id	sid	tid	token	lemma	pos
ejpam-6326	1	1	european	european	PROPN
ejpam-6326	1	2	journal	journal	PROPN
ejpam-6326	1	3	of	of	ADP
ejpam-6326	1	4	pure	pure	ADJ
ejpam-6326	1	5	and	and	CCONJ
ejpam-6326	1	6	applied	applied	ADJ
ejpam-6326	1	7	mathematics	mathematic	NOUN
ejpam-6326	1	8	2025	2025	NUM
ejpam-6326	1	9	,	,	PUNCT
ejpam-6326	1	10	vol	vol	NOUN
ejpam-6326	1	11	.	.	PROPN
ejpam-6326	1	12	18	18	NUM
ejpam-6326	1	13	,	,	PUNCT
ejpam-6326	1	14	issue	issue	NOUN
ejpam-6326	1	15	4	4	NUM
ejpam-6326	1	16	,	,	PUNCT
ejpam-6326	1	17	article	article	NOUN
ejpam-6326	1	18	number	number	NOUN
ejpam-6326	1	19	6326	6326	NUM
ejpam-6326	1	20	issn	issn	VERB
ejpam-6326	1	21	1307	1307	NUM
ejpam-6326	1	22	-	-	SYM
ejpam-6326	1	23	5543	5543	NUM
ejpam-6326	1	24	–	–	PUNCT
ejpam-6326	1	25	ejpam.com	ejpam.com	X
ejpam-6326	1	26	published	publish	VERB
ejpam-6326	1	27	by	by	ADP
ejpam-6326	1	28	new	new	PROPN
ejpam-6326	1	29	york	york	PROPN
ejpam-6326	1	30	business	business	PROPN
ejpam-6326	1	31	global	global	ADJ
ejpam-6326	1	32	friendly	friendly	ADJ
ejpam-6326	1	33	domination	domination	NOUN
ejpam-6326	1	34	in	in	ADP
ejpam-6326	1	35	graph	graph	NOUN
ejpam-6326	1	36	isagani	isagani	PROPN
ejpam-6326	1	37	s.	s.	PROPN
ejpam-6326	1	38	cabahug	cabahug	PROPN
ejpam-6326	1	39	,	,	PUNCT
ejpam-6326	1	40	jr.1,∗	jr.1,∗	PROPN
ejpam-6326	1	41	,	,	PUNCT
ejpam-6326	1	42	rolito	rolito	PROPN
ejpam-6326	1	43	g.	g.	PROPN
ejpam-6326	1	44	eballe1	eballe1	PROPN
ejpam-6326	1	45	,	,	PUNCT
ejpam-6326	1	46	reynard	reynard	PROPN
ejpam-6326	1	47	t.	t.	PROPN
ejpam-6326	1	48	fernandez1	fernandez1	PROPN
ejpam-6326	2	1	1	1	NUM
ejpam-6326	2	2	department	department	NOUN
ejpam-6326	2	3	of	of	ADP
ejpam-6326	2	4	mathematics	mathematic	NOUN
ejpam-6326	2	5	,	,	PUNCT
ejpam-6326	2	6	college	college	NOUN
ejpam-6326	2	7	of	of	ADP
ejpam-6326	2	8	arts	art	NOUN
ejpam-6326	2	9	and	and	CCONJ
ejpam-6326	2	10	sciences	science	NOUN
ejpam-6326	2	11	,	,	PUNCT
ejpam-6326	2	12	central	central	ADJ
ejpam-6326	2	13	mindanao	mindanao	PROPN
ejpam-6326	2	14	university	university	PROPN
ejpam-6326	2	15	,	,	PUNCT
ejpam-6326	2	16	8714	8714	NUM
ejpam-6326	2	17	bukidnon	bukidnon	NOUN
ejpam-6326	2	18	,	,	PUNCT
ejpam-6326	2	19	philippines	philippine	NOUN
ejpam-6326	2	20	abstract	abstract	ADJ
ejpam-6326	2	21	.	.	PUNCT
ejpam-6326	3	1	friendly	friendly	ADJ
ejpam-6326	3	2	domination	domination	NOUN
ejpam-6326	3	3	combines	combine	VERB
ejpam-6326	3	4	two	two	NUM
ejpam-6326	3	5	ideas	idea	NOUN
ejpam-6326	3	6	in	in	ADP
ejpam-6326	3	7	graph	graph	NOUN
ejpam-6326	3	8	theory	theory	NOUN
ejpam-6326	3	9	:	:	PUNCT
ejpam-6326	3	10	domination	domination	NOUN
ejpam-6326	3	11	,	,	PUNCT
ejpam-6326	3	12	which	which	PRON
ejpam-6326	3	13	captures	capture	VERB
ejpam-6326	3	14	influence	influence	NOUN
ejpam-6326	3	15	,	,	PUNCT
ejpam-6326	3	16	and	and	CCONJ
ejpam-6326	3	17	friendly	friendly	ADJ
ejpam-6326	3	18	sets	set	NOUN
ejpam-6326	3	19	,	,	PUNCT
ejpam-6326	3	20	which	which	PRON
ejpam-6326	3	21	balance	balance	VERB
ejpam-6326	3	22	that	that	DET
ejpam-6326	3	23	influence	influence	NOUN
ejpam-6326	3	24	.	.	PUNCT
ejpam-6326	4	1	we	we	PRON
ejpam-6326	4	2	study	study	VERB
ejpam-6326	4	3	the	the	DET
ejpam-6326	4	4	friendly	friendly	ADJ
ejpam-6326	4	5	dominating	dominating	NOUN
ejpam-6326	4	6	set	set	NOUN
ejpam-6326	4	7	and	and	CCONJ
ejpam-6326	4	8	the	the	DET
ejpam-6326	4	9	friendly	friendly	ADJ
