id	sid	tid	token	lemma	pos
ejpam-6123	1	1	european	european	PROPN
ejpam-6123	1	2	journal	journal	PROPN
ejpam-6123	1	3	of	of	ADP
ejpam-6123	1	4	pure	pure	ADJ
ejpam-6123	1	5	and	and	CCONJ
ejpam-6123	1	6	applied	applied	ADJ
ejpam-6123	1	7	mathematics	mathematic	NOUN
ejpam-6123	1	8	2025	2025	NUM
ejpam-6123	1	9	,	,	PUNCT
ejpam-6123	1	10	vol	vol	NOUN
ejpam-6123	1	11	.	.	PROPN
ejpam-6123	1	12	18	18	NUM
ejpam-6123	1	13	,	,	PUNCT
ejpam-6123	1	14	issue	issue	NOUN
ejpam-6123	1	15	3	3	NUM
ejpam-6123	1	16	,	,	PUNCT
ejpam-6123	1	17	article	article	NOUN
ejpam-6123	1	18	number	number	NOUN
ejpam-6123	1	19	6123	6123	NUM
ejpam-6123	1	20	issn	issn	VERB
ejpam-6123	1	21	1307	1307	NUM
ejpam-6123	1	22	-	-	SYM
ejpam-6123	1	23	5543	5543	NUM
ejpam-6123	1	24	–	–	PUNCT
ejpam-6123	1	25	ejpam.com	ejpam.com	X
ejpam-6123	1	26	published	publish	VERB
ejpam-6123	1	27	by	by	ADP
ejpam-6123	1	28	new	new	PROPN
ejpam-6123	1	29	york	york	PROPN
ejpam-6123	1	30	business	business	PROPN
ejpam-6123	1	31	global	global	ADJ
ejpam-6123	1	32	connected	connect	VERB
ejpam-6123	1	33	degree	degree	NOUN
ejpam-6123	1	34	equitable	equitable	ADJ
ejpam-6123	1	35	domination	domination	NOUN
ejpam-6123	1	36	in	in	ADP
ejpam-6123	1	37	graphs	graph	NOUN
ejpam-6123	1	38	hearty	hearty	ADJ
ejpam-6123	1	39	m.	m.	NOUN
ejpam-6123	1	40	nuenaymaglanque1	nuenaymaglanque1	PROPN
ejpam-6123	1	41	1	1	NUM
ejpam-6123	1	42	department	department	NOUN
ejpam-6123	1	43	of	of	ADP
ejpam-6123	1	44	applied	apply	VERB
ejpam-6123	1	45	mathematics	mathematic	NOUN
ejpam-6123	1	46	,	,	PUNCT
ejpam-6123	1	47	college	college	NOUN
ejpam-6123	1	48	of	of	ADP
ejpam-6123	1	49	science	science	NOUN
ejpam-6123	1	50	and	and	CCONJ
ejpam-6123	1	51	mathematics	mathematic	NOUN
ejpam-6123	1	52	,	,	PUNCT
ejpam-6123	1	53	university	university	NOUN
ejpam-6123	1	54	of	of	ADP
ejpam-6123	1	55	science	science	NOUN
ejpam-6123	1	56	and	and	CCONJ
ejpam-6123	1	57	technology	technology	NOUN
ejpam-6123	1	58	of	of	ADP
ejpam-6123	1	59	southern	southern	ADJ
ejpam-6123	1	60	philippines	philippine	NOUN
ejpam-6123	1	61	,	,	PUNCT
ejpam-6123	1	62	9000	9000	NUM
ejpam-6123	1	63	cagayan	cagayan	PROPN
ejpam-6123	1	64	de	de	PROPN
ejpam-6123	1	65	oro	oro	PROPN
ejpam-6123	1	66	city	city	NOUN
ejpam-6123	1	67	,	,	PUNCT
ejpam-6123	1	68	philippines	philippine	NOUN
ejpam-6123	1	69	abstract	abstract	ADJ
ejpam-6123	1	70	.	.	PUNCT
ejpam-6123	2	1	let	let	VERB
ejpam-6123	2	2	g	g	PRON
ejpam-6123	2	3	be	be	AUX
ejpam-6123	2	4	a	a	DET
ejpam-6123	2	5	connected	connected	ADJ
ejpam-6123	2	6	graph	graph	NOUN
ejpam-6123	2	7	.	.	PUNCT
ejpam-6123	3	1	a	a	DET
ejpam-6123	3	2	subset	subset	NOUN
ejpam-6123	3	3	s	s	VERB
ejpam-6123	3	4	⊆	⊆	NUM
ejpam-6123	3	5	v	v	NOUN
ejpam-6123	3	6	(	(	PUNCT
ejpam-6123	3	7	g	g	NOUN
ejpam-6123	3	8	)	)	PUNCT
ejpam-6123	3	9	is	be	AUX
ejpam-6123	3	10	an	an	DET
ejpam-6123	3	11	equitable	equitable	ADJ
ejpam-6123	3	12	dominating	dominating	NOUN
ejpam-6123	3	13	set	set	VERB
ejpam-6123	3	14	in	in	ADP
ejpam-6123	3	15	g	g	PROPN
ejpam-6123	3	16	if	if	SCONJ
ejpam-6123	3	17	for	for	ADP
ejpam-6123	3	18	each	each	DET
ejpam-6123	3	19	vertex	vertex	NOUN
ejpam-6123	3	20	not	not	PART
ejpam-6123	3	21	in	in	ADP
ejpam-6123	3	22	s	s	PRON
ejpam-6123	3	23	there	there	PRON
ejpam-6123	3	24	exists	exist	VERB
ejpam-6123	3	25	u	u	PROPN
ejpam-6123	3	26	∈	∈	PROPN
ejpam-6123	3	27	s	s	VERB
ejpam-6123	3	28	such	such	ADJ
ejpam-6123	3	29	that	that	SCONJ
ejpam-6123	3	30	uv	uv	PROPN
ejpam-6123	3	31	∈	∈	PROPN
ejpam-6123	3	32	e(g	e(g	PROPN
ejpam-6123	3	33	)	)	PUNCT
ejpam-6123	3	34	and	and	CCONJ
ejpam-6123	3	35	|degg	|degg	VERB
ejpam-6123	3	36	u	u	PRON
ejpam-6123	3	37	−	−	PROPN
ejpam-6123	3	38	degg	degg	NOUN
ejpam-6123	3	39	v|	v|	ADV
ejpam-6123	3	40	≤	≤	NUM
ejpam-6123	3	41	1	1	NUM
ejpam-6123	3	42	.	.	PUNCT
ejpam-6123	4	1	an	an	DET
ejpam-6123	4	2	equitable	equitable	ADJ
ejpam-6123	4	3	dominating	dominating	NOUN
ejpam-6123	4	4	set	set	NOUN
ejpam-6123	4	5	s	s	PROPN
ejpam-6123	4	6	⊆	⊆	NUM
ejpam-6123	4	7	v	v	NOUN
ejpam-6123	4	8	(	(	PUNCT
ejpam-6123	4	9	g	g	NOUN
ejpam-6123	4	10	)	)	PUNCT
ejpam-6123	4	11	is	be	AUX
ejpam-6123	4	12	called	call	VERB
ejpam-6123	4	13	a	a	DET
ejpam-6123	4	14	connected	connect	VERB
ejpam-6123	4	15	equitable	equitable	ADJ
ejpam-6123	4	16	dominating	dominating	NOUN
ejpam-6123	4	17	set	set	NOUN
ejpam-6123	4	18	of	of	ADP
ejpam-6123	4	19	g	g	PROPN
ejpam-6123	4	20	if	if	SCONJ
ejpam-6123	4	21	the	the	DET
ejpam-6123	4	22	subgraph	subgraph	NOUN
ejpam-6123	4	23	⟨s⟩	⟨s⟩	PROPN
ejpam-6123	4	24	induced	induce	VERB
ejpam-6123	4	25	by	by	ADP
ejpam-6123	4	26	s	s	PROPN
ejpam-6123	4	27	is	be	AUX
ejpam-6123	4	28	connected	connect	VERB
ejpam-6123	4	29	.	.	PUNCT
ejpam-6123	5	1	the	the	DET
ejpam-6123	5	2	minimum	minimum	ADJ
ejpam-6123	5	3	cardinality	cardinality	NOUN
ejpam-6123	5	4	of	of	ADP
ejpam-6123	5	5	such	such	ADJ
ejpam-6123	5	6	connected	connected	ADJ
ejpam-6123	5	7	equitable	equitable	ADJ
ejpam-6123	5	8	dominating	dominating	NOUN
ejpam-6123	5	9	sets	set	NOUN
ejpam-6123	5	10	in	in	ADP
ejpam-6123	5	11	g	g	PROPN
ejpam-6123	5	12	is	be	AUX
ejpam-6123	5	13	called	call	VERB
ejpam-6123	5	14	the	the	DET
ejpam-6123	5	15	connected	connect	VERB
ejpam-6123	5	16	equitable	equitable	ADJ
ejpam-6123	5	17	domination	domination	NOUN
ejpam-6123	5	18	number	number	NOUN
ejpam-6123	5	19	of	of	ADP
ejpam-6123	5	20	g	g	NOUN
ejpam-6123	5	21	and	and	CCONJ
ejpam-6123	5	22	is	be	AUX
ejpam-6123	5	23	denoted	denote	VERB
ejpam-6123	5	24	by	by	ADP
ejpam-6123	5	25	γce(g	γce(g	PROPN
ejpam-6123	5	26	)	)	PUNCT
ejpam-6123	5	27	.	.	PUNCT
ejpam-6123	6	1	this	this	DET
ejpam-6123	6	2	paper	paper	NOUN
ejpam-6123	6	3	investigates	investigate	VERB
ejpam-6123	6	4	the	the	DET
ejpam-6123	6	5	connected	connect	VERB
ejpam-6123	6	6	equitable	equitable	ADJ
ejpam-6123	6	7	domination	domination	NOUN
ejpam-6123	6	8	in	in	ADP
ejpam-6123	6	9	the	the	DET
ejpam-6123	6	10	join	join	NOUN
ejpam-6123	6	11	and	and	CCONJ
ejpam-6123	6	12	corona	corona	NOUN
ejpam-6123	6	13	of	of	ADP
ejpam-6123	6	14	graphs	graph	NOUN
ejpam-6123	6	15	.	.	PUNCT
ejpam-6123	7	1	the	the	DET
ejpam-6123	7	2	connected	connect	VERB
ejpam-6123	7	3	equitable	equitable	ADJ
ejpam-6123	7	4	dominating	dominating	NOUN
ejpam-6123	7	5	sets	set	NOUN
ejpam-6123	7	6	in	in	ADP
ejpam-6123	7	7	the	the	DET
ejpam-6123	7	8	join	join	NOUN
ejpam-6123	7	9	and	and	CCONJ
ejpam-6123	7	10	corona	corona	NOUN
ejpam-6123	7	11	of	of	ADP
ejpam-6123	7	12	graphs	graph	NOUN
ejpam-6123	7	13	are	be	AUX
ejpam-6123	7	14	characterized	characterize	VERB
ejpam-6123	7	15	and	and	CCONJ
ejpam-6123	7	16	,	,	PUNCT
ejpam-6123	7	17	as	as	ADP
ejpam-6123	7	18	direct	direct	ADJ
ejpam-6123	7	19	consequences	consequence	NOUN
ejpam-6123	7	20	,	,	PUNCT
ejpam-6123	7	21	the	the	DET
ejpam-6123	7	22	connected	connect	VERB
ejpam-6123	7	23	equitable	equitable	ADJ
ejpam-6123	7	24	domination	domination	NOUN
ejpam-6123	7	25	numbers	number	NOUN
ejpam-6123	7	26	of	of	ADP
ejpam-6123	7	27	these	these	DET
ejpam-6123	7	28	graphs	graph	NOUN
ejpam-6123	7	29	are	be	AUX
ejpam-6123	7	30	obtained	obtain	VERB
ejpam-6123	7	31	.	.	PUNCT
ejpam-6123	8	1	in	in	ADP
ejpam-6123	8	2	addition	addition	NOUN
ejpam-6123	8	3	,	,	PUNCT
ejpam-6123	8	4	a	a	DET
ejpam-6123	8	5	n	n	CCONJ
ejpam-6123	8	6	exact	exact	ADJ
ejpam-6123	8	7	value	value	NOUN
ejpam-6123	8	8	of	of	ADP
ejpam-6123	8	9	some	some	DET
ejpam-6123	8	10	families	family	NOUN
ejpam-6123	8	11	of	of	ADP
ejpam-6123	8	12	graphs	graph	NOUN
ejpam-6123	8	13	and	and	CCONJ
ejpam-6123	8	14	a	a	DET
ejpam-6123	8	15	realization	realization	NOUN
ejpam-6123	8	16	problem	problem	NOUN
ejpam-6123	8	17	are	be	AUX
ejpam-6123	8	18	established	establish	VERB
ejpam-6123	8	19	.	.	PUNCT
ejpam-6123	9	1	2020	2020	NUM
ejpam-6123	9	2	mathematics	mathematics	PROPN
ejpam-6123	9	3	subject	subject	NOUN
ejpam-6123	9	4	classifications	classification	NOUN
ejpam-6123	9	5	:	:	PUNCT
ejpam-6123	9	6	05c69	05c69	X
ejpam-6123	9	7	key	key	ADJ
ejpam-6123	9	8	words	word	NOUN
ejpam-6123	9	9	and	and	CCONJ
ejpam-6123	9	10	phrases	phrase	NOUN
ejpam-6123	9	11	:	:	PUNCT
ejpam-6123	9	12	equitable	equitable	ADJ
ejpam-6123	9	13	dominating	dominating	NOUN
ejpam-6123	9	14	set	set	NOUN
ejpam-6123	9	15	,	,	PUNCT
ejpam-6123	9	16	connected	connect	VERB
ejpam-6123	9	17	equitable	equitable	ADJ
ejpam-6123	9	18	dominating	dominating	NOUN
ejpam-6123	9	19	set	set	NOUN
ejpam-6123	9	20	,	,	PUNCT
ejpam-6123	9	21	equitable	equitable	ADJ
ejpam-6123	9	22	domination	domination	NOUN
ejpam-6123	9	23	number	number	NOUN
ejpam-6123	9	24	,	,	PUNCT
ejpam-6123	9	25	connected	connect	VERB
ejpam-6123	9	26	equitable	equitable	ADJ
ejpam-6123	9	27	domination	domination	NOUN
ejpam-6123	9	28	number	number	NOUN
ejpam-6123	9	29	1	1	NUM
ejpam-6123	9	30	.	.	PUNCT
ejpam-6123	9	31	introduction	introduction	NOUN
ejpam-6123	9	32	throughout	throughout	ADP
ejpam-6123	9	33	this	this	DET
ejpam-6123	9	34	paper	paper	NOUN
ejpam-6123	9	35	,	,	PUNCT
ejpam-6123	9	36	we	we	PRON
ejpam-6123	9	37	consider	consider	VERB
ejpam-6123	9	38	simple	simple	ADJ
ejpam-6123	9	39	,	,	PUNCT
ejpam-6123	9	40	finite	finite	ADJ
ejpam-6123	9	41	and	and	CCONJ
ejpam-6123	9	42	undirected	undirected	ADJ
ejpam-6123	9	43	graphs	graph	NOUN
ejpam-6123	9	44	g	g	NOUN
ejpam-6123	9	45	=	=	SYM
ejpam-6123	9	46	(	(	PUNCT
ejpam-6123	9	47	v	v	NOUN
ejpam-6123	9	48	(	(	PUNCT
ejpam-6123	9	49	g	g	NOUN
ejpam-6123	9	50	)	)	PUNCT
ejpam-6123	9	51	,	,	PUNCT
ejpam-6123	9	52	e(g	e(g	PROPN
ejpam-6123	9	53	)	)	PUNCT
ejpam-6123	9	54	)	)	PUNCT
ejpam-6123	9	55	.	.	PUNCT
ejpam-6123	10	1	for	for	ADP
ejpam-6123	10	2	a	a	DET
ejpam-6123	10	3	subset	subset	NOUN
ejpam-6123	10	4	s	s	VERB
ejpam-6123	10	5	⊆	⊆	NUM
ejpam-6123	10	6	v	v	NOUN
ejpam-6123	10	7	(	(	PUNCT
ejpam-6123	10	8	g	g	NOUN
ejpam-6123	10	9	)	)	PUNCT
ejpam-6123	10	10	,	,	PUNCT
ejpam-6123	10	11	the	the	DET
ejpam-6123	10	12	symbol	symbol	NOUN
ejpam-6123	10	13	|s|	|s|	NOUN
ejpam-6123	10	14	refers	refer	VERB
ejpam-6123	10	15	to	to	ADP
ejpam-6123	10	16	the	the	DET
ejpam-6123	10	17	cardinality	cardinality	NOUN
ejpam-6123	10	18	of	of	ADP
ejpam-6123	10	19	s.	s.	PROPN
ejpam-6123	10	20	in	in	ADP
ejpam-6123	10	21	particular	particular	ADJ
ejpam-6123	10	22	,	,	PUNCT
ejpam-6123	10	23	|v	|v	PROPN
ejpam-6123	10	24	(	(	PUNCT
ejpam-6123	10	25	g)|	g)|	PROPN
ejpam-6123	10	26	is	be	AUX
ejpam-6123	10	27	the	the	DET
ejpam-6123	10	28	order	order	NOUN
ejpam-6123	10	29	of	of	ADP
ejpam-6123	10	30	g.	g.	PROPN
ejpam-6123	10	31	let	let	VERB
ejpam-6123	10	32	g	g	NOUN
ejpam-6123	10	33	be	be	AUX
ejpam-6123	10	34	a	a	DET
ejpam-6123	10	35	connected	connected	ADJ
ejpam-6123	10	36	graph	graph	NOUN
ejpam-6123	10	37	.	.	PUNCT
ejpam-6123	11	1	if	if	SCONJ
ejpam-6123	11	2	the	the	DET
ejpam-6123	11	3	pair	pair	NOUN
ejpam-6123	11	4	e	e	X
ejpam-6123	11	5	=	=	PUNCT
ejpam-6123	11	6	[	[	X
ejpam-6123	11	7	u	u	NOUN
ejpam-6123	11	8	,	,	PUNCT
ejpam-6123	11	9	v	v	NOUN
ejpam-6123	11	10	]	]	PUNCT
ejpam-6123	11	11	is	be	AUX
ejpam-6123	11	12	in	in	ADP
ejpam-6123	11	13	e(g	e(g	PROPN
ejpam-6123	11	14	)	)	PUNCT
ejpam-6123	11	15	,	,	PUNCT
ejpam-6123	11	16	then	then	ADV
ejpam-6123	11	17	e	e	PROPN
ejpam-6123	11	18	is	be	AUX
ejpam-6123	11	19	an	an	DET
ejpam-6123	11	20	edge	edge	NOUN
ejpam-6123	11	21	in	in	ADP
ejpam-6123	11	22	g	g	NOUN
ejpam-6123	11	23	,	,	PUNCT
ejpam-6123	11	24	and	and	CCONJ
ejpam-6123	11	25	e	e	NOUN
ejpam-6123	11	26	is	be	AUX
ejpam-6123	11	27	said	say	VERB
ejpam-6123	11	28	to	to	PART
ejpam-6123	11	29	join	join	VERB
ejpam-6123	11	30	u	u	NOUN
ejpam-6123	11	31	and	and	CCONJ
ejpam-6123	11	32	v.	v.	ADP
ejpam-6123	11	33	in	in	ADP
ejpam-6123	11	34	this	this	DET
ejpam-6123	11	35	case	case	NOUN
ejpam-6123	11	36	,	,	PUNCT
ejpam-6123	11	37	it	it	PRON
ejpam-6123	11	38	is	be	AUX
ejpam-6123	11	39	customary	customary	ADJ
ejpam-6123	11	40	to	to	PART
ejpam-6123	11	41	write	write	VERB
ejpam-6123	11	42	e	e	NOUN
ejpam-6123	11	43	=	=	NOUN
ejpam-6123	11	44	uv	uv	NOUN
ejpam-6123	11	45	and	and	CCONJ
ejpam-6123	11	46	say	say	VERB
ejpam-6123	11	47	that	that	SCONJ
ejpam-6123	11	48	u	u	PROPN
ejpam-6123	11	49	and	and	CCONJ
ejpam-6123	11	50	v	v	NOUN
ejpam-6123	11	51	are	be	AUX
ejpam-6123	11	52	adjacent	adjacent	ADJ
ejpam-6123	11	53	,	,	PUNCT
ejpam-6123	11	54	while	while	SCONJ
ejpam-6123	11	55	u	u	PRON
ejpam-6123	11	56	and	and	CCONJ
ejpam-6123	11	57	e	e	PROPN
ejpam-6123	11	58	are	be	AUX
ejpam-6123	11	59	incident	incident	NOUN
ejpam-6123	11	60	,	,	PUNCT
ejpam-6123	11	61	as	as	SCONJ
ejpam-6123	11	62	v	v	NOUN
ejpam-6123	11	63	and	and	CCONJ
ejpam-6123	11	64	e	e	NOUN
ejpam-6123	11	65	are	be	AUX
ejpam-6123	11	66	.	.	PUNCT
ejpam-6123	12	1	the	the	DET
ejpam-6123	12	2	degree	degree	NOUN
ejpam-6123	12	3	of	of	ADP
ejpam-6123	12	4	a	a	DET
ejpam-6123	12	5	vertex	vertex	NOUN
ejpam-6123	12	6	v	v	ADP
ejpam-6123	12	7	∈	∈	NOUN
ejpam-6123	12	8	v	v	NOUN
ejpam-6123	12	9	(	(	PUNCT
ejpam-6123	12	10	g	g	NOUN
ejpam-6123	12	11	)	)	PUNCT
ejpam-6123	12	12	,	,	PUNCT
ejpam-6123	12	13	denoted	denote	VERB
ejpam-6123	12	14	by	by	ADP
ejpam-6123	12	15	degg(v	degg(v	PROPN
ejpam-6123	12	16	)	)	PUNCT
ejpam-6123	12	17	,	,	PUNCT
ejpam-6123	12	18	is	be	AUX
ejpam-6123	12	19	the	the	DET
ejpam-6123	12	20	number	number	NOUN
ejpam-6123	12	21	of	of	ADP
ejpam-6123	12	22	incident	incident	NOUN
ejpam-6123	12	23	edges	edge	NOUN
ejpam-6123	12	24	to	to	ADP
ejpam-6123	12	25	v.	v.	ADP
ejpam-6123	12	26	adjacent	adjacent	ADJ
ejpam-6123	12	27	vertices	vertex	NOUN
ejpam-6123	12	28	are	be	AUX
ejpam-6123	12	29	also	also	ADV
ejpam-6123	12	30	called	call	VERB
ejpam-6123	12	31	neighbors	neighbor	NOUN
ejpam-6123	12	32	.	.	PUNCT
ejpam-6123	13	1	for	for	ADP
ejpam-6123	13	2	v	v	NUM
ejpam-6123	13	3	∈	∈	PROPN
ejpam-6123	13	4	v	v	NOUN
ejpam-6123	13	5	(	(	PUNCT
ejpam-6123	13	6	g	g	NOUN
ejpam-6123	13	7	)	)	PUNCT
ejpam-6123	13	8	,	,	PUNCT
ejpam-6123	13	9	the	the	DET
ejpam-6123	13	10	open	open	ADJ
ejpam-6123	13	11	neighborhood	neighborhood	NOUN
ejpam-6123	13	12	of	of	ADP
ejpam-6123	13	13	v	v	NOUN
ejpam-6123	13	14	is	be	AUX
ejpam-6123	13	15	the	the	DET
ejpam-6123	13	16	set	set	NOUN
ejpam-6123	13	17	ng(v	ng(v	PUNCT
ejpam-6123	13	18	)	)	PUNCT
ejpam-6123	13	19	consisting	consist	VERB
ejpam-6123	13	20	of	of	ADP
ejpam-6123	13	21	all	all	DET
ejpam-6123	13	22	vertices	vertex	NOUN
ejpam-6123	13	23	adjacent	adjacent	ADJ
ejpam-6123	13	24	to	to	ADP
ejpam-6123	13	25	v.	v.	ADP
ejpam-6123	13	26	thus	thus	ADV
ejpam-6123	13	27	,	,	PUNCT
ejpam-6123	13	28	degg(v	degg(v	PROPN
ejpam-6123	13	29	)	)	PUNCT
ejpam-6123	13	30	=	=	PUNCT
ejpam-6123	13	31	|ng(v)|	|ng(v)|	NOUN
ejpam-6123	13	32	.	.	PUNCT
ejpam-6123	14	1	the	the	DET
ejpam-6123	14	2	maximum	maximum	ADJ
ejpam-6123	14	3	and	and	CCONJ
ejpam-6123	14	4	minimum	minimum	NOUN
ejpam-6123	14	5	degree	degree	NOUN
ejpam-6123	14	6	of	of	ADP
ejpam-6123	14	7	g	g	NOUN
ejpam-6123	14	8	are	be	AUX
ejpam-6123	14	9	denoted	denote	VERB
ejpam-6123	14	10	by	by	ADP
ejpam-6123	14	11	∆(g	∆(g	PROPN
ejpam-6123	14	12	)	)	PUNCT
ejpam-6123	14	13	and	and	CCONJ
ejpam-6123	14	14	δ(g	δ(g	NOUN
ejpam-6123	14	15	)	)	PUNCT
ejpam-6123	14	16	,	,	PUNCT
ejpam-6123	14	17	that	that	ADV
ejpam-6123	14	18	is	is	ADV
ejpam-6123	14	19	,	,	PUNCT
ejpam-6123	14	20	∆(g	∆(g	NOUN
ejpam-6123	14	21	)	)	PUNCT
ejpam-6123	14	22	=	=	PUNCT
ejpam-6123	14	23	max{degg(u	max{degg(u	PROPN
ejpam-6123	14	24	)	)	PUNCT
ejpam-6123	14	25	:	:	PUNCT
ejpam-6123	15	1	u	u	PROPN
ejpam-6123	15	2	∈	∈	PROPN
ejpam-6123	15	3	v	v	ADP
ejpam-6123	15	4	(	(	PUNCT
ejpam-6123	15	5	g	g	NOUN
ejpam-6123	15	6	)	)	PUNCT
ejpam-6123	15	7	}	}	PUNCT
ejpam-6123	15	8	and	and	CCONJ
ejpam-6123	15	9	δ(g	δ(g	PROPN
ejpam-6123	15	10	)	)	PUNCT
ejpam-6123	15	11	=	=	SYM
ejpam-6123	15	12	min{degg(u	min{degg(u	PROPN
ejpam-6123	15	13	)	)	PUNCT
ejpam-6123	15	14	:	:	PUNCT
ejpam-6123	15	15	u	u	PROPN
ejpam-6123	15	16	∈	∈	PROPN
ejpam-6123	15	17	v	v	ADP
ejpam-6123	15	18	(	(	PUNCT
ejpam-6123	15	19	g	g	NOUN
ejpam-6123	15	20	)	)	PUNCT
ejpam-6123	15	21	}	}	PUNCT
ejpam-6123	15	22	.	.	PUNCT
ejpam-6123	16	1	the	the	DET
ejpam-6123	16	2	subgraph	subgraph	NOUN
ejpam-6123	16	3	⟨s⟩	⟨s⟩	PROPN
ejpam-6123	16	4	of	of	ADP
ejpam-6123	16	5	g	g	PROPN
ejpam-6123	16	6	induced	induce	VERB
ejpam-6123	16	7	by	by	ADP
ejpam-6123	16	8	a	a	DET
ejpam-6123	16	9	subset	subset	NOUN
ejpam-6123	16	10	s	s	NOUN
ejpam-6123	16	11	of	of	ADP
ejpam-6123	16	12	v	v	NOUN
ejpam-6123	16	13	(	(	PUNCT
ejpam-6123	16	14	g	g	NOUN
ejpam-6123	16	15	)	)	PUNCT
ejpam-6123	16	16	is	be	AUX
ejpam-6123	16	17	the	the	DET
ejpam-6123	16	18	graph	graph	NOUN
ejpam-6123	16	19	having	have	VERB
ejpam-6123	16	20	vertex	vertex	NOUN
ejpam-6123	16	21	set	set	NOUN
ejpam-6123	16	22	s	s	VERB
ejpam-6123	16	23	doi	doi	NOUN
ejpam-6123	16	24	:	:	PUNCT
ejpam-6123	16	25	https://doi.org/10.29020/nybg.ejpam.v18i3.6123	https://doi.org/10.29020/nybg.ejpam.v18i3.6123	NOUN
ejpam-6123	16	26	email	email	NOUN
ejpam-6123	16	27	addresses	address	VERB
ejpam-6123	16	28	:	:	PUNCT
ejpam-6123	17	1	hearty.maglanque@ustp.edu.ph	hearty.maglanque@ustp.edu.ph	PROPN
ejpam-6123	17	2	(	(	PUNCT
ejpam-6123	17	3	h.	h.	PROPN
ejpam-6123	17	4	nuenay	nuenay	PROPN
ejpam-6123	17	5	-	-	PUNCT
ejpam-6123	17	6	maglanque	maglanque	ADJ
ejpam-6123	17	7	)	)	PUNCT
ejpam-6123	17	8	https://www.ejpam.com	https://www.ejpam.com	NOUN
ejpam-6123	17	9	1	1	NUM
ejpam-6123	17	10	copyright	copyright	NOUN
ejpam-6123	17	11	:	:	PUNCT
ejpam-6123	17	12	©	©	PROPN
ejpam-6123	17	13	2025	2025	NUM
ejpam-6123	17	14	the	the	DET
ejpam-6123	17	15	author(s	author(s	NOUN
ejpam-6123	17	16	)	)	PUNCT
ejpam-6123	17	17	.	.	PUNCT
ejpam-6123	18	1	(	(	PUNCT
ejpam-6123	18	2	cc	cc	NOUN
ejpam-6123	18	3	by	by	ADP
ejpam-6123	18	4	-	-	PUNCT
ejpam-6123	18	5	nc	nc	PROPN
ejpam-6123	18	6	4.0	4.0	NUM
ejpam-6123	18	7	)	)	PUNCT
ejpam-6123	18	8	h.	h.	PROPN
ejpam-6123	18	9	nuenay	nuenay	PROPN
ejpam-6123	18	10	-	-	PUNCT
ejpam-6123	18	11	maglanquel	maglanquel	PROPN
ejpam-6123	18	12	/	/	SYM
ejpam-6123	18	13	eur	eur	PROPN
ejpam-6123	18	14	.	.	PUNCT
ejpam-6123	19	1	j.	j.	PROPN
ejpam-6123	19	2	pure	pure	PROPN
ejpam-6123	19	3	appl	appl	PROPN
ejpam-6123	19	4	.	.	PROPN
ejpam-6123	19	5	math	math	PROPN
ejpam-6123	19	6	,	,	PUNCT
ejpam-6123	19	7	18	18	NUM
ejpam-6123	19	8	(	(	PUNCT
ejpam-6123	19	9	3	3	NUM
ejpam-6123	19	10	)	)	PUNCT
ejpam-6123	19	11	(	(	PUNCT
ejpam-6123	19	12	2025	2025	NUM
ejpam-6123	19	13	)	)	PUNCT
ejpam-6123	19	14	,	,	PUNCT
ejpam-6123	19	15	6123	6123	NUM
ejpam-6123	19	16	2	2	NUM
ejpam-6123	19	17	of	of	ADP
ejpam-6123	19	18	14	14	NUM
ejpam-6123	19	19	and	and	CCONJ
ejpam-6123	19	20	whose	whose	DET
ejpam-6123	19	21	edge	edge	NOUN
ejpam-6123	19	22	set	set	VERB
ejpam-6123	19	23	consists	consist	VERB
ejpam-6123	19	24	of	of	ADP
ejpam-6123	19	25	those	those	DET
ejpam-6123	19	26	edges	edge	NOUN
ejpam-6123	19	27	of	of	ADP
ejpam-6123	19	28	g	g	PROPN
ejpam-6123	19	29	incident	incident	NOUN
ejpam-6123	19	30	with	with	ADP
ejpam-6123	19	31	two	two	NUM
ejpam-6123	19	32	elements	element	NOUN
ejpam-6123	19	33	of	of	ADP
ejpam-6123	19	34	g.	g.	PROPN
ejpam-6123	19	35	a	a	DET
ejpam-6123	19	36	graph	graph	NOUN
ejpam-6123	19	37	is	be	AUX
ejpam-6123	19	38	called	call	VERB
ejpam-6123	19	39	connected	connect	VERB
ejpam-6123	19	40	if	if	SCONJ
ejpam-6123	19	41	every	every	DET
ejpam-6123	19	42	two	two	NUM
ejpam-6123	19	43	vertices	vertex	NOUN
ejpam-6123	19	44	are	be	AUX
ejpam-6123	19	45	joined	join	VERB
ejpam-6123	19	46	by	by	ADP
ejpam-6123	19	47	a	a	DET
ejpam-6123	19	48	path	path	NOUN
ejpam-6123	19	49	;	;	PUNCT
ejpam-6123	19	50	otherwise	otherwise	ADV
ejpam-6123	19	51	,	,	PUNCT
ejpam-6123	19	52	it	it	PRON
ejpam-6123	19	53	is	be	AUX
ejpam-6123	19	54	disconnected	disconnect	VERB
ejpam-6123	19	55	.	.	PUNCT
ejpam-6123	20	1	in	in	ADP
ejpam-6123	20	2	graph	graph	NOUN
ejpam-6123	20	3	theory	theory	NOUN
ejpam-6123	20	4	,	,	PUNCT
ejpam-6123	20	5	domination	domination	NOUN
ejpam-6123	20	6	in	in	ADP
ejpam-6123	20	7	graphs	graph	NOUN
ejpam-6123	20	8	is	be	AUX
ejpam-6123	20	9	one	one	NUM
ejpam-6123	20	10	of	of	ADP
ejpam-6123	20	11	the	the	DET
ejpam-6123	20	12	most	most	ADV
ejpam-6123	20	13	extensively	extensively	ADV
ejpam-6123	20	14	studied	study	VERB
ejpam-6123	20	15	concepts	concept	NOUN
ejpam-6123	20	16	in	in	ADP
ejpam-6123	20	17	graph	graph	NOUN
ejpam-6123	20	18	theory	theory	NOUN
ejpam-6123	20	19	with	with	ADP
ejpam-6123	20	20	numerous	numerous	ADJ
ejpam-6123	20	21	variants	variant	NOUN
ejpam-6123	20	22	that	that	PRON
ejpam-6123	20	23	model	model	NOUN
ejpam-6123	20	24	influence	influence	NOUN
ejpam-6123	20	25	,	,	PUNCT
ejpam-6123	20	26	control	control	NOUN
ejpam-6123	20	27	and	and	CCONJ
ejpam-6123	20	28	coverage	coverage	NOUN
ejpam-6123	20	29	,	,	PUNCT
ejpam-6123	20	30	in	in	ADP
ejpam-6123	20	31	networked	networked	ADJ
ejpam-6123	20	32	systems	system	NOUN
ejpam-6123	20	33	.	.	PUNCT
ejpam-6123	21	1	we	we	PRON
ejpam-6123	21	2	may	may	AUX
ejpam-6123	21	3	refer	refer	VERB
ejpam-6123	21	4	to	to	ADP
ejpam-6123	21	5	[	[	X
ejpam-6123	21	6	1–5	1–5	X
ejpam-6123	21	7	]	]	PUNCT
ejpam-6123	21	8	to	to	PART
ejpam-6123	21	9	see	see	VERB
ejpam-6123	21	10	some	some	PRON
ejpam-6123	21	11	of	of	ADP
ejpam-6123	21	12	the	the	DET
ejpam-6123	21	13	recent	recent	ADJ
ejpam-6123	21	14	studies	study	NOUN
ejpam-6123	21	15	in	in	ADP
ejpam-6123	21	16	domination	domination	NOUN
ejpam-6123	21	17	,	,	PUNCT
ejpam-6123	21	18	and	and	CCONJ
ejpam-6123	21	19	to	to	ADP
ejpam-6123	21	20	[	[	X
ejpam-6123	21	21	6–8	6–8	X
ejpam-6123	21	22	]	]	X
ejpam-6123	21	23	for	for	ADP
ejpam-6123	21	24	its	its	PRON
ejpam-6123	21	25	various	various	ADJ
ejpam-6123	21	26	applications	application	NOUN
ejpam-6123	21	27	,	,	PUNCT
ejpam-6123	21	28	such	such	ADJ
ejpam-6123	21	29	as	as	ADP
ejpam-6123	21	30	electrical	electrical	ADJ
ejpam-6123	21	31	power	power	NOUN
ejpam-6123	21	32	networks	network	NOUN
ejpam-6123	21	33	,	,	PUNCT
ejpam-6123	21	34	communication	communication	NOUN
ejpam-6123	21	35	network	network	NOUN
ejpam-6123	21	36	,	,	PUNCT
ejpam-6123	21	37	facility	facility	NOUN
ejpam-6123	21	38	location	location	NOUN
ejpam-6123	21	39	problem	problem	NOUN
ejpam-6123	21	40	,	,	PUNCT
ejpam-6123	21	41	land	land	NOUN
ejpam-6123	21	42	surveying	surveying	NOUN
ejpam-6123	21	43	,	,	PUNCT
ejpam-6123	21	44	and	and	CCONJ
ejpam-6123	21	45	routings	routing	NOUN
ejpam-6123	21	46	.	.	PUNCT
ejpam-6123	22	1	a	a	DET
ejpam-6123	22	2	subset	subset	NOUN
ejpam-6123	22	3	s	s	VERB
ejpam-6123	22	4	⊆	⊆	NUM
ejpam-6123	22	5	v	v	NOUN
ejpam-6123	22	6	(	(	PUNCT
ejpam-6123	22	7	g	g	NOUN
ejpam-6123	22	8	)	)	PUNCT
ejpam-6123	22	9	is	be	AUX
ejpam-6123	22	10	a	a	DET
ejpam-6123	22	11	dominating	dominating	NOUN
ejpam-6123	22	12	set	set	NOUN
ejpam-6123	22	13	of	of	ADP
ejpam-6123	22	14	g	g	PROPN
ejpam-6123	22	15	if	if	SCONJ
ejpam-6123	22	16	for	for	ADP
ejpam-6123	22	17	every	every	PRON
ejpam-6123	22	18	v	v	NUM
ejpam-6123	22	19	∈	∈	NOUN
ejpam-6123	22	20	v	v	NOUN
ejpam-6123	22	21	(	(	PUNCT
ejpam-6123	22	22	g	g	NOUN
ejpam-6123	22	23	)	)	PUNCT
ejpam-6123	22	24	\	\	PROPN
ejpam-6123	23	1	s	s	X
ejpam-6123	23	2	,	,	PUNCT
ejpam-6123	23	3	there	there	PRON
ejpam-6123	23	4	exists	exist	VERB
ejpam-6123	23	5	u	u	PROPN
ejpam-6123	23	6	∈	∈	PROPN
ejpam-6123	23	7	s	s	VERB
ejpam-6123	23	8	such	such	ADJ
ejpam-6123	23	9	that	that	DET
ejpam-6123	23	10	uv	uv	PROPN
ejpam-6123	23	11	∈	∈	PROPN
ejpam-6123	23	12	e(g	e(g	PROPN
ejpam-6123	23	13	)	)	PUNCT
ejpam-6123	23	14	.	.	PUNCT
ejpam-6123	24	1	the	the	DET
ejpam-6123	24	2	domination	domination	NOUN
ejpam-6123	24	3	number	number	NOUN
ejpam-6123	24	4	of	of	ADP
ejpam-6123	24	5	g	g	NOUN
ejpam-6123	24	6	,	,	PUNCT
ejpam-6123	24	7	denoted	denote	VERB
ejpam-6123	24	8	by	by	ADP
ejpam-6123	24	9	γ(g	γ(g	PROPN
ejpam-6123	24	10	)	)	PUNCT
ejpam-6123	24	11	,	,	PUNCT
ejpam-6123	24	12	is	be	AUX
ejpam-6123	24	13	the	the	DET
ejpam-6123	24	14	smallest	small	ADJ
ejpam-6123	24	15	cardinality	cardinality	NOUN
ejpam-6123	24	16	of	of	ADP
ejpam-6123	24	17	a	a	DET
ejpam-6123	24	18	dominating	dominating	NOUN
ejpam-6123	24	19	set	set	NOUN
ejpam-6123	24	20	of	of	ADP
ejpam-6123	24	21	g.	g.	PROPN
ejpam-6123	24	22	traditional	traditional	ADJ
ejpam-6123	24	23	domination	domination	NOUN
ejpam-6123	24	24	parameters	parameter	NOUN
ejpam-6123	24	25	focus	focus	VERB
ejpam-6123	24	26	on	on	ADP
ejpam-6123	24	27	identifying	identify	VERB
ejpam-6123	24	28	minimal	minimal	ADJ
ejpam-6123	24	29	sets	set	NOUN
ejpam-6123	24	30	of	of	ADP
ejpam-6123	24	31	vertices	vertex	NOUN
ejpam-6123	24	32	that	that	PRON
ejpam-6123	24	33	dominate	dominate	VERB
ejpam-6123	24	34	or	or	CCONJ
ejpam-6123	24	35	control	control	VERB
ejpam-6123	24	36	the	the	DET
ejpam-6123	24	37	entire	entire	ADJ
ejpam-6123	24	38	graph	graph	NOUN
ejpam-6123	24	39	structure	structure	NOUN
ejpam-6123	24	40	.	.	PUNCT
ejpam-6123	25	1	however	however	ADV
ejpam-6123	25	2	,	,	PUNCT
ejpam-6123	25	3	in	in	ADP
ejpam-6123	25	4	many	many	ADJ
ejpam-6123	25	5	real	real	ADJ
ejpam-6123	25	6	-	-	PUNCT
ejpam-6123	25	7	world	world	NOUN
ejpam-6123	25	8	contexts	context	NOUN
ejpam-6123	25	9	—	—	PUNCT
ejpam-6123	25	10	such	such	ADJ
ejpam-6123	25	11	as	as	ADP
ejpam-6123	25	12	sensor	sensor	NOUN
ejpam-6123	25	13	networks	network	NOUN
ejpam-6123	25	14	,	,	PUNCT
ejpam-6123	25	15	facility	facility	NOUN
ejpam-6123	25	16	location	location	NOUN
ejpam-6123	25	17	,	,	PUNCT
ejpam-6123	25	18	and	and	CCONJ
ejpam-6123	25	19	communication	communication	NOUN
ejpam-6123	25	20	systems	system	NOUN
ejpam-6123	25	21	—	—	PUNCT
ejpam-6123	25	22	there	there	PRON
ejpam-6123	25	23	is	be	VERB
ejpam-6123	25	24	often	often	ADV
ejpam-6123	25	25	a	a	DET
ejpam-6123	25	26	need	need	NOUN
ejpam-6123	25	27	to	to	PART
ejpam-6123	25	28	ensure	ensure	VERB
ejpam-6123	25	29	both	both	DET
ejpam-6123	25	30	fairness	fairness	NOUN
ejpam-6123	25	31	in	in	ADP
ejpam-6123	25	32	vertex	vertex	NOUN
ejpam-6123	25	33	degrees	degree	NOUN
ejpam-6123	25	34	and	and	CCONJ
ejpam-6123	25	35	connectivity	connectivity	NOUN
ejpam-6123	25	36	among	among	ADP
ejpam-6123	25	37	dominating	dominating	NOUN
ejpam-6123	25	38	vertices	vertex	NOUN
ejpam-6123	25	39	.	.	PUNCT
ejpam-6123	26	1	to	to	PART
ejpam-6123	26	2	address	address	VERB
ejpam-6123	26	3	this	this	PRON
ejpam-6123	26	4	,	,	PUNCT
ejpam-6123	26	5	the	the	DET
ejpam-6123	26	6	notion	notion	NOUN
ejpam-6123	26	7	of	of	ADP
ejpam-6123	26	8	equitable	equitable	ADJ
ejpam-6123	26	9	domination	domination	NOUN
ejpam-6123	26	10	has	have	AUX
ejpam-6123	26	11	been	be	AUX
ejpam-6123	26	12	introduced	introduce	VERB
ejpam-6123	26	13	.	.	PUNCT
ejpam-6123	27	1	we	we	PRON
ejpam-6123	27	2	may	may	AUX
ejpam-6123	27	3	refer	refer	VERB
ejpam-6123	27	4	to	to	ADP
ejpam-6123	27	5	[	[	X
ejpam-6123	27	6	9–13	9–13	NOUN
ejpam-6123	27	7	]	]	PUNCT
ejpam-6123	27	8	to	to	PART
ejpam-6123	27	9	see	see	VERB
ejpam-6123	27	10	some	some	PRON
ejpam-6123	27	11	of	of	ADP
ejpam-6123	27	12	the	the	DET
ejpam-6123	27	13	studies	study	NOUN
ejpam-6123	27	14	in	in	ADP
ejpam-6123	27	15	equitable	equitable	ADJ
ejpam-6123	27	16	domination	domination	NOUN
ejpam-6123	27	17	.	.	PUNCT
ejpam-6123	28	1	a	a	DET
ejpam-6123	28	2	dominating	dominating	NOUN
ejpam-6123	28	3	set	set	NOUN
ejpam-6123	28	4	s	s	PROPN
ejpam-6123	28	5	⊆	⊆	NUM
ejpam-6123	28	6	v	v	NOUN
ejpam-6123	28	7	(	(	PUNCT
ejpam-6123	28	8	g	g	NOUN
ejpam-6123	28	9	)	)	PUNCT
ejpam-6123	28	10	is	be	AUX
ejpam-6123	28	11	called	call	VERB
ejpam-6123	28	12	a	a	DET
ejpam-6123	28	13	(	(	PUNCT
ejpam-6123	28	14	degree	degree	NOUN
ejpam-6123	28	15	)	)	PUNCT
ejpam-6123	28	16	equitable	equitable	ADJ
ejpam-6123	28	17	dominating	dominating	NOUN
ejpam-6123	28	18	set	set	VERB
ejpam-6123	28	19	in	in	ADP
ejpam-6123	28	20	g	g	PROPN
ejpam-6123	28	21	if	if	SCONJ
ejpam-6123	28	22	for	for	ADP
ejpam-6123	28	23	every	every	DET
ejpam-6123	28	24	v	v	NUM
ejpam-6123	28	25	∈	∈	NOUN
ejpam-6123	28	26	v	v	NOUN
ejpam-6123	28	27	(	(	PUNCT
ejpam-6123	28	28	g	g	NOUN
ejpam-6123	28	29	)	)	PUNCT
ejpam-6123	28	30	\	\	PROPN
ejpam-6123	29	1	s	s	VERB
ejpam-6123	29	2	there	there	PRON
ejpam-6123	29	3	exists	exist	VERB
ejpam-6123	29	4	u	u	PROPN
ejpam-6123	29	5	∈	∈	PROPN
ejpam-6123	29	6	s	s	VERB
ejpam-6123	29	7	such	such	ADJ
ejpam-6123	29	8	that	that	SCONJ
ejpam-6123	29	9	|degg(u	|degg(u	PROPN
ejpam-6123	29	10	)	)	PUNCT
ejpam-6123	29	11	−	−	NOUN
ejpam-6123	30	1	degg(v)|	degg(v)|	NOUN
ejpam-6123	30	2	≤	≤	NOUN
ejpam-6123	30	3	1	1	NUM
ejpam-6123	30	4	.	.	PUNCT
ejpam-6123	31	1	the	the	DET
ejpam-6123	31	2	equitable	equitable	ADJ
ejpam-6123	31	3	domination	domination	NOUN
ejpam-6123	31	4	number	number	NOUN
ejpam-6123	31	5	of	of	ADP
ejpam-6123	31	6	g	g	NOUN
ejpam-6123	31	7	,	,	PUNCT
ejpam-6123	31	8	denoted	denote	VERB
ejpam-6123	31	9	by	by	ADP
ejpam-6123	31	10	γe(g	γe(g	NUM
ejpam-6123	31	11	)	)	PUNCT
ejpam-6123	31	12	,	,	PUNCT
ejpam-6123	31	13	is	be	AUX
ejpam-6123	31	14	the	the	DET
ejpam-6123	31	15	smallest	small	ADJ
ejpam-6123	31	16	cardinality	cardinality	NOUN
ejpam-6123	31	17	among	among	ADP
ejpam-6123	31	18	all	all	PRON
ejpam-6123	31	19	equitable	equitable	ADJ
ejpam-6123	31	20	dominating	dominating	NOUN
ejpam-6123	31	21	set	set	VERB
ejpam-6123	31	22	in	in	ADP
ejpam-6123	31	23	g.	g.	PROPN
ejpam-6123	31	24	the	the	DET
ejpam-6123	31	25	notion	notion	NOUN
ejpam-6123	31	26	of	of	ADP
ejpam-6123	31	27	equitable	equitable	ADJ
ejpam-6123	31	28	domination	domination	NOUN
ejpam-6123	31	29	imposed	impose	VERB
ejpam-6123	31	30	a	a	DET
ejpam-6123	31	31	balance	balance	NOUN
ejpam-6123	31	32	condition	condition	NOUN
ejpam-6123	31	33	,	,	PUNCT
ejpam-6123	31	34	that	that	ADV
ejpam-6123	31	35	is	is	ADV
ejpam-6123	31	36	,	,	PUNCT
ejpam-6123	31	37	the	the	DET
ejpam-6123	31	38	sizes	size	NOUN
ejpam-6123	31	39	of	of	ADP
ejpam-6123	31	40	the	the	DET
ejpam-6123	31	41	open	open	ADJ
ejpam-6123	31	42	neighborhoods	neighborhood	NOUN
ejpam-6123	31	43	of	of	ADP
ejpam-6123	31	44	the	the	DET
ejpam-6123	31	45	dominating	dominating	NOUN
ejpam-6123	31	46	vertices	vertex	NOUN
ejpam-6123	31	47	must	must	AUX
ejpam-6123	31	48	be	be	AUX
ejpam-6123	31	49	as	as	ADV
ejpam-6123	31	50	equal	equal	ADJ
ejpam-6123	31	51	as	as	ADP
ejpam-6123	31	52	possible	possible	ADJ
ejpam-6123	31	53	,	,	PUNCT
ejpam-6123	31	54	minimizing	minimize	VERB
ejpam-6123	31	55	the	the	DET
ejpam-6123	31	56	difference	difference	NOUN
ejpam-6123	31	57	in	in	ADP
ejpam-6123	31	58	coverage	coverage	NOUN
ejpam-6123	31	59	.	.	PUNCT
ejpam-6123	32	1	some	some	PRON
ejpam-6123	32	2	of	of	ADP
ejpam-6123	32	3	the	the	DET
ejpam-6123	32	4	studies	study	NOUN
ejpam-6123	32	5	among	among	ADP
ejpam-6123	32	6	the	the	DET
ejpam-6123	32	7	many	many	ADJ
ejpam-6123	32	8	variations	variation	NOUN
ejpam-6123	32	9	of	of	ADP
ejpam-6123	32	10	domination	domination	NOUN
ejpam-6123	32	11	is	be	AUX
ejpam-6123	32	12	the	the	DET
ejpam-6123	32	13	connected	connected	ADJ
ejpam-6123	32	14	domination	domination	NOUN
ejpam-6123	32	15	which	which	PRON
ejpam-6123	32	16	requires	require	VERB
ejpam-6123	32	17	that	that	SCONJ
ejpam-6123	32	18	the	the	DET
ejpam-6123	32	19	subgraph	subgraph	NOUN
ejpam-6123	32	20	⟨s⟩	⟨s⟩	PROPN
ejpam-6123	32	21	induced	induce	VERB
ejpam-6123	32	22	by	by	ADP
ejpam-6123	32	23	dominating	dominate	VERB
ejpam-6123	32	24	set	set	NOUN
ejpam-6123	32	25	s	s	PART
ejpam-6123	32	26	is	be	AUX
ejpam-6123	32	27	connected	connect	VERB
ejpam-6123	32	28	,	,	PUNCT
ejpam-6123	32	29	reflecting	reflect	VERB
ejpam-6123	32	30	situations	situation	NOUN
ejpam-6123	32	31	where	where	SCONJ
ejpam-6123	32	32	communication	communication	NOUN
ejpam-6123	32	33	or	or	CCONJ
ejpam-6123	32	34	cohesion	cohesion	NOUN
ejpam-6123	32	35	among	among	ADP
ejpam-6123	32	36	controlling	control	VERB
ejpam-6123	32	37	elements	element	NOUN
ejpam-6123	32	38	is	be	AUX
ejpam-6123	32	39	essential	essential	ADJ
ejpam-6123	32	40	.	.	PUNCT
ejpam-6123	33	1	we	we	PRON
ejpam-6123	33	2	may	may	AUX
ejpam-6123	33	3	refer	refer	VERB
ejpam-6123	33	4	to	to	ADP
ejpam-6123	33	5	[	[	X
ejpam-6123	33	6	14	14	NUM
ejpam-6123	33	7	,	,	PUNCT
ejpam-6123	33	8	15	15	NUM
ejpam-6123	33	9	]	]	PUNCT
ejpam-6123	33	10	to	to	PART
ejpam-6123	33	11	see	see	VERB
ejpam-6123	33	12	some	some	PRON
ejpam-6123	33	13	of	of	ADP
ejpam-6123	33	14	the	the	DET
ejpam-6123	33	15	studies	study	NOUN
ejpam-6123	33	16	in	in	ADP
ejpam-6123	33	17	connected	connected	ADJ
ejpam-6123	33	18	domination	domination	NOUN
ejpam-6123	33	19	.	.	PUNCT
ejpam-6123	34	1	combining	combine	VERB
ejpam-6123	34	2	these	these	DET
ejpam-6123	34	3	two	two	NUM
ejpam-6123	34	4	constraints	constraint	NOUN
ejpam-6123	34	5	”	"	PUNCT
ejpam-6123	34	6	connectedness	connectedness	NOUN
ejpam-6123	34	7	”	"	PUNCT
ejpam-6123	34	8	and	and	CCONJ
ejpam-6123	34	9	equitability	equitability	NOUN
ejpam-6123	34	10	or	or	CCONJ
ejpam-6123	34	11	fairness	fairness	NOUN
ejpam-6123	34	12	gives	give	VERB
ejpam-6123	34	13	rise	rise	NOUN
ejpam-6123	34	14	to	to	ADP
ejpam-6123	34	15	the	the	DET
ejpam-6123	34	16	concept	concept	NOUN
ejpam-6123	34	17	of	of	ADP
ejpam-6123	34	18	connected	connect	VERB
ejpam-6123	34	19	equitable	equitable	ADJ
ejpam-6123	34	20	domination	domination	NOUN
ejpam-6123	34	21	.	.	PUNCT
ejpam-6123	35	1	the	the	DET
ejpam-6123	35	2	notion	notion	NOUN
ejpam-6123	35	3	of	of	ADP
ejpam-6123	35	4	connected	connected	ADJ
ejpam-6123	35	5	equitable	equitable	ADJ
ejpam-6123	35	6	domination	domination	NOUN
ejpam-6123	35	7	has	have	AUX
ejpam-6123	35	8	been	be	AUX
ejpam-6123	35	9	introduced	introduce	VERB
ejpam-6123	35	10	where	where	SCONJ
ejpam-6123	35	11	dominating	dominating	NOUN
ejpam-6123	35	12	sets	set	NOUN
ejpam-6123	35	13	are	be	AUX
ejpam-6123	35	14	not	not	PART
ejpam-6123	35	15	only	only	ADV
ejpam-6123	35	16	degree	degree	NOUN
ejpam-6123	35	17	-	-	PUNCT
ejpam-6123	35	18	balanced	balanced	ADJ
ejpam-6123	35	19	but	but	CCONJ
ejpam-6123	35	20	also	also	ADV
ejpam-6123	35	21	induce	induce	VERB
ejpam-6123	35	22	connected	connected	ADJ
ejpam-6123	35	23	subgraphs	subgraph	NOUN
ejpam-6123	35	24	.	.	PUNCT
ejpam-6123	36	1	an	an	DET
ejpam-6123	36	2	equitable	equitable	ADJ
ejpam-6123	36	3	dominating	dominating	NOUN
ejpam-6123	36	4	set	set	NOUN
ejpam-6123	36	5	s	s	PROPN
ejpam-6123	36	6	⊆	⊆	NUM
ejpam-6123	36	7	v	v	NOUN
ejpam-6123	36	8	(	(	PUNCT
ejpam-6123	36	9	g	g	NOUN
ejpam-6123	36	10	)	)	PUNCT
ejpam-6123	36	11	is	be	AUX
ejpam-6123	36	12	called	call	VERB
ejpam-6123	36	13	a	a	DET
ejpam-6123	36	14	connected	connect	VERB
ejpam-6123	36	15	equitable	equitable	ADJ
ejpam-6123	36	16	dominating	dominating	NOUN
ejpam-6123	36	17	set	set	NOUN
ejpam-6123	36	18	of	of	ADP
ejpam-6123	36	19	g	g	PROPN
ejpam-6123	36	20	if	if	SCONJ
ejpam-6123	36	21	the	the	DET
ejpam-6123	36	22	subgraph	subgraph	NOUN
ejpam-6123	36	23	⟨s⟩	⟨s⟩	PROPN
ejpam-6123	36	24	induced	induce	VERB
ejpam-6123	36	25	by	by	ADP
ejpam-6123	36	26	s	s	PROPN
ejpam-6123	36	27	is	be	AUX
ejpam-6123	36	28	connected	connect	VERB
ejpam-6123	36	29	.	.	PUNCT
ejpam-6123	37	1	the	the	DET
ejpam-6123	37	2	minimum	minimum	ADJ
ejpam-6123	37	3	cardinality	cardinality	NOUN
ejpam-6123	37	4	among	among	ADP
ejpam-6123	37	5	all	all	DET
ejpam-6123	37	6	connected	connect	VERB
ejpam-6123	37	7	equitable	equitable	ADJ
ejpam-6123	37	8	dominating	dominating	NOUN
ejpam-6123	37	9	sets	set	NOUN
ejpam-6123	37	10	in	in	ADP
ejpam-6123	37	11	g	g	PROPN
ejpam-6123	37	12	is	be	AUX
ejpam-6123	37	13	called	call	VERB
ejpam-6123	37	14	the	the	DET
ejpam-6123	37	15	connected	connect	VERB
ejpam-6123	37	16	equitable	equitable	ADJ
ejpam-6123	37	17	domination	domination	NOUN
ejpam-6123	37	18	number	number	NOUN
ejpam-6123	37	19	of	of	ADP
ejpam-6123	37	20	g	g	NOUN
ejpam-6123	37	21	,	,	PUNCT
ejpam-6123	37	22	and	and	CCONJ
ejpam-6123	37	23	is	be	AUX
ejpam-6123	37	24	denoted	denote	VERB
ejpam-6123	37	25	by	by	ADP
ejpam-6123	37	26	γce(g	γce(g	PROPN
ejpam-6123	37	27	)	)	PUNCT
ejpam-6123	37	28	.	.	PUNCT
ejpam-6123	38	1	a	a	DET
ejpam-6123	38	2	subset	subset	NOUN
ejpam-6123	38	3	s	s	VERB
ejpam-6123	38	4	⊆	⊆	NUM
ejpam-6123	38	5	v	v	NOUN
