id	sid	tid	token	lemma	pos
ejpam-6117	1	1	european	european	PROPN
ejpam-6117	1	2	journal	journal	PROPN
ejpam-6117	1	3	of	of	ADP
ejpam-6117	1	4	pure	pure	ADJ
ejpam-6117	1	5	and	and	CCONJ
ejpam-6117	1	6	applied	applied	ADJ
ejpam-6117	1	7	mathematics	mathematic	NOUN
ejpam-6117	1	8	2025	2025	NUM
ejpam-6117	1	9	,	,	PUNCT
ejpam-6117	1	10	vol	vol	NOUN
ejpam-6117	1	11	.	.	PROPN
ejpam-6117	1	12	18	18	NUM
ejpam-6117	1	13	,	,	PUNCT
ejpam-6117	1	14	issue	issue	NOUN
ejpam-6117	1	15	2	2	NUM
ejpam-6117	1	16	,	,	PUNCT
ejpam-6117	1	17	article	article	NOUN
ejpam-6117	1	18	number	number	NOUN
ejpam-6117	1	19	6117	6117	NUM
ejpam-6117	1	20	issn	issn	PROPN
ejpam-6117	1	21	1307	1307	NUM
ejpam-6117	1	22	-	-	SYM
ejpam-6117	1	23	5543	5543	NUM
ejpam-6117	1	24	–	–	PUNCT
ejpam-6117	1	25	ejpam.com	ejpam.com	X
ejpam-6117	1	26	published	publish	VERB
ejpam-6117	1	27	by	by	ADP
ejpam-6117	1	28	new	new	PROPN
ejpam-6117	1	29	york	york	PROPN
ejpam-6117	1	30	business	business	PROPN
ejpam-6117	1	31	global	global	ADJ
ejpam-6117	1	32	weakly	weakly	ADJ
ejpam-6117	1	33	connected	connected	ADJ
ejpam-6117	1	34	independence	independence	NOUN
ejpam-6117	1	35	number	number	NOUN
ejpam-6117	1	36	of	of	ADP
ejpam-6117	1	37	a	a	DET
ejpam-6117	1	38	graph	graph	NOUN
ejpam-6117	1	39	reignver	reignver	ADP
ejpam-6117	1	40	merontos1,∗	merontos1,∗	NOUN
ejpam-6117	1	41	,	,	PUNCT
ejpam-6117	1	42	imelda	imelda	PROPN
ejpam-6117	1	43	s.	s.	PROPN
ejpam-6117	1	44	aniversario1,2	aniversario1,2	PROPN
ejpam-6117	1	45	,	,	PUNCT
ejpam-6117	1	46	michael	michael	PROPN
ejpam-6117	1	47	b.	b.	PROPN
ejpam-6117	1	48	frondoza1,2	frondoza1,2	PROPN
ejpam-6117	1	49	,	,	PUNCT
ejpam-6117	1	50	sergio	sergio	PROPN
ejpam-6117	1	51	r.	r.	PROPN
ejpam-6117	1	52	canoy	canoy	PROPN
ejpam-6117	1	53	,	,	PUNCT
ejpam-6117	1	54	jr.1,2	jr.1,2	ADJ
ejpam-6117	1	55	1	1	NUM
ejpam-6117	1	56	department	department	NOUN
ejpam-6117	1	57	of	of	ADP
ejpam-6117	1	58	mathematics	mathematic	NOUN
ejpam-6117	1	59	and	and	CCONJ
ejpam-6117	1	60	statistics	statistic	NOUN
ejpam-6117	1	61	,	,	PUNCT
ejpam-6117	1	62	college	college	NOUN
ejpam-6117	1	63	of	of	ADP
ejpam-6117	1	64	science	science	NOUN
ejpam-6117	1	65	and	and	CCONJ
ejpam-6117	1	66	mathematics	mathematic	NOUN
ejpam-6117	1	67	,	,	PUNCT
ejpam-6117	1	68	msu	msu	PROPN
ejpam-6117	1	69	-	-	PUNCT
ejpam-6117	1	70	iligan	iligan	PROPN
ejpam-6117	1	71	institute	institute	PROPN
ejpam-6117	1	72	of	of	ADP
ejpam-6117	1	73	technology	technology	PROPN
ejpam-6117	1	74	,	,	PUNCT
ejpam-6117	1	75	iligan	iligan	PROPN
ejpam-6117	1	76	city	city	PROPN
ejpam-6117	1	77	,	,	PUNCT
ejpam-6117	1	78	philippines	philippine	NOUN
ejpam-6117	1	79	2	2	NUM
ejpam-6117	1	80	center	center	NOUN
ejpam-6117	1	81	of	of	ADP
ejpam-6117	1	82	mathematical	mathematical	ADJ
ejpam-6117	1	83	and	and	CCONJ
ejpam-6117	1	84	theoretical	theoretical	ADJ
ejpam-6117	1	85	physical	physical	ADJ
ejpam-6117	1	86	sciences	science	NOUN
ejpam-6117	1	87	prism	prism	NOUN
ejpam-6117	1	88	,	,	PUNCT
ejpam-6117	1	89	msu	msu	PROPN
ejpam-6117	1	90	-	-	PUNCT
ejpam-6117	1	91	iligan	iligan	PROPN
ejpam-6117	1	92	institute	institute	PROPN
ejpam-6117	1	93	of	of	ADP
ejpam-6117	1	94	technology	technology	PROPN
ejpam-6117	1	95	,	,	PUNCT
ejpam-6117	1	96	iligan	iligan	PROPN
ejpam-6117	1	97	city	city	PROPN
ejpam-6117	1	98	,	,	PUNCT
ejpam-6117	1	99	philippines	philippine	NOUN
ejpam-6117	1	100	abstract	abstract	ADJ
ejpam-6117	1	101	.	.	PUNCT
ejpam-6117	2	1	let	let	VERB
ejpam-6117	2	2	g	g	PRON
ejpam-6117	2	3	be	be	AUX
ejpam-6117	2	4	a	a	DET
ejpam-6117	2	5	simple	simple	ADJ
ejpam-6117	2	6	undirected	undirected	ADJ
ejpam-6117	2	7	connected	connected	ADJ
ejpam-6117	2	8	graph	graph	NOUN
ejpam-6117	2	9	with	with	ADP
ejpam-6117	2	10	vertex	vertex	NOUN
ejpam-6117	2	11	and	and	CCONJ
ejpam-6117	2	12	edge	edge	NOUN
ejpam-6117	2	13	sets	set	NOUN
ejpam-6117	2	14	v	v	ADP
ejpam-6117	2	15	(	(	PUNCT
ejpam-6117	2	16	g	g	NOUN
ejpam-6117	2	17	)	)	PUNCT
ejpam-6117	2	18	and	and	CCONJ
ejpam-6117	2	19	e(g	e(g	PROPN
ejpam-6117	2	20	)	)	PUNCT
ejpam-6117	2	21	,	,	PUNCT
ejpam-6117	2	22	respectively	respectively	ADV
ejpam-6117	2	23	.	.	PUNCT
ejpam-6117	3	1	the	the	DET
ejpam-6117	3	2	subgraph	subgraph	PROPN
ejpam-6117	3	3	⟨s⟩w	⟨s⟩w	PROPN
ejpam-6117	3	4	of	of	ADP
ejpam-6117	3	5	s	s	NOUN
ejpam-6117	3	6	⊆	⊆	NUM
ejpam-6117	3	7	v	v	NOUN
ejpam-6117	3	8	(	(	PUNCT
ejpam-6117	3	9	g	g	NOUN
ejpam-6117	3	10	)	)	PUNCT
ejpam-6117	3	11	is	be	AUX
ejpam-6117	3	12	the	the	DET
ejpam-6117	3	13	graph	graph	NOUN
ejpam-6117	3	14	whose	whose	DET
ejpam-6117	3	15	vertex	vertex	NOUN
ejpam-6117	3	16	set	set	NOUN
ejpam-6117	3	17	is	be	AUX
ejpam-6117	3	18	n	n	PRON
ejpam-6117	3	19	[	[	X
ejpam-6117	3	20	s	s	X
ejpam-6117	3	21	]	]	X
ejpam-6117	3	22	and	and	CCONJ
ejpam-6117	3	23	whose	whose	DET
ejpam-6117	3	24	edge	edge	NOUN
ejpam-6117	3	25	set	set	VERB
ejpam-6117	3	26	ew	ew	INTJ
ejpam-6117	3	27	consists	consist	NOUN
ejpam-6117	3	28	of	of	ADP
ejpam-6117	3	29	edges	edge	NOUN
ejpam-6117	3	30	in	in	ADP
ejpam-6117	3	31	e(g	e(g	PROPN
ejpam-6117	3	32	)	)	PUNCT
ejpam-6117	3	33	incident	incident	NOUN
ejpam-6117	3	34	to	to	ADP
ejpam-6117	3	35	some	some	DET
ejpam-6117	3	36	vertex	vertex	NOUN
ejpam-6117	3	37	in	in	ADP
ejpam-6117	3	38	s.	s.	PROPN
ejpam-6117	3	39	a	a	DET
ejpam-6117	3	40	subset	subset	NOUN
ejpam-6117	3	41	s	s	X
ejpam-6117	3	42	of	of	ADP
ejpam-6117	3	43	v	v	NOUN
ejpam-6117	3	44	(	(	PUNCT
ejpam-6117	3	45	g	g	NOUN
ejpam-6117	3	46	)	)	PUNCT
ejpam-6117	3	47	is	be	AUX
ejpam-6117	3	48	a	a	DET
ejpam-6117	3	49	weakly	weakly	ADV
ejpam-6117	3	50	connected	connected	ADJ
ejpam-6117	3	51	set	set	NOUN
ejpam-6117	3	52	of	of	ADP
ejpam-6117	3	53	g	g	PROPN
ejpam-6117	3	54	if	if	SCONJ
ejpam-6117	3	55	⟨s⟩w	⟨s⟩w	NOUN
ejpam-6117	3	56	is	be	AUX
ejpam-6117	3	57	connected	connect	VERB
ejpam-6117	3	58	.	.	PUNCT
ejpam-6117	4	1	s	s	X
ejpam-6117	4	2	is	be	AUX
ejpam-6117	4	3	called	call	VERB
ejpam-6117	4	4	a	a	DET
ejpam-6117	4	5	weakly	weakly	ADV
ejpam-6117	4	6	connected	connected	ADJ
ejpam-6117	4	7	independent	independent	ADJ
ejpam-6117	4	8	set	set	NOUN
ejpam-6117	4	9	(	(	PUNCT
ejpam-6117	4	10	wcis	wcis	NOUN
ejpam-6117	4	11	)	)	PUNCT
ejpam-6117	4	12	of	of	ADP
ejpam-6117	4	13	g	g	PRON
ejpam-6117	4	14	if	if	SCONJ
ejpam-6117	4	15	it	it	PRON
ejpam-6117	4	16	is	be	AUX
ejpam-6117	4	17	both	both	PRON
ejpam-6117	4	18	weakly	weakly	ADV
ejpam-6117	4	19	connected	connected	ADJ
ejpam-6117	4	20	and	and	CCONJ
ejpam-6117	4	21	independent	independent	ADJ
ejpam-6117	4	22	.	.	PUNCT
ejpam-6117	5	1	in	in	ADP
ejpam-6117	5	2	this	this	DET
ejpam-6117	5	3	paper	paper	NOUN
ejpam-6117	5	4	,	,	PUNCT
ejpam-6117	5	5	we	we	PRON
ejpam-6117	5	6	characterize	characterize	VERB
ejpam-6117	5	7	the	the	DET
ejpam-6117	5	8	weakly	weakly	ADV
ejpam-6117	5	9	connected	connected	ADJ
ejpam-6117	5	10	independent	independent	ADJ
ejpam-6117	5	11	sets	set	NOUN
ejpam-6117	5	12	in	in	ADP
ejpam-6117	5	13	the	the	DET
ejpam-6117	5	14	join	join	NOUN
ejpam-6117	5	15	,	,	PUNCT
ejpam-6117	5	16	corona	corona	PROPN
ejpam-6117	5	17	,	,	PUNCT
ejpam-6117	5	18	and	and	CCONJ
ejpam-6117	5	19	the	the	DET
ejpam-6117	5	20	lexicographic	lexicographic	ADJ
ejpam-6117	5	21	product	product	NOUN
ejpam-6117	5	22	of	of	ADP
ejpam-6117	5	23	two	two	NUM
ejpam-6117	5	24	graphs	graph	NOUN
ejpam-6117	5	25	.	.	PUNCT
ejpam-6117	6	1	from	from	ADP
ejpam-6117	6	2	these	these	DET
ejpam-6117	6	3	characterizations	characterization	NOUN
ejpam-6117	6	4	the	the	DET
ejpam-6117	6	5	weakly	weakly	ADV
ejpam-6117	6	6	connected	connected	ADJ
ejpam-6117	6	7	independence	independence	NOUN
ejpam-6117	6	8	numbers	number	NOUN
ejpam-6117	6	9	of	of	ADP
ejpam-6117	6	10	the	the	DET
ejpam-6117	6	11	corresponding	correspond	VERB
ejpam-6117	6	12	graphs	graph	NOUN
ejpam-6117	6	13	are	be	AUX
ejpam-6117	6	14	easily	easily	ADV
ejpam-6117	6	15	determined	determine	VERB
ejpam-6117	6	16	.	.	PUNCT
ejpam-6117	7	1	also	also	ADV
ejpam-6117	7	2	,	,	PUNCT
ejpam-6117	7	3	characterization	characterization	NOUN
ejpam-6117	7	4	of	of	ADP
ejpam-6117	7	5	graphs	graph	NOUN
ejpam-6117	7	6	g	g	NOUN
ejpam-6117	7	7	with	with	ADP
ejpam-6117	7	8	weakly	weakly	ADJ
ejpam-6117	7	9	connected	connected	ADJ
ejpam-6117	7	10	independence	independence	NOUN
ejpam-6117	7	11	numbers	number	NOUN
ejpam-6117	7	12	αw(g	αw(g	NOUN
ejpam-6117	7	13	)	)	PUNCT
ejpam-6117	7	14	equal	equal	ADJ
ejpam-6117	7	15	to	to	ADP
ejpam-6117	7	16	1	1	NUM
ejpam-6117	7	17	,	,	PUNCT
ejpam-6117	7	18	n−1	n−1	PROPN
ejpam-6117	7	19	and	and	CCONJ
ejpam-6117	7	20	n	n	PROPN
ejpam-6117	7	21	are	be	AUX
ejpam-6117	7	22	given	give	VERB
ejpam-6117	7	23	.	.	PUNCT
ejpam-6117	8	1	it	it	PRON
ejpam-6117	8	2	is	be	AUX
ejpam-6117	8	3	also	also	ADV
ejpam-6117	8	4	shown	show	VERB
ejpam-6117	8	5	that	that	SCONJ
ejpam-6117	8	6	for	for	ADP
ejpam-6117	8	7	any	any	DET
ejpam-6117	8	8	non	non	ADJ
ejpam-6117	8	9	-	-	ADJ
ejpam-6117	8	10	negative	negative	ADJ
ejpam-6117	8	11	integers	integer	NOUN
ejpam-6117	8	12	k	k	PROPN
ejpam-6117	8	13	,	,	PUNCT
ejpam-6117	8	14	m	m	PROPN
ejpam-6117	8	15	,	,	PUNCT
ejpam-6117	8	16	and	and	CCONJ
ejpam-6117	8	17	n	n	X
ejpam-6117	8	18	with	with	ADP
ejpam-6117	8	19	k	k	PROPN
ejpam-6117	8	20	>	>	X
ejpam-6117	8	21	m+1	m+1	PROPN
ejpam-6117	8	22	and	and	CCONJ
ejpam-6117	8	23	n	n	PRON
ejpam-6117	8	24	≥	≥	NOUN
ejpam-6117	8	25	k+m+2	k+m+2	NOUN
ejpam-6117	8	26	,	,	PUNCT
ejpam-6117	8	27	there	there	PRON
ejpam-6117	8	28	exists	exist	VERB
ejpam-6117	8	29	a	a	DET
ejpam-6117	8	30	connected	connected	ADJ
ejpam-6117	8	31	graph	graph	NOUN
ejpam-6117	8	32	g	g	ADP
ejpam-6117	8	33	such	such	ADJ
ejpam-6117	8	34	that	that	SCONJ
ejpam-6117	8	35	|v	|v	PROPN
ejpam-6117	8	36	(	(	PUNCT
ejpam-6117	8	37	g)|	g)|	NOUN
ejpam-6117	8	38	=	=	PUNCT
ejpam-6117	8	39	n	n	CCONJ
ejpam-6117	8	40	,	,	PUNCT
ejpam-6117	8	41	αw(g	αw(g	NUM
ejpam-6117	8	42	)	)	PUNCT
ejpam-6117	8	43	=	=	SYM
ejpam-6117	8	44	k	k	PROPN
ejpam-6117	8	45	and	and	CCONJ
ejpam-6117	8	46	α(g	α(g	NUM
ejpam-6117	8	47	)	)	PUNCT
ejpam-6117	9	1	=	=	SYM
ejpam-6117	9	2	k	k	X
ejpam-6117	10	1	+	+	PUNCT
ejpam-6117	10	2	m.	m.	NOUN
ejpam-6117	10	3	2020	2020	NUM
ejpam-6117	10	4	mathematics	mathematics	PROPN
ejpam-6117	10	5	subject	subject	NOUN
ejpam-6117	10	6	classifications	classification	NOUN
ejpam-6117	10	7	:	:	PUNCT
ejpam-6117	10	8	05c69	05c69	X
ejpam-6117	10	9	key	key	ADJ
ejpam-6117	10	10	words	word	NOUN
ejpam-6117	10	11	and	and	CCONJ
ejpam-6117	10	12	phrases	phrase	NOUN
ejpam-6117	10	13	:	:	PUNCT
ejpam-6117	10	14	independent	independent	ADJ
ejpam-6117	10	15	,	,	PUNCT
ejpam-6117	10	16	weakly	weakly	ADV
ejpam-6117	10	17	connected	connected	ADJ
ejpam-6117	10	18	set	set	NOUN
ejpam-6117	10	19	,	,	PUNCT
ejpam-6117	10	20	weakly	weakly	ADV
ejpam-6117	10	21	connected	connected	ADJ
ejpam-6117	10	22	independence	independence	NOUN
ejpam-6117	10	23	number	number	NOUN
ejpam-6117	10	24	,	,	PUNCT
ejpam-6117	10	25	join	join	NOUN
ejpam-6117	10	26	,	,	PUNCT
ejpam-6117	10	27	corona	corona	PROPN
ejpam-6117	10	28	,	,	PUNCT
ejpam-6117	10	29	lexicographic	lexicographic	ADJ
ejpam-6117	10	30	1	1	NUM
ejpam-6117	10	31	.	.	PUNCT
ejpam-6117	10	32	introduction	introduction	NOUN
ejpam-6117	10	33	the	the	DET
ejpam-6117	10	34	concept	concept	NOUN
ejpam-6117	10	35	of	of	ADP
ejpam-6117	10	36	weakly	weakly	ADJ
ejpam-6117	10	37	connected	connected	ADJ
ejpam-6117	10	38	domination	domination	NOUN
ejpam-6117	10	39	was	be	AUX
ejpam-6117	10	40	introduced	introduce	VERB
ejpam-6117	10	41	by	by	ADP
ejpam-6117	10	42	grossman	grossman	NOUN
ejpam-6117	11	1	[	[	X
ejpam-6117	11	2	1	1	NUM
ejpam-6117	11	3	]	]	PUNCT
ejpam-6117	11	4	and	and	CCONJ
ejpam-6117	11	5	was	be	AUX
ejpam-6117	11	6	studied	study	VERB
ejpam-6117	11	7	by	by	ADP
ejpam-6117	11	8	dunbar	dunbar	PROPN
ejpam-6117	11	9	et.al	et.al	PROPN
ejpam-6117	12	1	[	[	X
ejpam-6117	12	2	2	2	NUM
ejpam-6117	12	3	]	]	PUNCT
ejpam-6117	12	4	where	where	SCONJ
ejpam-6117	12	5	upper	upper	ADJ
ejpam-6117	12	6	and	and	CCONJ
ejpam-6117	12	7	lower	low	ADJ
ejpam-6117	12	8	bounds	bound	NOUN
ejpam-6117	12	9	for	for	ADP
ejpam-6117	12	10	γw(g	γw(g	NOUN
ejpam-6117	12	11	)	)	PUNCT
ejpam-6117	12	12	were	be	AUX
ejpam-6117	12	13	obtained	obtain	VERB
ejpam-6117	12	14	.	.	PUNCT
ejpam-6117	13	1	this	this	DET
ejpam-6117	13	2	parameter	parameter	NOUN
ejpam-6117	13	3	extends	extend	VERB
ejpam-6117	13	4	domination	domination	NOUN
ejpam-6117	13	5	by	by	ADP
ejpam-6117	13	6	ensuring	ensure	VERB
ejpam-6117	13	7	weak	weak	ADJ
ejpam-6117	13	8	connectivity	connectivity	NOUN
ejpam-6117	13	9	within	within	ADP
ejpam-6117	13	10	the	the	DET
ejpam-6117	13	11	dominating	dominating	NOUN
ejpam-6117	13	12	set	set	NOUN
ejpam-6117	13	13	.	.	PUNCT
ejpam-6117	14	1	graph	graph	NOUN
ejpam-6117	14	2	theory	theory	NOUN
ejpam-6117	14	3	plays	play	VERB
ejpam-6117	14	4	a	a	DET
ejpam-6117	14	5	vital	vital	ADJ
ejpam-6117	14	6	role	role	NOUN
ejpam-6117	14	7	in	in	ADP
ejpam-6117	14	8	modeling	modeling	NOUN
ejpam-6117	14	9	networks	network	NOUN
ejpam-6117	14	10	,	,	PUNCT
ejpam-6117	14	11	where	where	SCONJ
ejpam-6117	14	12	balancing	balance	VERB
ejpam-6117	14	13	independence	independence	NOUN
ejpam-6117	14	14	and	and	CCONJ
ejpam-6117	14	15	connectivity	connectivity	NOUN
ejpam-6117	14	16	is	be	AUX
ejpam-6117	14	17	crucial	crucial	ADJ
ejpam-6117	14	18	.	.	PUNCT
ejpam-6117	15	1	weakly	weakly	ADJ
ejpam-6117	15	2	connected	connected	ADJ
ejpam-6117	15	3	domination	domination	NOUN
ejpam-6117	15	4	has	have	AUX
ejpam-6117	15	5	been	be	AUX
ejpam-6117	15	6	widely	widely	ADV
ejpam-6117	15	7	studied	study	VERB
ejpam-6117	15	8	,	,	PUNCT
ejpam-6117	15	9	with	with	ADP
ejpam-6117	15	10	works	work	NOUN
ejpam-6117	15	11	by	by	ADP
ejpam-6117	15	12	alzoubi	alzoubi	PROPN
ejpam-6117	15	13	et	et	PROPN
ejpam-6117	15	14	al	al	PROPN
ejpam-6117	15	15	.	.	PUNCT
ejpam-6117	16	1	[	[	X
ejpam-6117	16	2	3	3	X
ejpam-6117	16	3	]	]	PUNCT
ejpam-6117	16	4	on	on	ADP
ejpam-6117	16	5	minimal	minimal	ADJ
ejpam-6117	16	6	sets	set	NOUN
ejpam-6117	16	7	and	and	CCONJ
ejpam-6117	16	8	bendali	bendali	PROPN
ejpam-6117	16	9	et	et	PROPN
ejpam-6117	16	10	al	al	PROPN
ejpam-6117	16	11	.	.	PUNCT
ejpam-6117	17	1	[	[	X
ejpam-6117	17	2	4	4	X
ejpam-6117	17	3	]	]	PUNCT
ejpam-6117	17	4	on	on	ADP
ejpam-6117	17	5	computational	computational	ADJ
ejpam-6117	17	6	∗corresponding	∗corresponding	NOUN
ejpam-6117	17	7	author	author	NOUN
ejpam-6117	17	8	.	.	PUNCT
ejpam-6117	18	1	doi	doi	NOUN
ejpam-6117	18	2	:	:	PUNCT
ejpam-6117	18	3	https://doi.org/10.29020/nybg.ejpam.v18i2.6117	https://doi.org/10.29020/nybg.ejpam.v18i2.6117	PROPN
ejpam-6117	18	4	email	email	NOUN
ejpam-6117	18	5	addresses	address	NOUN
ejpam-6117	18	6	:	:	PUNCT
ejpam-6117	19	1	reignver.merontos@g.msuiit.edu.ph	reignver.merontos@g.msuiit.edu.ph	PROPN
ejpam-6117	19	2	(	(	PUNCT
ejpam-6117	19	3	r.	r.	NOUN
ejpam-6117	19	4	merontos	merontos	NOUN
ejpam-6117	19	5	)	)	PUNCT
ejpam-6117	19	6	imelda.aniversario@g.msuiit.edu.ph	imelda.aniversario@g.msuiit.edu.ph	PROPN
ejpam-6117	19	7	(	(	PUNCT
ejpam-6117	19	8	i.	i.	PROPN
ejpam-6117	19	9	aniversario	aniversario	PROPN
ejpam-6117	19	10	)	)	PUNCT
ejpam-6117	20	1	michael.frondoza@g.msuiit.edu.ph	michael.frondoza@g.msuiit.edu.ph	PROPN
ejpam-6117	20	2	(	(	PUNCT
ejpam-6117	20	3	m.	m.	NOUN
ejpam-6117	20	4	frondoza	frondoza	PROPN
ejpam-6117	20	5	)	)	PUNCT
ejpam-6117	20	6	sergio.canoy@g.msuiit.edu.ph	sergio.canoy@g.msuiit.edu.ph	PROPN
ejpam-6117	20	7	(	(	PUNCT
ejpam-6117	20	8	s.	s.	PROPN
ejpam-6117	20	9	canoy	canoy	PROPN
ejpam-6117	20	10	jr	jr	PROPN
ejpam-6117	20	11	.	.	PUNCT
ejpam-6117	20	12	)	)	PUNCT
ejpam-6117	20	13	https://www.ejpam.com	https://www.ejpam.com	X
ejpam-6117	21	1	1	1	NUM
ejpam-6117	21	2	copyright	copyright	NOUN
ejpam-6117	21	3	:	:	PUNCT
ejpam-6117	21	4	©	©	PROPN
ejpam-6117	21	5	2025	2025	NUM
ejpam-6117	21	6	the	the	DET
ejpam-6117	21	7	author(s	author(s	NOUN
ejpam-6117	21	8	)	)	PUNCT
ejpam-6117	21	9	.	.	PUNCT
ejpam-6117	22	1	(	(	PUNCT
ejpam-6117	22	2	cc	cc	NOUN
ejpam-6117	22	3	by	by	ADP
ejpam-6117	22	4	-	-	PUNCT
ejpam-6117	22	5	nc	nc	PROPN
ejpam-6117	22	6	4.0	4.0	NUM
ejpam-6117	22	7	)	)	PUNCT
ejpam-6117	22	8	r.	r.	NOUN
ejpam-6117	22	9	merontos	merontos	PROPN
ejpam-6117	22	10	et	et	PROPN
ejpam-6117	22	11	al	al	PROPN
ejpam-6117	22	12	.	.	PUNCT
ejpam-6117	22	13	/	/	SYM
ejpam-6117	22	14	eur	eur	PROPN
ejpam-6117	22	15	.	.	PUNCT
ejpam-6117	23	1	j.	j.	PROPN
ejpam-6117	23	2	pure	pure	PROPN
ejpam-6117	23	3	appl	appl	PROPN
ejpam-6117	23	4	.	.	PROPN
ejpam-6117	23	5	math	math	PROPN
ejpam-6117	23	6	,	,	PUNCT
ejpam-6117	23	7	18	18	NUM
ejpam-6117	23	8	(	(	PUNCT
ejpam-6117	23	9	2	2	NUM
ejpam-6117	23	10	)	)	PUNCT
ejpam-6117	23	11	(	(	PUNCT
ejpam-6117	23	12	2025	2025	NUM
ejpam-6117	23	13	)	)	PUNCT
ejpam-6117	23	14	,	,	PUNCT
ejpam-6117	23	15	6117	6117	NUM
ejpam-6117	23	16	2	2	NUM
ejpam-6117	23	17	of	of	ADP
ejpam-6117	23	18	9	9	NUM
ejpam-6117	23	19	complexity	complexity	NOUN
ejpam-6117	23	20	.	.	PUNCT
ejpam-6117	24	1	recent	recent	ADJ
ejpam-6117	24	2	studies	study	NOUN
ejpam-6117	24	3	,	,	PUNCT
ejpam-6117	24	4	such	such	ADJ
ejpam-6117	24	5	as	as	ADP
ejpam-6117	24	6	those	those	PRON
ejpam-6117	24	7	by	by	ADP
ejpam-6117	24	8	hamja	hamja	PROPN
ejpam-6117	24	9	et	et	PROPN
ejpam-6117	24	10	al	al	PROPN
ejpam-6117	24	11	.	.	PUNCT
ejpam-6117	25	1	[	[	X
ejpam-6117	25	2	5	5	NUM
ejpam-6117	25	3	]	]	PUNCT
ejpam-6117	25	4	and	and	CCONJ
ejpam-6117	25	5	militante	militante	NOUN
ejpam-6117	25	6	and	and	CCONJ
ejpam-6117	25	7	eballe	eballe	NOUN
ejpam-6117	26	1	[	[	X
ejpam-6117	26	2	6	6	NUM
ejpam-6117	26	3	]	]	PUNCT
ejpam-6117	26	4	,	,	PUNCT
ejpam-6117	26	5	further	far	ADV
ejpam-6117	26	6	explored	explore	VERB
ejpam-6117	26	7	variations	variation	NOUN
ejpam-6117	26	8	in	in	ADP
ejpam-6117	26	9	weakly	weakly	ADJ
ejpam-6117	26	10	connected	connected	ADJ
ejpam-6117	26	11	parameters	parameter	NOUN
ejpam-6117	26	12	.	.	PUNCT
ejpam-6117	26	13	motivated	motivate	VERB
ejpam-6117	26	14	by	by	ADP
ejpam-6117	26	15	the	the	DET
ejpam-6117	26	16	growing	grow	VERB
ejpam-6117	26	17	interest	interest	NOUN
ejpam-6117	26	18	in	in	ADP
ejpam-6117	26	19	weak	weak	ADJ
ejpam-6117	26	20	connectivity	connectivity	NOUN
ejpam-6117	26	21	and	and	CCONJ
ejpam-6117	26	22	its	its	PRON
ejpam-6117	26	23	role	role	NOUN
ejpam-6117	26	24	in	in	ADP
ejpam-6117	26	25	network	network	NOUN
ejpam-6117	26	26	theory	theory	NOUN
ejpam-6117	26	27	,	,	PUNCT
ejpam-6117	26	28	this	this	DET
ejpam-6117	26	29	paper	paper	NOUN
ejpam-6117	26	30	introduces	introduce	NOUN
ejpam-6117	26	31	and	and	CCONJ
ejpam-6117	26	32	investigates	investigate	VERB
ejpam-6117	26	33	the	the	DET
ejpam-6117	26	34	weakly	weakly	ADV
ejpam-6117	26	35	connected	connected	ADJ
ejpam-6117	26	36	independence	independence	NOUN
ejpam-6117	26	37	number	number	NOUN
ejpam-6117	26	38	.	.	PUNCT
ejpam-6117	27	1	the	the	DET
ejpam-6117	27	2	study	study	NOUN
ejpam-6117	27	3	of	of	ADP
ejpam-6117	27	4	this	this	DET
ejpam-6117	27	5	new	new	ADJ
ejpam-6117	27	6	parameter	parameter	NOUN
ejpam-6117	27	7	involves	involve	VERB
ejpam-6117	27	8	addressing	address	VERB
ejpam-6117	27	9	the	the	DET
ejpam-6117	27	10	inherent	inherent	ADJ
ejpam-6117	27	11	difficulty	difficulty	NOUN
ejpam-6117	27	12	of	of	ADP
ejpam-6117	27	13	combining	combine	VERB
ejpam-6117	27	14	independence	independence	NOUN
ejpam-6117	27	15	and	and	CCONJ
ejpam-6117	27	16	connectivity	connectivity	NOUN
ejpam-6117	27	17	,	,	PUNCT
ejpam-6117	27	18	two	two	NUM
ejpam-6117	27	19	properties	property	NOUN
ejpam-6117	27	20	that	that	PRON
ejpam-6117	27	21	often	often	ADV
ejpam-6117	27	22	counteract	counteract	VERB
ejpam-6117	27	23	each	each	DET
ejpam-6117	27	24	other	other	ADJ
ejpam-6117	27	25	in	in	ADP
ejpam-6117	27	26	graph	graph	NOUN
ejpam-6117	27	27	structures	structure	NOUN
ejpam-6117	27	28	.	.	PUNCT
ejpam-6117	28	1	similar	similar	ADJ
ejpam-6117	28	2	to	to	ADP
ejpam-6117	28	3	weakly	weakly	ADJ
ejpam-6117	28	4	connected	connected	ADJ
ejpam-6117	28	5	domination	domination	NOUN
ejpam-6117	28	6	,	,	PUNCT
ejpam-6117	28	7	we	we	PRON
ejpam-6117	28	8	believe	believe	VERB
ejpam-6117	28	9	that	that	SCONJ
ejpam-6117	28	10	understanding	understand	VERB
ejpam-6117	28	11	the	the	DET
ejpam-6117	28	12	weakly	weakly	ADV
ejpam-6117	28	13	connected	connected	ADJ
ejpam-6117	28	14	independence	independence	NOUN
ejpam-6117	28	15	number	number	NOUN
ejpam-6117	28	16	will	will	AUX
ejpam-6117	28	17	yield	yield	VERB
ejpam-6117	28	18	significant	significant	ADJ
ejpam-6117	28	19	contributions	contribution	NOUN
ejpam-6117	28	20	to	to	ADP
ejpam-6117	28	21	independence	independence	NOUN
ejpam-6117	28	22	theory	theory	NOUN
ejpam-6117	28	23	and	and	CCONJ
ejpam-6117	28	24	stimulate	stimulate	VERB
ejpam-6117	28	25	further	further	ADJ
ejpam-6117	28	26	research	research	NOUN
ejpam-6117	28	27	in	in	ADP
ejpam-6117	28	28	graph	graph	NOUN
ejpam-6117	28	29	operations	operation	NOUN
ejpam-6117	28	30	,	,	PUNCT
ejpam-6117	28	31	and	and	CCONJ
ejpam-6117	28	32	combinatorial	combinatorial	ADJ
ejpam-6117	28	33	optimization	optimization	NOUN
ejpam-6117	28	34	.	.	PUNCT
ejpam-6117	29	1	2	2	X
ejpam-6117	29	2	.	.	NOUN
ejpam-6117	29	3	terminologies	terminology	NOUN
ejpam-6117	29	4	and	and	CCONJ
ejpam-6117	29	5	notations	notation	NOUN
ejpam-6117	29	6	let	let	VERB
ejpam-6117	29	7	g	g	NOUN
ejpam-6117	29	8	=	=	SYM
ejpam-6117	29	9	(	(	PUNCT
ejpam-6117	29	10	v	v	NOUN
ejpam-6117	29	11	(	(	PUNCT
ejpam-6117	29	12	g	g	NOUN
ejpam-6117	29	13	)	)	PUNCT
ejpam-6117	29	14	,	,	PUNCT
ejpam-6117	29	15	e(g	e(g	PROPN
ejpam-6117	29	16	)	)	PUNCT
ejpam-6117	29	17	)	)	PUNCT
ejpam-6117	29	18	be	be	AUX
ejpam-6117	29	19	a	a	DET
ejpam-6117	29	20	simple	simple	ADJ
ejpam-6117	29	21	undirected	undirected	ADJ
ejpam-6117	29	22	graph	graph	NOUN
ejpam-6117	29	23	.	.	PUNCT
ejpam-6117	30	1	the	the	DET
ejpam-6117	30	2	distance	distance	NOUN
ejpam-6117	30	3	between	between	ADP
ejpam-6117	30	4	two	two	NUM
ejpam-6117	30	5	vertices	vertex	NOUN
ejpam-6117	30	6	v	v	ADP
ejpam-6117	30	7	,	,	PUNCT
ejpam-6117	30	8	w	w	PROPN
ejpam-6117	30	9	∈	∈	PROPN
ejpam-6117	30	10	v	v	ADP
ejpam-6117	30	11	(	(	PUNCT
ejpam-6117	30	12	g	g	NOUN
ejpam-6117	30	13	)	)	PUNCT
ejpam-6117	30	14	,	,	PUNCT
ejpam-6117	30	15	denoted	denote	VERB
ejpam-6117	30	16	dg(v	dg(v	NOUN
ejpam-6117	30	17	,	,	PUNCT
ejpam-6117	30	18	w	w	NOUN
ejpam-6117	30	19	)	)	PUNCT
ejpam-6117	30	20	,	,	PUNCT
ejpam-6117	30	21	is	be	AUX
ejpam-6117	30	22	the	the	DET
ejpam-6117	30	23	length	length	NOUN
ejpam-6117	30	24	of	of	ADP
ejpam-6117	30	25	a	a	DET
ejpam-6117	30	26	shortest	short	ADJ
ejpam-6117	30	27	v	v	NOUN
ejpam-6117	30	28	-	-	PUNCT
ejpam-6117	30	29	w	w	NOUN
ejpam-6117	30	30	path	path	NOUN
ejpam-6117	30	31	connecting	connect	VERB
ejpam-6117	30	32	v	v	NOUN
ejpam-6117	30	33	and	and	CCONJ
ejpam-6117	30	34	w.	w.	NOUN
ejpam-6117	30	35	any	any	DET
ejpam-6117	30	36	v	v	PROPN
ejpam-6117	30	37	-	-	PUNCT
ejpam-6117	30	38	w	w	NOUN
ejpam-6117	30	39	path	path	NOUN
ejpam-6117	30	40	of	of	ADP
ejpam-6117	30	41	length	length	NOUN
ejpam-6117	30	42	dg(v	dg(v	X
ejpam-6117	30	43	,	,	PUNCT
ejpam-6117	30	44	w	w	NOUN
ejpam-6117	30	45	)	)	PUNCT
ejpam-6117	30	46	is	be	AUX
ejpam-6117	30	47	called	call	VERB
ejpam-6117	30	48	a	a	DET
ejpam-6117	30	49	v	v	NOUN
ejpam-6117	30	50	-	-	PUNCT
ejpam-6117	30	51	w	w	NOUN
ejpam-6117	30	52	geodesic	geodesic	NOUN
ejpam-6117	30	53	.	.	PUNCT
ejpam-6117	31	1	the	the	DET
ejpam-6117	31	2	open	open	ADJ
ejpam-6117	31	3	neighborhood	neighborhood	NOUN
ejpam-6117	31	4	of	of	ADP
ejpam-6117	31	5	a	a	DET
ejpam-6117	31	6	vertex	vertex	NOUN
ejpam-6117	31	7	v	v	NOUN
ejpam-6117	31	8	of	of	ADP
ejpam-6117	31	9	g	g	PROPN
ejpam-6117	31	10	is	be	AUX
ejpam-6117	31	11	the	the	DET
ejpam-6117	31	12	set	set	NOUN
ejpam-6117	31	13	ng(v	ng(v	PUNCT
ejpam-6117	31	14	)	)	PUNCT
ejpam-6117	31	15	=	=	SYM
ejpam-6117	32	1	{	{	PUNCT
ejpam-6117	32	2	u	u	NOUN
ejpam-6117	32	3	∈	∈	PROPN
ejpam-6117	32	4	v	v	NOUN
ejpam-6117	32	5	(	(	PUNCT
ejpam-6117	32	6	g	g	NOUN
ejpam-6117	32	7	)	)	PUNCT
ejpam-6117	32	8	:	:	PUNCT
ejpam-6117	32	9	uv	uv	PROPN
ejpam-6117	32	10	∈	∈	PROPN
ejpam-6117	32	11	e(g	e(g	PROPN
ejpam-6117	32	12	)	)	PUNCT
ejpam-6117	32	13	}	}	PUNCT
ejpam-6117	32	14	,	,	PUNCT
ejpam-6117	32	15	while	while	SCONJ
ejpam-6117	32	16	its	its	PRON
ejpam-6117	32	17	closed	closed	ADJ
ejpam-6117	32	18	neighborhood	neighborhood	NOUN
ejpam-6117	32	19	is	be	AUX
ejpam-6117	32	20	the	the	DET
ejpam-6117	32	21	set	set	NOUN
ejpam-6117	32	22	ng[v	ng[v	NOUN
ejpam-6117	32	23	]	]	X
ejpam-6117	32	24	=	=	SYM
ejpam-6117	32	25	ng(v	ng(v	X
ejpam-6117	32	26	)	)	PUNCT
ejpam-6117	32	27	∪	∪	ADP
ejpam-6117	32	28	{	{	PUNCT
ejpam-6117	32	29	v	v	NOUN
