id	sid	tid	token	lemma	pos
ejpam-6125	1	1	european	european	PROPN
ejpam-6125	1	2	journal	journal	PROPN
ejpam-6125	1	3	of	of	ADP
ejpam-6125	1	4	pure	pure	ADJ
ejpam-6125	1	5	and	and	CCONJ
ejpam-6125	1	6	applied	applied	ADJ
ejpam-6125	1	7	mathematics	mathematic	NOUN
ejpam-6125	1	8	2025	2025	NUM
ejpam-6125	1	9	,	,	PUNCT
ejpam-6125	1	10	vol	vol	NOUN
ejpam-6125	1	11	.	.	PROPN
ejpam-6125	1	12	18	18	NUM
ejpam-6125	1	13	,	,	PUNCT
ejpam-6125	1	14	issue	issue	NOUN
ejpam-6125	1	15	3	3	NUM
ejpam-6125	1	16	,	,	PUNCT
ejpam-6125	1	17	article	article	NOUN
ejpam-6125	1	18	number	number	NOUN
ejpam-6125	1	19	6125	6125	NUM
ejpam-6125	1	20	issn	issn	VERB
ejpam-6125	1	21	1307	1307	NUM
ejpam-6125	1	22	-	-	SYM
ejpam-6125	1	23	5543	5543	NUM
ejpam-6125	1	24	–	–	PUNCT
ejpam-6125	1	25	ejpam.com	ejpam.com	X
ejpam-6125	1	26	published	publish	VERB
ejpam-6125	1	27	by	by	ADP
ejpam-6125	1	28	new	new	PROPN
ejpam-6125	1	29	york	york	PROPN
ejpam-6125	1	30	business	business	PROPN
ejpam-6125	1	31	global	global	ADJ
ejpam-6125	1	32	legal	legal	ADJ
ejpam-6125	1	33	closed	close	VERB
ejpam-6125	1	34	hop	hop	NOUN
ejpam-6125	1	35	neighborhood	neighborhood	NOUN
ejpam-6125	1	36	independent	independent	ADJ
ejpam-6125	1	37	sequences	sequence	NOUN
ejpam-6125	1	38	in	in	ADP
ejpam-6125	1	39	graphs	graph	NOUN
ejpam-6125	1	40	javier	javier	PROPN
ejpam-6125	1	41	a.	a.	PROPN
ejpam-6125	1	42	hassan1,2,∗	hassan1,2,∗	PROPN
ejpam-6125	1	43	,	,	PUNCT
ejpam-6125	1	44	farhadz	farhadz	VERB
ejpam-6125	1	45	a.	a.	NOUN
ejpam-6125	1	46	aripin1	aripin1	PROPN
ejpam-6125	1	47	,	,	PUNCT
ejpam-6125	1	48	maria	maria	PROPN
ejpam-6125	1	49	andrea	andrea	PROPN
ejpam-6125	1	50	o.	o.	PROPN
ejpam-6125	1	51	bonsocan3	bonsocan3	PROPN
ejpam-6125	1	52	,	,	PUNCT
ejpam-6125	1	53	kimberly	kimberly	PROPN
ejpam-6125	1	54	jane	jane	PROPN
ejpam-6125	1	55	pon1	pon1	PROPN
ejpam-6125	1	56	,	,	PUNCT
ejpam-6125	1	57	vergel	vergel	NOUN
ejpam-6125	1	58	t.	t.	NOUN
ejpam-6125	1	59	bilar3	bilar3	NOUN
ejpam-6125	1	60	1department	1department	NUM
ejpam-6125	1	61	of	of	ADP
ejpam-6125	1	62	mathematics	mathematic	NOUN
ejpam-6125	1	63	,	,	PUNCT
ejpam-6125	1	64	college	college	NOUN
ejpam-6125	1	65	of	of	ADP
ejpam-6125	1	66	arts	art	NOUN
ejpam-6125	1	67	and	and	CCONJ
ejpam-6125	1	68	sciences	science	NOUN
ejpam-6125	1	69	,	,	PUNCT
ejpam-6125	1	70	msu	msu	PROPN
ejpam-6125	1	71	tawi	tawi	PROPN
ejpam-6125	1	72	-	-	PUNCT
ejpam-6125	1	73	tawi	tawi	PROPN
ejpam-6125	1	74	college	college	PROPN
ejpam-6125	1	75	of	of	ADP
ejpam-6125	1	76	technology	technology	NOUN
ejpam-6125	1	77	and	and	CCONJ
ejpam-6125	1	78	oceanography	oceanography	NOUN
ejpam-6125	1	79	,	,	PUNCT
ejpam-6125	1	80	bongao	bongao	NOUN
ejpam-6125	1	81	,	,	PUNCT
ejpam-6125	1	82	tawi	tawi	NOUN
ejpam-6125	1	83	-	-	PUNCT
ejpam-6125	1	84	tawi	tawi	NOUN
ejpam-6125	1	85	,	,	PUNCT
ejpam-6125	1	86	philippines	philippine	NOUN
ejpam-6125	1	87	2department	2department	NUM
ejpam-6125	1	88	of	of	ADP
ejpam-6125	1	89	mathematics	mathematic	NOUN
ejpam-6125	1	90	,	,	PUNCT
ejpam-6125	1	91	college	college	NOUN
ejpam-6125	1	92	of	of	ADP
ejpam-6125	1	93	science	science	PROPN
ejpam-6125	1	94	,	,	PUNCT
ejpam-6125	1	95	korea	korea	PROPN
ejpam-6125	1	96	university	university	PROPN
ejpam-6125	1	97	,	,	PUNCT
ejpam-6125	1	98	seoul	seoul	PROPN
ejpam-6125	1	99	,	,	PUNCT
ejpam-6125	1	100	south	south	PROPN
ejpam-6125	1	101	korea	korea	PROPN
ejpam-6125	2	1	3department	3department	NUM
ejpam-6125	2	2	of	of	ADP
ejpam-6125	2	3	mathematics	mathematic	NOUN
ejpam-6125	2	4	,	,	PUNCT
ejpam-6125	2	5	ateneo	ateneo	X
ejpam-6125	2	6	de	de	PROPN
ejpam-6125	2	7	davao	davao	PROPN
ejpam-6125	2	8	university	university	PROPN
ejpam-6125	2	9	,	,	PUNCT
ejpam-6125	2	10	davao	davao	PROPN
ejpam-6125	2	11	city	city	PROPN
ejpam-6125	2	12	,	,	PUNCT
ejpam-6125	2	13	philippines	philippine	NOUN
ejpam-6125	2	14	abstract	abstract	ADJ
ejpam-6125	2	15	.	.	PUNCT
ejpam-6125	3	1	let	let	VERB
ejpam-6125	3	2	g	g	PRON
ejpam-6125	3	3	be	be	AUX
ejpam-6125	3	4	a	a	DET
ejpam-6125	3	5	graph	graph	NOUN
ejpam-6125	3	6	.	.	PUNCT
ejpam-6125	4	1	a	a	DET
ejpam-6125	4	2	sequence	sequence	NOUN
ejpam-6125	4	3	q	q	NOUN
ejpam-6125	5	1	=	=	SYM
ejpam-6125	5	2	(	(	PUNCT
ejpam-6125	5	3	x1	x1	PROPN
ejpam-6125	5	4	,	,	PUNCT
ejpam-6125	5	5	x2	x2	PROPN
ejpam-6125	5	6	,	,	PUNCT
ejpam-6125	5	7	...	...	PUNCT
ejpam-6125	5	8	,	,	PUNCT
ejpam-6125	5	9	xk	xk	PROPN
ejpam-6125	5	10	)	)	PUNCT
ejpam-6125	5	11	of	of	ADP
ejpam-6125	5	12	distinct	distinct	ADJ
ejpam-6125	5	13	vertices	vertex	NOUN
ejpam-6125	5	14	of	of	ADP
ejpam-6125	5	15	g	g	PROPN
ejpam-6125	5	16	is	be	AUX
ejpam-6125	5	17	called	call	VERB
ejpam-6125	5	18	a	a	DET
ejpam-6125	5	19	legal	legal	ADJ
ejpam-6125	5	20	closed	closed	ADJ
ejpam-6125	5	21	hop	hop	NOUN
ejpam-6125	5	22	neighborhood	neighborhood	NOUN
ejpam-6125	5	23	independent	independent	ADJ
ejpam-6125	5	24	sequence	sequence	NOUN
ejpam-6125	5	25	(	(	PUNCT
ejpam-6125	5	26	lchni	lchni	PROPN
ejpam-6125	5	27	sequence	sequence	PROPN
ejpam-6125	5	28	)	)	PUNCT
ejpam-6125	5	29	if	if	SCONJ
ejpam-6125	5	30	it	it	PRON
ejpam-6125	5	31	satisfies	satisfy	VERB
ejpam-6125	5	32	the	the	DET
ejpam-6125	5	33	following	follow	VERB
ejpam-6125	5	34	two	two	NUM
ejpam-6125	5	35	conditions	condition	NOUN
ejpam-6125	5	36	:	:	PUNCT
ejpam-6125	6	1	[	[	X
ejpam-6125	6	2	(	(	PUNCT
ejpam-6125	6	3	i	i	NOUN
ejpam-6125	6	4	)	)	PUNCT
ejpam-6125	6	5	]	]	PUNCT
ejpam-6125	6	6	n2	n2	PROPN
ejpam-6125	6	7	g[xi	g[xi	PROPN
ejpam-6125	6	8	]	]	PUNCT
ejpam-6125	6	9	\	\	PROPN
ejpam-6125	6	10	⋃i−1	⋃i−1	NOUN
ejpam-6125	6	11	j=1	j=1	PROPN
ejpam-6125	6	12	n	n	CCONJ
ejpam-6125	6	13	2	2	NUM
ejpam-6125	6	14	g[xj	g[xj	NOUN
ejpam-6125	6	15	]	]	PUNCT
ejpam-6125	7	1	̸=	̸=	PROPN
ejpam-6125	7	2	for	for	ADP
ejpam-6125	7	3	each	each	DET
ejpam-6125	7	4	i	i	PRON
ejpam-6125	7	5	∈	∈	PROPN
ejpam-6125	7	6	{	{	PUNCT
ejpam-6125	7	7	2	2	NUM
ejpam-6125	7	8	,	,	PUNCT
ejpam-6125	7	9	3	3	NUM
ejpam-6125	7	10	,	,	PUNCT
ejpam-6125	7	11	...	...	PUNCT
ejpam-6125	7	12	,	,	PUNCT
ejpam-6125	7	13	k	k	NOUN
ejpam-6125	7	14	}	}	PUNCT
ejpam-6125	7	15	,	,	PUNCT
ejpam-6125	7	16	and	and	CCONJ
ejpam-6125	7	17	[	[	X
ejpam-6125	7	18	(	(	PUNCT
ejpam-6125	7	19	ii	ii	NOUN
ejpam-6125	7	20	)	)	PUNCT
ejpam-6125	7	21	]	]	PUNCT
ejpam-6125	8	1	dg(xs	dg(xs	NOUN
ejpam-6125	8	2	,	,	PUNCT
ejpam-6125	8	3	xt	xt	ADJ
ejpam-6125	8	4	)	)	PUNCT
ejpam-6125	8	5	̸=	̸=	PROPN
ejpam-6125	8	6	1	1	NUM
ejpam-6125	8	7	for	for	ADP
ejpam-6125	8	8	each	each	DET
ejpam-6125	8	9	s	s	PROPN
ejpam-6125	8	10	,	,	PUNCT
ejpam-6125	8	11	t	t	PROPN
ejpam-6125	8	12	∈	∈	PROPN
ejpam-6125	8	13	{	{	PUNCT
ejpam-6125	8	14	1	1	NUM
ejpam-6125	8	15	,	,	PUNCT
ejpam-6125	8	16	2	2	NUM
ejpam-6125	8	17	,	,	PUNCT
ejpam-6125	8	18	...	...	PUNCT
ejpam-6125	8	19	,	,	PUNCT
ejpam-6125	8	20	k	k	X
ejpam-6125	8	21	}	}	PUNCT
ejpam-6125	8	22	,	,	PUNCT
ejpam-6125	8	23	where	where	SCONJ
ejpam-6125	8	24	s	s	AUX
ejpam-6125	8	25	̸=	̸=	PROPN
ejpam-6125	8	26	t.	t.	NOUN
ejpam-6125	8	27	the	the	DET
ejpam-6125	8	28	legal	legal	ADJ
ejpam-6125	8	29	closed	close	VERB
ejpam-6125	8	30	hop	hop	NOUN
ejpam-6125	8	31	neighborhood	neighborhood	NOUN
ejpam-6125	8	32	independence	independence	NOUN
ejpam-6125	8	33	number	number	NOUN
ejpam-6125	8	34	(	(	PUNCT
ejpam-6125	8	35	lchni	lchni	PROPN
ejpam-6125	8	36	number	number	PROPN
ejpam-6125	8	37	)	)	PUNCT
ejpam-6125	8	38	of	of	ADP
ejpam-6125	8	39	g	g	PROPN
ejpam-6125	8	40	is	be	AUX
ejpam-6125	8	41	the	the	DET
ejpam-6125	8	42	maximum	maximum	ADJ
ejpam-6125	8	43	length	length	NOUN
ejpam-6125	8	44	of	of	ADP
ejpam-6125	8	45	an	an	DET
ejpam-6125	8	46	lchni	lchni	ADJ
ejpam-6125	8	47	sequence	sequence	NOUN
ejpam-6125	8	48	of	of	ADP
ejpam-6125	8	49	g	g	NOUN
ejpam-6125	8	50	,	,	PUNCT
ejpam-6125	8	51	and	and	CCONJ
ejpam-6125	8	52	this	this	PRON
ejpam-6125	8	53	is	be	AUX
ejpam-6125	8	54	denoted	denote	VERB
ejpam-6125	8	55	by	by	ADP
ejpam-6125	8	56	θ(g	θ(g	NOUN
ejpam-6125	8	57	)	)	PUNCT
ejpam-6125	8	58	.	.	PUNCT
ejpam-6125	9	1	in	in	ADP
ejpam-6125	9	2	this	this	DET
ejpam-6125	9	3	paper	paper	NOUN
ejpam-6125	9	4	,	,	PUNCT
ejpam-6125	9	5	the	the	DET
ejpam-6125	9	6	authors	author	NOUN
ejpam-6125	9	7	initiate	initiate	VERB
ejpam-6125	9	8	the	the	DET
ejpam-6125	9	9	study	study	NOUN
ejpam-6125	9	10	of	of	ADP
ejpam-6125	9	11	a	a	DET
ejpam-6125	9	12	legal	legal	ADJ
ejpam-6125	9	13	closed	closed	ADJ
ejpam-6125	9	14	hop	hop	NOUN
ejpam-6125	9	15	neighborhood	neighborhood	NOUN
ejpam-6125	9	16	independent	independent	ADJ
ejpam-6125	9	17	sequence	sequence	NOUN
ejpam-6125	9	18	in	in	ADP
ejpam-6125	9	19	some	some	DET
ejpam-6125	9	20	special	special	ADJ
ejpam-6125	9	21	graphs	graph	NOUN
ejpam-6125	9	22	,	,	PUNCT
ejpam-6125	9	23	shadow	shadow	NOUN
ejpam-6125	9	24	graphs	graph	NOUN
ejpam-6125	9	25	,	,	PUNCT
ejpam-6125	9	26	and	and	CCONJ
ejpam-6125	9	27	the	the	DET
ejpam-6125	9	28	join	join	NOUN
ejpam-6125	9	29	of	of	ADP
ejpam-6125	9	30	two	two	NUM
ejpam-6125	9	31	graphs	graph	NOUN
ejpam-6125	9	32	.	.	PUNCT
ejpam-6125	10	1	in	in	ADP
ejpam-6125	10	2	particular	particular	ADJ
ejpam-6125	10	3	,	,	PUNCT
ejpam-6125	10	4	the	the	DET
ejpam-6125	10	5	authors	author	NOUN
ejpam-6125	10	6	determine	determine	VERB
ejpam-6125	10	7	the	the	DET
ejpam-6125	10	8	corresponding	corresponding	ADJ
ejpam-6125	10	9	legal	legal	ADJ
ejpam-6125	10	10	closed	closed	ADJ
ejpam-6125	10	11	neighborhood	neighborhood	NOUN
ejpam-6125	10	12	independence	independence	NOUN
ejpam-6125	10	13	numbers	number	NOUN
ejpam-6125	10	14	of	of	ADP
ejpam-6125	10	15	these	these	DET
ejpam-6125	10	16	graphs	graph	NOUN
ejpam-6125	10	17	.	.	PUNCT
ejpam-6125	11	1	2020	2020	NUM
ejpam-6125	11	2	mathematics	mathematic	NOUN
ejpam-6125	11	3	subject	subject	NOUN
ejpam-6125	11	4	classifications	classification	NOUN
ejpam-6125	11	5	:	:	PUNCT
ejpam-6125	11	6	05c69	05c69	X
ejpam-6125	11	7	key	key	ADJ
ejpam-6125	11	8	words	word	NOUN
ejpam-6125	11	9	and	and	CCONJ
ejpam-6125	11	10	phrases	phrase	NOUN
ejpam-6125	11	11	:	:	PUNCT
ejpam-6125	11	12	independent	independent	ADJ
ejpam-6125	11	13	set	set	NOUN
ejpam-6125	11	14	,	,	PUNCT
ejpam-6125	11	15	legal	legal	ADJ
ejpam-6125	11	16	closed	close	VERB
ejpam-6125	11	17	hop	hop	NOUN
ejpam-6125	11	18	neighborhood	neighborhood	NOUN
ejpam-6125	11	19	independent	independent	ADJ
ejpam-6125	11	20	sequence	sequence	NOUN
ejpam-6125	11	21	,	,	PUNCT
ejpam-6125	11	22	legal	legal	ADJ
ejpam-6125	11	23	closed	close	VERB
ejpam-6125	11	24	hop	hop	NOUN
ejpam-6125	11	25	neighborhood	neighborhood	NOUN
ejpam-6125	11	26	independence	independence	NOUN
ejpam-6125	11	27	number	number	NOUN
ejpam-6125	11	28	,	,	PUNCT
ejpam-6125	11	29	co	co	ADJ
ejpam-6125	11	30	-	-	ADJ
ejpam-6125	11	31	legal	legal	ADJ
ejpam-6125	11	32	closed	closed	ADJ
ejpam-6125	11	33	neighborhood	neighborhood	NOUN
ejpam-6125	11	34	independent	independent	ADJ
ejpam-6125	11	35	sequence	sequence	NOUN
ejpam-6125	11	36	1	1	NUM
ejpam-6125	11	37	.	.	PUNCT
ejpam-6125	11	38	introduction	introduction	NOUN
ejpam-6125	11	39	an	an	DET
ejpam-6125	11	40	independent	independent	ADJ
ejpam-6125	11	41	set	set	NOUN
ejpam-6125	11	42	in	in	ADP
ejpam-6125	11	43	a	a	DET
ejpam-6125	11	44	graph	graph	NOUN
ejpam-6125	11	45	is	be	AUX
ejpam-6125	11	46	a	a	DET
ejpam-6125	11	47	set	set	NOUN
ejpam-6125	11	48	of	of	ADP
ejpam-6125	11	49	vertices	vertex	NOUN
ejpam-6125	11	50	such	such	ADJ
ejpam-6125	11	51	that	that	SCONJ
ejpam-6125	11	52	no	no	DET
ejpam-6125	11	53	two	two	NUM
ejpam-6125	11	54	vertices	vertex	NOUN
ejpam-6125	11	55	in	in	ADP
ejpam-6125	11	56	the	the	DET
ejpam-6125	11	57	set	set	NOUN
ejpam-6125	11	58	are	be	AUX
ejpam-6125	11	59	adjacent	adjacent	ADJ
ejpam-6125	11	60	to	to	ADP
ejpam-6125	11	61	each	each	DET
ejpam-6125	11	62	other	other	ADJ
ejpam-6125	11	63	.	.	PUNCT
ejpam-6125	12	1	this	this	DET
ejpam-6125	12	2	idea	idea	NOUN
ejpam-6125	12	3	is	be	AUX
ejpam-6125	12	4	crucial	crucial	ADJ
ejpam-6125	12	5	in	in	ADP
ejpam-6125	12	6	various	various	ADJ
ejpam-6125	12	7	areas	area	NOUN
ejpam-6125	12	8	of	of	ADP
ejpam-6125	12	9	graph	graph	NOUN
ejpam-6125	12	10	theory	theory	NOUN
ejpam-6125	12	11	and	and	CCONJ
ejpam-6125	12	12	its	its	PRON
ejpam-6125	12	13	applications	application	NOUN
ejpam-6125	12	14	,	,	PUNCT
ejpam-6125	12	15	including	include	VERB
ejpam-6125	12	16	network	network	NOUN
ejpam-6125	12	17	design	design	NOUN
ejpam-6125	12	18	,	,	PUNCT
ejpam-6125	12	19	scheduling	scheduling	NOUN
ejpam-6125	12	20	problems	problem	NOUN
ejpam-6125	12	21	,	,	PUNCT
ejpam-6125	12	22	and	and	CCONJ
ejpam-6125	12	23	resource	resource	NOUN
ejpam-6125	12	24	allocation	allocation	NOUN
ejpam-6125	12	25	.	.	PUNCT
ejpam-6125	13	1	independent	independent	ADJ
ejpam-6125	13	2	sets	set	NOUN
ejpam-6125	13	3	in	in	ADP
ejpam-6125	13	4	graphs	graph	NOUN
ejpam-6125	13	5	have	have	AUX
ejpam-6125	13	6	been	be	AUX
ejpam-6125	13	7	studied	study	VERB
ejpam-6125	13	8	on	on	ADP
ejpam-6125	13	9	various	various	ADJ
ejpam-6125	13	10	types	type	NOUN
ejpam-6125	13	11	of	of	ADP
ejpam-6125	13	12	graphs	graph	NOUN
ejpam-6125	13	13	(	(	PUNCT
ejpam-6125	13	14	see	see	VERB
ejpam-6125	13	15	[	[	X
ejpam-6125	13	16	1–9	1–9	NOUN
ejpam-6125	13	17	]	]	PUNCT
ejpam-6125	13	18	)	)	PUNCT
ejpam-6125	13	19	.	.	PUNCT
ejpam-6125	14	1	in	in	ADP
ejpam-6125	14	2	2022	2022	NUM
ejpam-6125	14	3	,	,	PUNCT
ejpam-6125	14	4	hop	hop	NOUN
ejpam-6125	14	5	independent	independent	ADJ
ejpam-6125	14	6	set	set	NOUN
ejpam-6125	14	7	in	in	ADP
ejpam-6125	14	8	a	a	DET
ejpam-6125	14	9	graph	graph	NOUN
ejpam-6125	14	10	and	and	CCONJ
ejpam-6125	14	11	its	its	PRON
ejpam-6125	14	12	corresponding	corresponding	ADJ
ejpam-6125	14	13	parameter	parameter	NOUN
ejpam-6125	14	14	were	be	AUX
ejpam-6125	14	15	introduced	introduce	VERB
ejpam-6125	14	16	and	and	CCONJ
ejpam-6125	14	17	investigated	investigate	VERB
ejpam-6125	14	18	by	by	ADP
ejpam-6125	14	19	j.	j.	PROPN
ejpam-6125	14	20	hassan	hassan	PROPN
ejpam-6125	14	21	et	et	PROPN
ejpam-6125	14	22	al	al	PROPN
ejpam-6125	14	23	.	.	PUNCT
ejpam-6125	15	1	[	[	X
ejpam-6125	15	2	10	10	NUM
ejpam-6125	15	3	]	]	PUNCT
ejpam-6125	15	4	.	.	PUNCT
ejpam-6125	16	1	this	this	DET
ejpam-6125	16	2	defined	define	VERB
ejpam-6125	16	3	set	set	NOUN
ejpam-6125	16	4	stated	state	VERB
ejpam-6125	16	5	that	that	SCONJ
ejpam-6125	16	6	no	no	DET
ejpam-6125	16	7	two	two	NUM
ejpam-6125	16	8	distinct	distinct	ADJ
ejpam-6125	16	9	∗corresponding	∗corresponding	NOUN
ejpam-6125	16	10	author	author	NOUN
ejpam-6125	16	11	.	.	PUNCT
ejpam-6125	17	1	doi	doi	NOUN
ejpam-6125	17	2	:	:	PUNCT
ejpam-6125	17	3	https://doi.org/10.29020/nybg.ejpam.v18i3.6125	https://doi.org/10.29020/nybg.ejpam.v18i3.6125	DET
ejpam-6125	17	4	email	email	NOUN
ejpam-6125	17	5	addresses	address	NOUN
ejpam-6125	17	6	:	:	PUNCT
ejpam-6125	17	7	javierhassan@msutawi-tawi.edu.ph	javierhassan@msutawi-tawi.edu.ph	PROPN
ejpam-6125	17	8	(	(	PUNCT
ejpam-6125	17	9	j.	j.	PROPN
ejpam-6125	17	10	a.	a.	PROPN
ejpam-6125	17	11	hassan	hassan	PROPN
ejpam-6125	17	12	)	)	PUNCT
ejpam-6125	17	13	,	,	PUNCT
ejpam-6125	17	14	farhadzaripin@msutawi-tawi.edu.ph	farhadzaripin@msutawi-tawi.edu.ph	PROPN
ejpam-6125	17	15	(	(	PUNCT
ejpam-6125	17	16	f.	f.	PROPN
ejpam-6125	17	17	aripin	aripin	PROPN
ejpam-6125	17	18	)	)	PUNCT
ejpam-6125	17	19	,	,	PUNCT
ejpam-6125	17	20	maobonsocan@addu.edu.ph	maobonsocan@addu.edu.ph	NOUN
ejpam-6125	17	21	(	(	PUNCT
ejpam-6125	17	22	m.	m.	NOUN
ejpam-6125	17	23	a.	a.	PROPN
ejpam-6125	17	24	bonsocan	bonsocan	PROPN
ejpam-6125	17	25	)	)	PUNCT
ejpam-6125	17	26	,	,	PUNCT
ejpam-6125	17	27	kimberlyjanepon@msutawi-tawi.edu.ph	kimberlyjanepon@msutawi-tawi.edu.ph	PROPN
ejpam-6125	17	28	(	(	PUNCT
ejpam-6125	17	29	k.	k.	PROPN
ejpam-6125	17	30	j.	j.	PROPN
ejpam-6125	17	31	pon	pon	PROPN
ejpam-6125	17	32	)	)	PUNCT
ejpam-6125	17	33	,	,	PUNCT
ejpam-6125	17	34	vtbilar@addu.edu.ph	vtbilar@addu.edu.ph	PROPN
ejpam-6125	17	35	(	(	PUNCT
ejpam-6125	17	36	v.	v.	X
ejpam-6125	17	37	bilar	bilar	PROPN
ejpam-6125	17	38	)	)	PUNCT
ejpam-6125	17	39	https://www.ejpam.com	https://www.ejpam.com	NOUN
ejpam-6125	18	1	1	1	NUM
ejpam-6125	18	2	copyright	copyright	NOUN
ejpam-6125	18	3	:	:	PUNCT
ejpam-6125	18	4	©	©	PROPN
ejpam-6125	18	5	2025	2025	NUM
ejpam-6125	18	6	the	the	DET
ejpam-6125	18	7	author(s	author(s	NOUN
ejpam-6125	18	8	)	)	PUNCT
ejpam-6125	18	9	.	.	PUNCT
ejpam-6125	19	1	(	(	PUNCT
ejpam-6125	19	2	cc	cc	NOUN
ejpam-6125	19	3	by	by	ADP
ejpam-6125	19	4	-	-	PUNCT
ejpam-6125	19	5	nc	nc	PROPN
ejpam-6125	19	6	4.0	4.0	NUM
ejpam-6125	19	7	)	)	PUNCT
ejpam-6125	19	8	j.	j.	PROPN
ejpam-6125	19	9	a.	a.	PROPN
ejpam-6125	19	10	hassan	hassan	PROPN
ejpam-6125	19	11	et	et	PROPN
ejpam-6125	19	12	al	al	PROPN
ejpam-6125	19	13	.	.	PUNCT
ejpam-6125	19	14	/	/	SYM
ejpam-6125	19	15	eur	eur	PROPN
ejpam-6125	19	16	.	.	PUNCT
ejpam-6125	20	1	j.	j.	PROPN
ejpam-6125	20	2	pure	pure	PROPN
ejpam-6125	20	3	appl	appl	PROPN
ejpam-6125	20	4	.	.	PROPN
ejpam-6125	20	5	math	math	PROPN
ejpam-6125	20	6	,	,	PUNCT
ejpam-6125	20	7	18	18	NUM
ejpam-6125	20	8	(	(	PUNCT
ejpam-6125	20	9	3	3	NUM
ejpam-6125	20	10	)	)	PUNCT
ejpam-6125	20	11	(	(	PUNCT
ejpam-6125	20	12	2025	2025	NUM
ejpam-6125	20	13	)	)	PUNCT
ejpam-6125	20	14	,	,	PUNCT
ejpam-6125	20	15	6125	6125	NUM
ejpam-6125	20	16	2	2	NUM
ejpam-6125	20	17	of	of	ADP
ejpam-6125	20	18	8	8	NUM
ejpam-6125	20	19	vertices	vertex	NOUN
ejpam-6125	20	20	in	in	ADP
ejpam-6125	20	21	the	the	DET
ejpam-6125	20	22	set	set	NOUN
ejpam-6125	20	23	have	have	VERB
ejpam-6125	20	24	a	a	DET
ejpam-6125	20	25	distance	distance	NOUN
ejpam-6125	20	26	of	of	ADP
ejpam-6125	20	27	two	two	NUM
ejpam-6125	20	28	from	from	ADP
ejpam-6125	20	29	each	each	DET
ejpam-6125	20	30	other	other	ADJ
ejpam-6125	20	31	.	.	PUNCT
ejpam-6125	21	1	this	this	PRON
ejpam-6125	21	2	was	be	AUX
ejpam-6125	21	3	further	far	ADV
ejpam-6125	21	4	investigated	investigate	VERB
ejpam-6125	21	5	on	on	ADP
ejpam-6125	21	6	different	different	ADJ
ejpam-6125	21	7	types	type	NOUN
ejpam-6125	21	8	of	of	ADP
ejpam-6125	21	9	graphs	graph	NOUN
ejpam-6125	21	10	where	where	SCONJ
ejpam-6125	21	11	they	they	PRON
ejpam-6125	21	12	obtained	obtain	VERB
ejpam-6125	21	13	some	some	DET
ejpam-6125	21	14	interesting	interesting	ADJ
ejpam-6125	21	15	results	result	NOUN
ejpam-6125	21	16	.	.	PUNCT
ejpam-6125	22	1	several	several	ADJ
ejpam-6125	22	2	variants	variant	NOUN
ejpam-6125	22	3	of	of	ADP
ejpam-6125	22	4	this	this	DET
ejpam-6125	22	5	concept	concept	NOUN
ejpam-6125	22	6	have	have	AUX
ejpam-6125	22	7	been	be	AUX
ejpam-6125	22	8	introduced	introduce	VERB
ejpam-6125	22	9	and	and	CCONJ
ejpam-6125	22	10	studied	study	VERB
ejpam-6125	22	11	(	(	PUNCT
ejpam-6125	22	12	see	see	VERB
ejpam-6125	22	13	[	[	X
ejpam-6125	22	14	11–14	11–14	NUM
ejpam-6125	22	15	]	]	PUNCT
ejpam-6125	22	16	)	)	PUNCT
ejpam-6125	22	17	.	.	PUNCT
