id	sid	tid	token	lemma	pos
ejpam-4478	1	1	european	european	PROPN
ejpam-4478	1	2	journal	journal	PROPN
ejpam-4478	1	3	of	of	ADP
ejpam-4478	1	4	pure	pure	ADJ
ejpam-4478	1	5	and	and	CCONJ
ejpam-4478	1	6	applied	apply	VERB
ejpam-4478	1	7	mathematics	mathematic	NOUN
ejpam-4478	1	8	vol	vol	NOUN
ejpam-4478	1	9	.	.	PUNCT
ejpam-4478	2	1	16	16	NUM
ejpam-4478	2	2	,	,	PUNCT
ejpam-4478	2	3	no	no	INTJ
ejpam-4478	2	4	.	.	NOUN
ejpam-4478	2	5	1	1	NUM
ejpam-4478	2	6	,	,	PUNCT
ejpam-4478	2	7	2023	2023	NUM
ejpam-4478	2	8	,	,	PUNCT
ejpam-4478	2	9	44	44	NUM
ejpam-4478	2	10	-	-	SYM
ejpam-4478	2	11	61	61	NUM
ejpam-4478	2	12	issn	issn	PROPN
ejpam-4478	2	13	1307	1307	NUM
ejpam-4478	2	14	-	-	SYM
ejpam-4478	2	15	5543	5543	NUM
ejpam-4478	2	16	–	–	PUNCT
ejpam-4478	3	1	ejpam.com	ejpam.com	X
ejpam-4478	3	2	published	publish	VERB
ejpam-4478	3	3	by	by	ADP
ejpam-4478	3	4	new	new	PROPN
ejpam-4478	3	5	york	york	PROPN
ejpam-4478	3	6	business	business	PROPN
ejpam-4478	3	7	global	global	PROPN
ejpam-4478	3	8	on	on	ADP
ejpam-4478	3	9	the	the	DET
ejpam-4478	3	10	global	global	ADJ
ejpam-4478	3	11	distance	distance	NOUN
ejpam-4478	3	12	roman	roman	ADJ
ejpam-4478	3	13	domination	domination	NOUN
ejpam-4478	3	14	of	of	ADP
ejpam-4478	3	15	some	some	DET
ejpam-4478	3	16	graphs	graph	NOUN
ejpam-4478	3	17	giovannie	giovannie	PROPN
ejpam-4478	3	18	entero1,∗	entero1,∗	NOUN
ejpam-4478	3	19	,	,	PUNCT
ejpam-4478	3	20	stephanie	stephanie	PROPN
ejpam-4478	3	21	espinola2	espinola2	NOUN
ejpam-4478	3	22	1	1	NUM
ejpam-4478	3	23	department	department	NOUN
ejpam-4478	3	24	of	of	ADP
ejpam-4478	3	25	mathematics	mathematic	NOUN
ejpam-4478	3	26	and	and	CCONJ
ejpam-4478	3	27	statistics	statistic	NOUN
ejpam-4478	3	28	,	,	PUNCT
ejpam-4478	3	29	college	college	NOUN
ejpam-4478	3	30	of	of	ADP
ejpam-4478	3	31	arts	art	NOUN
ejpam-4478	3	32	and	and	CCONJ
ejpam-4478	3	33	sciences	science	NOUN
ejpam-4478	3	34	,	,	PUNCT
ejpam-4478	3	35	university	university	NOUN
ejpam-4478	3	36	of	of	ADP
ejpam-4478	3	37	southeastern	southeastern	ADJ
ejpam-4478	3	38	philippines	philippine	NOUN
ejpam-4478	3	39	,	,	PUNCT
ejpam-4478	3	40	bo	bo	PROPN
ejpam-4478	3	41	.	.	PROPN
ejpam-4478	3	42	obrero	obrero	PROPN
ejpam-4478	3	43	,	,	PUNCT
ejpam-4478	3	44	davao	davao	PROPN
ejpam-4478	3	45	city	city	PROPN
ejpam-4478	3	46	,	,	PUNCT
ejpam-4478	3	47	davao	davao	PROPN
ejpam-4478	3	48	del	del	PROPN
ejpam-4478	3	49	sur	sur	PROPN
ejpam-4478	3	50	,	,	PUNCT
ejpam-4478	3	51	philippines	philippines	PROPN
ejpam-4478	3	52	2	2	NUM
ejpam-4478	3	53	department	department	NOUN
ejpam-4478	3	54	of	of	ADP
ejpam-4478	3	55	mathematics	mathematic	NOUN
ejpam-4478	3	56	and	and	CCONJ
ejpam-4478	3	57	statistics	statistic	NOUN
ejpam-4478	3	58	,	,	PUNCT
ejpam-4478	3	59	faculty	faculty	NOUN
ejpam-4478	3	60	,	,	PUNCT
ejpam-4478	3	61	college	college	NOUN
ejpam-4478	3	62	of	of	ADP
ejpam-4478	3	63	arts	art	NOUN
ejpam-4478	3	64	and	and	CCONJ
ejpam-4478	3	65	sciences	science	NOUN
ejpam-4478	3	66	,	,	PUNCT
ejpam-4478	3	67	university	university	NOUN
ejpam-4478	3	68	of	of	ADP
ejpam-4478	3	69	southeastern	southeastern	ADJ
ejpam-4478	3	70	philippines	philippine	NOUN
ejpam-4478	3	71	,	,	PUNCT
ejpam-4478	3	72	bo	bo	PROPN
ejpam-4478	3	73	.	.	PROPN
ejpam-4478	3	74	obrero	obrero	PROPN
ejpam-4478	3	75	,	,	PUNCT
ejpam-4478	3	76	davao	davao	PROPN
ejpam-4478	3	77	city	city	PROPN
ejpam-4478	3	78	,	,	PUNCT
ejpam-4478	3	79	davao	davao	PROPN
ejpam-4478	3	80	del	del	PROPN
ejpam-4478	3	81	sur	sur	PROPN
ejpam-4478	3	82	,	,	PUNCT
ejpam-4478	3	83	philippines	philippine	NOUN
ejpam-4478	3	84	abstract	abstract	ADJ
ejpam-4478	3	85	.	.	PUNCT
ejpam-4478	4	1	let	let	VERB
ejpam-4478	4	2	k	k	PROPN
ejpam-4478	4	3	∈	∈	PROPN
ejpam-4478	4	4	z+	z+	PUNCT
ejpam-4478	4	5	.	.	PUNCT
ejpam-4478	5	1	a	a	DET
ejpam-4478	5	2	k	k	NOUN
ejpam-4478	5	3	−	−	PROPN
ejpam-4478	5	4	distance	distance	NOUN
ejpam-4478	5	5	roman	roman	ADJ
ejpam-4478	5	6	dominating	dominating	NOUN
ejpam-4478	5	7	function	function	NOUN
ejpam-4478	5	8	(	(	PUNCT
ejpam-4478	5	9	kdrdf	kdrdf	NOUN
ejpam-4478	5	10	)	)	PUNCT
ejpam-4478	5	11	on	on	ADP
ejpam-4478	5	12	g	g	PROPN
ejpam-4478	5	13	=	=	SYM
ejpam-4478	5	14	(	(	PUNCT
ejpam-4478	5	15	v	v	NOUN
ejpam-4478	5	16	,	,	PUNCT
ejpam-4478	5	17	e	e	NOUN
ejpam-4478	5	18	)	)	PUNCT
ejpam-4478	5	19	is	be	AUX
ejpam-4478	5	20	a	a	DET
ejpam-4478	5	21	function	function	NOUN
ejpam-4478	5	22	f	f	NOUN
ejpam-4478	5	23	:	:	PUNCT
ejpam-4478	5	24	v	v	X
ejpam-4478	5	25	→	→	SYM
ejpam-4478	5	26	{	{	PUNCT
ejpam-4478	5	27	0	0	NUM
ejpam-4478	5	28	,	,	PUNCT
ejpam-4478	5	29	1	1	NUM
ejpam-4478	5	30	,	,	PUNCT
ejpam-4478	5	31	2	2	NUM
ejpam-4478	5	32	}	}	PUNCT
ejpam-4478	5	33	such	such	ADJ
ejpam-4478	5	34	that	that	PRON
ejpam-4478	5	35	for	for	ADP
ejpam-4478	5	36	every	every	DET
ejpam-4478	5	37	vertex	vertex	NOUN
ejpam-4478	5	38	v	v	NOUN
ejpam-4478	5	39	with	with	ADP
ejpam-4478	5	40	f(v	f(v	NOUN
ejpam-4478	5	41	)	)	PUNCT
ejpam-4478	5	42	=	=	SYM
ejpam-4478	5	43	0	0	NUM
ejpam-4478	5	44	,	,	PUNCT
ejpam-4478	5	45	there	there	PRON
ejpam-4478	5	46	is	be	VERB
ejpam-4478	5	47	a	a	DET
ejpam-4478	5	48	vertex	vertex	NOUN
ejpam-4478	5	49	u	u	NOUN
ejpam-4478	5	50	with	with	ADP
ejpam-4478	5	51	f(u	f(u	PROPN
ejpam-4478	5	52	)	)	PUNCT
ejpam-4478	5	53	=	=	SYM
ejpam-4478	5	54	2	2	NUM
ejpam-4478	5	55	with	with	ADP
ejpam-4478	5	56	d(u	d(u	PROPN
ejpam-4478	5	57	,	,	PUNCT
ejpam-4478	5	58	v	v	NOUN
ejpam-4478	5	59	)	)	PUNCT
ejpam-4478	5	60	≤	≤	NOUN
ejpam-4478	5	61	k.	k.	VERB
ejpam-4478	6	1	the	the	DET
ejpam-4478	6	2	function	function	NOUN
ejpam-4478	6	3	f	f	PROPN
ejpam-4478	6	4	is	be	AUX
ejpam-4478	6	5	a	a	DET
ejpam-4478	6	6	global	global	ADJ
ejpam-4478	6	7	k	k	NOUN
ejpam-4478	6	8	−	−	PROPN
ejpam-4478	6	9	distance	distance	NOUN
ejpam-4478	6	10	roman	roman	ADJ
ejpam-4478	6	11	dominating	dominating	NOUN
ejpam-4478	6	12	function	function	NOUN
ejpam-4478	6	13	(	(	PUNCT
ejpam-4478	6	14	gkdrdf	gkdrdf	PROPN
ejpam-4478	6	15	)	)	PUNCT
ejpam-4478	6	16	on	on	ADP
ejpam-4478	6	17	g	g	PROPN
ejpam-4478	7	1	if	if	SCONJ
ejpam-4478	8	1	and	and	CCONJ
ejpam-4478	8	2	only	only	ADV
ejpam-4478	8	3	if	if	SCONJ
ejpam-4478	8	4	f	f	PROPN
ejpam-4478	8	5	is	be	AUX
ejpam-4478	8	6	a	a	DET
ejpam-4478	8	7	k	k	NOUN
ejpam-4478	8	8	−	−	PROPN
ejpam-4478	8	9	distance	distance	NOUN
ejpam-4478	8	10	roman	roman	ADJ
ejpam-4478	8	11	dominating	dominating	NOUN
ejpam-4478	8	12	function	function	NOUN
ejpam-4478	8	13	(	(	PUNCT
ejpam-4478	8	14	kdrdf	kdrdf	NOUN
ejpam-4478	8	15	)	)	PUNCT
ejpam-4478	8	16	on	on	ADP
ejpam-4478	8	17	g	g	PROPN
ejpam-4478	8	18	and	and	CCONJ
ejpam-4478	8	19	on	on	ADP
ejpam-4478	8	20	its	its	PRON
ejpam-4478	8	21	complement	complement	NOUN
ejpam-4478	8	22	g.	g.	NOUN
ejpam-4478	8	23	the	the	DET
ejpam-4478	8	24	weight	weight	NOUN
ejpam-4478	8	25	of	of	ADP
ejpam-4478	8	26	the	the	DET
ejpam-4478	8	27	global	global	ADJ
ejpam-4478	8	28	k	k	PROPN
ejpam-4478	8	29	−	−	PROPN
ejpam-4478	8	30	distance	distance	NOUN
ejpam-4478	8	31	roman	roman	ADJ
ejpam-4478	8	32	dominating	dominating	NOUN
ejpam-4478	8	33	function	function	NOUN
ejpam-4478	8	34	(	(	PUNCT
ejpam-4478	8	35	gkdrdf	gkdrdf	PROPN
ejpam-4478	8	36	)	)	PUNCT
ejpam-4478	9	1	f	f	PROPN
ejpam-4478	9	2	is	be	AUX
ejpam-4478	9	3	the	the	DET
ejpam-4478	9	4	value	value	NOUN
ejpam-4478	9	5	w(f	w(f	NOUN
ejpam-4478	9	6	)	)	PUNCT
ejpam-4478	10	1	=	=	SYM
ejpam-4478	10	2	∑	∑	PUNCT
ejpam-4478	10	3	x∈v	x∈v	PROPN
ejpam-4478	10	4	f(x	f(x	PROPN
ejpam-4478	10	5	)	)	PUNCT
ejpam-4478	10	6	.	.	PUNCT
ejpam-4478	11	1	the	the	DET
ejpam-4478	11	2	minimum	minimum	ADJ
ejpam-4478	11	3	weight	weight	NOUN
ejpam-4478	11	4	of	of	ADP
ejpam-4478	11	5	the	the	DET
ejpam-4478	11	6	global	global	ADJ
ejpam-4478	11	7	k	k	PROPN
ejpam-4478	11	8	−	−	PROPN
ejpam-4478	11	9	distance	distance	NOUN
ejpam-4478	11	10	roman	roman	ADJ
ejpam-4478	11	11	dominating	dominating	NOUN
ejpam-4478	11	12	function	function	NOUN
ejpam-4478	11	13	(	(	PUNCT
ejpam-4478	11	14	gkdrdf	gkdrdf	PROPN
ejpam-4478	11	15	)	)	PUNCT
ejpam-4478	11	16	on	on	ADP
ejpam-4478	11	17	the	the	DET
ejpam-4478	11	18	graph	graph	NOUN
ejpam-4478	11	19	g	g	NOUN
ejpam-4478	11	20	is	be	AUX
ejpam-4478	11	21	called	call	VERB
ejpam-4478	11	22	the	the	DET
ejpam-4478	11	23	global	global	ADJ
ejpam-4478	11	24	k	k	PROPN
ejpam-4478	11	25	−	−	PROPN
ejpam-4478	11	26	distance	distance	NOUN
ejpam-4478	11	27	roman	roman	ADJ
ejpam-4478	11	28	domination	domination	NOUN
ejpam-4478	11	29	number	number	NOUN
ejpam-4478	11	30	of	of	ADP
ejpam-4478	11	31	g	g	NOUN
ejpam-4478	11	32	and	and	CCONJ
ejpam-4478	11	33	is	be	AUX
ejpam-4478	11	34	denoted	denote	VERB
ejpam-4478	11	35	as	as	ADP
ejpam-4478	11	36	γk	γk	PROPN
ejpam-4478	11	37	gr(g	gr(g	PROPN
ejpam-4478	11	38	)	)	PUNCT
ejpam-4478	11	39	.	.	PUNCT
ejpam-4478	12	1	a	a	DET
ejpam-4478	12	2	γk	γk	NOUN
ejpam-4478	12	3	gr(g	gr(g	PUNCT
ejpam-4478	12	4	)	)	PUNCT
ejpam-4478	13	1	−	−	PRON
ejpam-4478	13	2	function	function	NOUN
ejpam-4478	13	3	is	be	AUX
ejpam-4478	13	4	the	the	DET
ejpam-4478	13	5	global	global	ADJ
ejpam-4478	13	6	k	k	PROPN
ejpam-4478	13	7	−	−	PROPN
ejpam-4478	13	8	distance	distance	NOUN
ejpam-4478	13	9	roman	roman	ADJ
ejpam-4478	13	10	dominating	dominating	NOUN
ejpam-4478	13	11	function	function	NOUN
ejpam-4478	13	12	on	on	ADP
ejpam-4478	13	13	g	g	PROPN
ejpam-4478	13	14	with	with	ADP
ejpam-4478	13	15	weight	weight	NOUN
ejpam-4478	13	16	γk	γk	PROPN
ejpam-4478	13	17	gr(g	gr(g	PROPN
ejpam-4478	13	18	)	)	PUNCT
ejpam-4478	13	19	.	.	PUNCT
ejpam-4478	14	1	note	note	VERB
ejpam-4478	14	2	that	that	SCONJ
ejpam-4478	14	3	,	,	PUNCT
ejpam-4478	14	4	the	the	DET
ejpam-4478	14	5	global	global	ADJ
ejpam-4478	14	6	1	1	NUM
ejpam-4478	14	7	−	−	NOUN
ejpam-4478	14	8	distance	distance	NOUN
ejpam-4478	14	9	roman	roman	ADJ
ejpam-4478	14	10	domination	domination	NOUN
ejpam-4478	14	11	number	number	NOUN
ejpam-4478	14	12	γ1	γ1	PROPN
ejpam-4478	14	13	gr(g	gr(g	PROPN
ejpam-4478	14	14	)	)	PUNCT
ejpam-4478	14	15	is	be	AUX
ejpam-4478	14	16	the	the	DET
ejpam-4478	14	17	usual	usual	ADJ
ejpam-4478	14	18	global	global	ADJ
ejpam-4478	14	19	roman	roman	ADJ
ejpam-4478	14	20	domination	domination	NOUN
ejpam-4478	14	21	number	number	NOUN
ejpam-4478	14	22	γgr(g	γgr(g	PROPN
ejpam-4478	14	23	)	)	PUNCT
ejpam-4478	14	24	,	,	PUNCT
ejpam-4478	14	25	that	that	ADV
ejpam-4478	14	26	is	is	ADV
ejpam-4478	14	27	,	,	PUNCT
ejpam-4478	14	28	γ1	γ1	PROPN
ejpam-4478	14	29	gr(g	gr(g	X
ejpam-4478	14	30	)	)	PUNCT
ejpam-4478	15	1	=	=	SYM
ejpam-4478	15	2	γgr(g	γgr(g	PROPN
ejpam-4478	15	3	)	)	PUNCT
ejpam-4478	15	4	.	.	PUNCT
ejpam-4478	16	1	the	the	DET
ejpam-4478	16	2	authors	author	NOUN
ejpam-4478	16	3	initiated	initiate	VERB
ejpam-4478	16	4	this	this	DET
ejpam-4478	16	5	study	study	NOUN
ejpam-4478	16	6	.	.	PUNCT
ejpam-4478	17	1	in	in	ADP
ejpam-4478	17	2	this	this	DET
ejpam-4478	17	3	paper	paper	NOUN
ejpam-4478	17	4	,	,	PUNCT
ejpam-4478	17	5	the	the	DET
ejpam-4478	17	6	authors	author	NOUN
ejpam-4478	17	7	obtained	obtain	VERB
ejpam-4478	17	8	and	and	CCONJ
ejpam-4478	17	9	established	establish	VERB
ejpam-4478	17	10	the	the	DET
ejpam-4478	17	11	following	follow	VERB
ejpam-4478	17	12	results	result	NOUN
ejpam-4478	17	13	:	:	PUNCT
ejpam-4478	17	14	preliminary	preliminary	ADJ
ejpam-4478	17	15	results	result	NOUN
ejpam-4478	17	16	on	on	ADP
ejpam-4478	17	17	global	global	ADJ
ejpam-4478	17	18	distance	distance	NOUN
ejpam-4478	17	19	roman	roman	ADJ
ejpam-4478	17	20	domination	domination	NOUN
ejpam-4478	17	21	;	;	PUNCT
ejpam-4478	17	22	the	the	DET
ejpam-4478	17	23	global	global	ADJ
ejpam-4478	17	24	distance	distance	NOUN
ejpam-4478	17	25	roman	roman	ADJ
ejpam-4478	17	26	domination	domination	NOUN
ejpam-4478	17	27	onkn	onkn	PROPN
ejpam-4478	17	28	,	,	PUNCT
ejpam-4478	17	29	kn	kn	PROPN
ejpam-4478	17	30	,	,	PUNCT
ejpam-4478	17	31	pn	pn	PROPN
ejpam-4478	17	32	,	,	PUNCT
ejpam-4478	17	33	and	and	CCONJ
ejpam-4478	17	34	cn	cn	INTJ
ejpam-4478	17	35	;	;	PUNCT
ejpam-4478	17	36	and	and	CCONJ
ejpam-4478	17	37	,	,	PUNCT
ejpam-4478	17	38	some	some	DET
ejpam-4478	17	39	bounds	bound	NOUN
ejpam-4478	17	40	and	and	CCONJ
ejpam-4478	17	41	characterizations	characterization	NOUN
ejpam-4478	17	42	of	of	ADP
ejpam-4478	17	43	global	global	ADJ
ejpam-4478	17	44	distance	distance	NOUN
ejpam-4478	17	45	roman	roman	ADJ
ejpam-4478	17	46	domination	domination	NOUN
ejpam-4478	17	47	over	over	ADP
ejpam-4478	17	48	any	any	DET
ejpam-4478	17	49	graphs	graph	NOUN
ejpam-4478	17	50	.	.	PUNCT
ejpam-4478	18	1	2020	2020	NUM
ejpam-4478	18	2	mathematics	mathematic	NOUN
ejpam-4478	18	3	subject	subject	NOUN
ejpam-4478	18	4	classifications	classification	NOUN
ejpam-4478	18	5	:	:	PUNCT
ejpam-4478	18	6	05a18	05a18	NUM
ejpam-4478	18	7	,	,	PUNCT
ejpam-4478	18	8	05c07	05c07	NOUN
ejpam-4478	18	9	,	,	PUNCT
ejpam-4478	18	10	05c12	05c12	NOUN
ejpam-4478	18	11	,	,	PUNCT
ejpam-4478	18	12	05c22	05c22	NOUN
ejpam-4478	18	13	,	,	PUNCT
ejpam-4478	18	14	05c38	05c38	NOUN
ejpam-4478	18	15	,	,	PUNCT
ejpam-4478	18	16	05c40	05c40	NOUN
ejpam-4478	18	17	,	,	PUNCT
ejpam-4478	18	18	05c62	05c62	NUM
ejpam-4478	18	19	,	,	PUNCT
ejpam-4478	18	20	05c69	05c69	NUM
ejpam-4478	18	21	,	,	PUNCT
ejpam-4478	18	22	05c70	05c70	NUM
ejpam-4478	18	23	,	,	PUNCT
ejpam-4478	18	24	05c76	05c76	PRON
ejpam-4478	18	25	,	,	PUNCT
ejpam-4478	18	26	05c78	05c78	NUM
ejpam-4478	18	27	,	,	PUNCT
ejpam-4478	18	28	97k30	97k30	NUM
ejpam-4478	18	29	key	key	ADJ
ejpam-4478	18	30	words	word	NOUN
ejpam-4478	18	31	and	and	CCONJ
ejpam-4478	18	32	phrases	phrase	NOUN
ejpam-4478	18	33	:	:	PUNCT
ejpam-4478	18	34	classical	classical	ADJ
ejpam-4478	18	35	domination	domination	NOUN
ejpam-4478	18	36	,	,	PUNCT
ejpam-4478	18	37	roman	roman	ADJ
ejpam-4478	18	38	domination	domination	NOUN
ejpam-4478	18	39	,	,	PUNCT
ejpam-4478	18	40	distance	distance	NOUN
ejpam-4478	18	41	domination	domination	NOUN
ejpam-4478	18	42	,	,	PUNCT
ejpam-4478	18	43	global	global	ADJ
ejpam-4478	18	44	domination	domination	NOUN
ejpam-4478	18	45	,	,	PUNCT
ejpam-4478	18	46	distance	distance	NOUN
ejpam-4478	18	47	roman	roman	ADJ
ejpam-4478	18	48	domination	domination	NOUN
ejpam-4478	18	49	,	,	PUNCT
ejpam-4478	18	50	global	global	ADJ
ejpam-4478	18	51	roman	roman	ADJ
ejpam-4478	18	52	domination	domination	NOUN
ejpam-4478	18	53	1	1	NUM
ejpam-4478	18	54	.	.	PUNCT
ejpam-4478	19	1	introduction	introduction	NOUN
ejpam-4478	19	2	mathematics	mathematics	PROPN
ejpam-4478	19	3	plays	play	VERB
ejpam-4478	19	4	a	a	DET
ejpam-4478	19	5	vital	vital	ADJ
ejpam-4478	19	6	role	role	NOUN
ejpam-4478	19	7	in	in	ADP
ejpam-4478	19	8	various	various	ADJ
ejpam-4478	19	9	fields	field	NOUN
ejpam-4478	19	10	.	.	PUNCT
ejpam-4478	20	1	one	one	NUM
ejpam-4478	20	2	of	of	ADP
ejpam-4478	20	3	the	the	DET
ejpam-4478	20	4	important	important	ADJ
ejpam-4478	20	5	areas	area	NOUN
ejpam-4478	20	6	in	in	ADP
ejpam-4478	20	7	mathematics	mathematics	PROPN
ejpam-4478	20	8	is	be	AUX
ejpam-4478	20	9	graph	graph	NOUN
ejpam-4478	20	10	theory	theory	NOUN
ejpam-4478	20	11	which	which	PRON
ejpam-4478	20	12	is	be	AUX
ejpam-4478	20	13	mainly	mainly	ADV
ejpam-4478	20	14	used	use	VERB
ejpam-4478	20	15	in	in	ADP
ejpam-4478	20	16	structural	structural	ADJ
ejpam-4478	20	17	models	model	NOUN
ejpam-4478	20	18	.	.	PUNCT
ejpam-4478	21	1	graph	graph	NOUN
ejpam-4478	21	2	theory	theory	NOUN
ejpam-4478	21	3	is	be	AUX
ejpam-4478	21	4	an	an	DET
ejpam-4478	21	5	interesting	interesting	ADJ
ejpam-4478	21	6	branch	branch	NOUN
ejpam-4478	21	7	of	of	ADP
ejpam-4478	21	8	mathematics	mathematic	NOUN
ejpam-4478	21	9	when	when	SCONJ
ejpam-4478	21	10	it	it	PRON
ejpam-4478	21	11	comes	come	VERB
ejpam-4478	21	12	to	to	ADP
ejpam-4478	21	13	research	research	NOUN
ejpam-4478	21	14	.	.	PUNCT
ejpam-4478	22	1	in	in	ADP
ejpam-4478	22	2	mathematics	mathematic	NOUN
ejpam-4478	22	3	and	and	CCONJ
ejpam-4478	22	4	computer	computer	NOUN
ejpam-4478	22	5	science	science	NOUN
ejpam-4478	22	6	,	,	PUNCT
ejpam-4478	22	7	graph	graph	NOUN
ejpam-4478	22	8	theory	theory	NOUN
ejpam-4478	22	9	is	be	AUX
ejpam-4478	22	10	the	the	DET
ejpam-4478	22	11	study	study	NOUN
ejpam-4478	22	12	of	of	ADP
ejpam-4478	22	13	graphs	graph	NOUN
ejpam-4478	22	14	which	which	PRON
ejpam-4478	22	15	are	be	AUX
ejpam-4478	22	16	mathematical	mathematical	ADJ
ejpam-4478	22	17	structures	structure	NOUN
ejpam-4478	22	18	used	use	VERB
ejpam-4478	22	19	∗corresponding	∗corresponde	VERB
ejpam-4478	22	20	author	author	NOUN
ejpam-4478	22	21	.	.	PUNCT
ejpam-4478	23	1	doi	doi	NOUN
ejpam-4478	23	2	:	:	PUNCT
ejpam-4478	23	3	https://doi.org/10.29020/nybg.ejpam.v16i1.4478	https://doi.org/10.29020/nybg.ejpam.v16i1.4478	PROPN
ejpam-4478	23	4	email	email	NOUN
ejpam-4478	23	5	addresses	address	NOUN
ejpam-4478	23	6	:	:	PUNCT
ejpam-4478	23	7	giovannieentero.ge@gmail.com	giovannieentero.ge@gmail.com	PROPN
ejpam-4478	23	8	(	(	PUNCT
ejpam-4478	23	9	g.	g.	PROPN
ejpam-4478	23	10	entero	entero	PROPN
ejpam-4478	23	11	)	)	PUNCT
ejpam-4478	23	12	,	,	PUNCT
ejpam-4478	23	13	saomega@usep.edu.ph	saomega@usep.edu.ph	PROPN
ejpam-4478	23	14	(	(	PUNCT
ejpam-4478	23	15	s.	s.	PROPN
ejpam-4478	23	16	espinola	espinola	PROPN
ejpam-4478	23	17	)	)	PUNCT
ejpam-4478	23	18	https://www.ejpam.com	https://www.ejpam.com	X
ejpam-4478	23	19	44	44	NUM
ejpam-4478	24	1	©	©	PROPN
ejpam-4478	24	2	2023	2023	NUM
ejpam-4478	24	3	ejpam	ejpam	NOUN
ejpam-4478	24	4	all	all	DET
ejpam-4478	24	5	rights	right	NOUN
ejpam-4478	24	6	reserved	reserve	VERB
ejpam-4478	24	7	.	.	PUNCT
ejpam-4478	25	1	g.	g.	PROPN
ejpam-4478	25	2	entero	entero	PROPN
ejpam-4478	25	3	,	,	PUNCT
ejpam-4478	25	4	s.	s.	PROPN
ejpam-4478	25	5	espinola	espinola	PROPN
ejpam-4478	25	6	/	/	SYM
ejpam-4478	25	7	eur	eur	PROPN
ejpam-4478	25	8	.	.	PUNCT
ejpam-4478	26	1	j.	j.	PROPN
ejpam-4478	26	2	pure	pure	PROPN
ejpam-4478	26	3	appl	appl	PROPN
ejpam-4478	26	4	.	.	PROPN
ejpam-4478	26	5	math	math	PROPN
ejpam-4478	26	6	,	,	PUNCT
ejpam-4478	26	7	16	16	NUM
ejpam-4478	26	8	(	(	PUNCT
ejpam-4478	26	9	1	1	NUM
ejpam-4478	26	10	)	)	PUNCT
ejpam-4478	26	11	(	(	PUNCT
ejpam-4478	26	12	2023	2023	NUM
ejpam-4478	26	13	)	)	PUNCT
ejpam-4478	26	14	,	,	PUNCT
ejpam-4478	26	15	44	44	NUM
ejpam-4478	26	16	-	-	SYM
ejpam-4478	26	17	61	61	NUM
ejpam-4478	26	18	45	45	NUM
ejpam-4478	26	19	to	to	PART
ejpam-4478	26	20	model	model	VERB
ejpam-4478	26	21	pairwise	pairwise	NOUN
ejpam-4478	26	22	relations	relation	NOUN
ejpam-4478	26	23	between	between	ADP
ejpam-4478	26	24	objects	object	NOUN
ejpam-4478	26	25	.	.	PUNCT
ejpam-4478	27	1	there	there	PRON
ejpam-4478	27	2	are	be	VERB
ejpam-4478	27	3	several	several	ADJ
ejpam-4478	27	4	areas	area	NOUN
ejpam-4478	27	5	in	in	ADP
ejpam-4478	27	6	graph	graph	NOUN
ejpam-4478	27	7	theory	theory	NOUN
ejpam-4478	27	8	in	in	ADP
ejpam-4478	27	9	which	which	PRON
ejpam-4478	27	10	extensive	extensive	ADJ
ejpam-4478	27	11	research	research	NOUN
ejpam-4478	27	12	activities	activity	NOUN
ejpam-4478	27	13	grow	grow	VERB
ejpam-4478	27	14	fast−one	fast−one	PROPN
ejpam-4478	27	15	of	of	ADP
ejpam-4478	27	16	which	which	PRON
ejpam-4478	27	17	is	be	AUX
ejpam-4478	27	18	domination	domination	NOUN
ejpam-4478	27	19	.	.	PUNCT
ejpam-4478	28	1	domination	domination	NOUN
ejpam-4478	28	2	is	be	AUX
ejpam-4478	28	3	a	a	DET
ejpam-4478	28	4	classical	classical	ADJ
ejpam-4478	28	5	and	and	CCONJ
ejpam-4478	28	6	an	an	DET
ejpam-4478	28	7	interesting	interesting	ADJ
ejpam-4478	28	8	topic	topic	NOUN
ejpam-4478	28	9	in	in	ADP
ejpam-4478	28	10	the	the	DET
ejpam-4478	28	11	theory	theory	NOUN
ejpam-4478	28	12	of	of	ADP
ejpam-4478	28	13	graphs	graph	NOUN
ejpam-4478	28	14	as	as	ADV
ejpam-4478	28	15	well	well	ADV
ejpam-4478	28	16	as	as	ADP
ejpam-4478	28	17	one	one	NUM
ejpam-4478	28	18	of	of	ADP
ejpam-4478	28	19	the	the	DET
ejpam-4478	28	20	most	most	ADV
ejpam-4478	28	21	active	active	ADJ
ejpam-4478	28	22	areas	area	NOUN
ejpam-4478	28	23	of	of	ADP
ejpam-4478	28	24	research	research	NOUN
ejpam-4478	28	25	in	in	ADP
ejpam-4478	28	26	this	this	DET
ejpam-4478	28	27	discipline	discipline	NOUN
ejpam-4478	28	28	.	.	PUNCT
ejpam-4478	29	1	the	the	DET
ejpam-4478	29	2	increasing	increase	VERB
ejpam-4478	29	3	interest	interest	NOUN
ejpam-4478	29	4	in	in	ADP
ejpam-4478	29	5	this	this	DET
ejpam-4478	29	6	area	area	NOUN
ejpam-4478	29	7	is	be	AUX
ejpam-4478	29	8	partly	partly	ADV
ejpam-4478	29	9	explained	explain	VERB
ejpam-4478	29	10	by	by	ADP
ejpam-4478	29	11	the	the	DET
ejpam-4478	29	12	diversity	diversity	NOUN
ejpam-4478	29	13	of	of	ADP
ejpam-4478	29	14	its	its	PRON
ejpam-4478	29	15	applications	application	NOUN
ejpam-4478	29	16	to	to	ADP
ejpam-4478	29	17	both	both	CCONJ
ejpam-4478	29	18	theoretical	theoretical	ADJ
ejpam-4478	29	19	and	and	CCONJ
ejpam-4478	29	20	real	real	ADJ
ejpam-4478	29	21	-	-	PUNCT
ejpam-4478	29	22	world	world	NOUN
ejpam-4478	29	23	problems	problem	NOUN
ejpam-4478	29	24	.	.	PUNCT
ejpam-4478	30	1	domination	domination	NOUN
ejpam-4478	30	2	comes	come	VERB
ejpam-4478	30	3	with	with	ADP
ejpam-4478	30	4	one	one	NUM
ejpam-4478	30	5	of	of	ADP
ejpam-4478	30	6	its	its	PRON
ejpam-4478	30	7	most	most	ADV
ejpam-4478	30	8	famous	famous	ADJ
ejpam-4478	30	9	variants	variant	NOUN
ejpam-4478	30	10	called	call	VERB
ejpam-4478	30	11	roman	roman	ADJ
ejpam-4478	30	12	domination−the	domination−the	DET
ejpam-4478	30	13	defense	defense	NOUN
ejpam-4478	30	14	strategy	strategy	NOUN
ejpam-4478	30	15	(	(	PUNCT
ejpam-4478	30	16	a.k.a	a.k.a	INTJ
ejpam-4478	30	17	.	.	PROPN
ejpam-4478	30	18	protection	protection	NOUN
ejpam-4478	30	19	strategy	strategy	NOUN
ejpam-4478	30	20	)	)	PUNCT
ejpam-4478	30	21	employed	employ	VERB
ejpam-4478	30	22	by	by	ADP
ejpam-4478	30	23	emperor	emperor	NOUN
ejpam-4478	30	24	constantine	constantine	PROPN
ejpam-4478	30	25	the	the	DET
ejpam-4478	30	26	great	great	ADJ
ejpam-4478	30	27	to	to	PART
ejpam-4478	30	28	defend	defend	VERB
ejpam-4478	30	29	the	the	DET
ejpam-4478	30	30	roman	roman	ADJ
ejpam-4478	30	31	empire	empire	NOUN
ejpam-4478	30	32	when	when	SCONJ
ejpam-4478	30	33	it	it	PRON
ejpam-4478	30	34	was	be	AUX
ejpam-4478	30	35	under	under	ADP
ejpam-4478	30	36	a	a	DET
ejpam-4478	30	37	certain	certain	ADJ
ejpam-4478	30	38	attack	attack	NOUN
ejpam-4478	30	39	.	.	PUNCT
ejpam-4478	31	1	it	it	PRON
ejpam-4478	31	2	was	be	AUX
ejpam-4478	31	3	traced	trace	VERB
ejpam-4478	31	4	back	back	ADV
ejpam-4478	31	5	that	that	SCONJ
ejpam-4478	31	6	,	,	PUNCT
ejpam-4478	31	7	in	in	ADP
ejpam-4478	31	8	the	the	DET
ejpam-4478	31	9	4th	4th	ADJ
ejpam-4478	31	10	century	century	NOUN
ejpam-4478	31	11	a.	a.	NOUN
ejpam-4478	31	12	d.	d.	PROPN
ejpam-4478	31	13	,	,	PUNCT
ejpam-4478	31	14	when	when	SCONJ
ejpam-4478	31	15	the	the	DET
ejpam-4478	31	16	roman	roman	ADJ
ejpam-4478	31	17	empire	empire	NOUN
ejpam-4478	31	18	was	be	AUX
ejpam-4478	31	19	under	under	ADP
ejpam-4478	31	20	attack	attack	NOUN
ejpam-4478	31	21	during	during	ADP
ejpam-4478	31	22	the	the	DET
ejpam-4478	31	23	period	period	NOUN
ejpam-4478	31	24	of	of	ADP
ejpam-4478	31	25	emperor	emperor	NOUN
ejpam-4478	31	26	constantine	constantine	PROPN
ejpam-4478	31	27	the	the	DET
ejpam-4478	31	28	great	great	ADJ
ejpam-4478	31	29	,	,	PUNCT
ejpam-4478	31	30	he	he	PRON
ejpam-4478	31	31	had	have	VERB
ejpam-4478	31	32	the	the	DET
ejpam-4478	31	33	requirement	requirement	NOUN
ejpam-4478	31	34	that	that	SCONJ
ejpam-4478	31	35	any	any	DET
ejpam-4478	31	36	army	army	NOUN
ejpam-4478	31	37	or	or	CCONJ
ejpam-4478	31	38	a	a	DET
ejpam-4478	31	39	legion	legion	NOUN
ejpam-4478	31	40	could	could	AUX
ejpam-4478	31	41	be	be	AUX
ejpam-4478	31	42	sent	send	VERB
ejpam-4478	31	43	from	from	ADP
ejpam-4478	31	44	its	its	PRON
ejpam-4478	31	45	home	home	NOUN
ejpam-4478	31	46	to	to	PART
ejpam-4478	31	47	defend	defend	VERB
ejpam-4478	31	48	a	a	DET
ejpam-4478	31	49	neighboring	neighboring	NOUN
ejpam-4478	31	50	location	location	NOUN
ejpam-4478	31	51	only	only	ADV
ejpam-4478	31	52	if	if	SCONJ
ejpam-4478	31	53	there	there	PRON
ejpam-4478	31	54	was	be	VERB
ejpam-4478	31	55	a	a	DET
ejpam-4478	31	56	second	second	ADJ
ejpam-4478	31	57	army	army	NOUN
ejpam-4478	31	58	which	which	PRON
ejpam-4478	31	59	would	would	AUX
ejpam-4478	31	60	stay	stay	VERB
ejpam-4478	31	61	and	and	CCONJ
ejpam-4478	31	62	protect	protect	VERB
ejpam-4478	31	63	the	the	DET
ejpam-4478	31	64	home	home	NOUN
ejpam-4478	32	1	[	[	X
ejpam-4478	32	2	12	12	NUM
ejpam-4478	32	3	]	]	PUNCT
ejpam-4478	32	4	.	.	PUNCT
ejpam-4478	33	1	thus	thus	ADV
ejpam-4478	33	2	,	,	PUNCT
ejpam-4478	33	3	there	there	PRON
ejpam-4478	33	4	are	be	VERB
ejpam-4478	33	5	two	two	NUM
ejpam-4478	33	6	types	type	NOUN
ejpam-4478	33	7	of	of	ADP
ejpam-4478	33	8	armies	army	NOUN
ejpam-4478	33	9	−	−	NOUN
ejpam-4478	33	10	traveling	travel	VERB
ejpam-4478	33	11	and	and	CCONJ
ejpam-4478	33	12	stationary	stationary	NOUN
ejpam-4478	33	13	.	.	PUNCT
ejpam-4478	34	1	the	the	DET
ejpam-4478	34	2	first	first	ADJ
ejpam-4478	34	3	type	type	NOUN
ejpam-4478	34	4	of	of	ADP
ejpam-4478	34	5	legion	legion	NOUN
ejpam-4478	34	6	was	be	AUX
ejpam-4478	34	7	particularly	particularly	ADV
ejpam-4478	34	8	skilled	skilled	ADJ
ejpam-4478	34	9	agile	agile	ADJ
ejpam-4478	34	10	combatants	combatant	NOUN
ejpam-4478	34	11	who	who	PRON
ejpam-4478	34	12	could	could	AUX
ejpam-4478	34	13	be	be	AUX
ejpam-4478	34	14	promptly	promptly	ADV
ejpam-4478	34	15	deployed	deploy	VERB
ejpam-4478	34	16	to	to	ADP
ejpam-4478	34	17	an	an	DET
ejpam-4478	34	18	adjacent	adjacent	ADJ
ejpam-4478	34	19	province	province	NOUN
ejpam-4478	34	20	for	for	ADP
ejpam-4478	34	21	defending	defend	VERB
ejpam-4478	34	22	against	against	ADP
ejpam-4478	34	23	any	any	DET
ejpam-4478	34	24	potential	potential	ADJ
ejpam-4478	34	25	attack	attack	NOUN
ejpam-4478	34	26	.	.	PUNCT
ejpam-4478	35	1	the	the	DET
ejpam-4478	35	2	latter	latter	ADJ
ejpam-4478	35	3	would	would	AUX
ejpam-4478	35	4	behave	behave	VERB
ejpam-4478	35	5	as	as	ADP
ejpam-4478	35	6	a	a	DET
ejpam-4478	35	7	local	local	ADJ
ejpam-4478	35	8	force	force	NOUN
ejpam-4478	35	9	permanently	permanently	ADV
ejpam-4478	35	10	located	locate	VERB
ejpam-4478	35	11	in	in	ADP
ejpam-4478	35	12	the	the	DET
ejpam-4478	35	13	given	give	VERB
ejpam-4478	35	14	province	province	NOUN
ejpam-4478	35	15	.	.	PUNCT
ejpam-4478	36	1	in	in	ADP
ejpam-4478	36	2	addition	addition	NOUN
ejpam-4478	36	3	,	,	PUNCT
ejpam-4478	36	4	no	no	DET
ejpam-4478	36	5	legion	legion	NOUN
ejpam-4478	36	6	could	could	AUX
ejpam-4478	36	7	ever	ever	ADV
ejpam-4478	36	8	depart	depart	VERB
ejpam-4478	36	9	a	a	DET
ejpam-4478	36	10	province	province	NOUN
ejpam-4478	36	11	in	in	ADP
ejpam-4478	36	12	order	order	NOUN
ejpam-4478	36	13	to	to	PART
ejpam-4478	36	14	defend	defend	VERB
ejpam-4478	36	15	another	another	DET
ejpam-4478	36	16	one	one	NOUN
ejpam-4478	36	17	if	if	SCONJ
ejpam-4478	36	18	such	such	ADJ
ejpam-4478	36	19	action	action	NOUN
ejpam-4478	36	20	leaves	leave	VERB
ejpam-4478	36	21	the	the	DET
ejpam-4478	36	22	base	base	NOUN
ejpam-4478	36	23	province	province	NOUN
ejpam-4478	36	24	unprotected	unprotecte	VERB
ejpam-4478	37	1	[	[	X
ejpam-4478	37	2	7	7	NUM
ejpam-4478	37	3	]	]	PUNCT
ejpam-4478	37	4	.	.	PUNCT
ejpam-4478	38	1	translating	translate	VERB
ejpam-4478	38	2	this	this	DET
ejpam-4478	38	3	strategy	strategy	NOUN
ejpam-4478	38	4	into	into	ADP
ejpam-4478	38	5	the	the	DET
ejpam-4478	38	6	language	language	NOUN
ejpam-4478	38	7	of	of	ADP
ejpam-4478	38	8	graph	graph	NOUN
ejpam-4478	38	9	theory	theory	NOUN
ejpam-4478	38	10	,	,	PUNCT
ejpam-4478	38	11	each	each	DET
ejpam-4478	38	12	vertex	vertex	NOUN
ejpam-4478	38	13	with	with	ADP
ejpam-4478	38	14	no	no	DET
ejpam-4478	38	15	army	army	NOUN
ejpam-4478	38	16	must	must	AUX
ejpam-4478	38	17	have	have	VERB
ejpam-4478	38	18	a	a	DET
ejpam-4478	38	19	neighboring	neighboring	NOUN
ejpam-4478	38	20	vertex	vertex	NOUN
ejpam-4478	38	21	with	with	ADP
ejpam-4478	38	22	a	a	DET
ejpam-4478	38	23	traveling	travel	VERB
ejpam-4478	38	24	army	army	NOUN
ejpam-4478	38	25	.	.	PUNCT
ejpam-4478	39	1	stationary	stationary	ADJ
ejpam-4478	39	2	armies	army	NOUN
ejpam-4478	39	3	then	then	ADV
ejpam-4478	39	4	dominate	dominate	VERB
ejpam-4478	39	5	their	their	PRON
ejpam-4478	39	6	own	own	ADJ
ejpam-4478	39	7	vertices	vertex	NOUN
ejpam-4478	39	8	and	and	CCONJ
ejpam-4478	39	9	a	a	DET
ejpam-4478	39	10	vertex	vertex	NOUN
ejpam-4478	39	11	with	with	ADP
ejpam-4478	39	12	two	two	NUM
ejpam-4478	39	13	armies	army	NOUN
ejpam-4478	39	14	is	be	AUX
ejpam-4478	39	15	dominated	dominate	VERB
ejpam-4478	39	16	by	by	ADP
ejpam-4478	39	17	its	its	PRON
ejpam-4478	39	18	stationary	stationary	ADJ
ejpam-4478	39	19	army	army	NOUN
ejpam-4478	39	20	and	and	CCONJ
ejpam-4478	39	21	its	its	PRON
ejpam-4478	39	22	open	open	ADJ
ejpam-4478	39	23	neighborhood	neighborhood	NOUN
ejpam-4478	39	24	is	be	AUX
ejpam-4478	39	25	dominated	dominate	VERB
ejpam-4478	39	26	by	by	ADP
ejpam-4478	39	27	the	the	DET
ejpam-4478	39	28	traveling	travel	VERB
ejpam-4478	39	29	army	army	NOUN
ejpam-4478	40	1	[	[	X
ejpam-4478	40	2	12	12	NUM
ejpam-4478	40	3	]	]	PUNCT
ejpam-4478	40	4	.	.	PUNCT
ejpam-4478	41	1	cockayne	cockayne	NOUN
ejpam-4478	41	2	et	et	PROPN
ejpam-4478	41	3	al	al	PROPN
ejpam-4478	41	4	.	.	PUNCT
ejpam-4478	42	1	[	[	X
ejpam-4478	42	2	4	4	X
ejpam-4478	42	3	]	]	PUNCT
ejpam-4478	42	4	introduced	introduce	VERB
ejpam-4478	42	5	a	a	DET
ejpam-4478	42	6	variant	variant	NOUN
ejpam-4478	42	7	of	of	ADP
ejpam-4478	42	8	domination	domination	NOUN
ejpam-4478	42	9	called	call	VERB
ejpam-4478	42	10	roman	roman	ADJ
ejpam-4478	42	11	domination	domination	NOUN
ejpam-4478	42	12	suggested	suggest	VERB
ejpam-4478	42	13	by	by	ADP
ejpam-4478	42	14	the	the	DET
ejpam-4478	42	15	recent	recent	ADJ
ejpam-4478	42	16	article	article	NOUN
ejpam-4478	42	17	in	in	ADP
ejpam-4478	42	18	scientific	scientific	ADJ
ejpam-4478	42	19	american	american	PROPN
ejpam-4478	42	20	by	by	ADP
ejpam-4478	42	21	ian	ian	PROPN
ejpam-4478	42	22	stewart	stewart	PROPN
ejpam-4478	42	23	,	,	PUNCT
ejpam-4478	42	24	entitled	entitle	VERB
ejpam-4478	42	25	“	"	PUNCT
ejpam-4478	42	26	defend	defend	VERB
ejpam-4478	42	27	the	the	DET
ejpam-4478	42	28	roman	roman	ADJ
ejpam-4478	42	29	empire	empire	NOUN
ejpam-4478	42	30	”	"	PUNCT
ejpam-4478	42	31	as	as	SCONJ
ejpam-4478	42	32	mentioned	mention	VERB
ejpam-4478	42	33	in	in	ADP
ejpam-4478	42	34	the	the	DET
ejpam-4478	42	35	previous	previous	ADJ
ejpam-4478	42	36	paragraph	paragraph	NOUN
ejpam-4478	42	37	.	.	PUNCT
ejpam-4478	43	1	according	accord	VERB
ejpam-4478	43	2	to	to	ADP
ejpam-4478	43	3	the	the	DET
ejpam-4478	43	4	mentioned	mention	VERB
ejpam-4478	43	5	authors	author	NOUN
ejpam-4478	43	6	,	,	PUNCT
ejpam-4478	43	7	the	the	DET
ejpam-4478	43	8	roman	roman	ADJ
ejpam-4478	43	9	dominating	dominating	NOUN
ejpam-4478	43	10	function	function	NOUN
ejpam-4478	43	11	(	(	PUNCT
ejpam-4478	43	12	rdf	rdf	NOUN
ejpam-4478	43	13	)	)	PUNCT
ejpam-4478	43	14	on	on	ADP
ejpam-4478	43	15	the	the	DET
ejpam-4478	43	16	graph	graph	NOUN
ejpam-4478	43	17	g	g	PROPN
ejpam-4478	43	18	=	=	SYM
ejpam-4478	43	19	(	(	PUNCT
ejpam-4478	43	20	v	v	NOUN
ejpam-4478	43	21	,	,	PUNCT
ejpam-4478	43	22	e	e	NOUN
ejpam-4478	43	23	)	)	PUNCT
ejpam-4478	43	24	is	be	AUX
ejpam-4478	43	25	the	the	DET
ejpam-4478	43	26	function	function	NOUN
ejpam-4478	43	27	f	f	NOUN
ejpam-4478	43	28	:	:	PUNCT
ejpam-4478	43	29	v	v	X
ejpam-4478	43	30	→	→	SYM
ejpam-4478	43	31	{	{	PUNCT
ejpam-4478	43	32	0	0	NUM
ejpam-4478	43	33	,	,	PUNCT
ejpam-4478	43	34	1	1	NUM
ejpam-4478	43	35	,	,	PUNCT
ejpam-4478	43	36	2	2	NUM
ejpam-4478	43	37	}	}	PUNCT
ejpam-4478	43	38	satisfying	satisfy	VERB
ejpam-4478	43	39	the	the	DET
ejpam-4478	43	40	condition	condition	NOUN
ejpam-4478	43	41	that	that	SCONJ
ejpam-4478	43	42	every	every	DET
ejpam-4478	43	43	vertex	vertex	NOUN
ejpam-4478	43	44	v	v	NOUN
ejpam-4478	43	45	for	for	ADP
ejpam-4478	43	46	which	which	PRON
ejpam-4478	43	47	f(v	f(v	NOUN
ejpam-4478	43	48	)	)	PUNCT
ejpam-4478	43	49	=	=	SYM
ejpam-4478	43	50	0	0	PUNCT
ejpam-4478	43	51	is	be	AUX
ejpam-4478	43	52	adjacent	adjacent	ADJ
ejpam-4478	43	53	to	to	ADP
ejpam-4478	43	54	at	at	ADV
ejpam-4478	43	55	least	least	ADV
ejpam-4478	43	56	one	one	NUM
ejpam-4478	43	57	vertex	vertex	NOUN
ejpam-4478	43	58	u	u	NOUN
ejpam-4478	43	59	for	for	ADP
ejpam-4478	43	60	which	which	PRON
ejpam-4478	43	61	f(u	f(u	PROPN
ejpam-4478	43	62	)	)	PUNCT
ejpam-4478	43	63	=	=	SYM
ejpam-4478	43	64	2	2	X
ejpam-4478	43	65	.	.	X
ejpam-4478	43	66	roman	roman	ADJ
ejpam-4478	43	67	domination	domination	NOUN
ejpam-4478	43	68	also	also	ADV
ejpam-4478	43	69	comes	come	VERB
ejpam-4478	43	70	with	with	ADP
ejpam-4478	43	71	various	various	ADJ
ejpam-4478	43	72	varieties	variety	NOUN
ejpam-4478	43	73	,	,	PUNCT
ejpam-4478	43	74	one	one	NUM
ejpam-4478	43	75	of	of	ADP
ejpam-4478	43	76	its	its	PRON
ejpam-4478	43	77	variants	variant	NOUN
ejpam-4478	43	78	that	that	PRON
ejpam-4478	43	79	caught	catch	VERB
ejpam-4478	43	80	the	the	DET
ejpam-4478	43	81	authors	author	NOUN
ejpam-4478	43	82	’	’	PART
ejpam-4478	43	83	attention	attention	NOUN
ejpam-4478	43	84	is	be	AUX
ejpam-4478	43	85	distance	distance	NOUN
ejpam-4478	43	86	roman	roman	ADJ
ejpam-4478	43	87	domination	domination	NOUN
ejpam-4478	43	88	initiated	initiate	VERB
ejpam-4478	43	89	by	by	ADP
ejpam-4478	43	90	aram	aram	PROPN
ejpam-4478	43	91	et	et	PROPN
ejpam-4478	43	92	al	al	PROPN
ejpam-4478	43	93	.	.	PUNCT
ejpam-4478	44	1	[	[	X
ejpam-4478	44	2	2	2	NUM
ejpam-4478	44	3	]	]	PUNCT
ejpam-4478	44	4	.	.	PUNCT
ejpam-4478	45	1	aram	aram	PROPN
ejpam-4478	45	2	et	et	PROPN
ejpam-4478	45	3	al	al	PROPN
ejpam-4478	45	4	.	.	PUNCT
ejpam-4478	46	1	[	[	X
ejpam-4478	46	2	2	2	X
ejpam-4478	46	3	]	]	PUNCT
ejpam-4478	46	4	defined	define	VERB
ejpam-4478	46	5	that	that	SCONJ
ejpam-4478	46	6	the	the	DET
ejpam-4478	46	7	k	k	PROPN
ejpam-4478	46	8	−	−	PROPN
ejpam-4478	46	9	distance	distance	NOUN
ejpam-4478	46	10	roman	roman	ADJ
ejpam-4478	46	11	dominating	dominating	NOUN
ejpam-4478	46	12	function	function	NOUN
ejpam-4478	46	13	(	(	PUNCT
ejpam-4478	46	14	kdrdf	kdrdf	PROPN
ejpam-4478	46	15	)	)	PUNCT
ejpam-4478	46	16	on	on	ADP
ejpam-4478	46	17	the	the	DET
ejpam-4478	46	18	graph	graph	NOUN
ejpam-4478	46	19	g	g	PROPN
ejpam-4478	46	20	=	=	SYM
ejpam-4478	46	21	(	(	PUNCT
ejpam-4478	46	22	v	v	NOUN
ejpam-4478	46	23	,	,	PUNCT
ejpam-4478	46	24	e	e	NOUN
ejpam-4478	46	25	)	)	PUNCT
ejpam-4478	46	26	is	be	AUX
ejpam-4478	46	27	the	the	DET
ejpam-4478	46	28	function	function	NOUN
ejpam-4478	46	29	f	f	NOUN
ejpam-4478	46	30	:	:	PUNCT
ejpam-4478	46	31	v	v	X
ejpam-4478	46	32	→	→	SYM
ejpam-4478	46	33	{	{	PUNCT
ejpam-4478	46	34	0	0	NUM
ejpam-4478	46	35	,	,	PUNCT
ejpam-4478	46	36	1	1	NUM
ejpam-4478	46	37	,	,	PUNCT
ejpam-4478	46	38	2	2	NUM
ejpam-4478	46	39	}	}	PUNCT
ejpam-4478	46	40	satisfying	satisfy	VERB
ejpam-4478	46	41	the	the	DET
ejpam-4478	46	42	conditions	condition	NOUN
ejpam-4478	46	43	that	that	PRON
ejpam-4478	46	44	for	for	ADP
ejpam-4478	46	45	every	every	DET
ejpam-4478	46	46	vertex	vertex	NOUN
ejpam-4478	46	47	v	v	NOUN
ejpam-4478	46	48	for	for	ADP
ejpam-4478	46	49	which	which	PRON
ejpam-4478	46	50	f(v	f(v	NOUN
ejpam-4478	46	51	)	)	PUNCT
ejpam-4478	46	52	=	=	SYM
ejpam-4478	46	53	0	0	NUM
ejpam-4478	46	54	,	,	PUNCT
ejpam-4478	46	55	there	there	PRON
ejpam-4478	46	56	is	be	VERB
ejpam-4478	46	57	a	a	DET
ejpam-4478	46	58	vertex	vertex	NOUN
ejpam-4478	46	59	u	u	NOUN
ejpam-4478	46	60	for	for	ADP
ejpam-4478	46	61	which	which	PRON
ejpam-4478	46	62	f(u	f(u	PROPN
ejpam-4478	46	63	)	)	PUNCT
ejpam-4478	46	64	=	=	SYM
ejpam-4478	46	65	2	2	NUM
ejpam-4478	46	66	and	and	CCONJ
ejpam-4478	46	67	d(u	d(u	PROPN
ejpam-4478	46	68	,	,	PUNCT
ejpam-4478	46	69	v	v	NOUN
ejpam-4478	46	70	)	)	PUNCT
ejpam-4478	46	71	≤	≤	NOUN
ejpam-4478	47	1	k	k	PROPN
ejpam-4478	47	2	,	,	PUNCT
ejpam-4478	47	3	where	where	SCONJ
ejpam-4478	47	4	d(u	d(u	PROPN
ejpam-4478	47	5	,	,	PUNCT
ejpam-4478	47	6	v	v	NOUN
ejpam-4478	47	7	)	)	PUNCT
ejpam-4478	47	8	is	be	AUX
ejpam-4478	47	9	the	the	DET
ejpam-4478	47	10	distance	distance	NOUN
ejpam-4478	47	11	from	from	ADP
ejpam-4478	47	12	u	u	PRON
ejpam-4478	47	13	to	to	ADP
ejpam-4478	47	14	v.	v.	CCONJ
ejpam-4478	47	15	additionally	additionally	ADV
ejpam-4478	47	16	,	,	PUNCT
ejpam-4478	47	17	the	the	DET
ejpam-4478	47	18	weight	weight	NOUN
ejpam-4478	47	19	of	of	ADP
ejpam-4478	47	20	kdrdf	kdrdf	NOUN
ejpam-4478	47	21	f	f	PROPN
ejpam-4478	47	22	is	be	AUX
ejpam-4478	47	23	the	the	DET
ejpam-4478	47	24	value	value	NOUN
ejpam-4478	47	25	w(f	w(f	NOUN
ejpam-4478	47	26	)	)	PUNCT
ejpam-4478	47	27	=	=	SYM
ejpam-4478	48	1	∑	∑	PUNCT
ejpam-4478	48	2	u∈v	u∈v	NOUN
ejpam-4478	48	3	f(u	f(u	PROPN
ejpam-4478	48	4	)	)	PUNCT
ejpam-4478	48	5	and	and	CCONJ
ejpam-4478	48	6	the	the	DET
ejpam-4478	48	7	minimum	minimum	ADJ
ejpam-4478	48	8	weight	weight	NOUN
ejpam-4478	48	9	of	of	ADP
ejpam-4478	48	10	the	the	DET
ejpam-4478	48	11	kdrdf	kdrdf	NOUN
ejpam-4478	48	12	on	on	ADP
ejpam-4478	48	13	g	g	PROPN
ejpam-4478	48	14	will	will	AUX
ejpam-4478	48	15	be	be	AUX
ejpam-4478	48	16	the	the	DET
ejpam-4478	48	17	k	k	PROPN
ejpam-4478	48	18	−	−	PROPN
ejpam-4478	48	19	distance	distance	NOUN
ejpam-4478	48	20	roman	roman	ADJ
ejpam-4478	48	21	domination	domination	NOUN
ejpam-4478	48	22	number	number	NOUN
ejpam-4478	48	23	and	and	CCONJ
ejpam-4478	48	24	is	be	AUX
ejpam-4478	48	25	denoted	denote	VERB
ejpam-4478	48	26	by	by	ADP
ejpam-4478	48	27	γkr(g	γkr(g	PROPN
ejpam-4478	48	28	)	)	PUNCT
ejpam-4478	48	29	,	,	PUNCT
ejpam-4478	48	30	where	where	SCONJ
ejpam-4478	48	31	k	k	PROPN
ejpam-4478	48	32	∈	∈	PROPN
ejpam-4478	48	33	z+	z+	PUNCT
ejpam-4478	48	34	.	.	PUNCT
ejpam-4478	49	1	all	all	DET
ejpam-4478	49	2	graphs	graph	NOUN
ejpam-4478	49	3	considered	consider	VERB
ejpam-4478	49	4	in	in	ADP
ejpam-4478	49	5	this	this	DET
ejpam-4478	49	6	paper	paper	NOUN
ejpam-4478	49	7	are	be	AUX
ejpam-4478	49	8	all	all	PRON
ejpam-4478	49	9	finite	finite	ADJ
ejpam-4478	49	10	,	,	PUNCT
ejpam-4478	49	11	simple	simple	ADJ
ejpam-4478	49	12	,	,	PUNCT
ejpam-4478	49	13	and	and	CCONJ
ejpam-4478	49	14	undirected	undirected	ADJ
ejpam-4478	49	15	.	.	PUNCT
ejpam-4478	50	1	let	let	VERB
ejpam-4478	50	2	g	g	PROPN
ejpam-4478	50	3	=	=	SYM
ejpam-4478	50	4	(	(	PUNCT
ejpam-4478	50	5	v	v	NOUN
ejpam-4478	50	6	,	,	PUNCT
ejpam-4478	50	7	e	e	NOUN
ejpam-4478	50	8	)	)	PUNCT
ejpam-4478	50	9	be	be	AUX
ejpam-4478	50	10	a	a	DET
ejpam-4478	50	11	finite	finite	NOUN
ejpam-4478	50	12	,	,	PUNCT
ejpam-4478	50	13	simple	simple	ADJ
ejpam-4478	50	14	,	,	PUNCT
ejpam-4478	50	15	and	and	CCONJ
ejpam-4478	50	16	undirected	undirected	ADJ
ejpam-4478	50	17	graph	graph	NOUN
ejpam-4478	50	18	.	.	PUNCT
ejpam-4478	51	1	the	the	DET
ejpam-4478	51	2	graphg	graphg	NOUN
ejpam-4478	51	3	has	have	VERB
ejpam-4478	51	4	vertex	vertex	NOUN
ejpam-4478	51	5	set	set	VERB
ejpam-4478	51	6	v	v	ADP
ejpam-4478	51	7	=	=	SYM
ejpam-4478	51	8	v	v	NOUN
ejpam-4478	51	9	(	(	PUNCT
ejpam-4478	51	10	g	g	NOUN
ejpam-4478	51	11	)	)	PUNCT
ejpam-4478	51	12	and	and	CCONJ
ejpam-4478	51	13	edge	edge	VERB
ejpam-4478	51	14	set	set	VERB
ejpam-4478	51	15	e	e	NOUN
ejpam-4478	51	16	=	=	PROPN
ejpam-4478	51	17	e(g	e(g	PROPN
ejpam-4478	51	18	)	)	PUNCT
ejpam-4478	51	19	.	.	PUNCT
ejpam-4478	52	1	further	far	ADV
ejpam-4478	52	2	,	,	PUNCT
ejpam-4478	52	3	let	let	VERB
ejpam-4478	52	4	the	the	DET
ejpam-4478	52	5	order	order	NOUN
ejpam-4478	52	6	of	of	ADP
ejpam-4478	52	7	the	the	DET
ejpam-4478	52	8	graph	graph	NOUN
ejpam-4478	52	9	g	g	PROPN
ejpam-4478	52	10	be	be	AUX
ejpam-4478	52	11	p	p	NOUN
ejpam-4478	52	12	,	,	PUNCT
ejpam-4478	52	13	that	that	ADV
ejpam-4478	52	14	is	is	ADV
ejpam-4478	52	15	,	,	PUNCT
ejpam-4478	52	16	|v	|v	PROPN
ejpam-4478	52	17	|	|	ADV
ejpam-4478	52	18	=	=	SYM
ejpam-4478	52	19	|v	|v	PROPN
ejpam-4478	52	20	(	(	PUNCT
ejpam-4478	52	21	g)|	g)|	NOUN
ejpam-4478	52	22	=	=	PUNCT
ejpam-4478	52	23	p	p	PROPN
ejpam-4478	52	24	and	and	CCONJ
ejpam-4478	52	25	the	the	DET
ejpam-4478	52	26	size	size	NOUN
ejpam-4478	52	27	be	be	VERB
ejpam-4478	52	28	q	q	ADJ
ejpam-4478	52	29	,	,	PUNCT
ejpam-4478	52	30	that	that	ADV
ejpam-4478	52	31	is	is	ADV
ejpam-4478	52	32	,	,	PUNCT
ejpam-4478	52	33	|e|	|e|	DET
ejpam-4478	52	34	=	=	PUNCT
ejpam-4478	52	35	|e(g)|	|e(g)|	PROPN
ejpam-4478	52	36	=	=	NOUN
ejpam-4478	52	37	q.	q.	NOUN
ejpam-4478	53	1	the	the	DET
ejpam-4478	53	2	authors	author	NOUN
ejpam-4478	53	3	defined	define	VERB
ejpam-4478	53	4	the	the	DET
ejpam-4478	53	5	global	global	ADJ
ejpam-4478	53	6	distance	distance	NOUN
ejpam-4478	53	7	roman	roman	ADJ
ejpam-4478	53	8	domination	domination	NOUN
ejpam-4478	53	9	on	on	ADP
ejpam-4478	53	10	graphs	graph	NOUN
ejpam-4478	53	11	as	as	SCONJ
ejpam-4478	53	12	follows	follow	VERB
ejpam-4478	53	13	:	:	PUNCT
ejpam-4478	53	14	the	the	DET
ejpam-4478	53	15	function	function	NOUN
ejpam-4478	53	16	f	f	PROPN
ejpam-4478	53	17	is	be	AUX
ejpam-4478	53	18	a	a	DET
ejpam-4478	53	19	global	global	ADJ
ejpam-4478	53	20	k	k	NOUN
ejpam-4478	53	21	−	−	PROPN
ejpam-4478	53	22	distance	distance	NOUN
ejpam-4478	53	23	roman	roman	ADJ
ejpam-4478	53	24	dominating	dominating	NOUN
ejpam-4478	53	25	function	function	NOUN
ejpam-4478	53	26	(	(	PUNCT
ejpam-4478	53	27	gkdrdf	gkdrdf	PROPN
ejpam-4478	53	28	)	)	PUNCT
ejpam-4478	53	29	on	on	ADP
ejpam-4478	53	30	g	g	PROPN
ejpam-4478	53	31	if	if	SCONJ
ejpam-4478	54	1	and	and	CCONJ
ejpam-4478	54	2	only	only	ADV
ejpam-4478	54	3	if	if	SCONJ
ejpam-4478	54	4	f	f	PROPN
ejpam-4478	54	5	is	be	AUX
ejpam-4478	54	6	a	a	DET
ejpam-4478	54	7	k	k	NOUN
ejpam-4478	54	8	−	−	PROPN
ejpam-4478	54	9	distance	distance	NOUN
ejpam-4478	54	10	roman	roman	ADJ
ejpam-4478	54	11	dominating	dominating	NOUN
ejpam-4478	54	12	function	function	NOUN
ejpam-4478	54	13	(	(	PUNCT
ejpam-4478	54	14	kdrdf	kdrdf	NOUN
ejpam-4478	54	15	)	)	PUNCT
ejpam-4478	54	16	on	on	ADP
ejpam-4478	54	17	g	g	PROPN
ejpam-4478	54	18	and	and	CCONJ
ejpam-4478	54	19	on	on	ADP
ejpam-4478	54	20	its	its	PRON
ejpam-4478	54	21	complement	complement	NOUN
ejpam-4478	54	22	g.	g.	NOUN
ejpam-4478	54	23	the	the	DET
ejpam-4478	54	24	weight	weight	NOUN
ejpam-4478	54	25	of	of	ADP
ejpam-4478	54	26	the	the	DET
ejpam-4478	54	27	global	global	ADJ
ejpam-4478	54	28	k	k	PROPN
ejpam-4478	54	29	−	−	PROPN
ejpam-4478	54	30	distance	distance	NOUN
ejpam-4478	54	31	roman	roman	ADJ
ejpam-4478	54	32	dominating	dominating	NOUN
ejpam-4478	54	33	function	function	NOUN
ejpam-4478	54	34	g.	g.	PROPN
ejpam-4478	54	35	entero	entero	PROPN
ejpam-4478	54	36	,	,	PUNCT
ejpam-4478	54	37	s.	s.	PROPN
ejpam-4478	54	38	espinola	espinola	PROPN
ejpam-4478	54	39	/	/	SYM
ejpam-4478	54	40	eur	eur	PROPN
ejpam-4478	54	41	.	.	PUNCT
ejpam-4478	55	1	j.	j.	PROPN
ejpam-4478	55	2	pure	pure	PROPN
ejpam-4478	55	3	appl	appl	PROPN
ejpam-4478	55	4	.	.	PROPN
ejpam-4478	55	5	math	math	PROPN
ejpam-4478	55	6	,	,	PUNCT
ejpam-4478	55	7	16	16	NUM
ejpam-4478	55	8	(	(	PUNCT
ejpam-4478	55	9	1	1	NUM
ejpam-4478	55	10	)	)	PUNCT
ejpam-4478	55	11	(	(	PUNCT
ejpam-4478	55	12	2023	2023	NUM
ejpam-4478	55	13	)	)	PUNCT
ejpam-4478	55	14	,	,	PUNCT
ejpam-4478	55	15	44	44	NUM
ejpam-4478	55	16	-	-	SYM
ejpam-4478	55	17	61	61	NUM
ejpam-4478	55	18	46	46	NUM
ejpam-4478	55	19	(	(	PUNCT
ejpam-4478	55	20	gkdrdf	gkdrdf	PROPN
ejpam-4478	55	21	)	)	PUNCT
ejpam-4478	55	22	f	f	PROPN
ejpam-4478	55	23	is	be	AUX
ejpam-4478	55	24	the	the	DET
ejpam-4478	55	25	value	value	NOUN
ejpam-4478	55	26	w(f	w(f	NOUN
ejpam-4478	55	27	)	)	PUNCT
ejpam-4478	55	28	=	=	SYM
ejpam-4478	56	1	∑	∑	PUNCT
ejpam-4478	56	2	x∈v	x∈v	PROPN
ejpam-4478	56	3	f(x	f(x	PROPN
ejpam-4478	56	4	)	)	PUNCT
ejpam-4478	56	5	.	.	PUNCT
ejpam-4478	57	1	the	the	DET
ejpam-4478	57	2	minimum	minimum	ADJ
ejpam-4478	57	3	weight	weight	NOUN
ejpam-4478	57	4	of	of	ADP
ejpam-4478	57	5	the	the	DET
ejpam-4478	57	6	global	global	ADJ
ejpam-4478	57	7	k	k	PROPN
ejpam-4478	57	8	−	−	PROPN
ejpam-4478	57	9	distance	distance	NOUN
ejpam-4478	57	10	roman	roman	ADJ
ejpam-4478	57	11	dominating	dominating	NOUN
ejpam-4478	57	12	function	function	NOUN
ejpam-4478	57	13	(	(	PUNCT
ejpam-4478	57	14	gkdrdf	gkdrdf	PROPN
ejpam-4478	57	15	)	)	PUNCT
ejpam-4478	57	16	on	on	ADP
ejpam-4478	57	17	the	the	DET
ejpam-4478	57	18	graph	graph	NOUN
ejpam-4478	57	19	g	g	NOUN
ejpam-4478	57	20	is	be	AUX
ejpam-4478	57	21	called	call	VERB
ejpam-4478	57	22	the	the	DET
ejpam-4478	57	23	global	global	ADJ
ejpam-4478	57	24	k	k	PROPN
ejpam-4478	57	25	−	−	PROPN
ejpam-4478	57	26	distance	distance	NOUN
ejpam-4478	57	27	roman	roman	ADJ
ejpam-4478	57	28	domination	domination	NOUN
ejpam-4478	57	29	number	number	NOUN
ejpam-4478	57	30	of	of	ADP
ejpam-4478	57	31	g	g	NOUN
ejpam-4478	57	32	and	and	CCONJ
ejpam-4478	57	33	is	be	AUX
ejpam-4478	57	34	denoted	denote	VERB
ejpam-4478	57	35	by	by	ADP
ejpam-4478	57	36	γkgr(g	γkgr(g	PROPN
ejpam-4478	57	37	)	)	PUNCT
ejpam-4478	57	38	.	.	PUNCT
ejpam-4478	58	1	a	a	DET
ejpam-4478	58	2	γkgr(g)−function	γkgr(g)−function	NOUN
ejpam-4478	58	3	is	be	AUX
ejpam-4478	58	4	the	the	DET
ejpam-4478	58	5	gkdrdf	gkdrdf	NOUN
ejpam-4478	58	6	on	on	ADP
ejpam-4478	58	7	g	g	NOUN
ejpam-4478	58	8	with	with	ADP
ejpam-4478	58	9	weight	weight	NOUN
ejpam-4478	58	10	γkgr(g	γkgr(g	PROPN
ejpam-4478	58	11	)	)	PUNCT
ejpam-4478	58	12	.	.	PUNCT
ejpam-4478	59	1	the	the	DET
ejpam-4478	59	2	gkdrdf	gkdrdf	PROPN
ejpam-4478	59	3	f	f	PROPN
ejpam-4478	59	4	:	:	PUNCT
ejpam-4478	59	5	v	v	X
ejpam-4478	59	6	→	→	SYM
ejpam-4478	59	7	{	{	PUNCT
ejpam-4478	59	8	0	0	NUM
ejpam-4478	59	9	,	,	PUNCT
ejpam-4478	59	10	1	1	NUM
ejpam-4478	59	11	,	,	PUNCT
ejpam-4478	59	12	2	2	NUM
ejpam-4478	59	13	}	}	PUNCT
ejpam-4478	59	14	can	can	AUX
ejpam-4478	59	15	be	be	AUX
ejpam-4478	59	16	represented	represent	VERB
ejpam-4478	59	17	by	by	ADP
ejpam-4478	59	18	the	the	DET
ejpam-4478	59	19	ordered	order	VERB
ejpam-4478	59	20	partition	partition	NOUN
ejpam-4478	59	21	(	(	PUNCT
ejpam-4478	59	22	v	v	NOUN
ejpam-4478	59	23	f	f	PROPN
