id	sid	tid	token	lemma	pos
ejpam-1506	1	1	3_yi.dvi	3_yi.dvi	NUM
ejpam-1506	1	2	european	european	ADJ
ejpam-1506	1	3	journal	journal	NOUN
ejpam-1506	1	4	of	of	ADP
ejpam-1506	1	5	pure	pure	ADJ
ejpam-1506	1	6	and	and	CCONJ
ejpam-1506	1	7	applied	apply	VERB
ejpam-1506	1	8	mathematics	mathematic	NOUN
ejpam-1506	1	9	vol	vol	NOUN
ejpam-1506	1	10	.	.	PROPN
ejpam-1506	1	11	5	5	NUM
ejpam-1506	1	12	,	,	PUNCT
ejpam-1506	1	13	no	no	INTJ
ejpam-1506	1	14	.	.	NOUN
ejpam-1506	1	15	3	3	NUM
ejpam-1506	1	16	,	,	PUNCT
ejpam-1506	1	17	2012	2012	NUM
ejpam-1506	1	18	,	,	PUNCT
ejpam-1506	1	19	302	302	NUM
ejpam-1506	1	20	-	-	SYM
ejpam-1506	1	21	316	316	NUM
ejpam-1506	1	22	issn	issn	PROPN
ejpam-1506	1	23	1307	1307	NUM
ejpam-1506	1	24	-	-	SYM
ejpam-1506	1	25	5543	5543	NUM
ejpam-1506	1	26	–	–	PUNCT
ejpam-1506	1	27	www.ejpam.com	www.ejpam.com	X
ejpam-1506	1	28	a	a	DET
ejpam-1506	1	29	comparison	comparison	NOUN
ejpam-1506	1	30	on	on	ADP
ejpam-1506	1	31	metric	metric	ADJ
ejpam-1506	1	32	dimension	dimension	NOUN
ejpam-1506	1	33	of	of	ADP
ejpam-1506	1	34	graphs	graph	NOUN
ejpam-1506	1	35	,	,	PUNCT
ejpam-1506	1	36	line	line	NOUN
ejpam-1506	1	37	graphs	graph	NOUN
ejpam-1506	1	38	,	,	PUNCT
ejpam-1506	1	39	and	and	CCONJ
ejpam-1506	1	40	line	line	NOUN
ejpam-1506	1	41	graphs	graph	NOUN
ejpam-1506	1	42	of	of	ADP
ejpam-1506	1	43	the	the	DET
ejpam-1506	1	44	subdivision	subdivision	NOUN
ejpam-1506	1	45	graphs	graph	VERB
ejpam-1506	1	46	douglas	douglas	PROPN
ejpam-1506	1	47	j.	j.	PROPN
ejpam-1506	1	48	klein	klein	PROPN
ejpam-1506	1	49	,	,	PUNCT
ejpam-1506	1	50	eunjeong	eunjeong	PROPN
ejpam-1506	1	51	yi∗	yi∗	NOUN
ejpam-1506	1	52	texas	texas	PROPN
ejpam-1506	1	53	a&m	a&m	PROPN
ejpam-1506	1	54	university	university	PROPN
ejpam-1506	1	55	at	at	ADP
ejpam-1506	1	56	galveston	galveston	PROPN
ejpam-1506	1	57	,	,	PUNCT
ejpam-1506	1	58	galveston	galveston	PROPN
ejpam-1506	1	59	,	,	PUNCT
ejpam-1506	1	60	tx	tx	ADP
ejpam-1506	1	61	77553	77553	NUM
ejpam-1506	1	62	,	,	PUNCT
ejpam-1506	1	63	usa	usa	PROPN
ejpam-1506	1	64	abstract	abstract	PROPN
ejpam-1506	1	65	.	.	PUNCT
ejpam-1506	2	1	the	the	DET
ejpam-1506	2	2	line	line	NOUN
ejpam-1506	2	3	graph	graph	NOUN
ejpam-1506	2	4	l(g	l(g	NOUN
ejpam-1506	2	5	)	)	PUNCT
ejpam-1506	2	6	of	of	ADP
ejpam-1506	2	7	a	a	DET
ejpam-1506	2	8	simple	simple	ADJ
ejpam-1506	2	9	graph	graph	NOUN
ejpam-1506	2	10	g	g	PROPN
ejpam-1506	2	11	is	be	AUX
ejpam-1506	2	12	the	the	DET
ejpam-1506	2	13	graph	graph	NOUN
ejpam-1506	2	14	whose	whose	DET
ejpam-1506	2	15	vertices	vertex	NOUN
ejpam-1506	2	16	are	be	AUX
ejpam-1506	2	17	in	in	ADP
ejpam-1506	2	18	one	one	NUM
ejpam-1506	2	19	-	-	PUNCT
ejpam-1506	2	20	to	to	ADP
ejpam-1506	2	21	-	-	PUNCT
ejpam-1506	2	22	one	one	NUM
ejpam-1506	2	23	correspondence	correspondence	NOUN
ejpam-1506	2	24	with	with	ADP
ejpam-1506	2	25	the	the	DET
ejpam-1506	2	26	edges	edge	NOUN
ejpam-1506	2	27	of	of	ADP
ejpam-1506	2	28	g	g	NOUN
ejpam-1506	2	29	;	;	PUNCT
ejpam-1506	2	30	two	two	NUM
ejpam-1506	2	31	vertices	vertex	NOUN
ejpam-1506	2	32	of	of	ADP
ejpam-1506	2	33	l(g	l(g	NOUN
ejpam-1506	2	34	)	)	PUNCT
ejpam-1506	2	35	are	be	AUX
ejpam-1506	2	36	adjacent	adjacent	ADJ
ejpam-1506	2	37	if	if	SCONJ
ejpam-1506	3	1	and	and	CCONJ
ejpam-1506	3	2	only	only	ADV
ejpam-1506	3	3	if	if	SCONJ
ejpam-1506	3	4	the	the	DET
ejpam-1506	3	5	corresponding	corresponding	ADJ
ejpam-1506	3	6	edges	edge	NOUN
ejpam-1506	3	7	of	of	ADP
ejpam-1506	3	8	g	g	NOUN
ejpam-1506	3	9	are	be	AUX
ejpam-1506	3	10	adjacent	adjacent	ADJ
ejpam-1506	3	11	.	.	PUNCT
ejpam-1506	4	1	if	if	SCONJ
ejpam-1506	4	2	s(g	s(g	PROPN
ejpam-1506	4	3	)	)	PUNCT
ejpam-1506	4	4	is	be	AUX
ejpam-1506	4	5	the	the	DET
ejpam-1506	4	6	subdivision	subdivision	NOUN
ejpam-1506	4	7	graph	graph	NOUN
ejpam-1506	4	8	of	of	ADP
ejpam-1506	4	9	a	a	DET
ejpam-1506	4	10	graph	graph	NOUN
ejpam-1506	4	11	g	g	NOUN
ejpam-1506	4	12	,	,	PUNCT
ejpam-1506	4	13	then	then	ADV
ejpam-1506	4	14	the	the	DET
ejpam-1506	4	15	para	para	ADJ
ejpam-1506	4	16	-	-	PUNCT
ejpam-1506	4	17	line	line	NOUN
ejpam-1506	4	18	graph	graph	NOUN
ejpam-1506	4	19	g∗	g∗	NOUN
ejpam-1506	4	20	of	of	ADP
ejpam-1506	4	21	g	g	PROPN
ejpam-1506	4	22	is	be	AUX
ejpam-1506	4	23	l(s(g	l(s(g	ADV
ejpam-1506	4	24	)	)	PUNCT
ejpam-1506	4	25	)	)	PUNCT
ejpam-1506	4	26	.	.	PUNCT
ejpam-1506	5	1	the	the	DET
ejpam-1506	5	2	metric	metric	ADJ
ejpam-1506	5	3	dimension	dimension	NOUN
ejpam-1506	5	4	dim(g	dim(g	PROPN
ejpam-1506	5	5	)	)	PUNCT
ejpam-1506	5	6	of	of	ADP
ejpam-1506	5	7	a	a	DET
ejpam-1506	5	8	graph	graph	NOUN
ejpam-1506	5	9	g	g	NOUN
ejpam-1506	5	10	is	be	AUX
ejpam-1506	5	11	the	the	DET
ejpam-1506	5	12	minimum	minimum	ADJ
ejpam-1506	5	13	cardinality	cardinality	NOUN
ejpam-1506	5	14	of	of	ADP
ejpam-1506	5	15	a	a	DET
ejpam-1506	5	16	set	set	NOUN
ejpam-1506	5	17	of	of	ADP
ejpam-1506	5	18	vertices	vertex	NOUN
ejpam-1506	5	19	such	such	ADJ
ejpam-1506	5	20	that	that	SCONJ
ejpam-1506	5	21	every	every	DET
ejpam-1506	5	22	vertex	vertex	NOUN
ejpam-1506	5	23	of	of	ADP
ejpam-1506	5	24	g	g	PROPN
ejpam-1506	5	25	is	be	AUX
ejpam-1506	5	26	uniquely	uniquely	ADV
ejpam-1506	5	27	determined	determine	VERB
ejpam-1506	5	28	by	by	ADP
ejpam-1506	5	29	its	its	PRON
ejpam-1506	5	30	vector	vector	NOUN
ejpam-1506	5	31	of	of	ADP
ejpam-1506	5	32	distances	distance	NOUN
ejpam-1506	5	33	to	to	ADP
ejpam-1506	5	34	the	the	DET
ejpam-1506	5	35	chosen	choose	VERB
ejpam-1506	5	36	vertices	vertex	NOUN
ejpam-1506	5	37	.	.	PUNCT
ejpam-1506	6	1	in	in	ADP
ejpam-1506	6	2	this	this	DET
ejpam-1506	6	3	paper	paper	NOUN
ejpam-1506	6	4	,	,	PUNCT
ejpam-1506	6	5	we	we	PRON
ejpam-1506	6	6	study	study	VERB
ejpam-1506	6	7	metric	metric	ADJ
ejpam-1506	6	8	dimension	dimension	NOUN
ejpam-1506	6	9	of	of	ADP
ejpam-1506	6	10	para	para	ADJ
ejpam-1506	6	11	-	-	PUNCT
ejpam-1506	6	12	line	line	NOUN
ejpam-1506	6	13	graphs	graph	NOUN
ejpam-1506	6	14	;	;	PUNCT
ejpam-1506	6	15	we	we	PRON
ejpam-1506	6	16	also	also	ADV
ejpam-1506	6	17	compare	compare	VERB
ejpam-1506	6	18	metric	metric	ADJ
ejpam-1506	6	19	dimension	dimension	NOUN
ejpam-1506	6	20	of	of	ADP
ejpam-1506	6	21	graphs	graph	NOUN
ejpam-1506	6	22	,	,	PUNCT
ejpam-1506	6	23	line	line	NOUN
ejpam-1506	6	24	graphs	graph	NOUN
ejpam-1506	6	25	,	,	PUNCT
ejpam-1506	6	26	and	and	CCONJ
ejpam-1506	6	27	para	para	NOUN
ejpam-1506	6	28	-	-	PUNCT
ejpam-1506	6	29	line	line	NOUN
ejpam-1506	6	30	graphs	graph	NOUN
ejpam-1506	6	31	.	.	PUNCT
ejpam-1506	7	1	first	first	ADV
ejpam-1506	7	2	,	,	PUNCT
ejpam-1506	7	3	we	we	PRON
ejpam-1506	7	4	show	show	VERB
ejpam-1506	7	5	that	that	SCONJ
ejpam-1506	7	6	⌈log2∆(g)⌉	⌈log2∆(g)⌉	PROPN
ejpam-1506	7	7	≤	≤	NUM
ejpam-1506	7	8	dim(g⋆	dim(g⋆	NOUN
ejpam-1506	7	9	)	)	PUNCT
ejpam-1506	7	10	≤	≤	NUM
ejpam-1506	7	11	n−	n−	NOUN
ejpam-1506	7	12	1	1	NUM
ejpam-1506	7	13	,	,	PUNCT
ejpam-1506	7	14	for	for	ADP
ejpam-1506	7	15	a	a	DET
ejpam-1506	7	16	simple	simple	ADJ
ejpam-1506	7	17	and	and	CCONJ
ejpam-1506	7	18	connected	connected	ADJ
ejpam-1506	7	19	graph	graph	NOUN
ejpam-1506	7	20	g	g	NOUN
ejpam-1506	7	21	of	of	ADP
ejpam-1506	7	22	order	order	NOUN
ejpam-1506	7	23	n	n	PRON
ejpam-1506	7	24	≥	≥	NOUN
ejpam-1506	7	25	2	2	NUM
ejpam-1506	7	26	with	with	ADP
ejpam-1506	7	27	the	the	DET
ejpam-1506	7	28	maximum	maximum	PROPN
ejpam-1506	7	29	degree	degree	NOUN
ejpam-1506	7	30	∆(g	∆(g	PROPN
ejpam-1506	7	31	)	)	PUNCT
ejpam-1506	7	32	,	,	PUNCT
ejpam-1506	7	33	where	where	SCONJ
ejpam-1506	7	34	both	both	DET
ejpam-1506	7	35	bounds	bound	NOUN
ejpam-1506	7	36	are	be	AUX
ejpam-1506	7	37	sharp	sharp	ADJ
ejpam-1506	7	38	.	.	PUNCT
ejpam-1506	8	1	second	second	ADJ
ejpam-1506	8	2	,	,	PUNCT
ejpam-1506	8	3	we	we	PRON
ejpam-1506	8	4	determine	determine	VERB
ejpam-1506	8	5	the	the	DET
ejpam-1506	8	6	metric	metric	ADJ
ejpam-1506	8	7	dimension	dimension	NOUN
ejpam-1506	8	8	of	of	ADP
ejpam-1506	8	9	para	para	NOUN
ejpam-1506	8	10	-	-	PUNCT
ejpam-1506	8	11	line	line	NOUN
ejpam-1506	8	12	graphs	graph	NOUN
ejpam-1506	8	13	for	for	ADP
ejpam-1506	8	14	some	some	DET
ejpam-1506	8	15	classes	class	NOUN
ejpam-1506	8	16	of	of	ADP
ejpam-1506	8	17	graphs	graph	NOUN
ejpam-1506	8	18	;	;	PUNCT
ejpam-1506	8	19	further	far	ADV
ejpam-1506	8	20	,	,	PUNCT
ejpam-1506	8	21	we	we	PRON
ejpam-1506	8	22	give	give	VERB
ejpam-1506	8	23	an	an	DET
ejpam-1506	8	24	example	example	NOUN
ejpam-1506	8	25	of	of	ADP
ejpam-1506	8	26	a	a	DET
ejpam-1506	8	27	graph	graph	NOUN
ejpam-1506	8	28	g	g	ADP
ejpam-1506	8	29	such	such	ADJ
ejpam-1506	8	30	that	that	DET
ejpam-1506	8	31	max{dim(g	max{dim(g	NOUN
ejpam-1506	8	32	)	)	PUNCT
ejpam-1506	8	33	,	,	PUNCT
ejpam-1506	8	34	dim(l(g	dim(l(g	NOUN
ejpam-1506	8	35	)	)	PUNCT
ejpam-1506	8	36	)	)	PUNCT
ejpam-1506	8	37	,	,	PUNCT
ejpam-1506	8	38	dim(g⋆	dim(g⋆	NOUN
ejpam-1506	8	39	)	)	PUNCT
ejpam-1506	8	40	}	}	PUNCT
ejpam-1506	8	41	equals	equal	VERB
ejpam-1506	8	42	dim(g	dim(g	PROPN
ejpam-1506	8	43	)	)	PUNCT
ejpam-1506	8	44	,	,	PUNCT
ejpam-1506	8	45	dim(l(g	dim(l(g	NOUN
ejpam-1506	8	46	)	)	PUNCT
ejpam-1506	8	47	)	)	PUNCT
ejpam-1506	8	48	,	,	PUNCT
ejpam-1506	8	49	and	and	CCONJ
ejpam-1506	8	50	dim(g⋆	dim(g⋆	NOUN
ejpam-1506	8	51	)	)	PUNCT
ejpam-1506	8	52	,	,	PUNCT
ejpam-1506	8	53	respectively	respectively	ADV
ejpam-1506	8	54	.	.	PUNCT
ejpam-1506	9	1	we	we	PRON
ejpam-1506	9	2	conclude	conclude	VERB
ejpam-1506	9	3	this	this	DET
ejpam-1506	9	4	paper	paper	NOUN
ejpam-1506	9	5	with	with	ADP
ejpam-1506	9	6	some	some	DET
ejpam-1506	9	7	open	open	ADJ
ejpam-1506	9	8	problems	problem	NOUN
ejpam-1506	9	9	.	.	PUNCT
ejpam-1506	10	1	2010	2010	NUM
ejpam-1506	10	2	mathematics	mathematic	NOUN
ejpam-1506	10	3	subject	subject	NOUN
ejpam-1506	10	4	classifications	classification	NOUN
ejpam-1506	10	5	:	:	PUNCT
ejpam-1506	10	6	05c12	05c12	X
ejpam-1506	10	7	key	key	ADJ
ejpam-1506	10	8	words	word	NOUN
ejpam-1506	10	9	and	and	CCONJ
ejpam-1506	10	10	phrases	phrase	NOUN
ejpam-1506	10	11	:	:	PUNCT
ejpam-1506	10	12	line	line	NOUN
ejpam-1506	10	13	graph	graph	NOUN
ejpam-1506	10	14	,	,	PUNCT
ejpam-1506	10	15	para	para	NOUN
ejpam-1506	10	16	-	-	PUNCT
ejpam-1506	10	17	line	line	NOUN
ejpam-1506	10	18	graph	graph	NOUN
ejpam-1506	10	19	,	,	PUNCT
ejpam-1506	10	20	line	line	NOUN
ejpam-1506	10	21	graph	graph	NOUN
ejpam-1506	10	22	of	of	ADP
ejpam-1506	10	23	the	the	DET
ejpam-1506	10	24	subdivision	subdivision	NOUN
ejpam-1506	10	25	graph	graph	NOUN
ejpam-1506	10	26	,	,	PUNCT
ejpam-1506	10	27	distance	distance	NOUN
ejpam-1506	10	28	,	,	PUNCT
ejpam-1506	10	29	resolving	resolve	VERB
ejpam-1506	10	30	set	set	NOUN
ejpam-1506	10	31	,	,	PUNCT
ejpam-1506	10	32	metric	metric	ADJ
ejpam-1506	10	33	dimension	dimension	NOUN
ejpam-1506	10	34	1	1	NUM
ejpam-1506	10	35	.	.	PUNCT
ejpam-1506	11	1	introduction	introduction	NOUN
ejpam-1506	11	2	let	let	VERB
ejpam-1506	11	3	g	g	NOUN
ejpam-1506	11	4	=	=	SYM
ejpam-1506	11	5	(	(	PUNCT
ejpam-1506	11	6	v	v	NOUN
ejpam-1506	11	7	(	(	PUNCT
ejpam-1506	11	8	g	g	NOUN
ejpam-1506	11	9	)	)	PUNCT
ejpam-1506	11	10	,	,	PUNCT
ejpam-1506	11	11	e(g	e(g	PROPN
ejpam-1506	11	12	)	)	PUNCT
ejpam-1506	11	13	)	)	PUNCT
ejpam-1506	11	14	be	be	AUX
ejpam-1506	11	15	a	a	DET
ejpam-1506	11	16	finite	finite	NOUN
ejpam-1506	11	17	,	,	PUNCT
ejpam-1506	11	18	simple	simple	ADJ
ejpam-1506	11	19	,	,	PUNCT
ejpam-1506	11	20	undirected	undirected	ADJ
ejpam-1506	11	21	,	,	PUNCT
ejpam-1506	11	22	and	and	CCONJ
ejpam-1506	11	23	connected	connected	ADJ
ejpam-1506	11	24	graph	graph	NOUN
ejpam-1506	11	25	of	of	ADP
ejpam-1506	11	26	order	order	NOUN
ejpam-1506	11	27	|v	|v	X
ejpam-1506	11	28	(	(	PUNCT
ejpam-1506	11	29	g)|	g)|	NOUN
ejpam-1506	11	30	=	=	PUNCT
ejpam-1506	11	31	n	n	NOUN
ejpam-1506	11	32	≥	≥	NOUN
ejpam-1506	11	33	2	2	NUM
ejpam-1506	11	34	.	.	X
ejpam-1506	11	35	for	for	ADP
ejpam-1506	11	36	a	a	DET
ejpam-1506	11	37	graph	graph	NOUN
ejpam-1506	11	38	g	g	NOUN
ejpam-1506	11	39	and	and	CCONJ
ejpam-1506	11	40	w	w	ADP
ejpam-1506	11	41	⊆	⊆	NUM
ejpam-1506	11	42	v	v	NOUN
ejpam-1506	11	43	(	(	PUNCT
ejpam-1506	11	44	g	g	NOUN
ejpam-1506	11	45	)	)	PUNCT
ejpam-1506	11	46	,	,	PUNCT
ejpam-1506	11	47	we	we	PRON
ejpam-1506	11	48	denote	denote	VERB
ejpam-1506	11	49	by	by	ADP
ejpam-1506	11	50	〈	〈	PROPN
ejpam-1506	11	51	w	w	PROPN
ejpam-1506	11	52	〉	〉	NOUN
ejpam-1506	11	53	the	the	DET
ejpam-1506	11	54	subgraph	subgraph	NOUN
ejpam-1506	11	55	induced	induce	VERB
ejpam-1506	11	56	by	by	ADP
ejpam-1506	11	57	w	w	PROPN
ejpam-1506	11	58	.	.	PUNCT
ejpam-1506	12	1	for	for	ADP
ejpam-1506	12	2	a	a	DET
ejpam-1506	12	3	vertex	vertex	NOUN
ejpam-1506	12	4	v	v	ADP
ejpam-1506	12	5	∈	∈	NOUN
ejpam-1506	12	6	v	v	NOUN
ejpam-1506	12	7	(	(	PUNCT
ejpam-1506	12	8	g	g	NOUN
ejpam-1506	12	9	)	)	PUNCT
ejpam-1506	12	10	,	,	PUNCT
ejpam-1506	12	11	the	the	DET
ejpam-1506	12	12	open	open	ADJ
ejpam-1506	12	13	neighborhood	neighborhood	NOUN
ejpam-1506	12	14	of	of	ADP
ejpam-1506	12	15	v	v	NOUN
ejpam-1506	12	16	is	be	AUX
ejpam-1506	12	17	the	the	DET
ejpam-1506	12	18	set	set	NOUN
ejpam-1506	12	19	ng(v	ng(v	PUNCT
ejpam-1506	12	20	)	)	PUNCT
ejpam-1506	12	21	=	=	PRON
ejpam-1506	13	1	{	{	PUNCT
ejpam-1506	13	2	u	u	NOUN
ejpam-1506	13	3	|	|	ADV
ejpam-1506	13	4	uv	uv	PROPN
ejpam-1506	13	5	∈	∈	PROPN
ejpam-1506	13	6	e(g	e(g	PROPN
ejpam-1506	13	7	)	)	PUNCT
ejpam-1506	13	8	}	}	PUNCT
ejpam-1506	13	9	,	,	PUNCT
ejpam-1506	13	10	and	and	CCONJ
ejpam-1506	13	11	the	the	DET
ejpam-1506	13	12	closed	closed	ADJ
ejpam-1506	13	13	neighborhood	neighborhood	NOUN
ejpam-1506	13	14	of	of	ADP
ejpam-1506	13	15	v	v	NOUN
ejpam-1506	13	16	is	be	AUX
ejpam-1506	13	17	the	the	DET
ejpam-1506	13	18	set	set	NOUN
ejpam-1506	13	19	ng[v	ng[v	NOUN
ejpam-1506	13	20	]	]	X
ejpam-1506	13	21	=	=	SYM
ejpam-1506	13	22	ng(v	ng(v	X
ejpam-1506	13	23	)	)	PUNCT
ejpam-1506	13	24	∪	∪	ADP
ejpam-1506	13	25	{	{	PUNCT
ejpam-1506	13	26	v	v	NOUN
ejpam-1506	13	27	}	}	PUNCT
ejpam-1506	13	28	;	;	PUNCT
ejpam-1506	13	29	for	for	ADP
ejpam-1506	13	30	s	s	PROPN
ejpam-1506	13	31	⊆	⊆	NUM
ejpam-1506	13	32	v	v	NOUN
ejpam-1506	13	33	(	(	PUNCT
ejpam-1506	13	34	g	g	NOUN
ejpam-1506	13	35	)	)	PUNCT
ejpam-1506	13	36	,	,	PUNCT
ejpam-1506	13	37	the	the	DET
ejpam-1506	13	38	open	open	ADJ
ejpam-1506	13	39	neighborhood	neighborhood	NOUN
ejpam-1506	13	40	of	of	ADP
ejpam-1506	13	41	s	s	NOUN
ejpam-1506	13	42	is	be	AUX
ejpam-1506	13	43	the	the	DET
ejpam-1506	13	44	set	set	NOUN
ejpam-1506	13	45	ng(s	ng(s	NOUN
ejpam-1506	13	46	)	)	PUNCT
ejpam-1506	13	47	=	=	SYM
ejpam-1506	13	48	∪v∈sng(v	∪v∈sng(v	PROPN
ejpam-1506	13	49	)	)	PUNCT
ejpam-1506	13	50	.	.	PUNCT
ejpam-1506	14	1	the	the	DET
ejpam-1506	14	2	degree	degree	NOUN
ejpam-1506	14	3	of	of	ADP
ejpam-1506	14	4	a	a	DET
ejpam-1506	14	5	vertex	vertex	NOUN
ejpam-1506	14	6	v	v	ADP
ejpam-1506	14	7	∈	∈	NOUN
ejpam-1506	14	8	v	v	NOUN
ejpam-1506	14	9	(	(	PUNCT
ejpam-1506	14	10	g	g	NOUN
ejpam-1506	14	11	)	)	PUNCT
ejpam-1506	14	12	,	,	PUNCT
ejpam-1506	14	13	denoted	denote	VERB
ejpam-1506	14	14	by	by	ADP
ejpam-1506	14	15	degg(v	degg(v	PROPN
ejpam-1506	14	16	)	)	PUNCT
ejpam-1506	14	17	,	,	PUNCT
ejpam-1506	14	18	is	be	AUX
ejpam-1506	14	19	the	the	DET
ejpam-1506	14	20	the	the	DET
ejpam-1506	14	21	number	number	NOUN
ejpam-1506	14	22	of	of	ADP
ejpam-1506	14	23	edges	edge	NOUN
ejpam-1506	14	24	incident	incident	NOUN
ejpam-1506	14	25	to	to	ADP
ejpam-1506	14	26	the	the	DET
ejpam-1506	14	27	vertex	vertex	NOUN
ejpam-1506	14	28	v	v	NOUN
ejpam-1506	14	29	in	in	ADP
ejpam-1506	14	30	g	g	PROPN
ejpam-1506	14	31	;	;	PUNCT
ejpam-1506	14	32	an	an	DET
ejpam-1506	14	33	end	end	NOUN
ejpam-1506	14	34	-	-	PUNCT
ejpam-1506	14	35	vertex	vertex	NOUN
ejpam-1506	14	36	is	be	AUX
ejpam-1506	14	37	a	a	DET
ejpam-1506	14	38	vertex	vertex	NOUN
ejpam-1506	14	39	of	of	ADP
ejpam-1506	14	40	degree	degree	NOUN
ejpam-1506	14	41	one	one	NUM
ejpam-1506	14	42	.	.	PUNCT
ejpam-1506	15	1	we	we	PRON
ejpam-1506	15	2	denote	denote	VERB
ejpam-1506	15	3	by	by	ADP
ejpam-1506	15	4	∆(g	∆(g	PROPN
ejpam-1506	15	5	)	)	PUNCT
ejpam-1506	15	6	the	the	DET
ejpam-1506	15	7	maximum	maximum	ADJ
ejpam-1506	15	8	degree	degree	NOUN
ejpam-1506	15	9	of	of	ADP
ejpam-1506	15	10	a	a	DET
ejpam-1506	15	11	graph	graph	NOUN
ejpam-1506	15	12	g.	g.	NOUN
ejpam-1506	15	13	the	the	DET
ejpam-1506	15	14	distance	distance	NOUN
ejpam-1506	15	15	between	between	ADP
ejpam-1506	15	16	two	two	NUM
ejpam-1506	15	17	vertices	vertex	NOUN
ejpam-1506	15	18	u	u	NOUN
ejpam-1506	15	19	,	,	PUNCT
ejpam-1506	15	20	v	v	NOUN
ejpam-1506	15	21	∈	∈	PROPN
ejpam-1506	15	22	v	v	NOUN
ejpam-1506	15	23	(	(	PUNCT
ejpam-1506	15	24	g	g	NOUN
ejpam-1506	15	25	)	)	PUNCT
ejpam-1506	15	26	,	,	PUNCT
ejpam-1506	15	27	denoted	denote	VERB
ejpam-1506	15	28	by	by	ADP
ejpam-1506	15	29	dg(u	dg(u	NOUN
ejpam-1506	15	30	,	,	PUNCT
ejpam-1506	15	31	v	v	NOUN
ejpam-1506	15	32	)	)	PUNCT
ejpam-1506	15	33	,	,	PUNCT
ejpam-1506	15	34	is	be	AUX
ejpam-1506	15	35	the	the	DET
ejpam-1506	15	36	length	length	NOUN
ejpam-1506	15	37	of	of	ADP
ejpam-1506	15	38	the	the	DET
ejpam-1506	15	39	shortest	short	ADJ
ejpam-1506	15	40	path	path	NOUN
ejpam-1506	15	41	in	in	ADP
ejpam-1506	15	42	g	g	NOUN
ejpam-1506	15	43	between	between	ADP
ejpam-1506	15	44	u	u	PROPN
ejpam-1506	15	45	and	and	CCONJ
ejpam-1506	15	46	v	v	NOUN
ejpam-1506	15	47	;	;	PUNCT
ejpam-1506	15	48	we	we	PRON
ejpam-1506	15	49	omit	omit	VERB
ejpam-1506	15	50	g	g	NOUN
ejpam-1506	15	51	when	when	SCONJ
ejpam-1506	15	52	ambiguity	ambiguity	NOUN
ejpam-1506	15	53	is	be	AUX
ejpam-1506	15	54	not	not	PART
ejpam-1506	15	55	a	a	DET
ejpam-1506	15	56	concern	concern	NOUN
ejpam-1506	15	57	.	.	PUNCT
ejpam-1506	16	1	the	the	DET
ejpam-1506	16	2	diameter	diameter	NOUN
ejpam-1506	16	3	,	,	PUNCT
ejpam-1506	16	4	diam(g	diam(g	PROPN
ejpam-1506	16	5	)	)	PUNCT
ejpam-1506	16	6	,	,	PUNCT
ejpam-1506	16	7	of	of	ADP
ejpam-1506	16	8	a	a	DET
ejpam-1506	16	9	graph	graph	NOUN
ejpam-1506	16	10	g	g	NOUN
ejpam-1506	16	11	is	be	AUX
ejpam-1506	16	12	given	give	VERB
ejpam-1506	16	13	by	by	ADP
ejpam-1506	16	14	max{d(u	max{d(u	PROPN
ejpam-1506	16	15	,	,	PUNCT
ejpam-1506	16	16	v	v	NOUN
ejpam-1506	16	17	)	)	PUNCT
ejpam-1506	16	18	|	|	ADV
ejpam-1506	16	19	u	u	NOUN
ejpam-1506	16	20	,	,	PUNCT
ejpam-1506	16	21	v	v	PROPN
ejpam-1506	16	22	∈	∈	PROPN
ejpam-1506	16	23	v	v	NOUN
ejpam-1506	16	24	(	(	PUNCT
ejpam-1506	16	25	g	g	NOUN
ejpam-1506	16	26	)	)	PUNCT
ejpam-1506	16	27	}	}	PUNCT
ejpam-1506	16	28	.	.	PUNCT
ejpam-1506	17	1	we	we	PRON
ejpam-1506	17	2	denote	denote	VERB
ejpam-1506	17	3	by	by	ADP
ejpam-1506	17	4	kn	kn	PROPN
ejpam-1506	17	5	,	,	PUNCT
ejpam-1506	17	6	cn	cn	PROPN
ejpam-1506	17	7	,	,	PUNCT
ejpam-1506	17	8	and	and	CCONJ
ejpam-1506	17	9	pn	pn	VERB
ejpam-1506	17	10	the	the	DET
ejpam-1506	17	11	complete	complete	ADJ
ejpam-1506	17	12	graph	graph	NOUN
ejpam-1506	17	13	,	,	PUNCT
ejpam-1506	17	14	the	the	DET
ejpam-1506	17	15	∗corresponding	∗corresponde	VERB
ejpam-1506	17	16	author	author	NOUN
ejpam-1506	17	17	.	.	PUNCT
ejpam-1506	18	1	email	email	NOUN
ejpam-1506	18	2	addresses	address	NOUN
ejpam-1506	18	3	:	:	PUNCT
ejpam-1506	18	4	kleind�tamug.edu	kleind�tamug.edu	PROPN
ejpam-1506	18	5	(	(	PUNCT
ejpam-1506	18	6	d.	d.	PROPN
ejpam-1506	18	7	klein	klein	PROPN
ejpam-1506	18	8	)	)	PUNCT
ejpam-1506	18	9	,	,	PUNCT
ejpam-1506	18	10	yie�tamug.edu	yie�tamug.edu	PROPN
ejpam-1506	18	11	(	(	PUNCT
ejpam-1506	18	12	e.	e.	PROPN
ejpam-1506	18	13	yi	yi	PROPN
ejpam-1506	18	14	)	)	PUNCT
ejpam-1506	18	15	http://www.ejpam.com	http://www.ejpam.com	X
ejpam-1506	19	1	302	302	NUM
ejpam-1506	20	1	c	c	X
ejpam-1506	20	2	©	©	PROPN
ejpam-1506	20	3	2012	2012	NUM
ejpam-1506	20	4	ejpam	ejpam	VERB
ejpam-1506	20	5	all	all	DET
ejpam-1506	20	6	rights	right	NOUN
ejpam-1506	20	7	reserved	reserve	VERB
ejpam-1506	20	8	.	.	PUNCT
ejpam-1506	21	1	d.	d.	PROPN
ejpam-1506	21	2	klein	klein	PROPN
ejpam-1506	21	3	,	,	PUNCT
ejpam-1506	21	4	e.	e.	PROPN
ejpam-1506	21	5	yi	yi	PROPN
ejpam-1506	21	6	/	/	SYM
ejpam-1506	21	7	eur	eur	PROPN
ejpam-1506	21	8	.	.	PUNCT
ejpam-1506	22	1	j.	j.	PROPN
ejpam-1506	22	2	pure	pure	PROPN
ejpam-1506	22	3	appl	appl	PROPN
ejpam-1506	22	4	.	.	PROPN
ejpam-1506	22	5	math	math	PROPN
ejpam-1506	22	6	,	,	PUNCT
ejpam-1506	22	7	5	5	NUM
ejpam-1506	22	8	(	(	PUNCT
ejpam-1506	22	9	2012	2012	NUM
ejpam-1506	22	10	)	)	PUNCT
ejpam-1506	22	11	,	,	PUNCT
ejpam-1506	22	12	302	302	NUM
ejpam-1506	22	13	-	-	SYM
ejpam-1506	22	14	316	316	NUM
ejpam-1506	22	15	303	303	NUM
ejpam-1506	22	16	cycle	cycle	NOUN
ejpam-1506	22	17	,	,	PUNCT
ejpam-1506	22	18	and	and	CCONJ
ejpam-1506	22	19	the	the	DET
ejpam-1506	22	20	path	path	NOUN
ejpam-1506	22	21	on	on	ADP
ejpam-1506	22	22	n	n	DET
ejpam-1506	22	23	vertices	vertex	NOUN
ejpam-1506	22	24	,	,	PUNCT
ejpam-1506	22	25	respectively	respectively	ADV
ejpam-1506	22	26	.	.	PUNCT
ejpam-1506	23	1	for	for	ADP
ejpam-1506	23	2	other	other	ADJ
ejpam-1506	23	3	terminologies	terminology	NOUN
ejpam-1506	23	4	in	in	ADP
ejpam-1506	23	5	graph	graph	NOUN
ejpam-1506	23	6	theory	theory	NOUN
ejpam-1506	23	7	,	,	PUNCT
ejpam-1506	23	8	refer	refer	VERB
ejpam-1506	23	9	to	to	ADP
ejpam-1506	23	10	[	[	X
ejpam-1506	23	11	6	6	NUM
ejpam-1506	23	12	]	]	PUNCT
ejpam-1506	23	13	.	.	PUNCT
ejpam-1506	24	1	the	the	DET
ejpam-1506	24	2	subdivision	subdivision	NOUN
ejpam-1506	24	3	graph	graph	NOUN
ejpam-1506	24	4	s(g	s(g	PROPN
ejpam-1506	24	5	)	)	PUNCT
ejpam-1506	24	6	of	of	ADP
ejpam-1506	24	7	a	a	DET
ejpam-1506	24	8	graph	graph	NOUN
ejpam-1506	24	9	g	g	NOUN
ejpam-1506	24	10	is	be	AUX
ejpam-1506	24	11	obtained	obtain	VERB
ejpam-1506	24	12	from	from	ADP
ejpam-1506	24	13	g	g	NOUN
ejpam-1506	24	14	by	by	ADP
ejpam-1506	24	15	deleting	delete	VERB
ejpam-1506	24	16	every	every	DET
ejpam-1506	24	17	edge	edge	NOUN
ejpam-1506	24	18	uv	uv	NOUN
ejpam-1506	24	19	of	of	ADP
ejpam-1506	24	20	g	g	NOUN
ejpam-1506	24	21	and	and	CCONJ
ejpam-1506	24	22	replacing	replace	VERB
ejpam-1506	24	23	it	it	PRON
ejpam-1506	24	24	by	by	ADP
ejpam-1506	24	25	a	a	DET
ejpam-1506	24	26	vertex	vertex	NOUN
ejpam-1506	24	27	w	w	NOUN
ejpam-1506	24	28	of	of	ADP
ejpam-1506	24	29	degree	degree	NOUN
ejpam-1506	24	30	2	2	NUM
ejpam-1506	24	31	that	that	PRON
ejpam-1506	24	32	is	be	AUX
ejpam-1506	24	33	joined	join	VERB
ejpam-1506	24	34	to	to	ADP
ejpam-1506	24	35	u	u	NOUN
ejpam-1506	24	36	and	and	CCONJ
ejpam-1506	24	37	v	v	ADP
ejpam-1506	24	38	[	[	PUNCT
ejpam-1506	24	39	see	see	VERB
ejpam-1506	24	40	p.151	p.151	NOUN
ejpam-1506	24	41	of	of	ADP
ejpam-1506	24	42	6	6	NUM
ejpam-1506	24	43	]	]	PUNCT
ejpam-1506	24	44	.	.	PUNCT
ejpam-1506	25	1	the	the	DET
ejpam-1506	25	2	line	line	NOUN
ejpam-1506	25	3	graph	graph	NOUN
ejpam-1506	25	4	l(g	l(g	NOUN
ejpam-1506	25	5	)	)	PUNCT
ejpam-1506	25	6	of	of	ADP
ejpam-1506	25	7	a	a	DET
ejpam-1506	25	8	simple	simple	ADJ
ejpam-1506	25	9	graph	graph	NOUN
ejpam-1506	25	10	g	g	PROPN
ejpam-1506	25	11	is	be	AUX
ejpam-1506	25	12	the	the	DET
ejpam-1506	25	13	graph	graph	NOUN
ejpam-1506	25	14	whose	whose	DET
ejpam-1506	25	15	vertices	vertex	NOUN
ejpam-1506	25	16	are	be	AUX
ejpam-1506	25	17	in	in	ADP
ejpam-1506	25	18	one	one	NUM
ejpam-1506	25	19	-	-	PUNCT
ejpam-1506	25	20	to	to	ADP
ejpam-1506	25	21	-	-	PUNCT
ejpam-1506	25	22	one	one	NUM
ejpam-1506	25	23	correspondence	correspondence	NOUN
ejpam-1506	25	24	with	with	ADP
ejpam-1506	25	25	the	the	DET
ejpam-1506	25	26	edges	edge	NOUN
ejpam-1506	25	27	of	of	ADP
ejpam-1506	25	28	g	g	NOUN
ejpam-1506	25	29	;	;	PUNCT
ejpam-1506	25	30	two	two	NUM
ejpam-1506	25	31	vertices	vertex	NOUN
ejpam-1506	25	32	of	of	ADP
ejpam-1506	25	33	l(g	l(g	NOUN
ejpam-1506	25	34	)	)	PUNCT
ejpam-1506	25	35	are	be	AUX
ejpam-1506	25	36	adjacent	adjacent	ADJ
ejpam-1506	25	37	if	if	SCONJ
ejpam-1506	26	1	and	and	CCONJ
ejpam-1506	26	2	only	only	ADV
ejpam-1506	26	3	if	if	SCONJ
ejpam-1506	26	4	the	the	DET
ejpam-1506	26	5	corresponding	corresponding	ADJ
ejpam-1506	26	6	edges	edge	NOUN
ejpam-1506	26	7	of	of	ADP
ejpam-1506	26	8	g	g	NOUN
ejpam-1506	26	9	are	be	AUX
ejpam-1506	26	10	adjacent	adjacent	ADJ
ejpam-1506	26	11	[	[	X
ejpam-1506	26	12	see	see	ADJ
ejpam-1506	26	13	2	2	NUM
ejpam-1506	26	14	,	,	PUNCT
ejpam-1506	26	15	18	18	NUM
ejpam-1506	26	16	,	,	PUNCT
ejpam-1506	26	17	28	28	NUM
ejpam-1506	26	18	,	,	PUNCT
ejpam-1506	26	19	29	29	NUM
ejpam-1506	26	20	]	]	PUNCT
ejpam-1506	26	21	.	.	PUNCT
ejpam-1506	27	1	following	follow	VERB
ejpam-1506	27	2	[	[	X
ejpam-1506	27	3	25	25	NUM
ejpam-1506	27	4	]	]	PUNCT
ejpam-1506	27	5	,	,	PUNCT
ejpam-1506	27	6	we	we	PRON
ejpam-1506	27	7	define	define	VERB
ejpam-1506	27	8	the	the	DET
ejpam-1506	27	9	para	para	NOUN
ejpam-1506	27	10	-	-	PUNCT
ejpam-1506	27	11	line	line	NOUN
ejpam-1506	27	12	graph	graph	NOUN
ejpam-1506	27	13	of	of	ADP
ejpam-1506	27	14	g	g	NOUN
ejpam-1506	27	15	to	to	PART
ejpam-1506	27	16	be	be	AUX
ejpam-1506	27	17	l(s(g	l(s(g	ADV
ejpam-1506	27	18	)	)	PUNCT
ejpam-1506	27	19	)	)	PUNCT
ejpam-1506	27	20	,	,	PUNCT
ejpam-1506	27	21	which	which	PRON
ejpam-1506	27	22	we	we	PRON
ejpam-1506	27	23	will	will	AUX
ejpam-1506	27	24	denote	denote	VERB
ejpam-1506	27	25	by	by	ADP
ejpam-1506	27	26	g⋆.	g⋆.	NOUN
ejpam-1506	27	27	alternatively	alternatively	ADV
ejpam-1506	27	28	,	,	PUNCT
ejpam-1506	27	29	we	we	PRON
ejpam-1506	27	30	can	can	AUX
ejpam-1506	27	31	construct	construct	VERB
ejpam-1506	27	32	g⋆	g⋆	NOUN
ejpam-1506	27	33	from	from	ADP
ejpam-1506	27	34	g	g	PROPN
ejpam-1506	27	35	as	as	SCONJ
ejpam-1506	27	36	follows	follow	VERB
ejpam-1506	27	37	:	:	PUNCT
ejpam-1506	27	38	(	(	PUNCT
ejpam-1506	27	39	i	i	NOUN
ejpam-1506	27	40	)	)	PUNCT
ejpam-1506	27	41	replace	replace	VERB
ejpam-1506	27	42	each	each	DET
ejpam-1506	27	43	vertex	vertex	NOUN
ejpam-1506	27	44	u	u	NOUN
ejpam-1506	27	45	∈	∈	PROPN
ejpam-1506	27	46	v	v	ADP
ejpam-1506	27	47	(	(	PUNCT
ejpam-1506	27	48	g	g	NOUN
ejpam-1506	27	49	)	)	PUNCT
ejpam-1506	27	50	by	by	ADP
ejpam-1506	27	51	k(u	k(u	NOUN
ejpam-1506	27	52	)	)	PUNCT
ejpam-1506	27	53	,	,	PUNCT
ejpam-1506	27	54	the	the	DET
ejpam-1506	27	55	complete	complete	ADJ
ejpam-1506	27	56	graph	graph	NOUN
ejpam-1506	27	57	on	on	ADP
ejpam-1506	27	58	degg(u	degg(u	NUM
ejpam-1506	27	59	)	)	PUNCT
ejpam-1506	27	60	vertices	vertex	NOUN
ejpam-1506	27	61	;	;	PUNCT
ejpam-1506	27	62	(	(	PUNCT
ejpam-1506	27	63	ii	ii	NOUN
ejpam-1506	27	64	)	)	PUNCT
ejpam-1506	27	65	there	there	PRON
ejpam-1506	27	66	is	be	VERB
ejpam-1506	27	67	an	an	DET
ejpam-1506	27	68	edge	edge	NOUN
ejpam-1506	27	69	joining	join	VERB
ejpam-1506	27	70	a	a	DET
ejpam-1506	27	71	vertex	vertex	NOUN
ejpam-1506	27	72	of	of	ADP
ejpam-1506	27	73	k(u1	k(u1	NOUN
ejpam-1506	27	74	)	)	PUNCT
ejpam-1506	27	75	and	and	CCONJ
ejpam-1506	27	76	a	a	DET
ejpam-1506	27	77	vertex	vertex	NOUN
ejpam-1506	27	78	of	of	ADP
ejpam-1506	27	79	k(u2	k(u2	NOUN
ejpam-1506	27	80	)	)	PUNCT
ejpam-1506	27	81	in	in	ADP
ejpam-1506	27	82	g⋆	g⋆	NOUN
ejpam-1506	27	83	if	if	SCONJ
ejpam-1506	28	1	and	and	CCONJ
ejpam-1506	28	2	only	only	ADV
ejpam-1506	28	3	if	if	SCONJ
ejpam-1506	28	4	there	there	PRON
ejpam-1506	28	5	is	be	VERB
ejpam-1506	28	6	an	an	DET
ejpam-1506	28	7	edge	edge	NOUN
ejpam-1506	28	8	joining	join	VERB
ejpam-1506	28	9	u1	u1	NOUN
ejpam-1506	28	10	and	and	CCONJ
ejpam-1506	28	11	u2	u2	NOUN
ejpam-1506	28	12	in	in	ADP
ejpam-1506	28	13	g	g	PROPN
ejpam-1506	28	14	;	;	PUNCT
ejpam-1506	28	15	(	(	PUNCT
ejpam-1506	28	16	iii	iii	NOUN
ejpam-1506	28	17	)	)	PUNCT
ejpam-1506	28	18	for	for	ADP
ejpam-1506	28	19	each	each	DET
ejpam-1506	28	20	vertex	vertex	NOUN
ejpam-1506	28	21	v	v	NOUN
ejpam-1506	28	22	of	of	ADP
ejpam-1506	28	23	k(u	k(u	NOUN
ejpam-1506	28	24	)	)	PUNCT
ejpam-1506	28	25	,	,	PUNCT
ejpam-1506	28	26	degg⋆(v	degg⋆(v	PROPN
ejpam-1506	28	27	)	)	PUNCT
ejpam-1506	28	28	=	=	SYM
ejpam-1506	28	29	degg(u	degg(u	PROPN
ejpam-1506	28	30	)	)	PUNCT
ejpam-1506	28	31	.	.	PUNCT
ejpam-1506	29	1	a	a	DET
ejpam-1506	29	2	vertex	vertex	NOUN
ejpam-1506	29	3	x	x	SYM
ejpam-1506	29	4	∈	∈	NOUN
ejpam-1506	29	5	v	v	ADP
ejpam-1506	29	6	(	(	PUNCT
ejpam-1506	29	7	g	g	NOUN
ejpam-1506	29	8	)	)	PUNCT
ejpam-1506	29	9	resolves	resolve	VERB
ejpam-1506	29	10	a	a	DET
ejpam-1506	29	11	pair	pair	NOUN
ejpam-1506	29	12	of	of	ADP
ejpam-1506	29	13	vertices	vertex	NOUN
ejpam-1506	29	14	u	u	NOUN
ejpam-1506	29	15	,	,	PUNCT
ejpam-1506	29	16	v	v	NOUN
ejpam-1506	29	17	∈	∈	PROPN
ejpam-1506	29	18	v	v	NOUN
ejpam-1506	29	19	(	(	PUNCT
ejpam-1506	29	20	g	g	NOUN
ejpam-1506	29	21	)	)	PUNCT
ejpam-1506	29	22	if	if	SCONJ
ejpam-1506	29	23	d(u	d(u	PROPN
ejpam-1506	29	24	,	,	PUNCT
ejpam-1506	29	25	x	x	X
ejpam-1506	29	26	)	)	PUNCT
ejpam-1506	29	27	6=	6=	ADP
ejpam-1506	30	1	d(v	d(v	PROPN
ejpam-1506	30	2	,	,	PUNCT
ejpam-1506	30	3	x	x	NOUN
ejpam-1506	30	4	)	)	PUNCT
ejpam-1506	30	5	.	.	PUNCT
ejpam-1506	31	1	a	a	DET
ejpam-1506	31	2	set	set	NOUN
ejpam-1506	31	3	of	of	ADP
ejpam-1506	31	4	vertices	vertex	NOUN
ejpam-1506	31	5	s	s	PART
ejpam-1506	31	6	⊆	⊆	NUM
ejpam-1506	31	7	v	v	NOUN
ejpam-1506	31	8	(	(	PUNCT
ejpam-1506	31	9	g	g	NOUN
ejpam-1506	31	10	)	)	PUNCT
ejpam-1506	31	11	resolves	resolve	VERB
ejpam-1506	31	12	g	g	NOUN
ejpam-1506	31	13	if	if	SCONJ
ejpam-1506	31	14	every	every	DET
ejpam-1506	31	15	pair	pair	NOUN
ejpam-1506	31	16	of	of	ADP
ejpam-1506	31	17	distinct	distinct	ADJ
ejpam-1506	31	18	vertices	vertex	NOUN
ejpam-1506	31	19	of	of	ADP
ejpam-1506	31	20	g	g	PROPN
ejpam-1506	31	21	is	be	AUX
ejpam-1506	31	22	resolved	resolve	VERB
ejpam-1506	31	23	by	by	ADP
ejpam-1506	31	24	some	some	DET
ejpam-1506	31	25	vertex	vertex	NOUN
ejpam-1506	31	26	in	in	ADP
ejpam-1506	31	27	s	s	PROPN
ejpam-1506	31	28	;	;	PUNCT
ejpam-1506	31	29	then	then	ADV
ejpam-1506	31	30	s	s	VERB
ejpam-1506	31	31	is	be	AUX
ejpam-1506	31	32	called	call	VERB
ejpam-1506	31	33	a	a	DET
ejpam-1506	31	34	resolving	resolving	NOUN
ejpam-1506	31	35	set	set	NOUN
ejpam-1506	31	36	of	of	ADP
ejpam-1506	31	37	g.	g.	PROPN
ejpam-1506	31	38	for	for	ADP
ejpam-1506	31	39	an	an	DET
ejpam-1506	31	40	ordered	order	VERB
ejpam-1506	31	41	set	set	NOUN
ejpam-1506	31	42	s	s	PART
ejpam-1506	31	43	=	=	SYM
ejpam-1506	31	44	{	{	PUNCT
ejpam-1506	31	45	w1	w1	NOUN
ejpam-1506	31	46	,	,	PUNCT
ejpam-1506	31	47	w2	w2	NOUN
ejpam-1506	31	48	,	,	PUNCT
ejpam-1506	31	49	.	.	PUNCT
ejpam-1506	31	50	.	.	PUNCT
ejpam-1506	32	1	.	.	PUNCT
ejpam-1506	33	1	,	,	PUNCT
ejpam-1506	33	2	wk	wk	X
ejpam-1506	33	3	}	}	PUNCT
ejpam-1506	33	4	⊆	⊆	NUM
ejpam-1506	33	5	v	v	NOUN
ejpam-1506	33	6	(	(	PUNCT
ejpam-1506	33	7	g	g	NOUN
ejpam-1506	33	8	)	)	PUNCT
ejpam-1506	33	9	of	of	ADP
ejpam-1506	33	10	distinct	distinct	ADJ
ejpam-1506	33	11	vertices	vertex	NOUN
ejpam-1506	33	12	,	,	PUNCT
ejpam-1506	33	13	the	the	DET
ejpam-1506	33	14	metric	metric	ADJ
ejpam-1506	33	15	code	code	NOUN
ejpam-1506	33	16	(	(	PUNCT
ejpam-1506	33	17	or	or	CCONJ
ejpam-1506	33	18	code	code	NOUN
ejpam-1506	33	19	,	,	PUNCT
ejpam-1506	33	20	for	for	ADP
ejpam-1506	33	21	short	short	ADJ
ejpam-1506	33	22	)	)	PUNCT
ejpam-1506	33	23	of	of	ADP
ejpam-1506	33	24	v	v	NUM
ejpam-1506	33	25	∈	∈	NOUN
ejpam-1506	33	26	v	v	NOUN
ejpam-1506	33	27	(	(	PUNCT
ejpam-1506	33	28	g	g	NOUN
ejpam-1506	33	29	)	)	PUNCT
ejpam-1506	33	30	with	with	ADP
ejpam-1506	33	31	respect	respect	NOUN
ejpam-1506	33	32	to	to	ADP
ejpam-1506	33	33	s	s	PRON
ejpam-1506	33	34	,	,	PUNCT
ejpam-1506	33	35	denoted	denote	VERB
ejpam-1506	33	36	by	by	ADP
ejpam-1506	33	37	codes(v	codes(v	PROPN
ejpam-1506	33	38	)	)	PUNCT
ejpam-1506	33	39	,	,	PUNCT
ejpam-1506	33	40	is	be	AUX
ejpam-1506	33	41	the	the	DET
ejpam-1506	33	42	k	k	NOUN
ejpam-1506	33	43	-	-	NOUN
ejpam-1506	33	44	vector	vector	NOUN
ejpam-1506	33	45	(	(	PUNCT
ejpam-1506	33	46	d(v	d(v	PROPN
ejpam-1506	33	47	,	,	PUNCT
ejpam-1506	33	48	w1	w1	NOUN
ejpam-1506	33	49	)	)	PUNCT
ejpam-1506	33	50	,	,	PUNCT
ejpam-1506	33	51	d(v	d(v	PROPN
ejpam-1506	33	52	,	,	PUNCT
ejpam-1506	33	53	w2	w2	NOUN
ejpam-1506	33	54	)	)	PUNCT
ejpam-1506	33	55	,	,	PUNCT
ejpam-1506	33	56	.	.	PUNCT
ejpam-1506	33	57	.	.	PUNCT
ejpam-1506	34	1	.	.	PUNCT
ejpam-1506	35	1	,	,	PUNCT
ejpam-1506	35	2	d(v	d(v	PROPN
ejpam-1506	35	3	,	,	PUNCT
ejpam-1506	35	4	wk	wk	NOUN
ejpam-1506	35	5	)	)	PUNCT
ejpam-1506	35	6	)	)	PUNCT
ejpam-1506	35	7	.	.	PUNCT
ejpam-1506	36	1	the	the	DET
ejpam-1506	36	2	metric	metric	ADJ
ejpam-1506	36	3	dimension	dimension	NOUN
ejpam-1506	36	4	of	of	ADP
ejpam-1506	36	5	g	g	NOUN
ejpam-1506	36	6	,	,	PUNCT
ejpam-1506	36	7	denoted	denote	VERB
ejpam-1506	36	8	by	by	ADP
ejpam-1506	36	9	dim(g	dim(g	PROPN
ejpam-1506	36	10	)	)	PUNCT
ejpam-1506	36	11	,	,	PUNCT
ejpam-1506	36	12	is	be	AUX
ejpam-1506	36	13	the	the	DET
ejpam-1506	36	14	minimum	minimum	ADJ
ejpam-1506	36	15	cardinality	cardinality	NOUN
ejpam-1506	36	16	over	over	ADP
ejpam-1506	36	17	all	all	PRON
ejpam-1506	36	18	resolving	resolve	VERB
ejpam-1506	36	19	sets	set	NOUN
ejpam-1506	36	20	of	of	ADP
ejpam-1506	36	21	g.	g.	PROPN
ejpam-1506	36	22	slater	slater	PROPN
ejpam-1506	37	1	[	[	X
ejpam-1506	37	2	26	26	NUM
ejpam-1506	37	3	,	,	PUNCT
ejpam-1506	37	4	27	27	NUM
ejpam-1506	37	5	]	]	PUNCT
ejpam-1506	37	6	introduced	introduce	VERB
ejpam-1506	37	7	the	the	DET
ejpam-1506	37	8	concept	concept	NOUN
ejpam-1506	37	9	of	of	ADP
ejpam-1506	37	10	a	a	DET
ejpam-1506	37	11	resolving	resolving	NOUN
ejpam-1506	37	12	set	set	VERB
ejpam-1506	37	13	for	for	ADP
ejpam-1506	37	14	a	a	DET
ejpam-1506	37	15	connected	connected	ADJ
ejpam-1506	37	16	graph	graph	NOUN
ejpam-1506	37	17	under	under	ADP
ejpam-1506	37	18	the	the	DET
ejpam-1506	37	19	term	term	NOUN
ejpam-1506	37	20	locating	locate	VERB
ejpam-1506	37	21	set	set	NOUN
ejpam-1506	37	22	.	.	PUNCT
ejpam-1506	38	1	he	he	PRON
ejpam-1506	38	2	referred	refer	VERB
ejpam-1506	38	3	to	to	ADP
ejpam-1506	38	4	a	a	DET
ejpam-1506	38	5	minimum	minimum	ADJ
ejpam-1506	38	6	resolving	resolving	NOUN
ejpam-1506	38	7	set	set	VERB
ejpam-1506	38	8	as	as	ADP
ejpam-1506	38	9	a	a	DET
ejpam-1506	38	10	reference	reference	NOUN
ejpam-1506	38	11	set	set	NOUN
ejpam-1506	38	12	,	,	PUNCT
ejpam-1506	38	13	and	and	CCONJ
ejpam-1506	38	14	the	the	DET
ejpam-1506	38	15	cardinality	cardinality	NOUN
ejpam-1506	38	16	of	of	ADP
ejpam-1506	38	17	a	a	DET
ejpam-1506	38	18	minimum	minimum	ADJ
ejpam-1506	38	19	resolving	resolving	NOUN
ejpam-1506	38	20	set	set	VERB
ejpam-1506	38	21	as	as	ADP
ejpam-1506	38	22	the	the	DET
ejpam-1506	38	23	location	location	NOUN
ejpam-1506	38	24	number	number	NOUN
ejpam-1506	38	25	of	of	ADP
ejpam-1506	38	26	a	a	DET
ejpam-1506	38	27	graph	graph	NOUN
ejpam-1506	38	28	.	.	PUNCT
ejpam-1506	39	1	independently	independently	ADV
ejpam-1506	39	2	,	,	PUNCT
ejpam-1506	39	3	harary	harary	NOUN
ejpam-1506	39	4	and	and	CCONJ
ejpam-1506	39	5	melter	melter	NOUN
ejpam-1506	39	6	[	[	X
ejpam-1506	39	7	13	13	NUM
ejpam-1506	39	8	]	]	PUNCT
ejpam-1506	39	9	studied	study	VERB
ejpam-1506	39	10	these	these	DET
ejpam-1506	39	11	concepts	concept	NOUN
ejpam-1506	39	12	under	under	ADP
ejpam-1506	39	13	the	the	DET
ejpam-1506	39	14	term	term	NOUN
ejpam-1506	39	15	metric	metric	ADJ
ejpam-1506	39	16	dimension	dimension	NOUN
ejpam-1506	39	17	.	.	PUNCT
ejpam-1506	40	1	since	since	SCONJ
ejpam-1506	40	2	metric	metric	ADJ
ejpam-1506	40	3	dimension	dimension	NOUN
ejpam-1506	40	4	is	be	AUX
ejpam-1506	40	5	suggestive	suggestive	ADJ
ejpam-1506	40	6	of	of	ADP
ejpam-1506	40	7	the	the	DET
ejpam-1506	40	8	dimension	dimension	NOUN
ejpam-1506	40	9	of	of	ADP
ejpam-1506	40	10	a	a	DET
ejpam-1506	40	11	vector	vector	NOUN
ejpam-1506	40	12	space	space	NOUN
ejpam-1506	40	13	in	in	ADP
ejpam-1506	40	14	linear	linear	PROPN
ejpam-1506	40	15	algebra	algebra	NOUN
ejpam-1506	40	16	,	,	PUNCT
ejpam-1506	40	17	sometimes	sometimes	ADV
ejpam-1506	40	18	a	a	DET
ejpam-1506	40	19	minimum	minimum	ADJ
ejpam-1506	40	20	resolving	resolving	NOUN
ejpam-1506	40	21	set	set	NOUN
ejpam-1506	40	22	of	of	ADP
ejpam-1506	40	23	g	g	PROPN