ejpam-6125	23	1	more	more	ADV
ejpam-6125	23	2	recently	recently	ADV
ejpam-6125	23	3	,	,	PUNCT
ejpam-6125	23	4	inspired	inspire	VERB
ejpam-6125	23	5	by	by	ADP
ejpam-6125	23	6	numerous	numerous	ADJ
ejpam-6125	23	7	articles	article	NOUN
ejpam-6125	23	8	on	on	ADP
ejpam-6125	23	9	independent	independent	ADJ
ejpam-6125	23	10	sets	set	NOUN
ejpam-6125	23	11	of	of	ADP
ejpam-6125	23	12	graphs	graph	NOUN
ejpam-6125	23	13	,	,	PUNCT
ejpam-6125	23	14	j.	j.	PROPN
ejpam-6125	23	15	hassan	hassan	PROPN
ejpam-6125	23	16	et	et	PROPN
ejpam-6125	23	17	al	al	PROPN
ejpam-6125	23	18	.	.	PUNCT
ejpam-6125	24	1	[	[	X
ejpam-6125	24	2	11	11	NUM
ejpam-6125	24	3	]	]	PUNCT
ejpam-6125	24	4	,	,	PUNCT
ejpam-6125	24	5	introduced	introduce	VERB
ejpam-6125	24	6	another	another	DET
ejpam-6125	24	7	variant	variant	NOUN
ejpam-6125	24	8	of	of	ADP
ejpam-6125	24	9	independent	independent	ADJ
ejpam-6125	24	10	set	set	NOUN
ejpam-6125	24	11	called	call	VERB
ejpam-6125	24	12	legal	legal	ADJ
ejpam-6125	24	13	hop	hop	NOUN
ejpam-6125	24	14	independent	independent	ADJ
ejpam-6125	24	15	sequence	sequence	NOUN
ejpam-6125	24	16	in	in	ADP
ejpam-6125	24	17	a	a	DET
ejpam-6125	24	18	graph	graph	NOUN
ejpam-6125	24	19	.	.	PUNCT
ejpam-6125	25	1	this	this	DET
ejpam-6125	25	2	new	new	ADJ
ejpam-6125	25	3	variant	variant	NOUN
ejpam-6125	25	4	added	add	VERB
ejpam-6125	25	5	another	another	DET
ejpam-6125	25	6	property	property	NOUN
ejpam-6125	25	7	wherein	wherein	SCONJ
ejpam-6125	25	8	the	the	DET
ejpam-6125	25	9	order	order	NOUN
ejpam-6125	25	10	of	of	ADP
ejpam-6125	25	11	vertices	vertex	NOUN
ejpam-6125	25	12	in	in	ADP
ejpam-6125	25	13	the	the	DET
ejpam-6125	25	14	set	set	NOUN
ejpam-6125	25	15	of	of	ADP
ejpam-6125	25	16	a	a	DET
ejpam-6125	25	17	graph	graph	NOUN
ejpam-6125	25	18	matters	matter	NOUN
ejpam-6125	25	19	,	,	PUNCT
ejpam-6125	25	20	and	and	CCONJ
ejpam-6125	25	21	is	be	AUX
ejpam-6125	25	22	defined	define	VERB
ejpam-6125	25	23	as	as	SCONJ
ejpam-6125	25	24	follows	follow	VERB
ejpam-6125	25	25	:	:	PUNCT
ejpam-6125	25	26	a	a	DET
ejpam-6125	25	27	sequence	sequence	NOUN
ejpam-6125	25	28	l	l	NOUN
ejpam-6125	25	29	=	=	SYM
ejpam-6125	25	30	(	(	PUNCT
ejpam-6125	25	31	w1	w1	NOUN
ejpam-6125	25	32	,	,	PUNCT
ejpam-6125	25	33	·	·	PUNCT
ejpam-6125	25	34	·	·	PUNCT
ejpam-6125	25	35	·	·	PUNCT
ejpam-6125	25	36	,	,	PUNCT
ejpam-6125	25	37	wk	wk	X
ejpam-6125	25	38	)	)	PUNCT
ejpam-6125	25	39	of	of	ADP
ejpam-6125	25	40	distinct	distinct	ADJ
ejpam-6125	25	41	vertices	vertex	NOUN
ejpam-6125	25	42	of	of	ADP
ejpam-6125	25	43	g	g	PROPN
ejpam-6125	25	44	is	be	AUX
ejpam-6125	25	45	called	call	VERB
ejpam-6125	25	46	a	a	DET
ejpam-6125	25	47	legal	legal	ADJ
ejpam-6125	25	48	hop	hop	NOUN
ejpam-6125	25	49	independent	independent	ADJ
ejpam-6125	25	50	sequence	sequence	NOUN
ejpam-6125	25	51	if	if	SCONJ
ejpam-6125	25	52	k	k	PROPN
ejpam-6125	25	53	=	=	SYM
ejpam-6125	25	54	1	1	NUM
ejpam-6125	25	55	or	or	CCONJ
ejpam-6125	25	56	l	l	NOUN
ejpam-6125	25	57	is	be	AUX
ejpam-6125	25	58	a	a	DET
ejpam-6125	25	59	hop	hop	NOUN
ejpam-6125	25	60	independent	independent	ADJ
ejpam-6125	25	61	and	and	CCONJ
ejpam-6125	25	62	ng[wi	ng[wi	PROPN
ejpam-6125	25	63	]	]	X
ejpam-6125	25	64	\	\	PROPN
ejpam-6125	25	65	⋃i−1	⋃i−1	NOUN
ejpam-6125	25	66	j=1ng[wj	j=1ng[wj	PROPN
ejpam-6125	25	67	]	]	PUNCT
ejpam-6125	25	68	̸=	̸=	PROPN
ejpam-6125	25	69	∅	∅	NOUN
ejpam-6125	25	70	for	for	ADP
ejpam-6125	25	71	every	every	DET
ejpam-6125	25	72	i	i	PROPN
ejpam-6125	25	73	∈	∈	PROPN
ejpam-6125	25	74	{	{	PUNCT
ejpam-6125	25	75	2	2	NUM
ejpam-6125	25	76	,	,	PUNCT
ejpam-6125	25	77	·	·	PUNCT
ejpam-6125	25	78	·	·	PUNCT
ejpam-6125	25	79	·	·	PUNCT
ejpam-6125	25	80	,	,	PUNCT
ejpam-6125	25	81	k	k	X
ejpam-6125	25	82	}	}	PUNCT
ejpam-6125	25	83	.	.	PUNCT
ejpam-6125	26	1	the	the	DET
ejpam-6125	26	2	maximum	maximum	ADJ
ejpam-6125	26	3	length	length	NOUN
ejpam-6125	26	4	of	of	ADP
ejpam-6125	26	5	a	a	DET
ejpam-6125	26	6	legal	legal	ADJ
ejpam-6125	26	7	hop	hop	NOUN
ejpam-6125	26	8	independent	independent	ADJ
ejpam-6125	26	9	sequence	sequence	NOUN
ejpam-6125	26	10	in	in	ADP
ejpam-6125	26	11	g	g	NOUN
ejpam-6125	26	12	,	,	PUNCT
ejpam-6125	26	13	denoted	denote	VERB
ejpam-6125	26	14	by	by	ADP
ejpam-6125	26	15	αlh(g),is	αlh(g),is	PRON
ejpam-6125	26	16	called	call	VERB
ejpam-6125	26	17	the	the	DET
ejpam-6125	26	18	legal	legal	ADJ
ejpam-6125	26	19	hop	hop	NOUN
ejpam-6125	26	20	independence	independence	NOUN
ejpam-6125	26	21	number	number	NOUN
ejpam-6125	26	22	of	of	ADP
ejpam-6125	26	23	g.	g.	PROPN
ejpam-6125	27	1	the	the	PRON
ejpam-6125	27	2	said	say	VERB
ejpam-6125	27	3	authors	author	NOUN
ejpam-6125	27	4	have	have	VERB
ejpam-6125	27	5	formulated	formulate	VERB
ejpam-6125	27	6	characterizations	characterization	NOUN
ejpam-6125	27	7	and	and	CCONJ
ejpam-6125	27	8	formulas	formula	NOUN
ejpam-6125	27	9	of	of	ADP
ejpam-6125	27	10	this	this	DET
ejpam-6125	27	11	parameter	parameter	NOUN
ejpam-6125	27	12	on	on	ADP
ejpam-6125	27	13	different	different	ADJ
ejpam-6125	27	14	types	type	NOUN
ejpam-6125	27	15	of	of	ADP
ejpam-6125	27	16	graphs	graph	NOUN
ejpam-6125	27	17	.	.	PUNCT
ejpam-6125	28	1	in	in	ADP
ejpam-6125	28	2	this	this	DET
ejpam-6125	28	3	paper	paper	NOUN
ejpam-6125	28	4	,	,	PUNCT
ejpam-6125	28	5	a	a	DET
ejpam-6125	28	6	new	new	ADJ
ejpam-6125	28	7	variant	variant	NOUN
ejpam-6125	28	8	of	of	ADP
ejpam-6125	28	9	independence	independence	NOUN
ejpam-6125	28	10	parameter	parameter	NOUN
ejpam-6125	28	11	is	be	AUX
ejpam-6125	28	12	introduced	introduce	VERB
ejpam-6125	28	13	and	and	CCONJ
ejpam-6125	28	14	initially	initially	ADV
ejpam-6125	28	15	investigated	investigate	VERB
ejpam-6125	28	16	,	,	PUNCT
ejpam-6125	28	17	and	and	CCONJ
ejpam-6125	28	18	we	we	PRON
ejpam-6125	28	19	call	call	VERB
ejpam-6125	28	20	this	this	DET
ejpam-6125	28	21	legal	legal	ADJ
ejpam-6125	28	22	closed	closed	ADJ
ejpam-6125	28	23	hop	hop	NOUN
ejpam-6125	28	24	neighborhood	neighborhood	NOUN
ejpam-6125	28	25	independent	independent	ADJ
ejpam-6125	28	26	sequence	sequence	NOUN
ejpam-6125	28	27	(	(	PUNCT
ejpam-6125	28	28	lchni	lchni	PROPN
ejpam-6125	28	29	sequence	sequence	PROPN
ejpam-6125	28	30	)	)	PUNCT
ejpam-6125	28	31	of	of	ADP
ejpam-6125	28	32	a	a	DET
ejpam-6125	28	33	graph	graph	NOUN
ejpam-6125	28	34	.	.	PUNCT
ejpam-6125	29	1	in	in	ADP
ejpam-6125	29	2	this	this	DET
ejpam-6125	29	3	parameter	parameter	NOUN
ejpam-6125	29	4	,	,	PUNCT
ejpam-6125	29	5	the	the	DET
ejpam-6125	29	6	authors	author	NOUN
ejpam-6125	29	7	put	put	VERB
ejpam-6125	29	8	some	some	DET
ejpam-6125	29	9	restrictions	restriction	NOUN
ejpam-6125	29	10	on	on	ADP
ejpam-6125	29	11	the	the	DET
ejpam-6125	29	12	usual	usual	ADJ
ejpam-6125	29	13	independent	independent	ADJ
ejpam-6125	29	14	set	set	NOUN
ejpam-6125	29	15	in	in	ADP
ejpam-6125	29	16	a	a	DET
ejpam-6125	29	17	graph	graph	NOUN
ejpam-6125	29	18	where	where	SCONJ
ejpam-6125	29	19	the	the	DET
ejpam-6125	29	20	order	order	NOUN
ejpam-6125	29	21	of	of	ADP
ejpam-6125	29	22	choosing	choose	VERB
ejpam-6125	29	23	vertices	vertex	NOUN
ejpam-6125	29	24	as	as	ADV
ejpam-6125	29	25	well	well	ADV
ejpam-6125	29	26	as	as	ADP
ejpam-6125	29	27	the	the	DET
ejpam-6125	29	28	behavior	behavior	NOUN
ejpam-6125	29	29	of	of	ADP
ejpam-6125	29	30	its	its	PRON
ejpam-6125	29	31	closed	closed	ADJ
ejpam-6125	29	32	hop	hop	NOUN
ejpam-6125	29	33	neighborhoods	neighborhood	NOUN
ejpam-6125	29	34	are	be	AUX
ejpam-6125	29	35	important	important	ADJ
ejpam-6125	29	36	.	.	PUNCT
ejpam-6125	30	1	the	the	DET
ejpam-6125	30	2	authors	author	NOUN
ejpam-6125	30	3	believe	believe	VERB
ejpam-6125	30	4	that	that	SCONJ
ejpam-6125	30	5	this	this	DET
ejpam-6125	30	6	newly	newly	ADV
ejpam-6125	30	7	defined	define	VERB
ejpam-6125	30	8	parameter	parameter	NOUN
ejpam-6125	30	9	would	would	AUX
ejpam-6125	30	10	open	open	VERB
ejpam-6125	30	11	more	more	ADV
ejpam-6125	30	12	interesting	interesting	ADJ
ejpam-6125	30	13	studies	study	NOUN
ejpam-6125	30	14	and	and	CCONJ
ejpam-6125	30	15	applications	application	NOUN
ejpam-6125	30	16	in	in	ADP
ejpam-6125	30	17	the	the	DET
ejpam-6125	30	18	future	future	NOUN
ejpam-6125	30	19	.	.	PUNCT
ejpam-6125	31	1	2	2	X
ejpam-6125	31	2	.	.	NOUN
ejpam-6125	31	3	terminologies	terminology	NOUN
ejpam-6125	31	4	and	and	CCONJ
ejpam-6125	31	5	notations	notation	NOUN
ejpam-6125	31	6	let	let	VERB
ejpam-6125	31	7	g	g	NOUN
ejpam-6125	31	8	be	be	AUX
ejpam-6125	31	9	an	an	DET
ejpam-6125	31	10	undirected	undirected	ADJ
ejpam-6125	31	11	graph	graph	NOUN
ejpam-6125	31	12	.	.	PUNCT
ejpam-6125	32	1	a	a	DET
ejpam-6125	32	2	subset	subset	NOUN
ejpam-6125	32	3	a	a	PRON
ejpam-6125	32	4	of	of	ADP
ejpam-6125	32	5	v	v	NOUN
ejpam-6125	32	6	(	(	PUNCT
ejpam-6125	32	7	g	g	NOUN
ejpam-6125	32	8	)	)	PUNCT
ejpam-6125	32	9	is	be	AUX
ejpam-6125	32	10	an	an	DET
ejpam-6125	32	11	independent	independent	ADJ
ejpam-6125	32	12	set	set	NOUN
ejpam-6125	32	13	if	if	SCONJ
ejpam-6125	32	14	for	for	SCONJ
ejpam-6125	32	15	every	every	DET
ejpam-6125	32	16	pair	pair	NOUN
ejpam-6125	32	17	of	of	ADP
ejpam-6125	32	18	distinct	distinct	ADJ
ejpam-6125	32	19	vertices	vertex	NOUN
ejpam-6125	32	20	in	in	ADP
ejpam-6125	32	21	a	a	DET
ejpam-6125	32	22	do	do	AUX
ejpam-6125	32	23	not	not	PART
ejpam-6125	32	24	form	form	VERB
ejpam-6125	32	25	an	an	DET
ejpam-6125	32	26	edge	edge	NOUN
ejpam-6125	32	27	.	.	PUNCT
ejpam-6125	33	1	the	the	DET
ejpam-6125	33	2	maximum	maximum	ADJ
ejpam-6125	33	3	cardinality	cardinality	NOUN
ejpam-6125	33	4	of	of	ADP
ejpam-6125	33	5	an	an	DET
ejpam-6125	33	6	independent	independent	ADJ
ejpam-6125	33	7	set	set	NOUN
ejpam-6125	33	8	in	in	ADP
ejpam-6125	33	9	g	g	NOUN
ejpam-6125	33	10	,	,	PUNCT
ejpam-6125	33	11	denoted	denote	VERB
ejpam-6125	33	12	by	by	ADP
ejpam-6125	33	13	α(g	α(g	NOUN
ejpam-6125	33	14	)	)	PUNCT
ejpam-6125	33	15	,	,	PUNCT
ejpam-6125	33	16	is	be	AUX
ejpam-6125	33	17	called	call	VERB
ejpam-6125	33	18	the	the	DET
ejpam-6125	33	19	independence	independence	NOUN
ejpam-6125	33	20	number	number	NOUN
ejpam-6125	33	21	of	of	ADP
ejpam-6125	33	22	g.	g.	PROPN
ejpam-6125	33	23	any	any	DET
ejpam-6125	33	24	independent	independent	ADJ
ejpam-6125	33	25	set	set	NOUN
ejpam-6125	33	26	with	with	ADP
ejpam-6125	33	27	cardinality	cardinality	NOUN
ejpam-6125	33	28	equal	equal	ADJ
ejpam-6125	33	29	to	to	ADP
ejpam-6125	33	30	α(g	α(g	NUM
ejpam-6125	33	31	)	)	PUNCT
ejpam-6125	33	32	is	be	AUX
ejpam-6125	33	33	called	call	VERB
ejpam-6125	33	34	an	an	DET
ejpam-6125	33	35	α	α	NOUN
ejpam-6125	33	36	-	-	PUNCT
ejpam-6125	33	37	set	set	VERB
ejpam-6125	33	38	in	in	ADP
ejpam-6125	33	39	g.	g.	PROPN
ejpam-6125	33	40	let	let	VERB
ejpam-6125	33	41	g	g	NOUN
ejpam-6125	33	42	be	be	AUX
ejpam-6125	33	43	an	an	DET
ejpam-6125	33	44	undirected	undirected	ADJ
ejpam-6125	33	45	graph	graph	NOUN
ejpam-6125	33	46	.	.	PUNCT
ejpam-6125	34	1	let	let	VERB
ejpam-6125	34	2	s	s	PRON
ejpam-6125	34	3	=	=	PUNCT
ejpam-6125	34	4	(	(	PUNCT
ejpam-6125	34	5	v1	v1	PROPN
ejpam-6125	34	6	,	,	PUNCT
ejpam-6125	34	7	v2	v2	PROPN
ejpam-6125	34	8	,	,	PUNCT
ejpam-6125	34	9	·	·	PUNCT
ejpam-6125	34	10	·	·	PUNCT
ejpam-6125	34	11	·	·	PUNCT
ejpam-6125	34	12	,	,	PUNCT
ejpam-6125	34	13	vk	vk	AUX
ejpam-6125	34	14	)	)	PUNCT
ejpam-6125	34	15	be	be	AUX
ejpam-6125	34	16	a	a	DET
ejpam-6125	34	17	sequence	sequence	NOUN
ejpam-6125	34	18	of	of	ADP
ejpam-6125	34	19	distinct	distinct	ADJ
ejpam-6125	34	20	vertices	vertex	NOUN
ejpam-6125	34	21	of	of	ADP
ejpam-6125	34	22	g	g	NOUN
ejpam-6125	34	23	and	and	CCONJ
ejpam-6125	34	24	let	let	VERB
ejpam-6125	34	25	ŝ	ŝ	X
ejpam-6125	34	26	=	=	SYM
ejpam-6125	34	27	{	{	PUNCT
ejpam-6125	34	28	v1	v1	PROPN
ejpam-6125	34	29	,	,	PUNCT
ejpam-6125	34	30	v2	v2	PROPN
ejpam-6125	34	31	,	,	PUNCT
ejpam-6125	34	32	.	.	PUNCT
ejpam-6125	34	33	.	.	PUNCT
ejpam-6125	35	1	.	.	PUNCT
ejpam-6125	36	1	,	,	PUNCT
ejpam-6125	36	2	vk	vk	ADP
ejpam-6125	36	3	}	}	PUNCT
ejpam-6125	36	4	.	.	PUNCT
ejpam-6125	37	1	then	then	ADV
ejpam-6125	37	2	s	s	VERB
ejpam-6125	37	3	is	be	AUX
ejpam-6125	37	4	a	a	DET
ejpam-6125	37	5	legal	legal	ADJ
ejpam-6125	37	6	closed	close	VERB
ejpam-6125	37	7	hop	hop	NOUN
ejpam-6125	37	8	neighborhood	neighborhood	NOUN
ejpam-6125	37	9	sequence	sequence	NOUN
ejpam-6125	37	10	of	of	ADP
ejpam-6125	37	11	g	g	PROPN
ejpam-6125	37	12	if	if	SCONJ
ejpam-6125	37	13	n2	n2	PROPN
ejpam-6125	37	14	g[vi	g[vi	PROPN
ejpam-6125	37	15	]	]	PUNCT
ejpam-6125	37	16	\	\	NOUN
ejpam-6125	37	17	∪	∪	X
ejpam-6125	37	18	i−1	i−1	PROPN
ejpam-6125	37	19	j=1n	j=1n	PROPN
ejpam-6125	37	20	2	2	NUM
ejpam-6125	37	21	g[vj	g[vj	PROPN
ejpam-6125	37	22	]	]	PUNCT
ejpam-6125	37	23	̸=	̸=	PROPN
ejpam-6125	37	24	∅	∅	NOUN
ejpam-6125	37	25	for	for	ADP
ejpam-6125	37	26	each	each	DET
ejpam-6125	37	27	i	i	PRON
ejpam-6125	37	28	∈	∈	PROPN
ejpam-6125	37	29	{	{	PUNCT
ejpam-6125	37	30	2	2	NUM
ejpam-6125	37	31	,	,	PUNCT
ejpam-6125	37	32	·	·	PUNCT
ejpam-6125	37	33	·	·	PUNCT
ejpam-6125	37	34	·	·	PUNCT
ejpam-6125	37	35	,	,	PUNCT
ejpam-6125	37	36	k	k	X
ejpam-6125	37	37	}	}	PUNCT
ejpam-6125	37	38	.	.	PUNCT
ejpam-6125	38	1	let	let	VERB
ejpam-6125	38	2	g	g	PRON
ejpam-6125	38	3	be	be	AUX
ejpam-6125	38	4	a	a	DET
ejpam-6125	38	5	graph	graph	NOUN
ejpam-6125	38	6	.	.	PUNCT
ejpam-6125	39	1	a	a	DET
ejpam-6125	39	2	sequence	sequence	NOUN
ejpam-6125	39	3	q	q	NOUN
ejpam-6125	40	1	=	=	SYM
ejpam-6125	40	2	(	(	PUNCT
ejpam-6125	40	3	x1	x1	PROPN
ejpam-6125	40	4	,	,	PUNCT
ejpam-6125	40	5	x2	x2	PROPN
ejpam-6125	40	6	,	,	PUNCT
ejpam-6125	40	7	...	...	PUNCT
ejpam-6125	40	8	,	,	PUNCT
ejpam-6125	40	9	xk	xk	PROPN
ejpam-6125	40	10	)	)	PUNCT
ejpam-6125	40	11	of	of	ADP
ejpam-6125	40	12	distinct	distinct	ADJ
ejpam-6125	40	13	vertices	vertex	NOUN
ejpam-6125	40	14	of	of	ADP
ejpam-6125	40	15	g	g	PROPN
ejpam-6125	40	16	is	be	AUX
ejpam-6125	40	17	called	call	VERB
ejpam-6125	40	18	a	a	DET
ejpam-6125	40	19	legal	legal	ADJ
ejpam-6125	40	20	closed	closed	ADJ
ejpam-6125	40	21	hop	hop	NOUN
ejpam-6125	40	22	neighborhood	neighborhood	NOUN
ejpam-6125	40	23	independent	independent	ADJ
ejpam-6125	40	24	sequence	sequence	NOUN
ejpam-6125	40	25	(	(	PUNCT
ejpam-6125	40	26	lchni	lchni	PROPN
ejpam-6125	40	27	sequence	sequence	PROPN
ejpam-6125	40	28	)	)	PUNCT
ejpam-6125	40	29	if	if	SCONJ
ejpam-6125	40	30	it	it	PRON
ejpam-6125	40	31	is	be	AUX
ejpam-6125	40	32	satisfies	satisfie	NOUN
ejpam-6125	40	33	the	the	DET
ejpam-6125	40	34	following	follow	VERB
ejpam-6125	40	35	two	two	NUM
ejpam-6125	40	36	conditions	condition	NOUN
ejpam-6125	40	37	:	:	PUNCT
ejpam-6125	40	38	(	(	PUNCT
ejpam-6125	40	39	1	1	NUM
ejpam-6125	40	40	.	.	PUNCT
ejpam-6125	40	41	)	)	PUNCT
ejpam-6125	41	1	n2	n2	PROPN
ejpam-6125	41	2	g[xi	g[xi	PROPN
ejpam-6125	41	3	]	]	PUNCT
ejpam-6125	41	4	\	\	PROPN
ejpam-6125	41	5	⋃i−1	⋃i−1	NOUN
ejpam-6125	41	6	j=1n	j=1n	PROPN
ejpam-6125	41	7	2	2	NUM
ejpam-6125	41	8	g[xj	g[xj	NOUN
ejpam-6125	41	9	]	]	PUNCT
ejpam-6125	42	1	̸=	̸=	PROPN
ejpam-6125	42	2	for	for	ADP
ejpam-6125	42	3	each	each	DET
ejpam-6125	42	4	i	i	PRON
ejpam-6125	42	5	∈	∈	PROPN
ejpam-6125	42	6	{	{	PUNCT
ejpam-6125	42	7	2	2	NUM
ejpam-6125	42	8	,	,	PUNCT
ejpam-6125	42	9	3	3	NUM
ejpam-6125	42	10	,	,	PUNCT
ejpam-6125	42	11	...	...	PUNCT
ejpam-6125	42	12	,	,	PUNCT
ejpam-6125	42	13	k	k	NOUN
ejpam-6125	42	14	}	}	PUNCT
ejpam-6125	42	15	.	.	PUNCT
ejpam-6125	43	1	(	(	PUNCT
ejpam-6125	43	2	2	2	NUM
ejpam-6125	43	3	.	.	NUM
ejpam-6125	43	4	)	)	PUNCT
ejpam-6125	43	5	dg(xs	dg(xs	NOUN
ejpam-6125	43	6	,	,	PUNCT
ejpam-6125	43	7	xt	xt	ADJ
ejpam-6125	43	8	)	)	PUNCT
ejpam-6125	43	9	̸=	̸=	PROPN
ejpam-6125	43	10	1	1	NUM
ejpam-6125	43	11	for	for	ADP
ejpam-6125	43	12	each	each	DET
ejpam-6125	43	13	s	s	PROPN
ejpam-6125	43	14	,	,	PUNCT
ejpam-6125	43	15	t	t	PROPN
ejpam-6125	43	16	∈	∈	PROPN
ejpam-6125	43	17	{	{	PUNCT
ejpam-6125	43	18	1	1	NUM
ejpam-6125	43	19	,	,	PUNCT
ejpam-6125	43	20	2	2	NUM
ejpam-6125	43	21	,	,	PUNCT
ejpam-6125	43	22	...	...	PUNCT
ejpam-6125	43	23	,	,	PUNCT
ejpam-6125	43	24	k	k	X
ejpam-6125	43	25	}	}	PUNCT
ejpam-6125	43	26	,	,	PUNCT
ejpam-6125	43	27	where	where	SCONJ
ejpam-6125	43	28	s	s	AUX
ejpam-6125	43	29	̸=	̸=	PROPN
ejpam-6125	43	30	t.	t.	NOUN
ejpam-6125	43	31	the	the	DET
ejpam-6125	43	32	legal	legal	ADJ
ejpam-6125	43	33	closed	close	VERB
ejpam-6125	43	34	hop	hop	NOUN
ejpam-6125	43	35	neighborhood	neighborhood	NOUN
ejpam-6125	43	36	independent	independent	ADJ
ejpam-6125	43	37	number	number	NOUN
ejpam-6125	43	38	(	(	PUNCT
ejpam-6125	43	39	lchni	lchni	PROPN
ejpam-6125	43	40	number	number	PROPN
ejpam-6125	43	41	)	)	PUNCT
ejpam-6125	43	42	of	of	ADP
ejpam-6125	43	43	g	g	PROPN
ejpam-6125	43	44	is	be	AUX
ejpam-6125	43	45	the	the	DET
ejpam-6125	43	46	maximum	maximum	ADJ
ejpam-6125	43	47	length	length	NOUN
ejpam-6125	43	48	of	of	ADP
ejpam-6125	43	49	an	an	DET
ejpam-6125	43	50	lchni	lchni	ADJ
ejpam-6125	43	51	sequence	sequence	NOUN
ejpam-6125	43	52	in	in	ADP
ejpam-6125	43	53	g.	g.	PROPN
ejpam-6125	44	1	the	the	DET
ejpam-6125	44	2	said	say	VERB
ejpam-6125	44	3	number	number	NOUN
ejpam-6125	44	4	is	be	AUX
ejpam-6125	44	5	denoted	denote	VERB
ejpam-6125	44	6	by	by	ADP
ejpam-6125	44	7	θ(g	θ(g	NOUN
ejpam-6125	44	8	)	)	PUNCT
ejpam-6125	44	9	.	.	PUNCT
ejpam-6125	45	1	we	we	PRON
ejpam-6125	45	2	call	call	VERB
ejpam-6125	45	3	the	the	DET
ejpam-6125	45	4	corresponding	correspond	VERB
ejpam-6125	45	5	set	set	VERB
ejpam-6125	45	6	q̂	q̂	PUNCT
ejpam-6125	45	7	an	an	DET
ejpam-6125	45	8	lchni	lchni	PROPN
ejpam-6125	45	9	set	set	NOUN
ejpam-6125	45	10	of	of	ADP
ejpam-6125	45	11	g.	g.	PROPN
ejpam-6125	45	12	we	we	PRON
ejpam-6125	45	13	call	call	VERB
ejpam-6125	45	14	the	the	DET
ejpam-6125	45	15	corresponding	correspond	VERB
ejpam-6125	45	16	set	set	VERB
ejpam-6125	45	17	q̂	q̂	X
ejpam-6125	45	18	of	of	ADP
ejpam-6125	45	19	q	q	PROPN
ejpam-6125	45	20	an	an	DET
ejpam-6125	45	21	lchni	lchni	PROPN
ejpam-6125	45	22	set	set	NOUN
ejpam-6125	45	23	of	of	ADP
ejpam-6125	45	24	g.	g.	PROPN
ejpam-6125	45	25	let	let	VERB
ejpam-6125	45	26	q1	q1	PROPN
ejpam-6125	45	27	=	=	SYM
ejpam-6125	45	28	(	(	PUNCT
ejpam-6125	45	29	v1	v1	PROPN
ejpam-6125	45	30	,	,	PUNCT
ejpam-6125	45	31	·	·	PUNCT
ejpam-6125	45	32	·	·	PUNCT
ejpam-6125	45	33	·	·	PUNCT
ejpam-6125	45	34	,	,	PUNCT
ejpam-6125	45	35	vn	vn	PROPN
ejpam-6125	45	36	)	)	PUNCT
ejpam-6125	45	37	and	and	CCONJ
ejpam-6125	45	38	q2	q2	NOUN
ejpam-6125	45	39	=	=	SYM
ejpam-6125	45	40	(	(	PUNCT
ejpam-6125	45	41	u1	u1	PROPN
ejpam-6125	45	42	,	,	PUNCT
ejpam-6125	45	43	·	·	PUNCT
ejpam-6125	45	44	·	·	PUNCT
ejpam-6125	45	45	·	·	PUNCT
ejpam-6125	45	46	,	,	PUNCT
ejpam-6125	45	47	um	um	INTJ
ejpam-6125	45	48	)	)	PUNCT
ejpam-6125	45	49	,	,	PUNCT
ejpam-6125	45	50	n	n	CCONJ
ejpam-6125	45	51	,	,	PUNCT
ejpam-6125	45	52	m	m	VERB
ejpam-6125	45	53	≥	≥	NOUN
ejpam-6125	45	54	1	1	NUM
ejpam-6125	45	55	be	be	AUX
ejpam-6125	45	56	two	two	NUM