ejpam-6117	32	30	}	}	PUNCT
ejpam-6117	32	31	.	.	PUNCT
ejpam-6117	33	1	the	the	DET
ejpam-6117	33	2	open	open	ADJ
ejpam-6117	33	3	neighborhood	neighborhood	NOUN
ejpam-6117	33	4	of	of	ADP
ejpam-6117	33	5	a	a	DET
ejpam-6117	33	6	set	set	NOUN
ejpam-6117	33	7	s	s	NOUN
ejpam-6117	33	8	⊆	⊆	NUM
ejpam-6117	33	9	v	v	NOUN
ejpam-6117	33	10	(	(	PUNCT
ejpam-6117	33	11	g	g	NOUN
ejpam-6117	33	12	)	)	PUNCT
ejpam-6117	33	13	is	be	AUX
ejpam-6117	33	14	the	the	DET
ejpam-6117	33	15	set	set	NOUN
ejpam-6117	33	16	ng(s	ng(s	NOUN
ejpam-6117	33	17	)	)	PUNCT
ejpam-6117	33	18	=	=	SYM
ejpam-6117	33	19	∪v∈sng(v	∪v∈sng(v	PROPN
ejpam-6117	33	20	)	)	PUNCT
ejpam-6117	33	21	and	and	CCONJ
ejpam-6117	33	22	its	its	PRON
ejpam-6117	33	23	closed	closed	ADJ
ejpam-6117	33	24	neighborhood	neighborhood	NOUN
ejpam-6117	33	25	is	be	AUX
ejpam-6117	33	26	the	the	DET
ejpam-6117	33	27	set	set	VERB
ejpam-6117	33	28	ng[s	ng[	NOUN
ejpam-6117	33	29	]	]	PUNCT
ejpam-6117	33	30	=	=	SYM
ejpam-6117	33	31	s	s	X
ejpam-6117	33	32	∪	∪	NOUN
ejpam-6117	33	33	ng(s	ng(s	NUM
ejpam-6117	33	34	)	)	PUNCT
ejpam-6117	33	35	.	.	PUNCT
ejpam-6117	34	1	any	any	DET
ejpam-6117	34	2	v	v	NUM
ejpam-6117	34	3	∈	∈	PROPN
ejpam-6117	34	4	v	v	NOUN
ejpam-6117	34	5	(	(	PUNCT
ejpam-6117	34	6	g	g	NOUN
ejpam-6117	34	7	)	)	PUNCT
ejpam-6117	34	8	with	with	ADP
ejpam-6117	34	9	|ng(v)|	|ng(v)|	NOUN
ejpam-6117	34	10	=	=	SYM
ejpam-6117	34	11	0	0	NUM
ejpam-6117	34	12	is	be	AUX
ejpam-6117	34	13	called	call	VERB
ejpam-6117	34	14	an	an	DET
ejpam-6117	34	15	isolated	isolated	ADJ
ejpam-6117	34	16	vertex	vertex	NOUN
ejpam-6117	34	17	.	.	PUNCT
ejpam-6117	35	1	vertex	vertex	NOUN
ejpam-6117	35	2	v	v	NOUN
ejpam-6117	35	3	is	be	AUX
ejpam-6117	35	4	a	a	DET
ejpam-6117	35	5	leaf	leaf	NOUN
ejpam-6117	35	6	or	or	CCONJ
ejpam-6117	35	7	an	an	DET
ejpam-6117	35	8	endvertex	endvertex	NOUN
ejpam-6117	35	9	if	if	SCONJ
ejpam-6117	35	10	|ng(v)|	|ng(v)|	VERB
ejpam-6117	35	11	=	=	SYM
ejpam-6117	35	12	1	1	X
ejpam-6117	35	13	.	.	PUNCT
ejpam-6117	36	1	a	a	DET
ejpam-6117	36	2	vertex	vertex	NOUN
ejpam-6117	36	3	w	w	NOUN
ejpam-6117	36	4	of	of	ADP
ejpam-6117	36	5	g	g	PROPN
ejpam-6117	36	6	is	be	AUX
ejpam-6117	36	7	a	a	DET
ejpam-6117	36	8	support	support	NOUN
ejpam-6117	36	9	vertex	vertex	NOUN
ejpam-6117	36	10	if	if	SCONJ
ejpam-6117	36	11	wv	wv	PROPN
ejpam-6117	36	12	∈	∈	PROPN
ejpam-6117	36	13	e(g	e(g	PROPN
ejpam-6117	36	14	)	)	PUNCT
ejpam-6117	36	15	for	for	ADP
ejpam-6117	36	16	some	some	DET
ejpam-6117	36	17	leaf	leaf	NOUN
ejpam-6117	36	18	v	v	NOUN
ejpam-6117	36	19	in	in	ADP
ejpam-6117	36	20	g.	g.	PROPN
ejpam-6117	36	21	the	the	DET
ejpam-6117	36	22	sets	set	NOUN
ejpam-6117	36	23	i(g	i(g	ADV
ejpam-6117	36	24	)	)	PUNCT
ejpam-6117	36	25	,	,	PUNCT
ejpam-6117	36	26	l(g	l(g	PROPN
ejpam-6117	36	27	)	)	PUNCT
ejpam-6117	36	28	,	,	PUNCT
ejpam-6117	36	29	and	and	CCONJ
ejpam-6117	36	30	s(g	s(g	PROPN
ejpam-6117	36	31	)	)	PUNCT
ejpam-6117	36	32	will	will	AUX
ejpam-6117	36	33	,	,	PUNCT
ejpam-6117	36	34	respectively	respectively	ADV
ejpam-6117	36	35	,	,	PUNCT
ejpam-6117	36	36	denote	denote	VERB
ejpam-6117	36	37	the	the	DET
ejpam-6117	36	38	sets	set	NOUN
ejpam-6117	36	39	containing	contain	VERB
ejpam-6117	36	40	all	all	DET
ejpam-6117	36	41	the	the	DET
ejpam-6117	36	42	isolated	isolated	ADJ
ejpam-6117	36	43	vertices	vertex	NOUN
ejpam-6117	36	44	,	,	PUNCT
ejpam-6117	36	45	leaves	leave	NOUN
ejpam-6117	36	46	,	,	PUNCT
ejpam-6117	36	47	and	and	CCONJ
ejpam-6117	36	48	support	support	NOUN
ejpam-6117	36	49	vertices	vertex	NOUN
ejpam-6117	36	50	in	in	ADP
ejpam-6117	36	51	g.	g.	PROPN
ejpam-6117	36	52	a	a	DET
ejpam-6117	36	53	subset	subset	VERB
ejpam-6117	36	54	a	a	PRON
ejpam-6117	36	55	of	of	ADP
ejpam-6117	36	56	v	v	NOUN
ejpam-6117	36	57	(	(	PUNCT
ejpam-6117	36	58	g	g	NOUN
ejpam-6117	36	59	)	)	PUNCT
ejpam-6117	36	60	is	be	AUX
ejpam-6117	36	61	an	an	DET
ejpam-6117	36	62	independent	independent	ADJ
ejpam-6117	36	63	set	set	NOUN
ejpam-6117	36	64	if	if	SCONJ
ejpam-6117	36	65	for	for	SCONJ
ejpam-6117	36	66	every	every	DET
ejpam-6117	36	67	pair	pair	NOUN
ejpam-6117	36	68	of	of	ADP
ejpam-6117	36	69	distinct	distinct	ADJ
ejpam-6117	36	70	vertices	vertex	NOUN
ejpam-6117	36	71	in	in	ADP
ejpam-6117	36	72	g	g	PROPN
ejpam-6117	36	73	do	do	AUX
ejpam-6117	36	74	not	not	PART
ejpam-6117	36	75	form	form	VERB
ejpam-6117	36	76	an	an	DET
ejpam-6117	36	77	edge	edge	NOUN
ejpam-6117	36	78	.	.	PUNCT
ejpam-6117	37	1	the	the	DET
ejpam-6117	37	2	maximum	maximum	ADJ
ejpam-6117	37	3	cardinality	cardinality	NOUN
ejpam-6117	37	4	of	of	ADP
ejpam-6117	37	5	an	an	DET
ejpam-6117	37	6	independent	independent	ADJ
ejpam-6117	37	7	set	set	NOUN
ejpam-6117	37	8	in	in	ADP
ejpam-6117	37	9	g	g	NOUN
ejpam-6117	37	10	,	,	PUNCT
ejpam-6117	37	11	denoted	denote	VERB
ejpam-6117	37	12	by	by	ADP
ejpam-6117	37	13	α(g	α(g	NOUN
ejpam-6117	37	14	)	)	PUNCT
ejpam-6117	37	15	,	,	PUNCT
ejpam-6117	37	16	is	be	AUX
ejpam-6117	37	17	called	call	VERB
ejpam-6117	37	18	the	the	DET
ejpam-6117	37	19	independence	independence	NOUN
ejpam-6117	37	20	number	number	NOUN
ejpam-6117	37	21	of	of	ADP
ejpam-6117	37	22	g.	g.	PROPN
ejpam-6117	37	23	any	any	DET
ejpam-6117	37	24	independent	independent	ADJ
ejpam-6117	37	25	set	set	NOUN
ejpam-6117	37	26	with	with	ADP
ejpam-6117	37	27	cardinality	cardinality	NOUN
ejpam-6117	37	28	equal	equal	ADJ
ejpam-6117	37	29	to	to	ADP
ejpam-6117	37	30	α(g	α(g	NUM
ejpam-6117	37	31	)	)	PUNCT
ejpam-6117	37	32	is	be	AUX
ejpam-6117	37	33	called	call	VERB
ejpam-6117	37	34	an	an	DET
ejpam-6117	37	35	α	α	NOUN
ejpam-6117	37	36	-	-	PUNCT
ejpam-6117	37	37	set	set	VERB
ejpam-6117	37	38	in	in	ADP
ejpam-6117	37	39	g.	g.	PROPN
ejpam-6117	37	40	a	a	DET
ejpam-6117	37	41	set	set	NOUN
ejpam-6117	37	42	s	s	PROPN
ejpam-6117	37	43	⊆	⊆	NUM
ejpam-6117	37	44	v	v	NOUN
ejpam-6117	37	45	(	(	PUNCT
ejpam-6117	37	46	g	g	NOUN
ejpam-6117	37	47	)	)	PUNCT
ejpam-6117	37	48	is	be	AUX
ejpam-6117	37	49	a	a	DET
ejpam-6117	37	50	dominating	dominating	NOUN
ejpam-6117	37	51	set	set	VERB
ejpam-6117	37	52	in	in	ADP
ejpam-6117	37	53	g	g	PROPN
ejpam-6117	37	54	if	if	SCONJ
ejpam-6117	37	55	ng[s	ng[	NOUN
ejpam-6117	37	56	]	]	PUNCT
ejpam-6117	37	57	=	=	SYM
ejpam-6117	37	58	v	v	NOUN
ejpam-6117	37	59	(	(	PUNCT
ejpam-6117	37	60	g	g	NOUN
ejpam-6117	37	61	)	)	PUNCT
ejpam-6117	37	62	.	.	PUNCT
ejpam-6117	38	1	it	it	PRON
ejpam-6117	38	2	is	be	AUX
ejpam-6117	38	3	a	a	DET
ejpam-6117	38	4	super	super	ADJ
ejpam-6117	38	5	dominating	dominating	NOUN
ejpam-6117	38	6	set	set	NOUN
ejpam-6117	38	7	if	if	SCONJ
ejpam-6117	38	8	for	for	ADP
ejpam-6117	38	9	every	every	PRON
ejpam-6117	38	10	v	v	NUM
ejpam-6117	38	11	∈	∈	NOUN
ejpam-6117	38	12	v	v	NOUN
ejpam-6117	38	13	(	(	PUNCT
ejpam-6117	38	14	g	g	NOUN
ejpam-6117	38	15	)	)	PUNCT
ejpam-6117	38	16	\s	\s	NOUN
ejpam-6117	38	17	there	there	PRON
ejpam-6117	38	18	exists	exist	VERB
ejpam-6117	38	19	w	w	PROPN
ejpam-6117	38	20	∈	∈	PROPN
ejpam-6117	38	21	s	s	VERB
ejpam-6117	38	22	such	such	ADJ
ejpam-6117	38	23	that	that	SCONJ
ejpam-6117	38	24	ng(w)∩	ng(w)∩	PUNCT
ejpam-6117	39	1	[	[	X
ejpam-6117	39	2	v	v	X
ejpam-6117	39	3	(	(	PUNCT
ejpam-6117	39	4	g	g	NOUN
ejpam-6117	39	5	)	)	PUNCT
ejpam-6117	39	6	\s	\s	NOUN
ejpam-6117	39	7	]	]	PUNCT
ejpam-6117	39	8	=	=	PUNCT
ejpam-6117	39	9	{	{	PUNCT
ejpam-6117	39	10	v	v	NOUN
ejpam-6117	39	11	}	}	PUNCT
ejpam-6117	39	12	.	.	PUNCT
ejpam-6117	40	1	the	the	DET
ejpam-6117	40	2	domination	domination	NOUN
ejpam-6117	40	3	number	number	NOUN
ejpam-6117	40	4	(	(	PUNCT
ejpam-6117	40	5	super	super	ADJ
ejpam-6117	40	6	domination	domination	NOUN
ejpam-6117	40	7	number	number	NOUN
ejpam-6117	40	8	)	)	PUNCT
ejpam-6117	40	9	of	of	ADP
ejpam-6117	40	10	g	g	NOUN
ejpam-6117	40	11	,	,	PUNCT
ejpam-6117	40	12	denoted	denote	VERB
ejpam-6117	40	13	γ(g	γ(g	PROPN
ejpam-6117	40	14	)	)	PUNCT
ejpam-6117	40	15	(	(	PUNCT
ejpam-6117	40	16	resp	resp	NOUN
ejpam-6117	40	17	.	.	PUNCT
ejpam-6117	41	1	γsp(g	γsp(g	NOUN
ejpam-6117	41	2	)	)	PUNCT
ejpam-6117	41	3	)	)	PUNCT
ejpam-6117	41	4	is	be	AUX
ejpam-6117	41	5	the	the	DET
ejpam-6117	41	6	minimum	minimum	ADJ
ejpam-6117	41	7	cardinality	cardinality	NOUN
ejpam-6117	41	8	of	of	ADP
ejpam-6117	41	9	a	a	DET
ejpam-6117	41	10	dominating	dominating	NOUN
ejpam-6117	41	11	(	(	PUNCT
ejpam-6117	41	12	resp	resp	NOUN
ejpam-6117	41	13	.	.	PUNCT
ejpam-6117	42	1	super	super	ADJ
ejpam-6117	42	2	dominating	dominating	NOUN
ejpam-6117	42	3	)	)	PUNCT
ejpam-6117	42	4	set	set	VERB
ejpam-6117	42	5	in	in	ADP
ejpam-6117	42	6	g.	g.	PROPN
ejpam-6117	42	7	any	any	DET
ejpam-6117	42	8	dominating	dominating	NOUN
ejpam-6117	42	9	set	set	NOUN
ejpam-6117	42	10	(	(	PUNCT
ejpam-6117	42	11	super	super	ADJ
ejpam-6117	42	12	dominating	dominating	NOUN
ejpam-6117	42	13	set	set	NOUN
ejpam-6117	42	14	)	)	PUNCT
ejpam-6117	42	15	with	with	ADP
ejpam-6117	42	16	cardinality	cardinality	PROPN
ejpam-6117	42	17	γ(g	γ(g	PROPN
ejpam-6117	42	18	)	)	PUNCT
ejpam-6117	42	19	(	(	PUNCT
ejpam-6117	42	20	resp	resp	NOUN
ejpam-6117	42	21	.	.	PUNCT
ejpam-6117	43	1	γsp(g	γsp(g	NOUN
ejpam-6117	43	2	)	)	PUNCT
ejpam-6117	43	3	)	)	PUNCT
ejpam-6117	43	4	is	be	AUX
ejpam-6117	43	5	called	call	VERB
ejpam-6117	43	6	a	a	DET
ejpam-6117	43	7	γ	γ	NOUN
ejpam-6117	43	8	-	-	PUNCT
ejpam-6117	43	9	set	set	ADJ
ejpam-6117	43	10	(	(	PUNCT
ejpam-6117	43	11	resp	resp	NOUN
ejpam-6117	43	12	.	.	PUNCT
ejpam-6117	44	1	γsp	γsp	PROPN
ejpam-6117	44	2	-	-	PUNCT
ejpam-6117	44	3	set	set	NOUN
ejpam-6117	44	4	)	)	PUNCT
ejpam-6117	44	5	.	.	PUNCT
ejpam-6117	45	1	the	the	DET
ejpam-6117	45	2	subgraph	subgraph	PROPN
ejpam-6117	45	3	⟨s⟩w	⟨s⟩w	PROPN
ejpam-6117	45	4	of	of	ADP
ejpam-6117	45	5	s	s	NOUN
ejpam-6117	45	6	⊆	⊆	NUM
ejpam-6117	45	7	v	v	NOUN
ejpam-6117	45	8	(	(	PUNCT
ejpam-6117	45	9	g	g	NOUN
ejpam-6117	45	10	)	)	PUNCT
ejpam-6117	45	11	is	be	AUX
ejpam-6117	45	12	the	the	DET
ejpam-6117	45	13	graph	graph	NOUN
ejpam-6117	45	14	whose	whose	DET
ejpam-6117	45	15	vertex	vertex	NOUN
ejpam-6117	45	16	set	set	NOUN
ejpam-6117	45	17	is	be	AUX
ejpam-6117	45	18	ng[s	ng[	NOUN
ejpam-6117	45	19	]	]	PUNCT
ejpam-6117	45	20	and	and	CCONJ
ejpam-6117	45	21	whose	whose	DET
ejpam-6117	45	22	edge	edge	NOUN
ejpam-6117	45	23	set	set	VERB
ejpam-6117	45	24	ew	ew	INTJ
ejpam-6117	45	25	consists	consist	NOUN
ejpam-6117	45	26	of	of	ADP
ejpam-6117	45	27	edges	edge	NOUN
ejpam-6117	45	28	in	in	ADP
ejpam-6117	45	29	e(g	e(g	PROPN
ejpam-6117	45	30	)	)	PUNCT
ejpam-6117	45	31	incident	incident	NOUN
ejpam-6117	45	32	to	to	ADP
ejpam-6117	45	33	some	some	DET
ejpam-6117	45	34	vertex	vertex	NOUN
ejpam-6117	45	35	in	in	ADP
ejpam-6117	45	36	s.	s.	PROPN
ejpam-6117	45	37	a	a	DET
ejpam-6117	45	38	subset	subset	NOUN
ejpam-6117	45	39	s	s	X
ejpam-6117	45	40	of	of	ADP
ejpam-6117	45	41	v	v	NOUN
ejpam-6117	45	42	(	(	PUNCT
ejpam-6117	45	43	g	g	NOUN
ejpam-6117	45	44	)	)	PUNCT
ejpam-6117	45	45	is	be	AUX
ejpam-6117	45	46	a	a	DET
ejpam-6117	45	47	weakly	weakly	ADV
ejpam-6117	45	48	connected	connected	ADJ
ejpam-6117	45	49	set	set	NOUN
ejpam-6117	45	50	of	of	ADP
ejpam-6117	45	51	g	g	PROPN
ejpam-6117	45	52	if	if	SCONJ
ejpam-6117	45	53	⟨s⟩w	⟨s⟩w	NOUN
ejpam-6117	45	54	is	be	AUX
ejpam-6117	45	55	connected	connect	VERB
ejpam-6117	45	56	.	.	PUNCT
ejpam-6117	46	1	s	s	X
ejpam-6117	46	2	is	be	AUX
ejpam-6117	46	3	called	call	VERB
ejpam-6117	46	4	a	a	DET
ejpam-6117	46	5	weakly	weakly	ADV
ejpam-6117	46	6	connected	connected	ADJ
ejpam-6117	46	7	independent	independent	ADJ
ejpam-6117	46	8	set	set	NOUN
ejpam-6117	46	9	(	(	PUNCT
ejpam-6117	46	10	wcis	wcis	NOUN
ejpam-6117	46	11	)	)	PUNCT
ejpam-6117	46	12	ofg	ofg	NOUN
ejpam-6117	46	13	if	if	SCONJ
ejpam-6117	46	14	it	it	PRON
ejpam-6117	46	15	is	be	AUX
ejpam-6117	46	16	both	both	PRON
ejpam-6117	46	17	weakly	weakly	ADV
ejpam-6117	46	18	connected	connected	ADJ
ejpam-6117	46	19	and	and	CCONJ
ejpam-6117	46	20	independent	independent	ADJ
ejpam-6117	46	21	.	.	PUNCT
ejpam-6117	47	1	the	the	DET
ejpam-6117	47	2	maximum	maximum	ADJ
ejpam-6117	47	3	cardinality	cardinality	NOUN
ejpam-6117	47	4	of	of	ADP
ejpam-6117	47	5	a	a	DET
ejpam-6117	47	6	wcis	wcis	NOUN
ejpam-6117	47	7	of	of	ADP
ejpam-6117	47	8	g	g	PROPN
ejpam-6117	47	9	is	be	AUX
ejpam-6117	47	10	called	call	VERB
ejpam-6117	47	11	the	the	DET
ejpam-6117	47	12	weakly	weakly	ADV
ejpam-6117	47	13	connected	connected	ADJ
ejpam-6117	47	14	independence	independence	NOUN
ejpam-6117	47	15	number	number	NOUN
ejpam-6117	47	16	of	of	ADP
ejpam-6117	47	17	g	g	NOUN
ejpam-6117	47	18	and	and	CCONJ
ejpam-6117	47	19	is	be	AUX
ejpam-6117	47	20	denoted	denote	VERB
ejpam-6117	47	21	by	by	ADP
ejpam-6117	47	22	αw(g	αw(g	NOUN
ejpam-6117	47	23	)	)	PUNCT
ejpam-6117	47	24	.	.	PUNCT
ejpam-6117	48	1	a	a	DET
ejpam-6117	48	2	wcis	wcis	NOUN
ejpam-6117	48	3	of	of	ADP
ejpam-6117	48	4	g	g	NOUN
ejpam-6117	48	5	having	have	VERB
ejpam-6117	48	6	cardinality	cardinality	NOUN
ejpam-6117	48	7	αw(g	αw(g	NUM
ejpam-6117	48	8	)	)	PUNCT
ejpam-6117	48	9	is	be	AUX
ejpam-6117	48	10	called	call	VERB
ejpam-6117	48	11	a	a	DET
ejpam-6117	48	12	maximum	maximum	ADJ
ejpam-6117	48	13	wcis	wcis	NOUN
ejpam-6117	48	14	of	of	ADP
ejpam-6117	48	15	g.	g.	PROPN
ejpam-6117	48	16	a	a	DET
ejpam-6117	48	17	set	set	NOUN
ejpam-6117	48	18	s	s	NOUN
ejpam-6117	48	19	that	that	PRON
ejpam-6117	48	20	is	be	AUX
ejpam-6117	48	21	both	both	CCONJ
ejpam-6117	48	22	a	a	DET
ejpam-6117	48	23	wcis	wcis	NOUN
ejpam-6117	48	24	and	and	CCONJ
ejpam-6117	48	25	a	a	DET
ejpam-6117	48	26	dominating	dominating	NOUN
ejpam-6117	48	27	set	set	NOUN
ejpam-6117	48	28	of	of	ADP
ejpam-6117	48	29	g	g	PROPN
ejpam-6117	48	30	is	be	AUX
ejpam-6117	48	31	called	call	VERB
ejpam-6117	48	32	a	a	DET
ejpam-6117	48	33	weakly	weakly	ADV
ejpam-6117	48	34	connected	connected	ADJ
ejpam-6117	48	35	independent	independent	ADJ
ejpam-6117	48	36	dominating	dominating	NOUN
ejpam-6117	48	37	set	set	NOUN
ejpam-6117	48	38	(	(	PUNCT
ejpam-6117	48	39	wcids	wcid	NOUN
ejpam-6117	48	40	)	)	PUNCT
ejpam-6117	48	41	of	of	ADP
ejpam-6117	48	42	g.	g.	PROPN
ejpam-6117	48	43	note	note	VERB
ejpam-6117	48	44	that	that	SCONJ
ejpam-6117	48	45	a	a	DET
ejpam-6117	48	46	minimum	minimum	NOUN
ejpam-6117	48	47	wcids	wcid	NOUN
ejpam-6117	48	48	of	of	ADP
ejpam-6117	48	49	g	g	PROPN
ejpam-6117	48	50	always	always	ADV
ejpam-6117	48	51	exists	exist	VERB
ejpam-6117	48	52	(	(	PUNCT
ejpam-6117	48	53	see	see	VERB
ejpam-6117	48	54	dunbar	dunbar	PROPN
ejpam-6117	48	55	et	et	PROPN
ejpam-6117	48	56	al	al	PROPN
ejpam-6117	48	57	.	.	PUNCT
ejpam-6117	49	1	[	[	X
ejpam-6117	49	2	2	2	NUM
ejpam-6117	49	3	]	]	PUNCT
ejpam-6117	49	4	)	)	PUNCT
ejpam-6117	49	5	.	.	PUNCT
ejpam-6117	49	6	denote	denote	VERB
ejpam-6117	49	7	by	by	ADP
ejpam-6117	49	8	ιc(g	ιc(g	NOUN
ejpam-6117	49	9	)	)	PUNCT
ejpam-6117	49	10	the	the	DET
ejpam-6117	49	11	cardinality	cardinality	NOUN
ejpam-6117	49	12	of	of	ADP
ejpam-6117	49	13	a	a	DET
ejpam-6117	49	14	minimum	minimum	ADJ
ejpam-6117	49	15	wcids	wcid	NOUN
ejpam-6117	49	16	(	(	PUNCT
ejpam-6117	49	17	or	or	CCONJ
ejpam-6117	49	18	ιg	ιg	ADV
ejpam-6117	49	19	-	-	PUNCT
ejpam-6117	49	20	set	set	NOUN
ejpam-6117	49	21	)	)	PUNCT
ejpam-6117	49	22	of	of	ADP
ejpam-6117	49	23	g.	g.	PROPN
ejpam-6117	49	24	let	let	VERB
ejpam-6117	49	25	g	g	NOUN
ejpam-6117	49	26	and	and	CCONJ
ejpam-6117	49	27	h	h	NOUN
ejpam-6117	49	28	be	be	VERB
ejpam-6117	49	29	any	any	DET
ejpam-6117	49	30	two	two	NUM
ejpam-6117	49	31	graphs	graph	NOUN
ejpam-6117	49	32	.	.	PUNCT
ejpam-6117	50	1	the	the	DET
ejpam-6117	50	2	join	join	NOUN
ejpam-6117	50	3	g	g	PROPN
ejpam-6117	50	4	+	+	CCONJ
ejpam-6117	50	5	h	h	NOUN
ejpam-6117	50	6	is	be	AUX
ejpam-6117	50	7	the	the	DET
ejpam-6117	50	8	graph	graph	NOUN
ejpam-6117	50	9	with	with	ADP
ejpam-6117	50	10	vertex	vertex	NOUN
ejpam-6117	51	1	set	set	VERB
ejpam-6117	51	2	r.	r.	PROPN
ejpam-6117	51	3	merontos	merontos	PROPN
ejpam-6117	51	4	et	et	PROPN
ejpam-6117	51	5	al	al	PROPN
ejpam-6117	51	6	.	.	PUNCT
ejpam-6117	51	7	/	/	SYM
ejpam-6117	51	8	eur	eur	PROPN
ejpam-6117	51	9	.	.	PUNCT
ejpam-6117	52	1	j.	j.	PROPN
ejpam-6117	52	2	pure	pure	PROPN
ejpam-6117	52	3	appl	appl	PROPN
ejpam-6117	52	4	.	.	PROPN
ejpam-6117	52	5	math	math	PROPN
ejpam-6117	52	6	,	,	PUNCT
ejpam-6117	52	7	18	18	NUM
ejpam-6117	52	8	(	(	PUNCT
ejpam-6117	52	9	2	2	NUM
ejpam-6117	52	10	)	)	PUNCT
ejpam-6117	52	11	(	(	PUNCT
ejpam-6117	52	12	2025	2025	NUM
ejpam-6117	52	13	)	)	PUNCT
ejpam-6117	52	14	,	,	PUNCT
ejpam-6117	52	15	6117	6117	NUM
ejpam-6117	52	16	3	3	NUM
ejpam-6117	52	17	of	of	ADP
ejpam-6117	52	18	9	9	NUM
ejpam-6117	52	19	v	v	NOUN
ejpam-6117	52	20	(	(	PUNCT
ejpam-6117	52	21	g+h	g+h	NOUN
ejpam-6117	52	22	)	)	PUNCT
ejpam-6117	52	23	=	=	SYM
ejpam-6117	52	24	v	v	NOUN
ejpam-6117	52	25	(	(	PUNCT
ejpam-6117	52	26	g)∪	g)∪	VERB
ejpam-6117	52	27	v	v	NUM
ejpam-6117	52	28	(	(	PUNCT
ejpam-6117	52	29	h	h	NOUN
ejpam-6117	52	30	)	)	PUNCT
ejpam-6117	52	31	and	and	CCONJ
ejpam-6117	52	32	edge	edge	NOUN
ejpam-6117	52	33	set	set	VERB
ejpam-6117	52	34	e(g+h	e(g+h	NUM
ejpam-6117	52	35	)	)	PUNCT
ejpam-6117	53	1	=	=	SYM
ejpam-6117	53	2	e(g)∪e(h)∪	e(g)∪e(h)∪	NOUN
ejpam-6117	53	3	{	{	PUNCT
ejpam-6117	53	4	uv	uv	NOUN
ejpam-6117	53	5	:	:	PUNCT
ejpam-6117	53	6	u	u	PROPN
ejpam-6117	53	7	∈	∈	PROPN
ejpam-6117	53	8	v	v	ADP
ejpam-6117	53	9	(	(	PUNCT
ejpam-6117	53	10	g	g	NOUN
ejpam-6117	53	11	)	)	PUNCT
ejpam-6117	53	12	,	,	PUNCT
ejpam-6117	53	13	v	v	X
ejpam-6117	53	14	∈	∈	PROPN
ejpam-6117	53	15	v	v	NOUN
ejpam-6117	53	16	(	(	PUNCT
ejpam-6117	53	17	h	h	NOUN
ejpam-6117	53	18	)	)	PUNCT
ejpam-6117	53	19	}	}	PUNCT
ejpam-6117	53	20	.	.	PUNCT
ejpam-6117	54	1	the	the	DET
ejpam-6117	54	2	corona	corona	NOUN
ejpam-6117	54	3	g	g	ADP
ejpam-6117	54	4	◦	◦	NOUN
ejpam-6117	54	5	h	h	NOUN
ejpam-6117	54	6	of	of	ADP
ejpam-6117	54	7	two	two	NUM
ejpam-6117	54	8	graphs	graph	NOUN
ejpam-6117	54	9	g	g	NOUN
ejpam-6117	54	10	and	and	CCONJ
ejpam-6117	54	11	h	h	NOUN
ejpam-6117	54	12	is	be	AUX
ejpam-6117	54	13	the	the	DET
ejpam-6117	54	14	graph	graph	NOUN
ejpam-6117	54	15	obtained	obtain	VERB
ejpam-6117	54	16	by	by	ADP
ejpam-6117	54	17	taking	take	VERB
ejpam-6117	54	18	one	one	NUM
ejpam-6117	54	19	copy	copy	NOUN
ejpam-6117	54	20	of	of	ADP
ejpam-6117	54	21	g	g	PROPN
ejpam-6117	54	22	and	and	CCONJ
ejpam-6117	54	23	|v	|v	PROPN
ejpam-6117	54	24	(	(	PUNCT
ejpam-6117	54	25	g)|	g)|	NOUN
ejpam-6117	54	26	copies	copy	NOUN
ejpam-6117	54	27	of	of	ADP
ejpam-6117	54	28	h	h	NOUN
ejpam-6117	54	29	,	,	PUNCT
ejpam-6117	54	30	and	and	CCONJ
ejpam-6117	54	31	then	then	ADV
ejpam-6117	54	32	joining	join	VERB
ejpam-6117	54	33	the	the	DET
ejpam-6117	54	34	ith	ith	PROPN
ejpam-6117	54	35	vertex	vertex	NOUN
ejpam-6117	54	36	of	of	ADP
ejpam-6117	54	37	g	g	NOUN
ejpam-6117	54	38	to	to	ADP
ejpam-6117	54	39	every	every	DET
ejpam-6117	54	40	vertex	vertex	NOUN
ejpam-6117	54	41	of	of	ADP
ejpam-6117	54	42	the	the	DET
ejpam-6117	54	43	ith	ith	PROPN
ejpam-6117	54	44	copy	copy	NOUN
ejpam-6117	54	45	of	of	ADP
ejpam-6117	54	46	h.	h.	PROPN
ejpam-6117	54	47	denote	denote	PROPN
ejpam-6117	54	48	by	by	ADP
ejpam-6117	54	49	hv	hv	PROPN
ejpam-6117	54	50	the	the	DET
ejpam-6117	54	51	copy	copy	NOUN
ejpam-6117	54	52	of	of	ADP
ejpam-6117	54	53	h	h	NOUN
ejpam-6117	54	54	in	in	ADP
ejpam-6117	54	55	g	g	PROPN
ejpam-6117	54	56	◦	◦	NOUN
ejpam-6117	54	57	h	h	NOUN
ejpam-6117	54	58	,	,	PUNCT
ejpam-6117	54	59	every	every	DET
ejpam-6117	54	60	vertex	vertex	NOUN
ejpam-6117	54	61	of	of	ADP
ejpam-6117	54	62	which	which	PRON
ejpam-6117	54	63	is	be	AUX
ejpam-6117	54	64	adjacent	adjacent	ADJ
ejpam-6117	54	65	to	to	ADP
ejpam-6117	54	66	a	a	DET
ejpam-6117	54	67	unique	unique	ADJ
ejpam-6117	54	68	vertex	vertex	NOUN
ejpam-6117	54	69	v	v	ADP
ejpam-6117	54	70	∈	∈	NOUN
ejpam-6117	54	71	g.	g.	NOUN
ejpam-6117	55	1	the	the	DET
ejpam-6117	55	2	lexicographic	lexicographic	ADJ
ejpam-6117	55	3	product	product	NOUN
ejpam-6117	55	4	g[h	g[h	PROPN
ejpam-6117	55	5	]	]	PUNCT
ejpam-6117	55	6	of	of	ADP
ejpam-6117	55	7	two	two	NUM
ejpam-6117	55	8	graphs	graph	NOUN
ejpam-6117	55	9	g	g	NOUN
ejpam-6117	55	10	and	and	CCONJ
ejpam-6117	55	11	h	h	NOUN
ejpam-6117	55	12	is	be	AUX
ejpam-6117	55	13	the	the	DET
ejpam-6117	55	14	graph	graph	NOUN
ejpam-6117	55	15	with	with	ADP
ejpam-6117	55	16	v	v	NOUN
ejpam-6117	55	17	(	(	PUNCT
ejpam-6117	55	18	g[h	g[h	PROPN
ejpam-6117	55	19	]	]	PUNCT
ejpam-6117	55	20	)	)	PUNCT
ejpam-6117	55	21	=	=	SYM
ejpam-6117	55	22	v	v	X
ejpam-6117	55	23	(	(	PUNCT
ejpam-6117	55	24	g)×v	g)×v	PROPN
ejpam-6117	55	25	(	(	PUNCT
ejpam-6117	55	26	h	h	NOUN
ejpam-6117	55	27	)	)	PUNCT
ejpam-6117	55	28	,	,	PUNCT
ejpam-6117	55	29	and	and	CCONJ
ejpam-6117	55	30	(	(	PUNCT
ejpam-6117	55	31	u	u	NOUN
ejpam-6117	55	32	,	,	PUNCT
ejpam-6117	55	33	u′)(v	u′)(v	NOUN
ejpam-6117	55	34	,	,	PUNCT
ejpam-6117	55	35	v′	v′	NOUN
ejpam-6117	55	36	)	)	PUNCT
ejpam-6117	55	37	∈	∈	NOUN
ejpam-6117	55	38	e(g[h	e(g[h	NOUN
ejpam-6117	55	39	]	]	PUNCT
ejpam-6117	55	40	)	)	PUNCT
ejpam-6117	56	1	if	if	SCONJ
ejpam-6117	56	2	and	and	CCONJ
ejpam-6117	56	3	only	only	ADV
ejpam-6117	56	4	if	if	SCONJ
ejpam-6117	56	5	either	either	DET
ejpam-6117	56	6	uv	uv	PROPN
ejpam-6117	56	7	∈	∈	PROPN
ejpam-6117	56	8	e(g	e(g	PROPN
ejpam-6117	56	9	)	)	PUNCT
ejpam-6117	56	10	or	or	CCONJ
ejpam-6117	56	11	u	u	X
ejpam-6117	56	12	=	=	NOUN
ejpam-6117	56	13	v	v	PROPN
ejpam-6117	56	14	and	and	CCONJ
ejpam-6117	56	15	u′v′	u′v′	PROPN
ejpam-6117	56	16	∈	∈	PROPN
ejpam-6117	56	17	e(h	e(h	PROPN
ejpam-6117	56	18	)	)	PUNCT
ejpam-6117	56	19	.	.	PUNCT
ejpam-6117	57	1	observe	observe	VERB
ejpam-6117	57	2	that	that	SCONJ
ejpam-6117	57	3	any	any	DET
ejpam-6117	57	4	non	non	ADJ
ejpam-6117	57	5	-	-	ADJ
ejpam-6117	57	6	empty	empty	ADJ
ejpam-6117	57	7	subset	subset	NOUN
ejpam-6117	57	8	c	c	NOUN
ejpam-6117	57	9	of	of	ADP
ejpam-6117	57	10	v	v	PROPN
ejpam-6117	57	11	(	(	PUNCT
ejpam-6117	57	12	g	g	NOUN
ejpam-6117	57	13	)	)	PUNCT
ejpam-6117	57	14	×	×	NOUN
ejpam-6117	57	15	v	v	NOUN
ejpam-6117	57	16	(	(	PUNCT
ejpam-6117	57	17	h	h	NOUN
ejpam-6117	57	18	)	)	PUNCT
ejpam-6117	57	19	(	(	PUNCT
ejpam-6117	57	20	in	in	ADP
ejpam-6117	57	21	fact	fact	NOUN
ejpam-6117	57	22	,	,	PUNCT
ejpam-6117	57	23	any	any	DET
ejpam-6117	57	24	set	set	NOUN
ejpam-6117	57	25	of	of	ADP
ejpam-6117	57	26	ordered	order	VERB
ejpam-6117	57	27	pairs	pair	NOUN
ejpam-6117	57	28	)	)	PUNCT
ejpam-6117	57	29	can	can	AUX
ejpam-6117	57	30	be	be	AUX
ejpam-6117	57	31	written	write	VERB
ejpam-6117	57	32	as	as	ADP
ejpam-6117	57	33	c	c	NOUN
ejpam-6117	57	34	=	=	SYM
ejpam-6117	57	35	∪x∈s({x	∪x∈s({x	NOUN
ejpam-6117	57	36	}	}	PUNCT
ejpam-6117	57	37	×	×	NOUN
ejpam-6117	57	38	tx	tx	PROPN
ejpam-6117	57	39	)	)	PUNCT
ejpam-6117	57	40	⊆	⊆	NUM
ejpam-6117	57	41	v	v	NOUN
ejpam-6117	57	42	(	(	PUNCT
ejpam-6117	57	43	g[h	g[h	PROPN
ejpam-6117	57	44	]	]	PUNCT
ejpam-6117	57	45	)	)	PUNCT
ejpam-6117	57	46	,	,	PUNCT
ejpam-6117	57	47	where	where	SCONJ
ejpam-6117	57	48	s	s	VERB
ejpam-6117	57	49	⊆	⊆	NUM
ejpam-6117	57	50	v	v	NOUN
ejpam-6117	57	51	(	(	PUNCT
ejpam-6117	57	52	g	g	NOUN
ejpam-6117	57	53	)	)	PUNCT
ejpam-6117	57	54	and	and	CCONJ
ejpam-6117	57	55	tx	tx	VERB
ejpam-6117	57	56	⊆	⊆	NUM
ejpam-6117	57	57	v	v	NOUN
ejpam-6117	57	58	(	(	PUNCT
ejpam-6117	57	59	h	h	NOUN
ejpam-6117	57	60	)	)	PUNCT
ejpam-6117	57	61	for	for	ADP
ejpam-6117	57	62	all	all	DET
ejpam-6117	57	63	x	x	PROPN
ejpam-6117	57	64	∈	∈	PROPN
ejpam-6117	57	65	s.	s.	PROPN
ejpam-6117	57	66	henceforth	henceforth	ADV
ejpam-6117	57	67	,	,	PUNCT
ejpam-6117	57	68	we	we	PRON
ejpam-6117	57	69	shall	shall	AUX
ejpam-6117	57	70	use	use	VERB
ejpam-6117	57	71	this	this	DET
ejpam-6117	57	72	form	form	NOUN
ejpam-6117	57	73	to	to	PART
ejpam-6117	57	74	denote	denote	VERB
ejpam-6117	57	75	any	any	DET
ejpam-6117	57	76	subset	subset	NOUN
ejpam-6117	57	77	c	c	NOUN
ejpam-6117	57	78	of	of	ADP
ejpam-6117	57	79	v	v	PROPN
ejpam-6117	57	80	(	(	PUNCT
ejpam-6117	57	81	g)×	g)×	NOUN
ejpam-6117	57	82	v	v	NOUN
ejpam-6117	57	83	(	(	PUNCT
ejpam-6117	57	84	h	h	NOUN
ejpam-6117	57	85	)	)	PUNCT
ejpam-6117	57	86	.	.	PUNCT
ejpam-6117	58	1	readers	reader	NOUN
ejpam-6117	58	2	are	be	AUX
ejpam-6117	58	3	referred	refer	VERB
ejpam-6117	58	4	to	to	ADP
ejpam-6117	58	5	[	[	X
ejpam-6117	58	6	7	7	X
ejpam-6117	58	7	]	]	PUNCT
ejpam-6117	58	8	for	for	ADP
ejpam-6117	58	9	other	other	ADJ
ejpam-6117	58	10	basic	basic	ADJ
ejpam-6117	58	11	definitions	definition	NOUN
ejpam-6117	58	12	that	that	PRON
ejpam-6117	58	13	are	be	AUX
ejpam-6117	58	14	not	not	PART
ejpam-6117	58	15	given	give	VERB
ejpam-6117	58	16	here	here	ADV
ejpam-6117	58	17	.	.	PUNCT
ejpam-6117	59	1	3	3	X
ejpam-6117	59	2	.	.	X
ejpam-6117	59	3	results	result	NOUN
ejpam-6117	59	4	it	it	PRON
ejpam-6117	59	5	is	be	AUX
ejpam-6117	59	6	worth	worth	ADJ