ejpam-1506	40	24	is	be	AUX
ejpam-1506	40	25	called	call	VERB
ejpam-1506	40	26	a	a	DET
ejpam-1506	40	27	basis	basis	NOUN
ejpam-1506	40	28	of	of	ADP
ejpam-1506	40	29	g.	g.	PROPN
ejpam-1506	40	30	metric	metric	PROPN
ejpam-1506	40	31	dimension	dimension	NOUN
ejpam-1506	40	32	as	as	ADP
ejpam-1506	40	33	a	a	DET
ejpam-1506	40	34	graph	graph	NOUN
ejpam-1506	40	35	parameter	parameter	NOUN
ejpam-1506	40	36	has	have	VERB
ejpam-1506	40	37	numerous	numerous	ADJ
ejpam-1506	40	38	applications	application	NOUN
ejpam-1506	40	39	,	,	PUNCT
ejpam-1506	40	40	among	among	ADP
ejpam-1506	40	41	them	they	PRON
ejpam-1506	40	42	are	be	AUX
ejpam-1506	40	43	robot	robot	NOUN
ejpam-1506	40	44	navigation	navigation	NOUN
ejpam-1506	40	45	[	[	X
ejpam-1506	40	46	17	17	NUM
ejpam-1506	40	47	]	]	PUNCT
ejpam-1506	40	48	,	,	PUNCT
ejpam-1506	40	49	sonar	sonar	NOUN
ejpam-1506	40	50	[	[	X
ejpam-1506	40	51	27	27	NUM
ejpam-1506	40	52	]	]	PUNCT
ejpam-1506	40	53	,	,	PUNCT
ejpam-1506	40	54	combinatorial	combinatorial	ADJ
ejpam-1506	40	55	optimization	optimization	NOUN
ejpam-1506	40	56	[	[	X
ejpam-1506	40	57	23	23	NUM
ejpam-1506	40	58	]	]	PUNCT
ejpam-1506	40	59	,	,	PUNCT
ejpam-1506	40	60	and	and	CCONJ
ejpam-1506	40	61	pharmaceutical	pharmaceutical	ADJ
ejpam-1506	40	62	chemistry	chemistry	NOUN
ejpam-1506	40	63	[	[	X
ejpam-1506	40	64	5	5	NUM
ejpam-1506	40	65	]	]	PUNCT
ejpam-1506	40	66	.	.	PUNCT
ejpam-1506	41	1	it	it	PRON
ejpam-1506	41	2	is	be	AUX
ejpam-1506	41	3	noted	note	VERB
ejpam-1506	41	4	in	in	ADP
ejpam-1506	41	5	[	[	X
ejpam-1506	41	6	12	12	NUM
ejpam-1506	41	7	]	]	PUNCT
ejpam-1506	41	8	that	that	SCONJ
ejpam-1506	41	9	determining	determine	VERB
ejpam-1506	41	10	the	the	DET
ejpam-1506	41	11	metric	metric	ADJ
ejpam-1506	41	12	dimension	dimension	NOUN
ejpam-1506	41	13	of	of	ADP
ejpam-1506	41	14	a	a	DET
ejpam-1506	41	15	graph	graph	NOUN
ejpam-1506	41	16	is	be	AUX
ejpam-1506	41	17	an	an	DET
ejpam-1506	41	18	np	np	NOUN
ejpam-1506	41	19	-	-	PUNCT
ejpam-1506	41	20	hard	hard	ADJ
ejpam-1506	41	21	problem	problem	NOUN
ejpam-1506	41	22	.	.	PUNCT
ejpam-1506	42	1	metric	metric	ADJ
ejpam-1506	42	2	dimension	dimension	NOUN
ejpam-1506	42	3	has	have	AUX
ejpam-1506	42	4	been	be	AUX
ejpam-1506	42	5	heavily	heavily	ADV
ejpam-1506	42	6	studied	study	VERB
ejpam-1506	42	7	;	;	PUNCT
ejpam-1506	42	8	for	for	ADP
ejpam-1506	42	9	surveys	survey	NOUN
ejpam-1506	42	10	,	,	PUNCT
ejpam-1506	42	11	see	see	VERB
ejpam-1506	42	12	[	[	X
ejpam-1506	42	13	1	1	X
ejpam-1506	42	14	]	]	PUNCT
ejpam-1506	42	15	and	and	CCONJ
ejpam-1506	42	16	[	[	X
ejpam-1506	42	17	7	7	NUM
ejpam-1506	42	18	]	]	PUNCT
ejpam-1506	42	19	.	.	PUNCT
ejpam-1506	43	1	for	for	ADP
ejpam-1506	43	2	more	more	ADJ
ejpam-1506	43	3	articles	article	NOUN
ejpam-1506	43	4	on	on	ADP
ejpam-1506	43	5	metric	metric	ADJ
ejpam-1506	43	6	dimension	dimension	NOUN
ejpam-1506	43	7	in	in	ADP
ejpam-1506	43	8	graphs	graph	NOUN
ejpam-1506	43	9	,	,	PUNCT
ejpam-1506	43	10	see	see	VERB
ejpam-1506	43	11	[	[	X
ejpam-1506	43	12	3	3	NUM
ejpam-1506	43	13	,	,	PUNCT
ejpam-1506	43	14	4	4	NUM
ejpam-1506	43	15	,	,	PUNCT
ejpam-1506	43	16	8	8	NUM
ejpam-1506	43	17	,	,	PUNCT
ejpam-1506	43	18	9	9	NUM
ejpam-1506	43	19	,	,	PUNCT
ejpam-1506	43	20	14	14	NUM
ejpam-1506	43	21	,	,	PUNCT
ejpam-1506	43	22	16	16	NUM
ejpam-1506	43	23	,	,	PUNCT
ejpam-1506	43	24	19	19	NUM
ejpam-1506	43	25	,	,	PUNCT
ejpam-1506	43	26	24	24	NUM
ejpam-1506	43	27	]	]	PUNCT
ejpam-1506	43	28	.	.	PUNCT
ejpam-1506	44	1	in	in	ADP
ejpam-1506	44	2	this	this	DET
ejpam-1506	44	3	paper	paper	NOUN
ejpam-1506	44	4	,	,	PUNCT
ejpam-1506	44	5	we	we	PRON
ejpam-1506	44	6	study	study	VERB
ejpam-1506	44	7	metric	metric	ADJ
ejpam-1506	44	8	dimension	dimension	NOUN
ejpam-1506	44	9	of	of	ADP
ejpam-1506	44	10	para	para	ADJ
ejpam-1506	44	11	-	-	PUNCT
ejpam-1506	44	12	line	line	NOUN
ejpam-1506	44	13	graphs	graph	NOUN
ejpam-1506	44	14	;	;	PUNCT
ejpam-1506	44	15	we	we	PRON
ejpam-1506	44	16	also	also	ADV
ejpam-1506	44	17	compare	compare	VERB
ejpam-1506	44	18	metric	metric	ADJ
ejpam-1506	44	19	dimension	dimension	NOUN
ejpam-1506	44	20	of	of	ADP
ejpam-1506	44	21	graphs	graph	NOUN
ejpam-1506	44	22	,	,	PUNCT
ejpam-1506	44	23	line	line	NOUN
ejpam-1506	44	24	graphs	graph	NOUN
ejpam-1506	44	25	,	,	PUNCT
ejpam-1506	44	26	and	and	CCONJ
ejpam-1506	44	27	para	para	NOUN
ejpam-1506	44	28	-	-	PUNCT
ejpam-1506	44	29	line	line	NOUN
ejpam-1506	44	30	graphs	graph	NOUN
ejpam-1506	44	31	.	.	PUNCT
ejpam-1506	45	1	for	for	ADP
ejpam-1506	45	2	a	a	DET
ejpam-1506	45	3	simple	simple	ADJ
ejpam-1506	45	4	and	and	CCONJ
ejpam-1506	45	5	connected	connected	ADJ
ejpam-1506	45	6	graph	graph	NOUN
ejpam-1506	45	7	g	g	NOUN
ejpam-1506	45	8	of	of	ADP
ejpam-1506	45	9	order	order	NOUN
ejpam-1506	45	10	n	n	PRON
ejpam-1506	45	11	≥	≥	NOUN
ejpam-1506	45	12	2	2	NUM
ejpam-1506	45	13	,	,	PUNCT
ejpam-1506	45	14	we	we	PRON
ejpam-1506	45	15	show	show	VERB
ejpam-1506	45	16	that	that	SCONJ
ejpam-1506	45	17	⌈log2∆(g)⌉	⌈log2∆(g)⌉	PROPN
ejpam-1506	45	18	≤	≤	NUM
ejpam-1506	45	19	dim(g⋆	dim(g⋆	NOUN
ejpam-1506	45	20	)	)	PUNCT
ejpam-1506	45	21	≤	≤	NUM
ejpam-1506	45	22	n−	n−	NOUN
ejpam-1506	45	23	1	1	NUM
ejpam-1506	45	24	,	,	PUNCT
ejpam-1506	45	25	and	and	CCONJ
ejpam-1506	45	26	the	the	DET
ejpam-1506	45	27	bounds	bound	NOUN
ejpam-1506	45	28	are	be	AUX
ejpam-1506	45	29	sharp	sharp	ADJ
ejpam-1506	45	30	.	.	PUNCT
ejpam-1506	46	1	we	we	PRON
ejpam-1506	46	2	also	also	ADV
ejpam-1506	46	3	determine	determine	VERB
ejpam-1506	46	4	the	the	DET
ejpam-1506	46	5	metric	metric	ADJ
ejpam-1506	46	6	dimension	dimension	NOUN
ejpam-1506	46	7	of	of	ADP
ejpam-1506	46	8	para	para	NOUN
ejpam-1506	46	9	-	-	PUNCT
ejpam-1506	46	10	line	line	NOUN
ejpam-1506	46	11	graphs	graph	NOUN
ejpam-1506	46	12	for	for	ADP
ejpam-1506	46	13	some	some	DET
ejpam-1506	46	14	classes	class	NOUN
ejpam-1506	46	15	of	of	ADP
ejpam-1506	46	16	graphs	graph	NOUN
ejpam-1506	46	17	;	;	PUNCT
ejpam-1506	46	18	further	far	ADV
ejpam-1506	46	19	,	,	PUNCT
ejpam-1506	46	20	we	we	PRON
ejpam-1506	46	21	give	give	VERB
ejpam-1506	46	22	an	an	DET
ejpam-1506	46	23	example	example	NOUN
ejpam-1506	46	24	of	of	ADP
ejpam-1506	46	25	a	a	DET
ejpam-1506	46	26	graph	graph	NOUN
ejpam-1506	46	27	g	g	ADP
ejpam-1506	46	28	such	such	ADJ
ejpam-1506	46	29	that	that	DET
ejpam-1506	46	30	max{dim(g	max{dim(g	NOUN
ejpam-1506	46	31	)	)	PUNCT
ejpam-1506	46	32	,	,	PUNCT
ejpam-1506	46	33	dim(l(g	dim(l(g	NOUN
ejpam-1506	46	34	)	)	PUNCT
ejpam-1506	46	35	)	)	PUNCT
ejpam-1506	46	36	,	,	PUNCT
ejpam-1506	46	37	dim(g⋆	dim(g⋆	NOUN
ejpam-1506	46	38	)	)	PUNCT
ejpam-1506	46	39	}	}	PUNCT
ejpam-1506	46	40	equals	equal	VERB
ejpam-1506	46	41	dim(g	dim(g	PROPN
ejpam-1506	46	42	)	)	PUNCT
ejpam-1506	46	43	,	,	PUNCT
ejpam-1506	46	44	dim(l(g	dim(l(g	NOUN
ejpam-1506	46	45	)	)	PUNCT
ejpam-1506	46	46	)	)	PUNCT
ejpam-1506	46	47	,	,	PUNCT
ejpam-1506	46	48	and	and	CCONJ
ejpam-1506	46	49	dim(g⋆	dim(g⋆	NOUN
ejpam-1506	46	50	)	)	PUNCT
ejpam-1506	46	51	,	,	PUNCT
ejpam-1506	46	52	respectively	respectively	ADV
ejpam-1506	46	53	.	.	PUNCT
ejpam-1506	47	1	we	we	PRON
ejpam-1506	47	2	conclude	conclude	VERB
ejpam-1506	47	3	this	this	DET
ejpam-1506	47	4	paper	paper	NOUN
ejpam-1506	47	5	with	with	ADP
ejpam-1506	47	6	some	some	DET
ejpam-1506	47	7	open	open	ADJ
ejpam-1506	47	8	problems	problem	NOUN
ejpam-1506	47	9	.	.	PUNCT
ejpam-1506	48	1	2	2	X
ejpam-1506	48	2	.	.	X
ejpam-1506	48	3	para	para	NOUN
ejpam-1506	48	4	-	-	PUNCT
ejpam-1506	48	5	line	line	NOUN
ejpam-1506	48	6	graphs	graph	NOUN
ejpam-1506	48	7	:	:	PUNCT
ejpam-1506	48	8	applications	application	NOUN
ejpam-1506	48	9	and	and	CCONJ
ejpam-1506	48	10	basic	basic	ADJ
ejpam-1506	48	11	properties	property	NOUN
ejpam-1506	48	12	para	para	ADJ
ejpam-1506	48	13	-	-	PUNCT
ejpam-1506	48	14	line	line	NOUN
ejpam-1506	48	15	graphs	graph	NOUN
ejpam-1506	48	16	are	be	AUX
ejpam-1506	48	17	chemically	chemically	ADV
ejpam-1506	48	18	relevant	relevant	ADJ
ejpam-1506	48	19	,	,	PUNCT
ejpam-1506	48	20	though	though	SCONJ
ejpam-1506	48	21	little	little	ADJ
ejpam-1506	48	22	considered	consider	VERB
ejpam-1506	48	23	in	in	ADP
ejpam-1506	48	24	the	the	DET
ejpam-1506	48	25	surge	surge	NOUN
ejpam-1506	48	26	of	of	ADP
ejpam-1506	48	27	chemical	chemical	NOUN
ejpam-1506	48	28	graph	graph	NOUN
ejpam-1506	48	29	theory	theory	NOUN
ejpam-1506	48	30	of	of	ADP
ejpam-1506	48	31	the	the	DET
ejpam-1506	48	32	last	last	ADJ
ejpam-1506	48	33	few	few	ADJ
ejpam-1506	48	34	decades	decade	NOUN
ejpam-1506	48	35	.	.	PUNCT
ejpam-1506	49	1	often	often	ADV
ejpam-1506	49	2	a	a	DET
ejpam-1506	49	3	(	(	PUNCT
ejpam-1506	49	4	usually	usually	ADV
ejpam-1506	49	5	“	"	PUNCT
ejpam-1506	49	6	organic	organic	ADJ
ejpam-1506	49	7	”	"	PUNCT
ejpam-1506	49	8	)	)	PUNCT
ejpam-1506	49	9	molecule	molecule	NOUN
ejpam-1506	49	10	is	be	AUX
ejpam-1506	49	11	represented	represent	VERB
ejpam-1506	49	12	by	by	ADP
ejpam-1506	49	13	d.	d.	PROPN
ejpam-1506	49	14	klein	klein	PROPN
ejpam-1506	49	15	,	,	PUNCT
ejpam-1506	49	16	e.	e.	PROPN
ejpam-1506	49	17	yi	yi	PROPN
ejpam-1506	49	18	/	/	SYM
ejpam-1506	49	19	eur	eur	PROPN
ejpam-1506	49	20	.	.	PUNCT
ejpam-1506	50	1	j.	j.	PROPN
ejpam-1506	50	2	pure	pure	PROPN
ejpam-1506	50	3	appl	appl	PROPN
ejpam-1506	50	4	.	.	PROPN
ejpam-1506	50	5	math	math	PROPN
ejpam-1506	50	6	,	,	PUNCT
ejpam-1506	50	7	5	5	NUM
ejpam-1506	50	8	(	(	PUNCT
ejpam-1506	50	9	2012	2012	NUM
ejpam-1506	50	10	)	)	PUNCT
ejpam-1506	50	11	,	,	PUNCT
ejpam-1506	50	12	302	302	NUM
ejpam-1506	50	13	-	-	SYM
ejpam-1506	50	14	316	316	NUM
ejpam-1506	50	15	304	304	NUM
ejpam-1506	50	16	a	a	DET
ejpam-1506	50	17	graph	graph	NOUN
ejpam-1506	50	18	whose	whose	DET
ejpam-1506	50	19	vertices	vertex	NOUN
ejpam-1506	50	20	correspond	correspond	VERB
ejpam-1506	50	21	to	to	ADP
ejpam-1506	50	22	atoms	atom	NOUN
ejpam-1506	50	23	other	other	ADJ
ejpam-1506	50	24	than	than	ADP
ejpam-1506	50	25	hydrogen	hydrogen	NOUN
ejpam-1506	50	26	,	,	PUNCT
ejpam-1506	50	27	and	and	CCONJ
ejpam-1506	50	28	whose	whose	DET
ejpam-1506	50	29	edges	edge	NOUN
ejpam-1506	50	30	correspond	correspond	VERB
ejpam-1506	50	31	to	to	ADP
ejpam-1506	50	32	bonded	bond	VERB
ejpam-1506	50	33	pairs	pair	NOUN
ejpam-1506	50	34	of	of	ADP
ejpam-1506	50	35	such	such	ADJ
ejpam-1506	50	36	atoms	atom	NOUN
ejpam-1506	50	37	.	.	PUNCT
ejpam-1506	51	1	for	for	ADP
ejpam-1506	51	2	example	example	NOUN
ejpam-1506	51	3	,	,	PUNCT
ejpam-1506	51	4	for	for	ADP
ejpam-1506	51	5	stable	stable	ADJ
ejpam-1506	51	6	hydrocarbon	hydrocarbon	NOUN
ejpam-1506	51	7	molecules	molecule	NOUN
ejpam-1506	51	8	,	,	PUNCT
ejpam-1506	51	9	it	it	PRON
ejpam-1506	51	10	is	be	AUX
ejpam-1506	51	11	understood	understand	VERB
ejpam-1506	51	12	that	that	SCONJ
ejpam-1506	51	13	the	the	DET
ejpam-1506	51	14	carbon	carbon	NOUN
ejpam-1506	51	15	atoms	atom	NOUN
ejpam-1506	51	16	have	have	VERB
ejpam-1506	51	17	valence	valence	NOUN
ejpam-1506	51	18	4	4	NUM
ejpam-1506	51	19	counting	counting	NOUN
ejpam-1506	51	20	connections	connection	NOUN
ejpam-1506	51	21	to	to	ADP
ejpam-1506	51	22	h	h	NOUN
ejpam-1506	51	23	atoms	atom	NOUN
ejpam-1506	51	24	,	,	PUNCT
ejpam-1506	51	25	so	so	SCONJ
ejpam-1506	51	26	that	that	SCONJ
ejpam-1506	51	27	this	this	DET
ejpam-1506	51	28	h	h	NOUN
ejpam-1506	51	29	-	-	PUNCT
ejpam-1506	51	30	deleted	delete	VERB
ejpam-1506	51	31	graph	graph	NOUN
ejpam-1506	51	32	determines	determine	VERB
ejpam-1506	51	33	the	the	DET
ejpam-1506	51	34	molecular	molecular	ADJ
ejpam-1506	51	35	structure	structure	NOUN
ejpam-1506	51	36	(	(	PUNCT
ejpam-1506	51	37	see	see	VERB
ejpam-1506	51	38	figure	figure	NOUN
ejpam-1506	51	39	1	1	NUM
ejpam-1506	51	40	)	)	PUNCT
ejpam-1506	51	41	.	.	PUNCT
ejpam-1506	52	1	but	but	CCONJ
ejpam-1506	52	2	there	there	PRON
ejpam-1506	52	3	are	be	VERB
ejpam-1506	52	4	other	other	ADJ
ejpam-1506	52	5	ways	way	NOUN
ejpam-1506	52	6	to	to	PART
ejpam-1506	52	7	represent	represent	VERB
ejpam-1506	52	8	molecules	molecule	NOUN
ejpam-1506	52	9	in	in	ADP
ejpam-1506	52	10	terms	term	NOUN
ejpam-1506	52	11	of	of	ADP
ejpam-1506	52	12	graphs	graph	NOUN
ejpam-1506	52	13	,	,	PUNCT
ejpam-1506	52	14	say	say	VERB
ejpam-1506	52	15	in	in	ADP
ejpam-1506	52	16	terms	term	NOUN
ejpam-1506	52	17	of	of	ADP
ejpam-1506	52	18	a	a	DET
ejpam-1506	52	19	minimal	minimal	ADJ
ejpam-1506	52	20	set	set	NOUN
ejpam-1506	52	21	of	of	ADP
ejpam-1506	52	22	localized	localized	ADJ
ejpam-1506	52	23	“	"	PUNCT
ejpam-1506	52	24	orbitals	orbital	NOUN
ejpam-1506	52	25	”	"	PUNCT
ejpam-1506	52	26	each	each	PRON
ejpam-1506	52	27	taken	take	VERB
ejpam-1506	52	28	as	as	ADP
ejpam-1506	52	29	a	a	DET
ejpam-1506	52	30	vertex	vertex	NOUN
ejpam-1506	52	31	,	,	PUNCT
ejpam-1506	52	32	with	with	ADP
ejpam-1506	52	33	edges	edge	NOUN
ejpam-1506	52	34	identified	identify	VERB
ejpam-1506	52	35	to	to	ADP
ejpam-1506	52	36	stronger	strong	ADJ
ejpam-1506	52	37	interactions	interaction	NOUN
ejpam-1506	52	38	between	between	ADP
ejpam-1506	52	39	pairs	pair	NOUN
ejpam-1506	52	40	of	of	ADP
ejpam-1506	52	41	orbitals	orbital	NOUN
ejpam-1506	52	42	.	.	PUNCT
ejpam-1506	53	1	indeed	indeed	ADV
ejpam-1506	53	2	such	such	DET
ejpam-1506	53	3	a	a	DET
ejpam-1506	53	4	graph	graph	NOUN
ejpam-1506	53	5	was	be	AUX
ejpam-1506	53	6	implicit	implicit	ADJ
ejpam-1506	53	7	in	in	ADP
ejpam-1506	53	8	several	several	ADJ
ejpam-1506	53	9	early	early	ADJ
ejpam-1506	53	10	quantum	quantum	ADJ
ejpam-1506	53	11	chemical	chemical	NOUN
ejpam-1506	53	12	works	work	NOUN
ejpam-1506	53	13	presaging	presage	VERB
ejpam-1506	53	14	the	the	DET
ejpam-1506	53	15	first	first	ADJ
ejpam-1506	53	16	-	-	PUNCT
ejpam-1506	53	17	principles	principle	NOUN
ejpam-1506	53	18	quantum	quantum	NOUN
ejpam-1506	53	19	chemistry	chemistry	NOUN
ejpam-1506	53	20	mediated	mediate	VERB
ejpam-1506	53	21	by	by	ADP
ejpam-1506	53	22	way	way	NOUN
ejpam-1506	53	23	of	of	ADP
ejpam-1506	53	24	large	large	ADJ
ejpam-1506	53	25	computers	computer	NOUN
ejpam-1506	53	26	.	.	PUNCT
ejpam-1506	54	1	this	this	DET
ejpam-1506	54	2	now	now	ADV
ejpam-1506	54	3	standard	standard	ADJ
ejpam-1506	54	4	quantum	quantum	ADJ
ejpam-1506	54	5	chemical	chemical	NOUN
ejpam-1506	54	6	approach	approach	NOUN
ejpam-1506	54	7	has	have	AUX
ejpam-1506	54	8	been	be	AUX
ejpam-1506	54	9	tremendously	tremendously	ADV
ejpam-1506	54	10	successful	successful	ADJ
ejpam-1506	54	11	in	in	ADP
ejpam-1506	54	12	treating	treat	VERB
ejpam-1506	54	13	molecules	molecule	NOUN
ejpam-1506	54	14	one	one	NUM
ejpam-1506	54	15	by	by	ADP
ejpam-1506	54	16	one	one	NUM
ejpam-1506	54	17	.	.	PUNCT
ejpam-1506	55	1	but	but	CCONJ
ejpam-1506	55	2	the	the	DET
ejpam-1506	55	3	simpler	simple	ADJ
ejpam-1506	55	4	orbital	orbital	ADJ
ejpam-1506	55	5	model	model	NOUN
ejpam-1506	55	6	offers	offer	VERB
ejpam-1506	55	7	a	a	DET
ejpam-1506	55	8	potential	potential	ADJ
ejpam-1506	55	9	advantage	advantage	NOUN
ejpam-1506	55	10	of	of	ADP
ejpam-1506	55	11	general	general	ADJ
ejpam-1506	55	12	meaningful	meaningful	ADJ
ejpam-1506	55	13	theorems	theorem	NOUN
ejpam-1506	55	14	applying	apply	VERB
ejpam-1506	55	15	to	to	ADP
ejpam-1506	55	16	whole	whole	ADJ
ejpam-1506	55	17	classes	class	NOUN
ejpam-1506	55	18	of	of	ADP
ejpam-1506	55	19	molecules	molecule	NOUN
ejpam-1506	55	20	.	.	PUNCT
ejpam-1506	56	1	one	one	PRON
ejpam-1506	56	2	can	can	AUX
ejpam-1506	56	3	view	view	VERB
ejpam-1506	56	4	para	para	ADJ
ejpam-1506	56	5	-	-	PUNCT
ejpam-1506	56	6	line	line	NOUN
ejpam-1506	56	7	graphs	graph	NOUN
ejpam-1506	56	8	with	with	ADP
ejpam-1506	56	9	vertices	vertex	NOUN
ejpam-1506	56	10	corresponding	correspond	VERB
ejpam-1506	56	11	to	to	ADP
ejpam-1506	56	12	atomic	atomic	ADJ
ejpam-1506	56	13	“	"	PUNCT
ejpam-1506	56	14	hybrid	hybrid	ADJ
ejpam-1506	56	15	”	"	PUNCT
ejpam-1506	56	16	orbitals	orbital	NOUN
ejpam-1506	56	17	,	,	PUNCT
ejpam-1506	56	18	and	and	CCONJ
ejpam-1506	56	19	edges	edge	NOUN
ejpam-1506	56	20	corresponding	correspond	VERB
ejpam-1506	56	21	to	to	ADP
ejpam-1506	56	22	stronger	strong	ADJ
ejpam-1506	56	23	interactions	interaction	NOUN
ejpam-1506	56	24	between	between	ADP
ejpam-1506	56	25	pairs	pair	NOUN
ejpam-1506	56	26	of	of	ADP
ejpam-1506	56	27	such	such	ADJ
ejpam-1506	56	28	orbitals	orbital	NOUN
ejpam-1506	56	29	.	.	PUNCT
ejpam-1506	57	1	given	give	VERB
ejpam-1506	57	2	a	a	DET
ejpam-1506	57	3	traditional	traditional	ADJ
ejpam-1506	57	4	molecular	molecular	ADJ
ejpam-1506	57	5	structural	structural	ADJ
ejpam-1506	57	6	formula	formula	NOUN
ejpam-1506	57	7	for	for	ADP
ejpam-1506	57	8	a	a	DET
ejpam-1506	57	9	hydrocarbon	hydrocarbon	NOUN
ejpam-1506	57	10	without	without	ADP
ejpam-1506	57	11	multiple	multiple	ADJ
ejpam-1506	57	12	bonds	bond	NOUN
ejpam-1506	57	13	,	,	PUNCT
ejpam-1506	57	14	there	there	PRON
ejpam-1506	57	15	corresponds	correspond	VERB
ejpam-1506	57	16	a	a	DET
ejpam-1506	57	17	graph	graph	NOUN
ejpam-1506	57	18	g	g	NOUN
ejpam-1506	57	19	with	with	ADP
ejpam-1506	57	20	one	one	NUM
ejpam-1506	57	21	vertex	vertex	NOUN
ejpam-1506	57	22	corresponding	correspond	VERB
ejpam-1506	57	23	to	to	ADP
ejpam-1506	57	24	each	each	DET
ejpam-1506	57	25	atom	atom	NOUN
ejpam-1506	57	26	(	(	PUNCT
ejpam-1506	57	27	either	either	CCONJ
ejpam-1506	57	28	c	c	PROPN
ejpam-1506	57	29	or	or	CCONJ
ejpam-1506	57	30	h	h	NOUN
ejpam-1506	57	31	)	)	PUNCT
ejpam-1506	57	32	,	,	PUNCT
ejpam-1506	57	33	and	and	CCONJ
ejpam-1506	57	34	an	an	DET
ejpam-1506	57	35	edge	edge	NOUN
ejpam-1506	57	36	corresponding	correspond	VERB
ejpam-1506	57	37	to	to	ADP
ejpam-1506	57	38	each	each	DET
ejpam-1506	57	39	bond	bond	NOUN
ejpam-1506	57	40	.	.	PUNCT
ejpam-1506	58	1	(	(	PUNCT
ejpam-1506	58	2	a	a	X
ejpam-1506	58	3	)	)	PUNCT
ejpam-1506	58	4	hydrocarbon	hydrocarbon	NOUN
ejpam-1506	58	5	c2h6	c2h6	PROPN
ejpam-1506	58	6	(	(	PUNCT
ejpam-1506	58	7	b	b	NOUN
ejpam-1506	58	8	)	)	PUNCT
ejpam-1506	58	9	hydrocarbon	hydrocarbon	NOUN
ejpam-1506	58	10	graph	graph	NOUN
ejpam-1506	58	11	g	g	PROPN
ejpam-1506	58	12	(	(	PUNCT
ejpam-1506	58	13	c	c	X
ejpam-1506	58	14	)	)	PUNCT
ejpam-1506	58	15	its	its	PRON
ejpam-1506	58	16	para	para	NOUN
ejpam-1506	58	17	-	-	PUNCT
ejpam-1506	58	18	line	line	NOUN
ejpam-1506	58	19	graph	graph	NOUN
ejpam-1506	58	20	g⋆	g⋆	NOUN
ejpam-1506	58	21	figure	figure	VERB
ejpam-1506	58	22	1	1	NUM
ejpam-1506	58	23	if	if	SCONJ
ejpam-1506	58	24	we	we	PRON
ejpam-1506	58	25	let	let	VERB
ejpam-1506	58	26	e⋆in	e⋆in	PRON
ejpam-1506	58	27	=	=	VERB
ejpam-1506	58	28	∪u∈v	∪u∈v	PROPN
ejpam-1506	58	29	(	(	PUNCT
ejpam-1506	58	30	g)e(k(u	g)e(k(u	NOUN
ejpam-1506	58	31	)	)	PUNCT
ejpam-1506	58	32	)	)	PUNCT
ejpam-1506	58	33	,	,	PUNCT
ejpam-1506	58	34	then	then	ADV
ejpam-1506	58	35	the	the	DET
ejpam-1506	58	36	usual	usual	ADJ
ejpam-1506	58	37	chemical	chemical	NOUN
ejpam-1506	58	38	bonds	bond	NOUN
ejpam-1506	58	39	are	be	AUX
ejpam-1506	58	40	manifested	manifest	VERB
ejpam-1506	58	41	in	in	ADP
ejpam-1506	58	42	e⋆	e⋆	NUM
ejpam-1506	58	43	bond	bond	NOUN
ejpam-1506	58	44	=	=	NOUN
ejpam-1506	58	45	e(g⋆)−	e(g⋆)−	X
ejpam-1506	58	46	e⋆in	e⋆in	X
ejpam-1506	58	47	.	.	PUNCT
ejpam-1506	59	1	for	for	ADP
ejpam-1506	59	2	a	a	DET
ejpam-1506	59	3	hydrocarbon	hydrocarbon	NOUN
ejpam-1506	59	4	,	,	PUNCT
ejpam-1506	59	5	each	each	DET
ejpam-1506	59	6	c	c	NOUN
ejpam-1506	59	7	atom	atom	NOUN
ejpam-1506	59	8	is	be	AUX
ejpam-1506	59	9	represented	represent	VERB
ejpam-1506	59	10	by	by	ADP
ejpam-1506	59	11	a	a	DET
ejpam-1506	59	12	tetrahedral	tetrahedral	ADJ
ejpam-1506	59	13	quartet	quartet	NOUN
ejpam-1506	59	14	of	of	ADP
ejpam-1506	59	15	interconnected	interconnected	ADJ
ejpam-1506	59	16	sites	site	NOUN
ejpam-1506	59	17	(	(	PUNCT
ejpam-1506	59	18	a	a	DET
ejpam-1506	59	19	4	4	NUM
ejpam-1506	59	20	-	-	PUNCT
ejpam-1506	59	21	vertex	vertex	NOUN
ejpam-1506	59	22	clique	clique	NOUN
ejpam-1506	59	23	)	)	PUNCT
ejpam-1506	59	24	,	,	PUNCT
ejpam-1506	59	25	each	each	DET
ejpam-1506	59	26	vertex	vertex	NOUN
ejpam-1506	59	27	corresponding	correspond	VERB
ejpam-1506	59	28	to	to	ADP
ejpam-1506	59	29	a	a	DET
ejpam-1506	59	30	different	different	ADJ
ejpam-1506	59	31	(	(	PUNCT
ejpam-1506	59	32	so	so	ADV
ejpam-1506	59	33	-	-	PUNCT
ejpam-1506	59	34	called	call	VERB
ejpam-1506	59	35	sp	sp	NOUN
ejpam-1506	59	36	)	)	PUNCT
ejpam-1506	59	37	hybrid	hybrid	ADJ
ejpam-1506	59	38	orbital	orbital	NOUN
ejpam-1506	59	39	;	;	PUNCT
ejpam-1506	59	40	each	each	DET
ejpam-1506	59	41	h	h	NOUN
ejpam-1506	59	42	atom	atom	NOUN
ejpam-1506	59	43	by	by	ADP
ejpam-1506	59	44	a	a	DET
ejpam-1506	59	45	single	single	ADJ
ejpam-1506	59	46	atomic	atomic	NOUN
ejpam-1506	59	47	(	(	PUNCT
ejpam-1506	59	48	1s	1s	NUM
ejpam-1506	59	49	)	)	PUNCT
ejpam-1506	59	50	orbital	orbital	NOUN
ejpam-1506	59	51	;	;	PUNCT
ejpam-1506	59	52	each	each	DET
ejpam-1506	59	53	edge	edge	NOUN
ejpam-1506	59	54	of	of	ADP
ejpam-1506	59	55	e⋆	e⋆	X
ejpam-1506	59	56	bond	bond	NOUN
ejpam-1506	59	57	corresponding	correspond	VERB
ejpam-1506	59	58	to	to	ADP
ejpam-1506	59	59	an	an	DET
ejpam-1506	59	60	interatomic	interatomic	ADJ
ejpam-1506	59	61	chemical	chemical	NOUN
ejpam-1506	59	62	molecular	molecular	ADJ
ejpam-1506	59	63	bond	bond	NOUN
ejpam-1506	59	64	;	;	PUNCT
ejpam-1506	59	65	and	and	CCONJ
ejpam-1506	59	66	each	each	DET
ejpam-1506	59	67	edge	edge	NOUN
ejpam-1506	59	68	of	of	ADP
ejpam-1506	59	69	e⋆in	e⋆in	NOUN
ejpam-1506	59	70	by	by	ADP
ejpam-1506	59	71	an	an	DET
ejpam-1506	59	72	intra	intra	ADJ
ejpam-1506	59	73	-	-	ADJ
ejpam-1506	59	74	atomic	atomic	ADJ
ejpam-1506	59	75	interaction	interaction	NOUN
ejpam-1506	59	76	.	.	PUNCT
ejpam-1506	60	1	often	often	ADV
ejpam-1506	60	2	,	,	PUNCT
ejpam-1506	60	3	the	the	DET
ejpam-1506	60	4	vertices	vertex	NOUN
ejpam-1506	60	5	of	of	ADP
ejpam-1506	60	6	the	the	DET
ejpam-1506	60	7	resulting	result	VERB
ejpam-1506	60	8	para	para	ADJ
ejpam-1506	60	9	-	-	PUNCT
ejpam-1506	60	10	line	line	NOUN
ejpam-1506	60	11	graph	graph	NOUN
ejpam-1506	60	12	are	be	AUX
ejpam-1506	60	13	assigned	assign	VERB
ejpam-1506	60	14	different	different	ADJ
ejpam-1506	60	15	weights	weight	NOUN
ejpam-1506	60	16	corresponding	correspond	VERB
ejpam-1506	60	17	to	to	ADP
ejpam-1506	60	18	c	c	NOUN
ejpam-1506	60	19	or	or	CCONJ
ejpam-1506	60	20	h	h	NOUN
ejpam-1506	60	21	atoms	atom	NOUN
ejpam-1506	60	22	;	;	PUNCT
ejpam-1506	60	23	also	also	ADV
ejpam-1506	60	24	,	,	PUNCT
ejpam-1506	60	25	it	it	PRON
ejpam-1506	60	26	’s	’s	AUX
ejpam-1506	60	27	typically	typically	ADV
ejpam-1506	60	28	given	give	VERB
ejpam-1506	60	29	different	different	ADJ
ejpam-1506	60	30	edge	edge	NOUN
ejpam-1506	60	31	weights	weight	NOUN
ejpam-1506	60	32	corresponding	correspond	VERB
ejpam-1506	60	33	to	to	ADP
ejpam-1506	60	34	the	the	DET
ejpam-1506	60	35	c−c	c−c	NOUN
ejpam-1506	60	36	and	and	CCONJ
ejpam-1506	60	37	c−h	c−h	NOUN
ejpam-1506	60	38	bonds	bond	NOUN
ejpam-1506	60	39	.	.	PUNCT
ejpam-1506	61	1	there	there	PRON
ejpam-1506	61	2	are	be	VERB
ejpam-1506	61	3	also	also	ADV
ejpam-1506	61	4	the	the	DET
ejpam-1506	61	5	intra	intra	ADJ
ejpam-1506	61	6	-	-	ADJ
ejpam-1506	61	7	atomic	atomic	ADJ
ejpam-1506	61	8	interactions	interaction	NOUN
ejpam-1506	61	9	between	between	ADP
ejpam-1506	61	10	different	different	ADJ
ejpam-1506	61	11	hybrid	hybrid	NOUN
ejpam-1506	61	12	orbitals	orbital	NOUN
ejpam-1506	61	13	on	on	ADP
ejpam-1506	61	14	the	the	DET
ejpam-1506	61	15	same	same	ADJ
ejpam-1506	61	16	atom	atom	NOUN
ejpam-1506	61	17	,	,	PUNCT
ejpam-1506	61	18	and	and	CCONJ
ejpam-1506	61	19	then	then	ADV
ejpam-1506	61	20	the	the	DET
ejpam-1506	61	21	intra	intra	ADJ
ejpam-1506	61	22	-	-	ADJ
ejpam-1506	61	23	atomic	atomic	ADJ
ejpam-1506	61	24	edge	edge	NOUN
ejpam-1506	61	25	weights	weight	NOUN
ejpam-1506	61	26	would	would	AUX
ejpam-1506	61	27	be	be	AUX
ejpam-1506	61	28	somewhat	somewhat	ADV
ejpam-1506	61	29	less	less	ADJ
ejpam-1506	61	30	than	than	ADP
ejpam-1506	61	31	the	the	DET
ejpam-1506	61	32	inter	inter	ADJ
ejpam-1506	61	33	-	-	ADJ
ejpam-1506	61	34	atomic	atomic	ADJ
ejpam-1506	61	35	weights	weight	NOUN
ejpam-1506	61	36	.	.	PUNCT
ejpam-1506	62	1	for	for	ADP
ejpam-1506	62	2	articles	article	NOUN
ejpam-1506	62	3	on	on	ADP
ejpam-1506	62	4	chemical	chemical	NOUN
ejpam-1506	62	5	graphs	graph	NOUN
ejpam-1506	62	6	,	,	PUNCT
ejpam-1506	62	7	see	see	VERB
ejpam-1506	62	8	[	[	X
ejpam-1506	62	9	11	11	NUM
ejpam-1506	62	10	,	,	PUNCT
ejpam-1506	62	11	20	20	NUM
ejpam-1506	62	12	,	,	PUNCT
ejpam-1506	62	13	21	21	NUM
ejpam-1506	62	14	,	,	PUNCT
ejpam-1506	62	15	22	22	NUM
ejpam-1506	62	16	]	]	PUNCT
ejpam-1506	62	17	.	.	PUNCT
ejpam-1506	63	1	next	next	ADV
ejpam-1506	63	2	,	,	PUNCT
ejpam-1506	63	3	we	we	PRON
ejpam-1506	63	4	recall	recall	VERB
ejpam-1506	63	5	the	the	DET
ejpam-1506	63	6	following	follow	VERB
ejpam-1506	63	7	theorem	theorem	NOUN
ejpam-1506	63	8	1	1	NUM
ejpam-1506	63	9	.	.	PUNCT
ejpam-1506	64	1	[	[	X
ejpam-1506	64	2	29	29	NUM
ejpam-1506	64	3	]	]	PUNCT
ejpam-1506	64	4	let	let	VERB
ejpam-1506	64	5	g1	g1	PROPN
ejpam-1506	64	6	and	and	CCONJ
ejpam-1506	64	7	g2	g2	PROPN
ejpam-1506	64	8	be	be	AUX
ejpam-1506	64	9	connected	connect	VERB
ejpam-1506	64	10	graphs	graph	NOUN
ejpam-1506	64	11	with	with	ADP
ejpam-1506	64	12	g1	g1	PROPN
ejpam-1506	64	13	6∼=	6∼=	NUM
ejpam-1506	64	14	g2	g2	PROPN
ejpam-1506	64	15	.	.	PUNCT
ejpam-1506	65	1	then	then	ADV
ejpam-1506	65	2	l(g1	l(g1	ADV
ejpam-1506	65	3	)	)	PUNCT
ejpam-1506	65	4	∼=	∼=	VERB
ejpam-1506	65	5	l(g2	l(g2	ADJ
ejpam-1506	65	6	)	)	PUNCT
ejpam-1506	66	1	if	if	SCONJ
ejpam-1506	66	2	and	and	CCONJ
ejpam-1506	66	3	only	only	ADV
ejpam-1506	66	4	if	if	SCONJ
ejpam-1506	66	5	{	{	PUNCT
ejpam-1506	66	6	g1	g1	X
ejpam-1506	66	7	,	,	PUNCT
ejpam-1506	66	8	g2	g2	PROPN
ejpam-1506	66	9	}	}	PUNCT
ejpam-1506	66	10	=	=	SYM
ejpam-1506	66	11	{	{	PUNCT
ejpam-1506	66	12	c3	c3	NOUN
ejpam-1506	66	13	,	,	PUNCT
ejpam-1506	66	14	k1,3	k1,3	PROPN
ejpam-1506	66	15	}	}	PUNCT
ejpam-1506	66	16	,	,	PUNCT
ejpam-1506	66	17	where	where	SCONJ
ejpam-1506	66	18	k1,3	k1,3	PROPN
ejpam-1506	66	19	is	be	AUX
ejpam-1506	66	20	the	the	DET
ejpam-1506	66	21	star	star	NOUN
ejpam-1506	66	22	on	on	ADP
ejpam-1506	66	23	4	4	NUM
ejpam-1506	66	24	vertices	vertex	NOUN
ejpam-1506	66	25	.	.	PUNCT
ejpam-1506	67	1	in	in	ADP
ejpam-1506	67	2	[	[	X
ejpam-1506	67	3	15	15	NUM
ejpam-1506	67	4	]	]	PUNCT
ejpam-1506	67	5	,	,	PUNCT
ejpam-1506	67	6	it	it	PRON
ejpam-1506	67	7	is	be	AUX
ejpam-1506	67	8	shown	show	VERB
ejpam-1506	67	9	that	that	SCONJ
ejpam-1506	67	10	g⋆1	g⋆1	PROPN
ejpam-1506	67	11	∼=	∼=	PART
ejpam-1506	67	12	g⋆2	g⋆2	NOUN
ejpam-1506	67	13	if	if	SCONJ
ejpam-1506	67	14	and	and	CCONJ
ejpam-1506	67	15	only	only	ADV
ejpam-1506	67	16	if	if	SCONJ
ejpam-1506	67	17	g1	g1	NOUN
ejpam-1506	67	18	∼=	∼=	PART
ejpam-1506	67	19	g2	g2	NOUN
ejpam-1506	67	20	.	.	PUNCT
ejpam-1506	68	1	in	in	ADP
ejpam-1506	68	2	order	order	NOUN
ejpam-1506	68	3	to	to	PART
ejpam-1506	68	4	be	be	AUX
ejpam-1506	68	5	self	self	NOUN
ejpam-1506	68	6	-	-	PUNCT
ejpam-1506	68	7	contained	contain	VERB
ejpam-1506	68	8	,	,	PUNCT
ejpam-1506	68	9	we	we	PRON
ejpam-1506	68	10	include	include	VERB
ejpam-1506	68	11	a	a	DET
ejpam-1506	68	12	proof	proof	NOUN
ejpam-1506	68	13	here	here	ADV
ejpam-1506	68	14	.	.	PUNCT
ejpam-1506	69	1	d.	d.	PROPN
ejpam-1506	69	2	klein	klein	PROPN
ejpam-1506	69	3	,	,	PUNCT
ejpam-1506	69	4	e.	e.	PROPN
ejpam-1506	69	5	yi	yi	PROPN
ejpam-1506	69	6	/	/	SYM
ejpam-1506	69	7	eur	eur	PROPN
ejpam-1506	69	8	.	.	PUNCT
ejpam-1506	70	1	j.	j.	PROPN
ejpam-1506	70	2	pure	pure	PROPN
ejpam-1506	70	3	appl	appl	PROPN
ejpam-1506	70	4	.	.	PROPN
ejpam-1506	70	5	math	math	PROPN
ejpam-1506	70	6	,	,	PUNCT
ejpam-1506	70	7	5	5	NUM
ejpam-1506	70	8	(	(	PUNCT
ejpam-1506	70	9	2012	2012	NUM
ejpam-1506	70	10	)	)	PUNCT
ejpam-1506	70	11	,	,	PUNCT
ejpam-1506	70	12	302	302	NUM
ejpam-1506	70	13	-	-	SYM
ejpam-1506	70	14	316	316	NUM
ejpam-1506	70	15	305	305	NUM
ejpam-1506	70	16	theorem	theorem	NOUN
ejpam-1506	70	17	2	2	NUM
ejpam-1506	70	18	.	.	PUNCT
ejpam-1506	71	1	[	[	X
ejpam-1506	71	2	15	15	NUM
ejpam-1506	71	3	]	]	PUNCT
ejpam-1506	71	4	let	let	VERB
ejpam-1506	71	5	g1	g1	PROPN
ejpam-1506	71	6	and	and	CCONJ
ejpam-1506	71	7	g2	g2	PROPN
ejpam-1506	71	8	be	be	AUX
ejpam-1506	71	9	connected	connect	VERB
ejpam-1506	71	10	graphs	graph	NOUN
ejpam-1506	71	11	.	.	PUNCT
ejpam-1506	72	1	then	then	ADV
ejpam-1506	72	2	g⋆1	g⋆1	PROPN
ejpam-1506	72	3	∼=	∼=	PROPN
ejpam-1506	72	4	g⋆2	g⋆2	NOUN
ejpam-1506	72	5	if	if	SCONJ
ejpam-1506	72	6	and	and	CCONJ
ejpam-1506	72	7	only	only	ADV
ejpam-1506	72	8	if	if	SCONJ
ejpam-1506	72	9	g1	g1	NOUN
ejpam-1506	72	10	∼=	∼=	PART
ejpam-1506	72	11	g2	g2	NOUN
ejpam-1506	72	12	.	.	PUNCT
ejpam-1506	73	1	proof	proof	NOUN
ejpam-1506	73	2	.	.	PUNCT
ejpam-1506	74	1	(	(	PUNCT
ejpam-1506	74	2	⇐	⇐	ADP
ejpam-1506	74	3	=)	=)	PROPN
ejpam-1506	74	4	it	it	PRON
ejpam-1506	74	5	is	be	AUX
ejpam-1506	74	6	obvious	obvious	ADJ
ejpam-1506	74	7	.	.	PUNCT
ejpam-1506	75	1	(=	(=	AUX
ejpam-1506	75	2	⇒	⇒	NOUN
ejpam-1506	75	3	)	)	PUNCT
ejpam-1506	75	4	let	let	VERB
ejpam-1506	75	5	s(g	s(g	PROPN
ejpam-1506	75	6	)	)	PUNCT
ejpam-1506	75	7	denote	denote	VERB
ejpam-1506	75	8	the	the	DET
ejpam-1506	75	9	subdivision	subdivision	NOUN
ejpam-1506	75	10	graph	graph	NOUN
ejpam-1506	75	11	of	of	ADP
ejpam-1506	75	12	a	a	DET
ejpam-1506	75	13	graph	graph	NOUN
ejpam-1506	75	14	g	g	NOUN
ejpam-1506	75	15	;	;	PUNCT
ejpam-1506	75	16	notice	notice	VERB
ejpam-1506	75	17	that	that	SCONJ
ejpam-1506	75	18	|e(s(g))|=	|e(s(g))|=	ADJ
ejpam-1506	75	19	2|e(g)|	2|e(g)|	NUM
ejpam-1506	75	20	,	,	PUNCT
ejpam-1506	75	21	an	an	DET
ejpam-1506	75	22	even	even	ADJ
ejpam-1506	75	23	number	number	NOUN
ejpam-1506	75	24	.	.	PUNCT
ejpam-1506	76	1	then	then	ADV
ejpam-1506	76	2	g⋆1	g⋆1	PROPN
ejpam-1506	76	3	∼=	∼=	PROPN
ejpam-1506	76	4	g⋆2	g⋆2	NOUN
ejpam-1506	76	5	implies	imply	VERB
ejpam-1506	76	6	l(s(g1	l(s(g1	ADJ
ejpam-1506	76	7	)	)	PUNCT
ejpam-1506	76	8	)	)	PUNCT
ejpam-1506	77	1	∼=	∼=	VERB
ejpam-1506	77	2	l(s(g2	l(s(g2	ADJ
ejpam-1506	77	3	)	)	PUNCT
ejpam-1506	77	4	)	)	PUNCT
ejpam-1506	77	5	.	.	PUNCT
ejpam-1506	78	1	if	if	SCONJ
ejpam-1506	78	2	s(g1	s(g1	NOUN
ejpam-1506	78	3	)	)	PUNCT
ejpam-1506	78	4	6∼=	6∼=	NUM
ejpam-1506	78	5	s(g2	s(g2	NOUN
ejpam-1506	78	6	)	)	PUNCT
ejpam-1506	78	7	,	,	PUNCT
ejpam-1506	78	8	then	then	ADV
ejpam-1506	78	9	s(g1	s(g1	VERB
ejpam-1506	78	10	)	)	PUNCT
ejpam-1506	79	1	=	=	SYM
ejpam-1506	79	2	c3	c3	PROPN
ejpam-1506	79	3	or	or	CCONJ
ejpam-1506	79	4	s(g1	s(g1	NOUN
ejpam-1506	79	5	)	)	PUNCT
ejpam-1506	80	1	=	=	SYM
ejpam-1506	80	2	k1,3	k1,3	X
ejpam-1506	80	3	by	by	ADP
ejpam-1506	80	4	theorem	theorem	NOUN
ejpam-1506	80	5	1	1	NUM
ejpam-1506	80	6	;	;	PUNCT
ejpam-1506	80	7	in	in	ADP
ejpam-1506	80	8	each	each	DET
ejpam-1506	80	9	case	case	NOUN
ejpam-1506	80	10	,	,	PUNCT
ejpam-1506	80	11	|e(s(g1)|	|e(s(g1)|	NOUN
ejpam-1506	80	12	=	=	SYM
ejpam-1506	80	13	3	3	NUM
ejpam-1506	80	14	,	,	PUNCT
ejpam-1506	80	15	an	an	DET
ejpam-1506	80	16	odd	odd	ADJ
ejpam-1506	80	17	number	number	NOUN
ejpam-1506	80	18	,	,	PUNCT
ejpam-1506	80	19	which	which	PRON
ejpam-1506	80	20	is	be	AUX
ejpam-1506	80	21	impossible	impossible	ADJ
ejpam-1506	80	22	.	.	PUNCT
ejpam-1506	81	1	thus	thus	ADV
ejpam-1506	81	2	,	,	PUNCT
ejpam-1506	81	3	s(g1	s(g1	NOUN
ejpam-1506	81	4	)	)	PUNCT
ejpam-1506	81	5	∼=	∼=	PROPN
ejpam-1506	81	6	s(g2	s(g2	NOUN
ejpam-1506	81	7	)	)	PUNCT
ejpam-1506	81	8	.	.	PUNCT
ejpam-1506	82	1	it	it	PRON
ejpam-1506	82	2	remains	remain	VERB
ejpam-1506	82	3	to	to	PART
ejpam-1506	82	4	show	show	VERB
ejpam-1506	82	5	that	that	SCONJ
ejpam-1506	82	6	g1	g1	VERB
ejpam-1506	82	7	∼=	∼=	PROPN
ejpam-1506	82	8	g2	g2	PROPN
ejpam-1506	82	9	.	.	PUNCT
ejpam-1506	83	1	notice	notice	VERB
ejpam-1506	83	2	that	that	SCONJ
ejpam-1506	83	3	s(gi	s(gi	NOUN
ejpam-1506	83	4	)	)	PUNCT
ejpam-1506	83	5	is	be	AUX
ejpam-1506	83	6	a	a	DET
ejpam-1506	83	7	bi	bi	ADJ
ejpam-1506	83	8	-	-	ADJ
ejpam-1506	83	9	partite	partite	ADJ
ejpam-1506	83	10	graph	graph	NOUN
ejpam-1506	83	11	with	with	ADP
ejpam-1506	83	12	a	a	DET
ejpam-1506	83	13	unique	unique	ADJ
ejpam-1506	83	14	bi	bi	NOUN
ejpam-1506	83	15	-	-	NOUN
ejpam-1506	83	16	partition	partition	NOUN
ejpam-1506	83	17	of	of	ADP
ejpam-1506	83	18	the	the	DET
ejpam-1506	83	19	vertex	vertex	NOUN
ejpam-1506	83	20	set	set	NOUN
ejpam-1506	83	21	of	of	ADP
ejpam-1506	83	22	s(gi	s(gi	PROPN
ejpam-1506	83	23	)	)	PUNCT
ejpam-1506	83	24	by	by	ADP
ejpam-1506	83	25	the	the	DET
ejpam-1506	83	26	connectedness	connectedness	NOUN
ejpam-1506	83	27	of	of	ADP
ejpam-1506	83	28	s(gi	s(gi	PROPN
ejpam-1506	83	29	)	)	PUNCT
ejpam-1506	83	30	,	,	PUNCT
ejpam-1506	83	31	where	where	SCONJ
ejpam-1506	83	32	i	i	PRON
ejpam-1506	83	33	=	=	SYM
ejpam-1506	83	34	1,2	1,2	X
ejpam-1506	83	35	.	.	PUNCT
ejpam-1506	84	1	let	let	VERB
ejpam-1506	84	2	vi,1	vi,1	NOUN
ejpam-1506	84	3	=	=	SYM
ejpam-1506	84	4	v	v	PROPN
ejpam-1506	84	5	(	(	PUNCT
ejpam-1506	84	6	gi	gi	NOUN
ejpam-1506	84	7	)	)	PUNCT
ejpam-1506	84	8	and	and	CCONJ
ejpam-1506	84	9	vi,2	vi,2	PROPN
ejpam-1506	84	10	=	=	SYM
ejpam-1506	84	11	v	v	PROPN
ejpam-1506	84	12	(	(	PUNCT
ejpam-1506	84	13	s(gi))−	s(gi))−	NOUN
ejpam-1506	84	14	v	v	NOUN
ejpam-1506	84	15	(	(	PUNCT
ejpam-1506	84	16	gi	gi	INTJ
ejpam-1506	84	17	)	)	PUNCT
ejpam-1506	84	18	be	be	AUX
ejpam-1506	84	19	the	the	DET
ejpam-1506	84	20	bi	bi	ADJ
ejpam-1506	84	21	-	-	ADJ
ejpam-1506	84	22	partite	partite	ADJ
ejpam-1506	84	23	sets	set	NOUN
ejpam-1506	84	24	of	of	ADP
ejpam-1506	84	25	s(gi	s(gi	NOUN
ejpam-1506	84	26	)	)	PUNCT
ejpam-1506	84	27	,	,	PUNCT
ejpam-1506	84	28	where	where	SCONJ
ejpam-1506	84	29	i	i	PRON
ejpam-1506	84	30	=	=	SYM
ejpam-1506	84	31	1,2	1,2	NUM
ejpam-1506	84	32	.	.	PUNCT
ejpam-1506	85	1	for	for	ADP
ejpam-1506	85	2	each	each	DET
ejpam-1506	85	3	p3	p3	NOUN
ejpam-1506	85	4	of	of	ADP
ejpam-1506	85	5	s(gi	s(gi	PROPN
ejpam-1506	85	6	)	)	PUNCT
ejpam-1506	85	7	,	,	PUNCT
ejpam-1506	85	8	say	say	VERB
ejpam-1506	85	9	uvu′	uvu′	PROPN
ejpam-1506	85	10	,	,	PUNCT
ejpam-1506	85	11	such	such	ADJ
ejpam-1506	85	12	that	that	SCONJ
ejpam-1506	85	13	u	u	NOUN
ejpam-1506	85	14	,	,	PUNCT
ejpam-1506	85	15	u′	u′	PROPN
ejpam-1506	85	16	∈	∈	PROPN
ejpam-1506	85	17	vi,1	vi,1	PROPN
ejpam-1506	85	18	and	and	CCONJ
ejpam-1506	85	19	v	v	ADP
ejpam-1506	85	20	∈	∈	PROPN
ejpam-1506	85	21	vi,2	vi,2	PROPN
ejpam-1506	85	22	(	(	PUNCT
ejpam-1506	85	23	i	i	NOUN
ejpam-1506	85	24	=	=	SYM
ejpam-1506	85	25	1,2	1,2	NUM
ejpam-1506	85	26	)	)	PUNCT
ejpam-1506	85	27	,	,	PUNCT
ejpam-1506	85	28	we	we	PRON
ejpam-1506	85	29	replace	replace	VERB
ejpam-1506	85	30	uvu′	uvu′	NOUN
ejpam-1506	85	31	by	by	ADP
ejpam-1506	85	32	uu′.	uu′.	PROPN
ejpam-1506	85	33	the	the	DET
ejpam-1506	85	34	resulting	result	VERB
ejpam-1506	85	35	graph	graph	NOUN
ejpam-1506	85	36	is	be	AUX
ejpam-1506	85	37	g1	g1	VERB
ejpam-1506	85	38	∼=	∼=	PROPN
ejpam-1506	85	39	g2	g2	NOUN
ejpam-1506	85	40	.	.	PUNCT
ejpam-1506	86	1	3	3	X
ejpam-1506	86	2	.	.	NUM
ejpam-1506	86	3	bounds	bound	NOUN
ejpam-1506	86	4	of	of	ADP
ejpam-1506	86	5	metric	metric	ADJ
ejpam-1506	86	6	dimension	dimension	NOUN
ejpam-1506	86	7	on	on	ADP
ejpam-1506	86	8	para	para	ADJ
ejpam-1506	86	9	-	-	PUNCT
ejpam-1506	86	10	line	line	NOUN
ejpam-1506	86	11	graphs	graph	NOUN
ejpam-1506	86	12	in	in	ADP
ejpam-1506	86	13	this	this	DET
ejpam-1506	86	14	section	section	NOUN
ejpam-1506	86	15	,	,	PUNCT
ejpam-1506	86	16	we	we	PRON
ejpam-1506	86	17	obtain	obtain	VERB
ejpam-1506	86	18	general	general	ADJ
ejpam-1506	86	19	bounds	bound	NOUN
ejpam-1506	86	20	of	of	ADP
ejpam-1506	86	21	the	the	DET
ejpam-1506	86	22	metric	metric	ADJ
ejpam-1506	86	23	dimension	dimension	NOUN
ejpam-1506	86	24	of	of	ADP
ejpam-1506	86	25	para	para	NOUN
ejpam-1506	86	26	-	-	PUNCT
ejpam-1506	86	27	line	line	NOUN
ejpam-1506	86	28	graphs	graph	NOUN