ejpam-6125	45	57	sequences	sequence	NOUN
ejpam-6125	45	58	of	of	ADP
ejpam-6125	45	59	distinct	distinct	ADJ
ejpam-6125	45	60	vertices	vertex	NOUN
ejpam-6125	45	61	of	of	ADP
ejpam-6125	45	62	g.	g.	PROPN
ejpam-6125	45	63	the	the	DET
ejpam-6125	45	64	concatenation	concatenation	NOUN
ejpam-6125	45	65	of	of	ADP
ejpam-6125	45	66	q1	q1	PROPN
ejpam-6125	45	67	and	and	CCONJ
ejpam-6125	45	68	q2	q2	NOUN
ejpam-6125	45	69	,	,	PUNCT
ejpam-6125	45	70	denoted	denote	VERB
ejpam-6125	45	71	by	by	ADP
ejpam-6125	45	72	q1⊕q2	q1⊕q2	PROPN
ejpam-6125	45	73	,	,	PUNCT
ejpam-6125	45	74	is	be	AUX
ejpam-6125	45	75	the	the	DET
ejpam-6125	45	76	sequence	sequence	NOUN
ejpam-6125	45	77	given	give	VERB
ejpam-6125	45	78	by	by	ADP
ejpam-6125	45	79	q1	q1	PROPN
ejpam-6125	45	80	⊕q2	⊕q2	PROPN
ejpam-6125	46	1	=	=	PRON
ejpam-6125	46	2	(	(	PUNCT
ejpam-6125	46	3	v1	v1	PROPN
ejpam-6125	46	4	,	,	PUNCT
ejpam-6125	46	5	·	·	PUNCT
ejpam-6125	46	6	·	·	PUNCT
ejpam-6125	46	7	·	·	PUNCT
ejpam-6125	46	8	,	,	PUNCT
ejpam-6125	46	9	vn	vn	PROPN
ejpam-6125	46	10	,	,	PUNCT
ejpam-6125	46	11	u1	u1	NOUN
ejpam-6125	46	12	,	,	PUNCT
ejpam-6125	46	13	·	·	PUNCT
ejpam-6125	46	14	·	·	PUNCT
ejpam-6125	46	15	·	·	PUNCT
ejpam-6125	46	16	,	,	PUNCT
ejpam-6125	46	17	um	um	INTJ
ejpam-6125	46	18	)	)	PUNCT
ejpam-6125	46	19	.	.	PUNCT
ejpam-6125	47	1	j.	j.	PROPN
ejpam-6125	47	2	a.	a.	PROPN
ejpam-6125	47	3	hassan	hassan	PROPN
ejpam-6125	47	4	et	et	PROPN
ejpam-6125	47	5	al	al	PROPN
ejpam-6125	47	6	.	.	PUNCT
ejpam-6125	47	7	/	/	SYM
ejpam-6125	47	8	eur	eur	PROPN
ejpam-6125	47	9	.	.	PUNCT
ejpam-6125	48	1	j.	j.	PROPN
ejpam-6125	48	2	pure	pure	PROPN
ejpam-6125	48	3	appl	appl	PROPN
ejpam-6125	48	4	.	.	PROPN
ejpam-6125	48	5	math	math	PROPN
ejpam-6125	48	6	,	,	PUNCT
ejpam-6125	48	7	18	18	NUM
ejpam-6125	48	8	(	(	PUNCT
ejpam-6125	48	9	3	3	NUM
ejpam-6125	48	10	)	)	PUNCT
ejpam-6125	48	11	(	(	PUNCT
ejpam-6125	48	12	2025	2025	NUM
ejpam-6125	48	13	)	)	PUNCT
ejpam-6125	48	14	,	,	PUNCT
ejpam-6125	48	15	6125	6125	NUM
ejpam-6125	48	16	3	3	NUM
ejpam-6125	48	17	of	of	ADP
ejpam-6125	48	18	8	8	NUM
ejpam-6125	48	19	let	let	VERB
ejpam-6125	48	20	g	g	NOUN
ejpam-6125	48	21	and	and	CCONJ
ejpam-6125	48	22	h	h	NOUN
ejpam-6125	48	23	be	be	VERB
ejpam-6125	48	24	any	any	DET
ejpam-6125	48	25	two	two	NUM
ejpam-6125	48	26	graphs	graph	NOUN
ejpam-6125	48	27	.	.	PUNCT
ejpam-6125	49	1	the	the	DET
ejpam-6125	49	2	join	join	NOUN
ejpam-6125	49	3	g	g	PROPN
ejpam-6125	49	4	+	+	CCONJ
ejpam-6125	49	5	h	h	NOUN
ejpam-6125	49	6	is	be	AUX
ejpam-6125	49	7	the	the	DET
ejpam-6125	49	8	graph	graph	NOUN
ejpam-6125	49	9	with	with	ADP
ejpam-6125	49	10	vertex	vertex	NOUN
ejpam-6125	49	11	set	set	VERB
ejpam-6125	49	12	v	v	NOUN
ejpam-6125	49	13	(	(	PUNCT
ejpam-6125	49	14	g+h	g+h	NOUN
ejpam-6125	49	15	)	)	PUNCT
ejpam-6125	49	16	=	=	SYM
ejpam-6125	49	17	v	v	NOUN
ejpam-6125	49	18	(	(	PUNCT
ejpam-6125	49	19	g)∪	g)∪	VERB
ejpam-6125	49	20	v	v	NUM
ejpam-6125	49	21	(	(	PUNCT
ejpam-6125	49	22	h	h	NOUN
ejpam-6125	49	23	)	)	PUNCT
ejpam-6125	49	24	and	and	CCONJ
ejpam-6125	49	25	edge	edge	NOUN
ejpam-6125	49	26	set	set	VERB
ejpam-6125	49	27	e(g+h	e(g+h	NUM
ejpam-6125	49	28	)	)	PUNCT
ejpam-6125	50	1	=	=	SYM
ejpam-6125	50	2	e(g)∪e(h)∪	e(g)∪e(h)∪	NOUN
ejpam-6125	50	3	{	{	PUNCT
ejpam-6125	50	4	uv	uv	NOUN
ejpam-6125	50	5	:	:	PUNCT
ejpam-6125	50	6	u	u	PROPN
ejpam-6125	50	7	∈	∈	PROPN
ejpam-6125	50	8	v	v	ADP
ejpam-6125	50	9	(	(	PUNCT
ejpam-6125	50	10	g	g	NOUN
ejpam-6125	50	11	)	)	PUNCT
ejpam-6125	50	12	,	,	PUNCT
ejpam-6125	50	13	v	v	X
ejpam-6125	50	14	∈	∈	PROPN
ejpam-6125	50	15	v	v	NOUN
ejpam-6125	50	16	(	(	PUNCT
ejpam-6125	50	17	h	h	NOUN
ejpam-6125	50	18	)	)	PUNCT
ejpam-6125	50	19	}	}	PUNCT
ejpam-6125	50	20	.	.	PUNCT
ejpam-6125	51	1	the	the	DET
ejpam-6125	51	2	corona	corona	NOUN
ejpam-6125	51	3	g	g	PROPN
ejpam-6125	51	4	◦	◦	NOUN
ejpam-6125	51	5	h	h	NOUN
ejpam-6125	51	6	is	be	AUX
ejpam-6125	51	7	the	the	DET
ejpam-6125	51	8	graph	graph	NOUN
ejpam-6125	51	9	obtained	obtain	VERB
ejpam-6125	51	10	by	by	ADP
ejpam-6125	51	11	taking	take	VERB
ejpam-6125	51	12	one	one	NUM
ejpam-6125	51	13	copy	copy	NOUN
ejpam-6125	51	14	of	of	ADP
ejpam-6125	51	15	g	g	PROPN
ejpam-6125	51	16	and	and	CCONJ
ejpam-6125	51	17	|v	|v	PROPN
ejpam-6125	51	18	(	(	PUNCT
ejpam-6125	51	19	g)|	g)|	NOUN
ejpam-6125	51	20	copies	copy	NOUN
ejpam-6125	51	21	of	of	ADP
ejpam-6125	51	22	h	h	NOUN
ejpam-6125	51	23	,	,	PUNCT
ejpam-6125	51	24	and	and	CCONJ
ejpam-6125	51	25	then	then	ADV
ejpam-6125	51	26	joining	join	VERB
ejpam-6125	51	27	the	the	DET
ejpam-6125	51	28	ith	ith	PROPN
ejpam-6125	51	29	vertex	vertex	NOUN
ejpam-6125	51	30	of	of	ADP
ejpam-6125	51	31	g	g	NOUN
ejpam-6125	51	32	to	to	ADP
ejpam-6125	51	33	every	every	DET
ejpam-6125	51	34	vertex	vertex	NOUN
ejpam-6125	51	35	of	of	ADP
ejpam-6125	51	36	the	the	DET
ejpam-6125	51	37	ith	ith	PROPN
ejpam-6125	51	38	copy	copy	NOUN
ejpam-6125	51	39	of	of	ADP
ejpam-6125	51	40	h.	h.	PROPN
ejpam-6125	51	41	we	we	PRON
ejpam-6125	51	42	denote	denote	VERB
ejpam-6125	51	43	by	by	ADP
ejpam-6125	51	44	hv	hv	PROPN
ejpam-6125	52	1	the	the	DET
ejpam-6125	52	2	copy	copy	NOUN
ejpam-6125	52	3	of	of	ADP
ejpam-6125	52	4	h	h	NOUN
ejpam-6125	52	5	in	in	ADP
ejpam-6125	52	6	g	g	PROPN
ejpam-6125	52	7	◦	◦	NOUN
ejpam-6125	52	8	h	h	NOUN
ejpam-6125	52	9	corresponding	correspond	VERB
ejpam-6125	52	10	to	to	ADP
ejpam-6125	52	11	the	the	DET
ejpam-6125	52	12	vertex	vertex	NOUN
ejpam-6125	52	13	v	v	ADP
ejpam-6125	52	14	∈	∈	PROPN
ejpam-6125	52	15	g	g	NOUN
ejpam-6125	52	16	and	and	CCONJ
ejpam-6125	52	17	write	write	VERB
ejpam-6125	52	18	v+hv	v+hv	PROPN
ejpam-6125	52	19	for	for	ADP
ejpam-6125	52	20	⟨{v}⟩+hv	⟨{v}⟩+hv	PROPN
ejpam-6125	52	21	.	.	PUNCT
ejpam-6125	53	1	the	the	DET
ejpam-6125	53	2	shadow	shadow	NOUN
ejpam-6125	53	3	graph	graph	NOUN
ejpam-6125	53	4	s(g	s(g	PROPN
ejpam-6125	53	5	)	)	PUNCT
ejpam-6125	53	6	of	of	ADP
ejpam-6125	53	7	graph	graph	NOUN
ejpam-6125	53	8	g	g	PROPN
ejpam-6125	53	9	is	be	AUX
ejpam-6125	53	10	constructed	construct	VERB
ejpam-6125	53	11	by	by	ADP
ejpam-6125	53	12	taking	take	VERB
ejpam-6125	53	13	two	two	NUM
ejpam-6125	53	14	copies	copy	NOUN
ejpam-6125	53	15	of	of	ADP
ejpam-6125	53	16	g	g	NOUN
ejpam-6125	53	17	,	,	PUNCT
ejpam-6125	53	18	say	say	VERB
ejpam-6125	53	19	g1	g1	PROPN
ejpam-6125	53	20	and	and	CCONJ
ejpam-6125	53	21	g2	g2	PROPN
ejpam-6125	53	22	,	,	PUNCT
ejpam-6125	53	23	and	and	CCONJ
ejpam-6125	53	24	then	then	ADV
ejpam-6125	53	25	joining	join	VERB
ejpam-6125	53	26	each	each	DET
ejpam-6125	53	27	vertex	vertex	NOUN
ejpam-6125	53	28	u	u	NOUN
ejpam-6125	53	29	∈	∈	PROPN
ejpam-6125	53	30	v	v	NOUN
ejpam-6125	53	31	(	(	PUNCT
ejpam-6125	53	32	g1	g1	PROPN
ejpam-6125	53	33	)	)	PUNCT
ejpam-6125	53	34	to	to	ADP
ejpam-6125	53	35	the	the	DET
ejpam-6125	53	36	neighbors	neighbor	NOUN
ejpam-6125	53	37	of	of	ADP
ejpam-6125	53	38	its	its	PRON
ejpam-6125	53	39	corresponding	correspond	VERB
ejpam-6125	53	40	vertex	vertex	NOUN
ejpam-6125	53	41	u′	u′	PROPN
ejpam-6125	53	42	∈	∈	PROPN
ejpam-6125	53	43	v	v	NOUN
ejpam-6125	53	44	(	(	PUNCT
ejpam-6125	53	45	g2	g2	PROPN
ejpam-6125	53	46	)	)	PUNCT
ejpam-6125	53	47	.	.	PUNCT
ejpam-6125	54	1	3	3	X
ejpam-6125	54	2	.	.	X
ejpam-6125	54	3	results	result	NOUN
ejpam-6125	54	4	remark	remark	VERB
ejpam-6125	54	5	1	1	NUM
ejpam-6125	54	6	.	.	PUNCT
ejpam-6125	55	1	let	let	VERB
ejpam-6125	55	2	g	g	PRON
ejpam-6125	55	3	be	be	AUX
ejpam-6125	55	4	a	a	DET
ejpam-6125	55	5	graph	graph	NOUN
ejpam-6125	55	6	.	.	PUNCT
ejpam-6125	56	1	then	then	ADV
ejpam-6125	56	2	each	each	PRON
ejpam-6125	56	3	of	of	ADP
ejpam-6125	56	4	the	the	DET
ejpam-6125	56	5	following	follow	VERB
ejpam-6125	56	6	holds	hold	VERB
ejpam-6125	56	7	:	:	PUNCT
ejpam-6125	56	8	(	(	PUNCT
ejpam-6125	56	9	i	i	NOUN
ejpam-6125	56	10	)	)	PUNCT
ejpam-6125	56	11	a	a	DET
ejpam-6125	56	12	legal	legal	ADJ
ejpam-6125	56	13	closed	close	VERB
ejpam-6125	56	14	hop	hop	NOUN
ejpam-6125	56	15	neighborhood	neighborhood	NOUN
ejpam-6125	56	16	sequence	sequence	NOUN
ejpam-6125	56	17	of	of	ADP
ejpam-6125	56	18	g	g	NOUN
ejpam-6125	56	19	need	need	AUX
ejpam-6125	56	20	not	not	PART
ejpam-6125	56	21	form	form	VERB
ejpam-6125	56	22	an	an	DET
ejpam-6125	56	23	independent	independent	ADJ
ejpam-6125	56	24	set	set	NOUN
ejpam-6125	56	25	of	of	ADP
ejpam-6125	56	26	g.	g.	PROPN
ejpam-6125	56	27	(	(	PUNCT
ejpam-6125	56	28	ii	ii	PROPN
ejpam-6125	56	29	)	)	PUNCT
ejpam-6125	56	30	an	an	DET
ejpam-6125	56	31	independent	independent	ADJ
ejpam-6125	56	32	set	set	NOUN
ejpam-6125	56	33	of	of	ADP
ejpam-6125	56	34	g	g	NOUN
ejpam-6125	56	35	need	need	AUX
ejpam-6125	56	36	not	not	PART
ejpam-6125	56	37	form	form	VERB
ejpam-6125	56	38	a	a	DET
ejpam-6125	56	39	legal	legal	ADJ
ejpam-6125	56	40	closed	closed	ADJ
ejpam-6125	56	41	hop	hop	NOUN
ejpam-6125	56	42	neighborhood	neighborhood	NOUN
ejpam-6125	56	43	sequence	sequence	NOUN
ejpam-6125	56	44	of	of	ADP
ejpam-6125	56	45	g.	g.	PROPN
ejpam-6125	56	46	(	(	PUNCT
ejpam-6125	56	47	iii	iii	X
ejpam-6125	56	48	)	)	PUNCT
ejpam-6125	56	49	a	a	DET
ejpam-6125	56	50	legal	legal	ADJ
ejpam-6125	56	51	closed	close	VERB
ejpam-6125	56	52	hop	hop	NOUN
ejpam-6125	56	53	neighborhood	neighborhood	NOUN
ejpam-6125	56	54	independent	independent	ADJ
ejpam-6125	56	55	sequence	sequence	NOUN
ejpam-6125	56	56	of	of	ADP
ejpam-6125	56	57	g	g	PROPN
ejpam-6125	56	58	induces	induce	VERB
ejpam-6125	56	59	an	an	DET
ejpam-6125	56	60	independent	independent	ADJ
ejpam-6125	56	61	set	set	NOUN
ejpam-6125	56	62	of	of	ADP
ejpam-6125	56	63	g.	g.	PROPN
ejpam-6125	56	64	(	(	PUNCT
ejpam-6125	56	65	iv	iv	X
ejpam-6125	56	66	)	)	PUNCT
ejpam-6125	56	67	1	1	NUM
ejpam-6125	56	68	≤	≤	NOUN
ejpam-6125	56	69	θ(g	θ(g	NUM
ejpam-6125	56	70	)	)	PUNCT
ejpam-6125	56	71	≤	≤	NOUN
ejpam-6125	56	72	α(g	α(g	NUM
ejpam-6125	56	73	)	)	PUNCT
ejpam-6125	56	74	≤	≤	NOUN
ejpam-6125	56	75	|v	|v	X
ejpam-6125	56	76	(	(	PUNCT
ejpam-6125	56	77	g)|	g)|	PROPN
ejpam-6125	56	78	.	.	PUNCT
ejpam-6125	56	79	theorem	theorem	NOUN
ejpam-6125	56	80	1	1	NUM
ejpam-6125	56	81	.	.	PUNCT
ejpam-6125	57	1	let	let	VERB
ejpam-6125	57	2	g	g	PRON
ejpam-6125	57	3	be	be	AUX
ejpam-6125	57	4	a	a	DET
ejpam-6125	57	5	graph	graph	NOUN
ejpam-6125	57	6	of	of	ADP
ejpam-6125	57	7	order	order	NOUN
ejpam-6125	57	8	n.	n.	NOUN
ejpam-6125	57	9	then	then	ADV
ejpam-6125	57	10	θ(g	θ(g	NUM
ejpam-6125	57	11	)	)	PUNCT
ejpam-6125	57	12	=	=	SYM
ejpam-6125	57	13	|v	|v	X
ejpam-6125	57	14	(	(	PUNCT
ejpam-6125	57	15	g)|	g)|	VERB
ejpam-6125	57	16	if	if	SCONJ
ejpam-6125	57	17	and	and	CCONJ
ejpam-6125	57	18	only	only	ADV
ejpam-6125	57	19	if	if	SCONJ
ejpam-6125	57	20	g	g	PROPN
ejpam-6125	57	21	=	=	PROPN
ejpam-6125	57	22	kn	kn	PROPN
ejpam-6125	57	23	.	.	PUNCT
ejpam-6125	57	24	proof	proof	PROPN
ejpam-6125	57	25	.	.	PUNCT
ejpam-6125	58	1	suppose	suppose	VERB
ejpam-6125	58	2	that	that	SCONJ
ejpam-6125	58	3	g	g	PROPN
ejpam-6125	58	4	=	=	PROPN
ejpam-6125	58	5	kn	kn	PROPN
ejpam-6125	58	6	.	.	PUNCT
ejpam-6125	59	1	let	let	VERB
ejpam-6125	59	2	v	v	X
ejpam-6125	59	3	(	(	PUNCT
ejpam-6125	59	4	kn	kn	PROPN
ejpam-6125	59	5	)	)	PUNCT
ejpam-6125	59	6	=	=	PRON
ejpam-6125	59	7	{	{	PUNCT
ejpam-6125	59	8	c1	c1	PROPN
ejpam-6125	59	9	,	,	PUNCT
ejpam-6125	59	10	c2	c2	PROPN
ejpam-6125	59	11	,	,	PUNCT
ejpam-6125	59	12	...	...	PUNCT
ejpam-6125	59	13	,	,	PUNCT
ejpam-6125	59	14	cn	cn	ADJ
ejpam-6125	59	15	}	}	PUNCT
ejpam-6125	59	16	=	=	SYM
ejpam-6125	59	17	q̂.	q̂.	NOUN
ejpam-6125	59	18	then	then	ADV
ejpam-6125	59	19	,	,	PUNCT
ejpam-6125	59	20	dkn	dkn	PROPN
ejpam-6125	59	21	(	(	PUNCT
ejpam-6125	59	22	ci	ci	PROPN
ejpam-6125	59	23	,	,	PUNCT
ejpam-6125	59	24	cj	cj	X
ejpam-6125	59	25	)	)	PUNCT
ejpam-6125	59	26	=	=	SYM
ejpam-6125	60	1	∞	∞	NUM
ejpam-6125	60	2	=	=	SYM
ejpam-6125	60	3	̸	̸	NOUN
ejpam-6125	60	4	1	1	NUM
ejpam-6125	60	5	for	for	ADP
ejpam-6125	60	6	each	each	DET
ejpam-6125	60	7	i	i	PRON
ejpam-6125	60	8	̸=	̸=	PROPN
ejpam-6125	60	9	j	j	PROPN
ejpam-6125	60	10	where	where	SCONJ
ejpam-6125	60	11	i	i	PRON
ejpam-6125	60	12	,	,	PUNCT
ejpam-6125	60	13	j	j	PROPN
ejpam-6125	60	14	∈	∈	PROPN
ejpam-6125	60	15	{	{	PUNCT
ejpam-6125	60	16	1	1	NUM
ejpam-6125	60	17	,	,	PUNCT
ejpam-6125	60	18	2	2	NUM
ejpam-6125	60	19	,	,	PUNCT
ejpam-6125	60	20	...	...	PUNCT
ejpam-6125	60	21	,	,	PUNCT
ejpam-6125	60	22	n	n	CCONJ
ejpam-6125	60	23	}	}	PUNCT
ejpam-6125	60	24	.	.	PUNCT
ejpam-6125	61	1	this	this	PRON
ejpam-6125	61	2	means	mean	VERB
ejpam-6125	61	3	that	that	SCONJ
ejpam-6125	61	4	q̂	q̂	PRON
ejpam-6125	61	5	is	be	AUX
ejpam-6125	61	6	the	the	DET
ejpam-6125	61	7	maximum	maximum	ADJ
ejpam-6125	61	8	independent	independent	ADJ
ejpam-6125	61	9	set	set	NOUN
ejpam-6125	61	10	of	of	ADP
ejpam-6125	61	11	kn	kn	PROPN
ejpam-6125	61	12	.	.	PROPN
ejpam-6125	61	13	notice	notice	VERB
ejpam-6125	61	14	that	that	SCONJ
ejpam-6125	61	15	ci	ci	PROPN
ejpam-6125	61	16	∈	∈	PROPN
ejpam-6125	61	17	n2	n2	NOUN
ejpam-6125	61	18	kn	kn	PROPN
ejpam-6125	62	1	[	[	X
ejpam-6125	62	2	ci	ci	X
ejpam-6125	62	3	]	]	PUNCT
ejpam-6125	62	4	\	\	PROPN
ejpam-6125	62	5	i−1⋃	i−1⋃	PROPN
ejpam-6125	62	6	j=1	j=1	PROPN
ejpam-6125	62	7	n2	n2	PROPN
ejpam-6125	62	8	k̄n	k̄n	PROPN
ejpam-6125	63	1	[	[	X
ejpam-6125	63	2	cj	cj	X
ejpam-6125	63	3	]	]	PUNCT
ejpam-6125	63	4	for	for	ADP
ejpam-6125	63	5	each	each	DET
ejpam-6125	63	6	i	i	PRON
ejpam-6125	63	7	∈	∈	PROPN
ejpam-6125	63	8	{	{	PUNCT
ejpam-6125	63	9	2	2	NUM
ejpam-6125	63	10	,	,	PUNCT
ejpam-6125	63	11	3	3	NUM
ejpam-6125	63	12	,	,	PUNCT
ejpam-6125	63	13	...	...	PUNCT
ejpam-6125	63	14	,	,	PUNCT
ejpam-6125	63	15	n	n	CCONJ
ejpam-6125	63	16	}	}	PUNCT
ejpam-6125	63	17	.	.	PUNCT
ejpam-6125	64	1	it	it	PRON
ejpam-6125	64	2	follows	follow	VERB
ejpam-6125	64	3	that	that	PRON
ejpam-6125	64	4	q	q	PROPN
ejpam-6125	64	5	=	=	SYM
ejpam-6125	64	6	(	(	PUNCT
ejpam-6125	64	7	c1	c1	PROPN
ejpam-6125	64	8	,	,	PUNCT
ejpam-6125	64	9	c2	c2	PROPN
ejpam-6125	64	10	,	,	PUNCT
ejpam-6125	64	11	...	...	PUNCT
ejpam-6125	64	12	,	,	PUNCT
ejpam-6125	64	13	cn	cn	PROPN
ejpam-6125	64	14	)	)	PUNCT
ejpam-6125	64	15	is	be	AUX
ejpam-6125	64	16	a	a	DET
ejpam-6125	64	17	legal	legal	ADJ
ejpam-6125	64	18	closed	close	VERB
ejpam-6125	64	19	hop	hop	NOUN
ejpam-6125	64	20	neighborhood	neighborhood	NOUN
ejpam-6125	64	21	sequence	sequence	NOUN
ejpam-6125	64	22	of	of	ADP
ejpam-6125	64	23	kn	kn	PROPN
ejpam-6125	64	24	.	.	PUNCT
ejpam-6125	65	1	consequently	consequently	ADV
ejpam-6125	65	2	,	,	PUNCT
ejpam-6125	65	3	q	q	X
ejpam-6125	65	4	is	be	AUX
ejpam-6125	65	5	the	the	DET
ejpam-6125	65	6	maximum	maximum	ADJ
ejpam-6125	65	7	legal	legal	ADJ
ejpam-6125	65	8	closed	closed	ADJ
ejpam-6125	65	9	hop	hop	NOUN
ejpam-6125	65	10	neighborhood	neighborhood	NOUN
ejpam-6125	65	11	independent	independent	ADJ
ejpam-6125	65	12	sequence	sequence	NOUN
ejpam-6125	65	13	of	of	ADP
ejpam-6125	65	14	kn	kn	PROPN
ejpam-6125	65	15	,	,	PUNCT
ejpam-6125	65	16	and	and	CCONJ
ejpam-6125	65	17	so	so	ADV
ejpam-6125	65	18	θ(kn	θ(kn	NOUN
ejpam-6125	65	19	)	)	PUNCT
ejpam-6125	65	20	=	=	SYM
ejpam-6125	65	21	|v	|v	X
ejpam-6125	65	22	(	(	PUNCT
ejpam-6125	65	23	k̄n)|	k̄n)|	PROPN
ejpam-6125	65	24	.	.	PUNCT
ejpam-6125	66	1	conversely	conversely	ADV
ejpam-6125	66	2	,	,	PUNCT
ejpam-6125	66	3	assume	assume	VERB
ejpam-6125	66	4	that	that	SCONJ
ejpam-6125	66	5	θ(g	θ(g	VERB
ejpam-6125	66	6	)	)	PUNCT
ejpam-6125	66	7	=	=	SYM
ejpam-6125	66	8	|v	|v	X
ejpam-6125	66	9	(	(	PUNCT
ejpam-6125	66	10	g	g	NOUN
ejpam-6125	66	11	)	)	PUNCT
ejpam-6125	66	12	.	.	PUNCT
ejpam-6125	67	1	suppose	suppose	VERB
ejpam-6125	67	2	further	far	ADV
ejpam-6125	67	3	that	that	SCONJ
ejpam-6125	67	4	g	g	PROPN
ejpam-6125	67	5	̸=	̸=	PROPN
ejpam-6125	67	6	kn	kn	PROPN
ejpam-6125	67	7	.	.	PUNCT
ejpam-6125	68	1	then	then	ADV
ejpam-6125	68	2	there	there	PRON
ejpam-6125	68	3	exist	exist	VERB
ejpam-6125	68	4	u	u	NOUN
ejpam-6125	68	5	,	,	PUNCT
ejpam-6125	68	6	v	v	NOUN
ejpam-6125	68	7	∈	∈	PROPN
ejpam-6125	68	8	v	v	NOUN
ejpam-6125	68	9	(	(	PUNCT
ejpam-6125	68	10	g	g	NOUN
ejpam-6125	68	11	)	)	PUNCT
ejpam-6125	68	12	such	such	ADJ
ejpam-6125	68	13	that	that	SCONJ
ejpam-6125	68	14	dg(u	dg(u	ADJ
ejpam-6125	68	15	,	,	PUNCT
ejpam-6125	68	16	v	v	NOUN
ejpam-6125	68	17	)	)	PUNCT
ejpam-6125	69	1	=	=	SYM
ejpam-6125	69	2	1	1	X
ejpam-6125	69	3	.	.	PUNCT
ejpam-6125	70	1	hence	hence	ADV
ejpam-6125	70	2	,	,	PUNCT
ejpam-6125	70	3	either	either	CCONJ
ejpam-6125	70	4	u	u	NOUN
ejpam-6125	70	5	or	or	CCONJ
ejpam-6125	70	6	v	v	NOUN
ejpam-6125	70	7	is	be	AUX
ejpam-6125	70	8	not	not	PART
ejpam-6125	70	9	in	in	ADP
ejpam-6125	70	10	any	any	DET
ejpam-6125	70	11	lchni	lchni	PROPN
ejpam-6125	70	12	set	set	NOUN
ejpam-6125	70	13	of	of	ADP
ejpam-6125	70	14	g	g	PROPN
ejpam-6125	70	15	,	,	PUNCT
ejpam-6125	70	16	a	a	DET
ejpam-6125	70	17	contradiction	contradiction	NOUN
ejpam-6125	70	18	to	to	ADP
ejpam-6125	70	19	the	the	DET
ejpam-6125	70	20	assumption	assumption	NOUN
ejpam-6125	70	21	that	that	SCONJ
ejpam-6125	70	22	α(g	α(g	NUM
ejpam-6125	70	23	)	)	PUNCT
ejpam-6125	70	24	=	=	SYM
ejpam-6125	70	25	|v	|v	PROPN
ejpam-6125	70	26	(	(	PUNCT
ejpam-6125	70	27	g)|	g)|	PROPN
ejpam-6125	70	28	.	.	PUNCT
ejpam-6125	71	1	therefore	therefore	ADV
ejpam-6125	71	2	,	,	PUNCT
ejpam-6125	71	3	g	g	PROPN
ejpam-6125	71	4	must	must	AUX
ejpam-6125	71	5	be	be	AUX
ejpam-6125	71	6	equal	equal	ADJ
ejpam-6125	71	7	to	to	ADP
ejpam-6125	71	8	kn	kn	PROPN
ejpam-6125	71	9	.	.	PUNCT
ejpam-6125	72	1	j.	j.	PROPN
ejpam-6125	72	2	a.	a.	PROPN
ejpam-6125	72	3	hassan	hassan	PROPN
ejpam-6125	72	4	et	et	PROPN
ejpam-6125	72	5	al	al	PROPN
ejpam-6125	72	6	.	.	PUNCT
ejpam-6125	72	7	/	/	SYM
ejpam-6125	72	8	eur	eur	PROPN
ejpam-6125	72	9	.	.	PUNCT
ejpam-6125	73	1	j.	j.	PROPN
ejpam-6125	73	2	pure	pure	PROPN
ejpam-6125	73	3	appl	appl	PROPN
ejpam-6125	73	4	.	.	PROPN
ejpam-6125	73	5	math	math	PROPN
ejpam-6125	73	6	,	,	PUNCT
ejpam-6125	73	7	18	18	NUM
ejpam-6125	73	8	(	(	PUNCT
ejpam-6125	73	9	3	3	NUM
ejpam-6125	73	10	)	)	PUNCT
ejpam-6125	73	11	(	(	PUNCT
ejpam-6125	73	12	2025	2025	NUM
ejpam-6125	73	13	)	)	PUNCT
ejpam-6125	73	14	,	,	PUNCT
ejpam-6125	73	15	6125	6125	NUM
ejpam-6125	73	16	4	4	NUM
ejpam-6125	73	17	of	of	ADP
ejpam-6125	73	18	8	8	NUM
ejpam-6125	73	19	theorem	theorem	NOUN
ejpam-6125	73	20	2	2	NUM
ejpam-6125	73	21	.	.	PUNCT
ejpam-6125	74	1	let	let	VERB
ejpam-6125	74	2	n	n	PRON
ejpam-6125	74	3	be	be	AUX
ejpam-6125	74	4	any	any	DET
ejpam-6125	74	5	positive	positive	ADJ
ejpam-6125	74	6	integer	integer	NOUN
ejpam-6125	74	7	.	.	PUNCT
ejpam-6125	75	1	then	then	ADV
ejpam-6125	75	2	θ(pn	θ(pn	NUM
ejpam-6125	75	3	)	)	PUNCT
ejpam-6125	75	4	=	=	SYM
ejpam-6125	76	1			NOUN
ejpam-6125	76	2	1	1	NUM
ejpam-6125	76	3	,	,	PUNCT
ejpam-6125	76	4	if	if	SCONJ
ejpam-6125	76	5	n=1	n=1	PROPN
ejpam-6125	76	6	n	n	ADV
ejpam-6125	76	7	2	2	NUM
ejpam-6125	76	8	,	,	PUNCT
ejpam-6125	76	9	if	if	SCONJ
ejpam-6125	76	10	n	n	PRON
