id	sid	tid	token	lemma	pos
ejpam-4687	1	1	european	european	PROPN
ejpam-4687	1	2	journal	journal	PROPN
ejpam-4687	1	3	of	of	ADP
ejpam-4687	1	4	pure	pure	ADJ
ejpam-4687	1	5	and	and	CCONJ
ejpam-4687	1	6	applied	apply	VERB
ejpam-4687	1	7	mathematics	mathematic	NOUN
ejpam-4687	1	8	vol	vol	NOUN
ejpam-4687	1	9	.	.	PUNCT
ejpam-4687	2	1	16	16	NUM
ejpam-4687	2	2	,	,	PUNCT
ejpam-4687	2	3	no	no	INTJ
ejpam-4687	2	4	.	.	NOUN
ejpam-4687	2	5	2	2	NUM
ejpam-4687	2	6	,	,	PUNCT
ejpam-4687	2	7	2023	2023	NUM
ejpam-4687	2	8	,	,	PUNCT
ejpam-4687	2	9	1212	1212	NUM
ejpam-4687	2	10	-	-	SYM
ejpam-4687	2	11	1227	1227	NUM
ejpam-4687	2	12	issn	issn	PROPN
ejpam-4687	2	13	1307	1307	NUM
ejpam-4687	2	14	-	-	SYM
ejpam-4687	2	15	5543	5543	NUM
ejpam-4687	2	16	–	–	PUNCT
ejpam-4687	2	17	ejpam.com	ejpam.com	X
ejpam-4687	2	18	published	publish	VERB
ejpam-4687	2	19	by	by	ADP
ejpam-4687	2	20	new	new	PROPN
ejpam-4687	2	21	york	york	PROPN
ejpam-4687	2	22	business	business	PROPN
ejpam-4687	2	23	global	global	PROPN
ejpam-4687	2	24	connected	connect	VERB
ejpam-4687	2	25	grundy	grundy	PROPN
ejpam-4687	2	26	hop	hop	NOUN
ejpam-4687	2	27	dominating	dominate	VERB
ejpam-4687	2	28	sequences	sequence	NOUN
ejpam-4687	2	29	in	in	ADP
ejpam-4687	2	30	graphs	graph	NOUN
ejpam-4687	2	31	javier	javier	PROPN
ejpam-4687	2	32	a.	a.	PROPN
ejpam-4687	2	33	hassan1,∗	hassan1,∗	PROPN
ejpam-4687	2	34	,	,	PUNCT
ejpam-4687	2	35	sergio	sergio	PROPN
ejpam-4687	2	36	r.	r.	PROPN
ejpam-4687	2	37	canoy	canoy	PROPN
ejpam-4687	2	38	,	,	PUNCT
ejpam-4687	2	39	jr.2,3	jr.2,3	PROPN
ejpam-4687	2	40	1mathematics	1mathematics	NUM
ejpam-4687	2	41	and	and	CCONJ
ejpam-4687	2	42	sciences	sciences	PROPN
ejpam-4687	2	43	department	department	PROPN
ejpam-4687	2	44	,	,	PUNCT
ejpam-4687	2	45	college	college	NOUN
ejpam-4687	2	46	of	of	ADP
ejpam-4687	2	47	arts	art	NOUN
ejpam-4687	2	48	and	and	CCONJ
ejpam-4687	2	49	sciences	science	NOUN
ejpam-4687	2	50	,	,	PUNCT
ejpam-4687	2	51	msu	msu	PROPN
ejpam-4687	2	52	tawi	tawi	PROPN
ejpam-4687	2	53	-	-	PUNCT
ejpam-4687	2	54	tawi	tawi	PROPN
ejpam-4687	2	55	college	college	PROPN
ejpam-4687	2	56	of	of	ADP
ejpam-4687	2	57	technology	technology	NOUN
ejpam-4687	2	58	and	and	CCONJ
ejpam-4687	2	59	oceanography	oceanography	NOUN
ejpam-4687	2	60	,	,	PUNCT
ejpam-4687	2	61	bongao	bongao	NOUN
ejpam-4687	2	62	,	,	PUNCT
ejpam-4687	2	63	tawi	tawi	NOUN
ejpam-4687	2	64	-	-	PUNCT
ejpam-4687	2	65	tawi	tawi	NOUN
ejpam-4687	2	66	,	,	PUNCT
ejpam-4687	2	67	philippines	philippine	NOUN
ejpam-4687	2	68	2department	2department	NUM
ejpam-4687	2	69	of	of	ADP
ejpam-4687	2	70	mathematics	mathematic	NOUN
ejpam-4687	2	71	and	and	CCONJ
ejpam-4687	2	72	statistics	statistic	NOUN
ejpam-4687	2	73	,	,	PUNCT
ejpam-4687	2	74	college	college	NOUN
ejpam-4687	2	75	of	of	ADP
ejpam-4687	2	76	science	science	NOUN
ejpam-4687	2	77	and	and	CCONJ
ejpam-4687	2	78	mathematics	mathematic	NOUN
ejpam-4687	2	79	3center	3center	NUM
ejpam-4687	2	80	for	for	ADP
ejpam-4687	2	81	mathematical	mathematical	ADJ
ejpam-4687	2	82	and	and	CCONJ
ejpam-4687	2	83	theoretical	theoretical	ADJ
ejpam-4687	2	84	physical	physical	ADJ
ejpam-4687	2	85	sciences	science	NOUN
ejpam-4687	2	86	,	,	PUNCT
ejpam-4687	2	87	premier	premier	PROPN
ejpam-4687	2	88	research	research	PROPN
ejpam-4687	2	89	institute	institute	PROPN
ejpam-4687	2	90	of	of	ADP
ejpam-4687	2	91	science	science	NOUN
ejpam-4687	2	92	and	and	CCONJ
ejpam-4687	2	93	mathematics	mathematic	NOUN
ejpam-4687	2	94	,	,	PUNCT
ejpam-4687	2	95	msu	msu	PROPN
ejpam-4687	2	96	-	-	PUNCT
ejpam-4687	2	97	iligan	iligan	PROPN
ejpam-4687	2	98	institute	institute	PROPN
ejpam-4687	2	99	of	of	ADP
ejpam-4687	2	100	technology	technology	PROPN
ejpam-4687	2	101	,	,	PUNCT
ejpam-4687	2	102	9200	9200	NUM
ejpam-4687	2	103	iligan	iligan	ADJ
ejpam-4687	2	104	city	city	NOUN
ejpam-4687	2	105	,	,	PUNCT
ejpam-4687	2	106	philippines	philippine	NOUN
ejpam-4687	2	107	abstract	abstract	ADJ
ejpam-4687	2	108	.	.	PUNCT
ejpam-4687	3	1	in	in	ADP
ejpam-4687	3	2	this	this	DET
ejpam-4687	3	3	paper	paper	NOUN
ejpam-4687	3	4	,	,	PUNCT
ejpam-4687	3	5	we	we	PRON
ejpam-4687	3	6	introduce	introduce	VERB
ejpam-4687	3	7	and	and	CCONJ
ejpam-4687	3	8	do	do	VERB
ejpam-4687	3	9	an	an	DET
ejpam-4687	3	10	initial	initial	ADJ
ejpam-4687	3	11	investigation	investigation	NOUN
ejpam-4687	3	12	of	of	ADP
ejpam-4687	3	13	a	a	DET
ejpam-4687	3	14	variant	variant	NOUN
ejpam-4687	3	15	of	of	ADP
ejpam-4687	3	16	grundy	grundy	PROPN
ejpam-4687	3	17	hop	hop	PROPN
ejpam-4687	3	18	domination	domination	NOUN
ejpam-4687	3	19	in	in	ADP
ejpam-4687	3	20	a	a	DET
ejpam-4687	3	21	graph	graph	NOUN
ejpam-4687	3	22	called	call	VERB
ejpam-4687	3	23	the	the	DET
ejpam-4687	3	24	connected	connect	VERB
ejpam-4687	3	25	grundy	grundy	PROPN
ejpam-4687	3	26	hop	hop	PROPN
ejpam-4687	3	27	domination	domination	NOUN
ejpam-4687	3	28	.	.	PUNCT
ejpam-4687	4	1	we	we	PRON
ejpam-4687	4	2	show	show	VERB
ejpam-4687	4	3	that	that	SCONJ
ejpam-4687	4	4	the	the	DET
ejpam-4687	4	5	connected	connected	ADJ
ejpam-4687	4	6	grundy	grundy	PROPN
ejpam-4687	4	7	hop	hop	PROPN
ejpam-4687	4	8	domination	domination	NOUN
ejpam-4687	4	9	number	number	NOUN
ejpam-4687	4	10	lies	lie	VERB
ejpam-4687	4	11	between	between	ADP
ejpam-4687	4	12	the	the	DET
ejpam-4687	4	13	connected	connect	VERB
ejpam-4687	4	14	hop	hop	NOUN
ejpam-4687	4	15	domination	domination	NOUN
ejpam-4687	4	16	and	and	CCONJ
ejpam-4687	4	17	grundy	grundy	PROPN
ejpam-4687	4	18	hop	hop	PROPN
ejpam-4687	4	19	domination	domination	NOUN
ejpam-4687	4	20	number	number	NOUN
ejpam-4687	4	21	of	of	ADP
ejpam-4687	4	22	a	a	DET
ejpam-4687	4	23	graph	graph	NOUN
ejpam-4687	4	24	.	.	PUNCT
ejpam-4687	5	1	in	in	ADP
ejpam-4687	5	2	particular	particular	ADJ
ejpam-4687	5	3	,	,	PUNCT
ejpam-4687	5	4	we	we	PRON
ejpam-4687	5	5	give	give	VERB
ejpam-4687	5	6	realization	realization	NOUN
ejpam-4687	5	7	results	result	NOUN
ejpam-4687	5	8	involving	involve	VERB
ejpam-4687	5	9	connected	connect	VERB
ejpam-4687	5	10	hop	hop	NOUN
ejpam-4687	5	11	domination	domination	NOUN
ejpam-4687	5	12	,	,	PUNCT
ejpam-4687	5	13	connected	connect	VERB
ejpam-4687	5	14	grundy	grundy	PROPN
ejpam-4687	5	15	hop	hop	PROPN
ejpam-4687	5	16	domination	domination	NOUN
ejpam-4687	5	17	,	,	PUNCT
ejpam-4687	5	18	and	and	CCONJ
ejpam-4687	5	19	grundy	grundy	PROPN
ejpam-4687	5	20	hop	hop	PROPN
ejpam-4687	5	21	domination	domination	NOUN
ejpam-4687	5	22	numbers	number	NOUN
ejpam-4687	5	23	.	.	PUNCT
ejpam-4687	6	1	moreover	moreover	ADV
ejpam-4687	6	2	,	,	PUNCT
ejpam-4687	6	3	we	we	PRON
ejpam-4687	6	4	determine	determine	VERB
ejpam-4687	6	5	the	the	DET
ejpam-4687	6	6	connected	connect	VERB
ejpam-4687	6	7	grundy	grundy	PROPN
ejpam-4687	6	8	hop	hop	PROPN
ejpam-4687	6	9	domination	domination	NOUN
ejpam-4687	6	10	numbers	number	NOUN
ejpam-4687	6	11	of	of	ADP
ejpam-4687	6	12	some	some	DET
ejpam-4687	6	13	graphs	graph	NOUN
ejpam-4687	6	14	.	.	PUNCT
ejpam-4687	7	1	2020	2020	NUM
ejpam-4687	7	2	mathematics	mathematic	NOUN
ejpam-4687	7	3	subject	subject	NOUN
ejpam-4687	7	4	classifications	classification	NOUN
ejpam-4687	7	5	:	:	PUNCT
ejpam-4687	7	6	05c69	05c69	X
ejpam-4687	7	7	key	key	ADJ
ejpam-4687	7	8	words	word	NOUN
ejpam-4687	7	9	and	and	CCONJ
ejpam-4687	7	10	phrases	phrase	NOUN
ejpam-4687	7	11	:	:	PUNCT
ejpam-4687	7	12	connected	connect	VERB
ejpam-4687	7	13	hop	hop	NOUN
ejpam-4687	7	14	domination	domination	NOUN
ejpam-4687	7	15	,	,	PUNCT
ejpam-4687	7	16	closed	close	VERB
ejpam-4687	7	17	hop	hop	NOUN
ejpam-4687	7	18	neighborhood	neighborhood	NOUN
ejpam-4687	7	19	sequence	sequence	NOUN
ejpam-4687	7	20	,	,	PUNCT
ejpam-4687	7	21	connected	connect	VERB
ejpam-4687	7	22	grundy	grundy	PROPN
ejpam-4687	7	23	hop	hop	NOUN
ejpam-4687	7	24	dominating	dominating	NOUN
ejpam-4687	7	25	sequence	sequence	NOUN
ejpam-4687	7	26	,	,	PUNCT
ejpam-4687	7	27	connected	connect	VERB
ejpam-4687	7	28	grundy	grundy	PROPN
ejpam-4687	7	29	hop	hop	PROPN
ejpam-4687	7	30	domination	domination	NOUN
ejpam-4687	7	31	number	number	NOUN
ejpam-4687	7	32	1	1	NUM
ejpam-4687	7	33	.	.	PUNCT
ejpam-4687	7	34	introduction	introduction	NOUN
ejpam-4687	7	35	in	in	ADP
ejpam-4687	7	36	2014	2014	NUM
ejpam-4687	7	37	,	,	PUNCT
ejpam-4687	7	38	bresar	bresar	VERB
ejpam-4687	7	39	et	et	PROPN
ejpam-4687	7	40	al	al	PROPN
ejpam-4687	7	41	.	.	PUNCT
ejpam-4687	8	1	[	[	X
ejpam-4687	8	2	6	6	NUM
ejpam-4687	8	3	]	]	PUNCT
ejpam-4687	8	4	introduced	introduce	VERB
ejpam-4687	8	5	grundy	grundy	PROPN
ejpam-4687	8	6	domination	domination	NOUN
ejpam-4687	8	7	in	in	ADP
ejpam-4687	8	8	a	a	DET
ejpam-4687	8	9	graph	graph	NOUN
ejpam-4687	8	10	and	and	CCONJ
ejpam-4687	8	11	made	make	VERB
ejpam-4687	8	12	an	an	DET
ejpam-4687	8	13	initial	initial	ADJ
ejpam-4687	8	14	study	study	NOUN
ejpam-4687	8	15	of	of	ADP
ejpam-4687	8	16	the	the	DET
ejpam-4687	8	17	concept	concept	NOUN
ejpam-4687	8	18	.	.	PUNCT
ejpam-4687	9	1	subsequent	subsequent	ADJ
ejpam-4687	9	2	studies	study	NOUN
ejpam-4687	9	3	on	on	ADP
ejpam-4687	9	4	this	this	DET
ejpam-4687	9	5	newly	newly	ADV
ejpam-4687	9	6	defined	define	VERB
ejpam-4687	9	7	parameter	parameter	NOUN
ejpam-4687	9	8	can	can	AUX
ejpam-4687	9	9	be	be	AUX
ejpam-4687	9	10	found	find	VERB
ejpam-4687	9	11	in	in	ADP
ejpam-4687	9	12	[	[	X
ejpam-4687	9	13	3	3	NUM
ejpam-4687	9	14	]	]	PUNCT
ejpam-4687	9	15	,	,	PUNCT
ejpam-4687	9	16	[	[	X
ejpam-4687	9	17	4	4	NUM
ejpam-4687	9	18	]	]	PUNCT
ejpam-4687	9	19	,	,	PUNCT
ejpam-4687	9	20	[	[	X
ejpam-4687	9	21	5	5	NUM
ejpam-4687	9	22	]	]	PUNCT
ejpam-4687	9	23	,	,	PUNCT
ejpam-4687	9	24	[	[	X
ejpam-4687	9	25	7	7	NUM
ejpam-4687	9	26	]	]	PUNCT
ejpam-4687	9	27	,	,	PUNCT
ejpam-4687	9	28	and	and	CCONJ
ejpam-4687	9	29	[	[	X
ejpam-4687	9	30	12	12	NUM
ejpam-4687	9	31	]	]	PUNCT
ejpam-4687	9	32	.	.	PUNCT
ejpam-4687	10	1	the	the	DET
ejpam-4687	10	2	study	study	NOUN
ejpam-4687	10	3	in	in	ADP
ejpam-4687	10	4	[	[	X
ejpam-4687	10	5	5	5	NUM
ejpam-4687	10	6	]	]	PUNCT
ejpam-4687	10	7	specifically	specifically	ADV
ejpam-4687	10	8	gave	give	VERB
ejpam-4687	10	9	exact	exact	ADJ
ejpam-4687	10	10	formulas	formula	NOUN
ejpam-4687	10	11	for	for	ADP
ejpam-4687	10	12	the	the	DET
ejpam-4687	10	13	grundy	grundy	PROPN
ejpam-4687	10	14	domination	domination	NOUN
ejpam-4687	10	15	numbers	number	NOUN
ejpam-4687	10	16	of	of	ADP
ejpam-4687	10	17	sierpinski	sierpinski	ADJ
ejpam-4687	10	18	graphs	graph	NOUN
ejpam-4687	10	19	where	where	SCONJ
ejpam-4687	10	20	the	the	DET
ejpam-4687	10	21	authors	author	NOUN
ejpam-4687	10	22	provided	provide	VERB
ejpam-4687	10	23	a	a	DET
ejpam-4687	10	24	linear	linear	ADJ
ejpam-4687	10	25	algorithm	algorithm	NOUN
ejpam-4687	10	26	for	for	ADP
ejpam-4687	10	27	determining	determine	VERB
ejpam-4687	10	28	these	these	DET
ejpam-4687	10	29	numbers	number	NOUN
ejpam-4687	10	30	in	in	ADP
ejpam-4687	10	31	arbitrary	arbitrary	ADJ
ejpam-4687	10	32	interval	interval	NOUN
ejpam-4687	10	33	graphs	graph	NOUN
ejpam-4687	10	34	.	.	PUNCT
ejpam-4687	11	1	it	it	PRON
ejpam-4687	11	2	is	be	AUX
ejpam-4687	11	3	without	without	ADP
ejpam-4687	11	4	a	a	DET
ejpam-4687	11	5	doubt	doubt	NOUN
ejpam-4687	11	6	that	that	SCONJ
ejpam-4687	11	7	the	the	DET
ejpam-4687	11	8	concept	concept	NOUN
ejpam-4687	11	9	of	of	ADP
ejpam-4687	11	10	hop	hop	NOUN
ejpam-4687	11	11	domination	domination	NOUN
ejpam-4687	11	12	,	,	PUNCT
ejpam-4687	11	13	just	just	ADV
ejpam-4687	11	14	like	like	ADP
ejpam-4687	11	15	domination	domination	NOUN
ejpam-4687	11	16	,	,	PUNCT
ejpam-4687	11	17	has	have	AUX
ejpam-4687	11	18	ably	ably	ADV
ejpam-4687	11	19	attracted	attract	VERB
ejpam-4687	11	20	a	a	DET
ejpam-4687	11	21	lot	lot	NOUN
ejpam-4687	11	22	of	of	ADP
ejpam-4687	11	23	researchers	researcher	NOUN
ejpam-4687	11	24	to	to	PART
ejpam-4687	11	25	study	study	VERB
ejpam-4687	11	26	it	it	PRON
ejpam-4687	11	27	.	.	PUNCT
ejpam-4687	12	1	some	some	PRON
ejpam-4687	12	2	of	of	ADP
ejpam-4687	12	3	these	these	DET
ejpam-4687	12	4	researchers	researcher	NOUN
ejpam-4687	12	5	have	have	AUX
ejpam-4687	12	6	actually	actually	ADV
ejpam-4687	12	7	introduced	introduce	VERB
ejpam-4687	12	8	some	some	DET
ejpam-4687	12	9	variants	variant	NOUN
ejpam-4687	12	10	of	of	ADP
ejpam-4687	12	11	the	the	DET
ejpam-4687	12	12	concept	concept	NOUN
ejpam-4687	12	13	by	by	ADP
ejpam-4687	12	14	imposing	impose	VERB
ejpam-4687	12	15	additional	additional	ADJ
ejpam-4687	12	16	properties	property	NOUN
ejpam-4687	12	17	on	on	ADP
ejpam-4687	12	18	the	the	DET
ejpam-4687	12	19	standard	standard	ADJ
ejpam-4687	12	20	definition	definition	NOUN
ejpam-4687	12	21	(	(	PUNCT
ejpam-4687	12	22	see	see	VERB
ejpam-4687	12	23	[	[	X
ejpam-4687	12	24	1	1	NUM
ejpam-4687	12	25	]	]	PUNCT
ejpam-4687	12	26	,	,	PUNCT
ejpam-4687	12	27	[	[	X
ejpam-4687	12	28	2	2	NUM
ejpam-4687	12	29	]	]	PUNCT
ejpam-4687	12	30	,	,	PUNCT
ejpam-4687	12	31	[	[	X
ejpam-4687	12	32	9	9	NUM
ejpam-4687	12	33	]	]	PUNCT
ejpam-4687	12	34	,	,	PUNCT
ejpam-4687	12	35	[	[	X
ejpam-4687	12	36	10	10	NUM
ejpam-4687	12	37	]	]	PUNCT
ejpam-4687	12	38	,	,	PUNCT
ejpam-4687	12	39	[	[	X
ejpam-4687	12	40	11	11	NUM
ejpam-4687	12	41	]	]	PUNCT
ejpam-4687	12	42	,	,	PUNCT
ejpam-4687	12	43	[	[	X
ejpam-4687	12	44	13	13	NUM
ejpam-4687	12	45	]	]	NUM
ejpam-4687	12	46	)	)	PUNCT
ejpam-4687	12	47	.	.	PUNCT
ejpam-4687	13	1	∗corresponding	∗corresponde	VERB
ejpam-4687	13	2	author	author	NOUN
ejpam-4687	13	3	.	.	PUNCT
ejpam-4687	14	1	doi	doi	NOUN
ejpam-4687	14	2	:	:	PUNCT
ejpam-4687	14	3	https://doi.org/10.29020/nybg.ejpam.v16i2.4687	https://doi.org/10.29020/nybg.ejpam.v16i2.4687	NOUN
ejpam-4687	14	4	email	email	NOUN
ejpam-4687	14	5	addresses	address	VERB
ejpam-4687	14	6	:	:	PUNCT
ejpam-4687	14	7	javierhassan@msutawi-tawi.edu.ph	javierhassan@msutawi-tawi.edu.ph	PROPN
ejpam-4687	14	8	(	(	PUNCT
ejpam-4687	14	9	j.	j.	PROPN
ejpam-4687	14	10	hassan	hassan	PROPN
ejpam-4687	14	11	)	)	PUNCT
ejpam-4687	14	12	,	,	PUNCT
ejpam-4687	14	13	sergio.canoy@g.msuiit.edu.ph	sergio.canoy@g.msuiit.edu.ph	PROPN
ejpam-4687	14	14	(	(	PUNCT
ejpam-4687	14	15	s.	s.	PROPN
ejpam-4687	14	16	canoy	canoy	PROPN
ejpam-4687	14	17	)	)	PUNCT
ejpam-4687	14	18	https://www.ejpam.com	https://www.ejpam.com	X
ejpam-4687	14	19	1212	1212	NUM
ejpam-4687	14	20	©	©	PROPN
ejpam-4687	14	21	2023	2023	NUM
ejpam-4687	14	22	ejpam	ejpam	NOUN
ejpam-4687	14	23	all	all	DET
ejpam-4687	14	24	rights	right	NOUN
ejpam-4687	14	25	reserved	reserve	VERB
ejpam-4687	14	26	.	.	PUNCT
ejpam-4687	15	1	j.	j.	PROPN
ejpam-4687	15	2	hassan	hassan	PROPN
ejpam-4687	15	3	,	,	PUNCT
ejpam-4687	15	4	s.	s.	PROPN
ejpam-4687	15	5	canoy	canoy	PROPN
ejpam-4687	15	6	jr	jr	PROPN
ejpam-4687	15	7	.	.	PROPN
ejpam-4687	15	8	/	/	SYM
ejpam-4687	15	9	eur	eur	PROPN
ejpam-4687	15	10	.	.	PUNCT
ejpam-4687	16	1	j.	j.	PROPN
ejpam-4687	16	2	pure	pure	PROPN
ejpam-4687	16	3	appl	appl	PROPN
ejpam-4687	16	4	.	.	PROPN
ejpam-4687	16	5	math	math	PROPN
ejpam-4687	16	6	,	,	PUNCT
ejpam-4687	16	7	16	16	NUM
ejpam-4687	16	8	(	(	PUNCT
ejpam-4687	16	9	2	2	NUM
ejpam-4687	16	10	)	)	PUNCT
ejpam-4687	16	11	(	(	PUNCT
ejpam-4687	16	12	2023	2023	NUM
ejpam-4687	16	13	)	)	PUNCT
ejpam-4687	16	14	,	,	PUNCT
ejpam-4687	16	15	1212	1212	NUM
ejpam-4687	16	16	-	-	SYM
ejpam-4687	16	17	1227	1227	NUM
ejpam-4687	16	18	1213	1213	NUM
ejpam-4687	16	19	following	follow	VERB
ejpam-4687	16	20	the	the	DET
ejpam-4687	16	21	definition	definition	NOUN
ejpam-4687	16	22	of	of	ADP
ejpam-4687	16	23	grundy	grundy	PROPN
ejpam-4687	16	24	domination	domination	NOUN
ejpam-4687	16	25	(	(	PUNCT
ejpam-4687	16	26	as	as	ADP
ejpam-4687	16	27	a	a	DET
ejpam-4687	16	28	variant	variant	NOUN
ejpam-4687	16	29	of	of	ADP
ejpam-4687	16	30	the	the	DET
ejpam-4687	16	31	standard	standard	ADJ
ejpam-4687	16	32	domination	domination	NOUN
ejpam-4687	16	33	concept	concept	NOUN
ejpam-4687	16	34	)	)	PUNCT
ejpam-4687	16	35	,	,	PUNCT
ejpam-4687	16	36	hassan	hassan	PROPN
ejpam-4687	16	37	et	et	PROPN
ejpam-4687	16	38	al	al	PROPN
ejpam-4687	16	39	.	.	PUNCT
ejpam-4687	17	1	in	in	ADP
ejpam-4687	17	2	[	[	X
ejpam-4687	17	3	8	8	NUM
ejpam-4687	17	4	]	]	PUNCT
ejpam-4687	17	5	introduced	introduce	VERB
ejpam-4687	17	6	and	and	CCONJ
ejpam-4687	17	7	studied	study	VERB
ejpam-4687	17	8	grundy	grundy	PROPN
ejpam-4687	17	9	hop	hop	PROPN
ejpam-4687	17	10	domination	domination	NOUN
ejpam-4687	17	11	as	as	ADP
ejpam-4687	17	12	a	a	DET
ejpam-4687	17	13	variation	variation	NOUN
ejpam-4687	17	14	of	of	ADP
ejpam-4687	17	15	hop	hop	NOUN
ejpam-4687	17	16	domination	domination	NOUN
ejpam-4687	17	17	.	.	PUNCT
ejpam-4687	18	1	it	it	PRON
ejpam-4687	18	2	was	be	AUX
ejpam-4687	18	3	shown	show	VERB
ejpam-4687	18	4	that	that	SCONJ
ejpam-4687	18	5	difference	difference	NOUN
ejpam-4687	18	6	of	of	ADP
ejpam-4687	18	7	the	the	DET
ejpam-4687	18	8	grundy	grundy	PROPN
ejpam-4687	18	9	hop	hop	PROPN
ejpam-4687	18	10	domination	domination	NOUN
ejpam-4687	18	11	number	number	NOUN
ejpam-4687	18	12	and	and	CCONJ
ejpam-4687	18	13	the	the	DET
ejpam-4687	18	14	hop	hop	NOUN
ejpam-4687	18	15	domination	domination	NOUN
ejpam-4687	18	16	number	number	NOUN
ejpam-4687	18	17	can	can	AUX
ejpam-4687	18	18	be	be	AUX
ejpam-4687	18	19	made	make	VERB
ejpam-4687	18	20	arbitrarily	arbitrarily	ADV
ejpam-4687	18	21	large	large	ADJ
ejpam-4687	18	22	.	.	PUNCT
ejpam-4687	19	1	values	value	NOUN
ejpam-4687	19	2	of	of	ADP
ejpam-4687	19	3	the	the	DET
ejpam-4687	19	4	grundy	grundy	PROPN
ejpam-4687	19	5	hop	hop	PROPN
ejpam-4687	19	6	domination	domination	NOUN
ejpam-4687	19	7	numbers	number	NOUN
ejpam-4687	19	8	had	have	AUX
ejpam-4687	19	9	also	also	ADV
ejpam-4687	19	10	been	be	AUX
ejpam-4687	19	11	determined	determine	VERB
ejpam-4687	19	12	for	for	ADP
ejpam-4687	19	13	some	some	DET
ejpam-4687	19	14	graphs	graph	NOUN
ejpam-4687	19	15	under	under	ADP
ejpam-4687	19	16	some	some	DET
ejpam-4687	19	17	binary	binary	ADJ
ejpam-4687	19	18	operations	operation	NOUN
ejpam-4687	19	19	.	.	PUNCT
ejpam-4687	20	1	in	in	ADP
ejpam-4687	20	2	this	this	DET
ejpam-4687	20	3	study	study	NOUN
ejpam-4687	20	4	,	,	PUNCT
ejpam-4687	20	5	the	the	DET
ejpam-4687	20	6	concept	concept	NOUN
ejpam-4687	20	7	of	of	ADP
ejpam-4687	20	8	connected	connected	ADJ
ejpam-4687	20	9	grundy	grundy	PROPN
ejpam-4687	20	10	hop	hop	PROPN
ejpam-4687	20	11	domination	domination	NOUN
ejpam-4687	20	12	in	in	ADP
ejpam-4687	20	13	a	a	DET
ejpam-4687	20	14	graph	graph	NOUN
ejpam-4687	20	15	will	will	AUX
ejpam-4687	20	16	be	be	AUX
ejpam-4687	20	17	introduced	introduce	VERB
ejpam-4687	20	18	.	.	PUNCT
ejpam-4687	21	1	realization	realization	NOUN
ejpam-4687	21	2	results	result	NOUN
ejpam-4687	21	3	involving	involve	VERB
ejpam-4687	21	4	connected	connect	VERB
ejpam-4687	21	5	hop	hop	NOUN
ejpam-4687	21	6	domination	domination	NOUN
ejpam-4687	21	7	,	,	PUNCT
ejpam-4687	21	8	grundy	grundy	PROPN
ejpam-4687	21	9	hop	hop	PROPN
ejpam-4687	21	10	domination	domination	NOUN
ejpam-4687	21	11	,	,	PUNCT
ejpam-4687	21	12	and	and	CCONJ
ejpam-4687	21	13	connected	connect	VERB
ejpam-4687	21	14	grundy	grundy	PROPN
ejpam-4687	21	15	hop	hop	PROPN
ejpam-4687	21	16	domination	domination	NOUN
ejpam-4687	21	17	numbers	number	NOUN
ejpam-4687	21	18	are	be	AUX
ejpam-4687	21	19	given	give	VERB
ejpam-4687	21	20	.	.	PUNCT
ejpam-4687	22	1	moreover	moreover	ADV
ejpam-4687	22	2	,	,	PUNCT
ejpam-4687	22	3	graphs	graph	NOUN
ejpam-4687	22	4	that	that	PRON
ejpam-4687	22	5	attain	attain	VERB
ejpam-4687	22	6	some	some	DET
ejpam-4687	22	7	specific	specific	ADJ
ejpam-4687	22	8	values	value	NOUN
ejpam-4687	22	9	for	for	ADP
ejpam-4687	22	10	the	the	DET
ejpam-4687	22	11	parameter	parameter	NOUN
ejpam-4687	22	12	are	be	AUX
ejpam-4687	22	13	characterized	characterize	VERB
ejpam-4687	22	14	.	.	PUNCT
ejpam-4687	23	1	2	2	X
ejpam-4687	23	2	.	.	X
ejpam-4687	23	3	terminology	terminology	NOUN
ejpam-4687	23	4	and	and	CCONJ
ejpam-4687	23	5	notation	notation	NOUN
ejpam-4687	23	6	let	let	VERB
ejpam-4687	23	7	g	g	PRON
ejpam-4687	23	8	be	be	AUX
ejpam-4687	23	9	a	a	DET
ejpam-4687	23	10	simple	simple	ADJ
ejpam-4687	23	11	undirected	undirected	ADJ
ejpam-4687	23	12	graph	graph	NOUN
ejpam-4687	23	13	.	.	PUNCT
ejpam-4687	24	1	two	two	NUM
ejpam-4687	24	2	vertices	vertex	NOUN
ejpam-4687	24	3	a	a	PRON
ejpam-4687	24	4	and	and	CCONJ
ejpam-4687	24	5	b	b	NOUN
ejpam-4687	24	6	of	of	ADP
ejpam-4687	24	7	g	g	PROPN
ejpam-4687	24	8	are	be	AUX
ejpam-4687	24	9	adjacent	adjacent	ADJ
ejpam-4687	24	10	,	,	PUNCT
ejpam-4687	24	11	or	or	CCONJ
ejpam-4687	24	12	neighbors	neighbor	NOUN
ejpam-4687	24	13	,	,	PUNCT
ejpam-4687	24	14	if	if	SCONJ
ejpam-4687	24	15	ab	ab	PROPN
ejpam-4687	24	16	is	be	AUX
ejpam-4687	24	17	an	an	DET
ejpam-4687	24	18	edge	edge	NOUN
ejpam-4687	24	19	of	of	ADP
ejpam-4687	24	20	g.	g.	PROPN
ejpam-4687	24	21	the	the	DET
ejpam-4687	24	22	set	set	NOUN
ejpam-4687	24	23	of	of	ADP
ejpam-4687	24	24	neighbors	neighbor	NOUN
ejpam-4687	24	25	of	of	ADP
ejpam-4687	24	26	a	a	DET
ejpam-4687	24	27	vertex	vertex	NOUN
ejpam-4687	24	28	u	u	NOUN
ejpam-4687	24	29	in	in	ADP
ejpam-4687	24	30	g	g	NOUN
ejpam-4687	24	31	,	,	PUNCT
ejpam-4687	24	32	denoted	denote	VERB
ejpam-4687	24	33	by	by	ADP
ejpam-4687	24	34	ng(u	ng(u	NOUN
ejpam-4687	24	35	)	)	PUNCT
ejpam-4687	24	36	,	,	PUNCT
ejpam-4687	24	37	is	be	AUX
ejpam-4687	24	38	called	call	VERB
ejpam-4687	24	39	the	the	DET
ejpam-4687	24	40	open	open	ADJ
ejpam-4687	24	41	neighborhood	neighborhood	NOUN
ejpam-4687	24	42	of	of	ADP
ejpam-4687	24	43	u	u	PROPN
ejpam-4687	24	44	in	in	ADP
ejpam-4687	24	45	g.	g.	PROPN
ejpam-4687	24	46	the	the	DET
ejpam-4687	24	47	closed	close	VERB
ejpam-4687	24	48	neighborhood	neighborhood	NOUN
ejpam-4687	24	49	of	of	ADP
ejpam-4687	24	50	u	u	NOUN
ejpam-4687	24	51	in	in	ADP
ejpam-4687	24	52	g	g	PROPN
ejpam-4687	24	53	is	be	AUX
ejpam-4687	24	54	the	the	DET
ejpam-4687	24	55	set	set	NOUN
ejpam-4687	24	56	ng[u	ng[u	PROPN
ejpam-4687	24	57	]	]	X
ejpam-4687	24	58	=	=	SYM
ejpam-4687	24	59	ng(u	ng(u	PROPN
ejpam-4687	24	60	)	)	PUNCT
ejpam-4687	24	61	∪	∪	NOUN
ejpam-4687	24	62	{	{	PUNCT
ejpam-4687	24	63	u	u	NOUN
ejpam-4687	24	64	}	}	PUNCT
ejpam-4687	24	65	.	.	PUNCT
ejpam-4687	25	1	if	if	SCONJ
ejpam-4687	25	2	x	x	PROPN
ejpam-4687	25	3	⊆	⊆	NUM
ejpam-4687	25	4	v	v	X
ejpam-4687	25	5	(	(	PUNCT
ejpam-4687	25	6	g	g	NOUN
ejpam-4687	25	7	)	)	PUNCT
ejpam-4687	25	8	,	,	PUNCT
ejpam-4687	25	9	the	the	DET
ejpam-4687	25	10	open	open	ADJ
ejpam-4687	25	11	neighborhood	neighborhood	NOUN
ejpam-4687	25	12	of	of	ADP
ejpam-4687	25	13	x	x	PUNCT
ejpam-4687	25	14	in	in	ADP
ejpam-4687	25	15	g	g	PROPN
ejpam-4687	25	16	is	be	AUX
ejpam-4687	25	17	the	the	DET
ejpam-4687	25	18	set	set	NOUN
ejpam-4687	25	19	ng(x	ng(x	NUM
ejpam-4687	25	20	)	)	PUNCT
ejpam-4687	26	1	=	=	SYM
ejpam-4687	26	2	⋃	⋃	NOUN
ejpam-4687	26	3	u∈x	u∈x	NOUN
ejpam-4687	26	4	ng(u	ng(u	NOUN
ejpam-4687	26	5	)	)	PUNCT
ejpam-4687	26	6	.	.	PUNCT
ejpam-4687	27	1	the	the	DET
ejpam-4687	27	2	closed	closed	ADJ
ejpam-4687	27	3	neighborhood	neighborhood	NOUN
ejpam-4687	27	4	of	of	ADP
ejpam-4687	27	5	x	x	PUNCT
ejpam-4687	27	6	in	in	ADP
ejpam-4687	27	7	g	g	PROPN
ejpam-4687	27	8	is	be	AUX
ejpam-4687	27	9	the	the	DET
ejpam-4687	27	10	set	set	NOUN
ejpam-4687	27	11	ng[x	ng[x	PROPN
ejpam-4687	27	12	]	]	X
ejpam-4687	27	13	=	=	SYM
ejpam-4687	27	14	ng(x)∪x	ng(x)∪x	PROPN
ejpam-4687	27	15	.	.	PUNCT
ejpam-4687	28	1	a	a	DET
ejpam-4687	28	2	set	set	NOUN
ejpam-4687	28	3	d	d	NOUN
ejpam-4687	28	4	⊆	⊆	NUM
ejpam-4687	28	5	v	v	ADP
ejpam-4687	28	6	(	(	PUNCT
ejpam-4687	28	7	g	g	NOUN
ejpam-4687	28	8	)	)	PUNCT
ejpam-4687	28	9	is	be	AUX
ejpam-4687	28	10	a	a	DET
ejpam-4687	28	11	dominating	dominating	NOUN
ejpam-4687	28	12	of	of	ADP
ejpam-4687	28	13	g	g	PROPN
ejpam-4687	28	14	if	if	SCONJ
ejpam-4687	28	15	for	for	ADP
ejpam-4687	28	16	every	every	DET
ejpam-4687	28	17	v	v	NUM
ejpam-4687	28	18	∈	∈	NOUN
ejpam-4687	28	19	v	v	NOUN
ejpam-4687	28	20	(	(	PUNCT
ejpam-4687	28	21	g	g	NOUN
ejpam-4687	28	22	)	)	PUNCT
ejpam-4687	28	23	\d	\d	NOUN
ejpam-4687	28	24	,	,	PUNCT
ejpam-4687	28	25	there	there	PRON
ejpam-4687	28	26	exists	exist	VERB
ejpam-4687	28	27	u	u	NOUN
ejpam-4687	28	28	∈	∈	PROPN
ejpam-4687	28	29	d	d	ADP
ejpam-4687	28	30	such	such	ADJ
ejpam-4687	28	31	that	that	DET
ejpam-4687	28	32	uv	uv	PROPN
ejpam-4687	28	33	∈	∈	PROPN
ejpam-4687	28	34	e(g	e(g	PROPN
ejpam-4687	28	35	)	)	PUNCT
ejpam-4687	28	36	,	,	PUNCT
ejpam-4687	28	37	that	that	ADV
ejpam-4687	28	38	is	is	ADV
ejpam-4687	28	39	,	,	PUNCT
ejpam-4687	28	40	ng[d	ng[d	PROPN
ejpam-4687	28	41	]	]	PUNCT
ejpam-4687	28	42	=	=	SYM
ejpam-4687	28	43	v	v	X
ejpam-4687	28	44	(	(	PUNCT
ejpam-4687	28	45	g	g	NOUN
ejpam-4687	28	46	)	)	PUNCT
ejpam-4687	28	47	.	.	PUNCT
ejpam-4687	29	1	the	the	DET
ejpam-4687	29	2	domination	domination	NOUN
ejpam-4687	29	3	number	number	NOUN
ejpam-4687	29	4	of	of	ADP
ejpam-4687	29	5	g	g	NOUN
ejpam-4687	29	6	,	,	PUNCT
ejpam-4687	29	7	denoted	denote	VERB
ejpam-4687	29	8	by	by	ADP
ejpam-4687	29	9	γ(g	γ(g	PROPN
ejpam-4687	29	10	)	)	PUNCT
ejpam-4687	29	11	,	,	PUNCT
ejpam-4687	29	12	is	be	AUX
ejpam-4687	29	13	the	the	DET
ejpam-4687	29	14	minimum	minimum	ADJ
ejpam-4687	29	15	cardinality	cardinality	NOUN
ejpam-4687	29	16	of	of	ADP
ejpam-4687	29	17	a	a	DET
ejpam-4687	29	18	dominating	dominating	NOUN
ejpam-4687	29	19	set	set	NOUN
ejpam-4687	29	20	of	of	ADP
ejpam-4687	29	21	g.	g.	PROPN
ejpam-4687	29	22	let	let	VERB
ejpam-4687	29	23	s	s	AUX
ejpam-4687	29	24	=	=	PUNCT
ejpam-4687	29	25	(	(	PUNCT
ejpam-4687	29	26	v1	v1	PROPN
ejpam-4687	29	27	,	,	PUNCT
ejpam-4687	29	28	v2	v2	PROPN
ejpam-4687	29	29	,	,	PUNCT
ejpam-4687	29	30	·	·	PUNCT
ejpam-4687	29	31	·	·	PUNCT
ejpam-4687	29	32	·	·	PUNCT
ejpam-4687	29	33	,	,	PUNCT
ejpam-4687	29	34	vk	vk	AUX
ejpam-4687	29	35	)	)	PUNCT
ejpam-4687	29	36	be	be	AUX
ejpam-4687	29	37	a	a	DET
ejpam-4687	29	38	sequence	sequence	NOUN
ejpam-4687	29	39	of	of	ADP
ejpam-4687	29	40	distinct	distinct	ADJ
ejpam-4687	29	41	vertices	vertex	NOUN
ejpam-4687	29	42	of	of	ADP
ejpam-4687	29	43	a	a	DET
ejpam-4687	29	44	graph	graph	NOUN
ejpam-4687	29	45	g	g	NOUN
ejpam-4687	29	46	,	,	PUNCT
ejpam-4687	29	47	and	and	CCONJ
ejpam-4687	29	48	let	let	VERB
ejpam-4687	29	49	ŝ	ŝ	X
ejpam-4687	29	50	=	=	SYM
ejpam-4687	29	51	{	{	PUNCT
ejpam-4687	29	52	v1	v1	PROPN
ejpam-4687	29	53	,	,	PUNCT
ejpam-4687	29	54	v2	v2	PROPN
ejpam-4687	29	55	,	,	PUNCT
ejpam-4687	29	56	·	·	PUNCT
ejpam-4687	29	57	·	·	PUNCT
ejpam-4687	29	58	·	·	PUNCT
ejpam-4687	29	59	,	,	PUNCT
ejpam-4687	29	60	vk	vk	ADP
ejpam-4687	29	61	}	}	PUNCT
ejpam-4687	29	62	.	.	PUNCT
ejpam-4687	30	1	then	then	ADV
ejpam-4687	30	2	s	s	VERB
ejpam-4687	30	3	is	be	AUX
ejpam-4687	30	4	a	a	DET
ejpam-4687	30	5	legal	legal	ADJ
ejpam-4687	30	6	closed	closed	ADJ
ejpam-4687	30	7	neighborhood	neighborhood	NOUN
ejpam-4687	30	8	sequence	sequence	NOUN
ejpam-4687	30	9	if	if	SCONJ
ejpam-4687	30	10	ng[vi]\	ng[vi]\	PROPN
ejpam-4687	30	11	⋃i−1	⋃i−1	PROPN
ejpam-4687	30	12	j=1ng[vj	j=1ng[vj	PROPN
ejpam-4687	30	13	]	]	PUNCT
ejpam-4687	30	14	̸=	̸=	PROPN
ejpam-4687	30	15	∅	∅	NOUN
ejpam-4687	30	16	for	for	ADP
ejpam-4687	30	17	every	every	DET
ejpam-4687	30	18	i	i	PROPN
ejpam-4687	30	19	∈	∈	PROPN
ejpam-4687	30	20	{	{	PUNCT
ejpam-4687	30	21	2	2	NUM
ejpam-4687	30	22	,	,	PUNCT
ejpam-4687	30	23	·	·	PUNCT
ejpam-4687	30	24	·	·	PUNCT
ejpam-4687	30	25	·	·	PUNCT
ejpam-4687	30	26	,	,	PUNCT
ejpam-4687	30	27	k	k	NOUN
ejpam-4687	30	28	}	}	PUNCT
ejpam-4687	30	29	.	.	PUNCT
ejpam-4687	31	1	if	if	SCONJ
ejpam-4687	31	2	,	,	PUNCT
ejpam-4687	31	3	in	in	ADP
ejpam-4687	31	4	addition	addition	NOUN
ejpam-4687	31	5	,	,	PUNCT
ejpam-4687	31	6	ŝ	ŝ	X
ejpam-4687	31	7	is	be	AUX
ejpam-4687	31	8	a	a	DET
ejpam-4687	31	9	dominating	dominating	NOUN
ejpam-4687	31	10	set	set	NOUN
ejpam-4687	31	11	of	of	ADP
ejpam-4687	31	12	g	g	NOUN
ejpam-4687	31	13	,	,	PUNCT
ejpam-4687	31	14	then	then	ADV
ejpam-4687	31	15	s	s	VERB
ejpam-4687	31	16	is	be	AUX
ejpam-4687	31	17	called	call	VERB
ejpam-4687	31	18	a	a	DET
ejpam-4687	31	19	grundy	grundy	PROPN
ejpam-4687	31	20	dominating	dominating	NOUN
ejpam-4687	31	21	sequence	sequence	NOUN
ejpam-4687	31	22	.	.	PUNCT
ejpam-4687	32	1	the	the	DET
ejpam-4687	32	2	maximum	maximum	ADJ
ejpam-4687	32	3	length	length	NOUN
ejpam-4687	32	4	of	of	ADP
ejpam-4687	32	5	a	a	DET
ejpam-4687	32	6	grundy	grundy	PROPN
ejpam-4687	32	7	dominating	dominating	NOUN
ejpam-4687	32	8	sequence	sequence	NOUN
ejpam-4687	32	9	in	in	ADP
ejpam-4687	32	10	a	a	DET
ejpam-4687	32	11	graph	graph	NOUN
ejpam-4687	32	12	g	g	NOUN
ejpam-4687	32	13	is	be	AUX
ejpam-4687	32	14	called	call	VERB
ejpam-4687	32	15	the	the	DET
ejpam-4687	32	16	grundy	grundy	PROPN
ejpam-4687	32	17	domination	domination	NOUN
ejpam-4687	32	18	number	number	NOUN
ejpam-4687	32	19	of	of	ADP
ejpam-4687	32	20	g	g	NOUN
ejpam-4687	32	21	,	,	PUNCT
ejpam-4687	32	22	and	and	CCONJ
ejpam-4687	32	23	is	be	AUX
ejpam-4687	32	24	denoted	denote	VERB
ejpam-4687	32	25	by	by	ADP
ejpam-4687	32	26	γgr(g	γgr(g	PROPN
ejpam-4687	32	27	)	)	PUNCT
ejpam-4687	32	28	.	.	PUNCT
ejpam-4687	33	1	we	we	PRON
ejpam-4687	33	2	say	say	VERB
ejpam-4687	33	3	that	that	SCONJ
ejpam-4687	33	4	vertex	vertex	PROPN
ejpam-4687	33	5	vi	vi	PROPN
ejpam-4687	33	6	footprints	footprint	NOUN
ejpam-4687	33	7	the	the	DET
ejpam-4687	33	8	vertices	vertex	NOUN
ejpam-4687	33	9	from	from	ADP
ejpam-4687	33	10	ng[vi	ng[vi	PROPN
ejpam-4687	33	11	]	]	PUNCT
ejpam-4687	33	12	\	\	PROPN
ejpam-4687	33	13	∪i	∪i	PROPN
ejpam-4687	33	14	j=1ng[vj	j=1ng[vj	PROPN
ejpam-4687	33	15	]	]	PUNCT
ejpam-4687	33	16	,	,	PUNCT
ejpam-4687	33	17	and	and	CCONJ
ejpam-4687	33	18	that	that	DET
ejpam-4687	33	19	vi	vi	PROPN
ejpam-4687	33	20	is	be	AUX
ejpam-4687	33	21	their	their	PRON
ejpam-4687	33	22	footprinter	footprinter	NOUN
ejpam-4687	33	23	.	.	PUNCT
ejpam-4687	34	1	any	any	DET
ejpam-4687	34	2	grundy	grundy	PROPN
ejpam-4687	34	3	dominating	dominating	NOUN
ejpam-4687	34	4	sequence	sequence	NOUN
ejpam-4687	34	5	s	s	PART
ejpam-4687	34	6	with	with	ADP
ejpam-4687	34	7	|ŝ|	|ŝ|	PROPN
ejpam-4687	34	8	=	=	SYM
ejpam-4687	34	9	γgr(g	γgr(g	PROPN
ejpam-4687	34	10	)	)	PUNCT
ejpam-4687	34	11	is	be	AUX
ejpam-4687	34	12	called	call	VERB
ejpam-4687	34	13	a	a	DET
ejpam-4687	34	14	maximum	maximum	ADJ
ejpam-4687	34	15	grundy	grundy	PROPN
ejpam-4687	34	16	dominating	dominating	NOUN
ejpam-4687	34	17	sequence	sequence	NOUN
ejpam-4687	34	18	or	or	CCONJ
ejpam-4687	34	19	a	a	DET
ejpam-4687	34	20	γgr	γgr	NOUN
ejpam-4687	34	21	-	-	PUNCT
ejpam-4687	34	22	sequence	sequence	NOUN
ejpam-4687	34	23	of	of	ADP
ejpam-4687	34	24	g.	g.	PROPN
ejpam-4687	34	25	in	in	ADP
ejpam-4687	34	26	this	this	DET
ejpam-4687	34	27	case	case	NOUN
ejpam-4687	34	28	,	,	PUNCT
ejpam-4687	34	29	we	we	PRON
ejpam-4687	34	30	call	call	VERB
ejpam-4687	34	31	ŝ	ŝ	NOUN
ejpam-4687	34	32	a	a	DET
ejpam-4687	34	33	γgr	γgr	NOUN
ejpam-4687	34	34	-	-	PUNCT
ejpam-4687	34	35	set	set	NOUN
ejpam-4687	34	36	of	of	ADP
ejpam-4687	34	37	g.	g.	PROPN
ejpam-4687	34	38	a	a	DET
ejpam-4687	34	39	vertex	vertex	NOUN
ejpam-4687	34	40	v	v	NOUN
ejpam-4687	34	41	in	in	ADP
ejpam-4687	34	42	g	g	PROPN
ejpam-4687	34	43	is	be	AUX
ejpam-4687	34	44	a	a	DET
ejpam-4687	34	45	hop	hop	NOUN
ejpam-4687	34	46	neighbor	neighbor	NOUN
ejpam-4687	34	47	of	of	ADP
ejpam-4687	34	48	vertex	vertex	NOUN
ejpam-4687	34	49	u	u	NOUN
ejpam-4687	34	50	in	in	ADP
ejpam-4687	34	51	g	g	PROPN
ejpam-4687	34	52	if	if	SCONJ
ejpam-4687	34	53	dg(u	dg(u	NOUN
ejpam-4687	34	54	,	,	PUNCT
ejpam-4687	34	55	v	v	NOUN
ejpam-4687	34	56	)	)	PUNCT
ejpam-4687	35	1	=	=	SYM
ejpam-4687	35	2	2	2	X
ejpam-4687	35	3	.	.	X
ejpam-4687	36	1	the	the	DET
ejpam-4687	36	2	set	set	ADJ
ejpam-4687	36	3	n2	n2	ADJ
ejpam-4687	36	4	g(u	g(u	PROPN
ejpam-4687	36	5	)	)	PUNCT
ejpam-4687	36	6	=	=	PRON
ejpam-4687	36	7	{	{	PUNCT
ejpam-4687	36	8	v	v	NUM
ejpam-4687	36	9	∈	∈	NOUN
ejpam-4687	36	10	v	v	NOUN
ejpam-4687	36	11	(	(	PUNCT
ejpam-4687	36	12	g	g	NOUN
ejpam-4687	36	13	)	)	PUNCT
ejpam-4687	36	14	:	:	PUNCT
ejpam-4687	36	15	dg(v	dg(v	X
ejpam-4687	36	16	,	,	PUNCT
ejpam-4687	36	17	u	u	NOUN
ejpam-4687	36	18	)	)	PUNCT
ejpam-4687	36	19	=	=	SYM
ejpam-4687	36	20	2	2	X
ejpam-4687	36	21	}	}	PUNCT
ejpam-4687	36	22	is	be	AUX
ejpam-4687	36	23	called	call	VERB
ejpam-4687	36	24	the	the	DET
ejpam-4687	36	25	open	open	ADJ
ejpam-4687	36	26	hop	hop	NOUN
ejpam-4687	36	27	neighborhood	neighborhood	NOUN
ejpam-4687	36	28	of	of	ADP
ejpam-4687	36	29	u.	u.	PROPN
ejpam-4687	36	30	the	the	DET
ejpam-4687	36	31	closed	closed	ADJ
ejpam-4687	36	32	hop	hop	NOUN
ejpam-4687	36	33	neighborhood	neighborhood	NOUN
ejpam-4687	36	34	of	of	ADP
ejpam-4687	36	35	u	u	PROPN
ejpam-4687	36	36	in	in	ADP
ejpam-4687	36	37	g	g	PROPN
ejpam-4687	36	38	is	be	AUX
ejpam-4687	36	39	given	give	VERB
ejpam-4687	36	40	by	by	ADP
ejpam-4687	36	41	n2	n2	PROPN
ejpam-4687	36	42	g[u	g[u	PROPN
ejpam-4687	36	43	]	]	X
ejpam-4687	36	44	=	=	SYM
ejpam-4687	36	45	n2	n2	ADJ
ejpam-4687	36	46	g(u	g(u	PROPN
ejpam-4687	36	47	)	)	PUNCT
ejpam-4687	36	48	∪	∪	NOUN
ejpam-4687	36	49	{	{	PUNCT
ejpam-4687	36	50	u	u	NOUN
ejpam-4687	36	51	}	}	PUNCT
ejpam-4687	36	52	.	.	PUNCT
ejpam-4687	37	1	the	the	DET
ejpam-4687	37	2	open	open	ADJ
ejpam-4687	37	3	hop	hop	NOUN
ejpam-4687	37	4	neighborhood	neighborhood	NOUN
ejpam-4687	37	5	of	of	ADP
ejpam-4687	37	6	x	x	PROPN
ejpam-4687	37	7	⊆	⊆	NUM
ejpam-4687	37	8	v	v	ADP
ejpam-4687	37	9	(	(	PUNCT
ejpam-4687	37	10	g	g	NOUN
ejpam-4687	37	11	)	)	PUNCT
ejpam-4687	37	12	is	be	AUX
ejpam-4687	37	13	the	the	DET
ejpam-4687	37	14	set	set	ADJ
ejpam-4687	37	15	n2	n2	ADJ
ejpam-4687	37	16	g(x	g(x	NOUN
ejpam-4687	37	17	)	)	PUNCT
ejpam-4687	38	1	=	=	SYM
ejpam-4687	38	2	⋃	⋃	NOUN
ejpam-4687	38	3	u∈x	u∈x	ADJ
ejpam-4687	38	4	n2	n2	NOUN
ejpam-4687	38	5	g(u	g(u	PROPN
ejpam-4687	38	6	)	)	PUNCT
ejpam-4687	38	7	.	.	PUNCT
ejpam-4687	39	1	the	the	DET
ejpam-4687	39	2	closed	closed	ADJ
ejpam-4687	39	3	hop	hop	NOUN
ejpam-4687	39	4	neighborhood	neighborhood	NOUN
ejpam-4687	39	5	of	of	ADP
ejpam-4687	39	6	x	x	PUNCT
ejpam-4687	39	7	in	in	ADP
ejpam-4687	39	8	g	g	PROPN
ejpam-4687	39	9	is	be	AUX
ejpam-4687	39	10	the	the	DET
ejpam-4687	39	11	set	set	ADJ
ejpam-4687	39	12	n2	n2	NOUN
ejpam-4687	39	13	g[x	g[x	PROPN
ejpam-4687	39	14	]	]	X
ejpam-4687	39	15	=	=	SYM
ejpam-4687	39	16	n2	n2	PROPN
ejpam-4687	39	17	g(x	g(x	NOUN
ejpam-4687	39	18	)	)	PUNCT
ejpam-4687	39	19	∪x	∪x	NUM
ejpam-4687	39	20	.	.	PUNCT
ejpam-4687	40	1	a	a	DET
ejpam-4687	40	2	set	set	NOUN
ejpam-4687	40	3	d	d	NOUN
ejpam-4687	40	4	⊆	⊆	NUM
ejpam-4687	40	5	v	v	ADP
ejpam-4687	40	6	(	(	PUNCT
ejpam-4687	40	7	g	g	NOUN
ejpam-4687	40	8	)	)	PUNCT
ejpam-4687	40	9	is	be	AUX
ejpam-4687	40	10	a	a	DET
ejpam-4687	40	11	hop	hop	NOUN
ejpam-4687	40	12	dominating	dominating	NOUN
ejpam-4687	40	13	set	set	NOUN
ejpam-4687	40	14	of	of	ADP
ejpam-4687	40	15	g	g	PROPN
ejpam-4687	40	16	if	if	SCONJ
ejpam-4687	40	17	n2	n2	PROPN
ejpam-4687	40	18	g[d	g[d	PROPN
ejpam-4687	40	19	]	]	X
ejpam-4687	40	20	=	=	SYM
ejpam-4687	40	21	v	v	NOUN
ejpam-4687	40	22	(	(	PUNCT
ejpam-4687	40	23	g	g	NOUN
ejpam-4687	40	24	)	)	PUNCT
ejpam-4687	40	25	,	,	PUNCT
ejpam-4687	40	26	that	that	ADV
ejpam-4687	40	27	is	is	ADV
ejpam-4687	40	28	,	,	PUNCT
ejpam-4687	40	29	for	for	ADP
ejpam-4687	40	30	every	every	DET
ejpam-4687	40	31	v	v	NUM
ejpam-4687	40	32	∈	∈	PROPN
ejpam-4687	40	33	v	v	NOUN
ejpam-4687	40	34	(	(	PUNCT
ejpam-4687	40	35	g)\d	g)\d	NOUN
ejpam-4687	40	36	,	,	PUNCT
ejpam-4687	40	37	there	there	PRON
ejpam-4687	40	38	exists	exist	VERB
ejpam-4687	40	39	u	u	NOUN
ejpam-4687	40	40	∈	∈	PROPN
ejpam-4687	40	41	d	d	ADP
ejpam-4687	40	42	such	such	ADJ
ejpam-4687	40	43	that	that	DET
ejpam-4687	40	44	dg(u	dg(u	ADJ
ejpam-4687	40	45	,	,	PUNCT
ejpam-4687	40	46	v	v	NOUN
ejpam-4687	40	47	)	)	PUNCT
ejpam-4687	41	1	=	=	SYM
ejpam-4687	41	2	2	2	X
ejpam-4687	41	3	.	.	PUNCT
ejpam-4687	42	1	the	the	DET
ejpam-4687	42	2	minimum	minimum	ADJ
ejpam-4687	42	3	cardinality	cardinality	NOUN
ejpam-4687	42	4	among	among	ADP
ejpam-4687	42	5	all	all	DET
ejpam-4687	42	6	hop	hop	NOUN
ejpam-4687	42	7	dominating	dominating	NOUN
ejpam-4687	42	8	sets	set	NOUN
ejpam-4687	42	9	of	of	ADP
ejpam-4687	42	10	g	g	NOUN
ejpam-4687	42	11	,	,	PUNCT
ejpam-4687	42	12	denoted	denote	VERB
ejpam-4687	42	13	by	by	ADP
ejpam-4687	42	14	γh(g	γh(g	NOUN
ejpam-4687	42	15	)	)	PUNCT
ejpam-4687	42	16	,	,	PUNCT
ejpam-4687	42	17	is	be	AUX
ejpam-4687	42	18	called	call	VERB
ejpam-4687	42	19	the	the	DET
ejpam-4687	42	20	hop	hop	NOUN
ejpam-4687	42	21	domination	domination	NOUN