ejpam-1506	86	29	.	.	PUNCT
ejpam-1506	87	1	first	first	ADV
ejpam-1506	87	2	,	,	PUNCT
ejpam-1506	87	3	we	we	PRON
ejpam-1506	87	4	recall	recall	VERB
ejpam-1506	87	5	the	the	DET
ejpam-1506	87	6	bounds	bound	NOUN
ejpam-1506	87	7	of	of	ADP
ejpam-1506	87	8	the	the	DET
ejpam-1506	87	9	metric	metric	ADJ
ejpam-1506	87	10	dimension	dimension	NOUN
ejpam-1506	87	11	of	of	ADP
ejpam-1506	87	12	graphs	graph	NOUN
ejpam-1506	87	13	and	and	CCONJ
ejpam-1506	87	14	its	its	PRON
ejpam-1506	87	15	line	line	NOUN
ejpam-1506	87	16	graphs	graph	NOUN
ejpam-1506	87	17	.	.	PUNCT
ejpam-1506	88	1	theorem	theorem	VERB
ejpam-1506	88	2	3	3	NUM
ejpam-1506	88	3	.	.	PUNCT
ejpam-1506	89	1	[	[	X
ejpam-1506	89	2	5	5	X
ejpam-1506	89	3	]	]	PUNCT
ejpam-1506	89	4	if	if	SCONJ
ejpam-1506	89	5	g	g	PROPN
ejpam-1506	89	6	is	be	AUX
ejpam-1506	89	7	a	a	DET
ejpam-1506	89	8	connected	connected	ADJ
ejpam-1506	89	9	graph	graph	NOUN
ejpam-1506	89	10	of	of	ADP
ejpam-1506	89	11	order	order	NOUN
ejpam-1506	89	12	n≥	n≥	NOUN
ejpam-1506	89	13	2	2	NUM
ejpam-1506	89	14	and	and	CCONJ
ejpam-1506	89	15	diameter	diameter	NOUN
ejpam-1506	89	16	d	d	NOUN
ejpam-1506	89	17	,	,	PUNCT
ejpam-1506	89	18	then	then	ADV
ejpam-1506	89	19	f	f	PROPN
ejpam-1506	89	20	(	(	PUNCT
ejpam-1506	89	21	n	n	CCONJ
ejpam-1506	89	22	,	,	PUNCT
ejpam-1506	89	23	d)≤	d)≤	NOUN
ejpam-1506	89	24	dim(g)≤	dim(g)≤	NOUN
ejpam-1506	89	25	n−	n−	PROPN
ejpam-1506	89	26	d	d	NOUN
ejpam-1506	89	27	,	,	PUNCT
ejpam-1506	89	28	where	where	SCONJ
ejpam-1506	89	29	f	f	PROPN
ejpam-1506	89	30	(	(	PUNCT
ejpam-1506	89	31	n	n	CCONJ
ejpam-1506	89	32	,	,	PUNCT
ejpam-1506	89	33	d	d	X
ejpam-1506	89	34	)	)	PUNCT
ejpam-1506	89	35	is	be	AUX
ejpam-1506	89	36	the	the	DET
ejpam-1506	89	37	least	least	ADV
ejpam-1506	89	38	positive	positive	ADJ
ejpam-1506	89	39	integer	integer	NOUN
ejpam-1506	89	40	k	k	PROPN
ejpam-1506	89	41	for	for	ADP
ejpam-1506	89	42	which	which	PRON
ejpam-1506	89	43	k+	k+	NOUN
ejpam-1506	89	44	dk	dk	PROPN
ejpam-1506	89	45	≥	≥	PROPN
ejpam-1506	89	46	n.	n.	VERB
ejpam-1506	89	47	a	a	DET
ejpam-1506	89	48	generalization	generalization	NOUN
ejpam-1506	89	49	of	of	ADP
ejpam-1506	89	50	theorem	theorem	ADJ
ejpam-1506	89	51	3	3	NUM
ejpam-1506	89	52	has	have	AUX
ejpam-1506	89	53	been	be	AUX
ejpam-1506	89	54	given	give	VERB
ejpam-1506	89	55	in	in	ADP
ejpam-1506	89	56	[	[	NOUN
ejpam-1506	89	57	14	14	NUM
ejpam-1506	89	58	]	]	PUNCT
ejpam-1506	89	59	by	by	ADP
ejpam-1506	89	60	hernando	hernando	PROPN
ejpam-1506	89	61	et	et	PROPN
ejpam-1506	89	62	al	al	PROPN
ejpam-1506	89	63	.	.	PROPN
ejpam-1506	89	64	theorem	theorem	VERB
ejpam-1506	89	65	4	4	NUM
ejpam-1506	89	66	.	.	PUNCT
ejpam-1506	90	1	[	[	X
ejpam-1506	90	2	14	14	NUM
ejpam-1506	90	3	]	]	PUNCT
ejpam-1506	90	4	let	let	VERB
ejpam-1506	90	5	g	g	PRON
ejpam-1506	90	6	be	be	AUX
ejpam-1506	90	7	a	a	DET
ejpam-1506	90	8	graph	graph	NOUN
ejpam-1506	90	9	of	of	ADP
ejpam-1506	90	10	order	order	NOUN
ejpam-1506	90	11	n	n	CCONJ
ejpam-1506	90	12	,	,	PUNCT
ejpam-1506	90	13	diameter	diameter	NOUN
ejpam-1506	90	14	d	d	PROPN
ejpam-1506	90	15	≥	≥	NUM
ejpam-1506	90	16	2	2	NUM
ejpam-1506	90	17	,	,	PUNCT
ejpam-1506	90	18	and	and	CCONJ
ejpam-1506	90	19	metric	metric	ADJ
ejpam-1506	90	20	dimension	dimension	NOUN
ejpam-1506	91	1	k.	k.	PROPN
ejpam-1506	92	1	then	then	ADV
ejpam-1506	92	2	n≤	n≤	PRON
ejpam-1506	92	3	�	�	PROPN
ejpam-1506	92	4	�	�	PROPN
ejpam-1506	92	5	2d	2d	PROPN
ejpam-1506	92	6	3	3	NUM
ejpam-1506	92	7	�	�	PROPN
ejpam-1506	92	8	+	+	CCONJ
ejpam-1506	92	9	1	1	NUM
ejpam-1506	92	10	�	�	NOUN
ejpam-1506	92	11	k	k	PROPN
ejpam-1506	92	12	+	+	CCONJ
ejpam-1506	93	1	k	k	PROPN
ejpam-1506	93	2	⌈	⌈	PROPN
ejpam-1506	93	3	d	d	PROPN
ejpam-1506	93	4	3	3	NUM
ejpam-1506	93	5	⌉	⌉	NOUN
ejpam-1506	93	6	∑	∑	ADP
ejpam-1506	93	7	i=1	i=1	PROPN
ejpam-1506	93	8	(	(	PUNCT
ejpam-1506	93	9	2i−	2i−	PROPN
ejpam-1506	93	10	1)k−1	1)k−1	NUM
ejpam-1506	93	11	.	.	PUNCT
ejpam-1506	94	1	theorem	theorem	VERB
ejpam-1506	94	2	5	5	NUM
ejpam-1506	94	3	.	.	PUNCT
ejpam-1506	95	1	[	[	X
ejpam-1506	95	2	10	10	NUM
ejpam-1506	95	3	]	]	X
ejpam-1506	95	4	if	if	SCONJ
ejpam-1506	95	5	g	g	PROPN
ejpam-1506	95	6	is	be	AUX
ejpam-1506	95	7	a	a	DET
ejpam-1506	95	8	connected	connected	ADJ
ejpam-1506	95	9	graph	graph	NOUN
ejpam-1506	95	10	of	of	ADP
ejpam-1506	95	11	order	order	NOUN
ejpam-1506	95	12	n≥	n≥	NOUN
ejpam-1506	95	13	5	5	NUM
ejpam-1506	95	14	,	,	PUNCT
ejpam-1506	95	15	then	then	ADV
ejpam-1506	95	16	⌈log2∆(g)⌉	⌈log2∆(g)⌉	PROPN
ejpam-1506	95	17	≤	≤	PROPN
ejpam-1506	95	18	dim(l(g))≤	dim(l(g))≤	ADJ
ejpam-1506	95	19	n−	n−	NOUN
ejpam-1506	95	20	2	2	NUM
ejpam-1506	95	21	.	.	PUNCT
ejpam-1506	96	1	next	next	ADV
ejpam-1506	96	2	,	,	PUNCT
ejpam-1506	96	3	we	we	PRON
ejpam-1506	96	4	recall	recall	VERB
ejpam-1506	96	5	the	the	DET
ejpam-1506	96	6	following	follow	VERB
ejpam-1506	96	7	definitions	definition	NOUN
ejpam-1506	96	8	that	that	PRON
ejpam-1506	96	9	are	be	AUX
ejpam-1506	96	10	stated	state	VERB
ejpam-1506	96	11	in	in	ADP
ejpam-1506	96	12	[	[	X
ejpam-1506	96	13	14	14	NUM
ejpam-1506	96	14	]	]	PUNCT
ejpam-1506	96	15	.	.	PUNCT
ejpam-1506	97	1	two	two	NUM
ejpam-1506	97	2	distinct	distinct	ADJ
ejpam-1506	97	3	vertices	vertex	NOUN
ejpam-1506	97	4	u	u	NOUN
ejpam-1506	97	5	,	,	PUNCT
ejpam-1506	97	6	v	v	NOUN
ejpam-1506	97	7	of	of	ADP
ejpam-1506	97	8	a	a	DET
ejpam-1506	97	9	graph	graph	NOUN
ejpam-1506	97	10	g	g	NOUN
ejpam-1506	97	11	are	be	AUX
ejpam-1506	97	12	adjacent	adjacent	ADJ
ejpam-1506	97	13	twins	twin	NOUN
ejpam-1506	97	14	if	if	SCONJ
ejpam-1506	97	15	ng[u	ng[u	PROPN
ejpam-1506	97	16	]	]	X
ejpam-1506	97	17	=	=	PUNCT
ejpam-1506	97	18	ng[v	ng[v	X
ejpam-1506	97	19	]	]	PUNCT
ejpam-1506	97	20	,	,	PUNCT
ejpam-1506	97	21	and	and	CCONJ
ejpam-1506	97	22	non	non	ADJ
ejpam-1506	97	23	-	-	ADJ
ejpam-1506	97	24	adjacent	adjacent	ADJ
ejpam-1506	97	25	twins	twin	NOUN
ejpam-1506	97	26	if	if	SCONJ
ejpam-1506	97	27	ng(u	ng(u	NOUN
ejpam-1506	97	28	)	)	PUNCT
ejpam-1506	97	29	=	=	PUNCT
ejpam-1506	97	30	ng(v	ng(v	X
ejpam-1506	97	31	)	)	PUNCT
ejpam-1506	97	32	.	.	PUNCT
ejpam-1506	98	1	observe	observe	VERB
ejpam-1506	98	2	that	that	SCONJ
ejpam-1506	98	3	if	if	SCONJ
ejpam-1506	98	4	u	u	NOUN
ejpam-1506	98	5	,	,	PUNCT
ejpam-1506	98	6	v	v	NOUN
ejpam-1506	98	7	are	be	AUX
ejpam-1506	98	8	adjacent	adjacent	ADJ
ejpam-1506	98	9	twins	twin	NOUN
ejpam-1506	98	10	then	then	ADV
ejpam-1506	98	11	uv	uv	PROPN
ejpam-1506	98	12	∈	∈	PROPN
ejpam-1506	98	13	e(g	e(g	PROPN
ejpam-1506	98	14	)	)	PUNCT
ejpam-1506	98	15	,	,	PUNCT
ejpam-1506	98	16	and	and	CCONJ
ejpam-1506	98	17	if	if	SCONJ
ejpam-1506	98	18	u	u	NOUN
ejpam-1506	98	19	,	,	PUNCT
ejpam-1506	98	20	v	v	NOUN
ejpam-1506	98	21	are	be	AUX
ejpam-1506	98	22	non	non	ADJ
ejpam-1506	98	23	-	-	ADJ
ejpam-1506	98	24	adjacent	adjacent	ADJ
ejpam-1506	98	25	twins	twin	NOUN
ejpam-1506	98	26	then	then	ADV
ejpam-1506	98	27	uv	uv	PROPN
ejpam-1506	98	28	6∈	6∈	PROPN
ejpam-1506	98	29	e(g	e(g	PROPN
ejpam-1506	98	30	)	)	PUNCT
ejpam-1506	98	31	.	.	PUNCT
ejpam-1506	99	1	if	if	SCONJ
ejpam-1506	99	2	u	u	NOUN
ejpam-1506	99	3	,	,	PUNCT
ejpam-1506	99	4	v	v	NOUN
ejpam-1506	99	5	are	be	AUX
ejpam-1506	99	6	adjacent	adjacent	ADJ
ejpam-1506	99	7	or	or	CCONJ
ejpam-1506	99	8	non	non	ADJ
ejpam-1506	99	9	-	-	ADJ
ejpam-1506	99	10	adjacent	adjacent	ADJ
ejpam-1506	99	11	twins	twin	NOUN
ejpam-1506	99	12	,	,	PUNCT
ejpam-1506	99	13	then	then	ADV
ejpam-1506	99	14	u	u	NOUN
ejpam-1506	99	15	,	,	PUNCT
ejpam-1506	99	16	v	v	NOUN
ejpam-1506	99	17	are	be	AUX
ejpam-1506	99	18	twins	twin	NOUN
ejpam-1506	99	19	;	;	PUNCT
ejpam-1506	99	20	if	if	SCONJ
ejpam-1506	99	21	s	s	NOUN
ejpam-1506	99	22	is	be	AUX
ejpam-1506	99	23	a	a	DET
ejpam-1506	99	24	resolving	resolving	NOUN
ejpam-1506	99	25	set	set	NOUN
ejpam-1506	99	26	of	of	ADP
ejpam-1506	99	27	g	g	NOUN
ejpam-1506	99	28	,	,	PUNCT
ejpam-1506	99	29	then	then	ADV
ejpam-1506	99	30	u	u	PROPN
ejpam-1506	99	31	∈	∈	PROPN
ejpam-1506	99	32	s	s	X
ejpam-1506	99	33	or	or	CCONJ
ejpam-1506	99	34	v	v	ADP
ejpam-1506	99	35	∈	∈	PROPN
ejpam-1506	99	36	s.	s.	PROPN
ejpam-1506	99	37	theorem	theorem	VERB
ejpam-1506	99	38	6	6	NUM
ejpam-1506	99	39	.	.	PUNCT
ejpam-1506	100	1	let	let	VERB
ejpam-1506	100	2	g	g	PRON
ejpam-1506	100	3	be	be	AUX
ejpam-1506	100	4	a	a	DET
ejpam-1506	100	5	connected	connected	ADJ
ejpam-1506	100	6	graph	graph	NOUN
ejpam-1506	100	7	of	of	ADP
ejpam-1506	100	8	order	order	NOUN
ejpam-1506	100	9	n≥	n≥	NOUN
ejpam-1506	100	10	2	2	X
ejpam-1506	100	11	.	.	PUNCT
ejpam-1506	101	1	then	then	ADV
ejpam-1506	101	2	⌈log2∆(g)⌉	⌈log2∆(g)⌉	PROPN
ejpam-1506	101	3	≤	≤	PUNCT
ejpam-1506	101	4	dim(g⋆)≤	dim(g⋆)≤	PRON
ejpam-1506	101	5	n−	n−	NOUN
ejpam-1506	101	6	1	1	NUM
ejpam-1506	101	7	,	,	PUNCT
ejpam-1506	101	8	(	(	PUNCT
ejpam-1506	101	9	1	1	NUM
ejpam-1506	101	10	)	)	PUNCT
ejpam-1506	101	11	and	and	CCONJ
ejpam-1506	101	12	both	both	DET
ejpam-1506	101	13	bounds	bound	NOUN
ejpam-1506	101	14	are	be	AUX
ejpam-1506	101	15	sharp	sharp	ADJ
ejpam-1506	101	16	.	.	PUNCT
ejpam-1506	102	1	d.	d.	PROPN
ejpam-1506	102	2	klein	klein	PROPN
ejpam-1506	102	3	,	,	PUNCT
ejpam-1506	102	4	e.	e.	PROPN
ejpam-1506	102	5	yi	yi	PROPN
ejpam-1506	102	6	/	/	SYM
ejpam-1506	102	7	eur	eur	PROPN
ejpam-1506	102	8	.	.	PUNCT
ejpam-1506	103	1	j.	j.	PROPN
ejpam-1506	103	2	pure	pure	PROPN
ejpam-1506	103	3	appl	appl	PROPN
ejpam-1506	103	4	.	.	PROPN
ejpam-1506	103	5	math	math	PROPN
ejpam-1506	103	6	,	,	PUNCT
ejpam-1506	103	7	5	5	NUM
ejpam-1506	103	8	(	(	PUNCT
ejpam-1506	103	9	2012	2012	NUM
ejpam-1506	103	10	)	)	PUNCT
ejpam-1506	103	11	,	,	PUNCT
ejpam-1506	103	12	302	302	NUM
ejpam-1506	103	13	-	-	SYM
ejpam-1506	103	14	316	316	NUM
ejpam-1506	103	15	306	306	NUM
ejpam-1506	103	16	proof	proof	NOUN
ejpam-1506	103	17	.	.	PUNCT
ejpam-1506	104	1	first	first	ADV
ejpam-1506	104	2	,	,	PUNCT
ejpam-1506	104	3	we	we	PRON
ejpam-1506	104	4	consider	consider	VERB
ejpam-1506	104	5	for	for	ADP
ejpam-1506	104	6	2≤	2≤	NUM
ejpam-1506	104	7	n≤	n≤	PRON
ejpam-1506	104	8	4	4	NUM
ejpam-1506	104	9	.	.	PUNCT
ejpam-1506	105	1	if	if	SCONJ
ejpam-1506	105	2	n=	n=	ADJ
ejpam-1506	105	3	2	2	NUM
ejpam-1506	105	4	,	,	PUNCT
ejpam-1506	105	5	then	then	ADV
ejpam-1506	105	6	∆(g	∆(g	NOUN
ejpam-1506	105	7	)	)	PUNCT
ejpam-1506	105	8	=	=	SYM
ejpam-1506	105	9	1	1	NUM
ejpam-1506	105	10	and	and	CCONJ
ejpam-1506	105	11	dim(g⋆	dim(g⋆	NOUN
ejpam-1506	105	12	)	)	PUNCT
ejpam-1506	105	13	=	=	SYM
ejpam-1506	106	1	1	1	X
ejpam-1506	106	2	.	.	PUNCT
ejpam-1506	106	3	if	if	SCONJ
ejpam-1506	106	4	n=	n=	ADJ
ejpam-1506	106	5	3	3	NUM
ejpam-1506	106	6	,	,	PUNCT
ejpam-1506	106	7	then	then	ADV
ejpam-1506	106	8	∆(g	∆(g	NOUN
ejpam-1506	106	9	)	)	PUNCT
ejpam-1506	106	10	=	=	SYM
ejpam-1506	106	11	2	2	NUM
ejpam-1506	106	12	and	and	CCONJ
ejpam-1506	106	13	1	1	NUM
ejpam-1506	106	14	≤	≤	NUM
ejpam-1506	106	15	dim(g⋆	dim(g⋆	NOUN
ejpam-1506	106	16	)	)	PUNCT
ejpam-1506	106	17	≤	≤	NOUN
ejpam-1506	106	18	2	2	NUM
ejpam-1506	106	19	.	.	PUNCT
ejpam-1506	107	1	if	if	SCONJ
ejpam-1506	107	2	n	n	NOUN
ejpam-1506	107	3	=	=	SYM
ejpam-1506	107	4	4	4	NUM
ejpam-1506	107	5	,	,	PUNCT
ejpam-1506	107	6	then	then	ADV
ejpam-1506	107	7	2	2	NUM
ejpam-1506	107	8	≤	≤	PROPN
ejpam-1506	107	9	∆(g	∆(g	NOUN
ejpam-1506	107	10	)	)	PUNCT
ejpam-1506	107	11	≤	≤	NOUN
ejpam-1506	107	12	3	3	NUM
ejpam-1506	107	13	and	and	CCONJ
ejpam-1506	107	14	1	1	NUM
ejpam-1506	107	15	≤	≤	NUM
ejpam-1506	107	16	dim(g⋆	dim(g⋆	NOUN
ejpam-1506	107	17	)	)	PUNCT
ejpam-1506	107	18	≤	≤	NOUN
ejpam-1506	108	1	3	3	NUM
ejpam-1506	108	2	;	;	PUNCT
ejpam-1506	108	3	here	here	ADV
ejpam-1506	108	4	,	,	PUNCT
ejpam-1506	108	5	dim(g⋆	dim(g⋆	NOUN
ejpam-1506	108	6	)	)	PUNCT
ejpam-1506	108	7	=	=	SYM
ejpam-1506	108	8	1	1	NUM
ejpam-1506	108	9	if	if	SCONJ
ejpam-1506	108	10	and	and	CCONJ
ejpam-1506	108	11	only	only	ADV
ejpam-1506	108	12	if	if	SCONJ
ejpam-1506	108	13	g	g	NOUN
ejpam-1506	108	14	=	=	SYM
ejpam-1506	108	15	p4	p4	ADJ
ejpam-1506	108	16	,	,	PUNCT
ejpam-1506	108	17	and	and	CCONJ
ejpam-1506	108	18	dim(g⋆	dim(g⋆	NOUN
ejpam-1506	108	19	)	)	PUNCT
ejpam-1506	108	20	=	=	SYM
ejpam-1506	108	21	3	3	NUM
ejpam-1506	108	22	if	if	SCONJ
ejpam-1506	108	23	and	and	CCONJ
ejpam-1506	108	24	only	only	ADV
ejpam-1506	108	25	if	if	SCONJ
ejpam-1506	108	26	g	g	PROPN
ejpam-1506	108	27	=	=	SYM
ejpam-1506	108	28	k4	k4	PROPN
ejpam-1506	108	29	.	.	PUNCT
ejpam-1506	109	1	so	so	ADV
ejpam-1506	109	2	,	,	PUNCT
ejpam-1506	109	3	(	(	PUNCT
ejpam-1506	109	4	1	1	X
ejpam-1506	109	5	)	)	PUNCT
ejpam-1506	109	6	holds	hold	VERB
ejpam-1506	109	7	for	for	ADP
ejpam-1506	109	8	2≤	2≤	NUM
ejpam-1506	109	9	n≤	n≤	PRON
ejpam-1506	109	10	4	4	NUM
ejpam-1506	109	11	.	.	PUNCT
ejpam-1506	110	1	next	next	ADV
ejpam-1506	110	2	,	,	PUNCT
ejpam-1506	110	3	we	we	PRON
ejpam-1506	110	4	consider	consider	VERB
ejpam-1506	110	5	n≥	n≥	NOUN
ejpam-1506	110	6	5	5	NUM
ejpam-1506	110	7	.	.	PUNCT
ejpam-1506	111	1	the	the	DET
ejpam-1506	111	2	lower	lower	ADV
ejpam-1506	111	3	bound	bind	VERB
ejpam-1506	111	4	of	of	ADP
ejpam-1506	111	5	(	(	PUNCT
ejpam-1506	111	6	1	1	X
ejpam-1506	111	7	)	)	PUNCT
ejpam-1506	111	8	follows	follow	VERB
ejpam-1506	111	9	by	by	ADP
ejpam-1506	111	10	theorem	theorem	NOUN
ejpam-1506	111	11	5	5	NUM
ejpam-1506	111	12	,	,	PUNCT
ejpam-1506	111	13	since	since	SCONJ
ejpam-1506	111	14	∆(g	∆(g	NOUN
ejpam-1506	111	15	)	)	PUNCT
ejpam-1506	111	16	=	=	SYM
ejpam-1506	111	17	∆(s(g	∆(s(g	NUM
ejpam-1506	111	18	)	)	PUNCT
ejpam-1506	111	19	)	)	PUNCT
ejpam-1506	111	20	.	.	PUNCT
ejpam-1506	112	1	for	for	ADP
ejpam-1506	112	2	the	the	DET
ejpam-1506	112	3	sharpness	sharpness	NOUN
ejpam-1506	112	4	of	of	ADP
ejpam-1506	112	5	the	the	DET
ejpam-1506	112	6	lower	lower	ADV
ejpam-1506	112	7	bound	bind	VERB
ejpam-1506	112	8	,	,	PUNCT
ejpam-1506	112	9	take	take	VERB
ejpam-1506	112	10	g	g	NOUN
ejpam-1506	112	11	=	=	SYM
ejpam-1506	112	12	pn	pn	PROPN
ejpam-1506	112	13	;	;	PUNCT
ejpam-1506	112	14	then	then	ADV
ejpam-1506	112	15	∆(g	∆(g	PROPN
ejpam-1506	112	16	)	)	PUNCT
ejpam-1506	112	17	=	=	SYM
ejpam-1506	112	18	2	2	NUM
ejpam-1506	112	19	and	and	CCONJ
ejpam-1506	112	20	dim(g⋆	dim(g⋆	NOUN
ejpam-1506	112	21	)	)	PUNCT
ejpam-1506	112	22	=	=	SYM
ejpam-1506	112	23	1	1	NUM
ejpam-1506	112	24	(	(	PUNCT
ejpam-1506	112	25	see	see	VERB
ejpam-1506	112	26	(	(	PUNCT
ejpam-1506	112	27	a	a	NOUN
ejpam-1506	112	28	)	)	PUNCT
ejpam-1506	112	29	of	of	ADP
ejpam-1506	112	30	theorem	theorem	NOUN
ejpam-1506	112	31	8)	8)	NUM
ejpam-1506	112	32	.	.	PUNCT
ejpam-1506	113	1	next	next	ADV
ejpam-1506	113	2	,	,	PUNCT
ejpam-1506	113	3	we	we	PRON
ejpam-1506	113	4	prove	prove	VERB
ejpam-1506	113	5	the	the	DET
ejpam-1506	113	6	upper	upper	ADJ
ejpam-1506	113	7	bound	bind	VERB
ejpam-1506	113	8	of	of	ADP
ejpam-1506	113	9	(	(	PUNCT
ejpam-1506	113	10	1	1	NUM
ejpam-1506	113	11	)	)	PUNCT
ejpam-1506	113	12	.	.	PUNCT
ejpam-1506	114	1	let	let	VERB
ejpam-1506	114	2	v	v	X
ejpam-1506	114	3	(	(	PUNCT
ejpam-1506	114	4	g	g	NOUN
ejpam-1506	114	5	)	)	PUNCT
ejpam-1506	114	6	=	=	SYM
ejpam-1506	114	7	{	{	PUNCT
ejpam-1506	114	8	v1	v1	PROPN
ejpam-1506	114	9	,	,	PUNCT
ejpam-1506	114	10	v2	v2	PROPN
ejpam-1506	114	11	,	,	PUNCT
ejpam-1506	114	12	.	.	PUNCT
ejpam-1506	114	13	.	.	PUNCT
ejpam-1506	115	1	.	.	PUNCT
ejpam-1506	116	1	,	,	PUNCT
ejpam-1506	116	2	vn	vn	PROPN
ejpam-1506	116	3	}	}	PUNCT
ejpam-1506	116	4	,	,	PUNCT
ejpam-1506	116	5	and	and	CCONJ
ejpam-1506	116	6	let	let	VERB
ejpam-1506	116	7	degg(vi	degg(vi	NOUN
ejpam-1506	116	8	)	)	PUNCT
ejpam-1506	116	9	=	=	SYM
ejpam-1506	116	10	di	di	X
ejpam-1506	116	11	(	(	PUNCT
ejpam-1506	116	12	1	1	NUM
ejpam-1506	116	13	≤	≤	NUM
ejpam-1506	116	14	i	i	NOUN
ejpam-1506	116	15	≤	≤	NOUN
ejpam-1506	116	16	n	n	CCONJ
ejpam-1506	116	17	)	)	PUNCT
ejpam-1506	116	18	.	.	PUNCT
ejpam-1506	117	1	following	follow	VERB
ejpam-1506	117	2	the	the	DET
ejpam-1506	117	3	construction	construction	NOUN
ejpam-1506	117	4	of	of	ADP
ejpam-1506	117	5	g⋆	g⋆	NOUN
ejpam-1506	117	6	from	from	ADP
ejpam-1506	117	7	g	g	NOUN
ejpam-1506	117	8	,	,	PUNCT
ejpam-1506	117	9	let	let	VERB
ejpam-1506	117	10	each	each	DET
ejpam-1506	117	11	vertex	vertex	NOUN
ejpam-1506	117	12	vi	vi	NOUN
ejpam-1506	117	13	be	be	AUX
ejpam-1506	117	14	replaced	replace	VERB
ejpam-1506	117	15	by	by	ADP
ejpam-1506	117	16	k(vi	k(vi	PROPN
ejpam-1506	117	17	)	)	PUNCT
ejpam-1506	117	18	∼=	∼=	PROPN
ejpam-1506	117	19	kdi	kdi	NOUN
ejpam-1506	117	20	;	;	PUNCT
ejpam-1506	117	21	here	here	ADV
ejpam-1506	117	22	,	,	PUNCT
ejpam-1506	117	23	we	we	PRON
ejpam-1506	117	24	denote	denote	VERB
ejpam-1506	117	25	by	by	ADP
ejpam-1506	117	26	ui	ui	PROPN
ejpam-1506	117	27	the	the	DET
ejpam-1506	117	28	vertex	vertex	NOUN
ejpam-1506	117	29	set	set	VERB
ejpam-1506	117	30	v	v	NOUN
ejpam-1506	117	31	(	(	PUNCT
ejpam-1506	117	32	k(vi	k(vi	PROPN
ejpam-1506	117	33	)	)	PUNCT
ejpam-1506	117	34	)	)	PUNCT
ejpam-1506	118	1	=	=	PUNCT
ejpam-1506	118	2	{	{	PUNCT
ejpam-1506	118	3	ui,1,ui,2	ui,1,ui,2	PROPN
ejpam-1506	118	4	,	,	PUNCT
ejpam-1506	118	5	.	.	PUNCT
ejpam-1506	118	6	.	.	PUNCT
ejpam-1506	119	1	.	.	PUNCT
ejpam-1506	120	1	,	,	PUNCT
ejpam-1506	120	2	ui	ui	PROPN
ejpam-1506	120	3	,	,	PUNCT
ejpam-1506	120	4	di	di	NOUN
ejpam-1506	120	5	}	}	PUNCT
ejpam-1506	120	6	⊆	⊆	NUM
ejpam-1506	120	7	v	v	NOUN
ejpam-1506	120	8	(	(	PUNCT
ejpam-1506	120	9	g⋆	g⋆	NOUN
ejpam-1506	120	10	)	)	PUNCT
ejpam-1506	120	11	for	for	ADP
ejpam-1506	120	12	each	each	DET
ejpam-1506	120	13	i	i	PRON
ejpam-1506	120	14	(	(	PUNCT
ejpam-1506	120	15	1	1	NUM
ejpam-1506	120	16	≤	≤	NUM
ejpam-1506	120	17	i	i	NOUN
ejpam-1506	120	18	≤	≤	NOUN
ejpam-1506	120	19	n	n	CCONJ
ejpam-1506	120	20	)	)	PUNCT
ejpam-1506	120	21	.	.	PUNCT
ejpam-1506	121	1	let	let	VERB
ejpam-1506	121	2	s	s	PRON
ejpam-1506	121	3	=	=	PUNCT
ejpam-1506	121	4	{	{	PUNCT
ejpam-1506	121	5	u1,a1	u1,a1	PROPN
ejpam-1506	121	6	,	,	PUNCT
ejpam-1506	121	7	u2,a2	u2,a2	PROPN
ejpam-1506	121	8	,	,	PUNCT
ejpam-1506	121	9	.	.	PUNCT
ejpam-1506	121	10	.	.	PUNCT
ejpam-1506	121	11	.	.	PUNCT
ejpam-1506	122	1	,	,	PUNCT
ejpam-1506	122	2	un−1,an−1	un−1,an−1	ADJ
ejpam-1506	122	3	}	}	PUNCT
ejpam-1506	122	4	with	with	ADP
ejpam-1506	122	5	|s|	|s|	PROPN
ejpam-1506	122	6	=	=	SYM
ejpam-1506	122	7	n−	n−	NOUN
ejpam-1506	122	8	1	1	NUM
ejpam-1506	122	9	such	such	ADJ
ejpam-1506	123	1	that	that	SCONJ
ejpam-1506	123	2	|s	|s	PROPN
ejpam-1506	123	3	∩ui|	∩ui|	NOUN
ejpam-1506	123	4	=	=	SYM
ejpam-1506	123	5	1	1	NUM
ejpam-1506	123	6	for	for	ADP
ejpam-1506	123	7	each	each	DET
ejpam-1506	123	8	i	i	PRON
ejpam-1506	123	9	(	(	PUNCT
ejpam-1506	123	10	1	1	NUM
ejpam-1506	123	11	≤	≤	NUM
ejpam-1506	123	12	i	i	PRON
ejpam-1506	123	13	≤	≤	ADJ
ejpam-1506	123	14	n−	n−	NOUN
ejpam-1506	123	15	1	1	NUM
ejpam-1506	123	16	)	)	PUNCT
ejpam-1506	123	17	and	and	CCONJ
ejpam-1506	123	18	that	that	SCONJ
ejpam-1506	123	19	no	no	DET
ejpam-1506	123	20	two	two	NUM
ejpam-1506	123	21	vertices	vertex	NOUN
ejpam-1506	123	22	in	in	ADP
ejpam-1506	123	23	s	s	NOUN
ejpam-1506	123	24	are	be	AUX
ejpam-1506	123	25	adjacent	adjacent	ADJ
ejpam-1506	123	26	in	in	ADP
ejpam-1506	123	27	g⋆.	g⋆.	NOUN
ejpam-1506	123	28	we	we	PRON
ejpam-1506	123	29	will	will	AUX
ejpam-1506	123	30	show	show	VERB
ejpam-1506	123	31	that	that	SCONJ
ejpam-1506	123	32	s	s	VERB
ejpam-1506	123	33	is	be	AUX
ejpam-1506	123	34	a	a	DET
ejpam-1506	123	35	resolving	resolving	NOUN
ejpam-1506	123	36	set	set	VERB
ejpam-1506	123	37	for	for	ADP
ejpam-1506	123	38	g⋆.	g⋆.	NOUN
ejpam-1506	123	39	it	it	PRON
ejpam-1506	123	40	suffices	suffice	VERB
ejpam-1506	123	41	to	to	PART
ejpam-1506	123	42	show	show	VERB
ejpam-1506	123	43	that	that	SCONJ
ejpam-1506	123	44	,	,	PUNCT
ejpam-1506	123	45	for	for	ADP
ejpam-1506	123	46	any	any	DET
ejpam-1506	123	47	two	two	NUM
ejpam-1506	123	48	vertices	vertex	NOUN
ejpam-1506	123	49	ux	ux	ADV
ejpam-1506	123	50	,	,	PUNCT
ejpam-1506	123	51	uy	uy	PROPN
ejpam-1506	123	52	∈	∈	PROPN
ejpam-1506	123	53	v	v	NOUN
ejpam-1506	123	54	(	(	PUNCT
ejpam-1506	123	55	g⋆)−	g⋆)−	PROPN
ejpam-1506	123	56	s	s	SYM
ejpam-1506	123	57	,	,	PUNCT
ejpam-1506	123	58	dg⋆(ux	dg⋆(ux	NOUN
ejpam-1506	123	59	,	,	PUNCT
ejpam-1506	123	60	ui	ui	NOUN
ejpam-1506	123	61	,	,	PUNCT
ejpam-1506	123	62	ai	ai	VERB
ejpam-1506	123	63	)	)	PUNCT
ejpam-1506	123	64	6=	6=	NUM
ejpam-1506	123	65	dg⋆(uy	dg⋆(uy	PROPN
ejpam-1506	123	66	,	,	PUNCT
ejpam-1506	123	67	ui	ui	PROPN
ejpam-1506	123	68	,	,	PUNCT
ejpam-1506	123	69	ai	ai	VERB
ejpam-1506	123	70	)	)	PUNCT
ejpam-1506	123	71	for	for	ADP
ejpam-1506	123	72	some	some	DET
ejpam-1506	123	73	ui	ui	NOUN
ejpam-1506	123	74	,	,	PUNCT
ejpam-1506	123	75	ai	ai	VERB
ejpam-1506	123	76	∈	∈	PROPN
ejpam-1506	123	77	s.	s.	PROPN
ejpam-1506	123	78	(	(	PUNCT
ejpam-1506	123	79	2	2	X
ejpam-1506	123	80	)	)	PUNCT
ejpam-1506	123	81	we	we	PRON
ejpam-1506	123	82	consider	consider	VERB
ejpam-1506	123	83	two	two	NUM
ejpam-1506	123	84	cases	case	NOUN
ejpam-1506	123	85	.	.	PUNCT
ejpam-1506	124	1	case	case	NOUN
ejpam-1506	124	2	1	1	NUM
ejpam-1506	124	3	:	:	SYM
ejpam-1506	124	4	ux	ux	INTJ
ejpam-1506	124	5	,	,	PUNCT
ejpam-1506	124	6	uy	uy	PROPN
ejpam-1506	124	7	∈	∈	PROPN
ejpam-1506	124	8	ui	ui	PROPN
ejpam-1506	124	9	for	for	ADP
ejpam-1506	124	10	some	some	DET
ejpam-1506	124	11	i	i	PRON
ejpam-1506	124	12	(	(	PUNCT
ejpam-1506	124	13	1≤	1≤	INTJ
ejpam-1506	124	14	i	i	NOUN
ejpam-1506	124	15	≤	≤	PROPN
ejpam-1506	124	16	n	n	CCONJ
ejpam-1506	124	17	)	)	PUNCT
ejpam-1506	124	18	.	.	PUNCT
ejpam-1506	125	1	we	we	PRON
ejpam-1506	125	2	consider	consider	VERB
ejpam-1506	125	3	two	two	NUM
ejpam-1506	125	4	subcases	subcase	NOUN
ejpam-1506	125	5	.	.	PUNCT
ejpam-1506	126	1	subcase	subcase	VERB
ejpam-1506	126	2	1.1	1.1	NUM
ejpam-1506	126	3	:	:	PUNCT
ejpam-1506	126	4	1≤	1≤	NUM
ejpam-1506	127	1	i	i	PROPN
ejpam-1506	127	2	≤	≤	PUNCT
ejpam-1506	128	1	n−1	n−1	PROPN
ejpam-1506	128	2	.	.	PUNCT
ejpam-1506	129	1	first	first	ADV
ejpam-1506	129	2	,	,	PUNCT
ejpam-1506	129	3	suppose	suppose	VERB
ejpam-1506	129	4	that	that	SCONJ
ejpam-1506	129	5	ng⋆(ux	ng⋆(ux	NOUN
ejpam-1506	129	6	)	)	PUNCT
ejpam-1506	129	7	∩un	∩un	NOUN
ejpam-1506	129	8	=	=	SYM
ejpam-1506	129	9	;	;	PUNCT
ejpam-1506	129	10	=	=	SYM
ejpam-1506	129	11	ng⋆(uy)∩un	ng⋆(uy)∩un	NOUN
ejpam-1506	129	12	.	.	PUNCT
ejpam-1506	130	1	since	since	SCONJ
ejpam-1506	130	2	at	at	ADP
ejpam-1506	130	3	most	most	ADV
ejpam-1506	130	4	one	one	NUM
ejpam-1506	130	5	vertex	vertex	NOUN
ejpam-1506	130	6	of	of	ADP
ejpam-1506	130	7	ui	ui	PROPN
ejpam-1506	130	8	−	−	PROPN
ejpam-1506	130	9	{	{	PUNCT
ejpam-1506	130	10	ui	ui	PROPN
ejpam-1506	130	11	,	,	PUNCT
ejpam-1506	130	12	ai	ai	AUX
ejpam-1506	130	13	}	}	PUNCT
ejpam-1506	130	14	is	be	AUX
ejpam-1506	130	15	adjacent	adjacent	ADJ
ejpam-1506	130	16	to	to	ADP
ejpam-1506	130	17	at	at	ADP
ejpam-1506	130	18	most	most	ADV
ejpam-1506	130	19	one	one	NUM
ejpam-1506	130	20	vertex	vertex	NOUN
ejpam-1506	130	21	of	of	ADP
ejpam-1506	130	22	uk	uk	PROPN
ejpam-1506	130	23	for	for	ADP
ejpam-1506	130	24	k	k	PROPN
ejpam-1506	130	25	6=	6=	PROPN
ejpam-1506	130	26	i	i	PRON
ejpam-1506	130	27	(	(	PUNCT
ejpam-1506	130	28	1	1	NUM
ejpam-1506	130	29	≤	≤	NUM
ejpam-1506	130	30	k	k	X
ejpam-1506	130	31	≤	≤	PROPN
ejpam-1506	130	32	n−	n−	NOUN
ejpam-1506	130	33	1	1	NUM
ejpam-1506	130	34	)	)	PUNCT
ejpam-1506	130	35	,	,	PUNCT
ejpam-1506	130	36	say	say	VERB
ejpam-1506	130	37	ng⋆(ux	ng⋆(ux	NOUN
ejpam-1506	130	38	)	)	PUNCT
ejpam-1506	130	39	∩	∩	NOUN
ejpam-1506	130	40	uk	uk	PROPN
ejpam-1506	130	41	6=	6=	PROPN
ejpam-1506	130	42	;	;	PUNCT
ejpam-1506	130	43	,	,	PUNCT
ejpam-1506	130	44	we	we	PRON
ejpam-1506	130	45	have	have	VERB
ejpam-1506	130	46	dg⋆(ux	dg⋆(ux	NOUN
ejpam-1506	130	47	,	,	PUNCT
ejpam-1506	130	48	uk	uk	PROPN
ejpam-1506	130	49	,	,	PUNCT
ejpam-1506	130	50	ak	ak	PROPN
ejpam-1506	130	51	)	)	PUNCT
ejpam-1506	130	52	<	<	X
ejpam-1506	131	1	dg⋆(uy	dg⋆(uy	PROPN
ejpam-1506	131	2	,	,	PUNCT
ejpam-1506	131	3	uk	uk	PROPN
ejpam-1506	131	4	,	,	PUNCT
ejpam-1506	131	5	ak	ak	PROPN
ejpam-1506	131	6	)	)	PUNCT
ejpam-1506	131	7	.	.	PUNCT
ejpam-1506	132	1	second	second	ADV
ejpam-1506	132	2	,	,	PUNCT
ejpam-1506	132	3	suppose	suppose	VERB
ejpam-1506	132	4	that	that	SCONJ
ejpam-1506	132	5	ng⋆(ux	ng⋆(ux	NOUN
ejpam-1506	132	6	)	)	PUNCT
ejpam-1506	132	7	∩	∩	NOUN
ejpam-1506	132	8	un	un	PROPN
ejpam-1506	132	9	6=	6=	PROPN
ejpam-1506	132	10	;	;	PUNCT
ejpam-1506	132	11	or	or	CCONJ
ejpam-1506	132	12	ng⋆(uy	ng⋆(uy	NUM
ejpam-1506	132	13	)	)	PUNCT
ejpam-1506	132	14	∩	∩	NOUN
ejpam-1506	132	15	un	un	PROPN
ejpam-1506	132	16	6=	6=	PROPN
ejpam-1506	132	17	;	;	PUNCT
ejpam-1506	132	18	,	,	PUNCT
ejpam-1506	132	19	say	say	VERB
ejpam-1506	132	20	the	the	DET
ejpam-1506	132	21	former	former	ADJ
ejpam-1506	132	22	;	;	PUNCT
ejpam-1506	132	23	notice	notice	VERB
ejpam-1506	132	24	that	that	SCONJ
ejpam-1506	132	25	not	not	PART
ejpam-1506	132	26	both	both	PRON
ejpam-1506	132	27	ux	ux	ADJ
ejpam-1506	132	28	and	and	CCONJ
ejpam-1506	132	29	uy	uy	PROPN
ejpam-1506	132	30	can	can	AUX
ejpam-1506	132	31	have	have	VERB
ejpam-1506	132	32	a	a	DET
ejpam-1506	132	33	neighbor	neighbor	NOUN
ejpam-1506	132	34	in	in	ADP
ejpam-1506	132	35	un	un	PROPN
ejpam-1506	132	36	by	by	ADP
ejpam-1506	132	37	the	the	DET
ejpam-1506	132	38	construction	construction	NOUN
ejpam-1506	132	39	of	of	ADP
ejpam-1506	132	40	g⋆.	g⋆.	NOUN
ejpam-1506	132	41	then	then	ADV
ejpam-1506	132	42	dg⋆(ux	dg⋆(ux	NOUN
ejpam-1506	132	43	,	,	PUNCT
ejpam-1506	132	44	uk	uk	PROPN
ejpam-1506	132	45	,	,	PUNCT
ejpam-1506	132	46	ak	ak	PROPN
ejpam-1506	132	47	)	)	PUNCT
ejpam-1506	132	48	>	>	X
ejpam-1506	133	1	dg⋆(uy	dg⋆(uy	PROPN
ejpam-1506	133	2	,	,	PUNCT
ejpam-1506	133	3	uk	uk	PROPN
ejpam-1506	133	4	,	,	PUNCT
ejpam-1506	133	5	ak	ak	PROPN
ejpam-1506	133	6	)	)	PUNCT
ejpam-1506	133	7	for	for	ADP
ejpam-1506	133	8	some	some	DET
ejpam-1506	133	9	k	k	PROPN
ejpam-1506	134	1	6=	6=	PROPN
ejpam-1506	134	2	i	i	PRON
ejpam-1506	134	3	(	(	PUNCT
ejpam-1506	134	4	1	1	NUM
ejpam-1506	134	5	≤	≤	NUM
ejpam-1506	134	6	k	k	X
ejpam-1506	134	7	≤	≤	PROPN
ejpam-1506	134	8	n−	n−	NOUN
ejpam-1506	134	9	1	1	NUM
ejpam-1506	134	10	)	)	PUNCT
ejpam-1506	134	11	satisfying	satisfy	VERB
ejpam-1506	134	12	ng⋆(uy)∩uk	ng⋆(uy)∩uk	PROPN
ejpam-1506	134	13	6=	6=	PROPN
ejpam-1506	134	14	;	;	PUNCT
ejpam-1506	134	15	.	.	PUNCT
ejpam-1506	135	1	so	so	ADV
ejpam-1506	135	2	,	,	PUNCT
ejpam-1506	135	3	(	(	PUNCT
ejpam-1506	135	4	2	2	X
ejpam-1506	135	5	)	)	PUNCT
ejpam-1506	135	6	holds	hold	VERB
ejpam-1506	135	7	for	for	ADP
ejpam-1506	135	8	each	each	DET
ejpam-1506	135	9	case	case	NOUN
ejpam-1506	135	10	.	.	PUNCT
ejpam-1506	136	1	subcase	subcase	PROPN
ejpam-1506	136	2	1.2	1.2	NUM
ejpam-1506	136	3	:	:	PUNCT
ejpam-1506	136	4	i	i	PRON
ejpam-1506	136	5	=	=	SYM
ejpam-1506	136	6	n.	n.	NOUN
ejpam-1506	136	7	since	since	SCONJ
ejpam-1506	136	8	dg⋆(ux	dg⋆(ux	NOUN
ejpam-1506	136	9	,	,	PUNCT
ejpam-1506	136	10	uk	uk	PROPN
ejpam-1506	136	11	,	,	PUNCT
ejpam-1506	136	12	ak	ak	PROPN
ejpam-1506	136	13	)	)	PUNCT
ejpam-1506	136	14	<	<	X
ejpam-1506	136	15	dg⋆(uy	dg⋆(uy	PROPN
ejpam-1506	136	16	,	,	PUNCT
ejpam-1506	136	17	uk	uk	PROPN
ejpam-1506	136	18	,	,	PUNCT
ejpam-1506	136	19	ak	ak	PROPN
ejpam-1506	136	20	)	)	PUNCT
ejpam-1506	136	21	for	for	ADP
ejpam-1506	136	22	some	some	DET
ejpam-1506	136	23	k	k	PROPN
ejpam-1506	136	24	(	(	PUNCT
ejpam-1506	136	25	1	1	NUM
ejpam-1506	136	26	≤	≤	NUM
ejpam-1506	136	27	k	k	X
ejpam-1506	136	28	≤	≤	PROPN
ejpam-1506	136	29	n−	n−	NOUN
ejpam-1506	136	30	1	1	NUM
ejpam-1506	136	31	)	)	PUNCT
ejpam-1506	136	32	satisfying	satisfy	VERB
ejpam-1506	136	33	ng⋆(ux	ng⋆(ux	NOUN
ejpam-1506	136	34	)	)	PUNCT
ejpam-1506	136	35	∩uk	∩uk	ADJ
ejpam-1506	136	36	6=	6=	NUM
ejpam-1506	136	37	;	;	PUNCT
ejpam-1506	136	38	,	,	PUNCT
ejpam-1506	136	39	(	(	PUNCT
ejpam-1506	136	40	2	2	X
ejpam-1506	136	41	)	)	PUNCT
ejpam-1506	136	42	holds	hold	VERB
ejpam-1506	136	43	.	.	PUNCT
ejpam-1506	137	1	case	case	NOUN
ejpam-1506	137	2	2	2	NUM
ejpam-1506	137	3	:	:	PUNCT
ejpam-1506	137	4	ux	ux	PROPN
ejpam-1506	137	5	∈	∈	PROPN
ejpam-1506	137	6	ui	ui	PROPN
ejpam-1506	138	1	and	and	CCONJ
ejpam-1506	138	2	uy	uy	PROPN
ejpam-1506	138	3	∈	∈	PROPN
ejpam-1506	138	4	u	u	PROPN
ejpam-1506	138	5	j	j	PROPN
ejpam-1506	138	6	for	for	ADP
ejpam-1506	138	7	i	i	PROPN
ejpam-1506	138	8	6=	6=	PROPN
ejpam-1506	138	9	j	j	PROPN
ejpam-1506	138	10	(	(	PUNCT
ejpam-1506	138	11	1≤	1≤	INTJ
ejpam-1506	138	12	i	i	PROPN
ejpam-1506	138	13	,	,	PUNCT
ejpam-1506	138	14	j	j	PROPN
ejpam-1506	138	15	≤	≤	PROPN
ejpam-1506	138	16	n	n	CCONJ
ejpam-1506	138	17	)	)	PUNCT
ejpam-1506	138	18	.	.	PUNCT
ejpam-1506	139	1	we	we	PRON
ejpam-1506	139	2	consider	consider	VERB
ejpam-1506	139	3	two	two	NUM
ejpam-1506	139	4	subcases	subcase	NOUN
ejpam-1506	139	5	.	.	PUNCT
ejpam-1506	140	1	subcase	subcase	PROPN
ejpam-1506	140	2	2.1	2.1	NUM
ejpam-1506	140	3	:	:	SYM
ejpam-1506	140	4	1≤	1≤	NUM
ejpam-1506	141	1	i	i	PRON
ejpam-1506	141	2	,	,	PUNCT
ejpam-1506	141	3	j	j	PROPN
ejpam-1506	141	4	≤	≤	PROPN
ejpam-1506	141	5	n−	n−	PROPN
ejpam-1506	141	6	1	1	NUM
ejpam-1506	141	7	.	.	PUNCT
ejpam-1506	142	1	if	if	SCONJ
ejpam-1506	142	2	ui	ui	PROPN
ejpam-1506	142	3	,	,	PUNCT
ejpam-1506	142	4	ai	ai	VERB
ejpam-1506	142	5	uy	uy	PROPN
ejpam-1506	142	6	6∈	6∈	PROPN
ejpam-1506	142	7	e(g⋆	e(g⋆	NOUN
ejpam-1506	142	8	)	)	PUNCT
ejpam-1506	142	9	,	,	PUNCT
ejpam-1506	142	10	then	then	ADV
ejpam-1506	142	11	dg⋆(ux	dg⋆(ux	NOUN
ejpam-1506	142	12	,	,	PUNCT
ejpam-1506	142	13	ui	ui	NOUN
ejpam-1506	142	14	,	,	PUNCT
ejpam-1506	142	15	ai	ai	VERB
ejpam-1506	142	16	)	)	PUNCT
ejpam-1506	142	17	=	=	SYM
ejpam-1506	142	18	1	1	NUM
ejpam-1506	142	19	and	and	CCONJ
ejpam-1506	142	20	dg⋆(uy	dg⋆(uy	PROPN
ejpam-1506	142	21	,	,	PUNCT
ejpam-1506	143	1	ui	ui	PROPN
ejpam-1506	143	2	,	,	PUNCT
ejpam-1506	143	3	ai	ai	VERB
ejpam-1506	143	4	)	)	PUNCT
ejpam-1506	143	5	≥	≥	NOUN
ejpam-1506	143	6	2	2	NUM
ejpam-1506	143	7	,	,	PUNCT
ejpam-1506	143	8	and	and	CCONJ
ejpam-1506	143	9	thus	thus	ADV
ejpam-1506	143	10	(	(	PUNCT
ejpam-1506	143	11	2	2	X
ejpam-1506	143	12	)	)	PUNCT
ejpam-1506	143	13	holds	hold	VERB
ejpam-1506	143	14	.	.	PUNCT
ejpam-1506	144	1	so	so	ADV
ejpam-1506	144	2	,	,	PUNCT
ejpam-1506	144	3	we	we	PRON
ejpam-1506	144	4	consider	consider	VERB
ejpam-1506	144	5	ui	ui	PROPN
ejpam-1506	144	6	,	,	PUNCT
ejpam-1506	144	7	ai	ai	VERB
ejpam-1506	144	8	uy	uy	PROPN
ejpam-1506	144	9	∈	∈	PROPN
ejpam-1506	144	10	e(g⋆	e(g⋆	NOUN
ejpam-1506	144	11	)	)	PUNCT
ejpam-1506	144	12	;	;	PUNCT
ejpam-1506	144	13	notice	notice	VERB
ejpam-1506	144	14	that	that	SCONJ
ejpam-1506	144	15	dg⋆(ux	dg⋆(ux	NOUN
ejpam-1506	144	16	,	,	PUNCT
ejpam-1506	144	17	ui	ui	NOUN
ejpam-1506	144	18	,	,	PUNCT
ejpam-1506	144	19	ai	ai	VERB
ejpam-1506	144	20	)	)	PUNCT
ejpam-1506	144	21	=	=	PUNCT
ejpam-1506	144	22	1=	1=	NUM
ejpam-1506	144	23	dg⋆(uy	dg⋆(uy	NUM
ejpam-1506	144	24	,	,	PUNCT
ejpam-1506	144	25	ui	ui	PROPN
ejpam-1506	144	26	,	,	PUNCT
ejpam-1506	144	27	ai	ai	INTJ
ejpam-1506	144	28	)	)	PUNCT
ejpam-1506	144	29	.	.	PUNCT
ejpam-1506	145	1	but	but	CCONJ
ejpam-1506	145	2	dg⋆(uy	dg⋆(uy	X
ejpam-1506	145	3	,	,	PUNCT
ejpam-1506	145	4	u	u	PROPN
ejpam-1506	145	5	j	j	PROPN
ejpam-1506	145	6	,	,	PUNCT
ejpam-1506	145	7	a	a	DET
ejpam-1506	145	8	j	j	NOUN
ejpam-1506	145	9	)	)	PUNCT
ejpam-1506	146	1	=	=	SYM
ejpam-1506	146	2	1	1	NUM
ejpam-1506	146	3	and	and	CCONJ
ejpam-1506	146	4	dg⋆(ux	dg⋆(ux	NOUN
ejpam-1506	146	5	,	,	PUNCT
ejpam-1506	146	6	u	u	PROPN
ejpam-1506	146	7	j	j	PROPN
ejpam-1506	146	8	,	,	PUNCT
ejpam-1506	146	9	a	a	DET
ejpam-1506	146	10	j	j	PROPN
ejpam-1506	146	11	)	)	PUNCT
ejpam-1506	146	12	≥	≥	PROPN
ejpam-1506	146	13	2	2	NUM
ejpam-1506	146	14	,	,	PUNCT
ejpam-1506	146	15	and	and	CCONJ
ejpam-1506	146	16	thus	thus	ADV
ejpam-1506	146	17	(	(	PUNCT
ejpam-1506	146	18	2	2	X
ejpam-1506	146	19	)	)	PUNCT
ejpam-1506	146	20	holds	hold	VERB
ejpam-1506	146	21	.	.	PUNCT
ejpam-1506	147	1	subcase	subcase	PROPN
ejpam-1506	147	2	2.2	2.2	NUM
ejpam-1506	147	3	:	:	SYM
ejpam-1506	147	4	1	1	NUM
ejpam-1506	147	5	≤	≤	NUM
ejpam-1506	148	1	i	i	PRON
ejpam-1506	148	2	≤	≤	ADJ
ejpam-1506	148	3	n−	n−	NOUN
ejpam-1506	148	4	1	1	NUM
ejpam-1506	148	5	and	and	CCONJ
ejpam-1506	148	6	j	j	PROPN
ejpam-1506	148	7	=	=	SYM
ejpam-1506	148	8	n	n	CCONJ
ejpam-1506	148	9	,	,	PUNCT
ejpam-1506	148	10	or	or	CCONJ
ejpam-1506	148	11	1	1	NUM
ejpam-1506	148	12	≤	≤	NUM
ejpam-1506	148	13	j	j	PROPN
ejpam-1506	148	14	≤	≤	PROPN
ejpam-1506	148	15	n	n	CCONJ
ejpam-1506	148	16	−	−	PROPN
ejpam-1506	148	17	1	1	NUM
ejpam-1506	148	18	and	and	CCONJ
ejpam-1506	148	19	i	i	PRON
ejpam-1506	148	20	=	=	SYM
ejpam-1506	148	21	n	n	CCONJ
ejpam-1506	148	22	,	,	PUNCT
ejpam-1506	148	23	say	say	VERB
ejpam-1506	148	24	the	the	DET
ejpam-1506	148	25	former	former	ADJ
ejpam-1506	148	26	.	.	PUNCT
ejpam-1506	149	1	if	if	SCONJ
ejpam-1506	149	2	ng⋆(uy	ng⋆(uy	NOUN
ejpam-1506	149	3	)	)	PUNCT
ejpam-1506	150	1	∩	∩	NOUN
ejpam-1506	150	2	ui	ui	PROPN
ejpam-1506	150	3	=	=	PUNCT
ejpam-1506	150	4	;	;	PUNCT
ejpam-1506	150	5	,	,	PUNCT
ejpam-1506	150	6	then	then	ADV
ejpam-1506	150	7	dg⋆(ux	dg⋆(ux	NOUN
ejpam-1506	150	8	,	,	PUNCT
ejpam-1506	150	9	ui	ui	NOUN
ejpam-1506	150	10	,	,	PUNCT
ejpam-1506	150	11	ai	ai	VERB
ejpam-1506	150	12	)	)	PUNCT
ejpam-1506	150	13	=	=	SYM
ejpam-1506	150	14	1	1	NUM
ejpam-1506	150	15	<	<	X
ejpam-1506	150	16	dg⋆(uy	dg⋆(uy	PROPN
ejpam-1506	150	17	,	,	PUNCT
ejpam-1506	150	18	ui	ui	PROPN
ejpam-1506	150	19	,	,	PUNCT
ejpam-1506	150	20	ai	ai	VERB
ejpam-1506	150	21	)	)	PUNCT
ejpam-1506	150	22	,	,	PUNCT
ejpam-1506	150	23	and	and	CCONJ
ejpam-1506	150	24	thus	thus	ADV
ejpam-1506	150	25	(	(	PUNCT
ejpam-1506	150	26	2	2	X
ejpam-1506	150	27	)	)	PUNCT
ejpam-1506	150	28	holds	hold	VERB
ejpam-1506	150	29	.	.	PUNCT
ejpam-1506	151	1	so	so	ADV
ejpam-1506	151	2	,	,	PUNCT
ejpam-1506	151	3	suppose	suppose	VERB
ejpam-1506	151	4	that	that	SCONJ
ejpam-1506	151	5	ng⋆(uy	ng⋆(uy	NOUN
ejpam-1506	151	6	)	)	PUNCT
ejpam-1506	151	7	∩ui	∩ui	NOUN
ejpam-1506	151	8	6=	6=	NUM
ejpam-1506	151	9	;	;	PUNCT
ejpam-1506	151	10	.	.	PUNCT
ejpam-1506	152	1	if	if	SCONJ
ejpam-1506	152	2	uyui	uyui	ADJ
ejpam-1506	152	3	,	,	PUNCT
ejpam-1506	152	4	ai	ai	VERB
ejpam-1506	152	5	6∈	6∈	NOUN
ejpam-1506	152	6	e(g⋆	e(g⋆	NOUN
ejpam-1506	152	7	)	)	PUNCT
ejpam-1506	152	8	,	,	PUNCT
ejpam-1506	152	9	then	then	ADV
ejpam-1506	152	10	dg⋆(ux	dg⋆(ux	NOUN
ejpam-1506	152	11	,	,	PUNCT
ejpam-1506	152	12	ui	ui	NOUN
ejpam-1506	152	13	,	,	PUNCT
ejpam-1506	152	14	ai	ai	VERB
ejpam-1506	152	15	)	)	PUNCT
ejpam-1506	152	16	=	=	SYM
ejpam-1506	152	17	1	1	NUM
ejpam-1506	152	18	and	and	CCONJ
ejpam-1506	152	19	dg⋆(uy	dg⋆(uy	PROPN
ejpam-1506	152	20	,	,	PUNCT
ejpam-1506	152	21	ui	ui	PROPN
ejpam-1506	152	22	,	,	PUNCT
ejpam-1506	152	23	ai	ai	VERB
ejpam-1506	152	24	)	)	PUNCT
ejpam-1506	152	25	=	=	SYM
ejpam-1506	152	26	2	2	NUM
ejpam-1506	152	27	,	,	PUNCT
ejpam-1506	152	28	and	and	CCONJ
ejpam-1506	152	29	thus	thus	ADV
ejpam-1506	152	30	(	(	PUNCT
ejpam-1506	152	31	2	2	X
ejpam-1506	152	32	)	)	PUNCT
ejpam-1506	152	33	holds	hold	VERB
ejpam-1506	152	34	.	.	PUNCT
ejpam-1506	153	1	if	if	SCONJ
ejpam-1506	153	2	uyui	uyui	ADJ
ejpam-1506	153	3	,	,	PUNCT
ejpam-1506	153	4	ai	ai	VERB
ejpam-1506	153	5	∈	∈	NOUN
ejpam-1506	153	6	e(g⋆	e(g⋆	NOUN
ejpam-1506	153	7	)	)	PUNCT
ejpam-1506	153	8	,	,	PUNCT
ejpam-1506	153	9	then	then	ADV
ejpam-1506	153	10	dg⋆(ux	dg⋆(ux	NOUN
ejpam-1506	153	11	,	,	PUNCT
ejpam-1506	153	12	ui	ui	NOUN
ejpam-1506	153	13	,	,	PUNCT
ejpam-1506	153	14	ai	ai	VERB
ejpam-1506	153	15	)	)	PUNCT
ejpam-1506	154	1	=	=	PUNCT
ejpam-1506	154	2	1=	1=	NUM
ejpam-1506	154	3	dg⋆(uy	dg⋆(uy	NUM
ejpam-1506	154	4	,	,	PUNCT
ejpam-1506	154	5	ui	ui	PROPN
ejpam-1506	154	6	,	,	PUNCT
ejpam-1506	154	7	ai	ai	INTJ
ejpam-1506	154	8	)	)	PUNCT
ejpam-1506	154	9	;	;	PUNCT
ejpam-1506	154	10	further	far	ADV