ejpam-6123	38	6	(	(	PUNCT
ejpam-6123	38	7	g	g	NOUN
ejpam-6123	38	8	)	)	PUNCT
ejpam-6123	38	9	is	be	AUX
ejpam-6123	38	10	called	call	VERB
ejpam-6123	38	11	minimal	minimal	ADJ
ejpam-6123	38	12	connected	connect	VERB
ejpam-6123	38	13	equitable	equitable	ADJ
ejpam-6123	38	14	dominating	dominating	NOUN
ejpam-6123	38	15	set	set	VERB
ejpam-6123	38	16	in	in	ADP
ejpam-6123	38	17	g	g	PROPN
ejpam-6123	38	18	if	if	SCONJ
ejpam-6123	38	19	no	no	DET
ejpam-6123	38	20	proper	proper	ADJ
ejpam-6123	38	21	subset	subset	NOUN
ejpam-6123	38	22	of	of	ADP
ejpam-6123	38	23	s	s	PROPN
ejpam-6123	38	24	is	be	AUX
ejpam-6123	38	25	a	a	DET
ejpam-6123	38	26	connected	connect	VERB
ejpam-6123	38	27	equitable	equitable	ADJ
ejpam-6123	38	28	dominating	dominating	NOUN
ejpam-6123	38	29	set	set	VERB
ejpam-6123	38	30	in	in	ADP
ejpam-6123	38	31	g.	g.	PROPN
ejpam-6123	38	32	the	the	DET
ejpam-6123	38	33	concept	concept	NOUN
ejpam-6123	38	34	of	of	ADP
ejpam-6123	38	35	connected	connect	VERB
ejpam-6123	38	36	equitable	equitable	ADJ
ejpam-6123	38	37	domination	domination	NOUN
ejpam-6123	38	38	has	have	VERB
ejpam-6123	38	39	many	many	ADJ
ejpam-6123	38	40	real	real	ADJ
ejpam-6123	38	41	life	life	NOUN
ejpam-6123	38	42	applications	application	NOUN
ejpam-6123	38	43	where	where	SCONJ
ejpam-6123	38	44	both	both	DET
ejpam-6123	38	45	connectivity	connectivity	NOUN
ejpam-6123	38	46	(	(	PUNCT
ejpam-6123	38	47	connectedness	connectedness	NOUN
ejpam-6123	38	48	)	)	PUNCT
ejpam-6123	38	49	and	and	CCONJ
ejpam-6123	38	50	fairness	fairness	NOUN
ejpam-6123	38	51	(	(	PUNCT
ejpam-6123	38	52	equitability	equitability	NOUN
ejpam-6123	38	53	)	)	PUNCT
ejpam-6123	38	54	in	in	ADP
ejpam-6123	38	55	workload	workload	NOUN
ejpam-6123	38	56	distribution	distribution	NOUN
ejpam-6123	38	57	or	or	CCONJ
ejpam-6123	38	58	influence	influence	NOUN
ejpam-6123	38	59	distribution	distribution	NOUN
ejpam-6123	38	60	are	be	AUX
ejpam-6123	38	61	essential	essential	ADJ
ejpam-6123	38	62	.	.	PUNCT
ejpam-6123	39	1	such	such	ADJ
ejpam-6123	39	2	theoretical	theoretical	ADJ
ejpam-6123	39	3	and	and	CCONJ
ejpam-6123	39	4	practical	practical	ADJ
ejpam-6123	39	5	applications	application	NOUN
ejpam-6123	39	6	includes	include	VERB
ejpam-6123	39	7	(	(	PUNCT
ejpam-6123	39	8	1	1	NUM
ejpam-6123	39	9	)	)	PUNCT
ejpam-6123	39	10	distributed	distribute	VERB
ejpam-6123	39	11	computing	computing	NOUN
ejpam-6123	39	12	and	and	CCONJ
ejpam-6123	39	13	load	load	NOUN
ejpam-6123	39	14	balancing	balancing	NOUN
ejpam-6123	39	15	:	:	PUNCT
ejpam-6123	39	16	in	in	ADP
ejpam-6123	39	17	distributed	distribute	VERB
ejpam-6123	39	18	systems	system	NOUN
ejpam-6123	39	19	,	,	PUNCT
ejpam-6123	39	20	assigning	assign	VERB
ejpam-6123	39	21	tasks	task	NOUN
ejpam-6123	39	22	to	to	ADP
ejpam-6123	39	23	processing	processing	NOUN
ejpam-6123	39	24	units	unit	NOUN
ejpam-6123	39	25	can	can	AUX
ejpam-6123	39	26	be	be	AUX
ejpam-6123	39	27	modeled	model	VERB
ejpam-6123	39	28	as	as	ADP
ejpam-6123	39	29	a	a	DET
ejpam-6123	39	30	connected	connected	ADJ
ejpam-6123	39	31	equitable	equitable	ADJ
ejpam-6123	39	32	dominating	dominating	NOUN
ejpam-6123	39	33	set	set	NOUN
ejpam-6123	39	34	which	which	PRON
ejpam-6123	39	35	ensures	ensure	VERB
ejpam-6123	39	36	that	that	SCONJ
ejpam-6123	39	37	workload	workload	NOUN
ejpam-6123	39	38	distribution	distribution	NOUN
ejpam-6123	39	39	is	be	AUX
ejpam-6123	39	40	equitable	equitable	ADJ
ejpam-6123	39	41	,	,	PUNCT
ejpam-6123	39	42	that	that	ADV
ejpam-6123	39	43	is	is	ADV
ejpam-6123	39	44	,	,	PUNCT
ejpam-6123	39	45	nearly	nearly	ADV
ejpam-6123	39	46	equal	equal	ADJ
ejpam-6123	39	47	,	,	PUNCT
ejpam-6123	39	48	and	and	CCONJ
ejpam-6123	39	49	all	all	DET
ejpam-6123	39	50	coordinating	coordinate	VERB
ejpam-6123	39	51	vertices	vertex	NOUN
ejpam-6123	39	52	can	can	AUX
ejpam-6123	39	53	communicate	communicate	VERB
ejpam-6123	39	54	efficiently	efficiently	ADV
ejpam-6123	39	55	;	;	PUNCT
ejpam-6123	39	56	(	(	PUNCT
ejpam-6123	39	57	2	2	X
ejpam-6123	39	58	)	)	PUNCT
ejpam-6123	39	59	social	social	ADJ
ejpam-6123	39	60	networks	network	NOUN
ejpam-6123	39	61	and	and	CCONJ
ejpam-6123	39	62	influence	influence	NOUN
ejpam-6123	39	63	modeling	modeling	NOUN
ejpam-6123	39	64	:	:	PUNCT
ejpam-6123	39	65	in	in	ADP
ejpam-6123	39	66	scenarios	scenario	NOUN
ejpam-6123	39	67	involving	involve	VERB
ejpam-6123	39	68	opinion	opinion	NOUN
ejpam-6123	39	69	formation	formation	NOUN
ejpam-6123	39	70	or	or	CCONJ
ejpam-6123	39	71	influence	influence	NOUN
ejpam-6123	39	72	spread	spread	VERB
ejpam-6123	39	73	,	,	PUNCT
ejpam-6123	39	74	a	a	DET
ejpam-6123	39	75	connected	connect	VERB
ejpam-6123	39	76	equitable	equitable	ADJ
ejpam-6123	39	77	dominating	dominating	NOUN
ejpam-6123	39	78	set	set	NOUN
ejpam-6123	39	79	can	can	AUX
ejpam-6123	39	80	represent	represent	VERB
ejpam-6123	39	81	a	a	DET
ejpam-6123	39	82	group	group	NOUN
ejpam-6123	39	83	of	of	ADP
ejpam-6123	39	84	influential	influential	ADJ
ejpam-6123	39	85	individuals	individual	NOUN
ejpam-6123	39	86	who	who	PRON
ejpam-6123	39	87	are	be	AUX
ejpam-6123	39	88	both	both	PRON
ejpam-6123	39	89	interconnected	interconnected	ADJ
ejpam-6123	39	90	and	and	CCONJ
ejpam-6123	39	91	exert	exert	VERB
ejpam-6123	39	92	roughly	roughly	ADV
ejpam-6123	39	93	equal	equal	ADJ
ejpam-6123	39	94	influence	influence	NOUN
ejpam-6123	39	95	over	over	ADP
ejpam-6123	39	96	the	the	DET
ejpam-6123	39	97	population	population	NOUN
ejpam-6123	39	98	;	;	PUNCT
ejpam-6123	39	99	(	(	PUNCT
ejpam-6123	39	100	3	3	X
ejpam-6123	39	101	)	)	PUNCT
ejpam-6123	39	102	urban	urban	ADJ
ejpam-6123	39	103	planning	planning	NOUN
ejpam-6123	39	104	and	and	CCONJ
ejpam-6123	39	105	facility	facility	NOUN
ejpam-6123	39	106	location	location	NOUN
ejpam-6123	39	107	:	:	PUNCT
ejpam-6123	39	108	h.	h.	PROPN
ejpam-6123	39	109	nuenay	nuenay	PROPN
ejpam-6123	39	110	-	-	PUNCT
ejpam-6123	39	111	maglanquel	maglanquel	PROPN
ejpam-6123	39	112	/	/	SYM
ejpam-6123	39	113	eur	eur	PROPN
ejpam-6123	39	114	.	.	PUNCT
ejpam-6123	40	1	j.	j.	PROPN
ejpam-6123	40	2	pure	pure	PROPN
ejpam-6123	40	3	appl	appl	PROPN
ejpam-6123	40	4	.	.	PROPN
ejpam-6123	40	5	math	math	PROPN
ejpam-6123	40	6	,	,	PUNCT
ejpam-6123	40	7	18	18	NUM
ejpam-6123	40	8	(	(	PUNCT
ejpam-6123	40	9	3	3	NUM
ejpam-6123	40	10	)	)	PUNCT
ejpam-6123	40	11	(	(	PUNCT
ejpam-6123	40	12	2025	2025	NUM
ejpam-6123	40	13	)	)	PUNCT
ejpam-6123	40	14	,	,	PUNCT
ejpam-6123	40	15	6123	6123	NUM
ejpam-6123	40	16	3	3	NUM
ejpam-6123	40	17	of	of	ADP
ejpam-6123	40	18	14	14	NUM
ejpam-6123	40	19	when	when	SCONJ
ejpam-6123	40	20	placing	place	VERB
ejpam-6123	40	21	essential	essential	ADJ
ejpam-6123	40	22	facilities	facility	NOUN
ejpam-6123	40	23	such	such	ADJ
ejpam-6123	40	24	as	as	ADP
ejpam-6123	40	25	hospitals	hospital	NOUN
ejpam-6123	40	26	,	,	PUNCT
ejpam-6123	40	27	fire	fire	NOUN
ejpam-6123	40	28	stations	station	NOUN
ejpam-6123	40	29	and	and	CCONJ
ejpam-6123	40	30	etc	etc	X
ejpam-6123	40	31	.	.	X
ejpam-6123	41	1	in	in	ADP
ejpam-6123	41	2	a	a	DET
ejpam-6123	41	3	city	city	NOUN
ejpam-6123	41	4	modeled	model	VERB
ejpam-6123	41	5	as	as	ADP
ejpam-6123	41	6	a	a	DET
ejpam-6123	41	7	graph	graph	NOUN
ejpam-6123	41	8	,	,	PUNCT
ejpam-6123	41	9	a	a	DET
ejpam-6123	41	10	connected	connect	VERB
ejpam-6123	41	11	equitable	equitable	ADJ
ejpam-6123	41	12	dominating	dominating	NOUN
ejpam-6123	41	13	set	set	NOUN
ejpam-6123	41	14	ensures	ensure	VERB
ejpam-6123	41	15	that	that	SCONJ
ejpam-6123	41	16	all	all	DET
ejpam-6123	41	17	regions	region	NOUN
ejpam-6123	41	18	are	be	AUX
ejpam-6123	41	19	covered	cover	VERB
ejpam-6123	41	20	by	by	ADP
ejpam-6123	41	21	a	a	DET
ejpam-6123	41	22	connected	connected	ADJ
ejpam-6123	41	23	and	and	CCONJ
ejpam-6123	41	24	fairly	fairly	ADV
ejpam-6123	41	25	distributed	distribute	VERB
ejpam-6123	41	26	set	set	NOUN
ejpam-6123	41	27	of	of	ADP
ejpam-6123	41	28	facilities	facility	NOUN
ejpam-6123	41	29	.	.	PUNCT
ejpam-6123	42	1	by	by	ADP
ejpam-6123	42	2	combining	combine	VERB
ejpam-6123	42	3	the	the	DET
ejpam-6123	42	4	structural	structural	ADJ
ejpam-6123	42	5	benefits	benefit	NOUN
ejpam-6123	42	6	of	of	ADP
ejpam-6123	42	7	connectedness	connectedness	NOUN
ejpam-6123	42	8	with	with	ADP
ejpam-6123	42	9	the	the	DET
ejpam-6123	42	10	fairness	fairness	NOUN
ejpam-6123	42	11	of	of	ADP
ejpam-6123	42	12	equitable	equitable	ADJ
ejpam-6123	42	13	load	load	NOUN
ejpam-6123	42	14	distribution	distribution	NOUN
ejpam-6123	42	15	,	,	PUNCT
ejpam-6123	42	16	connected	connect	VERB
ejpam-6123	42	17	equitable	equitable	ADJ
ejpam-6123	42	18	domination	domination	NOUN
ejpam-6123	42	19	provides	provide	VERB
ejpam-6123	42	20	a	a	DET
ejpam-6123	42	21	robust	robust	ADJ
ejpam-6123	42	22	and	and	CCONJ
ejpam-6123	42	23	efficient	efficient	ADJ
ejpam-6123	42	24	framework	framework	NOUN
ejpam-6123	42	25	for	for	ADP
ejpam-6123	42	26	designing	design	VERB
ejpam-6123	42	27	and	and	CCONJ
ejpam-6123	42	28	analyzing	analyze	VERB
ejpam-6123	42	29	network	network	NOUN
ejpam-6123	42	30	systems	system	NOUN
ejpam-6123	42	31	;	;	PUNCT
ejpam-6123	42	32	and	and	CCONJ
ejpam-6123	42	33	(	(	PUNCT
ejpam-6123	42	34	4	4	X
ejpam-6123	42	35	)	)	PUNCT
ejpam-6123	42	36	telecommunication	telecommunication	NOUN
ejpam-6123	42	37	and	and	CCONJ
ejpam-6123	42	38	sensor	sensor	NOUN
ejpam-6123	42	39	networks	network	NOUN
ejpam-6123	42	40	:	:	PUNCT
ejpam-6123	42	41	in	in	ADP
ejpam-6123	42	42	wireless	wireless	ADJ
ejpam-6123	42	43	sensor	sensor	NOUN
ejpam-6123	42	44	networks	network	NOUN
ejpam-6123	42	45	or	or	CCONJ
ejpam-6123	42	46	ad	ad	X
ejpam-6123	42	47	hoc	hoc	X
ejpam-6123	42	48	communication	communication	NOUN
ejpam-6123	42	49	networks	network	NOUN
ejpam-6123	42	50	,	,	PUNCT
ejpam-6123	42	51	a	a	DET
ejpam-6123	42	52	connected	connect	VERB
ejpam-6123	42	53	equitable	equitable	ADJ
ejpam-6123	42	54	dominating	dominating	NOUN
ejpam-6123	42	55	set	set	NOUN
ejpam-6123	42	56	can	can	AUX
ejpam-6123	42	57	represent	represent	VERB
ejpam-6123	42	58	as	as	ADP
ejpam-6123	42	59	a	a	DET
ejpam-6123	42	60	core	core	NOUN
ejpam-6123	42	61	framework	framework	NOUN
ejpam-6123	42	62	where	where	SCONJ
ejpam-6123	42	63	central	central	ADJ
ejpam-6123	42	64	vertices	vertex	NOUN
ejpam-6123	42	65	(	(	PUNCT
ejpam-6123	42	66	dominating	dominating	NOUN
ejpam-6123	42	67	vertices	vertex	NOUN
ejpam-6123	42	68	)	)	PUNCT
ejpam-6123	42	69	are	be	AUX
ejpam-6123	42	70	connected	connect	VERB
ejpam-6123	42	71	and	and	CCONJ
ejpam-6123	42	72	share	share	VERB
ejpam-6123	42	73	the	the	DET
ejpam-6123	42	74	communication	communication	NOUN
ejpam-6123	42	75	or	or	CCONJ
ejpam-6123	42	76	monitoring	monitoring	NOUN
ejpam-6123	42	77	load	load	NOUN
ejpam-6123	42	78	equitably	equitably	ADV
ejpam-6123	42	79	.	.	PUNCT
ejpam-6123	43	1	this	this	PRON
ejpam-6123	43	2	ensures	ensure	VERB
ejpam-6123	43	3	both	both	DET
ejpam-6123	43	4	robustness	robustness	NOUN
ejpam-6123	43	5	(	(	PUNCT
ejpam-6123	43	6	connectivity	connectivity	NOUN
ejpam-6123	43	7	)	)	PUNCT
ejpam-6123	43	8	and	and	CCONJ
ejpam-6123	43	9	balanced	balanced	ADJ
ejpam-6123	43	10	energy	energy	NOUN
ejpam-6123	43	11	usage	usage	NOUN
ejpam-6123	43	12	or	or	CCONJ
ejpam-6123	43	13	data	datum	NOUN
ejpam-6123	43	14	handling	handling	NOUN
ejpam-6123	43	15	.	.	PUNCT
ejpam-6123	44	1	in	in	ADP
ejpam-6123	44	2	this	this	DET
ejpam-6123	44	3	paper	paper	NOUN
ejpam-6123	44	4	,	,	PUNCT
ejpam-6123	44	5	we	we	PRON
ejpam-6123	44	6	investigate	investigate	VERB
ejpam-6123	44	7	this	this	DET
ejpam-6123	44	8	parameter	parameter	NOUN
ejpam-6123	44	9	in	in	ADP
ejpam-6123	44	10	graphs	graph	NOUN
ejpam-6123	44	11	formed	form	VERB
ejpam-6123	44	12	through	through	ADP
ejpam-6123	44	13	two	two	NUM
ejpam-6123	44	14	fundamental	fundamental	ADJ
ejpam-6123	44	15	operations	operation	NOUN
ejpam-6123	44	16	:	:	PUNCT
ejpam-6123	44	17	the	the	DET
ejpam-6123	44	18	join	join	NOUN
ejpam-6123	44	19	and	and	CCONJ
ejpam-6123	44	20	the	the	DET
ejpam-6123	44	21	corona	corona	NOUN
ejpam-6123	44	22	.	.	PUNCT
ejpam-6123	45	1	we	we	PRON
ejpam-6123	45	2	provide	provide	VERB
ejpam-6123	45	3	structural	structural	ADJ
ejpam-6123	45	4	characterizations	characterization	NOUN
ejpam-6123	45	5	of	of	ADP
ejpam-6123	45	6	such	such	ADJ
ejpam-6123	45	7	dominating	dominating	NOUN
ejpam-6123	45	8	sets	set	NOUN
ejpam-6123	45	9	in	in	ADP
ejpam-6123	45	10	these	these	DET
ejpam-6123	45	11	graphs	graph	NOUN
ejpam-6123	45	12	and	and	CCONJ
ejpam-6123	45	13	determine	determine	VERB
ejpam-6123	45	14	exact	exact	ADJ
ejpam-6123	45	15	values	value	NOUN
ejpam-6123	45	16	of	of	ADP
ejpam-6123	45	17	γce(g	γce(g	NOUN
ejpam-6123	45	18	)	)	PUNCT
ejpam-6123	45	19	for	for	ADP
ejpam-6123	45	20	various	various	ADJ
ejpam-6123	45	21	graph	graph	NOUN
ejpam-6123	45	22	families	family	NOUN
ejpam-6123	45	23	.	.	PUNCT
ejpam-6123	46	1	a	a	DET
ejpam-6123	46	2	realization	realization	NOUN
ejpam-6123	46	3	problem	problem	NOUN
ejpam-6123	46	4	is	be	AUX
ejpam-6123	46	5	also	also	ADV
ejpam-6123	46	6	addressed	address	VERB
ejpam-6123	46	7	,	,	PUNCT
ejpam-6123	46	8	offering	offer	VERB
ejpam-6123	46	9	insight	insight	NOUN
ejpam-6123	46	10	into	into	ADP
ejpam-6123	46	11	how	how	SCONJ
ejpam-6123	46	12	graphs	graph	NOUN
ejpam-6123	46	13	can	can	AUX
ejpam-6123	46	14	be	be	AUX
ejpam-6123	46	15	constructed	construct	VERB
ejpam-6123	46	16	to	to	PART
ejpam-6123	46	17	attain	attain	VERB
ejpam-6123	46	18	prescribed	prescribed	ADJ
ejpam-6123	46	19	values	value	NOUN
ejpam-6123	46	20	of	of	ADP
ejpam-6123	46	21	the	the	DET
ejpam-6123	46	22	parameter	parameter	NOUN
ejpam-6123	46	23	.	.	PUNCT
ejpam-6123	47	1	for	for	ADP
ejpam-6123	47	2	simplicity	simplicity	NOUN
ejpam-6123	47	3	,	,	PUNCT
ejpam-6123	47	4	we	we	PRON
ejpam-6123	47	5	use	use	VERB
ejpam-6123	47	6	the	the	DET
ejpam-6123	47	7	terms	term	NOUN
ejpam-6123	47	8	γce	γce	NOUN
ejpam-6123	47	9	-	-	PUNCT
ejpam-6123	47	10	set	set	VERB
ejpam-6123	47	11	,	,	PUNCT
ejpam-6123	47	12	γc	γc	NOUN
ejpam-6123	47	13	-	-	PUNCT
ejpam-6123	47	14	set	set	NOUN
ejpam-6123	47	15	,	,	PUNCT
ejpam-6123	47	16	and	and	CCONJ
ejpam-6123	47	17	γe	γe	VERB
ejpam-6123	47	18	to	to	PART
ejpam-6123	47	19	refer	refer	VERB
ejpam-6123	47	20	to	to	ADP
ejpam-6123	47	21	the	the	DET
ejpam-6123	47	22	connected	connect	VERB
ejpam-6123	47	23	equitable	equitable	ADJ
ejpam-6123	47	24	dominating	dominating	NOUN
ejpam-6123	47	25	set	set	VERB
ejpam-6123	47	26	with	with	ADP
ejpam-6123	47	27	cardinality	cardinality	PROPN
ejpam-6123	47	28	γce(g	γce(g	PROPN
ejpam-6123	47	29	)	)	PUNCT
ejpam-6123	47	30	,	,	PUNCT
ejpam-6123	47	31	connected	connect	VERB
ejpam-6123	47	32	dominating	dominating	NOUN
ejpam-6123	47	33	set	set	VERB
ejpam-6123	47	34	with	with	ADP
ejpam-6123	47	35	cardinality	cardinality	NOUN
ejpam-6123	47	36	γc(g	γc(g	PUNCT
ejpam-6123	47	37	)	)	PUNCT
ejpam-6123	47	38	and	and	CCONJ
ejpam-6123	47	39	equitable	equitable	ADJ
ejpam-6123	47	40	dominating	dominating	NOUN
ejpam-6123	47	41	set	set	VERB
ejpam-6123	47	42	with	with	ADP
ejpam-6123	47	43	cardinality	cardinality	NOUN
ejpam-6123	47	44	γe(g	γe(g	NUM
ejpam-6123	47	45	)	)	PUNCT
ejpam-6123	47	46	,	,	PUNCT
ejpam-6123	47	47	respectively	respectively	ADV
ejpam-6123	47	48	.	.	PUNCT
ejpam-6123	48	1	the	the	DET
ejpam-6123	48	2	following	follow	VERB
ejpam-6123	48	3	results	result	NOUN
ejpam-6123	48	4	are	be	AUX
ejpam-6123	48	5	due	due	ADJ
ejpam-6123	48	6	to	to	ADP
ejpam-6123	48	7	v.	v.	ADP
ejpam-6123	48	8	swaminathan	swaminathan	ADV
ejpam-6123	48	9	et.al	et.al	PROPN
ejpam-6123	48	10	and	and	CCONJ
ejpam-6123	48	11	sivakumar	sivakumar	PROPN
ejpam-6123	48	12	et.al	et.al	PROPN
ejpam-6123	48	13	theorem	theorem	NOUN
ejpam-6123	48	14	1	1	NUM
ejpam-6123	48	15	.	.	PUNCT
ejpam-6123	49	1	[	[	X
ejpam-6123	49	2	15	15	NUM
ejpam-6123	49	3	]	]	PUNCT
ejpam-6123	49	4	the	the	DET
ejpam-6123	49	5	connected	connect	VERB
ejpam-6123	49	6	equitable	equitable	ADJ
ejpam-6123	49	7	domination	domination	NOUN
ejpam-6123	49	8	number	number	NOUN
ejpam-6123	49	9	of	of	ADP
ejpam-6123	49	10	some	some	DET
ejpam-6123	49	11	standard	standard	ADJ
ejpam-6123	49	12	graphs	graph	NOUN
ejpam-6123	49	13	are	be	AUX
ejpam-6123	49	14	(	(	PUNCT
ejpam-6123	49	15	i	i	NOUN
ejpam-6123	49	16	)	)	PUNCT
ejpam-6123	49	17	for	for	ADP
ejpam-6123	49	18	a	a	DET
ejpam-6123	49	19	complete	complete	ADJ
ejpam-6123	49	20	graph	graph	NOUN
ejpam-6123	49	21	kn	kn	PROPN
ejpam-6123	49	22	on	on	ADP
ejpam-6123	49	23	n	n	PRON
ejpam-6123	49	24	vertices	vertex	NOUN
ejpam-6123	49	25	,	,	PUNCT
ejpam-6123	49	26	γce(g	γce(g	NOUN
ejpam-6123	49	27	)	)	PUNCT
ejpam-6123	49	28	=	=	SYM
ejpam-6123	50	1	1	1	X
ejpam-6123	50	2	.	.	PUNCT
ejpam-6123	50	3	(	(	PUNCT
ejpam-6123	50	4	ii	ii	NOUN
ejpam-6123	50	5	)	)	PUNCT
ejpam-6123	50	6	for	for	ADP
ejpam-6123	50	7	the	the	DET
ejpam-6123	50	8	paths	path	NOUN
ejpam-6123	50	9	pn	pn	NOUN
ejpam-6123	50	10	and	and	CCONJ
ejpam-6123	50	11	the	the	DET
ejpam-6123	50	12	cycles	cycle	NOUN
ejpam-6123	50	13	cn	cn	PROPN
ejpam-6123	50	14	on	on	ADP
ejpam-6123	50	15	n	n	PRON
ejpam-6123	50	16	vertices	vertex	NOUN
ejpam-6123	50	17	,	,	PUNCT
ejpam-6123	50	18	γce(cn	γce(cn	NOUN
ejpam-6123	50	19	)	)	PUNCT
ejpam-6123	50	20	=	=	SYM
ejpam-6123	50	21	γce(pn	γce(pn	NOUN
ejpam-6123	50	22	)	)	PUNCT
ejpam-6123	50	23	=	=	SYM
ejpam-6123	50	24	n−	n−	NOUN
ejpam-6123	50	25	2	2	NUM
ejpam-6123	50	26	.	.	PUNCT
ejpam-6123	50	27	(	(	PUNCT
ejpam-6123	50	28	iii	iii	X
ejpam-6123	50	29	)	)	PUNCT
ejpam-6123	50	30	if	if	SCONJ
ejpam-6123	50	31	wn	wn	PROPN
ejpam-6123	50	32	denotes	denote	VERB
ejpam-6123	50	33	the	the	DET
ejpam-6123	50	34	wheel	wheel	NOUN
ejpam-6123	50	35	on	on	ADP
ejpam-6123	50	36	n	n	PRON
ejpam-6123	50	37	vertices	vertex	NOUN
ejpam-6123	50	38	,	,	PUNCT
ejpam-6123	50	39	then	then	ADV
ejpam-6123	50	40	γce(wn	γce(wn	NOUN
ejpam-6123	50	41	)	)	PUNCT
ejpam-6123	50	42	=	=	SYM
ejpam-6123	50	43	{	{	PUNCT
ejpam-6123	50	44	⌈	⌈	NOUN
ejpam-6123	50	45	n+3	n+3	PROPN
ejpam-6123	50	46	3	3	NUM
ejpam-6123	50	47	⌉	⌉	X
ejpam-6123	50	48	,	,	PUNCT
ejpam-6123	50	49	if	if	SCONJ
ejpam-6123	50	50	n	n	PRON
ejpam-6123	50	51	≥	≥	NOUN
ejpam-6123	50	52	6	6	NUM
ejpam-6123	50	53	1	1	NUM
ejpam-6123	50	54	,	,	PUNCT
ejpam-6123	50	55	otherwise	otherwise	ADV
ejpam-6123	50	56	.	.	PUNCT
ejpam-6123	51	1	(	(	PUNCT
ejpam-6123	51	2	iv	iv	X
ejpam-6123	51	3	)	)	PUNCT
ejpam-6123	51	4	for	for	ADP
ejpam-6123	51	5	the	the	DET
ejpam-6123	51	6	complete	complete	ADJ
ejpam-6123	51	7	bipartite	bipartite	PROPN
ejpam-6123	51	8	graph	graph	NOUN
ejpam-6123	51	9	km	km	PROPN
ejpam-6123	51	10	,	,	PUNCT
ejpam-6123	51	11	n	n	CCONJ
ejpam-6123	51	12	,	,	PUNCT
ejpam-6123	51	13	we	we	PRON
ejpam-6123	51	14	have	have	VERB
ejpam-6123	51	15	,	,	PUNCT
ejpam-6123	51	16	γce(km	γce(km	PROPN
ejpam-6123	51	17	,	,	PUNCT
ejpam-6123	51	18	n	n	CCONJ
ejpam-6123	51	19	)	)	PUNCT
ejpam-6123	51	20	=	=	PUNCT
ejpam-6123	52	1			PROPN
ejpam-6123	52	2	1	1	NUM
ejpam-6123	52	3	,	,	PUNCT
ejpam-6123	52	4	either	either	CCONJ
ejpam-6123	52	5	m	m	PROPN
ejpam-6123	52	6	=	=	SYM
ejpam-6123	52	7	1	1	NUM
ejpam-6123	52	8	or	or	CCONJ
ejpam-6123	52	9	n	n	CCONJ
ejpam-6123	52	10	=	=	SYM
ejpam-6123	52	11	1	1	NUM
ejpam-6123	52	12	2	2	NUM
ejpam-6123	52	13	,	,	PUNCT
ejpam-6123	52	14	if	if	SCONJ
ejpam-6123	52	15	|m−	|m−	PROPN
ejpam-6123	52	16	n|	n|	NOUN
ejpam-6123	52	17	≤	≤	PROPN
ejpam-6123	52	18	1	1	NUM
ejpam-6123	52	19	and	and	CCONJ
ejpam-6123	52	20	m	m	PROPN
ejpam-6123	52	21	,	,	PUNCT
ejpam-6123	52	22	n	n	PRON
ejpam-6123	52	23	≥	≥	NUM
ejpam-6123	52	24	2	2	NUM
ejpam-6123	52	25	m+	m+	NUM
ejpam-6123	52	26	n	n	CCONJ
ejpam-6123	52	27	,	,	PUNCT
ejpam-6123	52	28	if	if	SCONJ
ejpam-6123	52	29	|m−	|m−	PROPN
ejpam-6123	52	30	n|	n|	X
ejpam-6123	52	31	≥	≥	NOUN
ejpam-6123	52	32	2	2	NUM
ejpam-6123	52	33	,	,	PUNCT
ejpam-6123	52	34	and	and	CCONJ
ejpam-6123	52	35	m	m	PROPN
ejpam-6123	52	36	,	,	PUNCT
ejpam-6123	52	37	n	n	PRON
ejpam-6123	52	38	≥	≥	NOUN
ejpam-6123	52	39	2	2	NUM
ejpam-6123	52	40	.	.	NOUN
ejpam-6123	52	41	2	2	NUM
ejpam-6123	52	42	.	.	NOUN
ejpam-6123	52	43	results	result	VERB
ejpam-6123	52	44	the	the	DET
ejpam-6123	52	45	following	follow	VERB
ejpam-6123	52	46	result	result	NOUN
ejpam-6123	52	47	theorem	theorem	VERB
ejpam-6123	52	48	2	2	NUM
ejpam-6123	52	49	is	be	AUX
ejpam-6123	52	50	a	a	DET
ejpam-6123	52	51	correction	correction	NOUN
ejpam-6123	52	52	of	of	ADP
ejpam-6123	52	53	theorem	theorem	ADJ
ejpam-6123	52	54	1	1	NUM
ejpam-6123	52	55	(	(	PUNCT
ejpam-6123	52	56	iv	iv	X
ejpam-6123	52	57	)	)	PUNCT
ejpam-6123	52	58	found	find	VERB
ejpam-6123	52	59	in	in	ADP
ejpam-6123	52	60	[	[	X
ejpam-6123	52	61	15	15	NUM
ejpam-6123	52	62	]	]	PUNCT
ejpam-6123	52	63	.	.	PUNCT
ejpam-6123	53	1	theorem	theorem	NOUN
ejpam-6123	53	2	2	2	NUM
ejpam-6123	53	3	.	.	X
ejpam-6123	53	4	for	for	ADP
ejpam-6123	53	5	any	any	DET
ejpam-6123	53	6	complete	complete	ADJ
ejpam-6123	53	7	bipartite	bipartite	PROPN
ejpam-6123	53	8	km	km	PROPN
ejpam-6123	53	9	,	,	PUNCT
ejpam-6123	53	10	n	n	CCONJ
ejpam-6123	53	11	with	with	ADP
ejpam-6123	53	12	m	m	PROPN
ejpam-6123	53	13	,	,	PUNCT
ejpam-6123	53	14	n	n	PRON
ejpam-6123	53	15	≥	≥	NOUN
ejpam-6123	53	16	1	1	NUM
ejpam-6123	53	17	,	,	PUNCT
ejpam-6123	53	18	γce(km	γce(km	NUM
ejpam-6123	53	19	,	,	PUNCT
ejpam-6123	53	20	n	n	CCONJ
ejpam-6123	53	21	)	)	PUNCT
ejpam-6123	54	1	=	=	NOUN
ejpam-6123	54	2	{	{	PUNCT
ejpam-6123	54	3	2	2	NUM
ejpam-6123	54	4	,	,	PUNCT
ejpam-6123	54	5	if	if	SCONJ
ejpam-6123	54	6	|m−	|m−	PROPN
ejpam-6123	54	7	n|	n|	NOUN
ejpam-6123	54	8	≤	≤	PROPN
ejpam-6123	54	9	1	1	NUM
ejpam-6123	54	10	m+	m+	NUM
ejpam-6123	54	11	n	n	CCONJ
ejpam-6123	54	12	,	,	PUNCT
ejpam-6123	54	13	if	if	SCONJ
ejpam-6123	54	14	|m−	|m−	PROPN
ejpam-6123	54	15	n|	n|	X
ejpam-6123	54	16	≥	≥	NOUN
ejpam-6123	54	17	2	2	NUM
ejpam-6123	54	18	.	.	PUNCT
ejpam-6123	55	1	h.	h.	PROPN
ejpam-6123	55	2	nuenay	nuenay	PROPN
ejpam-6123	55	3	-	-	PUNCT
ejpam-6123	55	4	maglanquel	maglanquel	PROPN
ejpam-6123	55	5	/	/	SYM
ejpam-6123	55	6	eur	eur	PROPN
ejpam-6123	55	7	.	.	PUNCT
ejpam-6123	56	1	j.	j.	PROPN
ejpam-6123	56	2	pure	pure	PROPN
ejpam-6123	56	3	appl	appl	PROPN
ejpam-6123	56	4	.	.	PROPN
ejpam-6123	56	5	math	math	PROPN
ejpam-6123	56	6	,	,	PUNCT
ejpam-6123	56	7	18	18	NUM
ejpam-6123	56	8	(	(	PUNCT
ejpam-6123	56	9	3	3	NUM
ejpam-6123	56	10	)	)	PUNCT
ejpam-6123	56	11	(	(	PUNCT
ejpam-6123	56	12	2025	2025	NUM
ejpam-6123	56	13	)	)	PUNCT
ejpam-6123	56	14	,	,	PUNCT
ejpam-6123	56	15	6123	6123	NUM
ejpam-6123	56	16	4	4	NUM
ejpam-6123	56	17	of	of	ADP
ejpam-6123	56	18	14	14	NUM
ejpam-6123	56	19	proof	proof	NOUN
ejpam-6123	56	20	.	.	PUNCT
ejpam-6123	57	1	let	let	VERB
ejpam-6123	57	2	km	km	PROPN
ejpam-6123	57	3	,	,	PUNCT
ejpam-6123	57	4	n	n	PRON
ejpam-6123	57	5	be	be	VERB
ejpam-6123	57	6	a	a	DET
ejpam-6123	57	7	complete	complete	ADJ
ejpam-6123	57	8	bipartite	bipartite	NOUN
ejpam-6123	57	9	graph	graph	NOUN
ejpam-6123	57	10	with	with	ADP
ejpam-6123	57	11	m	m	PROPN
ejpam-6123	57	12	vertices	vertex	NOUN
ejpam-6123	57	13	in	in	ADP
ejpam-6123	57	14	one	one	NUM
ejpam-6123	57	15	partition	partition	NOUN
ejpam-6123	57	16	say	say	VERB
ejpam-6123	57	17	a	a	DET
ejpam-6123	57	18	and	and	CCONJ
ejpam-6123	57	19	n	n	PRON
ejpam-6123	57	20	vertices	vertex	NOUN
ejpam-6123	57	21	in	in	ADP
ejpam-6123	57	22	another	another	DET
ejpam-6123	57	23	partition	partition	NOUN
ejpam-6123	57	24	say	say	VERB
ejpam-6123	57	25	b.	b.	PROPN
ejpam-6123	57	26	then	then	ADV
ejpam-6123	57	27	,	,	PUNCT
ejpam-6123	57	28	deg(u	deg(u	PROPN
ejpam-6123	57	29	)	)	PUNCT
ejpam-6123	57	30	=	=	NUM
ejpam-6123	57	31	{	{	PUNCT
ejpam-6123	57	32	n	n	CCONJ
ejpam-6123	57	33	,	,	PUNCT
ejpam-6123	57	34	if	if	SCONJ
ejpam-6123	57	35	u	u	PROPN
ejpam-6123	57	36	∈	∈	VERB
ejpam-6123	57	37	a	a	DET
ejpam-6123	57	38	m	m	PROPN
ejpam-6123	57	39	,	,	PUNCT
ejpam-6123	57	40	ifu	ifu	VERB
ejpam-6123	57	41	∈	∈	PROPN
ejpam-6123	57	42	b.	b.	PROPN
ejpam-6123	58	1	if	if	SCONJ
ejpam-6123	58	2	|m	|m	NOUN
ejpam-6123	58	3	−	−	PROPN
ejpam-6123	58	4	n|	n|	NOUN
ejpam-6123	58	5	≤	≤	NOUN
ejpam-6123	58	6	1	1	NUM
ejpam-6123	58	7	,	,	PUNCT
ejpam-6123	58	8	then	then	ADV
ejpam-6123	58	9	any	any	DET
ejpam-6123	58	10	set	set	NOUN
ejpam-6123	58	11	{	{	PUNCT
ejpam-6123	58	12	x	x	NOUN
ejpam-6123	58	13	,	,	PUNCT
ejpam-6123	58	14	y	y	NOUN
ejpam-6123	58	15	}	}	PUNCT
ejpam-6123	58	16	with	with	ADP
ejpam-6123	58	17	x	x	PROPN
ejpam-6123	58	18	∈	∈	PROPN
ejpam-6123	58	19	a	a	PRON
ejpam-6123	58	20	and	and	CCONJ
ejpam-6123	58	21	y	y	PROPN
ejpam-6123	58	22	∈	∈	PROPN
ejpam-6123	58	23	b	b	PROPN
ejpam-6123	58	24	is	be	AUX
ejpam-6123	58	25	a	a	DET
ejpam-6123	58	26	connected	connect	VERB
ejpam-6123	58	27	equitable	equitable	ADJ
ejpam-6123	58	28	dominating	dominating	NOUN
ejpam-6123	58	29	set	set	VERB
ejpam-6123	58	30	in	in	ADP
ejpam-6123	58	31	km	km	PROPN
ejpam-6123	58	32	,	,	PUNCT
ejpam-6123	58	33	n.	n.	PROPN
ejpam-6123	58	34	thus	thus	ADV
ejpam-6123	58	35	,	,	PUNCT
ejpam-6123	58	36	γ	γ	PROPN
ejpam-6123	58	37	ce(km	ce(km	PROPN
ejpam-6123	58	38	,	,	PUNCT
ejpam-6123	58	39	n	n	CCONJ
ejpam-6123	58	40	)	)	PUNCT
ejpam-6123	58	41	=	=	SYM
ejpam-6123	58	42	2	2	X
ejpam-6123	58	43	.	.	PUNCT
ejpam-6123	58	44	suppose	suppose	VERB
ejpam-6123	58	45	that	that	SCONJ
ejpam-6123	58	46	|m	|m	NOUN
ejpam-6123	58	47	−	−	PROPN
ejpam-6123	58	48	n|	n|	X
ejpam-6123	58	49	≥	≥	NOUN
ejpam-6123	58	50	2	2	NUM
ejpam-6123	58	51	.	.	PUNCT
ejpam-6123	59	1	let	let	VERB
ejpam-6123	59	2	s	s	PRON
ejpam-6123	59	3	be	be	AUX
ejpam-6123	59	4	a	a	DET
ejpam-6123	59	5	minimum	minimum	NOUN
ejpam-6123	59	6	connected	connect	VERB
ejpam-6123	59	7	equitable	equitable	ADJ
ejpam-6123	59	8	dominating	dominating	NOUN
ejpam-6123	59	9	set	set	VERB
ejpam-6123	59	10	in	in	ADP
ejpam-6123	59	11	km	km	PROPN
ejpam-6123	59	12	,	,	PUNCT
ejpam-6123	59	13	n	n	PUNCT
ejpam-6123	59	14	and	and	CCONJ
ejpam-6123	59	15	suppose	suppose	VERB
ejpam-6123	59	16	that	that	SCONJ
ejpam-6123	59	17	|s|	|s|	VERB
ejpam-6123	59	18	<	<	X
ejpam-6123	59	19	m	m	PROPN
ejpam-6123	59	20	+	+	X
ejpam-6123	59	21	n.	n.	NOUN
ejpam-6123	59	22	then	then	ADV
ejpam-6123	59	23	,	,	PUNCT
ejpam-6123	59	24	there	there	PRON
ejpam-6123	59	25	exists	exist	VERB
ejpam-6123	59	26	u	u	PROPN
ejpam-6123	59	27	∈	∈	PROPN
ejpam-6123	59	28	v	v	NOUN
ejpam-6123	59	29	which	which	PRON
ejpam-6123	59	30	is	be	AUX
ejpam-6123	59	31	not	not	PART
ejpam-6123	59	32	in	in	ADP
ejpam-6123	59	33	s.	s.	PROPN
ejpam-6123	59	34	wlog	wlog	PROPN
ejpam-6123	59	35	,	,	PUNCT
ejpam-6123	59	36	let	let	VERB
ejpam-6123	59	37	u	u	PRON
ejpam-6123	59	38	∈	∈	PROPN
ejpam-6123	59	39	b.	b.	PROPN
ejpam-6123	59	40	then	then	ADV
ejpam-6123	59	41	,	,	PUNCT
ejpam-6123	59	42	degg(u	degg(u	PROPN
ejpam-6123	59	43	)	)	PUNCT
ejpam-6123	59	44	=	=	SYM
ejpam-6123	59	45	m.	m.	NOUN
ejpam-6123	59	46	since	since	SCONJ
ejpam-6123	59	47	s	s	PROPN
ejpam-6123	59	48	is	be	AUX
ejpam-6123	59	49	a	a	DET
ejpam-6123	59	50	connected	connect	VERB
ejpam-6123	59	51	equitable	equitable	ADJ
ejpam-6123	59	52	dominating	dominating	NOUN
ejpam-6123	59	53	set	set	VERB
ejpam-6123	59	54	in	in	ADP
ejpam-6123	59	55	km	km	PROPN
ejpam-6123	59	56	,	,	PUNCT
ejpam-6123	59	57	n	n	CCONJ
ejpam-6123	59	58	,	,	PUNCT
ejpam-6123	59	59	there	there	PRON
ejpam-6123	59	60	exists	exist	VERB
ejpam-6123	59	61	v	v	ADP
ejpam-6123	59	62	∈	∈	PROPN
ejpam-6123	59	63	s	s	VERB
ejpam-6123	59	64	such	such	ADJ
ejpam-6123	59	65	that	that	SCONJ
ejpam-6123	59	66	u	u	NOUN
ejpam-6123	59	67	is	be	AUX
ejpam-6123	59	68	adjacent	adjacent	ADJ
ejpam-6123	59	69	with	with	ADP
ejpam-6123	59	70	v	v	NOUN
ejpam-6123	59	71	and	and	CCONJ
ejpam-6123	59	72	|degg(u	|degg(u	ADJ
ejpam-6123	59	73	)	)	PUNCT
ejpam-6123	59	74	−	−	NOUN
ejpam-6123	60	1	degg(v)|	degg(v)|	NOUN
ejpam-6123	60	2	≤	≤	NOUN
ejpam-6123	60	3	1	1	NUM
ejpam-6123	60	4	.	.	PUNCT
ejpam-6123	61	1	clearly	clearly	ADV
ejpam-6123	61	2	,	,	PUNCT
ejpam-6123	61	3	v	v	PROPN
ejpam-6123	61	4	∈	∈	PROPN
ejpam-6123	61	5	a.	a.	NOUN
ejpam-6123	61	6	therefore	therefore	ADV
ejpam-6123	61	7	degg(v	degg(v	PROPN
ejpam-6123	61	8	)	)	PUNCT
ejpam-6123	61	9	=	=	VERB
ejpam-6123	62	1	n.	n.	PROPN
ejpam-6123	62	2	thus	thus	ADV
ejpam-6123	62	3	,	,	PUNCT
ejpam-6123	62	4	|degg(u	|degg(u	ADJ
ejpam-6123	62	5	)	)	PUNCT
ejpam-6123	62	6	−	−	NOUN
ejpam-6123	62	7	degg(v)|	degg(v)|	NOUN
ejpam-6123	62	8	=	=	NOUN
ejpam-6123	62	9	|m	|m	NOUN
ejpam-6123	62	10	−	−	PROPN
ejpam-6123	62	11	n|	n|	X
ejpam-6123	62	12	≥	≥	NOUN
ejpam-6123	62	13	2	2	NUM
ejpam-6123	62	14	,	,	PUNCT
ejpam-6123	62	15	a	a	DET
ejpam-6123	62	16	contradiction	contradiction	NOUN
ejpam-6123	62	17	.	.	PUNCT
ejpam-6123	63	1	therefore	therefore	ADV
ejpam-6123	63	2	,	,	PUNCT
ejpam-6123	63	3	|s|	|s|	PROPN
ejpam-6123	63	4	=	=	SYM
ejpam-6123	63	5	m	m	PROPN
ejpam-6123	63	6	+	+	X
ejpam-6123	63	7	n.	n.	NOUN
ejpam-6123	63	8	consequently	consequently	ADV
ejpam-6123	63	9	,	,	PUNCT
ejpam-6123	63	10	γce(km	γce(km	PROPN
ejpam-6123	63	11	,	,	PUNCT
ejpam-6123	63	12	n	n	CCONJ
ejpam-6123	63	13	)	)	PUNCT
ejpam-6123	63	14	=	=	SYM
ejpam-6123	63	15	m+	m+	NUM
ejpam-6123	63	16	n	n	PROPN
ejpam-6123	63	17	if	if	SCONJ
ejpam-6123	63	18	|m−	|m−	PROPN
ejpam-6123	63	19	n|	n|	X
ejpam-6123	63	20	≥	≥	NOUN
ejpam-6123	63	21	2	2	NUM
ejpam-6123	63	22	.	.	PUNCT
ejpam-6123	63	23	theorem	theorem	NOUN
ejpam-6123	63	24	3	3	NUM
ejpam-6123	63	25	.	.	X
ejpam-6123	63	26	for	for	ADP
ejpam-6123	63	27	a	a	DET
ejpam-6123	63	28	fan	fan	NOUN
ejpam-6123	63	29	graph	graph	NOUN
ejpam-6123	63	30	fn	fn	NOUN
ejpam-6123	63	31	with	with	ADP
ejpam-6123	63	32	n	n	NOUN
ejpam-6123	63	33	vertices	vertex	NOUN
ejpam-6123	63	34	on	on	ADP
ejpam-6123	63	35	a	a	DET
ejpam-6123	63	36	path	path	NOUN
ejpam-6123	63	37	and	and	CCONJ
ejpam-6123	63	38	a	a	DET
ejpam-6123	63	39	single	single	ADJ
ejpam-6123	63	40	vertex	vertex	NOUN
ejpam-6123	63	41	u	u	NOUN
ejpam-6123	63	42	connected	connect	VERB
ejpam-6123	63	43	to	to	ADP
ejpam-6123	63	44	each	each	DET
ejpam-6123	63	45	vertex	vertex	NOUN
ejpam-6123	63	46	in	in	ADP
ejpam-6123	63	47	pn	pn	PROPN
ejpam-6123	63	48	,	,	PUNCT
ejpam-6123	63	49	γce(fn	γce(fn	PROPN
ejpam-6123	63	50	)	)	PUNCT
ejpam-6123	63	51	=	=	PUNCT
ejpam-6123	64	1			PUNCT
ejpam-6123	64	2	1	1	NUM
ejpam-6123	64	3	,	,	PUNCT
ejpam-6123	64	4	if	if	SCONJ
ejpam-6123	64	5	n	n	PRON
ejpam-6123	64	6	=	=	NOUN
ejpam-6123	64	7	:	:	PUNCT
ejpam-6123	64	8	2	2	NUM
ejpam-6123	64	9	,	,	PUNCT
ejpam-6123	64	10	3	3	NUM
ejpam-6123	64	11	2	2	NUM
ejpam-6123	64	12	,	,	PUNCT
ejpam-6123	64	13	if	if	SCONJ
ejpam-6123	64	14	n	n	NOUN
ejpam-6123	64	15	=	=	SYM
ejpam-6123	64	16	4⌈	4⌈	NUM
ejpam-6123	64	17	n	n	PRON
ejpam-6123	64	18	3	3	NUM
ejpam-6123	64	19	⌉	⌉	NOUN
ejpam-6123	64	20	+	+	ADJ
ejpam-6123	64	21	1	1	NUM
ejpam-6123	64	22	,	,	PUNCT
ejpam-6123	64	23	if	if	SCONJ
ejpam-6123	64	24	n	n	PRON
ejpam-6123	64	25	≥	≥	NOUN
ejpam-6123	64	26	5	5	NUM
ejpam-6123	64	27	.	.	PUNCT
ejpam-6123	65	1	proof	proof	NOUN
ejpam-6123	65	2	.	.	PUNCT
ejpam-6123	66	1	let	let	VERB
ejpam-6123	66	2	g	g	NOUN
ejpam-6123	66	3	=	=	PUNCT
ejpam-6123	66	4	fn	fn	VERB
ejpam-6123	66	5	be	be	AUX
ejpam-6123	66	6	a	a	DET
ejpam-6123	66	7	fan	fan	NOUN
ejpam-6123	66	8	with	with	ADP
ejpam-6123	66	9	n	n	NOUN
ejpam-6123	66	10	vertices	vertex	NOUN
ejpam-6123	66	11	on	on	ADP
ejpam-6123	66	12	a	a	DET
ejpam-6123	66	13	path	path	NOUN
ejpam-6123	66	14	and	and	CCONJ
ejpam-6123	66	15	a	a	DET
ejpam-6123	66	16	single	single	ADJ
ejpam-6123	66	17	vertex	vertex	NOUN
ejpam-6123	66	18	u	u	NOUN
ejpam-6123	66	19	connected	connect	VERB
ejpam-6123	66	20	to	to	ADP
ejpam-6123	66	21	each	each	DET
ejpam-6123	66	22	vertex	vertex	NOUN
ejpam-6123	66	23	in	in	ADP
ejpam-6123	66	24	pn	pn	PROPN
ejpam-6123	66	25	.	.	PUNCT
ejpam-6123	67	1	let	let	VERB
ejpam-6123	67	2	v	v	X
ejpam-6123	67	3	(	(	PUNCT
ejpam-6123	67	4	fn	fn	NOUN
ejpam-6123	67	5	)	)	PUNCT
ejpam-6123	67	6	=	=	PRON
ejpam-6123	67	7	{	{	PUNCT
ejpam-6123	67	8	u	u	NOUN
ejpam-6123	67	9	,	,	PUNCT
ejpam-6123	67	10	v1	v1	NOUN
ejpam-6123	67	11	,	,	PUNCT
ejpam-6123	67	12	v2	v2	NOUN
ejpam-6123	67	13	,	,	PUNCT
ejpam-6123	67	14	.	.	PUNCT
ejpam-6123	67	15	.	.	PUNCT
ejpam-6123	68	1	.	.	PUNCT
ejpam-6123	69	1	,	,	PUNCT
ejpam-6123	69	2	vn	vn	PROPN
ejpam-6123	69	3	}	}	PUNCT
ejpam-6123	69	4	where	where	SCONJ
ejpam-6123	69	5	u	u	NOUN
ejpam-6123	69	6	is	be	AUX
ejpam-6123	69	7	adjacent	adjacent	ADJ
ejpam-6123	69	8	to	to	ADP
ejpam-6123	69	9	every	every	DET
ejpam-6123	69	10	vertex	vertex	NOUN
ejpam-6123	69	11	vi	vi	PROPN
ejpam-6123	69	12	(	(	PUNCT
ejpam-6123	69	13	1	1	NUM
ejpam-6123	69	14	≤	≤	NUM
ejpam-6123	69	15	i	i	NOUN
ejpam-6123	69	16	≤	≤	NOUN
ejpam-6123	69	17	n	n	CCONJ
ejpam-6123	69	18	)	)	PUNCT
ejpam-6123	69	19	on	on	ADP
ejpam-6123	69	20	path	path	NOUN
ejpam-6123	69	21	pn	pn	PROPN
ejpam-6123	69	22	.	.	PROPN
ejpam-6123	70	1	when	when	SCONJ
ejpam-6123	70	2	n	n	X
ejpam-6123	70	3	=	=	SYM
ejpam-6123	70	4	2	2	NUM
ejpam-6123	70	5	,	,	PUNCT
ejpam-6123	70	6	{	{	PUNCT
ejpam-6123	70	7	u	u	NOUN
ejpam-6123	70	8	}	}	PUNCT
ejpam-6123	70	9	,	,	PUNCT
ejpam-6123	70	10	{	{	PUNCT
ejpam-6123	70	11	v1	v1	NOUN
ejpam-6123	70	12	}	}	PUNCT
ejpam-6123	70	13	and	and	CCONJ
ejpam-6123	70	14	{	{	PUNCT
ejpam-6123	70	15	v2	v2	NOUN
ejpam-6123	70	16	}	}	PUNCT
ejpam-6123	70	17	are	be	AUX
ejpam-6123	70	18	minimal	minimal	ADJ
ejpam-6123	70	19	connected	connect	VERB
ejpam-6123	70	20	equitable	equitable	ADJ
ejpam-6123	70	21	dominating	dominating	NOUN
ejpam-6123	70	22	sets	set	NOUN
ejpam-6123	70	23	in	in	ADP
ejpam-6123	70	24	g.	g.	PROPN
ejpam-6123	70	25	thus	thus	ADV
ejpam-6123	70	26	,	,	PUNCT
ejpam-6123	70	27	γce(fn	γce(fn	PROPN
ejpam-6123	70	28	)	)	PUNCT
ejpam-6123	71	1	=	=	SYM
ejpam-6123	71	2	1	1	X
ejpam-6123	71	3	.	.	PUNCT
ejpam-6123	71	4	when	when	SCONJ
ejpam-6123	71	5	n	n	X
ejpam-6123	71	6	=	=	SYM
ejpam-6123	71	7	3	3	NUM
ejpam-6123	71	8	,	,	PUNCT
ejpam-6123	71	9	since	since	SCONJ
ejpam-6123	71	10	degg(u	degg(u	PROPN
ejpam-6123	71	11	)	)	PUNCT
ejpam-6123	71	12	=	=	NOUN
ejpam-6123	71	13	degg(v2	degg(v2	NOUN
ejpam-6123	71	14	)	)	PUNCT
ejpam-6123	71	15	=	=	SYM
ejpam-6123	71	16	3	3	NUM
ejpam-6123	71	17	and	and	CCONJ
ejpam-6123	71	18	degg(v1	degg(v1	NOUN
ejpam-6123	71	19	)	)	PUNCT
ejpam-6123	71	20	=	=	SYM
ejpam-6123	71	21	degg(v3	degg(v3	NOUN
ejpam-6123	71	22	)	)	PUNCT
ejpam-6123	71	23	=	=	SYM
ejpam-6123	72	1	2	2	NUM
ejpam-6123	72	2	,	,	PUNCT
ejpam-6123	72	3	the	the	DET
ejpam-6123	72	4	sets	set	NOUN
ejpam-6123	72	5	{	{	PUNCT
ejpam-6123	72	6	u	u	NOUN
ejpam-6123	72	7	}	}	PUNCT
ejpam-6123	72	8	and	and	CCONJ
ejpam-6123	72	9	{	{	PUNCT
ejpam-6123	72	10	v2	v2	NOUN
ejpam-6123	72	11	}	}	PUNCT
ejpam-6123	72	12	are	be	AUX
ejpam-6123	72	13	the	the	DET
ejpam-6123	72	14	only	only	ADJ
ejpam-6123	72	15	minimal	minimal	ADJ
ejpam-6123	72	16	connected	connect	VERB
ejpam-6123	72	17	equitable	equitable	ADJ
ejpam-6123	72	18	dominating	dominating	NOUN
ejpam-6123	72	19	sets	set	NOUN
ejpam-6123	72	20	in	in	ADP
ejpam-6123	72	21	g.	g.	PROPN
ejpam-6123	72	22	thus	thus	ADV
ejpam-6123	72	23	,	,	PUNCT
ejpam-6123	72	24	γce(fn	γce(fn	PROPN
ejpam-6123	72	25	)	)	PUNCT
ejpam-6123	72	26	=	=	SYM
ejpam-6123	73	1	1	1	X
ejpam-6123	73	2	.	.	X
ejpam-6123	73	3	for	for	ADP
ejpam-6123	73	4	n	n	NOUN
ejpam-6123	73	5	=	=	SYM
ejpam-6123	73	6	4	4	NUM
ejpam-6123	73	7	,	,	PUNCT
ejpam-6123	73	8	since	since	SCONJ
ejpam-6123	73	9	the	the	DET
ejpam-6123	73	10	deg(vi	deg(vi	NOUN
ejpam-6123	73	11	)	)	PUNCT
ejpam-6123	73	12	of	of	ADP
ejpam-6123	73	13	any	any	DET
ejpam-6123	73	14	vertices	vertex	NOUN
ejpam-6123	73	15	vi	vi	X
ejpam-6123	73	16	lying	lie	VERB
ejpam-6123	73	17	on	on	ADP
ejpam-6123	73	18	p4	p4	ADJ
ejpam-6123	73	19	is	be	AUX
ejpam-6123	73	20	either	either	CCONJ
ejpam-6123	73	21	2	2	NUM
ejpam-6123	73	22	or	or	CCONJ
ejpam-6123	73	23	3	3	NUM
ejpam-6123	73	24	and	and	CCONJ
ejpam-6123	73	25	degg(u	degg(u	NUM
ejpam-6123	73	26	)	)	PUNCT
ejpam-6123	73	27	=	=	SYM
ejpam-6123	73	28	4	4	NUM
ejpam-6123	73	29	,	,	PUNCT
ejpam-6123	73	30	any	any	DET
ejpam-6123	73	31	connected	connected	ADJ
ejpam-6123	73	32	dominating	dominating	NOUN
ejpam-6123	73	33	set	set	VERB