ejpam-4687	42	22	number	number	NOUN
ejpam-4687	42	23	of	of	ADP
ejpam-4687	42	24	g.	g.	PROPN
ejpam-4687	42	25	any	any	DET
ejpam-4687	42	26	hop	hop	NOUN
ejpam-4687	42	27	dominating	dominating	NOUN
ejpam-4687	42	28	set	set	VERB
ejpam-4687	42	29	with	with	ADP
ejpam-4687	42	30	cardinality	cardinality	NOUN
ejpam-4687	42	31	equal	equal	ADJ
ejpam-4687	42	32	to	to	ADP
ejpam-4687	42	33	γh(g	γh(g	NOUN
ejpam-4687	42	34	)	)	PUNCT
ejpam-4687	42	35	is	be	AUX
ejpam-4687	42	36	called	call	VERB
ejpam-4687	42	37	a	a	DET
ejpam-4687	42	38	γh	γh	ADV
ejpam-4687	42	39	-	-	PUNCT
ejpam-4687	42	40	set	set	NOUN
ejpam-4687	42	41	.	.	PUNCT
ejpam-4687	43	1	a	a	DET
ejpam-4687	43	2	hop	hop	NOUN
ejpam-4687	43	3	dominating	dominating	NOUN
ejpam-4687	43	4	set	set	NOUN
ejpam-4687	43	5	d	d	NOUN
ejpam-4687	43	6	is	be	AUX
ejpam-4687	43	7	called	call	VERB
ejpam-4687	43	8	a	a	DET
ejpam-4687	43	9	connected	connected	ADJ
ejpam-4687	43	10	hop	hop	NOUN
ejpam-4687	43	11	dominating	dominating	NOUN
ejpam-4687	43	12	set	set	NOUN
ejpam-4687	43	13	if	if	SCONJ
ejpam-4687	43	14	⟨d⟩	⟨d⟩	PROPN
ejpam-4687	43	15	is	be	AUX
ejpam-4687	43	16	connected	connect	VERB
ejpam-4687	43	17	.	.	PUNCT
ejpam-4687	44	1	the	the	DET
ejpam-4687	44	2	minimum	minimum	ADJ
ejpam-4687	44	3	cardinality	cardinality	NOUN
ejpam-4687	44	4	among	among	ADP
ejpam-4687	44	5	all	all	DET
ejpam-4687	44	6	connected	connect	VERB
ejpam-4687	44	7	hop	hop	NOUN
ejpam-4687	44	8	dominating	dominating	NOUN
ejpam-4687	44	9	sets	set	NOUN
ejpam-4687	44	10	of	of	ADP
ejpam-4687	44	11	g	g	NOUN
ejpam-4687	44	12	,	,	PUNCT
ejpam-4687	44	13	denoted	denote	VERB
ejpam-4687	44	14	by	by	ADP
ejpam-4687	44	15	j.	j.	PROPN
ejpam-4687	44	16	hassan	hassan	PROPN
ejpam-4687	44	17	,	,	PUNCT
ejpam-4687	44	18	s.	s.	PROPN
ejpam-4687	44	19	canoy	canoy	PROPN
ejpam-4687	44	20	jr	jr	PROPN
ejpam-4687	44	21	.	.	PROPN
ejpam-4687	44	22	/	/	SYM
ejpam-4687	44	23	eur	eur	PROPN
ejpam-4687	44	24	.	.	PUNCT
ejpam-4687	45	1	j.	j.	PROPN
ejpam-4687	45	2	pure	pure	PROPN
ejpam-4687	45	3	appl	appl	PROPN
ejpam-4687	45	4	.	.	PROPN
ejpam-4687	45	5	math	math	PROPN
ejpam-4687	45	6	,	,	PUNCT
ejpam-4687	45	7	16	16	NUM
ejpam-4687	45	8	(	(	PUNCT
ejpam-4687	45	9	2	2	NUM
ejpam-4687	45	10	)	)	PUNCT
ejpam-4687	45	11	(	(	PUNCT
ejpam-4687	45	12	2023	2023	NUM
ejpam-4687	45	13	)	)	PUNCT
ejpam-4687	45	14	,	,	PUNCT
ejpam-4687	45	15	1212	1212	NUM
ejpam-4687	45	16	-	-	SYM
ejpam-4687	45	17	1227	1227	NUM
ejpam-4687	45	18	1214	1214	NUM
ejpam-4687	45	19	γch(g	γch(g	NOUN
ejpam-4687	45	20	)	)	PUNCT
ejpam-4687	45	21	,	,	PUNCT
ejpam-4687	45	22	is	be	AUX
ejpam-4687	45	23	called	call	VERB
ejpam-4687	45	24	the	the	DET
ejpam-4687	45	25	connected	connect	VERB
ejpam-4687	45	26	hop	hop	NOUN
ejpam-4687	45	27	domination	domination	NOUN
ejpam-4687	45	28	number	number	NOUN
ejpam-4687	45	29	of	of	ADP
ejpam-4687	45	30	g.	g.	PROPN
ejpam-4687	45	31	any	any	DET
ejpam-4687	45	32	connected	connect	VERB
ejpam-4687	45	33	hop	hop	NOUN
ejpam-4687	45	34	dominating	dominating	NOUN
ejpam-4687	45	35	set	set	VERB
ejpam-4687	45	36	with	with	ADP
ejpam-4687	45	37	cardinality	cardinality	NOUN
ejpam-4687	45	38	equal	equal	ADJ
ejpam-4687	45	39	to	to	ADP
ejpam-4687	45	40	γch(g	γch(g	NOUN
ejpam-4687	45	41	)	)	PUNCT
ejpam-4687	45	42	is	be	AUX
ejpam-4687	45	43	called	call	VERB
ejpam-4687	45	44	a	a	DET
ejpam-4687	45	45	γch	γch	NOUN
ejpam-4687	45	46	-	-	PUNCT
ejpam-4687	45	47	set	set	NOUN
ejpam-4687	45	48	.	.	PUNCT
ejpam-4687	46	1	let	let	VERB
ejpam-4687	46	2	s	s	PRON
ejpam-4687	46	3	=	=	PUNCT
ejpam-4687	46	4	(	(	PUNCT
ejpam-4687	46	5	v1	v1	PROPN
ejpam-4687	46	6	,	,	PUNCT
ejpam-4687	46	7	v2	v2	PROPN
ejpam-4687	46	8	,	,	PUNCT
ejpam-4687	46	9	·	·	PUNCT
ejpam-4687	46	10	·	·	PUNCT
ejpam-4687	46	11	·	·	PUNCT
ejpam-4687	46	12	,	,	PUNCT
ejpam-4687	46	13	vk	vk	AUX
ejpam-4687	46	14	)	)	PUNCT
ejpam-4687	46	15	be	be	AUX
ejpam-4687	46	16	a	a	DET
ejpam-4687	46	17	sequence	sequence	NOUN
ejpam-4687	46	18	of	of	ADP
ejpam-4687	46	19	distinct	distinct	ADJ
ejpam-4687	46	20	vertices	vertex	NOUN
ejpam-4687	46	21	of	of	ADP
ejpam-4687	46	22	g	g	NOUN
ejpam-4687	46	23	and	and	CCONJ
ejpam-4687	46	24	let	let	VERB
ejpam-4687	46	25	ŝ	ŝ	X
ejpam-4687	46	26	=	=	SYM
ejpam-4687	46	27	{	{	PUNCT
ejpam-4687	46	28	v1	v1	PROPN
ejpam-4687	46	29	,	,	PUNCT
ejpam-4687	46	30	·	·	PUNCT
ejpam-4687	46	31	·	·	PUNCT
ejpam-4687	46	32	·	·	PUNCT
ejpam-4687	46	33	,	,	PUNCT
ejpam-4687	46	34	vk	vk	ADP
ejpam-4687	46	35	}	}	PUNCT
ejpam-4687	46	36	.	.	PUNCT
ejpam-4687	47	1	then	then	ADV
ejpam-4687	47	2	s	s	VERB
ejpam-4687	47	3	is	be	AUX
ejpam-4687	47	4	a	a	DET
ejpam-4687	47	5	legal	legal	ADJ
ejpam-4687	47	6	closed	close	VERB
ejpam-4687	47	7	hop	hop	NOUN
ejpam-4687	47	8	neighborhood	neighborhood	NOUN
ejpam-4687	47	9	sequence	sequence	NOUN
ejpam-4687	47	10	of	of	ADP
ejpam-4687	47	11	g	g	PROPN
ejpam-4687	47	12	if	if	SCONJ
ejpam-4687	47	13	n2	n2	PROPN
ejpam-4687	47	14	g[vi	g[vi	PROPN
ejpam-4687	47	15	]	]	PUNCT
ejpam-4687	47	16	\	\	X
ejpam-4687	48	1	∪i−1	∪i−1	PUNCT
ejpam-4687	48	2	j=1n	j=1n	PROPN
ejpam-4687	48	3	2	2	NUM
ejpam-4687	48	4	g[vj	g[vj	PROPN
ejpam-4687	48	5	]	]	PUNCT
ejpam-4687	48	6	̸=	̸=	PROPN
ejpam-4687	48	7	∅	∅	NOUN
ejpam-4687	48	8	for	for	ADP
ejpam-4687	48	9	each	each	DET
ejpam-4687	48	10	i	i	PRON
ejpam-4687	48	11	∈	∈	PROPN
ejpam-4687	48	12	{	{	PUNCT
ejpam-4687	48	13	2	2	NUM
ejpam-4687	48	14	,	,	PUNCT
ejpam-4687	48	15	·	·	PUNCT
ejpam-4687	48	16	·	·	PUNCT
ejpam-4687	48	17	·	·	PUNCT
ejpam-4687	48	18	,	,	PUNCT
ejpam-4687	48	19	k	k	NOUN
ejpam-4687	48	20	}	}	PUNCT
ejpam-4687	48	21	.	.	PUNCT
ejpam-4687	49	1	if	if	SCONJ
ejpam-4687	49	2	,	,	PUNCT
ejpam-4687	49	3	in	in	ADP
ejpam-4687	49	4	addition	addition	NOUN
ejpam-4687	49	5	,	,	PUNCT
ejpam-4687	49	6	ŝ	ŝ	X
ejpam-4687	49	7	is	be	AUX
ejpam-4687	49	8	a	a	DET
ejpam-4687	49	9	hop	hop	NOUN
ejpam-4687	49	10	dominating	dominating	NOUN
ejpam-4687	49	11	set	set	NOUN
ejpam-4687	49	12	of	of	ADP
ejpam-4687	49	13	g	g	PROPN
ejpam-4687	49	14	,	,	PUNCT
ejpam-4687	49	15	then	then	ADV
ejpam-4687	49	16	s	s	VERB
ejpam-4687	49	17	is	be	AUX
ejpam-4687	49	18	called	call	VERB
ejpam-4687	49	19	a	a	DET
ejpam-4687	49	20	grundy	grundy	PROPN
ejpam-4687	49	21	hop	hop	NOUN
ejpam-4687	49	22	dominating	dominating	NOUN
ejpam-4687	49	23	sequence	sequence	NOUN
ejpam-4687	49	24	.	.	PUNCT
ejpam-4687	50	1	the	the	DET
ejpam-4687	50	2	maximum	maximum	ADJ
ejpam-4687	50	3	length	length	NOUN
ejpam-4687	50	4	of	of	ADP
ejpam-4687	50	5	a	a	DET
ejpam-4687	50	6	grundy	grundy	PROPN
ejpam-4687	50	7	hop	hop	NOUN
ejpam-4687	50	8	dominating	dominating	NOUN
ejpam-4687	50	9	sequence	sequence	NOUN
ejpam-4687	50	10	in	in	ADP
ejpam-4687	50	11	a	a	DET
ejpam-4687	50	12	graph	graph	NOUN
ejpam-4687	50	13	g	g	NOUN
ejpam-4687	50	14	,	,	PUNCT
ejpam-4687	50	15	denoted	denote	VERB
ejpam-4687	50	16	by	by	ADP
ejpam-4687	50	17	γhgr(g	γhgr(g	PROPN
ejpam-4687	50	18	)	)	PUNCT
ejpam-4687	50	19	,	,	PUNCT
ejpam-4687	50	20	is	be	AUX
ejpam-4687	50	21	called	call	VERB
ejpam-4687	50	22	the	the	DET
ejpam-4687	50	23	grundy	grundy	PROPN
ejpam-4687	50	24	hop	hop	PROPN
ejpam-4687	50	25	domination	domination	NOUN
ejpam-4687	50	26	number	number	NOUN
ejpam-4687	50	27	of	of	ADP
ejpam-4687	50	28	g.	g.	PROPN
ejpam-4687	50	29	we	we	PRON
ejpam-4687	50	30	say	say	VERB
ejpam-4687	50	31	that	that	DET
ejpam-4687	50	32	vertex	vertex	NOUN
ejpam-4687	50	33	vi	vi	PROPN
ejpam-4687	50	34	hop	hop	NOUN
ejpam-4687	50	35	-	-	PUNCT
ejpam-4687	50	36	footprints	footprint	NOUN
ejpam-4687	50	37	the	the	DET
ejpam-4687	50	38	vertices	vertex	NOUN
ejpam-4687	50	39	from	from	ADP
ejpam-4687	50	40	n2	n2	ADJ
ejpam-4687	51	1	g[vi]\∪i	g[vi]\∪i	PROPN
ejpam-4687	51	2	j=1n	j=1n	PROPN
ejpam-4687	51	3	2	2	NUM
ejpam-4687	51	4	g[vj	g[vj	PROPN
ejpam-4687	51	5	]	]	PUNCT
ejpam-4687	51	6	,	,	PUNCT
ejpam-4687	51	7	and	and	CCONJ
ejpam-4687	51	8	that	that	DET
ejpam-4687	51	9	vi	vi	PROPN
ejpam-4687	51	10	is	be	AUX
ejpam-4687	51	11	their	their	PRON
ejpam-4687	51	12	hop	hop	NOUN
ejpam-4687	51	13	-	-	PUNCT
ejpam-4687	51	14	footprinter	footprinter	NOUN
ejpam-4687	51	15	.	.	PUNCT
ejpam-4687	52	1	any	any	DET
ejpam-4687	52	2	grundy	grundy	PROPN
ejpam-4687	52	3	hop	hop	NOUN
ejpam-4687	52	4	dominating	dominating	NOUN
ejpam-4687	52	5	sequence	sequence	NOUN
ejpam-4687	52	6	s	s	PART
ejpam-4687	52	7	with	with	ADP
ejpam-4687	52	8	|ŝ|	|ŝ|	PROPN
ejpam-4687	52	9	=	=	SYM
ejpam-4687	52	10	γhgr(g	γhgr(g	PROPN
ejpam-4687	52	11	)	)	PUNCT
ejpam-4687	52	12	is	be	AUX
ejpam-4687	52	13	called	call	VERB
ejpam-4687	52	14	a	a	DET
ejpam-4687	52	15	maximum	maximum	ADJ
ejpam-4687	52	16	grundy	grundy	PROPN
ejpam-4687	52	17	hop	hop	NOUN
ejpam-4687	52	18	dominating	dominating	NOUN
ejpam-4687	52	19	sequence	sequence	NOUN
ejpam-4687	52	20	or	or	CCONJ
ejpam-4687	52	21	a	a	DET
ejpam-4687	52	22	γhgr	γhgr	ADJ
ejpam-4687	52	23	-	-	PUNCT
ejpam-4687	52	24	sequence	sequence	NOUN
ejpam-4687	52	25	of	of	ADP
ejpam-4687	52	26	g.	g.	PROPN
ejpam-4687	52	27	in	in	ADP
ejpam-4687	52	28	this	this	DET
ejpam-4687	52	29	case	case	NOUN
ejpam-4687	52	30	,	,	PUNCT
ejpam-4687	52	31	we	we	PRON
ejpam-4687	52	32	call	call	VERB
ejpam-4687	52	33	ŝ	ŝ	NOUN
ejpam-4687	52	34	a	a	DET
ejpam-4687	52	35	γhgr	γhgr	VERB
ejpam-4687	52	36	-	-	PUNCT
ejpam-4687	52	37	set	set	NOUN
ejpam-4687	52	38	of	of	ADP
ejpam-4687	52	39	g.	g.	PROPN
ejpam-4687	52	40	a	a	DET
ejpam-4687	52	41	grundy	grundy	PROPN
ejpam-4687	52	42	hop	hop	NOUN
ejpam-4687	52	43	dominating	dominating	NOUN
ejpam-4687	52	44	sequence	sequence	NOUN
ejpam-4687	52	45	s	s	PART
ejpam-4687	52	46	is	be	AUX
ejpam-4687	52	47	called	call	VERB
ejpam-4687	52	48	a	a	DET
ejpam-4687	52	49	connected	connect	VERB
ejpam-4687	52	50	grundy	grundy	PROPN
ejpam-4687	52	51	hop	hop	NOUN
ejpam-4687	52	52	dominating	dominating	NOUN
ejpam-4687	52	53	sequence	sequence	NOUN
ejpam-4687	52	54	if	if	SCONJ
ejpam-4687	52	55	⟨ŝ⟩	⟨ŝ⟩	PRON
ejpam-4687	52	56	is	be	AUX
ejpam-4687	52	57	connected	connect	VERB
ejpam-4687	52	58	.	.	PUNCT
ejpam-4687	53	1	the	the	DET
ejpam-4687	53	2	maximum	maximum	ADJ
ejpam-4687	53	3	length	length	NOUN
ejpam-4687	53	4	of	of	ADP
ejpam-4687	53	5	a	a	DET
ejpam-4687	53	6	connected	connect	VERB
ejpam-4687	53	7	grundy	grundy	PROPN
ejpam-4687	53	8	hop	hop	NOUN
ejpam-4687	53	9	dominating	dominating	NOUN
ejpam-4687	53	10	sequence	sequence	NOUN
ejpam-4687	53	11	in	in	ADP
ejpam-4687	53	12	a	a	DET
ejpam-4687	53	13	graph	graph	NOUN
ejpam-4687	53	14	g	g	NOUN
ejpam-4687	53	15	,	,	PUNCT
ejpam-4687	53	16	denoted	denote	VERB
ejpam-4687	53	17	by	by	ADP
ejpam-4687	53	18	γchgr	γchgr	PROPN
ejpam-4687	53	19	(	(	PUNCT
ejpam-4687	53	20	g	g	NOUN
ejpam-4687	53	21	)	)	PUNCT
ejpam-4687	53	22	,	,	PUNCT
ejpam-4687	53	23	is	be	AUX
ejpam-4687	53	24	called	call	VERB
ejpam-4687	53	25	the	the	DET
ejpam-4687	53	26	connected	connect	VERB
ejpam-4687	53	27	grundy	grundy	PROPN
ejpam-4687	53	28	hop	hop	PROPN
ejpam-4687	53	29	domination	domination	NOUN
ejpam-4687	53	30	number	number	NOUN
ejpam-4687	53	31	of	of	ADP
ejpam-4687	53	32	g.	g.	PROPN
ejpam-4687	53	33	we	we	PRON
ejpam-4687	53	34	say	say	VERB
ejpam-4687	53	35	that	that	DET
ejpam-4687	53	36	vertex	vertex	NOUN
ejpam-4687	53	37	vi	vi	PROPN
ejpam-4687	53	38	hop	hop	NOUN
ejpam-4687	53	39	-	-	PUNCT
ejpam-4687	53	40	footprints	footprint	NOUN
ejpam-4687	53	41	the	the	DET
ejpam-4687	53	42	vertices	vertex	NOUN
ejpam-4687	53	43	from	from	ADP
ejpam-4687	53	44	n2	n2	PROPN
ejpam-4687	53	45	g[vi	g[vi	PROPN
ejpam-4687	53	46	]	]	PUNCT
ejpam-4687	53	47	\	\	NOUN
ejpam-4687	54	1	∪i	∪i	PUNCT
ejpam-4687	54	2	j=1n	j=1n	PROPN
ejpam-4687	54	3	2	2	NUM
ejpam-4687	54	4	g[vj	g[vj	PROPN
ejpam-4687	54	5	]	]	PUNCT
ejpam-4687	54	6	,	,	PUNCT
ejpam-4687	54	7	and	and	CCONJ
ejpam-4687	54	8	that	that	DET
ejpam-4687	54	9	vi	vi	PROPN
ejpam-4687	54	10	is	be	AUX
ejpam-4687	54	11	their	their	PRON
ejpam-4687	54	12	hop	hop	NOUN
ejpam-4687	54	13	-	-	PUNCT
ejpam-4687	54	14	footprinter	footprinter	NOUN
ejpam-4687	54	15	.	.	PUNCT
ejpam-4687	55	1	any	any	DET
ejpam-4687	55	2	connected	connected	ADJ
ejpam-4687	55	3	grundy	grundy	PROPN
ejpam-4687	55	4	hop	hop	NOUN
ejpam-4687	55	5	dominating	dominating	NOUN
ejpam-4687	55	6	sequence	sequence	NOUN
ejpam-4687	55	7	s	s	PART
ejpam-4687	55	8	with	with	ADP
ejpam-4687	55	9	|ŝ|	|ŝ|	PROPN
ejpam-4687	55	10	=	=	SYM
ejpam-4687	55	11	γchgr	γchgr	X
ejpam-4687	55	12	(	(	PUNCT
ejpam-4687	55	13	g	g	NOUN
ejpam-4687	55	14	)	)	PUNCT
ejpam-4687	55	15	is	be	AUX
ejpam-4687	55	16	called	call	VERB
ejpam-4687	55	17	a	a	DET
ejpam-4687	55	18	maximum	maximum	ADV
ejpam-4687	55	19	connected	connect	VERB
ejpam-4687	55	20	grundy	grundy	PROPN
ejpam-4687	55	21	hop	hop	NOUN
ejpam-4687	55	22	dominating	dominating	NOUN
ejpam-4687	55	23	sequence	sequence	NOUN
ejpam-4687	55	24	or	or	CCONJ
ejpam-4687	55	25	a	a	DET
ejpam-4687	55	26	γchgr	γchgr	ADJ
ejpam-4687	55	27	-sequence	-sequence	NOUN
ejpam-4687	55	28	of	of	ADP
ejpam-4687	55	29	g.	g.	NOUN
ejpam-4687	55	30	in	in	ADP
ejpam-4687	55	31	this	this	DET
ejpam-4687	55	32	case	case	NOUN
ejpam-4687	55	33	,	,	PUNCT
ejpam-4687	55	34	we	we	PRON
ejpam-4687	55	35	call	call	VERB
ejpam-4687	55	36	ŝ	ŝ	NOUN
ejpam-4687	55	37	a	a	DET
ejpam-4687	55	38	γchgr	γchgr	ADJ
ejpam-4687	55	39	-set	-set	ADJ
ejpam-4687	55	40	of	of	ADP
ejpam-4687	55	41	g.	g.	PROPN
ejpam-4687	55	42	a	a	DET
ejpam-4687	55	43	sequence	sequence	NOUN
ejpam-4687	55	44	s	s	PART
ejpam-4687	55	45	=	=	PUNCT
ejpam-4687	55	46	(	(	PUNCT
ejpam-4687	55	47	v1	v1	PROPN
ejpam-4687	55	48	,	,	PUNCT
ejpam-4687	55	49	v2	v2	PROPN
ejpam-4687	55	50	,	,	PUNCT
ejpam-4687	55	51	·	·	PUNCT
ejpam-4687	55	52	·	·	PUNCT
ejpam-4687	55	53	·	·	PUNCT
ejpam-4687	55	54	,	,	PUNCT
ejpam-4687	55	55	vk	vk	PROPN
ejpam-4687	55	56	)	)	PUNCT
ejpam-4687	55	57	of	of	ADP
ejpam-4687	55	58	distinct	distinct	ADJ
ejpam-4687	55	59	vertices	vertex	NOUN
ejpam-4687	55	60	of	of	ADP
ejpam-4687	55	61	a	a	DET
ejpam-4687	55	62	graph	graph	NOUN
ejpam-4687	55	63	g	g	NOUN
ejpam-4687	55	64	is	be	AUX
ejpam-4687	55	65	a	a	DET
ejpam-4687	55	66	co	co	ADJ
ejpam-4687	55	67	-	-	ADJ
ejpam-4687	55	68	legal	legal	ADJ
ejpam-4687	55	69	closed	closed	ADJ
ejpam-4687	55	70	neighborhood	neighborhood	NOUN
ejpam-4687	55	71	sequence	sequence	NOUN
ejpam-4687	55	72	in	in	ADP
ejpam-4687	55	73	g	g	PROPN
ejpam-4687	55	74	if	if	SCONJ
ejpam-4687	55	75	[	[	X
ejpam-4687	55	76	v	v	X
ejpam-4687	55	77	(	(	PUNCT
ejpam-4687	55	78	g	g	NOUN
ejpam-4687	55	79	)	)	PUNCT
ejpam-4687	55	80	\	\	NOUN
ejpam-4687	55	81	ng(vi	ng(vi	NOUN
ejpam-4687	55	82	)	)	PUNCT
ejpam-4687	55	83	]	]	PUNCT
ejpam-4687	55	84	\	\	X
ejpam-4687	56	1	∪i−1	∪i−1	PROPN
ejpam-4687	56	2	j=1[v	j=1[v	X
ejpam-4687	56	3	(	(	PUNCT
ejpam-4687	56	4	g	g	NOUN
ejpam-4687	56	5	)	)	PUNCT
ejpam-4687	56	6	\	\	NOUN
ejpam-4687	56	7	ng(vj	ng(vj	NOUN
ejpam-4687	56	8	)	)	PUNCT
ejpam-4687	56	9	]	]	PUNCT
ejpam-4687	57	1	̸=	̸=	NOUN
ejpam-4687	57	2	∅	∅	NOUN
ejpam-4687	57	3	for	for	ADP
ejpam-4687	57	4	each	each	DET
ejpam-4687	57	5	i	i	PRON
ejpam-4687	57	6	∈	∈	PROPN
ejpam-4687	57	7	{	{	PUNCT
ejpam-4687	57	8	2	2	NUM
ejpam-4687	57	9	,	,	PUNCT
ejpam-4687	57	10	.	.	PUNCT
ejpam-4687	57	11	.	.	PUNCT
ejpam-4687	57	12	.	.	PUNCT
ejpam-4687	57	13	,	,	PUNCT
ejpam-4687	57	14	k	k	X
ejpam-4687	57	15	}	}	PUNCT
ejpam-4687	57	16	,	,	PUNCT
ejpam-4687	57	17	i.e.	i.e.	X
ejpam-4687	57	18	,	,	PUNCT
ejpam-4687	57	19	s	s	X
ejpam-4687	57	20	is	be	AUX
ejpam-4687	57	21	legal	legal	ADJ
ejpam-4687	57	22	closed	closed	ADJ
ejpam-4687	57	23	neighborhood	neighborhood	NOUN
ejpam-4687	57	24	sequence	sequence	NOUN
ejpam-4687	57	25	in	in	ADP
ejpam-4687	57	26	g.	g.	PROPN
ejpam-4687	57	27	a	a	DET
ejpam-4687	57	28	co	co	ADJ
ejpam-4687	57	29	-	-	ADJ
ejpam-4687	57	30	legal	legal	ADJ
ejpam-4687	57	31	closed	closed	ADJ
ejpam-4687	57	32	neighborhood	neighborhood	NOUN
ejpam-4687	57	33	sequence	sequence	NOUN
ejpam-4687	57	34	s	s	PART
ejpam-4687	57	35	=	=	PUNCT
ejpam-4687	57	36	(	(	PUNCT
ejpam-4687	57	37	v1	v1	PROPN
ejpam-4687	57	38	,	,	PUNCT
ejpam-4687	57	39	v2	v2	NOUN
ejpam-4687	57	40	,	,	PUNCT
ejpam-4687	57	41	.	.	PUNCT
ejpam-4687	57	42	.	.	PUNCT
ejpam-4687	58	1	.	.	PUNCT
ejpam-4687	59	1	,	,	PUNCT
ejpam-4687	59	2	vk	vk	PROPN
ejpam-4687	59	3	)	)	PUNCT
ejpam-4687	59	4	is	be	AUX
ejpam-4687	59	5	a	a	DET
ejpam-4687	59	6	co	co	ADJ
ejpam-4687	59	7	-	-	ADJ
ejpam-4687	59	8	grundy	grundy	ADJ
ejpam-4687	59	9	dominating	dominating	NOUN
ejpam-4687	59	10	sequence	sequence	NOUN
ejpam-4687	59	11	if	if	SCONJ
ejpam-4687	59	12	v	v	X
ejpam-4687	59	13	(	(	PUNCT
ejpam-4687	59	14	g	g	NOUN
ejpam-4687	59	15	)	)	PUNCT
ejpam-4687	59	16	=	=	PUNCT
ejpam-4687	60	1	∪k	∪k	NUM
ejpam-4687	60	2	i=1[v	i=1[v	X
ejpam-4687	60	3	(	(	PUNCT
ejpam-4687	60	4	g	g	NOUN
ejpam-4687	60	5	)	)	PUNCT
ejpam-4687	60	6	\ng(vi	\ng(vi	NOUN
ejpam-4687	60	7	)	)	PUNCT
ejpam-4687	60	8	]	]	PUNCT
ejpam-4687	60	9	,	,	PUNCT
ejpam-4687	60	10	i.e.	i.e.	X
ejpam-4687	60	11	,	,	PUNCT
ejpam-4687	60	12	s	s	X
ejpam-4687	60	13	is	be	AUX
ejpam-4687	60	14	grundy	grundy	PROPN
ejpam-4687	60	15	dominating	dominating	NOUN
ejpam-4687	60	16	sequence	sequence	NOUN
ejpam-4687	60	17	in	in	ADP
ejpam-4687	60	18	g.	g.	PROPN
ejpam-4687	60	19	the	the	DET
ejpam-4687	60	20	maximum	maximum	ADJ
ejpam-4687	60	21	length	length	NOUN
ejpam-4687	60	22	of	of	ADP
ejpam-4687	60	23	a	a	DET
ejpam-4687	60	24	co	co	ADJ
ejpam-4687	60	25	-	-	ADJ
ejpam-4687	60	26	grundy	grundy	ADJ
ejpam-4687	60	27	dominating	dominating	NOUN
ejpam-4687	60	28	sequence	sequence	NOUN
ejpam-4687	60	29	in	in	ADP
ejpam-4687	60	30	a	a	DET
ejpam-4687	60	31	graph	graph	NOUN
ejpam-4687	60	32	g	g	NOUN
ejpam-4687	60	33	is	be	AUX
ejpam-4687	60	34	called	call	VERB
ejpam-4687	60	35	the	the	DET
ejpam-4687	60	36	co	co	ADJ
ejpam-4687	60	37	-	-	ADJ
ejpam-4687	60	38	grundy	grundy	ADJ
ejpam-4687	60	39	domination	domination	NOUN
ejpam-4687	60	40	number	number	NOUN
ejpam-4687	60	41	of	of	ADP
ejpam-4687	60	42	g	g	NOUN
ejpam-4687	60	43	,	,	PUNCT
ejpam-4687	60	44	and	and	CCONJ
ejpam-4687	60	45	is	be	AUX
ejpam-4687	60	46	denoted	denote	VERB
ejpam-4687	60	47	by	by	ADP
ejpam-4687	60	48	γcogr(g	γcogr(g	PROPN
ejpam-4687	60	49	)	)	PUNCT
ejpam-4687	60	50	.	.	PUNCT
ejpam-4687	61	1	clearly	clearly	ADV
ejpam-4687	61	2	,	,	PUNCT
ejpam-4687	61	3	γcogr(g	γcogr(g	NOUN
ejpam-4687	61	4	)	)	PUNCT
ejpam-4687	61	5	=	=	SYM
ejpam-4687	61	6	γgr(g	γgr(g	PROPN
ejpam-4687	61	7	)	)	PUNCT
ejpam-4687	61	8	.	.	PUNCT
ejpam-4687	62	1	the	the	DET
ejpam-4687	62	2	shadow	shadow	NOUN
ejpam-4687	62	3	graph	graph	NOUN
ejpam-4687	62	4	s(g	s(g	PROPN
ejpam-4687	62	5	)	)	PUNCT
ejpam-4687	62	6	of	of	ADP
ejpam-4687	62	7	a	a	DET
ejpam-4687	62	8	graph	graph	NOUN
ejpam-4687	62	9	g	g	NOUN
ejpam-4687	62	10	is	be	AUX
ejpam-4687	62	11	constructed	construct	VERB
ejpam-4687	62	12	by	by	ADP
ejpam-4687	62	13	taking	take	VERB
ejpam-4687	62	14	two	two	NUM
ejpam-4687	62	15	copies	copy	NOUN
ejpam-4687	62	16	of	of	ADP
ejpam-4687	62	17	g	g	NOUN
ejpam-4687	62	18	,	,	PUNCT
ejpam-4687	62	19	say	say	VERB
ejpam-4687	62	20	g1	g1	PROPN
ejpam-4687	62	21	and	and	CCONJ
ejpam-4687	62	22	g2	g2	PROPN
ejpam-4687	62	23	,	,	PUNCT
ejpam-4687	62	24	and	and	CCONJ
ejpam-4687	62	25	then	then	ADV
ejpam-4687	62	26	joining	join	VERB
ejpam-4687	62	27	each	each	DET
ejpam-4687	62	28	vertex	vertex	NOUN
ejpam-4687	62	29	u	u	PROPN
ejpam-4687	62	30	∈	∈	PROPN
ejpam-4687	62	31	g1	g1	PROPN
ejpam-4687	62	32	to	to	ADP
ejpam-4687	62	33	the	the	DET
ejpam-4687	62	34	neighbors	neighbor	NOUN
ejpam-4687	62	35	of	of	ADP
ejpam-4687	62	36	its	its	PRON
ejpam-4687	62	37	corresponding	corresponding	ADJ
ejpam-4687	62	38	vertex	vertex	NOUN
ejpam-4687	62	39	u′	u′	PROPN
ejpam-4687	62	40	∈	∈	PROPN
ejpam-4687	62	41	g2	g2	PROPN
ejpam-4687	62	42	.	.	PUNCT
ejpam-4687	63	1	let	let	VERB
ejpam-4687	63	2	g	g	NOUN
ejpam-4687	63	3	and	and	CCONJ
ejpam-4687	63	4	h	h	NOUN
ejpam-4687	63	5	be	be	VERB
ejpam-4687	63	6	two	two	NUM
ejpam-4687	63	7	graphs	graph	NOUN
ejpam-4687	63	8	.	.	PUNCT
ejpam-4687	64	1	the	the	DET
ejpam-4687	64	2	join	join	NOUN
ejpam-4687	64	3	of	of	ADP
ejpam-4687	64	4	g	g	PROPN
ejpam-4687	64	5	and	and	CCONJ
ejpam-4687	64	6	h	h	NOUN
ejpam-4687	64	7	,	,	PUNCT
ejpam-4687	64	8	denoted	denote	VERB
ejpam-4687	64	9	by	by	ADP
ejpam-4687	64	10	g	g	PROPN
ejpam-4687	64	11	+	+	CCONJ
ejpam-4687	64	12	h	h	NOUN
ejpam-4687	64	13	is	be	AUX
ejpam-4687	64	14	the	the	DET
ejpam-4687	64	15	graph	graph	NOUN
ejpam-4687	64	16	with	with	ADP
ejpam-4687	64	17	vertex	vertex	NOUN
ejpam-4687	64	18	set	set	VERB
ejpam-4687	64	19	v	v	NOUN
ejpam-4687	64	20	(	(	PUNCT
ejpam-4687	64	21	g+h	g+h	NOUN
ejpam-4687	64	22	)	)	PUNCT
ejpam-4687	64	23	=	=	SYM
ejpam-4687	64	24	v	v	X
ejpam-4687	64	25	(	(	PUNCT
ejpam-4687	64	26	g)∪v	g)∪v	NOUN
ejpam-4687	64	27	(	(	PUNCT
ejpam-4687	64	28	h	h	NOUN
ejpam-4687	64	29	)	)	PUNCT
ejpam-4687	64	30	and	and	CCONJ
ejpam-4687	64	31	edge	edge	NOUN
ejpam-4687	64	32	set	set	VERB
ejpam-4687	64	33	e(g+h	e(g+h	NUM
ejpam-4687	64	34	)	)	PUNCT
ejpam-4687	65	1	=	=	SYM
ejpam-4687	65	2	e(g)∪e(h)∪{uv	e(g)∪e(h)∪{uv	X
ejpam-4687	65	3	:	:	PUNCT
ejpam-4687	65	4	u	u	PROPN
ejpam-4687	65	5	∈	∈	PROPN
ejpam-4687	65	6	v	v	NOUN
ejpam-4687	65	7	(	(	PUNCT
ejpam-4687	65	8	g	g	NOUN
ejpam-4687	65	9	)	)	PUNCT
ejpam-4687	65	10	,	,	PUNCT
ejpam-4687	65	11	v	v	X
ejpam-4687	65	12	∈	∈	PROPN
ejpam-4687	65	13	v	v	NOUN
ejpam-4687	65	14	(	(	PUNCT
ejpam-4687	65	15	h	h	NOUN
ejpam-4687	65	16	)	)	PUNCT
ejpam-4687	65	17	}	}	PUNCT
ejpam-4687	65	18	.	.	PUNCT
ejpam-4687	66	1	3	3	X
ejpam-4687	66	2	.	.	X
ejpam-4687	66	3	results	result	NOUN
ejpam-4687	66	4	remark	remark	VERB
ejpam-4687	66	5	1	1	NUM
ejpam-4687	66	6	.	.	PUNCT
ejpam-4687	67	1	let	let	VERB
ejpam-4687	67	2	g	g	PRON
ejpam-4687	67	3	be	be	AUX
ejpam-4687	67	4	a	a	DET
ejpam-4687	67	5	connected	connected	ADJ
ejpam-4687	67	6	graph	graph	NOUN
ejpam-4687	67	7	.	.	PUNCT
ejpam-4687	68	1	then	then	ADV
ejpam-4687	68	2	each	each	PRON
ejpam-4687	68	3	of	of	ADP
ejpam-4687	68	4	the	the	DET
ejpam-4687	68	5	following	following	NOUN
ejpam-4687	68	6	is	be	AUX
ejpam-4687	68	7	true	true	ADJ
ejpam-4687	68	8	.	.	PUNCT
ejpam-4687	69	1	(	(	PUNCT
ejpam-4687	69	2	i	i	NOUN
ejpam-4687	69	3	)	)	PUNCT
ejpam-4687	69	4	the	the	DET
ejpam-4687	69	5	vertex	vertex	NOUN
ejpam-4687	69	6	set	set	VERB
ejpam-4687	69	7	v	v	NOUN
ejpam-4687	69	8	(	(	PUNCT
ejpam-4687	69	9	g	g	NOUN
ejpam-4687	69	10	)	)	PUNCT
ejpam-4687	69	11	may	may	AUX
ejpam-4687	69	12	not	not	PART
ejpam-4687	69	13	form	form	VERB
ejpam-4687	69	14	a	a	DET
ejpam-4687	69	15	connected	connect	VERB
ejpam-4687	69	16	grundy	grundy	PROPN
ejpam-4687	69	17	hop	hop	NOUN
ejpam-4687	69	18	dominating	dominating	NOUN
ejpam-4687	69	19	sequence	sequence	NOUN
ejpam-4687	69	20	.	.	PUNCT
ejpam-4687	70	1	(	(	PUNCT
ejpam-4687	70	2	ii	ii	X
ejpam-4687	70	3	)	)	PUNCT
ejpam-4687	70	4	a	a	DET
ejpam-4687	70	5	proper	proper	ADJ
ejpam-4687	70	6	connected	connected	ADJ
ejpam-4687	70	7	hop	hop	NOUN
ejpam-4687	70	8	dominating	dominating	NOUN
ejpam-4687	70	9	set	set	NOUN
ejpam-4687	70	10	may	may	AUX
ejpam-4687	70	11	not	not	PART
ejpam-4687	70	12	form	form	VERB
ejpam-4687	70	13	a	a	DET
ejpam-4687	70	14	connected	connect	VERB
ejpam-4687	70	15	grundy	grundy	PROPN
ejpam-4687	70	16	hop	hop	NOUN
ejpam-4687	70	17	dominating	dominating	NOUN
ejpam-4687	70	18	sequence	sequence	NOUN
ejpam-4687	70	19	.	.	PUNCT
ejpam-4687	71	1	(	(	PUNCT
ejpam-4687	71	2	iii	iii	X
ejpam-4687	71	3	)	)	PUNCT
ejpam-4687	71	4	a	a	DET
ejpam-4687	71	5	grundy	grundy	PROPN
ejpam-4687	71	6	hop	hop	NOUN
ejpam-4687	71	7	dominating	dominating	NOUN
ejpam-4687	71	8	sequence	sequence	NOUN
ejpam-4687	71	9	need	need	AUX
ejpam-4687	71	10	not	not	PART
ejpam-4687	71	11	be	be	AUX
ejpam-4687	71	12	a	a	DET
ejpam-4687	71	13	connected	connected	ADJ
ejpam-4687	71	14	grundy	grundy	PROPN
ejpam-4687	71	15	hop	hop	NOUN
ejpam-4687	71	16	dominating	dominating	NOUN
ejpam-4687	71	17	sequence	sequence	NOUN
ejpam-4687	71	18	.	.	PUNCT
ejpam-4687	72	1	j.	j.	PROPN
ejpam-4687	72	2	hassan	hassan	PROPN
ejpam-4687	72	3	,	,	PUNCT
ejpam-4687	72	4	s.	s.	PROPN
ejpam-4687	72	5	canoy	canoy	PROPN
ejpam-4687	72	6	jr	jr	PROPN
ejpam-4687	72	7	.	.	PROPN
ejpam-4687	72	8	/	/	SYM
ejpam-4687	72	9	eur	eur	PROPN
ejpam-4687	72	10	.	.	PUNCT
ejpam-4687	73	1	j.	j.	PROPN
ejpam-4687	73	2	pure	pure	PROPN
ejpam-4687	73	3	appl	appl	PROPN
ejpam-4687	73	4	.	.	PROPN
ejpam-4687	73	5	math	math	PROPN
ejpam-4687	73	6	,	,	PUNCT
ejpam-4687	73	7	16	16	NUM
ejpam-4687	73	8	(	(	PUNCT
ejpam-4687	73	9	2	2	NUM
ejpam-4687	73	10	)	)	PUNCT
ejpam-4687	73	11	(	(	PUNCT
ejpam-4687	73	12	2023	2023	NUM
ejpam-4687	73	13	)	)	PUNCT
ejpam-4687	73	14	,	,	PUNCT
ejpam-4687	73	15	1212	1212	NUM
ejpam-4687	73	16	-	-	SYM
ejpam-4687	73	17	1227	1227	NUM
ejpam-4687	73	18	1215	1215	NUM
ejpam-4687	73	19	to	to	PART
ejpam-4687	73	20	see	see	VERB
ejpam-4687	73	21	(	(	PUNCT
ejpam-4687	73	22	i	i	NOUN
ejpam-4687	73	23	)	)	PUNCT
ejpam-4687	73	24	,	,	PUNCT
ejpam-4687	73	25	consider	consider	VERB
ejpam-4687	73	26	the	the	DET
ejpam-4687	73	27	graph	graph	NOUN
ejpam-4687	73	28	g	g	NOUN
ejpam-4687	73	29	in	in	ADP
ejpam-4687	73	30	figure	figure	NOUN
ejpam-4687	73	31	1	1	NUM
ejpam-4687	73	32	.	.	PUNCT
ejpam-4687	74	1	let	let	VERB
ejpam-4687	74	2	s	s	PRON
ejpam-4687	74	3	=	=	PUNCT
ejpam-4687	74	4	(	(	PUNCT
ejpam-4687	74	5	v1	v1	PROPN
ejpam-4687	74	6	,	,	PUNCT
ejpam-4687	74	7	v2	v2	PROPN
ejpam-4687	74	8	,	,	PUNCT
ejpam-4687	74	9	v3	v3	PROPN
ejpam-4687	74	10	,	,	PUNCT
ejpam-4687	74	11	v4	v4	PROPN
ejpam-4687	74	12	,	,	PUNCT
ejpam-4687	74	13	v5	v5	PROPN
ejpam-4687	74	14	,	,	PUNCT
ejpam-4687	74	15	v6	v6	NOUN
ejpam-4687	74	16	)	)	PUNCT
ejpam-4687	74	17	.	.	PUNCT
ejpam-4687	75	1	then	then	ADV
ejpam-4687	75	2	ŝ	ŝ	PROPN
ejpam-4687	75	3	is	be	AUX
ejpam-4687	75	4	a	a	DET
ejpam-4687	75	5	connected	connected	ADJ
ejpam-4687	75	6	hop	hop	NOUN
ejpam-4687	75	7	dominating	dominating	NOUN
ejpam-4687	75	8	set	set	NOUN
ejpam-4687	75	9	of	of	ADP
ejpam-4687	75	10	g.	g.	PROPN
ejpam-4687	75	11	observe	observe	VERB
ejpam-4687	75	12	that	that	SCONJ
ejpam-4687	75	13	n2	n2	ADJ
ejpam-4687	75	14	g[v6	g[v6	NOUN
ejpam-4687	75	15	]	]	X
ejpam-4687	75	16	=	=	SYM
ejpam-4687	75	17	{	{	PUNCT
ejpam-4687	75	18	v3	v3	PROPN
ejpam-4687	75	19	,	,	PUNCT
ejpam-4687	75	20	v6	v6	NOUN
ejpam-4687	75	21	}	}	PUNCT
ejpam-4687	75	22	⊆	⊆	NUM
ejpam-4687	75	23	ng[v3	ng[v3	PROPN
ejpam-4687	75	24	]	]	PUNCT
ejpam-4687	75	25	.	.	PUNCT
ejpam-4687	76	1	it	it	PRON
ejpam-4687	76	2	follows	follow	VERB
ejpam-4687	76	3	that	that	DET
ejpam-4687	76	4	n2	n2	ADJ
ejpam-4687	76	5	g[v6	g[v6	NOUN
ejpam-4687	76	6	]	]	PUNCT
ejpam-4687	76	7	\	\	NOUN
ejpam-4687	76	8	⋃5	⋃5	NOUN
ejpam-4687	76	9	j=1n	j=1n	VERB
ejpam-4687	76	10	2	2	NUM
ejpam-4687	76	11	g[vj	g[vj	PROPN
ejpam-4687	76	12	]	]	PUNCT
ejpam-4687	76	13	=	=	PUNCT
ejpam-4687	76	14	∅.	∅.	VERB
ejpam-4687	76	15	hence	hence	ADV
ejpam-4687	76	16	,	,	PUNCT
ejpam-4687	76	17	s	s	VERB
ejpam-4687	76	18	is	be	AUX
ejpam-4687	76	19	not	not	PART
ejpam-4687	76	20	a	a	DET
ejpam-4687	76	21	legal	legal	ADJ
ejpam-4687	76	22	closed	close	VERB
ejpam-4687	76	23	hop	hop	NOUN
ejpam-4687	76	24	neighborhood	neighborhood	NOUN
ejpam-4687	76	25	sequence	sequence	NOUN
ejpam-4687	76	26	of	of	ADP
ejpam-4687	76	27	g.	g.	PROPN
ejpam-4687	76	28	consequently	consequently	ADV
ejpam-4687	76	29	,	,	PUNCT
ejpam-4687	76	30	s	s	VERB
ejpam-4687	76	31	is	be	AUX
ejpam-4687	76	32	not	not	PART
ejpam-4687	76	33	connected	connect	VERB
ejpam-4687	76	34	grundy	grundy	PROPN
ejpam-4687	76	35	hop	hop	NOUN
ejpam-4687	76	36	dominating	dominating	NOUN
ejpam-4687	76	37	sequence	sequence	NOUN
ejpam-4687	76	38	of	of	ADP
ejpam-4687	76	39	g.	g.	PROPN
ejpam-4687	76	40	in	in	ADP
ejpam-4687	76	41	fact	fact	NOUN
ejpam-4687	76	42	(	(	PUNCT
ejpam-4687	76	43	as	as	SCONJ
ejpam-4687	76	44	to	to	PART
ejpam-4687	76	45	be	be	AUX
ejpam-4687	76	46	shown	show	VERB
ejpam-4687	76	47	later	later	ADV
ejpam-4687	76	48	)	)	PUNCT
ejpam-4687	76	49	,	,	PUNCT
ejpam-4687	76	50	γchgr	γchgr	X
ejpam-4687	76	51	(	(	PUNCT
ejpam-4687	76	52	g	g	NOUN
ejpam-4687	76	53	)	)	PUNCT
ejpam-4687	76	54	̸=	̸=	PROPN
ejpam-4687	76	55	6	6	NUM
ejpam-4687	76	56	.	.	PUNCT
ejpam-4687	77	1	g	g	NOUN
ejpam-4687	77	2	:	:	PUNCT
ejpam-4687	77	3	v1	v1	VERB
ejpam-4687	77	4	v2	v2	PROPN
ejpam-4687	77	5	v3	v3	PROPN
ejpam-4687	77	6	v4	v4	PROPN
ejpam-4687	77	7	v5	v5	PROPN
ejpam-4687	77	8	v6	v6	PROPN
ejpam-4687	77	9	figure	figure	NOUN
ejpam-4687	77	10	1	1	NUM
ejpam-4687	77	11	:	:	PUNCT
ejpam-4687	77	12	a	a	DET
ejpam-4687	77	13	graph	graph	NOUN
ejpam-4687	77	14	g	g	NOUN
ejpam-4687	77	15	where	where	SCONJ
ejpam-4687	77	16	vertex	vertex	NOUN
ejpam-4687	77	17	set	set	NOUN
ejpam-4687	77	18	does	do	AUX
ejpam-4687	77	19	not	not	PART
ejpam-4687	77	20	form	form	VERB
ejpam-4687	77	21	a	a	DET
ejpam-4687	77	22	connected	connect	VERB
ejpam-4687	77	23	grundy	grundy	PROPN
ejpam-4687	77	24	hop	hop	NOUN
ejpam-4687	77	25	dominating	dominating	NOUN
ejpam-4687	77	26	sequence	sequence	NOUN
ejpam-4687	77	27	for	for	ADP
ejpam-4687	77	28	(	(	PUNCT
ejpam-4687	77	29	ii	ii	NOUN
ejpam-4687	77	30	)	)	PUNCT
ejpam-4687	77	31	,	,	PUNCT
ejpam-4687	77	32	consider	consider	VERB
ejpam-4687	77	33	the	the	DET
ejpam-4687	77	34	graph	graph	NOUN
ejpam-4687	77	35	g	g	PROPN
ejpam-4687	77	36	=	=	PUNCT
ejpam-4687	77	37	c5	c5	PROPN
ejpam-4687	77	38	=	=	PUNCT
ejpam-4687	78	1	[	[	X
ejpam-4687	78	2	x	x	X
ejpam-4687	78	3	,	,	PUNCT
ejpam-4687	78	4	z	z	PROPN
ejpam-4687	78	5	,	,	PUNCT
ejpam-4687	78	6	y	y	PROPN
ejpam-4687	78	7	,	,	PUNCT
ejpam-4687	78	8	w	w	PROPN
ejpam-4687	78	9	,	,	PUNCT
ejpam-4687	78	10	v	v	NOUN
ejpam-4687	78	11	,	,	PUNCT
ejpam-4687	78	12	x	x	NOUN
ejpam-4687	78	13	]	]	X
ejpam-4687	78	14	.	.	PUNCT
ejpam-4687	79	1	clearly	clearly	ADV
ejpam-4687	79	2	,	,	PUNCT
ejpam-4687	79	3	ŝ	ŝ	X
ejpam-4687	79	4	=	=	PUNCT
ejpam-4687	79	5	{	{	PUNCT
ejpam-4687	79	6	x	x	NOUN
ejpam-4687	79	7	,	,	PUNCT
ejpam-4687	79	8	y	y	PROPN
ejpam-4687	79	9	,	,	PUNCT
ejpam-4687	79	10	z	z	PROPN
ejpam-4687	79	11	,	,	PUNCT
ejpam-4687	79	12	w	w	NOUN
ejpam-4687	79	13	}	}	PUNCT
ejpam-4687	79	14	is	be	AUX
ejpam-4687	79	15	a	a	DET
ejpam-4687	79	16	connected	connected	ADJ
ejpam-4687	79	17	hop	hop	NOUN
ejpam-4687	79	18	dominating	dominating	NOUN
ejpam-4687	79	19	set	set	NOUN
ejpam-4687	79	20	of	of	ADP
ejpam-4687	79	21	g.	g.	PROPN
ejpam-4687	79	22	observe	observe	VERB
ejpam-4687	79	23	that	that	DET
ejpam-4687	79	24	n2	n2	ADJ
ejpam-4687	79	25	g[w	g[w	PROPN
ejpam-4687	79	26	]	]	PUNCT
ejpam-4687	79	27	⊆	⊆	NUM
ejpam-4687	79	28	n2	n2	PROPN
ejpam-4687	79	29	g[x	g[x	PROPN
ejpam-4687	79	30	]	]	PUNCT
ejpam-4687	79	31	∪	∪	ADP
ejpam-4687	79	32	n2	n2	ADJ
ejpam-4687	79	33	g[y	g[y	NOUN
ejpam-4687	79	34	]	]	PUNCT
ejpam-4687	79	35	∪	∪	PROPN
ejpam-4687	79	36	n2	n2	PROPN
ejpam-4687	79	37	g[z	g[z	PROPN
ejpam-4687	79	38	]	]	PUNCT
ejpam-4687	79	39	.	.	PUNCT
ejpam-4687	80	1	it	it	PRON
ejpam-4687	80	2	follows	follow	VERB
ejpam-4687	80	3	that	that	DET
ejpam-4687	80	4	n2	n2	ADJ
ejpam-4687	80	5	g[w	g[w	PROPN
ejpam-4687	80	6	]	]	PUNCT
ejpam-4687	80	7	\	\	PUNCT
ejpam-4687	81	1	[	[	X
ejpam-4687	81	2	n2	n2	NOUN
ejpam-4687	81	3	g[x	g[x	NOUN
ejpam-4687	81	4	]	]	PUNCT
ejpam-4687	81	5	∪	∪	ADP
ejpam-4687	81	6	n2	n2	ADJ
ejpam-4687	81	7	g[y	g[y	NOUN
ejpam-4687	81	8	]	]	PUNCT
ejpam-4687	81	9	∪	∪	PROPN
ejpam-4687	81	10	n2	n2	PROPN
ejpam-4687	81	11	g[z	g[z	PROPN
ejpam-4687	81	12	]	]	X
ejpam-4687	81	13	]	]	X
ejpam-4687	81	14	=	=	PUNCT
ejpam-4687	81	15	∅.	∅.	VERB
ejpam-4687	81	16	thus	thus	ADV
ejpam-4687	81	17	,	,	PUNCT
ejpam-4687	81	18	s	s	VERB
ejpam-4687	81	19	is	be	AUX
ejpam-4687	81	20	not	not	PART
ejpam-4687	81	21	a	a	DET
ejpam-4687	81	22	legal	legal	ADJ
ejpam-4687	81	23	closed	close	VERB
ejpam-4687	81	24	hop	hop	NOUN
ejpam-4687	81	25	neighborhood	neighborhood	NOUN
ejpam-4687	81	26	sequence	sequence	NOUN
ejpam-4687	81	27	of	of	ADP
ejpam-4687	81	28	g.	g.	PROPN
ejpam-4687	81	29	therefore	therefore	ADV
ejpam-4687	81	30	,	,	PUNCT
ejpam-4687	81	31	s	s	VERB
ejpam-4687	81	32	is	be	AUX
ejpam-4687	81	33	not	not	PART
ejpam-4687	81	34	a	a	DET
ejpam-4687	81	35	connected	connect	VERB
ejpam-4687	81	36	grundy	grundy	PROPN
ejpam-4687	81	37	hop	hop	NOUN
ejpam-4687	81	38	dominating	dominating	NOUN
ejpam-4687	81	39	sequence	sequence	NOUN
ejpam-4687	81	40	of	of	ADP
ejpam-4687	81	41	g.	g.	PROPN
ejpam-4687	81	42	finally	finally	ADV
ejpam-4687	81	43	,	,	PUNCT
ejpam-4687	81	44	for	for	ADP
ejpam-4687	81	45	(	(	PUNCT
ejpam-4687	81	46	iii	iii	NOUN
ejpam-4687	81	47	)	)	PUNCT
ejpam-4687	81	48	,	,	PUNCT
ejpam-4687	81	49	consider	consider	VERB
ejpam-4687	81	50	g	g	NOUN
ejpam-4687	81	51	=	=	SYM
ejpam-4687	81	52	p5	p5	PROPN
ejpam-4687	81	53	=	=	PUNCT
ejpam-4687	82	1	[	[	X
ejpam-4687	82	2	v1	v1	NOUN
ejpam-4687	82	3	,	,	PUNCT
ejpam-4687	82	4	v2	v2	PROPN
ejpam-4687	82	5	,	,	PUNCT
ejpam-4687	82	6	v3	v3	PROPN
ejpam-4687	82	7	,	,	PUNCT
ejpam-4687	82	8	v4	v4	PROPN
ejpam-4687	82	9	,	,	PUNCT
ejpam-4687	82	10	v5	v5	PROPN
ejpam-4687	82	11	]	]	PUNCT
ejpam-4687	82	12	and	and	CCONJ
ejpam-4687	82	13	let	let	VERB
ejpam-4687	82	14	s′	s′	ADJ
ejpam-4687	82	15	=	=	SYM
ejpam-4687	82	16	(	(	PUNCT
ejpam-4687	82	17	v1	v1	PROPN
ejpam-4687	82	18	,	,	PUNCT
ejpam-4687	82	19	v3	v3	PROPN
ejpam-4687	82	20	,	,	PUNCT
ejpam-4687	82	21	v4	v4	PROPN
ejpam-4687	82	22	)	)	PUNCT
ejpam-4687	82	23	.	.	PUNCT
ejpam-4687	83	1	clearly	clearly	ADV
ejpam-4687	83	2	,	,	PUNCT
ejpam-4687	83	3	ŝ′	ŝ′	PROPN
ejpam-4687	83	4	is	be	AUX
ejpam-4687	83	5	a	a	DET
ejpam-4687	83	6	hop	hop	NOUN
ejpam-4687	83	7	dominating	dominating	NOUN
ejpam-4687	83	8	set	set	NOUN
ejpam-4687	83	9	of	of	ADP
ejpam-4687	83	10	g.	g.	PROPN
ejpam-4687	83	11	observe	observe	VERB
ejpam-4687	83	12	that	that	SCONJ
ejpam-4687	83	13	v5	v5	PROPN
ejpam-4687	83	14	∈	∈	PROPN
ejpam-4687	83	15	n2	n2	NOUN
ejpam-4687	83	16	g[v3	g[v3	NOUN
ejpam-4687	83	17	]	]	PUNCT
ejpam-4687	84	1	\n2	\n2	PROPN
ejpam-4687	84	2	g[v1	g[v1	X
ejpam-4687	84	3	]	]	PUNCT
ejpam-4687	84	4	and	and	CCONJ
ejpam-4687	84	5	v2	v2	PROPN
ejpam-4687	84	6	,	,	PUNCT
ejpam-4687	84	7	v4	v4	NOUN
ejpam-4687	84	8	∈	∈	PROPN
ejpam-4687	84	9	ng[v4	ng[v4	PROPN
ejpam-4687	84	10	]	]	PUNCT
ejpam-4687	84	11	\	\	PUNCT
ejpam-4687	84	12	(	(	PUNCT
ejpam-4687	84	13	n2	n2	PROPN
ejpam-4687	84	14	g[v1	g[v1	PROPN
ejpam-4687	84	15	]	]	PUNCT
ejpam-4687	84	16	∪	∪	ADP
ejpam-4687	84	17	n2	n2	ADJ
ejpam-4687	84	18	g[v3	g[v3	NOUN
ejpam-4687	84	19	]	]	PUNCT
ejpam-4687	84	20	)	)	PUNCT
ejpam-4687	84	21	.	.	PUNCT
ejpam-4687	85	1	it	it	PRON
ejpam-4687	85	2	follows	follow	VERB
ejpam-4687	85	3	that	that	SCONJ
ejpam-4687	85	4	s′	s′	ADJ
ejpam-4687	85	5	is	be	AUX
ejpam-4687	85	6	a	a	DET
ejpam-4687	85	7	legal	legal	ADJ
ejpam-4687	85	8	closed	close	VERB
ejpam-4687	85	9	hop	hop	NOUN
ejpam-4687	85	10	neighborhood	neighborhood	NOUN
ejpam-4687	85	11	sequence	sequence	NOUN
ejpam-4687	85	12	of	of	ADP
ejpam-4687	85	13	g.	g.	PROPN
ejpam-4687	85	14	thus	thus	ADV
ejpam-4687	85	15	,	,	PUNCT
ejpam-4687	85	16	s′	s′	PROPN
ejpam-4687	85	17	is	be	AUX
ejpam-4687	85	18	a	a	DET
ejpam-4687	85	19	grundy	grundy	PROPN
ejpam-4687	85	20	hop	hop	NOUN
ejpam-4687	85	21	dominating	dominating	NOUN
ejpam-4687	85	22	sequence	sequence	NOUN
ejpam-4687	85	23	of	of	ADP
ejpam-4687	85	24	g.	g.	PROPN
ejpam-4687	85	25	however	however	ADV
ejpam-4687	85	26	,	,	PUNCT
ejpam-4687	85	27	⟨ŝ⟩	⟨ŝ⟩	PRON
ejpam-4687	85	28	is	be	AUX
ejpam-4687	85	29	not	not	PART
ejpam-4687	85	30	connected	connect	VERB
ejpam-4687	85	31	.	.	PUNCT
ejpam-4687	86	1	hence	hence	ADV
ejpam-4687	86	2	,	,	PUNCT
ejpam-4687	86	3	s′	s′	PROPN
ejpam-4687	86	4	is	be	AUX
ejpam-4687	86	5	not	not	PART
ejpam-4687	86	6	a	a	DET
ejpam-4687	86	7	connected	connect	VERB
ejpam-4687	86	8	grundy	grundy	PROPN
ejpam-4687	86	9	hop	hop	NOUN
ejpam-4687	86	10	dominating	dominating	NOUN
ejpam-4687	86	11	sequence	sequence	NOUN
ejpam-4687	86	12	of	of	ADP
ejpam-4687	86	13	g.	g.	PROPN
ejpam-4687	86	14	remark	remark	PROPN
ejpam-4687	86	15	2	2	NUM
ejpam-4687	86	16	.	.	PUNCT
ejpam-4687	87	1	let	let	VERB
ejpam-4687	87	2	g	g	PRON
ejpam-4687	87	3	be	be	AUX
ejpam-4687	87	4	a	a	DET
ejpam-4687	87	5	connected	connected	ADJ
ejpam-4687	87	6	graph	graph	NOUN
ejpam-4687	87	7	.	.	PUNCT
ejpam-4687	88	1	then	then	ADV
ejpam-4687	88	2	γch(g	γch(g	NOUN
ejpam-4687	88	3	)	)	PUNCT
ejpam-4687	88	4	≤	≤	NOUN
ejpam-4687	88	5	γchgr	γchgr	NOUN
ejpam-4687	88	6	(	(	PUNCT
ejpam-4687	88	7	g	g	NOUN
ejpam-4687	88	8	)	)	PUNCT
