	265	7	v1	v1	NOUN
ejpam-4894	265	8	}	}	PUNCT
ejpam-4894	265	9	⊆	⊆	NUM
ejpam-4894	265	10	v	v	NOUN
ejpam-4894	265	11	(	(	PUNCT
ejpam-4894	265	12	km	km	PROPN
ejpam-4894	265	13	,	,	PUNCT
ejpam-4894	265	14	n	n	CCONJ
ejpam-4894	265	15	)	)	PUNCT
ejpam-4894	265	16	,	,	PUNCT
ejpam-4894	265	17	where	where	SCONJ
ejpam-4894	265	18	u1	u1	PROPN
ejpam-4894	265	19	∈	∈	PROPN
ejpam-4894	265	20	u	u	NOUN
ejpam-4894	265	21	and	and	CCONJ
ejpam-4894	265	22	v1	v1	PROPN
ejpam-4894	265	23	∈	∈	PROPN
ejpam-4894	265	24	v	v	NOUN
ejpam-4894	265	25	.	.	PUNCT
ejpam-4894	266	1	observe	observe	VERB
ejpam-4894	266	2	that	that	SCONJ
ejpam-4894	266	3	for	for	ADP
ejpam-4894	266	4	every	every	DET
ejpam-4894	266	5	vertex	vertex	NOUN
ejpam-4894	266	6	ui	ui	NOUN
ejpam-4894	266	7	∈	∈	PROPN
ejpam-4894	266	8	u	u	NOUN
ejpam-4894	266	9	\	\	PROPN
ejpam-4894	266	10	s	s	PROPN
ejpam-4894	266	11	,	,	PUNCT
ejpam-4894	266	12	i	i	PRON
ejpam-4894	266	13	∈	∈	PROPN
ejpam-4894	266	14	{	{	PUNCT
ejpam-4894	266	15	1	1	NUM
ejpam-4894	266	16	,	,	PUNCT
ejpam-4894	266	17	2	2	NUM
ejpam-4894	266	18	,	,	PUNCT
ejpam-4894	266	19	.	.	PUNCT
ejpam-4894	266	20	.	.	PUNCT
ejpam-4894	266	21	.	.	PUNCT
ejpam-4894	267	1	,	,	PUNCT
ejpam-4894	267	2	m	m	VERB
ejpam-4894	267	3	−	−	PROPN
ejpam-4894	267	4	1,m	1,m	ADJ
ejpam-4894	267	5	}	}	PUNCT
ejpam-4894	267	6	is	be	AUX
ejpam-4894	267	7	dominated	dominate	VERB
ejpam-4894	267	8	by	by	ADP
ejpam-4894	267	9	exactly	exactly	ADV
ejpam-4894	267	10	one	one	NUM
ejpam-4894	267	11	vertex	vertex	NOUN
ejpam-4894	267	12	v1	v1	NOUN
ejpam-4894	267	13	∈	∈	PROPN
ejpam-4894	267	14	s	s	PART
ejpam-4894	267	15	and	and	CCONJ
ejpam-4894	267	16	v1	v1	NOUN
ejpam-4894	267	17	has	have	VERB
ejpam-4894	267	18	at	at	ADV
ejpam-4894	267	19	least	least	ADV
ejpam-4894	267	20	two	two	NUM
ejpam-4894	267	21	neighbors	neighbor	NOUN
ejpam-4894	267	22	in	in	ADP
ejpam-4894	267	23	v	v	PROPN
ejpam-4894	267	24	(	(	PUNCT
ejpam-4894	267	25	km	km	NOUN
ejpam-4894	267	26	,	,	PUNCT
ejpam-4894	267	27	n	n	CCONJ
ejpam-4894	267	28	)	)	PUNCT
ejpam-4894	268	1	\	\	PROPN
ejpam-4894	268	2	s	s	PART
ejpam-4894	268	3	since	since	SCONJ
ejpam-4894	268	4	m	m	PROPN
ejpam-4894	268	5	≥	≥	NOUN
ejpam-4894	268	6	3	3	NUM
ejpam-4894	268	7	.	.	PUNCT
ejpam-4894	268	8	similarly	similarly	ADV
ejpam-4894	268	9	,	,	PUNCT
ejpam-4894	268	10	for	for	ADP
ejpam-4894	268	11	every	every	DET
ejpam-4894	268	12	vertex	vertex	NOUN
ejpam-4894	268	13	vj	vj	ADP
ejpam-4894	268	14	∈	∈	PROPN
ejpam-4894	268	15	v	v	ADP
ejpam-4894	268	16	\	\	PROPN
ejpam-4894	268	17	s	s	PROPN
ejpam-4894	268	18	,	,	PUNCT
ejpam-4894	268	19	j	j	PROPN
ejpam-4894	268	20	∈	∈	PROPN
ejpam-4894	268	21	{	{	PUNCT
ejpam-4894	268	22	1	1	NUM
ejpam-4894	268	23	,	,	PUNCT
ejpam-4894	268	24	2	2	NUM
ejpam-4894	268	25	,	,	PUNCT
ejpam-4894	268	26	.	.	PUNCT
ejpam-4894	268	27	.	.	PUNCT
ejpam-4894	269	1	.	.	PUNCT
ejpam-4894	270	1	,	,	PUNCT
ejpam-4894	270	2	n−1	n−1	PROPN
ejpam-4894	270	3	,	,	PUNCT
ejpam-4894	270	4	n}is	n}i	NOUN
ejpam-4894	270	5	dominated	dominate	VERB
ejpam-4894	270	6	by	by	ADP
ejpam-4894	270	7	exactly	exactly	ADV
ejpam-4894	270	8	one	one	NUM
ejpam-4894	270	9	u1	u1	NOUN
ejpam-4894	270	10	∈	∈	PROPN
ejpam-4894	270	11	s	s	PART
ejpam-4894	270	12	and	and	CCONJ
ejpam-4894	270	13	u1	u1	NOUN
ejpam-4894	270	14	has	have	VERB
ejpam-4894	270	15	at	at	ADV
ejpam-4894	270	16	least	least	ADV
ejpam-4894	270	17	two	two	NUM
ejpam-4894	270	18	neighbors	neighbor	NOUN
ejpam-4894	270	19	in	in	ADP
ejpam-4894	270	20	v	v	PROPN
ejpam-4894	270	21	(	(	PUNCT
ejpam-4894	270	22	km	km	NOUN
ejpam-4894	270	23	,	,	PUNCT
ejpam-4894	270	24	n	n	CCONJ
ejpam-4894	270	25	)	)	PUNCT
ejpam-4894	270	26	\	\	PROPN
ejpam-4894	271	1	s	s	PART
ejpam-4894	271	2	since	since	SCONJ
ejpam-4894	271	3	n	n	PROPN
ejpam-4894	271	4	≥	≥	NOUN
ejpam-4894	271	5	3	3	NUM
ejpam-4894	271	6	.	.	PUNCT
ejpam-4894	272	1	furthermore	furthermore	ADV
ejpam-4894	272	2	,	,	PUNCT
ejpam-4894	272	3	ng[s	ng[s	PROPN
ejpam-4894	272	4	]	]	PUNCT
ejpam-4894	272	5	=	=	SYM
ejpam-4894	272	6	v	v	X
ejpam-4894	272	7	(	(	PUNCT
ejpam-4894	272	8	km	km	PROPN
ejpam-4894	272	9	,	,	PUNCT
ejpam-4894	272	10	n	n	CCONJ
ejpam-4894	272	11	)	)	PUNCT
ejpam-4894	272	12	.	.	PUNCT
ejpam-4894	273	1	therefore	therefore	ADV
ejpam-4894	273	2	,	,	PUNCT
ejpam-4894	273	3	γcerp(km	γcerp(km	PROPN
ejpam-4894	273	4	,	,	PUNCT
ejpam-4894	273	5	n	n	CCONJ
ejpam-4894	273	6	)	)	PUNCT
ejpam-4894	273	7	=	=	SYM
ejpam-4894	273	8	|s|	|s|	NOUN
ejpam-4894	273	9	=	=	SYM
ejpam-4894	273	10	2	2	NUM
ejpam-4894	273	11	.	.	NOUN
ejpam-4894	273	12	4	4	NUM
ejpam-4894	273	13	.	.	NUM
ejpam-4894	273	14	certified	certify	VERB
ejpam-4894	273	15	perfect	perfect	ADJ
ejpam-4894	273	16	domination	domination	NOUN
ejpam-4894	273	17	number	number	NOUN
ejpam-4894	273	18	in	in	ADP
ejpam-4894	273	19	the	the	DET
ejpam-4894	273	20	join	join	NOUN
ejpam-4894	273	21	of	of	ADP
ejpam-4894	273	22	two	two	NUM
ejpam-4894	273	23	graphs	graph	NOUN
ejpam-4894	273	24	this	this	DET
ejpam-4894	273	25	section	section	NOUN
ejpam-4894	273	26	presents	present	VERB
ejpam-4894	273	27	the	the	DET
ejpam-4894	273	28	outcomes	outcome	NOUN
ejpam-4894	273	29	obtained	obtain	VERB
ejpam-4894	273	30	when	when	SCONJ
ejpam-4894	273	31	the	the	DET
ejpam-4894	273	32	graph	graph	NOUN
ejpam-4894	273	33	g+h	g+h	PROPN
ejpam-4894	273	34	possesses	possess	VERB
ejpam-4894	273	35	a	a	DET
ejpam-4894	273	36	γcerp	γcerp	NOUN
ejpam-4894	273	37	-	-	PUNCT
ejpam-4894	273	38	set	set	NOUN
ejpam-4894	273	39	along	along	ADP
ejpam-4894	273	40	with	with	ADP
ejpam-4894	273	41	its	its	PRON
ejpam-4894	273	42	certified	certify	VERB
ejpam-4894	273	43	perfect	perfect	ADJ
ejpam-4894	273	44	domination	domination	NOUN
ejpam-4894	273	45	number	number	NOUN
ejpam-4894	273	46	.	.	PUNCT
ejpam-4894	274	1	theorem	theorem	VERB
ejpam-4894	274	2	8	8	NUM
ejpam-4894	274	3	.	.	PUNCT
ejpam-4894	275	1	let	let	VERB
ejpam-4894	275	2	g	g	PRON
ejpam-4894	275	3	be	be	AUX
ejpam-4894	275	4	a	a	DET
ejpam-4894	275	5	graph	graph	NOUN
ejpam-4894	275	6	of	of	ADP
ejpam-4894	275	7	order	order	NOUN
ejpam-4894	275	8	n	n	PRON
ejpam-4894	275	9	≥	≥	NOUN
ejpam-4894	275	10	3	3	NUM
ejpam-4894	275	11	.	.	PUNCT
ejpam-4894	275	12	then	then	ADV
ejpam-4894	275	13	γcerp(g	γcerp(g	PROPN
ejpam-4894	275	14	)	)	PUNCT
ejpam-4894	275	15	=	=	SYM
ejpam-4894	275	16	1	1	NUM
ejpam-4894	276	1	if	if	SCONJ
ejpam-4894	276	2	and	and	CCONJ
ejpam-4894	276	3	only	only	ADV
ejpam-4894	276	4	if	if	SCONJ
ejpam-4894	276	5	g	g	NOUN
ejpam-4894	276	6	=	=	PROPN
ejpam-4894	276	7	k1	k1	PROPN
ejpam-4894	277	1	+	+	NOUN
ejpam-4894	277	2	h	h	NOUN
ejpam-4894	277	3	for	for	ADP
ejpam-4894	277	4	some	some	DET
ejpam-4894	277	5	graph	graph	NOUN
ejpam-4894	277	6	h	h	NOUN
ejpam-4894	277	7	of	of	ADP
ejpam-4894	277	8	order	order	NOUN
ejpam-4894	277	9	n	n	PRON
ejpam-4894	277	10	≥	≥	NOUN
ejpam-4894	277	11	2	2	NUM
ejpam-4894	277	12	.	.	PUNCT
ejpam-4894	278	1	j.	j.	PROPN
ejpam-4894	278	2	hamja	hamja	PROPN
ejpam-4894	278	3	/	/	SYM
ejpam-4894	278	4	eur	eur	PROPN
ejpam-4894	278	5	.	.	PUNCT
ejpam-4894	279	1	j.	j.	PROPN
ejpam-4894	279	2	pure	pure	PROPN
ejpam-4894	279	3	appl	appl	PROPN
ejpam-4894	279	4	.	.	PROPN
ejpam-4894	279	5	math	math	PROPN
ejpam-4894	279	6	,	,	PUNCT
ejpam-4894	279	7	16	16	NUM
ejpam-4894	279	8	(	(	PUNCT
ejpam-4894	279	9	4	4	NUM
ejpam-4894	279	10	)	)	PUNCT
ejpam-4894	279	11	(	(	PUNCT
ejpam-4894	279	12	2023	2023	NUM
ejpam-4894	279	13	)	)	PUNCT
ejpam-4894	279	14	,	,	PUNCT
ejpam-4894	279	15	2763	2763	NUM