ejpam-6125	76	11	is	be	AUX
ejpam-6125	76	12	even	even	ADV
ejpam-6125	76	13	n−1	n−1	PROPN
ejpam-6125	76	14	2	2	NUM
ejpam-6125	76	15	,	,	PUNCT
ejpam-6125	76	16	if	if	SCONJ
ejpam-6125	76	17	n	n	PRON
ejpam-6125	76	18	≥	≥	NOUN
ejpam-6125	76	19	3	3	NUM
ejpam-6125	76	20	and	and	CCONJ
ejpam-6125	76	21	odd	odd	ADJ
ejpam-6125	76	22	proof	proof	NOUN
ejpam-6125	76	23	.	.	PUNCT
ejpam-6125	77	1	by	by	ADP
ejpam-6125	77	2	theorem	theorem	NOUN
ejpam-6125	77	3	1	1	NUM
ejpam-6125	77	4	,	,	PUNCT
ejpam-6125	77	5	θ(p1	θ(p1	NOUN
ejpam-6125	77	6	)	)	PUNCT
ejpam-6125	77	7	=	=	SYM
ejpam-6125	78	1	1	1	X
ejpam-6125	78	2	.	.	PUNCT
ejpam-6125	78	3	suppose	suppose	VERB
ejpam-6125	78	4	that	that	SCONJ
ejpam-6125	78	5	n	n	PRON
ejpam-6125	78	6	is	be	AUX
ejpam-6125	78	7	even	even	ADV
ejpam-6125	78	8	.	.	PUNCT
ejpam-6125	79	1	clearly	clearly	ADV
ejpam-6125	79	2	,	,	PUNCT
ejpam-6125	79	3	θ(p2	θ(p2	ADJ
ejpam-6125	79	4	)	)	PUNCT
ejpam-6125	79	5	=	=	SYM
ejpam-6125	80	1	1	1	X
ejpam-6125	80	2	.	.	PUNCT
ejpam-6125	80	3	now	now	ADV
ejpam-6125	80	4	,	,	PUNCT
ejpam-6125	80	5	for	for	ADP
ejpam-6125	80	6	n	n	PRON
ejpam-6125	80	7	≥	≥	NOUN
ejpam-6125	80	8	4	4	NUM
ejpam-6125	80	9	,	,	PUNCT
ejpam-6125	80	10	consider	consider	VERB
ejpam-6125	80	11	v	v	NOUN
ejpam-6125	80	12	(	(	PUNCT
ejpam-6125	80	13	pn	pn	NOUN
ejpam-6125	80	14	)	)	PUNCT
ejpam-6125	80	15	=	=	SYM
ejpam-6125	80	16	{	{	PUNCT
ejpam-6125	80	17	a1	a1	PROPN
ejpam-6125	80	18	,	,	PUNCT
ejpam-6125	80	19	a2	a2	PROPN
ejpam-6125	80	20	,	,	PUNCT
ejpam-6125	80	21	...	...	PUNCT
ejpam-6125	80	22	,	,	PUNCT
ejpam-6125	80	23	an	an	PRON
ejpam-6125	80	24	}	}	PUNCT
ejpam-6125	80	25	and	and	CCONJ
ejpam-6125	80	26	q	q	ADJ
ejpam-6125	80	27	=	=	NOUN
ejpam-6125	80	28	{	{	PUNCT
ejpam-6125	80	29	a1	a1	NOUN
ejpam-6125	80	30	,	,	PUNCT
ejpam-6125	80	31	a3	a3	NOUN
ejpam-6125	80	32	,	,	PUNCT
ejpam-6125	80	33	...	...	PUNCT
ejpam-6125	80	34	,	,	PUNCT
ejpam-6125	80	35	an−3	an−3	PROPN
ejpam-6125	80	36	,	,	PUNCT
ejpam-6125	80	37	an	an	PRON
ejpam-6125	80	38	}	}	PUNCT
ejpam-6125	80	39	.	.	PUNCT
ejpam-6125	81	1	then	then	ADV
ejpam-6125	81	2	q̂	q̂	PRON
ejpam-6125	81	3	is	be	AUX
ejpam-6125	81	4	a	a	DET
ejpam-6125	81	5	maximum	maximum	ADJ
ejpam-6125	81	6	independent	independent	ADJ
ejpam-6125	81	7	set	set	NOUN
ejpam-6125	81	8	of	of	ADP
ejpam-6125	81	9	pn	pn	PROPN
ejpam-6125	81	10	.	.	PROPN
ejpam-6125	81	11	observe	observe	VERB
ejpam-6125	81	12	that	that	SCONJ
ejpam-6125	81	13	ai+2	ai+2	NUM
ejpam-6125	81	14	∈	∈	PROPN
ejpam-6125	81	15	n2	n2	NOUN
ejpam-6125	81	16	g[ai	g[ai	PROPN
ejpam-6125	81	17	]	]	PUNCT
ejpam-6125	81	18	\	\	PUNCT
ejpam-6125	82	1	⋃	⋃	PUNCT
ejpam-6125	82	2	j	j	PROPN
ejpam-6125	82	3	<	<	X
ejpam-6125	82	4	in	in	ADP
ejpam-6125	82	5	2	2	NUM
ejpam-6125	82	6	g[aj	g[aj	PROPN
ejpam-6125	82	7	]	]	PUNCT
ejpam-6125	82	8	for	for	ADP
ejpam-6125	82	9	each	each	DET
ejpam-6125	82	10	i	i	PRON
ejpam-6125	82	11	∈	∈	PROPN
ejpam-6125	82	12	{	{	PUNCT
ejpam-6125	82	13	3	3	NUM
ejpam-6125	82	14	,	,	PUNCT
ejpam-6125	82	15	5	5	NUM
ejpam-6125	82	16	,	,	PUNCT
ejpam-6125	82	17	...	...	PUNCT
ejpam-6125	82	18	,	,	PUNCT
ejpam-6125	82	19	n	n	CCONJ
ejpam-6125	82	20	−	−	PROPN
ejpam-6125	82	21	3	3	NUM
ejpam-6125	82	22	}	}	PUNCT
ejpam-6125	82	23	and	and	CCONJ
ejpam-6125	82	24	an	an	DET
ejpam-6125	82	25	∈	∈	PROPN
ejpam-6125	82	26	n2	n2	NOUN
ejpam-6125	82	27	g[an	g[an	PROPN
ejpam-6125	82	28	]	]	PUNCT
ejpam-6125	82	29	\	\	PUNCT
ejpam-6125	82	30	⋃	⋃	PROPN
ejpam-6125	82	31	m	m	NOUN
ejpam-6125	82	32	<	<	X
ejpam-6125	82	33	nn	nn	X
ejpam-6125	82	34	2	2	NUM
ejpam-6125	82	35	g[am	g[am	NOUN
ejpam-6125	82	36	]	]	PUNCT
ejpam-6125	82	37	.	.	PUNCT
ejpam-6125	83	1	it	it	PRON
ejpam-6125	83	2	follows	follow	VERB
ejpam-6125	83	3	that	that	SCONJ
ejpam-6125	83	4	q	q	NOUN
ejpam-6125	83	5	is	be	AUX
ejpam-6125	83	6	an	an	DET
ejpam-6125	83	7	lchni	lchni	ADJ
ejpam-6125	83	8	sequence	sequence	NOUN
ejpam-6125	83	9	of	of	ADP
ejpam-6125	83	10	pn	pn	PROPN
ejpam-6125	83	11	.	.	PROPN
ejpam-6125	84	1	therefore	therefore	ADV
ejpam-6125	84	2	,	,	PUNCT
ejpam-6125	84	3	q	q	X
ejpam-6125	84	4	is	be	AUX
ejpam-6125	84	5	a	a	DET
ejpam-6125	84	6	maximum	maximum	ADJ
ejpam-6125	84	7	lchni	lchni	ADJ
ejpam-6125	84	8	sequence	sequence	NOUN
ejpam-6125	84	9	of	of	ADP
ejpam-6125	84	10	pn	pn	PROPN
ejpam-6125	84	11	,	,	PUNCT
ejpam-6125	84	12	and	and	CCONJ
ejpam-6125	84	13	so	so	ADV
ejpam-6125	84	14	θ(pn	θ(pn	ADJ
ejpam-6125	84	15	)	)	PUNCT
ejpam-6125	84	16	=	=	SYM
ejpam-6125	84	17	n	n	DET
ejpam-6125	84	18	2	2	NUM
ejpam-6125	84	19	for	for	ADP
ejpam-6125	84	20	all	all	DET
ejpam-6125	84	21	even	even	ADJ
ejpam-6125	84	22	numbers	number	NOUN
ejpam-6125	84	23	n	n	PRON
ejpam-6125	84	24	≥	≥	NOUN
ejpam-6125	84	25	2	2	NUM
ejpam-6125	84	26	.	.	PUNCT
ejpam-6125	85	1	next	next	ADV
ejpam-6125	85	2	,	,	PUNCT
ejpam-6125	85	3	suppose	suppose	VERB
ejpam-6125	85	4	that	that	SCONJ
ejpam-6125	85	5	n	n	PRON
ejpam-6125	85	6	is	be	AUX
ejpam-6125	85	7	an	an	DET
ejpam-6125	85	8	odd	odd	ADJ
ejpam-6125	85	9	positive	positive	ADJ
ejpam-6125	85	10	integer	integer	NOUN
ejpam-6125	85	11	.	.	PUNCT
ejpam-6125	86	1	for	for	ADP
ejpam-6125	86	2	n	n	NOUN
ejpam-6125	86	3	=	=	SYM
ejpam-6125	86	4	3	3	NUM
ejpam-6125	86	5	,	,	PUNCT
ejpam-6125	86	6	let	let	VERB
ejpam-6125	86	7	v	v	NOUN
ejpam-6125	86	8	(	(	PUNCT
ejpam-6125	86	9	p3	p3	PROPN
ejpam-6125	86	10	)	)	PUNCT
ejpam-6125	87	1	=	=	PRON
ejpam-6125	87	2	{	{	PUNCT
ejpam-6125	87	3	b1	b1	NOUN
ejpam-6125	87	4	,	,	PUNCT
ejpam-6125	87	5	b2	b2	NOUN
ejpam-6125	87	6	,	,	PUNCT
ejpam-6125	87	7	b3	b3	PROPN
ejpam-6125	87	8	}	}	PUNCT
ejpam-6125	87	9	.	.	PUNCT
ejpam-6125	88	1	since	since	SCONJ
ejpam-6125	88	2	n2	n2	PROPN
ejpam-6125	88	3	p3	p3	PROPN
ejpam-6125	88	4	[	[	X
ejpam-6125	88	5	b1	b1	X
ejpam-6125	88	6	]	]	X
ejpam-6125	88	7	=	=	SYM
ejpam-6125	88	8	n2	n2	PROPN
ejpam-6125	88	9	p3	p3	PROPN
ejpam-6125	88	10	[	[	X
ejpam-6125	88	11	b3	b3	PROPN
ejpam-6125	88	12	]	]	PUNCT
ejpam-6125	88	13	,	,	PUNCT
ejpam-6125	88	14	it	it	PRON
ejpam-6125	88	15	follows	follow	VERB
ejpam-6125	88	16	that	that	PRON
ejpam-6125	88	17	θ(p3	θ(p3	NOUN
ejpam-6125	88	18	)	)	PUNCT
ejpam-6125	89	1	=	=	SYM
ejpam-6125	89	2	1	1	X
ejpam-6125	89	3	.	.	PUNCT
ejpam-6125	89	4	now	now	ADV
ejpam-6125	89	5	,	,	PUNCT
ejpam-6125	89	6	assume	assume	VERB
ejpam-6125	89	7	that	that	SCONJ
ejpam-6125	89	8	n	n	PRON
ejpam-6125	89	9	≥	≥	NOUN
ejpam-6125	89	10	5	5	NUM
ejpam-6125	89	11	and	and	CCONJ
ejpam-6125	89	12	odd	odd	ADJ
ejpam-6125	89	13	.	.	PUNCT
ejpam-6125	90	1	let	let	VERB
ejpam-6125	90	2	v	v	X
ejpam-6125	90	3	(	(	PUNCT
ejpam-6125	90	4	pn	pn	NOUN
ejpam-6125	90	5	)	)	PUNCT
ejpam-6125	90	6	=	=	SYM
ejpam-6125	90	7	{	{	PUNCT
ejpam-6125	90	8	v1	v1	PROPN
ejpam-6125	90	9	,	,	PUNCT
ejpam-6125	90	10	v2	v2	PROPN
ejpam-6125	90	11	,	,	PUNCT
ejpam-6125	90	12	...	...	PUNCT
ejpam-6125	90	13	,	,	PUNCT
ejpam-6125	90	14	vn	vn	PROPN
ejpam-6125	90	15	}	}	PUNCT
ejpam-6125	90	16	,	,	PUNCT
ejpam-6125	90	17	and	and	CCONJ
ejpam-6125	90	18	consider	consider	VERB
ejpam-6125	90	19	t	t	NOUN
ejpam-6125	90	20	=	=	SYM
ejpam-6125	90	21	{	{	PUNCT
ejpam-6125	90	22	v1	v1	PROPN
ejpam-6125	90	23	,	,	PUNCT
ejpam-6125	90	24	v3	v3	PROPN
ejpam-6125	90	25	,	,	PUNCT
ejpam-6125	90	26	...	...	PUNCT
ejpam-6125	90	27	,	,	PUNCT
ejpam-6125	90	28	vn−2	vn−2	PROPN
ejpam-6125	90	29	}	}	PUNCT
ejpam-6125	90	30	.	.	PUNCT
ejpam-6125	91	1	then	then	ADV
ejpam-6125	91	2	t̂	t̂	PRON
ejpam-6125	91	3	is	be	AUX
ejpam-6125	91	4	an	an	DET
ejpam-6125	91	5	independent	independent	ADJ
ejpam-6125	91	6	set	set	NOUN
ejpam-6125	91	7	of	of	ADP
ejpam-6125	91	8	pn	pn	PROPN
ejpam-6125	91	9	and	and	CCONJ
ejpam-6125	91	10	vi+2	vi+2	NUM
ejpam-6125	91	11	∈	∈	PROPN
ejpam-6125	91	12	n2	n2	PROPN
ejpam-6125	91	13	g[vi	g[vi	PROPN
ejpam-6125	91	14	]	]	PUNCT
ejpam-6125	91	15	\	\	PUNCT
ejpam-6125	92	1	⋃	⋃	SCONJ
ejpam-6125	92	2	j	j	PROPN
ejpam-6125	92	3	<	<	X
ejpam-6125	92	4	in	in	ADP
ejpam-6125	92	5	2	2	NUM
ejpam-6125	92	6	g[vj	g[vj	PROPN
ejpam-6125	92	7	]	]	PUNCT
ejpam-6125	92	8	for	for	ADP
ejpam-6125	92	9	all	all	PRON
ejpam-6125	92	10	i	i	PRON
ejpam-6125	92	11	∈	∈	PROPN
ejpam-6125	92	12	{	{	PUNCT
ejpam-6125	92	13	3	3	NUM
ejpam-6125	92	14	,	,	PUNCT
ejpam-6125	92	15	5	5	NUM
ejpam-6125	92	16	,	,	PUNCT
ejpam-6125	92	17	...	...	PUNCT
ejpam-6125	92	18	,	,	PUNCT
ejpam-6125	92	19	n	n	CCONJ
ejpam-6125	92	20	−	−	PROPN
ejpam-6125	92	21	2	2	NUM
ejpam-6125	92	22	}	}	PUNCT
ejpam-6125	92	23	.	.	PUNCT
ejpam-6125	93	1	thus	thus	ADV
ejpam-6125	93	2	,	,	PUNCT
ejpam-6125	93	3	t	t	PROPN
ejpam-6125	93	4	is	be	AUX
ejpam-6125	93	5	an	an	DET
ejpam-6125	93	6	lchni	lchni	ADJ
ejpam-6125	93	7	sequence	sequence	NOUN
ejpam-6125	93	8	of	of	ADP
ejpam-6125	93	9	pn	pn	PROPN
ejpam-6125	93	10	.	.	PUNCT
ejpam-6125	94	1	since	since	SCONJ
ejpam-6125	94	2	n	n	ADV
ejpam-6125	94	3	2	2	NUM
ejpam-6125	94	4	g[vn	g[vn	PROPN
ejpam-6125	94	5	]	]	PUNCT
ejpam-6125	94	6	⊆	⊆	NUM
ejpam-6125	94	7	n2	n2	NOUN
ejpam-6125	94	8	g[vn−2	g[vn−2	PROPN
ejpam-6125	94	9	]	]	PUNCT
ejpam-6125	94	10	,	,	PUNCT
ejpam-6125	94	11	it	it	PRON
ejpam-6125	94	12	follows	follow	VERB
ejpam-6125	94	13	that	that	SCONJ
ejpam-6125	94	14	t	t	PROPN
ejpam-6125	94	15	is	be	AUX
ejpam-6125	94	16	a	a	DET
ejpam-6125	94	17	maximum	maximum	ADJ
ejpam-6125	94	18	lchni	lchni	ADJ
ejpam-6125	94	19	sequence	sequence	NOUN
ejpam-6125	94	20	of	of	ADP
ejpam-6125	94	21	pn	pn	PROPN
ejpam-6125	94	22	.	.	PUNCT
ejpam-6125	95	1	consequently	consequently	ADV
ejpam-6125	95	2	,	,	PUNCT
ejpam-6125	95	3	θ(pn	θ(pn	NOUN
ejpam-6125	95	4	)	)	PUNCT
ejpam-6125	95	5	=	=	SYM
ejpam-6125	95	6	n−1	n−1	PROPN
ejpam-6125	95	7	2	2	NUM
ejpam-6125	95	8	for	for	ADP
ejpam-6125	95	9	all	all	DET
ejpam-6125	95	10	odd	odd	ADJ
ejpam-6125	95	11	positive	positive	ADJ
ejpam-6125	95	12	integers	integer	NOUN
ejpam-6125	95	13	n	n	PRON
ejpam-6125	95	14	≥	≥	NUM
ejpam-6125	95	15	3	3	NUM
ejpam-6125	95	16	.	.	PUNCT
ejpam-6125	95	17	theorem	theorem	NOUN
ejpam-6125	95	18	3	3	X
ejpam-6125	95	19	.	.	PUNCT
ejpam-6125	96	1	let	let	VERB
ejpam-6125	96	2	n	n	PRON
ejpam-6125	96	3	≥	≥	X
ejpam-6125	96	4	3	3	NUM
ejpam-6125	96	5	be	be	AUX
ejpam-6125	96	6	any	any	DET
ejpam-6125	96	7	positive	positive	ADJ
ejpam-6125	96	8	integer	integer	NOUN
ejpam-6125	96	9	.	.	PUNCT
ejpam-6125	97	1	then	then	ADV
ejpam-6125	97	2	θ(cn	θ(cn	PROPN
ejpam-6125	97	3	)	)	PUNCT
ejpam-6125	97	4	=	=	PRON
ejpam-6125	97	5	{	{	PUNCT
ejpam-6125	97	6	α(cn	α(cn	NOUN
ejpam-6125	97	7	)	)	PUNCT
ejpam-6125	97	8	,	,	PUNCT
ejpam-6125	97	9	if	if	SCONJ
ejpam-6125	97	10	n	n	PRON
ejpam-6125	97	11	is	be	AUX
ejpam-6125	97	12	odd	odd	ADJ
ejpam-6125	97	13	α(cn)−	α(cn)−	NOUN
ejpam-6125	97	14	1	1	NUM
ejpam-6125	97	15	,	,	PUNCT
ejpam-6125	97	16	if	if	SCONJ
ejpam-6125	97	17	n	n	PRON
ejpam-6125	97	18	is	be	AUX
ejpam-6125	97	19	even	even	ADV
ejpam-6125	97	20	proof	proof	ADJ
ejpam-6125	97	21	.	.	PUNCT
ejpam-6125	98	1	suppose	suppose	VERB
ejpam-6125	98	2	that	that	SCONJ
ejpam-6125	98	3	n	n	PRON
ejpam-6125	98	4	is	be	AUX
ejpam-6125	98	5	odd	odd	ADJ
ejpam-6125	98	6	.	.	PUNCT
ejpam-6125	99	1	clearly	clearly	ADV
ejpam-6125	99	2	,	,	PUNCT
ejpam-6125	99	3	for	for	ADP
ejpam-6125	99	4	n	n	NOUN
ejpam-6125	99	5	=	=	SYM
ejpam-6125	99	6	3	3	NUM
ejpam-6125	99	7	,	,	PUNCT
ejpam-6125	99	8	θ(cn	θ(cn	NOUN
ejpam-6125	99	9	)	)	PUNCT
ejpam-6125	99	10	=	=	SYM
ejpam-6125	100	1	1	1	X
ejpam-6125	100	2	.	.	PUNCT
ejpam-6125	100	3	now	now	ADV
ejpam-6125	100	4	for	for	ADP
ejpam-6125	100	5	n	n	X
ejpam-6125	100	6	≥	≥	NUM
ejpam-6125	100	7	5	5	NUM
ejpam-6125	100	8	,	,	PUNCT
ejpam-6125	100	9	let	let	VERB
ejpam-6125	100	10	v	v	X
ejpam-6125	100	11	(	(	PUNCT
ejpam-6125	100	12	cn	cn	PROPN
ejpam-6125	100	13	)	)	PUNCT
ejpam-6125	100	14	=	=	SYM
ejpam-6125	100	15	{	{	PUNCT
ejpam-6125	100	16	c1	c1	PROPN
ejpam-6125	100	17	,	,	PUNCT
ejpam-6125	100	18	c2	c2	PROPN
ejpam-6125	100	19	,	,	PUNCT
ejpam-6125	100	20	...	...	PUNCT
ejpam-6125	100	21	,	,	PUNCT
ejpam-6125	100	22	cn	cn	ADJ
ejpam-6125	100	23	}	}	PUNCT
ejpam-6125	100	24	and	and	CCONJ
ejpam-6125	100	25	consider	consider	VERB
ejpam-6125	100	26	s	s	PRON
ejpam-6125	100	27	=	=	PUNCT
ejpam-6125	100	28	(	(	PUNCT
ejpam-6125	100	29	c1	c1	PROPN
ejpam-6125	100	30	,	,	PUNCT
ejpam-6125	100	31	c3	c3	PROPN
ejpam-6125	100	32	,	,	PUNCT
ejpam-6125	100	33	...	...	PUNCT
ejpam-6125	100	34	,	,	PUNCT
ejpam-6125	100	35	cn−2	cn−2	PROPN
ejpam-6125	100	36	)	)	PUNCT
ejpam-6125	100	37	.	.	PUNCT
ejpam-6125	101	1	then	then	ADV
ejpam-6125	101	2	ŝ	ŝ	PROPN
ejpam-6125	101	3	is	be	AUX
ejpam-6125	101	4	a	a	DET
ejpam-6125	101	5	maximum	maximum	ADJ
ejpam-6125	101	6	independent	independent	ADJ
ejpam-6125	101	7	set	set	NOUN
ejpam-6125	101	8	of	of	ADP
ejpam-6125	101	9	cn	cn	PROPN
ejpam-6125	101	10	.	.	PROPN
ejpam-6125	101	11	notice	notice	VERB
ejpam-6125	101	12	that	that	SCONJ
ejpam-6125	101	13	ci+2	ci+2	PROPN
ejpam-6125	101	14	∈	∈	PROPN
ejpam-6125	101	15	n2	n2	NOUN
ejpam-6125	101	16	g[ci]\	g[ci]\	NOUN
ejpam-6125	101	17	⋃	⋃	PROPN
ejpam-6125	101	18	j	j	NOUN
ejpam-6125	101	19	<	<	X
ejpam-6125	101	20	in	in	ADP
ejpam-6125	101	21	2	2	NUM
ejpam-6125	101	22	g[cj	g[cj	NOUN
ejpam-6125	101	23	]	]	PUNCT
ejpam-6125	101	24	for	for	ADP
ejpam-6125	101	25	each	each	DET
ejpam-6125	101	26	i	i	PRON
ejpam-6125	101	27	∈	∈	PROPN
ejpam-6125	101	28	{	{	PUNCT
ejpam-6125	101	29	3	3	NUM
ejpam-6125	101	30	,	,	PUNCT
ejpam-6125	101	31	5	5	NUM
ejpam-6125	101	32	,	,	PUNCT
ejpam-6125	101	33	...	...	PUNCT
ejpam-6125	101	34	,	,	PUNCT
ejpam-6125	101	35	n−2	n−2	PROPN
ejpam-6125	101	36	}	}	PUNCT
ejpam-6125	101	37	.	.	PUNCT
ejpam-6125	102	1	this	this	PRON
ejpam-6125	102	2	means	mean	VERB
ejpam-6125	102	3	that	that	SCONJ
ejpam-6125	102	4	s	s	VERB
ejpam-6125	102	5	is	be	AUX
ejpam-6125	102	6	a	a	DET
ejpam-6125	102	7	legal	legal	ADJ
ejpam-6125	102	8	closed	close	VERB
ejpam-6125	102	9	hop	hop	NOUN
ejpam-6125	102	10	neighborhood	neighborhood	NOUN
ejpam-6125	102	11	sequence	sequence	NOUN
ejpam-6125	102	12	of	of	ADP
ejpam-6125	102	13	cn	cn	PROPN
ejpam-6125	102	14	.	.	PUNCT
ejpam-6125	103	1	thus	thus	ADV
ejpam-6125	103	2	,	,	PUNCT
ejpam-6125	103	3	s	s	VERB
ejpam-6125	103	4	is	be	AUX
ejpam-6125	103	5	a	a	DET
ejpam-6125	103	6	maximum	maximum	ADJ
ejpam-6125	103	7	lchni	lchni	ADJ
ejpam-6125	103	8	sequence	sequence	NOUN
ejpam-6125	103	9	of	of	ADP
ejpam-6125	103	10	cn	cn	PROPN
ejpam-6125	103	11	,	,	PUNCT
ejpam-6125	103	12	and	and	CCONJ
ejpam-6125	103	13	so	so	ADV
ejpam-6125	103	14	θ(cn	θ(cn	NOUN
ejpam-6125	103	15	)	)	PUNCT
ejpam-6125	103	16	=	=	SYM
ejpam-6125	103	17	α(cn	α(cn	NOUN
ejpam-6125	103	18	)	)	PUNCT
ejpam-6125	103	19	for	for	ADP
ejpam-6125	103	20	all	all	DET
ejpam-6125	103	21	odd	odd	ADJ
ejpam-6125	103	22	numbers	number	NOUN
ejpam-6125	103	23	n	n	PRON
ejpam-6125	103	24	≥	≥	NOUN
ejpam-6125	103	25	3	3	NUM
ejpam-6125	103	26	.	.	PUNCT
ejpam-6125	104	1	next	next	ADV
ejpam-6125	104	2	,	,	PUNCT
ejpam-6125	104	3	suppose	suppose	VERB
ejpam-6125	104	4	that	that	SCONJ
ejpam-6125	104	5	n	n	PRON
ejpam-6125	104	6	is	be	AUX
ejpam-6125	104	7	even	even	ADV
ejpam-6125	104	8	.	.	PUNCT
ejpam-6125	105	1	for	for	ADP
ejpam-6125	105	2	n	n	NOUN
ejpam-6125	105	3	=	=	SYM
ejpam-6125	105	4	4	4	NUM
ejpam-6125	105	5	,	,	PUNCT
ejpam-6125	105	6	let	let	VERB
ejpam-6125	105	7	v	v	X
ejpam-6125	105	8	(	(	PUNCT
ejpam-6125	105	9	c4	c4	NOUN
ejpam-6125	105	10	)	)	PUNCT
ejpam-6125	105	11	=	=	SYM
ejpam-6125	105	12	{	{	PUNCT
ejpam-6125	105	13	a1	a1	PROPN
ejpam-6125	105	14	,	,	PUNCT
ejpam-6125	105	15	a2	a2	PROPN
ejpam-6125	105	16	,	,	PUNCT
ejpam-6125	105	17	a3	a3	NOUN
ejpam-6125	105	18	,	,	PUNCT
ejpam-6125	105	19	a4	a4	PROPN
ejpam-6125	105	20	}	}	PUNCT
ejpam-6125	105	21	.	.	PUNCT
ejpam-6125	106	1	then	then	ADV
ejpam-6125	106	2	n2	n2	PROPN
ejpam-6125	106	3	c4	c4	NOUN
ejpam-6125	106	4	[	[	X
ejpam-6125	106	5	a1	a1	NOUN
ejpam-6125	106	6	]	]	X
ejpam-6125	106	7	=	=	SYM
ejpam-6125	106	8	n2	n2	PROPN
ejpam-6125	106	9	c4	c4	NOUN
ejpam-6125	106	10	[	[	X
ejpam-6125	106	11	a3	a3	NOUN
ejpam-6125	106	12	]	]	PUNCT
ejpam-6125	106	13	and	and	CCONJ
ejpam-6125	106	14	n2	n2	ADJ
ejpam-6125	106	15	c4	c4	NOUN
ejpam-6125	106	16	[	[	X
ejpam-6125	106	17	a2	a2	X
ejpam-6125	106	18	]	]	X
ejpam-6125	106	19	=	=	SYM
ejpam-6125	106	20	n2	n2	PROPN
ejpam-6125	106	21	c4	c4	NOUN
ejpam-6125	106	22	[	[	X
ejpam-6125	106	23	a4	a4	NOUN
ejpam-6125	106	24	]	]	PUNCT
ejpam-6125	106	25	,	,	PUNCT
ejpam-6125	106	26	where	where	SCONJ
ejpam-6125	106	27	dc4(a1	dc4(a1	NOUN
ejpam-6125	106	28	,	,	PUNCT
ejpam-6125	106	29	a3	a3	NOUN
ejpam-6125	106	30	)	)	PUNCT
ejpam-6125	106	31	=	=	SYM
ejpam-6125	106	32	2	2	NUM
ejpam-6125	106	33	=	=	SYM
ejpam-6125	106	34	dc4(a2	dc4(a2	NOUN
ejpam-6125	106	35	,	,	PUNCT
ejpam-6125	106	36	a4	a4	NOUN
ejpam-6125	106	37	)	)	PUNCT
ejpam-6125	106	38	.	.	PUNCT
ejpam-6125	107	1	it	it	PRON
ejpam-6125	107	2	follows	follow	VERB
ejpam-6125	107	3	that	that	DET
ejpam-6125	107	4	θ(c4	θ(c4	NOUN
ejpam-6125	107	5	)	)	PUNCT
ejpam-6125	107	6	=	=	SYM
ejpam-6125	108	1	1	1	X
ejpam-6125	108	2	.	.	X
ejpam-6125	108	3	for	for	ADP
ejpam-6125	108	4	n	n	NOUN
ejpam-6125	108	5	=	=	SYM
ejpam-6125	108	6	6	6	NUM
ejpam-6125	108	7	,	,	PUNCT
ejpam-6125	108	8	let	let	VERB
ejpam-6125	108	9	v	v	X
ejpam-6125	108	10	(	(	PUNCT
ejpam-6125	108	11	c6	c6	PROPN
ejpam-6125	108	12	)	)	PUNCT
ejpam-6125	108	13	=	=	PRON
ejpam-6125	108	14	{	{	PUNCT
ejpam-6125	108	15	b1	b1	NOUN
ejpam-6125	108	16	,	,	PUNCT
ejpam-6125	108	17	b2	b2	NOUN
ejpam-6125	108	18	,	,	PUNCT
ejpam-6125	108	19	...	...	PUNCT
ejpam-6125	108	20	,	,	PUNCT
ejpam-6125	108	21	b6}.notice	b6}.notice	VERB
ejpam-6125	108	22	that	that	DET
ejpam-6125	108	23	n2	n2	PROPN
ejpam-6125	108	24	c6	c6	PROPN
ejpam-6125	108	25	[	[	X
ejpam-6125	108	26	b1	b1	X
ejpam-6125	108	27	]	]	X
ejpam-6125	108	28	=	=	SYM
ejpam-6125	108	29	n2	n2	PROPN
ejpam-6125	108	30	c6	c6	PROPN
ejpam-6125	109	1	[	[	X
ejpam-6125	109	2	b3	b3	PROPN
ejpam-6125	109	3	]	]	PUNCT
ejpam-6125	109	4	=	=	SYM
ejpam-6125	109	5	n2	n2	PROPN
ejpam-6125	109	6	c6	c6	PROPN
ejpam-6125	110	1	[	[	X
ejpam-6125	110	2	b4	b4	X
ejpam-6125	110	3	]	]	PUNCT
ejpam-6125	110	4	and	and	CCONJ
ejpam-6125	110	5	n2	n2	PROPN
ejpam-6125	110	6	c6	c6	PROPN
ejpam-6125	111	1	[	[	X
ejpam-6125	111	2	b2	b2	X
ejpam-6125	111	3	]	]	X
ejpam-6125	111	4	=	=	SYM
ejpam-6125	111	5	n2	n2	PROPN
ejpam-6125	111	6	c6	c6	PROPN
ejpam-6125	111	7	[	[	X
ejpam-6125	111	8	b4	b4	X
ejpam-6125	111	9	]	]	PUNCT
ejpam-6125	111	10	=	=	SYM
ejpam-6125	111	11	n2	n2	PROPN
ejpam-6125	111	12	c6	c6	PROPN
ejpam-6125	112	1	[	[	X
ejpam-6125	112	2	b6	b6	NOUN
ejpam-6125	112	3	]	]	PUNCT
ejpam-6125	112	4	,	,	PUNCT
ejpam-6125	112	5	where	where	SCONJ
ejpam-6125	112	6	dc6(b1	dc6(b1	NOUN
ejpam-6125	112	7	,	,	PUNCT
ejpam-6125	112	8	b3	b3	NOUN
ejpam-6125	112	9	)	)	PUNCT
ejpam-6125	112	10	=	=	PUNCT
ejpam-6125	112	11	dc6(b3	dc6(b3	NOUN
ejpam-6125	112	12	,	,	PUNCT
ejpam-6125	112	13	b5	b5	PROPN