ejpam-1506	154	11	,	,	PUNCT
ejpam-1506	154	12	one	one	PRON
ejpam-1506	154	13	can	can	AUX
ejpam-1506	154	14	easily	easily	ADV
ejpam-1506	154	15	verify	verify	VERB
ejpam-1506	154	16	that	that	SCONJ
ejpam-1506	154	17	codes(ux	codes(ux	NOUN
ejpam-1506	154	18	)	)	PUNCT
ejpam-1506	154	19	=	=	SYM
ejpam-1506	154	20	codes(uy	codes(uy	NOUN
ejpam-1506	154	21	)	)	PUNCT
ejpam-1506	154	22	implies	imply	VERB
ejpam-1506	154	23	that	that	SCONJ
ejpam-1506	154	24	vi	vi	PROPN
ejpam-1506	154	25	and	and	CCONJ
ejpam-1506	154	26	vn	vn	PROPN
ejpam-1506	154	27	are	be	AUX
ejpam-1506	154	28	adjacent	adjacent	ADJ
ejpam-1506	154	29	twins	twin	NOUN
ejpam-1506	154	30	in	in	ADP
ejpam-1506	154	31	g.	g.	PROPN
ejpam-1506	154	32	if	if	SCONJ
ejpam-1506	154	33	every	every	DET
ejpam-1506	154	34	pair	pair	NOUN
ejpam-1506	154	35	of	of	ADP
ejpam-1506	154	36	vertices	vertex	NOUN
ejpam-1506	154	37	in	in	ADP
ejpam-1506	154	38	g	g	PROPN
ejpam-1506	154	39	are	be	AUX
ejpam-1506	154	40	twins	twin	NOUN
ejpam-1506	154	41	in	in	ADP
ejpam-1506	154	42	g	g	PROPN
ejpam-1506	154	43	,	,	PUNCT
ejpam-1506	154	44	then	then	ADV
ejpam-1506	154	45	g	g	PROPN
ejpam-1506	154	46	∼=	∼=	PROPN
ejpam-1506	154	47	kn	kn	PROPN
ejpam-1506	154	48	,	,	PUNCT
ejpam-1506	154	49	and	and	CCONJ
ejpam-1506	154	50	dim(k⋆n	dim(k⋆n	PROPN
ejpam-1506	154	51	)	)	PUNCT
ejpam-1506	154	52	=	=	SYM
ejpam-1506	155	1	n	n	CCONJ
ejpam-1506	155	2	−	−	PROPN
ejpam-1506	155	3	1	1	NUM
ejpam-1506	155	4	(	(	PUNCT
ejpam-1506	155	5	see	see	VERB
ejpam-1506	155	6	theorem	theorem	NOUN
ejpam-1506	155	7	12	12	NUM
ejpam-1506	155	8	)	)	PUNCT
ejpam-1506	155	9	.	.	PUNCT
ejpam-1506	156	1	d.	d.	PROPN
ejpam-1506	156	2	klein	klein	PROPN
ejpam-1506	156	3	,	,	PUNCT
ejpam-1506	156	4	e.	e.	PROPN
ejpam-1506	156	5	yi	yi	PROPN
ejpam-1506	156	6	/	/	SYM
ejpam-1506	156	7	eur	eur	PROPN
ejpam-1506	156	8	.	.	PUNCT
ejpam-1506	157	1	j.	j.	PROPN
ejpam-1506	157	2	pure	pure	PROPN
ejpam-1506	157	3	appl	appl	PROPN
ejpam-1506	157	4	.	.	PROPN
ejpam-1506	157	5	math	math	PROPN
ejpam-1506	157	6	,	,	PUNCT
ejpam-1506	157	7	5	5	NUM
ejpam-1506	157	8	(	(	PUNCT
ejpam-1506	157	9	2012	2012	NUM
ejpam-1506	157	10	)	)	PUNCT
ejpam-1506	157	11	,	,	PUNCT
ejpam-1506	157	12	302	302	NUM
ejpam-1506	157	13	-	-	SYM
ejpam-1506	157	14	316	316	NUM
ejpam-1506	157	15	307	307	NUM
ejpam-1506	157	16	in	in	ADP
ejpam-1506	157	17	each	each	DET
ejpam-1506	157	18	case	case	NOUN
ejpam-1506	157	19	,	,	PUNCT
ejpam-1506	157	20	s	s	NOUN
ejpam-1506	157	21	forms	form	VERB
ejpam-1506	157	22	a	a	DET
ejpam-1506	157	23	resolving	resolving	NOUN
ejpam-1506	157	24	set	set	VERB
ejpam-1506	157	25	for	for	ADP
ejpam-1506	157	26	g⋆	g⋆	NOUN
ejpam-1506	157	27	with	with	ADP
ejpam-1506	157	28	|s|	|s|	NOUN
ejpam-1506	157	29	=	=	SYM
ejpam-1506	157	30	n−1	n−1	PROPN
ejpam-1506	157	31	.	.	PUNCT
ejpam-1506	158	1	so	so	ADV
ejpam-1506	158	2	,	,	PUNCT
ejpam-1506	158	3	the	the	DET
ejpam-1506	158	4	upper	upper	ADJ
ejpam-1506	158	5	bound	bound	NOUN
ejpam-1506	158	6	of	of	ADP
ejpam-1506	158	7	(	(	PUNCT
ejpam-1506	158	8	1	1	X
ejpam-1506	158	9	)	)	PUNCT
ejpam-1506	158	10	follows	follow	VERB
ejpam-1506	158	11	.	.	PUNCT
ejpam-1506	159	1	for	for	ADP
ejpam-1506	159	2	the	the	DET
ejpam-1506	159	3	sharpness	sharpness	NOUN
ejpam-1506	159	4	of	of	ADP
ejpam-1506	159	5	the	the	DET
ejpam-1506	159	6	upper	upper	ADJ
ejpam-1506	159	7	bound	bind	VERB
ejpam-1506	159	8	,	,	PUNCT
ejpam-1506	159	9	take	take	VERB
ejpam-1506	159	10	g	g	NOUN
ejpam-1506	159	11	=	=	SYM
ejpam-1506	159	12	kn	kn	PROPN
ejpam-1506	159	13	;	;	PUNCT
ejpam-1506	159	14	then	then	ADV
ejpam-1506	159	15	dim(g⋆	dim(g⋆	NOUN
ejpam-1506	159	16	)	)	PUNCT
ejpam-1506	160	1	=	=	SYM
ejpam-1506	160	2	n−1	n−1	PROPN
ejpam-1506	160	3	by	by	ADP
ejpam-1506	160	4	theorem	theorem	NOUN
ejpam-1506	160	5	12	12	NUM
ejpam-1506	160	6	.	.	PUNCT
ejpam-1506	161	1	the	the	DET
ejpam-1506	161	2	authors	author	NOUN
ejpam-1506	161	3	of	of	ADP
ejpam-1506	161	4	[	[	X
ejpam-1506	161	5	5	5	NUM
ejpam-1506	161	6	]	]	PUNCT
ejpam-1506	161	7	characterized	characterize	VERB
ejpam-1506	161	8	connected	connected	ADJ
ejpam-1506	161	9	graphs	graph	NOUN
ejpam-1506	161	10	of	of	ADP
ejpam-1506	161	11	order	order	NOUN
ejpam-1506	161	12	n	n	PRON
ejpam-1506	161	13	with	with	ADP
ejpam-1506	161	14	metric	metric	ADJ
ejpam-1506	161	15	dimension	dimension	NOUN
ejpam-1506	161	16	1	1	NUM
ejpam-1506	161	17	,	,	PUNCT
ejpam-1506	161	18	n−	n−	NOUN
ejpam-1506	161	19	2	2	NUM
ejpam-1506	161	20	,	,	PUNCT
ejpam-1506	161	21	and	and	CCONJ
ejpam-1506	161	22	n−	n−	NOUN
ejpam-1506	161	23	1	1	NUM
ejpam-1506	161	24	,	,	PUNCT
ejpam-1506	161	25	respectively	respectively	ADV
ejpam-1506	161	26	.	.	PUNCT
ejpam-1506	162	1	theorem	theorem	VERB
ejpam-1506	162	2	7	7	NUM
ejpam-1506	162	3	.	.	PUNCT
ejpam-1506	163	1	[	[	X
ejpam-1506	163	2	5	5	NUM
ejpam-1506	163	3	]	]	PUNCT
ejpam-1506	163	4	let	let	VERB
ejpam-1506	163	5	g	g	PRON
ejpam-1506	163	6	be	be	AUX
ejpam-1506	163	7	a	a	DET
ejpam-1506	163	8	connected	connected	ADJ
ejpam-1506	163	9	graph	graph	NOUN
ejpam-1506	163	10	of	of	ADP
ejpam-1506	163	11	order	order	NOUN
ejpam-1506	163	12	n≥	n≥	NOUN
ejpam-1506	163	13	2	2	X
ejpam-1506	163	14	.	.	PUNCT
ejpam-1506	164	1	then	then	ADV
ejpam-1506	164	2	(	(	PUNCT
ejpam-1506	164	3	a	a	X
ejpam-1506	164	4	)	)	PUNCT
ejpam-1506	164	5	dim(g	dim(g	PROPN
ejpam-1506	164	6	)	)	PUNCT
ejpam-1506	164	7	=	=	SYM
ejpam-1506	164	8	1	1	NUM
ejpam-1506	164	9	if	if	SCONJ
ejpam-1506	164	10	and	and	CCONJ
ejpam-1506	164	11	only	only	ADV
ejpam-1506	164	12	if	if	SCONJ
ejpam-1506	164	13	g	g	PROPN
ejpam-1506	164	14	=	=	SYM
ejpam-1506	164	15	pn	pn	PROPN
ejpam-1506	164	16	,	,	PUNCT
ejpam-1506	164	17	(	(	PUNCT
ejpam-1506	164	18	b	b	X
ejpam-1506	164	19	)	)	PUNCT
ejpam-1506	164	20	dim(g	dim(g	PROPN
ejpam-1506	164	21	)	)	PUNCT
ejpam-1506	164	22	=	=	PUNCT
ejpam-1506	165	1	n−	n−	NOUN
ejpam-1506	165	2	1	1	NUM
ejpam-1506	165	3	if	if	SCONJ
ejpam-1506	166	1	and	and	CCONJ
ejpam-1506	166	2	only	only	ADV
ejpam-1506	166	3	if	if	SCONJ
ejpam-1506	166	4	g	g	PROPN
ejpam-1506	166	5	=	=	SYM
ejpam-1506	166	6	kn	kn	PROPN
ejpam-1506	166	7	,	,	PUNCT
ejpam-1506	166	8	(	(	PUNCT
ejpam-1506	166	9	c	c	NOUN
ejpam-1506	166	10	)	)	PUNCT
ejpam-1506	166	11	for	for	ADP
ejpam-1506	166	12	n≥	n≥	PROPN
ejpam-1506	166	13	4	4	NUM
ejpam-1506	166	14	,	,	PUNCT
ejpam-1506	166	15	dim(g	dim(g	PROPN
ejpam-1506	166	16	)	)	PUNCT
ejpam-1506	167	1	=	=	PUNCT
ejpam-1506	167	2	n−	n−	NOUN
ejpam-1506	167	3	2	2	NUM
ejpam-1506	167	4	if	if	SCONJ
ejpam-1506	167	5	and	and	CCONJ
ejpam-1506	167	6	only	only	ADV
ejpam-1506	167	7	if	if	SCONJ
ejpam-1506	167	8	g	g	PROPN
ejpam-1506	167	9	=	=	SYM
ejpam-1506	167	10	ks	ks	PROPN
ejpam-1506	167	11	,	,	PUNCT
ejpam-1506	167	12	t	t	PROPN
ejpam-1506	167	13	(	(	PUNCT
ejpam-1506	167	14	s	s	PROPN
ejpam-1506	167	15	,	,	PUNCT
ejpam-1506	167	16	t	t	PROPN
ejpam-1506	167	17	≥	≥	NUM
ejpam-1506	167	18	1	1	NUM
ejpam-1506	167	19	)	)	PUNCT
ejpam-1506	167	20	,	,	PUNCT
ejpam-1506	167	21	g	g	NOUN
ejpam-1506	167	22	=	=	SYM
ejpam-1506	167	23	ks+	ks+	PROPN
ejpam-1506	167	24	k	k	PROPN
ejpam-1506	167	25	t	t	PROPN
ejpam-1506	167	26	(	(	PUNCT
ejpam-1506	167	27	s	s	X
ejpam-1506	167	28	≥	≥	NOUN
ejpam-1506	167	29	1	1	NUM
ejpam-1506	167	30	,	,	PUNCT
ejpam-1506	167	31	t	t	PROPN
ejpam-1506	167	32	≥	≥	NUM
ejpam-1506	167	33	2	2	NUM
ejpam-1506	167	34	)	)	PUNCT
ejpam-1506	167	35	,	,	PUNCT
ejpam-1506	167	36	or	or	CCONJ
ejpam-1506	167	37	g	g	PROPN
ejpam-1506	167	38	=	=	PROPN
ejpam-1506	167	39	ks+(k1∪kt	ks+(k1∪kt	PROPN
ejpam-1506	167	40	)	)	PUNCT
ejpam-1506	167	41	(	(	PUNCT
ejpam-1506	167	42	s	s	PROPN
ejpam-1506	167	43	,	,	PUNCT
ejpam-1506	167	44	t	t	PROPN
ejpam-1506	167	45	≥	≥	NUM
ejpam-1506	167	46	1	1	NUM
ejpam-1506	167	47	)	)	PUNCT
ejpam-1506	167	48	;	;	PUNCT
ejpam-1506	167	49	here	here	ADV
ejpam-1506	167	50	,	,	PUNCT
ejpam-1506	167	51	a+b	a+b	NUM
ejpam-1506	167	52	denotes	denote	VERB
ejpam-1506	167	53	the	the	DET
ejpam-1506	167	54	graph	graph	NOUN
ejpam-1506	167	55	obtained	obtain	VERB
ejpam-1506	167	56	from	from	ADP
ejpam-1506	167	57	the	the	DET
ejpam-1506	167	58	disjoint	disjoint	PROPN
ejpam-1506	167	59	union	union	NOUN
ejpam-1506	167	60	of	of	ADP
ejpam-1506	167	61	graphs	graph	NOUN
ejpam-1506	167	62	a	a	PRON
ejpam-1506	167	63	and	and	CCONJ
ejpam-1506	167	64	b	b	NOUN
ejpam-1506	167	65	by	by	ADP
ejpam-1506	167	66	joining	join	VERB
ejpam-1506	167	67	every	every	DET
ejpam-1506	167	68	vertex	vertex	NOUN
ejpam-1506	167	69	of	of	ADP
ejpam-1506	167	70	a	a	PRON
ejpam-1506	167	71	with	with	ADP
ejpam-1506	167	72	every	every	DET
ejpam-1506	167	73	vertex	vertex	NOUN
ejpam-1506	167	74	of	of	ADP
ejpam-1506	167	75	b	b	NOUN
ejpam-1506	167	76	,	,	PUNCT
ejpam-1506	167	77	and	and	CCONJ
ejpam-1506	167	78	h	h	NOUN
ejpam-1506	167	79	denotes	denote	VERB
ejpam-1506	167	80	the	the	DET
ejpam-1506	167	81	graph	graph	NOUN
ejpam-1506	167	82	whose	whose	DET
ejpam-1506	167	83	vertex	vertex	NOUN
ejpam-1506	167	84	set	set	NOUN
ejpam-1506	167	85	is	be	AUX
ejpam-1506	167	86	v	v	NOUN
ejpam-1506	167	87	(	(	PUNCT
ejpam-1506	167	88	h	h	NOUN
ejpam-1506	167	89	)	)	PUNCT
ejpam-1506	167	90	and	and	CCONJ
ejpam-1506	167	91	uv	uv	NOUN
ejpam-1506	167	92	∈	∈	PROPN
ejpam-1506	167	93	e(h	e(h	PROPN
ejpam-1506	167	94	)	)	PUNCT
ejpam-1506	168	1	if	if	SCONJ
ejpam-1506	168	2	and	and	CCONJ
ejpam-1506	168	3	only	only	ADV
ejpam-1506	168	4	if	if	SCONJ
ejpam-1506	168	5	uv	uv	PROPN
ejpam-1506	168	6	6∈	6∈	NOUN
ejpam-1506	168	7	e(h	e(h	PROPN
ejpam-1506	168	8	)	)	PUNCT
ejpam-1506	168	9	for	for	ADP
ejpam-1506	168	10	u	u	NOUN
ejpam-1506	168	11	,	,	PUNCT
ejpam-1506	168	12	v	v	PROPN
ejpam-1506	168	13	∈	∈	PROPN
ejpam-1506	168	14	v	v	NOUN
ejpam-1506	168	15	(	(	PUNCT
ejpam-1506	168	16	h	h	NOUN
ejpam-1506	168	17	)	)	PUNCT
ejpam-1506	168	18	.	.	PUNCT
ejpam-1506	169	1	next	next	ADV
ejpam-1506	169	2	,	,	PUNCT
ejpam-1506	169	3	we	we	PRON
ejpam-1506	169	4	characterize	characterize	VERB
ejpam-1506	169	5	graphs	graph	NOUN
ejpam-1506	169	6	g	g	ADP
ejpam-1506	169	7	satisfying	satisfy	VERB
ejpam-1506	169	8	dim(g⋆	dim(g⋆	NOUN
ejpam-1506	169	9	)	)	PUNCT
ejpam-1506	169	10	=	=	SYM
ejpam-1506	169	11	1	1	NUM
ejpam-1506	169	12	and	and	CCONJ
ejpam-1506	169	13	dim(g⋆	dim(g⋆	NOUN
ejpam-1506	169	14	)	)	PUNCT
ejpam-1506	169	15	=	=	SYM
ejpam-1506	169	16	|v	|v	PROPN
ejpam-1506	169	17	(	(	PUNCT
ejpam-1506	169	18	g)|	g)|	NOUN
ejpam-1506	169	19	−	−	NOUN
ejpam-1506	169	20	1	1	NUM
ejpam-1506	169	21	,	,	PUNCT
ejpam-1506	169	22	respectively	respectively	ADV
ejpam-1506	169	23	.	.	PUNCT
ejpam-1506	170	1	theorem	theorem	VERB
ejpam-1506	170	2	8	8	NUM
ejpam-1506	170	3	.	.	PUNCT
ejpam-1506	171	1	let	let	VERB
ejpam-1506	171	2	g	g	PRON
ejpam-1506	171	3	be	be	AUX
ejpam-1506	171	4	a	a	DET
ejpam-1506	171	5	connected	connected	ADJ
ejpam-1506	171	6	graph	graph	NOUN
ejpam-1506	171	7	of	of	ADP
ejpam-1506	171	8	order	order	NOUN
ejpam-1506	171	9	n≥	n≥	NOUN
ejpam-1506	171	10	2	2	X
ejpam-1506	171	11	.	.	PUNCT
ejpam-1506	172	1	then	then	ADV
ejpam-1506	172	2	(	(	PUNCT
ejpam-1506	172	3	a	a	X
ejpam-1506	172	4	)	)	PUNCT
ejpam-1506	172	5	dim(g⋆	dim(g⋆	NOUN
ejpam-1506	172	6	)	)	PUNCT
ejpam-1506	172	7	=	=	SYM
ejpam-1506	172	8	1	1	NUM
ejpam-1506	172	9	if	if	SCONJ
ejpam-1506	172	10	and	and	CCONJ
ejpam-1506	172	11	only	only	ADV
ejpam-1506	172	12	if	if	SCONJ
ejpam-1506	172	13	g	g	PROPN
ejpam-1506	172	14	=	=	SYM
ejpam-1506	172	15	pn	pn	PROPN
ejpam-1506	172	16	,	,	PUNCT
ejpam-1506	172	17	(	(	PUNCT
ejpam-1506	172	18	b	b	NOUN
ejpam-1506	172	19	)	)	PUNCT
ejpam-1506	172	20	for	for	ADP
ejpam-1506	172	21	n≥	n≥	PROPN
ejpam-1506	172	22	4	4	NUM
ejpam-1506	172	23	,	,	PUNCT
ejpam-1506	172	24	dim(g⋆	dim(g⋆	NOUN
ejpam-1506	172	25	)	)	PUNCT
ejpam-1506	172	26	=	=	PUNCT
ejpam-1506	173	1	n−	n−	NOUN
ejpam-1506	173	2	1	1	NUM
ejpam-1506	173	3	if	if	SCONJ
ejpam-1506	173	4	and	and	CCONJ
ejpam-1506	173	5	only	only	ADV
ejpam-1506	173	6	if	if	SCONJ
ejpam-1506	173	7	g	g	PROPN
ejpam-1506	173	8	=	=	PROPN
ejpam-1506	173	9	kn	kn	PROPN
ejpam-1506	173	10	.	.	PUNCT
ejpam-1506	173	11	proof	proof	NOUN
ejpam-1506	173	12	.	.	PUNCT
ejpam-1506	174	1	let	let	VERB
ejpam-1506	174	2	g	g	PRON
ejpam-1506	174	3	be	be	AUX
ejpam-1506	174	4	a	a	DET
ejpam-1506	174	5	connected	connected	ADJ
ejpam-1506	174	6	graph	graph	NOUN
ejpam-1506	174	7	of	of	ADP
ejpam-1506	174	8	order	order	NOUN
ejpam-1506	174	9	n≥	n≥	NOUN
ejpam-1506	174	10	2	2	NUM
ejpam-1506	174	11	.	.	PUNCT
ejpam-1506	175	1	(	(	PUNCT
ejpam-1506	175	2	a	a	X
ejpam-1506	175	3	)	)	PUNCT
ejpam-1506	175	4	(	(	PUNCT
ejpam-1506	175	5	⇐	⇐	NOUN
ejpam-1506	175	6	=)	=)	PROPN
ejpam-1506	175	7	if	if	SCONJ
ejpam-1506	175	8	g	g	PROPN
ejpam-1506	175	9	=	=	SYM
ejpam-1506	175	10	pn	pn	PROPN
ejpam-1506	175	11	,	,	PUNCT
ejpam-1506	175	12	then	then	ADV
ejpam-1506	175	13	g⋆	g⋆	VERB
ejpam-1506	175	14	=	=	SYM
ejpam-1506	175	15	p2n−2	p2n−2	PROPN
ejpam-1506	175	16	,	,	PUNCT
ejpam-1506	175	17	and	and	CCONJ
ejpam-1506	175	18	thus	thus	ADV
ejpam-1506	175	19	dim(g⋆	dim(g⋆	NOUN
ejpam-1506	175	20	)	)	PUNCT
ejpam-1506	175	21	=	=	SYM
ejpam-1506	176	1	1	1	NUM
ejpam-1506	176	2	by	by	ADP
ejpam-1506	176	3	(	(	PUNCT
ejpam-1506	176	4	a	a	NOUN
ejpam-1506	176	5	)	)	PUNCT
ejpam-1506	176	6	of	of	ADP
ejpam-1506	176	7	theorem	theorem	NOUN
ejpam-1506	176	8	7	7	NUM
ejpam-1506	176	9	.	.	PUNCT
ejpam-1506	176	10	(=	(=	NOUN
ejpam-1506	176	11	⇒	⇒	NOUN
ejpam-1506	176	12	)	)	PUNCT
ejpam-1506	176	13	suppose	suppose	VERB
ejpam-1506	176	14	dim(g⋆	dim(g⋆	NOUN
ejpam-1506	176	15	)	)	PUNCT
ejpam-1506	176	16	=	=	SYM
ejpam-1506	177	1	1	1	X
ejpam-1506	177	2	.	.	X
ejpam-1506	177	3	then	then	ADV
ejpam-1506	177	4	∆(g	∆(g	PROPN
ejpam-1506	177	5	)	)	PUNCT
ejpam-1506	177	6	≤	≤	NOUN
ejpam-1506	177	7	2	2	NUM
ejpam-1506	177	8	;	;	PUNCT
ejpam-1506	177	9	otherwise	otherwise	ADV
ejpam-1506	177	10	,	,	PUNCT
ejpam-1506	177	11	g⋆	g⋆	NOUN
ejpam-1506	177	12	contains	contain	VERB
ejpam-1506	177	13	k3	k3	ADJ
ejpam-1506	177	14	as	as	ADP
ejpam-1506	177	15	a	a	DET
ejpam-1506	177	16	subgraph	subgraph	NOUN
ejpam-1506	177	17	,	,	PUNCT
ejpam-1506	177	18	and	and	CCONJ
ejpam-1506	177	19	thus	thus	ADV
ejpam-1506	177	20	dim(g⋆	dim(g⋆	NOUN
ejpam-1506	177	21	)	)	PUNCT
ejpam-1506	177	22	≥	≥	NOUN
ejpam-1506	177	23	2	2	NUM
ejpam-1506	177	24	by	by	ADP
ejpam-1506	177	25	(	(	PUNCT
ejpam-1506	177	26	a	a	NOUN
ejpam-1506	177	27	)	)	PUNCT
ejpam-1506	177	28	of	of	ADP
ejpam-1506	177	29	theorem	theorem	NOUN
ejpam-1506	177	30	7	7	NUM
ejpam-1506	177	31	.	.	PUNCT
ejpam-1506	177	32	since	since	SCONJ
ejpam-1506	177	33	∆(g	∆(g	NOUN
ejpam-1506	177	34	)	)	PUNCT
ejpam-1506	177	35	≤	≤	NOUN
ejpam-1506	177	36	2	2	NUM
ejpam-1506	177	37	,	,	PUNCT
ejpam-1506	177	38	g	g	PROPN
ejpam-1506	177	39	is	be	AUX
ejpam-1506	177	40	either	either	CCONJ
ejpam-1506	177	41	a	a	DET
ejpam-1506	177	42	path	path	NOUN
ejpam-1506	177	43	or	or	CCONJ
ejpam-1506	177	44	a	a	DET
ejpam-1506	177	45	cycle	cycle	NOUN
ejpam-1506	177	46	.	.	PUNCT
ejpam-1506	178	1	since	since	SCONJ
ejpam-1506	178	2	c⋆n	c⋆n	VERB
ejpam-1506	178	3	∼=	∼=	PROPN
ejpam-1506	178	4	c2n	c2n	NOUN
ejpam-1506	178	5	and	and	CCONJ
ejpam-1506	178	6	dim(c⋆n	dim(c⋆n	NOUN
ejpam-1506	178	7	)	)	PUNCT
ejpam-1506	178	8	=	=	SYM
ejpam-1506	178	9	2	2	NUM
ejpam-1506	178	10	,	,	PUNCT
ejpam-1506	178	11	g	g	PROPN
ejpam-1506	178	12	∼=	∼=	PROPN
ejpam-1506	178	13	pn	pn	PROPN
ejpam-1506	178	14	.	.	PUNCT
ejpam-1506	179	1	(	(	PUNCT
ejpam-1506	179	2	b	b	X
ejpam-1506	179	3	)	)	PUNCT
ejpam-1506	179	4	(	(	PUNCT
ejpam-1506	179	5	⇐	⇐	ADJ
ejpam-1506	179	6	=)	=)	PROPN
ejpam-1506	179	7	see	see	VERB
ejpam-1506	179	8	theorem	theorem	NOUN
ejpam-1506	179	9	12	12	NUM
ejpam-1506	179	10	.	.	PUNCT
ejpam-1506	180	1	(=	(=	AUX
ejpam-1506	180	2	⇒	⇒	NOUN
ejpam-1506	180	3	)	)	PUNCT
ejpam-1506	180	4	suppose	suppose	VERB
ejpam-1506	180	5	that	that	SCONJ
ejpam-1506	180	6	g	g	PROPN
ejpam-1506	180	7	6=	6=	PROPN
ejpam-1506	180	8	kn	kn	PROPN
ejpam-1506	180	9	for	for	ADP
ejpam-1506	180	10	n	n	X
ejpam-1506	180	11	≥	≥	NOUN
ejpam-1506	180	12	4	4	NUM
ejpam-1506	180	13	,	,	PUNCT
ejpam-1506	180	14	and	and	CCONJ
ejpam-1506	180	15	let	let	VERB
ejpam-1506	180	16	v	v	NOUN
ejpam-1506	180	17	(	(	PUNCT
ejpam-1506	180	18	g	g	NOUN
ejpam-1506	180	19	)	)	PUNCT
ejpam-1506	180	20	=	=	SYM
ejpam-1506	180	21	{	{	PUNCT
ejpam-1506	180	22	v1	v1	PROPN
ejpam-1506	180	23	,	,	PUNCT
ejpam-1506	180	24	v2	v2	PROPN
ejpam-1506	180	25	,	,	PUNCT
ejpam-1506	180	26	.	.	PUNCT
ejpam-1506	180	27	.	.	PUNCT
ejpam-1506	180	28	.	.	PUNCT
ejpam-1506	181	1	,	,	PUNCT
ejpam-1506	181	2	vn	vn	VERB
ejpam-1506	181	3	}	}	PUNCT
ejpam-1506	181	4	with	with	ADP
ejpam-1506	181	5	degg(vi	degg(vi	PROPN
ejpam-1506	181	6	)	)	PUNCT
ejpam-1506	181	7	=	=	SYM
ejpam-1506	181	8	di	di	X
ejpam-1506	181	9	(	(	PUNCT
ejpam-1506	181	10	1	1	NUM
ejpam-1506	181	11	≤	≤	NUM
ejpam-1506	181	12	i	i	NOUN
ejpam-1506	181	13	≤	≤	NOUN
ejpam-1506	181	14	n	n	CCONJ
ejpam-1506	181	15	)	)	PUNCT
ejpam-1506	181	16	.	.	PUNCT
ejpam-1506	182	1	without	without	ADP
ejpam-1506	182	2	loss	loss	NOUN
ejpam-1506	182	3	of	of	ADP
ejpam-1506	182	4	generality	generality	NOUN
ejpam-1506	182	5	,	,	PUNCT
ejpam-1506	182	6	assume	assume	VERB
ejpam-1506	182	7	that	that	SCONJ
ejpam-1506	182	8	v1v2	v1v2	VERB
ejpam-1506	182	9	6∈	6∈	PROPN
ejpam-1506	182	10	e(g	e(g	PROPN
ejpam-1506	182	11	)	)	PUNCT
ejpam-1506	182	12	.	.	PUNCT
ejpam-1506	183	1	following	follow	VERB
ejpam-1506	183	2	the	the	DET
ejpam-1506	183	3	construction	construction	NOUN
ejpam-1506	183	4	of	of	ADP
ejpam-1506	183	5	g⋆	g⋆	NOUN
ejpam-1506	183	6	from	from	ADP
ejpam-1506	183	7	g	g	NOUN
ejpam-1506	183	8	,	,	PUNCT
ejpam-1506	183	9	let	let	VERB
ejpam-1506	183	10	each	each	DET
ejpam-1506	183	11	vertex	vertex	NOUN
ejpam-1506	183	12	vi	vi	NOUN
ejpam-1506	183	13	be	be	AUX
ejpam-1506	183	14	replaced	replace	VERB
ejpam-1506	183	15	by	by	ADP
ejpam-1506	183	16	k(vi	k(vi	PROPN
ejpam-1506	183	17	)	)	PUNCT
ejpam-1506	183	18	∼=	∼=	PROPN
ejpam-1506	183	19	kdi	kdi	NOUN
ejpam-1506	183	20	;	;	PUNCT
ejpam-1506	183	21	here	here	ADV
ejpam-1506	183	22	,	,	PUNCT
ejpam-1506	183	23	we	we	PRON
ejpam-1506	183	24	denote	denote	VERB
ejpam-1506	183	25	by	by	ADP
ejpam-1506	183	26	ui	ui	PROPN
ejpam-1506	183	27	the	the	DET
ejpam-1506	183	28	vertex	vertex	NOUN
ejpam-1506	183	29	set	set	VERB
ejpam-1506	183	30	v	v	NOUN
ejpam-1506	183	31	(	(	PUNCT
ejpam-1506	183	32	k(vi	k(vi	PROPN
ejpam-1506	183	33	)	)	PUNCT
ejpam-1506	183	34	)	)	PUNCT
ejpam-1506	184	1	=	=	PUNCT
ejpam-1506	184	2	{	{	PUNCT
ejpam-1506	184	3	ui,1,ui,2	ui,1,ui,2	PROPN
ejpam-1506	184	4	,	,	PUNCT
ejpam-1506	184	5	.	.	PUNCT
ejpam-1506	184	6	.	.	PUNCT
ejpam-1506	185	1	.	.	PUNCT
ejpam-1506	186	1	,	,	PUNCT
ejpam-1506	186	2	ui	ui	PROPN
ejpam-1506	186	3	,	,	PUNCT
ejpam-1506	186	4	di	di	NOUN
ejpam-1506	186	5	}	}	PUNCT
ejpam-1506	186	6	⊆	⊆	NUM
ejpam-1506	186	7	v	v	NOUN
ejpam-1506	186	8	(	(	PUNCT
ejpam-1506	186	9	g⋆	g⋆	NOUN
ejpam-1506	186	10	)	)	PUNCT
ejpam-1506	186	11	for	for	ADP
ejpam-1506	186	12	each	each	DET
ejpam-1506	186	13	i	i	PRON
ejpam-1506	186	14	(	(	PUNCT
ejpam-1506	186	15	1	1	NUM
ejpam-1506	186	16	≤	≤	NUM
ejpam-1506	186	17	i	i	NOUN
ejpam-1506	186	18	≤	≤	NOUN
ejpam-1506	186	19	n	n	CCONJ
ejpam-1506	186	20	)	)	PUNCT
ejpam-1506	186	21	.	.	PUNCT
ejpam-1506	187	1	let	let	VERB
ejpam-1506	187	2	s	s	PRON
ejpam-1506	187	3	=	=	X
ejpam-1506	187	4	{	{	PUNCT
ejpam-1506	187	5	u2,a2	u2,a2	PROPN
ejpam-1506	187	6	,	,	PUNCT
ejpam-1506	187	7	u3,a3	u3,a3	PROPN
ejpam-1506	187	8	,	,	PUNCT
ejpam-1506	187	9	.	.	PUNCT
ejpam-1506	187	10	.	.	PUNCT
ejpam-1506	187	11	.	.	PUNCT
ejpam-1506	188	1	,	,	PUNCT
ejpam-1506	188	2	un−1,an−1	un−1,an−1	ADJ
ejpam-1506	188	3	}	}	PUNCT
ejpam-1506	188	4	with	with	ADP
ejpam-1506	188	5	|s|=	|s|=	DET
ejpam-1506	188	6	n−2	n−2	PROPN
ejpam-1506	189	1	such	such	ADJ
ejpam-1506	189	2	that	that	PRON
ejpam-1506	189	3	|s∩ui|	|s∩ui|	PROPN
ejpam-1506	189	4	=	=	SYM
ejpam-1506	189	5	1	1	NUM
ejpam-1506	189	6	for	for	ADP
ejpam-1506	189	7	each	each	DET
ejpam-1506	189	8	i	i	PRON
ejpam-1506	189	9	(	(	PUNCT
ejpam-1506	189	10	2≤	2≤	NUM
ejpam-1506	189	11	i	i	PRON
ejpam-1506	189	12	≤	≤	PUNCT
ejpam-1506	189	13	n−1	n−1	PROPN
ejpam-1506	189	14	)	)	PUNCT
ejpam-1506	189	15	and	and	CCONJ
ejpam-1506	189	16	that	that	SCONJ
ejpam-1506	189	17	no	no	DET
ejpam-1506	189	18	two	two	NUM
ejpam-1506	189	19	vertices	vertex	NOUN
ejpam-1506	189	20	in	in	ADP
ejpam-1506	189	21	s	s	NOUN
ejpam-1506	189	22	are	be	AUX
ejpam-1506	189	23	adjacent	adjacent	ADJ
ejpam-1506	189	24	in	in	ADP
ejpam-1506	189	25	g⋆.	g⋆.	NOUN
ejpam-1506	189	26	we	we	PRON
ejpam-1506	189	27	will	will	AUX
ejpam-1506	189	28	show	show	VERB
ejpam-1506	189	29	that	that	SCONJ
ejpam-1506	189	30	s	s	VERB
ejpam-1506	189	31	is	be	AUX
ejpam-1506	189	32	a	a	DET
ejpam-1506	189	33	resolving	resolving	NOUN
ejpam-1506	189	34	set	set	VERB
ejpam-1506	189	35	for	for	ADP
ejpam-1506	189	36	g⋆.	g⋆.	NOUN
ejpam-1506	189	37	it	it	PRON
ejpam-1506	189	38	suffices	suffice	VERB
ejpam-1506	189	39	to	to	PART
ejpam-1506	189	40	show	show	VERB
ejpam-1506	189	41	that	that	SCONJ
ejpam-1506	189	42	,	,	PUNCT
ejpam-1506	189	43	for	for	ADP
ejpam-1506	189	44	any	any	DET
ejpam-1506	189	45	two	two	NUM
ejpam-1506	189	46	vertices	vertex	NOUN
ejpam-1506	189	47	ux	ux	ADV
ejpam-1506	189	48	,	,	PUNCT
ejpam-1506	189	49	uy	uy	PROPN
ejpam-1506	189	50	∈	∈	PROPN
ejpam-1506	189	51	v	v	ADP
ejpam-1506	189	52	(	(	PUNCT
ejpam-1506	189	53	g⋆)−s	g⋆)−s	ADJ
ejpam-1506	189	54	,	,	PUNCT
ejpam-1506	189	55	dg⋆(ux	dg⋆(ux	NOUN
ejpam-1506	189	56	,	,	PUNCT
ejpam-1506	189	57	ui	ui	NOUN
ejpam-1506	189	58	,	,	PUNCT
ejpam-1506	189	59	ai	ai	VERB
ejpam-1506	189	60	)	)	PUNCT
ejpam-1506	189	61	6=	6=	NUM
ejpam-1506	189	62	dg⋆(uy	dg⋆(uy	PROPN
ejpam-1506	189	63	,	,	PUNCT
ejpam-1506	189	64	ui	ui	PROPN
ejpam-1506	189	65	,	,	PUNCT
ejpam-1506	189	66	ai	ai	VERB
ejpam-1506	189	67	)	)	PUNCT
ejpam-1506	189	68	for	for	ADP
ejpam-1506	189	69	some	some	DET
ejpam-1506	189	70	ui	ui	NOUN
ejpam-1506	189	71	,	,	PUNCT
ejpam-1506	189	72	ai	ai	VERB
ejpam-1506	189	73	∈	∈	PROPN
ejpam-1506	189	74	s.	s.	PROPN
ejpam-1506	189	75	as	as	ADP
ejpam-1506	189	76	in	in	ADP
ejpam-1506	189	77	the	the	DET
ejpam-1506	189	78	proof	proof	NOUN
ejpam-1506	189	79	of	of	ADP
ejpam-1506	189	80	theorem	theorem	ADJ
ejpam-1506	189	81	6	6	NUM
ejpam-1506	189	82	,	,	PUNCT
ejpam-1506	189	83	s	s	PART
ejpam-1506	189	84	forms	form	NOUN
ejpam-1506	189	85	a	a	DET
ejpam-1506	189	86	resolving	resolving	NOUN
ejpam-1506	189	87	set	set	VERB
ejpam-1506	189	88	for	for	ADP
ejpam-1506	189	89	〈	〈	PROPN
ejpam-1506	189	90	∪n−1	∪n−1	PROPN
ejpam-1506	189	91	i=1	i=1	PROPN
ejpam-1506	189	92	mathcalui	mathcalui	PRON
ejpam-1506	189	93	〉	〉	NOUN
ejpam-1506	189	94	⊆	⊆	NUM
ejpam-1506	189	95	g⋆	g⋆	NOUN
ejpam-1506	189	96	and	and	CCONJ
ejpam-1506	189	97	〈	〈	NOUN
ejpam-1506	189	98	∪n	∪n	PROPN
ejpam-1506	189	99	i=2	i=2	PROPN
ejpam-1506	189	100	ui	ui	NOUN
ejpam-1506	189	101	〉	〉	NOUN
ejpam-1506	189	102	⊆	⊆	NUM
ejpam-1506	189	103	g⋆.	g⋆.	NOUN
ejpam-1506	189	104	so	so	ADV
ejpam-1506	189	105	,	,	PUNCT
ejpam-1506	189	106	it	it	PRON
ejpam-1506	189	107	remains	remain	VERB
ejpam-1506	189	108	to	to	PART
ejpam-1506	189	109	show	show	VERB
ejpam-1506	189	110	that	that	SCONJ
ejpam-1506	189	111	,	,	PUNCT
ejpam-1506	189	112	for	for	ADP
ejpam-1506	189	113	ux	ux	PROPN
ejpam-1506	189	114	∈	∈	PROPN
ejpam-1506	189	115	u1	u1	NOUN
ejpam-1506	189	116	and	and	CCONJ
ejpam-1506	189	117	uy	uy	PROPN
ejpam-1506	189	118	∈	∈	PROPN
ejpam-1506	189	119	un	un	PROPN
ejpam-1506	189	120	,	,	PUNCT
ejpam-1506	189	121	dg⋆(ux	dg⋆(ux	NOUN
ejpam-1506	189	122	,	,	PUNCT
ejpam-1506	189	123	ui	ui	NOUN
ejpam-1506	189	124	,	,	PUNCT
ejpam-1506	189	125	ai	ai	VERB
ejpam-1506	189	126	)	)	PUNCT
ejpam-1506	189	127	6=	6=	NUM
ejpam-1506	190	1	dg⋆(uy	dg⋆(uy	PROPN
ejpam-1506	190	2	,	,	PUNCT
ejpam-1506	190	3	ui	ui	PROPN
ejpam-1506	190	4	,	,	PUNCT
ejpam-1506	190	5	ai	ai	VERB
ejpam-1506	190	6	)	)	PUNCT
ejpam-1506	190	7	for	for	ADP
ejpam-1506	190	8	some	some	DET
ejpam-1506	190	9	ui	ui	NOUN
ejpam-1506	190	10	,	,	PUNCT
ejpam-1506	190	11	ai	ai	VERB
ejpam-1506	190	12	∈	∈	PROPN
ejpam-1506	190	13	s.	s.	PROPN
ejpam-1506	190	14	if	if	SCONJ
ejpam-1506	190	15	uxui	uxui	NOUN
ejpam-1506	190	16	,	,	PUNCT
ejpam-1506	190	17	ai	ai	VERB
ejpam-1506	190	18	∈	∈	NOUN
ejpam-1506	190	19	e(g⋆	e(g⋆	NOUN
ejpam-1506	190	20	)	)	PUNCT
ejpam-1506	190	21	or	or	CCONJ
ejpam-1506	190	22	uy	uy	PROPN
ejpam-1506	190	23	ui	ui	PROPN
ejpam-1506	190	24	,	,	PUNCT
ejpam-1506	190	25	ai	ai	VERB
ejpam-1506	190	26	∈	∈	NOUN
ejpam-1506	190	27	e(g⋆	e(g⋆	NOUN
ejpam-1506	190	28	)	)	PUNCT
ejpam-1506	190	29	for	for	ADP
ejpam-1506	190	30	some	some	DET
ejpam-1506	190	31	i	i	NOUN
ejpam-1506	190	32	(	(	PUNCT
ejpam-1506	190	33	2≤	2≤	NUM
ejpam-1506	190	34	i	i	PRON
ejpam-1506	190	35	≤	≤	ADV
ejpam-1506	190	36	n−	n−	NOUN
ejpam-1506	190	37	1	1	NUM
ejpam-1506	190	38	)	)	PUNCT
ejpam-1506	190	39	,	,	PUNCT
ejpam-1506	190	40	say	say	VERB
ejpam-1506	190	41	the	the	DET
ejpam-1506	190	42	former	former	ADJ
ejpam-1506	190	43	,	,	PUNCT
ejpam-1506	190	44	then	then	ADV
ejpam-1506	190	45	dg⋆(ux	dg⋆(ux	NOUN
ejpam-1506	190	46	,	,	PUNCT
ejpam-1506	190	47	ui	ui	NOUN
ejpam-1506	190	48	,	,	PUNCT
ejpam-1506	190	49	ai	ai	VERB
ejpam-1506	190	50	)	)	PUNCT
ejpam-1506	190	51	=	=	PUNCT
ejpam-1506	190	52	1	1	NUM
ejpam-1506	190	53	<	<	X
ejpam-1506	190	54	dg⋆(uy	dg⋆(uy	PROPN
ejpam-1506	190	55	,	,	PUNCT
ejpam-1506	190	56	ui	ui	PROPN
ejpam-1506	190	57	,	,	PUNCT
ejpam-1506	190	58	ai	ai	INTJ
ejpam-1506	190	59	)	)	PUNCT
ejpam-1506	190	60	.	.	PUNCT
ejpam-1506	191	1	if	if	SCONJ
ejpam-1506	191	2	uxui	uxui	NOUN
ejpam-1506	191	3	,	,	PUNCT
ejpam-1506	191	4	ai	ai	VERB
ejpam-1506	191	5	6∈	6∈	NOUN
ejpam-1506	191	6	e(g⋆	e(g⋆	NOUN
ejpam-1506	191	7	)	)	PUNCT
ejpam-1506	191	8	and	and	CCONJ
ejpam-1506	191	9	uyui	uyui	PROPN
ejpam-1506	191	10	,	,	PUNCT
ejpam-1506	191	11	ai	ai	VERB
ejpam-1506	191	12	6∈	6∈	NOUN
ejpam-1506	191	13	e(g⋆	e(g⋆	NOUN
ejpam-1506	191	14	)	)	PUNCT
ejpam-1506	191	15	for	for	ADP
ejpam-1506	191	16	each	each	DET
ejpam-1506	191	17	i	i	PRON
ejpam-1506	191	18	(	(	PUNCT
ejpam-1506	191	19	2≤	2≤	NUM
ejpam-1506	191	20	i	i	PRON
ejpam-1506	191	21	≤	≤	ADV
ejpam-1506	191	22	n−	n−	NOUN
ejpam-1506	191	23	1	1	NUM
ejpam-1506	191	24	)	)	PUNCT
ejpam-1506	191	25	,	,	PUNCT
ejpam-1506	191	26	then	then	ADV
ejpam-1506	191	27	codes(ux	codes(ux	VERB
ejpam-1506	191	28	)	)	PUNCT
ejpam-1506	191	29	=	=	SYM
ejpam-1506	191	30	codes(uy	codes(uy	NOUN
ejpam-1506	191	31	)	)	PUNCT
ejpam-1506	191	32	implies	imply	VERB
ejpam-1506	191	33	that	that	SCONJ
ejpam-1506	191	34	v1	v1	NOUN
ejpam-1506	191	35	and	and	CCONJ
ejpam-1506	191	36	vn	vn	PROPN
ejpam-1506	191	37	are	be	AUX
ejpam-1506	191	38	non	non	ADJ
ejpam-1506	191	39	-	-	ADJ
ejpam-1506	191	40	adjacent	adjacent	ADJ
ejpam-1506	191	41	twins	twin	NOUN
ejpam-1506	191	42	in	in	ADP
ejpam-1506	191	43	g.	g.	PROPN
ejpam-1506	191	44	if	if	SCONJ
ejpam-1506	191	45	there	there	PRON
ejpam-1506	191	46	exist	exist	VERB
ejpam-1506	191	47	a	a	DET
ejpam-1506	191	48	pair	pair	NOUN
ejpam-1506	191	49	of	of	ADP
ejpam-1506	191	50	vertices	vertex	NOUN
ejpam-1506	191	51	in	in	ADP
ejpam-1506	191	52	g	g	PROPN
ejpam-1506	191	53	that	that	PRON
ejpam-1506	191	54	are	be	AUX
ejpam-1506	191	55	not	not	PART
ejpam-1506	191	56	non	non	ADJ
ejpam-1506	191	57	-	-	ADJ
ejpam-1506	191	58	adjacent	adjacent	ADJ
ejpam-1506	191	59	twin	twin	PROPN
ejpam-1506	191	60	d.	d.	PROPN
ejpam-1506	191	61	klein	klein	PROPN
ejpam-1506	191	62	,	,	PUNCT
ejpam-1506	191	63	e.	e.	PROPN
ejpam-1506	191	64	yi	yi	PROPN
ejpam-1506	191	65	/	/	SYM
ejpam-1506	191	66	eur	eur	PROPN
ejpam-1506	191	67	.	.	PUNCT
ejpam-1506	192	1	j.	j.	PROPN
ejpam-1506	192	2	pure	pure	PROPN
ejpam-1506	192	3	appl	appl	PROPN
ejpam-1506	192	4	.	.	PROPN
ejpam-1506	192	5	math	math	PROPN
ejpam-1506	192	6	,	,	PUNCT
ejpam-1506	192	7	5	5	NUM
ejpam-1506	192	8	(	(	PUNCT
ejpam-1506	192	9	2012	2012	NUM
ejpam-1506	192	10	)	)	PUNCT
ejpam-1506	192	11	,	,	PUNCT
ejpam-1506	192	12	302	302	NUM
ejpam-1506	192	13	-	-	SYM
ejpam-1506	192	14	316	316	NUM
ejpam-1506	192	15	308	308	NUM
ejpam-1506	192	16	in	in	ADP
ejpam-1506	192	17	g	g	PROPN
ejpam-1506	192	18	,	,	PUNCT
ejpam-1506	192	19	we	we	PRON
ejpam-1506	192	20	are	be	AUX
ejpam-1506	192	21	done	do	VERB
ejpam-1506	192	22	.	.	PUNCT
ejpam-1506	193	1	otherwise	otherwise	ADV
ejpam-1506	193	2	,	,	PUNCT
ejpam-1506	193	3	every	every	DET
ejpam-1506	193	4	pair	pair	NOUN
ejpam-1506	193	5	of	of	ADP
ejpam-1506	193	6	vertices	vertex	NOUN
ejpam-1506	193	7	must	must	AUX
ejpam-1506	193	8	be	be	AUX
ejpam-1506	193	9	non	non	ADJ
ejpam-1506	193	10	-	-	ADJ
ejpam-1506	193	11	adjacent	adjacent	ADJ
ejpam-1506	193	12	twins	twin	NOUN
ejpam-1506	193	13	in	in	ADP
ejpam-1506	193	14	g	g	PROPN
ejpam-1506	193	15	,	,	PUNCT
ejpam-1506	193	16	but	but	CCONJ
ejpam-1506	193	17	this	this	PRON
ejpam-1506	193	18	is	be	AUX
ejpam-1506	193	19	impossible	impossible	ADJ
ejpam-1506	193	20	:	:	PUNCT
ejpam-1506	193	21	if	if	SCONJ
ejpam-1506	193	22	w1	w1	NOUN
ejpam-1506	193	23	and	and	CCONJ
ejpam-1506	193	24	w2	w2	NOUN
ejpam-1506	193	25	are	be	AUX
ejpam-1506	193	26	non	non	ADJ
ejpam-1506	193	27	-	-	ADJ
ejpam-1506	193	28	adjacent	adjacent	ADJ
ejpam-1506	193	29	twins	twin	NOUN
ejpam-1506	193	30	in	in	ADP
ejpam-1506	193	31	g	g	PROPN
ejpam-1506	193	32	satisfying	satisfy	VERB
ejpam-1506	193	33	wk	wk	ADP
ejpam-1506	193	34	∈	∈	PROPN
ejpam-1506	193	35	ng(w1)∩ng(w2	ng(w1)∩ng(w2	NUM
ejpam-1506	193	36	)	)	PUNCT
ejpam-1506	193	37	,	,	PUNCT
ejpam-1506	193	38	then	then	ADV
ejpam-1506	193	39	w1wk	w1wk	PUNCT
ejpam-1506	193	40	∈	∈	PROPN
ejpam-1506	193	41	e(g	e(g	PROPN
ejpam-1506	193	42	)	)	PUNCT
ejpam-1506	193	43	.	.	PUNCT
ejpam-1506	194	1	so	so	ADV
ejpam-1506	194	2	,	,	PUNCT
ejpam-1506	194	3	s	s	VERB
ejpam-1506	194	4	is	be	AUX
ejpam-1506	194	5	a	a	DET
ejpam-1506	194	6	resolving	resolving	NOUN
ejpam-1506	194	7	set	set	VERB
ejpam-1506	194	8	for	for	ADP
ejpam-1506	194	9	g⋆	g⋆	NOUN
ejpam-1506	194	10	with	with	ADP
ejpam-1506	194	11	|s|	|s|	NOUN
ejpam-1506	194	12	=	=	SYM
ejpam-1506	194	13	n−	n−	NOUN
ejpam-1506	194	14	2	2	NUM
ejpam-1506	194	15	,	,	PUNCT
ejpam-1506	194	16	and	and	CCONJ
ejpam-1506	194	17	hence	hence	ADV
ejpam-1506	194	18	dim(g⋆	dim(g⋆	NOUN
ejpam-1506	194	19	)	)	PUNCT
ejpam-1506	194	20	≤	≤	NUM
ejpam-1506	194	21	n−	n−	NOUN
ejpam-1506	194	22	2	2	NUM
ejpam-1506	194	23	.	.	NOUN
ejpam-1506	194	24	4	4	NUM
ejpam-1506	194	25	.	.	X
ejpam-1506	194	26	metric	metric	ADJ
ejpam-1506	194	27	dimension	dimension	NOUN
ejpam-1506	194	28	of	of	ADP
ejpam-1506	194	29	para	para	NOUN
ejpam-1506	194	30	-	-	PUNCT
ejpam-1506	194	31	line	line	NOUN
ejpam-1506	194	32	graphs	graph	NOUN
ejpam-1506	194	33	for	for	ADP
ejpam-1506	194	34	trees	tree	NOUN
ejpam-1506	194	35	,	,	PUNCT
ejpam-1506	194	36	complete	complete	ADJ
ejpam-1506	194	37	graphs	graph	NOUN
ejpam-1506	194	38	,	,	PUNCT
ejpam-1506	194	39	complete	complete	ADJ
ejpam-1506	194	40	bi	bi	ADJ
ejpam-1506	194	41	-	-	ADJ
ejpam-1506	194	42	partite	partite	ADJ
ejpam-1506	194	43	graphs	graph	NOUN
ejpam-1506	194	44	,	,	PUNCT
ejpam-1506	194	45	wheel	wheel	NOUN
ejpam-1506	194	46	graphs	graph	NOUN
ejpam-1506	194	47	,	,	PUNCT
ejpam-1506	194	48	and	and	CCONJ
ejpam-1506	194	49	bouquet	bouquet	NOUN
ejpam-1506	194	50	of	of	ADP
ejpam-1506	194	51	circles	circle	NOUN
ejpam-1506	194	52	in	in	ADP
ejpam-1506	194	53	this	this	DET
ejpam-1506	194	54	section	section	NOUN
ejpam-1506	194	55	,	,	PUNCT
ejpam-1506	194	56	we	we	PRON
ejpam-1506	194	57	determine	determine	VERB
ejpam-1506	194	58	the	the	DET
ejpam-1506	194	59	metric	metric	ADJ
ejpam-1506	194	60	dimension	dimension	NOUN
ejpam-1506	194	61	of	of	ADP
ejpam-1506	194	62	some	some	DET
ejpam-1506	194	63	classes	class	NOUN
ejpam-1506	194	64	of	of	ADP
ejpam-1506	194	65	para	para	NOUN
ejpam-1506	194	66	-	-	PUNCT
ejpam-1506	194	67	line	line	NOUN
ejpam-1506	194	68	graphs	graph	NOUN
ejpam-1506	194	69	.	.	PUNCT
ejpam-1506	195	1	first	first	ADV
ejpam-1506	195	2	,	,	PUNCT
ejpam-1506	195	3	we	we	PRON
ejpam-1506	195	4	determine	determine	VERB
ejpam-1506	195	5	the	the	DET
ejpam-1506	195	6	metric	metric	ADJ
ejpam-1506	195	7	dimension	dimension	NOUN
ejpam-1506	195	8	of	of	ADP
ejpam-1506	195	9	t	t	PROPN
ejpam-1506	195	10	⋆	⋆	VERB
ejpam-1506	195	11	for	for	ADP
ejpam-1506	195	12	a	a	DET
ejpam-1506	195	13	tree	tree	NOUN
ejpam-1506	195	14	t	t	NOUN
ejpam-1506	195	15	that	that	PRON
ejpam-1506	195	16	is	be	AUX
ejpam-1506	195	17	not	not	PART
ejpam-1506	195	18	a	a	DET
ejpam-1506	195	19	path	path	NOUN
ejpam-1506	195	20	.	.	PUNCT
ejpam-1506	196	1	the	the	DET
ejpam-1506	196	2	following	follow	VERB
ejpam-1506	196	3	definitions	definition	NOUN
ejpam-1506	196	4	are	be	AUX
ejpam-1506	196	5	stated	state	VERB
ejpam-1506	196	6	in	in	ADP
ejpam-1506	196	7	[	[	X
ejpam-1506	196	8	5	5	NUM
ejpam-1506	196	9	]	]	PUNCT
ejpam-1506	196	10	.	.	PUNCT
ejpam-1506	197	1	fix	fix	VERB
ejpam-1506	197	2	a	a	DET
ejpam-1506	197	3	graph	graph	NOUN
ejpam-1506	197	4	g.	g.	NOUN
ejpam-1506	197	5	a	a	DET
ejpam-1506	197	6	vertex	vertex	NOUN
ejpam-1506	197	7	of	of	ADP
ejpam-1506	197	8	degree	degree	NOUN
ejpam-1506	197	9	at	at	ADV
ejpam-1506	197	10	least	least	ADJ
ejpam-1506	197	11	three	three	NUM
ejpam-1506	197	12	is	be	AUX
ejpam-1506	197	13	called	call	VERB
ejpam-1506	197	14	a	a	DET
ejpam-1506	197	15	major	major	ADJ
ejpam-1506	197	16	vertex	vertex	NOUN
ejpam-1506	197	17	.	.	PUNCT
ejpam-1506	198	1	an	an	DET
ejpam-1506	198	2	end	end	NOUN
ejpam-1506	198	3	-	-	PUNCT
ejpam-1506	198	4	vertex	vertex	NOUN
ejpam-1506	198	5	u	u	NOUN
ejpam-1506	198	6	is	be	AUX
ejpam-1506	198	7	called	call	VERB
ejpam-1506	198	8	a	a	DET
ejpam-1506	198	9	terminal	terminal	ADJ
ejpam-1506	198	10	vertex	vertex	NOUN
ejpam-1506	198	11	of	of	ADP
ejpam-1506	198	12	a	a	DET
ejpam-1506	198	13	major	major	ADJ
ejpam-1506	198	14	vertex	vertex	NOUN
ejpam-1506	198	15	v	v	NOUN
ejpam-1506	198	16	if	if	SCONJ
ejpam-1506	198	17	d(u	d(u	PROPN
ejpam-1506	198	18	,	,	PUNCT
ejpam-1506	198	19	v	v	NOUN
ejpam-1506	198	20	)	)	PUNCT
ejpam-1506	198	21	<	<	X
ejpam-1506	198	22	d(u	d(u	PROPN
ejpam-1506	198	23	,	,	PUNCT
ejpam-1506	198	24	w	w	NOUN
ejpam-1506	198	25	)	)	PUNCT
ejpam-1506	198	26	for	for	ADP
ejpam-1506	198	27	every	every	DET
ejpam-1506	198	28	other	other	ADJ
ejpam-1506	198	29	major	major	ADJ
ejpam-1506	198	30	vertex	vertex	NOUN
ejpam-1506	198	31	w.	w.	NOUN
ejpam-1506	198	32	the	the	DET
ejpam-1506	198	33	terminal	terminal	ADJ
ejpam-1506	198	34	degree	degree	NOUN
ejpam-1506	198	35	of	of	ADP
ejpam-1506	198	36	a	a	DET
ejpam-1506	198	37	major	major	ADJ
ejpam-1506	198	38	vertex	vertex	NOUN
ejpam-1506	198	39	v	v	NOUN
ejpam-1506	198	40	is	be	AUX
ejpam-1506	198	41	the	the	DET
ejpam-1506	198	42	number	number	NOUN
ejpam-1506	198	43	of	of	ADP
ejpam-1506	198	44	terminal	terminal	ADJ
ejpam-1506	198	45	vertices	vertex	NOUN
ejpam-1506	198	46	of	of	ADP
ejpam-1506	198	47	v.	v.	ADP
ejpam-1506	198	48	a	a	DET
ejpam-1506	198	49	major	major	ADJ
ejpam-1506	198	50	vertex	vertex	NOUN
ejpam-1506	198	51	v	v	NOUN
ejpam-1506	198	52	is	be	AUX
ejpam-1506	198	53	an	an	DET
ejpam-1506	198	54	exterior	exterior	ADJ
ejpam-1506	198	55	major	major	ADJ
ejpam-1506	198	56	vertex	vertex	NOUN
ejpam-1506	198	57	if	if	SCONJ
ejpam-1506	198	58	it	it	PRON
ejpam-1506	198	59	has	have	VERB
ejpam-1506	198	60	positive	positive	ADJ
ejpam-1506	198	61	terminal	terminal	ADJ
ejpam-1506	198	62	degree	degree	NOUN
ejpam-1506	198	63	.	.	PUNCT
ejpam-1506	199	1	let	let	VERB
ejpam-1506	199	2	σ(g	σ(g	NOUN
ejpam-1506	199	3	)	)	PUNCT
ejpam-1506	199	4	denote	denote	VERB
ejpam-1506	199	5	the	the	DET
ejpam-1506	199	6	sum	sum	NOUN
ejpam-1506	199	7	of	of	ADP
ejpam-1506	199	8	terminal	terminal	ADJ