ejpam-6117	59	7	noting	note	VERB
ejpam-6117	59	8	that	that	SCONJ
ejpam-6117	59	9	if	if	SCONJ
ejpam-6117	59	10	g	g	PROPN
ejpam-6117	59	11	is	be	AUX
ejpam-6117	59	12	a	a	DET
ejpam-6117	59	13	graph	graph	NOUN
ejpam-6117	59	14	and	and	CCONJ
ejpam-6117	59	15	s	s	VERB
ejpam-6117	59	16	⊆	⊆	NUM
ejpam-6117	59	17	v	v	NOUN
ejpam-6117	59	18	(	(	PUNCT
ejpam-6117	59	19	g	g	NOUN
ejpam-6117	59	20	)	)	PUNCT
ejpam-6117	59	21	,	,	PUNCT
ejpam-6117	59	22	then	then	ADV
ejpam-6117	59	23	⟨s⟩w	⟨s⟩w	NOUN
ejpam-6117	59	24	is	be	AUX
ejpam-6117	59	25	simply	simply	ADV
ejpam-6117	59	26	obtained	obtain	VERB
ejpam-6117	59	27	from	from	ADP
ejpam-6117	59	28	the	the	DET
ejpam-6117	59	29	⟨n	⟨n	NUM
ejpam-6117	59	30	[	[	PUNCT
ejpam-6117	59	31	s]⟩	s]⟩	VERB
ejpam-6117	59	32	by	by	ADP
ejpam-6117	59	33	deleting	delete	VERB
ejpam-6117	59	34	all	all	DET
ejpam-6117	59	35	the	the	DET
ejpam-6117	59	36	edges	edge	NOUN
ejpam-6117	59	37	e	e	NOUN
ejpam-6117	59	38	=	=	PUNCT
ejpam-6117	59	39	xy	xy	PROPN
ejpam-6117	59	40	in	in	ADP
ejpam-6117	59	41	e(g	e(g	PROPN
ejpam-6117	59	42	)	)	PUNCT
ejpam-6117	59	43	with	with	ADP
ejpam-6117	59	44	x	x	PROPN
ejpam-6117	59	45	,	,	PUNCT
ejpam-6117	59	46	y	y	PROPN
ejpam-6117	59	47	∈	∈	PROPN
ejpam-6117	59	48	n(s	n(s	PROPN
ejpam-6117	59	49	)	)	PUNCT
ejpam-6117	59	50	\	\	PUNCT
ejpam-6117	60	1	s.	s.	PROPN
ejpam-6117	60	2	the	the	DET
ejpam-6117	60	3	first	first	ADJ
ejpam-6117	60	4	result	result	NOUN
ejpam-6117	60	5	is	be	AUX
ejpam-6117	60	6	easy	easy	ADJ
ejpam-6117	60	7	and	and	CCONJ
ejpam-6117	60	8	almost	almost	ADV
ejpam-6117	60	9	follows	follow	VERB
ejpam-6117	60	10	from	from	ADP
ejpam-6117	60	11	the	the	DET
ejpam-6117	60	12	definitions	definition	NOUN
ejpam-6117	60	13	.	.	PUNCT
ejpam-6117	61	1	theorem	theorem	NOUN
ejpam-6117	61	2	1	1	NUM
ejpam-6117	61	3	.	.	X
ejpam-6117	61	4	for	for	ADP
ejpam-6117	61	5	any	any	DET
ejpam-6117	61	6	graph	graph	NOUN
ejpam-6117	61	7	g	g	NOUN
ejpam-6117	61	8	of	of	ADP
ejpam-6117	61	9	order	order	NOUN
ejpam-6117	61	10	n	n	CCONJ
ejpam-6117	61	11	,	,	PUNCT
ejpam-6117	61	12	1	1	NUM
ejpam-6117	61	13	≤	≤	NOUN
ejpam-6117	61	14	αw(g	αw(g	NOUN
ejpam-6117	61	15	)	)	PUNCT
ejpam-6117	61	16	≤	≤	NOUN
ejpam-6117	61	17	α(g	α(g	NUM
ejpam-6117	61	18	)	)	PUNCT
ejpam-6117	61	19	.	.	PUNCT
ejpam-6117	62	1	moreover	moreover	ADV
ejpam-6117	62	2	,	,	PUNCT
ejpam-6117	62	3	(	(	PUNCT
ejpam-6117	62	4	i	i	NOUN
ejpam-6117	62	5	)	)	PUNCT
ejpam-6117	62	6	αw(g	αw(g	NUM
ejpam-6117	62	7	)	)	PUNCT
ejpam-6117	62	8	=	=	SYM
ejpam-6117	62	9	1	1	NUM
ejpam-6117	62	10	if	if	SCONJ
ejpam-6117	62	11	and	and	CCONJ
ejpam-6117	62	12	only	only	ADV
ejpam-6117	62	13	if	if	SCONJ
ejpam-6117	62	14	every	every	DET
ejpam-6117	62	15	component	component	NOUN
ejpam-6117	62	16	h	h	NOUN
ejpam-6117	62	17	of	of	ADP
ejpam-6117	62	18	g	g	PROPN
ejpam-6117	62	19	is	be	AUX
ejpam-6117	62	20	complete	complete	ADJ
ejpam-6117	62	21	;	;	PUNCT
ejpam-6117	62	22	and	and	CCONJ
ejpam-6117	62	23	(	(	PUNCT
ejpam-6117	62	24	ii	ii	NOUN
ejpam-6117	62	25	)	)	PUNCT
ejpam-6117	62	26	αw(g	αw(g	NUM
ejpam-6117	62	27	)	)	PUNCT
ejpam-6117	62	28	=	=	SYM
ejpam-6117	62	29	α(g	α(g	NUM
ejpam-6117	62	30	)	)	PUNCT
ejpam-6117	62	31	if	if	SCONJ
ejpam-6117	62	32	and	and	CCONJ
ejpam-6117	62	33	only	only	ADV
ejpam-6117	62	34	if	if	SCONJ
ejpam-6117	62	35	⟨s⟩w	⟨s⟩w	NOUN
ejpam-6117	62	36	is	be	AUX
ejpam-6117	62	37	connected	connect	VERB
ejpam-6117	62	38	for	for	ADP
ejpam-6117	62	39	some	some	DET
ejpam-6117	62	40	α	α	NOUN
ejpam-6117	62	41	-	-	PUNCT
ejpam-6117	62	42	set	set	VERB
ejpam-6117	62	43	sin	sin	NOUN
ejpam-6117	62	44	g.	g.	NOUN
ejpam-6117	62	45	proof	proof	NOUN
ejpam-6117	62	46	.	.	PUNCT
ejpam-6117	63	1	clearly	clearly	ADV
ejpam-6117	63	2	,	,	PUNCT
ejpam-6117	63	3	1	1	NUM
ejpam-6117	63	4	≤	≤	NOUN
ejpam-6117	63	5	αw(g	αw(g	NUM
ejpam-6117	63	6	)	)	PUNCT
ejpam-6117	63	7	.	.	PUNCT
ejpam-6117	64	1	since	since	SCONJ
ejpam-6117	64	2	every	every	DET
ejpam-6117	64	3	weakly	weakly	ADV
ejpam-6117	64	4	connected	connected	ADJ
ejpam-6117	64	5	independent	independent	ADJ
ejpam-6117	64	6	set	set	NOUN
ejpam-6117	64	7	is	be	AUX
ejpam-6117	64	8	independent	independent	ADJ
ejpam-6117	64	9	,	,	PUNCT
ejpam-6117	64	10	it	it	PRON
ejpam-6117	64	11	follows	follow	VERB
ejpam-6117	64	12	that	that	SCONJ
ejpam-6117	64	13	αw(g	αw(g	NOUN
ejpam-6117	64	14	)	)	PUNCT
ejpam-6117	64	15	≤	≤	NOUN
ejpam-6117	64	16	α(g	α(g	NUM
ejpam-6117	64	17	)	)	PUNCT
ejpam-6117	64	18	.	.	PUNCT
ejpam-6117	65	1	(	(	PUNCT
ejpam-6117	65	2	i	i	NOUN
ejpam-6117	65	3	)	)	PUNCT
ejpam-6117	65	4	suppose	suppose	VERB
ejpam-6117	65	5	that	that	SCONJ
ejpam-6117	65	6	αw(g	αw(g	NUM
ejpam-6117	65	7	)	)	PUNCT
ejpam-6117	65	8	=	=	SYM
ejpam-6117	65	9	1	1	NUM
ejpam-6117	65	10	and	and	CCONJ
ejpam-6117	65	11	assume	assume	VERB
ejpam-6117	65	12	on	on	ADP
ejpam-6117	65	13	the	the	DET
ejpam-6117	65	14	contrary	contrary	NOUN
ejpam-6117	65	15	that	that	SCONJ
ejpam-6117	65	16	g	g	PROPN
ejpam-6117	65	17	has	have	VERB
ejpam-6117	65	18	a	a	DET
ejpam-6117	65	19	component	component	NOUN
ejpam-6117	65	20	h	h	NOUN
ejpam-6117	65	21	which	which	PRON
ejpam-6117	65	22	is	be	AUX
ejpam-6117	65	23	not	not	PART
ejpam-6117	65	24	complete	complete	ADJ
ejpam-6117	65	25	.	.	PUNCT
ejpam-6117	66	1	then	then	ADV
ejpam-6117	66	2	there	there	PRON
ejpam-6117	66	3	exist	exist	VERB
ejpam-6117	66	4	distinct	distinct	ADJ
ejpam-6117	66	5	vertices	vertex	NOUN
ejpam-6117	66	6	v	v	NOUN
ejpam-6117	66	7	and	and	CCONJ
ejpam-6117	66	8	w	w	NOUN
ejpam-6117	66	9	of	of	ADP
ejpam-6117	66	10	h	h	NOUN
ejpam-6117	66	11	such	such	ADJ
ejpam-6117	66	12	that	that	SCONJ
ejpam-6117	66	13	dh(v	dh(v	NOUN
ejpam-6117	66	14	,	,	PUNCT
ejpam-6117	66	15	w	w	NOUN
ejpam-6117	66	16	)	)	PUNCT
ejpam-6117	66	17	=	=	SYM
ejpam-6117	66	18	dg(v	dg(v	X
ejpam-6117	66	19	,	,	PUNCT
ejpam-6117	66	20	w	w	NOUN
ejpam-6117	66	21	)	)	PUNCT
ejpam-6117	66	22	=	=	SYM
ejpam-6117	66	23	2	2	X
ejpam-6117	66	24	.	.	X
ejpam-6117	66	25	let	let	VERB
ejpam-6117	66	26	s	s	AUX
ejpam-6117	66	27	=	=	PUNCT
ejpam-6117	66	28	{	{	PUNCT
ejpam-6117	66	29	v	v	NOUN
ejpam-6117	66	30	,	,	PUNCT
ejpam-6117	66	31	w	w	NOUN
ejpam-6117	66	32	}	}	PUNCT
ejpam-6117	66	33	and	and	CCONJ
ejpam-6117	66	34	let	let	VERB
ejpam-6117	66	35	u	u	PRON
ejpam-6117	66	36	∈	∈	PROPN
ejpam-6117	66	37	ng(v	ng(v	NOUN
ejpam-6117	66	38	)	)	PUNCT
ejpam-6117	66	39	∩	∩	NOUN
ejpam-6117	66	40	ng(w	ng(w	NOUN
ejpam-6117	66	41	)	)	PUNCT
ejpam-6117	66	42	.	.	PUNCT
ejpam-6117	67	1	then	then	ADV
ejpam-6117	67	2	s	s	VERB
ejpam-6117	67	3	is	be	AUX
ejpam-6117	67	4	independent	independent	ADJ
ejpam-6117	67	5	and	and	CCONJ
ejpam-6117	67	6	⟨s⟩w	⟨s⟩w	NOUN
ejpam-6117	67	7	is	be	AUX
ejpam-6117	67	8	connected	connect	VERB
ejpam-6117	67	9	.	.	PUNCT
ejpam-6117	68	1	this	this	PRON
ejpam-6117	68	2	implies	imply	VERB
ejpam-6117	68	3	that	that	SCONJ
ejpam-6117	68	4	αw(g	αw(g	NUM
ejpam-6117	68	5	)	)	PUNCT
ejpam-6117	68	6	≥	≥	NOUN
ejpam-6117	68	7	|s|	|s|	NOUN
ejpam-6117	68	8	=	=	SYM
ejpam-6117	68	9	2	2	NUM
ejpam-6117	68	10	,	,	PUNCT
ejpam-6117	68	11	contrary	contrary	ADJ
ejpam-6117	68	12	to	to	ADP
ejpam-6117	68	13	our	our	PRON
ejpam-6117	68	14	assumption	assumption	NOUN
ejpam-6117	68	15	that	that	SCONJ
ejpam-6117	68	16	αw(g	αw(g	X
ejpam-6117	68	17	)	)	PUNCT
ejpam-6117	68	18	=	=	SYM
ejpam-6117	68	19	1	1	X
ejpam-6117	68	20	.	.	PUNCT
ejpam-6117	69	1	thus	thus	ADV
ejpam-6117	69	2	,	,	PUNCT
ejpam-6117	69	3	every	every	DET
ejpam-6117	69	4	component	component	NOUN
ejpam-6117	69	5	h	h	NOUN
ejpam-6117	69	6	of	of	ADP
ejpam-6117	69	7	g	g	PROPN
ejpam-6117	69	8	is	be	AUX
ejpam-6117	69	9	complete	complete	ADJ
ejpam-6117	69	10	.	.	PUNCT
ejpam-6117	70	1	for	for	ADP
ejpam-6117	70	2	the	the	DET
ejpam-6117	70	3	converse	converse	NOUN
ejpam-6117	70	4	,	,	PUNCT
ejpam-6117	70	5	suppose	suppose	VERB
ejpam-6117	70	6	that	that	SCONJ
ejpam-6117	70	7	every	every	DET
ejpam-6117	70	8	component	component	NOUN
ejpam-6117	70	9	of	of	ADP
ejpam-6117	70	10	g	g	PROPN
ejpam-6117	70	11	is	be	AUX
ejpam-6117	70	12	complete	complete	ADJ
ejpam-6117	70	13	.	.	PUNCT
ejpam-6117	71	1	let	let	VERB
ejpam-6117	71	2	s	s	PRON
ejpam-6117	71	3	be	be	AUX
ejpam-6117	71	4	an	an	DET
ejpam-6117	71	5	αw	αw	NOUN
ejpam-6117	71	6	-	-	PUNCT
ejpam-6117	71	7	set	set	VERB
ejpam-6117	71	8	in	in	ADP
ejpam-6117	71	9	g.	g.	PROPN
ejpam-6117	71	10	since	since	SCONJ
ejpam-6117	71	11	⟨s⟩w	⟨s⟩w	PROPN
ejpam-6117	71	12	is	be	AUX
ejpam-6117	71	13	connected	connect	VERB
ejpam-6117	71	14	,	,	PUNCT
ejpam-6117	71	15	s	s	VERB
ejpam-6117	71	16	⊆	⊆	NUM
ejpam-6117	71	17	v	v	NOUN
ejpam-6117	71	18	(	(	PUNCT
ejpam-6117	71	19	h	h	NOUN
ejpam-6117	71	20	)	)	PUNCT
ejpam-6117	71	21	for	for	ADP
ejpam-6117	71	22	a	a	DET
ejpam-6117	71	23	unique	unique	ADJ
ejpam-6117	71	24	complete	complete	ADJ
ejpam-6117	71	25	component	component	NOUN
ejpam-6117	71	26	h	h	NOUN
ejpam-6117	71	27	of	of	ADP
ejpam-6117	71	28	g.	g.	PROPN
ejpam-6117	71	29	since	since	SCONJ
ejpam-6117	71	30	s	s	PROPN
ejpam-6117	71	31	is	be	AUX
ejpam-6117	71	32	an	an	DET
ejpam-6117	71	33	independent	independent	ADJ
ejpam-6117	71	34	set	set	NOUN
ejpam-6117	71	35	in	in	ADP
ejpam-6117	71	36	h	h	NOUN
ejpam-6117	71	37	,	,	PUNCT
ejpam-6117	71	38	it	it	PRON
ejpam-6117	71	39	follows	follow	VERB
ejpam-6117	71	40	that	that	SCONJ
ejpam-6117	71	41	|s|	|s|	NOUN
ejpam-6117	71	42	=	=	SYM
ejpam-6117	71	43	1	1	NUM
ejpam-6117	71	44	.	.	PUNCT
ejpam-6117	72	1	thus	thus	ADV
ejpam-6117	72	2	,	,	PUNCT
ejpam-6117	72	3	αw(g	αw(g	X
ejpam-6117	72	4	)	)	PUNCT
ejpam-6117	72	5	=	=	SYM
ejpam-6117	72	6	|s|	|s|	NOUN
ejpam-6117	72	7	=	=	SYM
ejpam-6117	72	8	1	1	PROPN
ejpam-6117	72	9	.	.	PUNCT
ejpam-6117	72	10	(	(	PUNCT
ejpam-6117	72	11	ii	ii	NOUN
ejpam-6117	72	12	)	)	PUNCT
ejpam-6117	72	13	suppose	suppose	VERB
ejpam-6117	72	14	that	that	SCONJ
ejpam-6117	72	15	αw(g	αw(g	NUM
ejpam-6117	72	16	)	)	PUNCT
ejpam-6117	72	17	=	=	SYM
ejpam-6117	73	1	α(g	α(g	NUM
ejpam-6117	73	2	)	)	PUNCT
ejpam-6117	73	3	,	,	PUNCT
ejpam-6117	73	4	say	say	VERB
ejpam-6117	73	5	s	s	NOUN
ejpam-6117	73	6	is	be	AUX
ejpam-6117	73	7	an	an	DET
ejpam-6117	73	8	αw	αw	NOUN
ejpam-6117	73	9	-	-	PUNCT
ejpam-6117	73	10	set	set	VERB
ejpam-6117	73	11	in	in	ADP
ejpam-6117	73	12	g.	g.	PROPN
ejpam-6117	74	1	then	then	ADV
ejpam-6117	74	2	s	s	VERB
ejpam-6117	74	3	is	be	AUX
ejpam-6117	74	4	an	an	DET
ejpam-6117	74	5	independent	independent	ADJ
ejpam-6117	74	6	set	set	NOUN
ejpam-6117	74	7	and	and	CCONJ
ejpam-6117	74	8	and	and	CCONJ
ejpam-6117	74	9	⟨s⟩w	⟨s⟩w	NOUN
ejpam-6117	74	10	is	be	AUX
ejpam-6117	74	11	connected	connect	VERB
ejpam-6117	74	12	.	.	PUNCT
ejpam-6117	75	1	since	since	SCONJ
ejpam-6117	75	2	|s|	|s|	NOUN
ejpam-6117	75	3	=	=	SYM
ejpam-6117	75	4	α(g	α(g	PROPN
ejpam-6117	75	5	)	)	PUNCT
ejpam-6117	75	6	,	,	PUNCT
ejpam-6117	75	7	it	it	PRON
ejpam-6117	75	8	follows	follow	VERB
ejpam-6117	75	9	that	that	SCONJ
ejpam-6117	75	10	s	s	VERB
ejpam-6117	75	11	is	be	AUX
ejpam-6117	75	12	an	an	DET
ejpam-6117	75	13	α	α	NOUN
ejpam-6117	75	14	-	-	PUNCT
ejpam-6117	75	15	set	set	VERB
ejpam-6117	75	16	in	in	ADP
ejpam-6117	75	17	g.	g.	NOUN
ejpam-6117	75	18	conversely	conversely	ADV
ejpam-6117	75	19	,	,	PUNCT
ejpam-6117	75	20	suppose	suppose	VERB
ejpam-6117	75	21	⟨s⟩w	⟨s⟩w	NOUN
ejpam-6117	75	22	is	be	AUX
ejpam-6117	75	23	connected	connect	VERB
ejpam-6117	75	24	for	for	ADP
ejpam-6117	75	25	some	some	DET
ejpam-6117	75	26	α	α	NOUN
ejpam-6117	75	27	-	-	PUNCT
ejpam-6117	75	28	set	set	VERB
ejpam-6117	75	29	s	s	PROPN
ejpam-6117	75	30	in	in	ADP
ejpam-6117	75	31	g.	g.	PROPN
ejpam-6117	76	1	then	then	ADV
ejpam-6117	76	2	s	s	VERB
ejpam-6117	76	3	is	be	AUX
ejpam-6117	76	4	a	a	DET
ejpam-6117	76	5	weakly	weakly	ADV
ejpam-6117	76	6	connected	connected	ADJ
ejpam-6117	76	7	independent	independent	ADJ
ejpam-6117	76	8	set	set	NOUN
ejpam-6117	76	9	.	.	PUNCT
ejpam-6117	77	1	therefore	therefore	ADV
ejpam-6117	77	2	,	,	PUNCT
ejpam-6117	77	3	αw(g	αw(g	NUM
ejpam-6117	77	4	)	)	PUNCT
ejpam-6117	77	5	=	=	SYM
ejpam-6117	77	6	|s|	|s|	NOUN
ejpam-6117	77	7	=	=	SYM
ejpam-6117	77	8	α(g	α(g	PROPN
ejpam-6117	77	9	)	)	PUNCT
ejpam-6117	77	10	.	.	PUNCT
ejpam-6117	78	1	the	the	DET
ejpam-6117	78	2	next	next	ADJ
ejpam-6117	78	3	result	result	NOUN
ejpam-6117	78	4	follows	follow	VERB
ejpam-6117	78	5	from	from	ADP
ejpam-6117	78	6	theorem	theorem	ADJ
ejpam-6117	78	7	1	1	NUM
ejpam-6117	78	8	.	.	PUNCT
ejpam-6117	78	9	corollary	corollary	ADJ
ejpam-6117	78	10	1	1	NUM
ejpam-6117	78	11	.	.	PUNCT
ejpam-6117	79	1	let	let	VERB
ejpam-6117	79	2	g	g	PRON
ejpam-6117	79	3	be	be	AUX
ejpam-6117	79	4	a	a	DET
ejpam-6117	79	5	connected	connected	ADJ
ejpam-6117	79	6	graph	graph	NOUN
ejpam-6117	79	7	of	of	ADP
ejpam-6117	79	8	order	order	NOUN
ejpam-6117	79	9	n.	n.	NOUN
ejpam-6117	79	10	then	then	ADV
ejpam-6117	79	11	αw(g	αw(g	X
ejpam-6117	79	12	)	)	PUNCT
ejpam-6117	79	13	=	=	SYM
ejpam-6117	79	14	1	1	NUM
ejpam-6117	79	15	if	if	SCONJ
ejpam-6117	79	16	and	and	CCONJ
ejpam-6117	79	17	only	only	ADV
ejpam-6117	79	18	if	if	SCONJ
ejpam-6117	79	19	g	g	PROPN
ejpam-6117	79	20	=	=	PROPN
ejpam-6117	79	21	kn	kn	PROPN
ejpam-6117	79	22	.	.	PUNCT
ejpam-6117	80	1	we	we	PRON
ejpam-6117	80	2	now	now	ADV
ejpam-6117	80	3	characterize	characterize	VERB
ejpam-6117	80	4	all	all	DET
ejpam-6117	80	5	connected	connected	ADJ
ejpam-6117	80	6	graphs	graph	NOUN
ejpam-6117	80	7	g	g	ADP
ejpam-6117	80	8	of	of	ADP
ejpam-6117	80	9	order	order	NOUN
ejpam-6117	80	10	n	n	PRON
ejpam-6117	80	11	≥	≥	NUM
ejpam-6117	80	12	2	2	NUM
ejpam-6117	80	13	such	such	ADJ
ejpam-6117	80	14	that	that	PRON
ejpam-6117	80	15	αw(g	αw(g	NUM
ejpam-6117	80	16	)	)	PUNCT
ejpam-6117	80	17	=	=	SYM
ejpam-6117	80	18	n−	n−	NOUN
ejpam-6117	80	19	1	1	NUM
ejpam-6117	80	20	.	.	PUNCT
ejpam-6117	80	21	r.	r.	PROPN
ejpam-6117	80	22	merontos	merontos	PROPN
ejpam-6117	80	23	et	et	PROPN
ejpam-6117	80	24	al	al	PROPN
ejpam-6117	80	25	.	.	PUNCT
ejpam-6117	80	26	/	/	SYM
ejpam-6117	80	27	eur	eur	PROPN
ejpam-6117	80	28	.	.	PUNCT
ejpam-6117	81	1	j.	j.	PROPN
ejpam-6117	81	2	pure	pure	PROPN
ejpam-6117	81	3	appl	appl	PROPN
ejpam-6117	81	4	.	.	PROPN
ejpam-6117	81	5	math	math	PROPN
ejpam-6117	81	6	,	,	PUNCT
ejpam-6117	81	7	18	18	NUM
ejpam-6117	81	8	(	(	PUNCT
ejpam-6117	81	9	2	2	NUM
ejpam-6117	81	10	)	)	PUNCT
ejpam-6117	81	11	(	(	PUNCT
ejpam-6117	81	12	2025	2025	NUM
ejpam-6117	81	13	)	)	PUNCT
ejpam-6117	81	14	,	,	PUNCT
ejpam-6117	81	15	6117	6117	NUM
ejpam-6117	81	16	4	4	NUM
ejpam-6117	81	17	of	of	ADP
ejpam-6117	81	18	9	9	NUM
ejpam-6117	81	19	theorem	theorem	NOUN
ejpam-6117	81	20	2	2	NUM
ejpam-6117	81	21	.	.	PUNCT
ejpam-6117	82	1	let	let	VERB
ejpam-6117	82	2	g	g	PRON
ejpam-6117	82	3	be	be	AUX
ejpam-6117	82	4	a	a	DET
ejpam-6117	82	5	connected	connected	ADJ
ejpam-6117	82	6	graph	graph	NOUN
ejpam-6117	82	7	of	of	ADP
ejpam-6117	82	8	order	order	NOUN
ejpam-6117	82	9	n	n	PRON
ejpam-6117	82	10	≥	≥	NOUN
ejpam-6117	82	11	2	2	NUM
ejpam-6117	82	12	.	.	PUNCT
ejpam-6117	83	1	then	then	ADV
ejpam-6117	83	2	αw(g	αw(g	NOUN
ejpam-6117	83	3	)	)	PUNCT
ejpam-6117	83	4	=	=	SYM
ejpam-6117	84	1	n	n	CCONJ
ejpam-6117	84	2	−	−	PROPN
ejpam-6117	84	3	1	1	NUM
ejpam-6117	85	1	if	if	SCONJ
ejpam-6117	85	2	and	and	CCONJ
ejpam-6117	85	3	only	only	ADV
ejpam-6117	85	4	if	if	SCONJ
ejpam-6117	85	5	g	g	NOUN
ejpam-6117	85	6	=	=	SYM
ejpam-6117	85	7	k1,n−1	k1,n−1	ADJ
ejpam-6117	85	8	.	.	PUNCT
ejpam-6117	86	1	proof	proof	NOUN
ejpam-6117	86	2	.	.	PUNCT
ejpam-6117	87	1	assume	assume	VERB
ejpam-6117	87	2	that	that	SCONJ
ejpam-6117	87	3	g	g	PROPN
ejpam-6117	87	4	is	be	AUX
ejpam-6117	87	5	a	a	DET
ejpam-6117	87	6	connected	connected	ADJ
ejpam-6117	87	7	graph	graph	NOUN
ejpam-6117	87	8	of	of	ADP
ejpam-6117	87	9	order	order	NOUN
ejpam-6117	87	10	n	n	PRON
ejpam-6117	87	11	≥	≥	NOUN
ejpam-6117	87	12	2	2	NUM
ejpam-6117	87	13	and	and	CCONJ
ejpam-6117	87	14	αw(g	αw(g	NUM
ejpam-6117	87	15	)	)	PUNCT
ejpam-6117	87	16	=	=	SYM
ejpam-6117	88	1	n	n	CCONJ
ejpam-6117	88	2	−	−	NOUN
ejpam-6117	89	1	1	1	X
ejpam-6117	89	2	.	.	PUNCT
ejpam-6117	90	1	if	if	SCONJ
ejpam-6117	90	2	n	n	NOUN
ejpam-6117	90	3	=	=	SYM
ejpam-6117	90	4	2	2	NUM
ejpam-6117	90	5	,	,	PUNCT
ejpam-6117	90	6	then	then	ADV
ejpam-6117	90	7	αw(g	αw(g	NUM
ejpam-6117	90	8	)	)	PUNCT
ejpam-6117	90	9	=	=	SYM
ejpam-6117	91	1	1	1	X
ejpam-6117	91	2	.	.	PUNCT
ejpam-6117	91	3	by	by	ADP
ejpam-6117	91	4	corollary	corollary	ADJ
ejpam-6117	91	5	1	1	NUM
ejpam-6117	91	6	,	,	PUNCT
ejpam-6117	91	7	g	g	PROPN
ejpam-6117	91	8	=	=	SYM
ejpam-6117	91	9	k2	k2	PROPN
ejpam-6117	91	10	=	=	PROPN
ejpam-6117	91	11	k1,1	k1,1	PROPN
ejpam-6117	91	12	.	.	PUNCT
ejpam-6117	91	13	suppose	suppose	VERB
ejpam-6117	91	14	n	n	PRON
ejpam-6117	91	15	≥	≥	NUM
ejpam-6117	91	16	3	3	X
ejpam-6117	91	17	.	.	PUNCT
ejpam-6117	92	1	let	let	VERB
ejpam-6117	92	2	s	s	PRON
ejpam-6117	92	3	be	be	AUX
ejpam-6117	92	4	an	an	DET
ejpam-6117	92	5	αw	αw	NOUN
ejpam-6117	92	6	-	-	PUNCT
ejpam-6117	92	7	set	set	NOUN
ejpam-6117	92	8	of	of	ADP
ejpam-6117	92	9	g	g	NOUN
ejpam-6117	92	10	,	,	PUNCT
ejpam-6117	92	11	say	say	VERB
ejpam-6117	92	12	s	s	VERB
ejpam-6117	92	13	=	=	SYM
ejpam-6117	92	14	v	v	PROPN
ejpam-6117	92	15	(	(	PUNCT
ejpam-6117	92	16	g	g	NOUN
ejpam-6117	92	17	)	)	PUNCT
ejpam-6117	92	18	∖	∖	NOUN
ejpam-6117	92	19	{	{	PUNCT
ejpam-6117	92	20	w	w	NOUN
ejpam-6117	92	21	}	}	PUNCT
ejpam-6117	92	22	.	.	PUNCT
ejpam-6117	93	1	since	since	SCONJ
ejpam-6117	93	2	s	s	PROPN
ejpam-6117	93	3	is	be	AUX
ejpam-6117	93	4	independent	independent	ADJ
ejpam-6117	93	5	,	,	PUNCT
ejpam-6117	93	6	it	it	PRON
ejpam-6117	93	7	follows	follow	VERB
ejpam-6117	93	8	that	that	PRON
ejpam-6117	93	9	uv	uv	PROPN
ejpam-6117	93	10	/∈	/∈	PUNCT
ejpam-6117	93	11	e(g	e(g	PROPN
ejpam-6117	93	12	)	)	PUNCT
ejpam-6117	93	13	for	for	ADP
ejpam-6117	93	14	every	every	DET
ejpam-6117	93	15	pair	pair	NOUN
ejpam-6117	93	16	of	of	ADP
ejpam-6117	93	17	vertices	vertex	NOUN
ejpam-6117	93	18	u	u	NOUN
ejpam-6117	93	19	,	,	PUNCT
ejpam-6117	93	20	v	v	PROPN
ejpam-6117	93	21	∈	∈	NOUN
ejpam-6117	93	22	s.	s.	PROPN
ejpam-6117	93	23	now	now	ADV
ejpam-6117	93	24	,	,	PUNCT
ejpam-6117	93	25	since	since	SCONJ
ejpam-6117	93	26	s	s	NOUN
ejpam-6117	93	27	is	be	AUX
ejpam-6117	93	28	weakly	weakly	ADV
ejpam-6117	93	29	connected	connected	ADJ
ejpam-6117	93	30	in	in	ADP
ejpam-6117	93	31	g	g	PROPN
ejpam-6117	93	32	,	,	PUNCT
ejpam-6117	93	33	it	it	PRON
ejpam-6117	93	34	follows	follow	VERB
ejpam-6117	93	35	that	that	SCONJ
ejpam-6117	93	36	w	w	PROPN
ejpam-6117	93	37	∈	∈	PROPN
ejpam-6117	93	38	ng(v	ng(v	PUNCT
ejpam-6117	93	39	)	)	PUNCT
ejpam-6117	93	40	for	for	ADP
ejpam-6117	93	41	each	each	DET
ejpam-6117	93	42	v	v	NOUN
ejpam-6117	93	43	∈	∈	PROPN
ejpam-6117	93	44	s.	s.	PROPN
ejpam-6117	93	45	consequently	consequently	ADV
ejpam-6117	93	46	,	,	PUNCT
ejpam-6117	93	47	g	g	PROPN
ejpam-6117	93	48	=	=	PUNCT
ejpam-6117	93	49	k1,n−1	k1,n−1	PROPN
ejpam-6117	93	50	.	.	PUNCT
ejpam-6117	94	1	for	for	ADP
ejpam-6117	94	2	the	the	DET
ejpam-6117	94	3	converse	converse	NOUN
ejpam-6117	94	4	,	,	PUNCT
ejpam-6117	94	5	suppose	suppose	VERB
ejpam-6117	94	6	that	that	SCONJ
ejpam-6117	94	7	g	g	PROPN
ejpam-6117	94	8	=	=	SYM
ejpam-6117	94	9	k1,n−1	k1,n−1	PROPN
ejpam-6117	94	10	.	.	PUNCT
ejpam-6117	95	1	let	let	VERB
ejpam-6117	95	2	w	w	NOUN
ejpam-6117	95	3	be	be	AUX
ejpam-6117	95	4	the	the	DET
ejpam-6117	95	5	central	central	ADJ
ejpam-6117	95	6	vertex	vertex	NOUN
ejpam-6117	95	7	of	of	ADP
ejpam-6117	95	8	g.	g.	PROPN
ejpam-6117	95	9	then	then	ADV
ejpam-6117	95	10	,	,	PUNCT
ejpam-6117	95	11	clearly	clearly	ADV
ejpam-6117	95	12	,	,	PUNCT
ejpam-6117	95	13	s	s	NOUN
ejpam-6117	95	14	=	=	SYM
ejpam-6117	95	15	v	v	X
ejpam-6117	95	16	(	(	PUNCT
ejpam-6117	95	17	g	g	NOUN
ejpam-6117	95	18	)	)	PUNCT
ejpam-6117	95	19	∖	∖	NOUN
ejpam-6117	95	20	{	{	PUNCT
ejpam-6117	95	21	w	w	NOUN
ejpam-6117	95	22	}	}	PUNCT
ejpam-6117	95	23	is	be	AUX
ejpam-6117	95	24	a	a	DET
ejpam-6117	95	25	weakly	weakly	ADV
ejpam-6117	95	26	connected	connected	ADJ
ejpam-6117	95	27	independent	independent	ADJ
ejpam-6117	95	28	set	set	NOUN
ejpam-6117	95	29	in	in	ADP
ejpam-6117	95	30	g.	g.	PROPN
ejpam-6117	95	31	since	since	SCONJ
ejpam-6117	95	32	s	s	PROPN
ejpam-6117	95	33	is	be	AUX
ejpam-6117	95	34	also	also	ADV
ejpam-6117	95	35	an	an	DET
ejpam-6117	95	36	α	α	NOUN
ejpam-6117	95	37	-	-	PUNCT
ejpam-6117	95	38	set	set	NOUN
ejpam-6117	95	39	in	in	ADP
ejpam-6117	95	40	g	g	NOUN
ejpam-6117	95	41	,	,	PUNCT
ejpam-6117	95	42	it	it	PRON
ejpam-6117	95	43	follows	follow	VERB
ejpam-6117	95	44	from	from	ADP
ejpam-6117	95	45	theorem	theorem	ADJ
ejpam-6117	95	46	1(ii	1(ii	NUM
ejpam-6117	95	47	)	)	PUNCT
ejpam-6117	95	48	that	that	SCONJ
ejpam-6117	95	49	αw(g	αw(g	NUM
ejpam-6117	95	50	)	)	PUNCT
ejpam-6117	95	51	=	=	SYM
ejpam-6117	95	52	n−	n−	NOUN
ejpam-6117	95	53	1	1	NUM
ejpam-6117	95	54	.	.	PUNCT
ejpam-6117	96	1	theorem	theorem	NOUN
ejpam-6117	96	2	3	3	X
ejpam-6117	96	3	.	.	PUNCT
ejpam-6117	97	1	let	let	VERB
ejpam-6117	97	2	k	k	PROPN
ejpam-6117	97	3	,	,	PUNCT
ejpam-6117	97	4	m	m	PROPN
ejpam-6117	97	5	,	,	PUNCT
ejpam-6117	97	6	and	and	CCONJ
ejpam-6117	97	7	n	n	CCONJ
ejpam-6117	97	8	be	be	AUX
ejpam-6117	97	9	non	non	ADJ
ejpam-6117	97	10	-	-	ADJ
ejpam-6117	97	11	negative	negative	ADJ
ejpam-6117	97	12	integers	integer	NOUN
ejpam-6117	97	13	with	with	ADP
ejpam-6117	97	14	k	k	PROPN
ejpam-6117	97	15	>	>	X
ejpam-6117	97	16	m+1	m+1	PROPN
ejpam-6117	97	17	and	and	CCONJ
ejpam-6117	97	18	n	n	PRON
ejpam-6117	97	19	≥	≥	NOUN
ejpam-6117	97	20	k+m+2	k+m+2	PROPN
ejpam-6117	97	21	.	.	PUNCT
ejpam-6117	98	1	then	then	ADV
ejpam-6117	98	2	there	there	PRON
ejpam-6117	98	3	exists	exist	VERB
ejpam-6117	98	4	a	a	DET
ejpam-6117	98	5	connected	connected	ADJ
ejpam-6117	98	6	graph	graph	NOUN
ejpam-6117	98	7	g	g	ADP
ejpam-6117	98	8	such	such	ADJ
ejpam-6117	98	9	that	that	SCONJ
ejpam-6117	98	10	|v	|v	PROPN
ejpam-6117	98	11	(	(	PUNCT
ejpam-6117	98	12	g)|	g)|	NOUN
ejpam-6117	98	13	=	=	PUNCT
ejpam-6117	98	14	n	n	CCONJ
ejpam-6117	98	15	,	,	PUNCT
ejpam-6117	98	16	αw(g	αw(g	NUM
ejpam-6117	98	17	)	)	PUNCT
ejpam-6117	98	18	=	=	SYM
ejpam-6117	98	19	k	k	PROPN
ejpam-6117	98	20	and	and	CCONJ
ejpam-6117	98	21	α(g	α(g	NUM
ejpam-6117	98	22	)	)	PUNCT
ejpam-6117	98	23	=	=	PUNCT
ejpam-6117	99	1	k	k	X
ejpam-6117	100	1	+	+	ADJ
ejpam-6117	100	2	m.	m.	NOUN
ejpam-6117	100	3	proof	proof	NOUN
ejpam-6117	100	4	.	.	PUNCT
ejpam-6117	101	1	let	let	VERB
ejpam-6117	101	2	us	we	PRON
ejpam-6117	101	3	consider	consider	VERB
ejpam-6117	101	4	the	the	DET
ejpam-6117	101	5	following	follow	VERB
ejpam-6117	101	6	cases	case	NOUN
ejpam-6117	101	7	:	:	PUNCT
ejpam-6117	101	8	case	case	NOUN
ejpam-6117	101	9	1	1	NUM
ejpam-6117	101	10	.	.	PUNCT
ejpam-6117	101	11	suppose	suppose	VERB
ejpam-6117	101	12	that	that	SCONJ
ejpam-6117	101	13	n	n	NOUN
ejpam-6117	101	14	=	=	SYM
ejpam-6117	102	1	k	k	PROPN
ejpam-6117	103	1	+	+	PROPN
ejpam-6117	103	2	m+	m+	NUM
ejpam-6117	103	3	2	2	NUM
ejpam-6117	103	4	.	.	PUNCT
ejpam-6117	103	5	suppose	suppose	VERB
ejpam-6117	103	6	first	first	ADV
ejpam-6117	103	7	that	that	SCONJ
ejpam-6117	103	8	m	m	VERB
ejpam-6117	103	9	=	=	ADJ
ejpam-6117	103	10	0	0	X
ejpam-6117	103	11	.	.	PUNCT
ejpam-6117	104	1	let	let	VERB
ejpam-6117	104	2	h1	h1	PROPN
ejpam-6117	104	3	=	=	PROPN
ejpam-6117	104	4	k1,k	k1,k	PROPN
ejpam-6117	104	5	,	,	PUNCT
ejpam-6117	104	6	where	where	SCONJ
ejpam-6117	104	7	u	u	NOUN
ejpam-6117	104	8	is	be	AUX
ejpam-6117	104	9	the	the	DET
ejpam-6117	104	10	central	central	ADJ
ejpam-6117	104	11	vertex	vertex	NOUN
ejpam-6117	104	12	of	of	ADP
ejpam-6117	104	13	h1	h1	NOUN
ejpam-6117	104	14	and	and	CCONJ
ejpam-6117	104	15	v1	v1	NOUN
ejpam-6117	104	16	,	,	PUNCT
ejpam-6117	104	17	v2	v2	NOUN
ejpam-6117	104	18	,	,	PUNCT
ejpam-6117	104	19	.	.	PUNCT
ejpam-6117	104	20	.	.	PUNCT
ejpam-6117	105	1	.	.	PUNCT
ejpam-6117	106	1	,	,	PUNCT
ejpam-6117	106	2	vk	vk	NOUN
ejpam-6117	106	3	are	be	AUX
ejpam-6117	106	4	the	the	DET
ejpam-6117	106	5	remaining	remain	VERB
ejpam-6117	106	6	vertices	vertex	NOUN
ejpam-6117	106	7	(	(	PUNCT
ejpam-6117	106	8	see	see	VERB
ejpam-6117	106	9	figure	figure	NOUN
ejpam-6117	106	10	1	1	NUM
ejpam-6117	106	11	)	)	PUNCT
ejpam-6117	106	12	.	.	PUNCT
ejpam-6117	107	1	let	let	VERB
ejpam-6117	107	2	g	g	NOUN
ejpam-6117	107	3	be	be	AUX
ejpam-6117	107	4	the	the	DET
ejpam-6117	107	5	graph	graph	NOUN
ejpam-6117	107	6	obtained	obtain	VERB
ejpam-6117	107	7	from	from	ADP
ejpam-6117	107	8	h1	h1	NOUN
ejpam-6117	107	9	by	by	ADP
ejpam-6117	107	10	adding	add	VERB
ejpam-6117	107	11	the	the	DET
ejpam-6117	107	12	vertex	vertex	NOUN
ejpam-6117	107	13	x	x	PUNCT
ejpam-6117	107	14	and	and	CCONJ
ejpam-6117	107	15	the	the	DET
ejpam-6117	107	16	edges	edge	NOUN
ejpam-6117	107	17	xu	xu	PROPN
ejpam-6117	107	18	and	and	CCONJ