ejpam-6326	4	10	domination	domination	NOUN
ejpam-6326	4	11	number	number	NOUN
ejpam-6326	4	12	,	,	PUNCT
ejpam-6326	4	13	the	the	DET
ejpam-6326	4	14	smallest	small	ADJ
ejpam-6326	4	15	size	size	NOUN
ejpam-6326	4	16	of	of	ADP
ejpam-6326	4	17	a	a	DET
ejpam-6326	4	18	set	set	NOUN
ejpam-6326	4	19	that	that	PRON
ejpam-6326	4	20	is	be	AUX
ejpam-6326	4	21	both	both	PRON
ejpam-6326	4	22	dominating	dominate	VERB
ejpam-6326	4	23	and	and	CCONJ
ejpam-6326	4	24	friendly	friendly	ADJ
ejpam-6326	4	25	.	.	PUNCT
ejpam-6326	5	1	first	first	ADV
ejpam-6326	5	2	,	,	PUNCT
ejpam-6326	5	3	we	we	PRON
ejpam-6326	5	4	prove	prove	VERB
ejpam-6326	5	5	that	that	SCONJ
ejpam-6326	5	6	for	for	ADP
ejpam-6326	5	7	every	every	DET
ejpam-6326	5	8	graph	graph	NOUN
ejpam-6326	5	9	the	the	DET
ejpam-6326	5	10	usual	usual	ADJ
ejpam-6326	5	11	domination	domination	NOUN
ejpam-6326	5	12	number	number	NOUN
ejpam-6326	5	13	is	be	AUX
ejpam-6326	5	14	never	never	ADV
ejpam-6326	5	15	larger	large	ADJ
ejpam-6326	5	16	than	than	ADP
ejpam-6326	5	17	its	its	PRON
ejpam-6326	5	18	friendly	friendly	ADJ
ejpam-6326	5	19	domination	domination	NOUN
ejpam-6326	5	20	number	number	NOUN
ejpam-6326	5	21	,	,	PUNCT
ejpam-6326	5	22	and	and	CCONJ
ejpam-6326	5	23	we	we	PRON
ejpam-6326	5	24	identify	identify	VERB
ejpam-6326	5	25	all	all	DET
ejpam-6326	5	26	graphs	graph	NOUN
ejpam-6326	5	27	whose	whose	DET
ejpam-6326	5	28	friendly	friendly	ADJ
ejpam-6326	5	29	domination	domination	NOUN
ejpam-6326	5	30	number	number	NOUN
ejpam-6326	5	31	equals	equal	VERB
ejpam-6326	5	32	one	one	NUM
ejpam-6326	5	33	or	or	CCONJ
ejpam-6326	5	34	two	two	NUM
ejpam-6326	5	35	.	.	PUNCT
ejpam-6326	6	1	we	we	PRON
ejpam-6326	6	2	then	then	ADV
ejpam-6326	6	3	show	show	VERB
ejpam-6326	6	4	that	that	SCONJ
ejpam-6326	6	5	the	the	DET
ejpam-6326	6	6	gap	gap	NOUN
ejpam-6326	6	7	between	between	ADP
ejpam-6326	6	8	these	these	DET
ejpam-6326	6	9	two	two	NUM
ejpam-6326	6	10	parameters	parameter	NOUN
ejpam-6326	6	11	can	can	AUX
ejpam-6326	6	12	be	be	AUX
ejpam-6326	6	13	made	make	VERB
ejpam-6326	6	14	arbitrarily	arbitrarily	ADV
ejpam-6326	6	15	large	large	ADJ
ejpam-6326	6	16	.	.	PUNCT
ejpam-6326	7	1	exact	exact	ADJ
ejpam-6326	7	2	formulas	formula	NOUN
ejpam-6326	7	3	are	be	AUX
ejpam-6326	7	4	derived	derive	VERB
ejpam-6326	7	5	for	for	ADP
ejpam-6326	7	6	paths	path	NOUN
ejpam-6326	7	7	and	and	CCONJ
ejpam-6326	7	8	cycles	cycle	NOUN
ejpam-6326	7	9	.	.	PUNCT
ejpam-6326	8	1	sharp	sharp	PROPN
ejpam-6326	8	2	nordhaus	nordhaus	PROPN
ejpam-6326	8	3	–	–	PUNCT
ejpam-6326	8	4	gaddum	gaddum	PROPN
ejpam-6326	8	5	bounds	bound	NOUN
ejpam-6326	8	6	are	be	AUX
ejpam-6326	8	7	obtained	obtain	VERB
ejpam-6326	8	8	for	for	ADP
ejpam-6326	8	9	the	the	DET
ejpam-6326	8	10	sum	sum	NOUN
ejpam-6326	8	11	and	and	CCONJ
ejpam-6326	8	12	product	product	NOUN
ejpam-6326	8	13	of	of	ADP
ejpam-6326	8	14	the	the	DET
ejpam-6326	8	15	friendly	friendly	ADJ
ejpam-6326	8	16	domination	domination	NOUN
ejpam-6326	8	17	number	number	NOUN
ejpam-6326	8	18	of	of	ADP
ejpam-6326	8	19	a	a	DET
ejpam-6326	8	20	graph	graph	NOUN
ejpam-6326	8	21	and	and	CCONJ
ejpam-6326	8	22	its	its	PRON
ejpam-6326	8	23	complement	complement	NOUN
ejpam-6326	8	24	.	.	PUNCT
ejpam-6326	9	1	finally	finally	ADV
ejpam-6326	9	2	,	,	PUNCT
ejpam-6326	9	3	we	we	PRON
ejpam-6326	9	4	give	give	VERB
ejpam-6326	9	5	complete	complete	ADJ
ejpam-6326	9	6	structural	structural	ADJ