ejpam-6123	73	34	in	in	ADP
ejpam-6123	73	35	p4	p4	NOUN
ejpam-6123	73	36	is	be	AUX
ejpam-6123	73	37	equitable	equitable	ADJ
ejpam-6123	73	38	in	in	ADP
ejpam-6123	73	39	f4	f4	PROPN
ejpam-6123	73	40	.	.	PUNCT
ejpam-6123	74	1	thus	thus	ADV
ejpam-6123	74	2	,	,	PUNCT
ejpam-6123	74	3	γc(p4	γc(p4	X
ejpam-6123	74	4	)	)	PUNCT
ejpam-6123	74	5	=	=	SYM
ejpam-6123	74	6	⌈	⌈	PROPN
ejpam-6123	74	7	n	n	CCONJ
ejpam-6123	74	8	3	3	NUM
ejpam-6123	74	9	⌉	⌉	NOUN
ejpam-6123	74	10	=	=	SYM
ejpam-6123	74	11	2	2	NUM
ejpam-6123	74	12	=	=	SYM
ejpam-6123	74	13	γce(f4	γce(f4	NOUN
ejpam-6123	74	14	)	)	PUNCT
ejpam-6123	74	15	.	.	PUNCT
ejpam-6123	75	1	suppose	suppose	VERB
ejpam-6123	75	2	that	that	SCONJ
ejpam-6123	75	3	n	n	PROPN
ejpam-6123	75	4	≥	≥	NUM
ejpam-6123	75	5	5	5	NUM
ejpam-6123	75	6	.	.	PUNCT
ejpam-6123	75	7	then	then	ADV
ejpam-6123	75	8	deg(u	deg(u	PROPN
ejpam-6123	75	9	)	)	PUNCT
ejpam-6123	75	10	=	=	SYM
ejpam-6123	75	11	n	n	NOUN
ejpam-6123	75	12	and	and	CCONJ
ejpam-6123	75	13	degg(vi	degg(vi	PROPN
ejpam-6123	75	14	)	)	PUNCT
ejpam-6123	75	15	=	=	NOUN
ejpam-6123	75	16	{	{	PUNCT
ejpam-6123	75	17	2	2	NUM
ejpam-6123	75	18	,	,	PUNCT
ejpam-6123	75	19	if	if	SCONJ
ejpam-6123	75	20	i	i	PRON
ejpam-6123	75	21	=	=	NOUN
ejpam-6123	75	22	:	:	PUNCT
ejpam-6123	75	23	1	1	NUM
ejpam-6123	75	24	,	,	PUNCT
ejpam-6123	75	25	n	n	PRON
ejpam-6123	75	26	3	3	NUM
ejpam-6123	75	27	,	,	PUNCT
ejpam-6123	75	28	if	if	SCONJ
ejpam-6123	75	29	2	2	NUM
ejpam-6123	75	30	≤	≤	NOUN
ejpam-6123	75	31	i	i	PRON
ejpam-6123	75	32	<	<	X
ejpam-6123	75	33	n.	n.	PROPN
ejpam-6123	75	34	thus	thus	ADV
ejpam-6123	75	35	,	,	PUNCT
ejpam-6123	75	36	any	any	DET
ejpam-6123	75	37	dominating	dominating	NOUN
ejpam-6123	75	38	set	set	VERB
ejpam-6123	75	39	in	in	ADP
ejpam-6123	75	40	pn	pn	PROPN
ejpam-6123	75	41	is	be	AUX
ejpam-6123	75	42	not	not	PART
ejpam-6123	75	43	equitable	equitable	ADJ
ejpam-6123	75	44	in	in	ADP
ejpam-6123	75	45	fn	fn	NOUN
ejpam-6123	75	46	since	since	SCONJ
ejpam-6123	75	47	|degg(u	|degg(u	PROPN
ejpam-6123	75	48	)	)	PUNCT
ejpam-6123	75	49	−	−	NOUN
ejpam-6123	76	1	degg(vi)|	degg(vi)|	INTJ
ejpam-6123	76	2	>	>	X
ejpam-6123	76	3	2	2	NUM
ejpam-6123	76	4	for	for	ADP
ejpam-6123	76	5	all	all	DET
ejpam-6123	76	6	i	i	PRON
ejpam-6123	76	7	=	=	NOUN
ejpam-6123	76	8	1	1	NUM
ejpam-6123	76	9	,	,	PUNCT
ejpam-6123	76	10	2	2	NUM
ejpam-6123	76	11	,	,	PUNCT
ejpam-6123	76	12	.	.	PUNCT
ejpam-6123	76	13	.	.	PUNCT
ejpam-6123	76	14	.	.	PUNCT
ejpam-6123	77	1	,	,	PUNCT
ejpam-6123	77	2	n.	n.	PROPN
ejpam-6123	77	3	now	now	ADV
ejpam-6123	77	4	,	,	PUNCT
ejpam-6123	77	5	consider	consider	VERB
ejpam-6123	77	6	s	s	PRON
ejpam-6123	77	7	=	=	NOUN
ejpam-6123	77	8	d	d	X
ejpam-6123	77	9	∪	∪	X
ejpam-6123	77	10	{	{	PUNCT
ejpam-6123	77	11	u	u	NOUN
ejpam-6123	77	12	}	}	PUNCT
ejpam-6123	77	13	where	where	SCONJ
ejpam-6123	77	14	d	d	NOUN
ejpam-6123	77	15	is	be	AUX
ejpam-6123	77	16	a	a	DET
ejpam-6123	77	17	γeset	γeset	NOUN
ejpam-6123	77	18	in	in	ADP
ejpam-6123	77	19	pn	pn	PROPN
ejpam-6123	77	20	.	.	PUNCT
ejpam-6123	77	21	clearly	clearly	ADV
ejpam-6123	77	22	,	,	PUNCT
ejpam-6123	77	23	s	s	VERB
ejpam-6123	77	24	is	be	AUX
ejpam-6123	77	25	a	a	DET
ejpam-6123	77	26	connected	connect	VERB
ejpam-6123	77	27	equitable	equitable	ADJ
ejpam-6123	77	28	dominating	dominating	NOUN
ejpam-6123	77	29	set	set	VERB
ejpam-6123	77	30	in	in	ADP
ejpam-6123	77	31	fn	fn	NOUN
ejpam-6123	77	32	of	of	ADP
ejpam-6123	77	33	minimum	minimum	ADJ
ejpam-6123	77	34	cardinality	cardinality	NOUN
ejpam-6123	77	35	.	.	PUNCT
ejpam-6123	78	1	therefore	therefore	ADV
ejpam-6123	78	2	,	,	PUNCT
ejpam-6123	78	3	γce(fn	γce(fn	NOUN
ejpam-6123	78	4	)	)	PUNCT
ejpam-6123	78	5	=	=	SYM
ejpam-6123	78	6	γe(pn	γe(pn	PROPN
ejpam-6123	78	7	)	)	PUNCT
ejpam-6123	79	1	+	+	CCONJ
ejpam-6123	79	2	1	1	NUM
ejpam-6123	79	3	=	=	SYM
ejpam-6123	79	4	⌈	⌈	NOUN
ejpam-6123	79	5	n	n	CCONJ
ejpam-6123	79	6	3	3	NUM
ejpam-6123	79	7	⌉	⌉	NOUN
ejpam-6123	79	8	+	+	CCONJ
ejpam-6123	79	9	1	1	X
ejpam-6123	79	10	.	.	X
ejpam-6123	79	11	theorem	theorem	VERB
ejpam-6123	79	12	4	4	NUM
ejpam-6123	79	13	.	.	X
ejpam-6123	80	1	for	for	ADP
ejpam-6123	80	2	any	any	DET
ejpam-6123	80	3	double	double	ADJ
ejpam-6123	80	4	star	star	NOUN
ejpam-6123	80	5	graph	graph	NOUN
ejpam-6123	80	6	sr	sr	PROPN
ejpam-6123	80	7	,	,	PUNCT
ejpam-6123	80	8	s	s	PROPN
ejpam-6123	80	9	,	,	PUNCT
ejpam-6123	80	10	where	where	SCONJ
ejpam-6123	80	11	r	r	NOUN
ejpam-6123	80	12	,	,	PUNCT
ejpam-6123	80	13	s	s	PART
ejpam-6123	80	14	≥	≥	NOUN
ejpam-6123	80	15	1	1	NUM
ejpam-6123	80	16	,	,	PUNCT
ejpam-6123	80	17	γce(sr	γce(sr	NUM
ejpam-6123	80	18	,	,	PUNCT
ejpam-6123	80	19	s	s	X
ejpam-6123	80	20	)	)	PUNCT
ejpam-6123	80	21	=	=	SYM
ejpam-6123	80	22			NUM
ejpam-6123	80	23	2	2	NUM
ejpam-6123	80	24	,	,	PUNCT
ejpam-6123	80	25	if	if	SCONJ
ejpam-6123	80	26	r	r	NOUN
ejpam-6123	80	27	=	=	SYM
ejpam-6123	80	28	s	s	NOUN
ejpam-6123	80	29	=	=	SYM
ejpam-6123	80	30	1	1	NUM
ejpam-6123	80	31	s+	s+	NUM
ejpam-6123	80	32	2	2	NUM
ejpam-6123	80	33	,	,	PUNCT
ejpam-6123	80	34	if	if	SCONJ
ejpam-6123	80	35	r	r	NOUN
ejpam-6123	80	36	=	=	SYM
ejpam-6123	80	37	1	1	NUM
ejpam-6123	80	38	and	and	CCONJ
ejpam-6123	80	39	s	s	X
ejpam-6123	80	40	≥	≥	NOUN
ejpam-6123	80	41	2	2	NUM
ejpam-6123	80	42	r	r	NOUN
ejpam-6123	80	43	+	+	NOUN
ejpam-6123	80	44	2	2	NUM
ejpam-6123	80	45	,	,	PUNCT
ejpam-6123	80	46	if	if	SCONJ
ejpam-6123	80	47	s	s	VERB
ejpam-6123	80	48	=	=	SYM
ejpam-6123	80	49	1	1	NUM
ejpam-6123	80	50	and	and	CCONJ
ejpam-6123	80	51	r	r	NOUN
ejpam-6123	80	52	≥	≥	NUM
ejpam-6123	80	53	2	2	NUM
ejpam-6123	80	54	r	r	NOUN
ejpam-6123	80	55	+	+	CCONJ
ejpam-6123	80	56	s+	s+	NUM
ejpam-6123	80	57	2	2	NUM
ejpam-6123	80	58	,	,	PUNCT
ejpam-6123	80	59	if	if	SCONJ
ejpam-6123	80	60	r	r	NOUN
ejpam-6123	80	61	,	,	PUNCT
ejpam-6123	80	62	s	s	PART
ejpam-6123	80	63	≥	≥	NOUN
ejpam-6123	80	64	2	2	NUM
ejpam-6123	80	65	proof	proof	NOUN
ejpam-6123	80	66	.	.	PUNCT
ejpam-6123	81	1	let	let	VERB
ejpam-6123	81	2	u	u	PRON
ejpam-6123	81	3	and	and	CCONJ
ejpam-6123	81	4	v	v	NOUN
ejpam-6123	81	5	be	be	AUX
ejpam-6123	81	6	the	the	DET
ejpam-6123	81	7	two	two	NUM
ejpam-6123	81	8	central	central	ADJ
ejpam-6123	81	9	vertices	vertex	NOUN
ejpam-6123	81	10	of	of	ADP
ejpam-6123	81	11	g	g	PROPN
ejpam-6123	81	12	=	=	SYM
ejpam-6123	81	13	sr	sr	PROPN
ejpam-6123	81	14	,	,	PUNCT
ejpam-6123	81	15	s	s	PART
ejpam-6123	81	16	,	,	PUNCT
ejpam-6123	81	17	and	and	CCONJ
ejpam-6123	81	18	let	let	VERB
ejpam-6123	81	19	u	u	PRON
ejpam-6123	81	20	and	and	CCONJ
ejpam-6123	81	21	v	v	NOUN
ejpam-6123	81	22	be	be	AUX
ejpam-6123	81	23	the	the	DET
ejpam-6123	81	24	sets	set	NOUN
ejpam-6123	81	25	of	of	ADP
ejpam-6123	81	26	all	all	DET
ejpam-6123	81	27	leaves	leave	NOUN
ejpam-6123	81	28	adjacent	adjacent	ADJ
ejpam-6123	81	29	to	to	ADP
ejpam-6123	81	30	u	u	NOUN
ejpam-6123	81	31	and	and	CCONJ
ejpam-6123	81	32	v	v	NOUN
ejpam-6123	81	33	,	,	PUNCT
ejpam-6123	81	34	respectively	respectively	ADV
ejpam-6123	81	35	,	,	PUNCT
ejpam-6123	81	36	with	with	ADP
ejpam-6123	81	37	|u	|u	ADJ
ejpam-6123	81	38	|	|	NOUN
ejpam-6123	81	39	=	=	SYM
ejpam-6123	81	40	r	r	NOUN
ejpam-6123	81	41	and	and	CCONJ
ejpam-6123	81	42	|v	|v	ADJ
ejpam-6123	81	43	|	|	ADV
ejpam-6123	81	44	=	=	SYM
ejpam-6123	81	45	s.	s.	PROPN
ejpam-6123	81	46	then	then	ADV
ejpam-6123	81	47	h.	h.	PROPN
ejpam-6123	81	48	nuenay	nuenay	PROPN
ejpam-6123	81	49	-	-	PUNCT
ejpam-6123	81	50	maglanquel	maglanquel	PROPN
ejpam-6123	81	51	/	/	SYM
ejpam-6123	81	52	eur	eur	PROPN
ejpam-6123	81	53	.	.	PUNCT
ejpam-6123	82	1	j.	j.	PROPN
ejpam-6123	82	2	pure	pure	PROPN
ejpam-6123	82	3	appl	appl	PROPN
ejpam-6123	82	4	.	.	PROPN
ejpam-6123	82	5	math	math	PROPN
ejpam-6123	82	6	,	,	PUNCT
ejpam-6123	82	7	18	18	NUM
ejpam-6123	82	8	(	(	PUNCT
ejpam-6123	82	9	3	3	NUM
ejpam-6123	82	10	)	)	PUNCT
ejpam-6123	82	11	(	(	PUNCT
ejpam-6123	82	12	2025	2025	NUM
ejpam-6123	82	13	)	)	PUNCT
ejpam-6123	82	14	,	,	PUNCT
ejpam-6123	82	15	6123	6123	NUM
ejpam-6123	82	16	5	5	NUM
ejpam-6123	82	17	of	of	ADP
ejpam-6123	82	18	14	14	NUM
ejpam-6123	82	19	ng({u	ng({u	NUM
ejpam-6123	82	20	}	}	PUNCT
ejpam-6123	82	21	)	)	PUNCT
ejpam-6123	83	1	=	=	PRON
ejpam-6123	83	2	{	{	PUNCT
ejpam-6123	83	3	v	v	NOUN
ejpam-6123	83	4	}	}	PUNCT
ejpam-6123	83	5	∪	∪	NOUN
ejpam-6123	83	6	u	u	NOUN
ejpam-6123	83	7	and	and	CCONJ
ejpam-6123	83	8	ng({v	ng({v	NUM
ejpam-6123	83	9	}	}	PUNCT
ejpam-6123	83	10	)	)	PUNCT
ejpam-6123	84	1	=	=	PRON
ejpam-6123	84	2	{	{	PUNCT
ejpam-6123	84	3	u	u	NOUN
ejpam-6123	84	4	}	}	PUNCT
ejpam-6123	84	5	∪	∪	VERB
ejpam-6123	84	6	v	v	NOUN
ejpam-6123	84	7	.if	.if	PUNCT
ejpam-6123	84	8	r	r	NOUN
ejpam-6123	84	9	=	=	PUNCT
ejpam-6123	84	10	s	s	NOUN
ejpam-6123	84	11	=	=	SYM
ejpam-6123	84	12	1	1	NUM
ejpam-6123	84	13	,	,	PUNCT
ejpam-6123	84	14	then	then	ADV
ejpam-6123	84	15	the	the	DET
ejpam-6123	84	16	only	only	ADJ
ejpam-6123	84	17	connected	connected	ADJ
ejpam-6123	84	18	equitable	equitable	ADJ
ejpam-6123	84	19	dominating	dominating	NOUN
ejpam-6123	84	20	set	set	VERB
ejpam-6123	84	21	in	in	ADP
ejpam-6123	84	22	g	g	PROPN
ejpam-6123	84	23	is	be	AUX
ejpam-6123	84	24	the	the	DET
ejpam-6123	84	25	set	set	NOUN
ejpam-6123	84	26	{	{	PUNCT
ejpam-6123	84	27	u	u	NOUN
ejpam-6123	84	28	,	,	PUNCT
ejpam-6123	84	29	v	v	NOUN
ejpam-6123	84	30	}	}	PUNCT
ejpam-6123	84	31	.	.	PUNCT
ejpam-6123	85	1	thus	thus	ADV
ejpam-6123	85	2	γce(g	γce(g	NUM
ejpam-6123	85	3	)	)	PUNCT
ejpam-6123	85	4	=	=	SYM
ejpam-6123	86	1	2	2	X
ejpam-6123	86	2	.	.	PUNCT
ejpam-6123	86	3	suppose	suppose	VERB
ejpam-6123	86	4	that	that	SCONJ
ejpam-6123	86	5	r	r	NOUN
ejpam-6123	86	6	=	=	SYM
ejpam-6123	86	7	1	1	NUM
ejpam-6123	86	8	and	and	CCONJ
ejpam-6123	86	9	s	s	PRON
ejpam-6123	86	10	≥	≥	NOUN
ejpam-6123	86	11	2	2	NUM
ejpam-6123	86	12	.	.	PUNCT
ejpam-6123	87	1	since	since	SCONJ
ejpam-6123	87	2	every	every	DET
ejpam-6123	87	3	y	y	PROPN
ejpam-6123	87	4	∈	∈	PROPN
ejpam-6123	87	5	u	u	NOUN
ejpam-6123	87	6	∪v	∪v	PUNCT
ejpam-6123	87	7	is	be	AUX
ejpam-6123	87	8	of	of	ADP
ejpam-6123	87	9	degree	degree	NOUN
ejpam-6123	87	10	1	1	NUM
ejpam-6123	87	11	and	and	CCONJ
ejpam-6123	87	12	degg(u	degg(u	NUM
ejpam-6123	87	13	)	)	PUNCT
ejpam-6123	87	14	=	=	SYM
ejpam-6123	87	15	2	2	NUM
ejpam-6123	87	16	,	,	PUNCT
ejpam-6123	87	17	the	the	DET
ejpam-6123	87	18	only	only	ADJ
ejpam-6123	87	19	connected	connected	ADJ
ejpam-6123	87	20	equitable	equitable	ADJ
ejpam-6123	87	21	dominating	dominating	NOUN
ejpam-6123	87	22	set	set	VERB
ejpam-6123	87	23	in	in	ADP
ejpam-6123	87	24	g	g	PROPN
ejpam-6123	87	25	is	be	AUX
ejpam-6123	87	26	the	the	DET
ejpam-6123	87	27	{	{	PUNCT
ejpam-6123	87	28	u	u	NOUN
ejpam-6123	87	29	,	,	PUNCT
ejpam-6123	87	30	v}∪v	v}∪v	X
ejpam-6123	87	31	.	.	PUNCT
ejpam-6123	88	1	thus	thus	ADV
ejpam-6123	88	2	,	,	PUNCT
ejpam-6123	88	3	γce(g	γce(g	PROPN
ejpam-6123	88	4	)	)	PUNCT
ejpam-6123	88	5	=	=	PUNCT
ejpam-6123	89	1	s+2	s+2	X
ejpam-6123	89	2	.	.	X
ejpam-6123	90	1	similarly	similarly	ADV
ejpam-6123	90	2	,	,	PUNCT
ejpam-6123	90	3	if	if	SCONJ
ejpam-6123	90	4	s	s	VERB
ejpam-6123	90	5	=	=	SYM
ejpam-6123	90	6	1	1	NUM
ejpam-6123	90	7	and	and	CCONJ
ejpam-6123	90	8	r	r	NOUN
ejpam-6123	90	9	≥	≥	NUM
ejpam-6123	90	10	2	2	NUM
ejpam-6123	90	11	,	,	PUNCT
ejpam-6123	90	12	γce(g	γce(g	PROPN
ejpam-6123	90	13	)	)	PUNCT
ejpam-6123	91	1	=	=	SYM
ejpam-6123	91	2	r	r	NOUN
ejpam-6123	91	3	+	+	NOUN
ejpam-6123	91	4	2	2	NUM
ejpam-6123	91	5	.	.	PUNCT
ejpam-6123	91	6	suppose	suppose	VERB
ejpam-6123	91	7	that	that	SCONJ
ejpam-6123	91	8	r	r	NOUN
ejpam-6123	91	9	,	,	PUNCT
ejpam-6123	91	10	s	s	PART
ejpam-6123	91	11	≥	≥	NOUN
ejpam-6123	91	12	2	2	NUM
ejpam-6123	91	13	.	.	PUNCT
ejpam-6123	91	14	let	let	VERB
ejpam-6123	91	15	s	s	PRON
ejpam-6123	91	16	be	be	AUX
ejpam-6123	91	17	a	a	DET
ejpam-6123	91	18	connected	connect	VERB
ejpam-6123	91	19	equitable	equitable	ADJ
ejpam-6123	91	20	dominating	dominating	NOUN
ejpam-6123	91	21	set	set	VERB
ejpam-6123	91	22	in	in	ADP
ejpam-6123	91	23	g	g	NOUN
ejpam-6123	91	24	with	with	ADP
ejpam-6123	91	25	|s|	|s|	PROPN
ejpam-6123	91	26	=	=	SYM
ejpam-6123	91	27	γce(g	γce(g	PROPN
ejpam-6123	91	28	)	)	PUNCT
ejpam-6123	91	29	.	.	PUNCT
ejpam-6123	92	1	suppose	suppose	VERB
ejpam-6123	92	2	that	that	SCONJ
ejpam-6123	92	3	|s|	|s|	VERB
ejpam-6123	92	4	<	<	X
ejpam-6123	92	5	r	r	NOUN
ejpam-6123	92	6	+	+	SYM
ejpam-6123	92	7	s	s	NOUN
ejpam-6123	92	8	+	+	CCONJ
ejpam-6123	92	9	2	2	NUM
ejpam-6123	92	10	=	=	SYM
ejpam-6123	92	11	|v	|v	X
ejpam-6123	92	12	(	(	PUNCT
ejpam-6123	92	13	g)|	g)|	VERB
ejpam-6123	92	14	and	and	CCONJ
ejpam-6123	92	15	let	let	VERB
ejpam-6123	92	16	x	x	PUNCT
ejpam-6123	92	17	∈	∈	PROPN
ejpam-6123	92	18	v	v	ADP
ejpam-6123	92	19	\	\	PROPN
ejpam-6123	92	20	s.	s.	PROPN
ejpam-6123	92	21	then	then	ADV
ejpam-6123	92	22	,	,	PUNCT
ejpam-6123	92	23	either	either	CCONJ
ejpam-6123	92	24	x	x	X
ejpam-6123	92	25	∈	∈	PROPN
ejpam-6123	92	26	{	{	PUNCT
ejpam-6123	92	27	u	u	NOUN
ejpam-6123	92	28	,	,	PUNCT
ejpam-6123	92	29	v	v	NOUN
ejpam-6123	92	30	}	}	PUNCT
ejpam-6123	92	31	,	,	PUNCT
ejpam-6123	92	32	x	x	PUNCT
ejpam-6123	92	33	∈	∈	PROPN
ejpam-6123	92	34	u	u	NOUN
ejpam-6123	92	35	,	,	PUNCT
ejpam-6123	92	36	or	or	CCONJ
ejpam-6123	92	37	x	x	X
ejpam-6123	92	38	∈	∈	NOUN
ejpam-6123	92	39	v	v	NOUN
ejpam-6123	92	40	.	.	PUNCT
ejpam-6123	92	41	suppose	suppose	VERB
ejpam-6123	92	42	that	that	SCONJ
ejpam-6123	92	43	x	x	PRON
ejpam-6123	92	44	is	be	AUX
ejpam-6123	92	45	any	any	PRON
ejpam-6123	92	46	of	of	ADP
ejpam-6123	92	47	the	the	DET
ejpam-6123	92	48	central	central	ADJ
ejpam-6123	92	49	vertices	vertex	NOUN
ejpam-6123	92	50	of	of	ADP
ejpam-6123	92	51	g	g	NOUN
ejpam-6123	92	52	,	,	PUNCT
ejpam-6123	92	53	say	say	VERB
ejpam-6123	92	54	u.	u.	NOUN
ejpam-6123	92	55	this	this	PRON
ejpam-6123	92	56	means	mean	VERB
ejpam-6123	92	57	that	that	SCONJ
ejpam-6123	92	58	there	there	PRON
ejpam-6123	92	59	exists	exist	VERB
ejpam-6123	92	60	uj	uj	PROPN
ejpam-6123	92	61	∈	∈	PROPN
ejpam-6123	92	62	u	u	NOUN
ejpam-6123	92	63	which	which	PRON
ejpam-6123	92	64	is	be	AUX
ejpam-6123	92	65	an	an	DET
ejpam-6123	92	66	element	element	NOUN
ejpam-6123	92	67	of	of	ADP
ejpam-6123	92	68	s	s	PROPN
ejpam-6123	92	69	,	,	PUNCT
ejpam-6123	92	70	a	a	DET
ejpam-6123	92	71	contradiction	contradiction	NOUN
ejpam-6123	92	72	to	to	ADP
ejpam-6123	92	73	the	the	DET
ejpam-6123	92	74	assumption	assumption	NOUN
ejpam-6123	92	75	that	that	SCONJ
ejpam-6123	92	76	s	s	VERB
ejpam-6123	92	77	is	be	AUX
ejpam-6123	92	78	connected	connect	VERB
ejpam-6123	92	79	.	.	PUNCT
ejpam-6123	93	1	thus	thus	ADV
ejpam-6123	93	2	,	,	PUNCT
ejpam-6123	93	3	x	x	PRON
ejpam-6123	93	4	must	must	AUX
ejpam-6123	93	5	not	not	PART
ejpam-6123	93	6	be	be	AUX
ejpam-6123	93	7	any	any	PRON
ejpam-6123	93	8	of	of	ADP
ejpam-6123	93	9	the	the	DET
ejpam-6123	93	10	central	central	ADJ
ejpam-6123	93	11	vertices	vertex	NOUN
ejpam-6123	93	12	.	.	PUNCT
ejpam-6123	94	1	now	now	ADV
ejpam-6123	94	2	,	,	PUNCT
ejpam-6123	94	3	either	either	CCONJ
ejpam-6123	94	4	x	x	X
ejpam-6123	94	5	∈	∈	PROPN
ejpam-6123	94	6	u	u	NOUN
ejpam-6123	94	7	or	or	CCONJ
ejpam-6123	94	8	x	x	NOUN
ejpam-6123	94	9	∈	∈	PROPN
ejpam-6123	94	10	v	v	NOUN
ejpam-6123	94	11	.	.	PUNCT
ejpam-6123	95	1	without	without	ADP
ejpam-6123	95	2	loss	loss	NOUN
ejpam-6123	95	3	of	of	ADP
ejpam-6123	95	4	generality	generality	NOUN
ejpam-6123	95	5	,	,	PUNCT
ejpam-6123	95	6	let	let	VERB
ejpam-6123	95	7	x	x	PUNCT
ejpam-6123	95	8	∈	∈	PROPN
ejpam-6123	95	9	u	u	NOUN
ejpam-6123	95	10	.	.	PUNCT
ejpam-6123	96	1	since	since	SCONJ
ejpam-6123	96	2	s	s	PROPN
ejpam-6123	96	3	is	be	AUX
ejpam-6123	96	4	a	a	DET
ejpam-6123	96	5	connected	connect	VERB
ejpam-6123	96	6	equitable	equitable	ADJ
ejpam-6123	96	7	dominating	dominating	NOUN
ejpam-6123	96	8	set	set	VERB
ejpam-6123	96	9	in	in	ADP
ejpam-6123	96	10	g	g	NOUN
ejpam-6123	96	11	,	,	PUNCT
ejpam-6123	96	12	there	there	PRON
ejpam-6123	96	13	exists	exist	VERB
ejpam-6123	96	14	y	y	PROPN
ejpam-6123	96	15	∈	∈	PROPN
ejpam-6123	96	16	s	s	VERB
ejpam-6123	96	17	such	such	ADJ
ejpam-6123	96	18	that	that	SCONJ
ejpam-6123	96	19	xy	xy	PROPN
ejpam-6123	96	20	∈	∈	PROPN
ejpam-6123	96	21	e(g	e(g	PROPN
ejpam-6123	96	22	)	)	PUNCT
ejpam-6123	96	23	and	and	CCONJ
ejpam-6123	96	24	|degg(x)−degg(y)|	|degg(x)−degg(y)|	NOUN
ejpam-6123	96	25	≤	≤	NUM
ejpam-6123	96	26	1	1	NUM
ejpam-6123	96	27	.	.	PUNCT
ejpam-6123	97	1	note	note	VERB
ejpam-6123	97	2	that	that	SCONJ
ejpam-6123	97	3	the	the	DET
ejpam-6123	97	4	only	only	ADJ
ejpam-6123	97	5	adjacent	adjacent	ADJ
ejpam-6123	97	6	vertex	vertex	NOUN
ejpam-6123	97	7	of	of	ADP
ejpam-6123	97	8	x	x	PRON
ejpam-6123	97	9	in	in	ADP
ejpam-6123	97	10	g	g	PROPN
ejpam-6123	97	11	is	be	AUX
ejpam-6123	97	12	the	the	DET
ejpam-6123	97	13	central	central	ADJ
ejpam-6123	97	14	vertex	vertex	NOUN
ejpam-6123	97	15	u.	u.	PROPN
ejpam-6123	98	1	thus	thus	ADV
ejpam-6123	98	2	,	,	PUNCT
ejpam-6123	98	3	y	y	PROPN
ejpam-6123	98	4	=	=	SYM
ejpam-6123	98	5	u	u	PROPN
ejpam-6123	98	6	and	and	CCONJ
ejpam-6123	98	7	|degg(x	|degg(x	NUM
ejpam-6123	98	8	)	)	PUNCT
ejpam-6123	98	9	−	−	PROPN
ejpam-6123	98	10	degg(y)|	degg(y)|	PROPN
ejpam-6123	98	11	=	=	PROPN
ejpam-6123	98	12	|degg(x	|degg(x	PROPN
ejpam-6123	98	13	)	)	PUNCT
ejpam-6123	98	14	−	−	PROPN
ejpam-6123	98	15	degg(u)|	degg(u)|	NOUN
ejpam-6123	98	16	=	=	PUNCT
ejpam-6123	98	17	|1	|1	NUM
ejpam-6123	99	1	−	−	PROPN
ejpam-6123	99	2	(	(	PUNCT
ejpam-6123	99	3	r	r	NOUN
ejpam-6123	99	4	+	+	NUM
ejpam-6123	99	5	1)|	1)|	NUM
ejpam-6123	99	6	=	=	SYM
ejpam-6123	99	7	r	r	NOUN
ejpam-6123	99	8	>	>	SYM
ejpam-6123	99	9	1	1	NUM
ejpam-6123	99	10	,	,	PUNCT
ejpam-6123	99	11	a	a	DET
ejpam-6123	99	12	contradiction	contradiction	NOUN
ejpam-6123	99	13	.	.	PUNCT
ejpam-6123	100	1	therefore	therefore	ADV
ejpam-6123	100	2	,	,	PUNCT
ejpam-6123	100	3	s	s	NOUN
ejpam-6123	100	4	=	=	SYM
ejpam-6123	100	5	v	v	X
ejpam-6123	100	6	(	(	PUNCT
ejpam-6123	100	7	g	g	NOUN
ejpam-6123	100	8	)	)	PUNCT
ejpam-6123	100	9	.	.	PUNCT
ejpam-6123	101	1	hence	hence	ADV
ejpam-6123	101	2	,	,	PUNCT
ejpam-6123	101	3	γce(g	γce(g	PROPN
ejpam-6123	101	4	)	)	PUNCT
ejpam-6123	101	5	=	=	PUNCT
ejpam-6123	101	6	|s|	|s|	NOUN
ejpam-6123	101	7	=	=	SYM
ejpam-6123	101	8	r	r	NOUN
ejpam-6123	101	9	+	+	CCONJ
ejpam-6123	101	10	s+	s+	NUM
ejpam-6123	101	11	2	2	NUM
ejpam-6123	101	12	.	.	X
ejpam-6123	101	13	theorem	theorem	NOUN
ejpam-6123	101	14	5	5	NUM
ejpam-6123	101	15	.	.	NUM
ejpam-6123	101	16	γce(g	γce(g	NOUN
ejpam-6123	101	17	)	)	PUNCT
ejpam-6123	102	1	=	=	SYM
ejpam-6123	102	2	1	1	NUM
ejpam-6123	102	3	if	if	SCONJ
ejpam-6123	102	4	and	and	CCONJ
ejpam-6123	102	5	only	only	ADV
ejpam-6123	102	6	if	if	SCONJ
ejpam-6123	102	7	there	there	PRON
ejpam-6123	102	8	exists	exist	VERB
ejpam-6123	102	9	v	v	ADP
ejpam-6123	102	10	∈	∈	PROPN
ejpam-6123	102	11	v	v	NOUN
ejpam-6123	102	12	(	(	PUNCT
ejpam-6123	102	13	g	g	NOUN
ejpam-6123	102	14	)	)	PUNCT
ejpam-6123	102	15	such	such	ADJ
ejpam-6123	102	16	that	that	DET
ejpam-6123	102	17	degg(v	degg(v	PROPN
ejpam-6123	102	18	)	)	PUNCT
ejpam-6123	102	19	=	=	SYM
ejpam-6123	102	20	|v	|v	PROPN
ejpam-6123	103	1	(	(	PUNCT
ejpam-6123	103	2	g)|	g)|	INTJ
ejpam-6123	103	3	−	−	PROPN
ejpam-6123	103	4	1	1	NUM
ejpam-6123	103	5	and	and	CCONJ
ejpam-6123	103	6	degg(u	degg(u	PROPN
ejpam-6123	103	7	)	)	PUNCT
ejpam-6123	103	8	≥	≥	NOUN
ejpam-6123	103	9	|v	|v	NOUN
ejpam-6123	103	10	(	(	PUNCT
ejpam-6123	103	11	g)|	g)|	INTJ
ejpam-6123	103	12	−	−	NOUN
ejpam-6123	103	13	2	2	NUM
ejpam-6123	103	14	for	for	ADP
ejpam-6123	103	15	all	all	DET
ejpam-6123	103	16	u	u	NOUN
ejpam-6123	103	17	̸=	̸=	PROPN
ejpam-6123	103	18	v	v	NOUN
ejpam-6123	103	19	in	in	ADP
ejpam-6123	103	20	g.	g.	PROPN
ejpam-6123	103	21	proof	proof	PROPN
ejpam-6123	103	22	.	.	PUNCT
ejpam-6123	104	1	suppose	suppose	VERB
ejpam-6123	104	2	that	that	SCONJ
ejpam-6123	104	3	γce(g	γce(g	PROPN
ejpam-6123	104	4	)	)	PUNCT
ejpam-6123	104	5	=	=	SYM
ejpam-6123	105	1	1	1	X
ejpam-6123	105	2	.	.	PUNCT
ejpam-6123	106	1	then	then	ADV
ejpam-6123	106	2	,	,	PUNCT
ejpam-6123	106	3	there	there	PRON
ejpam-6123	106	4	exists	exist	VERB
ejpam-6123	106	5	v	v	NOUN
ejpam-6123	106	6	in	in	ADP
ejpam-6123	106	7	g	g	PROPN
ejpam-6123	106	8	such	such	ADJ
ejpam-6123	106	9	that	that	SCONJ
ejpam-6123	106	10	{	{	PUNCT
ejpam-6123	106	11	v	v	NOUN
ejpam-6123	106	12	}	}	PUNCT
ejpam-6123	106	13	is	be	AUX
ejpam-6123	106	14	a	a	DET
ejpam-6123	106	15	connected	connect	VERB
ejpam-6123	106	16	equitable	equitable	ADJ
ejpam-6123	106	17	dominating	dominating	NOUN
ejpam-6123	106	18	set	set	VERB
ejpam-6123	106	19	in	in	ADP
ejpam-6123	106	20	g	g	PROPN
ejpam-6123	106	21	with	with	ADP
ejpam-6123	106	22	degg(v	degg(v	PROPN
ejpam-6123	106	23	)	)	PUNCT
ejpam-6123	106	24	=	=	SYM
ejpam-6123	106	25	|v	|v	PROPN
ejpam-6123	106	26	(	(	PUNCT
ejpam-6123	106	27	g)|−	g)|−	PRON
ejpam-6123	106	28	1	1	X
ejpam-6123	106	29	.	.	PUNCT
ejpam-6123	107	1	now	now	ADV
ejpam-6123	107	2	,	,	PUNCT
ejpam-6123	107	3	let	let	VERB
ejpam-6123	107	4	u	u	PRON
ejpam-6123	107	5	∈	∈	PROPN
ejpam-6123	107	6	v	v	ADP
ejpam-6123	107	7	(	(	PUNCT
ejpam-6123	107	8	g	g	NOUN
ejpam-6123	107	9	)	)	PUNCT
ejpam-6123	107	10	\	\	NOUN
ejpam-6123	107	11	{	{	PUNCT
ejpam-6123	107	12	v	v	NOUN
ejpam-6123	107	13	}	}	PUNCT
ejpam-6123	107	14	.	.	PUNCT
ejpam-6123	108	1	since	since	SCONJ
ejpam-6123	108	2	{	{	PUNCT
ejpam-6123	108	3	v	v	NOUN
ejpam-6123	108	4	}	}	PUNCT
ejpam-6123	108	5	is	be	AUX
ejpam-6123	108	6	a	a	DET
ejpam-6123	108	7	γce	γce	NOUN
ejpam-6123	108	8	-	-	PUNCT
ejpam-6123	108	9	set	set	VERB
ejpam-6123	108	10	in	in	ADP
ejpam-6123	108	11	g	g	NOUN
ejpam-6123	108	12	,	,	PUNCT
ejpam-6123	108	13	uv	uv	PROPN
ejpam-6123	108	14	∈	∈	PROPN
ejpam-6123	108	15	e(g	e(g	PROPN
ejpam-6123	108	16	)	)	PUNCT
ejpam-6123	108	17	and	and	CCONJ
ejpam-6123	108	18	|	|	ADV
ejpam-6123	108	19	degg(u)−	degg(u)−	VERB
ejpam-6123	108	20	degg(v)|	degg(v)|	VERB
ejpam-6123	108	21	≤	≤	NOUN
ejpam-6123	108	22	1	1	NUM
ejpam-6123	108	23	.	.	PUNCT
ejpam-6123	109	1	that	that	PRON
ejpam-6123	109	2	is	be	AUX
ejpam-6123	109	3	,	,	PUNCT
ejpam-6123	109	4	|degg(u)−	|degg(u)−	NOUN
ejpam-6123	109	5	degg(v)|	degg(v)|	VERB
ejpam-6123	109	6	≤	≤	NOUN
ejpam-6123	109	7	1	1	NUM
ejpam-6123	109	8	=	=	NOUN
ejpam-6123	109	9	⇒	⇒	NOUN
ejpam-6123	109	10	|	|	ADV
ejpam-6123	109	11	degg(u)−	degg(u)−	PROPN
ejpam-6123	109	12	(	(	PUNCT
ejpam-6123	109	13	|v	|v	X
ejpam-6123	109	14	(	(	PUNCT
ejpam-6123	109	15	g)|	g)|	NOUN
ejpam-6123	109	16	−	−	PROPN
ejpam-6123	109	17	1	1	NUM
ejpam-6123	109	18	)	)	PUNCT
ejpam-6123	109	19	≤	≤	NOUN
ejpam-6123	109	20	1	1	NUM
ejpam-6123	109	21	=	=	NOUN
ejpam-6123	109	22	⇒	⇒	NOUN
ejpam-6123	109	23	−1	−1	NOUN
ejpam-6123	109	24	≤	≤	NUM
ejpam-6123	109	25	degg(u)−	degg(u)−	PROPN
ejpam-6123	109	26	|v	|v	PROPN
ejpam-6123	109	27	(	(	PUNCT
ejpam-6123	109	28	g)|+	g)|+	NOUN
ejpam-6123	109	29	1	1	NUM
ejpam-6123	109	30	≤	≤	NUM
ejpam-6123	109	31	1	1	NUM
ejpam-6123	109	32	=	=	NOUN
ejpam-6123	109	33	⇒	⇒	NOUN
ejpam-6123	109	34	|v	|v	NOUN
ejpam-6123	109	35	(	(	PUNCT
ejpam-6123	109	36	g)|	g)|	NOUN
ejpam-6123	109	37	−	−	PROPN
ejpam-6123	109	38	2	2	NUM
ejpam-6123	109	39	≤	≤	NUM
ejpam-6123	109	40	degg(u	degg(u	PROPN
ejpam-6123	109	41	)	)	PUNCT
ejpam-6123	109	42	≤	≤	NOUN
ejpam-6123	109	43	|v	|v	X
ejpam-6123	109	44	(	(	PUNCT
ejpam-6123	109	45	g)|	g)|	PROPN
ejpam-6123	109	46	thus	thus	ADV
ejpam-6123	109	47	,	,	PUNCT
ejpam-6123	109	48	degg(u	degg(u	PROPN
ejpam-6123	109	49	)	)	PUNCT
ejpam-6123	109	50	≥	≥	NOUN
ejpam-6123	109	51	|v	|v	NOUN
ejpam-6123	109	52	(	(	PUNCT
ejpam-6123	109	53	g)|	g)|	INTJ
ejpam-6123	109	54	−	−	NOUN
ejpam-6123	109	55	2	2	NUM
ejpam-6123	109	56	for	for	ADP
ejpam-6123	109	57	all	all	DET
ejpam-6123	109	58	u	u	NOUN
ejpam-6123	109	59	̸=	̸=	PROPN
ejpam-6123	109	60	v	v	NOUN
ejpam-6123	109	61	in	in	ADP
ejpam-6123	109	62	g.	g.	NOUN
ejpam-6123	109	63	conversely	conversely	ADV
ejpam-6123	109	64	suppose	suppose	VERB
ejpam-6123	109	65	that	that	SCONJ
ejpam-6123	109	66	there	there	PRON
ejpam-6123	109	67	exists	exist	VERB
ejpam-6123	109	68	v	v	ADP
ejpam-6123	109	69	∈	∈	PROPN
ejpam-6123	109	70	v	v	NOUN
ejpam-6123	109	71	(	(	PUNCT
ejpam-6123	109	72	g	g	NOUN
ejpam-6123	109	73	)	)	PUNCT
ejpam-6123	109	74	such	such	ADJ
ejpam-6123	109	75	that	that	DET
ejpam-6123	109	76	degg(v	degg(v	PROPN
ejpam-6123	109	77	)	)	PUNCT
ejpam-6123	109	78	=	=	SYM
ejpam-6123	109	79	|v	|v	PROPN
ejpam-6123	109	80	(	(	PUNCT
ejpam-6123	109	81	g)|	g)|	INTJ
ejpam-6123	109	82	−	−	PROPN
ejpam-6123	109	83	1	1	NUM
ejpam-6123	109	84	and	and	CCONJ
ejpam-6123	109	85	degg(u	degg(u	PROPN
ejpam-6123	109	86	)	)	PUNCT
ejpam-6123	109	87	≥	≥	NOUN
ejpam-6123	109	88	|v	|v	NOUN
ejpam-6123	109	89	(	(	PUNCT
ejpam-6123	109	90	g)|	g)|	INTJ
ejpam-6123	109	91	−	−	NOUN
ejpam-6123	109	92	2	2	NUM
ejpam-6123	109	93	for	for	ADP
ejpam-6123	109	94	all	all	DET
ejpam-6123	109	95	u	u	NOUN
ejpam-6123	109	96	̸=	̸=	PROPN
ejpam-6123	109	97	v	v	NOUN
ejpam-6123	109	98	in	in	ADP
ejpam-6123	109	99	g.	g.	PROPN
ejpam-6123	109	100	then	then	ADV
ejpam-6123	109	101	,	,	PUNCT
ejpam-6123	109	102	{	{	PUNCT
ejpam-6123	109	103	v	v	NOUN
ejpam-6123	109	104	}	}	PUNCT
ejpam-6123	109	105	is	be	AUX
ejpam-6123	109	106	a	a	DET
ejpam-6123	109	107	dominating	dominating	NOUN
ejpam-6123	109	108	set	set	NOUN
ejpam-6123	109	109	and	and	CCONJ
ejpam-6123	109	110	|	|	ADV
ejpam-6123	109	111	degg(v)−	degg(v)−	PROPN
ejpam-6123	109	112	degg(u)|	degg(u)|	PROPN
ejpam-6123	109	113	≤	≤	PROPN
ejpam-6123	109	114	|v	|v	PROPN
ejpam-6123	109	115	(	(	PUNCT
ejpam-6123	109	116	g)|	g)|	PROPN
ejpam-6123	109	117	−	−	PROPN
ejpam-6123	109	118	1−	1−	NUM
ejpam-6123	109	119	(	(	PUNCT
ejpam-6123	109	120	|v	|v	X
ejpam-6123	109	121	(	(	PUNCT
ejpam-6123	109	122	g)|	g)|	NOUN
ejpam-6123	109	123	−	−	NOUN
ejpam-6123	109	124	2	2	NUM
ejpam-6123	109	125	)	)	PUNCT
ejpam-6123	109	126	≤	≤	NUM
ejpam-6123	109	127	1	1	NUM
ejpam-6123	109	128	for	for	ADP
ejpam-6123	109	129	all	all	DET
ejpam-6123	109	130	u	u	NOUN
ejpam-6123	109	131	̸=	̸=	PROPN
ejpam-6123	109	132	v	v	NOUN
ejpam-6123	109	133	in	in	ADP
ejpam-6123	109	134	g.	g.	PROPN
ejpam-6123	109	135	thus	thus	ADV
ejpam-6123	109	136	,	,	PUNCT
ejpam-6123	109	137	{	{	PUNCT
ejpam-6123	109	138	v	v	NOUN
ejpam-6123	109	139	}	}	PUNCT
ejpam-6123	109	140	is	be	AUX
ejpam-6123	109	141	a	a	DET
ejpam-6123	109	142	connected	connect	VERB
ejpam-6123	109	143	equitable	equitable	ADJ
ejpam-6123	109	144	dominating	dominating	NOUN
ejpam-6123	109	145	set	set	VERB
ejpam-6123	109	146	in	in	ADP
ejpam-6123	109	147	g.	g.	PROPN
ejpam-6123	109	148	consequently	consequently	ADV
ejpam-6123	109	149	,	,	PUNCT
ejpam-6123	109	150	γce(g	γce(g	PROPN
ejpam-6123	109	151	)	)	PUNCT
ejpam-6123	109	152	=	=	SYM
ejpam-6123	110	1	1	1	X
ejpam-6123	110	2	.	.	PUNCT
ejpam-6123	110	3	remark	remark	NOUN
ejpam-6123	110	4	1	1	NUM
ejpam-6123	110	5	.	.	PUNCT
ejpam-6123	111	1	if	if	SCONJ
ejpam-6123	111	2	a	a	DET
ejpam-6123	111	3	subset	subset	NOUN
ejpam-6123	111	4	s	s	VERB
ejpam-6123	111	5	⊆	⊆	NUM
ejpam-6123	111	6	v	v	NOUN
ejpam-6123	111	7	(	(	PUNCT
ejpam-6123	111	8	g	g	NOUN
ejpam-6123	111	9	)	)	PUNCT
ejpam-6123	111	10	is	be	AUX
ejpam-6123	111	11	a	a	DET
ejpam-6123	111	12	connected	connect	VERB
ejpam-6123	111	13	equitable	equitable	ADJ
ejpam-6123	111	14	dominating	dominating	NOUN
ejpam-6123	111	15	set	set	VERB
ejpam-6123	111	16	in	in	ADP
ejpam-6123	111	17	g	g	PROPN
ejpam-6123	111	18	and	and	CCONJ
ejpam-6123	111	19	|degg(u)−	|degg(u)−	PROPN
ejpam-6123	111	20	degg(v)|	degg(v)|	NOUN
ejpam-6123	111	21	≥	≥	NOUN
ejpam-6123	111	22	2	2	NUM
ejpam-6123	111	23	for	for	ADP
ejpam-6123	111	24	all	all	DET
ejpam-6123	111	25	v	v	ADP
ejpam-6123	111	26	∈	∈	NOUN
ejpam-6123	111	27	ng(u	ng(u	NOUN
ejpam-6123	111	28	)	)	PUNCT
ejpam-6123	111	29	,	,	PUNCT
ejpam-6123	111	30	then	then	ADV
ejpam-6123	111	31	u	u	PROPN
ejpam-6123	111	32	∈	∈	PROPN
ejpam-6123	111	33	s.	s.	PROPN
ejpam-6123	111	34	remark	remark	PROPN
ejpam-6123	111	35	2	2	NUM
ejpam-6123	111	36	.	.	PUNCT
ejpam-6123	112	1	every	every	DET
ejpam-6123	112	2	equitable	equitable	ADJ
ejpam-6123	112	3	dominating	dominating	NOUN
ejpam-6123	112	4	set	set	NOUN
ejpam-6123	112	5	is	be	AUX
ejpam-6123	112	6	a	a	DET
ejpam-6123	112	7	dominating	dominating	NOUN
ejpam-6123	112	8	set	set	NOUN
ejpam-6123	112	9	.	.	PUNCT
ejpam-6123	113	1	thus	thus	ADV
ejpam-6123	113	2	,	,	PUNCT
ejpam-6123	113	3	γ(g	γ(g	PROPN
ejpam-6123	113	4	)	)	PUNCT
ejpam-6123	113	5	≤	≤	NOUN
ejpam-6123	113	6	γe(g	γe(g	NUM
ejpam-6123	113	7	)	)	PUNCT
ejpam-6123	113	8	.	.	PUNCT
ejpam-6123	114	1	remark	remark	NOUN
ejpam-6123	114	2	3	3	NUM
ejpam-6123	114	3	.	.	PUNCT
ejpam-6123	115	1	every	every	DET
ejpam-6123	115	2	connected	connect	VERB
ejpam-6123	115	3	dominating	dominating	NOUN
ejpam-6123	115	4	set	set	NOUN
ejpam-6123	115	5	is	be	AUX
ejpam-6123	115	6	a	a	DET
ejpam-6123	115	7	dominating	dominating	NOUN
ejpam-6123	115	8	set	set	NOUN
ejpam-6123	115	9	.	.	PUNCT
ejpam-6123	116	1	thus	thus	ADV
ejpam-6123	116	2	,	,	PUNCT
ejpam-6123	116	3	γ(g	γ(g	PROPN
ejpam-6123	116	4	)	)	PUNCT
ejpam-6123	116	5	≤	≤	NOUN
ejpam-6123	116	6	γc(g	γc(g	NUM
ejpam-6123	116	7	)	)	PUNCT
ejpam-6123	116	8	.	.	PUNCT
ejpam-6123	117	1	remark	remark	PROPN
ejpam-6123	117	2	4	4	NUM
ejpam-6123	117	3	.	.	PUNCT
ejpam-6123	118	1	every	every	DET
ejpam-6123	118	2	equitable	equitable	ADJ
ejpam-6123	118	3	connected	connected	ADJ
ejpam-6123	118	4	dominating	dominating	NOUN
ejpam-6123	118	5	set	set	NOUN
ejpam-6123	118	6	is	be	AUX
ejpam-6123	118	7	a	a	DET
ejpam-6123	118	8	connected	connected	ADJ
ejpam-6123	118	9	dominating	dominating	NOUN
ejpam-6123	118	10	set	set	NOUN
ejpam-6123	118	11	.	.	PUNCT
ejpam-6123	119	1	thus	thus	ADV
ejpam-6123	119	2	,	,	PUNCT
ejpam-6123	119	3	γc(g	γc(g	NUM
ejpam-6123	119	4	)	)	PUNCT
ejpam-6123	119	5	≤	≤	NUM
ejpam-6123	119	6	γce(g	γce(g	PROPN
ejpam-6123	119	7	)	)	PUNCT
ejpam-6123	119	8	.	.	PUNCT
ejpam-6123	120	1	theorem	theorem	VERB
ejpam-6123	120	2	6	6	NUM
ejpam-6123	120	3	.	.	PUNCT
ejpam-6123	120	4	for	for	ADP
ejpam-6123	120	5	any	any	DET
ejpam-6123	120	6	graph	graph	NOUN
ejpam-6123	120	7	g	g	PROPN
ejpam-6123	120	8	,	,	PUNCT
ejpam-6123	120	9	γ(g	γ(g	PROPN
ejpam-6123	120	10	)	)	PUNCT
ejpam-6123	120	11	≤	≤	NOUN
ejpam-6123	120	12	γc(g	γc(g	PUNCT
ejpam-6123	120	13	)	)	PUNCT
ejpam-6123	120	14	≤	≤	NUM
ejpam-6123	120	15	γce(g	γce(g	PROPN
ejpam-6123	120	16	)	)	PUNCT
ejpam-6123	120	17	.	.	PUNCT
ejpam-6123	121	1	h.	h.	PROPN
ejpam-6123	121	2	nuenay	nuenay	PROPN
ejpam-6123	121	3	-	-	PUNCT
ejpam-6123	121	4	maglanquel	maglanquel	PROPN
ejpam-6123	121	5	/	/	SYM
ejpam-6123	121	6	eur	eur	PROPN
ejpam-6123	121	7	.	.	PUNCT
ejpam-6123	122	1	j.	j.	PROPN
ejpam-6123	122	2	pure	pure	PROPN
ejpam-6123	122	3	appl	appl	PROPN
ejpam-6123	122	4	.	.	PROPN
ejpam-6123	122	5	math	math	PROPN
ejpam-6123	122	6	,	,	PUNCT
ejpam-6123	122	7	18	18	NUM
ejpam-6123	122	8	(	(	PUNCT
ejpam-6123	122	9	3	3	NUM
ejpam-6123	122	10	)	)	PUNCT
ejpam-6123	122	11	(	(	PUNCT
ejpam-6123	122	12	2025	2025	NUM
ejpam-6123	122	13	)	)	PUNCT
ejpam-6123	122	14	,	,	PUNCT
ejpam-6123	122	15	6123	6123	NUM
ejpam-6123	122	16	6	6	NUM
ejpam-6123	122	17	of	of	ADP
ejpam-6123	122	18	14	14	NUM
ejpam-6123	122	19	proof	proof	NOUN
ejpam-6123	122	20	.	.	PUNCT
ejpam-6123	123	1	let	let	VERB
ejpam-6123	123	2	s	s	PRON
ejpam-6123	123	3	be	be	AUX
ejpam-6123	123	4	a	a	DET
ejpam-6123	123	5	γce	γce	NOUN
ejpam-6123	123	6	-	-	PUNCT
ejpam-6123	123	7	set	set	VERB
ejpam-6123	123	8	in	in	ADP
ejpam-6123	123	9	g.	g.	PROPN
ejpam-6123	124	1	then	then	ADV
ejpam-6123	124	2	s	s	VERB
ejpam-6123	124	3	is	be	AUX
ejpam-6123	124	4	a	a	DET
ejpam-6123	124	5	connected	connected	ADJ
ejpam-6123	124	6	dominating	dominating	NOUN
ejpam-6123	124	7	set	set	VERB
ejpam-6123	124	8	in	in	ADP
ejpam-6123	124	9	g.	g.	PROPN
ejpam-6123	124	10	thus	thus	ADV
ejpam-6123	124	11	,	,	PUNCT
ejpam-6123	124	12	by	by	ADP
ejpam-6123	124	13	remark	remark	NOUN
ejpam-6123	124	14	4	4	NUM
ejpam-6123	124	15	,	,	PUNCT
ejpam-6123	124	16	γc(g	γc(g	NUM
ejpam-6123	124	17	)	)	PUNCT
ejpam-6123	124	18	≤	≤	NUM
ejpam-6123	124	19	|s|	|s|	PROPN
ejpam-6123	124	20	=	=	SYM
ejpam-6123	124	21	γce(g	γce(g	PROPN
ejpam-6123	124	22	)	)	PUNCT
ejpam-6123	124	23	.	.	PUNCT
ejpam-6123	125	1	furthermore	furthermore	ADV
ejpam-6123	125	2	,	,	PUNCT
ejpam-6123	125	3	s	s	VERB
ejpam-6123	125	4	is	be	AUX
ejpam-6123	125	5	a	a	DET
ejpam-6123	125	6	dominating	dominating	NOUN
ejpam-6123	125	7	set	set	VERB
ejpam-6123	125	8	in	in	ADP
ejpam-6123	125	9	g.	g.	PROPN
ejpam-6123	125	10	by	by	ADP
ejpam-6123	125	11	remark	remark	NOUN
ejpam-6123	125	12	3	3	NUM
ejpam-6123	125	13	,	,	PUNCT
ejpam-6123	125	14	the	the	DET
ejpam-6123	125	15	result	result	NOUN
ejpam-6123	125	16	follows	follow	VERB
ejpam-6123	125	17	.	.	PUNCT
ejpam-6123	126	1	the	the	DET
ejpam-6123	126	2	next	next	ADJ
ejpam-6123	126	3	result	result	NOUN
ejpam-6123	126	4	shows	show	VERB
ejpam-6123	126	5	that	that	SCONJ
ejpam-6123	126	6	every	every	DET
ejpam-6123	126	7	pair	pair	NOUN
ejpam-6123	126	8	of	of	ADP
ejpam-6123	126	9	positive	positive	ADJ
ejpam-6123	126	10	integers	integer	NOUN
ejpam-6123	126	11	are	be	AUX
ejpam-6123	126	12	realizable	realizable	ADJ
ejpam-6123	126	13	as	as	ADP
ejpam-6123	126	14	the	the	DET
ejpam-6123	126	15	connected	connected	ADJ
ejpam-6123	126	16	domination	domination	NOUN
ejpam-6123	126	17	number	number	NOUN
ejpam-6123	126	18	and	and	CCONJ
ejpam-6123	126	19	connected	connect	VERB
ejpam-6123	126	20	equitable	equitable	ADJ
ejpam-6123	126	21	domination	domination	NOUN
ejpam-6123	126	22	number	number	NOUN
ejpam-6123	126	23	of	of	ADP
ejpam-6123	126	24	a	a	DET
ejpam-6123	126	25	connected	connected	ADJ
ejpam-6123	126	26	graph	graph	NOUN
ejpam-6123	126	27	.	.	PUNCT
ejpam-6123	127	1	theorem	theorem	ADJ
ejpam-6123	127	2	7	7	NUM
ejpam-6123	127	3	.	.	X
ejpam-6123	128	1	for	for	ADP
ejpam-6123	128	2	every	every	DET
ejpam-6123	128	3	positive	positive	ADJ
ejpam-6123	128	4	integers	integer	NOUN
ejpam-6123	128	5	a	a	PRON
ejpam-6123	128	6	and	and	CCONJ
ejpam-6123	128	7	b	b	NOUN
ejpam-6123	128	8	with	with	ADP
ejpam-6123	128	9	1	1	NUM
ejpam-6123	128	10	≤	≤	NOUN
ejpam-6123	128	11	a	a	DET
ejpam-6123	128	12	≤	≤	PROPN
ejpam-6123	128	13	b	b	NOUN
ejpam-6123	128	14	there	there	PRON
ejpam-6123	128	15	exists	exist	VERB
ejpam-6123	128	16	a	a	DET
ejpam-6123	128	17	connected	connected	ADJ
ejpam-6123	128	18	graph	graph	NOUN
ejpam-6123	128	19	g	g	ADP
ejpam-6123	128	20	such	such	ADJ
ejpam-6123	128	21	that	that	PRON
ejpam-6123	128	22	γc(g	γc(g	NUM
ejpam-6123	128	23	)	)	PUNCT
ejpam-6123	128	24	≤	≤	NUM
ejpam-6123	128	25	γce(g	γce(g	PROPN
ejpam-6123	128	26	)	)	PUNCT
ejpam-6123	128	27	.	.	PUNCT
ejpam-6123	129	1	proof	proof	NOUN
ejpam-6123	129	2	.	.	PUNCT
ejpam-6123	130	1	suppose	suppose	VERB
ejpam-6123	130	2	that	that	SCONJ
ejpam-6123	130	3	a	a	DET
ejpam-6123	130	4	=	=	X
ejpam-6123	130	5	b.	b.	PROPN
ejpam-6123	130	6	write	write	NOUN
ejpam-6123	130	7	v	v	PROPN
ejpam-6123	130	8	(	(	PUNCT
ejpam-6123	130	9	k1,a−1	k1,a−1	PROPN
ejpam-6123	130	10	)	)	PUNCT
ejpam-6123	130	11	=	=	PRON
ejpam-6123	130	12	{	{	PUNCT
ejpam-6123	130	13	x	x	NOUN
ejpam-6123	130	14	,	,	PUNCT
ejpam-6123	130	15	u1	u1	NOUN
ejpam-6123	130	16	,	,	PUNCT
ejpam-6123	130	17	u2	u2	NOUN
ejpam-6123	130	18	,	,	PUNCT
ejpam-6123	130	19	.	.	PUNCT
ejpam-6123	130	20	.	.	PUNCT
ejpam-6123	131	1	.	.	PUNCT
ejpam-6123	132	1	,	,	PUNCT
ejpam-6123	132	2	ua−1	ua−1	NOUN
ejpam-6123	132	3	}	}	PUNCT
ejpam-6123	132	4	as	as	ADP
ejpam-6123	132	5	in	in	ADP
ejpam-6123	132	6	figure	figure	NOUN
ejpam-6123	132	7	1	1	NUM
ejpam-6123	132	8	.	.	PUNCT
ejpam-6123	132	9	obtain	obtain	VERB
ejpam-6123	132	10	g	g	NOUN
ejpam-6123	132	11	from	from	ADP
ejpam-6123	132	12	k1,a−1	k1,a−1	NOUN
ejpam-6123	132	13	by	by	ADP
ejpam-6123	132	14	adding	add	VERB
ejpam-6123	132	15	pendant	pendant	ADJ
ejpam-6123	132	16	edges	edge	NOUN
ejpam-6123	132	17	vjuj	vjuj	ADV
ejpam-6123	132	18	,	,	PUNCT
ejpam-6123	132	19	j	j	PROPN
ejpam-6123	132	20	=	=	SYM
ejpam-6123	132	21	1	1	NUM
ejpam-6123	132	22	,	,	PUNCT
ejpam-6123	132	23	2	2	NUM
ejpam-6123	132	24	,	,	PUNCT