ejpam-6117	107	19	xvk	xvk	PROPN
ejpam-6117	107	20	.	.	PUNCT
ejpam-6117	108	1	clearly	clearly	ADV
ejpam-6117	108	2	,	,	PUNCT
ejpam-6117	108	3	the	the	DET
ejpam-6117	108	4	set	set	NOUN
ejpam-6117	108	5	s1	s1	NOUN
ejpam-6117	108	6	=	=	SYM
ejpam-6117	108	7	{	{	PUNCT
ejpam-6117	108	8	v1	v1	PROPN
ejpam-6117	108	9	,	,	PUNCT
ejpam-6117	108	10	v2	v2	PROPN
ejpam-6117	108	11	,	,	PUNCT
ejpam-6117	108	12	.	.	PUNCT
ejpam-6117	108	13	.	.	PUNCT
ejpam-6117	108	14	.	.	PUNCT
ejpam-6117	109	1	,	,	PUNCT
ejpam-6117	109	2	vk	vk	PROPN
ejpam-6117	109	3	}	}	PUNCT
ejpam-6117	109	4	is	be	AUX
ejpam-6117	109	5	both	both	PRON
ejpam-6117	109	6	an	an	DET
ejpam-6117	109	7	α	α	NOUN
ejpam-6117	109	8	-	-	PUNCT
ejpam-6117	109	9	set	set	VERB
ejpam-6117	109	10	and	and	CCONJ
ejpam-6117	109	11	an	an	DET
ejpam-6117	109	12	αw	αw	NOUN
ejpam-6117	109	13	-	-	PUNCT
ejpam-6117	109	14	set	set	NOUN
ejpam-6117	109	15	of	of	ADP
ejpam-6117	109	16	g.	g.	PROPN
ejpam-6117	109	17	thus	thus	ADV
ejpam-6117	109	18	,	,	PUNCT
ejpam-6117	109	19	|v	|v	PROPN
ejpam-6117	109	20	(	(	PUNCT
ejpam-6117	109	21	g)|	g)|	NOUN
ejpam-6117	109	22	=	=	SYM
ejpam-6117	109	23	k	k	PROPN
ejpam-6117	110	1	+	+	CCONJ
ejpam-6117	110	2	2	2	NUM
ejpam-6117	110	3	=	=	SYM
ejpam-6117	110	4	n	n	CCONJ
ejpam-6117	110	5	,	,	PUNCT
ejpam-6117	110	6	αw(g	αw(g	NUM
ejpam-6117	110	7	)	)	PUNCT
ejpam-6117	110	8	=	=	SYM
ejpam-6117	110	9	α(g	α(g	NUM
ejpam-6117	110	10	)	)	PUNCT
ejpam-6117	110	11	=	=	PUNCT
ejpam-6117	110	12	k.	k.	PROPN
ejpam-6117	110	13	....................................	....................................	PUNCT
ejpam-6117	110	14	....................................	....................................	PUNCT
ejpam-6117	111	1	....................................	....................................	PUNCT
ejpam-6117	111	2	....................................	....................................	PUNCT
ejpam-6117	112	1	....................................	....................................	PUNCT
ejpam-6117	112	2	....................................	....................................	PUNCT
ejpam-6117	112	3	...............................................	...............................................	PUNCT
ejpam-6117	112	4	........	........	PUNCT
ejpam-6117	112	5	........	........	PUNCT
ejpam-6117	112	6	........	........	PUNCT
ejpam-6117	112	7	........	........	PUNCT
ejpam-6117	112	8	........	........	PUNCT
ejpam-6117	113	1	........	........	PUNCT
ejpam-6117	113	2	...	...	PUNCT
ejpam-6117	113	3	............	............	PUNCT
ejpam-6117	113	4	...........	...........	PUNCT
ejpam-6117	113	5	...........	...........	PUNCT
ejpam-6117	113	6	.....	.....	PUNCT
ejpam-6117	113	7	............	............	PUNCT
ejpam-6117	113	8	...........	...........	PUNCT
ejpam-6117	113	9	...........	...........	PUNCT
ejpam-6117	113	10	.....	.....	PUNCT
ejpam-6117	113	11	.......................................	.......................................	PUNCT
ejpam-6117	114	1	u	u	PROPN
ejpam-6117	114	2	v3	v3	PROPN
ejpam-6117	114	3	v2	v2	PROPN
ejpam-6117	114	4	v1	v1	NOUN
ejpam-6117	114	5	vkv4	vkv4	NOUN
ejpam-6117	114	6	.	.	PUNCT
ejpam-6117	114	7	.	.	PUNCT
ejpam-6117	114	8	.	.	PUNCT
ejpam-6117	115	1	h1	h1	PROPN
ejpam-6117	115	2	:	:	PUNCT
ejpam-6117	115	3	....................................	....................................	PUNCT
ejpam-6117	115	4	....................................	....................................	PUNCT
ejpam-6117	115	5	....................................	....................................	PUNCT
ejpam-6117	116	1	....................................	....................................	PUNCT
ejpam-6117	116	2	....................................	....................................	PUNCT
ejpam-6117	117	1	....................................	....................................	PUNCT
ejpam-6117	117	2	....................................	....................................	PUNCT
ejpam-6117	117	3	...............................................	...............................................	PUNCT
ejpam-6117	117	4	........	........	PUNCT
ejpam-6117	117	5	........	........	PUNCT
ejpam-6117	117	6	........	........	PUNCT
ejpam-6117	117	7	........	........	PUNCT
ejpam-6117	117	8	........	........	PUNCT
ejpam-6117	118	1	........	........	PUNCT
ejpam-6117	118	2	...	...	PUNCT
ejpam-6117	118	3	............	............	PUNCT
ejpam-6117	118	4	...........	...........	PUNCT
ejpam-6117	118	5	...........	...........	PUNCT
ejpam-6117	118	6	.....	.....	PUNCT
ejpam-6117	118	7	............	............	PUNCT
ejpam-6117	118	8	...........	...........	PUNCT
ejpam-6117	118	9	...........	...........	PUNCT
ejpam-6117	119	1	.....	.....	PUNCT
ejpam-6117	119	2	.......................................	.......................................	PUNCT
ejpam-6117	119	3	...........................................................	...........................................................	PUNCT
ejpam-6117	119	4	............	............	PUNCT
ejpam-6117	119	5	...........	...........	PUNCT
ejpam-6117	119	6	...........	...........	PUNCT
ejpam-6117	119	7	.....	.....	PUNCT
ejpam-6117	120	1	xu	xu	PROPN
ejpam-6117	121	1	v3	v3	PROPN
ejpam-6117	121	2	v2	v2	PROPN
ejpam-6117	121	3	v1	v1	NOUN
ejpam-6117	121	4	vkv4	vkv4	NOUN
ejpam-6117	121	5	.	.	PUNCT
ejpam-6117	121	6	.	.	PUNCT
ejpam-6117	121	7	.	.	PUNCT
ejpam-6117	122	1	g	g	NOUN
ejpam-6117	122	2	:	:	PUNCT
ejpam-6117	122	3	figure	figure	NOUN
ejpam-6117	122	4	1	1	NUM
ejpam-6117	122	5	:	:	PUNCT
ejpam-6117	122	6	graph	graph	VERB
ejpam-6117	122	7	g	g	NOUN
ejpam-6117	122	8	with	with	ADP
ejpam-6117	122	9	αw(g	αw(g	NUM
ejpam-6117	122	10	)	)	PUNCT
ejpam-6117	122	11	=	=	SYM
ejpam-6117	123	1	α(g	α(g	NUM
ejpam-6117	123	2	)	)	PUNCT
ejpam-6117	123	3	=	=	NOUN
ejpam-6117	124	1	k	k	PROPN
ejpam-6117	124	2	next	next	ADV
ejpam-6117	124	3	,	,	PUNCT
ejpam-6117	124	4	suppose	suppose	VERB
ejpam-6117	124	5	that	that	SCONJ
ejpam-6117	124	6	m	m	VERB
ejpam-6117	124	7	>	>	X
ejpam-6117	124	8	0	0	X
ejpam-6117	124	9	.	.	PUNCT
ejpam-6117	125	1	let	let	VERB
ejpam-6117	125	2	h2	h2	PROPN
ejpam-6117	125	3	be	be	AUX
ejpam-6117	125	4	the	the	DET
ejpam-6117	125	5	union	union	NOUN
ejpam-6117	125	6	of	of	ADP
ejpam-6117	125	7	k1,k−1	k1,k−1	PROPN
ejpam-6117	125	8	and	and	CCONJ
ejpam-6117	125	9	k1,m+1	k1,m+1	PROPN
ejpam-6117	125	10	,	,	PUNCT
ejpam-6117	125	11	where	where	SCONJ
ejpam-6117	125	12	u	u	NOUN
ejpam-6117	125	13	is	be	AUX
ejpam-6117	125	14	the	the	DET
ejpam-6117	125	15	central	central	ADJ
ejpam-6117	125	16	vertex	vertex	NOUN
ejpam-6117	125	17	of	of	ADP
ejpam-6117	125	18	k1,k−1	k1,k−1	PROPN
ejpam-6117	125	19	and	and	CCONJ
ejpam-6117	125	20	x1	x1	PROPN
ejpam-6117	125	21	,	,	PUNCT
ejpam-6117	125	22	x2	x2	PROPN
ejpam-6117	125	23	,	,	PUNCT
ejpam-6117	125	24	.	.	PUNCT
ejpam-6117	125	25	.	.	PUNCT
ejpam-6117	126	1	.	.	PUNCT
ejpam-6117	127	1	,	,	PUNCT
ejpam-6117	127	2	xk−1	xk−1	PROPN
ejpam-6117	127	3	are	be	AUX
ejpam-6117	127	4	its	its	PRON
ejpam-6117	127	5	remaining	remain	VERB
ejpam-6117	127	6	vertices	vertex	NOUN
ejpam-6117	127	7	,	,	PUNCT
ejpam-6117	127	8	v	v	NOUN
ejpam-6117	127	9	is	be	AUX
ejpam-6117	127	10	the	the	DET
ejpam-6117	127	11	central	central	ADJ
ejpam-6117	127	12	vertex	vertex	NOUN
ejpam-6117	127	13	of	of	ADP
ejpam-6117	127	14	k1,m+1	k1,m+1	PROPN
ejpam-6117	127	15	and	and	CCONJ
ejpam-6117	127	16	y1	y1	PROPN
ejpam-6117	127	17	,	,	PUNCT
ejpam-6117	127	18	y2	y2	INTJ
ejpam-6117	127	19	,	,	PUNCT
ejpam-6117	127	20	.	.	PUNCT
ejpam-6117	127	21	.	.	PUNCT
ejpam-6117	128	1	.	.	PUNCT
ejpam-6117	129	1	,	,	PUNCT
ejpam-6117	129	2	ym+1	ym+1	PROPN
ejpam-6117	129	3	are	be	AUX
ejpam-6117	129	4	its	its	PRON
ejpam-6117	129	5	remaining	remain	VERB
ejpam-6117	129	6	vertices	vertex	NOUN
ejpam-6117	129	7	.	.	PUNCT
ejpam-6117	130	1	let	let	VERB
ejpam-6117	130	2	g	g	NOUN
ejpam-6117	130	3	be	be	AUX
ejpam-6117	130	4	the	the	DET
ejpam-6117	130	5	graph	graph	NOUN
ejpam-6117	130	6	obtained	obtain	VERB
ejpam-6117	130	7	from	from	ADP
ejpam-6117	130	8	h2	h2	NOUN
ejpam-6117	130	9	by	by	ADP
ejpam-6117	130	10	adding	add	VERB
ejpam-6117	130	11	the	the	DET
ejpam-6117	130	12	edge	edge	NOUN
ejpam-6117	130	13	uv	uv	INTJ
ejpam-6117	130	14	(	(	PUNCT
ejpam-6117	130	15	see	see	VERB
ejpam-6117	130	16	figure	figure	NOUN
ejpam-6117	130	17	2	2	NUM
ejpam-6117	130	18	)	)	PUNCT
ejpam-6117	130	19	.	.	PUNCT
ejpam-6117	131	1	the	the	DET
ejpam-6117	131	2	set	set	VERB
ejpam-6117	131	3	s2	s2	NOUN
ejpam-6117	131	4	=	=	SYM
ejpam-6117	131	5	{	{	PUNCT
ejpam-6117	131	6	x1	x1	PROPN
ejpam-6117	131	7	,	,	PUNCT
ejpam-6117	131	8	x2	x2	PROPN
ejpam-6117	131	9	,	,	PUNCT
ejpam-6117	131	10	.	.	PUNCT
ejpam-6117	131	11	.	.	PUNCT
ejpam-6117	131	12	.	.	PUNCT
ejpam-6117	132	1	,	,	PUNCT
ejpam-6117	132	2	xk−1}∪	xk−1}∪	PROPN
ejpam-6117	132	3	{	{	PUNCT
ejpam-6117	132	4	y1	y1	PROPN
ejpam-6117	132	5	,	,	PUNCT
ejpam-6117	132	6	y2	y2	PROPN
ejpam-6117	132	7	,	,	PUNCT
ejpam-6117	132	8	.	.	PUNCT
ejpam-6117	132	9	.	.	PUNCT
ejpam-6117	133	1	.	.	PUNCT
ejpam-6117	134	1	,	,	PUNCT
ejpam-6117	134	2	ym+1	ym+1	X
ejpam-6117	134	3	}	}	PUNCT
ejpam-6117	134	4	is	be	AUX
ejpam-6117	134	5	the	the	DET
ejpam-6117	134	6	unique	unique	ADJ
ejpam-6117	134	7	α	α	NOUN
ejpam-6117	134	8	-	-	NOUN
ejpam-6117	134	9	set	set	NOUN
ejpam-6117	134	10	of	of	ADP
ejpam-6117	134	11	g	g	NOUN
ejpam-6117	134	12	and	and	CCONJ
ejpam-6117	134	13	s3	s3	PROPN
ejpam-6117	134	14	=	=	SYM
ejpam-6117	134	15	{	{	PUNCT
ejpam-6117	134	16	x1	x1	PROPN
ejpam-6117	134	17	,	,	PUNCT
ejpam-6117	134	18	x2	x2	PROPN
ejpam-6117	134	19	,	,	PUNCT
ejpam-6117	134	20	.	.	PUNCT
ejpam-6117	134	21	.	.	PUNCT
ejpam-6117	135	1	.	.	PUNCT
ejpam-6117	136	1	,	,	PUNCT
ejpam-6117	136	2	xk−1	xk−1	PROPN
ejpam-6117	136	3	,	,	PUNCT
ejpam-6117	136	4	v	v	NOUN
ejpam-6117	136	5	}	}	PUNCT
ejpam-6117	136	6	is	be	AUX
ejpam-6117	136	7	an	an	DET
ejpam-6117	136	8	αw	αw	NOUN
ejpam-6117	136	9	-	-	PUNCT
ejpam-6117	136	10	set	set	NOUN
ejpam-6117	136	11	of	of	ADP
ejpam-6117	136	12	g.	g.	PROPN
ejpam-6117	136	13	hence	hence	ADV
ejpam-6117	136	14	,	,	PUNCT
ejpam-6117	136	15	|v	|v	PROPN
ejpam-6117	136	16	(	(	PUNCT
ejpam-6117	136	17	g)|	g)|	NOUN
ejpam-6117	136	18	=	=	PUNCT
ejpam-6117	136	19	k	k	PROPN
ejpam-6117	137	1	+	+	PROPN
ejpam-6117	137	2	m+	m+	NUM
ejpam-6117	137	3	2	2	NUM
ejpam-6117	137	4	=	=	SYM
ejpam-6117	137	5	n	n	CCONJ
ejpam-6117	137	6	,	,	PUNCT
ejpam-6117	137	7	αw(g	αw(g	NUM
ejpam-6117	137	8	)	)	PUNCT
ejpam-6117	137	9	=	=	SYM
ejpam-6117	137	10	|s3|	|s3|	NOUN
ejpam-6117	137	11	=	=	SYM
ejpam-6117	137	12	k	k	PROPN
ejpam-6117	137	13	and	and	CCONJ
ejpam-6117	137	14	α(g	α(g	NUM
ejpam-6117	137	15	)	)	PUNCT
ejpam-6117	138	1	=	=	SYM
ejpam-6117	138	2	|s2|	|s2|	NOUN
ejpam-6117	138	3	=	=	SYM
ejpam-6117	138	4	k	k	PROPN
ejpam-6117	139	1	+	+	PROPN
ejpam-6117	139	2	m.	m.	NOUN
ejpam-6117	139	3	....................................	....................................	PUNCT
ejpam-6117	139	4	....................................	....................................	PUNCT
ejpam-6117	139	5	....................................	....................................	PUNCT
ejpam-6117	139	6	....................................	....................................	PUNCT
ejpam-6117	140	1	....................................	....................................	PUNCT
ejpam-6117	140	2	....................................	....................................	PUNCT
ejpam-6117	141	1	....................................	....................................	PUNCT
ejpam-6117	141	2	....................................	....................................	PUNCT
ejpam-6117	142	1	....................................	....................................	PUNCT
ejpam-6117	142	2	....................................	....................................	PUNCT
ejpam-6117	143	1	....................................	....................................	PUNCT
ejpam-6117	143	2	....................................	....................................	PUNCT
ejpam-6117	144	1	...............................................................................................................................	...............................................................................................................................	PUNCT
ejpam-6117	144	2	...............................................	...............................................	PUNCT
ejpam-6117	144	3	........	........	PUNCT
ejpam-6117	144	4	........	........	PUNCT
ejpam-6117	144	5	........	........	PUNCT
ejpam-6117	144	6	........	........	PUNCT
ejpam-6117	144	7	........	........	PUNCT
ejpam-6117	144	8	........	........	PUNCT
ejpam-6117	144	9	...	...	PUNCT
ejpam-6117	144	10	............	............	PUNCT
ejpam-6117	144	11	...........	...........	PUNCT
ejpam-6117	144	12	...........	...........	PUNCT
ejpam-6117	144	13	.....	.....	PUNCT
ejpam-6117	144	14	............	............	PUNCT
ejpam-6117	144	15	...........	...........	PUNCT
ejpam-6117	144	16	...........	...........	PUNCT
ejpam-6117	144	17	.....	.....	PUNCT
ejpam-6117	144	18	.......................................	.......................................	PUNCT
ejpam-6117	144	19	...............................................	...............................................	PUNCT
ejpam-6117	144	20	........	........	PUNCT
ejpam-6117	144	21	........	........	PUNCT
ejpam-6117	144	22	........	........	PUNCT
ejpam-6117	144	23	........	........	PUNCT
ejpam-6117	144	24	........	........	PUNCT
ejpam-6117	144	25	........	........	PUNCT
ejpam-6117	144	26	...	...	PUNCT
ejpam-6117	144	27	............	............	PUNCT
ejpam-6117	144	28	...........	...........	PUNCT
ejpam-6117	144	29	...........	...........	PUNCT
ejpam-6117	144	30	.....	.....	PUNCT
ejpam-6117	144	31	............	............	PUNCT
ejpam-6117	144	32	...........	...........	PUNCT
ejpam-6117	144	33	...........	...........	PUNCT
ejpam-6117	144	34	.....	.....	PUNCT
ejpam-6117	144	35	.......................................	.......................................	PUNCT
ejpam-6117	145	1	uv	uv	NOUN
ejpam-6117	145	2	y3	y3	NOUN
ejpam-6117	145	3	y2	y2	PROPN
ejpam-6117	145	4	y1	y1	PROPN
ejpam-6117	145	5	ym+1y4	ym+1y4	PROPN
ejpam-6117	145	6	x3	x3	PROPN
ejpam-6117	146	1	x2	x2	PROPN
ejpam-6117	147	1	x1	x1	ADJ
ejpam-6117	147	2	xk−1x4	xk−1x4	X
ejpam-6117	147	3	.	.	PUNCT
ejpam-6117	147	4	.	.	PUNCT
ejpam-6117	147	5	.	.	PUNCT
ejpam-6117	147	6	.	.	PUNCT
ejpam-6117	147	7	.	.	PUNCT
ejpam-6117	147	8	.	.	PUNCT
ejpam-6117	148	1	g	g	NOUN
ejpam-6117	148	2	:	:	PUNCT
ejpam-6117	148	3	figure	figure	NOUN
ejpam-6117	148	4	2	2	NUM
ejpam-6117	148	5	:	:	PUNCT
ejpam-6117	148	6	graph	graph	VERB
ejpam-6117	148	7	g	g	NOUN
ejpam-6117	148	8	with	with	ADP
ejpam-6117	148	9	αw(g	αw(g	NUM
ejpam-6117	148	10	)	)	PUNCT
ejpam-6117	148	11	=	=	SYM
ejpam-6117	149	1	k	k	PROPN
ejpam-6117	149	2	and	and	CCONJ
ejpam-6117	149	3	α(g	α(g	NUM
ejpam-6117	149	4	)	)	PUNCT
ejpam-6117	150	1	=	=	PUNCT
ejpam-6117	150	2	k	k	X
ejpam-6117	151	1	+	+	PROPN
ejpam-6117	151	2	m	m	PROPN
ejpam-6117	151	3	r.	r.	NOUN
ejpam-6117	151	4	merontos	merontos	PROPN
ejpam-6117	151	5	et	et	PROPN
ejpam-6117	151	6	al	al	PROPN
ejpam-6117	151	7	.	.	PUNCT
ejpam-6117	151	8	/	/	SYM
ejpam-6117	151	9	eur	eur	PROPN
ejpam-6117	151	10	.	.	PUNCT
ejpam-6117	152	1	j.	j.	PROPN
ejpam-6117	152	2	pure	pure	PROPN
ejpam-6117	152	3	appl	appl	PROPN
ejpam-6117	152	4	.	.	PROPN
ejpam-6117	152	5	math	math	PROPN
ejpam-6117	152	6	,	,	PUNCT
ejpam-6117	152	7	18	18	NUM
ejpam-6117	152	8	(	(	PUNCT
ejpam-6117	152	9	2	2	NUM
ejpam-6117	152	10	)	)	PUNCT
ejpam-6117	152	11	(	(	PUNCT
ejpam-6117	152	12	2025	2025	NUM
ejpam-6117	152	13	)	)	PUNCT
ejpam-6117	152	14	,	,	PUNCT
ejpam-6117	152	15	6117	6117	NUM
ejpam-6117	152	16	5	5	NUM
ejpam-6117	152	17	of	of	ADP
ejpam-6117	152	18	9	9	NUM
ejpam-6117	152	19	case	case	NOUN
ejpam-6117	152	20	2	2	NUM
ejpam-6117	152	21	.	.	PUNCT
ejpam-6117	152	22	suppose	suppose	VERB
ejpam-6117	152	23	that	that	SCONJ
ejpam-6117	152	24	n	n	PROPN
ejpam-6117	152	25	>	>	X
ejpam-6117	152	26	k	k	PROPN
ejpam-6117	153	1	+	+	PROPN
ejpam-6117	153	2	m+	m+	NUM
ejpam-6117	153	3	2	2	NUM
ejpam-6117	153	4	.	.	PUNCT
ejpam-6117	153	5	consider	consider	VERB
ejpam-6117	153	6	the	the	DET
ejpam-6117	153	7	graphs	graph	NOUN
ejpam-6117	153	8	g1	g1	VERB
ejpam-6117	153	9	and	and	CCONJ
ejpam-6117	153	10	g2	g2	PROPN
ejpam-6117	153	11	in	in	ADP
ejpam-6117	153	12	figure	figure	NOUN
ejpam-6117	153	13	3	3	NUM
ejpam-6117	153	14	.	.	PUNCT
ejpam-6117	153	15	suppose	suppose	VERB
ejpam-6117	153	16	that	that	SCONJ
ejpam-6117	153	17	m	m	VERB
ejpam-6117	153	18	=	=	NOUN
ejpam-6117	153	19	0	0	PROPN
ejpam-6117	153	20	.	.	PUNCT
ejpam-6117	153	21	....................................	....................................	PUNCT
ejpam-6117	154	1	....................................	....................................	PUNCT
ejpam-6117	154	2	....................................	....................................	PUNCT
ejpam-6117	155	1	....................................	....................................	PUNCT
ejpam-6117	155	2	....................................	....................................	PUNCT
ejpam-6117	156	1	....................................	....................................	PUNCT
ejpam-6117	156	2	....................................	....................................	PUNCT
ejpam-6117	157	1	....................................	....................................	PUNCT
ejpam-6117	157	2	....................................	....................................	PUNCT
ejpam-6117	158	1	....................................	....................................	PUNCT
ejpam-6117	158	2	...............................................	...............................................	PUNCT
ejpam-6117	158	3	........	........	PUNCT
ejpam-6117	158	4	........	........	PUNCT
ejpam-6117	158	5	........	........	PUNCT
ejpam-6117	158	6	........	........	PUNCT
ejpam-6117	158	7	........	........	PUNCT
ejpam-6117	158	8	........	........	PUNCT
ejpam-6117	158	9	...	...	PUNCT
ejpam-6117	158	10	............	............	PUNCT
ejpam-6117	158	11	...........	...........	PUNCT
ejpam-6117	158	12	...........	...........	PUNCT
ejpam-6117	158	13	.....	.....	PUNCT
ejpam-6117	158	14	............	............	PUNCT
ejpam-6117	158	15	...........	...........	PUNCT
ejpam-6117	158	16	...........	...........	PUNCT
ejpam-6117	159	1	.....	.....	PUNCT
ejpam-6117	159	2	.......................................	.......................................	PUNCT
ejpam-6117	159	3	..........	..........	PUNCT
ejpam-6117	160	1	..........	..........	PUNCT
ejpam-6117	160	2	..........	..........	PUNCT
ejpam-6117	161	1	..........	..........	PUNCT
ejpam-6117	161	2	..........	..........	PUNCT
ejpam-6117	162	1	..........	..........	PUNCT
ejpam-6117	162	2	..........	..........	PUNCT
ejpam-6117	163	1	..........	..........	PUNCT
ejpam-6117	163	2	..........	..........	PUNCT
ejpam-6117	164	1	........	........	PUNCT
ejpam-6117	164	2	................	................	PUNCT
ejpam-6117	164	3	...............	...............	PUNCT
ejpam-6117	164	4	...............	...............	PUNCT
ejpam-6117	164	5	...............	...............	PUNCT
ejpam-6117	164	6	...............	...............	PUNCT
ejpam-6117	164	7	...............	...............	PUNCT
ejpam-6117	164	8	.......	.......	PUNCT
ejpam-6117	164	9	................................................................	................................................................	PUNCT
ejpam-6117	165	1	..................	..................	PUNCT
ejpam-6117	165	2	...............................................................	...............................................................	PUNCT
ejpam-6117	165	3	.......................................	.......................................	PUNCT
ejpam-6117	165	4	...........	...........	PUNCT
ejpam-6117	166	1	..........	..........	PUNCT
ejpam-6117	166	2	..........	..........	PUNCT
ejpam-6117	167	1	..........	..........	PUNCT
ejpam-6117	167	2	..........	..........	PUNCT
ejpam-6117	168	1	......	......	PUNCT
ejpam-6117	168	2	.............................................................................	.............................................................................	PUNCT
ejpam-6117	168	3	.........	.........	PUNCT
ejpam-6117	168	4	........	........	PUNCT
ejpam-6117	168	5	........	........	PUNCT
ejpam-6117	168	6	........	........	PUNCT
ejpam-6117	168	7	........	........	PUNCT
ejpam-6117	168	8	........	........	PUNCT
ejpam-6117	168	9	........	........	PUNCT
ejpam-6117	168	10	........	........	PUNCT
ejpam-6117	168	11	........	........	PUNCT
ejpam-6117	168	12	........	........	PUNCT
ejpam-6117	168	13	........	........	PUNCT
ejpam-6117	168	14	........	........	PUNCT
ejpam-6117	168	15	........	........	PUNCT
ejpam-6117	168	16	........	........	PUNCT
ejpam-6117	168	17	........	........	PUNCT
ejpam-6117	169	1	......	......	PUNCT
ejpam-6117	169	2	..........	..........	PUNCT
ejpam-6117	170	1	.........	.........	PUNCT
ejpam-6117	170	2	.........	.........	PUNCT
ejpam-6117	171	1	.........	.........	PUNCT
ejpam-6117	171	2	.........	.........	PUNCT
ejpam-6117	172	1	.........	.........	PUNCT
ejpam-6117	172	2	.........	.........	PUNCT
ejpam-6117	173	1	.........	.........	PUNCT
ejpam-6117	173	2	.........	.........	PUNCT
ejpam-6117	173	3	.........	.........	PUNCT
ejpam-6117	173	4	........	........	PUNCT
ejpam-6117	173	5	.........	.........	PUNCT
ejpam-6117	173	6	........	........	PUNCT
ejpam-6117	173	7	........	........	PUNCT
ejpam-6117	173	8	........	........	PUNCT
ejpam-6117	173	9	...	...	PUNCT
ejpam-6117	174	1	..........................	..........................	PUNCT
ejpam-6117	174	2	...	...	PUNCT
ejpam-6117	175	1	u	u	PRON
ejpam-6117	176	1	x3	x3	NOUN
ejpam-6117	176	2	x2	x2	PROPN
ejpam-6117	177	1	x1	x1	ADJ
ejpam-6117	177	2	xkx4	xkx4	PROPN
ejpam-6117	177	3	.	.	PUNCT
ejpam-6117	177	4	.	.	PUNCT
ejpam-6117	177	5	.	.	PUNCT
ejpam-6117	178	1	g1	g1	PROPN
ejpam-6117	178	2	:	:	PUNCT
ejpam-6117	178	3	z1	z1	PROPN
ejpam-6117	178	4	z2	z2	PROPN
ejpam-6117	178	5	z3	z3	PROPN
ejpam-6117	178	6	zt	zt	PROPN
ejpam-6117	178	7	....................................	....................................	PUNCT
ejpam-6117	178	8	....................................	....................................	PUNCT
ejpam-6117	178	9	....................................	....................................	PUNCT
ejpam-6117	178	10	....................................	....................................	PUNCT
ejpam-6117	178	11	....................................	....................................	PUNCT
ejpam-6117	178	12	....................................	....................................	PUNCT
ejpam-6117	178	13	....................................	....................................	PUNCT
ejpam-6117	178	14	....................................	....................................	PUNCT
ejpam-6117	178	15	....................................	....................................	PUNCT
ejpam-6117	178	16	....................................	....................................	PUNCT
ejpam-6117	178	17	....................................	....................................	PUNCT
ejpam-6117	178	18	....................................	....................................	PUNCT
ejpam-6117	178	19	....................................	....................................	PUNCT
ejpam-6117	178	20	....................................	....................................	PUNCT
ejpam-6117	178	21	....................................	....................................	PUNCT
ejpam-6117	178	22	....................................	....................................	PUNCT
ejpam-6117	178	23	...........	...........	PUNCT
ejpam-6117	178	24	..........	..........	PUNCT
ejpam-6117	179	1	..........	..........	PUNCT
ejpam-6117	179	2	..........	..........	PUNCT
ejpam-6117	180	1	..........	..........	PUNCT
ejpam-6117	180	2	..........	..........	PUNCT
ejpam-6117	181	1	..........	..........	PUNCT
ejpam-6117	181	2	..........	..........	PUNCT
ejpam-6117	182	1	..........	..........	PUNCT
ejpam-6117	182	2	.......	.......	PUNCT
ejpam-6117	183	1	................	................	PUNCT
ejpam-6117	183	2	...............	...............	PUNCT
ejpam-6117	183	3	...............	...............	PUNCT
ejpam-6117	183	4	...............	...............	PUNCT
ejpam-6117	183	5	...............	...............	PUNCT