ejpam-4894	279	16	-	-	SYM
ejpam-4894	279	17	2774	2774	NUM
ejpam-4894	279	18	2772	2772	NUM
ejpam-4894	279	19	proof	proof	NOUN
ejpam-4894	279	20	.	.	PUNCT
ejpam-4894	280	1	suppose	suppose	VERB
ejpam-4894	280	2	γcerp(g	γcerp(g	NUM
ejpam-4894	280	3	)	)	PUNCT
ejpam-4894	280	4	=	=	SYM
ejpam-4894	281	1	1	1	X
ejpam-4894	281	2	.	.	PUNCT
ejpam-4894	281	3	then	then	ADV
ejpam-4894	281	4	there	there	PRON
ejpam-4894	281	5	exists	exist	VERB
ejpam-4894	281	6	a	a	DET
ejpam-4894	281	7	dominating	dominating	NOUN
ejpam-4894	281	8	set	set	NOUN
ejpam-4894	281	9	s	s	PROPN
ejpam-4894	281	10	⊆	⊆	NUM
ejpam-4894	281	11	v	v	NOUN
ejpam-4894	281	12	(	(	PUNCT
ejpam-4894	281	13	g	g	NOUN
ejpam-4894	281	14	)	)	PUNCT
ejpam-4894	281	15	consisting	consist	VERB
ejpam-4894	281	16	of	of	ADP
ejpam-4894	281	17	a	a	DET
ejpam-4894	281	18	single	single	ADJ
ejpam-4894	281	19	vertex	vertex	NOUN
ejpam-4894	281	20	,	,	PUNCT
ejpam-4894	281	21	that	that	ADV
ejpam-4894	281	22	is	be	AUX
ejpam-4894	281	23	,	,	PUNCT
ejpam-4894	281	24	s	s	PART
ejpam-4894	281	25	=	=	PUNCT
ejpam-4894	281	26	{	{	PUNCT
ejpam-4894	281	27	v	v	NOUN
ejpam-4894	281	28	}	}	PUNCT
ejpam-4894	281	29	.	.	PUNCT
ejpam-4894	282	1	consequently	consequently	ADV
ejpam-4894	282	2	,	,	PUNCT
ejpam-4894	282	3	every	every	DET
ejpam-4894	282	4	vertex	vertex	NOUN
ejpam-4894	282	5	in	in	ADP
ejpam-4894	282	6	v	v	NOUN
ejpam-4894	282	7	(	(	PUNCT
ejpam-4894	282	8	g	g	NOUN
ejpam-4894	282	9	)	)	PUNCT
ejpam-4894	282	10	\	\	PROPN
ejpam-4894	283	1	s	s	PART
ejpam-4894	283	2	is	be	AUX
ejpam-4894	283	3	dominated	dominate	VERB
ejpam-4894	283	4	by	by	ADP
ejpam-4894	283	5	v	v	NUM
ejpam-4894	283	6	∈	∈	PROPN
ejpam-4894	283	7	s.	s.	PROPN
ejpam-4894	283	8	therefore	therefore	ADV
ejpam-4894	283	9	,	,	PUNCT
ejpam-4894	283	10	s	s	VERB
ejpam-4894	283	11	is	be	AUX
ejpam-4894	283	12	a	a	DET
ejpam-4894	283	13	perfect	perfect	ADJ
ejpam-4894	283	14	dominating	dominating	NOUN
ejpam-4894	283	15	set	set	NOUN
ejpam-4894	283	16	of	of	ADP
ejpam-4894	283	17	g.	g.	PROPN
ejpam-4894	283	18	this	this	PRON
ejpam-4894	283	19	means	mean	VERB
ejpam-4894	283	20	that	that	SCONJ
ejpam-4894	283	21	ng(v	ng(v	PUNCT
ejpam-4894	283	22	)	)	PUNCT
ejpam-4894	283	23	=	=	SYM
ejpam-4894	283	24	|v	|v	PROPN
ejpam-4894	283	25	(	(	PUNCT
ejpam-4894	283	26	h)|	h)|	VERB
ejpam-4894	283	27	for	for	ADP
ejpam-4894	283	28	some	some	DET
ejpam-4894	283	29	graph	graph	NOUN
ejpam-4894	283	30	h	h	NOUN
ejpam-4894	283	31	of	of	ADP
ejpam-4894	283	32	order	order	NOUN
ejpam-4894	283	33	n	n	PRON
ejpam-4894	283	34	≥	≥	NOUN
ejpam-4894	283	35	2	2	NUM
ejpam-4894	283	36	.	.	PUNCT
ejpam-4894	284	1	thus	thus	ADV
ejpam-4894	284	2	,	,	PUNCT
ejpam-4894	284	3	s	s	VERB
ejpam-4894	284	4	is	be	AUX
ejpam-4894	284	5	a	a	DET
ejpam-4894	284	6	certified	certify	VERB
ejpam-4894	284	7	dominating	dominating	NOUN
ejpam-4894	284	8	set	set	NOUN
ejpam-4894	284	9	of	of	ADP
ejpam-4894	284	10	g.	g.	PROPN
ejpam-4894	284	11	from	from	ADP
ejpam-4894	284	12	these	these	DET
ejpam-4894	284	13	observations	observation	NOUN
ejpam-4894	284	14	,	,	PUNCT
ejpam-4894	284	15	we	we	PRON
ejpam-4894	284	16	conclude	conclude	VERB
ejpam-4894	284	17	that	that	SCONJ
ejpam-4894	284	18	s	s	VERB
ejpam-4894	284	19	is	be	AUX
ejpam-4894	284	20	a	a	DET
ejpam-4894	284	21	certified	certify	VERB
ejpam-4894	284	22	perfect	perfect	ADJ
ejpam-4894	284	23	dominating	dominating	NOUN
ejpam-4894	284	24	set	set	NOUN
ejpam-4894	284	25	of	of	ADP
ejpam-4894	284	26	g	g	NOUN
ejpam-4894	284	27	=	=	PROPN
ejpam-4894	284	28	k1	k1	PROPN
ejpam-4894	284	29	+	+	NOUN
ejpam-4894	284	30	h.	h.	NOUN
ejpam-4894	284	31	conversely	conversely	ADV
ejpam-4894	284	32	,	,	PUNCT
ejpam-4894	284	33	suppose	suppose	VERB
ejpam-4894	284	34	that	that	SCONJ
ejpam-4894	284	35	g	g	PROPN
ejpam-4894	284	36	=	=	SYM
ejpam-4894	284	37	k1	k1	PROPN
ejpam-4894	284	38	+	+	CCONJ
ejpam-4894	284	39	h	h	NOUN
ejpam-4894	284	40	for	for	ADP
ejpam-4894	284	41	some	some	DET
ejpam-4894	284	42	h	h	NOUN
ejpam-4894	284	43	is	be	AUX
ejpam-4894	284	44	a	a	DET
ejpam-4894	284	45	graph	graph	NOUN
ejpam-4894	284	46	of	of	ADP
ejpam-4894	284	47	order	order	NOUN
ejpam-4894	284	48	n	n	PRON
ejpam-4894	284	49	≥	≥	NOUN
ejpam-4894	284	50	2	2	X
ejpam-4894	284	51	.	.	PUNCT
ejpam-4894	285	1	let	let	VERB
ejpam-4894	285	2	s	s	NOUN
ejpam-4894	285	3	=	=	X
ejpam-4894	285	4	v	v	PROPN
ejpam-4894	285	5	(	(	PUNCT
ejpam-4894	285	6	k1	k1	NOUN
ejpam-4894	285	7	)	)	PUNCT
ejpam-4894	285	8	.	.	PUNCT
ejpam-4894	286	1	then	then	ADV
ejpam-4894	286	2	s	s	VERB
ejpam-4894	286	3	=	=	PUNCT
ejpam-4894	286	4	{	{	PUNCT
ejpam-4894	286	5	v	v	NOUN
ejpam-4894	286	6	}	}	PUNCT
ejpam-4894	286	7	.	.	PUNCT
ejpam-4894	287	1	since	since	SCONJ
ejpam-4894	287	2	every	every	DET
ejpam-4894	287	3	vertex	vertex	NOUN
ejpam-4894	287	4	in	in	ADP
ejpam-4894	287	5	v	v	NUM
ejpam-4894	287	6	(	(	PUNCT
ejpam-4894	287	7	h	h	NOUN
ejpam-4894	287	8	)	)	PUNCT
ejpam-4894	287	9	is	be	AUX
ejpam-4894	287	10	dominated	dominate	VERB
ejpam-4894	287	11	by	by	ADP
ejpam-4894	287	12	exactly	exactly	ADV
ejpam-4894	287	13	one	one	NUM
ejpam-4894	287	14	vertex	vertex	NOUN
ejpam-4894	287	15	v	v	ADP
ejpam-4894	287	16	∈	∈	PROPN
ejpam-4894	287	17	s	s	PART
ejpam-4894	287	18	and	and	CCONJ
ejpam-4894	287	19	vertex	vertex	NOUN
ejpam-4894	287	20	v	v	ADP
ejpam-4894	287	21	∈	∈	NOUN
ejpam-4894	287	22	s	s	PART
ejpam-4894	287	23	has	have	VERB
ejpam-4894	287	24	at	at	ADV
ejpam-4894	287	25	least	least	ADV
ejpam-4894	287	26	two	two	NUM
ejpam-4894	287	27	neighbors	neighbor	NOUN
ejpam-4894	287	28	.	.	PUNCT
ejpam-4894	288	1	therefore	therefore	ADV
ejpam-4894	288	2	,	,	PUNCT
ejpam-4894	288	3	γcerp(g	γcerp(g	PROPN
ejpam-4894	288	4	)	)	PUNCT
ejpam-4894	288	5	=	=	SYM
ejpam-4894	288	6	|s|	|s|	NOUN
ejpam-4894	288	7	=	=	SYM
ejpam-4894	288	8	1	1	X
ejpam-4894	288	9	.	.	PUNCT
ejpam-4894	288	10	proposition	proposition	NOUN
ejpam-4894	288	11	2	2	NUM
ejpam-4894	288	12	.	.	PUNCT
ejpam-4894	289	1	if	if	SCONJ
ejpam-4894	289	2	s	s	NOUN
ejpam-4894	289	3	is	be	AUX
ejpam-4894	289	4	a	a	DET
ejpam-4894	289	5	dominating	dominating	NOUN
ejpam-4894	289	6	set	set	NOUN
ejpam-4894	289	7	or	or	CCONJ
ejpam-4894	289	8	a	a	DET
ejpam-4894	289	9	perfect	perfect	ADJ
ejpam-4894	289	10	dominating	dominating	NOUN
ejpam-4894	289	11	set	set	NOUN
ejpam-4894	289	12	of	of	ADP
ejpam-4894	289	13	g	g	NOUN
ejpam-4894	289	14	with	with	ADP
ejpam-4894	289	15	|s|	|s|	NOUN
ejpam-4894	289	16	=	=	SYM
ejpam-4894	289	17	1	1	NUM
ejpam-4894	289	18	,	,	PUNCT
ejpam-4894	289	19	then	then	ADV
ejpam-4894	289	20	s	s	VERB
ejpam-4894	289	21	is	be	AUX
ejpam-4894	289	22	a	a	DET
ejpam-4894	289	23	certified	certify	VERB
ejpam-4894	289	24	perfect	perfect	ADJ
ejpam-4894	289	25	dominating	dominating	NOUN
ejpam-4894	289	26	set	set	NOUN
ejpam-4894	289	27	of	of	ADP
ejpam-4894	289	28	g.	g.	PROPN
ejpam-4894	289	29	in	in	ADP
ejpam-4894	289	30	particular	particular	ADJ
ejpam-4894	289	31	,	,	PUNCT
ejpam-4894	289	32	γ(g	γ(g	PROPN
ejpam-4894	289	33	)	)	PUNCT
ejpam-4894	289	34	=	=	SYM
ejpam-4894	289	35	γp(g	γp(g	X
ejpam-4894	289	36	)	)	PUNCT
ejpam-4894	289	37	if	if	SCONJ
ejpam-4894	289	38	and	and	CCONJ
ejpam-4894	289	39	only	only	ADV
ejpam-4894	289	40	if	if	SCONJ
ejpam-4894	289	41	γcerp(g	γcerp(g	PROPN
ejpam-4894	289	42	)	)	PUNCT
ejpam-4894	289	43	=	=	SYM
ejpam-4894	290	1	1	1	X
ejpam-4894	290	2	.	.	PUNCT
ejpam-4894	291	1	the	the	DET
ejpam-4894	291	2	next	next	ADJ
ejpam-4894	291	3	result	result	NOUN
ejpam-4894	291	4	follows	follow	VERB