ejpam-6123	132	25	.	.	PUNCT
ejpam-6123	132	26	.	.	PUNCT
ejpam-6123	133	1	.	.	PUNCT
ejpam-6123	134	1	,	,	PUNCT
ejpam-6123	134	2	a−1	a−1	PROPN
ejpam-6123	134	3	as	as	SCONJ
ejpam-6123	134	4	shown	show	VERB
ejpam-6123	134	5	in	in	ADP
ejpam-6123	134	6	figure	figure	NOUN
ejpam-6123	134	7	1	1	NUM
ejpam-6123	134	8	.	.	PUNCT
ejpam-6123	134	9	....................................	....................................	PUNCT
ejpam-6123	134	10	....................................	....................................	PUNCT
ejpam-6123	135	1	....................................	....................................	PUNCT
ejpam-6123	135	2	....................................	....................................	PUNCT
ejpam-6123	136	1	........................................................................	........................................................................	PUNCT
ejpam-6123	136	2	.....................................................................................................................................................................	.....................................................................................................................................................................	PUNCT
ejpam-6123	136	3	.....................................	.....................................	PUNCT
ejpam-6123	136	4	.........	.........	PUNCT
ejpam-6123	136	5	........	........	PUNCT
ejpam-6123	136	6	........	........	PUNCT
ejpam-6123	136	7	........	........	PUNCT
ejpam-6123	136	8	........	........	PUNCT
ejpam-6123	136	9	........	........	PUNCT
ejpam-6123	136	10	........	........	PUNCT
ejpam-6123	136	11	........	........	PUNCT
ejpam-6123	136	12	........	........	PUNCT
ejpam-6123	136	13	........	........	PUNCT
ejpam-6123	137	1	.....	.....	PUNCT
ejpam-6123	137	2	..........	..........	PUNCT
ejpam-6123	138	1	.........	.........	PUNCT
ejpam-6123	138	2	.........	.........	PUNCT
ejpam-6123	139	1	.........	.........	PUNCT
ejpam-6123	139	2	.........	.........	PUNCT
ejpam-6123	140	1	.........	.........	PUNCT
ejpam-6123	140	2	.........	.........	PUNCT
ejpam-6123	141	1	.........	.........	PUNCT
ejpam-6123	141	2	.........	.........	PUNCT
ejpam-6123	142	1	.........	.........	PUNCT
ejpam-6123	142	2	.........	.........	PUNCT
ejpam-6123	142	3	........................................................................................................................	........................................................................................................................	PUNCT
ejpam-6123	142	4	.	.	PUNCT
ejpam-6123	142	5	.	.	PUNCT
ejpam-6123	142	6	.	.	PUNCT
ejpam-6123	143	1	ua−1	ua−1	NOUN
ejpam-6123	143	2	u1	u1	PROPN
ejpam-6123	143	3	u2u3	u2u3	PROPN
ejpam-6123	143	4	u4	u4	PROPN
ejpam-6123	143	5	x	x	PROPN
ejpam-6123	143	6	k1,a−1	k1,a−1	PROPN
ejpam-6123	143	7	....................................	....................................	PUNCT
ejpam-6123	143	8	....................................	....................................	PUNCT
ejpam-6123	144	1	....................................	....................................	PUNCT
ejpam-6123	144	2	....................................	....................................	PUNCT
ejpam-6123	145	1	........................................................................	........................................................................	PUNCT
ejpam-6123	145	2	.....................................................................................................................................................................	.....................................................................................................................................................................	PUNCT
ejpam-6123	145	3	.....................................	.....................................	PUNCT
ejpam-6123	145	4	.........	.........	PUNCT
ejpam-6123	145	5	........	........	PUNCT
ejpam-6123	145	6	........	........	PUNCT
ejpam-6123	145	7	........	........	PUNCT
ejpam-6123	145	8	........	........	PUNCT
ejpam-6123	145	9	........	........	PUNCT
ejpam-6123	145	10	........	........	PUNCT
ejpam-6123	145	11	........	........	PUNCT
ejpam-6123	145	12	........	........	PUNCT
ejpam-6123	145	13	........	........	PUNCT
ejpam-6123	146	1	.....	.....	PUNCT
ejpam-6123	146	2	..........	..........	PUNCT
ejpam-6123	147	1	.........	.........	PUNCT
ejpam-6123	147	2	.........	.........	PUNCT
ejpam-6123	148	1	.........	.........	PUNCT
ejpam-6123	148	2	.........	.........	PUNCT
ejpam-6123	149	1	.........	.........	PUNCT
ejpam-6123	149	2	.........	.........	PUNCT
ejpam-6123	150	1	.........	.........	PUNCT
ejpam-6123	150	2	.........	.........	PUNCT
ejpam-6123	151	1	.........	.........	PUNCT
ejpam-6123	151	2	.........	.........	PUNCT
ejpam-6123	151	3	........................................................................................................................	........................................................................................................................	PUNCT
ejpam-6123	151	4	.	.	PUNCT
ejpam-6123	151	5	.	.	PUNCT
ejpam-6123	151	6	.	.	PUNCT
ejpam-6123	152	1	....................................	....................................	PUNCT
ejpam-6123	152	2	........................................................................	........................................................................	PUNCT
ejpam-6123	153	1	....................................	....................................	PUNCT
ejpam-6123	153	2	....................................	....................................	PUNCT
ejpam-6123	154	1	...........................................................	...........................................................	PUNCT
ejpam-6123	154	2	...........................................................	...........................................................	PUNCT
ejpam-6123	154	3	.........	.........	PUNCT
ejpam-6123	154	4	........	........	PUNCT
ejpam-6123	154	5	........	........	PUNCT
ejpam-6123	154	6	........	........	PUNCT
ejpam-6123	154	7	........	........	PUNCT
ejpam-6123	154	8	........	........	PUNCT
ejpam-6123	155	1	........	........	PUNCT
ejpam-6123	155	2	..	..	PUNCT
ejpam-6123	155	3	.............................................................	.............................................................	PUNCT
ejpam-6123	156	1	.........	.........	PUNCT
ejpam-6123	156	2	.........	.........	PUNCT
ejpam-6123	157	1	.........	.........	PUNCT
ejpam-6123	157	2	.........	.........	PUNCT
ejpam-6123	157	3	...	...	PUNCT
ejpam-6123	158	1	ua−1	ua−1	NOUN
ejpam-6123	158	2	x	x	SYM
ejpam-6123	158	3	va−1	va−1	PROPN
ejpam-6123	158	4	u1	u1	NOUN
ejpam-6123	158	5	v1	v1	NOUN
ejpam-6123	158	6	u2	u2	PROPN
ejpam-6123	158	7	v2	v2	PROPN
ejpam-6123	158	8	u3	u3	NOUN
ejpam-6123	158	9	v3	v3	PROPN
ejpam-6123	158	10	u4v4	u4v4	NOUN
ejpam-6123	158	11	figure	figure	NOUN
ejpam-6123	158	12	1	1	NUM
ejpam-6123	158	13	:	:	PUNCT
ejpam-6123	158	14	g	g	PROPN
ejpam-6123	158	15	obtained	obtain	VERB
ejpam-6123	158	16	from	from	ADP
ejpam-6123	158	17	k1,a−1	k1,a−1	PROPN
ejpam-6123	158	18	observe	observe	VERB
ejpam-6123	158	19	that	that	SCONJ
ejpam-6123	158	20	the	the	DET
ejpam-6123	158	21	set	set	NOUN
ejpam-6123	158	22	s	s	PART
ejpam-6123	158	23	=	=	PUNCT
ejpam-6123	158	24	{	{	PUNCT
ejpam-6123	158	25	x	x	NOUN
ejpam-6123	158	26	,	,	PUNCT
ejpam-6123	158	27	u1	u1	NOUN
ejpam-6123	158	28	,	,	PUNCT
ejpam-6123	158	29	u2	u2	NOUN
ejpam-6123	158	30	,	,	PUNCT
ejpam-6123	158	31	.	.	PUNCT
ejpam-6123	158	32	.	.	PUNCT
ejpam-6123	158	33	.	.	PUNCT
ejpam-6123	159	1	,	,	PUNCT
ejpam-6123	159	2	ua−1	ua−1	PROPN
ejpam-6123	159	3	}	}	PUNCT
ejpam-6123	159	4	is	be	AUX
ejpam-6123	159	5	the	the	DET
ejpam-6123	159	6	only	only	ADJ
ejpam-6123	159	7	minimal	minimal	ADJ
ejpam-6123	159	8	connected	connect	VERB
ejpam-6123	159	9	equitable	equitable	ADJ
ejpam-6123	159	10	dominating	dominating	NOUN
ejpam-6123	159	11	set	set	NOUN
ejpam-6123	159	12	,	,	PUNCT
ejpam-6123	159	13	and	and	CCONJ
ejpam-6123	159	14	a	a	DET
ejpam-6123	159	15	minimal	minimal	ADJ
ejpam-6123	159	16	connected	connected	ADJ
ejpam-6123	159	17	dominating	dominating	NOUN
ejpam-6123	159	18	set	set	NOUN
ejpam-6123	159	19	of	of	ADP
ejpam-6123	159	20	g.	g.	PROPN
ejpam-6123	159	21	thus	thus	ADV
ejpam-6123	159	22	,	,	PUNCT
ejpam-6123	159	23	γc(g	γc(g	NUM
ejpam-6123	159	24	)	)	PUNCT
ejpam-6123	159	25	=	=	SYM
ejpam-6123	159	26	|s|	|s|	NOUN
ejpam-6123	159	27	=	=	SYM
ejpam-6123	159	28	a−	a−	PROPN
ejpam-6123	159	29	1	1	NUM
ejpam-6123	159	30	+	+	CCONJ
ejpam-6123	159	31	1	1	NUM
ejpam-6123	159	32	=	=	SYM
ejpam-6123	159	33	a	a	DET
ejpam-6123	159	34	=	=	SYM
ejpam-6123	159	35	b	b	PROPN
ejpam-6123	159	36	=	=	SYM
ejpam-6123	159	37	γce(g	γce(g	PROPN
ejpam-6123	159	38	)	)	PUNCT
ejpam-6123	159	39	.	.	PUNCT
ejpam-6123	160	1	suppose	suppose	VERB
ejpam-6123	160	2	that	that	SCONJ
ejpam-6123	160	3	b	b	X
ejpam-6123	160	4	=	=	PUNCT
ejpam-6123	160	5	a	a	DET
ejpam-6123	160	6	+	+	NUM
ejpam-6123	160	7	1	1	NUM
ejpam-6123	160	8	.	.	X
ejpam-6123	160	9	obtain	obtain	VERB
ejpam-6123	160	10	g	g	NOUN
ejpam-6123	160	11	=	=	PUNCT
ejpam-6123	160	12	g1	g1	PROPN
ejpam-6123	160	13	from	from	ADP
ejpam-6123	160	14	k1,a−1	k1,a−1	NOUN
ejpam-6123	160	15	by	by	ADP
ejpam-6123	160	16	adding	add	VERB
ejpam-6123	160	17	pendant	pendant	ADJ
ejpam-6123	160	18	edges	edge	NOUN
ejpam-6123	160	19	vjuj	vjuj	ADV
ejpam-6123	160	20	,	,	PUNCT
ejpam-6123	160	21	j	j	PROPN
ejpam-6123	160	22	=	=	SYM
ejpam-6123	160	23	2	2	NUM
ejpam-6123	160	24	,	,	PUNCT
ejpam-6123	160	25	.	.	PUNCT
ejpam-6123	160	26	.	.	PUNCT
ejpam-6123	161	1	.	.	PUNCT
ejpam-6123	162	1	,	,	PUNCT
ejpam-6123	162	2	a−	a−	PROPN
ejpam-6123	162	3	1	1	NUM
ejpam-6123	162	4	,	,	PUNCT
ejpam-6123	162	5	and	and	CCONJ
ejpam-6123	162	6	joining	join	VERB
ejpam-6123	162	7	the	the	DET
ejpam-6123	162	8	path	path	NOUN
ejpam-6123	162	9	p3	p3	PROPN
ejpam-6123	162	10	=	=	PUNCT
ejpam-6123	163	1	[	[	X
ejpam-6123	163	2	w1	w1	NOUN
ejpam-6123	163	3	,	,	PUNCT
ejpam-6123	163	4	v1	v1	NOUN
ejpam-6123	163	5	,	,	PUNCT
ejpam-6123	163	6	h1	h1	NOUN
ejpam-6123	163	7	]	]	PUNCT
ejpam-6123	163	8	to	to	ADP
ejpam-6123	163	9	vertex	vertex	NOUN
ejpam-6123	163	10	u1	u1	NOUN
ejpam-6123	163	11	of	of	ADP
ejpam-6123	163	12	k1,a−1	k1,a−1	PROPN
ejpam-6123	163	13	as	as	ADP
ejpam-6123	163	14	in	in	ADP
ejpam-6123	163	15	figure	figure	NOUN
ejpam-6123	163	16	2	2	NUM
ejpam-6123	163	17	.	.	X
ejpam-6123	163	18	observe	observe	VERB
ejpam-6123	163	19	that	that	SCONJ
ejpam-6123	163	20	the	the	DET
ejpam-6123	163	21	set	set	NOUN
ejpam-6123	163	22	{	{	PUNCT
ejpam-6123	163	23	x	x	NOUN
ejpam-6123	163	24	,	,	PUNCT
ejpam-6123	163	25	u1	u1	NOUN
ejpam-6123	163	26	,	,	PUNCT
ejpam-6123	163	27	u2	u2	NOUN
ejpam-6123	163	28	,	,	PUNCT
ejpam-6123	163	29	.	.	PUNCT
ejpam-6123	163	30	.	.	PUNCT
ejpam-6123	164	1	.	.	PUNCT
ejpam-6123	165	1	,	,	PUNCT
ejpam-6123	165	2	ua−1	ua−1	PROPN
ejpam-6123	165	3	}	}	PUNCT
ejpam-6123	165	4	is	be	AUX
ejpam-6123	165	5	the	the	DET
ejpam-6123	165	6	only	only	ADJ
ejpam-6123	165	7	minimal	minimal	ADJ
ejpam-6123	165	8	connected	connected	ADJ
ejpam-6123	165	9	dominating	dominating	NOUN
ejpam-6123	165	10	set	set	NOUN
ejpam-6123	165	11	of	of	ADP
ejpam-6123	165	12	g.	g.	PROPN
ejpam-6123	165	13	thus	thus	ADV
ejpam-6123	165	14	γc(g	γc(g	PUNCT
ejpam-6123	165	15	)	)	PUNCT
ejpam-6123	166	1	=	=	SYM
ejpam-6123	166	2	a−1	a−1	NOUN
ejpam-6123	166	3	+	+	NOUN
ejpam-6123	166	4	1	1	NUM
ejpam-6123	166	5	=	=	NOUN
ejpam-6123	166	6	a.	a.	NOUN
ejpam-6123	166	7	also	also	ADV
ejpam-6123	166	8	,	,	PUNCT
ejpam-6123	166	9	note	note	VERB
ejpam-6123	166	10	that	that	SCONJ
ejpam-6123	166	11	the	the	DET
ejpam-6123	166	12	sets	set	NOUN
ejpam-6123	166	13	s1	s1	NOUN
ejpam-6123	166	14	=	=	PUNCT
ejpam-6123	166	15	{	{	PUNCT
ejpam-6123	166	16	x	x	NOUN
ejpam-6123	166	17	,	,	PUNCT
ejpam-6123	166	18	u1	u1	NOUN
ejpam-6123	166	19	,	,	PUNCT
ejpam-6123	166	20	u2	u2	NOUN
ejpam-6123	166	21	,	,	PUNCT
ejpam-6123	166	22	.	.	PUNCT
ejpam-6123	166	23	.	.	PUNCT
ejpam-6123	167	1	.	.	PUNCT
ejpam-6123	168	1	,	,	PUNCT
ejpam-6123	168	2	ua−1	ua−1	PROPN
ejpam-6123	168	3	,	,	PUNCT
ejpam-6123	168	4	v1	v1	PROPN
ejpam-6123	168	5	}	}	PUNCT
ejpam-6123	168	6	and	and	CCONJ
ejpam-6123	168	7	s2	s2	VERB
ejpam-6123	168	8	=	=	SYM
ejpam-6123	168	9	{	{	PUNCT
ejpam-6123	168	10	x	x	NOUN
ejpam-6123	168	11	,	,	PUNCT
ejpam-6123	168	12	u1	u1	NOUN
ejpam-6123	168	13	,	,	PUNCT
ejpam-6123	168	14	u2	u2	NOUN
ejpam-6123	168	15	,	,	PUNCT
ejpam-6123	168	16	.	.	PUNCT
ejpam-6123	168	17	.	.	PUNCT
ejpam-6123	169	1	.	.	PUNCT
ejpam-6123	170	1	,	,	PUNCT
ejpam-6123	170	2	ua−1	ua−1	PROPN
ejpam-6123	170	3	,	,	PUNCT
ejpam-6123	170	4	w1	w1	PROPN
ejpam-6123	170	5	,	,	PUNCT
ejpam-6123	170	6	h1	h1	PROPN
ejpam-6123	170	7	}	}	PUNCT
ejpam-6123	170	8	are	be	AUX
ejpam-6123	170	9	the	the	DET
ejpam-6123	170	10	only	only	ADJ
ejpam-6123	170	11	minimal	minimal	ADJ
ejpam-6123	170	12	connected	connect	VERB
ejpam-6123	170	13	equitable	equitable	ADJ
ejpam-6123	170	14	dominating	dominating	NOUN
ejpam-6123	170	15	sets	set	NOUN
ejpam-6123	170	16	of	of	ADP
ejpam-6123	170	17	g.	g.	PROPN
ejpam-6123	170	18	since	since	SCONJ
ejpam-6123	170	19	s1	s1	PROPN
ejpam-6123	170	20	=	=	PROPN
ejpam-6123	170	21	a	a	DET
ejpam-6123	170	22	−	−	PROPN
ejpam-6123	170	23	1	1	NUM
ejpam-6123	170	24	+	+	NUM
ejpam-6123	170	25	1	1	NUM
ejpam-6123	170	26	+	+	SYM
ejpam-6123	170	27	1	1	NUM
ejpam-6123	170	28	=	=	SYM
ejpam-6123	170	29	a	a	DET
ejpam-6123	170	30	+	+	NUM
ejpam-6123	170	31	1	1	NUM
ejpam-6123	170	32	and	and	CCONJ
ejpam-6123	170	33	|s2|	|s2|	NOUN
ejpam-6123	170	34	=	=	NOUN
ejpam-6123	170	35	a	a	DET
ejpam-6123	170	36	−	−	PROPN
ejpam-6123	170	37	1	1	NUM
ejpam-6123	171	1	+	+	NUM
ejpam-6123	171	2	1	1	NUM
ejpam-6123	171	3	+	+	NUM
ejpam-6123	171	4	2	2	NUM
ejpam-6123	171	5	=	=	SYM
ejpam-6123	171	6	a	a	DET
ejpam-6123	171	7	+	+	NUM
ejpam-6123	171	8	2	2	NUM
ejpam-6123	171	9	,	,	PUNCT
ejpam-6123	171	10	we	we	PRON
ejpam-6123	171	11	have	have	VERB
ejpam-6123	171	12	γce(g	γce(g	NOUN
ejpam-6123	171	13	)	)	PUNCT
ejpam-6123	172	1	=	=	SYM
ejpam-6123	172	2	|s1|	|s1|	NOUN
ejpam-6123	172	3	=	=	SYM
ejpam-6123	172	4	a−	a−	PROPN
ejpam-6123	172	5	1	1	NUM
ejpam-6123	172	6	+	+	CCONJ
ejpam-6123	172	7	1	1	NUM
ejpam-6123	172	8	+	+	SYM
ejpam-6123	172	9	1	1	NUM
ejpam-6123	172	10	=	=	SYM
ejpam-6123	172	11	a+	a+	PUNCT
ejpam-6123	172	12	1	1	NUM
ejpam-6123	172	13	=	=	SYM
ejpam-6123	172	14	b	b	PROPN
ejpam-6123	172	15	=	=	SYM
ejpam-6123	172	16	γce(g	γce(g	PROPN
ejpam-6123	172	17	)	)	PUNCT
ejpam-6123	172	18	.	.	PUNCT
ejpam-6123	173	1	suppose	suppose	VERB
ejpam-6123	173	2	that	that	SCONJ
ejpam-6123	173	3	b	b	X
ejpam-6123	173	4	=	=	PUNCT
ejpam-6123	173	5	a	a	PROPN
ejpam-6123	174	1	+	+	X
ejpam-6123	174	2	k	k	PROPN
ejpam-6123	174	3	for	for	ADP
ejpam-6123	174	4	k	k	PROPN
ejpam-6123	174	5	≥	≥	PROPN
ejpam-6123	174	6	2	2	NUM
ejpam-6123	174	7	.	.	NUM
ejpam-6123	174	8	obtained	obtain	VERB
ejpam-6123	174	9	g	g	PROPN
ejpam-6123	174	10	=	=	PROPN
ejpam-6123	174	11	g2	g2	PROPN
ejpam-6123	174	12	from	from	ADP
ejpam-6123	174	13	g1	g1	PROPN
ejpam-6123	174	14	by	by	ADP
ejpam-6123	174	15	joining	join	VERB
ejpam-6123	174	16	k	k	PROPN
ejpam-6123	174	17	−	−	PROPN
ejpam-6123	174	18	1	1	NUM
ejpam-6123	174	19	path	path	NOUN
ejpam-6123	174	20	p3j	p3j	NOUN
ejpam-6123	174	21	=	=	PUNCT
ejpam-6123	175	1	[	[	X
ejpam-6123	175	2	wj	wj	X
ejpam-6123	175	3	,	,	PUNCT
ejpam-6123	175	4	vj	vj	INTJ
ejpam-6123	175	5	,	,	PUNCT
ejpam-6123	175	6	hj	hj	PROPN
ejpam-6123	175	7	]	]	PUNCT
ejpam-6123	175	8	to	to	ADP
ejpam-6123	175	9	vertex	vertex	NOUN
ejpam-6123	175	10	u1	u1	NOUN
ejpam-6123	175	11	of	of	ADP
ejpam-6123	175	12	g1	g1	PROPN
ejpam-6123	175	13	where	where	SCONJ
ejpam-6123	175	14	2	2	NUM
ejpam-6123	175	15	≤	≤	NUM
ejpam-6123	175	16	j	j	PROPN
ejpam-6123	175	17	≤	≤	PROPN
ejpam-6123	175	18	k	k	PROPN
ejpam-6123	175	19	as	as	ADP
ejpam-6123	175	20	in	in	ADP
ejpam-6123	175	21	figure	figure	NOUN
ejpam-6123	175	22	3	3	NUM
ejpam-6123	175	23	.	.	X
ejpam-6123	175	24	observe	observe	VERB
ejpam-6123	175	25	that	that	SCONJ
ejpam-6123	175	26	the	the	DET
ejpam-6123	175	27	set	set	NOUN
ejpam-6123	175	28	{	{	PUNCT
ejpam-6123	175	29	x	x	NOUN
ejpam-6123	175	30	,	,	PUNCT
ejpam-6123	175	31	u1	u1	NOUN
ejpam-6123	175	32	,	,	PUNCT
ejpam-6123	175	33	u2	u2	NOUN
ejpam-6123	175	34	,	,	PUNCT
ejpam-6123	175	35	.	.	PUNCT
ejpam-6123	175	36	.	.	PUNCT
ejpam-6123	176	1	.	.	PUNCT
ejpam-6123	177	1	,	,	PUNCT
ejpam-6123	177	2	ua−1	ua−1	PROPN
ejpam-6123	177	3	}	}	PUNCT
ejpam-6123	177	4	is	be	AUX
ejpam-6123	177	5	the	the	DET
ejpam-6123	177	6	only	only	ADJ
ejpam-6123	177	7	minimal	minimal	ADJ
ejpam-6123	177	8	connected	connected	ADJ
ejpam-6123	177	9	dominating	dominating	NOUN
ejpam-6123	177	10	set	set	NOUN
ejpam-6123	177	11	of	of	ADP
ejpam-6123	177	12	g.	g.	PROPN
ejpam-6123	177	13	thus	thus	ADV
ejpam-6123	177	14	γc(g	γc(g	PUNCT
ejpam-6123	177	15	)	)	PUNCT
ejpam-6123	178	1	=	=	SYM
ejpam-6123	178	2	a−1	a−1	NOUN
ejpam-6123	178	3	+	+	NOUN
ejpam-6123	178	4	1	1	NUM
ejpam-6123	178	5	=	=	NOUN
ejpam-6123	178	6	a.	a.	NOUN
ejpam-6123	178	7	also	also	ADV
ejpam-6123	178	8	,	,	PUNCT
ejpam-6123	178	9	note	note	VERB
ejpam-6123	178	10	that	that	SCONJ
ejpam-6123	178	11	the	the	DET
ejpam-6123	178	12	sets	set	NOUN
ejpam-6123	178	13	s1	s1	NOUN
ejpam-6123	178	14	=	=	PUNCT
ejpam-6123	178	15	{	{	PUNCT
ejpam-6123	178	16	x	x	NOUN
ejpam-6123	178	17	,	,	PUNCT
ejpam-6123	178	18	u1	u1	NOUN
ejpam-6123	178	19	,	,	PUNCT
ejpam-6123	178	20	u2	u2	NOUN
ejpam-6123	178	21	,	,	PUNCT
ejpam-6123	178	22	.	.	PUNCT
ejpam-6123	178	23	.	.	PUNCT
ejpam-6123	179	1	.	.	PUNCT
ejpam-6123	180	1	,	,	PUNCT
ejpam-6123	180	2	ua−1}∪	ua−1}∪	PROPN
ejpam-6123	180	3	(	(	PUNCT
ejpam-6123	180	4	k⋃	k⋃	X
ejpam-6123	180	5	j=1	j=1	PROPN
ejpam-6123	180	6	{	{	PUNCT
ejpam-6123	180	7	vj	vj	PROPN
ejpam-6123	180	8	}	}	PUNCT
ejpam-6123	180	9	)	)	PUNCT
ejpam-6123	180	10	and	and	CCONJ
ejpam-6123	180	11	s2	s2	VERB
ejpam-6123	180	12	=	=	SYM
ejpam-6123	180	13	{	{	PUNCT
ejpam-6123	180	14	x	x	NOUN
ejpam-6123	180	15	,	,	PUNCT
ejpam-6123	180	16	u1	u1	NOUN
ejpam-6123	180	17	,	,	PUNCT
ejpam-6123	180	18	u2	u2	NOUN
ejpam-6123	180	19	,	,	PUNCT
ejpam-6123	180	20	.	.	PUNCT
ejpam-6123	180	21	.	.	PUNCT
ejpam-6123	180	22	.	.	PUNCT
ejpam-6123	181	1	,	,	PUNCT
ejpam-6123	181	2	ua−1}∪	ua−1}∪	PROPN
ejpam-6123	181	3	(	(	PUNCT
ejpam-6123	181	4	k⋃	k⋃	X
ejpam-6123	181	5	j=1	j=1	PROPN
ejpam-6123	181	6	{	{	PUNCT
ejpam-6123	181	7	wj	wj	PROPN
ejpam-6123	181	8	,	,	PUNCT
ejpam-6123	181	9	hj	hj	PROPN
ejpam-6123	181	10	}	}	PUNCT
ejpam-6123	181	11	)	)	PUNCT
ejpam-6123	181	12	are	be	AUX
ejpam-6123	181	13	the	the	DET
ejpam-6123	181	14	only	only	ADJ
ejpam-6123	181	15	minimal	minimal	ADJ
ejpam-6123	181	16	connected	connected	ADJ
ejpam-6123	181	17	h.	h.	PROPN
ejpam-6123	181	18	nuenay	nuenay	PROPN
ejpam-6123	181	19	-	-	PUNCT
ejpam-6123	181	20	maglanquel	maglanquel	PROPN
ejpam-6123	181	21	/	/	SYM
ejpam-6123	181	22	eur	eur	PROPN
ejpam-6123	181	23	.	.	PUNCT
ejpam-6123	182	1	j.	j.	PROPN
ejpam-6123	182	2	pure	pure	PROPN
ejpam-6123	182	3	appl	appl	PROPN
ejpam-6123	182	4	.	.	PROPN
ejpam-6123	182	5	math	math	PROPN
ejpam-6123	182	6	,	,	PUNCT
ejpam-6123	182	7	18	18	NUM
ejpam-6123	182	8	(	(	PUNCT
ejpam-6123	182	9	3	3	NUM
ejpam-6123	182	10	)	)	PUNCT
ejpam-6123	182	11	(	(	PUNCT
ejpam-6123	182	12	2025	2025	NUM
ejpam-6123	182	13	)	)	PUNCT
ejpam-6123	182	14	,	,	PUNCT
ejpam-6123	182	15	6123	6123	NUM
ejpam-6123	182	16	7	7	NUM
ejpam-6123	182	17	of	of	ADP
ejpam-6123	182	18	14	14	NUM
ejpam-6123	182	19	....................................	....................................	PUNCT
ejpam-6123	182	20	....................................	....................................	PUNCT
ejpam-6123	183	1	....................................	....................................	PUNCT
ejpam-6123	183	2	....................................	....................................	PUNCT
ejpam-6123	184	1	........................................................................	........................................................................	PUNCT
ejpam-6123	184	2	................................................................................................................................................................	................................................................................................................................................................	PUNCT
ejpam-6123	184	3	................................	................................	PUNCT
ejpam-6123	184	4	.........	.........	PUNCT
ejpam-6123	184	5	........	........	PUNCT
ejpam-6123	184	6	........	........	PUNCT
ejpam-6123	184	7	........	........	PUNCT
ejpam-6123	184	8	........	........	PUNCT
ejpam-6123	184	9	........	........	PUNCT
ejpam-6123	184	10	........	........	PUNCT
ejpam-6123	184	11	........	........	PUNCT
ejpam-6123	184	12	........	........	PUNCT
ejpam-6123	184	13	........	........	PUNCT
ejpam-6123	184	14	.	.	PUNCT
ejpam-6123	185	1	..........	..........	PUNCT
ejpam-6123	185	2	.........	.........	PUNCT
ejpam-6123	186	1	.........	.........	PUNCT
ejpam-6123	186	2	.........	.........	PUNCT
ejpam-6123	187	1	.........	.........	PUNCT
ejpam-6123	187	2	.........	.........	PUNCT
ejpam-6123	188	1	.........	.........	PUNCT
ejpam-6123	188	2	.........	.........	PUNCT
ejpam-6123	189	1	.........	.........	PUNCT
ejpam-6123	189	2	.........	.........	PUNCT
ejpam-6123	190	1	.........	.........	PUNCT
ejpam-6123	190	2	..............................................................................................................	..............................................................................................................	PUNCT
ejpam-6123	190	3	.	.	PUNCT
ejpam-6123	190	4	.	.	PUNCT
ejpam-6123	190	5	.	.	PUNCT
ejpam-6123	191	1	ua−1	ua−1	NOUN
ejpam-6123	191	2	u1	u1	PROPN
ejpam-6123	191	3	u2u3	u2u3	PROPN
ejpam-6123	191	4	u4	u4	PROPN
ejpam-6123	191	5	x	x	PROPN
ejpam-6123	191	6	k1,a−1	k1,a−1	PROPN
ejpam-6123	191	7	....................................	....................................	PUNCT
ejpam-6123	191	8	....................................	....................................	PUNCT
ejpam-6123	192	1	....................................	....................................	PUNCT
ejpam-6123	192	2	....................................	....................................	PUNCT
ejpam-6123	193	1	........................................................................	........................................................................	PUNCT
ejpam-6123	193	2	................................................................................................................................................................	................................................................................................................................................................	PUNCT
ejpam-6123	193	3	................................	................................	PUNCT
ejpam-6123	193	4	.........	.........	PUNCT
ejpam-6123	193	5	........	........	PUNCT
ejpam-6123	193	6	........	........	PUNCT
ejpam-6123	193	7	........	........	PUNCT
ejpam-6123	193	8	........	........	PUNCT
ejpam-6123	193	9	........	........	PUNCT
ejpam-6123	193	10	........	........	PUNCT
ejpam-6123	193	11	........	........	PUNCT
ejpam-6123	193	12	........	........	PUNCT
ejpam-6123	193	13	........	........	PUNCT
ejpam-6123	193	14	.	.	PUNCT
ejpam-6123	194	1	..........	..........	PUNCT
ejpam-6123	194	2	.........	.........	PUNCT
ejpam-6123	195	1	.........	.........	PUNCT
ejpam-6123	195	2	.........	.........	PUNCT
ejpam-6123	196	1	.........	.........	PUNCT
ejpam-6123	196	2	.........	.........	PUNCT
ejpam-6123	197	1	.........	.........	PUNCT
ejpam-6123	197	2	.........	.........	PUNCT
ejpam-6123	198	1	.........	.........	PUNCT
ejpam-6123	198	2	.........	.........	PUNCT
ejpam-6123	199	1	.........	.........	PUNCT
ejpam-6123	199	2	..............................................................................................................	..............................................................................................................	PUNCT
ejpam-6123	199	3	.	.	PUNCT
ejpam-6123	199	4	.	.	PUNCT
ejpam-6123	199	5	.	.	PUNCT
ejpam-6123	200	1	....................................	....................................	PUNCT
ejpam-6123	200	2	........................................................................	........................................................................	PUNCT
ejpam-6123	201	1	....................................	....................................	PUNCT
ejpam-6123	201	2	....................................	....................................	PUNCT
ejpam-6123	201	3	........................................................	........................................................	PUNCT
ejpam-6123	201	4	........................................................	........................................................	PUNCT
ejpam-6123	202	1	.........	.........	PUNCT
ejpam-6123	202	2	........	........	PUNCT
ejpam-6123	202	3	........	........	PUNCT
ejpam-6123	202	4	........	........	PUNCT
ejpam-6123	202	5	........	........	PUNCT
ejpam-6123	203	1	........	........	PUNCT
ejpam-6123	203	2	.......	.......	PUNCT
ejpam-6123	203	3	...........................................................	...........................................................	PUNCT
ejpam-6123	204	1	.........	.........	PUNCT
ejpam-6123	204	2	.........	.........	PUNCT
ejpam-6123	205	1	.........	.........	PUNCT
ejpam-6123	205	2	.........	.........	PUNCT
ejpam-6123	206	1	....................................	....................................	PUNCT
ejpam-6123	206	2	....................................	....................................	PUNCT
ejpam-6123	206	3	..................................................................................................................................	..................................................................................................................................	PUNCT
ejpam-6123	206	4	........	........	PUNCT
ejpam-6123	206	5	........	........	PUNCT
ejpam-6123	206	6	.......	.......	PUNCT
ejpam-6123	206	7	........	........	PUNCT
ejpam-6123	206	8	........	........	PUNCT
ejpam-6123	206	9	........	........	PUNCT
ejpam-6123	207	1	.........	.........	PUNCT
ejpam-6123	207	2	.........	.........	PUNCT
ejpam-6123	208	1	..........	..........	PUNCT
ejpam-6123	208	2	............	............	PUNCT
ejpam-6123	208	3	.............................................................................................................................	.............................................................................................................................	PUNCT
ejpam-6123	209	1	ua−1	ua−1	NOUN
ejpam-6123	209	2	x	x	SYM
ejpam-6123	209	3	va−1	va−1	PROPN
ejpam-6123	209	4	u1	u1	NOUN
ejpam-6123	209	5	v1	v1	NOUN
ejpam-6123	209	6	u2	u2	PROPN
ejpam-6123	209	7	v2	v2	PROPN
ejpam-6123	209	8	u3	u3	NOUN
ejpam-6123	209	9	v3	v3	PROPN
ejpam-6123	209	10	u4v4v4	u4v4v4	PROPN
ejpam-6123	209	11	w1	w1	PROPN
ejpam-6123	209	12	h1	h1	NOUN
ejpam-6123	209	13	g	g	PROPN
ejpam-6123	209	14	=	=	PROPN
ejpam-6123	209	15	g1	g1	PROPN
ejpam-6123	209	16	figure	figure	NOUN
ejpam-6123	209	17	2	2	NUM
ejpam-6123	209	18	:	:	PUNCT
ejpam-6123	209	19	g1	g1	NOUN
ejpam-6123	209	20	obtained	obtain	VERB
ejpam-6123	209	21	from	from	ADP
ejpam-6123	209	22	k1,a−1	k1,a−1	PROPN
ejpam-6123	209	23	....................................	....................................	PUNCT
ejpam-6123	210	1	....................................	....................................	PUNCT
ejpam-6123	210	2	....................................	....................................	PUNCT
ejpam-6123	211	1	....................................	....................................	PUNCT
ejpam-6123	211	2	........................................................................	........................................................................	PUNCT
ejpam-6123	211	3	................................................................................................................................................................	................................................................................................................................................................	PUNCT
ejpam-6123	211	4	................................	................................	PUNCT
ejpam-6123	211	5	.........	.........	PUNCT
ejpam-6123	211	6	........	........	PUNCT
ejpam-6123	211	7	........	........	PUNCT
ejpam-6123	211	8	........	........	PUNCT
ejpam-6123	211	9	........	........	PUNCT
ejpam-6123	211	10	........	........	PUNCT
ejpam-6123	211	11	........	........	PUNCT
ejpam-6123	211	12	........	........	PUNCT
ejpam-6123	211	13	........	........	PUNCT
ejpam-6123	211	14	........	........	PUNCT
ejpam-6123	211	15	.	.	PUNCT
ejpam-6123	212	1	..........	..........	PUNCT
ejpam-6123	212	2	.........	.........	PUNCT
ejpam-6123	213	1	.........	.........	PUNCT
ejpam-6123	213	2	.........	.........	PUNCT
ejpam-6123	214	1	.........	.........	PUNCT
ejpam-6123	214	2	.........	.........	PUNCT
ejpam-6123	215	1	.........	.........	PUNCT
ejpam-6123	215	2	.........	.........	PUNCT
ejpam-6123	216	1	.........	.........	PUNCT
ejpam-6123	216	2	.........	.........	PUNCT
ejpam-6123	217	1	.........	.........	PUNCT
ejpam-6123	217	2	..............................................................................................................	..............................................................................................................	PUNCT
ejpam-6123	217	3	.	.	PUNCT
ejpam-6123	217	4	.	.	PUNCT
ejpam-6123	217	5	.	.	PUNCT
ejpam-6123	218	1	....................................	....................................	PUNCT
ejpam-6123	218	2	........................................................................	........................................................................	PUNCT
ejpam-6123	219	1	....................................	....................................	PUNCT
ejpam-6123	219	2	....................................	....................................	PUNCT
ejpam-6123	219	3	........................................................	........................................................	PUNCT
ejpam-6123	219	4	........................................................	........................................................	PUNCT
ejpam-6123	220	1	.........	.........	PUNCT
ejpam-6123	220	2	........	........	PUNCT
ejpam-6123	220	3	........	........	PUNCT
ejpam-6123	220	4	........	........	PUNCT
ejpam-6123	220	5	........	........	PUNCT
ejpam-6123	221	1	........	........	PUNCT
ejpam-6123	221	2	.......	.......	PUNCT
ejpam-6123	221	3	...........................................................	...........................................................	PUNCT
ejpam-6123	222	1	.........	.........	PUNCT
ejpam-6123	222	2	.........	.........	PUNCT
ejpam-6123	223	1	.........	.........	PUNCT
ejpam-6123	223	2	.........	.........	PUNCT
ejpam-6123	224	1	....................................	....................................	PUNCT
ejpam-6123	224	2	....................................	....................................	PUNCT
ejpam-6123	224	3	..................................................................................................................................	..................................................................................................................................	PUNCT
ejpam-6123	224	4	........	........	PUNCT
ejpam-6123	224	5	........	........	PUNCT
ejpam-6123	224	6	.......	.......	PUNCT
ejpam-6123	224	7	........	........	PUNCT
ejpam-6123	224	8	........	........	PUNCT
ejpam-6123	224	9	........	........	PUNCT
ejpam-6123	225	1	.........	.........	PUNCT
ejpam-6123	225	2	.........	.........	PUNCT
ejpam-6123	226	1	..........	..........	PUNCT
ejpam-6123	226	2	............	............	PUNCT
ejpam-6123	226	3	.............................................................................................................................	.............................................................................................................................	PUNCT
ejpam-6123	227	1	ua−1	ua−1	NOUN
ejpam-6123	227	2	x	x	SYM
ejpam-6123	227	3	va−1	va−1	PROPN
ejpam-6123	227	4	u1	u1	NOUN
ejpam-6123	227	5	v1	v1	NOUN
ejpam-6123	227	6	u2	u2	PROPN
ejpam-6123	227	7	v2	v2	PROPN
ejpam-6123	227	8	u3	u3	NOUN
ejpam-6123	227	9	v3	v3	PROPN
ejpam-6123	227	10	u4v4v4	u4v4v4	PROPN
ejpam-6123	227	11	w1	w1	PROPN
ejpam-6123	227	12	h1	h1	PROPN
ejpam-6123	227	13	g1	g1	PROPN
ejpam-6123	227	14	....................................	....................................	PUNCT
ejpam-6123	227	15	....................................	....................................	PUNCT
ejpam-6123	227	16	....................................	....................................	PUNCT
ejpam-6123	227	17	....................................	....................................	PUNCT
ejpam-6123	227	18	........................................................................	........................................................................	PUNCT
ejpam-6123	227	19	................................................................................................................................................................	................................................................................................................................................................	PUNCT
ejpam-6123	227	20	................................	................................	PUNCT
ejpam-6123	227	21	.........	.........	PUNCT
ejpam-6123	227	22	........	........	PUNCT
ejpam-6123	227	23	........	........	PUNCT
ejpam-6123	227	24	........	........	PUNCT
ejpam-6123	227	25	........	........	PUNCT
ejpam-6123	227	26	........	........	PUNCT
ejpam-6123	227	27	........	........	PUNCT
ejpam-6123	227	28	........	........	PUNCT
ejpam-6123	227	29	........	........	PUNCT
ejpam-6123	227	30	........	........	PUNCT
ejpam-6123	227	31	.	.	PUNCT
ejpam-6123	228	1	..........	..........	PUNCT
ejpam-6123	228	2	.........	.........	PUNCT
ejpam-6123	229	1	.........	.........	PUNCT
ejpam-6123	229	2	.........	.........	PUNCT
ejpam-6123	230	1	.........	.........	PUNCT
ejpam-6123	230	2	.........	.........	PUNCT
ejpam-6123	231	1	.........	.........	PUNCT
ejpam-6123	231	2	.........	.........	PUNCT
ejpam-6123	232	1	.........	.........	PUNCT
ejpam-6123	232	2	.........	.........	PUNCT
ejpam-6123	233	1	.........	.........	PUNCT
ejpam-6123	233	2	..............................................................................................................	..............................................................................................................	PUNCT
ejpam-6123	233	3	.	.	PUNCT
ejpam-6123	233	4	.	.	PUNCT
ejpam-6123	233	5	.	.	PUNCT
ejpam-6123	234	1	....................................	....................................	PUNCT
ejpam-6123	234	2	....................................	....................................	PUNCT
ejpam-6123	235	1	....................................	....................................	PUNCT
ejpam-6123	235	2	....................................	....................................	PUNCT
ejpam-6123	235	3	........................................................	........................................................	PUNCT