ejpam-6125	112	14	)	)	PUNCT
ejpam-6125	113	1	=	=	PUNCT
ejpam-6125	113	2	dc6(b1	dc6(b1	PROPN
ejpam-6125	113	3	,	,	PUNCT
ejpam-6125	113	4	b5	b5	PROPN
ejpam-6125	113	5	)	)	PUNCT
ejpam-6125	113	6	=	=	SYM
ejpam-6125	113	7	2	2	NUM
ejpam-6125	113	8	and	and	CCONJ
ejpam-6125	113	9	dc6(b2	dc6(b2	NOUN
ejpam-6125	113	10	,	,	PUNCT
ejpam-6125	113	11	b4	b4	NOUN
ejpam-6125	113	12	)	)	PUNCT
ejpam-6125	113	13	=	=	SYM
ejpam-6125	113	14	dc6(b4	dc6(b4	NOUN
ejpam-6125	113	15	,	,	PUNCT
ejpam-6125	113	16	b6	b6	NOUN
ejpam-6125	113	17	)	)	PUNCT
ejpam-6125	113	18	=	=	SYM
ejpam-6125	113	19	dc6(b2	dc6(b2	NOUN
ejpam-6125	113	20	,	,	PUNCT
ejpam-6125	113	21	b6	b6	NOUN
ejpam-6125	113	22	)	)	PUNCT
ejpam-6125	113	23	=	=	SYM
ejpam-6125	114	1	2	2	X
ejpam-6125	114	2	.	.	PUNCT
ejpam-6125	114	3	this	this	PRON
ejpam-6125	114	4	means	mean	VERB
ejpam-6125	114	5	that	that	SCONJ
ejpam-6125	114	6	only	only	ADV
ejpam-6125	114	7	one	one	NUM
ejpam-6125	114	8	element	element	NOUN
ejpam-6125	114	9	in	in	ADP
ejpam-6125	114	10	{	{	PUNCT
ejpam-6125	114	11	b1	b1	NOUN
ejpam-6125	114	12	,	,	PUNCT
ejpam-6125	114	13	b3	b3	PROPN
ejpam-6125	114	14	,	,	PUNCT
ejpam-6125	114	15	b5	b5	PROPN
ejpam-6125	114	16	}	}	PUNCT
ejpam-6125	114	17	could	could	AUX
ejpam-6125	114	18	be	be	AUX
ejpam-6125	114	19	in	in	ADP
ejpam-6125	114	20	any	any	DET
ejpam-6125	114	21	legal	legal	ADJ
ejpam-6125	114	22	closed	close	VERB
ejpam-6125	114	23	hop	hop	NOUN
ejpam-6125	114	24	neighborhood	neighborhood	NOUN
ejpam-6125	114	25	independent	independent	ADJ
ejpam-6125	114	26	sequence	sequence	NOUN
ejpam-6125	114	27	of	of	ADP
ejpam-6125	114	28	c6	c6	PROPN
ejpam-6125	114	29	.	.	PUNCT
ejpam-6125	115	1	similarly	similarly	ADV
ejpam-6125	115	2	,	,	PUNCT
ejpam-6125	115	3	for	for	ADP
ejpam-6125	115	4	the	the	DET
ejpam-6125	115	5	set	set	NOUN
ejpam-6125	115	6	{	{	PUNCT
ejpam-6125	115	7	b2	b2	NOUN
ejpam-6125	115	8	,	,	PUNCT
ejpam-6125	115	9	b4	b4	NOUN
ejpam-6125	115	10	,	,	PUNCT
ejpam-6125	115	11	b6	b6	NOUN
ejpam-6125	115	12	}	}	PUNCT
ejpam-6125	115	13	.	.	PUNCT
ejpam-6125	116	1	since	since	SCONJ
ejpam-6125	116	2	dc6(b1	dc6(b1	NOUN
ejpam-6125	116	3	,	,	PUNCT
ejpam-6125	116	4	b1	b1	NOUN
ejpam-6125	116	5	)	)	PUNCT
ejpam-6125	116	6	=	=	SYM
ejpam-6125	116	7	1	1	NUM
ejpam-6125	116	8	=	=	SYM
ejpam-6125	116	9	dc6(b1	dc6(b1	PROPN
ejpam-6125	116	10	,	,	PUNCT
ejpam-6125	116	11	b6	b6	NOUN
ejpam-6125	116	12	)	)	PUNCT
ejpam-6125	116	13	,	,	PUNCT
ejpam-6125	116	14	dc6(b1	dc6(b1	NOUN
ejpam-6125	116	15	,	,	PUNCT
ejpam-6125	116	16	b4	b4	NOUN
ejpam-6125	116	17	)	)	PUNCT
ejpam-6125	116	18	=	=	SYM
ejpam-6125	116	19	3	3	NUM
ejpam-6125	116	20	,	,	PUNCT
ejpam-6125	116	21	and	and	CCONJ
ejpam-6125	116	22	n2	n2	PROPN
ejpam-6125	116	23	c6	c6	PROPN
ejpam-6125	116	24	[	[	X
ejpam-6125	116	25	b4	b4	X
ejpam-6125	116	26	]	]	PUNCT
ejpam-6125	116	27	\	\	PROPN
ejpam-6125	116	28	n2	n2	PROPN
ejpam-6125	116	29	c6	c6	PROPN
ejpam-6125	117	1	[	[	X
ejpam-6125	117	2	b1	b1	X
ejpam-6125	117	3	]	]	X
ejpam-6125	117	4	=	=	SYM
ejpam-6125	117	5	{	{	PUNCT
ejpam-6125	117	6	b2	b2	NOUN
ejpam-6125	117	7	,	,	PUNCT
ejpam-6125	117	8	b4	b4	NOUN
ejpam-6125	117	9	,	,	PUNCT
ejpam-6125	117	10	b6	b6	NOUN
ejpam-6125	117	11	}	}	PUNCT
ejpam-6125	117	12	,	,	PUNCT
ejpam-6125	117	13	it	it	PRON
ejpam-6125	117	14	follows	follow	VERB
ejpam-6125	117	15	that	that	PRON
ejpam-6125	117	16	t	t	NOUN
ejpam-6125	117	17	=	=	PUNCT
ejpam-6125	117	18	(	(	PUNCT
ejpam-6125	117	19	b1	b1	NOUN
ejpam-6125	117	20	,	,	PUNCT
ejpam-6125	117	21	b4	b4	NOUN
ejpam-6125	117	22	)	)	PUNCT
ejpam-6125	117	23	is	be	AUX
ejpam-6125	117	24	a	a	DET
ejpam-6125	117	25	maximum	maximum	ADJ
ejpam-6125	117	26	lchni	lchni	ADJ
ejpam-6125	117	27	sequence	sequence	NOUN
ejpam-6125	117	28	of	of	ADP
ejpam-6125	117	29	c6	c6	PROPN
ejpam-6125	117	30	,	,	PUNCT
ejpam-6125	117	31	and	and	CCONJ
ejpam-6125	117	32	so	so	ADV
ejpam-6125	117	33	θ(c6	θ(c6	PROPN
ejpam-6125	117	34	)	)	PUNCT
ejpam-6125	117	35	=	=	SYM
ejpam-6125	118	1	2	2	X
ejpam-6125	118	2	.	.	PUNCT
ejpam-6125	118	3	now	now	ADV
ejpam-6125	118	4	for	for	ADP
ejpam-6125	118	5	n	n	PROPN
ejpam-6125	118	6	≥	≥	NOUN
ejpam-6125	118	7	8	8	NUM
ejpam-6125	118	8	,	,	PUNCT
ejpam-6125	118	9	let	let	VERB
ejpam-6125	118	10	v	v	X
ejpam-6125	118	11	(	(	PUNCT
ejpam-6125	118	12	cn	cn	PROPN
ejpam-6125	118	13	)	)	PUNCT
ejpam-6125	118	14	=	=	PRON
ejpam-6125	118	15	{	{	PUNCT
ejpam-6125	118	16	x1	x1	PROPN
ejpam-6125	118	17	,	,	PUNCT
ejpam-6125	118	18	x2	x2	PROPN
ejpam-6125	118	19	,	,	PUNCT
ejpam-6125	118	20	...	...	PUNCT
ejpam-6125	118	21	,	,	PUNCT
ejpam-6125	118	22	xn	xn	PROPN
ejpam-6125	118	23	}	}	PUNCT
ejpam-6125	118	24	and	and	CCONJ
ejpam-6125	118	25	consider	consider	VERB
ejpam-6125	118	26	w	w	NOUN
ejpam-6125	118	27	=	=	PUNCT
ejpam-6125	118	28	{	{	PUNCT
ejpam-6125	118	29	x1	x1	PROPN
ejpam-6125	118	30	,	,	PUNCT
ejpam-6125	118	31	x3	x3	ADJ
ejpam-6125	118	32	,	,	PUNCT
ejpam-6125	118	33	...	...	PUNCT
ejpam-6125	118	34	,	,	PUNCT
ejpam-6125	118	35	xn−5	xn−5	PROPN
ejpam-6125	118	36	,	,	PUNCT
ejpam-6125	118	37	xn−2	xn−2	PROPN
ejpam-6125	118	38	}	}	PUNCT
ejpam-6125	118	39	.	.	PUNCT
ejpam-6125	119	1	then	then	ADV
ejpam-6125	119	2	ŵ	ŵ	PROPN
ejpam-6125	119	3	is	be	AUX
ejpam-6125	119	4	an	an	DET
ejpam-6125	119	5	independent	independent	ADJ
ejpam-6125	119	6	set	set	NOUN
ejpam-6125	119	7	of	of	ADP
ejpam-6125	119	8	cn	cn	PROPN
ejpam-6125	119	9	with	with	ADP
ejpam-6125	119	10	|ŵ	|ŵ	NOUN
ejpam-6125	119	11	|	|	ADV
ejpam-6125	119	12	=	=	SYM
ejpam-6125	119	13	α(cn	α(cn	NOUN
ejpam-6125	119	14	)	)	PUNCT
ejpam-6125	119	15	−	−	NUM
ejpam-6125	120	1	1	1	X
ejpam-6125	120	2	.	.	X
ejpam-6125	120	3	notice	notice	VERB
ejpam-6125	120	4	that	that	SCONJ
ejpam-6125	120	5	xi+2	xi+2	PROPN
ejpam-6125	120	6	∈	∈	PROPN
ejpam-6125	120	7	j.	j.	PROPN
ejpam-6125	120	8	a.	a.	PROPN
ejpam-6125	120	9	hassan	hassan	PROPN
ejpam-6125	120	10	et	et	PROPN
ejpam-6125	120	11	al	al	PROPN
ejpam-6125	120	12	.	.	PUNCT
ejpam-6125	120	13	/	/	SYM
ejpam-6125	120	14	eur	eur	PROPN
ejpam-6125	120	15	.	.	PUNCT
ejpam-6125	121	1	j.	j.	PROPN
ejpam-6125	121	2	pure	pure	PROPN
ejpam-6125	121	3	appl	appl	PROPN
ejpam-6125	121	4	.	.	PROPN
ejpam-6125	121	5	math	math	PROPN
ejpam-6125	121	6	,	,	PUNCT
ejpam-6125	121	7	18	18	NUM
ejpam-6125	121	8	(	(	PUNCT
ejpam-6125	121	9	3	3	NUM
ejpam-6125	121	10	)	)	PUNCT
ejpam-6125	121	11	(	(	PUNCT
ejpam-6125	121	12	2025	2025	NUM
ejpam-6125	121	13	)	)	PUNCT
ejpam-6125	121	14	,	,	PUNCT
ejpam-6125	121	15	6125	6125	NUM
ejpam-6125	121	16	5	5	NUM
ejpam-6125	121	17	of	of	ADP
ejpam-6125	121	18	8	8	NUM
ejpam-6125	121	19	n2	n2	NOUN
ejpam-6125	121	20	cn	cn	PROPN
ejpam-6125	122	1	[	[	X
ejpam-6125	122	2	xi	xi	X
ejpam-6125	122	3	]	]	PUNCT
ejpam-6125	122	4	\	\	PUNCT
ejpam-6125	122	5	⋃	⋃	PUNCT
ejpam-6125	122	6	j	j	PROPN
ejpam-6125	122	7	<	<	X
ejpam-6125	122	8	in	in	ADP
ejpam-6125	122	9	2	2	NUM
ejpam-6125	122	10	cn	cn	NOUN
ejpam-6125	122	11	[	[	X
ejpam-6125	122	12	xj	xj	X
ejpam-6125	122	13	]	]	PUNCT
ejpam-6125	122	14	for	for	ADP
ejpam-6125	122	15	all	all	PRON
ejpam-6125	122	16	i	i	PRON
ejpam-6125	122	17	∈	∈	PROPN
ejpam-6125	122	18	{	{	PUNCT
ejpam-6125	122	19	3	3	NUM
ejpam-6125	122	20	,	,	PUNCT
ejpam-6125	122	21	5	5	NUM
ejpam-6125	122	22	,	,	PUNCT
ejpam-6125	122	23	...	...	PUNCT
ejpam-6125	122	24	,	,	PUNCT
ejpam-6125	122	25	n	n	CCONJ
ejpam-6125	122	26	−	−	PROPN
ejpam-6125	122	27	5	5	NUM
ejpam-6125	122	28	}	}	PUNCT
ejpam-6125	122	29	and	and	CCONJ
ejpam-6125	122	30	xn	xn	PROPN
ejpam-6125	122	31	,	,	PUNCT
ejpam-6125	122	32	xn−2	xn−2	PROPN
ejpam-6125	122	33	,	,	PUNCT
ejpam-6125	122	34	xn−4	xn−4	PROPN
ejpam-6125	122	35	∈	∈	PROPN
ejpam-6125	122	36	n2	n2	PROPN
ejpam-6125	122	37	cn	cn	PROPN
ejpam-6125	123	1	[	[	X
ejpam-6125	123	2	xn−2	xn−2	PROPN
ejpam-6125	123	3	]	]	X
ejpam-6125	123	4	\⋃	\⋃	NOUN
ejpam-6125	123	5	j∈{1,3,	j∈{1,3,	PROPN
ejpam-6125	123	6	...	...	PUNCT
ejpam-6125	123	7	,n−5}n	,n−5}n	PUNCT
ejpam-6125	123	8	2	2	NUM
ejpam-6125	123	9	cn	cn	NOUN
ejpam-6125	123	10	[	[	X
ejpam-6125	123	11	xj	xj	X
ejpam-6125	123	12	]	]	PUNCT
ejpam-6125	123	13	.	.	PUNCT
ejpam-6125	124	1	thus	thus	ADV
ejpam-6125	124	2	,	,	PUNCT
ejpam-6125	124	3	w	w	PROPN
ejpam-6125	124	4	is	be	AUX
ejpam-6125	124	5	a	a	DET
ejpam-6125	124	6	lchni	lchni	ADJ
ejpam-6125	124	7	sequence	sequence	NOUN
ejpam-6125	124	8	of	of	ADP
ejpam-6125	124	9	cn	cn	PROPN
ejpam-6125	124	10	.	.	PUNCT
ejpam-6125	125	1	since	since	SCONJ
ejpam-6125	125	2	n2	n2	PROPN
ejpam-6125	125	3	cn	cn	PROPN
ejpam-6125	126	1	[	[	X
ejpam-6125	126	2	xn−3	xn−3	PROPN
ejpam-6125	126	3	]	]	X
ejpam-6125	126	4	⊆	⊆	NUM
ejpam-6125	126	5	n2	n2	NOUN
ejpam-6125	126	6	cn	cn	PROPN
ejpam-6125	127	1	[	[	X
ejpam-6125	127	2	xn−5	xn−5	PROPN
ejpam-6125	127	3	]	]	PUNCT
ejpam-6125	127	4	⋃	⋃	PROPN
ejpam-6125	127	5	n2	n2	NOUN
ejpam-6125	127	6	cn	cn	PROPN
ejpam-6125	128	1	[	[	X
ejpam-6125	128	2	xi	xi	X
ejpam-6125	128	3	]	]	X
ejpam-6125	128	4	and	and	CCONJ
ejpam-6125	128	5	n2	n2	PROPN
ejpam-6125	128	6	cn	cn	PROPN
ejpam-6125	129	1	[	[	X
ejpam-6125	129	2	xn−1	xn−1	X
ejpam-6125	129	3	]	]	X
ejpam-6125	129	4	⊆	⊆	NUM
ejpam-6125	129	5	n2	n2	NOUN
ejpam-6125	129	6	cn	cn	PROPN
ejpam-6125	130	1	[	[	X
ejpam-6125	130	2	xn−5	xn−5	PROPN
ejpam-6125	130	3	]	]	PUNCT
ejpam-6125	130	4	⋃	⋃	PROPN
ejpam-6125	130	5	n2	n2	NOUN
ejpam-6125	130	6	cn	cn	PROPN
ejpam-6125	131	1	[	[	X
ejpam-6125	131	2	xi	xi	X
ejpam-6125	131	3	]	]	PUNCT
ejpam-6125	131	4	,	,	PUNCT
ejpam-6125	131	5	it	it	PRON
ejpam-6125	131	6	follows	follow	VERB
ejpam-6125	131	7	that	that	SCONJ
ejpam-6125	131	8	w	w	NOUN
ejpam-6125	131	9	is	be	AUX
ejpam-6125	131	10	a	a	DET
ejpam-6125	131	11	maximum	maximum	ADJ
ejpam-6125	131	12	lchni	lchni	ADJ
ejpam-6125	131	13	sequence	sequence	NOUN
ejpam-6125	131	14	of	of	ADP
ejpam-6125	131	15	cn	cn	PROPN
ejpam-6125	131	16	.	.	PUNCT
ejpam-6125	132	1	since	since	SCONJ
ejpam-6125	132	2	m	m	PROPN
ejpam-6125	132	3	=	=	SYM
ejpam-6125	132	4	{	{	PUNCT
ejpam-6125	132	5	x1	x1	PROPN
ejpam-6125	132	6	,	,	PUNCT
ejpam-6125	132	7	x3	x3	ADJ
ejpam-6125	132	8	,	,	PUNCT
ejpam-6125	132	9	.	.	PUNCT
ejpam-6125	132	10	.	.	PUNCT
ejpam-6125	132	11	.	.	PUNCT
ejpam-6125	133	1	,	,	PUNCT
ejpam-6125	133	2	xn−1	xn−1	PROPN
ejpam-6125	133	3	}	}	PUNCT
ejpam-6125	133	4	is	be	AUX
ejpam-6125	133	5	a	a	DET
ejpam-6125	133	6	maximum	maximum	ADJ
ejpam-6125	133	7	independent	independent	ADJ
ejpam-6125	133	8	set	set	NOUN
ejpam-6125	133	9	of	of	ADP
ejpam-6125	133	10	cn	cn	PROPN
ejpam-6125	133	11	for	for	ADP
ejpam-6125	133	12	all	all	PRON
ejpam-6125	133	13	positive	positive	ADJ
ejpam-6125	133	14	even	even	ADV
ejpam-6125	133	15	integers	integer	NOUN
ejpam-6125	133	16	n	n	PRON
ejpam-6125	133	17	≥	≥	NOUN
ejpam-6125	133	18	4	4	NUM
ejpam-6125	133	19	.	.	PUNCT
ejpam-6125	134	1	it	it	PRON
ejpam-6125	134	2	follows	follow	VERB
ejpam-6125	134	3	that	that	SCONJ
ejpam-6125	134	4	θ(cn	θ(cn	NOUN
ejpam-6125	134	5	)	)	PUNCT
ejpam-6125	134	6	=	=	SYM
ejpam-6125	134	7	α(cn)−	α(cn)−	NOUN
ejpam-6125	134	8	1	1	NUM
ejpam-6125	134	9	for	for	ADP
ejpam-6125	134	10	all	all	DET
ejpam-6125	134	11	positive	positive	ADJ
ejpam-6125	134	12	even	even	ADJ
ejpam-6125	134	13	numbers	number	NOUN
ejpam-6125	134	14	n	n	PRON
ejpam-6125	134	15	≥	≥	NUM
ejpam-6125	134	16	4	4	NUM
ejpam-6125	134	17	.	.	PUNCT
ejpam-6125	135	1	the	the	DET
ejpam-6125	135	2	following	follow	VERB
ejpam-6125	135	3	concept	concept	NOUN
ejpam-6125	135	4	will	will	AUX
ejpam-6125	135	5	be	be	AUX
ejpam-6125	135	6	used	use	VERB
ejpam-6125	135	7	to	to	PART
ejpam-6125	135	8	study	study	VERB
ejpam-6125	135	9	the	the	DET
ejpam-6125	135	10	behavior	behavior	NOUN
ejpam-6125	135	11	of	of	ADP
ejpam-6125	135	12	lchni	lchni	PROPN
ejpam-6125	135	13	sequences	sequence	NOUN
ejpam-6125	135	14	in	in	ADP
ejpam-6125	135	15	the	the	DET
ejpam-6125	135	16	join	join	NOUN
ejpam-6125	135	17	of	of	ADP
ejpam-6125	135	18	any	any	DET
ejpam-6125	135	19	two	two	NUM
ejpam-6125	135	20	graphs	graph	NOUN
ejpam-6125	135	21	.	.	PUNCT
ejpam-6125	136	1	definition	definition	NOUN
ejpam-6125	136	2	1	1	NUM
ejpam-6125	136	3	.	.	PUNCT
ejpam-6125	137	1	let	let	VERB
ejpam-6125	137	2	g	g	PRON
ejpam-6125	137	3	be	be	AUX
ejpam-6125	137	4	a	a	DET
ejpam-6125	137	5	graph	graph	NOUN
ejpam-6125	137	6	.	.	PUNCT
ejpam-6125	138	1	then	then	ADV
ejpam-6125	138	2	a	a	DET
ejpam-6125	138	3	sequence	sequence	NOUN
ejpam-6125	138	4	p	p	NOUN
ejpam-6125	138	5	of	of	ADP
ejpam-6125	138	6	distinct	distinct	ADJ
ejpam-6125	138	7	vertices	vertex	NOUN
ejpam-6125	138	8	of	of	ADP
ejpam-6125	138	9	g	g	PROPN
ejpam-6125	138	10	is	be	AUX
ejpam-6125	138	11	called	call	VERB
ejpam-6125	138	12	a	a	DET
ejpam-6125	138	13	co	co	ADJ
ejpam-6125	138	14	-	-	ADJ
ejpam-6125	138	15	legal	legal	ADJ
ejpam-6125	138	16	closed	closed	ADJ
ejpam-6125	138	17	neighborhood	neighborhood	NOUN
ejpam-6125	138	18	independent	independent	ADJ
ejpam-6125	138	19	sequence	sequence	NOUN
ejpam-6125	138	20	(	(	PUNCT
ejpam-6125	138	21	clcni	clcni	NOUN
ejpam-6125	138	22	sequence	sequence	NOUN
ejpam-6125	138	23	)	)	PUNCT
ejpam-6125	138	24	of	of	ADP
ejpam-6125	138	25	g	g	PROPN
ejpam-6125	138	26	if	if	SCONJ
ejpam-6125	138	27	p	p	NOUN
ejpam-6125	138	28	is	be	AUX
ejpam-6125	138	29	a	a	DET
ejpam-6125	138	30	legal	legal	ADJ
ejpam-6125	138	31	closed	closed	ADJ
ejpam-6125	138	32	neighborhood	neighborhood	NOUN
ejpam-6125	138	33	sequence	sequence	NOUN
ejpam-6125	138	34	in	in	ADP
ejpam-6125	138	35	ḡ	ḡ	VERB
ejpam-6125	138	36	and	and	CCONJ
ejpam-6125	138	37	p	p	NOUN
ejpam-6125	138	38	is	be	AUX
ejpam-6125	138	39	an	an	DET
ejpam-6125	138	40	independent	independent	ADJ
ejpam-6125	138	41	set	set	NOUN
ejpam-6125	138	42	of	of	ADP
ejpam-6125	138	43	g.	g.	PROPN
ejpam-6125	138	44	the	the	DET
ejpam-6125	138	45	co	co	ADJ
ejpam-6125	138	46	-	-	ADJ
ejpam-6125	138	47	legal	legal	ADJ
ejpam-6125	138	48	closed	closed	ADJ
ejpam-6125	138	49	neighborhood	neighborhood	NOUN
ejpam-6125	138	50	independence	independence	NOUN
ejpam-6125	138	51	number	number	NOUN
ejpam-6125	138	52	(	(	PUNCT
ejpam-6125	138	53	clcni	clcni	NOUN
ejpam-6125	138	54	number	number	NOUN
ejpam-6125	138	55	)	)	PUNCT
ejpam-6125	138	56	of	of	ADP
ejpam-6125	138	57	g	g	PROPN
ejpam-6125	138	58	is	be	AUX
ejpam-6125	138	59	the	the	DET
ejpam-6125	138	60	maximum	maximum	ADJ
ejpam-6125	138	61	length	length	NOUN
ejpam-6125	138	62	of	of	ADP
ejpam-6125	138	63	a	a	DET
ejpam-6125	138	64	clcni	clcni	NOUN
ejpam-6125	138	65	sequence	sequence	NOUN
ejpam-6125	138	66	of	of	ADP
ejpam-6125	138	67	g	g	NOUN
ejpam-6125	138	68	,	,	PUNCT
ejpam-6125	138	69	and	and	CCONJ
ejpam-6125	138	70	is	be	AUX
ejpam-6125	138	71	denoted	denote	VERB
ejpam-6125	138	72	by	by	ADP
ejpam-6125	138	73	αcl(g	αcl(g	PROPN
ejpam-6125	138	74	)	)	PUNCT
ejpam-6125	138	75	.	.	PUNCT
ejpam-6125	139	1	the	the	DET
ejpam-6125	139	2	following	follow	VERB
ejpam-6125	139	3	theorem	theorem	NOUN
ejpam-6125	139	4	will	will	AUX
ejpam-6125	139	5	be	be	AUX
ejpam-6125	139	6	used	use	VERB
ejpam-6125	139	7	to	to	PART
ejpam-6125	139	8	prove	prove	VERB
ejpam-6125	139	9	the	the	DET
ejpam-6125	139	10	characterization	characterization	NOUN
ejpam-6125	139	11	of	of	ADP
ejpam-6125	139	12	an	an	DET
ejpam-6125	139	13	lchni	lchni	ADJ
ejpam-6125	139	14	sequence	sequence	NOUN
ejpam-6125	139	15	on	on	ADP
ejpam-6125	139	16	the	the	DET
ejpam-6125	139	17	join	join	NOUN
ejpam-6125	139	18	of	of	ADP
ejpam-6125	139	19	two	two	NUM
ejpam-6125	139	20	graphs	graph	NOUN
ejpam-6125	139	21	.	.	PUNCT
ejpam-6125	140	1	theorem	theorem	ADJ
ejpam-6125	140	2	4	4	NUM
ejpam-6125	140	3	.	.	PUNCT
ejpam-6125	141	1	[	[	X
ejpam-6125	141	2	15	15	NUM
ejpam-6125	141	3	]	]	PUNCT
ejpam-6125	141	4	let	let	VERB
ejpam-6125	141	5	g	g	NOUN
ejpam-6125	141	6	and	and	CCONJ
ejpam-6125	141	7	h	h	NOUN
ejpam-6125	141	8	be	be	VERB
ejpam-6125	141	9	any	any	DET
ejpam-6125	141	10	two	two	NUM
ejpam-6125	141	11	graphs	graph	NOUN
ejpam-6125	141	12	.	.	PUNCT
ejpam-6125	142	1	a	a	DET
ejpam-6125	142	2	sequence	sequence	NOUN
ejpam-6125	142	3	s	s	VERB
ejpam-6125	142	4	of	of	ADP
ejpam-6125	142	5	distinct	distinct	ADJ
ejpam-6125	142	6	vertices	vertex	NOUN
ejpam-6125	142	7	of	of	ADP
ejpam-6125	142	8	g+h	g+h	PROPN
ejpam-6125	142	9	is	be	AUX
ejpam-6125	142	10	a	a	DET
ejpam-6125	142	11	legal	legal	ADJ
ejpam-6125	142	12	closed	closed	ADJ
ejpam-6125	142	13	hop	hop	NOUN
ejpam-6125	142	14	neighborhood	neighborhood	NOUN
ejpam-6125	142	15	sequence	sequence	NOUN
ejpam-6125	142	16	if	if	SCONJ
ejpam-6125	142	17	and	and	CCONJ
ejpam-6125	142	18	only	only	ADV
ejpam-6125	142	19	if	if	SCONJ
ejpam-6125	142	20	one	one	NUM
ejpam-6125	142	21	of	of	ADP
ejpam-6125	142	22	the	the	DET
ejpam-6125	142	23	following	follow	VERB
ejpam-6125	142	24	holds	hold	VERB
ejpam-6125	142	25	:	:	PUNCT
ejpam-6125	142	26	(	(	PUNCT
ejpam-6125	142	27	i	i	NOUN
ejpam-6125	142	28	)	)	PUNCT
ejpam-6125	142	29	s	s	VERB
ejpam-6125	142	30	is	be	AUX
ejpam-6125	142	31	a	a	DET
ejpam-6125	142	32	co	co	ADJ
ejpam-6125	142	33	-	-	ADJ
ejpam-6125	142	34	legal	legal	ADJ
ejpam-6125	142	35	closed	closed	ADJ
ejpam-6125	142	36	neighborhood	neighborhood	NOUN
ejpam-6125	142	37	sequence	sequence	NOUN
ejpam-6125	142	38	in	in	ADP
ejpam-6125	142	39	g	g	PROPN
ejpam-6125	142	40	(	(	PUNCT
ejpam-6125	142	41	legal	legal	ADJ
ejpam-6125	142	42	closed	close	VERB
ejpam-6125	142	43	neighborhood	neighborhood	NOUN
ejpam-6125	142	44	sequence	sequence	NOUN
ejpam-6125	142	45	in	in	ADP
ejpam-6125	142	46	g	g	NOUN
ejpam-6125	142	47	)	)	PUNCT
ejpam-6125	142	48	.	.	PUNCT
ejpam-6125	143	1	(	(	PUNCT
ejpam-6125	143	2	ii	ii	X
ejpam-6125	143	3	)	)	PUNCT
ejpam-6125	143	4	s	s	VERB
ejpam-6125	143	5	is	be	AUX
ejpam-6125	143	6	a	a	DET
ejpam-6125	143	7	co	co	ADJ
ejpam-6125	143	8	-	-	ADJ
ejpam-6125	143	9	legal	legal	ADJ
ejpam-6125	143	10	closed	closed	ADJ
ejpam-6125	143	11	neighborhood	neighborhood	NOUN
ejpam-6125	143	12	sequence	sequence	NOUN
ejpam-6125	143	13	in	in	ADP
ejpam-6125	143	14	h	h	PROPN
ejpam-6125	143	15	(	(	PUNCT
ejpam-6125	143	16	legal	legal	ADJ
ejpam-6125	143	17	closed	close	VERB
ejpam-6125	143	18	neighborhood	neighborhood	NOUN
ejpam-6125	143	19	sequence	sequence	NOUN
ejpam-6125	143	20	in	in	ADP
ejpam-6125	143	21	h	h	NOUN
ejpam-6125	143	22	)	)	PUNCT
ejpam-6125	143	23	.	.	PUNCT
ejpam-6125	144	1	(	(	PUNCT
ejpam-6125	144	2	iii	iii	X
ejpam-6125	144	3	)	)	PUNCT
ejpam-6125	144	4	s	s	AUX
ejpam-6125	144	5	is	be	AUX
ejpam-6125	144	6	a	a	DET
ejpam-6125	144	7	concatenation	concatenation	NOUN
ejpam-6125	144	8	sg	sg	ADP
ejpam-6125	144	9	⊕	⊕	PROPN
ejpam-6125	144	10	sh	sh	INTJ
ejpam-6125	144	11	,	,	PUNCT
ejpam-6125	144	12	where	where	SCONJ
ejpam-6125	144	13	sg	sg	PROPN
ejpam-6125	144	14	and	and	CCONJ
ejpam-6125	144	15	sh	sh	PROPN
ejpam-6125	144	16	are	be	AUX
ejpam-6125	144	17	co	co	ADJ
ejpam-6125	144	18	-	-	ADJ
ejpam-6125	144	19	legal	legal	ADJ
ejpam-6125	144	20	closed	closed	ADJ