ejpam-4894	291	5	from	from	ADP
ejpam-4894	291	6	theorem	theorem	ADJ
ejpam-4894	291	7	8	8	NUM
ejpam-4894	291	8	and	and	CCONJ
ejpam-4894	291	9	proposition	proposition	NOUN
ejpam-4894	291	10	2	2	NUM
ejpam-4894	291	11	corollary	corollary	NOUN
ejpam-4894	291	12	1	1	NUM
ejpam-4894	291	13	.	.	PUNCT
ejpam-4894	292	1	the	the	DET
ejpam-4894	292	2	following	follow	VERB
ejpam-4894	292	3	are	be	AUX
ejpam-4894	292	4	graphs	graph	NOUN
ejpam-4894	292	5	having	have	VERB
ejpam-4894	292	6	γcerp(g	γcerp(g	VERB
ejpam-4894	292	7	)	)	PUNCT
ejpam-4894	292	8	=	=	SYM
ejpam-4894	292	9	1	1	NUM
ejpam-4894	292	10	:	:	PUNCT
ejpam-4894	292	11	i.	i.	PROPN
ejpam-4894	292	12	star	star	PROPN
ejpam-4894	292	13	graph	graph	VERB
ejpam-4894	292	14	sn	sn	PROPN
ejpam-4894	292	15	=	=	SYM
ejpam-4894	292	16	k1	k1	PROPN
ejpam-4894	293	1	+	+	PROPN
ejpam-4894	293	2	kn	kn	PROPN
ejpam-4894	293	3	,	,	PUNCT
ejpam-4894	293	4	n	n	PRON
ejpam-4894	294	1	≥	≥	NOUN
ejpam-4894	294	2	2	2	NUM
ejpam-4894	294	3	.	.	X
ejpam-4894	294	4	ii	ii	PROPN
ejpam-4894	294	5	.	.	PUNCT
ejpam-4894	295	1	fan	fan	PROPN
ejpam-4894	295	2	graph	graph	NOUN
ejpam-4894	295	3	fn	fn	NOUN
ejpam-4894	295	4	=	=	SYM
ejpam-4894	295	5	k1	k1	PROPN
ejpam-4894	295	6	+	+	CCONJ
ejpam-4894	295	7	pn	pn	PROPN
ejpam-4894	295	8	,	,	PUNCT
ejpam-4894	295	9	n	n	PRON
ejpam-4894	295	10	≥	≥	NOUN
ejpam-4894	295	11	2	2	NUM
ejpam-4894	295	12	.	.	X
ejpam-4894	295	13	iii	iii	PROPN
ejpam-4894	295	14	.	.	PROPN
ejpam-4894	295	15	wheel	wheel	PROPN
ejpam-4894	296	1	wn	wn	PROPN
ejpam-4894	296	2	=	=	PROPN
ejpam-4894	296	3	k1	k1	PROPN
ejpam-4894	296	4	+	+	CCONJ
ejpam-4894	296	5	cn	cn	PROPN
ejpam-4894	296	6	,	,	PUNCT
ejpam-4894	296	7	n	n	PRON
ejpam-4894	296	8	≥	≥	NOUN
ejpam-4894	296	9	3	3	NUM
ejpam-4894	296	10	.	.	NUM
ejpam-4894	297	1	iv	iv	PROPN
ejpam-4894	297	2	.	.	PUNCT
ejpam-4894	297	3	friendship	friendship	NOUN
ejpam-4894	297	4	graph	graph	NOUN
ejpam-4894	297	5	frn	frn	PROPN
ejpam-4894	297	6	=	=	PROPN
ejpam-4894	297	7	k1	k1	PROPN
ejpam-4894	297	8	+	+	CCONJ
ejpam-4894	297	9	np1	np1	PROPN
ejpam-4894	297	10	,	,	PUNCT
ejpam-4894	297	11	n	n	PRON
ejpam-4894	297	12	≥	≥	NOUN
ejpam-4894	297	13	2	2	NUM
ejpam-4894	297	14	.	.	X
ejpam-4894	298	1	v.	v.	ADP
ejpam-4894	298	2	windmill	windmill	NOUN
ejpam-4894	298	3	graph	graph	NOUN
ejpam-4894	298	4	wm	wm	PROPN
ejpam-4894	298	5	n	n	PROPN
ejpam-4894	298	6	=	=	PROPN
ejpam-4894	298	7	k1	k1	PROPN
ejpam-4894	299	1	+	+	PROPN
ejpam-4894	299	2	mkn−1	mkn−1	PROPN
ejpam-4894	299	3	,	,	PUNCT
ejpam-4894	299	4	m	m	VERB
ejpam-4894	299	5	≥	≥	NOUN
ejpam-4894	299	6	2	2	NUM
ejpam-4894	299	7	and	and	CCONJ
ejpam-4894	299	8	n	n	PRON
ejpam-4894	299	9	≥	≥	NOUN
ejpam-4894	299	10	3	3	NUM
ejpam-4894	299	11	.	.	NOUN
ejpam-4894	299	12	vi	vi	PROPN
ejpam-4894	299	13	.	.	NOUN
ejpam-4894	299	14	complete	complete	ADJ
ejpam-4894	299	15	bipartite	bipartite	PROPN
ejpam-4894	299	16	graph	graph	NOUN
ejpam-4894	299	17	km	km	PROPN
ejpam-4894	299	18	,	,	PUNCT
ejpam-4894	299	19	n	n	NOUN
ejpam-4894	300	1	=	=	SYM
ejpam-4894	300	2	km	km	PROPN
ejpam-4894	300	3	+	+	PROPN
ejpam-4894	300	4	kn	kn	PROPN
ejpam-4894	300	5	,	,	PUNCT
ejpam-4894	300	6	m	m	VERB
ejpam-4894	300	7	=	=	ADJ
ejpam-4894	300	8	1	1	NUM
ejpam-4894	300	9	or	or	CCONJ
ejpam-4894	300	10	n	n	NOUN
ejpam-4894	300	11	=	=	SYM
ejpam-4894	300	12	1	1	X
ejpam-4894	300	13	.	.	PUNCT
ejpam-4894	300	14	corollary	corollary	ADJ
ejpam-4894	300	15	2	2	NUM
ejpam-4894	300	16	.	.	PUNCT
ejpam-4894	301	1	let	let	VERB
ejpam-4894	301	2	g	g	NOUN
ejpam-4894	301	3	and	and	CCONJ
ejpam-4894	301	4	h	h	NOUN
ejpam-4894	301	5	be	be	VERB
ejpam-4894	301	6	any	any	DET
ejpam-4894	301	7	graph	graph	NOUN
ejpam-4894	301	8	of	of	ADP
ejpam-4894	301	9	order	order	NOUN
ejpam-4894	301	10	m	m	VERB
ejpam-4894	301	11	and	and	CCONJ
ejpam-4894	301	12	n	n	CCONJ
ejpam-4894	301	13	,	,	PUNCT
ejpam-4894	301	14	respectively	respectively	ADV
ejpam-4894	301	15	with	with	ADP
ejpam-4894	301	16	γ(g	γ(g	PROPN
ejpam-4894	301	17	)	)	PUNCT
ejpam-4894	301	18	=	=	SYM
ejpam-4894	301	19	1	1	NUM
ejpam-4894	301	20	or	or	CCONJ
ejpam-4894	301	21	γ(h	γ(h	NOUN
ejpam-4894	301	22	)	)	PUNCT
ejpam-4894	301	23	=	=	SYM
ejpam-4894	302	1	1	1	X
ejpam-4894	302	2	.	.	X
ejpam-4894	302	3	then	then	ADV
ejpam-4894	302	4	γcerp(g+h	γcerp(g+h	PROPN
ejpam-4894	302	5	)	)	PUNCT
ejpam-4894	302	6	=	=	SYM
ejpam-4894	303	1	1	1	X
ejpam-4894	303	2	.	.	X
ejpam-4894	303	3	proposition	proposition	NOUN
ejpam-4894	303	4	3	3	X
ejpam-4894	303	5	.	.	PUNCT
ejpam-4894	304	1	let	let	VERB
ejpam-4894	304	2	g	g	NOUN
ejpam-4894	305	1	and	and	CCONJ
ejpam-4894	305	2	h	h	NOUN
ejpam-4894	305	3	be	be	VERB
ejpam-4894	305	4	a	a	DET
ejpam-4894	305	5	trivial	trivial	ADJ
ejpam-4894	305	6	graphs	graph	NOUN
ejpam-4894	305	7	.	.	PUNCT
ejpam-4894	306	1	then	then	ADV
ejpam-4894	306	2	γcerp(g+h	γcerp(g+h	PROPN
ejpam-4894	306	3	)	)	PUNCT
ejpam-4894	306	4	=	=	SYM
ejpam-4894	307	1	2	2	X
ejpam-4894	307	2	.	.	X
ejpam-4894	307	3	proof	proof	NOUN
ejpam-4894	307	4	.	.	PUNCT
ejpam-4894	308	1	clearly	clearly	ADV
ejpam-4894	308	2	,	,	PUNCT
ejpam-4894	308	3	if	if	SCONJ
ejpam-4894	308	4	g	g	PROPN
ejpam-4894	308	5	and	and	CCONJ
ejpam-4894	308	6	h	h	NOUN
ejpam-4894	308	7	are	be	AUX
ejpam-4894	308	8	graphs	graph	NOUN
ejpam-4894	308	9	of	of	ADP
ejpam-4894	308	10	order	order	NOUN
ejpam-4894	308	11	m	m	VERB
ejpam-4894	308	12	=	=	SYM
ejpam-4894	308	13	1	1	NUM
ejpam-4894	308	14	and	and	CCONJ
ejpam-4894	308	15	n	n	CCONJ
ejpam-4894	308	16	=	=	SYM
ejpam-4894	308	17	1	1	NUM
ejpam-4894	308	18	,	,	PUNCT
ejpam-4894	308	19	respectively	respectively	ADV
ejpam-4894	308	20	.	.	PUNCT
ejpam-4894	309	1	then	then	ADV
ejpam-4894	309	2	γcerp(g+h	γcerp(g+h	PROPN
ejpam-4894	309	3	)	)	PUNCT
ejpam-4894	309	4	=	=	SYM
ejpam-4894	310	1	2	2	X
ejpam-4894	310	2	.	.	X
ejpam-4894	310	3	proposition	proposition	NOUN
ejpam-4894	310	4	4	4	NUM
ejpam-4894	310	5	.	.	PUNCT
ejpam-4894	311	1	let	let	VERB
ejpam-4894	311	2	g	g	NOUN
ejpam-4894	311	3	and	and	CCONJ
ejpam-4894	311	4	h	h	NOUN
ejpam-4894	311	5	be	be	VERB
ejpam-4894	311	6	any	any	DET
ejpam-4894	311	7	connected	connected	ADJ
ejpam-4894	311	8	non	non	ADJ
ejpam-4894	311	9	-	-	ADJ
ejpam-4894	311	10	trivial	trivial	ADJ
ejpam-4894	311	11	graph	graph	NOUN
ejpam-4894	311	12	of	of	ADP
ejpam-4894	311	13	order	order	NOUN
ejpam-4894	311	14	m	m	VERB
ejpam-4894	311	15	and	and	CCONJ
ejpam-4894	311	16	n	n	CCONJ
ejpam-4894	311	17	,	,	PUNCT
ejpam-4894	311	18	respectively	respectively	ADV
ejpam-4894	311	19	with	with	ADP
ejpam-4894	311	20	γ(g	γ(g	NOUN
ejpam-4894	311	21	)	)	PUNCT
ejpam-4894	311	22	̸=	̸=	PROPN
ejpam-4894	311	23	1	1	NUM
ejpam-4894	311	24	or	or	CCONJ
ejpam-4894	311	25	γ(h	γ(h	NOUN
ejpam-4894	311	26	)	)	PUNCT
ejpam-4894	311	27	̸=	̸=	PROPN
ejpam-4894	311	28	1	1	NUM
ejpam-4894	311	29	.	.	PUNCT
ejpam-4894	312	1	then	then	ADV
ejpam-4894	312	2	γcerp(g+h	γcerp(g+h	PROPN
ejpam-4894	312	3	)	)	PUNCT
ejpam-4894	313	1	=	=	SYM
ejpam-4894	313	2	|v	|v	X
ejpam-4894	313	3	(	(	PUNCT
ejpam-4894	313	4	g+h)|	g+h)|	PROPN
ejpam-4894	313	5	.	.	PROPN
ejpam-4894	313	6	5	5	NUM
ejpam-4894	313	7	.	.	PUNCT
ejpam-4894	314	1	the	the	DET
ejpam-4894	314	2	certified	certify	VERB
ejpam-4894	314	3	perfect	perfect	ADJ
ejpam-4894	314	4	domination	domination	NOUN
ejpam-4894	314	5	in	in	ADP
ejpam-4894	314	6	the	the	DET
ejpam-4894	314	7	corona	corona	NOUN
ejpam-4894	314	8	of	of	ADP
ejpam-4894	314	9	graphs	graph	NOUN
ejpam-4894	314	10	in	in	ADP
ejpam-4894	314	11	this	this	DET