ejpam-6123	236	1	..........................................................................................................................	..........................................................................................................................	PUNCT
ejpam-6123	236	2	.........	.........	PUNCT
ejpam-6123	236	3	........	........	PUNCT
ejpam-6123	236	4	........	........	PUNCT
ejpam-6123	236	5	........	........	PUNCT
ejpam-6123	236	6	........	........	PUNCT
ejpam-6123	236	7	........	........	PUNCT
ejpam-6123	236	8	.......	.......	PUNCT
ejpam-6123	236	9	...........................................................	...........................................................	PUNCT
ejpam-6123	237	1	.........	.........	PUNCT
ejpam-6123	237	2	.........	.........	PUNCT
ejpam-6123	238	1	.........	.........	PUNCT
ejpam-6123	238	2	.........	.........	PUNCT
ejpam-6123	239	1	....................................	....................................	PUNCT
ejpam-6123	239	2	....................................	....................................	PUNCT
ejpam-6123	240	1	....................................	....................................	PUNCT
ejpam-6123	240	2	............	............	PUNCT
ejpam-6123	240	3	...........	...........	PUNCT
ejpam-6123	240	4	...........	...........	PUNCT
ejpam-6123	240	5	...........	...........	PUNCT
ejpam-6123	240	6	...........	...........	PUNCT
ejpam-6123	240	7	...........	...........	PUNCT
ejpam-6123	240	8	...........	...........	PUNCT
ejpam-6123	240	9	...........	...........	PUNCT
ejpam-6123	240	10	...........	...........	PUNCT
ejpam-6123	240	11	...........	...........	PUNCT
ejpam-6123	240	12	...........	...........	PUNCT
ejpam-6123	240	13	...........	...........	PUNCT
ejpam-6123	240	14	...........	...........	PUNCT
ejpam-6123	240	15	...........	...........	PUNCT
ejpam-6123	240	16	...........	...........	PUNCT
ejpam-6123	240	17	...........	...........	PUNCT
ejpam-6123	240	18	...........	...........	PUNCT
ejpam-6123	240	19	.......	.......	PUNCT
ejpam-6123	240	20	.............	.............	PUNCT
ejpam-6123	240	21	............	............	PUNCT
ejpam-6123	240	22	............	............	PUNCT
ejpam-6123	240	23	............	............	PUNCT
ejpam-6123	240	24	............	............	PUNCT
ejpam-6123	240	25	............	............	PUNCT
ejpam-6123	240	26	............	............	PUNCT
ejpam-6123	240	27	............	............	PUNCT
ejpam-6123	240	28	............	............	PUNCT
ejpam-6123	240	29	............	............	PUNCT
ejpam-6123	240	30	............	............	PUNCT
ejpam-6123	240	31	............	............	PUNCT
ejpam-6123	240	32	............	............	PUNCT
ejpam-6123	240	33	..........	..........	PUNCT
ejpam-6123	240	34	........................................................	........................................................	PUNCT
ejpam-6123	240	35	.........	.........	PUNCT
ejpam-6123	241	1	........	........	PUNCT
ejpam-6123	241	2	........	........	PUNCT
ejpam-6123	242	1	.....	.....	PUNCT
ejpam-6123	242	2	.........	.........	PUNCT
ejpam-6123	243	1	.........	.........	PUNCT
ejpam-6123	243	2	.........	.........	PUNCT
ejpam-6123	244	1	.........	.........	PUNCT
ejpam-6123	244	2	...........	...........	PUNCT
ejpam-6123	244	3	.............................................................................................................................................................................................................................................................................................	.............................................................................................................................................................................................................................................................................................	PUNCT
ejpam-6123	244	4	....................................	....................................	PUNCT
ejpam-6123	245	1	....................................	....................................	PUNCT
ejpam-6123	245	2	....................................	....................................	PUNCT
ejpam-6123	246	1	......................................................................................................................................................	......................................................................................................................................................	PUNCT
ejpam-6123	246	2	...........................	...........................	PUNCT
ejpam-6123	246	3	...........................	...........................	PUNCT
ejpam-6123	246	4	...........................	...........................	PUNCT
ejpam-6123	246	5	..................	..................	PUNCT
ejpam-6123	246	6	........................................................	........................................................	PUNCT
ejpam-6123	246	7	.........	.........	PUNCT
ejpam-6123	247	1	........	........	PUNCT
ejpam-6123	247	2	........	........	PUNCT
ejpam-6123	248	1	.....	.....	PUNCT
ejpam-6123	248	2	.........	.........	PUNCT
ejpam-6123	249	1	.........	.........	PUNCT
ejpam-6123	249	2	...........	...........	PUNCT
ejpam-6123	249	3	..............	..............	PUNCT
ejpam-6123	249	4	............................................................................................................................................................................................	............................................................................................................................................................................................	PUNCT
ejpam-6123	250	1	....................................	....................................	PUNCT
ejpam-6123	250	2	....................................	....................................	PUNCT
ejpam-6123	251	1	....................................	....................................	PUNCT
ejpam-6123	251	2	...................................................................................................................................................................................................	...................................................................................................................................................................................................	PUNCT
ejpam-6123	251	3	...................................................................................................................................................................................................................................	...................................................................................................................................................................................................................................	PUNCT
ejpam-6123	251	4	........................................................	........................................................	PUNCT
ejpam-6123	252	1	.........	.........	PUNCT
ejpam-6123	252	2	........	........	PUNCT
ejpam-6123	253	1	........	........	PUNCT
ejpam-6123	253	2	.....	.....	PUNCT
ejpam-6123	254	1	.........	.........	PUNCT
ejpam-6123	254	2	.........	.........	PUNCT
ejpam-6123	255	1	..........	..........	PUNCT
ejpam-6123	255	2	..........	..........	PUNCT
ejpam-6123	255	3	...........	...........	PUNCT
ejpam-6123	255	4	...........	...........	PUNCT
ejpam-6123	255	5	............	............	PUNCT
ejpam-6123	255	6	............	............	PUNCT
ejpam-6123	255	7	.............	.............	PUNCT
ejpam-6123	255	8	.............	.............	PUNCT
ejpam-6123	256	1	..............	..............	PUNCT
ejpam-6123	256	2	..............	..............	PUNCT
ejpam-6123	256	3	...............	...............	PUNCT
ejpam-6123	256	4	...............	...............	PUNCT
ejpam-6123	257	1	................	................	PUNCT
ejpam-6123	257	2	................	................	PUNCT
ejpam-6123	257	3	.................	.................	PUNCT
ejpam-6123	258	1	.................	.................	PUNCT
ejpam-6123	258	2	.............	.............	PUNCT
ejpam-6123	258	3	ua−1	ua−1	NOUN
ejpam-6123	258	4	x	x	SYM
ejpam-6123	258	5	va−1	va−1	PROPN
ejpam-6123	258	6	u1	u1	NOUN
ejpam-6123	258	7	u2	u2	PROPN
ejpam-6123	258	8	v2	v2	PROPN
ejpam-6123	258	9	u3	u3	NOUN
ejpam-6123	258	10	v3	v3	PROPN
ejpam-6123	258	11	u4v4v4	u4v4v4	PROPN
ejpam-6123	258	12	v1	v1	PROPN
ejpam-6123	258	13	v2	v2	PROPN
ejpam-6123	258	14	vk	vk	NOUN
ejpam-6123	258	15	h1	h1	PROPN
ejpam-6123	258	16	h2	h2	PROPN
ejpam-6123	258	17	hk	hk	PROPN
ejpam-6123	258	18	w1	w1	PROPN
ejpam-6123	258	19	w2	w2	PROPN
ejpam-6123	258	20	wk	wk	INTJ
ejpam-6123	258	21	...	...	PUNCT
ejpam-6123	258	22	g	g	PROPN
ejpam-6123	258	23	=	=	PUNCT
ejpam-6123	258	24	g2	g2	PROPN
ejpam-6123	258	25	figure	figure	NOUN
ejpam-6123	258	26	3	3	NUM
ejpam-6123	258	27	:	:	PUNCT
ejpam-6123	258	28	g2	g2	PROPN
ejpam-6123	258	29	obtained	obtain	VERB
ejpam-6123	258	30	from	from	ADP
ejpam-6123	258	31	g1	g1	PROPN
ejpam-6123	258	32	equitable	equitable	ADJ
ejpam-6123	258	33	dominating	dominating	NOUN
ejpam-6123	258	34	sets	set	NOUN
ejpam-6123	258	35	of	of	ADP
ejpam-6123	258	36	g.	g.	PROPN
ejpam-6123	258	37	now	now	ADV
ejpam-6123	258	38	,	,	PUNCT
ejpam-6123	258	39	|s1|	|s1|	NOUN
ejpam-6123	258	40	=	=	SYM
ejpam-6123	258	41	a−	a−	PROPN
ejpam-6123	258	42	1	1	NUM
ejpam-6123	258	43	+	+	NOUN
ejpam-6123	258	44	1	1	NUM
ejpam-6123	258	45	+	+	NOUN
ejpam-6123	258	46	k	k	NOUN
ejpam-6123	258	47	=	=	SYM
ejpam-6123	258	48	a+	a+	PUNCT
ejpam-6123	258	49	k	k	PROPN
ejpam-6123	258	50	and	and	CCONJ
ejpam-6123	258	51	|s2|	|s2|	NOUN
ejpam-6123	258	52	=	=	SYM
ejpam-6123	258	53	a−	a−	PROPN
ejpam-6123	258	54	1	1	NUM
ejpam-6123	259	1	+	+	NOUN
ejpam-6123	259	2	2k	2k	NOUN
ejpam-6123	259	3	=	=	SYM
ejpam-6123	259	4	a+	a+	PUNCT
ejpam-6123	259	5	2k	2k	NOUN
ejpam-6123	259	6	−	−	NOUN
ejpam-6123	259	7	1	1	X
ejpam-6123	259	8	.	.	PUNCT
ejpam-6123	260	1	since	since	SCONJ
ejpam-6123	260	2	k	k	PROPN
ejpam-6123	260	3	>	>	X
ejpam-6123	260	4	1	1	NUM
ejpam-6123	260	5	,	,	PUNCT
ejpam-6123	260	6	|s2|	|s2|	NOUN
ejpam-6123	260	7	=	=	SYM
ejpam-6123	260	8	a+	a+	PUNCT
ejpam-6123	260	9	2k	2k	NOUN
ejpam-6123	260	10	−	−	NOUN
ejpam-6123	260	11	1	1	NUM
ejpam-6123	260	12	>	>	PUNCT
ejpam-6123	260	13	a+	a+	PUNCT
ejpam-6123	260	14	k	k	PROPN
ejpam-6123	260	15	=	=	NOUN
ejpam-6123	260	16	|s1|	|s1|	NOUN
ejpam-6123	260	17	.	.	PUNCT
ejpam-6123	261	1	thus	thus	ADV
ejpam-6123	261	2	,	,	PUNCT
ejpam-6123	261	3	γce(g	γce(g	PROPN
ejpam-6123	261	4	)	)	PUNCT
ejpam-6123	261	5	=	=	SYM
ejpam-6123	261	6	|s1|	|s1|	NOUN
ejpam-6123	261	7	=	=	SYM
ejpam-6123	261	8	a+	a+	PUNCT
ejpam-6123	261	9	k	k	PROPN
ejpam-6123	261	10	=	=	SYM
ejpam-6123	261	11	b	b	PROPN
ejpam-6123	261	12	>	>	X
ejpam-6123	261	13	γc(g	γc(g	PROPN
ejpam-6123	261	14	)	)	PUNCT
ejpam-6123	261	15	=	=	SYM
ejpam-6123	261	16	a.	a.	NOUN
ejpam-6123	261	17	theorem	theorem	NOUN
ejpam-6123	261	18	8	8	NUM
ejpam-6123	261	19	.	.	PUNCT
ejpam-6123	262	1	for	for	ADP
ejpam-6123	262	2	any	any	DET
ejpam-6123	262	3	regular	regular	ADJ
ejpam-6123	262	4	graph	graph	NOUN
ejpam-6123	262	5	g	g	NOUN
ejpam-6123	262	6	,	,	PUNCT
ejpam-6123	262	7	γce(g	γce(g	PROPN
ejpam-6123	262	8	)	)	PUNCT
ejpam-6123	262	9	=	=	SYM
ejpam-6123	262	10	γc(g	γc(g	X
ejpam-6123	262	11	)	)	PUNCT
ejpam-6123	262	12	.	.	PUNCT
ejpam-6123	263	1	proof	proof	NOUN
ejpam-6123	263	2	.	.	PUNCT
ejpam-6123	264	1	let	let	VERB
ejpam-6123	264	2	s	s	PRON
ejpam-6123	264	3	⊆	⊆	NUM
ejpam-6123	264	4	v	v	NOUN
ejpam-6123	264	5	(	(	PUNCT
ejpam-6123	264	6	g	g	NOUN
ejpam-6123	264	7	)	)	PUNCT
ejpam-6123	264	8	be	be	AUX
ejpam-6123	264	9	a	a	DET
ejpam-6123	264	10	γc	γc	NOUN
ejpam-6123	264	11	-	-	PUNCT
ejpam-6123	264	12	set	set	NOUN
ejpam-6123	264	13	of	of	ADP
ejpam-6123	264	14	g	g	NOUN
ejpam-6123	264	15	and	and	CCONJ
ejpam-6123	265	1	u	u	PROPN
ejpam-6123	265	2	∈	∈	PROPN
ejpam-6123	265	3	v	v	ADP
ejpam-6123	265	4	(	(	PUNCT
ejpam-6123	265	5	g	g	NOUN
ejpam-6123	265	6	)	)	PUNCT
ejpam-6123	265	7	\	\	PUNCT
ejpam-6123	265	8	s.	s.	PROPN
ejpam-6123	265	9	since	since	SCONJ
ejpam-6123	265	10	s	s	PROPN
ejpam-6123	265	11	is	be	AUX
ejpam-6123	265	12	a	a	DET
ejpam-6123	265	13	dominating	dominating	NOUN
ejpam-6123	265	14	set	set	NOUN
ejpam-6123	265	15	in	in	ADP
ejpam-6123	265	16	g	g	NOUN
ejpam-6123	265	17	,	,	PUNCT
ejpam-6123	265	18	there	there	PRON
ejpam-6123	265	19	exists	exist	VERB
ejpam-6123	265	20	v	v	ADP
ejpam-6123	265	21	∈	∈	PROPN
ejpam-6123	265	22	s	s	VERB
ejpam-6123	265	23	such	such	ADJ
ejpam-6123	265	24	that	that	DET
ejpam-6123	265	25	uv	uv	PROPN
ejpam-6123	265	26	∈	∈	PROPN
ejpam-6123	265	27	e(g	e(g	PROPN
ejpam-6123	265	28	)	)	PUNCT
ejpam-6123	265	29	.	.	PUNCT
ejpam-6123	266	1	since	since	SCONJ
ejpam-6123	266	2	g	g	PROPN
ejpam-6123	266	3	is	be	AUX
ejpam-6123	266	4	a	a	DET
ejpam-6123	266	5	regular	regular	ADJ
ejpam-6123	266	6	graph	graph	NOUN
ejpam-6123	266	7	,	,	PUNCT
ejpam-6123	266	8	say	say	VERB
ejpam-6123	266	9	n	n	ADV
ejpam-6123	266	10	regular	regular	ADJ
ejpam-6123	266	11	,	,	PUNCT
ejpam-6123	266	12	degg(x	degg(x	NOUN
ejpam-6123	266	13	)	)	PUNCT
ejpam-6123	266	14	=	=	SYM
ejpam-6123	266	15	n	n	PROPN
ejpam-6123	266	16	for	for	ADP
ejpam-6123	266	17	all	all	PRON
ejpam-6123	266	18	x	x	SYM
ejpam-6123	266	19	∈	∈	NOUN
ejpam-6123	266	20	v	v	NOUN
ejpam-6123	266	21	(	(	PUNCT
ejpam-6123	266	22	g	g	NOUN
ejpam-6123	266	23	)	)	PUNCT
ejpam-6123	266	24	.	.	PUNCT
ejpam-6123	267	1	thus	thus	ADV
ejpam-6123	267	2	,	,	PUNCT
ejpam-6123	267	3	|	|	ADV
ejpam-6123	267	4	degg(u)−	degg(u)−	VERB
ejpam-6123	267	5	degg(v)|	degg(v)|	PROPN
ejpam-6123	267	6	=	=	SYM
ejpam-6123	267	7	0	0	PUNCT
ejpam-6123	267	8	<	<	X
ejpam-6123	267	9	1	1	NUM
ejpam-6123	267	10	implying	imply	VERB
ejpam-6123	267	11	that	that	SCONJ
ejpam-6123	267	12	s	s	VERB
ejpam-6123	267	13	is	be	AUX
ejpam-6123	267	14	an	an	DET
ejpam-6123	267	15	equitable	equitable	ADJ
ejpam-6123	267	16	set	set	NOUN
ejpam-6123	267	17	in	in	ADP
ejpam-6123	267	18	g.	g.	PROPN
ejpam-6123	267	19	thus	thus	ADV
ejpam-6123	267	20	s	s	VERB
ejpam-6123	267	21	is	be	AUX
ejpam-6123	267	22	a	a	DET
ejpam-6123	267	23	connected	connect	VERB
ejpam-6123	267	24	equitable	equitable	ADJ
ejpam-6123	267	25	dominating	dominating	NOUN
ejpam-6123	267	26	set	set	VERB
ejpam-6123	267	27	in	in	ADP
ejpam-6123	267	28	g.	g.	PROPN
ejpam-6123	267	29	hence	hence	ADV
ejpam-6123	267	30	,	,	PUNCT
ejpam-6123	267	31	γce(g	γce(g	PROPN
ejpam-6123	267	32	)	)	PUNCT
ejpam-6123	267	33	≤	≤	NUM
ejpam-6123	267	34	|s|	|s|	PROPN
ejpam-6123	267	35	=	=	NOUN
ejpam-6123	267	36	γc(g	γc(g	NUM
ejpam-6123	267	37	)	)	PUNCT
ejpam-6123	267	38	.	.	PUNCT
ejpam-6123	268	1	from	from	ADP
ejpam-6123	268	2	remark	remark	NOUN
ejpam-6123	268	3	4	4	NUM
ejpam-6123	268	4	,	,	PUNCT
ejpam-6123	268	5	γce(g	γce(g	PROPN
ejpam-6123	268	6	)	)	PUNCT
ejpam-6123	268	7	=	=	SYM
ejpam-6123	268	8	γc(g	γc(g	X
ejpam-6123	268	9	)	)	PUNCT
ejpam-6123	268	10	.	.	PUNCT
ejpam-6123	269	1	h.	h.	PROPN
ejpam-6123	269	2	nuenay	nuenay	PROPN
ejpam-6123	269	3	-	-	PUNCT
ejpam-6123	269	4	maglanquel	maglanquel	PROPN
ejpam-6123	269	5	/	/	SYM
ejpam-6123	269	6	eur	eur	PROPN
ejpam-6123	269	7	.	.	PUNCT
ejpam-6123	270	1	j.	j.	PROPN
ejpam-6123	270	2	pure	pure	PROPN
ejpam-6123	270	3	appl	appl	PROPN
ejpam-6123	270	4	.	.	PROPN
ejpam-6123	270	5	math	math	PROPN
ejpam-6123	270	6	,	,	PUNCT
ejpam-6123	270	7	18	18	NUM
ejpam-6123	270	8	(	(	PUNCT
ejpam-6123	270	9	3	3	NUM
ejpam-6123	270	10	)	)	PUNCT
ejpam-6123	270	11	(	(	PUNCT
ejpam-6123	270	12	2025	2025	NUM
ejpam-6123	270	13	)	)	PUNCT
ejpam-6123	270	14	,	,	PUNCT
ejpam-6123	270	15	6123	6123	NUM
ejpam-6123	270	16	8	8	NUM
ejpam-6123	270	17	of	of	ADP
ejpam-6123	270	18	14	14	NUM
ejpam-6123	270	19	3	3	NUM
ejpam-6123	270	20	.	.	PUNCT
ejpam-6123	271	1	the	the	DET
ejpam-6123	271	2	join	join	NOUN
ejpam-6123	271	3	of	of	ADP
ejpam-6123	271	4	graphs	graph	NOUN
ejpam-6123	271	5	the	the	DET
ejpam-6123	271	6	join	join	NOUN
ejpam-6123	271	7	of	of	ADP
ejpam-6123	271	8	two	two	NUM
ejpam-6123	271	9	graphs	graph	NOUN
ejpam-6123	271	10	g	g	NOUN
ejpam-6123	271	11	and	and	CCONJ
ejpam-6123	271	12	h	h	NOUN
ejpam-6123	271	13	,	,	PUNCT
ejpam-6123	271	14	denoted	denote	VERB
ejpam-6123	271	15	by	by	ADP
ejpam-6123	271	16	g	g	PROPN
ejpam-6123	271	17	+	+	PROPN
ejpam-6123	271	18	h	h	NOUN
ejpam-6123	271	19	,	,	PUNCT
ejpam-6123	271	20	is	be	AUX
ejpam-6123	271	21	the	the	DET
ejpam-6123	271	22	graph	graph	NOUN
ejpam-6123	271	23	with	with	ADP
ejpam-6123	271	24	vertex	vertex	NOUN
ejpam-6123	271	25	-	-	PUNCT
ejpam-6123	271	26	set	set	VERB
ejpam-6123	271	27	v	v	NOUN
ejpam-6123	271	28	(	(	PUNCT
ejpam-6123	271	29	g+h	g+h	NOUN
ejpam-6123	271	30	)	)	PUNCT
ejpam-6123	271	31	=	=	SYM
ejpam-6123	271	32	v	v	NOUN
ejpam-6123	271	33	(	(	PUNCT
ejpam-6123	271	34	g)∪	g)∪	VERB
ejpam-6123	271	35	v	v	NUM
ejpam-6123	271	36	(	(	PUNCT
ejpam-6123	271	37	h	h	NOUN
ejpam-6123	271	38	)	)	PUNCT
ejpam-6123	271	39	and	and	CCONJ
ejpam-6123	271	40	edge	edge	NOUN
ejpam-6123	271	41	-	-	PUNCT
ejpam-6123	271	42	set	set	VERB
ejpam-6123	271	43	e(g+h	e(g+h	NUM
ejpam-6123	271	44	)	)	PUNCT
ejpam-6123	271	45	=	=	SYM
ejpam-6123	272	1	e(g)∪e(h)∪	e(g)∪e(h)∪	NOUN
ejpam-6123	272	2	{	{	PUNCT
ejpam-6123	272	3	uv	uv	NOUN
ejpam-6123	272	4	:	:	PUNCT
ejpam-6123	272	5	u	u	PROPN
ejpam-6123	272	6	∈	∈	PROPN
ejpam-6123	272	7	v	v	ADP
ejpam-6123	272	8	(	(	PUNCT
ejpam-6123	272	9	g	g	NOUN
ejpam-6123	272	10	)	)	PUNCT
ejpam-6123	272	11	,	,	PUNCT
ejpam-6123	272	12	v	v	X
ejpam-6123	272	13	∈	∈	PROPN
ejpam-6123	272	14	v	v	NOUN
ejpam-6123	272	15	(	(	PUNCT
ejpam-6123	272	16	h	h	NOUN
ejpam-6123	272	17	)	)	PUNCT
ejpam-6123	272	18	}	}	PUNCT
ejpam-6123	272	19	.	.	PUNCT
ejpam-6123	273	1	theorem	theorem	NOUN
ejpam-6123	273	2	9	9	NUM
ejpam-6123	273	3	.	.	PUNCT
ejpam-6123	274	1	let	let	VERB
ejpam-6123	274	2	g	g	PRON
ejpam-6123	274	3	be	be	AUX
ejpam-6123	274	4	a	a	DET
ejpam-6123	274	5	graph	graph	NOUN
ejpam-6123	274	6	such	such	ADJ
ejpam-6123	274	7	that	that	DET
ejpam-6123	274	8	∆(g	∆(g	NOUN
ejpam-6123	274	9	)	)	PUNCT
ejpam-6123	274	10	≤	≤	NOUN
ejpam-6123	274	11	|v	|v	X
ejpam-6123	274	12	(	(	PUNCT
ejpam-6123	274	13	g)−	g)−	PROPN
ejpam-6123	274	14	3|	3|	PROPN
ejpam-6123	274	15	.	.	PUNCT
ejpam-6123	275	1	then	then	ADV
ejpam-6123	275	2	s	s	VERB
ejpam-6123	275	3	⊆	⊆	NUM
ejpam-6123	275	4	v	v	NOUN
ejpam-6123	275	5	(	(	PUNCT
ejpam-6123	275	6	k1	k1	NOUN
ejpam-6123	275	7	+	+	NOUN
ejpam-6123	275	8	g	g	NOUN
ejpam-6123	275	9	)	)	PUNCT
ejpam-6123	275	10	is	be	AUX
ejpam-6123	275	11	a	a	DET
ejpam-6123	275	12	connected	connect	VERB
ejpam-6123	275	13	equitable	equitable	ADJ
ejpam-6123	275	14	dominating	dominating	NOUN
ejpam-6123	275	15	set	set	NOUN
ejpam-6123	275	16	of	of	ADP
ejpam-6123	275	17	k1	k1	NOUN
ejpam-6123	276	1	+	+	ADP
ejpam-6123	276	2	g	g	PROPN
ejpam-6123	276	3	if	if	SCONJ
ejpam-6123	276	4	and	and	CCONJ
ejpam-6123	276	5	only	only	ADV
ejpam-6123	276	6	if	if	SCONJ
ejpam-6123	276	7	s	s	VERB
ejpam-6123	276	8	=	=	SYM
ejpam-6123	276	9	v	v	PROPN
ejpam-6123	276	10	(	(	PUNCT
ejpam-6123	276	11	k1	k1	NOUN
ejpam-6123	276	12	)	)	PUNCT
ejpam-6123	276	13	∪	∪	NOUN
ejpam-6123	276	14	sg	sg	ADP
ejpam-6123	276	15	where	where	SCONJ
ejpam-6123	276	16	sg	sg	PROPN
ejpam-6123	276	17	is	be	AUX
ejpam-6123	276	18	an	an	DET
ejpam-6123	276	19	equitable	equitable	ADJ
ejpam-6123	276	20	dominating	dominating	NOUN
ejpam-6123	276	21	set	set	NOUN
ejpam-6123	276	22	of	of	ADP
ejpam-6123	276	23	g.	g.	PROPN
ejpam-6123	276	24	proof	proof	PROPN
ejpam-6123	276	25	.	.	PUNCT
ejpam-6123	277	1	suppose	suppose	VERB
ejpam-6123	277	2	s	s	PRON
ejpam-6123	277	3	is	be	AUX
ejpam-6123	277	4	a	a	DET
ejpam-6123	277	5	connected	connect	VERB
ejpam-6123	277	6	equitable	equitable	ADJ
ejpam-6123	277	7	dominating	dominating	NOUN
ejpam-6123	277	8	set	set	NOUN
ejpam-6123	277	9	of	of	ADP
ejpam-6123	277	10	k1	k1	PROPN
ejpam-6123	277	11	+	+	CCONJ
ejpam-6123	277	12	g.	g.	PROPN
ejpam-6123	277	13	since	since	SCONJ
ejpam-6123	277	14	∆(g	∆(g	PROPN
ejpam-6123	277	15	)	)	PUNCT
ejpam-6123	277	16	≤	≤	NOUN
ejpam-6123	277	17	|v	|v	X
ejpam-6123	277	18	(	(	PUNCT
ejpam-6123	277	19	g	g	NOUN
ejpam-6123	277	20	)	)	PUNCT
ejpam-6123	277	21	−	−	PROPN
ejpam-6123	277	22	3|	3|	NUM
ejpam-6123	277	23	and	and	CCONJ
ejpam-6123	277	24	degk1+g(v	degk1+g(v	ADV
ejpam-6123	277	25	)	)	PUNCT
ejpam-6123	278	1	=	=	SYM
ejpam-6123	278	2	|v	|v	PROPN
ejpam-6123	278	3	(	(	PUNCT
ejpam-6123	278	4	g)|	g)|	NOUN
ejpam-6123	278	5	,	,	PUNCT
ejpam-6123	278	6	where	where	SCONJ
ejpam-6123	278	7	v	v	X
ejpam-6123	278	8	∈	∈	PROPN
ejpam-6123	278	9	v	v	NOUN
ejpam-6123	278	10	(	(	PUNCT
ejpam-6123	278	11	k1	k1	NOUN
ejpam-6123	278	12	)	)	PUNCT
ejpam-6123	278	13	,	,	PUNCT
ejpam-6123	278	14	it	it	PRON
ejpam-6123	278	15	follows	follow	VERB
ejpam-6123	278	16	that	that	SCONJ
ejpam-6123	278	17	v	v	X
ejpam-6123	278	18	∈	∈	PROPN
ejpam-6123	278	19	s.	s.	PROPN
ejpam-6123	278	20	let	let	VERB
ejpam-6123	278	21	sg	sg	PROPN
ejpam-6123	278	22	=	=	SYM
ejpam-6123	278	23	v	v	PROPN
ejpam-6123	278	24	(	(	PUNCT
ejpam-6123	278	25	g	g	NOUN
ejpam-6123	278	26	)	)	PUNCT
ejpam-6123	278	27	∩	∩	NOUN
ejpam-6123	278	28	s.	s.	PROPN
ejpam-6123	278	29	since	since	SCONJ
ejpam-6123	278	30	{	{	PUNCT
ejpam-6123	278	31	v	v	NOUN
ejpam-6123	278	32	}	}	PUNCT
ejpam-6123	278	33	is	be	AUX
ejpam-6123	278	34	not	not	PART
ejpam-6123	278	35	an	an	DET
ejpam-6123	278	36	equitable	equitable	ADJ
ejpam-6123	278	37	dominating	dominating	NOUN
ejpam-6123	278	38	set	set	NOUN
ejpam-6123	278	39	of	of	ADP
ejpam-6123	278	40	k1	k1	NOUN
ejpam-6123	278	41	+	+	CCONJ
ejpam-6123	278	42	g	g	NOUN
ejpam-6123	278	43	,	,	PUNCT
ejpam-6123	278	44	sg	sg	AUX
ejpam-6123	278	45	̸=	̸=	PROPN
ejpam-6123	278	46	∅.	∅.	ADV
ejpam-6123	278	47	let	let	VERB
ejpam-6123	278	48	u	u	PRON
ejpam-6123	278	49	∈	∈	PROPN
ejpam-6123	278	50	v	v	ADP
ejpam-6123	278	51	(	(	PUNCT
ejpam-6123	278	52	g	g	NOUN
ejpam-6123	278	53	)	)	PUNCT
ejpam-6123	278	54	\	\	PROPN
ejpam-6123	278	55	sg	sg	PROPN
ejpam-6123	278	56	.	.	PUNCT
ejpam-6123	279	1	then	then	ADV
ejpam-6123	279	2	∃w	∃w	PROPN
ejpam-6123	279	3	∈	∈	PROPN
ejpam-6123	279	4	s	s	VERB
ejpam-6123	279	5	such	such	ADJ
ejpam-6123	279	6	that	that	DET
ejpam-6123	279	7	|degk1+g(w	|degk1+g(w	ADJ
ejpam-6123	279	8	)	)	PUNCT
ejpam-6123	279	9	−	−	NOUN
ejpam-6123	279	10	degk1+g(u)|	degk1+g(u)|	NOUN
ejpam-6123	279	11	≤	≤	NOUN
ejpam-6123	279	12	1	1	NUM
ejpam-6123	279	13	.	.	PUNCT
ejpam-6123	279	14	since	since	SCONJ
ejpam-6123	279	15	degk1+g(u	degk1+g(u	PROPN
ejpam-6123	279	16	)	)	PUNCT
ejpam-6123	279	17	=	=	SYM
ejpam-6123	279	18	1	1	NUM
ejpam-6123	279	19	+	+	NUM
ejpam-6123	279	20	degg(u	degg(u	PROPN
ejpam-6123	279	21	)	)	PUNCT
ejpam-6123	279	22	≤	≤	NOUN
ejpam-6123	279	23	1	1	NUM
ejpam-6123	280	1	+	+	CCONJ
ejpam-6123	280	2	|v	|v	X
ejpam-6123	280	3	(	(	PUNCT
ejpam-6123	280	4	g)|	g)|	NOUN
ejpam-6123	280	5	−	−	PROPN
ejpam-6123	280	6	3	3	NUM
ejpam-6123	280	7	=	=	SYM
ejpam-6123	280	8	|v	|v	X
ejpam-6123	280	9	(	(	PUNCT
ejpam-6123	280	10	g)|	g)|	INTJ
ejpam-6123	280	11	−	−	PROPN
ejpam-6123	280	12	2	2	NUM
ejpam-6123	280	13	and	and	CCONJ
ejpam-6123	280	14	degk1+g(v	degk1+g(v	ADV
ejpam-6123	280	15	)	)	PUNCT
ejpam-6123	280	16	=	=	SYM
ejpam-6123	280	17	|v	|v	PROPN
ejpam-6123	280	18	(	(	PUNCT
ejpam-6123	280	19	g)|	g)|	PROPN
ejpam-6123	280	20	,	,	PUNCT
ejpam-6123	280	21	w	w	PROPN
ejpam-6123	280	22	̸=	̸=	PROPN
ejpam-6123	280	23	v.	v.	CCONJ
ejpam-6123	280	24	thus	thus	ADV
ejpam-6123	280	25	,	,	PUNCT
ejpam-6123	280	26	w	w	PROPN
ejpam-6123	280	27	∈	∈	PROPN
ejpam-6123	280	28	sg	sg	NOUN
ejpam-6123	280	29	and	and	CCONJ
ejpam-6123	280	30	|degg(w)−	|degg(w)−	NUM
ejpam-6123	280	31	degg(u)|	degg(u)|	NOUN
ejpam-6123	280	32	≤	≤	NOUN
ejpam-6123	280	33	1	1	NUM
ejpam-6123	280	34	.	.	PUNCT
ejpam-6123	281	1	this	this	PRON
ejpam-6123	281	2	implies	imply	VERB
ejpam-6123	281	3	that	that	SCONJ
ejpam-6123	281	4	sg	sg	PROPN
ejpam-6123	281	5	is	be	AUX
ejpam-6123	281	6	an	an	DET
ejpam-6123	281	7	equitable	equitable	ADJ
ejpam-6123	281	8	dominating	dominating	NOUN
ejpam-6123	281	9	set	set	NOUN
ejpam-6123	281	10	of	of	ADP
ejpam-6123	281	11	g.	g.	PROPN
ejpam-6123	281	12	the	the	DET
ejpam-6123	281	13	converse	converse	NOUN
ejpam-6123	281	14	is	be	AUX
ejpam-6123	281	15	clear	clear	ADJ
ejpam-6123	281	16	.	.	PUNCT
ejpam-6123	282	1	corollary	corollary	ADJ
ejpam-6123	282	2	1	1	NUM
ejpam-6123	282	3	.	.	PUNCT
ejpam-6123	283	1	let	let	VERB
ejpam-6123	283	2	g	g	PRON
ejpam-6123	283	3	be	be	AUX
ejpam-6123	283	4	a	a	DET
ejpam-6123	283	5	graph	graph	NOUN
ejpam-6123	283	6	such	such	ADJ
ejpam-6123	283	7	that	that	DET
ejpam-6123	283	8	∆(g	∆(g	NOUN
ejpam-6123	283	9	)	)	PUNCT
ejpam-6123	283	10	≤	≤	NOUN
ejpam-6123	283	11	|v	|v	X
ejpam-6123	283	12	(	(	PUNCT
ejpam-6123	283	13	g)|	g)|	INTJ
ejpam-6123	283	14	−	−	NOUN
ejpam-6123	283	15	3	3	NUM
ejpam-6123	283	16	.	.	PUNCT
ejpam-6123	283	17	then	then	ADV
ejpam-6123	283	18	γce(k1	γce(k1	PROPN
ejpam-6123	283	19	+	+	PROPN
ejpam-6123	283	20	g	g	NOUN
ejpam-6123	283	21	)	)	PUNCT
ejpam-6123	283	22	=	=	SYM
ejpam-6123	283	23	1	1	NUM
ejpam-6123	283	24	+	+	CCONJ
ejpam-6123	283	25	γe(g	γe(g	NUM
ejpam-6123	283	26	)	)	PUNCT
ejpam-6123	283	27	.	.	PUNCT
ejpam-6123	284	1	theorem	theorem	ADJ
ejpam-6123	284	2	10	10	NUM
ejpam-6123	284	3	.	.	PUNCT
ejpam-6123	285	1	let	let	VERB
ejpam-6123	285	2	g	g	PRON
ejpam-6123	285	3	be	be	AUX
ejpam-6123	285	4	a	a	DET
ejpam-6123	285	5	graph	graph	NOUN
ejpam-6123	285	6	such	such	ADJ
ejpam-6123	285	7	that	that	DET
ejpam-6123	285	8	∆(g	∆(g	PROPN
ejpam-6123	285	9	)	)	PUNCT
ejpam-6123	285	10	≥	≥	NOUN
ejpam-6123	285	11	n−2	n−2	PROPN
ejpam-6123	285	12	,	,	PUNCT
ejpam-6123	285	13	where	where	SCONJ
ejpam-6123	285	14	n	n	X
ejpam-6123	285	15	=	=	SYM
ejpam-6123	285	16	|v	|v	X
ejpam-6123	285	17	(	(	PUNCT
ejpam-6123	285	18	g)|	g)|	NOUN
ejpam-6123	285	19	.	.	PUNCT
ejpam-6123	286	1	then	then	ADV
ejpam-6123	286	2	s	s	VERB
ejpam-6123	286	3	is	be	AUX
ejpam-6123	286	4	an	an	DET
ejpam-6123	286	5	equitable	equitable	ADJ
ejpam-6123	286	6	connected	connected	ADJ
ejpam-6123	286	7	dominating	dominating	NOUN
ejpam-6123	286	8	set	set	NOUN
ejpam-6123	286	9	of	of	ADP
ejpam-6123	286	10	k1+g	k1+g	NOUN
ejpam-6123	286	11	if	if	SCONJ
ejpam-6123	286	12	and	and	CCONJ
ejpam-6123	286	13	only	only	ADV
ejpam-6123	286	14	if	if	SCONJ
ejpam-6123	286	15	it	it	PRON
ejpam-6123	286	16	satisfies	satisfy	VERB
ejpam-6123	286	17	one	one	NUM
ejpam-6123	286	18	of	of	ADP
ejpam-6123	286	19	the	the	DET
ejpam-6123	286	20	following	following	ADJ
ejpam-6123	286	21	statements	statement	NOUN
ejpam-6123	286	22	:	:	PUNCT
ejpam-6123	286	23	(	(	PUNCT
ejpam-6123	286	24	i.	i.	NOUN
ejpam-6123	286	25	)	)	PUNCT
ejpam-6123	286	26	s	s	PART
ejpam-6123	286	27	⊆	⊆	NUM
ejpam-6123	286	28	v	v	NOUN
ejpam-6123	286	29	(	(	PUNCT
ejpam-6123	286	30	g	g	NOUN
ejpam-6123	286	31	)	)	PUNCT
ejpam-6123	286	32	and	and	CCONJ
ejpam-6123	286	33	is	be	AUX
ejpam-6123	286	34	a	a	DET
ejpam-6123	286	35	connected	connect	VERB
ejpam-6123	286	36	equitable	equitable	ADJ
ejpam-6123	286	37	dominating	dominating	NOUN
ejpam-6123	286	38	set	set	NOUN
ejpam-6123	286	39	of	of	ADP
ejpam-6123	286	40	g.	g.	PROPN
ejpam-6123	286	41	(	(	PUNCT
ejpam-6123	286	42	ii	ii	PROPN
ejpam-6123	286	43	.	.	PUNCT
ejpam-6123	286	44	)	)	PUNCT
ejpam-6123	287	1	s	s	PART
ejpam-6123	287	2	=	=	SYM
ejpam-6123	287	3	v	v	NOUN
ejpam-6123	287	4	(	(	PUNCT
ejpam-6123	287	5	k1)∪sg	k1)∪sg	NOUN
ejpam-6123	287	6	such	such	ADJ
ejpam-6123	287	7	that	that	SCONJ
ejpam-6123	287	8	sg	sg	ADP
ejpam-6123	287	9	⊆	⊆	NUM
ejpam-6123	287	10	v	v	NOUN
ejpam-6123	287	11	(	(	PUNCT
ejpam-6123	287	12	g	g	NOUN
ejpam-6123	287	13	)	)	PUNCT
ejpam-6123	287	14	equitably	equitably	ADV
ejpam-6123	287	15	dominates	dominate	VERB
ejpam-6123	287	16	v	v	NOUN
ejpam-6123	287	17	(	(	PUNCT
ejpam-6123	287	18	g	g	NOUN
ejpam-6123	287	19	)	)	PUNCT
ejpam-6123	287	20	\	\	PUNCT
ejpam-6123	288	1	(	(	PUNCT
ejpam-6123	288	2	dg	dg	PROPN
ejpam-6123	288	3	∪sg	∪sg	PROPN
ejpam-6123	288	4	)	)	PUNCT
ejpam-6123	288	5	where	where	SCONJ
ejpam-6123	288	6	dg	dg	VERB
ejpam-6123	288	7	=	=	PUNCT
ejpam-6123	288	8	{	{	PUNCT
ejpam-6123	288	9	z	z	NOUN
ejpam-6123	288	10	∈	∈	PROPN
ejpam-6123	288	11	v	v	ADP
ejpam-6123	288	12	(	(	PUNCT
ejpam-6123	288	13	g	g	NOUN
ejpam-6123	288	14	)	)	PUNCT
ejpam-6123	288	15	\	\	PROPN
ejpam-6123	289	1	sg	sg	PROPN
ejpam-6123	289	2	:	:	PUNCT
ejpam-6123	289	3	degg(z	degg(z	PROPN
ejpam-6123	289	4	)	)	PUNCT
ejpam-6123	289	5	≥	≥	NOUN
ejpam-6123	289	6	n−	n−	NOUN
ejpam-6123	289	7	2	2	NUM
ejpam-6123	289	8	}	}	PUNCT
ejpam-6123	289	9	.	.	PUNCT
ejpam-6123	290	1	proof	proof	NOUN
ejpam-6123	290	2	.	.	PUNCT
ejpam-6123	291	1	suppose	suppose	VERB
ejpam-6123	291	2	s	s	NOUN
ejpam-6123	291	3	is	be	AUX
ejpam-6123	291	4	a	a	DET
ejpam-6123	291	5	connected	connect	VERB
ejpam-6123	291	6	equitable	equitable	ADJ
ejpam-6123	291	7	dominating	dominating	NOUN
ejpam-6123	291	8	set	set	NOUN
ejpam-6123	291	9	of	of	ADP
ejpam-6123	291	10	k1	k1	PROPN
ejpam-6123	292	1	+	+	PROPN
ejpam-6123	292	2	g.	g.	PROPN
ejpam-6123	292	3	let	let	VERB
ejpam-6123	292	4	v	v	NUM
ejpam-6123	292	5	∈	∈	PROPN
ejpam-6123	292	6	v	v	NOUN
ejpam-6123	292	7	(	(	PUNCT
ejpam-6123	292	8	k1	k1	NOUN
ejpam-6123	292	9	)	)	PUNCT
ejpam-6123	292	10	and	and	CCONJ
ejpam-6123	292	11	set	set	VERB
ejpam-6123	292	12	sg	sg	PROPN
ejpam-6123	292	13	=	=	SYM
ejpam-6123	292	14	v	v	PROPN
ejpam-6123	292	15	(	(	PUNCT
ejpam-6123	292	16	g)∩s	g)∩s	PROPN
ejpam-6123	292	17	.	.	PUNCT
ejpam-6123	292	18	suppose	suppose	VERB
ejpam-6123	292	19	first	first	ADV
ejpam-6123	292	20	that	that	PRON
ejpam-6123	292	21	v	v	NOUN
ejpam-6123	292	22	/∈	/∈	PUNCT
ejpam-6123	293	1	s.	s.	PROPN
ejpam-6123	293	2	then	then	ADV
ejpam-6123	293	3	s	s	VERB
ejpam-6123	293	4	⊆	⊆	NUM
ejpam-6123	293	5	v	v	NOUN
ejpam-6123	293	6	(	(	PUNCT
ejpam-6123	293	7	g	g	NOUN
ejpam-6123	293	8	)	)	PUNCT
ejpam-6123	293	9	.	.	PUNCT
ejpam-6123	294	1	since	since	SCONJ
ejpam-6123	294	2	s	s	PROPN
ejpam-6123	294	3	is	be	AUX
ejpam-6123	294	4	a	a	DET
ejpam-6123	294	5	connected	connect	VERB
ejpam-6123	294	6	equitable	equitable	ADJ
ejpam-6123	294	7	dominating	dominating	NOUN
ejpam-6123	294	8	set	set	NOUN
ejpam-6123	294	9	of	of	ADP
ejpam-6123	294	10	k1+g	k1+g	PROPN
ejpam-6123	294	11	,	,	PUNCT
ejpam-6123	294	12	s	s	PART
ejpam-6123	294	13	is	be	AUX
ejpam-6123	294	14	a	a	DET
ejpam-6123	294	15	connected	connect	VERB
ejpam-6123	294	16	equitable	equitable	ADJ
ejpam-6123	294	17	dominating	dominating	NOUN
ejpam-6123	294	18	set	set	NOUN
ejpam-6123	294	19	of	of	ADP
ejpam-6123	294	20	g.	g.	PROPN
ejpam-6123	294	21	next	next	ADV
ejpam-6123	294	22	,	,	PUNCT
ejpam-6123	294	23	suppose	suppose	VERB
ejpam-6123	294	24	that	that	SCONJ
ejpam-6123	294	25	v	v	X
ejpam-6123	294	26	∈	∈	PROPN
ejpam-6123	294	27	s.	s.	PROPN
ejpam-6123	294	28	let	let	VERB
ejpam-6123	294	29	z	z	PROPN
ejpam-6123	294	30	∈	∈	PROPN
ejpam-6123	294	31	v	v	X
ejpam-6123	294	32	(	(	PUNCT
ejpam-6123	294	33	g)\dg	g)\dg	PROPN
ejpam-6123	294	34	.	.	PROPN
ejpam-6123	294	35	then	then	ADV
ejpam-6123	294	36	degg(z	degg(z	PROPN
ejpam-6123	294	37	)	)	PUNCT
ejpam-6123	294	38	≤	≤	NOUN
ejpam-6123	295	1	n−3	n−3	PROPN
ejpam-6123	295	2	,	,	PUNCT
ejpam-6123	295	3	that	that	ADV
ejpam-6123	295	4	is	is	ADV
ejpam-6123	295	5	,	,	PUNCT
ejpam-6123	295	6	degk1+g(z	degk1+g(z	NUM
ejpam-6123	295	7	)	)	PUNCT
ejpam-6123	295	8	≤	≤	PROPN
ejpam-6123	296	1	n−2	n−2	PROPN
ejpam-6123	296	2	.	.	PUNCT
ejpam-6123	297	1	since	since	SCONJ
ejpam-6123	297	2	degk1+g(v	degk1+g(v	PART
ejpam-6123	297	3	)	)	PUNCT
ejpam-6123	297	4	=	=	SYM
ejpam-6123	297	5	n	n	PROPN
ejpam-6123	297	6	and	and	CCONJ
ejpam-6123	297	7	s	s	VERB
ejpam-6123	297	8	is	be	AUX
ejpam-6123	297	9	a	a	DET
ejpam-6123	297	10	connected	connect	VERB
ejpam-6123	297	11	equitable	equitable	ADJ
ejpam-6123	297	12	dominating	dominating	NOUN
ejpam-6123	297	13	set	set	NOUN
ejpam-6123	297	14	ofk1+g	ofk1+g	NOUN
ejpam-6123	297	15	,	,	PUNCT
ejpam-6123	297	16	there	there	PRON
ejpam-6123	297	17	exists	exist	VERB
ejpam-6123	297	18	y	y	PROPN
ejpam-6123	297	19	∈	∈	PROPN
ejpam-6123	297	20	sg	sg	ADP
ejpam-6123	297	21	such	such	ADJ
ejpam-6123	297	22	that	that	SCONJ
ejpam-6123	297	23	|	|	INTJ
ejpam-6123	297	24	degg(y)−	degg(y)−	PROPN
ejpam-6123	297	25	degg(z)|	degg(z)|	PROPN
ejpam-6123	297	26	≤	≤	ADJ
ejpam-6123	297	27	1	1	NUM
ejpam-6123	297	28	.	.	PUNCT
ejpam-6123	298	1	this	this	PRON
ejpam-6123	298	2	implies	imply	VERB
ejpam-6123	298	3	that	that	SCONJ
ejpam-6123	298	4	every	every	DET
ejpam-6123	298	5	element	element	NOUN
ejpam-6123	298	6	of	of	ADP
ejpam-6123	298	7	v	v	NOUN
ejpam-6123	298	8	(	(	PUNCT
ejpam-6123	298	9	g	g	NOUN
ejpam-6123	298	10	)	)	PUNCT
ejpam-6123	298	11	\dg	\dg	PROPN
ejpam-6123	298	12	is	be	AUX
ejpam-6123	298	13	equitably	equitably	ADV
ejpam-6123	298	14	dominated	dominate	VERB
ejpam-6123	298	15	by	by	ADP
ejpam-6123	298	16	sg	sg	PROPN
ejpam-6123	298	17	.	.	PROPN
ejpam-6123	299	1	for	for	ADP
ejpam-6123	299	2	the	the	DET
ejpam-6123	299	3	converse	converse	NOUN
ejpam-6123	299	4	,	,	PUNCT
ejpam-6123	299	5	suppose	suppose	VERB
ejpam-6123	299	6	that	that	SCONJ
ejpam-6123	299	7	(	(	PUNCT
ejpam-6123	299	8	i	i	NOUN
ejpam-6123	299	9	)	)	PUNCT
ejpam-6123	299	10	holds	hold	VERB
ejpam-6123	299	11	.	.	PUNCT
ejpam-6123	300	1	then	then	ADV
ejpam-6123	300	2	s	s	VERB
ejpam-6123	300	3	contains	contain	VERB
ejpam-6123	300	4	a	a	DET
ejpam-6123	300	5	vertex	vertex	NOUN
ejpam-6123	300	6	u	u	NOUN
ejpam-6123	300	7	∈	∈	PROPN
ejpam-6123	300	8	v	v	ADP
ejpam-6123	300	9	(	(	PUNCT
ejpam-6123	300	10	g	g	NOUN
ejpam-6123	300	11	)	)	PUNCT
ejpam-6123	300	12	with	with	ADP
ejpam-6123	300	13	degg(u	degg(u	PROPN
ejpam-6123	300	14	)	)	PUNCT
ejpam-6123	300	15	≥	≥	NOUN
ejpam-6123	300	16	n−2	n−2	PROPN
ejpam-6123	300	17	.	.	PUNCT
ejpam-6123	301	1	hence	hence	ADV
ejpam-6123	301	2	,	,	PUNCT
ejpam-6123	301	3	|	|	ADV
ejpam-6123	301	4	degk1+g(v)−degk1+g(u)|	degk1+g(v)−degk1+g(u)|	NOUN
ejpam-6123	301	5	≤	≤	ADV
ejpam-6123	301	6	1	1	NUM
ejpam-6123	301	7	.	.	PUNCT
ejpam-6123	301	8	further	far	ADV
ejpam-6123	301	9	,	,	PUNCT
ejpam-6123	301	10	since	since	SCONJ
ejpam-6123	301	11	s	s	NOUN
ejpam-6123	301	12	is	be	AUX
ejpam-6123	301	13	a	a	DET
ejpam-6123	301	14	connected	connect	VERB
ejpam-6123	301	15	equitable	equitable	ADJ
ejpam-6123	301	16	dominating	dominating	NOUN
ejpam-6123	301	17	set	set	NOUN
ejpam-6123	301	18	of	of	ADP
ejpam-6123	301	19	g	g	NOUN
ejpam-6123	301	20	,	,	PUNCT
ejpam-6123	301	21	s	s	PART
ejpam-6123	301	22	is	be	AUX
ejpam-6123	301	23	a	a	DET
ejpam-6123	301	24	connected	connect	VERB
ejpam-6123	301	25	equitable	equitable	ADJ
ejpam-6123	301	26	dominating	dominating	NOUN
ejpam-6123	301	27	set	set	NOUN
ejpam-6123	301	28	of	of	ADP
ejpam-6123	301	29	k1	k1	PROPN
ejpam-6123	301	30	+	+	PROPN
ejpam-6123	301	31	g.	g.	PROPN
ejpam-6123	301	32	next	next	ADV
ejpam-6123	301	33	,	,	PUNCT
ejpam-6123	301	34	suppose	suppose	VERB
ejpam-6123	301	35	that	that	SCONJ
ejpam-6123	301	36	(	(	PUNCT
ejpam-6123	301	37	ii	ii	NOUN
ejpam-6123	301	38	)	)	PUNCT
ejpam-6123	301	39	holds	hold	VERB
ejpam-6123	301	40	.	.	PUNCT
ejpam-6123	302	1	let	let	VERB
ejpam-6123	302	2	z	z	NOUN
ejpam-6123	302	3	∈	∈	PROPN
ejpam-6123	302	4	dg	dg	NOUN
ejpam-6123	302	5	.	.	PUNCT
ejpam-6123	303	1	then	then	ADV
ejpam-6123	303	2	v	v	X
ejpam-6123	303	3	equitably	equitably	ADV
ejpam-6123	303	4	dominates	dominate	VERB
ejpam-6123	303	5	z.	z.	PROPN
ejpam-6123	303	6	moreover	moreover	ADV
ejpam-6123	303	7	,	,	PUNCT
ejpam-6123	303	8	since	since	SCONJ
ejpam-6123	303	9	sg	sg	PROPN
ejpam-6123	303	10	equitably	equitably	ADV
ejpam-6123	303	11	dominates	dominate	VERB
ejpam-6123	303	12	v	v	PROPN
ejpam-6123	303	13	(	(	PUNCT
ejpam-6123	303	14	g)\dg	g)\dg	PROPN
ejpam-6123	303	15	,	,	PUNCT
ejpam-6123	303	16	s	s	PART
ejpam-6123	303	17	is	be	AUX
ejpam-6123	303	18	a	a	DET
ejpam-6123	303	19	connected	connect	VERB