ejpam-6117	183	6	...............	...............	PUNCT
ejpam-6117	183	7	.......	.......	PUNCT
ejpam-6117	184	1	...............................................................	...............................................................	PUNCT
ejpam-6117	184	2	.......................................	.......................................	PUNCT
ejpam-6117	184	3	.........	.........	PUNCT
ejpam-6117	184	4	........	........	PUNCT
ejpam-6117	184	5	........	........	PUNCT
ejpam-6117	184	6	........	........	PUNCT
ejpam-6117	184	7	........	........	PUNCT
ejpam-6117	184	8	........	........	PUNCT
ejpam-6117	184	9	........	........	PUNCT
ejpam-6117	184	10	........	........	PUNCT
ejpam-6117	184	11	........	........	PUNCT
ejpam-6117	184	12	........	........	PUNCT
ejpam-6117	184	13	........	........	PUNCT
ejpam-6117	184	14	........	........	PUNCT
ejpam-6117	184	15	........	........	PUNCT
ejpam-6117	184	16	........	........	PUNCT
ejpam-6117	185	1	........	........	PUNCT
ejpam-6117	185	2	......	......	PUNCT
ejpam-6117	186	1	..........	..........	PUNCT
ejpam-6117	186	2	.........	.........	PUNCT
ejpam-6117	187	1	.........	.........	PUNCT
ejpam-6117	187	2	.........	.........	PUNCT
ejpam-6117	188	1	.........	.........	PUNCT
ejpam-6117	188	2	.........	.........	PUNCT
ejpam-6117	189	1	.........	.........	PUNCT
ejpam-6117	189	2	.........	.........	PUNCT
ejpam-6117	190	1	.........	.........	PUNCT
ejpam-6117	190	2	.........	.........	PUNCT
ejpam-6117	190	3	........	........	PUNCT
ejpam-6117	190	4	...........	...........	PUNCT
ejpam-6117	190	5	..........	..........	PUNCT
ejpam-6117	191	1	..........	..........	PUNCT
ejpam-6117	191	2	..........	..........	PUNCT
ejpam-6117	192	1	..........	..........	PUNCT
ejpam-6117	192	2	......	......	PUNCT
ejpam-6117	192	3	.........	.........	PUNCT
ejpam-6117	192	4	........	........	PUNCT
ejpam-6117	192	5	........	........	PUNCT
ejpam-6117	192	6	........	........	PUNCT
ejpam-6117	192	7	...	...	PUNCT
ejpam-6117	192	8	....................................................................................................................	....................................................................................................................	PUNCT
ejpam-6117	192	9	...............................................	...............................................	PUNCT
ejpam-6117	192	10	........	........	PUNCT
ejpam-6117	192	11	........	........	PUNCT
ejpam-6117	192	12	........	........	PUNCT
ejpam-6117	192	13	........	........	PUNCT
ejpam-6117	192	14	........	........	PUNCT
ejpam-6117	192	15	........	........	PUNCT
ejpam-6117	192	16	...	...	PUNCT
ejpam-6117	192	17	............	............	PUNCT
ejpam-6117	192	18	...........	...........	PUNCT
ejpam-6117	192	19	...........	...........	PUNCT
ejpam-6117	192	20	.....	.....	PUNCT
ejpam-6117	192	21	............	............	PUNCT
ejpam-6117	192	22	...........	...........	PUNCT
ejpam-6117	192	23	...........	...........	PUNCT
ejpam-6117	192	24	.....	.....	PUNCT
ejpam-6117	192	25	.......................................	.......................................	PUNCT
ejpam-6117	192	26	...........................................	...........................................	PUNCT
ejpam-6117	192	27	........	........	PUNCT
ejpam-6117	192	28	........	........	PUNCT
ejpam-6117	192	29	........	........	PUNCT
ejpam-6117	192	30	........	........	PUNCT
ejpam-6117	192	31	........	........	PUNCT
ejpam-6117	192	32	........	........	PUNCT
ejpam-6117	192	33	...	...	PUNCT
ejpam-6117	192	34	............	............	PUNCT
ejpam-6117	192	35	...........	...........	PUNCT
ejpam-6117	192	36	...........	...........	PUNCT
ejpam-6117	192	37	.....	.....	PUNCT
ejpam-6117	192	38	............	............	PUNCT
ejpam-6117	192	39	...........	...........	PUNCT
ejpam-6117	192	40	...........	...........	PUNCT
ejpam-6117	192	41	.	.	PUNCT
ejpam-6117	192	42	.......................................	.......................................	PUNCT
ejpam-6117	192	43	................................................................	................................................................	PUNCT
ejpam-6117	193	1	..................	..................	PUNCT
ejpam-6117	193	2	.............................................................................	.............................................................................	PUNCT
ejpam-6117	194	1	..........................	..........................	PUNCT
ejpam-6117	195	1	z1	z1	PROPN
ejpam-6117	195	2	z2	z2	PROPN
ejpam-6117	195	3	z3	z3	PROPN
ejpam-6117	195	4	zr	zr	PROPN
ejpam-6117	195	5	uv	uv	NOUN
ejpam-6117	195	6	y3	y3	NOUN
ejpam-6117	195	7	y2	y2	PROPN
ejpam-6117	195	8	y1	y1	PROPN
ejpam-6117	195	9	ym+1y4	ym+1y4	PROPN
ejpam-6117	195	10	x3	x3	PROPN
ejpam-6117	195	11	x2	x2	PROPN
ejpam-6117	195	12	x1	x1	ADJ
ejpam-6117	195	13	xk−1x4	xk−1x4	X
ejpam-6117	195	14	...	...	PUNCT
ejpam-6117	195	15	.	.	PUNCT
ejpam-6117	195	16	.	.	PUNCT
ejpam-6117	195	17	.	.	PUNCT
ejpam-6117	195	18	.	.	PUNCT
ejpam-6117	195	19	.	.	PUNCT
ejpam-6117	195	20	.	.	PUNCT
ejpam-6117	196	1	g2	g2	PROPN
ejpam-6117	196	2	:	:	PUNCT
ejpam-6117	196	3	figure	figure	VERB
ejpam-6117	196	4	3	3	NUM
ejpam-6117	196	5	:	:	PUNCT
ejpam-6117	196	6	graph	graph	VERB
ejpam-6117	196	7	g	g	NOUN
ejpam-6117	196	8	with	with	ADP
ejpam-6117	196	9	αw(g	αw(g	NUM
ejpam-6117	196	10	)	)	PUNCT
ejpam-6117	196	11	=	=	SYM
ejpam-6117	197	1	k	k	PROPN
ejpam-6117	197	2	and	and	CCONJ
ejpam-6117	197	3	α(g	α(g	NUM
ejpam-6117	197	4	)	)	PUNCT
ejpam-6117	198	1	=	=	SYM
ejpam-6117	198	2	k	k	X
ejpam-6117	199	1	+	+	ADV
ejpam-6117	199	2	m	m	AUX
ejpam-6117	199	3	let	let	VERB
ejpam-6117	199	4	t	t	NOUN
ejpam-6117	199	5	=	=	PUNCT
ejpam-6117	199	6	n	n	PROPN
ejpam-6117	199	7	−	−	NOUN
ejpam-6117	199	8	k	k	NOUN
ejpam-6117	200	1	−	−	PROPN
ejpam-6117	200	2	1	1	NUM
ejpam-6117	201	1	and	and	CCONJ
ejpam-6117	201	2	take	take	VERB
ejpam-6117	201	3	g	g	NOUN
ejpam-6117	201	4	=	=	SYM
ejpam-6117	201	5	g1	g1	PROPN
ejpam-6117	201	6	.	.	PUNCT
ejpam-6117	202	1	then	then	ADV
ejpam-6117	202	2	,	,	PUNCT
ejpam-6117	202	3	clearly	clearly	ADV
ejpam-6117	202	4	,	,	PUNCT
ejpam-6117	202	5	s4	s4	PROPN
ejpam-6117	202	6	=	=	SYM
ejpam-6117	202	7	{	{	PUNCT
ejpam-6117	202	8	x1	x1	PROPN
ejpam-6117	202	9	,	,	PUNCT
ejpam-6117	202	10	x2	x2	PROPN
ejpam-6117	202	11	,	,	PUNCT
ejpam-6117	202	12	.	.	PUNCT
ejpam-6117	202	13	.	.	PUNCT
ejpam-6117	202	14	.	.	PUNCT
ejpam-6117	203	1	,	,	PUNCT
ejpam-6117	203	2	xk	xk	PROPN
ejpam-6117	203	3	}	}	PUNCT
ejpam-6117	203	4	is	be	AUX
ejpam-6117	203	5	a	a	DET
ejpam-6117	203	6	both	both	CCONJ
ejpam-6117	203	7	an	an	DET
ejpam-6117	203	8	α	α	NOUN
ejpam-6117	203	9	-	-	PUNCT
ejpam-6117	203	10	set	set	VERB
ejpam-6117	203	11	and	and	CCONJ
ejpam-6117	203	12	an	an	DET
ejpam-6117	203	13	αw	αw	NOUN
ejpam-6117	203	14	-	-	PUNCT
ejpam-6117	203	15	set	set	NOUN
ejpam-6117	203	16	of	of	ADP
ejpam-6117	203	17	g.	g.	PROPN
ejpam-6117	203	18	thus	thus	ADV
ejpam-6117	203	19	,	,	PUNCT
ejpam-6117	203	20	|v	|v	PROPN
ejpam-6117	203	21	(	(	PUNCT
ejpam-6117	203	22	g)|	g)|	NOUN
ejpam-6117	203	23	=	=	PUNCT
ejpam-6117	203	24	k	k	PROPN
ejpam-6117	204	1	+	+	PROPN
ejpam-6117	204	2	t	t	PROPN
ejpam-6117	204	3	+	+	CCONJ
ejpam-6117	204	4	1	1	NUM
ejpam-6117	204	5	=	=	SYM
ejpam-6117	204	6	n	n	CCONJ
ejpam-6117	204	7	,	,	PUNCT
ejpam-6117	204	8	αw(g	αw(g	NUM
ejpam-6117	204	9	)	)	PUNCT
ejpam-6117	204	10	=	=	SYM
ejpam-6117	204	11	α(g	α(g	NUM
ejpam-6117	204	12	)	)	PUNCT
ejpam-6117	205	1	=	=	VERB
ejpam-6117	205	2	k.	k.	NOUN
ejpam-6117	206	1	if	if	SCONJ
ejpam-6117	206	2	m	m	VERB
ejpam-6117	206	3	>	>	X
ejpam-6117	206	4	0	0	NUM
ejpam-6117	206	5	,	,	PUNCT
ejpam-6117	206	6	then	then	ADV
ejpam-6117	206	7	set	set	VERB
ejpam-6117	206	8	r	r	NOUN
ejpam-6117	206	9	=	=	PUNCT
ejpam-6117	206	10	n	n	PROPN
ejpam-6117	206	11	−	−	PROPN
ejpam-6117	206	12	k	k	NOUN
ejpam-6117	207	1	−	−	PROPN
ejpam-6117	207	2	m	m	NOUN
ejpam-6117	207	3	−	−	NOUN
ejpam-6117	207	4	2	2	NUM
ejpam-6117	207	5	and	and	CCONJ
ejpam-6117	207	6	take	take	VERB
ejpam-6117	207	7	g	g	NOUN
ejpam-6117	207	8	=	=	PUNCT
ejpam-6117	207	9	g2	g2	PROPN
ejpam-6117	207	10	.	.	PUNCT
ejpam-6117	208	1	it	it	PRON
ejpam-6117	208	2	can	can	AUX
ejpam-6117	208	3	easily	easily	ADV
ejpam-6117	208	4	be	be	AUX
ejpam-6117	208	5	verified	verify	VERB
ejpam-6117	208	6	that	that	SCONJ
ejpam-6117	208	7	|v	|v	PROPN
ejpam-6117	208	8	(	(	PUNCT
ejpam-6117	208	9	g)|	g)|	NOUN
ejpam-6117	208	10	=	=	SYM
ejpam-6117	208	11	(	(	PUNCT
ejpam-6117	208	12	k	k	NOUN
ejpam-6117	208	13	−	−	PROPN
ejpam-6117	208	14	1	1	NUM
ejpam-6117	208	15	)	)	PUNCT
ejpam-6117	208	16	+	+	CCONJ
ejpam-6117	208	17	(	(	PUNCT
ejpam-6117	208	18	m+	m+	NUM
ejpam-6117	208	19	1	1	NUM
ejpam-6117	208	20	)	)	PUNCT
ejpam-6117	209	1	+	+	CCONJ
ejpam-6117	209	2	r	r	NOUN
ejpam-6117	209	3	+	+	CCONJ
ejpam-6117	209	4	2	2	NUM
ejpam-6117	209	5	=	=	SYM
ejpam-6117	209	6	n	n	CCONJ
ejpam-6117	209	7	,	,	PUNCT
ejpam-6117	209	8	αw(g	αw(g	NUM
ejpam-6117	209	9	)	)	PUNCT
ejpam-6117	209	10	=	=	SYM
ejpam-6117	209	11	k	k	PROPN
ejpam-6117	209	12	and	and	CCONJ
ejpam-6117	209	13	α(g	α(g	NUM
ejpam-6117	209	14	)	)	PUNCT
ejpam-6117	209	15	=	=	SYM
ejpam-6117	210	1	k	k	X
ejpam-6117	211	1	+	+	NOUN
ejpam-6117	211	2	m.	m.	NOUN
ejpam-6117	211	3	the	the	DET
ejpam-6117	211	4	next	next	ADJ
ejpam-6117	211	5	result	result	NOUN
ejpam-6117	211	6	is	be	AUX
ejpam-6117	211	7	a	a	DET
ejpam-6117	211	8	consequence	consequence	NOUN
ejpam-6117	211	9	of	of	ADP
ejpam-6117	211	10	theorem	theorem	ADJ
ejpam-6117	211	11	3	3	NUM
ejpam-6117	211	12	.	.	PUNCT
ejpam-6117	211	13	corollary	corollary	ADJ
ejpam-6117	211	14	2	2	NUM
ejpam-6117	211	15	.	.	PUNCT
ejpam-6117	212	1	the	the	DET
ejpam-6117	212	2	difference	difference	NOUN
ejpam-6117	212	3	α(g)−	α(g)−	NOUN
ejpam-6117	212	4	αw(g	αw(g	NUM
ejpam-6117	212	5	)	)	PUNCT
ejpam-6117	212	6	can	can	AUX
ejpam-6117	212	7	be	be	AUX
ejpam-6117	212	8	made	make	VERB
ejpam-6117	212	9	arbitrarily	arbitrarily	ADV
ejpam-6117	212	10	large	large	ADJ
ejpam-6117	212	11	.	.	PUNCT
ejpam-6117	213	1	proof	proof	NOUN
ejpam-6117	213	2	.	.	PUNCT
ejpam-6117	214	1	let	let	VERB
ejpam-6117	214	2	m	m	PRON
ejpam-6117	214	3	,	,	PUNCT
ejpam-6117	214	4	n	n	CCONJ
ejpam-6117	214	5	,	,	PUNCT
ejpam-6117	214	6	and	and	CCONJ
ejpam-6117	214	7	k	k	PROPN
ejpam-6117	214	8	be	be	AUX
ejpam-6117	214	9	positive	positive	ADJ
ejpam-6117	214	10	integers	integer	NOUN
ejpam-6117	214	11	such	such	ADJ
ejpam-6117	214	12	that	that	SCONJ
ejpam-6117	214	13	k	k	PROPN
ejpam-6117	214	14	>	>	X
ejpam-6117	214	15	m+1	m+1	PROPN
ejpam-6117	214	16	and	and	CCONJ
ejpam-6117	214	17	n	n	NOUN
ejpam-6117	214	18	=	=	PROPN
ejpam-6117	214	19	k+m+2	k+m+2	PROPN
ejpam-6117	214	20	.	.	PUNCT
ejpam-6117	215	1	by	by	ADP
ejpam-6117	215	2	theorem	theorem	NOUN
ejpam-6117	215	3	3	3	NUM
ejpam-6117	215	4	,	,	PUNCT
ejpam-6117	215	5	there	there	PRON
ejpam-6117	215	6	exists	exist	VERB
ejpam-6117	215	7	a	a	DET
ejpam-6117	215	8	connected	connected	ADJ
ejpam-6117	215	9	graph	graph	NOUN
ejpam-6117	215	10	with	with	ADP
ejpam-6117	215	11	|v	|v	PROPN
ejpam-6117	215	12	(	(	PUNCT
ejpam-6117	215	13	g)|	g)|	NOUN
ejpam-6117	215	14	=	=	PUNCT
ejpam-6117	215	15	n	n	CCONJ
ejpam-6117	215	16	,	,	PUNCT
ejpam-6117	215	17	αw(g	αw(g	NUM
ejpam-6117	215	18	)	)	PUNCT
ejpam-6117	215	19	=	=	SYM
ejpam-6117	215	20	k	k	PROPN
ejpam-6117	215	21	and	and	CCONJ
ejpam-6117	215	22	α(g	α(g	NUM
ejpam-6117	215	23	)	)	PUNCT
ejpam-6117	216	1	=	=	SYM
ejpam-6117	216	2	k+m	k+m	PROPN
ejpam-6117	216	3	.	.	PUNCT
ejpam-6117	217	1	therefore	therefore	ADV
ejpam-6117	217	2	,	,	PUNCT
ejpam-6117	217	3	α(g)−	α(g)−	NOUN
ejpam-6117	217	4	αw(g	αw(g	NUM
ejpam-6117	217	5	)	)	PUNCT
ejpam-6117	217	6	=	=	VERB
ejpam-6117	217	7	m.	m.	NOUN
ejpam-6117	217	8	next	next	ADV
ejpam-6117	217	9	,	,	PUNCT
ejpam-6117	217	10	we	we	PRON
ejpam-6117	217	11	give	give	VERB
ejpam-6117	217	12	the	the	DET
ejpam-6117	217	13	weakly	weakly	ADJ
ejpam-6117	217	14	connected	connected	ADJ
ejpam-6117	217	15	independence	independence	NOUN
ejpam-6117	217	16	number	number	NOUN
ejpam-6117	217	17	of	of	ADP
ejpam-6117	217	18	paths	path	NOUN
ejpam-6117	217	19	and	and	CCONJ
ejpam-6117	217	20	cycles	cycle	NOUN
ejpam-6117	217	21	.	.	PUNCT
ejpam-6117	218	1	theorem	theorem	ADJ
ejpam-6117	218	2	4	4	NUM
ejpam-6117	218	3	.	.	PUNCT
ejpam-6117	219	1	(	(	PUNCT
ejpam-6117	219	2	i	i	NOUN
ejpam-6117	219	3	)	)	PUNCT
ejpam-6117	219	4	αw(pn	αw(pn	PROPN
ejpam-6117	219	5	)	)	PUNCT
ejpam-6117	219	6	=	=	PUNCT
ejpam-6117	220	1	⌈n	⌈n	NOUN
ejpam-6117	220	2	2	2	NUM
ejpam-6117	220	3	⌉	⌉	NOUN
ejpam-6117	220	4	=	=	SYM
ejpam-6117	220	5	α(pn	α(pn	NOUN
ejpam-6117	220	6	)	)	PUNCT
ejpam-6117	220	7	for	for	ADP
ejpam-6117	220	8	all	all	DET
ejpam-6117	220	9	n	n	PRON
ejpam-6117	220	10	≥	≥	NOUN
ejpam-6117	220	11	1	1	NUM
ejpam-6117	220	12	.	.	PUNCT
ejpam-6117	220	13	(	(	PUNCT
ejpam-6117	220	14	ii	ii	NOUN
ejpam-6117	220	15	)	)	PUNCT
ejpam-6117	220	16	αw(cn	αw(cn	PROPN
ejpam-6117	220	17	)	)	PUNCT
ejpam-6117	221	1	=	=	PUNCT
ejpam-6117	221	2	⌊n	⌊n	NUM
ejpam-6117	221	3	2	2	NUM
ejpam-6117	221	4	⌋	⌋	NOUN
ejpam-6117	221	5	=	=	SYM
ejpam-6117	221	6	α(cn	α(cn	NOUN
ejpam-6117	221	7	)	)	PUNCT
ejpam-6117	221	8	for	for	ADP
ejpam-6117	221	9	all	all	DET
ejpam-6117	221	10	n	n	PRON
ejpam-6117	221	11	≥	≥	NOUN
ejpam-6117	221	12	3	3	NUM
ejpam-6117	221	13	.	.	PUNCT
ejpam-6117	222	1	proof	proof	NOUN
ejpam-6117	222	2	.	.	PUNCT
ejpam-6117	223	1	(	(	PUNCT
ejpam-6117	223	2	i	i	NOUN
ejpam-6117	223	3	)	)	PUNCT
ejpam-6117	223	4	let	let	VERB
ejpam-6117	223	5	pn	pn	NOUN
ejpam-6117	223	6	=	=	PUNCT
ejpam-6117	224	1	[	[	X
ejpam-6117	224	2	v1	v1	NOUN
ejpam-6117	224	3	,	,	PUNCT
ejpam-6117	224	4	v2	v2	NOUN
ejpam-6117	224	5	,	,	PUNCT
ejpam-6117	224	6	.	.	PUNCT
ejpam-6117	224	7	.	.	PUNCT
ejpam-6117	224	8	.	.	PUNCT
ejpam-6117	225	1	,	,	PUNCT
ejpam-6117	225	2	vn	vn	X
ejpam-6117	225	3	]	]	PUNCT
ejpam-6117	225	4	.	.	PUNCT
ejpam-6117	226	1	if	if	SCONJ
ejpam-6117	226	2	n	n	PRON
ejpam-6117	226	3	is	be	AUX
ejpam-6117	226	4	even	even	ADV
ejpam-6117	226	5	,	,	PUNCT
ejpam-6117	226	6	then	then	ADV
ejpam-6117	226	7	se	se	ADV
ejpam-6117	226	8	=	=	PRON
ejpam-6117	226	9	{	{	PUNCT
ejpam-6117	226	10	vi	vi	NOUN
ejpam-6117	226	11	∈	∈	PROPN
ejpam-6117	226	12	v	v	NOUN
ejpam-6117	226	13	(	(	PUNCT
ejpam-6117	226	14	pn	pn	NOUN
ejpam-6117	226	15	)	)	PUNCT
ejpam-6117	226	16	:	:	PUNCT
ejpam-6117	226	17	i	i	PRON
ejpam-6117	226	18	is	be	AUX
ejpam-6117	226	19	even	even	ADV
ejpam-6117	226	20	}	}	PUNCT
ejpam-6117	226	21	and	and	CCONJ
ejpam-6117	226	22	so	so	ADV
ejpam-6117	226	23	=	=	SYM
ejpam-6117	226	24	{	{	PUNCT
ejpam-6117	226	25	vj	vj	INTJ
ejpam-6117	226	26	∈	∈	PROPN
ejpam-6117	226	27	v	v	PROPN
ejpam-6117	226	28	(	(	PUNCT
ejpam-6117	226	29	pn	pn	NOUN
ejpam-6117	226	30	)	)	PUNCT
ejpam-6117	226	31	:	:	PUNCT
ejpam-6117	226	32	j	j	PROPN
ejpam-6117	226	33	is	be	AUX
ejpam-6117	226	34	odd	odd	ADJ
ejpam-6117	226	35	}	}	PUNCT
ejpam-6117	226	36	are	be	AUX
ejpam-6117	226	37	α	α	NOUN
ejpam-6117	226	38	-	-	PUNCT
ejpam-6117	226	39	sets	set	NOUN
ejpam-6117	226	40	in	in	ADP
ejpam-6117	226	41	pn	pn	PROPN
ejpam-6117	226	42	.	.	PUNCT
ejpam-6117	227	1	since	since	SCONJ
ejpam-6117	227	2	⟨ng[se	⟨ng[se	PROPN
ejpam-6117	227	3	]	]	X
ejpam-6117	227	4	⟩	⟩	NOUN
ejpam-6117	227	5	=	=	SYM
ejpam-6117	227	6	⟨ng[so]⟩	⟨ng[so]⟩	PROPN
ejpam-6117	227	7	=	=	SYM
ejpam-6117	227	8	pn	pn	PROPN
ejpam-6117	227	9	,	,	PUNCT
ejpam-6117	227	10	it	it	PRON
ejpam-6117	227	11	follows	follow	VERB
ejpam-6117	227	12	that	that	SCONJ
ejpam-6117	227	13	se	se	PROPN
ejpam-6117	228	1	and	and	CCONJ
ejpam-6117	228	2	so	so	ADV
ejpam-6117	228	3	are	be	AUX
ejpam-6117	228	4	weakly	weakly	ADV
ejpam-6117	228	5	connected	connected	ADJ
ejpam-6117	228	6	independent	independent	ADJ
ejpam-6117	228	7	sets	set	NOUN
ejpam-6117	228	8	in	in	ADP
ejpam-6117	228	9	pn	pn	PROPN
ejpam-6117	228	10	.	.	PUNCT
ejpam-6117	228	11	by	by	ADP
ejpam-6117	228	12	theorem	theorem	NOUN
ejpam-6117	228	13	1(ii	1(ii	NUM
ejpam-6117	228	14	)	)	PUNCT
ejpam-6117	228	15	,	,	PUNCT
ejpam-6117	228	16	we	we	PRON
ejpam-6117	228	17	have	have	VERB
ejpam-6117	228	18	αw(pn	αw(pn	NOUN
ejpam-6117	228	19	)	)	PUNCT
ejpam-6117	228	20	=	=	SYM
ejpam-6117	228	21	α(pn	α(pn	NOUN
ejpam-6117	228	22	)	)	PUNCT
ejpam-6117	228	23	=	=	NOUN
ejpam-6117	228	24	|se	|se	PUNCT
ejpam-6117	228	25	|	|	NOUN
ejpam-6117	228	26	=	=	SYM
ejpam-6117	228	27	n	n	PRON
ejpam-6117	228	28	2	2	NUM
ejpam-6117	228	29	.	.	PUNCT
ejpam-6117	229	1	if	if	SCONJ
ejpam-6117	229	2	n	n	NOUN
ejpam-6117	229	3	is	be	AUX
ejpam-6117	229	4	odd	odd	ADJ
ejpam-6117	229	5	,	,	PUNCT
ejpam-6117	229	6	then	then	ADV
ejpam-6117	229	7	s	s	VERB
ejpam-6117	229	8	=	=	PUNCT
ejpam-6117	229	9	{	{	PUNCT
ejpam-6117	229	10	vj	vj	INTJ
ejpam-6117	229	11	∈	∈	PROPN
ejpam-6117	229	12	v	v	PROPN
ejpam-6117	229	13	(	(	PUNCT
ejpam-6117	229	14	pn	pn	NOUN
ejpam-6117	229	15	)	)	PUNCT
ejpam-6117	229	16	:	:	PUNCT
ejpam-6117	230	1	j	j	PROPN
ejpam-6117	230	2	is	be	AUX
ejpam-6117	230	3	odd	odd	ADJ
ejpam-6117	230	4	}	}	PUNCT
ejpam-6117	230	5	is	be	AUX
ejpam-6117	230	6	the	the	DET
ejpam-6117	230	7	unique	unique	ADJ
ejpam-6117	230	8	α	α	NOUN
ejpam-6117	230	9	-	-	PUNCT
ejpam-6117	230	10	set	set	VERB
ejpam-6117	230	11	in	in	ADP
ejpam-6117	230	12	pn	pn	PROPN
ejpam-6117	230	13	.	.	PUNCT
ejpam-6117	230	14	again	again	ADV
ejpam-6117	230	15	,	,	PUNCT
ejpam-6117	230	16	since	since	SCONJ
ejpam-6117	230	17	⟨ng[s]⟩	⟨ng[s]⟩	NOUN
ejpam-6117	230	18	=	=	SYM
ejpam-6117	230	19	pn	pn	NOUN
ejpam-6117	230	20	,	,	PUNCT
ejpam-6117	230	21	it	it	PRON
ejpam-6117	230	22	follows	follow	VERB
ejpam-6117	230	23	that	that	SCONJ
ejpam-6117	230	24	s	s	VERB
ejpam-6117	230	25	is	be	AUX
ejpam-6117	230	26	a	a	DET
ejpam-6117	230	27	weakly	weakly	ADV
ejpam-6117	230	28	connected	connected	ADJ
ejpam-6117	230	29	independent	independent	ADJ
ejpam-6117	230	30	set	set	NOUN
ejpam-6117	230	31	in	in	ADP
ejpam-6117	230	32	pn	pn	PROPN
ejpam-6117	230	33	.	.	PUNCT
ejpam-6117	231	1	by	by	ADP
ejpam-6117	231	2	theorem	theorem	NOUN
ejpam-6117	231	3	1(ii	1(ii	NUM
ejpam-6117	231	4	)	)	PUNCT
ejpam-6117	231	5	,	,	PUNCT
ejpam-6117	231	6	we	we	PRON
ejpam-6117	231	7	have	have	VERB
ejpam-6117	231	8	αw(pn	αw(pn	NOUN
ejpam-6117	231	9	)	)	PUNCT
ejpam-6117	231	10	=	=	SYM
ejpam-6117	231	11	|s|	|s|	NOUN
ejpam-6117	231	12	=	=	SYM
ejpam-6117	231	13	n+1	n+1	PROPN
ejpam-6117	231	14	2	2	NUM
ejpam-6117	231	15	.	.	PUNCT
ejpam-6117	232	1	(	(	PUNCT
ejpam-6117	232	2	ii	ii	NOUN
ejpam-6117	232	3	)	)	PUNCT
ejpam-6117	232	4	let	let	VERB
ejpam-6117	232	5	cn	cn	PROPN
ejpam-6117	232	6	=	=	PUNCT
ejpam-6117	233	1	[	[	X
ejpam-6117	233	2	v1	v1	NOUN
ejpam-6117	233	3	,	,	PUNCT
ejpam-6117	233	4	v2	v2	NOUN
ejpam-6117	233	5	,	,	PUNCT
ejpam-6117	233	6	.	.	PUNCT
ejpam-6117	233	7	.	.	PUNCT
ejpam-6117	233	8	.	.	PUNCT
ejpam-6117	234	1	,	,	PUNCT
ejpam-6117	234	2	vn	vn	X
ejpam-6117	234	3	,	,	PUNCT
ejpam-6117	234	4	v1	v1	PROPN
ejpam-6117	234	5	]	]	PUNCT
ejpam-6117	234	6	.	.	PUNCT
ejpam-6117	235	1	if	if	SCONJ
ejpam-6117	235	2	n	n	PRON
ejpam-6117	235	3	is	be	AUX
ejpam-6117	235	4	even	even	ADV
ejpam-6117	235	5	,	,	PUNCT
ejpam-6117	235	6	the	the	DET
ejpam-6117	235	7	s1	s1	NOUN
ejpam-6117	235	8	=	=	PUNCT
ejpam-6117	235	9	{	{	PUNCT
ejpam-6117	235	10	v1	v1	PROPN
ejpam-6117	235	11	,	,	PUNCT
ejpam-6117	235	12	v3	v3	PROPN
ejpam-6117	235	13	,	,	PUNCT
ejpam-6117	235	14	·	·	PUNCT
ejpam-6117	235	15	·	·	PUNCT
ejpam-6117	235	16	·	·	PUNCT
ejpam-6117	235	17	,	,	PUNCT
ejpam-6117	235	18	vn−1	vn−1	PROPN
ejpam-6117	235	19	}	}	PUNCT
ejpam-6117	235	20	is	be	AUX
ejpam-6117	235	21	an	an	DET
ejpam-6117	235	22	αset	αset	NOUN
ejpam-6117	235	23	in	in	ADP
ejpam-6117	235	24	cn	cn	PROPN
ejpam-6117	235	25	.	.	PUNCT
ejpam-6117	236	1	since	since	SCONJ
ejpam-6117	236	2	⟨ng[s1]⟩	⟨ng[s1]⟩	PROPN
ejpam-6117	236	3	=	=	SYM
ejpam-6117	236	4	cn	cn	PROPN
ejpam-6117	236	5	,	,	PUNCT
ejpam-6117	236	6	it	it	PRON
ejpam-6117	236	7	follows	follow	VERB
ejpam-6117	236	8	that	that	SCONJ
ejpam-6117	236	9	s1	s1	NOUN
ejpam-6117	236	10	is	be	AUX
ejpam-6117	236	11	a	a	DET
ejpam-6117	236	12	weakly	weakly	ADV
ejpam-6117	236	13	connected	connected	ADJ
ejpam-6117	236	14	independent	independent	ADJ
ejpam-6117	236	15	set	set	NOUN
ejpam-6117	236	16	in	in	ADP
ejpam-6117	236	17	cn	cn	PROPN
ejpam-6117	236	18	.	.	PUNCT
ejpam-6117	237	1	by	by	ADP
ejpam-6117	237	2	theorem	theorem	NOUN
ejpam-6117	237	3	1(ii	1(ii	NUM
ejpam-6117	237	4	)	)	PUNCT
ejpam-6117	237	5	,	,	PUNCT
ejpam-6117	237	6	we	we	PRON
ejpam-6117	237	7	have	have	VERB
ejpam-6117	237	8	αw(cn	αw(cn	NOUN
ejpam-6117	237	9	)	)	PUNCT
ejpam-6117	238	1	=	=	SYM
ejpam-6117	238	2	α(cn	α(cn	NOUN
ejpam-6117	238	3	)	)	PUNCT
ejpam-6117	238	4	=	=	NOUN
ejpam-6117	238	5	|s1|	|s1|	NOUN
ejpam-6117	238	6	=	=	SYM
ejpam-6117	238	7	n	n	PRON
ejpam-6117	238	8	2	2	NUM
ejpam-6117	238	9	.	.	PUNCT
ejpam-6117	239	1	if	if	SCONJ
ejpam-6117	239	2	n	n	NOUN
ejpam-6117	239	3	is	be	AUX
ejpam-6117	239	4	odd	odd	ADJ
ejpam-6117	239	5	,	,	PUNCT
ejpam-6117	239	6	then	then	ADV
ejpam-6117	239	7	s2	s2	VERB
ejpam-6117	239	8	=	=	SYM
ejpam-6117	239	9	{	{	PUNCT
ejpam-6117	239	10	v1	v1	PROPN
ejpam-6117	239	11	,	,	PUNCT
ejpam-6117	239	12	v3	v3	PROPN
ejpam-6117	239	13	,	,	PUNCT
ejpam-6117	239	14	·	·	PUNCT
ejpam-6117	239	15	·	·	PUNCT
ejpam-6117	239	16	·	·	PUNCT
ejpam-6117	239	17	,	,	PUNCT
ejpam-6117	239	18	vn−2	vn−2	AUX
ejpam-6117	239	19	}	}	PUNCT
ejpam-6117	239	20	is	be	AUX
ejpam-6117	239	21	an	an	DET
ejpam-6117	239	22	α	α	NOUN
ejpam-6117	239	23	-	-	PUNCT
ejpam-6117	239	24	set	set	NOUN
ejpam-6117	239	25	in	in	ADP
ejpam-6117	239	26	cn	cn	PROPN
ejpam-6117	239	27	.	.	PUNCT
ejpam-6117	240	1	since	since	SCONJ
ejpam-6117	240	2	⟨ng[s]⟩	⟨ng[s]⟩	NOUN
ejpam-6117	240	3	=	=	SYM
ejpam-6117	240	4	pn	pn	NOUN
ejpam-6117	240	5	,	,	PUNCT
ejpam-6117	240	6	it	it	PRON
ejpam-6117	240	7	follows	follow	VERB
ejpam-6117	240	8	that	that	SCONJ
ejpam-6117	240	9	s2	s2	NOUN
ejpam-6117	240	10	is	be	AUX
ejpam-6117	240	11	a	a	DET
ejpam-6117	240	12	weakly	weakly	ADV
ejpam-6117	240	13	connected	connected	ADJ
ejpam-6117	240	14	independent	independent	ADJ
ejpam-6117	240	15	set	set	NOUN
ejpam-6117	240	16	in	in	ADP
ejpam-6117	240	17	cn	cn	PROPN
ejpam-6117	240	18	.	.	PUNCT
ejpam-6117	241	1	by	by	ADP
ejpam-6117	241	2	theorem	theorem	NOUN
ejpam-6117	241	3	1(ii	1(ii	NUM
ejpam-6117	241	4	)	)	PUNCT
ejpam-6117	241	5	,	,	PUNCT
ejpam-6117	241	6	we	we	PRON
ejpam-6117	241	7	have	have	VERB
ejpam-6117	241	8	αw(cn	αw(cn	NOUN
ejpam-6117	241	9	)	)	PUNCT
ejpam-6117	242	1	=	=	SYM
ejpam-6117	242	2	α(cn	α(cn	NOUN
ejpam-6117	242	3	)	)	PUNCT
ejpam-6117	242	4	=	=	SYM
ejpam-6117	242	5	|s|	|s|	NOUN
ejpam-6117	242	6	=	=	SYM
ejpam-6117	242	7	n−1	n−1	PROPN
ejpam-6117	242	8	2	2	NUM
ejpam-6117	242	9	.	.	PUNCT
ejpam-6117	243	1	in	in	ADP
ejpam-6117	243	2	what	what	PRON
ejpam-6117	243	3	follows	follow	VERB
ejpam-6117	243	4	,	,	PUNCT
ejpam-6117	243	5	we	we	PRON
ejpam-6117	243	6	characterize	characterize	VERB
ejpam-6117	243	7	the	the	DET
ejpam-6117	243	8	wcis	wcis	NOUN
ejpam-6117	243	9	in	in	ADP
ejpam-6117	243	10	g	g	PROPN
ejpam-6117	243	11	+	+	NOUN
ejpam-6117	243	12	h	h	NOUN
ejpam-6117	243	13	and	and	CCONJ
ejpam-6117	243	14	determine	determine	VERB
ejpam-6117	243	15	the	the	DET
ejpam-6117	243	16	weakly	weakly	ADV
ejpam-6117	243	17	connected	connected	ADJ
ejpam-6117	243	18	independent	independent	ADJ
ejpam-6117	243	19	number	number	NOUN
ejpam-6117	243	20	of	of	ADP
ejpam-6117	243	21	g+h	g+h	PROPN
ejpam-6117	243	22	.	.	PUNCT
ejpam-6117	244	1	r.	r.	PROPN
ejpam-6117	244	2	merontos	merontos	PROPN
ejpam-6117	244	3	et	et	PROPN
ejpam-6117	244	4	al	al	PROPN
ejpam-6117	244	5	.	.	PUNCT
ejpam-6117	244	6	/	/	SYM
ejpam-6117	244	7	eur	eur	PROPN
ejpam-6117	244	8	.	.	PUNCT
ejpam-6117	245	1	j.	j.	PROPN
ejpam-6117	245	2	pure	pure	PROPN
ejpam-6117	245	3	appl	appl	PROPN
ejpam-6117	245	4	.	.	PROPN
ejpam-6117	245	5	math	math	PROPN
ejpam-6117	245	6	,	,	PUNCT
ejpam-6117	245	7	18	18	NUM
ejpam-6117	245	8	(	(	PUNCT
ejpam-6117	245	9	2	2	NUM
ejpam-6117	245	10	)	)	PUNCT
ejpam-6117	245	11	(	(	PUNCT
ejpam-6117	245	12	2025	2025	NUM
ejpam-6117	245	13	)	)	PUNCT
ejpam-6117	245	14	,	,	PUNCT
ejpam-6117	245	15	6117	6117	NUM
ejpam-6117	245	16	6	6	NUM
ejpam-6117	245	17	of	of	ADP
ejpam-6117	245	18	9	9	NUM
ejpam-6117	245	19	theorem	theorem	NOUN
ejpam-6117	245	20	5	5	NUM
ejpam-6117	245	21	.	.	PUNCT
ejpam-6117	246	1	let	let	VERB
ejpam-6117	246	2	g	g	NOUN
ejpam-6117	246	3	and	and	CCONJ
ejpam-6117	246	4	h	h	NOUN
ejpam-6117	246	5	be	be	VERB
ejpam-6117	246	6	any	any	DET
ejpam-6117	246	7	two	two	NUM
ejpam-6117	246	8	graphs	graph	NOUN