ejpam-4478	59	24	0	0	NUM
ejpam-4478	59	25	,	,	PUNCT
ejpam-4478	59	26	v	v	NOUN
ejpam-4478	59	27	f	f	PROPN
ejpam-4478	59	28	1	1	NUM
ejpam-4478	59	29	,	,	PUNCT
ejpam-4478	59	30	v	v	NOUN
ejpam-4478	59	31	f	f	PROPN
ejpam-4478	59	32	2	2	NUM
ejpam-4478	59	33	)	)	PUNCT
ejpam-4478	59	34	of	of	ADP
ejpam-4478	59	35	v	v	NUM
ejpam-4478	59	36	induced	induce	VERB
ejpam-4478	59	37	by	by	ADP
ejpam-4478	59	38	f	f	PROPN
ejpam-4478	59	39	,	,	PUNCT
ejpam-4478	59	40	where	where	SCONJ
ejpam-4478	59	41	f	f	PROPN
ejpam-4478	59	42	is	be	AUX
ejpam-4478	59	43	the	the	DET
ejpam-4478	59	44	given	give	VERB
ejpam-4478	59	45	function	function	NOUN
ejpam-4478	59	46	and	and	CCONJ
ejpam-4478	59	47	v	v	NOUN
ejpam-4478	59	48	f	f	NOUN
ejpam-4478	59	49	i	i	PRON
ejpam-4478	59	50	=	=	PUNCT
ejpam-4478	59	51	{	{	PUNCT
ejpam-4478	59	52	v	v	NUM
ejpam-4478	59	53	∈	∈	NOUN
ejpam-4478	59	54	v	v	X
ejpam-4478	59	55	|f(v	|f(v	PROPN
ejpam-4478	59	56	)	)	PUNCT
ejpam-4478	60	1	=	=	PUNCT
ejpam-4478	61	1	i	i	PRON
ejpam-4478	61	2	and	and	CCONJ
ejpam-4478	61	3	i	i	PRON
ejpam-4478	61	4	=	=	NOUN
ejpam-4478	61	5	0	0	NUM
ejpam-4478	61	6	,	,	PUNCT
ejpam-4478	61	7	1	1	NUM
ejpam-4478	61	8	,	,	PUNCT
ejpam-4478	61	9	2	2	NUM
ejpam-4478	61	10	}	}	PUNCT
ejpam-4478	61	11	.	.	PUNCT
ejpam-4478	62	1	observe	observe	VERB
ejpam-4478	62	2	that	that	SCONJ
ejpam-4478	62	3	,	,	PUNCT
ejpam-4478	62	4	there	there	PRON
ejpam-4478	62	5	is	be	VERB
ejpam-4478	62	6	a	a	DET
ejpam-4478	62	7	one	one	NUM
ejpam-4478	62	8	-	-	PUNCT
ejpam-4478	62	9	to	to	ADP
ejpam-4478	62	10	-	-	PUNCT
ejpam-4478	62	11	one	one	NUM
ejpam-4478	62	12	correspondence	correspondence	NOUN
ejpam-4478	62	13	between	between	ADP
ejpam-4478	62	14	the	the	DET
ejpam-4478	62	15	function	function	NOUN
ejpam-4478	62	16	f	f	NOUN
ejpam-4478	62	17	:	:	PUNCT
ejpam-4478	62	18	v	v	X
ejpam-4478	62	19	→	→	SYM
ejpam-4478	62	20	{	{	PUNCT
ejpam-4478	62	21	0	0	NUM
ejpam-4478	62	22	,	,	PUNCT
ejpam-4478	62	23	1	1	NUM
ejpam-4478	62	24	,	,	PUNCT
ejpam-4478	62	25	2	2	NUM
ejpam-4478	62	26	}	}	PUNCT
ejpam-4478	62	27	and	and	CCONJ
ejpam-4478	62	28	the	the	DET
ejpam-4478	62	29	ordered	order	VERB
ejpam-4478	62	30	partition	partition	NOUN
ejpam-4478	62	31	(	(	PUNCT
ejpam-4478	62	32	v	v	NOUN
ejpam-4478	62	33	f	f	PROPN
ejpam-4478	62	34	0	0	NUM
ejpam-4478	62	35	,	,	PUNCT
ejpam-4478	62	36	v	v	NOUN
ejpam-4478	62	37	f	f	PROPN
ejpam-4478	62	38	1	1	NUM
ejpam-4478	62	39	,	,	PUNCT
ejpam-4478	62	40	v	v	NOUN
ejpam-4478	62	41	f	f	PROPN
ejpam-4478	62	42	2	2	NUM
ejpam-4478	62	43	)	)	PUNCT
ejpam-4478	62	44	of	of	ADP
ejpam-4478	62	45	v	v	NUM
ejpam-4478	62	46	induced	induce	VERB
ejpam-4478	62	47	by	by	ADP
ejpam-4478	62	48	f	f	PROPN
ejpam-4478	62	49	.	.	PUNCT
ejpam-4478	63	1	hence	hence	ADV
ejpam-4478	63	2	,	,	PUNCT
ejpam-4478	63	3	we	we	PRON
ejpam-4478	63	4	may	may	AUX
ejpam-4478	63	5	write	write	VERB
ejpam-4478	63	6	f	f	PROPN
ejpam-4478	63	7	=	=	SYM
ejpam-4478	63	8	(	(	PUNCT
ejpam-4478	63	9	v	v	NOUN
ejpam-4478	63	10	f	f	PROPN
ejpam-4478	63	11	0	0	NUM
ejpam-4478	63	12	,	,	PUNCT
ejpam-4478	63	13	v	v	NOUN
ejpam-4478	63	14	f	f	PROPN
ejpam-4478	63	15	1	1	NUM
ejpam-4478	63	16	,	,	PUNCT
ejpam-4478	63	17	v	v	NOUN
ejpam-4478	63	18	f	f	PROPN
ejpam-4478	63	19	2	2	NUM
ejpam-4478	63	20	)	)	PUNCT
ejpam-4478	63	21	.	.	PUNCT
ejpam-4478	64	1	in	in	ADP
ejpam-4478	64	2	this	this	DET
ejpam-4478	64	3	representation	representation	NOUN
ejpam-4478	64	4	,	,	PUNCT
ejpam-4478	64	5	its	its	PRON
ejpam-4478	64	6	weight	weight	NOUN
ejpam-4478	64	7	can	can	AUX
ejpam-4478	64	8	be	be	AUX
ejpam-4478	64	9	computed	compute	VERB
ejpam-4478	64	10	as	as	ADP
ejpam-4478	64	11	w(f	w(f	PROPN
ejpam-4478	64	12	)	)	PUNCT
ejpam-4478	64	13	=	=	SYM
ejpam-4478	65	1	|v	|v	PROPN
ejpam-4478	65	2	f	f	NOUN
ejpam-4478	65	3	1	1	NUM
ejpam-4478	65	4	|+	|+	NOUN
ejpam-4478	65	5	2|v	2|v	NOUN
ejpam-4478	66	1	f	f	NOUN
ejpam-4478	66	2	2	2	NUM
ejpam-4478	66	3	|	|	NOUN
ejpam-4478	66	4	.	.	PUNCT
ejpam-4478	67	1	note	note	VERB
ejpam-4478	67	2	that	that	SCONJ
ejpam-4478	67	3	,	,	PUNCT
ejpam-4478	67	4	the	the	DET
ejpam-4478	67	5	global	global	ADJ
ejpam-4478	67	6	1	1	NUM
ejpam-4478	67	7	−	−	NOUN
ejpam-4478	67	8	distance	distance	NOUN
ejpam-4478	67	9	roman	roman	ADJ
ejpam-4478	67	10	domination	domination	NOUN
ejpam-4478	67	11	number	number	NOUN
ejpam-4478	67	12	γ1gr(g	γ1gr(g	PROPN
ejpam-4478	67	13	)	)	PUNCT
ejpam-4478	67	14	is	be	AUX
ejpam-4478	67	15	the	the	DET
ejpam-4478	67	16	usual	usual	ADJ
ejpam-4478	67	17	global	global	ADJ
ejpam-4478	67	18	roman	roman	ADJ
ejpam-4478	67	19	domination	domination	NOUN
ejpam-4478	67	20	number	number	NOUN
ejpam-4478	67	21	γgr(g	γgr(g	PROPN
ejpam-4478	67	22	)	)	PUNCT
ejpam-4478	67	23	,	,	PUNCT
ejpam-4478	67	24	that	that	ADV
ejpam-4478	67	25	is	is	ADV
ejpam-4478	67	26	,	,	PUNCT
ejpam-4478	67	27	γ1gr(g	γ1gr(g	X
ejpam-4478	67	28	)	)	PUNCT
ejpam-4478	68	1	=	=	SYM
ejpam-4478	68	2	γgr(g	γgr(g	PROPN
ejpam-4478	68	3	)	)	PUNCT
ejpam-4478	68	4	.	.	PUNCT
ejpam-4478	69	1	it	it	PRON
ejpam-4478	69	2	is	be	AUX
ejpam-4478	69	3	worth	worth	ADJ
ejpam-4478	69	4	noting	note	VERB
ejpam-4478	69	5	that	that	SCONJ
ejpam-4478	69	6	,	,	PUNCT
ejpam-4478	69	7	since	since	SCONJ
ejpam-4478	69	8	we	we	PRON
ejpam-4478	69	9	are	be	AUX
ejpam-4478	69	10	dealing	deal	VERB
ejpam-4478	69	11	with	with	ADP
ejpam-4478	69	12	simple	simple	ADJ
ejpam-4478	69	13	graphs	graph	NOUN
ejpam-4478	69	14	,	,	PUNCT
ejpam-4478	69	15	the	the	DET
ejpam-4478	69	16	distance	distance	NOUN
ejpam-4478	69	17	of	of	ADP
ejpam-4478	69	18	each	each	DET
ejpam-4478	69	19	vertex	vertex	NOUN
ejpam-4478	69	20	,	,	PUNCT
ejpam-4478	69	21	say	say	VERB
ejpam-4478	69	22	u	u	PROPN
ejpam-4478	69	23	∈	∈	PROPN
ejpam-4478	69	24	v	v	NOUN
ejpam-4478	69	25	,	,	PUNCT
ejpam-4478	69	26	to	to	ADP
ejpam-4478	69	27	itself	itself	PRON
ejpam-4478	69	28	is	be	AUX
ejpam-4478	69	29	zero	zero	NUM
ejpam-4478	69	30	,	,	PUNCT
ejpam-4478	69	31	that	that	ADV
ejpam-4478	69	32	is	be	AUX
ejpam-4478	69	33	,	,	PUNCT
ejpam-4478	69	34	d(u	d(u	PROPN
ejpam-4478	69	35	,	,	PUNCT
ejpam-4478	69	36	u	u	NOUN
ejpam-4478	69	37	)	)	PUNCT
ejpam-4478	69	38	=	=	SYM
ejpam-4478	69	39	0	0	PUNCT
ejpam-4478	70	1	while	while	SCONJ
ejpam-4478	70	2	the	the	DET
ejpam-4478	70	3	distance	distance	NOUN
ejpam-4478	70	4	of	of	ADP
ejpam-4478	70	5	two	two	NUM
ejpam-4478	70	6	different	different	ADJ
ejpam-4478	70	7	vertices	vertex	NOUN
ejpam-4478	70	8	say	say	VERB
ejpam-4478	70	9	u	u	NOUN
ejpam-4478	70	10	,	,	PUNCT
ejpam-4478	70	11	v	v	PROPN
ejpam-4478	70	12	∈	∈	PROPN
ejpam-4478	70	13	v	v	NOUN
ejpam-4478	70	14	,	,	PUNCT
ejpam-4478	70	15	coming	come	VERB
ejpam-4478	70	16	from	from	ADP
ejpam-4478	70	17	different	different	ADJ
ejpam-4478	70	18	components	component	NOUN
ejpam-4478	70	19	of	of	ADP
ejpam-4478	70	20	graph	graph	NOUN
ejpam-4478	70	21	g	g	PROPN
ejpam-4478	70	22	is	be	AUX
ejpam-4478	70	23	assigned	assign	VERB
ejpam-4478	70	24	to	to	PART
ejpam-4478	70	25	be	be	AUX
ejpam-4478	70	26	∞	∞	PROPN
ejpam-4478	70	27	,	,	PUNCT
ejpam-4478	70	28	that	that	ADV
ejpam-4478	70	29	is	be	AUX
ejpam-4478	70	30	,	,	PUNCT
ejpam-4478	70	31	d(u	d(u	PROPN
ejpam-4478	70	32	,	,	PUNCT
ejpam-4478	70	33	v	v	NOUN
ejpam-4478	70	34	)	)	PUNCT
ejpam-4478	71	1	=	=	SYM
ejpam-4478	71	2	∞	∞	PROPN
ejpam-4478	71	3	,	,	PUNCT
ejpam-4478	71	4	where	where	SCONJ
ejpam-4478	71	5	u	u	NOUN
ejpam-4478	71	6	and	and	CCONJ
ejpam-4478	71	7	v	v	NOUN
ejpam-4478	71	8	belong	belong	VERB
ejpam-4478	71	9	to	to	ADP
ejpam-4478	71	10	different	different	ADJ
ejpam-4478	71	11	components	component	NOUN
ejpam-4478	71	12	of	of	ADP
ejpam-4478	71	13	g.	g.	PROPN
ejpam-4478	71	14	2	2	NUM
ejpam-4478	71	15	.	.	PUNCT
ejpam-4478	71	16	terminologies	terminology	NOUN
ejpam-4478	71	17	and	and	CCONJ
ejpam-4478	71	18	notations	notation	NOUN
ejpam-4478	71	19	to	to	PART
ejpam-4478	71	20	better	well	ADV
ejpam-4478	71	21	understand	understand	VERB
ejpam-4478	71	22	the	the	DET
ejpam-4478	71	23	scope	scope	NOUN
ejpam-4478	71	24	of	of	ADP
ejpam-4478	71	25	this	this	DET
ejpam-4478	71	26	study	study	NOUN
ejpam-4478	71	27	,	,	PUNCT
ejpam-4478	71	28	we	we	PRON
ejpam-4478	71	29	will	will	AUX
ejpam-4478	71	30	be	be	AUX
ejpam-4478	71	31	needing	need	VERB
ejpam-4478	71	32	the	the	DET
ejpam-4478	71	33	following	follow	VERB
ejpam-4478	71	34	definitions	definition	NOUN
ejpam-4478	71	35	and	and	CCONJ
ejpam-4478	71	36	some	some	DET
ejpam-4478	71	37	related	related	ADJ
ejpam-4478	71	38	literature	literature	NOUN
ejpam-4478	71	39	.	.	PUNCT
ejpam-4478	72	1	the	the	DET
ejpam-4478	72	2	distance	distance	NOUN
ejpam-4478	72	3	between	between	ADP
ejpam-4478	72	4	vertices	vertex	NOUN
ejpam-4478	72	5	u	u	NOUN
ejpam-4478	72	6	and	and	CCONJ
ejpam-4478	72	7	v	v	NOUN
ejpam-4478	72	8	in	in	ADP
ejpam-4478	72	9	graph	graph	NOUN
ejpam-4478	72	10	g	g	NOUN
ejpam-4478	72	11	,	,	PUNCT
ejpam-4478	72	12	denoted	denote	VERB
ejpam-4478	72	13	by	by	ADP
ejpam-4478	72	14	d(u	d(u	PROPN
ejpam-4478	72	15	,	,	PUNCT
ejpam-4478	72	16	v	v	NOUN
ejpam-4478	72	17	)	)	PUNCT
ejpam-4478	72	18	,	,	PUNCT
ejpam-4478	72	19	is	be	AUX
ejpam-4478	72	20	the	the	DET
ejpam-4478	72	21	length	length	NOUN
ejpam-4478	72	22	of	of	ADP
ejpam-4478	72	23	the	the	DET
ejpam-4478	72	24	shortest	short	ADJ
ejpam-4478	72	25	path	path	NOUN
ejpam-4478	72	26	from	from	ADP
ejpam-4478	72	27	vertex	vertex	NOUN
ejpam-4478	72	28	u	u	NOUN
ejpam-4478	72	29	to	to	PART
ejpam-4478	72	30	vertex	vertex	VERB
ejpam-4478	72	31	v	v	NOUN
ejpam-4478	72	32	in	in	ADP
ejpam-4478	72	33	graph	graph	NOUN
ejpam-4478	72	34	g.	g.	NOUN
ejpam-4478	73	1	the	the	DET
ejpam-4478	73	2	eccentricity	eccentricity	NOUN
ejpam-4478	73	3	of	of	ADP
ejpam-4478	73	4	vertex	vertex	NOUN
ejpam-4478	73	5	u	u	NOUN
ejpam-4478	73	6	on	on	ADP
ejpam-4478	73	7	graph	graph	NOUN
ejpam-4478	73	8	g	g	PROPN
ejpam-4478	73	9	is	be	AUX
ejpam-4478	73	10	the	the	DET
ejpam-4478	73	11	maximum	maximum	ADJ
ejpam-4478	73	12	distance	distance	NOUN
ejpam-4478	73	13	from	from	ADP
ejpam-4478	73	14	vertex	vertex	NOUN
ejpam-4478	73	15	u	u	NOUN
ejpam-4478	73	16	to	to	ADP
ejpam-4478	73	17	any	any	DET
ejpam-4478	73	18	other	other	ADJ
ejpam-4478	73	19	vertex	vertex	NOUN
ejpam-4478	73	20	,	,	PUNCT
ejpam-4478	73	21	say	say	VERB
ejpam-4478	73	22	vertex	vertex	NOUN
ejpam-4478	73	23	v	v	NOUN
ejpam-4478	73	24	,	,	PUNCT
ejpam-4478	73	25	in	in	ADP
ejpam-4478	73	26	graph	graph	NOUN
ejpam-4478	73	27	g	g	NOUN
ejpam-4478	73	28	and	and	CCONJ
ejpam-4478	73	29	is	be	AUX
ejpam-4478	73	30	denoted	denote	VERB
ejpam-4478	73	31	by	by	ADP
ejpam-4478	73	32	ecc(u	ecc(u	PROPN
ejpam-4478	73	33	)	)	PUNCT
ejpam-4478	73	34	=	=	SYM
ejpam-4478	73	35	max{d(u	max{d(u	PROPN
ejpam-4478	73	36	,	,	PUNCT
ejpam-4478	73	37	v	v	NOUN
ejpam-4478	73	38	)	)	PUNCT
ejpam-4478	73	39	:	:	PUNCT
ejpam-4478	73	40	v	v	X
ejpam-4478	73	41	∈	∈	PROPN
ejpam-4478	73	42	v	v	NOUN
ejpam-4478	73	43	(	(	PUNCT
ejpam-4478	73	44	g	g	NOUN
ejpam-4478	73	45	)	)	PUNCT
ejpam-4478	73	46	}	}	PUNCT
ejpam-4478	73	47	.	.	PUNCT
ejpam-4478	74	1	the	the	DET
ejpam-4478	74	2	radius	radius	NOUN
ejpam-4478	74	3	of	of	ADP
ejpam-4478	74	4	graph	graph	NOUN
ejpam-4478	74	5	g	g	PROPN
ejpam-4478	74	6	is	be	AUX
ejpam-4478	74	7	the	the	DET
ejpam-4478	74	8	minimum	minimum	ADJ
ejpam-4478	74	9	eccentricity	eccentricity	NOUN
ejpam-4478	74	10	taken	take	VERB
ejpam-4478	74	11	over	over	ADP
ejpam-4478	74	12	all	all	DET
ejpam-4478	74	13	vertices	vertex	NOUN
ejpam-4478	74	14	of	of	ADP
ejpam-4478	74	15	graph	graph	NOUN
ejpam-4478	74	16	g	g	PROPN
ejpam-4478	74	17	and	and	CCONJ
ejpam-4478	74	18	is	be	AUX
ejpam-4478	74	19	denoted	denote	VERB
ejpam-4478	74	20	as	as	ADP
ejpam-4478	74	21	rad(g	rad(g	NOUN
ejpam-4478	74	22	)	)	PUNCT
ejpam-4478	74	23	=	=	SYM
ejpam-4478	74	24	min{ecc(u	min{ecc(u	PROPN
ejpam-4478	74	25	)	)	PUNCT
ejpam-4478	74	26	:	:	PUNCT
ejpam-4478	74	27	u	u	PROPN
ejpam-4478	74	28	∈	∈	PROPN
ejpam-4478	74	29	v	v	ADP
ejpam-4478	74	30	(	(	PUNCT
ejpam-4478	74	31	g	g	NOUN
ejpam-4478	74	32	)	)	PUNCT
ejpam-4478	74	33	}	}	PUNCT
ejpam-4478	74	34	and	and	CCONJ
ejpam-4478	74	35	the	the	DET
ejpam-4478	74	36	diameter	diameter	NOUN
ejpam-4478	74	37	of	of	ADP
ejpam-4478	74	38	graph	graph	NOUN
ejpam-4478	74	39	g	g	PROPN
ejpam-4478	74	40	is	be	AUX
ejpam-4478	74	41	the	the	DET
ejpam-4478	74	42	maximum	maximum	ADJ
ejpam-4478	74	43	eccentricity	eccentricity	NOUN
ejpam-4478	74	44	taken	take	VERB
ejpam-4478	74	45	over	over	ADP
ejpam-4478	74	46	all	all	DET
ejpam-4478	74	47	vertices	vertex	NOUN
ejpam-4478	74	48	of	of	ADP
ejpam-4478	74	49	graph	graph	NOUN
ejpam-4478	74	50	g	g	PROPN
ejpam-4478	74	51	and	and	CCONJ
ejpam-4478	74	52	is	be	AUX
ejpam-4478	74	53	denoted	denote	VERB
ejpam-4478	74	54	as	as	ADP
ejpam-4478	74	55	diam(g	diam(g	NOUN
ejpam-4478	74	56	)	)	PUNCT
ejpam-4478	74	57	=	=	SYM
ejpam-4478	74	58	max{ecc(u	max{ecc(u	PROPN
ejpam-4478	74	59	)	)	PUNCT
ejpam-4478	74	60	:	:	PUNCT
ejpam-4478	75	1	u	u	PROPN
ejpam-4478	75	2	∈	∈	PROPN
ejpam-4478	75	3	v	v	ADP
ejpam-4478	75	4	(	(	PUNCT
ejpam-4478	75	5	g	g	NOUN
ejpam-4478	75	6	)	)	PUNCT
ejpam-4478	75	7	}	}	PUNCT
ejpam-4478	75	8	.	.	PUNCT
ejpam-4478	76	1	[	[	X
ejpam-4478	76	2	8	8	X
ejpam-4478	76	3	]	]	X
ejpam-4478	76	4	the	the	DET
ejpam-4478	76	5	degree	degree	NOUN
ejpam-4478	76	6	of	of	ADP
ejpam-4478	76	7	a	a	DET
ejpam-4478	76	8	vertex	vertex	NOUN
ejpam-4478	76	9	v	v	NOUN
ejpam-4478	76	10	of	of	ADP
ejpam-4478	76	11	the	the	DET
ejpam-4478	76	12	graph	graph	NOUN
ejpam-4478	76	13	g	g	PROPN
ejpam-4478	76	14	is	be	AUX
ejpam-4478	76	15	the	the	DET
ejpam-4478	76	16	number	number	NOUN
ejpam-4478	76	17	of	of	ADP
ejpam-4478	76	18	edges	edge	NOUN
ejpam-4478	76	19	incident	incident	NOUN
ejpam-4478	76	20	with	with	ADP
ejpam-4478	76	21	v	v	NOUN
ejpam-4478	76	22	in	in	ADP
ejpam-4478	76	23	g	g	NOUN
ejpam-4478	76	24	and	and	CCONJ
ejpam-4478	76	25	is	be	AUX
ejpam-4478	76	26	denoted	denote	VERB
ejpam-4478	76	27	by	by	ADP
ejpam-4478	76	28	deg(v	deg(v	PROPN
ejpam-4478	76	29	)	)	PUNCT
ejpam-4478	76	30	.	.	PUNCT
ejpam-4478	77	1	the	the	DET
ejpam-4478	77	2	maximum	maximum	ADJ
ejpam-4478	77	3	degree	degree	NOUN
ejpam-4478	77	4	of	of	ADP
ejpam-4478	77	5	the	the	DET
ejpam-4478	77	6	graph	graph	NOUN
ejpam-4478	77	7	g	g	NOUN
ejpam-4478	77	8	,	,	PUNCT
ejpam-4478	77	9	denoted	denote	VERB
ejpam-4478	77	10	by	by	ADP
ejpam-4478	77	11	∆(g	∆(g	PROPN
ejpam-4478	77	12	)	)	PUNCT
ejpam-4478	77	13	,	,	PUNCT
ejpam-4478	77	14	is	be	AUX
ejpam-4478	77	15	the	the	DET
ejpam-4478	77	16	maximum	maximum	ADJ
ejpam-4478	77	17	degree	degree	NOUN
ejpam-4478	77	18	for	for	ADP
ejpam-4478	77	19	every	every	DET
ejpam-4478	77	20	vertex	vertex	NOUN
ejpam-4478	77	21	in	in	ADP
ejpam-4478	77	22	g	g	NOUN
ejpam-4478	77	23	,	,	PUNCT
ejpam-4478	77	24	that	that	ADV
ejpam-4478	77	25	is	is	ADV
ejpam-4478	77	26	,	,	PUNCT
ejpam-4478	77	27	∆(g	∆(g	NOUN
ejpam-4478	77	28	)	)	PUNCT
ejpam-4478	77	29	=	=	SYM
ejpam-4478	77	30	max{deg(v	max{deg(v	PROPN
ejpam-4478	77	31	)	)	PUNCT
ejpam-4478	77	32	:	:	PUNCT
ejpam-4478	77	33	v	v	X
ejpam-4478	77	34	∈	∈	PROPN
ejpam-4478	77	35	v	v	NOUN
ejpam-4478	77	36	(	(	PUNCT
ejpam-4478	77	37	g	g	NOUN
ejpam-4478	77	38	)	)	PUNCT
ejpam-4478	77	39	}	}	PUNCT
ejpam-4478	77	40	.	.	PUNCT
ejpam-4478	78	1	the	the	DET
ejpam-4478	78	2	minimum	minimum	NOUN
ejpam-4478	78	3	degree	degree	NOUN
ejpam-4478	78	4	of	of	ADP
ejpam-4478	78	5	the	the	DET
ejpam-4478	78	6	graph	graph	NOUN
ejpam-4478	78	7	g	g	NOUN
ejpam-4478	78	8	,	,	PUNCT
ejpam-4478	78	9	denoted	denote	VERB
ejpam-4478	78	10	by	by	ADP
ejpam-4478	78	11	δ(g	δ(g	PROPN
ejpam-4478	78	12	)	)	PUNCT
ejpam-4478	78	13	,	,	PUNCT
ejpam-4478	78	14	is	be	AUX
ejpam-4478	78	15	the	the	DET
ejpam-4478	78	16	minimum	minimum	NOUN
ejpam-4478	78	17	degree	degree	NOUN
ejpam-4478	78	18	for	for	ADP
ejpam-4478	78	19	every	every	DET
ejpam-4478	78	20	vertex	vertex	NOUN
ejpam-4478	78	21	in	in	ADP
ejpam-4478	78	22	g	g	NOUN
ejpam-4478	78	23	,	,	PUNCT
ejpam-4478	78	24	that	that	ADV
ejpam-4478	78	25	is	is	ADV
ejpam-4478	78	26	,	,	PUNCT
ejpam-4478	78	27	δ(g	δ(g	ADJ
ejpam-4478	78	28	)	)	PUNCT
ejpam-4478	78	29	=	=	SYM
ejpam-4478	78	30	min{deg(v	min{deg(v	PROPN
ejpam-4478	78	31	)	)	PUNCT
ejpam-4478	78	32	:	:	PUNCT
ejpam-4478	79	1	v	v	X
ejpam-4478	79	2	∈	∈	PROPN
ejpam-4478	79	3	v	v	NOUN
ejpam-4478	79	4	(	(	PUNCT
ejpam-4478	79	5	g	g	NOUN
ejpam-4478	79	6	)	)	PUNCT
ejpam-4478	79	7	}	}	PUNCT
ejpam-4478	79	8	.	.	PUNCT
ejpam-4478	80	1	[	[	X
ejpam-4478	80	2	3	3	X
ejpam-4478	80	3	]	]	PUNCT
ejpam-4478	80	4	the	the	DET
ejpam-4478	80	5	neighbourhood	neighbourhood	NOUN
ejpam-4478	80	6	(	(	PUNCT
ejpam-4478	80	7	or	or	CCONJ
ejpam-4478	80	8	open	open	ADJ
ejpam-4478	80	9	neighbourhood	neighbourhood	NOUN
ejpam-4478	80	10	)	)	PUNCT
ejpam-4478	80	11	of	of	ADP
ejpam-4478	80	12	a	a	DET
ejpam-4478	80	13	vertex	vertex	NOUN
ejpam-4478	80	14	v	v	NOUN
ejpam-4478	80	15	,	,	PUNCT
ejpam-4478	80	16	denoted	denote	VERB
ejpam-4478	80	17	by	by	ADP
ejpam-4478	80	18	n(v	n(v	PROPN
ejpam-4478	80	19	)	)	PUNCT
ejpam-4478	80	20	,	,	PUNCT
ejpam-4478	80	21	is	be	AUX
ejpam-4478	80	22	the	the	DET
ejpam-4478	80	23	set	set	NOUN
ejpam-4478	80	24	of	of	ADP
ejpam-4478	80	25	vertices	vertex	NOUN
ejpam-4478	80	26	adjacent	adjacent	ADJ
ejpam-4478	80	27	to	to	ADP
ejpam-4478	80	28	v	v	NOUN
ejpam-4478	80	29	,	,	PUNCT
ejpam-4478	80	30	that	that	ADV
ejpam-4478	80	31	is	is	ADV
ejpam-4478	80	32	,	,	PUNCT
ejpam-4478	80	33	n(v	n(v	PROPN
ejpam-4478	80	34	)	)	PUNCT
ejpam-4478	80	35	=	=	PRON
ejpam-4478	81	1	{	{	PUNCT
ejpam-4478	81	2	x	x	PUNCT
ejpam-4478	81	3	∈	∈	PROPN
ejpam-4478	81	4	v	v	NOUN
ejpam-4478	81	5	(	(	PUNCT
ejpam-4478	81	6	g	g	NOUN
ejpam-4478	81	7	)	)	PUNCT
ejpam-4478	81	8	:	:	PUNCT
ejpam-4478	81	9	vx	vx	PROPN
ejpam-4478	81	10	∈	∈	PROPN
ejpam-4478	81	11	e(g	e(g	PROPN
ejpam-4478	81	12	)	)	PUNCT
ejpam-4478	81	13	}	}	PUNCT
ejpam-4478	81	14	.	.	PUNCT
ejpam-4478	82	1	the	the	DET
ejpam-4478	82	2	closed	closed	ADJ
ejpam-4478	82	3	neighbourhood	neighbourhood	NOUN
ejpam-4478	82	4	of	of	ADP
ejpam-4478	82	5	a	a	DET
ejpam-4478	82	6	vertex	vertex	NOUN
ejpam-4478	82	7	v	v	NOUN
ejpam-4478	82	8	,	,	PUNCT
ejpam-4478	82	9	denoted	denote	VERB
ejpam-4478	82	10	by	by	ADP
ejpam-4478	82	11	n	n	PRON
ejpam-4478	82	12	[	[	X
ejpam-4478	82	13	v	v	NOUN
ejpam-4478	82	14	]	]	PUNCT
ejpam-4478	82	15	,	,	PUNCT
ejpam-4478	82	16	is	be	AUX
ejpam-4478	82	17	simply	simply	ADV
ejpam-4478	82	18	the	the	DET
ejpam-4478	82	19	set	set	NOUN
ejpam-4478	82	20	{	{	PUNCT
ejpam-4478	82	21	v	v	NOUN
ejpam-4478	82	22	}	}	PUNCT
ejpam-4478	82	23	∪n(v	∪n(v	PROPN
ejpam-4478	82	24	)	)	PUNCT
ejpam-4478	82	25	.	.	PUNCT
ejpam-4478	83	1	given	give	VERB
ejpam-4478	83	2	a	a	DET
ejpam-4478	83	3	set	set	NOUN
ejpam-4478	83	4	s	s	NOUN
ejpam-4478	83	5	of	of	ADP
ejpam-4478	83	6	vertices	vertex	NOUN
ejpam-4478	83	7	,	,	PUNCT
ejpam-4478	83	8	we	we	PRON
ejpam-4478	83	9	define	define	VERB
ejpam-4478	83	10	the	the	DET
ejpam-4478	83	11	neighbourhood	neighbourhood	NOUN
ejpam-4478	83	12	of	of	ADP
ejpam-4478	83	13	s	s	PROPN
ejpam-4478	83	14	,	,	PUNCT
ejpam-4478	83	15	denoted	denote	VERB
ejpam-4478	83	16	by	by	ADP
ejpam-4478	83	17	n(s	n(s	PROPN
ejpam-4478	83	18	)	)	PUNCT
ejpam-4478	83	19	,	,	PUNCT
ejpam-4478	83	20	to	to	PART
ejpam-4478	83	21	be	be	AUX
ejpam-4478	83	22	the	the	DET
ejpam-4478	83	23	union	union	NOUN
ejpam-4478	83	24	of	of	ADP
ejpam-4478	83	25	the	the	DET
ejpam-4478	83	26	neighbourhoods	neighbourhood	NOUN
ejpam-4478	83	27	of	of	ADP
ejpam-4478	83	28	the	the	DET
ejpam-4478	83	29	vertices	vertex	NOUN
ejpam-4478	83	30	in	in	ADP
ejpam-4478	83	31	s.	s.	PROPN
ejpam-4478	83	32	similarly	similarly	ADV
ejpam-4478	83	33	,	,	PUNCT
ejpam-4478	83	34	the	the	DET
ejpam-4478	83	35	closed	closed	ADJ
ejpam-4478	83	36	neighbourhood	neighbourhood	NOUN
ejpam-4478	83	37	of	of	ADP
ejpam-4478	83	38	s	s	PROPN
ejpam-4478	83	39	,	,	PUNCT
ejpam-4478	83	40	denoted	denote	VERB
ejpam-4478	83	41	by	by	ADP
ejpam-4478	83	42	n	n	PRON
ejpam-4478	83	43	[	[	X
ejpam-4478	83	44	s	s	X
ejpam-4478	83	45	]	]	X
ejpam-4478	83	46	,	,	PUNCT
ejpam-4478	83	47	is	be	AUX
ejpam-4478	83	48	defined	define	VERB
ejpam-4478	83	49	to	to	PART
ejpam-4478	83	50	be	be	AUX
ejpam-4478	83	51	s	s	NOUN
ejpam-4478	83	52	∪n(s	∪n(s	NOUN
ejpam-4478	83	53	)	)	PUNCT
ejpam-4478	83	54	.	.	PUNCT
ejpam-4478	84	1	[	[	X
ejpam-4478	84	2	11	11	NUM
ejpam-4478	84	3	]	]	PUNCT
ejpam-4478	84	4	let	let	VERB
ejpam-4478	84	5	k	k	PROPN
ejpam-4478	84	6	∈	∈	PROPN
ejpam-4478	84	7	z+	z+	PUNCT
ejpam-4478	84	8	.	.	PUNCT
ejpam-4478	85	1	the	the	DET
ejpam-4478	85	2	k	k	PROPN
ejpam-4478	85	3	−	−	PROPN
ejpam-4478	85	4	degree	degree	NOUN
ejpam-4478	85	5	of	of	ADP
ejpam-4478	85	6	a	a	DET
ejpam-4478	85	7	vertex	vertex	NOUN
ejpam-4478	85	8	v	v	NOUN
ejpam-4478	85	9	in	in	ADP
ejpam-4478	85	10	graph	graph	NOUN
ejpam-4478	85	11	g	g	NOUN
ejpam-4478	85	12	,	,	PUNCT
ejpam-4478	85	13	denoted	denote	VERB
ejpam-4478	85	14	as	as	ADP
ejpam-4478	85	15	degk	degk	PROPN
ejpam-4478	85	16	,	,	PUNCT
ejpam-4478	85	17	g(v	g(v	PROPN
ejpam-4478	85	18	)	)	PUNCT
ejpam-4478	85	19	,	,	PUNCT
ejpam-4478	85	20	is	be	AUX
ejpam-4478	85	21	defined	define	VERB
ejpam-4478	85	22	to	to	PART
ejpam-4478	85	23	be	be	AUX
ejpam-4478	85	24	degk	degk	ADJ
ejpam-4478	85	25	,	,	PUNCT
ejpam-4478	85	26	g(v	g(v	X
ejpam-4478	85	27	)	)	PUNCT
ejpam-4478	85	28	=	=	PRON
ejpam-4478	85	29	|{u	|{u	VERB
ejpam-4478	85	30	∈	∈	NOUN
ejpam-4478	85	31	v	v	X
ejpam-4478	85	32	(	(	PUNCT
ejpam-4478	85	33	g)|u	g)|u	PROPN
ejpam-4478	85	34	̸=	̸=	PROPN
ejpam-4478	85	35	v	v	NOUN
ejpam-4478	85	36	and	and	CCONJ
ejpam-4478	85	37	d(u	d(u	PROPN
ejpam-4478	85	38	,	,	PUNCT
ejpam-4478	85	39	v	v	NOUN
ejpam-4478	85	40	)	)	PUNCT
ejpam-4478	85	41	≤	≤	NUM
ejpam-4478	85	42	k}|	k}|	PUNCT
ejpam-4478	85	43	.	.	PUNCT
ejpam-4478	86	1	let	let	VERB
ejpam-4478	86	2	k	k	PROPN
ejpam-4478	86	3	∈	∈	PROPN
ejpam-4478	86	4	z+	z+	PUNCT
ejpam-4478	86	5	.	.	PUNCT
ejpam-4478	87	1	the	the	DET
ejpam-4478	87	2	maximum	maximum	ADJ
ejpam-4478	87	3	k	k	PROPN
ejpam-4478	87	4	−	−	NOUN
ejpam-4478	87	5	degree	degree	NOUN
ejpam-4478	87	6	of	of	ADP
ejpam-4478	87	7	the	the	DET
ejpam-4478	87	8	graph	graph	NOUN
ejpam-4478	87	9	g	g	NOUN
ejpam-4478	87	10	,	,	PUNCT
ejpam-4478	87	11	denoted	denote	VERB
ejpam-4478	87	12	by	by	ADP
ejpam-4478	87	13	∆k(g	∆k(g	NOUN
ejpam-4478	87	14	)	)	PUNCT
ejpam-4478	87	15	,	,	PUNCT
ejpam-4478	87	16	is	be	AUX
ejpam-4478	87	17	the	the	DET
ejpam-4478	87	18	maximum	maximum	ADJ
ejpam-4478	87	19	k	k	PROPN
ejpam-4478	87	20	−	−	NOUN
ejpam-4478	87	21	degree	degree	NOUN
ejpam-4478	87	22	taken	take	VERB
ejpam-4478	87	23	over	over	ADP
ejpam-4478	87	24	all	all	DET
ejpam-4478	87	25	vertices	vertex	NOUN
ejpam-4478	87	26	of	of	ADP
ejpam-4478	87	27	graph	graph	NOUN
ejpam-4478	87	28	g	g	PROPN
ejpam-4478	87	29	,	,	PUNCT
ejpam-4478	87	30	that	that	ADV
ejpam-4478	87	31	is	is	ADV
ejpam-4478	87	32	,	,	PUNCT
ejpam-4478	87	33	∆k(g	∆k(g	PROPN
ejpam-4478	87	34	)	)	PUNCT
ejpam-4478	87	35	=	=	SYM
ejpam-4478	87	36	max{degk	max{degk	NOUN
ejpam-4478	87	37	,	,	PUNCT
ejpam-4478	87	38	g(v	g(v	PROPN
ejpam-4478	87	39	)	)	PUNCT
ejpam-4478	87	40	:	:	PUNCT
ejpam-4478	87	41	v	v	X
ejpam-4478	87	42	∈	∈	PROPN
ejpam-4478	87	43	v	v	NOUN
ejpam-4478	87	44	(	(	PUNCT
ejpam-4478	87	45	g	g	NOUN
ejpam-4478	87	46	)	)	PUNCT
ejpam-4478	87	47	}	}	PUNCT
ejpam-4478	87	48	.	.	PUNCT
ejpam-4478	88	1	the	the	DET
ejpam-4478	88	2	minimum	minimum	PROPN
ejpam-4478	88	3	g.	g.	PROPN
ejpam-4478	88	4	entero	entero	PROPN
ejpam-4478	88	5	,	,	PUNCT
ejpam-4478	88	6	s.	s.	PROPN
ejpam-4478	88	7	espinola	espinola	PROPN
ejpam-4478	88	8	/	/	SYM
ejpam-4478	88	9	eur	eur	PROPN
ejpam-4478	88	10	.	.	PUNCT
ejpam-4478	89	1	j.	j.	PROPN
ejpam-4478	89	2	pure	pure	PROPN
ejpam-4478	89	3	appl	appl	PROPN
ejpam-4478	89	4	.	.	PROPN
ejpam-4478	89	5	math	math	PROPN
ejpam-4478	89	6	,	,	PUNCT
ejpam-4478	89	7	16	16	NUM
ejpam-4478	89	8	(	(	PUNCT
ejpam-4478	89	9	1	1	NUM
ejpam-4478	89	10	)	)	PUNCT
ejpam-4478	89	11	(	(	PUNCT
ejpam-4478	89	12	2023	2023	NUM
ejpam-4478	89	13	)	)	PUNCT
ejpam-4478	89	14	,	,	PUNCT
ejpam-4478	89	15	44	44	NUM
ejpam-4478	89	16	-	-	SYM
ejpam-4478	89	17	61	61	NUM
ejpam-4478	89	18	47	47	NUM
ejpam-4478	89	19	k	k	NOUN
ejpam-4478	89	20	−	−	NOUN
ejpam-4478	89	21	degree	degree	NOUN
ejpam-4478	89	22	of	of	ADP
ejpam-4478	89	23	the	the	DET
ejpam-4478	89	24	graph	graph	NOUN
ejpam-4478	89	25	g	g	NOUN
ejpam-4478	89	26	,	,	PUNCT
ejpam-4478	89	27	denoted	denote	VERB
ejpam-4478	89	28	by	by	ADP
ejpam-4478	89	29	δk(g	δk(g	NOUN
ejpam-4478	89	30	)	)	PUNCT
ejpam-4478	89	31	,	,	PUNCT
ejpam-4478	89	32	is	be	AUX
ejpam-4478	89	33	the	the	DET
ejpam-4478	89	34	minimum	minimum	ADJ
ejpam-4478	89	35	k	k	PROPN
ejpam-4478	89	36	−	−	NOUN
ejpam-4478	89	37	degree	degree	NOUN
ejpam-4478	89	38	taken	take	VERB
ejpam-4478	89	39	over	over	ADP
ejpam-4478	89	40	all	all	DET
ejpam-4478	89	41	vertices	vertex	NOUN
ejpam-4478	89	42	of	of	ADP
ejpam-4478	89	43	graph	graph	NOUN
ejpam-4478	89	44	g	g	PROPN
ejpam-4478	89	45	,	,	PUNCT
ejpam-4478	89	46	that	that	ADV
ejpam-4478	89	47	is	is	ADV
ejpam-4478	89	48	,	,	PUNCT
ejpam-4478	89	49	δk(g	δk(g	ADJ
ejpam-4478	89	50	)	)	PUNCT
ejpam-4478	89	51	=	=	SYM
ejpam-4478	89	52	min{degk	min{degk	ADJ
ejpam-4478	89	53	,	,	PUNCT
ejpam-4478	89	54	g(v	g(v	PROPN
ejpam-4478	89	55	)	)	PUNCT
ejpam-4478	89	56	:	:	PUNCT
ejpam-4478	89	57	v	v	X
ejpam-4478	89	58	∈	∈	PROPN
ejpam-4478	89	59	v	v	NOUN
ejpam-4478	89	60	(	(	PUNCT
ejpam-4478	89	61	g	g	NOUN
ejpam-4478	89	62	)	)	PUNCT
ejpam-4478	89	63	}	}	PUNCT
ejpam-4478	89	64	.	.	PUNCT
ejpam-4478	90	1	[	[	X
ejpam-4478	90	2	2	2	X
ejpam-4478	90	3	]	]	PUNCT
ejpam-4478	90	4	the	the	DET
ejpam-4478	90	5	k	k	PROPN
ejpam-4478	90	6	−	−	PROPN
ejpam-4478	90	7	neighbourhood	neighbourhood	NOUN
ejpam-4478	90	8	(	(	PUNCT
ejpam-4478	90	9	or	or	CCONJ
ejpam-4478	90	10	open	open	VERB
ejpam-4478	90	11	k	k	PRON
ejpam-4478	90	12	−	−	NOUN
ejpam-4478	90	13	neighbourhood	neighbourhood	NOUN
ejpam-4478	90	14	)	)	PUNCT
ejpam-4478	90	15	of	of	ADP
ejpam-4478	90	16	a	a	DET
ejpam-4478	90	17	vertex	vertex	NOUN
ejpam-4478	90	18	v	v	NOUN
ejpam-4478	90	19	in	in	ADP
ejpam-4478	90	20	graph	graph	NOUN
ejpam-4478	90	21	g	g	NOUN
ejpam-4478	90	22	,	,	PUNCT
ejpam-4478	90	23	denoted	denote	VERB
ejpam-4478	90	24	by	by	ADP
ejpam-4478	90	25	nk	nk	PROPN
ejpam-4478	90	26	,	,	PUNCT
ejpam-4478	90	27	g(v	g(v	PROPN
ejpam-4478	90	28	)	)	PUNCT
ejpam-4478	90	29	,	,	PUNCT
ejpam-4478	90	30	is	be	AUX
ejpam-4478	90	31	the	the	DET
ejpam-4478	90	32	set	set	NOUN
ejpam-4478	90	33	of	of	ADP
ejpam-4478	90	34	vertices	vertex	NOUN
ejpam-4478	90	35	(	(	PUNCT
ejpam-4478	90	36	different	different	ADJ
ejpam-4478	90	37	from	from	ADP
ejpam-4478	90	38	v	v	NOUN
ejpam-4478	90	39	)	)	PUNCT
ejpam-4478	90	40	adjacent	adjacent	ADJ
ejpam-4478	90	41	to	to	ADP
ejpam-4478	90	42	v	v	NOUN
ejpam-4478	90	43	in	in	ADP
ejpam-4478	90	44	g	g	NOUN
ejpam-4478	90	45	within	within	ADP
ejpam-4478	90	46	distance	distance	NOUN
ejpam-4478	90	47	k	k	NOUN
ejpam-4478	90	48	,	,	PUNCT
ejpam-4478	90	49	that	that	ADV
ejpam-4478	90	50	is	is	ADV
ejpam-4478	90	51	,	,	PUNCT
ejpam-4478	90	52	nk	nk	PROPN
ejpam-4478	90	53	,	,	PUNCT
ejpam-4478	90	54	g(v	g(v	PROPN
ejpam-4478	90	55	)	)	PUNCT
ejpam-4478	90	56	=	=	SYM
ejpam-4478	91	1	{	{	PUNCT
ejpam-4478	91	2	u	u	NOUN
ejpam-4478	91	3	∈	∈	PROPN
ejpam-4478	91	4	v	v	NOUN
ejpam-4478	91	5	(	(	PUNCT
ejpam-4478	91	6	g	g	NOUN
ejpam-4478	91	7	)	)	PUNCT
ejpam-4478	91	8	:	:	PUNCT
ejpam-4478	91	9	u	u	PROPN
ejpam-4478	91	10	̸=	̸=	PROPN
ejpam-4478	91	11	v	v	NOUN
ejpam-4478	91	12	and	and	CCONJ
ejpam-4478	91	13	d(u	d(u	PROPN
ejpam-4478	91	14	,	,	PUNCT
ejpam-4478	91	15	v	v	NOUN
ejpam-4478	91	16	)	)	PUNCT
ejpam-4478	91	17	≤	≤	NOUN
ejpam-4478	92	1	k	k	X
ejpam-4478	92	2	}	}	PUNCT
ejpam-4478	92	3	,	,	PUNCT
ejpam-4478	92	4	where	where	SCONJ
ejpam-4478	92	5	k	k	PROPN
ejpam-4478	92	6	∈	∈	PROPN
ejpam-4478	92	7	z+	z+	PUNCT
ejpam-4478	92	8	.	.	PUNCT
ejpam-4478	93	1	the	the	DET
ejpam-4478	93	2	closed	closed	ADJ
ejpam-4478	93	3	k	k	PROPN
ejpam-4478	93	4	−	−	PROPN
ejpam-4478	93	5	neighbourhood	neighbourhood	NOUN
ejpam-4478	93	6	of	of	ADP
ejpam-4478	93	7	a	a	DET
ejpam-4478	93	8	vertex	vertex	NOUN
ejpam-4478	93	9	v	v	NOUN
ejpam-4478	93	10	,	,	PUNCT
ejpam-4478	93	11	denoted	denote	VERB
ejpam-4478	93	12	by	by	ADP
ejpam-4478	93	13	nk	nk	PROPN
ejpam-4478	93	14	,	,	PUNCT
ejpam-4478	93	15	g[v	g[v	PROPN
ejpam-4478	93	16	]	]	PUNCT
ejpam-4478	93	17	,	,	PUNCT
ejpam-4478	93	18	is	be	AUX
ejpam-4478	93	19	simply	simply	ADV
ejpam-4478	93	20	the	the	DET
ejpam-4478	93	21	set	set	NOUN
ejpam-4478	93	22	{	{	PUNCT
ejpam-4478	93	23	v	v	NOUN
ejpam-4478	93	24	}	}	PUNCT
ejpam-4478	93	25	∪	∪	NOUN
ejpam-4478	93	26	nk	nk	PROPN
ejpam-4478	93	27	,	,	PUNCT
ejpam-4478	93	28	g(v	g(v	PROPN
ejpam-4478	93	29	)	)	PUNCT
ejpam-4478	93	30	.	.	PUNCT
ejpam-4478	94	1	given	give	VERB
ejpam-4478	94	2	a	a	DET
ejpam-4478	94	3	set	set	NOUN
ejpam-4478	94	4	s	s	NOUN
ejpam-4478	94	5	of	of	ADP
ejpam-4478	94	6	vertices	vertex	NOUN
ejpam-4478	94	7	,	,	PUNCT
ejpam-4478	94	8	we	we	PRON
ejpam-4478	94	9	define	define	VERB
ejpam-4478	94	10	the	the	DET
ejpam-4478	94	11	k	k	PROPN
ejpam-4478	94	12	−	−	PROPN
ejpam-4478	94	13	neighbourhood	neighbourhood	NOUN
ejpam-4478	94	14	of	of	ADP
ejpam-4478	94	15	s	s	PROPN
ejpam-4478	94	16	,	,	PUNCT
ejpam-4478	94	17	denoted	denote	VERB
ejpam-4478	94	18	by	by	ADP
ejpam-4478	94	19	nk	nk	PROPN
ejpam-4478	94	20	,	,	PUNCT
ejpam-4478	94	21	g(s	g(s	PROPN
ejpam-4478	94	22	)	)	PUNCT
ejpam-4478	94	23	,	,	PUNCT
ejpam-4478	94	24	to	to	PART
ejpam-4478	94	25	be	be	AUX
ejpam-4478	94	26	the	the	DET
ejpam-4478	94	27	union	union	NOUN
ejpam-4478	94	28	of	of	ADP
ejpam-4478	94	29	the	the	DET
ejpam-4478	94	30	k	k	PROPN
ejpam-4478	94	31	−	−	PROPN
ejpam-4478	94	32	neighbourhoods	neighbourhood	NOUN
ejpam-4478	94	33	of	of	ADP
ejpam-4478	94	34	the	the	DET
ejpam-4478	94	35	vertices	vertex	NOUN
ejpam-4478	94	36	in	in	ADP
ejpam-4478	94	37	s	s	PRON
ejpam-4478	94	38	within	within	ADP
ejpam-4478	94	39	distance	distance	NOUN
ejpam-4478	94	40	k	k	PROPN
ejpam-4478	94	41	with	with	ADP
ejpam-4478	94	42	respect	respect	NOUN
ejpam-4478	94	43	to	to	AUX
ejpam-4478	94	44	graph	graph	VERB
ejpam-4478	94	45	g	g	PROPN
ejpam-4478	94	46	,	,	PUNCT
ejpam-4478	94	47	where	where	SCONJ
ejpam-4478	94	48	k	k	PROPN
ejpam-4478	94	49	∈	∈	PROPN
ejpam-4478	94	50	z+	z+	PUNCT
ejpam-4478	94	51	.	.	PUNCT
ejpam-4478	95	1	similarly	similarly	ADV
ejpam-4478	95	2	,	,	PUNCT
ejpam-4478	95	3	the	the	DET
ejpam-4478	95	4	closed	closed	ADJ
ejpam-4478	95	5	k	k	PROPN
ejpam-4478	95	6	−	−	PROPN
ejpam-4478	95	7	neighbourhood	neighbourhood	NOUN
ejpam-4478	95	8	of	of	ADP
ejpam-4478	95	9	s	s	PROPN
ejpam-4478	95	10	,	,	PUNCT
ejpam-4478	95	11	denoted	denote	VERB
ejpam-4478	95	12	by	by	ADP
ejpam-4478	95	13	nk	nk	PROPN
ejpam-4478	95	14	,	,	PUNCT
ejpam-4478	95	15	g[s	g[s	PROPN
ejpam-4478	95	16	]	]	PUNCT
ejpam-4478	95	17	,	,	PUNCT
ejpam-4478	95	18	is	be	AUX
ejpam-4478	95	19	defined	define	VERB
ejpam-4478	95	20	to	to	PART
ejpam-4478	95	21	be	be	AUX
ejpam-4478	95	22	s	s	PROPN
ejpam-4478	95	23	∪nk	∪nk	NOUN
ejpam-4478	95	24	,	,	PUNCT
ejpam-4478	95	25	g(s	g(s	NOUN
ejpam-4478	95	26	)	)	PUNCT
ejpam-4478	95	27	.	.	PUNCT
ejpam-4478	96	1	[	[	X
ejpam-4478	96	2	2	2	X
ejpam-4478	96	3	]	]	PUNCT
ejpam-4478	96	4	let	let	VERB
ejpam-4478	96	5	v	v	NUM
ejpam-4478	96	6	∈	∈	NOUN
ejpam-4478	96	7	s	s	VERB
ejpam-4478	96	8	⊆	⊆	NUM
ejpam-4478	96	9	v	v	NOUN
ejpam-4478	96	10	.	.	PUNCT
ejpam-4478	97	1	then	then	ADV
ejpam-4478	97	2	,	,	PUNCT
ejpam-4478	97	3	u	u	NOUN
ejpam-4478	97	4	is	be	AUX
ejpam-4478	97	5	called	call	VERB
ejpam-4478	97	6	a	a	DET
ejpam-4478	97	7	private	private	ADJ
ejpam-4478	97	8	neighbour	neighbour	NOUN
ejpam-4478	97	9	of	of	ADP
ejpam-4478	97	10	v	v	NOUN
ejpam-4478	97	11	with	with	ADP
ejpam-4478	97	12	respect	respect	NOUN
ejpam-4478	97	13	to	to	ADP
ejpam-4478	97	14	s	s	PRON
ejpam-4478	97	15	,	,	PUNCT
ejpam-4478	97	16	denoted	denote	VERB
ejpam-4478	97	17	by	by	ADP
ejpam-4478	97	18	u	u	NOUN
ejpam-4478	97	19	is	be	AUX
ejpam-4478	97	20	an	an	DET
ejpam-4478	97	21	s−pn	s−pn	NOUN
ejpam-4478	97	22	of	of	ADP
ejpam-4478	97	23	v	v	NOUN
ejpam-4478	97	24	,	,	PUNCT
ejpam-4478	97	25	if	if	SCONJ
ejpam-4478	97	26	u	u	PROPN
ejpam-4478	97	27	∈	∈	PROPN
ejpam-4478	97	28	n(v)−n(s−{v	n(v)−n(s−{v	X
ejpam-4478	97	29	}	}	PUNCT
ejpam-4478	97	30	)	)	PUNCT
ejpam-4478	97	31	.	.	PUNCT
ejpam-4478	98	1	an	an	DET
ejpam-4478	98	2	s−pn	s−pn	NOUN
ejpam-4478	98	3	of	of	ADP
ejpam-4478	98	4	v	v	NOUN
ejpam-4478	98	5	is	be	AUX
ejpam-4478	98	6	external	external	ADJ
ejpam-4478	98	7	if	if	SCONJ
ejpam-4478	98	8	it	it	PRON
ejpam-4478	98	9	is	be	AUX
ejpam-4478	98	10	in	in	ADP
ejpam-4478	98	11	v	v	NUM
ejpam-4478	98	12	−s	−s	NOUN
ejpam-4478	98	13	.	.	PUNCT
ejpam-4478	99	1	the	the	DET
ejpam-4478	99	2	pn(v	pn(v	PROPN
ejpam-4478	99	3	,	,	PUNCT
ejpam-4478	99	4	s	s	NOUN
ejpam-4478	99	5	)	)	PUNCT
ejpam-4478	99	6	=	=	SYM
ejpam-4478	99	7	n(v)−n(s−{v	n(v)−n(s−{v	X
ejpam-4478	99	8	}	}	PUNCT
ejpam-4478	99	9	)	)	PUNCT
ejpam-4478	99	10	of	of	ADP
ejpam-4478	99	11	all	all	DET
ejpam-4478	99	12	s−	s−	PROPN
ejpam-4478	99	13	pn	pn	PROPN
ejpam-4478	99	14	’s	’s	ADV
ejpam-4478	99	15	of	of	ADP
ejpam-4478	99	16	v	v	NOUN
ejpam-4478	99	17	is	be	AUX
ejpam-4478	99	18	called	call	VERB
ejpam-4478	99	19	the	the	DET
ejpam-4478	99	20	private	private	ADJ
ejpam-4478	99	21	neighbourhood	neighbourhood	NOUN
ejpam-4478	99	22	set	set	VERB
ejpam-4478	99	23	of	of	ADP
ejpam-4478	99	24	v	v	NOUN
ejpam-4478	99	25	with	with	ADP
ejpam-4478	99	26	respect	respect	NOUN
ejpam-4478	99	27	to	to	ADP
ejpam-4478	99	28	s.	s.	PROPN
ejpam-4478	99	29	equivalently	equivalently	PROPN
ejpam-4478	99	30	,	,	PUNCT
ejpam-4478	99	31	pn(v	pn(v	X
ejpam-4478	99	32	,	,	PUNCT
ejpam-4478	99	33	s	s	X
ejpam-4478	99	34	)	)	PUNCT
ejpam-4478	99	35	=	=	SYM
ejpam-4478	99	36	{	{	PUNCT
ejpam-4478	99	37	u	u	NOUN
ejpam-4478	99	38	∈	∈	PROPN
ejpam-4478	99	39	v	v	ADP
ejpam-4478	99	40	|n(u	|n(u	NOUN
ejpam-4478	99	41	)	)	PUNCT
ejpam-4478	99	42	∩	∩	PROPN
ejpam-4478	99	43	s	s	PART
ejpam-4478	99	44	=	=	X
ejpam-4478	99	45	{	{	PUNCT
ejpam-4478	99	46	v	v	NOUN
ejpam-4478	99	47	}	}	PUNCT
ejpam-4478	99	48	}	}	PUNCT
ejpam-4478	99	49	.	.	PUNCT
ejpam-4478	100	1	[	[	X
ejpam-4478	100	2	4	4	X
ejpam-4478	100	3	]	]	PUNCT
ejpam-4478	100	4	the	the	DET
ejpam-4478	100	5	complement	complement	NOUN
ejpam-4478	100	6	g	g	NOUN
ejpam-4478	100	7	of	of	ADP
ejpam-4478	100	8	a	a	DET
ejpam-4478	100	9	graph	graph	NOUN
ejpam-4478	100	10	g	g	NOUN
ejpam-4478	100	11	is	be	AUX
ejpam-4478	100	12	the	the	DET
ejpam-4478	100	13	graph	graph	NOUN
ejpam-4478	100	14	with	with	ADP
ejpam-4478	100	15	vertex	vertex	NOUN
ejpam-4478	100	16	set	set	VERB
ejpam-4478	100	17	v	v	NOUN
ejpam-4478	100	18	(	(	PUNCT
ejpam-4478	100	19	g	g	NOUN
ejpam-4478	100	20	)	)	PUNCT
ejpam-4478	100	21	such	such	ADJ
ejpam-4478	100	22	that	that	SCONJ
ejpam-4478	100	23	two	two	NUM
ejpam-4478	100	24	vertices	vertex	NOUN
ejpam-4478	100	25	are	be	AUX
ejpam-4478	100	26	adjacent	adjacent	ADJ
ejpam-4478	100	27	in	in	ADP
ejpam-4478	100	28	g	g	PROPN
ejpam-4478	100	29	if	if	SCONJ
ejpam-4478	100	30	and	and	CCONJ
ejpam-4478	100	31	only	only	ADV
ejpam-4478	100	32	if	if	SCONJ
ejpam-4478	100	33	these	these	DET
ejpam-4478	100	34	vertices	vertex	NOUN
ejpam-4478	100	35	are	be	AUX
ejpam-4478	100	36	not	not	PART
ejpam-4478	100	37	adjacent	adjacent	ADJ
ejpam-4478	100	38	in	in	ADP
ejpam-4478	100	39	g.	g.	PROPN
ejpam-4478	100	40	this	this	PRON
ejpam-4478	100	41	means	mean	VERB
ejpam-4478	100	42	that	that	SCONJ
ejpam-4478	100	43	both	both	CCONJ
ejpam-4478	100	44	g	g	PROPN
ejpam-4478	100	45	and	and	CCONJ
ejpam-4478	100	46	its	its	PRON
ejpam-4478	100	47	complement	complement	NOUN
ejpam-4478	100	48	g	g	NOUN
ejpam-4478	100	49	have	have	VERB
ejpam-4478	100	50	the	the	DET
ejpam-4478	100	51	same	same	ADJ
ejpam-4478	100	52	vertices	vertex	NOUN
ejpam-4478	100	53	,	,	PUNCT
ejpam-4478	100	54	but	but	CCONJ
ejpam-4478	100	55	g	g	NOUN
ejpam-4478	100	56	has	have	VERB
ejpam-4478	100	57	precisely	precisely	ADV
ejpam-4478	100	58	the	the	DET
ejpam-4478	100	59	edges	edge	NOUN
ejpam-4478	100	60	that	that	PRON
ejpam-4478	100	61	g	g	PROPN
ejpam-4478	100	62	lacks	lack	VERB
ejpam-4478	100	63	.	.	PUNCT
ejpam-4478	101	1	[	[	X
ejpam-4478	101	2	8	8	NUM
ejpam-4478	101	3	]	]	X
ejpam-4478	101	4	the	the	DET
ejpam-4478	101	5	join	join	NOUN
ejpam-4478	101	6	of	of	ADP
ejpam-4478	101	7	two	two	NUM
ejpam-4478	101	8	graphs	graph	NOUN
ejpam-4478	101	9	g	g	NOUN
ejpam-4478	101	10	and	and	CCONJ
ejpam-4478	101	11	h	h	NOUN
ejpam-4478	101	12	,	,	PUNCT
ejpam-4478	101	13	denoted	denote	VERB
ejpam-4478	101	14	by	by	ADP
ejpam-4478	101	15	g	g	PROPN
ejpam-4478	101	16	+	+	PROPN
ejpam-4478	101	17	h	h	NOUN
ejpam-4478	101	18	,	,	PUNCT
ejpam-4478	101	19	is	be	AUX
ejpam-4478	101	20	the	the	DET
ejpam-4478	101	21	graph	graph	NOUN
ejpam-4478	101	22	with	with	ADP
ejpam-4478	101	23	vertex	vertex	NOUN
ejpam-4478	101	24	set	set	VERB
ejpam-4478	101	25	v	v	NOUN
ejpam-4478	101	26	(	(	PUNCT
ejpam-4478	101	27	g	g	PROPN
ejpam-4478	101	28	+	+	NOUN
ejpam-4478	101	29	h	h	NOUN
ejpam-4478	101	30	)	)	PUNCT
ejpam-4478	101	31	=	=	NOUN
ejpam-4478	101	32	v	v	X
ejpam-4478	101	33	(	(	PUNCT
ejpam-4478	101	34	g	g	NOUN
ejpam-4478	101	35	)	)	PUNCT
ejpam-4478	101	36	∪	∪	NOUN
ejpam-4478	101	37	v	v	NOUN
ejpam-4478	101	38	(	(	PUNCT
ejpam-4478	101	39	h	h	NOUN
ejpam-4478	101	40	)	)	PUNCT
ejpam-4478	101	41	and	and	CCONJ
ejpam-4478	101	42	the	the	DET
ejpam-4478	101	43	edge	edge	NOUN
ejpam-4478	101	44	set	set	VERB
ejpam-4478	101	45	e(g	e(g	PROPN
ejpam-4478	101	46	+	+	CCONJ
ejpam-4478	101	47	h	h	NOUN
ejpam-4478	101	48	)	)	PUNCT
ejpam-4478	101	49	=	=	SYM
ejpam-4478	101	50	e(g	e(g	PROPN
ejpam-4478	101	51	)	)	PUNCT
ejpam-4478	101	52	∪	∪	ADP
ejpam-4478	101	53	e(h	e(h	PROPN
ejpam-4478	101	54	)	)	PUNCT
ejpam-4478	101	55	∪	∪	NOUN
ejpam-4478	101	56	{	{	PUNCT
ejpam-4478	101	57	uv	uv	NOUN
ejpam-4478	101	58	:	:	PUNCT
ejpam-4478	101	59	u	u	PROPN
ejpam-4478	101	60	∈	∈	PROPN
ejpam-4478	101	61	v	v	ADP
ejpam-4478	101	62	(	(	PUNCT
ejpam-4478	101	63	g	g	NOUN
ejpam-4478	101	64	)	)	PUNCT
ejpam-4478	101	65	and	and	CCONJ
ejpam-4478	101	66	v	v	ADP
ejpam-4478	101	67	∈	∈	PROPN
ejpam-4478	101	68	v	v	NOUN
ejpam-4478	101	69	(	(	PUNCT
ejpam-4478	101	70	h	h	NOUN
ejpam-4478	101	71	)	)	PUNCT
ejpam-4478	101	72	}	}	PUNCT
ejpam-4478	101	73	.	.	PUNCT
ejpam-4478	102	1	[	[	X
ejpam-4478	102	2	8	8	NUM
ejpam-4478	102	3	]	]	PUNCT
ejpam-4478	102	4	let	let	VERB
ejpam-4478	102	5	g	g	NOUN
ejpam-4478	102	6	and	and	CCONJ
ejpam-4478	102	7	h	h	NOUN
ejpam-4478	102	8	be	be	AUX
ejpam-4478	102	9	graphs	graph	NOUN
ejpam-4478	102	10	of	of	ADP
ejpam-4478	102	11	orders	order	NOUN
ejpam-4478	102	12	p1	p1	NOUN
ejpam-4478	102	13	and	and	CCONJ
ejpam-4478	102	14	p2	p2	NOUN
ejpam-4478	102	15	,	,	PUNCT
ejpam-4478	102	16	respectively	respectively	ADV
ejpam-4478	102	17	.	.	PUNCT
ejpam-4478	103	1	then	then	ADV
ejpam-4478	103	2	the	the	DET
ejpam-4478	103	3	graph	graph	NOUN
ejpam-4478	103	4	obtained	obtain	VERB
ejpam-4478	103	5	by	by	ADP
ejpam-4478	103	6	taking	take	VERB
ejpam-4478	103	7	one	one	NUM
ejpam-4478	103	8	copy	copy	NOUN
ejpam-4478	103	9	of	of	ADP
ejpam-4478	103	10	g	g	NOUN
ejpam-4478	103	11	of	of	ADP
ejpam-4478	103	12	order	order	NOUN
ejpam-4478	103	13	p1	p1	NOUN
ejpam-4478	103	14	and	and	CCONJ
ejpam-4478	103	15	p1	p1	PROPN
ejpam-4478	103	16	copies	copy	NOUN
ejpam-4478	103	17	of	of	ADP
ejpam-4478	103	18	h	h	NOUN
ejpam-4478	103	19	and	and	CCONJ
ejpam-4478	103	20	then	then	ADV
ejpam-4478	103	21	connecting	connect	VERB
ejpam-4478	103	22	the	the	DET
ejpam-4478	103	23	ith	ith	PROPN
ejpam-4478	103	24	vertex	vertex	NOUN
ejpam-4478	103	25	of	of	ADP
ejpam-4478	103	26	g	g	NOUN
ejpam-4478	103	27	to	to	ADP
ejpam-4478	103	28	every	every	DET
ejpam-4478	103	29	vertex	vertex	NOUN
ejpam-4478	103	30	of	of	ADP
ejpam-4478	103	31	the	the	DET
ejpam-4478	103	32	ith	ith	PROPN
ejpam-4478	103	33	copy	copy	NOUN
ejpam-4478	103	34	of	of	ADP
ejpam-4478	103	35	h	h	PROPN
ejpam-4478	103	36	(	(	PUNCT
ejpam-4478	103	37	ith	ith	PROPN
ejpam-4478	103	38	means	mean	VERB
ejpam-4478	103	39	first	first	ADJ
ejpam-4478	103	40	,	,	PUNCT
ejpam-4478	103	41	second	second	ADJ
ejpam-4478	103	42	,	,	PUNCT
ejpam-4478	103	43	third	third	ADJ
ejpam-4478	103	44	and	and	CCONJ
ejpam-4478	103	45	so	so	ADV
ejpam-4478	103	46	on	on	ADV
ejpam-4478	103	47	)	)	PUNCT
ejpam-4478	103	48	is	be	AUX
ejpam-4478	103	49	called	call	VERB
ejpam-4478	103	50	corona	corona	NOUN
ejpam-4478	103	51	,	,	PUNCT
ejpam-4478	103	52	denoted	denote	VERB
ejpam-4478	103	53	by	by	ADP
ejpam-4478	103	54	g	g	PROPN
ejpam-4478	103	55	◦	◦	NOUN
ejpam-4478	103	56	h.	h.	NOUN
ejpam-4478	103	57	the	the	DET
ejpam-4478	103	58	order	order	NOUN
ejpam-4478	103	59	and	and	CCONJ
ejpam-4478	103	60	the	the	DET
ejpam-4478	103	61	size	size	NOUN
ejpam-4478	103	62	of	of	ADP
ejpam-4478	103	63	the	the	DET
ejpam-4478	103	64	corona	corona	NOUN
ejpam-4478	103	65	g	g	PROPN
ejpam-4478	103	66	◦	◦	NOUN
ejpam-4478	103	67	h	h	NOUN
ejpam-4478	103	68	are	be	AUX
ejpam-4478	103	69	p1	p1	PROPN
ejpam-4478	103	70	+	+	CCONJ
ejpam-4478	103	71	p1p2	p1p2	PROPN
ejpam-4478	103	72	and	and	CCONJ
ejpam-4478	103	73	q1+p1q2+p1p2	q1+p1q2+p1p2	PROPN
ejpam-4478	103	74	,	,	PUNCT
ejpam-4478	103	75	respectively	respectively	ADV
ejpam-4478	103	76	,	,	PUNCT
ejpam-4478	103	77	where	where	SCONJ
ejpam-4478	103	78	q1	q1	PROPN
ejpam-4478	103	79	and	and	CCONJ
ejpam-4478	103	80	q2	q2	NOUN
ejpam-4478	103	81	are	be	AUX
ejpam-4478	103	82	the	the	DET
ejpam-4478	103	83	sizes	size	NOUN
ejpam-4478	103	84	of	of	ADP
ejpam-4478	103	85	graphs	graph	NOUN
ejpam-4478	103	86	g	g	PROPN
ejpam-4478	103	87	and	and	CCONJ
ejpam-4478	103	88	h	h	NOUN
ejpam-4478	103	89	,	,	PUNCT
ejpam-4478	103	90	respectively	respectively	ADV
ejpam-4478	103	91	.	.	PUNCT
ejpam-4478	104	1	[	[	X
ejpam-4478	104	2	5	5	NUM
ejpam-4478	104	3	]	]	PUNCT
ejpam-4478	104	4	a	a	DET
ejpam-4478	104	5	cartesian	cartesian	ADJ
ejpam-4478	104	6	product	product	NOUN
ejpam-4478	104	7	,	,	PUNCT
ejpam-4478	104	8	denoted	denote	VERB
ejpam-4478	104	9	by	by	ADP
ejpam-4478	104	10	g	g	PROPN
ejpam-4478	104	11	×	×	PROPN
ejpam-4478	104	12	h	h	NOUN
ejpam-4478	104	13	,	,	PUNCT
ejpam-4478	104	14	of	of	ADP
ejpam-4478	104	15	two	two	NUM
ejpam-4478	104	16	graphs	graph	NOUN
ejpam-4478	104	17	g	g	NOUN
ejpam-4478	104	18	and	and	CCONJ
ejpam-4478	104	19	h	h	NOUN
ejpam-4478	104	20	,	,	PUNCT
ejpam-4478	104	21	is	be	AUX
ejpam-4478	104	22	the	the	DET
ejpam-4478	104	23	graph	graph	NOUN
ejpam-4478	104	24	with	with	ADP
ejpam-4478	104	25	vertex	vertex	NOUN
ejpam-4478	104	26	set	set	VERB
ejpam-4478	104	27	v	v	NOUN
ejpam-4478	104	28	(	(	PUNCT
ejpam-4478	104	29	g	g	PROPN
ejpam-4478	104	30	×	×	PROPN
ejpam-4478	104	31	h	h	NOUN
ejpam-4478	104	32	)	)	PUNCT
ejpam-4478	105	1	=	=	NOUN
ejpam-4478	105	2	v	v	X
ejpam-4478	105	3	(	(	PUNCT
ejpam-4478	105	4	g	g	NOUN
ejpam-4478	105	5	)	)	PUNCT
ejpam-4478	105	6	×	×	NOUN
ejpam-4478	105	7	v	v	NOUN
ejpam-4478	105	8	(	(	PUNCT
ejpam-4478	105	9	h	h	NOUN
ejpam-4478	105	10	)	)	PUNCT
ejpam-4478	105	11	and	and	CCONJ
ejpam-4478	105	12	edge	edge	NOUN
ejpam-4478	105	13	set	set	VERB
ejpam-4478	105	14	e(g	e(g	PROPN
ejpam-4478	105	15	×	×	PROPN
ejpam-4478	105	16	h	h	NOUN
ejpam-4478	105	17	)	)	PUNCT
ejpam-4478	105	18	satisfying	satisfy	VERB
ejpam-4478	105	19	the	the	DET
ejpam-4478	105	20	following	follow	VERB
ejpam-4478	105	21	conditions	condition	NOUN
ejpam-4478	105	22	:	:	PUNCT
ejpam-4478	105	23	(	(	PUNCT
ejpam-4478	105	24	u1	u1	PROPN
ejpam-4478	105	25	,	,	PUNCT
ejpam-4478	105	26	v1)(u2	v1)(u2	PROPN
ejpam-4478	105	27	,	,	PUNCT
ejpam-4478	105	28	v2	v2	PROPN
ejpam-4478	105	29	)	)	PUNCT
ejpam-4478	105	30	∈	∈	NOUN
ejpam-4478	105	31	e(g×h	e(g×h	NOUN
ejpam-4478	105	32	)	)	PUNCT
ejpam-4478	105	33	if	if	SCONJ
ejpam-4478	105	34	and	and	CCONJ
ejpam-4478	105	35	only	only	ADV
ejpam-4478	105	36	if	if	SCONJ
ejpam-4478	105	37	either	either	PRON
ejpam-4478	105	38	u1	u1	NOUN
ejpam-4478	105	39	=	=	SYM
ejpam-4478	105	40	u2	u2	PROPN
ejpam-4478	105	41	and	and	CCONJ
ejpam-4478	105	42	v1v2	v1v2	PUNCT
ejpam-4478	105	43	∈	∈	PROPN
ejpam-4478	105	44	e(h	e(h	PROPN
ejpam-4478	105	45	)	)	PUNCT
ejpam-4478	105	46	or	or	CCONJ
ejpam-4478	105	47	v1	v1	NOUN
ejpam-4478	105	48	=	=	SYM
ejpam-4478	105	49	v2	v2	PROPN
ejpam-4478	105	50	and	and	CCONJ
ejpam-4478	105	51	u1u2	u1u2	NOUN
ejpam-4478	105	52	∈	∈	PROPN
ejpam-4478	105	53	e(g	e(g	PROPN
ejpam-4478	105	54	)	)	PUNCT
ejpam-4478	105	55	.	.	PUNCT
ejpam-4478	106	1	[	[	X
ejpam-4478	106	2	3	3	X
ejpam-4478	106	3	]	]	X
ejpam-4478	106	4	the	the	DET
ejpam-4478	106	5	lexicographic	lexicographic	ADJ
ejpam-4478	106	6	product	product	NOUN
ejpam-4478	106	7	of	of	ADP
ejpam-4478	106	8	graphs	graph	NOUN
ejpam-4478	106	9	g	g	NOUN
ejpam-4478	106	10	and	and	CCONJ
ejpam-4478	106	11	h	h	NOUN
ejpam-4478	106	12	is	be	AUX
ejpam-4478	106	13	the	the	DET
ejpam-4478	106	14	graph	graph	NOUN
ejpam-4478	106	15	g[h	g[h	PROPN
ejpam-4478	106	16	]	]	PUNCT
ejpam-4478	106	17	with	with	ADP
ejpam-4478	106	18	a	a	DET
ejpam-4478	106	19	vertex	vertex	NOUN
ejpam-4478	106	20	set	set	VERB
ejpam-4478	106	21	v	v	NOUN
ejpam-4478	106	22	(	(	PUNCT
ejpam-4478	106	23	g[h	g[h	PROPN
ejpam-4478	106	24	]	]	PUNCT
ejpam-4478	106	25	)	)	PUNCT
ejpam-4478	106	26	=	=	SYM
ejpam-4478	106	27	{	{	PUNCT
ejpam-4478	106	28	(	(	PUNCT
ejpam-4478	106	29	u	u	NOUN
ejpam-4478	106	30	,	,	PUNCT
ejpam-4478	106	31	v	v	NOUN
ejpam-4478	106	32	)	)	PUNCT
ejpam-4478	106	33	:	:	PUNCT
ejpam-4478	106	34	u	u	PROPN
ejpam-4478	106	35	∈	∈	PROPN
ejpam-4478	106	36	v	v	ADP
ejpam-4478	106	37	(	(	PUNCT
ejpam-4478	106	38	g	g	NOUN
ejpam-4478	106	39	)	)	PUNCT
ejpam-4478	106	40	and	and	CCONJ
ejpam-4478	106	41	v	v	ADP
ejpam-4478	106	42	∈	∈	PROPN
ejpam-4478	106	43	v	v	NOUN