ejpam-4687	88	9	≤	≤	NOUN
ejpam-4687	89	1	γhgr(g	γhgr(g	PROPN
ejpam-4687	89	2	)	)	PUNCT
ejpam-4687	89	3	and	and	CCONJ
ejpam-4687	89	4	these	these	DET
ejpam-4687	89	5	bounds	bound	NOUN
ejpam-4687	89	6	are	be	AUX
ejpam-4687	89	7	tight	tight	ADJ
ejpam-4687	89	8	.	.	PUNCT
ejpam-4687	90	1	moreover	moreover	ADV
ejpam-4687	90	2	,	,	PUNCT
ejpam-4687	90	3	both	both	DET
ejpam-4687	90	4	strict	strict	ADJ
ejpam-4687	90	5	inequalities	inequality	NOUN
ejpam-4687	90	6	are	be	AUX
ejpam-4687	90	7	attainable	attainable	ADJ
ejpam-4687	90	8	.	.	PUNCT
ejpam-4687	91	1	note	note	VERB
ejpam-4687	91	2	that	that	SCONJ
ejpam-4687	91	3	the	the	DET
ejpam-4687	91	4	first	first	ADJ
ejpam-4687	91	5	inequality	inequality	NOUN
ejpam-4687	91	6	follows	follow	VERB
ejpam-4687	91	7	from	from	ADP
ejpam-4687	91	8	the	the	DET
ejpam-4687	91	9	fact	fact	NOUN
ejpam-4687	91	10	that	that	SCONJ
ejpam-4687	91	11	every	every	DET
ejpam-4687	91	12	connected	connect	VERB
ejpam-4687	91	13	grundy	grundy	PROPN
ejpam-4687	91	14	hop	hop	NOUN
ejpam-4687	91	15	dominating	dominating	NOUN
ejpam-4687	91	16	sequence	sequence	NOUN
ejpam-4687	91	17	induces	induce	VERB
ejpam-4687	91	18	a	a	DET
ejpam-4687	91	19	connected	connected	ADJ
ejpam-4687	91	20	hop	hop	NOUN
ejpam-4687	91	21	dominating	dominating	NOUN
ejpam-4687	91	22	set	set	NOUN
ejpam-4687	91	23	(	(	PUNCT
ejpam-4687	91	24	by	by	ADP
ejpam-4687	91	25	definition	definition	NOUN
ejpam-4687	91	26	)	)	PUNCT
ejpam-4687	91	27	.	.	PUNCT
ejpam-4687	92	1	moreover	moreover	ADV
ejpam-4687	92	2	,	,	PUNCT
ejpam-4687	92	3	since	since	SCONJ
ejpam-4687	92	4	every	every	DET
ejpam-4687	92	5	connected	connect	VERB
ejpam-4687	92	6	grundy	grundy	PROPN
ejpam-4687	92	7	hop	hop	NOUN
ejpam-4687	92	8	dominating	dominating	NOUN
ejpam-4687	92	9	sequence	sequence	NOUN
ejpam-4687	92	10	is	be	AUX
ejpam-4687	92	11	a	a	DET
ejpam-4687	92	12	grundy	grundy	PROPN
ejpam-4687	92	13	hop	hop	NOUN
ejpam-4687	92	14	dominating	dominating	NOUN
ejpam-4687	92	15	sequence	sequence	NOUN
ejpam-4687	92	16	,	,	PUNCT
ejpam-4687	92	17	the	the	DET
ejpam-4687	92	18	second	second	ADJ
ejpam-4687	92	19	inequality	inequality	NOUN
ejpam-4687	92	20	follows	follow	VERB
ejpam-4687	92	21	.	.	PUNCT
ejpam-4687	93	1	to	to	PART
ejpam-4687	93	2	see	see	VERB
ejpam-4687	93	3	that	that	SCONJ
ejpam-4687	93	4	the	the	DET
ejpam-4687	93	5	bounds	bound	NOUN
ejpam-4687	93	6	are	be	AUX
ejpam-4687	93	7	tight	tight	ADJ
ejpam-4687	93	8	,	,	PUNCT
ejpam-4687	93	9	consider	consider	VERB
ejpam-4687	93	10	g	g	NOUN
ejpam-4687	93	11	=	=	SYM
ejpam-4687	93	12	k4	k4	PROPN
ejpam-4687	93	13	.	.	PUNCT
ejpam-4687	94	1	then	then	ADV
ejpam-4687	94	2	γch(g	γch(g	NOUN
ejpam-4687	94	3	)	)	PUNCT
ejpam-4687	94	4	=	=	VERB
ejpam-4687	95	1	γchgr	γchgr	X
ejpam-4687	95	2	(	(	PUNCT
ejpam-4687	95	3	g	g	NOUN
ejpam-4687	95	4	)	)	PUNCT
ejpam-4687	95	5	=	=	SYM
ejpam-4687	96	1	γhgr(g	γhgr(g	X
ejpam-4687	96	2	)	)	PUNCT
ejpam-4687	96	3	=	=	PUNCT
ejpam-4687	97	1	4	4	X
ejpam-4687	97	2	.	.	X
ejpam-4687	97	3	for	for	ADP
ejpam-4687	97	4	strict	strict	ADJ
ejpam-4687	97	5	inequalities	inequality	NOUN
ejpam-4687	97	6	,	,	PUNCT
ejpam-4687	97	7	consider	consider	VERB
ejpam-4687	97	8	first	first	ADV
ejpam-4687	97	9	the	the	DET
ejpam-4687	97	10	graph	graph	NOUN
ejpam-4687	97	11	g	g	NOUN
ejpam-4687	97	12	in	in	ADP
ejpam-4687	97	13	figure	figure	NOUN
ejpam-4687	97	14	2	2	NUM
ejpam-4687	97	15	.	.	PUNCT
ejpam-4687	98	1	let	let	VERB
ejpam-4687	98	2	s1	s1	PROPN
ejpam-4687	98	3	=	=	SYM
ejpam-4687	98	4	{	{	PUNCT
ejpam-4687	98	5	s3	s3	PROPN
ejpam-4687	98	6	,	,	PUNCT
ejpam-4687	98	7	s4	s4	PROPN
ejpam-4687	98	8	,	,	PUNCT
ejpam-4687	98	9	s5	s5	PROPN
ejpam-4687	98	10	,	,	PUNCT
ejpam-4687	98	11	s6	s6	PROPN
ejpam-4687	98	12	}	}	PUNCT
ejpam-4687	98	13	and	and	CCONJ
ejpam-4687	98	14	s2	s2	PROPN
ejpam-4687	98	15	=	=	SYM
ejpam-4687	98	16	(	(	PUNCT
ejpam-4687	98	17	s1	s1	PROPN
ejpam-4687	98	18	,	,	PUNCT
ejpam-4687	98	19	s2	s2	PROPN
ejpam-4687	98	20	,	,	PUNCT
ejpam-4687	98	21	s3	s3	PROPN
ejpam-4687	98	22	,	,	PUNCT
ejpam-4687	98	23	s4	s4	PROPN
ejpam-4687	98	24	,	,	PUNCT
ejpam-4687	98	25	s5	s5	PROPN
ejpam-4687	98	26	,	,	PUNCT
ejpam-4687	98	27	s6	s6	PROPN
ejpam-4687	98	28	)	)	PUNCT
ejpam-4687	98	29	.	.	PUNCT
ejpam-4687	99	1	then	then	ADV
ejpam-4687	99	2	it	it	PRON
ejpam-4687	99	3	can	can	AUX
ejpam-4687	99	4	be	be	AUX
ejpam-4687	99	5	verified	verify	VERB
ejpam-4687	99	6	that	that	SCONJ
ejpam-4687	99	7	s1	s1	NOUN
ejpam-4687	99	8	and	and	CCONJ
ejpam-4687	99	9	s2	s2	PROPN
ejpam-4687	99	10	are	be	AUX
ejpam-4687	99	11	γchand	γchand	ADV
ejpam-4687	99	12	γchgr	γchgr	ADJ
ejpam-4687	99	13	sequences	sequence	NOUN
ejpam-4687	99	14	of	of	ADP
ejpam-4687	99	15	g	g	NOUN
ejpam-4687	99	16	,	,	PUNCT
ejpam-4687	99	17	respectively	respectively	ADV
ejpam-4687	99	18	.	.	PUNCT
ejpam-4687	100	1	hence	hence	ADV
ejpam-4687	100	2	,	,	PUNCT
ejpam-4687	100	3	γch(g	γch(g	NOUN
ejpam-4687	100	4	)	)	PUNCT
ejpam-4687	100	5	=	=	PUNCT
ejpam-4687	100	6	4	4	NUM
ejpam-4687	100	7	<	<	SYM
ejpam-4687	100	8	6	6	NUM
ejpam-4687	100	9	=	=	SYM
ejpam-4687	100	10	γchgr	γchgr	NOUN
ejpam-4687	100	11	(	(	PUNCT
ejpam-4687	100	12	g	g	NOUN
ejpam-4687	100	13	)	)	PUNCT
ejpam-4687	100	14	=	=	SYM
ejpam-4687	100	15	6	6	X
ejpam-4687	100	16	.	.	PUNCT
ejpam-4687	101	1	j.	j.	PROPN
ejpam-4687	101	2	hassan	hassan	PROPN
ejpam-4687	101	3	,	,	PUNCT
ejpam-4687	101	4	s.	s.	PROPN
ejpam-4687	101	5	canoy	canoy	PROPN
ejpam-4687	101	6	jr	jr	PROPN
ejpam-4687	101	7	.	.	PROPN
ejpam-4687	101	8	/	/	SYM
ejpam-4687	101	9	eur	eur	PROPN
ejpam-4687	101	10	.	.	PUNCT
ejpam-4687	102	1	j.	j.	PROPN
ejpam-4687	102	2	pure	pure	PROPN
ejpam-4687	102	3	appl	appl	PROPN
ejpam-4687	102	4	.	.	PROPN
ejpam-4687	102	5	math	math	PROPN
ejpam-4687	102	6	,	,	PUNCT
ejpam-4687	102	7	16	16	NUM
ejpam-4687	102	8	(	(	PUNCT
ejpam-4687	102	9	2	2	NUM
ejpam-4687	102	10	)	)	PUNCT
ejpam-4687	102	11	(	(	PUNCT
ejpam-4687	102	12	2023	2023	NUM
ejpam-4687	102	13	)	)	PUNCT
ejpam-4687	102	14	,	,	PUNCT
ejpam-4687	102	15	1212	1212	NUM
ejpam-4687	102	16	-	-	SYM
ejpam-4687	102	17	1227	1227	NUM
ejpam-4687	102	18	1216	1216	NUM
ejpam-4687	102	19	s2	s2	PROPN
ejpam-4687	102	20	s5	s5	PROPN
ejpam-4687	102	21	s1	s1	PROPN
ejpam-4687	102	22	g	g	PROPN
ejpam-4687	102	23	:	:	PUNCT
ejpam-4687	102	24	s4s3	s4s3	PROPN
ejpam-4687	102	25	s6	s6	PROPN
ejpam-4687	102	26	figure	figure	NOUN
ejpam-4687	102	27	2	2	NUM
ejpam-4687	102	28	:	:	PUNCT
ejpam-4687	102	29	a	a	DET
ejpam-4687	102	30	graph	graph	NOUN
ejpam-4687	102	31	g	g	NOUN
ejpam-4687	102	32	with	with	ADP
ejpam-4687	102	33	γch(g	γch(g	NOUN
ejpam-4687	102	34	)	)	PUNCT
ejpam-4687	102	35	<	<	X
ejpam-4687	102	36	γch	γch	X
ejpam-4687	102	37	gr	gr	X
ejpam-4687	102	38	(	(	PUNCT
ejpam-4687	102	39	g	g	NOUN
ejpam-4687	102	40	)	)	PUNCT
ejpam-4687	102	41	lastly	lastly	ADV
ejpam-4687	102	42	,	,	PUNCT
ejpam-4687	102	43	consider	consider	VERB
ejpam-4687	102	44	the	the	DET
ejpam-4687	102	45	graphg′	graphg′	NOUN
ejpam-4687	102	46	in	in	ADP
ejpam-4687	102	47	figure	figure	NOUN
ejpam-4687	102	48	3	3	NUM
ejpam-4687	102	49	.	.	PUNCT
ejpam-4687	103	1	let	let	VERB
ejpam-4687	103	2	s′	s′	ADJ
ejpam-4687	103	3	=	=	SYM
ejpam-4687	103	4	(	(	PUNCT
ejpam-4687	103	5	u3	u3	PROPN
ejpam-4687	103	6	,	,	PUNCT
ejpam-4687	103	7	u4	u4	PROPN
ejpam-4687	103	8	,	,	PUNCT
ejpam-4687	103	9	u5	u5	PROPN
ejpam-4687	103	10	)	)	PUNCT
ejpam-4687	103	11	and	and	CCONJ
ejpam-4687	103	12	s′′	s′′	PROPN
ejpam-4687	103	13	=	=	SYM
ejpam-4687	103	14	(	(	PUNCT
ejpam-4687	103	15	u1	u1	PROPN
ejpam-4687	103	16	,	,	PUNCT
ejpam-4687	103	17	u2	u2	NOUN
ejpam-4687	103	18	,	,	PUNCT
ejpam-4687	103	19	u6	u6	PROPN
ejpam-4687	103	20	,	,	PUNCT
ejpam-4687	103	21	u7	u7	PROPN
ejpam-4687	103	22	)	)	PUNCT
ejpam-4687	103	23	.	.	PUNCT
ejpam-4687	104	1	then	then	ADV
ejpam-4687	104	2	s′	s′	NUM
ejpam-4687	104	3	and	and	CCONJ
ejpam-4687	104	4	s′′	s′′	PROPN
ejpam-4687	104	5	are	be	AUX
ejpam-4687	104	6	γchgr	γchgr	ADJ
ejpam-4687	104	7	-sequence	-sequence	NOUN
ejpam-4687	104	8	and	and	CCONJ
ejpam-4687	104	9	γhgr	γhgr	ADJ
ejpam-4687	104	10	-	-	PUNCT
ejpam-4687	104	11	sequence	sequence	NOUN
ejpam-4687	104	12	of	of	ADP
ejpam-4687	104	13	g′	g′	NOUN
ejpam-4687	104	14	,	,	PUNCT
ejpam-4687	104	15	respectively	respectively	ADV
ejpam-4687	104	16	.	.	PUNCT
ejpam-4687	105	1	hence	hence	ADV
ejpam-4687	105	2	,	,	PUNCT
ejpam-4687	105	3	γchgr	γchgr	X
ejpam-4687	105	4	(	(	PUNCT
ejpam-4687	105	5	g	g	PROPN
ejpam-4687	105	6	′	′	NUM
ejpam-4687	105	7	)	)	PUNCT
ejpam-4687	105	8	=	=	SYM
ejpam-4687	105	9	3	3	NUM
ejpam-4687	105	10	and	and	CCONJ
ejpam-4687	105	11	γhgr(g	γhgr(g	PRON
ejpam-4687	105	12	′	′	NUM
ejpam-4687	105	13	)	)	PUNCT
ejpam-4687	105	14	=	=	SYM
ejpam-4687	106	1	4	4	NUM
ejpam-4687	106	2	,	,	PUNCT
ejpam-4687	106	3	that	that	ADV
ejpam-4687	106	4	is	is	ADV
ejpam-4687	106	5	,	,	PUNCT
ejpam-4687	106	6	γchgr	γchgr	ADJ
ejpam-4687	106	7	(	(	PUNCT
ejpam-4687	106	8	g	g	PROPN
ejpam-4687	106	9	′	′	NUM
ejpam-4687	106	10	)	)	PUNCT
ejpam-4687	106	11	=	=	PUNCT
ejpam-4687	106	12	3	3	NUM
ejpam-4687	106	13	<	<	SYM
ejpam-4687	106	14	4	4	NUM
ejpam-4687	106	15	=	=	NOUN
ejpam-4687	106	16	γhgr(g	γhgr(g	NOUN
ejpam-4687	106	17	′	′	NUM
ejpam-4687	106	18	)	)	PUNCT
ejpam-4687	106	19	.	.	PUNCT
ejpam-4687	107	1	u1	u1	PROPN
ejpam-4687	107	2	u2	u2	PROPN
ejpam-4687	107	3	u6	u6	PROPN
ejpam-4687	107	4	u7	u7	PROPN
ejpam-4687	107	5	u3	u3	PROPN
ejpam-4687	107	6	u5	u5	PROPN
ejpam-4687	107	7	u4	u4	PROPN
ejpam-4687	107	8	g′	g′	PROPN
ejpam-4687	107	9	:	:	PUNCT
ejpam-4687	107	10	figure	figure	VERB
ejpam-4687	107	11	3	3	NUM
ejpam-4687	107	12	:	:	PUNCT
ejpam-4687	107	13	a	a	DET
ejpam-4687	107	14	graph	graph	NOUN
ejpam-4687	107	15	g′	g′	NOUN
ejpam-4687	107	16	with	with	ADP
ejpam-4687	107	17	γch	γch	NOUN
ejpam-4687	107	18	gr	gr	PROPN
ejpam-4687	107	19	(	(	PUNCT
ejpam-4687	107	20	g	g	PROPN
ejpam-4687	107	21	′	′	NUM
ejpam-4687	107	22	)	)	PUNCT
ejpam-4687	107	23	<	<	X
ejpam-4687	107	24	γh	γh	PROPN
ejpam-4687	107	25	gr(g	gr(g	PROPN
ejpam-4687	107	26	′	′	NUM
ejpam-4687	107	27	)	)	PUNCT
ejpam-4687	107	28	proposition	proposition	NOUN
ejpam-4687	107	29	1	1	NUM
ejpam-4687	107	30	.	.	PUNCT
ejpam-4687	108	1	let	let	VERB
ejpam-4687	108	2	g	g	PRON
ejpam-4687	108	3	be	be	AUX
ejpam-4687	108	4	a	a	DET
ejpam-4687	108	5	connected	connected	ADJ
ejpam-4687	108	6	graph	graph	NOUN
ejpam-4687	108	7	.	.	PUNCT
ejpam-4687	109	1	then	then	ADV
ejpam-4687	109	2	s	s	VERB
ejpam-4687	109	3	=	=	SYM
ejpam-4687	109	4	(	(	PUNCT
ejpam-4687	109	5	s1	s1	PROPN
ejpam-4687	109	6	,	,	PUNCT
ejpam-4687	109	7	s2	s2	PROPN
ejpam-4687	109	8	,	,	PUNCT
ejpam-4687	109	9	·	·	PUNCT
ejpam-4687	109	10	·	·	PUNCT
ejpam-4687	109	11	·	·	PUNCT
ejpam-4687	109	12	,	,	PUNCT
ejpam-4687	109	13	sk	sk	PROPN
ejpam-4687	109	14	)	)	PUNCT
ejpam-4687	109	15	is	be	AUX
ejpam-4687	109	16	a	a	DET
ejpam-4687	109	17	legal	legal	ADJ
ejpam-4687	109	18	closed	close	VERB
ejpam-4687	109	19	hop	hop	NOUN
ejpam-4687	109	20	neighborhood	neighborhood	NOUN
ejpam-4687	109	21	sequence	sequence	NOUN
ejpam-4687	109	22	of	of	ADP
ejpam-4687	109	23	g	g	NOUN
ejpam-4687	109	24	with	with	ADP
ejpam-4687	109	25	maximum	maximum	ADJ
ejpam-4687	109	26	length	length	NOUN
ejpam-4687	109	27	and	and	CCONJ
ejpam-4687	109	28	⟨ŝ⟩	⟨ŝ⟩	PRON
ejpam-4687	109	29	connected	connect	VERB
ejpam-4687	109	30	if	if	SCONJ
ejpam-4687	109	31	and	and	CCONJ
ejpam-4687	109	32	only	only	ADV
ejpam-4687	109	33	if	if	SCONJ
ejpam-4687	109	34	s	s	NOUN
ejpam-4687	109	35	is	be	AUX
ejpam-4687	109	36	a	a	DET
ejpam-4687	109	37	connected	connect	VERB
ejpam-4687	109	38	grundy	grundy	PROPN
ejpam-4687	109	39	hop	hop	NOUN
ejpam-4687	109	40	dominating	dominating	NOUN
ejpam-4687	109	41	sequence	sequence	NOUN
ejpam-4687	109	42	of	of	ADP
ejpam-4687	109	43	g	g	NOUN
ejpam-4687	109	44	with	with	ADP
ejpam-4687	109	45	γchgr	γchgr	ADJ
ejpam-4687	109	46	(	(	PUNCT
ejpam-4687	109	47	g	g	NOUN
ejpam-4687	109	48	)	)	PUNCT
ejpam-4687	109	49	=	=	PUNCT
ejpam-4687	110	1	k.	k.	NOUN
ejpam-4687	110	2	proof	proof	NOUN
ejpam-4687	110	3	.	.	PUNCT
ejpam-4687	111	1	let	let	VERB
ejpam-4687	111	2	s	s	PRON
ejpam-4687	111	3	=	=	PUNCT
ejpam-4687	111	4	(	(	PUNCT
ejpam-4687	111	5	s1	s1	PROPN
ejpam-4687	111	6	,	,	PUNCT
ejpam-4687	111	7	s2	s2	PROPN
ejpam-4687	111	8	,	,	PUNCT
ejpam-4687	111	9	·	·	PUNCT
ejpam-4687	111	10	·	·	PUNCT
ejpam-4687	111	11	·	·	PUNCT
ejpam-4687	111	12	,	,	PUNCT
ejpam-4687	111	13	sk	sk	PART
ejpam-4687	111	14	)	)	PUNCT
ejpam-4687	111	15	be	be	AUX
ejpam-4687	111	16	a	a	DET
ejpam-4687	111	17	legal	legal	ADJ
ejpam-4687	111	18	closed	close	VERB
ejpam-4687	111	19	hop	hop	NOUN
ejpam-4687	111	20	neighborhood	neighborhood	NOUN
ejpam-4687	111	21	sequence	sequence	NOUN
ejpam-4687	111	22	of	of	ADP
ejpam-4687	111	23	g	g	NOUN
ejpam-4687	111	24	with	with	ADP
ejpam-4687	111	25	maximum	maximum	ADJ
ejpam-4687	111	26	length	length	NOUN
ejpam-4687	111	27	k	k	PROPN
ejpam-4687	111	28	and	and	CCONJ
ejpam-4687	111	29	⟨ŝ⟩	⟨ŝ⟩	PRON
ejpam-4687	111	30	connected	connect	VERB
ejpam-4687	111	31	.	.	PUNCT
ejpam-4687	112	1	suppose	suppose	VERB
ejpam-4687	112	2	ŝ	ŝ	NOUN
ejpam-4687	112	3	is	be	AUX
ejpam-4687	112	4	not	not	PART
ejpam-4687	112	5	a	a	DET
ejpam-4687	112	6	connected	connected	ADJ
ejpam-4687	112	7	hop	hop	NOUN
ejpam-4687	112	8	dominating	dominating	NOUN
ejpam-4687	112	9	set	set	NOUN
ejpam-4687	112	10	of	of	ADP
ejpam-4687	112	11	g.	g.	PROPN
ejpam-4687	112	12	then	then	ADV
ejpam-4687	112	13	there	there	PRON
ejpam-4687	112	14	exists	exist	VERB
ejpam-4687	112	15	v	v	ADP
ejpam-4687	112	16	∈	∈	PROPN
ejpam-4687	112	17	v	v	NOUN
ejpam-4687	112	18	(	(	PUNCT
ejpam-4687	112	19	g)\n2	g)\n2	PROPN
ejpam-4687	112	20	g[ŝ	g[ŝ	PROPN
ejpam-4687	112	21	]	]	PUNCT
ejpam-4687	112	22	.	.	PUNCT
ejpam-4687	113	1	this	this	PRON
ejpam-4687	113	2	implies	imply	VERB
ejpam-4687	113	3	that	that	SCONJ
ejpam-4687	113	4	v	v	X
ejpam-4687	113	5	/∈	/∈	PUNCT
ejpam-4687	113	6	n2	n2	PROPN
ejpam-4687	113	7	g[u	g[u	PROPN
ejpam-4687	113	8	]	]	PUNCT
ejpam-4687	113	9	for	for	SCONJ
ejpam-4687	113	10	every	every	DET
ejpam-4687	113	11	u	u	PROPN
ejpam-4687	113	12	∈	∈	PROPN
ejpam-4687	113	13	ŝ.	ŝ.	NOUN
ejpam-4687	113	14	pick	pick	VERB
ejpam-4687	113	15	u0	u0	PROPN
ejpam-4687	113	16	=	=	PROPN
ejpam-4687	113	17	st	st	PROPN
ejpam-4687	113	18	∈	∈	PROPN
ejpam-4687	113	19	ŝ	ŝ	VERB
ejpam-4687	113	20	such	such	ADJ
ejpam-4687	113	21	that	that	PRON
ejpam-4687	113	22	dg(v	dg(v	ADJ
ejpam-4687	113	23	,	,	PUNCT
ejpam-4687	113	24	u0	u0	ADJ
ejpam-4687	113	25	)	)	PUNCT
ejpam-4687	113	26	≤	≤	NOUN
ejpam-4687	113	27	dg(v	dg(v	PUNCT
ejpam-4687	113	28	,	,	PUNCT
ejpam-4687	113	29	sj	sj	INTJ
ejpam-4687	113	30	)	)	PUNCT
ejpam-4687	113	31	for	for	ADP
ejpam-4687	113	32	all	all	DET
ejpam-4687	113	33	j	j	PROPN
ejpam-4687	113	34	∈	∈	PROPN
ejpam-4687	113	35	{	{	PUNCT
ejpam-4687	113	36	1	1	NUM
ejpam-4687	113	37	,	,	PUNCT
ejpam-4687	113	38	2	2	NUM
ejpam-4687	113	39	,	,	PUNCT
ejpam-4687	113	40	.	.	PUNCT
ejpam-4687	113	41	.	.	PUNCT
ejpam-4687	114	1	.	.	PUNCT
ejpam-4687	115	1	,	,	PUNCT
ejpam-4687	115	2	k	k	X
ejpam-4687	115	3	}	}	PUNCT
ejpam-4687	115	4	.	.	PUNCT
ejpam-4687	116	1	let	let	VERB
ejpam-4687	116	2	[	[	X
ejpam-4687	116	3	q1	q1	ADP
ejpam-4687	116	4	,	,	PUNCT
ejpam-4687	116	5	q2	q2	NOUN
ejpam-4687	116	6	,	,	PUNCT
ejpam-4687	116	7	.	.	PUNCT
ejpam-4687	116	8	.	.	PUNCT
ejpam-4687	117	1	.	.	PUNCT
ejpam-4687	118	1	,	,	PUNCT
ejpam-4687	118	2	qm	qm	PROPN
ejpam-4687	118	3	]	]	X
ejpam-4687	118	4	,	,	PUNCT
ejpam-4687	118	5	where	where	SCONJ
ejpam-4687	118	6	q1	q1	PROPN
ejpam-4687	118	7	=	=	PUNCT
ejpam-4687	118	8	u0	u0	PROPN
ejpam-4687	118	9	and	and	CCONJ
ejpam-4687	118	10	qm	qm	PROPN
ejpam-4687	118	11	=	=	PROPN
ejpam-4687	118	12	v	v	PROPN
ejpam-4687	118	13	,	,	PUNCT
ejpam-4687	118	14	be	be	AUX
ejpam-4687	118	15	a	a	DET
ejpam-4687	118	16	u0	u0	ADJ
ejpam-4687	118	17	-	-	PUNCT
ejpam-4687	118	18	v	v	NOUN
ejpam-4687	118	19	geodesic	geodesic	NOUN
ejpam-4687	118	20	.	.	PUNCT
ejpam-4687	119	1	thenm	thenm	PROPN
ejpam-4687	119	2	≥	≥	NUM
ejpam-4687	119	3	4	4	NUM
ejpam-4687	119	4	and	and	CCONJ
ejpam-4687	119	5	q4	q4	PROPN
ejpam-4687	119	6	/∈	/∈	PROPN
ejpam-4687	119	7	n2	n2	PROPN
ejpam-4687	119	8	g[u	g[u	PROPN
ejpam-4687	119	9	]	]	PUNCT
ejpam-4687	119	10	for	for	ADP
ejpam-4687	119	11	every	every	DET
ejpam-4687	119	12	u	u	PROPN
ejpam-4687	119	13	∈	∈	PROPN
ejpam-4687	119	14	ŝ.	ŝ.	NOUN
ejpam-4687	119	15	let	let	VERB
ejpam-4687	119	16	s∗	s∗	PROPN
ejpam-4687	119	17	=	=	SYM
ejpam-4687	119	18	(	(	PUNCT
ejpam-4687	119	19	s1	s1	NOUN
ejpam-4687	119	20	,	,	PUNCT
ejpam-4687	119	21	s2	s2	PROPN
ejpam-4687	119	22	,	,	PUNCT
ejpam-4687	119	23	·	·	PUNCT
ejpam-4687	119	24	·	·	PUNCT
ejpam-4687	119	25	·	·	PUNCT
ejpam-4687	119	26	,	,	PUNCT
ejpam-4687	119	27	sk	sk	ADP
ejpam-4687	119	28	,	,	PUNCT
ejpam-4687	119	29	q2	q2	NOUN
ejpam-4687	119	30	)	)	PUNCT
ejpam-4687	119	31	.	.	PUNCT
ejpam-4687	120	1	then	then	ADV
ejpam-4687	120	2	⟨ŝ∗⟩	⟨ŝ∗⟩	PROPN
ejpam-4687	120	3	is	be	AUX
ejpam-4687	120	4	connected	connect	VERB
ejpam-4687	120	5	and	and	CCONJ
ejpam-4687	120	6	q4	q4	PROPN
ejpam-4687	120	7	∈	∈	PROPN
ejpam-4687	120	8	n2	n2	PROPN
ejpam-4687	120	9	g[q2	g[q2	PROPN
ejpam-4687	120	10	]	]	PUNCT
ejpam-4687	120	11	\	\	PROPN
ejpam-4687	121	1	∪k	∪k	PROPN
ejpam-4687	121	2	j=1n	j=1n	PROPN
ejpam-4687	121	3	2	2	NUM
ejpam-4687	121	4	g[si	g[si	NOUN
ejpam-4687	121	5	]	]	PUNCT
ejpam-4687	121	6	̸=	̸=	PROPN
ejpam-4687	121	7	∅.	∅.	NOUN
ejpam-4687	121	8	it	it	PRON
ejpam-4687	121	9	follows	follow	VERB
ejpam-4687	121	10	that	that	SCONJ
ejpam-4687	121	11	s∗	s∗	PROPN
ejpam-4687	121	12	is	be	AUX
ejpam-4687	121	13	a	a	DET
ejpam-4687	121	14	legal	legal	ADJ
ejpam-4687	121	15	closed	close	VERB
ejpam-4687	121	16	hop	hop	NOUN
ejpam-4687	121	17	neighborhood	neighborhood	NOUN
ejpam-4687	121	18	sequence	sequence	NOUN
ejpam-4687	121	19	of	of	ADP
ejpam-4687	121	20	g	g	NOUN
ejpam-4687	121	21	,	,	PUNCT
ejpam-4687	121	22	a	a	DET
ejpam-4687	121	23	contradiction	contradiction	NOUN
ejpam-4687	121	24	to	to	ADP
ejpam-4687	121	25	the	the	DET
ejpam-4687	121	26	maximality	maximality	NOUN
ejpam-4687	121	27	of	of	ADP
ejpam-4687	121	28	s.	s.	PROPN
ejpam-4687	121	29	thus	thus	ADV
ejpam-4687	121	30	,	,	PUNCT
ejpam-4687	121	31	ŝ	ŝ	X
ejpam-4687	121	32	is	be	AUX
ejpam-4687	121	33	a	a	DET
ejpam-4687	121	34	connected	connected	ADJ
ejpam-4687	121	35	dominating	dominating	NOUN
ejpam-4687	121	36	set	set	NOUN
ejpam-4687	121	37	of	of	ADP
ejpam-4687	121	38	g.	g.	PROPN
ejpam-4687	121	39	therefore	therefore	ADV
ejpam-4687	121	40	,	,	PUNCT
ejpam-4687	121	41	by	by	ADP
ejpam-4687	121	42	assumption	assumption	NOUN
ejpam-4687	121	43	,	,	PUNCT
ejpam-4687	121	44	s	s	PART
ejpam-4687	121	45	is	be	AUX
ejpam-4687	121	46	a	a	DET
ejpam-4687	121	47	connected	connect	VERB
ejpam-4687	121	48	grundy	grundy	PROPN
ejpam-4687	121	49	hop	hop	NOUN
ejpam-4687	121	50	dominating	dominating	NOUN
ejpam-4687	121	51	sequence	sequence	NOUN
ejpam-4687	121	52	of	of	ADP
ejpam-4687	121	53	g	g	NOUN
ejpam-4687	121	54	and	and	CCONJ
ejpam-4687	121	55	γchgr	γchgr	ADJ
ejpam-4687	121	56	(	(	PUNCT
ejpam-4687	121	57	g	g	NOUN
ejpam-4687	121	58	)	)	PUNCT
ejpam-4687	121	59	=	=	VERB
ejpam-4687	122	1	k.	k.	NOUN
ejpam-4687	123	1	the	the	DET
ejpam-4687	123	2	converse	converse	NOUN
ejpam-4687	123	3	is	be	AUX
ejpam-4687	123	4	clear	clear	ADJ
ejpam-4687	123	5	.	.	PUNCT
ejpam-4687	124	1	the	the	DET
ejpam-4687	124	2	next	next	ADJ
ejpam-4687	124	3	result	result	NOUN
ejpam-4687	124	4	follows	follow	VERB
ejpam-4687	124	5	from	from	ADP
ejpam-4687	124	6	proposition	proposition	NOUN
ejpam-4687	124	7	1	1	NUM
ejpam-4687	124	8	j.	j.	PROPN
ejpam-4687	124	9	hassan	hassan	PROPN
ejpam-4687	124	10	,	,	PUNCT
ejpam-4687	124	11	s.	s.	PROPN
ejpam-4687	124	12	canoy	canoy	PROPN
ejpam-4687	124	13	jr	jr	PROPN
ejpam-4687	124	14	.	.	PROPN
ejpam-4687	124	15	/	/	SYM
ejpam-4687	124	16	eur	eur	PROPN
ejpam-4687	124	17	.	.	PUNCT
ejpam-4687	125	1	j.	j.	PROPN
ejpam-4687	125	2	pure	pure	PROPN
ejpam-4687	125	3	appl	appl	PROPN
ejpam-4687	125	4	.	.	PROPN
ejpam-4687	125	5	math	math	PROPN
ejpam-4687	125	6	,	,	PUNCT
ejpam-4687	125	7	16	16	NUM
ejpam-4687	125	8	(	(	PUNCT
ejpam-4687	125	9	2	2	NUM
ejpam-4687	125	10	)	)	PUNCT
ejpam-4687	125	11	(	(	PUNCT
ejpam-4687	125	12	2023	2023	NUM
ejpam-4687	125	13	)	)	PUNCT
ejpam-4687	125	14	,	,	PUNCT
ejpam-4687	125	15	1212	1212	NUM
ejpam-4687	125	16	-	-	SYM
ejpam-4687	125	17	1227	1227	NUM
ejpam-4687	125	18	1217	1217	NUM
ejpam-4687	125	19	corollary	corollary	ADJ
ejpam-4687	125	20	1	1	NUM
ejpam-4687	125	21	.	.	PUNCT
ejpam-4687	126	1	let	let	VERB
ejpam-4687	126	2	g	g	PRON
ejpam-4687	126	3	be	be	AUX
ejpam-4687	126	4	a	a	DET
ejpam-4687	126	5	connected	connected	ADJ
ejpam-4687	126	6	graph	graph	NOUN
ejpam-4687	126	7	and	and	CCONJ
ejpam-4687	126	8	let	let	VERB
ejpam-4687	126	9	s	s	PRON
ejpam-4687	126	10	=	=	PUNCT
ejpam-4687	126	11	(	(	PUNCT
ejpam-4687	126	12	s1	s1	PROPN
ejpam-4687	126	13	,	,	PUNCT
ejpam-4687	126	14	s2	s2	NOUN
ejpam-4687	126	15	,	,	PUNCT
ejpam-4687	126	16	.	.	PUNCT
ejpam-4687	126	17	.	.	PUNCT
ejpam-4687	127	1	.	.	PUNCT
ejpam-4687	128	1	,	,	PUNCT
ejpam-4687	128	2	sm	sm	X
ejpam-4687	128	3	)	)	PUNCT
ejpam-4687	128	4	be	be	VERB
ejpam-4687	128	5	a	a	DET
ejpam-4687	128	6	legal	legal	ADJ
ejpam-4687	128	7	closed	close	VERB
ejpam-4687	128	8	hop	hop	NOUN
ejpam-4687	128	9	neighborhood	neighborhood	NOUN
ejpam-4687	128	10	sequence	sequence	NOUN
ejpam-4687	128	11	of	of	ADP
ejpam-4687	128	12	g	g	NOUN
ejpam-4687	128	13	such	such	ADJ
ejpam-4687	128	14	that	that	SCONJ
ejpam-4687	128	15	⟨ŝ⟩	⟨ŝ⟩	PROPN
ejpam-4687	128	16	is	be	AUX
ejpam-4687	128	17	connected	connect	VERB
ejpam-4687	128	18	.	.	PUNCT
ejpam-4687	129	1	then	then	ADV
ejpam-4687	129	2	|ŝ|	|ŝ|	PROPN
ejpam-4687	129	3	=	=	SYM
ejpam-4687	129	4	m	m	VERB
ejpam-4687	129	5	≤	≤	NOUN
ejpam-4687	129	6	γchgr	γchgr	NOUN
ejpam-4687	129	7	(	(	PUNCT
ejpam-4687	129	8	g	g	NOUN
ejpam-4687	129	9	)	)	PUNCT
ejpam-4687	129	10	.	.	PUNCT
ejpam-4687	130	1	theorem	theorem	NOUN
ejpam-4687	130	2	1	1	NUM
ejpam-4687	130	3	.	.	PUNCT
ejpam-4687	131	1	[	[	X
ejpam-4687	131	2	8	8	NUM
ejpam-4687	131	3	]	]	PUNCT
ejpam-4687	131	4	let	let	VERB
ejpam-4687	131	5	g	g	NOUN
ejpam-4687	131	6	be	be	AUX
ejpam-4687	131	7	any	any	DET
ejpam-4687	131	8	graph	graph	NOUN
ejpam-4687	131	9	on	on	ADP
ejpam-4687	131	10	n	n	PROPN
ejpam-4687	131	11	(	(	PUNCT
ejpam-4687	131	12	n	n	CCONJ
ejpam-4687	131	13	≥	≥	NOUN
ejpam-4687	131	14	2	2	NUM
ejpam-4687	131	15	)	)	PUNCT
ejpam-4687	131	16	vertices	vertex	NOUN
ejpam-4687	131	17	.	.	PUNCT
ejpam-4687	132	1	then	then	ADV
ejpam-4687	132	2	γhgr(g	γhgr(g	X
ejpam-4687	132	3	)	)	PUNCT
ejpam-4687	132	4	=	=	SYM
ejpam-4687	133	1	n	n	NOUN
ejpam-4687	133	2	if	if	SCONJ
ejpam-4687	133	3	and	and	CCONJ
ejpam-4687	133	4	only	only	ADV
ejpam-4687	133	5	if	if	SCONJ
ejpam-4687	133	6	every	every	DET
ejpam-4687	133	7	component	component	NOUN
ejpam-4687	133	8	c	c	NOUN
ejpam-4687	133	9	of	of	ADP
ejpam-4687	133	10	g	g	PROPN
ejpam-4687	133	11	is	be	AUX
ejpam-4687	133	12	complete	complete	ADJ
ejpam-4687	133	13	.	.	PUNCT
ejpam-4687	134	1	theorem	theorem	NOUN
ejpam-4687	134	2	2	2	NUM
ejpam-4687	134	3	.	.	PUNCT
ejpam-4687	135	1	let	let	VERB
ejpam-4687	135	2	g	g	PRON
ejpam-4687	135	3	be	be	AUX
ejpam-4687	135	4	a	a	DET
ejpam-4687	135	5	connected	connected	ADJ
ejpam-4687	135	6	graph	graph	NOUN
ejpam-4687	135	7	on	on	ADP
ejpam-4687	135	8	n	n	DET
ejpam-4687	135	9	vertices	vertex	NOUN
ejpam-4687	135	10	.	.	PUNCT
ejpam-4687	136	1	then	then	ADV
ejpam-4687	136	2	1	1	NUM
ejpam-4687	136	3	≤	≤	NOUN
ejpam-4687	136	4	γchgr	γchgr	NOUN
ejpam-4687	136	5	(	(	PUNCT
ejpam-4687	136	6	g	g	NOUN
ejpam-4687	136	7	)	)	PUNCT
ejpam-4687	136	8	≤	≤	NOUN
ejpam-4687	136	9	n.	n.	NOUN
ejpam-4687	136	10	moreover	moreover	ADV
ejpam-4687	136	11	,	,	PUNCT
ejpam-4687	136	12	each	each	PRON
ejpam-4687	136	13	of	of	ADP
ejpam-4687	136	14	the	the	DET
ejpam-4687	136	15	following	following	ADJ
ejpam-4687	136	16	statements	statement	NOUN
ejpam-4687	136	17	holds	hold	VERB
ejpam-4687	136	18	.	.	PUNCT
ejpam-4687	137	1	(	(	PUNCT
ejpam-4687	137	2	i	i	NOUN
ejpam-4687	137	3	)	)	PUNCT
ejpam-4687	137	4	γchgr	γchgr	NOUN
ejpam-4687	137	5	(	(	PUNCT
ejpam-4687	137	6	g	g	NOUN
ejpam-4687	137	7	)	)	PUNCT
ejpam-4687	137	8	=	=	SYM
ejpam-4687	137	9	1	1	NUM
ejpam-4687	137	10	if	if	SCONJ
ejpam-4687	137	11	and	and	CCONJ
ejpam-4687	137	12	only	only	ADV
ejpam-4687	137	13	if	if	SCONJ
ejpam-4687	137	14	g	g	PROPN
ejpam-4687	137	15	is	be	AUX
ejpam-4687	137	16	trivial	trivial	ADJ
ejpam-4687	137	17	.	.	PUNCT
ejpam-4687	138	1	(	(	PUNCT
ejpam-4687	138	2	ii	ii	NOUN
ejpam-4687	138	3	)	)	PUNCT
ejpam-4687	138	4	γchgr	γchgr	NOUN
ejpam-4687	138	5	(	(	PUNCT
ejpam-4687	138	6	g	g	NOUN
ejpam-4687	138	7	)	)	PUNCT
ejpam-4687	138	8	=	=	SYM
ejpam-4687	138	9	2	2	NUM
ejpam-4687	138	10	if	if	SCONJ
ejpam-4687	138	11	and	and	CCONJ
ejpam-4687	138	12	only	only	ADV
ejpam-4687	138	13	if	if	SCONJ
ejpam-4687	138	14	g	g	PROPN
ejpam-4687	138	15	a	a	DET
ejpam-4687	138	16	non	non	ADJ
ejpam-4687	138	17	-	-	ADJ
ejpam-4687	138	18	trivial	trivial	ADJ
ejpam-4687	138	19	graph	graph	NOUN
ejpam-4687	138	20	,	,	PUNCT
ejpam-4687	138	21	has	have	VERB
ejpam-4687	138	22	no	no	DET
ejpam-4687	138	23	induced	induce	VERB
ejpam-4687	138	24	cycles	cycle	NOUN
ejpam-4687	138	25	c3	c3	PROPN
ejpam-4687	138	26	and	and	CCONJ
ejpam-4687	138	27	c5	c5	PROPN
ejpam-4687	138	28	,	,	PUNCT
ejpam-4687	138	29	and	and	CCONJ
ejpam-4687	138	30	{	{	PUNCT
ejpam-4687	138	31	a	a	DET
ejpam-4687	138	32	,	,	PUNCT
ejpam-4687	138	33	b	b	NOUN
ejpam-4687	138	34	}	}	PUNCT
ejpam-4687	138	35	is	be	AUX
ejpam-4687	138	36	a	a	DET
ejpam-4687	138	37	(	(	PUNCT
ejpam-4687	138	38	connected	connected	ADJ
ejpam-4687	138	39	)	)	PUNCT
ejpam-4687	138	40	hop	hop	NOUN
ejpam-4687	138	41	dominating	dominating	NOUN
ejpam-4687	138	42	set	set	NOUN
ejpam-4687	138	43	for	for	ADP
ejpam-4687	138	44	each	each	DET
ejpam-4687	138	45	pair	pair	NOUN
ejpam-4687	138	46	of	of	ADP
ejpam-4687	138	47	adjacent	adjacent	ADJ
ejpam-4687	138	48	vertices	vertex	NOUN
ejpam-4687	138	49	a	a	DET
ejpam-4687	138	50	,	,	PUNCT
ejpam-4687	138	51	b	b	PROPN
ejpam-4687	138	52	∈	∈	PROPN
ejpam-4687	138	53	v	v	NOUN
ejpam-4687	138	54	(	(	PUNCT
ejpam-4687	138	55	g	g	NOUN
ejpam-4687	138	56	)	)	PUNCT
ejpam-4687	138	57	.	.	PUNCT
ejpam-4687	139	1	(	(	PUNCT
ejpam-4687	139	2	iii	iii	NOUN
ejpam-4687	139	3	)	)	PUNCT
ejpam-4687	139	4	γchgr	γchgr	NOUN
ejpam-4687	139	5	(	(	PUNCT
ejpam-4687	139	6	g	g	NOUN
ejpam-4687	139	7	)	)	PUNCT
ejpam-4687	139	8	=	=	SYM
ejpam-4687	140	1	n	n	NOUN
ejpam-4687	140	2	if	if	SCONJ
ejpam-4687	140	3	and	and	CCONJ
ejpam-4687	140	4	only	only	ADV
ejpam-4687	140	5	if	if	SCONJ
ejpam-4687	140	6	g	g	PROPN
ejpam-4687	140	7	is	be	AUX
ejpam-4687	140	8	complete	complete	ADJ
ejpam-4687	140	9	.	.	PUNCT
ejpam-4687	141	1	proof	proof	NOUN
ejpam-4687	141	2	.	.	PUNCT
ejpam-4687	142	1	clearly	clearly	ADV
ejpam-4687	142	2	,	,	PUNCT
ejpam-4687	142	3	1	1	NUM
ejpam-4687	142	4	≤	≤	NOUN
ejpam-4687	142	5	γchgr	γchgr	NOUN
ejpam-4687	142	6	(	(	PUNCT
ejpam-4687	142	7	g	g	NOUN
ejpam-4687	142	8	)	)	PUNCT
ejpam-4687	142	9	≤	≤	NOUN
ejpam-4687	142	10	n.	n.	NOUN
ejpam-4687	142	11	(	(	PUNCT
ejpam-4687	142	12	i	i	NOUN
ejpam-4687	142	13	)	)	PUNCT
ejpam-4687	142	14	assume	assume	VERB
ejpam-4687	142	15	that	that	SCONJ
ejpam-4687	142	16	γhgr(g	γhgr(g	NOUN
ejpam-4687	142	17	)	)	PUNCT
ejpam-4687	142	18	=	=	SYM
ejpam-4687	143	1	1	1	X
ejpam-4687	143	2	.	.	PUNCT
ejpam-4687	143	3	suppose	suppose	VERB
ejpam-4687	143	4	on	on	ADP
ejpam-4687	143	5	the	the	DET
ejpam-4687	143	6	contrary	contrary	NOUN
ejpam-4687	143	7	that	that	SCONJ
ejpam-4687	143	8	g	g	PROPN
ejpam-4687	143	9	is	be	AUX
ejpam-4687	143	10	non	non	ADJ
ejpam-4687	143	11	-	-	ADJ
ejpam-4687	143	12	trivial	trivial	ADJ
ejpam-4687	143	13	.	.	PUNCT
ejpam-4687	144	1	then	then	ADV
ejpam-4687	144	2	γch(g	γch(g	NOUN
ejpam-4687	144	3	)	)	PUNCT
ejpam-4687	144	4	≥	≥	NOUN
ejpam-4687	144	5	2	2	NUM
ejpam-4687	144	6	.	.	PUNCT
ejpam-4687	145	1	by	by	ADP
ejpam-4687	145	2	proposition	proposition	NOUN
ejpam-4687	145	3	2	2	NUM
ejpam-4687	145	4	,	,	PUNCT
ejpam-4687	145	5	γchgr	γchgr	X
ejpam-4687	145	6	(	(	PUNCT
ejpam-4687	145	7	g	g	NOUN
ejpam-4687	145	8	)	)	PUNCT
ejpam-4687	145	9	≥	≥	NOUN
ejpam-4687	145	10	2	2	NUM
ejpam-4687	145	11	,	,	PUNCT
ejpam-4687	145	12	a	a	DET
ejpam-4687	145	13	contradiction	contradiction	NOUN
ejpam-4687	145	14	.	.	PUNCT
ejpam-4687	146	1	therefore	therefore	ADV
ejpam-4687	146	2	,	,	PUNCT
ejpam-4687	146	3	g	g	PROPN
ejpam-4687	146	4	is	be	AUX
ejpam-4687	146	5	trivial	trivial	ADJ
ejpam-4687	146	6	.	.	PUNCT
ejpam-4687	147	1	the	the	DET
ejpam-4687	147	2	converse	converse	NOUN
ejpam-4687	147	3	is	be	AUX
ejpam-4687	147	4	clear	clear	ADJ
ejpam-4687	147	5	.	.	PUNCT
ejpam-4687	148	1	(	(	PUNCT
ejpam-4687	148	2	ii	ii	NOUN
ejpam-4687	148	3	)	)	PUNCT
ejpam-4687	148	4	suppose	suppose	VERB
ejpam-4687	149	1	γchgr	γchgr	ADJ
ejpam-4687	149	2	(	(	PUNCT
ejpam-4687	149	3	g	g	NOUN
ejpam-4687	149	4	)	)	PUNCT
ejpam-4687	149	5	=	=	SYM
ejpam-4687	150	1	2	2	X
ejpam-4687	150	2	.	.	PUNCT
ejpam-4687	150	3	then	then	ADV
ejpam-4687	150	4	g	g	PROPN
ejpam-4687	150	5	is	be	AUX
ejpam-4687	150	6	non	non	ADJ
ejpam-4687	150	7	-	-	ADJ
ejpam-4687	150	8	trivial	trivial	ADJ
ejpam-4687	150	9	by	by	ADP
ejpam-4687	150	10	(	(	PUNCT
ejpam-4687	150	11	i	i	NOUN
ejpam-4687	150	12	)	)	PUNCT
ejpam-4687	150	13	.	.	PUNCT
ejpam-4687	151	1	suppose	suppose	VERB
ejpam-4687	151	2	g	g	PROPN
ejpam-4687	151	3	has	have	VERB
ejpam-4687	151	4	a	a	DET
ejpam-4687	151	5	triangle	triangle	NOUN
ejpam-4687	151	6	,	,	PUNCT
ejpam-4687	151	7	say	say	VERB
ejpam-4687	151	8	c3	c3	NOUN
ejpam-4687	151	9	=	=	PUNCT
ejpam-4687	152	1	[	[	X
ejpam-4687	152	2	x	x	X
ejpam-4687	152	3	,	,	PUNCT
ejpam-4687	152	4	y	y	PROPN
ejpam-4687	152	5	,	,	PUNCT
ejpam-4687	152	6	z	z	PROPN
ejpam-4687	152	7	,	,	PUNCT
ejpam-4687	152	8	x	x	NOUN
ejpam-4687	152	9	]	]	X
ejpam-4687	152	10	.	.	PUNCT
ejpam-4687	153	1	then	then	ADV
ejpam-4687	153	2	⟨{x	⟨{x	PROPN
ejpam-4687	153	3	,	,	PUNCT
ejpam-4687	153	4	y	y	PROPN
ejpam-4687	153	5	,	,	PUNCT
ejpam-4687	153	6	z}⟩	z}⟩	PROPN
ejpam-4687	153	7	is	be	AUX
ejpam-4687	153	8	connected	connect	VERB
ejpam-4687	153	9	and	and	CCONJ
ejpam-4687	153	10	(	(	PUNCT
ejpam-4687	153	11	x	x	X
ejpam-4687	153	12	,	,	PUNCT
ejpam-4687	153	13	y	y	PROPN
ejpam-4687	153	14	,	,	PUNCT
ejpam-4687	153	15	z	z	NOUN
ejpam-4687	153	16	)	)	PUNCT
ejpam-4687	153	17	is	be	AUX
ejpam-4687	153	18	a	a	DET
ejpam-4687	153	19	legal	legal	ADJ
ejpam-4687	153	20	closed	close	VERB
ejpam-4687	153	21	hop	hop	NOUN
ejpam-4687	153	22	neighborhood	neighborhood	NOUN
ejpam-4687	153	23	sequence	sequence	NOUN
ejpam-4687	153	24	of	of	ADP
ejpam-4687	153	25	g.	g.	PROPN
ejpam-4687	153	26	by	by	ADP
ejpam-4687	153	27	corollary	corollary	ADJ
ejpam-4687	153	28	1	1	NUM
ejpam-4687	153	29	,	,	PUNCT
ejpam-4687	153	30	γchgr	γchgr	X
ejpam-4687	153	31	(	(	PUNCT
ejpam-4687	153	32	g	g	NOUN
ejpam-4687	153	33	)	)	PUNCT
ejpam-4687	153	34	≥	≥	NOUN
ejpam-4687	153	35	3	3	NUM
ejpam-4687	153	36	,	,	PUNCT
ejpam-4687	153	37	a	a	DET
ejpam-4687	153	38	contradiction	contradiction	NOUN
ejpam-4687	153	39	.	.	PUNCT
ejpam-4687	154	1	hence	hence	ADV
ejpam-4687	154	2	,	,	PUNCT
ejpam-4687	154	3	g	g	PROPN
ejpam-4687	154	4	is	be	AUX
ejpam-4687	154	5	triangle	triangle	NOUN
ejpam-4687	154	6	-	-	PUNCT
ejpam-4687	154	7	free	free	ADJ
ejpam-4687	154	8	.	.	PUNCT
ejpam-4687	155	1	next	next	ADV
ejpam-4687	155	2	,	,	PUNCT
ejpam-4687	155	3	suppose	suppose	VERB
ejpam-4687	155	4	that	that	SCONJ
ejpam-4687	155	5	g	g	PROPN
ejpam-4687	155	6	has	have	VERB
ejpam-4687	155	7	an	an	DET
ejpam-4687	155	8	induced	induced	ADJ
ejpam-4687	155	9	cycle	cycle	NOUN
ejpam-4687	155	10	c5	c5	PROPN
ejpam-4687	155	11	=	=	PUNCT
ejpam-4687	156	1	[	[	X
ejpam-4687	156	2	p1	p1	NOUN
ejpam-4687	156	3	,	,	PUNCT
ejpam-4687	156	4	p2	p2	NOUN
ejpam-4687	156	5	,	,	PUNCT
ejpam-4687	156	6	.	.	PUNCT
ejpam-4687	156	7	.	.	PUNCT
ejpam-4687	156	8	.	.	PUNCT
ejpam-4687	157	1	,	,	PUNCT
ejpam-4687	157	2	p5	p5	VERB
ejpam-4687	157	3	,	,	PUNCT
ejpam-4687	157	4	p1	p1	NOUN
ejpam-4687	157	5	]	]	PUNCT
ejpam-4687	157	6	.	.	PUNCT
ejpam-4687	158	1	then	then	ADV
ejpam-4687	158	2	(	(	PUNCT
ejpam-4687	158	3	p1	p1	PROPN
ejpam-4687	158	4	,	,	PUNCT
ejpam-4687	158	5	p3	p3	PROPN
ejpam-4687	158	6	,	,	PUNCT
ejpam-4687	158	7	p2	p2	PROPN
ejpam-4687	158	8	)	)	PUNCT
ejpam-4687	158	9	is	be	AUX
ejpam-4687	158	10	a	a	DET
ejpam-4687	158	11	legal	legal	ADJ
ejpam-4687	158	12	closed	close	VERB
ejpam-4687	158	13	hop	hop	NOUN
ejpam-4687	158	14	neighborhood	neighborhood	NOUN
ejpam-4687	158	15	sequence	sequence	NOUN
ejpam-4687	158	16	and	and	CCONJ
ejpam-4687	158	17	⟨{p1	⟨{p1	PROPN
ejpam-4687	158	18	,	,	PUNCT
ejpam-4687	158	19	p3	p3	PROPN
ejpam-4687	158	20	,	,	PUNCT
ejpam-4687	158	21	p2}⟩	p2}⟩	ADV
ejpam-4687	158	22	is	be	AUX
ejpam-4687	158	23	connected	connect	VERB
ejpam-4687	158	24	.	.	PUNCT
ejpam-4687	159	1	again	again	ADV
ejpam-4687	159	2	,	,	PUNCT
ejpam-4687	159	3	by	by	ADP
ejpam-4687	159	4	corollary	corollary	ADJ
ejpam-4687	159	5	1	1	NUM
ejpam-4687	159	6	,	,	PUNCT
ejpam-4687	159	7	this	this	PRON
ejpam-4687	159	8	implies	imply	VERB
ejpam-4687	159	9	that	that	SCONJ
ejpam-4687	159	10	γchgr	γchgr	NOUN
ejpam-4687	159	11	(	(	PUNCT
ejpam-4687	159	12	g	g	NOUN
ejpam-4687	159	13	)	)	PUNCT
ejpam-4687	159	14	≥	≥	NOUN
ejpam-4687	159	15	3	3	NUM
ejpam-4687	159	16	,	,	PUNCT
ejpam-4687	159	17	a	a	DET
ejpam-4687	159	18	contradiction	contradiction	NOUN
ejpam-4687	159	19	.	.	PUNCT
ejpam-4687	160	1	thus	thus	ADV
ejpam-4687	160	2	,	,	PUNCT
ejpam-4687	160	3	g	g	PROPN
ejpam-4687	160	4	does	do	AUX
ejpam-4687	160	5	not	not	PART
ejpam-4687	160	6	have	have	VERB
ejpam-4687	160	7	an	an	DET
ejpam-4687	160	8	induced	induced	ADJ
ejpam-4687	160	9	cycle	cycle	NOUN
ejpam-4687	160	10	of	of	ADP
ejpam-4687	160	11	order	order	NOUN
ejpam-4687	160	12	5	5	X
ejpam-4687	160	13	.	.	PUNCT
ejpam-4687	161	1	now	now	ADV
ejpam-4687	161	2	let	let	VERB
ejpam-4687	161	3	u	u	PRON
ejpam-4687	161	4	and	and	CCONJ
ejpam-4687	161	5	v	v	NOUN
ejpam-4687	161	6	be	be	AUX
ejpam-4687	161	7	adjacent	adjacent	ADJ
ejpam-4687	161	8	vertices	vertex	NOUN
ejpam-4687	161	9	.	.	PUNCT
ejpam-4687	162	1	then	then	ADV
ejpam-4687	162	2	(	(	PUNCT
ejpam-4687	162	3	u	u	NOUN
ejpam-4687	162	4	,	,	PUNCT
ejpam-4687	162	5	v	v	NOUN
ejpam-4687	162	6	)	)	PUNCT
ejpam-4687	162	7	is	be	AUX
ejpam-4687	162	8	a	a	DET
ejpam-4687	162	9	legal	legal	ADJ
ejpam-4687	162	10	closed	close	VERB
ejpam-4687	162	11	hop	hop	NOUN
ejpam-4687	162	12	neighborhood	neighborhood	NOUN
ejpam-4687	162	13	sequence	sequence	NOUN
ejpam-4687	162	14	of	of	ADP
ejpam-4687	162	15	g.	g.	PROPN
ejpam-4687	162	16	by	by	ADP
ejpam-4687	162	17	assumption	assumption	NOUN
ejpam-4687	162	18	and	and	CCONJ
ejpam-4687	162	19	proposition	proposition	NOUN
ejpam-4687	162	20	1	1	NUM
ejpam-4687	162	21	,	,	PUNCT
ejpam-4687	162	22	{	{	PUNCT
ejpam-4687	162	23	a	a	DET
ejpam-4687	162	24	,	,	PUNCT
ejpam-4687	162	25	b	b	NOUN
ejpam-4687	162	26	}	}	PUNCT
ejpam-4687	162	27	is	be	AUX
ejpam-4687	162	28	a	a	DET
ejpam-4687	162	29	hop	hop	NOUN
ejpam-4687	162	30	dominating	dominating	NOUN
ejpam-4687	162	31	set	set	NOUN
ejpam-4687	162	32	of	of	ADP
ejpam-4687	162	33	g.	g.	PROPN
ejpam-4687	162	34	for	for	ADP
ejpam-4687	162	35	the	the	DET
ejpam-4687	162	36	converse	converse	NOUN
ejpam-4687	162	37	,	,	PUNCT
ejpam-4687	162	38	suppose	suppose	VERB
ejpam-4687	162	39	that	that	SCONJ
ejpam-4687	162	40	g	g	PROPN
ejpam-4687	162	41	satisfies	satisfy	VERB
ejpam-4687	162	42	the	the	DET
ejpam-4687	162	43	given	give	VERB
ejpam-4687	162	44	conditions	condition	NOUN
ejpam-4687	162	45	.	.	PUNCT
ejpam-4687	163	1	let	let	VERB
ejpam-4687	163	2	(	(	PUNCT
ejpam-4687	163	3	v1	v1	VERB
ejpam-4687	163	4	,	,	PUNCT
ejpam-4687	163	5	v2	v2	NOUN
ejpam-4687	163	6	,	,	PUNCT
ejpam-4687	163	7	.	.	PUNCT
ejpam-4687	163	8	.	.	PUNCT
ejpam-4687	164	1	.	.	PUNCT
ejpam-4687	165	1	,	,	PUNCT
ejpam-4687	165	2	vk	vk	AUX
ejpam-4687	165	3	)	)	PUNCT
ejpam-4687	165	4	be	be	AUX
ejpam-4687	165	5	a	a	DET
ejpam-4687	165	6	γchgr	γchgr	ADJ
ejpam-4687	165	7	-sequence	-sequence	NOUN
ejpam-4687	165	8	of	of	ADP
ejpam-4687	165	9	g.	g.	PROPN
ejpam-4687	165	10	suppose	suppose	VERB
ejpam-4687	165	11	further	far	ADV
ejpam-4687	165	12	that	that	SCONJ
ejpam-4687	165	13	k	k	PROPN
ejpam-4687	165	14	≥	≥	NUM