ejpam-6326	9	7	characterizations	characterization	NOUN
ejpam-6326	9	8	of	of	ADP
ejpam-6326	9	9	friendly	friendly	ADJ
ejpam-6326	9	10	dominating	dominating	NOUN
ejpam-6326	9	11	sets	set	NOUN
ejpam-6326	9	12	in	in	ADP
ejpam-6326	9	13	the	the	DET
ejpam-6326	9	14	join	join	NOUN
ejpam-6326	9	15	and	and	CCONJ
ejpam-6326	9	16	corona	corona	NOUN
ejpam-6326	9	17	of	of	ADP
ejpam-6326	9	18	two	two	NUM
ejpam-6326	9	19	graphs	graph	NOUN
ejpam-6326	9	20	.	.	PUNCT
ejpam-6326	10	1	2020	2020	NUM
ejpam-6326	10	2	mathematics	mathematic	NOUN
ejpam-6326	10	3	subject	subject	NOUN
ejpam-6326	10	4	classifications	classification	NOUN
ejpam-6326	10	5	:	:	PUNCT
ejpam-6326	10	6	05c69	05c69	X
ejpam-6326	10	7	key	key	ADJ
ejpam-6326	10	8	words	word	NOUN
ejpam-6326	10	9	and	and	CCONJ
ejpam-6326	10	10	phrases	phrase	NOUN
ejpam-6326	10	11	:	:	PUNCT
ejpam-6326	10	12	friendly	friendly	ADJ
ejpam-6326	10	13	sets	set	NOUN
ejpam-6326	10	14	,	,	PUNCT
ejpam-6326	10	15	friendly	friendly	ADJ
ejpam-6326	10	16	domination	domination	NOUN
ejpam-6326	10	17	,	,	PUNCT
ejpam-6326	10	18	domination	domination	NOUN
ejpam-6326	10	19	1	1	NUM
ejpam-6326	10	20	.	.	PUNCT
ejpam-6326	11	1	introduction	introduction	NOUN
ejpam-6326	11	2	graph	graph	NOUN
ejpam-6326	11	3	theory	theory	NOUN
ejpam-6326	11	4	has	have	AUX
ejpam-6326	11	5	been	be	AUX
ejpam-6326	11	6	around	around	ADV
ejpam-6326	11	7	since	since	SCONJ
ejpam-6326	11	8	the	the	DET
ejpam-6326	11	9	eighteenth	eighteenth	ADJ
ejpam-6326	11	10	century	century	NOUN
ejpam-6326	11	11	and	and	CCONJ
ejpam-6326	11	12	has	have	AUX
ejpam-6326	11	13	grown	grow	VERB
ejpam-6326	11	14	into	into	ADP
ejpam-6326	11	15	an	an	DET
ejpam-6326	11	16	important	important	ADJ
ejpam-6326	11	17	part	part	NOUN
ejpam-6326	11	18	of	of	ADP
ejpam-6326	11	19	mathematics	mathematic	NOUN
ejpam-6326	11	20	.	.	PUNCT
ejpam-6326	12	1	it	it	PRON
ejpam-6326	12	2	helps	help	VERB
ejpam-6326	12	3	us	we	PRON
ejpam-6326	12	4	understand	understand	VERB
ejpam-6326	12	5	communication	communication	NOUN
ejpam-6326	12	6	networks	network	NOUN
ejpam-6326	12	7	,	,	PUNCT
ejpam-6326	12	8	chemical	chemical	NOUN
ejpam-6326	12	9	structures	structure	NOUN
ejpam-6326	12	10	,	,	PUNCT
ejpam-6326	12	11	disease	disease	NOUN
ejpam-6326	12	12	spread	spread	VERB
ejpam-6326	12	13	,	,	PUNCT
ejpam-6326	12	14	and	and	CCONJ
ejpam-6326	12	15	social	social	ADJ
ejpam-6326	12	16	connections	connection	NOUN
ejpam-6326	12	17	[	[	X
ejpam-6326	12	18	1	1	NUM
ejpam-6326	12	19	,	,	PUNCT
ejpam-6326	12	20	2	2	NUM
ejpam-6326	12	21	]	]	PUNCT
ejpam-6326	12	22	.	.	PUNCT
ejpam-6326	13	1	one	one	NUM
ejpam-6326	13	2	of	of	ADP
ejpam-6326	13	3	the	the	DET
ejpam-6326	13	4	key	key	ADJ
ejpam-6326	13	5	ideas	idea	NOUN
ejpam-6326	13	6	studied	study	VERB
ejpam-6326	13	7	is	be	AUX
ejpam-6326	13	8	domination	domination	NOUN
ejpam-6326	13	9	,	,	PUNCT
ejpam-6326	13	10	where	where	SCONJ
ejpam-6326	13	11	a	a	DET
ejpam-6326	13	12	small	small	ADJ
ejpam-6326	13	13	group	group	NOUN
ejpam-6326	13	14	of	of	ADP
ejpam-6326	13	15	vertices	vertex	NOUN
ejpam-6326	13	16	can	can	AUX
ejpam-6326	13	17	control	control	VERB
ejpam-6326	13	18	or	or	CCONJ
ejpam-6326	13	19	influence	influence	VERB
ejpam-6326	13	20	the	the	DET
ejpam-6326	13	21	entire	entire	ADJ
ejpam-6326	13	22	graph	graph	NOUN
ejpam-6326	13	23	.	.	PUNCT
ejpam-6326	14	1	many	many	ADJ
ejpam-6326	14	2	researchers	researcher	NOUN