ejpam-4894	314	12	section	section	NOUN
ejpam-4894	314	13	presents	present	VERB
ejpam-4894	314	14	the	the	DET
ejpam-4894	314	15	γcerp	γcerp	NOUN
ejpam-4894	314	16	-	-	PUNCT
ejpam-4894	314	17	set	set	NOUN
ejpam-4894	314	18	of	of	ADP
ejpam-4894	314	19	g	g	PROPN
ejpam-4894	314	20	◦	◦	NOUN
ejpam-4894	314	21	h	h	NOUN
ejpam-4894	314	22	and	and	CCONJ
ejpam-4894	314	23	its	its	PRON
ejpam-4894	314	24	certified	certify	VERB
ejpam-4894	314	25	perfect	perfect	ADJ
ejpam-4894	314	26	domination	domination	NOUN
ejpam-4894	314	27	number	number	NOUN
ejpam-4894	314	28	.	.	PUNCT
ejpam-4894	315	1	j.	j.	PROPN
ejpam-4894	315	2	hamja	hamja	PROPN
ejpam-4894	315	3	/	/	SYM
ejpam-4894	315	4	eur	eur	PROPN
ejpam-4894	315	5	.	.	PUNCT
ejpam-4894	316	1	j.	j.	PROPN
ejpam-4894	316	2	pure	pure	PROPN
ejpam-4894	316	3	appl	appl	PROPN
ejpam-4894	316	4	.	.	PROPN
ejpam-4894	316	5	math	math	PROPN
ejpam-4894	316	6	,	,	PUNCT
ejpam-4894	316	7	16	16	NUM
ejpam-4894	316	8	(	(	PUNCT
ejpam-4894	316	9	4	4	NUM
ejpam-4894	316	10	)	)	PUNCT
ejpam-4894	316	11	(	(	PUNCT
ejpam-4894	316	12	2023	2023	NUM
ejpam-4894	316	13	)	)	PUNCT
ejpam-4894	316	14	,	,	PUNCT
ejpam-4894	316	15	2763	2763	NUM
ejpam-4894	316	16	-	-	SYM
ejpam-4894	316	17	2774	2774	NUM
ejpam-4894	316	18	2773	2773	NUM
ejpam-4894	316	19	theorem	theorem	NOUN
ejpam-4894	316	20	9	9	NUM
ejpam-4894	316	21	.	.	PUNCT
ejpam-4894	317	1	let	let	VERB
ejpam-4894	317	2	g	g	PRON
ejpam-4894	317	3	be	be	AUX
ejpam-4894	317	4	a	a	DET
ejpam-4894	317	5	connected	connected	ADJ
ejpam-4894	317	6	graph	graph	NOUN
ejpam-4894	317	7	of	of	ADP
ejpam-4894	317	8	order	order	NOUN
ejpam-4894	317	9	m	m	VERB
ejpam-4894	317	10	and	and	CCONJ
ejpam-4894	317	11	h	h	NOUN
ejpam-4894	317	12	be	be	VERB
ejpam-4894	317	13	any	any	DET
ejpam-4894	317	14	graph	graph	NOUN
ejpam-4894	317	15	of	of	ADP
ejpam-4894	317	16	order	order	NOUN
ejpam-4894	317	17	n	n	PRON
ejpam-4894	317	18	≥	≥	NOUN
ejpam-4894	317	19	2	2	NUM
ejpam-4894	317	20	.	.	PUNCT
ejpam-4894	318	1	then	then	ADV
ejpam-4894	318	2	a	a	DET
ejpam-4894	318	3	subset	subset	NOUN
ejpam-4894	318	4	s	s	X
ejpam-4894	318	5	of	of	ADP
ejpam-4894	318	6	v	v	NOUN
ejpam-4894	318	7	(	(	PUNCT
ejpam-4894	318	8	g	g	PROPN
ejpam-4894	318	9	◦	◦	NOUN
ejpam-4894	318	10	h	h	NOUN
ejpam-4894	318	11	)	)	PUNCT
ejpam-4894	318	12	is	be	AUX
ejpam-4894	318	13	a	a	DET
ejpam-4894	318	14	certified	certify	VERB
ejpam-4894	318	15	perfect	perfect	ADJ
ejpam-4894	318	16	dominating	dominating	NOUN
ejpam-4894	318	17	set	set	NOUN
ejpam-4894	318	18	of	of	ADP
ejpam-4894	318	19	g	g	PROPN
ejpam-4894	318	20	◦	◦	NOUN
ejpam-4894	318	21	h	h	NOUN
ejpam-4894	318	22	if	if	SCONJ
ejpam-4894	319	1	and	and	CCONJ
ejpam-4894	319	2	only	only	ADV
ejpam-4894	319	3	if	if	SCONJ
ejpam-4894	319	4	s	s	ADP
ejpam-4894	319	5	∩	∩	ADJ
ejpam-4894	319	6	v	v	ADJ
ejpam-4894	319	7	(	(	PUNCT
ejpam-4894	319	8	v	v	PROPN
ejpam-4894	319	9	+	+	NOUN
ejpam-4894	319	10	hv	hv	NOUN
ejpam-4894	319	11	)	)	PUNCT
ejpam-4894	319	12	is	be	AUX
ejpam-4894	319	13	a	a	DET
ejpam-4894	319	14	certified	certify	VERB
ejpam-4894	319	15	perfect	perfect	ADJ
ejpam-4894	319	16	dominating	dominating	NOUN
ejpam-4894	319	17	set	set	NOUN
ejpam-4894	319	18	of	of	ADP
ejpam-4894	319	19	v	v	DET
ejpam-4894	319	20	+	+	NOUN
ejpam-4894	319	21	hv	hv	NOUN
ejpam-4894	319	22	for	for	ADP
ejpam-4894	319	23	every	every	DET
ejpam-4894	319	24	v	v	NUM
ejpam-4894	319	25	∈	∈	PROPN
ejpam-4894	319	26	v	v	NOUN
ejpam-4894	319	27	(	(	PUNCT
ejpam-4894	319	28	g	g	NOUN
ejpam-4894	319	29	)	)	PUNCT
ejpam-4894	319	30	.	.	PUNCT
ejpam-4894	320	1	proof	proof	NOUN
ejpam-4894	320	2	.	.	PUNCT
ejpam-4894	321	1	let	let	VERB
ejpam-4894	321	2	s	s	PRON
ejpam-4894	321	3	⊆	⊆	NUM
ejpam-4894	321	4	g	g	NOUN
ejpam-4894	321	5	◦	◦	NOUN
ejpam-4894	321	6	h	h	NOUN
ejpam-4894	321	7	be	be	VERB
ejpam-4894	321	8	a	a	DET
ejpam-4894	321	9	certified	certify	VERB
ejpam-4894	321	10	perfect	perfect	ADJ
ejpam-4894	321	11	dominating	dominating	NOUN
ejpam-4894	321	12	set	set	NOUN
ejpam-4894	321	13	of	of	ADP
ejpam-4894	321	14	g	g	PROPN
ejpam-4894	321	15	◦	◦	NOUN
ejpam-4894	321	16	h	h	NOUN
ejpam-4894	321	17	and	and	CCONJ
ejpam-4894	321	18	let	let	VERB
ejpam-4894	321	19	v	v	NUM
ejpam-4894	321	20	∈	∈	PROPN
ejpam-4894	321	21	v	v	NOUN
ejpam-4894	321	22	(	(	PUNCT
ejpam-4894	321	23	g	g	NOUN
ejpam-4894	321	24	)	)	PUNCT
ejpam-4894	321	25	.	.	PUNCT
ejpam-4894	322	1	if	if	SCONJ
ejpam-4894	322	2	v	v	NUM
ejpam-4894	322	3	∈	∈	PROPN
ejpam-4894	322	4	s	s	NOUN
ejpam-4894	322	5	,	,	PUNCT
ejpam-4894	322	6	then	then	ADV
ejpam-4894	322	7	v	v	NOUN
ejpam-4894	322	8	is	be	AUX
ejpam-4894	322	9	a	a	DET
ejpam-4894	322	10	certified	certify	VERB
ejpam-4894	322	11	perfect	perfect	ADJ
ejpam-4894	322	12	dominating	dominating	NOUN
ejpam-4894	322	13	set	set	NOUN
ejpam-4894	322	14	of	of	ADP
ejpam-4894	322	15	v	v	PRON
ejpam-4894	322	16	+	+	NOUN
ejpam-4894	322	17	hv	hv	NOUN
ejpam-4894	322	18	since	since	SCONJ
ejpam-4894	322	19	h	h	PROPN
ejpam-4894	322	20	is	be	AUX
ejpam-4894	322	21	any	any	DET
ejpam-4894	322	22	graph	graph	NOUN
ejpam-4894	322	23	with	with	ADP
ejpam-4894	322	24	vertices	vertex	NOUN
ejpam-4894	322	25	n	n	PRON
ejpam-4894	322	26	≥	≥	NUM
ejpam-4894	322	27	2	2	NUM
ejpam-4894	322	28	.	.	PUNCT
ejpam-4894	322	29	suppose	suppose	VERB
ejpam-4894	322	30	that	that	SCONJ
ejpam-4894	322	31	v	v	NOUN
ejpam-4894	322	32	/∈	/∈	PUNCT
ejpam-4894	322	33	s.	s.	PROPN
ejpam-4894	322	34	let	let	VERB
ejpam-4894	322	35	a	a	DET
ejpam-4894	322	36	∈	∈	PROPN
ejpam-4894	322	37	v	v	NOUN
ejpam-4894	322	38	(	(	PUNCT
ejpam-4894	322	39	v	v	NOUN
ejpam-4894	322	40	+	+	CCONJ
ejpam-4894	322	41	hv	hv	NOUN
ejpam-4894	322	42	)	)	PUNCT
ejpam-4894	322	43	\	\	PROPN
ejpam-4894	323	1	s	s	PROPN
ejpam-4894	323	2	,	,	PUNCT
ejpam-4894	323	3	where	where	SCONJ
ejpam-4894	323	4	a	a	DET
ejpam-4894	323	5	̸=	̸=	PROPN
ejpam-4894	323	6	v.	v.	CCONJ
ejpam-4894	323	7	since	since	SCONJ
ejpam-4894	323	8	s	s	PROPN
ejpam-4894	323	9	is	be	AUX
ejpam-4894	323	10	a	a	DET
ejpam-4894	323	11	certified	certify	VERB
ejpam-4894	323	12	perfect	perfect	ADJ
ejpam-4894	323	13	dominating	dominating	NOUN
ejpam-4894	323	14	set	set	NOUN
ejpam-4894	323	15	of	of	ADP
ejpam-4894	323	16	g	g	PROPN
ejpam-4894	323	17	◦	◦	NOUN
ejpam-4894	323	18	h	h	NOUN
ejpam-4894	323	19	,	,	PUNCT
ejpam-4894	323	20	there	there	PRON
ejpam-4894	323	21	exist	exist	VERB
ejpam-4894	323	22	b	b	PROPN
ejpam-4894	323	23	∈	∈	NOUN
ejpam-4894	323	24	s	s	VERB
ejpam-4894	323	25	such	such	ADJ
ejpam-4894	323	26	that	that	SCONJ
ejpam-4894	323	27	ab	ab	PROPN
ejpam-4894	323	28	∈	∈	PROPN
ejpam-4894	323	29	e(g	e(g	PROPN
ejpam-4894	323	30	◦	◦	PROPN
ejpam-4894	323	31	h	h	NOUN
ejpam-4894	323	32	)	)	PUNCT
ejpam-4894	323	33	.	.	PUNCT
ejpam-4894	324	1	this	this	PRON
ejpam-4894	324	2	means	mean	VERB
ejpam-4894	324	3	that	that	SCONJ
ejpam-4894	324	4	b	b	PROPN
ejpam-4894	324	5	∈	∈	PROPN
ejpam-4894	324	6	v	v	NOUN
ejpam-4894	324	7	(	(	PUNCT
ejpam-4894	324	8	v	v	PROPN
ejpam-4894	324	9	+	+	NOUN
ejpam-4894	324	10	hv	hv	NOUN
ejpam-4894	324	11	)	)	PUNCT
ejpam-4894	324	12	∩	∩	PROPN
ejpam-4894	324	13	s	s	PART
ejpam-4894	324	14	and	and	CCONJ
ejpam-4894	324	15	ab	ab	PROPN
ejpam-4894	324	16	∈	∈	PROPN
ejpam-4894	324	17	e(v	e(v	PROPN
ejpam-4894	324	18	+	+	NOUN
ejpam-4894	324	19	hv	hv	NOUN
ejpam-4894	324	20	)	)	PUNCT
ejpam-4894	324	21	.	.	PUNCT
ejpam-4894	325	1	this	this	PRON
ejpam-4894	325	2	proves	prove	VERB
ejpam-4894	325	3	that	that	SCONJ
ejpam-4894	325	4	s	s	VERB