ejpam-4687	165	15	2	2	X
ejpam-4687	165	16	.	.	PUNCT
ejpam-4687	166	1	let	let	VERB
ejpam-4687	166	2	i	i	PRON
ejpam-4687	166	3	∈	∈	PROPN
ejpam-4687	166	4	{	{	PUNCT
ejpam-4687	166	5	1	1	NUM
ejpam-4687	166	6	,	,	PUNCT
ejpam-4687	166	7	2	2	NUM
ejpam-4687	166	8	,	,	PUNCT
ejpam-4687	166	9	.	.	PUNCT
ejpam-4687	166	10	.	.	PUNCT
ejpam-4687	167	1	.	.	PUNCT
ejpam-4687	168	1	,	,	PUNCT
ejpam-4687	169	1	k	k	PROPN
ejpam-4687	170	1	−	−	PROPN
ejpam-4687	171	1	1	1	NUM
ejpam-4687	171	2	}	}	PUNCT
ejpam-4687	171	3	.	.	PUNCT
ejpam-4687	172	1	since	since	SCONJ
ejpam-4687	172	2	⟨ŝ⟩	⟨ŝ⟩	PROPN
ejpam-4687	172	3	is	be	AUX
ejpam-4687	172	4	connected	connect	VERB
ejpam-4687	172	5	,	,	PUNCT
ejpam-4687	172	6	there	there	PRON
ejpam-4687	172	7	exists	exist	VERB
ejpam-4687	173	1	1	1	NUM
ejpam-4687	173	2	≤	≤	NUM
ejpam-4687	173	3	j	j	PROPN
ejpam-4687	173	4	≤	≤	PROPN
ejpam-4687	173	5	k	k	PROPN
ejpam-4687	173	6	,	,	PUNCT
ejpam-4687	173	7	where	where	SCONJ
ejpam-4687	173	8	j	j	PROPN
ejpam-4687	173	9	̸=	̸=	PROPN
ejpam-4687	173	10	i	i	PROPN
ejpam-4687	173	11	,	,	PUNCT
ejpam-4687	173	12	such	such	ADJ
ejpam-4687	173	13	that	that	SCONJ
ejpam-4687	173	14	vivj	vivj	PROPN
ejpam-4687	173	15	∈	∈	PROPN
ejpam-4687	173	16	e(g	e(g	PROPN
ejpam-4687	173	17	)	)	PUNCT
ejpam-4687	173	18	.	.	PUNCT
ejpam-4687	174	1	if	if	SCONJ
ejpam-4687	174	2	j	j	PROPN
ejpam-4687	174	3	<	<	X
ejpam-4687	174	4	k	k	X
ejpam-4687	174	5	,	,	PUNCT
ejpam-4687	174	6	then	then	ADV
ejpam-4687	174	7	n2	n2	PROPN
ejpam-4687	174	8	g[vk	g[vk	PROPN
ejpam-4687	174	9	]	]	PUNCT
ejpam-4687	174	10	⊆	⊆	NUM
ejpam-4687	174	11	n2	n2	ADJ
ejpam-4687	174	12	g[vi	g[vi	PROPN
ejpam-4687	174	13	]	]	PUNCT
ejpam-4687	174	14	∪	∪	X
ejpam-4687	174	15	n2	n2	PROPN
ejpam-4687	174	16	g[vj	g[vj	PROPN
ejpam-4687	174	17	]	]	PUNCT
ejpam-4687	174	18	because	because	SCONJ
ejpam-4687	174	19	{	{	PUNCT
ejpam-4687	174	20	vi	vi	NOUN
ejpam-4687	174	21	,	,	PUNCT
ejpam-4687	174	22	vj	vj	ADJ
ejpam-4687	174	23	}	}	PUNCT
ejpam-4687	174	24	is	be	AUX
ejpam-4687	174	25	a	a	DET
ejpam-4687	174	26	hop	hop	NOUN
ejpam-4687	174	27	dominating	dominating	NOUN
ejpam-4687	174	28	set	set	NOUN
ejpam-4687	174	29	of	of	ADP
ejpam-4687	174	30	g	g	NOUN
ejpam-4687	174	31	by	by	ADP
ejpam-4687	174	32	assumption	assumption	NOUN
ejpam-4687	174	33	.	.	PUNCT
ejpam-4687	175	1	thus	thus	ADV
ejpam-4687	175	2	,	,	PUNCT
ejpam-4687	175	3	n2	n2	ADJ
ejpam-4687	175	4	g[vk]\∪	g[vk]\∪	NOUN
ejpam-4687	175	5	k−1	k−1	PROPN
ejpam-4687	175	6	r=1n	r=1n	VERB
ejpam-4687	175	7	2	2	NUM
ejpam-4687	175	8	g[vr	g[vr	NOUN
ejpam-4687	175	9	]	]	X
ejpam-4687	175	10	=	=	SYM
ejpam-4687	175	11	∅	∅	NOUN
ejpam-4687	175	12	,	,	PUNCT
ejpam-4687	175	13	a	a	DET
ejpam-4687	175	14	contradiction	contradiction	NOUN
ejpam-4687	175	15	.	.	PUNCT
ejpam-4687	176	1	therefore	therefore	ADV
ejpam-4687	176	2	,	,	PUNCT
ejpam-4687	176	3	j	j	PROPN
ejpam-4687	176	4	=	=	SYM
ejpam-4687	176	5	k	k	PROPN
ejpam-4687	176	6	and	and	CCONJ
ejpam-4687	176	7	vivk	vivk	PROPN
ejpam-4687	176	8	∈	∈	PROPN
ejpam-4687	176	9	e(g	e(g	PROPN
ejpam-4687	176	10	)	)	PUNCT
ejpam-4687	176	11	for	for	ADP
ejpam-4687	176	12	all	all	PRON
ejpam-4687	176	13	i	i	PRON
ejpam-4687	176	14	∈	∈	PROPN
ejpam-4687	176	15	{	{	PUNCT
ejpam-4687	176	16	1	1	NUM
ejpam-4687	176	17	,	,	PUNCT
ejpam-4687	176	18	2	2	NUM
ejpam-4687	176	19	,	,	PUNCT
ejpam-4687	176	20	.	.	PUNCT
ejpam-4687	176	21	.	.	PUNCT
ejpam-4687	176	22	.	.	PUNCT
ejpam-4687	177	1	,	,	PUNCT
ejpam-4687	177	2	k−	k−	NOUN
ejpam-4687	177	3	1	1	NUM
ejpam-4687	177	4	}	}	PUNCT
ejpam-4687	177	5	.	.	PUNCT
ejpam-4687	178	1	moreover	moreover	ADV
ejpam-4687	178	2	,	,	PUNCT
ejpam-4687	178	3	since	since	SCONJ
ejpam-4687	178	4	g	g	PROPN
ejpam-4687	178	5	is	be	AUX
ejpam-4687	178	6	triangle	triangle	NOUN
ejpam-4687	178	7	-	-	PUNCT
ejpam-4687	178	8	free	free	ADJ
ejpam-4687	178	9	,	,	PUNCT
ejpam-4687	178	10	⟨ŝ⟩	⟨ŝ⟩	PRON
ejpam-4687	178	11	is	be	AUX
ejpam-4687	178	12	a	a	DET
ejpam-4687	178	13	star	star	NOUN
ejpam-4687	178	14	.	.	PUNCT
ejpam-4687	179	1	now	now	ADV
ejpam-4687	179	2	let	let	VERB
ejpam-4687	179	3	w	w	PROPN
ejpam-4687	179	4	∈	∈	PROPN
ejpam-4687	179	5	n2	n2	PROPN
ejpam-4687	179	6	g[v2	g[v2	NOUN
ejpam-4687	179	7	]	]	PUNCT
ejpam-4687	179	8	.	.	PUNCT
ejpam-4687	180	1	suppose	suppose	VERB
ejpam-4687	180	2	w	w	PROPN
ejpam-4687	180	3	/∈	/∈	PROPN
ejpam-4687	180	4	n2	n2	PROPN
ejpam-4687	180	5	g[v1	g[v1	PROPN
ejpam-4687	180	6	]	]	PUNCT
ejpam-4687	180	7	.	.	PUNCT
ejpam-4687	181	1	let	let	VERB
ejpam-4687	181	2	[	[	X
ejpam-4687	181	3	v2	v2	VERB
ejpam-4687	181	4	,	,	PUNCT
ejpam-4687	181	5	z	z	PROPN
ejpam-4687	181	6	,	,	PUNCT
ejpam-4687	181	7	w	w	PROPN
ejpam-4687	181	8	]	]	PUNCT
ejpam-4687	181	9	be	be	AUX
ejpam-4687	181	10	a	a	DET
ejpam-4687	181	11	v2	v2	PROPN
ejpam-4687	181	12	-	-	PUNCT
ejpam-4687	181	13	w	w	NOUN
ejpam-4687	181	14	geodesic	geodesic	NOUN
ejpam-4687	181	15	.	.	PUNCT
ejpam-4687	182	1	since	since	SCONJ
ejpam-4687	182	2	{	{	PUNCT
ejpam-4687	182	3	v1	v1	NOUN
ejpam-4687	182	4	,	,	PUNCT
ejpam-4687	182	5	vk	vk	PROPN
ejpam-4687	182	6	}	}	PUNCT
ejpam-4687	182	7	is	be	AUX
ejpam-4687	182	8	a	a	DET
ejpam-4687	182	9	hop	hop	NOUN
ejpam-4687	182	10	dominating	dominating	NOUN
ejpam-4687	182	11	set	set	NOUN
ejpam-4687	182	12	of	of	ADP
ejpam-4687	182	13	g	g	PROPN
ejpam-4687	182	14	and	and	CCONJ
ejpam-4687	182	15	w	w	PROPN
ejpam-4687	182	16	∈	∈	PROPN
ejpam-4687	182	17	n2	n2	PROPN
ejpam-4687	182	18	g[v1	g[v1	PROPN
ejpam-4687	182	19	]	]	PUNCT
ejpam-4687	182	20	,	,	PUNCT
ejpam-4687	182	21	it	it	PRON
ejpam-4687	182	22	follows	follow	VERB
ejpam-4687	182	23	that	that	PRON
ejpam-4687	182	24	w	w	PROPN
ejpam-4687	182	25	/∈	/∈	PUNCT
ejpam-4687	182	26	n2	n2	PROPN
ejpam-4687	182	27	g[vk	g[vk	PROPN
ejpam-4687	182	28	]	]	PUNCT
ejpam-4687	182	29	.	.	PUNCT
ejpam-4687	183	1	let	let	VERB
ejpam-4687	183	2	[	[	X
ejpam-4687	183	3	vk	vk	ADP
ejpam-4687	183	4	,	,	PUNCT
ejpam-4687	183	5	y	y	PROPN
ejpam-4687	183	6	,	,	PUNCT
ejpam-4687	183	7	w	w	PROPN
ejpam-4687	183	8	]	]	PUNCT
ejpam-4687	183	9	be	be	AUX
ejpam-4687	183	10	a	a	DET
ejpam-4687	183	11	vk	vk	NOUN
ejpam-4687	183	12	-	-	PUNCT
ejpam-4687	183	13	w	w	NOUN
ejpam-4687	183	14	geodesic	geodesic	NOUN
ejpam-4687	183	15	.	.	PUNCT
ejpam-4687	184	1	since	since	SCONJ
ejpam-4687	184	2	v2vk	v2vk	PROPN
ejpam-4687	184	3	∈	∈	PROPN
ejpam-4687	184	4	e(g	e(g	PROPN
ejpam-4687	184	5	)	)	PUNCT
ejpam-4687	184	6	and	and	CCONJ
ejpam-4687	184	7	g	g	PROPN
ejpam-4687	184	8	is	be	AUX
ejpam-4687	184	9	tringle	tringle	NOUN
ejpam-4687	184	10	-	-	PUNCT
ejpam-4687	184	11	free	free	ADJ
ejpam-4687	184	12	,	,	PUNCT
ejpam-4687	184	13	zvk	zvk	PROPN
ejpam-4687	184	14	/∈	/∈	PUNCT
ejpam-4687	185	1	e(g	e(g	PROPN
ejpam-4687	185	2	)	)	PUNCT
ejpam-4687	185	3	;	;	PUNCT
ejpam-4687	185	4	hence	hence	ADV
ejpam-4687	185	5	,	,	PUNCT
ejpam-4687	185	6	y	y	PROPN
ejpam-4687	185	7	̸=	̸=	PROPN
ejpam-4687	185	8	z.	z.	PROPN
ejpam-4687	186	1	this	this	PRON
ejpam-4687	186	2	implies	imply	VERB
ejpam-4687	186	3	that	that	SCONJ
ejpam-4687	186	4	[	[	X
ejpam-4687	186	5	vk	vk	X
ejpam-4687	186	6	,	,	PUNCT
ejpam-4687	186	7	v2	v2	PROPN
ejpam-4687	186	8	,	,	PUNCT
ejpam-4687	186	9	z	z	PROPN
ejpam-4687	186	10	,	,	PUNCT
ejpam-4687	186	11	w	w	PROPN
ejpam-4687	186	12	,	,	PUNCT
ejpam-4687	186	13	y	y	PROPN
ejpam-4687	186	14	,	,	PUNCT
ejpam-4687	186	15	vk	vk	PROPN
ejpam-4687	186	16	]	]	PUNCT
ejpam-4687	186	17	is	be	AUX
ejpam-4687	186	18	an	an	DET
ejpam-4687	186	19	induced	induced	ADJ
ejpam-4687	186	20	cycle	cycle	NOUN
ejpam-4687	186	21	of	of	ADP
ejpam-4687	186	22	g	g	NOUN
ejpam-4687	186	23	of	of	ADP
ejpam-4687	186	24	order	order	NOUN
ejpam-4687	186	25	5	5	NUM
ejpam-4687	186	26	,	,	PUNCT
ejpam-4687	186	27	a	a	DET
ejpam-4687	186	28	contradiction	contradiction	NOUN
ejpam-4687	186	29	to	to	ADP
ejpam-4687	186	30	an	an	DET
ejpam-4687	186	31	assumption	assumption	NOUN
ejpam-4687	186	32	.	.	PUNCT
ejpam-4687	187	1	thus	thus	ADV
ejpam-4687	187	2	,	,	PUNCT
ejpam-4687	187	3	w	w	PROPN
ejpam-4687	187	4	∈	∈	PROPN
ejpam-4687	187	5	n2	n2	PROPN
ejpam-4687	187	6	g[v1	g[v1	PROPN
ejpam-4687	187	7	]	]	PUNCT
ejpam-4687	187	8	,	,	PUNCT
ejpam-4687	187	9	implying	imply	VERB
ejpam-4687	187	10	that	that	DET
ejpam-4687	187	11	n2	n2	PROPN
ejpam-4687	187	12	g[v2	g[v2	PROPN
ejpam-4687	187	13	]	]	PUNCT
ejpam-4687	187	14	⊆	⊆	NUM
ejpam-4687	187	15	n2	n2	ADJ
ejpam-4687	187	16	g[v1	g[v1	PROPN
ejpam-4687	187	17	]	]	PUNCT
ejpam-4687	187	18	.	.	PUNCT
ejpam-4687	188	1	this	this	PRON
ejpam-4687	188	2	contradicts	contradict	VERB
ejpam-4687	188	3	the	the	DET
ejpam-4687	188	4	legality	legality	NOUN
ejpam-4687	188	5	property	property	NOUN
ejpam-4687	188	6	of	of	ADP
ejpam-4687	188	7	s.	s.	PROPN
ejpam-4687	188	8	therefore	therefore	ADV
ejpam-4687	188	9	,	,	PUNCT
ejpam-4687	188	10	k	k	PROPN
ejpam-4687	188	11	=	=	SYM
ejpam-4687	188	12	2	2	NUM
ejpam-4687	188	13	,	,	PUNCT
ejpam-4687	188	14	i.e.	i.e.	X
ejpam-4687	188	15	,	,	PUNCT
ejpam-4687	188	16	γchgr	γchgr	X
ejpam-4687	188	17	(	(	PUNCT
ejpam-4687	188	18	g	g	NOUN
ejpam-4687	188	19	)	)	PUNCT
ejpam-4687	188	20	=	=	SYM
ejpam-4687	188	21	2	2	X
ejpam-4687	188	22	.	.	PUNCT
ejpam-4687	188	23	(	(	PUNCT
ejpam-4687	188	24	iii	iii	NOUN
ejpam-4687	188	25	)	)	PUNCT
ejpam-4687	188	26	suppose	suppose	VERB
ejpam-4687	188	27	γchgr	γchgr	ADJ
ejpam-4687	188	28	(	(	PUNCT
ejpam-4687	188	29	g	g	NOUN
ejpam-4687	188	30	)	)	PUNCT
ejpam-4687	188	31	=	=	VERB
ejpam-4687	188	32	n.	n.	NOUN
ejpam-4687	188	33	then	then	ADV
ejpam-4687	188	34	by	by	ADP
ejpam-4687	188	35	remark	remark	NOUN
ejpam-4687	188	36	2	2	NUM
ejpam-4687	188	37	,	,	PUNCT
ejpam-4687	188	38	γhgr(g	γhgr(g	NOUN
ejpam-4687	188	39	)	)	PUNCT
ejpam-4687	189	1	=	=	VERB
ejpam-4687	189	2	n.	n.	NOUN
ejpam-4687	189	3	since	since	SCONJ
ejpam-4687	189	4	g	g	PROPN
ejpam-4687	189	5	is	be	AUX
ejpam-4687	189	6	connected	connect	VERB
ejpam-4687	189	7	,	,	PUNCT
ejpam-4687	189	8	it	it	PRON
ejpam-4687	189	9	follows	follow	VERB
ejpam-4687	189	10	that	that	SCONJ
ejpam-4687	189	11	g	g	PROPN
ejpam-4687	189	12	is	be	AUX
ejpam-4687	189	13	complete	complete	ADJ
ejpam-4687	189	14	by	by	ADP
ejpam-4687	189	15	theorem	theorem	NOUN
ejpam-4687	189	16	1	1	NUM
ejpam-4687	189	17	.	.	PUNCT
ejpam-4687	189	18	j.	j.	PROPN
ejpam-4687	189	19	hassan	hassan	PROPN
ejpam-4687	189	20	,	,	PUNCT
ejpam-4687	189	21	s.	s.	PROPN
ejpam-4687	189	22	canoy	canoy	PROPN
ejpam-4687	189	23	jr	jr	PROPN
ejpam-4687	189	24	.	.	PROPN
ejpam-4687	189	25	/	/	SYM
ejpam-4687	189	26	eur	eur	PROPN
ejpam-4687	189	27	.	.	PUNCT
ejpam-4687	190	1	j.	j.	PROPN
ejpam-4687	190	2	pure	pure	PROPN
ejpam-4687	190	3	appl	appl	PROPN
ejpam-4687	190	4	.	.	PROPN
ejpam-4687	190	5	math	math	PROPN
ejpam-4687	190	6	,	,	PUNCT
ejpam-4687	190	7	16	16	NUM
ejpam-4687	190	8	(	(	PUNCT
ejpam-4687	190	9	2	2	NUM
ejpam-4687	190	10	)	)	PUNCT
ejpam-4687	190	11	(	(	PUNCT
ejpam-4687	190	12	2023	2023	NUM
ejpam-4687	190	13	)	)	PUNCT
ejpam-4687	190	14	,	,	PUNCT
ejpam-4687	190	15	1212	1212	NUM
ejpam-4687	190	16	-	-	SYM
ejpam-4687	190	17	1227	1227	NUM
ejpam-4687	190	18	1218	1218	NUM
ejpam-4687	190	19	conversely	conversely	ADV
ejpam-4687	190	20	,	,	PUNCT
ejpam-4687	190	21	suppose	suppose	VERB
ejpam-4687	190	22	that	that	SCONJ
ejpam-4687	190	23	g	g	PROPN
ejpam-4687	190	24	is	be	AUX
ejpam-4687	190	25	complete	complete	ADJ
ejpam-4687	190	26	.	.	PUNCT
ejpam-4687	191	1	then	then	ADV
ejpam-4687	191	2	n2	n2	PROPN
ejpam-4687	191	3	g[u	g[u	PROPN
ejpam-4687	191	4	]	]	X
ejpam-4687	191	5	=	=	SYM
ejpam-4687	191	6	{	{	PUNCT
ejpam-4687	191	7	u	u	NOUN
ejpam-4687	191	8	}	}	PUNCT
ejpam-4687	191	9	for	for	ADP
ejpam-4687	191	10	each	each	DET
ejpam-4687	191	11	u	u	PROPN
ejpam-4687	191	12	∈	∈	PROPN
ejpam-4687	191	13	v	v	NOUN
ejpam-4687	191	14	(	(	PUNCT
ejpam-4687	191	15	g	g	NOUN
ejpam-4687	191	16	)	)	PUNCT
ejpam-4687	191	17	.	.	PUNCT
ejpam-4687	192	1	let	let	VERB
ejpam-4687	192	2	v	v	X
ejpam-4687	192	3	(	(	PUNCT
ejpam-4687	192	4	g	g	NOUN
ejpam-4687	192	5	)	)	PUNCT
ejpam-4687	192	6	=	=	SYM
ejpam-4687	192	7	{	{	PUNCT
ejpam-4687	192	8	a1	a1	PROPN
ejpam-4687	192	9	,	,	PUNCT
ejpam-4687	192	10	a2	a2	PROPN
ejpam-4687	192	11	,	,	PUNCT
ejpam-4687	192	12	.	.	PUNCT
ejpam-4687	192	13	.	.	PUNCT
ejpam-4687	193	1	.	.	PUNCT
ejpam-4687	194	1	,	,	PUNCT
ejpam-4687	194	2	an	an	PRON
ejpam-4687	194	3	}	}	PUNCT
ejpam-4687	194	4	.	.	PUNCT
ejpam-4687	195	1	then	then	ADV
ejpam-4687	195	2	n2	n2	PROPN
ejpam-4687	195	3	g[ai	g[ai	PROPN
ejpam-4687	195	4	]	]	PUNCT
ejpam-4687	195	5	\	\	X
ejpam-4687	195	6	∪i−1	∪i−1	PROPN
ejpam-4687	195	7	j=1n	j=1n	PROPN
ejpam-4687	195	8	2	2	NUM
ejpam-4687	195	9	g[aj	g[aj	PROPN
ejpam-4687	195	10	]	]	X
ejpam-4687	196	1	=	=	X
ejpam-4687	196	2	{	{	PUNCT
ejpam-4687	196	3	ai	ai	PROPN
ejpam-4687	196	4	}	}	PUNCT
ejpam-4687	196	5	\	\	NOUN
ejpam-4687	196	6	{	{	PUNCT
ejpam-4687	196	7	aj	aj	PROPN
ejpam-4687	196	8	:	:	PUNCT
ejpam-4687	196	9	j	j	PROPN
ejpam-4687	196	10	̸=	̸=	PROPN
ejpam-4687	196	11	i	i	PRON
ejpam-4687	196	12	}	}	PUNCT
ejpam-4687	196	13	=	=	PRON
ejpam-4687	196	14	{	{	PUNCT
ejpam-4687	196	15	ai	ai	NOUN
ejpam-4687	196	16	}	}	PUNCT
ejpam-4687	196	17	=	=	NOUN
ejpam-4687	196	18	̸	̸	ADJ
ejpam-4687	196	19	∅	∅	NOUN
ejpam-4687	196	20	for	for	ADP
ejpam-4687	196	21	each	each	DET
ejpam-4687	196	22	i	i	PRON
ejpam-4687	196	23	∈	∈	PROPN
ejpam-4687	196	24	{	{	PUNCT
ejpam-4687	196	25	2	2	NUM
ejpam-4687	196	26	,	,	PUNCT
ejpam-4687	196	27	3	3	NUM
ejpam-4687	196	28	,	,	PUNCT
ejpam-4687	196	29	.	.	PUNCT
ejpam-4687	196	30	.	.	PUNCT
ejpam-4687	197	1	.	.	PUNCT
ejpam-4687	197	2	,	,	PUNCT
ejpam-4687	198	1	n	n	CCONJ
ejpam-4687	198	2	}	}	PUNCT
ejpam-4687	198	3	.	.	PUNCT
ejpam-4687	199	1	it	it	PRON
ejpam-4687	199	2	follows	follow	VERB
ejpam-4687	199	3	that	that	SCONJ
ejpam-4687	199	4	(	(	PUNCT
ejpam-4687	199	5	a1	a1	PROPN
ejpam-4687	199	6	,	,	PUNCT
ejpam-4687	199	7	a2	a2	PROPN
ejpam-4687	199	8	,	,	PUNCT
ejpam-4687	199	9	·	·	PUNCT
ejpam-4687	199	10	·	·	PUNCT
ejpam-4687	199	11	·	·	PUNCT
ejpam-4687	199	12	,	,	PUNCT
ejpam-4687	199	13	an	an	PRON
ejpam-4687	199	14	)	)	PUNCT
ejpam-4687	199	15	is	be	AUX
ejpam-4687	199	16	a	a	DET
ejpam-4687	199	17	grundy	grundy	PROPN
ejpam-4687	199	18	hop	hop	NOUN
ejpam-4687	199	19	dominating	dominating	NOUN
ejpam-4687	199	20	sequence	sequence	NOUN
ejpam-4687	199	21	of	of	ADP
ejpam-4687	199	22	g.	g.	PROPN
ejpam-4687	199	23	since	since	SCONJ
ejpam-4687	199	24	g	g	PROPN
ejpam-4687	199	25	is	be	AUX
ejpam-4687	199	26	connected	connect	VERB
ejpam-4687	199	27	,	,	PUNCT
ejpam-4687	199	28	it	it	PRON
ejpam-4687	199	29	follows	follow	VERB
ejpam-4687	199	30	that	that	SCONJ
ejpam-4687	199	31	γchgr	γchgr	NOUN
ejpam-4687	199	32	(	(	PUNCT
ejpam-4687	199	33	g	g	NOUN
ejpam-4687	199	34	)	)	PUNCT
ejpam-4687	199	35	=	=	VERB
ejpam-4687	199	36	n.	n.	NOUN
ejpam-4687	199	37	the	the	DET
ejpam-4687	199	38	next	next	ADJ
ejpam-4687	199	39	results	result	NOUN
ejpam-4687	199	40	are	be	AUX
ejpam-4687	199	41	immediate	immediate	ADJ
ejpam-4687	199	42	from	from	ADP
ejpam-4687	199	43	theorem	theorem	ADJ
ejpam-4687	199	44	2	2	NUM
ejpam-4687	199	45	.	.	PUNCT
ejpam-4687	199	46	corollary	corollary	ADJ
ejpam-4687	199	47	2	2	NUM
ejpam-4687	199	48	.	.	PUNCT
ejpam-4687	200	1	let	let	VERB
ejpam-4687	200	2	t	t	PROPN
ejpam-4687	200	3	be	be	AUX
ejpam-4687	200	4	a	a	DET
ejpam-4687	200	5	non	non	ADJ
ejpam-4687	200	6	-	-	ADJ
ejpam-4687	200	7	trivial	trivial	ADJ
ejpam-4687	200	8	tree	tree	NOUN
ejpam-4687	200	9	.	.	PUNCT
ejpam-4687	201	1	then	then	ADV
ejpam-4687	201	2	γchgr	γchgr	X
ejpam-4687	201	3	(	(	PUNCT
ejpam-4687	201	4	g	g	NOUN
ejpam-4687	201	5	)	)	PUNCT
ejpam-4687	201	6	=	=	SYM
ejpam-4687	201	7	2	2	NUM
ejpam-4687	201	8	if	if	SCONJ
ejpam-4687	201	9	and	and	CCONJ
ejpam-4687	201	10	only	only	ADV
ejpam-4687	201	11	if	if	SCONJ
ejpam-4687	201	12	{	{	PUNCT
ejpam-4687	201	13	a	a	PRON
ejpam-4687	201	14	,	,	PUNCT
ejpam-4687	201	15	b	b	NOUN
ejpam-4687	201	16	}	}	PUNCT
ejpam-4687	201	17	is	be	AUX
ejpam-4687	201	18	a	a	DET
ejpam-4687	201	19	(	(	PUNCT
ejpam-4687	201	20	connected	connected	ADJ
ejpam-4687	201	21	)	)	PUNCT
ejpam-4687	201	22	hop	hop	NOUN
ejpam-4687	201	23	dominating	dominating	NOUN
ejpam-4687	201	24	set	set	NOUN
ejpam-4687	201	25	for	for	ADP
ejpam-4687	201	26	each	each	DET
ejpam-4687	201	27	pair	pair	NOUN
ejpam-4687	201	28	of	of	ADP
ejpam-4687	201	29	adjacent	adjacent	ADJ
ejpam-4687	201	30	vertices	vertex	NOUN
ejpam-4687	201	31	a	a	DET
ejpam-4687	201	32	,	,	PUNCT
ejpam-4687	201	33	b	b	PROPN
ejpam-4687	201	34	∈	∈	PROPN
ejpam-4687	201	35	v	v	NOUN
ejpam-4687	201	36	(	(	PUNCT
ejpam-4687	201	37	t	t	PROPN
ejpam-4687	201	38	)	)	PUNCT
ejpam-4687	201	39	.	.	PUNCT
ejpam-4687	202	1	corollary	corollary	ADJ
ejpam-4687	202	2	3	3	X
ejpam-4687	202	3	.	.	PUNCT
ejpam-4687	203	1	let	let	VERB
ejpam-4687	203	2	g	g	PRON
ejpam-4687	203	3	be	be	AUX
ejpam-4687	203	4	a	a	DET
ejpam-4687	203	5	connected	connected	ADJ
ejpam-4687	203	6	graph	graph	NOUN
ejpam-4687	203	7	on	on	ADP
ejpam-4687	203	8	n	n	DET
ejpam-4687	203	9	vertices	vertex	NOUN
ejpam-4687	203	10	.	.	PUNCT
ejpam-4687	204	1	then	then	ADV
ejpam-4687	204	2	γchgr	γchgr	X
ejpam-4687	204	3	(	(	PUNCT
ejpam-4687	204	4	g	g	NOUN
ejpam-4687	204	5	)	)	PUNCT
ejpam-4687	204	6	≤	≤	NUM
ejpam-4687	204	7	n−	n−	NOUN
ejpam-4687	204	8	1	1	NUM
ejpam-4687	204	9	if	if	SCONJ
ejpam-4687	204	10	and	and	CCONJ
ejpam-4687	204	11	only	only	ADV
ejpam-4687	204	12	if	if	SCONJ
ejpam-4687	204	13	g	g	PROPN
ejpam-4687	204	14	is	be	AUX
ejpam-4687	204	15	non	non	ADJ
ejpam-4687	204	16	-	-	ADJ
ejpam-4687	204	17	complete	complete	ADJ
ejpam-4687	204	18	.	.	PUNCT
ejpam-4687	205	1	proposition	proposition	NOUN
ejpam-4687	205	2	2	2	NUM
ejpam-4687	205	3	.	.	PUNCT
ejpam-4687	206	1	let	let	VERB
ejpam-4687	206	2	n	n	PRON
ejpam-4687	206	3	be	be	AUX
ejpam-4687	206	4	any	any	DET
ejpam-4687	206	5	positive	positive	ADJ
ejpam-4687	206	6	integer	integer	NOUN
ejpam-4687	206	7	.	.	PUNCT
ejpam-4687	207	1	then	then	ADV
ejpam-4687	207	2	each	each	PRON
ejpam-4687	207	3	of	of	ADP
ejpam-4687	207	4	the	the	DET
ejpam-4687	207	5	following	follow	VERB
ejpam-4687	207	6	holds	hold	NOUN
ejpam-4687	207	7	.	.	PUNCT
ejpam-4687	208	1	(	(	PUNCT
ejpam-4687	208	2	i	i	NOUN
ejpam-4687	208	3	)	)	PUNCT
ejpam-4687	208	4	there	there	PRON
ejpam-4687	208	5	exists	exist	VERB
ejpam-4687	208	6	a	a	DET
ejpam-4687	208	7	connected	connected	ADJ
ejpam-4687	208	8	graph	graph	NOUN
ejpam-4687	208	9	g	g	ADP
ejpam-4687	208	10	such	such	ADJ
ejpam-4687	208	11	that	that	SCONJ
ejpam-4687	208	12	γchgr	γchgr	NOUN
ejpam-4687	208	13	(	(	PUNCT
ejpam-4687	208	14	g)−	g)−	PROPN
ejpam-4687	208	15	γch(g	γch(g	PROPN
ejpam-4687	208	16	)	)	PUNCT
ejpam-4687	208	17	=	=	SYM
ejpam-4687	209	1	n.	n.	NOUN
ejpam-4687	209	2	(	(	PUNCT
ejpam-4687	209	3	ii	ii	NOUN
ejpam-4687	209	4	)	)	PUNCT
ejpam-4687	209	5	there	there	PRON
ejpam-4687	209	6	exists	exist	VERB
ejpam-4687	209	7	a	a	DET
ejpam-4687	209	8	connected	connected	ADJ
ejpam-4687	209	9	graph	graph	NOUN
ejpam-4687	209	10	h	h	NOUN
ejpam-4687	210	1	such	such	ADJ
ejpam-4687	210	2	that	that	SCONJ
ejpam-4687	210	3	γhgr(h)−	γhgr(h)−	PROPN
ejpam-4687	210	4	γchgr	γchgr	PROPN
ejpam-4687	210	5	(	(	PUNCT
ejpam-4687	210	6	h	h	NOUN
ejpam-4687	210	7	)	)	PUNCT
ejpam-4687	210	8	=	=	SYM
ejpam-4687	210	9	n.	n.	NOUN
ejpam-4687	210	10	proof	proof	NOUN
ejpam-4687	210	11	.	.	PUNCT
ejpam-4687	211	1	for	for	ADP
ejpam-4687	211	2	(	(	PUNCT
ejpam-4687	211	3	i	i	NOUN
ejpam-4687	211	4	)	)	PUNCT
ejpam-4687	211	5	,	,	PUNCT
ejpam-4687	211	6	consider	consider	VERB
ejpam-4687	211	7	the	the	DET
ejpam-4687	211	8	graph	graph	NOUN
ejpam-4687	211	9	g	g	NOUN
ejpam-4687	211	10	given	give	VERB
ejpam-4687	211	11	in	in	ADP
ejpam-4687	211	12	figure	figure	NOUN
ejpam-4687	211	13	4	4	NUM
ejpam-4687	211	14	.	.	PUNCT
ejpam-4687	212	1	let	let	VERB
ejpam-4687	212	2	s1	s1	PROPN
ejpam-4687	212	3	=	=	PUNCT
ejpam-4687	212	4	{	{	PUNCT
ejpam-4687	212	5	u	u	NOUN
ejpam-4687	212	6	,	,	PUNCT
ejpam-4687	212	7	vn+2	vn+2	PROPN
ejpam-4687	212	8	}	}	PUNCT
ejpam-4687	212	9	and	and	CCONJ
ejpam-4687	212	10	s2	s2	VERB
ejpam-4687	212	11	=	=	SYM
ejpam-4687	212	12	(	(	PUNCT
ejpam-4687	212	13	v1	v1	PROPN
ejpam-4687	212	14	,	,	PUNCT
ejpam-4687	212	15	v2	v2	PROPN
ejpam-4687	212	16	,	,	PUNCT
ejpam-4687	212	17	·	·	PUNCT
ejpam-4687	212	18	·	·	PUNCT
ejpam-4687	212	19	·	·	PUNCT
ejpam-4687	212	20	,	,	PUNCT
ejpam-4687	212	21	vn+2	vn+2	NUM
ejpam-4687	212	22	)	)	PUNCT
ejpam-4687	212	23	.	.	PUNCT
ejpam-4687	213	1	then	then	ADV
ejpam-4687	213	2	s1	s1	PROPN
ejpam-4687	213	3	and	and	CCONJ
ejpam-4687	213	4	s2	s2	PROPN
ejpam-4687	213	5	are	be	AUX
ejpam-4687	213	6	γch	γch	VERB
ejpam-4687	213	7	-	-	PUNCT
ejpam-4687	213	8	set	set	VERB
ejpam-4687	213	9	and	and	CCONJ
ejpam-4687	213	10	γchgr	γchgr	ADJ
ejpam-4687	213	11	-sequence	-sequence	NOUN
ejpam-4687	213	12	of	of	ADP
ejpam-4687	213	13	g	g	NOUN
ejpam-4687	213	14	,	,	PUNCT
ejpam-4687	213	15	respectively	respectively	ADV
ejpam-4687	213	16	.	.	PUNCT
ejpam-4687	214	1	hence	hence	ADV
ejpam-4687	214	2	,	,	PUNCT
ejpam-4687	214	3	γch(g	γch(g	PROPN
ejpam-4687	214	4	)	)	PUNCT
ejpam-4687	214	5	=	=	SYM
ejpam-4687	214	6	2	2	NUM
ejpam-4687	214	7	and	and	CCONJ
ejpam-4687	214	8	γchgr	γchgr	ADJ
ejpam-4687	214	9	(	(	PUNCT
ejpam-4687	214	10	g	g	NOUN
ejpam-4687	214	11	)	)	PUNCT
ejpam-4687	214	12	=	=	PUNCT
ejpam-4687	214	13	n+	n+	PUNCT
ejpam-4687	215	1	2	2	X
ejpam-4687	215	2	.	.	PUNCT
ejpam-4687	215	3	consequently	consequently	ADV
ejpam-4687	215	4	,	,	PUNCT
ejpam-4687	215	5	γchgr	γchgr	PROPN
ejpam-4687	215	6	(	(	PUNCT
ejpam-4687	215	7	g)−	g)−	PROPN
ejpam-4687	215	8	γch(g	γch(g	NOUN
ejpam-4687	215	9	)	)	PUNCT
ejpam-4687	215	10	=	=	SYM
ejpam-4687	215	11	n+	n+	ADP
ejpam-4687	215	12	2−	2−	NUM
ejpam-4687	215	13	2	2	NUM
ejpam-4687	215	14	=	=	NOUN
ejpam-4687	215	15	n.	n.	NOUN
ejpam-4687	215	16	v1	v1	PROPN
ejpam-4687	215	17	v3	v3	PROPN
ejpam-4687	215	18	v2	v2	PROPN
ejpam-4687	215	19	vn+2	vn+2	PROPN
ejpam-4687	215	20	g	g	NOUN
ejpam-4687	215	21	:	:	PUNCT
ejpam-4687	215	22	u	u	PROPN
ejpam-4687	215	23	.	.	PUNCT
ejpam-4687	215	24	.	.	PUNCT
ejpam-4687	215	25	.	.	PUNCT
ejpam-4687	216	1	figure	figure	VERB
ejpam-4687	216	2	4	4	NUM
ejpam-4687	216	3	:	:	PUNCT
ejpam-4687	216	4	a	a	DET
ejpam-4687	216	5	graph	graph	NOUN
ejpam-4687	216	6	g	g	NOUN
ejpam-4687	216	7	with	with	ADP
ejpam-4687	216	8	γch	γch	NOUN
ejpam-4687	216	9	gr	gr	X
ejpam-4687	216	10	(	(	PUNCT
ejpam-4687	216	11	g)−	g)−	PROPN
ejpam-4687	216	12	γch(g	γch(g	NOUN
ejpam-4687	216	13	)	)	PUNCT
ejpam-4687	216	14	=	=	SYM
ejpam-4687	216	15	n	n	CCONJ
ejpam-4687	216	16	for	for	ADP
ejpam-4687	216	17	(	(	PUNCT
ejpam-4687	216	18	ii	ii	NOUN
ejpam-4687	216	19	)	)	PUNCT
ejpam-4687	216	20	,	,	PUNCT
ejpam-4687	216	21	consider	consider	VERB
ejpam-4687	216	22	the	the	DET
ejpam-4687	216	23	graph	graph	NOUN
ejpam-4687	216	24	h	h	NOUN
ejpam-4687	216	25	given	give	VERB
ejpam-4687	216	26	in	in	ADP
ejpam-4687	216	27	figure	figure	NOUN
ejpam-4687	216	28	5	5	NUM
ejpam-4687	216	29	.	.	PUNCT
ejpam-4687	217	1	let	let	VERB
ejpam-4687	217	2	s′	s′	ADJ
ejpam-4687	217	3	=	=	SYM
ejpam-4687	217	4	(	(	PUNCT
ejpam-4687	217	5	v1	v1	PROPN
ejpam-4687	217	6	,	,	PUNCT
ejpam-4687	217	7	u	u	NOUN
ejpam-4687	217	8	,	,	PUNCT
ejpam-4687	217	9	w	w	NOUN
ejpam-4687	217	10	)	)	PUNCT
ejpam-4687	217	11	and	and	CCONJ
ejpam-4687	217	12	s′′	s′′	PROPN
ejpam-4687	217	13	=	=	SYM
ejpam-4687	217	14	(	(	PUNCT
ejpam-4687	217	15	v1	v1	PROPN
ejpam-4687	217	16	,	,	PUNCT
ejpam-4687	217	17	v2	v2	NOUN
ejpam-4687	217	18	,	,	PUNCT
ejpam-4687	217	19	.	.	PUNCT
ejpam-4687	217	20	.	.	PUNCT
ejpam-4687	218	1	.	.	PUNCT
ejpam-4687	219	1	,	,	PUNCT
ejpam-4687	219	2	vn+2	vn+2	X
ejpam-4687	219	3	,	,	PUNCT
ejpam-4687	219	4	u	u	NOUN
ejpam-4687	219	5	)	)	PUNCT
ejpam-4687	219	6	.	.	PUNCT
ejpam-4687	220	1	then	then	ADV
ejpam-4687	220	2	s′	s′	NUM
ejpam-4687	220	3	and	and	CCONJ
ejpam-4687	220	4	s′′	s′′	PROPN
ejpam-4687	220	5	are	be	AUX
ejpam-4687	220	6	γchgr	γchgr	ADJ
ejpam-4687	220	7	and	and	CCONJ
ejpam-4687	220	8	γhgr	γhgr	ADJ
ejpam-4687	220	9	-	-	PUNCT
ejpam-4687	220	10	sequences	sequence	NOUN
ejpam-4687	220	11	of	of	ADP
ejpam-4687	220	12	h	h	NOUN
ejpam-4687	220	13	,	,	PUNCT
ejpam-4687	220	14	respectively	respectively	ADV
ejpam-4687	220	15	.	.	PUNCT
ejpam-4687	221	1	therefore	therefore	ADV
ejpam-4687	221	2	,	,	PUNCT
ejpam-4687	221	3	γchgr	γchgr	X
ejpam-4687	221	4	(	(	PUNCT
ejpam-4687	221	5	h	h	NOUN
ejpam-4687	221	6	)	)	PUNCT
ejpam-4687	221	7	=	=	SYM
ejpam-4687	221	8	3	3	NUM
ejpam-4687	221	9	and	and	CCONJ
ejpam-4687	221	10	γhgr(h	γhgr(h	NOUN
ejpam-4687	221	11	)	)	PUNCT
ejpam-4687	221	12	=	=	SYM
ejpam-4687	221	13	n+3	n+3	PROPN
ejpam-4687	221	14	.	.	PUNCT
ejpam-4687	222	1	consequently	consequently	ADV
ejpam-4687	222	2	,	,	PUNCT
ejpam-4687	222	3	γhgr(h)−γchgr	γhgr(h)−γchgr	NUM
ejpam-4687	222	4	(	(	PUNCT
ejpam-4687	222	5	h	h	NOUN
ejpam-4687	222	6	)	)	PUNCT
ejpam-4687	222	7	=	=	PUNCT
ejpam-4687	223	1	n+3−3	n+3−3	PROPN
ejpam-4687	223	2	=	=	PUNCT
ejpam-4687	223	3	n.	n.	PROPN
ejpam-4687	223	4	j.	j.	PROPN
ejpam-4687	223	5	hassan	hassan	PROPN
ejpam-4687	223	6	,	,	PUNCT
ejpam-4687	223	7	s.	s.	PROPN
ejpam-4687	223	8	canoy	canoy	PROPN
ejpam-4687	223	9	jr	jr	PROPN
ejpam-4687	223	10	.	.	PROPN
ejpam-4687	223	11	/	/	SYM
ejpam-4687	223	12	eur	eur	PROPN
ejpam-4687	223	13	.	.	PUNCT
ejpam-4687	224	1	j.	j.	PROPN
ejpam-4687	224	2	pure	pure	PROPN
ejpam-4687	224	3	appl	appl	PROPN
ejpam-4687	224	4	.	.	PROPN
ejpam-4687	224	5	math	math	PROPN
ejpam-4687	224	6	,	,	PUNCT
ejpam-4687	224	7	16	16	NUM
ejpam-4687	224	8	(	(	PUNCT
ejpam-4687	224	9	2	2	NUM
ejpam-4687	224	10	)	)	PUNCT
ejpam-4687	224	11	(	(	PUNCT
ejpam-4687	224	12	2023	2023	NUM
ejpam-4687	224	13	)	)	PUNCT
ejpam-4687	224	14	,	,	PUNCT
ejpam-4687	224	15	1212	1212	NUM
ejpam-4687	224	16	-	-	SYM
ejpam-4687	224	17	1227	1227	NUM
ejpam-4687	224	18	1219	1219	NUM
ejpam-4687	224	19	h	h	NOUN
ejpam-4687	224	20	:	:	PUNCT
ejpam-4687	224	21	.	.	PUNCT
ejpam-4687	224	22	.	.	PUNCT
ejpam-4687	224	23	.	.	PUNCT
ejpam-4687	225	1	v1	v1	PROPN
ejpam-4687	225	2	v2	v2	PROPN
ejpam-4687	225	3	v3	v3	PROPN
ejpam-4687	225	4	vn+2	vn+2	PROPN
ejpam-4687	225	5	vn+1	vn+1	PROPN
ejpam-4687	225	6	u	u	PROPN
ejpam-4687	225	7	w	w	NOUN
ejpam-4687	225	8	figure	figure	NOUN
ejpam-4687	225	9	5	5	NUM
ejpam-4687	225	10	:	:	PUNCT
ejpam-4687	225	11	a	a	DET
ejpam-4687	225	12	graph	graph	NOUN
ejpam-4687	225	13	h	h	NOUN
ejpam-4687	225	14	with	with	ADP
ejpam-4687	225	15	γh	γh	PROPN
ejpam-4687	225	16	gr(h)−	gr(h)−	PROPN
ejpam-4687	225	17	γch	γch	PUNCT
ejpam-4687	225	18	gr	gr	PROPN
ejpam-4687	225	19	(	(	PUNCT
ejpam-4687	225	20	h	h	NOUN
ejpam-4687	225	21	)	)	PUNCT
ejpam-4687	225	22	=	=	SYM
ejpam-4687	226	1	n	n	CCONJ
ejpam-4687	226	2	this	this	PRON
ejpam-4687	226	3	proves	prove	VERB
ejpam-4687	226	4	the	the	DET
ejpam-4687	226	5	assertion	assertion	NOUN
ejpam-4687	226	6	.	.	PUNCT
ejpam-4687	227	1	corollary	corollary	ADJ
ejpam-4687	227	2	4	4	NUM
ejpam-4687	227	3	.	.	PUNCT
ejpam-4687	228	1	let	let	VERB
ejpam-4687	228	2	g	g	PRON
ejpam-4687	228	3	be	be	AUX
ejpam-4687	228	4	a	a	DET
ejpam-4687	228	5	connected	connected	ADJ
ejpam-4687	228	6	graph	graph	NOUN
ejpam-4687	228	7	.	.	PUNCT
ejpam-4687	229	1	then	then	ADV
ejpam-4687	229	2	γchgr	γchgr	X
ejpam-4687	229	3	(	(	PUNCT
ejpam-4687	229	4	g	g	NOUN
ejpam-4687	229	5	)	)	PUNCT
ejpam-4687	229	6	−	−	PROPN
ejpam-4687	229	7	γch(g	γch(g	NOUN
ejpam-4687	229	8	)	)	PUNCT
ejpam-4687	229	9	and	and	CCONJ
ejpam-4687	229	10	γhgr(g	γhgr(g	NUM
ejpam-4687	229	11	)	)	PUNCT
ejpam-4687	229	12	−	−	PROPN
ejpam-4687	230	1	γchgr	γchgr	ADJ
ejpam-4687	230	2	(	(	PUNCT
ejpam-4687	230	3	g	g	NOUN
ejpam-4687	230	4	)	)	PUNCT
ejpam-4687	230	5	can	can	AUX
ejpam-4687	230	6	be	be	AUX
ejpam-4687	230	7	made	make	VERB
ejpam-4687	230	8	arbitrarily	arbitrarily	ADV
ejpam-4687	230	9	large	large	ADJ
ejpam-4687	230	10	.	.	PUNCT
ejpam-4687	231	1	next	next	ADV
ejpam-4687	231	2	,	,	PUNCT
ejpam-4687	231	3	we	we	PRON
ejpam-4687	231	4	give	give	VERB
ejpam-4687	231	5	a	a	DET
ejpam-4687	231	6	more	more	ADV
ejpam-4687	231	7	general	general	ADJ
ejpam-4687	231	8	result	result	NOUN
ejpam-4687	231	9	(	(	PUNCT
ejpam-4687	231	10	than	than	ADP
ejpam-4687	231	11	proposition	proposition	NOUN
ejpam-4687	231	12	2	2	NUM
ejpam-4687	231	13	)	)	PUNCT
ejpam-4687	231	14	involving	involve	VERB
ejpam-4687	231	15	the	the	DET
ejpam-4687	231	16	connected	connected	ADJ
ejpam-4687	231	17	hop	hop	NOUN
ejpam-4687	231	18	domination	domination	NOUN
ejpam-4687	231	19	,	,	PUNCT
ejpam-4687	231	20	connected	connect	VERB
ejpam-4687	231	21	grundy	grundy	PROPN
ejpam-4687	231	22	hop	hop	PROPN
ejpam-4687	231	23	domination	domination	PROPN
ejpam-4687	231	24	and	and	CCONJ
ejpam-4687	231	25	grundy	grundy	PROPN
ejpam-4687	231	26	hop	hop	PROPN
ejpam-4687	231	27	domination	domination	PROPN
ejpam-4687	231	28	parameters	parameter	NOUN
ejpam-4687	231	29	.	.	PUNCT
ejpam-4687	232	1	theorem	theorem	NOUN
ejpam-4687	232	2	3	3	X
ejpam-4687	232	3	.	.	PUNCT
ejpam-4687	233	1	let	let	VERB
ejpam-4687	233	2	a	a	PRON
ejpam-4687	233	3	and	and	CCONJ
ejpam-4687	233	4	b	b	NOUN
ejpam-4687	233	5	be	be	AUX
ejpam-4687	233	6	positive	positive	ADJ
ejpam-4687	233	7	integers	integer	NOUN
ejpam-4687	233	8	such	such	ADJ
ejpam-4687	233	9	that	that	SCONJ
ejpam-4687	233	10	3	3	NUM
ejpam-4687	233	11	≤	≤	NOUN
ejpam-4687	233	12	a	a	DET
ejpam-4687	233	13	≤	≤	PROPN
ejpam-4687	233	14	b.	b.	NOUN
ejpam-4687	233	15	then	then	ADV
ejpam-4687	233	16	each	each	PRON
ejpam-4687	233	17	of	of	ADP
ejpam-4687	233	18	the	the	DET
ejpam-4687	233	19	following	follow	VERB
ejpam-4687	233	20	holds	hold	NOUN
ejpam-4687	233	21	.	.	PUNCT
ejpam-4687	234	1	(	(	PUNCT
ejpam-4687	234	2	i	i	NOUN
ejpam-4687	234	3	)	)	PUNCT
ejpam-4687	234	4	there	there	PRON
ejpam-4687	234	5	exists	exist	VERB
ejpam-4687	234	6	a	a	DET
ejpam-4687	234	7	connected	connected	ADJ
ejpam-4687	234	8	graph	graph	NOUN
ejpam-4687	234	9	g	g	ADP
ejpam-4687	234	10	such	such	ADJ
ejpam-4687	234	11	that	that	DET
ejpam-4687	234	12	γch(g	γch(g	NOUN
ejpam-4687	234	13	)	)	PUNCT
ejpam-4687	234	14	=	=	SYM
ejpam-4687	234	15	a	a	PRON
ejpam-4687	234	16	and	and	CCONJ
ejpam-4687	234	17	γchgr	γchgr	ADJ
ejpam-4687	234	18	(	(	PUNCT
ejpam-4687	234	19	g	g	NOUN
ejpam-4687	234	20	)	)	PUNCT
ejpam-4687	234	21	=	=	SYM
ejpam-4687	234	22	b.	b.	PROPN
ejpam-4687	234	23	(	(	PUNCT
ejpam-4687	234	24	ii	ii	PROPN
ejpam-4687	234	25	)	)	PUNCT
ejpam-4687	234	26	there	there	PRON
ejpam-4687	234	27	exists	exist	VERB
ejpam-4687	234	28	a	a	DET
ejpam-4687	234	29	connected	connected	ADJ
ejpam-4687	234	30	graph	graph	NOUN
ejpam-4687	234	31	g′	g′	NOUN
ejpam-4687	234	32	such	such	ADJ
ejpam-4687	234	33	that	that	SCONJ
ejpam-4687	234	34	γchgr	γchgr	NOUN
ejpam-4687	234	35	(	(	PUNCT
ejpam-4687	234	36	g	g	PROPN
ejpam-4687	234	37	′	′	NUM
ejpam-4687	234	38	)	)	PUNCT
ejpam-4687	234	39	=	=	PUNCT
ejpam-4687	235	1	a	a	PRON
ejpam-4687	235	2	and	and	CCONJ
ejpam-4687	235	3	γhgr(g	γhgr(g	NUM
ejpam-4687	235	4	′	′	NUM
ejpam-4687	235	5	)	)	PUNCT
ejpam-4687	236	1	=	=	SYM
ejpam-4687	236	2	b.	b.	PROPN
ejpam-4687	236	3	proof	proof	NOUN
ejpam-4687	236	4	.	.	PUNCT
ejpam-4687	237	1	for	for	ADP
ejpam-4687	237	2	a	a	DET
ejpam-4687	237	3	=	=	SYM
ejpam-4687	237	4	b	b	NOUN
ejpam-4687	237	5	,	,	PUNCT
ejpam-4687	237	6	consider	consider	VERB
ejpam-4687	237	7	g	g	PROPN
ejpam-4687	237	8	=	=	SYM
ejpam-4687	237	9	ka	ka	PROPN
ejpam-4687	237	10	.	.	PUNCT
ejpam-4687	238	1	then	then	ADV
ejpam-4687	238	2	γch(g	γch(g	NOUN
ejpam-4687	238	3	)	)	PUNCT
ejpam-4687	238	4	=	=	PUNCT
ejpam-4687	239	1	a	a	DET
ejpam-4687	239	2	=	=	X
ejpam-4687	239	3	γchgr	γchgr	ADJ
ejpam-4687	239	4	(	(	PUNCT
ejpam-4687	239	5	g	g	NOUN
ejpam-4687	239	6	)	)	PUNCT
ejpam-4687	239	7	=	=	SYM
ejpam-4687	240	1	γhgr(g	γhgr(g	NOUN
ejpam-4687	240	2	)	)	PUNCT
ejpam-4687	240	3	.	.	PUNCT
ejpam-4687	241	1	suppose	suppose	VERB
ejpam-4687	241	2	now	now	ADV
ejpam-4687	241	3	that	that	SCONJ
ejpam-4687	241	4	a	a	DET
ejpam-4687	241	5	<	<	X
ejpam-4687	241	6	b.	b.	PROPN
ejpam-4687	241	7	for	for	ADP
ejpam-4687	241	8	(	(	PUNCT
ejpam-4687	241	9	i	i	NOUN
ejpam-4687	241	10	)	)	PUNCT
ejpam-4687	241	11	,	,	PUNCT
ejpam-4687	241	12	let	let	VERB
ejpam-4687	241	13	m	m	VERB
ejpam-4687	241	14	=	=	VERB
ejpam-4687	241	15	b−	b−	PROPN
ejpam-4687	241	16	a	a	PRON
ejpam-4687	241	17	and	and	CCONJ
ejpam-4687	241	18	consider	consider	VERB
ejpam-4687	241	19	the	the	DET
ejpam-4687	241	20	following	follow	VERB
ejpam-4687	241	21	cases	case	NOUN
ejpam-4687	241	22	:	:	PUNCT
ejpam-4687	241	23	case	case	NOUN
ejpam-4687	241	24	1	1	NUM
ejpam-4687	241	25	:	:	PUNCT
ejpam-4687	241	26	a	a	PRON
ejpam-4687	241	27	is	be	AUX
ejpam-4687	241	28	odd	odd	ADJ
ejpam-4687	241	29	.	.	PUNCT
ejpam-4687	242	1	consider	consider	VERB
ejpam-4687	242	2	the	the	DET
ejpam-4687	242	3	graph	graph	NOUN
ejpam-4687	242	4	g	g	NOUN
ejpam-4687	242	5	given	give	VERB
ejpam-4687	242	6	in	in	ADP
ejpam-4687	242	7	figure	figure	NOUN
ejpam-4687	242	8	6	6	NUM
ejpam-4687	242	9	,	,	PUNCT
ejpam-4687	242	10	where	where	SCONJ
ejpam-4687	242	11	⟨{y1	⟨{y1	PROPN
ejpam-4687	242	12	,	,	PUNCT
ejpam-4687	242	13	y2	y2	PROPN
ejpam-4687	242	14	,	,	PUNCT
ejpam-4687	242	15	·	·	PUNCT
ejpam-4687	242	16	·	·	PUNCT
ejpam-4687	242	17	·	·	PUNCT
ejpam-4687	242	18	,	,	PUNCT
ejpam-4687	242	19	ym−1	ym−1	PROPN
ejpam-4687	242	20	,	,	PUNCT
ejpam-4687	242	21	w}⟩	w}⟩	PROPN
ejpam-4687	242	22	is	be	AUX
ejpam-4687	242	23	complete	complete	ADJ
ejpam-4687	242	24	.	.	PUNCT
ejpam-4687	243	1	one	one	PRON
ejpam-4687	243	2	can	can	AUX
ejpam-4687	243	3	verify	verify	VERB
ejpam-4687	243	4	that	that	DET
ejpam-4687	243	5	s1	s1	NOUN
ejpam-4687	243	6	=	=	SYM
ejpam-4687	243	7	(	(	PUNCT
ejpam-4687	243	8	v1	v1	PROPN
ejpam-4687	243	9	,	,	PUNCT
ejpam-4687	243	10	v2	v2	NOUN
ejpam-4687	243	11	,	,	PUNCT
ejpam-4687	243	12	.	.	PUNCT
ejpam-4687	243	13	.	.	PUNCT
ejpam-4687	244	1	.	.	PUNCT
ejpam-4687	245	1	,	,	PUNCT
ejpam-4687	245	2	va	va	NOUN
ejpam-4687	245	3	}	}	PUNCT
ejpam-4687	245	4	and	and	CCONJ
ejpam-4687	245	5	s2	s2	PROPN
ejpam-4687	245	6	=	=	SYM
ejpam-4687	245	7	(	(	PUNCT
ejpam-4687	245	8	y1	y1	PROPN
ejpam-4687	245	9	,	,	PUNCT
ejpam-4687	245	10	y2	y2	PROPN
ejpam-4687	245	11	,	,	PUNCT
ejpam-4687	245	12	·	·	PUNCT
ejpam-4687	245	13	·	·	PUNCT
ejpam-4687	245	14	·	·	PUNCT
ejpam-4687	245	15	,	,	PUNCT
ejpam-4687	245	16	ym−1	ym−1	PROPN
ejpam-4687	245	17	,	,	PUNCT
ejpam-4687	245	18	w	w	PROPN
ejpam-4687	245	19	,	,	PUNCT
ejpam-4687	245	20	va	va	NOUN
ejpam-4687	245	21	,	,	PUNCT
ejpam-4687	245	22	va−1	va−1	NOUN
ejpam-4687	245	23	·	·	PUNCT
ejpam-4687	245	24	·	·	PUNCT
ejpam-4687	245	25	·	·	PUNCT
ejpam-4687	245	26	,	,	PUNCT
ejpam-4687	245	27	v1	v1	NOUN
ejpam-4687	245	28	)	)	PUNCT
ejpam-4687	245	29	are	be	AUX
ejpam-4687	245	30	γchand	γchand	X
ejpam-4687	245	31	γchgr	γchgr	ADJ
ejpam-4687	245	32	-sequences	-sequence	NOUN
ejpam-4687	245	33	of	of	ADP
ejpam-4687	245	34	g	g	NOUN
ejpam-4687	245	35	,	,	PUNCT
ejpam-4687	245	36	respectively	respectively	ADV
ejpam-4687	245	37	.	.	PUNCT
ejpam-4687	246	1	hence	hence	ADV
ejpam-4687	246	2	,	,	PUNCT
ejpam-4687	246	3	γch(g	γch(g	NOUN
ejpam-4687	246	4	)	)	PUNCT
ejpam-4687	246	5	=	=	SYM
ejpam-4687	246	6	a	a	PRON
ejpam-4687	246	7	and	and	CCONJ
ejpam-4687	246	8	γchgr	γchgr	ADJ
ejpam-4687	246	9	(	(	PUNCT
ejpam-4687	246	10	g	g	NOUN
ejpam-4687	246	11	)	)	PUNCT
ejpam-4687	246	12	=	=	NOUN
ejpam-4687	246	13	m+a	m+a	PROPN
ejpam-4687	246	14	=	=	SYM
ejpam-4687	246	15	b.	b.	PROPN
ejpam-4687	246	16	j.	j.	PROPN
ejpam-4687	246	17	hassan	hassan	PROPN
ejpam-4687	246	18	,	,	PUNCT
ejpam-4687	246	19	s.	s.	PROPN
ejpam-4687	246	20	canoy	canoy	PROPN
ejpam-4687	246	21	jr	jr	PROPN
ejpam-4687	246	22	.	.	PROPN
ejpam-4687	246	23	/	/	SYM
ejpam-4687	246	24	eur	eur	PROPN
ejpam-4687	246	25	.	.	PUNCT
ejpam-4687	247	1	j.	j.	PROPN
ejpam-4687	247	2	pure	pure	PROPN
ejpam-4687	247	3	appl	appl	PROPN
ejpam-4687	247	4	.	.	PROPN
ejpam-4687	247	5	math	math	PROPN
ejpam-4687	247	6	,	,	PUNCT
ejpam-4687	247	7	16	16	NUM
ejpam-4687	247	8	(	(	PUNCT
ejpam-4687	247	9	2	2	NUM
ejpam-4687	247	10	)	)	PUNCT
ejpam-4687	247	11	(	(	PUNCT
ejpam-4687	247	12	2023	2023	NUM
ejpam-4687	247	13	)	)	PUNCT
ejpam-4687	247	14	,	,	PUNCT
ejpam-4687	247	15	1212	1212	NUM
ejpam-4687	247	16	-	-	SYM
ejpam-4687	247	17	1227	1227	NUM
ejpam-4687	247	18	1220	1220	NUM
ejpam-4687	247	19	.	.	PUNCT
ejpam-4687	247	20	.	.	PUNCT
ejpam-4687	248	1	.v1	.v1	PUNCT
ejpam-4687	249	1	v2	v2	NOUN