ejpam-6125	144	21	neighborhood	neighborhood	NOUN
ejpam-6125	144	22	sequences	sequence	NOUN
ejpam-6125	144	23	in	in	ADP
ejpam-6125	144	24	g	g	PROPN
ejpam-6125	144	25	and	and	CCONJ
ejpam-6125	144	26	h	h	NOUN
ejpam-6125	144	27	,	,	PUNCT
ejpam-6125	144	28	respectively	respectively	ADV
ejpam-6125	144	29	.	.	PUNCT
ejpam-6125	145	1	theorem	theorem	NOUN
ejpam-6125	145	2	5	5	NUM
ejpam-6125	145	3	.	.	PUNCT
ejpam-6125	146	1	let	let	VERB
ejpam-6125	146	2	g	g	NOUN
ejpam-6125	146	3	and	and	CCONJ
ejpam-6125	146	4	h	h	NOUN
ejpam-6125	146	5	be	be	VERB
ejpam-6125	146	6	any	any	DET
ejpam-6125	146	7	two	two	NUM
ejpam-6125	146	8	graphs	graph	NOUN
ejpam-6125	146	9	.	.	PUNCT
ejpam-6125	147	1	then	then	ADV
ejpam-6125	147	2	a	a	DET
ejpam-6125	147	3	sequence	sequence	NOUN
ejpam-6125	147	4	f	f	NOUN
ejpam-6125	147	5	of	of	ADP
ejpam-6125	147	6	distinct	distinct	ADJ
ejpam-6125	147	7	vertices	vertex	NOUN
ejpam-6125	147	8	of	of	ADP
ejpam-6125	147	9	g	g	PROPN
ejpam-6125	147	10	+	+	NOUN
ejpam-6125	147	11	h	h	NOUN
ejpam-6125	147	12	is	be	AUX
ejpam-6125	147	13	an	an	DET
ejpam-6125	147	14	lchni	lchni	ADJ
ejpam-6125	147	15	sequence	sequence	NOUN
ejpam-6125	147	16	of	of	ADP
ejpam-6125	147	17	g	g	PROPN
ejpam-6125	148	1	+	+	PROPN
ejpam-6125	148	2	h	h	NOUN
ejpam-6125	148	3	if	if	SCONJ
ejpam-6125	148	4	and	and	CCONJ
ejpam-6125	148	5	only	only	ADV
ejpam-6125	148	6	if	if	SCONJ
ejpam-6125	148	7	f	f	PROPN
ejpam-6125	148	8	satisfies	satisfy	VERB
ejpam-6125	148	9	any	any	PRON
ejpam-6125	148	10	of	of	ADP
ejpam-6125	148	11	the	the	DET
ejpam-6125	148	12	following	follow	VERB
ejpam-6125	148	13	conditions	condition	NOUN
ejpam-6125	148	14	:	:	PUNCT
ejpam-6125	148	15	(	(	PUNCT
ejpam-6125	148	16	i	i	NOUN
ejpam-6125	148	17	)	)	PUNCT
ejpam-6125	148	18	f	f	PROPN
ejpam-6125	148	19	is	be	AUX
ejpam-6125	148	20	a	a	DET
ejpam-6125	148	21	clcni	clcni	NOUN
ejpam-6125	148	22	sequence	sequence	NOUN
ejpam-6125	148	23	of	of	ADP
ejpam-6125	148	24	g.	g.	PROPN
ejpam-6125	148	25	(	(	PUNCT
ejpam-6125	148	26	ii	ii	PROPN
ejpam-6125	148	27	)	)	PUNCT
ejpam-6125	148	28	f	f	PROPN
ejpam-6125	148	29	is	be	AUX
ejpam-6125	148	30	a	a	DET
ejpam-6125	148	31	clcni	clcni	ADJ
ejpam-6125	148	32	sequence	sequence	NOUN
ejpam-6125	148	33	of	of	ADP
ejpam-6125	148	34	h.	h.	PROPN
ejpam-6125	148	35	proof	proof	NOUN
ejpam-6125	148	36	.	.	PUNCT
ejpam-6125	149	1	suppose	suppose	VERB
ejpam-6125	149	2	that	that	SCONJ
ejpam-6125	149	3	f	f	PROPN
ejpam-6125	149	4	is	be	AUX
ejpam-6125	149	5	an	an	DET
ejpam-6125	149	6	lchni	lchni	ADJ
ejpam-6125	149	7	sequence	sequence	NOUN
ejpam-6125	149	8	of	of	ADP
ejpam-6125	149	9	g	g	PROPN
ejpam-6125	149	10	+	+	CCONJ
ejpam-6125	149	11	h.	h.	PROPN
ejpam-6125	149	12	then	then	ADV
ejpam-6125	149	13	f̂	f̂	PROPN
ejpam-6125	149	14	is	be	AUX
ejpam-6125	149	15	an	an	DET
ejpam-6125	149	16	independent	independent	ADJ
ejpam-6125	149	17	set	set	NOUN
ejpam-6125	149	18	of	of	ADP
ejpam-6125	149	19	g	g	PROPN
ejpam-6125	149	20	+	+	PROPN
ejpam-6125	149	21	h.	h.	PROPN
ejpam-6125	149	22	thus	thus	ADV
ejpam-6125	149	23	,	,	PUNCT
ejpam-6125	149	24	either	either	CCONJ
ejpam-6125	149	25	f̂	f̂	NUM
ejpam-6125	149	26	is	be	AUX
ejpam-6125	149	27	an	an	DET
ejpam-6125	149	28	independent	independent	ADJ
ejpam-6125	149	29	set	set	NOUN
ejpam-6125	149	30	of	of	ADP
ejpam-6125	149	31	g	g	NOUN
ejpam-6125	149	32	or	or	CCONJ
ejpam-6125	149	33	f̂	f̂	NUM
ejpam-6125	149	34	is	be	AUX
ejpam-6125	149	35	an	an	DET
ejpam-6125	149	36	independent	independent	ADJ
ejpam-6125	149	37	set	set	NOUN
ejpam-6125	149	38	of	of	ADP
ejpam-6125	149	39	h.	h.	PROPN
ejpam-6125	149	40	by	by	ADP
ejpam-6125	149	41	theorem	theorem	NOUN
ejpam-6125	149	42	4	4	NUM
ejpam-6125	149	43	,	,	PUNCT
ejpam-6125	149	44	f	f	PROPN
ejpam-6125	149	45	is	be	AUX
ejpam-6125	149	46	either	either	CCONJ
ejpam-6125	149	47	a	a	DET
ejpam-6125	149	48	clcni	clcni	NOUN
ejpam-6125	149	49	sequence	sequence	NOUN
ejpam-6125	149	50	of	of	ADP
ejpam-6125	149	51	g	g	PROPN
ejpam-6125	149	52	or	or	CCONJ
ejpam-6125	149	53	h.	h.	PROPN
ejpam-6125	149	54	hence	hence	ADV
ejpam-6125	149	55	,	,	PUNCT
ejpam-6125	149	56	f	f	PROPN
ejpam-6125	149	57	is	be	AUX
ejpam-6125	149	58	either	either	CCONJ
ejpam-6125	149	59	a	a	DET
ejpam-6125	149	60	clcni	clcni	NOUN
ejpam-6125	149	61	sequence	sequence	NOUN
ejpam-6125	149	62	of	of	ADP
ejpam-6125	149	63	g	g	PROPN
ejpam-6125	149	64	or	or	CCONJ
ejpam-6125	149	65	h.	h.	PROPN
ejpam-6125	149	66	j.	j.	PROPN
ejpam-6125	149	67	a.	a.	PROPN
ejpam-6125	149	68	hassan	hassan	PROPN
ejpam-6125	150	1	et	et	PROPN
ejpam-6125	150	2	al	al	PROPN
ejpam-6125	150	3	.	.	PUNCT
ejpam-6125	150	4	/	/	SYM
ejpam-6125	150	5	eur	eur	PROPN
ejpam-6125	150	6	.	.	PUNCT
ejpam-6125	151	1	j.	j.	PROPN
ejpam-6125	151	2	pure	pure	PROPN
ejpam-6125	151	3	appl	appl	PROPN
ejpam-6125	151	4	.	.	PROPN
ejpam-6125	151	5	math	math	PROPN
ejpam-6125	151	6	,	,	PUNCT
ejpam-6125	151	7	18	18	NUM
ejpam-6125	151	8	(	(	PUNCT
ejpam-6125	151	9	3	3	NUM
ejpam-6125	151	10	)	)	PUNCT
ejpam-6125	151	11	(	(	PUNCT
ejpam-6125	151	12	2025	2025	NUM
ejpam-6125	151	13	)	)	PUNCT
ejpam-6125	151	14	,	,	PUNCT
ejpam-6125	151	15	6125	6125	NUM
ejpam-6125	151	16	6	6	NUM
ejpam-6125	151	17	of	of	ADP
ejpam-6125	151	18	8	8	NUM
ejpam-6125	151	19	conversely	conversely	ADV
ejpam-6125	151	20	,	,	PUNCT
ejpam-6125	151	21	suppose	suppose	VERB
ejpam-6125	151	22	that	that	SCONJ
ejpam-6125	151	23	(	(	PUNCT
ejpam-6125	151	24	i	i	NOUN
ejpam-6125	151	25	)	)	PUNCT
ejpam-6125	151	26	holds	hold	VERB
ejpam-6125	151	27	.	.	PUNCT
ejpam-6125	152	1	then	then	ADV
ejpam-6125	152	2	by	by	ADP
ejpam-6125	152	3	theorem	theorem	NOUN
ejpam-6125	152	4	4	4	NUM
ejpam-6125	152	5	,	,	PUNCT
ejpam-6125	152	6	f	f	PROPN
ejpam-6125	152	7	is	be	AUX
ejpam-6125	152	8	a	a	DET
ejpam-6125	152	9	legal	legal	ADJ
ejpam-6125	152	10	closed	close	VERB
ejpam-6125	152	11	hop	hop	NOUN
ejpam-6125	152	12	neighborhood	neighborhood	NOUN
ejpam-6125	152	13	sequence	sequence	NOUN
ejpam-6125	152	14	of	of	ADP
ejpam-6125	152	15	g+h	g+h	PROPN
ejpam-6125	152	16	.	.	PUNCT
ejpam-6125	153	1	since	since	SCONJ
ejpam-6125	153	2	f̂	f̂	PROPN
ejpam-6125	153	3	is	be	AUX
ejpam-6125	153	4	an	an	DET
ejpam-6125	153	5	independent	independent	ADJ
ejpam-6125	153	6	set	set	NOUN
ejpam-6125	153	7	of	of	ADP
ejpam-6125	153	8	g	g	NOUN
ejpam-6125	153	9	,	,	PUNCT
ejpam-6125	153	10	it	it	PRON
ejpam-6125	153	11	follows	follow	VERB
ejpam-6125	153	12	that	that	SCONJ
ejpam-6125	153	13	f	f	PROPN
ejpam-6125	153	14	is	be	AUX
ejpam-6125	153	15	a	a	DET
ejpam-6125	153	16	lchni	lchni	ADJ
ejpam-6125	153	17	sequence	sequence	NOUN
ejpam-6125	153	18	of	of	ADP
ejpam-6125	153	19	g+h	g+h	PROPN
ejpam-6125	153	20	.	.	PUNCT
ejpam-6125	154	1	similarly	similarly	ADV
ejpam-6125	154	2	,	,	PUNCT
ejpam-6125	154	3	the	the	DET
ejpam-6125	154	4	assertion	assertion	NOUN
ejpam-6125	154	5	holds	hold	VERB
ejpam-6125	154	6	when	when	SCONJ
ejpam-6125	154	7	(	(	PUNCT
ejpam-6125	154	8	ii	ii	NOUN
ejpam-6125	154	9	)	)	PUNCT
ejpam-6125	154	10	is	be	AUX
ejpam-6125	154	11	true	true	ADJ
ejpam-6125	154	12	.	.	PUNCT
ejpam-6125	155	1	theorem	theorem	ADJ
ejpam-6125	155	2	6	6	NUM
ejpam-6125	155	3	.	.	PUNCT
ejpam-6125	156	1	let	let	VERB
ejpam-6125	156	2	g	g	NOUN
ejpam-6125	156	3	and	and	CCONJ
ejpam-6125	156	4	h	h	NOUN
ejpam-6125	156	5	be	be	VERB
ejpam-6125	156	6	any	any	DET
ejpam-6125	156	7	two	two	NUM
ejpam-6125	156	8	graphs	graph	NOUN
ejpam-6125	156	9	.	.	PUNCT
ejpam-6125	157	1	then	then	ADV
ejpam-6125	157	2	θ(g+h	θ(g+h	NUM
ejpam-6125	157	3	)	)	PUNCT
ejpam-6125	157	4	=	=	SYM
ejpam-6125	157	5	max{αcl(g	max{αcl(g	PROPN
ejpam-6125	157	6	)	)	PUNCT
ejpam-6125	157	7	,	,	PUNCT
ejpam-6125	157	8	αcl(h	αcl(h	PROPN
ejpam-6125	157	9	)	)	PUNCT
ejpam-6125	157	10	}	}	PUNCT
ejpam-6125	157	11	proof	proof	NOUN
ejpam-6125	157	12	.	.	PUNCT
ejpam-6125	158	1	we	we	PRON
ejpam-6125	158	2	may	may	AUX
ejpam-6125	158	3	assume	assume	VERB
ejpam-6125	158	4	that	that	SCONJ
ejpam-6125	158	5	αcl(g	αcl(g	NUM
ejpam-6125	158	6	)	)	PUNCT
ejpam-6125	158	7	≥	≥	NOUN
ejpam-6125	158	8	αcl(h	αcl(h	NUM
ejpam-6125	158	9	)	)	PUNCT
ejpam-6125	158	10	.	.	PUNCT
ejpam-6125	159	1	first	first	ADV
ejpam-6125	159	2	,	,	PUNCT
ejpam-6125	159	3	let	let	VERB
ejpam-6125	159	4	p	p	PRON
ejpam-6125	159	5	be	be	AUX
ejpam-6125	159	6	a	a	DET
ejpam-6125	159	7	maximum	maximum	ADJ
ejpam-6125	159	8	lchni	lchni	ADJ
ejpam-6125	159	9	sequence	sequence	NOUN
ejpam-6125	159	10	of	of	ADP
ejpam-6125	159	11	g+h	g+h	PROPN
ejpam-6125	159	12	.	.	PUNCT
ejpam-6125	160	1	then	then	ADV
ejpam-6125	160	2	by	by	ADP
ejpam-6125	160	3	theorem	theorem	NOUN
ejpam-6125	160	4	3	3	NUM
ejpam-6125	160	5	,	,	PUNCT
ejpam-6125	160	6	p	p	PRON
ejpam-6125	160	7	is	be	AUX
ejpam-6125	160	8	a	a	DET
ejpam-6125	160	9	clcni	clcni	NOUN
ejpam-6125	160	10	sequence	sequence	NOUN
ejpam-6125	160	11	of	of	ADP
ejpam-6125	160	12	g.	g.	PROPN
ejpam-6125	160	13	hence	hence	ADV
ejpam-6125	160	14	,	,	PUNCT
ejpam-6125	160	15	θ(g+h	θ(g+h	PROPN
ejpam-6125	160	16	)	)	PUNCT
ejpam-6125	161	1	=	=	PRON
ejpam-6125	161	2	|p̂	|p̂	VERB
ejpam-6125	161	3	|	|	ADV
ejpam-6125	161	4	≤	≤	NUM
ejpam-6125	161	5	αcl(g	αcl(g	NUM
ejpam-6125	161	6	)	)	PUNCT
ejpam-6125	161	7	.	.	PUNCT
ejpam-6125	162	1	next	next	ADV
ejpam-6125	162	2	,	,	PUNCT
ejpam-6125	162	3	suppose	suppose	VERB
ejpam-6125	162	4	that	that	SCONJ
ejpam-6125	162	5	f	f	PROPN
ejpam-6125	162	6	is	be	AUX
ejpam-6125	162	7	a	a	DET
ejpam-6125	162	8	maximum	maximum	ADJ
ejpam-6125	162	9	clcni	clcni	NOUN
ejpam-6125	162	10	sequence	sequence	NOUN
ejpam-6125	162	11	of	of	ADP
ejpam-6125	162	12	g.	g.	PROPN
ejpam-6125	162	13	then	then	ADV
ejpam-6125	162	14	f	f	PROPN
ejpam-6125	162	15	is	be	AUX
ejpam-6125	162	16	an	an	DET
ejpam-6125	162	17	lchni	lchni	ADJ
ejpam-6125	162	18	sequence	sequence	NOUN
ejpam-6125	162	19	of	of	ADP
ejpam-6125	162	20	g	g	PROPN
ejpam-6125	162	21	+	+	CCONJ
ejpam-6125	162	22	h	h	NOUN
ejpam-6125	162	23	by	by	ADP
ejpam-6125	162	24	theorem	theorem	NOUN
ejpam-6125	162	25	3	3	X
ejpam-6125	162	26	.	.	PUNCT
ejpam-6125	163	1	it	it	PRON
ejpam-6125	163	2	follows	follow	VERB
ejpam-6125	163	3	that	that	SCONJ
ejpam-6125	163	4	θ(g	θ(g	PROPN
ejpam-6125	163	5	+	+	SYM
ejpam-6125	163	6	h	h	NOUN
ejpam-6125	163	7	)	)	PUNCT
ejpam-6125	163	8	≥	≥	NOUN
ejpam-6125	163	9	|f̂	|f̂	INTJ
ejpam-6125	163	10	|	|	ADV
ejpam-6125	163	11	=	=	NOUN
ejpam-6125	163	12	αcl(g	αcl(g	PROPN
ejpam-6125	163	13	)	)	PUNCT
ejpam-6125	163	14	.	.	PUNCT
ejpam-6125	164	1	this	this	PRON
ejpam-6125	164	2	establishes	establish	VERB
ejpam-6125	164	3	the	the	DET
ejpam-6125	164	4	desired	desire	VERB
ejpam-6125	164	5	equality	equality	NOUN
ejpam-6125	164	6	.	.	PUNCT
ejpam-6125	165	1	corollary	corollary	ADJ
ejpam-6125	165	2	1	1	NUM
ejpam-6125	165	3	.	.	PUNCT
ejpam-6125	166	1	let	let	VERB
ejpam-6125	166	2	n	n	PRON
ejpam-6125	166	3	and	and	CCONJ
ejpam-6125	166	4	m	m	AUX
ejpam-6125	166	5	be	be	AUX
ejpam-6125	166	6	any	any	DET
ejpam-6125	166	7	two	two	NUM
ejpam-6125	166	8	positive	positive	ADJ
ejpam-6125	166	9	natural	natural	ADJ
ejpam-6125	166	10	numbers	number	NOUN
ejpam-6125	166	11	.	.	PUNCT
ejpam-6125	167	1	then	then	ADV
ejpam-6125	167	2	each	each	PRON
ejpam-6125	167	3	of	of	ADP
ejpam-6125	167	4	the	the	DET
ejpam-6125	167	5	following	follow	VERB
ejpam-6125	167	6	holds	hold	VERB
ejpam-6125	167	7	:	:	PUNCT
ejpam-6125	167	8	(	(	PUNCT
ejpam-6125	167	9	i	i	NOUN
ejpam-6125	167	10	)	)	PUNCT
ejpam-6125	167	11	θ(fn	θ(fn	NOUN
ejpam-6125	167	12	)	)	PUNCT
ejpam-6125	167	13	=	=	SYM
ejpam-6125	167	14	αcl(pn	αcl(pn	NOUN
ejpam-6125	167	15	)	)	PUNCT
ejpam-6125	167	16	(	(	PUNCT
ejpam-6125	167	17	ii	ii	NOUN
ejpam-6125	167	18	)	)	PUNCT
ejpam-6125	167	19	θ(wn	θ(wn	PROPN
ejpam-6125	167	20	)	)	PUNCT
ejpam-6125	167	21	=	=	SYM
ejpam-6125	167	22	αcl(cn	αcl(cn	NUM
ejpam-6125	167	23	)	)	PUNCT
ejpam-6125	167	24	(	(	PUNCT
ejpam-6125	167	25	iii	iii	NOUN
ejpam-6125	167	26	)	)	PUNCT
ejpam-6125	167	27	θ(sn	θ(sn	PROPN
ejpam-6125	167	28	)	)	PUNCT
ejpam-6125	167	29	=	=	SYM
ejpam-6125	167	30	αcl(k̄n	αcl(k̄n	PROPN
ejpam-6125	167	31	)	)	PUNCT
ejpam-6125	167	32	=	=	SYM
ejpam-6125	167	33	1	1	NUM
ejpam-6125	167	34	(	(	PUNCT
ejpam-6125	167	35	iv	iv	X
ejpam-6125	167	36	)	)	PUNCT
ejpam-6125	167	37	θ(pn	θ(pn	PROPN
ejpam-6125	167	38	+	+	CCONJ
ejpam-6125	167	39	pm	pm	NOUN
ejpam-6125	167	40	)	)	PUNCT
ejpam-6125	167	41	=	=	SYM
ejpam-6125	167	42	max{αcl(pn	max{αcl(pn	NOUN
ejpam-6125	167	43	)	)	PUNCT
ejpam-6125	167	44	,	,	PUNCT
ejpam-6125	167	45	αcl(pm	αcl(pm	NOUN
ejpam-6125	167	46	)	)	PUNCT
ejpam-6125	167	47	}	}	PUNCT
ejpam-6125	167	48	(	(	PUNCT
ejpam-6125	167	49	v	v	NOUN
ejpam-6125	167	50	)	)	PUNCT
ejpam-6125	167	51	θ(cn	θ(cn	NOUN
ejpam-6125	167	52	+	+	CCONJ
ejpam-6125	167	53	cm	cm	NOUN
ejpam-6125	167	54	)	)	PUNCT
ejpam-6125	167	55	=	=	SYM
ejpam-6125	167	56	max{αcl(cn	max{αcl(cn	NOUN
ejpam-6125	167	57	)	)	PUNCT
ejpam-6125	167	58	,	,	PUNCT
ejpam-6125	167	59	αcl(cm	αcl(cm	NOUN
ejpam-6125	167	60	)	)	PUNCT
ejpam-6125	167	61	}	}	PUNCT
ejpam-6125	167	62	(	(	PUNCT
ejpam-6125	167	63	vi	vi	NOUN
ejpam-6125	167	64	)	)	PUNCT
ejpam-6125	167	65	θ(kn	θ(kn	NOUN
ejpam-6125	167	66	+	+	NOUN
ejpam-6125	167	67	km	km	NOUN
ejpam-6125	167	68	)	)	PUNCT
ejpam-6125	167	69	=	=	SYM
ejpam-6125	167	70	max{αcl(kn	max{αcl(kn	NOUN
ejpam-6125	167	71	)	)	PUNCT
ejpam-6125	167	72	,	,	PUNCT
ejpam-6125	167	73	αcl(km	αcl(km	NOUN
ejpam-6125	167	74	)	)	PUNCT
ejpam-6125	167	75	}	}	PUNCT
ejpam-6125	167	76	=	=	SYM
ejpam-6125	167	77	1	1	NUM
ejpam-6125	167	78	now	now	ADV
ejpam-6125	167	79	,	,	PUNCT
ejpam-6125	167	80	let	let	VERB
ejpam-6125	167	81	’s	’s	PRON
ejpam-6125	167	82	study	study	VERB
ejpam-6125	167	83	the	the	DET
ejpam-6125	167	84	behavior	behavior	NOUN
ejpam-6125	167	85	of	of	ADP
ejpam-6125	167	86	an	an	DET
ejpam-6125	167	87	lchni	lchni	ADJ
ejpam-6125	167	88	sequence	sequence	NOUN
ejpam-6125	167	89	in	in	ADP
ejpam-6125	167	90	the	the	DET
ejpam-6125	167	91	shadow	shadow	NOUN
ejpam-6125	167	92	graphs	graph	NOUN
ejpam-6125	167	93	.	.	PUNCT
ejpam-6125	168	1	we	we	PRON
ejpam-6125	168	2	shall	shall	AUX
ejpam-6125	168	3	note	note	VERB
ejpam-6125	168	4	the	the	DET
ejpam-6125	168	5	following	follow	VERB
ejpam-6125	168	6	notations	notation	NOUN
ejpam-6125	168	7	:	:	PUNCT
ejpam-6125	168	8	let	let	VERB
ejpam-6125	168	9	g1	g1	PROPN
ejpam-6125	168	10	and	and	CCONJ
ejpam-6125	168	11	g2	g2	PROPN
ejpam-6125	168	12	be	be	VERB
ejpam-6125	168	13	two	two	NUM
ejpam-6125	168	14	copies	copy	NOUN
ejpam-6125	168	15	of	of	ADP
ejpam-6125	168	16	a	a	DET
ejpam-6125	168	17	graph	graph	NOUN
ejpam-6125	168	18	g	g	NOUN
ejpam-6125	168	19	in	in	ADP
ejpam-6125	168	20	the	the	DET
ejpam-6125	168	21	definition	definition	NOUN
ejpam-6125	168	22	of	of	ADP
ejpam-6125	168	23	the	the	DET
ejpam-6125	168	24	shadow	shadow	NOUN
ejpam-6125	168	25	graph	graph	NOUN
ejpam-6125	168	26	s(g	s(g	PROPN
ejpam-6125	168	27	)	)	PUNCT
ejpam-6125	168	28	.	.	PUNCT
ejpam-6125	169	1	if	if	SCONJ
ejpam-6125	169	2	qg1	qg1	ADV
ejpam-6125	169	3	⊆	⊆	NUM
ejpam-6125	169	4	v	v	NOUN
ejpam-6125	169	5	(	(	PUNCT
ejpam-6125	169	6	g1	g1	PROPN
ejpam-6125	169	7	)	)	PUNCT
ejpam-6125	169	8	and	and	CCONJ
ejpam-6125	169	9	qg2	qg2	NOUN
ejpam-6125	169	10	⊆	⊆	NUM
ejpam-6125	169	11	v	v	NOUN
ejpam-6125	169	12	(	(	PUNCT
ejpam-6125	169	13	g2	g2	PROPN
ejpam-6125	169	14	)	)	PUNCT
ejpam-6125	169	15	,	,	PUNCT
ejpam-6125	169	16	then	then	ADV
ejpam-6125	169	17	the	the	DET
ejpam-6125	169	18	sets	set	NOUN
ejpam-6125	169	19	q′	q′	NOUN
ejpam-6125	169	20	g1	g1	PROPN
ejpam-6125	169	21	and	and	CCONJ
ejpam-6125	169	22	q′	q′	NOUN
ejpam-6125	169	23	g2	g2	PROPN
ejpam-6125	169	24	are	be	AUX
ejpam-6125	169	25	defined	define	VERB
ejpam-6125	169	26	as	as	SCONJ
ejpam-6125	169	27	follows	follow	VERB
ejpam-6125	169	28	:	:	PUNCT
ejpam-6125	169	29	q′	q′	NOUN
ejpam-6125	169	30	g1	g1	PROPN
ejpam-6125	169	31	=	=	PRON
ejpam-6125	169	32	{	{	PUNCT
ejpam-6125	169	33	x′	x′	PROPN
ejpam-6125	169	34	∈	∈	PROPN
ejpam-6125	169	35	v	v	PROPN
ejpam-6125	169	36	(	(	PUNCT
ejpam-6125	169	37	g2	g2	PROPN
ejpam-6125	169	38	)	)	PUNCT
ejpam-6125	169	39	:	:	PUNCT
ejpam-6125	170	1	x	x	X
ejpam-6125	170	2	∈	∈	NOUN
ejpam-6125	170	3	qg1	qg1	ADV
ejpam-6125	170	4	}	}	PUNCT
ejpam-6125	170	5	and	and	CCONJ
ejpam-6125	170	6	q′	q′	NOUN
ejpam-6125	170	7	g2	g2	PROPN
ejpam-6125	170	8	=	=	PRON
ejpam-6125	171	1	{	{	PUNCT
ejpam-6125	171	2	b	b	PROPN
ejpam-6125	171	3	∈	∈	PROPN
ejpam-6125	171	4	v	v	NOUN
ejpam-6125	171	5	(	(	PUNCT
ejpam-6125	171	6	g1	g1	PROPN
ejpam-6125	171	7	)	)	PUNCT
ejpam-6125	171	8	:	:	PUNCT
ejpam-6125	172	1	b	b	X
ejpam-6125	172	2	′	′	NUM
ejpam-6125	172	3	∈	∈	PROPN
ejpam-6125	172	4	qg2	qg2	NOUN
ejpam-6125	172	5	}	}	PUNCT
ejpam-6125	172	6	.	.	PUNCT
ejpam-6125	173	1	theorem	theorem	VERB
ejpam-6125	173	2	7	7	NUM
ejpam-6125	173	3	.	.	PUNCT
ejpam-6125	174	1	let	let	VERB
ejpam-6125	174	2	g	g	PRON
ejpam-6125	174	3	be	be	AUX
ejpam-6125	174	4	a	a	DET
ejpam-6125	174	5	non	non	ADJ
ejpam-6125	174	6	-	-	ADJ
ejpam-6125	174	7	trivial	trivial	ADJ
ejpam-6125	174	8	connected	connected	ADJ
ejpam-6125	174	9	graph	graph	NOUN
ejpam-6125	174	10	.	.	PUNCT
ejpam-6125	175	1	then	then	ADV
ejpam-6125	175	2	q	q	X
ejpam-6125	175	3	is	be	AUX
ejpam-6125	175	4	an	an	DET
ejpam-6125	175	5	lchni	lchni	ADJ
ejpam-6125	175	6	sequence	sequence	NOUN
ejpam-6125	175	7	in	in	ADP
ejpam-6125	175	8	s(g	s(g	PROPN
ejpam-6125	175	9	)	)	PUNCT
ejpam-6125	175	10	if	if	SCONJ
ejpam-6125	175	11	and	and	CCONJ
ejpam-6125	175	12	only	only	ADV
ejpam-6125	175	13	if	if	SCONJ
ejpam-6125	175	14	one	one	NUM
ejpam-6125	175	15	of	of	ADP
ejpam-6125	175	16	the	the	DET
ejpam-6125	175	17	following	follow	VERB
ejpam-6125	175	18	conditions	condition	NOUN
ejpam-6125	175	19	holds	hold	VERB
ejpam-6125	175	20	:	:	PUNCT
ejpam-6125	175	21	(	(	PUNCT
ejpam-6125	175	22	i	i	NOUN
ejpam-6125	175	23	)	)	PUNCT
ejpam-6125	175	24	q	q	PUNCT
ejpam-6125	175	25	is	be	AUX
ejpam-6125	175	26	an	an	DET
ejpam-6125	175	27	lchni	lchni	ADJ
ejpam-6125	175	28	sequence	sequence	NOUN
ejpam-6125	175	29	in	in	ADP
ejpam-6125	175	30	g1	g1	PROPN
ejpam-6125	175	31	.	.	PUNCT
ejpam-6125	176	1	(	(	PUNCT
ejpam-6125	176	2	ii	ii	NOUN
ejpam-6125	176	3	)	)	PUNCT
ejpam-6125	176	4	q	q	PUNCT
ejpam-6125	176	5	is	be	AUX
ejpam-6125	176	6	an	an	DET
ejpam-6125	176	7	lchni	lchni	ADJ
ejpam-6125	176	8	sequence	sequence	NOUN
ejpam-6125	176	9	in	in	ADP
ejpam-6125	176	10	g2	g2	PROPN
ejpam-6125	176	11	.	.	PUNCT
ejpam-6125	177	1	(	(	PUNCT
ejpam-6125	177	2	iii	iii	X
ejpam-6125	177	3	)	)	PUNCT
ejpam-6125	177	4	q	q	NOUN
ejpam-6125	178	1	=	=	PUNCT
ejpam-6125	178	2	qg1	qg1	PRON
ejpam-6125	178	3	⊕qg2	⊕qg2	NOUN
ejpam-6125	178	4	such	such	ADJ
ejpam-6125	178	5	that	that	SCONJ
ejpam-6125	178	6	q̂g1	q̂g1	ADJ
ejpam-6125	178	7	∪	∪	ADP
ejpam-6125	178	8	q̂′	q̂′	PROPN
ejpam-6125	178	9	g2	g2	PROPN
ejpam-6125	178	10	and	and	CCONJ