ejpam-4478	106	44	(	(	PUNCT
ejpam-4478	106	45	h	h	NOUN
ejpam-4478	106	46	)	)	PUNCT
ejpam-4478	106	47	}	}	PUNCT
ejpam-4478	106	48	and	and	CCONJ
ejpam-4478	106	49	with	with	ADP
ejpam-4478	106	50	edge	edge	NOUN
ejpam-4478	106	51	set	set	VERB
ejpam-4478	106	52	e(g[h	e(g[h	NOUN
ejpam-4478	106	53	]	]	PUNCT
ejpam-4478	106	54	)	)	PUNCT
ejpam-4478	106	55	=	=	SYM
ejpam-4478	106	56	{	{	PUNCT
ejpam-4478	106	57	(	(	PUNCT
ejpam-4478	106	58	u	u	NOUN
ejpam-4478	106	59	,	,	PUNCT
ejpam-4478	106	60	v)(w	v)(w	NOUN
ejpam-4478	106	61	,	,	PUNCT
ejpam-4478	106	62	x	x	NOUN
ejpam-4478	106	63	)	)	PUNCT
ejpam-4478	106	64	:	:	PUNCT
ejpam-4478	106	65	uw	uw	PROPN
ejpam-4478	106	66	∈	∈	PROPN
ejpam-4478	106	67	e(g	e(g	PROPN
ejpam-4478	106	68	)	)	PUNCT
ejpam-4478	106	69	or	or	CCONJ
ejpam-4478	106	70	u	u	X
ejpam-4478	106	71	=	=	PROPN
ejpam-4478	106	72	w	w	PROPN
ejpam-4478	106	73	and	and	CCONJ
ejpam-4478	106	74	vx	vx	PROPN
ejpam-4478	106	75	∈	∈	PROPN
ejpam-4478	106	76	e(h	e(h	PROPN
ejpam-4478	106	77	)	)	PUNCT
ejpam-4478	106	78	}	}	PUNCT
ejpam-4478	106	79	.	.	PUNCT
ejpam-4478	107	1	[	[	X
ejpam-4478	107	2	6	6	NUM
ejpam-4478	107	3	]	]	SYM
ejpam-4478	107	4	3	3	NUM
ejpam-4478	107	5	.	.	PUNCT
ejpam-4478	107	6	basic	basic	ADJ
ejpam-4478	107	7	concepts	concept	NOUN
ejpam-4478	107	8	definition	definition	NOUN
ejpam-4478	107	9	3.1	3.1	NUM
ejpam-4478	107	10	.	.	PUNCT
ejpam-4478	108	1	[	[	X
ejpam-4478	108	2	5	5	NUM
ejpam-4478	108	3	]	]	PUNCT
ejpam-4478	108	4	a	a	DET
ejpam-4478	108	5	set	set	NOUN
ejpam-4478	108	6	of	of	ADP
ejpam-4478	108	7	vertices	vertex	NOUN
ejpam-4478	108	8	s	s	PART
ejpam-4478	108	9	⊆	⊆	NUM
ejpam-4478	108	10	v	v	NOUN
ejpam-4478	108	11	(	(	PUNCT
ejpam-4478	108	12	g	g	NOUN
ejpam-4478	108	13	)	)	PUNCT
ejpam-4478	108	14	is	be	AUX
ejpam-4478	108	15	a	a	DET
ejpam-4478	108	16	dominating	dominating	NOUN
ejpam-4478	108	17	set	set	NOUN
ejpam-4478	108	18	for	for	ADP
ejpam-4478	108	19	graph	graph	NOUN
ejpam-4478	108	20	g	g	PROPN
ejpam-4478	108	21	=	=	PUNCT
ejpam-4478	108	22	(	(	PUNCT
ejpam-4478	108	23	v	v	NOUN
ejpam-4478	108	24	(	(	PUNCT
ejpam-4478	108	25	g	g	NOUN
ejpam-4478	108	26	)	)	PUNCT
ejpam-4478	108	27	,	,	PUNCT
ejpam-4478	108	28	e(g	e(g	PROPN
ejpam-4478	108	29	)	)	PUNCT
ejpam-4478	108	30	)	)	PUNCT
ejpam-4478	109	1	if	if	SCONJ
ejpam-4478	109	2	every	every	DET
ejpam-4478	109	3	vertex	vertex	NOUN
ejpam-4478	109	4	not	not	PART
ejpam-4478	109	5	in	in	ADP
ejpam-4478	109	6	s	s	PROPN
ejpam-4478	109	7	is	be	AUX
ejpam-4478	109	8	adjacent	adjacent	ADJ
ejpam-4478	109	9	to	to	ADP
ejpam-4478	109	10	at	at	ADV
ejpam-4478	109	11	least	least	ADV
ejpam-4478	109	12	one	one	NUM
ejpam-4478	109	13	vertex	vertex	NOUN
ejpam-4478	109	14	in	in	ADP
ejpam-4478	109	15	s.	s.	PROPN
ejpam-4478	109	16	the	the	DET
ejpam-4478	109	17	domination	domination	NOUN
ejpam-4478	109	18	number	number	NOUN
ejpam-4478	109	19	of	of	ADP
ejpam-4478	109	20	graph	graph	NOUN
ejpam-4478	109	21	g	g	PROPN
ejpam-4478	109	22	is	be	AUX
ejpam-4478	109	23	the	the	DET
ejpam-4478	109	24	cardinality	cardinality	NOUN
ejpam-4478	109	25	of	of	ADP
ejpam-4478	109	26	any	any	DET
ejpam-4478	109	27	minimum	minimum	NOUN
ejpam-4478	109	28	(	(	PUNCT
ejpam-4478	109	29	smallest	small	ADJ
ejpam-4478	109	30	)	)	PUNCT
ejpam-4478	109	31	dominating	dominating	NOUN
ejpam-4478	109	32	set	set	VERB
ejpam-4478	109	33	in	in	ADP
ejpam-4478	109	34	g	g	NOUN
ejpam-4478	109	35	and	and	CCONJ
ejpam-4478	109	36	is	be	AUX
ejpam-4478	109	37	denoted	denote	VERB
ejpam-4478	109	38	by	by	ADP
ejpam-4478	109	39	γ(g	γ(g	PROPN
ejpam-4478	109	40	)	)	PUNCT
ejpam-4478	109	41	.	.	PUNCT
ejpam-4478	110	1	definition	definition	NOUN
ejpam-4478	110	2	3.2	3.2	NUM
ejpam-4478	110	3	.	.	PUNCT
ejpam-4478	111	1	[	[	X
ejpam-4478	111	2	14	14	NUM
ejpam-4478	111	3	]	]	PUNCT
ejpam-4478	111	4	let	let	VERB
ejpam-4478	111	5	k	k	PROPN
ejpam-4478	111	6	∈	∈	PROPN
ejpam-4478	111	7	z+	z+	PUNCT
ejpam-4478	111	8	.	.	PUNCT
ejpam-4478	112	1	a	a	DET
ejpam-4478	112	2	set	set	NOUN
ejpam-4478	112	3	d	d	NOUN
ejpam-4478	112	4	⊆	⊆	NUM
ejpam-4478	112	5	v	v	ADP
ejpam-4478	112	6	(	(	PUNCT
ejpam-4478	112	7	g	g	NOUN
ejpam-4478	112	8	)	)	PUNCT
ejpam-4478	112	9	is	be	AUX
ejpam-4478	112	10	a	a	DET
ejpam-4478	112	11	distance	distance	NOUN
ejpam-4478	113	1	k	k	NOUN
ejpam-4478	113	2	−	−	NOUN
ejpam-4478	113	3	dominating	dominating	NOUN
ejpam-4478	113	4	set	set	NOUN
ejpam-4478	113	5	of	of	ADP
ejpam-4478	113	6	g	g	PROPN
ejpam-4478	113	7	if	if	SCONJ
ejpam-4478	113	8	each	each	DET
ejpam-4478	113	9	x	x	SYM
ejpam-4478	113	10	∈	∈	PROPN
ejpam-4478	113	11	v	v	NOUN
ejpam-4478	113	12	(	(	PUNCT
ejpam-4478	113	13	g)\d	g)\d	NOUN
ejpam-4478	113	14	is	be	AUX
ejpam-4478	113	15	within	within	ADP
ejpam-4478	113	16	distance	distance	NOUN
ejpam-4478	113	17	k	k	NOUN
ejpam-4478	113	18	from	from	ADP
ejpam-4478	113	19	some	some	DET
ejpam-4478	113	20	vertex	vertex	NOUN
ejpam-4478	113	21	of	of	ADP
ejpam-4478	113	22	d.	d.	PROPN
ejpam-4478	113	23	the	the	DET
ejpam-4478	113	24	minimum	minimum	PROPN
ejpam-4478	113	25	g.	g.	PROPN
ejpam-4478	113	26	entero	entero	PROPN
ejpam-4478	113	27	,	,	PUNCT
ejpam-4478	113	28	s.	s.	PROPN
ejpam-4478	113	29	espinola	espinola	PROPN
ejpam-4478	113	30	/	/	SYM
ejpam-4478	113	31	eur	eur	PROPN
ejpam-4478	113	32	.	.	PUNCT
ejpam-4478	114	1	j.	j.	PROPN
ejpam-4478	114	2	pure	pure	PROPN
ejpam-4478	114	3	appl	appl	PROPN
ejpam-4478	114	4	.	.	PROPN
ejpam-4478	114	5	math	math	PROPN
ejpam-4478	114	6	,	,	PUNCT
ejpam-4478	114	7	16	16	NUM
ejpam-4478	114	8	(	(	PUNCT
ejpam-4478	114	9	1	1	NUM
ejpam-4478	114	10	)	)	PUNCT
ejpam-4478	114	11	(	(	PUNCT
ejpam-4478	114	12	2023	2023	NUM
ejpam-4478	114	13	)	)	PUNCT
ejpam-4478	114	14	,	,	PUNCT
ejpam-4478	114	15	44	44	NUM
ejpam-4478	114	16	-	-	SYM
ejpam-4478	114	17	61	61	NUM
ejpam-4478	114	18	48	48	NUM
ejpam-4478	114	19	cardinality	cardinality	NOUN
ejpam-4478	114	20	taken	take	VERB
ejpam-4478	114	21	over	over	ADP
ejpam-4478	114	22	all	all	DET
ejpam-4478	114	23	distance	distance	NOUN
ejpam-4478	115	1	k	k	NOUN
ejpam-4478	115	2	−	−	NOUN
ejpam-4478	115	3	dominating	dominating	NOUN
ejpam-4478	115	4	sets	set	NOUN
ejpam-4478	115	5	of	of	ADP
ejpam-4478	115	6	graph	graph	NOUN
ejpam-4478	115	7	g	g	PROPN
ejpam-4478	115	8	is	be	AUX
ejpam-4478	115	9	called	call	VERB
ejpam-4478	115	10	the	the	DET
ejpam-4478	115	11	distance	distance	NOUN
ejpam-4478	115	12	k	k	PROPN
ejpam-4478	115	13	−	−	PROPN
ejpam-4478	115	14	domination	domination	NOUN
ejpam-4478	115	15	number	number	NOUN
ejpam-4478	115	16	of	of	ADP
ejpam-4478	115	17	g	g	NOUN
ejpam-4478	115	18	and	and	CCONJ
ejpam-4478	115	19	is	be	AUX
ejpam-4478	115	20	denoted	denote	VERB
ejpam-4478	115	21	by	by	ADP
ejpam-4478	115	22	γk(g	γk(g	NOUN
ejpam-4478	115	23	)	)	PUNCT
ejpam-4478	115	24	.	.	PUNCT
ejpam-4478	116	1	definition	definition	NOUN
ejpam-4478	116	2	3.3	3.3	NUM
ejpam-4478	116	3	.	.	PUNCT
ejpam-4478	117	1	[	[	X
ejpam-4478	117	2	4	4	X
ejpam-4478	117	3	]	]	PUNCT
ejpam-4478	117	4	a	a	DET
ejpam-4478	117	5	roman	roman	ADJ
ejpam-4478	117	6	dominating	dominating	NOUN
ejpam-4478	117	7	function	function	NOUN
ejpam-4478	117	8	(	(	PUNCT
ejpam-4478	117	9	rdf	rdf	VERB
ejpam-4478	117	10	)	)	PUNCT
ejpam-4478	117	11	on	on	ADP
ejpam-4478	117	12	a	a	DET
ejpam-4478	117	13	graph	graph	NOUN
ejpam-4478	117	14	g	g	NOUN
ejpam-4478	117	15	=	=	PUNCT
ejpam-4478	117	16	(	(	PUNCT
ejpam-4478	117	17	v	v	NOUN
ejpam-4478	117	18	,	,	PUNCT
ejpam-4478	117	19	e	e	NOUN
ejpam-4478	117	20	)	)	PUNCT
ejpam-4478	117	21	is	be	AUX
ejpam-4478	117	22	a	a	DET
ejpam-4478	117	23	function	function	NOUN
ejpam-4478	117	24	f	f	NOUN
ejpam-4478	117	25	:	:	PUNCT
ejpam-4478	117	26	v	v	X
ejpam-4478	117	27	→	→	SYM
ejpam-4478	117	28	{	{	PUNCT
ejpam-4478	117	29	0	0	NUM
ejpam-4478	117	30	,	,	PUNCT
ejpam-4478	117	31	1	1	NUM
ejpam-4478	117	32	,	,	PUNCT
ejpam-4478	117	33	2	2	NUM
ejpam-4478	117	34	}	}	PUNCT
ejpam-4478	117	35	satisfying	satisfy	VERB
ejpam-4478	117	36	the	the	DET
ejpam-4478	117	37	condition	condition	NOUN
ejpam-4478	117	38	that	that	SCONJ
ejpam-4478	117	39	every	every	DET
ejpam-4478	117	40	vertex	vertex	NOUN
ejpam-4478	117	41	v	v	NOUN
ejpam-4478	117	42	for	for	ADP
ejpam-4478	117	43	which	which	PRON
ejpam-4478	117	44	f(v	f(v	NOUN
ejpam-4478	117	45	)	)	PUNCT
ejpam-4478	118	1	=	=	SYM
ejpam-4478	118	2	0	0	PUNCT
ejpam-4478	118	3	is	be	AUX
ejpam-4478	118	4	adjacent	adjacent	ADJ
ejpam-4478	118	5	to	to	ADP
ejpam-4478	118	6	at	at	ADV
ejpam-4478	118	7	least	least	ADV
ejpam-4478	118	8	one	one	NUM
ejpam-4478	118	9	vertex	vertex	NOUN
ejpam-4478	118	10	u	u	NOUN
ejpam-4478	118	11	for	for	ADP
ejpam-4478	118	12	which	which	PRON
ejpam-4478	118	13	f(u	f(u	PROPN
ejpam-4478	118	14	)	)	PUNCT
ejpam-4478	118	15	=	=	SYM
ejpam-4478	119	1	2	2	X
ejpam-4478	119	2	.	.	PUNCT
ejpam-4478	119	3	the	the	DET
ejpam-4478	119	4	weight	weight	NOUN
ejpam-4478	119	5	of	of	ADP
ejpam-4478	119	6	the	the	DET
ejpam-4478	119	7	roman	roman	ADJ
ejpam-4478	119	8	dominating	dominating	NOUN
ejpam-4478	119	9	function	function	NOUN
ejpam-4478	119	10	(	(	PUNCT
ejpam-4478	119	11	rdf	rdf	VERB
ejpam-4478	119	12	)	)	PUNCT
ejpam-4478	119	13	f	f	PROPN
ejpam-4478	119	14	is	be	AUX
ejpam-4478	119	15	the	the	DET
ejpam-4478	119	16	value	value	NOUN
ejpam-4478	119	17	w(f	w(f	NOUN
ejpam-4478	119	18	)	)	PUNCT
ejpam-4478	120	1	=	=	SYM
ejpam-4478	120	2	∑	∑	PUNCT
ejpam-4478	120	3	x∈v	x∈v	PROPN
ejpam-4478	120	4	f(x	f(x	PROPN
ejpam-4478	120	5	)	)	PUNCT
ejpam-4478	120	6	.	.	PUNCT
ejpam-4478	121	1	the	the	DET
ejpam-4478	121	2	minimum	minimum	ADJ
ejpam-4478	121	3	weight	weight	NOUN
ejpam-4478	121	4	of	of	ADP
ejpam-4478	121	5	the	the	DET
ejpam-4478	121	6	roman	roman	ADJ
ejpam-4478	121	7	dominating	dominating	NOUN
ejpam-4478	121	8	function	function	NOUN
ejpam-4478	121	9	of	of	ADP
ejpam-4478	121	10	the	the	DET
ejpam-4478	121	11	graph	graph	NOUN
ejpam-4478	121	12	g	g	NOUN
ejpam-4478	121	13	is	be	AUX
ejpam-4478	121	14	called	call	VERB
ejpam-4478	121	15	the	the	DET
ejpam-4478	121	16	roman	roman	ADJ
ejpam-4478	121	17	domination	domination	NOUN
ejpam-4478	121	18	number	number	NOUN
ejpam-4478	121	19	of	of	ADP
ejpam-4478	121	20	g	g	NOUN
ejpam-4478	121	21	and	and	CCONJ
ejpam-4478	121	22	is	be	AUX
ejpam-4478	121	23	denoted	denote	VERB
ejpam-4478	121	24	as	as	ADP
ejpam-4478	121	25	γr(g	γr(g	PROPN
ejpam-4478	121	26	)	)	PUNCT
ejpam-4478	121	27	.	.	PUNCT
ejpam-4478	122	1	definition	definition	NOUN
ejpam-4478	122	2	3.4	3.4	NUM
ejpam-4478	122	3	.	.	PUNCT
ejpam-4478	123	1	[	[	X
ejpam-4478	123	2	9	9	NUM
ejpam-4478	123	3	]	]	PUNCT
ejpam-4478	123	4	a	a	DET
ejpam-4478	123	5	dominating	dominating	NOUN
ejpam-4478	123	6	set	set	NOUN
ejpam-4478	123	7	d	d	NOUN
ejpam-4478	123	8	of	of	ADP
ejpam-4478	123	9	g	g	PROPN
ejpam-4478	123	10	=	=	SYM
ejpam-4478	123	11	(	(	PUNCT
ejpam-4478	123	12	v	v	NOUN
ejpam-4478	123	13	,	,	PUNCT
ejpam-4478	123	14	e	e	NOUN
ejpam-4478	123	15	)	)	PUNCT
ejpam-4478	123	16	is	be	AUX
ejpam-4478	123	17	a	a	DET
ejpam-4478	123	18	global	global	ADJ
ejpam-4478	123	19	dominating	dominating	NOUN
ejpam-4478	123	20	set	set	NOUN
ejpam-4478	123	21	if	if	SCONJ
ejpam-4478	123	22	d	d	NOUN
ejpam-4478	123	23	is	be	AUX
ejpam-4478	123	24	also	also	ADV
ejpam-4478	123	25	a	a	DET
ejpam-4478	123	26	dominating	dominating	NOUN
ejpam-4478	123	27	set	set	NOUN
ejpam-4478	123	28	of	of	ADP
ejpam-4478	123	29	the	the	DET
ejpam-4478	123	30	complement	complement	NOUN
ejpam-4478	123	31	g	g	PROPN
ejpam-4478	123	32	of	of	ADP
ejpam-4478	123	33	g.	g.	PROPN
ejpam-4478	123	34	the	the	DET
ejpam-4478	123	35	minimum	minimum	ADJ
ejpam-4478	123	36	cardinality	cardinality	NOUN
ejpam-4478	123	37	taken	take	VERB
ejpam-4478	123	38	over	over	ADP
ejpam-4478	123	39	all	all	DET
ejpam-4478	123	40	global	global	ADJ
ejpam-4478	123	41	dominating	dominating	NOUN
ejpam-4478	123	42	sets	set	NOUN
ejpam-4478	123	43	of	of	ADP
ejpam-4478	123	44	g	g	PROPN
ejpam-4478	123	45	is	be	AUX
ejpam-4478	123	46	called	call	VERB
ejpam-4478	123	47	the	the	DET
ejpam-4478	123	48	global	global	ADJ
ejpam-4478	123	49	domination	domination	NOUN
ejpam-4478	123	50	number	number	NOUN
ejpam-4478	123	51	of	of	ADP
ejpam-4478	123	52	g	g	NOUN
ejpam-4478	123	53	and	and	CCONJ
ejpam-4478	123	54	is	be	AUX
ejpam-4478	123	55	denoted	denote	VERB
ejpam-4478	123	56	by	by	ADP
ejpam-4478	123	57	γg(g	γg(g	NOUN
ejpam-4478	123	58	)	)	PUNCT
ejpam-4478	123	59	.	.	PUNCT
ejpam-4478	124	1	definition	definition	NOUN
ejpam-4478	124	2	3.5	3.5	NUM
ejpam-4478	124	3	.	.	PUNCT
ejpam-4478	125	1	[	[	X
ejpam-4478	125	2	1	1	X
ejpam-4478	125	3	]	]	PUNCT
ejpam-4478	125	4	a	a	DET
ejpam-4478	125	5	k	k	NOUN
ejpam-4478	125	6	−	−	PROPN
ejpam-4478	125	7	distance	distance	NOUN
ejpam-4478	125	8	roman	roman	ADJ
ejpam-4478	125	9	dominating	dominating	NOUN
ejpam-4478	125	10	function	function	NOUN
ejpam-4478	125	11	(	(	PUNCT
ejpam-4478	125	12	kdrdf	kdrdf	NOUN
ejpam-4478	125	13	)	)	PUNCT
ejpam-4478	125	14	on	on	ADP
ejpam-4478	125	15	g	g	PROPN
ejpam-4478	125	16	=	=	SYM
ejpam-4478	125	17	(	(	PUNCT
ejpam-4478	125	18	v	v	NOUN
ejpam-4478	125	19	,	,	PUNCT
ejpam-4478	125	20	e	e	NOUN
ejpam-4478	125	21	)	)	PUNCT
ejpam-4478	125	22	is	be	AUX
ejpam-4478	125	23	a	a	DET
ejpam-4478	125	24	function	function	NOUN
ejpam-4478	125	25	f	f	NOUN
ejpam-4478	125	26	:	:	PUNCT
ejpam-4478	125	27	v	v	X
ejpam-4478	125	28	→	→	SYM
ejpam-4478	125	29	{	{	PUNCT
ejpam-4478	125	30	0	0	NUM
ejpam-4478	125	31	,	,	PUNCT
ejpam-4478	125	32	1	1	NUM
ejpam-4478	125	33	,	,	PUNCT
ejpam-4478	125	34	2	2	NUM
ejpam-4478	125	35	}	}	PUNCT
ejpam-4478	125	36	such	such	ADJ
ejpam-4478	125	37	that	that	PRON
ejpam-4478	125	38	for	for	ADP
ejpam-4478	125	39	every	every	DET
ejpam-4478	125	40	vertex	vertex	NOUN
ejpam-4478	125	41	v	v	NOUN
ejpam-4478	125	42	with	with	ADP
ejpam-4478	125	43	f(v	f(v	NOUN
ejpam-4478	125	44	)	)	PUNCT
ejpam-4478	125	45	=	=	SYM
ejpam-4478	125	46	0	0	NUM
ejpam-4478	125	47	,	,	PUNCT
ejpam-4478	125	48	there	there	PRON
ejpam-4478	125	49	is	be	VERB
ejpam-4478	125	50	a	a	DET
ejpam-4478	125	51	vertex	vertex	NOUN
ejpam-4478	125	52	u	u	NOUN
ejpam-4478	125	53	with	with	ADP
ejpam-4478	125	54	f(u	f(u	PROPN
ejpam-4478	125	55	)	)	PUNCT
ejpam-4478	125	56	=	=	SYM
ejpam-4478	125	57	2	2	NUM
ejpam-4478	125	58	with	with	ADP
ejpam-4478	125	59	distance	distance	NOUN
ejpam-4478	125	60	of	of	ADP
ejpam-4478	125	61	at	at	ADP
ejpam-4478	125	62	most	most	ADV
ejpam-4478	125	63	k	k	NOUN
ejpam-4478	125	64	from	from	ADP
ejpam-4478	125	65	each	each	DET
ejpam-4478	125	66	other	other	ADJ
ejpam-4478	125	67	.	.	PUNCT
ejpam-4478	126	1	the	the	DET
ejpam-4478	126	2	weight	weight	NOUN
ejpam-4478	126	3	of	of	ADP
ejpam-4478	126	4	the	the	DET
ejpam-4478	126	5	distance	distance	NOUN
ejpam-4478	126	6	roman	roman	ADJ
ejpam-4478	126	7	dominating	dominating	NOUN
ejpam-4478	126	8	function	function	NOUN
ejpam-4478	126	9	f	f	PROPN
ejpam-4478	126	10	is	be	AUX
ejpam-4478	126	11	the	the	DET
ejpam-4478	126	12	value	value	NOUN
ejpam-4478	126	13	w(f	w(f	NOUN
ejpam-4478	126	14	)	)	PUNCT
ejpam-4478	127	1	=	=	SYM
ejpam-4478	127	2	∑	∑	PUNCT
ejpam-4478	127	3	x∈v	x∈v	PROPN
ejpam-4478	127	4	f(x	f(x	PROPN
ejpam-4478	127	5	)	)	PUNCT
ejpam-4478	127	6	.	.	PUNCT
ejpam-4478	128	1	the	the	DET
ejpam-4478	128	2	minimum	minimum	ADJ
ejpam-4478	128	3	weight	weight	NOUN
ejpam-4478	128	4	of	of	ADP
ejpam-4478	128	5	a	a	DET
ejpam-4478	128	6	k	k	NOUN
ejpam-4478	128	7	−	−	PROPN
ejpam-4478	128	8	distance	distance	NOUN
ejpam-4478	128	9	roman	roman	ADJ
ejpam-4478	128	10	dominating	dominating	NOUN
ejpam-4478	128	11	function	function	NOUN
ejpam-4478	128	12	on	on	ADP
ejpam-4478	128	13	the	the	DET
ejpam-4478	128	14	graph	graph	NOUN
ejpam-4478	128	15	g	g	NOUN
ejpam-4478	128	16	is	be	AUX
ejpam-4478	128	17	called	call	VERB
ejpam-4478	128	18	the	the	DET
ejpam-4478	128	19	k	k	PROPN
ejpam-4478	128	20	−	−	PROPN
ejpam-4478	128	21	distance	distance	NOUN
ejpam-4478	128	22	roman	roman	ADJ
ejpam-4478	128	23	domination	domination	NOUN
ejpam-4478	128	24	number	number	NOUN
ejpam-4478	128	25	of	of	ADP
ejpam-4478	128	26	g	g	NOUN
ejpam-4478	128	27	and	and	CCONJ
ejpam-4478	128	28	is	be	AUX
ejpam-4478	128	29	denoted	denote	VERB
ejpam-4478	128	30	as	as	ADP
ejpam-4478	128	31	γkr(g	γkr(g	PROPN
ejpam-4478	128	32	)	)	PUNCT
ejpam-4478	128	33	.	.	PUNCT
ejpam-4478	129	1	definition	definition	NOUN
ejpam-4478	129	2	3.6	3.6	NUM
ejpam-4478	129	3	.	.	PUNCT
ejpam-4478	130	1	[	[	X
ejpam-4478	130	2	12	12	NUM
ejpam-4478	130	3	]	]	PUNCT
ejpam-4478	130	4	a	a	DET
ejpam-4478	130	5	global	global	ADJ
ejpam-4478	130	6	roman	roman	ADJ
ejpam-4478	130	7	dominating	dominating	NOUN
ejpam-4478	130	8	function	function	NOUN
ejpam-4478	130	9	(	(	PUNCT
ejpam-4478	130	10	grdf	grdf	PROPN
ejpam-4478	130	11	)	)	PUNCT
ejpam-4478	130	12	on	on	ADP
ejpam-4478	130	13	the	the	DET
ejpam-4478	130	14	graph	graph	NOUN
ejpam-4478	130	15	g	g	PROPN
ejpam-4478	130	16	=	=	SYM
ejpam-4478	130	17	(	(	PUNCT
ejpam-4478	130	18	v	v	NOUN
ejpam-4478	130	19	,	,	PUNCT
ejpam-4478	130	20	e	e	NOUN
ejpam-4478	130	21	)	)	PUNCT
ejpam-4478	130	22	is	be	AUX
ejpam-4478	130	23	a	a	DET
ejpam-4478	130	24	function	function	NOUN
ejpam-4478	130	25	f	f	NOUN
ejpam-4478	130	26	:	:	PUNCT
ejpam-4478	130	27	v	v	X
ejpam-4478	130	28	→	→	SYM
ejpam-4478	130	29	{	{	PUNCT
ejpam-4478	130	30	0	0	NUM
ejpam-4478	130	31	,	,	PUNCT
ejpam-4478	130	32	1	1	NUM
ejpam-4478	130	33	,	,	PUNCT
ejpam-4478	130	34	2	2	NUM
ejpam-4478	130	35	}	}	PUNCT
ejpam-4478	130	36	such	such	ADJ
ejpam-4478	130	37	that	that	SCONJ
ejpam-4478	130	38	f	f	PROPN
ejpam-4478	130	39	is	be	AUX
ejpam-4478	130	40	an	an	DET
ejpam-4478	130	41	rdf	rdf	NOUN
ejpam-4478	130	42	for	for	ADP
ejpam-4478	130	43	both	both	CCONJ
ejpam-4478	130	44	g	g	NOUN
ejpam-4478	130	45	and	and	CCONJ
ejpam-4478	130	46	its	its	PRON
ejpam-4478	130	47	complement	complement	NOUN
ejpam-4478	130	48	g.	g.	NOUN
ejpam-4478	130	49	the	the	DET
ejpam-4478	130	50	weight	weight	NOUN
ejpam-4478	130	51	of	of	ADP
ejpam-4478	130	52	the	the	DET
ejpam-4478	130	53	global	global	ADJ
ejpam-4478	130	54	roman	roman	ADJ
ejpam-4478	130	55	dominating	dominating	NOUN
ejpam-4478	130	56	function	function	NOUN
ejpam-4478	130	57	f	f	PROPN
ejpam-4478	130	58	is	be	AUX
ejpam-4478	130	59	the	the	DET
ejpam-4478	130	60	value	value	NOUN
ejpam-4478	130	61	w(f	w(f	NOUN
ejpam-4478	130	62	)	)	PUNCT
ejpam-4478	131	1	=	=	SYM
ejpam-4478	131	2	∑	∑	PUNCT
ejpam-4478	131	3	x∈v	x∈v	PROPN
ejpam-4478	131	4	f(x	f(x	PROPN
ejpam-4478	131	5	)	)	PUNCT
ejpam-4478	131	6	.	.	PUNCT
ejpam-4478	132	1	the	the	DET
ejpam-4478	132	2	minimum	minimum	ADJ
ejpam-4478	132	3	weight	weight	NOUN
ejpam-4478	132	4	of	of	ADP
ejpam-4478	132	5	the	the	DET
ejpam-4478	132	6	global	global	ADJ
ejpam-4478	132	7	roman	roman	ADJ
ejpam-4478	132	8	dominating	dominating	NOUN
ejpam-4478	132	9	function	function	NOUN
ejpam-4478	132	10	on	on	ADP
ejpam-4478	132	11	the	the	DET
ejpam-4478	132	12	graph	graph	NOUN
ejpam-4478	132	13	g	g	NOUN
ejpam-4478	132	14	is	be	AUX
ejpam-4478	132	15	called	call	VERB
ejpam-4478	132	16	the	the	DET
ejpam-4478	132	17	global	global	ADJ
ejpam-4478	132	18	roman	roman	ADJ
ejpam-4478	132	19	domination	domination	NOUN
ejpam-4478	132	20	number	number	NOUN
ejpam-4478	132	21	of	of	ADP
ejpam-4478	132	22	g	g	NOUN
ejpam-4478	132	23	and	and	CCONJ
ejpam-4478	132	24	is	be	AUX
ejpam-4478	132	25	denoted	denote	VERB
ejpam-4478	132	26	as	as	ADP
ejpam-4478	132	27	γgr(g	γgr(g	PROPN
ejpam-4478	132	28	)	)	PUNCT
ejpam-4478	132	29	.	.	PUNCT
ejpam-4478	133	1	definition	definition	NOUN
ejpam-4478	133	2	3.7	3.7	NUM
ejpam-4478	133	3	.	.	PUNCT
ejpam-4478	134	1	let	let	VERB
ejpam-4478	134	2	k	k	PROPN
ejpam-4478	134	3	∈	∈	PROPN
ejpam-4478	134	4	z+	z+	PUNCT
ejpam-4478	134	5	.	.	PUNCT
ejpam-4478	135	1	a	a	DET
ejpam-4478	135	2	k	k	NOUN
ejpam-4478	135	3	−	−	PROPN
ejpam-4478	135	4	distance	distance	NOUN
ejpam-4478	135	5	roman	roman	ADJ
ejpam-4478	135	6	dominating	dominating	NOUN
ejpam-4478	135	7	function	function	NOUN
ejpam-4478	135	8	(	(	PUNCT
ejpam-4478	135	9	kdrdf	kdrdf	NOUN
ejpam-4478	135	10	)	)	PUNCT
ejpam-4478	135	11	on	on	ADP
ejpam-4478	135	12	g	g	PROPN
ejpam-4478	135	13	=	=	SYM
ejpam-4478	135	14	(	(	PUNCT
ejpam-4478	135	15	v	v	NOUN
ejpam-4478	135	16	,	,	PUNCT
ejpam-4478	135	17	e	e	NOUN
ejpam-4478	135	18	)	)	PUNCT
ejpam-4478	135	19	is	be	AUX
ejpam-4478	135	20	a	a	DET
ejpam-4478	135	21	function	function	NOUN
ejpam-4478	135	22	f	f	NOUN
ejpam-4478	135	23	:	:	PUNCT
ejpam-4478	135	24	v	v	X
ejpam-4478	135	25	→	→	SYM
ejpam-4478	135	26	{	{	PUNCT
ejpam-4478	135	27	0	0	NUM
ejpam-4478	135	28	,	,	PUNCT
ejpam-4478	135	29	1	1	NUM
ejpam-4478	135	30	,	,	PUNCT
ejpam-4478	135	31	2	2	NUM
ejpam-4478	135	32	}	}	PUNCT
ejpam-4478	135	33	such	such	ADJ
ejpam-4478	135	34	that	that	PRON
ejpam-4478	135	35	for	for	ADP
ejpam-4478	135	36	every	every	DET
ejpam-4478	135	37	vertex	vertex	NOUN
ejpam-4478	135	38	v	v	NOUN
ejpam-4478	135	39	with	with	ADP
ejpam-4478	135	40	f(v	f(v	NOUN
ejpam-4478	135	41	)	)	PUNCT
ejpam-4478	135	42	=	=	SYM
ejpam-4478	135	43	0	0	NUM
ejpam-4478	135	44	,	,	PUNCT
ejpam-4478	135	45	there	there	PRON
ejpam-4478	135	46	is	be	VERB
ejpam-4478	135	47	a	a	DET
ejpam-4478	135	48	vertex	vertex	NOUN
ejpam-4478	135	49	u	u	NOUN
ejpam-4478	135	50	with	with	ADP
ejpam-4478	135	51	f(u	f(u	PROPN
ejpam-4478	135	52	)	)	PUNCT
ejpam-4478	135	53	=	=	SYM
ejpam-4478	135	54	2	2	NUM
ejpam-4478	135	55	such	such	ADJ
ejpam-4478	135	56	that	that	SCONJ
ejpam-4478	135	57	d(u	d(u	PROPN
ejpam-4478	135	58	,	,	PUNCT
ejpam-4478	135	59	v	v	NOUN
ejpam-4478	135	60	)	)	PUNCT
ejpam-4478	135	61	≤	≤	NOUN
ejpam-4478	135	62	k.	k.	VERB
ejpam-4478	136	1	the	the	DET
ejpam-4478	136	2	function	function	NOUN
ejpam-4478	136	3	f	f	PROPN
ejpam-4478	136	4	is	be	AUX
ejpam-4478	136	5	a	a	DET
ejpam-4478	136	6	global	global	ADJ
ejpam-4478	136	7	k	k	NOUN
ejpam-4478	136	8	−	−	PROPN
ejpam-4478	136	9	distance	distance	NOUN
ejpam-4478	136	10	roman	roman	ADJ
ejpam-4478	136	11	dominating	dominating	NOUN
ejpam-4478	136	12	function	function	NOUN
ejpam-4478	136	13	(	(	PUNCT
ejpam-4478	136	14	gkdrdf	gkdrdf	PROPN
ejpam-4478	136	15	)	)	PUNCT
ejpam-4478	136	16	on	on	ADP
ejpam-4478	136	17	g	g	PROPN
ejpam-4478	137	1	if	if	SCONJ
ejpam-4478	138	1	and	and	CCONJ
ejpam-4478	138	2	only	only	ADV
ejpam-4478	138	3	if	if	SCONJ
ejpam-4478	138	4	f	f	PROPN
ejpam-4478	138	5	is	be	AUX
ejpam-4478	138	6	a	a	DET
ejpam-4478	138	7	k	k	NOUN
ejpam-4478	138	8	−	−	PROPN
ejpam-4478	138	9	distance	distance	NOUN
ejpam-4478	138	10	roman	roman	ADJ
ejpam-4478	138	11	dominating	dominating	NOUN
ejpam-4478	138	12	function	function	NOUN
ejpam-4478	138	13	(	(	PUNCT
ejpam-4478	138	14	kdrdf	kdrdf	NOUN
ejpam-4478	138	15	)	)	PUNCT
ejpam-4478	138	16	on	on	ADP
ejpam-4478	138	17	g	g	PROPN
ejpam-4478	138	18	and	and	CCONJ
ejpam-4478	138	19	on	on	ADP
ejpam-4478	138	20	its	its	PRON
ejpam-4478	138	21	complement	complement	NOUN
ejpam-4478	138	22	g.	g.	NOUN
ejpam-4478	138	23	definition	definition	NOUN
ejpam-4478	138	24	3.8	3.8	NUM
ejpam-4478	138	25	.	.	PUNCT
ejpam-4478	139	1	let	let	VERB
ejpam-4478	139	2	k	k	PROPN
ejpam-4478	139	3	∈	∈	PROPN
ejpam-4478	139	4	z+	z+	PUNCT
ejpam-4478	139	5	.	.	PUNCT
ejpam-4478	140	1	the	the	DET
ejpam-4478	140	2	weight	weight	NOUN
ejpam-4478	140	3	of	of	ADP
ejpam-4478	140	4	the	the	DET
ejpam-4478	140	5	global	global	ADJ
ejpam-4478	140	6	k	k	PROPN
ejpam-4478	140	7	−	−	PROPN
ejpam-4478	140	8	distance	distance	NOUN
ejpam-4478	140	9	roman	roman	ADJ
ejpam-4478	140	10	dominating	dominating	NOUN
ejpam-4478	140	11	function	function	NOUN
ejpam-4478	140	12	(	(	PUNCT
ejpam-4478	140	13	gkdrdf	gkdrdf	PROPN
ejpam-4478	140	14	)	)	PUNCT
ejpam-4478	140	15	f	f	PROPN
ejpam-4478	140	16	is	be	AUX
ejpam-4478	140	17	the	the	DET
ejpam-4478	140	18	value	value	NOUN
ejpam-4478	140	19	w(f	w(f	NOUN
ejpam-4478	140	20	)	)	PUNCT
ejpam-4478	141	1	=	=	SYM
ejpam-4478	141	2	∑	∑	PUNCT
ejpam-4478	141	3	x∈v	x∈v	PROPN
ejpam-4478	141	4	f(x	f(x	PROPN
ejpam-4478	141	5	)	)	PUNCT
ejpam-4478	141	6	.	.	PUNCT
ejpam-4478	142	1	the	the	DET
ejpam-4478	142	2	minimum	minimum	ADJ
ejpam-4478	142	3	weight	weight	NOUN
ejpam-4478	142	4	of	of	ADP
ejpam-4478	142	5	the	the	DET
ejpam-4478	142	6	global	global	ADJ
ejpam-4478	142	7	k	k	PROPN
ejpam-4478	142	8	−	−	PROPN
ejpam-4478	142	9	distance	distance	NOUN
ejpam-4478	142	10	roman	roman	ADJ
ejpam-4478	142	11	dominating	dominating	NOUN
ejpam-4478	142	12	function	function	NOUN
ejpam-4478	142	13	(	(	PUNCT
ejpam-4478	142	14	gkdrdf	gkdrdf	PROPN
ejpam-4478	142	15	)	)	PUNCT
ejpam-4478	142	16	on	on	ADP
ejpam-4478	142	17	the	the	DET
ejpam-4478	142	18	graph	graph	NOUN
ejpam-4478	142	19	g	g	NOUN
ejpam-4478	142	20	is	be	AUX
ejpam-4478	142	21	called	call	VERB
ejpam-4478	142	22	the	the	DET
ejpam-4478	142	23	global	global	ADJ
ejpam-4478	142	24	k	k	PROPN
ejpam-4478	142	25	−	−	PROPN
ejpam-4478	142	26	distance	distance	NOUN
ejpam-4478	142	27	roman	roman	ADJ
ejpam-4478	142	28	domination	domination	NOUN
ejpam-4478	142	29	number	number	NOUN
ejpam-4478	142	30	of	of	ADP
ejpam-4478	142	31	g	g	NOUN
ejpam-4478	142	32	and	and	CCONJ
ejpam-4478	142	33	is	be	AUX
ejpam-4478	142	34	denoted	denote	VERB
ejpam-4478	142	35	as	as	ADP
ejpam-4478	142	36	γkgr(g	γkgr(g	NOUN
ejpam-4478	142	37	)	)	PUNCT
ejpam-4478	142	38	.	.	PUNCT
ejpam-4478	143	1	a	a	DET
ejpam-4478	143	2	γkgr(g	γkgr(g	NOUN
ejpam-4478	143	3	)	)	PUNCT
ejpam-4478	144	1	−	−	PROPN
ejpam-4478	144	2	function	function	NOUN
ejpam-4478	144	3	is	be	AUX
ejpam-4478	144	4	a	a	DET
ejpam-4478	144	5	gkdrdf	gkdrdf	NOUN
ejpam-4478	144	6	with	with	ADP
ejpam-4478	144	7	weight	weight	NOUN
ejpam-4478	144	8	γkgr(g	γkgr(g	PROPN
ejpam-4478	144	9	)	)	PUNCT
ejpam-4478	144	10	.	.	PUNCT
ejpam-4478	145	1	remark	remark	VERB
ejpam-4478	145	2	3.9	3.9	NUM
ejpam-4478	145	3	.	.	PUNCT
ejpam-4478	146	1	[	[	X
ejpam-4478	146	2	12	12	NUM
ejpam-4478	146	3	]	]	PUNCT
ejpam-4478	146	4	for	for	ADP
ejpam-4478	146	5	any	any	DET
ejpam-4478	146	6	n	n	DET
ejpam-4478	146	7	−	−	PROPN
ejpam-4478	146	8	vertex	vertex	NOUN
ejpam-4478	146	9	graph	graph	NOUN
ejpam-4478	146	10	g	g	NOUN
ejpam-4478	146	11	,	,	PUNCT
ejpam-4478	146	12	2	2	NUM
ejpam-4478	146	13	≤	≤	NUM
ejpam-4478	146	14	γgr(g	γgr(g	PROPN
ejpam-4478	146	15	)	)	PUNCT
ejpam-4478	146	16	≤	≤	PROPN
ejpam-4478	146	17	n.	n.	NOUN
ejpam-4478	146	18	theorem	theorem	VERB
ejpam-4478	146	19	3.10	3.10	NUM
ejpam-4478	146	20	.	.	PUNCT
ejpam-4478	147	1	[	[	X
ejpam-4478	147	2	10	10	NUM
ejpam-4478	147	3	]	]	PUNCT
ejpam-4478	147	4	let	let	VERB
ejpam-4478	147	5	g	g	PRON
ejpam-4478	147	6	be	be	AUX
ejpam-4478	147	7	a	a	DET
ejpam-4478	147	8	simple	simple	ADJ
ejpam-4478	147	9	graph	graph	NOUN
ejpam-4478	147	10	of	of	ADP
ejpam-4478	147	11	order	order	NOUN
ejpam-4478	147	12	n.	n.	NOUN
ejpam-4478	147	13	then	then	ADV
ejpam-4478	147	14	γ(g	γ(g	PROPN
ejpam-4478	147	15	)	)	PUNCT
ejpam-4478	148	1	=	=	SYM
ejpam-4478	149	1	n	n	NOUN
ejpam-4478	149	2	if	if	SCONJ
ejpam-4478	149	3	and	and	CCONJ
ejpam-4478	149	4	only	only	ADV
ejpam-4478	149	5	if	if	SCONJ
ejpam-4478	149	6	g	g	PROPN
ejpam-4478	149	7	≡	≡	PROPN
ejpam-4478	149	8	kn	kn	PROPN
ejpam-4478	149	9	.	.	PUNCT
ejpam-4478	149	10	proposition	proposition	PROPN
ejpam-4478	149	11	3.11	3.11	NUM
ejpam-4478	149	12	.	.	PUNCT
ejpam-4478	150	1	[	[	X
ejpam-4478	150	2	15	15	NUM
ejpam-4478	150	3	]	]	PUNCT
ejpam-4478	150	4	for	for	ADP
ejpam-4478	150	5	any	any	DET
ejpam-4478	150	6	graph	graph	NOUN
ejpam-4478	150	7	g	g	NOUN
ejpam-4478	150	8	of	of	ADP
ejpam-4478	150	9	order	order	NOUN
ejpam-4478	150	10	n	n	CCONJ
ejpam-4478	150	11	,	,	PUNCT
ejpam-4478	150	12	γ(g	γ(g	PROPN
ejpam-4478	150	13	)	)	PUNCT
ejpam-4478	150	14	=	=	SYM
ejpam-4478	150	15	γk(g	γk(g	X
ejpam-4478	150	16	)	)	PUNCT
ejpam-4478	150	17	if	if	SCONJ
ejpam-4478	150	18	and	and	CCONJ
ejpam-4478	150	19	only	only	ADV
ejpam-4478	150	20	if	if	SCONJ
ejpam-4478	150	21	every	every	DET
ejpam-4478	150	22	vertex	vertex	NOUN
ejpam-4478	150	23	in	in	ADP
ejpam-4478	150	24	g	g	PROPN
ejpam-4478	150	25	has	have	VERB
ejpam-4478	150	26	degree	degree	NOUN
ejpam-4478	150	27	0	0	NUM
ejpam-4478	150	28	.	.	PUNCT
ejpam-4478	151	1	(	(	PUNCT
ejpam-4478	151	2	such	such	DET
ejpam-4478	151	3	a	a	DET
ejpam-4478	151	4	graph	graph	NOUN
ejpam-4478	151	5	is	be	AUX
ejpam-4478	151	6	denoted	denote	VERB
ejpam-4478	151	7	as	as	ADP
ejpam-4478	151	8	g	g	PROPN
ejpam-4478	151	9	=	=	PROPN
ejpam-4478	151	10	kn	kn	PROPN
ejpam-4478	151	11	,	,	PUNCT
ejpam-4478	151	12	the	the	DET
ejpam-4478	151	13	complement	complement	NOUN
ejpam-4478	151	14	of	of	ADP
ejpam-4478	151	15	the	the	DET
ejpam-4478	151	16	complete	complete	ADJ
ejpam-4478	151	17	graph	graph	NOUN
ejpam-4478	151	18	of	of	ADP
ejpam-4478	151	19	order	order	NOUN
ejpam-4478	151	20	n	n	CCONJ
ejpam-4478	151	21	)	)	PUNCT
ejpam-4478	151	22	g.	g.	PROPN
ejpam-4478	151	23	entero	entero	PROPN
ejpam-4478	151	24	,	,	PUNCT
ejpam-4478	151	25	s.	s.	PROPN
ejpam-4478	151	26	espinola	espinola	PROPN
ejpam-4478	151	27	/	/	SYM
ejpam-4478	151	28	eur	eur	PROPN
ejpam-4478	151	29	.	.	PUNCT
ejpam-4478	152	1	j.	j.	PROPN
ejpam-4478	152	2	pure	pure	PROPN
ejpam-4478	152	3	appl	appl	PROPN
ejpam-4478	152	4	.	.	PROPN
ejpam-4478	152	5	math	math	PROPN
ejpam-4478	152	6	,	,	PUNCT
ejpam-4478	152	7	16	16	NUM
ejpam-4478	152	8	(	(	PUNCT
ejpam-4478	152	9	1	1	NUM
ejpam-4478	152	10	)	)	PUNCT
ejpam-4478	152	11	(	(	PUNCT
ejpam-4478	152	12	2023	2023	NUM
ejpam-4478	152	13	)	)	PUNCT
ejpam-4478	152	14	,	,	PUNCT
ejpam-4478	152	15	44	44	NUM
ejpam-4478	152	16	-	-	SYM
ejpam-4478	152	17	61	61	NUM
ejpam-4478	152	18	49	49	NUM
ejpam-4478	152	19	remark	remark	NOUN
ejpam-4478	152	20	3.12	3.12	NUM
ejpam-4478	152	21	.	.	PUNCT
ejpam-4478	153	1	[	[	X
ejpam-4478	153	2	2	2	X
ejpam-4478	153	3	]	]	X
ejpam-4478	153	4	let	let	VERB
ejpam-4478	153	5	k	k	PROPN
ejpam-4478	153	6	≥	≥	NUM
ejpam-4478	153	7	1	1	NUM
ejpam-4478	153	8	be	be	AUX
ejpam-4478	153	9	an	an	DET
ejpam-4478	153	10	integer	integer	NOUN
ejpam-4478	153	11	.	.	PUNCT
ejpam-4478	154	1	for	for	ADP
ejpam-4478	154	2	n	n	CCONJ
ejpam-4478	154	3	−	−	PROPN
ejpam-4478	154	4	vertex	vertex	NOUN
ejpam-4478	154	5	graphs	graph	NOUN
ejpam-4478	154	6	,	,	PUNCT
ejpam-4478	154	7	always	always	ADV
ejpam-4478	154	8	γkr(g	γkr(g	PROPN
ejpam-4478	154	9	)	)	PUNCT
ejpam-4478	154	10	≤	≤	NOUN
ejpam-4478	154	11	n	n	CCONJ
ejpam-4478	154	12	,	,	PUNCT
ejpam-4478	154	13	with	with	ADP
ejpam-4478	154	14	equality	equality	NOUN
ejpam-4478	154	15	when	when	SCONJ
ejpam-4478	154	16	g	g	PROPN
ejpam-4478	154	17	∼=	∼=	PROPN
ejpam-4478	154	18	kn	kn	PROPN
ejpam-4478	154	19	.	.	PUNCT
ejpam-4478	154	20	proposition	proposition	PROPN
ejpam-4478	154	21	3.13	3.13	NUM
ejpam-4478	154	22	.	.	PUNCT
ejpam-4478	155	1	[	[	X
ejpam-4478	155	2	13	13	NUM
ejpam-4478	155	3	]	]	PUNCT
ejpam-4478	155	4	(	(	PUNCT
ejpam-4478	155	5	i	i	NOUN
ejpam-4478	155	6	)	)	PUNCT
ejpam-4478	155	7	for	for	ADP
ejpam-4478	155	8	a	a	DET
ejpam-4478	155	9	graph	graph	NOUN
ejpam-4478	155	10	g	g	NOUN
ejpam-4478	155	11	with	with	ADP
ejpam-4478	155	12	p	p	NOUN
ejpam-4478	155	13	vertices	vertex	NOUN
ejpam-4478	155	14	,	,	PUNCT
ejpam-4478	155	15	γg(g	γg(g	NOUN
ejpam-4478	155	16	)	)	PUNCT
ejpam-4478	155	17	=	=	NOUN
ejpam-4478	156	1	p	p	NOUN
ejpam-4478	156	2	if	if	SCONJ
ejpam-4478	156	3	and	and	CCONJ
ejpam-4478	156	4	only	only	ADV
ejpam-4478	156	5	if	if	SCONJ
ejpam-4478	156	6	g	g	PROPN
ejpam-4478	156	7	=	=	VERB
ejpam-4478	156	8	kp	kp	PROPN
ejpam-4478	156	9	or	or	CCONJ
ejpam-4478	156	10	kp	kp	PROPN
ejpam-4478	156	11	.	.	PUNCT
ejpam-4478	156	12	(	(	PUNCT
ejpam-4478	156	13	ii	ii	NOUN
ejpam-4478	156	14	)	)	PUNCT
ejpam-4478	156	15	γg(km	γg(km	PROPN
ejpam-4478	156	16	,	,	PUNCT
ejpam-4478	156	17	n	n	CCONJ
ejpam-4478	156	18	)	)	PUNCT
ejpam-4478	156	19	=	=	SYM
ejpam-4478	156	20	2	2	NUM
ejpam-4478	156	21	for	for	ADP
ejpam-4478	156	22	all	all	DET
ejpam-4478	156	23	m	m	PROPN
ejpam-4478	156	24	,	,	PUNCT
ejpam-4478	156	25	n	n	PRON
ejpam-4478	156	26	≥	≥	NOUN
ejpam-4478	156	27	1	1	NUM
ejpam-4478	156	28	.	.	PUNCT
ejpam-4478	156	29	(	(	PUNCT
ejpam-4478	156	30	iii	iii	NOUN
ejpam-4478	156	31	)	)	PUNCT
ejpam-4478	156	32	γg(c4	γg(c4	NOUN
ejpam-4478	156	33	)	)	PUNCT
ejpam-4478	156	34	=	=	SYM
ejpam-4478	156	35	2	2	NUM
ejpam-4478	156	36	,	,	PUNCT
ejpam-4478	156	37	γg(c5	γg(c5	NOUN
ejpam-4478	156	38	)	)	PUNCT
ejpam-4478	156	39	=	=	SYM
ejpam-4478	156	40	3	3	NUM
ejpam-4478	156	41	and	and	CCONJ
ejpam-4478	156	42	γg(cn	γg(cn	NOUN
ejpam-4478	156	43	)	)	PUNCT
ejpam-4478	156	44	=	=	PRON
ejpam-4478	156	45	{	{	PUNCT
ejpam-4478	156	46	n	n	NOUN
ejpam-4478	156	47	3	3	NUM
ejpam-4478	156	48	}	}	PUNCT
ejpam-4478	156	49	,	,	PUNCT
ejpam-4478	156	50	for	for	ADP
ejpam-4478	156	51	n	n	PRON
ejpam-4478	156	52	≥	≥	NUM
ejpam-4478	156	53	6	6	NUM
ejpam-4478	156	54	.	.	PUNCT
ejpam-4478	156	55	(	(	PUNCT
ejpam-4478	156	56	vi	vi	NOUN
ejpam-4478	156	57	)	)	PUNCT
ejpam-4478	156	58	γg(pn	γg(pn	NOUN
ejpam-4478	156	59	)	)	PUNCT
ejpam-4478	156	60	=	=	SYM
ejpam-4478	156	61	2	2	NUM
ejpam-4478	156	62	for	for	ADP
ejpam-4478	156	63	n	n	NOUN
ejpam-4478	156	64	=	=	SYM
ejpam-4478	156	65	2	2	NUM
ejpam-4478	156	66	,	,	PUNCT
ejpam-4478	156	67	3	3	NUM
ejpam-4478	156	68	and	and	CCONJ
ejpam-4478	156	69	γg(pn	γg(pn	NOUN
ejpam-4478	156	70	)	)	PUNCT
ejpam-4478	156	71	=	=	PRON
ejpam-4478	156	72	{	{	PUNCT
ejpam-4478	157	1	n	n	PROPN
ejpam-4478	157	2	3	3	NUM
ejpam-4478	157	3	}	}	PUNCT
ejpam-4478	157	4	for	for	ADP
ejpam-4478	157	5	n	n	X
ejpam-4478	157	6	≥	≥	NUM
ejpam-4478	157	7	4	4	NUM
ejpam-4478	157	8	.	.	PUNCT
ejpam-4478	157	9	proposition	proposition	NOUN
ejpam-4478	157	10	3.14	3.14	NUM
ejpam-4478	157	11	.	.	PUNCT
ejpam-4478	158	1	[	[	X
ejpam-4478	158	2	12	12	NUM
ejpam-4478	158	3	]	]	PUNCT
ejpam-4478	158	4	let	let	VERB
ejpam-4478	158	5	g	g	NOUN
ejpam-4478	158	6	be	be	AUX
ejpam-4478	158	7	any	any	DET
ejpam-4478	158	8	graph	graph	NOUN
ejpam-4478	158	9	.	.	PUNCT
ejpam-4478	159	1	then	then	ADV
ejpam-4478	159	2	γg(g	γg(g	NUM
ejpam-4478	159	3	)	)	PUNCT
ejpam-4478	159	4	=	=	SYM
ejpam-4478	159	5	γgr(g	γgr(g	PROPN
ejpam-4478	159	6	)	)	PUNCT
ejpam-4478	160	1	if	if	SCONJ
ejpam-4478	160	2	and	and	CCONJ
ejpam-4478	160	3	only	only	ADV
ejpam-4478	160	4	if	if	SCONJ
ejpam-4478	160	5	g	g	PROPN
ejpam-4478	160	6	=	=	PROPN
ejpam-4478	160	7	kn	kn	PROPN
ejpam-4478	160	8	.	.	PROPN
ejpam-4478	160	9	4	4	NUM
ejpam-4478	160	10	.	.	NOUN
ejpam-4478	160	11	results	result	NOUN
ejpam-4478	160	12	and	and	CCONJ
ejpam-4478	160	13	discussions	discussion	NOUN
ejpam-4478	160	14	all	all	PRON
ejpam-4478	160	15	throughout	throughout	ADP
ejpam-4478	160	16	this	this	DET
ejpam-4478	160	17	paper	paper	NOUN
ejpam-4478	160	18	,	,	PUNCT
ejpam-4478	160	19	we	we	PRON
ejpam-4478	160	20	will	will	AUX
ejpam-4478	160	21	be	be	AUX
ejpam-4478	160	22	using	use	VERB
ejpam-4478	160	23	either	either	PRON
ejpam-4478	160	24	of	of	ADP
ejpam-4478	160	25	the	the	DET
ejpam-4478	160	26	notations	notation	NOUN
ejpam-4478	160	27	fk	fk	INTJ
ejpam-4478	160	28	or	or	CCONJ
ejpam-4478	160	29	f	f	PROPN
ejpam-4478	160	30	to	to	PART
ejpam-4478	160	31	denote	denote	VERB
ejpam-4478	160	32	a	a	DET
ejpam-4478	160	33	function	function	NOUN
ejpam-4478	160	34	.	.	PUNCT
ejpam-4478	161	1	however	however	ADV
ejpam-4478	161	2	,	,	PUNCT
ejpam-4478	161	3	for	for	ADP
ejpam-4478	161	4	general	general	ADJ
ejpam-4478	161	5	cases	case	NOUN
ejpam-4478	161	6	where	where	SCONJ
ejpam-4478	161	7	the	the	DET
ejpam-4478	161	8	distance	distance	NOUN
ejpam-4478	161	9	k	k	PROPN
ejpam-4478	161	10	∈	∈	PROPN
ejpam-4478	161	11	z+	z+	NUM
ejpam-4478	161	12	is	be	AUX
ejpam-4478	161	13	explicit	explicit	ADJ
ejpam-4478	161	14	,	,	PUNCT
ejpam-4478	161	15	we	we	PRON
ejpam-4478	161	16	will	will	AUX
ejpam-4478	161	17	be	be	AUX
ejpam-4478	161	18	using	use	VERB
ejpam-4478	161	19	the	the	DET
ejpam-4478	161	20	notation	notation	NOUN
ejpam-4478	161	21	f	f	PROPN
ejpam-4478	161	22	to	to	PART
ejpam-4478	161	23	denote	denote	VERB
ejpam-4478	161	24	a	a	DET
ejpam-4478	161	25	function	function	NOUN
ejpam-4478	161	26	.	.	PUNCT
ejpam-4478	162	1	nevertheless	nevertheless	ADV
ejpam-4478	162	2	,	,	PUNCT
ejpam-4478	162	3	for	for	ADP
ejpam-4478	162	4	the	the	DET
ejpam-4478	162	5	cases	case	NOUN
ejpam-4478	162	6	where	where	SCONJ
ejpam-4478	162	7	the	the	DET
ejpam-4478	162	8	distance	distance	NOUN
ejpam-4478	162	9	k	k	PROPN
ejpam-4478	162	10	∈	∈	PROPN
ejpam-4478	162	11	z+	z+	NUM
ejpam-4478	162	12	must	must	AUX
ejpam-4478	162	13	be	be	AUX
ejpam-4478	162	14	specified	specify	VERB
ejpam-4478	162	15	,	,	PUNCT
ejpam-4478	162	16	we	we	PRON
ejpam-4478	162	17	will	will	AUX
ejpam-4478	162	18	be	be	AUX
ejpam-4478	162	19	using	use	VERB
ejpam-4478	162	20	the	the	DET
ejpam-4478	162	21	notations	notation	NOUN
ejpam-4478	162	22	fk	fk	INTJ
ejpam-4478	162	23	to	to	PART
ejpam-4478	162	24	refer	refer	VERB
ejpam-4478	162	25	to	to	ADP
ejpam-4478	162	26	a	a	DET
ejpam-4478	162	27	function	function	NOUN
ejpam-4478	162	28	with	with	ADP
ejpam-4478	162	29	respect	respect	NOUN
ejpam-4478	162	30	to	to	ADP
ejpam-4478	162	31	such	such	ADJ
ejpam-4478	162	32	distances	distance	NOUN
ejpam-4478	162	33	.	.	PUNCT
ejpam-4478	163	1	additionally	additionally	ADV
ejpam-4478	163	2	,	,	PUNCT
ejpam-4478	163	3	for	for	ADP
ejpam-4478	163	4	each	each	DET
ejpam-4478	163	5	k	k	PROPN
ejpam-4478	163	6	∈	∈	PROPN
ejpam-4478	163	7	z+	z+	X
ejpam-4478	163	8	,	,	PUNCT
ejpam-4478	163	9	we	we	PRON
ejpam-4478	163	10	may	may	AUX
ejpam-4478	163	11	have	have	VERB
ejpam-4478	163	12	at	at	ADV
ejpam-4478	163	13	least	least	ADV
ejpam-4478	163	14	one	one	NUM
ejpam-4478	163	15	fk	fk	INTJ
ejpam-4478	163	16	(	(	PUNCT
ejpam-4478	163	17	resp	resp	PROPN
ejpam-4478	163	18	.	.	PUNCT
ejpam-4478	163	19	,	,	PUNCT
ejpam-4478	163	20	f	f	X
ejpam-4478	163	21	)	)	PUNCT
ejpam-4478	163	22	on	on	ADP
ejpam-4478	163	23	just	just	ADV
ejpam-4478	163	24	a	a	DET
ejpam-4478	163	25	single	single	ADJ
ejpam-4478	163	26	graph	graph	NOUN
ejpam-4478	163	27	,	,	PUNCT
ejpam-4478	163	28	that	that	ADV
ejpam-4478	163	29	is	is	ADV
ejpam-4478	163	30	,	,	PUNCT
ejpam-4478	163	31	for	for	ADP
ejpam-4478	163	32	a	a	DET
ejpam-4478	163	33	particular	particular	ADJ
ejpam-4478	163	34	distance	distance	NOUN
ejpam-4478	163	35	,	,	PUNCT
ejpam-4478	163	36	we	we	PRON
ejpam-4478	163	37	can	can	AUX
ejpam-4478	163	38	define	define	VERB
ejpam-4478	163	39	several	several	ADJ
ejpam-4478	163	40	global	global	ADJ
ejpam-4478	163	41	distance	distance	NOUN
ejpam-4478	163	42	roman	roman	ADJ
ejpam-4478	163	43	dominating	dominating	NOUN
ejpam-4478	163	44	functions	function	NOUN
ejpam-4478	163	45	over	over	ADP
ejpam-4478	163	46	a	a	DET
ejpam-4478	163	47	given	give	VERB
ejpam-4478	163	48	graph	graph	NOUN
ejpam-4478	163	49	.	.	PUNCT
ejpam-4478	164	1	4.1	4.1	NUM
ejpam-4478	164	2	.	.	PUNCT
ejpam-4478	165	1	preliminary	preliminary	ADJ
ejpam-4478	165	2	results	result	NOUN
ejpam-4478	165	3	on	on	ADP
ejpam-4478	165	4	global	global	ADJ
ejpam-4478	165	5	distance	distance	NOUN
ejpam-4478	165	6	roman	roman	ADJ
ejpam-4478	165	7	domination	domination	NOUN
ejpam-4478	165	8	remark	remark	NOUN
ejpam-4478	165	9	4.1	4.1	NUM
ejpam-4478	165	10	.	.	PUNCT
ejpam-4478	166	1	let	let	VERB
ejpam-4478	166	2	k	k	PROPN
ejpam-4478	166	3	∈	∈	PROPN
ejpam-4478	166	4	z+	z+	PUNCT
ejpam-4478	166	5	.	.	PUNCT
ejpam-4478	167	1	given	give	VERB
ejpam-4478	167	2	the	the	DET
ejpam-4478	167	3	global	global	ADJ
ejpam-4478	167	4	k	k	PROPN
ejpam-4478	167	5	−	−	PROPN
ejpam-4478	167	6	distance	distance	NOUN
ejpam-4478	167	7	roman	roman	ADJ
ejpam-4478	167	8	dominating	dominating	NOUN
ejpam-4478	167	9	function	function	NOUN
ejpam-4478	167	10	(	(	PUNCT
ejpam-4478	167	11	gkdrdf	gkdrdf	PROPN
ejpam-4478	167	12	)	)	PUNCT
ejpam-4478	167	13	fk	fk	INTJ
ejpam-4478	167	14	:	:	PUNCT
ejpam-4478	167	15	v	v	X
ejpam-4478	167	16	→	→	SYM
ejpam-4478	167	17	{	{	PUNCT
ejpam-4478	167	18	0	0	NUM
ejpam-4478	167	19	,	,	PUNCT
ejpam-4478	167	20	1	1	NUM
ejpam-4478	167	21	,	,	PUNCT
ejpam-4478	167	22	2	2	NUM
ejpam-4478	167	23	}	}	PUNCT
ejpam-4478	167	24	on	on	ADP
ejpam-4478	167	25	graph	graph	NOUN
ejpam-4478	167	26	g	g	PROPN
ejpam-4478	167	27	=	=	SYM
ejpam-4478	167	28	(	(	PUNCT
ejpam-4478	167	29	v	v	NOUN
ejpam-4478	167	30	,	,	PUNCT
ejpam-4478	167	31	e	e	NOUN
ejpam-4478	167	32	)	)	PUNCT
ejpam-4478	167	33	,	,	PUNCT
ejpam-4478	167	34	for	for	ADP
ejpam-4478	167	35	all	all	DET
ejpam-4478	167	36	k	k	PROPN
ejpam-4478	167	37	∈	∈	PROPN
ejpam-4478	167	38	z+	z+	X
ejpam-4478	167	39	,	,	PUNCT
ejpam-4478	167	40	we	we	PRON
ejpam-4478	167	41	have	have	VERB
ejpam-4478	167	42	the	the	DET
ejpam-4478	167	43	following	follow	VERB
ejpam-4478	167	44	facts	fact	NOUN
ejpam-4478	167	45	:	:	PUNCT
ejpam-4478	167	46	(	(	PUNCT
ejpam-4478	167	47	i	i	NOUN
ejpam-4478	167	48	)	)	PUNCT
ejpam-4478	167	49	function	function	NOUN
ejpam-4478	167	50	fk	fk	INTJ
ejpam-4478	167	51	can	can	AUX
ejpam-4478	167	52	be	be	AUX
ejpam-4478	167	53	represented	represent	VERB
ejpam-4478	167	54	by	by	ADP
ejpam-4478	167	55	the	the	DET
ejpam-4478	167	56	ordered	order	VERB
ejpam-4478	167	57	partition	partition	NOUN
ejpam-4478	167	58	(	(	PUNCT
ejpam-4478	167	59	v	v	NOUN
ejpam-4478	167	60	fk	fk	INTJ
ejpam-4478	167	61	0	0	NUM
ejpam-4478	167	62	,	,	PUNCT
ejpam-4478	167	63	v	v	NOUN
ejpam-4478	167	64	fk	fk	INTJ
ejpam-4478	167	65	1	1	NUM
ejpam-4478	167	66	,	,	PUNCT
ejpam-4478	167	67	v	v	X
ejpam-4478	167	68	fk	fk	INTJ
ejpam-4478	167	69	2	2	NUM
ejpam-4478	167	70	)	)	PUNCT
ejpam-4478	167	71	of	of	ADP
ejpam-4478	167	72	v	v	NOUN
ejpam-4478	167	73	induced	induce	VERB
ejpam-4478	167	74	by	by	ADP
ejpam-4478	167	75	fk	fk	INTJ
ejpam-4478	167	76	,	,	PUNCT
ejpam-4478	167	77	where	where	SCONJ
ejpam-4478	167	78	v	v	NOUN
ejpam-4478	167	79	fk	fk	INTJ
ejpam-4478	167	80	i	i	NOUN
ejpam-4478	167	81	=	=	PUNCT
ejpam-4478	167	82	{	{	PUNCT
ejpam-4478	167	83	v	v	NUM
ejpam-4478	167	84	∈	∈	NOUN
ejpam-4478	167	85	v	v	ADP
ejpam-4478	167	86	|fk(v	|fk(v	PROPN
ejpam-4478	167	87	)	)	PUNCT
ejpam-4478	168	1	=	=	SYM
ejpam-4478	169	1	i	i	PRON
ejpam-4478	169	2	and	and	CCONJ
ejpam-4478	169	3	i	i	PRON
ejpam-4478	169	4	=	=	NOUN
ejpam-4478	169	5	0	0	NUM
ejpam-4478	169	6	,	,	PUNCT
ejpam-4478	169	7	1	1	NUM
ejpam-4478	169	8	,	,	PUNCT
ejpam-4478	169	9	2	2	NUM
ejpam-4478	169	10	}	}	PUNCT
ejpam-4478	169	11	;	;	PUNCT
ejpam-4478	169	12	(	(	PUNCT
ejpam-4478	169	13	ii	ii	NOUN
ejpam-4478	169	14	)	)	PUNCT
ejpam-4478	169	15	from	from	ADP
ejpam-4478	169	16	(	(	PUNCT
ejpam-4478	169	17	i	i	NOUN
ejpam-4478	169	18	)	)	PUNCT
ejpam-4478	169	19	,	,	PUNCT
ejpam-4478	169	20	there	there	PRON
ejpam-4478	169	21	is	be	VERB
ejpam-4478	169	22	a	a	DET
ejpam-4478	169	23	one	one	NUM
ejpam-4478	169	24	-	-	PUNCT
ejpam-4478	169	25	to	to	ADP
ejpam-4478	169	26	-	-	PUNCT
ejpam-4478	169	27	one	one	NUM
ejpam-4478	169	28	correspondence	correspondence	NOUN
ejpam-4478	169	29	between	between	ADP
ejpam-4478	169	30	fk	fk	INTJ
ejpam-4478	169	31	:	:	PUNCT
ejpam-4478	169	32	v	v	X
ejpam-4478	169	33	→	→	SYM
ejpam-4478	169	34	{	{	PUNCT
ejpam-4478	169	35	0	0	NUM
ejpam-4478	169	36	,	,	PUNCT
ejpam-4478	169	37	1	1	NUM
ejpam-4478	169	38	,	,	PUNCT
ejpam-4478	169	39	2	2	NUM
ejpam-4478	169	40	}	}	PUNCT
ejpam-4478	169	41	and	and	CCONJ
ejpam-4478	169	42	the	the	DET
ejpam-4478	169	43	ordered	order	VERB
ejpam-4478	169	44	partition	partition	NOUN
ejpam-4478	169	45	(	(	PUNCT
ejpam-4478	169	46	v	v	NOUN
ejpam-4478	169	47	fk	fk	INTJ
ejpam-4478	169	48	0	0	NUM
ejpam-4478	169	49	,	,	PUNCT
ejpam-4478	169	50	v	v	NOUN
ejpam-4478	169	51	fk	fk	INTJ
ejpam-4478	169	52	1	1	NUM
ejpam-4478	169	53	,	,	PUNCT
ejpam-4478	169	54	v	v	X
ejpam-4478	169	55	fk	fk	INTJ
ejpam-4478	169	56	2	2	NUM
ejpam-4478	169	57	)	)	PUNCT
ejpam-4478	169	58	of	of	ADP
ejpam-4478	169	59	v	v	NOUN
ejpam-4478	169	60	induced	induce	VERB
ejpam-4478	169	61	by	by	ADP
ejpam-4478	169	62	fk	fk	INTJ
ejpam-4478	169	63	;	;	PUNCT
ejpam-4478	169	64	and	and	CCONJ
ejpam-4478	169	65	,	,	PUNCT
ejpam-4478	169	66	(	(	PUNCT
ejpam-4478	169	67	iii	iii	NOUN
ejpam-4478	169	68	)	)	PUNCT
ejpam-4478	169	69	from	from	ADP
ejpam-4478	169	70	(	(	PUNCT
ejpam-4478	169	71	ii	ii	NOUN
ejpam-4478	169	72	)	)	PUNCT
ejpam-4478	169	73	,	,	PUNCT
ejpam-4478	169	74	fk	fk	INTJ
ejpam-4478	169	75	can	can	AUX
ejpam-4478	169	76	be	be	AUX
ejpam-4478	169	77	written	write	VERB
ejpam-4478	169	78	as	as	ADP
ejpam-4478	169	79	fk	fk	INTJ
ejpam-4478	169	80	=	=	PUNCT
ejpam-4478	169	81	(	(	PUNCT
ejpam-4478	169	82	v	v	NUM
ejpam-4478	169	83	fk	fk	INTJ
ejpam-4478	169	84	0	0	NUM
ejpam-4478	169	85	,	,	PUNCT
ejpam-4478	169	86	v	v	NOUN
ejpam-4478	169	87	fk	fk	INTJ
ejpam-4478	169	88	1	1	NUM
ejpam-4478	169	89	,	,	PUNCT
ejpam-4478	169	90	v	v	NOUN
ejpam-4478	169	91	fk	fk	INTJ
ejpam-4478	169	92	2	2	NUM
ejpam-4478	169	93	)	)	PUNCT
ejpam-4478	169	94	.	.	PUNCT
ejpam-4478	170	1	proof	proof	NOUN
ejpam-4478	170	2	.	.	PUNCT
ejpam-4478	171	1	let	let	VERB
ejpam-4478	171	2	k	k	PROPN
ejpam-4478	171	3	∈	∈	PROPN
ejpam-4478	171	4	z+	z+	PUNCT
ejpam-4478	171	5	.	.	PUNCT
ejpam-4478	171	6	suppose	suppose	VERB
ejpam-4478	171	7	we	we	PRON
ejpam-4478	171	8	have	have	VERB
ejpam-4478	171	9	the	the	DET
ejpam-4478	171	10	global	global	ADJ
ejpam-4478	171	11	k	k	PROPN
ejpam-4478	171	12	−	−	PROPN
ejpam-4478	171	13	distance	distance	NOUN
ejpam-4478	171	14	roman	roman	ADJ
ejpam-4478	171	15	dominating	dominating	NOUN
ejpam-4478	171	16	function	function	NOUN
ejpam-4478	171	17	(	(	PUNCT
ejpam-4478	171	18	gkdrdf	gkdrdf	PROPN
ejpam-4478	171	19	)	)	PUNCT
ejpam-4478	172	1	fk	fk	INTJ
ejpam-4478	172	2	:	:	PUNCT
ejpam-4478	172	3	v	v	X
ejpam-4478	172	4	→	→	SYM
ejpam-4478	172	5	{	{	PUNCT
ejpam-4478	172	6	0	0	NUM
ejpam-4478	172	7	,	,	PUNCT
ejpam-4478	172	8	1	1	NUM
ejpam-4478	172	9	,	,	PUNCT
ejpam-4478	172	10	2	2	NUM
ejpam-4478	172	11	}	}	PUNCT
ejpam-4478	172	12	on	on	ADP
ejpam-4478	172	13	graph	graph	NOUN
ejpam-4478	172	14	g	g	PROPN
ejpam-4478	172	15	=	=	SYM
ejpam-4478	172	16	(	(	PUNCT