ejpam-6117	246	9	.	.	PUNCT
ejpam-6117	247	1	then	then	ADV
ejpam-6117	247	2	s	s	VERB
ejpam-6117	247	3	is	be	AUX
ejpam-6117	247	4	a	a	DET
ejpam-6117	247	5	wcis	wcis	NOUN
ejpam-6117	247	6	in	in	ADP
ejpam-6117	247	7	g+h	g+h	PROPN
ejpam-6117	247	8	if	if	SCONJ
ejpam-6117	247	9	and	and	CCONJ
ejpam-6117	247	10	only	only	ADV
ejpam-6117	247	11	if	if	SCONJ
ejpam-6117	247	12	either	either	PRON
ejpam-6117	247	13	s	s	VERB
ejpam-6117	247	14	is	be	AUX
ejpam-6117	247	15	an	an	DET
ejpam-6117	247	16	independent	independent	ADJ
ejpam-6117	247	17	set	set	NOUN
ejpam-6117	247	18	in	in	ADP
ejpam-6117	247	19	g	g	PROPN
ejpam-6117	247	20	or	or	CCONJ
ejpam-6117	247	21	s	s	NOUN
ejpam-6117	247	22	is	be	AUX
ejpam-6117	247	23	an	an	DET
ejpam-6117	247	24	independent	independent	ADJ
ejpam-6117	247	25	set	set	NOUN
ejpam-6117	247	26	in	in	ADP
ejpam-6117	247	27	h.	h.	PROPN
ejpam-6117	247	28	proof	proof	PROPN
ejpam-6117	247	29	.	.	PUNCT
ejpam-6117	248	1	assume	assume	VERB
ejpam-6117	248	2	that	that	SCONJ
ejpam-6117	248	3	s	s	VERB
ejpam-6117	248	4	is	be	AUX
ejpam-6117	248	5	a	a	DET
ejpam-6117	248	6	wcis	wcis	NOUN
ejpam-6117	248	7	in	in	ADP
ejpam-6117	248	8	g+h	g+h	PROPN
ejpam-6117	248	9	.	.	PUNCT
ejpam-6117	249	1	since	since	SCONJ
ejpam-6117	249	2	s	s	PROPN
ejpam-6117	249	3	is	be	AUX
ejpam-6117	249	4	an	an	DET
ejpam-6117	249	5	independent	independent	ADJ
ejpam-6117	249	6	set	set	NOUN
ejpam-6117	249	7	in	in	ADP
ejpam-6117	249	8	g+h	g+h	PROPN
ejpam-6117	249	9	,	,	PUNCT
ejpam-6117	249	10	it	it	PRON
ejpam-6117	249	11	follows	follow	VERB
ejpam-6117	249	12	that	that	SCONJ
ejpam-6117	249	13	either	either	PRON
ejpam-6117	249	14	s	s	VERB
ejpam-6117	249	15	⊆	⊆	NUM
ejpam-6117	249	16	v	v	NOUN
ejpam-6117	249	17	(	(	PUNCT
ejpam-6117	249	18	g	g	NOUN
ejpam-6117	249	19	)	)	PUNCT
ejpam-6117	249	20	or	or	CCONJ
ejpam-6117	249	21	s	s	PRON
ejpam-6117	249	22	⊆	⊆	NUM
ejpam-6117	249	23	v	v	NOUN
ejpam-6117	249	24	(	(	PUNCT
ejpam-6117	249	25	h	h	NOUN
ejpam-6117	249	26	)	)	PUNCT
ejpam-6117	249	27	.	.	PUNCT
ejpam-6117	250	1	thus	thus	ADV
ejpam-6117	250	2	,	,	PUNCT
ejpam-6117	250	3	s	s	VERB
ejpam-6117	250	4	is	be	AUX
ejpam-6117	250	5	an	an	DET
ejpam-6117	250	6	independent	independent	ADJ
ejpam-6117	250	7	set	set	NOUN
ejpam-6117	250	8	in	in	ADP
ejpam-6117	250	9	g	g	PROPN
ejpam-6117	250	10	or	or	CCONJ
ejpam-6117	250	11	an	an	DET
ejpam-6117	250	12	independent	independent	ADJ
ejpam-6117	250	13	set	set	NOUN
ejpam-6117	250	14	in	in	ADP
ejpam-6117	250	15	h.	h.	NOUN
ejpam-6117	250	16	conversely	conversely	ADV
ejpam-6117	250	17	,	,	PUNCT
ejpam-6117	250	18	let	let	VERB
ejpam-6117	250	19	s	s	PRON
ejpam-6117	250	20	be	be	AUX
ejpam-6117	250	21	an	an	DET
ejpam-6117	250	22	independent	independent	ADJ
ejpam-6117	250	23	set	set	NOUN
ejpam-6117	250	24	in	in	ADP
ejpam-6117	250	25	g.	g.	PROPN
ejpam-6117	250	26	clearly	clearly	ADV
ejpam-6117	250	27	,	,	PUNCT
ejpam-6117	250	28	s	s	VERB
ejpam-6117	250	29	is	be	AUX
ejpam-6117	250	30	an	an	DET
ejpam-6117	250	31	independent	independent	ADJ
ejpam-6117	250	32	set	set	NOUN
ejpam-6117	250	33	of	of	ADP
ejpam-6117	250	34	g+h	g+h	PROPN
ejpam-6117	250	35	.	.	PUNCT
ejpam-6117	251	1	since	since	SCONJ
ejpam-6117	251	2	v	v	NOUN
ejpam-6117	251	3	(	(	PUNCT
ejpam-6117	251	4	h	h	NOUN
ejpam-6117	251	5	)	)	PUNCT
ejpam-6117	251	6	⊆	⊆	NUM
ejpam-6117	251	7	ng+h(x	ng+h(x	PROPN
ejpam-6117	251	8	)	)	PUNCT
ejpam-6117	251	9	for	for	ADP
ejpam-6117	251	10	every	every	DET
ejpam-6117	251	11	x	x	SYM
ejpam-6117	251	12	∈	∈	PROPN
ejpam-6117	251	13	ng[s	ng[s	PROPN
ejpam-6117	251	14	]	]	PUNCT
ejpam-6117	251	15	,	,	PUNCT
ejpam-6117	251	16	it	it	PRON
ejpam-6117	251	17	follows	follow	VERB
ejpam-6117	251	18	that	that	PRON
ejpam-6117	251	19	⟨ng+h	⟨ng+h	PROPN
ejpam-6117	252	1	[	[	X
ejpam-6117	252	2	s]⟩	s]⟩	VERB
ejpam-6117	252	3	is	be	AUX
ejpam-6117	252	4	connected	connect	VERB
ejpam-6117	252	5	.	.	PUNCT
ejpam-6117	253	1	this	this	PRON
ejpam-6117	253	2	implies	imply	VERB
ejpam-6117	253	3	that	that	SCONJ
ejpam-6117	253	4	⟨s⟩w	⟨s⟩w	NOUN
ejpam-6117	253	5	is	be	AUX
ejpam-6117	253	6	weakly	weakly	ADV
ejpam-6117	253	7	connected	connected	ADJ
ejpam-6117	253	8	in	in	ADP
ejpam-6117	253	9	g	g	PROPN
ejpam-6117	253	10	+	+	PROPN
ejpam-6117	253	11	h.	h.	PROPN
ejpam-6117	253	12	similarly	similarly	ADV
ejpam-6117	253	13	,	,	PUNCT
ejpam-6117	253	14	⟨s⟩w	⟨s⟩w	PROPN
ejpam-6117	253	15	is	be	AUX
ejpam-6117	253	16	weakly	weakly	ADV
ejpam-6117	253	17	connected	connected	ADJ
ejpam-6117	253	18	in	in	ADP
ejpam-6117	253	19	g+h	g+h	PROPN
ejpam-6117	253	20	if	if	SCONJ
ejpam-6117	253	21	s	s	AUX
ejpam-6117	253	22	be	be	AUX
ejpam-6117	253	23	an	an	DET
ejpam-6117	253	24	independent	independent	ADJ
ejpam-6117	253	25	set	set	NOUN
ejpam-6117	253	26	in	in	ADP
ejpam-6117	253	27	g.	g.	PROPN
ejpam-6117	253	28	corollary	corollary	PROPN
ejpam-6117	253	29	3	3	X
ejpam-6117	253	30	.	.	PUNCT
ejpam-6117	254	1	let	let	VERB
ejpam-6117	254	2	g	g	NOUN
ejpam-6117	254	3	and	and	CCONJ
ejpam-6117	254	4	h	h	PROPN
ejpam-6117	254	5	be	be	AUX
ejpam-6117	254	6	graphs	graph	NOUN
ejpam-6117	254	7	.	.	PUNCT
ejpam-6117	255	1	then	then	ADV
ejpam-6117	255	2	αw(g+h	αw(g+h	X
ejpam-6117	255	3	)	)	PUNCT
ejpam-6117	255	4	=	=	PUNCT
ejpam-6117	255	5	max{α(g	max{α(g	PROPN
ejpam-6117	255	6	)	)	PUNCT
ejpam-6117	255	7	,	,	PUNCT
ejpam-6117	255	8	α(h	α(h	NOUN
ejpam-6117	255	9	)	)	PUNCT
ejpam-6117	255	10	}	}	PUNCT
ejpam-6117	255	11	=	=	SYM
ejpam-6117	255	12	α(g+h	α(g+h	NUM
ejpam-6117	255	13	)	)	PUNCT
ejpam-6117	255	14	.	.	PUNCT
ejpam-6117	256	1	example	example	NOUN
ejpam-6117	257	1	1	1	X
ejpam-6117	257	2	.	.	PUNCT
ejpam-6117	257	3	let	let	VERB
ejpam-6117	257	4	g	g	NOUN
ejpam-6117	257	5	be	be	AUX
ejpam-6117	257	6	any	any	DET
ejpam-6117	257	7	graph	graph	NOUN
ejpam-6117	257	8	and	and	CCONJ
ejpam-6117	257	9	m	m	AUX
ejpam-6117	257	10	be	be	AUX
ejpam-6117	257	11	a	a	DET
ejpam-6117	257	12	positive	positive	ADJ
ejpam-6117	257	13	integer	integer	NOUN
ejpam-6117	257	14	.	.	PUNCT
ejpam-6117	258	1	then	then	ADV
ejpam-6117	258	2	(	(	PUNCT
ejpam-6117	258	3	i	i	NOUN
ejpam-6117	258	4	)	)	PUNCT
ejpam-6117	258	5	αw(km	αw(km	PROPN
ejpam-6117	258	6	+	+	PROPN
ejpam-6117	258	7	g	g	NOUN
ejpam-6117	258	8	)	)	PUNCT
ejpam-6117	258	9	=	=	SYM
ejpam-6117	258	10	max{m	max{m	NOUN
ejpam-6117	258	11	,	,	PUNCT
ejpam-6117	258	12	α(g	α(g	NUM
ejpam-6117	258	13	)	)	PUNCT
ejpam-6117	258	14	}	}	PUNCT
ejpam-6117	258	15	,	,	PUNCT
ejpam-6117	258	16	(	(	PUNCT
ejpam-6117	258	17	ii	ii	NOUN
ejpam-6117	258	18	)	)	PUNCT
ejpam-6117	258	19	αw(km	αw(km	PROPN
ejpam-6117	259	1	+	+	PROPN
ejpam-6117	259	2	g	g	NOUN
ejpam-6117	259	3	)	)	PUNCT
ejpam-6117	259	4	=	=	SYM
ejpam-6117	259	5	α(g	α(g	NUM
ejpam-6117	259	6	)	)	PUNCT
ejpam-6117	259	7	,	,	PUNCT
ejpam-6117	259	8	(	(	PUNCT
ejpam-6117	259	9	iii	iii	X
ejpam-6117	259	10	)	)	PUNCT
ejpam-6117	259	11	αw(km	αw(km	PROPN
ejpam-6117	259	12	,	,	PUNCT
ejpam-6117	259	13	n	n	CCONJ
ejpam-6117	259	14	)	)	PUNCT
ejpam-6117	259	15	=	=	SYM
ejpam-6117	260	1	αw(km	αw(km	PROPN
ejpam-6117	260	2	+	+	PROPN
ejpam-6117	260	3	kn	kn	PROPN
ejpam-6117	260	4	)	)	PUNCT
ejpam-6117	260	5	=	=	PUNCT
ejpam-6117	260	6	max{m	max{m	PROPN
ejpam-6117	260	7	,	,	PUNCT
ejpam-6117	260	8	n	n	CCONJ
ejpam-6117	260	9	}	}	PUNCT
ejpam-6117	260	10	.	.	PUNCT
ejpam-6117	261	1	the	the	DET
ejpam-6117	261	2	next	next	ADJ
ejpam-6117	261	3	result	result	NOUN
ejpam-6117	261	4	characterizes	characterize	VERB
ejpam-6117	261	5	the	the	DET
ejpam-6117	261	6	wcis	wcis	NOUN
ejpam-6117	261	7	of	of	ADP
ejpam-6117	261	8	g	g	PROPN
ejpam-6117	261	9	◦	◦	PROPN
ejpam-6117	261	10	h.	h.	NOUN
ejpam-6117	261	11	theorem	theorem	NOUN
ejpam-6117	261	12	6	6	NUM
ejpam-6117	261	13	.	.	PUNCT
ejpam-6117	262	1	let	let	VERB
ejpam-6117	262	2	g	g	PRON
ejpam-6117	262	3	be	be	AUX
ejpam-6117	262	4	a	a	DET
ejpam-6117	262	5	connected	connected	ADJ
ejpam-6117	262	6	graph	graph	NOUN
ejpam-6117	262	7	and	and	CCONJ
ejpam-6117	262	8	h	h	NOUN
ejpam-6117	262	9	be	be	AUX
ejpam-6117	262	10	any	any	DET
ejpam-6117	262	11	graph	graph	NOUN
ejpam-6117	262	12	.	.	PUNCT
ejpam-6117	263	1	a	a	DET
ejpam-6117	263	2	subset	subset	NOUN
ejpam-6117	263	3	s	s	NOUN
ejpam-6117	263	4	of	of	ADP
ejpam-6117	263	5	v	v	NOUN
ejpam-6117	263	6	(	(	PUNCT
ejpam-6117	263	7	g	g	PROPN
ejpam-6117	263	8	◦	◦	NOUN
ejpam-6117	263	9	h	h	NOUN
ejpam-6117	263	10	)	)	PUNCT
ejpam-6117	263	11	is	be	AUX
ejpam-6117	263	12	a	a	DET
ejpam-6117	263	13	wcis	wcis	NOUN
ejpam-6117	263	14	in	in	ADP
ejpam-6117	263	15	g	g	PROPN
ejpam-6117	263	16	◦	◦	NOUN
ejpam-6117	263	17	h	h	NOUN
ejpam-6117	263	18	if	if	SCONJ
ejpam-6117	264	1	and	and	CCONJ
ejpam-6117	264	2	only	only	ADV
ejpam-6117	264	3	if	if	SCONJ
ejpam-6117	264	4	one	one	NUM
ejpam-6117	264	5	of	of	ADP
ejpam-6117	264	6	the	the	DET
ejpam-6117	264	7	following	follow	VERB
ejpam-6117	264	8	holds	hold	VERB
ejpam-6117	264	9	:	:	PUNCT
ejpam-6117	264	10	(	(	PUNCT
ejpam-6117	264	11	i	i	NOUN
ejpam-6117	264	12	)	)	PUNCT
ejpam-6117	264	13	s	s	VERB
ejpam-6117	264	14	is	be	AUX
ejpam-6117	264	15	an	an	DET
ejpam-6117	264	16	independent	independent	ADJ
ejpam-6117	264	17	set	set	NOUN
ejpam-6117	264	18	in	in	ADP
ejpam-6117	264	19	hv	hv	NOUN
ejpam-6117	264	20	for	for	ADP
ejpam-6117	264	21	some	some	DET
ejpam-6117	264	22	v	v	NUM
ejpam-6117	264	23	∈	∈	PROPN
ejpam-6117	264	24	v	v	NOUN
ejpam-6117	264	25	(	(	PUNCT
ejpam-6117	264	26	g	g	NOUN
ejpam-6117	264	27	)	)	PUNCT
ejpam-6117	264	28	.	.	PUNCT
ejpam-6117	265	1	(	(	PUNCT
ejpam-6117	265	2	ii	ii	X
ejpam-6117	265	3	)	)	PUNCT
ejpam-6117	265	4	s	s	PART
ejpam-6117	266	1	=	=	X
ejpam-6117	266	2	c	c	X
ejpam-6117	266	3	∪	∪	X
ejpam-6117	266	4	(	(	PUNCT
ejpam-6117	266	5	∪v∈ng(c)sv	∪v∈ng(c)sv	NOUN
ejpam-6117	266	6	)	)	PUNCT
ejpam-6117	266	7	,	,	PUNCT
ejpam-6117	266	8	where	where	SCONJ
ejpam-6117	266	9	(	(	PUNCT
ejpam-6117	266	10	a	a	X
ejpam-6117	266	11	)	)	PUNCT
ejpam-6117	266	12	c	c	NOUN
ejpam-6117	266	13	is	be	AUX
ejpam-6117	266	14	a	a	DET
ejpam-6117	266	15	wcis	wcis	NOUN
ejpam-6117	266	16	in	in	ADP
ejpam-6117	266	17	g	g	NOUN
ejpam-6117	266	18	,	,	PUNCT
ejpam-6117	266	19	and	and	CCONJ
ejpam-6117	266	20	(	(	PUNCT
ejpam-6117	266	21	b	b	X
ejpam-6117	266	22	)	)	PUNCT
ejpam-6117	266	23	sv	sv	PROPN
ejpam-6117	266	24	is	be	AUX
ejpam-6117	266	25	an	an	DET
ejpam-6117	266	26	independent	independent	ADJ
ejpam-6117	266	27	set	set	NOUN
ejpam-6117	266	28	(	(	PUNCT
ejpam-6117	266	29	may	may	AUX
ejpam-6117	266	30	be	be	AUX
ejpam-6117	266	31	empty	empty	ADJ
ejpam-6117	266	32	)	)	PUNCT
ejpam-6117	266	33	in	in	ADP
ejpam-6117	266	34	hv	hv	PROPN
ejpam-6117	266	35	for	for	ADP
ejpam-6117	266	36	each	each	DET
ejpam-6117	266	37	v	v	ADP
ejpam-6117	266	38	∈	∈	PROPN
ejpam-6117	266	39	ng(c	ng(c	NUM
ejpam-6117	266	40	)	)	PUNCT
ejpam-6117	266	41	.	.	PUNCT
ejpam-6117	267	1	proof	proof	NOUN
ejpam-6117	267	2	.	.	PUNCT
ejpam-6117	268	1	suppose	suppose	VERB
ejpam-6117	268	2	s	s	PRON
ejpam-6117	268	3	is	be	AUX
ejpam-6117	268	4	a	a	DET
ejpam-6117	268	5	wcis	wcis	NOUN
ejpam-6117	268	6	in	in	ADP
ejpam-6117	268	7	g	g	PROPN
ejpam-6117	268	8	◦	◦	PROPN
ejpam-6117	269	1	h.	h.	NOUN
ejpam-6117	270	1	then	then	ADV
ejpam-6117	270	2	s	s	VERB
ejpam-6117	270	3	is	be	AUX
ejpam-6117	270	4	an	an	DET
ejpam-6117	270	5	independent	independent	ADJ
ejpam-6117	270	6	set	set	NOUN
ejpam-6117	270	7	in	in	ADP
ejpam-6117	270	8	g	g	PROPN
ejpam-6117	270	9	◦	◦	NOUN
ejpam-6117	270	10	h	h	NOUN
ejpam-6117	270	11	and	and	CCONJ
ejpam-6117	270	12	⟨s⟩w	⟨s⟩w	NOUN
ejpam-6117	270	13	is	be	AUX
ejpam-6117	270	14	connected	connect	VERB
ejpam-6117	270	15	.	.	PUNCT
ejpam-6117	271	1	consider	consider	VERB
ejpam-6117	271	2	the	the	DET
ejpam-6117	271	3	following	follow	VERB
ejpam-6117	271	4	cases	case	NOUN
ejpam-6117	271	5	:	:	PUNCT
ejpam-6117	271	6	case	case	NOUN
ejpam-6117	271	7	1	1	NUM
ejpam-6117	271	8	:	:	SYM
ejpam-6117	271	9	v	v	NOUN
ejpam-6117	271	10	(	(	PUNCT
ejpam-6117	271	11	g	g	NOUN
ejpam-6117	271	12	)	)	PUNCT
ejpam-6117	272	1	∩	∩	NOUN
ejpam-6117	272	2	s	s	PART
ejpam-6117	272	3	=	=	X
ejpam-6117	272	4	∅.	∅.	X
ejpam-6117	272	5	then	then	ADV
ejpam-6117	272	6	s	s	VERB
ejpam-6117	272	7	⊆	⊆	NUM
ejpam-6117	272	8	∪u∈v	∪u∈v	PROPN
ejpam-6117	272	9	(	(	PUNCT
ejpam-6117	272	10	g)v	g)v	X
ejpam-6117	272	11	(	(	PUNCT
ejpam-6117	272	12	hu	hu	PROPN
ejpam-6117	272	13	)	)	PUNCT
ejpam-6117	272	14	.	.	PUNCT
ejpam-6117	273	1	since	since	SCONJ
ejpam-6117	273	2	⟨s⟩w	⟨s⟩w	PROPN
ejpam-6117	273	3	is	be	AUX
ejpam-6117	273	4	connected	connect	VERB
ejpam-6117	273	5	,	,	PUNCT
ejpam-6117	273	6	it	it	PRON
ejpam-6117	273	7	follows	follow	VERB
ejpam-6117	273	8	that	that	SCONJ
ejpam-6117	273	9	s	s	VERB
ejpam-6117	273	10	is	be	AUX
ejpam-6117	273	11	in	in	ADP
ejpam-6117	273	12	exactly	exactly	ADV
ejpam-6117	273	13	one	one	NUM
ejpam-6117	273	14	of	of	ADP
ejpam-6117	273	15	the	the	DET
ejpam-6117	273	16	components	component	NOUN
ejpam-6117	273	17	of	of	ADP
ejpam-6117	273	18	⟨∪u∈v	⟨∪u∈v	PROPN
ejpam-6117	273	19	(	(	PUNCT
ejpam-6117	273	20	g)v	g)v	X
ejpam-6117	273	21	(	(	PUNCT
ejpam-6117	273	22	hu)⟩	hu)⟩	NOUN
ejpam-6117	273	23	,	,	PUNCT
ejpam-6117	273	24	that	that	ADV
ejpam-6117	273	25	is	be	AUX
ejpam-6117	273	26	,	,	PUNCT
ejpam-6117	273	27	s	s	VERB
ejpam-6117	273	28	⊆	⊆	NUM
ejpam-6117	273	29	v	v	NOUN
ejpam-6117	273	30	(	(	PUNCT
ejpam-6117	273	31	hv	hv	PROPN
ejpam-6117	273	32	)	)	PUNCT
ejpam-6117	273	33	for	for	ADP
ejpam-6117	273	34	a	a	DET
ejpam-6117	273	35	unique	unique	ADJ
ejpam-6117	273	36	vertex	vertex	NOUN
ejpam-6117	273	37	v	v	ADP
ejpam-6117	273	38	∈	∈	NOUN
ejpam-6117	273	39	v	v	NOUN
ejpam-6117	273	40	(	(	PUNCT
ejpam-6117	273	41	g	g	NOUN
ejpam-6117	273	42	)	)	PUNCT
ejpam-6117	273	43	.	.	PUNCT
ejpam-6117	274	1	consequently	consequently	ADV
ejpam-6117	274	2	,	,	PUNCT
ejpam-6117	274	3	s	s	VERB
ejpam-6117	274	4	is	be	AUX
ejpam-6117	274	5	an	an	DET
ejpam-6117	274	6	independent	independent	ADJ
ejpam-6117	274	7	subset	subset	NOUN
ejpam-6117	274	8	of	of	ADP
ejpam-6117	274	9	v	v	PROPN
ejpam-6117	274	10	(	(	PUNCT
ejpam-6117	274	11	hv	hv	PROPN
ejpam-6117	274	12	)	)	PUNCT
ejpam-6117	274	13	.	.	PUNCT
ejpam-6117	275	1	this	this	PRON
ejpam-6117	275	2	shows	show	VERB
ejpam-6117	275	3	that	that	SCONJ
ejpam-6117	275	4	(	(	PUNCT
ejpam-6117	275	5	i	i	NOUN
ejpam-6117	275	6	)	)	PUNCT
ejpam-6117	275	7	holds	hold	VERB
ejpam-6117	275	8	.	.	PUNCT
ejpam-6117	276	1	case	case	NOUN
ejpam-6117	276	2	2	2	NUM
ejpam-6117	276	3	:	:	SYM
ejpam-6117	276	4	v	v	NOUN
ejpam-6117	276	5	(	(	PUNCT
ejpam-6117	276	6	g	g	NOUN
ejpam-6117	276	7	)	)	PUNCT
ejpam-6117	276	8	∩	∩	PROPN
ejpam-6117	276	9	s	s	PART
ejpam-6117	276	10	̸=	̸=	PROPN
ejpam-6117	276	11	∅.	∅.	ADV
ejpam-6117	276	12	let	let	VERB
ejpam-6117	276	13	c	c	NOUN
ejpam-6117	276	14	=	=	SYM
ejpam-6117	276	15	v	v	PROPN
ejpam-6117	276	16	(	(	PUNCT
ejpam-6117	276	17	g	g	NOUN
ejpam-6117	276	18	)	)	PUNCT
ejpam-6117	276	19	∩	∩	PROPN
ejpam-6117	276	20	s	s	NOUN
ejpam-6117	276	21	and	and	CCONJ
ejpam-6117	276	22	sv	sv	X
ejpam-6117	276	23	=	=	SYM
ejpam-6117	276	24	s	s	PROPN
ejpam-6117	276	25	∩	∩	ADJ
ejpam-6117	276	26	v	v	X
ejpam-6117	276	27	(	(	PUNCT
ejpam-6117	276	28	hv	hv	PROPN
ejpam-6117	276	29	)	)	PUNCT
ejpam-6117	276	30	for	for	ADP
ejpam-6117	276	31	each	each	DET
ejpam-6117	276	32	v	v	NUM
ejpam-6117	276	33	∈	∈	PROPN
ejpam-6117	276	34	v	v	NOUN
ejpam-6117	276	35	(	(	PUNCT
ejpam-6117	276	36	g	g	NOUN
ejpam-6117	276	37	)	)	PUNCT
ejpam-6117	276	38	.	.	PUNCT
ejpam-6117	277	1	since	since	SCONJ
ejpam-6117	277	2	s	s	PROPN
ejpam-6117	277	3	is	be	AUX
ejpam-6117	277	4	a	a	DET
ejpam-6117	277	5	wcis	wcis	NOUN
ejpam-6117	277	6	in	in	ADP
ejpam-6117	277	7	g	g	PROPN
ejpam-6117	277	8	◦	◦	NOUN
ejpam-6117	277	9	h	h	NOUN
ejpam-6117	277	10	,	,	PUNCT
ejpam-6117	277	11	c	c	PROPN
ejpam-6117	277	12	is	be	AUX
ejpam-6117	277	13	a	a	DET
ejpam-6117	277	14	wcis	wcis	NOUN
ejpam-6117	277	15	in	in	ADP
ejpam-6117	277	16	g.	g.	PROPN
ejpam-6117	277	17	let	let	VERB
ejpam-6117	277	18	v	v	NUM
ejpam-6117	277	19	∈	∈	PROPN
ejpam-6117	277	20	v	v	NOUN
ejpam-6117	277	21	(	(	PUNCT
ejpam-6117	277	22	g	g	NOUN
ejpam-6117	277	23	)	)	PUNCT
ejpam-6117	277	24	such	such	ADJ
ejpam-6117	277	25	that	that	SCONJ
ejpam-6117	277	26	sv	sv	PROPN
ejpam-6117	278	1	̸=	̸=	PROPN
ejpam-6117	278	2	∅.	∅.	NOUN
ejpam-6117	278	3	then	then	ADV
ejpam-6117	278	4	sv	sv	PROPN
ejpam-6117	278	5	is	be	AUX
ejpam-6117	278	6	an	an	DET
ejpam-6117	278	7	independent	independent	ADJ
ejpam-6117	278	8	set	set	NOUN
ejpam-6117	278	9	in	in	ADP
ejpam-6117	278	10	hv	hv	PROPN
ejpam-6117	278	11	r.	r.	PROPN
ejpam-6117	278	12	merontos	merontos	PROPN
ejpam-6117	278	13	et	et	PROPN
ejpam-6117	278	14	al	al	PROPN
ejpam-6117	278	15	.	.	PUNCT
ejpam-6117	278	16	/	/	SYM
ejpam-6117	278	17	eur	eur	PROPN
ejpam-6117	278	18	.	.	PUNCT
ejpam-6117	279	1	j.	j.	PROPN
ejpam-6117	279	2	pure	pure	PROPN
ejpam-6117	279	3	appl	appl	PROPN
ejpam-6117	279	4	.	.	PROPN
ejpam-6117	279	5	math	math	PROPN
ejpam-6117	279	6	,	,	PUNCT
ejpam-6117	279	7	18	18	NUM
ejpam-6117	279	8	(	(	PUNCT
ejpam-6117	279	9	2	2	NUM
ejpam-6117	279	10	)	)	PUNCT
ejpam-6117	279	11	(	(	PUNCT
ejpam-6117	279	12	2025	2025	NUM
ejpam-6117	279	13	)	)	PUNCT
ejpam-6117	279	14	,	,	PUNCT
ejpam-6117	279	15	6117	6117	NUM
ejpam-6117	279	16	7	7	NUM
ejpam-6117	279	17	of	of	ADP
ejpam-6117	279	18	9	9	NUM
ejpam-6117	279	19	and	and	CCONJ
ejpam-6117	279	20	v	v	NOUN
ejpam-6117	279	21	/∈	/∈	PUNCT
ejpam-6117	279	22	c	c	NOUN
ejpam-6117	280	1	because	because	SCONJ
ejpam-6117	280	2	s	s	NOUN
ejpam-6117	280	3	is	be	AUX
ejpam-6117	280	4	an	an	DET
ejpam-6117	280	5	independent	independent	ADJ
ejpam-6117	280	6	set	set	NOUN
ejpam-6117	280	7	in	in	ADP
ejpam-6117	280	8	g	g	PROPN
ejpam-6117	280	9	◦	◦	NOUN
ejpam-6117	280	10	h.	h.	NOUN
ejpam-6117	280	11	moreover	moreover	ADV
ejpam-6117	280	12	,	,	PUNCT
ejpam-6117	280	13	since	since	SCONJ
ejpam-6117	280	14	⟨s⟩w	⟨s⟩w	NOUN
ejpam-6117	280	15	is	be	AUX
ejpam-6117	280	16	connected	connect	VERB
ejpam-6117	280	17	,	,	PUNCT
ejpam-6117	280	18	v	v	ADP
ejpam-6117	280	19	∈	∈	PROPN
ejpam-6117	280	20	ng(c	ng(c	NUM
ejpam-6117	280	21	)	)	PUNCT
ejpam-6117	280	22	.	.	PUNCT
ejpam-6117	281	1	thus	thus	ADV
ejpam-6117	281	2	,	,	PUNCT
ejpam-6117	281	3	s	s	VERB
ejpam-6117	281	4	=	=	PUNCT
ejpam-6117	281	5	c	c	X
ejpam-6117	281	6	∪	∪	X
ejpam-6117	281	7	(	(	PUNCT
ejpam-6117	281	8	∪v∈sv	∪v∈sv	ADJ
ejpam-6117	281	9	)	)	PUNCT
ejpam-6117	281	10	and	and	CCONJ
ejpam-6117	281	11	(	(	PUNCT
ejpam-6117	281	12	a	a	X
ejpam-6117	281	13	)	)	PUNCT
ejpam-6117	281	14	and	and	CCONJ
ejpam-6117	281	15	(	(	PUNCT
ejpam-6117	281	16	b	b	NOUN
ejpam-6117	281	17	)	)	PUNCT
ejpam-6117	281	18	hold	hold	NOUN
ejpam-6117	281	19	.	.	PUNCT
ejpam-6117	282	1	for	for	ADP
ejpam-6117	282	2	the	the	DET
ejpam-6117	282	3	converse	converse	NOUN
ejpam-6117	282	4	,	,	PUNCT
ejpam-6117	282	5	suppose	suppose	VERB
ejpam-6117	282	6	first	first	ADV
ejpam-6117	282	7	that	that	SCONJ
ejpam-6117	282	8	(	(	PUNCT
ejpam-6117	282	9	i	i	NOUN
ejpam-6117	282	10	)	)	PUNCT
ejpam-6117	282	11	holds	hold	VERB
ejpam-6117	282	12	,	,	PUNCT
ejpam-6117	282	13	that	that	ADV
ejpam-6117	282	14	is	is	ADV
ejpam-6117	282	15	,	,	PUNCT
ejpam-6117	282	16	suppose	suppose	VERB
ejpam-6117	282	17	that	that	SCONJ
ejpam-6117	282	18	s	s	VERB
ejpam-6117	282	19	is	be	AUX
ejpam-6117	282	20	an	an	DET
ejpam-6117	282	21	independent	independent	ADJ
ejpam-6117	282	22	set	set	NOUN
ejpam-6117	282	23	of	of	ADP
ejpam-6117	282	24	hv	hv	PROPN
ejpam-6117	282	25	for	for	ADP
ejpam-6117	282	26	a	a	DET
ejpam-6117	282	27	unique	unique	ADJ
ejpam-6117	282	28	v	v	ADP
ejpam-6117	282	29	∈	∈	NOUN
ejpam-6117	282	30	v	v	NOUN
ejpam-6117	282	31	(	(	PUNCT
ejpam-6117	282	32	g	g	NOUN
ejpam-6117	282	33	)	)	PUNCT
ejpam-6117	282	34	.	.	PUNCT
ejpam-6117	283	1	then	then	ADV
ejpam-6117	283	2	,	,	PUNCT
ejpam-6117	283	3	clearly	clearly	ADV
ejpam-6117	283	4	,	,	PUNCT
ejpam-6117	283	5	s	s	VERB
ejpam-6117	283	6	is	be	AUX
ejpam-6117	283	7	a	a	DET
ejpam-6117	283	8	wcis	wcis	NOUN
ejpam-6117	283	9	of	of	ADP
ejpam-6117	283	10	g	g	PROPN
ejpam-6117	283	11	◦	◦	NOUN
ejpam-6117	283	12	h.	h.	NOUN
ejpam-6117	283	13	next	next	ADV
ejpam-6117	283	14	,	,	PUNCT
ejpam-6117	283	15	suppose	suppose	VERB
ejpam-6117	283	16	that	that	SCONJ
ejpam-6117	283	17	(	(	PUNCT
ejpam-6117	283	18	ii	ii	NOUN
ejpam-6117	283	19	)	)	PUNCT
ejpam-6117	283	20	holds	hold	NOUN
ejpam-6117	283	21	,	,	PUNCT
ejpam-6117	283	22	i.e.	i.e.	X
ejpam-6117	283	23	,	,	PUNCT
ejpam-6117	283	24	s	s	PART
ejpam-6117	283	25	=	=	PUNCT
ejpam-6117	283	26	c	c	X
ejpam-6117	283	27	∪	∪	X
ejpam-6117	283	28	(	(	PUNCT
ejpam-6117	283	29	∪v∈x⊆ng(c)sv	∪v∈x⊆ng(c)sv	NUM
ejpam-6117	283	30	)	)	PUNCT
ejpam-6117	283	31	and	and	CCONJ
ejpam-6117	283	32	satisfies	satisfie	NOUN
ejpam-6117	283	33	(	(	PUNCT
ejpam-6117	283	34	a	a	X
ejpam-6117	283	35	)	)	PUNCT
ejpam-6117	283	36	and	and	CCONJ
ejpam-6117	283	37	(	(	PUNCT
ejpam-6117	283	38	b	b	NOUN
ejpam-6117	283	39	)	)	PUNCT
ejpam-6117	283	40	.	.	PUNCT
ejpam-6117	284	1	since	since	SCONJ
ejpam-6117	284	2	c	c	PROPN
ejpam-6117	284	3	∩ng(c	∩ng(c	PROPN
ejpam-6117	284	4	)	)	PUNCT
ejpam-6117	284	5	=	=	NOUN
ejpam-6117	284	6	∅	∅	NOUN
ejpam-6117	284	7	and	and	CCONJ
ejpam-6117	284	8	c	c	PROPN
ejpam-6117	284	9	and	and	CCONJ
ejpam-6117	284	10	sv	sv	PROPN
ejpam-6117	284	11	and	and	CCONJ
ejpam-6117	284	12	are	be	AUX
ejpam-6117	284	13	independent	independent	ADJ
ejpam-6117	284	14	sets	set	NOUN
ejpam-6117	284	15	,	,	PUNCT
ejpam-6117	284	16	s	s	VERB
ejpam-6117	284	17	is	be	AUX
ejpam-6117	284	18	an	an	DET
ejpam-6117	284	19	independent	independent	ADJ
ejpam-6117	284	20	set	set	NOUN
ejpam-6117	284	21	in	in	ADP
ejpam-6117	284	22	g	g	PROPN
ejpam-6117	284	23	◦	◦	NOUN
ejpam-6117	284	24	h.	h.	NOUN
ejpam-6117	284	25	moreover	moreover	ADV
ejpam-6117	284	26	,	,	PUNCT
ejpam-6117	284	27	since	since	SCONJ
ejpam-6117	284	28	⟨c⟩w	⟨c⟩w	NOUN
ejpam-6117	284	29	is	be	AUX
ejpam-6117	284	30	connected	connect	VERB
ejpam-6117	284	31	in	in	ADP
ejpam-6117	284	32	g	g	PROPN
ejpam-6117	284	33	and	and	CCONJ
ejpam-6117	284	34	v	v	ADP
ejpam-6117	284	35	∈	∈	PROPN
ejpam-6117	284	36	ng(c	ng(c	NUM
ejpam-6117	284	37	)	)	PUNCT
ejpam-6117	284	38	for	for	ADP
ejpam-6117	284	39	each	each	DET
ejpam-6117	284	40	non	non	ADJ
ejpam-6117	284	41	-	-	ADJ
ejpam-6117	284	42	empty	empty	ADJ
ejpam-6117	284	43	set	set	NOUN
ejpam-6117	284	44	sv	sv	PROPN
ejpam-6117	284	45	,	,	PUNCT
ejpam-6117	284	46	it	it	PRON
ejpam-6117	284	47	follows	follow	VERB
ejpam-6117	284	48	that	that	SCONJ
ejpam-6117	284	49	⟨s⟩w	⟨s⟩w	NOUN
ejpam-6117	284	50	is	be	AUX
ejpam-6117	284	51	connected	connect	VERB
ejpam-6117	284	52	in	in	ADP
ejpam-6117	284	53	g	g	PROPN
ejpam-6117	284	54	◦	◦	NOUN
ejpam-6117	284	55	h.	h.	PROPN
ejpam-6117	284	56	thus	thus	ADV
ejpam-6117	284	57	,	,	PUNCT
ejpam-6117	284	58	s	s	VERB
ejpam-6117	284	59	is	be	AUX
ejpam-6117	284	60	a	a	DET
ejpam-6117	284	61	wcis	wcis	NOUN
ejpam-6117	284	62	in	in	ADP
ejpam-6117	284	63	g	g	PROPN
ejpam-6117	284	64	◦	◦	PROPN
ejpam-6117	284	65	h.	h.	NOUN
ejpam-6117	284	66	αw(g	αw(g	PUNCT
ejpam-6117	284	67	◦	◦	NOUN
ejpam-6117	284	68	h	h	NOUN
ejpam-6117	284	69	)	)	PUNCT
ejpam-6117	284	70	≥	≥	NOUN
ejpam-6117	284	71	|s′|	|s′|	NOUN
ejpam-6117	284	72	=	=	SYM
ejpam-6117	284	73	≤	≤	X
ejpam-6117	284	74	α(h)|v	α(h)|v	PROPN
ejpam-6117	284	75	(	(	PUNCT
ejpam-6117	284	76	g)|+	g)|+	PROPN
ejpam-6117	284	77	(	(	PUNCT
ejpam-6117	284	78	1−	1−	NUM
ejpam-6117	284	79	α(h))ιc(g	α(h))ιc(g	NUM
ejpam-6117	284	80	)	)	PUNCT
ejpam-6117	284	81	.	.	PUNCT
ejpam-6117	285	1	corollary	corollary	ADJ
ejpam-6117	285	2	4	4	NUM
ejpam-6117	285	3	.	.	PUNCT
ejpam-6117	286	1	let	let	VERB
ejpam-6117	286	2	g	g	PRON
ejpam-6117	286	3	be	be	AUX
ejpam-6117	286	4	a	a	DET
ejpam-6117	286	5	connected	connected	ADJ
ejpam-6117	286	6	graph	graph	NOUN
ejpam-6117	286	7	and	and	CCONJ
ejpam-6117	286	8	h	h	NOUN
ejpam-6117	286	9	be	be	AUX
ejpam-6117	286	10	any	any	DET
ejpam-6117	286	11	graph	graph	NOUN
ejpam-6117	286	12	.	.	PUNCT
ejpam-6117	287	1	then	then	ADV
ejpam-6117	287	2	αw(g	αw(g	VERB
ejpam-6117	287	3	◦	◦	NOUN
ejpam-6117	287	4	h	h	NOUN
ejpam-6117	287	5	)	)	PUNCT
ejpam-6117	287	6	=	=	SYM
ejpam-6117	287	7	α(h	α(h	NOUN
ejpam-6117	287	8	)	)	PUNCT
ejpam-6117	287	9	if	if	SCONJ
ejpam-6117	287	10	g	g	PROPN
ejpam-6117	287	11	=	=	SYM
ejpam-6117	287	12	k1	k1	PROPN
ejpam-6117	287	13	.	.	PUNCT
ejpam-6117	288	1	otherwise	otherwise	ADV
ejpam-6117	288	2	,	,	PUNCT
ejpam-6117	288	3	αw(g	αw(g	X
ejpam-6117	288	4	◦	◦	NOUN
ejpam-6117	288	5	h	h	NOUN
ejpam-6117	288	6	)	)	PUNCT
ejpam-6117	289	1	=	=	SYM
ejpam-6117	289	2	α(h)(|v	α(h)(|v	NUM
ejpam-6117	289	3	(	(	PUNCT
ejpam-6117	289	4	g)|+	g)|+	PROPN
ejpam-6117	289	5	(	(	PUNCT
ejpam-6117	289	6	1−	1−	NUM
ejpam-6117	289	7	α(h))ιc(g	α(h))ιc(g	NUM