ejpam-4894	325	5	∩	∩	ADJ
ejpam-4894	325	6	v	v	ADJ
ejpam-4894	325	7	(	(	PUNCT
ejpam-4894	325	8	v	v	PROPN
ejpam-4894	325	9	+	+	NOUN
ejpam-4894	325	10	hv	hv	NOUN
ejpam-4894	325	11	)	)	PUNCT
ejpam-4894	325	12	is	be	AUX
ejpam-4894	325	13	a	a	DET
ejpam-4894	325	14	certified	certify	VERB
ejpam-4894	325	15	perfect	perfect	ADJ
ejpam-4894	325	16	dominating	dominating	NOUN
ejpam-4894	325	17	set	set	NOUN
ejpam-4894	325	18	of	of	ADP
ejpam-4894	325	19	v	v	DET
ejpam-4894	325	20	+	+	NOUN
ejpam-4894	325	21	hv	hv	PROPN
ejpam-4894	325	22	.	.	PUNCT
ejpam-4894	326	1	conversely	conversely	ADV
ejpam-4894	326	2	,	,	PUNCT
ejpam-4894	326	3	suppose	suppose	VERB
ejpam-4894	326	4	that	that	SCONJ
ejpam-4894	326	5	s	s	VERB
ejpam-4894	326	6	∩	∩	ADJ
ejpam-4894	326	7	v	v	X
ejpam-4894	326	8	(	(	PUNCT
ejpam-4894	326	9	v+hv	v+hv	NOUN
ejpam-4894	326	10	)	)	PUNCT
ejpam-4894	326	11	is	be	AUX
ejpam-4894	326	12	a	a	DET
ejpam-4894	326	13	certified	certify	VERB
ejpam-4894	326	14	perfect	perfect	ADJ
ejpam-4894	326	15	dominating	dominating	NOUN
ejpam-4894	326	16	set	set	NOUN
ejpam-4894	326	17	of	of	ADP
ejpam-4894	326	18	v+hv	v+hv	NOUN
ejpam-4894	326	19	for	for	ADP
ejpam-4894	326	20	every	every	DET
ejpam-4894	326	21	v	v	NUM
ejpam-4894	326	22	∈	∈	PROPN
ejpam-4894	326	23	v	v	NOUN
ejpam-4894	326	24	(	(	PUNCT
ejpam-4894	326	25	g	g	NOUN
ejpam-4894	326	26	)	)	PUNCT
ejpam-4894	326	27	.	.	PUNCT
ejpam-4894	327	1	indeed	indeed	ADV
ejpam-4894	327	2	,	,	PUNCT
ejpam-4894	327	3	s	s	VERB
ejpam-4894	327	4	is	be	AUX
ejpam-4894	327	5	a	a	DET
ejpam-4894	327	6	certified	certify	VERB
ejpam-4894	327	7	perfect	perfect	ADJ
ejpam-4894	327	8	dominating	dominating	NOUN
ejpam-4894	327	9	set	set	NOUN
ejpam-4894	327	10	of	of	ADP
ejpam-4894	327	11	g	g	PROPN
ejpam-4894	327	12	◦	◦	NOUN
ejpam-4894	327	13	h.	h.	PROPN
ejpam-4894	327	14	corollary	corollary	ADJ
ejpam-4894	327	15	3	3	X
ejpam-4894	327	16	.	.	PUNCT
ejpam-4894	328	1	if	if	SCONJ
ejpam-4894	328	2	g	g	PROPN
ejpam-4894	328	3	is	be	AUX
ejpam-4894	328	4	a	a	DET
ejpam-4894	328	5	connected	connected	ADJ
ejpam-4894	328	6	graph	graph	NOUN
ejpam-4894	328	7	of	of	ADP
ejpam-4894	328	8	order	order	NOUN
ejpam-4894	328	9	m	m	VERB
ejpam-4894	328	10	and	and	CCONJ
ejpam-4894	328	11	h	h	NOUN
ejpam-4894	328	12	be	be	VERB
ejpam-4894	328	13	any	any	DET
ejpam-4894	328	14	graph	graph	NOUN
ejpam-4894	328	15	of	of	ADP
ejpam-4894	328	16	order	order	NOUN
ejpam-4894	328	17	n	n	PRON
ejpam-4894	328	18	≥	≥	NOUN
ejpam-4894	328	19	2	2	NUM
ejpam-4894	328	20	.	.	PUNCT
ejpam-4894	329	1	then	then	ADV
ejpam-4894	329	2	γcerp(g	γcerp(g	VERB
ejpam-4894	329	3	◦	◦	NOUN
ejpam-4894	329	4	h	h	NOUN
ejpam-4894	329	5	)	)	PUNCT
ejpam-4894	329	6	=	=	NOUN
ejpam-4894	329	7	m.	m.	NOUN
ejpam-4894	329	8	proof	proof	NOUN
ejpam-4894	329	9	.	.	PUNCT
ejpam-4894	330	1	let	let	VERB
ejpam-4894	330	2	s	s	PRON
ejpam-4894	330	3	⊆	⊆	NUM
ejpam-4894	330	4	v	v	NOUN
ejpam-4894	330	5	(	(	PUNCT
ejpam-4894	330	6	g	g	PROPN
ejpam-4894	330	7	◦	◦	NOUN
ejpam-4894	330	8	h	h	NOUN
ejpam-4894	330	9	)	)	PUNCT
ejpam-4894	330	10	.	.	PUNCT
ejpam-4894	331	1	suppose	suppose	VERB
ejpam-4894	331	2	that	that	SCONJ
ejpam-4894	331	3	s	s	VERB
ejpam-4894	331	4	=	=	SYM
ejpam-4894	331	5	v	v	X
ejpam-4894	331	6	(	(	PUNCT
ejpam-4894	331	7	g	g	NOUN
ejpam-4894	331	8	)	)	PUNCT
ejpam-4894	331	9	.	.	PUNCT
ejpam-4894	332	1	then	then	ADV
ejpam-4894	332	2	s	s	VERB
ejpam-4894	332	3	∩	∩	ADJ
ejpam-4894	332	4	v	v	ADJ
ejpam-4894	332	5	(	(	PUNCT
ejpam-4894	332	6	v	v	NOUN
ejpam-4894	332	7	+	+	CCONJ
ejpam-4894	332	8	hv	hv	NOUN
ejpam-4894	332	9	)	)	PUNCT
ejpam-4894	332	10	=	=	NOUN
ejpam-4894	333	1	v	v	NOUN
ejpam-4894	333	2	is	be	AUX
ejpam-4894	333	3	a	a	DET
ejpam-4894	333	4	certified	certify	VERB
ejpam-4894	333	5	perfect	perfect	ADJ
ejpam-4894	333	6	dominating	dominating	NOUN
ejpam-4894	333	7	set	set	NOUN
ejpam-4894	333	8	of	of	ADP
ejpam-4894	333	9	v	v	NOUN
ejpam-4894	333	10	+	+	CCONJ
ejpam-4894	333	11	hv	hv	NOUN
ejpam-4894	333	12	for	for	ADP
ejpam-4894	333	13	every	every	DET
ejpam-4894	333	14	v	v	NUM
ejpam-4894	333	15	∈	∈	PROPN
ejpam-4894	333	16	v	v	NOUN
ejpam-4894	333	17	(	(	PUNCT
ejpam-4894	333	18	g	g	NOUN
ejpam-4894	333	19	)	)	PUNCT
ejpam-4894	333	20	since	since	SCONJ
ejpam-4894	333	21	h	h	NOUN
ejpam-4894	333	22	is	be	AUX
ejpam-4894	333	23	any	any	DET
ejpam-4894	333	24	graph	graph	NOUN
ejpam-4894	333	25	with	with	ADP
ejpam-4894	333	26	vertices	vertex	NOUN
ejpam-4894	333	27	n	n	PRON
ejpam-4894	333	28	≥	≥	NOUN
ejpam-4894	333	29	2	2	NUM
ejpam-4894	333	30	.	.	PUNCT
ejpam-4894	334	1	this	this	PRON
ejpam-4894	334	2	implies	imply	VERB
ejpam-4894	334	3	that	that	SCONJ
ejpam-4894	334	4	s	s	VERB
ejpam-4894	334	5	is	be	AUX
ejpam-4894	334	6	a	a	DET
ejpam-4894	334	7	certified	certify	VERB
ejpam-4894	334	8	perfect	perfect	ADJ
ejpam-4894	334	9	dominating	dominating	NOUN
ejpam-4894	334	10	set	set	NOUN
ejpam-4894	334	11	of	of	ADP
ejpam-4894	334	12	g	g	PROPN
ejpam-4894	334	13	◦	◦	NOUN
ejpam-4894	334	14	h	h	NOUN
ejpam-4894	334	15	by	by	ADP
ejpam-4894	334	16	theorem	theorem	NOUN
ejpam-4894	334	17	9	9	NUM
ejpam-4894	334	18	.	.	PUNCT
ejpam-4894	334	19	consequently	consequently	ADV
ejpam-4894	334	20	,	,	PUNCT
ejpam-4894	334	21	γcerp(g	γcerp(g	PROPN
ejpam-4894	334	22	◦	◦	NOUN
ejpam-4894	334	23	h	h	NOUN
ejpam-4894	334	24	)	)	PUNCT
ejpam-4894	334	25	≤	≤	NUM
ejpam-4894	334	26	|s|	|s|	PROPN
ejpam-4894	334	27	=	=	SYM
ejpam-4894	334	28	m.	m.	NOUN
ejpam-4894	334	29	moreover	moreover	ADV
ejpam-4894	334	30	,	,	PUNCT
ejpam-4894	334	31	if	if	SCONJ
ejpam-4894	334	32	s∗	s∗	PROPN
ejpam-4894	334	33	is	be	AUX
ejpam-4894	334	34	a	a	DET
ejpam-4894	334	35	minimum	minimum	NOUN
ejpam-4894	334	36	certified	certify	VERB
ejpam-4894	334	37	perfect	perfect	ADJ
ejpam-4894	334	38	dominating	dominating	NOUN
ejpam-4894	334	39	set	set	NOUN
ejpam-4894	334	40	of	of	ADP
ejpam-4894	334	41	g	g	PROPN
ejpam-4894	334	42	◦	◦	NOUN
ejpam-4894	334	43	h	h	NOUN
ejpam-4894	334	44	,	,	PUNCT
ejpam-4894	334	45	then	then	ADV
ejpam-4894	334	46	v	v	X
ejpam-4894	334	47	(	(	PUNCT
ejpam-4894	334	48	s∗	s∗	PROPN
ejpam-4894	334	49	∩	∩	X
ejpam-4894	334	50	v+hv	v+hv	PROPN
ejpam-4894	334	51	)	)	PUNCT
ejpam-4894	334	52	is	be	AUX
ejpam-4894	334	53	a	a	DET
ejpam-4894	334	54	certified	certify	VERB
ejpam-4894	334	55	perfect	perfect	ADJ
ejpam-4894	334	56	dominating	dominating	NOUN
ejpam-4894	334	57	set	set	NOUN
ejpam-4894	334	58	of	of	ADP
ejpam-4894	334	59	v	v	NOUN
ejpam-4894	334	60	+	+	CCONJ
ejpam-4894	334	61	hv	hv	NOUN
ejpam-4894	334	62	for	for	ADP
ejpam-4894	334	63	every	every	DET
ejpam-4894	334	64	v	v	NUM
ejpam-4894	334	65	∈	∈	PROPN
ejpam-4894	334	66	v	v	NOUN
ejpam-4894	334	67	(	(	PUNCT
ejpam-4894	334	68	g	g	NOUN
ejpam-4894	334	69	)	)	PUNCT
ejpam-4894	334	70	since	since	SCONJ
ejpam-4894	334	71	h	h	NOUN
ejpam-4894	334	72	is	be	AUX
ejpam-4894	334	73	any	any	DET
ejpam-4894	334	74	graph	graph	NOUN
ejpam-4894	334	75	with	with	ADP
ejpam-4894	334	76	vertices	vertex	NOUN
ejpam-4894	334	77	n	n	PRON
ejpam-4894	334	78	≥	≥	NOUN
ejpam-4894	334	79	2	2	NUM
ejpam-4894	334	80	by	by	ADP
ejpam-4894	334	81	theorem	theorem	NOUN
ejpam-4894	334	82	9	9	NUM
ejpam-4894	334	83	.	.	PUNCT
ejpam-4894	335	1	this	this	PRON
ejpam-4894	335	2	means	mean	VERB
ejpam-4894	335	3	that	that	SCONJ
ejpam-4894	335	4	γcerp(g	γcerp(g	PROPN
ejpam-4894	335	5	◦	◦	NOUN
ejpam-4894	335	6	h	h	NOUN
ejpam-4894	335	7	)	)	PUNCT
ejpam-4894	335	8	=	=	NOUN
ejpam-4894	335	9	|s∗|	|s∗|	NUM
ejpam-4894	335	10	≥	≥	NOUN