ejpam-4478	172	17	v	v	NOUN
ejpam-4478	172	18	,	,	PUNCT
ejpam-4478	172	19	e	e	NOUN
ejpam-4478	172	20	)	)	PUNCT
ejpam-4478	172	21	.	.	PUNCT
ejpam-4478	173	1	•	•	INTJ
ejpam-4478	173	2	there	there	PRON
ejpam-4478	173	3	is	be	VERB
ejpam-4478	173	4	nothing	nothing	PRON
ejpam-4478	173	5	to	to	PART
ejpam-4478	173	6	prove	prove	VERB
ejpam-4478	173	7	in	in	ADP
ejpam-4478	173	8	part	part	NOUN
ejpam-4478	173	9	(	(	PUNCT
ejpam-4478	173	10	i	i	NOUN
ejpam-4478	173	11	)	)	PUNCT
ejpam-4478	173	12	.	.	PUNCT
ejpam-4478	174	1	•	•	NOUN
ejpam-4478	174	2	for	for	ADP
ejpam-4478	174	3	part	part	NOUN
ejpam-4478	174	4	(	(	PUNCT
ejpam-4478	174	5	ii	ii	NOUN
ejpam-4478	174	6	)	)	PUNCT
ejpam-4478	174	7	,	,	PUNCT
ejpam-4478	174	8	we	we	PRON
ejpam-4478	174	9	note	note	VERB
ejpam-4478	174	10	that	that	SCONJ
ejpam-4478	174	11	fk	fk	INTJ
ejpam-4478	174	12	can	can	AUX
ejpam-4478	174	13	be	be	AUX
ejpam-4478	174	14	expressed	express	VERB
ejpam-4478	174	15	as	as	SCONJ
ejpam-4478	174	16	follows	follow	VERB
ejpam-4478	174	17	fk	fk	INTJ
ejpam-4478	174	18	=	=	SYM
ejpam-4478	174	19	{	{	PUNCT
ejpam-4478	174	20	(	(	PUNCT
ejpam-4478	174	21	uj	uj	PROPN
ejpam-4478	174	22	,	,	PUNCT
ejpam-4478	174	23	f(uj	f(uj	PROPN
ejpam-4478	174	24	)	)	PUNCT
ejpam-4478	174	25	)	)	PUNCT
ejpam-4478	174	26	:	:	PUNCT
ejpam-4478	175	1	uj	uj	PROPN
ejpam-4478	175	2	∈	∈	PROPN
ejpam-4478	175	3	v	v	PROPN
ejpam-4478	175	4	and	and	CCONJ
ejpam-4478	175	5	f(uj	f(uj	PROPN
ejpam-4478	175	6	)	)	PUNCT
ejpam-4478	175	7	∈	∈	PROPN
ejpam-4478	175	8	{	{	PUNCT
ejpam-4478	175	9	0	0	NUM
ejpam-4478	175	10	,	,	PUNCT
ejpam-4478	175	11	1	1	NUM
ejpam-4478	175	12	,	,	PUNCT
ejpam-4478	175	13	2	2	NUM
ejpam-4478	175	14	}	}	PUNCT
ejpam-4478	175	15	}	}	PUNCT
ejpam-4478	175	16	.	.	PUNCT
ejpam-4478	176	1	g.	g.	PROPN
ejpam-4478	176	2	entero	entero	PROPN
ejpam-4478	176	3	,	,	PUNCT
ejpam-4478	176	4	s.	s.	PROPN
ejpam-4478	176	5	espinola	espinola	PROPN
ejpam-4478	176	6	/	/	SYM
ejpam-4478	176	7	eur	eur	PROPN
ejpam-4478	176	8	.	.	PUNCT
ejpam-4478	177	1	j.	j.	PROPN
ejpam-4478	177	2	pure	pure	PROPN
ejpam-4478	177	3	appl	appl	PROPN
ejpam-4478	177	4	.	.	PROPN
ejpam-4478	177	5	math	math	PROPN
ejpam-4478	177	6	,	,	PUNCT
ejpam-4478	177	7	16	16	NUM
ejpam-4478	177	8	(	(	PUNCT
ejpam-4478	177	9	1	1	NUM
ejpam-4478	177	10	)	)	PUNCT
ejpam-4478	177	11	(	(	PUNCT
ejpam-4478	177	12	2023	2023	NUM
ejpam-4478	177	13	)	)	PUNCT
ejpam-4478	177	14	,	,	PUNCT
ejpam-4478	177	15	44	44	NUM
ejpam-4478	177	16	-	-	SYM
ejpam-4478	177	17	61	61	NUM
ejpam-4478	177	18	50	50	NUM
ejpam-4478	177	19	now	now	ADV
ejpam-4478	177	20	,	,	PUNCT
ejpam-4478	177	21	let	let	VERB
ejpam-4478	177	22	us	we	PRON
ejpam-4478	177	23	partition	partition	VERB
ejpam-4478	177	24	(	(	PUNCT
ejpam-4478	177	25	ordered	order	VERB
ejpam-4478	177	26	partition	partition	NOUN
ejpam-4478	177	27	)	)	PUNCT
ejpam-4478	177	28	the	the	DET
ejpam-4478	177	29	function	function	NOUN
ejpam-4478	177	30	fk	fk	INTJ
ejpam-4478	177	31	in	in	ADP
ejpam-4478	177	32	terms	term	NOUN
ejpam-4478	177	33	of	of	ADP
ejpam-4478	177	34	images	image	NOUN
ejpam-4478	177	35	of	of	ADP
ejpam-4478	177	36	each	each	DET
ejpam-4478	177	37	uj	uj	PROPN
ejpam-4478	177	38	∈	∈	PROPN
ejpam-4478	177	39	v	v	ADP
ejpam-4478	177	40	treating	treat	VERB
ejpam-4478	177	41	fk	fk	INTJ
ejpam-4478	177	42	as	as	ADP
ejpam-4478	177	43	the	the	DET
ejpam-4478	177	44	set	set	NOUN
ejpam-4478	177	45	of	of	ADP
ejpam-4478	177	46	ordered	order	VERB
ejpam-4478	177	47	pairs	pair	NOUN
ejpam-4478	177	48	.	.	PUNCT
ejpam-4478	178	1	so	so	ADV
ejpam-4478	178	2	,	,	PUNCT
ejpam-4478	178	3	we	we	PRON
ejpam-4478	178	4	can	can	AUX
ejpam-4478	178	5	have	have	VERB
ejpam-4478	178	6	the	the	DET
ejpam-4478	178	7	cells	cell	NOUN
ejpam-4478	178	8	f0	f0	PROPN
ejpam-4478	178	9	k	k	PROPN
ejpam-4478	178	10	,	,	PUNCT
ejpam-4478	178	11	f	f	PROPN
ejpam-4478	178	12	1	1	NUM
ejpam-4478	178	13	k	k	NOUN
ejpam-4478	178	14	,	,	PUNCT
ejpam-4478	178	15	and	and	CCONJ
ejpam-4478	178	16	f2	f2	PROPN
ejpam-4478	179	1	k	k	PROPN
ejpam-4478	179	2	in	in	ADP
ejpam-4478	179	3	which	which	PRON
ejpam-4478	179	4	fk	fk	INTJ
ejpam-4478	179	5	=	=	PUNCT
ejpam-4478	179	6	f0	f0	PROPN
ejpam-4478	179	7	k	k	PROPN
ejpam-4478	179	8	∪	∪	PROPN
ejpam-4478	179	9	f1	f1	PROPN
ejpam-4478	179	10	k	k	PROPN
ejpam-4478	179	11	∪	∪	PROPN
ejpam-4478	179	12	f2	f2	PROPN
ejpam-4478	179	13	k	k	PROPN
ejpam-4478	179	14	,	,	PUNCT
ejpam-4478	179	15	where	where	SCONJ
ejpam-4478	179	16	f0	f0	PROPN
ejpam-4478	179	17	k	k	PROPN
ejpam-4478	179	18	=	=	PRON
ejpam-4478	179	19	{	{	PUNCT
ejpam-4478	179	20	(	(	PUNCT
ejpam-4478	179	21	uj	uj	PROPN
ejpam-4478	179	22	,	,	PUNCT
ejpam-4478	179	23	0	0	NUM
ejpam-4478	179	24	)	)	PUNCT
ejpam-4478	179	25	:	:	PUNCT
ejpam-4478	180	1	uj	uj	PROPN
ejpam-4478	180	2	∈	∈	PROPN
ejpam-4478	180	3	v	v	NOUN
ejpam-4478	180	4	}	}	PUNCT
ejpam-4478	180	5	,	,	PUNCT
ejpam-4478	180	6	f1	f1	PROPN
ejpam-4478	180	7	k	k	PROPN
ejpam-4478	181	1	=	=	PRON
ejpam-4478	182	1	{	{	PUNCT
ejpam-4478	183	1	(	(	PUNCT
ejpam-4478	183	2	uj	uj	PROPN
ejpam-4478	183	3	,	,	PUNCT
ejpam-4478	183	4	1	1	NUM
ejpam-4478	183	5	)	)	PUNCT
ejpam-4478	183	6	:	:	PUNCT
ejpam-4478	184	1	uj	uj	PROPN
ejpam-4478	184	2	∈	∈	PROPN
ejpam-4478	184	3	v	v	NOUN
ejpam-4478	184	4	}	}	PUNCT
ejpam-4478	184	5	,	,	PUNCT
ejpam-4478	184	6	and	and	CCONJ
ejpam-4478	184	7	f2	f2	PROPN
ejpam-4478	184	8	k	k	PROPN
ejpam-4478	184	9	=	=	PRON
ejpam-4478	184	10	{	{	PUNCT
ejpam-4478	184	11	(	(	PUNCT
ejpam-4478	184	12	uj	uj	PROPN
ejpam-4478	184	13	,	,	PUNCT
ejpam-4478	184	14	2	2	NUM
ejpam-4478	184	15	)	)	PUNCT
ejpam-4478	184	16	:	:	PUNCT
ejpam-4478	184	17	uj	uj	PROPN
ejpam-4478	184	18	∈	∈	PROPN
ejpam-4478	184	19	v	v	NOUN
ejpam-4478	184	20	}	}	PUNCT
ejpam-4478	184	21	.	.	PUNCT
ejpam-4478	185	1	and	and	CCONJ
ejpam-4478	185	2	f	f	X
ejpam-4478	185	3	i	i	PROPN
ejpam-4478	185	4	k	k	PROPN
ejpam-4478	185	5	∩	∩	PROPN
ejpam-4478	185	6	f	f	PROPN
ejpam-4478	185	7	l	l	NOUN
ejpam-4478	185	8	k	k	X
ejpam-4478	185	9	=	=	NOUN
ejpam-4478	185	10	∅	∅	NOUN
ejpam-4478	185	11	for	for	ADP
ejpam-4478	185	12	all	all	PRON
ejpam-4478	185	13	i	i	PRON
ejpam-4478	185	14	̸=	̸=	PROPN
ejpam-4478	185	15	l	l	NOUN
ejpam-4478	185	16	and	and	CCONJ
ejpam-4478	185	17	i	i	PRON
ejpam-4478	185	18	,	,	PUNCT
ejpam-4478	185	19	l	l	PROPN
ejpam-4478	185	20	∈	∈	PROPN
ejpam-4478	185	21	{	{	PUNCT
ejpam-4478	185	22	0	0	NUM
ejpam-4478	185	23	,	,	PUNCT
ejpam-4478	185	24	1	1	NUM
ejpam-4478	185	25	,	,	PUNCT
ejpam-4478	185	26	2	2	NUM
ejpam-4478	185	27	}	}	PUNCT
ejpam-4478	185	28	which	which	PRON
ejpam-4478	185	29	means	mean	VERB
ejpam-4478	185	30	that	that	SCONJ
ejpam-4478	185	31	f0	f0	PROPN
ejpam-4478	185	32	k	k	PROPN
ejpam-4478	185	33	∩	∩	PROPN
ejpam-4478	185	34	f1	f1	PROPN
ejpam-4478	185	35	k	k	PROPN
ejpam-4478	185	36	∩	∩	PROPN
ejpam-4478	185	37	f2	f2	PROPN
ejpam-4478	185	38	k	k	NOUN
ejpam-4478	186	1	=	=	PUNCT
ejpam-4478	186	2	∅.	∅.	VERB
ejpam-4478	186	3	now	now	ADV
ejpam-4478	186	4	,	,	PUNCT
ejpam-4478	186	5	for	for	ADP
ejpam-4478	186	6	the	the	DET
ejpam-4478	186	7	ordered	order	VERB
ejpam-4478	186	8	partition	partition	NOUN
ejpam-4478	186	9	(	(	PUNCT
ejpam-4478	186	10	v	v	NOUN
ejpam-4478	186	11	fk	fk	INTJ
ejpam-4478	186	12	0	0	NUM
ejpam-4478	186	13	,	,	PUNCT
ejpam-4478	186	14	v	v	NOUN
ejpam-4478	186	15	fk	fk	INTJ
ejpam-4478	186	16	1	1	NUM
ejpam-4478	186	17	,	,	PUNCT
ejpam-4478	186	18	v	v	X
ejpam-4478	186	19	fk	fk	INTJ
ejpam-4478	186	20	2	2	NUM
ejpam-4478	186	21	)	)	PUNCT
ejpam-4478	186	22	of	of	ADP
ejpam-4478	186	23	v	v	NOUN
ejpam-4478	186	24	induced	induce	VERB
ejpam-4478	186	25	by	by	ADP
ejpam-4478	186	26	fk	fk	INTJ
ejpam-4478	186	27	,	,	PUNCT
ejpam-4478	186	28	where	where	SCONJ
ejpam-4478	186	29	v	v	NOUN
ejpam-4478	186	30	fk	fk	INTJ
ejpam-4478	186	31	i	i	NOUN
ejpam-4478	186	32	=	=	PUNCT
ejpam-4478	186	33	{	{	PUNCT
ejpam-4478	186	34	v	v	NUM
ejpam-4478	186	35	∈	∈	NOUN
ejpam-4478	186	36	v	v	ADP
ejpam-4478	186	37	|fk(v	|fk(v	PROPN
ejpam-4478	186	38	)	)	PUNCT
ejpam-4478	187	1	=	=	SYM
ejpam-4478	188	1	i	i	PRON
ejpam-4478	188	2	and	and	CCONJ
ejpam-4478	188	3	i	i	PRON
ejpam-4478	188	4	=	=	NOUN
ejpam-4478	188	5	0	0	NUM
ejpam-4478	188	6	,	,	PUNCT
ejpam-4478	188	7	1	1	NUM
ejpam-4478	188	8	,	,	PUNCT
ejpam-4478	188	9	2	2	NUM
ejpam-4478	188	10	}	}	PUNCT
ejpam-4478	188	11	,	,	PUNCT
ejpam-4478	188	12	we	we	PRON
ejpam-4478	188	13	have	have	VERB
ejpam-4478	188	14	v	v	NUM
ejpam-4478	188	15	fk	fk	INTJ
ejpam-4478	188	16	0	0	NUM
ejpam-4478	189	1	=	=	SYM
ejpam-4478	189	2	{	{	PUNCT
ejpam-4478	189	3	uj	uj	PROPN
ejpam-4478	189	4	∈	∈	PROPN
ejpam-4478	189	5	v	v	NOUN
ejpam-4478	189	6	:	:	PUNCT
ejpam-4478	189	7	fk(uj	fk(uj	NOUN
ejpam-4478	189	8	)	)	PUNCT
ejpam-4478	189	9	=	=	PUNCT
ejpam-4478	190	1	0	0	NUM
ejpam-4478	190	2	}	}	PUNCT
ejpam-4478	190	3	,	,	PUNCT
ejpam-4478	190	4	v	v	X
ejpam-4478	190	5	fk	fk	INTJ
ejpam-4478	190	6	1	1	NUM
ejpam-4478	190	7	=	=	SYM
ejpam-4478	190	8	{	{	PUNCT
ejpam-4478	190	9	uj	uj	PROPN
ejpam-4478	190	10	∈	∈	PROPN
ejpam-4478	190	11	v	v	NOUN
ejpam-4478	190	12	:	:	PUNCT
ejpam-4478	190	13	fk(uj	fk(uj	NOUN
ejpam-4478	190	14	)	)	PUNCT
ejpam-4478	190	15	=	=	PUNCT
ejpam-4478	191	1	1	1	NUM
ejpam-4478	191	2	}	}	PUNCT
ejpam-4478	191	3	,	,	PUNCT
ejpam-4478	191	4	and	and	CCONJ
ejpam-4478	191	5	v	v	ADP
ejpam-4478	191	6	fk	fk	INTJ
ejpam-4478	191	7	2	2	NUM
ejpam-4478	191	8	=	=	SYM
ejpam-4478	191	9	{	{	PUNCT
ejpam-4478	191	10	uj	uj	PROPN
ejpam-4478	191	11	∈	∈	PROPN
ejpam-4478	191	12	v	v	NOUN
ejpam-4478	191	13	:	:	PUNCT
ejpam-4478	191	14	fk(uj	fk(uj	NOUN
ejpam-4478	191	15	)	)	PUNCT
ejpam-4478	191	16	=	=	PUNCT
ejpam-4478	192	1	2	2	NUM
ejpam-4478	192	2	}	}	PUNCT
ejpam-4478	192	3	.	.	PUNCT
ejpam-4478	193	1	which	which	PRON
ejpam-4478	193	2	also	also	ADV
ejpam-4478	193	3	means	mean	VERB
ejpam-4478	193	4	that	that	SCONJ
ejpam-4478	193	5	v	v	INTJ
ejpam-4478	193	6	fk	fk	INTJ
ejpam-4478	193	7	i	i	PROPN
ejpam-4478	193	8	∩	∩	PROPN
ejpam-4478	193	9	v	v	ADP
ejpam-4478	193	10	fk	fk	INTJ
ejpam-4478	193	11	l	l	NOUN
ejpam-4478	193	12	=	=	NOUN
ejpam-4478	193	13	∅	∅	NOUN
ejpam-4478	193	14	for	for	ADP
ejpam-4478	193	15	all	all	PRON
ejpam-4478	193	16	i	i	PRON
ejpam-4478	193	17	̸=	̸=	PROPN
ejpam-4478	193	18	l	l	NOUN
ejpam-4478	193	19	and	and	CCONJ
ejpam-4478	193	20	i	i	PRON
ejpam-4478	193	21	,	,	PUNCT
ejpam-4478	193	22	l	l	PROPN
ejpam-4478	193	23	∈	∈	PROPN
ejpam-4478	193	24	{	{	PUNCT
ejpam-4478	193	25	0	0	NUM
ejpam-4478	193	26	,	,	PUNCT
ejpam-4478	193	27	1	1	NUM
ejpam-4478	193	28	,	,	PUNCT
ejpam-4478	193	29	2	2	NUM
ejpam-4478	193	30	}	}	PUNCT
ejpam-4478	193	31	and	and	CCONJ
ejpam-4478	193	32	v	v	ADP
ejpam-4478	193	33	fk	fk	INTJ
ejpam-4478	193	34	0	0	NUM
ejpam-4478	193	35	∩	∩	PROPN
ejpam-4478	193	36	v	v	X
ejpam-4478	193	37	fk	fk	INTJ
ejpam-4478	193	38	1	1	NUM
ejpam-4478	193	39	∩	∩	X
ejpam-4478	193	40	v	v	X
ejpam-4478	193	41	fk	fk	INTJ
ejpam-4478	193	42	2	2	NUM
ejpam-4478	193	43	=	=	NOUN
ejpam-4478	193	44	∅.	∅.	VERB
ejpam-4478	193	45	hence	hence	ADV
ejpam-4478	193	46	,	,	PUNCT
ejpam-4478	193	47	by	by	ADP
ejpam-4478	193	48	matching	match	VERB
ejpam-4478	193	49	class	class	NOUN
ejpam-4478	193	50	f0	f0	PROPN
ejpam-4478	193	51	k	k	PROPN
ejpam-4478	193	52	to	to	ADP
ejpam-4478	193	53	class	class	NOUN
ejpam-4478	193	54	v	v	NOUN
ejpam-4478	193	55	fk	fk	INTJ
ejpam-4478	193	56	0	0	NUM
ejpam-4478	193	57	,	,	PUNCT
ejpam-4478	193	58	class	class	NOUN
ejpam-4478	193	59	f1	f1	NOUN
ejpam-4478	193	60	k	k	PROPN
ejpam-4478	193	61	to	to	ADP
ejpam-4478	193	62	class	class	NOUN
ejpam-4478	193	63	v	v	NOUN
ejpam-4478	193	64	fk	fk	INTJ
ejpam-4478	193	65	1	1	NUM
ejpam-4478	193	66	,	,	PUNCT
ejpam-4478	193	67	and	and	CCONJ
ejpam-4478	193	68	class	class	NOUN
ejpam-4478	193	69	f2	f2	PROPN
ejpam-4478	193	70	k	k	PROPN
ejpam-4478	193	71	to	to	ADP
ejpam-4478	193	72	class	class	NOUN
ejpam-4478	193	73	v	v	NOUN
ejpam-4478	193	74	fk	fk	INTJ
ejpam-4478	193	75	2	2	NUM
ejpam-4478	193	76	,	,	PUNCT
ejpam-4478	193	77	we	we	PRON
ejpam-4478	193	78	can	can	AUX
ejpam-4478	193	79	see	see	VERB
ejpam-4478	193	80	that	that	SCONJ
ejpam-4478	193	81	there	there	PRON
ejpam-4478	193	82	is	be	VERB
ejpam-4478	193	83	a	a	DET
ejpam-4478	193	84	one	one	NUM
ejpam-4478	193	85	-	-	PUNCT
ejpam-4478	193	86	to	to	ADP
ejpam-4478	193	87	-	-	PUNCT
ejpam-4478	193	88	one	one	NUM
ejpam-4478	193	89	correspondence	correspondence	NOUN
ejpam-4478	193	90	between	between	ADP
ejpam-4478	193	91	fk	fk	INTJ
ejpam-4478	193	92	:	:	PUNCT
ejpam-4478	193	93	v	v	X
ejpam-4478	193	94	→	→	SYM
ejpam-4478	193	95	{	{	PUNCT
ejpam-4478	193	96	0	0	NUM
ejpam-4478	193	97	,	,	PUNCT
ejpam-4478	193	98	1	1	NUM
ejpam-4478	193	99	,	,	PUNCT
ejpam-4478	193	100	2	2	NUM
ejpam-4478	193	101	}	}	PUNCT
ejpam-4478	193	102	and	and	CCONJ
ejpam-4478	193	103	the	the	DET
ejpam-4478	193	104	ordered	order	VERB
ejpam-4478	193	105	partition	partition	NOUN
ejpam-4478	193	106	(	(	PUNCT
ejpam-4478	193	107	v	v	NOUN
ejpam-4478	193	108	fk	fk	INTJ
ejpam-4478	193	109	0	0	NUM
ejpam-4478	193	110	,	,	PUNCT
ejpam-4478	193	111	v	v	NOUN
ejpam-4478	193	112	fk	fk	INTJ
ejpam-4478	193	113	1	1	NUM
ejpam-4478	193	114	,	,	PUNCT
ejpam-4478	193	115	v	v	X
ejpam-4478	193	116	fk	fk	INTJ
ejpam-4478	193	117	2	2	NUM
ejpam-4478	193	118	)	)	PUNCT
ejpam-4478	193	119	of	of	ADP
ejpam-4478	193	120	v	v	NOUN
ejpam-4478	193	121	induced	induce	VERB
ejpam-4478	193	122	by	by	ADP
ejpam-4478	193	123	fk	fk	INTJ
ejpam-4478	193	124	.	.	PUNCT
ejpam-4478	194	1	this	this	PRON
ejpam-4478	194	2	concludes	conclude	VERB
ejpam-4478	194	3	the	the	DET
ejpam-4478	194	4	proof	proof	NOUN
ejpam-4478	194	5	of	of	ADP
ejpam-4478	194	6	part	part	NOUN
ejpam-4478	194	7	(	(	PUNCT
ejpam-4478	194	8	ii	ii	NOUN
ejpam-4478	194	9	)	)	PUNCT
ejpam-4478	194	10	.	.	PUNCT
ejpam-4478	195	1	•	•	NOUN
ejpam-4478	195	2	for	for	ADP
ejpam-4478	195	3	part	part	NOUN
ejpam-4478	195	4	(	(	PUNCT
ejpam-4478	195	5	iii	iii	NOUN
ejpam-4478	195	6	)	)	PUNCT
ejpam-4478	195	7	,	,	PUNCT
ejpam-4478	195	8	the	the	DET
ejpam-4478	195	9	proof	proof	NOUN
ejpam-4478	195	10	is	be	AUX
ejpam-4478	195	11	straightforward	straightforward	ADJ
ejpam-4478	195	12	.	.	PUNCT
ejpam-4478	196	1	therefore	therefore	ADV
ejpam-4478	196	2	,	,	PUNCT
ejpam-4478	196	3	we	we	PRON
ejpam-4478	196	4	can	can	AUX
ejpam-4478	196	5	say	say	VERB
ejpam-4478	196	6	now	now	ADV
ejpam-4478	196	7	that	that	DET
ejpam-4478	196	8	remark	remark	NOUN
ejpam-4478	196	9	4.1	4.1	NUM
ejpam-4478	196	10	is	be	AUX
ejpam-4478	196	11	true	true	ADJ
ejpam-4478	196	12	and	and	CCONJ
ejpam-4478	196	13	valid	valid	ADJ
ejpam-4478	196	14	.	.	PUNCT
ejpam-4478	197	1	remark	remark	PROPN
ejpam-4478	197	2	4.2	4.2	NUM
ejpam-4478	197	3	.	.	PUNCT
ejpam-4478	198	1	for	for	ADP
ejpam-4478	198	2	any	any	DET
ejpam-4478	198	3	graph	graph	NOUN
ejpam-4478	198	4	g	g	NOUN
ejpam-4478	198	5	=	=	SYM
ejpam-4478	198	6	(	(	PUNCT
ejpam-4478	198	7	v	v	NOUN
ejpam-4478	198	8	,	,	PUNCT
ejpam-4478	198	9	e	e	NOUN
ejpam-4478	198	10	)	)	PUNCT
ejpam-4478	198	11	of	of	ADP
ejpam-4478	198	12	order	order	NOUN
ejpam-4478	198	13	n	n	CCONJ
ejpam-4478	198	14	,	,	PUNCT
ejpam-4478	198	15	there	there	PRON
ejpam-4478	198	16	exists	exist	VERB
ejpam-4478	198	17	a	a	DET
ejpam-4478	198	18	function	function	NOUN
ejpam-4478	198	19	f	f	NOUN
ejpam-4478	198	20	:	:	PUNCT
ejpam-4478	198	21	v	v	X
ejpam-4478	198	22	→	→	SYM
ejpam-4478	198	23	{	{	PUNCT
ejpam-4478	198	24	0	0	NUM
ejpam-4478	198	25	,	,	PUNCT
ejpam-4478	198	26	1	1	NUM
ejpam-4478	198	27	,	,	PUNCT
ejpam-4478	198	28	2	2	NUM
ejpam-4478	198	29	}	}	PUNCT
ejpam-4478	198	30	satisfying	satisfy	VERB
ejpam-4478	198	31	the	the	DET
ejpam-4478	198	32	conditions	condition	NOUN
ejpam-4478	199	1	that	that	PRON
ejpam-4478	199	2	for	for	ADP
ejpam-4478	199	3	every	every	DET
ejpam-4478	199	4	vertex	vertex	NOUN
ejpam-4478	199	5	v	v	NOUN
ejpam-4478	199	6	for	for	ADP
ejpam-4478	199	7	which	which	PRON
ejpam-4478	199	8	f(v	f(v	NOUN
ejpam-4478	199	9	)	)	PUNCT
ejpam-4478	199	10	=	=	PUNCT
ejpam-4478	199	11	0	0	NUM
ejpam-4478	199	12	there	there	PRON
ejpam-4478	199	13	exists	exist	VERB
ejpam-4478	199	14	at	at	ADV
ejpam-4478	199	15	least	least	ADV
ejpam-4478	199	16	one	one	NUM
ejpam-4478	199	17	vertex	vertex	NOUN
ejpam-4478	199	18	u	u	NOUN
ejpam-4478	199	19	for	for	ADP
ejpam-4478	199	20	which	which	PRON
ejpam-4478	199	21	f(u	f(u	PROPN
ejpam-4478	199	22	)	)	PUNCT
ejpam-4478	199	23	=	=	SYM
ejpam-4478	199	24	2	2	NUM
ejpam-4478	199	25	with	with	ADP
ejpam-4478	199	26	d(u	d(u	PROPN
ejpam-4478	199	27	,	,	PUNCT
ejpam-4478	199	28	v	v	NOUN
ejpam-4478	199	29	)	)	PUNCT
ejpam-4478	199	30	≤	≤	NOUN
ejpam-4478	200	1	k	k	ADP
ejpam-4478	200	2	such	such	ADJ
ejpam-4478	200	3	that	that	SCONJ
ejpam-4478	200	4	f	f	PROPN
ejpam-4478	200	5	is	be	AUX
ejpam-4478	200	6	a	a	DET
ejpam-4478	200	7	k	k	NOUN
ejpam-4478	200	8	−	−	PROPN
ejpam-4478	200	9	distance	distance	NOUN
ejpam-4478	200	10	roman	roman	ADJ
ejpam-4478	200	11	dominating	dominating	NOUN
ejpam-4478	200	12	function	function	NOUN
ejpam-4478	200	13	(	(	PUNCT
ejpam-4478	200	14	kdrdf	kdrdf	NOUN
ejpam-4478	200	15	)	)	PUNCT
ejpam-4478	200	16	on	on	ADP
ejpam-4478	200	17	g	g	PROPN
ejpam-4478	200	18	and	and	CCONJ
ejpam-4478	200	19	on	on	ADP
ejpam-4478	200	20	its	its	PRON
ejpam-4478	200	21	complement	complement	NOUN
ejpam-4478	200	22	g	g	NOUN
ejpam-4478	200	23	,	,	PUNCT
ejpam-4478	200	24	where	where	SCONJ
ejpam-4478	200	25	k	k	PROPN
ejpam-4478	200	26	∈	∈	PROPN
ejpam-4478	200	27	z+	z+	PUNCT
ejpam-4478	200	28	.	.	PUNCT
ejpam-4478	201	1	then	then	ADV
ejpam-4478	201	2	such	such	DET
ejpam-4478	201	3	a	a	DET
ejpam-4478	201	4	function	function	NOUN
ejpam-4478	201	5	f	f	NOUN
ejpam-4478	201	6	on	on	ADP
ejpam-4478	201	7	g	g	PROPN
ejpam-4478	201	8	with	with	ADP
ejpam-4478	201	9	minimum	minimum	ADJ
ejpam-4478	201	10	weight	weight	NOUN
ejpam-4478	201	11	also	also	ADV
ejpam-4478	201	12	exists	exist	VERB
ejpam-4478	201	13	.	.	PUNCT
ejpam-4478	202	1	we	we	PRON
ejpam-4478	202	2	call	call	VERB
ejpam-4478	202	3	the	the	DET
ejpam-4478	202	4	function	function	NOUN
ejpam-4478	202	5	f	f	NOUN
ejpam-4478	202	6	:	:	PUNCT
ejpam-4478	202	7	v	v	X
ejpam-4478	202	8	→	→	SYM
ejpam-4478	202	9	{	{	PUNCT
ejpam-4478	202	10	0	0	NUM
ejpam-4478	202	11	,	,	PUNCT
ejpam-4478	202	12	1	1	NUM
ejpam-4478	202	13	,	,	PUNCT
ejpam-4478	202	14	2	2	NUM
ejpam-4478	202	15	}	}	PUNCT
ejpam-4478	202	16	as	as	ADP
ejpam-4478	202	17	the	the	DET
ejpam-4478	202	18	global	global	ADJ
ejpam-4478	202	19	k	k	PROPN
ejpam-4478	202	20	−	−	PROPN
ejpam-4478	202	21	distance	distance	NOUN
ejpam-4478	202	22	roman	roman	ADJ
ejpam-4478	202	23	dominating	dominating	NOUN
ejpam-4478	202	24	function	function	NOUN
ejpam-4478	202	25	(	(	PUNCT
ejpam-4478	202	26	gkdrdf	gkdrdf	PROPN
ejpam-4478	202	27	)	)	PUNCT
ejpam-4478	202	28	on	on	ADP
ejpam-4478	202	29	g.	g.	PROPN
ejpam-4478	202	30	proof	proof	PROPN
ejpam-4478	202	31	.	.	PUNCT
ejpam-4478	203	1	suppose	suppose	VERB
ejpam-4478	203	2	that	that	SCONJ
ejpam-4478	203	3	g	g	PROPN
ejpam-4478	203	4	is	be	AUX
ejpam-4478	203	5	a	a	DET
ejpam-4478	203	6	graph	graph	NOUN
ejpam-4478	203	7	of	of	ADP
ejpam-4478	203	8	order	order	NOUN
ejpam-4478	203	9	n.	n.	NOUN
ejpam-4478	203	10	assume	assume	VERB
ejpam-4478	203	11	that	that	SCONJ
ejpam-4478	203	12	the	the	DET
ejpam-4478	203	13	vertex	vertex	NOUN
ejpam-4478	203	14	set	set	NOUN
ejpam-4478	203	15	of	of	ADP
ejpam-4478	203	16	graph	graph	NOUN
ejpam-4478	203	17	g	g	PROPN
ejpam-4478	203	18	is	be	AUX
ejpam-4478	203	19	v	v	NOUN
ejpam-4478	203	20	(	(	PUNCT
ejpam-4478	203	21	g	g	NOUN
ejpam-4478	203	22	)	)	PUNCT
ejpam-4478	203	23	=	=	SYM
ejpam-4478	203	24	{	{	PUNCT
ejpam-4478	203	25	u1	u1	NOUN
ejpam-4478	203	26	,	,	PUNCT
ejpam-4478	203	27	u2	u2	NOUN
ejpam-4478	203	28	,	,	PUNCT
ejpam-4478	203	29	u3	u3	NOUN
ejpam-4478	203	30	,	,	PUNCT
ejpam-4478	203	31	.	.	PUNCT
ejpam-4478	203	32	.	.	PUNCT
ejpam-4478	204	1	.	.	PUNCT
ejpam-4478	205	1	,	,	PUNCT
ejpam-4478	205	2	un−1	un−1	PROPN
ejpam-4478	205	3	,	,	PUNCT
ejpam-4478	205	4	un	un	ADJ
ejpam-4478	205	5	}	}	PUNCT
ejpam-4478	205	6	.	.	PUNCT
ejpam-4478	206	1	let	let	VERB
ejpam-4478	206	2	f	f	NOUN
ejpam-4478	206	3	:	:	PUNCT
ejpam-4478	206	4	v	v	X
ejpam-4478	206	5	(	(	PUNCT
ejpam-4478	206	6	g	g	NOUN
ejpam-4478	206	7	)	)	PUNCT
ejpam-4478	206	8	→	→	SYM
ejpam-4478	206	9	{	{	PUNCT
ejpam-4478	206	10	0	0	NUM
ejpam-4478	206	11	,	,	PUNCT
ejpam-4478	206	12	1	1	NUM
ejpam-4478	206	13	,	,	PUNCT
ejpam-4478	206	14	2	2	NUM
ejpam-4478	206	15	}	}	PUNCT
ejpam-4478	206	16	be	be	AUX
ejpam-4478	206	17	a	a	DET
ejpam-4478	206	18	function	function	NOUN
ejpam-4478	206	19	on	on	ADP
ejpam-4478	206	20	g	g	PROPN
ejpam-4478	206	21	and	and	CCONJ
ejpam-4478	206	22	let	let	VERB
ejpam-4478	206	23	k	k	PROPN
ejpam-4478	206	24	∈	∈	PROPN
ejpam-4478	206	25	z+	z+	PUNCT
ejpam-4478	206	26	.	.	PUNCT
ejpam-4478	207	1	for	for	ADP
ejpam-4478	207	2	all	all	PRON
ejpam-4478	207	3	k	k	PROPN
ejpam-4478	207	4	∈	∈	PROPN
ejpam-4478	207	5	z+	z+	NOUN
ejpam-4478	207	6	,	,	PUNCT
ejpam-4478	207	7	since	since	SCONJ
ejpam-4478	207	8	f	f	PROPN
ejpam-4478	207	9	=	=	SYM
ejpam-4478	207	10	(	(	PUNCT
ejpam-4478	207	11	v	v	NOUN
ejpam-4478	207	12	f	f	NOUN
ejpam-4478	207	13	0	0	PUNCT
ejpam-4478	207	14	(	(	PUNCT
ejpam-4478	207	15	g	g	NOUN
ejpam-4478	207	16	)	)	PUNCT
ejpam-4478	207	17	,	,	PUNCT
ejpam-4478	207	18	v	v	NOUN
ejpam-4478	207	19	f	f	PROPN
ejpam-4478	207	20	1	1	NUM
ejpam-4478	207	21	(	(	PUNCT
ejpam-4478	207	22	g	g	NOUN
ejpam-4478	207	23	)	)	PUNCT
ejpam-4478	207	24	,	,	PUNCT
ejpam-4478	207	25	v	v	X
ejpam-4478	207	26	f	f	PROPN
ejpam-4478	207	27	2	2	NUM
ejpam-4478	207	28	(	(	PUNCT
ejpam-4478	207	29	g	g	NOUN
ejpam-4478	207	30	)	)	PUNCT
ejpam-4478	207	31	)	)	PUNCT
ejpam-4478	207	32	,	,	PUNCT
ejpam-4478	207	33	we	we	PRON
ejpam-4478	207	34	let	let	VERB
ejpam-4478	207	35	v	v	ADP
ejpam-4478	207	36	f	f	PROPN
ejpam-4478	207	37	0	0	PUNCT
ejpam-4478	208	1	(	(	PUNCT
ejpam-4478	208	2	g	g	NOUN
ejpam-4478	208	3	)	)	PUNCT
ejpam-4478	208	4	=	=	PUNCT
ejpam-4478	209	1	v	v	ADP
ejpam-4478	209	2	f	f	PROPN
ejpam-4478	209	3	2	2	NUM
ejpam-4478	209	4	(	(	PUNCT
ejpam-4478	209	5	g	g	NOUN
ejpam-4478	209	6	)	)	PUNCT
ejpam-4478	209	7	=	=	NOUN
ejpam-4478	209	8	∅	∅	NOUN
ejpam-4478	209	9	and	and	CCONJ
ejpam-4478	209	10	v	v	X
ejpam-4478	209	11	f	f	PROPN
ejpam-4478	209	12	1	1	NUM
ejpam-4478	209	13	(	(	PUNCT
ejpam-4478	209	14	g	g	NOUN
ejpam-4478	209	15	)	)	PUNCT
ejpam-4478	209	16	=	=	SYM
ejpam-4478	209	17	{	{	PUNCT
ejpam-4478	209	18	u1	u1	NOUN
ejpam-4478	209	19	,	,	PUNCT
ejpam-4478	209	20	u2	u2	NOUN
ejpam-4478	209	21	,	,	PUNCT
ejpam-4478	209	22	u3	u3	NOUN
ejpam-4478	209	23	,	,	PUNCT
ejpam-4478	209	24	.	.	PUNCT
ejpam-4478	209	25	.	.	PUNCT
ejpam-4478	210	1	.	.	PUNCT
ejpam-4478	211	1	,	,	PUNCT
ejpam-4478	211	2	un−1	un−1	PROPN
ejpam-4478	211	3	,	,	PUNCT
ejpam-4478	211	4	un	un	ADJ
ejpam-4478	211	5	}	}	PUNCT
ejpam-4478	211	6	=	=	SYM
ejpam-4478	211	7	v	v	NOUN
ejpam-4478	211	8	(	(	PUNCT
ejpam-4478	211	9	g	g	NOUN
ejpam-4478	211	10	)	)	PUNCT
ejpam-4478	211	11	,	,	PUNCT
ejpam-4478	211	12	which	which	PRON
ejpam-4478	211	13	means	mean	VERB
ejpam-4478	211	14	that	that	SCONJ
ejpam-4478	211	15	,	,	PUNCT
ejpam-4478	211	16	f(ui	f(ui	PROPN
ejpam-4478	211	17	)	)	PUNCT
ejpam-4478	211	18	=	=	SYM
ejpam-4478	211	19	1	1	NUM
ejpam-4478	211	20	,	,	PUNCT
ejpam-4478	211	21	where	where	SCONJ
ejpam-4478	211	22	ui	ui	PROPN
ejpam-4478	211	23	∈	∈	PROPN
ejpam-4478	211	24	v	v	ADP
ejpam-4478	211	25	(	(	PUNCT
ejpam-4478	211	26	g	g	NOUN
ejpam-4478	211	27	)	)	PUNCT
ejpam-4478	211	28	and	and	CCONJ
ejpam-4478	211	29	i	i	NOUN
ejpam-4478	211	30	=	=	NOUN
ejpam-4478	211	31	1	1	NUM
ejpam-4478	211	32	,	,	PUNCT
ejpam-4478	211	33	2	2	NUM
ejpam-4478	211	34	,	,	PUNCT
ejpam-4478	211	35	3	3	NUM
ejpam-4478	211	36	,	,	PUNCT
ejpam-4478	211	37	.	.	PUNCT
ejpam-4478	211	38	.	.	PUNCT
ejpam-4478	211	39	.	.	PUNCT
ejpam-4478	212	1	,	,	PUNCT
ejpam-4478	212	2	n−1	n−1	PROPN
ejpam-4478	212	3	,	,	PUNCT
ejpam-4478	212	4	n	n	CCONJ
ejpam-4478	212	5	,	,	PUNCT
ejpam-4478	212	6	for	for	ADP
ejpam-4478	212	7	all	all	PRON
ejpam-4478	212	8	k	k	PROPN
ejpam-4478	212	9	∈	∈	PROPN
ejpam-4478	212	10	z+	z+	PUNCT
ejpam-4478	212	11	.	.	PUNCT
ejpam-4478	213	1	hence	hence	ADV
ejpam-4478	213	2	,	,	PUNCT
ejpam-4478	213	3	by	by	ADP
ejpam-4478	213	4	definition	definition	NOUN
ejpam-4478	213	5	3.7	3.7	NUM
ejpam-4478	213	6	,	,	PUNCT
ejpam-4478	213	7	the	the	DET
ejpam-4478	213	8	function	function	NOUN
ejpam-4478	213	9	f	f	PROPN
ejpam-4478	213	10	is	be	AUX
ejpam-4478	213	11	a	a	DET
ejpam-4478	213	12	global	global	ADJ
ejpam-4478	213	13	k	k	NOUN
ejpam-4478	213	14	−	−	PROPN
ejpam-4478	213	15	distance	distance	NOUN
ejpam-4478	213	16	roman	roman	ADJ
ejpam-4478	213	17	dominating	dominating	NOUN
ejpam-4478	213	18	function	function	NOUN
ejpam-4478	213	19	(	(	PUNCT
ejpam-4478	213	20	gkdrdf	gkdrdf	PROPN
ejpam-4478	213	21	)	)	PUNCT
ejpam-4478	213	22	on	on	ADP
ejpam-4478	213	23	g.	g.	PROPN
ejpam-4478	213	24	furthermore	furthermore	ADV
ejpam-4478	213	25	,	,	PUNCT
ejpam-4478	213	26	since	since	SCONJ
ejpam-4478	213	27	the	the	DET
ejpam-4478	213	28	existence	existence	NOUN
ejpam-4478	213	29	of	of	ADP
ejpam-4478	213	30	the	the	DET
ejpam-4478	213	31	gkdrdf	gkdrdf	NOUN
ejpam-4478	213	32	on	on	ADP
ejpam-4478	213	33	any	any	DET
ejpam-4478	213	34	graph	graph	NOUN
ejpam-4478	213	35	is	be	AUX
ejpam-4478	213	36	now	now	ADV
ejpam-4478	213	37	guaranteed	guarantee	VERB
ejpam-4478	213	38	and	and	CCONJ
ejpam-4478	213	39	since	since	SCONJ
ejpam-4478	213	40	the	the	DET
ejpam-4478	213	41	order	order	NOUN
ejpam-4478	213	42	of	of	ADP
ejpam-4478	213	43	graph	graph	NOUN
ejpam-4478	213	44	g	g	PROPN
ejpam-4478	213	45	is	be	AUX
ejpam-4478	213	46	finite	finite	ADJ
ejpam-4478	213	47	,	,	PUNCT
ejpam-4478	213	48	it	it	PRON
ejpam-4478	213	49	follows	follow	VERB
ejpam-4478	213	50	that	that	SCONJ
ejpam-4478	213	51	the	the	DET
ejpam-4478	213	52	existence	existence	NOUN
ejpam-4478	213	53	of	of	ADP
ejpam-4478	213	54	the	the	DET
ejpam-4478	213	55	gkdrdf	gkdrdf	NOUN
ejpam-4478	213	56	with	with	ADP
ejpam-4478	213	57	minimum	minimum	ADJ
ejpam-4478	213	58	weight	weight	NOUN
ejpam-4478	213	59	on	on	ADP
ejpam-4478	213	60	any	any	DET
ejpam-4478	213	61	graph	graph	NOUN
ejpam-4478	213	62	is	be	AUX
ejpam-4478	213	63	also	also	ADV
ejpam-4478	213	64	guaranteed	guarantee	VERB
ejpam-4478	213	65	.	.	PUNCT
ejpam-4478	214	1	this	this	PRON
ejpam-4478	214	2	proves	prove	VERB
ejpam-4478	214	3	remark	remark	NOUN
ejpam-4478	214	4	4.2	4.2	NUM
ejpam-4478	214	5	.	.	PUNCT
ejpam-4478	214	6	remark	remark	PROPN
ejpam-4478	214	7	4.3	4.3	NUM
ejpam-4478	214	8	.	.	PUNCT
ejpam-4478	215	1	for	for	ADP
ejpam-4478	215	2	all	all	PRON
ejpam-4478	215	3	k	k	PROPN
ejpam-4478	215	4	∈	∈	PROPN
ejpam-4478	215	5	z+	z+	X
ejpam-4478	215	6	,	,	PUNCT
ejpam-4478	215	7	if	if	SCONJ
ejpam-4478	215	8	fk	fk	PRON
ejpam-4478	215	9	can	can	AUX
ejpam-4478	215	10	be	be	AUX
ejpam-4478	215	11	written	write	VERB
ejpam-4478	215	12	as	as	ADP
ejpam-4478	215	13	fk	fk	INTJ
ejpam-4478	215	14	=	=	PUNCT
ejpam-4478	215	15	(	(	PUNCT
ejpam-4478	215	16	v	v	NUM
ejpam-4478	215	17	fk	fk	INTJ
ejpam-4478	215	18	0	0	NUM
ejpam-4478	215	19	,	,	PUNCT
ejpam-4478	215	20	v	v	NOUN
ejpam-4478	215	21	fk	fk	INTJ
ejpam-4478	215	22	1	1	NUM
ejpam-4478	215	23	,	,	PUNCT
ejpam-4478	215	24	v	v	NOUN
ejpam-4478	215	25	fk	fk	INTJ
ejpam-4478	215	26	2	2	NUM
ejpam-4478	215	27	)	)	PUNCT
ejpam-4478	215	28	,	,	PUNCT
ejpam-4478	215	29	then	then	ADV
ejpam-4478	215	30	its	its	PRON
ejpam-4478	215	31	weight	weight	NOUN
ejpam-4478	215	32	w(fk	w(fk	NOUN
ejpam-4478	215	33	)	)	PUNCT
ejpam-4478	215	34	can	can	AUX
ejpam-4478	215	35	be	be	AUX
ejpam-4478	215	36	computed	compute	VERB
ejpam-4478	215	37	as	as	ADP
ejpam-4478	215	38	w(fk	w(fk	NOUN
ejpam-4478	215	39	)	)	PUNCT
ejpam-4478	215	40	=	=	SYM
ejpam-4478	216	1	|v	|v	PROPN
ejpam-4478	216	2	fk	fk	INTJ
ejpam-4478	216	3	1	1	NUM
ejpam-4478	216	4	|+	|+	NOUN
ejpam-4478	217	1	2|v	2|v	NOUN
ejpam-4478	218	1	fk	fk	INTJ
ejpam-4478	218	2	2	2	NUM
ejpam-4478	218	3	|	|	NOUN
ejpam-4478	218	4	.	.	PUNCT
ejpam-4478	219	1	g.	g.	PROPN
ejpam-4478	219	2	entero	entero	PROPN
ejpam-4478	219	3	,	,	PUNCT
ejpam-4478	219	4	s.	s.	PROPN
ejpam-4478	219	5	espinola	espinola	PROPN
ejpam-4478	219	6	/	/	SYM
ejpam-4478	219	7	eur	eur	PROPN
ejpam-4478	219	8	.	.	PUNCT
ejpam-4478	220	1	j.	j.	PROPN
ejpam-4478	220	2	pure	pure	PROPN
ejpam-4478	220	3	appl	appl	PROPN
ejpam-4478	220	4	.	.	PROPN
ejpam-4478	220	5	math	math	PROPN
ejpam-4478	220	6	,	,	PUNCT
ejpam-4478	220	7	16	16	NUM
ejpam-4478	220	8	(	(	PUNCT
ejpam-4478	220	9	1	1	NUM
ejpam-4478	220	10	)	)	PUNCT
ejpam-4478	220	11	(	(	PUNCT
ejpam-4478	220	12	2023	2023	NUM
ejpam-4478	220	13	)	)	PUNCT
ejpam-4478	220	14	,	,	PUNCT
ejpam-4478	220	15	44	44	NUM
ejpam-4478	220	16	-	-	SYM
ejpam-4478	220	17	61	61	NUM
ejpam-4478	220	18	51	51	NUM
ejpam-4478	220	19	proof	proof	NOUN
ejpam-4478	220	20	.	.	PUNCT
ejpam-4478	221	1	let	let	VERB
ejpam-4478	221	2	k	k	PROPN
ejpam-4478	221	3	∈	∈	PROPN
ejpam-4478	221	4	z+	z+	PUNCT
ejpam-4478	221	5	.	.	PUNCT
ejpam-4478	222	1	we	we	PRON
ejpam-4478	222	2	suppose	suppose	VERB
ejpam-4478	222	3	that	that	SCONJ
ejpam-4478	222	4	fk	fk	PRON
ejpam-4478	222	5	can	can	AUX
ejpam-4478	222	6	be	be	AUX
ejpam-4478	222	7	written	write	VERB
ejpam-4478	222	8	as	as	ADP
ejpam-4478	222	9	fk	fk	INTJ
ejpam-4478	222	10	=	=	PUNCT
ejpam-4478	222	11	(	(	PUNCT
ejpam-4478	222	12	v	v	NUM
ejpam-4478	222	13	fk	fk	INTJ
ejpam-4478	222	14	0	0	NUM
ejpam-4478	222	15	,	,	PUNCT
ejpam-4478	222	16	v	v	NOUN
ejpam-4478	222	17	fk	fk	INTJ
ejpam-4478	222	18	1	1	NUM
ejpam-4478	222	19	,	,	PUNCT
ejpam-4478	222	20	v	v	NOUN
ejpam-4478	222	21	fk	fk	INTJ
ejpam-4478	222	22	2	2	NUM
ejpam-4478	222	23	)	)	PUNCT
ejpam-4478	222	24	.	.	PUNCT
ejpam-4478	223	1	now	now	ADV
ejpam-4478	223	2	,	,	PUNCT
ejpam-4478	223	3	for	for	ADP
ejpam-4478	223	4	all	all	PRON
ejpam-4478	223	5	k	k	PROPN
ejpam-4478	223	6	∈	∈	PROPN
ejpam-4478	223	7	z+	z+	X
ejpam-4478	223	8	,	,	PUNCT
ejpam-4478	223	9	since	since	SCONJ
ejpam-4478	223	10	v	v	NUM
ejpam-4478	223	11	=	=	SYM
ejpam-4478	223	12	v	v	NOUN
ejpam-4478	223	13	fk	fk	INTJ
ejpam-4478	223	14	0	0	SYM
ejpam-4478	223	15	∪v	∪v	NOUN
ejpam-4478	223	16	fk	fk	INTJ
ejpam-4478	223	17	1	1	NUM
ejpam-4478	223	18	∪v	∪v	NOUN
ejpam-4478	223	19	fk	fk	INTJ
ejpam-4478	223	20	2	2	NUM
ejpam-4478	223	21	with	with	ADP
ejpam-4478	223	22	|v	|v	PROPN
ejpam-4478	223	23	|	|	ADV
ejpam-4478	223	24	=	=	SYM
ejpam-4478	223	25	|v	|v	PROPN
ejpam-4478	223	26	fk	fk	INTJ
ejpam-4478	223	27	0	0	NUM
ejpam-4478	223	28	|+	|+	NOUN
ejpam-4478	223	29	|v	|v	PROPN
ejpam-4478	223	30	fk	fk	INTJ
ejpam-4478	223	31	1	1	NUM
ejpam-4478	223	32	|+	|+	NOUN
ejpam-4478	223	33	|v	|v	PROPN
ejpam-4478	223	34	fk	fk	INTJ
ejpam-4478	223	35	2	2	NUM
ejpam-4478	223	36	|	|	ADV
ejpam-4478	223	37	and	and	CCONJ
ejpam-4478	223	38	since	since	ADV
ejpam-4478	223	39	,	,	PUNCT
ejpam-4478	223	40	from	from	ADP
ejpam-4478	223	41	definition	definition	NOUN
ejpam-4478	223	42	3.8	3.8	NUM
ejpam-4478	223	43	,	,	PUNCT
ejpam-4478	223	44	w(fk	w(fk	NOUN
ejpam-4478	223	45	)	)	PUNCT
ejpam-4478	223	46	=	=	SYM
ejpam-4478	223	47	∑	∑	PUNCT
ejpam-4478	223	48	uj∈v	uj∈v	NOUN
ejpam-4478	223	49	fk(uj	fk(uj	PROPN
ejpam-4478	223	50	)	)	PUNCT
ejpam-4478	223	51	,	,	PUNCT
ejpam-4478	223	52	we	we	PRON
ejpam-4478	223	53	have	have	VERB
ejpam-4478	223	54	w(fk	w(fk	NOUN
ejpam-4478	223	55	)	)	PUNCT
ejpam-4478	223	56	=	=	SYM
ejpam-4478	224	1	∑	∑	PUNCT
ejpam-4478	224	2	uj∈v	uj∈v	NOUN
ejpam-4478	224	3	fk(uj	fk(uj	NOUN
ejpam-4478	224	4	)	)	PUNCT
ejpam-4478	224	5	=	=	PUNCT
ejpam-4478	225	1	∑	∑	PUNCT
ejpam-4478	225	2	uj∈v	uj∈v	X
ejpam-4478	226	1	fk	fk	INTJ
ejpam-4478	226	2	0	0	PUNCT
ejpam-4478	226	3	fk(uj	fk(uj	NOUN
ejpam-4478	226	4	)	)	PUNCT
ejpam-4478	227	1	+	+	CCONJ
ejpam-4478	227	2	∑	∑	PUNCT
ejpam-4478	227	3	uj∈v	uj∈v	NOUN
ejpam-4478	227	4	fk	fk	INTJ
ejpam-4478	227	5	1	1	NUM
ejpam-4478	227	6	fk(uj	fk(uj	NOUN
ejpam-4478	227	7	)	)	PUNCT
ejpam-4478	228	1	+	+	CCONJ
ejpam-4478	228	2	∑	∑	PUNCT
ejpam-4478	228	3	uj∈v	uj∈v	NOUN
ejpam-4478	228	4	fk	fk	INTJ
ejpam-4478	228	5	2	2	NUM
ejpam-4478	228	6	fk(uj	fk(uj	NOUN
ejpam-4478	228	7	)	)	PUNCT
ejpam-4478	229	1	=	=	SYM
ejpam-4478	229	2	0	0	PUNCT
ejpam-4478	230	1	+	+	CCONJ
ejpam-4478	230	2	(	(	PUNCT
ejpam-4478	230	3	1)(|v	1)(|v	NUM
ejpam-4478	230	4	fk	fk	INTJ
ejpam-4478	230	5	1	1	NUM
ejpam-4478	230	6	|	|	ADV
ejpam-4478	230	7	)	)	PUNCT
ejpam-4478	230	8	+	+	CCONJ
ejpam-4478	230	9	(	(	PUNCT
ejpam-4478	230	10	2)(|v	2)(|v	NOUN
ejpam-4478	230	11	fk	fk	VERB
ejpam-4478	230	12	2	2	NUM
ejpam-4478	230	13	|	|	NOUN
ejpam-4478	230	14	)	)	PUNCT
ejpam-4478	230	15	=	=	SYM
ejpam-4478	231	1	|v	|v	PROPN
ejpam-4478	231	2	fk	fk	INTJ
ejpam-4478	231	3	1	1	NUM
ejpam-4478	231	4	|+	|+	NOUN
ejpam-4478	232	1	2|v	2|v	NOUN
ejpam-4478	233	1	fk	fk	INTJ
ejpam-4478	233	2	2	2	NUM
ejpam-4478	233	3	|	|	NOUN
ejpam-4478	233	4	.	.	PUNCT
ejpam-4478	234	1	that	that	PRON
ejpam-4478	234	2	is	be	AUX
ejpam-4478	234	3	,	,	PUNCT
ejpam-4478	234	4	w(fk	w(fk	NOUN
ejpam-4478	234	5	)	)	PUNCT
ejpam-4478	234	6	=	=	SYM
ejpam-4478	235	1	|v	|v	PROPN
ejpam-4478	235	2	fk	fk	INTJ
ejpam-4478	235	3	1	1	NUM
ejpam-4478	235	4	|+2|v	|+2|v	NOUN
ejpam-4478	235	5	fk	fk	INTJ
ejpam-4478	235	6	2	2	NUM
ejpam-4478	235	7	|	|	ADV
ejpam-4478	235	8	,	,	PUNCT
ejpam-4478	235	9	for	for	ADP
ejpam-4478	235	10	all	all	DET
ejpam-4478	235	11	k	k	PROPN
ejpam-4478	235	12	∈	∈	PROPN
ejpam-4478	235	13	z+	z+	PUNCT
ejpam-4478	235	14	.	.	PUNCT
ejpam-4478	236	1	therefore	therefore	ADV
ejpam-4478	236	2	,	,	PUNCT
ejpam-4478	236	3	for	for	ADP
ejpam-4478	236	4	all	all	DET
ejpam-4478	236	5	k	k	PROPN
ejpam-4478	236	6	∈	∈	PROPN
ejpam-4478	236	7	z+	z+	X
ejpam-4478	236	8	,	,	PUNCT
ejpam-4478	236	9	if	if	SCONJ
ejpam-4478	236	10	fk	fk	PRON
ejpam-4478	236	11	can	can	AUX
ejpam-4478	236	12	be	be	AUX
ejpam-4478	236	13	written	write	VERB
ejpam-4478	236	14	as	as	ADP
ejpam-4478	236	15	fk	fk	INTJ
ejpam-4478	236	16	=	=	PUNCT
ejpam-4478	236	17	(	(	PUNCT
ejpam-4478	236	18	v	v	NUM
ejpam-4478	236	19	fk	fk	INTJ
ejpam-4478	236	20	0	0	NUM
ejpam-4478	236	21	,	,	PUNCT
ejpam-4478	236	22	v	v	NOUN
ejpam-4478	236	23	fk	fk	INTJ
ejpam-4478	236	24	1	1	NUM
ejpam-4478	236	25	,	,	PUNCT
ejpam-4478	236	26	v	v	NOUN
ejpam-4478	236	27	fk	fk	INTJ
ejpam-4478	236	28	2	2	NUM
ejpam-4478	236	29	)	)	PUNCT
ejpam-4478	236	30	,	,	PUNCT
ejpam-4478	236	31	then	then	ADV
ejpam-4478	236	32	its	its	PRON
ejpam-4478	236	33	weight	weight	NOUN
ejpam-4478	236	34	w(fk	w(fk	NOUN
ejpam-4478	236	35	)	)	PUNCT
ejpam-4478	236	36	can	can	AUX
ejpam-4478	236	37	be	be	AUX
ejpam-4478	236	38	computed	compute	VERB
ejpam-4478	236	39	as	as	ADP
ejpam-4478	236	40	w(fk	w(fk	NOUN
ejpam-4478	236	41	)	)	PUNCT
ejpam-4478	236	42	=	=	SYM
ejpam-4478	236	43	|v	|v	PROPN
ejpam-4478	236	44	fk	fk	INTJ
ejpam-4478	236	45	1	1	NUM
ejpam-4478	236	46	|+2|v	|+2|v	NOUN
ejpam-4478	236	47	fk	fk	INTJ
ejpam-4478	236	48	2	2	NUM
ejpam-4478	236	49	|	|	NOUN
ejpam-4478	236	50	.	.	PUNCT
ejpam-4478	237	1	remark	remark	PROPN
ejpam-4478	237	2	4.4	4.4	NUM
ejpam-4478	237	3	.	.	PUNCT
ejpam-4478	238	1	let	let	VERB
ejpam-4478	238	2	k	k	PROPN
ejpam-4478	238	3	∈	∈	PROPN
ejpam-4478	238	4	z+	z+	PUNCT
ejpam-4478	238	5	.	.	PUNCT
ejpam-4478	239	1	for	for	ADP
ejpam-4478	239	2	any	any	DET
ejpam-4478	239	3	graph	graph	NOUN
ejpam-4478	239	4	g	g	PROPN
ejpam-4478	239	5	,	,	PUNCT
ejpam-4478	239	6	γkr(g	γkr(g	PROPN
ejpam-4478	239	7	)	)	PUNCT
ejpam-4478	239	8	≤	≤	NOUN
ejpam-4478	239	9	γkgr(g	γkgr(g	NOUN
ejpam-4478	239	10	)	)	PUNCT
ejpam-4478	239	11	.	.	PUNCT
ejpam-4478	240	1	proof	proof	NOUN
ejpam-4478	240	2	.	.	PUNCT
ejpam-4478	241	1	let	let	VERB
ejpam-4478	241	2	f	f	PRON
ejpam-4478	241	3	be	be	AUX
ejpam-4478	241	4	a	a	DET
ejpam-4478	241	5	γkgr(g	γkgr(g	NOUN
ejpam-4478	241	6	)	)	PUNCT
ejpam-4478	241	7	−	−	PROPN
ejpam-4478	242	1	function	function	NOUN
ejpam-4478	242	2	of	of	ADP
ejpam-4478	242	3	g	g	NOUN
ejpam-4478	243	1	and	and	CCONJ
ejpam-4478	243	2	let	let	VERB
ejpam-4478	243	3	k	k	PROPN
ejpam-4478	243	4	∈	∈	PROPN
ejpam-4478	243	5	z+	z+	PUNCT
ejpam-4478	243	6	.	.	PUNCT
ejpam-4478	244	1	then	then	ADV
ejpam-4478	244	2	f	f	PROPN
ejpam-4478	244	3	is	be	AUX
ejpam-4478	244	4	a	a	DET
ejpam-4478	244	5	k	k	NOUN
ejpam-4478	244	6	−	−	PROPN
ejpam-4478	244	7	distance	distance	NOUN
ejpam-4478	244	8	roman	roman	ADJ
ejpam-4478	244	9	dominating	dominating	NOUN
ejpam-4478	244	10	function	function	NOUN
ejpam-4478	244	11	of	of	ADP
ejpam-4478	244	12	g.	g.	PROPN
ejpam-4478	244	13	thus	thus	ADV
ejpam-4478	244	14	,	,	PUNCT
ejpam-4478	244	15	γkr(g	γkr(g	PROPN
ejpam-4478	244	16	)	)	PUNCT
ejpam-4478	244	17	≤	≤	NOUN
ejpam-4478	244	18	γkgr(g	γkgr(g	NOUN
ejpam-4478	244	19	)	)	PUNCT
ejpam-4478	244	20	for	for	ADP
ejpam-4478	244	21	all	all	PRON
ejpam-4478	244	22	k	k	PROPN
ejpam-4478	244	23	∈	∈	PROPN
ejpam-4478	244	24	z+	z+	PUNCT
ejpam-4478	244	25	.	.	PUNCT
ejpam-4478	244	26	remark	remark	PROPN
ejpam-4478	244	27	4.5	4.5	NUM
ejpam-4478	244	28	.	.	PUNCT
ejpam-4478	245	1	for	for	ADP
ejpam-4478	245	2	any	any	DET
ejpam-4478	245	3	graph	graph	NOUN
ejpam-4478	245	4	g	g	NOUN
ejpam-4478	245	5	,	,	PUNCT
ejpam-4478	245	6	γkgr(g	γkgr(g	NOUN
ejpam-4478	245	7	)	)	PUNCT
ejpam-4478	245	8	=	=	SYM
ejpam-4478	245	9	γkgr(g	γkgr(g	NOUN
ejpam-4478	245	10	)	)	PUNCT
ejpam-4478	245	11	,	,	PUNCT
ejpam-4478	245	12	for	for	ADP
ejpam-4478	245	13	all	all	DET
ejpam-4478	245	14	k	k	PROPN
ejpam-4478	245	15	∈	∈	PROPN
ejpam-4478	245	16	z+	z+	PUNCT
ejpam-4478	245	17	.	.	PUNCT
ejpam-4478	245	18	proof	proof	NOUN
ejpam-4478	245	19	.	.	PUNCT
ejpam-4478	246	1	by	by	ADP
ejpam-4478	246	2	saying	say	VERB
ejpam-4478	246	3	global	global	ADJ
ejpam-4478	246	4	,	,	PUNCT
ejpam-4478	246	5	it	it	PRON
ejpam-4478	246	6	constitutes	constitute	VERB
ejpam-4478	246	7	the	the	DET
ejpam-4478	246	8	given	give	VERB
ejpam-4478	246	9	graph	graph	NOUN
ejpam-4478	246	10	,	,	PUNCT
ejpam-4478	246	11	say	say	VERB
ejpam-4478	246	12	graph	graph	NOUN
ejpam-4478	246	13	g	g	NOUN
ejpam-4478	246	14	,	,	PUNCT
ejpam-4478	246	15	together	together	ADV
ejpam-4478	246	16	with	with	ADP
ejpam-4478	246	17	its	its	PRON
ejpam-4478	246	18	complement	complement	NOUN
ejpam-4478	246	19	g.	g.	NOUN
ejpam-4478	246	20	thus	thus	ADV
ejpam-4478	246	21	,	,	PUNCT
ejpam-4478	246	22	for	for	ADP
ejpam-4478	246	23	all	all	DET
ejpam-4478	246	24	k	k	PROPN
ejpam-4478	246	25	∈	∈	PROPN
ejpam-4478	246	26	z+	z+	X
ejpam-4478	246	27	,	,	PUNCT
ejpam-4478	246	28	γkgr(g	γkgr(g	NOUN
ejpam-4478	246	29	)	)	PUNCT
ejpam-4478	246	30	accounts	account	VERB
ejpam-4478	246	31	γkr(g	γkr(g	NUM
ejpam-4478	246	32	)	)	PUNCT
ejpam-4478	246	33	and	and	CCONJ
ejpam-4478	246	34	γkr(g	γkr(g	NUM
ejpam-4478	246	35	)	)	PUNCT
ejpam-4478	246	36	simultaneously	simultaneously	ADV
ejpam-4478	246	37	and	and	CCONJ
ejpam-4478	246	38	also	also	ADV
ejpam-4478	246	39	,	,	PUNCT
ejpam-4478	246	40	since	since	SCONJ
ejpam-4478	246	41	g	g	PROPN
ejpam-4478	246	42	=	=	SYM
ejpam-4478	246	43	g	g	PROPN
ejpam-4478	246	44	,	,	PUNCT
ejpam-4478	246	45	γkgr(g	γkgr(g	NOUN
ejpam-4478	246	46	)	)	PUNCT
ejpam-4478	246	47	accounts	account	VERB
ejpam-4478	246	48	γkr(g	γkr(g	NUM
ejpam-4478	246	49	)	)	PUNCT
ejpam-4478	246	50	and	and	CCONJ
ejpam-4478	246	51	γkr(g	γkr(g	NUM
ejpam-4478	246	52	)	)	PUNCT
ejpam-4478	246	53	=	=	SYM
ejpam-4478	246	54	γkr(g	γkr(g	PROPN
ejpam-4478	246	55	)	)	PUNCT
ejpam-4478	246	56	simultaneously	simultaneously	ADV
ejpam-4478	246	57	.	.	PUNCT
ejpam-4478	247	1	therefore	therefore	ADV
ejpam-4478	247	2	,	,	PUNCT
ejpam-4478	247	3	for	for	ADP
ejpam-4478	247	4	any	any	DET
ejpam-4478	247	5	graph	graph	NOUN
ejpam-4478	247	6	g	g	NOUN
ejpam-4478	247	7	,	,	PUNCT
ejpam-4478	247	8	γkgr(g	γkgr(g	NOUN
ejpam-4478	247	9	)	)	PUNCT
ejpam-4478	247	10	=	=	SYM
ejpam-4478	247	11	γkgr(g	γkgr(g	NOUN
ejpam-4478	247	12	)	)	PUNCT
ejpam-4478	247	13	,	,	PUNCT
ejpam-4478	247	14	for	for	ADP
ejpam-4478	247	15	all	all	DET
ejpam-4478	247	16	k	k	PROPN
ejpam-4478	247	17	∈	∈	PROPN
ejpam-4478	247	18	z+	z+	X
ejpam-4478	247	19	.	.	PROPN
ejpam-4478	247	20	5	5	NUM
ejpam-4478	247	21	.	.	PUNCT
ejpam-4478	248	1	the	the	DET
ejpam-4478	248	2	global	global	ADJ
ejpam-4478	248	3	distance	distance	NOUN
ejpam-4478	248	4	roman	roman	ADJ
ejpam-4478	248	5	domination	domination	NOUN
ejpam-4478	248	6	on	on	ADP
ejpam-4478	248	7	special	special	ADJ
ejpam-4478	248	8	graphs	graph	NOUN
ejpam-4478	248	9	5.1	5.1	NUM
ejpam-4478	248	10	.	.	PUNCT
ejpam-4478	249	1	the	the	DET
ejpam-4478	249	2	global	global	ADJ
ejpam-4478	249	3	distance	distance	NOUN
ejpam-4478	249	4	roman	roman	ADJ
ejpam-4478	249	5	domination	domination	NOUN
ejpam-4478	249	6	on	on	ADP
ejpam-4478	249	7	empty	empty	ADJ
ejpam-4478	249	8	graph	graph	NOUN
ejpam-4478	249	9	kn	kn	PROPN
ejpam-4478	249	10	and	and	CCONJ
ejpam-4478	249	11	on	on	ADP
ejpam-4478	249	12	complete	complete	ADJ
ejpam-4478	249	13	graph	graph	NOUN
ejpam-4478	249	14	kn	kn	PROPN
ejpam-4478	249	15	theorem	theorem	VERB
ejpam-4478	249	16	5.1	5.1	NUM
ejpam-4478	249	17	.	.	PUNCT
ejpam-4478	250	1	let	let	VERB
ejpam-4478	250	2	g	g	PROPN
ejpam-4478	250	3	∼=	∼=	PROPN
ejpam-4478	250	4	kn	kn	PROPN
ejpam-4478	250	5	,	,	PUNCT
ejpam-4478	250	6	where	where	SCONJ
ejpam-4478	250	7	kn	kn	PROPN
ejpam-4478	250	8	is	be	AUX
ejpam-4478	250	9	the	the	DET
ejpam-4478	250	10	null	null	ADJ
ejpam-4478	250	11	graph	graph	NOUN
ejpam-4478	250	12	(	(	PUNCT
ejpam-4478	250	13	empty	empty	ADJ
ejpam-4478	250	14	graph	graph	NOUN
ejpam-4478	250	15	)	)	PUNCT
ejpam-4478	250	16	of	of	ADP
ejpam-4478	250	17	order	order	NOUN
ejpam-4478	250	18	n.	n.	NOUN
ejpam-4478	250	19	then	then	ADV
ejpam-4478	250	20	,	,	PUNCT
ejpam-4478	250	21	for	for	ADP
ejpam-4478	250	22	all	all	PRON
ejpam-4478	250	23	k	k	PROPN
ejpam-4478	250	24	∈	∈	PROPN
ejpam-4478	250	25	z+	z+	X
ejpam-4478	250	26	,	,	PUNCT
ejpam-4478	250	27	γkgr(g	γkgr(g	NOUN
ejpam-4478	250	28	)	)	PUNCT
ejpam-4478	250	29	=	=	SYM
ejpam-4478	250	30	n.	n.	NOUN
ejpam-4478	250	31	proof	proof	NOUN
ejpam-4478	250	32	.	.	PUNCT
ejpam-4478	251	1	assume	assume	VERB
ejpam-4478	251	2	that	that	SCONJ
ejpam-4478	251	3	g	g	PROPN
ejpam-4478	251	4	∼=	∼=	PROPN
ejpam-4478	251	5	kn	kn	NOUN
ejpam-4478	251	6	is	be	AUX
ejpam-4478	251	7	of	of	ADP
ejpam-4478	251	8	order	order	NOUN
ejpam-4478	251	9	n.	n.	NOUN
ejpam-4478	251	10	suppose	suppose	VERB
ejpam-4478	251	11	that	that	SCONJ
ejpam-4478	251	12	we	we	PRON
ejpam-4478	251	13	have	have	VERB
ejpam-4478	251	14	a	a	DET
ejpam-4478	251	15	function	function	NOUN
ejpam-4478	251	16	fk	fk	INTJ
ejpam-4478	251	17	mapping	map	VERB
ejpam-4478	251	18	the	the	DET
ejpam-4478	251	19	vertex	vertex	NOUN
ejpam-4478	251	20	set	set	VERB
ejpam-4478	251	21	v	v	NOUN
ejpam-4478	251	22	(	(	PUNCT
ejpam-4478	251	23	g	g	NOUN
ejpam-4478	251	24	)	)	PUNCT
ejpam-4478	251	25	of	of	ADP
ejpam-4478	251	26	graph	graph	NOUN
ejpam-4478	251	27	g	g	PROPN
ejpam-4478	251	28	to	to	ADP
ejpam-4478	251	29	the	the	DET
ejpam-4478	251	30	set	set	NOUN
ejpam-4478	251	31	{	{	PUNCT
ejpam-4478	251	32	0	0	NUM
ejpam-4478	251	33	,	,	PUNCT
ejpam-4478	251	34	1	1	NUM
ejpam-4478	251	35	,	,	PUNCT
ejpam-4478	251	36	2	2	NUM
ejpam-4478	251	37	}	}	PUNCT
ejpam-4478	251	38	,	,	PUNCT
ejpam-4478	251	39	that	that	ADV
ejpam-4478	251	40	is	is	ADV
ejpam-4478	251	41	,	,	PUNCT
ejpam-4478	251	42	fk	fk	INTJ
ejpam-4478	251	43	:	:	PUNCT
ejpam-4478	251	44	v	v	X
ejpam-4478	251	45	(	(	PUNCT
ejpam-4478	251	46	g	g	NOUN
ejpam-4478	251	47	)	)	PUNCT
ejpam-4478	251	48	−→	−→	NOUN
ejpam-4478	251	49	{	{	PUNCT
ejpam-4478	251	50	0	0	NUM
ejpam-4478	251	51	,	,	PUNCT
ejpam-4478	251	52	1	1	NUM
ejpam-4478	251	53	,	,	PUNCT
ejpam-4478	251	54	2	2	NUM
ejpam-4478	251	55	}	}	PUNCT
ejpam-4478	251	56	,	,	PUNCT
ejpam-4478	251	57	for	for	ADP
ejpam-4478	251	58	all	all	PRON
ejpam-4478	251	59	k	k	PROPN
ejpam-4478	251	60	∈	∈	PROPN
ejpam-4478	251	61	z+	z+	PUNCT
ejpam-4478	251	62	.	.	PUNCT
ejpam-4478	252	1	let	let	VERB
ejpam-4478	252	2	fk	fk	INTJ
ejpam-4478	252	3	be	be	AUX
ejpam-4478	252	4	defined	define	VERB
ejpam-4478	252	5	by	by	ADP
ejpam-4478	252	6	fk(ui	fk(ui	PROPN
ejpam-4478	252	7	)	)	PUNCT
ejpam-4478	253	1	=	=	SYM
ejpam-4478	253	2	1	1	NUM
ejpam-4478	253	3	,	,	PUNCT
ejpam-4478	253	4	for	for	ADP
ejpam-4478	253	5	all	all	DET
ejpam-4478	253	6	ui	ui	NOUN
ejpam-4478	253	7	∈	∈	PROPN
ejpam-4478	253	8	v	v	NOUN
ejpam-4478	253	9	(	(	PUNCT
ejpam-4478	253	10	g	g	NOUN
ejpam-4478	253	11	)	)	PUNCT
ejpam-4478	253	12	.	.	PUNCT
ejpam-4478	254	1	this	this	DET
ejpam-4478	254	2	mapping	mapping	NOUN
ejpam-4478	254	3	will	will	AUX
ejpam-4478	254	4	give	give	VERB
ejpam-4478	254	5	a	a	DET
ejpam-4478	254	6	weight	weight	NOUN
ejpam-4478	254	7	of	of	ADP
ejpam-4478	254	8	w(fk	w(fk	NOUN
ejpam-4478	254	9	)	)	PUNCT
ejpam-4478	254	10	=	=	SYM
ejpam-4478	254	11	n	n	CCONJ
ejpam-4478	254	12	,	,	PUNCT
ejpam-4478	254	13	for	for	ADP
ejpam-4478	254	14	all	all	DET
ejpam-4478	254	15	k	k	PROPN
ejpam-4478	254	16	∈	∈	PROPN
ejpam-4478	254	17	z+	z+	NUM
ejpam-4478	254	18	and	and	CCONJ
ejpam-4478	254	19	this	this	PRON
ejpam-4478	254	20	is	be	AUX
ejpam-4478	254	21	minimum	minimum	ADJ
ejpam-4478	254	22	.	.	PUNCT
ejpam-4478	255	1	this	this	PRON
ejpam-4478	255	2	can	can	AUX
ejpam-4478	255	3	be	be	AUX
ejpam-4478	255	4	easily	easily	ADV
ejpam-4478	255	5	verified	verify	VERB
ejpam-4478	255	6	.	.	PUNCT
ejpam-4478	256	1	corollary	corollary	ADJ
ejpam-4478	256	2	5.2	5.2	NUM
ejpam-4478	256	3	.	.	PUNCT
ejpam-4478	257	1	let	let	VERB
ejpam-4478	257	2	g	g	PROPN
ejpam-4478	257	3	∼=	∼=	PROPN
ejpam-4478	257	4	kn	kn	PROPN
ejpam-4478	257	5	,	,	PUNCT
ejpam-4478	257	6	where	where	SCONJ
ejpam-4478	257	7	kn	kn	PROPN
ejpam-4478	257	8	is	be	AUX
ejpam-4478	257	9	the	the	DET
ejpam-4478	257	10	complete	complete	ADJ
ejpam-4478	257	11	graph	graph	NOUN
ejpam-4478	257	12	of	of	ADP
ejpam-4478	257	13	order	order	NOUN
ejpam-4478	257	14	n.	n.	NOUN