ejpam-6117	289	8	)	)	PUNCT
ejpam-6117	289	9	)	)	PUNCT
ejpam-6117	289	10	.	.	PUNCT
ejpam-6117	290	1	proof	proof	NOUN
ejpam-6117	290	2	.	.	PUNCT
ejpam-6117	291	1	clearly	clearly	ADV
ejpam-6117	291	2	,	,	PUNCT
ejpam-6117	291	3	αw(k1	αw(k1	PROPN
ejpam-6117	291	4	◦	◦	NOUN
ejpam-6117	291	5	h	h	NOUN
ejpam-6117	291	6	)	)	PUNCT
ejpam-6117	291	7	=	=	SYM
ejpam-6117	291	8	αw(k1+h	αw(k1+h	NUM
ejpam-6117	291	9	)	)	PUNCT
ejpam-6117	291	10	=	=	SYM
ejpam-6117	291	11	α(h	α(h	NOUN
ejpam-6117	291	12	)	)	PUNCT
ejpam-6117	291	13	(	(	PUNCT
ejpam-6117	291	14	see	see	VERB
ejpam-6117	291	15	example	example	NOUN
ejpam-6117	291	16	1(ii	1(ii	NUM
ejpam-6117	291	17	)	)	PUNCT
ejpam-6117	291	18	)	)	PUNCT
ejpam-6117	291	19	.	.	PUNCT
ejpam-6117	291	20	.	.	PUNCT
ejpam-6117	292	1	so	so	ADV
ejpam-6117	292	2	suppose	suppose	VERB
ejpam-6117	292	3	g	g	PROPN
ejpam-6117	292	4	̸=	̸=	PROPN
ejpam-6117	292	5	k1	k1	NOUN
ejpam-6117	292	6	and	and	CCONJ
ejpam-6117	292	7	let	let	VERB
ejpam-6117	292	8	s	s	PRON
ejpam-6117	292	9	be	be	AUX
ejpam-6117	292	10	an	an	DET
ejpam-6117	292	11	αw	αw	NOUN
ejpam-6117	292	12	-	-	VERB
ejpam-6117	292	13	set	set	VERB
ejpam-6117	292	14	in	in	ADP
ejpam-6117	292	15	g	g	PROPN
ejpam-6117	292	16	◦	◦	NOUN
ejpam-6117	292	17	h.	h.	NOUN
ejpam-6117	292	18	then	then	ADV
ejpam-6117	292	19	s	s	VERB
ejpam-6117	292	20	=	=	SYM
ejpam-6117	292	21	c∪	c∪	NOUN
ejpam-6117	292	22	(	(	PUNCT
ejpam-6117	292	23	∪v∈ng(c)sv	∪v∈ng(c)sv	NOUN
ejpam-6117	292	24	)	)	PUNCT
ejpam-6117	292	25	,	,	PUNCT
ejpam-6117	292	26	where	where	SCONJ
ejpam-6117	292	27	c	c	PROPN
ejpam-6117	292	28	is	be	AUX
ejpam-6117	292	29	a	a	DET
ejpam-6117	292	30	wcis	wcis	NOUN
ejpam-6117	292	31	in	in	ADP
ejpam-6117	292	32	g	g	PROPN
ejpam-6117	292	33	and	and	CCONJ
ejpam-6117	292	34	sv	sv	PROPN
ejpam-6117	292	35	is	be	AUX
ejpam-6117	292	36	an	an	DET
ejpam-6117	292	37	independent	independent	ADJ
ejpam-6117	292	38	set	set	NOUN
ejpam-6117	292	39	of	of	ADP
ejpam-6117	292	40	hv	hv	PROPN
ejpam-6117	292	41	for	for	ADP
ejpam-6117	292	42	each	each	DET
ejpam-6117	292	43	v	v	ADP
ejpam-6117	292	44	∈	∈	PROPN
ejpam-6117	292	45	ng(c	ng(c	NUM
ejpam-6117	292	46	)	)	PUNCT
ejpam-6117	292	47	by	by	ADP
ejpam-6117	292	48	theorem	theorem	NOUN
ejpam-6117	292	49	6	6	NUM
ejpam-6117	292	50	.	.	PUNCT
ejpam-6117	292	51	suppose	suppose	VERB
ejpam-6117	292	52	c	c	NOUN
ejpam-6117	292	53	is	be	AUX
ejpam-6117	292	54	not	not	PART
ejpam-6117	292	55	a	a	DET
ejpam-6117	292	56	dominating	dominating	NOUN
ejpam-6117	292	57	set	set	NOUN
ejpam-6117	292	58	of	of	ADP
ejpam-6117	292	59	g.	g.	PROPN
ejpam-6117	293	1	then	then	ADV
ejpam-6117	293	2	v	v	X
ejpam-6117	293	3	(	(	PUNCT
ejpam-6117	293	4	g	g	NOUN
ejpam-6117	293	5	)	)	PUNCT
ejpam-6117	293	6	\	\	NOUN
ejpam-6117	293	7	ng[c	ng[c	PROPN
ejpam-6117	293	8	]	]	PUNCT
ejpam-6117	293	9	̸=	̸=	PROPN
ejpam-6117	293	10	∅.	∅.	ADV
ejpam-6117	293	11	choose	choose	VERB
ejpam-6117	293	12	w	w	PROPN
ejpam-6117	293	13	∈	∈	PROPN
ejpam-6117	293	14	v	v	ADP
ejpam-6117	293	15	(	(	PUNCT
ejpam-6117	293	16	g	g	NOUN
ejpam-6117	293	17	)	)	PUNCT
ejpam-6117	293	18	\	\	NOUN
ejpam-6117	293	19	ng[c	ng[c	PROPN
ejpam-6117	293	20	]	]	PUNCT
ejpam-6117	293	21	such	such	ADJ
ejpam-6117	293	22	that	that	SCONJ
ejpam-6117	293	23	wx	wx	PROPN
ejpam-6117	293	24	∈	∈	PROPN
ejpam-6117	293	25	e(g	e(g	PROPN
ejpam-6117	293	26	)	)	PUNCT
ejpam-6117	293	27	for	for	ADP
ejpam-6117	293	28	some	some	DET
ejpam-6117	293	29	x	x	SYM
ejpam-6117	293	30	∈	∈	PROPN
ejpam-6117	293	31	ng(c	ng(c	NUM
ejpam-6117	293	32	)	)	PUNCT
ejpam-6117	293	33	.	.	PUNCT
ejpam-6117	294	1	then	then	ADV
ejpam-6117	294	2	c∗	c∗	PROPN
ejpam-6117	294	3	=	=	PUNCT
ejpam-6117	294	4	c	c	NOUN
ejpam-6117	294	5	∪	∪	X
ejpam-6117	294	6	{	{	PUNCT
ejpam-6117	294	7	w	w	NOUN
ejpam-6117	294	8	}	}	PUNCT
ejpam-6117	294	9	is	be	AUX
ejpam-6117	294	10	a	a	DET
ejpam-6117	294	11	wcis	wcis	NOUN
ejpam-6117	294	12	in	in	ADP
ejpam-6117	294	13	g	g	PROPN
ejpam-6117	294	14	and	and	CCONJ
ejpam-6117	294	15	ng(c	ng(c	NUM
ejpam-6117	294	16	∗	∗	NOUN
ejpam-6117	294	17	)	)	PUNCT
ejpam-6117	294	18	=	=	SYM
ejpam-6117	294	19	ng(c	ng(c	X
ejpam-6117	294	20	)	)	PUNCT
ejpam-6117	294	21	∪	∪	ADP
ejpam-6117	294	22	ng(w	ng(w	NOUN
ejpam-6117	294	23	)	)	PUNCT
ejpam-6117	294	24	.	.	PUNCT
ejpam-6117	295	1	let	let	VERB
ejpam-6117	295	2	lv	lv	PROPN
ejpam-6117	295	3	be	be	AUX
ejpam-6117	295	4	an	an	DET
ejpam-6117	295	5	α	α	NOUN
ejpam-6117	295	6	-	-	PUNCT
ejpam-6117	295	7	set	set	VERB
ejpam-6117	295	8	in	in	ADP
ejpam-6117	295	9	hv	hv	PROPN
ejpam-6117	295	10	for	for	ADP
ejpam-6117	295	11	each	each	DET
ejpam-6117	295	12	v	v	PROPN
ejpam-6117	295	13	∈	∈	PROPN
ejpam-6117	295	14	ng(c	ng(c	PUNCT
ejpam-6117	295	15	∗	∗	NOUN
ejpam-6117	295	16	)	)	PUNCT
ejpam-6117	295	17	.	.	PUNCT
ejpam-6117	296	1	then	then	ADV
ejpam-6117	296	2	,	,	PUNCT
ejpam-6117	296	3	by	by	ADP
ejpam-6117	296	4	theorem	theorem	NOUN
ejpam-6117	296	5	6	6	NUM
ejpam-6117	296	6	,	,	PUNCT
ejpam-6117	296	7	s∗	s∗	PROPN
ejpam-6117	296	8	=	=	SYM
ejpam-6117	296	9	c∗	c∗	PROPN
ejpam-6117	296	10	∪	∪	X
ejpam-6117	296	11	(	(	PUNCT
ejpam-6117	296	12	∪v∈ng(c∗)lv	∪v∈ng(c∗)lv	NOUN
ejpam-6117	296	13	)	)	PUNCT
ejpam-6117	296	14	is	be	AUX
ejpam-6117	296	15	a	a	DET
ejpam-6117	296	16	wcis	wcis	NOUN
ejpam-6117	296	17	of	of	ADP
ejpam-6117	296	18	g	g	PROPN
ejpam-6117	296	19	◦	◦	NOUN
ejpam-6117	296	20	h.	h.	NOUN
ejpam-6117	296	21	this	this	PRON
ejpam-6117	296	22	implies	imply	VERB
ejpam-6117	296	23	that	that	SCONJ
ejpam-6117	296	24	αw(g	αw(g	NUM
ejpam-6117	296	25	◦	◦	NOUN
ejpam-6117	296	26	h	h	NOUN
ejpam-6117	296	27	)	)	PUNCT
ejpam-6117	296	28	=	=	PUNCT
ejpam-6117	296	29	|s|	|s|	PROPN
ejpam-6117	296	30	<	<	X
ejpam-6117	296	31	|s∗|	|s∗|	NUM
ejpam-6117	296	32	which	which	PRON
ejpam-6117	296	33	is	be	AUX
ejpam-6117	296	34	not	not	PART
ejpam-6117	296	35	possible	possible	ADJ
ejpam-6117	296	36	.	.	PUNCT
ejpam-6117	297	1	therefore	therefore	ADV
ejpam-6117	297	2	,	,	PUNCT
ejpam-6117	297	3	c	c	PROPN
ejpam-6117	297	4	is	be	AUX
ejpam-6117	297	5	a	a	DET
ejpam-6117	297	6	wcids	wcid	NOUN
ejpam-6117	297	7	of	of	ADP
ejpam-6117	297	8	g.	g.	PROPN
ejpam-6117	297	9	from	from	ADP
ejpam-6117	297	10	this	this	PRON
ejpam-6117	297	11	and	and	CCONJ
ejpam-6117	297	12	the	the	DET
ejpam-6117	297	13	fact	fact	NOUN
ejpam-6117	297	14	that	that	SCONJ
ejpam-6117	297	15	1−	1−	NUM
ejpam-6117	297	16	α(h	α(h	NOUN
ejpam-6117	297	17	)	)	PUNCT
ejpam-6117	297	18	≤	≤	NOUN
ejpam-6117	297	19	0	0	NUM
ejpam-6117	297	20	,	,	PUNCT
ejpam-6117	297	21	we	we	PRON
ejpam-6117	297	22	have	have	AUX
ejpam-6117	297	23	αw(g	αw(g	NUM
ejpam-6117	297	24	◦	◦	NOUN
ejpam-6117	297	25	h	h	NOUN
ejpam-6117	297	26	)	)	PUNCT
ejpam-6117	297	27	=	=	PUNCT
ejpam-6117	297	28	|s|	|s|	NOUN
ejpam-6117	297	29	≤	≤	NUM
ejpam-6117	297	30	|c|+	|c|+	NOUN
ejpam-6117	297	31	α(h)(|v	α(h)(|v	NUM
ejpam-6117	297	32	(	(	PUNCT
ejpam-6117	297	33	g)|	g)|	NOUN
ejpam-6117	297	34	−	−	PROPN
ejpam-6117	297	35	|c|	|c|	PROPN
ejpam-6117	297	36	)	)	PUNCT
ejpam-6117	297	37	=	=	SYM
ejpam-6117	297	38	α(h)|v	α(h)|v	PROPN
ejpam-6117	297	39	(	(	PUNCT
ejpam-6117	297	40	g)|+	g)|+	NOUN
ejpam-6117	297	41	(	(	PUNCT
ejpam-6117	297	42	1−	1−	NUM
ejpam-6117	297	43	α(h))|c|	α(h))|c|	NUM
ejpam-6117	297	44	≤	≤	X
ejpam-6117	298	1	α(h)|v	α(h)|v	PROPN
ejpam-6117	298	2	(	(	PUNCT
ejpam-6117	298	3	g)|+	g)|+	PROPN
ejpam-6117	298	4	(	(	PUNCT
ejpam-6117	298	5	1−	1−	NUM
ejpam-6117	298	6	α(h))ιc(g	α(h))ιc(g	NUM
ejpam-6117	298	7	)	)	PUNCT
ejpam-6117	298	8	.	.	PUNCT
ejpam-6117	299	1	next	next	ADV
ejpam-6117	299	2	,	,	PUNCT
ejpam-6117	299	3	let	let	VERB
ejpam-6117	299	4	c0	c0	NOUN
ejpam-6117	299	5	be	be	AUX
ejpam-6117	299	6	a	a	DET
ejpam-6117	299	7	minimum	minimum	ADJ
ejpam-6117	299	8	wcids	wcid	NOUN
ejpam-6117	299	9	of	of	ADP
ejpam-6117	299	10	g	g	NOUN
ejpam-6117	299	11	and	and	CCONJ
ejpam-6117	299	12	let	let	VERB
ejpam-6117	299	13	sv	sv	INTJ
ejpam-6117	299	14	be	be	AUX
ejpam-6117	299	15	an	an	DET
ejpam-6117	299	16	α	α	NOUN
ejpam-6117	299	17	-	-	PUNCT
ejpam-6117	299	18	set	set	VERB
ejpam-6117	299	19	in	in	ADP
ejpam-6117	299	20	hv	hv	PROPN
ejpam-6117	299	21	for	for	ADP
ejpam-6117	299	22	each	each	DET
ejpam-6117	299	23	v	v	ADP
ejpam-6117	299	24	∈	∈	PROPN
ejpam-6117	299	25	ng(c0	ng(c0	NOUN
ejpam-6117	299	26	)	)	PUNCT
ejpam-6117	299	27	.	.	PUNCT
ejpam-6117	300	1	then	then	ADV
ejpam-6117	300	2	,	,	PUNCT
ejpam-6117	300	3	by	by	ADP
ejpam-6117	300	4	theorem	theorem	NOUN
ejpam-6117	300	5	6	6	NUM
ejpam-6117	300	6	,	,	PUNCT
ejpam-6117	300	7	s′	s′	ADJ
ejpam-6117	300	8	=	=	PUNCT
ejpam-6117	300	9	c0	c0	NOUN
ejpam-6117	300	10	∪	∪	X
ejpam-6117	300	11	(	(	PUNCT
ejpam-6117	300	12	∪v∈ng(c0)sv	∪v∈ng(c0)sv	NOUN
ejpam-6117	300	13	)	)	PUNCT
ejpam-6117	300	14	is	be	AUX
ejpam-6117	300	15	a	a	DET
ejpam-6117	300	16	wcis	wcis	NOUN
ejpam-6117	300	17	of	of	ADP
ejpam-6117	300	18	g	g	PROPN
ejpam-6117	300	19	◦	◦	NOUN
ejpam-6117	300	20	h.	h.	PROPN
ejpam-6117	300	21	hence	hence	ADV
ejpam-6117	300	22	,	,	PUNCT
ejpam-6117	300	23	αw(g	αw(g	X
ejpam-6117	300	24	◦	◦	NOUN
ejpam-6117	300	25	h	h	NOUN
ejpam-6117	300	26	)	)	PUNCT
ejpam-6117	300	27	≥	≥	NOUN
ejpam-6117	300	28	|c|	|c|	PROPN
ejpam-6117	300	29	=	=	SYM
ejpam-6117	300	30	≤	≤	X
ejpam-6117	300	31	α(h)|v	α(h)|v	PROPN
ejpam-6117	300	32	(	(	PUNCT
ejpam-6117	300	33	g)|+	g)|+	PROPN
ejpam-6117	300	34	(	(	PUNCT
ejpam-6117	300	35	1−	1−	NUM
ejpam-6117	300	36	α(h))ιc(g	α(h))ιc(g	NUM
ejpam-6117	300	37	)	)	PUNCT
ejpam-6117	300	38	.	.	PUNCT
ejpam-6117	301	1	this	this	PRON
ejpam-6117	301	2	proves	prove	VERB
ejpam-6117	301	3	the	the	DET
ejpam-6117	301	4	desired	desire	VERB
ejpam-6117	301	5	equality	equality	NOUN
ejpam-6117	301	6	.	.	PUNCT
ejpam-6117	302	1	4	4	X
ejpam-6117	302	2	.	.	NOUN
ejpam-6117	302	3	lexicographic	lexicographic	ADJ
ejpam-6117	302	4	product	product	NOUN
ejpam-6117	302	5	of	of	ADP
ejpam-6117	302	6	graphs	graph	NOUN
ejpam-6117	302	7	sandueta	sandueta	ADJ
ejpam-6117	302	8	and	and	CCONJ
ejpam-6117	302	9	canoy	canoy	NOUN
ejpam-6117	302	10	characterized	characterize	VERB
ejpam-6117	302	11	the	the	DET
ejpam-6117	302	12	weakly	weakly	ADV
ejpam-6117	302	13	connected	connected	ADJ
ejpam-6117	302	14	sets	set	NOUN
ejpam-6117	302	15	in	in	ADP
ejpam-6117	302	16	the	the	DET
ejpam-6117	302	17	lexicographic	lexicographic	ADJ
ejpam-6117	302	18	product	product	NOUN
ejpam-6117	302	19	of	of	ADP
ejpam-6117	302	20	graphs	graph	NOUN
ejpam-6117	302	21	.	.	PUNCT
ejpam-6117	303	1	r.	r.	PROPN
ejpam-6117	303	2	merontos	merontos	PROPN
ejpam-6117	303	3	et	et	PROPN
ejpam-6117	303	4	al	al	PROPN
ejpam-6117	303	5	.	.	PUNCT
ejpam-6117	303	6	/	/	SYM
ejpam-6117	303	7	eur	eur	PROPN
ejpam-6117	303	8	.	.	PUNCT
ejpam-6117	304	1	j.	j.	PROPN
ejpam-6117	304	2	pure	pure	PROPN
ejpam-6117	304	3	appl	appl	PROPN
ejpam-6117	304	4	.	.	PROPN
ejpam-6117	304	5	math	math	PROPN
ejpam-6117	304	6	,	,	PUNCT
ejpam-6117	304	7	18	18	NUM
ejpam-6117	304	8	(	(	PUNCT
ejpam-6117	304	9	2	2	NUM
ejpam-6117	304	10	)	)	PUNCT
ejpam-6117	304	11	(	(	PUNCT
ejpam-6117	304	12	2025	2025	NUM
ejpam-6117	304	13	)	)	PUNCT
ejpam-6117	304	14	,	,	PUNCT
ejpam-6117	304	15	6117	6117	NUM
ejpam-6117	304	16	8	8	NUM
ejpam-6117	304	17	of	of	ADP
ejpam-6117	304	18	9	9	NUM
ejpam-6117	304	19	theorem	theorem	VERB
ejpam-6117	304	20	7	7	NUM
ejpam-6117	304	21	.	.	PUNCT
ejpam-6117	305	1	[	[	X
ejpam-6117	305	2	8	8	NUM
ejpam-6117	305	3	]	]	PUNCT
ejpam-6117	305	4	let	let	VERB
ejpam-6117	305	5	g	g	NOUN
ejpam-6117	305	6	and	and	CCONJ
ejpam-6117	305	7	h	h	NOUN
ejpam-6117	305	8	be	be	AUX
ejpam-6117	305	9	connected	connect	VERB
ejpam-6117	305	10	non	non	ADJ
ejpam-6117	305	11	-	-	ADJ
ejpam-6117	305	12	trivial	trivial	ADJ
ejpam-6117	305	13	graphs	graph	NOUN
ejpam-6117	305	14	and	and	CCONJ
ejpam-6117	305	15	let	let	VERB
ejpam-6117	305	16	c	c	NOUN
ejpam-6117	305	17	=	=	SYM
ejpam-6117	305	18	∪x∈s({x	∪x∈s({x	ADJ
ejpam-6117	305	19	}	}	PUNCT
ejpam-6117	305	20	×	×	NOUN
ejpam-6117	305	21	tx	tx	PROPN
ejpam-6117	305	22	)	)	PUNCT
ejpam-6117	305	23	⊆	⊆	NUM
ejpam-6117	305	24	v	v	NOUN
ejpam-6117	305	25	(	(	PUNCT
ejpam-6117	305	26	g[h	g[h	PROPN
ejpam-6117	305	27	]	]	PUNCT
ejpam-6117	305	28	)	)	PUNCT
ejpam-6117	305	29	,	,	PUNCT
ejpam-6117	305	30	where	where	SCONJ
ejpam-6117	305	31	s	s	VERB
ejpam-6117	305	32	⊆	⊆	NUM
ejpam-6117	305	33	v	v	NOUN
ejpam-6117	305	34	(	(	PUNCT
ejpam-6117	305	35	g	g	NOUN
ejpam-6117	305	36	)	)	PUNCT
ejpam-6117	305	37	and	and	CCONJ
ejpam-6117	305	38	tx	tx	VERB
ejpam-6117	305	39	⊆	⊆	NUM
ejpam-6117	305	40	v	v	NOUN
ejpam-6117	305	41	(	(	PUNCT
ejpam-6117	305	42	h	h	NOUN
ejpam-6117	305	43	)	)	PUNCT
ejpam-6117	305	44	for	for	ADP
ejpam-6117	305	45	all	all	DET
ejpam-6117	305	46	x	x	SYM
ejpam-6117	305	47	∈	∈	PROPN
ejpam-6117	305	48	s.	s.	PROPN
ejpam-6117	305	49	then	then	ADV
ejpam-6117	305	50	c	c	PROPN
ejpam-6117	305	51	is	be	AUX
ejpam-6117	305	52	weakly	weakly	ADV
ejpam-6117	305	53	connected	connected	ADJ
ejpam-6117	305	54	in	in	ADP
ejpam-6117	305	55	g[h	g[h	NOUN
ejpam-6117	305	56	]	]	PUNCT
ejpam-6117	305	57	if	if	SCONJ
ejpam-6117	305	58	and	and	CCONJ
ejpam-6117	305	59	only	only	ADV
ejpam-6117	305	60	if	if	SCONJ
ejpam-6117	305	61	s	s	NOUN
ejpam-6117	305	62	is	be	AUX
ejpam-6117	305	63	a	a	DET
ejpam-6117	305	64	weakly	weakly	ADV
ejpam-6117	305	65	connected	connect	VERB
ejpam-6117	305	66	in	in	ADP
ejpam-6117	305	67	g	g	PROPN
ejpam-6117	305	68	the	the	DET
ejpam-6117	305	69	next	next	ADJ
ejpam-6117	305	70	result	result	NOUN
ejpam-6117	305	71	characterizes	characterize	VERB
ejpam-6117	305	72	wcis	wcis	NOUN
ejpam-6117	305	73	in	in	ADP
ejpam-6117	305	74	g[h	g[h	PROPN
ejpam-6117	305	75	]	]	PUNCT
ejpam-6117	305	76	.	.	PUNCT
ejpam-6117	306	1	theorem	theorem	ADJ
ejpam-6117	306	2	8	8	NUM
ejpam-6117	306	3	.	.	PUNCT
ejpam-6117	307	1	let	let	VERB
ejpam-6117	307	2	g	g	NOUN
ejpam-6117	307	3	and	and	CCONJ
ejpam-6117	307	4	h	h	NOUN
ejpam-6117	307	5	be	be	AUX
ejpam-6117	307	6	connected	connect	VERB
ejpam-6117	307	7	non	non	ADJ
ejpam-6117	307	8	-	-	ADJ
ejpam-6117	307	9	trivial	trivial	ADJ
ejpam-6117	307	10	graphs	graph	NOUN
ejpam-6117	307	11	and	and	CCONJ
ejpam-6117	307	12	let	let	VERB
ejpam-6117	307	13	c	c	NOUN
ejpam-6117	307	14	=	=	SYM
ejpam-6117	307	15	∪x∈s({x}×tx	∪x∈s({x}×tx	PROPN
ejpam-6117	307	16	)	)	PUNCT
ejpam-6117	307	17	⊆	⊆	NUM
ejpam-6117	307	18	v	v	NOUN
ejpam-6117	307	19	(	(	PUNCT
ejpam-6117	307	20	g[h	g[h	PROPN
ejpam-6117	307	21	]	]	PUNCT
ejpam-6117	307	22	)	)	PUNCT
ejpam-6117	307	23	,	,	PUNCT
ejpam-6117	307	24	where	where	SCONJ
ejpam-6117	307	25	s	s	VERB
ejpam-6117	307	26	⊆	⊆	NUM
ejpam-6117	307	27	v	v	NOUN
ejpam-6117	307	28	(	(	PUNCT
ejpam-6117	307	29	g	g	NOUN
ejpam-6117	307	30	)	)	PUNCT
ejpam-6117	307	31	and	and	CCONJ
ejpam-6117	307	32	tx	tx	VERB
ejpam-6117	307	33	⊆	⊆	NUM
ejpam-6117	307	34	v	v	NOUN
ejpam-6117	307	35	(	(	PUNCT
ejpam-6117	307	36	h	h	NOUN
ejpam-6117	307	37	)	)	PUNCT
ejpam-6117	307	38	for	for	ADP
ejpam-6117	307	39	all	all	DET
ejpam-6117	307	40	x	x	SYM
ejpam-6117	307	41	∈	∈	PROPN
ejpam-6117	307	42	s.	s.	PROPN
ejpam-6117	308	1	then	then	ADV
ejpam-6117	308	2	c	c	PROPN
ejpam-6117	308	3	is	be	AUX
ejpam-6117	308	4	wcis	wcis	NOUN
ejpam-6117	308	5	in	in	ADP
ejpam-6117	308	6	g[h	g[h	PROPN
ejpam-6117	308	7	]	]	PUNCT
ejpam-6117	308	8	if	if	SCONJ
ejpam-6117	308	9	and	and	CCONJ
ejpam-6117	308	10	only	only	ADV
ejpam-6117	308	11	if	if	SCONJ
ejpam-6117	308	12	s	s	NOUN
ejpam-6117	308	13	is	be	AUX
ejpam-6117	308	14	a	a	DET
ejpam-6117	308	15	wcis	wcis	NOUN
ejpam-6117	308	16	of	of	ADP
ejpam-6117	308	17	g	g	PROPN
ejpam-6117	308	18	and	and	CCONJ
ejpam-6117	308	19	tx	tx	PROPN
ejpam-6117	308	20	is	be	AUX
ejpam-6117	308	21	an	an	DET
ejpam-6117	308	22	independent	independent	ADJ
ejpam-6117	308	23	set	set	NOUN
ejpam-6117	308	24	of	of	ADP
ejpam-6117	308	25	h	h	NOUN
ejpam-6117	308	26	for	for	ADP
ejpam-6117	308	27	each	each	DET
ejpam-6117	308	28	x	x	SYM
ejpam-6117	308	29	∈	∈	PROPN
ejpam-6117	308	30	s.	s.	PROPN
ejpam-6117	308	31	proof	proof	PROPN
ejpam-6117	308	32	.	.	PUNCT
ejpam-6117	309	1	suppose	suppose	VERB
ejpam-6117	309	2	c	c	NOUN
ejpam-6117	309	3	=	=	SYM
ejpam-6117	309	4	∪x∈s({x	∪x∈s({x	PROPN
ejpam-6117	309	5	}	}	PUNCT
ejpam-6117	309	6	×	×	NOUN
ejpam-6117	309	7	tx	tx	PROPN
ejpam-6117	309	8	)	)	PUNCT
ejpam-6117	309	9	⊆	⊆	NUM
ejpam-6117	309	10	v	v	NOUN
ejpam-6117	309	11	(	(	PUNCT
ejpam-6117	309	12	g[h	g[h	PROPN
ejpam-6117	309	13	]	]	PUNCT
ejpam-6117	309	14	)	)	PUNCT
ejpam-6117	309	15	,	,	PUNCT
ejpam-6117	309	16	where	where	SCONJ
ejpam-6117	309	17	s	s	VERB
ejpam-6117	309	18	⊆	⊆	NUM
ejpam-6117	309	19	v	v	NOUN
ejpam-6117	309	20	(	(	PUNCT
ejpam-6117	309	21	g	g	NOUN
ejpam-6117	309	22	)	)	PUNCT
ejpam-6117	309	23	and	and	CCONJ
ejpam-6117	309	24	tx	tx	VERB
ejpam-6117	309	25	⊆	⊆	NUM
ejpam-6117	309	26	v	v	NOUN
ejpam-6117	309	27	(	(	PUNCT
ejpam-6117	309	28	h	h	NOUN
ejpam-6117	309	29	)	)	PUNCT
ejpam-6117	309	30	for	for	ADP
ejpam-6117	309	31	all	all	DET
ejpam-6117	309	32	x	x	SYM
ejpam-6117	309	33	∈	∈	PROPN
ejpam-6117	309	34	s	s	NOUN
ejpam-6117	309	35	and	and	CCONJ
ejpam-6117	309	36	that	that	SCONJ
ejpam-6117	309	37	c	c	PROPN
ejpam-6117	309	38	is	be	AUX
ejpam-6117	309	39	a	a	DET
ejpam-6117	309	40	wcis	wcis	NOUN
ejpam-6117	309	41	in	in	ADP
ejpam-6117	309	42	g[h	g[h	PROPN
ejpam-6117	309	43	]	]	PUNCT
ejpam-6117	309	44	.	.	PUNCT
ejpam-6117	310	1	by	by	ADP
ejpam-6117	310	2	theorem	theorem	NOUN
ejpam-6117	310	3	7	7	NUM
ejpam-6117	310	4	,	,	PUNCT
ejpam-6117	310	5	s	s	VERB
ejpam-6117	310	6	is	be	AUX
ejpam-6117	310	7	weakly	weakly	ADV
ejpam-6117	310	8	connected	connected	ADJ
ejpam-6117	310	9	in	in	ADP
ejpam-6117	310	10	g.	g.	PROPN
ejpam-6117	310	11	next	next	ADV
ejpam-6117	310	12	,	,	PUNCT
ejpam-6117	310	13	let	let	VERB
ejpam-6117	310	14	x	x	PRON
ejpam-6117	310	15	,	,	PUNCT
ejpam-6117	310	16	y	y	PROPN
ejpam-6117	310	17	∈	∈	PROPN
ejpam-6117	310	18	s	s	VERB
ejpam-6117	310	19	such	such	ADJ
ejpam-6117	310	20	that	that	SCONJ
ejpam-6117	310	21	x	x	SYM
ejpam-6117	310	22	̸=	̸=	PROPN
ejpam-6117	310	23	y.	y.	NOUN
ejpam-6117	310	24	pick	pick	VERB
ejpam-6117	310	25	a	a	DET
ejpam-6117	310	26	∈	∈	PROPN
ejpam-6117	310	27	tx	tx	NOUN
ejpam-6117	310	28	and	and	CCONJ
ejpam-6117	310	29	b	b	X
ejpam-6117	310	30	∈	∈	PROPN
ejpam-6117	311	1	ty	ty	INTJ
ejpam-6117	311	2	.	.	PUNCT
ejpam-6117	312	1	then	then	ADV
ejpam-6117	312	2	(	(	PUNCT
ejpam-6117	312	3	x	x	X
ejpam-6117	312	4	,	,	PUNCT
ejpam-6117	312	5	a	a	PRON
ejpam-6117	312	6	)	)	PUNCT
ejpam-6117	312	7	,	,	PUNCT
ejpam-6117	312	8	(	(	PUNCT
ejpam-6117	312	9	y	y	PROPN
ejpam-6117	312	10	,	,	PUNCT
ejpam-6117	312	11	b	b	NOUN
ejpam-6117	312	12	)	)	PUNCT
ejpam-6117	312	13	∈	∈	PROPN
ejpam-6117	312	14	c	c	NOUN
ejpam-6117	312	15	and	and	CCONJ
ejpam-6117	312	16	(	(	PUNCT
ejpam-6117	312	17	x	x	NOUN
ejpam-6117	312	18	,	,	PUNCT
ejpam-6117	312	19	a	a	PRON
ejpam-6117	312	20	)	)	PUNCT
ejpam-6117	312	21	̸=	̸=	PROPN
ejpam-6117	312	22	(	(	PUNCT
ejpam-6117	312	23	y	y	PROPN
ejpam-6117	312	24	,	,	PUNCT
ejpam-6117	312	25	b	b	NOUN
ejpam-6117	312	26	)	)	PUNCT
ejpam-6117	312	27	.	.	PUNCT
ejpam-6117	313	1	since	since	SCONJ
ejpam-6117	313	2	c	c	PROPN
ejpam-6117	313	3	is	be	AUX
ejpam-6117	313	4	independent	independent	ADJ
ejpam-6117	313	5	,	,	PUNCT
ejpam-6117	313	6	(	(	PUNCT
ejpam-6117	313	7	x	x	X
ejpam-6117	313	8	,	,	PUNCT
ejpam-6117	313	9	a)(y	a)(y	PROPN
ejpam-6117	313	10	,	,	PUNCT
ejpam-6117	313	11	b	b	NOUN
ejpam-6117	313	12	)	)	PUNCT
ejpam-6117	313	13	/∈	/∈	PUNCT
ejpam-6117	313	14	e(g[h	e(g[h	NOUN
ejpam-6117	313	15	]	]	PUNCT
ejpam-6117	313	16	)	)	PUNCT
ejpam-6117	313	17	.	.	PUNCT
ejpam-6117	314	1	this	this	PRON
ejpam-6117	314	2	implies	imply	VERB
ejpam-6117	314	3	that	that	SCONJ
ejpam-6117	314	4	xy	xy	PROPN
ejpam-6117	314	5	/∈	/∈	PUNCT
ejpam-6117	314	6	e(g	e(g	PROPN
ejpam-6117	314	7	)	)	PUNCT
ejpam-6117	314	8	.	.	PUNCT
ejpam-6117	315	1	thus	thus	ADV
ejpam-6117	315	2	,	,	PUNCT
ejpam-6117	315	3	s	s	VERB
ejpam-6117	315	4	is	be	AUX
ejpam-6117	315	5	independent	independent	ADJ
ejpam-6117	315	6	in	in	ADP
ejpam-6117	315	7	g.	g.	PROPN
ejpam-6117	315	8	hence	hence	ADV
ejpam-6117	315	9	,	,	PUNCT
ejpam-6117	315	10	s	s	VERB
ejpam-6117	315	11	a	a	DET
ejpam-6117	315	12	wcis	wcis	NOUN
ejpam-6117	315	13	in	in	ADP
ejpam-6117	315	14	g.	g.	PROPN
ejpam-6117	315	15	now	now	ADV
ejpam-6117	315	16	,	,	PUNCT
ejpam-6117	315	17	let	let	VERB
ejpam-6117	315	18	x	x	PUNCT
ejpam-6117	315	19	∈	∈	NOUN
ejpam-6117	315	20	s	s	PART
ejpam-6117	315	21	and	and	CCONJ
ejpam-6117	315	22	let	let	VERB
ejpam-6117	315	23	c	c	NOUN
ejpam-6117	315	24	,	,	PUNCT
ejpam-6117	315	25	d	d	PROPN
ejpam-6117	315	26	∈	∈	PROPN
ejpam-6117	315	27	tx	tx	VERB
ejpam-6117	315	28	such	such	ADJ
ejpam-6117	315	29	that	that	SCONJ
ejpam-6117	315	30	c	c	PROPN
ejpam-6117	315	31	̸=	̸=	PROPN
ejpam-6117	315	32	d.	d.	PROPN
ejpam-6117	315	33	then	then	ADV
ejpam-6117	315	34	(	(	PUNCT
ejpam-6117	315	35	x	x	X
ejpam-6117	315	36	,	,	PUNCT
ejpam-6117	315	37	c	c	NOUN
ejpam-6117	315	38	)	)	PUNCT
ejpam-6117	315	39	,	,	PUNCT
ejpam-6117	315	40	(	(	PUNCT
ejpam-6117	315	41	x	x	X
ejpam-6117	315	42	,	,	PUNCT
ejpam-6117	315	43	d	d	NOUN
ejpam-6117	315	44	)	)	PUNCT
ejpam-6117	315	45	∈	∈	PROPN
ejpam-6117	315	46	c	c	NOUN
ejpam-6117	315	47	and	and	CCONJ
ejpam-6117	315	48	(	(	PUNCT
ejpam-6117	315	49	x	x	NOUN
ejpam-6117	315	50	,	,	PUNCT
ejpam-6117	315	51	c	c	NOUN
ejpam-6117	315	52	)	)	PUNCT
ejpam-6117	315	53	̸=	̸=	PROPN
ejpam-6117	315	54	(	(	PUNCT
ejpam-6117	315	55	x	x	X
ejpam-6117	315	56	,	,	PUNCT
ejpam-6117	315	57	d	d	NOUN
ejpam-6117	315	58	)	)	PUNCT
ejpam-6117	315	59	.	.	PUNCT
ejpam-6117	316	1	since	since	SCONJ
ejpam-6117	316	2	c	c	PROPN
ejpam-6117	316	3	is	be	AUX
ejpam-6117	316	4	an	an	DET
ejpam-6117	316	5	independent	independent	ADJ
ejpam-6117	316	6	set	set	NOUN
ejpam-6117	316	7	of	of	ADP
ejpam-6117	316	8	g[h	g[h	PROPN
ejpam-6117	316	9	]	]	PUNCT
ejpam-6117	316	10	,	,	PUNCT
ejpam-6117	316	11	we	we	PRON
ejpam-6117	316	12	have	have	VERB
ejpam-6117	316	13	(	(	PUNCT
ejpam-6117	316	14	x	x	NOUN
ejpam-6117	316	15	,	,	PUNCT
ejpam-6117	316	16	c)(x	c)(x	PROPN
ejpam-6117	316	17	,	,	PUNCT
ejpam-6117	316	18	d	d	NOUN
ejpam-6117	316	19	)	)	PUNCT
ejpam-6117	316	20	/∈	/∈	PUNCT
ejpam-6117	316	21	e(g[h	e(g[h	NOUN
ejpam-6117	316	22	]	]	PUNCT
ejpam-6117	316	23	)	)	PUNCT
ejpam-6117	316	24	.	.	PUNCT
ejpam-6117	317	1	this	this	PRON
ejpam-6117	317	2	implies	imply	VERB
ejpam-6117	317	3	that	that	DET
ejpam-6117	317	4	cd	cd	PROPN
ejpam-6117	317	5	/∈	/∈	PUNCT
ejpam-6117	317	6	e(h	e(h	PROPN
ejpam-6117	317	7	)	)	PUNCT
ejpam-6117	317	8	.	.	PUNCT
ejpam-6117	318	1	thus	thus	ADV
ejpam-6117	318	2	,	,	PUNCT
ejpam-6117	318	3	tx	tx	PROPN
ejpam-6117	318	4	is	be	AUX
ejpam-6117	318	5	an	an	DET
ejpam-6117	318	6	independent	independent	ADJ
ejpam-6117	318	7	set	set	NOUN
ejpam-6117	318	8	of	of	ADP
ejpam-6117	318	9	h.	h.	NOUN
ejpam-6117	318	10	conversely	conversely	ADV
ejpam-6117	318	11	,	,	PUNCT
ejpam-6117	318	12	assume	assume	VERB
ejpam-6117	318	13	that	that	SCONJ
ejpam-6117	318	14	s	s	VERB
ejpam-6117	318	15	is	be	AUX
ejpam-6117	318	16	wcis	wcis	NOUN
ejpam-6117	318	17	in	in	ADP
ejpam-6117	318	18	g	g	PROPN
ejpam-6117	318	19	and	and	CCONJ
ejpam-6117	318	20	tx	tx	PROPN
ejpam-6117	318	21	is	be	AUX
ejpam-6117	318	22	independent	independent	ADJ
ejpam-6117	318	23	in	in	ADP
ejpam-6117	318	24	h	h	NOUN
ejpam-6117	318	25	for	for	ADP
ejpam-6117	318	26	all	all	DET
ejpam-6117	318	27	x	x	SYM
ejpam-6117	318	28	∈	∈	PROPN
ejpam-6117	318	29	s.	s.	PROPN
ejpam-6117	318	30	by	by	ADP
ejpam-6117	318	31	theorem	theorem	NOUN
ejpam-6117	318	32	7	7	NUM
ejpam-6117	318	33	,	,	PUNCT
ejpam-6117	318	34	c	c	PROPN
ejpam-6117	318	35	is	be	AUX
ejpam-6117	318	36	a	a	DET
ejpam-6117	318	37	weakly	weakly	ADV
ejpam-6117	318	38	connected	connected	ADJ
ejpam-6117	318	39	set	set	VERB
ejpam-6117	318	40	in	in	ADP
ejpam-6117	318	41	g[h	g[h	PROPN
ejpam-6117	318	42	]	]	PUNCT
ejpam-6117	318	43	.	.	PUNCT
ejpam-6117	319	1	let	let	VERB
ejpam-6117	319	2	(	(	PUNCT
ejpam-6117	319	3	x	x	X
ejpam-6117	319	4	,	,	PUNCT
ejpam-6117	319	5	a	a	PRON
ejpam-6117	319	6	)	)	PUNCT
ejpam-6117	319	7	,	,	PUNCT
ejpam-6117	319	8	(	(	PUNCT
ejpam-6117	319	9	y	y	PROPN
ejpam-6117	319	10	,	,	PUNCT
ejpam-6117	319	11	b	b	NOUN
ejpam-6117	319	12	)	)	PUNCT
ejpam-6117	319	13	∈	∈	PROPN
ejpam-6117	319	14	c	c	NOUN
ejpam-6117	319	15	such	such	ADJ
ejpam-6117	319	16	that	that	PRON
ejpam-6117	319	17	(	(	PUNCT
ejpam-6117	319	18	x	x	NOUN
ejpam-6117	319	19	,	,	PUNCT
ejpam-6117	319	20	a	a	PRON
ejpam-6117	319	21	)	)	PUNCT
ejpam-6117	319	22	̸=	̸=	PROPN
ejpam-6117	319	23	(	(	PUNCT
ejpam-6117	319	24	y	y	PROPN
ejpam-6117	319	25	,	,	PUNCT
ejpam-6117	319	26	b	b	NOUN
ejpam-6117	319	27	)	)	PUNCT
ejpam-6117	319	28	.	.	PUNCT