ejpam-6125	178	11	q̂′	q̂′	PROPN
ejpam-6125	178	12	g1	g1	PROPN
ejpam-6125	178	13	∪	∪	ADJ
ejpam-6125	178	14	q̂g2	q̂g2	NOUN
ejpam-6125	178	15	are	be	AUX
ejpam-6125	178	16	lchni	lchni	VERB
ejpam-6125	178	17	sets	set	NOUN
ejpam-6125	178	18	in	in	ADP
ejpam-6125	178	19	g1	g1	PROPN
ejpam-6125	178	20	and	and	CCONJ
ejpam-6125	178	21	g2	g2	PROPN
ejpam-6125	178	22	,	,	PUNCT
ejpam-6125	178	23	respectively	respectively	ADV
ejpam-6125	178	24	.	.	PUNCT
ejpam-6125	179	1	j.	j.	PROPN
ejpam-6125	179	2	a.	a.	PROPN
ejpam-6125	179	3	hassan	hassan	PROPN
ejpam-6125	179	4	et	et	PROPN
ejpam-6125	179	5	al	al	PROPN
ejpam-6125	179	6	.	.	PUNCT
ejpam-6125	179	7	/	/	SYM
ejpam-6125	179	8	eur	eur	PROPN
ejpam-6125	179	9	.	.	PUNCT
ejpam-6125	180	1	j.	j.	PROPN
ejpam-6125	180	2	pure	pure	PROPN
ejpam-6125	180	3	appl	appl	PROPN
ejpam-6125	180	4	.	.	PROPN
ejpam-6125	180	5	math	math	PROPN
ejpam-6125	180	6	,	,	PUNCT
ejpam-6125	180	7	18	18	NUM
ejpam-6125	180	8	(	(	PUNCT
ejpam-6125	180	9	3	3	NUM
ejpam-6125	180	10	)	)	PUNCT
ejpam-6125	180	11	(	(	PUNCT
ejpam-6125	180	12	2025	2025	NUM
ejpam-6125	180	13	)	)	PUNCT
ejpam-6125	180	14	,	,	PUNCT
ejpam-6125	180	15	6125	6125	NUM
ejpam-6125	180	16	7	7	NUM
ejpam-6125	180	17	of	of	ADP
ejpam-6125	180	18	8	8	NUM
ejpam-6125	180	19	proof	proof	NOUN
ejpam-6125	180	20	.	.	PUNCT
ejpam-6125	181	1	let	let	VERB
ejpam-6125	181	2	q	q	PRON
ejpam-6125	181	3	be	be	AUX
ejpam-6125	181	4	an	an	DET
ejpam-6125	181	5	lchni	lchni	ADJ
ejpam-6125	181	6	sequence	sequence	NOUN
ejpam-6125	181	7	in	in	ADP
ejpam-6125	181	8	s(g	s(g	PROPN
ejpam-6125	181	9	)	)	PUNCT
ejpam-6125	181	10	.	.	PUNCT
ejpam-6125	182	1	let	let	VERB
ejpam-6125	182	2	q̂g1	q̂g1	ADJ
ejpam-6125	182	3	=	=	SYM
ejpam-6125	182	4	q̂	q̂	X
ejpam-6125	182	5	∩	∩	X
ejpam-6125	182	6	v	v	X
ejpam-6125	182	7	(	(	PUNCT
ejpam-6125	182	8	g1	g1	PROPN
ejpam-6125	182	9	)	)	PUNCT
ejpam-6125	182	10	and	and	CCONJ
ejpam-6125	182	11	q̂g2	q̂g2	NOUN
ejpam-6125	182	12	=	=	SYM
ejpam-6125	182	13	q̂	q̂	NUM
ejpam-6125	182	14	∩	∩	X
ejpam-6125	182	15	v	v	X
ejpam-6125	182	16	(	(	PUNCT
ejpam-6125	182	17	g2	g2	PROPN
ejpam-6125	182	18	)	)	PUNCT
ejpam-6125	182	19	.	.	PUNCT
ejpam-6125	183	1	if	if	SCONJ
ejpam-6125	183	2	q̂g2	q̂g2	NOUN
ejpam-6125	183	3	=	=	SYM
ejpam-6125	183	4	∅	∅	NOUN
ejpam-6125	183	5	,	,	PUNCT
ejpam-6125	183	6	then	then	ADV
ejpam-6125	183	7	q	q	NOUN
ejpam-6125	183	8	=	=	PUNCT
ejpam-6125	183	9	qg1	qg1	ADV
ejpam-6125	183	10	is	be	AUX
ejpam-6125	183	11	an	an	DET
ejpam-6125	183	12	lchni	lchni	ADJ
ejpam-6125	183	13	sequence	sequence	NOUN
ejpam-6125	183	14	in	in	ADP
ejpam-6125	183	15	g1	g1	PROPN
ejpam-6125	183	16	.	.	PUNCT
ejpam-6125	184	1	moreover	moreover	ADV
ejpam-6125	184	2	,	,	PUNCT
ejpam-6125	184	3	if	if	SCONJ
ejpam-6125	184	4	q̂g1	q̂g1	ADJ
ejpam-6125	184	5	=	=	SYM
ejpam-6125	184	6	∅	∅	NOUN
ejpam-6125	184	7	,	,	PUNCT
ejpam-6125	184	8	then	then	ADV
ejpam-6125	184	9	q	q	NOUN
ejpam-6125	184	10	=	=	NOUN
ejpam-6125	184	11	qg2	qg2	NOUN
ejpam-6125	184	12	is	be	AUX
ejpam-6125	184	13	an	an	DET
ejpam-6125	184	14	lchni	lchni	ADJ
ejpam-6125	184	15	sequence	sequence	NOUN
ejpam-6125	184	16	in	in	ADP
ejpam-6125	184	17	g2	g2	PROPN
ejpam-6125	184	18	.	.	PUNCT
ejpam-6125	185	1	therefore	therefore	ADV
ejpam-6125	185	2	,	,	PUNCT
ejpam-6125	185	3	either	either	CCONJ
ejpam-6125	185	4	(	(	PUNCT
ejpam-6125	185	5	i	i	NOUN
ejpam-6125	185	6	)	)	PUNCT
ejpam-6125	185	7	or	or	CCONJ
ejpam-6125	185	8	(	(	PUNCT
ejpam-6125	185	9	ii	ii	NOUN
ejpam-6125	185	10	)	)	PUNCT
ejpam-6125	185	11	holds	hold	VERB
ejpam-6125	185	12	.	.	PUNCT
ejpam-6125	186	1	now	now	ADV
ejpam-6125	186	2	assume	assume	VERB
ejpam-6125	186	3	that	that	SCONJ
ejpam-6125	186	4	q̂g1	q̂g1	ADJ
ejpam-6125	186	5	̸=	̸=	PROPN
ejpam-6125	186	6	∅	∅	NOUN
ejpam-6125	186	7	and	and	CCONJ
ejpam-6125	186	8	q̂g2	q̂g2	NOUN
ejpam-6125	186	9	̸=	̸=	PROPN
ejpam-6125	186	10	∅.	∅.	ADV
ejpam-6125	186	11	let	let	VERB
ejpam-6125	186	12	q̂g1	q̂g1	ADJ
ejpam-6125	186	13	∪	∪	VERB
ejpam-6125	186	14	q̂′	q̂′	NOUN
ejpam-6125	186	15	g2	g2	PROPN
ejpam-6125	186	16	=	=	SYM
ejpam-6125	186	17	{	{	PUNCT
ejpam-6125	186	18	v1	v1	PROPN
ejpam-6125	186	19	,	,	PUNCT
ejpam-6125	186	20	v2	v2	PROPN
ejpam-6125	186	21	,	,	PUNCT
ejpam-6125	186	22	.	.	PUNCT
ejpam-6125	186	23	.	.	PUNCT
ejpam-6125	187	1	.	.	PUNCT
ejpam-6125	188	1	,	,	PUNCT
ejpam-6125	188	2	vk	vk	ADP
ejpam-6125	188	3	}	}	PUNCT
ejpam-6125	188	4	,	,	PUNCT
ejpam-6125	188	5	and	and	CCONJ
ejpam-6125	188	6	consider	consider	VERB
ejpam-6125	188	7	vi	vi	PROPN
ejpam-6125	188	8	,	,	PUNCT
ejpam-6125	188	9	vj	vj	PROPN
ejpam-6125	188	10	∈	∈	PROPN
ejpam-6125	188	11	q̂g1	q̂g1	ADJ
ejpam-6125	188	12	∪	∪	ADP
ejpam-6125	188	13	q̂′	q̂′	PROPN
ejpam-6125	188	14	g2	g2	PROPN
ejpam-6125	188	15	and	and	CCONJ
ejpam-6125	188	16	consider	consider	VERB
ejpam-6125	188	17	the	the	DET
ejpam-6125	188	18	following	follow	VERB
ejpam-6125	188	19	cases	case	NOUN
ejpam-6125	188	20	:	:	PUNCT
ejpam-6125	188	21	case	case	NOUN
ejpam-6125	188	22	1	1	NUM
ejpam-6125	188	23	:	:	PUNCT
ejpam-6125	188	24	if	if	SCONJ
ejpam-6125	188	25	vi	vi	PROPN
ejpam-6125	188	26	,	,	PUNCT
ejpam-6125	188	27	vj	vj	X
ejpam-6125	188	28	∈	∈	PROPN
ejpam-6125	188	29	q̂g1	q̂g1	NOUN
ejpam-6125	188	30	=	=	PRON
ejpam-6125	189	1	q̂∩v	q̂∩v	VERB
ejpam-6125	189	2	(	(	PUNCT
ejpam-6125	189	3	g	g	NOUN
ejpam-6125	189	4	)	)	PUNCT
ejpam-6125	189	5	,	,	PUNCT
ejpam-6125	189	6	then	then	ADV
ejpam-6125	189	7	dg1(vi	dg1(vi	PROPN
ejpam-6125	189	8	,	,	PUNCT
ejpam-6125	189	9	vj	vj	INTJ
ejpam-6125	189	10	)	)	PUNCT
ejpam-6125	189	11	̸=	̸=	PROPN
ejpam-6125	189	12	1	1	NUM
ejpam-6125	189	13	since	since	SCONJ
ejpam-6125	189	14	q̂	q̂	NUM
ejpam-6125	189	15	is	be	AUX
ejpam-6125	189	16	an	an	DET
ejpam-6125	189	17	independent	independent	ADJ
ejpam-6125	189	18	set	set	NOUN
ejpam-6125	189	19	in	in	ADP
ejpam-6125	189	20	s(g	s(g	PROPN
ejpam-6125	189	21	)	)	PUNCT
ejpam-6125	189	22	with	with	ADP
ejpam-6125	189	23	q̂g1	q̂g1	ADJ
ejpam-6125	189	24	⊆	⊆	NUM
ejpam-6125	189	25	q̂.	q̂.	ADJ
ejpam-6125	189	26	case	case	NOUN
ejpam-6125	189	27	2	2	NUM
ejpam-6125	189	28	:	:	PUNCT
ejpam-6125	189	29	if	if	SCONJ
ejpam-6125	189	30	vi	vi	PROPN
ejpam-6125	189	31	,	,	PUNCT
ejpam-6125	189	32	vj	vj	PROPN
ejpam-6125	189	33	∈	∈	PROPN
ejpam-6125	189	34	q̂′	q̂′	PROPN
ejpam-6125	189	35	g2	g2	PROPN
ejpam-6125	189	36	,	,	PUNCT
ejpam-6125	189	37	then	then	ADV
ejpam-6125	189	38	v′i	v′i	NOUN
ejpam-6125	189	39	,	,	PUNCT
ejpam-6125	189	40	v	v	NOUN
ejpam-6125	189	41	′	′	NUM
ejpam-6125	189	42	j	j	PROPN
ejpam-6125	189	43	∈	∈	PROPN
ejpam-6125	189	44	q̂g2	q̂g2	NOUN
ejpam-6125	189	45	.	.	PUNCT
ejpam-6125	190	1	since	since	SCONJ
ejpam-6125	190	2	q̂g2	q̂g2	NOUN
ejpam-6125	190	3	⊆	⊆	NUM
ejpam-6125	190	4	q̂	q̂	NUM
ejpam-6125	190	5	and	and	CCONJ
ejpam-6125	190	6	q̂	q̂	NUM
ejpam-6125	190	7	is	be	AUX
ejpam-6125	190	8	an	an	DET
ejpam-6125	190	9	independent	independent	ADJ
ejpam-6125	190	10	,	,	PUNCT
ejpam-6125	190	11	it	it	PRON
ejpam-6125	190	12	follows	follow	VERB
ejpam-6125	190	13	that	that	SCONJ
ejpam-6125	190	14	dg2(vi	dg2(vi	PROPN
ejpam-6125	190	15	,	,	PUNCT
ejpam-6125	190	16	vj	vj	ADJ
ejpam-6125	190	17	)	)	PUNCT
ejpam-6125	190	18	̸=	̸=	PROPN
ejpam-6125	190	19	1	1	NUM
ejpam-6125	190	20	,	,	PUNCT
ejpam-6125	190	21	and	and	CCONJ
ejpam-6125	190	22	we	we	PRON
ejpam-6125	190	23	are	be	AUX
ejpam-6125	190	24	done	do	VERB
ejpam-6125	190	25	.	.	PUNCT
ejpam-6125	191	1	case	case	NOUN
ejpam-6125	191	2	3	3	X
ejpam-6125	191	3	:	:	PUNCT
ejpam-6125	191	4	assume	assume	VERB
ejpam-6125	191	5	that	that	SCONJ
ejpam-6125	191	6	vi	vi	PROPN
ejpam-6125	191	7	∈	∈	NOUN
ejpam-6125	191	8	q̂g1	q̂g1	NOUN
ejpam-6125	191	9	and	and	CCONJ
ejpam-6125	191	10	vj	vj	DET
ejpam-6125	191	11	∈	∈	PROPN
ejpam-6125	191	12	q̂′	q̂′	PROPN
ejpam-6125	191	13	g2	g2	PROPN
ejpam-6125	191	14	.	.	PUNCT
ejpam-6125	192	1	then	then	ADV
ejpam-6125	192	2	v′j	v′j	PROPN
ejpam-6125	192	3	∈	∈	PROPN
ejpam-6125	192	4	q̂g2	q̂g2	NOUN
ejpam-6125	192	5	⊆	⊆	NUM
ejpam-6125	192	6	q̂.	q̂.	NOUN
ejpam-6125	192	7	thus	thus	ADV
ejpam-6125	192	8	,	,	PUNCT
ejpam-6125	192	9	ds(g)(vi	ds(g)(vi	PROPN
ejpam-6125	192	10	,	,	PUNCT
ejpam-6125	192	11	vj	vj	ADJ
ejpam-6125	192	12	)	)	PUNCT
ejpam-6125	192	13	̸=	̸=	NOUN
ejpam-6125	192	14	1	1	NUM
ejpam-6125	192	15	implying	imply	VERB
ejpam-6125	192	16	that	that	SCONJ
ejpam-6125	192	17	dg1(vi	dg1(vi	PROPN
ejpam-6125	192	18	,	,	PUNCT
ejpam-6125	192	19	vj	vj	ADJ
ejpam-6125	192	20	)	)	PUNCT
ejpam-6125	192	21	̸=	̸=	PROPN
ejpam-6125	192	22	1	1	NUM
ejpam-6125	192	23	.	.	PUNCT
ejpam-6125	193	1	similarly	similarly	ADV
ejpam-6125	193	2	,	,	PUNCT
ejpam-6125	193	3	the	the	DET
ejpam-6125	193	4	same	same	ADJ
ejpam-6125	193	5	result	result	NOUN
ejpam-6125	193	6	follows	follow	VERB
ejpam-6125	193	7	when	when	SCONJ
ejpam-6125	193	8	vj	vj	X
ejpam-6125	193	9	∈	∈	PROPN
ejpam-6125	193	10	q̂g1	q̂g1	NOUN
ejpam-6125	193	11	and	and	CCONJ
ejpam-6125	193	12	vi	vi	NOUN
ejpam-6125	193	13	∈	∈	PROPN
ejpam-6125	193	14	q̂′	q̂′	PROPN
ejpam-6125	193	15	g2	g2	PROPN
ejpam-6125	193	16	.	.	PUNCT
ejpam-6125	194	1	now	now	ADV
ejpam-6125	194	2	,	,	PUNCT
ejpam-6125	194	3	let	let	VERB
ejpam-6125	194	4	q̂g1	q̂g1	ADJ
ejpam-6125	194	5	=	=	PUNCT
ejpam-6125	194	6	{	{	PUNCT
ejpam-6125	194	7	vs	vs	ADP
ejpam-6125	194	8	:	:	PUNCT
ejpam-6125	194	9	s	s	X
ejpam-6125	194	10	∈	∈	PROPN
ejpam-6125	194	11	s	s	PART
ejpam-6125	194	12	⊆	⊆	NUM
ejpam-6125	194	13	{	{	PUNCT
ejpam-6125	194	14	1	1	NUM
ejpam-6125	194	15	,	,	PUNCT
ejpam-6125	194	16	2	2	NUM
ejpam-6125	194	17	,	,	PUNCT
ejpam-6125	194	18	.	.	PUNCT
ejpam-6125	194	19	.	.	PUNCT
ejpam-6125	195	1	.	.	PUNCT
ejpam-6125	196	1	,	,	PUNCT
ejpam-6125	196	2	k	k	X
ejpam-6125	196	3	}	}	PUNCT
ejpam-6125	196	4	}	}	PUNCT
ejpam-6125	196	5	and	and	CCONJ
ejpam-6125	196	6	q̂′	q̂′	PROPN
ejpam-6125	196	7	g2	g2	PROPN
ejpam-6125	196	8	=	=	PRON
ejpam-6125	196	9	{	{	PUNCT
ejpam-6125	196	10	vt	vt	NOUN
ejpam-6125	196	11	:	:	PUNCT
ejpam-6125	196	12	t	t	PROPN
ejpam-6125	196	13	∈	∈	PROPN
ejpam-6125	196	14	{	{	PUNCT
ejpam-6125	196	15	1	1	NUM
ejpam-6125	196	16	,	,	PUNCT
ejpam-6125	196	17	2	2	NUM
ejpam-6125	196	18	,	,	PUNCT
ejpam-6125	196	19	.	.	PUNCT
ejpam-6125	196	20	.	.	PUNCT
ejpam-6125	196	21	.	.	PUNCT
ejpam-6125	197	1	,	,	PUNCT
ejpam-6125	197	2	k	k	X
ejpam-6125	197	3	}	}	PUNCT
ejpam-6125	197	4	\	\	NOUN
ejpam-6125	197	5	s	s	X
ejpam-6125	197	6	}	}	PUNCT
ejpam-6125	197	7	.	.	PUNCT
ejpam-6125	198	1	then	then	ADV
ejpam-6125	198	2	v′t	v′t	PROPN
ejpam-6125	198	3	∈	∈	PROPN
ejpam-6125	198	4	q̂g2	q̂g2	NOUN
ejpam-6125	198	5	for	for	ADP
ejpam-6125	198	6	all	all	DET
ejpam-6125	198	7	t	t	NOUN
ejpam-6125	198	8	∈	∈	PROPN
ejpam-6125	198	9	{	{	PUNCT
ejpam-6125	198	10	1	1	NUM
ejpam-6125	198	11	,	,	PUNCT
ejpam-6125	198	12	2	2	NUM
ejpam-6125	198	13	,	,	PUNCT
ejpam-6125	198	14	.	.	PUNCT
ejpam-6125	198	15	.	.	PUNCT
ejpam-6125	198	16	.	.	PUNCT
ejpam-6125	199	1	,	,	PUNCT
ejpam-6125	199	2	vk	vk	VERB
ejpam-6125	199	3	}	}	PUNCT
ejpam-6125	199	4	\	\	NOUN
ejpam-6125	199	5	s.	s.	PROPN
ejpam-6125	199	6	since	since	SCONJ
ejpam-6125	199	7	n2	n2	PROPN
ejpam-6125	199	8	s(g)[v	s(g)[v	PROPN
ejpam-6125	199	9	′	′	NUM
ejpam-6125	199	10	t	t	PROPN
ejpam-6125	199	11	]	]	X
ejpam-6125	199	12	=	=	SYM
ejpam-6125	199	13	n2	n2	PROPN
ejpam-6125	199	14	s(g)[vt	s(g)[vt	PROPN
ejpam-6125	199	15	]	]	PUNCT
ejpam-6125	199	16	and	and	CCONJ
ejpam-6125	199	17	q	q	NOUN
ejpam-6125	199	18	is	be	AUX
ejpam-6125	199	19	a	a	DET
ejpam-6125	199	20	legal	legal	ADJ
ejpam-6125	199	21	closed	closed	ADJ
ejpam-6125	199	22	neighborhood	neighborhood	NOUN
ejpam-6125	199	23	sequence	sequence	NOUN
ejpam-6125	199	24	of	of	ADP
ejpam-6125	199	25	s(g	s(g	PROPN
ejpam-6125	199	26	)	)	PUNCT
ejpam-6125	199	27	,	,	PUNCT
ejpam-6125	199	28	it	it	PRON
ejpam-6125	199	29	follows	follow	VERB
ejpam-6125	199	30	that	that	SCONJ
ejpam-6125	199	31	n2	n2	PROPN
ejpam-6125	199	32	g[vi	g[vi	PROPN
ejpam-6125	199	33	]	]	PUNCT
ejpam-6125	199	34	\	\	PROPN
ejpam-6125	199	35	⋃i−1	⋃i−1	NOUN
ejpam-6125	199	36	j=1n	j=1n	PROPN
ejpam-6125	199	37	2	2	NUM
ejpam-6125	199	38	g[vj	g[vj	PROPN
ejpam-6125	199	39	]	]	PUNCT
ejpam-6125	199	40	̸=	̸=	PROPN
ejpam-6125	199	41	∅	∅	NOUN
ejpam-6125	199	42	for	for	ADP
ejpam-6125	199	43	all	all	PRON
ejpam-6125	199	44	i	i	PRON
ejpam-6125	199	45	∈	∈	PROPN
ejpam-6125	199	46	{	{	PUNCT
ejpam-6125	199	47	2	2	NUM
ejpam-6125	199	48	,	,	PUNCT
ejpam-6125	199	49	3	3	NUM
ejpam-6125	199	50	,	,	PUNCT
ejpam-6125	199	51	.	.	PUNCT
ejpam-6125	199	52	.	.	PUNCT
ejpam-6125	200	1	.	.	PUNCT
ejpam-6125	201	1	,	,	PUNCT
ejpam-6125	201	2	k	k	X
ejpam-6125	201	3	}	}	PUNCT
ejpam-6125	201	4	.	.	PUNCT
ejpam-6125	202	1	thus	thus	ADV
ejpam-6125	202	2	,	,	PUNCT
ejpam-6125	202	3	q̂g1	q̂g1	ADV
ejpam-6125	202	4	∪	∪	ADP
ejpam-6125	202	5	q̂′	q̂′	PROPN
ejpam-6125	202	6	g2	g2	PROPN
ejpam-6125	202	7	is	be	AUX
ejpam-6125	202	8	an	an	DET
ejpam-6125	202	9	lchni	lchni	PROPN
ejpam-6125	202	10	set	set	NOUN
ejpam-6125	202	11	of	of	ADP
ejpam-6125	202	12	g1	g1	PROPN
ejpam-6125	202	13	.	.	PUNCT
ejpam-6125	203	1	similarly	similarly	ADV
ejpam-6125	203	2	,	,	PUNCT
ejpam-6125	203	3	q̂′	q̂′	NOUN
ejpam-6125	203	4	g1	g1	PROPN
ejpam-6125	203	5	∪	∪	ADJ
ejpam-6125	203	6	q̂g2	q̂g2	NOUN
ejpam-6125	203	7	is	be	AUX
ejpam-6125	203	8	an	an	DET
ejpam-6125	203	9	lchni	lchni	VERB
ejpam-6125	203	10	set	set	NOUN
ejpam-6125	203	11	of	of	ADP
ejpam-6125	203	12	g2	g2	PROPN
ejpam-6125	203	13	.	.	PUNCT
ejpam-6125	204	1	the	the	DET
ejpam-6125	204	2	converse	converse	NOUN
ejpam-6125	204	3	is	be	AUX
ejpam-6125	204	4	clear	clear	ADJ
ejpam-6125	204	5	.	.	PUNCT
ejpam-6125	205	1	the	the	DET
ejpam-6125	205	2	following	following	ADJ
ejpam-6125	205	3	result	result	NOUN
ejpam-6125	205	4	follows	follow	VERB
ejpam-6125	205	5	from	from	ADP
ejpam-6125	205	6	theorem	theorem	ADJ
ejpam-6125	205	7	7	7	NUM
ejpam-6125	205	8	.	.	PUNCT
ejpam-6125	205	9	corollary	corollary	ADJ
ejpam-6125	205	10	2	2	NUM
ejpam-6125	205	11	.	.	PUNCT
ejpam-6125	206	1	let	let	VERB
ejpam-6125	206	2	g	g	PRON
ejpam-6125	206	3	be	be	AUX
ejpam-6125	206	4	a	a	DET
ejpam-6125	206	5	non	non	ADJ
ejpam-6125	206	6	-	-	ADJ
ejpam-6125	206	7	trivial	trivial	ADJ
ejpam-6125	206	8	connected	connected	ADJ
ejpam-6125	206	9	graph	graph	NOUN
ejpam-6125	206	10	.	.	PUNCT
ejpam-6125	207	1	then	then	ADV
ejpam-6125	207	2	θ(s(g	θ(s(g	PROPN
ejpam-6125	207	3	)	)	PUNCT
ejpam-6125	207	4	)	)	PUNCT
ejpam-6125	208	1	=	=	PUNCT
ejpam-6125	208	2	θ(g	θ(g	NUM
ejpam-6125	208	3	)	)	PUNCT
ejpam-6125	208	4	.	.	PUNCT
ejpam-6125	209	1	4	4	X
ejpam-6125	209	2	.	.	X
ejpam-6125	209	3	conclusion	conclusion	VERB
ejpam-6125	209	4	the	the	DET
ejpam-6125	209	5	concept	concept	NOUN
ejpam-6125	209	6	of	of	ADP
ejpam-6125	209	7	legal	legal	ADJ
ejpam-6125	209	8	closed	closed	ADJ
ejpam-6125	209	9	hop	hop	NOUN
ejpam-6125	209	10	neighborhood	neighborhood	NOUN
ejpam-6125	209	11	independent	independent	ADJ
ejpam-6125	209	12	sequence	sequence	NOUN
ejpam-6125	209	13	has	have	AUX
ejpam-6125	209	14	been	be	AUX
ejpam-6125	209	15	introduced	introduce	VERB
ejpam-6125	209	16	and	and	CCONJ
ejpam-6125	209	17	initially	initially	ADV
ejpam-6125	209	18	investigated	investigate	VERB
ejpam-6125	209	19	in	in	ADP
ejpam-6125	209	20	this	this	DET
ejpam-6125	209	21	study	study	NOUN
ejpam-6125	209	22	.	.	PUNCT
ejpam-6125	210	1	characterizations	characterization	NOUN
ejpam-6125	210	2	and	and	CCONJ
ejpam-6125	210	3	formulas	formula	NOUN
ejpam-6125	210	4	for	for	ADP
ejpam-6125	210	5	the	the	DET
ejpam-6125	210	6	legal	legal	ADJ
ejpam-6125	210	7	closed	close	VERB
ejpam-6125	210	8	hop	hop	NOUN
ejpam-6125	210	9	neighborhood	neighborhood	NOUN
ejpam-6125	210	10	independence	independence	NOUN
ejpam-6125	210	11	have	have	AUX
ejpam-6125	210	12	been	be	AUX
ejpam-6125	210	13	obtained	obtain	VERB
ejpam-6125	210	14	on	on	ADP
ejpam-6125	210	15	some	some	DET
ejpam-6125	210	16	special	special	ADJ
ejpam-6125	210	17	graphs	graph	NOUN
ejpam-6125	210	18	,	,	PUNCT
ejpam-6125	210	19	shadow	shadow	NOUN
ejpam-6125	210	20	graphs	graph	NOUN
ejpam-6125	210	21	,	,	PUNCT
ejpam-6125	210	22	and	and	CCONJ
ejpam-6125	210	23	on	on	ADP
ejpam-6125	210	24	the	the	DET
ejpam-6125	210	25	join	join	NOUN
ejpam-6125	210	26	of	of	ADP
ejpam-6125	210	27	any	any	DET
ejpam-6125	210	28	two	two	NUM
ejpam-6125	210	29	graphs	graph	NOUN
ejpam-6125	210	30	.	.	PUNCT
ejpam-6125	211	1	further	further	ADJ
ejpam-6125	211	2	exploration	exploration	NOUN
ejpam-6125	211	3	of	of	ADP
ejpam-6125	211	4	this	this	DET
ejpam-6125	211	5	parameter	parameter	NOUN
ejpam-6125	211	6	on	on	ADP
ejpam-6125	211	7	other	other	ADJ
ejpam-6125	211	8	graphs	graph	NOUN
ejpam-6125	211	9	under	under	ADP
ejpam-6125	211	10	some	some	DET
ejpam-6125	211	11	operations	operation	NOUN
ejpam-6125	211	12	,	,	PUNCT
ejpam-6125	211	13	as	as	ADV
ejpam-6125	211	14	well	well	ADV
ejpam-6125	211	15	as	as	ADP
ejpam-6125	211	16	its	its	PRON
ejpam-6125	211	17	relationships	relationship	NOUN
ejpam-6125	211	18	with	with	ADP
ejpam-6125	211	19	other	other	ADJ
ejpam-6125	211	20	parameters	parameter	NOUN
ejpam-6125	211	21	may	may	AUX
ejpam-6125	211	22	lead	lead	VERB
ejpam-6125	211	23	to	to	ADP
ejpam-6125	211	24	deep	deep	ADJ
ejpam-6125	211	25	insights	insight	NOUN
ejpam-6125	211	26	and	and	CCONJ
ejpam-6125	211	27	may	may	AUX
ejpam-6125	211	28	show	show	VERB
ejpam-6125	211	29	connections	connection	NOUN
ejpam-6125	211	30	with	with	ADP
ejpam-6125	211	31	other	other	ADJ
ejpam-6125	211	32	graph	graph	NOUN
ejpam-6125	211	33	properties	property	NOUN
ejpam-6125	211	34	.	.	PUNCT
ejpam-6125	212	1	interested	interested	ADJ
ejpam-6125	212	2	researchers	researcher	NOUN
ejpam-6125	212	3	may	may	AUX
ejpam-6125	212	4	provide	provide	VERB
ejpam-6125	212	5	a	a	DET
ejpam-6125	212	6	real	real	ADJ
ejpam-6125	212	7	-	-	PUNCT
ejpam-6125	212	8	world	world	NOUN
ejpam-6125	212	9	application	application	NOUN
ejpam-6125	212	10	of	of	ADP
ejpam-6125	212	11	the	the	DET
ejpam-6125	212	12	legal	legal	ADJ
ejpam-6125	212	13	closed	close	VERB
ejpam-6125	212	14	hop	hop	NOUN
ejpam-6125	212	15	neighborhood	neighborhood	NOUN
ejpam-6125	212	16	independent	independent	ADJ
ejpam-6125	212	17	,	,	PUNCT
ejpam-6125	212	18	and	and	CCONJ
ejpam-6125	212	19	may	may	AUX
ejpam-6125	212	20	investigate	investigate	VERB
ejpam-6125	212	21	the	the	DET
ejpam-6125	212	22	complexity	complexity	NOUN
ejpam-6125	212	23	of	of	ADP
ejpam-6125	212	24	this	this	DET
ejpam-6125	212	25	concept	concept	NOUN
ejpam-6125	212	26	.	.	PUNCT
ejpam-6125	213	1	acknowledgements	acknowledgement	NOUN
ejpam-6125	213	2	the	the	DET
ejpam-6125	213	3	authors	author	NOUN
ejpam-6125	213	4	would	would	AUX
ejpam-6125	213	5	like	like	VERB
ejpam-6125	213	6	to	to	PART
ejpam-6125	213	7	thank	thank	VERB
ejpam-6125	213	8	the	the	DET
ejpam-6125	213	9	referees	referee	NOUN
ejpam-6125	213	10	for	for	ADP
ejpam-6125	213	11	their	their	PRON