ejpam-1506	199	9	degrees	degree	NOUN
ejpam-1506	199	10	of	of	ADP
ejpam-1506	199	11	all	all	DET
ejpam-1506	199	12	major	major	ADJ
ejpam-1506	199	13	vertices	vertex	NOUN
ejpam-1506	199	14	of	of	ADP
ejpam-1506	199	15	g	g	NOUN
ejpam-1506	199	16	,	,	PUNCT
ejpam-1506	199	17	and	and	CCONJ
ejpam-1506	199	18	let	let	VERB
ejpam-1506	199	19	ex(g	ex(g	NOUN
ejpam-1506	199	20	)	)	PUNCT
ejpam-1506	199	21	denote	denote	VERB
ejpam-1506	199	22	the	the	DET
ejpam-1506	199	23	number	number	NOUN
ejpam-1506	199	24	of	of	ADP
ejpam-1506	199	25	exterior	exterior	ADJ
ejpam-1506	199	26	major	major	ADJ
ejpam-1506	199	27	vertices	vertex	NOUN
ejpam-1506	199	28	of	of	ADP
ejpam-1506	199	29	g.	g.	PROPN
ejpam-1506	199	30	theorem	theorem	VERB
ejpam-1506	199	31	9	9	NUM
ejpam-1506	199	32	.	.	PUNCT
ejpam-1506	200	1	[	[	X
ejpam-1506	200	2	5	5	NUM
ejpam-1506	200	3	,	,	PUNCT
ejpam-1506	200	4	17	17	NUM
ejpam-1506	200	5	,	,	PUNCT
ejpam-1506	200	6	19	19	NUM
ejpam-1506	200	7	]	]	PUNCT
ejpam-1506	200	8	if	if	SCONJ
ejpam-1506	200	9	t	t	PROPN
ejpam-1506	200	10	is	be	AUX
ejpam-1506	200	11	a	a	DET
ejpam-1506	200	12	tree	tree	NOUN
ejpam-1506	200	13	that	that	PRON
ejpam-1506	200	14	is	be	AUX
ejpam-1506	200	15	not	not	PART
ejpam-1506	200	16	a	a	DET
ejpam-1506	200	17	path	path	NOUN
ejpam-1506	200	18	,	,	PUNCT
ejpam-1506	200	19	then	then	ADV
ejpam-1506	200	20	dim(t	dim(t	NOUN
ejpam-1506	200	21	)	)	PUNCT
ejpam-1506	201	1	=	=	SYM
ejpam-1506	201	2	σ(t	σ(t	PROPN
ejpam-1506	201	3	)	)	PUNCT
ejpam-1506	201	4	−	−	PROPN
ejpam-1506	201	5	ex(t	ex(t	NUM
ejpam-1506	201	6	)	)	PUNCT
ejpam-1506	201	7	.	.	PUNCT
ejpam-1506	202	1	theorem	theorem	ADJ
ejpam-1506	202	2	10	10	NUM
ejpam-1506	202	3	.	.	PUNCT
ejpam-1506	203	1	[	[	X
ejpam-1506	203	2	10	10	NUM
ejpam-1506	203	3	]	]	X
ejpam-1506	203	4	if	if	SCONJ
ejpam-1506	203	5	t	t	PROPN
ejpam-1506	203	6	is	be	AUX
ejpam-1506	203	7	a	a	DET
ejpam-1506	203	8	tree	tree	NOUN
ejpam-1506	203	9	that	that	PRON
ejpam-1506	203	10	is	be	AUX
ejpam-1506	203	11	not	not	PART
ejpam-1506	203	12	a	a	DET
ejpam-1506	203	13	path	path	NOUN
ejpam-1506	203	14	,	,	PUNCT
ejpam-1506	203	15	then	then	ADV
ejpam-1506	203	16	dim(l(t	dim(l(t	VERB
ejpam-1506	203	17	)	)	PUNCT
ejpam-1506	203	18	)	)	PUNCT
ejpam-1506	204	1	=	=	SYM
ejpam-1506	204	2	σ(t	σ(t	PROPN
ejpam-1506	204	3	)	)	PUNCT
ejpam-1506	204	4	−	−	PROPN
ejpam-1506	204	5	ex(t	ex(t	NUM
ejpam-1506	204	6	)	)	PUNCT
ejpam-1506	204	7	.	.	PUNCT
ejpam-1506	205	1	as	as	ADP
ejpam-1506	205	2	an	an	DET
ejpam-1506	205	3	immediate	immediate	ADJ
ejpam-1506	205	4	consequence	consequence	NOUN
ejpam-1506	205	5	of	of	ADP
ejpam-1506	205	6	theorem	theorem	NOUN
ejpam-1506	205	7	10	10	NUM
ejpam-1506	205	8	,	,	PUNCT
ejpam-1506	205	9	we	we	PRON
ejpam-1506	205	10	have	have	VERB
ejpam-1506	205	11	the	the	DET
ejpam-1506	205	12	following	follow	VERB
ejpam-1506	205	13	corollary	corollary	ADJ
ejpam-1506	205	14	1	1	NUM
ejpam-1506	205	15	.	.	PUNCT
ejpam-1506	206	1	if	if	SCONJ
ejpam-1506	206	2	t	t	PROPN
ejpam-1506	206	3	is	be	AUX
ejpam-1506	206	4	a	a	DET
ejpam-1506	206	5	tree	tree	NOUN
ejpam-1506	206	6	that	that	PRON
ejpam-1506	206	7	is	be	AUX
ejpam-1506	206	8	not	not	PART
ejpam-1506	206	9	a	a	DET
ejpam-1506	206	10	path	path	NOUN
ejpam-1506	206	11	,	,	PUNCT
ejpam-1506	206	12	then	then	ADV
ejpam-1506	206	13	dim(t	dim(t	NOUN
ejpam-1506	206	14	⋆	⋆	VERB
ejpam-1506	206	15	)	)	PUNCT
ejpam-1506	206	16	=	=	SYM
ejpam-1506	206	17	σ(t	σ(t	PROPN
ejpam-1506	206	18	)	)	PUNCT
ejpam-1506	206	19	−	−	PROPN
ejpam-1506	206	20	ex(t	ex(t	NUM
ejpam-1506	206	21	)	)	PUNCT
ejpam-1506	206	22	.	.	PUNCT
ejpam-1506	207	1	second	second	ADJ
ejpam-1506	207	2	,	,	PUNCT
ejpam-1506	207	3	we	we	PRON
ejpam-1506	207	4	determine	determine	VERB
ejpam-1506	207	5	the	the	DET
ejpam-1506	207	6	metric	metric	ADJ
ejpam-1506	207	7	dimension	dimension	NOUN
ejpam-1506	207	8	of	of	ADP
ejpam-1506	207	9	k⋆n	k⋆n	NOUN
ejpam-1506	207	10	for	for	ADP
ejpam-1506	207	11	the	the	DET
ejpam-1506	207	12	complete	complete	ADJ
ejpam-1506	207	13	graph	graph	NOUN
ejpam-1506	207	14	kn	kn	NOUN
ejpam-1506	207	15	of	of	ADP
ejpam-1506	207	16	order	order	NOUN
ejpam-1506	207	17	n≥	n≥	NOUN
ejpam-1506	207	18	2	2	X
ejpam-1506	207	19	.	.	X
ejpam-1506	207	20	we	we	PRON
ejpam-1506	207	21	first	first	ADV
ejpam-1506	207	22	recall	recall	VERB
ejpam-1506	207	23	the	the	DET
ejpam-1506	207	24	metric	metric	ADJ
ejpam-1506	207	25	dimension	dimension	NOUN
ejpam-1506	207	26	of	of	ADP
ejpam-1506	207	27	l(kn	l(kn	PROPN
ejpam-1506	207	28	)	)	PUNCT
ejpam-1506	207	29	.	.	PUNCT
ejpam-1506	208	1	theorem	theorem	VERB
ejpam-1506	208	2	11	11	NUM
ejpam-1506	208	3	.	.	PUNCT
ejpam-1506	209	1	[	[	X
ejpam-1506	209	2	1	1	X
ejpam-1506	209	3	]	]	PUNCT
ejpam-1506	209	4	for	for	ADP
ejpam-1506	209	5	the	the	DET
ejpam-1506	209	6	complete	complete	ADJ
ejpam-1506	209	7	graph	graph	NOUN
ejpam-1506	209	8	kn	kn	NOUN
ejpam-1506	209	9	of	of	ADP
ejpam-1506	209	10	order	order	NOUN
ejpam-1506	209	11	n≥	n≥	PROPN
ejpam-1506	209	12	6	6	NUM
ejpam-1506	209	13	,	,	PUNCT
ejpam-1506	209	14	dim(l(kn	dim(l(kn	NUM
ejpam-1506	209	15	)	)	PUNCT
ejpam-1506	209	16	)	)	PUNCT
ejpam-1506	210	1	=	=	PUNCT
ejpam-1506	210	2	⌈	⌈	NUM
ejpam-1506	210	3	2n	2n	NUM
ejpam-1506	210	4	3	3	NUM
ejpam-1506	210	5	⌉.	⌉.	ADV
ejpam-1506	210	6	theorem	theorem	VERB
ejpam-1506	210	7	12	12	NUM
ejpam-1506	210	8	.	.	PUNCT
ejpam-1506	211	1	if	if	SCONJ
ejpam-1506	211	2	kn	kn	PROPN
ejpam-1506	211	3	is	be	AUX
ejpam-1506	211	4	the	the	DET
ejpam-1506	211	5	complete	complete	ADJ
ejpam-1506	211	6	graph	graph	NOUN
ejpam-1506	211	7	of	of	ADP
ejpam-1506	211	8	order	order	NOUN
ejpam-1506	211	9	n≥	n≥	NOUN
ejpam-1506	211	10	2	2	NUM
ejpam-1506	211	11	,	,	PUNCT
ejpam-1506	211	12	then	then	ADV
ejpam-1506	211	13	dim(k⋆n	dim(k⋆n	PROPN
ejpam-1506	211	14	)	)	PUNCT
ejpam-1506	212	1	=	=	PUNCT
ejpam-1506	212	2	n−	n−	NOUN
ejpam-1506	212	3	1	1	NUM
ejpam-1506	212	4	.	.	PUNCT
ejpam-1506	213	1	proof	proof	NOUN
ejpam-1506	213	2	.	.	PUNCT
ejpam-1506	214	1	let	let	VERB
ejpam-1506	214	2	g	g	PROPN
ejpam-1506	214	3	=	=	PROPN
ejpam-1506	214	4	kn	kn	PROPN
ejpam-1506	214	5	for	for	ADP
ejpam-1506	214	6	n	n	X
ejpam-1506	214	7	≥	≥	NOUN
ejpam-1506	214	8	2	2	NUM
ejpam-1506	214	9	.	.	PUNCT
ejpam-1506	215	1	if	if	SCONJ
ejpam-1506	215	2	n	n	NOUN
ejpam-1506	215	3	=	=	SYM
ejpam-1506	215	4	2	2	NUM
ejpam-1506	215	5	,	,	PUNCT
ejpam-1506	215	6	then	then	ADV
ejpam-1506	215	7	g	g	PROPN
ejpam-1506	215	8	∼=	∼=	PROPN
ejpam-1506	215	9	p2	p2	NOUN
ejpam-1506	215	10	;	;	PUNCT
ejpam-1506	215	11	thus	thus	ADV
ejpam-1506	215	12	dim(g⋆	dim(g⋆	NOUN
ejpam-1506	215	13	)	)	PUNCT
ejpam-1506	215	14	=	=	SYM
ejpam-1506	216	1	1	1	X
ejpam-1506	216	2	.	.	PUNCT
ejpam-1506	217	1	if	if	SCONJ
ejpam-1506	217	2	n	n	NOUN
ejpam-1506	217	3	=	=	SYM
ejpam-1506	217	4	3	3	NUM
ejpam-1506	217	5	,	,	PUNCT
ejpam-1506	217	6	then	then	ADV
ejpam-1506	217	7	g	g	PROPN
ejpam-1506	217	8	∼=	∼=	PROPN
ejpam-1506	217	9	c3	c3	NOUN
ejpam-1506	217	10	;	;	PUNCT
ejpam-1506	217	11	thus	thus	ADV
ejpam-1506	217	12	dim(g⋆	dim(g⋆	NOUN
ejpam-1506	217	13	)	)	PUNCT
ejpam-1506	217	14	=	=	SYM
ejpam-1506	218	1	2	2	X
ejpam-1506	218	2	.	.	X
ejpam-1506	219	1	if	if	SCONJ
ejpam-1506	219	2	n	n	NOUN
ejpam-1506	219	3	=	=	SYM
ejpam-1506	219	4	4	4	NUM
ejpam-1506	219	5	,	,	PUNCT
ejpam-1506	219	6	then	then	ADV
ejpam-1506	219	7	dim(g⋆	dim(g⋆	NOUN
ejpam-1506	219	8	)	)	PUNCT
ejpam-1506	219	9	≥	≥	NOUN
ejpam-1506	219	10	3	3	NUM
ejpam-1506	219	11	by	by	ADP
ejpam-1506	219	12	theorem	theorem	NOUN
ejpam-1506	219	13	3	3	NUM
ejpam-1506	219	14	,	,	PUNCT
ejpam-1506	219	15	since	since	SCONJ
ejpam-1506	219	16	diam(g⋆	diam(g⋆	PROPN
ejpam-1506	219	17	)	)	PUNCT
ejpam-1506	219	18	=	=	SYM
ejpam-1506	219	19	3	3	NUM
ejpam-1506	219	20	and	and	CCONJ
ejpam-1506	219	21	|v	|v	PROPN
ejpam-1506	219	22	(	(	PUNCT
ejpam-1506	219	23	g⋆)|	g⋆)|	NOUN
ejpam-1506	219	24	=	=	PROPN
ejpam-1506	219	25	12	12	NUM
ejpam-1506	219	26	;	;	PUNCT
ejpam-1506	219	27	thus	thus	ADV
ejpam-1506	219	28	dim(g⋆	dim(g⋆	NOUN
ejpam-1506	219	29	)	)	PUNCT
ejpam-1506	219	30	=	=	SYM
ejpam-1506	219	31	3	3	NUM
ejpam-1506	219	32	by	by	ADP
ejpam-1506	219	33	theorem	theorem	NOUN
ejpam-1506	219	34	6	6	NUM
ejpam-1506	219	35	.	.	PUNCT
ejpam-1506	220	1	so	so	ADV
ejpam-1506	220	2	,	,	PUNCT
ejpam-1506	220	3	let	let	VERB
ejpam-1506	220	4	n	n	PRON
ejpam-1506	220	5	≥	≥	NOUN
ejpam-1506	220	6	5	5	NUM
ejpam-1506	220	7	.	.	PUNCT
ejpam-1506	221	1	let	let	VERB
ejpam-1506	221	2	v	v	X
ejpam-1506	221	3	(	(	PUNCT
ejpam-1506	221	4	g	g	NOUN
ejpam-1506	221	5	)	)	PUNCT
ejpam-1506	221	6	=	=	SYM
ejpam-1506	221	7	{	{	PUNCT
ejpam-1506	221	8	v1	v1	PROPN
ejpam-1506	221	9	,	,	PUNCT
ejpam-1506	221	10	v2	v2	PROPN
ejpam-1506	221	11	,	,	PUNCT
ejpam-1506	221	12	.	.	PUNCT
ejpam-1506	221	13	.	.	PUNCT
ejpam-1506	222	1	.	.	PUNCT
ejpam-1506	223	1	,	,	PUNCT
ejpam-1506	223	2	vn	vn	VERB
ejpam-1506	223	3	}	}	PUNCT
ejpam-1506	223	4	with	with	ADP
ejpam-1506	223	5	degg(vi	degg(vi	PROPN
ejpam-1506	223	6	)	)	PUNCT
ejpam-1506	223	7	=	=	SYM
ejpam-1506	223	8	n−1	n−1	PROPN
ejpam-1506	223	9	(	(	PUNCT
ejpam-1506	223	10	1≤	1≤	NUM
ejpam-1506	223	11	i	i	NOUN
ejpam-1506	223	12	≤	≤	PROPN
ejpam-1506	223	13	n	n	CCONJ
ejpam-1506	223	14	)	)	PUNCT
ejpam-1506	223	15	.	.	PUNCT
ejpam-1506	224	1	following	follow	VERB
ejpam-1506	224	2	the	the	DET
ejpam-1506	224	3	construction	construction	NOUN
ejpam-1506	224	4	of	of	ADP
ejpam-1506	224	5	g⋆	g⋆	NOUN
ejpam-1506	224	6	from	from	ADP
ejpam-1506	224	7	g	g	NOUN
ejpam-1506	224	8	,	,	PUNCT
ejpam-1506	224	9	let	let	VERB
ejpam-1506	224	10	each	each	DET
ejpam-1506	224	11	vertex	vertex	NOUN
ejpam-1506	224	12	vi	vi	NOUN
ejpam-1506	224	13	be	be	AUX
ejpam-1506	224	14	replaced	replace	VERB
ejpam-1506	224	15	by	by	ADP
ejpam-1506	224	16	k(vi	k(vi	PROPN
ejpam-1506	224	17	)	)	PUNCT
ejpam-1506	224	18	∼=	∼=	PROPN
ejpam-1506	224	19	kn−1	kn−1	PROPN
ejpam-1506	224	20	;	;	PUNCT
ejpam-1506	224	21	here	here	ADV
ejpam-1506	224	22	,	,	PUNCT
ejpam-1506	224	23	we	we	PRON
ejpam-1506	224	24	denote	denote	VERB
ejpam-1506	224	25	by	by	ADP
ejpam-1506	224	26	ui	ui	PROPN
ejpam-1506	224	27	the	the	DET
ejpam-1506	224	28	vertex	vertex	NOUN
ejpam-1506	224	29	set	set	VERB
ejpam-1506	224	30	v	v	NOUN
ejpam-1506	224	31	(	(	PUNCT
ejpam-1506	224	32	k(vi	k(vi	PROPN
ejpam-1506	224	33	)	)	PUNCT
ejpam-1506	224	34	)	)	PUNCT
ejpam-1506	225	1	=	=	PUNCT
ejpam-1506	225	2	{	{	PUNCT
ejpam-1506	225	3	ui,1,ui,2	ui,1,ui,2	PROPN
ejpam-1506	225	4	,	,	PUNCT
ejpam-1506	225	5	.	.	PUNCT
ejpam-1506	225	6	.	.	PUNCT
ejpam-1506	225	7	.	.	PUNCT
ejpam-1506	226	1	,	,	PUNCT
ejpam-1506	226	2	ui	ui	PROPN
ejpam-1506	226	3	,	,	PUNCT
ejpam-1506	226	4	n	n	CCONJ
ejpam-1506	226	5	}	}	PUNCT
ejpam-1506	226	6	−	−	PROPN
ejpam-1506	226	7	{	{	PUNCT
ejpam-1506	226	8	ui	ui	PROPN
ejpam-1506	226	9	,	,	PUNCT
ejpam-1506	226	10	i	i	NOUN
ejpam-1506	226	11	}	}	PUNCT
ejpam-1506	226	12	⊆	⊆	NUM
ejpam-1506	226	13	v	v	NOUN
ejpam-1506	226	14	(	(	PUNCT
ejpam-1506	226	15	g⋆	g⋆	NOUN
ejpam-1506	226	16	)	)	PUNCT
ejpam-1506	226	17	for	for	ADP
ejpam-1506	226	18	each	each	DET
ejpam-1506	226	19	i	i	PRON
ejpam-1506	226	20	(	(	PUNCT
ejpam-1506	226	21	1	1	NUM
ejpam-1506	226	22	≤	≤	NUM
ejpam-1506	226	23	i	i	NOUN
ejpam-1506	226	24	≤	≤	NOUN
ejpam-1506	226	25	n	n	CCONJ
ejpam-1506	226	26	)	)	PUNCT
ejpam-1506	226	27	.	.	PUNCT
ejpam-1506	227	1	see	see	VERB
ejpam-1506	227	2	figure	figure	NOUN
ejpam-1506	227	3	2	2	NUM
ejpam-1506	227	4	for	for	ADP
ejpam-1506	227	5	the	the	DET
ejpam-1506	227	6	labelings	labeling	NOUN
ejpam-1506	227	7	of	of	ADP
ejpam-1506	227	8	kn	kn	PROPN
ejpam-1506	227	9	and	and	CCONJ
ejpam-1506	227	10	k⋆n	k⋆n	PROPN
ejpam-1506	227	11	;	;	PUNCT
ejpam-1506	227	12	here	here	ADV
ejpam-1506	227	13	,	,	PUNCT
ejpam-1506	227	14	the	the	DET
ejpam-1506	227	15	solid	solid	ADJ
ejpam-1506	227	16	vertices	vertex	NOUN
ejpam-1506	227	17	form	form	VERB
ejpam-1506	227	18	a	a	DET
ejpam-1506	227	19	minimum	minimum	NOUN
ejpam-1506	227	20	resolving	resolving	NOUN
ejpam-1506	227	21	set	set	VERB
ejpam-1506	227	22	for	for	ADP
ejpam-1506	227	23	k6	k6	NOUN
ejpam-1506	227	24	and	and	CCONJ
ejpam-1506	227	25	k⋆	k⋆	PROPN
ejpam-1506	227	26	6	6	NUM
ejpam-1506	227	27	,	,	PUNCT
ejpam-1506	227	28	respectively	respectively	ADV
ejpam-1506	227	29	.	.	PUNCT
ejpam-1506	228	1	let	let	VERB
ejpam-1506	228	2	s	s	PRON
ejpam-1506	228	3	be	be	AUX
ejpam-1506	228	4	a	a	DET
ejpam-1506	228	5	resolving	resolving	NOUN
ejpam-1506	228	6	set	set	VERB
ejpam-1506	228	7	for	for	ADP
ejpam-1506	228	8	k⋆n	k⋆n	NOUN
ejpam-1506	228	9	.	.	PUNCT
ejpam-1506	229	1	by	by	ADP
ejpam-1506	229	2	theorem	theorem	ADJ
ejpam-1506	229	3	6	6	NUM
ejpam-1506	229	4	,	,	PUNCT
ejpam-1506	229	5	dim(k⋆n	dim(k⋆n	PROPN
ejpam-1506	229	6	)	)	PUNCT
ejpam-1506	229	7	≤	≤	NUM
ejpam-1506	229	8	n−	n−	PROPN
ejpam-1506	229	9	1†.	1†.	NOUN
ejpam-1506	229	10	it	it	PRON
ejpam-1506	229	11	remains	remain	VERB
ejpam-1506	229	12	to	to	PART
ejpam-1506	229	13	show	show	VERB
ejpam-1506	229	14	that	that	SCONJ
ejpam-1506	229	15	dim(k⋆n	dim(k⋆n	NOUN
ejpam-1506	229	16	)	)	PUNCT
ejpam-1506	229	17	≥	≥	PROPN
ejpam-1506	229	18	n−	n−	NOUN
ejpam-1506	229	19	1	1	NUM
ejpam-1506	229	20	.	.	PUNCT
ejpam-1506	230	1	assume	assume	VERB
ejpam-1506	230	2	,	,	PUNCT
ejpam-1506	230	3	to	to	ADP
ejpam-1506	230	4	the	the	DET
ejpam-1506	230	5	contrary	contrary	NOUN
ejpam-1506	230	6	,	,	PUNCT
ejpam-1506	230	7	that	that	DET
ejpam-1506	230	8	dim(k⋆n	dim(k⋆n	NOUN
ejpam-1506	230	9	)	)	PUNCT
ejpam-1506	230	10	≤	≤	NUM
ejpam-1506	230	11	n−	n−	NOUN
ejpam-1506	230	12	2	2	NUM
ejpam-1506	230	13	;	;	PUNCT
ejpam-1506	230	14	then	then	ADV
ejpam-1506	230	15	|s|	|s|	NOUN
ejpam-1506	230	16	≤	≤	PROPN
ejpam-1506	230	17	n−	n−	PROPN
ejpam-1506	230	18	2	2	NUM
ejpam-1506	230	19	and	and	CCONJ
ejpam-1506	230	20	there	there	PRON
ejpam-1506	230	21	are	be	VERB
ejpam-1506	230	22	two	two	NUM
ejpam-1506	230	23	or	or	CCONJ
ejpam-1506	230	24	more	more	ADJ
ejpam-1506	230	25	ui	ui	PROPN
ejpam-1506	230	26	’s	’	VERB
ejpam-1506	230	27	satisfying	satisfy	VERB
ejpam-1506	230	28	s	s	X
ejpam-1506	230	29	∩ui	∩ui	NOUN
ejpam-1506	230	30	=	=	PUNCT
ejpam-1506	230	31	;	;	PUNCT
ejpam-1506	230	32	.	.	PUNCT
ejpam-1506	231	1	we	we	PRON
ejpam-1506	231	2	may	may	AUX
ejpam-1506	231	3	assume	assume	VERB
ejpam-1506	231	4	that	that	SCONJ
ejpam-1506	231	5	|s	|s	PROPN
ejpam-1506	231	6	∩u1|	∩u1|	PROPN
ejpam-1506	231	7	≥	≥	NUM
ejpam-1506	231	8	|s	|s	PROPN
ejpam-1506	231	9	∩u2|	∩u2|	NUM
ejpam-1506	231	10	≥	≥	NUM
ejpam-1506	231	11	.	.	PUNCT
ejpam-1506	231	12	.	.	PUNCT
ejpam-1506	232	1	.	.	PUNCT
ejpam-1506	233	1	≥	≥	PROPN
ejpam-1506	233	2	|s	|s	PROPN
ejpam-1506	233	3	∩un−2|	∩un−2|	PROPN
ejpam-1506	233	4	≥	≥	X
ejpam-1506	233	5	0	0	PUNCT
ejpam-1506	234	1	and	and	CCONJ
ejpam-1506	234	2	that	that	PRON
ejpam-1506	234	3	s	s	VERB
ejpam-1506	234	4	∩ui	∩ui	NOUN
ejpam-1506	234	5	=	=	PUNCT
ejpam-1506	234	6	;	;	PUNCT
ejpam-1506	234	7	for	for	ADP
ejpam-1506	234	8	i	i	PRON
ejpam-1506	234	9	∈	∈	PROPN
ejpam-1506	234	10	{	{	PUNCT
ejpam-1506	234	11	n−	n−	NOUN
ejpam-1506	234	12	1	1	NUM
ejpam-1506	234	13	,	,	PUNCT
ejpam-1506	234	14	n	n	CCONJ
ejpam-1506	234	15	}	}	PUNCT
ejpam-1506	234	16	,	,	PUNCT
ejpam-1506	234	17	by	by	ADP
ejpam-1506	234	18	relabeling	relabele	VERB
ejpam-1506	234	19	if	if	SCONJ
ejpam-1506	234	20	necessary	necessary	ADJ
ejpam-1506	234	21	.	.	PUNCT
ejpam-1506	235	1	if	if	SCONJ
ejpam-1506	235	2	|s	|s	PROPN
ejpam-1506	235	3	∩	∩	PROPN
ejpam-1506	235	4	u1|	u1|	PROPN
ejpam-1506	235	5	≥	≥	NOUN
ejpam-1506	235	6	n	n	CCONJ
ejpam-1506	235	7	−	−	PROPN
ejpam-1506	235	8	3	3	NUM
ejpam-1506	235	9	,	,	PUNCT
ejpam-1506	235	10	then	then	ADV
ejpam-1506	235	11	one	one	PRON
ejpam-1506	235	12	can	can	AUX
ejpam-1506	235	13	easily	easily	ADV
ejpam-1506	235	14	see	see	VERB
ejpam-1506	235	15	that	that	PRON
ejpam-1506	235	16	s	s	VERB
ejpam-1506	235	17	fails	fail	VERB
ejpam-1506	235	18	to	to	PART
ejpam-1506	235	19	resolve	resolve	VERB
ejpam-1506	235	20	k⋆n	k⋆n	NOUN
ejpam-1506	235	21	.	.	PUNCT
ejpam-1506	236	1	so	so	ADV
ejpam-1506	236	2	,	,	PUNCT
ejpam-1506	236	3	†one	†one	PRON
ejpam-1506	236	4	can	can	AUX
ejpam-1506	236	5	readily	readily	ADV
ejpam-1506	236	6	check	check	VERB
ejpam-1506	236	7	that	that	PRON
ejpam-1506	236	8	s	s	VERB
ejpam-1506	236	9	=	=	PUNCT
ejpam-1506	236	10	{	{	PUNCT
ejpam-1506	236	11	ui	ui	NOUN
ejpam-1506	236	12	,	,	PUNCT
ejpam-1506	236	13	i+1	i+1	PUNCT
ejpam-1506	236	14	|	|	ADV
ejpam-1506	236	15	1	1	NUM
ejpam-1506	236	16	≤	≤	NUM
ejpam-1506	237	1	i	i	PRON
ejpam-1506	237	2	l	l	NOUN
ejpam-1506	237	3	en−	en−	PUNCT
ejpam-1506	237	4	1	1	NUM
ejpam-1506	237	5	}	}	PUNCT
ejpam-1506	237	6	is	be	AUX
ejpam-1506	237	7	a	a	DET
ejpam-1506	237	8	resolving	resolving	NOUN
ejpam-1506	237	9	set	set	VERB
ejpam-1506	237	10	for	for	ADP
ejpam-1506	237	11	k⋆	k⋆	PROPN
ejpam-1506	237	12	n	n	PROPN
ejpam-1506	237	13	d.	d.	PROPN
ejpam-1506	237	14	klein	klein	PROPN
ejpam-1506	237	15	,	,	PUNCT
ejpam-1506	237	16	e.	e.	PROPN
ejpam-1506	237	17	yi	yi	PROPN
ejpam-1506	237	18	/	/	SYM
ejpam-1506	237	19	eur	eur	PROPN
ejpam-1506	237	20	.	.	PUNCT
ejpam-1506	238	1	j.	j.	PROPN
ejpam-1506	238	2	pure	pure	PROPN
ejpam-1506	238	3	appl	appl	PROPN
ejpam-1506	238	4	.	.	PROPN
ejpam-1506	238	5	math	math	PROPN
ejpam-1506	238	6	,	,	PUNCT
ejpam-1506	238	7	5	5	NUM
ejpam-1506	238	8	(	(	PUNCT
ejpam-1506	238	9	2012	2012	NUM
ejpam-1506	238	10	)	)	PUNCT
ejpam-1506	238	11	,	,	PUNCT
ejpam-1506	238	12	302	302	NUM
ejpam-1506	238	13	-	-	SYM
ejpam-1506	238	14	316	316	NUM
ejpam-1506	238	15	309	309	NUM
ejpam-1506	238	16	|s	|s	PROPN
ejpam-1506	238	17	∩ui|	∩ui|	NOUN
ejpam-1506	238	18	≤	≤	NUM
ejpam-1506	238	19	n−	n−	NOUN
ejpam-1506	238	20	4	4	NUM
ejpam-1506	238	21	each	each	PRON
ejpam-1506	238	22	i	i	PRON
ejpam-1506	238	23	(	(	PUNCT
ejpam-1506	238	24	1	1	NUM
ejpam-1506	238	25	≤	≤	NUM
ejpam-1506	238	26	i	i	NOUN
ejpam-1506	238	27	≤	≤	NOUN
ejpam-1506	238	28	n	n	CCONJ
ejpam-1506	238	29	)	)	PUNCT
ejpam-1506	238	30	,	,	PUNCT
ejpam-1506	238	31	and	and	CCONJ
ejpam-1506	238	32	we	we	PRON
ejpam-1506	238	33	may	may	AUX
ejpam-1506	238	34	assume	assume	VERB
ejpam-1506	238	35	that	that	SCONJ
ejpam-1506	238	36	s	s	VERB
ejpam-1506	238	37	∩	∩	NOUN
ejpam-1506	238	38	{	{	PUNCT
ejpam-1506	238	39	u1,n−2,u1,n−1,u1,n	u1,n−2,u1,n−1,u1,n	NOUN
ejpam-1506	238	40	}	}	PUNCT
ejpam-1506	238	41	=	=	SYM
ejpam-1506	238	42	;	;	PUNCT
ejpam-1506	238	43	by	by	ADP
ejpam-1506	238	44	relabeling	relabele	VERB
ejpam-1506	238	45	if	if	SCONJ
ejpam-1506	238	46	necessary	necessary	ADJ
ejpam-1506	238	47	.	.	PUNCT
ejpam-1506	239	1	then	then	ADV
ejpam-1506	239	2	codes(u1,n−1	codes(u1,n−1	ADJ
ejpam-1506	239	3	)	)	PUNCT
ejpam-1506	240	1	=	=	SYM
ejpam-1506	240	2	codes(u1,n	codes(u1,n	PROPN
ejpam-1506	240	3	)	)	PUNCT
ejpam-1506	240	4	,	,	PUNCT
ejpam-1506	240	5	contradicting	contradict	VERB
ejpam-1506	240	6	the	the	DET
ejpam-1506	240	7	assumption	assumption	NOUN
ejpam-1506	240	8	that	that	SCONJ
ejpam-1506	240	9	s	s	VERB
ejpam-1506	240	10	is	be	AUX
ejpam-1506	240	11	a	a	DET
ejpam-1506	240	12	resolving	resolving	NOUN
ejpam-1506	240	13	set	set	VERB
ejpam-1506	240	14	for	for	ADP
ejpam-1506	240	15	k⋆n	k⋆n	NOUN
ejpam-1506	240	16	.	.	PUNCT
ejpam-1506	241	1	thus	thus	ADV
ejpam-1506	241	2	,	,	PUNCT
ejpam-1506	241	3	any	any	DET
ejpam-1506	241	4	resolving	resolving	NOUN
ejpam-1506	241	5	set	set	VERB
ejpam-1506	241	6	s	s	PRON
ejpam-1506	241	7	for	for	ADP
ejpam-1506	241	8	k⋆n	k⋆n	NOUN
ejpam-1506	241	9	must	must	AUX
ejpam-1506	241	10	satisfy	satisfy	VERB
ejpam-1506	241	11	|s|	|s|	PROPN
ejpam-1506	241	12	≥	≥	PROPN
ejpam-1506	241	13	n−1	n−1	PROPN
ejpam-1506	241	14	.	.	PROPN
ejpam-1506	242	1	therefore	therefore	ADV
ejpam-1506	242	2	,	,	PUNCT
ejpam-1506	242	3	dim(k⋆n	dim(k⋆n	PROPN
ejpam-1506	242	4	)	)	PUNCT
ejpam-1506	243	1	=	=	PUNCT
ejpam-1506	243	2	n−	n−	NOUN
ejpam-1506	243	3	1	1	NUM
ejpam-1506	243	4	.	.	PUNCT
ejpam-1506	244	1	(	(	PUNCT
ejpam-1506	244	2	a	a	X
ejpam-1506	244	3	)	)	PUNCT
ejpam-1506	244	4	k6	k6	NOUN
ejpam-1506	244	5	(	(	PUNCT
ejpam-1506	244	6	b	b	NOUN
ejpam-1506	244	7	)	)	PUNCT
ejpam-1506	244	8	k⋆	k⋆	NOUN
ejpam-1506	244	9	6	6	NUM
ejpam-1506	244	10	figure	figure	NOUN
ejpam-1506	244	11	2	2	NUM
ejpam-1506	244	12	:	:	PUNCT
ejpam-1506	244	13	labeling	labeling	NOUN
ejpam-1506	244	14	of	of	ADP
ejpam-1506	244	15	kn	kn	PROPN
ejpam-1506	244	16	and	and	CCONJ
ejpam-1506	244	17	k⋆n	k⋆n	PROPN
ejpam-1506	244	18	third	third	ADJ
ejpam-1506	244	19	,	,	PUNCT
ejpam-1506	244	20	we	we	PRON
ejpam-1506	244	21	consider	consider	VERB
ejpam-1506	244	22	the	the	DET
ejpam-1506	244	23	metric	metric	ADJ
ejpam-1506	244	24	dimension	dimension	NOUN
ejpam-1506	244	25	of	of	ADP
ejpam-1506	244	26	k⋆s	k⋆s	PROPN
ejpam-1506	244	27	,	,	PUNCT
ejpam-1506	244	28	t	t	PROPN
ejpam-1506	244	29	for	for	ADP
ejpam-1506	244	30	the	the	DET
ejpam-1506	244	31	complete	complete	ADJ
ejpam-1506	244	32	bi	bi	ADJ
ejpam-1506	244	33	-	-	ADJ
ejpam-1506	244	34	partite	partite	ADJ
ejpam-1506	244	35	graph	graph	NOUN
ejpam-1506	244	36	ks	ks	PROPN
ejpam-1506	244	37	,	,	PUNCT
ejpam-1506	244	38	t	t	PROPN
ejpam-1506	244	39	of	of	ADP
ejpam-1506	244	40	order	order	NOUN
ejpam-1506	244	41	s+	s+	PUNCT
ejpam-1506	244	42	t	t	PROPN
ejpam-1506	244	43	≥	≥	NUM
ejpam-1506	244	44	4	4	NUM
ejpam-1506	244	45	,	,	PUNCT
ejpam-1506	244	46	where	where	SCONJ
ejpam-1506	244	47	s	s	X
ejpam-1506	244	48	,	,	PUNCT
ejpam-1506	244	49	t	t	PROPN
ejpam-1506	244	50	≥	≥	NUM
ejpam-1506	244	51	2	2	X
ejpam-1506	244	52	.	.	PUNCT
ejpam-1506	245	1	we	we	PRON
ejpam-1506	245	2	recall	recall	VERB
ejpam-1506	245	3	the	the	DET
ejpam-1506	245	4	metric	metric	ADJ
ejpam-1506	245	5	dimension	dimension	NOUN
ejpam-1506	245	6	of	of	ADP
ejpam-1506	245	7	l(ks	l(ks	PROPN
ejpam-1506	245	8	,	,	PUNCT
ejpam-1506	245	9	t	t	PROPN
ejpam-1506	245	10	)	)	PUNCT
ejpam-1506	245	11	first	first	ADV
ejpam-1506	245	12	.	.	PUNCT
ejpam-1506	246	1	theorem	theorem	VERB
ejpam-1506	246	2	13	13	NUM
ejpam-1506	246	3	.	.	PUNCT
ejpam-1506	247	1	[	[	X
ejpam-1506	247	2	4	4	X
ejpam-1506	247	3	]	]	X
ejpam-1506	247	4	let	let	VERB
ejpam-1506	247	5	ks	ks	PROPN
ejpam-1506	247	6	,	,	PUNCT
ejpam-1506	247	7	t	t	PROPN
ejpam-1506	247	8	be	be	AUX
ejpam-1506	247	9	the	the	DET
ejpam-1506	247	10	complete	complete	ADJ
ejpam-1506	247	11	bi	bi	ADJ
ejpam-1506	247	12	-	-	ADJ
ejpam-1506	247	13	partite	partite	ADJ
ejpam-1506	247	14	graph	graph	NOUN
ejpam-1506	247	15	,	,	PUNCT
ejpam-1506	247	16	where	where	SCONJ
ejpam-1506	247	17	t	t	PROPN
ejpam-1506	247	18	≥	≥	NOUN
ejpam-1506	247	19	s	s	PART
ejpam-1506	247	20	≥	≥	NOUN
ejpam-1506	247	21	1	1	NUM
ejpam-1506	247	22	.	.	PUNCT
ejpam-1506	248	1	then	then	ADV
ejpam-1506	248	2	dim(l(ks	dim(l(ks	PROPN
ejpam-1506	248	3	,	,	PUNCT
ejpam-1506	248	4	t	t	PROPN
ejpam-1506	248	5	)	)	PUNCT
ejpam-1506	248	6	)	)	PUNCT
ejpam-1506	249	1	=	=	PRON
ejpam-1506	249	2	(	(	PUNCT
ejpam-1506	249	3	⌊2(s+t−1	⌊2(s+t−1	PROPN
ejpam-1506	249	4	)	)	PUNCT
ejpam-1506	249	5	3	3	NUM
ejpam-1506	249	6	⌋	⌋	NOUN
ejpam-1506	249	7	if	if	SCONJ
ejpam-1506	249	8	s	s	VERB
ejpam-1506	249	9	≤	≤	X
ejpam-1506	249	10	t	t	NOUN
ejpam-1506	249	11	≤	≤	NUM
ejpam-1506	249	12	2s−	2s−	NUM
ejpam-1506	249	13	1	1	NUM
ejpam-1506	249	14	,	,	PUNCT
ejpam-1506	249	15	t	t	NOUN
ejpam-1506	249	16	−	−	PROPN
ejpam-1506	249	17	1	1	NUM
ejpam-1506	249	18	if	if	SCONJ
ejpam-1506	249	19	t	t	PROPN
ejpam-1506	249	20	≥	≥	X
ejpam-1506	249	21	2s	2s	X
ejpam-1506	249	22	.	.	PUNCT
ejpam-1506	250	1	since	since	SCONJ
ejpam-1506	250	2	k2,2	k2,2	PROPN
ejpam-1506	250	3	∼=	∼=	PART
ejpam-1506	250	4	c4	c4	NOUN
ejpam-1506	250	5	and	and	CCONJ
ejpam-1506	250	6	c⋆4	c⋆4	VERB
ejpam-1506	250	7	∼=	∼=	PROPN
ejpam-1506	250	8	c8	c8	NOUN
ejpam-1506	250	9	,	,	PUNCT
ejpam-1506	250	10	dim(k⋆2,2	dim(k⋆2,2	PROPN
ejpam-1506	250	11	)	)	PUNCT
ejpam-1506	250	12	=	=	SYM
ejpam-1506	250	13	2	2	X
ejpam-1506	250	14	.	.	PUNCT
ejpam-1506	251	1	so	so	ADV
ejpam-1506	251	2	,	,	PUNCT
ejpam-1506	251	3	we	we	PRON
ejpam-1506	251	4	consider	consider	VERB
ejpam-1506	251	5	for	for	ADP
ejpam-1506	251	6	s	s	PROPN
ejpam-1506	251	7	,	,	PUNCT
ejpam-1506	251	8	t	t	PROPN
ejpam-1506	251	9	≥	≥	NUM
ejpam-1506	251	10	2	2	NUM
ejpam-1506	251	11	excluding	exclude	VERB
ejpam-1506	251	12	s	s	NOUN
ejpam-1506	251	13	=	=	X
ejpam-1506	251	14	t	t	NOUN
ejpam-1506	251	15	=	=	SYM
ejpam-1506	251	16	2	2	X
ejpam-1506	251	17	.	.	PUNCT
ejpam-1506	251	18	theorem	theorem	NOUN
ejpam-1506	251	19	14	14	NUM
ejpam-1506	251	20	.	.	PUNCT
ejpam-1506	252	1	let	let	VERB
ejpam-1506	252	2	ks	ks	PROPN
ejpam-1506	252	3	,	,	PUNCT
ejpam-1506	252	4	t	t	PROPN
ejpam-1506	252	5	be	be	AUX
ejpam-1506	252	6	the	the	DET
ejpam-1506	252	7	complete	complete	ADJ
ejpam-1506	252	8	bi	bi	ADJ
ejpam-1506	252	9	-	-	ADJ
ejpam-1506	252	10	partite	partite	ADJ
ejpam-1506	252	11	graph	graph	NOUN
ejpam-1506	252	12	of	of	ADP
ejpam-1506	252	13	order	order	NOUN
ejpam-1506	252	14	s+	s+	PUNCT
ejpam-1506	252	15	t	t	PROPN
ejpam-1506	252	16	≥	≥	NUM
ejpam-1506	252	17	4	4	NUM
ejpam-1506	252	18	.	.	PUNCT
ejpam-1506	253	1	for	for	ADP
ejpam-1506	253	2	s	s	PROPN
ejpam-1506	253	3	,	,	PUNCT
ejpam-1506	253	4	t	t	PROPN
ejpam-1506	253	5	≥	≥	NUM
ejpam-1506	253	6	2	2	NUM
ejpam-1506	253	7	,	,	PUNCT
ejpam-1506	253	8	excluding	exclude	VERB
ejpam-1506	253	9	s	s	NOUN
ejpam-1506	253	10	=	=	X
ejpam-1506	253	11	t	t	NOUN
ejpam-1506	253	12	=	=	SYM
ejpam-1506	253	13	2	2	NUM
ejpam-1506	253	14	,	,	PUNCT
ejpam-1506	253	15	dim(k⋆s	dim(k⋆s	PROPN
ejpam-1506	253	16	,	,	PUNCT
ejpam-1506	253	17	t)≤	t)≤	PRON
ejpam-1506	253	18	s+	s+	ADJ
ejpam-1506	253	19	t	t	PROPN
ejpam-1506	253	20	−	−	PROPN
ejpam-1506	253	21	3	3	X
ejpam-1506	253	22	.	.	PUNCT
ejpam-1506	253	23	proof	proof	NOUN
ejpam-1506	253	24	.	.	PUNCT
ejpam-1506	254	1	for	for	ADP
ejpam-1506	254	2	t	t	PROPN
ejpam-1506	254	3	≥	≥	NOUN
ejpam-1506	254	4	s	s	PART
ejpam-1506	254	5	≥	≥	NOUN
ejpam-1506	254	6	2	2	NUM
ejpam-1506	254	7	excluding	exclude	VERB
ejpam-1506	254	8	s	s	NOUN
ejpam-1506	254	9	=	=	X
ejpam-1506	254	10	t	t	NOUN
ejpam-1506	254	11	=	=	SYM
ejpam-1506	254	12	2	2	NUM
ejpam-1506	254	13	,	,	PUNCT
ejpam-1506	254	14	let	let	VERB
ejpam-1506	254	15	g	g	PROPN
ejpam-1506	254	16	=	=	SYM
ejpam-1506	254	17	ks	ks	PROPN
ejpam-1506	254	18	,	,	PUNCT
ejpam-1506	254	19	t	t	PROPN
ejpam-1506	254	20	.	.	PUNCT
ejpam-1506	255	1	let	let	VERB
ejpam-1506	255	2	v	v	NOUN
ejpam-1506	255	3	and	and	CCONJ
ejpam-1506	255	4	w	w	PROPN
ejpam-1506	255	5	be	be	AUX
ejpam-1506	255	6	the	the	DET
ejpam-1506	255	7	bi	bi	ADJ
ejpam-1506	255	8	-	-	ADJ
ejpam-1506	255	9	partite	partite	ADJ
ejpam-1506	255	10	sets	set	NOUN
ejpam-1506	255	11	of	of	ADP
ejpam-1506	255	12	g	g	NOUN
ejpam-1506	255	13	,	,	PUNCT
ejpam-1506	255	14	where	where	SCONJ
ejpam-1506	255	15	v	v	NOUN
ejpam-1506	255	16	=	=	SYM
ejpam-1506	255	17	{	{	PUNCT
ejpam-1506	255	18	v1	v1	PROPN
ejpam-1506	255	19	,	,	PUNCT
ejpam-1506	255	20	v2	v2	PROPN
ejpam-1506	255	21	,	,	PUNCT
ejpam-1506	255	22	.	.	PUNCT
ejpam-1506	255	23	.	.	PUNCT
ejpam-1506	256	1	.	.	PUNCT
ejpam-1506	257	1	,	,	PUNCT
ejpam-1506	257	2	vs	vs	ADP
ejpam-1506	257	3	}	}	PUNCT
ejpam-1506	257	4	and	and	CCONJ
ejpam-1506	257	5	w	w	NOUN
ejpam-1506	257	6	=	=	SYM
ejpam-1506	257	7	{	{	PUNCT
ejpam-1506	257	8	w1	w1	NOUN
ejpam-1506	257	9	,	,	PUNCT
ejpam-1506	257	10	w2	w2	NOUN
ejpam-1506	257	11	,	,	PUNCT
ejpam-1506	257	12	.	.	PUNCT
ejpam-1506	257	13	.	.	PUNCT
ejpam-1506	258	1	.	.	PUNCT
ejpam-1506	259	1	,	,	PUNCT
ejpam-1506	259	2	wt	wt	ADP
ejpam-1506	259	3	}	}	PUNCT
ejpam-1506	259	4	.	.	PUNCT
ejpam-1506	260	1	following	follow	VERB
ejpam-1506	260	2	the	the	DET
ejpam-1506	260	3	construction	construction	NOUN
ejpam-1506	260	4	of	of	ADP
ejpam-1506	260	5	g⋆	g⋆	NOUN
ejpam-1506	260	6	from	from	ADP
ejpam-1506	260	7	g	g	NOUN
ejpam-1506	260	8	,	,	PUNCT
ejpam-1506	260	9	let	let	VERB
ejpam-1506	260	10	each	each	DET
ejpam-1506	260	11	vertex	vertex	NOUN
ejpam-1506	260	12	vi	vi	X
ejpam-1506	260	13	(	(	PUNCT
ejpam-1506	260	14	w	w	PROPN
ejpam-1506	260	15	j	j	PROPN
ejpam-1506	260	16	,	,	PUNCT
ejpam-1506	260	17	respectively	respectively	ADV
ejpam-1506	260	18	)	)	PUNCT
ejpam-1506	260	19	be	be	AUX
ejpam-1506	260	20	replaced	replace	VERB
ejpam-1506	260	21	by	by	ADP
ejpam-1506	260	22	k(vi	k(vi	PROPN
ejpam-1506	260	23	)	)	PUNCT
ejpam-1506	260	24	∼=	∼=	PROPN
ejpam-1506	260	25	kt	kt	PROPN
ejpam-1506	260	26	(	(	PUNCT
ejpam-1506	260	27	k(w	k(w	PROPN
ejpam-1506	260	28	j	j	PROPN
ejpam-1506	260	29	)	)	PUNCT
ejpam-1506	260	30	∼=	∼=	PROPN
ejpam-1506	260	31	ks	k	NOUN
ejpam-1506	260	32	,	,	PUNCT
ejpam-1506	260	33	respectively	respectively	ADV
ejpam-1506	260	34	)	)	PUNCT
ejpam-1506	260	35	;	;	PUNCT
ejpam-1506	260	36	here	here	ADV
ejpam-1506	260	37	,	,	PUNCT
ejpam-1506	260	38	we	we	PRON
ejpam-1506	260	39	denote	denote	VERB
ejpam-1506	260	40	by	by	ADP
ejpam-1506	260	41	ui	ui	PROPN
ejpam-1506	260	42	the	the	DET
ejpam-1506	260	43	vertex	vertex	NOUN
ejpam-1506	260	44	set	set	VERB
ejpam-1506	260	45	v	v	NOUN
ejpam-1506	260	46	(	(	PUNCT
ejpam-1506	260	47	k(vi	k(vi	PROPN
ejpam-1506	260	48	)	)	PUNCT
ejpam-1506	260	49	)	)	PUNCT
ejpam-1506	261	1	=	=	PUNCT
ejpam-1506	261	2	{	{	PUNCT
ejpam-1506	261	3	ui,1,ui,2	ui,1,ui,2	PROPN
ejpam-1506	261	4	,	,	PUNCT
ejpam-1506	261	5	.	.	PUNCT
ejpam-1506	261	6	.	.	PUNCT
ejpam-1506	261	7	.	.	PUNCT
ejpam-1506	262	1	,	,	PUNCT
ejpam-1506	262	2	ui	ui	PROPN
ejpam-1506	262	3	,	,	PUNCT
ejpam-1506	262	4	t	t	PROPN
ejpam-1506	262	5	}	}	PUNCT
ejpam-1506	262	6	⊆	⊆	NUM
ejpam-1506	262	7	v	v	NOUN
ejpam-1506	262	8	(	(	PUNCT
ejpam-1506	262	9	g⋆	g⋆	NOUN
ejpam-1506	262	10	)	)	PUNCT
ejpam-1506	262	11	and	and	CCONJ
ejpam-1506	262	12	we	we	PRON
ejpam-1506	262	13	denote	denote	VERB
ejpam-1506	262	14	by	by	ADP
ejpam-1506	262	15	u	u	NOUN
ejpam-1506	262	16	′j	′j	VERB
ejpam-1506	262	17	the	the	DET
ejpam-1506	262	18	vertex	vertex	NOUN
ejpam-1506	262	19	set	set	VERB
ejpam-1506	262	20	v	v	NOUN
ejpam-1506	262	21	(	(	PUNCT
ejpam-1506	262	22	k(w	k(w	PROPN
ejpam-1506	262	23	j	j	PROPN
ejpam-1506	262	24	)	)	PUNCT
ejpam-1506	262	25	)	)	PUNCT
ejpam-1506	263	1	=	=	PRON
ejpam-1506	263	2	{	{	PUNCT
ejpam-1506	263	3	u′j,1,u′j,2	u′j,1,u′j,2	NOUN
ejpam-1506	263	4	,	,	PUNCT
ejpam-1506	263	5	.	.	PUNCT
ejpam-1506	263	6	.	.	PUNCT
ejpam-1506	264	1	.	.	PUNCT
ejpam-1506	265	1	,	,	PUNCT
ejpam-1506	265	2	u′j	u′j	PRON
ejpam-1506	265	3	,	,	PUNCT
ejpam-1506	265	4	s	s	NOUN
ejpam-1506	265	5	}	}	PUNCT
ejpam-1506	265	6	⊆	⊆	NUM
ejpam-1506	265	7	v	v	NOUN
ejpam-1506	265	8	(	(	PUNCT
ejpam-1506	265	9	g⋆	g⋆	NOUN
ejpam-1506	265	10	)	)	PUNCT
ejpam-1506	265	11	for	for	ADP
ejpam-1506	265	12	each	each	DET
ejpam-1506	265	13	i	i	PROPN
ejpam-1506	265	14	,	,	PUNCT
ejpam-1506	265	15	j	j	PROPN
ejpam-1506	265	16	(	(	PUNCT
ejpam-1506	265	17	1	1	NUM
ejpam-1506	265	18	≤	≤	NUM
ejpam-1506	265	19	i	i	PRON
ejpam-1506	265	20	≤	≤	PROPN
ejpam-1506	265	21	s	s	X
ejpam-1506	265	22	and	and	CCONJ
ejpam-1506	265	23	1≤	1≤	NUM
ejpam-1506	265	24	j	j	PROPN
ejpam-1506	265	25	≤	≤	PROPN
ejpam-1506	265	26	t	t	PROPN
ejpam-1506	265	27	)	)	PUNCT
ejpam-1506	265	28	.	.	PUNCT
ejpam-1506	266	1	let	let	VERB
ejpam-1506	266	2	ui	ui	PROPN
ejpam-1506	266	3	,	,	PUNCT
ejpam-1506	266	4	ku′	ku′	PROPN
ejpam-1506	267	1	k	k	PROPN
ejpam-1506	267	2	,	,	PUNCT
ejpam-1506	267	3	i	i	PROPN
ejpam-1506	267	4	∈	∈	PROPN
ejpam-1506	267	5	e(g	e(g	PROPN
ejpam-1506	267	6	)	)	PUNCT
ejpam-1506	267	7	for	for	ADP
ejpam-1506	267	8	ui	ui	PROPN
ejpam-1506	267	9	,	,	PUNCT
ejpam-1506	267	10	k	k	PROPN
ejpam-1506	267	11	∈	∈	PROPN
ejpam-1506	267	12	ui	ui	PROPN
ejpam-1506	267	13	and	and	CCONJ
ejpam-1506	267	14	u′	u′	PROPN
ejpam-1506	267	15	k	k	NOUN
ejpam-1506	267	16	,	,	PUNCT
ejpam-1506	267	17	i	i	PRON
ejpam-1506	267	18	∈	∈	VERB
ejpam-1506	267	19	u	u	NOUN
ejpam-1506	267	20	′	′	NOUN
ejpam-1506	268	1	k	k	NOUN
ejpam-1506	268	2	,	,	PUNCT
ejpam-1506	268	3	where	where	SCONJ
ejpam-1506	268	4	1≤	1≤	X
ejpam-1506	268	5	i	i	PROPN
ejpam-1506	268	6	≤	≤	X
ejpam-1506	268	7	s	s	X
ejpam-1506	268	8	and	and	CCONJ
ejpam-1506	268	9	1≤	1≤	NUM
ejpam-1506	268	10	k	k	PROPN
ejpam-1506	268	11	≤	≤	PROPN
ejpam-1506	269	1	t.	t.	NOUN
ejpam-1506	269	2	see	see	VERB
ejpam-1506	269	3	figure	figure	NOUN
ejpam-1506	269	4	3	3	NUM
ejpam-1506	269	5	for	for	ADP
ejpam-1506	269	6	the	the	DET
ejpam-1506	269	7	labelings	labeling	NOUN
ejpam-1506	269	8	of	of	ADP
ejpam-1506	269	9	ks	ks	PROPN
ejpam-1506	269	10	,	,	PUNCT
ejpam-1506	269	11	t	t	PROPN
ejpam-1506	269	12	and	and	CCONJ
ejpam-1506	269	13	k⋆s	k⋆s	PROPN
ejpam-1506	269	14	,	,	PUNCT
ejpam-1506	269	15	t	t	PROPN
ejpam-1506	269	16	;	;	PUNCT
ejpam-1506	269	17	here	here	ADV
ejpam-1506	269	18	,	,	PUNCT
ejpam-1506	269	19	the	the	DET
ejpam-1506	269	20	solid	solid	ADJ
ejpam-1506	269	21	vertices	vertex	NOUN
ejpam-1506	269	22	form	form	VERB
ejpam-1506	269	23	a	a	DET
ejpam-1506	269	24	resolving	resolving	NOUN
ejpam-1506	269	25	set	set	VERB
ejpam-1506	269	26	for	for	ADP
ejpam-1506	269	27	k3,4	k3,4	ADJ
ejpam-1506	269	28	and	and	CCONJ
ejpam-1506	269	29	k⋆3,4	k⋆3,4	PROPN
ejpam-1506	269	30	,	,	PUNCT
ejpam-1506	269	31	respectively	respectively	ADV
ejpam-1506	269	32	.	.	PUNCT
ejpam-1506	270	1	we	we	PRON
ejpam-1506	270	2	will	will	AUX
ejpam-1506	270	3	show	show	VERB
ejpam-1506	270	4	that	that	PRON
ejpam-1506	270	5	s	s	VERB
ejpam-1506	270	6	=	=	PUNCT
ejpam-1506	270	7	{	{	PUNCT
ejpam-1506	270	8	u1,a	u1,a	PROPN
ejpam-1506	270	9	|	|	ADV
ejpam-1506	270	10	1	1	NUM
ejpam-1506	270	11	≤	≤	ADV
ejpam-1506	270	12	a	a	DET
ejpam-1506	270	13	≤	≤	NUM
ejpam-1506	270	14	t	t	NOUN
ejpam-1506	270	15	−	−	PROPN
ejpam-1506	270	16	1	1	NUM
ejpam-1506	270	17	}	}	PUNCT
ejpam-1506	270	18	∪	∪	X
ejpam-1506	270	19	(	(	PUNCT
ejpam-1506	270	20	∪s−1	∪s−1	X
ejpam-1506	270	21	b=2	b=2	PROPN
ejpam-1506	270	22	{	{	PUNCT
ejpam-1506	270	23	ub,1	ub,1	PROPN
ejpam-1506	270	24	}	}	PUNCT
ejpam-1506	270	25	)	)	PUNCT
ejpam-1506	270	26	forms	form	VERB
ejpam-1506	270	27	a	a	DET
ejpam-1506	270	28	resolving	resolving	NOUN
ejpam-1506	270	29	set	set	VERB
ejpam-1506	270	30	for	for	ADP
ejpam-1506	270	31	k⋆s	k⋆s	PROPN
ejpam-1506	270	32	,	,	PUNCT
ejpam-1506	270	33	t	t	PROPN
ejpam-1506	270	34	with	with	ADP
ejpam-1506	270	35	|s|	|s|	PROPN
ejpam-1506	270	36	=	=	PROPN
ejpam-1506	270	37	s+	s+	PROPN
ejpam-1506	270	38	t	t	PROPN
ejpam-1506	270	39	−	−	PROPN
ejpam-1506	270	40	3	3	NUM
ejpam-1506	270	41	,	,	PUNCT
ejpam-1506	270	42	and	and	CCONJ
ejpam-1506	270	43	thus	thus	ADV
ejpam-1506	270	44	dim(k⋆s	dim(k⋆s	PROPN
ejpam-1506	270	45	,	,	PUNCT
ejpam-1506	270	46	t	t	PROPN
ejpam-1506	270	47	)	)	PUNCT
ejpam-1506	270	48	≤	≤	NOUN
ejpam-1506	270	49	s+	s+	PUNCT
ejpam-1506	270	50	t	t	PROPN
ejpam-1506	270	51	−	−	PROPN
ejpam-1506	270	52	3	3	X
ejpam-1506	270	53	.	.	PUNCT
ejpam-1506	271	1	it	it	PRON
ejpam-1506	271	2	suffices	suffice	VERB
ejpam-1506	271	3	to	to	PART
ejpam-1506	271	4	show	show	VERB
ejpam-1506	271	5	that	that	SCONJ
ejpam-1506	271	6	,	,	PUNCT
ejpam-1506	271	7	for	for	ADP
ejpam-1506	271	8	any	any	DET
ejpam-1506	271	9	two	two	NUM
ejpam-1506	271	10	vertices	vertex	NOUN
ejpam-1506	271	11	ux	ux	ADV
ejpam-1506	271	12	,	,	PUNCT
ejpam-1506	271	13	uy	uy	PROPN
ejpam-1506	271	14	∈	∈	PROPN
ejpam-1506	271	15	v	v	NOUN
ejpam-1506	271	16	(	(	PUNCT
ejpam-1506	271	17	g⋆)−	g⋆)−	PROPN
ejpam-1506	271	18	s	s	NOUN
ejpam-1506	271	19	,	,	PUNCT
ejpam-1506	271	20	dg⋆(ux	dg⋆(ux	NOUN
ejpam-1506	271	21	,	,	PUNCT
ejpam-1506	271	22	z	z	NOUN
ejpam-1506	271	23	)	)	PUNCT
ejpam-1506	271	24	6=	6=	NUM
ejpam-1506	272	1	dg⋆(uy	dg⋆(uy	PROPN
ejpam-1506	272	2	,	,	PUNCT
ejpam-1506	272	3	z	z	NOUN
ejpam-1506	272	4	)	)	PUNCT
ejpam-1506	272	5	for	for	ADP
ejpam-1506	272	6	some	some	DET
ejpam-1506	272	7	z	z	PROPN
ejpam-1506	272	8	∈	∈	PROPN
ejpam-1506	272	9	s.	s.	PROPN
ejpam-1506	272	10	(	(	PUNCT