ejpam-6117	320	1	suppose	suppose	VERB
ejpam-6117	320	2	x	x	PUNCT
ejpam-6117	320	3	̸=	̸=	PROPN
ejpam-6117	320	4	y.	y.	NOUN
ejpam-6117	320	5	since	since	SCONJ
ejpam-6117	320	6	s	s	PROPN
ejpam-6117	320	7	is	be	AUX
ejpam-6117	320	8	an	an	DET
ejpam-6117	320	9	independent	independent	ADJ
ejpam-6117	320	10	set	set	NOUN
ejpam-6117	320	11	in	in	ADP
ejpam-6117	320	12	g	g	PROPN
ejpam-6117	320	13	,	,	PUNCT
ejpam-6117	320	14	xy	xy	PROPN
ejpam-6117	320	15	/∈	/∈	PUNCT
ejpam-6117	320	16	e(g	e(g	PROPN
ejpam-6117	320	17	)	)	PUNCT
ejpam-6117	320	18	.	.	PUNCT
ejpam-6117	321	1	thus	thus	ADV
ejpam-6117	321	2	,	,	PUNCT
ejpam-6117	321	3	(	(	PUNCT
ejpam-6117	321	4	x	x	X
ejpam-6117	321	5	,	,	PUNCT
ejpam-6117	321	6	a)(y	a)(y	PROPN
ejpam-6117	321	7	,	,	PUNCT
ejpam-6117	321	8	b	b	NOUN
ejpam-6117	321	9	)	)	PUNCT
ejpam-6117	321	10	/∈	/∈	PUNCT
ejpam-6117	321	11	e(g[h	e(g[h	NOUN
ejpam-6117	321	12	]	]	PUNCT
ejpam-6117	321	13	)	)	PUNCT
ejpam-6117	321	14	.	.	PUNCT
ejpam-6117	322	1	now	now	ADV
ejpam-6117	322	2	,	,	PUNCT
ejpam-6117	322	3	assume	assume	VERB
ejpam-6117	322	4	x	x	X
ejpam-6117	322	5	=	=	PUNCT
ejpam-6117	322	6	y.	y.	NOUN
ejpam-6117	322	7	then	then	ADV
ejpam-6117	322	8	,	,	PUNCT
ejpam-6117	322	9	a	a	DET
ejpam-6117	322	10	,	,	PUNCT
ejpam-6117	322	11	b	b	PROPN
ejpam-6117	322	12	∈	∈	PROPN
ejpam-6117	322	13	tx	tx	PROPN
ejpam-6117	322	14	and	and	CCONJ
ejpam-6117	322	15	a	a	DET
ejpam-6117	322	16	̸=	̸=	PROPN
ejpam-6117	322	17	b.	b.	NOUN
ejpam-6117	322	18	since	since	SCONJ
ejpam-6117	322	19	tx	tx	PROPN
ejpam-6117	322	20	is	be	AUX
ejpam-6117	322	21	an	an	DET
ejpam-6117	322	22	independent	independent	ADJ
ejpam-6117	322	23	set	set	NOUN
ejpam-6117	322	24	of	of	ADP
ejpam-6117	322	25	h	h	NOUN
ejpam-6117	322	26	,	,	PUNCT
ejpam-6117	322	27	ab	ab	PROPN
ejpam-6117	322	28	/∈	/∈	PUNCT
ejpam-6117	322	29	e(h	e(h	PROPN
ejpam-6117	322	30	)	)	PUNCT
ejpam-6117	322	31	.	.	PUNCT
ejpam-6117	323	1	hence	hence	ADV
ejpam-6117	323	2	,	,	PUNCT
ejpam-6117	323	3	(	(	PUNCT
ejpam-6117	323	4	x	x	X
ejpam-6117	323	5	,	,	PUNCT
ejpam-6117	323	6	a)(y	a)(y	PROPN
ejpam-6117	323	7	,	,	PUNCT
ejpam-6117	323	8	b	b	NOUN
ejpam-6117	323	9	)	)	PUNCT
ejpam-6117	323	10	/∈	/∈	PUNCT
ejpam-6117	323	11	e(g[h	e(g[h	NOUN
ejpam-6117	323	12	]	]	PUNCT
ejpam-6117	323	13	)	)	PUNCT
ejpam-6117	323	14	.	.	PUNCT
ejpam-6117	324	1	thus	thus	ADV
ejpam-6117	324	2	,	,	PUNCT
ejpam-6117	324	3	c	c	PROPN
ejpam-6117	324	4	is	be	AUX
ejpam-6117	324	5	an	an	DET
ejpam-6117	324	6	independent	independent	ADJ
ejpam-6117	324	7	set	set	NOUN
ejpam-6117	324	8	of	of	ADP
ejpam-6117	324	9	g[h	g[h	NOUN
ejpam-6117	324	10	]	]	PUNCT
ejpam-6117	324	11	.	.	PUNCT
ejpam-6117	325	1	therefore	therefore	ADV
ejpam-6117	325	2	,	,	PUNCT
ejpam-6117	325	3	c	c	PROPN
ejpam-6117	325	4	is	be	AUX
ejpam-6117	325	5	a	a	DET
ejpam-6117	325	6	weakly	weakly	ADV
ejpam-6117	325	7	connected	connected	ADJ
ejpam-6117	325	8	independent	independent	ADJ
ejpam-6117	325	9	set	set	NOUN
ejpam-6117	325	10	in	in	ADP
ejpam-6117	325	11	g[h	g[h	PROPN
ejpam-6117	325	12	]	]	PUNCT
ejpam-6117	325	13	.	.	PUNCT
ejpam-6117	326	1	corollary	corollary	ADJ
ejpam-6117	326	2	5	5	NUM
ejpam-6117	326	3	.	.	PUNCT
ejpam-6117	327	1	let	let	VERB
ejpam-6117	327	2	g	g	NOUN
ejpam-6117	327	3	and	and	CCONJ
ejpam-6117	327	4	h	h	NOUN
ejpam-6117	327	5	be	be	AUX
ejpam-6117	327	6	connected	connect	VERB
ejpam-6117	327	7	non	non	ADJ
ejpam-6117	327	8	-	-	ADJ
ejpam-6117	327	9	trivial	trivial	ADJ
ejpam-6117	327	10	graphs	graph	NOUN
ejpam-6117	327	11	.	.	PUNCT
ejpam-6117	328	1	then	then	ADV
ejpam-6117	328	2	αw(g[h	αw(g[h	VERB
ejpam-6117	328	3	]	]	X
ejpam-6117	328	4	)	)	PUNCT
ejpam-6117	328	5	=	=	SYM
ejpam-6117	328	6	αw(g)α(h	αw(g)α(h	NOUN
ejpam-6117	328	7	)	)	PUNCT
ejpam-6117	328	8	.	.	PUNCT
ejpam-6117	329	1	proof	proof	NOUN
ejpam-6117	329	2	.	.	PUNCT
ejpam-6117	330	1	let	let	VERB
ejpam-6117	330	2	c	c	NOUN
ejpam-6117	330	3	=	=	SYM
ejpam-6117	330	4	∪x∈s({x	∪x∈s({x	ADJ
ejpam-6117	330	5	}	}	PUNCT
ejpam-6117	330	6	×	×	NOUN
ejpam-6117	330	7	tx	tx	PROPN
ejpam-6117	330	8	)	)	PUNCT
ejpam-6117	330	9	⊆	⊆	NUM
ejpam-6117	330	10	v	v	NOUN
ejpam-6117	330	11	(	(	PUNCT
ejpam-6117	330	12	g[h	g[h	PROPN
ejpam-6117	330	13	]	]	PUNCT
ejpam-6117	330	14	)	)	PUNCT
ejpam-6117	330	15	,	,	PUNCT
ejpam-6117	330	16	where	where	SCONJ
ejpam-6117	330	17	s	s	VERB
ejpam-6117	330	18	⊆	⊆	NUM
ejpam-6117	330	19	v	v	NOUN
ejpam-6117	330	20	(	(	PUNCT
ejpam-6117	330	21	g	g	NOUN
ejpam-6117	330	22	)	)	PUNCT
ejpam-6117	330	23	and	and	CCONJ
ejpam-6117	330	24	tx	tx	VERB
ejpam-6117	330	25	⊆	⊆	NUM
ejpam-6117	330	26	v	v	NOUN
ejpam-6117	330	27	(	(	PUNCT
ejpam-6117	330	28	h	h	NOUN
ejpam-6117	330	29	)	)	PUNCT
ejpam-6117	330	30	for	for	ADP
ejpam-6117	330	31	all	all	PRON
ejpam-6117	330	32	x	x	SYM
ejpam-6117	330	33	∈	∈	PROPN
ejpam-6117	330	34	s	s	AUX
ejpam-6117	330	35	,	,	PUNCT
ejpam-6117	330	36	be	be	AUX
ejpam-6117	330	37	an	an	DET
ejpam-6117	330	38	αw	αw	NOUN
ejpam-6117	330	39	-	-	ADJ
ejpam-6117	330	40	set	set	VERB
ejpam-6117	330	41	in	in	ADP
ejpam-6117	330	42	g[h	g[h	NOUN
ejpam-6117	330	43	]	]	PUNCT
ejpam-6117	330	44	.	.	PUNCT
ejpam-6117	331	1	by	by	ADP
ejpam-6117	331	2	theorem	theorem	NOUN
ejpam-6117	331	3	8	8	NUM
ejpam-6117	331	4	,	,	PUNCT
ejpam-6117	331	5	s	s	VERB
ejpam-6117	331	6	is	be	AUX
ejpam-6117	331	7	a	a	DET
ejpam-6117	331	8	wcis	wcis	NOUN
ejpam-6117	331	9	in	in	ADP
ejpam-6117	331	10	g	g	PROPN
ejpam-6117	331	11	and	and	CCONJ
ejpam-6117	331	12	tx	tx	PROPN
ejpam-6117	331	13	is	be	AUX
ejpam-6117	331	14	an	an	DET
ejpam-6117	331	15	independent	independent	ADJ
ejpam-6117	331	16	set	set	NOUN
ejpam-6117	331	17	in	in	ADP
ejpam-6117	331	18	h	h	NOUN
ejpam-6117	331	19	for	for	ADP
ejpam-6117	331	20	every	every	DET
ejpam-6117	331	21	x	x	PROPN
ejpam-6117	331	22	∈	∈	PROPN
ejpam-6117	331	23	s.	s.	PROPN
ejpam-6117	331	24	hence	hence	ADV
ejpam-6117	331	25	,	,	PUNCT
ejpam-6117	331	26	αw(g[h	αw(g[h	NOUN
ejpam-6117	331	27	]	]	X
ejpam-6117	331	28	)	)	PUNCT
ejpam-6117	332	1	=	=	SYM
ejpam-6117	332	2	|c|	|c|	PROPN
ejpam-6117	332	3	=	=	PUNCT
ejpam-6117	332	4	|σx∈s({x	|σx∈s({x	PRON
ejpam-6117	332	5	}	}	PUNCT
ejpam-6117	332	6	×	×	NOUN
ejpam-6117	332	7	tx)|	tx)|	NOUN
ejpam-6117	332	8	≤	≤	ADJ
ejpam-6117	332	9	αw(g)α(h	αw(g)α(h	NOUN
ejpam-6117	332	10	)	)	PUNCT
ejpam-6117	332	11	.	.	PUNCT
ejpam-6117	333	1	next	next	ADV
ejpam-6117	333	2	,	,	PUNCT
ejpam-6117	333	3	let	let	VERB
ejpam-6117	333	4	s	s	PRON
ejpam-6117	333	5	be	be	AUX
ejpam-6117	333	6	an	an	DET
ejpam-6117	333	7	αw	αw	NOUN
ejpam-6117	333	8	-	-	VERB
ejpam-6117	333	9	set	set	VERB
ejpam-6117	333	10	in	in	ADP
ejpam-6117	333	11	g	g	PROPN
ejpam-6117	333	12	and	and	CCONJ
ejpam-6117	333	13	a	a	DET
ejpam-6117	333	14	be	be	AUX
ejpam-6117	333	15	an	an	DET
ejpam-6117	333	16	α	α	NOUN
ejpam-6117	333	17	-	-	PUNCT
ejpam-6117	333	18	set	set	VERB
ejpam-6117	333	19	in	in	ADP
ejpam-6117	333	20	h.	h.	NOUN
ejpam-6117	333	21	for	for	ADP
ejpam-6117	333	22	each	each	DET
ejpam-6117	333	23	x	x	SYM
ejpam-6117	333	24	∈	∈	PROPN
ejpam-6117	333	25	s	s	NOUN
ejpam-6117	333	26	,	,	PUNCT
ejpam-6117	333	27	let	let	VERB
ejpam-6117	333	28	tx	tx	VERB
ejpam-6117	333	29	=	=	VERB
ejpam-6117	333	30	a.	a.	NOUN
ejpam-6117	333	31	by	by	ADP
ejpam-6117	333	32	theorem	theorem	NOUN
ejpam-6117	333	33	8	8	NUM
ejpam-6117	333	34	,	,	PUNCT
ejpam-6117	333	35	c	c	NOUN
ejpam-6117	333	36	=	=	SYM
ejpam-6117	333	37	∪x∈s({x	∪x∈s({x	ADJ
ejpam-6117	333	38	}	}	PUNCT
ejpam-6117	333	39	×	×	NOUN
ejpam-6117	333	40	tx	tx	PROPN
ejpam-6117	333	41	)	)	PUNCT
ejpam-6117	333	42	is	be	AUX
ejpam-6117	333	43	a	a	DET
ejpam-6117	333	44	wcis	wcis	NOUN
ejpam-6117	333	45	in	in	ADP
ejpam-6117	333	46	g[h	g[h	PROPN
ejpam-6117	333	47	]	]	PUNCT
ejpam-6117	333	48	.	.	PUNCT
ejpam-6117	334	1	consequently	consequently	ADV
ejpam-6117	334	2	,	,	PUNCT
ejpam-6117	334	3	αw(g[h	αw(g[h	NOUN
ejpam-6117	334	4	]	]	PUNCT
ejpam-6117	334	5	)	)	PUNCT
ejpam-6117	334	6	≥	≥	NOUN
ejpam-6117	334	7	|c|	|c|	PROPN
ejpam-6117	334	8	=	=	SYM
ejpam-6117	334	9	|σx∈s({x	|σx∈s({x	PRON
ejpam-6117	334	10	}	}	PUNCT
ejpam-6117	334	11	×	×	ADJ
ejpam-6117	334	12	tx)|	tx)|	NOUN
ejpam-6117	334	13	=	=	SYM
ejpam-6117	334	14	αw(g)α(h	αw(g)α(h	NOUN
ejpam-6117	334	15	)	)	PUNCT
ejpam-6117	334	16	.	.	PUNCT
ejpam-6117	335	1	therefore	therefore	ADV
ejpam-6117	335	2	,	,	PUNCT
ejpam-6117	335	3	αw(g[h	αw(g[h	NOUN
ejpam-6117	335	4	]	]	X
ejpam-6117	335	5	)	)	PUNCT
ejpam-6117	335	6	=	=	SYM
ejpam-6117	335	7	αw(g)α(h	αw(g)α(h	NOUN
ejpam-6117	335	8	)	)	PUNCT
ejpam-6117	335	9	.	.	PUNCT
ejpam-6117	336	1	5	5	X
ejpam-6117	336	2	.	.	X
ejpam-6117	336	3	conclusion	conclusion	NOUN
ejpam-6117	336	4	the	the	DET
ejpam-6117	336	5	concept	concept	NOUN
ejpam-6117	336	6	of	of	ADP
ejpam-6117	336	7	weakly	weakly	ADJ
ejpam-6117	336	8	connected	connected	ADJ
ejpam-6117	336	9	independent	independent	ADJ
ejpam-6117	336	10	set	set	NOUN
ejpam-6117	336	11	as	as	ADV
ejpam-6117	336	12	well	well	ADV
ejpam-6117	336	13	as	as	ADP
ejpam-6117	336	14	the	the	DET
ejpam-6117	336	15	parameter	parameter	NOUN
ejpam-6117	336	16	weakly	weakly	ADV
ejpam-6117	336	17	connected	connected	ADJ
ejpam-6117	336	18	independence	independence	NOUN
ejpam-6117	336	19	number	number	NOUN
ejpam-6117	336	20	were	be	AUX
ejpam-6117	336	21	introduced	introduce	VERB
ejpam-6117	336	22	and	and	CCONJ
ejpam-6117	336	23	initially	initially	ADV
ejpam-6117	336	24	investigated	investigate	VERB
ejpam-6117	336	25	in	in	ADP
ejpam-6117	336	26	this	this	DET
ejpam-6117	336	27	study	study	NOUN
ejpam-6117	336	28	.	.	PUNCT
ejpam-6117	337	1	r.	r.	PROPN
ejpam-6117	337	2	merontos	merontos	PROPN
ejpam-6117	337	3	et	et	PROPN
ejpam-6117	337	4	al	al	PROPN
ejpam-6117	337	5	.	.	PUNCT
ejpam-6117	337	6	/	/	SYM
ejpam-6117	337	7	eur	eur	PROPN
ejpam-6117	337	8	.	.	PUNCT
ejpam-6117	338	1	j.	j.	PROPN
ejpam-6117	338	2	pure	pure	PROPN
ejpam-6117	338	3	appl	appl	PROPN
ejpam-6117	338	4	.	.	PROPN
ejpam-6117	338	5	math	math	PROPN
ejpam-6117	338	6	,	,	PUNCT
ejpam-6117	338	7	18	18	NUM
ejpam-6117	338	8	(	(	PUNCT
ejpam-6117	338	9	2	2	NUM
ejpam-6117	338	10	)	)	PUNCT
ejpam-6117	338	11	(	(	PUNCT
ejpam-6117	338	12	2025	2025	NUM
ejpam-6117	338	13	)	)	PUNCT
ejpam-6117	338	14	,	,	PUNCT
ejpam-6117	338	15	6117	6117	NUM
ejpam-6117	338	16	9	9	NUM
ejpam-6117	338	17	of	of	ADP
ejpam-6117	338	18	9	9	NUM
ejpam-6117	338	19	graphs	graph	NOUN
ejpam-6117	338	20	for	for	ADP
ejpam-6117	338	21	which	which	PRON
ejpam-6117	338	22	the	the	DET
ejpam-6117	338	23	weakly	weakly	ADJ
ejpam-6117	338	24	connected	connected	ADJ
ejpam-6117	338	25	independence	independence	NOUN
ejpam-6117	338	26	number	number	NOUN
ejpam-6117	338	27	and	and	CCONJ
ejpam-6117	338	28	independence	independence	NOUN
ejpam-6117	338	29	number	number	NOUN
ejpam-6117	338	30	are	be	AUX
ejpam-6117	338	31	equal	equal	ADJ
ejpam-6117	338	32	were	be	AUX
ejpam-6117	338	33	characterized	characterize	VERB
ejpam-6117	338	34	.	.	PUNCT
ejpam-6117	339	1	it	it	PRON
ejpam-6117	339	2	was	be	AUX
ejpam-6117	339	3	shown	show	VERB
ejpam-6117	339	4	that	that	SCONJ
ejpam-6117	339	5	the	the	DET
ejpam-6117	339	6	difference	difference	NOUN
ejpam-6117	339	7	between	between	ADP
ejpam-6117	339	8	these	these	DET
ejpam-6117	339	9	two	two	NUM
ejpam-6117	339	10	parameters	parameter	NOUN
ejpam-6117	339	11	can	can	AUX
ejpam-6117	339	12	be	be	AUX
ejpam-6117	339	13	made	make	VERB
ejpam-6117	339	14	arbitrarily	arbitrarily	ADV
ejpam-6117	339	15	large	large	ADJ
ejpam-6117	339	16	.	.	PUNCT
ejpam-6117	340	1	we	we	PRON
ejpam-6117	340	2	also	also	ADV
ejpam-6117	340	3	characterized	characterize	VERB
ejpam-6117	340	4	the	the	DET
ejpam-6117	340	5	weakly	weakly	ADV
ejpam-6117	340	6	connected	connected	ADJ
ejpam-6117	340	7	independent	independent	ADJ
ejpam-6117	340	8	sets	set	NOUN
ejpam-6117	340	9	in	in	ADP
ejpam-6117	340	10	the	the	DET
ejpam-6117	340	11	join	join	NOUN
ejpam-6117	340	12	,	,	PUNCT
ejpam-6117	340	13	corona	corona	PROPN
ejpam-6117	340	14	,	,	PUNCT
ejpam-6117	340	15	and	and	CCONJ
ejpam-6117	340	16	lexicographic	lexicographic	ADJ
ejpam-6117	340	17	product	product	NOUN
ejpam-6117	340	18	of	of	ADP
ejpam-6117	340	19	graphs	graph	NOUN
ejpam-6117	340	20	and	and	CCONJ
ejpam-6117	340	21	determined	determine	VERB
ejpam-6117	340	22	their	their	PRON
ejpam-6117	340	23	respective	respective	ADJ
ejpam-6117	340	24	weakly	weakly	ADJ
ejpam-6117	340	25	connected	connected	ADJ
ejpam-6117	340	26	independence	independence	NOUN
ejpam-6117	340	27	number	number	NOUN
ejpam-6117	340	28	.	.	PUNCT
ejpam-6117	341	1	the	the	DET
ejpam-6117	341	2	newly	newly	ADV
ejpam-6117	341	3	defined	define	VERB
ejpam-6117	341	4	parameter	parameter	NOUN
ejpam-6117	341	5	may	may	AUX
ejpam-6117	341	6	be	be	AUX
ejpam-6117	341	7	explored	explore	VERB
ejpam-6117	341	8	further	far	ADV
ejpam-6117	341	9	for	for	ADP
ejpam-6117	341	10	many	many	ADJ
ejpam-6117	341	11	other	other	ADJ
ejpam-6117	341	12	graphs	graph	NOUN
ejpam-6117	341	13	.	.	PUNCT
ejpam-6117	342	1	interested	interested	ADJ
ejpam-6117	342	2	readers	reader	NOUN
ejpam-6117	342	3	may	may	AUX
ejpam-6117	342	4	also	also	ADV
ejpam-6117	342	5	investigate	investigate	VERB
ejpam-6117	342	6	the	the	DET
ejpam-6117	342	7	complexity	complexity	NOUN
ejpam-6117	342	8	of	of	ADP
ejpam-6117	342	9	the	the	DET
ejpam-6117	342	10	wcis	wcis	NOUN
ejpam-6117	342	11	problem	problem	NOUN
ejpam-6117	342	12	.	.	PUNCT
ejpam-6117	343	1	acknowledgements	acknowledgement	NOUN
ejpam-6117	343	2	the	the	DET
ejpam-6117	343	3	authors	author	NOUN
ejpam-6117	343	4	would	would	AUX
ejpam-6117	343	5	like	like	VERB
ejpam-6117	343	6	to	to	PART
ejpam-6117	343	7	thank	thank	VERB
ejpam-6117	343	8	the	the	DET
ejpam-6117	343	9	referees	referee	NOUN
ejpam-6117	343	10	for	for	ADP
ejpam-6117	343	11	reading	read	VERB
ejpam-6117	343	12	the	the	DET
ejpam-6117	343	13	paper	paper	NOUN
ejpam-6117	343	14	and	and	CCONJ
ejpam-6117	343	15	giving	give	VERB
ejpam-6117	343	16	their	their	PRON
ejpam-6117	343	17	respective	respective	ADJ
ejpam-6117	343	18	comments	comment	NOUN
ejpam-6117	343	19	and	and	CCONJ
ejpam-6117	343	20	suggestions	suggestion	NOUN
ejpam-6117	343	21	.	.	PUNCT
ejpam-6117	344	1	the	the	DET
ejpam-6117	344	2	authors	author	NOUN
ejpam-6117	344	3	are	be	AUX
ejpam-6117	344	4	grateful	grateful	ADJ
ejpam-6117	344	5	to	to	ADP
ejpam-6117	344	6	department	department	NOUN
ejpam-6117	344	7	of	of	ADP
ejpam-6117	344	8	science	science	NOUN
ejpam-6117	344	9	and	and	CCONJ
ejpam-6117	344	10	technology	technology	NOUN
ejpam-6117	344	11	accelerated	accelerate	VERB
ejpam-6117	344	12	science	science	NOUN
ejpam-6117	344	13	and	and	CCONJ
ejpam-6117	344	14	technology	technology	NOUN
ejpam-6117	344	15	human	human	ADJ
ejpam-6117	344	16	resource	resource	NOUN
ejpam-6117	344	17	development	development	NOUN
ejpam-6117	344	18	program	program	NOUN
ejpam-6117	344	19	(	(	PUNCT
ejpam-6117	344	20	dost	dost	NOUN
ejpam-6117	344	21	-	-	PUNCT
ejpam-6117	344	22	asthrdp)-philippines	asthrdp)-philippine	NOUN
ejpam-6117	344	23	,	,	PUNCT
ejpam-6117	344	24	and	and	CCONJ
ejpam-6117	344	25	msu	msu	PROPN
ejpam-6117	344	26	-	-	PUNCT
ejpam-6117	344	27	iligan	iligan	PROPN
ejpam-6117	344	28	institute	institute	PROPN
ejpam-6117	344	29	of	of	ADP
ejpam-6117	344	30	technology	technology	NOUN
ejpam-6117	344	31	for	for	ADP
ejpam-6117	344	32	the	the	DET
ejpam-6117	344	33	financial	financial	ADJ
ejpam-6117	344	34	support	support	NOUN
ejpam-6117	344	35	they	they	PRON
ejpam-6117	344	36	have	have	AUX
ejpam-6117	344	37	extended	extend	VERB
ejpam-6117	344	38	for	for	ADP
ejpam-6117	344	39	the	the	DET
ejpam-6117	344	40	conduct	conduct	NOUN
ejpam-6117	344	41	of	of	ADP
ejpam-6117	344	42	this	this	DET
ejpam-6117	344	43	research	research	NOUN
ejpam-6117	344	44	.	.	PUNCT
ejpam-6117	345	1	references	reference	NOUN
ejpam-6117	345	2	[	[	X
ejpam-6117	345	3	1	1	X
ejpam-6117	345	4	]	]	PUNCT
ejpam-6117	345	5	j.	j.	PROPN
ejpam-6117	345	6	w.	w.	PROPN
ejpam-6117	345	7	grossman	grossman	PROPN
ejpam-6117	345	8	.	.	PUNCT
ejpam-6117	346	1	dominating	dominating	NOUN
ejpam-6117	346	2	sets	set	NOUN
ejpam-6117	346	3	whose	whose	DET
ejpam-6117	346	4	closed	close	VERB
ejpam-6117	346	5	stars	star	NOUN
ejpam-6117	346	6	form	form	VERB
ejpam-6117	346	7	spanning	span	VERB
ejpam-6117	346	8	trees	tree	NOUN
ejpam-6117	346	9	.	.	PUNCT
ejpam-6117	347	1	discrete	discrete	ADJ
ejpam-6117	347	2	mathematics	mathematic	NOUN
ejpam-6117	347	3	,	,	PUNCT
ejpam-6117	347	4	169:83–94	169:83–94	NUM
ejpam-6117	347	5	,	,	PUNCT
ejpam-6117	347	6	1997	1997	NUM
ejpam-6117	347	7	.	.	PUNCT
ejpam-6117	348	1	[	[	X
ejpam-6117	348	2	2	2	X
ejpam-6117	348	3	]	]	PUNCT
ejpam-6117	348	4	j.	j.	PROPN
ejpam-6117	348	5	e.	e.	PROPN
ejpam-6117	348	6	dunbar	dunbar	PROPN
ejpam-6117	348	7	,	,	PUNCT
ejpam-6117	348	8	j.	j.	PROPN
ejpam-6117	348	9	w.	w.	PROPN
ejpam-6117	348	10	grossman	grossman	PROPN
ejpam-6117	348	11	,	,	PUNCT
ejpam-6117	348	12	j.	j.	PROPN
ejpam-6117	348	13	h.	h.	PROPN
ejpam-6117	348	14	hattingh	hattingh	PROPN
ejpam-6117	348	15	,	,	PUNCT
ejpam-6117	348	16	s.	s.	PROPN
ejpam-6117	348	17	t.	t.	PROPN
ejpam-6117	348	18	hedetniemi	hedetniemi	PROPN
ejpam-6117	348	19	,	,	PUNCT
ejpam-6117	348	20	and	and	CCONJ
ejpam-6117	348	21	a.	a.	NOUN
ejpam-6117	348	22	a.	a.	PROPN
ejpam-6117	348	23	mcrae	mcrae	PROPN
ejpam-6117	348	24	.	.	PUNCT
ejpam-6117	349	1	on	on	ADP
ejpam-6117	349	2	weakly	weakly	ADJ
ejpam-6117	349	3	connected	connected	ADJ
ejpam-6117	349	4	domination	domination	NOUN
ejpam-6117	349	5	in	in	ADP
ejpam-6117	349	6	graphs	graph	NOUN
ejpam-6117	349	7	.	.	PUNCT
ejpam-6117	350	1	discrete	discrete	ADJ
ejpam-6117	350	2	mathematics	mathematic	NOUN
ejpam-6117	350	3	,	,	PUNCT
ejpam-6117	350	4	167/168:261–269	167/168:261–269	NUM
ejpam-6117	350	5	,	,	PUNCT
ejpam-6117	350	6	1997	1997	NUM
ejpam-6117	350	7	.	.	PUNCT
ejpam-6117	351	1	[	[	X
ejpam-6117	351	2	3	3	X
ejpam-6117	351	3	]	]	PUNCT
ejpam-6117	351	4	k.	k.	PROPN
ejpam-6117	351	5	m.	m.	PROPN
ejpam-6117	351	6	alzoubi	alzoubi	PROPN
ejpam-6117	351	7	,	,	PUNCT
ejpam-6117	351	8	p.	p.	PROPN
ejpam-6117	351	9	j.	j.	PROPN
ejpam-6117	351	10	wan	wan	PROPN
ejpam-6117	351	11	,	,	PUNCT
ejpam-6117	351	12	and	and	CCONJ
ejpam-6117	351	13	o.	o.	PROPN
ejpam-6117	351	14	frieder	frieder	NOUN
ejpam-6117	351	15	.	.	PUNCT
ejpam-6117	352	1	maximal	maximal	ADJ
ejpam-6117	352	2	independent	independent	ADJ
ejpam-6117	352	3	set	set	NOUN
ejpam-6117	352	4	,	,	PUNCT
ejpam-6117	352	5	weakly	weakly	ADV
ejpam-6117	352	6	connected	connected	ADJ
ejpam-6117	352	7	dominating	dominating	NOUN
ejpam-6117	352	8	set	set	NOUN
ejpam-6117	352	9	,	,	PUNCT
ejpam-6117	352	10	and	and	CCONJ
ejpam-6117	352	11	induced	induced	ADJ
ejpam-6117	352	12	spanners	spanner	NOUN
ejpam-6117	352	13	in	in	ADP
ejpam-6117	352	14	wireless	wireless	ADJ
ejpam-6117	352	15	ad	ad	X
ejpam-6117	352	16	hoc	hoc	X
ejpam-6117	352	17	networks	network	NOUN
ejpam-6117	352	18	.	.	PUNCT
ejpam-6117	353	1	international	international	ADJ
ejpam-6117	353	2	journal	journal	NOUN
ejpam-6117	353	3	of	of	ADP
ejpam-6117	353	4	foundations	foundation	NOUN
ejpam-6117	353	5	of	of	ADP
ejpam-6117	353	6	computer	computer	NOUN
ejpam-6117	353	7	science	science	NOUN
ejpam-6117	353	8	,	,	PUNCT
ejpam-6117	353	9	14(3):287–303	14(3):287–303	PROPN
ejpam-6117	353	10	,	,	PUNCT
ejpam-6117	353	11	2003	2003	NUM
ejpam-6117	353	12	.	.	PUNCT
ejpam-6117	354	1	[	[	X
ejpam-6117	354	2	4	4	NUM
ejpam-6117	354	3	]	]	X
ejpam-6117	354	4	f.	f.	PROPN
ejpam-6117	354	5	bendali	bendali	PROPN
ejpam-6117	354	6	,	,	PUNCT
ejpam-6117	354	7	j.	j.	PROPN
ejpam-6117	354	8	mailfert	mailfert	PROPN
ejpam-6117	354	9	,	,	PUNCT
ejpam-6117	354	10	and	and	CCONJ
ejpam-6117	354	11	d.	d.	PROPN
ejpam-6117	354	12	mameri	mameri	PROPN
ejpam-6117	354	13	.	.	PUNCT
ejpam-6117	355	1	on	on	ADP
ejpam-6117	355	2	minimum	minimum	ADJ
ejpam-6117	355	3	weakly	weakly	ADJ
ejpam-6117	355	4	connected	connected	ADJ
ejpam-6117	355	5	independent	independent	ADJ
ejpam-6117	355	6	sets	set	NOUN
ejpam-6117	355	7	for	for	ADP
ejpam-6117	355	8	wireless	wireless	ADJ
ejpam-6117	355	9	sensor	sensor	NOUN
ejpam-6117	355	10	networks	network	NOUN
ejpam-6117	355	11	:	:	PUNCT
ejpam-6117	355	12	properties	property	NOUN
ejpam-6117	355	13	and	and	CCONJ
ejpam-6117	355	14	enumeration	enumeration	NOUN
ejpam-6117	355	15	algorithm	algorithm	NOUN
ejpam-6117	355	16	.	.	PUNCT
ejpam-6117	356	1	rairooperations	rairooperation	NOUN
ejpam-6117	356	2	research	research	NOUN
ejpam-6117	356	3	,	,	PUNCT
ejpam-6117	356	4	49(2):313–334	49(2):313–334	PROPN
ejpam-6117	356	5	,	,	PUNCT
ejpam-6117	356	6	2015	2015	NUM
ejpam-6117	356	7	.	.	PUNCT
ejpam-6117	357	1	[	[	X
ejpam-6117	357	2	5	5	X
ejpam-6117	357	3	]	]	PUNCT
ejpam-6117	357	4	j.	j.	PROPN
ejpam-6117	357	5	hamja	hamja	PROPN
ejpam-6117	357	6	,	,	PUNCT
ejpam-6117	357	7	i.	i.	PROPN
ejpam-6117	357	8	aniversario	aniversario	PROPN
ejpam-6117	357	9	,	,	PUNCT
ejpam-6117	357	10	and	and	CCONJ
ejpam-6117	357	11	h.	h.	PROPN
ejpam-6117	357	12	rara	rara	PROPN
ejpam-6117	357	13	.	.	PUNCT
ejpam-6117	358	1	on	on	ADP
ejpam-6117	358	2	weakly	weakly	ADJ
ejpam-6117	358	3	connected	connect	VERB
ejpam-6117	358	4	closed	close	VERB
ejpam-6117	358	5	geodetic	geodetic	ADJ
ejpam-6117	358	6	domination	domination	NOUN
ejpam-6117	358	7	in	in	ADP
ejpam-6117	358	8	graphs	graph	NOUN
ejpam-6117	358	9	under	under	ADP
ejpam-6117	358	10	some	some	DET
ejpam-6117	358	11	binary	binary	ADJ
ejpam-6117	358	12	operations	operation	NOUN
ejpam-6117	358	13	.	.	PUNCT
ejpam-6117	359	1	european	european	ADJ
ejpam-6117	359	2	journal	journal	PROPN
ejpam-6117	359	3	of	of	ADP
ejpam-6117	359	4	pure	pure	ADJ
ejpam-6117	359	5	and	and	CCONJ
ejpam-6117	359	6	applied	applied	ADJ
ejpam-6117	359	7	mathematics	mathematic	NOUN
ejpam-6117	359	8	,	,	PUNCT
ejpam-6117	359	9	15(2):736–752	15(2):736–752	PROPN
ejpam-6117	359	10	,	,	PUNCT
ejpam-6117	359	11	2022	2022	NUM
ejpam-6117	359	12	.	.	PUNCT
ejpam-6117	360	1	[	[	X
ejpam-6117	360	2	6	6	NUM
ejpam-6117	360	3	]	]	PUNCT
ejpam-6117	360	4	m.	m.	NOUN
ejpam-6117	360	5	militante	militante	NOUN
ejpam-6117	360	6	and	and	CCONJ
ejpam-6117	360	7	r.	r.	PROPN
ejpam-6117	360	8	eballe	eballe	PROPN
ejpam-6117	360	9	.	.	PUNCT
ejpam-6117	361	1	weakly	weakly	ADV
ejpam-6117	361	2	connected	connected	ADJ
ejpam-6117	361	3	2	2	NUM
ejpam-6117	361	4	-	-	PUNCT
ejpam-6117	361	5	domination	domination	NOUN
ejpam-6117	361	6	in	in	ADP
ejpam-6117	361	7	the	the	DET
ejpam-6117	361	8	lexicographic	lexicographic	ADJ
ejpam-6117	361	9	product	product	NOUN
ejpam-6117	361	10	of	of	ADP
ejpam-6117	361	11	graphs	graph	NOUN
ejpam-6117	361	12	.	.	PUNCT
ejpam-6117	362	1	international	international	ADJ
ejpam-6117	362	2	journal	journal	PROPN
ejpam-6117	362	3	of	of	ADP
ejpam-6117	362	4	mathematical	mathematical	ADJ
ejpam-6117	362	5	analysis	analysis	NOUN
ejpam-6117	362	6	,	,	PUNCT
ejpam-6117	362	7	16:125–132	16:125–132	PROPN
ejpam-6117	362	8	,	,	PUNCT
ejpam-6117	362	9	2022	2022	NUM
ejpam-6117	362	10	.	.	PUNCT
ejpam-6117	363	1	[	[	X
ejpam-6117	363	2	7	7	X
ejpam-6117	363	3	]	]	X
ejpam-6117	363	4	f.	f.	PROPN
ejpam-6117	363	5	buckley	buckley	PROPN
ejpam-6117	363	6	and	and	CCONJ
ejpam-6117	363	7	f.	f.	PROPN
ejpam-6117	363	8	harary	harary	PROPN
ejpam-6117	363	9	.	.	PUNCT
ejpam-6117	364	1	distance	distance	NOUN
ejpam-6117	364	2	in	in	ADP
ejpam-6117	364	3	graphs	graph	NOUN
ejpam-6117	364	4	.	.	PUNCT
ejpam-6117	365	1	addison	addison	PROPN
ejpam-6117	365	2	-	-	PUNCT
ejpam-6117	365	3	wesley	wesley	PROPN
ejpam-6117	365	4	,	,	PUNCT
ejpam-6117	365	5	1990	1990	NUM
ejpam-6117	365	6	.	.	PUNCT
ejpam-6117	366	1	[	[	X
ejpam-6117	366	2	8	8	X
ejpam-6117	366	3	]	]	X
ejpam-6117	366	4	e.	e.	PROPN
ejpam-6117	366	5	sandueta	sandueta	PROPN
ejpam-6117	366	6	and	and	CCONJ
ejpam-6117	366	7	s.	s.	PROPN
ejpam-6117	366	8	r.	r.	PROPN
ejpam-6117	366	9	canoy	canoy	PROPN
ejpam-6117	366	10	jr	jr	PROPN
ejpam-6117	366	11	.	.	PROPN
ejpam-6117	366	12	weakly	weakly	ADV
ejpam-6117	366	13	connected	connected	ADJ
ejpam-6117	366	14	dominating	dominating	NOUN
ejpam-6117	366	15	sets	set	NOUN
ejpam-6117	366	16	in	in	ADP
ejpam-6117	366	17	the	the	DET
ejpam-6117	366	18	lexicographic	lexicographic	ADJ
ejpam-6117	366	19	product	product	NOUN
ejpam-6117	366	20	of	of	ADP
ejpam-6117	366	21	graphs	graph	NOUN
ejpam-6117	366	22	.	.	PUNCT
ejpam-6117	367	1	international	international	ADJ
ejpam-6117	367	2	journal	journal	PROPN
ejpam-6117	367	3	of	of	ADP
ejpam-6117	367	4	mathematical	mathematical	ADJ
ejpam-6117	367	5	analysis	analysis	NOUN
ejpam-6117	367	6	,	,	PUNCT
ejpam-6117	367	7	8(4):1973–1980	8(4):1973–1980	NUM
ejpam-6117	367	8	,	,	PUNCT
ejpam-6117	367	9	2014	2014	NUM
ejpam-6117	367	10	.	.	PUNCT