ejpam-4478	257	15	then	then	ADV
ejpam-4478	257	16	,	,	PUNCT
ejpam-4478	257	17	for	for	ADP
ejpam-4478	257	18	any	any	DET
ejpam-4478	257	19	k	k	PROPN
ejpam-4478	257	20	∈	∈	PROPN
ejpam-4478	257	21	z+	z+	X
ejpam-4478	257	22	,	,	PUNCT
ejpam-4478	257	23	γkgr(g	γkgr(g	NOUN
ejpam-4478	257	24	)	)	PUNCT
ejpam-4478	257	25	=	=	SYM
ejpam-4478	258	1	n.	n.	NOUN
ejpam-4478	258	2	proof	proof	NOUN
ejpam-4478	258	3	.	.	PUNCT
ejpam-4478	259	1	the	the	DET
ejpam-4478	259	2	proof	proof	NOUN
ejpam-4478	259	3	follows	follow	VERB
ejpam-4478	259	4	from	from	ADP
ejpam-4478	259	5	theorem	theorem	ADJ
ejpam-4478	259	6	5.1	5.1	NUM
ejpam-4478	259	7	.	.	PUNCT
ejpam-4478	260	1	g.	g.	PROPN
ejpam-4478	260	2	entero	entero	PROPN
ejpam-4478	260	3	,	,	PUNCT
ejpam-4478	260	4	s.	s.	PROPN
ejpam-4478	260	5	espinola	espinola	PROPN
ejpam-4478	260	6	/	/	SYM
ejpam-4478	260	7	eur	eur	PROPN
ejpam-4478	260	8	.	.	PUNCT
ejpam-4478	261	1	j.	j.	PROPN
ejpam-4478	261	2	pure	pure	PROPN
ejpam-4478	261	3	appl	appl	PROPN
ejpam-4478	261	4	.	.	PROPN
ejpam-4478	261	5	math	math	PROPN
ejpam-4478	261	6	,	,	PUNCT
ejpam-4478	261	7	16	16	NUM
ejpam-4478	261	8	(	(	PUNCT
ejpam-4478	261	9	1	1	NUM
ejpam-4478	261	10	)	)	PUNCT
ejpam-4478	261	11	(	(	PUNCT
ejpam-4478	261	12	2023	2023	NUM
ejpam-4478	261	13	)	)	PUNCT
ejpam-4478	261	14	,	,	PUNCT
ejpam-4478	261	15	44	44	NUM
ejpam-4478	261	16	-	-	SYM
ejpam-4478	261	17	61	61	NUM
ejpam-4478	261	18	52	52	NUM
ejpam-4478	261	19	5.2	5.2	NUM
ejpam-4478	261	20	.	.	PUNCT
ejpam-4478	262	1	the	the	DET
ejpam-4478	262	2	global	global	ADJ
ejpam-4478	262	3	distance	distance	NOUN
ejpam-4478	262	4	roman	roman	ADJ
ejpam-4478	262	5	domination	domination	NOUN
ejpam-4478	262	6	on	on	ADP
ejpam-4478	262	7	path	path	NOUN
ejpam-4478	262	8	graph	graph	NOUN
ejpam-4478	262	9	pn	pn	PROPN
ejpam-4478	262	10	proposition	proposition	NOUN
ejpam-4478	262	11	5.3	5.3	NUM
ejpam-4478	262	12	.	.	PUNCT
ejpam-4478	263	1	let	let	VERB
ejpam-4478	263	2	g	g	PRON
ejpam-4478	263	3	∼=	∼=	PROPN
ejpam-4478	263	4	pn	pn	NOUN
ejpam-4478	263	5	,	,	PUNCT
ejpam-4478	263	6	where	where	SCONJ
ejpam-4478	263	7	pn	pn	PROPN
ejpam-4478	263	8	is	be	AUX
ejpam-4478	263	9	the	the	DET
ejpam-4478	263	10	path	path	NOUN
ejpam-4478	263	11	graph	graph	NOUN
ejpam-4478	263	12	of	of	ADP
ejpam-4478	263	13	order	order	NOUN
ejpam-4478	263	14	n.	n.	NOUN
ejpam-4478	263	15	for	for	ADP
ejpam-4478	263	16	all	all	DET
ejpam-4478	263	17	k	k	PROPN
ejpam-4478	263	18	∈	∈	PROPN
ejpam-4478	263	19	z+	z+	X
ejpam-4478	263	20	,	,	PUNCT
ejpam-4478	263	21	γkgr(g	γkgr(g	NOUN
ejpam-4478	263	22	)	)	PUNCT
ejpam-4478	263	23	=	=	SYM
ejpam-4478	263	24			VERB
ejpam-4478	263	25	for	for	ADP
ejpam-4478	263	26	n	n	DET
ejpam-4478	263	27	≤	≤	NOUN
ejpam-4478	263	28	4	4	NUM
ejpam-4478	263	29	:	:	PUNCT
ejpam-4478	263	30	{	{	PUNCT
ejpam-4478	263	31	n	n	CCONJ
ejpam-4478	263	32	,	,	PUNCT
ejpam-4478	263	33	if	if	SCONJ
ejpam-4478	263	34	n	n	CCONJ
ejpam-4478	263	35	=	=	SYM
ejpam-4478	263	36	1	1	NUM
ejpam-4478	263	37	,	,	PUNCT
ejpam-4478	263	38	2	2	NUM
ejpam-4478	263	39	,	,	PUNCT
ejpam-4478	263	40	3	3	NUM
ejpam-4478	263	41	and	and	CCONJ
ejpam-4478	263	42	∀	∀	NOUN
ejpam-4478	264	1	k	k	X
ejpam-4478	264	2	∈	∈	PROPN
ejpam-4478	264	3	z+	z+	NUM
ejpam-4478	264	4	⌊	⌊	PROPN
ejpam-4478	264	5	4	4	NUM
ejpam-4478	264	6	2k⌋+	2k⌋+	NUM
ejpam-4478	264	7	2	2	NUM
ejpam-4478	264	8	,	,	PUNCT
ejpam-4478	264	9	if	if	SCONJ
ejpam-4478	264	10	n	n	NOUN
ejpam-4478	264	11	=	=	SYM
ejpam-4478	264	12	4	4	NUM
ejpam-4478	264	13	and	and	CCONJ
ejpam-4478	264	14	∀	∀	NOUN
ejpam-4478	264	15	k	k	PROPN
ejpam-4478	264	16	∈	∈	PROPN
ejpam-4478	264	17	z+	z+	NUM
ejpam-4478	264	18	for	for	ADP
ejpam-4478	264	19	n	n	X
ejpam-4478	264	20	>	>	X
ejpam-4478	264	21	4	4	NUM
ejpam-4478	264	22	:	:	PUNCT
ejpam-4478	264	23			NOUN
ejpam-4478	264	24	2⌊	2⌊	NUM
ejpam-4478	264	25	n	n	DET
ejpam-4478	264	26	∆k+1⌋	∆k+1⌋	NOUN
ejpam-4478	264	27	,	,	PUNCT
ejpam-4478	264	28	if	if	SCONJ
ejpam-4478	264	29	n	n	PRON
ejpam-4478	264	30	≡	≡	PROPN
ejpam-4478	264	31	0	0	PUNCT
ejpam-4478	264	32	(	(	PUNCT
ejpam-4478	264	33	mod	mod	PROPN
ejpam-4478	264	34	(	(	PUNCT
ejpam-4478	264	35	∆k	∆k	PROPN
ejpam-4478	264	36	+	+	PROPN
ejpam-4478	264	37	1	1	NUM
ejpam-4478	264	38	)	)	PUNCT
ejpam-4478	264	39	)	)	PUNCT
ejpam-4478	264	40	and	and	CCONJ
ejpam-4478	264	41	1	1	NUM
ejpam-4478	264	42	≤	≤	NUM
ejpam-4478	265	1	k	k	X
ejpam-4478	265	2	<	<	X
ejpam-4478	265	3	rad(g	rad(g	PROPN
ejpam-4478	265	4	)	)	PUNCT
ejpam-4478	265	5	2⌊	2⌊	NUM
ejpam-4478	266	1	n	n	NUM
ejpam-4478	266	2	∆k+1⌋+	∆k+1⌋+	PROPN
ejpam-4478	266	3	1	1	NUM
ejpam-4478	266	4	,	,	PUNCT
ejpam-4478	266	5	if	if	SCONJ
ejpam-4478	266	6	n	n	PRON
ejpam-4478	266	7	≡	≡	PROPN
ejpam-4478	266	8	1	1	NUM
ejpam-4478	266	9	(	(	PUNCT
ejpam-4478	266	10	mod	mod	NOUN
ejpam-4478	266	11	(	(	PUNCT
ejpam-4478	266	12	∆k	∆k	PROPN
ejpam-4478	266	13	+	+	PROPN
ejpam-4478	266	14	1	1	NUM
ejpam-4478	266	15	)	)	PUNCT
ejpam-4478	266	16	)	)	PUNCT
ejpam-4478	266	17	and	and	CCONJ
ejpam-4478	266	18	1	1	NUM
ejpam-4478	266	19	≤	≤	NUM
ejpam-4478	267	1	k	k	X
ejpam-4478	267	2	<	<	X
ejpam-4478	267	3	rad(g	rad(g	PROPN
ejpam-4478	267	4	)	)	PUNCT
ejpam-4478	267	5	2⌊	2⌊	NUM
ejpam-4478	268	1	n	n	NUM
ejpam-4478	268	2	∆k+1⌋+	∆k+1⌋+	PROPN
ejpam-4478	268	3	2	2	NUM
ejpam-4478	268	4	,	,	PUNCT
ejpam-4478	268	5	if	if	SCONJ
ejpam-4478	268	6	n	n	NOUN
ejpam-4478	268	7	̸≡	̸≡	VERB
ejpam-4478	268	8	0	0	NUM
ejpam-4478	268	9	,	,	PUNCT
ejpam-4478	268	10	1	1	NUM
ejpam-4478	268	11	(	(	PUNCT
ejpam-4478	268	12	mod	mod	NOUN
ejpam-4478	268	13	(	(	PUNCT
ejpam-4478	268	14	∆k	∆k	PROPN
ejpam-4478	268	15	+	+	PROPN
ejpam-4478	268	16	1	1	NUM
ejpam-4478	268	17	)	)	PUNCT
ejpam-4478	268	18	)	)	PUNCT
ejpam-4478	268	19	and	and	CCONJ
ejpam-4478	268	20	1	1	NUM
ejpam-4478	268	21	≤	≤	NUM
ejpam-4478	269	1	k	k	X
ejpam-4478	269	2	<	<	X
ejpam-4478	269	3	rad(g	rad(g	PROPN
ejpam-4478	269	4	)	)	PUNCT
ejpam-4478	269	5	2	2	NUM
ejpam-4478	269	6	,	,	PUNCT
ejpam-4478	269	7	if	if	SCONJ
ejpam-4478	269	8	k	k	PROPN
ejpam-4478	269	9	≥	≥	NUM
ejpam-4478	269	10	rad(g	rad(g	NUM
ejpam-4478	269	11	)	)	PUNCT
ejpam-4478	269	12	.	.	PUNCT
ejpam-4478	270	1	5.3	5.3	NUM
ejpam-4478	270	2	.	.	PUNCT
ejpam-4478	271	1	the	the	DET
ejpam-4478	271	2	global	global	ADJ
ejpam-4478	271	3	distance	distance	NOUN
ejpam-4478	271	4	roman	roman	ADJ
ejpam-4478	271	5	domination	domination	NOUN
ejpam-4478	271	6	on	on	ADP
ejpam-4478	271	7	cycle	cycle	NOUN
ejpam-4478	271	8	graph	graph	NOUN
ejpam-4478	271	9	cn	cn	PROPN
ejpam-4478	271	10	proposition	proposition	NOUN
ejpam-4478	271	11	5.4	5.4	NUM
ejpam-4478	271	12	.	.	PUNCT
ejpam-4478	272	1	let	let	VERB
ejpam-4478	272	2	g	g	PRON
ejpam-4478	272	3	∼=	∼=	PROPN
ejpam-4478	272	4	cn	cn	PROPN
ejpam-4478	272	5	,	,	PUNCT
ejpam-4478	272	6	where	where	SCONJ
ejpam-4478	272	7	cn	cn	PROPN
ejpam-4478	272	8	is	be	AUX
ejpam-4478	272	9	the	the	DET
ejpam-4478	272	10	cycle	cycle	NOUN
ejpam-4478	272	11	graph	graph	NOUN
ejpam-4478	272	12	of	of	ADP
ejpam-4478	272	13	order	order	NOUN
ejpam-4478	272	14	n	n	PRON
ejpam-4478	272	15	≥	≥	NOUN
ejpam-4478	272	16	3	3	NUM
ejpam-4478	272	17	.	.	PUNCT
ejpam-4478	273	1	for	for	ADP
ejpam-4478	273	2	all	all	DET
ejpam-4478	273	3	k	k	PROPN
ejpam-4478	273	4	∈	∈	PROPN
ejpam-4478	273	5	z+	z+	X
ejpam-4478	273	6	,	,	PUNCT
ejpam-4478	273	7	γkgr(g	γkgr(g	NOUN
ejpam-4478	273	8	)	)	PUNCT
ejpam-4478	273	9	=	=	PUNCT
ejpam-4478	273	10			NUM
ejpam-4478	273	11	for	for	ADP
ejpam-4478	273	12	n	n	DET
ejpam-4478	273	13	≤	≤	NOUN
ejpam-4478	273	14	5	5	NUM
ejpam-4478	273	15	:	:	PUNCT
ejpam-4478	273	16			PROPN
ejpam-4478	273	17	n	n	CCONJ
ejpam-4478	273	18	,	,	PUNCT
ejpam-4478	273	19	if	if	SCONJ
ejpam-4478	273	20	n	n	X
ejpam-4478	273	21	=	=	SYM
ejpam-4478	273	22	3	3	NUM
ejpam-4478	273	23	,	,	PUNCT
ejpam-4478	273	24	4	4	NUM
ejpam-4478	273	25	and	and	CCONJ
ejpam-4478	273	26	∀	∀	NUM
ejpam-4478	273	27	k	k	PROPN
ejpam-4478	273	28	∈	∈	PROPN
ejpam-4478	273	29	z+	z+	NUM
ejpam-4478	273	30	n	n	CCONJ
ejpam-4478	273	31	,	,	PUNCT
ejpam-4478	273	32	if	if	SCONJ
ejpam-4478	273	33	n	n	CCONJ
ejpam-4478	273	34	=	=	SYM
ejpam-4478	273	35	5	5	NUM
ejpam-4478	273	36	and	and	CCONJ
ejpam-4478	273	37	k	k	NOUN
ejpam-4478	273	38	=	=	NOUN
ejpam-4478	273	39	1	1	NUM
ejpam-4478	273	40	2	2	NUM
ejpam-4478	273	41	,	,	PUNCT
ejpam-4478	273	42	otherwise	otherwise	ADV
ejpam-4478	273	43	for	for	ADP
ejpam-4478	273	44	n	n	X
ejpam-4478	273	45	>	>	SYM
ejpam-4478	273	46	5	5	NUM
ejpam-4478	273	47	:	:	PUNCT
ejpam-4478	273	48			NOUN
ejpam-4478	273	49	2n	2n	NUM
ejpam-4478	273	50	2k+1	2k+1	NOUN
ejpam-4478	273	51	,	,	PUNCT
ejpam-4478	273	52	if	if	SCONJ
ejpam-4478	273	53	n	n	PRON
ejpam-4478	273	54	≡	≡	PROPN
ejpam-4478	273	55	0	0	PUNCT
ejpam-4478	273	56	(	(	PUNCT
ejpam-4478	273	57	mod	mod	X
ejpam-4478	273	58	(	(	PUNCT
ejpam-4478	273	59	2k	2k	NOUN
ejpam-4478	273	60	+	+	CCONJ
ejpam-4478	273	61	1	1	NUM
ejpam-4478	273	62	)	)	PUNCT
ejpam-4478	273	63	)	)	PUNCT
ejpam-4478	273	64	and	and	CCONJ
ejpam-4478	273	65	1	1	NUM
ejpam-4478	273	66	≤	≤	NUM
ejpam-4478	273	67	k	k	X
ejpam-4478	273	68	<	<	X
ejpam-4478	273	69	diam(g	diam(g	PROPN
ejpam-4478	273	70	)	)	PUNCT
ejpam-4478	273	71	2⌊	2⌊	NUM
ejpam-4478	273	72	n	n	CCONJ
ejpam-4478	273	73	2k+1⌋+	2k+1⌋+	NUM
ejpam-4478	273	74	1	1	NUM
ejpam-4478	273	75	,	,	PUNCT
ejpam-4478	273	76	if	if	SCONJ
ejpam-4478	273	77	n	n	PRON
ejpam-4478	273	78	≡	≡	PROPN
ejpam-4478	273	79	1	1	NUM
ejpam-4478	273	80	(	(	PUNCT
ejpam-4478	273	81	mod	mod	X
ejpam-4478	273	82	(	(	PUNCT
ejpam-4478	273	83	2k	2k	NOUN
ejpam-4478	273	84	+	+	CCONJ
ejpam-4478	273	85	1	1	NUM
ejpam-4478	273	86	)	)	PUNCT
ejpam-4478	273	87	)	)	PUNCT
ejpam-4478	273	88	and	and	CCONJ
ejpam-4478	273	89	1	1	NUM
ejpam-4478	273	90	≤	≤	NUM
ejpam-4478	273	91	k	k	X
ejpam-4478	273	92	<	<	X
ejpam-4478	273	93	diam(g	diam(g	PROPN
ejpam-4478	273	94	)	)	PUNCT
ejpam-4478	273	95	2⌊	2⌊	NUM
ejpam-4478	273	96	n	n	CCONJ
ejpam-4478	273	97	2k+1⌋+	2k+1⌋+	NUM
ejpam-4478	273	98	2	2	NUM
ejpam-4478	273	99	,	,	PUNCT
ejpam-4478	273	100	if	if	SCONJ
ejpam-4478	273	101	n	n	NOUN
ejpam-4478	273	102	̸≡	̸≡	VERB
ejpam-4478	273	103	0	0	NUM
ejpam-4478	273	104	,	,	PUNCT
ejpam-4478	273	105	1	1	NUM
ejpam-4478	273	106	(	(	PUNCT
ejpam-4478	273	107	mod	mod	X
ejpam-4478	273	108	(	(	PUNCT
ejpam-4478	273	109	2k	2k	NOUN
ejpam-4478	273	110	+	+	CCONJ
ejpam-4478	273	111	1	1	NUM
ejpam-4478	273	112	)	)	PUNCT
ejpam-4478	273	113	)	)	PUNCT
ejpam-4478	273	114	and	and	CCONJ
ejpam-4478	273	115	1	1	NUM
ejpam-4478	273	116	≤	≤	NUM
ejpam-4478	273	117	k	k	X
ejpam-4478	273	118	<	<	X
ejpam-4478	273	119	diam(g	diam(g	PROPN
ejpam-4478	273	120	)	)	PUNCT
ejpam-4478	273	121	2	2	NUM
ejpam-4478	273	122	,	,	PUNCT
ejpam-4478	273	123	if	if	SCONJ
ejpam-4478	273	124	k	k	PROPN
ejpam-4478	273	125	≥	≥	NUM
ejpam-4478	273	126	diam(g	diam(g	PROPN
ejpam-4478	273	127	)	)	PUNCT
ejpam-4478	273	128	.	.	PUNCT
ejpam-4478	274	1	6	6	X
ejpam-4478	274	2	.	.	X
ejpam-4478	275	1	some	some	DET
ejpam-4478	275	2	bounds	bound	NOUN
ejpam-4478	275	3	of	of	ADP
ejpam-4478	275	4	the	the	DET
ejpam-4478	275	5	global	global	ADJ
ejpam-4478	275	6	distance	distance	NOUN
ejpam-4478	275	7	roman	roman	ADJ
ejpam-4478	275	8	domination	domination	NOUN
ejpam-4478	275	9	lemma	lemma	PROPN
ejpam-4478	275	10	6.1	6.1	NUM
ejpam-4478	275	11	.	.	PUNCT
ejpam-4478	276	1	let	let	VERB
ejpam-4478	276	2	k	k	PROPN
ejpam-4478	276	3	∈	∈	PROPN
ejpam-4478	276	4	z+	z+	PUNCT
ejpam-4478	276	5	.	.	PUNCT
ejpam-4478	277	1	given	give	VERB
ejpam-4478	277	2	any	any	DET
ejpam-4478	277	3	graph	graph	NOUN
ejpam-4478	277	4	g	g	NOUN
ejpam-4478	277	5	,	,	PUNCT
ejpam-4478	277	6	as	as	SCONJ
ejpam-4478	277	7	k	k	PROPN
ejpam-4478	277	8	−→	−→	NOUN
ejpam-4478	277	9	∞	∞	PROPN
ejpam-4478	277	10	,	,	PUNCT
ejpam-4478	277	11	γkgr(g	γkgr(g	NOUN
ejpam-4478	277	12	)	)	PUNCT
ejpam-4478	277	13	is	be	AUX
ejpam-4478	277	14	decreasing	decrease	VERB
ejpam-4478	277	15	,	,	PUNCT
ejpam-4478	277	16	that	that	ADV
ejpam-4478	277	17	is	is	ADV
ejpam-4478	277	18	,	,	PUNCT
ejpam-4478	277	19	γ1gr(g	γ1gr(g	PROPN
ejpam-4478	277	20	)	)	PUNCT
ejpam-4478	277	21	≥	≥	NOUN
ejpam-4478	277	22	γ2gr(g	γ2gr(g	NOUN
ejpam-4478	277	23	)	)	PUNCT
ejpam-4478	277	24	≥	≥	X
ejpam-4478	277	25	γ3gr(g	γ3gr(g	PROPN
ejpam-4478	277	26	)	)	PUNCT
ejpam-4478	277	27	≥	≥	X
ejpam-4478	277	28	·	·	PUNCT
ejpam-4478	277	29	·	·	PUNCT
ejpam-4478	277	30	·	·	PUNCT
ejpam-4478	277	31	≥	≥	PUNCT
ejpam-4478	277	32	γk−1	γk−1	ADJ
ejpam-4478	277	33	gr	gr	X
ejpam-4478	277	34	(	(	PUNCT
ejpam-4478	277	35	g	g	NOUN
ejpam-4478	277	36	)	)	PUNCT
ejpam-4478	277	37	≥	≥	NOUN
ejpam-4478	277	38	γkgr(g	γkgr(g	NOUN
ejpam-4478	277	39	)	)	PUNCT
ejpam-4478	277	40	≥	≥	NUM
ejpam-4478	277	41	γk+1	γk+1	X
ejpam-4478	278	1	gr	gr	INTJ
ejpam-4478	278	2	(	(	PUNCT
ejpam-4478	278	3	g	g	NOUN
ejpam-4478	278	4	)	)	PUNCT
ejpam-4478	278	5	≥	≥	NOUN
ejpam-4478	278	6	.	.	PUNCT
ejpam-4478	278	7	.	.	PUNCT
ejpam-4478	278	8	.	.	PUNCT
ejpam-4478	278	9	.	.	PUNCT
ejpam-4478	279	1	proof	proof	NOUN
ejpam-4478	279	2	.	.	PUNCT
ejpam-4478	280	1	let	let	VERB
ejpam-4478	280	2	k	k	PROPN
ejpam-4478	280	3	∈	∈	PROPN
ejpam-4478	280	4	z+	z+	PUNCT
ejpam-4478	280	5	.	.	PUNCT
ejpam-4478	281	1	note	note	VERB
ejpam-4478	281	2	that	that	SCONJ
ejpam-4478	281	3	,	,	PUNCT
ejpam-4478	281	4	if	if	SCONJ
ejpam-4478	281	5	k	k	PROPN
ejpam-4478	281	6	=	=	SYM
ejpam-4478	281	7	1	1	NUM
ejpam-4478	281	8	,	,	PUNCT
ejpam-4478	281	9	then	then	ADV
ejpam-4478	281	10	it	it	PRON
ejpam-4478	281	11	is	be	AUX
ejpam-4478	281	12	clear	clear	ADJ
ejpam-4478	281	13	that	that	SCONJ
ejpam-4478	281	14	γ1gr(g	γ1gr(g	PROPN
ejpam-4478	281	15	)	)	PUNCT
ejpam-4478	281	16	=	=	SYM
ejpam-4478	281	17	γgr(g	γgr(g	PROPN
ejpam-4478	281	18	)	)	PUNCT
ejpam-4478	281	19	.	.	PUNCT
ejpam-4478	282	1	now	now	ADV
ejpam-4478	282	2	,	,	PUNCT
ejpam-4478	282	3	given	give	VERB
ejpam-4478	282	4	k	k	PROPN
ejpam-4478	282	5	>	>	X
ejpam-4478	282	6	k−1	k−1	PROPN
ejpam-4478	282	7	,	,	PUNCT
ejpam-4478	282	8	we	we	PRON
ejpam-4478	282	9	have	have	VERB
ejpam-4478	282	10	∆k(g	∆k(g	NOUN
ejpam-4478	282	11	)	)	PUNCT
ejpam-4478	282	12	≥	≥	NOUN
ejpam-4478	282	13	∆k−1(g	∆k−1(g	PROPN
ejpam-4478	282	14	)	)	PUNCT
ejpam-4478	282	15	and	and	CCONJ
ejpam-4478	282	16	it	it	PRON
ejpam-4478	282	17	follows	follow	VERB
ejpam-4478	282	18	that	that	SCONJ
ejpam-4478	282	19	,	,	PUNCT
ejpam-4478	282	20	γkgr(g	γkgr(g	NOUN
ejpam-4478	282	21	)	)	PUNCT
ejpam-4478	282	22	≤	≤	NOUN
ejpam-4478	283	1	γk−1	γk−1	PROPN
ejpam-4478	283	2	gr	gr	X
ejpam-4478	283	3	(	(	PUNCT
ejpam-4478	283	4	g	g	NOUN
ejpam-4478	283	5	)	)	PUNCT
ejpam-4478	283	6	.	.	PUNCT
ejpam-4478	284	1	hence	hence	ADV
ejpam-4478	284	2	,	,	PUNCT
ejpam-4478	284	3	in	in	ADP
ejpam-4478	284	4	general	general	ADJ
ejpam-4478	284	5	,	,	PUNCT
ejpam-4478	284	6	given	give	VERB
ejpam-4478	284	7	·	·	PUNCT
ejpam-4478	284	8	·	·	PUNCT
ejpam-4478	284	9	·	·	PUNCT
ejpam-4478	284	10	>	>	PUNCT
ejpam-4478	285	1	k	k	X
ejpam-4478	286	1	+	+	CCONJ
ejpam-4478	286	2	1	1	NUM
ejpam-4478	286	3	>	>	X
ejpam-4478	286	4	k	k	X
ejpam-4478	286	5	>	>	X
ejpam-4478	287	1	k	k	PROPN
ejpam-4478	288	1	−	−	PROPN
ejpam-4478	288	2	1	1	NUM
ejpam-4478	288	3	>	>	PUNCT
ejpam-4478	288	4	·	·	PUNCT
ejpam-4478	288	5	·	·	PUNCT
ejpam-4478	288	6	·	·	PUNCT
ejpam-4478	288	7	>	>	PUNCT
ejpam-4478	288	8	3	3	NUM
ejpam-4478	288	9	>	>	SYM
ejpam-4478	288	10	2	2	NUM
ejpam-4478	288	11	>	>	SYM
ejpam-4478	288	12	1	1	NUM
ejpam-4478	288	13	,	,	PUNCT
ejpam-4478	288	14	we	we	PRON
ejpam-4478	288	15	have	have	AUX
ejpam-4478	288	16	·	·	PUNCT
ejpam-4478	288	17	·	·	PUNCT
ejpam-4478	288	18	·	·	PUNCT
ejpam-4478	288	19	>	>	X
ejpam-4478	288	20	∆k+1(g	∆k+1(g	PROPN
ejpam-4478	288	21	)	)	PUNCT
ejpam-4478	288	22	≥	≥	NOUN
ejpam-4478	288	23	∆k(g	∆k(g	PROPN
ejpam-4478	288	24	)	)	PUNCT
ejpam-4478	288	25	≥	≥	NOUN
ejpam-4478	288	26	∆k−1(g	∆k−1(g	PROPN
ejpam-4478	288	27	)	)	PUNCT
ejpam-4478	288	28	>	>	X
ejpam-4478	288	29	·	·	PUNCT
ejpam-4478	288	30	·	·	PUNCT
ejpam-4478	288	31	·	·	PUNCT
ejpam-4478	288	32	>	>	PUNCT
ejpam-4478	288	33	∆3(g	∆3(g	PROPN
ejpam-4478	288	34	)	)	PUNCT
ejpam-4478	288	35	≥	≥	NOUN
ejpam-4478	288	36	∆2(g	∆2(g	NOUN
ejpam-4478	288	37	)	)	PUNCT
ejpam-4478	288	38	≥	≥	NOUN
ejpam-4478	288	39	∆1(g	∆1(g	NUM
ejpam-4478	288	40	)	)	PUNCT
ejpam-4478	288	41	≥	≥	NOUN
ejpam-4478	288	42	and	and	CCONJ
ejpam-4478	288	43	so	so	ADV
ejpam-4478	288	44	,	,	PUNCT
ejpam-4478	288	45	we	we	PRON
ejpam-4478	288	46	can	can	AUX
ejpam-4478	288	47	say	say	VERB
ejpam-4478	288	48	that	that	SCONJ
ejpam-4478	288	49	,	,	PUNCT
ejpam-4478	288	50	γ1gr(g	γ1gr(g	PROPN
ejpam-4478	288	51	)	)	PUNCT
ejpam-4478	288	52	≥	≥	NOUN
ejpam-4478	288	53	γ2gr(g	γ2gr(g	NOUN
ejpam-4478	288	54	)	)	PUNCT
ejpam-4478	288	55	≥	≥	X
ejpam-4478	288	56	γ3gr(g	γ3gr(g	PROPN
ejpam-4478	288	57	)	)	PUNCT
ejpam-4478	288	58	≥	≥	X
ejpam-4478	288	59	·	·	PUNCT
ejpam-4478	288	60	·	·	PUNCT
ejpam-4478	288	61	·	·	PUNCT
ejpam-4478	289	1	≥	≥	PUNCT
ejpam-4478	289	2	γk−1	γk−1	ADJ
ejpam-4478	289	3	gr	gr	X
ejpam-4478	289	4	(	(	PUNCT
ejpam-4478	289	5	g	g	NOUN
ejpam-4478	289	6	)	)	PUNCT
ejpam-4478	289	7	≥	≥	NOUN
ejpam-4478	289	8	γkgr(g	γkgr(g	NOUN
ejpam-4478	289	9	)	)	PUNCT
ejpam-4478	289	10	≥	≥	NUM
ejpam-4478	289	11	γk+1	γk+1	X
ejpam-4478	290	1	gr	gr	INTJ
ejpam-4478	290	2	(	(	PUNCT
ejpam-4478	290	3	g	g	NOUN
ejpam-4478	290	4	)	)	PUNCT
ejpam-4478	290	5	≥	≥	NOUN
ejpam-4478	290	6	.	.	PUNCT
ejpam-4478	290	7	.	.	PUNCT
ejpam-4478	290	8	.	.	PUNCT
ejpam-4478	290	9	.	.	PUNCT
ejpam-4478	291	1	this	this	PRON
ejpam-4478	291	2	completes	complete	VERB
ejpam-4478	291	3	the	the	DET
ejpam-4478	291	4	proof	proof	NOUN
ejpam-4478	291	5	.	.	PUNCT
ejpam-4478	292	1	g.	g.	PROPN
ejpam-4478	292	2	entero	entero	PROPN
ejpam-4478	292	3	,	,	PUNCT
ejpam-4478	292	4	s.	s.	PROPN
ejpam-4478	292	5	espinola	espinola	PROPN
ejpam-4478	292	6	/	/	SYM
ejpam-4478	292	7	eur	eur	PROPN
ejpam-4478	292	8	.	.	PUNCT
ejpam-4478	293	1	j.	j.	PROPN
ejpam-4478	293	2	pure	pure	PROPN
ejpam-4478	293	3	appl	appl	PROPN
ejpam-4478	293	4	.	.	PROPN
ejpam-4478	293	5	math	math	PROPN
ejpam-4478	293	6	,	,	PUNCT
ejpam-4478	293	7	16	16	NUM
ejpam-4478	293	8	(	(	PUNCT
ejpam-4478	293	9	1	1	NUM
ejpam-4478	293	10	)	)	PUNCT
ejpam-4478	293	11	(	(	PUNCT
ejpam-4478	293	12	2023	2023	NUM
ejpam-4478	293	13	)	)	PUNCT
ejpam-4478	293	14	,	,	PUNCT
ejpam-4478	293	15	44	44	NUM
ejpam-4478	293	16	-	-	SYM
ejpam-4478	293	17	61	61	NUM
ejpam-4478	293	18	53	53	NUM
ejpam-4478	293	19	theorem	theorem	NOUN
ejpam-4478	293	20	6.2	6.2	NUM
ejpam-4478	293	21	.	.	PUNCT
ejpam-4478	294	1	let	let	VERB
ejpam-4478	294	2	k	k	PROPN
ejpam-4478	294	3	∈	∈	PROPN
ejpam-4478	294	4	z+	z+	PUNCT
ejpam-4478	294	5	.	.	PUNCT
ejpam-4478	295	1	for	for	ADP
ejpam-4478	295	2	any	any	DET
ejpam-4478	295	3	graph	graph	NOUN
ejpam-4478	295	4	g	g	NOUN
ejpam-4478	295	5	of	of	ADP
ejpam-4478	295	6	order	order	NOUN
ejpam-4478	295	7	n	n	PRON
ejpam-4478	295	8	≥	≥	NOUN
ejpam-4478	295	9	1	1	NUM
ejpam-4478	295	10	,	,	PUNCT
ejpam-4478	295	11	γkgr(g	γkgr(g	NOUN
ejpam-4478	295	12	)	)	PUNCT
ejpam-4478	295	13	≤	≤	PROPN
ejpam-4478	295	14	γgr(g	γgr(g	PROPN
ejpam-4478	295	15	)	)	PUNCT
ejpam-4478	295	16	.	.	PUNCT
ejpam-4478	296	1	proof	proof	NOUN
ejpam-4478	296	2	.	.	PUNCT
ejpam-4478	297	1	suppose	suppose	VERB
ejpam-4478	297	2	that	that	SCONJ
ejpam-4478	297	3	k	k	PROPN
ejpam-4478	297	4	∈	∈	PROPN
ejpam-4478	297	5	z+	z+	NUM
ejpam-4478	297	6	and	and	CCONJ
ejpam-4478	297	7	let	let	VERB
ejpam-4478	297	8	g	g	NOUN
ejpam-4478	297	9	be	be	AUX
ejpam-4478	297	10	any	any	DET
ejpam-4478	297	11	graph	graph	NOUN
ejpam-4478	297	12	of	of	ADP
ejpam-4478	297	13	order	order	NOUN
ejpam-4478	297	14	n	n	PRON
ejpam-4478	297	15	≥	≥	NOUN
ejpam-4478	297	16	1	1	NUM
ejpam-4478	297	17	.	.	PUNCT
ejpam-4478	297	18	since	since	SCONJ
ejpam-4478	297	19	γ1gr(g	γ1gr(g	NUM
ejpam-4478	297	20	)	)	PUNCT
ejpam-4478	297	21	=	=	SYM
ejpam-4478	297	22	γgr(g	γgr(g	PROPN
ejpam-4478	297	23	)	)	PUNCT
ejpam-4478	297	24	and	and	CCONJ
ejpam-4478	297	25	by	by	ADP
ejpam-4478	297	26	lemma	lemma	PROPN
ejpam-4478	297	27	6.1	6.1	NUM
ejpam-4478	297	28	,	,	PUNCT
ejpam-4478	297	29	we	we	PRON
ejpam-4478	297	30	have	have	VERB
ejpam-4478	297	31	·	·	PUNCT
ejpam-4478	297	32	·	·	PUNCT
ejpam-4478	297	33	·	·	PUNCT
ejpam-4478	297	34	≤	≤	NUM
ejpam-4478	297	35	γkgr(g	γkgr(g	NUM
ejpam-4478	297	36	)	)	PUNCT
ejpam-4478	297	37	≤	≤	PUNCT
ejpam-4478	297	38	γk−1	γk−1	PROPN
ejpam-4478	297	39	gr	gr	X
ejpam-4478	297	40	(	(	PUNCT
ejpam-4478	297	41	g	g	NOUN
ejpam-4478	297	42	)	)	PUNCT
ejpam-4478	297	43	≤	≤	NOUN
ejpam-4478	297	44	·	·	PUNCT
ejpam-4478	297	45	·	·	PUNCT
ejpam-4478	297	46	·	·	PUNCT
ejpam-4478	297	47	≤	≤	NUM
ejpam-4478	297	48	γ3gr(g	γ3gr(g	PROPN
ejpam-4478	297	49	)	)	PUNCT
ejpam-4478	297	50	≤	≤	NOUN
ejpam-4478	297	51	γ2gr(g	γ2gr(g	PROPN
ejpam-4478	297	52	)	)	PUNCT
ejpam-4478	297	53	≤	≤	NUM
ejpam-4478	297	54	γ1gr(g	γ1gr(g	PROPN
ejpam-4478	297	55	)	)	PUNCT
ejpam-4478	297	56	=	=	SYM
ejpam-4478	297	57	γgr(g	γgr(g	PROPN
ejpam-4478	297	58	)	)	PUNCT
ejpam-4478	297	59	,	,	PUNCT
ejpam-4478	297	60	that	that	ADV
ejpam-4478	297	61	is	be	AUX
ejpam-4478	297	62	,	,	PUNCT
ejpam-4478	297	63	·	·	PUNCT
ejpam-4478	297	64	·	·	PUNCT
ejpam-4478	297	65	·	·	PUNCT
ejpam-4478	298	1	≤	≤	NUM
ejpam-4478	298	2	γkgr(g	γkgr(g	NUM
ejpam-4478	298	3	)	)	PUNCT
ejpam-4478	298	4	≤	≤	PUNCT
ejpam-4478	299	1	γk−1	γk−1	PROPN
ejpam-4478	299	2	gr	gr	X
ejpam-4478	299	3	(	(	PUNCT
ejpam-4478	299	4	g	g	NOUN
ejpam-4478	299	5	)	)	PUNCT
ejpam-4478	299	6	≤	≤	NOUN
ejpam-4478	299	7	·	·	PUNCT
ejpam-4478	299	8	·	·	PUNCT
ejpam-4478	299	9	·	·	PUNCT
ejpam-4478	299	10	≤	≤	NUM
ejpam-4478	299	11	γ3gr(g	γ3gr(g	PROPN
ejpam-4478	299	12	)	)	PUNCT
ejpam-4478	299	13	≤	≤	NOUN
ejpam-4478	299	14	γ2gr(g	γ2gr(g	PROPN
ejpam-4478	299	15	)	)	PUNCT
ejpam-4478	299	16	≤	≤	PUNCT
ejpam-4478	300	1	γgr(g	γgr(g	PROPN
ejpam-4478	300	2	)	)	PUNCT
ejpam-4478	300	3	,	,	PUNCT
ejpam-4478	300	4	and	and	CCONJ
ejpam-4478	300	5	hence	hence	ADV
ejpam-4478	300	6	,	,	PUNCT
ejpam-4478	300	7	by	by	ADP
ejpam-4478	300	8	transitivity	transitivity	NOUN
ejpam-4478	300	9	,	,	PUNCT
ejpam-4478	300	10	we	we	PRON
ejpam-4478	300	11	have	have	VERB
ejpam-4478	300	12	γkgr(g	γkgr(g	NOUN
ejpam-4478	300	13	)	)	PUNCT
ejpam-4478	300	14	≤	≤	PROPN
ejpam-4478	300	15	γgr(g	γgr(g	PROPN
ejpam-4478	300	16	)	)	PUNCT
ejpam-4478	300	17	.	.	PUNCT
ejpam-4478	301	1	so	so	ADV
ejpam-4478	301	2	,	,	PUNCT
ejpam-4478	301	3	for	for	ADP
ejpam-4478	301	4	all	all	DET
ejpam-4478	301	5	k	k	PROPN
ejpam-4478	301	6	∈	∈	PROPN
ejpam-4478	301	7	z+	z+	X
ejpam-4478	301	8	,	,	PUNCT
ejpam-4478	301	9	γkgr(g	γkgr(g	NOUN
ejpam-4478	301	10	)	)	PUNCT
ejpam-4478	301	11	≤	≤	PROPN
ejpam-4478	301	12	γgr(g	γgr(g	PROPN
ejpam-4478	301	13	)	)	PUNCT
ejpam-4478	301	14	.	.	PUNCT
ejpam-4478	302	1	this	this	PRON
ejpam-4478	302	2	completes	complete	VERB
ejpam-4478	302	3	the	the	DET
ejpam-4478	302	4	proof	proof	NOUN
ejpam-4478	302	5	.	.	PUNCT
ejpam-4478	303	1	remark	remark	VERB
ejpam-4478	303	2	6.3	6.3	NUM
ejpam-4478	303	3	.	.	PUNCT
ejpam-4478	304	1	the	the	DET
ejpam-4478	304	2	upper	upper	ADJ
ejpam-4478	304	3	bound	bind	VERB
ejpam-4478	304	4	in	in	ADP
ejpam-4478	304	5	theorem	theorem	ADJ
ejpam-4478	304	6	6.2	6.2	NUM
ejpam-4478	304	7	is	be	AUX
ejpam-4478	304	8	sharp	sharp	ADJ
ejpam-4478	304	9	and	and	CCONJ
ejpam-4478	304	10	the	the	DET
ejpam-4478	304	11	strict	strict	ADJ
ejpam-4478	304	12	inequality	inequality	NOUN
ejpam-4478	304	13	is	be	AUX
ejpam-4478	304	14	attained	attain	VERB
ejpam-4478	304	15	.	.	PUNCT
ejpam-4478	305	1	proof	proof	NOUN
ejpam-4478	305	2	.	.	PUNCT
ejpam-4478	306	1	the	the	DET
ejpam-4478	306	2	proof	proof	NOUN
ejpam-4478	306	3	is	be	AUX
ejpam-4478	306	4	clear	clear	ADJ
ejpam-4478	306	5	.	.	PUNCT
ejpam-4478	307	1	theorem	theorem	VERB
ejpam-4478	307	2	6.4	6.4	NUM
ejpam-4478	307	3	.	.	PUNCT
ejpam-4478	308	1	let	let	VERB
ejpam-4478	308	2	k	k	PROPN
ejpam-4478	308	3	∈	∈	PROPN
ejpam-4478	308	4	z+	z+	PUNCT
ejpam-4478	308	5	.	.	PUNCT
ejpam-4478	309	1	for	for	ADP
ejpam-4478	309	2	any	any	DET
ejpam-4478	309	3	graph	graph	NOUN
ejpam-4478	309	4	g	g	NOUN
ejpam-4478	309	5	=	=	PUNCT
ejpam-4478	309	6	(	(	PUNCT
ejpam-4478	309	7	v	v	NOUN
ejpam-4478	309	8	(	(	PUNCT
ejpam-4478	309	9	g	g	NOUN
ejpam-4478	309	10	)	)	PUNCT
ejpam-4478	309	11	,	,	PUNCT
ejpam-4478	309	12	e(g	e(g	PROPN
ejpam-4478	309	13	)	)	PUNCT
ejpam-4478	309	14	)	)	PUNCT
ejpam-4478	309	15	of	of	ADP
ejpam-4478	309	16	order	order	NOUN
ejpam-4478	309	17	|v	|v	X
ejpam-4478	309	18	(	(	PUNCT
ejpam-4478	309	19	g)|	g)|	X
ejpam-4478	309	20	≥	≥	NOUN
ejpam-4478	309	21	2	2	NUM
ejpam-4478	309	22	,	,	PUNCT
ejpam-4478	309	23	2	2	NUM
ejpam-4478	309	24	≤	≤	NUM
ejpam-4478	309	25	γkgr(g	γkgr(g	NUM
ejpam-4478	309	26	)	)	PUNCT
ejpam-4478	309	27	≤	≤	NUM
ejpam-4478	309	28	2|v	2|v	PROPN
ejpam-4478	309	29	(	(	PUNCT
ejpam-4478	309	30	g)|	g)|	NOUN
ejpam-4478	309	31	.	.	PUNCT
ejpam-4478	310	1	proof	proof	NOUN
ejpam-4478	310	2	.	.	PUNCT
ejpam-4478	311	1	let	let	VERB
ejpam-4478	311	2	g	g	PROPN
ejpam-4478	311	3	=	=	SYM
ejpam-4478	311	4	(	(	PUNCT
ejpam-4478	311	5	v	v	NOUN
ejpam-4478	311	6	(	(	PUNCT
ejpam-4478	311	7	g	g	NOUN
ejpam-4478	311	8	)	)	PUNCT
ejpam-4478	311	9	,	,	PUNCT
ejpam-4478	311	10	e(g	e(g	PROPN
ejpam-4478	311	11	)	)	PUNCT
ejpam-4478	311	12	)	)	PUNCT
ejpam-4478	312	1	be	be	AUX
ejpam-4478	312	2	any	any	DET
ejpam-4478	312	3	graph	graph	NOUN
ejpam-4478	312	4	of	of	ADP
ejpam-4478	312	5	order	order	NOUN
ejpam-4478	312	6	|v	|v	X
ejpam-4478	312	7	(	(	PUNCT
ejpam-4478	312	8	g)|	g)|	X
ejpam-4478	312	9	≥	≥	NOUN
ejpam-4478	312	10	2	2	NUM
ejpam-4478	312	11	and	and	CCONJ
ejpam-4478	312	12	let	let	VERB
ejpam-4478	312	13	k	k	PROPN
ejpam-4478	312	14	∈	∈	PROPN
ejpam-4478	312	15	z+	z+	PUNCT
ejpam-4478	312	16	.	.	PUNCT
ejpam-4478	313	1	we	we	PRON
ejpam-4478	313	2	let	let	VERB
ejpam-4478	313	3	fk	fk	INTJ
ejpam-4478	313	4	to	to	PART
ejpam-4478	313	5	be	be	AUX
ejpam-4478	313	6	an	an	DET
ejpam-4478	313	7	arbitrary	arbitrary	ADJ
ejpam-4478	313	8	global	global	ADJ
ejpam-4478	313	9	k	k	NOUN
ejpam-4478	313	10	−	−	PROPN
ejpam-4478	313	11	distance	distance	NOUN
ejpam-4478	313	12	roman	roman	ADJ
ejpam-4478	313	13	dominating	dominating	NOUN
ejpam-4478	313	14	function	function	NOUN
ejpam-4478	313	15	(	(	PUNCT
ejpam-4478	313	16	gkdrdf	gkdrdf	PROPN
ejpam-4478	313	17	)	)	PUNCT
ejpam-4478	313	18	on	on	ADP
ejpam-4478	313	19	graph	graph	NOUN
ejpam-4478	313	20	g	g	PROPN
ejpam-4478	313	21	,	,	PUNCT
ejpam-4478	313	22	where	where	SCONJ
ejpam-4478	313	23	k	k	PROPN
ejpam-4478	313	24	∈	∈	PROPN
ejpam-4478	313	25	z+	z+	PUNCT
ejpam-4478	313	26	.	.	PUNCT
ejpam-4478	314	1	for	for	ADP
ejpam-4478	314	2	all	all	PRON
ejpam-4478	314	3	k	k	PROPN
ejpam-4478	314	4	∈	∈	PROPN
ejpam-4478	314	5	z+	z+	NUM
ejpam-4478	314	6	,	,	PUNCT
ejpam-4478	314	7	by	by	ADP
ejpam-4478	314	8	remark	remark	NOUN
ejpam-4478	314	9	4.3	4.3	NUM
ejpam-4478	314	10	,	,	PUNCT
ejpam-4478	314	11	w(fk	w(fk	NOUN
ejpam-4478	314	12	)	)	PUNCT
ejpam-4478	314	13	=	=	SYM
ejpam-4478	314	14	|v	|v	PROPN
ejpam-4478	314	15	fk	fk	INTJ
ejpam-4478	314	16	1	1	NUM
ejpam-4478	314	17	(	(	PUNCT
ejpam-4478	314	18	g)|	g)|	NOUN
ejpam-4478	314	19	+	+	CCONJ
ejpam-4478	314	20	2|v	2|v	PROPN
ejpam-4478	314	21	fk	fk	INTJ
ejpam-4478	314	22	2	2	NUM
ejpam-4478	314	23	(	(	PUNCT
ejpam-4478	314	24	g)|	g)|	NOUN
ejpam-4478	314	25	.	.	PUNCT
ejpam-4478	315	1	since	since	SCONJ
ejpam-4478	315	2	we	we	PRON
ejpam-4478	315	3	are	be	AUX
ejpam-4478	315	4	talking	talk	VERB
ejpam-4478	315	5	about	about	ADP
ejpam-4478	315	6	number	number	NOUN
ejpam-4478	315	7	of	of	ADP
ejpam-4478	315	8	elements	element	NOUN
ejpam-4478	315	9	,	,	PUNCT
ejpam-4478	315	10	|v	|v	PROPN
ejpam-4478	315	11	(	(	PUNCT
ejpam-4478	315	12	g)|	g)|	PROPN
ejpam-4478	315	13	,	,	PUNCT
ejpam-4478	315	14	|v	|v	PROPN
ejpam-4478	315	15	fk	fk	INTJ
ejpam-4478	315	16	0	0	NUM
ejpam-4478	316	1	(	(	PUNCT
ejpam-4478	316	2	g)|	g)|	PROPN
ejpam-4478	316	3	,	,	PUNCT
ejpam-4478	316	4	|v	|v	PROPN
ejpam-4478	316	5	fk	fk	INTJ
ejpam-4478	316	6	1	1	NUM
ejpam-4478	316	7	(	(	PUNCT
ejpam-4478	316	8	g)|	g)|	PROPN
ejpam-4478	316	9	,	,	PUNCT
ejpam-4478	316	10	|v	|v	PROPN
ejpam-4478	316	11	fk	fk	INTJ
ejpam-4478	316	12	2	2	NUM
ejpam-4478	316	13	(	(	PUNCT
ejpam-4478	316	14	g)|	g)|	VERB
ejpam-4478	316	15	≥	≥	NOUN
ejpam-4478	316	16	0	0	NUM
ejpam-4478	317	1	and	and	CCONJ
ejpam-4478	317	2	so	so	ADV
ejpam-4478	317	3	as	as	ADP
ejpam-4478	317	4	the	the	DET
ejpam-4478	317	5	weight	weight	NOUN
ejpam-4478	317	6	w(fk	w(fk	NOUN
ejpam-4478	317	7	)	)	PUNCT
ejpam-4478	317	8	of	of	ADP
ejpam-4478	317	9	fk	fk	INTJ
ejpam-4478	317	10	.	.	PUNCT
ejpam-4478	318	1	now	now	ADV
ejpam-4478	318	2	,	,	PUNCT
ejpam-4478	318	3	since	since	SCONJ
ejpam-4478	318	4	|v	|v	PROPN
ejpam-4478	318	5	(	(	PUNCT
ejpam-4478	318	6	g)|	g)|	X
ejpam-4478	318	7	≥	≥	NOUN
ejpam-4478	318	8	2	2	NUM
ejpam-4478	318	9	,	,	PUNCT
ejpam-4478	318	10	it	it	PRON
ejpam-4478	318	11	follows	follow	VERB
ejpam-4478	318	12	that	that	SCONJ
ejpam-4478	318	13	,	,	PUNCT
ejpam-4478	318	14	γkgr(g	γkgr(g	NUM
ejpam-4478	318	15	)	)	PUNCT
ejpam-4478	318	16	≥	≥	NOUN
ejpam-4478	318	17	2	2	NUM
ejpam-4478	318	18	(	(	PUNCT
ejpam-4478	318	19	1	1	NUM
ejpam-4478	318	20	)	)	PUNCT
ejpam-4478	318	21	and	and	CCONJ
ejpam-4478	318	22	since	since	SCONJ
ejpam-4478	318	23	γkgr(g	γkgr(g	NUM
ejpam-4478	318	24	)	)	PUNCT
ejpam-4478	318	25	is	be	AUX
ejpam-4478	318	26	the	the	DET
ejpam-4478	318	27	minimum	minimum	ADJ
ejpam-4478	318	28	weight	weight	NOUN
ejpam-4478	318	29	taken	take	VERB
ejpam-4478	318	30	over	over	ADP
ejpam-4478	318	31	all	all	DET
ejpam-4478	318	32	gkdrdf	gkdrdf	NOUN
ejpam-4478	318	33	on	on	ADP
ejpam-4478	318	34	graph	graph	NOUN
ejpam-4478	318	35	g	g	PROPN
ejpam-4478	318	36	,	,	PUNCT
ejpam-4478	318	37	for	for	ADP
ejpam-4478	318	38	all	all	DET
ejpam-4478	318	39	fk	fk	NOUN
ejpam-4478	318	40	on	on	ADP
ejpam-4478	318	41	g	g	PROPN
ejpam-4478	318	42	and	and	CCONJ
ejpam-4478	318	43	for	for	ADP
ejpam-4478	318	44	all	all	PRON
ejpam-4478	318	45	k	k	PROPN
ejpam-4478	318	46	∈	∈	PROPN
ejpam-4478	318	47	z+	z+	X
ejpam-4478	318	48	,	,	PUNCT
ejpam-4478	318	49	we	we	PRON
ejpam-4478	318	50	have	have	VERB
ejpam-4478	318	51	γkgr(g	γkgr(g	NOUN
ejpam-4478	318	52	)	)	PUNCT
ejpam-4478	318	53	≤	≤	NOUN
ejpam-4478	318	54	w(fk	w(fk	NOUN
ejpam-4478	318	55	)	)	PUNCT
ejpam-4478	318	56	≤	≤	NUM
ejpam-4478	319	1	2|v	2|v	PROPN
ejpam-4478	319	2	(	(	PUNCT
ejpam-4478	319	3	g)|	g)|	NOUN
ejpam-4478	319	4	and	and	CCONJ
ejpam-4478	319	5	thus	thus	ADV
ejpam-4478	319	6	,	,	PUNCT
ejpam-4478	319	7	by	by	ADP
ejpam-4478	319	8	transitivity	transitivity	NOUN
ejpam-4478	319	9	,	,	PUNCT
ejpam-4478	319	10	we	we	PRON
ejpam-4478	319	11	have	have	VERB
ejpam-4478	319	12	γkgr(g	γkgr(g	NOUN
ejpam-4478	319	13	)	)	PUNCT
ejpam-4478	319	14	≤	≤	NUM
ejpam-4478	320	1	2|v	2|v	PROPN
ejpam-4478	320	2	(	(	PUNCT
ejpam-4478	320	3	g)|	g)|	X
ejpam-4478	320	4	(	(	PUNCT
ejpam-4478	320	5	2	2	NUM
ejpam-4478	320	6	)	)	PUNCT
ejpam-4478	320	7	hence	hence	ADV
ejpam-4478	320	8	,	,	PUNCT
ejpam-4478	320	9	from	from	ADP
ejpam-4478	320	10	(	(	PUNCT
ejpam-4478	320	11	1	1	NUM
ejpam-4478	320	12	)	)	PUNCT
ejpam-4478	320	13	and	and	CCONJ
ejpam-4478	320	14	(	(	PUNCT
ejpam-4478	320	15	2	2	NUM
ejpam-4478	320	16	)	)	PUNCT
ejpam-4478	320	17	,	,	PUNCT
ejpam-4478	320	18	for	for	ADP
ejpam-4478	320	19	all	all	DET
ejpam-4478	320	20	k	k	PROPN
ejpam-4478	320	21	∈	∈	PROPN
ejpam-4478	320	22	z+	z+	X
ejpam-4478	320	23	,	,	PUNCT
ejpam-4478	320	24	we	we	PRON
ejpam-4478	320	25	have	have	VERB
ejpam-4478	320	26	2	2	NUM
ejpam-4478	320	27	≤	≤	NUM
ejpam-4478	320	28	γkgr(g	γkgr(g	NUM
ejpam-4478	320	29	)	)	PUNCT
ejpam-4478	320	30	≤	≤	NUM
ejpam-4478	321	1	2|v	2|v	PROPN
ejpam-4478	322	1	(	(	PUNCT
ejpam-4478	322	2	g)|	g)|	NOUN
ejpam-4478	322	3	.	.	PUNCT
ejpam-4478	323	1	this	this	PRON
ejpam-4478	323	2	completes	complete	VERB
ejpam-4478	323	3	the	the	DET
ejpam-4478	323	4	proof	proof	NOUN
ejpam-4478	323	5	.	.	PUNCT
ejpam-4478	324	1	theorem	theorem	VERB
ejpam-4478	324	2	6.5	6.5	NUM
ejpam-4478	324	3	.	.	PUNCT
ejpam-4478	325	1	let	let	VERB
ejpam-4478	325	2	k	k	PROPN
ejpam-4478	325	3	∈	∈	PROPN
ejpam-4478	325	4	z+	z+	PUNCT
ejpam-4478	325	5	.	.	PUNCT
ejpam-4478	326	1	for	for	ADP
ejpam-4478	326	2	any	any	DET
ejpam-4478	326	3	graph	graph	NOUN
ejpam-4478	326	4	g	g	NOUN
ejpam-4478	326	5	of	of	ADP
ejpam-4478	326	6	order	order	NOUN
ejpam-4478	326	7	n	n	CCONJ
ejpam-4478	326	8	,	,	PUNCT
ejpam-4478	326	9	γkgr(g	γkgr(g	NOUN
ejpam-4478	326	10	)	)	PUNCT
ejpam-4478	326	11	≤	≤	NOUN
ejpam-4478	326	12	n	n	CCONJ
ejpam-4478	326	13	,	,	PUNCT
ejpam-4478	326	14	with	with	ADP
ejpam-4478	326	15	the	the	DET
ejpam-4478	326	16	equality	equality	NOUN
ejpam-4478	326	17	when	when	SCONJ
ejpam-4478	326	18	g	g	PROPN
ejpam-4478	326	19	∼=	∼=	PROPN
ejpam-4478	326	20	kn	kn	PROPN
ejpam-4478	326	21	,	,	PUNCT
ejpam-4478	326	22	kn	kn	PROPN
ejpam-4478	326	23	.	.	PUNCT
ejpam-4478	326	24	g.	g.	PROPN
ejpam-4478	326	25	entero	entero	PROPN
ejpam-4478	326	26	,	,	PUNCT
ejpam-4478	326	27	s.	s.	PROPN
ejpam-4478	326	28	espinola	espinola	PROPN
ejpam-4478	326	29	/	/	SYM
ejpam-4478	326	30	eur	eur	PROPN
ejpam-4478	326	31	.	.	PUNCT
ejpam-4478	327	1	j.	j.	PROPN
ejpam-4478	327	2	pure	pure	PROPN
ejpam-4478	327	3	appl	appl	PROPN
ejpam-4478	327	4	.	.	PROPN
ejpam-4478	327	5	math	math	PROPN
ejpam-4478	327	6	,	,	PUNCT
ejpam-4478	327	7	16	16	NUM
ejpam-4478	327	8	(	(	PUNCT
ejpam-4478	327	9	1	1	NUM
ejpam-4478	327	10	)	)	PUNCT
ejpam-4478	327	11	(	(	PUNCT
ejpam-4478	327	12	2023	2023	NUM
ejpam-4478	327	13	)	)	PUNCT
ejpam-4478	327	14	,	,	PUNCT
ejpam-4478	327	15	44	44	NUM
ejpam-4478	327	16	-	-	SYM
ejpam-4478	327	17	61	61	NUM
ejpam-4478	327	18	54	54	NUM
ejpam-4478	327	19	proof	proof	NOUN
ejpam-4478	327	20	.	.	PUNCT
ejpam-4478	328	1	let	let	VERB
ejpam-4478	328	2	k	k	PROPN
ejpam-4478	328	3	∈	∈	PROPN
ejpam-4478	328	4	z+	z+	NUM
ejpam-4478	328	5	and	and	CCONJ
ejpam-4478	328	6	let	let	VERB
ejpam-4478	328	7	g	g	NOUN
ejpam-4478	328	8	be	be	AUX
ejpam-4478	328	9	any	any	DET
ejpam-4478	328	10	graph	graph	NOUN
ejpam-4478	328	11	of	of	ADP
ejpam-4478	328	12	order	order	NOUN
ejpam-4478	328	13	n.	n.	NOUN
ejpam-4478	328	14	for	for	ADP
ejpam-4478	328	15	all	all	PRON
ejpam-4478	328	16	k	k	PROPN
ejpam-4478	328	17	∈	∈	PROPN
ejpam-4478	328	18	z+	z+	NUM
ejpam-4478	328	19	,	,	PUNCT
ejpam-4478	328	20	by	by	ADP
ejpam-4478	328	21	theorem	theorem	NOUN
ejpam-4478	328	22	6.2	6.2	NUM
ejpam-4478	328	23	,	,	PUNCT
ejpam-4478	328	24	we	we	PRON
ejpam-4478	328	25	have	have	VERB
ejpam-4478	328	26	γkgr(g	γkgr(g	NOUN
ejpam-4478	328	27	)	)	PUNCT
ejpam-4478	328	28	≤	≤	NUM
ejpam-4478	328	29	γgr(g	γgr(g	PROPN
ejpam-4478	328	30	)	)	PUNCT
ejpam-4478	328	31	(	(	PUNCT
ejpam-4478	328	32	1	1	NUM
ejpam-4478	328	33	)	)	PUNCT
ejpam-4478	328	34	and	and	CCONJ
ejpam-4478	328	35	by	by	ADP
ejpam-4478	328	36	remark	remark	NOUN
ejpam-4478	328	37	3.9	3.9	NUM
ejpam-4478	328	38	,	,	PUNCT
ejpam-4478	328	39	we	we	PRON
ejpam-4478	328	40	have	have	VERB
ejpam-4478	328	41	2	2	NUM
ejpam-4478	328	42	≤	≤	NUM
ejpam-4478	328	43	γgr(g	γgr(g	PROPN
ejpam-4478	328	44	)	)	PUNCT
ejpam-4478	328	45	≤	≤	NUM
ejpam-4478	329	1	n	n	CCONJ
ejpam-4478	329	2	(	(	PUNCT
ejpam-4478	329	3	2	2	NUM
ejpam-4478	329	4	)	)	PUNCT
ejpam-4478	329	5	thus	thus	ADV
ejpam-4478	329	6	,	,	PUNCT
ejpam-4478	329	7	for	for	ADP
ejpam-4478	329	8	all	all	PRON
ejpam-4478	329	9	k	k	PROPN
ejpam-4478	329	10	∈	∈	PROPN
ejpam-4478	329	11	z+	z+	X
ejpam-4478	329	12	,	,	PUNCT
ejpam-4478	329	13	from	from	ADP
ejpam-4478	329	14	(	(	PUNCT
ejpam-4478	329	15	1	1	NUM
ejpam-4478	329	16	)	)	PUNCT
ejpam-4478	329	17	and	and	CCONJ
ejpam-4478	329	18	(	(	PUNCT
ejpam-4478	329	19	2	2	NUM
ejpam-4478	329	20	)	)	PUNCT
ejpam-4478	329	21	,	,	PUNCT
ejpam-4478	329	22	we	we	PRON
ejpam-4478	329	23	obtain	obtain	VERB
ejpam-4478	329	24	γkgr(g	γkgr(g	NOUN
ejpam-4478	329	25	)	)	PUNCT
ejpam-4478	329	26	≤	≤	PUNCT
ejpam-4478	329	27	γgr(g	γgr(g	PROPN
ejpam-4478	329	28	)	)	PUNCT
ejpam-4478	329	29	≤	≤	NOUN
ejpam-4478	329	30	n	n	CCONJ
ejpam-4478	329	31	,	,	PUNCT
ejpam-4478	329	32	and	and	CCONJ
ejpam-4478	329	33	by	by	ADP
ejpam-4478	329	34	transitivity	transitivity	NOUN
ejpam-4478	329	35	,	,	PUNCT
ejpam-4478	329	36	we	we	PRON
ejpam-4478	329	37	get	get	VERB
ejpam-4478	329	38	γkgr(g	γkgr(g	NOUN
ejpam-4478	329	39	)	)	PUNCT
ejpam-4478	329	40	≤	≤	NOUN
ejpam-4478	329	41	n.	n.	NOUN
ejpam-4478	329	42	moreover	moreover	ADV
ejpam-4478	329	43	,	,	PUNCT
ejpam-4478	329	44	the	the	DET
ejpam-4478	329	45	equality	equality	NOUN
ejpam-4478	329	46	γkgr(g	γkgr(g	NUM
ejpam-4478	329	47	)	)	PUNCT
ejpam-4478	329	48	=	=	SYM
ejpam-4478	330	1	n	n	CCONJ
ejpam-4478	330	2	,	,	PUNCT
ejpam-4478	330	3	when	when	SCONJ
ejpam-4478	330	4	g	g	PROPN
ejpam-4478	330	5	∼=	∼=	PROPN
ejpam-4478	330	6	kn	kn	PROPN
ejpam-4478	330	7	,	,	PUNCT
ejpam-4478	330	8	kn	kn	PROPN
ejpam-4478	330	9	,	,	PUNCT
ejpam-4478	330	10	is	be	AUX
ejpam-4478	330	11	an	an	DET
ejpam-4478	330	12	immediate	immediate	ADJ
ejpam-4478	330	13	consequence	consequence	NOUN
ejpam-4478	330	14	of	of	ADP
ejpam-4478	330	15	theorem	theorem	ADJ
ejpam-4478	330	16	5.1	5.1	NUM
ejpam-4478	330	17	and	and	CCONJ
ejpam-4478	330	18	corollary	corollary	ADJ
ejpam-4478	330	19	5.2	5.2	NUM
ejpam-4478	330	20	,	,	PUNCT
ejpam-4478	330	21	and	and	CCONJ
ejpam-4478	330	22	hence	hence	ADV
ejpam-4478	330	23	,	,	PUNCT
ejpam-4478	330	24	the	the	DET
ejpam-4478	330	25	upper	upper	ADJ
ejpam-4478	330	26	bound	bind	VERB
ejpam-4478	330	27	n	n	NOUN
ejpam-4478	330	28	is	be	AUX
ejpam-4478	330	29	sharp	sharp	ADJ
ejpam-4478	330	30	.	.	PUNCT
ejpam-4478	331	1	this	this	PRON
ejpam-4478	331	2	completes	complete	VERB
ejpam-4478	331	3	the	the	DET
ejpam-4478	331	4	proof	proof	NOUN
ejpam-4478	331	5	.	.	PUNCT
ejpam-4478	332	1	7	7	X
ejpam-4478	332	2	.	.	X
ejpam-4478	332	3	some	some	DET
ejpam-4478	332	4	characterizations	characterization	NOUN
ejpam-4478	332	5	of	of	ADP
ejpam-4478	332	6	the	the	DET
ejpam-4478	332	7	global	global	ADJ
ejpam-4478	332	8	distance	distance	NOUN
ejpam-4478	332	9	roman	roman	ADJ
ejpam-4478	332	10	domination	domination	NOUN
ejpam-4478	332	11	theorem	theorem	VERB
ejpam-4478	332	12	7.1	7.1	NUM
ejpam-4478	332	13	.	.	PUNCT
ejpam-4478	333	1	let	let	VERB
ejpam-4478	333	2	g	g	NOUN
ejpam-4478	333	3	be	be	AUX
ejpam-4478	333	4	the	the	DET
ejpam-4478	333	5	class	class	NOUN
ejpam-4478	333	6	of	of	ADP
ejpam-4478	333	7	connected	connected	ADJ
ejpam-4478	333	8	graphs	graph	NOUN
ejpam-4478	333	9	whose	whose	DET
ejpam-4478	333	10	complements	complement	NOUN
ejpam-4478	333	11	are	be	AUX
ejpam-4478	333	12	also	also	ADV
ejpam-4478	333	13	connected	connect	VERB
ejpam-4478	333	14	.	.	PUNCT
ejpam-4478	334	1	for	for	ADP
ejpam-4478	334	2	any	any	DET
ejpam-4478	334	3	graph	graph	NOUN
ejpam-4478	334	4	g	g	ADP
ejpam-4478	334	5	∈	∈	PROPN
ejpam-4478	334	6	g	g	NOUN
ejpam-4478	334	7	of	of	ADP
ejpam-4478	334	8	order	order	NOUN
ejpam-4478	334	9	n	n	PRON
ejpam-4478	334	10	≥	≥	NOUN
ejpam-4478	334	11	4	4	NUM
ejpam-4478	334	12	,	,	PUNCT
ejpam-4478	334	13	γkgr(g	γkgr(g	NOUN
ejpam-4478	334	14	)	)	PUNCT
ejpam-4478	334	15	=	=	SYM
ejpam-4478	334	16	2	2	NUM
ejpam-4478	334	17	if	if	SCONJ
ejpam-4478	334	18	and	and	CCONJ
ejpam-4478	334	19	only	only	ADV
ejpam-4478	334	20	if	if	SCONJ
ejpam-4478	334	21	k	k	PROPN
ejpam-4478	334	22	≥	≥	X
ejpam-4478	334	23	max{diam(g	max{diam(g	PROPN
ejpam-4478	334	24	)	)	PUNCT
ejpam-4478	334	25	,	,	PUNCT
ejpam-4478	334	26	diam(g	diam(g	NOUN
ejpam-4478	334	27	)	)	PUNCT
ejpam-4478	334	28	}	}	PUNCT
ejpam-4478	334	29	,	,	PUNCT
ejpam-4478	334	30	where	where	SCONJ
ejpam-4478	334	31	k	k	PROPN
ejpam-4478	334	32	∈	∈	PROPN
ejpam-4478	334	33	z+	z+	PUNCT
ejpam-4478	334	34	.	.	PUNCT
ejpam-4478	335	1	proof	proof	NOUN
ejpam-4478	335	2	.	.	PUNCT
ejpam-4478	336	1	suppose	suppose	VERB
ejpam-4478	336	2	that	that	SCONJ
ejpam-4478	336	3	g	g	PROPN
ejpam-4478	336	4	denote	denote	VERB
ejpam-4478	336	5	the	the	DET
ejpam-4478	336	6	class	class	NOUN
ejpam-4478	336	7	of	of	ADP
ejpam-4478	336	8	connected	connected	ADJ
ejpam-4478	336	9	graphs	graph	NOUN
ejpam-4478	336	10	whose	whose	DET
ejpam-4478	336	11	complements	complement	NOUN
ejpam-4478	336	12	are	be	AUX
ejpam-4478	336	13	also	also	ADV
ejpam-4478	336	14	connected	connect	VERB
ejpam-4478	336	15	.	.	PUNCT
ejpam-4478	337	1	let	let	VERB
ejpam-4478	337	2	g	g	PROPN
ejpam-4478	337	3	∈	∈	PROPN
ejpam-4478	337	4	g	g	PROPN
ejpam-4478	337	5	be	be	AUX
ejpam-4478	337	6	any	any	DET
ejpam-4478	337	7	graph	graph	NOUN
ejpam-4478	337	8	of	of	ADP
ejpam-4478	337	9	order	order	NOUN
ejpam-4478	337	10	n	n	PRON
ejpam-4478	337	11	≥	≥	NOUN
ejpam-4478	337	12	4	4	NUM
ejpam-4478	337	13	and	and	CCONJ
ejpam-4478	337	14	let	let	VERB
ejpam-4478	337	15	its	its	PRON
ejpam-4478	337	16	vertex	vertex	NOUN
ejpam-4478	337	17	set	set	NOUN
ejpam-4478	337	18	be	be	AUX
ejpam-4478	337	19	v	v	ADP
ejpam-4478	337	20	(	(	PUNCT
ejpam-4478	337	21	g	g	NOUN
ejpam-4478	337	22	)	)	PUNCT
ejpam-4478	337	23	=	=	SYM
ejpam-4478	337	24	{	{	PUNCT
ejpam-4478	337	25	u1	u1	NOUN
ejpam-4478	337	26	,	,	PUNCT
ejpam-4478	337	27	u2	u2	NOUN
ejpam-4478	337	28	,	,	PUNCT
ejpam-4478	337	29	u3	u3	NOUN
ejpam-4478	337	30	,	,	PUNCT
ejpam-4478	337	31	.	.	PUNCT
ejpam-4478	337	32	.	.	PUNCT
ejpam-4478	338	1	.	.	PUNCT
ejpam-4478	339	1	,	,	PUNCT
ejpam-4478	339	2	un−1	un−1	PROPN
ejpam-4478	339	3	,	,	PUNCT
ejpam-4478	339	4	un	un	ADJ
ejpam-4478	339	5	}	}	PUNCT
ejpam-4478	339	6	.	.	PUNCT
ejpam-4478	340	1	since	since	SCONJ
ejpam-4478	340	2	g	g	PROPN
ejpam-4478	340	3	∈	∈	PROPN
ejpam-4478	340	4	g	g	PROPN
ejpam-4478	340	5	,	,	PUNCT
ejpam-4478	340	6	with	with	ADP
ejpam-4478	340	7	n	n	NOUN
ejpam-4478	340	8	=	=	SYM
ejpam-4478	340	9	|v	|v	X
ejpam-4478	340	10	(	(	PUNCT
ejpam-4478	340	11	g)|	g)|	X
ejpam-4478	340	12	≥	≥	NOUN
ejpam-4478	340	13	4	4	NUM
ejpam-4478	340	14	,	,	PUNCT
ejpam-4478	340	15	the	the	DET
ejpam-4478	340	16	distances	distance	NOUN
ejpam-4478	340	17	and	and	CCONJ
ejpam-4478	340	18	eccentricities	eccentricity	NOUN
ejpam-4478	340	19	in	in	ADP
ejpam-4478	340	20	graph	graph	NOUN
ejpam-4478	340	21	g	g	PROPN
ejpam-4478	340	22	are	be	AUX
ejpam-4478	340	23	finite	finite	ADJ
ejpam-4478	340	24	.	.	PUNCT
ejpam-4478	341	1	thus	thus	ADV
ejpam-4478	341	2	,	,	PUNCT
ejpam-4478	341	3	diam(g	diam(g	PROPN
ejpam-4478	341	4	)	)	PUNCT
ejpam-4478	341	5	,	,	PUNCT
ejpam-4478	341	6	diam(g	diam(g	NOUN
ejpam-4478	341	7	)	)	PUNCT
ejpam-4478	341	8	<	<	X
ejpam-4478	341	9	∞.	∞.	PROPN
ejpam-4478	341	10	now	now	ADV
ejpam-4478	341	11	,	,	PUNCT
ejpam-4478	341	12	let	let	VERB
ejpam-4478	341	13	us	we	PRON
ejpam-4478	341	14	consider	consider	VERB
ejpam-4478	341	15	the	the	DET
ejpam-4478	341	16	following	following	NOUN
ejpam-4478	341	17	:	:	PUNCT
ejpam-4478	341	18	(	(	PUNCT
ejpam-4478	341	19	=	=	NOUN
ejpam-4478	341	20	⇒	⇒	NOUN
ejpam-4478	341	21	)	)	PUNCT
ejpam-4478	341	22	assume	assume	VERB
ejpam-4478	341	23	that	that	SCONJ
ejpam-4478	341	24	γkgr(g	γkgr(g	NOUN
ejpam-4478	341	25	)	)	PUNCT
ejpam-4478	341	26	=	=	SYM
ejpam-4478	342	1	2	2	X
ejpam-4478	342	2	.	.	X
ejpam-4478	342	3	we	we	PRON
ejpam-4478	342	4	let	let	VERB
ejpam-4478	342	5	fk	fk	INTJ
ejpam-4478	342	6	:	:	PUNCT
ejpam-4478	342	7	v	v	X
ejpam-4478	342	8	(	(	PUNCT
ejpam-4478	342	9	g	g	NOUN
ejpam-4478	342	10	)	)	PUNCT
ejpam-4478	342	11	−→	−→	NOUN
ejpam-4478	342	12	{	{	PUNCT
ejpam-4478	342	13	0	0	NUM
ejpam-4478	342	14	,	,	PUNCT
ejpam-4478	342	15	1	1	NUM
ejpam-4478	342	16	,	,	PUNCT
ejpam-4478	342	17	2	2	NUM
ejpam-4478	342	18	}	}	PUNCT
ejpam-4478	342	19	be	be	AUX
ejpam-4478	342	20	a	a	DET
ejpam-4478	342	21	γkgr(g)−	γkgr(g)−	PROPN
ejpam-4478	342	22	function	function	NOUN
ejpam-4478	342	23	,	,	PUNCT
ejpam-4478	342	24	where	where	SCONJ
ejpam-4478	342	25	k	k	PROPN
ejpam-4478	342	26	∈	∈	PROPN
ejpam-4478	342	27	z+	z+	PUNCT
ejpam-4478	342	28	.	.	PUNCT
ejpam-4478	343	1	this	this	PRON
ejpam-4478	343	2	implies	imply	VERB
ejpam-4478	343	3	that	that	SCONJ
ejpam-4478	343	4	,	,	PUNCT
ejpam-4478	343	5	w(fk	w(fk	NOUN
ejpam-4478	343	6	)	)	PUNCT
ejpam-4478	343	7	=	=	PUNCT
ejpam-4478	344	1	∑	∑	PUNCT
ejpam-4478	344	2	ui∈v	ui∈v	X
ejpam-4478	344	3	(	(	PUNCT
ejpam-4478	344	4	g	g	NOUN
ejpam-4478	344	5	)	)	PUNCT
ejpam-4478	344	6	fk(ui	fk(ui	PROPN
ejpam-4478	344	7	)	)	PUNCT
ejpam-4478	345	1	=	=	SYM
ejpam-4478	345	2	γkgr(g	γkgr(g	NUM
ejpam-4478	345	3	)	)	PUNCT
ejpam-4478	345	4	=	=	SYM
ejpam-4478	345	5	2	2	NUM
ejpam-4478	345	6	,	,	PUNCT
ejpam-4478	345	7	where	where	SCONJ
ejpam-4478	345	8	i	i	PRON
ejpam-4478	345	9	=	=	NOUN
ejpam-4478	345	10	1	1	NUM
ejpam-4478	345	11	,	,	PUNCT
ejpam-4478	345	12	2	2	NUM
ejpam-4478	345	13	,	,	PUNCT
ejpam-4478	345	14	3	3	NUM
ejpam-4478	345	15	,	,	PUNCT
ejpam-4478	345	16	.	.	PUNCT
ejpam-4478	345	17	.	.	PUNCT
ejpam-4478	345	18	.	.	PUNCT
ejpam-4478	346	1	,	,	PUNCT
ejpam-4478	346	2	n	n	CCONJ
ejpam-4478	346	3	−	−	PROPN
ejpam-4478	346	4	1	1	NUM
ejpam-4478	346	5	,	,	PUNCT
ejpam-4478	346	6	n.	n.	NOUN
ejpam-4478	346	7	this	this	PRON
ejpam-4478	346	8	implies	imply	VERB
ejpam-4478	346	9	further	far	ADV
ejpam-4478	346	10	that	that	SCONJ
ejpam-4478	346	11	,	,	PUNCT
ejpam-4478	346	12	for	for	ADP
ejpam-4478	346	13	k	k	PROPN
ejpam-4478	346	14	∈	∈	PROPN
ejpam-4478	346	15	z+	z+	NUM
ejpam-4478	346	16	,	,	PUNCT
ejpam-4478	346	17	we	we	PRON
ejpam-4478	346	18	may	may	AUX
ejpam-4478	346	19	define	define	VERB
ejpam-4478	346	20	the	the	DET
ejpam-4478	346	21	γkgr(g)−	γkgr(g)−	PROPN
ejpam-4478	346	22	function	function	NOUN
ejpam-4478	346	23	fk	fk	INTJ
ejpam-4478	346	24	as	as	ADV
ejpam-4478	346	25	fk	fk	INTJ
ejpam-4478	346	26	=	=	SYM
ejpam-4478	346	27	(	(	PUNCT
ejpam-4478	346	28	v	v	NUM
ejpam-4478	346	29	fk	fk	INTJ
ejpam-4478	346	30	0	0	NUM
ejpam-4478	347	1	(	(	PUNCT
ejpam-4478	347	2	g	g	NOUN
ejpam-4478	347	3	)	)	PUNCT
ejpam-4478	347	4	,	,	PUNCT
ejpam-4478	347	5	v	v	X
ejpam-4478	347	6	fk	fk	INTJ
ejpam-4478	347	7	1	1	NUM
ejpam-4478	347	8	(	(	PUNCT