ejpam-1506	272	11	3	3	X
ejpam-1506	272	12	)	)	PUNCT
ejpam-1506	272	13	d.	d.	PROPN
ejpam-1506	272	14	klein	klein	PROPN
ejpam-1506	272	15	,	,	PUNCT
ejpam-1506	272	16	e.	e.	PROPN
ejpam-1506	272	17	yi	yi	PROPN
ejpam-1506	272	18	/	/	SYM
ejpam-1506	272	19	eur	eur	PROPN
ejpam-1506	272	20	.	.	PUNCT
ejpam-1506	273	1	j.	j.	PROPN
ejpam-1506	273	2	pure	pure	PROPN
ejpam-1506	273	3	appl	appl	PROPN
ejpam-1506	273	4	.	.	PROPN
ejpam-1506	273	5	math	math	PROPN
ejpam-1506	273	6	,	,	PUNCT
ejpam-1506	273	7	5	5	NUM
ejpam-1506	273	8	(	(	PUNCT
ejpam-1506	273	9	2012	2012	NUM
ejpam-1506	273	10	)	)	PUNCT
ejpam-1506	273	11	,	,	PUNCT
ejpam-1506	273	12	302	302	NUM
ejpam-1506	273	13	-	-	SYM
ejpam-1506	273	14	316	316	NUM
ejpam-1506	273	15	310	310	NUM
ejpam-1506	273	16	(	(	PUNCT
ejpam-1506	273	17	a	a	NOUN
ejpam-1506	273	18	)	)	PUNCT
ejpam-1506	273	19	k3,4	k3,4	ADJ
ejpam-1506	273	20	(	(	PUNCT
ejpam-1506	273	21	b	b	NOUN
ejpam-1506	273	22	)	)	PUNCT
ejpam-1506	273	23	k⋆	k⋆	NOUN
ejpam-1506	274	1	3,4	3,4	NUM
ejpam-1506	274	2	figure	figure	NOUN
ejpam-1506	274	3	3	3	NUM
ejpam-1506	274	4	:	:	PUNCT
ejpam-1506	274	5	labeling	labeling	NOUN
ejpam-1506	274	6	of	of	ADP
ejpam-1506	274	7	ks	ks	PROPN
ejpam-1506	274	8	,	,	PUNCT
ejpam-1506	274	9	t	t	PROPN
ejpam-1506	274	10	and	and	CCONJ
ejpam-1506	274	11	k⋆s	k⋆s	PROPN
ejpam-1506	274	12	,	,	PUNCT
ejpam-1506	274	13	t	t	PROPN
ejpam-1506	274	14	we	we	PRON
ejpam-1506	274	15	consider	consider	VERB
ejpam-1506	274	16	three	three	NUM
ejpam-1506	274	17	cases	case	NOUN
ejpam-1506	274	18	.	.	PUNCT
ejpam-1506	275	1	case	case	NOUN
ejpam-1506	275	2	1	1	NUM
ejpam-1506	275	3	:	:	PUNCT
ejpam-1506	275	4	ux	ux	PROPN
ejpam-1506	275	5	∈	∈	PROPN
ejpam-1506	275	6	ui	ui	PROPN
ejpam-1506	275	7	and	and	CCONJ
ejpam-1506	275	8	uy	uy	PROPN
ejpam-1506	275	9	∈	∈	PROPN
ejpam-1506	275	10	u	u	PROPN
ejpam-1506	275	11	j	j	PROPN
ejpam-1506	275	12	,	,	PUNCT
ejpam-1506	275	13	where	where	SCONJ
ejpam-1506	275	14	1≤	1≤	X
ejpam-1506	275	15	i	i	PROPN
ejpam-1506	275	16	,	,	PUNCT
ejpam-1506	275	17	j	j	PROPN
ejpam-1506	275	18	≤	≤	PROPN
ejpam-1506	275	19	s.	s.	PROPN
ejpam-1506	275	20	we	we	PRON
ejpam-1506	275	21	consider	consider	VERB
ejpam-1506	275	22	two	two	NUM
ejpam-1506	275	23	subcases	subcase	NOUN
ejpam-1506	275	24	.	.	PUNCT
ejpam-1506	276	1	subcase	subcase	PROPN
ejpam-1506	276	2	1.1	1.1	NUM
ejpam-1506	276	3	:	:	PUNCT
ejpam-1506	276	4	i	i	PROPN
ejpam-1506	276	5	6=	6=	PROPN
ejpam-1506	276	6	j.	j.	PROPN
ejpam-1506	277	1	if	if	SCONJ
ejpam-1506	277	2	i	i	PRON
ejpam-1506	277	3	=	=	VERB
ejpam-1506	277	4	1	1	NUM
ejpam-1506	277	5	or	or	CCONJ
ejpam-1506	277	6	j	j	NOUN
ejpam-1506	277	7	=	=	SYM
ejpam-1506	277	8	1	1	NUM
ejpam-1506	277	9	,	,	PUNCT
ejpam-1506	277	10	say	say	VERB
ejpam-1506	277	11	the	the	DET
ejpam-1506	277	12	former	former	ADJ
ejpam-1506	277	13	,	,	PUNCT
ejpam-1506	277	14	then	then	ADV
ejpam-1506	277	15	1=	1=	NUM
ejpam-1506	277	16	dg⋆(ux	dg⋆(ux	NOUN
ejpam-1506	277	17	,	,	PUNCT
ejpam-1506	277	18	u1,1	u1,1	ADJ
ejpam-1506	277	19	)	)	PUNCT
ejpam-1506	277	20	<	<	X
ejpam-1506	277	21	dg⋆(uy	dg⋆(uy	X
ejpam-1506	277	22	,	,	PUNCT
ejpam-1506	277	23	u1,1	u1,1	PROPN
ejpam-1506	277	24	)	)	PUNCT
ejpam-1506	277	25	.	.	PUNCT
ejpam-1506	278	1	if	if	SCONJ
ejpam-1506	278	2	i	i	PRON
ejpam-1506	278	3	=	=	SYM
ejpam-1506	278	4	s	s	X
ejpam-1506	278	5	or	or	CCONJ
ejpam-1506	278	6	j	j	PROPN
ejpam-1506	278	7	=	=	SYM
ejpam-1506	278	8	s	s	PROPN
ejpam-1506	278	9	,	,	PUNCT
ejpam-1506	278	10	say	say	VERB
ejpam-1506	278	11	the	the	DET
ejpam-1506	278	12	former	former	ADJ
ejpam-1506	278	13	,	,	PUNCT
ejpam-1506	278	14	then	then	ADV
ejpam-1506	278	15	dg⋆(uy	dg⋆(uy	ADJ
ejpam-1506	278	16	,	,	PUNCT
ejpam-1506	278	17	u	u	NOUN
ejpam-1506	278	18	j,1	j,1	NOUN
ejpam-1506	278	19	)	)	PUNCT
ejpam-1506	278	20	=	=	PUNCT
ejpam-1506	279	1	1	1	NUM
ejpam-1506	279	2	<	<	X
ejpam-1506	279	3	dg⋆(ux	dg⋆(ux	NOUN
ejpam-1506	279	4	,	,	PUNCT
ejpam-1506	279	5	u	u	NOUN
ejpam-1506	279	6	j,1	j,1	NOUN
ejpam-1506	279	7	)	)	PUNCT
ejpam-1506	279	8	.	.	PUNCT
ejpam-1506	280	1	if	if	SCONJ
ejpam-1506	280	2	2≤	2≤	NUM
ejpam-1506	280	3	i	i	PRON
ejpam-1506	280	4	,	,	PUNCT
ejpam-1506	280	5	j	j	PROPN
ejpam-1506	280	6	≤	≤	PROPN
ejpam-1506	280	7	s−	s−	PROPN
ejpam-1506	280	8	1	1	NUM
ejpam-1506	280	9	,	,	PUNCT
ejpam-1506	280	10	then	then	ADV
ejpam-1506	280	11	1=	1=	NUM
ejpam-1506	280	12	dg⋆(ux	dg⋆(ux	NOUN
ejpam-1506	280	13	,	,	PUNCT
ejpam-1506	280	14	ui,1	ui,1	PROPN
ejpam-1506	280	15	)	)	PUNCT
ejpam-1506	280	16	<	<	X
ejpam-1506	281	1	dg⋆(uy	dg⋆(uy	X
ejpam-1506	281	2	,	,	PUNCT
ejpam-1506	281	3	ui,1	ui,1	PROPN
ejpam-1506	281	4	)	)	PUNCT
ejpam-1506	281	5	.	.	PUNCT
ejpam-1506	282	1	so	so	ADV
ejpam-1506	282	2	,	,	PUNCT
ejpam-1506	282	3	(	(	PUNCT
ejpam-1506	282	4	3	3	X
ejpam-1506	282	5	)	)	PUNCT
ejpam-1506	282	6	holds	hold	VERB
ejpam-1506	282	7	in	in	ADP
ejpam-1506	282	8	each	each	DET
ejpam-1506	282	9	case	case	NOUN
ejpam-1506	282	10	.	.	PUNCT
ejpam-1506	283	1	subcase	subcase	PROPN
ejpam-1506	283	2	1.2	1.2	NUM
ejpam-1506	283	3	:	:	PUNCT
ejpam-1506	284	1	i	i	PRON
ejpam-1506	284	2	=	=	PUNCT
ejpam-1506	284	3	j.	j.	PROPN
ejpam-1506	284	4	notice	notice	VERB
ejpam-1506	284	5	that	that	SCONJ
ejpam-1506	284	6	2	2	NUM
ejpam-1506	284	7	≤	≤	NUM
ejpam-1506	284	8	i	i	PRON
ejpam-1506	284	9	≤	≤	PROPN
ejpam-1506	284	10	s	s	VERB
ejpam-1506	284	11	in	in	ADP
ejpam-1506	284	12	this	this	DET
ejpam-1506	284	13	case	case	NOUN
ejpam-1506	284	14	.	.	PUNCT
ejpam-1506	285	1	write	write	VERB
ejpam-1506	285	2	ux	ux	PROPN
ejpam-1506	286	1	=	=	SYM
ejpam-1506	286	2	ui	ui	PROPN
ejpam-1506	286	3	,	,	PUNCT
ejpam-1506	286	4	α	α	PROPN
ejpam-1506	286	5	and	and	CCONJ
ejpam-1506	286	6	uy	uy	PROPN
ejpam-1506	286	7	=	=	SYM
ejpam-1506	286	8	ui	ui	PROPN
ejpam-1506	286	9	,	,	PUNCT
ejpam-1506	286	10	β	β	PROPN
ejpam-1506	286	11	for	for	ADP
ejpam-1506	286	12	α	α	PROPN
ejpam-1506	286	13	6=	6=	ADP
ejpam-1506	286	14	β	β	X
ejpam-1506	286	15	(	(	PUNCT
ejpam-1506	286	16	1	1	NUM
ejpam-1506	286	17	≤	≤	NUM
ejpam-1506	286	18	α	α	NOUN
ejpam-1506	286	19	,	,	PUNCT
ejpam-1506	286	20	β	β	X
ejpam-1506	286	21	≤	≤	NUM
ejpam-1506	286	22	t	t	PROPN
ejpam-1506	286	23	)	)	PUNCT
ejpam-1506	286	24	;	;	PUNCT
ejpam-1506	286	25	then	then	ADV
ejpam-1506	286	26	dg⋆(ui	dg⋆(ui	X
ejpam-1506	286	27	,	,	PUNCT
ejpam-1506	286	28	α	α	NOUN
ejpam-1506	286	29	,	,	PUNCT
ejpam-1506	286	30	u1,α	u1,α	PROPN
ejpam-1506	286	31	)	)	PUNCT
ejpam-1506	286	32	=	=	SYM
ejpam-1506	286	33	3	3	NUM
ejpam-1506	286	34	and	and	CCONJ
ejpam-1506	286	35	dg⋆(ui	dg⋆(ui	PROPN
ejpam-1506	286	36	,	,	PUNCT
ejpam-1506	286	37	β	β	X
ejpam-1506	286	38	,	,	PUNCT
ejpam-1506	286	39	u1,α	u1,α	PROPN
ejpam-1506	286	40	)	)	PUNCT
ejpam-1506	286	41	=	=	SYM
ejpam-1506	286	42	4	4	NUM
ejpam-1506	286	43	for	for	ADP
ejpam-1506	286	44	each	each	DET
ejpam-1506	286	45	α	α	NOUN
ejpam-1506	286	46	(	(	PUNCT
ejpam-1506	286	47	1≤	1≤	NUM
ejpam-1506	286	48	α	α	PROPN
ejpam-1506	286	49	≤	≤	PROPN
ejpam-1506	286	50	t	t	NOUN
ejpam-1506	286	51	−	−	PROPN
ejpam-1506	286	52	1	1	NUM
ejpam-1506	286	53	)	)	PUNCT
ejpam-1506	286	54	,	,	PUNCT
ejpam-1506	286	55	and	and	CCONJ
ejpam-1506	286	56	thus	thus	ADV
ejpam-1506	286	57	(	(	PUNCT
ejpam-1506	286	58	3	3	X
ejpam-1506	286	59	)	)	PUNCT
ejpam-1506	286	60	holds	hold	VERB
ejpam-1506	286	61	.	.	PUNCT
ejpam-1506	287	1	case	case	NOUN
ejpam-1506	287	2	2	2	NUM
ejpam-1506	287	3	:	:	PUNCT
ejpam-1506	287	4	ux	ux	PROPN
ejpam-1506	287	5	∈	∈	PROPN
ejpam-1506	287	6	ui	ui	PROPN
ejpam-1506	287	7	and	and	CCONJ
ejpam-1506	287	8	uy	uy	PROPN
ejpam-1506	287	9	∈	∈	PROPN
ejpam-1506	287	10	u	u	NOUN
ejpam-1506	287	11	′	′	PROPN
ejpam-1506	287	12	j	j	PROPN
ejpam-1506	287	13	,	,	PUNCT
ejpam-1506	287	14	where	where	SCONJ
ejpam-1506	287	15	1≤	1≤	X
ejpam-1506	287	16	i	i	PROPN
ejpam-1506	287	17	≤	≤	X
ejpam-1506	287	18	s	s	X
ejpam-1506	287	19	and	and	CCONJ
ejpam-1506	287	20	1≤	1≤	NUM
ejpam-1506	288	1	j	j	PROPN
ejpam-1506	288	2	≤	≤	X
ejpam-1506	288	3	t.	t.	NOUN
ejpam-1506	289	1	if	if	SCONJ
ejpam-1506	289	2	i	i	PRON
ejpam-1506	289	3	=	=	NOUN
ejpam-1506	289	4	1	1	NUM
ejpam-1506	289	5	,	,	PUNCT
ejpam-1506	289	6	then	then	ADV
ejpam-1506	289	7	1	1	NUM
ejpam-1506	289	8	=	=	NOUN
ejpam-1506	289	9	dg⋆(ux	dg⋆(ux	NOUN
ejpam-1506	289	10	,	,	PUNCT
ejpam-1506	289	11	u1,1	u1,1	ADJ
ejpam-1506	289	12	)	)	PUNCT
ejpam-1506	289	13	<	<	X
ejpam-1506	289	14	dg⋆(uy	dg⋆(uy	X
ejpam-1506	289	15	,	,	PUNCT
ejpam-1506	289	16	u1,1	u1,1	ADJ
ejpam-1506	289	17	)	)	PUNCT
ejpam-1506	289	18	or	or	CCONJ
ejpam-1506	289	19	1	1	NUM
ejpam-1506	289	20	=	=	NOUN
ejpam-1506	289	21	dg⋆(ux	dg⋆(ux	NOUN
ejpam-1506	289	22	,	,	PUNCT
ejpam-1506	289	23	u1,2	u1,2	NUM
ejpam-1506	289	24	)	)	PUNCT
ejpam-1506	289	25	<	<	X
ejpam-1506	290	1	dg⋆(uy	dg⋆(uy	X
ejpam-1506	290	2	,	,	PUNCT
ejpam-1506	290	3	u1,2	u1,2	NUM
ejpam-1506	290	4	)	)	PUNCT
ejpam-1506	290	5	.	.	PUNCT
ejpam-1506	291	1	if	if	SCONJ
ejpam-1506	291	2	i	i	PRON
ejpam-1506	291	3	=	=	SYM
ejpam-1506	291	4	s	s	PROPN
ejpam-1506	291	5	,	,	PUNCT
ejpam-1506	291	6	then	then	ADV
ejpam-1506	291	7	3	3	NUM
ejpam-1506	291	8	≤	≤	NUM
ejpam-1506	291	9	dg⋆(ux	dg⋆(ux	NOUN
ejpam-1506	291	10	,	,	PUNCT
ejpam-1506	291	11	u1,1	u1,1	ADJ
ejpam-1506	291	12	)	)	PUNCT
ejpam-1506	291	13	≤	≤	NUM
ejpam-1506	291	14	4	4	NUM
ejpam-1506	291	15	and	and	CCONJ
ejpam-1506	291	16	1	1	NUM
ejpam-1506	291	17	≤	≤	NUM
ejpam-1506	291	18	dg⋆(uy	dg⋆(uy	NOUN
ejpam-1506	291	19	,	,	PUNCT
ejpam-1506	291	20	u1,1	u1,1	NOUN
ejpam-1506	291	21	)	)	PUNCT
ejpam-1506	291	22	≤	≤	NOUN
ejpam-1506	291	23	3	3	NUM
ejpam-1506	291	24	;	;	PUNCT
ejpam-1506	291	25	for	for	ADP
ejpam-1506	291	26	dg⋆(ux	dg⋆(ux	NOUN
ejpam-1506	291	27	,	,	PUNCT
ejpam-1506	291	28	u1,1	u1,1	ADJ
ejpam-1506	291	29	)	)	PUNCT
ejpam-1506	291	30	=	=	SYM
ejpam-1506	291	31	3	3	NUM
ejpam-1506	291	32	(	(	PUNCT
ejpam-1506	291	33	i.e.	i.e.	X
ejpam-1506	291	34	,	,	PUNCT
ejpam-1506	291	35	ux	ux	PROPN
ejpam-1506	291	36	=	=	SYM
ejpam-1506	291	37	us,1	us,1	PROPN
ejpam-1506	291	38	)	)	PUNCT
ejpam-1506	291	39	,	,	PUNCT
ejpam-1506	291	40	dg⋆(ux	dg⋆(ux	NOUN
ejpam-1506	291	41	,	,	PUNCT
ejpam-1506	291	42	u1,2	u1,2	NUM
ejpam-1506	291	43	)	)	PUNCT
ejpam-1506	291	44	=	=	SYM
ejpam-1506	292	1	4	4	NUM
ejpam-1506	292	2	>	>	SYM
ejpam-1506	292	3	dg⋆(uy	dg⋆(uy	X
ejpam-1506	292	4	,	,	PUNCT
ejpam-1506	292	5	u1,2	u1,2	NUM
ejpam-1506	292	6	)	)	PUNCT
ejpam-1506	292	7	.	.	PUNCT
ejpam-1506	293	1	if	if	SCONJ
ejpam-1506	293	2	2	2	NUM
ejpam-1506	293	3	≤	≤	NUM
ejpam-1506	293	4	i	i	NOUN
ejpam-1506	293	5	≤	≤	PROPN
ejpam-1506	293	6	s	s	VERB
ejpam-1506	293	7	−	−	PROPN
ejpam-1506	293	8	1	1	NUM
ejpam-1506	293	9	,	,	PUNCT
ejpam-1506	293	10	then	then	ADV
ejpam-1506	293	11	dg⋆(ux	dg⋆(ux	NOUN
ejpam-1506	293	12	,	,	PUNCT
ejpam-1506	293	13	ui,1	ui,1	PROPN
ejpam-1506	293	14	)	)	PUNCT
ejpam-1506	293	15	=	=	PUNCT
ejpam-1506	293	16	1	1	NUM
ejpam-1506	293	17	<	<	X
ejpam-1506	293	18	dg⋆(uy	dg⋆(uy	PROPN
ejpam-1506	293	19	,	,	PUNCT
ejpam-1506	293	20	ui,1	ui,1	PROPN
ejpam-1506	293	21	)	)	PUNCT
ejpam-1506	293	22	for	for	ADP
ejpam-1506	293	23	uy	uy	PROPN
ejpam-1506	293	24	6=	6=	PROPN
ejpam-1506	293	25	u′1,i	u′1,i	PROPN
ejpam-1506	293	26	;	;	PUNCT
ejpam-1506	293	27	dg⋆(ux	dg⋆(ux	NOUN
ejpam-1506	293	28	,	,	PUNCT
ejpam-1506	293	29	u1,1	u1,1	ADJ
ejpam-1506	293	30	)	)	PUNCT
ejpam-1506	293	31	=	=	SYM
ejpam-1506	293	32	4	4	NUM
ejpam-1506	293	33	>	>	SYM
ejpam-1506	293	34	dg⋆(uy	dg⋆(uy	X
ejpam-1506	293	35	,	,	PUNCT
ejpam-1506	293	36	u1,1	u1,1	PROPN
ejpam-1506	293	37	)	)	PUNCT
ejpam-1506	293	38	for	for	ADP
ejpam-1506	293	39	uy	uy	NOUN
ejpam-1506	293	40	=	=	SYM
ejpam-1506	293	41	u′1,i	u′1,i	NOUN
ejpam-1506	293	42	.	.	PUNCT
ejpam-1506	294	1	so	so	ADV
ejpam-1506	294	2	,	,	PUNCT
ejpam-1506	294	3	(	(	PUNCT
ejpam-1506	294	4	3	3	X
ejpam-1506	294	5	)	)	PUNCT
ejpam-1506	294	6	holds	hold	VERB
ejpam-1506	294	7	in	in	ADP
ejpam-1506	294	8	each	each	DET
ejpam-1506	294	9	case	case	NOUN
ejpam-1506	294	10	.	.	PUNCT
ejpam-1506	295	1	case	case	NOUN
ejpam-1506	295	2	3	3	NUM
ejpam-1506	295	3	:	:	PUNCT
ejpam-1506	296	1	ux	ux	PROPN
ejpam-1506	296	2	∈	∈	PROPN
ejpam-1506	296	3	u	u	NOUN
ejpam-1506	296	4	′	′	NOUN
ejpam-1506	297	1	i	i	PRON
ejpam-1506	297	2	and	and	CCONJ
ejpam-1506	297	3	uy	uy	PROPN
ejpam-1506	297	4	∈	∈	PROPN
ejpam-1506	297	5	u	u	NOUN
ejpam-1506	297	6	′	′	PROPN
ejpam-1506	297	7	j	j	PROPN
ejpam-1506	297	8	,	,	PUNCT
ejpam-1506	297	9	where	where	SCONJ
ejpam-1506	297	10	1≤	1≤	X
ejpam-1506	297	11	i	i	PROPN
ejpam-1506	297	12	,	,	PUNCT
ejpam-1506	297	13	j	j	PROPN
ejpam-1506	297	14	≤	≤	PROPN
ejpam-1506	297	15	t.	t.	NOUN
ejpam-1506	297	16	we	we	PRON
ejpam-1506	297	17	consider	consider	VERB
ejpam-1506	297	18	two	two	NUM
ejpam-1506	297	19	subcases	subcase	NOUN
ejpam-1506	297	20	.	.	PUNCT
ejpam-1506	298	1	subcase	subcase	PROPN
ejpam-1506	298	2	3.1	3.1	NUM
ejpam-1506	298	3	:	:	PUNCT
ejpam-1506	298	4	i	i	PROPN
ejpam-1506	298	5	6=	6=	PROPN
ejpam-1506	298	6	j.	j.	PROPN
ejpam-1506	299	1	if	if	SCONJ
ejpam-1506	299	2	ux	ux	PROPN
ejpam-1506	299	3	∈	∈	PROPN
ejpam-1506	299	4	ng⋆(s	ng⋆(	NOUN
ejpam-1506	299	5	)	)	PUNCT
ejpam-1506	299	6	or	or	CCONJ
ejpam-1506	299	7	uy	uy	PROPN
ejpam-1506	299	8	∈	∈	PROPN
ejpam-1506	299	9	ng⋆(s	ng⋆(	NOUN
ejpam-1506	299	10	)	)	PUNCT
ejpam-1506	299	11	,	,	PUNCT
ejpam-1506	299	12	say	say	VERB
ejpam-1506	299	13	the	the	DET
ejpam-1506	299	14	former	former	ADJ
ejpam-1506	299	15	,	,	PUNCT
ejpam-1506	299	16	then	then	ADV
ejpam-1506	299	17	dg⋆(ux	dg⋆(ux	NOUN
ejpam-1506	299	18	,	,	PUNCT
ejpam-1506	299	19	z	z	NOUN
ejpam-1506	299	20	)	)	PUNCT
ejpam-1506	299	21	=	=	SYM
ejpam-1506	299	22	1	1	NUM
ejpam-1506	299	23	<	<	X
ejpam-1506	299	24	dg⋆(ux	dg⋆(ux	NOUN
ejpam-1506	299	25	,	,	PUNCT
ejpam-1506	299	26	z	z	NOUN
ejpam-1506	299	27	)	)	PUNCT
ejpam-1506	299	28	for	for	ADP
ejpam-1506	299	29	z	z	PROPN
ejpam-1506	299	30	∈	∈	PROPN
ejpam-1506	299	31	s.	s.	PROPN
ejpam-1506	300	1	so	so	ADV
ejpam-1506	300	2	,	,	PUNCT
ejpam-1506	300	3	suppose	suppose	VERB
ejpam-1506	300	4	that	that	SCONJ
ejpam-1506	300	5	ux	ux	PROPN
ejpam-1506	300	6	6∈	6∈	PROPN
ejpam-1506	300	7	ng⋆(s	ng⋆(	NOUN
ejpam-1506	300	8	)	)	PUNCT
ejpam-1506	300	9	and	and	CCONJ
ejpam-1506	300	10	uy	uy	PROPN
ejpam-1506	300	11	6∈	6∈	NOUN
ejpam-1506	300	12	ng⋆(s	ng⋆(	NOUN
ejpam-1506	300	13	)	)	PUNCT
ejpam-1506	300	14	;	;	PUNCT
ejpam-1506	300	15	we	we	PRON
ejpam-1506	300	16	write	write	VERB
ejpam-1506	300	17	ux	ux	PROPN
ejpam-1506	301	1	=	=	SYM
ejpam-1506	301	2	u′	u′	PROPN
ejpam-1506	302	1	i	i	PRON
ejpam-1506	302	2	,	,	PUNCT
ejpam-1506	302	3	α	α	PROPN
ejpam-1506	302	4	and	and	CCONJ
ejpam-1506	302	5	uy	uy	NOUN
ejpam-1506	302	6	=	=	PUNCT
ejpam-1506	302	7	u′	u′	PROPN
ejpam-1506	302	8	j	j	PROPN
ejpam-1506	302	9	,	,	PUNCT
ejpam-1506	302	10	β	β	PROPN
ejpam-1506	302	11	,	,	PUNCT
ejpam-1506	302	12	where	where	SCONJ
ejpam-1506	302	13	2	2	NUM
ejpam-1506	302	14	≤	≤	NOUN
ejpam-1506	302	15	α	α	X
ejpam-1506	302	16	,	,	PUNCT
ejpam-1506	302	17	β	β	X
ejpam-1506	302	18	≤	≤	ADJ
ejpam-1506	302	19	s.	s.	PROPN
ejpam-1506	303	1	if	if	SCONJ
ejpam-1506	303	2	i	i	PRON
ejpam-1506	303	3	=	=	VERB
ejpam-1506	303	4	1	1	NUM
ejpam-1506	303	5	or	or	CCONJ
ejpam-1506	303	6	j	j	NOUN
ejpam-1506	303	7	=	=	SYM
ejpam-1506	303	8	1	1	NUM
ejpam-1506	303	9	,	,	PUNCT
ejpam-1506	303	10	say	say	VERB
ejpam-1506	303	11	the	the	DET
ejpam-1506	303	12	former	former	ADJ
ejpam-1506	303	13	,	,	PUNCT
ejpam-1506	303	14	then	then	ADV
ejpam-1506	303	15	ux	ux	INTJ
ejpam-1506	303	16	=	=	SYM
ejpam-1506	303	17	u′1,s	u′1,s	PROPN
ejpam-1506	303	18	:	:	PUNCT
ejpam-1506	303	19	dg⋆(u	dg⋆(u	PROPN
ejpam-1506	303	20	′	′	NUM
ejpam-1506	303	21	1,s	1,s	NUM
ejpam-1506	303	22	,	,	PUNCT
ejpam-1506	303	23	u1,1	u1,1	ADJ
ejpam-1506	303	24	)	)	PUNCT
ejpam-1506	303	25	=	=	SYM
ejpam-1506	303	26	2	2	NUM
ejpam-1506	303	27	and	and	CCONJ
ejpam-1506	303	28	dg⋆(u	dg⋆(u	PROPN
ejpam-1506	303	29	′	′	NUM
ejpam-1506	303	30	j	j	PROPN
ejpam-1506	303	31	,	,	PUNCT
ejpam-1506	303	32	β	β	X
ejpam-1506	303	33	,	,	PUNCT
ejpam-1506	303	34	u1,1	u1,1	ADJ
ejpam-1506	303	35	)	)	PUNCT
ejpam-1506	303	36	=	=	SYM
ejpam-1506	304	1	3	3	X
ejpam-1506	304	2	.	.	PUNCT
ejpam-1506	305	1	so	so	ADV
ejpam-1506	305	2	,	,	PUNCT
ejpam-1506	305	3	we	we	PRON
ejpam-1506	305	4	consider	consider	VERB
ejpam-1506	305	5	2	2	NUM
ejpam-1506	305	6	≤	≤	NOUN
ejpam-1506	306	1	i	i	PRON
ejpam-1506	306	2	,	,	PUNCT
ejpam-1506	306	3	j	j	PROPN
ejpam-1506	306	4	≤	≤	PROPN
ejpam-1506	306	5	t.	t.	NOUN
ejpam-1506	306	6	first	first	ADV
ejpam-1506	306	7	,	,	PUNCT
ejpam-1506	306	8	suppose	suppose	VERB
ejpam-1506	306	9	dg⋆(u	dg⋆(u	PROPN
ejpam-1506	306	10	′	′	NUM
ejpam-1506	307	1	i	i	PROPN
ejpam-1506	307	2	,	,	PUNCT
ejpam-1506	307	3	α	α	X
ejpam-1506	307	4	,	,	PUNCT
ejpam-1506	307	5	z	z	NOUN
ejpam-1506	307	6	)	)	PUNCT
ejpam-1506	307	7	=	=	SYM
ejpam-1506	307	8	2	2	NUM
ejpam-1506	307	9	for	for	ADP
ejpam-1506	307	10	some	some	DET
ejpam-1506	307	11	z	z	PROPN
ejpam-1506	307	12	∈	∈	PROPN
ejpam-1506	307	13	s.	s.	PROPN
ejpam-1506	308	1	then	then	ADV
ejpam-1506	308	2	u′i,1	u′i,1	PROPN
ejpam-1506	308	3	∈	∈	PROPN
ejpam-1506	308	4	n(s	n(s	PROPN
ejpam-1506	308	5	)	)	PUNCT
ejpam-1506	308	6	(	(	PUNCT
ejpam-1506	308	7	i.e.	i.e.	X
ejpam-1506	308	8	,	,	PUNCT
ejpam-1506	308	9	z	z	NOUN
ejpam-1506	308	10	=	=	SYM
ejpam-1506	308	11	u1,i	u1,i	PROPN
ejpam-1506	308	12	for	for	ADP
ejpam-1506	308	13	i	i	PROPN
ejpam-1506	308	14	6=	6=	PROPN
ejpam-1506	308	15	t	t	PROPN
ejpam-1506	308	16	)	)	PUNCT
ejpam-1506	308	17	or	or	CCONJ
ejpam-1506	308	18	z	z	NOUN
ejpam-1506	308	19	=	=	SYM
ejpam-1506	308	20	uα,1	uα,1	PROPN
ejpam-1506	308	21	,	,	PUNCT
ejpam-1506	309	1	where	where	SCONJ
ejpam-1506	309	2	2	2	NUM
ejpam-1506	309	3	≤	≤	NOUN
ejpam-1506	309	4	i	i	PRON
ejpam-1506	309	5	,	,	PUNCT
ejpam-1506	309	6	α	α	PROPN
ejpam-1506	309	7	≤	≤	PUNCT
ejpam-1506	309	8	s−	s−	PROPN
ejpam-1506	309	9	1	1	NUM
ejpam-1506	309	10	:	:	PUNCT
ejpam-1506	309	11	if	if	SCONJ
ejpam-1506	309	12	z	z	NOUN
ejpam-1506	309	13	=	=	SYM
ejpam-1506	309	14	u1,i	u1,i	PROPN
ejpam-1506	309	15	,	,	PUNCT
ejpam-1506	309	16	then	then	ADV
ejpam-1506	309	17	dg⋆(u	dg⋆(u	PROPN
ejpam-1506	309	18	′	′	NUM
ejpam-1506	309	19	j	j	PROPN
ejpam-1506	309	20	,	,	PUNCT
ejpam-1506	309	21	β	β	X
ejpam-1506	309	22	,	,	PUNCT
ejpam-1506	309	23	u1,i	u1,i	PROPN
ejpam-1506	309	24	)	)	PUNCT
ejpam-1506	309	25	=	=	SYM
ejpam-1506	310	1	3	3	NUM
ejpam-1506	310	2	;	;	PUNCT
ejpam-1506	310	3	if	if	SCONJ
ejpam-1506	310	4	z	z	NOUN
ejpam-1506	310	5	=	=	SYM
ejpam-1506	310	6	uα,1	uα,1	PROPN
ejpam-1506	310	7	,	,	PUNCT
ejpam-1506	310	8	then	then	ADV
ejpam-1506	310	9	dg⋆(u	dg⋆(u	PROPN
ejpam-1506	310	10	′	′	NUM
ejpam-1506	310	11	j	j	PROPN
ejpam-1506	310	12	,	,	PUNCT
ejpam-1506	310	13	β	β	X
ejpam-1506	310	14	,	,	PUNCT
ejpam-1506	310	15	uα,1	uα,1	PROPN
ejpam-1506	310	16	)	)	PUNCT
ejpam-1506	310	17	=	=	SYM
ejpam-1506	310	18	2	2	NUM
ejpam-1506	310	19	implies	imply	VERB
ejpam-1506	310	20	α	α	X
ejpam-1506	310	21	=	=	SYM
ejpam-1506	310	22	β	β	NOUN
ejpam-1506	310	23	,	,	PUNCT
ejpam-1506	310	24	but	but	CCONJ
ejpam-1506	310	25	dg⋆(u	dg⋆(u	PROPN
ejpam-1506	310	26	′	′	NUM
ejpam-1506	311	1	i	i	PROPN
ejpam-1506	311	2	,	,	PUNCT
ejpam-1506	311	3	α	α	NOUN
ejpam-1506	311	4	,	,	PUNCT
ejpam-1506	311	5	u1,i	u1,i	PROPN
ejpam-1506	311	6	)	)	PUNCT
ejpam-1506	311	7	=	=	SYM
ejpam-1506	311	8	2	2	NUM
ejpam-1506	311	9	and	and	CCONJ
ejpam-1506	311	10	dg⋆(u	dg⋆(u	PROPN
ejpam-1506	311	11	′	′	NUM
ejpam-1506	311	12	j	j	PROPN
ejpam-1506	311	13	,	,	PUNCT
ejpam-1506	311	14	α	α	NOUN
ejpam-1506	311	15	,	,	PUNCT
ejpam-1506	311	16	u1,i	u1,i	PROPN
ejpam-1506	311	17	)	)	PUNCT
ejpam-1506	311	18	=	=	SYM
ejpam-1506	312	1	3	3	X
ejpam-1506	312	2	.	.	X
ejpam-1506	312	3	second	second	ADV
ejpam-1506	312	4	,	,	PUNCT
ejpam-1506	312	5	suppose	suppose	VERB
ejpam-1506	312	6	dg⋆(u	dg⋆(u	PROPN
ejpam-1506	312	7	′	′	NUM
ejpam-1506	313	1	i	i	PROPN
ejpam-1506	313	2	,	,	PUNCT
ejpam-1506	313	3	alpha	alpha	NOUN
ejpam-1506	313	4	,	,	PUNCT
ejpam-1506	313	5	z	z	NOUN
ejpam-1506	313	6	)	)	PUNCT
ejpam-1506	313	7	=	=	SYM
ejpam-1506	313	8	3	3	NUM
ejpam-1506	313	9	for	for	ADP
ejpam-1506	313	10	all	all	DET
ejpam-1506	313	11	z	z	NOUN
ejpam-1506	313	12	∈	∈	PROPN
ejpam-1506	313	13	s.	s.	PROPN
ejpam-1506	313	14	then	then	ADV
ejpam-1506	313	15	u′i	u′i	PROPN
ejpam-1506	313	16	,	,	PUNCT
ejpam-1506	313	17	α	α	NOUN
ejpam-1506	313	18	=	=	SYM
ejpam-1506	313	19	u′t	u′t	PROPN
ejpam-1506	313	20	,	,	PUNCT
ejpam-1506	313	21	s	s	PART
ejpam-1506	313	22	,	,	PUNCT
ejpam-1506	313	23	the	the	DET
ejpam-1506	313	24	unique	unique	ADJ
ejpam-1506	313	25	vertex	vertex	NOUN
ejpam-1506	313	26	of	of	ADP
ejpam-1506	313	27	∪t	∪t	NUM
ejpam-1506	313	28	j=1	j=1	PROPN
ejpam-1506	313	29	u	u	NOUN
ejpam-1506	313	30	′j	′j	VERB
ejpam-1506	313	31	with	with	ADP
ejpam-1506	313	32	3	3	NUM
ejpam-1506	313	33	in	in	ADP
ejpam-1506	313	34	each	each	DET
ejpam-1506	313	35	entry	entry	NOUN
ejpam-1506	313	36	of	of	ADP
ejpam-1506	313	37	its	its	PRON
ejpam-1506	313	38	code	code	NOUN
ejpam-1506	313	39	.	.	PUNCT
ejpam-1506	314	1	so	so	ADV
ejpam-1506	314	2	,	,	PUNCT
ejpam-1506	314	3	(	(	PUNCT
ejpam-1506	314	4	3	3	X
ejpam-1506	314	5	)	)	PUNCT
ejpam-1506	314	6	holds	hold	VERB
ejpam-1506	314	7	in	in	ADP
ejpam-1506	314	8	each	each	DET
ejpam-1506	314	9	case	case	NOUN
ejpam-1506	314	10	.	.	PUNCT
ejpam-1506	315	1	d.	d.	PROPN
ejpam-1506	315	2	klein	klein	PROPN
ejpam-1506	315	3	,	,	PUNCT
ejpam-1506	315	4	e.	e.	PROPN
ejpam-1506	315	5	yi	yi	PROPN
ejpam-1506	315	6	/	/	SYM
ejpam-1506	315	7	eur	eur	PROPN
ejpam-1506	315	8	.	.	PUNCT
ejpam-1506	316	1	j.	j.	PROPN
ejpam-1506	316	2	pure	pure	PROPN
ejpam-1506	316	3	appl	appl	PROPN
ejpam-1506	316	4	.	.	PROPN
ejpam-1506	316	5	math	math	PROPN
ejpam-1506	316	6	,	,	PUNCT
ejpam-1506	316	7	5	5	NUM
ejpam-1506	316	8	(	(	PUNCT
ejpam-1506	316	9	2012	2012	NUM
ejpam-1506	316	10	)	)	PUNCT
ejpam-1506	316	11	,	,	PUNCT
ejpam-1506	316	12	302	302	NUM
ejpam-1506	316	13	-	-	SYM
ejpam-1506	316	14	316	316	NUM
ejpam-1506	316	15	311	311	NUM
ejpam-1506	316	16	subcase	subcase	NOUN
ejpam-1506	316	17	3.2	3.2	NUM
ejpam-1506	316	18	:	:	PUNCT
ejpam-1506	317	1	i	i	PRON
ejpam-1506	317	2	=	=	PUNCT
ejpam-1506	317	3	j.	j.	PROPN
ejpam-1506	318	1	if	if	SCONJ
ejpam-1506	318	2	ux	ux	PROPN
ejpam-1506	318	3	∈	∈	PROPN
ejpam-1506	318	4	ng⋆(s	ng⋆(	NOUN
ejpam-1506	318	5	)	)	PUNCT
ejpam-1506	318	6	or	or	CCONJ
ejpam-1506	318	7	uy	uy	PROPN
ejpam-1506	318	8	∈	∈	PROPN
ejpam-1506	318	9	ng⋆(s	ng⋆(	NOUN
ejpam-1506	318	10	)	)	PUNCT
ejpam-1506	318	11	,	,	PUNCT
ejpam-1506	318	12	say	say	VERB
ejpam-1506	318	13	the	the	DET
ejpam-1506	318	14	former	former	ADJ
ejpam-1506	318	15	,	,	PUNCT
ejpam-1506	318	16	then	then	ADV
ejpam-1506	318	17	dg⋆(ux	dg⋆(ux	NOUN
ejpam-1506	318	18	,	,	PUNCT
ejpam-1506	318	19	z	z	NOUN
ejpam-1506	318	20	)	)	PUNCT
ejpam-1506	318	21	=	=	SYM
ejpam-1506	318	22	1	1	NUM
ejpam-1506	318	23	<	<	X
ejpam-1506	318	24	dg⋆(ux	dg⋆(ux	NOUN
ejpam-1506	318	25	,	,	PUNCT
ejpam-1506	318	26	z	z	NOUN
ejpam-1506	318	27	)	)	PUNCT
ejpam-1506	318	28	for	for	ADP
ejpam-1506	318	29	z	z	PROPN
ejpam-1506	318	30	∈	∈	PROPN
ejpam-1506	318	31	s.	s.	PROPN
ejpam-1506	319	1	so	so	ADV
ejpam-1506	319	2	,	,	PUNCT
ejpam-1506	319	3	suppose	suppose	VERB
ejpam-1506	319	4	that	that	SCONJ
ejpam-1506	319	5	ux	ux	PROPN
ejpam-1506	319	6	6∈	6∈	PROPN
ejpam-1506	319	7	ng⋆(s	ng⋆(	NOUN
ejpam-1506	319	8	)	)	PUNCT
ejpam-1506	319	9	and	and	CCONJ
ejpam-1506	319	10	uy	uy	PROPN
ejpam-1506	319	11	6∈	6∈	NOUN
ejpam-1506	319	12	ng⋆(s	ng⋆(	NOUN
ejpam-1506	319	13	)	)	PUNCT
ejpam-1506	319	14	;	;	PUNCT
ejpam-1506	319	15	notice	notice	VERB
ejpam-1506	319	16	that	that	SCONJ
ejpam-1506	319	17	2	2	NUM
ejpam-1506	319	18	≤	≤	NUM
ejpam-1506	319	19	j	j	PROPN
ejpam-1506	319	20	≤	≤	PROPN
ejpam-1506	319	21	t	t	PROPN
ejpam-1506	319	22	in	in	ADP
ejpam-1506	319	23	this	this	DET
ejpam-1506	319	24	case	case	NOUN
ejpam-1506	319	25	.	.	PUNCT
ejpam-1506	320	1	we	we	PRON
ejpam-1506	320	2	write	write	VERB
ejpam-1506	320	3	ux	ux	PROPN
ejpam-1506	320	4	=	=	SYM
ejpam-1506	320	5	u′j	u′j	PROPN
ejpam-1506	320	6	,	,	PUNCT
ejpam-1506	320	7	α	α	NOUN
ejpam-1506	320	8	and	and	CCONJ
ejpam-1506	320	9	uy	uy	NOUN
ejpam-1506	320	10	=	=	PUNCT
ejpam-1506	320	11	u′	u′	PROPN
ejpam-1506	320	12	j	j	PROPN
ejpam-1506	320	13	,	,	PUNCT
ejpam-1506	320	14	β	β	PROPN
ejpam-1506	320	15	.	.	PUNCT
ejpam-1506	321	1	then	then	ADV
ejpam-1506	321	2	dg⋆(u	dg⋆(u	PROPN
ejpam-1506	321	3	′	′	NUM
ejpam-1506	321	4	j	j	PROPN
ejpam-1506	321	5	,	,	PUNCT
ejpam-1506	321	6	α	α	PROPN
ejpam-1506	321	7	,	,	PUNCT
ejpam-1506	321	8	uα,1	uα,1	PROPN
ejpam-1506	321	9	)	)	PUNCT
ejpam-1506	321	10	=	=	SYM
ejpam-1506	321	11	2	2	NUM
ejpam-1506	321	12	and	and	CCONJ
ejpam-1506	321	13	dg⋆(u	dg⋆(u	PROPN
ejpam-1506	321	14	′	′	NUM
ejpam-1506	321	15	j	j	PROPN
ejpam-1506	321	16	,	,	PUNCT
ejpam-1506	321	17	β	β	X
ejpam-1506	321	18	,	,	PUNCT
ejpam-1506	321	19	uα,1	uα,1	PROPN
ejpam-1506	321	20	)	)	PUNCT
ejpam-1506	321	21	=	=	SYM
ejpam-1506	321	22	3	3	NUM
ejpam-1506	321	23	and	and	CCONJ
ejpam-1506	321	24	for	for	ADP
ejpam-1506	321	25	α	α	PROPN
ejpam-1506	321	26	6=	6=	ADP
ejpam-1506	321	27	β	β	NOUN
ejpam-1506	321	28	,	,	PUNCT
ejpam-1506	321	29	where	where	SCONJ
ejpam-1506	321	30	1≤	1≤	X
ejpam-1506	321	31	α≤	α≤	NOUN
ejpam-1506	321	32	s−1	s−1	PROPN
ejpam-1506	321	33	and	and	CCONJ
ejpam-1506	321	34	1≤	1≤	NUM
ejpam-1506	322	1	β	β	X
ejpam-1506	322	2	≤	≤	ADJ
ejpam-1506	322	3	s.	s.	PROPN
ejpam-1506	322	4	in	in	ADP
ejpam-1506	322	5	each	each	DET
ejpam-1506	322	6	case	case	NOUN
ejpam-1506	322	7	,	,	PUNCT
ejpam-1506	322	8	(	(	PUNCT
ejpam-1506	322	9	3	3	X
ejpam-1506	322	10	)	)	PUNCT
ejpam-1506	322	11	holds	hold	VERB
ejpam-1506	322	12	.	.	PUNCT
ejpam-1506	323	1	next	next	ADV
ejpam-1506	323	2	,	,	PUNCT
ejpam-1506	323	3	we	we	PRON
ejpam-1506	323	4	determine	determine	VERB
ejpam-1506	323	5	the	the	DET
ejpam-1506	323	6	metric	metric	ADJ
ejpam-1506	323	7	dimension	dimension	NOUN
ejpam-1506	323	8	of	of	ADP
ejpam-1506	323	9	w	w	PROPN
ejpam-1506	323	10	⋆	⋆	NOUN
ejpam-1506	323	11	1,n	1,n	PROPN
ejpam-1506	323	12	for	for	ADP
ejpam-1506	323	13	the	the	DET
ejpam-1506	323	14	wheel	wheel	NOUN
ejpam-1506	323	15	graph	graph	NOUN
ejpam-1506	323	16	w1,n	w1,n	PROPN
ejpam-1506	323	17	=	=	SYM
ejpam-1506	323	18	k1	k1	PROPN
ejpam-1506	323	19	+	+	CCONJ
ejpam-1506	323	20	cn	cn	PROPN
ejpam-1506	323	21	,	,	PUNCT
ejpam-1506	323	22	where	where	SCONJ
ejpam-1506	323	23	n≥	n≥	PROPN
ejpam-1506	323	24	3	3	X
ejpam-1506	323	25	.	.	X
ejpam-1506	324	1	we	we	PRON
ejpam-1506	324	2	recall	recall	VERB
ejpam-1506	324	3	the	the	DET
ejpam-1506	324	4	metric	metric	ADJ
ejpam-1506	324	5	dimension	dimension	NOUN
ejpam-1506	324	6	of	of	ADP
ejpam-1506	324	7	the	the	DET
ejpam-1506	324	8	wheel	wheel	NOUN
ejpam-1506	324	9	graph	graph	NOUN
ejpam-1506	324	10	and	and	CCONJ
ejpam-1506	324	11	its	its	PRON
ejpam-1506	324	12	line	line	NOUN
ejpam-1506	324	13	graph	graph	NOUN
ejpam-1506	324	14	.	.	PUNCT
ejpam-1506	325	1	theorem	theorem	NOUN
ejpam-1506	325	2	15	15	NUM
ejpam-1506	325	3	.	.	PUNCT
ejpam-1506	326	1	[	[	X
ejpam-1506	326	2	3	3	NUM
ejpam-1506	326	3	,	,	PUNCT
ejpam-1506	326	4	24	24	NUM
ejpam-1506	326	5	]	]	PUNCT
ejpam-1506	326	6	for	for	ADP
ejpam-1506	326	7	n≥	n≥	PROPN
ejpam-1506	326	8	3	3	NUM
ejpam-1506	326	9	,	,	PUNCT
ejpam-1506	326	10	let	let	VERB
ejpam-1506	326	11	w1,n	w1,n	PROPN
ejpam-1506	326	12	=	=	SYM
ejpam-1506	326	13	k1	k1	PROPN
ejpam-1506	326	14	+	+	CCONJ
ejpam-1506	326	15	cn	cn	PROPN
ejpam-1506	326	16	be	be	VERB
ejpam-1506	326	17	the	the	DET
ejpam-1506	326	18	wheel	wheel	NOUN
ejpam-1506	326	19	graph	graph	NOUN
ejpam-1506	326	20	on	on	ADP
ejpam-1506	326	21	n+	n+	ADP
ejpam-1506	326	22	1	1	NUM
ejpam-1506	326	23	vertices	vertex	NOUN
ejpam-1506	326	24	.	.	PUNCT
ejpam-1506	327	1	then	then	ADV
ejpam-1506	327	2	dim(w1,n	dim(w1,n	PROPN
ejpam-1506	327	3	)	)	PUNCT
ejpam-1506	327	4	=	=	PUNCT
ejpam-1506	328	1	(	(	PUNCT
ejpam-1506	328	2	3	3	NUM
ejpam-1506	328	3	if	if	SCONJ
ejpam-1506	328	4	n=	n=	ADJ
ejpam-1506	328	5	3	3	NUM
ejpam-1506	328	6	or	or	CCONJ
ejpam-1506	328	7	n=	n=	ADJ
ejpam-1506	328	8	6	6	NUM
ejpam-1506	328	9	,	,	PUNCT
ejpam-1506	328	10	⌊2n+2	⌊2n+2	PROPN
ejpam-1506	328	11	5	5	NUM
ejpam-1506	328	12	⌋	⌋	NOUN
ejpam-1506	328	13	otherwise	otherwise	ADV
ejpam-1506	328	14	.	.	PUNCT
ejpam-1506	329	1	theorem	theorem	VERB
ejpam-1506	329	2	16	16	NUM
ejpam-1506	329	3	.	.	PUNCT
ejpam-1506	330	1	[	[	X
ejpam-1506	330	2	8	8	NUM
ejpam-1506	330	3	]	]	PUNCT
ejpam-1506	330	4	for	for	ADP
ejpam-1506	330	5	n≥	n≥	PROPN
ejpam-1506	330	6	3	3	NUM
ejpam-1506	330	7	,	,	PUNCT
ejpam-1506	330	8	dim(l(w1,n	dim(l(w1,n	NOUN
ejpam-1506	330	9	)	)	PUNCT
ejpam-1506	330	10	)	)	PUNCT
ejpam-1506	331	1	=	=	PUNCT
ejpam-1506	331	2			PROPN
ejpam-1506	331	3			X
ejpam-1506	331	4			NOUN
ejpam-1506	331	5	3	3	NUM
ejpam-1506	331	6	if	if	SCONJ
ejpam-1506	331	7	n=	n=	ADJ
ejpam-1506	331	8	3,4	3,4	NUM
ejpam-1506	331	9	,	,	PUNCT
ejpam-1506	331	10	4	4	NUM
ejpam-1506	331	11	if	if	SCONJ
ejpam-1506	331	12	n=	n=	ADJ
ejpam-1506	331	13	5	5	NUM
ejpam-1506	331	14	,	,	PUNCT
ejpam-1506	331	15	n−	n−	NOUN
ejpam-1506	331	16	⌈	⌈	NOUN
ejpam-1506	331	17	n	n	CCONJ
ejpam-1506	331	18	3	3	NUM
ejpam-1506	331	19	⌉	⌉	X
ejpam-1506	331	20	otherwise	otherwise	ADV
ejpam-1506	331	21	.	.	PUNCT
ejpam-1506	332	1	theorem	theorem	VERB
ejpam-1506	332	2	17	17	NUM
ejpam-1506	332	3	.	.	PUNCT
ejpam-1506	333	1	for	for	ADP
ejpam-1506	333	2	n≥	n≥	PROPN
ejpam-1506	333	3	3	3	NUM
ejpam-1506	333	4	,	,	PUNCT
ejpam-1506	333	5	dim(w	dim(w	X
ejpam-1506	333	6	⋆	⋆	X
ejpam-1506	333	7	1,n	1,n	NUM
ejpam-1506	333	8	)	)	PUNCT
ejpam-1506	333	9	=	=	NOUN
ejpam-1506	333	10	(	(	PUNCT
ejpam-1506	333	11	3	3	NUM
ejpam-1506	333	12	if	if	SCONJ
ejpam-1506	333	13	n=	n=	ADJ
ejpam-1506	333	14	3	3	NUM
ejpam-1506	333	15	,	,	PUNCT
ejpam-1506	333	16	n−	n−	NOUN
ejpam-1506	333	17	1	1	NUM
ejpam-1506	333	18	if	if	SCONJ
ejpam-1506	333	19	n≥	n≥	PROPN
ejpam-1506	333	20	4	4	NUM
ejpam-1506	333	21	.	.	PUNCT
ejpam-1506	334	1	proof	proof	NOUN
ejpam-1506	334	2	.	.	PUNCT
ejpam-1506	335	1	for	for	ADP
ejpam-1506	335	2	n	n	PRON
ejpam-1506	335	3	≥	≥	NUM
ejpam-1506	335	4	3	3	NUM
ejpam-1506	335	5	,	,	PUNCT
ejpam-1506	335	6	let	let	VERB
ejpam-1506	335	7	g	g	PROPN
ejpam-1506	335	8	=	=	SYM
ejpam-1506	335	9	w1,n	w1,n	PROPN
ejpam-1506	335	10	,	,	PUNCT
ejpam-1506	335	11	and	and	CCONJ
ejpam-1506	335	12	let	let	VERB
ejpam-1506	335	13	v	v	NOUN
ejpam-1506	335	14	(	(	PUNCT
ejpam-1506	335	15	g	g	NOUN
ejpam-1506	335	16	)	)	PUNCT
ejpam-1506	335	17	=	=	SYM
ejpam-1506	335	18	{	{	PUNCT
ejpam-1506	335	19	v0	v0	NOUN
ejpam-1506	335	20	,	,	PUNCT
ejpam-1506	335	21	v1	v1	NOUN
ejpam-1506	335	22	,	,	PUNCT
ejpam-1506	335	23	v2	v2	NOUN
ejpam-1506	335	24	,	,	PUNCT
ejpam-1506	335	25	.	.	PUNCT
ejpam-1506	335	26	.	.	PUNCT
ejpam-1506	336	1	.	.	PUNCT
ejpam-1506	337	1	,	,	PUNCT
ejpam-1506	337	2	vn	vn	VERB
ejpam-1506	337	3	}	}	PUNCT
ejpam-1506	337	4	with	with	ADP
ejpam-1506	337	5	degg(v0	degg(v0	ADJ
ejpam-1506	337	6	)	)	PUNCT
ejpam-1506	337	7	=	=	SYM
ejpam-1506	337	8	n.	n.	NOUN
ejpam-1506	337	9	following	follow	VERB
ejpam-1506	337	10	the	the	DET
ejpam-1506	337	11	construction	construction	NOUN
ejpam-1506	337	12	of	of	ADP
ejpam-1506	337	13	g⋆	g⋆	NOUN
ejpam-1506	337	14	from	from	ADP
ejpam-1506	337	15	g	g	NOUN
ejpam-1506	337	16	,	,	PUNCT
ejpam-1506	337	17	let	let	VERB
ejpam-1506	337	18	the	the	DET
ejpam-1506	337	19	vertex	vertex	NOUN
ejpam-1506	337	20	v0	v0	NOUN
ejpam-1506	337	21	be	be	AUX
ejpam-1506	337	22	replaced	replace	VERB
ejpam-1506	337	23	by	by	ADP
ejpam-1506	337	24	k(v0	k(v0	VERB
ejpam-1506	337	25	)	)	PUNCT
ejpam-1506	338	1	∼=	∼=	PROPN
ejpam-1506	338	2	kn	kn	NOUN
ejpam-1506	338	3	and	and	CCONJ
ejpam-1506	338	4	let	let	VERB
ejpam-1506	338	5	each	each	DET
ejpam-1506	338	6	vertex	vertex	NOUN
ejpam-1506	338	7	vi	vi	X
ejpam-1506	338	8	(	(	PUNCT
ejpam-1506	338	9	1	1	NUM
ejpam-1506	338	10	≤	≤	NUM
ejpam-1506	338	11	i	i	NOUN
ejpam-1506	338	12	≤	≤	NOUN
ejpam-1506	338	13	n	n	CCONJ
ejpam-1506	338	14	)	)	PUNCT
ejpam-1506	338	15	be	be	AUX
ejpam-1506	338	16	replaced	replace	VERB
ejpam-1506	338	17	by	by	ADP
ejpam-1506	338	18	k(vi	k(vi	PROPN
ejpam-1506	338	19	)	)	PUNCT
ejpam-1506	338	20	∼=	∼=	NOUN
ejpam-1506	338	21	k3	k3	VERB
ejpam-1506	338	22	;	;	PUNCT
ejpam-1506	338	23	here	here	ADV
ejpam-1506	338	24	,	,	PUNCT
ejpam-1506	338	25	we	we	PRON
ejpam-1506	338	26	denote	denote	VERB
ejpam-1506	338	27	by	by	ADP
ejpam-1506	338	28	u0	u0	ADJ
ejpam-1506	338	29	the	the	DET
ejpam-1506	338	30	vertex	vertex	NOUN
ejpam-1506	338	31	set	set	VERB
ejpam-1506	338	32	v	v	NOUN
ejpam-1506	338	33	(	(	PUNCT
ejpam-1506	338	34	k(v0	k(v0	VERB
ejpam-1506	338	35	)	)	PUNCT
ejpam-1506	338	36	)	)	PUNCT
ejpam-1506	339	1	=	=	PRON
ejpam-1506	339	2	{	{	PUNCT
ejpam-1506	339	3	u0,1,u0,2	u0,1,u0,2	NOUN
ejpam-1506	339	4	,	,	PUNCT
ejpam-1506	339	5	.	.	PUNCT
ejpam-1506	339	6	.	.	PUNCT
ejpam-1506	339	7	.	.	PUNCT
ejpam-1506	340	1	,	,	PUNCT
ejpam-1506	340	2	u0,n	u0,n	VERB
ejpam-1506	340	3	}	}	PUNCT
ejpam-1506	340	4	⊆	⊆	NUM
ejpam-1506	340	5	v	v	NOUN
ejpam-1506	340	6	(	(	PUNCT
ejpam-1506	340	7	g⋆	g⋆	NOUN
ejpam-1506	340	8	)	)	PUNCT
ejpam-1506	340	9	and	and	CCONJ
ejpam-1506	340	10	we	we	PRON
ejpam-1506	340	11	denote	denote	VERB
ejpam-1506	340	12	by	by	ADP
ejpam-1506	340	13	ui	ui	PROPN
ejpam-1506	340	14	(	(	PUNCT
ejpam-1506	340	15	1	1	NUM
ejpam-1506	340	16	≤	≤	NUM
ejpam-1506	340	17	i	i	NOUN
ejpam-1506	340	18	≤	≤	NOUN
ejpam-1506	340	19	n	n	CCONJ
ejpam-1506	340	20	)	)	PUNCT
ejpam-1506	340	21	the	the	DET
ejpam-1506	340	22	vertex	vertex	NOUN
ejpam-1506	340	23	set	set	VERB
ejpam-1506	340	24	v	v	NOUN
ejpam-1506	340	25	(	(	PUNCT
ejpam-1506	340	26	k(vi	k(vi	PROPN
ejpam-1506	340	27	)	)	PUNCT
ejpam-1506	340	28	)	)	PUNCT
ejpam-1506	341	1	=	=	PUNCT
ejpam-1506	341	2	{	{	PUNCT
ejpam-1506	341	3	ui,0,ui	ui,0,ui	NOUN
ejpam-1506	341	4	,	,	PUNCT
ejpam-1506	341	5	i−1,ui	i−1,ui	PROPN
ejpam-1506	341	6	,	,	PUNCT
ejpam-1506	341	7	i+1	i+1	PRON
ejpam-1506	341	8	}	}	PUNCT
ejpam-1506	341	9	⊆	⊆	NUM
ejpam-1506	341	10	v	v	NOUN
ejpam-1506	341	11	(	(	PUNCT
ejpam-1506	341	12	g⋆	g⋆	NOUN
ejpam-1506	341	13	)	)	PUNCT
ejpam-1506	341	14	,	,	PUNCT
ejpam-1506	341	15	where	where	SCONJ
ejpam-1506	341	16	the	the	DET
ejpam-1506	341	17	subscript	subscript	NOUN
ejpam-1506	341	18	of	of	ADP
ejpam-1506	341	19	u	u	NOUN
ejpam-1506	341	20	is	be	AUX
ejpam-1506	341	21	taken	take	VERB
ejpam-1506	341	22	modulo	modulo	NOUN
ejpam-1506	341	23	n	n	CCONJ
ejpam-1506	341	24	if	if	SCONJ
ejpam-1506	341	25	the	the	DET
ejpam-1506	341	26	subscript	subscript	NOUN
ejpam-1506	341	27	is	be	AUX
ejpam-1506	341	28	bigger	big	ADJ
ejpam-1506	341	29	than	than	ADP
ejpam-1506	341	30	n	n	PRON
ejpam-1506	341	31	except	except	SCONJ
ejpam-1506	341	32	when	when	SCONJ
ejpam-1506	341	33	i	i	PRON
ejpam-1506	341	34	=	=	VERB
ejpam-1506	341	35	1	1	NUM
ejpam-1506	341	36	(	(	PUNCT
ejpam-1506	341	37	we	we	PRON
ejpam-1506	341	38	take	take	VERB
ejpam-1506	341	39	u1,n	u1,n	PROPN
ejpam-1506	341	40	in	in	ADP
ejpam-1506	341	41	place	place	NOUN
ejpam-1506	341	42	of	of	ADP
ejpam-1506	341	43	ui	ui	PROPN
ejpam-1506	341	44	,	,	PUNCT
ejpam-1506	341	45	i−1	i−1	PROPN
ejpam-1506	341	46	if	if	SCONJ
ejpam-1506	341	47	i	i	PRON
ejpam-1506	341	48	=	=	NOUN
ejpam-1506	341	49	1	1	NUM
ejpam-1506	341	50	)	)	PUNCT
ejpam-1506	341	51	.	.	PUNCT
ejpam-1506	342	1	see	see	VERB
ejpam-1506	342	2	figure	figure	NOUN
ejpam-1506	342	3	4	4	NUM
ejpam-1506	342	4	for	for	ADP
ejpam-1506	342	5	the	the	DET
ejpam-1506	342	6	labelings	labeling	NOUN
ejpam-1506	342	7	of	of	ADP
ejpam-1506	342	8	w1,n	w1,n	PROPN
ejpam-1506	342	9	and	and	CCONJ
ejpam-1506	342	10	w	w	NOUN
ejpam-1506	342	11	⋆	⋆	NOUN
ejpam-1506	342	12	1,n	1,n	NOUN