ejpam-4894	335	11	m.	m.	NOUN
ejpam-4894	335	12	consequently	consequently	ADV
ejpam-4894	335	13	,	,	PUNCT
ejpam-4894	335	14	γcerp(g	γcerp(g	PROPN
ejpam-4894	335	15	◦	◦	NOUN
ejpam-4894	335	16	h	h	NOUN
ejpam-4894	335	17	)	)	PUNCT
ejpam-4894	335	18	=	=	SYM
ejpam-4894	335	19	m.	m.	NOUN
ejpam-4894	335	20	theorem	theorem	VERB
ejpam-4894	335	21	10	10	NUM
ejpam-4894	335	22	.	.	PUNCT
ejpam-4894	336	1	let	let	VERB
ejpam-4894	336	2	g	g	PRON
ejpam-4894	336	3	be	be	AUX
ejpam-4894	336	4	a	a	DET
ejpam-4894	336	5	connected	connected	ADJ
ejpam-4894	336	6	graph	graph	NOUN
ejpam-4894	336	7	of	of	ADP
ejpam-4894	336	8	order	order	NOUN
ejpam-4894	336	9	m	m	VERB
ejpam-4894	336	10	and	and	CCONJ
ejpam-4894	336	11	h	h	NOUN
ejpam-4894	336	12	be	be	VERB
ejpam-4894	336	13	a	a	DET
ejpam-4894	336	14	trivial	trivial	ADJ
ejpam-4894	336	15	graph	graph	NOUN
ejpam-4894	336	16	.	.	PUNCT
ejpam-4894	337	1	then	then	ADV
ejpam-4894	337	2	γcerp(g	γcerp(g	VERB
ejpam-4894	337	3	◦	◦	NOUN
ejpam-4894	337	4	h	h	NOUN
ejpam-4894	337	5	)	)	PUNCT
ejpam-4894	337	6	=	=	SYM
ejpam-4894	337	7	2	2	NUM
ejpam-4894	337	8	m.	m.	NOUN
ejpam-4894	337	9	proof	proof	NOUN
ejpam-4894	337	10	.	.	PUNCT
ejpam-4894	338	1	let	let	VERB
ejpam-4894	338	2	s1	s1	NOUN
ejpam-4894	338	3	be	be	AUX
ejpam-4894	338	4	a	a	DET
ejpam-4894	338	5	vertex	vertex	NOUN
ejpam-4894	338	6	set	set	NOUN
ejpam-4894	338	7	of	of	ADP
ejpam-4894	338	8	a	a	DET
ejpam-4894	338	9	connected	connected	ADJ
ejpam-4894	338	10	graph	graph	NOUN
ejpam-4894	338	11	g	g	NOUN
ejpam-4894	338	12	of	of	ADP
ejpam-4894	338	13	order	order	NOUN
ejpam-4894	338	14	m	m	VERB
ejpam-4894	338	15	and	and	CCONJ
ejpam-4894	338	16	s2	s2	PROPN
ejpam-4894	338	17	be	be	AUX
ejpam-4894	338	18	a	a	DET
ejpam-4894	338	19	vertex	vertex	NOUN
ejpam-4894	338	20	set	set	NOUN
ejpam-4894	338	21	of	of	ADP
ejpam-4894	338	22	a	a	DET
ejpam-4894	338	23	trivial	trivial	ADJ
ejpam-4894	338	24	graph	graph	NOUN
ejpam-4894	338	25	h.	h.	NOUN
ejpam-4894	338	26	since	since	SCONJ
ejpam-4894	338	27	h	h	NOUN
ejpam-4894	338	28	∼=	∼=	PROPN
ejpam-4894	338	29	k1	k1	NOUN
ejpam-4894	338	30	,	,	PUNCT
ejpam-4894	338	31	v	v	NOUN
ejpam-4894	338	32	(	(	PUNCT
ejpam-4894	338	33	h	h	NOUN
ejpam-4894	338	34	)	)	PUNCT
ejpam-4894	338	35	are	be	AUX
ejpam-4894	338	36	pendant	pendant	ADJ
ejpam-4894	338	37	vertices	vertex	NOUN
ejpam-4894	338	38	of	of	ADP
ejpam-4894	338	39	g	g	PROPN
ejpam-4894	338	40	◦	◦	NOUN
ejpam-4894	338	41	h.	h.	NOUN
ejpam-4894	338	42	this	this	PRON
ejpam-4894	338	43	means	mean	VERB
ejpam-4894	338	44	that	that	SCONJ
ejpam-4894	338	45	|s2|	|s2|	NOUN
ejpam-4894	338	46	=	=	NOUN
ejpam-4894	338	47	|s1|	|s1|	NOUN
ejpam-4894	338	48	=	=	SYM
ejpam-4894	338	49	m.	m.	NOUN
ejpam-4894	338	50	furthermore	furthermore	ADV
ejpam-4894	338	51	,	,	PUNCT
ejpam-4894	338	52	every	every	DET
ejpam-4894	338	53	vertex	vertex	NOUN
ejpam-4894	338	54	in	in	ADP
ejpam-4894	338	55	v	v	NUM
ejpam-4894	338	56	∈	∈	NOUN
ejpam-4894	338	57	v	v	NOUN
ejpam-4894	338	58	(	(	PUNCT
ejpam-4894	338	59	v	v	PROPN
ejpam-4894	338	60	+	+	NOUN
ejpam-4894	338	61	hv	hv	X
ejpam-4894	338	62	)	)	PUNCT
ejpam-4894	338	63	has	have	VERB
ejpam-4894	338	64	only	only	ADV
ejpam-4894	338	65	one	one	NUM
ejpam-4894	338	66	neighbor	neighbor	NOUN
ejpam-4894	338	67	in	in	ADP
ejpam-4894	338	68	v	v	PROPN
ejpam-4894	338	69	(	(	PUNCT
ejpam-4894	338	70	v	v	PROPN
ejpam-4894	338	71	+	+	NOUN
ejpam-4894	338	72	hv	hv	NOUN
ejpam-4894	338	73	)	)	PUNCT
ejpam-4894	338	74	\	\	PROPN
ejpam-4894	338	75	v	v	NOUN
ejpam-4894	338	76	for	for	ADP
ejpam-4894	338	77	every	every	DET
ejpam-4894	338	78	v	v	NOUN
ejpam-4894	338	79	∈	∈	PROPN
ejpam-4894	338	80	s1	s1	NOUN
ejpam-4894	338	81	.	.	PUNCT
ejpam-4894	339	1	hence	hence	ADV
ejpam-4894	339	2	,	,	PUNCT
ejpam-4894	339	3	s1	s1	PROPN
ejpam-4894	339	4	∪	∪	NOUN
ejpam-4894	339	5	s2	s2	NOUN
ejpam-4894	339	6	is	be	AUX
ejpam-4894	339	7	a	a	DET
ejpam-4894	339	8	certified	certify	VERB
ejpam-4894	339	9	perfect	perfect	ADJ
ejpam-4894	339	10	dominating	dominating	NOUN
ejpam-4894	339	11	set	set	NOUN
ejpam-4894	339	12	of	of	ADP
ejpam-4894	339	13	g	g	PROPN
ejpam-4894	339	14	◦	◦	NOUN
ejpam-4894	339	15	h.	h.	NOUN
ejpam-4894	339	16	therefore	therefore	ADV
ejpam-4894	339	17	γcerp(g	γcerp(g	VERB
ejpam-4894	339	18	◦	◦	NOUN
ejpam-4894	339	19	h	h	NOUN
ejpam-4894	339	20	)	)	PUNCT
ejpam-4894	340	1	=	=	SYM
ejpam-4894	340	2	|v	|v	PROPN
ejpam-4894	340	3	(	(	PUNCT
ejpam-4894	340	4	g	g	PROPN
ejpam-4894	340	5	◦	◦	NOUN
ejpam-4894	340	6	h)|	h)|	NOUN
ejpam-4894	340	7	=	=	SYM
ejpam-4894	340	8	|s1|+	|s1|+	NOUN
ejpam-4894	340	9	|s2|	|s2|	NOUN
ejpam-4894	340	10	=	=	SYM
ejpam-4894	340	11	m+m	m+m	NOUN
ejpam-4894	340	12	=	=	SYM
ejpam-4894	340	13	2	2	NUM
ejpam-4894	340	14	m	m	NOUN
ejpam-4894	340	15	6	6	NUM
ejpam-4894	340	16	.	.	PUNCT
ejpam-4894	341	1	conclusion	conclusion	NOUN
ejpam-4894	341	2	and	and	CCONJ
ejpam-4894	341	3	recommendation	recommendation	NOUN
ejpam-4894	341	4	the	the	DET
ejpam-4894	341	5	study	study	NOUN
ejpam-4894	341	6	has	have	AUX
ejpam-4894	341	7	introduced	introduce	VERB
ejpam-4894	341	8	and	and	CCONJ
ejpam-4894	341	9	examined	examine	VERB
ejpam-4894	341	10	the	the	DET
ejpam-4894	341	11	concept	concept	NOUN
ejpam-4894	341	12	of	of	ADP
ejpam-4894	341	13	a	a	DET
ejpam-4894	341	14	certified	certify	VERB
ejpam-4894	341	15	perfect	perfect	ADJ
ejpam-4894	341	16	dominating	dominating	NOUN
ejpam-4894	341	17	set	set	NOUN
ejpam-4894	341	18	,	,	PUNCT
ejpam-4894	341	19	denoted	denote	VERB
ejpam-4894	341	20	as	as	ADP
ejpam-4894	341	21	s	s	PROPN
ejpam-4894	341	22	,	,	PUNCT
ejpam-4894	341	23	within	within	ADP
ejpam-4894	341	24	the	the	DET
ejpam-4894	341	25	context	context	NOUN
ejpam-4894	341	26	of	of	ADP
ejpam-4894	341	27	graph	graph	NOUN
ejpam-4894	341	28	g.	g.	PROPN
ejpam-4894	341	29	the	the	DET
ejpam-4894	341	30	author	author	NOUN
ejpam-4894	341	31	’s	’s	PART
ejpam-4894	341	32	primary	primary	ADJ
ejpam-4894	341	33	focus	focus	NOUN
ejpam-4894	341	34	has	have	AUX
ejpam-4894	341	35	been	be	AUX
ejpam-4894	341	36	on	on	ADP
ejpam-4894	341	37	several	several	ADJ
ejpam-4894	341	38	critical	critical	ADJ
ejpam-4894	341	39	areas	area	NOUN
ejpam-4894	341	40	:	:	PUNCT
ejpam-4894	341	41	characterizing	characterize	VERB
ejpam-4894	341	42	the	the	DET
ejpam-4894	341	43	certified	certify	VERB
ejpam-4894	341	44	perfect	perfect	ADJ
ejpam-4894	341	45	dominating	dominating	NOUN
ejpam-4894	341	46	set	set	NOUN
ejpam-4894	341	47	,	,	PUNCT
ejpam-4894	341	48	determining	determine	VERB
ejpam-4894	341	49	precise	precise	ADJ
ejpam-4894	341	50	values	value	NOUN
ejpam-4894	341	51	for	for	ADP
ejpam-4894	341	52	the	the	DET
ejpam-4894	341	53	certified	certify	VERB
ejpam-4894	341	54	perfect	perfect	ADJ
ejpam-4894	341	55	domination	domination	NOUN
ejpam-4894	341	56	number	number	NOUN
ejpam-4894	341	57	in	in	ADP
ejpam-4894	341	58	specific	specific	ADJ
ejpam-4894	341	59	graphs	graph	NOUN
ejpam-4894	341	60	,	,	PUNCT
ejpam-4894	341	61	and	and	CCONJ
ejpam-4894	341	62	exploring	explore	VERB
ejpam-4894	341	63	the	the	DET
ejpam-4894	341	64	certified	certify	VERB
ejpam-4894	341	65	perfect	perfect	ADJ
ejpam-4894	341	66	domination	domination	NOUN
ejpam-4894	341	67	number	number	NOUN
ejpam-4894	341	68	in	in	ADP
ejpam-4894	341	69	graphs	graph	NOUN
ejpam-4894	341	70	resulting	result	VERB
ejpam-4894	341	71	from	from	ADP
ejpam-4894	341	72	join	join	NOUN
ejpam-4894	341	73	and	and	CCONJ
ejpam-4894	341	74	corona	corona	NOUN
ejpam-4894	341	75	operations	operation	NOUN