ejpam-4478	347	9	g	g	NOUN
ejpam-4478	347	10	)	)	PUNCT
ejpam-4478	347	11	,	,	PUNCT
ejpam-4478	348	1	v	v	X
ejpam-4478	348	2	fk	fk	INTJ
ejpam-4478	348	3	2	2	NUM
ejpam-4478	348	4	(	(	PUNCT
ejpam-4478	348	5	g	g	NOUN
ejpam-4478	348	6	)	)	PUNCT
ejpam-4478	348	7	)	)	PUNCT
ejpam-4478	348	8	,	,	PUNCT
ejpam-4478	348	9	where	where	SCONJ
ejpam-4478	348	10	there	there	PRON
ejpam-4478	348	11	exists	exist	VERB
ejpam-4478	348	12	a	a	DET
ejpam-4478	348	13	vertex	vertex	NOUN
ejpam-4478	348	14	uj	uj	X
ejpam-4478	348	15	∈	∈	PROPN
ejpam-4478	348	16	v	v	ADP
ejpam-4478	348	17	(	(	PUNCT
ejpam-4478	348	18	g	g	NOUN
ejpam-4478	348	19	)	)	PUNCT
ejpam-4478	348	20	=	=	NOUN
ejpam-4478	348	21	v	v	X
ejpam-4478	348	22	(	(	PUNCT
ejpam-4478	348	23	g	g	NOUN
ejpam-4478	348	24	)	)	PUNCT
ejpam-4478	349	1	such	such	ADJ
ejpam-4478	349	2	that	that	PRON
ejpam-4478	349	3	v	v	NUM
ejpam-4478	350	1	fk	fk	INTJ
ejpam-4478	350	2	0	0	NUM
ejpam-4478	351	1	(	(	PUNCT
ejpam-4478	351	2	g	g	NOUN
ejpam-4478	351	3	)	)	PUNCT
ejpam-4478	351	4	=	=	SYM
ejpam-4478	351	5	{	{	PUNCT
ejpam-4478	351	6	ui	ui	PROPN
ejpam-4478	351	7	̸=j	̸=j	PROPN
ejpam-4478	351	8	∈	∈	PROPN
ejpam-4478	351	9	v	v	NOUN
ejpam-4478	351	10	(	(	PUNCT
ejpam-4478	351	11	g	g	NOUN
ejpam-4478	351	12	)	)	PUNCT
ejpam-4478	351	13	:	:	PUNCT
ejpam-4478	352	1	i	i	PRON
ejpam-4478	352	2	∈	∈	PROPN
ejpam-4478	352	3	{	{	PUNCT
ejpam-4478	352	4	1	1	NUM
ejpam-4478	352	5	,	,	PUNCT
ejpam-4478	352	6	2	2	NUM
ejpam-4478	352	7	,	,	PUNCT
ejpam-4478	352	8	3	3	NUM
ejpam-4478	352	9	,	,	PUNCT
ejpam-4478	352	10	.	.	PUNCT
ejpam-4478	352	11	.	.	PUNCT
ejpam-4478	352	12	.	.	PUNCT
ejpam-4478	352	13	,	,	PUNCT
ejpam-4478	352	14	n	n	CCONJ
ejpam-4478	352	15	}	}	PUNCT
ejpam-4478	352	16	}	}	PUNCT
ejpam-4478	352	17	=	=	SYM
ejpam-4478	352	18	v	v	X
ejpam-4478	352	19	(	(	PUNCT
ejpam-4478	352	20	g)\{uj	g)\{uj	NOUN
ejpam-4478	352	21	}	}	PUNCT
ejpam-4478	352	22	,	,	PUNCT
ejpam-4478	352	23	v	v	X
ejpam-4478	352	24	fk	fk	INTJ
ejpam-4478	352	25	1	1	NUM
ejpam-4478	352	26	(	(	PUNCT
ejpam-4478	352	27	g	g	NOUN
ejpam-4478	352	28	)	)	PUNCT
ejpam-4478	352	29	=	=	NOUN
ejpam-4478	352	30	∅	∅	NOUN
ejpam-4478	352	31	,	,	PUNCT
ejpam-4478	352	32	and	and	CCONJ
ejpam-4478	352	33	v	v	ADP
ejpam-4478	352	34	fk	fk	INTJ
ejpam-4478	352	35	2	2	NUM
ejpam-4478	352	36	(	(	PUNCT
ejpam-4478	352	37	g	g	NOUN
ejpam-4478	352	38	)	)	PUNCT
ejpam-4478	352	39	=	=	SYM
ejpam-4478	352	40	{	{	PUNCT
ejpam-4478	352	41	uj	uj	PROPN
ejpam-4478	352	42	}	}	PUNCT
ejpam-4478	352	43	,	,	PUNCT
ejpam-4478	352	44	for	for	ADP
ejpam-4478	352	45	all	all	DET
ejpam-4478	352	46	k	k	PROPN
ejpam-4478	352	47	∈	∈	PROPN
ejpam-4478	352	48	z+	z+	NUM
ejpam-4478	352	49	.	.	PUNCT
ejpam-4478	353	1	thus	thus	ADV
ejpam-4478	353	2	,	,	PUNCT
ejpam-4478	353	3	for	for	ADP
ejpam-4478	353	4	all	all	DET
ejpam-4478	353	5	ui	ui	PROPN
ejpam-4478	353	6	̸=j	̸=j	PROPN
ejpam-4478	353	7	∈	∈	PROPN
ejpam-4478	353	8	v	v	NOUN
ejpam-4478	353	9	(	(	PUNCT
ejpam-4478	353	10	g	g	NOUN
ejpam-4478	353	11	)	)	PUNCT
ejpam-4478	353	12	=	=	NOUN
ejpam-4478	353	13	v	v	X
ejpam-4478	353	14	(	(	PUNCT
ejpam-4478	353	15	g	g	NOUN
ejpam-4478	353	16	)	)	PUNCT
ejpam-4478	353	17	,	,	PUNCT
ejpam-4478	353	18	where	where	SCONJ
ejpam-4478	353	19	i	i	PRON
ejpam-4478	353	20	∈	∈	PROPN
ejpam-4478	353	21	{	{	PUNCT
ejpam-4478	353	22	1	1	NUM
ejpam-4478	353	23	,	,	PUNCT
ejpam-4478	353	24	2	2	NUM
ejpam-4478	353	25	,	,	PUNCT
ejpam-4478	353	26	3	3	NUM
ejpam-4478	353	27	,	,	PUNCT
ejpam-4478	353	28	.	.	PUNCT
ejpam-4478	353	29	.	.	PUNCT
ejpam-4478	353	30	.	.	PUNCT
ejpam-4478	353	31	,	,	PUNCT
ejpam-4478	353	32	n	n	CCONJ
ejpam-4478	353	33	}	}	PUNCT
ejpam-4478	353	34	,	,	PUNCT
ejpam-4478	353	35	and	and	CCONJ
ejpam-4478	353	36	for	for	ADP
ejpam-4478	353	37	all	all	DET
ejpam-4478	353	38	k	k	PROPN
ejpam-4478	353	39	∈	∈	PROPN
ejpam-4478	353	40	z+	z+	X
ejpam-4478	353	41	,	,	PUNCT
ejpam-4478	353	42	we	we	PRON
ejpam-4478	353	43	have	have	VERB
ejpam-4478	353	44	fk(ui	fk(ui	PROPN
ejpam-4478	353	45	̸=j	̸=j	NOUN
ejpam-4478	353	46	)	)	PUNCT
ejpam-4478	353	47	=	=	SYM
ejpam-4478	353	48	0	0	NUM
ejpam-4478	353	49	and	and	CCONJ
ejpam-4478	353	50	fk(uj	fk(uj	NOUN
ejpam-4478	353	51	)	)	PUNCT
ejpam-4478	354	1	=	=	SYM
ejpam-4478	354	2	2	2	NUM
ejpam-4478	354	3	and	and	CCONJ
ejpam-4478	354	4	this	this	PRON
ejpam-4478	354	5	implies	imply	VERB
ejpam-4478	354	6	further	far	ADV
ejpam-4478	354	7	that	that	SCONJ
ejpam-4478	354	8	,	,	PUNCT
ejpam-4478	354	9	for	for	ADP
ejpam-4478	354	10	a	a	DET
ejpam-4478	354	11	distance	distance	NOUN
ejpam-4478	354	12	k	k	PROPN
ejpam-4478	354	13	∈	∈	PROPN
ejpam-4478	354	14	z+	z+	X
ejpam-4478	354	15	,	,	PUNCT
ejpam-4478	354	16	we	we	PRON
ejpam-4478	354	17	have	have	VERB
ejpam-4478	354	18	d(ui	d(ui	PROPN
ejpam-4478	354	19	̸=j	̸=j	PROPN
ejpam-4478	354	20	,	,	PUNCT
ejpam-4478	354	21	uj	uj	PROPN
ejpam-4478	354	22	)	)	PUNCT
ejpam-4478	354	23	≤	≤	PUNCT
ejpam-4478	355	1	k	k	X
ejpam-4478	355	2	(	(	PUNCT
ejpam-4478	355	3	1	1	X
ejpam-4478	355	4	)	)	PUNCT
ejpam-4478	355	5	which	which	PRON
ejpam-4478	355	6	means	mean	VERB
ejpam-4478	355	7	that	that	SCONJ
ejpam-4478	355	8	the	the	DET
ejpam-4478	355	9	distances	distance	NOUN
ejpam-4478	355	10	of	of	ADP
ejpam-4478	355	11	vertices	vertex	NOUN
ejpam-4478	355	12	ui	ui	PROPN
ejpam-4478	355	13	̸=j	̸=j	PROPN
ejpam-4478	355	14	from	from	ADP
ejpam-4478	355	15	vertex	vertex	NOUN
ejpam-4478	355	16	uj	uj	PROPN
ejpam-4478	355	17	is	be	AUX
ejpam-4478	355	18	at	at	ADP
ejpam-4478	355	19	most	most	ADJ
ejpam-4478	355	20	k.	k.	NOUN
ejpam-4478	355	21	rewriting	rewrite	VERB
ejpam-4478	355	22	(	(	PUNCT
ejpam-4478	355	23	1	1	NUM
ejpam-4478	355	24	)	)	PUNCT
ejpam-4478	355	25	,	,	PUNCT
ejpam-4478	355	26	we	we	PRON
ejpam-4478	355	27	get	get	VERB
ejpam-4478	355	28	d(uj	d(uj	PRON
ejpam-4478	355	29	,	,	PUNCT
ejpam-4478	355	30	ui	ui	PROPN
ejpam-4478	355	31	̸=j	̸=j	PROPN
ejpam-4478	355	32	)	)	PUNCT
ejpam-4478	355	33	≤	≤	PUNCT
ejpam-4478	356	1	k	k	X
ejpam-4478	356	2	(	(	PUNCT
ejpam-4478	356	3	2	2	NUM
ejpam-4478	356	4	)	)	PUNCT
ejpam-4478	356	5	g.	g.	PROPN
ejpam-4478	356	6	entero	entero	PROPN
ejpam-4478	356	7	,	,	PUNCT
ejpam-4478	356	8	s.	s.	PROPN
ejpam-4478	356	9	espinola	espinola	PROPN
ejpam-4478	356	10	/	/	SYM
ejpam-4478	356	11	eur	eur	PROPN
ejpam-4478	356	12	.	.	PUNCT
ejpam-4478	357	1	j.	j.	PROPN
ejpam-4478	357	2	pure	pure	PROPN
ejpam-4478	357	3	appl	appl	PROPN
ejpam-4478	357	4	.	.	PROPN
ejpam-4478	357	5	math	math	PROPN
ejpam-4478	357	6	,	,	PUNCT
ejpam-4478	357	7	16	16	NUM
ejpam-4478	357	8	(	(	PUNCT
ejpam-4478	357	9	1	1	NUM
ejpam-4478	357	10	)	)	PUNCT
ejpam-4478	357	11	(	(	PUNCT
ejpam-4478	357	12	2023	2023	NUM
ejpam-4478	357	13	)	)	PUNCT
ejpam-4478	357	14	,	,	PUNCT
ejpam-4478	357	15	44	44	NUM
ejpam-4478	357	16	-	-	SYM
ejpam-4478	357	17	61	61	NUM
ejpam-4478	357	18	55	55	NUM
ejpam-4478	357	19	since	since	SCONJ
ejpam-4478	357	20	we	we	PRON
ejpam-4478	357	21	are	be	AUX
ejpam-4478	357	22	talking	talk	VERB
ejpam-4478	357	23	about	about	ADP
ejpam-4478	357	24	distances	distance	NOUN
ejpam-4478	357	25	here	here	ADV
ejpam-4478	357	26	,	,	PUNCT
ejpam-4478	357	27	we	we	PRON
ejpam-4478	357	28	have	have	AUX
ejpam-4478	357	29	d(uj	d(uj	PROPN
ejpam-4478	357	30	,	,	PUNCT
ejpam-4478	357	31	ui	ui	PROPN
ejpam-4478	357	32	̸=j	̸=j	NOUN
ejpam-4478	357	33	)	)	PUNCT
ejpam-4478	357	34	,	,	PUNCT
ejpam-4478	358	1	k	k	X
ejpam-4478	358	2	>	>	X
ejpam-4478	358	3	0	0	NUM
ejpam-4478	358	4	,	,	PUNCT
ejpam-4478	358	5	and	and	CCONJ
ejpam-4478	358	6	so	so	ADV
ejpam-4478	358	7	,	,	PUNCT
ejpam-4478	358	8	from	from	ADP
ejpam-4478	358	9	(	(	PUNCT
ejpam-4478	358	10	2	2	NUM
ejpam-4478	358	11	)	)	PUNCT
ejpam-4478	358	12	,	,	PUNCT
ejpam-4478	358	13	with	with	ADP
ejpam-4478	358	14	respect	respect	NOUN
ejpam-4478	358	15	to	to	ADP
ejpam-4478	358	16	g	g	NOUN
ejpam-4478	358	17	and	and	CCONJ
ejpam-4478	358	18	g	g	NOUN
ejpam-4478	358	19	,	,	PUNCT
ejpam-4478	358	20	we	we	PRON
ejpam-4478	358	21	have	have	VERB
ejpam-4478	358	22	max{d(uj	max{d(uj	PROPN
ejpam-4478	358	23	,	,	PUNCT
ejpam-4478	358	24	ui	ui	PROPN
ejpam-4478	358	25	̸=j	̸=j	PROPN
ejpam-4478	358	26	)	)	PUNCT
ejpam-4478	358	27	:	:	PUNCT
ejpam-4478	358	28	ui	ui	PROPN
ejpam-4478	358	29	∈	∈	PROPN
ejpam-4478	358	30	v	v	ADP
ejpam-4478	358	31	(	(	PUNCT
ejpam-4478	358	32	g	g	NOUN
ejpam-4478	358	33	)	)	PUNCT
ejpam-4478	358	34	where	where	SCONJ
ejpam-4478	358	35	i	i	PRON
ejpam-4478	358	36	∈	∈	PROPN
ejpam-4478	358	37	{	{	PUNCT
ejpam-4478	358	38	1	1	NUM
ejpam-4478	358	39	,	,	PUNCT
ejpam-4478	358	40	2	2	NUM
ejpam-4478	358	41	,	,	PUNCT
ejpam-4478	358	42	3	3	NUM
ejpam-4478	358	43	,	,	PUNCT
ejpam-4478	358	44	.	.	PUNCT
ejpam-4478	358	45	.	.	PUNCT
ejpam-4478	358	46	.	.	PUNCT
ejpam-4478	358	47	,	,	PUNCT
ejpam-4478	358	48	n	n	CCONJ
ejpam-4478	358	49	}	}	PUNCT
ejpam-4478	358	50	}	}	PUNCT
ejpam-4478	358	51	≤	≤	PROPN
ejpam-4478	358	52	max{k	max{k	NOUN
ejpam-4478	358	53	}	}	PUNCT
ejpam-4478	358	54	which	which	PRON
ejpam-4478	358	55	implies	imply	VERB
ejpam-4478	358	56	that	that	SCONJ
ejpam-4478	358	57	max{d(uj	max{d(uj	PROPN
ejpam-4478	358	58	,	,	PUNCT
ejpam-4478	358	59	ui	ui	PROPN
ejpam-4478	358	60	̸=j	̸=j	PROPN
ejpam-4478	358	61	)	)	PUNCT
ejpam-4478	358	62	:	:	PUNCT
ejpam-4478	358	63	ui	ui	PROPN
ejpam-4478	358	64	∈	∈	PROPN
ejpam-4478	358	65	v	v	ADP
ejpam-4478	358	66	(	(	PUNCT
ejpam-4478	358	67	g	g	NOUN
ejpam-4478	358	68	)	)	PUNCT
ejpam-4478	358	69	where	where	SCONJ
ejpam-4478	358	70	i	i	PRON
ejpam-4478	358	71	∈	∈	PROPN
ejpam-4478	358	72	{	{	PUNCT
ejpam-4478	358	73	1	1	NUM
ejpam-4478	358	74	,	,	PUNCT
ejpam-4478	358	75	2	2	NUM
ejpam-4478	358	76	,	,	PUNCT
ejpam-4478	358	77	3	3	NUM
ejpam-4478	358	78	,	,	PUNCT
ejpam-4478	358	79	.	.	PUNCT
ejpam-4478	358	80	.	.	PUNCT
ejpam-4478	358	81	.	.	PUNCT
ejpam-4478	358	82	,	,	PUNCT
ejpam-4478	358	83	n	n	CCONJ
ejpam-4478	358	84	}	}	PUNCT
ejpam-4478	358	85	}	}	PUNCT
ejpam-4478	358	86	≤	≤	NUM
ejpam-4478	358	87	k	k	X
ejpam-4478	358	88	(	(	PUNCT
ejpam-4478	358	89	3	3	NUM
ejpam-4478	358	90	)	)	PUNCT
ejpam-4478	358	91	from	from	ADP
ejpam-4478	358	92	(	(	PUNCT
ejpam-4478	358	93	3	3	NUM
ejpam-4478	358	94	)	)	PUNCT
ejpam-4478	358	95	,	,	PUNCT
ejpam-4478	358	96	ecc(uj	ecc(uj	NOUN
ejpam-4478	358	97	)	)	PUNCT
ejpam-4478	358	98	≤	≤	PUNCT
ejpam-4478	358	99	k.	k.	NOUN
ejpam-4478	359	1	(	(	PUNCT
ejpam-4478	359	2	4	4	NUM
ejpam-4478	359	3	)	)	PUNCT
ejpam-4478	359	4	since	since	SCONJ
ejpam-4478	359	5	uj	uj	PROPN
ejpam-4478	359	6	is	be	AUX
ejpam-4478	359	7	the	the	DET
ejpam-4478	359	8	only	only	ADJ
ejpam-4478	359	9	vertex	vertex	NOUN
ejpam-4478	359	10	in	in	ADP
ejpam-4478	359	11	the	the	DET
ejpam-4478	359	12	set	set	ADJ
ejpam-4478	359	13	partition	partition	NOUN
ejpam-4478	359	14	v	v	ADP
ejpam-4478	359	15	fk	fk	INTJ
ejpam-4478	359	16	2	2	NUM
ejpam-4478	359	17	(	(	PUNCT
ejpam-4478	359	18	g	g	NOUN
ejpam-4478	359	19	)	)	PUNCT
ejpam-4478	359	20	(	(	PUNCT
ejpam-4478	359	21	note	note	VERB
ejpam-4478	359	22	that	that	SCONJ
ejpam-4478	359	23	,	,	PUNCT
ejpam-4478	359	24	v	v	X
ejpam-4478	359	25	fk	fk	INTJ
ejpam-4478	359	26	2	2	NUM
ejpam-4478	359	27	(	(	PUNCT
ejpam-4478	359	28	g)=v	g)=v	X
ejpam-4478	359	29	fk	fk	INTJ
ejpam-4478	359	30	2	2	NUM
ejpam-4478	359	31	(	(	PUNCT
ejpam-4478	359	32	g	g	NOUN
ejpam-4478	359	33	)	)	PUNCT
ejpam-4478	359	34	)	)	PUNCT
ejpam-4478	359	35	at	at	ADP
ejpam-4478	359	36	distance	distance	NOUN
ejpam-4478	359	37	k	k	PROPN
ejpam-4478	359	38	∈	∈	PROPN
ejpam-4478	359	39	z+	z+	X
ejpam-4478	359	40	,	,	PUNCT
ejpam-4478	359	41	for	for	ADP
ejpam-4478	359	42	all	all	DET
ejpam-4478	359	43	i	i	PRON
ejpam-4478	359	44	̸=	̸=	PROPN
ejpam-4478	359	45	j	j	PROPN
ejpam-4478	359	46	,	,	PUNCT
ejpam-4478	359	47	ecc(ui	ecc(ui	PROPN
ejpam-4478	359	48	̸=j	̸=j	NOUN
ejpam-4478	359	49	)	)	PUNCT
ejpam-4478	359	50	≤	≤	NOUN
ejpam-4478	359	51	ecc(uj	ecc(uj	NOUN
ejpam-4478	359	52	)	)	PUNCT
ejpam-4478	359	53	(	(	PUNCT
ejpam-4478	359	54	5	5	NUM
ejpam-4478	359	55	)	)	PUNCT
ejpam-4478	359	56	hence	hence	ADV
ejpam-4478	359	57	,	,	PUNCT
ejpam-4478	359	58	from	from	ADP
ejpam-4478	359	59	(	(	PUNCT
ejpam-4478	359	60	4	4	NUM
ejpam-4478	359	61	)	)	PUNCT
ejpam-4478	359	62	and	and	CCONJ
ejpam-4478	359	63	(	(	PUNCT
ejpam-4478	359	64	5	5	NUM
ejpam-4478	359	65	)	)	PUNCT
ejpam-4478	359	66	,	,	PUNCT
ejpam-4478	359	67	we	we	PRON
ejpam-4478	359	68	have	have	VERB
ejpam-4478	359	69	ecc(ui	ecc(ui	NOUN
ejpam-4478	359	70	̸=j	̸=j	NOUN
ejpam-4478	359	71	)	)	PUNCT
ejpam-4478	359	72	≤	≤	PUNCT
ejpam-4478	360	1	k	k	X
ejpam-4478	360	2	(	(	PUNCT
ejpam-4478	360	3	6	6	NUM
ejpam-4478	360	4	)	)	PUNCT
ejpam-4478	360	5	thus	thus	ADV
ejpam-4478	360	6	,	,	PUNCT
ejpam-4478	360	7	for	for	ADP
ejpam-4478	360	8	all	all	DET
ejpam-4478	360	9	ui	ui	NOUN
ejpam-4478	360	10	∈	∈	PROPN
ejpam-4478	360	11	v	v	ADP
ejpam-4478	360	12	(	(	PUNCT
ejpam-4478	360	13	g	g	NOUN
ejpam-4478	360	14	)	)	PUNCT
ejpam-4478	360	15	=	=	NOUN
ejpam-4478	360	16	v	v	X
ejpam-4478	360	17	(	(	PUNCT
ejpam-4478	360	18	g	g	NOUN
ejpam-4478	360	19	)	)	PUNCT
ejpam-4478	360	20	,	,	PUNCT
ejpam-4478	360	21	including	include	VERB
ejpam-4478	360	22	vertex	vertex	NOUN
ejpam-4478	360	23	uj	uj	PROPN
ejpam-4478	360	24	,	,	PUNCT
ejpam-4478	360	25	from	from	ADP
ejpam-4478	360	26	(	(	PUNCT
ejpam-4478	360	27	4	4	NUM
ejpam-4478	360	28	)	)	PUNCT
ejpam-4478	360	29	and	and	CCONJ
ejpam-4478	360	30	(	(	PUNCT
ejpam-4478	360	31	6	6	NUM
ejpam-4478	360	32	)	)	PUNCT
ejpam-4478	360	33	,	,	PUNCT
ejpam-4478	360	34	(	(	PUNCT
ejpam-4478	360	35	ui	ui	NOUN
ejpam-4478	360	36	)	)	PUNCT
ejpam-4478	360	37	≤	≤	PUNCT
ejpam-4478	361	1	k	k	NOUN
ejpam-4478	361	2	,	,	PUNCT
ejpam-4478	361	3	that	that	ADV
ejpam-4478	361	4	is	is	ADV
ejpam-4478	361	5	,	,	PUNCT
ejpam-4478	361	6	with	with	ADP
ejpam-4478	361	7	respect	respect	NOUN
ejpam-4478	361	8	to	to	AUX
ejpam-4478	361	9	graph	graph	VERB
ejpam-4478	361	10	g	g	PROPN
ejpam-4478	361	11	and	and	CCONJ
ejpam-4478	361	12	g	g	NOUN
ejpam-4478	361	13	,	,	PUNCT
ejpam-4478	361	14	ecc(u1	ecc(u1	PROPN
ejpam-4478	361	15	)	)	PUNCT
ejpam-4478	361	16	,	,	PUNCT
ejpam-4478	361	17	ecc(u2	ecc(u2	PROPN
ejpam-4478	361	18	)	)	PUNCT
ejpam-4478	361	19	,	,	PUNCT
ejpam-4478	361	20	ecc(u3	ecc(u3	NOUN
ejpam-4478	361	21	)	)	PUNCT
ejpam-4478	361	22	,	,	PUNCT
ejpam-4478	361	23	.	.	PUNCT
ejpam-4478	361	24	.	.	PUNCT
ejpam-4478	361	25	.	.	PUNCT
ejpam-4478	362	1	,	,	PUNCT
ejpam-4478	362	2	ecc(un	ecc(un	NUM
ejpam-4478	362	3	)	)	PUNCT
ejpam-4478	362	4	≤	≤	PUNCT
ejpam-4478	363	1	k	k	X
ejpam-4478	363	2	(	(	PUNCT
ejpam-4478	363	3	7	7	X
ejpam-4478	363	4	)	)	PUNCT
ejpam-4478	363	5	taking	take	VERB
ejpam-4478	363	6	the	the	DET
ejpam-4478	363	7	maximum	maximum	NOUN
ejpam-4478	363	8	of	of	ADP
ejpam-4478	363	9	both	both	DET
ejpam-4478	363	10	sides	side	NOUN
ejpam-4478	363	11	of	of	ADP
ejpam-4478	363	12	(	(	PUNCT
ejpam-4478	363	13	7	7	NUM
ejpam-4478	363	14	)	)	PUNCT
ejpam-4478	363	15	,	,	PUNCT
ejpam-4478	363	16	we	we	PRON
ejpam-4478	363	17	get	get	VERB
ejpam-4478	363	18	max{ecc(u1	max{ecc(u1	PROPN
ejpam-4478	363	19	)	)	PUNCT
ejpam-4478	363	20	,	,	PUNCT
ejpam-4478	363	21	ecc(u2	ecc(u2	PROPN
ejpam-4478	363	22	)	)	PUNCT
ejpam-4478	363	23	,	,	PUNCT
ejpam-4478	363	24	ecc(u3	ecc(u3	NOUN
ejpam-4478	363	25	)	)	PUNCT
ejpam-4478	363	26	,	,	PUNCT
ejpam-4478	363	27	.	.	PUNCT
ejpam-4478	363	28	.	.	PUNCT
ejpam-4478	364	1	.	.	PUNCT
ejpam-4478	365	1	,	,	PUNCT
ejpam-4478	365	2	ecc(un	ecc(un	NUM
ejpam-4478	365	3	)	)	PUNCT
ejpam-4478	365	4	}	}	PUNCT
ejpam-4478	365	5	≤	≤	NUM
ejpam-4478	365	6	max{k	max{k	NOUN
ejpam-4478	365	7	}	}	PUNCT
ejpam-4478	365	8	which	which	PRON
ejpam-4478	365	9	implies	imply	VERB
ejpam-4478	365	10	that	that	SCONJ
ejpam-4478	365	11	max{ecc(u1	max{ecc(u1	PROPN
ejpam-4478	365	12	)	)	PUNCT
ejpam-4478	365	13	,	,	PUNCT
ejpam-4478	365	14	ecc(u2	ecc(u2	PROPN
ejpam-4478	365	15	)	)	PUNCT
ejpam-4478	365	16	,	,	PUNCT
ejpam-4478	365	17	ecc(u3	ecc(u3	NOUN
ejpam-4478	365	18	)	)	PUNCT
ejpam-4478	365	19	,	,	PUNCT
ejpam-4478	365	20	.	.	PUNCT
ejpam-4478	365	21	.	.	PUNCT
ejpam-4478	366	1	.	.	PUNCT
ejpam-4478	367	1	,	,	PUNCT
ejpam-4478	367	2	ecc(un	ecc(un	NUM
ejpam-4478	367	3	)	)	PUNCT
ejpam-4478	367	4	}	}	PUNCT
ejpam-4478	367	5	≤	≤	NUM
ejpam-4478	368	1	k	k	X
ejpam-4478	368	2	(	(	PUNCT
ejpam-4478	368	3	8)	8)	NUM
ejpam-4478	368	4	now	now	ADV
ejpam-4478	368	5	,	,	PUNCT
ejpam-4478	368	6	from	from	ADP
ejpam-4478	368	7	(	(	PUNCT
ejpam-4478	368	8	8)	8)	NUM
ejpam-4478	368	9	,	,	PUNCT
ejpam-4478	368	10	we	we	PRON
ejpam-4478	368	11	know	know	VERB
ejpam-4478	368	12	that	that	SCONJ
ejpam-4478	368	13	max{ecc(u1	max{ecc(u1	PROPN
ejpam-4478	368	14	)	)	PUNCT
ejpam-4478	368	15	,	,	PUNCT
ejpam-4478	368	16	ecc(u2	ecc(u2	PROPN
ejpam-4478	368	17	)	)	PUNCT
ejpam-4478	368	18	,	,	PUNCT
ejpam-4478	368	19	ecc(u3	ecc(u3	NOUN
ejpam-4478	368	20	)	)	PUNCT
ejpam-4478	368	21	,	,	PUNCT
ejpam-4478	368	22	.	.	PUNCT
ejpam-4478	368	23	.	.	PUNCT
ejpam-4478	369	1	.	.	PUNCT
ejpam-4478	370	1	,	,	PUNCT
ejpam-4478	370	2	ecc(un	ecc(un	NUM
ejpam-4478	370	3	)	)	PUNCT
ejpam-4478	370	4	}	}	PUNCT
ejpam-4478	371	1	=	=	SYM
ejpam-4478	371	2	max{diam(g	max{diam(g	PROPN
ejpam-4478	371	3	)	)	PUNCT
ejpam-4478	371	4	,	,	PUNCT
ejpam-4478	371	5	diam(g	diam(g	NOUN
ejpam-4478	371	6	)	)	PUNCT
ejpam-4478	371	7	}	}	PUNCT
ejpam-4478	371	8	(	(	PUNCT
ejpam-4478	371	9	9	9	NUM
ejpam-4478	371	10	)	)	PUNCT
ejpam-4478	371	11	and	and	CCONJ
ejpam-4478	371	12	hence	hence	ADV
ejpam-4478	371	13	,	,	PUNCT
ejpam-4478	371	14	from	from	ADP
ejpam-4478	371	15	(	(	PUNCT
ejpam-4478	371	16	8)	8)	NUM
ejpam-4478	371	17	and	and	CCONJ
ejpam-4478	371	18	(	(	PUNCT
ejpam-4478	371	19	9	9	NUM
ejpam-4478	371	20	)	)	PUNCT
ejpam-4478	371	21	,	,	PUNCT
ejpam-4478	371	22	we	we	PRON
ejpam-4478	371	23	have	have	VERB
ejpam-4478	371	24	max{diam(g	max{diam(g	PROPN
ejpam-4478	371	25	)	)	PUNCT
ejpam-4478	371	26	,	,	PUNCT
ejpam-4478	371	27	diam(g	diam(g	NOUN
ejpam-4478	371	28	)	)	PUNCT
ejpam-4478	371	29	}	}	PUNCT
ejpam-4478	371	30	≤	≤	NUM
ejpam-4478	372	1	k	k	NOUN
ejpam-4478	372	2	which	which	PRON
ejpam-4478	372	3	can	can	AUX
ejpam-4478	372	4	be	be	AUX
ejpam-4478	372	5	rewritten	rewrite	VERB
ejpam-4478	372	6	as	as	ADP
ejpam-4478	372	7	k	k	PROPN
ejpam-4478	372	8	≥	≥	X
ejpam-4478	372	9	max{diam(g	max{diam(g	PROPN
ejpam-4478	372	10	)	)	PUNCT
ejpam-4478	372	11	,	,	PUNCT
ejpam-4478	372	12	diam(g	diam(g	NOUN
ejpam-4478	372	13	)	)	PUNCT
ejpam-4478	372	14	}	}	PUNCT
ejpam-4478	372	15	.	.	PUNCT
ejpam-4478	373	1	hence	hence	ADV
ejpam-4478	373	2	,	,	PUNCT
ejpam-4478	373	3	if	if	SCONJ
ejpam-4478	373	4	γkgr(g	γkgr(g	NUM
ejpam-4478	373	5	)	)	PUNCT
ejpam-4478	373	6	=	=	SYM
ejpam-4478	374	1	2	2	NUM
ejpam-4478	374	2	,	,	PUNCT
ejpam-4478	374	3	then	then	ADV
ejpam-4478	374	4	k	k	PROPN
ejpam-4478	374	5	≥	≥	PROPN
ejpam-4478	374	6	max{diam(g	max{diam(g	PROPN
ejpam-4478	374	7	)	)	PUNCT
ejpam-4478	374	8	,	,	PUNCT
ejpam-4478	374	9	diam(g	diam(g	NOUN
ejpam-4478	374	10	)	)	PUNCT
ejpam-4478	374	11	}	}	PUNCT
ejpam-4478	374	12	,	,	PUNCT
ejpam-4478	374	13	where	where	SCONJ
ejpam-4478	374	14	k	k	PROPN
ejpam-4478	374	15	∈	∈	PROPN
ejpam-4478	374	16	z+	z+	PUNCT
ejpam-4478	374	17	.	.	PUNCT
ejpam-4478	375	1	this	this	PRON
ejpam-4478	375	2	proves	prove	VERB
ejpam-4478	375	3	the	the	DET
ejpam-4478	375	4	forward	forward	ADJ
ejpam-4478	375	5	part	part	NOUN
ejpam-4478	375	6	of	of	ADP
ejpam-4478	375	7	the	the	DET
ejpam-4478	375	8	theorem	theorem	NOUN
ejpam-4478	375	9	.	.	PUNCT
ejpam-4478	376	1	(	(	PUNCT
ejpam-4478	376	2	⇐	⇐	ADJ
ejpam-4478	376	3	=)	=)	PROPN
ejpam-4478	376	4	assume	assume	VERB
ejpam-4478	376	5	that	that	SCONJ
ejpam-4478	376	6	k	k	PROPN
ejpam-4478	376	7	≥	≥	NOUN
ejpam-4478	376	8	max{diam(g	max{diam(g	PROPN
ejpam-4478	376	9	)	)	PUNCT
ejpam-4478	376	10	,	,	PUNCT
ejpam-4478	376	11	diam(g	diam(g	NOUN
ejpam-4478	376	12	)	)	PUNCT
ejpam-4478	376	13	}	}	PUNCT
ejpam-4478	376	14	,	,	PUNCT
ejpam-4478	376	15	where	where	SCONJ
ejpam-4478	376	16	k	k	PROPN
ejpam-4478	376	17	∈	∈	PROPN
ejpam-4478	376	18	z+	z+	PUNCT
ejpam-4478	376	19	.	.	PUNCT
ejpam-4478	377	1	hence	hence	ADV
ejpam-4478	377	2	,	,	PUNCT
ejpam-4478	377	3	if	if	SCONJ
ejpam-4478	377	4	k	k	PROPN
ejpam-4478	377	5	=	=	SYM
ejpam-4478	377	6	max{diam(g	max{diam(g	PROPN
ejpam-4478	377	7	)	)	PUNCT
ejpam-4478	377	8	,	,	PUNCT
ejpam-4478	377	9	diam(g	diam(g	NOUN
ejpam-4478	377	10	)	)	PUNCT
ejpam-4478	377	11	}	}	PUNCT
ejpam-4478	377	12	∈	∈	PROPN
ejpam-4478	377	13	z+	z+	NUM
ejpam-4478	377	14	,	,	PUNCT
ejpam-4478	377	15	then	then	ADV
ejpam-4478	377	16	there	there	PRON
ejpam-4478	377	17	exists	exist	VERB
ejpam-4478	377	18	at	at	ADV
ejpam-4478	377	19	least	least	ADV
ejpam-4478	377	20	one	one	NUM
ejpam-4478	377	21	vertex	vertex	NOUN
ejpam-4478	377	22	,	,	PUNCT
ejpam-4478	377	23	say	say	VERB
ejpam-4478	377	24	uj	uj	PROPN
ejpam-4478	377	25	∈	∈	PROPN
ejpam-4478	377	26	v	v	ADP
ejpam-4478	377	27	(	(	PUNCT
ejpam-4478	377	28	g	g	NOUN
ejpam-4478	377	29	)	)	PUNCT
ejpam-4478	377	30	=	=	NOUN
ejpam-4478	377	31	v	v	X
ejpam-4478	377	32	(	(	PUNCT
ejpam-4478	377	33	g	g	NOUN
ejpam-4478	377	34	)	)	PUNCT
ejpam-4478	377	35	,	,	PUNCT
ejpam-4478	377	36	such	such	ADJ
ejpam-4478	377	37	that	that	SCONJ
ejpam-4478	377	38	nk	nk	PROPN
ejpam-4478	377	39	,	,	PUNCT
ejpam-4478	377	40	g(uj	g(uj	PROPN
ejpam-4478	377	41	)	)	PUNCT
ejpam-4478	377	42	=	=	VERB
ejpam-4478	378	1	⋃	⋃	VERB
ejpam-4478	378	2	ui∈v	ui∈v	NOUN
ejpam-4478	378	3	(	(	PUNCT
ejpam-4478	378	4	g	g	NOUN
ejpam-4478	378	5	)	)	PUNCT
ejpam-4478	378	6	i∈{1,2,	i∈{1,2,	NOUN
ejpam-4478	378	7	...	...	PUNCT
ejpam-4478	378	8	,n	,n	PUNCT
ejpam-4478	378	9	}	}	PUNCT
ejpam-4478	378	10	nk	nk	PROPN
ejpam-4478	378	11	,	,	PUNCT
ejpam-4478	378	12	g(ui	g(ui	PROPN
ejpam-4478	378	13	̸=j	̸=j	PROPN
ejpam-4478	378	14	)	)	PUNCT
ejpam-4478	378	15	=	=	SYM
ejpam-4478	378	16	v	v	X
ejpam-4478	378	17	(	(	PUNCT
ejpam-4478	378	18	g)\{uj	g)\{uj	PROPN
ejpam-4478	378	19	}	}	PUNCT
ejpam-4478	378	20	g.	g.	NOUN
ejpam-4478	378	21	entero	entero	PROPN
ejpam-4478	378	22	,	,	PUNCT
ejpam-4478	378	23	s.	s.	PROPN
ejpam-4478	378	24	espinola	espinola	PROPN
ejpam-4478	378	25	/	/	SYM
ejpam-4478	378	26	eur	eur	PROPN
ejpam-4478	378	27	.	.	PUNCT
ejpam-4478	379	1	j.	j.	PROPN
ejpam-4478	379	2	pure	pure	PROPN
ejpam-4478	379	3	appl	appl	PROPN
ejpam-4478	379	4	.	.	PROPN
ejpam-4478	379	5	math	math	PROPN
ejpam-4478	379	6	,	,	PUNCT
ejpam-4478	379	7	16	16	NUM
ejpam-4478	379	8	(	(	PUNCT
ejpam-4478	379	9	1	1	NUM
ejpam-4478	379	10	)	)	PUNCT
ejpam-4478	379	11	(	(	PUNCT
ejpam-4478	379	12	2023	2023	NUM
ejpam-4478	379	13	)	)	PUNCT
ejpam-4478	379	14	,	,	PUNCT
ejpam-4478	379	15	44	44	NUM
ejpam-4478	379	16	-	-	SYM
ejpam-4478	379	17	61	61	NUM
ejpam-4478	379	18	56	56	NUM
ejpam-4478	379	19	and	and	CCONJ
ejpam-4478	379	20	nk	nk	PROPN
ejpam-4478	379	21	,	,	PUNCT
ejpam-4478	379	22	g(uj	g(uj	PROPN
ejpam-4478	379	23	)	)	PUNCT
ejpam-4478	379	24	=	=	VERB
ejpam-4478	380	1	⋃	⋃	VERB
ejpam-4478	380	2	ui∈v	ui∈v	NOUN
ejpam-4478	380	3	(	(	PUNCT
ejpam-4478	380	4	g	g	NOUN
ejpam-4478	380	5	)	)	PUNCT
ejpam-4478	380	6	i∈{1,2,	i∈{1,2,	NOUN
ejpam-4478	380	7	...	...	PUNCT
ejpam-4478	380	8	,n	,n	PUNCT
ejpam-4478	380	9	}	}	PUNCT
ejpam-4478	380	10	nk	nk	PROPN
ejpam-4478	380	11	,	,	PUNCT
ejpam-4478	380	12	g(ui	g(ui	PROPN
ejpam-4478	380	13	̸=j	̸=j	PROPN
ejpam-4478	380	14	)	)	PUNCT
ejpam-4478	380	15	=	=	SYM
ejpam-4478	380	16	v	v	X
ejpam-4478	380	17	(	(	PUNCT
ejpam-4478	380	18	g)\{uj	g)\{uj	NOUN
ejpam-4478	380	19	}	}	PUNCT
ejpam-4478	380	20	.	.	PUNCT
ejpam-4478	381	1	thus	thus	ADV
ejpam-4478	381	2	,	,	PUNCT
ejpam-4478	381	3	for	for	ADP
ejpam-4478	381	4	this	this	DET
ejpam-4478	381	5	case	case	NOUN
ejpam-4478	381	6	,	,	PUNCT
ejpam-4478	381	7	we	we	PRON
ejpam-4478	381	8	will	will	AUX
ejpam-4478	381	9	define	define	VERB
ejpam-4478	381	10	a	a	DET
ejpam-4478	381	11	function	function	NOUN
ejpam-4478	381	12	fk	fk	INTJ
ejpam-4478	381	13	over	over	ADP
ejpam-4478	381	14	g	g	PROPN
ejpam-4478	381	15	and	and	CCONJ
ejpam-4478	381	16	over	over	ADP
ejpam-4478	381	17	g	g	NOUN
ejpam-4478	381	18	,	,	PUNCT
ejpam-4478	381	19	respectively	respectively	ADV
ejpam-4478	381	20	,	,	PUNCT
ejpam-4478	381	21	as	as	ADP
ejpam-4478	381	22	fk	fk	INTJ
ejpam-4478	381	23	=	=	SYM
ejpam-4478	381	24	(	(	PUNCT
ejpam-4478	381	25	v	v	NUM
ejpam-4478	381	26	fk	fk	INTJ
ejpam-4478	381	27	0	0	NUM
ejpam-4478	381	28	(	(	PUNCT
ejpam-4478	381	29	g	g	NOUN
ejpam-4478	381	30	)	)	PUNCT
ejpam-4478	381	31	,	,	PUNCT
ejpam-4478	381	32	v	v	X
ejpam-4478	381	33	fk	fk	INTJ
ejpam-4478	381	34	1	1	NUM
ejpam-4478	381	35	(	(	PUNCT
ejpam-4478	381	36	g	g	NOUN
ejpam-4478	381	37	)	)	PUNCT
ejpam-4478	381	38	,	,	PUNCT
ejpam-4478	381	39	v	v	X
ejpam-4478	381	40	fk	fk	INTJ
ejpam-4478	381	41	2	2	NUM
ejpam-4478	381	42	(	(	PUNCT
ejpam-4478	381	43	g	g	NOUN
ejpam-4478	381	44	)	)	PUNCT
ejpam-4478	381	45	)	)	PUNCT
ejpam-4478	382	1	and	and	CCONJ
ejpam-4478	382	2	fk	fk	INTJ
ejpam-4478	382	3	=	=	PUNCT
ejpam-4478	382	4	(	(	PUNCT
ejpam-4478	382	5	v	v	NUM
ejpam-4478	382	6	fk	fk	INTJ
ejpam-4478	382	7	0	0	NUM
ejpam-4478	383	1	(	(	PUNCT
ejpam-4478	384	1	g	g	NOUN
ejpam-4478	384	2	)	)	PUNCT
ejpam-4478	384	3	,	,	PUNCT
ejpam-4478	384	4	v	v	X
ejpam-4478	384	5	fk	fk	INTJ
ejpam-4478	384	6	1	1	NUM
ejpam-4478	384	7	(	(	PUNCT
ejpam-4478	384	8	g	g	NOUN
ejpam-4478	384	9	)	)	PUNCT
ejpam-4478	384	10	,	,	PUNCT
ejpam-4478	384	11	v	v	X
ejpam-4478	384	12	fk	fk	INTJ
ejpam-4478	384	13	2	2	NUM
ejpam-4478	384	14	(	(	PUNCT
ejpam-4478	384	15	g	g	NOUN
ejpam-4478	384	16	)	)	PUNCT
ejpam-4478	384	17	)	)	PUNCT
ejpam-4478	384	18	where	where	SCONJ
ejpam-4478	384	19	v	v	NOUN
ejpam-4478	384	20	fk	fk	INTJ
ejpam-4478	384	21	0	0	NUM
ejpam-4478	385	1	(	(	PUNCT
ejpam-4478	385	2	g	g	NOUN
ejpam-4478	385	3	)	)	PUNCT
ejpam-4478	385	4	=	=	SYM
ejpam-4478	385	5	nk	nk	PROPN
ejpam-4478	385	6	,	,	PUNCT
ejpam-4478	385	7	g(uj)\{uj	g(uj)\{uj	PROPN
ejpam-4478	385	8	}	}	PUNCT
ejpam-4478	385	9	=	=	SYM
ejpam-4478	385	10	v	v	X
ejpam-4478	385	11	(	(	PUNCT
ejpam-4478	385	12	g)\{uj	g)\{uj	NOUN
ejpam-4478	385	13	}	}	PUNCT
ejpam-4478	385	14	,	,	PUNCT
ejpam-4478	385	15	v	v	X
ejpam-4478	385	16	fk	fk	INTJ
ejpam-4478	385	17	1	1	NUM
ejpam-4478	385	18	(	(	PUNCT
ejpam-4478	385	19	g	g	NOUN
ejpam-4478	385	20	)	)	PUNCT
ejpam-4478	385	21	=	=	NOUN
ejpam-4478	385	22	∅	∅	NOUN
ejpam-4478	385	23	,	,	PUNCT
ejpam-4478	385	24	and	and	CCONJ
ejpam-4478	386	1	v	v	ADP
ejpam-4478	386	2	fk	fk	INTJ
ejpam-4478	386	3	2	2	NUM
ejpam-4478	386	4	(	(	PUNCT
ejpam-4478	386	5	g	g	NOUN
ejpam-4478	386	6	)	)	PUNCT
ejpam-4478	386	7	=	=	SYM
ejpam-4478	386	8	{	{	PUNCT
ejpam-4478	386	9	uj	uj	NOUN
ejpam-4478	386	10	}	}	PUNCT
ejpam-4478	386	11	,	,	PUNCT
ejpam-4478	386	12	and	and	CCONJ
ejpam-4478	387	1	v	v	ADP
ejpam-4478	387	2	fk	fk	INTJ
ejpam-4478	387	3	0	0	NUM
ejpam-4478	388	1	(	(	PUNCT
ejpam-4478	388	2	g	g	NOUN
ejpam-4478	388	3	)	)	PUNCT
ejpam-4478	388	4	=	=	SYM
ejpam-4478	388	5	nk	nk	PROPN
ejpam-4478	388	6	,	,	PUNCT
ejpam-4478	388	7	g(uj)\{uj	g(uj)\{uj	PROPN
ejpam-4478	388	8	}	}	PUNCT
ejpam-4478	388	9	=	=	SYM
ejpam-4478	388	10	v	v	X
ejpam-4478	388	11	(	(	PUNCT
ejpam-4478	388	12	g)\{uj	g)\{uj	NOUN
ejpam-4478	388	13	}	}	PUNCT
ejpam-4478	388	14	,	,	PUNCT
ejpam-4478	388	15	v	v	X
ejpam-4478	388	16	fk	fk	INTJ
ejpam-4478	388	17	1	1	NUM
ejpam-4478	388	18	(	(	PUNCT
ejpam-4478	388	19	g	g	NOUN
ejpam-4478	388	20	)	)	PUNCT
ejpam-4478	388	21	=	=	NOUN
ejpam-4478	388	22	∅	∅	NOUN
ejpam-4478	388	23	,	,	PUNCT
ejpam-4478	388	24	and	and	CCONJ
ejpam-4478	389	1	v	v	ADP
ejpam-4478	389	2	fk	fk	INTJ
ejpam-4478	389	3	2	2	NUM
ejpam-4478	389	4	(	(	PUNCT
ejpam-4478	389	5	g	g	NOUN
ejpam-4478	389	6	)	)	PUNCT
ejpam-4478	389	7	=	=	SYM
ejpam-4478	389	8	{	{	PUNCT
ejpam-4478	389	9	uj	uj	PROPN
ejpam-4478	389	10	}	}	PUNCT
ejpam-4478	389	11	,	,	PUNCT
ejpam-4478	389	12	which	which	PRON
ejpam-4478	389	13	means	mean	VERB
ejpam-4478	389	14	that	that	SCONJ
ejpam-4478	389	15	,	,	PUNCT
ejpam-4478	389	16	with	with	ADP
ejpam-4478	389	17	respect	respect	NOUN
ejpam-4478	389	18	to	to	ADP
ejpam-4478	389	19	g	g	NOUN
ejpam-4478	389	20	and	and	CCONJ
ejpam-4478	389	21	g	g	NOUN
ejpam-4478	389	22	,	,	PUNCT
ejpam-4478	389	23	for	for	ADP
ejpam-4478	389	24	i	i	PROPN
ejpam-4478	389	25	̸=	̸=	PROPN
ejpam-4478	389	26	j	j	PROPN
ejpam-4478	389	27	,	,	PUNCT
ejpam-4478	389	28	fk(ui	fk(ui	PROPN
ejpam-4478	389	29	̸=j	̸=j	PROPN
ejpam-4478	389	30	)	)	PUNCT
ejpam-4478	389	31	=	=	SYM
ejpam-4478	389	32	0	0	NUM
ejpam-4478	389	33	and	and	CCONJ
ejpam-4478	389	34	fk(uj	fk(uj	NOUN
ejpam-4478	389	35	)	)	PUNCT
ejpam-4478	389	36	=	=	SYM
ejpam-4478	390	1	2	2	X
ejpam-4478	390	2	.	.	PUNCT
ejpam-4478	390	3	since	since	SCONJ
ejpam-4478	390	4	all	all	DET
ejpam-4478	390	5	vertices	vertex	NOUN
ejpam-4478	390	6	ui	ui	PROPN
ejpam-4478	390	7	̸=j	̸=j	PROPN
ejpam-4478	390	8	∈	∈	PROPN
ejpam-4478	390	9	v	v	NOUN
ejpam-4478	390	10	(	(	PUNCT
ejpam-4478	390	11	g	g	NOUN
ejpam-4478	390	12	)	)	PUNCT
ejpam-4478	390	13	=	=	NOUN
ejpam-4478	390	14	v	v	X
ejpam-4478	390	15	(	(	PUNCT
ejpam-4478	390	16	g	g	NOUN
ejpam-4478	390	17	)	)	PUNCT
ejpam-4478	390	18	,	,	PUNCT
ejpam-4478	390	19	for	for	ADP
ejpam-4478	390	20	i	i	PRON
ejpam-4478	390	21	∈	∈	PROPN
ejpam-4478	390	22	{	{	PUNCT
ejpam-4478	390	23	1	1	NUM
ejpam-4478	390	24	,	,	PUNCT
ejpam-4478	390	25	2	2	NUM
ejpam-4478	390	26	,	,	PUNCT
ejpam-4478	390	27	.	.	PUNCT
ejpam-4478	390	28	.	.	PUNCT
ejpam-4478	390	29	.	.	PUNCT
ejpam-4478	390	30	,	,	PUNCT
ejpam-4478	390	31	n	n	CCONJ
ejpam-4478	390	32	}	}	PUNCT
ejpam-4478	390	33	,	,	PUNCT
ejpam-4478	390	34	is	be	AUX
ejpam-4478	390	35	open	open	ADJ
ejpam-4478	390	36	neighbours	neighbour	NOUN
ejpam-4478	390	37	of	of	ADP
ejpam-4478	390	38	uj	uj	PROPN
ejpam-4478	390	39	∈	∈	PROPN
ejpam-4478	390	40	v	v	ADP
ejpam-4478	390	41	(	(	PUNCT
ejpam-4478	390	42	g	g	NOUN
ejpam-4478	390	43	)	)	PUNCT
ejpam-4478	390	44	=	=	NOUN
ejpam-4478	390	45	v	v	X
ejpam-4478	390	46	(	(	PUNCT
ejpam-4478	390	47	g	g	NOUN
ejpam-4478	390	48	)	)	PUNCT
ejpam-4478	390	49	,	,	PUNCT
ejpam-4478	390	50	and	and	CCONJ
ejpam-4478	390	51	,	,	PUNCT
ejpam-4478	390	52	with	with	ADP
ejpam-4478	390	53	respect	respect	NOUN
ejpam-4478	390	54	to	to	ADP
ejpam-4478	390	55	g	g	NOUN
ejpam-4478	390	56	and	and	CCONJ
ejpam-4478	390	57	g	g	NOUN
ejpam-4478	390	58	,	,	PUNCT
ejpam-4478	390	59	since	since	SCONJ
ejpam-4478	390	60	fk(uj	fk(uj	NOUN
ejpam-4478	390	61	)	)	PUNCT
ejpam-4478	391	1	=	=	SYM
ejpam-4478	391	2	2	2	NUM
ejpam-4478	391	3	,	,	PUNCT
ejpam-4478	391	4	fk(ui	fk(ui	NOUN
ejpam-4478	391	5	̸=j	̸=j	NOUN
ejpam-4478	391	6	)	)	PUNCT
ejpam-4478	391	7	=	=	SYM
ejpam-4478	391	8	0	0	NUM
ejpam-4478	391	9	is	be	AUX
ejpam-4478	391	10	permissible	permissible	ADJ
ejpam-4478	391	11	for	for	SCONJ
ejpam-4478	391	12	fk	fk	INTJ
ejpam-4478	391	13	to	to	PART
ejpam-4478	391	14	be	be	AUX
ejpam-4478	391	15	called	call	VERB
ejpam-4478	391	16	as	as	ADP
ejpam-4478	391	17	k	k	PROPN
ejpam-4478	391	18	−	−	PROPN
ejpam-4478	391	19	distance	distance	NOUN
ejpam-4478	391	20	roman	roman	ADJ
ejpam-4478	391	21	dominating	dominating	NOUN
ejpam-4478	391	22	function	function	NOUN
ejpam-4478	391	23	for	for	ADP
ejpam-4478	391	24	g	g	PROPN
ejpam-4478	391	25	and	and	CCONJ
ejpam-4478	391	26	for	for	ADP
ejpam-4478	391	27	g.	g.	PROPN
ejpam-4478	391	28	so	so	ADV
ejpam-4478	391	29	,	,	PUNCT
ejpam-4478	391	30	by	by	ADP
ejpam-4478	391	31	definition	definition	NOUN
ejpam-4478	391	32	3.7	3.7	NUM
ejpam-4478	391	33	,	,	PUNCT
ejpam-4478	391	34	the	the	DET
ejpam-4478	391	35	function	function	NOUN
ejpam-4478	391	36	fk	fk	INTJ
ejpam-4478	391	37	is	be	AUX
ejpam-4478	391	38	a	a	DET
ejpam-4478	391	39	global	global	ADJ
ejpam-4478	391	40	k	k	NOUN
ejpam-4478	391	41	−	−	PROPN
ejpam-4478	391	42	distance	distance	NOUN
ejpam-4478	391	43	roman	roman	ADJ
ejpam-4478	391	44	dominating	dominating	NOUN
ejpam-4478	391	45	function	function	NOUN
ejpam-4478	391	46	on	on	ADP
ejpam-4478	391	47	graph	graph	NOUN
ejpam-4478	391	48	g	g	PROPN
ejpam-4478	391	49	with	with	ADP
ejpam-4478	391	50	k	k	PROPN
ejpam-4478	391	51	=	=	PUNCT
ejpam-4478	391	52	max{diam(g	max{diam(g	PROPN
ejpam-4478	391	53	)	)	PUNCT
ejpam-4478	391	54	,	,	PUNCT
ejpam-4478	391	55	diam(g	diam(g	NOUN
ejpam-4478	391	56	)	)	PUNCT
ejpam-4478	391	57	}	}	PUNCT
ejpam-4478	391	58	∈	∈	PROPN
ejpam-4478	391	59	z+	z+	NUM
ejpam-4478	391	60	.	.	PUNCT
ejpam-4478	392	1	now	now	ADV
ejpam-4478	392	2	,	,	PUNCT
ejpam-4478	392	3	using	use	VERB
ejpam-4478	392	4	definition	definition	NOUN
ejpam-4478	392	5	3.8	3.8	NUM
ejpam-4478	392	6	to	to	PART
ejpam-4478	392	7	compute	compute	VERB
ejpam-4478	392	8	the	the	DET
ejpam-4478	392	9	weight	weight	NOUN
ejpam-4478	392	10	of	of	ADP
ejpam-4478	392	11	fk	fk	INTJ
ejpam-4478	392	12	,	,	PUNCT
ejpam-4478	392	13	we	we	PRON
ejpam-4478	392	14	have	have	VERB
ejpam-4478	392	15	w(fk	w(fk	NOUN
ejpam-4478	392	16	)	)	PUNCT
ejpam-4478	392	17	=	=	PUNCT
ejpam-4478	393	1	∑	∑	PUNCT
ejpam-4478	393	2	ui∈v	ui∈v	X
ejpam-4478	393	3	(	(	PUNCT
ejpam-4478	393	4	g	g	NOUN
ejpam-4478	393	5	)	)	PUNCT
ejpam-4478	393	6	i∈{1,2,	i∈{1,2,	NOUN
ejpam-4478	393	7	...	...	PUNCT
ejpam-4478	393	8	,n	,n	PRON
ejpam-4478	393	9	}	}	PUNCT
ejpam-4478	393	10	fk(ui	fk(ui	PROPN
ejpam-4478	393	11	)	)	PUNCT
ejpam-4478	393	12	=	=	SYM
ejpam-4478	393	13	fk(u1	fk(u1	NOUN
ejpam-4478	393	14	)	)	PUNCT
ejpam-4478	393	15	+	+	CCONJ
ejpam-4478	393	16	fk(u2	fk(u2	X
ejpam-4478	393	17	)	)	PUNCT
ejpam-4478	394	1	+	+	CCONJ
ejpam-4478	394	2	·	·	PUNCT
ejpam-4478	394	3	·	·	PUNCT
ejpam-4478	394	4	·	·	PUNCT
ejpam-4478	394	5	+	+	NUM
ejpam-4478	394	6	fk(uj−1	fk(uj−1	NOUN
ejpam-4478	394	7	)	)	PUNCT
ejpam-4478	394	8	+	+	CCONJ
ejpam-4478	394	9	fk(uj	fk(uj	NOUN
ejpam-4478	394	10	)	)	PUNCT
ejpam-4478	395	1	+	+	SYM
ejpam-4478	395	2	fk(uj+1	fk(uj+1	NOUN
ejpam-4478	395	3	)	)	PUNCT
ejpam-4478	395	4	+	+	NUM
ejpam-4478	395	5	·	·	PUNCT
ejpam-4478	395	6	·	·	PUNCT
ejpam-4478	395	7	·	·	PUNCT
ejpam-4478	395	8	+	+	NUM
ejpam-4478	395	9	fk(un	fk(un	NOUN
ejpam-4478	395	10	)	)	PUNCT
ejpam-4478	395	11	=	=	PUNCT
ejpam-4478	395	12	0	0	PUNCT
ejpam-4478	396	1	+	+	CCONJ
ejpam-4478	396	2	0	0	NUM
ejpam-4478	397	1	+	+	CCONJ
ejpam-4478	397	2	·	·	PUNCT
ejpam-4478	397	3	·	·	PUNCT
ejpam-4478	397	4	·	·	PUNCT
ejpam-4478	397	5	+	+	SYM
ejpam-4478	397	6	0	0	NUM
ejpam-4478	398	1	+	+	NUM
ejpam-4478	398	2	2	2	NUM
ejpam-4478	398	3	+	+	SYM
ejpam-4478	398	4	0	0	NUM
ejpam-4478	399	1	+	+	CCONJ
ejpam-4478	399	2	·	·	PUNCT
ejpam-4478	399	3	·	·	PUNCT
ejpam-4478	399	4	·	·	PUNCT
ejpam-4478	399	5	+	+	SYM
ejpam-4478	399	6	0	0	NUM
ejpam-4478	399	7	=	=	SYM
ejpam-4478	399	8	2	2	NUM
ejpam-4478	399	9	.	.	PUNCT
ejpam-4478	400	1	hence	hence	ADV
ejpam-4478	400	2	,	,	PUNCT
ejpam-4478	400	3	if	if	SCONJ
ejpam-4478	400	4	k	k	PROPN
ejpam-4478	400	5	=	=	SYM
ejpam-4478	400	6	max{diam(g	max{diam(g	PROPN
ejpam-4478	400	7	)	)	PUNCT
ejpam-4478	400	8	,	,	PUNCT
ejpam-4478	400	9	diam(g	diam(g	NOUN
ejpam-4478	400	10	)	)	PUNCT
ejpam-4478	400	11	}	}	PUNCT
ejpam-4478	400	12	,	,	PUNCT
ejpam-4478	400	13	then	then	ADV
ejpam-4478	400	14	γkgr(g	γkgr(g	NUM
ejpam-4478	400	15	)	)	PUNCT
ejpam-4478	400	16	=	=	SYM
ejpam-4478	400	17	2	2	NUM
ejpam-4478	400	18	and	and	CCONJ
ejpam-4478	400	19	this	this	PRON
ejpam-4478	400	20	follows	follow	VERB
ejpam-4478	400	21	from	from	ADP
ejpam-4478	400	22	theorem	theorem	ADJ
ejpam-4478	400	23	6.4	6.4	NUM
ejpam-4478	400	24	.	.	PUNCT
ejpam-4478	401	1	moreover	moreover	ADV
ejpam-4478	401	2	,	,	PUNCT
ejpam-4478	401	3	by	by	ADP
ejpam-4478	401	4	just	just	ADV
ejpam-4478	401	5	following	follow	VERB
ejpam-4478	401	6	the	the	DET
ejpam-4478	401	7	same	same	ADJ
ejpam-4478	401	8	arguments	argument	NOUN
ejpam-4478	401	9	,	,	PUNCT
ejpam-4478	401	10	we	we	PRON
ejpam-4478	401	11	may	may	AUX
ejpam-4478	401	12	generalize	generalize	VERB
ejpam-4478	401	13	this	this	DET
ejpam-4478	401	14	fact	fact	NOUN
ejpam-4478	401	15	as	as	SCONJ
ejpam-4478	401	16	k	k	PROPN
ejpam-4478	401	17	≥	≥	X
ejpam-4478	401	18	max{diam(g	max{diam(g	PROPN
ejpam-4478	401	19	)	)	PUNCT
ejpam-4478	401	20	,	,	PUNCT
ejpam-4478	401	21	diam(g	diam(g	NOUN
ejpam-4478	401	22	)	)	PUNCT
ejpam-4478	401	23	}	}	PUNCT
ejpam-4478	401	24	,	,	PUNCT
ejpam-4478	401	25	where	where	SCONJ
ejpam-4478	401	26	k	k	PROPN
ejpam-4478	401	27	∈	∈	PROPN
ejpam-4478	401	28	z+	z+	NUM
ejpam-4478	401	29	,	,	PUNCT
ejpam-4478	401	30	and	and	CCONJ
ejpam-4478	401	31	still	still	ADV
ejpam-4478	401	32	get	get	VERB
ejpam-4478	401	33	γkgr(g	γkgr(g	NOUN
ejpam-4478	401	34	)	)	PUNCT
ejpam-4478	401	35	=	=	SYM
ejpam-4478	401	36	2	2	NUM
ejpam-4478	401	37	,	,	PUNCT
ejpam-4478	401	38	for	for	ADP
ejpam-4478	401	39	all	all	DET
ejpam-4478	401	40	k	k	PROPN
ejpam-4478	401	41	≥	≥	NUM
ejpam-4478	401	42	max{diam(g	max{diam(g	PROPN
ejpam-4478	401	43	)	)	PUNCT
ejpam-4478	401	44	,	,	PUNCT
ejpam-4478	401	45	diam(g	diam(g	NOUN
ejpam-4478	401	46	)	)	PUNCT
ejpam-4478	401	47	}	}	PUNCT
ejpam-4478	401	48	.	.	PUNCT
ejpam-4478	402	1	thus	thus	ADV
ejpam-4478	402	2	,	,	PUNCT
ejpam-4478	402	3	if	if	SCONJ
ejpam-4478	402	4	k	k	PROPN
ejpam-4478	402	5	≥	≥	X
ejpam-4478	402	6	max{diam(g	max{diam(g	PROPN
ejpam-4478	402	7	)	)	PUNCT
ejpam-4478	402	8	,	,	PUNCT
ejpam-4478	402	9	diam(g	diam(g	NOUN
ejpam-4478	402	10	)	)	PUNCT
ejpam-4478	402	11	}	}	PUNCT
ejpam-4478	402	12	,	,	PUNCT
ejpam-4478	402	13	then	then	ADV
ejpam-4478	402	14	γkgr(g	γkgr(g	NUM
ejpam-4478	402	15	)	)	PUNCT
ejpam-4478	402	16	=	=	SYM
ejpam-4478	402	17	2	2	X
ejpam-4478	402	18	.	.	PUNCT
ejpam-4478	402	19	this	this	PRON
ejpam-4478	402	20	proves	prove	VERB
ejpam-4478	402	21	the	the	DET
ejpam-4478	402	22	backward	backward	ADJ
ejpam-4478	402	23	part	part	NOUN
ejpam-4478	402	24	of	of	ADP
ejpam-4478	402	25	the	the	DET
ejpam-4478	402	26	theorem	theorem	NOUN
ejpam-4478	402	27	.	.	PUNCT
ejpam-4478	403	1	therefore	therefore	ADV
ejpam-4478	403	2	,	,	PUNCT
ejpam-4478	403	3	given	give	VERB
ejpam-4478	403	4	the	the	DET
ejpam-4478	403	5	class	class	NOUN
ejpam-4478	403	6	g	g	NOUN
ejpam-4478	403	7	of	of	ADP
ejpam-4478	403	8	connected	connected	ADJ
ejpam-4478	403	9	graphs	graph	NOUN
ejpam-4478	403	10	whose	whose	DET
ejpam-4478	403	11	complement	complement	NOUN
ejpam-4478	403	12	are	be	AUX
ejpam-4478	403	13	also	also	ADV
ejpam-4478	403	14	connected	connect	VERB
ejpam-4478	403	15	,	,	PUNCT
ejpam-4478	403	16	for	for	ADP
ejpam-4478	403	17	any	any	DET
ejpam-4478	403	18	graph	graph	NOUN
ejpam-4478	403	19	g	g	ADP
ejpam-4478	403	20	∈	∈	PROPN
ejpam-4478	403	21	g	g	NOUN
ejpam-4478	403	22	of	of	ADP
ejpam-4478	403	23	order	order	NOUN
ejpam-4478	403	24	n	n	PRON
ejpam-4478	403	25	≥	≥	NOUN
ejpam-4478	403	26	4	4	NUM
ejpam-4478	403	27	,	,	PUNCT
ejpam-4478	403	28	γkgr(g	γkgr(g	NOUN
ejpam-4478	403	29	)	)	PUNCT
ejpam-4478	403	30	=	=	SYM
ejpam-4478	403	31	2	2	NUM
ejpam-4478	403	32	if	if	SCONJ
ejpam-4478	403	33	and	and	CCONJ
ejpam-4478	403	34	only	only	ADV
ejpam-4478	403	35	if	if	SCONJ
ejpam-4478	403	36	k	k	PROPN
ejpam-4478	403	37	≥	≥	X
ejpam-4478	403	38	max{diam(g	max{diam(g	PROPN
ejpam-4478	403	39	)	)	PUNCT
ejpam-4478	403	40	,	,	PUNCT
ejpam-4478	403	41	diam(g	diam(g	NOUN
ejpam-4478	403	42	)	)	PUNCT
ejpam-4478	403	43	}	}	PUNCT
ejpam-4478	403	44	,	,	PUNCT
ejpam-4478	403	45	where	where	SCONJ
ejpam-4478	403	46	k	k	PROPN
ejpam-4478	403	47	∈	∈	PROPN
ejpam-4478	403	48	z+	z+	PUNCT
ejpam-4478	403	49	.	.	PUNCT
ejpam-4478	403	50	theorem	theorem	VERB
ejpam-4478	403	51	7.2	7.2	NUM
ejpam-4478	403	52	.	.	PUNCT
ejpam-4478	404	1	let	let	VERB
ejpam-4478	404	2	k	k	PROPN
ejpam-4478	404	3	∈	∈	PROPN
ejpam-4478	404	4	z+	z+	PUNCT
ejpam-4478	404	5	.	.	PUNCT
ejpam-4478	405	1	for	for	ADP
ejpam-4478	405	2	any	any	DET
ejpam-4478	405	3	graph	graph	NOUN
ejpam-4478	405	4	g	g	NOUN
ejpam-4478	405	5	=	=	PUNCT
ejpam-4478	405	6	(	(	PUNCT
ejpam-4478	405	7	v	v	NOUN
ejpam-4478	405	8	(	(	PUNCT
ejpam-4478	405	9	g	g	NOUN
ejpam-4478	405	10	)	)	PUNCT
ejpam-4478	405	11	,	,	PUNCT
ejpam-4478	405	12	e(g	e(g	PROPN
ejpam-4478	405	13	)	)	PUNCT
ejpam-4478	405	14	)	)	PUNCT
ejpam-4478	405	15	of	of	ADP
ejpam-4478	405	16	order	order	NOUN
ejpam-4478	405	17	n	n	CCONJ
ejpam-4478	405	18	,	,	PUNCT
ejpam-4478	405	19	γkgr(g	γkgr(g	NOUN
ejpam-4478	405	20	)	)	PUNCT
ejpam-4478	405	21	=	=	PUNCT
ejpam-4478	405	22	γ(g	γ(g	PROPN
ejpam-4478	405	23	)	)	PUNCT
ejpam-4478	405	24	if	if	SCONJ
ejpam-4478	405	25	and	and	CCONJ
ejpam-4478	405	26	only	only	ADV
ejpam-4478	405	27	if	if	SCONJ
ejpam-4478	405	28	g	g	PROPN
ejpam-4478	405	29	∼=	∼=	PROPN
ejpam-4478	405	30	kn	kn	PROPN
ejpam-4478	405	31	.	.	PUNCT
ejpam-4478	405	32	g.	g.	PROPN
ejpam-4478	405	33	entero	entero	PROPN
ejpam-4478	405	34	,	,	PUNCT
ejpam-4478	405	35	s.	s.	PROPN
ejpam-4478	405	36	espinola	espinola	PROPN
ejpam-4478	405	37	/	/	SYM
ejpam-4478	405	38	eur	eur	PROPN
ejpam-4478	405	39	.	.	PUNCT
ejpam-4478	406	1	j.	j.	PROPN
ejpam-4478	406	2	pure	pure	PROPN
ejpam-4478	406	3	appl	appl	PROPN
ejpam-4478	406	4	.	.	PROPN
ejpam-4478	406	5	math	math	PROPN
ejpam-4478	406	6	,	,	PUNCT
ejpam-4478	406	7	16	16	NUM
ejpam-4478	406	8	(	(	PUNCT
ejpam-4478	406	9	1	1	NUM
ejpam-4478	406	10	)	)	PUNCT
ejpam-4478	406	11	(	(	PUNCT
ejpam-4478	406	12	2023	2023	NUM
ejpam-4478	406	13	)	)	PUNCT
ejpam-4478	406	14	,	,	PUNCT
ejpam-4478	406	15	44	44	NUM
ejpam-4478	406	16	-	-	SYM
ejpam-4478	406	17	61	61	NUM
ejpam-4478	406	18	57	57	NUM
ejpam-4478	406	19	proof	proof	NOUN
ejpam-4478	406	20	.	.	PUNCT
ejpam-4478	407	1	let	let	VERB
ejpam-4478	407	2	k	k	PROPN
ejpam-4478	407	3	∈	∈	PROPN
ejpam-4478	407	4	z+	z+	NUM
ejpam-4478	407	5	and	and	CCONJ
ejpam-4478	407	6	let	let	VERB
ejpam-4478	407	7	g	g	NOUN
ejpam-4478	407	8	=	=	SYM
ejpam-4478	407	9	(	(	PUNCT
ejpam-4478	407	10	v	v	NOUN
ejpam-4478	407	11	(	(	PUNCT
ejpam-4478	407	12	g	g	NOUN
ejpam-4478	407	13	)	)	PUNCT
ejpam-4478	407	14	,	,	PUNCT
ejpam-4478	407	15	e(g	e(g	PROPN
ejpam-4478	407	16	)	)	PUNCT
ejpam-4478	407	17	)	)	PUNCT
ejpam-4478	407	18	be	be	AUX
ejpam-4478	407	19	any	any	DET
ejpam-4478	407	20	graph	graph	NOUN
ejpam-4478	407	21	of	of	ADP
ejpam-4478	407	22	order	order	NOUN
ejpam-4478	407	23	n.	n.	NOUN
ejpam-4478	407	24	suppose	suppose	VERB
ejpam-4478	407	25	that	that	SCONJ
ejpam-4478	407	26	γkgr(g	γkgr(g	NOUN
ejpam-4478	407	27	)	)	PUNCT
ejpam-4478	407	28	=	=	PUNCT
ejpam-4478	407	29	γ(g	γ(g	PROPN
ejpam-4478	407	30	)	)	PUNCT
ejpam-4478	407	31	and	and	CCONJ
ejpam-4478	407	32	let	let	VERB
ejpam-4478	407	33	f	f	PROPN
ejpam-4478	407	34	=	=	SYM
ejpam-4478	407	35	(	(	PUNCT
ejpam-4478	407	36	v	v	NOUN
ejpam-4478	407	37	f	f	NOUN
ejpam-4478	407	38	0	0	PUNCT
ejpam-4478	407	39	(	(	PUNCT
ejpam-4478	407	40	g	g	NOUN
ejpam-4478	407	41	)	)	PUNCT
ejpam-4478	407	42	,	,	PUNCT
ejpam-4478	407	43	v	v	NOUN
ejpam-4478	407	44	f	f	PROPN
ejpam-4478	407	45	1	1	NUM
ejpam-4478	407	46	(	(	PUNCT
ejpam-4478	407	47	g	g	NOUN
ejpam-4478	407	48	)	)	PUNCT
ejpam-4478	407	49	,	,	PUNCT
ejpam-4478	407	50	v	v	X
ejpam-4478	407	51	f	f	PROPN
ejpam-4478	407	52	2	2	NUM
ejpam-4478	407	53	(	(	PUNCT
ejpam-4478	407	54	g	g	NOUN
ejpam-4478	407	55	)	)	PUNCT
ejpam-4478	407	56	)	)	PUNCT
ejpam-4478	407	57	be	be	AUX
ejpam-4478	407	58	a	a	DET
ejpam-4478	407	59	γkgr(g	γkgr(g	NOUN
ejpam-4478	407	60	)	)	PUNCT
ejpam-4478	407	61	−	−	PROPN
ejpam-4478	407	62	function	function	NOUN
ejpam-4478	407	63	of	of	ADP
ejpam-4478	407	64	g	g	NOUN
ejpam-4478	407	65	,	,	PUNCT
ejpam-4478	407	66	where	where	SCONJ
ejpam-4478	407	67	v	v	ADP
ejpam-4478	407	68	f	f	NOUN
ejpam-4478	407	69	i	i	PRON
ejpam-4478	407	70	(	(	PUNCT
ejpam-4478	407	71	g	g	NOUN
ejpam-4478	407	72	)	)	PUNCT
ejpam-4478	407	73	=	=	SYM
ejpam-4478	407	74	{	{	PUNCT
ejpam-4478	407	75	v	v	NUM
ejpam-4478	407	76	∈	∈	NOUN
ejpam-4478	407	77	v	v	NOUN
ejpam-4478	407	78	(	(	PUNCT
ejpam-4478	407	79	g	g	NOUN
ejpam-4478	407	80	)	)	PUNCT
ejpam-4478	407	81	:	:	PUNCT
ejpam-4478	407	82	f(v	f(v	NOUN
ejpam-4478	407	83	)	)	PUNCT
ejpam-4478	408	1	=	=	SYM
ejpam-4478	408	2	i	i	PRON
ejpam-4478	408	3	for	for	ADP
ejpam-4478	408	4	i	i	PRON
ejpam-4478	408	5	=	=	NOUN
ejpam-4478	408	6	0	0	NUM
ejpam-4478	408	7	,	,	PUNCT
ejpam-4478	408	8	1	1	NUM
ejpam-4478	408	9	,	,	PUNCT
ejpam-4478	408	10	2	2	NUM
ejpam-4478	408	11	}	}	PUNCT
ejpam-4478	408	12	is	be	AUX
ejpam-4478	408	13	the	the	DET
ejpam-4478	408	14	partition	partition	NOUN
ejpam-4478	408	15	of	of	ADP
ejpam-4478	408	16	the	the	DET
ejpam-4478	408	17	vertex	vertex	NOUN
ejpam-4478	408	18	set	set	VERB
ejpam-4478	408	19	v	v	NOUN
ejpam-4478	408	20	(	(	PUNCT
ejpam-4478	408	21	g	g	NOUN
ejpam-4478	408	22	)	)	PUNCT
ejpam-4478	408	23	of	of	ADP
ejpam-4478	408	24	g	g	NOUN
ejpam-4478	408	25	induced	induce	VERB
ejpam-4478	408	26	by	by	ADP
ejpam-4478	408	27	the	the	DET
ejpam-4478	408	28	function	function	NOUN
ejpam-4478	408	29	f	f	PROPN
ejpam-4478	408	30	and	and	CCONJ
ejpam-4478	408	31	v	v	PROPN
ejpam-4478	408	32	(	(	PUNCT
ejpam-4478	408	33	g	g	NOUN
ejpam-4478	408	34	)	)	PUNCT
ejpam-4478	408	35	=	=	PUNCT
ejpam-4478	409	1	v	v	ADP
ejpam-4478	409	2	f	f	PROPN
ejpam-4478	409	3	0	0	PUNCT
ejpam-4478	410	1	(	(	PUNCT
ejpam-4478	410	2	g	g	NOUN
ejpam-4478	410	3	)	)	PUNCT
ejpam-4478	410	4	∪	∪	NOUN
ejpam-4478	410	5	v	v	ADP
ejpam-4478	410	6	f	f	PROPN
ejpam-4478	410	7	1	1	NUM
ejpam-4478	410	8	(	(	PUNCT
ejpam-4478	410	9	g	g	NOUN
ejpam-4478	410	10	)	)	PUNCT
ejpam-4478	410	11	∪	∪	NOUN
ejpam-4478	410	12	v	v	PROPN
ejpam-4478	410	13	f	f	PROPN
ejpam-4478	410	14	2	2	NUM
ejpam-4478	410	15	(	(	PUNCT
ejpam-4478	410	16	g	g	NOUN
ejpam-4478	410	17	)	)	PUNCT
ejpam-4478	410	18	.	.	PUNCT
ejpam-4478	411	1	note	note	VERB
ejpam-4478	411	2	that	that	SCONJ
ejpam-4478	411	3	,	,	PUNCT
ejpam-4478	411	4	|v	|v	PROPN
ejpam-4478	411	5	f	f	PROPN
ejpam-4478	411	6	0	0	PUNCT
ejpam-4478	411	7	(	(	PUNCT
ejpam-4478	411	8	g)|	g)|	PROPN
ejpam-4478	411	9	,	,	PUNCT
ejpam-4478	411	10	|v	|v	PROPN
ejpam-4478	411	11	f	f	PROPN
ejpam-4478	411	12	1	1	NUM
ejpam-4478	411	13	(	(	PUNCT
ejpam-4478	411	14	g)|	g)|	PROPN
ejpam-4478	411	15	,	,	PUNCT
ejpam-4478	411	16	|v	|v	PROPN
ejpam-4478	411	17	f	f	PROPN
ejpam-4478	411	18	2	2	NUM