ejpam-1506	342	13	;	;	PUNCT
ejpam-1506	342	14	here	here	ADV
ejpam-1506	342	15	,	,	PUNCT
ejpam-1506	342	16	the	the	DET
ejpam-1506	342	17	solid	solid	ADJ
ejpam-1506	342	18	vertices	vertex	NOUN
ejpam-1506	342	19	form	form	VERB
ejpam-1506	342	20	a	a	DET
ejpam-1506	342	21	minimum	minimum	NOUN
ejpam-1506	342	22	resolving	resolving	NOUN
ejpam-1506	342	23	set	set	VERB
ejpam-1506	342	24	for	for	ADP
ejpam-1506	342	25	w1,6	w1,6	PROPN
ejpam-1506	342	26	,	,	PUNCT
ejpam-1506	342	27	l(w1,6	l(w1,6	PROPN
ejpam-1506	342	28	)	)	PUNCT
ejpam-1506	342	29	,	,	PUNCT
ejpam-1506	342	30	and	and	CCONJ
ejpam-1506	342	31	w	w	NOUN
ejpam-1506	342	32	⋆	⋆	CCONJ
ejpam-1506	342	33	1,6	1,6	NUM
ejpam-1506	342	34	,	,	PUNCT
ejpam-1506	342	35	respectively	respectively	ADV
ejpam-1506	342	36	.	.	PUNCT
ejpam-1506	343	1	if	if	SCONJ
ejpam-1506	343	2	n	n	NUM
ejpam-1506	343	3	=	=	SYM
ejpam-1506	343	4	3	3	NUM
ejpam-1506	343	5	,	,	PUNCT
ejpam-1506	343	6	noting	note	VERB
ejpam-1506	343	7	that	that	SCONJ
ejpam-1506	343	8	w1,3	w1,3	PROPN
ejpam-1506	343	9	∼=	∼=	PART
ejpam-1506	343	10	k4	k4	NOUN
ejpam-1506	343	11	,	,	PUNCT
ejpam-1506	343	12	dim(w	dim(w	X
ejpam-1506	343	13	⋆	⋆	PUNCT
ejpam-1506	343	14	1,3	1,3	NUM
ejpam-1506	343	15	)	)	PUNCT
ejpam-1506	343	16	=	=	SYM
ejpam-1506	343	17	3	3	NUM
ejpam-1506	343	18	by	by	ADP
ejpam-1506	343	19	theorem	theorem	NOUN
ejpam-1506	343	20	12	12	NUM
ejpam-1506	343	21	.	.	PUNCT
ejpam-1506	344	1	so	so	ADV
ejpam-1506	344	2	,	,	PUNCT
ejpam-1506	344	3	we	we	PRON
ejpam-1506	344	4	consider	consider	VERB
ejpam-1506	344	5	n	n	PRON
ejpam-1506	344	6	≥	≥	NOUN
ejpam-1506	344	7	4	4	NUM
ejpam-1506	344	8	;	;	PUNCT
ejpam-1506	344	9	let	let	VERB
ejpam-1506	344	10	s	s	PRON
ejpam-1506	344	11	be	be	AUX
ejpam-1506	344	12	a	a	DET
ejpam-1506	344	13	resolving	resolving	NOUN
ejpam-1506	344	14	set	set	VERB
ejpam-1506	344	15	for	for	ADP
ejpam-1506	344	16	w	w	NOUN
ejpam-1506	344	17	⋆	⋆	NOUN
ejpam-1506	344	18	1,n	1,n	PROPN
ejpam-1506	344	19	.	.	PUNCT
ejpam-1506	345	1	we	we	PRON
ejpam-1506	345	2	make	make	VERB
ejpam-1506	345	3	the	the	DET
ejpam-1506	345	4	following	follow	VERB
ejpam-1506	345	5	claim	claim	NOUN
ejpam-1506	345	6	:	:	PUNCT
ejpam-1506	345	7	for	for	ADP
ejpam-1506	345	8	n≥	n≥	NOUN
ejpam-1506	345	9	4	4	NUM
ejpam-1506	345	10	,	,	PUNCT
ejpam-1506	345	11	there	there	PRON
ejpam-1506	345	12	exists	exist	VERB
ejpam-1506	345	13	at	at	ADP
ejpam-1506	345	14	most	most	ADJ
ejpam-1506	345	15	one	one	NUM
ejpam-1506	345	16	i	i	PRON
ejpam-1506	345	17	such	such	VERB
ejpam-1506	345	18	that	that	DET
ejpam-1506	345	19	s∩(ui∪{u0,i	s∩(ui∪{u0,i	NOUN
ejpam-1506	345	20	}	}	PUNCT
ejpam-1506	345	21	)	)	PUNCT
ejpam-1506	345	22	6=	6=	NUM
ejpam-1506	345	23	;	;	PUNCT
ejpam-1506	345	24	,	,	PUNCT
ejpam-1506	345	25	where	where	SCONJ
ejpam-1506	345	26	1≤	1≤	X
ejpam-1506	345	27	i	i	NOUN
ejpam-1506	345	28	≤	≤	X
ejpam-1506	345	29	n.	n.	NOUN
ejpam-1506	345	30	proof	proof	NOUN
ejpam-1506	345	31	of	of	ADP
ejpam-1506	345	32	claim	claim	NOUN
ejpam-1506	345	33	.	.	PUNCT
ejpam-1506	346	1	assume	assume	VERB
ejpam-1506	346	2	,	,	PUNCT
ejpam-1506	346	3	to	to	ADP
ejpam-1506	346	4	the	the	DET
ejpam-1506	346	5	contrary	contrary	NOUN
ejpam-1506	346	6	,	,	PUNCT
ejpam-1506	346	7	that	that	PRON
ejpam-1506	346	8	s	s	VERB
ejpam-1506	346	9	∩	∩	NOUN
ejpam-1506	346	10	(	(	PUNCT
ejpam-1506	346	11	ux	ux	ADV
ejpam-1506	346	12	∪	∪	X
ejpam-1506	346	13	{	{	PUNCT
ejpam-1506	346	14	u0,x	u0,x	PROPN
ejpam-1506	346	15	}	}	PUNCT
ejpam-1506	346	16	)	)	PUNCT
ejpam-1506	346	17	=	=	PUNCT
ejpam-1506	346	18	;	;	PUNCT
ejpam-1506	346	19	=	=	SYM
ejpam-1506	346	20	s	s	NOUN
ejpam-1506	346	21	∩	∩	NOUN
ejpam-1506	346	22	(	(	PUNCT
ejpam-1506	346	23	uy	uy	NOUN
ejpam-1506	346	24	∪	∪	PROPN
ejpam-1506	346	25	{	{	PUNCT
ejpam-1506	346	26	u0,y	u0,y	PROPN
ejpam-1506	346	27	}	}	PUNCT
ejpam-1506	346	28	)	)	PUNCT
ejpam-1506	346	29	for	for	ADP
ejpam-1506	346	30	two	two	NUM
ejpam-1506	346	31	distinct	distinct	ADJ
ejpam-1506	346	32	x	x	SYM
ejpam-1506	346	33	,	,	PUNCT
ejpam-1506	346	34	y	y	PROPN
ejpam-1506	346	35	,	,	PUNCT
ejpam-1506	346	36	where	where	SCONJ
ejpam-1506	346	37	1	1	NUM
ejpam-1506	346	38	≤	≤	NOUN
ejpam-1506	346	39	x	x	X
ejpam-1506	346	40	,	,	PUNCT
ejpam-1506	346	41	y	y	PROPN
ejpam-1506	346	42	≤	≤	PROPN
ejpam-1506	346	43	n.	n.	NOUN
ejpam-1506	346	44	then	then	ADV
ejpam-1506	346	45	codes(u0,x	codes(u0,x	ADJ
ejpam-1506	346	46	)	)	PUNCT
ejpam-1506	346	47	=	=	SYM
ejpam-1506	346	48	codes(u0,y	codes(u0,y	PROPN
ejpam-1506	346	49	)	)	PUNCT
ejpam-1506	346	50	,	,	PUNCT
ejpam-1506	346	51	contradicting	contradict	VERB
ejpam-1506	346	52	the	the	DET
ejpam-1506	346	53	assumption	assumption	NOUN
ejpam-1506	346	54	that	that	SCONJ
ejpam-1506	346	55	s	s	VERB
ejpam-1506	346	56	is	be	AUX
ejpam-1506	346	57	a	a	DET
ejpam-1506	346	58	resolving	resolving	NOUN
ejpam-1506	346	59	set	set	VERB
ejpam-1506	346	60	for	for	ADP
ejpam-1506	346	61	w	w	NOUN
ejpam-1506	346	62	⋆	⋆	NOUN
ejpam-1506	346	63	1,n	1,n	PROPN
ejpam-1506	346	64	.	.	PUNCT
ejpam-1506	347	1	so	so	ADV
ejpam-1506	347	2	,	,	PUNCT
ejpam-1506	347	3	there	there	PRON
ejpam-1506	347	4	exists	exist	VERB
ejpam-1506	347	5	at	at	ADP
ejpam-1506	347	6	most	most	ADJ
ejpam-1506	347	7	one	one	NUM
ejpam-1506	347	8	i	i	PRON
ejpam-1506	347	9	such	such	ADJ
ejpam-1506	347	10	that	that	DET
ejpam-1506	347	11	s	s	NOUN
ejpam-1506	347	12	∩	∩	NOUN
ejpam-1506	347	13	(	(	PUNCT
ejpam-1506	347	14	ui	ui	NOUN
ejpam-1506	347	15	∪	∪	NOUN
ejpam-1506	347	16	{	{	PUNCT
ejpam-1506	347	17	u0,i	u0,i	PROPN
ejpam-1506	347	18	}	}	PUNCT
ejpam-1506	347	19	)	)	PUNCT
ejpam-1506	347	20	6=	6=	NUM
ejpam-1506	347	21	;	;	PUNCT
ejpam-1506	347	22	.	.	PUNCT
ejpam-1506	348	1	d.	d.	PROPN
ejpam-1506	348	2	klein	klein	PROPN
ejpam-1506	348	3	,	,	PUNCT
ejpam-1506	348	4	e.	e.	PROPN
ejpam-1506	348	5	yi	yi	PROPN
ejpam-1506	348	6	/	/	SYM
ejpam-1506	348	7	eur	eur	PROPN
ejpam-1506	348	8	.	.	PUNCT
ejpam-1506	349	1	j.	j.	PROPN
ejpam-1506	349	2	pure	pure	PROPN
ejpam-1506	349	3	appl	appl	PROPN
ejpam-1506	349	4	.	.	PROPN
ejpam-1506	349	5	math	math	PROPN
ejpam-1506	349	6	,	,	PUNCT
ejpam-1506	349	7	5	5	NUM
ejpam-1506	349	8	(	(	PUNCT
ejpam-1506	349	9	2012	2012	NUM
ejpam-1506	349	10	)	)	PUNCT
ejpam-1506	349	11	,	,	PUNCT
ejpam-1506	349	12	302	302	NUM
ejpam-1506	349	13	-	-	SYM
ejpam-1506	349	14	316	316	NUM
ejpam-1506	349	15	312	312	NUM
ejpam-1506	349	16	(	(	PUNCT
ejpam-1506	349	17	a	a	NOUN
ejpam-1506	349	18	)	)	PUNCT
ejpam-1506	349	19	w1,6	w1,6	PROPN
ejpam-1506	349	20	(	(	PUNCT
ejpam-1506	349	21	b	b	NOUN
ejpam-1506	349	22	)	)	PUNCT
ejpam-1506	349	23	l(w1,6	l(w1,6	PROPN
ejpam-1506	349	24	)	)	PUNCT
ejpam-1506	349	25	(	(	PUNCT
ejpam-1506	349	26	c	c	X
ejpam-1506	349	27	)	)	PUNCT
ejpam-1506	349	28	w	w	NOUN
ejpam-1506	349	29	⋆	⋆	VERB
ejpam-1506	349	30	1,6	1,6	NUM
ejpam-1506	349	31	figure	figure	NOUN
ejpam-1506	349	32	4	4	NUM
ejpam-1506	349	33	:	:	PUNCT
ejpam-1506	349	34	labelings	labeling	NOUN
ejpam-1506	349	35	of	of	ADP
ejpam-1506	349	36	w1,n	w1,n	PROPN
ejpam-1506	349	37	and	and	CCONJ
ejpam-1506	349	38	w	w	NOUN
ejpam-1506	349	39	⋆	⋆	NOUN
ejpam-1506	349	40	1,n	1,n	NUM
ejpam-1506	349	41	;	;	PUNCT
ejpam-1506	349	42	dim(w1,6	dim(w1,6	PROPN
ejpam-1506	349	43	)	)	PUNCT
ejpam-1506	349	44	=	=	SYM
ejpam-1506	349	45	3,dim(l(w1,6	3,dim(l(w1,6	NUM
ejpam-1506	349	46	)	)	PUNCT
ejpam-1506	349	47	)	)	PUNCT
ejpam-1506	350	1	=	=	SYM
ejpam-1506	351	1	4,dim(w	4,dim(w	NUM
ejpam-1506	351	2	⋆	⋆	VERB
ejpam-1506	351	3	1,6	1,6	NUM
ejpam-1506	351	4	)	)	PUNCT
ejpam-1506	351	5	=	=	SYM
ejpam-1506	351	6	5	5	NUM
ejpam-1506	351	7	by	by	ADP
ejpam-1506	351	8	claim	claim	NOUN
ejpam-1506	351	9	,	,	PUNCT
ejpam-1506	351	10	|s|	|s|	PROPN
ejpam-1506	351	11	≥	≥	NOUN
ejpam-1506	351	12	n	n	CCONJ
ejpam-1506	351	13	−	−	PROPN
ejpam-1506	351	14	1	1	NUM
ejpam-1506	351	15	,	,	PUNCT
ejpam-1506	351	16	and	and	CCONJ
ejpam-1506	351	17	thus	thus	ADV
ejpam-1506	351	18	dim(w	dim(w	X
ejpam-1506	351	19	⋆	⋆	X
ejpam-1506	351	20	1,n	1,n	NUM
ejpam-1506	351	21	)	)	PUNCT
ejpam-1506	351	22	≥	≥	NOUN
ejpam-1506	351	23	n	n	CCONJ
ejpam-1506	351	24	−	−	PROPN
ejpam-1506	351	25	1	1	NUM
ejpam-1506	351	26	for	for	ADP
ejpam-1506	351	27	n	n	X
ejpam-1506	351	28	≥	≥	NOUN
ejpam-1506	351	29	4	4	NUM
ejpam-1506	351	30	.	.	PUNCT
ejpam-1506	352	1	on	on	ADP
ejpam-1506	352	2	the	the	DET
ejpam-1506	352	3	other	other	ADJ
ejpam-1506	352	4	hand	hand	NOUN
ejpam-1506	352	5	,	,	PUNCT
ejpam-1506	352	6	dim(w	dim(w	X
ejpam-1506	352	7	⋆	⋆	VERB
ejpam-1506	352	8	1,n)≤	1,n)≤	PROPN
ejpam-1506	352	9	n−	n−	NOUN
ejpam-1506	352	10	1	1	NUM
ejpam-1506	352	11	by	by	ADP
ejpam-1506	352	12	(	(	PUNCT
ejpam-1506	352	13	b	b	NOUN
ejpam-1506	352	14	)	)	PUNCT
ejpam-1506	352	15	of	of	ADP
ejpam-1506	352	16	theorem	theorem	ADJ
ejpam-1506	352	17	8‡.	8‡.	NUM
ejpam-1506	352	18	therefore	therefore	ADV
ejpam-1506	352	19	,	,	PUNCT
ejpam-1506	352	20	dim(w	dim(w	X
ejpam-1506	352	21	⋆	⋆	X
ejpam-1506	352	22	1,n	1,n	NUM
ejpam-1506	352	23	)	)	PUNCT
ejpam-1506	352	24	=	=	PUNCT
ejpam-1506	352	25	n−	n−	NOUN
ejpam-1506	352	26	1	1	NUM
ejpam-1506	352	27	for	for	ADP
ejpam-1506	352	28	n≥	n≥	NOUN
ejpam-1506	352	29	4	4	NUM
ejpam-1506	352	30	.	.	PUNCT
ejpam-1506	353	1	it	it	PRON
ejpam-1506	353	2	is	be	AUX
ejpam-1506	353	3	well	well	ADV
ejpam-1506	353	4	known	know	VERB
ejpam-1506	353	5	that	that	SCONJ
ejpam-1506	353	6	dim(cn	dim(cn	NOUN
ejpam-1506	353	7	)	)	PUNCT
ejpam-1506	353	8	=	=	SYM
ejpam-1506	353	9	2	2	NUM
ejpam-1506	353	10	for	for	ADP
ejpam-1506	353	11	n≥	n≥	NOUN
ejpam-1506	353	12	3	3	NUM
ejpam-1506	353	13	.	.	PUNCT
ejpam-1506	354	1	let	let	VERB
ejpam-1506	354	2	bn	bn	INTJ
ejpam-1506	354	3	=	=	SYM
ejpam-1506	354	4	(	(	PUNCT
ejpam-1506	354	5	k1	k1	PROPN
ejpam-1506	354	6	,	,	PUNCT
ejpam-1506	354	7	k2	k2	NOUN
ejpam-1506	354	8	,	,	PUNCT
ejpam-1506	354	9	.	.	PUNCT
ejpam-1506	354	10	.	.	PUNCT
ejpam-1506	355	1	.	.	PUNCT
ejpam-1506	356	1	,	,	PUNCT
ejpam-1506	356	2	kn	kn	PROPN
ejpam-1506	356	3	)	)	PUNCT
ejpam-1506	356	4	be	be	VERB
ejpam-1506	356	5	a	a	DET
ejpam-1506	356	6	bouquet	bouquet	NOUN
ejpam-1506	356	7	of	of	ADP
ejpam-1506	356	8	n≥	n≥	NOUN
ejpam-1506	356	9	2	2	NUM
ejpam-1506	356	10	circles	circle	NOUN
ejpam-1506	356	11	c1	c1	NOUN
ejpam-1506	356	12	,	,	PUNCT
ejpam-1506	356	13	c2	c2	PROPN
ejpam-1506	356	14	,	,	PUNCT
ejpam-1506	356	15	.	.	PUNCT
ejpam-1506	356	16	.	.	PUNCT
ejpam-1506	357	1	.	.	PUNCT
ejpam-1506	358	1	,	,	PUNCT
ejpam-1506	358	2	cn	cn	PROPN
ejpam-1506	358	3	,	,	PUNCT
ejpam-1506	358	4	with	with	ADP
ejpam-1506	358	5	a	a	DET
ejpam-1506	358	6	cut	cut	NOUN
ejpam-1506	358	7	-	-	PUNCT
ejpam-1506	358	8	vertex	vertex	NOUN
ejpam-1506	358	9	,	,	PUNCT
ejpam-1506	358	10	where	where	SCONJ
ejpam-1506	358	11	ki	ki	PROPN
ejpam-1506	358	12	is	be	AUX
ejpam-1506	358	13	the	the	DET
ejpam-1506	358	14	number	number	NOUN
ejpam-1506	358	15	of	of	ADP
ejpam-1506	358	16	vertices	vertex	NOUN
ejpam-1506	358	17	of	of	ADP
ejpam-1506	358	18	c	c	NOUN
ejpam-1506	358	19	i	i	PRON
ejpam-1506	358	20	(	(	PUNCT
ejpam-1506	358	21	1≤	1≤	INTJ
ejpam-1506	358	22	i	i	NOUN
ejpam-1506	358	23	≤	≤	PROPN
ejpam-1506	358	24	n	n	CCONJ
ejpam-1506	358	25	)	)	PUNCT
ejpam-1506	358	26	.	.	PUNCT
ejpam-1506	359	1	see	see	VERB
ejpam-1506	359	2	figure	figure	NOUN
ejpam-1506	359	3	5	5	NUM
ejpam-1506	359	4	for	for	ADP
ejpam-1506	359	5	b4	b4	NOUN
ejpam-1506	359	6	=	=	SYM
ejpam-1506	359	7	(	(	PUNCT
ejpam-1506	359	8	3,4,5,6	3,4,5,6	NUM
ejpam-1506	359	9	)	)	PUNCT
ejpam-1506	359	10	and	and	CCONJ
ejpam-1506	359	11	its	its	PRON
ejpam-1506	359	12	line	line	NOUN
ejpam-1506	359	13	graph	graph	NOUN
ejpam-1506	359	14	.	.	PUNCT
ejpam-1506	360	1	we	we	PRON
ejpam-1506	360	2	recall	recall	VERB
ejpam-1506	360	3	the	the	DET
ejpam-1506	360	4	metric	metric	ADJ
ejpam-1506	360	5	dimension	dimension	NOUN
ejpam-1506	360	6	of	of	ADP
ejpam-1506	360	7	a	a	DET
ejpam-1506	360	8	bouquet	bouquet	NOUN
ejpam-1506	360	9	of	of	ADP
ejpam-1506	360	10	circles	circle	NOUN
ejpam-1506	360	11	and	and	CCONJ
ejpam-1506	360	12	its	its	PRON
ejpam-1506	360	13	line	line	NOUN
ejpam-1506	360	14	graph	graph	NOUN
ejpam-1506	360	15	.	.	PUNCT
ejpam-1506	361	1	(	(	PUNCT
ejpam-1506	361	2	a	a	X
ejpam-1506	361	3	)	)	PUNCT
ejpam-1506	361	4	b4	b4	NOUN
ejpam-1506	361	5	=	=	SYM
ejpam-1506	361	6	(	(	PUNCT
ejpam-1506	361	7	3	3	NUM
ejpam-1506	361	8	,	,	PUNCT
ejpam-1506	361	9	4	4	NUM
ejpam-1506	361	10	,	,	PUNCT
ejpam-1506	361	11	5	5	NUM
ejpam-1506	361	12	,	,	PUNCT
ejpam-1506	361	13	6	6	NUM
ejpam-1506	361	14	)	)	PUNCT
ejpam-1506	361	15	(	(	PUNCT
ejpam-1506	361	16	b	b	X
ejpam-1506	361	17	)	)	PUNCT
ejpam-1506	361	18	l(b4	l(b4	ADJ
ejpam-1506	361	19	)	)	PUNCT
ejpam-1506	361	20	figure	figure	NOUN
ejpam-1506	361	21	5	5	NUM
ejpam-1506	361	22	:	:	PUNCT
ejpam-1506	361	23	a	a	DET
ejpam-1506	361	24	bouquet	bouquet	NOUN
ejpam-1506	361	25	of	of	ADP
ejpam-1506	361	26	four	four	NUM
ejpam-1506	361	27	circles	circle	NOUN
ejpam-1506	361	28	b4	b4	NOUN
ejpam-1506	361	29	=	=	SYM
ejpam-1506	361	30	(	(	PUNCT
ejpam-1506	361	31	3,4,5,6	3,4,5,6	NUM
ejpam-1506	361	32	)	)	PUNCT
ejpam-1506	361	33	and	and	CCONJ
ejpam-1506	361	34	its	its	PRON
ejpam-1506	361	35	line	line	NOUN
ejpam-1506	361	36	graph	graph	NOUN
ejpam-1506	361	37	theorem	theorem	VERB
ejpam-1506	361	38	18	18	NUM
ejpam-1506	361	39	.	.	PUNCT
ejpam-1506	362	1	[	[	X
ejpam-1506	362	2	16	16	NUM
ejpam-1506	362	3	]	]	PUNCT
ejpam-1506	362	4	let	let	VERB
ejpam-1506	362	5	bn	bn	INTJ
ejpam-1506	362	6	=	=	SYM
ejpam-1506	362	7	(	(	PUNCT
ejpam-1506	362	8	k1	k1	PROPN
ejpam-1506	362	9	,	,	PUNCT
ejpam-1506	362	10	k2	k2	NOUN
ejpam-1506	362	11	,	,	PUNCT
ejpam-1506	362	12	.	.	PUNCT
ejpam-1506	362	13	.	.	PUNCT
ejpam-1506	362	14	.	.	PUNCT
ejpam-1506	363	1	,	,	PUNCT
ejpam-1506	363	2	kn	kn	PROPN
ejpam-1506	363	3	)	)	PUNCT
ejpam-1506	363	4	be	be	VERB
ejpam-1506	363	5	a	a	DET
ejpam-1506	363	6	bouquet	bouquet	NOUN
ejpam-1506	363	7	of	of	ADP
ejpam-1506	363	8	n≥	n≥	NOUN
ejpam-1506	363	9	2	2	NUM
ejpam-1506	363	10	circles	circle	NOUN
ejpam-1506	363	11	with	with	ADP
ejpam-1506	363	12	a	a	DET
ejpam-1506	363	13	cut	cut	NOUN
ejpam-1506	363	14	-	-	PUNCT
ejpam-1506	363	15	vertex	vertex	NOUN
ejpam-1506	363	16	.	.	PUNCT
ejpam-1506	364	1	if	if	SCONJ
ejpam-1506	364	2	x	x	PRON
ejpam-1506	364	3	is	be	AUX
ejpam-1506	364	4	the	the	DET
ejpam-1506	364	5	number	number	NOUN
ejpam-1506	364	6	of	of	ADP
ejpam-1506	364	7	even	even	ADJ
ejpam-1506	364	8	cycles	cycle	NOUN
ejpam-1506	364	9	of	of	ADP
ejpam-1506	364	10	bn	bn	NOUN
ejpam-1506	364	11	,	,	PUNCT
ejpam-1506	364	12	then	then	ADV
ejpam-1506	364	13	dim(bn	dim(bn	X
ejpam-1506	364	14	)	)	PUNCT
ejpam-1506	364	15	=	=	PUNCT
ejpam-1506	365	1	(	(	PUNCT
ejpam-1506	365	2	n	n	CCONJ
ejpam-1506	365	3	if	if	SCONJ
ejpam-1506	365	4	x	x	SYM
ejpam-1506	365	5	=	=	SYM
ejpam-1506	365	6	0	0	NUM
ejpam-1506	365	7	n+	n+	NOUN
ejpam-1506	365	8	x	x	PUNCT
ejpam-1506	365	9	−	−	NOUN
ejpam-1506	365	10	1	1	NUM
ejpam-1506	365	11	if	if	SCONJ
ejpam-1506	365	12	x	x	X
ejpam-1506	365	13	≥	≥	NUM
ejpam-1506	365	14	1	1	NUM
ejpam-1506	365	15	.	.	PUNCT
ejpam-1506	365	16	theorem	theorem	VERB
ejpam-1506	365	17	19	19	NUM
ejpam-1506	365	18	.	.	PUNCT
ejpam-1506	366	1	[	[	X
ejpam-1506	366	2	8	8	NUM
ejpam-1506	366	3	]	]	PUNCT
ejpam-1506	366	4	let	let	VERB
ejpam-1506	366	5	bn	bn	INTJ
ejpam-1506	366	6	=	=	SYM
ejpam-1506	366	7	(	(	PUNCT
ejpam-1506	366	8	k1	k1	PROPN
ejpam-1506	366	9	,	,	PUNCT
ejpam-1506	366	10	k2	k2	NOUN
ejpam-1506	366	11	,	,	PUNCT
ejpam-1506	366	12	.	.	PUNCT
ejpam-1506	366	13	.	.	PUNCT
ejpam-1506	367	1	.	.	PUNCT
ejpam-1506	368	1	,	,	PUNCT
ejpam-1506	368	2	kn	kn	PROPN
ejpam-1506	368	3	)	)	PUNCT
ejpam-1506	368	4	be	be	VERB
ejpam-1506	368	5	a	a	DET
ejpam-1506	368	6	bouquet	bouquet	NOUN
ejpam-1506	368	7	of	of	ADP
ejpam-1506	368	8	n≥	n≥	NOUN
ejpam-1506	368	9	2	2	NUM
ejpam-1506	368	10	circles	circle	NOUN
ejpam-1506	368	11	with	with	ADP
ejpam-1506	368	12	a	a	DET
ejpam-1506	368	13	cut	cut	NOUN
ejpam-1506	368	14	-	-	PUNCT
ejpam-1506	368	15	vertex	vertex	NOUN
ejpam-1506	368	16	.	.	PUNCT
ejpam-1506	369	1	then	then	ADV
ejpam-1506	369	2	dim(l(bn	dim(l(bn	NOUN
ejpam-1506	369	3	)	)	PUNCT
ejpam-1506	369	4	)	)	PUNCT
ejpam-1506	370	1	=	=	PUNCT
ejpam-1506	371	1	2n−	2n−	NUM
ejpam-1506	371	2	1	1	NUM
ejpam-1506	371	3	.	.	PUNCT
ejpam-1506	372	1	as	as	ADP
ejpam-1506	372	2	an	an	DET
ejpam-1506	372	3	immediate	immediate	ADJ
ejpam-1506	372	4	consequence	consequence	NOUN
ejpam-1506	372	5	of	of	ADP
ejpam-1506	372	6	theorem	theorem	NOUN
ejpam-1506	372	7	19	19	NUM
ejpam-1506	372	8	,	,	PUNCT
ejpam-1506	372	9	we	we	PRON
ejpam-1506	372	10	have	have	AUX
ejpam-1506	372	11	the	the	DET
ejpam-1506	372	12	following	follow	VERB
ejpam-1506	372	13	‡one	‡one	NOUN
ejpam-1506	372	14	can	can	AUX
ejpam-1506	372	15	readily	readily	ADV
ejpam-1506	372	16	check	check	VERB
ejpam-1506	372	17	that	that	PRON
ejpam-1506	372	18	s	s	VERB
ejpam-1506	372	19	=	=	PUNCT
ejpam-1506	372	20	{	{	PUNCT
ejpam-1506	372	21	ui	ui	NOUN
ejpam-1506	372	22	,	,	PUNCT
ejpam-1506	372	23	i+1	i+1	PUNCT
ejpam-1506	372	24	|	|	ADV
ejpam-1506	372	25	1	1	NUM
ejpam-1506	372	26	≤	≤	NUM
ejpam-1506	372	27	i	i	PRON
ejpam-1506	372	28	≤	≤	ADJ
ejpam-1506	372	29	n−	n−	NOUN
ejpam-1506	372	30	1	1	NUM
ejpam-1506	372	31	}	}	PUNCT
ejpam-1506	372	32	forms	form	VERB
ejpam-1506	372	33	a	a	DET
ejpam-1506	372	34	resolving	resolving	NOUN
ejpam-1506	372	35	set	set	VERB
ejpam-1506	372	36	for	for	ADP
ejpam-1506	372	37	w	w	PROPN
ejpam-1506	372	38	⋆	⋆	NOUN
ejpam-1506	372	39	1,n	1,n	PROPN
ejpam-1506	372	40	.	.	PUNCT
ejpam-1506	373	1	d.	d.	PROPN
ejpam-1506	373	2	klein	klein	PROPN
ejpam-1506	373	3	,	,	PUNCT
ejpam-1506	373	4	e.	e.	PROPN
ejpam-1506	373	5	yi	yi	PROPN
ejpam-1506	373	6	/	/	SYM
ejpam-1506	373	7	eur	eur	PROPN
ejpam-1506	373	8	.	.	PUNCT
ejpam-1506	374	1	j.	j.	PROPN
ejpam-1506	374	2	pure	pure	PROPN
ejpam-1506	374	3	appl	appl	PROPN
ejpam-1506	374	4	.	.	PROPN
ejpam-1506	374	5	math	math	PROPN
ejpam-1506	374	6	,	,	PUNCT
ejpam-1506	374	7	5	5	NUM
ejpam-1506	374	8	(	(	PUNCT
ejpam-1506	374	9	2012	2012	NUM
ejpam-1506	374	10	)	)	PUNCT
ejpam-1506	374	11	,	,	PUNCT
ejpam-1506	374	12	302	302	NUM
ejpam-1506	374	13	-	-	SYM
ejpam-1506	374	14	316	316	NUM
ejpam-1506	374	15	313	313	NUM
ejpam-1506	374	16	corollary	corollary	NOUN
ejpam-1506	374	17	2	2	NUM
ejpam-1506	374	18	.	.	PUNCT
ejpam-1506	375	1	let	let	VERB
ejpam-1506	375	2	bn	bn	INTJ
ejpam-1506	375	3	=	=	SYM
ejpam-1506	375	4	(	(	PUNCT
ejpam-1506	375	5	k1	k1	PROPN
ejpam-1506	375	6	,	,	PUNCT
ejpam-1506	375	7	k2	k2	NOUN
ejpam-1506	375	8	,	,	PUNCT
ejpam-1506	375	9	.	.	PUNCT
ejpam-1506	375	10	.	.	PUNCT
ejpam-1506	376	1	.	.	PUNCT
ejpam-1506	377	1	,	,	PUNCT
ejpam-1506	377	2	kn	kn	PROPN
ejpam-1506	377	3	)	)	PUNCT
ejpam-1506	377	4	be	be	VERB
ejpam-1506	377	5	a	a	DET
ejpam-1506	377	6	bouquet	bouquet	NOUN
ejpam-1506	377	7	of	of	ADP
ejpam-1506	377	8	n≥	n≥	NOUN
ejpam-1506	377	9	2	2	NUM
ejpam-1506	377	10	circles	circle	NOUN
ejpam-1506	377	11	with	with	ADP
ejpam-1506	377	12	a	a	DET
ejpam-1506	377	13	cut	cut	NOUN
ejpam-1506	377	14	-	-	PUNCT
ejpam-1506	377	15	vertex	vertex	NOUN
ejpam-1506	377	16	.	.	PUNCT
ejpam-1506	378	1	then	then	ADV
ejpam-1506	378	2	dim(b⋆n	dim(b⋆n	NOUN
ejpam-1506	378	3	)	)	PUNCT
ejpam-1506	378	4	=	=	PUNCT
ejpam-1506	379	1	2n−	2n−	NUM
ejpam-1506	379	2	1	1	NUM
ejpam-1506	379	3	.	.	NOUN
ejpam-1506	379	4	5	5	NUM
ejpam-1506	379	5	.	.	X
ejpam-1506	379	6	metric	metric	ADJ
ejpam-1506	379	7	dimension	dimension	NOUN
ejpam-1506	379	8	of	of	ADP
ejpam-1506	379	9	graphs	graph	NOUN
ejpam-1506	379	10	,	,	PUNCT
ejpam-1506	379	11	line	line	NOUN
ejpam-1506	379	12	graphs	graph	NOUN
ejpam-1506	379	13	,	,	PUNCT
ejpam-1506	379	14	and	and	CCONJ
ejpam-1506	379	15	para	para	NOUN
ejpam-1506	379	16	-	-	PUNCT
ejpam-1506	379	17	line	line	NOUN
ejpam-1506	379	18	graphs	graph	NOUN
ejpam-1506	379	19	in	in	ADP
ejpam-1506	379	20	this	this	DET
ejpam-1506	379	21	section	section	NOUN
ejpam-1506	379	22	,	,	PUNCT
ejpam-1506	379	23	we	we	PRON
ejpam-1506	379	24	compare	compare	VERB
ejpam-1506	379	25	metric	metric	ADJ
ejpam-1506	379	26	dimension	dimension	NOUN
ejpam-1506	379	27	of	of	ADP
ejpam-1506	379	28	a	a	DET
ejpam-1506	379	29	connected	connected	ADJ
ejpam-1506	379	30	graph	graph	NOUN
ejpam-1506	379	31	g	g	NOUN
ejpam-1506	379	32	,	,	PUNCT
ejpam-1506	379	33	its	its	PRON
ejpam-1506	379	34	line	line	NOUN
ejpam-1506	379	35	graph	graph	NOUN
ejpam-1506	379	36	l(g	l(g	NOUN
ejpam-1506	379	37	)	)	PUNCT
ejpam-1506	379	38	,	,	PUNCT
ejpam-1506	379	39	and	and	CCONJ
ejpam-1506	379	40	its	its	PRON
ejpam-1506	379	41	para	para	ADJ
ejpam-1506	379	42	-	-	PUNCT
ejpam-1506	379	43	line	line	NOUN
ejpam-1506	379	44	graph	graph	NOUN
ejpam-1506	379	45	g⋆	g⋆	NOUN
ejpam-1506	379	46	:	:	PUNCT
ejpam-1506	379	47	we	we	PRON
ejpam-1506	379	48	give	give	VERB
ejpam-1506	379	49	an	an	DET
ejpam-1506	379	50	example	example	NOUN
ejpam-1506	379	51	of	of	ADP
ejpam-1506	379	52	a	a	DET
ejpam-1506	379	53	graph	graph	NOUN
ejpam-1506	379	54	g	g	ADP
ejpam-1506	379	55	such	such	ADJ
ejpam-1506	379	56	that	that	DET
ejpam-1506	379	57	max{dim(g	max{dim(g	NOUN
ejpam-1506	379	58	)	)	PUNCT
ejpam-1506	379	59	,	,	PUNCT
ejpam-1506	379	60	dim(l(g	dim(l(g	NOUN
ejpam-1506	379	61	)	)	PUNCT
ejpam-1506	379	62	)	)	PUNCT
ejpam-1506	379	63	,	,	PUNCT
ejpam-1506	379	64	dim(g⋆	dim(g⋆	NOUN
ejpam-1506	379	65	)	)	PUNCT
ejpam-1506	379	66	}	}	PUNCT
ejpam-1506	379	67	equals	equal	VERB
ejpam-1506	379	68	dim(g	dim(g	PROPN
ejpam-1506	379	69	)	)	PUNCT
ejpam-1506	379	70	,	,	PUNCT
ejpam-1506	379	71	dim(l(g	dim(l(g	NOUN
ejpam-1506	379	72	)	)	PUNCT
ejpam-1506	379	73	)	)	PUNCT
ejpam-1506	379	74	,	,	PUNCT
ejpam-1506	379	75	and	and	CCONJ
ejpam-1506	379	76	dim(g⋆	dim(g⋆	NOUN
ejpam-1506	379	77	)	)	PUNCT
ejpam-1506	379	78	,	,	PUNCT
ejpam-1506	379	79	respectively	respectively	ADV
ejpam-1506	379	80	.	.	PUNCT
ejpam-1506	380	1	remark	remark	PROPN
ejpam-1506	380	2	1	1	NUM
ejpam-1506	380	3	.	.	PUNCT
ejpam-1506	381	1	there	there	PRON
ejpam-1506	381	2	exists	exist	VERB
ejpam-1506	381	3	a	a	DET
ejpam-1506	381	4	graph	graph	NOUN
ejpam-1506	381	5	g	g	NOUN
ejpam-1506	381	6	with	with	ADP
ejpam-1506	381	7	max{dim(g	max{dim(g	PROPN
ejpam-1506	381	8	)	)	PUNCT
ejpam-1506	381	9	,	,	PUNCT
ejpam-1506	381	10	dim(l(g	dim(l(g	NOUN
ejpam-1506	381	11	)	)	PUNCT
ejpam-1506	381	12	)	)	PUNCT
ejpam-1506	381	13	,	,	PUNCT
ejpam-1506	381	14	dim(g⋆	dim(g⋆	NOUN
ejpam-1506	381	15	)	)	PUNCT
ejpam-1506	381	16	}	}	PUNCT
ejpam-1506	381	17	=	=	SYM
ejpam-1506	381	18	dim(g	dim(g	PROPN
ejpam-1506	381	19	)	)	PUNCT
ejpam-1506	381	20	.	.	PUNCT
ejpam-1506	382	1	if	if	SCONJ
ejpam-1506	382	2	g	g	NOUN
ejpam-1506	382	3	=	=	SYM
ejpam-1506	382	4	ks,2s	ks,2s	PROPN
ejpam-1506	382	5	for	for	ADP
ejpam-1506	382	6	s	s	PRON
ejpam-1506	382	7	≥	≥	NOUN
ejpam-1506	382	8	2	2	NUM
ejpam-1506	382	9	,	,	PUNCT
ejpam-1506	382	10	then	then	ADV
ejpam-1506	382	11	dim(g	dim(g	PROPN
ejpam-1506	382	12	)	)	PUNCT
ejpam-1506	382	13	=	=	PUNCT
ejpam-1506	383	1	3s−	3s−	NUM
ejpam-1506	383	2	2	2	NUM
ejpam-1506	383	3	,	,	PUNCT
ejpam-1506	383	4	dim(l(g	dim(l(g	NOUN
ejpam-1506	383	5	)	)	PUNCT
ejpam-1506	383	6	)	)	PUNCT
ejpam-1506	384	1	=	=	PUNCT
ejpam-1506	385	1	2s−	2s−	NUM
ejpam-1506	385	2	1	1	NUM
ejpam-1506	385	3	,	,	PUNCT
ejpam-1506	385	4	and	and	CCONJ
ejpam-1506	385	5	dim(g⋆	dim(g⋆	NOUN
ejpam-1506	385	6	)	)	PUNCT
ejpam-1506	385	7	≤	≤	NUM
ejpam-1506	386	1	3s−	3s−	NUM
ejpam-1506	386	2	3	3	NUM
ejpam-1506	386	3	;	;	PUNCT
ejpam-1506	386	4	further	far	ADV
ejpam-1506	386	5	,	,	PUNCT
ejpam-1506	386	6	notice	notice	VERB
ejpam-1506	386	7	that	that	SCONJ
ejpam-1506	386	8	dim(g)−	dim(g)−	ADJ
ejpam-1506	386	9	dim(l(g	dim(l(g	NOUN
ejpam-1506	386	10	)	)	PUNCT
ejpam-1506	386	11	)	)	PUNCT
ejpam-1506	386	12	can	can	AUX
ejpam-1506	386	13	be	be	AUX
ejpam-1506	386	14	arbitrarily	arbitrarily	ADV
ejpam-1506	386	15	large	large	ADJ
ejpam-1506	386	16	.	.	PUNCT
ejpam-1506	387	1	remark	remark	NOUN
ejpam-1506	387	2	2	2	NUM
ejpam-1506	387	3	.	.	PUNCT
ejpam-1506	388	1	there	there	PRON
ejpam-1506	388	2	exists	exist	VERB
ejpam-1506	388	3	a	a	DET
ejpam-1506	388	4	graph	graph	NOUN
ejpam-1506	388	5	g	g	NOUN
ejpam-1506	388	6	with	with	ADP
ejpam-1506	388	7	max{dim(g	max{dim(g	PROPN
ejpam-1506	388	8	)	)	PUNCT
ejpam-1506	388	9	,	,	PUNCT
ejpam-1506	388	10	dim(l(g	dim(l(g	NOUN
ejpam-1506	388	11	)	)	PUNCT
ejpam-1506	388	12	)	)	PUNCT
ejpam-1506	388	13	,	,	PUNCT
ejpam-1506	388	14	dim(g⋆	dim(g⋆	NOUN
ejpam-1506	388	15	)	)	PUNCT
ejpam-1506	388	16	}	}	PUNCT
ejpam-1506	388	17	=	=	SYM
ejpam-1506	388	18	dim(l(g	dim(l(g	NOUN
ejpam-1506	388	19	)	)	PUNCT
ejpam-1506	388	20	)	)	PUNCT
ejpam-1506	388	21	.	.	PUNCT
ejpam-1506	389	1	if	if	SCONJ
ejpam-1506	389	2	g	g	PROPN
ejpam-1506	389	3	=	=	SYM
ejpam-1506	389	4	bn	bn	PROPN
ejpam-1506	389	5	,	,	PUNCT
ejpam-1506	389	6	a	a	DET
ejpam-1506	389	7	bouquet	bouquet	NOUN
ejpam-1506	389	8	of	of	ADP
ejpam-1506	389	9	n≥	n≥	NOUN
ejpam-1506	389	10	2	2	NUM
ejpam-1506	389	11	circles	circle	NOUN
ejpam-1506	389	12	,	,	PUNCT
ejpam-1506	389	13	containing	contain	VERB
ejpam-1506	389	14	no	no	DET
ejpam-1506	389	15	even	even	ADV
ejpam-1506	389	16	cycles	cycle	NOUN
ejpam-1506	389	17	,	,	PUNCT
ejpam-1506	389	18	then	then	ADV
ejpam-1506	389	19	dim(g	dim(g	PROPN
ejpam-1506	389	20	)	)	PUNCT
ejpam-1506	389	21	=	=	SYM
ejpam-1506	389	22	n	n	NOUN
ejpam-1506	389	23	and	and	CCONJ
ejpam-1506	389	24	dim(l(g	dim(l(g	NOUN
ejpam-1506	389	25	)	)	PUNCT
ejpam-1506	389	26	)	)	PUNCT
ejpam-1506	390	1	=	=	SYM
ejpam-1506	390	2	2n−1=	2n−1=	NUM
ejpam-1506	390	3	dim(g⋆	dim(g⋆	NOUN
ejpam-1506	390	4	)	)	PUNCT
ejpam-1506	390	5	;	;	PUNCT
ejpam-1506	390	6	further	far	ADV
ejpam-1506	390	7	,	,	PUNCT
ejpam-1506	390	8	notice	notice	VERB
ejpam-1506	390	9	that	that	SCONJ
ejpam-1506	390	10	dim(l(g))−dim(g	dim(l(g))−dim(g	PROPN
ejpam-1506	390	11	)	)	PUNCT
ejpam-1506	390	12	and	and	CCONJ
ejpam-1506	390	13	dim(g⋆)−dim(g	dim(g⋆)−dim(g	PROPN
ejpam-1506	390	14	)	)	PUNCT
ejpam-1506	390	15	can	can	AUX
ejpam-1506	390	16	be	be	AUX
ejpam-1506	390	17	arbitrarily	arbitrarily	ADV
ejpam-1506	390	18	large	large	ADJ
ejpam-1506	390	19	.	.	PUNCT
ejpam-1506	391	1	remark	remark	NOUN
ejpam-1506	391	2	3	3	NUM
ejpam-1506	391	3	.	.	PUNCT
ejpam-1506	392	1	there	there	PRON
ejpam-1506	392	2	exists	exist	VERB
ejpam-1506	392	3	a	a	DET
ejpam-1506	392	4	graph	graph	NOUN
ejpam-1506	392	5	g	g	NOUN
ejpam-1506	392	6	with	with	ADP
ejpam-1506	392	7	max{dim(g	max{dim(g	PROPN
ejpam-1506	392	8	)	)	PUNCT
ejpam-1506	392	9	,	,	PUNCT
ejpam-1506	392	10	dim(l(g	dim(l(g	NOUN
ejpam-1506	392	11	)	)	PUNCT
ejpam-1506	392	12	)	)	PUNCT
ejpam-1506	392	13	,	,	PUNCT
ejpam-1506	392	14	dim(g⋆	dim(g⋆	NOUN
ejpam-1506	392	15	)	)	PUNCT
ejpam-1506	392	16	}	}	PUNCT
ejpam-1506	392	17	=	=	SYM
ejpam-1506	392	18	dim(g⋆	dim(g⋆	NOUN
ejpam-1506	392	19	)	)	PUNCT
ejpam-1506	392	20	.	.	PUNCT
ejpam-1506	393	1	if	if	SCONJ
ejpam-1506	393	2	g	g	PROPN
ejpam-1506	393	3	=	=	SYM
ejpam-1506	393	4	w1,n	w1,n	PROPN
ejpam-1506	393	5	for	for	ADP
ejpam-1506	393	6	n	n	PRON
ejpam-1506	393	7	≥	≥	NUM
ejpam-1506	393	8	7	7	NUM
ejpam-1506	393	9	,	,	PUNCT
ejpam-1506	393	10	then	then	ADV
ejpam-1506	393	11	dim(g	dim(g	PROPN
ejpam-1506	393	12	)	)	PUNCT
ejpam-1506	394	1	=	=	SYM
ejpam-1506	394	2	⌊2n+2	⌊2n+2	PROPN
ejpam-1506	394	3	5	5	NUM
ejpam-1506	394	4	⌋	⌋	NOUN
ejpam-1506	394	5	and	and	CCONJ
ejpam-1506	394	6	dim(l(g	dim(l(g	NOUN
ejpam-1506	394	7	)	)	PUNCT
ejpam-1506	394	8	)	)	PUNCT
ejpam-1506	395	1	=	=	PUNCT
ejpam-1506	395	2	n−	n−	NOUN
ejpam-1506	395	3	⌈	⌈	NOUN
ejpam-1506	395	4	n	n	CCONJ
ejpam-1506	395	5	3	3	NUM
ejpam-1506	395	6	⌉	⌉	NOUN
ejpam-1506	395	7	,	,	PUNCT
ejpam-1506	395	8	and	and	CCONJ
ejpam-1506	395	9	dim(g⋆	dim(g⋆	NOUN
ejpam-1506	395	10	)	)	PUNCT
ejpam-1506	395	11	=	=	NUM
ejpam-1506	395	12	n−	n−	NOUN
ejpam-1506	395	13	1	1	NUM
ejpam-1506	395	14	;	;	PUNCT
ejpam-1506	395	15	further	far	ADV
ejpam-1506	395	16	,	,	PUNCT
ejpam-1506	395	17	notice	notice	VERB
ejpam-1506	395	18	that	that	SCONJ
ejpam-1506	395	19	dim(g⋆)−	dim(g⋆)−	PROPN
ejpam-1506	395	20	dim(g	dim(g	PROPN
ejpam-1506	395	21	)	)	PUNCT
ejpam-1506	395	22	and	and	CCONJ
ejpam-1506	395	23	dim(g⋆)−	dim(g⋆)−	NOUN
ejpam-1506	395	24	dim(l(g	dim(l(g	NOUN
ejpam-1506	395	25	)	)	PUNCT
ejpam-1506	395	26	)	)	PUNCT
ejpam-1506	395	27	can	can	AUX
ejpam-1506	395	28	be	be	AUX
ejpam-1506	395	29	arbitrarily	arbitrarily	ADV
ejpam-1506	395	30	large	large	ADJ
ejpam-1506	395	31	.	.	PUNCT
ejpam-1506	396	1	6	6	X
ejpam-1506	396	2	.	.	X
ejpam-1506	396	3	summary	summary	NOUN
ejpam-1506	396	4	and	and	CCONJ
ejpam-1506	396	5	open	open	ADJ
ejpam-1506	396	6	problems	problem	NOUN
ejpam-1506	396	7	in	in	ADP
ejpam-1506	396	8	table	table	NOUN
ejpam-1506	396	9	1	1	NUM
ejpam-1506	396	10	,	,	PUNCT
ejpam-1506	396	11	we	we	PRON
ejpam-1506	396	12	summarize	summarize	VERB
ejpam-1506	396	13	metric	metric	ADJ
ejpam-1506	396	14	dimension	dimension	NOUN
ejpam-1506	396	15	of	of	ADP
ejpam-1506	396	16	some	some	DET
ejpam-1506	396	17	graphs	graph	NOUN
ejpam-1506	396	18	g	g	ADP
ejpam-1506	396	19	,	,	PUNCT
ejpam-1506	396	20	the	the	DET
ejpam-1506	396	21	line	line	NOUN
ejpam-1506	396	22	graphs	graph	VERB
ejpam-1506	396	23	l(g	l(g	NOUN
ejpam-1506	396	24	)	)	PUNCT
ejpam-1506	396	25	,	,	PUNCT
ejpam-1506	396	26	and	and	CCONJ
ejpam-1506	396	27	the	the	DET
ejpam-1506	396	28	para	para	NOUN
ejpam-1506	396	29	-	-	PUNCT
ejpam-1506	396	30	line	line	NOUN
ejpam-1506	396	31	graphs	graph	NOUN
ejpam-1506	396	32	g⋆.	g⋆.	NOUN
ejpam-1506	396	33	here	here	ADV
ejpam-1506	396	34	,	,	PUNCT
ejpam-1506	396	35	x	x	PRON
ejpam-1506	396	36	denotes	denote	VERB
ejpam-1506	396	37	the	the	DET
ejpam-1506	396	38	number	number	NOUN
ejpam-1506	396	39	of	of	ADP
ejpam-1506	396	40	even	even	ADV
ejpam-1506	396	41	cycles	cycle	NOUN
ejpam-1506	396	42	of	of	ADP
ejpam-1506	396	43	a	a	DET
ejpam-1506	396	44	bouquet	bouquet	NOUN
ejpam-1506	396	45	of	of	ADP
ejpam-1506	396	46	circles	circle	NOUN
ejpam-1506	396	47	bn	bn	VERB
ejpam-1506	396	48	;	;	PUNCT
ejpam-1506	396	49	for	for	ADP
ejpam-1506	396	50	the	the	DET
ejpam-1506	396	51	complete	complete	ADJ
ejpam-1506	396	52	bi	bi	ADJ
ejpam-1506	396	53	-	-	ADJ
ejpam-1506	396	54	partite	partite	ADJ
ejpam-1506	396	55	graphs	graph	NOUN
ejpam-1506	396	56	ks	ks	PROPN
ejpam-1506	396	57	,	,	PUNCT
ejpam-1506	396	58	t	t	PROPN
ejpam-1506	396	59	we	we	PRON
ejpam-1506	396	60	consider	consider	VERB
ejpam-1506	396	61	s	s	PROPN
ejpam-1506	396	62	,	,	PUNCT
ejpam-1506	396	63	t	t	PROPN
ejpam-1506	396	64	≥	≥	NUM
ejpam-1506	396	65	2	2	NUM
ejpam-1506	396	66	excluding	exclude	VERB
ejpam-1506	396	67	s	s	NOUN
ejpam-1506	396	68	=	=	X
ejpam-1506	396	69	t	t	NOUN
ejpam-1506	396	70	=	=	SYM
ejpam-1506	396	71	2	2	X
ejpam-1506	396	72	.	.	X
ejpam-1506	397	1	we	we	PRON
ejpam-1506	397	2	conclude	conclude	VERB
ejpam-1506	397	3	this	this	DET
ejpam-1506	397	4	paper	paper	NOUN
ejpam-1506	397	5	with	with	ADP
ejpam-1506	397	6	some	some	DET
ejpam-1506	397	7	open	open	ADJ
ejpam-1506	397	8	problems	problem	NOUN
ejpam-1506	397	9	.	.	PUNCT
ejpam-1506	398	1	problems	problem	NOUN
ejpam-1506	398	2	.	.	PUNCT
ejpam-1506	399	1	for	for	ADP
ejpam-1506	399	2	a	a	DET
ejpam-1506	399	3	connected	connected	ADJ
ejpam-1506	399	4	graph	graph	NOUN
ejpam-1506	399	5	g	g	NOUN
ejpam-1506	399	6	,	,	PUNCT
ejpam-1506	399	7	let	let	VERB
ejpam-1506	399	8	l(g	l(g	X
ejpam-1506	399	9	)	)	PUNCT
ejpam-1506	399	10	be	be	AUX
ejpam-1506	399	11	the	the	DET
ejpam-1506	399	12	line	line	NOUN
ejpam-1506	399	13	graph	graph	NOUN
ejpam-1506	399	14	of	of	ADP
ejpam-1506	399	15	g	g	NOUN
ejpam-1506	399	16	and	and	CCONJ
ejpam-1506	399	17	let	let	VERB
ejpam-1506	399	18	g⋆	g⋆	PRON
ejpam-1506	399	19	be	be	AUX
ejpam-1506	399	20	the	the	DET
ejpam-1506	399	21	para	para	NOUN
ejpam-1506	399	22	-	-	PUNCT
ejpam-1506	399	23	line	line	NOUN
ejpam-1506	399	24	graph	graph	NOUN
ejpam-1506	399	25	of	of	ADP
ejpam-1506	399	26	g.	g.	PROPN
ejpam-1506	399	27	q1	q1	PROPN
ejpam-1506	399	28	.	.	PUNCT
ejpam-1506	400	1	[	[	X
ejpam-1506	400	2	10	10	NUM
ejpam-1506	400	3	]	]	PUNCT
ejpam-1506	400	4	can	can	AUX
ejpam-1506	400	5	we	we	PRON
ejpam-1506	400	6	characterize	characterize	VERB
ejpam-1506	400	7	graphs	graph	NOUN
ejpam-1506	400	8	g	g	ADP
ejpam-1506	400	9	such	such	ADJ
ejpam-1506	400	10	that	that	SCONJ
ejpam-1506	400	11	dim(g	dim(g	PROPN
ejpam-1506	400	12	)	)	PUNCT
ejpam-1506	400	13	=	=	SYM
ejpam-1506	400	14	dim(l(g	dim(l(g	NOUN
ejpam-1506	400	15	)	)	PUNCT
ejpam-1506	400	16	)	)	PUNCT
ejpam-1506	400	17	?	?	PUNCT
ejpam-1506	401	1	q2	q2	NOUN
ejpam-1506	401	2	.	.	PUNCT
ejpam-1506	402	1	can	can	AUX
ejpam-1506	402	2	we	we	PRON
ejpam-1506	402	3	characterize	characterize	VERB
ejpam-1506	402	4	graphs	graph	NOUN
ejpam-1506	402	5	g	g	ADP
ejpam-1506	402	6	such	such	ADJ
ejpam-1506	402	7	that	that	SCONJ
ejpam-1506	402	8	dim(g	dim(g	PROPN
ejpam-1506	402	9	)	)	PUNCT
ejpam-1506	402	10	=	=	SYM
ejpam-1506	402	11	dim(g⋆	dim(g⋆	NOUN
ejpam-1506	402	12	)	)	PUNCT
ejpam-1506	402	13	?	?	PUNCT
ejpam-1506	403	1	q3	q3	PROPN
ejpam-1506	403	2	.	.	PUNCT
ejpam-1506	404	1	can	can	AUX
ejpam-1506	404	2	we	we	PRON
ejpam-1506	404	3	characterize	characterize	VERB
ejpam-1506	404	4	graphs	graph	NOUN
ejpam-1506	404	5	g	g	ADP
ejpam-1506	404	6	such	such	ADJ
ejpam-1506	404	7	that	that	DET
ejpam-1506	404	8	dim(l(g	dim(l(g	NOUN
ejpam-1506	404	9	)	)	PUNCT
ejpam-1506	404	10	)	)	PUNCT
ejpam-1506	405	1	=	=	SYM
ejpam-1506	405	2	dim(g⋆	dim(g⋆	NOUN
ejpam-1506	405	3	)	)	PUNCT
ejpam-1506	405	4	?	?	PUNCT
ejpam-1506	406	1	references	reference	NOUN
ejpam-1506	406	2	314	314	NUM
ejpam-1506	406	3	table	table	NOUN
ejpam-1506	406	4	1	1	NUM
ejpam-1506	406	5	:	:	PUNCT
ejpam-1506	406	6	metric	metric	ADJ
ejpam-1506	406	7	dimension	dimension	NOUN
ejpam-1506	406	8	of	of	ADP
ejpam-1506	406	9	some	some	DET
ejpam-1506	406	10	graphs	graph	NOUN
ejpam-1506	406	11	g	g	ADP
ejpam-1506	406	12	,	,	PUNCT
ejpam-1506	406	13	line	line	NOUN
ejpam-1506	406	14	graphs	graph	NOUN
ejpam-1506	406	15	l(g	l(g	NOUN
ejpam-1506	406	16	)	)	PUNCT
ejpam-1506	406	17	,	,	PUNCT
ejpam-1506	406	18	and	and	CCONJ
ejpam-1506	406	19	para	para	NOUN
ejpam-1506	406	20	-	-	PUNCT
ejpam-1506	406	21	line	line	NOUN
ejpam-1506	406	22	graphs	graph	NOUN
ejpam-1506	406	23	g⋆.	g⋆.	NOUN
ejpam-1506	406	24	g	g	PROPN
ejpam-1506	406	25	dim(g	dim(g	PROPN
ejpam-1506	406	26	)	)	PUNCT
ejpam-1506	406	27	dim(l(g	dim(l(g	NOUN
ejpam-1506	406	28	)	)	PUNCT
ejpam-1506	406	29	)	)	PUNCT
ejpam-1506	406	30	dim(g⋆	dim(g⋆	NOUN
ejpam-1506	406	31	)	)	PUNCT
ejpam-1506	406	32	pn	pn	NOUN
ejpam-1506	406	33	,	,	PUNCT
ejpam-1506	406	34	n≥	n≥	NOUN
ejpam-1506	406	35	2	2	NUM
ejpam-1506	406	36	1	1	NUM
ejpam-1506	406	37	1	1	NUM
ejpam-1506	406	38	1	1	NUM
ejpam-1506	406	39	cn	cn	PROPN
ejpam-1506	406	40	,	,	PUNCT
ejpam-1506	406	41	n≥	n≥	NOUN
ejpam-1506	406	42	3	3	NUM
ejpam-1506	406	43	2	2	NUM
ejpam-1506	406	44	2	2	NUM
ejpam-1506	406	45	2	2	NUM
ejpam-1506	406	46	t	t	NOUN
ejpam-1506	406	47	(	(	PUNCT
ejpam-1506	406	48	6=	6=	ADP
ejpam-1506	406	49	pn	pn	NOUN
ejpam-1506	406	50	)	)	PUNCT
ejpam-1506	406	51	σ(t	σ(t	PROPN
ejpam-1506	406	52	)	)	PUNCT
ejpam-1506	406	53	−	−	PROPN
ejpam-1506	406	54	ex(t	ex(t	NUM
ejpam-1506	406	55	)	)	PUNCT
ejpam-1506	406	56	σ(t	σ(t	PROPN
ejpam-1506	406	57	)	)	PUNCT
ejpam-1506	406	58	−	−	PROPN
ejpam-1506	406	59	ex(t	ex(t	NUM
ejpam-1506	406	60	)	)	PUNCT
ejpam-1506	406	61	σ(t	σ(t	PROPN
ejpam-1506	406	62	)	)	PUNCT
ejpam-1506	406	63	−	−	PROPN
ejpam-1506	406	64	ex(t	ex(t	PUNCT
ejpam-1506	406	65	)	)	PUNCT
ejpam-1506	406	66	kn	kn	PROPN
ejpam-1506	406	67	,	,	PUNCT
ejpam-1506	406	68	n≥	n≥	PROPN
ejpam-1506	406	69	6	6	NUM
ejpam-1506	406	70	n−	n−	NOUN
ejpam-1506	406	71	1	1	NUM
ejpam-1506	406	72	⌈2n	⌈2n	NUM
ejpam-1506	406	73	3	3	NUM