ejpam-6326	14	3	,	,	PUNCT
ejpam-6326	14	4	like	like	ADP
ejpam-6326	14	5	henning	henning	NOUN
ejpam-6326	14	6	and	and	CCONJ
ejpam-6326	14	7	yeo	yeo	PROPN
ejpam-6326	15	1	[	[	X
ejpam-6326	15	2	3	3	NUM
ejpam-6326	15	3	]	]	PUNCT
ejpam-6326	15	4	,	,	PUNCT
ejpam-6326	15	5	have	have	AUX
ejpam-6326	15	6	expanded	expand	VERB
ejpam-6326	15	7	these	these	DET
ejpam-6326	15	8	ideas	idea	NOUN
ejpam-6326	15	9	and	and	CCONJ
ejpam-6326	15	10	connected	connect	VERB
ejpam-6326	15	11	them	they	PRON
ejpam-6326	15	12	to	to	ADP
ejpam-6326	15	13	other	other	ADJ
ejpam-6326	15	14	topics	topic	NOUN
ejpam-6326	15	15	like	like	ADP
ejpam-6326	15	16	hypergraphs	hypergraph	NOUN
ejpam-6326	15	17	,	,	PUNCT
ejpam-6326	15	18	graphs	graph	NOUN
ejpam-6326	15	19	with	with	ADP
ejpam-6326	15	20	special	special	ADJ
ejpam-6326	15	21	distance	distance	NOUN
ejpam-6326	15	22	properties	property	NOUN
ejpam-6326	15	23	,	,	PUNCT
ejpam-6326	15	24	and	and	CCONJ
ejpam-6326	15	25	computer	computer	NOUN
ejpam-6326	15	26	algorithms	algorithm	NOUN
ejpam-6326	15	27	.	.	PUNCT
ejpam-6326	16	1	recently	recently	ADV
ejpam-6326	16	2	,	,	PUNCT
ejpam-6326	16	3	the	the	DET
ejpam-6326	16	4	idea	idea	NOUN
ejpam-6326	16	5	of	of	ADP
ejpam-6326	16	6	a	a	DET
ejpam-6326	16	7	friendly	friendly	ADJ
ejpam-6326	16	8	set	set	NOUN
ejpam-6326	16	9	,	,	PUNCT
ejpam-6326	16	10	where	where	SCONJ
ejpam-6326	16	11	each	each	DET
ejpam-6326	16	12	vertex	vertex	NOUN
ejpam-6326	16	13	outside	outside	ADP
ejpam-6326	16	14	the	the	DET
ejpam-6326	16	15	set	set	NOUN
ejpam-6326	16	16	has	have	VERB
ejpam-6326	16	17	at	at	ADV
ejpam-6326	16	18	least	least	ADJ
ejpam-6326	16	19	as	as	ADP
ejpam-6326	16	20	many	many	ADJ
ejpam-6326	16	21	neighbors	neighbor	NOUN
ejpam-6326	16	22	outside	outside	ADV
ejpam-6326	16	23	as	as	ADP
ejpam-6326	16	24	inside	inside	ADV
ejpam-6326	16	25	,	,	PUNCT
ejpam-6326	16	26	was	be	AUX
ejpam-6326	16	27	introduced	introduce	VERB
ejpam-6326	16	28	by	by	ADP
ejpam-6326	16	29	haynes	hayne	NOUN
ejpam-6326	16	30	,	,	PUNCT
ejpam-6326	16	31	hedetniemi	hedetniemi	ADV
ejpam-6326	16	32	,	,	PUNCT
ejpam-6326	16	33	and	and	CCONJ
ejpam-6326	16	34	henning	henne	VERB
ejpam-6326	17	1	[	[	X
ejpam-6326	17	2	4	4	NUM
ejpam-6326	17	3	]	]	PUNCT
ejpam-6326	17	4	.	.	PUNCT
ejpam-6326	18	1	building	build	VERB
ejpam-6326	18	2	on	on	ADP
ejpam-6326	18	3	this	this	DET
ejpam-6326	18	4	concept	concept	NOUN
ejpam-6326	18	5	,	,	PUNCT
ejpam-6326	18	6	we	we	PRON
ejpam-6326	18	7	define	define	VERB
ejpam-6326	18	8	friendly	friendly	ADJ
ejpam-6326	18	9	domination	domination	NOUN
ejpam-6326	18	10	as	as	ADP
ejpam-6326	18	11	a	a	DET
ejpam-6326	18	12	dominating	dominating	NOUN
ejpam-6326	18	13	set	set	NOUN
ejpam-6326	18	14	that	that	PRON
ejpam-6326	18	15	is	be	AUX
ejpam-6326	18	16	also	also	ADV
ejpam-6326	18	17	a	a	DET
ejpam-6326	18	18	friendly	friendly	ADJ
ejpam-6326	18	19	set	set	NOUN
ejpam-6326	18	20	.	.	PUNCT
ejpam-6326	19	1	this	this	DET
ejpam-6326	19	2	new	new	ADJ
ejpam-6326	19	3	concept	concept	NOUN
ejpam-6326	19	4	reflects	reflect	VERB
ejpam-6326	19	5	situations	situation	NOUN
ejpam-6326	19	6	where	where	SCONJ
ejpam-6326	19	7	influence	influence	NOUN
ejpam-6326	19	8	needs	need	VERB
ejpam-6326	19	9	to	to	PART
ejpam-6326	19	10	be	be	AUX
ejpam-6326	19	11	applied	apply	VERB
ejpam-6326	19	12	carefully	carefully	ADV
ejpam-6326	19	13	without	without	ADP