ejpam-4478	411	19	(	(	PUNCT
ejpam-4478	411	20	g)|	g)|	VERB
ejpam-4478	411	21	≥	≥	NOUN
ejpam-4478	411	22	0	0	NUM
ejpam-4478	411	23	.	.	PUNCT
ejpam-4478	412	1	clearly	clearly	ADV
ejpam-4478	412	2	,	,	PUNCT
ejpam-4478	412	3	v	v	PROPN
ejpam-4478	412	4	f	f	PROPN
ejpam-4478	412	5	1	1	NUM
ejpam-4478	412	6	(	(	PUNCT
ejpam-4478	412	7	g	g	NOUN
ejpam-4478	412	8	)	)	PUNCT
ejpam-4478	412	9	∪	∪	NOUN
ejpam-4478	412	10	v	v	PROPN
ejpam-4478	412	11	f	f	PROPN
ejpam-4478	412	12	2	2	NUM
ejpam-4478	412	13	(	(	PUNCT
ejpam-4478	412	14	g	g	NOUN
ejpam-4478	412	15	)	)	PUNCT
ejpam-4478	412	16	is	be	AUX
ejpam-4478	412	17	a	a	DET
ejpam-4478	412	18	dominating	dominating	NOUN
ejpam-4478	412	19	set	set	NOUN
ejpam-4478	412	20	of	of	ADP
ejpam-4478	412	21	g.	g.	PROPN
ejpam-4478	412	22	thus	thus	ADV
ejpam-4478	412	23	,	,	PUNCT
ejpam-4478	412	24	since	since	SCONJ
ejpam-4478	412	25	γ(g	γ(g	PROPN
ejpam-4478	412	26	)	)	PUNCT
ejpam-4478	412	27	is	be	AUX
ejpam-4478	412	28	the	the	DET
ejpam-4478	412	29	minimum	minimum	ADJ
ejpam-4478	412	30	cardinality	cardinality	NOUN
ejpam-4478	412	31	taken	take	VERB
ejpam-4478	412	32	over	over	ADP
ejpam-4478	412	33	all	all	DET
ejpam-4478	412	34	dominating	dominating	NOUN
ejpam-4478	412	35	sets	set	NOUN
ejpam-4478	412	36	of	of	ADP
ejpam-4478	412	37	g	g	PROPN
ejpam-4478	412	38	,	,	PUNCT
ejpam-4478	412	39	γ(g	γ(g	PROPN
ejpam-4478	412	40	)	)	PUNCT
ejpam-4478	412	41	≤	≤	PUNCT
ejpam-4478	413	1	|v	|v	PROPN
ejpam-4478	413	2	f	f	NOUN
ejpam-4478	413	3	1	1	NUM
ejpam-4478	413	4	(	(	PUNCT
ejpam-4478	413	5	g	g	NOUN
ejpam-4478	413	6	)	)	PUNCT
ejpam-4478	413	7	∪	∪	NOUN
ejpam-4478	413	8	v	v	ADP
ejpam-4478	413	9	f	f	PROPN
ejpam-4478	413	10	2	2	NUM
ejpam-4478	413	11	(	(	PUNCT
ejpam-4478	413	12	g)|	g)|	NOUN
ejpam-4478	413	13	=	=	SYM
ejpam-4478	413	14	|v	|v	PROPN
ejpam-4478	413	15	f	f	PROPN
ejpam-4478	413	16	1	1	NUM
ejpam-4478	413	17	(	(	PUNCT
ejpam-4478	413	18	g)|+	g)|+	PROPN
ejpam-4478	413	19	|v	|v	PROPN
ejpam-4478	413	20	f	f	PROPN
ejpam-4478	413	21	2	2	NUM
ejpam-4478	413	22	(	(	PUNCT
ejpam-4478	413	23	g)|	g)|	NOUN
ejpam-4478	413	24	,	,	PUNCT
ejpam-4478	413	25	and	and	CCONJ
ejpam-4478	413	26	so	so	ADV
ejpam-4478	413	27	,	,	PUNCT
ejpam-4478	413	28	we	we	PRON
ejpam-4478	413	29	have	have	VERB
ejpam-4478	413	30	γ(g	γ(g	NOUN
ejpam-4478	413	31	)	)	PUNCT
ejpam-4478	413	32	≤	≤	NOUN
ejpam-4478	413	33	|v	|v	PROPN
ejpam-4478	413	34	f	f	PROPN
ejpam-4478	413	35	1	1	NUM
ejpam-4478	413	36	(	(	PUNCT
ejpam-4478	413	37	g)|+	g)|+	PROPN
ejpam-4478	413	38	|v	|v	PROPN
ejpam-4478	413	39	f	f	PROPN
ejpam-4478	413	40	2	2	NUM
ejpam-4478	413	41	(	(	PUNCT
ejpam-4478	413	42	g)|	g)|	NOUN
ejpam-4478	413	43	.	.	PUNCT
ejpam-4478	414	1	(	(	PUNCT
ejpam-4478	414	2	1	1	X
ejpam-4478	414	3	)	)	PUNCT
ejpam-4478	414	4	since	since	SCONJ
ejpam-4478	414	5	f	f	PROPN
ejpam-4478	414	6	=	=	PUNCT
ejpam-4478	414	7	(	(	PUNCT
ejpam-4478	414	8	v	v	NOUN
ejpam-4478	414	9	f	f	NOUN
ejpam-4478	414	10	0	0	PUNCT
ejpam-4478	414	11	(	(	PUNCT
ejpam-4478	414	12	g	g	NOUN
ejpam-4478	414	13	)	)	PUNCT
ejpam-4478	414	14	,	,	PUNCT
ejpam-4478	414	15	v	v	NOUN
ejpam-4478	414	16	f	f	PROPN
ejpam-4478	414	17	1	1	NUM
ejpam-4478	414	18	(	(	PUNCT
ejpam-4478	414	19	g	g	NOUN
ejpam-4478	414	20	)	)	PUNCT
ejpam-4478	414	21	,	,	PUNCT
ejpam-4478	414	22	v	v	X
ejpam-4478	414	23	f	f	PROPN
ejpam-4478	414	24	2	2	NUM
ejpam-4478	414	25	(	(	PUNCT
ejpam-4478	414	26	g	g	NOUN
ejpam-4478	414	27	)	)	PUNCT
ejpam-4478	414	28	)	)	PUNCT
ejpam-4478	414	29	is	be	AUX
ejpam-4478	414	30	a	a	DET
ejpam-4478	414	31	γkgr(g	γkgr(g	NOUN
ejpam-4478	414	32	)	)	PUNCT
ejpam-4478	414	33	−	−	PROPN
ejpam-4478	414	34	function	function	NOUN
ejpam-4478	414	35	of	of	ADP
ejpam-4478	414	36	g	g	PROPN
ejpam-4478	414	37	,	,	PUNCT
ejpam-4478	414	38	γkgr(g	γkgr(g	NOUN
ejpam-4478	414	39	)	)	PUNCT
ejpam-4478	415	1	=	=	PUNCT
ejpam-4478	415	2	|v	|v	PROPN
ejpam-4478	415	3	f	f	PROPN
ejpam-4478	415	4	1	1	NUM
ejpam-4478	415	5	(	(	PUNCT
ejpam-4478	415	6	g)|+	g)|+	NOUN
ejpam-4478	415	7	2|v	2|v	PROPN
ejpam-4478	415	8	f	f	NOUN
ejpam-4478	415	9	2	2	NUM
ejpam-4478	415	10	(	(	PUNCT
ejpam-4478	415	11	g)|	g)|	NOUN
ejpam-4478	415	12	.	.	PUNCT
ejpam-4478	416	1	(	(	PUNCT
ejpam-4478	416	2	2	2	NUM
ejpam-4478	416	3	)	)	PUNCT
ejpam-4478	416	4	thus	thus	ADV
ejpam-4478	416	5	,	,	PUNCT
ejpam-4478	416	6	since	since	SCONJ
ejpam-4478	416	7	we	we	PRON
ejpam-4478	416	8	assumed	assume	VERB
ejpam-4478	416	9	that	that	SCONJ
ejpam-4478	416	10	γkgr(g	γkgr(g	NOUN
ejpam-4478	416	11	)	)	PUNCT
ejpam-4478	416	12	=	=	SYM
ejpam-4478	416	13	γ(g	γ(g	PROPN
ejpam-4478	416	14	)	)	PUNCT
ejpam-4478	416	15	,	,	PUNCT
ejpam-4478	416	16	from	from	ADP
ejpam-4478	416	17	(	(	PUNCT
ejpam-4478	416	18	1	1	NUM
ejpam-4478	416	19	)	)	PUNCT
ejpam-4478	416	20	and	and	CCONJ
ejpam-4478	416	21	(	(	PUNCT
ejpam-4478	416	22	2	2	NUM
ejpam-4478	416	23	)	)	PUNCT
ejpam-4478	416	24	,	,	PUNCT
ejpam-4478	416	25	we	we	PRON
ejpam-4478	416	26	have	have	VERB
ejpam-4478	416	27	|v	|v	PROPN
ejpam-4478	416	28	f	f	PROPN
ejpam-4478	416	29	1	1	NUM
ejpam-4478	416	30	(	(	PUNCT
ejpam-4478	416	31	g)|	g)|	NOUN
ejpam-4478	416	32	+	+	CCONJ
ejpam-4478	416	33	2|v	2|v	PROPN
ejpam-4478	416	34	f	f	NOUN
ejpam-4478	416	35	2	2	NUM
ejpam-4478	416	36	(	(	PUNCT
ejpam-4478	416	37	g)|	g)|	NOUN
ejpam-4478	416	38	=	=	SYM
ejpam-4478	416	39	γkgr(g	γkgr(g	PROPN
ejpam-4478	416	40	)	)	PUNCT
ejpam-4478	416	41	=	=	SYM
ejpam-4478	416	42	γ(g	γ(g	PROPN
ejpam-4478	416	43	)	)	PUNCT
ejpam-4478	416	44	≤	≤	PUNCT
ejpam-4478	416	45	|v	|v	PROPN
ejpam-4478	416	46	f	f	PROPN
ejpam-4478	416	47	1	1	NUM
ejpam-4478	416	48	(	(	PUNCT
ejpam-4478	416	49	g)|+	g)|+	PROPN
ejpam-4478	416	50	|v	|v	PROPN
ejpam-4478	416	51	f	f	PROPN
ejpam-4478	416	52	2	2	NUM
ejpam-4478	416	53	(	(	PUNCT
ejpam-4478	416	54	g)|	g)|	NOUN
ejpam-4478	416	55	,	,	PUNCT
ejpam-4478	416	56	and	and	CCONJ
ejpam-4478	416	57	hence	hence	ADV
ejpam-4478	416	58	,	,	PUNCT
ejpam-4478	416	59	we	we	PRON
ejpam-4478	416	60	have	have	VERB
ejpam-4478	416	61	|v	|v	PROPN
ejpam-4478	416	62	f	f	PROPN
ejpam-4478	416	63	1	1	NUM
ejpam-4478	416	64	(	(	PUNCT
ejpam-4478	416	65	g)|+	g)|+	NOUN
ejpam-4478	417	1	2|v	2|v	PROPN
ejpam-4478	417	2	f	f	NOUN
ejpam-4478	417	3	2	2	NUM
ejpam-4478	417	4	(	(	PUNCT
ejpam-4478	417	5	g)|	g)|	VERB
ejpam-4478	417	6	≤	≤	NUM
ejpam-4478	417	7	|v	|v	ADP
ejpam-4478	417	8	f	f	NOUN
ejpam-4478	417	9	1	1	NUM
ejpam-4478	417	10	(	(	PUNCT
ejpam-4478	417	11	g)|+	g)|+	PROPN
ejpam-4478	417	12	|v	|v	PROPN
ejpam-4478	417	13	f	f	PROPN
ejpam-4478	417	14	2	2	NUM
ejpam-4478	417	15	(	(	PUNCT
ejpam-4478	417	16	g)|	g)|	NOUN
ejpam-4478	417	17	.	.	PUNCT
ejpam-4478	417	18	(	(	PUNCT
ejpam-4478	417	19	3	3	X
ejpam-4478	417	20	)	)	PUNCT
ejpam-4478	417	21	since	since	SCONJ
ejpam-4478	417	22	|v	|v	PROPN
ejpam-4478	417	23	f	f	PROPN
ejpam-4478	417	24	0	0	PUNCT
ejpam-4478	417	25	(	(	PUNCT
ejpam-4478	417	26	g)|	g)|	PROPN
ejpam-4478	417	27	,	,	PUNCT
ejpam-4478	417	28	|v	|v	PROPN
ejpam-4478	417	29	f	f	PROPN
ejpam-4478	417	30	1	1	NUM
ejpam-4478	417	31	(	(	PUNCT
ejpam-4478	417	32	g)|	g)|	PROPN
ejpam-4478	417	33	,	,	PUNCT
ejpam-4478	417	34	|v	|v	PROPN
ejpam-4478	417	35	f	f	PROPN
ejpam-4478	417	36	2	2	NUM
ejpam-4478	417	37	(	(	PUNCT
ejpam-4478	417	38	g)|	g)|	VERB
ejpam-4478	417	39	≥	≥	NOUN
ejpam-4478	417	40	0	0	NUM
ejpam-4478	417	41	,	,	PUNCT
ejpam-4478	417	42	by	by	ADP
ejpam-4478	417	43	addition	addition	NOUN
ejpam-4478	417	44	and	and	CCONJ
ejpam-4478	417	45	subtraction	subtraction	NOUN
ejpam-4478	417	46	properties	property	NOUN
ejpam-4478	417	47	of	of	ADP
ejpam-4478	417	48	equality	equality	NOUN
ejpam-4478	417	49	,	,	PUNCT
ejpam-4478	417	50	we	we	PRON
ejpam-4478	417	51	have	have	VERB
ejpam-4478	417	52	|v	|v	PROPN
ejpam-4478	417	53	f	f	PROPN
ejpam-4478	417	54	2	2	NUM
ejpam-4478	417	55	(	(	PUNCT
ejpam-4478	417	56	g)|	g)|	VERB
ejpam-4478	417	57	≤	≤	NOUN
ejpam-4478	417	58	0	0	NUM
ejpam-4478	417	59	,	,	PUNCT
ejpam-4478	417	60	that	that	ADV
ejpam-4478	417	61	is	is	ADV
ejpam-4478	417	62	,	,	PUNCT
ejpam-4478	417	63	|v	|v	PROPN
ejpam-4478	417	64	f	f	PROPN
ejpam-4478	417	65	2	2	NUM
ejpam-4478	417	66	(	(	PUNCT
ejpam-4478	417	67	g)|	g)|	VERB
ejpam-4478	417	68	≤	≤	NOUN
ejpam-4478	417	69	0	0	PUNCT
ejpam-4478	418	1	and	and	CCONJ
ejpam-4478	418	2	since	since	SCONJ
ejpam-4478	418	3	|v	|v	PROPN
ejpam-4478	418	4	f	f	PROPN
ejpam-4478	418	5	2	2	NUM
ejpam-4478	418	6	(	(	PUNCT
ejpam-4478	418	7	g)|	g)|	VERB
ejpam-4478	418	8	≥	≥	NOUN
ejpam-4478	418	9	0	0	NUM
ejpam-4478	418	10	,	,	PUNCT
ejpam-4478	418	11	we	we	PRON
ejpam-4478	418	12	may	may	AUX
ejpam-4478	418	13	conclude	conclude	VERB
ejpam-4478	418	14	that	that	SCONJ
ejpam-4478	418	15	|v	|v	PROPN
ejpam-4478	418	16	f	f	X
ejpam-4478	418	17	2	2	NUM
ejpam-4478	418	18	(	(	PUNCT
ejpam-4478	418	19	g)|	g)|	NOUN
ejpam-4478	418	20	=	=	NOUN
ejpam-4478	418	21	0	0	NUM
ejpam-4478	418	22	.	.	PUNCT
ejpam-4478	419	1	thus	thus	ADV
ejpam-4478	419	2	,	,	PUNCT
ejpam-4478	419	3	v	v	PROPN
ejpam-4478	419	4	f	f	PROPN
ejpam-4478	419	5	0	0	PUNCT
ejpam-4478	419	6	(	(	PUNCT
ejpam-4478	419	7	g	g	NOUN
ejpam-4478	419	8	)	)	PUNCT
ejpam-4478	419	9	=	=	NOUN
ejpam-4478	419	10	∅	∅	NOUN
ejpam-4478	419	11	,	,	PUNCT
ejpam-4478	419	12	and	and	CCONJ
ejpam-4478	419	13	so	so	ADV
ejpam-4478	419	14	,	,	PUNCT
ejpam-4478	419	15	we	we	PRON
ejpam-4478	419	16	are	be	AUX
ejpam-4478	419	17	forced	force	VERB
ejpam-4478	419	18	to	to	PART
ejpam-4478	419	19	have	have	VERB
ejpam-4478	419	20	v	v	NUM
ejpam-4478	419	21	f	f	PROPN
ejpam-4478	419	22	1	1	NUM
ejpam-4478	419	23	(	(	PUNCT
ejpam-4478	419	24	g	g	NOUN
ejpam-4478	419	25	)	)	PUNCT
ejpam-4478	419	26	=	=	NOUN
ejpam-4478	419	27	v	v	X
ejpam-4478	419	28	(	(	PUNCT
ejpam-4478	419	29	g	g	NOUN
ejpam-4478	419	30	)	)	PUNCT
ejpam-4478	419	31	which	which	PRON
ejpam-4478	419	32	implies	imply	VERB
ejpam-4478	419	33	that	that	SCONJ
ejpam-4478	419	34	|v	|v	PROPN
ejpam-4478	419	35	f	f	PROPN
ejpam-4478	419	36	1	1	NUM
ejpam-4478	419	37	(	(	PUNCT
ejpam-4478	419	38	g)|	g)|	NOUN
ejpam-4478	419	39	=	=	PUNCT
ejpam-4478	419	40	|v	|v	X
ejpam-4478	419	41	(	(	PUNCT
ejpam-4478	419	42	g)|	g)|	NOUN
ejpam-4478	419	43	=	=	SYM
ejpam-4478	419	44	n.	n.	NOUN
ejpam-4478	419	45	since	since	SCONJ
ejpam-4478	419	46	f	f	PROPN
ejpam-4478	419	47	is	be	AUX
ejpam-4478	419	48	a	a	DET
ejpam-4478	419	49	γkgr(g	γkgr(g	NOUN
ejpam-4478	419	50	)	)	PUNCT
ejpam-4478	419	51	−	−	PROPN
ejpam-4478	419	52	function	function	NOUN
ejpam-4478	419	53	of	of	ADP
ejpam-4478	419	54	g	g	PROPN
ejpam-4478	419	55	,	,	PUNCT
ejpam-4478	419	56	γkgr(g	γkgr(g	NOUN
ejpam-4478	419	57	)	)	PUNCT
ejpam-4478	419	58	=	=	PUNCT
ejpam-4478	420	1	|v	|v	PROPN
ejpam-4478	420	2	f	f	PROPN
ejpam-4478	420	3	1	1	NUM
ejpam-4478	420	4	(	(	PUNCT
ejpam-4478	420	5	g)|	g)|	NOUN
ejpam-4478	420	6	+	+	CCONJ
ejpam-4478	420	7	2|v	2|v	PROPN
ejpam-4478	420	8	f	f	NOUN
ejpam-4478	420	9	2	2	NUM
ejpam-4478	420	10	(	(	PUNCT
ejpam-4478	420	11	g)|	g)|	NOUN
ejpam-4478	420	12	=	=	PUNCT
ejpam-4478	420	13	n	n	PROPN
ejpam-4478	420	14	+	+	NOUN
ejpam-4478	420	15	0	0	NUM
ejpam-4478	420	16	=	=	SYM
ejpam-4478	420	17	n	n	NOUN
ejpam-4478	420	18	which	which	PRON
ejpam-4478	420	19	implies	imply	VERB
ejpam-4478	420	20	that	that	SCONJ
ejpam-4478	420	21	γ(g	γ(g	PROPN
ejpam-4478	420	22	)	)	PUNCT
ejpam-4478	420	23	=	=	SYM
ejpam-4478	420	24	n	n	PROPN
ejpam-4478	420	25	for	for	SCONJ
ejpam-4478	420	26	we	we	PRON
ejpam-4478	420	27	assumed	assume	VERB
ejpam-4478	420	28	that	that	SCONJ
ejpam-4478	420	29	γkgr(g	γkgr(g	NOUN
ejpam-4478	420	30	)	)	PUNCT
ejpam-4478	420	31	=	=	SYM
ejpam-4478	420	32	γ(g	γ(g	PROPN
ejpam-4478	420	33	)	)	PUNCT
ejpam-4478	420	34	.	.	PUNCT
ejpam-4478	421	1	thus	thus	ADV
ejpam-4478	421	2	,	,	PUNCT
ejpam-4478	421	3	in	in	ADP
ejpam-4478	421	4	reference	reference	NOUN
ejpam-4478	421	5	to	to	AUX
ejpam-4478	421	6	theorem	theorem	VERB
ejpam-4478	421	7	3.10	3.10	NUM
ejpam-4478	421	8	,	,	PUNCT
ejpam-4478	421	9	this	this	PRON
ejpam-4478	421	10	shows	show	VERB
ejpam-4478	421	11	that	that	SCONJ
ejpam-4478	421	12	g	g	PROPN
ejpam-4478	421	13	∼=	∼=	PROPN
ejpam-4478	421	14	kn	kn	PROPN
ejpam-4478	421	15	.	.	PUNCT
ejpam-4478	422	1	conversely	conversely	ADV
ejpam-4478	422	2	,	,	PUNCT
ejpam-4478	422	3	assume	assume	VERB
ejpam-4478	422	4	that	that	SCONJ
ejpam-4478	422	5	g	g	PROPN
ejpam-4478	422	6	∼=	∼=	PROPN
ejpam-4478	422	7	kn	kn	PROPN
ejpam-4478	422	8	.	.	PUNCT
ejpam-4478	423	1	by	by	ADP
ejpam-4478	423	2	theorem	theorem	NOUN
ejpam-4478	423	3	5.1	5.1	NUM
ejpam-4478	423	4	,	,	PUNCT
ejpam-4478	423	5	γkgr(g	γkgr(g	NOUN
ejpam-4478	423	6	)	)	PUNCT
ejpam-4478	423	7	=	=	SYM
ejpam-4478	423	8	γkgr(kn	γkgr(kn	NOUN
ejpam-4478	423	9	)	)	PUNCT
ejpam-4478	423	10	=	=	SYM
ejpam-4478	424	1	n	n	NOUN
ejpam-4478	424	2	and	and	CCONJ
ejpam-4478	424	3	since	since	SCONJ
ejpam-4478	424	4	g	g	PROPN
ejpam-4478	424	5	∼=	∼=	PROPN
ejpam-4478	424	6	kn	kn	NOUN
ejpam-4478	424	7	,	,	PUNCT
ejpam-4478	424	8	it	it	PRON
ejpam-4478	424	9	follows	follow	VERB
ejpam-4478	424	10	that	that	SCONJ
ejpam-4478	424	11	,	,	PUNCT
ejpam-4478	424	12	γ(g	γ(g	PROPN
ejpam-4478	424	13	)	)	PUNCT
ejpam-4478	424	14	=	=	SYM
ejpam-4478	424	15	γ(kn	γ(kn	PROPN
ejpam-4478	424	16	)	)	PUNCT
ejpam-4478	424	17	=	=	SYM
ejpam-4478	424	18	n.	n.	PROPN
ejpam-4478	424	19	thus	thus	ADV
ejpam-4478	424	20	,	,	PUNCT
ejpam-4478	424	21	since	since	SCONJ
ejpam-4478	424	22	γkgr(g	γkgr(g	NUM
ejpam-4478	424	23	)	)	PUNCT
ejpam-4478	424	24	=	=	SYM
ejpam-4478	425	1	n	n	PROPN
ejpam-4478	425	2	and	and	CCONJ
ejpam-4478	425	3	γ(g	γ(g	PROPN
ejpam-4478	425	4	)	)	PUNCT
ejpam-4478	426	1	=	=	SYM
ejpam-4478	426	2	n	n	CCONJ
ejpam-4478	426	3	,	,	PUNCT
ejpam-4478	426	4	γkgr(g	γkgr(g	NOUN
ejpam-4478	426	5	)	)	PUNCT
ejpam-4478	426	6	=	=	SYM
ejpam-4478	426	7	γ(g	γ(g	PROPN
ejpam-4478	426	8	)	)	PUNCT
ejpam-4478	426	9	.	.	PUNCT
ejpam-4478	427	1	this	this	PRON
ejpam-4478	427	2	completes	complete	VERB
ejpam-4478	427	3	the	the	DET
ejpam-4478	427	4	proof	proof	NOUN
ejpam-4478	427	5	.	.	PUNCT
ejpam-4478	428	1	theorem	theorem	VERB
ejpam-4478	428	2	7.3	7.3	NUM
ejpam-4478	428	3	.	.	PUNCT
ejpam-4478	429	1	let	let	VERB
ejpam-4478	429	2	k	k	PROPN
ejpam-4478	429	3	∈	∈	PROPN
ejpam-4478	429	4	z+	z+	PUNCT
ejpam-4478	429	5	.	.	PUNCT
ejpam-4478	430	1	for	for	ADP
ejpam-4478	430	2	any	any	DET
ejpam-4478	430	3	graph	graph	NOUN
ejpam-4478	430	4	g	g	NOUN
ejpam-4478	430	5	=	=	PUNCT
ejpam-4478	430	6	(	(	PUNCT
ejpam-4478	430	7	v	v	NOUN
ejpam-4478	430	8	(	(	PUNCT
ejpam-4478	430	9	g	g	NOUN
ejpam-4478	430	10	)	)	PUNCT
ejpam-4478	430	11	,	,	PUNCT
ejpam-4478	430	12	e(g	e(g	PROPN
ejpam-4478	430	13	)	)	PUNCT
ejpam-4478	430	14	)	)	PUNCT
ejpam-4478	430	15	of	of	ADP
ejpam-4478	430	16	order	order	NOUN
ejpam-4478	430	17	n	n	CCONJ
ejpam-4478	430	18	,	,	PUNCT
ejpam-4478	430	19	γkgr(g	γkgr(g	NOUN
ejpam-4478	430	20	)	)	PUNCT
ejpam-4478	430	21	=	=	SYM
ejpam-4478	430	22	γk(g	γk(g	NOUN
ejpam-4478	430	23	)	)	PUNCT
ejpam-4478	430	24	if	if	SCONJ
ejpam-4478	430	25	and	and	CCONJ
ejpam-4478	430	26	only	only	ADV
ejpam-4478	430	27	if	if	SCONJ
ejpam-4478	430	28	g	g	PROPN
ejpam-4478	430	29	∼=	∼=	PROPN
ejpam-4478	430	30	kn	kn	PROPN
ejpam-4478	430	31	.	.	PUNCT
ejpam-4478	430	32	proof	proof	NOUN
ejpam-4478	430	33	.	.	PUNCT
ejpam-4478	431	1	let	let	VERB
ejpam-4478	431	2	k	k	PROPN
ejpam-4478	431	3	∈	∈	PROPN
ejpam-4478	431	4	z+	z+	NUM
ejpam-4478	431	5	and	and	CCONJ
ejpam-4478	431	6	let	let	VERB
ejpam-4478	431	7	g	g	NOUN
ejpam-4478	431	8	=	=	SYM
ejpam-4478	431	9	(	(	PUNCT
ejpam-4478	431	10	v	v	NOUN
ejpam-4478	431	11	(	(	PUNCT
ejpam-4478	431	12	g	g	NOUN
ejpam-4478	431	13	)	)	PUNCT
ejpam-4478	431	14	,	,	PUNCT
ejpam-4478	431	15	e(g	e(g	PROPN
ejpam-4478	431	16	)	)	PUNCT
ejpam-4478	431	17	)	)	PUNCT
ejpam-4478	431	18	be	be	AUX
ejpam-4478	431	19	any	any	DET
ejpam-4478	431	20	graph	graph	NOUN
ejpam-4478	431	21	of	of	ADP
ejpam-4478	431	22	order	order	NOUN
ejpam-4478	431	23	n.	n.	NOUN
ejpam-4478	431	24	assume	assume	VERB
ejpam-4478	431	25	that	that	SCONJ
ejpam-4478	431	26	γkgr(g	γkgr(g	NOUN
ejpam-4478	431	27	)	)	PUNCT
ejpam-4478	431	28	=	=	SYM
ejpam-4478	431	29	γk(g	γk(g	NOUN
ejpam-4478	431	30	)	)	PUNCT
ejpam-4478	431	31	and	and	CCONJ
ejpam-4478	431	32	let	let	VERB
ejpam-4478	431	33	f	f	PROPN
ejpam-4478	431	34	=	=	SYM
ejpam-4478	431	35	(	(	PUNCT
ejpam-4478	431	36	v	v	NOUN
ejpam-4478	431	37	f	f	NOUN
ejpam-4478	431	38	0	0	PUNCT
ejpam-4478	431	39	(	(	PUNCT
ejpam-4478	431	40	g	g	NOUN
ejpam-4478	431	41	)	)	PUNCT
ejpam-4478	431	42	,	,	PUNCT
ejpam-4478	431	43	v	v	NOUN
ejpam-4478	431	44	f	f	PROPN
ejpam-4478	431	45	1	1	NUM
ejpam-4478	431	46	(	(	PUNCT
ejpam-4478	431	47	g	g	NOUN
ejpam-4478	431	48	)	)	PUNCT
ejpam-4478	431	49	,	,	PUNCT
ejpam-4478	431	50	v	v	X
ejpam-4478	431	51	f	f	PROPN
ejpam-4478	431	52	2	2	NUM
ejpam-4478	431	53	(	(	PUNCT
ejpam-4478	431	54	g	g	NOUN
ejpam-4478	431	55	)	)	PUNCT
ejpam-4478	431	56	)	)	PUNCT
ejpam-4478	431	57	be	be	AUX
ejpam-4478	431	58	a	a	DET
ejpam-4478	431	59	γkgr(g	γkgr(g	NOUN
ejpam-4478	431	60	)	)	PUNCT
ejpam-4478	431	61	−	−	PROPN
ejpam-4478	431	62	function	function	NOUN
ejpam-4478	431	63	of	of	ADP
ejpam-4478	431	64	g	g	NOUN
ejpam-4478	431	65	,	,	PUNCT
ejpam-4478	431	66	where	where	SCONJ
ejpam-4478	431	67	v	v	ADP
ejpam-4478	431	68	f	f	NOUN
ejpam-4478	431	69	i	i	PRON
ejpam-4478	431	70	(	(	PUNCT
ejpam-4478	431	71	g	g	NOUN
ejpam-4478	431	72	)	)	PUNCT
ejpam-4478	431	73	=	=	SYM
ejpam-4478	431	74	{	{	PUNCT
ejpam-4478	431	75	v	v	NUM
ejpam-4478	431	76	∈	∈	NOUN
ejpam-4478	431	77	v	v	NOUN
ejpam-4478	431	78	(	(	PUNCT
ejpam-4478	431	79	g	g	NOUN
ejpam-4478	431	80	)	)	PUNCT
ejpam-4478	431	81	:	:	PUNCT
ejpam-4478	431	82	f(v	f(v	NOUN
ejpam-4478	431	83	)	)	PUNCT
ejpam-4478	432	1	=	=	SYM
ejpam-4478	432	2	i	i	PRON
ejpam-4478	432	3	for	for	ADP
ejpam-4478	432	4	i	i	PRON
ejpam-4478	432	5	=	=	NOUN
ejpam-4478	432	6	0	0	NUM
ejpam-4478	432	7	,	,	PUNCT
ejpam-4478	432	8	1	1	NUM
ejpam-4478	432	9	,	,	PUNCT
ejpam-4478	432	10	2	2	NUM
ejpam-4478	432	11	}	}	PUNCT
ejpam-4478	432	12	is	be	AUX
ejpam-4478	432	13	the	the	DET
ejpam-4478	432	14	partition	partition	NOUN
ejpam-4478	432	15	of	of	ADP
ejpam-4478	432	16	the	the	DET
ejpam-4478	432	17	vertex	vertex	NOUN
ejpam-4478	432	18	set	set	VERB
ejpam-4478	432	19	v	v	NOUN
ejpam-4478	432	20	(	(	PUNCT
ejpam-4478	432	21	g	g	NOUN
ejpam-4478	432	22	)	)	PUNCT
ejpam-4478	432	23	of	of	ADP
ejpam-4478	432	24	g	g	NOUN
ejpam-4478	432	25	induced	induce	VERB
ejpam-4478	432	26	by	by	ADP
ejpam-4478	432	27	the	the	DET
ejpam-4478	432	28	function	function	NOUN
ejpam-4478	432	29	f	f	PROPN
ejpam-4478	432	30	and	and	CCONJ
ejpam-4478	432	31	v	v	PROPN
ejpam-4478	432	32	(	(	PUNCT
ejpam-4478	432	33	g	g	NOUN
ejpam-4478	432	34	)	)	PUNCT
ejpam-4478	432	35	=	=	PUNCT
ejpam-4478	433	1	v	v	ADP
ejpam-4478	433	2	f	f	PROPN
ejpam-4478	433	3	0	0	PUNCT
ejpam-4478	434	1	(	(	PUNCT
ejpam-4478	434	2	g	g	NOUN
ejpam-4478	434	3	)	)	PUNCT
ejpam-4478	434	4	∪	∪	NOUN
ejpam-4478	434	5	v	v	ADP
ejpam-4478	434	6	f	f	PROPN
ejpam-4478	434	7	1	1	NUM
ejpam-4478	434	8	(	(	PUNCT
ejpam-4478	434	9	g	g	NOUN
ejpam-4478	434	10	)	)	PUNCT
ejpam-4478	434	11	∪	∪	NOUN
ejpam-4478	434	12	v	v	PROPN
ejpam-4478	434	13	f	f	PROPN
ejpam-4478	434	14	2	2	NUM
ejpam-4478	434	15	(	(	PUNCT
ejpam-4478	434	16	g	g	NOUN
ejpam-4478	434	17	)	)	PUNCT
ejpam-4478	434	18	.	.	PUNCT
ejpam-4478	435	1	note	note	VERB
ejpam-4478	435	2	that	that	SCONJ
ejpam-4478	435	3	,	,	PUNCT
ejpam-4478	435	4	|v	|v	PROPN
ejpam-4478	435	5	f	f	PROPN
ejpam-4478	435	6	0	0	PUNCT
ejpam-4478	435	7	(	(	PUNCT
ejpam-4478	435	8	g)|	g)|	PROPN
ejpam-4478	435	9	,	,	PUNCT
ejpam-4478	435	10	|v	|v	PROPN
ejpam-4478	435	11	f	f	PROPN
ejpam-4478	435	12	1	1	NUM
ejpam-4478	435	13	(	(	PUNCT
ejpam-4478	435	14	g)|	g)|	PROPN
ejpam-4478	435	15	,	,	PUNCT
ejpam-4478	435	16	|v	|v	PROPN
ejpam-4478	435	17	f	f	PROPN
ejpam-4478	435	18	2	2	NUM
ejpam-4478	435	19	(	(	PUNCT
ejpam-4478	435	20	g)|	g)|	VERB
ejpam-4478	435	21	≥	≥	NOUN
ejpam-4478	435	22	0	0	NUM
ejpam-4478	435	23	.	.	PUNCT
ejpam-4478	436	1	clearly	clearly	ADV
ejpam-4478	436	2	,	,	PUNCT
ejpam-4478	436	3	v	v	PROPN
ejpam-4478	436	4	f	f	PROPN
ejpam-4478	436	5	1	1	NUM
ejpam-4478	436	6	(	(	PUNCT
ejpam-4478	436	7	g	g	NOUN
ejpam-4478	436	8	)	)	PUNCT
ejpam-4478	436	9	∪	∪	NOUN
ejpam-4478	436	10	v	v	PROPN
ejpam-4478	436	11	f	f	PROPN
ejpam-4478	436	12	2	2	NUM
ejpam-4478	436	13	(	(	PUNCT
ejpam-4478	436	14	g	g	NOUN
ejpam-4478	436	15	)	)	PUNCT
ejpam-4478	436	16	is	be	AUX
ejpam-4478	436	17	a	a	DET
ejpam-4478	436	18	k	k	NOUN
ejpam-4478	436	19	−	−	NOUN
ejpam-4478	436	20	distance	distance	NOUN
ejpam-4478	436	21	dominating	dominating	NOUN
ejpam-4478	436	22	set	set	NOUN
ejpam-4478	436	23	of	of	ADP
ejpam-4478	436	24	g	g	NOUN
ejpam-4478	436	25	,	,	PUNCT
ejpam-4478	436	26	where	where	SCONJ
ejpam-4478	436	27	k	k	PROPN
ejpam-4478	436	28	∈	∈	PROPN
ejpam-4478	436	29	z+	z+	PUNCT
ejpam-4478	436	30	.	.	PUNCT
ejpam-4478	437	1	hence	hence	ADV
ejpam-4478	437	2	,	,	PUNCT
ejpam-4478	437	3	since	since	SCONJ
ejpam-4478	437	4	γk(g	γk(g	NOUN
ejpam-4478	437	5	)	)	PUNCT
ejpam-4478	437	6	is	be	AUX
ejpam-4478	437	7	the	the	DET
ejpam-4478	437	8	minimum	minimum	ADJ
ejpam-4478	437	9	cardinality	cardinality	NOUN
ejpam-4478	437	10	taken	take	VERB
ejpam-4478	437	11	over	over	ADP
ejpam-4478	437	12	all	all	PRON
ejpam-4478	437	13	k	k	NOUN
ejpam-4478	437	14	−	−	NOUN
ejpam-4478	438	1	distance	distance	NOUN
ejpam-4478	439	1	dominating	dominating	NOUN
ejpam-4478	439	2	sets	set	NOUN
ejpam-4478	439	3	of	of	ADP
ejpam-4478	439	4	g	g	NOUN
ejpam-4478	439	5	,	,	PUNCT
ejpam-4478	439	6	for	for	ADP
ejpam-4478	439	7	each	each	DET
ejpam-4478	439	8	k	k	PROPN
ejpam-4478	439	9	∈	∈	PROPN
ejpam-4478	439	10	z+	z+	X
ejpam-4478	439	11	,	,	PUNCT
ejpam-4478	439	12	γk(g	γk(g	NOUN
ejpam-4478	439	13	)	)	PUNCT
ejpam-4478	439	14	≤	≤	NOUN
ejpam-4478	439	15	|v	|v	ADP
ejpam-4478	439	16	f	f	NOUN
ejpam-4478	439	17	1	1	NUM
ejpam-4478	439	18	(	(	PUNCT
ejpam-4478	439	19	g	g	NOUN
ejpam-4478	439	20	)	)	PUNCT
ejpam-4478	439	21	∪	∪	NOUN
ejpam-4478	439	22	v	v	ADP
ejpam-4478	439	23	f	f	PROPN
ejpam-4478	439	24	2	2	NUM
ejpam-4478	439	25	(	(	PUNCT
ejpam-4478	439	26	g)|	g)|	NOUN
ejpam-4478	439	27	=	=	SYM
ejpam-4478	439	28	|v	|v	PROPN
ejpam-4478	439	29	f	f	PROPN
ejpam-4478	439	30	1	1	NUM
ejpam-4478	439	31	(	(	PUNCT
ejpam-4478	439	32	g)|+	g)|+	PROPN
ejpam-4478	439	33	|v	|v	PROPN
ejpam-4478	439	34	f	f	PROPN
ejpam-4478	439	35	2	2	NUM
ejpam-4478	439	36	(	(	PUNCT
ejpam-4478	439	37	g)|	g)|	PROPN
ejpam-4478	439	38	,	,	PUNCT
ejpam-4478	439	39	g.	g.	PROPN
ejpam-4478	439	40	entero	entero	PROPN
ejpam-4478	439	41	,	,	PUNCT
ejpam-4478	439	42	s.	s.	PROPN
ejpam-4478	439	43	espinola	espinola	PROPN
ejpam-4478	439	44	/	/	SYM
ejpam-4478	439	45	eur	eur	PROPN
ejpam-4478	439	46	.	.	PUNCT
ejpam-4478	440	1	j.	j.	PROPN
ejpam-4478	440	2	pure	pure	PROPN
ejpam-4478	440	3	appl	appl	PROPN
ejpam-4478	440	4	.	.	PROPN
ejpam-4478	440	5	math	math	PROPN
ejpam-4478	440	6	,	,	PUNCT
ejpam-4478	440	7	16	16	NUM
ejpam-4478	440	8	(	(	PUNCT
ejpam-4478	440	9	1	1	NUM
ejpam-4478	440	10	)	)	PUNCT
ejpam-4478	440	11	(	(	PUNCT
ejpam-4478	440	12	2023	2023	NUM
ejpam-4478	440	13	)	)	PUNCT
ejpam-4478	440	14	,	,	PUNCT
ejpam-4478	440	15	44	44	NUM
ejpam-4478	440	16	-	-	SYM
ejpam-4478	440	17	61	61	NUM
ejpam-4478	440	18	58	58	NUM
ejpam-4478	440	19	that	that	PRON
ejpam-4478	440	20	is	be	AUX
ejpam-4478	440	21	,	,	PUNCT
ejpam-4478	440	22	we	we	PRON
ejpam-4478	440	23	have	have	AUX
ejpam-4478	440	24	γk(g	γk(g	NOUN
ejpam-4478	440	25	)	)	PUNCT
ejpam-4478	440	26	≤	≤	NOUN
ejpam-4478	440	27	|v	|v	ADP
ejpam-4478	440	28	f	f	PROPN
ejpam-4478	440	29	1	1	NUM
ejpam-4478	440	30	(	(	PUNCT
ejpam-4478	440	31	g)|+	g)|+	PROPN
ejpam-4478	440	32	|v	|v	PROPN
ejpam-4478	440	33	f	f	PROPN
ejpam-4478	440	34	2	2	NUM
ejpam-4478	440	35	(	(	PUNCT
ejpam-4478	440	36	g)|	g)|	NOUN
ejpam-4478	440	37	.	.	PUNCT
ejpam-4478	441	1	(	(	PUNCT
ejpam-4478	441	2	1	1	X
ejpam-4478	441	3	)	)	PUNCT
ejpam-4478	441	4	since	since	SCONJ
ejpam-4478	441	5	f	f	PROPN
ejpam-4478	441	6	=	=	PUNCT
ejpam-4478	441	7	(	(	PUNCT
ejpam-4478	441	8	v	v	NOUN
ejpam-4478	441	9	f	f	NOUN
ejpam-4478	441	10	0	0	PUNCT
ejpam-4478	441	11	(	(	PUNCT
ejpam-4478	441	12	g	g	NOUN
ejpam-4478	441	13	)	)	PUNCT
ejpam-4478	441	14	,	,	PUNCT
ejpam-4478	441	15	v	v	NOUN
ejpam-4478	441	16	f	f	PROPN
ejpam-4478	441	17	1	1	NUM
ejpam-4478	441	18	(	(	PUNCT
ejpam-4478	441	19	g	g	NOUN
ejpam-4478	441	20	)	)	PUNCT
ejpam-4478	441	21	,	,	PUNCT
ejpam-4478	441	22	v	v	X
ejpam-4478	441	23	f	f	PROPN
ejpam-4478	441	24	2	2	NUM
ejpam-4478	441	25	(	(	PUNCT
ejpam-4478	441	26	g	g	NOUN
ejpam-4478	441	27	)	)	PUNCT
ejpam-4478	441	28	)	)	PUNCT
ejpam-4478	441	29	is	be	AUX
ejpam-4478	441	30	a	a	DET
ejpam-4478	441	31	γkgr(g	γkgr(g	NOUN
ejpam-4478	441	32	)	)	PUNCT
ejpam-4478	441	33	−	−	PROPN
ejpam-4478	441	34	function	function	NOUN
ejpam-4478	441	35	of	of	ADP
ejpam-4478	441	36	g	g	PROPN
ejpam-4478	441	37	,	,	PUNCT
ejpam-4478	441	38	γkgr(g	γkgr(g	NOUN
ejpam-4478	441	39	)	)	PUNCT
ejpam-4478	442	1	=	=	PUNCT
ejpam-4478	442	2	|v	|v	PROPN
ejpam-4478	442	3	f	f	PROPN
ejpam-4478	442	4	1	1	NUM
ejpam-4478	442	5	(	(	PUNCT
ejpam-4478	442	6	g)|+	g)|+	NOUN
ejpam-4478	442	7	2|v	2|v	PROPN
ejpam-4478	442	8	f	f	NOUN
ejpam-4478	442	9	2	2	NUM
ejpam-4478	442	10	(	(	PUNCT
ejpam-4478	442	11	g)|	g)|	NOUN
ejpam-4478	442	12	.	.	PUNCT
ejpam-4478	443	1	(	(	PUNCT
ejpam-4478	443	2	2	2	NUM
ejpam-4478	443	3	)	)	PUNCT
ejpam-4478	443	4	thus	thus	ADV
ejpam-4478	443	5	,	,	PUNCT
ejpam-4478	443	6	since	since	SCONJ
ejpam-4478	443	7	we	we	PRON
ejpam-4478	443	8	assumed	assume	VERB
ejpam-4478	443	9	that	that	SCONJ
ejpam-4478	443	10	γkgr(g	γkgr(g	NOUN
ejpam-4478	443	11	)	)	PUNCT
ejpam-4478	443	12	=	=	SYM
ejpam-4478	443	13	γk(g	γk(g	NOUN
ejpam-4478	443	14	)	)	PUNCT
ejpam-4478	443	15	,	,	PUNCT
ejpam-4478	443	16	from	from	ADP
ejpam-4478	443	17	(	(	PUNCT
ejpam-4478	443	18	1	1	NUM
ejpam-4478	443	19	)	)	PUNCT
ejpam-4478	443	20	and	and	CCONJ
ejpam-4478	443	21	(	(	PUNCT
ejpam-4478	443	22	2	2	NUM
ejpam-4478	443	23	)	)	PUNCT
ejpam-4478	443	24	,	,	PUNCT
ejpam-4478	443	25	we	we	PRON
ejpam-4478	443	26	have	have	VERB
ejpam-4478	443	27	|v	|v	PROPN
ejpam-4478	443	28	f	f	PROPN
ejpam-4478	443	29	1	1	NUM
ejpam-4478	443	30	(	(	PUNCT
ejpam-4478	443	31	g)|	g)|	NOUN
ejpam-4478	443	32	+	+	CCONJ
ejpam-4478	443	33	2|v	2|v	PROPN
ejpam-4478	443	34	f	f	NOUN
ejpam-4478	443	35	2	2	NUM
ejpam-4478	443	36	(	(	PUNCT
ejpam-4478	443	37	g)|	g)|	NOUN
ejpam-4478	443	38	=	=	SYM
ejpam-4478	443	39	γkgr(g	γkgr(g	PROPN
ejpam-4478	443	40	)	)	PUNCT
ejpam-4478	443	41	=	=	SYM
ejpam-4478	444	1	γk(g	γk(g	X
ejpam-4478	444	2	)	)	PUNCT
ejpam-4478	444	3	≤	≤	NOUN
ejpam-4478	444	4	|v	|v	ADP
ejpam-4478	444	5	f	f	PROPN
ejpam-4478	444	6	1	1	NUM
ejpam-4478	444	7	(	(	PUNCT
ejpam-4478	444	8	g)|+	g)|+	PROPN
ejpam-4478	444	9	|v	|v	PROPN
ejpam-4478	444	10	f	f	PROPN
ejpam-4478	444	11	2	2	NUM
ejpam-4478	444	12	(	(	PUNCT
ejpam-4478	444	13	g)|	g)|	NOUN
ejpam-4478	444	14	,	,	PUNCT
ejpam-4478	444	15	and	and	CCONJ
ejpam-4478	444	16	hence	hence	ADV
ejpam-4478	444	17	,	,	PUNCT
ejpam-4478	444	18	we	we	PRON
ejpam-4478	444	19	have	have	VERB
ejpam-4478	444	20	|v	|v	PROPN
ejpam-4478	444	21	f	f	PROPN
ejpam-4478	444	22	1	1	NUM
ejpam-4478	444	23	(	(	PUNCT
ejpam-4478	444	24	g)|+	g)|+	NOUN
ejpam-4478	445	1	2|v	2|v	PROPN
ejpam-4478	445	2	f	f	NOUN
ejpam-4478	445	3	2	2	NUM
ejpam-4478	445	4	(	(	PUNCT
ejpam-4478	445	5	g)|	g)|	VERB
ejpam-4478	445	6	≤	≤	NUM
ejpam-4478	445	7	|v	|v	ADP
ejpam-4478	445	8	f	f	NOUN
ejpam-4478	445	9	1	1	NUM
ejpam-4478	445	10	(	(	PUNCT
ejpam-4478	445	11	g)|+	g)|+	PROPN
ejpam-4478	445	12	|v	|v	PROPN
ejpam-4478	445	13	f	f	PROPN
ejpam-4478	445	14	2	2	NUM
ejpam-4478	445	15	(	(	PUNCT
ejpam-4478	445	16	g)|	g)|	NOUN
ejpam-4478	445	17	.	.	PUNCT
ejpam-4478	445	18	(	(	PUNCT
ejpam-4478	445	19	3	3	X
ejpam-4478	445	20	)	)	PUNCT
ejpam-4478	445	21	since	since	SCONJ
ejpam-4478	445	22	|v	|v	PROPN
ejpam-4478	445	23	f	f	PROPN
ejpam-4478	445	24	0	0	PUNCT
ejpam-4478	445	25	(	(	PUNCT
ejpam-4478	445	26	g)|	g)|	PROPN
ejpam-4478	445	27	,	,	PUNCT
ejpam-4478	445	28	|v	|v	PROPN
ejpam-4478	445	29	f	f	PROPN
ejpam-4478	445	30	1	1	NUM
ejpam-4478	445	31	(	(	PUNCT
ejpam-4478	445	32	g)|	g)|	PROPN
ejpam-4478	445	33	,	,	PUNCT
ejpam-4478	445	34	|v	|v	PROPN
ejpam-4478	445	35	f	f	PROPN
ejpam-4478	445	36	2	2	NUM
ejpam-4478	445	37	(	(	PUNCT
ejpam-4478	445	38	g)|	g)|	VERB
ejpam-4478	445	39	≥	≥	NOUN
ejpam-4478	445	40	0	0	NUM
ejpam-4478	445	41	,	,	PUNCT
ejpam-4478	445	42	by	by	ADP
ejpam-4478	445	43	addition	addition	NOUN
ejpam-4478	445	44	and	and	CCONJ
ejpam-4478	445	45	subtraction	subtraction	NOUN
ejpam-4478	445	46	properties	property	NOUN
ejpam-4478	445	47	of	of	ADP
ejpam-4478	445	48	equality	equality	NOUN
ejpam-4478	445	49	,	,	PUNCT
ejpam-4478	445	50	we	we	PRON
ejpam-4478	445	51	have	have	VERB
ejpam-4478	445	52	|v	|v	PROPN
ejpam-4478	445	53	f	f	PROPN
ejpam-4478	445	54	2	2	NUM
ejpam-4478	445	55	(	(	PUNCT
ejpam-4478	445	56	g)|	g)|	VERB
ejpam-4478	445	57	≤	≤	NOUN
ejpam-4478	445	58	0	0	NUM
ejpam-4478	445	59	,	,	PUNCT
ejpam-4478	445	60	that	that	ADV
ejpam-4478	445	61	is	is	ADV
ejpam-4478	445	62	,	,	PUNCT
ejpam-4478	445	63	|v	|v	PROPN
ejpam-4478	445	64	f	f	PROPN
ejpam-4478	445	65	2	2	NUM
ejpam-4478	445	66	(	(	PUNCT
ejpam-4478	445	67	g)|	g)|	VERB
ejpam-4478	445	68	≤	≤	NOUN
ejpam-4478	445	69	0	0	PUNCT
ejpam-4478	446	1	and	and	CCONJ
ejpam-4478	446	2	since	since	SCONJ
ejpam-4478	446	3	|v	|v	PROPN
ejpam-4478	446	4	f	f	PROPN
ejpam-4478	446	5	2	2	NUM
ejpam-4478	446	6	(	(	PUNCT
ejpam-4478	446	7	g)|	g)|	VERB
ejpam-4478	446	8	≥	≥	NOUN
ejpam-4478	446	9	0	0	NUM
ejpam-4478	446	10	,	,	PUNCT
ejpam-4478	446	11	we	we	PRON
ejpam-4478	446	12	may	may	AUX
ejpam-4478	446	13	conclude	conclude	VERB
ejpam-4478	446	14	that	that	SCONJ
ejpam-4478	446	15	|v	|v	PROPN
ejpam-4478	446	16	f	f	X
ejpam-4478	446	17	2	2	NUM
ejpam-4478	446	18	(	(	PUNCT
ejpam-4478	446	19	g)|	g)|	NOUN
ejpam-4478	446	20	=	=	NOUN
ejpam-4478	446	21	0	0	NUM
ejpam-4478	446	22	.	.	PUNCT
ejpam-4478	447	1	hence	hence	ADV
ejpam-4478	447	2	,	,	PUNCT
ejpam-4478	447	3	v	v	PROPN
ejpam-4478	447	4	f	f	PROPN
ejpam-4478	447	5	0	0	PUNCT
ejpam-4478	447	6	(	(	PUNCT
ejpam-4478	447	7	g	g	NOUN
ejpam-4478	447	8	)	)	PUNCT
ejpam-4478	447	9	=	=	NOUN
ejpam-4478	447	10	∅	∅	NOUN
ejpam-4478	447	11	,	,	PUNCT
ejpam-4478	447	12	and	and	CCONJ
ejpam-4478	447	13	so	so	ADV
ejpam-4478	447	14	,	,	PUNCT
ejpam-4478	447	15	we	we	PRON
ejpam-4478	447	16	are	be	AUX
ejpam-4478	447	17	forced	force	VERB
ejpam-4478	447	18	to	to	PART
ejpam-4478	447	19	have	have	VERB
ejpam-4478	447	20	v	v	NUM
ejpam-4478	447	21	f	f	PROPN
ejpam-4478	447	22	1	1	NUM
ejpam-4478	447	23	(	(	PUNCT
ejpam-4478	447	24	g	g	NOUN
ejpam-4478	447	25	)	)	PUNCT
ejpam-4478	447	26	=	=	NOUN
ejpam-4478	447	27	v	v	X
ejpam-4478	447	28	(	(	PUNCT
ejpam-4478	447	29	g	g	NOUN
ejpam-4478	447	30	)	)	PUNCT
ejpam-4478	447	31	which	which	PRON
ejpam-4478	447	32	implies	imply	VERB
ejpam-4478	447	33	that	that	SCONJ
ejpam-4478	447	34	|v	|v	PROPN
ejpam-4478	447	35	f	f	PROPN
ejpam-4478	447	36	1	1	NUM
ejpam-4478	447	37	(	(	PUNCT
ejpam-4478	447	38	g)|	g)|	NOUN
ejpam-4478	447	39	=	=	PUNCT
ejpam-4478	447	40	|v	|v	X
ejpam-4478	447	41	(	(	PUNCT
ejpam-4478	447	42	g)|	g)|	NOUN
ejpam-4478	447	43	=	=	SYM
ejpam-4478	447	44	n.	n.	NOUN
ejpam-4478	447	45	since	since	SCONJ
ejpam-4478	447	46	f	f	PROPN
ejpam-4478	447	47	is	be	AUX
ejpam-4478	447	48	a	a	DET
ejpam-4478	447	49	γkgr(g	γkgr(g	NOUN
ejpam-4478	447	50	)	)	PUNCT
ejpam-4478	447	51	−	−	PROPN
ejpam-4478	447	52	function	function	NOUN
ejpam-4478	447	53	of	of	ADP
ejpam-4478	447	54	g	g	PROPN
ejpam-4478	447	55	,	,	PUNCT
ejpam-4478	447	56	γkgr(g	γkgr(g	NOUN
ejpam-4478	447	57	)	)	PUNCT
ejpam-4478	447	58	=	=	PUNCT
ejpam-4478	448	1	|v	|v	PROPN
ejpam-4478	448	2	f	f	PROPN
ejpam-4478	448	3	1	1	NUM
ejpam-4478	448	4	(	(	PUNCT
ejpam-4478	448	5	g)|	g)|	NOUN
ejpam-4478	448	6	+	+	CCONJ
ejpam-4478	448	7	2|v	2|v	PROPN
ejpam-4478	448	8	f	f	NOUN
ejpam-4478	448	9	2	2	NUM
ejpam-4478	448	10	(	(	PUNCT
ejpam-4478	448	11	g)|	g)|	NOUN
ejpam-4478	448	12	=	=	PUNCT
ejpam-4478	448	13	n	n	PROPN
ejpam-4478	448	14	+	+	NOUN
ejpam-4478	448	15	0	0	NUM
ejpam-4478	448	16	=	=	SYM
ejpam-4478	448	17	n	n	NOUN
ejpam-4478	448	18	which	which	PRON
ejpam-4478	448	19	implies	imply	VERB
ejpam-4478	448	20	that	that	PRON
ejpam-4478	448	21	γk(g	γk(g	PUNCT
ejpam-4478	448	22	)	)	PUNCT
ejpam-4478	448	23	=	=	SYM
ejpam-4478	448	24	n	n	PROPN
ejpam-4478	448	25	for	for	SCONJ
ejpam-4478	448	26	we	we	PRON
ejpam-4478	448	27	assumed	assume	VERB
ejpam-4478	448	28	that	that	SCONJ
ejpam-4478	448	29	γkgr(g	γkgr(g	NOUN
ejpam-4478	448	30	)	)	PUNCT
ejpam-4478	448	31	=	=	SYM
ejpam-4478	448	32	γk(g	γk(g	NOUN
ejpam-4478	448	33	)	)	PUNCT
ejpam-4478	448	34	.	.	PUNCT
ejpam-4478	449	1	hence	hence	ADV
ejpam-4478	449	2	,	,	PUNCT
ejpam-4478	449	3	in	in	ADP
ejpam-4478	449	4	reference	reference	NOUN
ejpam-4478	449	5	to	to	PART
ejpam-4478	449	6	proposition	proposition	NOUN
ejpam-4478	449	7	3.11	3.11	NUM
ejpam-4478	449	8	,	,	PUNCT
ejpam-4478	449	9	this	this	PRON
ejpam-4478	449	10	shows	show	VERB
ejpam-4478	449	11	that	that	SCONJ
ejpam-4478	449	12	g	g	PROPN
ejpam-4478	449	13	∼=	∼=	PROPN
ejpam-4478	449	14	kn	kn	PROPN
ejpam-4478	449	15	.	.	PUNCT
ejpam-4478	450	1	conversely	conversely	ADV
ejpam-4478	450	2	,	,	PUNCT
ejpam-4478	450	3	suppose	suppose	VERB
ejpam-4478	450	4	that	that	SCONJ
ejpam-4478	450	5	g	g	PROPN
ejpam-4478	450	6	∼=	∼=	PROPN
ejpam-4478	450	7	kn	kn	PROPN
ejpam-4478	450	8	.	.	PUNCT
ejpam-4478	451	1	by	by	ADP
ejpam-4478	451	2	theorem	theorem	NOUN
ejpam-4478	451	3	5.1	5.1	NUM
ejpam-4478	451	4	,	,	PUNCT
ejpam-4478	451	5	γkgr(g	γkgr(g	NOUN
ejpam-4478	451	6	)	)	PUNCT
ejpam-4478	451	7	=	=	SYM
ejpam-4478	451	8	γkgr(kn	γkgr(kn	NOUN
ejpam-4478	451	9	)	)	PUNCT
ejpam-4478	451	10	=	=	SYM
ejpam-4478	452	1	n	n	NOUN
ejpam-4478	452	2	and	and	CCONJ
ejpam-4478	452	3	since	since	SCONJ
ejpam-4478	452	4	g	g	PROPN
ejpam-4478	452	5	∼=	∼=	PROPN
ejpam-4478	452	6	kn	kn	NOUN
ejpam-4478	452	7	,	,	PUNCT
ejpam-4478	452	8	it	it	PRON
ejpam-4478	452	9	follows	follow	VERB
ejpam-4478	452	10	that	that	SCONJ
ejpam-4478	452	11	,	,	PUNCT
ejpam-4478	452	12	γk(g	γk(g	PUNCT
ejpam-4478	452	13	)	)	PUNCT
ejpam-4478	452	14	=	=	SYM
ejpam-4478	452	15	γk(kn	γk(kn	PROPN
ejpam-4478	452	16	)	)	PUNCT
ejpam-4478	453	1	=	=	SYM
ejpam-4478	453	2	n.	n.	PROPN
ejpam-4478	453	3	thus	thus	ADV
ejpam-4478	453	4	,	,	PUNCT
ejpam-4478	453	5	since	since	SCONJ
ejpam-4478	453	6	γkgr(g	γkgr(g	NUM
ejpam-4478	453	7	)	)	PUNCT
ejpam-4478	453	8	=	=	SYM
ejpam-4478	453	9	n	n	NOUN
ejpam-4478	453	10	and	and	CCONJ
ejpam-4478	453	11	γk(g	γk(g	PUNCT
ejpam-4478	453	12	)	)	PUNCT
ejpam-4478	453	13	=	=	SYM
ejpam-4478	453	14	n	n	CCONJ
ejpam-4478	453	15	,	,	PUNCT
ejpam-4478	453	16	γkgr(g	γkgr(g	NOUN
ejpam-4478	453	17	)	)	PUNCT
ejpam-4478	453	18	=	=	SYM
ejpam-4478	453	19	γk(g	γk(g	NOUN
ejpam-4478	453	20	)	)	PUNCT
ejpam-4478	453	21	.	.	PUNCT
ejpam-4478	454	1	this	this	PRON
ejpam-4478	454	2	completes	complete	VERB
ejpam-4478	454	3	the	the	DET
ejpam-4478	454	4	proof	proof	NOUN
ejpam-4478	454	5	.	.	PUNCT
ejpam-4478	455	1	theorem	theorem	VERB
ejpam-4478	455	2	7.4	7.4	NUM
ejpam-4478	455	3	.	.	PUNCT
ejpam-4478	456	1	let	let	VERB
ejpam-4478	456	2	k	k	PROPN
ejpam-4478	456	3	∈	∈	PROPN
ejpam-4478	456	4	z+	z+	PUNCT
ejpam-4478	456	5	.	.	PUNCT
ejpam-4478	457	1	for	for	ADP
ejpam-4478	457	2	any	any	DET
ejpam-4478	457	3	graph	graph	NOUN
ejpam-4478	457	4	g	g	NOUN
ejpam-4478	457	5	=	=	PUNCT
ejpam-4478	457	6	(	(	PUNCT
ejpam-4478	457	7	v	v	NOUN
ejpam-4478	457	8	(	(	PUNCT
ejpam-4478	457	9	g	g	NOUN
ejpam-4478	457	10	)	)	PUNCT
ejpam-4478	457	11	,	,	PUNCT
ejpam-4478	457	12	e(g	e(g	PROPN
ejpam-4478	457	13	)	)	PUNCT
ejpam-4478	457	14	)	)	PUNCT
ejpam-4478	457	15	of	of	ADP
ejpam-4478	457	16	order	order	NOUN
ejpam-4478	457	17	n	n	CCONJ
ejpam-4478	457	18	,	,	PUNCT
ejpam-4478	457	19	γkgr(g	γkgr(g	NOUN
ejpam-4478	457	20	)	)	PUNCT
ejpam-4478	457	21	=	=	SYM
ejpam-4478	457	22	γr(g	γr(g	X
ejpam-4478	457	23	)	)	PUNCT
ejpam-4478	457	24	if	if	SCONJ
ejpam-4478	457	25	and	and	CCONJ
ejpam-4478	457	26	only	only	ADV
ejpam-4478	457	27	if	if	SCONJ
ejpam-4478	457	28	g	g	PROPN
ejpam-4478	457	29	∼=	∼=	PROPN
ejpam-4478	457	30	kn	kn	PROPN
ejpam-4478	457	31	.	.	PUNCT
ejpam-4478	457	32	proof	proof	NOUN
ejpam-4478	457	33	.	.	PUNCT
ejpam-4478	458	1	let	let	VERB
ejpam-4478	458	2	k	k	PROPN
ejpam-4478	458	3	∈	∈	PROPN
ejpam-4478	458	4	z+	z+	NUM
ejpam-4478	458	5	and	and	CCONJ
ejpam-4478	458	6	let	let	VERB
ejpam-4478	458	7	g	g	NOUN
ejpam-4478	458	8	=	=	SYM
ejpam-4478	458	9	(	(	PUNCT
ejpam-4478	458	10	v	v	NOUN
ejpam-4478	458	11	(	(	PUNCT
ejpam-4478	458	12	g	g	NOUN
ejpam-4478	458	13	)	)	PUNCT
ejpam-4478	458	14	,	,	PUNCT
ejpam-4478	458	15	e(g	e(g	PROPN
ejpam-4478	458	16	)	)	PUNCT
ejpam-4478	458	17	)	)	PUNCT
ejpam-4478	458	18	be	be	AUX
ejpam-4478	458	19	any	any	DET
ejpam-4478	458	20	graph	graph	NOUN
ejpam-4478	458	21	of	of	ADP
ejpam-4478	458	22	order	order	NOUN
ejpam-4478	458	23	n.	n.	NOUN
ejpam-4478	458	24	suppose	suppose	VERB
ejpam-4478	458	25	that	that	SCONJ
ejpam-4478	458	26	γkgr(g	γkgr(g	NOUN
ejpam-4478	458	27	)	)	PUNCT
ejpam-4478	458	28	=	=	SYM
ejpam-4478	458	29	γr(g	γr(g	NOUN
ejpam-4478	458	30	)	)	PUNCT
ejpam-4478	458	31	.	.	PUNCT
ejpam-4478	459	1	by	by	ADP
ejpam-4478	459	2	remark	remark	NOUN
ejpam-4478	459	3	3.12	3.12	NUM
ejpam-4478	459	4	,	,	PUNCT
ejpam-4478	459	5	we	we	PRON
ejpam-4478	459	6	have	have	VERB
ejpam-4478	459	7	γkr(g	γkr(g	PROPN
ejpam-4478	459	8	)	)	PUNCT
ejpam-4478	459	9	≤	≤	NOUN
ejpam-4478	459	10	n	n	ADP
ejpam-4478	459	11	and	and	CCONJ
ejpam-4478	459	12	γkr(g	γkr(g	NUM
ejpam-4478	459	13	)	)	PUNCT
ejpam-4478	460	1	=	=	SYM
ejpam-4478	460	2	n	n	NOUN
ejpam-4478	460	3	if	if	SCONJ
ejpam-4478	460	4	and	and	CCONJ
ejpam-4478	460	5	only	only	ADV
ejpam-4478	460	6	if	if	SCONJ
ejpam-4478	460	7	g	g	PROPN
ejpam-4478	460	8	∼=	∼=	PROPN
ejpam-4478	460	9	kn	kn	NOUN
ejpam-4478	460	10	and	and	CCONJ
ejpam-4478	460	11	this	this	PRON
ejpam-4478	460	12	is	be	AUX
ejpam-4478	460	13	true	true	ADJ
ejpam-4478	460	14	for	for	ADP
ejpam-4478	460	15	all	all	DET
ejpam-4478	460	16	k	k	PROPN
ejpam-4478	460	17	∈	∈	PROPN
ejpam-4478	460	18	z+	z+	PUNCT
ejpam-4478	460	19	.	.	PUNCT
ejpam-4478	461	1	so	so	ADV
ejpam-4478	461	2	,	,	PUNCT
ejpam-4478	461	3	γ1r(g	γ1r(g	PROPN
ejpam-4478	461	4	)	)	PUNCT
ejpam-4478	461	5	=	=	SYM
ejpam-4478	461	6	γr(g	γr(g	X
ejpam-4478	461	7	)	)	PUNCT
ejpam-4478	461	8	≤	≤	NOUN
ejpam-4478	462	1	n	n	ADP
ejpam-4478	462	2	and	and	CCONJ
ejpam-4478	462	3	γ1r(g	γ1r(g	NUM
ejpam-4478	462	4	)	)	PUNCT
ejpam-4478	463	1	=	=	SYM
ejpam-4478	463	2	γr(g	γr(g	NOUN
ejpam-4478	463	3	)	)	PUNCT
ejpam-4478	464	1	=	=	SYM
ejpam-4478	465	1	n	n	NOUN
ejpam-4478	465	2	if	if	SCONJ
ejpam-4478	466	1	and	and	CCONJ
ejpam-4478	466	2	only	only	ADV
ejpam-4478	466	3	if	if	SCONJ
ejpam-4478	466	4	g	g	PROPN
ejpam-4478	466	5	∼=	∼=	PROPN
ejpam-4478	466	6	kn	kn	PROPN
ejpam-4478	466	7	.	.	PUNCT
ejpam-4478	467	1	thus	thus	ADV
ejpam-4478	467	2	,	,	PUNCT
ejpam-4478	467	3	since	since	SCONJ
ejpam-4478	467	4	γkgr(g	γkgr(g	NUM
ejpam-4478	467	5	)	)	PUNCT
ejpam-4478	467	6	=	=	SYM
ejpam-4478	467	7	γr(g	γr(g	PROPN
ejpam-4478	467	8	)	)	PUNCT
ejpam-4478	467	9	and	and	CCONJ
ejpam-4478	467	10	γr(g	γr(g	NUM
ejpam-4478	467	11	)	)	PUNCT
ejpam-4478	467	12	=	=	SYM
ejpam-4478	468	1	n	n	NOUN
ejpam-4478	468	2	if	if	SCONJ
ejpam-4478	469	1	and	and	CCONJ
ejpam-4478	469	2	only	only	ADV
ejpam-4478	469	3	if	if	SCONJ
ejpam-4478	469	4	g	g	PROPN
ejpam-4478	469	5	∼=	∼=	PROPN
ejpam-4478	469	6	kn	kn	NOUN
ejpam-4478	469	7	,	,	PUNCT
ejpam-4478	469	8	it	it	PRON
ejpam-4478	469	9	follows	follow	VERB
ejpam-4478	469	10	that	that	SCONJ
ejpam-4478	469	11	γkgr(g	γkgr(g	NOUN
ejpam-4478	469	12	)	)	PUNCT
ejpam-4478	469	13	=	=	SYM
ejpam-4478	470	1	n	n	NOUN
ejpam-4478	470	2	if	if	SCONJ
ejpam-4478	470	3	and	and	CCONJ
ejpam-4478	470	4	only	only	ADV
ejpam-4478	470	5	if	if	SCONJ
ejpam-4478	470	6	g	g	PROPN
ejpam-4478	470	7	∼=	∼=	PROPN
ejpam-4478	470	8	kn	kn	PROPN
ejpam-4478	470	9	.	.	PUNCT
ejpam-4478	471	1	hence	hence	ADV
ejpam-4478	471	2	,	,	PUNCT
ejpam-4478	471	3	by	by	ADP
ejpam-4478	471	4	remark	remark	NOUN
ejpam-4478	471	5	3.12	3.12	NUM
ejpam-4478	471	6	,	,	PUNCT
ejpam-4478	471	7	γkgr(g	γkgr(g	NOUN
ejpam-4478	471	8	)	)	PUNCT
ejpam-4478	471	9	=	=	SYM
ejpam-4478	471	10	γr(g	γr(g	NOUN
ejpam-4478	471	11	)	)	PUNCT
ejpam-4478	471	12	=	=	SYM
ejpam-4478	472	1	n	n	NOUN
ejpam-4478	472	2	if	if	SCONJ
ejpam-4478	473	1	and	and	CCONJ
ejpam-4478	473	2	only	only	ADV
ejpam-4478	473	3	if	if	SCONJ
ejpam-4478	473	4	g	g	PROPN
ejpam-4478	473	5	∼=	∼=	PROPN
ejpam-4478	473	6	kn	kn	NOUN
ejpam-4478	473	7	and	and	CCONJ
ejpam-4478	473	8	thus	thus	ADV
ejpam-4478	473	9	,	,	PUNCT
ejpam-4478	473	10	the	the	DET
ejpam-4478	473	11	converse	converse	NOUN
ejpam-4478	473	12	will	will	AUX
ejpam-4478	473	13	then	then	ADV
ejpam-4478	473	14	follows	follow	VERB
ejpam-4478	473	15	.	.	PUNCT
ejpam-4478	474	1	this	this	PRON
ejpam-4478	474	2	completes	complete	VERB
ejpam-4478	474	3	the	the	DET
ejpam-4478	474	4	proof	proof	NOUN
ejpam-4478	474	5	.	.	PUNCT
ejpam-4478	475	1	theorem	theorem	VERB
ejpam-4478	475	2	7.5	7.5	NUM
ejpam-4478	475	3	.	.	PUNCT
ejpam-4478	476	1	let	let	VERB
ejpam-4478	476	2	k	k	PROPN
ejpam-4478	476	3	∈	∈	PROPN
ejpam-4478	476	4	z+	z+	PUNCT
ejpam-4478	476	5	.	.	PUNCT
ejpam-4478	477	1	for	for	ADP
ejpam-4478	477	2	any	any	DET
ejpam-4478	477	3	graph	graph	NOUN
ejpam-4478	477	4	g	g	NOUN
ejpam-4478	477	5	=	=	PUNCT
ejpam-4478	477	6	(	(	PUNCT
ejpam-4478	477	7	v	v	NOUN
ejpam-4478	477	8	(	(	PUNCT
ejpam-4478	477	9	g	g	NOUN
ejpam-4478	477	10	)	)	PUNCT
ejpam-4478	477	11	,	,	PUNCT
ejpam-4478	477	12	e(g	e(g	PROPN
ejpam-4478	477	13	)	)	PUNCT
ejpam-4478	477	14	)	)	PUNCT
ejpam-4478	477	15	of	of	ADP
ejpam-4478	477	16	order	order	NOUN
ejpam-4478	477	17	n	n	CCONJ
ejpam-4478	477	18	,	,	PUNCT
ejpam-4478	477	19	γkgr(g	γkgr(g	NOUN
ejpam-4478	477	20	)	)	PUNCT
ejpam-4478	477	21	=	=	PUNCT
ejpam-4478	477	22	γg(g	γg(g	X
ejpam-4478	477	23	)	)	PUNCT
ejpam-4478	477	24	if	if	SCONJ
ejpam-4478	477	25	and	and	CCONJ
ejpam-4478	477	26	only	only	ADV
ejpam-4478	477	27	if	if	SCONJ
ejpam-4478	477	28	g	g	PROPN
ejpam-4478	477	29	∼=	∼=	PROPN
ejpam-4478	477	30	kn	kn	NOUN
ejpam-4478	477	31	or	or	CCONJ
ejpam-4478	477	32	g	g	PROPN
ejpam-4478	477	33	∼=	∼=	PROPN
ejpam-4478	477	34	kn	kn	PROPN
ejpam-4478	477	35	.	.	PUNCT
ejpam-4478	477	36	proof	proof	NOUN
ejpam-4478	477	37	.	.	PUNCT
ejpam-4478	478	1	let	let	VERB
ejpam-4478	478	2	k	k	PROPN
ejpam-4478	478	3	∈	∈	PROPN
ejpam-4478	478	4	z+	z+	NUM
ejpam-4478	478	5	and	and	CCONJ
ejpam-4478	478	6	let	let	VERB
ejpam-4478	478	7	g	g	NOUN
ejpam-4478	478	8	=	=	SYM
ejpam-4478	478	9	(	(	PUNCT
ejpam-4478	478	10	v	v	NOUN
ejpam-4478	478	11	(	(	PUNCT
ejpam-4478	478	12	g	g	NOUN
ejpam-4478	478	13	)	)	PUNCT
ejpam-4478	478	14	,	,	PUNCT
ejpam-4478	478	15	e(g	e(g	PROPN
ejpam-4478	478	16	)	)	PUNCT
ejpam-4478	478	17	)	)	PUNCT
ejpam-4478	478	18	be	be	AUX
ejpam-4478	478	19	any	any	DET
ejpam-4478	478	20	graph	graph	NOUN
ejpam-4478	478	21	of	of	ADP
ejpam-4478	478	22	order	order	NOUN
ejpam-4478	478	23	n.	n.	NOUN
ejpam-4478	478	24	suppose	suppose	VERB
ejpam-4478	478	25	that	that	SCONJ
ejpam-4478	478	26	γkgr(g	γkgr(g	NOUN
ejpam-4478	478	27	)	)	PUNCT
ejpam-4478	478	28	=	=	SYM
ejpam-4478	478	29	γg(g	γg(g	X
ejpam-4478	478	30	)	)	PUNCT
ejpam-4478	478	31	and	and	CCONJ
ejpam-4478	478	32	let	let	VERB
ejpam-4478	478	33	f	f	PROPN
ejpam-4478	478	34	=	=	SYM
ejpam-4478	478	35	(	(	PUNCT
ejpam-4478	478	36	v	v	NOUN
ejpam-4478	478	37	f	f	NOUN
ejpam-4478	478	38	0	0	PUNCT
ejpam-4478	478	39	(	(	PUNCT
ejpam-4478	478	40	g	g	NOUN
ejpam-4478	478	41	)	)	PUNCT
ejpam-4478	478	42	,	,	PUNCT
ejpam-4478	478	43	v	v	NOUN
ejpam-4478	478	44	f	f	PROPN
ejpam-4478	478	45	1	1	NUM
ejpam-4478	478	46	(	(	PUNCT
ejpam-4478	478	47	g	g	NOUN
ejpam-4478	478	48	)	)	PUNCT
ejpam-4478	478	49	,	,	PUNCT
ejpam-4478	478	50	v	v	X
ejpam-4478	478	51	f	f	PROPN
ejpam-4478	478	52	2	2	NUM
ejpam-4478	478	53	(	(	PUNCT
ejpam-4478	478	54	g	g	NOUN
ejpam-4478	478	55	)	)	PUNCT
ejpam-4478	478	56	)	)	PUNCT
ejpam-4478	478	57	be	be	AUX
ejpam-4478	478	58	a	a	DET
ejpam-4478	478	59	γkgr(g	γkgr(g	NOUN
ejpam-4478	478	60	)	)	PUNCT
ejpam-4478	478	61	−	−	PROPN
ejpam-4478	478	62	function	function	NOUN
ejpam-4478	478	63	of	of	ADP
ejpam-4478	478	64	g	g	NOUN
ejpam-4478	478	65	,	,	PUNCT
ejpam-4478	478	66	where	where	SCONJ
ejpam-4478	478	67	v	v	ADP
ejpam-4478	478	68	f	f	NOUN
ejpam-4478	478	69	i	i	PRON
ejpam-4478	478	70	(	(	PUNCT
ejpam-4478	478	71	g	g	NOUN
ejpam-4478	478	72	)	)	PUNCT
ejpam-4478	478	73	=	=	SYM
ejpam-4478	478	74	{	{	PUNCT
ejpam-4478	478	75	v	v	NUM
ejpam-4478	478	76	∈	∈	NOUN
ejpam-4478	478	77	v	v	NOUN
ejpam-4478	478	78	(	(	PUNCT
ejpam-4478	478	79	g	g	NOUN
ejpam-4478	478	80	)	)	PUNCT
ejpam-4478	478	81	:	:	PUNCT
ejpam-4478	478	82	f(v	f(v	NOUN
ejpam-4478	478	83	)	)	PUNCT
ejpam-4478	479	1	=	=	SYM
ejpam-4478	479	2	i	i	PRON
ejpam-4478	479	3	for	for	ADP
ejpam-4478	479	4	i	i	PRON
ejpam-4478	479	5	=	=	NOUN
ejpam-4478	479	6	0	0	NUM
ejpam-4478	479	7	,	,	PUNCT
ejpam-4478	479	8	1	1	NUM
ejpam-4478	479	9	,	,	PUNCT
ejpam-4478	479	10	2	2	NUM
ejpam-4478	479	11	}	}	PUNCT
ejpam-4478	479	12	is	be	AUX
ejpam-4478	479	13	the	the	DET
ejpam-4478	479	14	partition	partition	NOUN
ejpam-4478	479	15	of	of	ADP
ejpam-4478	479	16	the	the	DET
ejpam-4478	479	17	vertex	vertex	NOUN
ejpam-4478	479	18	set	set	VERB
ejpam-4478	479	19	v	v	NOUN
ejpam-4478	479	20	(	(	PUNCT
ejpam-4478	479	21	g	g	NOUN
ejpam-4478	479	22	)	)	PUNCT
ejpam-4478	479	23	of	of	ADP
ejpam-4478	479	24	g	g	NOUN
ejpam-4478	479	25	induced	induce	VERB
ejpam-4478	479	26	by	by	ADP
ejpam-4478	479	27	the	the	DET
ejpam-4478	479	28	function	function	NOUN
ejpam-4478	479	29	f	f	PROPN
ejpam-4478	479	30	and	and	CCONJ