ejpam-4894	341	76	.	.	PUNCT
ejpam-4894	342	1	furthermore	furthermore	ADV
ejpam-4894	342	2	,	,	PUNCT
ejpam-4894	342	3	the	the	DET
ejpam-4894	342	4	study	study	NOUN
ejpam-4894	342	5	has	have	AUX
ejpam-4894	342	6	established	establish	VERB
ejpam-4894	342	7	relationships	relationship	NOUN
ejpam-4894	342	8	between	between	ADP
ejpam-4894	342	9	the	the	DET
ejpam-4894	342	10	perfect	perfect	ADJ
ejpam-4894	342	11	dominating	dominating	NOUN
ejpam-4894	342	12	set	set	NOUN
ejpam-4894	342	13	and	and	CCONJ
ejpam-4894	342	14	the	the	DET
ejpam-4894	342	15	certified	certify	VERB
ejpam-4894	342	16	perfect	perfect	ADJ
ejpam-4894	342	17	dominating	dominating	NOUN
ejpam-4894	342	18	set	set	NOUN
ejpam-4894	342	19	of	of	ADP
ejpam-4894	342	20	graph	graph	NOUN
ejpam-4894	342	21	g.	g.	NOUN
ejpam-4894	342	22	references	reference	NOUN
ejpam-4894	342	23	2774	2774	NUM
ejpam-4894	342	24	researchers	researcher	NOUN
ejpam-4894	342	25	interested	interested	ADJ
ejpam-4894	342	26	in	in	ADP
ejpam-4894	342	27	this	this	DET
ejpam-4894	342	28	concept	concept	NOUN
ejpam-4894	342	29	can	can	AUX
ejpam-4894	342	30	further	far	ADV
ejpam-4894	342	31	investigate	investigate	VERB
ejpam-4894	342	32	it	it	PRON
ejpam-4894	342	33	in	in	ADP
ejpam-4894	342	34	various	various	ADJ
ejpam-4894	342	35	graph	graph	NOUN
ejpam-4894	342	36	products	product	NOUN
ejpam-4894	342	37	that	that	PRON
ejpam-4894	342	38	were	be	AUX
ejpam-4894	342	39	not	not	PART
ejpam-4894	342	40	addressed	address	VERB
ejpam-4894	342	41	in	in	ADP
ejpam-4894	342	42	this	this	DET
ejpam-4894	342	43	paper	paper	NOUN
ejpam-4894	342	44	.	.	PUNCT
ejpam-4894	343	1	additionally	additionally	ADV
ejpam-4894	343	2	,	,	PUNCT
ejpam-4894	343	3	they	they	PRON
ejpam-4894	343	4	may	may	AUX
ejpam-4894	343	5	explore	explore	VERB
ejpam-4894	343	6	and	and	CCONJ
ejpam-4894	343	7	analyze	analyze	VERB
ejpam-4894	343	8	its	its	PRON
ejpam-4894	343	9	bounds	bound	NOUN
ejpam-4894	343	10	in	in	ADP
ejpam-4894	343	11	relation	relation	NOUN
ejpam-4894	343	12	to	to	ADP
ejpam-4894	343	13	other	other	ADJ
ejpam-4894	343	14	well	well	ADV
ejpam-4894	343	15	-	-	PUNCT
ejpam-4894	343	16	established	establish	VERB
ejpam-4894	343	17	parameters	parameter	NOUN
ejpam-4894	343	18	in	in	ADP
ejpam-4894	343	19	graph	graph	NOUN
ejpam-4894	343	20	theory	theory	NOUN
ejpam-4894	343	21	.	.	PUNCT
ejpam-4894	344	1	acknowledgements	acknowledgement	NOUN
ejpam-4894	344	2	the	the	DET
ejpam-4894	344	3	author	author	NOUN
ejpam-4894	344	4	would	would	AUX
ejpam-4894	344	5	like	like	VERB
ejpam-4894	344	6	to	to	PART
ejpam-4894	344	7	thank	thank	VERB
ejpam-4894	344	8	the	the	DET
ejpam-4894	344	9	anonymous	anonymous	ADJ
ejpam-4894	344	10	referees	referee	NOUN
ejpam-4894	344	11	for	for	ADP
ejpam-4894	344	12	their	their	PRON
ejpam-4894	344	13	comments	comment	NOUN
ejpam-4894	344	14	and	and	CCONJ
ejpam-4894	344	15	suggestions	suggestion	NOUN
ejpam-4894	344	16	which	which	PRON
ejpam-4894	344	17	led	lead	VERB
ejpam-4894	344	18	to	to	ADP
ejpam-4894	344	19	the	the	DET
ejpam-4894	344	20	betterment	betterment	NOUN
ejpam-4894	344	21	of	of	ADP
ejpam-4894	344	22	the	the	DET
ejpam-4894	344	23	paper	paper	NOUN
ejpam-4894	344	24	.	.	PUNCT
ejpam-4894	345	1	this	this	DET
ejpam-4894	345	2	study	study	NOUN
ejpam-4894	345	3	has	have	AUX
ejpam-4894	345	4	been	be	AUX
ejpam-4894	345	5	supported	support	VERB
ejpam-4894	345	6	by	by	ADP
ejpam-4894	345	7	mindanao	mindanao	PROPN
ejpam-4894	345	8	state	state	PROPN
ejpam-4894	345	9	university	university	PROPN
ejpam-4894	345	10	tawi	tawi	PROPN
ejpam-4894	345	11	-	-	PUNCT
ejpam-4894	345	12	tawi	tawi	PROPN
ejpam-4894	345	13	college	college	PROPN
ejpam-4894	345	14	of	of	ADP
ejpam-4894	345	15	technology	technology	NOUN
ejpam-4894	345	16	and	and	CCONJ
ejpam-4894	345	17	oceanography	oceanography	NOUN
ejpam-4894	345	18	.	.	PUNCT
ejpam-4894	346	1	references	reference	NOUN
ejpam-4894	346	2	[	[	X
ejpam-4894	346	3	1	1	X
ejpam-4894	346	4	]	]	PUNCT
ejpam-4894	346	5	v.	v.	PROPN
ejpam-4894	346	6	g.	g.	PROPN
ejpam-4894	346	7	bhagavathi	bhagavathi	PROPN
ejpam-4894	346	8	ammal	ammal	PROPN
ejpam-4894	346	9	and	and	CCONJ
ejpam-4894	346	10	r.	r.	PROPN
ejpam-4894	346	11	louisa	louisa	PROPN
ejpam-4894	346	12	dickfania	dickfania	PROPN
ejpam-4894	346	13	.	.	PUNCT
ejpam-4894	347	1	accurate	accurate	ADJ
ejpam-4894	347	2	certified	certify	VERB
ejpam-4894	347	3	domination	domination	NOUN
ejpam-4894	347	4	number	number	NOUN
ejpam-4894	347	5	of	of	ADP
ejpam-4894	347	6	graphs	graph	NOUN
ejpam-4894	347	7	.	.	PUNCT
ejpam-4894	348	1	international	international	ADJ
ejpam-4894	348	2	journal	journal	PROPN
ejpam-4894	348	3	of	of	ADP
ejpam-4894	348	4	mathematics	mathematics	NOUN
ejpam-4894	348	5	trends	trend	NOUN
ejpam-4894	348	6	and	and	CCONJ
ejpam-4894	348	7	technology	technology	NOUN
ejpam-4894	348	8	,	,	PUNCT
ejpam-4894	348	9	66(05):90–98	66(05):90–98	NUM
ejpam-4894	348	10	,	,	PUNCT
ejpam-4894	348	11	2020	2020	NUM
ejpam-4894	348	12	.	.	PUNCT
ejpam-4894	349	1	[	[	X
ejpam-4894	349	2	2	2	NUM
ejpam-4894	349	3	]	]	PUNCT
ejpam-4894	349	4	c.	c.	PROPN
ejpam-4894	349	5	l.	l.	PROPN
ejpam-4894	349	6	armada	armada	PROPN
ejpam-4894	349	7	and	and	CCONJ
ejpam-4894	349	8	j.	j.	PROPN
ejpam-4894	349	9	j.	j.	PROPN
ejpam-4894	349	10	hamja	hamja	PROPN
ejpam-4894	349	11	.	.	PUNCT
ejpam-4894	350	1	perfect	perfect	PROPN
ejpam-4894	350	2	isolate	isolate	NOUN
ejpam-4894	350	3	domination	domination	NOUN
ejpam-4894	350	4	in	in	ADP
ejpam-4894	350	5	graphs	graph	NOUN
ejpam-4894	350	6	.	.	PUNCT
ejpam-4894	351	1	europian	europian	ADJ
ejpam-4894	351	2	journal	journal	NOUN
ejpam-4894	351	3	of	of	ADP
ejpam-4894	351	4	pure	pure	ADJ
ejpam-4894	351	5	and	and	CCONJ
ejpam-4894	351	6	applied	applied	ADJ
ejpam-4894	351	7	mathematics	mathematic	NOUN
ejpam-4894	351	8	.	.	PUNCT
ejpam-4894	351	9	,	,	PUNCT
ejpam-4894	351	10	16:1326–1341	16:1326–1341	NUM
ejpam-4894	351	11	,	,	PUNCT
ejpam-4894	351	12	2023	2023	NUM
ejpam-4894	351	13	.	.	PUNCT
ejpam-4894	352	1	[	[	X
ejpam-4894	352	2	3	3	X
ejpam-4894	352	3	]	]	X
ejpam-4894	352	4	b.	b.	PROPN
ejpam-4894	352	5	chaluvaraju	chaluvaraju	PROPN
ejpam-4894	352	6	,	,	PUNCT
ejpam-4894	352	7	m.	m.	NOUN
ejpam-4894	352	8	chellali	chellali	PROPN
ejpam-4894	352	9	,	,	PUNCT
ejpam-4894	352	10	and	and	CCONJ
ejpam-4894	352	11	k.	k.	PROPN
ejpam-4894	352	12	a.	a.	PROPN
ejpam-4894	352	13	vidya	vidya	PROPN
ejpam-4894	352	14	.	.	PUNCT
ejpam-4894	353	1	perfect	perfect	ADJ
ejpam-4894	353	2	k	k	NOUN
ejpam-4894	353	3	-	-	NOUN
ejpam-4894	353	4	domination	domination	NOUN
ejpam-4894	353	5	in	in	ADP
ejpam-4894	353	6	graphs	graph	NOUN
ejpam-4894	353	7	.	.	PUNCT
ejpam-4894	354	1	autraliasian	autraliasian	ADJ
ejpam-4894	354	2	journal	journal	PROPN
ejpam-4894	354	3	of	of	ADP
ejpam-4894	354	4	combinatorics	combinatorics	PROPN
ejpam-4894	354	5	,	,	PUNCT
ejpam-4894	354	6	48:175–184	48:175–184	PROPN
ejpam-4894	354	7	,	,	PUNCT
ejpam-4894	354	8	2010	2010	NUM
ejpam-4894	354	9	.	.	PUNCT
ejpam-4894	355	1	[	[	X
ejpam-4894	355	2	4	4	NUM
ejpam-4894	355	3	]	]	X
ejpam-4894	355	4	m.	m.	NOUN
ejpam-4894	355	5	dettlaff	dettlaff	NOUN
ejpam-4894	355	6	,	,	PUNCT
ejpam-4894	355	7	m.	m.	NOUN
ejpam-4894	355	8	lemanska	lemanska	PROPN
ejpam-4894	355	9	,	,	PUNCT
ejpam-4894	355	10	r.	r.	PROPN
ejpam-4894	355	11	ziemann	ziemann	PROPN
ejpam-4894	355	12	j.	j.	PROPN
ejpam-4894	355	13	topp	topp	PROPN
ejpam-4894	355	14	,	,	PUNCT
ejpam-4894	355	15	and	and	CCONJ