ejpam-4687	249	2	va−1	va−1	PROPN
ejpam-4687	249	3	va	va	PROPN
ejpam-4687	249	4	y1	y1	PROPN
ejpam-4687	249	5	w	w	PROPN
ejpam-4687	249	6	ym−1	ym−1	PROPN
ejpam-4687	249	7	...	...	PUNCT
ejpam-4687	249	8	g	g	NOUN
ejpam-4687	249	9	:	:	PUNCT
ejpam-4687	250	1	y2	y2	PROPN
ejpam-4687	250	2	v3	v3	PROPN
ejpam-4687	250	3	figure	figure	NOUN
ejpam-4687	250	4	6	6	NUM
ejpam-4687	250	5	:	:	PUNCT
ejpam-4687	250	6	a	a	DET
ejpam-4687	250	7	graph	graph	NOUN
ejpam-4687	250	8	g	g	NOUN
ejpam-4687	250	9	with	with	ADP
ejpam-4687	250	10	γch(g	γch(g	NOUN
ejpam-4687	250	11	)	)	PUNCT
ejpam-4687	250	12	<	<	X
ejpam-4687	251	1	γch	γch	X
ejpam-4687	251	2	gr	gr	X
ejpam-4687	251	3	(	(	PUNCT
ejpam-4687	251	4	g	g	NOUN
ejpam-4687	251	5	)	)	PUNCT
ejpam-4687	251	6	case	case	NOUN
ejpam-4687	251	7	2	2	NUM
ejpam-4687	251	8	:	:	PUNCT
ejpam-4687	251	9	a	a	PRON
ejpam-4687	251	10	is	be	AUX
ejpam-4687	251	11	even	even	ADV
ejpam-4687	251	12	.	.	PUNCT
ejpam-4687	252	1	consider	consider	VERB
ejpam-4687	252	2	the	the	DET
ejpam-4687	252	3	graph	graph	NOUN
ejpam-4687	252	4	g′	g′	NOUN
ejpam-4687	252	5	given	give	VERB
ejpam-4687	252	6	in	in	ADP
ejpam-4687	252	7	figure	figure	NOUN
ejpam-4687	252	8	7	7	NUM
ejpam-4687	252	9	,	,	PUNCT
ejpam-4687	252	10	where	where	SCONJ
ejpam-4687	252	11	⟨{y1	⟨{y1	PROPN
ejpam-4687	252	12	,	,	PUNCT
ejpam-4687	252	13	y2	y2	PROPN
ejpam-4687	252	14	,	,	PUNCT
ejpam-4687	252	15	·	·	PUNCT
ejpam-4687	252	16	·	·	PUNCT
ejpam-4687	252	17	·	·	PUNCT
ejpam-4687	252	18	,	,	PUNCT
ejpam-4687	252	19	ym−1	ym−1	PROPN
ejpam-4687	252	20	,	,	PUNCT
ejpam-4687	252	21	w}⟩	w}⟩	PROPN
ejpam-4687	252	22	is	be	AUX
ejpam-4687	252	23	complete	complete	ADJ
ejpam-4687	252	24	.	.	PUNCT
ejpam-4687	253	1	one	one	PRON
ejpam-4687	253	2	can	can	AUX
ejpam-4687	253	3	verify	verify	VERB
ejpam-4687	253	4	that	that	DET
ejpam-4687	253	5	s′	s′	ADJ
ejpam-4687	253	6	=	=	PUNCT
ejpam-4687	253	7	{	{	PUNCT
ejpam-4687	253	8	v1	v1	NOUN
ejpam-4687	253	9	,	,	PUNCT
ejpam-4687	253	10	v2	v2	PROPN
ejpam-4687	253	11	,	,	PUNCT
ejpam-4687	253	12	.	.	PUNCT
ejpam-4687	253	13	.	.	PUNCT
ejpam-4687	254	1	.	.	PUNCT
ejpam-4687	255	1	,	,	PUNCT
ejpam-4687	255	2	va	va	NOUN
ejpam-4687	255	3	}	}	PUNCT
ejpam-4687	255	4	and	and	CCONJ
ejpam-4687	255	5	s′′	s′′	PROPN
ejpam-4687	255	6	=	=	SYM
ejpam-4687	255	7	(	(	PUNCT
ejpam-4687	255	8	y1	y1	PROPN
ejpam-4687	255	9	,	,	PUNCT
ejpam-4687	255	10	y2	y2	PROPN
ejpam-4687	255	11	,	,	PUNCT
ejpam-4687	255	12	.	.	PUNCT
ejpam-4687	255	13	.	.	PUNCT
ejpam-4687	256	1	.	.	PUNCT
ejpam-4687	257	1	,	,	PUNCT
ejpam-4687	257	2	ym−1	ym−1	PROPN
ejpam-4687	257	3	,	,	PUNCT
ejpam-4687	257	4	w	w	PROPN
ejpam-4687	257	5	,	,	PUNCT
ejpam-4687	257	6	va	va	NOUN
ejpam-4687	257	7	,	,	PUNCT
ejpam-4687	257	8	va−1	va−1	PROPN
ejpam-4687	257	9	.	.	PUNCT
ejpam-4687	257	10	.	.	PUNCT
ejpam-4687	258	1	.	.	PUNCT
ejpam-4687	259	1	,	,	PUNCT
ejpam-4687	259	2	v1	v1	NOUN
ejpam-4687	259	3	)	)	PUNCT
ejpam-4687	259	4	are	be	AUX
ejpam-4687	259	5	γchand	γchand	X
ejpam-4687	259	6	γchgr	γchgr	ADJ
ejpam-4687	259	7	-sequences	-sequence	NOUN
ejpam-4687	259	8	of	of	ADP
ejpam-4687	259	9	g	g	NOUN
ejpam-4687	259	10	′	′	NOUN
ejpam-4687	259	11	,	,	PUNCT
ejpam-4687	259	12	respectively	respectively	ADV
ejpam-4687	259	13	.	.	PUNCT
ejpam-4687	260	1	thus	thus	ADV
ejpam-4687	260	2	,	,	PUNCT
ejpam-4687	260	3	γch(g	γch(g	NOUN
ejpam-4687	260	4	′	′	NUM
ejpam-4687	260	5	)	)	PUNCT
ejpam-4687	260	6	=	=	PUNCT
ejpam-4687	260	7	a	a	PROPN
ejpam-4687	260	8	and	and	CCONJ
ejpam-4687	260	9	γchgr	γchgr	ADJ
ejpam-4687	260	10	(	(	PUNCT
ejpam-4687	260	11	g	g	NOUN
ejpam-4687	260	12	′	′	NUM
ejpam-4687	260	13	)	)	PUNCT
ejpam-4687	261	1	=	=	PRON
ejpam-4687	261	2	m+	m+	NUM
ejpam-4687	261	3	a	a	DET
ejpam-4687	261	4	=	=	X
ejpam-4687	261	5	b.	b.	PROPN
ejpam-4687	261	6	.	.	PUNCT
ejpam-4687	261	7	.	.	PUNCT
ejpam-4687	262	1	.v1	.v1	PUNCT
ejpam-4687	263	1	v2	v2	NOUN
ejpam-4687	263	2	va−1	va−1	PROPN
ejpam-4687	263	3	va	va	PROPN
ejpam-4687	263	4	y1	y1	PROPN
ejpam-4687	263	5	w	w	PROPN
ejpam-4687	263	6	ym−1	ym−1	PROPN
ejpam-4687	263	7	...	...	PUNCT
ejpam-4687	263	8	g′	g′	NOUN
ejpam-4687	263	9	:	:	PUNCT
ejpam-4687	264	1	y2	y2	PROPN
ejpam-4687	264	2	v3	v3	PROPN
ejpam-4687	264	3	va−2	va−2	PROPN
ejpam-4687	264	4	figure	figure	NOUN
ejpam-4687	264	5	7	7	NUM
ejpam-4687	264	6	:	:	PUNCT
ejpam-4687	264	7	a	a	DET
ejpam-4687	264	8	graph	graph	NOUN
ejpam-4687	264	9	g′	g′	NOUN
ejpam-4687	264	10	with	with	ADP
ejpam-4687	264	11	γch(g	γch(g	NOUN
ejpam-4687	264	12	′	′	NUM
ejpam-4687	264	13	)	)	PUNCT
ejpam-4687	264	14	<	<	X
ejpam-4687	265	1	γch	γch	X
ejpam-4687	265	2	gr	gr	X
ejpam-4687	265	3	(	(	PUNCT
ejpam-4687	265	4	g	g	PROPN
ejpam-4687	265	5	′	′	NUM
ejpam-4687	265	6	)	)	PUNCT
ejpam-4687	265	7	for	for	ADP
ejpam-4687	265	8	(	(	PUNCT
ejpam-4687	265	9	ii	ii	NOUN
ejpam-4687	265	10	)	)	PUNCT
ejpam-4687	265	11	,	,	PUNCT
ejpam-4687	265	12	let	let	VERB
ejpam-4687	265	13	m	m	VERB
ejpam-4687	265	14	=	=	VERB
ejpam-4687	265	15	b	b	X
ejpam-4687	265	16	−	−	PROPN
ejpam-4687	265	17	a	a	DET
ejpam-4687	265	18	+	+	NOUN
ejpam-4687	265	19	1	1	NUM
ejpam-4687	265	20	.	.	X
ejpam-4687	265	21	consider	consider	VERB
ejpam-4687	265	22	the	the	DET
ejpam-4687	265	23	graph	graph	NOUN
ejpam-4687	265	24	h	h	NOUN
ejpam-4687	265	25	given	give	VERB
ejpam-4687	265	26	in	in	ADP
ejpam-4687	265	27	figure	figure	NOUN
ejpam-4687	265	28	8	8	NUM
ejpam-4687	265	29	.	.	PUNCT
ejpam-4687	266	1	one	one	PRON
ejpam-4687	266	2	can	can	AUX
ejpam-4687	266	3	verify	verify	VERB
ejpam-4687	266	4	that	that	PRON
ejpam-4687	266	5	c	c	NOUN
ejpam-4687	266	6	′	′	NUM
ejpam-4687	267	1	=	=	PUNCT
ejpam-4687	268	1	{	{	PUNCT
ejpam-4687	268	2	x1	x1	PROPN
ejpam-4687	268	3	,	,	PUNCT
ejpam-4687	268	4	x2	x2	PROPN
ejpam-4687	268	5	,	,	PUNCT
ejpam-4687	268	6	·	·	PUNCT
ejpam-4687	268	7	·	·	PUNCT
ejpam-4687	268	8	·	·	PUNCT
ejpam-4687	268	9	,	,	PUNCT
ejpam-4687	268	10	xa	xa	PROPN
ejpam-4687	268	11	)	)	PUNCT
ejpam-4687	268	12	and	and	CCONJ
ejpam-4687	268	13	c	c	X
ejpam-4687	269	1	′′	′′	NOUN
ejpam-4687	269	2	=	=	SYM
ejpam-4687	269	3	(	(	PUNCT
ejpam-4687	269	4	x1	x1	PROPN
ejpam-4687	269	5	,	,	PUNCT
ejpam-4687	269	6	x2	x2	PROPN
ejpam-4687	269	7	,	,	PUNCT
ejpam-4687	269	8	·	·	PUNCT
ejpam-4687	269	9	·	·	PUNCT
ejpam-4687	269	10	·	·	PUNCT
ejpam-4687	269	11	,	,	PUNCT
ejpam-4687	269	12	xa−1	xa−1	PROPN
ejpam-4687	269	13	,	,	PUNCT
ejpam-4687	269	14	y1	y1	PROPN
ejpam-4687	269	15	,	,	PUNCT
ejpam-4687	269	16	y2	y2	PROPN
ejpam-4687	269	17	,	,	PUNCT
ejpam-4687	269	18	·	·	PUNCT
ejpam-4687	269	19	·	·	PUNCT
ejpam-4687	269	20	·	·	PUNCT
ejpam-4687	269	21	,	,	PUNCT
ejpam-4687	269	22	ym	ym	PROPN
ejpam-4687	269	23	)	)	PUNCT
ejpam-4687	269	24	are	be	AUX
ejpam-4687	269	25	γchgr	γchgr	ADJ
ejpam-4687	269	26	-sequence	-sequence	NOUN
ejpam-4687	269	27	and	and	CCONJ
ejpam-4687	269	28	γhgr	γhgr	ADJ
ejpam-4687	269	29	-	-	PUNCT
ejpam-4687	269	30	sequence	sequence	NOUN
ejpam-4687	269	31	of	of	ADP
ejpam-4687	269	32	h	h	NOUN
ejpam-4687	269	33	,	,	PUNCT
ejpam-4687	269	34	respectively	respectively	ADV
ejpam-4687	269	35	.	.	PUNCT
ejpam-4687	270	1	therefore	therefore	ADV
ejpam-4687	270	2	,	,	PUNCT
ejpam-4687	270	3	γchgr	γchgr	X
ejpam-4687	270	4	(	(	PUNCT
ejpam-4687	270	5	h	h	NOUN
ejpam-4687	270	6	)	)	PUNCT
ejpam-4687	270	7	=	=	PUNCT
ejpam-4687	270	8	a	a	PRON
ejpam-4687	270	9	and	and	CCONJ
ejpam-4687	270	10	γhgr(h	γhgr(h	NOUN
ejpam-4687	270	11	)	)	PUNCT
ejpam-4687	271	1	=	=	VERB
ejpam-4687	271	2	m+	m+	NUM
ejpam-4687	271	3	a−	a−	PROPN
ejpam-4687	271	4	1	1	NUM
ejpam-4687	271	5	=	=	SYM
ejpam-4687	271	6	b.	b.	PROPN
ejpam-4687	271	7	j.	j.	PROPN
ejpam-4687	271	8	hassan	hassan	PROPN
ejpam-4687	271	9	,	,	PUNCT
ejpam-4687	271	10	s.	s.	PROPN
ejpam-4687	271	11	canoy	canoy	PROPN
ejpam-4687	271	12	jr	jr	PROPN
ejpam-4687	271	13	.	.	PROPN
ejpam-4687	271	14	/	/	SYM
ejpam-4687	271	15	eur	eur	PROPN
ejpam-4687	271	16	.	.	PUNCT
ejpam-4687	272	1	j.	j.	PROPN
ejpam-4687	272	2	pure	pure	PROPN
ejpam-4687	272	3	appl	appl	PROPN
ejpam-4687	272	4	.	.	PROPN
ejpam-4687	272	5	math	math	PROPN
ejpam-4687	272	6	,	,	PUNCT
ejpam-4687	272	7	16	16	NUM
ejpam-4687	272	8	(	(	PUNCT
ejpam-4687	272	9	2	2	NUM
ejpam-4687	272	10	)	)	PUNCT
ejpam-4687	272	11	(	(	PUNCT
ejpam-4687	272	12	2023	2023	NUM
ejpam-4687	272	13	)	)	PUNCT
ejpam-4687	272	14	,	,	PUNCT
ejpam-4687	272	15	1212	1212	NUM
ejpam-4687	272	16	-	-	SYM
ejpam-4687	272	17	1227	1227	NUM
ejpam-4687	272	18	1221	1221	NUM
ejpam-4687	272	19	h	h	NOUN
ejpam-4687	272	20	:	:	PUNCT
ejpam-4687	272	21	.	.	PUNCT
ejpam-4687	272	22	.	.	PUNCT
ejpam-4687	272	23	.	.	PUNCT
ejpam-4687	273	1	y1	y1	INTJ
ejpam-4687	274	1	y2	y2	INTJ
ejpam-4687	274	2	ym	ym	NOUN
ejpam-4687	275	1	x1	x1	NUM
ejpam-4687	276	1	x2	x2	PROPN
ejpam-4687	276	2	x3	x3	PROPN
ejpam-4687	276	3	xa−1	xa−1	PROPN
ejpam-4687	276	4	xa	xa	PROPN
ejpam-4687	276	5	.	.	PUNCT
ejpam-4687	276	6	.	.	PUNCT
ejpam-4687	276	7	.	.	PUNCT
ejpam-4687	277	1	figure	figure	VERB
ejpam-4687	277	2	8	8	NUM
ejpam-4687	277	3	:	:	PUNCT
ejpam-4687	277	4	a	a	DET
ejpam-4687	277	5	graph	graph	NOUN
ejpam-4687	277	6	h	h	NOUN
ejpam-4687	277	7	with	with	ADP
ejpam-4687	277	8	γch	γch	NOUN
ejpam-4687	277	9	gr	gr	PROPN
ejpam-4687	277	10	(	(	PUNCT
ejpam-4687	277	11	h	h	NOUN
ejpam-4687	277	12	)	)	PUNCT
ejpam-4687	277	13	<	<	X
ejpam-4687	277	14	γh	γh	PROPN
ejpam-4687	277	15	gr(h	gr(h	NOUN
ejpam-4687	277	16	)	)	PUNCT
ejpam-4687	278	1	this	this	PRON
ejpam-4687	278	2	proves	prove	VERB
ejpam-4687	278	3	the	the	DET
ejpam-4687	278	4	assertion	assertion	NOUN
ejpam-4687	278	5	.	.	PUNCT
ejpam-4687	279	1	proposition	proposition	NOUN
ejpam-4687	279	2	3	3	NUM
ejpam-4687	279	3	.	.	PUNCT
ejpam-4687	280	1	[	[	X
ejpam-4687	280	2	8	8	NUM
ejpam-4687	280	3	]	]	PUNCT
ejpam-4687	280	4	for	for	ADP
ejpam-4687	280	5	any	any	DET
ejpam-4687	280	6	positive	positive	ADJ
ejpam-4687	280	7	integer	integer	NOUN
ejpam-4687	280	8	n	n	PRON
ejpam-4687	280	9	≥	≥	NUM
ejpam-4687	280	10	2	2	NUM
ejpam-4687	280	11	,	,	PUNCT
ejpam-4687	280	12	γhgr(pn	γhgr(pn	ADJ
ejpam-4687	280	13	)	)	PUNCT
ejpam-4687	280	14	=	=	SYM
ejpam-4687	280	15	{	{	PUNCT
ejpam-4687	280	16	2	2	NUM
ejpam-4687	280	17	if	if	SCONJ
ejpam-4687	280	18	n	n	X
ejpam-4687	280	19	=	=	SYM
ejpam-4687	280	20	2	2	NUM
ejpam-4687	280	21	,	,	PUNCT
ejpam-4687	280	22	3	3	NUM
ejpam-4687	280	23	n−	n−	NOUN
ejpam-4687	280	24	2	2	NUM
ejpam-4687	280	25	if	if	SCONJ
ejpam-4687	280	26	n	n	PRON
ejpam-4687	280	27	≥	≥	NOUN
ejpam-4687	280	28	4	4	NUM
ejpam-4687	280	29	.	.	PUNCT
ejpam-4687	280	30	proposition	proposition	NOUN
ejpam-4687	280	31	4	4	NUM
ejpam-4687	280	32	.	.	X
ejpam-4687	281	1	for	for	ADP
ejpam-4687	281	2	any	any	DET
ejpam-4687	281	3	positive	positive	ADJ
ejpam-4687	281	4	integer	integer	NOUN
ejpam-4687	281	5	n	n	PRON
ejpam-4687	281	6	≥	≥	NOUN
ejpam-4687	281	7	2	2	NUM
ejpam-4687	281	8	,	,	PUNCT
ejpam-4687	281	9	γchgr	γchgr	X
ejpam-4687	281	10	(	(	PUNCT
ejpam-4687	281	11	pn	pn	NOUN
ejpam-4687	281	12	)	)	PUNCT
ejpam-4687	281	13	=	=	PRON
ejpam-4687	281	14	{	{	PUNCT
ejpam-4687	281	15	2	2	NUM
ejpam-4687	281	16	if	if	SCONJ
ejpam-4687	281	17	n	n	X
ejpam-4687	281	18	=	=	SYM
ejpam-4687	281	19	2	2	NUM
ejpam-4687	281	20	,	,	PUNCT
ejpam-4687	281	21	3	3	NUM
ejpam-4687	281	22	n−	n−	NOUN
ejpam-4687	281	23	2	2	NUM
ejpam-4687	281	24	if	if	SCONJ
ejpam-4687	281	25	n	n	PRON
ejpam-4687	281	26	≥	≥	NOUN
ejpam-4687	281	27	4	4	NUM
ejpam-4687	281	28	.	.	PUNCT
ejpam-4687	282	1	proof	proof	NOUN
ejpam-4687	282	2	.	.	PUNCT
ejpam-4687	283	1	let	let	VERB
ejpam-4687	283	2	pn	pn	VERB
ejpam-4687	283	3	=	=	PUNCT
ejpam-4687	284	1	[	[	X
ejpam-4687	284	2	v1	v1	NOUN
ejpam-4687	284	3	,	,	PUNCT
ejpam-4687	284	4	v2	v2	PROPN
ejpam-4687	284	5	,	,	PUNCT
ejpam-4687	284	6	·	·	PUNCT
ejpam-4687	284	7	·	·	PUNCT
ejpam-4687	284	8	·	·	PUNCT
ejpam-4687	284	9	,	,	PUNCT
ejpam-4687	284	10	vn	vn	X
ejpam-4687	284	11	]	]	PUNCT
ejpam-4687	284	12	.	.	PUNCT
ejpam-4687	285	1	clearly	clearly	ADV
ejpam-4687	285	2	,	,	PUNCT
ejpam-4687	285	3	γchgr	γchgr	PROPN
ejpam-4687	285	4	(	(	PUNCT
ejpam-4687	285	5	pn	pn	NOUN
ejpam-4687	285	6	)	)	PUNCT
ejpam-4687	285	7	=	=	SYM
ejpam-4687	285	8	2	2	NUM
ejpam-4687	285	9	for	for	ADP
ejpam-4687	285	10	n	n	NOUN
ejpam-4687	285	11	=	=	SYM
ejpam-4687	285	12	2	2	NUM
ejpam-4687	285	13	,	,	PUNCT
ejpam-4687	285	14	3	3	NUM
ejpam-4687	285	15	.	.	PUNCT
ejpam-4687	286	1	next	next	ADV
ejpam-4687	286	2	,	,	PUNCT
ejpam-4687	286	3	suppose	suppose	VERB
ejpam-4687	286	4	that	that	SCONJ
ejpam-4687	286	5	n	n	PROPN
ejpam-4687	286	6	≥	≥	NUM
ejpam-4687	286	7	4	4	NUM
ejpam-4687	286	8	.	.	PUNCT
ejpam-4687	287	1	let	let	VERB
ejpam-4687	287	2	s′	s′	ADJ
ejpam-4687	287	3	=	=	SYM
ejpam-4687	287	4	(	(	PUNCT
ejpam-4687	287	5	v1	v1	NOUN
ejpam-4687	287	6	,	,	PUNCT
ejpam-4687	287	7	v2	v2	PROPN
ejpam-4687	287	8	·	·	PUNCT
ejpam-4687	287	9	·	·	PUNCT
ejpam-4687	287	10	·	·	PUNCT
ejpam-4687	287	11	,	,	PUNCT
ejpam-4687	287	12	vn−2	vn−2	PROPN
ejpam-4687	287	13	)	)	PUNCT
ejpam-4687	287	14	.	.	PUNCT
ejpam-4687	288	1	clearly	clearly	ADV
ejpam-4687	288	2	,	,	PUNCT
ejpam-4687	288	3	s′	s′	PROPN
ejpam-4687	288	4	is	be	AUX
ejpam-4687	288	5	a	a	DET
ejpam-4687	288	6	connected	connected	ADJ
ejpam-4687	288	7	grundy	grundy	PROPN
ejpam-4687	288	8	hop	hop	NOUN
ejpam-4687	288	9	dominating	dominating	NOUN
ejpam-4687	288	10	sequence	sequence	NOUN
ejpam-4687	288	11	of	of	ADP
ejpam-4687	288	12	pn	pn	PROPN
ejpam-4687	288	13	.	.	PUNCT
ejpam-4687	289	1	thus	thus	ADV
ejpam-4687	289	2	,	,	PUNCT
ejpam-4687	289	3	γchgr	γchgr	PROPN
ejpam-4687	289	4	(	(	PUNCT
ejpam-4687	289	5	pn	pn	PROPN
ejpam-4687	289	6	)	)	PUNCT
ejpam-4687	289	7	≥	≥	NOUN
ejpam-4687	289	8	n	n	CCONJ
ejpam-4687	289	9	−	−	PROPN
ejpam-4687	289	10	2	2	NUM
ejpam-4687	289	11	.	.	PUNCT
ejpam-4687	289	12	since	since	SCONJ
ejpam-4687	289	13	γhgr(pn	γhgr(pn	ADJ
ejpam-4687	289	14	)	)	PUNCT
ejpam-4687	289	15	=	=	SYM
ejpam-4687	289	16	n	n	CCONJ
ejpam-4687	289	17	−	−	NUM
ejpam-4687	289	18	2	2	NUM
ejpam-4687	289	19	for	for	ADP
ejpam-4687	289	20	all	all	DET
ejpam-4687	289	21	n	n	PRON
ejpam-4687	289	22	≥	≥	NOUN
ejpam-4687	289	23	4	4	NUM
ejpam-4687	289	24	,	,	PUNCT
ejpam-4687	289	25	it	it	PRON
ejpam-4687	289	26	follows	follow	VERB
ejpam-4687	289	27	that	that	SCONJ
ejpam-4687	289	28	γchgr	γchgr	NOUN
ejpam-4687	289	29	(	(	PUNCT
ejpam-4687	289	30	pn	pn	NOUN
ejpam-4687	289	31	)	)	PUNCT
ejpam-4687	289	32	≤	≤	NOUN
ejpam-4687	289	33	n	n	CCONJ
ejpam-4687	289	34	−	−	PROPN
ejpam-4687	289	35	2	2	NUM
ejpam-4687	289	36	for	for	ADP
ejpam-4687	289	37	all	all	DET
ejpam-4687	289	38	n	n	PRON
ejpam-4687	289	39	≥	≥	NOUN
ejpam-4687	289	40	4	4	NUM
ejpam-4687	289	41	by	by	ADP
ejpam-4687	289	42	remark	remark	NOUN
ejpam-4687	289	43	2	2	NUM
ejpam-4687	289	44	.	.	PUNCT
ejpam-4687	289	45	consequently	consequently	ADV
ejpam-4687	289	46	,	,	PUNCT
ejpam-4687	289	47	γchgr	γchgr	PROPN
ejpam-4687	289	48	(	(	PUNCT
ejpam-4687	289	49	pn	pn	NOUN
ejpam-4687	289	50	)	)	PUNCT
ejpam-4687	289	51	=	=	SYM
ejpam-4687	289	52	n	n	CCONJ
ejpam-4687	289	53	−	−	NUM
ejpam-4687	289	54	2	2	NUM
ejpam-4687	289	55	for	for	ADP
ejpam-4687	289	56	all	all	DET
ejpam-4687	289	57	n	n	PRON
ejpam-4687	289	58	≥	≥	NOUN
ejpam-4687	289	59	4	4	NUM
ejpam-4687	289	60	.	.	PUNCT
ejpam-4687	289	61	proposition	proposition	NOUN
ejpam-4687	289	62	5	5	NUM
ejpam-4687	289	63	.	.	PUNCT
ejpam-4687	290	1	let	let	VERB
ejpam-4687	290	2	g	g	PRON
ejpam-4687	290	3	be	be	AUX
ejpam-4687	290	4	a	a	DET
ejpam-4687	290	5	connected	connected	ADJ
ejpam-4687	290	6	graph	graph	NOUN
ejpam-4687	290	7	of	of	ADP
ejpam-4687	290	8	order	order	NOUN
ejpam-4687	290	9	n.	n.	NOUN
ejpam-4687	290	10	if	if	SCONJ
ejpam-4687	290	11	|n2	|n2	PROPN
ejpam-4687	290	12	g[v]|	g[v]|	PROPN
ejpam-4687	290	13	≥	≥	NUM
ejpam-4687	290	14	m	m	VERB
ejpam-4687	290	15	for	for	ADP
ejpam-4687	290	16	every	every	DET
ejpam-4687	290	17	v	v	NUM
ejpam-4687	290	18	∈	∈	PROPN
ejpam-4687	290	19	v	v	NOUN
ejpam-4687	290	20	(	(	PUNCT
ejpam-4687	290	21	g	g	NOUN
ejpam-4687	290	22	)	)	PUNCT
ejpam-4687	290	23	,	,	PUNCT
ejpam-4687	290	24	then	then	ADV
ejpam-4687	290	25	γchgr	γchgr	X
ejpam-4687	290	26	(	(	PUNCT
ejpam-4687	290	27	g	g	NOUN
ejpam-4687	290	28	)	)	PUNCT
ejpam-4687	290	29	≤	≤	NOUN
ejpam-4687	290	30	n−	n−	NOUN
ejpam-4687	290	31	(	(	PUNCT
ejpam-4687	290	32	m−	m−	PROPN
ejpam-4687	290	33	1	1	NUM
ejpam-4687	290	34	)	)	PUNCT
ejpam-4687	290	35	.	.	PUNCT
ejpam-4687	291	1	proof	proof	NOUN
ejpam-4687	291	2	.	.	PUNCT
ejpam-4687	292	1	let	let	VERB
ejpam-4687	292	2	v	v	X
ejpam-4687	292	3	(	(	PUNCT
ejpam-4687	292	4	g	g	NOUN
ejpam-4687	292	5	)	)	PUNCT
ejpam-4687	292	6	=	=	SYM
ejpam-4687	292	7	{	{	PUNCT
ejpam-4687	292	8	v1	v1	PROPN
ejpam-4687	292	9	,	,	PUNCT
ejpam-4687	292	10	v2	v2	PROPN
ejpam-4687	292	11	,	,	PUNCT
ejpam-4687	292	12	.	.	PUNCT
ejpam-4687	292	13	.	.	PUNCT
ejpam-4687	293	1	.	.	PUNCT
ejpam-4687	294	1	,	,	PUNCT
ejpam-4687	294	2	vn	vn	PROPN
ejpam-4687	294	3	}	}	PUNCT
ejpam-4687	294	4	.	.	PUNCT
ejpam-4687	295	1	suppose	suppose	VERB
ejpam-4687	296	1	γchgr	γchgr	ADJ
ejpam-4687	296	2	(	(	PUNCT
ejpam-4687	296	3	g	g	NOUN
ejpam-4687	296	4	)	)	PUNCT
ejpam-4687	296	5	=	=	SYM
ejpam-4687	297	1	k	k	NOUN
ejpam-4687	297	2	,	,	PUNCT
ejpam-4687	297	3	say	say	VERB
ejpam-4687	297	4	s	s	X
ejpam-4687	297	5	=	=	SYM
ejpam-4687	297	6	(	(	PUNCT
ejpam-4687	297	7	s1	s1	PROPN
ejpam-4687	297	8	,	,	PUNCT
ejpam-4687	297	9	s2	s2	PROPN
ejpam-4687	297	10	,	,	PUNCT
ejpam-4687	297	11	·	·	PUNCT
ejpam-4687	297	12	·	·	PUNCT
ejpam-4687	297	13	·	·	PUNCT
ejpam-4687	297	14	,	,	PUNCT
ejpam-4687	297	15	sk	sk	PROPN
ejpam-4687	297	16	)	)	PUNCT
ejpam-4687	297	17	is	be	AUX
ejpam-4687	297	18	a	a	DET
ejpam-4687	297	19	connected	connected	ADJ
ejpam-4687	297	20	grundy	grundy	PROPN
ejpam-4687	297	21	hop	hop	NOUN
ejpam-4687	297	22	dominating	dominating	NOUN
ejpam-4687	297	23	sequence	sequence	NOUN
ejpam-4687	297	24	of	of	ADP
ejpam-4687	297	25	g.	g.	PROPN
ejpam-4687	297	26	assume	assume	VERB
ejpam-4687	297	27	s1	s1	PROPN
ejpam-4687	297	28	=	=	SYM
ejpam-4687	297	29	vi	vi	PROPN
ejpam-4687	297	30	for	for	ADP
ejpam-4687	297	31	some	some	DET
ejpam-4687	297	32	i	i	PRON
ejpam-4687	297	33	∈	∈	PROPN
ejpam-4687	297	34	{	{	PUNCT
ejpam-4687	297	35	1	1	NUM
ejpam-4687	297	36	,	,	PUNCT
ejpam-4687	297	37	.	.	PUNCT
ejpam-4687	297	38	.	.	PUNCT
ejpam-4687	297	39	.	.	PUNCT
ejpam-4687	298	1	,	,	PUNCT
ejpam-4687	299	1	n	n	CCONJ
ejpam-4687	299	2	}	}	PUNCT
ejpam-4687	299	3	.	.	PUNCT
ejpam-4687	300	1	then	then	ADV
ejpam-4687	300	2	|n2	|n2	PROPN
ejpam-4687	300	3	g[s1]|	g[s1]|	PROPN
ejpam-4687	300	4	=	=	SYM
ejpam-4687	300	5	|n2	|n2	PROPN
ejpam-4687	300	6	g[vi]|	g[vi]|	PROPN
ejpam-4687	300	7	≥	≥	VERB
ejpam-4687	300	8	m	m	VERB
ejpam-4687	300	9	by	by	ADP
ejpam-4687	300	10	assumption	assumption	NOUN
ejpam-4687	300	11	.	.	PUNCT
ejpam-4687	301	1	it	it	PRON
ejpam-4687	301	2	follows	follow	VERB
ejpam-4687	301	3	that	that	SCONJ
ejpam-4687	301	4	there	there	PRON
ejpam-4687	301	5	are	be	VERB
ejpam-4687	301	6	at	at	ADP
ejpam-4687	301	7	most	most	ADJ
ejpam-4687	301	8	n	n	ADP
ejpam-4687	301	9	−	−	NOUN
ejpam-4687	301	10	m	m	VERB
ejpam-4687	301	11	remaining	remain	VERB
ejpam-4687	301	12	vertices	vertex	NOUN
ejpam-4687	301	13	of	of	ADP
ejpam-4687	301	14	g	g	NOUN
ejpam-4687	301	15	that	that	PRON
ejpam-4687	301	16	could	could	AUX
ejpam-4687	301	17	be	be	AUX
ejpam-4687	301	18	hop	hop	NOUN
ejpam-4687	301	19	footprinted	footprinte	VERB
ejpam-4687	301	20	by	by	ADP
ejpam-4687	301	21	the	the	DET
ejpam-4687	301	22	next	next	ADJ
ejpam-4687	301	23	terms	term	NOUN
ejpam-4687	301	24	of	of	ADP
ejpam-4687	301	25	s.	s.	PROPN
ejpam-4687	301	26	therefore	therefore	ADV
ejpam-4687	301	27	,	,	PUNCT
ejpam-4687	301	28	γchgr	γchgr	X
ejpam-4687	301	29	(	(	PUNCT
ejpam-4687	301	30	g	g	NOUN
ejpam-4687	301	31	)	)	PUNCT
ejpam-4687	301	32	=	=	SYM
ejpam-4687	302	1	k	k	PROPN
ejpam-4687	302	2	≤	≤	PROPN
ejpam-4687	302	3	n−m+	n−m+	PROPN
ejpam-4687	302	4	|{vi}|	|{vi}|	PROPN
ejpam-4687	302	5	=	=	PUNCT
ejpam-4687	302	6	n−m+	n−m+	PROPN
ejpam-4687	302	7	1	1	NUM
ejpam-4687	302	8	=	=	SYM
ejpam-4687	302	9	n−	n−	PROPN
ejpam-4687	302	10	(	(	PUNCT
ejpam-4687	302	11	m−	m−	PROPN
ejpam-4687	302	12	1	1	NUM
ejpam-4687	302	13	)	)	PUNCT
ejpam-4687	302	14	.	.	PUNCT
ejpam-4687	303	1	the	the	DET
ejpam-4687	303	2	next	next	ADJ
ejpam-4687	303	3	result	result	NOUN
ejpam-4687	303	4	follows	follow	VERB
ejpam-4687	303	5	immediately	immediately	ADV
ejpam-4687	303	6	form	form	VERB
ejpam-4687	303	7	proposition	proposition	NOUN
ejpam-4687	303	8	5	5	NUM
ejpam-4687	303	9	.	.	PUNCT
ejpam-4687	303	10	corollary	corollary	ADJ
ejpam-4687	303	11	5	5	NUM
ejpam-4687	303	12	.	.	PUNCT
ejpam-4687	304	1	let	let	VERB
ejpam-4687	304	2	g	g	PRON
ejpam-4687	304	3	be	be	AUX
ejpam-4687	304	4	a	a	DET
ejpam-4687	304	5	connected	connected	ADJ
ejpam-4687	304	6	graph	graph	NOUN
ejpam-4687	304	7	on	on	ADP
ejpam-4687	304	8	n	n	DET
ejpam-4687	304	9	vertices	vertex	NOUN
ejpam-4687	304	10	.	.	PUNCT
ejpam-4687	305	1	if	if	SCONJ
ejpam-4687	305	2	|n2	|n2	PROPN
ejpam-4687	305	3	g[u]|	g[u]|	PROPN
ejpam-4687	305	4	=	=	PUNCT
ejpam-4687	305	5	3	3	NUM
ejpam-4687	305	6	for	for	ADP
ejpam-4687	305	7	every	every	DET
ejpam-4687	305	8	u	u	PROPN
ejpam-4687	305	9	∈	∈	PROPN
ejpam-4687	305	10	v	v	NOUN
ejpam-4687	305	11	(	(	PUNCT
ejpam-4687	305	12	g	g	NOUN
ejpam-4687	305	13	)	)	PUNCT
ejpam-4687	305	14	,	,	PUNCT
ejpam-4687	305	15	then	then	ADV
ejpam-4687	305	16	γchgr	γchgr	X
ejpam-4687	305	17	(	(	PUNCT
ejpam-4687	305	18	g	g	NOUN
ejpam-4687	305	19	)	)	PUNCT
ejpam-4687	305	20	≤	≤	NUM
ejpam-4687	305	21	n−	n−	NOUN
ejpam-4687	305	22	2	2	NUM
ejpam-4687	305	23	.	.	PUNCT
ejpam-4687	305	24	j.	j.	PROPN
ejpam-4687	305	25	hassan	hassan	PROPN
ejpam-4687	305	26	,	,	PUNCT
ejpam-4687	305	27	s.	s.	PROPN
ejpam-4687	305	28	canoy	canoy	PROPN
ejpam-4687	305	29	jr	jr	PROPN
ejpam-4687	305	30	.	.	PROPN
ejpam-4687	305	31	/	/	SYM
ejpam-4687	305	32	eur	eur	PROPN
ejpam-4687	305	33	.	.	PUNCT
ejpam-4687	306	1	j.	j.	PROPN
ejpam-4687	306	2	pure	pure	PROPN
ejpam-4687	306	3	appl	appl	PROPN
ejpam-4687	306	4	.	.	PROPN
ejpam-4687	306	5	math	math	PROPN
ejpam-4687	306	6	,	,	PUNCT
ejpam-4687	306	7	16	16	NUM
ejpam-4687	306	8	(	(	PUNCT
ejpam-4687	306	9	2	2	NUM
ejpam-4687	306	10	)	)	PUNCT
ejpam-4687	306	11	(	(	PUNCT
ejpam-4687	306	12	2023	2023	NUM
ejpam-4687	306	13	)	)	PUNCT
ejpam-4687	306	14	,	,	PUNCT
ejpam-4687	306	15	1212	1212	NUM
ejpam-4687	306	16	-	-	SYM
ejpam-4687	306	17	1227	1227	NUM
ejpam-4687	306	18	1222	1222	NUM
ejpam-4687	306	19	proposition	proposition	NOUN
ejpam-4687	306	20	6	6	NUM
ejpam-4687	306	21	.	.	PUNCT
ejpam-4687	307	1	[	[	X
ejpam-4687	307	2	8	8	NUM
ejpam-4687	307	3	]	]	PUNCT
ejpam-4687	307	4	for	for	ADP
ejpam-4687	307	5	any	any	DET
ejpam-4687	307	6	positive	positive	ADJ
ejpam-4687	307	7	integer	integer	NOUN
ejpam-4687	307	8	n	n	PRON
ejpam-4687	307	9	≥	≥	NOUN
ejpam-4687	307	10	3	3	NUM
ejpam-4687	307	11	,	,	PUNCT
ejpam-4687	307	12	γhgr(cn	γhgr(cn	NOUN
ejpam-4687	307	13	)	)	PUNCT
ejpam-4687	307	14	=	=	PUNCT
ejpam-4687	308	1			NOUN
ejpam-4687	308	2	3	3	NUM
ejpam-4687	308	3	if	if	SCONJ
ejpam-4687	308	4	n	n	NOUN
ejpam-4687	308	5	=	=	SYM
ejpam-4687	308	6	3	3	NUM
ejpam-4687	308	7	2	2	NUM
ejpam-4687	308	8	if	if	SCONJ
ejpam-4687	308	9	n	n	NOUN
ejpam-4687	308	10	=	=	SYM
ejpam-4687	308	11	4	4	NUM
ejpam-4687	308	12	n−	n−	NOUN
ejpam-4687	308	13	4	4	NUM
ejpam-4687	308	14	if	if	SCONJ
ejpam-4687	308	15	n	n	PRON
ejpam-4687	308	16	≥	≥	NOUN
ejpam-4687	308	17	6	6	NUM
ejpam-4687	308	18	and	and	CCONJ
ejpam-4687	308	19	even	even	ADV
ejpam-4687	308	20	n−	n−	PROPN
ejpam-4687	308	21	2	2	NUM
ejpam-4687	308	22	if	if	SCONJ
ejpam-4687	308	23	n	n	PRON
ejpam-4687	308	24	≥	≥	NOUN
ejpam-4687	308	25	5	5	NUM
ejpam-4687	308	26	and	and	CCONJ
ejpam-4687	308	27	odd	odd	ADJ
ejpam-4687	308	28	.	.	PUNCT
ejpam-4687	309	1	proposition	proposition	NOUN
ejpam-4687	309	2	7	7	NUM
ejpam-4687	309	3	.	.	X
ejpam-4687	310	1	for	for	ADP
ejpam-4687	310	2	any	any	DET
ejpam-4687	310	3	positive	positive	ADJ
ejpam-4687	310	4	integer	integer	NOUN
ejpam-4687	310	5	n	n	PRON
ejpam-4687	310	6	≥	≥	NOUN
ejpam-4687	310	7	3	3	NUM
ejpam-4687	310	8	,	,	PUNCT
ejpam-4687	310	9	γchgr	γchgr	X
ejpam-4687	310	10	(	(	PUNCT
ejpam-4687	310	11	cn	cn	PROPN
ejpam-4687	310	12	)	)	PUNCT
ejpam-4687	310	13	=	=	PUNCT
ejpam-4687	310	14			NOUN
ejpam-4687	310	15	2	2	NUM
ejpam-4687	310	16	if	if	SCONJ
ejpam-4687	310	17	n	n	NOUN
ejpam-4687	310	18	=	=	SYM
ejpam-4687	310	19	4	4	NUM
ejpam-4687	310	20	3	3	NUM
ejpam-4687	310	21	if	if	SCONJ
ejpam-4687	310	22	n	n	NOUN
ejpam-4687	310	23	=	=	SYM
ejpam-4687	310	24	3	3	NUM
ejpam-4687	310	25	,	,	PUNCT
ejpam-4687	310	26	5	5	NUM
ejpam-4687	310	27	n−	n−	NOUN
ejpam-4687	310	28	4	4	NUM
ejpam-4687	310	29	if	if	SCONJ
ejpam-4687	310	30	n	n	PRON
ejpam-4687	310	31	≥	≥	NOUN
ejpam-4687	310	32	6	6	NUM
ejpam-4687	310	33	and	and	CCONJ
ejpam-4687	310	34	even	even	ADV
ejpam-4687	310	35	n−	n−	PROPN
ejpam-4687	310	36	3	3	NUM
ejpam-4687	310	37	if	if	SCONJ
ejpam-4687	310	38	n	n	PRON
ejpam-4687	310	39	≥	≥	VERB
ejpam-4687	310	40	7	7	NUM
ejpam-4687	310	41	and	and	CCONJ
ejpam-4687	310	42	odd	odd	ADJ
ejpam-4687	310	43	proof	proof	NOUN
ejpam-4687	310	44	.	.	PUNCT
ejpam-4687	311	1	let	let	VERB
ejpam-4687	311	2	g	g	NOUN
ejpam-4687	311	3	=	=	PUNCT
ejpam-4687	311	4	cn	cn	PROPN
ejpam-4687	311	5	=	=	PUNCT
ejpam-4687	312	1	[	[	X
ejpam-4687	312	2	v1	v1	NOUN
ejpam-4687	312	3	,	,	PUNCT
ejpam-4687	312	4	v2	v2	PROPN
ejpam-4687	312	5	,	,	PUNCT
ejpam-4687	312	6	·	·	PUNCT
ejpam-4687	312	7	·	·	PUNCT
ejpam-4687	312	8	·	·	PUNCT
ejpam-4687	312	9	,	,	PUNCT
ejpam-4687	312	10	vn	vn	X
ejpam-4687	312	11	,	,	PUNCT
ejpam-4687	312	12	v1	v1	PROPN
ejpam-4687	312	13	]	]	PUNCT
ejpam-4687	312	14	.	.	PUNCT
ejpam-4687	313	1	clearly	clearly	ADV
ejpam-4687	313	2	,	,	PUNCT
ejpam-4687	313	3	γchgr	γchgr	PROPN
ejpam-4687	313	4	(	(	PUNCT
ejpam-4687	313	5	c3	c3	PROPN
ejpam-4687	313	6	)	)	PUNCT
ejpam-4687	313	7	=	=	SYM
ejpam-4687	313	8	3	3	NUM
ejpam-4687	313	9	=	=	SYM
ejpam-4687	313	10	γchgr	γchgr	X
ejpam-4687	313	11	(	(	PUNCT
ejpam-4687	313	12	c5	c5	PROPN
ejpam-4687	313	13	)	)	PUNCT
ejpam-4687	313	14	and	and	CCONJ
ejpam-4687	313	15	γchgr	γchgr	ADJ
ejpam-4687	313	16	(	(	PUNCT
ejpam-4687	313	17	c4	c4	NOUN
ejpam-4687	313	18	)	)	PUNCT
ejpam-4687	313	19	=	=	SYM
ejpam-4687	313	20	2	2	X
ejpam-4687	313	21	.	.	PUNCT
ejpam-4687	313	22	suppose	suppose	VERB
ejpam-4687	313	23	that	that	SCONJ
ejpam-4687	313	24	n	n	PROPN
ejpam-4687	313	25	≥	≥	NUM
ejpam-4687	313	26	6	6	NUM
ejpam-4687	313	27	and	and	CCONJ
ejpam-4687	313	28	is	be	AUX
ejpam-4687	313	29	even	even	ADV
ejpam-4687	313	30	.	.	PUNCT
ejpam-4687	314	1	let	let	VERB
ejpam-4687	314	2	s	s	PRON
ejpam-4687	314	3	=	=	PUNCT
ejpam-4687	314	4	(	(	PUNCT
ejpam-4687	314	5	v1	v1	PROPN
ejpam-4687	314	6	,	,	PUNCT
ejpam-4687	314	7	v2	v2	PROPN
ejpam-4687	314	8	,	,	PUNCT
ejpam-4687	314	9	·	·	PUNCT
ejpam-4687	314	10	·	·	PUNCT
ejpam-4687	314	11	·	·	PUNCT
ejpam-4687	314	12	,	,	PUNCT
ejpam-4687	314	13	vn−4	vn−4	NOUN
ejpam-4687	314	14	)	)	PUNCT
ejpam-4687	314	15	.	.	PUNCT
ejpam-4687	315	1	then	then	ADV
ejpam-4687	315	2	n2	n2	PROPN
ejpam-4687	315	3	g[v2	g[v2	PROPN
ejpam-4687	315	4	]	]	PUNCT
ejpam-4687	315	5	\	\	PROPN
ejpam-4687	315	6	n2	n2	PROPN
ejpam-4687	315	7	g[v1	g[v1	PROPN
ejpam-4687	315	8	]	]	X
ejpam-4687	315	9	=	=	PRON
ejpam-4687	315	10	{	{	PUNCT
ejpam-4687	315	11	v2	v2	PROPN
ejpam-4687	315	12	,	,	PUNCT
ejpam-4687	315	13	v4	v4	NOUN
ejpam-4687	315	14	,	,	PUNCT
ejpam-4687	315	15	vn	vn	INTJ
ejpam-4687	315	16	}	}	PUNCT
ejpam-4687	315	17	=	=	NOUN
ejpam-4687	315	18	̸	̸	ADJ
ejpam-4687	315	19	∅	∅	NOUN
ejpam-4687	315	20	and	and	CCONJ
ejpam-4687	315	21	vi+2	vi+2	NUM
ejpam-4687	315	22	∈	∈	PROPN
ejpam-4687	315	23	n2	n2	NOUN
ejpam-4687	315	24	g[vi]\∪	g[vi]\∪	VERB
ejpam-4687	315	25	i−1	i−1	PROPN
ejpam-4687	315	26	j=1n	j=1n	PROPN
ejpam-4687	315	27	2	2	NUM
ejpam-4687	315	28	g[vj	g[vj	PROPN
ejpam-4687	315	29	]	]	PUNCT
ejpam-4687	315	30	for	for	ADP
ejpam-4687	315	31	all	all	PRON
ejpam-4687	315	32	i	i	PRON
ejpam-4687	315	33	∈	∈	PROPN
ejpam-4687	315	34	{	{	PUNCT
ejpam-4687	315	35	3	3	NUM
ejpam-4687	315	36	,	,	PUNCT
ejpam-4687	315	37	4	4	NUM
ejpam-4687	315	38	,	,	PUNCT
ejpam-4687	315	39	·	·	PUNCT
ejpam-4687	315	40	·	·	PUNCT
ejpam-4687	315	41	·	·	PUNCT
ejpam-4687	315	42	,	,	PUNCT
ejpam-4687	315	43	n−4	n−4	PROPN
ejpam-4687	315	44	}	}	PUNCT
ejpam-4687	315	45	.	.	PUNCT
ejpam-4687	316	1	it	it	PRON
ejpam-4687	316	2	follows	follow	VERB
ejpam-4687	316	3	that	that	SCONJ
ejpam-4687	316	4	s	s	VERB
ejpam-4687	316	5	is	be	AUX
ejpam-4687	316	6	a	a	DET
ejpam-4687	316	7	connected	connect	VERB
ejpam-4687	316	8	grundy	grundy	PROPN
ejpam-4687	316	9	hop	hop	NOUN
ejpam-4687	316	10	dominating	dominating	NOUN
ejpam-4687	316	11	sequence	sequence	NOUN
ejpam-4687	316	12	.	.	PUNCT
ejpam-4687	317	1	hence	hence	ADV
ejpam-4687	317	2	,	,	PUNCT
ejpam-4687	317	3	γchgr	γchgr	X
ejpam-4687	317	4	(	(	PUNCT
ejpam-4687	317	5	cn	cn	PROPN
ejpam-4687	317	6	)	)	PUNCT
ejpam-4687	317	7	≥	≥	NOUN
ejpam-4687	317	8	|ŝ|	|ŝ|	PROPN
ejpam-4687	317	9	=	=	SYM
ejpam-4687	317	10	n−4	n−4	PROPN
ejpam-4687	317	11	.	.	PUNCT
ejpam-4687	318	1	now	now	ADV
ejpam-4687	318	2	,	,	PUNCT
ejpam-4687	318	3	since	since	SCONJ
ejpam-4687	318	4	γhgr(cn	γhgr(cn	NOUN
ejpam-4687	318	5	)	)	PUNCT
ejpam-4687	318	6	=	=	SYM
ejpam-4687	318	7	n	n	CCONJ
ejpam-4687	318	8	−	−	NOUN
ejpam-4687	318	9	4	4	NUM
ejpam-4687	318	10	for	for	ADP
ejpam-4687	318	11	even	even	ADV
ejpam-4687	318	12	integers	integer	NOUN
ejpam-4687	318	13	n	n	PRON
ejpam-4687	318	14	≥	≥	NUM
ejpam-4687	318	15	6	6	NUM
ejpam-4687	318	16	,	,	PUNCT
ejpam-4687	318	17	it	it	PRON
ejpam-4687	318	18	follows	follow	VERB
ejpam-4687	318	19	that	that	SCONJ
ejpam-4687	318	20	γchgr	γchgr	PROPN
ejpam-4687	318	21	(	(	PUNCT
ejpam-4687	318	22	cn	cn	PROPN
ejpam-4687	318	23	)	)	PUNCT
ejpam-4687	318	24	≤	≤	NOUN
ejpam-4687	318	25	n	n	CCONJ
ejpam-4687	318	26	−	−	PROPN
ejpam-4687	318	27	4	4	NUM
ejpam-4687	318	28	by	by	ADP
ejpam-4687	318	29	proposition	proposition	NOUN
ejpam-4687	318	30	2	2	NUM
ejpam-4687	318	31	.	.	PUNCT
ejpam-4687	318	32	consequently	consequently	ADV
ejpam-4687	318	33	,	,	PUNCT
ejpam-4687	318	34	γchgr	γchgr	PROPN
ejpam-4687	318	35	(	(	PUNCT
ejpam-4687	318	36	cn	cn	NOUN
ejpam-4687	318	37	)	)	PUNCT
ejpam-4687	318	38	=	=	PUNCT
ejpam-4687	318	39	n−	n−	NOUN
ejpam-4687	318	40	4	4	NUM
ejpam-4687	318	41	for	for	ADP
ejpam-4687	318	42	all	all	DET
ejpam-4687	318	43	even	even	ADV
ejpam-4687	318	44	integers	integer	NOUN
ejpam-4687	318	45	n	n	PRON
ejpam-4687	318	46	≥	≥	NUM
ejpam-4687	318	47	6	6	NUM
ejpam-4687	318	48	.	.	PUNCT
ejpam-4687	319	1	next	next	ADV
ejpam-4687	319	2	,	,	PUNCT
ejpam-4687	319	3	suppose	suppose	VERB
ejpam-4687	319	4	that	that	SCONJ
ejpam-4687	319	5	n	n	PROPN
ejpam-4687	319	6	≥	≥	X
ejpam-4687	319	7	7	7	NUM
ejpam-4687	319	8	and	and	CCONJ
ejpam-4687	319	9	is	be	AUX
ejpam-4687	319	10	odd	odd	ADJ
ejpam-4687	319	11	.	.	PUNCT
ejpam-4687	320	1	let	let	VERB
ejpam-4687	320	2	s	s	PRON
ejpam-4687	320	3	=	=	PUNCT
ejpam-4687	320	4	(	(	PUNCT
ejpam-4687	320	5	v1	v1	PROPN
ejpam-4687	320	6	,	,	PUNCT
ejpam-4687	320	7	v3	v3	PROPN
ejpam-4687	320	8	,	,	PUNCT
ejpam-4687	320	9	·	·	PUNCT
ejpam-4687	320	10	·	·	PUNCT
ejpam-4687	320	11	·	·	PUNCT
ejpam-4687	320	12	,	,	PUNCT
ejpam-4687	320	13	vn−4	vn−4	NOUN
ejpam-4687	320	14	,	,	PUNCT
ejpam-4687	320	15	vn−3	vn−3	PROPN
ejpam-4687	320	16	,	,	PUNCT
ejpam-4687	320	17	vn−5	vn−5	PROPN
ejpam-4687	320	18	,	,	PUNCT
ejpam-4687	320	19	·	·	PUNCT
ejpam-4687	320	20	·	·	PUNCT
ejpam-4687	320	21	·	·	PUNCT
ejpam-4687	320	22	,	,	PUNCT
ejpam-4687	320	23	v2	v2	PROPN
ejpam-4687	320	24	)	)	PUNCT
ejpam-4687	320	25	.	.	PUNCT
ejpam-4687	321	1	then	then	ADV
ejpam-4687	321	2	s	s	VERB
ejpam-4687	321	3	is	be	AUX
ejpam-4687	321	4	a	a	DET
ejpam-4687	321	5	maximum	maximum	ADV
ejpam-4687	321	6	connected	connect	VERB
ejpam-4687	321	7	grundy	grundy	PROPN
ejpam-4687	321	8	hop	hop	NOUN
ejpam-4687	321	9	dominating	dominating	NOUN
ejpam-4687	321	10	sequence	sequence	NOUN
ejpam-4687	321	11	of	of	ADP
ejpam-4687	321	12	cn	cn	PROPN
ejpam-4687	321	13	.	.	PUNCT
ejpam-4687	321	14	hence	hence	ADV
ejpam-4687	321	15	,	,	PUNCT
ejpam-4687	321	16	γchgr	γchgr	X
ejpam-4687	321	17	(	(	PUNCT
ejpam-4687	321	18	g	g	NOUN
ejpam-4687	321	19	)	)	PUNCT
ejpam-4687	321	20	=	=	PUNCT
ejpam-4687	321	21	n−	n−	NOUN
ejpam-4687	321	22	3	3	NUM
ejpam-4687	321	23	for	for	ADP
ejpam-4687	321	24	all	all	DET
ejpam-4687	321	25	n	n	PRON
ejpam-4687	321	26	≥	≥	NOUN
ejpam-4687	321	27	7	7	NUM
ejpam-4687	321	28	and	and	CCONJ
ejpam-4687	321	29	odd	odd	ADJ
ejpam-4687	321	30	.	.	PUNCT
ejpam-4687	322	1	lemma	lemma	PROPN
ejpam-4687	322	2	1	1	NUM
ejpam-4687	322	3	.	.	PUNCT
ejpam-4687	323	1	[	[	X
ejpam-4687	323	2	11	11	NUM
ejpam-4687	323	3	]	]	PUNCT
ejpam-4687	323	4	let	let	VERB
ejpam-4687	323	5	g	g	PRON
ejpam-4687	323	6	be	be	AUX
ejpam-4687	323	7	a	a	DET
ejpam-4687	323	8	non	non	ADJ
ejpam-4687	323	9	-	-	ADJ
ejpam-4687	323	10	trivial	trivial	ADJ
ejpam-4687	323	11	connected	connected	ADJ
ejpam-4687	323	12	graph	graph	NOUN
ejpam-4687	323	13	and	and	CCONJ
ejpam-4687	323	14	let	let	VERB
ejpam-4687	323	15	g1	g1	PROPN
ejpam-4687	323	16	and	and	CCONJ
ejpam-4687	323	17	g2	g2	PROPN
ejpam-4687	323	18	be	be	VERB
ejpam-4687	323	19	two	two	NUM
ejpam-4687	323	20	copies	copy	NOUN
ejpam-4687	323	21	of	of	ADP
ejpam-4687	323	22	g	g	NOUN
ejpam-4687	323	23	in	in	ADP
ejpam-4687	323	24	the	the	DET
ejpam-4687	323	25	graph	graph	NOUN
ejpam-4687	323	26	s(g	s(g	PROPN
ejpam-4687	323	27	)	)	PUNCT
ejpam-4687	323	28	.	.	PUNCT
ejpam-4687	324	1	if	if	SCONJ
ejpam-4687	324	2	w	w	PROPN
ejpam-4687	324	3	∈	∈	PROPN
ejpam-4687	324	4	v	v	X
ejpam-4687	324	5	(	(	PUNCT
ejpam-4687	324	6	g1	g1	PROPN
ejpam-4687	324	7	)	)	PUNCT
ejpam-4687	324	8	and	and	CCONJ
ejpam-4687	324	9	w′	w′	PROPN
ejpam-4687	324	10	∈	∈	PROPN
ejpam-4687	324	11	v	v	X
ejpam-4687	324	12	(	(	PUNCT
ejpam-4687	324	13	g2	g2	PROPN
ejpam-4687	324	14	)	)	PUNCT
ejpam-4687	324	15	is	be	AUX
ejpam-4687	324	16	the	the	DET
ejpam-4687	324	17	corresponding	corresponding	ADJ
ejpam-4687	324	18	vertex	vertex	NOUN
ejpam-4687	324	19	of	of	ADP
ejpam-4687	324	20	w	w	PROPN
ejpam-4687	324	21	,	,	PUNCT
ejpam-4687	324	22	then	then	ADV
ejpam-4687	324	23	n2	n2	PROPN
ejpam-4687	324	24	s(g)[w	s(g)[w	PROPN
ejpam-4687	324	25	]	]	PUNCT
ejpam-4687	324	26	=	=	SYM
ejpam-4687	324	27	n2	n2	PROPN
ejpam-4687	324	28	g1	g1	PROPN
ejpam-4687	325	1	[	[	X
ejpam-4687	325	2	w	w	X
ejpam-4687	325	3	]	]	X
ejpam-4687	325	4	∪n2	∪n2	PROPN
ejpam-4687	325	5	g2	g2	PROPN
ejpam-4687	326	1	[	[	X
ejpam-4687	326	2	w′	w′	ADP
ejpam-4687	326	3	]	]	X
ejpam-4687	326	4	=	=	SYM
ejpam-4687	326	5	n2	n2	ADJ
ejpam-4687	326	6	s(g)[w	s(g)[w	NOUN
ejpam-4687	326	7	′	′	NOUN
ejpam-4687	326	8	]	]	PUNCT
ejpam-4687	326	9	.	.	PUNCT
ejpam-4687	327	1	in	in	ADP
ejpam-4687	327	2	what	what	PRON
ejpam-4687	327	3	follows	follow	VERB
ejpam-4687	327	4	,	,	PUNCT
ejpam-4687	327	5	if	if	SCONJ
ejpam-4687	327	6	g1	g1	PROPN
ejpam-4687	327	7	and	and	CCONJ
ejpam-4687	327	8	g2	g2	PROPN
ejpam-4687	327	9	are	be	AUX
ejpam-4687	327	10	copies	copy	NOUN
ejpam-4687	327	11	of	of	ADP
ejpam-4687	327	12	g	g	NOUN
ejpam-4687	327	13	in	in	ADP
ejpam-4687	327	14	the	the	DET
ejpam-4687	327	15	shadow	shadow	NOUN
ejpam-4687	327	16	graph	graph	NOUN
ejpam-4687	327	17	s(g	s(g	PROPN
ejpam-4687	327	18	)	)	PUNCT
ejpam-4687	327	19	,	,	PUNCT
ejpam-4687	327	20	and	and	CCONJ
ejpam-4687	327	21	d	d	X
ejpam-4687	327	22	⊆	⊆	NUM
ejpam-4687	327	23	v	v	NOUN
ejpam-4687	327	24	(	(	PUNCT
ejpam-4687	327	25	g1	g1	PROPN
ejpam-4687	327	26	)	)	PUNCT
ejpam-4687	327	27	,	,	PUNCT
ejpam-4687	327	28	and	and	CCONJ
ejpam-4687	327	29	q	q	PRON
ejpam-4687	327	30	⊆	⊆	NUM
ejpam-4687	327	31	v	v	NOUN
ejpam-4687	327	32	(	(	PUNCT
ejpam-4687	327	33	g2	g2	PROPN
ejpam-4687	327	34	)	)	PUNCT
ejpam-4687	327	35	,	,	PUNCT
ejpam-4687	327	36	then	then	ADV
ejpam-4687	327	37	the	the	DET
ejpam-4687	327	38	sets	set	NOUN
ejpam-4687	327	39	d′	d′	PRON
ejpam-4687	327	40	and	and	CCONJ
ejpam-4687	327	41	q′	q′	NOUN
ejpam-4687	327	42	are	be	AUX
ejpam-4687	327	43	given	give	VERB
ejpam-4687	327	44	by	by	ADP
ejpam-4687	327	45	d′	d′	X
ejpam-4687	327	46	=	=	PUNCT
ejpam-4687	327	47	{	{	PUNCT
ejpam-4687	327	48	v′	v′	NOUN
ejpam-4687	327	49	∈	∈	PROPN
ejpam-4687	327	50	v	v	NOUN
ejpam-4687	327	51	(	(	PUNCT
ejpam-4687	327	52	g2	g2	PROPN
ejpam-4687	327	53	)	)	PUNCT
ejpam-4687	327	54	:	:	PUNCT