ejpam-4478	479	31	v	v	PROPN
ejpam-4478	479	32	(	(	PUNCT
ejpam-4478	479	33	g	g	NOUN
ejpam-4478	479	34	)	)	PUNCT
ejpam-4478	479	35	=	=	PUNCT
ejpam-4478	480	1	v	v	ADP
ejpam-4478	480	2	f	f	PROPN
ejpam-4478	480	3	0	0	PUNCT
ejpam-4478	481	1	(	(	PUNCT
ejpam-4478	481	2	g	g	NOUN
ejpam-4478	481	3	)	)	PUNCT
ejpam-4478	481	4	∪	∪	NOUN
ejpam-4478	481	5	v	v	ADP
ejpam-4478	481	6	f	f	PROPN
ejpam-4478	481	7	1	1	NUM
ejpam-4478	481	8	(	(	PUNCT
ejpam-4478	481	9	g	g	NOUN
ejpam-4478	481	10	)	)	PUNCT
ejpam-4478	481	11	∪	∪	NOUN
ejpam-4478	481	12	v	v	PROPN
ejpam-4478	481	13	f	f	PROPN
ejpam-4478	481	14	2	2	NUM
ejpam-4478	481	15	(	(	PUNCT
ejpam-4478	481	16	g	g	NOUN
ejpam-4478	481	17	)	)	PUNCT
ejpam-4478	481	18	.	.	PUNCT
ejpam-4478	482	1	note	note	VERB
ejpam-4478	482	2	that	that	SCONJ
ejpam-4478	482	3	,	,	PUNCT
ejpam-4478	482	4	|v	|v	PROPN
ejpam-4478	482	5	f	f	PROPN
ejpam-4478	482	6	0	0	PUNCT
ejpam-4478	482	7	(	(	PUNCT
ejpam-4478	482	8	g)|	g)|	PROPN
ejpam-4478	482	9	,	,	PUNCT
ejpam-4478	482	10	|v	|v	PROPN
ejpam-4478	482	11	f	f	PROPN
ejpam-4478	482	12	1	1	NUM
ejpam-4478	482	13	(	(	PUNCT
ejpam-4478	482	14	g)|	g)|	PROPN
ejpam-4478	482	15	,	,	PUNCT
ejpam-4478	482	16	|v	|v	PROPN
ejpam-4478	482	17	f	f	PROPN
ejpam-4478	482	18	2	2	NUM
ejpam-4478	482	19	(	(	PUNCT
ejpam-4478	482	20	g)|	g)|	VERB
ejpam-4478	482	21	≥	≥	NOUN
ejpam-4478	482	22	0	0	NUM
ejpam-4478	482	23	.	.	PUNCT
ejpam-4478	483	1	clearly	clearly	ADV
ejpam-4478	483	2	,	,	PUNCT
ejpam-4478	483	3	v	v	PROPN
ejpam-4478	483	4	f	f	PROPN
ejpam-4478	483	5	1	1	NUM
ejpam-4478	483	6	(	(	PUNCT
ejpam-4478	483	7	g	g	NOUN
ejpam-4478	483	8	)	)	PUNCT
ejpam-4478	483	9	∪	∪	NOUN
ejpam-4478	483	10	v	v	PROPN
ejpam-4478	483	11	f	f	PROPN
ejpam-4478	483	12	2	2	NUM
ejpam-4478	483	13	(	(	PUNCT
ejpam-4478	483	14	g	g	NOUN
ejpam-4478	483	15	)	)	PUNCT
ejpam-4478	483	16	is	be	AUX
ejpam-4478	483	17	a	a	DET
ejpam-4478	483	18	global	global	ADJ
ejpam-4478	483	19	dominating	dominating	NOUN
ejpam-4478	483	20	set	set	NOUN
ejpam-4478	483	21	of	of	ADP
ejpam-4478	483	22	g.	g.	PROPN
ejpam-4478	484	1	thus	thus	ADV
ejpam-4478	484	2	,	,	PUNCT
ejpam-4478	484	3	since	since	SCONJ
ejpam-4478	484	4	γg(g	γg(g	NUM
ejpam-4478	484	5	)	)	PUNCT
ejpam-4478	484	6	is	be	AUX
ejpam-4478	484	7	the	the	DET
ejpam-4478	484	8	minimum	minimum	ADJ
ejpam-4478	484	9	cardinality	cardinality	NOUN
ejpam-4478	484	10	taken	take	VERB
ejpam-4478	484	11	over	over	ADP
ejpam-4478	484	12	all	all	DET
ejpam-4478	484	13	global	global	ADJ
ejpam-4478	484	14	dominating	dominating	NOUN
ejpam-4478	484	15	sets	set	NOUN
ejpam-4478	484	16	of	of	ADP
ejpam-4478	484	17	g	g	NOUN
ejpam-4478	484	18	,	,	PUNCT
ejpam-4478	484	19	γg(g	γg(g	NOUN
ejpam-4478	484	20	)	)	PUNCT
ejpam-4478	484	21	≤	≤	NOUN
ejpam-4478	484	22	|v	|v	X
ejpam-4478	484	23	f	f	NOUN
ejpam-4478	484	24	1	1	NUM
ejpam-4478	484	25	(	(	PUNCT
ejpam-4478	484	26	g	g	NOUN
ejpam-4478	484	27	)	)	PUNCT
ejpam-4478	484	28	∪	∪	NOUN
ejpam-4478	484	29	v	v	ADP
ejpam-4478	484	30	f	f	PROPN
ejpam-4478	484	31	2	2	NUM
ejpam-4478	484	32	(	(	PUNCT
ejpam-4478	484	33	g)|	g)|	NOUN
ejpam-4478	484	34	=	=	SYM
ejpam-4478	484	35	|v	|v	PROPN
ejpam-4478	484	36	f	f	PROPN
ejpam-4478	484	37	1	1	NUM
ejpam-4478	484	38	(	(	PUNCT
ejpam-4478	484	39	g)|+	g)|+	PROPN
ejpam-4478	484	40	|v	|v	PROPN
ejpam-4478	484	41	f	f	PROPN
ejpam-4478	484	42	2	2	NUM
ejpam-4478	484	43	(	(	PUNCT
ejpam-4478	484	44	g)|	g)|	PROPN
ejpam-4478	484	45	,	,	PUNCT
ejpam-4478	484	46	g.	g.	PROPN
ejpam-4478	484	47	entero	entero	PROPN
ejpam-4478	484	48	,	,	PUNCT
ejpam-4478	484	49	s.	s.	PROPN
ejpam-4478	484	50	espinola	espinola	PROPN
ejpam-4478	484	51	/	/	SYM
ejpam-4478	484	52	eur	eur	PROPN
ejpam-4478	484	53	.	.	PUNCT
ejpam-4478	485	1	j.	j.	PROPN
ejpam-4478	485	2	pure	pure	PROPN
ejpam-4478	485	3	appl	appl	PROPN
ejpam-4478	485	4	.	.	PROPN
ejpam-4478	485	5	math	math	PROPN
ejpam-4478	485	6	,	,	PUNCT
ejpam-4478	485	7	16	16	NUM
ejpam-4478	485	8	(	(	PUNCT
ejpam-4478	485	9	1	1	NUM
ejpam-4478	485	10	)	)	PUNCT
ejpam-4478	485	11	(	(	PUNCT
ejpam-4478	485	12	2023	2023	NUM
ejpam-4478	485	13	)	)	PUNCT
ejpam-4478	485	14	,	,	PUNCT
ejpam-4478	485	15	44	44	NUM
ejpam-4478	485	16	-	-	SYM
ejpam-4478	485	17	61	61	NUM
ejpam-4478	485	18	59	59	NUM
ejpam-4478	485	19	and	and	CCONJ
ejpam-4478	485	20	so	so	ADV
ejpam-4478	485	21	,	,	PUNCT
ejpam-4478	485	22	we	we	PRON
ejpam-4478	485	23	have	have	VERB
ejpam-4478	485	24	γg(g	γg(g	NOUN
ejpam-4478	485	25	)	)	PUNCT
ejpam-4478	485	26	≤	≤	NOUN
ejpam-4478	485	27	|v	|v	X
ejpam-4478	485	28	f	f	PROPN
ejpam-4478	485	29	1	1	NUM
ejpam-4478	485	30	(	(	PUNCT
ejpam-4478	485	31	g)|+	g)|+	PROPN
ejpam-4478	485	32	|v	|v	PROPN
ejpam-4478	485	33	f	f	PROPN
ejpam-4478	485	34	2	2	NUM
ejpam-4478	485	35	(	(	PUNCT
ejpam-4478	485	36	g)|	g)|	NOUN
ejpam-4478	485	37	.	.	PUNCT
ejpam-4478	486	1	(	(	PUNCT
ejpam-4478	486	2	1	1	X
ejpam-4478	486	3	)	)	PUNCT
ejpam-4478	486	4	since	since	SCONJ
ejpam-4478	486	5	f	f	PROPN
ejpam-4478	486	6	=	=	PUNCT
ejpam-4478	486	7	(	(	PUNCT
ejpam-4478	486	8	v	v	NOUN
ejpam-4478	486	9	f	f	NOUN
ejpam-4478	486	10	0	0	PUNCT
ejpam-4478	486	11	(	(	PUNCT
ejpam-4478	486	12	g	g	NOUN
ejpam-4478	486	13	)	)	PUNCT
ejpam-4478	486	14	,	,	PUNCT
ejpam-4478	486	15	v	v	NOUN
ejpam-4478	486	16	f	f	PROPN
ejpam-4478	486	17	1	1	NUM
ejpam-4478	486	18	(	(	PUNCT
ejpam-4478	486	19	g	g	NOUN
ejpam-4478	486	20	)	)	PUNCT
ejpam-4478	486	21	,	,	PUNCT
ejpam-4478	486	22	v	v	X
ejpam-4478	486	23	f	f	PROPN
ejpam-4478	486	24	2	2	NUM
ejpam-4478	486	25	(	(	PUNCT
ejpam-4478	486	26	g	g	NOUN
ejpam-4478	486	27	)	)	PUNCT
ejpam-4478	486	28	)	)	PUNCT
ejpam-4478	486	29	is	be	AUX
ejpam-4478	486	30	a	a	DET
ejpam-4478	486	31	γkgr(g	γkgr(g	NOUN
ejpam-4478	486	32	)	)	PUNCT
ejpam-4478	486	33	−	−	PROPN
ejpam-4478	486	34	function	function	NOUN
ejpam-4478	486	35	of	of	ADP
ejpam-4478	486	36	g	g	PROPN
ejpam-4478	486	37	,	,	PUNCT
ejpam-4478	486	38	γkgr(g	γkgr(g	NOUN
ejpam-4478	486	39	)	)	PUNCT
ejpam-4478	487	1	=	=	PUNCT
ejpam-4478	487	2	|v	|v	PROPN
ejpam-4478	487	3	f	f	PROPN
ejpam-4478	487	4	1	1	NUM
ejpam-4478	487	5	(	(	PUNCT
ejpam-4478	487	6	g)|+	g)|+	NOUN
ejpam-4478	487	7	2|v	2|v	PROPN
ejpam-4478	487	8	f	f	NOUN
ejpam-4478	487	9	2	2	NUM
ejpam-4478	487	10	(	(	PUNCT
ejpam-4478	487	11	g)|	g)|	NOUN
ejpam-4478	487	12	.	.	PUNCT
ejpam-4478	488	1	(	(	PUNCT
ejpam-4478	488	2	2	2	NUM
ejpam-4478	488	3	)	)	PUNCT
ejpam-4478	488	4	thus	thus	ADV
ejpam-4478	488	5	,	,	PUNCT
ejpam-4478	488	6	since	since	SCONJ
ejpam-4478	488	7	we	we	PRON
ejpam-4478	488	8	assumed	assume	VERB
ejpam-4478	488	9	that	that	SCONJ
ejpam-4478	488	10	γkgr(g	γkgr(g	NOUN
ejpam-4478	488	11	)	)	PUNCT
ejpam-4478	488	12	=	=	SYM
ejpam-4478	488	13	γg(g	γg(g	NOUN
ejpam-4478	488	14	)	)	PUNCT
ejpam-4478	488	15	,	,	PUNCT
ejpam-4478	488	16	from	from	ADP
ejpam-4478	488	17	(	(	PUNCT
ejpam-4478	488	18	1	1	NUM
ejpam-4478	488	19	)	)	PUNCT
ejpam-4478	488	20	and	and	CCONJ
ejpam-4478	488	21	(	(	PUNCT
ejpam-4478	488	22	2	2	NUM
ejpam-4478	488	23	)	)	PUNCT
ejpam-4478	488	24	,	,	PUNCT
ejpam-4478	488	25	we	we	PRON
ejpam-4478	488	26	have	have	VERB
ejpam-4478	488	27	|v	|v	PROPN
ejpam-4478	488	28	f	f	PROPN
ejpam-4478	488	29	1	1	NUM
ejpam-4478	488	30	(	(	PUNCT
ejpam-4478	488	31	g)|	g)|	NOUN
ejpam-4478	488	32	+	+	CCONJ
ejpam-4478	488	33	2|v	2|v	PROPN
ejpam-4478	488	34	f	f	NOUN
ejpam-4478	488	35	2	2	NUM
ejpam-4478	488	36	(	(	PUNCT
ejpam-4478	488	37	g)|	g)|	NOUN
ejpam-4478	488	38	=	=	SYM
ejpam-4478	488	39	γkgr(g	γkgr(g	PROPN
ejpam-4478	488	40	)	)	PUNCT
ejpam-4478	488	41	=	=	SYM
ejpam-4478	488	42	γg(g	γg(g	NOUN
ejpam-4478	488	43	)	)	PUNCT
ejpam-4478	488	44	≤	≤	NOUN
ejpam-4478	488	45	|v	|v	X
ejpam-4478	488	46	f	f	PROPN
ejpam-4478	488	47	1	1	NUM
ejpam-4478	488	48	(	(	PUNCT
ejpam-4478	488	49	g)|+	g)|+	PROPN
ejpam-4478	488	50	|v	|v	PROPN
ejpam-4478	488	51	f	f	PROPN
ejpam-4478	488	52	2	2	NUM
ejpam-4478	488	53	(	(	PUNCT
ejpam-4478	488	54	g)|	g)|	NOUN
ejpam-4478	488	55	,	,	PUNCT
ejpam-4478	488	56	and	and	CCONJ
ejpam-4478	488	57	hence	hence	ADV
ejpam-4478	488	58	,	,	PUNCT
ejpam-4478	488	59	we	we	PRON
ejpam-4478	488	60	have	have	VERB
ejpam-4478	488	61	|v	|v	PROPN
ejpam-4478	488	62	f	f	PROPN
ejpam-4478	488	63	1	1	NUM
ejpam-4478	488	64	(	(	PUNCT
ejpam-4478	488	65	g)|+	g)|+	NOUN
ejpam-4478	489	1	2|v	2|v	PROPN
ejpam-4478	489	2	f	f	NOUN
ejpam-4478	489	3	2	2	NUM
ejpam-4478	489	4	(	(	PUNCT
ejpam-4478	489	5	g)|	g)|	VERB
ejpam-4478	489	6	≤	≤	NUM
ejpam-4478	489	7	|v	|v	ADP
ejpam-4478	489	8	f	f	NOUN
ejpam-4478	489	9	1	1	NUM
ejpam-4478	489	10	(	(	PUNCT
ejpam-4478	489	11	g)|+	g)|+	PROPN
ejpam-4478	489	12	|v	|v	PROPN
ejpam-4478	489	13	f	f	PROPN
ejpam-4478	489	14	2	2	NUM
ejpam-4478	489	15	(	(	PUNCT
ejpam-4478	489	16	g)|	g)|	NOUN
ejpam-4478	489	17	.	.	PUNCT
ejpam-4478	489	18	(	(	PUNCT
ejpam-4478	489	19	3	3	X
ejpam-4478	489	20	)	)	PUNCT
ejpam-4478	489	21	since	since	SCONJ
ejpam-4478	489	22	|v	|v	PROPN
ejpam-4478	489	23	f	f	PROPN
ejpam-4478	489	24	0	0	PUNCT
ejpam-4478	489	25	(	(	PUNCT
ejpam-4478	489	26	g)|	g)|	PROPN
ejpam-4478	489	27	,	,	PUNCT
ejpam-4478	489	28	|v	|v	PROPN
ejpam-4478	489	29	f	f	PROPN
ejpam-4478	489	30	1	1	NUM
ejpam-4478	489	31	(	(	PUNCT
ejpam-4478	489	32	g)|	g)|	PROPN
ejpam-4478	489	33	,	,	PUNCT
ejpam-4478	489	34	|v	|v	PROPN
ejpam-4478	489	35	f	f	PROPN
ejpam-4478	489	36	2	2	NUM
ejpam-4478	489	37	(	(	PUNCT
ejpam-4478	489	38	g)|	g)|	VERB
ejpam-4478	489	39	≥	≥	NOUN
ejpam-4478	489	40	0	0	NUM
ejpam-4478	489	41	,	,	PUNCT
ejpam-4478	489	42	by	by	ADP
ejpam-4478	489	43	addition	addition	NOUN
ejpam-4478	489	44	and	and	CCONJ
ejpam-4478	489	45	subtraction	subtraction	NOUN
ejpam-4478	489	46	properties	property	NOUN
ejpam-4478	489	47	of	of	ADP
ejpam-4478	489	48	equality	equality	NOUN
ejpam-4478	489	49	,	,	PUNCT
ejpam-4478	489	50	we	we	PRON
ejpam-4478	489	51	have	have	VERB
ejpam-4478	489	52	|v	|v	PROPN
ejpam-4478	489	53	f	f	PROPN
ejpam-4478	489	54	2	2	NUM
ejpam-4478	489	55	(	(	PUNCT
ejpam-4478	489	56	g)|	g)|	VERB
ejpam-4478	489	57	≤	≤	NOUN
ejpam-4478	489	58	0	0	NUM
ejpam-4478	489	59	,	,	PUNCT
ejpam-4478	489	60	that	that	ADV
ejpam-4478	489	61	is	is	ADV
ejpam-4478	489	62	,	,	PUNCT
ejpam-4478	489	63	|v	|v	PROPN
ejpam-4478	489	64	f	f	PROPN
ejpam-4478	489	65	2	2	NUM
ejpam-4478	489	66	(	(	PUNCT
ejpam-4478	489	67	g)|	g)|	VERB
ejpam-4478	489	68	≤	≤	NOUN
ejpam-4478	489	69	0	0	PUNCT
ejpam-4478	490	1	and	and	CCONJ
ejpam-4478	490	2	since	since	SCONJ
ejpam-4478	490	3	|v	|v	PROPN
ejpam-4478	490	4	f	f	PROPN
ejpam-4478	490	5	2	2	NUM
ejpam-4478	490	6	(	(	PUNCT
ejpam-4478	490	7	g)|	g)|	VERB
ejpam-4478	490	8	≥	≥	NOUN
ejpam-4478	490	9	0	0	NUM
ejpam-4478	490	10	,	,	PUNCT
ejpam-4478	490	11	we	we	PRON
ejpam-4478	490	12	may	may	AUX
ejpam-4478	490	13	conclude	conclude	VERB
ejpam-4478	490	14	that	that	SCONJ
ejpam-4478	490	15	|v	|v	PROPN
ejpam-4478	490	16	f	f	X
ejpam-4478	490	17	2	2	NUM
ejpam-4478	490	18	(	(	PUNCT
ejpam-4478	490	19	g)|	g)|	NOUN
ejpam-4478	490	20	=	=	NOUN
ejpam-4478	490	21	0	0	NUM
ejpam-4478	490	22	.	.	PUNCT
ejpam-4478	491	1	hence	hence	ADV
ejpam-4478	491	2	,	,	PUNCT
ejpam-4478	491	3	v	v	PROPN
ejpam-4478	491	4	f	f	PROPN
ejpam-4478	491	5	0	0	PUNCT
ejpam-4478	491	6	(	(	PUNCT
ejpam-4478	491	7	g	g	NOUN
ejpam-4478	491	8	)	)	PUNCT
ejpam-4478	491	9	=	=	NOUN
ejpam-4478	491	10	∅	∅	NOUN
ejpam-4478	491	11	,	,	PUNCT
ejpam-4478	491	12	and	and	CCONJ
ejpam-4478	491	13	so	so	ADV
ejpam-4478	491	14	,	,	PUNCT
ejpam-4478	491	15	we	we	PRON
ejpam-4478	491	16	are	be	AUX
ejpam-4478	491	17	forced	force	VERB
ejpam-4478	491	18	to	to	PART
ejpam-4478	491	19	have	have	VERB
ejpam-4478	491	20	v	v	NUM
ejpam-4478	491	21	f	f	PROPN
ejpam-4478	491	22	1	1	NUM
ejpam-4478	491	23	(	(	PUNCT
ejpam-4478	491	24	g	g	NOUN
ejpam-4478	491	25	)	)	PUNCT
ejpam-4478	491	26	=	=	NOUN
ejpam-4478	491	27	v	v	X
ejpam-4478	491	28	(	(	PUNCT
ejpam-4478	491	29	g	g	NOUN
ejpam-4478	491	30	)	)	PUNCT
ejpam-4478	491	31	which	which	PRON
ejpam-4478	491	32	implies	imply	VERB
ejpam-4478	491	33	that	that	SCONJ
ejpam-4478	491	34	|v	|v	PROPN
ejpam-4478	491	35	f	f	PROPN
ejpam-4478	491	36	1	1	NUM
ejpam-4478	491	37	(	(	PUNCT
ejpam-4478	491	38	g)|	g)|	NOUN
ejpam-4478	491	39	=	=	PUNCT
ejpam-4478	491	40	|v	|v	X
ejpam-4478	491	41	(	(	PUNCT
ejpam-4478	491	42	g)|	g)|	NOUN
ejpam-4478	491	43	=	=	SYM
ejpam-4478	491	44	n.	n.	NOUN
ejpam-4478	491	45	since	since	SCONJ
ejpam-4478	491	46	f	f	PROPN
ejpam-4478	491	47	is	be	AUX
ejpam-4478	491	48	a	a	DET
ejpam-4478	491	49	γkgr(g	γkgr(g	NOUN
ejpam-4478	491	50	)	)	PUNCT
ejpam-4478	491	51	−	−	PROPN
ejpam-4478	491	52	function	function	NOUN
ejpam-4478	491	53	of	of	ADP
ejpam-4478	491	54	g	g	PROPN
ejpam-4478	491	55	,	,	PUNCT
ejpam-4478	491	56	γkgr(g	γkgr(g	NOUN
ejpam-4478	491	57	)	)	PUNCT
ejpam-4478	491	58	=	=	PUNCT
ejpam-4478	492	1	|v	|v	PROPN
ejpam-4478	492	2	f	f	PROPN
ejpam-4478	492	3	1	1	NUM
ejpam-4478	492	4	(	(	PUNCT
ejpam-4478	492	5	g)|	g)|	NOUN
ejpam-4478	492	6	+	+	CCONJ
ejpam-4478	492	7	2|v	2|v	PROPN
ejpam-4478	492	8	f	f	NOUN
ejpam-4478	492	9	2	2	NUM
ejpam-4478	492	10	(	(	PUNCT
ejpam-4478	492	11	g)|	g)|	NOUN
ejpam-4478	492	12	=	=	PUNCT
ejpam-4478	492	13	n	n	PROPN
ejpam-4478	492	14	+	+	NOUN
ejpam-4478	492	15	0	0	NUM
ejpam-4478	492	16	=	=	SYM
ejpam-4478	492	17	n	n	NOUN
ejpam-4478	492	18	which	which	PRON
ejpam-4478	492	19	implies	imply	VERB
ejpam-4478	492	20	that	that	PRON
ejpam-4478	492	21	γg(g	γg(g	PRON
ejpam-4478	492	22	)	)	PUNCT
ejpam-4478	492	23	=	=	SYM
ejpam-4478	492	24	n	n	CCONJ
ejpam-4478	492	25	for	for	SCONJ
ejpam-4478	492	26	we	we	PRON
ejpam-4478	492	27	assumed	assume	VERB
ejpam-4478	492	28	that	that	SCONJ
ejpam-4478	492	29	γkgr(g	γkgr(g	NOUN
ejpam-4478	492	30	)	)	PUNCT
ejpam-4478	492	31	=	=	PUNCT
ejpam-4478	492	32	γg(g	γg(g	NOUN
ejpam-4478	492	33	)	)	PUNCT
ejpam-4478	492	34	.	.	PUNCT
ejpam-4478	493	1	hence	hence	ADV
ejpam-4478	493	2	,	,	PUNCT
ejpam-4478	493	3	in	in	ADP
ejpam-4478	493	4	reference	reference	NOUN
ejpam-4478	493	5	to	to	ADP
ejpam-4478	493	6	proposition	proposition	NOUN
ejpam-4478	493	7	3.13	3.13	NUM
ejpam-4478	493	8	,	,	PUNCT
ejpam-4478	493	9	g	g	PROPN
ejpam-4478	493	10	∼=	∼=	PROPN
ejpam-4478	493	11	kn	kn	PROPN
ejpam-4478	493	12	.	.	PUNCT
ejpam-4478	494	1	conversely	conversely	ADV
ejpam-4478	494	2	,	,	PUNCT
ejpam-4478	494	3	assume	assume	VERB
ejpam-4478	494	4	that	that	SCONJ
ejpam-4478	494	5	g	g	PROPN
ejpam-4478	494	6	∼=	∼=	PROPN
ejpam-4478	494	7	kn	kn	NOUN
ejpam-4478	494	8	or	or	CCONJ
ejpam-4478	494	9	g	g	PROPN
ejpam-4478	494	10	∼=	∼=	PROPN
ejpam-4478	494	11	kn	kn	PROPN
ejpam-4478	494	12	.	.	PUNCT
ejpam-4478	495	1	by	by	ADP
ejpam-4478	495	2	theorem	theorem	ADJ
ejpam-4478	495	3	5.1	5.1	NUM
ejpam-4478	495	4	and	and	CCONJ
ejpam-4478	495	5	corollary	corollary	ADJ
ejpam-4478	495	6	5.2	5.2	NUM
ejpam-4478	495	7	,	,	PUNCT
ejpam-4478	495	8	γkgr(g	γkgr(g	NOUN
ejpam-4478	495	9	)	)	PUNCT
ejpam-4478	495	10	=	=	SYM
ejpam-4478	495	11	γkgr(kn	γkgr(kn	NOUN
ejpam-4478	495	12	)	)	PUNCT
ejpam-4478	495	13	=	=	SYM
ejpam-4478	495	14	n	n	PROPN
ejpam-4478	495	15	and	and	CCONJ
ejpam-4478	495	16	γkgr(g	γkgr(g	NUM
ejpam-4478	495	17	)	)	PUNCT
ejpam-4478	495	18	=	=	SYM
ejpam-4478	495	19	γkgr(kn	γkgr(kn	NOUN
ejpam-4478	495	20	)	)	PUNCT
ejpam-4478	495	21	=	=	SYM
ejpam-4478	495	22	n	n	CCONJ
ejpam-4478	495	23	,	,	PUNCT
ejpam-4478	495	24	respectively	respectively	ADV
ejpam-4478	495	25	.	.	PUNCT
ejpam-4478	496	1	also	also	ADV
ejpam-4478	496	2	,	,	PUNCT
ejpam-4478	496	3	by	by	ADP
ejpam-4478	496	4	proposition	proposition	NOUN
ejpam-4478	496	5	3.13	3.13	NUM
ejpam-4478	496	6	,	,	PUNCT
ejpam-4478	496	7	γg(g	γg(g	X
ejpam-4478	496	8	)	)	PUNCT
ejpam-4478	496	9	=	=	SYM
ejpam-4478	497	1	n	n	NOUN
ejpam-4478	497	2	if	if	SCONJ
ejpam-4478	498	1	and	and	CCONJ
ejpam-4478	498	2	only	only	ADV
ejpam-4478	498	3	if	if	SCONJ
ejpam-4478	498	4	g	g	PROPN
ejpam-4478	498	5	∼=	∼=	PROPN
ejpam-4478	498	6	kn	kn	NOUN
ejpam-4478	498	7	or	or	CCONJ
ejpam-4478	498	8	g	g	PROPN
ejpam-4478	498	9	∼=	∼=	PROPN
ejpam-4478	498	10	kn	kn	PROPN
ejpam-4478	498	11	.	.	PUNCT
ejpam-4478	499	1	thus	thus	ADV
ejpam-4478	499	2	,	,	PUNCT
ejpam-4478	499	3	the	the	DET
ejpam-4478	499	4	desired	desire	VERB
ejpam-4478	499	5	result	result	NOUN
ejpam-4478	499	6	follows	follow	VERB
ejpam-4478	499	7	.	.	PUNCT
ejpam-4478	500	1	theorem	theorem	NOUN
ejpam-4478	500	2	7.6	7.6	NUM
ejpam-4478	500	3	.	.	PUNCT
ejpam-4478	501	1	let	let	VERB
ejpam-4478	501	2	k	k	PROPN
ejpam-4478	501	3	∈	∈	PROPN
ejpam-4478	501	4	z+	z+	PUNCT
ejpam-4478	501	5	.	.	PUNCT
ejpam-4478	502	1	for	for	ADP
ejpam-4478	502	2	any	any	DET
ejpam-4478	502	3	graph	graph	NOUN
ejpam-4478	502	4	g	g	NOUN
ejpam-4478	502	5	=	=	PUNCT
ejpam-4478	502	6	(	(	PUNCT
ejpam-4478	502	7	v	v	NOUN
ejpam-4478	502	8	(	(	PUNCT
ejpam-4478	502	9	g	g	NOUN
ejpam-4478	502	10	)	)	PUNCT
ejpam-4478	502	11	,	,	PUNCT
ejpam-4478	502	12	e(g	e(g	PROPN
ejpam-4478	502	13	)	)	PUNCT
ejpam-4478	502	14	)	)	PUNCT
ejpam-4478	502	15	of	of	ADP
ejpam-4478	502	16	order	order	NOUN
ejpam-4478	502	17	n	n	CCONJ
ejpam-4478	502	18	,	,	PUNCT
ejpam-4478	502	19	γkgr(g	γkgr(g	NOUN
ejpam-4478	502	20	)	)	PUNCT
ejpam-4478	502	21	=	=	SYM
ejpam-4478	502	22	γkr(g	γkr(g	PROPN
ejpam-4478	502	23	)	)	PUNCT
ejpam-4478	503	1	if	if	SCONJ
ejpam-4478	503	2	and	and	CCONJ
ejpam-4478	503	3	only	only	ADV
ejpam-4478	503	4	if	if	SCONJ
ejpam-4478	503	5	g	g	PROPN
ejpam-4478	503	6	∼=	∼=	PROPN
ejpam-4478	503	7	kn	kn	PROPN
ejpam-4478	503	8	.	.	PUNCT
ejpam-4478	503	9	proof	proof	NOUN
ejpam-4478	503	10	.	.	PUNCT
ejpam-4478	504	1	let	let	VERB
ejpam-4478	504	2	k	k	PROPN
ejpam-4478	504	3	∈	∈	PROPN
ejpam-4478	504	4	z+	z+	NUM
ejpam-4478	504	5	and	and	CCONJ
ejpam-4478	504	6	let	let	VERB
ejpam-4478	504	7	g	g	NOUN
ejpam-4478	504	8	=	=	SYM
ejpam-4478	504	9	(	(	PUNCT
ejpam-4478	504	10	v	v	NOUN
ejpam-4478	504	11	(	(	PUNCT
ejpam-4478	504	12	g	g	NOUN
ejpam-4478	504	13	)	)	PUNCT
ejpam-4478	504	14	,	,	PUNCT
ejpam-4478	504	15	e(g	e(g	PROPN
ejpam-4478	504	16	)	)	PUNCT
ejpam-4478	504	17	)	)	PUNCT
ejpam-4478	504	18	be	be	AUX
ejpam-4478	504	19	any	any	DET
ejpam-4478	504	20	graph	graph	NOUN
ejpam-4478	504	21	of	of	ADP
ejpam-4478	504	22	order	order	NOUN
ejpam-4478	504	23	n.	n.	NOUN
ejpam-4478	504	24	suppose	suppose	VERB
ejpam-4478	504	25	that	that	SCONJ
ejpam-4478	504	26	γkgr(g	γkgr(g	NOUN
ejpam-4478	504	27	)	)	PUNCT
ejpam-4478	505	1	=	=	SYM
ejpam-4478	505	2	γkr(g	γkr(g	PROPN
ejpam-4478	505	3	)	)	PUNCT
ejpam-4478	505	4	.	.	PUNCT
ejpam-4478	506	1	by	by	ADP
ejpam-4478	506	2	remark	remark	NOUN
ejpam-4478	506	3	3.12	3.12	NUM
ejpam-4478	506	4	,	,	PUNCT
ejpam-4478	506	5	we	we	PRON
ejpam-4478	506	6	have	have	VERB
ejpam-4478	506	7	γkr(g	γkr(g	PROPN
ejpam-4478	506	8	)	)	PUNCT
ejpam-4478	506	9	≤	≤	NOUN
ejpam-4478	506	10	n	n	ADP
ejpam-4478	506	11	and	and	CCONJ
ejpam-4478	506	12	γkr(g	γkr(g	NUM
ejpam-4478	506	13	)	)	PUNCT
ejpam-4478	507	1	=	=	SYM
ejpam-4478	507	2	n	n	NOUN
ejpam-4478	507	3	if	if	SCONJ
ejpam-4478	507	4	and	and	CCONJ
ejpam-4478	507	5	only	only	ADV
ejpam-4478	507	6	if	if	SCONJ
ejpam-4478	507	7	g	g	PROPN
ejpam-4478	507	8	∼=	∼=	PROPN
ejpam-4478	507	9	kn	kn	PROPN
ejpam-4478	507	10	.	.	PUNCT
ejpam-4478	508	1	thus	thus	ADV
ejpam-4478	508	2	,	,	PUNCT
ejpam-4478	508	3	since	since	SCONJ
ejpam-4478	508	4	γkgr(g	γkgr(g	NUM
ejpam-4478	508	5	)	)	PUNCT
ejpam-4478	508	6	=	=	SYM
ejpam-4478	508	7	γkr(g	γkr(g	PROPN
ejpam-4478	508	8	)	)	PUNCT
ejpam-4478	508	9	and	and	CCONJ
ejpam-4478	508	10	γkr(g	γkr(g	NUM
ejpam-4478	508	11	)	)	PUNCT
ejpam-4478	508	12	=	=	SYM
ejpam-4478	509	1	n	n	NOUN
ejpam-4478	509	2	if	if	SCONJ
ejpam-4478	510	1	and	and	CCONJ
ejpam-4478	510	2	only	only	ADV
ejpam-4478	510	3	if	if	SCONJ
ejpam-4478	510	4	g	g	PROPN
ejpam-4478	510	5	∼=	∼=	PROPN
ejpam-4478	510	6	kn	kn	NOUN
ejpam-4478	510	7	,	,	PUNCT
ejpam-4478	510	8	it	it	PRON
ejpam-4478	510	9	follows	follow	VERB
ejpam-4478	510	10	that	that	SCONJ
ejpam-4478	510	11	γkgr(g	γkgr(g	NOUN
ejpam-4478	510	12	)	)	PUNCT
ejpam-4478	510	13	=	=	SYM
ejpam-4478	511	1	n	n	NOUN
ejpam-4478	511	2	if	if	SCONJ
ejpam-4478	511	3	and	and	CCONJ
ejpam-4478	511	4	only	only	ADV
ejpam-4478	511	5	if	if	SCONJ
ejpam-4478	511	6	g	g	PROPN
ejpam-4478	511	7	∼=	∼=	PROPN
ejpam-4478	511	8	kn	kn	PROPN
ejpam-4478	511	9	.	.	PUNCT
ejpam-4478	512	1	hence	hence	ADV
ejpam-4478	512	2	,	,	PUNCT
ejpam-4478	512	3	γkgr(g	γkgr(g	NOUN
ejpam-4478	512	4	)	)	PUNCT
ejpam-4478	512	5	=	=	SYM
ejpam-4478	512	6	γkr(g	γkr(g	PROPN
ejpam-4478	512	7	)	)	PUNCT
ejpam-4478	512	8	=	=	SYM
ejpam-4478	513	1	n	n	NOUN
ejpam-4478	513	2	if	if	SCONJ
ejpam-4478	514	1	and	and	CCONJ
ejpam-4478	514	2	only	only	ADV
ejpam-4478	514	3	if	if	SCONJ
ejpam-4478	514	4	g	g	PROPN
ejpam-4478	514	5	∼=	∼=	PROPN
ejpam-4478	514	6	kn	kn	NOUN
ejpam-4478	514	7	and	and	CCONJ
ejpam-4478	514	8	thus	thus	ADV
ejpam-4478	514	9	,	,	PUNCT
ejpam-4478	514	10	the	the	DET
ejpam-4478	514	11	converse	converse	NOUN
ejpam-4478	514	12	will	will	AUX
ejpam-4478	514	13	then	then	ADV
ejpam-4478	514	14	follows	follow	VERB
ejpam-4478	514	15	.	.	PUNCT
ejpam-4478	515	1	this	this	PRON
ejpam-4478	515	2	completes	complete	VERB
ejpam-4478	515	3	the	the	DET
ejpam-4478	515	4	proof	proof	NOUN
ejpam-4478	515	5	.	.	PUNCT
ejpam-4478	516	1	theorem	theorem	VERB
ejpam-4478	516	2	7.7	7.7	NUM
ejpam-4478	516	3	.	.	PUNCT
ejpam-4478	517	1	let	let	VERB
ejpam-4478	517	2	k	k	PROPN
ejpam-4478	517	3	∈	∈	PROPN
ejpam-4478	517	4	z+	z+	PUNCT
ejpam-4478	517	5	.	.	PUNCT
ejpam-4478	518	1	for	for	ADP
ejpam-4478	518	2	any	any	DET
ejpam-4478	518	3	graph	graph	NOUN
ejpam-4478	518	4	g	g	NOUN
ejpam-4478	518	5	=	=	PUNCT
ejpam-4478	518	6	(	(	PUNCT
ejpam-4478	518	7	v	v	NOUN
ejpam-4478	518	8	(	(	PUNCT
ejpam-4478	518	9	g	g	NOUN
ejpam-4478	518	10	)	)	PUNCT
ejpam-4478	518	11	,	,	PUNCT
ejpam-4478	518	12	e(g	e(g	PROPN
ejpam-4478	518	13	)	)	PUNCT
ejpam-4478	518	14	)	)	PUNCT
ejpam-4478	518	15	of	of	ADP
ejpam-4478	518	16	order	order	NOUN
ejpam-4478	518	17	n	n	CCONJ
ejpam-4478	518	18	,	,	PUNCT
ejpam-4478	518	19	γkgr(g	γkgr(g	NOUN
ejpam-4478	518	20	)	)	PUNCT
ejpam-4478	518	21	=	=	SYM
ejpam-4478	518	22	γgr(g	γgr(g	PROPN
ejpam-4478	518	23	)	)	PUNCT
ejpam-4478	518	24	if	if	SCONJ
ejpam-4478	518	25	and	and	CCONJ
ejpam-4478	518	26	only	only	ADV
ejpam-4478	518	27	if	if	SCONJ
ejpam-4478	518	28	g	g	PROPN
ejpam-4478	518	29	∼=	∼=	PROPN
ejpam-4478	518	30	kn	kn	NOUN
ejpam-4478	518	31	or	or	CCONJ
ejpam-4478	518	32	g	g	PROPN
ejpam-4478	518	33	∼=	∼=	PROPN
ejpam-4478	518	34	kn	kn	PROPN
ejpam-4478	518	35	.	.	PUNCT
ejpam-4478	518	36	proof	proof	NOUN
ejpam-4478	518	37	.	.	PUNCT
ejpam-4478	519	1	let	let	VERB
ejpam-4478	519	2	k	k	PROPN
ejpam-4478	519	3	∈	∈	PROPN
ejpam-4478	519	4	z+	z+	NUM
ejpam-4478	519	5	and	and	CCONJ
ejpam-4478	519	6	let	let	VERB
ejpam-4478	519	7	g	g	NOUN
ejpam-4478	519	8	=	=	SYM
ejpam-4478	519	9	(	(	PUNCT
ejpam-4478	519	10	v	v	NOUN
ejpam-4478	519	11	(	(	PUNCT
ejpam-4478	519	12	g	g	NOUN
ejpam-4478	519	13	)	)	PUNCT
ejpam-4478	519	14	,	,	PUNCT
ejpam-4478	519	15	e(g	e(g	PROPN
ejpam-4478	519	16	)	)	PUNCT
ejpam-4478	519	17	)	)	PUNCT
ejpam-4478	519	18	be	be	AUX
ejpam-4478	519	19	any	any	DET
ejpam-4478	519	20	graph	graph	NOUN
ejpam-4478	519	21	of	of	ADP
ejpam-4478	519	22	order	order	NOUN
ejpam-4478	519	23	n.	n.	NOUN
ejpam-4478	519	24	by	by	ADP
ejpam-4478	519	25	theorem	theorem	ADJ
ejpam-4478	519	26	7.5	7.5	NUM
ejpam-4478	519	27	,	,	PUNCT
ejpam-4478	519	28	γkgr(g	γkgr(g	NOUN
ejpam-4478	519	29	)	)	PUNCT
ejpam-4478	519	30	=	=	PUNCT
ejpam-4478	520	1	γg(g	γg(g	X
ejpam-4478	520	2	)	)	PUNCT
ejpam-4478	520	3	if	if	SCONJ
ejpam-4478	520	4	and	and	CCONJ
ejpam-4478	520	5	only	only	ADV
ejpam-4478	520	6	ifg	ifg	VERB
ejpam-4478	520	7	∼=	∼=	PROPN
ejpam-4478	520	8	kn	kn	NOUN
ejpam-4478	520	9	org	org	NOUN
ejpam-4478	520	10	∼=	∼=	PROPN
ejpam-4478	520	11	kn	kn	NOUN
ejpam-4478	520	12	and	and	CCONJ
ejpam-4478	520	13	by	by	ADP
ejpam-4478	520	14	proposition	proposition	NOUN
ejpam-4478	520	15	3.14	3.14	NUM
ejpam-4478	520	16	,	,	PUNCT
ejpam-4478	520	17	γg(g	γg(g	X
ejpam-4478	520	18	)	)	PUNCT
ejpam-4478	521	1	=	=	SYM
ejpam-4478	521	2	γgr(g	γgr(g	PROPN
ejpam-4478	521	3	)	)	PUNCT
ejpam-4478	521	4	if	if	SCONJ
ejpam-4478	521	5	and	and	CCONJ
ejpam-4478	521	6	only	only	ADV
ejpam-4478	521	7	if	if	SCONJ
ejpam-4478	521	8	g	g	PROPN
ejpam-4478	521	9	∼=	∼=	PROPN
ejpam-4478	521	10	kn	kn	PROPN
ejpam-4478	521	11	.	.	PUNCT
ejpam-4478	522	1	moreover	moreover	ADV
ejpam-4478	522	2	,	,	PUNCT
ejpam-4478	522	3	since	since	SCONJ
ejpam-4478	522	4	kn	kn	PROPN
ejpam-4478	522	5	=	=	PROPN
ejpam-4478	522	6	kn	kn	PROPN
ejpam-4478	522	7	,	,	PUNCT
ejpam-4478	522	8	by	by	ADP
ejpam-4478	522	9	proposition	proposition	NOUN
ejpam-4478	522	10	3.14	3.14	NUM
ejpam-4478	522	11	,	,	PUNCT
ejpam-4478	522	12	we	we	PRON
ejpam-4478	522	13	can	can	AUX
ejpam-4478	522	14	say	say	VERB
ejpam-4478	522	15	that	that	PRON
ejpam-4478	522	16	γg(g	γg(g	PUNCT
ejpam-4478	522	17	)	)	PUNCT
ejpam-4478	522	18	=	=	SYM
ejpam-4478	522	19	γgr(g	γgr(g	PROPN
ejpam-4478	522	20	)	)	PUNCT
ejpam-4478	522	21	if	if	SCONJ
ejpam-4478	522	22	and	and	CCONJ
ejpam-4478	522	23	only	only	ADV
ejpam-4478	522	24	if	if	SCONJ
ejpam-4478	522	25	g	g	PROPN
ejpam-4478	522	26	∼=	∼=	PROPN
ejpam-4478	522	27	kn	kn	PROPN
ejpam-4478	522	28	.	.	PUNCT
ejpam-4478	523	1	thus	thus	ADV
ejpam-4478	523	2	,	,	PUNCT
ejpam-4478	523	3	from	from	ADP
ejpam-4478	523	4	proposition	proposition	NOUN
ejpam-4478	523	5	3.14	3.14	NUM
ejpam-4478	523	6	,	,	PUNCT
ejpam-4478	523	7	γg(g	γg(g	X
ejpam-4478	523	8	)	)	PUNCT
ejpam-4478	523	9	=	=	SYM
ejpam-4478	523	10	γgr(g	γgr(g	PROPN
ejpam-4478	523	11	)	)	PUNCT
ejpam-4478	524	1	if	if	SCONJ
ejpam-4478	524	2	and	and	CCONJ
ejpam-4478	524	3	only	only	ADV
ejpam-4478	524	4	if	if	SCONJ
ejpam-4478	524	5	g	g	PROPN
ejpam-4478	524	6	∼=	∼=	PROPN
ejpam-4478	524	7	kn	kn	NOUN
ejpam-4478	524	8	or	or	CCONJ
ejpam-4478	524	9	g	g	PROPN
ejpam-4478	524	10	∼=	∼=	PROPN
ejpam-4478	524	11	kn	kn	PROPN
ejpam-4478	524	12	.	.	PUNCT
ejpam-4478	524	13	hence	hence	ADV
ejpam-4478	524	14	,	,	PUNCT
ejpam-4478	524	15	since	since	SCONJ
ejpam-4478	524	16	γkgr(g	γkgr(g	NUM
ejpam-4478	524	17	)	)	PUNCT
ejpam-4478	524	18	=	=	PUNCT
ejpam-4478	524	19	γg(g	γg(g	X
ejpam-4478	524	20	)	)	PUNCT
ejpam-4478	524	21	if	if	SCONJ
ejpam-4478	524	22	and	and	CCONJ
ejpam-4478	524	23	only	only	ADV
ejpam-4478	524	24	if	if	SCONJ
ejpam-4478	524	25	g	g	PROPN
ejpam-4478	524	26	∼=	∼=	PROPN
ejpam-4478	524	27	kn	kn	NOUN
ejpam-4478	524	28	or	or	CCONJ
ejpam-4478	524	29	g	g	NOUN
ejpam-4478	524	30	∼=	∼=	PROPN
ejpam-4478	524	31	kn	kn	NOUN
ejpam-4478	524	32	and	and	CCONJ
ejpam-4478	524	33	γg(g	γg(g	NOUN
ejpam-4478	524	34	)	)	PUNCT
ejpam-4478	524	35	=	=	SYM
ejpam-4478	524	36	γgr(g	γgr(g	PROPN
ejpam-4478	524	37	)	)	PUNCT
ejpam-4478	524	38	if	if	SCONJ
ejpam-4478	524	39	and	and	CCONJ
ejpam-4478	524	40	only	only	ADV
ejpam-4478	524	41	if	if	SCONJ
ejpam-4478	524	42	g	g	PROPN
ejpam-4478	524	43	∼=	∼=	PROPN
ejpam-4478	524	44	kn	kn	NOUN
ejpam-4478	524	45	or	or	CCONJ
ejpam-4478	524	46	g	g	PROPN
ejpam-4478	524	47	∼=	∼=	PROPN
ejpam-4478	524	48	kn	kn	NOUN
ejpam-4478	524	49	,	,	PUNCT
ejpam-4478	524	50	it	it	PRON
ejpam-4478	524	51	follows	follow	VERB
ejpam-4478	524	52	that	that	SCONJ
ejpam-4478	524	53	γkgr(g	γkgr(g	NOUN
ejpam-4478	524	54	)	)	PUNCT
ejpam-4478	524	55	=	=	SYM
ejpam-4478	524	56	γgr(g	γgr(g	PROPN
ejpam-4478	524	57	)	)	PUNCT
ejpam-4478	524	58	if	if	SCONJ
ejpam-4478	524	59	and	and	CCONJ
ejpam-4478	524	60	only	only	ADV
ejpam-4478	524	61	if	if	SCONJ
ejpam-4478	524	62	g	g	PROPN
ejpam-4478	524	63	∼=	∼=	PROPN
ejpam-4478	524	64	kn	kn	NOUN
ejpam-4478	524	65	or	or	CCONJ
ejpam-4478	524	66	g	g	PROPN
ejpam-4478	524	67	∼=	∼=	PROPN
ejpam-4478	524	68	kn	kn	PROPN
ejpam-4478	524	69	.	.	PUNCT
ejpam-4478	525	1	this	this	PRON
ejpam-4478	525	2	completes	complete	VERB
ejpam-4478	525	3	the	the	DET
ejpam-4478	525	4	proof	proof	NOUN
ejpam-4478	525	5	.	.	PUNCT
ejpam-4478	526	1	references	reference	NOUN
ejpam-4478	526	2	60	60	NUM
ejpam-4478	526	3	acknowledgements	acknowledgement	NOUN
ejpam-4478	526	4	the	the	DET
ejpam-4478	526	5	authors	author	NOUN
ejpam-4478	526	6	would	would	AUX
ejpam-4478	526	7	like	like	VERB
ejpam-4478	526	8	to	to	PART
ejpam-4478	526	9	thank	thank	VERB
ejpam-4478	526	10	the	the	DET
ejpam-4478	526	11	department	department	NOUN
ejpam-4478	526	12	of	of	ADP
ejpam-4478	526	13	science	science	NOUN
ejpam-4478	526	14	and	and	CCONJ
ejpam-4478	526	15	technology	technology	NOUN
ejpam-4478	526	16	−	−	PROPN
ejpam-4478	526	17	science	science	PROPN
ejpam-4478	526	18	education	education	PROPN
ejpam-4478	526	19	institute	institute	PROPN
ejpam-4478	526	20	(	(	PUNCT
ejpam-4478	526	21	dost−sei	dost−sei	PROPN
ejpam-4478	526	22	)	)	PUNCT
ejpam-4478	526	23	for	for	ADP
ejpam-4478	526	24	funding	fund	VERB
ejpam-4478	526	25	this	this	DET
ejpam-4478	526	26	publication	publication	NOUN
ejpam-4478	526	27	.	.	PUNCT
ejpam-4478	527	1	references	reference	NOUN
ejpam-4478	527	2	[	[	X
ejpam-4478	527	3	1	1	NUM
ejpam-4478	527	4	]	]	X
ejpam-4478	527	5	aram	aram	PROPN
ejpam-4478	527	6	,	,	PUNCT
ejpam-4478	527	7	h.	h.	PROPN
ejpam-4478	527	8	et	et	PROPN
ejpam-4478	527	9	al	al	PROPN
ejpam-4478	527	10	.	.	PROPN
ejpam-4478	527	11	,	,	PUNCT
ejpam-4478	527	12	the	the	DET
ejpam-4478	527	13	distance	distance	NOUN
ejpam-4478	527	14	roman	roman	ADJ
ejpam-4478	527	15	domatic	domatic	ADJ
ejpam-4478	527	16	number	number	NOUN
ejpam-4478	527	17	of	of	ADP
ejpam-4478	527	18	a	a	DET
ejpam-4478	527	19	graph	graph	NOUN
ejpam-4478	527	20	,	,	PUNCT
ejpam-4478	527	21	akce	akce	ADJ
ejpam-4478	527	22	international	international	ADJ
ejpam-4478	527	23	journal	journal	NOUN
ejpam-4478	527	24	of	of	ADP
ejpam-4478	527	25	graphs	graph	NOUN
ejpam-4478	527	26	and	and	CCONJ
ejpam-4478	527	27	combinatorics	combinatoric	NOUN
ejpam-4478	527	28	,	,	PUNCT
ejpam-4478	527	29	vol	vol	NOUN
ejpam-4478	527	30	.	.	PROPN
ejpam-4478	527	31	9	9	NUM
ejpam-4478	527	32	,	,	PUNCT
ejpam-4478	527	33	no	no	INTJ
ejpam-4478	527	34	.	.	NOUN
ejpam-4478	527	35	2	2	NUM
ejpam-4478	527	36	,	,	PUNCT
ejpam-4478	527	37	2012	2012	NUM
ejpam-4478	527	38	,	,	PUNCT
ejpam-4478	527	39	pp	pp	ADJ
ejpam-4478	527	40	.	.	PUNCT
ejpam-4478	528	1	205	205	NUM
ejpam-4478	528	2	-	-	SYM
ejpam-4478	528	3	212	212	NUM
ejpam-4478	528	4	.	.	PUNCT
ejpam-4478	529	1	[	[	X
ejpam-4478	529	2	2	2	NUM
ejpam-4478	529	3	]	]	X
ejpam-4478	529	4	aram	aram	PROPN
ejpam-4478	529	5	,	,	PUNCT
ejpam-4478	529	6	h.	h.	PROPN
ejpam-4478	529	7	et	et	PROPN
ejpam-4478	529	8	al	al	PROPN
ejpam-4478	529	9	.	.	PROPN
ejpam-4478	529	10	,	,	PUNCT
ejpam-4478	529	11	the	the	DET
ejpam-4478	529	12	distance	distance	NOUN
ejpam-4478	529	13	roman	roman	ADJ
ejpam-4478	529	14	domination	domination	NOUN
ejpam-4478	529	15	numbers	number	NOUN
ejpam-4478	529	16	of	of	ADP
ejpam-4478	529	17	graphs	graph	NOUN
ejpam-4478	529	18	,	,	PUNCT
ejpam-4478	529	19	discussiones	discussione	NOUN
ejpam-4478	529	20	mathematicae	mathematicae	PROPN
ejpam-4478	529	21	,	,	PUNCT
ejpam-4478	529	22	january	january	PROPN
ejpam-4478	529	23	2013	2013	NUM
ejpam-4478	529	24	.	.	PUNCT
ejpam-4478	530	1	[	[	X
ejpam-4478	530	2	3	3	NUM
ejpam-4478	530	3	]	]	SYM
ejpam-4478	530	4	chartrand	chartrand	NOUN
ejpam-4478	530	5	,	,	PUNCT
ejpam-4478	530	6	g.	g.	PROPN
ejpam-4478	530	7	and	and	CCONJ
ejpam-4478	530	8	zhang	zhang	PROPN
ejpam-4478	530	9	,	,	PUNCT
ejpam-4478	530	10	p.	p.	PROPN
ejpam-4478	530	11	,	,	PUNCT
ejpam-4478	530	12	chromatic	chromatic	ADJ
ejpam-4478	530	13	graph	graph	NOUN
ejpam-4478	530	14	theory	theory	NOUN
ejpam-4478	530	15	,	,	PUNCT
ejpam-4478	530	16	discrete	discrete	ADJ
ejpam-4478	530	17	mathematics	mathematic	NOUN
ejpam-4478	530	18	and	and	CCONJ
ejpam-4478	530	19	its	its	PRON
ejpam-4478	530	20	applications	application	NOUN
ejpam-4478	530	21	,	,	PUNCT
ejpam-4478	530	22	taylor	taylor	PROPN
ejpam-4478	530	23	&	&	CCONJ
ejpam-4478	530	24	francis	francis	PROPN
ejpam-4478	530	25	group	group	PROPN
ejpam-4478	530	26	,	,	PUNCT
ejpam-4478	530	27	llc	llc	PROPN
ejpam-4478	530	28	−	−	PROPN
ejpam-4478	530	29	crc	crc	PROPN
ejpam-4478	530	30	press	press	PROPN
ejpam-4478	530	31	,	,	PUNCT
ejpam-4478	530	32	boca	boca	PROPN
ejpam-4478	530	33	raton	raton	PROPN
ejpam-4478	530	34	,	,	PUNCT
ejpam-4478	530	35	fl	fl	PROPN
ejpam-4478	530	36	,	,	PUNCT
ejpam-4478	530	37	2009	2009	NUM
ejpam-4478	530	38	.	.	PUNCT
ejpam-4478	531	1	[	[	X
ejpam-4478	531	2	4	4	NUM
ejpam-4478	531	3	]	]	PUNCT
ejpam-4478	531	4	cockayne	cockayne	NOUN
ejpam-4478	531	5	,	,	PUNCT
ejpam-4478	531	6	e.	e.	PROPN
ejpam-4478	531	7	j.	j.	PROPN
ejpam-4478	531	8	et	et	PROPN
ejpam-4478	531	9	al	al	PROPN
ejpam-4478	531	10	.	.	PROPN
ejpam-4478	531	11	,	,	PUNCT
ejpam-4478	531	12	roman	roman	ADJ
ejpam-4478	531	13	domination	domination	NOUN
ejpam-4478	531	14	in	in	ADP
ejpam-4478	531	15	graphs	graph	NOUN
ejpam-4478	531	16	,	,	PUNCT
ejpam-4478	531	17	discrete	discrete	ADJ
ejpam-4478	531	18	mathematics	mathematic	NOUN
ejpam-4478	531	19	,	,	PUNCT
ejpam-4478	531	20	vol	vol	NOUN
ejpam-4478	531	21	.	.	PROPN
ejpam-4478	531	22	278	278	NUM
ejpam-4478	531	23	,	,	PUNCT
ejpam-4478	531	24	march	march	PROPN
ejpam-4478	531	25	2004	2004	NUM
ejpam-4478	531	26	,	,	PUNCT
ejpam-4478	531	27	pp	pp	ADJ
ejpam-4478	531	28	.	.	PUNCT
ejpam-4478	532	1	11	11	NUM
ejpam-4478	532	2	-	-	SYM
ejpam-4478	532	3	12	12	NUM
ejpam-4478	532	4	.	.	PUNCT
ejpam-4478	533	1	[	[	X
ejpam-4478	533	2	5	5	NUM
ejpam-4478	533	3	]	]	SYM
ejpam-4478	533	4	entero	entero	X
ejpam-4478	533	5	,	,	PUNCT
ejpam-4478	533	6	g.	g.	PROPN
ejpam-4478	533	7	m.	m.	NOUN
ejpam-4478	533	8	and	and	CCONJ
ejpam-4478	533	9	pedrano	pedrano	PROPN
ejpam-4478	533	10	,	,	PUNCT
ejpam-4478	533	11	a.	a.	PROPN
ejpam-4478	533	12	c.	c.	PROPN
ejpam-4478	533	13	,	,	PUNCT
ejpam-4478	533	14	on	on	ADP
ejpam-4478	533	15	connected	connected	ADJ
ejpam-4478	533	16	total	total	ADJ
ejpam-4478	533	17	dominating	dominating	NOUN
ejpam-4478	533	18	sets	set	NOUN
ejpam-4478	533	19	and	and	CCONJ
ejpam-4478	533	20	connected	connect	VERB
ejpam-4478	533	21	total	total	ADJ
ejpam-4478	533	22	domination	domination	NOUN
ejpam-4478	533	23	polynomial	polynomial	NOUN
ejpam-4478	533	24	of	of	ADP
ejpam-4478	533	25	corona	corona	NOUN
ejpam-4478	533	26	graphs	graph	NOUN
ejpam-4478	533	27	,	,	PUNCT
ejpam-4478	533	28	advances	advance	NOUN
ejpam-4478	533	29	and	and	CCONJ
ejpam-4478	533	30	applications	application	NOUN
ejpam-4478	533	31	in	in	ADP
ejpam-4478	533	32	discrete	discrete	ADJ
ejpam-4478	533	33	mathematics	mathematic	NOUN
ejpam-4478	533	34	,	,	PUNCT
ejpam-4478	533	35	pushpa	pushpa	NOUN
ejpam-4478	533	36	publishing	publishing	PROPN
ejpam-4478	533	37	house	house	PROPN
ejpam-4478	533	38	,	,	PUNCT
ejpam-4478	533	39	prayagraj	prayagraj	PROPN
ejpam-4478	533	40	,	,	PUNCT
ejpam-4478	533	41	india	india	PROPN
ejpam-4478	533	42	,	,	PUNCT
ejpam-4478	533	43	vol	vol	NOUN
ejpam-4478	533	44	.	.	PROPN
ejpam-4478	533	45	22	22	NUM
ejpam-4478	533	46	,	,	PUNCT
ejpam-4478	533	47	no	no	INTJ
ejpam-4478	533	48	.	.	NOUN
ejpam-4478	533	49	1	1	NUM
ejpam-4478	533	50	,	,	PUNCT
ejpam-4478	533	51	september	september	PROPN
ejpam-4478	533	52	2019	2019	NUM
ejpam-4478	533	53	,	,	PUNCT
ejpam-4478	533	54	pp	pp	ADP
ejpam-4478	533	55	.	.	PUNCT
ejpam-4478	533	56	103	103	NUM
ejpam-4478	533	57	-	-	SYM
ejpam-4478	533	58	115	115	NUM
ejpam-4478	533	59	.	.	PUNCT
ejpam-4478	534	1	[	[	X
ejpam-4478	534	2	6	6	NUM
ejpam-4478	534	3	]	]	SYM
ejpam-4478	534	4	entero	entero	X
ejpam-4478	534	5	,	,	PUNCT
ejpam-4478	534	6	g.	g.	PROPN
ejpam-4478	534	7	m.	m.	NOUN
ejpam-4478	534	8	and	and	CCONJ
ejpam-4478	534	9	pedrano	pedrano	PROPN
ejpam-4478	534	10	,	,	PUNCT
ejpam-4478	534	11	a.	a.	PROPN
ejpam-4478	534	12	c.	c.	PROPN
ejpam-4478	534	13	,	,	PUNCT
ejpam-4478	534	14	on	on	ADP
ejpam-4478	534	15	connected	connected	ADJ
ejpam-4478	534	16	total	total	ADJ
ejpam-4478	534	17	domination	domination	NOUN
ejpam-4478	534	18	polynomial	polynomial	NOUN
ejpam-4478	534	19	of	of	ADP
ejpam-4478	534	20	some	some	DET
ejpam-4478	534	21	lexicographic	lexicographic	ADJ
ejpam-4478	534	22	product	product	NOUN
ejpam-4478	534	23	graphs	graph	NOUN
ejpam-4478	534	24	,	,	PUNCT
ejpam-4478	534	25	advances	advance	NOUN
ejpam-4478	534	26	and	and	CCONJ
ejpam-4478	534	27	applications	application	NOUN
ejpam-4478	534	28	in	in	ADP
ejpam-4478	534	29	discrete	discrete	ADJ
ejpam-4478	534	30	mathematics	mathematic	NOUN
ejpam-4478	534	31	,	,	PUNCT
ejpam-4478	534	32	pushpa	pushpa	NOUN
ejpam-4478	534	33	publishing	publishing	PROPN
ejpam-4478	534	34	house	house	PROPN
ejpam-4478	534	35	,	,	PUNCT
ejpam-4478	534	36	prayagraj	prayagraj	PROPN
ejpam-4478	534	37	,	,	PUNCT
ejpam-4478	534	38	india	india	PROPN
ejpam-4478	534	39	,	,	PUNCT
ejpam-4478	534	40	vol	vol	NOUN
ejpam-4478	534	41	.	.	PROPN
ejpam-4478	534	42	27	27	NUM
ejpam-4478	534	43	,	,	PUNCT
ejpam-4478	534	44	issue	issue	NOUN
ejpam-4478	534	45	1	1	NUM
ejpam-4478	534	46	,	,	PUNCT
ejpam-4478	534	47	may	may	AUX
ejpam-4478	534	48	2021	2021	NUM
ejpam-4478	534	49	,	,	PUNCT
ejpam-4478	534	50	pp	pp	ADV
ejpam-4478	534	51	.	.	PUNCT
ejpam-4478	534	52	147	147	NUM
ejpam-4478	534	53	-	-	SYM
ejpam-4478	534	54	155	155	NUM
ejpam-4478	534	55	.	.	PUNCT
ejpam-4478	535	1	[	[	X
ejpam-4478	535	2	7	7	X
ejpam-4478	535	3	]	]	X
ejpam-4478	535	4	ghaffari	ghaffari	ADJ
ejpam-4478	535	5	-	-	PUNCT
ejpam-4478	535	6	hadigheh	hadigheh	ADJ
ejpam-4478	535	7	,	,	PUNCT
ejpam-4478	535	8	a.	a.	NOUN
ejpam-4478	535	9	,	,	PUNCT
ejpam-4478	535	10	roman	roman	ADJ
ejpam-4478	535	11	domination	domination	NOUN
ejpam-4478	535	12	problem	problem	NOUN
ejpam-4478	535	13	with	with	ADP
ejpam-4478	535	14	uncertain	uncertain	ADJ
ejpam-4478	535	15	positioning	positioning	NOUN
ejpam-4478	535	16	and	and	CCONJ
ejpam-4478	535	17	deployment	deployment	NOUN
ejpam-4478	535	18	costs	cost	NOUN
ejpam-4478	535	19	,	,	PUNCT
ejpam-4478	535	20	soft	soft	ADJ
ejpam-4478	535	21	computing	computing	NOUN
ejpam-4478	535	22	,	,	PUNCT
ejpam-4478	535	23	vol	vol	NOUN
ejpam-4478	535	24	.	.	PROPN
ejpam-4478	535	25	24	24	NUM
ejpam-4478	535	26	,	,	PUNCT
ejpam-4478	535	27	2020	2020	NUM
ejpam-4478	535	28	,	,	PUNCT
ejpam-4478	535	29	pp	pp	ADV
ejpam-4478	535	30	.	.	PUNCT
ejpam-4478	536	1	2637	2637	NUM
ejpam-4478	536	2	-	-	SYM
ejpam-4478	536	3	2645	2645	NUM
ejpam-4478	536	4	.	.	PUNCT
ejpam-4478	537	1	[	[	X
ejpam-4478	537	2	8	8	NUM
ejpam-4478	537	3	]	]	X
ejpam-4478	537	4	harary	harary	NOUN
ejpam-4478	537	5	,	,	PUNCT
ejpam-4478	537	6	f.	f.	PROPN
ejpam-4478	537	7	,	,	PUNCT
ejpam-4478	537	8	graph	graph	NOUN
ejpam-4478	537	9	theory	theory	NOUN
ejpam-4478	537	10	,	,	PUNCT
ejpam-4478	537	11	addison	addison	PROPN
ejpam-4478	537	12	wesly	wesly	ADV
ejpam-4478	537	13	,	,	PUNCT
ejpam-4478	537	14	reading	read	VERB
ejpam-4478	537	15	mass	mass	PROPN
ejpam-4478	537	16	,	,	PUNCT
ejpam-4478	537	17	1969	1969	NUM
ejpam-4478	537	18	.	.	PUNCT
ejpam-4478	538	1	[	[	X
ejpam-4478	538	2	9	9	NUM
ejpam-4478	538	3	]	]	X
ejpam-4478	538	4	harris	harris	PROPN
ejpam-4478	538	5	,	,	PUNCT
ejpam-4478	538	6	e.	e.	PROPN
ejpam-4478	538	7	m.	m.	PROPN
ejpam-4478	538	8	,	,	PUNCT
ejpam-4478	538	9	global	global	ADJ
ejpam-4478	538	10	domination	domination	NOUN
ejpam-4478	538	11	stable	stable	ADJ
ejpam-4478	538	12	graphs	graph	NOUN
ejpam-4478	538	13	,	,	PUNCT
ejpam-4478	538	14	electronic	electronic	ADJ
ejpam-4478	538	15	theses	thesis	NOUN
ejpam-4478	538	16	and	and	CCONJ
ejpam-4478	538	17	dissertations	dissertation	NOUN
ejpam-4478	538	18	,	,	PUNCT
ejpam-4478	538	19	2012	2012	NUM
ejpam-4478	538	20	.	.	PUNCT
ejpam-4478	539	1	[	[	X
ejpam-4478	539	2	10	10	NUM
ejpam-4478	539	3	]	]	X
ejpam-4478	539	4	haynes	hayne	NOUN
ejpam-4478	539	5	,	,	PUNCT
ejpam-4478	539	6	t.w	t.w	PROPN
ejpam-4478	539	7	.	.	PROPN
ejpam-4478	539	8	,	,	PUNCT
ejpam-4478	539	9	hedetniemi	hedetniemi	PROPN
ejpam-4478	539	10	,	,	PUNCT
ejpam-4478	539	11	s.t	s.t	PROPN
ejpam-4478	539	12	.	.	PROPN
ejpam-4478	539	13	,	,	PUNCT
ejpam-4478	539	14	and	and	CCONJ
ejpam-4478	539	15	slater	slater	NOUN
ejpam-4478	539	16	,	,	PUNCT
ejpam-4478	539	17	p.j	p.j	PROPN
ejpam-4478	539	18	.	.	PROPN
ejpam-4478	539	19	,	,	PUNCT
ejpam-4478	539	20	fundamentals	fundamental	NOUN
ejpam-4478	539	21	of	of	ADP
ejpam-4478	539	22	domination	domination	NOUN
ejpam-4478	539	23	in	in	ADP
ejpam-4478	539	24	graphs	graph	NOUN
ejpam-4478	539	25	,	,	PUNCT
ejpam-4478	539	26	pure	pure	ADJ
ejpam-4478	539	27	and	and	CCONJ
ejpam-4478	539	28	applied	applied	ADJ
ejpam-4478	539	29	mathematics	mathematic	NOUN
ejpam-4478	539	30	,	,	PUNCT
ejpam-4478	539	31	marcel	marcel	PROPN
ejpam-4478	539	32	dekker	dekker	PROPN
ejpam-4478	539	33	,	,	PUNCT
ejpam-4478	539	34	inc	inc	PROPN
ejpam-4478	539	35	.	.	PROPN
ejpam-4478	539	36	,	,	PUNCT
ejpam-4478	539	37	new	new	PROPN
ejpam-4478	539	38	york	york	PROPN
ejpam-4478	539	39	,	,	PUNCT
ejpam-4478	539	40	1998	1998	NUM
ejpam-4478	539	41	.	.	PUNCT
ejpam-4478	540	1	[	[	X
ejpam-4478	540	2	11	11	NUM
ejpam-4478	540	3	]	]	SYM
ejpam-4478	540	4	hedetniemi	hedetniemi	ADV
ejpam-4478	540	5	,	,	PUNCT
ejpam-4478	540	6	s.	s.	PROPN
ejpam-4478	540	7	t.	t.	PROPN
ejpam-4478	540	8	and	and	CCONJ
ejpam-4478	540	9	laskar	laskar	PROPN
ejpam-4478	540	10	,	,	PUNCT
ejpam-4478	540	11	r.	r.	PROPN
ejpam-4478	540	12	c.	c.	PROPN
ejpam-4478	540	13	,	,	PUNCT
ejpam-4478	540	14	bibliography	bibliography	NOUN
ejpam-4478	540	15	on	on	ADP
ejpam-4478	540	16	domination	domination	NOUN
ejpam-4478	540	17	in	in	ADP
ejpam-4478	540	18	graphs	graph	NOUN
ejpam-4478	540	19	and	and	CCONJ
ejpam-4478	540	20	some	some	DET
ejpam-4478	540	21	basic	basic	ADJ
ejpam-4478	540	22	definitions	definition	NOUN
ejpam-4478	540	23	of	of	ADP
ejpam-4478	540	24	domination	domination	NOUN
ejpam-4478	540	25	parameters	parameter	NOUN
ejpam-4478	540	26	,	,	PUNCT
ejpam-4478	540	27	discrete	discrete	ADJ
ejpam-4478	540	28	mathematics	mathematic	NOUN
ejpam-4478	540	29	,	,	PUNCT
ejpam-4478	540	30	north	north	NOUN
ejpam-4478	540	31	-	-	PUNCT
ejpam-4478	540	32	holland	holland	PROPN
ejpam-4478	540	33	,	,	PUNCT
ejpam-4478	540	34	vol	vol	NOUN
ejpam-4478	540	35	.	.	PROPN
ejpam-4478	540	36	86	86	NUM
ejpam-4478	540	37	,	,	PUNCT
ejpam-4478	540	38	1990	1990	NUM
ejpam-4478	540	39	,	,	PUNCT
ejpam-4478	540	40	pp	pp	ADJ
ejpam-4478	540	41	.	.	PUNCT
ejpam-4478	541	1	257	257	NUM
ejpam-4478	541	2	-	-	SYM
ejpam-4478	541	3	277	277	NUM
ejpam-4478	541	4	.	.	PUNCT
ejpam-4478	542	1	[	[	X
ejpam-4478	542	2	12	12	NUM
ejpam-4478	542	3	]	]	PUNCT
ejpam-4478	542	4	pushpam	pushpam	NOUN
ejpam-4478	542	5	,	,	PUNCT
ejpam-4478	542	6	l.	l.	PROPN
ejpam-4478	542	7	r.	r.	PROPN
ejpam-4478	542	8	p.	p.	PROPN
ejpam-4478	542	9	and	and	CCONJ
ejpam-4478	542	10	padmapriea	padmapriea	PROPN
ejpam-4478	542	11	,	,	PUNCT
ejpam-4478	542	12	s.	s.	PROPN
ejpam-4478	542	13	,	,	PUNCT
ejpam-4478	542	14	global	global	ADJ
ejpam-4478	542	15	roman	roman	ADJ
ejpam-4478	542	16	domination	domination	NOUN
ejpam-4478	542	17	in	in	ADP
ejpam-4478	542	18	graphs	graph	NOUN
ejpam-4478	542	19	,	,	PUNCT
ejpam-4478	542	20	discrete	discrete	ADJ
ejpam-4478	542	21	applied	apply	VERB
ejpam-4478	542	22	mathematics	mathematic	NOUN
ejpam-4478	542	23	,	,	PUNCT
ejpam-4478	542	24	elsevier	elsevier	PROPN
ejpam-4478	542	25	b.	b.	PROPN
ejpam-4478	543	1	v.	v.	PROPN
ejpam-4478	543	2	,	,	PUNCT
ejpam-4478	543	3	august	august	PROPN
ejpam-4478	543	4	2015	2015	NUM
ejpam-4478	543	5	.	.	PUNCT
ejpam-4478	544	1	references	reference	NOUN
ejpam-4478	544	2	61	61	NUM
ejpam-4478	544	3	[	[	SYM
ejpam-4478	544	4	13	13	NUM
ejpam-4478	544	5	]	]	X
ejpam-4478	544	6	sampathkumar	sampathkumar	PROPN
ejpam-4478	544	7	,	,	PUNCT
ejpam-4478	544	8	e.	e.	PROPN
ejpam-4478	544	9	,	,	PUNCT
ejpam-4478	544	10	the	the	DET
ejpam-4478	544	11	global	global	ADJ
ejpam-4478	544	12	domination	domination	NOUN
ejpam-4478	544	13	number	number	NOUN
ejpam-4478	544	14	of	of	ADP
ejpam-4478	544	15	a	a	DET
ejpam-4478	544	16	graph	graph	NOUN
ejpam-4478	544	17	,	,	PUNCT
ejpam-4478	544	18	journal	journal	NOUN
ejpam-4478	544	19	of	of	ADP
ejpam-4478	544	20	math	math	NOUN
ejpam-4478	544	21	.	.	PUNCT
ejpam-4478	545	1	phy	phy	PROPN
ejpam-4478	545	2	.	.	PUNCT
ejpam-4478	546	1	sci	sci	PROPN
ejpam-4478	546	2	.	.	PROPN
ejpam-4478	546	3	,	,	PUNCT
ejpam-4478	546	4	vol	vol	NOUN
ejpam-4478	546	5	.	.	PROPN
ejpam-4478	547	1	23	23	NUM
ejpam-4478	547	2	,	,	PUNCT
ejpam-4478	548	1	no	no	INTJ
ejpam-4478	548	2	.	.	NOUN
ejpam-4478	548	3	5	5	NUM
ejpam-4478	548	4	,	,	PUNCT
ejpam-4478	548	5	october	october	PROPN
ejpam-4478	548	6	1989	1989	NUM
ejpam-4478	548	7	.	.	PUNCT
ejpam-4478	549	1	[	[	X
ejpam-4478	549	2	14	14	NUM
ejpam-4478	549	3	]	]	X
ejpam-4478	549	4	tian	tian	PROPN
ejpam-4478	549	5	,	,	PUNCT
ejpam-4478	549	6	f.	f.	PROPN
ejpam-4478	549	7	,	,	PUNCT
ejpam-4478	549	8	a	a	DET
ejpam-4478	549	9	note	note	NOUN
ejpam-4478	549	10	on	on	ADP
ejpam-4478	549	11	distance	distance	NOUN
ejpam-4478	549	12	domination	domination	NOUN
ejpam-4478	549	13	numbers	number	NOUN
ejpam-4478	549	14	of	of	ADP
ejpam-4478	549	15	graphs	graph	NOUN
ejpam-4478	549	16	,	,	PUNCT
ejpam-4478	549	17	australian	australian	ADJ
ejpam-4478	549	18	journal	journal	NOUN
ejpam-4478	549	19	of	of	ADP
ejpam-4478	549	20	combinatorics	combinatorics	PROPN
ejpam-4478	549	21	,	,	PUNCT
ejpam-4478	549	22	vol	vol	NOUN
ejpam-4478	549	23	.	.	PROPN
ejpam-4478	549	24	43	43	NUM
ejpam-4478	549	25	,	,	PUNCT
ejpam-4478	549	26	2009	2009	NUM
ejpam-4478	549	27	,	,	PUNCT
ejpam-4478	549	28	pp	pp	ADJ
ejpam-4478	549	29	.	.	PUNCT
ejpam-4478	550	1	181	181	NUM
ejpam-4478	550	2	-	-	SYM
ejpam-4478	550	3	190	190	NUM
ejpam-4478	550	4	.	.	PUNCT
ejpam-4478	551	1	[	[	X
ejpam-4478	551	2	15	15	NUM
ejpam-4478	551	3	]	]	X
ejpam-4478	551	4	xu	xu	PROPN
ejpam-4478	551	5	,	,	PUNCT
ejpam-4478	551	6	l.	l.	PROPN
ejpam-4478	551	7	,	,	PUNCT
ejpam-4478	551	8	roman	roman	ADJ
ejpam-4478	551	9	domination	domination	NOUN
ejpam-4478	551	10	,	,	PUNCT
ejpam-4478	551	11	2015	2015	NUM
ejpam-4478	551	12	.	.	PUNCT