ejpam-6123	303	20	equitable	equitable	ADJ
ejpam-6123	303	21	dominating	dominating	NOUN
ejpam-6123	303	22	set	set	NOUN
ejpam-6123	303	23	of	of	ADP
ejpam-6123	303	24	k1	k1	PROPN
ejpam-6123	303	25	+	+	PROPN
ejpam-6123	303	26	g.	g.	PROPN
ejpam-6123	303	27	corollary	corollary	NOUN
ejpam-6123	303	28	2	2	PROPN
ejpam-6123	303	29	.	.	PUNCT
ejpam-6123	304	1	let	let	VERB
ejpam-6123	304	2	g	g	PRON
ejpam-6123	304	3	be	be	AUX
ejpam-6123	304	4	a	a	DET
ejpam-6123	304	5	graph	graph	NOUN
ejpam-6123	304	6	such	such	ADJ
ejpam-6123	304	7	that	that	DET
ejpam-6123	304	8	∆(g	∆(g	NOUN
ejpam-6123	304	9	)	)	PUNCT
ejpam-6123	304	10	≥	≥	NOUN
ejpam-6123	304	11	n−	n−	NOUN
ejpam-6123	304	12	2	2	NUM
ejpam-6123	304	13	,	,	PUNCT
ejpam-6123	304	14	where	where	SCONJ
ejpam-6123	304	15	n	n	NOUN
ejpam-6123	304	16	=	=	SYM
ejpam-6123	304	17	|v	|v	X
ejpam-6123	304	18	(	(	PUNCT
ejpam-6123	304	19	g)|	g)|	PROPN
ejpam-6123	304	20	.	.	PUNCT
ejpam-6123	305	1	then	then	ADV
ejpam-6123	305	2	γce(k1	γce(k1	PROPN
ejpam-6123	305	3	+	+	PROPN
ejpam-6123	305	4	g	g	NOUN
ejpam-6123	305	5	)	)	PUNCT
ejpam-6123	305	6	=	=	SYM
ejpam-6123	305	7	min{γce(g	min{γce(g	PROPN
ejpam-6123	305	8	)	)	PUNCT
ejpam-6123	305	9	,	,	PUNCT
ejpam-6123	305	10	1	1	NUM
ejpam-6123	305	11	+	+	NUM
ejpam-6123	305	12	γe(⟨v	γe(⟨v	NOUN
ejpam-6123	305	13	(	(	PUNCT
ejpam-6123	305	14	g	g	NOUN
ejpam-6123	305	15	)	)	PUNCT
ejpam-6123	305	16	\d∗	\d∗	NOUN
ejpam-6123	305	17	g⟩	g⟩	VERB
ejpam-6123	305	18	)	)	PUNCT
ejpam-6123	305	19	}	}	PUNCT
ejpam-6123	305	20	where	where	SCONJ
ejpam-6123	305	21	d∗	d∗	VERB
ejpam-6123	305	22	g	g	PROPN
ejpam-6123	305	23	=	=	SYM
ejpam-6123	305	24	{	{	PUNCT
ejpam-6123	305	25	z	z	NOUN
ejpam-6123	305	26	∈	∈	PROPN
ejpam-6123	305	27	v	v	NOUN
ejpam-6123	305	28	(	(	PUNCT
ejpam-6123	305	29	g	g	NOUN
ejpam-6123	305	30	)	)	PUNCT
ejpam-6123	305	31	:	:	PUNCT
ejpam-6123	305	32	degg(z	degg(z	X
ejpam-6123	305	33	)	)	PUNCT
ejpam-6123	305	34	≥	≥	NOUN
ejpam-6123	305	35	n−	n−	NOUN
ejpam-6123	305	36	2	2	NUM
ejpam-6123	305	37	}	}	PUNCT
ejpam-6123	305	38	.	.	PUNCT
ejpam-6123	306	1	h.	h.	PROPN
ejpam-6123	306	2	nuenay	nuenay	PROPN
ejpam-6123	306	3	-	-	PUNCT
ejpam-6123	306	4	maglanquel	maglanquel	PROPN
ejpam-6123	306	5	/	/	SYM
ejpam-6123	306	6	eur	eur	PROPN
ejpam-6123	306	7	.	.	PUNCT
ejpam-6123	307	1	j.	j.	PROPN
ejpam-6123	307	2	pure	pure	PROPN
ejpam-6123	307	3	appl	appl	PROPN
ejpam-6123	307	4	.	.	PROPN
ejpam-6123	307	5	math	math	PROPN
ejpam-6123	307	6	,	,	PUNCT
ejpam-6123	307	7	18	18	NUM
ejpam-6123	307	8	(	(	PUNCT
ejpam-6123	307	9	3	3	NUM
ejpam-6123	307	10	)	)	PUNCT
ejpam-6123	307	11	(	(	PUNCT
ejpam-6123	307	12	2025	2025	NUM
ejpam-6123	307	13	)	)	PUNCT
ejpam-6123	307	14	,	,	PUNCT
ejpam-6123	307	15	6123	6123	NUM
ejpam-6123	307	16	9	9	NUM
ejpam-6123	307	17	of	of	ADP
ejpam-6123	307	18	14	14	NUM
ejpam-6123	307	19	theorem	theorem	VERB
ejpam-6123	307	20	11	11	NUM
ejpam-6123	307	21	.	.	PUNCT
ejpam-6123	308	1	γce(g+k1	γce(g+k1	PROPN
ejpam-6123	308	2	)	)	PUNCT
ejpam-6123	309	1	=	=	SYM
ejpam-6123	309	2	1	1	NUM
ejpam-6123	309	3	if	if	SCONJ
ejpam-6123	309	4	and	and	CCONJ
ejpam-6123	309	5	only	only	ADV
ejpam-6123	309	6	if	if	SCONJ
ejpam-6123	309	7	∆(g+k1	∆(g+k1	ADJ
ejpam-6123	309	8	)	)	PUNCT
ejpam-6123	309	9	≤	≤	NUM
ejpam-6123	309	10	degg(v	degg(v	PROPN
ejpam-6123	309	11	)	)	PUNCT
ejpam-6123	309	12	+	+	CCONJ
ejpam-6123	309	13	2	2	NUM
ejpam-6123	309	14	for	for	ADP
ejpam-6123	309	15	all	all	DET
ejpam-6123	309	16	v	v	NOUN
ejpam-6123	309	17	in	in	ADP
ejpam-6123	309	18	g.	g.	PROPN
ejpam-6123	309	19	proof	proof	NOUN
ejpam-6123	309	20	.	.	PUNCT
ejpam-6123	310	1	let	let	VERB
ejpam-6123	310	2	g	g	PROPN
ejpam-6123	310	3	a	a	DET
ejpam-6123	310	4	connected	connected	ADJ
ejpam-6123	310	5	graph	graph	NOUN
ejpam-6123	310	6	of	of	ADP
ejpam-6123	310	7	order	order	NOUN
ejpam-6123	310	8	n.	n.	NOUN
ejpam-6123	310	9	let	let	VERB
ejpam-6123	310	10	v1	v1	NOUN
ejpam-6123	310	11	,	,	PUNCT
ejpam-6123	310	12	v2	v2	PROPN
ejpam-6123	310	13	,	,	PUNCT
ejpam-6123	310	14	.	.	PUNCT
ejpam-6123	310	15	.	.	PUNCT
ejpam-6123	311	1	.	.	PUNCT
ejpam-6123	312	1	,	,	PUNCT
ejpam-6123	312	2	vn	vn	PROPN
ejpam-6123	312	3	ne	ne	PROPN
ejpam-6123	312	4	vertices	vertice	VERB
ejpam-6123	312	5	in	in	ADP
ejpam-6123	312	6	g.	g.	PROPN
ejpam-6123	312	7	suppose	suppose	VERB
ejpam-6123	312	8	that	that	SCONJ
ejpam-6123	312	9	γce(g	γce(g	PROPN
ejpam-6123	313	1	+	+	NUM
ejpam-6123	313	2	k1	k1	NOUN
ejpam-6123	313	3	)	)	PUNCT
ejpam-6123	313	4	=	=	SYM
ejpam-6123	314	1	1	1	X
ejpam-6123	314	2	.	.	PUNCT
ejpam-6123	314	3	then	then	ADV
ejpam-6123	314	4	there	there	PRON
ejpam-6123	314	5	exists	exist	VERB
ejpam-6123	314	6	a	a	DET
ejpam-6123	314	7	vertex	vertex	NOUN
ejpam-6123	314	8	u	u	NOUN
ejpam-6123	314	9	in	in	ADP
ejpam-6123	314	10	g	g	PROPN
ejpam-6123	314	11	+	+	CCONJ
ejpam-6123	314	12	k1	k1	NOUN
ejpam-6123	314	13	which	which	PRON
ejpam-6123	314	14	is	be	AUX
ejpam-6123	314	15	an	an	DET
ejpam-6123	314	16	equitable	equitable	ADJ
ejpam-6123	314	17	dominating	dominating	NOUN
ejpam-6123	314	18	in	in	ADP
ejpam-6123	314	19	g+k1	g+k1	PROPN
ejpam-6123	314	20	.	.	PUNCT
ejpam-6123	315	1	consider	consider	VERB
ejpam-6123	315	2	the	the	DET
ejpam-6123	315	3	following	follow	VERB
ejpam-6123	315	4	cases	case	NOUN
ejpam-6123	315	5	:	:	PUNCT
ejpam-6123	315	6	case	case	NOUN
ejpam-6123	315	7	1	1	NUM
ejpam-6123	315	8	:	:	PUNCT
ejpam-6123	315	9	u	u	PROPN
ejpam-6123	315	10	∈	∈	PROPN
ejpam-6123	315	11	v	v	ADP
ejpam-6123	315	12	(	(	PUNCT
ejpam-6123	315	13	k1	k1	NOUN
ejpam-6123	315	14	)	)	PUNCT
ejpam-6123	315	15	.	.	PUNCT
ejpam-6123	316	1	then	then	ADV
ejpam-6123	316	2	1	1	NUM
ejpam-6123	316	3	≥	≥	NOUN
ejpam-6123	316	4	|degg+k1	|degg+k1	X
ejpam-6123	316	5	(	(	PUNCT
ejpam-6123	316	6	u)−	u)−	PROPN
ejpam-6123	316	7	degg+k1	degg+k1	X
ejpam-6123	316	8	(	(	PUNCT
ejpam-6123	316	9	vi)|	vi)|	NOUN
ejpam-6123	316	10	=	=	SYM
ejpam-6123	316	11	|n−	|n−	NOUN
ejpam-6123	316	12	(	(	PUNCT
ejpam-6123	316	13	degg(vi	degg(vi	PROPN
ejpam-6123	316	14	)	)	PUNCT
ejpam-6123	316	15	+	+	NOUN
ejpam-6123	316	16	1)|	1)|	NUM
ejpam-6123	316	17	=	=	NOUN
ejpam-6123	316	18	|(n−	|(n−	ADJ
ejpam-6123	316	19	1)−	1)−	NUM
ejpam-6123	316	20	degg(vi)|	degg(vi)|	NOUN
ejpam-6123	316	21	.	.	PUNCT
ejpam-6123	317	1	thus	thus	ADV
ejpam-6123	317	2	,	,	PUNCT
ejpam-6123	317	3	degg(vi	degg(vi	PROPN
ejpam-6123	317	4	)	)	PUNCT
ejpam-6123	317	5	≥	≥	NOUN
ejpam-6123	317	6	n	n	CCONJ
ejpam-6123	317	7	−	−	PROPN
ejpam-6123	317	8	2	2	NUM
ejpam-6123	317	9	for	for	ADP
ejpam-6123	317	10	all	all	DET
ejpam-6123	317	11	i	i	PRON
ejpam-6123	317	12	=	=	NOUN
ejpam-6123	317	13	1	1	NUM
ejpam-6123	317	14	,	,	PUNCT
ejpam-6123	317	15	2	2	NUM
ejpam-6123	317	16	,	,	PUNCT
ejpam-6123	317	17	.	.	PUNCT
ejpam-6123	317	18	.	.	PUNCT
ejpam-6123	318	1	.	.	PUNCT
ejpam-6123	319	1	,	,	PUNCT
ejpam-6123	319	2	n.	n.	PROPN
ejpam-6123	319	3	hence	hence	ADV
ejpam-6123	319	4	,	,	PUNCT
ejpam-6123	319	5	∆(g	∆(g	PROPN
ejpam-6123	319	6	+	+	CCONJ
ejpam-6123	319	7	k1	k1	NOUN
ejpam-6123	319	8	)	)	PUNCT
ejpam-6123	319	9	=	=	PUNCT
ejpam-6123	320	1	degg+k1	degg+k1	X
ejpam-6123	320	2	(	(	PUNCT
ejpam-6123	320	3	u	u	NOUN
ejpam-6123	320	4	)	)	PUNCT
ejpam-6123	320	5	=	=	SYM
ejpam-6123	320	6	n	n	CCONJ
ejpam-6123	320	7	≤	≤	NUM
ejpam-6123	320	8	degg(vi	degg(vi	NOUN
ejpam-6123	320	9	)	)	PUNCT
ejpam-6123	320	10	+	+	CCONJ
ejpam-6123	320	11	2	2	NUM
ejpam-6123	320	12	for	for	ADP
ejpam-6123	320	13	all	all	DET
ejpam-6123	320	14	vi	vi	PROPN
ejpam-6123	320	15	in	in	ADP
ejpam-6123	320	16	g.	g.	NOUN
ejpam-6123	320	17	case	case	NOUN
ejpam-6123	320	18	2	2	NUM
ejpam-6123	320	19	:	:	PUNCT
ejpam-6123	320	20	if	if	SCONJ
ejpam-6123	320	21	u	u	PROPN
ejpam-6123	320	22	∈	∈	PROPN
ejpam-6123	320	23	v	v	X
ejpam-6123	320	24	(	(	PUNCT
ejpam-6123	320	25	g	g	NOUN
ejpam-6123	320	26	)	)	PUNCT
ejpam-6123	320	27	.	.	PUNCT
ejpam-6123	321	1	,	,	PUNCT
ejpam-6123	321	2	then	then	ADV
ejpam-6123	321	3	|degg+k1	|degg+k1	PRON
ejpam-6123	321	4	(	(	PUNCT
ejpam-6123	321	5	u)−	u)−	PROPN
ejpam-6123	321	6	degg+k1	degg+k1	NOUN
ejpam-6123	321	7	(	(	PUNCT
ejpam-6123	321	8	v)|	v)|	NOUN
ejpam-6123	321	9	≤	≤	ADJ
ejpam-6123	321	10	1	1	NUM
ejpam-6123	321	11	for	for	ADP
ejpam-6123	321	12	all	all	PRON
ejpam-6123	321	13	v	v	ADP
ejpam-6123	321	14	̸=	̸=	PROPN
ejpam-6123	321	15	u	u	NOUN
ejpam-6123	321	16	in	in	ADP
ejpam-6123	321	17	g+k1	g+k1	NOUN
ejpam-6123	321	18	.	.	PUNCT
ejpam-6123	322	1	this	this	PRON
ejpam-6123	322	2	means	mean	VERB
ejpam-6123	322	3	that	that	SCONJ
ejpam-6123	323	1	degg+k1	degg+k1	X
ejpam-6123	323	2	(	(	PUNCT
ejpam-6123	323	3	u	u	NOUN
ejpam-6123	323	4	)	)	PUNCT
ejpam-6123	323	5	≤	≤	NUM
ejpam-6123	323	6	degg+k1	degg+k1	X
ejpam-6123	323	7	(	(	PUNCT
ejpam-6123	323	8	v	v	NOUN
ejpam-6123	323	9	)	)	PUNCT
ejpam-6123	323	10	+	+	NOUN
ejpam-6123	323	11	1	1	X
ejpam-6123	323	12	.	.	PUNCT
ejpam-6123	323	13	thus	thus	ADV
ejpam-6123	323	14	,	,	PUNCT
ejpam-6123	323	15	∆(g+k1	∆(g+k1	ADJ
ejpam-6123	323	16	)	)	PUNCT
ejpam-6123	323	17	=	=	SYM
ejpam-6123	324	1	degg+k1	degg+k1	X
ejpam-6123	324	2	(	(	PUNCT
ejpam-6123	324	3	u	u	NOUN
ejpam-6123	324	4	)	)	PUNCT
ejpam-6123	324	5	≤	≤	NUM
ejpam-6123	324	6	degg+k1	degg+k1	X
ejpam-6123	324	7	(	(	PUNCT
ejpam-6123	324	8	v	v	NOUN
ejpam-6123	324	9	)	)	PUNCT
ejpam-6123	324	10	+	+	CCONJ
ejpam-6123	324	11	1	1	NUM
ejpam-6123	324	12	,	,	PUNCT
ejpam-6123	324	13	v	v	NOUN
ejpam-6123	324	14	∈	∈	PROPN
ejpam-6123	324	15	v	v	NOUN
ejpam-6123	324	16	(	(	PUNCT
ejpam-6123	324	17	g+k1	g+k1	NOUN
ejpam-6123	324	18	)	)	PUNCT
ejpam-6123	324	19	\	\	NOUN
ejpam-6123	324	20	{	{	PUNCT
ejpam-6123	324	21	u	u	NOUN
ejpam-6123	324	22	}	}	PUNCT
ejpam-6123	324	23	=	=	SYM
ejpam-6123	324	24	degg(v	degg(v	PROPN
ejpam-6123	324	25	)	)	PUNCT
ejpam-6123	324	26	+	+	CCONJ
ejpam-6123	324	27	1	1	NUM
ejpam-6123	324	28	+	+	NUM
ejpam-6123	324	29	1	1	NUM
ejpam-6123	324	30	,	,	PUNCT
ejpam-6123	324	31	v	v	ADP
ejpam-6123	324	32	̸=	̸=	PROPN
ejpam-6123	324	33	u	u	NOUN
ejpam-6123	324	34	in	in	ADP
ejpam-6123	324	35	g	g	PROPN
ejpam-6123	324	36	=	=	SYM
ejpam-6123	324	37	degg(v	degg(v	PROPN
ejpam-6123	324	38	)	)	PUNCT
ejpam-6123	324	39	+	+	NOUN
ejpam-6123	324	40	2	2	X
ejpam-6123	324	41	.	.	X
ejpam-6123	324	42	for	for	ADP
ejpam-6123	324	43	the	the	DET
ejpam-6123	324	44	converse	converse	NOUN
ejpam-6123	324	45	,	,	PUNCT
ejpam-6123	324	46	suppose	suppose	VERB
ejpam-6123	324	47	that	that	SCONJ
ejpam-6123	324	48	∆(g+k1	∆(g+k1	ADJ
ejpam-6123	324	49	)	)	PUNCT
ejpam-6123	324	50	≤	≤	NUM
ejpam-6123	324	51	degg(v	degg(v	PROPN
ejpam-6123	324	52	)	)	PUNCT
ejpam-6123	324	53	+	+	CCONJ
ejpam-6123	324	54	2	2	NUM
ejpam-6123	324	55	for	for	ADP
ejpam-6123	324	56	all	all	DET
ejpam-6123	324	57	v	v	NOUN
ejpam-6123	324	58	in	in	ADP
ejpam-6123	324	59	g.	g.	PROPN
ejpam-6123	324	60	let	let	VERB
ejpam-6123	324	61	u	u	PRON
ejpam-6123	324	62	in	in	ADP
ejpam-6123	324	63	g+k1	g+k1	PROPN
ejpam-6123	324	64	be	be	VERB
ejpam-6123	324	65	of	of	ADP
ejpam-6123	324	66	maximum	maximum	ADJ
ejpam-6123	324	67	degree	degree	NOUN
ejpam-6123	324	68	.	.	PUNCT
ejpam-6123	325	1	if	if	SCONJ
ejpam-6123	325	2	x	x	SYM
ejpam-6123	325	3	∈	∈	PROPN
ejpam-6123	325	4	v	v	X
ejpam-6123	325	5	(	(	PUNCT
ejpam-6123	325	6	k1	k1	NOUN
ejpam-6123	325	7	)	)	PUNCT
ejpam-6123	325	8	,	,	PUNCT
ejpam-6123	325	9	then	then	ADV
ejpam-6123	325	10	degg+k1	degg+k1	X
ejpam-6123	325	11	(	(	PUNCT
ejpam-6123	325	12	u	u	NOUN
ejpam-6123	325	13	)	)	PUNCT
ejpam-6123	325	14	=	=	SYM
ejpam-6123	325	15	∆(g+k1	∆(g+k1	X
ejpam-6123	325	16	)	)	PUNCT
ejpam-6123	325	17	degg+k1	degg+k1	X
ejpam-6123	325	18	(	(	PUNCT
ejpam-6123	325	19	v	v	NOUN
ejpam-6123	325	20	)	)	PUNCT
ejpam-6123	325	21	+	+	CCONJ
ejpam-6123	325	22	1	1	NUM
ejpam-6123	325	23	for	for	ADP
ejpam-6123	325	24	all	all	DET
ejpam-6123	325	25	v	v	NOUN
ejpam-6123	325	26	in	in	ADP
ejpam-6123	325	27	g.	g.	PROPN
ejpam-6123	325	28	that	that	PRON
ejpam-6123	325	29	is	be	AUX
ejpam-6123	325	30	,	,	PUNCT
ejpam-6123	325	31	degg+k1	degg+k1	X
ejpam-6123	325	32	(	(	PUNCT
ejpam-6123	325	33	v	v	NOUN
ejpam-6123	325	34	)	)	PUNCT
ejpam-6123	325	35	≥	≥	NOUN
ejpam-6123	325	36	degg+k1	degg+k1	X
ejpam-6123	325	37	(	(	PUNCT
ejpam-6123	325	38	u)−	u)−	PROPN
ejpam-6123	325	39	1	1	NUM
ejpam-6123	325	40	for	for	ADP
ejpam-6123	325	41	all	all	DET
ejpam-6123	325	42	v	v	NOUN
ejpam-6123	325	43	in	in	ADP
ejpam-6123	325	44	g.	g.	PROPN
ejpam-6123	325	45	this	this	PRON
ejpam-6123	325	46	implies	imply	VERB
ejpam-6123	325	47	that	that	SCONJ
ejpam-6123	325	48	g	g	PROPN
ejpam-6123	325	49	is	be	AUX
ejpam-6123	325	50	a	a	DET
ejpam-6123	325	51	(	(	PUNCT
ejpam-6123	325	52	t	t	PROPN
ejpam-6123	325	53	,	,	PUNCT
ejpam-6123	325	54	t+1	t+1	NOUN
ejpam-6123	325	55	)	)	PUNCT
ejpam-6123	325	56	bi	bi	ADJ
ejpam-6123	325	57	-	-	ADJ
ejpam-6123	325	58	regular	regular	ADJ
ejpam-6123	325	59	graph	graph	NOUN
ejpam-6123	325	60	.	.	PUNCT
ejpam-6123	326	1	take	take	VERB
ejpam-6123	326	2	{	{	PUNCT
ejpam-6123	326	3	u	u	NOUN
ejpam-6123	326	4	}	}	PUNCT
ejpam-6123	326	5	.	.	PUNCT
ejpam-6123	327	1	observe	observe	VERB
ejpam-6123	327	2	that	that	SCONJ
ejpam-6123	327	3	u	u	NOUN
ejpam-6123	327	4	is	be	AUX
ejpam-6123	327	5	an	an	DET
ejpam-6123	327	6	equitable	equitable	ADJ
ejpam-6123	327	7	dominating	dominating	NOUN
ejpam-6123	327	8	in	in	ADP
ejpam-6123	327	9	g+k1	g+k1	PROPN
ejpam-6123	327	10	.	.	PUNCT
ejpam-6123	328	1	thus	thus	ADV
ejpam-6123	328	2	,	,	PUNCT
ejpam-6123	328	3	γce(g	γce(g	PROPN
ejpam-6123	328	4	+	+	NOUN
ejpam-6123	328	5	k1	k1	NOUN
ejpam-6123	328	6	)	)	PUNCT
ejpam-6123	328	7	=	=	SYM
ejpam-6123	328	8	1	1	X
ejpam-6123	328	9	.	.	X
ejpam-6123	328	10	similarly	similarly	ADV
ejpam-6123	328	11	,	,	PUNCT
ejpam-6123	328	12	if	if	SCONJ
ejpam-6123	328	13	u	u	PROPN
ejpam-6123	328	14	∈	∈	PROPN
ejpam-6123	328	15	v	v	X
ejpam-6123	328	16	(	(	PUNCT
ejpam-6123	328	17	g	g	NOUN
ejpam-6123	328	18	)	)	PUNCT
ejpam-6123	328	19	,	,	PUNCT
ejpam-6123	328	20	degg+k1	degg+k1	X
ejpam-6123	328	21	(	(	PUNCT
ejpam-6123	328	22	v	v	NOUN
ejpam-6123	328	23	)	)	PUNCT
ejpam-6123	328	24	≥	≥	NOUN
ejpam-6123	328	25	degg+k1	degg+k1	X
ejpam-6123	328	26	(	(	PUNCT
ejpam-6123	328	27	u	u	NOUN
ejpam-6123	328	28	)	)	PUNCT
ejpam-6123	328	29	−	−	PROPN
ejpam-6123	328	30	1	1	NUM
ejpam-6123	328	31	for	for	ADP
ejpam-6123	328	32	all	all	DET
ejpam-6123	328	33	v	v	NOUN
ejpam-6123	328	34	in	in	ADP
ejpam-6123	328	35	g.	g.	PROPN
ejpam-6123	328	36	taking	take	VERB
ejpam-6123	328	37	{	{	PUNCT
ejpam-6123	328	38	u	u	NOUN
ejpam-6123	328	39	}	}	PUNCT
ejpam-6123	328	40	which	which	PRON
ejpam-6123	328	41	is	be	AUX
ejpam-6123	328	42	a	a	DET
ejpam-6123	328	43	connected	connect	VERB
ejpam-6123	328	44	equitable	equitable	ADJ
ejpam-6123	328	45	dominating	dominating	NOUN
ejpam-6123	328	46	set	set	VERB
ejpam-6123	328	47	in	in	ADP
ejpam-6123	328	48	g	g	PROPN
ejpam-6123	328	49	+	+	CCONJ
ejpam-6123	328	50	k1	k1	NOUN
ejpam-6123	328	51	.	.	PUNCT
ejpam-6123	329	1	therefore	therefore	ADV
ejpam-6123	329	2	,	,	PUNCT
ejpam-6123	329	3	γce(g+k1	γce(g+k1	PROPN
ejpam-6123	329	4	)	)	PUNCT
ejpam-6123	330	1	=	=	SYM
ejpam-6123	330	2	1	1	X
ejpam-6123	330	3	.	.	PUNCT
ejpam-6123	330	4	theorem	theorem	NOUN
ejpam-6123	330	5	12	12	NUM
ejpam-6123	330	6	.	.	PUNCT
ejpam-6123	331	1	let	let	VERB
ejpam-6123	331	2	g	g	PRON
ejpam-6123	331	3	be	be	AUX
ejpam-6123	331	4	a	a	DET
ejpam-6123	331	5	complete	complete	ADJ
ejpam-6123	331	6	graph	graph	NOUN
ejpam-6123	331	7	of	of	ADP
ejpam-6123	331	8	order	order	NOUN
ejpam-6123	331	9	n	n	NOUN
ejpam-6123	332	1	and	and	CCONJ
ejpam-6123	332	2	h	h	NOUN
ejpam-6123	332	3	be	be	AUX
ejpam-6123	332	4	any	any	DET
ejpam-6123	332	5	graph	graph	NOUN
ejpam-6123	332	6	of	of	ADP
ejpam-6123	332	7	order	order	NOUN
ejpam-6123	332	8	m	m	VERB
ejpam-6123	332	9	with	with	ADP
ejpam-6123	332	10	n−m	n−m	PROPN
ejpam-6123	332	11	=	=	SYM
ejpam-6123	332	12	k	k	X
ejpam-6123	332	13	>	>	X
ejpam-6123	333	1	0	0	X
ejpam-6123	333	2	.	.	PUNCT
ejpam-6123	333	3	then	then	ADV
ejpam-6123	333	4	γce(g+h	γce(g+h	NUM
ejpam-6123	333	5	)	)	PUNCT
ejpam-6123	333	6	=	=	SYM
ejpam-6123	333	7	1	1	NUM
ejpam-6123	333	8	+	+	NUM
ejpam-6123	333	9	γe(⟨v	γe(⟨v	NOUN
ejpam-6123	333	10	(	(	PUNCT
ejpam-6123	333	11	h	h	NOUN
ejpam-6123	333	12	\dh⟩	\dh⟩	NOUN
ejpam-6123	333	13	)	)	PUNCT
ejpam-6123	333	14	where	where	SCONJ
ejpam-6123	333	15	v	v	X
ejpam-6123	333	16	(	(	PUNCT
ejpam-6123	333	17	h	h	NOUN
ejpam-6123	333	18	)	)	PUNCT
ejpam-6123	333	19	\	\	NOUN
ejpam-6123	333	20	dh	dh	NOUN
ejpam-6123	333	21	=	=	PUNCT
ejpam-6123	333	22	{	{	PUNCT
ejpam-6123	333	23	v	v	NUM
ejpam-6123	333	24	∈	∈	NOUN
ejpam-6123	333	25	v	v	NOUN
ejpam-6123	333	26	(	(	PUNCT
ejpam-6123	333	27	h	h	NOUN
ejpam-6123	333	28	)	)	PUNCT
ejpam-6123	333	29	:	:	PUNCT
ejpam-6123	333	30	degh(v	degh(v	NOUN
ejpam-6123	333	31	)	)	PUNCT
ejpam-6123	333	32	<	<	X
ejpam-6123	333	33	∆(g	∆(g	PROPN
ejpam-6123	333	34	)	)	PUNCT
ejpam-6123	333	35	−	−	PROPN
ejpam-6123	334	1	(	(	PUNCT
ejpam-6123	334	2	k	k	NOUN
ejpam-6123	334	3	+	+	PROPN
ejpam-6123	334	4	1	1	X
ejpam-6123	334	5	)	)	PUNCT
ejpam-6123	334	6	=	=	SYM
ejpam-6123	335	1	n	n	PRON
ejpam-6123	335	2	−	−	NOUN
ejpam-6123	336	1	k	k	NOUN
ejpam-6123	336	2	−	−	PROPN
ejpam-6123	336	3	2	2	NUM
ejpam-6123	336	4	}	}	PUNCT
ejpam-6123	336	5	and	and	CCONJ
ejpam-6123	336	6	dh	dh	NOUN
ejpam-6123	336	7	=	=	PUNCT
ejpam-6123	336	8	{	{	PUNCT
ejpam-6123	336	9	z	z	NOUN
ejpam-6123	336	10	∈	∈	PROPN
ejpam-6123	336	11	v	v	NOUN
ejpam-6123	336	12	(	(	PUNCT
ejpam-6123	336	13	h	h	NOUN
ejpam-6123	336	14	)	)	PUNCT
ejpam-6123	336	15	:	:	PUNCT
ejpam-6123	336	16	degh(z	degh(z	ADJ
ejpam-6123	336	17	)	)	PUNCT
ejpam-6123	336	18	≥	≥	NOUN
ejpam-6123	336	19	∆(g)−	∆(g)−	NOUN
ejpam-6123	336	20	(	(	PUNCT
ejpam-6123	336	21	k	k	X
ejpam-6123	336	22	+	+	PROPN
ejpam-6123	336	23	1	1	X
ejpam-6123	336	24	)	)	PUNCT
ejpam-6123	336	25	=	=	SYM
ejpam-6123	337	1	n−	n−	NOUN
ejpam-6123	337	2	k	k	NOUN
ejpam-6123	337	3	−	−	PROPN
ejpam-6123	337	4	2	2	NUM
ejpam-6123	337	5	}	}	PUNCT
ejpam-6123	337	6	.	.	PUNCT
ejpam-6123	338	1	proof	proof	NOUN
ejpam-6123	338	2	.	.	PUNCT
ejpam-6123	339	1	let	let	VERB
ejpam-6123	339	2	u	u	PRON
ejpam-6123	339	3	∈	∈	PROPN
ejpam-6123	339	4	v	v	ADP
ejpam-6123	339	5	(	(	PUNCT
ejpam-6123	339	6	g	g	NOUN
ejpam-6123	339	7	)	)	PUNCT
ejpam-6123	339	8	and	and	CCONJ
ejpam-6123	339	9	s	s	VERB
ejpam-6123	339	10	⊆	⊆	NUM
ejpam-6123	339	11	v	v	NOUN
ejpam-6123	339	12	(	(	PUNCT
ejpam-6123	339	13	h	h	NOUN
ejpam-6123	339	14	)	)	PUNCT
ejpam-6123	339	15	\	\	PROPN
ejpam-6123	339	16	d∗	d∗	PROPN
ejpam-6123	339	17	h	h	NOUN
ejpam-6123	339	18	be	be	VERB
ejpam-6123	339	19	a	a	DET
ejpam-6123	339	20	γe	γe	NOUN
ejpam-6123	339	21	-	-	PUNCT
ejpam-6123	339	22	set	set	NOUN
ejpam-6123	339	23	of	of	ADP
ejpam-6123	339	24	⟨v	⟨v	PROPN
ejpam-6123	339	25	(	(	PUNCT
ejpam-6123	339	26	h	h	NOUN
ejpam-6123	339	27	\	\	PROPN
ejpam-6123	339	28	dh⟩	dh⟩	PROPN
ejpam-6123	339	29	where	where	SCONJ
ejpam-6123	339	30	dh	dh	NOUN
ejpam-6123	339	31	=	=	PUNCT
ejpam-6123	339	32	{	{	PUNCT
ejpam-6123	339	33	z	z	NOUN
ejpam-6123	339	34	∈	∈	PROPN
ejpam-6123	339	35	v	v	NOUN
ejpam-6123	339	36	(	(	PUNCT
ejpam-6123	339	37	h	h	NOUN
ejpam-6123	339	38	)	)	PUNCT
ejpam-6123	339	39	:	:	PUNCT
ejpam-6123	339	40	degh(z	degh(z	ADJ
ejpam-6123	339	41	)	)	PUNCT
ejpam-6123	339	42	≥	≥	NOUN
ejpam-6123	339	43	∆(g	∆(g	NOUN
ejpam-6123	339	44	)	)	PUNCT
ejpam-6123	339	45	−	−	PROPN
ejpam-6123	340	1	(	(	PUNCT
ejpam-6123	340	2	k	k	NOUN
ejpam-6123	340	3	+	+	PROPN
ejpam-6123	340	4	1	1	X
ejpam-6123	340	5	)	)	PUNCT
ejpam-6123	340	6	=	=	SYM
ejpam-6123	341	1	n	n	PRON
ejpam-6123	341	2	−	−	NOUN
ejpam-6123	342	1	k	k	NOUN
ejpam-6123	342	2	−	−	PROPN
ejpam-6123	342	3	2	2	NUM
ejpam-6123	342	4	}	}	PUNCT
ejpam-6123	342	5	.	.	PUNCT
ejpam-6123	343	1	put	put	VERB
ejpam-6123	343	2	c	c	NOUN
ejpam-6123	343	3	=	=	SYM
ejpam-6123	343	4	{	{	PUNCT
ejpam-6123	343	5	u	u	NOUN
ejpam-6123	343	6	}	}	PUNCT
ejpam-6123	343	7	∪	∪	VERB
ejpam-6123	343	8	s.	s.	PROPN
ejpam-6123	343	9	we	we	PRON
ejpam-6123	343	10	will	will	AUX
ejpam-6123	343	11	show	show	VERB
ejpam-6123	343	12	that	that	SCONJ
ejpam-6123	343	13	c	c	PROPN
ejpam-6123	343	14	is	be	AUX
ejpam-6123	343	15	a	a	DET
ejpam-6123	343	16	γce	γce	NOUN
ejpam-6123	343	17	-	-	PUNCT
ejpam-6123	343	18	set	set	NOUN
ejpam-6123	343	19	of	of	ADP
ejpam-6123	343	20	g+h	g+h	PROPN
ejpam-6123	343	21	.	.	PUNCT
ejpam-6123	344	1	let	let	VERB
ejpam-6123	344	2	v	v	NUM
ejpam-6123	344	3	∈	∈	PROPN
ejpam-6123	344	4	v	v	NOUN
ejpam-6123	344	5	(	(	PUNCT
ejpam-6123	344	6	g+h	g+h	NOUN
ejpam-6123	344	7	)	)	PUNCT
ejpam-6123	344	8	\c	\c	NOUN
ejpam-6123	344	9	.	.	PUNCT
ejpam-6123	345	1	if	if	SCONJ
ejpam-6123	345	2	v	v	NUM
ejpam-6123	345	3	∈	∈	PROPN
ejpam-6123	345	4	v	v	NOUN
ejpam-6123	345	5	(	(	PUNCT
ejpam-6123	345	6	g	g	NOUN
ejpam-6123	345	7	)	)	PUNCT
ejpam-6123	345	8	,	,	PUNCT
ejpam-6123	345	9	then	then	ADV
ejpam-6123	345	10	v	v	X
ejpam-6123	345	11	∈	∈	PROPN
ejpam-6123	345	12	ng+h(u	ng+h(u	PROPN
ejpam-6123	345	13	)	)	PUNCT
ejpam-6123	345	14	and	and	CCONJ
ejpam-6123	345	15	|degg+h(u)−	|degg+h(u)−	VERB
ejpam-6123	345	16	degg+h(v)|	degg+h(v)|	PROPN
ejpam-6123	345	17	=	=	SYM
ejpam-6123	345	18	|degg(u	|degg(u	PROPN
ejpam-6123	345	19	)	)	PUNCT
ejpam-6123	345	20	+	+	CCONJ
ejpam-6123	345	21	|v	|v	X
ejpam-6123	345	22	(	(	PUNCT
ejpam-6123	345	23	h)|	h)|	NOUN
ejpam-6123	345	24	−	−	PROPN
ejpam-6123	345	25	(	(	PUNCT
ejpam-6123	345	26	degg(v	degg(v	PROPN
ejpam-6123	345	27	)	)	PUNCT
ejpam-6123	345	28	+	+	CCONJ
ejpam-6123	345	29	|v	|v	X
ejpam-6123	345	30	(	(	PUNCT
ejpam-6123	345	31	h)|)|	h)|)|	PROPN
ejpam-6123	345	32	=	=	SYM
ejpam-6123	345	33	|	|	NOUN
ejpam-6123	346	1	degg(u)−	degg(u)−	PROPN
ejpam-6123	347	1	degg(v)|	degg(v)|	PROPN
ejpam-6123	348	1	=	=	SYM
ejpam-6123	349	1	(	(	PUNCT
ejpam-6123	349	2	n−	n−	PROPN
ejpam-6123	349	3	1)−	1)−	PROPN
ejpam-6123	349	4	(	(	PUNCT
ejpam-6123	349	5	n−	n−	NOUN
ejpam-6123	349	6	1	1	NUM
ejpam-6123	349	7	)	)	PUNCT
ejpam-6123	349	8	=	=	PUNCT
ejpam-6123	349	9	0	0	PUNCT
ejpam-6123	349	10	<	<	X
ejpam-6123	349	11	1	1	NUM
ejpam-6123	349	12	.	.	PUNCT
ejpam-6123	349	13	h.	h.	PROPN
ejpam-6123	349	14	nuenay	nuenay	PROPN
ejpam-6123	349	15	-	-	PUNCT
ejpam-6123	349	16	maglanquel	maglanquel	PROPN
ejpam-6123	349	17	/	/	SYM
ejpam-6123	349	18	eur	eur	PROPN
ejpam-6123	349	19	.	.	PUNCT
ejpam-6123	350	1	j.	j.	PROPN
ejpam-6123	350	2	pure	pure	PROPN
ejpam-6123	350	3	appl	appl	PROPN
ejpam-6123	350	4	.	.	PROPN
ejpam-6123	350	5	math	math	PROPN
ejpam-6123	350	6	,	,	PUNCT
ejpam-6123	350	7	18	18	NUM
ejpam-6123	350	8	(	(	PUNCT
ejpam-6123	350	9	3	3	NUM
ejpam-6123	350	10	)	)	PUNCT
ejpam-6123	350	11	(	(	PUNCT
ejpam-6123	350	12	2025	2025	NUM
ejpam-6123	350	13	)	)	PUNCT
ejpam-6123	350	14	,	,	PUNCT
ejpam-6123	350	15	6123	6123	NUM
ejpam-6123	350	16	10	10	NUM
ejpam-6123	350	17	of	of	ADP
ejpam-6123	350	18	14	14	NUM
ejpam-6123	350	19	suppose	suppose	VERB
ejpam-6123	350	20	that	that	SCONJ
ejpam-6123	350	21	v	v	X
ejpam-6123	350	22	∈	∈	PROPN
ejpam-6123	350	23	v	v	NOUN
ejpam-6123	350	24	(	(	PUNCT
ejpam-6123	350	25	h	h	NOUN
ejpam-6123	350	26	)	)	PUNCT
ejpam-6123	350	27	.	.	PUNCT
ejpam-6123	351	1	if	if	SCONJ
ejpam-6123	351	2	degh(v	degh(v	PROPN
ejpam-6123	351	3	)	)	PUNCT
ejpam-6123	351	4	≥	≥	PROPN
ejpam-6123	352	1	n−	n−	NOUN
ejpam-6123	352	2	k	k	NOUN
ejpam-6123	352	3	−	−	PROPN
ejpam-6123	352	4	2	2	NUM
ejpam-6123	352	5	,	,	PUNCT
ejpam-6123	352	6	the	the	DET
ejpam-6123	352	7	v	v	NOUN
ejpam-6123	352	8	is	be	AUX
ejpam-6123	352	9	equitably	equitably	ADV
ejpam-6123	352	10	dominated	dominate	VERB
ejpam-6123	352	11	by	by	ADP
ejpam-6123	352	12	u	u	NOUN
ejpam-6123	352	13	,	,	PUNCT
ejpam-6123	352	14	that	that	ADV
ejpam-6123	352	15	is	is	ADV
ejpam-6123	352	16	,	,	PUNCT
ejpam-6123	352	17	v	v	ADP
ejpam-6123	352	18	∈	∈	PROPN
ejpam-6123	352	19	ng+h(u	ng+h(u	PROPN
ejpam-6123	352	20	)	)	PUNCT
ejpam-6123	352	21	and	and	CCONJ
ejpam-6123	352	22	|degg+h(u)−	|degg+h(u)−	NOUN
ejpam-6123	352	23	degg+h(v)|	degg+h(v)|	PROPN
ejpam-6123	352	24	=	=	SYM
ejpam-6123	352	25	|	|	ADV
ejpam-6123	352	26	degg(u	degg(u	NOUN
ejpam-6123	352	27	)	)	PUNCT
ejpam-6123	353	1	+	+	CCONJ
ejpam-6123	353	2	|v	|v	X
ejpam-6123	353	3	(	(	PUNCT
ejpam-6123	353	4	h)|	h)|	NOUN
ejpam-6123	353	5	−	−	PROPN
ejpam-6123	353	6	(	(	PUNCT
ejpam-6123	353	7	degh(v	degh(v	PROPN
ejpam-6123	353	8	)	)	PUNCT
ejpam-6123	353	9	+	+	CCONJ
ejpam-6123	353	10	|v	|v	PROPN
ejpam-6123	353	11	(	(	PUNCT
ejpam-6123	353	12	g)|)|	g)|)|	PROPN
ejpam-6123	353	13	=	=	PUNCT
ejpam-6123	353	14	|	|	ADV
ejpam-6123	353	15	degg(u)−	degg(u)−	PROPN
ejpam-6123	353	16	degh(v)−	degh(v)−	PROPN
ejpam-6123	353	17	(	(	PUNCT
ejpam-6123	353	18	|v	|v	X
ejpam-6123	353	19	(	(	PUNCT
ejpam-6123	353	20	g)|	g)|	PROPN
ejpam-6123	353	21	−	−	PROPN
ejpam-6123	353	22	|v	|v	NOUN
ejpam-6123	353	23	(	(	PUNCT
ejpam-6123	353	24	h)|)|	h)|)|	PROPN
ejpam-6123	353	25	=	=	SYM
ejpam-6123	354	1	|	|	ADV
ejpam-6123	354	2	degg(u)−	degg(u)−	PROPN
ejpam-6123	355	1	degh(v)−	degh(v)−	PROPN
ejpam-6123	355	2	(	(	PUNCT
ejpam-6123	355	3	n−m)|	n−m)|	SYM
ejpam-6123	355	4	=	=	SYM
ejpam-6123	355	5	|	|	ADV
ejpam-6123	355	6	degg(u)−	degg(u)−	VERB
ejpam-6123	355	7	degh(v)−	degh(v)−	PROPN
ejpam-6123	355	8	k|	k|	NOUN
ejpam-6123	355	9	<	<	X
ejpam-6123	355	10	|	|	NOUN
ejpam-6123	355	11	degg(u)−	degg(u)−	PROPN
ejpam-6123	355	12	(	(	PUNCT
ejpam-6123	355	13	n−	n−	NOUN
ejpam-6123	355	14	k	k	NOUN
ejpam-6123	355	15	−	−	PROPN
ejpam-6123	355	16	2)−	2)−	NUM
ejpam-6123	355	17	k|	k|	NOUN
ejpam-6123	355	18	=	=	SYM
ejpam-6123	355	19	|n−	|n−	NOUN
ejpam-6123	355	20	1−	1−	NUM
ejpam-6123	355	21	n+	n+	PUNCT
ejpam-6123	356	1	k	k	PROPN
ejpam-6123	357	1	+	+	CCONJ
ejpam-6123	357	2	2−	2−	NUM
ejpam-6123	357	3	k|	k|	NOUN
ejpam-6123	357	4	=	=	SYM
ejpam-6123	357	5	1	1	X
ejpam-6123	357	6	.	.	PUNCT
ejpam-6123	357	7	suppose	suppose	VERB
ejpam-6123	357	8	that	that	SCONJ
ejpam-6123	357	9	degh(v	degh(v	NOUN
ejpam-6123	357	10	)	)	PUNCT
ejpam-6123	357	11	<	<	X
ejpam-6123	357	12	n−	n−	PROPN
ejpam-6123	357	13	k−	k−	PROPN
ejpam-6123	357	14	2	2	NUM
ejpam-6123	357	15	.	.	PUNCT
ejpam-6123	358	1	since	since	SCONJ
ejpam-6123	358	2	s	s	PROPN
ejpam-6123	358	3	is	be	AUX
ejpam-6123	358	4	an	an	DET
ejpam-6123	358	5	equitable	equitable	ADJ
ejpam-6123	358	6	dominating	dominating	NOUN
ejpam-6123	358	7	set	set	VERB
ejpam-6123	358	8	in	in	ADP
ejpam-6123	358	9	⟨v	⟨v	PROPN
ejpam-6123	358	10	(	(	PUNCT
ejpam-6123	358	11	h	h	NOUN
ejpam-6123	358	12	\dh⟩	\dh⟩	NOUN
ejpam-6123	358	13	there	there	ADV
ejpam-6123	358	14	exists	exist	VERB
ejpam-6123	358	15	y	y	PROPN
ejpam-6123	358	16	∈	∈	PROPN
ejpam-6123	358	17	s	s	VERB
ejpam-6123	358	18	such	such	ADJ
ejpam-6123	358	19	that	that	PRON
ejpam-6123	358	20	v	v	ADP
ejpam-6123	358	21	∈	∈	PROPN
ejpam-6123	358	22	ng+h(y	ng+h(y	NUM
ejpam-6123	358	23	)	)	PUNCT
ejpam-6123	358	24	and	and	CCONJ
ejpam-6123	358	25	|	|	ADV
ejpam-6123	358	26	degg+h(y)−	degg+h(y)−	VERB
ejpam-6123	358	27	degg+h(v)|	degg+h(v)|	PROPN
ejpam-6123	358	28	=	=	SYM
ejpam-6123	358	29	|	|	ADV
ejpam-6123	358	30	degh(y	degh(y	ADJ
ejpam-6123	358	31	)	)	PUNCT
ejpam-6123	359	1	+	+	CCONJ
ejpam-6123	359	2	|v	|v	X
ejpam-6123	359	3	(	(	PUNCT
ejpam-6123	359	4	g)|	g)|	INTJ
ejpam-6123	359	5	−	−	PROPN
ejpam-6123	359	6	(	(	PUNCT
ejpam-6123	359	7	degh(v	degh(v	PROPN
ejpam-6123	359	8	)	)	PUNCT
ejpam-6123	359	9	+	+	CCONJ
ejpam-6123	359	10	|v	|v	PROPN
ejpam-6123	359	11	(	(	PUNCT
ejpam-6123	359	12	g)|)|	g)|)|	PROPN
ejpam-6123	359	13	=	=	PROPN
ejpam-6123	359	14	|	|	PROPN
ejpam-6123	359	15	degh(y)−	degh(y)−	PROPN
ejpam-6123	359	16	degh(v)|	degh(v)|	PROPN
ejpam-6123	359	17	<	<	X
ejpam-6123	359	18	1	1	NUM
ejpam-6123	359	19	.	.	PUNCT
ejpam-6123	360	1	thus	thus	ADV
ejpam-6123	360	2	,	,	PUNCT
ejpam-6123	360	3	c	c	PROPN
ejpam-6123	360	4	is	be	AUX
ejpam-6123	360	5	a	a	DET
ejpam-6123	360	6	connected	connect	VERB
ejpam-6123	360	7	equitable	equitable	ADJ
ejpam-6123	360	8	dominating	dominating	NOUN
ejpam-6123	360	9	set	set	VERB
ejpam-6123	360	10	in	in	ADP
ejpam-6123	360	11	g+h	g+h	PROPN
ejpam-6123	360	12	.	.	PUNCT
ejpam-6123	361	1	hence	hence	ADV
ejpam-6123	361	2	,	,	PUNCT
ejpam-6123	361	3	γce(g+h	γce(g+h	ADJ
ejpam-6123	361	4	)	)	PUNCT
ejpam-6123	361	5	≤	≤	NUM
ejpam-6123	361	6	|c|	|c|	PROPN
ejpam-6123	361	7	.	.	PUNCT
ejpam-6123	361	8	suppose	suppose	VERB
ejpam-6123	361	9	that	that	SCONJ
ejpam-6123	361	10	γce(g+h	γce(g+h	PROPN
ejpam-6123	361	11	)	)	PUNCT
ejpam-6123	361	12	<	<	X
ejpam-6123	361	13	|c|	|c|	PROPN
ejpam-6123	361	14	.	.	PUNCT
ejpam-6123	362	1	then	then	ADV
ejpam-6123	362	2	there	there	PRON
ejpam-6123	362	3	exists	exist	VERB
ejpam-6123	362	4	a	a	DET
ejpam-6123	362	5	γce	γce	NOUN
ejpam-6123	362	6	-	-	PUNCT
ejpam-6123	362	7	set	set	VERB
ejpam-6123	362	8	in	in	ADP
ejpam-6123	362	9	g+h	g+h	PROPN
ejpam-6123	362	10	say	say	VERB
ejpam-6123	362	11	a	a	PRON
ejpam-6123	362	12	with	with	ADP
ejpam-6123	362	13	|a|	|a|	NOUN
ejpam-6123	362	14	<	<	X
ejpam-6123	362	15	|c|	|c|	PROPN
ejpam-6123	362	16	.	.	PUNCT
ejpam-6123	362	17	suppose	suppose	VERB
ejpam-6123	362	18	that	that	SCONJ
ejpam-6123	362	19	a	a	DET
ejpam-6123	362	20	=	=	SYM
ejpam-6123	362	21	c	c	NOUN
ejpam-6123	362	22	\	\	X
ejpam-6123	362	23	{	{	PUNCT
ejpam-6123	362	24	x	x	NOUN
ejpam-6123	362	25	}	}	PUNCT
ejpam-6123	362	26	with	with	ADP
ejpam-6123	362	27	x	x	PROPN
ejpam-6123	362	28	∈	∈	PROPN
ejpam-6123	362	29	{	{	PUNCT
ejpam-6123	362	30	u	u	NOUN
ejpam-6123	362	31	}	}	PUNCT
ejpam-6123	362	32	∪	∪	VERB
ejpam-6123	362	33	s	s	PRON
ejpam-6123	362	34	with	with	ADP
ejpam-6123	362	35	s	s	PRON
ejpam-6123	362	36	a	a	DET
ejpam-6123	362	37	γe	γe	NOUN
ejpam-6123	362	38	-	-	PUNCT
ejpam-6123	362	39	set	set	NOUN
ejpam-6123	362	40	of	of	ADP
ejpam-6123	362	41	⟨v	⟨v	PROPN
ejpam-6123	362	42	(	(	PUNCT
ejpam-6123	362	43	h	h	NOUN
ejpam-6123	362	44	\d∗	\d∗	PROPN
ejpam-6123	362	45	h⟩	h⟩	PROPN
ejpam-6123	362	46	.	.	PUNCT
ejpam-6123	363	1	if	if	SCONJ
ejpam-6123	363	2	x	x	X
ejpam-6123	363	3	=	=	SYM
ejpam-6123	363	4	u	u	NOUN
ejpam-6123	363	5	,	,	PUNCT
ejpam-6123	363	6	then	then	ADV
ejpam-6123	363	7	for	for	ADP
ejpam-6123	363	8	all	all	DET
ejpam-6123	363	9	z	z	NOUN
ejpam-6123	363	10	∈	∈	PROPN
ejpam-6123	363	11	v	v	NOUN
ejpam-6123	363	12	(	(	PUNCT
ejpam-6123	363	13	g+h	g+h	PROPN
ejpam-6123	363	14	)	)	PUNCT
ejpam-6123	363	15	whose	whose	DET
ejpam-6123	363	16	degree	degree	NOUN
ejpam-6123	363	17	is	be	AUX
ejpam-6123	363	18	less	less	ADJ
ejpam-6123	363	19	than	than	ADP
ejpam-6123	363	20	n+	n+	PUNCT
ejpam-6123	364	1	k	k	NOUN
ejpam-6123	365	1	−	−	NOUN
ejpam-6123	365	2	2	2	NUM
ejpam-6123	365	3	is	be	AUX
ejpam-6123	365	4	not	not	PART
ejpam-6123	365	5	equitably	equitably	ADV
ejpam-6123	365	6	dominated	dominate	VERB
ejpam-6123	365	7	by	by	ADP
ejpam-6123	365	8	a	a	DET
ejpam-6123	365	9	,	,	PUNCT
ejpam-6123	365	10	a	a	DET
ejpam-6123	365	11	contradiction	contradiction	NOUN
ejpam-6123	365	12	.	.	PUNCT
ejpam-6123	366	1	if	if	SCONJ
ejpam-6123	366	2	x	x	PUNCT
ejpam-6123	366	3	∈	∈	PROPN
ejpam-6123	366	4	s	s	PROPN
ejpam-6123	366	5	,	,	PUNCT
ejpam-6123	366	6	then	then	ADV
ejpam-6123	366	7	there	there	PRON
ejpam-6123	366	8	exists	exist	VERB
ejpam-6123	366	9	an	an	DET
ejpam-6123	366	10	element	element	NOUN
ejpam-6123	366	11	w	w	PROPN
ejpam-6123	366	12	∈	∈	PROPN
ejpam-6123	366	13	⟨v	⟨v	PUNCT
ejpam-6123	366	14	(	(	PUNCT
ejpam-6123	366	15	h	h	NOUN
ejpam-6123	366	16	\	\	PROPN
ejpam-6123	366	17	dh⟩	dh⟩	PROPN
ejpam-6123	366	18	which	which	PRON
ejpam-6123	366	19	nor	nor	CCONJ
ejpam-6123	366	20	equitably	equitably	ADV
ejpam-6123	366	21	dominated	dominate	VERB
ejpam-6123	366	22	by	by	ADP
ejpam-6123	366	23	a	a	DET
ejpam-6123	366	24	,	,	PUNCT
ejpam-6123	366	25	a	a	DET
ejpam-6123	366	26	contradiction	contradiction	NOUN
ejpam-6123	366	27	.	.	PUNCT
ejpam-6123	367	1	thus	thus	ADV
ejpam-6123	367	2	,	,	PUNCT
ejpam-6123	367	3	γce(g	γce(g	PROPN
ejpam-6123	367	4	+	+	CCONJ
ejpam-6123	367	5	h	h	NOUN
ejpam-6123	367	6	)	)	PUNCT
ejpam-6123	367	7	=	=	SYM
ejpam-6123	367	8	|c|	|c|	PROPN
ejpam-6123	367	9	.	.	PUNCT
ejpam-6123	367	10	therefore	therefore	ADV
ejpam-6123	367	11	,	,	PUNCT
ejpam-6123	367	12	γce(g+h	γce(g+h	ADJ
ejpam-6123	367	13	)	)	PUNCT
ejpam-6123	367	14	=	=	SYM
ejpam-6123	367	15	1	1	NUM
ejpam-6123	367	16	+	+	NUM
ejpam-6123	367	17	γe(⟨v	γe(⟨v	NOUN
ejpam-6123	367	18	(	(	PUNCT
ejpam-6123	367	19	h	h	NOUN
ejpam-6123	367	20	\dh⟩	\dh⟩	NOUN
ejpam-6123	367	21	)	)	PUNCT
ejpam-6123	367	22	.	.	PUNCT
ejpam-6123	368	1	theorem	theorem	NOUN
ejpam-6123	368	2	13	13	NUM
ejpam-6123	368	3	.	.	PUNCT
ejpam-6123	369	1	let	let	VERB
ejpam-6123	369	2	g	g	PRON
ejpam-6123	369	3	be	be	AUX
ejpam-6123	369	4	a	a	DET
ejpam-6123	369	5	complete	complete	ADJ
ejpam-6123	369	6	graph	graph	NOUN
ejpam-6123	369	7	of	of	ADP
ejpam-6123	369	8	order	order	NOUN
ejpam-6123	369	9	n	n	NOUN
ejpam-6123	370	1	and	and	CCONJ
ejpam-6123	370	2	h	h	NOUN
ejpam-6123	370	3	be	be	AUX
ejpam-6123	370	4	any	any	DET
ejpam-6123	370	5	graph	graph	NOUN
ejpam-6123	370	6	of	of	ADP
ejpam-6123	370	7	order	order	NOUN
ejpam-6123	370	8	m	m	VERB
ejpam-6123	370	9	with	with	ADP
ejpam-6123	370	10	m−	m−	PROPN
ejpam-6123	370	11	n	n	PROPN
ejpam-6123	370	12	=	=	SYM
ejpam-6123	370	13	k	k	X
ejpam-6123	370	14	≥	≥	NUM
ejpam-6123	370	15	0	0	NUM
ejpam-6123	370	16	.	.	PUNCT
ejpam-6123	371	1	then	then	ADV
ejpam-6123	371	2	γce(g+h	γce(g+h	NUM
ejpam-6123	371	3	)	)	PUNCT
ejpam-6123	371	4	=	=	SYM
ejpam-6123	372	1	1	1	NUM
ejpam-6123	372	2	+	+	NUM
ejpam-6123	372	3	γe(⟨v	γe(⟨v	NOUN
ejpam-6123	372	4	(	(	PUNCT
ejpam-6123	372	5	h	h	NOUN
ejpam-6123	372	6	\d∗	\d∗	NOUN
ejpam-6123	372	7	h⟩	h⟩	PROPN
ejpam-6123	372	8	)	)	PUNCT
ejpam-6123	372	9	where	where	SCONJ
ejpam-6123	372	10	v	v	X
ejpam-6123	372	11	(	(	PUNCT
ejpam-6123	372	12	h	h	NOUN
ejpam-6123	372	13	)	)	PUNCT
ejpam-6123	372	14	\d∗	\d∗	X
ejpam-6123	372	15	h	h	NOUN
ejpam-6123	372	16	=	=	PRON
ejpam-6123	372	17	{	{	PUNCT
ejpam-6123	372	18	v	v	NUM
ejpam-6123	372	19	∈	∈	NOUN
ejpam-6123	372	20	v	v	NOUN
ejpam-6123	372	21	(	(	PUNCT
ejpam-6123	372	22	h	h	NOUN
ejpam-6123	372	23	)	)	PUNCT
ejpam-6123	372	24	:	:	PUNCT
ejpam-6123	372	25	degh(v	degh(v	NOUN
ejpam-6123	372	26	)	)	PUNCT
ejpam-6123	372	27	<	<	X
ejpam-6123	372	28	∆(g	∆(g	PROPN
ejpam-6123	372	29	)	)	PUNCT
ejpam-6123	372	30	+	+	CCONJ
ejpam-6123	372	31	(	(	PUNCT
ejpam-6123	372	32	k−	k−	NOUN
ejpam-6123	372	33	1	1	NUM
ejpam-6123	372	34	)	)	PUNCT
ejpam-6123	372	35	=	=	PRON
ejpam-6123	372	36	n+	n+	PART
ejpam-6123	372	37	k−	k−	PROPN
ejpam-6123	372	38	2	2	NUM
ejpam-6123	372	39	}	}	PUNCT
ejpam-6123	372	40	and	and	CCONJ
ejpam-6123	372	41	d∗	d∗	PROPN
ejpam-6123	372	42	h	h	NOUN
ejpam-6123	372	43	=	=	PRON
ejpam-6123	372	44	{	{	PUNCT
ejpam-6123	372	45	z	z	NOUN
ejpam-6123	372	46	∈	∈	PROPN
ejpam-6123	372	47	v	v	NOUN
ejpam-6123	372	48	(	(	PUNCT
ejpam-6123	372	49	h	h	NOUN
ejpam-6123	372	50	)	)	PUNCT
ejpam-6123	372	51	:	:	PUNCT
ejpam-6123	373	1	degh(z	degh(z	ADJ
ejpam-6123	373	2	)	)	PUNCT
ejpam-6123	373	3	≥	≥	NOUN
ejpam-6123	373	4	∆(g	∆(g	NOUN
ejpam-6123	373	5	)	)	PUNCT
ejpam-6123	374	1	+	+	CCONJ
ejpam-6123	374	2	(	(	PUNCT
ejpam-6123	374	3	k	k	NOUN
ejpam-6123	374	4	−	−	PROPN
ejpam-6123	374	5	1	1	NUM
ejpam-6123	374	6	)	)	PUNCT
ejpam-6123	374	7	=	=	PUNCT
ejpam-6123	374	8	n+	n+	PUNCT
ejpam-6123	375	1	k	k	X
ejpam-6123	375	2	−	−	NOUN
ejpam-6123	375	3	2	2	NUM
ejpam-6123	375	4	}	}	PUNCT
ejpam-6123	375	5	proof	proof	NOUN
ejpam-6123	375	6	.	.	PUNCT
ejpam-6123	376	1	let	let	VERB
ejpam-6123	376	2	u	u	PRON
ejpam-6123	376	3	∈	∈	PROPN
ejpam-6123	376	4	v	v	ADP
ejpam-6123	376	5	(	(	PUNCT
ejpam-6123	376	6	g	g	NOUN
ejpam-6123	376	7	)	)	PUNCT
ejpam-6123	376	8	and	and	CCONJ
ejpam-6123	376	9	s	s	VERB
ejpam-6123	376	10	⊆	⊆	NUM
ejpam-6123	376	11	v	v	NOUN
ejpam-6123	376	12	(	(	PUNCT
ejpam-6123	376	13	h	h	NOUN
ejpam-6123	376	14	)	)	PUNCT
ejpam-6123	376	15	\	\	PROPN
ejpam-6123	376	16	d∗	d∗	PROPN
ejpam-6123	376	17	h	h	NOUN
ejpam-6123	376	18	be	be	VERB
ejpam-6123	376	19	a	a	DET
ejpam-6123	376	20	γe	γe	NOUN
ejpam-6123	376	21	-	-	PUNCT
ejpam-6123	376	22	set	set	NOUN
ejpam-6123	376	23	of	of	ADP
ejpam-6123	376	24	⟨v	⟨v	PROPN