ejpam-4687	328	1	v	v	X
ejpam-4687	328	2	∈	∈	ADJ
ejpam-4687	328	3	d	d	NOUN
ejpam-4687	328	4	}	}	PUNCT
ejpam-4687	328	5	and	and	CCONJ
ejpam-4687	328	6	q′	q′	NOUN
ejpam-4687	328	7	=	=	SYM
ejpam-4687	328	8	{	{	PUNCT
ejpam-4687	328	9	w	w	PROPN
ejpam-4687	328	10	∈	∈	PROPN
ejpam-4687	328	11	v	v	NOUN
ejpam-4687	328	12	(	(	PUNCT
ejpam-4687	328	13	g1	g1	PROPN
ejpam-4687	328	14	)	)	PUNCT
ejpam-4687	328	15	:	:	PUNCT
ejpam-4687	328	16	w	w	X
ejpam-4687	328	17	∈	∈	PROPN
ejpam-4687	328	18	q	q	X
ejpam-4687	328	19	}	}	PUNCT
ejpam-4687	328	20	.	.	PUNCT
ejpam-4687	329	1	theorem	theorem	NOUN
ejpam-4687	329	2	4	4	NUM
ejpam-4687	329	3	.	.	PUNCT
ejpam-4687	330	1	let	let	VERB
ejpam-4687	330	2	g	g	PRON
ejpam-4687	330	3	be	be	AUX
ejpam-4687	330	4	a	a	DET
ejpam-4687	330	5	non	non	ADJ
ejpam-4687	330	6	-	-	ADJ
ejpam-4687	330	7	trivial	trivial	ADJ
ejpam-4687	330	8	connected	connected	ADJ
ejpam-4687	330	9	graph	graph	NOUN
ejpam-4687	330	10	and	and	CCONJ
ejpam-4687	330	11	let	let	VERB
ejpam-4687	330	12	g1	g1	PROPN
ejpam-4687	330	13	and	and	CCONJ
ejpam-4687	330	14	g2	g2	PROPN
ejpam-4687	330	15	be	be	VERB
ejpam-4687	330	16	copies	copy	NOUN
ejpam-4687	330	17	of	of	ADP
ejpam-4687	330	18	g	g	NOUN
ejpam-4687	330	19	in	in	ADP
ejpam-4687	330	20	the	the	DET
ejpam-4687	330	21	shadow	shadow	NOUN
ejpam-4687	330	22	graph	graph	NOUN
ejpam-4687	330	23	s(g	s(g	PROPN
ejpam-4687	330	24	)	)	PUNCT
ejpam-4687	330	25	.	.	PUNCT
ejpam-4687	331	1	then	then	ADV
ejpam-4687	331	2	c	c	PROPN
ejpam-4687	331	3	is	be	AUX
ejpam-4687	331	4	a	a	DET
ejpam-4687	331	5	connected	connected	ADJ
ejpam-4687	331	6	hop	hop	NOUN
ejpam-4687	331	7	dominating	dominating	NOUN
ejpam-4687	331	8	set	set	NOUN
ejpam-4687	331	9	of	of	ADP
ejpam-4687	331	10	s(g	s(g	PROPN
ejpam-4687	331	11	)	)	PUNCT
ejpam-4687	331	12	if	if	SCONJ
ejpam-4687	331	13	and	and	CCONJ
ejpam-4687	331	14	only	only	ADV
ejpam-4687	331	15	if	if	SCONJ
ejpam-4687	331	16	one	one	NUM
ejpam-4687	331	17	of	of	ADP
ejpam-4687	331	18	the	the	DET
ejpam-4687	331	19	following	follow	VERB
ejpam-4687	331	20	conditions	condition	NOUN
ejpam-4687	331	21	holds	hold	VERB
ejpam-4687	331	22	:	:	PUNCT
ejpam-4687	331	23	(	(	PUNCT
ejpam-4687	331	24	i	i	NOUN
ejpam-4687	331	25	)	)	PUNCT
ejpam-4687	331	26	c	c	PROPN
ejpam-4687	331	27	is	be	AUX
ejpam-4687	331	28	a	a	DET
ejpam-4687	331	29	connected	connected	ADJ
ejpam-4687	331	30	hop	hop	NOUN
ejpam-4687	331	31	dominating	dominating	NOUN
ejpam-4687	331	32	set	set	VERB
ejpam-4687	331	33	in	in	ADP
ejpam-4687	331	34	g1	g1	PROPN
ejpam-4687	331	35	.	.	PUNCT
ejpam-4687	332	1	(	(	PUNCT
ejpam-4687	332	2	ii	ii	NOUN
ejpam-4687	332	3	)	)	PUNCT
ejpam-4687	332	4	c	c	PROPN
ejpam-4687	332	5	is	be	AUX
ejpam-4687	332	6	a	a	DET
ejpam-4687	332	7	connected	connected	ADJ
ejpam-4687	332	8	hop	hop	NOUN
ejpam-4687	332	9	dominating	dominating	NOUN
ejpam-4687	332	10	set	set	NOUN
ejpam-4687	332	11	in	in	ADP
ejpam-4687	332	12	g2	g2	PROPN
ejpam-4687	332	13	.	.	PUNCT
ejpam-4687	333	1	(	(	PUNCT
ejpam-4687	333	2	iii	iii	X
ejpam-4687	333	3	)	)	PUNCT
ejpam-4687	333	4	c	c	NOUN
ejpam-4687	333	5	=	=	SYM
ejpam-4687	333	6	cg1	cg1	NOUN
ejpam-4687	333	7	∪	∪	NOUN
ejpam-4687	333	8	cg2	cg2	NOUN
ejpam-4687	333	9	such	such	ADJ
ejpam-4687	333	10	that	that	DET
ejpam-4687	333	11	cg1	cg1	NOUN
ejpam-4687	333	12	∪	∪	VERB
ejpam-4687	333	13	c	c	NOUN
ejpam-4687	333	14	′	′	NUM
ejpam-4687	334	1	g2	g2	PROPN
ejpam-4687	334	2	and	and	CCONJ
ejpam-4687	334	3	c	c	PROPN
ejpam-4687	334	4	′	′	PROPN
ejpam-4687	334	5	g1	g1	PROPN
ejpam-4687	334	6	∪	∪	ADJ
ejpam-4687	334	7	cg2	cg2	PROPN
ejpam-4687	334	8	are	be	AUX
ejpam-4687	334	9	connected	connect	VERB
ejpam-4687	334	10	hop	hop	ADJ
ejpam-4687	334	11	dominating	dominating	NOUN
ejpam-4687	334	12	sets	set	NOUN
ejpam-4687	334	13	of	of	ADP
ejpam-4687	334	14	g1	g1	NOUN
ejpam-4687	334	15	and	and	CCONJ
ejpam-4687	334	16	g2	g2	PROPN
ejpam-4687	334	17	,	,	PUNCT
ejpam-4687	334	18	respectively	respectively	ADV
ejpam-4687	334	19	.	.	PUNCT
ejpam-4687	335	1	j.	j.	PROPN
ejpam-4687	335	2	hassan	hassan	PROPN
ejpam-4687	335	3	,	,	PUNCT
ejpam-4687	335	4	s.	s.	PROPN
ejpam-4687	335	5	canoy	canoy	PROPN
ejpam-4687	335	6	jr	jr	PROPN
ejpam-4687	335	7	.	.	PROPN
ejpam-4687	335	8	/	/	SYM
ejpam-4687	335	9	eur	eur	PROPN
ejpam-4687	335	10	.	.	PUNCT
ejpam-4687	336	1	j.	j.	PROPN
ejpam-4687	336	2	pure	pure	PROPN
ejpam-4687	336	3	appl	appl	PROPN
ejpam-4687	336	4	.	.	PROPN
ejpam-4687	336	5	math	math	PROPN
ejpam-4687	336	6	,	,	PUNCT
ejpam-4687	336	7	16	16	NUM
ejpam-4687	336	8	(	(	PUNCT
ejpam-4687	336	9	2	2	NUM
ejpam-4687	336	10	)	)	PUNCT
ejpam-4687	336	11	(	(	PUNCT
ejpam-4687	336	12	2023	2023	NUM
ejpam-4687	336	13	)	)	PUNCT
ejpam-4687	336	14	,	,	PUNCT
ejpam-4687	336	15	1212	1212	NUM
ejpam-4687	336	16	-	-	SYM
ejpam-4687	336	17	1227	1227	NUM
ejpam-4687	336	18	1223	1223	NUM
ejpam-4687	336	19	proof	proof	NOUN
ejpam-4687	336	20	.	.	PUNCT
ejpam-4687	337	1	let	let	VERB
ejpam-4687	337	2	cg1	cg1	VERB
ejpam-4687	337	3	=	=	SYM
ejpam-4687	337	4	c	c	PROPN
ejpam-4687	337	5	∩	∩	X
ejpam-4687	337	6	v	v	X
ejpam-4687	337	7	(	(	PUNCT
ejpam-4687	337	8	g1	g1	PROPN
ejpam-4687	337	9	)	)	PUNCT
ejpam-4687	337	10	and	and	CCONJ
ejpam-4687	337	11	cg2	cg2	NOUN
ejpam-4687	337	12	=	=	SYM
ejpam-4687	337	13	c	c	PROPN
ejpam-4687	337	14	∩	∩	X
ejpam-4687	337	15	v	v	X
ejpam-4687	337	16	(	(	PUNCT
ejpam-4687	337	17	g2	g2	PROPN
ejpam-4687	337	18	)	)	PUNCT
ejpam-4687	337	19	.	.	PUNCT
ejpam-4687	338	1	if	if	SCONJ
ejpam-4687	338	2	cg2	cg2	NOUN
ejpam-4687	338	3	=	=	SYM
ejpam-4687	338	4	∅	∅	NOUN
ejpam-4687	338	5	,	,	PUNCT
ejpam-4687	338	6	then	then	ADV
ejpam-4687	338	7	c	c	NOUN
ejpam-4687	338	8	=	=	SYM
ejpam-4687	338	9	cg1	cg1	NOUN
ejpam-4687	338	10	is	be	AUX
ejpam-4687	338	11	a	a	DET
ejpam-4687	338	12	connected	connected	ADJ
ejpam-4687	338	13	hop	hop	NOUN
ejpam-4687	338	14	dominating	dominating	NOUN
ejpam-4687	338	15	set	set	NOUN
ejpam-4687	338	16	of	of	ADP
ejpam-4687	338	17	g1	g1	PROPN
ejpam-4687	338	18	.	.	PUNCT
ejpam-4687	339	1	if	if	SCONJ
ejpam-4687	339	2	cg1	cg1	NOUN
ejpam-4687	339	3	=	=	SYM
ejpam-4687	339	4	∅	∅	NOUN
ejpam-4687	339	5	,	,	PUNCT
ejpam-4687	339	6	then	then	ADV
ejpam-4687	339	7	c	c	NOUN
ejpam-4687	339	8	=	=	PUNCT
ejpam-4687	339	9	cg2	cg2	PROPN
ejpam-4687	339	10	is	be	AUX
ejpam-4687	339	11	a	a	DET
ejpam-4687	339	12	connected	connected	ADJ
ejpam-4687	339	13	hop	hop	NOUN
ejpam-4687	339	14	dominating	dominating	NOUN
ejpam-4687	339	15	set	set	NOUN
ejpam-4687	339	16	of	of	ADP
ejpam-4687	339	17	g2	g2	PROPN
ejpam-4687	339	18	.	.	PUNCT
ejpam-4687	340	1	hence	hence	ADV
ejpam-4687	340	2	,	,	PUNCT
ejpam-4687	340	3	(	(	PUNCT
ejpam-4687	340	4	i	i	NOUN
ejpam-4687	340	5	)	)	PUNCT
ejpam-4687	340	6	or	or	CCONJ
ejpam-4687	340	7	(	(	PUNCT
ejpam-4687	340	8	ii	ii	NOUN
ejpam-4687	340	9	)	)	PUNCT
ejpam-4687	340	10	holds	hold	VERB
ejpam-4687	340	11	.	.	PUNCT
ejpam-4687	341	1	next	next	ADV
ejpam-4687	341	2	,	,	PUNCT
ejpam-4687	341	3	suppose	suppose	VERB
ejpam-4687	341	4	cg1	cg1	NOUN
ejpam-4687	341	5	̸=	̸=	PROPN
ejpam-4687	341	6	∅	∅	NOUN
ejpam-4687	341	7	and	and	CCONJ
ejpam-4687	341	8	cg2	cg2	PROPN
ejpam-4687	341	9	̸=	̸=	PROPN
ejpam-4687	341	10	∅.	∅.	ADV
ejpam-4687	341	11	let	let	VERB
ejpam-4687	341	12	x	x	SYM
ejpam-4687	341	13	∈	∈	PROPN
ejpam-4687	341	14	v	v	X
ejpam-4687	341	15	(	(	PUNCT
ejpam-4687	341	16	g1	g1	PROPN
ejpam-4687	341	17	)	)	PUNCT
ejpam-4687	341	18	\	\	PUNCT
ejpam-4687	341	19	(	(	PUNCT
ejpam-4687	341	20	cg1	cg1	NOUN
ejpam-4687	341	21	∪	∪	ADP
ejpam-4687	341	22	c	c	NOUN
ejpam-4687	341	23	′	′	NUM
ejpam-4687	341	24	g2	g2	PROPN
ejpam-4687	341	25	)	)	PUNCT
ejpam-4687	341	26	.	.	PUNCT
ejpam-4687	342	1	then	then	ADV
ejpam-4687	342	2	x	x	SYM
ejpam-4687	342	3	∈	∈	PROPN
ejpam-4687	342	4	v	v	X
ejpam-4687	342	5	(	(	PUNCT
ejpam-4687	342	6	s(g	s(g	PROPN
ejpam-4687	342	7	)	)	PUNCT
ejpam-4687	342	8	)	)	PUNCT
ejpam-4687	342	9	\	\	PROPN
ejpam-4687	343	1	c.	c.	NOUN
ejpam-4687	343	2	since	since	SCONJ
ejpam-4687	343	3	c	c	PROPN
ejpam-4687	343	4	is	be	AUX
ejpam-4687	343	5	a	a	DET
ejpam-4687	343	6	hop	hop	NOUN
ejpam-4687	343	7	dominating	dominating	NOUN
ejpam-4687	343	8	of	of	ADP
ejpam-4687	343	9	s(g	s(g	PROPN
ejpam-4687	343	10	)	)	PUNCT
ejpam-4687	343	11	,	,	PUNCT
ejpam-4687	343	12	there	there	PRON
ejpam-4687	343	13	exists	exist	VERB
ejpam-4687	343	14	y	y	PROPN
ejpam-4687	343	15	∈	∈	PROPN
ejpam-4687	343	16	c	c	PROPN
ejpam-4687	343	17	such	such	ADJ
ejpam-4687	343	18	that	that	DET
ejpam-4687	343	19	ds(g)(x	ds(g)(x	NOUN
ejpam-4687	343	20	,	,	PUNCT
ejpam-4687	343	21	y	y	NOUN
ejpam-4687	343	22	)	)	PUNCT
ejpam-4687	343	23	=	=	SYM
ejpam-4687	344	1	2	2	X
ejpam-4687	344	2	.	.	X
ejpam-4687	345	1	if	if	SCONJ
ejpam-4687	345	2	y	y	PROPN
ejpam-4687	345	3	∈	∈	PROPN
ejpam-4687	345	4	cg1	cg1	INTJ
ejpam-4687	345	5	,	,	PUNCT
ejpam-4687	345	6	then	then	ADV
ejpam-4687	345	7	we	we	PRON
ejpam-4687	345	8	are	be	AUX
ejpam-4687	345	9	done	do	VERB
ejpam-4687	345	10	.	.	PUNCT
ejpam-4687	346	1	suppose	suppose	VERB
ejpam-4687	346	2	y	y	PROPN
ejpam-4687	346	3	∈	∈	PROPN
ejpam-4687	346	4	cg2	cg2	PROPN
ejpam-4687	346	5	,	,	PUNCT
ejpam-4687	346	6	say	say	VERB
ejpam-4687	346	7	y	y	PROPN
ejpam-4687	346	8	=	=	SYM
ejpam-4687	346	9	z′	z′	PROPN
ejpam-4687	346	10	,	,	PUNCT
ejpam-4687	346	11	where	where	SCONJ
ejpam-4687	346	12	z	z	PROPN
ejpam-4687	346	13	∈	∈	PROPN
ejpam-4687	346	14	v	v	NOUN
ejpam-4687	346	15	(	(	PUNCT
ejpam-4687	346	16	g1	g1	PROPN
ejpam-4687	346	17	)	)	PUNCT
ejpam-4687	346	18	.	.	PUNCT
ejpam-4687	347	1	then	then	ADV
ejpam-4687	347	2	z	z	PROPN
ejpam-4687	347	3	∈	∈	PROPN
ejpam-4687	347	4	c	c	NOUN
ejpam-4687	347	5	′	′	NUM
ejpam-4687	348	1	g2	g2	PROPN
ejpam-4687	348	2	and	and	CCONJ
ejpam-4687	348	3	ds(g)(x	ds(g)(x	PROPN
ejpam-4687	348	4	,	,	PUNCT
ejpam-4687	348	5	y	y	NOUN
ejpam-4687	348	6	)	)	PUNCT
ejpam-4687	348	7	=	=	SYM
ejpam-4687	348	8	dg1(x	dg1(x	PROPN
ejpam-4687	348	9	,	,	PUNCT
ejpam-4687	348	10	z	z	NOUN
ejpam-4687	348	11	)	)	PUNCT
ejpam-4687	348	12	=	=	SYM
ejpam-4687	348	13	2	2	NUM
ejpam-4687	348	14	by	by	ADP
ejpam-4687	348	15	lemma	lemma	PROPN
ejpam-4687	348	16	1	1	NUM
ejpam-4687	348	17	.	.	PUNCT
ejpam-4687	348	18	therefore	therefore	ADV
ejpam-4687	348	19	,	,	PUNCT
ejpam-4687	348	20	cg1	cg1	NOUN
ejpam-4687	348	21	∪	∪	ADP
ejpam-4687	348	22	c	c	NOUN
ejpam-4687	348	23	′	′	PUNCT
ejpam-4687	348	24	g2	g2	PROPN
ejpam-4687	348	25	is	be	AUX
ejpam-4687	348	26	a	a	DET
ejpam-4687	348	27	hop	hop	NOUN
ejpam-4687	348	28	dominating	dominating	NOUN
ejpam-4687	348	29	set	set	NOUN
ejpam-4687	348	30	of	of	ADP
ejpam-4687	348	31	g1	g1	PROPN
ejpam-4687	348	32	.	.	PUNCT
ejpam-4687	349	1	clearly	clearly	ADV
ejpam-4687	349	2	,	,	PUNCT
ejpam-4687	349	3	⟨cg1	⟨cg1	PROPN
ejpam-4687	349	4	∪	∪	VERB
ejpam-4687	349	5	c	c	PROPN
ejpam-4687	349	6	′	′	NUM
ejpam-4687	349	7	g2	g2	PROPN
ejpam-4687	349	8	⟩	⟩	PROPN
ejpam-4687	349	9	is	be	AUX
ejpam-4687	349	10	connected	connect	VERB
ejpam-4687	349	11	.	.	PUNCT
ejpam-4687	350	1	consequently	consequently	ADV
ejpam-4687	350	2	,	,	PUNCT
ejpam-4687	350	3	cg1	cg1	NOUN
ejpam-4687	350	4	∪	∪	ADP
ejpam-4687	350	5	c	c	NOUN
ejpam-4687	350	6	′	′	PUNCT
ejpam-4687	351	1	g2	g2	PROPN
ejpam-4687	351	2	is	be	AUX
ejpam-4687	351	3	a	a	DET
ejpam-4687	351	4	connected	connected	ADJ
ejpam-4687	351	5	hop	hop	NOUN
ejpam-4687	351	6	dominating	dominating	NOUN
ejpam-4687	351	7	set	set	NOUN
ejpam-4687	351	8	of	of	ADP
ejpam-4687	351	9	g1	g1	PROPN
ejpam-4687	351	10	.	.	PUNCT
ejpam-4687	352	1	similarly	similarly	ADV
ejpam-4687	352	2	,	,	PUNCT
ejpam-4687	352	3	c	c	NOUN
ejpam-4687	352	4	′	′	NUM
ejpam-4687	352	5	g1	g1	PROPN
ejpam-4687	352	6	∪	∪	ADP
ejpam-4687	352	7	cg2	cg2	PROPN
ejpam-4687	352	8	is	be	AUX
ejpam-4687	352	9	a	a	DET
ejpam-4687	352	10	connected	connected	ADJ
ejpam-4687	352	11	hop	hop	NOUN
ejpam-4687	352	12	dominating	dominating	NOUN
ejpam-4687	352	13	set	set	NOUN
ejpam-4687	352	14	of	of	ADP
ejpam-4687	352	15	g2	g2	PROPN
ejpam-4687	352	16	.	.	PUNCT
ejpam-4687	353	1	hence	hence	ADV
ejpam-4687	353	2	,	,	PUNCT
ejpam-4687	353	3	(	(	PUNCT
ejpam-4687	353	4	iii	iii	NOUN
ejpam-4687	353	5	)	)	PUNCT
ejpam-4687	353	6	holds	hold	VERB
ejpam-4687	353	7	.	.	PUNCT
ejpam-4687	354	1	conversely	conversely	ADV
ejpam-4687	354	2	,	,	PUNCT
ejpam-4687	354	3	suppose	suppose	VERB
ejpam-4687	354	4	(	(	PUNCT
ejpam-4687	354	5	i	i	NOUN
ejpam-4687	354	6	)	)	PUNCT
ejpam-4687	354	7	holds	hold	VERB
ejpam-4687	354	8	.	.	PUNCT
ejpam-4687	355	1	let	let	VERB
ejpam-4687	355	2	a	a	DET
ejpam-4687	355	3	∈	∈	PROPN
ejpam-4687	355	4	v	v	NOUN
ejpam-4687	355	5	(	(	PUNCT
ejpam-4687	355	6	s(g	s(g	PROPN
ejpam-4687	355	7	)	)	PUNCT
ejpam-4687	355	8	)	)	PUNCT
ejpam-4687	356	1	\	\	PROPN
ejpam-4687	356	2	c.	c.	NOUN
ejpam-4687	356	3	if	if	SCONJ
ejpam-4687	356	4	a	a	DET
ejpam-4687	356	5	∈	∈	PROPN
ejpam-4687	356	6	v	v	NOUN
ejpam-4687	356	7	(	(	PUNCT
ejpam-4687	356	8	g1	g1	PROPN
ejpam-4687	356	9	)	)	PUNCT
ejpam-4687	356	10	,	,	PUNCT
ejpam-4687	356	11	then	then	ADV
ejpam-4687	356	12	there	there	PRON
ejpam-4687	356	13	exists	exist	VERB
ejpam-4687	356	14	b	b	PROPN
ejpam-4687	356	15	∈	∈	PROPN
ejpam-4687	356	16	c	c	NOUN
ejpam-4687	356	17	such	such	ADJ
ejpam-4687	356	18	that	that	SCONJ
ejpam-4687	356	19	dg1(a	dg1(a	PROPN
ejpam-4687	356	20	,	,	PUNCT
ejpam-4687	356	21	b	b	NOUN
ejpam-4687	356	22	)	)	PUNCT
ejpam-4687	356	23	=	=	PUNCT
ejpam-4687	356	24	ds(g)(a	ds(g)(a	PROPN
ejpam-4687	356	25	,	,	PUNCT
ejpam-4687	356	26	b	b	NOUN
ejpam-4687	356	27	)	)	PUNCT
ejpam-4687	356	28	=	=	SYM
ejpam-4687	356	29	2	2	X
ejpam-4687	356	30	.	.	PUNCT
ejpam-4687	356	31	suppose	suppose	VERB
ejpam-4687	356	32	a	a	DET
ejpam-4687	356	33	∈	∈	PROPN
ejpam-4687	356	34	v	v	NOUN
ejpam-4687	356	35	(	(	PUNCT
ejpam-4687	356	36	g2	g2	PROPN
ejpam-4687	356	37	)	)	PUNCT
ejpam-4687	356	38	,	,	PUNCT
ejpam-4687	356	39	say	say	VERB
ejpam-4687	356	40	a	a	DET
ejpam-4687	356	41	=	=	SYM
ejpam-4687	356	42	u′	u′	PROPN
ejpam-4687	356	43	,	,	PUNCT
ejpam-4687	356	44	where	where	SCONJ
ejpam-4687	356	45	u	u	PROPN
ejpam-4687	356	46	∈	∈	PROPN
ejpam-4687	356	47	v	v	NOUN
ejpam-4687	356	48	(	(	PUNCT
ejpam-4687	356	49	g1	g1	PROPN
ejpam-4687	356	50	)	)	PUNCT
ejpam-4687	356	51	.	.	PUNCT
ejpam-4687	357	1	if	if	SCONJ
ejpam-4687	357	2	u	u	PROPN
ejpam-4687	357	3	∈	∈	PROPN
ejpam-4687	357	4	c	c	NOUN
ejpam-4687	357	5	,	,	PUNCT
ejpam-4687	357	6	then	then	ADV
ejpam-4687	357	7	dg1(a	dg1(a	PROPN
ejpam-4687	357	8	,	,	PUNCT
ejpam-4687	357	9	u	u	NOUN
ejpam-4687	357	10	)	)	PUNCT
ejpam-4687	357	11	=	=	SYM
ejpam-4687	357	12	ds(g)(a	ds(g)(a	PROPN
ejpam-4687	357	13	,	,	PUNCT
ejpam-4687	357	14	u	u	NOUN
ejpam-4687	357	15	)	)	PUNCT
ejpam-4687	357	16	=	=	SYM
ejpam-4687	358	1	2	2	X
ejpam-4687	358	2	.	.	X
ejpam-4687	359	1	if	if	SCONJ
ejpam-4687	359	2	u	u	PROPN
ejpam-4687	359	3	/∈	/∈	PUNCT
ejpam-4687	360	1	c	c	X
ejpam-4687	360	2	,	,	PUNCT
ejpam-4687	360	3	then	then	ADV
ejpam-4687	360	4	there	there	PRON
ejpam-4687	360	5	exists	exist	VERB
ejpam-4687	360	6	v	v	ADP
ejpam-4687	360	7	∈	∈	PROPN
ejpam-4687	360	8	c	c	NOUN
ejpam-4687	360	9	such	such	ADJ
ejpam-4687	360	10	that	that	DET
ejpam-4687	360	11	dg1(u	dg1(u	NOUN
ejpam-4687	360	12	,	,	PUNCT
ejpam-4687	360	13	v	v	NOUN
ejpam-4687	360	14	)	)	PUNCT
ejpam-4687	360	15	=	=	SYM
ejpam-4687	361	1	2	2	X
ejpam-4687	361	2	.	.	PUNCT
ejpam-4687	361	3	it	it	PRON
ejpam-4687	361	4	follows	follow	VERB
ejpam-4687	361	5	that	that	SCONJ
ejpam-4687	361	6	ds(g)(a	ds(g)(a	NOUN
ejpam-4687	361	7	,	,	PUNCT
ejpam-4687	361	8	u	u	NOUN
ejpam-4687	361	9	)	)	PUNCT
ejpam-4687	361	10	=	=	SYM
ejpam-4687	361	11	ds(g)(u	ds(g)(u	NUM
ejpam-4687	361	12	′	′	NUM
ejpam-4687	361	13	,	,	PUNCT
ejpam-4687	361	14	v	v	NOUN
ejpam-4687	361	15	)	)	PUNCT
ejpam-4687	362	1	=	=	SYM
ejpam-4687	362	2	2	2	X
ejpam-4687	362	3	.	.	X
ejpam-4687	362	4	therefore	therefore	ADV
ejpam-4687	362	5	,	,	PUNCT
ejpam-4687	362	6	c	c	PROPN
ejpam-4687	362	7	is	be	AUX
ejpam-4687	362	8	a	a	DET
ejpam-4687	362	9	hop	hop	NOUN
ejpam-4687	362	10	dominating	dominating	NOUN
ejpam-4687	362	11	set	set	NOUN
ejpam-4687	362	12	of	of	ADP
ejpam-4687	362	13	s(g	s(g	PROPN
ejpam-4687	362	14	)	)	PUNCT
ejpam-4687	362	15	.	.	PUNCT
ejpam-4687	363	1	clearly	clearly	ADV
ejpam-4687	363	2	,	,	PUNCT
ejpam-4687	363	3	⟨c⟩	⟨c⟩	PROPN
ejpam-4687	363	4	is	be	AUX
ejpam-4687	363	5	connected	connect	VERB
ejpam-4687	363	6	.	.	PUNCT
ejpam-4687	364	1	consequently	consequently	ADV
ejpam-4687	364	2	,	,	PUNCT
ejpam-4687	364	3	c	c	PROPN
ejpam-4687	364	4	is	be	AUX
ejpam-4687	364	5	a	a	DET
ejpam-4687	364	6	connected	connected	ADJ
ejpam-4687	364	7	hop	hop	NOUN
ejpam-4687	364	8	dominating	dominating	NOUN
ejpam-4687	364	9	set	set	NOUN
ejpam-4687	364	10	of	of	ADP
ejpam-4687	364	11	s(g	s(g	PROPN
ejpam-4687	364	12	)	)	PUNCT
ejpam-4687	364	13	.	.	PUNCT
ejpam-4687	365	1	similarly	similarly	ADV
ejpam-4687	365	2	,	,	PUNCT
ejpam-4687	365	3	if	if	SCONJ
ejpam-4687	365	4	(	(	PUNCT
ejpam-4687	365	5	ii	ii	NOUN
ejpam-4687	365	6	)	)	PUNCT
ejpam-4687	365	7	holds	hold	VERB
ejpam-4687	365	8	,	,	PUNCT
ejpam-4687	365	9	then	then	ADV
ejpam-4687	365	10	c	c	PROPN
ejpam-4687	365	11	is	be	AUX
ejpam-4687	365	12	a	a	DET
ejpam-4687	365	13	connected	connected	ADJ
ejpam-4687	365	14	hop	hop	NOUN
ejpam-4687	365	15	dominating	dominating	NOUN
ejpam-4687	365	16	set	set	NOUN
ejpam-4687	365	17	of	of	ADP
ejpam-4687	365	18	s(g	s(g	PROPN
ejpam-4687	365	19	)	)	PUNCT
ejpam-4687	365	20	.	.	PUNCT
ejpam-4687	366	1	next	next	ADV
ejpam-4687	366	2	,	,	PUNCT
ejpam-4687	366	3	suppose	suppose	VERB
ejpam-4687	366	4	that	that	SCONJ
ejpam-4687	366	5	(	(	PUNCT
ejpam-4687	366	6	iii	iii	NOUN
ejpam-4687	366	7	)	)	PUNCT
ejpam-4687	366	8	holds	hold	VERB
ejpam-4687	366	9	.	.	PUNCT
ejpam-4687	367	1	let	let	VERB
ejpam-4687	367	2	x	x	SYM
ejpam-4687	367	3	∈	∈	PROPN
ejpam-4687	367	4	v	v	X
ejpam-4687	367	5	(	(	PUNCT
ejpam-4687	367	6	s(g	s(g	PROPN
ejpam-4687	367	7	)	)	PUNCT
ejpam-4687	367	8	)	)	PUNCT
ejpam-4687	367	9	\	\	PROPN
ejpam-4687	368	1	c.	c.	NOUN
ejpam-4687	368	2	then	then	ADV
ejpam-4687	368	3	x	x	X
ejpam-4687	368	4	/∈	/∈	PUNCT
ejpam-4687	368	5	cg1	cg1	NOUN
ejpam-4687	368	6	∪	∪	NOUN
ejpam-4687	368	7	cg2	cg2	NOUN
ejpam-4687	368	8	.	.	PUNCT
ejpam-4687	369	1	suppose	suppose	VERB
ejpam-4687	369	2	x	x	SYM
ejpam-4687	369	3	∈	∈	PROPN
ejpam-4687	369	4	v	v	X
ejpam-4687	369	5	(	(	PUNCT
ejpam-4687	369	6	g2	g2	PROPN
ejpam-4687	369	7	)	)	PUNCT
ejpam-4687	369	8	\	\	PROPN
ejpam-4687	369	9	cg2	cg2	PROPN
ejpam-4687	369	10	,	,	PUNCT
ejpam-4687	369	11	say	say	VERB
ejpam-4687	369	12	x	x	X
ejpam-4687	369	13	=	=	SYM
ejpam-4687	369	14	y′	y′	NOUN
ejpam-4687	369	15	,	,	PUNCT
ejpam-4687	369	16	where	where	SCONJ
ejpam-4687	369	17	y	y	PROPN
ejpam-4687	369	18	∈	∈	PROPN
ejpam-4687	369	19	v	v	PROPN
ejpam-4687	369	20	(	(	PUNCT
ejpam-4687	369	21	g1	g1	PROPN
ejpam-4687	369	22	)	)	PUNCT
ejpam-4687	369	23	.	.	PUNCT
ejpam-4687	370	1	then	then	ADV
ejpam-4687	370	2	y	y	PROPN
ejpam-4687	370	3	/∈	/∈	PUNCT
ejpam-4687	371	1	c	c	NOUN
ejpam-4687	371	2	′	′	NUM
ejpam-4687	372	1	g2	g2	PROPN
ejpam-4687	372	2	.	.	PUNCT
ejpam-4687	373	1	if	if	SCONJ
ejpam-4687	373	2	y	y	PROPN
ejpam-4687	373	3	∈	∈	PROPN
ejpam-4687	373	4	cg1	cg1	INTJ
ejpam-4687	373	5	,	,	PUNCT
ejpam-4687	373	6	then	then	ADV
ejpam-4687	373	7	ds(g)(x	ds(g)(x	PROPN
ejpam-4687	373	8	,	,	PUNCT
ejpam-4687	373	9	y	y	NOUN
ejpam-4687	373	10	)	)	PUNCT
ejpam-4687	373	11	=	=	SYM
ejpam-4687	373	12	ds(g)(y	ds(g)(y	ADJ
ejpam-4687	373	13	′	′	NOUN
ejpam-4687	373	14	,	,	PUNCT
ejpam-4687	373	15	y	y	NOUN
ejpam-4687	373	16	)	)	PUNCT
ejpam-4687	373	17	=	=	SYM
ejpam-4687	373	18	2	2	X
ejpam-4687	373	19	.	.	X
ejpam-4687	373	20	suppose	suppose	VERB
ejpam-4687	373	21	y	y	PRON
ejpam-4687	373	22	/∈	/∈	PUNCT
ejpam-4687	373	23	cg1	cg1	INTJ
ejpam-4687	373	24	.	.	PUNCT
ejpam-4687	374	1	since	since	SCONJ
ejpam-4687	374	2	cg1	cg1	NOUN
ejpam-4687	374	3	∪	∪	VERB
ejpam-4687	374	4	c	c	NOUN
ejpam-4687	374	5	′	′	PUNCT
ejpam-4687	375	1	g2	g2	PROPN
ejpam-4687	375	2	is	be	AUX
ejpam-4687	375	3	a	a	DET
ejpam-4687	375	4	hop	hop	NOUN
ejpam-4687	375	5	dominating	dominating	NOUN
ejpam-4687	375	6	set	set	NOUN
ejpam-4687	375	7	of	of	ADP
ejpam-4687	375	8	g1	g1	NOUN
ejpam-4687	375	9	,	,	PUNCT
ejpam-4687	375	10	there	there	PRON
ejpam-4687	375	11	exists	exist	VERB
ejpam-4687	375	12	w	w	PROPN
ejpam-4687	375	13	∈	∈	NOUN
ejpam-4687	375	14	cg1	cg1	NOUN
ejpam-4687	375	15	∪	∪	PROPN
ejpam-4687	375	16	c	c	NOUN
ejpam-4687	375	17	′	′	NUM
ejpam-4687	375	18	g2	g2	PROPN
ejpam-4687	375	19	such	such	ADJ
ejpam-4687	375	20	that	that	SCONJ
ejpam-4687	375	21	dg1(w	dg1(w	PROPN
ejpam-4687	375	22	,	,	PUNCT
ejpam-4687	375	23	y	y	NOUN
ejpam-4687	375	24	)	)	PUNCT
ejpam-4687	375	25	=	=	SYM
ejpam-4687	375	26	2	2	NUM
ejpam-4687	375	27	=	=	SYM
ejpam-4687	375	28	ds(g)(w	ds(g)(w	PROPN
ejpam-4687	375	29	,	,	PUNCT
ejpam-4687	375	30	y	y	NOUN
ejpam-4687	375	31	)	)	PUNCT
ejpam-4687	375	32	.	.	PUNCT
ejpam-4687	376	1	if	if	SCONJ
ejpam-4687	376	2	w	w	PROPN
ejpam-4687	376	3	∈	∈	PROPN
ejpam-4687	376	4	cg1	cg1	NOUN
ejpam-4687	376	5	,	,	PUNCT
ejpam-4687	376	6	then	then	ADV
ejpam-4687	376	7	w	w	PROPN
ejpam-4687	376	8	∈	∈	PROPN
ejpam-4687	376	9	c	c	PROPN
ejpam-4687	376	10	and	and	CCONJ
ejpam-4687	376	11	ds(g)(w	ds(g)(w	NOUN
ejpam-4687	376	12	,	,	PUNCT
ejpam-4687	376	13	y	y	PROPN
ejpam-4687	376	14	′	′	NOUN
ejpam-4687	376	15	)	)	PUNCT
ejpam-4687	376	16	=	=	SYM
ejpam-4687	376	17	2	2	NUM
ejpam-4687	376	18	by	by	ADP
ejpam-4687	376	19	lemma	lemma	PROPN
ejpam-4687	376	20	1	1	NUM
ejpam-4687	376	21	.	.	PUNCT
ejpam-4687	377	1	if	if	SCONJ
ejpam-4687	377	2	w	w	PROPN
ejpam-4687	377	3	∈	∈	PROPN
ejpam-4687	377	4	c	c	NOUN
ejpam-4687	377	5	′	′	NUM
ejpam-4687	377	6	g2	g2	PROPN
ejpam-4687	377	7	,	,	PUNCT
ejpam-4687	377	8	then	then	ADV
ejpam-4687	377	9	w′	w′	PROPN
ejpam-4687	377	10	∈	∈	PROPN
ejpam-4687	377	11	cg2	cg2	VERB
ejpam-4687	377	12	⊆	⊆	NUM
ejpam-4687	377	13	c	c	PROPN
ejpam-4687	377	14	and	and	CCONJ
ejpam-4687	377	15	dg2(w	dg2(w	PROPN
ejpam-4687	377	16	′	′	NUM
ejpam-4687	377	17	,	,	PUNCT
ejpam-4687	377	18	y′	y′	NUM
ejpam-4687	377	19	)	)	PUNCT
ejpam-4687	378	1	=	=	SYM
ejpam-4687	378	2	ds(g)(w	ds(g)(w	NOUN
ejpam-4687	378	3	′	′	NOUN
ejpam-4687	378	4	,	,	PUNCT
ejpam-4687	378	5	y′	y′	NUM
ejpam-4687	378	6	)	)	PUNCT
ejpam-4687	378	7	=	=	SYM
ejpam-4687	378	8	2	2	NUM
ejpam-4687	378	9	by	by	ADP
ejpam-4687	378	10	lemma	lemma	PROPN
ejpam-4687	378	11	1	1	NUM
ejpam-4687	378	12	.	.	PUNCT
ejpam-4687	379	1	therefore	therefore	ADV
ejpam-4687	379	2	,	,	PUNCT
ejpam-4687	379	3	c	c	PROPN
ejpam-4687	379	4	is	be	AUX
ejpam-4687	379	5	a	a	DET
ejpam-4687	379	6	hop	hop	NOUN
ejpam-4687	379	7	dominating	dominating	NOUN
ejpam-4687	379	8	set	set	NOUN
ejpam-4687	379	9	of	of	ADP
ejpam-4687	379	10	s(g	s(g	PROPN
ejpam-4687	379	11	)	)	PUNCT
ejpam-4687	379	12	.	.	PUNCT
ejpam-4687	380	1	clearly	clearly	ADV
ejpam-4687	380	2	,	,	PUNCT
ejpam-4687	380	3	⟨c⟩	⟨c⟩	PROPN
ejpam-4687	380	4	is	be	AUX
ejpam-4687	380	5	connected	connect	VERB
ejpam-4687	380	6	.	.	PUNCT
ejpam-4687	381	1	consequently	consequently	ADV
ejpam-4687	381	2	,	,	PUNCT
ejpam-4687	381	3	c	c	PROPN
ejpam-4687	381	4	is	be	AUX
ejpam-4687	381	5	a	a	DET
ejpam-4687	381	6	connected	connected	ADJ
ejpam-4687	381	7	hop	hop	NOUN
ejpam-4687	381	8	dominating	dominating	NOUN
ejpam-4687	381	9	set	set	NOUN
ejpam-4687	381	10	of	of	ADP
ejpam-4687	381	11	s(g	s(g	PROPN
ejpam-4687	381	12	)	)	PUNCT
ejpam-4687	381	13	.	.	PUNCT
ejpam-4687	382	1	the	the	DET
ejpam-4687	382	2	next	next	ADJ
ejpam-4687	382	3	result	result	NOUN
ejpam-4687	382	4	follows	follow	VERB
ejpam-4687	382	5	from	from	ADP
ejpam-4687	382	6	theorem	theorem	ADJ
ejpam-4687	382	7	4	4	NUM
ejpam-4687	382	8	.	.	PUNCT
ejpam-4687	382	9	corollary	corollary	ADJ
ejpam-4687	382	10	6	6	NUM
ejpam-4687	382	11	.	.	PUNCT
ejpam-4687	383	1	let	let	VERB
ejpam-4687	383	2	g	g	PRON
ejpam-4687	383	3	be	be	AUX
ejpam-4687	383	4	a	a	DET
ejpam-4687	383	5	non	non	ADJ
ejpam-4687	383	6	-	-	ADJ
ejpam-4687	383	7	trivial	trivial	ADJ
ejpam-4687	383	8	connected	connected	ADJ
ejpam-4687	383	9	graph	graph	NOUN
ejpam-4687	383	10	and	and	CCONJ
ejpam-4687	383	11	let	let	VERB
ejpam-4687	383	12	g1	g1	PROPN
ejpam-4687	383	13	and	and	CCONJ
ejpam-4687	383	14	g2	g2	PROPN
ejpam-4687	383	15	be	be	VERB
ejpam-4687	383	16	copies	copy	NOUN
ejpam-4687	383	17	of	of	ADP
ejpam-4687	383	18	g	g	NOUN
ejpam-4687	383	19	in	in	ADP
ejpam-4687	383	20	the	the	DET
ejpam-4687	383	21	shadow	shadow	NOUN
ejpam-4687	383	22	graph	graph	NOUN
ejpam-4687	383	23	s(g	s(g	PROPN
ejpam-4687	383	24	)	)	PUNCT
ejpam-4687	383	25	.	.	PUNCT
ejpam-4687	384	1	then	then	ADV
ejpam-4687	384	2	γch(s(g	γch(s(g	NOUN
ejpam-4687	384	3	)	)	PUNCT
ejpam-4687	384	4	)	)	PUNCT
ejpam-4687	385	1	=	=	SYM
ejpam-4687	385	2	γch(g	γch(g	NOUN
ejpam-4687	385	3	)	)	PUNCT
ejpam-4687	385	4	.	.	PUNCT
ejpam-4687	386	1	theorem	theorem	NOUN
ejpam-4687	386	2	5	5	NUM
ejpam-4687	386	3	.	.	PUNCT
ejpam-4687	387	1	let	let	VERB
ejpam-4687	387	2	g	g	PRON
ejpam-4687	387	3	be	be	AUX
ejpam-4687	387	4	a	a	DET
ejpam-4687	387	5	non	non	ADJ
ejpam-4687	387	6	-	-	ADJ
ejpam-4687	387	7	trivial	trivial	ADJ
ejpam-4687	387	8	connected	connected	ADJ
ejpam-4687	387	9	graph	graph	NOUN
ejpam-4687	387	10	and	and	CCONJ
ejpam-4687	387	11	let	let	VERB
ejpam-4687	387	12	g1	g1	PROPN
ejpam-4687	387	13	and	and	CCONJ
ejpam-4687	387	14	g2	g2	PROPN
ejpam-4687	387	15	be	be	VERB
ejpam-4687	387	16	copies	copy	NOUN
ejpam-4687	387	17	of	of	ADP
ejpam-4687	387	18	g	g	NOUN
ejpam-4687	387	19	in	in	ADP
ejpam-4687	387	20	the	the	DET
ejpam-4687	387	21	shadow	shadow	NOUN
ejpam-4687	387	22	graph	graph	NOUN
ejpam-4687	387	23	s(g	s(g	PROPN
ejpam-4687	387	24	)	)	PUNCT
ejpam-4687	387	25	.	.	PUNCT
ejpam-4687	388	1	if	if	SCONJ
ejpam-4687	388	2	s	s	NOUN
ejpam-4687	388	3	is	be	AUX
ejpam-4687	388	4	a	a	DET
ejpam-4687	388	5	connected	connect	VERB
ejpam-4687	388	6	grundy	grundy	PROPN
ejpam-4687	388	7	hop	hop	NOUN
ejpam-4687	388	8	dominating	dominating	NOUN
ejpam-4687	388	9	sequence	sequence	NOUN
ejpam-4687	388	10	of	of	ADP
ejpam-4687	388	11	g1	g1	NOUN
ejpam-4687	388	12	or	or	CCONJ
ejpam-4687	388	13	g2	g2	PROPN
ejpam-4687	388	14	,	,	PUNCT
ejpam-4687	388	15	then	then	ADV
ejpam-4687	388	16	s	s	VERB
ejpam-4687	388	17	is	be	AUX
ejpam-4687	388	18	a	a	DET
ejpam-4687	388	19	connected	connect	VERB
ejpam-4687	388	20	grundy	grundy	PROPN
ejpam-4687	388	21	hop	hop	NOUN
ejpam-4687	388	22	dominating	dominating	NOUN
ejpam-4687	388	23	sequence	sequence	NOUN
ejpam-4687	388	24	of	of	ADP
ejpam-4687	388	25	s(g	s(g	PROPN
ejpam-4687	388	26	)	)	PUNCT
ejpam-4687	388	27	.	.	PUNCT
ejpam-4687	389	1	in	in	ADP
ejpam-4687	389	2	particular	particular	ADJ
ejpam-4687	389	3	,	,	PUNCT
ejpam-4687	389	4	γchgr	γchgr	X
ejpam-4687	389	5	(	(	PUNCT
ejpam-4687	389	6	g	g	NOUN
ejpam-4687	389	7	)	)	PUNCT
ejpam-4687	389	8	≤	≤	NOUN
ejpam-4687	389	9	γchgr	γchgr	NOUN
ejpam-4687	389	10	(	(	PUNCT
ejpam-4687	389	11	s(g	s(g	PROPN
ejpam-4687	389	12	)	)	PUNCT
ejpam-4687	389	13	)	)	PUNCT
ejpam-4687	389	14	.	.	PUNCT
ejpam-4687	390	1	proof	proof	NOUN
ejpam-4687	390	2	.	.	PUNCT
ejpam-4687	391	1	suppose	suppose	VERB
ejpam-4687	391	2	s	s	X
ejpam-4687	391	3	=	=	SYM
ejpam-4687	391	4	(	(	PUNCT
ejpam-4687	391	5	v1	v1	PROPN
ejpam-4687	391	6	,	,	PUNCT
ejpam-4687	391	7	v2	v2	NOUN
ejpam-4687	391	8	,	,	PUNCT
ejpam-4687	391	9	.	.	PUNCT
ejpam-4687	391	10	.	.	PUNCT
ejpam-4687	392	1	.	.	PUNCT
ejpam-4687	393	1	,	,	PUNCT
ejpam-4687	393	2	vk	vk	PROPN
ejpam-4687	393	3	)	)	PUNCT
ejpam-4687	393	4	is	be	AUX
ejpam-4687	393	5	a	a	DET
ejpam-4687	393	6	connected	connected	ADJ
ejpam-4687	393	7	grundy	grundy	PROPN
ejpam-4687	393	8	hop	hop	NOUN
ejpam-4687	393	9	dominating	dominating	NOUN
ejpam-4687	393	10	sequence	sequence	NOUN
ejpam-4687	393	11	of	of	ADP
ejpam-4687	393	12	g1	g1	PROPN
ejpam-4687	393	13	.	.	PUNCT
ejpam-4687	394	1	then	then	ADV
ejpam-4687	394	2	ŝ	ŝ	PROPN
ejpam-4687	394	3	is	be	AUX
ejpam-4687	394	4	a	a	DET
ejpam-4687	394	5	connected	connected	ADJ
ejpam-4687	394	6	hop	hop	NOUN
ejpam-4687	394	7	dominating	dominating	NOUN
ejpam-4687	394	8	set	set	NOUN
ejpam-4687	394	9	of	of	ADP
ejpam-4687	394	10	g1	g1	NOUN
ejpam-4687	394	11	.	.	PUNCT
ejpam-4687	395	1	hence	hence	ADV
ejpam-4687	395	2	,	,	PUNCT
ejpam-4687	395	3	ŝ	ŝ	X
ejpam-4687	395	4	is	be	AUX
ejpam-4687	395	5	a	a	DET
ejpam-4687	395	6	connected	connected	ADJ
ejpam-4687	395	7	hop	hop	NOUN
ejpam-4687	395	8	dominating	dominating	NOUN
ejpam-4687	395	9	set	set	NOUN
ejpam-4687	395	10	of	of	ADP
ejpam-4687	395	11	s(g	s(g	PROPN
ejpam-4687	395	12	)	)	PUNCT
ejpam-4687	395	13	by	by	ADP
ejpam-4687	395	14	theorem	theorem	NOUN
ejpam-4687	395	15	4	4	NUM
ejpam-4687	395	16	.	.	PUNCT
ejpam-4687	396	1	let	let	VERB
ejpam-4687	396	2	i	i	PRON
ejpam-4687	396	3	∈	∈	PROPN
ejpam-4687	396	4	{	{	PUNCT
ejpam-4687	396	5	2	2	NUM
ejpam-4687	396	6	,	,	PUNCT
ejpam-4687	396	7	3	3	NUM
ejpam-4687	396	8	,	,	PUNCT
ejpam-4687	396	9	.	.	PUNCT
ejpam-4687	396	10	.	.	PUNCT
ejpam-4687	397	1	.	.	PUNCT
ejpam-4687	398	1	,	,	PUNCT
ejpam-4687	398	2	k	k	X
ejpam-4687	398	3	}	}	PUNCT
ejpam-4687	398	4	.	.	PUNCT
ejpam-4687	399	1	since	since	SCONJ
ejpam-4687	399	2	n2	n2	ADJ
ejpam-4687	399	3	g1	g1	PROPN
ejpam-4687	399	4	[	[	X
ejpam-4687	399	5	vi]∩(∪i−1	vi]∩(∪i−1	PROPN
ejpam-4687	399	6	j=1n	j=1n	PROPN
ejpam-4687	399	7	2	2	NUM
ejpam-4687	399	8	g2	g2	PROPN
ejpam-4687	400	1	[	[	X
ejpam-4687	400	2	v′j	v′j	X
ejpam-4687	400	3	]	]	X
ejpam-4687	400	4	)	)	PUNCT
ejpam-4687	400	5	=	=	SYM
ejpam-4687	400	6	∅	∅	NOUN
ejpam-4687	400	7	and	and	CCONJ
ejpam-4687	400	8	n2	n2	ADJ
ejpam-4687	400	9	g2	g2	PROPN
ejpam-4687	401	1	[	[	X
ejpam-4687	401	2	v′i	v′i	X
ejpam-4687	401	3	]	]	X
ejpam-4687	401	4	∩	∩	NOUN
ejpam-4687	401	5	(	(	PUNCT
ejpam-4687	401	6	∪i−1	∪i−1	X
ejpam-4687	401	7	j=1n	j=1n	PROPN
ejpam-4687	401	8	2	2	NUM
ejpam-4687	401	9	g1	g1	NOUN
ejpam-4687	401	10	[	[	X
ejpam-4687	401	11	vj	vj	X
ejpam-4687	401	12	]	]	X
ejpam-4687	401	13	)	)	PUNCT
ejpam-4687	401	14	=	=	SYM
ejpam-4687	401	15	∅	∅	NOUN
ejpam-4687	401	16	,	,	PUNCT
ejpam-4687	401	17	lemma	lemma	PROPN
ejpam-4687	401	18	1	1	NUM
ejpam-4687	401	19	implies	imply	VERB
ejpam-4687	401	20	that	that	DET
ejpam-4687	401	21	n2	n2	PROPN
ejpam-4687	401	22	s(g)[vi	s(g)[vi	PROPN
ejpam-4687	401	23	]	]	X
ejpam-4687	401	24	\	\	PROPN
ejpam-4687	401	25	∪	∪	X
ejpam-4687	401	26	i−1	i−1	PROPN
ejpam-4687	401	27	j=1n	j=1n	PROPN
ejpam-4687	401	28	2	2	NUM
ejpam-4687	401	29	s(g)[vj	s(g)[vj	NOUN
ejpam-4687	401	30	]	]	PUNCT
ejpam-4687	401	31	=	=	SYM
ejpam-4687	401	32	(	(	PUNCT
ejpam-4687	401	33	n2	n2	PROPN
ejpam-4687	401	34	g1	g1	PROPN
ejpam-4687	401	35	[	[	X
ejpam-4687	401	36	vi	vi	X
ejpam-4687	401	37	]	]	X
ejpam-4687	401	38	\	\	PUNCT
ejpam-4687	402	1	(	(	PUNCT
ejpam-4687	402	2	∪i−1	∪i−1	PROPN
ejpam-4687	402	3	j=1n	j=1n	PROPN
ejpam-4687	402	4	2	2	NUM
ejpam-4687	402	5	g1	g1	NOUN
ejpam-4687	402	6	[	[	X
ejpam-4687	402	7	vj	vj	X
ejpam-4687	402	8	]	]	X
ejpam-4687	402	9	)	)	PUNCT
ejpam-4687	402	10	∪	∪	NOUN
ejpam-4687	402	11	(	(	PUNCT
ejpam-4687	402	12	n2	n2	PROPN
ejpam-4687	402	13	g2	g2	PROPN
ejpam-4687	403	1	[	[	X
ejpam-4687	403	2	v′i	v′i	X
ejpam-4687	403	3	]	]	X
ejpam-4687	403	4	\	\	PUNCT
ejpam-4687	403	5	(	(	PUNCT
ejpam-4687	403	6	∪i−1	∪i−1	PROPN
ejpam-4687	403	7	j=1n	j=1n	PROPN
ejpam-4687	403	8	2	2	NUM
ejpam-4687	403	9	g2	g2	PROPN
ejpam-4687	404	1	[	[	X
ejpam-4687	404	2	v′j	v′j	X
ejpam-4687	404	3	]	]	X
ejpam-4687	404	4	)	)	PUNCT
ejpam-4687	404	5	.	.	PUNCT
ejpam-4687	405	1	by	by	ADP
ejpam-4687	405	2	the	the	DET
ejpam-4687	405	3	legality	legality	NOUN
ejpam-4687	405	4	property	property	NOUN
ejpam-4687	405	5	of	of	ADP
ejpam-4687	405	6	s	s	NOUN
ejpam-4687	405	7	,	,	PUNCT
ejpam-4687	405	8	∅	∅	NOUN
ejpam-4687	405	9	̸=	̸=	PROPN
ejpam-4687	405	10	n2	n2	NOUN
ejpam-4687	405	11	g1	g1	PROPN
ejpam-4687	405	12	[	[	X
ejpam-4687	405	13	vi	vi	X
ejpam-4687	405	14	]	]	PUNCT
ejpam-4687	405	15	\	\	X
ejpam-4687	405	16	∪i−1	∪i−1	PROPN
ejpam-4687	405	17	j=1n	j=1n	PROPN
ejpam-4687	405	18	2	2	NUM
ejpam-4687	405	19	g1	g1	NOUN
ejpam-4687	405	20	[	[	X
ejpam-4687	405	21	vj	vj	X
ejpam-4687	405	22	]	]	PUNCT
ejpam-4687	405	23	⊆	⊆	NUM
ejpam-4687	405	24	n2	n2	NOUN
ejpam-4687	405	25	s(g)[vi	s(g)[vi	PROPN
ejpam-4687	405	26	]	]	X
ejpam-4687	405	27	\	\	PROPN
ejpam-4687	405	28	∪	∪	X
ejpam-4687	405	29	i−1	i−1	PROPN
ejpam-4687	405	30	j=1n	j=1n	PROPN
ejpam-4687	405	31	2	2	NUM
ejpam-4687	405	32	s(g)[vj	s(g)[vj	NOUN
ejpam-4687	405	33	]	]	PUNCT
ejpam-4687	405	34	.	.	PUNCT
ejpam-4687	406	1	j.	j.	PROPN
ejpam-4687	406	2	hassan	hassan	PROPN
ejpam-4687	406	3	,	,	PUNCT
ejpam-4687	406	4	s.	s.	PROPN
ejpam-4687	406	5	canoy	canoy	PROPN
ejpam-4687	406	6	jr	jr	PROPN
ejpam-4687	406	7	.	.	PROPN
ejpam-4687	406	8	/	/	SYM
ejpam-4687	406	9	eur	eur	PROPN
ejpam-4687	406	10	.	.	PUNCT
ejpam-4687	407	1	j.	j.	PROPN
ejpam-4687	407	2	pure	pure	PROPN
ejpam-4687	407	3	appl	appl	PROPN
ejpam-4687	407	4	.	.	PROPN
ejpam-4687	407	5	math	math	PROPN
ejpam-4687	407	6	,	,	PUNCT
ejpam-4687	407	7	16	16	NUM
ejpam-4687	407	8	(	(	PUNCT
ejpam-4687	407	9	2	2	NUM
ejpam-4687	407	10	)	)	PUNCT
ejpam-4687	407	11	(	(	PUNCT
ejpam-4687	407	12	2023	2023	NUM
ejpam-4687	407	13	)	)	PUNCT
ejpam-4687	407	14	,	,	PUNCT
ejpam-4687	407	15	1212	1212	NUM
ejpam-4687	407	16	-	-	SYM
ejpam-4687	407	17	1227	1227	NUM
ejpam-4687	407	18	1224	1224	NUM
ejpam-4687	407	19	thus	thus	ADV
ejpam-4687	407	20	,	,	PUNCT
ejpam-4687	407	21	s	s	VERB
ejpam-4687	407	22	is	be	AUX
ejpam-4687	407	23	a	a	DET
ejpam-4687	407	24	legal	legal	ADJ
ejpam-4687	407	25	closed	closed	ADJ
ejpam-4687	407	26	hop	hop	NOUN
ejpam-4687	407	27	neighborhood	neighborhood	NOUN
ejpam-4687	407	28	sequence	sequence	NOUN
ejpam-4687	407	29	in	in	ADP
ejpam-4687	407	30	s(g	s(g	PROPN
ejpam-4687	407	31	)	)	PUNCT
ejpam-4687	407	32	,	,	PUNCT
ejpam-4687	407	33	showing	show	VERB
ejpam-4687	407	34	that	that	SCONJ
ejpam-4687	407	35	s	s	VERB
ejpam-4687	407	36	is	be	AUX
ejpam-4687	407	37	a	a	DET
ejpam-4687	407	38	connected	connect	VERB
ejpam-4687	407	39	grundy	grundy	PROPN
ejpam-4687	407	40	hop	hop	NOUN
ejpam-4687	407	41	dominating	dominating	NOUN
ejpam-4687	407	42	sequence	sequence	NOUN
ejpam-4687	407	43	in	in	ADP
ejpam-4687	407	44	s(g	s(g	PROPN
ejpam-4687	407	45	)	)	PUNCT
ejpam-4687	407	46	.	.	PUNCT
ejpam-4687	408	1	similarly	similarly	ADV
ejpam-4687	408	2	,	,	PUNCT
ejpam-4687	408	3	s	s	VERB
ejpam-4687	408	4	is	be	AUX
ejpam-4687	408	5	a	a	DET
ejpam-4687	408	6	connected	connect	VERB
ejpam-4687	408	7	grundy	grundy	PROPN
ejpam-4687	408	8	hop	hop	NOUN
ejpam-4687	408	9	dominating	dominating	NOUN
ejpam-4687	408	10	sequence	sequence	NOUN
ejpam-4687	408	11	in	in	ADP
ejpam-4687	408	12	s(g	s(g	PROPN
ejpam-4687	408	13	)	)	PUNCT
ejpam-4687	408	14	if	if	SCONJ
ejpam-4687	408	15	s	s	NOUN
ejpam-4687	408	16	is	be	AUX
ejpam-4687	408	17	a	a	DET
ejpam-4687	408	18	connected	connect	VERB
ejpam-4687	408	19	grundy	grundy	PROPN
ejpam-4687	408	20	hop	hop	NOUN
ejpam-4687	408	21	dominating	dominating	NOUN
ejpam-4687	408	22	sequence	sequence	NOUN
ejpam-4687	408	23	in	in	ADP
ejpam-4687	408	24	g2	g2	PROPN
ejpam-4687	408	25	.	.	PUNCT
ejpam-4687	409	1	therefore	therefore	ADV
ejpam-4687	409	2	,	,	PUNCT
ejpam-4687	409	3	γchgr	γchgr	X
ejpam-4687	409	4	(	(	PUNCT
ejpam-4687	409	5	g	g	NOUN
ejpam-4687	409	6	)	)	PUNCT
ejpam-4687	409	7	≤	≤	NOUN
ejpam-4687	409	8	γchgr	γchgr	NOUN
ejpam-4687	409	9	(	(	PUNCT
ejpam-4687	409	10	s(g	s(g	PROPN
ejpam-4687	409	11	)	)	PUNCT
ejpam-4687	409	12	)	)	PUNCT
ejpam-4687	409	13	.	.	PUNCT
ejpam-4687	410	1	lemma	lemma	PROPN
ejpam-4687	410	2	2	2	X
ejpam-4687	410	3	.	.	PUNCT
ejpam-4687	411	1	let	let	VERB
ejpam-4687	411	2	g	g	PRON
ejpam-4687	411	3	be	be	AUX
ejpam-4687	411	4	a	a	DET
ejpam-4687	411	5	connected	connected	ADJ
ejpam-4687	411	6	graph	graph	NOUN
ejpam-4687	411	7	of	of	ADP
ejpam-4687	411	8	order	order	NOUN
ejpam-4687	411	9	n.	n.	NOUN
ejpam-4687	411	10	if	if	SCONJ
ejpam-4687	411	11	|ng[v]|	|ng[v]|	PROPN
ejpam-4687	411	12	≥	≥	PUNCT
ejpam-4687	411	13	k	k	NOUN
ejpam-4687	411	14	for	for	ADP
ejpam-4687	411	15	every	every	DET
ejpam-4687	411	16	v	v	NUM
ejpam-4687	411	17	∈	∈	PROPN
ejpam-4687	411	18	v	v	NOUN
ejpam-4687	411	19	(	(	PUNCT
ejpam-4687	411	20	g	g	NOUN
ejpam-4687	411	21	)	)	PUNCT
ejpam-4687	411	22	,	,	PUNCT
ejpam-4687	411	23	then	then	ADV
ejpam-4687	411	24	γgr(g	γgr(g	PROPN
ejpam-4687	411	25	)	)	PUNCT
ejpam-4687	411	26	≤	≤	NUM
ejpam-4687	411	27	n−	n−	NOUN
ejpam-4687	411	28	(	(	PUNCT
ejpam-4687	411	29	k	k	NOUN