ejpam-6326	19	14	overwhelming	overwhelm	VERB
ejpam-6326	19	15	any	any	DET
ejpam-6326	19	16	single	single	ADJ
ejpam-6326	19	17	vertex	vertex	NOUN
ejpam-6326	19	18	.	.	PUNCT
ejpam-6326	20	1	friendly	friendly	ADJ
ejpam-6326	20	2	domination	domination	NOUN
ejpam-6326	20	3	is	be	AUX
ejpam-6326	20	4	closely	closely	ADV
ejpam-6326	20	5	related	relate	VERB
ejpam-6326	20	6	to	to	ADP
ejpam-6326	20	7	other	other	ADJ
ejpam-6326	20	8	ideas	idea	NOUN
ejpam-6326	20	9	like	like	ADP
ejpam-6326	20	10	offensive	offensive	ADJ
ejpam-6326	20	11	alliance	alliance	NOUN
ejpam-6326	21	1	[	[	X
ejpam-6326	21	2	5–8	5–8	X
ejpam-6326	21	3	]	]	PUNCT
ejpam-6326	21	4	and	and	CCONJ
ejpam-6326	21	5	defensive	defensive	ADJ
ejpam-6326	21	6	alliances	alliance	NOUN
ejpam-6326	22	1	[	[	X
ejpam-6326	22	2	9–12	9–12	NOUN
ejpam-6326	22	3	]	]	PUNCT
ejpam-6326	22	4	.	.	PUNCT
ejpam-6326	23	1	∗corresponding	∗corresponde	VERB
ejpam-6326	23	2	author	author	NOUN
ejpam-6326	23	3	.	.	PUNCT
ejpam-6326	24	1	doi	doi	NOUN
ejpam-6326	24	2	:	:	PUNCT
ejpam-6326	24	3	https://doi.org/10.29020/nybg.ejpam.v18i4.6326	https://doi.org/10.29020/nybg.ejpam.v18i4.6326	NUM
ejpam-6326	24	4	email	email	NOUN
ejpam-6326	24	5	addresses	address	NOUN
ejpam-6326	24	6	:	:	PUNCT
ejpam-6326	24	7	isaganicabahugjr@cmu.edu.ph	isaganicabahugjr@cmu.edu.ph	PROPN
ejpam-6326	24	8	(	(	PUNCT
ejpam-6326	24	9	i.	i.	PROPN
ejpam-6326	24	10	s.	s.	PROPN
ejpam-6326	24	11	cabahug	cabahug	PROPN
ejpam-6326	24	12	,	,	PUNCT
ejpam-6326	24	13	jr	jr	PROPN
ejpam-6326	24	14	.	.	PROPN
ejpam-6326	24	15	)	)	PUNCT
ejpam-6326	24	16	,	,	PUNCT
ejpam-6326	24	17	rgeballe@cmu.edu.ph	rgeballe@cmu.edu.ph	PROPN
ejpam-6326	24	18	(	(	PUNCT
ejpam-6326	24	19	r.	r.	PROPN
ejpam-6326	24	20	g.	g.	PROPN
ejpam-6326	24	21	eballe	eballe	PROPN
ejpam-6326	24	22	)	)	PUNCT
ejpam-6326	24	23	,	,	PUNCT
ejpam-6326	24	24	f.reynard.fernandez@cmu.edu.ph	f.reynard.fernandez@cmu.edu.ph	PROPN
ejpam-6326	24	25	(	(	PUNCT
ejpam-6326	24	26	r.	r.	PROPN
ejpam-6326	24	27	t.	t.	PROPN
ejpam-6326	24	28	fernandez	fernandez	PROPN
ejpam-6326	24	29	)	)	PUNCT
ejpam-6326	24	30	https://www.ejpam.com	https://www.ejpam.com	NOUN
ejpam-6326	25	1	1	1	NUM
ejpam-6326	25	2	copyright	copyright	NOUN
ejpam-6326	25	3	:	:	PUNCT
ejpam-6326	25	4	©	©	PROPN
ejpam-6326	25	5	2025	2025	NUM
ejpam-6326	25	6	the	the	DET
ejpam-6326	25	7	author(s	author(s	NOUN
ejpam-6326	25	8	)	)	PUNCT
ejpam-6326	25	9	.	.	PUNCT
ejpam-6326	26	1	(	(	PUNCT
ejpam-6326	26	2	cc	cc	NOUN
ejpam-6326	26	3	by	by	ADP
ejpam-6326	26	4	-	-	PUNCT
ejpam-6326	26	5	nc	nc	PROPN
ejpam-6326	26	6	4.0	4.0	NUM
ejpam-6326	26	7	)	)	PUNCT
ejpam-6326	26	8	i.	i.	PROPN
ejpam-6326	26	9	s.	s.	PROPN
ejpam-6326	26	10	cabahug	cabahug	PROPN
ejpam-6326	26	11	,	,	PUNCT
ejpam-6326	26	12	jr	jr	PROPN
ejpam-6326	26	13	.	.	PROPN
ejpam-6326	26	14	,	,	PUNCT
ejpam-6326	26	15	r.	r.	PROPN
ejpam-6326	26	16	g.	g.	PROPN
ejpam-6326	26	17	eballe	eballe	PROPN
ejpam-6326	26	18	,	,	PUNCT
ejpam-6326	26	19	r.	r.	PROPN
ejpam-6326	26	20	t.	t.	PROPN
ejpam-6326	26	21	fernandez	fernandez	PROPN
ejpam-6326	26	22	/	/	SYM
ejpam-6326	26	23	eur	eur	PROPN
ejpam-6326	26	24	.	.	PUNCT
ejpam-6326	27	1	j.	j.	PROPN
ejpam-6326	27	2	pure	pure	PROPN
ejpam-6326	27	3	appl	appl	PROPN
ejpam-6326	27	4	.	.	PROPN
ejpam-6326	27	5	math	math	PROPN
ejpam-6326	27	6	,	,	PUNCT
ejpam-6326	27	7	18	18	NUM
ejpam-6326	27	8	(	(	PUNCT
ejpam-6326	27	9	4	4	NUM
ejpam-6326	27	10	)	)	PUNCT