ejpam-6123	376	25	(	(	PUNCT
ejpam-6123	376	26	h	h	NOUN
ejpam-6123	376	27	\	\	PROPN
ejpam-6123	376	28	d∗	d∗	PROPN
ejpam-6123	376	29	h⟩	h⟩	PROPN
ejpam-6123	376	30	where	where	SCONJ
ejpam-6123	376	31	d∗	d∗	PROPN
ejpam-6123	376	32	h	h	NOUN
ejpam-6123	376	33	=	=	PRON
ejpam-6123	376	34	{	{	PUNCT
ejpam-6123	376	35	z	z	NOUN
ejpam-6123	376	36	∈	∈	PROPN
ejpam-6123	376	37	v	v	NOUN
ejpam-6123	376	38	(	(	PUNCT
ejpam-6123	376	39	h	h	NOUN
ejpam-6123	376	40	)	)	PUNCT
ejpam-6123	376	41	:	:	PUNCT
ejpam-6123	376	42	degh(z	degh(z	ADJ
ejpam-6123	376	43	)	)	PUNCT
ejpam-6123	376	44	≥	≥	NOUN
ejpam-6123	376	45	∆(g	∆(g	NOUN
ejpam-6123	376	46	)	)	PUNCT
ejpam-6123	377	1	+	+	CCONJ
ejpam-6123	377	2	(	(	PUNCT
ejpam-6123	377	3	k	k	NOUN
ejpam-6123	377	4	−	−	PROPN
ejpam-6123	377	5	1	1	NUM
ejpam-6123	377	6	)	)	PUNCT
ejpam-6123	377	7	=	=	SYM
ejpam-6123	378	1	n	n	PRON
ejpam-6123	378	2	−	−	NOUN
ejpam-6123	379	1	k	k	NOUN
ejpam-6123	379	2	−	−	PROPN
ejpam-6123	379	3	2	2	NUM
ejpam-6123	379	4	}	}	PUNCT
ejpam-6123	379	5	.	.	PUNCT
ejpam-6123	380	1	put	put	VERB
ejpam-6123	380	2	c	c	NOUN
ejpam-6123	380	3	=	=	SYM
ejpam-6123	380	4	{	{	PUNCT
ejpam-6123	380	5	u	u	NOUN
ejpam-6123	380	6	}	}	PUNCT
ejpam-6123	380	7	∪	∪	VERB
ejpam-6123	380	8	s.	s.	PROPN
ejpam-6123	380	9	we	we	PRON
ejpam-6123	380	10	will	will	AUX
ejpam-6123	380	11	show	show	VERB
ejpam-6123	380	12	that	that	SCONJ
ejpam-6123	380	13	c	c	PROPN
ejpam-6123	380	14	is	be	AUX
ejpam-6123	380	15	a	a	DET
ejpam-6123	380	16	γce	γce	NOUN
ejpam-6123	380	17	-	-	PUNCT
ejpam-6123	380	18	set	set	NOUN
ejpam-6123	380	19	of	of	ADP
ejpam-6123	380	20	g+h	g+h	PROPN
ejpam-6123	380	21	.	.	PUNCT
ejpam-6123	381	1	let	let	VERB
ejpam-6123	381	2	v	v	NUM
ejpam-6123	381	3	∈	∈	PROPN
ejpam-6123	381	4	v	v	NOUN
ejpam-6123	381	5	(	(	PUNCT
ejpam-6123	381	6	g+h	g+h	NOUN
ejpam-6123	381	7	)	)	PUNCT
ejpam-6123	381	8	\c	\c	NOUN
ejpam-6123	381	9	.	.	PUNCT
ejpam-6123	382	1	if	if	SCONJ
ejpam-6123	382	2	v	v	NUM
ejpam-6123	382	3	∈	∈	PROPN
ejpam-6123	382	4	v	v	NOUN
ejpam-6123	382	5	(	(	PUNCT
ejpam-6123	382	6	g	g	NOUN
ejpam-6123	382	7	)	)	PUNCT
ejpam-6123	382	8	,	,	PUNCT
ejpam-6123	382	9	then	then	ADV
ejpam-6123	382	10	v	v	X
ejpam-6123	382	11	∈	∈	PROPN
ejpam-6123	382	12	ng+h(u	ng+h(u	PROPN
ejpam-6123	382	13	)	)	PUNCT
ejpam-6123	382	14	and	and	CCONJ
ejpam-6123	382	15	|degg+h(u)−	|degg+h(u)−	VERB
ejpam-6123	382	16	degg+h(v)|	degg+h(v)|	PROPN
ejpam-6123	382	17	=	=	SYM
ejpam-6123	382	18	|degg(u	|degg(u	PROPN
ejpam-6123	382	19	)	)	PUNCT
ejpam-6123	382	20	+	+	CCONJ
ejpam-6123	382	21	|v	|v	X
ejpam-6123	382	22	(	(	PUNCT
ejpam-6123	382	23	h)|	h)|	NOUN
ejpam-6123	382	24	−	−	PROPN
ejpam-6123	382	25	(	(	PUNCT
ejpam-6123	382	26	degg(v	degg(v	PROPN
ejpam-6123	382	27	)	)	PUNCT
ejpam-6123	382	28	+	+	CCONJ
ejpam-6123	382	29	|v	|v	X
ejpam-6123	382	30	(	(	PUNCT
ejpam-6123	382	31	h)|)|	h)|)|	PROPN
ejpam-6123	382	32	=	=	SYM
ejpam-6123	382	33	|	|	NOUN
ejpam-6123	383	1	degg(u)−	degg(u)−	PROPN
ejpam-6123	384	1	degg(v)|	degg(v)|	PROPN
ejpam-6123	385	1	=	=	SYM
ejpam-6123	386	1	(	(	PUNCT
ejpam-6123	386	2	n−	n−	PROPN
ejpam-6123	386	3	1)−	1)−	PROPN
ejpam-6123	386	4	(	(	PUNCT
ejpam-6123	386	5	n−	n−	NOUN
ejpam-6123	386	6	1	1	NUM
ejpam-6123	386	7	)	)	PUNCT
ejpam-6123	386	8	=	=	PUNCT
ejpam-6123	386	9	0	0	PUNCT
ejpam-6123	386	10	<	<	X
ejpam-6123	386	11	1	1	NUM
ejpam-6123	386	12	.	.	PUNCT
ejpam-6123	386	13	h.	h.	PROPN
ejpam-6123	386	14	nuenay	nuenay	PROPN
ejpam-6123	386	15	-	-	PUNCT
ejpam-6123	386	16	maglanquel	maglanquel	PROPN
ejpam-6123	386	17	/	/	SYM
ejpam-6123	386	18	eur	eur	PROPN
ejpam-6123	386	19	.	.	PUNCT
ejpam-6123	387	1	j.	j.	PROPN
ejpam-6123	387	2	pure	pure	PROPN
ejpam-6123	387	3	appl	appl	PROPN
ejpam-6123	387	4	.	.	PROPN
ejpam-6123	387	5	math	math	PROPN
ejpam-6123	387	6	,	,	PUNCT
ejpam-6123	387	7	18	18	NUM
ejpam-6123	387	8	(	(	PUNCT
ejpam-6123	387	9	3	3	NUM
ejpam-6123	387	10	)	)	PUNCT
ejpam-6123	387	11	(	(	PUNCT
ejpam-6123	387	12	2025	2025	NUM
ejpam-6123	387	13	)	)	PUNCT
ejpam-6123	387	14	,	,	PUNCT
ejpam-6123	387	15	6123	6123	NUM
ejpam-6123	387	16	11	11	NUM
ejpam-6123	387	17	of	of	ADP
ejpam-6123	387	18	14	14	NUM
ejpam-6123	387	19	suppose	suppose	VERB
ejpam-6123	387	20	that	that	SCONJ
ejpam-6123	387	21	v	v	X
ejpam-6123	387	22	∈	∈	PROPN
ejpam-6123	387	23	v	v	NOUN
ejpam-6123	387	24	(	(	PUNCT
ejpam-6123	387	25	h	h	NOUN
ejpam-6123	387	26	)	)	PUNCT
ejpam-6123	387	27	.	.	PUNCT
ejpam-6123	388	1	if	if	SCONJ
ejpam-6123	388	2	degh(v	degh(v	PROPN
ejpam-6123	388	3	)	)	PUNCT
ejpam-6123	388	4	≥	≥	NUM
ejpam-6123	388	5	n+	n+	PUNCT
ejpam-6123	389	1	k	k	X
ejpam-6123	390	1	−	−	PROPN
ejpam-6123	390	2	2	2	NUM
ejpam-6123	390	3	,	,	PUNCT
ejpam-6123	390	4	the	the	DET
ejpam-6123	390	5	v	v	NOUN
ejpam-6123	390	6	is	be	AUX
ejpam-6123	390	7	equitably	equitably	ADV
ejpam-6123	390	8	dominated	dominate	VERB
ejpam-6123	390	9	by	by	ADP
ejpam-6123	390	10	u	u	NOUN
ejpam-6123	390	11	,	,	PUNCT
ejpam-6123	390	12	that	that	ADV
ejpam-6123	390	13	is	is	ADV
ejpam-6123	390	14	,	,	PUNCT
ejpam-6123	390	15	v	v	ADP
ejpam-6123	390	16	∈	∈	PROPN
ejpam-6123	390	17	ng+h(u	ng+h(u	PROPN
ejpam-6123	390	18	)	)	PUNCT
ejpam-6123	390	19	and	and	CCONJ
ejpam-6123	390	20	|degg+h(u)−	|degg+h(u)−	NOUN
ejpam-6123	390	21	degg+h(v)|	degg+h(v)|	PROPN
ejpam-6123	390	22	=	=	SYM
ejpam-6123	390	23	|	|	ADV
ejpam-6123	390	24	degg(u	degg(u	NOUN
ejpam-6123	390	25	)	)	PUNCT
ejpam-6123	391	1	+	+	CCONJ
ejpam-6123	391	2	|v	|v	X
ejpam-6123	391	3	(	(	PUNCT
ejpam-6123	391	4	h)|	h)|	NOUN
ejpam-6123	391	5	−	−	PROPN
ejpam-6123	391	6	(	(	PUNCT
ejpam-6123	391	7	degh(v	degh(v	PROPN
ejpam-6123	391	8	)	)	PUNCT
ejpam-6123	391	9	+	+	CCONJ
ejpam-6123	391	10	|v	|v	PROPN
ejpam-6123	391	11	(	(	PUNCT
ejpam-6123	392	1	g)|)|	g)|)|	PROPN
ejpam-6123	392	2	=	=	SYM
ejpam-6123	392	3	|	|	NOUN
ejpam-6123	392	4	degg(u)−	degg(u)−	PROPN
ejpam-6123	392	5	degh(v	degh(v	PROPN
ejpam-6123	392	6	)	)	PUNCT
ejpam-6123	393	1	+	+	CCONJ
ejpam-6123	393	2	(	(	PUNCT
ejpam-6123	393	3	|v	|v	X
ejpam-6123	393	4	(	(	PUNCT
ejpam-6123	393	5	h)|	h)|	PROPN
ejpam-6123	393	6	−	−	PROPN
ejpam-6123	393	7	|v	|v	PROPN
ejpam-6123	393	8	(	(	PUNCT
ejpam-6123	393	9	g)|)|	g)|)|	PROPN
ejpam-6123	393	10	=	=	SYM
ejpam-6123	393	11	|	|	NOUN
ejpam-6123	393	12	degg(u)−	degg(u)−	PROPN
ejpam-6123	393	13	degh(v	degh(v	PROPN
ejpam-6123	393	14	)	)	PUNCT
ejpam-6123	393	15	+	+	CCONJ
ejpam-6123	393	16	(	(	PUNCT
ejpam-6123	393	17	n−m)|	n−m)|	SYM
ejpam-6123	393	18	=	=	SYM
ejpam-6123	393	19	|	|	NOUN
ejpam-6123	393	20	degg(u)−	degg(u)−	PROPN
ejpam-6123	393	21	degh(v	degh(v	PROPN
ejpam-6123	393	22	)	)	PUNCT
ejpam-6123	394	1	+	+	CCONJ
ejpam-6123	394	2	k|	k|	NOUN
ejpam-6123	394	3	<	<	X
ejpam-6123	394	4	|	|	NOUN
ejpam-6123	394	5	degg(u)−	degg(u)−	PROPN
ejpam-6123	394	6	(	(	PUNCT
ejpam-6123	394	7	n+	n+	NUM
ejpam-6123	394	8	k	k	NOUN
ejpam-6123	395	1	−	−	NOUN
ejpam-6123	395	2	2	2	NUM
ejpam-6123	395	3	)	)	PUNCT
ejpam-6123	395	4	+	+	CCONJ
ejpam-6123	395	5	k|	k|	NOUN
ejpam-6123	395	6	=	=	SYM
ejpam-6123	395	7	|n−	|n−	NOUN
ejpam-6123	395	8	1−	1−	NUM
ejpam-6123	395	9	n−	n−	NOUN
ejpam-6123	395	10	k	k	NOUN
ejpam-6123	396	1	+	+	CCONJ
ejpam-6123	396	2	2	2	NUM
ejpam-6123	396	3	+	+	NUM
ejpam-6123	396	4	k|	k|	NOUN
ejpam-6123	396	5	=	=	SYM
ejpam-6123	396	6	1	1	X
ejpam-6123	396	7	.	.	PUNCT
ejpam-6123	396	8	suppose	suppose	VERB
ejpam-6123	396	9	that	that	SCONJ
ejpam-6123	396	10	degh(v	degh(v	NOUN
ejpam-6123	396	11	)	)	PUNCT
ejpam-6123	396	12	<	<	X
ejpam-6123	396	13	n+	n+	PROPN
ejpam-6123	396	14	k−	k−	NOUN
ejpam-6123	396	15	2	2	NUM
ejpam-6123	396	16	.	.	PUNCT
ejpam-6123	397	1	since	since	SCONJ
ejpam-6123	397	2	s	s	PROPN
ejpam-6123	397	3	is	be	AUX
ejpam-6123	397	4	an	an	DET
ejpam-6123	397	5	equitable	equitable	ADJ
ejpam-6123	397	6	dominating	dominating	NOUN
ejpam-6123	397	7	set	set	VERB
ejpam-6123	397	8	in	in	ADP
ejpam-6123	397	9	⟨v	⟨v	PROPN
ejpam-6123	397	10	(	(	PUNCT
ejpam-6123	397	11	h	h	NOUN
ejpam-6123	397	12	\d∗	\d∗	NOUN
ejpam-6123	397	13	h⟩	h⟩	X
ejpam-6123	397	14	there	there	PRON
ejpam-6123	397	15	exists	exist	VERB
ejpam-6123	397	16	y	y	PROPN
ejpam-6123	397	17	∈	∈	PROPN
ejpam-6123	397	18	s	s	VERB
ejpam-6123	397	19	such	such	ADJ
ejpam-6123	397	20	that	that	PRON
ejpam-6123	397	21	v	v	ADP
ejpam-6123	397	22	∈	∈	PROPN
ejpam-6123	397	23	ng+h(y	ng+h(y	NUM
ejpam-6123	397	24	)	)	PUNCT
ejpam-6123	397	25	and	and	CCONJ
ejpam-6123	397	26	|	|	ADV
ejpam-6123	397	27	degg+h(y)−	degg+h(y)−	VERB
ejpam-6123	397	28	degg+h(v)|	degg+h(v)|	PROPN
ejpam-6123	397	29	=	=	SYM
ejpam-6123	397	30	|	|	ADV
ejpam-6123	397	31	degh(y	degh(y	ADJ
ejpam-6123	397	32	)	)	PUNCT
ejpam-6123	398	1	+	+	CCONJ
ejpam-6123	398	2	|v	|v	X
ejpam-6123	398	3	(	(	PUNCT
ejpam-6123	398	4	g)|	g)|	INTJ
ejpam-6123	398	5	−	−	PROPN
ejpam-6123	398	6	(	(	PUNCT
ejpam-6123	398	7	degh(v	degh(v	PROPN
ejpam-6123	398	8	)	)	PUNCT
ejpam-6123	398	9	+	+	CCONJ
ejpam-6123	398	10	|v	|v	PROPN
ejpam-6123	398	11	(	(	PUNCT
ejpam-6123	398	12	g)|)|	g)|)|	PROPN
ejpam-6123	398	13	=	=	PROPN
ejpam-6123	398	14	|	|	PROPN
ejpam-6123	398	15	degh(y)−	degh(y)−	PROPN
ejpam-6123	398	16	degh(v)|	degh(v)|	PROPN
ejpam-6123	398	17	<	<	X
ejpam-6123	398	18	1	1	NUM
ejpam-6123	398	19	.	.	PUNCT
ejpam-6123	399	1	thus	thus	ADV
ejpam-6123	399	2	,	,	PUNCT
ejpam-6123	399	3	c	c	PROPN
ejpam-6123	399	4	is	be	AUX
ejpam-6123	399	5	a	a	DET
ejpam-6123	399	6	connected	connect	VERB
ejpam-6123	399	7	equitable	equitable	ADJ
ejpam-6123	399	8	dominating	dominating	NOUN
ejpam-6123	399	9	set	set	VERB
ejpam-6123	399	10	in	in	ADP
ejpam-6123	399	11	g+h	g+h	PROPN
ejpam-6123	399	12	.	.	PUNCT
ejpam-6123	400	1	hence	hence	ADV
ejpam-6123	400	2	,	,	PUNCT
ejpam-6123	400	3	γce(g+h	γce(g+h	ADJ
ejpam-6123	400	4	)	)	PUNCT
ejpam-6123	400	5	≤	≤	NUM
ejpam-6123	400	6	|c|	|c|	PROPN
ejpam-6123	400	7	.	.	PUNCT
ejpam-6123	400	8	suppose	suppose	VERB
ejpam-6123	400	9	that	that	SCONJ
ejpam-6123	400	10	γce(g+h	γce(g+h	PROPN
ejpam-6123	400	11	)	)	PUNCT
ejpam-6123	400	12	<	<	X
ejpam-6123	400	13	|c|	|c|	PROPN
ejpam-6123	400	14	.	.	PUNCT
ejpam-6123	401	1	then	then	ADV
ejpam-6123	401	2	there	there	PRON
ejpam-6123	401	3	exists	exist	VERB
ejpam-6123	401	4	a	a	DET
ejpam-6123	401	5	γce	γce	NOUN
ejpam-6123	401	6	-	-	PUNCT
ejpam-6123	401	7	set	set	VERB
ejpam-6123	401	8	in	in	ADP
ejpam-6123	401	9	g+h	g+h	PROPN
ejpam-6123	401	10	say	say	VERB
ejpam-6123	401	11	a	a	PRON
ejpam-6123	401	12	with	with	ADP
ejpam-6123	401	13	|a|	|a|	NOUN
ejpam-6123	401	14	<	<	X
ejpam-6123	401	15	|c|	|c|	PROPN
ejpam-6123	401	16	.	.	PUNCT
ejpam-6123	401	17	suppose	suppose	VERB
ejpam-6123	401	18	that	that	SCONJ
ejpam-6123	401	19	a	a	DET
ejpam-6123	401	20	=	=	SYM
ejpam-6123	401	21	c	c	NOUN
ejpam-6123	401	22	\	\	X
ejpam-6123	401	23	{	{	PUNCT
ejpam-6123	401	24	x	x	NOUN
ejpam-6123	401	25	}	}	PUNCT
ejpam-6123	401	26	with	with	ADP
ejpam-6123	401	27	x	x	PROPN
ejpam-6123	401	28	∈	∈	PROPN
ejpam-6123	401	29	{	{	PUNCT
ejpam-6123	401	30	u	u	NOUN
ejpam-6123	401	31	}	}	PUNCT
ejpam-6123	401	32	∪	∪	VERB
ejpam-6123	401	33	s	s	PRON
ejpam-6123	401	34	with	with	ADP
ejpam-6123	401	35	s	s	PRON
ejpam-6123	401	36	a	a	DET
ejpam-6123	401	37	γe	γe	NOUN
ejpam-6123	401	38	-	-	PUNCT
ejpam-6123	401	39	set	set	NOUN
ejpam-6123	401	40	of	of	ADP
ejpam-6123	401	41	⟨v	⟨v	PROPN
ejpam-6123	401	42	(	(	PUNCT
ejpam-6123	401	43	h	h	NOUN
ejpam-6123	401	44	\d∗	\d∗	PROPN
ejpam-6123	401	45	h⟩	h⟩	PROPN
ejpam-6123	401	46	.	.	PUNCT
ejpam-6123	402	1	if	if	SCONJ
ejpam-6123	402	2	x	x	X
ejpam-6123	402	3	=	=	SYM
ejpam-6123	402	4	u	u	NOUN
ejpam-6123	402	5	,	,	PUNCT
ejpam-6123	402	6	then	then	ADV
ejpam-6123	402	7	for	for	ADP
ejpam-6123	402	8	all	all	DET
ejpam-6123	402	9	z	z	NOUN
ejpam-6123	402	10	∈	∈	PROPN
ejpam-6123	402	11	v	v	NOUN
ejpam-6123	402	12	(	(	PUNCT
ejpam-6123	402	13	g+h	g+h	PROPN
ejpam-6123	402	14	)	)	PUNCT
ejpam-6123	402	15	whose	whose	DET
ejpam-6123	402	16	degree	degree	NOUN
ejpam-6123	402	17	is	be	AUX
ejpam-6123	402	18	less	less	ADJ
ejpam-6123	402	19	than	than	ADP
ejpam-6123	402	20	n+	n+	PUNCT
ejpam-6123	403	1	k	k	NOUN
ejpam-6123	404	1	−	−	NOUN
ejpam-6123	404	2	2	2	NUM
ejpam-6123	404	3	is	be	AUX
ejpam-6123	404	4	not	not	PART
ejpam-6123	404	5	equitably	equitably	ADV
ejpam-6123	404	6	dominated	dominate	VERB
ejpam-6123	404	7	by	by	ADP
ejpam-6123	404	8	a	a	DET
ejpam-6123	404	9	,	,	PUNCT
ejpam-6123	404	10	a	a	DET
ejpam-6123	404	11	contradiction	contradiction	NOUN
ejpam-6123	404	12	.	.	PUNCT
ejpam-6123	405	1	if	if	SCONJ
ejpam-6123	405	2	x	x	PUNCT
ejpam-6123	405	3	∈	∈	PROPN
ejpam-6123	405	4	s	s	PROPN
ejpam-6123	405	5	,	,	PUNCT
ejpam-6123	405	6	then	then	ADV
ejpam-6123	405	7	there	there	PRON
ejpam-6123	405	8	exists	exist	VERB
ejpam-6123	405	9	an	an	DET
ejpam-6123	405	10	element	element	NOUN
ejpam-6123	405	11	w	w	PROPN
ejpam-6123	405	12	∈	∈	PROPN
ejpam-6123	405	13	⟨v	⟨v	PUNCT
ejpam-6123	405	14	(	(	PUNCT
ejpam-6123	405	15	h	h	NOUN
ejpam-6123	405	16	\	\	PROPN
ejpam-6123	405	17	d∗	d∗	PROPN
ejpam-6123	405	18	h⟩	h⟩	X
ejpam-6123	405	19	which	which	PRON
ejpam-6123	405	20	nor	nor	CCONJ
ejpam-6123	405	21	equitably	equitably	ADV
ejpam-6123	405	22	dominated	dominate	VERB
ejpam-6123	405	23	by	by	ADP
ejpam-6123	405	24	a	a	DET
ejpam-6123	405	25	,	,	PUNCT
ejpam-6123	405	26	a	a	DET
ejpam-6123	405	27	contradiction	contradiction	NOUN
ejpam-6123	405	28	.	.	PUNCT
ejpam-6123	406	1	thus	thus	ADV
ejpam-6123	406	2	,	,	PUNCT
ejpam-6123	406	3	γce(g	γce(g	PROPN
ejpam-6123	406	4	+	+	CCONJ
ejpam-6123	406	5	h	h	NOUN
ejpam-6123	406	6	)	)	PUNCT
ejpam-6123	406	7	=	=	SYM
ejpam-6123	406	8	|c|	|c|	PROPN
ejpam-6123	406	9	.	.	PUNCT
ejpam-6123	406	10	therefore	therefore	ADV
ejpam-6123	406	11	,	,	PUNCT
ejpam-6123	406	12	γce(g+h	γce(g+h	ADJ
ejpam-6123	406	13	)	)	PUNCT
ejpam-6123	406	14	=	=	SYM
ejpam-6123	406	15	1	1	NUM
ejpam-6123	406	16	+	+	NUM
ejpam-6123	406	17	γe(⟨v	γe(⟨v	NOUN
ejpam-6123	406	18	(	(	PUNCT
ejpam-6123	406	19	h	h	NOUN
ejpam-6123	406	20	\d∗	\d∗	NOUN
ejpam-6123	406	21	h⟩	h⟩	PROPN
ejpam-6123	406	22	)	)	PUNCT
ejpam-6123	406	23	.	.	PUNCT
ejpam-6123	407	1	4	4	X
ejpam-6123	407	2	.	.	X
ejpam-6123	407	3	corona	corona	NOUN
ejpam-6123	407	4	of	of	ADP
ejpam-6123	407	5	graphs	graph	NOUN
ejpam-6123	407	6	let	let	VERB
ejpam-6123	407	7	g	g	NOUN
ejpam-6123	407	8	and	and	CCONJ
ejpam-6123	407	9	h	h	NOUN
ejpam-6123	407	10	be	be	AUX
ejpam-6123	407	11	graphs	graph	NOUN
ejpam-6123	407	12	of	of	ADP
ejpam-6123	407	13	order	order	NOUN
ejpam-6123	407	14	m	m	VERB
ejpam-6123	407	15	and	and	CCONJ
ejpam-6123	407	16	n	n	CCONJ
ejpam-6123	407	17	,	,	PUNCT
ejpam-6123	407	18	respectively	respectively	ADV
ejpam-6123	407	19	.	.	PUNCT
ejpam-6123	408	1	the	the	DET
ejpam-6123	408	2	corona	corona	NOUN
ejpam-6123	408	3	g	g	PROPN
ejpam-6123	408	4	◦	◦	NOUN
ejpam-6123	408	5	h	h	NOUN
ejpam-6123	408	6	of	of	ADP
ejpam-6123	408	7	g	g	PROPN
ejpam-6123	408	8	and	and	CCONJ
ejpam-6123	408	9	h	h	NOUN
ejpam-6123	408	10	is	be	AUX
ejpam-6123	408	11	the	the	DET
ejpam-6123	408	12	graph	graph	NOUN
ejpam-6123	408	13	obtained	obtain	VERB
ejpam-6123	408	14	by	by	ADP
ejpam-6123	408	15	taking	take	VERB
ejpam-6123	408	16	one	one	NUM
ejpam-6123	408	17	copy	copy	NOUN
ejpam-6123	408	18	of	of	ADP
ejpam-6123	408	19	g	g	PROPN
ejpam-6123	408	20	and	and	CCONJ
ejpam-6123	408	21	m	m	PROPN
ejpam-6123	408	22	copies	copy	NOUN
ejpam-6123	408	23	of	of	ADP
ejpam-6123	408	24	h	h	NOUN
ejpam-6123	408	25	,	,	PUNCT
ejpam-6123	408	26	and	and	CCONJ
ejpam-6123	408	27	then	then	ADV
ejpam-6123	408	28	joining	join	VERB
ejpam-6123	408	29	the	the	DET
ejpam-6123	408	30	ith	ith	PROPN
ejpam-6123	408	31	vertex	vertex	NOUN
ejpam-6123	408	32	of	of	ADP
ejpam-6123	408	33	g	g	NOUN
ejpam-6123	408	34	to	to	ADP
ejpam-6123	408	35	every	every	DET
ejpam-6123	408	36	vertex	vertex	NOUN
ejpam-6123	408	37	of	of	ADP
ejpam-6123	408	38	the	the	DET
ejpam-6123	408	39	ith	ith	PROPN
ejpam-6123	408	40	copy	copy	NOUN
ejpam-6123	408	41	of	of	ADP
ejpam-6123	408	42	h.	h.	PROPN
ejpam-6123	408	43	for	for	ADP
ejpam-6123	408	44	everyv	everyv	ADJ
ejpam-6123	408	45	∈	∈	PROPN
ejpam-6123	408	46	v	v	NOUN
ejpam-6123	408	47	(	(	PUNCT
ejpam-6123	408	48	g	g	NOUN
ejpam-6123	408	49	)	)	PUNCT
ejpam-6123	408	50	,	,	PUNCT
ejpam-6123	408	51	denote	denote	VERB
ejpam-6123	408	52	by	by	ADP
ejpam-6123	408	53	hv	hv	PROPN
ejpam-6123	408	54	the	the	DET
ejpam-6123	408	55	copy	copy	NOUN
ejpam-6123	408	56	of	of	ADP
ejpam-6123	408	57	h	h	NOUN
ejpam-6123	408	58	whose	whose	DET
ejpam-6123	408	59	vertices	vertex	NOUN
ejpam-6123	408	60	are	be	AUX
ejpam-6123	408	61	attached	attach	VERB
ejpam-6123	408	62	one	one	NUM
ejpam-6123	408	63	by	by	ADP
ejpam-6123	408	64	one	one	NUM
ejpam-6123	408	65	to	to	ADP
ejpam-6123	408	66	the	the	DET
ejpam-6123	408	67	vertex	vertex	NOUN
ejpam-6123	408	68	v.	v.	ADP
ejpam-6123	408	69	denote	denote	VERB
ejpam-6123	408	70	by	by	ADP
ejpam-6123	408	71	v	v	PRON
ejpam-6123	408	72	+	+	NOUN
ejpam-6123	408	73	hv	hv	NOUN
ejpam-6123	408	74	the	the	DET
ejpam-6123	408	75	subgraph	subgraph	NOUN
ejpam-6123	408	76	of	of	ADP
ejpam-6123	408	77	the	the	DET
ejpam-6123	408	78	corona	corona	NOUN
ejpam-6123	408	79	g	g	PROPN
ejpam-6123	408	80	◦	◦	NOUN
ejpam-6123	408	81	h	h	NOUN
ejpam-6123	408	82	corresponding	correspond	VERB
ejpam-6123	408	83	to	to	ADP
ejpam-6123	408	84	the	the	DET
ejpam-6123	408	85	join	join	NOUN
ejpam-6123	408	86	⟨{v}⟩+hv	⟨{v}⟩+hv	PROPN
ejpam-6123	408	87	.	.	PUNCT
ejpam-6123	409	1	theorem	theorem	VERB
ejpam-6123	409	2	14	14	NUM
ejpam-6123	409	3	.	.	PUNCT
ejpam-6123	410	1	let	let	VERB
ejpam-6123	410	2	g	g	PRON
ejpam-6123	410	3	be	be	AUX
ejpam-6123	410	4	a	a	DET
ejpam-6123	410	5	connected	connected	ADJ
ejpam-6123	410	6	non	non	ADJ
ejpam-6123	410	7	-	-	ADJ
ejpam-6123	410	8	trivial	trivial	ADJ
ejpam-6123	410	9	graph	graph	NOUN
ejpam-6123	410	10	and	and	CCONJ
ejpam-6123	410	11	h	h	NOUN
ejpam-6123	410	12	be	be	AUX
ejpam-6123	410	13	any	any	DET
ejpam-6123	410	14	graph	graph	NOUN
ejpam-6123	410	15	of	of	ADP
ejpam-6123	410	16	order	order	NOUN
ejpam-6123	410	17	n.	n.	VERB
ejpam-6123	410	18	the	the	DET
ejpam-6123	410	19	c	c	PROPN
ejpam-6123	410	20	⊆	⊆	NUM
ejpam-6123	410	21	v	v	NOUN
ejpam-6123	410	22	(	(	PUNCT
ejpam-6123	410	23	g	g	PROPN
ejpam-6123	410	24	◦	◦	NOUN
ejpam-6123	410	25	h	h	NOUN
ejpam-6123	410	26	)	)	PUNCT
ejpam-6123	410	27	is	be	AUX
ejpam-6123	410	28	a	a	DET
ejpam-6123	410	29	connected	connect	VERB
ejpam-6123	410	30	equitable	equitable	ADJ
ejpam-6123	410	31	dominating	dominating	NOUN
ejpam-6123	410	32	set	set	NOUN
ejpam-6123	410	33	of	of	ADP
ejpam-6123	410	34	g	g	PROPN
ejpam-6123	410	35	◦	◦	NOUN
ejpam-6123	410	36	h	h	NOUN
ejpam-6123	411	1	if	if	SCONJ
ejpam-6123	411	2	and	and	CCONJ
ejpam-6123	411	3	only	only	ADV
ejpam-6123	411	4	if	if	SCONJ
ejpam-6123	411	5	c	c	PROPN
ejpam-6123	411	6	=	=	SYM
ejpam-6123	411	7	v	v	PROPN
ejpam-6123	411	8	(	(	PUNCT
ejpam-6123	411	9	g	g	NOUN
ejpam-6123	411	10	)	)	PUNCT
ejpam-6123	411	11	∪	∪	NOUN
ejpam-6123	411	12	(	(	PUNCT
ejpam-6123	411	13	⋃	⋃	ADJ
ejpam-6123	411	14	v∈v	v∈v	NOUN
ejpam-6123	411	15	(	(	PUNCT
ejpam-6123	411	16	g	g	NOUN
ejpam-6123	411	17	)	)	PUNCT
ejpam-6123	411	18	bv	bv	PROPN
ejpam-6123	411	19	)	)	PUNCT
ejpam-6123	411	20	,	,	PUNCT
ejpam-6123	411	21	where	where	SCONJ
ejpam-6123	411	22	for	for	ADP
ejpam-6123	411	23	each	each	DET
ejpam-6123	411	24	v	v	NUM
ejpam-6123	411	25	∈	∈	PROPN
ejpam-6123	411	26	v	v	NOUN
ejpam-6123	411	27	(	(	PUNCT
ejpam-6123	411	28	g	g	NOUN
ejpam-6123	411	29	)	)	PUNCT
ejpam-6123	411	30	,	,	PUNCT
ejpam-6123	411	31	(	(	PUNCT
ejpam-6123	411	32	i.	i.	PROPN
ejpam-6123	411	33	)	)	PUNCT
ejpam-6123	411	34	bv	bv	PROPN
ejpam-6123	411	35	is	be	AUX
ejpam-6123	411	36	an	an	DET
ejpam-6123	411	37	equitable	equitable	ADJ
ejpam-6123	411	38	dominating	dominating	NOUN
ejpam-6123	411	39	set	set	NOUN
ejpam-6123	411	40	of	of	ADP
ejpam-6123	411	41	hv	hv	PRON
ejpam-6123	411	42	whenever	whenever	SCONJ
ejpam-6123	411	43	∆(h	∆(h	VERB
ejpam-6123	411	44	)	)	PUNCT
ejpam-6123	411	45	≤	≤	NOUN
ejpam-6123	411	46	n	n	CCONJ
ejpam-6123	411	47	−	−	PROPN
ejpam-6123	411	48	2	2	NUM
ejpam-6123	411	49	or	or	CCONJ
ejpam-6123	411	50	degg(v	degg(v	PROPN
ejpam-6123	411	51	)	)	PUNCT
ejpam-6123	411	52	≥	≥	NOUN
ejpam-6123	411	53	2	2	NUM
ejpam-6123	411	54	and	and	CCONJ
ejpam-6123	411	55	(	(	PUNCT
ejpam-6123	411	56	ii	ii	NOUN
ejpam-6123	411	57	.	.	PUNCT
ejpam-6123	411	58	)	)	PUNCT
ejpam-6123	412	1	bv	bv	PROPN
ejpam-6123	412	2	dominates	dominate	VERB
ejpam-6123	412	3	equitably	equitably	ADV
ejpam-6123	412	4	v	v	ADP
ejpam-6123	412	5	(	(	PUNCT
ejpam-6123	412	6	hv)\sv	hv)\sv	PROPN
ejpam-6123	412	7	,	,	PUNCT
ejpam-6123	412	8	where	where	SCONJ
ejpam-6123	412	9	sv	sv	NOUN
ejpam-6123	412	10	=	=	PUNCT
ejpam-6123	412	11	bv∪{x	bv∪{x	PROPN
ejpam-6123	412	12	∈	∈	PROPN
ejpam-6123	412	13	v	v	ADP
ejpam-6123	412	14	(	(	PUNCT
ejpam-6123	412	15	hv	hv	PROPN
ejpam-6123	412	16	)	)	PUNCT
ejpam-6123	412	17	:	:	PUNCT
ejpam-6123	412	18	degh(x	degh(x	NOUN
ejpam-6123	412	19	)	)	PUNCT
ejpam-6123	412	20	=	=	SYM
ejpam-6123	412	21	n−1	n−1	ADJ
ejpam-6123	412	22	}	}	PUNCT
ejpam-6123	412	23	whenever	whenever	SCONJ
ejpam-6123	412	24	degg(v	degg(v	VERB
ejpam-6123	412	25	)	)	PUNCT
ejpam-6123	412	26	=	=	SYM
ejpam-6123	412	27	1	1	NUM
ejpam-6123	412	28	and	and	CCONJ
ejpam-6123	412	29	∆(h	∆(h	NOUN
ejpam-6123	412	30	)	)	PUNCT
ejpam-6123	412	31	=	=	SYM
ejpam-6123	412	32	n−	n−	NOUN
ejpam-6123	412	33	1	1	NUM
ejpam-6123	412	34	.	.	PUNCT
ejpam-6123	413	1	h.	h.	PROPN
ejpam-6123	413	2	nuenay	nuenay	PROPN
ejpam-6123	413	3	-	-	PUNCT
ejpam-6123	413	4	maglanquel	maglanquel	PROPN
ejpam-6123	413	5	/	/	SYM
ejpam-6123	413	6	eur	eur	PROPN
ejpam-6123	413	7	.	.	PUNCT
ejpam-6123	414	1	j.	j.	PROPN
ejpam-6123	414	2	pure	pure	PROPN
ejpam-6123	414	3	appl	appl	PROPN
ejpam-6123	414	4	.	.	PROPN
ejpam-6123	414	5	math	math	PROPN
ejpam-6123	414	6	,	,	PUNCT
ejpam-6123	414	7	18	18	NUM
ejpam-6123	414	8	(	(	PUNCT
ejpam-6123	414	9	3	3	NUM
ejpam-6123	414	10	)	)	PUNCT
ejpam-6123	414	11	(	(	PUNCT
ejpam-6123	414	12	2025	2025	NUM
ejpam-6123	414	13	)	)	PUNCT
ejpam-6123	414	14	,	,	PUNCT
ejpam-6123	414	15	6123	6123	NUM
ejpam-6123	414	16	12	12	NUM
ejpam-6123	414	17	of	of	ADP
ejpam-6123	414	18	14	14	NUM
ejpam-6123	414	19	proof	proof	NOUN
ejpam-6123	414	20	.	.	PUNCT
ejpam-6123	414	21	suppose	suppose	VERB
ejpam-6123	414	22	that	that	SCONJ
ejpam-6123	414	23	c	c	PROPN
ejpam-6123	414	24	is	be	AUX
ejpam-6123	414	25	a	a	DET
ejpam-6123	414	26	connected	connect	VERB
ejpam-6123	414	27	equitable	equitable	ADJ
ejpam-6123	414	28	dominating	dominating	NOUN
ejpam-6123	414	29	set	set	NOUN
ejpam-6123	414	30	of	of	ADP
ejpam-6123	414	31	g	g	PROPN
ejpam-6123	414	32	◦	◦	PROPN
ejpam-6123	414	33	h.	h.	PROPN
ejpam-6123	414	34	let	let	VERB
ejpam-6123	414	35	a	a	DET
ejpam-6123	414	36	=	=	SYM
ejpam-6123	414	37	c	c	NOUN
ejpam-6123	414	38	∩	∩	X
ejpam-6123	414	39	v	v	X
ejpam-6123	414	40	(	(	PUNCT
ejpam-6123	414	41	g	g	NOUN
ejpam-6123	414	42	)	)	PUNCT
ejpam-6123	414	43	and	and	CCONJ
ejpam-6123	414	44	let	let	VERB
ejpam-6123	414	45	bv	bv	PROPN
ejpam-6123	414	46	=	=	PROPN
ejpam-6123	414	47	c	c	PROPN
ejpam-6123	414	48	∩	∩	X
ejpam-6123	414	49	v	v	X
ejpam-6123	414	50	(	(	PUNCT
ejpam-6123	414	51	hv	hv	PROPN
ejpam-6123	414	52	)	)	PUNCT
ejpam-6123	414	53	for	for	ADP
ejpam-6123	414	54	each	each	DET
ejpam-6123	414	55	v	v	NUM
ejpam-6123	414	56	∈	∈	PROPN
ejpam-6123	414	57	v	v	NOUN
ejpam-6123	414	58	(	(	PUNCT
ejpam-6123	414	59	g	g	NOUN
ejpam-6123	414	60	)	)	PUNCT
ejpam-6123	414	61	.	.	PUNCT
ejpam-6123	415	1	since	since	SCONJ
ejpam-6123	415	2	⟨c⟩	⟨c⟩	PROPN
ejpam-6123	415	3	is	be	AUX
ejpam-6123	415	4	connected	connect	VERB
ejpam-6123	415	5	subgraph	subgraph	NOUN
ejpam-6123	415	6	of	of	ADP
ejpam-6123	415	7	g	g	PROPN
ejpam-6123	415	8	◦	◦	NOUN
ejpam-6123	415	9	h	h	NOUN
ejpam-6123	415	10	,	,	PUNCT
ejpam-6123	415	11	⟨a⟩	⟨a⟩	PROPN
ejpam-6123	415	12	is	be	AUX
ejpam-6123	415	13	a	a	DET
ejpam-6123	415	14	connected	connected	ADJ
ejpam-6123	415	15	subgraph	subgraph	NOUN
ejpam-6123	415	16	of	of	ADP
ejpam-6123	415	17	g.	g.	PROPN
ejpam-6123	415	18	if	if	SCONJ
ejpam-6123	415	19	there	there	PRON
ejpam-6123	415	20	exists	exist	VERB
ejpam-6123	415	21	v	v	ADP
ejpam-6123	415	22	∈	∈	PROPN
ejpam-6123	415	23	v	v	NOUN
ejpam-6123	415	24	(	(	PUNCT
ejpam-6123	415	25	g	g	NOUN
ejpam-6123	415	26	)	)	PUNCT
ejpam-6123	415	27	\	\	PROPN
ejpam-6123	416	1	a	a	DET
ejpam-6123	416	2	,	,	PUNCT
ejpam-6123	416	3	then	then	ADV
ejpam-6123	416	4	bv	bv	PROPN
ejpam-6123	416	5	̸=	̸=	PROPN
ejpam-6123	416	6	∅	∅	NOUN
ejpam-6123	416	7	since	since	SCONJ
ejpam-6123	416	8	c	c	PROPN
ejpam-6123	416	9	is	be	AUX
ejpam-6123	416	10	a	a	DET
ejpam-6123	416	11	dominating	dominating	NOUN
ejpam-6123	416	12	set	set	NOUN
ejpam-6123	416	13	of	of	ADP
ejpam-6123	416	14	g	g	PROPN
ejpam-6123	416	15	◦	◦	PROPN
ejpam-6123	416	16	h.	h.	NOUN
ejpam-6123	416	17	this	this	PRON
ejpam-6123	416	18	implies	imply	VERB
ejpam-6123	416	19	that	that	SCONJ
ejpam-6123	416	20	⟨c⟩	⟨c⟩	PROPN
ejpam-6123	416	21	is	be	AUX
ejpam-6123	416	22	not	not	PART
ejpam-6123	416	23	connected	connect	VERB
ejpam-6123	416	24	,	,	PUNCT
ejpam-6123	416	25	contrary	contrary	ADV
ejpam-6123	416	26	to	to	ADP
ejpam-6123	416	27	our	our	PRON
ejpam-6123	416	28	assumption	assumption	NOUN
ejpam-6123	416	29	that	that	SCONJ
ejpam-6123	416	30	c	c	PROPN
ejpam-6123	416	31	is	be	AUX
ejpam-6123	416	32	a	a	DET
ejpam-6123	416	33	connected	connected	ADJ
ejpam-6123	416	34	dominating	dominating	NOUN
ejpam-6123	416	35	set	set	NOUN
ejpam-6123	416	36	of	of	ADP
ejpam-6123	416	37	g	g	PROPN
ejpam-6123	416	38	◦	◦	PROPN
ejpam-6123	416	39	h.	h.	PROPN
ejpam-6123	417	1	thus	thus	ADV
ejpam-6123	417	2	,	,	PUNCT
ejpam-6123	417	3	a	a	DET
ejpam-6123	417	4	=	=	SYM
ejpam-6123	417	5	v	v	NOUN
ejpam-6123	417	6	(	(	PUNCT
ejpam-6123	417	7	g	g	NOUN
ejpam-6123	417	8	)	)	PUNCT
ejpam-6123	417	9	.	.	PUNCT
ejpam-6123	418	1	next	next	ADV
ejpam-6123	418	2	,	,	PUNCT
ejpam-6123	418	3	let	let	VERB
ejpam-6123	418	4	v	v	NUM
ejpam-6123	418	5	∈	∈	PROPN
ejpam-6123	418	6	v	v	NOUN
ejpam-6123	418	7	(	(	PUNCT
ejpam-6123	418	8	g	g	NOUN
ejpam-6123	418	9	)	)	PUNCT
ejpam-6123	418	10	and	and	CCONJ
ejpam-6123	418	11	let	let	VERB
ejpam-6123	418	12	q	q	PROPN
ejpam-6123	418	13	∈	∈	PROPN
ejpam-6123	418	14	v	v	ADP
ejpam-6123	418	15	(	(	PUNCT
ejpam-6123	418	16	hv	hv	PROPN
ejpam-6123	418	17	)	)	PUNCT
ejpam-6123	418	18	\	\	PROPN
ejpam-6123	418	19	bv	bv	PROPN
ejpam-6123	418	20	.	.	PROPN
ejpam-6123	418	21	then	then	ADV
ejpam-6123	418	22	there	there	PRON
ejpam-6123	418	23	exists	exist	VERB
ejpam-6123	418	24	w	w	PROPN
ejpam-6123	418	25	∈	∈	PROPN
ejpam-6123	418	26	c	c	PROPN
ejpam-6123	418	27	∩	∩	X
ejpam-6123	418	28	ng	ng	PROPN
ejpam-6123	418	29	◦	◦	NOUN
ejpam-6123	418	30	h(q	h(q	ADV
ejpam-6123	418	31	)	)	PUNCT
ejpam-6123	418	32	such	such	ADJ
ejpam-6123	418	33	that	that	SCONJ
ejpam-6123	418	34	|	|	ADV
ejpam-6123	418	35	degg	degg	NOUN
ejpam-6123	418	36	◦	◦	NOUN
ejpam-6123	418	37	h(w	h(w	PROPN
ejpam-6123	418	38	)	)	PUNCT
ejpam-6123	419	1	−	−	PROPN
ejpam-6123	419	2	degg	degg	NOUN
ejpam-6123	419	3	◦	◦	NOUN
ejpam-6123	419	4	h(q)|	h(q)|	X
ejpam-6123	419	5	≤	≤	NOUN
ejpam-6123	419	6	1	1	NUM
ejpam-6123	419	7	.	.	PUNCT
ejpam-6123	420	1	if	if	SCONJ
ejpam-6123	420	2	∆(h	∆(h	NOUN
ejpam-6123	420	3	)	)	PUNCT
ejpam-6123	420	4	≤	≤	NOUN
ejpam-6123	420	5	n	n	CCONJ
ejpam-6123	420	6	−	−	PROPN
ejpam-6123	420	7	2	2	NUM
ejpam-6123	420	8	,	,	PUNCT
ejpam-6123	420	9	then	then	ADV
ejpam-6123	420	10	degg	degg	NOUN
ejpam-6123	420	11	◦	◦	NOUN
ejpam-6123	420	12	h(v	h(v	PROPN
ejpam-6123	420	13	)	)	PUNCT
ejpam-6123	420	14	≥	≥	NOUN
ejpam-6123	421	1	n	n	NOUN
ejpam-6123	421	2	+	+	CCONJ
ejpam-6123	421	3	1	1	NUM
ejpam-6123	421	4	and	and	CCONJ
ejpam-6123	421	5	degg	degg	NOUN
ejpam-6123	421	6	◦	◦	NOUN
ejpam-6123	421	7	h(q	h(q	ADV
ejpam-6123	421	8	)	)	PUNCT
ejpam-6123	421	9	≤	≤	NOUN
ejpam-6123	421	10	n	n	CCONJ
ejpam-6123	421	11	−	−	PROPN
ejpam-6123	422	1	1	1	NUM
ejpam-6123	422	2	.	.	PUNCT
ejpam-6123	423	1	if	if	SCONJ
ejpam-6123	423	2	degg(v	degg(v	PROPN
ejpam-6123	423	3	)	)	PUNCT
ejpam-6123	423	4	≥	≥	NOUN
ejpam-6123	423	5	2	2	NUM
ejpam-6123	423	6	,	,	PUNCT
ejpam-6123	423	7	then	then	ADV
ejpam-6123	423	8	degg	degg	NOUN
ejpam-6123	423	9	◦	◦	NOUN
ejpam-6123	423	10	h(v	h(v	PROPN
ejpam-6123	423	11	)	)	PUNCT
ejpam-6123	423	12	≥	≥	NOUN
ejpam-6123	423	13	n	n	NOUN
ejpam-6123	423	14	+	+	CCONJ
ejpam-6123	423	15	2	2	NUM
ejpam-6123	423	16	and	and	CCONJ
ejpam-6123	423	17	degg	degg	NOUN
ejpam-6123	423	18	◦	◦	NOUN
ejpam-6123	423	19	h(q	h(q	ADV
ejpam-6123	423	20	)	)	PUNCT
ejpam-6123	423	21	≤	≤	NUM
ejpam-6123	423	22	n.	n.	NOUN
ejpam-6123	423	23	hence	hence	ADV
ejpam-6123	423	24	,	,	PUNCT
ejpam-6123	423	25	w	w	PROPN
ejpam-6123	423	26	̸=	̸=	PROPN
ejpam-6123	423	27	v	v	ADP
ejpam-6123	423	28	whenever	whenever	SCONJ
ejpam-6123	423	29	∆(h	∆(h	NOUN
ejpam-6123	423	30	)	)	PUNCT
ejpam-6123	423	31	≤	≤	NOUN
ejpam-6123	423	32	n	n	CCONJ
ejpam-6123	423	33	−	−	PROPN
ejpam-6123	423	34	2	2	NUM
ejpam-6123	423	35	or	or	CCONJ
ejpam-6123	423	36	degg(v	degg(v	PROPN
ejpam-6123	423	37	)	)	PUNCT
ejpam-6123	423	38	≥	≥	NOUN
ejpam-6123	423	39	2	2	NUM
ejpam-6123	423	40	.	.	PUNCT
ejpam-6123	424	1	it	it	PRON
ejpam-6123	424	2	follows	follow	VERB
ejpam-6123	424	3	that	that	SCONJ
ejpam-6123	424	4	w	w	PROPN
ejpam-6123	424	5	∈	∈	PROPN
ejpam-6123	424	6	bv	bv	PROPN
ejpam-6123	424	7	∩	∩	X
ejpam-6123	424	8	nhv(q	nhv(q	PROPN
ejpam-6123	424	9	)	)	PUNCT
ejpam-6123	424	10	and	and	CCONJ
ejpam-6123	424	11	|deghv(w	|deghv(w	NOUN
ejpam-6123	424	12	)	)	PUNCT
ejpam-6123	424	13	−	−	PUNCT
ejpam-6123	425	1	deghv(q)|	deghv(q)|	VERB
ejpam-6123	425	2	≤	≤	NOUN
ejpam-6123	425	3	1.this	1.this	NUM
ejpam-6123	425	4	shows	show	VERB
ejpam-6123	425	5	that	that	SCONJ
ejpam-6123	425	6	bv	bv	PROPN
ejpam-6123	425	7	is	be	AUX
ejpam-6123	425	8	an	an	DET
ejpam-6123	425	9	equitable	equitable	ADJ
ejpam-6123	425	10	dominating	dominating	NOUN
ejpam-6123	425	11	set	set	NOUN
ejpam-6123	425	12	of	of	ADP
ejpam-6123	425	13	hv	hv	PRON
ejpam-6123	425	14	whenever	whenever	SCONJ
ejpam-6123	425	15	∆(h	∆(h	VERB
ejpam-6123	425	16	)	)	PUNCT
ejpam-6123	425	17	≤	≤	NOUN
ejpam-6123	425	18	n	n	CCONJ
ejpam-6123	425	19	−	−	PROPN
ejpam-6123	425	20	2	2	NUM
ejpam-6123	425	21	or	or	CCONJ
ejpam-6123	425	22	degg(v	degg(v	PROPN
ejpam-6123	425	23	)	)	PUNCT
ejpam-6123	425	24	≥	≥	NOUN
ejpam-6123	425	25	2	2	NUM
ejpam-6123	425	26	.	.	PUNCT
ejpam-6123	425	27	suppose	suppose	VERB
ejpam-6123	425	28	that	that	SCONJ
ejpam-6123	425	29	degg(v	degg(v	PROPN
ejpam-6123	425	30	)	)	PUNCT
ejpam-6123	425	31	=	=	SYM
ejpam-6123	425	32	1	1	NUM
ejpam-6123	425	33	and	and	CCONJ
ejpam-6123	425	34	∆(h	∆(h	VERB
ejpam-6123	425	35	)	)	PUNCT
ejpam-6123	425	36	=	=	SYM
ejpam-6123	425	37	n	n	CCONJ
ejpam-6123	425	38	−	−	PROPN
ejpam-6123	425	39	1	1	X
ejpam-6123	425	40	.	.	PUNCT
ejpam-6123	425	41	suppose	suppose	VERB
ejpam-6123	425	42	further	far	ADV
ejpam-6123	425	43	that	that	DET
ejpam-6123	425	44	deghv(q	deghv(q	NOUN
ejpam-6123	425	45	)	)	PUNCT
ejpam-6123	425	46	̸=	̸=	PROPN
ejpam-6123	425	47	n−	n−	NOUN
ejpam-6123	425	48	1	1	NUM
ejpam-6123	425	49	.	.	PUNCT
ejpam-6123	426	1	then	then	ADV
ejpam-6123	426	2	degg	degg	NOUN
ejpam-6123	426	3	◦	◦	NOUN
ejpam-6123	426	4	h(v	h(v	NOUN
ejpam-6123	426	5	)	)	PUNCT
ejpam-6123	427	1	=	=	SYM
ejpam-6123	427	2	n+	n+	ADP
ejpam-6123	427	3	1	1	NUM
ejpam-6123	427	4	and	and	CCONJ
ejpam-6123	427	5	degg	degg	NOUN
ejpam-6123	427	6	◦	◦	NOUN
ejpam-6123	427	7	h(q	h(q	ADV
ejpam-6123	427	8	)	)	PUNCT
ejpam-6123	427	9	≤	≤	NUM
ejpam-6123	427	10	n−	n−	NOUN
ejpam-6123	427	11	1	1	NUM
ejpam-6123	427	12	.	.	PUNCT
ejpam-6123	428	1	this	this	PRON
ejpam-6123	428	2	implies	imply	VERB
ejpam-6123	428	3	that	that	SCONJ
ejpam-6123	428	4	w	w	PROPN
ejpam-6123	428	5	̸=	̸=	PROPN
ejpam-6123	428	6	v.	v.	CCONJ
ejpam-6123	428	7	hence	hence	ADV
ejpam-6123	428	8	,	,	PUNCT
ejpam-6123	428	9	w	w	PROPN
ejpam-6123	428	10	∈	∈	PROPN
ejpam-6123	428	11	bv	bv	PROPN
ejpam-6123	428	12	and	and	CCONJ
ejpam-6123	428	13	|	|	ADV
ejpam-6123	428	14	deghv(w	deghv(w	PROPN
ejpam-6123	428	15	)	)	PUNCT
ejpam-6123	428	16	−	−	NOUN
ejpam-6123	428	17	deghv(q)|	deghv(q)|	VERB
ejpam-6123	428	18	≤	≤	ADV
ejpam-6123	428	19	1	1	NUM
ejpam-6123	428	20	.	.	PUNCT
ejpam-6123	429	1	consequently	consequently	ADV
ejpam-6123	429	2	,	,	PUNCT
ejpam-6123	429	3	bv	bv	PROPN
ejpam-6123	429	4	dominates	dominate	VERB
ejpam-6123	429	5	equitably	equitably	ADV
ejpam-6123	429	6	v	v	ADP
ejpam-6123	429	7	(	(	PUNCT
ejpam-6123	429	8	hv	hv	PROPN
ejpam-6123	429	9	)	)	PUNCT
ejpam-6123	429	10	\	\	PROPN
ejpam-6123	430	1	sv	sv	PROPN
ejpam-6123	430	2	.	.	PUNCT
ejpam-6123	431	1	the	the	DET
ejpam-6123	431	2	converse	converse	NOUN
ejpam-6123	431	3	is	be	AUX
ejpam-6123	431	4	easy	easy	ADJ
ejpam-6123	431	5	.	.	PUNCT
ejpam-6123	432	1	the	the	DET
ejpam-6123	432	2	next	next	ADJ
ejpam-6123	432	3	result	result	NOUN
ejpam-6123	432	4	is	be	AUX
ejpam-6123	432	5	a	a	DET
ejpam-6123	432	6	consequence	consequence	NOUN
ejpam-6123	432	7	of	of	ADP
ejpam-6123	432	8	the	the	DET
ejpam-6123	432	9	theorem	theorem	NOUN
ejpam-6123	432	10	above	above	ADV
ejpam-6123	432	11	.	.	PUNCT
ejpam-6123	433	1	corollary	corollary	ADJ
ejpam-6123	433	2	3	3	X
ejpam-6123	433	3	.	.	PUNCT
ejpam-6123	434	1	let	let	VERB
ejpam-6123	434	2	g	g	PRON
ejpam-6123	434	3	be	be	AUX
ejpam-6123	434	4	a	a	DET
ejpam-6123	434	5	connected	connected	ADJ
ejpam-6123	434	6	graph	graph	NOUN
ejpam-6123	434	7	of	of	ADP
ejpam-6123	434	8	order	order	NOUN
ejpam-6123	434	9	m	m	VERB
ejpam-6123	434	10	≥	≥	NOUN
ejpam-6123	434	11	2	2	NUM
ejpam-6123	434	12	and	and	CCONJ
ejpam-6123	434	13	h	h	NOUN
ejpam-6123	434	14	be	be	VERB
ejpam-6123	434	15	any	any	DET
ejpam-6123	434	16	graph	graph	NOUN
ejpam-6123	434	17	of	of	ADP
ejpam-6123	434	18	order	order	NOUN
ejpam-6123	434	19	n.	n.	NOUN
ejpam-6123	434	20	if	if	SCONJ
ejpam-6123	434	21	δ(g	δ(g	PROPN
ejpam-6123	434	22	)	)	PUNCT
ejpam-6123	434	23	≥	≥	NOUN
ejpam-6123	434	24	2	2	NUM
ejpam-6123	434	25	or	or	CCONJ
ejpam-6123	434	26	∆(h	∆(h	NOUN
ejpam-6123	434	27	)	)	PUNCT
ejpam-6123	434	28	≤	≤	NUM
ejpam-6123	434	29	n−	n−	NOUN
ejpam-6123	434	30	2	2	NUM
ejpam-6123	434	31	,	,	PUNCT
ejpam-6123	434	32	then	then	ADV
ejpam-6123	434	33	γce(g	γce(g	PROPN
ejpam-6123	435	1	◦	◦	NOUN
ejpam-6123	435	2	h	h	NOUN
ejpam-6123	435	3	)	)	PUNCT
ejpam-6123	436	1	=	=	PUNCT
ejpam-6123	437	1	[	[	X
ejpam-6123	437	2	1	1	NUM
ejpam-6123	437	3	+	+	NUM
ejpam-6123	437	4	γe(h)]m	γe(h)]m	NOUN
ejpam-6123	437	5	.	.	PUNCT
ejpam-6123	438	1	proof	proof	NOUN
ejpam-6123	438	2	.	.	PUNCT
ejpam-6123	439	1	let	let	VERB
ejpam-6123	439	2	c	c	PRON
ejpam-6123	439	3	be	be	AUX
ejpam-6123	439	4	a	a	DET
ejpam-6123	439	5	γce	γce	NOUN
ejpam-6123	439	6	-	-	PUNCT
ejpam-6123	439	7	set	set	NOUN
ejpam-6123	439	8	of	of	ADP
ejpam-6123	439	9	g	g	PROPN
ejpam-6123	439	10	◦	◦	NOUN
ejpam-6123	439	11	h.	h.	NOUN
ejpam-6123	439	12	by	by	ADP
ejpam-6123	439	13	theorem	theorem	PROPN
ejpam-6123	439	14	14	14	NUM
ejpam-6123	439	15	,	,	PUNCT
ejpam-6123	439	16	c	c	NOUN
ejpam-6123	439	17	=	=	SYM
ejpam-6123	439	18	v	v	PROPN
ejpam-6123	439	19	(	(	PUNCT
ejpam-6123	439	20	g)∪	g)∪	NOUN
ejpam-6123	439	21	(	(	PUNCT
ejpam-6123	439	22	⋃	⋃	NOUN
ejpam-6123	439	23	v∈v	v∈v	NOUN
ejpam-6123	439	24	(	(	PUNCT
ejpam-6123	439	25	g	g	NOUN
ejpam-6123	439	26	)	)	PUNCT
ejpam-6123	439	27	bv	bv	PROPN
ejpam-6123	439	28	)	)	PUNCT
ejpam-6123	439	29	,	,	PUNCT
ejpam-6123	439	30	where	where	SCONJ
ejpam-6123	439	31	bv	bv	PROPN
ejpam-6123	439	32	is	be	AUX
ejpam-6123	439	33	a	a	DET
ejpam-6123	439	34	connected	connect	VERB
ejpam-6123	439	35	equitable	equitable	ADJ
ejpam-6123	439	36	dominating	dominating	NOUN
ejpam-6123	439	37	set	set	NOUN
ejpam-6123	439	38	of	of	ADP
ejpam-6123	439	39	hv	hv	PROPN
ejpam-6123	439	40	for	for	ADP
ejpam-6123	439	41	each	each	DET
ejpam-6123	439	42	v	v	NUM
ejpam-6123	439	43	∈	∈	PROPN
ejpam-6123	439	44	v	v	NOUN
ejpam-6123	439	45	(	(	PUNCT
ejpam-6123	439	46	g	g	NOUN
ejpam-6123	439	47	)	)	PUNCT
ejpam-6123	439	48	.	.	PUNCT
ejpam-6123	440	1	since	since	SCONJ