ejpam-4687	411	30	−	−	PROPN
ejpam-4687	411	31	1	1	NUM
ejpam-4687	411	32	)	)	PUNCT
ejpam-4687	411	33	.	.	PUNCT
ejpam-4687	412	1	proof	proof	NOUN
ejpam-4687	412	2	.	.	PUNCT
ejpam-4687	413	1	let	let	VERB
ejpam-4687	413	2	v	v	X
ejpam-4687	413	3	(	(	PUNCT
ejpam-4687	413	4	g	g	NOUN
ejpam-4687	413	5	)	)	PUNCT
ejpam-4687	413	6	=	=	SYM
ejpam-4687	413	7	{	{	PUNCT
ejpam-4687	413	8	v1	v1	PROPN
ejpam-4687	413	9	,	,	PUNCT
ejpam-4687	413	10	v2	v2	PROPN
ejpam-4687	413	11	,	,	PUNCT
ejpam-4687	413	12	.	.	PUNCT
ejpam-4687	413	13	.	.	PUNCT
ejpam-4687	414	1	.	.	PUNCT
ejpam-4687	415	1	,	,	PUNCT
ejpam-4687	415	2	vn	vn	PROPN
ejpam-4687	415	3	}	}	PUNCT
ejpam-4687	415	4	.	.	PUNCT
ejpam-4687	416	1	suppose	suppose	VERB
ejpam-4687	416	2	γgr(g	γgr(g	PROPN
ejpam-4687	416	3	)	)	PUNCT
ejpam-4687	416	4	=	=	SYM
ejpam-4687	416	5	t	t	PROPN
ejpam-4687	416	6	,	,	PUNCT
ejpam-4687	416	7	say	say	VERB
ejpam-4687	416	8	s	s	X
ejpam-4687	416	9	=	=	SYM
ejpam-4687	416	10	(	(	PUNCT
ejpam-4687	416	11	s1	s1	PROPN
ejpam-4687	416	12	,	,	PUNCT
ejpam-4687	416	13	s2	s2	PROPN
ejpam-4687	416	14	,	,	PUNCT
ejpam-4687	416	15	·	·	PUNCT
ejpam-4687	416	16	·	·	PUNCT
ejpam-4687	416	17	·	·	PUNCT
ejpam-4687	416	18	,	,	PUNCT
ejpam-4687	416	19	st	st	PROPN
ejpam-4687	416	20	)	)	PUNCT
ejpam-4687	416	21	is	be	AUX
ejpam-4687	416	22	a	a	DET
ejpam-4687	416	23	grundy	grundy	PROPN
ejpam-4687	416	24	dominating	dominating	NOUN
ejpam-4687	416	25	sequence	sequence	NOUN
ejpam-4687	416	26	of	of	ADP
ejpam-4687	416	27	g.	g.	PROPN
ejpam-4687	416	28	then	then	ADV
ejpam-4687	416	29	|ng[s1]|	|ng[s1]|	PROPN
ejpam-4687	416	30	≥	≥	PROPN
ejpam-4687	416	31	k	k	X
ejpam-4687	416	32	by	by	ADP
ejpam-4687	416	33	assumption	assumption	NOUN
ejpam-4687	416	34	.	.	PUNCT
ejpam-4687	417	1	it	it	PRON
ejpam-4687	417	2	follows	follow	VERB
ejpam-4687	417	3	that	that	SCONJ
ejpam-4687	417	4	there	there	PRON
ejpam-4687	417	5	are	be	VERB
ejpam-4687	417	6	at	at	ADP
ejpam-4687	417	7	most	most	ADJ
ejpam-4687	417	8	n	n	ADP
ejpam-4687	417	9	−	−	NOUN
ejpam-4687	417	10	k	k	NOUN
ejpam-4687	417	11	remaining	remain	VERB
ejpam-4687	417	12	vertices	vertex	NOUN
ejpam-4687	417	13	of	of	ADP
ejpam-4687	417	14	g	g	NOUN
ejpam-4687	417	15	that	that	PRON
ejpam-4687	417	16	could	could	AUX
ejpam-4687	417	17	be	be	AUX
ejpam-4687	417	18	footprinted	footprinte	VERB
ejpam-4687	417	19	by	by	ADP
ejpam-4687	417	20	the	the	DET
ejpam-4687	417	21	next	next	ADJ
ejpam-4687	417	22	terms	term	NOUN
ejpam-4687	417	23	of	of	ADP
ejpam-4687	417	24	s.	s.	PROPN
ejpam-4687	417	25	therefore	therefore	ADV
ejpam-4687	417	26	,	,	PUNCT
ejpam-4687	417	27	γcgr(g	γcgr(g	NOUN
ejpam-4687	417	28	)	)	PUNCT
ejpam-4687	417	29	=	=	PUNCT
ejpam-4687	417	30	t	t	NOUN
ejpam-4687	417	31	≤	≤	NUM
ejpam-4687	417	32	n−	n−	PROPN
ejpam-4687	417	33	k	k	PROPN
ejpam-4687	418	1	+	+	CCONJ
ejpam-4687	418	2	1	1	X
ejpam-4687	418	3	.	.	X
ejpam-4687	418	4	proposition	proposition	NOUN
ejpam-4687	418	5	8	8	NUM
ejpam-4687	418	6	.	.	PUNCT
ejpam-4687	419	1	let	let	VERB
ejpam-4687	419	2	n	n	PRON
ejpam-4687	419	3	be	be	AUX
ejpam-4687	419	4	a	a	DET
ejpam-4687	419	5	positive	positive	ADJ
ejpam-4687	419	6	integer	integer	NOUN
ejpam-4687	419	7	.	.	PUNCT
ejpam-4687	420	1	then	then	ADV
ejpam-4687	420	2	γgr(pn	γgr(pn	NOUN
ejpam-4687	420	3	)	)	PUNCT
ejpam-4687	420	4	=	=	SYM
ejpam-4687	421	1			NOUN
ejpam-4687	421	2	1	1	NUM
ejpam-4687	421	3	,	,	PUNCT
ejpam-4687	421	4	n	n	NOUN
ejpam-4687	421	5	=	=	SYM
ejpam-4687	421	6	1	1	NUM
ejpam-4687	421	7	,	,	PUNCT
ejpam-4687	421	8	2	2	NUM
ejpam-4687	421	9	,	,	PUNCT
ejpam-4687	421	10	n	n	NOUN
ejpam-4687	421	11	=	=	SYM
ejpam-4687	421	12	2	2	NUM
ejpam-4687	421	13	,	,	PUNCT
ejpam-4687	421	14	3	3	NUM
ejpam-4687	421	15	3	3	NUM
ejpam-4687	421	16	,	,	PUNCT
ejpam-4687	421	17	n	n	PRON
ejpam-4687	421	18	≥	≥	NOUN
ejpam-4687	421	19	4	4	NUM
ejpam-4687	421	20	.	.	PUNCT
ejpam-4687	422	1	proof	proof	NOUN
ejpam-4687	422	2	.	.	PUNCT
ejpam-4687	423	1	clearly	clearly	ADV
ejpam-4687	423	2	,	,	PUNCT
ejpam-4687	423	3	for	for	ADP
ejpam-4687	423	4	n	n	NOUN
ejpam-4687	423	5	=	=	SYM
ejpam-4687	423	6	1	1	NUM
ejpam-4687	423	7	and	and	CCONJ
ejpam-4687	423	8	n	n	CCONJ
ejpam-4687	423	9	=	=	SYM
ejpam-4687	423	10	2	2	NUM
ejpam-4687	423	11	,	,	PUNCT
ejpam-4687	423	12	3	3	NUM
ejpam-4687	423	13	,	,	PUNCT
ejpam-4687	423	14	γgr(pn	γgr(pn	NUM
ejpam-4687	423	15	)	)	PUNCT
ejpam-4687	423	16	=	=	SYM
ejpam-4687	423	17	1	1	NUM
ejpam-4687	423	18	and	and	CCONJ
ejpam-4687	423	19	γgr(pn	γgr(pn	NUM
ejpam-4687	423	20	)	)	PUNCT
ejpam-4687	423	21	=	=	SYM
ejpam-4687	423	22	2	2	NUM
ejpam-4687	423	23	,	,	PUNCT
ejpam-4687	423	24	respectively	respectively	ADV
ejpam-4687	423	25	.	.	PUNCT
ejpam-4687	424	1	let	let	AUX
ejpam-4687	424	2	{	{	PUNCT
ejpam-4687	424	3	v1	v1	VERB
ejpam-4687	424	4	,	,	PUNCT
ejpam-4687	424	5	v2	v2	PROPN
ejpam-4687	424	6	,	,	PUNCT
ejpam-4687	424	7	·	·	PUNCT
ejpam-4687	424	8	·	·	PUNCT
ejpam-4687	424	9	·	·	PUNCT
ejpam-4687	424	10	,	,	PUNCT
ejpam-4687	424	11	vn	vn	PART
ejpam-4687	424	12	}	}	PUNCT
ejpam-4687	424	13	be	be	AUX
ejpam-4687	424	14	a	a	DET
ejpam-4687	424	15	vertex	vertex	NOUN
ejpam-4687	424	16	set	set	NOUN
ejpam-4687	424	17	of	of	ADP
ejpam-4687	424	18	g	g	PROPN
ejpam-4687	424	19	=	=	SYM
ejpam-4687	424	20	pn	pn	PROPN
ejpam-4687	424	21	.	.	PUNCT
ejpam-4687	425	1	let	let	VERB
ejpam-4687	425	2	=	=	PRON
ejpam-4687	425	3	{	{	PUNCT
ejpam-4687	425	4	v2	v2	PROPN
ejpam-4687	425	5	,	,	PUNCT
ejpam-4687	425	6	v4	v4	NOUN
ejpam-4687	425	7	,	,	PUNCT
ejpam-4687	425	8	v3	v3	PROPN
ejpam-4687	425	9	}	}	PUNCT
ejpam-4687	425	10	.	.	PUNCT
ejpam-4687	426	1	notice	notice	VERB
ejpam-4687	426	2	that	that	SCONJ
ejpam-4687	426	3	s	s	VERB
ejpam-4687	426	4	is	be	AUX
ejpam-4687	426	5	a	a	DET
ejpam-4687	426	6	grundy	grundy	PROPN
ejpam-4687	426	7	dominating	dominating	NOUN
ejpam-4687	426	8	sequence	sequence	NOUN
ejpam-4687	426	9	of	of	ADP
ejpam-4687	426	10	g.	g.	PROPN
ejpam-4687	426	11	hence	hence	ADV
ejpam-4687	426	12	,	,	PUNCT
ejpam-4687	426	13	γgr(g	γgr(g	PROPN
ejpam-4687	426	14	)	)	PUNCT
ejpam-4687	426	15	≥	≥	NOUN
ejpam-4687	426	16	3	3	NUM
ejpam-4687	426	17	.	.	PUNCT
ejpam-4687	427	1	now	now	ADV
ejpam-4687	427	2	,	,	PUNCT
ejpam-4687	427	3	notice	notice	VERB
ejpam-4687	427	4	that	that	SCONJ
ejpam-4687	427	5	|ng[vi]|	|ng[vi]|	NOUN
ejpam-4687	427	6	=	=	PUNCT
ejpam-4687	427	7	n	n	CCONJ
ejpam-4687	427	8	−	−	NUM
ejpam-4687	427	9	2	2	NUM
ejpam-4687	427	10	for	for	ADP
ejpam-4687	427	11	every	every	DET
ejpam-4687	427	12	i	i	PROPN
ejpam-4687	427	13	,	,	PUNCT
ejpam-4687	427	14	where	where	SCONJ
ejpam-4687	427	15	i	i	PRON
ejpam-4687	427	16	̸=	̸=	PROPN
ejpam-4687	427	17	1	1	NUM
ejpam-4687	427	18	,	,	PUNCT
ejpam-4687	427	19	n.	n.	PROPN
ejpam-4687	427	20	fix	fix	PROPN
ejpam-4687	427	21	i.	i.	NOUN
ejpam-4687	427	22	if	if	SCONJ
ejpam-4687	427	23	we	we	PRON
ejpam-4687	427	24	let	let	VERB
ejpam-4687	427	25	vi	vi	NOUN
ejpam-4687	427	26	to	to	PART
ejpam-4687	427	27	be	be	AUX
ejpam-4687	427	28	the	the	DET
ejpam-4687	427	29	first	first	ADJ
ejpam-4687	427	30	element	element	NOUN
ejpam-4687	427	31	of	of	ADP
ejpam-4687	427	32	a	a	DET
ejpam-4687	427	33	grundy	grundy	PROPN
ejpam-4687	427	34	dominating	dominating	NOUN
ejpam-4687	427	35	sequence	sequence	NOUN
ejpam-4687	427	36	s′	s′	NOUN
ejpam-4687	427	37	,	,	PUNCT
ejpam-4687	427	38	then	then	ADV
ejpam-4687	427	39	there	there	PRON
ejpam-4687	427	40	are	be	VERB
ejpam-4687	427	41	only	only	ADV
ejpam-4687	427	42	two	two	NUM
ejpam-4687	427	43	remaining	remain	VERB
ejpam-4687	427	44	vertices	vertex	NOUN
ejpam-4687	427	45	which	which	PRON
ejpam-4687	427	46	are	be	AUX
ejpam-4687	427	47	not	not	PART
ejpam-4687	427	48	in	in	ADP
ejpam-4687	427	49	ng[vi	ng[vi	NOUN
ejpam-4687	427	50	]	]	PUNCT
ejpam-4687	427	51	.	.	PUNCT
ejpam-4687	428	1	thus	thus	ADV
ejpam-4687	428	2	,	,	PUNCT
ejpam-4687	428	3	γgr(g	γgr(g	PROPN
ejpam-4687	428	4	)	)	PUNCT
ejpam-4687	428	5	≤	≤	NOUN
ejpam-4687	428	6	3	3	NUM
ejpam-4687	428	7	.	.	PUNCT
ejpam-4687	429	1	therefore	therefore	ADV
ejpam-4687	429	2	,	,	PUNCT
ejpam-4687	429	3	γhgr(g	γhgr(g	NOUN
ejpam-4687	429	4	)	)	PUNCT
ejpam-4687	429	5	=	=	SYM
ejpam-4687	429	6	3	3	X
ejpam-4687	429	7	.	.	X
ejpam-4687	429	8	proposition	proposition	NOUN
ejpam-4687	429	9	9	9	NUM
ejpam-4687	429	10	.	.	PUNCT
ejpam-4687	430	1	let	let	VERB
ejpam-4687	430	2	n	n	PRON
ejpam-4687	430	3	be	be	AUX
ejpam-4687	430	4	a	a	DET
ejpam-4687	430	5	positve	positve	NOUN
ejpam-4687	430	6	integer	integer	NOUN
ejpam-4687	430	7	.	.	PUNCT
ejpam-4687	431	1	then	then	ADV
ejpam-4687	431	2	γgr(cn	γgr(cn	X
ejpam-4687	431	3	)	)	PUNCT
ejpam-4687	432	1	=	=	SYM
ejpam-4687	432	2			NOUN
ejpam-4687	432	3	3	3	NUM
ejpam-4687	432	4	,	,	PUNCT
ejpam-4687	432	5	n	n	NOUN
ejpam-4687	432	6	=	=	SYM
ejpam-4687	432	7	3	3	NUM
ejpam-4687	432	8	,	,	PUNCT
ejpam-4687	432	9	2	2	NUM
ejpam-4687	432	10	,	,	PUNCT
ejpam-4687	432	11	n	n	NOUN
ejpam-4687	432	12	=	=	SYM
ejpam-4687	432	13	4	4	NUM
ejpam-4687	432	14	3	3	NUM
ejpam-4687	432	15	,	,	PUNCT
ejpam-4687	432	16	n	n	PRON
ejpam-4687	432	17	≥	≥	NOUN
ejpam-4687	432	18	5	5	NUM
ejpam-4687	432	19	.	.	PUNCT
ejpam-4687	433	1	proof	proof	NOUN
ejpam-4687	433	2	.	.	PUNCT
ejpam-4687	434	1	clearly	clearly	ADV
ejpam-4687	434	2	,	,	PUNCT
ejpam-4687	434	3	for	for	ADP
ejpam-4687	434	4	n	n	NOUN
ejpam-4687	434	5	=	=	SYM
ejpam-4687	434	6	3	3	NUM
ejpam-4687	434	7	and	and	CCONJ
ejpam-4687	434	8	n	n	NOUN
ejpam-4687	434	9	=	=	SYM
ejpam-4687	434	10	4	4	NUM
ejpam-4687	434	11	,	,	PUNCT
ejpam-4687	434	12	γgr(cn	γgr(cn	ADV
ejpam-4687	434	13	)	)	PUNCT
ejpam-4687	434	14	=	=	SYM
ejpam-4687	434	15	3	3	NUM
ejpam-4687	434	16	and	and	CCONJ
ejpam-4687	434	17	γgr(cn	γgr(cn	NUM
ejpam-4687	434	18	)	)	PUNCT
ejpam-4687	435	1	=	=	SYM
ejpam-4687	435	2	2	2	NUM
ejpam-4687	435	3	,	,	PUNCT
ejpam-4687	435	4	respectively	respectively	ADV
ejpam-4687	435	5	.	.	PUNCT
ejpam-4687	436	1	let	let	AUX
ejpam-4687	436	2	{	{	PUNCT
ejpam-4687	436	3	v1	v1	VERB
ejpam-4687	436	4	,	,	PUNCT
ejpam-4687	436	5	v2	v2	PROPN
ejpam-4687	436	6	,	,	PUNCT
ejpam-4687	436	7	·	·	PUNCT
ejpam-4687	436	8	·	·	PUNCT
ejpam-4687	436	9	·	·	PUNCT
ejpam-4687	436	10	,	,	PUNCT
ejpam-4687	436	11	vn	vn	PART
ejpam-4687	436	12	}	}	PUNCT
ejpam-4687	436	13	be	be	AUX
ejpam-4687	436	14	a	a	DET
ejpam-4687	436	15	vertex	vertex	NOUN
ejpam-4687	436	16	set	set	NOUN
ejpam-4687	436	17	of	of	ADP
ejpam-4687	436	18	g	g	PROPN
ejpam-4687	436	19	=	=	SYM
ejpam-4687	436	20	cn	cn	PROPN
ejpam-4687	436	21	.	.	PUNCT
ejpam-4687	437	1	let	let	VERB
ejpam-4687	437	2	=	=	PRON
ejpam-4687	437	3	{	{	PUNCT
ejpam-4687	437	4	v1	v1	PROPN
ejpam-4687	437	5	,	,	PUNCT
ejpam-4687	437	6	v3	v3	PROPN
ejpam-4687	437	7	,	,	PUNCT
ejpam-4687	437	8	v2	v2	PROPN
ejpam-4687	437	9	}	}	PUNCT
ejpam-4687	437	10	.	.	PUNCT
ejpam-4687	438	1	notice	notice	VERB
ejpam-4687	438	2	that	that	SCONJ
ejpam-4687	438	3	s	s	VERB
ejpam-4687	438	4	is	be	AUX
ejpam-4687	438	5	a	a	DET
ejpam-4687	438	6	grundy	grundy	PROPN
ejpam-4687	438	7	dominating	dominating	NOUN
ejpam-4687	438	8	sequence	sequence	NOUN
ejpam-4687	438	9	of	of	ADP
ejpam-4687	438	10	g.	g.	PROPN
ejpam-4687	438	11	hence	hence	ADV
ejpam-4687	438	12	,	,	PUNCT
ejpam-4687	438	13	γgr(g	γgr(g	PROPN
ejpam-4687	438	14	)	)	PUNCT
ejpam-4687	438	15	≥	≥	NOUN
ejpam-4687	438	16	3	3	NUM
ejpam-4687	438	17	.	.	PUNCT
ejpam-4687	439	1	now	now	ADV
ejpam-4687	439	2	,	,	PUNCT
ejpam-4687	439	3	notice	notice	VERB
ejpam-4687	439	4	that	that	SCONJ
ejpam-4687	439	5	|ng[vi]|	|ng[vi]|	NOUN
ejpam-4687	439	6	=	=	PUNCT
ejpam-4687	439	7	n−2	n−2	PROPN
ejpam-4687	439	8	for	for	ADP
ejpam-4687	439	9	every	every	DET
ejpam-4687	439	10	i	i	PROPN
ejpam-4687	439	11	∈	∈	PROPN
ejpam-4687	439	12	{	{	PUNCT
ejpam-4687	439	13	1	1	NUM
ejpam-4687	439	14	,	,	PUNCT
ejpam-4687	439	15	·	·	PUNCT
ejpam-4687	439	16	·	·	PUNCT
ejpam-4687	439	17	·	·	PUNCT
ejpam-4687	439	18	,	,	PUNCT
ejpam-4687	439	19	n	n	CCONJ
ejpam-4687	439	20	}	}	PUNCT
ejpam-4687	439	21	.	.	PUNCT
ejpam-4687	440	1	fix	fix	NOUN
ejpam-4687	440	2	i.	i.	NOUN
ejpam-4687	440	3	if	if	SCONJ
ejpam-4687	440	4	we	we	PRON
ejpam-4687	440	5	let	let	VERB
ejpam-4687	440	6	vi	vi	NOUN
ejpam-4687	440	7	to	to	PART
ejpam-4687	440	8	be	be	AUX
ejpam-4687	440	9	the	the	DET
ejpam-4687	440	10	first	first	ADJ
ejpam-4687	440	11	element	element	NOUN
ejpam-4687	440	12	of	of	ADP
ejpam-4687	440	13	a	a	DET
ejpam-4687	440	14	grundy	grundy	PROPN
ejpam-4687	440	15	dominating	dominating	NOUN
ejpam-4687	440	16	sequence	sequence	NOUN
ejpam-4687	440	17	s′	s′	NOUN
ejpam-4687	440	18	,	,	PUNCT
ejpam-4687	440	19	then	then	ADV
ejpam-4687	440	20	there	there	PRON
ejpam-4687	440	21	are	be	VERB
ejpam-4687	440	22	only	only	ADV
ejpam-4687	440	23	two	two	NUM
ejpam-4687	440	24	remaining	remain	VERB
ejpam-4687	440	25	vertices	vertex	NOUN
ejpam-4687	440	26	which	which	PRON
ejpam-4687	440	27	are	be	AUX
ejpam-4687	440	28	not	not	PART
ejpam-4687	440	29	in	in	ADP
ejpam-4687	440	30	ng[vi	ng[vi	NOUN
ejpam-4687	440	31	]	]	PUNCT
ejpam-4687	440	32	.	.	PUNCT
ejpam-4687	441	1	thus	thus	ADV
ejpam-4687	441	2	,	,	PUNCT
ejpam-4687	441	3	γgr(g	γgr(g	PROPN
ejpam-4687	441	4	)	)	PUNCT
ejpam-4687	441	5	≤	≤	NOUN
ejpam-4687	441	6	3	3	NUM
ejpam-4687	441	7	.	.	PUNCT
ejpam-4687	442	1	therefore	therefore	ADV
ejpam-4687	442	2	,	,	PUNCT
ejpam-4687	442	3	γgr(g	γgr(g	PROPN
ejpam-4687	442	4	)	)	PUNCT
ejpam-4687	442	5	=	=	SYM
ejpam-4687	443	1	3	3	X
ejpam-4687	443	2	.	.	PUNCT
ejpam-4687	444	1	the	the	DET
ejpam-4687	444	2	next	next	ADJ
ejpam-4687	444	3	result	result	NOUN
ejpam-4687	444	4	is	be	AUX
ejpam-4687	444	5	the	the	DET
ejpam-4687	444	6	correction	correction	NOUN
ejpam-4687	444	7	of	of	ADP
ejpam-4687	444	8	a	a	DET
ejpam-4687	444	9	result	result	NOUN
ejpam-4687	444	10	found	find	VERB
ejpam-4687	444	11	in	in	ADP
ejpam-4687	444	12	an	an	DET
ejpam-4687	444	13	earlier	early	ADJ
ejpam-4687	444	14	paper	paper	NOUN
ejpam-4687	444	15	of	of	ADP
ejpam-4687	444	16	the	the	DET
ejpam-4687	444	17	authors	author	NOUN
ejpam-4687	444	18	in	in	ADP
ejpam-4687	444	19	[	[	X
ejpam-4687	444	20	8	8	NUM
ejpam-4687	444	21	]	]	PUNCT
ejpam-4687	444	22	.	.	PUNCT
ejpam-4687	445	1	theorem	theorem	ADJ
ejpam-4687	445	2	6	6	NUM
ejpam-4687	445	3	.	.	PUNCT
ejpam-4687	446	1	let	let	VERB
ejpam-4687	446	2	g	g	NOUN
ejpam-4687	446	3	and	and	CCONJ
ejpam-4687	446	4	h	h	NOUN
ejpam-4687	446	5	be	be	VERB
ejpam-4687	446	6	two	two	NUM
ejpam-4687	446	7	graphs	graph	NOUN
ejpam-4687	446	8	and	and	CCONJ
ejpam-4687	446	9	let	let	VERB
ejpam-4687	446	10	c	c	PRON
ejpam-4687	446	11	be	be	AUX
ejpam-4687	446	12	a	a	DET
ejpam-4687	446	13	sequence	sequence	NOUN
ejpam-4687	446	14	of	of	ADP
ejpam-4687	446	15	distinct	distinct	ADJ
ejpam-4687	446	16	vertices	vertex	NOUN
ejpam-4687	446	17	of	of	ADP
ejpam-4687	446	18	g	g	PROPN
ejpam-4687	446	19	+	+	PROPN
ejpam-4687	446	20	h.	h.	PROPN
ejpam-4687	446	21	then	then	ADV
ejpam-4687	446	22	c	c	PROPN
ejpam-4687	446	23	is	be	AUX
ejpam-4687	446	24	a	a	DET
ejpam-4687	446	25	legal	legal	ADJ
ejpam-4687	446	26	closed	closed	ADJ
ejpam-4687	446	27	hop	hop	NOUN
ejpam-4687	446	28	neighborhood	neighborhood	NOUN
ejpam-4687	446	29	sequence	sequence	NOUN
ejpam-4687	446	30	in	in	ADP
ejpam-4687	446	31	g	g	PROPN
ejpam-4687	446	32	+	+	NOUN
ejpam-4687	446	33	h	h	NOUN
ejpam-4687	446	34	if	if	SCONJ
ejpam-4687	447	1	and	and	CCONJ
ejpam-4687	447	2	only	only	ADV
ejpam-4687	447	3	if	if	SCONJ
ejpam-4687	447	4	one	one	NUM
ejpam-4687	447	5	of	of	ADP
ejpam-4687	447	6	the	the	DET
ejpam-4687	447	7	following	follow	VERB
ejpam-4687	447	8	holds	hold	VERB
ejpam-4687	447	9	:	:	PUNCT
ejpam-4687	447	10	(	(	PUNCT
ejpam-4687	447	11	i	i	NOUN
ejpam-4687	447	12	)	)	PUNCT
ejpam-4687	447	13	c	c	PROPN
ejpam-4687	447	14	is	be	AUX
ejpam-4687	447	15	a	a	DET
ejpam-4687	447	16	co	co	ADJ
ejpam-4687	447	17	-	-	ADJ
ejpam-4687	447	18	legal	legal	ADJ
ejpam-4687	447	19	closed	closed	ADJ
ejpam-4687	447	20	neighborhood	neighborhood	NOUN
ejpam-4687	447	21	sequence	sequence	NOUN
ejpam-4687	447	22	in	in	ADP
ejpam-4687	447	23	g.	g.	PROPN
ejpam-4687	447	24	j.	j.	PROPN
ejpam-4687	447	25	hassan	hassan	PROPN
ejpam-4687	447	26	,	,	PUNCT
ejpam-4687	447	27	s.	s.	PROPN
ejpam-4687	447	28	canoy	canoy	PROPN
ejpam-4687	447	29	jr	jr	PROPN
ejpam-4687	447	30	.	.	PROPN
ejpam-4687	447	31	/	/	SYM
ejpam-4687	447	32	eur	eur	PROPN
ejpam-4687	447	33	.	.	PUNCT
ejpam-4687	448	1	j.	j.	PROPN
ejpam-4687	448	2	pure	pure	PROPN
ejpam-4687	448	3	appl	appl	PROPN
ejpam-4687	448	4	.	.	PROPN
ejpam-4687	448	5	math	math	PROPN
ejpam-4687	448	6	,	,	PUNCT
ejpam-4687	448	7	16	16	NUM
ejpam-4687	448	8	(	(	PUNCT
ejpam-4687	448	9	2	2	NUM
ejpam-4687	448	10	)	)	PUNCT
ejpam-4687	448	11	(	(	PUNCT
ejpam-4687	448	12	2023	2023	NUM
ejpam-4687	448	13	)	)	PUNCT
ejpam-4687	448	14	,	,	PUNCT
ejpam-4687	448	15	1212	1212	NUM
ejpam-4687	448	16	-	-	SYM
ejpam-4687	448	17	1227	1227	NUM
ejpam-4687	448	18	1225	1225	NUM
ejpam-4687	448	19	(	(	PUNCT
ejpam-4687	448	20	ii	ii	NOUN
ejpam-4687	448	21	)	)	PUNCT
ejpam-4687	448	22	c	c	PROPN
ejpam-4687	448	23	is	be	AUX
ejpam-4687	448	24	a	a	DET
ejpam-4687	448	25	co	co	ADJ
ejpam-4687	448	26	-	-	ADJ
ejpam-4687	448	27	legal	legal	ADJ
ejpam-4687	448	28	closed	closed	ADJ
ejpam-4687	448	29	neighborhood	neighborhood	NOUN
ejpam-4687	448	30	sequence	sequence	NOUN
ejpam-4687	448	31	in	in	ADP
ejpam-4687	448	32	h.	h.	PROPN
ejpam-4687	448	33	(	(	PUNCT
ejpam-4687	448	34	iii	iii	X
ejpam-4687	448	35	)	)	PUNCT
ejpam-4687	448	36	the	the	DET
ejpam-4687	448	37	subsequences	subsequence	NOUN
ejpam-4687	448	38	cg	cg	NOUN
ejpam-4687	448	39	and	and	CCONJ
ejpam-4687	448	40	ch	ch	NOUN
ejpam-4687	448	41	of	of	ADP
ejpam-4687	448	42	c	c	PROPN
ejpam-4687	448	43	,	,	PUNCT
ejpam-4687	448	44	where	where	SCONJ
ejpam-4687	448	45	ĉ	ĉ	ADV
ejpam-4687	448	46	=	=	SYM
ejpam-4687	448	47	ĉg	ĉg	X
ejpam-4687	448	48	∪	∪	VERB
ejpam-4687	448	49	ĉh	ĉh	NOUN
ejpam-4687	448	50	,	,	PUNCT
ejpam-4687	448	51	are	be	AUX
ejpam-4687	448	52	co	co	ADJ
ejpam-4687	448	53	-	-	ADJ
ejpam-4687	448	54	legal	legal	ADJ
ejpam-4687	448	55	closed	closed	ADJ
ejpam-4687	448	56	neighborhood	neighborhood	NOUN
ejpam-4687	448	57	sequences	sequence	NOUN
ejpam-4687	448	58	of	of	ADP
ejpam-4687	448	59	g	g	PROPN
ejpam-4687	448	60	and	and	CCONJ
ejpam-4687	448	61	h	h	NOUN
ejpam-4687	448	62	,	,	PUNCT
ejpam-4687	448	63	respectively	respectively	ADV
ejpam-4687	448	64	.	.	PUNCT
ejpam-4687	449	1	proof	proof	NOUN
ejpam-4687	449	2	.	.	PUNCT
ejpam-4687	450	1	assume	assume	VERB
ejpam-4687	450	2	that	that	SCONJ
ejpam-4687	450	3	c	c	AUX
ejpam-4687	450	4	=	=	SYM
ejpam-4687	450	5	(	(	PUNCT
ejpam-4687	450	6	a1	a1	PROPN
ejpam-4687	450	7	,	,	PUNCT
ejpam-4687	450	8	a2	a2	PROPN
ejpam-4687	450	9	,	,	PUNCT
ejpam-4687	450	10	·	·	PUNCT
ejpam-4687	450	11	·	·	PUNCT
ejpam-4687	450	12	·	·	PUNCT
ejpam-4687	450	13	,	,	PUNCT
ejpam-4687	450	14	am	be	AUX
ejpam-4687	450	15	)	)	PUNCT
ejpam-4687	450	16	is	be	AUX
ejpam-4687	450	17	a	a	DET
ejpam-4687	450	18	legal	legal	ADJ
ejpam-4687	450	19	closed	close	VERB
ejpam-4687	450	20	hop	hop	NOUN
ejpam-4687	450	21	neighborhood	neighborhood	NOUN
ejpam-4687	450	22	sequence	sequence	NOUN
ejpam-4687	450	23	of	of	ADP
ejpam-4687	450	24	g+h	g+h	PROPN
ejpam-4687	450	25	.	.	PUNCT
ejpam-4687	451	1	if	if	SCONJ
ejpam-4687	451	2	ĉ	ĉ	ADV
ejpam-4687	451	3	⊆	⊆	NUM
ejpam-4687	451	4	v	v	NOUN
ejpam-4687	451	5	(	(	PUNCT
ejpam-4687	451	6	g	g	NOUN
ejpam-4687	451	7	)	)	PUNCT
ejpam-4687	451	8	,	,	PUNCT
ejpam-4687	451	9	then	then	ADV
ejpam-4687	451	10	n2	n2	PROPN
ejpam-4687	451	11	g[ai	g[ai	PROPN
ejpam-4687	451	12	]	]	PUNCT
ejpam-4687	451	13	=	=	SYM
ejpam-4687	451	14	n2	n2	PROPN
ejpam-4687	451	15	g+h	g+h	PROPN
ejpam-4687	452	1	[	[	X
ejpam-4687	452	2	ai	ai	X
ejpam-4687	452	3	]	]	X
ejpam-4687	452	4	=	=	SYM
ejpam-4687	452	5	v	v	X
ejpam-4687	452	6	(	(	PUNCT
ejpam-4687	452	7	g)\ng(ai	g)\ng(ai	PROPN
ejpam-4687	452	8	)	)	PUNCT
ejpam-4687	452	9	for	for	ADP
ejpam-4687	452	10	each	each	DET
ejpam-4687	452	11	i	i	PRON
ejpam-4687	452	12	∈	∈	PROPN
ejpam-4687	452	13	{	{	PUNCT
ejpam-4687	452	14	1	1	NUM
ejpam-4687	452	15	,	,	PUNCT
ejpam-4687	452	16	2	2	NUM
ejpam-4687	452	17	,	,	PUNCT
ejpam-4687	452	18	.	.	PUNCT
ejpam-4687	452	19	.	.	PUNCT
ejpam-4687	452	20	.	.	PUNCT
ejpam-4687	453	1	,	,	PUNCT
ejpam-4687	453	2	m	m	VERB
ejpam-4687	453	3	}	}	PUNCT
ejpam-4687	453	4	.	.	PUNCT
ejpam-4687	454	1	thus	thus	ADV
ejpam-4687	454	2	,	,	PUNCT
ejpam-4687	454	3	by	by	ADP
ejpam-4687	454	4	the	the	DET
ejpam-4687	454	5	legality	legality	NOUN
ejpam-4687	454	6	condition	condition	NOUN
ejpam-4687	454	7	property	property	NOUN
ejpam-4687	454	8	of	of	ADP
ejpam-4687	454	9	c	c	NOUN
ejpam-4687	454	10	,	,	PUNCT
ejpam-4687	454	11	[	[	X
ejpam-4687	454	12	v	v	X
ejpam-4687	454	13	(	(	PUNCT
ejpam-4687	454	14	g	g	NOUN
ejpam-4687	454	15	)	)	PUNCT
ejpam-4687	454	16	\ng(ai	\ng(ai	NOUN
ejpam-4687	454	17	)	)	PUNCT
ejpam-4687	454	18	]	]	PUNCT
ejpam-4687	454	19	\	\	X
ejpam-4687	454	20	∪i−1	∪i−1	PROPN
ejpam-4687	454	21	j=1[v	j=1[v	X
ejpam-4687	454	22	(	(	PUNCT
ejpam-4687	454	23	g	g	NOUN
ejpam-4687	454	24	)	)	PUNCT
ejpam-4687	454	25	\ng(aj	\ng(aj	NUM
ejpam-4687	454	26	)	)	PUNCT
ejpam-4687	454	27	]	]	PUNCT
ejpam-4687	454	28	̸=	̸=	NOUN
ejpam-4687	454	29	∅	∅	NOUN
ejpam-4687	454	30	for	for	ADP
ejpam-4687	454	31	each	each	DET
ejpam-4687	454	32	i	i	PRON
ejpam-4687	454	33	∈	∈	PROPN
ejpam-4687	454	34	{	{	PUNCT
ejpam-4687	454	35	2	2	NUM
ejpam-4687	454	36	,	,	PUNCT
ejpam-4687	454	37	3	3	NUM
ejpam-4687	454	38	,	,	PUNCT
ejpam-4687	454	39	.	.	PUNCT
ejpam-4687	454	40	.	.	PUNCT
ejpam-4687	454	41	.	.	PUNCT
ejpam-4687	455	1	,	,	PUNCT
ejpam-4687	455	2	m	m	VERB
ejpam-4687	455	3	}	}	PUNCT
ejpam-4687	455	4	.	.	PUNCT
ejpam-4687	456	1	it	it	PRON
ejpam-4687	456	2	follows	follow	VERB
ejpam-4687	456	3	that	that	SCONJ
ejpam-4687	456	4	c	c	PROPN
ejpam-4687	456	5	is	be	AUX
ejpam-4687	456	6	a	a	DET
ejpam-4687	456	7	co	co	ADJ
ejpam-4687	456	8	-	-	ADJ
ejpam-4687	456	9	legal	legal	ADJ
ejpam-4687	456	10	closed	closed	ADJ
ejpam-4687	456	11	neighborhood	neighborhood	NOUN
ejpam-4687	456	12	sequence	sequence	NOUN
ejpam-4687	456	13	in	in	ADP
ejpam-4687	456	14	g.	g.	PROPN
ejpam-4687	456	15	hence	hence	ADV
ejpam-4687	456	16	,	,	PUNCT
ejpam-4687	456	17	(	(	PUNCT
ejpam-4687	456	18	i	i	NOUN
ejpam-4687	456	19	)	)	PUNCT
ejpam-4687	456	20	holds	hold	VERB
ejpam-4687	456	21	.	.	PUNCT
ejpam-4687	457	1	similarly	similarly	ADV
ejpam-4687	457	2	,	,	PUNCT
ejpam-4687	457	3	if	if	SCONJ
ejpam-4687	457	4	ĉ	ĉ	ADV
ejpam-4687	457	5	⊆	⊆	NUM
ejpam-4687	457	6	v	v	NOUN
ejpam-4687	457	7	(	(	PUNCT
ejpam-4687	457	8	h	h	NOUN
ejpam-4687	457	9	)	)	PUNCT
ejpam-4687	457	10	,	,	PUNCT
ejpam-4687	457	11	then	then	ADV
ejpam-4687	457	12	(	(	PUNCT
ejpam-4687	457	13	ii	ii	NOUN
ejpam-4687	457	14	)	)	PUNCT
ejpam-4687	457	15	holds	hold	VERB
ejpam-4687	457	16	.	.	PUNCT
ejpam-4687	458	1	next	next	ADV
ejpam-4687	458	2	,	,	PUNCT
ejpam-4687	458	3	suppose	suppose	VERB
ejpam-4687	458	4	that	that	SCONJ
ejpam-4687	458	5	ĉ∩v	ĉ∩v	NOUN
ejpam-4687	458	6	(	(	PUNCT
ejpam-4687	458	7	g	g	NOUN
ejpam-4687	458	8	)	)	PUNCT
ejpam-4687	458	9	̸=	̸=	PROPN
ejpam-4687	458	10	∅	∅	NOUN
ejpam-4687	458	11	and	and	CCONJ
ejpam-4687	458	12	ĉ∩v	ĉ∩v	VERB
ejpam-4687	458	13	(	(	PUNCT
ejpam-4687	458	14	h	h	NOUN
ejpam-4687	458	15	)	)	PUNCT
ejpam-4687	458	16	̸=	̸=	PROPN
ejpam-4687	458	17	∅.	∅.	ADV
ejpam-4687	458	18	let	let	VERB
ejpam-4687	458	19	cg	cg	NOUN
ejpam-4687	458	20	and	and	CCONJ
ejpam-4687	458	21	ch	ch	NOUN
ejpam-4687	458	22	be	be	AUX
ejpam-4687	458	23	subsequences	subsequence	NOUN
ejpam-4687	458	24	of	of	ADP
ejpam-4687	458	25	c	c	NOUN
ejpam-4687	458	26	such	such	ADJ
ejpam-4687	458	27	that	that	PRON
ejpam-4687	458	28	ĉg	ĉg	VERB
ejpam-4687	458	29	=	=	NOUN
ejpam-4687	458	30	ĉ	ĉ	NOUN
ejpam-4687	458	31	∩v	∩v	NOUN
ejpam-4687	458	32	(	(	PUNCT
ejpam-4687	458	33	g	g	NOUN
ejpam-4687	458	34	)	)	PUNCT
ejpam-4687	458	35	and	and	CCONJ
ejpam-4687	458	36	ĉh	ĉh	NOUN
ejpam-4687	458	37	=	=	SYM
ejpam-4687	458	38	ĉ	ĉ	X
ejpam-4687	458	39	∩v	∩v	NOUN
ejpam-4687	458	40	(	(	PUNCT
ejpam-4687	458	41	h	h	NOUN
ejpam-4687	458	42	)	)	PUNCT
ejpam-4687	458	43	.	.	PUNCT
ejpam-4687	459	1	let	let	VERB
ejpam-4687	459	2	cg	cg	NOUN
ejpam-4687	460	1	=	=	PUNCT
ejpam-4687	461	1	(	(	PUNCT
ejpam-4687	461	2	an1	an1	PROPN
ejpam-4687	461	3	,	,	PUNCT
ejpam-4687	461	4	an2	an2	PROPN
ejpam-4687	461	5	,	,	PUNCT
ejpam-4687	461	6	.	.	PUNCT
ejpam-4687	461	7	.	.	PUNCT
ejpam-4687	462	1	.	.	PUNCT
ejpam-4687	463	1	,	,	PUNCT
ejpam-4687	463	2	ant	ant	NOUN
ejpam-4687	463	3	)	)	PUNCT
ejpam-4687	463	4	and	and	CCONJ
ejpam-4687	463	5	ch	ch	NOUN
ejpam-4687	463	6	=	=	SYM
ejpam-4687	463	7	(	(	PUNCT
ejpam-4687	463	8	am1	am1	PROPN
ejpam-4687	463	9	,	,	PUNCT
ejpam-4687	463	10	am2	am2	X
ejpam-4687	463	11	,	,	PUNCT
ejpam-4687	463	12	.	.	PUNCT
ejpam-4687	463	13	.	.	PUNCT
ejpam-4687	464	1	.	.	PUNCT
ejpam-4687	465	1	,	,	PUNCT
ejpam-4687	465	2	amr	amr	PROPN
ejpam-4687	465	3	)	)	PUNCT
ejpam-4687	465	4	.	.	PUNCT
ejpam-4687	466	1	note	note	VERB
ejpam-4687	466	2	that	that	SCONJ
ejpam-4687	466	3	n	n	PROPN
ejpam-4687	466	4	2	2	NUM
ejpam-4687	466	5	g+h	g+h	PROPN
ejpam-4687	467	1	[	[	X
ejpam-4687	467	2	anj	anj	X
ejpam-4687	467	3	]	]	PUNCT
ejpam-4687	467	4	⊆	⊆	NUM
ejpam-4687	467	5	v	v	ADP
ejpam-4687	467	6	(	(	PUNCT
ejpam-4687	467	7	g	g	NOUN
ejpam-4687	467	8	)	)	PUNCT
ejpam-4687	467	9	for	for	ADP
ejpam-4687	467	10	all	all	DET
ejpam-4687	467	11	j	j	PROPN
ejpam-4687	467	12	∈	∈	PROPN
ejpam-4687	467	13	{	{	PUNCT
ejpam-4687	467	14	1	1	NUM
ejpam-4687	467	15	,	,	PUNCT
ejpam-4687	467	16	2	2	NUM
ejpam-4687	467	17	,	,	PUNCT
ejpam-4687	467	18	.	.	PUNCT
ejpam-4687	467	19	.	.	PUNCT
ejpam-4687	467	20	.	.	PUNCT
ejpam-4687	467	21	,	,	PUNCT
ejpam-4687	467	22	t	t	PROPN
ejpam-4687	467	23	}	}	PUNCT
ejpam-4687	467	24	and	and	CCONJ
ejpam-4687	467	25	n2	n2	PROPN
ejpam-4687	467	26	g+h	g+h	PROPN
ejpam-4687	468	1	[	[	X
ejpam-4687	468	2	ams	ams	X
ejpam-4687	468	3	]	]	PUNCT
ejpam-4687	468	4	⊆	⊆	NUM
ejpam-4687	468	5	v	v	X
ejpam-4687	468	6	(	(	PUNCT
ejpam-4687	468	7	h	h	NOUN
ejpam-4687	468	8	)	)	PUNCT
ejpam-4687	468	9	for	for	ADP
ejpam-4687	468	10	all	all	PRON
ejpam-4687	468	11	s	s	PROPN
ejpam-4687	468	12	∈	∈	NOUN
ejpam-4687	468	13	{	{	PUNCT
ejpam-4687	468	14	1	1	NUM
ejpam-4687	468	15	,	,	PUNCT
ejpam-4687	468	16	2	2	NUM
ejpam-4687	468	17	,	,	PUNCT
ejpam-4687	468	18	.	.	PUNCT
ejpam-4687	468	19	.	.	PUNCT
ejpam-4687	468	20	.	.	PUNCT
ejpam-4687	468	21	,	,	PUNCT
ejpam-4687	468	22	r	r	NOUN
ejpam-4687	468	23	}	}	PUNCT
ejpam-4687	468	24	.	.	PUNCT
ejpam-4687	469	1	since	since	SCONJ
ejpam-4687	469	2	c	c	PROPN
ejpam-4687	469	3	is	be	AUX
ejpam-4687	469	4	a	a	DET
ejpam-4687	469	5	connected	connected	ADJ
ejpam-4687	469	6	legal	legal	ADJ
ejpam-4687	469	7	closed	close	VERB
ejpam-4687	469	8	hop	hop	NOUN
ejpam-4687	469	9	neighborhood	neighborhood	NOUN
ejpam-4687	469	10	sequence	sequence	NOUN
ejpam-4687	469	11	in	in	ADP
ejpam-4687	469	12	g+h	g+h	PROPN
ejpam-4687	469	13	,	,	PUNCT
ejpam-4687	469	14	it	it	PRON
ejpam-4687	469	15	follows	follow	VERB
ejpam-4687	469	16	that	that	SCONJ
ejpam-4687	469	17	[	[	X
ejpam-4687	469	18	v	v	X
ejpam-4687	469	19	(	(	PUNCT
ejpam-4687	469	20	g	g	NOUN
ejpam-4687	469	21	)	)	PUNCT
ejpam-4687	469	22	\ng(ani	\ng(ani	PROPN
ejpam-4687	469	23	)	)	PUNCT
ejpam-4687	469	24	]	]	PUNCT
ejpam-4687	469	25	\	\	X
ejpam-4687	470	1	∪i−1	∪i−1	PROPN
ejpam-4687	470	2	j=1[v	j=1[v	X
ejpam-4687	470	3	(	(	PUNCT
ejpam-4687	470	4	g	g	NOUN
ejpam-4687	470	5	)	)	PUNCT
ejpam-4687	470	6	\ng(anj	\ng(anj	PROPN
ejpam-4687	470	7	)	)	PUNCT
ejpam-4687	470	8	]	]	PUNCT
ejpam-4687	470	9	=	=	PUNCT
ejpam-4687	470	10	n2	n2	PROPN
ejpam-4687	470	11	g+h	g+h	PROPN
ejpam-4687	471	1	[	[	X
ejpam-4687	471	2	ani	ani	X
ejpam-4687	471	3	]	]	PUNCT
ejpam-4687	471	4	\	\	X
ejpam-4687	471	5	∪i−1	∪i−1	PUNCT
ejpam-4687	471	6	j=1n	j=1n	PROPN
ejpam-4687	471	7	2	2	NUM
ejpam-4687	471	8	g+h	g+h	PROPN
ejpam-4687	472	1	[	[	X
ejpam-4687	472	2	anj	anj	X
ejpam-4687	472	3	]	]	PUNCT
ejpam-4687	472	4	̸=	̸=	PROPN
ejpam-4687	472	5	∅	∅	NOUN
ejpam-4687	472	6	for	for	ADP
ejpam-4687	472	7	all	all	PRON
ejpam-4687	472	8	i	i	PRON
ejpam-4687	472	9	∈	∈	PROPN
ejpam-4687	472	10	{	{	PUNCT
ejpam-4687	472	11	2	2	NUM
ejpam-4687	472	12	,	,	PUNCT
ejpam-4687	472	13	3	3	NUM
ejpam-4687	472	14	,	,	PUNCT
ejpam-4687	472	15	.	.	PUNCT
ejpam-4687	472	16	.	.	PUNCT
ejpam-4687	472	17	.	.	PUNCT
ejpam-4687	472	18	,	,	PUNCT
ejpam-4687	472	19	t	t	PROPN
ejpam-4687	472	20	}	}	PUNCT
ejpam-4687	472	21	.	.	PUNCT
ejpam-4687	473	1	hence	hence	ADV
ejpam-4687	473	2	,	,	PUNCT
ejpam-4687	473	3	cg	cg	NOUN
ejpam-4687	473	4	is	be	AUX
ejpam-4687	473	5	a	a	DET
ejpam-4687	473	6	co	co	ADJ
ejpam-4687	473	7	-	-	ADJ
ejpam-4687	473	8	legal	legal	ADJ
ejpam-4687	473	9	closed	closed	ADJ
ejpam-4687	473	10	neighborhood	neighborhood	NOUN
ejpam-4687	473	11	sequence	sequence	NOUN
ejpam-4687	473	12	in	in	ADP
ejpam-4687	473	13	g.	g.	PROPN
ejpam-4687	473	14	similarly	similarly	ADV
ejpam-4687	473	15	,	,	PUNCT
ejpam-4687	473	16	ch	ch	PROPN
ejpam-4687	473	17	is	be	AUX
ejpam-4687	473	18	a	a	DET
ejpam-4687	473	19	co	co	ADJ
ejpam-4687	473	20	-	-	ADJ
ejpam-4687	473	21	legal	legal	ADJ
ejpam-4687	473	22	closed	closed	ADJ
ejpam-4687	473	23	neighborhood	neighborhood	NOUN
ejpam-4687	473	24	sequence	sequence	NOUN
ejpam-4687	473	25	in	in	ADP
ejpam-4687	473	26	h.	h.	PROPN
ejpam-4687	473	27	therefore	therefore	ADV
ejpam-4687	473	28	,	,	PUNCT
ejpam-4687	473	29	(	(	PUNCT
ejpam-4687	473	30	iii	iii	X
ejpam-4687	473	31	)	)	PUNCT
ejpam-4687	473	32	holds	hold	VERB
ejpam-4687	473	33	.	.	PUNCT
ejpam-4687	474	1	for	for	ADP
ejpam-4687	474	2	the	the	DET
ejpam-4687	474	3	converse	converse	NOUN
ejpam-4687	474	4	,	,	PUNCT
ejpam-4687	474	5	suppose	suppose	VERB
ejpam-4687	474	6	(	(	PUNCT
ejpam-4687	474	7	i	i	NOUN
ejpam-4687	474	8	)	)	PUNCT
ejpam-4687	474	9	or	or	CCONJ
ejpam-4687	474	10	(	(	PUNCT
ejpam-4687	474	11	ii	ii	NOUN
ejpam-4687	474	12	)	)	PUNCT
ejpam-4687	474	13	holds	hold	VERB
ejpam-4687	474	14	.	.	PUNCT
ejpam-4687	475	1	then	then	ADV
ejpam-4687	475	2	c	c	PROPN
ejpam-4687	475	3	is	be	AUX
ejpam-4687	475	4	a	a	DET
ejpam-4687	475	5	legal	legal	ADJ
ejpam-4687	475	6	closed	closed	ADJ
ejpam-4687	475	7	hop	hop	NOUN
ejpam-4687	475	8	neighborhood	neighborhood	NOUN
ejpam-4687	475	9	sequence	sequence	NOUN
ejpam-4687	475	10	in	in	ADP
ejpam-4687	475	11	g	g	PROPN
ejpam-4687	475	12	+	+	PROPN
ejpam-4687	475	13	h.	h.	PROPN
ejpam-4687	475	14	suppose	suppose	VERB
ejpam-4687	475	15	(	(	PUNCT
ejpam-4687	475	16	iii	iii	NOUN
ejpam-4687	475	17	)	)	PUNCT
ejpam-4687	475	18	holds	hold	VERB
ejpam-4687	475	19	.	.	PUNCT
ejpam-4687	476	1	let	let	VERB
ejpam-4687	476	2	c	c	NOUN
ejpam-4687	476	3	=	=	SYM
ejpam-4687	476	4	(	(	PUNCT
ejpam-4687	476	5	a1	a1	PROPN
ejpam-4687	476	6	,	,	PUNCT
ejpam-4687	476	7	a2	a2	PROPN
ejpam-4687	476	8	,	,	PUNCT
ejpam-4687	476	9	·	·	PUNCT
ejpam-4687	476	10	·	·	PUNCT
ejpam-4687	476	11	·	·	PUNCT
ejpam-4687	476	12	,	,	PUNCT
ejpam-4687	476	13	am	be	AUX
ejpam-4687	476	14	)	)	PUNCT
ejpam-4687	476	15	and	and	CCONJ
ejpam-4687	476	16	let	let	VERB
ejpam-4687	476	17	cg	cg	NOUN
ejpam-4687	476	18	=	=	PUNCT
ejpam-4687	476	19	(	(	PUNCT
ejpam-4687	476	20	an1	an1	PROPN
ejpam-4687	476	21	,	,	PUNCT
ejpam-4687	476	22	an2	an2	PROPN
ejpam-4687	476	23	,	,	PUNCT
ejpam-4687	476	24	.	.	PUNCT
ejpam-4687	476	25	.	.	PUNCT
ejpam-4687	477	1	.	.	PUNCT
ejpam-4687	478	1	,	,	PUNCT
ejpam-4687	478	2	ant	ant	NOUN
ejpam-4687	478	3	)	)	PUNCT
ejpam-4687	478	4	and	and	CCONJ
ejpam-4687	478	5	ch	ch	NOUN
ejpam-4687	478	6	=	=	SYM
ejpam-4687	478	7	(	(	PUNCT
ejpam-4687	478	8	am1	am1	PROPN
ejpam-4687	478	9	,	,	PUNCT
ejpam-4687	478	10	am2	am2	X
ejpam-4687	478	11	,	,	PUNCT
ejpam-4687	478	12	.	.	PUNCT
ejpam-4687	478	13	.	.	PUNCT
ejpam-4687	479	1	.	.	PUNCT
ejpam-4687	480	1	,	,	PUNCT
ejpam-4687	480	2	amr	amr	NOUN
ejpam-4687	480	3	)	)	PUNCT
ejpam-4687	480	4	.	.	PUNCT
ejpam-4687	481	1	let	let	VERB
ejpam-4687	481	2	i	i	PRON
ejpam-4687	481	3	∈	∈	PROPN
ejpam-4687	481	4	{	{	PUNCT
ejpam-4687	481	5	2	2	NUM
ejpam-4687	481	6	,	,	PUNCT
ejpam-4687	481	7	3	3	NUM
ejpam-4687	481	8	,	,	PUNCT
ejpam-4687	481	9	.	.	PUNCT
ejpam-4687	481	10	.	.	PUNCT
ejpam-4687	482	1	.	.	PUNCT
ejpam-4687	483	1	,	,	PUNCT
ejpam-4687	483	2	m	m	VERB
ejpam-4687	483	3	}	}	PUNCT
ejpam-4687	483	4	.	.	PUNCT
ejpam-4687	484	1	suppose	suppose	VERB
ejpam-4687	484	2	ai	ai	VERB
ejpam-4687	484	3	∈	∈	PROPN
ejpam-4687	484	4	ĉg	ĉg	PRON
ejpam-4687	484	5	.	.	PUNCT
ejpam-4687	485	1	then	then	ADV
ejpam-4687	485	2	ai	ai	VERB
ejpam-4687	485	3	=	=	SYM
ejpam-4687	485	4	ank	ank	PROPN
ejpam-4687	485	5	for	for	ADP
ejpam-4687	485	6	some	some	DET
ejpam-4687	485	7	k	k	PROPN
ejpam-4687	485	8	∈	∈	PROPN
ejpam-4687	485	9	{	{	PUNCT
ejpam-4687	485	10	1	1	NUM
ejpam-4687	485	11	,	,	PUNCT
ejpam-4687	485	12	2	2	NUM
ejpam-4687	485	13	,	,	PUNCT
ejpam-4687	485	14	.	.	PUNCT
ejpam-4687	485	15	.	.	PUNCT
ejpam-4687	486	1	.	.	PUNCT
ejpam-4687	487	1	,	,	PUNCT
ejpam-4687	487	2	t	t	PROPN
ejpam-4687	487	3	}	}	PUNCT
ejpam-4687	487	4	.	.	PUNCT
ejpam-4687	488	1	since	since	SCONJ
ejpam-4687	488	2	cg	cg	NOUN
ejpam-4687	488	3	is	be	AUX
ejpam-4687	488	4	a	a	DET
ejpam-4687	488	5	co	co	ADJ
ejpam-4687	488	6	-	-	ADJ
ejpam-4687	488	7	legal	legal	ADJ
ejpam-4687	488	8	closed	closed	ADJ
ejpam-4687	488	9	neighborhood	neighborhood	NOUN
ejpam-4687	488	10	sequence	sequence	NOUN
ejpam-4687	488	11	in	in	ADP
ejpam-4687	488	12	g	g	PROPN
ejpam-4687	488	13	,	,	PUNCT
ejpam-4687	488	14	n2	n2	NOUN
ejpam-4687	488	15	g+h	g+h	PROPN
ejpam-4687	489	1	[	[	X
ejpam-4687	489	2	ai	ai	ADP
ejpam-4687	489	3	]	]	PUNCT
ejpam-4687	489	4	\	\	X
ejpam-4687	489	5	∪i−1	∪i−1	PROPN
ejpam-4687	489	6	j=1n	j=1n	PROPN
ejpam-4687	489	7	2	2	NUM
ejpam-4687	489	8	g+h	g+h	PROPN
ejpam-4687	490	1	[	[	X
ejpam-4687	490	2	aj	aj	X
ejpam-4687	490	3	]	]	X
ejpam-4687	490	4	=	=	SYM
ejpam-4687	490	5	n2	n2	PROPN
ejpam-4687	490	6	g+h	g+h	PROPN
ejpam-4687	491	1	[	[	X
ejpam-4687	491	2	ank	ank	X
ejpam-4687	491	3	]	]	PUNCT
ejpam-4687	491	4	\	\	X
ejpam-4687	491	5	∪k−1	∪k−1	NUM
ejpam-4687	491	6	p=1n	p=1n	VERB
ejpam-4687	491	7	2	2	NUM
ejpam-4687	491	8	g+h	g+h	PROPN
ejpam-4687	492	1	[	[	X
ejpam-4687	492	2	anp	anp	X
ejpam-4687	492	3	]	]	PUNCT
ejpam-4687	492	4	=	=	PUNCT
ejpam-4687	493	1	[	[	X
ejpam-4687	493	2	v	v	X
ejpam-4687	493	3	(	(	PUNCT
ejpam-4687	493	4	g	g	NOUN
ejpam-4687	493	5	)	)	PUNCT
ejpam-4687	493	6	\ng(ank	\ng(ank	PROPN
ejpam-4687	493	7	)	)	PUNCT
ejpam-4687	493	8	]	]	PUNCT
ejpam-4687	494	1	\	\	X
ejpam-4687	494	2	∪k−1	∪k−1	NUM
ejpam-4687	494	3	p=1[v	p=1[v	NOUN
ejpam-4687	494	4	(	(	PUNCT
ejpam-4687	494	5	g	g	NOUN
ejpam-4687	494	6	)	)	PUNCT
ejpam-4687	494	7	\ng(anp	\ng(anp	NOUN
ejpam-4687	494	8	)	)	PUNCT
ejpam-4687	494	9	]	]	PUNCT
ejpam-4687	495	1	̸=	̸=	PROPN
ejpam-4687	495	2	∅.	∅.	ADV
ejpam-4687	495	3	if	if	SCONJ
ejpam-4687	495	4	ai	ai	VERB
ejpam-4687	495	5	∈	∈	PROPN
ejpam-4687	495	6	ĉh	ĉh	NOUN
ejpam-4687	495	7	,	,	PUNCT
ejpam-4687	495	8	then	then	ADV
ejpam-4687	495	9	ai	ai	VERB
ejpam-4687	495	10	=	=	PUNCT
ejpam-4687	495	11	amq	amq	INTJ
ejpam-4687	495	12	for	for	ADP
ejpam-4687	495	13	some	some	DET
ejpam-4687	495	14	q	q	NOUN
ejpam-4687	495	15	∈	∈	PROPN
ejpam-4687	495	16	{	{	PUNCT
ejpam-4687	495	17	1	1	NUM
ejpam-4687	495	18	,	,	PUNCT
ejpam-4687	495	19	2	2	NUM
ejpam-4687	495	20	,	,	PUNCT
ejpam-4687	495	21	.	.	PUNCT
ejpam-4687	495	22	.	.	PUNCT
ejpam-4687	495	23	.	.	PUNCT
ejpam-4687	496	1	,	,	PUNCT
ejpam-4687	496	2	m	m	VERB
ejpam-4687	496	3	}	}	PUNCT
ejpam-4687	496	4	.	.	PUNCT
ejpam-4687	497	1	since	since	SCONJ
ejpam-4687	497	2	ch	ch	NOUN
ejpam-4687	497	3	is	be	AUX
ejpam-4687	497	4	a	a	DET
ejpam-4687	497	5	co	co	ADJ
ejpam-4687	497	6	-	-	ADJ
ejpam-4687	497	7	legal	legal	ADJ
ejpam-4687	497	8	closed	closed	ADJ
ejpam-4687	497	9	neighborhood	neighborhood	NOUN
ejpam-4687	497	10	sequence	sequence	NOUN
ejpam-4687	497	11	in	in	ADP
ejpam-4687	497	12	h	h	NOUN
ejpam-4687	497	13	,	,	PUNCT
ejpam-4687	497	14	n2	n2	PROPN
ejpam-4687	497	15	g+h	g+h	PROPN
ejpam-4687	498	1	[	[	X
ejpam-4687	498	2	ai	ai	ADP
ejpam-4687	498	3	]	]	PUNCT
ejpam-4687	498	4	\	\	X
ejpam-4687	498	5	∪i−1	∪i−1	PROPN
ejpam-4687	498	6	j=1n	j=1n	PROPN
ejpam-4687	498	7	2	2	NUM
ejpam-4687	498	8	g+h	g+h	PROPN
ejpam-4687	499	1	[	[	X
ejpam-4687	499	2	aj	aj	X
ejpam-4687	499	3	]	]	X
ejpam-4687	499	4	=	=	SYM
ejpam-4687	499	5	n2	n2	PROPN
ejpam-4687	499	6	g+h	g+h	PROPN
ejpam-4687	500	1	[	[	X
ejpam-4687	500	2	amq	amq	X
ejpam-4687	500	3	]	]	PUNCT