ejpam-6326	27	11	(	(	PUNCT
ejpam-6326	27	12	2025	2025	NUM
ejpam-6326	27	13	)	)	PUNCT
ejpam-6326	27	14	,	,	PUNCT
ejpam-6326	27	15	6326	6326	NUM
ejpam-6326	27	16	2	2	NUM
ejpam-6326	27	17	of	of	ADP
ejpam-6326	27	18	15	15	NUM
ejpam-6326	27	19	at	at	ADP
ejpam-6326	27	20	the	the	DET
ejpam-6326	27	21	same	same	ADJ
ejpam-6326	27	22	time	time	NOUN
ejpam-6326	27	23	,	,	PUNCT
ejpam-6326	27	24	there	there	PRON
ejpam-6326	27	25	has	have	AUX
ejpam-6326	27	26	been	be	AUX
ejpam-6326	27	27	growing	grow	VERB
ejpam-6326	27	28	interest	interest	NOUN
ejpam-6326	27	29	in	in	ADP
ejpam-6326	27	30	other	other	ADJ
ejpam-6326	27	31	fairness	fairness	NOUN
ejpam-6326	27	32	conditions	condition	NOUN
ejpam-6326	27	33	on	on	ADP
ejpam-6326	27	34	dominating	dominating	NOUN
ejpam-6326	27	35	sets	set	NOUN
ejpam-6326	27	36	.	.	PUNCT
ejpam-6326	28	1	one	one	NUM
ejpam-6326	28	2	example	example	NOUN
ejpam-6326	28	3	is	be	AUX
ejpam-6326	28	4	fair	fair	ADJ
ejpam-6326	28	5	domination	domination	NOUN
ejpam-6326	28	6	,	,	PUNCT
ejpam-6326	28	7	where	where	SCONJ
ejpam-6326	28	8	each	each	DET
ejpam-6326	28	9	vertex	vertex	NOUN
ejpam-6326	28	10	outside	outside	ADP
ejpam-6326	28	11	the	the	DET
ejpam-6326	28	12	set	set	NOUN
ejpam-6326	28	13	must	must	AUX
ejpam-6326	28	14	have	have	VERB
ejpam-6326	28	15	the	the	DET
ejpam-6326	28	16	same	same	ADJ
ejpam-6326	28	17	number	number	NOUN
ejpam-6326	28	18	of	of	ADP
ejpam-6326	28	19	connections	connection	NOUN
ejpam-6326	28	20	to	to	ADP
ejpam-6326	28	21	the	the	DET
ejpam-6326	28	22	set	set	NOUN
ejpam-6326	28	23	[	[	X
ejpam-6326	28	24	13	13	NUM
ejpam-6326	28	25	]	]	PUNCT
ejpam-6326	28	26	.	.	PUNCT
ejpam-6326	29	1	these	these	DET
ejpam-6326	29	2	ideas	idea	NOUN
ejpam-6326	29	3	show	show	VERB
ejpam-6326	29	4	a	a	DET
ejpam-6326	29	5	trend	trend	NOUN
ejpam-6326	29	6	toward	toward	ADP
ejpam-6326	29	7	studying	study	VERB
ejpam-6326	29	8	domination	domination	NOUN
ejpam-6326	29	9	with	with	ADP
ejpam-6326	29	10	more	more	ADV
ejpam-6326	29	11	detailed	detailed	ADJ
ejpam-6326	29	12	and	and	CCONJ
ejpam-6326	29	13	realistic	realistic	ADJ
ejpam-6326	29	14	rules	rule	NOUN
ejpam-6326	29	15	compared	compare	VERB
ejpam-6326	29	16	to	to	ADP
ejpam-6326	29	17	traditional	traditional	ADJ
ejpam-6326	29	18	domination	domination	NOUN
ejpam-6326	29	19	.	.	PUNCT
ejpam-6326	30	1	even	even	ADV
ejpam-6326	30	2	with	with	ADP
ejpam-6326	30	3	these	these	DET
ejpam-6326	30	4	developments	development	NOUN
ejpam-6326	30	5	,	,	PUNCT
ejpam-6326	30	6	there	there	PRON
ejpam-6326	30	7	are	be	VERB
ejpam-6326	30	8	still	still	ADV
ejpam-6326	30	9	many	many	ADJ
ejpam-6326	30	10	open	open	ADJ
ejpam-6326	30	11	questions	question	NOUN
ejpam-6326	30	12	about	about	ADP
ejpam-6326	30	13	the	the	DET
ejpam-6326	30	14	friendly	friendly	ADJ
ejpam-6326	30	15	domination	domination	NOUN
ejpam-6326	30	16	.	.	PUNCT
ejpam-6326	31	1	basic	basic	ADJ
ejpam-6326	31	2	properties	property	NOUN
ejpam-6326	31	3	that	that	PRON
ejpam-6326	31	4	were	be	AUX
ejpam-6326	31	5	solved	solve	VERB
ejpam-6326	31	6	long	long	ADV
ejpam-6326	31	7	ago	ago	ADV
ejpam-6326	31	8	for	for	ADP
ejpam-6326	31	9	ordinary	ordinary	ADJ
ejpam-6326	31	10	domination	domination	NOUN
ejpam-6326	31	11	,	,	PUNCT
ejpam-6326	31	12	such	such	ADJ
ejpam-6326	31	13	as	as	ADP
ejpam-6326	31	14	nordhaus	nordhaus	NOUN
ejpam-6326	31	15	–	–	PUNCT
ejpam-6326	31	16	gaddum	gaddum	NOUN