ejpam-6125	213	12	respective	respective	ADJ
ejpam-6125	213	13	invaluable	invaluable	ADJ
ejpam-6125	213	14	comments	comment	NOUN
ejpam-6125	213	15	and	and	CCONJ
ejpam-6125	213	16	suggestions	suggestion	NOUN
ejpam-6125	213	17	.	.	PUNCT
ejpam-6125	214	1	the	the	DET
ejpam-6125	214	2	authors	author	NOUN
ejpam-6125	214	3	are	be	AUX
ejpam-6125	214	4	thankful	thankful	ADJ
ejpam-6125	214	5	to	to	AUX
ejpam-6125	214	6	msu	msu	PROPN
ejpam-6125	214	7	tawi	tawi	PROPN
ejpam-6125	214	8	-	-	PUNCT
ejpam-6125	214	9	tawi	tawi	PROPN
ejpam-6125	214	10	college	college	PROPN
ejpam-6125	214	11	of	of	ADP
ejpam-6125	214	12	technology	technology	PROPN
ejpam-6125	214	13	and	and	CCONJ
ejpam-6125	214	14	j.	j.	PROPN
ejpam-6125	214	15	a.	a.	PROPN
ejpam-6125	214	16	hassan	hassan	PROPN
ejpam-6125	214	17	et	et	PROPN
ejpam-6125	214	18	al	al	PROPN
ejpam-6125	214	19	.	.	PUNCT
ejpam-6125	214	20	/	/	SYM
ejpam-6125	214	21	eur	eur	PROPN
ejpam-6125	214	22	.	.	PUNCT
ejpam-6125	215	1	j.	j.	PROPN
ejpam-6125	215	2	pure	pure	PROPN
ejpam-6125	215	3	appl	appl	PROPN
ejpam-6125	215	4	.	.	PROPN
ejpam-6125	215	5	math	math	PROPN
ejpam-6125	215	6	,	,	PUNCT
ejpam-6125	215	7	18	18	NUM
ejpam-6125	215	8	(	(	PUNCT
ejpam-6125	215	9	3	3	NUM
ejpam-6125	215	10	)	)	PUNCT
ejpam-6125	215	11	(	(	PUNCT
ejpam-6125	215	12	2025	2025	NUM
ejpam-6125	215	13	)	)	PUNCT
ejpam-6125	215	14	,	,	PUNCT
ejpam-6125	215	15	6125	6125	NUM
ejpam-6125	215	16	8	8	NUM
ejpam-6125	215	17	of	of	ADP
ejpam-6125	215	18	8	8	NUM
ejpam-6125	215	19	oceanography	oceanography	NOUN
ejpam-6125	215	20	,	,	PUNCT
ejpam-6125	215	21	korea	korea	PROPN
ejpam-6125	215	22	university	university	PROPN
ejpam-6125	215	23	,	,	PUNCT
ejpam-6125	215	24	and	and	CCONJ
ejpam-6125	215	25	ateneo	ateneo	PROPN
ejpam-6125	215	26	de	de	PROPN
ejpam-6125	215	27	davao	davao	PROPN
ejpam-6125	215	28	university	university	PROPN
ejpam-6125	215	29	for	for	ADP
ejpam-6125	215	30	the	the	DET
ejpam-6125	215	31	financial	financial	ADJ
ejpam-6125	215	32	support	support	NOUN
ejpam-6125	215	33	they	they	PRON
ejpam-6125	215	34	have	have	AUX
ejpam-6125	215	35	extended	extend	VERB
ejpam-6125	215	36	.	.	PUNCT
ejpam-6125	216	1	references	reference	NOUN
ejpam-6125	216	2	[	[	X
ejpam-6125	216	3	1	1	NUM
ejpam-6125	216	4	]	]	PUNCT
ejpam-6125	216	5	e.	e.	PROPN
ejpam-6125	216	6	davies	davies	PROPN
ejpam-6125	216	7	,	,	PUNCT
ejpam-6125	216	8	m.	m.	PROPN
ejpam-6125	216	9	jenssen	jenssen	PROPN
ejpam-6125	216	10	,	,	PUNCT
ejpam-6125	216	11	w.	w.	PROPN
ejpam-6125	216	12	perkins	perkins	PROPN
ejpam-6125	216	13	,	,	PUNCT
ejpam-6125	216	14	and	and	CCONJ
ejpam-6125	216	15	b.	b.	PROPN
ejpam-6125	216	16	roberts	roberts	PROPN
ejpam-6125	216	17	.	.	PUNCT
ejpam-6125	217	1	independent	independent	ADJ
ejpam-6125	217	2	sets	set	NOUN
ejpam-6125	217	3	,	,	PUNCT
ejpam-6125	217	4	matchings	matching	NOUN
ejpam-6125	217	5	,	,	PUNCT
ejpam-6125	217	6	and	and	CCONJ
ejpam-6125	217	7	occupancy	occupancy	NOUN
ejpam-6125	217	8	fractions	fraction	NOUN
ejpam-6125	217	9	.	.	PUNCT
ejpam-6125	218	1	journal	journal	NOUN
ejpam-6125	218	2	of	of	ADP
ejpam-6125	218	3	the	the	DET
ejpam-6125	218	4	london	london	PROPN
ejpam-6125	218	5	mathematical	mathematical	ADJ
ejpam-6125	218	6	society	society	NOUN
ejpam-6125	218	7	,	,	PUNCT
ejpam-6125	218	8	96:211–220	96:211–220	NUM
ejpam-6125	218	9	,	,	PUNCT
ejpam-6125	218	10	2017	2017	NUM
ejpam-6125	218	11	.	.	PUNCT
ejpam-6125	219	1	[	[	X
ejpam-6125	219	2	2	2	X
ejpam-6125	219	3	]	]	PUNCT
ejpam-6125	219	4	e.	e.	PROPN
ejpam-6125	219	5	davies	davies	PROPN
ejpam-6125	219	6	,	,	PUNCT
ejpam-6125	219	7	m.	m.	PROPN
ejpam-6125	219	8	jenssen	jenssen	PROPN
ejpam-6125	219	9	,	,	PUNCT
ejpam-6125	219	10	w.	w.	PROPN
ejpam-6125	219	11	perkins	perkins	PROPN
ejpam-6125	219	12	,	,	PUNCT
ejpam-6125	219	13	and	and	CCONJ
ejpam-6125	219	14	b.	b.	PROPN
ejpam-6125	219	15	roberts	roberts	PROPN
ejpam-6125	219	16	.	.	PUNCT
ejpam-6125	220	1	on	on	ADP
ejpam-6125	220	2	the	the	DET
ejpam-6125	220	3	average	average	ADJ
ejpam-6125	220	4	size	size	NOUN
ejpam-6125	220	5	of	of	ADP
ejpam-6125	220	6	independent	independent	ADJ
ejpam-6125	220	7	sets	set	NOUN
ejpam-6125	220	8	in	in	ADP
ejpam-6125	220	9	triangle	triangle	NOUN
ejpam-6125	220	10	-	-	PUNCT
ejpam-6125	220	11	free	free	ADJ
ejpam-6125	220	12	graphs	graph	NOUN
ejpam-6125	220	13	.	.	PUNCT
ejpam-6125	221	1	proceedings	proceeding	NOUN
ejpam-6125	221	2	of	of	ADP
ejpam-6125	221	3	the	the	DET
ejpam-6125	221	4	american	american	PROPN
ejpam-6125	221	5	mathematical	mathematical	PROPN
ejpam-6125	221	6	society	society	NOUN
ejpam-6125	221	7	,	,	PUNCT
ejpam-6125	221	8	146(1):111–124	146(1):111–124	NUM
ejpam-6125	221	9	,	,	PUNCT
ejpam-6125	221	10	2018	2018	NUM
ejpam-6125	221	11	.	.	PUNCT
ejpam-6125	222	1	[	[	X
ejpam-6125	222	2	3	3	X
ejpam-6125	222	3	]	]	PUNCT
ejpam-6125	222	4	z.	z.	PROPN
ejpam-6125	222	5	füredi	füredi	PROPN
ejpam-6125	222	6	.	.	PUNCT
ejpam-6125	223	1	the	the	DET
ejpam-6125	223	2	number	number	NOUN
ejpam-6125	223	3	of	of	ADP
ejpam-6125	223	4	maximal	maximal	ADJ
ejpam-6125	223	5	independent	independent	ADJ
ejpam-6125	223	6	sets	set	NOUN
ejpam-6125	223	7	in	in	ADP
ejpam-6125	223	8	connected	connected	ADJ
ejpam-6125	223	9	graphs	graph	NOUN
ejpam-6125	223	10	.	.	PUNCT
ejpam-6125	224	1	journal	journal	NOUN
ejpam-6125	224	2	of	of	ADP
ejpam-6125	224	3	graph	graph	NOUN
ejpam-6125	224	4	theory	theory	NOUN
ejpam-6125	224	5	,	,	PUNCT
ejpam-6125	224	6	11(4):463–470	11(4):463–470	NOUN
ejpam-6125	224	7	,	,	PUNCT
ejpam-6125	224	8	1987	1987	NUM
ejpam-6125	224	9	.	.	PUNCT
ejpam-6125	225	1	[	[	X
ejpam-6125	225	2	4	4	X
ejpam-6125	225	3	]	]	PUNCT
ejpam-6125	225	4	j.	j.	PROPN
ejpam-6125	225	5	r.	r.	PROPN
ejpam-6125	225	6	griggs	griggs	PROPN
ejpam-6125	225	7	,	,	PUNCT
ejpam-6125	225	8	c.	c.	PROPN
ejpam-6125	225	9	m.	m.	PROPN
ejpam-6125	225	10	grinstead	grinstead	PROPN
ejpam-6125	225	11	,	,	PUNCT
ejpam-6125	225	12	and	and	CCONJ
ejpam-6125	225	13	d.	d.	PROPN
ejpam-6125	225	14	r.	r.	PROPN
ejpam-6125	225	15	guichard	guichard	PROPN
ejpam-6125	225	16	.	.	PUNCT
ejpam-6125	226	1	the	the	DET
ejpam-6125	226	2	number	number	NOUN
ejpam-6125	226	3	of	of	ADP
ejpam-6125	226	4	maximal	maximal	ADJ
ejpam-6125	226	5	independent	independent	ADJ
ejpam-6125	226	6	sets	set	NOUN
ejpam-6125	226	7	in	in	ADP
ejpam-6125	226	8	a	a	DET
ejpam-6125	226	9	connected	connected	ADJ
ejpam-6125	226	10	graph	graph	NOUN
ejpam-6125	226	11	.	.	PUNCT
ejpam-6125	226	12	discrete	discrete	ADJ
ejpam-6125	226	13	mathematics	mathematic	NOUN
ejpam-6125	226	14	,	,	PUNCT
ejpam-6125	226	15	68:211–220	68:211–220	PROPN
ejpam-6125	226	16	,	,	PUNCT
ejpam-6125	226	17	1988	1988	NUM
ejpam-6125	226	18	.	.	PUNCT
ejpam-6125	227	1	[	[	X
ejpam-6125	227	2	5	5	X
ejpam-6125	227	3	]	]	X
ejpam-6125	227	4	g.	g.	PROPN
ejpam-6125	227	5	hopkins	hopkins	PROPN
ejpam-6125	227	6	and	and	CCONJ
ejpam-6125	227	7	w.	w.	PROPN
ejpam-6125	227	8	staton	staton	PROPN
ejpam-6125	227	9	.	.	PUNCT
ejpam-6125	228	1	graphs	graph	NOUN
ejpam-6125	228	2	with	with	ADP
ejpam-6125	228	3	unique	unique	ADJ
ejpam-6125	228	4	maximum	maximum	ADJ
ejpam-6125	228	5	independent	independent	ADJ
ejpam-6125	228	6	sets	set	NOUN
ejpam-6125	228	7	.	.	PUNCT
ejpam-6125	229	1	discrete	discrete	ADJ
ejpam-6125	229	2	mathematics	mathematic	NOUN
ejpam-6125	229	3	,	,	PUNCT
ejpam-6125	229	4	57:245–251	57:245–251	PROPN
ejpam-6125	229	5	,	,	PUNCT
ejpam-6125	229	6	1985	1985	NUM
ejpam-6125	229	7	.	.	PUNCT
ejpam-6125	230	1	[	[	X
ejpam-6125	230	2	6	6	NUM
ejpam-6125	230	3	]	]	PUNCT
ejpam-6125	230	4	d.	d.	PROPN
ejpam-6125	230	5	g.	g.	PROPN
ejpam-6125	230	6	c.	c.	PROPN
ejpam-6125	230	7	horrocks	horrocks	PROPN
ejpam-6125	230	8	.	.	PUNCT
ejpam-6125	231	1	doubly	doubly	ADV
ejpam-6125	231	2	independent	independent	ADJ
ejpam-6125	231	3	sets	set	NOUN
ejpam-6125	231	4	in	in	ADP
ejpam-6125	231	5	graphs	graph	NOUN
ejpam-6125	231	6	.	.	PUNCT
ejpam-6125	232	1	australasian	australasian	ADJ
ejpam-6125	232	2	journal	journal	NOUN
ejpam-6125	232	3	of	of	ADP
ejpam-6125	232	4	combinatorics	combinatoric	NOUN
ejpam-6125	232	5	,	,	PUNCT
ejpam-6125	232	6	22:105–116	22:105–116	PROPN
ejpam-6125	232	7	,	,	PUNCT
ejpam-6125	232	8	2000	2000	NUM
ejpam-6125	232	9	.	.	PUNCT
ejpam-6125	233	1	[	[	X
ejpam-6125	233	2	7	7	X
ejpam-6125	233	3	]	]	X
ejpam-6125	233	4	d.	d.	PROPN
ejpam-6125	233	5	s.	s.	PROPN
ejpam-6125	233	6	johnson	johnson	PROPN
ejpam-6125	233	7	,	,	PUNCT
ejpam-6125	233	8	m.	m.	NOUN
ejpam-6125	233	9	yannakakis	yannakakis	PROPN
ejpam-6125	233	10	,	,	PUNCT
ejpam-6125	233	11	and	and	CCONJ
ejpam-6125	233	12	c.	c.	PROPN
ejpam-6125	233	13	j.	j.	PROPN
ejpam-6125	233	14	papadimitriou	papadimitriou	PROPN
ejpam-6125	233	15	.	.	PUNCT
ejpam-6125	234	1	on	on	ADP
ejpam-6125	234	2	generating	generate	VERB
ejpam-6125	234	3	all	all	DET
ejpam-6125	234	4	maximal	maximal	ADJ
ejpam-6125	234	5	independent	independent	ADJ
ejpam-6125	234	6	sets	set	NOUN
ejpam-6125	234	7	.	.	PUNCT
ejpam-6125	235	1	information	information	NOUN
ejpam-6125	235	2	processing	processing	NOUN
ejpam-6125	235	3	letters	letter	NOUN
ejpam-6125	235	4	,	,	PUNCT
ejpam-6125	235	5	27:119–123	27:119–123	NOUN
ejpam-6125	235	6	,	,	PUNCT
ejpam-6125	235	7	1988	1988	NUM
ejpam-6125	235	8	.	.	PUNCT
ejpam-6125	236	1	[	[	X
ejpam-6125	236	2	8	8	X
ejpam-6125	236	3	]	]	PUNCT
ejpam-6125	236	4	h.	h.	PROPN
ejpam-6125	236	5	s.	s.	PROPN
ejpam-6125	236	6	wilf	wilf	PROPN
ejpam-6125	236	7	.	.	PUNCT
ejpam-6125	237	1	the	the	DET
ejpam-6125	237	2	number	number	NOUN
ejpam-6125	237	3	of	of	ADP
ejpam-6125	237	4	maximal	maximal	ADJ
ejpam-6125	237	5	independent	independent	ADJ
ejpam-6125	237	6	sets	set	NOUN
ejpam-6125	237	7	in	in	ADP
ejpam-6125	237	8	a	a	DET
ejpam-6125	237	9	tree	tree	NOUN
ejpam-6125	237	10	.	.	PUNCT
ejpam-6125	238	1	siam	siam	PROPN
ejpam-6125	238	2	journal	journal	PROPN
ejpam-6125	238	3	on	on	ADP
ejpam-6125	238	4	algebraic	algebraic	ADJ
ejpam-6125	238	5	and	and	CCONJ
ejpam-6125	238	6	discrete	discrete	ADJ
ejpam-6125	238	7	methods	method	NOUN
ejpam-6125	238	8	,	,	PUNCT
ejpam-6125	238	9	7:125–130	7:125–130	NUM
ejpam-6125	238	10	,	,	PUNCT
ejpam-6125	238	11	1986	1986	NUM
ejpam-6125	238	12	.	.	PUNCT
ejpam-6125	239	1	[	[	X
ejpam-6125	239	2	9	9	NUM
ejpam-6125	239	3	]	]	PUNCT
ejpam-6125	239	4	j.	j.	PROPN
ejpam-6125	239	5	zito	zito	PROPN
ejpam-6125	239	6	.	.	PUNCT
ejpam-6125	240	1	the	the	DET
ejpam-6125	240	2	structure	structure	NOUN
ejpam-6125	240	3	and	and	CCONJ
ejpam-6125	240	4	maximum	maximum	ADJ
ejpam-6125	240	5	number	number	NOUN
ejpam-6125	240	6	of	of	ADP
ejpam-6125	240	7	maximum	maximum	ADJ
ejpam-6125	240	8	independent	independent	ADJ
ejpam-6125	240	9	sets	set	NOUN
ejpam-6125	240	10	in	in	ADP
ejpam-6125	240	11	trees	tree	NOUN
ejpam-6125	240	12	.	.	PUNCT
ejpam-6125	241	1	journal	journal	PROPN
ejpam-6125	241	2	of	of	ADP
ejpam-6125	241	3	graph	graph	NOUN
ejpam-6125	241	4	theory	theory	NOUN
ejpam-6125	241	5	,	,	PUNCT
ejpam-6125	241	6	15(2):207–221	15(2):207–221	NUM
ejpam-6125	241	7	,	,	PUNCT
ejpam-6125	241	8	1991	1991	NUM
ejpam-6125	241	9	.	.	PUNCT
ejpam-6125	242	1	[	[	X
ejpam-6125	242	2	10	10	NUM
ejpam-6125	242	3	]	]	X
ejpam-6125	242	4	j.	j.	PROPN
ejpam-6125	242	5	hassan	hassan	PROPN
ejpam-6125	242	6	,	,	PUNCT
ejpam-6125	242	7	s.	s.	PROPN
ejpam-6125	242	8	canoy	canoy	PROPN
ejpam-6125	242	9	jr	jr	PROPN
ejpam-6125	242	10	.	.	PROPN
ejpam-6125	242	11	,	,	PUNCT
ejpam-6125	242	12	and	and	CCONJ
ejpam-6125	242	13	a.	a.	PROPN
ejpam-6125	242	14	aradais	aradais	PROPN
ejpam-6125	242	15	.	.	PUNCT
ejpam-6125	243	1	hop	hop	PROPN
ejpam-6125	243	2	independent	independent	ADJ
ejpam-6125	243	3	sets	set	NOUN
ejpam-6125	243	4	in	in	ADP
ejpam-6125	243	5	graphs	graph	NOUN
ejpam-6125	243	6	.	.	PUNCT
ejpam-6125	244	1	european	european	ADJ
ejpam-6125	244	2	journal	journal	PROPN
ejpam-6125	244	3	of	of	ADP
ejpam-6125	244	4	pure	pure	ADJ
ejpam-6125	244	5	and	and	CCONJ
ejpam-6125	244	6	applied	applied	ADJ
ejpam-6125	244	7	mathematics	mathematic	NOUN
ejpam-6125	244	8	,	,	PUNCT
ejpam-6125	244	9	15(2):467–477	15(2):467–477	PROPN
ejpam-6125	244	10	,	,	PUNCT
ejpam-6125	244	11	2022	2022	NUM
ejpam-6125	244	12	.	.	PUNCT
ejpam-6125	245	1	[	[	X
ejpam-6125	245	2	11	11	NUM
ejpam-6125	245	3	]	]	PUNCT
ejpam-6125	245	4	j.	j.	PROPN
ejpam-6125	245	5	hassan	hassan	PROPN
ejpam-6125	245	6	,	,	PUNCT
ejpam-6125	245	7	m.	m.	NOUN
ejpam-6125	245	8	langamin	langamin	PROPN
ejpam-6125	245	9	,	,	PUNCT
ejpam-6125	245	10	a.	a.	NOUN
ejpam-6125	245	11	laja	laja	PROPN
ejpam-6125	245	12	,	,	PUNCT
ejpam-6125	245	13	b.	b.	PROPN
ejpam-6125	245	14	amiruddin	amiruddin	PROPN
ejpam-6125	245	15	-	-	PUNCT
ejpam-6125	245	16	rajik	rajik	NOUN
ejpam-6125	245	17	,	,	PUNCT
ejpam-6125	245	18	e.	e.	PROPN
ejpam-6125	245	19	ahmad	ahmad	PROPN
ejpam-6125	245	20	,	,	PUNCT
ejpam-6125	245	21	and	and	CCONJ
ejpam-6125	245	22	j.	j.	PROPN
ejpam-6125	245	23	manditong	manditong	PROPN
ejpam-6125	245	24	.	.	PUNCT
ejpam-6125	246	1	legal	legal	ADJ
ejpam-6125	246	2	hop	hop	NOUN
ejpam-6125	246	3	independent	independent	ADJ
ejpam-6125	246	4	sequences	sequence	NOUN
ejpam-6125	246	5	in	in	ADP
ejpam-6125	246	6	graphs	graph	NOUN
ejpam-6125	246	7	.	.	PUNCT
ejpam-6125	247	1	european	european	ADJ
ejpam-6125	247	2	journal	journal	PROPN
ejpam-6125	247	3	of	of	ADP
ejpam-6125	247	4	pure	pure	ADJ
ejpam-6125	247	5	and	and	CCONJ
ejpam-6125	247	6	applied	applied	ADJ
ejpam-6125	247	7	mathematics	mathematic	NOUN
ejpam-6125	247	8	,	,	PUNCT
ejpam-6125	247	9	17(2):725–735	17(2):725–735	NUM
ejpam-6125	247	10	,	,	PUNCT
ejpam-6125	247	11	2024	2024	NUM
ejpam-6125	247	12	.	.	PUNCT
ejpam-6125	248	1	[	[	X
ejpam-6125	248	2	12	12	NUM
ejpam-6125	248	3	]	]	PUNCT
ejpam-6125	248	4	j.	j.	PROPN
ejpam-6125	248	5	hassan	hassan	PROPN
ejpam-6125	248	6	,	,	PUNCT
ejpam-6125	248	7	a.	a.	NOUN
ejpam-6125	248	8	tapeing	tapeing	NOUN
ejpam-6125	248	9	,	,	PUNCT
ejpam-6125	248	10	h.	h.	PROPN
ejpam-6125	248	11	copel	copel	PROPN
ejpam-6125	248	12	,	,	PUNCT
ejpam-6125	248	13	a.	a.	NOUN
ejpam-6125	248	14	bakkang	bakkang	PROPN
ejpam-6125	248	15	,	,	PUNCT
ejpam-6125	248	16	and	and	CCONJ
ejpam-6125	248	17	s.	s.	PROPN
ejpam-6125	248	18	d.	d.	PROPN
ejpam-6125	248	19	aming	aming	PROPN
ejpam-6125	248	20	.	.	PUNCT
ejpam-6125	249	1	j2	j2	PROPN
ejpam-6125	249	2	-	-	PUNCT
ejpam-6125	249	3	independence	independence	NOUN
ejpam-6125	249	4	parameters	parameter	NOUN
ejpam-6125	249	5	of	of	ADP
ejpam-6125	249	6	some	some	DET
ejpam-6125	249	7	graphs	graph	NOUN
ejpam-6125	249	8	.	.	PUNCT
ejpam-6125	250	1	european	european	ADJ
ejpam-6125	250	2	journal	journal	PROPN
ejpam-6125	250	3	of	of	ADP
ejpam-6125	250	4	pure	pure	ADJ
ejpam-6125	250	5	and	and	CCONJ
ejpam-6125	250	6	applied	applied	ADJ
ejpam-6125	250	7	mathematics	mathematic	NOUN
ejpam-6125	250	8	,	,	PUNCT
ejpam-6125	250	9	17(1):124–134	17(1):124–134	NUM
ejpam-6125	250	10	,	,	PUNCT
ejpam-6125	250	11	2024	2024	NUM
ejpam-6125	250	12	.	.	PUNCT
ejpam-6125	251	1	[	[	X
ejpam-6125	251	2	13	13	NUM
ejpam-6125	251	3	]	]	PUNCT
ejpam-6125	251	4	j.	j.	PROPN
ejpam-6125	251	5	hassan	hassan	PROPN
ejpam-6125	251	6	,	,	PUNCT
ejpam-6125	251	7	n.	n.	PROPN
ejpam-6125	251	8	bakar	bakar	PROPN
ejpam-6125	251	9	,	,	PUNCT
ejpam-6125	251	10	n.	n.	PROPN
ejpam-6125	251	11	dagsaan	dagsaan	PROPN
ejpam-6125	251	12	,	,	PUNCT
ejpam-6125	251	13	m.	m.	NOUN
ejpam-6125	251	14	langamin	langamin	PROPN
ejpam-6125	251	15	,	,	PUNCT
ejpam-6125	251	16	and	and	CCONJ
ejpam-6125	251	17	n.	n.	PROPN
ejpam-6125	251	18	h.	h.	PROPN
ejpam-6125	251	19	mohammad	mohammad	PROPN
ejpam-6125	251	20	.	.	PUNCT
ejpam-6125	252	1	j	j	ADJ
ejpam-6125	252	2	-	-	ADJ
ejpam-6125	252	3	open	open	ADJ
ejpam-6125	252	4	independent	independent	ADJ
ejpam-6125	252	5	sets	set	NOUN
ejpam-6125	252	6	in	in	ADP
ejpam-6125	252	7	graphs	graph	NOUN
ejpam-6125	252	8	.	.	PUNCT
ejpam-6125	253	1	european	european	ADJ
ejpam-6125	253	2	journal	journal	PROPN
ejpam-6125	253	3	of	of	ADP
ejpam-6125	253	4	pure	pure	ADJ
ejpam-6125	253	5	and	and	CCONJ
ejpam-6125	253	6	applied	applied	ADJ
ejpam-6125	253	7	mathematics	mathematic	NOUN
ejpam-6125	253	8	,	,	PUNCT
ejpam-6125	253	9	17(2):922–930	17(2):922–930	NUM
ejpam-6125	253	10	,	,	PUNCT
ejpam-6125	253	11	2024	2024	NUM
ejpam-6125	253	12	.	.	PUNCT
ejpam-6125	254	1	[	[	X
ejpam-6125	254	2	14	14	NUM
ejpam-6125	254	3	]	]	X
ejpam-6125	254	4	s.	s.	PROPN
ejpam-6125	254	5	kaida	kaida	PROPN
ejpam-6125	254	6	,	,	PUNCT
ejpam-6125	254	7	k.	k.	PROPN
ejpam-6125	254	8	j.	j.	PROPN
ejpam-6125	254	9	maharajul	maharajul	PROPN
ejpam-6125	254	10	,	,	PUNCT
ejpam-6125	254	11	j.	j.	PROPN
ejpam-6125	254	12	hassan	hassan	PROPN
ejpam-6125	254	13	,	,	PUNCT
ejpam-6125	254	14	l.	l.	PROPN
ejpam-6125	254	15	laja	laja	PROPN
ejpam-6125	254	16	,	,	PUNCT
ejpam-6125	254	17	a.	a.	NOUN
ejpam-6125	254	18	lintasan	lintasan	NOUN
ejpam-6125	254	19	,	,	PUNCT
ejpam-6125	254	20	and	and	CCONJ
ejpam-6125	254	21	a.	a.	NOUN
ejpam-6125	254	22	pablo	pablo	PROPN
ejpam-6125	254	23	.	.	PUNCT
ejpam-6125	255	1	certified	certify	VERB
ejpam-6125	255	2	hop	hop	NOUN
ejpam-6125	255	3	independence	independence	NOUN
ejpam-6125	255	4	:	:	PUNCT
ejpam-6125	255	5	properties	property	NOUN
ejpam-6125	255	6	and	and	CCONJ
ejpam-6125	255	7	connections	connection	NOUN
ejpam-6125	255	8	with	with	ADP
ejpam-6125	255	9	other	other	ADJ
ejpam-6125	255	10	variants	variant	NOUN
ejpam-6125	255	11	of	of	ADP
ejpam-6125	255	12	independence	independence	NOUN
ejpam-6125	255	13	.	.	PUNCT
ejpam-6125	256	1	european	european	ADJ
ejpam-6125	256	2	journal	journal	PROPN
ejpam-6125	256	3	of	of	ADP
ejpam-6125	256	4	pure	pure	ADJ
ejpam-6125	256	5	and	and	CCONJ
ejpam-6125	256	6	applied	applied	ADJ
ejpam-6125	256	7	mathematics	mathematic	NOUN
ejpam-6125	256	8	,	,	PUNCT
ejpam-6125	256	9	17(1):435–444	17(1):435–444	NUM
ejpam-6125	256	10	,	,	PUNCT
ejpam-6125	256	11	2024	2024	NUM
ejpam-6125	256	12	.	.	PUNCT
ejpam-6125	257	1	[	[	X
ejpam-6125	257	2	15	15	NUM
ejpam-6125	257	3	]	]	X
ejpam-6125	257	4	j.	j.	PROPN
ejpam-6125	257	5	hassan	hassan	PROPN
ejpam-6125	257	6	and	and	CCONJ
ejpam-6125	257	7	s.	s.	PROPN
ejpam-6125	257	8	canoy	canoy	PROPN
ejpam-6125	257	9	jr	jr	PROPN
ejpam-6125	257	10	.	.	PUNCT
ejpam-6125	258	1	grundy	grundy	PROPN
ejpam-6125	258	2	hop	hop	PROPN
ejpam-6125	258	3	domination	domination	PROPN
ejpam-6125	258	4	in	in	ADP
ejpam-6125	258	5	graphs	graph	NOUN
ejpam-6125	258	6	.	.	PUNCT
ejpam-6125	259	1	european	european	ADJ
ejpam-6125	259	2	journal	journal	PROPN
ejpam-6125	259	3	of	of	ADP
ejpam-6125	259	4	pure	pure	ADJ
ejpam-6125	259	5	and	and	CCONJ
ejpam-6125	259	6	applied	applied	ADJ
ejpam-6125	259	7	mathematics	mathematic	NOUN
ejpam-6125	259	8	,	,	PUNCT
ejpam-6125	259	9	15(4):1623–1636	15(4):1623–1636	NUM
ejpam-6125	259	10	,	,	PUNCT
ejpam-6125	259	11	2022	2022	NUM
ejpam-6125	259	12	.	.	PUNCT