ejpam-4687	500	4	\	\	PROPN
ejpam-4687	500	5	∪	∪	X
ejpam-4687	500	6	q−1	q−1	PROPN
ejpam-4687	500	7	r=1n	r=1n	VERB
ejpam-4687	500	8	2	2	NUM
ejpam-4687	500	9	g+h	g+h	PROPN
ejpam-4687	501	1	[	[	X
ejpam-4687	501	2	amr	amr	X
ejpam-4687	501	3	]	]	X
ejpam-4687	501	4	=	=	PUNCT
ejpam-4687	502	1	[	[	X
ejpam-4687	502	2	v	v	X
ejpam-4687	502	3	(	(	PUNCT
ejpam-4687	502	4	h	h	NOUN
ejpam-4687	502	5	)	)	PUNCT
ejpam-4687	502	6	\nh(amq	\nh(amq	PUNCT
ejpam-4687	502	7	)	)	PUNCT
ejpam-4687	502	8	]	]	PUNCT
ejpam-4687	502	9	\	\	PROPN
ejpam-4687	502	10	∪	∪	X
ejpam-4687	502	11	q−1	q−1	PROPN
ejpam-4687	502	12	r=1[v	r=1[v	PROPN
ejpam-4687	502	13	(	(	PUNCT
ejpam-4687	502	14	h	h	NOUN
ejpam-4687	502	15	)	)	PUNCT
ejpam-4687	502	16	\nh(amr	\nh(amr	NOUN
ejpam-4687	502	17	)	)	PUNCT
ejpam-4687	502	18	]	]	PUNCT
ejpam-4687	503	1	̸=	̸=	PROPN
ejpam-4687	503	2	∅.	∅.	ADP
ejpam-4687	503	3	this	this	DET
ejpam-4687	503	4	shows	show	VERB
ejpam-4687	503	5	that	that	SCONJ
ejpam-4687	503	6	c	c	PROPN
ejpam-4687	503	7	is	be	AUX
ejpam-4687	503	8	a	a	DET
ejpam-4687	503	9	legal	legal	ADJ
ejpam-4687	503	10	closed	closed	ADJ
ejpam-4687	503	11	hop	hop	NOUN
ejpam-4687	503	12	neighborhood	neighborhood	NOUN
ejpam-4687	503	13	sequence	sequence	NOUN
ejpam-4687	503	14	in	in	ADP
ejpam-4687	503	15	g+h	g+h	PROPN
ejpam-4687	503	16	.	.	PUNCT
ejpam-4687	504	1	theorem	theorem	VERB
ejpam-4687	504	2	7	7	NUM
ejpam-4687	504	3	.	.	PUNCT
ejpam-4687	505	1	let	let	VERB
ejpam-4687	505	2	g	g	NOUN
ejpam-4687	505	3	and	and	CCONJ
ejpam-4687	505	4	h	h	NOUN
ejpam-4687	505	5	be	be	VERB
ejpam-4687	505	6	two	two	NUM
ejpam-4687	505	7	graphs	graph	NOUN
ejpam-4687	505	8	and	and	CCONJ
ejpam-4687	505	9	let	let	VERB
ejpam-4687	505	10	c	c	PRON
ejpam-4687	505	11	be	be	AUX
ejpam-4687	505	12	a	a	DET
ejpam-4687	505	13	sequence	sequence	NOUN
ejpam-4687	505	14	of	of	ADP
ejpam-4687	505	15	distinct	distinct	ADJ
ejpam-4687	505	16	vertices	vertex	NOUN
ejpam-4687	505	17	of	of	ADP
ejpam-4687	505	18	g	g	PROPN
ejpam-4687	505	19	+	+	CCONJ
ejpam-4687	505	20	h.	h.	PROPN
ejpam-4687	506	1	then	then	ADV
ejpam-4687	506	2	c	c	PROPN
ejpam-4687	506	3	is	be	AUX
ejpam-4687	506	4	a	a	DET
ejpam-4687	506	5	connected	connected	ADJ
ejpam-4687	506	6	grundy	grundy	PROPN
ejpam-4687	506	7	hop	hop	NOUN
ejpam-4687	506	8	dominating	dominating	NOUN
ejpam-4687	506	9	sequence	sequence	NOUN
ejpam-4687	506	10	in	in	ADP
ejpam-4687	506	11	g	g	PROPN
ejpam-4687	506	12	+	+	NOUN
ejpam-4687	506	13	h	h	NOUN
ejpam-4687	506	14	if	if	SCONJ
ejpam-4687	507	1	and	and	CCONJ
ejpam-4687	507	2	only	only	ADV
ejpam-4687	507	3	if	if	SCONJ
ejpam-4687	507	4	the	the	DET
ejpam-4687	507	5	subsequences	subsequence	NOUN
ejpam-4687	507	6	cg	cg	NOUN
ejpam-4687	507	7	and	and	CCONJ
ejpam-4687	507	8	ch	ch	NOUN
ejpam-4687	507	9	of	of	ADP
ejpam-4687	507	10	c	c	PROPN
ejpam-4687	507	11	,	,	PUNCT
ejpam-4687	507	12	where	where	SCONJ
ejpam-4687	507	13	ĉ	ĉ	ADV
ejpam-4687	507	14	=	=	SYM
ejpam-4687	507	15	ĉg	ĉg	X
ejpam-4687	507	16	∪	∪	VERB
ejpam-4687	507	17	ĉh	ĉh	NOUN
ejpam-4687	507	18	,	,	PUNCT
ejpam-4687	507	19	are	be	AUX
ejpam-4687	507	20	co	co	ADJ
ejpam-4687	507	21	-	-	ADJ
ejpam-4687	507	22	grundy	grundy	ADJ
ejpam-4687	507	23	dominating	dominating	NOUN
ejpam-4687	507	24	sequences	sequence	NOUN
ejpam-4687	507	25	in	in	ADP
ejpam-4687	507	26	g	g	PROPN
ejpam-4687	507	27	and	and	CCONJ
ejpam-4687	507	28	h	h	NOUN
ejpam-4687	507	29	,	,	PUNCT
ejpam-4687	507	30	respectively	respectively	ADV
ejpam-4687	507	31	.	.	PUNCT
ejpam-4687	508	1	proof	proof	NOUN
ejpam-4687	508	2	.	.	PUNCT
ejpam-4687	509	1	suppose	suppose	VERB
ejpam-4687	509	2	c	c	NOUN
ejpam-4687	509	3	is	be	AUX
ejpam-4687	509	4	a	a	DET
ejpam-4687	509	5	connected	connected	ADJ
ejpam-4687	509	6	grundy	grundy	PROPN
ejpam-4687	509	7	hop	hop	NOUN
ejpam-4687	509	8	dominating	dominating	NOUN
ejpam-4687	509	9	sequence	sequence	NOUN
ejpam-4687	509	10	in	in	ADP
ejpam-4687	509	11	g	g	PROPN
ejpam-4687	509	12	+	+	PROPN
ejpam-4687	509	13	h.	h.	PROPN
ejpam-4687	509	14	since	since	SCONJ
ejpam-4687	509	15	c	c	PROPN
ejpam-4687	509	16	is	be	AUX
ejpam-4687	509	17	a	a	DET
ejpam-4687	509	18	legal	legal	ADJ
ejpam-4687	509	19	closed	close	VERB
ejpam-4687	509	20	hop	hop	NOUN
ejpam-4687	509	21	neighborhood	neighborhood	NOUN
ejpam-4687	509	22	sequence	sequence	NOUN
ejpam-4687	509	23	and	and	CCONJ
ejpam-4687	509	24	ĉ	ĉ	X
ejpam-4687	509	25	is	be	AUX
ejpam-4687	509	26	a	a	DET
ejpam-4687	509	27	connected	connected	ADJ
ejpam-4687	509	28	hop	hop	NOUN
ejpam-4687	509	29	dominating	dominating	NOUN
ejpam-4687	509	30	set	set	VERB
ejpam-4687	509	31	j.	j.	PROPN
ejpam-4687	509	32	hassan	hassan	PROPN
ejpam-4687	509	33	,	,	PUNCT
ejpam-4687	509	34	s.	s.	PROPN
ejpam-4687	509	35	canoy	canoy	PROPN
ejpam-4687	509	36	jr	jr	PROPN
ejpam-4687	509	37	.	.	PROPN
ejpam-4687	509	38	/	/	SYM
ejpam-4687	509	39	eur	eur	PROPN
ejpam-4687	509	40	.	.	PUNCT
ejpam-4687	510	1	j.	j.	PROPN
ejpam-4687	510	2	pure	pure	PROPN
ejpam-4687	510	3	appl	appl	PROPN
ejpam-4687	510	4	.	.	PROPN
ejpam-4687	510	5	math	math	PROPN
ejpam-4687	510	6	,	,	PUNCT
ejpam-4687	510	7	16	16	NUM
ejpam-4687	510	8	(	(	PUNCT
ejpam-4687	510	9	2	2	NUM
ejpam-4687	510	10	)	)	PUNCT
ejpam-4687	510	11	(	(	PUNCT
ejpam-4687	510	12	2023	2023	NUM
ejpam-4687	510	13	)	)	PUNCT
ejpam-4687	510	14	,	,	PUNCT
ejpam-4687	510	15	1212	1212	NUM
ejpam-4687	510	16	-	-	SYM
ejpam-4687	510	17	1227	1227	NUM
ejpam-4687	510	18	1226	1226	NUM
ejpam-4687	510	19	in	in	ADP
ejpam-4687	510	20	g+h	g+h	PROPN
ejpam-4687	511	1	,	,	PUNCT
ejpam-4687	511	2	it	it	PRON
ejpam-4687	511	3	follows	follow	VERB
ejpam-4687	511	4	that	that	SCONJ
ejpam-4687	511	5	c	c	PROPN
ejpam-4687	511	6	satisfies	satisfie	NOUN
ejpam-4687	511	7	(	(	PUNCT
ejpam-4687	511	8	iii	iii	NOUN
ejpam-4687	511	9	)	)	PUNCT
ejpam-4687	511	10	of	of	ADP
ejpam-4687	511	11	theorem	theorem	NOUN
ejpam-4687	511	12	6	6	NUM
ejpam-4687	511	13	.	.	PUNCT
ejpam-4687	512	1	hence	hence	ADV
ejpam-4687	512	2	,	,	PUNCT
ejpam-4687	512	3	the	the	DET
ejpam-4687	512	4	subsequences	subsequence	NOUN
ejpam-4687	512	5	cg	cg	NOUN
ejpam-4687	512	6	and	and	CCONJ
ejpam-4687	512	7	ch	ch	NOUN
ejpam-4687	512	8	of	of	ADP
ejpam-4687	512	9	c	c	PROPN
ejpam-4687	512	10	,	,	PUNCT
ejpam-4687	512	11	where	where	SCONJ
ejpam-4687	512	12	ĉ	ĉ	ADV
ejpam-4687	512	13	=	=	SYM
ejpam-4687	512	14	ĉg	ĉg	X
ejpam-4687	512	15	∪	∪	VERB
ejpam-4687	512	16	ĉh	ĉh	NOUN
ejpam-4687	512	17	,	,	PUNCT
ejpam-4687	512	18	are	be	AUX
ejpam-4687	512	19	co	co	ADJ
ejpam-4687	512	20	-	-	ADJ
ejpam-4687	512	21	legal	legal	ADJ
ejpam-4687	512	22	closed	closed	ADJ
ejpam-4687	512	23	neighborhood	neighborhood	NOUN
ejpam-4687	512	24	sequences	sequence	NOUN
ejpam-4687	512	25	in	in	ADP
ejpam-4687	512	26	g	g	PROPN
ejpam-4687	512	27	and	and	CCONJ
ejpam-4687	512	28	h	h	NOUN
ejpam-4687	512	29	,	,	PUNCT
ejpam-4687	512	30	respectively	respectively	ADV
ejpam-4687	512	31	.	.	PUNCT
ejpam-4687	513	1	since	since	SCONJ
ejpam-4687	513	2	ĉ	ĉ	ADV
ejpam-4687	513	3	is	be	AUX
ejpam-4687	513	4	a	a	DET
ejpam-4687	513	5	hop	hop	NOUN
ejpam-4687	513	6	dominating	dominating	NOUN
ejpam-4687	513	7	set	set	VERB
ejpam-4687	513	8	in	in	ADP
ejpam-4687	513	9	g	g	PROPN
ejpam-4687	514	1	+	+	CCONJ
ejpam-4687	514	2	h	h	NOUN
ejpam-4687	514	3	,	,	PUNCT
ejpam-4687	514	4	v	v	ADJ
ejpam-4687	514	5	(	(	PUNCT
ejpam-4687	514	6	g	g	NOUN
ejpam-4687	514	7	)	)	PUNCT
ejpam-4687	515	1	=	=	PUNCT
ejpam-4687	515	2	∪v∈cg	∪v∈cg	PROPN
ejpam-4687	515	3	[	[	X
ejpam-4687	515	4	v	v	X
ejpam-4687	515	5	(	(	PUNCT
ejpam-4687	515	6	g	g	NOUN
ejpam-4687	515	7	)	)	PUNCT
ejpam-4687	515	8	\	\	NOUN
ejpam-4687	515	9	ng(v	ng(v	PUNCT
ejpam-4687	515	10	)	)	PUNCT
ejpam-4687	515	11	]	]	PUNCT
ejpam-4687	515	12	and	and	CCONJ
ejpam-4687	515	13	v	v	X
ejpam-4687	515	14	(	(	PUNCT
ejpam-4687	515	15	h	h	NOUN
ejpam-4687	515	16	)	)	PUNCT
ejpam-4687	516	1	=	=	SYM
ejpam-4687	516	2	∪w∈ch	∪w∈ch	NOUN
ejpam-4687	516	3	[	[	X
ejpam-4687	516	4	v	v	X
ejpam-4687	516	5	(	(	PUNCT
ejpam-4687	516	6	h	h	NOUN
ejpam-4687	516	7	)	)	PUNCT
ejpam-4687	516	8	\	\	NOUN
ejpam-4687	516	9	nh(w	nh(w	PUNCT
ejpam-4687	516	10	)	)	PUNCT
ejpam-4687	516	11	]	]	PUNCT
ejpam-4687	516	12	.	.	PUNCT
ejpam-4687	517	1	thus	thus	ADV
ejpam-4687	517	2	,	,	PUNCT
ejpam-4687	517	3	cg	cg	NOUN
ejpam-4687	517	4	and	and	CCONJ
ejpam-4687	517	5	and	and	CCONJ
ejpam-4687	517	6	ch	ch	NOUN
ejpam-4687	517	7	are	be	AUX
ejpam-4687	517	8	co	co	ADJ
ejpam-4687	517	9	-	-	ADJ
ejpam-4687	517	10	grundy	grundy	ADJ
ejpam-4687	517	11	dominating	dominating	NOUN
ejpam-4687	517	12	sets	set	NOUN
ejpam-4687	517	13	in	in	ADP
ejpam-4687	517	14	g	g	PROPN
ejpam-4687	517	15	and	and	CCONJ
ejpam-4687	517	16	h	h	NOUN
ejpam-4687	517	17	,	,	PUNCT
ejpam-4687	517	18	respectively	respectively	ADV
ejpam-4687	517	19	.	.	PUNCT
ejpam-4687	518	1	conversely	conversely	ADV
ejpam-4687	518	2	,	,	PUNCT
ejpam-4687	518	3	suppose	suppose	VERB
ejpam-4687	518	4	that	that	SCONJ
ejpam-4687	518	5	the	the	DET
ejpam-4687	518	6	subsequences	subsequence	NOUN
ejpam-4687	518	7	cg	cg	NOUN
ejpam-4687	518	8	and	and	CCONJ
ejpam-4687	518	9	ch	ch	NOUN
ejpam-4687	518	10	of	of	ADP
ejpam-4687	518	11	c	c	PROPN
ejpam-4687	518	12	,	,	PUNCT
ejpam-4687	518	13	where	where	SCONJ
ejpam-4687	518	14	ĉ	ĉ	ADV
ejpam-4687	518	15	=	=	SYM
ejpam-4687	518	16	ĉg	ĉg	X
ejpam-4687	518	17	∪	∪	VERB
ejpam-4687	518	18	ĉh	ĉh	NOUN
ejpam-4687	518	19	,	,	PUNCT
ejpam-4687	518	20	are	be	AUX
ejpam-4687	518	21	co	co	ADJ
ejpam-4687	518	22	-	-	ADJ
ejpam-4687	518	23	grundy	grundy	ADJ
ejpam-4687	518	24	dominating	dominating	NOUN
ejpam-4687	518	25	sequences	sequence	NOUN
ejpam-4687	518	26	in	in	ADP
ejpam-4687	518	27	g	g	PROPN
ejpam-4687	518	28	and	and	CCONJ
ejpam-4687	518	29	h	h	NOUN
ejpam-4687	518	30	,	,	PUNCT
ejpam-4687	518	31	respectively	respectively	ADV
ejpam-4687	518	32	.	.	PUNCT
ejpam-4687	519	1	by	by	ADP
ejpam-4687	519	2	theorem	theorem	NOUN
ejpam-4687	519	3	6	6	NUM
ejpam-4687	519	4	,	,	PUNCT
ejpam-4687	519	5	c	c	PROPN
ejpam-4687	519	6	is	be	AUX
ejpam-4687	519	7	a	a	DET
ejpam-4687	519	8	legal	legal	ADJ
ejpam-4687	519	9	closed	closed	ADJ
ejpam-4687	519	10	hop	hop	NOUN
ejpam-4687	519	11	neighborhood	neighborhood	NOUN
ejpam-4687	519	12	sequence	sequence	NOUN
ejpam-4687	519	13	in	in	ADP
ejpam-4687	519	14	g	g	PROPN
ejpam-4687	519	15	+	+	CCONJ
ejpam-4687	519	16	h.	h.	PROPN
ejpam-4687	519	17	since	since	SCONJ
ejpam-4687	519	18	cg	cg	NOUN
ejpam-4687	519	19	and	and	CCONJ
ejpam-4687	519	20	ch	ch	PROPN
ejpam-4687	519	21	co	co	PROPN
ejpam-4687	519	22	-	-	ADJ
ejpam-4687	519	23	grundy	grundy	ADJ
ejpam-4687	519	24	dominating	dominating	NOUN
ejpam-4687	519	25	sequences	sequence	NOUN
ejpam-4687	519	26	in	in	ADP
ejpam-4687	519	27	g	g	PROPN
ejpam-4687	519	28	and	and	CCONJ
ejpam-4687	519	29	h	h	NOUN
ejpam-4687	519	30	,	,	PUNCT
ejpam-4687	519	31	respectively	respectively	ADV
ejpam-4687	519	32	,	,	PUNCT
ejpam-4687	519	33	ĉ	ĉ	X
ejpam-4687	519	34	is	be	AUX
ejpam-4687	519	35	a	a	DET
ejpam-4687	519	36	hop	hop	NOUN
ejpam-4687	519	37	dominating	dominating	NOUN
ejpam-4687	519	38	set	set	NOUN
ejpam-4687	519	39	.	.	PUNCT
ejpam-4687	520	1	moreover	moreover	ADV
ejpam-4687	520	2	,	,	PUNCT
ejpam-4687	520	3	because	because	SCONJ
ejpam-4687	520	4	⟨ĉ⟩	⟨ĉ⟩	PROPN
ejpam-4687	520	5	is	be	AUX
ejpam-4687	520	6	connected	connect	VERB
ejpam-4687	520	7	,	,	PUNCT
ejpam-4687	520	8	c	c	PROPN
ejpam-4687	520	9	is	be	AUX
ejpam-4687	520	10	a	a	DET
ejpam-4687	520	11	connected	connected	ADJ
ejpam-4687	520	12	grundy	grundy	PROPN
ejpam-4687	520	13	hop	hop	NOUN
ejpam-4687	520	14	dominating	dominating	NOUN
ejpam-4687	520	15	sequence	sequence	NOUN
ejpam-4687	520	16	in	in	ADP
ejpam-4687	520	17	g+h	g+h	PROPN
ejpam-4687	520	18	.	.	PUNCT
ejpam-4687	521	1	the	the	DET
ejpam-4687	521	2	following	following	ADJ
ejpam-4687	521	3	result	result	NOUN
ejpam-4687	521	4	follows	follow	VERB
ejpam-4687	521	5	from	from	ADP
ejpam-4687	521	6	theorem	theorem	ADJ
ejpam-4687	521	7	7	7	NUM
ejpam-4687	521	8	,	,	PUNCT
ejpam-4687	521	9	proposition	proposition	NOUN
ejpam-4687	521	10	8	8	NUM
ejpam-4687	521	11	,	,	PUNCT
ejpam-4687	521	12	and	and	CCONJ
ejpam-4687	521	13	proposition	proposition	NOUN
ejpam-4687	521	14	9	9	NUM
ejpam-4687	521	15	.	.	PUNCT
ejpam-4687	521	16	corollary	corollary	ADJ
ejpam-4687	521	17	7	7	NUM
ejpam-4687	521	18	.	.	PUNCT
ejpam-4687	522	1	let	let	VERB
ejpam-4687	522	2	g	g	NOUN
ejpam-4687	522	3	and	and	CCONJ
ejpam-4687	522	4	h	h	PROPN
ejpam-4687	522	5	be	be	AUX
ejpam-4687	522	6	graphs	graph	NOUN
ejpam-4687	522	7	.	.	PUNCT
ejpam-4687	523	1	then	then	ADV
ejpam-4687	523	2	γchgr	γchgr	PROPN
ejpam-4687	523	3	(	(	PUNCT
ejpam-4687	523	4	g+h	g+h	PROPN
ejpam-4687	523	5	)	)	PUNCT
ejpam-4687	523	6	=	=	SYM
ejpam-4687	523	7	γcogr(g	γcogr(g	NOUN
ejpam-4687	523	8	)	)	PUNCT
ejpam-4687	524	1	+	+	CCONJ
ejpam-4687	524	2	γcogr(h	γcogr(h	NOUN
ejpam-4687	524	3	)	)	PUNCT
ejpam-4687	524	4	.	.	PUNCT
ejpam-4687	525	1	in	in	ADP
ejpam-4687	525	2	particular	particular	ADJ
ejpam-4687	525	3	,	,	PUNCT
ejpam-4687	525	4	each	each	PRON
ejpam-4687	525	5	of	of	ADP
ejpam-4687	525	6	the	the	DET
ejpam-4687	525	7	following	follow	VERB
ejpam-4687	525	8	holds	hold	NOUN
ejpam-4687	525	9	.	.	PUNCT
ejpam-4687	526	1	(	(	PUNCT
ejpam-4687	526	2	i	i	NOUN
ejpam-4687	526	3	)	)	PUNCT
ejpam-4687	526	4	γchgr	γchgr	PROPN
ejpam-4687	526	5	(	(	PUNCT
ejpam-4687	526	6	k1	k1	NOUN
ejpam-4687	526	7	+	+	NOUN
ejpam-4687	526	8	g	g	NOUN
ejpam-4687	526	9	)	)	PUNCT
ejpam-4687	526	10	=	=	SYM
ejpam-4687	526	11	1	1	NUM
ejpam-4687	526	12	+	+	NUM
ejpam-4687	526	13	γcogr(g	γcogr(g	NOUN
ejpam-4687	526	14	)	)	PUNCT
ejpam-4687	526	15	.	.	PUNCT
ejpam-4687	527	1	(	(	PUNCT
ejpam-4687	527	2	ii	ii	NOUN
ejpam-4687	527	3	)	)	PUNCT
ejpam-4687	527	4	γchgr	γchgr	NOUN
ejpam-4687	527	5	(	(	PUNCT
ejpam-4687	527	6	km	km	NOUN
ejpam-4687	527	7	,	,	PUNCT
ejpam-4687	527	8	n	n	CCONJ
ejpam-4687	527	9	)	)	PUNCT
ejpam-4687	527	10	=	=	SYM
ejpam-4687	527	11	2	2	NUM
ejpam-4687	527	12	for	for	ADP
ejpam-4687	527	13	m	m	PROPN
ejpam-4687	527	14	,	,	PUNCT
ejpam-4687	527	15	n	n	PRON
ejpam-4687	527	16	≥	≥	NOUN
ejpam-4687	527	17	1	1	NUM
ejpam-4687	527	18	.	.	PUNCT
ejpam-4687	527	19	(	(	PUNCT
ejpam-4687	527	20	iii	iii	NOUN
ejpam-4687	527	21	)	)	PUNCT
ejpam-4687	527	22	γchgr	γchgr	NOUN
ejpam-4687	527	23	(	(	PUNCT
ejpam-4687	527	24	wn	wn	PROPN
ejpam-4687	527	25	)	)	PUNCT
ejpam-4687	527	26	=	=	NOUN
ejpam-4687	527	27	4	4	NUM
ejpam-4687	527	28	for	for	ADP
ejpam-4687	527	29	all	all	DET
ejpam-4687	527	30	n	n	PRON
ejpam-4687	527	31	≥	≥	NOUN
ejpam-4687	527	32	5	5	NUM
ejpam-4687	527	33	.	.	PUNCT
ejpam-4687	527	34	(	(	PUNCT
ejpam-4687	527	35	iv	iv	X
ejpam-4687	527	36	)	)	PUNCT
ejpam-4687	527	37	γchgr	γchgr	NOUN
ejpam-4687	527	38	(	(	PUNCT
ejpam-4687	527	39	fn	fn	NOUN
ejpam-4687	527	40	)	)	PUNCT
ejpam-4687	527	41	=	=	SYM
ejpam-4687	527	42	4	4	NUM
ejpam-4687	527	43	for	for	ADP
ejpam-4687	527	44	all	all	DET
ejpam-4687	527	45	n	n	PRON
ejpam-4687	527	46	≥	≥	NOUN
ejpam-4687	527	47	4	4	NUM
ejpam-4687	527	48	.	.	PUNCT
ejpam-4687	527	49	(	(	PUNCT
ejpam-4687	527	50	v	v	NOUN
ejpam-4687	527	51	)	)	PUNCT
ejpam-4687	527	52	γchgr	γchgr	NOUN
ejpam-4687	527	53	(	(	PUNCT
ejpam-4687	527	54	pn	pn	NOUN
ejpam-4687	527	55	+	+	CCONJ
ejpam-4687	527	56	pm	pm	NOUN
ejpam-4687	527	57	)	)	PUNCT
ejpam-4687	527	58	=	=	SYM
ejpam-4687	527	59	6	6	NUM
ejpam-4687	527	60	for	for	ADP
ejpam-4687	527	61	all	all	DET
ejpam-4687	527	62	n	n	CCONJ
ejpam-4687	527	63	,	,	PUNCT
ejpam-4687	527	64	m	m	VERB
ejpam-4687	527	65	≥	≥	NOUN
ejpam-4687	527	66	4	4	NUM
ejpam-4687	527	67	.	.	PUNCT
ejpam-4687	527	68	(	(	PUNCT
ejpam-4687	527	69	vi	vi	NOUN
ejpam-4687	527	70	)	)	PUNCT
ejpam-4687	527	71	γchgr	γchgr	NOUN
ejpam-4687	527	72	(	(	PUNCT
ejpam-4687	527	73	cn	cn	X
ejpam-4687	527	74	+	+	NOUN
ejpam-4687	527	75	cm	cm	NOUN
ejpam-4687	527	76	)	)	PUNCT
ejpam-4687	527	77	=	=	PUNCT
ejpam-4687	527	78	6	6	NUM
ejpam-4687	527	79	for	for	ADP
ejpam-4687	527	80	all	all	DET
ejpam-4687	527	81	n	n	CCONJ
ejpam-4687	527	82	,	,	PUNCT
ejpam-4687	527	83	m	m	VERB
ejpam-4687	527	84	≥	≥	NOUN
ejpam-4687	527	85	5	5	NUM
ejpam-4687	527	86	.	.	PUNCT
ejpam-4687	527	87	(	(	PUNCT
ejpam-4687	527	88	vii	vii	PROPN
ejpam-4687	527	89	)	)	PUNCT
ejpam-4687	527	90	γchgr	γchgr	NOUN
ejpam-4687	527	91	(	(	PUNCT
ejpam-4687	527	92	pn	pn	X
ejpam-4687	527	93	+	+	CCONJ
ejpam-4687	527	94	cm	cm	NOUN
ejpam-4687	527	95	)	)	PUNCT
ejpam-4687	527	96	=	=	PUNCT
ejpam-4687	527	97	6	6	NUM
ejpam-4687	527	98	for	for	ADP
ejpam-4687	527	99	all	all	PRON
ejpam-4687	527	100	n	n	PRON
ejpam-4687	527	101	≥	≥	NOUN
ejpam-4687	527	102	4	4	NUM
ejpam-4687	527	103	and	and	CCONJ
ejpam-4687	527	104	m	m	PRON
ejpam-4687	527	105	≥	≥	NOUN
ejpam-4687	527	106	5	5	NUM
ejpam-4687	527	107	.	.	NOUN
ejpam-4687	527	108	4	4	NUM
ejpam-4687	527	109	.	.	X
ejpam-4687	527	110	conclusion	conclusion	NOUN
ejpam-4687	527	111	in	in	ADP
ejpam-4687	527	112	this	this	DET
ejpam-4687	527	113	study	study	NOUN
ejpam-4687	527	114	,	,	PUNCT
ejpam-4687	527	115	connected	connect	VERB
ejpam-4687	527	116	grundy	grundy	PROPN
ejpam-4687	527	117	hop	hop	PROPN
ejpam-4687	527	118	domination	domination	NOUN
ejpam-4687	527	119	numbers	number	NOUN
ejpam-4687	527	120	of	of	ADP
ejpam-4687	527	121	some	some	DET
ejpam-4687	527	122	graphs	graph	NOUN
ejpam-4687	527	123	are	be	AUX
ejpam-4687	527	124	determined	determine	VERB
ejpam-4687	527	125	.	.	PUNCT
ejpam-4687	528	1	realization	realization	NOUN
ejpam-4687	528	2	results	result	NOUN
ejpam-4687	528	3	involving	involve	VERB
ejpam-4687	528	4	connected	connect	VERB
ejpam-4687	528	5	hop	hop	NOUN
ejpam-4687	528	6	domination	domination	NOUN
ejpam-4687	528	7	,	,	PUNCT
ejpam-4687	528	8	grundy	grundy	PROPN
ejpam-4687	528	9	hop	hop	PROPN
ejpam-4687	528	10	domination	domination	NOUN
ejpam-4687	528	11	,	,	PUNCT
ejpam-4687	528	12	and	and	CCONJ
ejpam-4687	528	13	connected	connect	VERB
ejpam-4687	528	14	grundy	grundy	PROPN
ejpam-4687	528	15	hop	hop	PROPN
ejpam-4687	528	16	domination	domination	NOUN
ejpam-4687	528	17	numbers	number	NOUN
ejpam-4687	528	18	are	be	AUX
ejpam-4687	528	19	given	give	VERB
ejpam-4687	528	20	.	.	PUNCT
ejpam-4687	529	1	for	for	ADP
ejpam-4687	529	2	the	the	DET
ejpam-4687	529	3	shadow	shadow	NOUN
ejpam-4687	529	4	graph	graph	NOUN
ejpam-4687	529	5	s(g	s(g	PROPN
ejpam-4687	529	6	)	)	PUNCT
ejpam-4687	529	7	of	of	ADP
ejpam-4687	529	8	a	a	DET
ejpam-4687	529	9	non	non	ADJ
ejpam-4687	529	10	-	-	ADJ
ejpam-4687	529	11	trivial	trivial	ADJ
ejpam-4687	529	12	graph	graph	NOUN
ejpam-4687	529	13	g	g	NOUN
ejpam-4687	529	14	,	,	PUNCT
ejpam-4687	529	15	it	it	PRON
ejpam-4687	529	16	is	be	AUX
ejpam-4687	529	17	conjectured	conjecture	VERB
ejpam-4687	529	18	that	that	SCONJ
ejpam-4687	529	19	γchgr	γchgr	PROPN
ejpam-4687	529	20	(	(	PUNCT
ejpam-4687	529	21	s(g	s(g	PROPN
ejpam-4687	529	22	)	)	PUNCT
ejpam-4687	529	23	)	)	PUNCT
ejpam-4687	530	1	=	=	PUNCT
ejpam-4687	530	2	γchgr	γchgr	X
ejpam-4687	530	3	(	(	PUNCT
ejpam-4687	530	4	g	g	NOUN
ejpam-4687	530	5	)	)	PUNCT
ejpam-4687	530	6	.	.	PUNCT
ejpam-4687	531	1	connected	connect	VERB
ejpam-4687	531	2	grundy	grundy	PROPN
ejpam-4687	531	3	hop	hop	PROPN
ejpam-4687	531	4	domination	domination	NOUN
ejpam-4687	531	5	can	can	AUX
ejpam-4687	531	6	still	still	ADV
ejpam-4687	531	7	be	be	AUX
ejpam-4687	531	8	studied	study	VERB
ejpam-4687	531	9	further	far	ADV
ejpam-4687	531	10	.	.	PUNCT
ejpam-4687	532	1	acknowledgements	acknowledgement	NOUN
ejpam-4687	532	2	the	the	DET
ejpam-4687	532	3	authors	author	NOUN
ejpam-4687	532	4	would	would	AUX
ejpam-4687	532	5	like	like	VERB
ejpam-4687	532	6	to	to	PART
ejpam-4687	532	7	thank	thank	VERB
ejpam-4687	532	8	the	the	DET
ejpam-4687	532	9	department	department	NOUN
ejpam-4687	532	10	of	of	ADP
ejpam-4687	532	11	science	science	NOUN
ejpam-4687	532	12	and	and	CCONJ
ejpam-4687	532	13	technology	technology	NOUN
ejpam-4687	532	14	-	-	PUNCT
ejpam-4687	532	15	accelerated	accelerate	VERB
ejpam-4687	532	16	science	science	NOUN
ejpam-4687	532	17	and	and	CCONJ
ejpam-4687	532	18	technology	technology	NOUN
ejpam-4687	532	19	human	human	ADJ
ejpam-4687	532	20	resource	resource	NOUN
ejpam-4687	532	21	development	development	NOUN
ejpam-4687	532	22	program	program	NOUN
ejpam-4687	532	23	(	(	PUNCT
ejpam-4687	532	24	dost	dost	NOUN
ejpam-4687	532	25	-	-	PUNCT
ejpam-4687	532	26	asthrdp)philippines	asthrdp)philippine	NOUN
ejpam-4687	532	27	,	,	PUNCT
ejpam-4687	532	28	msu	msu	PROPN
ejpam-4687	532	29	-	-	PUNCT
ejpam-4687	532	30	iligan	iligan	PROPN
ejpam-4687	532	31	institute	institute	PROPN
ejpam-4687	532	32	of	of	ADP
ejpam-4687	532	33	technology	technology	NOUN
ejpam-4687	532	34	,	,	PUNCT
ejpam-4687	532	35	and	and	CCONJ
ejpam-4687	532	36	msu	msu	PROPN
ejpam-4687	532	37	tawi	tawi	PROPN
ejpam-4687	532	38	-	-	PUNCT
ejpam-4687	532	39	tawi	tawi	PROPN
ejpam-4687	532	40	college	college	PROPN
ejpam-4687	532	41	of	of	ADP
ejpam-4687	532	42	technology	technology	NOUN
ejpam-4687	532	43	and	and	CCONJ
ejpam-4687	532	44	oceanography	oceanography	NOUN
ejpam-4687	532	45	for	for	ADP
ejpam-4687	532	46	funding	fund	VERB
ejpam-4687	532	47	this	this	DET
ejpam-4687	532	48	research	research	NOUN
ejpam-4687	532	49	.	.	PUNCT
ejpam-4687	533	1	references	reference	NOUN
ejpam-4687	533	2	1227	1227	NUM
ejpam-4687	533	3	references	reference	NOUN
ejpam-4687	533	4	[	[	X
ejpam-4687	533	5	1	1	NUM
ejpam-4687	533	6	]	]	PUNCT
ejpam-4687	533	7	s.	s.	PROPN
ejpam-4687	533	8	ayyaswamy	ayyaswamy	PROPN
ejpam-4687	533	9	,	,	PUNCT
ejpam-4687	533	10	b.	b.	PROPN
ejpam-4687	533	11	krishnakumari	krishnakumari	PROPN
ejpam-4687	533	12	,	,	PUNCT
ejpam-4687	533	13	b.	b.	PROPN
ejpam-4687	533	14	natarjan	natarjan	PROPN
ejpam-4687	533	15	,	,	PUNCT
ejpam-4687	533	16	and	and	CCONJ
ejpam-4687	533	17	y.	y.	PROPN
ejpam-4687	533	18	venkatakrishnan	venkatakrishnan	PROPN
ejpam-4687	533	19	.	.	PUNCT
ejpam-4687	534	1	bounds	bound	NOUN
ejpam-4687	534	2	on	on	ADP
ejpam-4687	534	3	the	the	DET
ejpam-4687	534	4	hop	hop	NOUN
ejpam-4687	534	5	domination	domination	NOUN
ejpam-4687	534	6	number	number	NOUN
ejpam-4687	534	7	of	of	ADP
ejpam-4687	534	8	a	a	DET
ejpam-4687	534	9	tree	tree	NOUN
ejpam-4687	534	10	.	.	PUNCT
ejpam-4687	535	1	proceedings	proceeding	NOUN
ejpam-4687	535	2	-	-	PUNCT
ejpam-4687	535	3	mathematical	mathematical	ADJ
ejpam-4687	535	4	sciences	science	NOUN
ejpam-4687	535	5	.	.	PUNCT
ejpam-4687	535	6	,	,	PUNCT
ejpam-4687	535	7	125(4):449–455	125(4):449–455	ADP
ejpam-4687	535	8	,	,	PUNCT
ejpam-4687	535	9	2015	2015	NUM
ejpam-4687	535	10	.	.	PUNCT
ejpam-4687	536	1	[	[	X
ejpam-4687	536	2	2	2	NUM
ejpam-4687	536	3	]	]	PUNCT
ejpam-4687	536	4	s.	s.	PROPN
ejpam-4687	536	5	ayyaswamy	ayyaswamy	PROPN
ejpam-4687	536	6	,	,	PUNCT
ejpam-4687	536	7	c.	c.	PROPN
ejpam-4687	536	8	natarajan	natarajan	PROPN
ejpam-4687	536	9	,	,	PUNCT
ejpam-4687	536	10	and	and	CCONJ
ejpam-4687	536	11	g.	g.	PROPN
ejpam-4687	536	12	sathiamoorphy	sathiamoorphy	PROPN
ejpam-4687	536	13	.	.	PUNCT
ejpam-4687	537	1	a	a	DET
ejpam-4687	537	2	note	note	NOUN
ejpam-4687	537	3	on	on	ADP
ejpam-4687	537	4	hop	hop	NOUN
ejpam-4687	537	5	domination	domination	NOUN
ejpam-4687	537	6	number	number	NOUN
ejpam-4687	537	7	of	of	ADP
ejpam-4687	537	8	some	some	DET
ejpam-4687	537	9	special	special	ADJ
ejpam-4687	537	10	families	family	NOUN
ejpam-4687	537	11	of	of	ADP
ejpam-4687	537	12	graphs	graph	NOUN
ejpam-4687	537	13	.	.	PUNCT
ejpam-4687	538	1	international	international	ADJ
ejpam-4687	538	2	journal	journal	NOUN
ejpam-4687	538	3	of	of	ADP
ejpam-4687	538	4	pure	pure	ADJ
ejpam-4687	538	5	and	and	CCONJ
ejpam-4687	538	6	applied	applied	ADJ
ejpam-4687	538	7	mathematics	mathematic	NOUN
ejpam-4687	538	8	.	.	PUNCT
ejpam-4687	538	9	,	,	PUNCT
ejpam-4687	538	10	119(12):11465–14171	119(12):11465–14171	NUM
ejpam-4687	538	11	,	,	PUNCT
ejpam-4687	538	12	2018	2018	NUM
ejpam-4687	538	13	.	.	PUNCT
ejpam-4687	539	1	[	[	X
ejpam-4687	539	2	3	3	X
ejpam-4687	539	3	]	]	X
ejpam-4687	539	4	b.	b.	PROPN
ejpam-4687	539	5	bresar	bresar	PROPN
ejpam-4687	539	6	,	,	PUNCT
ejpam-4687	539	7	cs	cs	PROPN
ejpam-4687	539	8	.	.	PROPN
ejpam-4687	539	9	bujtas	bujtas	PROPN
ejpam-4687	539	10	,	,	PUNCT
ejpam-4687	539	11	t.	t.	NOUN
ejpam-4687	539	12	gologranc	gologranc	PROPN
ejpam-4687	539	13	,	,	PUNCT
ejpam-4687	539	14	s.	s.	PROPN
ejpam-4687	539	15	klavzar	klavzar	PROPN
ejpam-4687	539	16	,	,	PUNCT
ejpam-4687	539	17	g.	g.	PROPN
ejpam-4687	539	18	kosmrlj	kosmrlj	PROPN
ejpam-4687	539	19	,	,	PUNCT
ejpam-4687	539	20	b.	b.	PROPN
ejpam-4687	539	21	patkos	patkos	PROPN
ejpam-4687	539	22	,	,	PUNCT
ejpam-4687	539	23	zs	zs	PROPN
ejpam-4687	539	24	.	.	PUNCT
ejpam-4687	539	25	tuza	tuza	PROPN
ejpam-4687	539	26	,	,	PUNCT
ejpam-4687	539	27	and	and	CCONJ
ejpam-4687	539	28	m.	m.	NOUN
ejpam-4687	539	29	vizer	vizer	NOUN
ejpam-4687	539	30	.	.	PUNCT
ejpam-4687	540	1	dominating	dominate	VERB
ejpam-4687	540	2	sequence	sequence	NOUN
ejpam-4687	540	3	in	in	ADP
ejpam-4687	540	4	grid	grid	NOUN
ejpam-4687	540	5	-	-	PUNCT
ejpam-4687	540	6	like	like	ADJ
ejpam-4687	540	7	and	and	CCONJ
ejpam-4687	540	8	toroidal	toroidal	ADJ
ejpam-4687	540	9	graphs	graph	NOUN
ejpam-4687	540	10	.	.	PUNCT
ejpam-4687	541	1	electron	electron	PROPN
ejpam-4687	541	2	.	.	PUNCT
ejpam-4687	542	1	j.	j.	PROPN
ejpam-4687	542	2	combin	combin	PROPN
ejpam-4687	542	3	.	.	PROPN
ejpam-4687	542	4	,	,	PUNCT
ejpam-4687	542	5	(	(	PUNCT
ejpam-4687	542	6	23):1–17	23):1–17	NUM
ejpam-4687	542	7	,	,	PUNCT
ejpam-4687	542	8	2016	2016	NUM
ejpam-4687	542	9	.	.	PUNCT
ejpam-4687	543	1	[	[	X
ejpam-4687	543	2	4	4	X
ejpam-4687	543	3	]	]	X
ejpam-4687	543	4	b.	b.	PROPN
ejpam-4687	543	5	bresar	bresar	PROPN
ejpam-4687	543	6	,	,	PUNCT
ejpam-4687	543	7	cs	cs	PROPN
ejpam-4687	543	8	.	.	PROPN
ejpam-4687	543	9	bujtas	bujtas	PROPN
ejpam-4687	543	10	,	,	PUNCT
ejpam-4687	543	11	t.	t.	NOUN
ejpam-4687	543	12	gologranc	gologranc	PROPN
ejpam-4687	543	13	,	,	PUNCT
ejpam-4687	543	14	s.	s.	PROPN
ejpam-4687	543	15	klavzar	klavzar	PROPN
ejpam-4687	543	16	,	,	PUNCT
ejpam-4687	543	17	g.	g.	PROPN
ejpam-4687	543	18	kosmrlj	kosmrlj	PROPN
ejpam-4687	543	19	,	,	PUNCT
ejpam-4687	543	20	b.	b.	PROPN
ejpam-4687	543	21	patkos	patkos	PROPN
ejpam-4687	543	22	,	,	PUNCT
ejpam-4687	543	23	zs	zs	PROPN
ejpam-4687	543	24	.	.	PUNCT
ejpam-4687	543	25	tuza	tuza	PROPN
ejpam-4687	543	26	,	,	PUNCT
ejpam-4687	543	27	and	and	CCONJ
ejpam-4687	543	28	m.	m.	NOUN
ejpam-4687	543	29	vizer	vizer	NOUN
ejpam-4687	543	30	.	.	PUNCT
ejpam-4687	544	1	grundy	grundy	PROPN
ejpam-4687	544	2	dominating	dominating	NOUN
ejpam-4687	544	3	sequence	sequence	NOUN
ejpam-4687	544	4	and	and	CCONJ
ejpam-4687	544	5	zero	zero	NUM
ejpam-4687	544	6	forcing	forcing	NOUN
ejpam-4687	544	7	sets	set	NOUN
ejpam-4687	544	8	.	.	PUNCT
ejpam-4687	545	1	discrete	discrete	ADJ
ejpam-4687	545	2	optim	optim	ADJ
ejpam-4687	545	3	.	.	PUNCT
ejpam-4687	545	4	,	,	PUNCT
ejpam-4687	545	5	(	(	PUNCT
ejpam-4687	545	6	26):66–77	26):66–77	NUM
ejpam-4687	545	7	,	,	PUNCT
ejpam-4687	545	8	2017	2017	NUM
ejpam-4687	545	9	.	.	PUNCT
ejpam-4687	546	1	[	[	X
ejpam-4687	546	2	5	5	NUM
ejpam-4687	546	3	]	]	PUNCT
ejpam-4687	546	4	b.	b.	PROPN
ejpam-4687	546	5	bresar	bresar	PROPN
ejpam-4687	546	6	,	,	PUNCT
ejpam-4687	546	7	t.	t.	NOUN
ejpam-4687	546	8	gologranc	gologranc	PROPN
ejpam-4687	546	9	,	,	PUNCT
ejpam-4687	546	10	and	and	CCONJ
ejpam-4687	546	11	t.	t.	PROPN
ejpam-4687	546	12	kos	kos	PROPN
ejpam-4687	546	13	.	.	PUNCT
ejpam-4687	547	1	dominating	dominate	VERB
ejpam-4687	547	2	sequences	sequence	NOUN
ejpam-4687	547	3	under	under	ADP
ejpam-4687	547	4	atomic	atomic	ADJ
ejpam-4687	547	5	changes	change	NOUN
ejpam-4687	547	6	with	with	ADP
ejpam-4687	547	7	applications	application	NOUN
ejpam-4687	547	8	in	in	ADP
ejpam-4687	547	9	sierpinski	sierpinski	ADJ
ejpam-4687	547	10	and	and	CCONJ
ejpam-4687	547	11	interval	interval	NOUN
ejpam-4687	547	12	graphs	graph	NOUN
ejpam-4687	547	13	.	.	PUNCT
ejpam-4687	548	1	appl	appl	PROPN
ejpam-4687	548	2	.	.	PUNCT
ejpam-4687	549	1	anal	anal	PROPN
ejpam-4687	549	2	.	.	PUNCT
ejpam-4687	550	1	discrete	discrete	ADJ
ejpam-4687	550	2	math	math	NOUN
ejpam-4687	550	3	.	.	PUNCT
ejpam-4687	551	1	,	,	PUNCT
ejpam-4687	551	2	(	(	PUNCT
ejpam-4687	551	3	10):518–531	10):518–531	NUM
ejpam-4687	551	4	,	,	PUNCT
ejpam-4687	551	5	2016	2016	NUM
ejpam-4687	551	6	.	.	PUNCT
ejpam-4687	552	1	[	[	X
ejpam-4687	552	2	6	6	NUM
ejpam-4687	552	3	]	]	PUNCT
ejpam-4687	552	4	b.	b.	PROPN
ejpam-4687	552	5	bresar	bresar	PROPN
ejpam-4687	552	6	,	,	PUNCT
ejpam-4687	552	7	t.	t.	NOUN
ejpam-4687	552	8	gologranc	gologranc	PROPN
ejpam-4687	552	9	,	,	PUNCT
ejpam-4687	552	10	m.	m.	NOUN
ejpam-4687	552	11	milanic	milanic	PROPN
ejpam-4687	552	12	,	,	PUNCT
ejpam-4687	552	13	d.	d.	PROPN
ejpam-4687	552	14	rall	rall	PROPN
ejpam-4687	552	15	,	,	PUNCT
ejpam-4687	552	16	and	and	CCONJ
ejpam-4687	552	17	r.	r.	PROPN
ejpam-4687	552	18	rizzi	rizzi	PROPN
ejpam-4687	552	19	.	.	PUNCT
ejpam-4687	553	1	dominating	dominate	VERB
ejpam-4687	553	2	sequence	sequence	NOUN
ejpam-4687	553	3	in	in	ADP
ejpam-4687	553	4	graphs	graph	NOUN
ejpam-4687	553	5	.	.	PUNCT
ejpam-4687	554	1	discrete	discrete	ADJ
ejpam-4687	554	2	math	math	NOUN
ejpam-4687	554	3	.	.	PUNCT
ejpam-4687	555	1	,	,	PUNCT
ejpam-4687	555	2	(	(	PUNCT
ejpam-4687	555	3	336):22–36	336):22–36	NUM
ejpam-4687	555	4	,	,	PUNCT
ejpam-4687	555	5	2014	2014	NUM
ejpam-4687	555	6	.	.	PUNCT
ejpam-4687	556	1	[	[	X
ejpam-4687	556	2	7	7	X
ejpam-4687	556	3	]	]	X
ejpam-4687	556	4	b.	b.	PROPN
ejpam-4687	556	5	bresar	bresar	PROPN
ejpam-4687	556	6	,	,	PUNCT
ejpam-4687	556	7	t.	t.	PROPN
ejpam-4687	556	8	kos	kos	PROPN
ejpam-4687	556	9	,	,	PUNCT
ejpam-4687	556	10	and	and	CCONJ
ejpam-4687	556	11	p.	p.	NOUN
ejpam-4687	556	12	torres	torre	NOUN
ejpam-4687	556	13	.	.	PUNCT
ejpam-4687	557	1	grundy	grundy	PROPN
ejpam-4687	557	2	domination	domination	NOUN
ejpam-4687	557	3	and	and	CCONJ
ejpam-4687	557	4	zero	zero	NUM
ejpam-4687	557	5	forcing	force	VERB
ejpam-4687	557	6	in	in	ADP
ejpam-4687	557	7	kneser	kneser	NOUN
ejpam-4687	557	8	graphs	graph	NOUN
ejpam-4687	557	9	.	.	PUNCT
ejpam-4687	558	1	ars	ar	VERB
ejpam-4687	558	2	math	math	PROPN
ejpam-4687	558	3	.	.	PUNCT
ejpam-4687	559	1	contemp	contemp	NOUN
ejpam-4687	559	2	.	.	PUNCT
ejpam-4687	560	1	,	,	PUNCT
ejpam-4687	560	2	(	(	PUNCT
ejpam-4687	560	3	17):419–430	17):419–430	NUM
ejpam-4687	560	4	,	,	PUNCT
ejpam-4687	560	5	2019	2019	NUM
ejpam-4687	560	6	.	.	PUNCT
ejpam-4687	561	1	[	[	X
ejpam-4687	561	2	8	8	X
ejpam-4687	561	3	]	]	PUNCT
ejpam-4687	561	4	j.	j.	PROPN
ejpam-4687	561	5	hassan	hassan	PROPN
ejpam-4687	561	6	and	and	CCONJ
ejpam-4687	561	7	s.	s.	PROPN
ejpam-4687	561	8	canoy	canoy	PROPN
ejpam-4687	561	9	jr	jr	PROPN
ejpam-4687	561	10	.	.	PUNCT
ejpam-4687	562	1	grundy	grundy	PROPN
ejpam-4687	562	2	hop	hop	PROPN
ejpam-4687	562	3	domination	domination	PROPN
ejpam-4687	562	4	in	in	ADP
ejpam-4687	562	5	graphs	graph	NOUN
ejpam-4687	562	6	.	.	PUNCT
ejpam-4687	563	1	eur	eur	PROPN
ejpam-4687	563	2	.	.	PUNCT
ejpam-4687	564	1	j.	j.	PROPN
ejpam-4687	564	2	pure	pure	PROPN
ejpam-4687	564	3	appl	appl	PROPN
ejpam-4687	564	4	.	.	PUNCT
ejpam-4687	564	5	math	math	PROPN
ejpam-4687	564	6	.	.	PUNCT
ejpam-4687	564	7	,	,	PUNCT
ejpam-4687	564	8	15(4):1623–1636	15(4):1623–1636	NUM
ejpam-4687	564	9	,	,	PUNCT
ejpam-4687	564	10	2022	2022	NUM
ejpam-4687	564	11	.	.	PUNCT
ejpam-4687	565	1	[	[	X
ejpam-4687	565	2	9	9	NUM
ejpam-4687	565	3	]	]	PUNCT
ejpam-4687	565	4	j.	j.	PROPN
ejpam-4687	565	5	hassan	hassan	PROPN
ejpam-4687	565	6	and	and	CCONJ
ejpam-4687	565	7	s.	s.	PROPN
ejpam-4687	565	8	canoy	canoy	PROPN
ejpam-4687	565	9	jr	jr	PROPN
ejpam-4687	565	10	.	.	PROPN
ejpam-4687	565	11	hop	hop	PROPN
ejpam-4687	565	12	independent	independent	ADJ
ejpam-4687	565	13	hop	hop	NOUN
ejpam-4687	565	14	domination	domination	NOUN
ejpam-4687	565	15	in	in	ADP
ejpam-4687	565	16	graphs	graph	NOUN
ejpam-4687	565	17	.	.	PUNCT
ejpam-4687	566	1	eur	eur	PROPN
ejpam-4687	566	2	.	.	PUNCT
ejpam-4687	567	1	j.	j.	PROPN
ejpam-4687	567	2	pure	pure	PROPN
ejpam-4687	567	3	appl	appl	PROPN
ejpam-4687	567	4	.	.	PUNCT
ejpam-4687	567	5	math	math	PROPN
ejpam-4687	567	6	.	.	PUNCT
ejpam-4687	567	7	,	,	PUNCT
ejpam-4687	567	8	15(4):1783–1796	15(4):1783–1796	NUM
ejpam-4687	567	9	,	,	PUNCT
ejpam-4687	567	10	2022	2022	NUM
ejpam-4687	567	11	.	.	PUNCT
ejpam-4687	568	1	[	[	X
ejpam-4687	568	2	10	10	NUM
ejpam-4687	568	3	]	]	X
ejpam-4687	568	4	s.	s.	PROPN
ejpam-4687	568	5	canoy	canoy	PROPN
ejpam-4687	568	6	jr	jr	PROPN
ejpam-4687	568	7	.	.	PROPN
ejpam-4687	568	8	,	,	PUNCT
ejpam-4687	568	9	r.	r.	PROPN
ejpam-4687	568	10	mollejon	mollejon	NOUN
ejpam-4687	568	11	,	,	PUNCT
ejpam-4687	568	12	and	and	CCONJ
ejpam-4687	568	13	j.	j.	PROPN
ejpam-4687	568	14	g.	g.	PROPN
ejpam-4687	568	15	canoy	canoy	PROPN
ejpam-4687	568	16	.	.	PUNCT
ejpam-4687	569	1	hop	hop	PROPN
ejpam-4687	569	2	dominating	dominating	NOUN
ejpam-4687	569	3	sets	set	NOUN
ejpam-4687	569	4	in	in	ADP
ejpam-4687	569	5	graphs	graph	NOUN
ejpam-4687	569	6	under	under	ADP
ejpam-4687	569	7	binary	binary	ADJ
ejpam-4687	569	8	operations	operation	NOUN
ejpam-4687	569	9	.	.	PUNCT
ejpam-4687	570	1	eur	eur	PROPN
ejpam-4687	570	2	.	.	PUNCT
ejpam-4687	571	1	j.	j.	PROPN
ejpam-4687	571	2	pure	pure	PROPN
ejpam-4687	571	3	appl	appl	PROPN
ejpam-4687	571	4	.	.	PUNCT
ejpam-4687	571	5	math	math	PROPN
ejpam-4687	571	6	.	.	PUNCT
ejpam-4687	571	7	,	,	PUNCT
ejpam-4687	572	1	12(4):1455–1463	12(4):1455–1463	NUM
ejpam-4687	572	2	,	,	PUNCT
ejpam-4687	572	3	2019	2019	NUM
ejpam-4687	572	4	.	.	PUNCT
ejpam-4687	573	1	[	[	X
ejpam-4687	573	2	11	11	NUM
ejpam-4687	573	3	]	]	X
ejpam-4687	573	4	s.	s.	PROPN
ejpam-4687	573	5	canoy	canoy	PROPN
ejpam-4687	573	6	jr	jr	PROPN
ejpam-4687	573	7	.	.	PROPN
ejpam-4687	573	8	and	and	CCONJ
ejpam-4687	573	9	g.	g.	PROPN
ejpam-4687	573	10	salasalan	salasalan	NOUN
ejpam-4687	573	11	.	.	PUNCT
ejpam-4687	574	1	revisiting	revisit	VERB
ejpam-4687	574	2	domination	domination	NOUN
ejpam-4687	574	3	,	,	PUNCT
ejpam-4687	574	4	hop	hop	NOUN
ejpam-4687	574	5	domination	domination	NOUN
ejpam-4687	574	6	and	and	CCONJ
ejpam-4687	574	7	global	global	ADJ
ejpam-4687	574	8	hop	hop	NOUN
ejpam-4687	574	9	domination	domination	NOUN
ejpam-4687	574	10	in	in	ADP
ejpam-4687	574	11	graphs	graph	NOUN
ejpam-4687	574	12	.	.	PUNCT
ejpam-4687	575	1	eur	eur	PROPN
ejpam-4687	575	2	.	.	PUNCT
ejpam-4687	576	1	j.	j.	PROPN
ejpam-4687	576	2	pure	pure	PROPN
ejpam-4687	576	3	appl	appl	PROPN
ejpam-4687	576	4	.	.	PUNCT
ejpam-4687	576	5	math	math	PROPN
ejpam-4687	576	6	.	.	PUNCT
ejpam-4687	577	1	,	,	PUNCT
ejpam-4687	577	2	(	(	PUNCT
ejpam-4687	577	3	14):1415–1428	14):1415–1428	X
ejpam-4687	577	4	,	,	PUNCT
ejpam-4687	577	5	2021	2021	NUM
ejpam-4687	577	6	.	.	PUNCT
ejpam-4687	578	1	[	[	X
ejpam-4687	578	2	12	12	NUM
ejpam-4687	578	3	]	]	X
ejpam-4687	578	4	g.	g.	PROPN
ejpam-4687	578	5	nasini	nasini	PROPN
ejpam-4687	578	6	and	and	CCONJ
ejpam-4687	578	7	p.	p.	NOUN
ejpam-4687	578	8	torres	torre	NOUN
ejpam-4687	578	9	.	.	PUNCT
ejpam-4687	579	1	grundy	grundy	PROPN
ejpam-4687	579	2	dominating	dominate	VERB
ejpam-4687	579	3	sequences	sequence	NOUN
ejpam-4687	579	4	on	on	ADP
ejpam-4687	579	5	x	x	ADJ
ejpam-4687	579	6	-	-	ADJ
ejpam-4687	579	7	join	join	ADJ
ejpam-4687	579	8	product	product	NOUN
ejpam-4687	579	9	.	.	PUNCT
ejpam-4687	580	1	discrete	discrete	ADJ
ejpam-4687	580	2	applied	applied	ADJ
ejpam-4687	580	3	mathematics	mathematic	NOUN
ejpam-4687	580	4	.	.	PUNCT
ejpam-4687	580	5	,	,	PUNCT
ejpam-4687	580	6	(	(	PUNCT
ejpam-4687	580	7	284):138–149	284):138–149	NOUN
ejpam-4687	580	8	,	,	PUNCT
ejpam-4687	580	9	2020	2020	NUM
ejpam-4687	580	10	.	.	PUNCT
ejpam-4687	581	1	[	[	X
ejpam-4687	581	2	13	13	NUM
ejpam-4687	581	3	]	]	X
ejpam-4687	581	4	c.	c.	PROPN
ejpam-4687	581	5	natarajan	natarajan	PROPN
ejpam-4687	581	6	and	and	CCONJ
ejpam-4687	581	7	s.	s.	PROPN
ejpam-4687	581	8	ayyaswamy	ayyaswamy	PROPN
ejpam-4687	581	9	.	.	PUNCT
ejpam-4687	582	1	hop	hop	PROPN
ejpam-4687	582	2	domination	domination	NOUN
ejpam-4687	582	3	in	in	ADP
ejpam-4687	582	4	graphs	graphs	PROPN
ejpam-4687	582	5	ii	ii	PROPN
ejpam-4687	582	6	.	.	PUNCT
ejpam-4687	582	7	versita	versita	PROPN
ejpam-4687	582	8	,	,	PUNCT
ejpam-4687	582	9	23(2):187	23(2):187	NUM
ejpam-4687	582	10	–	–	PUNCT
ejpam-4687	582	11	199	199	NUM
ejpam-4687	582	12	,	,	PUNCT
ejpam-4687	582	13	2015	2015	NUM
ejpam-4687	582	14	.	.	PUNCT