ejpam-1506	406	74	⌉	⌉	PRON
ejpam-1506	406	75	n−	n−	PROPN
ejpam-1506	406	76	1	1	NUM
ejpam-1506	406	77	ks	k	NOUN
ejpam-1506	406	78	,	,	PUNCT
ejpam-1506	406	79	t	t	PROPN
ejpam-1506	406	80	s+	s+	PUNCT
ejpam-1506	406	81	t	t	PROPN
ejpam-1506	406	82	−	−	PROPN
ejpam-1506	406	83	2	2	NUM
ejpam-1506	406	84	(	(	PUNCT
ejpam-1506	406	85	⌊2(s+t−1	⌊2(s+t−1	PROPN
ejpam-1506	406	86	)	)	PUNCT
ejpam-1506	406	87	3	3	NUM
ejpam-1506	406	88	⌋	⌋	NOUN
ejpam-1506	406	89	if	if	SCONJ
ejpam-1506	406	90	s	s	VERB
ejpam-1506	406	91	≤	≤	X
ejpam-1506	406	92	t	t	NOUN
ejpam-1506	406	93	≤	≤	NUM
ejpam-1506	407	1	2s	2s	NUM
ejpam-1506	407	2	t	t	NOUN
ejpam-1506	407	3	−	−	NOUN
ejpam-1506	407	4	1	1	NUM
ejpam-1506	407	5	if	if	SCONJ
ejpam-1506	407	6	t	t	PROPN
ejpam-1506	407	7	≥	≥	VERB
ejpam-1506	407	8	2s	2s	PROPN
ejpam-1506	407	9	≤	≤	NUM
ejpam-1506	407	10	s+	s+	PUNCT
ejpam-1506	407	11	t	t	PROPN
ejpam-1506	407	12	−	−	PROPN
ejpam-1506	407	13	3	3	NUM
ejpam-1506	407	14	w1,n	w1,n	PROPN
ejpam-1506	407	15	,	,	PUNCT
ejpam-1506	407	16	n≥	n≥	PROPN
ejpam-1506	407	17	3	3	NUM
ejpam-1506	407	18	(	(	PUNCT
ejpam-1506	407	19	3	3	NUM
ejpam-1506	407	20	if	if	SCONJ
ejpam-1506	407	21	n=	n=	ADJ
ejpam-1506	407	22	3,6	3,6	NUM
ejpam-1506	407	23	⌊2n+2	⌊2n+2	PROPN
ejpam-1506	407	24	5	5	NUM
ejpam-1506	407	25	⌋	⌋	NOUN
ejpam-1506	407	26	otherwise	otherwise	ADV
ejpam-1506	407	27			VERB
ejpam-1506	407	28			ADJ
ejpam-1506	407	29			NOUN
ejpam-1506	407	30	3	3	NUM
ejpam-1506	407	31	if	if	SCONJ
ejpam-1506	407	32	n=	n=	ADJ
ejpam-1506	407	33	3,4	3,4	NUM
ejpam-1506	407	34	4	4	NUM
ejpam-1506	407	35	if	if	SCONJ
ejpam-1506	407	36	n=	n=	ADJ
ejpam-1506	407	37	5	5	NUM
ejpam-1506	407	38	n−	n−	PROPN
ejpam-1506	407	39	⌈	⌈	NOUN
ejpam-1506	407	40	n	n	CCONJ
ejpam-1506	407	41	3	3	NUM
ejpam-1506	407	42	⌉	⌉	X
ejpam-1506	407	43	if	if	SCONJ
ejpam-1506	407	44	n≥	n≥	PROPN
ejpam-1506	407	45	6	6	NUM
ejpam-1506	407	46	(	(	PUNCT
ejpam-1506	407	47	3	3	NUM
ejpam-1506	407	48	if	if	SCONJ
ejpam-1506	407	49	n=	n=	ADJ
ejpam-1506	407	50	3	3	NUM
ejpam-1506	407	51	n−	n−	NOUN
ejpam-1506	407	52	1	1	NUM
ejpam-1506	407	53	if	if	SCONJ
ejpam-1506	407	54	n≥	n≥	PROPN
ejpam-1506	407	55	4	4	NUM
ejpam-1506	407	56	bn	bn	NOUN
ejpam-1506	407	57	,	,	PUNCT
ejpam-1506	407	58	n≥	n≥	PROPN
ejpam-1506	407	59	2	2	NUM
ejpam-1506	407	60	(	(	PUNCT
ejpam-1506	407	61	n	n	NOUN
ejpam-1506	407	62	if	if	SCONJ
ejpam-1506	407	63	x	x	SYM
ejpam-1506	407	64	=	=	SYM
ejpam-1506	407	65	0	0	NUM
ejpam-1506	407	66	n+	n+	NOUN
ejpam-1506	407	67	x	x	PUNCT
ejpam-1506	407	68	−	−	NOUN
ejpam-1506	407	69	1	1	NUM
ejpam-1506	407	70	if	if	SCONJ
ejpam-1506	407	71	x	x	PRON
ejpam-1506	407	72	≥	≥	VERB
ejpam-1506	407	73	1	1	NUM
ejpam-1506	407	74	2n−	2n−	NUM
ejpam-1506	407	75	1	1	NUM
ejpam-1506	407	76	2n−	2n−	NUM
ejpam-1506	407	77	1	1	NUM
ejpam-1506	407	78	acknowledgements	acknowledgement	NOUN
ejpam-1506	407	79	the	the	DET
ejpam-1506	407	80	first	first	ADJ
ejpam-1506	407	81	author	author	NOUN
ejpam-1506	407	82	is	be	AUX
ejpam-1506	407	83	supported	support	VERB
ejpam-1506	407	84	by	by	ADP
ejpam-1506	407	85	the	the	DET
ejpam-1506	407	86	welch	welch	PROPN
ejpam-1506	407	87	foundation	foundation	PROPN
ejpam-1506	407	88	of	of	ADP
ejpam-1506	407	89	houston	houston	PROPN
ejpam-1506	407	90	,	,	PUNCT
ejpam-1506	407	91	texas	texas	PROPN
ejpam-1506	407	92	,	,	PUNCT
ejpam-1506	407	93	grant	grant	VERB
ejpam-1506	407	94	bd-0894	bd-0894	NOUN
ejpam-1506	407	95	.	.	PUNCT
ejpam-1506	408	1	the	the	DET
ejpam-1506	408	2	authors	author	NOUN
ejpam-1506	408	3	thank	thank	VERB
ejpam-1506	408	4	cong	cong	PROPN
ejpam-1506	408	5	x.	x.	PROPN
ejpam-1506	408	6	kang	kang	PROPN
ejpam-1506	408	7	for	for	ADP
ejpam-1506	408	8	his	his	PRON
ejpam-1506	408	9	helpful	helpful	ADJ
ejpam-1506	408	10	comments	comment	NOUN
ejpam-1506	408	11	and	and	CCONJ
ejpam-1506	408	12	suggestions	suggestion	NOUN
ejpam-1506	408	13	,	,	PUNCT
ejpam-1506	408	14	especially	especially	ADV
ejpam-1506	408	15	on	on	ADP
ejpam-1506	408	16	theorem	theorem	NOUN
ejpam-1506	408	17	6	6	NUM
ejpam-1506	408	18	.	.	PUNCT
ejpam-1506	408	19	references	reference	NOUN
ejpam-1506	408	20	[	[	X
ejpam-1506	408	21	1	1	NUM
ejpam-1506	408	22	]	]	X
ejpam-1506	408	23	r.f	r.f	PROPN
ejpam-1506	408	24	.	.	PROPN
ejpam-1506	408	25	bailey	bailey	PROPN
ejpam-1506	408	26	and	and	CCONJ
ejpam-1506	408	27	p.j	p.j	PROPN
ejpam-1506	408	28	.	.	PROPN
ejpam-1506	408	29	cameron	cameron	PROPN
ejpam-1506	408	30	.	.	PUNCT
ejpam-1506	409	1	base	base	PROPN
ejpam-1506	409	2	size	size	NOUN
ejpam-1506	409	3	,	,	PUNCT
ejpam-1506	409	4	metric	metric	ADJ
ejpam-1506	409	5	dimension	dimension	NOUN
ejpam-1506	409	6	and	and	CCONJ
ejpam-1506	409	7	other	other	ADJ
ejpam-1506	409	8	invariants	invariant	NOUN
ejpam-1506	409	9	of	of	ADP
ejpam-1506	409	10	groups	group	NOUN
ejpam-1506	409	11	and	and	CCONJ
ejpam-1506	409	12	graphs	graph	NOUN
ejpam-1506	409	13	.	.	PUNCT
ejpam-1506	410	1	bulletin	bulletin	NOUN
ejpam-1506	410	2	of	of	ADP
ejpam-1506	410	3	the	the	DET
ejpam-1506	410	4	london	london	PROPN
ejpam-1506	410	5	mathematical	mathematical	ADJ
ejpam-1506	410	6	society	society	NOUN
ejpam-1506	410	7	.	.	PUNCT
ejpam-1506	411	1	43(2	43(2	NUM
ejpam-1506	411	2	)	)	PUNCT
ejpam-1506	411	3	,	,	PUNCT
ejpam-1506	411	4	209	209	NUM
ejpam-1506	411	5	-	-	SYM
ejpam-1506	411	6	242	242	NUM
ejpam-1506	411	7	.	.	PUNCT
ejpam-1506	412	1	2011	2011	NUM
ejpam-1506	412	2	.	.	PUNCT
ejpam-1506	413	1	[	[	X
ejpam-1506	413	2	2	2	NUM
ejpam-1506	413	3	]	]	X
ejpam-1506	413	4	l.w	l.w	PROPN
ejpam-1506	413	5	.	.	PROPN
ejpam-1506	413	6	beineke	beineke	PROPN
ejpam-1506	413	7	.	.	PUNCT
ejpam-1506	414	1	characterizations	characterization	NOUN
ejpam-1506	414	2	of	of	ADP
ejpam-1506	414	3	derived	derive	VERB
ejpam-1506	414	4	graphs	graph	NOUN
ejpam-1506	414	5	.	.	PUNCT
ejpam-1506	415	1	journal	journal	NOUN
ejpam-1506	415	2	of	of	ADP
ejpam-1506	415	3	combinatorial	combinatorial	ADJ
ejpam-1506	415	4	theory	theory	NOUN
ejpam-1506	415	5	.	.	PUNCT
ejpam-1506	416	1	9	9	NUM
ejpam-1506	416	2	,	,	PUNCT
ejpam-1506	416	3	129	129	NUM
ejpam-1506	416	4	-	-	SYM
ejpam-1506	416	5	135	135	NUM
ejpam-1506	416	6	.	.	PUNCT
ejpam-1506	416	7	1970	1970	NUM
ejpam-1506	416	8	.	.	PUNCT
ejpam-1506	417	1	[	[	X
ejpam-1506	417	2	3	3	X
ejpam-1506	417	3	]	]	X
ejpam-1506	417	4	p.s	p.s	PROPN
ejpam-1506	417	5	.	.	PROPN
ejpam-1506	417	6	buczkowski	buczkowski	PROPN
ejpam-1506	417	7	,	,	PUNCT
ejpam-1506	417	8	g.	g.	PROPN
ejpam-1506	417	9	chartrand	chartrand	PROPN
ejpam-1506	417	10	,	,	PUNCT
ejpam-1506	417	11	c.	c.	PROPN
ejpam-1506	417	12	poisson	poisson	PROPN
ejpam-1506	417	13	,	,	PUNCT
ejpam-1506	417	14	and	and	CCONJ
ejpam-1506	417	15	p.	p.	PROPN
ejpam-1506	417	16	zhang	zhang	PROPN
ejpam-1506	417	17	.	.	PUNCT
ejpam-1506	418	1	on	on	ADP
ejpam-1506	418	2	k	k	ADJ
ejpam-1506	418	3	-	-	ADJ
ejpam-1506	418	4	dimensional	dimensional	ADJ
ejpam-1506	418	5	graphs	graph	NOUN
ejpam-1506	418	6	and	and	CCONJ
ejpam-1506	418	7	their	their	PRON
ejpam-1506	418	8	bases	basis	NOUN
ejpam-1506	418	9	.	.	PUNCT
ejpam-1506	419	1	periodica	periodica	PROPN
ejpam-1506	419	2	mathematica	mathematica	PROPN
ejpam-1506	419	3	hungarica	hungarica	PROPN
ejpam-1506	419	4	.	.	PUNCT
ejpam-1506	420	1	46(1	46(1	X
ejpam-1506	420	2	)	)	PUNCT
ejpam-1506	421	1	,	,	PUNCT
ejpam-1506	421	2	9	9	NUM
ejpam-1506	421	3	-	-	SYM
ejpam-1506	421	4	15	15	NUM
ejpam-1506	421	5	.	.	PUNCT
ejpam-1506	421	6	2003	2003	NUM
ejpam-1506	421	7	.	.	PUNCT
ejpam-1506	422	1	[	[	X
ejpam-1506	422	2	4	4	X
ejpam-1506	422	3	]	]	PUNCT
ejpam-1506	422	4	j.	j.	PROPN
ejpam-1506	422	5	cáceres	cáceres	PROPN
ejpam-1506	422	6	,	,	PUNCT
ejpam-1506	422	7	c.	c.	PROPN
ejpam-1506	422	8	hernado	hernado	PROPN
ejpam-1506	422	9	,	,	PUNCT
ejpam-1506	422	10	m.	m.	PROPN
ejpam-1506	422	11	mora	mora	PROPN
ejpam-1506	422	12	,	,	PUNCT
ejpam-1506	422	13	i.m	i.m	PROPN
ejpam-1506	422	14	.	.	PUNCT
ejpam-1506	422	15	pelayo	pelayo	PROPN
ejpam-1506	422	16	,	,	PUNCT
ejpam-1506	422	17	m.l	m.l	PROPN
ejpam-1506	422	18	.	.	PROPN
ejpam-1506	422	19	puertas	puertas	PROPN
ejpam-1506	422	20	,	,	PUNCT
ejpam-1506	422	21	c.	c.	PROPN
ejpam-1506	422	22	seara	seara	PROPN
ejpam-1506	422	23	,	,	PUNCT
ejpam-1506	422	24	and	and	CCONJ
ejpam-1506	422	25	d.r	d.r	PROPN
ejpam-1506	422	26	.	.	PROPN
ejpam-1506	422	27	wood	wood	NOUN
ejpam-1506	422	28	.	.	PUNCT
ejpam-1506	423	1	on	on	ADP
ejpam-1506	423	2	the	the	DET
ejpam-1506	423	3	metric	metric	ADJ
ejpam-1506	423	4	dimension	dimension	NOUN
ejpam-1506	423	5	of	of	ADP
ejpam-1506	423	6	cartesian	cartesian	ADJ
ejpam-1506	423	7	products	product	NOUN
ejpam-1506	423	8	of	of	ADP
ejpam-1506	423	9	graphs	graph	NOUN
ejpam-1506	423	10	.	.	PUNCT
ejpam-1506	424	1	siam	siam	PROPN
ejpam-1506	424	2	journal	journal	PROPN
ejpam-1506	424	3	on	on	ADP
ejpam-1506	424	4	discrete	discrete	ADJ
ejpam-1506	424	5	mathematics	mathematic	NOUN
ejpam-1506	424	6	.	.	PUNCT
ejpam-1506	425	1	21	21	NUM
ejpam-1506	425	2	,	,	PUNCT
ejpam-1506	425	3	issue	issue	NOUN
ejpam-1506	425	4	2	2	NUM
ejpam-1506	425	5	,	,	PUNCT
ejpam-1506	425	6	423	423	NUM
ejpam-1506	425	7	-	-	SYM
ejpam-1506	425	8	441	441	NUM
ejpam-1506	425	9	.	.	PUNCT
ejpam-1506	425	10	2007	2007	NUM
ejpam-1506	425	11	.	.	PUNCT
ejpam-1506	426	1	[	[	X
ejpam-1506	426	2	5	5	X
ejpam-1506	426	3	]	]	PUNCT
ejpam-1506	426	4	g.	g.	PROPN
ejpam-1506	426	5	chartrand	chartrand	PROPN
ejpam-1506	426	6	,	,	PUNCT
ejpam-1506	426	7	l.	l.	PROPN
ejpam-1506	426	8	eroh	eroh	PROPN
ejpam-1506	426	9	,	,	PUNCT
ejpam-1506	426	10	m.a	m.a	PROPN
ejpam-1506	426	11	.	.	PROPN
ejpam-1506	426	12	johnson	johnson	PROPN
ejpam-1506	426	13	,	,	PUNCT
ejpam-1506	426	14	o.r	o.r	PROPN
ejpam-1506	426	15	.	.	PROPN
ejpam-1506	426	16	oellermann	oellermann	PROPN
ejpam-1506	426	17	.	.	PUNCT
ejpam-1506	427	1	resolvability	resolvability	NOUN
ejpam-1506	427	2	in	in	ADP
ejpam-1506	427	3	graphs	graph	NOUN
ejpam-1506	427	4	and	and	CCONJ
ejpam-1506	427	5	the	the	DET
ejpam-1506	427	6	metric	metric	ADJ
ejpam-1506	427	7	dimension	dimension	NOUN
ejpam-1506	427	8	of	of	ADP
ejpam-1506	427	9	a	a	DET
ejpam-1506	427	10	graph	graph	NOUN
ejpam-1506	427	11	.	.	PUNCT
ejpam-1506	428	1	discrete	discrete	ADJ
ejpam-1506	428	2	applied	apply	VERB
ejpam-1506	428	3	mathematics	mathematic	NOUN
ejpam-1506	428	4	.	.	PUNCT
ejpam-1506	429	1	105	105	NUM
ejpam-1506	429	2	,	,	PUNCT
ejpam-1506	429	3	99	99	NUM
ejpam-1506	429	4	-	-	SYM
ejpam-1506	429	5	113	113	NUM
ejpam-1506	429	6	.	.	PUNCT
ejpam-1506	430	1	2000	2000	NUM
ejpam-1506	430	2	.	.	PUNCT
ejpam-1506	431	1	[	[	X
ejpam-1506	431	2	6	6	NUM
ejpam-1506	431	3	]	]	PUNCT
ejpam-1506	431	4	g.	g.	PROPN
ejpam-1506	431	5	chartrand	chartrand	PROPN
ejpam-1506	431	6	and	and	CCONJ
ejpam-1506	431	7	p.	p.	PROPN
ejpam-1506	431	8	zhang	zhang	PROPN
ejpam-1506	431	9	.	.	PUNCT
ejpam-1506	432	1	introduction	introduction	NOUN
ejpam-1506	432	2	to	to	AUX
ejpam-1506	432	3	graph	graph	NOUN
ejpam-1506	432	4	theory	theory	NOUN
ejpam-1506	432	5	.	.	PUNCT
ejpam-1506	433	1	mcgraw	mcgraw	PROPN
ejpam-1506	433	2	-	-	PUNCT
ejpam-1506	433	3	hill	hill	PROPN
ejpam-1506	433	4	,	,	PUNCT
ejpam-1506	433	5	kalamazoo	kalamazoo	PROPN
ejpam-1506	433	6	,	,	PUNCT
ejpam-1506	433	7	mi	mi	PROPN
ejpam-1506	433	8	.	.	PROPN
ejpam-1506	433	9	2004	2004	NUM
ejpam-1506	433	10	.	.	PUNCT
ejpam-1506	434	1	[	[	X
ejpam-1506	434	2	7	7	X
ejpam-1506	434	3	]	]	X
ejpam-1506	434	4	g.	g.	PROPN
ejpam-1506	434	5	chartrand	chartrand	PROPN
ejpam-1506	434	6	and	and	CCONJ
ejpam-1506	434	7	p.	p.	PROPN
ejpam-1506	434	8	zhang	zhang	PROPN
ejpam-1506	434	9	.	.	PUNCT
ejpam-1506	435	1	the	the	DET
ejpam-1506	435	2	theory	theory	NOUN
ejpam-1506	435	3	and	and	CCONJ
ejpam-1506	435	4	applications	application	NOUN
ejpam-1506	435	5	of	of	ADP
ejpam-1506	435	6	resolvability	resolvability	NOUN
ejpam-1506	435	7	in	in	ADP
ejpam-1506	435	8	graphs	graph	NOUN
ejpam-1506	435	9	.	.	PUNCT
ejpam-1506	436	1	a	a	DET
ejpam-1506	436	2	survey	survey	NOUN
ejpam-1506	436	3	.	.	PUNCT
ejpam-1506	437	1	congressus	congressus	PROPN
ejpam-1506	437	2	numerantium	numerantium	PROPN
ejpam-1506	437	3	.	.	PUNCT
ejpam-1506	438	1	160	160	NUM
ejpam-1506	438	2	,	,	PUNCT
ejpam-1506	438	3	47	47	NUM
ejpam-1506	438	4	-	-	SYM
ejpam-1506	438	5	68	68	NUM
ejpam-1506	438	6	.	.	PUNCT
ejpam-1506	438	7	2003	2003	NUM
ejpam-1506	438	8	.	.	PUNCT
ejpam-1506	439	1	references	reference	NOUN
ejpam-1506	439	2	315	315	NUM
ejpam-1506	439	3	[	[	SYM
ejpam-1506	439	4	8	8	NUM
ejpam-1506	439	5	]	]	PUNCT
ejpam-1506	439	6	l.	l.	PROPN
ejpam-1506	439	7	eroh	eroh	PROPN
ejpam-1506	439	8	,	,	PUNCT
ejpam-1506	439	9	c.x	c.x	PROPN
ejpam-1506	439	10	.	.	PROPN
ejpam-1506	439	11	kang	kang	PROPN
ejpam-1506	439	12	,	,	PUNCT
ejpam-1506	439	13	and	and	CCONJ
ejpam-1506	439	14	e.	e.	PROPN
ejpam-1506	439	15	yi	yi	PROPN
ejpam-1506	439	16	.	.	PUNCT
ejpam-1506	440	1	a	a	DET
ejpam-1506	440	2	comparison	comparison	NOUN
ejpam-1506	440	3	between	between	ADP
ejpam-1506	440	4	the	the	DET
ejpam-1506	440	5	metric	metric	ADJ
ejpam-1506	440	6	dimension	dimension	NOUN
ejpam-1506	440	7	and	and	CCONJ
ejpam-1506	440	8	zero	zero	NUM
ejpam-1506	440	9	forcing	force	VERB
ejpam-1506	440	10	number	number	NOUN
ejpam-1506	440	11	of	of	ADP
ejpam-1506	440	12	line	line	NOUN
ejpam-1506	440	13	graphs	graph	NOUN
ejpam-1506	440	14	.	.	PUNCT
ejpam-1506	441	1	arxiv:1207.6127v1	arxiv:1207.6127v1	ADJ
ejpam-1506	441	2	.	.	PUNCT
ejpam-1506	442	1	[	[	X
ejpam-1506	442	2	9	9	NUM
ejpam-1506	442	3	]	]	PUNCT
ejpam-1506	442	4	l.	l.	PROPN
ejpam-1506	442	5	eroh	eroh	PROPN
ejpam-1506	442	6	,	,	PUNCT
ejpam-1506	442	7	c.x	c.x	PROPN
ejpam-1506	442	8	.	.	PROPN
ejpam-1506	442	9	kang	kang	PROPN
ejpam-1506	442	10	,	,	PUNCT
ejpam-1506	442	11	and	and	CCONJ
ejpam-1506	442	12	e.	e.	PROPN
ejpam-1506	442	13	yi	yi	PROPN
ejpam-1506	442	14	.	.	PUNCT
ejpam-1506	443	1	on	on	ADP
ejpam-1506	443	2	metric	metric	ADJ
ejpam-1506	443	3	dimension	dimension	NOUN
ejpam-1506	443	4	of	of	ADP
ejpam-1506	443	5	graphs	graph	NOUN
ejpam-1506	443	6	and	and	CCONJ
ejpam-1506	443	7	their	their	PRON
ejpam-1506	443	8	complements	complement	NOUN
ejpam-1506	443	9	.	.	PUNCT
ejpam-1506	444	1	journal	journal	NOUN
ejpam-1506	444	2	of	of	ADP
ejpam-1506	444	3	combinatorial	combinatorial	ADJ
ejpam-1506	444	4	mathematics	mathematic	NOUN
ejpam-1506	444	5	and	and	CCONJ
ejpam-1506	444	6	combinatorial	combinatorial	ADJ
ejpam-1506	444	7	computing	computing	NOUN
ejpam-1506	444	8	.	.	PUNCT
ejpam-1506	445	1	to	to	PART
ejpam-1506	445	2	appear	appear	VERB
ejpam-1506	445	3	.	.	PUNCT
ejpam-1506	446	1	[	[	X
ejpam-1506	446	2	10	10	NUM
ejpam-1506	446	3	]	]	PUNCT
ejpam-1506	446	4	m.	m.	NOUN
ejpam-1506	446	5	feng	feng	PROPN
ejpam-1506	446	6	,	,	PUNCT
ejpam-1506	446	7	m.	m.	PROPN
ejpam-1506	446	8	xu	xu	PROPN
ejpam-1506	446	9	,	,	PUNCT
ejpam-1506	446	10	and	and	CCONJ
ejpam-1506	446	11	k.	k.	PROPN
ejpam-1506	446	12	wang	wang	PROPN
ejpam-1506	446	13	.	.	PUNCT
ejpam-1506	447	1	on	on	ADP
ejpam-1506	447	2	the	the	DET
ejpam-1506	447	3	metric	metric	ADJ
ejpam-1506	447	4	dimension	dimension	NOUN
ejpam-1506	447	5	of	of	ADP
ejpam-1506	447	6	line	line	NOUN
ejpam-1506	447	7	graphs	graph	NOUN
ejpam-1506	447	8	.	.	PUNCT
ejpam-1506	448	1	arxiv:1107.4140v1	arxiv:1107.4140v1	NOUN
ejpam-1506	448	2	.	.	PUNCT
ejpam-1506	449	1	[	[	X
ejpam-1506	449	2	11	11	NUM
ejpam-1506	449	3	]	]	PUNCT
ejpam-1506	449	4	k.	k.	PROPN
ejpam-1506	449	5	fukui	fukui	PROPN
ejpam-1506	449	6	,	,	PUNCT
ejpam-1506	449	7	h.	h.	PROPN
ejpam-1506	449	8	kato	kato	PROPN
ejpam-1506	449	9	,	,	PUNCT
ejpam-1506	449	10	and	and	CCONJ
ejpam-1506	449	11	t.	t.	PROPN
ejpam-1506	449	12	yonezawa	yonezawa	PROPN
ejpam-1506	449	13	.	.	PUNCT
ejpam-1506	450	1	a	a	DET
ejpam-1506	450	2	new	new	ADJ
ejpam-1506	450	3	quantum	quantum	ADJ
ejpam-1506	450	4	-	-	ADJ
ejpam-1506	450	5	mechanical	mechanical	ADJ
ejpam-1506	450	6	reactivity	reactivity	NOUN
ejpam-1506	450	7	index	index	NOUN
ejpam-1506	450	8	for	for	ADP
ejpam-1506	450	9	saturated	saturate	VERB
ejpam-1506	450	10	compounds	compound	NOUN
ejpam-1506	450	11	.	.	PUNCT
ejpam-1506	451	1	bulletin	bulletin	NOUN
ejpam-1506	451	2	of	of	ADP
ejpam-1506	451	3	the	the	DET
ejpam-1506	451	4	chemical	chemical	NOUN
ejpam-1506	451	5	society	society	NOUN
ejpam-1506	451	6	of	of	ADP
ejpam-1506	451	7	japan	japan	PROPN
ejpam-1506	451	8	.	.	PUNCT
ejpam-1506	452	1	34	34	NUM
ejpam-1506	452	2	,	,	PUNCT
ejpam-1506	452	3	1111	1111	NUM
ejpam-1506	452	4	-	-	SYM
ejpam-1506	452	5	1115	1115	NUM
ejpam-1506	452	6	.	.	PUNCT
ejpam-1506	453	1	1961	1961	NUM
ejpam-1506	453	2	.	.	PUNCT
ejpam-1506	454	1	[	[	X
ejpam-1506	454	2	12	12	NUM
ejpam-1506	454	3	]	]	X
ejpam-1506	454	4	m.r	m.r	PROPN
ejpam-1506	454	5	.	.	PROPN
ejpam-1506	454	6	garey	garey	PROPN
ejpam-1506	454	7	and	and	CCONJ
ejpam-1506	454	8	d.s	d.s	PROPN
ejpam-1506	454	9	.	.	PROPN
ejpam-1506	454	10	johnson	johnson	PROPN
ejpam-1506	454	11	.	.	PUNCT
ejpam-1506	455	1	computers	computer	NOUN
ejpam-1506	455	2	and	and	CCONJ
ejpam-1506	455	3	intractability	intractability	NOUN
ejpam-1506	455	4	:	:	PUNCT
ejpam-1506	455	5	a	a	DET
ejpam-1506	455	6	guide	guide	NOUN
ejpam-1506	455	7	to	to	ADP
ejpam-1506	455	8	the	the	DET
ejpam-1506	455	9	theory	theory	NOUN
ejpam-1506	455	10	of	of	ADP
ejpam-1506	455	11	np	np	NOUN
ejpam-1506	455	12	-	-	NOUN
ejpam-1506	455	13	completeness	completeness	NOUN
ejpam-1506	455	14	.	.	PUNCT
ejpam-1506	456	1	freeman	freeman	PROPN
ejpam-1506	456	2	,	,	PUNCT
ejpam-1506	456	3	new	new	PROPN
ejpam-1506	456	4	york	york	PROPN
ejpam-1506	456	5	.	.	PUNCT
ejpam-1506	457	1	1979	1979	NUM
ejpam-1506	457	2	.	.	PUNCT
ejpam-1506	458	1	[	[	X
ejpam-1506	458	2	13	13	NUM
ejpam-1506	458	3	]	]	X
ejpam-1506	458	4	f.	f.	PROPN
ejpam-1506	458	5	harary	harary	PROPN
ejpam-1506	458	6	and	and	CCONJ
ejpam-1506	458	7	r.a	r.a	PROPN
ejpam-1506	458	8	.	.	PROPN
ejpam-1506	458	9	melter	melter	NOUN
ejpam-1506	458	10	.	.	PUNCT
ejpam-1506	459	1	on	on	ADP
ejpam-1506	459	2	the	the	DET
ejpam-1506	459	3	metric	metric	ADJ
ejpam-1506	459	4	dimension	dimension	NOUN
ejpam-1506	459	5	of	of	ADP
ejpam-1506	459	6	a	a	DET
ejpam-1506	459	7	graph	graph	NOUN
ejpam-1506	459	8	.	.	PUNCT
ejpam-1506	459	9	ars	ars	PROPN
ejpam-1506	459	10	combinatoria	combinatoria	NOUN
ejpam-1506	459	11	.	.	PUNCT
ejpam-1506	460	1	2	2	NUM
ejpam-1506	460	2	,	,	PUNCT
ejpam-1506	460	3	191	191	NUM
ejpam-1506	460	4	-	-	SYM
ejpam-1506	460	5	195	195	NUM
ejpam-1506	460	6	.	.	NOUN
ejpam-1506	460	7	1976	1976	NUM
ejpam-1506	460	8	.	.	PUNCT
ejpam-1506	461	1	[	[	X
ejpam-1506	461	2	14	14	NUM
ejpam-1506	461	3	]	]	X
ejpam-1506	461	4	c.	c.	PROPN
ejpam-1506	461	5	hernando	hernando	PROPN
ejpam-1506	461	6	,	,	PUNCT
ejpam-1506	461	7	m.	m.	PROPN
ejpam-1506	461	8	mora	mora	PROPN
ejpam-1506	461	9	,	,	PUNCT
ejpam-1506	461	10	i.m	i.m	PROPN
ejpam-1506	461	11	.	.	PUNCT
ejpam-1506	461	12	pelayo	pelayo	PROPN
ejpam-1506	461	13	,	,	PUNCT
ejpam-1506	461	14	c.	c.	PROPN
ejpam-1506	461	15	seara	seara	PROPN
ejpam-1506	461	16	,	,	PUNCT
ejpam-1506	461	17	and	and	CCONJ
ejpam-1506	461	18	d.r	d.r	PROPN
ejpam-1506	461	19	.	.	PROPN
ejpam-1506	461	20	wood	wood	NOUN
ejpam-1506	461	21	.	.	PUNCT
ejpam-1506	462	1	extremal	extremal	ADJ
ejpam-1506	462	2	graph	graph	NOUN
ejpam-1506	462	3	theory	theory	NOUN
ejpam-1506	462	4	for	for	ADP
ejpam-1506	462	5	metric	metric	ADJ
ejpam-1506	462	6	dimension	dimension	NOUN
ejpam-1506	462	7	and	and	CCONJ
ejpam-1506	462	8	diameter	diameter	NOUN
ejpam-1506	462	9	.	.	PUNCT
ejpam-1506	463	1	electronic	electronic	ADJ
ejpam-1506	463	2	journal	journal	NOUN
ejpam-1506	463	3	of	of	ADP
ejpam-1506	463	4	combinatorics	combinatoric	NOUN
ejpam-1506	463	5	.	.	PUNCT
ejpam-1506	464	1	17	17	NUM
ejpam-1506	464	2	(	(	PUNCT
ejpam-1506	464	3	1	1	NUM
ejpam-1506	464	4	)	)	PUNCT
ejpam-1506	464	5	,	,	PUNCT
ejpam-1506	464	6	#	#	SYM
ejpam-1506	464	7	r30	r30	NOUN
ejpam-1506	464	8	.	.	PUNCT
ejpam-1506	465	1	2010	2010	NUM
ejpam-1506	465	2	.	.	PUNCT
ejpam-1506	466	1	[	[	X
ejpam-1506	466	2	15	15	X
ejpam-1506	466	3	]	]	X
ejpam-1506	466	4	y.	y.	PROPN
ejpam-1506	466	5	higuchi	higuchi	PROPN
ejpam-1506	466	6	and	and	CCONJ
ejpam-1506	466	7	t.	t.	PROPN
ejpam-1506	466	8	shirai	shirai	PROPN
ejpam-1506	466	9	.	.	PUNCT
ejpam-1506	467	1	some	some	DET
ejpam-1506	467	2	spectral	spectral	ADJ
ejpam-1506	467	3	and	and	CCONJ
ejpam-1506	467	4	geometric	geometric	ADJ
ejpam-1506	467	5	properties	property	NOUN
ejpam-1506	467	6	for	for	ADP
ejpam-1506	467	7	infinite	infinite	ADJ
ejpam-1506	467	8	graphs	graph	NOUN
ejpam-1506	467	9	.	.	PUNCT
ejpam-1506	468	1	discrete	discrete	ADJ
ejpam-1506	468	2	geometric	geometric	ADJ
ejpam-1506	468	3	analysis	analysis	NOUN
ejpam-1506	468	4	,	,	PUNCT
ejpam-1506	468	5	contemporary	contemporary	ADJ
ejpam-1506	468	6	mathematics	mathematic	NOUN
ejpam-1506	468	7	.	.	PUNCT
ejpam-1506	469	1	347	347	NUM
ejpam-1506	469	2	,	,	PUNCT
ejpam-1506	469	3	29	29	NUM
ejpam-1506	469	4	-	-	SYM
ejpam-1506	469	5	56	56	NUM
ejpam-1506	469	6	.	.	PUNCT
ejpam-1506	470	1	american	american	PROPN
ejpam-1506	470	2	mathematical	mathematical	PROPN
ejpam-1506	470	3	society	society	NOUN
ejpam-1506	470	4	,	,	PUNCT
ejpam-1506	470	5	providence	providence	NOUN
ejpam-1506	470	6	,	,	PUNCT
ejpam-1506	470	7	ri	ri	PROPN
ejpam-1506	470	8	.	.	NOUN
ejpam-1506	470	9	2004	2004	NUM
ejpam-1506	470	10	.	.	PUNCT
ejpam-1506	471	1	[	[	X
ejpam-1506	471	2	16	16	NUM
ejpam-1506	471	3	]	]	X
ejpam-1506	471	4	h.	h.	PROPN
ejpam-1506	471	5	iswadi	iswadi	PROPN
ejpam-1506	471	6	,	,	PUNCT
ejpam-1506	471	7	e.t	e.t	PROPN
ejpam-1506	471	8	.	.	PROPN
ejpam-1506	471	9	baskoro	baskoro	PROPN
ejpam-1506	471	10	,	,	PUNCT
ejpam-1506	471	11	a.n.m	a.n.m	PROPN
ejpam-1506	471	12	.	.	PUNCT
ejpam-1506	471	13	salman	salman	PROPN
ejpam-1506	471	14	,	,	PUNCT
ejpam-1506	471	15	and	and	CCONJ
ejpam-1506	471	16	r.	r.	PROPN
ejpam-1506	471	17	simanjuntak	simanjuntak	PROPN
ejpam-1506	471	18	.	.	PUNCT
ejpam-1506	472	1	the	the	DET
ejpam-1506	472	2	metric	metric	ADJ
ejpam-1506	472	3	dimension	dimension	NOUN
ejpam-1506	472	4	of	of	ADP
ejpam-1506	472	5	amalgamation	amalgamation	NOUN
ejpam-1506	472	6	of	of	ADP
ejpam-1506	472	7	cycles	cycle	NOUN
ejpam-1506	472	8	.	.	PUNCT
ejpam-1506	473	1	far	far	PROPN
ejpam-1506	473	2	east	east	PROPN
ejpam-1506	473	3	journal	journal	PROPN
ejpam-1506	473	4	of	of	ADP
ejpam-1506	473	5	mathematical	mathematical	ADJ
ejpam-1506	473	6	sciences	science	NOUN
ejpam-1506	473	7	.	.	PUNCT
ejpam-1506	474	1	41	41	NUM
ejpam-1506	474	2	,	,	PUNCT
ejpam-1506	474	3	number	number	NOUN
ejpam-1506	474	4	1	1	NUM
ejpam-1506	474	5	,	,	PUNCT
ejpam-1506	474	6	19	19	NUM
ejpam-1506	474	7	-	-	SYM
ejpam-1506	474	8	31	31	NUM
ejpam-1506	474	9	.	.	PUNCT
ejpam-1506	475	1	2010	2010	NUM
ejpam-1506	475	2	.	.	PUNCT
ejpam-1506	476	1	[	[	X
ejpam-1506	476	2	17	17	NUM
ejpam-1506	476	3	]	]	X
ejpam-1506	476	4	s.	s.	PROPN
ejpam-1506	476	5	khuller	khuller	PROPN
ejpam-1506	476	6	,	,	PUNCT
ejpam-1506	476	7	b.	b.	PROPN
ejpam-1506	476	8	raghavachari	raghavachari	PROPN
ejpam-1506	476	9	,	,	PUNCT
ejpam-1506	476	10	and	and	CCONJ
ejpam-1506	476	11	a.	a.	NOUN
ejpam-1506	476	12	rosenfeld	rosenfeld	PROPN
ejpam-1506	476	13	.	.	PUNCT
ejpam-1506	477	1	landmarks	landmark	NOUN
ejpam-1506	477	2	in	in	ADP
ejpam-1506	477	3	graphs	graph	NOUN
ejpam-1506	477	4	.	.	PUNCT
ejpam-1506	478	1	discrete	discrete	ADJ
ejpam-1506	478	2	applied	apply	VERB
ejpam-1506	478	3	mathematics	mathematic	NOUN
ejpam-1506	478	4	.	.	PUNCT
ejpam-1506	479	1	70	70	NUM
ejpam-1506	479	2	,	,	PUNCT
ejpam-1506	479	3	217	217	NUM
ejpam-1506	479	4	-	-	SYM
ejpam-1506	479	5	229	229	NUM
ejpam-1506	479	6	.	.	PUNCT
ejpam-1506	480	1	1996	1996	NUM
ejpam-1506	480	2	.	.	PUNCT
ejpam-1506	481	1	[	[	X
ejpam-1506	481	2	18	18	NUM
ejpam-1506	481	3	]	]	X
ejpam-1506	481	4	j.	j.	PROPN
ejpam-1506	481	5	krausz	krausz	PROPN
ejpam-1506	481	6	.	.	PUNCT
ejpam-1506	482	1	demonstration	demonstration	NOUN
ejpam-1506	482	2	nouvelle	nouvelle	PROPN
ejpam-1506	482	3	d’une	d’une	DET
ejpam-1506	482	4	théorème	théorème	PROPN
ejpam-1506	482	5	de	de	PROPN
ejpam-1506	482	6	whitney	whitney	PROPN
ejpam-1506	482	7	sur	sur	PROPN
ejpam-1506	482	8	les	les	X
ejpam-1506	482	9	réseaux	réseaux	PROPN
ejpam-1506	482	10	.	.	PUNCT
ejpam-1506	483	1	mat	mat	NOUN
ejpam-1506	483	2	.	.	PUNCT
ejpam-1506	483	3	fiz	fiz	PROPN
ejpam-1506	483	4	.	.	PUNCT
ejpam-1506	484	1	lapok	lapok	NOUN
ejpam-1506	484	2	50	50	NUM
ejpam-1506	484	3	,	,	PUNCT
ejpam-1506	484	4	75	75	NUM
ejpam-1506	484	5	-	-	SYM
ejpam-1506	484	6	85	85	NUM
ejpam-1506	484	7	.	.	PUNCT
ejpam-1506	484	8	1943	1943	NUM
ejpam-1506	484	9	.	.	PUNCT
ejpam-1506	485	1	[	[	X
ejpam-1506	485	2	19	19	NUM
ejpam-1506	485	3	]	]	X
ejpam-1506	485	4	c.	c.	NOUN
ejpam-1506	485	5	poisson	poisson	PROPN
ejpam-1506	485	6	and	and	CCONJ
ejpam-1506	485	7	p.	p.	PROPN
ejpam-1506	485	8	zhang	zhang	PROPN
ejpam-1506	485	9	.	.	PUNCT
ejpam-1506	486	1	the	the	DET
ejpam-1506	486	2	metric	metric	ADJ
ejpam-1506	486	3	dimension	dimension	NOUN
ejpam-1506	486	4	of	of	ADP
ejpam-1506	486	5	unicyclic	unicyclic	ADJ
ejpam-1506	486	6	graphs	graph	NOUN
ejpam-1506	486	7	.	.	PUNCT
ejpam-1506	487	1	journal	journal	NOUN
ejpam-1506	487	2	of	of	ADP
ejpam-1506	487	3	combinatorial	combinatorial	ADJ
ejpam-1506	487	4	mathematics	mathematic	NOUN
ejpam-1506	487	5	and	and	CCONJ
ejpam-1506	487	6	combinatorial	combinatorial	ADJ
ejpam-1506	487	7	computing	computing	NOUN
ejpam-1506	487	8	.	.	PUNCT
ejpam-1506	488	1	40	40	NUM
ejpam-1506	488	2	,	,	PUNCT
ejpam-1506	488	3	17	17	NUM
ejpam-1506	488	4	-	-	SYM
ejpam-1506	488	5	32	32	NUM
ejpam-1506	488	6	.	.	PUNCT
ejpam-1506	489	1	2002	2002	NUM
ejpam-1506	489	2	.	.	PUNCT
ejpam-1506	490	1	[	[	X
ejpam-1506	490	2	20	20	NUM
ejpam-1506	490	3	]	]	X
ejpam-1506	490	4	j.a	j.a	PROPN
ejpam-1506	490	5	.	.	PROPN
ejpam-1506	490	6	pople	pople	PROPN
ejpam-1506	490	7	and	and	CCONJ
ejpam-1506	490	8	d.p	d.p	PROPN
ejpam-1506	490	9	.	.	PROPN
ejpam-1506	490	10	santry	santry	PROPN
ejpam-1506	490	11	.	.	PUNCT
ejpam-1506	491	1	a	a	DET
ejpam-1506	491	2	molecular	molecular	ADJ
ejpam-1506	491	3	orbital	orbital	ADJ
ejpam-1506	491	4	theory	theory	NOUN
ejpam-1506	491	5	of	of	ADP
ejpam-1506	491	6	hydrocarbons	hydrocarbon	NOUN
ejpam-1506	491	7	.	.	PUNCT
ejpam-1506	492	1	molecular	molecular	ADJ
ejpam-1506	492	2	physics	physic	NOUN
ejpam-1506	492	3	.	.	PUNCT
ejpam-1506	493	1	7	7	NUM
ejpam-1506	493	2	issue	issue	NOUN
ejpam-1506	493	3	3	3	NUM
ejpam-1506	493	4	,	,	PUNCT
ejpam-1506	493	5	269	269	NUM
ejpam-1506	493	6	-	-	SYM
ejpam-1506	493	7	286	286	NUM
ejpam-1506	493	8	.	.	PUNCT
ejpam-1506	493	9	1964	1964	NUM
ejpam-1506	493	10	.	.	PUNCT
ejpam-1506	494	1	[	[	X
ejpam-1506	494	2	21	21	NUM
ejpam-1506	494	3	]	]	X
ejpam-1506	494	4	p.s	p.s	PROPN
ejpam-1506	494	5	.	.	PROPN
ejpam-1506	494	6	ranjini	ranjini	PROPN
ejpam-1506	494	7	,	,	PUNCT
ejpam-1506	494	8	v.	v.	ADP
ejpam-1506	494	9	lokesha	lokesha	PROPN
ejpam-1506	494	10	,	,	PUNCT
ejpam-1506	494	11	and	and	CCONJ
ejpam-1506	494	12	i.n	i.n	PROPN
ejpam-1506	494	13	.	.	PROPN
ejpam-1506	494	14	cangül	cangül	PROPN
ejpam-1506	494	15	.	.	PUNCT
ejpam-1506	495	1	on	on	ADP
ejpam-1506	495	2	the	the	DET
ejpam-1506	495	3	zagreb	zagreb	PROPN
ejpam-1506	495	4	indices	index	NOUN
ejpam-1506	495	5	of	of	ADP
ejpam-1506	495	6	the	the	DET
ejpam-1506	495	7	line	line	NOUN
ejpam-1506	495	8	graphs	graph	NOUN
ejpam-1506	495	9	of	of	ADP
ejpam-1506	495	10	the	the	DET
ejpam-1506	495	11	subdivision	subdivision	NOUN
ejpam-1506	495	12	graphs	graph	NOUN
ejpam-1506	495	13	.	.	PUNCT
ejpam-1506	496	1	applied	apply	VERB
ejpam-1506	496	2	mathematics	mathematic	NOUN
ejpam-1506	496	3	and	and	CCONJ
ejpam-1506	496	4	computation	computation	NOUN
ejpam-1506	496	5	.	.	PUNCT
ejpam-1506	497	1	218	218	NUM
ejpam-1506	497	2	,	,	PUNCT
ejpam-1506	497	3	issue	issue	NOUN
ejpam-1506	497	4	3	3	NUM
ejpam-1506	497	5	,	,	PUNCT
ejpam-1506	497	6	699	699	NUM
ejpam-1506	497	7	-	-	SYM
ejpam-1506	497	8	702	702	NUM
ejpam-1506	497	9	.	.	NOUN
ejpam-1506	497	10	2011	2011	NUM
ejpam-1506	497	11	.	.	PUNCT
ejpam-1506	498	1	[	[	X
ejpam-1506	498	2	22	22	NUM
ejpam-1506	498	3	]	]	PUNCT
ejpam-1506	498	4	c.	c.	PROPN
ejpam-1506	498	5	sandorfy	sandorfy	PROPN
ejpam-1506	498	6	.	.	PUNCT
ejpam-1506	499	1	lcao	lcao	PROPN
ejpam-1506	499	2	mo	mo	PROPN
ejpam-1506	499	3	calculations	calculation	NOUN
ejpam-1506	499	4	on	on	ADP
ejpam-1506	499	5	saturated	saturate	VERB
ejpam-1506	499	6	hydrocarbons	hydrocarbon	NOUN
ejpam-1506	499	7	and	and	CCONJ
ejpam-1506	499	8	their	their	PRON
ejpam-1506	499	9	substituted	substitute	VERB
ejpam-1506	499	10	derivatives	derivative	NOUN
ejpam-1506	499	11	.	.	PUNCT
ejpam-1506	500	1	canadian	canadian	ADJ
ejpam-1506	500	2	journal	journal	NOUN
ejpam-1506	500	3	of	of	ADP
ejpam-1506	500	4	chemistry	chemistry	NOUN
ejpam-1506	500	5	.	.	PUNCT
ejpam-1506	501	1	33	33	NUM
ejpam-1506	501	2	,	,	PUNCT
ejpam-1506	501	3	1337	1337	NUM
ejpam-1506	501	4	-	-	SYM
ejpam-1506	501	5	1351	1351	NUM
ejpam-1506	501	6	.	.	PUNCT
ejpam-1506	502	1	1955	1955	NUM
ejpam-1506	502	2	.	.	PUNCT
ejpam-1506	503	1	references	reference	NOUN
ejpam-1506	503	2	316	316	NUM
ejpam-1506	504	1	[	[	X
ejpam-1506	504	2	23	23	NUM
ejpam-1506	504	3	]	]	PUNCT
ejpam-1506	504	4	a.	a.	NOUN
ejpam-1506	504	5	sebö	sebö	PROPN
ejpam-1506	504	6	and	and	CCONJ
ejpam-1506	504	7	e.	e.	PROPN
ejpam-1506	504	8	tannier	tannier	PROPN
ejpam-1506	504	9	.	.	PUNCT
ejpam-1506	505	1	on	on	ADP
ejpam-1506	505	2	metric	metric	ADJ
ejpam-1506	505	3	generators	generator	NOUN
ejpam-1506	505	4	of	of	ADP
ejpam-1506	505	5	graphs	graph	NOUN
ejpam-1506	505	6	.	.	PUNCT
ejpam-1506	506	1	mathematics	mathematic	NOUN
ejpam-1506	506	2	of	of	ADP
ejpam-1506	506	3	operations	operation	NOUN
ejpam-1506	506	4	research	research	NOUN
ejpam-1506	506	5	.	.	PUNCT
ejpam-1506	507	1	29	29	NUM
ejpam-1506	507	2	,	,	PUNCT
ejpam-1506	507	3	383	383	NUM
ejpam-1506	507	4	-	-	SYM
ejpam-1506	507	5	393	393	NUM
ejpam-1506	507	6	.	.	PUNCT
ejpam-1506	508	1	2004	2004	NUM
ejpam-1506	508	2	.	.	PUNCT
ejpam-1506	509	1	[	[	X
ejpam-1506	509	2	24	24	NUM
ejpam-1506	509	3	]	]	PUNCT
ejpam-1506	509	4	b.	b.	PROPN
ejpam-1506	509	5	shanmukha	shanmukha	PROPN
ejpam-1506	509	6	,	,	PUNCT
ejpam-1506	509	7	b.	b.	PROPN
ejpam-1506	509	8	sooryanarayana	sooryanarayana	PROPN
ejpam-1506	509	9	,	,	PUNCT
ejpam-1506	509	10	and	and	CCONJ
ejpam-1506	509	11	k.s	k.s	PROPN
ejpam-1506	509	12	.	.	PROPN
ejpam-1506	509	13	harinath	harinath	PROPN
ejpam-1506	509	14	.	.	PUNCT
ejpam-1506	510	1	metric	metric	ADJ
ejpam-1506	510	2	dimension	dimension	NOUN
ejpam-1506	510	3	of	of	ADP
ejpam-1506	510	4	wheels	wheel	NOUN
ejpam-1506	510	5	.	.	PUNCT
ejpam-1506	511	1	far	far	PROPN
ejpam-1506	511	2	east	east	PROPN
ejpam-1506	511	3	journal	journal	PROPN
ejpam-1506	511	4	of	of	ADP
ejpam-1506	511	5	applied	apply	VERB
ejpam-1506	511	6	mathematics	mathematic	NOUN
ejpam-1506	511	7	.	.	PUNCT
ejpam-1506	512	1	8	8	NUM
ejpam-1506	512	2	(	(	PUNCT
ejpam-1506	512	3	3	3	NUM
ejpam-1506	512	4	)	)	PUNCT
ejpam-1506	512	5	,	,	PUNCT
ejpam-1506	512	6	217	217	NUM
ejpam-1506	512	7	-	-	SYM
ejpam-1506	512	8	229	229	NUM
ejpam-1506	512	9	.	.	PUNCT
ejpam-1506	512	10	2002	2002	NUM
ejpam-1506	512	11	.	.	PUNCT
ejpam-1506	513	1	[	[	X
ejpam-1506	513	2	25	25	NUM
ejpam-1506	513	3	]	]	PUNCT
ejpam-1506	513	4	t.	t.	PROPN
ejpam-1506	513	5	shiraithe	shiraithe	PROPN
ejpam-1506	513	6	.	.	PUNCT
ejpam-1506	514	1	spectrum	spectrum	NOUN
ejpam-1506	514	2	of	of	ADP
ejpam-1506	514	3	infinite	infinite	ADJ
ejpam-1506	514	4	regular	regular	ADJ
ejpam-1506	514	5	line	line	NOUN
ejpam-1506	514	6	graphs	graph	NOUN
ejpam-1506	514	7	.	.	PUNCT
ejpam-1506	515	1	transactions	transaction	NOUN
ejpam-1506	515	2	of	of	ADP
ejpam-1506	515	3	the	the	DET
ejpam-1506	515	4	american	american	PROPN
ejpam-1506	515	5	mathematical	mathematical	PROPN
ejpam-1506	515	6	society	society	NOUN
ejpam-1506	515	7	.	.	PUNCT
ejpam-1506	516	1	352	352	NUM
ejpam-1506	516	2	,	,	PUNCT
ejpam-1506	516	3	number	number	NOUN
ejpam-1506	516	4	1	1	NUM
ejpam-1506	516	5	,	,	PUNCT
ejpam-1506	516	6	115	115	NUM
ejpam-1506	516	7	-	-	SYM
ejpam-1506	516	8	132	132	NUM
ejpam-1506	516	9	.	.	NOUN
ejpam-1506	516	10	1999	1999	NUM
ejpam-1506	516	11	.	.	PUNCT
ejpam-1506	517	1	[	[	X
ejpam-1506	517	2	26	26	NUM
ejpam-1506	517	3	]	]	X
ejpam-1506	517	4	p.j	p.j	PROPN
ejpam-1506	517	5	.	.	PROPN
ejpam-1506	517	6	slater	slater	PROPN
ejpam-1506	517	7	.	.	PUNCT
ejpam-1506	518	1	dominating	dominating	NOUN
ejpam-1506	518	2	and	and	CCONJ
ejpam-1506	518	3	reference	reference	NOUN
ejpam-1506	518	4	sets	set	NOUN
ejpam-1506	518	5	in	in	ADP
ejpam-1506	518	6	a	a	DET
ejpam-1506	518	7	graph	graph	NOUN
ejpam-1506	518	8	.	.	PUNCT
ejpam-1506	519	1	journal	journal	PROPN
ejpam-1506	519	2	of	of	ADP
ejpam-1506	519	3	mathematical	mathematical	ADJ
ejpam-1506	519	4	and	and	CCONJ
ejpam-1506	519	5	physical	physical	ADJ
ejpam-1506	519	6	sciences	science	NOUN
ejpam-1506	519	7	.	.	PUNCT
ejpam-1506	520	1	22	22	NUM
ejpam-1506	520	2	,	,	PUNCT
ejpam-1506	520	3	445	445	NUM
ejpam-1506	520	4	-	-	SYM
ejpam-1506	520	5	455	455	NUM
ejpam-1506	520	6	.	.	PUNCT
ejpam-1506	521	1	1998	1998	NUM
ejpam-1506	521	2	.	.	PUNCT
ejpam-1506	522	1	[	[	X
ejpam-1506	522	2	27	27	NUM
ejpam-1506	522	3	]	]	X
ejpam-1506	522	4	p.j	p.j	PROPN
ejpam-1506	522	5	.	.	PROPN
ejpam-1506	522	6	slater	slater	PROPN
ejpam-1506	522	7	.	.	PUNCT
ejpam-1506	523	1	leaves	leave	NOUN
ejpam-1506	523	2	of	of	ADP
ejpam-1506	523	3	trees	tree	NOUN
ejpam-1506	523	4	.	.	PUNCT
ejpam-1506	524	1	congressus	congressus	PROPN
ejpam-1506	524	2	numerantium	numerantium	PROPN
ejpam-1506	524	3	.	.	PUNCT
ejpam-1506	525	1	14	14	NUM
ejpam-1506	525	2	,	,	PUNCT
ejpam-1506	525	3	549	549	NUM
ejpam-1506	525	4	-	-	SYM
ejpam-1506	525	5	559	559	NUM
ejpam-1506	525	6	.	.	PUNCT
ejpam-1506	526	1	1975	1975	NUM
ejpam-1506	526	2	.	.	PUNCT
ejpam-1506	527	1	[	[	X
ejpam-1506	527	2	28	28	NUM
ejpam-1506	527	3	]	]	X
ejpam-1506	527	4	a.	a.	NOUN
ejpam-1506	527	5	van	van	PROPN
ejpam-1506	527	6	rooij	rooij	PROPN
ejpam-1506	527	7	and	and	CCONJ
ejpam-1506	527	8	h.	h.	PROPN
ejpam-1506	527	9	wilf	wilf	PROPN
ejpam-1506	527	10	.	.	PUNCT
ejpam-1506	528	1	the	the	DET
ejpam-1506	528	2	interchange	interchange	NOUN
ejpam-1506	528	3	graph	graph	NOUN
ejpam-1506	528	4	of	of	ADP
ejpam-1506	528	5	a	a	DET
ejpam-1506	528	6	finite	finite	ADJ
ejpam-1506	528	7	graph	graph	NOUN
ejpam-1506	528	8	.	.	PUNCT
ejpam-1506	529	1	acta	acta	PROPN
ejpam-1506	529	2	mathematica	mathematica	PROPN
ejpam-1506	529	3	academiae	academiae	PROPN
ejpam-1506	529	4	scientiarum	scientiarum	PROPN
ejpam-1506	529	5	hungaricae	hungaricae	PROPN
ejpam-1506	529	6	.	.	PUNCT
ejpam-1506	530	1	16	16	NUM
ejpam-1506	530	2	,	,	PUNCT
ejpam-1506	530	3	263	263	NUM
ejpam-1506	530	4	-	-	SYM
ejpam-1506	530	5	269	269	NUM
ejpam-1506	530	6	.	.	PUNCT
ejpam-1506	530	7	1965	1965	NUM
ejpam-1506	530	8	.	.	PUNCT
ejpam-1506	531	1	[	[	X
ejpam-1506	531	2	29	29	NUM
ejpam-1506	531	3	]	]	X
ejpam-1506	531	4	h.	h.	PROPN
ejpam-1506	531	5	whitney	whitney	PROPN
ejpam-1506	531	6	.	.	PUNCT
ejpam-1506	532	1	congruent	congruent	ADJ
ejpam-1506	532	2	graphs	graph	NOUN
ejpam-1506	532	3	and	and	CCONJ
ejpam-1506	532	4	the	the	DET
ejpam-1506	532	5	connectivity	connectivity	NOUN
ejpam-1506	532	6	of	of	ADP
ejpam-1506	532	7	graphs	graph	NOUN
ejpam-1506	532	8	.	.	PUNCT
ejpam-1506	533	1	american	american	ADJ
ejpam-1506	533	2	journal	journal	PROPN
ejpam-1506	533	3	of	of	ADP
ejpam-1506	533	4	mathematics	mathematic	NOUN
ejpam-1506	533	5	.	.	PUNCT
ejpam-1506	534	1	54	54	NUM
ejpam-1506	534	2	,	,	PUNCT
ejpam-1506	534	3	150	150	NUM
ejpam-1506	534	4	-	-	SYM
ejpam-1506	534	5	168	168	NUM
ejpam-1506	534	6	.	.	PUNCT
ejpam-1506	535	1	1932	1932	NUM
ejpam-1506	535	2	.	.	PUNCT