ejpam-4894	355	16	p.	p.	PROPN
ejpam-4894	355	17	zylinski	zylinski	PROPN
ejpam-4894	355	18	.	.	PUNCT
ejpam-4894	356	1	certified	certified	ADJ
ejpam-4894	356	2	domination	domination	NOUN
ejpam-4894	356	3	.	.	PUNCT
ejpam-4894	357	1	akce	akce	PROPN
ejpam-4894	357	2	international	international	PROPN
ejpam-4894	357	3	journal	journal	NOUN
ejpam-4894	357	4	of	of	ADP
ejpam-4894	357	5	graphs	graph	NOUN
ejpam-4894	357	6	and	and	CCONJ
ejpam-4894	357	7	combinatirics	combinatiric	NOUN
ejpam-4894	357	8	,	,	PUNCT
ejpam-4894	357	9	09(004):1–12	09(004):1–12	PROPN
ejpam-4894	357	10	,	,	PUNCT
ejpam-4894	357	11	2018	2018	NUM
ejpam-4894	357	12	.	.	PUNCT
ejpam-4894	358	1	[	[	X
ejpam-4894	358	2	5	5	NUM
ejpam-4894	358	3	]	]	PUNCT
ejpam-4894	358	4	f.	f.	PROPN
ejpam-4894	358	5	harary	harary	PROPN
ejpam-4894	358	6	.	.	PUNCT
ejpam-4894	359	1	graph	graph	NOUN
ejpam-4894	359	2	theory	theory	NOUN
ejpam-4894	359	3	.	.	PUNCT
ejpam-4894	360	1	addison	addison	PROPN
ejpam-4894	360	2	-	-	PUNCT
ejpam-4894	360	3	wesley	wesley	PROPN
ejpam-4894	360	4	publishing	publishing	PROPN
ejpam-4894	360	5	company	company	PROPN
ejpam-4894	360	6	,	,	PUNCT
ejpam-4894	360	7	inc	inc	PROPN
ejpam-4894	360	8	.	.	PROPN
ejpam-4894	360	9	,	,	PUNCT
ejpam-4894	360	10	usa	usa	PROPN
ejpam-4894	360	11	,	,	PUNCT
ejpam-4894	360	12	1969	1969	NUM
ejpam-4894	360	13	.	.	PUNCT
ejpam-4894	361	1	[	[	X
ejpam-4894	361	2	6	6	NUM
ejpam-4894	361	3	]	]	X
ejpam-4894	361	4	y.	y.	PROPN
ejpam-4894	361	5	s.	s.	PROPN
ejpam-4894	361	6	kwon	kwon	PROPN
ejpam-4894	361	7	and	and	CCONJ
ejpam-4894	361	8	j.	j.	PROPN
ejpam-4894	361	9	lee	lee	PROPN
ejpam-4894	361	10	.	.	PROPN
ejpam-4894	361	11	perfect	perfect	ADJ
ejpam-4894	361	12	domination	domination	NOUN
ejpam-4894	361	13	sets	set	NOUN
ejpam-4894	361	14	in	in	ADP
ejpam-4894	361	15	cayley	cayley	ADJ
ejpam-4894	361	16	graphs	graph	NOUN
ejpam-4894	361	17	.	.	PUNCT
ejpam-4894	362	1	discrete	discrete	ADJ
ejpam-4894	362	2	applied	apply	VERB
ejpam-4894	362	3	mathematics	mathematic	NOUN
ejpam-4894	362	4	,	,	PUNCT
ejpam-4894	362	5	162:259–263	162:259–263	NUM
ejpam-4894	362	6	.	.	PUNCT
ejpam-4894	362	7	,	,	PUNCT
ejpam-4894	362	8	2014	2014	NUM
ejpam-4894	362	9	.	.	PUNCT
ejpam-4894	363	1	[	[	X
ejpam-4894	363	2	7	7	X
ejpam-4894	363	3	]	]	X
ejpam-4894	363	4	m.	m.	NOUN
ejpam-4894	363	5	livingston	livingston	PROPN
ejpam-4894	363	6	and	and	CCONJ
ejpam-4894	363	7	q.	q.	PROPN
ejpam-4894	363	8	f.	f.	PROPN
ejpam-4894	363	9	stout	stout	PROPN
ejpam-4894	363	10	.	.	PUNCT
ejpam-4894	364	1	perfect	perfect	ADJ
ejpam-4894	364	2	dominating	dominating	NOUN
ejpam-4894	364	3	sets	set	NOUN
ejpam-4894	364	4	.	.	PUNCT
ejpam-4894	365	1	congressus	congressus	PROPN
ejpam-4894	365	2	numerantium	numerantium	PROPN
ejpam-4894	365	3	.	.	PUNCT
ejpam-4894	366	1	,	,	PUNCT
ejpam-4894	366	2	79:187–203	79:187–203	NUM
ejpam-4894	366	3	.	.	PUNCT
ejpam-4894	366	4	,	,	PUNCT
ejpam-4894	366	5	1990	1990	NUM
ejpam-4894	366	6	.	.	PUNCT
ejpam-4894	367	1	[	[	X
ejpam-4894	367	2	8	8	NUM
ejpam-4894	367	3	]	]	PUNCT
ejpam-4894	367	4	s.	s.	PROPN
ejpam-4894	367	5	durai	durai	PROPN
ejpam-4894	367	6	raj	raj	PROPN
ejpam-4894	367	7	,	,	PUNCT
ejpam-4894	367	8	s.	s.	PROPN
ejpam-4894	367	9	g.	g.	PROPN
ejpam-4894	367	10	shiji	shiji	PROPN
ejpam-4894	367	11	kumari	kumari	PROPN
ejpam-4894	367	12	,	,	PUNCT
ejpam-4894	367	13	and	and	CCONJ
ejpam-4894	367	14	a.	a.	NOUN
ejpam-4894	367	15	m.	m.	NOUN
ejpam-4894	367	16	anto	anto	PROPN
ejpam-4894	367	17	.	.	PUNCT
ejpam-4894	368	1	certified	certify	VERB
ejpam-4894	368	2	domination	domination	NOUN
ejpam-4894	368	3	number	number	NOUN
ejpam-4894	368	4	in	in	ADP
ejpam-4894	368	5	corona	corona	NOUN
ejpam-4894	368	6	product	product	NOUN
ejpam-4894	368	7	of	of	ADP
ejpam-4894	368	8	graphs	graph	NOUN
ejpam-4894	368	9	.	.	PUNCT
ejpam-4894	369	1	malaya	malaya	PROPN
ejpam-4894	369	2	journal	journal	PROPN
ejpam-4894	369	3	of	of	ADP
ejpam-4894	369	4	matematik	matematik	PROPN
ejpam-4894	369	5	,	,	PUNCT
ejpam-4894	369	6	9(1):1080–1082	9(1):1080–1082	PROPN
ejpam-4894	369	7	,	,	PUNCT
ejpam-4894	369	8	2021	2021	NUM
ejpam-4894	369	9	.	.	PUNCT
ejpam-4894	370	1	[	[	X
ejpam-4894	370	2	9	9	NUM
ejpam-4894	370	3	]	]	PUNCT
ejpam-4894	370	4	s.	s.	PROPN
ejpam-4894	370	5	durai	durai	PROPN
ejpam-4894	370	6	raj	raj	PROPN
ejpam-4894	370	7	and	and	CCONJ
ejpam-4894	370	8	s.g	s.g	PROPN
ejpam-4894	370	9	.	.	PROPN
ejpam-4894	370	10	shiji	shiji	PROPN
ejpam-4894	370	11	kumari	kumari	PROPN
ejpam-4894	370	12	.	.	PUNCT
ejpam-4894	371	1	certified	certify	VERB
ejpam-4894	371	2	domination	domination	NOUN
ejpam-4894	371	3	number	number	NOUN
ejpam-4894	371	4	in	in	ADP
ejpam-4894	371	5	product	product	NOUN
ejpam-4894	371	6	of	of	ADP
ejpam-4894	371	7	graphs	graph	NOUN
ejpam-4894	371	8	.	.	PUNCT
ejpam-4894	372	1	turkish	turkish	ADJ
ejpam-4894	372	2	journal	journal	NOUN
ejpam-4894	372	3	of	of	ADP
ejpam-4894	372	4	computer	computer	NOUN
ejpam-4894	372	5	and	and	CCONJ
ejpam-4894	372	6	mathematics	mathematic	NOUN
ejpam-4894	372	7	education	education	NOUN
ejpam-4894	372	8	,	,	PUNCT
ejpam-4894	372	9	11(03):1166–1170	11(03):1166–1170	NUM
ejpam-4894	372	10	,	,	PUNCT
ejpam-4894	372	11	2020	2020	NUM
ejpam-4894	372	12	.	.	PUNCT
ejpam-4894	373	1	[	[	X
ejpam-4894	373	2	10	10	NUM
ejpam-4894	373	3	]	]	PUNCT
ejpam-4894	373	4	s.	s.	PROPN
ejpam-4894	373	5	durai	durai	PROPN
ejpam-4894	373	6	raj	raj	PROPN
ejpam-4894	373	7	,	,	PUNCT
ejpam-4894	373	8	s.g	s.g	PROPN
ejpam-4894	373	9	.	.	PROPN
ejpam-4894	373	10	shiji	shiji	PROPN
ejpam-4894	373	11	kumari	kumari	PROPN
ejpam-4894	373	12	,	,	PUNCT
ejpam-4894	373	13	and	and	CCONJ
ejpam-4894	373	14	a.m.	a.m.	PROPN
ejpam-4894	373	15	anto	anto	PROPN
ejpam-4894	373	16	.	.	PROPN
ejpam-4894	373	17	certified	certify	VERB
ejpam-4894	373	18	domination	domination	NOUN
ejpam-4894	373	19	number	number	NOUN
ejpam-4894	373	20	in	in	ADP
ejpam-4894	373	21	subdivision	subdivision	NOUN
ejpam-4894	373	22	of	of	ADP
ejpam-4894	373	23	graphs	graph	NOUN
ejpam-4894	373	24	.	.	PUNCT
ejpam-4894	374	1	international	international	ADJ
ejpam-4894	374	2	journal	journal	NOUN
ejpam-4894	374	3	of	of	ADP
ejpam-4894	374	4	mechanical	mechanical	ADJ
ejpam-4894	374	5	engineering	engineering	NOUN
ejpam-4894	374	6	.	.	PUNCT
ejpam-4894	374	7	,	,	PUNCT
ejpam-4894	374	8	06(03):4324–4327	06(03):4324–4327	PROPN
ejpam-4894	374	9	.	.	PROPN
ejpam-4894	374	10	,	,	PUNCT
ejpam-4894	374	11	2021	2021	NUM
ejpam-4894	374	12	.	.	PUNCT
ejpam-4894	375	1	[	[	X
ejpam-4894	375	2	11	11	NUM
ejpam-4894	375	3	]	]	PUNCT
ejpam-4894	375	4	r.	r.	PROPN
ejpam-4894	375	5	c.	c.	PROPN
ejpam-4894	375	6	rakim	rakim	PROPN
ejpam-4894	375	7	,	,	PUNCT
ejpam-4894	375	8	cj	cj	PROPN
ejpam-4894	375	9	c.	c.	PROPN
ejpam-4894	375	10	saromines	saromine	NOUN
ejpam-4894	375	11	,	,	PUNCT
ejpam-4894	375	12	and	and	CCONJ
ejpam-4894	375	13	h.	h.	PROPN
ejpam-4894	375	14	m.	m.	PROPN
ejpam-4894	375	15	rara	rara	PROPN
ejpam-4894	375	16	.	.	PUNCT
ejpam-4894	376	1	perfect	perfect	ADJ
ejpam-4894	376	2	hop	hop	NOUN
ejpam-4894	376	3	domination	domination	NOUN
ejpam-4894	376	4	in	in	ADP
ejpam-4894	376	5	graphs	graph	NOUN
ejpam-4894	376	6	.	.	PUNCT
ejpam-4894	377	1	applied	apply	VERB
ejpam-4894	377	2	mathematical	mathematical	ADJ
ejpam-4894	377	3	sciences	science	NOUN
ejpam-4894	377	4	,	,	PUNCT
ejpam-4894	377	5	12(13):635–649	12(13):635–649	NUM
ejpam-4894	377	6	.	.	NOUN
ejpam-4894	377	7	,	,	PUNCT
ejpam-4894	377	8	2018	2018	NUM
ejpam-4894	377	9	.	.	PUNCT