ejpam-6123	440	2	c	c	PROPN
ejpam-6123	440	3	is	be	AUX
ejpam-6123	440	4	a	a	DET
ejpam-6123	440	5	γce	γce	NOUN
ejpam-6123	440	6	-	-	PUNCT
ejpam-6123	440	7	set	set	VERB
ejpam-6123	440	8	,	,	PUNCT
ejpam-6123	440	9	bv	bv	PROPN
ejpam-6123	440	10	is	be	AUX
ejpam-6123	440	11	a	a	DET
ejpam-6123	440	12	γe	γe	NOUN
ejpam-6123	440	13	-	-	PUNCT
ejpam-6123	440	14	set	set	NOUN
ejpam-6123	440	15	of	of	ADP
ejpam-6123	440	16	hv	hv	PROPN
ejpam-6123	440	17	for	for	ADP
ejpam-6123	440	18	each	each	DET
ejpam-6123	440	19	v	v	NUM
ejpam-6123	440	20	∈	∈	PROPN
ejpam-6123	440	21	v	v	NOUN
ejpam-6123	440	22	(	(	PUNCT
ejpam-6123	440	23	g	g	NOUN
ejpam-6123	440	24	)	)	PUNCT
ejpam-6123	440	25	.	.	PUNCT
ejpam-6123	441	1	thus	thus	ADV
ejpam-6123	441	2	,	,	PUNCT
ejpam-6123	441	3	γce(g	γce(g	PROPN
ejpam-6123	441	4	◦	◦	NOUN
ejpam-6123	441	5	h	h	NOUN
ejpam-6123	441	6	)	)	PUNCT
ejpam-6123	441	7	=	=	SYM
ejpam-6123	441	8	|c|	|c|	PROPN
ejpam-6123	441	9	=	=	SYM
ejpam-6123	441	10	m+mγe(h	m+mγe(h	PROPN
ejpam-6123	441	11	)	)	PUNCT
ejpam-6123	441	12	=	=	SYM
ejpam-6123	441	13	m[1+γe(h	m[1+γe(h	PROPN
ejpam-6123	441	14	)	)	PUNCT
ejpam-6123	441	15	]	]	PUNCT
ejpam-6123	441	16	.	.	PUNCT
ejpam-6123	442	1	corollary	corollary	ADJ
ejpam-6123	442	2	4	4	NUM
ejpam-6123	442	3	.	.	PUNCT
ejpam-6123	443	1	for	for	ADP
ejpam-6123	443	2	any	any	DET
ejpam-6123	443	3	graph	graph	NOUN
ejpam-6123	443	4	g	g	NOUN
ejpam-6123	443	5	of	of	ADP
ejpam-6123	443	6	order	order	NOUN
ejpam-6123	443	7	n	n	CCONJ
ejpam-6123	443	8	,	,	PUNCT
ejpam-6123	443	9	γce(g	γce(g	PROPN
ejpam-6123	443	10	◦	◦	NOUN
ejpam-6123	443	11	kp	kp	NOUN
ejpam-6123	443	12	)	)	PUNCT
ejpam-6123	443	13	=	=	SYM
ejpam-6123	443	14	n+	n+	NOUN
ejpam-6123	443	15	|s|	|s|	NOUN
ejpam-6123	443	16	where	where	SCONJ
ejpam-6123	443	17	s	s	VERB
ejpam-6123	443	18	=	=	PUNCT
ejpam-6123	443	19	{	{	PUNCT
ejpam-6123	443	20	u	u	NOUN
ejpam-6123	443	21	∈	∈	PROPN
ejpam-6123	443	22	v	v	NOUN
ejpam-6123	443	23	(	(	PUNCT
ejpam-6123	443	24	g	g	NOUN
ejpam-6123	443	25	)	)	PUNCT
ejpam-6123	443	26	:	:	PUNCT
ejpam-6123	443	27	degg(u	degg(u	X
ejpam-6123	443	28	)	)	PUNCT
ejpam-6123	443	29	≥	≥	NOUN
ejpam-6123	443	30	2	2	NUM
ejpam-6123	443	31	}	}	PUNCT
ejpam-6123	443	32	.	.	PUNCT
ejpam-6123	444	1	proof	proof	NOUN
ejpam-6123	444	2	.	.	PUNCT
ejpam-6123	445	1	follows	follow	VERB
ejpam-6123	445	2	from	from	ADP
ejpam-6123	445	3	theorem	theorem	ADJ
ejpam-6123	445	4	14	14	NUM
ejpam-6123	445	5	and	and	CCONJ
ejpam-6123	445	6	corollary	corollary	ADJ
ejpam-6123	445	7	3	3	NUM
ejpam-6123	445	8	.	.	PUNCT
ejpam-6123	445	9	corollary	corollary	ADJ
ejpam-6123	445	10	5	5	NUM
ejpam-6123	445	11	.	.	PUNCT
ejpam-6123	446	1	for	for	ADP
ejpam-6123	446	2	any	any	DET
ejpam-6123	446	3	graph	graph	NOUN
ejpam-6123	446	4	g	g	NOUN
ejpam-6123	446	5	of	of	ADP
ejpam-6123	446	6	order	order	NOUN
ejpam-6123	446	7	n	n	CCONJ
ejpam-6123	446	8	,	,	PUNCT
ejpam-6123	446	9	and	and	CCONJ
ejpam-6123	446	10	a	a	DET
ejpam-6123	446	11	r	r	NOUN
ejpam-6123	446	12	-	-	PUNCT
ejpam-6123	446	13	regular	regular	ADJ
ejpam-6123	446	14	graph	graph	NOUN
ejpam-6123	446	15	h	h	NOUN
ejpam-6123	446	16	with	with	ADP
ejpam-6123	446	17	r	r	NOUN
ejpam-6123	446	18	̸=	̸=	PROPN
ejpam-6123	446	19	|v	|v	NOUN
ejpam-6123	446	20	(	(	PUNCT
ejpam-6123	446	21	h)|	h)|	NOUN
ejpam-6123	446	22	−	−	PROPN
ejpam-6123	446	23	1	1	NUM
ejpam-6123	446	24	,	,	PUNCT
ejpam-6123	446	25	γce(g	γce(g	PROPN
ejpam-6123	446	26	◦	◦	NOUN
ejpam-6123	446	27	h	h	NOUN
ejpam-6123	446	28	)	)	PUNCT
ejpam-6123	446	29	=	=	SYM
ejpam-6123	446	30	n+	n+	X
ejpam-6123	446	31	nγ(h	nγ(h	NOUN
ejpam-6123	446	32	)	)	PUNCT
ejpam-6123	446	33	.	.	PUNCT
ejpam-6123	447	1	proof	proof	NOUN
ejpam-6123	447	2	.	.	PUNCT
ejpam-6123	448	1	let	let	VERB
ejpam-6123	448	2	v	v	NUM
ejpam-6123	448	3	∈	∈	PROPN
ejpam-6123	448	4	v	v	NOUN
ejpam-6123	448	5	(	(	PUNCT
ejpam-6123	448	6	g	g	NOUN
ejpam-6123	448	7	)	)	PUNCT
ejpam-6123	448	8	and	and	CCONJ
ejpam-6123	448	9	cv	cv	PROPN
ejpam-6123	448	10	⊆	⊆	NUM
ejpam-6123	448	11	v	v	PROPN
ejpam-6123	448	12	(	(	PUNCT
ejpam-6123	448	13	hv	hv	NOUN
ejpam-6123	448	14	)	)	PUNCT
ejpam-6123	448	15	be	be	VERB
ejpam-6123	448	16	a	a	DET
ejpam-6123	448	17	γ	γ	NOUN
ejpam-6123	448	18	-	-	PUNCT
ejpam-6123	448	19	set	set	NOUN
ejpam-6123	448	20	in	in	ADP
ejpam-6123	448	21	hv	hv	PROPN
ejpam-6123	448	22	.	.	PUNCT
ejpam-6123	449	1	then	then	ADV
ejpam-6123	449	2	cv	cv	PROPN
ejpam-6123	449	3	is	be	AUX
ejpam-6123	449	4	an	an	DET
ejpam-6123	449	5	equitable	equitable	ADJ
ejpam-6123	449	6	dominating	dominating	NOUN
ejpam-6123	449	7	set	set	VERB
ejpam-6123	449	8	in	in	ADP
ejpam-6123	449	9	hv	hv	PROPN
ejpam-6123	449	10	since	since	SCONJ
ejpam-6123	449	11	h	h	PROPN
ejpam-6123	449	12	is	be	AUX
ejpam-6123	449	13	r	r	NOUN
ejpam-6123	449	14	-	-	ADJ
ejpam-6123	449	15	regular	regular	ADJ
ejpam-6123	449	16	.	.	PUNCT
ejpam-6123	450	1	furthermore	furthermore	ADV
ejpam-6123	450	2	,	,	PUNCT
ejpam-6123	450	3	since	since	SCONJ
ejpam-6123	450	4	r	r	PROPN
ejpam-6123	450	5	̸=	̸=	PROPN
ejpam-6123	450	6	|v	|v	NOUN
ejpam-6123	450	7	(	(	PUNCT
ejpam-6123	450	8	h)|	h)|	NOUN
ejpam-6123	450	9	−	−	PROPN
ejpam-6123	450	10	1	1	NUM
ejpam-6123	450	11	,	,	PUNCT
ejpam-6123	450	12	the	the	DET
ejpam-6123	450	13	set⋃	set⋃	NOUN
ejpam-6123	450	14	u∈v	u∈v	NOUN
ejpam-6123	450	15	(	(	PUNCT
ejpam-6123	450	16	g	g	NOUN
ejpam-6123	450	17	)	)	PUNCT
ejpam-6123	450	18	{	{	PUNCT
ejpam-6123	450	19	v	v	NOUN
ejpam-6123	450	20	}	}	PUNCT
ejpam-6123	450	21	∪cv	∪cv	PROPN
ejpam-6123	450	22	is	be	AUX
ejpam-6123	450	23	a	a	DET
ejpam-6123	450	24	connected	connect	VERB
ejpam-6123	450	25	equitable	equitable	ADJ
ejpam-6123	450	26	dominating	dominating	NOUN
ejpam-6123	450	27	set	set	VERB
ejpam-6123	450	28	in	in	ADP
ejpam-6123	450	29	g	g	PROPN
ejpam-6123	450	30	◦	◦	NOUN
ejpam-6123	450	31	h	h	NOUN
ejpam-6123	450	32	of	of	ADP
ejpam-6123	450	33	minimum	minimum	ADJ
ejpam-6123	450	34	cardinality	cardinality	NOUN
ejpam-6123	450	35	.	.	PUNCT
ejpam-6123	451	1	thus	thus	ADV
ejpam-6123	451	2	,	,	PUNCT
ejpam-6123	451	3	γce(g	γce(g	PROPN
ejpam-6123	451	4	◦	◦	NOUN
ejpam-6123	451	5	h	h	NOUN
ejpam-6123	451	6	)	)	PUNCT
ejpam-6123	451	7	=	=	SYM
ejpam-6123	451	8	n+	n+	PROPN
ejpam-6123	451	9	n|cv|	n|cv|	PROPN
ejpam-6123	451	10	=	=	SYM
ejpam-6123	451	11	n+	n+	X
ejpam-6123	451	12	nγ(h	nγ(h	NOUN
ejpam-6123	451	13	)	)	PUNCT
ejpam-6123	451	14	.	.	PUNCT
ejpam-6123	452	1	h.	h.	PROPN
ejpam-6123	452	2	nuenay	nuenay	PROPN
ejpam-6123	452	3	-	-	PUNCT
ejpam-6123	452	4	maglanquel	maglanquel	PROPN
ejpam-6123	452	5	/	/	SYM
ejpam-6123	452	6	eur	eur	PROPN
ejpam-6123	452	7	.	.	PUNCT
ejpam-6123	453	1	j.	j.	PROPN
ejpam-6123	453	2	pure	pure	PROPN
ejpam-6123	453	3	appl	appl	PROPN
ejpam-6123	453	4	.	.	PROPN
ejpam-6123	453	5	math	math	PROPN
ejpam-6123	453	6	,	,	PUNCT
ejpam-6123	453	7	18	18	NUM
ejpam-6123	453	8	(	(	PUNCT
ejpam-6123	453	9	3	3	NUM
ejpam-6123	453	10	)	)	PUNCT
ejpam-6123	453	11	(	(	PUNCT
ejpam-6123	453	12	2025	2025	NUM
ejpam-6123	453	13	)	)	PUNCT
ejpam-6123	453	14	,	,	PUNCT
ejpam-6123	453	15	6123	6123	NUM
ejpam-6123	453	16	13	13	NUM
ejpam-6123	453	17	of	of	ADP
ejpam-6123	453	18	14	14	NUM
ejpam-6123	453	19	5	5	NUM
ejpam-6123	453	20	.	.	PUNCT
ejpam-6123	454	1	conclusion	conclusion	NOUN
ejpam-6123	454	2	in	in	ADP
ejpam-6123	454	3	this	this	DET
ejpam-6123	454	4	paper	paper	NOUN
ejpam-6123	454	5	,	,	PUNCT
ejpam-6123	454	6	we	we	PRON
ejpam-6123	454	7	have	have	AUX
ejpam-6123	454	8	investigated	investigate	VERB
ejpam-6123	454	9	the	the	DET
ejpam-6123	454	10	concept	concept	NOUN
ejpam-6123	454	11	of	of	ADP
ejpam-6123	454	12	connected	connect	VERB
ejpam-6123	454	13	equitable	equitable	ADJ
ejpam-6123	454	14	domination	domination	NOUN
ejpam-6123	454	15	in	in	ADP
ejpam-6123	454	16	graphs	graph	NOUN
ejpam-6123	454	17	focusing	focus	VERB
ejpam-6123	454	18	on	on	ADP
ejpam-6123	454	19	some	some	DET
ejpam-6123	454	20	families	family	NOUN
ejpam-6123	454	21	of	of	ADP
ejpam-6123	454	22	graphs	graph	NOUN
ejpam-6123	454	23	,	,	PUNCT
ejpam-6123	454	24	and	and	CCONJ
ejpam-6123	454	25	on	on	ADP
ejpam-6123	454	26	graphs	graph	NOUN
ejpam-6123	454	27	arising	arise	VERB
ejpam-6123	454	28	from	from	ADP
ejpam-6123	454	29	the	the	DET
ejpam-6123	454	30	join	join	NOUN
ejpam-6123	454	31	and	and	CCONJ
ejpam-6123	454	32	corona	corona	NOUN
ejpam-6123	454	33	of	of	ADP
ejpam-6123	454	34	graphs	graph	NOUN
ejpam-6123	454	35	.	.	PUNCT
ejpam-6123	455	1	specifically	specifically	ADV
ejpam-6123	455	2	,	,	PUNCT
ejpam-6123	455	3	the	the	DET
ejpam-6123	455	4	connected	connect	VERB
ejpam-6123	455	5	equitable	equitable	ADJ
ejpam-6123	455	6	dominating	dominating	NOUN
ejpam-6123	455	7	sets	set	NOUN
ejpam-6123	455	8	in	in	ADP
ejpam-6123	455	9	the	the	DET
ejpam-6123	455	10	join	join	NOUN
ejpam-6123	455	11	and	and	CCONJ
ejpam-6123	455	12	corona	corona	NOUN
ejpam-6123	455	13	of	of	ADP
ejpam-6123	455	14	graphs	graph	NOUN
ejpam-6123	455	15	are	be	AUX
ejpam-6123	455	16	characterized	characterize	VERB
ejpam-6123	455	17	.	.	PUNCT
ejpam-6123	456	1	we	we	PRON
ejpam-6123	456	2	also	also	ADV
ejpam-6123	456	3	obtained	obtain	VERB
ejpam-6123	456	4	the	the	DET
ejpam-6123	456	5	exact	exact	ADJ
ejpam-6123	456	6	value	value	NOUN
ejpam-6123	456	7	of	of	ADP
ejpam-6123	456	8	γce(g	γce(g	PROPN
ejpam-6123	456	9	)	)	PUNCT
ejpam-6123	456	10	for	for	ADP
ejpam-6123	456	11	some	some	DET
ejpam-6123	456	12	families	family	NOUN
ejpam-6123	456	13	of	of	ADP
ejpam-6123	456	14	graphs	graph	NOUN
ejpam-6123	456	15	;	;	PUNCT
ejpam-6123	456	16	and	and	CCONJ
ejpam-6123	456	17	graphs	graph	NOUN
ejpam-6123	456	18	formed	form	VERB
ejpam-6123	456	19	in	in	ADP
ejpam-6123	456	20	the	the	DET
ejpam-6123	456	21	join	join	NOUN
ejpam-6123	456	22	and	and	CCONJ
ejpam-6123	456	23	corona	corona	NOUN
ejpam-6123	456	24	of	of	ADP
ejpam-6123	456	25	graphs	graph	NOUN
ejpam-6123	456	26	.	.	PUNCT
ejpam-6123	457	1	additionally	additionally	ADV
ejpam-6123	457	2	,	,	PUNCT
ejpam-6123	457	3	a	a	DET
ejpam-6123	457	4	realization	realization	NOUN
ejpam-6123	457	5	problem	problem	NOUN
ejpam-6123	457	6	is	be	AUX
ejpam-6123	457	7	established	establish	VERB
ejpam-6123	457	8	.	.	PUNCT
ejpam-6123	458	1	the	the	DET
ejpam-6123	458	2	significance	significance	NOUN
ejpam-6123	458	3	of	of	ADP
ejpam-6123	458	4	this	this	DET
ejpam-6123	458	5	study	study	NOUN
ejpam-6123	458	6	lie	lie	VERB
ejpam-6123	458	7	in	in	ADP
ejpam-6123	458	8	its	its	PRON
ejpam-6123	458	9	extension	extension	NOUN
ejpam-6123	458	10	of	of	ADP
ejpam-6123	458	11	domination	domination	NOUN
ejpam-6123	458	12	theory	theory	NOUN
ejpam-6123	458	13	to	to	ADP
ejpam-6123	458	14	connected	connect	VERB
ejpam-6123	458	15	equitable	equitable	ADJ
ejpam-6123	458	16	domination	domination	NOUN
ejpam-6123	458	17	through	through	ADP
ejpam-6123	458	18	combining	combine	VERB
ejpam-6123	458	19	both	both	DET
ejpam-6123	458	20	connectivity	connectivity	NOUN
ejpam-6123	458	21	and	and	CCONJ
ejpam-6123	458	22	equity	equity	NOUN
ejpam-6123	458	23	constraints	constraint	NOUN
ejpam-6123	458	24	,	,	PUNCT
ejpam-6123	458	25	reflecting	reflect	VERB
ejpam-6123	458	26	more	more	ADV
ejpam-6123	458	27	realistic	realistic	ADJ
ejpam-6123	458	28	models	model	NOUN
ejpam-6123	458	29	of	of	ADP
ejpam-6123	458	30	control	control	NOUN
ejpam-6123	458	31	,	,	PUNCT
ejpam-6123	458	32	influence	influence	NOUN
ejpam-6123	458	33	,	,	PUNCT
ejpam-6123	458	34	and	and	CCONJ
ejpam-6123	458	35	resource	resource	NOUN
ejpam-6123	458	36	distribution	distribution	NOUN
ejpam-6123	458	37	in	in	ADP
ejpam-6123	458	38	networked	networked	ADJ
ejpam-6123	458	39	systems	system	NOUN
ejpam-6123	458	40	;	;	PUNCT
ejpam-6123	458	41	and	and	CCONJ
ejpam-6123	458	42	focus	focus	VERB
ejpam-6123	458	43	on	on	ADP
ejpam-6123	458	44	more	more	ADJ
ejpam-6123	458	45	complex	complex	ADJ
ejpam-6123	458	46	structure	structure	NOUN
ejpam-6123	458	47	of	of	ADP
ejpam-6123	458	48	graphs	graph	NOUN
ejpam-6123	458	49	:	:	PUNCT
ejpam-6123	458	50	graphs	graph	NOUN
ejpam-6123	458	51	obtained	obtain	VERB
ejpam-6123	458	52	in	in	ADP
ejpam-6123	458	53	the	the	DET
ejpam-6123	458	54	join	join	NOUN
ejpam-6123	458	55	and	and	CCONJ
ejpam-6123	458	56	corona	corona	NOUN
ejpam-6123	458	57	of	of	ADP
ejpam-6123	458	58	graphs	graph	NOUN
ejpam-6123	458	59	.	.	PUNCT
ejpam-6123	459	1	unlike	unlike	ADP
ejpam-6123	459	2	classical	classical	ADJ
ejpam-6123	459	3	domination	domination	NOUN
ejpam-6123	459	4	parameters	parameter	NOUN
ejpam-6123	459	5	that	that	PRON
ejpam-6123	459	6	focus	focus	VERB
ejpam-6123	459	7	solely	solely	ADV
ejpam-6123	459	8	on	on	ADP
ejpam-6123	459	9	coverage	coverage	NOUN
ejpam-6123	459	10	or	or	CCONJ
ejpam-6123	459	11	connectivity	connectivity	NOUN
ejpam-6123	459	12	,	,	PUNCT
ejpam-6123	459	13	the	the	DET
ejpam-6123	459	14	connected	connect	VERB
ejpam-6123	459	15	equitable	equitable	ADJ
ejpam-6123	459	16	domination	domination	NOUN
ejpam-6123	459	17	framework	framework	NOUN
ejpam-6123	459	18	ensures	ensure	VERB
ejpam-6123	459	19	balanced	balanced	ADJ
ejpam-6123	459	20	distribution	distribution	NOUN
ejpam-6123	459	21	of	of	ADP
ejpam-6123	459	22	domination	domination	NOUN
ejpam-6123	459	23	responsibilities	responsibility	NOUN
ejpam-6123	459	24	among	among	ADP
ejpam-6123	459	25	vertices	vertex	NOUN
ejpam-6123	459	26	while	while	SCONJ
ejpam-6123	459	27	maintaining	maintain	VERB
ejpam-6123	459	28	network	network	NOUN
ejpam-6123	459	29	cohesion.this	cohesion.this	PRON
ejpam-6123	459	30	study	study	NOUN
ejpam-6123	459	31	provides	provide	VERB
ejpam-6123	459	32	a	a	DET
ejpam-6123	459	33	foundation	foundation	NOUN
ejpam-6123	459	34	for	for	ADP
ejpam-6123	459	35	efficient	efficient	ADJ
ejpam-6123	459	36	network	network	NOUN
ejpam-6123	459	37	design	design	NOUN
ejpam-6123	459	38	in	in	ADP
ejpam-6123	459	39	practical	practical	ADJ
ejpam-6123	459	40	domains	domain	NOUN
ejpam-6123	459	41	such	such	ADJ
ejpam-6123	459	42	as	as	ADP
ejpam-6123	459	43	sensor	sensor	NOUN
ejpam-6123	459	44	placement	placement	NOUN
ejpam-6123	459	45	,	,	PUNCT
ejpam-6123	459	46	facility	facility	NOUN
ejpam-6123	459	47	location	location	NOUN
ejpam-6123	459	48	,	,	PUNCT
ejpam-6123	459	49	communication	communication	NOUN
ejpam-6123	459	50	networks	network	NOUN
ejpam-6123	459	51	,	,	PUNCT
ejpam-6123	459	52	and	and	CCONJ
ejpam-6123	459	53	distributed	distributed	ADJ
ejpam-6123	459	54	computing	computing	NOUN
ejpam-6123	459	55	,	,	PUNCT
ejpam-6123	459	56	where	where	SCONJ
ejpam-6123	459	57	both	both	DET
ejpam-6123	459	58	connectivity	connectivity	NOUN
ejpam-6123	459	59	and	and	CCONJ
ejpam-6123	459	60	fairness	fairness	NOUN
ejpam-6123	459	61	are	be	AUX
ejpam-6123	459	62	critical	critical	ADJ
ejpam-6123	459	63	.	.	PUNCT
ejpam-6123	460	1	the	the	DET
ejpam-6123	460	2	exact	exact	ADJ
ejpam-6123	460	3	values	value	NOUN
ejpam-6123	460	4	and	and	CCONJ
ejpam-6123	460	5	characterizations	characterization	NOUN
ejpam-6123	460	6	obtained	obtain	VERB
ejpam-6123	460	7	here	here	ADV
ejpam-6123	460	8	can	can	AUX
ejpam-6123	460	9	also	also	ADV
ejpam-6123	460	10	serve	serve	VERB
ejpam-6123	460	11	as	as	ADP
ejpam-6123	460	12	benchmarks	benchmark	NOUN
ejpam-6123	460	13	for	for	ADP
ejpam-6123	460	14	algorithm	algorithm	NOUN
ejpam-6123	460	15	development	development	NOUN
ejpam-6123	460	16	,	,	PUNCT
ejpam-6123	460	17	particularly	particularly	ADV
ejpam-6123	460	18	in	in	ADP
ejpam-6123	460	19	designing	design	VERB
ejpam-6123	460	20	heuristics	heuristic	NOUN
ejpam-6123	460	21	and	and	CCONJ
ejpam-6123	460	22	approximation	approximation	NOUN
ejpam-6123	460	23	algorithms	algorithm	NOUN
ejpam-6123	460	24	for	for	ADP
ejpam-6123	460	25	complex	complex	ADJ
ejpam-6123	460	26	networks	network	NOUN
ejpam-6123	460	27	.	.	PUNCT
ejpam-6123	461	1	nonetheless	nonetheless	ADV
ejpam-6123	461	2	,	,	PUNCT
ejpam-6123	461	3	the	the	DET
ejpam-6123	461	4	work	work	NOUN
ejpam-6123	461	5	is	be	AUX
ejpam-6123	461	6	confined	confine	VERB
ejpam-6123	461	7	to	to	ADP
ejpam-6123	461	8	unweighted	unweighte	VERB
ejpam-6123	461	9	,	,	PUNCT
ejpam-6123	461	10	undirected	undirected	ADJ
ejpam-6123	461	11	,	,	PUNCT
ejpam-6123	461	12	and	and	CCONJ
ejpam-6123	461	13	simple	simple	ADJ
ejpam-6123	461	14	graphs	graph	NOUN
ejpam-6123	461	15	,	,	PUNCT
ejpam-6123	461	16	which	which	PRON
ejpam-6123	461	17	may	may	AUX
ejpam-6123	461	18	limit	limit	VERB
ejpam-6123	461	19	its	its	PRON
ejpam-6123	461	20	direct	direct	ADJ
ejpam-6123	461	21	applicability	applicability	NOUN
ejpam-6123	461	22	to	to	ADP
ejpam-6123	461	23	more	more	ADV
ejpam-6123	461	24	complex	complex	ADJ
ejpam-6123	461	25	or	or	CCONJ
ejpam-6123	461	26	dynamic	dynamic	ADJ
ejpam-6123	461	27	networks	network	NOUN
ejpam-6123	461	28	.	.	PUNCT
ejpam-6123	462	1	the	the	DET
ejpam-6123	462	2	analysis	analysis	NOUN
ejpam-6123	462	3	focuses	focus	VERB
ejpam-6123	462	4	specifically	specifically	ADV
ejpam-6123	462	5	on	on	ADP
ejpam-6123	462	6	the	the	DET
ejpam-6123	462	7	join	join	NOUN
ejpam-6123	462	8	and	and	CCONJ
ejpam-6123	462	9	corona	corona	NOUN
ejpam-6123	462	10	operations	operation	NOUN
ejpam-6123	462	11	,	,	PUNCT
ejpam-6123	462	12	leaving	leave	VERB
ejpam-6123	462	13	room	room	NOUN
ejpam-6123	462	14	to	to	PART
ejpam-6123	462	15	explore	explore	VERB
ejpam-6123	462	16	the	the	DET
ejpam-6123	462	17	concept	concept	NOUN
ejpam-6123	462	18	on	on	ADP
ejpam-6123	462	19	some	some	DET
ejpam-6123	462	20	other	other	ADJ
ejpam-6123	462	21	binary	binary	ADJ
ejpam-6123	462	22	operations	operation	NOUN
ejpam-6123	462	23	such	such	ADJ
ejpam-6123	462	24	as	as	ADP
ejpam-6123	462	25	lexicographic	lexicographic	ADJ
ejpam-6123	462	26	product	product	NOUN
ejpam-6123	462	27	or	or	CCONJ
ejpam-6123	462	28	cartesian	cartesian	ADJ
ejpam-6123	462	29	product	product	NOUN
ejpam-6123	462	30	of	of	ADP
ejpam-6123	462	31	graphs	graph	NOUN
ejpam-6123	462	32	,	,	PUNCT
ejpam-6123	462	33	on	on	ADP
ejpam-6123	462	34	weighted	weighted	ADJ
ejpam-6123	462	35	graphs	graph	NOUN
ejpam-6123	462	36	,	,	PUNCT
ejpam-6123	462	37	directed	direct	VERB
ejpam-6123	462	38	graphs	graph	NOUN
ejpam-6123	462	39	,	,	PUNCT
ejpam-6123	462	40	or	or	CCONJ
ejpam-6123	462	41	dynamic	dynamic	ADJ
ejpam-6123	462	42	environments	environment	NOUN
ejpam-6123	462	43	,	,	PUNCT
ejpam-6123	462	44	potentially	potentially	ADV
ejpam-6123	462	45	yielding	yield	VERB
ejpam-6123	462	46	deeper	deep	ADJ
ejpam-6123	462	47	insights	insight	NOUN
ejpam-6123	462	48	into	into	ADP
ejpam-6123	462	49	equitable	equitable	ADJ
ejpam-6123	462	50	network	network	NOUN
ejpam-6123	462	51	optimization	optimization	NOUN
ejpam-6123	462	52	under	under	ADP
ejpam-6123	462	53	evolving	evolve	VERB
ejpam-6123	462	54	constraints	constraint	NOUN
ejpam-6123	462	55	.	.	PUNCT
ejpam-6123	463	1	furthermore	furthermore	ADV
ejpam-6123	463	2	,	,	PUNCT
ejpam-6123	463	3	while	while	SCONJ
ejpam-6123	463	4	the	the	DET
ejpam-6123	463	5	results	result	NOUN
ejpam-6123	463	6	are	be	AUX
ejpam-6123	463	7	mathematically	mathematically	ADV
ejpam-6123	463	8	rigorous	rigorous	ADJ
ejpam-6123	463	9	,	,	PUNCT
ejpam-6123	463	10	the	the	DET
ejpam-6123	463	11	study	study	NOUN
ejpam-6123	463	12	does	do	AUX
ejpam-6123	463	13	not	not	PART
ejpam-6123	463	14	address	address	VERB
ejpam-6123	463	15	algorithmic	algorithmic	ADJ
ejpam-6123	463	16	complexity	complexity	NOUN
ejpam-6123	463	17	or	or	CCONJ
ejpam-6123	463	18	include	include	VERB
ejpam-6123	463	19	empirical	empirical	ADJ
ejpam-6123	463	20	validation	validation	NOUN
ejpam-6123	463	21	,	,	PUNCT
ejpam-6123	463	22	which	which	PRON
ejpam-6123	463	23	could	could	AUX
ejpam-6123	463	24	be	be	AUX
ejpam-6123	463	25	valuable	valuable	ADJ
ejpam-6123	463	26	extensions	extension	NOUN
ejpam-6123	463	27	in	in	ADP
ejpam-6123	463	28	future	future	ADJ
ejpam-6123	463	29	research	research	NOUN
ejpam-6123	463	30	.	.	PUNCT
ejpam-6123	464	1	references	reference	NOUN
ejpam-6123	464	2	[	[	X
ejpam-6123	464	3	1	1	NUM
ejpam-6123	464	4	]	]	PUNCT
ejpam-6123	464	5	r.	r.	PROPN
ejpam-6123	464	6	b.	b.	PROPN
ejpam-6123	464	7	allan	allan	PROPN
ejpam-6123	464	8	and	and	CCONJ
ejpam-6123	464	9	r.	r.	PROPN
ejpam-6123	464	10	laskar	laskar	PROPN
ejpam-6123	464	11	.	.	PUNCT
ejpam-6123	465	1	on	on	ADP
ejpam-6123	465	2	domination	domination	NOUN
ejpam-6123	465	3	and	and	CCONJ
ejpam-6123	465	4	independent	independent	ADJ
ejpam-6123	465	5	domination	domination	NOUN
ejpam-6123	465	6	numbers	number	NOUN
ejpam-6123	465	7	of	of	ADP
ejpam-6123	465	8	a	a	DET
ejpam-6123	465	9	graph	graph	NOUN
ejpam-6123	465	10	.	.	PUNCT
ejpam-6123	466	1	discrete	discrete	ADJ
ejpam-6123	466	2	mathematics	mathematic	NOUN
ejpam-6123	466	3	,	,	PUNCT
ejpam-6123	466	4	23(2):73–76	23(2):73–76	NUM
ejpam-6123	466	5	,	,	PUNCT
ejpam-6123	466	6	1978	1978	NUM
ejpam-6123	466	7	.	.	PUNCT
ejpam-6123	467	1	[	[	X
ejpam-6123	467	2	2	2	X
ejpam-6123	467	3	]	]	PUNCT
ejpam-6123	467	4	t.	t.	PROPN
ejpam-6123	467	5	haynes	haynes	PROPN
ejpam-6123	467	6	,	,	PUNCT
ejpam-6123	467	7	s.	s.	PROPN
ejpam-6123	467	8	hedetniemi	hedetniemi	PROPN
ejpam-6123	467	9	,	,	PUNCT
ejpam-6123	467	10	and	and	CCONJ
ejpam-6123	467	11	p.	p.	PROPN
ejpam-6123	467	12	slater	slater	PROPN
ejpam-6123	467	13	.	.	PUNCT
ejpam-6123	468	1	fundamentals	fundamental	NOUN
ejpam-6123	468	2	of	of	ADP
ejpam-6123	468	3	domination	domination	NOUN
ejpam-6123	468	4	in	in	ADP
ejpam-6123	468	5	graphs	graph	NOUN
ejpam-6123	468	6	.	.	PUNCT
ejpam-6123	469	1	marcel	marcel	PROPN
ejpam-6123	469	2	dekker	dekker	PROPN
ejpam-6123	469	3	,	,	PUNCT
ejpam-6123	469	4	new	new	PROPN
ejpam-6123	469	5	york	york	PROPN
ejpam-6123	469	6	,	,	PUNCT
ejpam-6123	469	7	1998	1998	NUM
ejpam-6123	469	8	.	.	PUNCT
ejpam-6123	470	1	[	[	X
ejpam-6123	470	2	3	3	X
ejpam-6123	470	3	]	]	X
ejpam-6123	470	4	f.	f.	PROPN
ejpam-6123	470	5	jamil	jamil	PROPN
ejpam-6123	470	6	and	and	CCONJ
ejpam-6123	470	7	h.	h.	PROPN
ejpam-6123	470	8	maglanque	maglanque	PROPN
ejpam-6123	470	9	.	.	PUNCT
ejpam-6123	471	1	on	on	ADP
ejpam-6123	471	2	cost	cost	NOUN
ejpam-6123	471	3	effective	effective	ADJ
ejpam-6123	471	4	domination	domination	NOUN
ejpam-6123	471	5	in	in	ADP
ejpam-6123	471	6	join	join	NOUN
ejpam-6123	471	7	,	,	PUNCT
ejpam-6123	471	8	corona	corona	NOUN
ejpam-6123	471	9	and	and	CCONJ
ejpam-6123	471	10	composition	composition	NOUN
ejpam-6123	471	11	of	of	ADP
ejpam-6123	471	12	graphs	graph	NOUN
ejpam-6123	471	13	.	.	PUNCT
ejpam-6123	472	1	european	european	ADJ
ejpam-6123	472	2	journal	journal	PROPN
ejpam-6123	472	3	of	of	ADP
ejpam-6123	472	4	pure	pure	ADJ
ejpam-6123	472	5	and	and	CCONJ
ejpam-6123	472	6	applied	applied	ADJ
ejpam-6123	472	7	mathematics	mathematic	NOUN
ejpam-6123	472	8	,	,	PUNCT
ejpam-6123	472	9	12(3):978–998	12(3):978–998	NUM
ejpam-6123	472	10	,	,	PUNCT
ejpam-6123	472	11	2019	2019	NUM
ejpam-6123	472	12	.	.	PUNCT
ejpam-6123	473	1	[	[	X
ejpam-6123	473	2	4	4	X
ejpam-6123	473	3	]	]	PUNCT
ejpam-6123	473	4	h.	h.	PROPN
ejpam-6123	473	5	nuenay	nuenay	PROPN
ejpam-6123	473	6	and	and	CCONJ
ejpam-6123	473	7	f.	f.	PROPN
ejpam-6123	473	8	jamil	jamil	PROPN
ejpam-6123	473	9	.	.	PUNCT
ejpam-6123	474	1	on	on	ADP
ejpam-6123	474	2	the	the	DET
ejpam-6123	474	3	minimal	minimal	ADJ
ejpam-6123	474	4	geodetic	geodetic	ADJ
ejpam-6123	474	5	domination	domination	NOUN
ejpam-6123	474	6	in	in	ADP
ejpam-6123	474	7	graphs	graph	NOUN
ejpam-6123	474	8	.	.	PUNCT
ejpam-6123	475	1	discussiones	discussione	NOUN
ejpam-6123	475	2	mathematicae	mathematicae	VERB
ejpam-6123	475	3	,	,	PUNCT
ejpam-6123	475	4	graph	graph	NOUN
ejpam-6123	475	5	theory	theory	NOUN
ejpam-6123	475	6	,	,	PUNCT
ejpam-6123	475	7	45:403–418	45:403–418	PROPN
ejpam-6123	475	8	,	,	PUNCT
ejpam-6123	475	9	2015	2015	NUM
ejpam-6123	475	10	.	.	PUNCT
ejpam-6123	476	1	[	[	X
ejpam-6123	476	2	5	5	X
ejpam-6123	476	3	]	]	PUNCT
ejpam-6123	476	4	h.	h.	PROPN
ejpam-6123	476	5	b.	b.	PROPN
ejpam-6123	476	6	walikar	walikar	PROPN
ejpam-6123	476	7	,	,	PUNCT
ejpam-6123	476	8	b.	b.	PROPN
ejpam-6123	476	9	d.	d.	PROPN
ejpam-6123	476	10	acharya	acharya	PROPN
ejpam-6123	476	11	,	,	PUNCT
ejpam-6123	476	12	and	and	CCONJ
ejpam-6123	476	13	e.	e.	PROPN
ejpam-6123	476	14	sampathkumar	sampathkumar	PROPN
ejpam-6123	476	15	.	.	PUNCT
ejpam-6123	477	1	recent	recent	ADJ
ejpam-6123	477	2	developments	development	NOUN
ejpam-6123	477	3	in	in	ADP
ejpam-6123	477	4	the	the	DET
ejpam-6123	477	5	theory	theory	NOUN
ejpam-6123	477	6	of	of	ADP
ejpam-6123	477	7	domination	domination	NOUN
ejpam-6123	477	8	in	in	ADP
ejpam-6123	477	9	graphs	graph	NOUN
ejpam-6123	477	10	and	and	CCONJ
ejpam-6123	477	11	its	its	PRON
ejpam-6123	477	12	applications	application	NOUN
ejpam-6123	477	13	.	.	PUNCT
ejpam-6123	478	1	unspecified	unspecified	ADJ
ejpam-6123	478	2	,	,	PUNCT
ejpam-6123	478	3	1979	1979	NUM
ejpam-6123	478	4	.	.	PUNCT
ejpam-6123	479	1	h.	h.	PROPN
ejpam-6123	479	2	nuenay	nuenay	PROPN
ejpam-6123	479	3	-	-	PUNCT
ejpam-6123	479	4	maglanquel	maglanquel	PROPN
ejpam-6123	479	5	/	/	SYM
ejpam-6123	479	6	eur	eur	PROPN
ejpam-6123	479	7	.	.	PUNCT
ejpam-6123	480	1	j.	j.	PROPN
ejpam-6123	480	2	pure	pure	PROPN
ejpam-6123	480	3	appl	appl	PROPN
ejpam-6123	480	4	.	.	PROPN
ejpam-6123	480	5	math	math	PROPN
ejpam-6123	480	6	,	,	PUNCT
ejpam-6123	480	7	18	18	NUM
ejpam-6123	480	8	(	(	PUNCT
ejpam-6123	480	9	3	3	NUM
ejpam-6123	480	10	)	)	PUNCT
ejpam-6123	480	11	(	(	PUNCT
ejpam-6123	480	12	2025	2025	NUM
ejpam-6123	480	13	)	)	PUNCT
ejpam-6123	480	14	,	,	PUNCT
ejpam-6123	480	15	6123	6123	NUM
ejpam-6123	480	16	14	14	NUM
ejpam-6123	480	17	of	of	ADP
ejpam-6123	480	18	14	14	NUM
ejpam-6123	481	1	[	[	SYM
ejpam-6123	481	2	6	6	NUM
ejpam-6123	481	3	]	]	PUNCT
ejpam-6123	482	1	p.	p.	NOUN
ejpam-6123	482	2	dreyer	dreyer	PROPN
ejpam-6123	482	3	jr	jr	PROPN
ejpam-6123	482	4	.	.	PROPN
ejpam-6123	482	5	applications	application	NOUN
ejpam-6123	482	6	and	and	CCONJ
ejpam-6123	482	7	variations	variation	NOUN
ejpam-6123	482	8	of	of	ADP
ejpam-6123	482	9	dominations	domination	NOUN
ejpam-6123	482	10	in	in	ADP
ejpam-6123	482	11	graphs	graph	NOUN
ejpam-6123	482	12	.	.	PUNCT
ejpam-6123	483	1	ph.d	ph.d	PROPN
ejpam-6123	483	2	.	.	PUNCT
ejpam-6123	484	1	dissertation	dissertation	NOUN
ejpam-6123	484	2	,	,	PUNCT
ejpam-6123	484	3	rutgers	rutger	NOUN
ejpam-6123	484	4	,	,	PUNCT
ejpam-6123	484	5	the	the	DET
ejpam-6123	484	6	state	state	PROPN
ejpam-6123	484	7	university	university	PROPN
ejpam-6123	484	8	of	of	ADP
ejpam-6123	484	9	new	new	PROPN
ejpam-6123	484	10	jersey	jersey	PROPN
ejpam-6123	484	11	,	,	PUNCT
ejpam-6123	484	12	new	new	PROPN
ejpam-6123	484	13	brunswick	brunswick	PROPN
ejpam-6123	484	14	,	,	PUNCT
ejpam-6123	484	15	2000	2000	NUM
ejpam-6123	484	16	.	.	PUNCT
ejpam-6123	485	1	[	[	X
ejpam-6123	485	2	7	7	X
ejpam-6123	485	3	]	]	X
ejpam-6123	485	4	p.	p.	PROPN
ejpam-6123	485	5	gupta	gupta	PROPN
ejpam-6123	485	6	.	.	PUNCT
ejpam-6123	486	1	domination	domination	NOUN
ejpam-6123	486	2	in	in	ADP
ejpam-6123	486	3	graph	graph	NOUN
ejpam-6123	486	4	with	with	ADP
ejpam-6123	486	5	application	application	NOUN
ejpam-6123	486	6	.	.	PUNCT
ejpam-6123	487	1	indian	indian	ADJ
ejpam-6123	487	2	journal	journal	PROPN
ejpam-6123	487	3	of	of	ADP
ejpam-6123	487	4	research	research	NOUN
ejpam-6123	487	5	,	,	PUNCT
ejpam-6123	487	6	2(3):115–117	2(3):115–117	NUM
ejpam-6123	487	7	,	,	PUNCT
ejpam-6123	487	8	2013	2013	NUM
ejpam-6123	487	9	.	.	PUNCT
ejpam-6123	488	1	[	[	X
ejpam-6123	488	2	8	8	X
ejpam-6123	488	3	]	]	PUNCT
ejpam-6123	488	4	t.	t.	PROPN
ejpam-6123	488	5	haynes	haynes	PROPN
ejpam-6123	488	6	,	,	PUNCT
ejpam-6123	488	7	s.	s.	PROPN
ejpam-6123	488	8	hedetniemi	hedetniemi	PROPN
ejpam-6123	488	9	,	,	PUNCT
ejpam-6123	488	10	and	and	CCONJ
ejpam-6123	488	11	m.	m.	PROPN
ejpam-6123	488	12	henning	henning	PROPN
ejpam-6123	488	13	.	.	PUNCT
ejpam-6123	489	1	domination	domination	NOUN
ejpam-6123	489	2	in	in	ADP
ejpam-6123	489	3	graphs	graph	NOUN
ejpam-6123	489	4	applied	apply	VERB
ejpam-6123	489	5	to	to	ADP
ejpam-6123	489	6	electrical	electrical	ADJ
ejpam-6123	489	7	power	power	NOUN
ejpam-6123	489	8	networks	network	NOUN
ejpam-6123	489	9	.	.	PUNCT
ejpam-6123	490	1	journal	journal	NOUN
ejpam-6123	490	2	of	of	ADP
ejpam-6123	490	3	discrete	discrete	ADJ
ejpam-6123	490	4	mathematics	mathematic	NOUN
ejpam-6123	490	5	,	,	PUNCT
ejpam-6123	490	6	15(4):519–529	15(4):519–529	NUM
ejpam-6123	490	7	,	,	PUNCT
ejpam-6123	490	8	2002	2002	NUM
ejpam-6123	490	9	.	.	PUNCT
ejpam-6123	491	1	[	[	X
ejpam-6123	491	2	9	9	NUM
ejpam-6123	491	3	]	]	PUNCT
ejpam-6123	491	4	a.	a.	NOUN
ejpam-6123	491	5	anitha	anitha	PROPN
ejpam-6123	491	6	,	,	PUNCT
ejpam-6123	491	7	s.	s.	PROPN
ejpam-6123	491	8	arumugam	arumugam	PROPN
ejpam-6123	491	9	,	,	PUNCT
ejpam-6123	491	10	and	and	CCONJ
ejpam-6123	491	11	e.	e.	PROPN
ejpam-6123	491	12	sampathkumar	sampathkumar	PROPN
ejpam-6123	491	13	.	.	PUNCT
ejpam-6123	491	14	degree	degree	NOUN
ejpam-6123	491	15	equitable	equitable	ADJ
ejpam-6123	491	16	sets	set	NOUN
ejpam-6123	491	17	in	in	ADP
ejpam-6123	491	18	a	a	DET
ejpam-6123	491	19	graph	graph	NOUN
ejpam-6123	491	20	.	.	PUNCT
ejpam-6123	492	1	international	international	ADJ
ejpam-6123	492	2	journal	journal	PROPN
ejpam-6123	492	3	of	of	ADP
ejpam-6123	492	4	mathematical	mathematical	ADJ
ejpam-6123	492	5	combinatorics	combinatoric	NOUN
ejpam-6123	492	6	,	,	PUNCT
ejpam-6123	492	7	3:32–47	3:32–47	NUM
ejpam-6123	492	8	,	,	PUNCT
ejpam-6123	492	9	2009	2009	NUM
ejpam-6123	492	10	.	.	PUNCT
ejpam-6123	493	1	[	[	X
ejpam-6123	493	2	10	10	NUM
ejpam-6123	493	3	]	]	PUNCT
ejpam-6123	493	4	a.	a.	PROPN
ejpam-6123	493	5	anitha	anitha	PROPN
ejpam-6123	493	6	,	,	PUNCT
ejpam-6123	493	7	s.	s.	PROPN
ejpam-6123	493	8	arumugam	arumugam	PROPN
ejpam-6123	493	9	,	,	PUNCT
ejpam-6123	493	10	and	and	CCONJ
ejpam-6123	493	11	mustapha	mustapha	PROPN
ejpam-6123	493	12	chellali	chellali	PROPN
ejpam-6123	493	13	.	.	PUNCT
ejpam-6123	494	1	equitable	equitable	ADJ
ejpam-6123	494	2	domination	domination	NOUN
ejpam-6123	494	3	in	in	ADP
ejpam-6123	494	4	graphs	graph	NOUN
ejpam-6123	494	5	.	.	PUNCT
ejpam-6123	495	1	discrete	discrete	ADJ
ejpam-6123	495	2	mathematics	mathematic	NOUN
ejpam-6123	495	3	,	,	PUNCT
ejpam-6123	495	4	algorithms	algorithm	NOUN
ejpam-6123	495	5	and	and	CCONJ
ejpam-6123	495	6	applications	application	NOUN
ejpam-6123	495	7	,	,	PUNCT
ejpam-6123	495	8	3(3):311–321	3(3):311–321	NUM
ejpam-6123	495	9	,	,	PUNCT
ejpam-6123	495	10	2011	2011	NUM
ejpam-6123	495	11	.	.	PUNCT
ejpam-6123	496	1	[	[	X
ejpam-6123	496	2	11	11	NUM
ejpam-6123	496	3	]	]	PUNCT
ejpam-6123	496	4	s.	s.	PROPN
ejpam-6123	496	5	hosamani	hosamani	PROPN
ejpam-6123	496	6	,	,	PUNCT
ejpam-6123	496	7	s.	s.	PROPN
ejpam-6123	496	8	shirkol	shirkol	PROPN
ejpam-6123	496	9	,	,	PUNCT
ejpam-6123	496	10	p.	p.	NOUN
ejpam-6123	496	11	jinagouda	jinagouda	NOUN
ejpam-6123	496	12	,	,	PUNCT
ejpam-6123	496	13	and	and	CCONJ
ejpam-6123	496	14	m.	m.	PROPN
ejpam-6123	496	15	krzywkowski	krzywkowski	PROPN
ejpam-6123	496	16	.	.	PUNCT
ejpam-6123	496	17	degree	degree	PROPN
ejpam-6123	496	18	equitable	equitable	ADJ
ejpam-6123	496	19	restrained	restrain	VERB
ejpam-6123	496	20	double	double	ADJ
ejpam-6123	496	21	domination	domination	NOUN
ejpam-6123	496	22	in	in	ADP
ejpam-6123	496	23	graphs	graph	NOUN
ejpam-6123	496	24	.	.	PUNCT
ejpam-6123	497	1	electronic	electronic	ADJ
ejpam-6123	497	2	journal	journal	NOUN
ejpam-6123	497	3	of	of	ADP
ejpam-6123	497	4	graph	graph	NOUN
ejpam-6123	497	5	theory	theory	NOUN
ejpam-6123	497	6	and	and	CCONJ
ejpam-6123	497	7	applications	application	NOUN
ejpam-6123	497	8	,	,	PUNCT
ejpam-6123	497	9	9(1	9(1	NUM
ejpam-6123	497	10	)	)	PUNCT
ejpam-6123	497	11	,	,	PUNCT
ejpam-6123	497	12	2021	2021	NUM
ejpam-6123	497	13	.	.	PUNCT
ejpam-6123	498	1	[	[	X
ejpam-6123	498	2	12	12	NUM
ejpam-6123	498	3	]	]	PUNCT
ejpam-6123	498	4	a.	a.	NOUN
ejpam-6123	498	5	nellai	nellai	PROPN
ejpam-6123	498	6	murugan	murugan	PROPN
ejpam-6123	498	7	and	and	CCONJ
ejpam-6123	498	8	g.	g.	PROPN
ejpam-6123	498	9	victor	victor	PROPN
ejpam-6123	498	10	emmanuel	emmanuel	PROPN
ejpam-6123	498	11	.	.	PUNCT
ejpam-6123	498	12	degree	degree	PROPN
ejpam-6123	498	13	equitable	equitable	ADJ
ejpam-6123	498	14	domination	domination	NOUN
ejpam-6123	498	15	number	number	NOUN
ejpam-6123	498	16	and	and	CCONJ
ejpam-6123	498	17	independent	independent	ADJ
ejpam-6123	498	18	domination	domination	NOUN
ejpam-6123	498	19	number	number	NOUN
ejpam-6123	498	20	of	of	ADP
ejpam-6123	498	21	a	a	DET
ejpam-6123	498	22	graph	graph	NOUN
ejpam-6123	498	23	.	.	PUNCT
ejpam-6123	499	1	international	international	ADJ
ejpam-6123	499	2	journal	journal	NOUN
ejpam-6123	499	3	of	of	ADP
ejpam-6123	499	4	innovative	innovative	ADJ
ejpam-6123	499	5	research	research	NOUN
ejpam-6123	499	6	in	in	ADP
ejpam-6123	499	7	science	science	NOUN
ejpam-6123	499	8	,	,	PUNCT
ejpam-6123	499	9	engineering	engineering	NOUN
ejpam-6123	499	10	and	and	CCONJ
ejpam-6123	499	11	technology	technology	NOUN
ejpam-6123	499	12	,	,	PUNCT
ejpam-6123	499	13	2(11	2(11	NUM
ejpam-6123	499	14	)	)	PUNCT
ejpam-6123	499	15	,	,	PUNCT
ejpam-6123	499	16	2013	2013	NUM
ejpam-6123	499	17	.	.	PUNCT
ejpam-6123	500	1	[	[	X
ejpam-6123	500	2	13	13	NUM
ejpam-6123	500	3	]	]	PUNCT
ejpam-6123	500	4	v.	v.	CCONJ
ejpam-6123	500	5	swaminathan	swaminathan	ADV
ejpam-6123	500	6	and	and	CCONJ
ejpam-6123	501	1	k.	k.	PROPN
ejpam-6123	501	2	m.	m.	PROPN
ejpam-6123	501	3	dharmalingam	dharmalingam	PROPN
ejpam-6123	501	4	.	.	PUNCT
ejpam-6123	502	1	degree	degree	NOUN
ejpam-6123	502	2	equitable	equitable	ADJ
ejpam-6123	502	3	domination	domination	NOUN
ejpam-6123	502	4	on	on	ADP
ejpam-6123	502	5	graphs	graph	NOUN
ejpam-6123	502	6	.	.	PUNCT
ejpam-6123	503	1	kragujevac	kragujevac	PROPN
ejpam-6123	503	2	journal	journal	PROPN
ejpam-6123	503	3	of	of	ADP
ejpam-6123	503	4	mathematics	mathematic	NOUN
ejpam-6123	503	5	,	,	PUNCT
ejpam-6123	503	6	35(1):191–197	35(1):191–197	PROPN
ejpam-6123	503	7	,	,	PUNCT
ejpam-6123	503	8	2011	2011	NUM
ejpam-6123	503	9	.	.	PUNCT
ejpam-6123	504	1	[	[	X
ejpam-6123	504	2	14	14	NUM
ejpam-6123	504	3	]	]	X
ejpam-6123	504	4	e.	e.	PROPN
ejpam-6123	504	5	sampathkumar	sampathkumar	PROPN
ejpam-6123	504	6	and	and	CCONJ
ejpam-6123	504	7	h.	h.	PROPN
ejpam-6123	504	8	b.	b.	PROPN
ejpam-6123	504	9	walikar	walikar	PROPN
ejpam-6123	504	10	.	.	PUNCT
ejpam-6123	505	1	the	the	DET
ejpam-6123	505	2	connected	connected	ADJ
ejpam-6123	505	3	domination	domination	NOUN
ejpam-6123	505	4	number	number	NOUN
ejpam-6123	505	5	of	of	ADP
ejpam-6123	505	6	a	a	DET
ejpam-6123	505	7	graph	graph	NOUN
ejpam-6123	505	8	.	.	PUNCT
ejpam-6123	505	9	journal	journal	NOUN
ejpam-6123	505	10	of	of	ADP
ejpam-6123	505	11	mathematics	mathematics	PROPN
ejpam-6123	505	12	and	and	CCONJ
ejpam-6123	505	13	physical	physical	ADJ
ejpam-6123	505	14	sciences	science	NOUN
ejpam-6123	505	15	,	,	PUNCT
ejpam-6123	505	16	13:607–613	13:607–613	NUM
ejpam-6123	505	17	,	,	PUNCT
ejpam-6123	505	18	1979	1979	NUM
ejpam-6123	505	19	.	.	PUNCT
ejpam-6123	506	1	[	[	X
ejpam-6123	506	2	15	15	NUM
ejpam-6123	506	3	]	]	X
ejpam-6123	506	4	s.	s.	PROPN
ejpam-6123	506	5	sivakumar	sivakumar	PROPN
ejpam-6123	506	6	,	,	PUNCT
ejpam-6123	506	7	n.	n.	PROPN
ejpam-6123	506	8	d.	d.	PROPN
ejpam-6123	506	9	soner	soner	PROPN
ejpam-6123	506	10	,	,	PUNCT
ejpam-6123	506	11	anwar	anwar	PROPN
ejpam-6123	506	12	alwardi	alwardi	PROPN
ejpam-6123	506	13	,	,	PUNCT
ejpam-6123	506	14	and	and	CCONJ
ejpam-6123	506	15	g.	g.	PROPN
ejpam-6123	506	16	deepak	deepak	PROPN
ejpam-6123	506	17	.	.	PUNCT
ejpam-6123	507	1	connected	connect	VERB
ejpam-6123	507	2	equitable	equitable	ADJ
ejpam-6123	507	3	domination	domination	NOUN
ejpam-6123	507	4	in	in	ADP
ejpam-6123	507	5	graphs	graph	NOUN
ejpam-6123	507	6	.	.	PUNCT
ejpam-6123	508	1	pure	pure	ADJ
ejpam-6123	508	2	mathematical	mathematical	ADJ
ejpam-6123	508	3	sciences	science	NOUN
ejpam-6123	508	4	,	,	PUNCT
ejpam-6123	508	5	1(3):123–130	1(3):123–130	NUM
ejpam-6123	508	6	,	,	PUNCT
ejpam-6123	508	7	2012	2012	NUM
ejpam-6123	508	8	.	.	PUNCT