ejpam-6326	31	17	-	-	PUNCT
ejpam-6326	31	18	type	type	NOUN
ejpam-6326	31	19	inequalities	inequality	NOUN
ejpam-6326	31	20	[	[	X
ejpam-6326	31	21	14	14	NUM
ejpam-6326	31	22	]	]	PUNCT
ejpam-6326	31	23	,	,	PUNCT
ejpam-6326	31	24	are	be	AUX
ejpam-6326	31	25	still	still	ADV
ejpam-6326	31	26	open	open	ADJ
ejpam-6326	31	27	for	for	ADP
ejpam-6326	31	28	friendly	friendly	ADJ
ejpam-6326	31	29	domination	domination	NOUN
ejpam-6326	31	30	.	.	PUNCT
ejpam-6326	32	1	also	also	ADV
ejpam-6326	32	2	,	,	PUNCT
ejpam-6326	32	3	exact	exact	ADJ
ejpam-6326	32	4	values	value	NOUN
ejpam-6326	32	5	and	and	CCONJ
ejpam-6326	32	6	sharp	sharp	ADJ
ejpam-6326	32	7	bounds	bound	NOUN
ejpam-6326	32	8	are	be	AUX
ejpam-6326	32	9	only	only	ADV
ejpam-6326	32	10	known	know	VERB
ejpam-6326	32	11	for	for	ADP
ejpam-6326	32	12	a	a	DET
ejpam-6326	32	13	few	few	ADJ
ejpam-6326	32	14	simple	simple	ADJ
ejpam-6326	32	15	types	type	NOUN
ejpam-6326	32	16	of	of	ADP
ejpam-6326	32	17	graphs	graph	NOUN
ejpam-6326	32	18	,	,	PUNCT
ejpam-6326	32	19	and	and	CCONJ
ejpam-6326	32	20	there	there	PRON
ejpam-6326	32	21	is	be	VERB
ejpam-6326	32	22	little	little	ADV
ejpam-6326	32	23	known	known	ADJ
ejpam-6326	32	24	about	about	ADP
ejpam-6326	32	25	how	how	SCONJ
ejpam-6326	32	26	friendly	friendly	ADJ
ejpam-6326	32	27	domination	domination	NOUN
ejpam-6326	32	28	behaves	behave	VERB
ejpam-6326	32	29	under	under	ADP
ejpam-6326	32	30	graph	graph	NOUN
ejpam-6326	32	31	operations	operation	NOUN
ejpam-6326	32	32	like	like	ADP
ejpam-6326	32	33	graph	graph	NOUN
ejpam-6326	32	34	products	product	NOUN
ejpam-6326	32	35	.	.	PUNCT
ejpam-6326	33	1	in	in	ADP
ejpam-6326	33	2	this	this	DET
ejpam-6326	33	3	paper	paper	NOUN
ejpam-6326	33	4	,	,	PUNCT
ejpam-6326	33	5	we	we	PRON
ejpam-6326	33	6	prove	prove	VERB
ejpam-6326	33	7	that	that	SCONJ
ejpam-6326	33	8	the	the	DET
ejpam-6326	33	9	domination	domination	NOUN
ejpam-6326	33	10	number	number	NOUN
ejpam-6326	33	11	is	be	AUX
ejpam-6326	33	12	always	always	ADV
ejpam-6326	33	13	less	less	ADJ
ejpam-6326	33	14	than	than	ADP
ejpam-6326	33	15	or	or	CCONJ
ejpam-6326	33	16	equal	equal	ADJ
ejpam-6326	33	17	to	to	ADP
ejpam-6326	33	18	the	the	DET
ejpam-6326	33	19	friendly	friendly	ADJ
ejpam-6326	33	20	domination	domination	NOUN
ejpam-6326	33	21	number	number	NOUN
ejpam-6326	33	22	,	,	PUNCT
ejpam-6326	33	23	and	and	CCONJ
ejpam-6326	33	24	we	we	PRON
ejpam-6326	33	25	describe	describe	VERB
ejpam-6326	33	26	graphs	graph	NOUN
ejpam-6326	33	27	where	where	SCONJ
ejpam-6326	33	28	the	the	DET
ejpam-6326	33	29	friendly	friendly	ADJ
ejpam-6326	33	30	domination	domination	NOUN
ejpam-6326	33	31	number	number	NOUN
ejpam-6326	33	32	equals	equal	VERB
ejpam-6326	33	33	one	one	NUM
ejpam-6326	33	34	or	or	CCONJ
ejpam-6326	33	35	two	two	NUM
ejpam-6326	33	36	.	.	PUNCT
ejpam-6326	34	1	we	we	PRON
ejpam-6326	34	2	also	also	ADV
ejpam-6326	34	3	show	show	VERB
ejpam-6326	34	4	that	that	SCONJ
ejpam-6326	34	5	the	the	DET
ejpam-6326	34	6	difference	difference	NOUN
ejpam-6326	34	7	between	between	ADP
ejpam-6326	34	8	the	the	DET
ejpam-6326	34	9	friendly	friendly	ADJ
ejpam-6326	34	10	domination	domination	NOUN
ejpam-6326	34	11	number	number	NOUN
ejpam-6326	34	12	and	and	CCONJ
ejpam-6326	34	13	the	the	DET
ejpam-6326	34	14	domination	domination	NOUN
ejpam-6326	34	15	number	number	NOUN
ejpam-6326	34	16	can	can	AUX
ejpam-6326	34	17	be	be	AUX
ejpam-6326	34	18	made	make	VERB
ejpam-6326	34	19	as	as	ADV
ejpam-6326	34	20	large	large	ADJ
ejpam-6326	34	21	as	as	SCONJ
ejpam-6326	34	22	desired	desire	VERB
ejpam-6326	34	23	by	by	ADP
ejpam-6326	34	24	constructing	construct	VERB
ejpam-6326	34	25	appropriate	appropriate	ADJ
ejpam-6326	34	26	graphs	graph	NOUN
ejpam-6326	34	27	.	.	PUNCT
ejpam-6326	35	1	furthermore	furthermore	ADV
ejpam-6326	35	2	,	,	PUNCT
ejpam-6326	35	3	we	we	PRON
ejpam-6326	35	4	find	find	VERB
ejpam-6326	35	5	exact	exact	ADJ
ejpam-6326	35	6	formulas	formula	NOUN
ejpam-6326	35	7	for	for	ADP
ejpam-6326	35	8	the	the	DET
ejpam-6326	35	9	friendly	friendly	ADJ
ejpam-6326	35	10	domination	domination	NOUN
ejpam-6326	35	11	number	number	NOUN
ejpam-6326	35	12	in	in	ADP
ejpam-6326	35	13	paths	path	NOUN
ejpam-6326	35	14	,	,	PUNCT
ejpam-6326	35	15	cycles	cycle	NOUN
ejpam-6326	35	16	,	,	PUNCT
ejpam-6326	35	17	and	and	CCONJ
ejpam-6326	35	18	graphs	graph	NOUN
ejpam-6326	35	19	with	with	ADP
ejpam-6326	35	20	maximum	maximum	ADJ
ejpam-6326	35	21	degree	degree	NOUN
ejpam-6326	35	22	two	two	NUM
ejpam-6326	35	23	.	.	PUNCT
ejpam-6326	36	1	in	in	ADP
ejpam-6326	36	2	addition	addition	NOUN
ejpam-6326	36	3	,	,	PUNCT
ejpam-6326	36	4	we	we	PRON
ejpam-6326	36	5	establish	establish	VERB
ejpam-6326	36	6	sharp	sharp	ADJ
ejpam-6326	36	7	nordhaus	nordhaus	NOUN
ejpam-6326	36	8	–	–	PUNCT
ejpam-6326	36	9	gaddum	gaddum	NOUN
ejpam-6326	36	10	bounds	bound	VERB
ejpam-6326	36	11	for	for	ADP
ejpam-6326	36	12	the	the	DET
ejpam-6326	36	13	sum	sum	NOUN
ejpam-6326	36	14	and	and	CCONJ
ejpam-6326	36	15	product	product	NOUN
ejpam-6326	36	16	of	of	ADP
ejpam-6326	36	17	the	the	DET
ejpam-6326	36	18	friendly	friendly	ADJ
ejpam-6326	36	19	domination	domination	NOUN
ejpam-6326	36	20	number	number	NOUN
ejpam-6326	36	21	of	of	ADP
ejpam-6326	36	22	a	a	DET
ejpam-6326	36	23	graph	graph	NOUN
ejpam-6326	36	24	and	and	CCONJ
ejpam-6326	36	25	its	its	PRON
ejpam-6326	36	26	complement	complement	NOUN
ejpam-6326	36	27	.	.	PUNCT
ejpam-6326	37	1	finally	finally	ADV
ejpam-6326	37	2	,	,	PUNCT
ejpam-6326	37	3	we	we	PRON
ejpam-6326	37	4	characterize	characterize	VERB
ejpam-6326	37	5	friendly	friendly	ADJ
ejpam-6326	37	6	dominating	dominating	NOUN
ejpam-6326	37	7	sets	set	NOUN
ejpam-6326	37	8	in	in	ADP
ejpam-6326	37	9	the	the	DET
ejpam-6326	37	10	join	join	NOUN
ejpam-6326	37	11	and	and	CCONJ
ejpam-6326	37	12	corona	corona	NOUN
ejpam-6326	37	13	of	of	ADP
ejpam-6326	37	14	graphs	graph	NOUN
ejpam-6326	37	15	.	.	PUNCT
ejpam-6326	38	1	2	2	X
ejpam-6326	38	2	.	.	X
ejpam-6326	38	3	terminology	terminology	NOUN
ejpam-6326	38	4	and	and	CCONJ
ejpam-6326	38	5	notation	notation	NOUN
ejpam-6326	38	6	given	give	VERB
ejpam-6326	38	7	a	a	DET
ejpam-6326	38	8	graph	graph	NOUN
ejpam-6326	38	9	g	g	NOUN
ejpam-6326	38	10	=	=	PUNCT
ejpam-6326	38	11	(	(	PUNCT
ejpam-6326	38	12	v	v	NOUN
ejpam-6326	38	13	(	(	PUNCT
ejpam-6326	38	14	g	g	NOUN
ejpam-6326	38	15	)	)	PUNCT
ejpam-6326	38	16	,	,	PUNCT
ejpam-6326	38	17	e(g	e(g	PROPN
ejpam-6326	38	18	)	)	PUNCT
ejpam-6326	38	19	)	)	PUNCT
ejpam-6326	38	20	,	,	PUNCT
ejpam-6326	38	21	the	the	DET
ejpam-6326	38	22	open	open	ADJ
ejpam-6326	38	23	neighborhood	neighborhood	NOUN
ejpam-6326	38	24	of	of	ADP
ejpam-6326	38	25	a	a	DET
ejpam-6326	38	26	vertex	vertex	NOUN
ejpam-6326	38	27	v	v	ADP
ejpam-6326	38	28	∈	∈	PROPN
ejpam-6326	38	29	v	v	NOUN
ejpam-6326	38	30	,	,	PUNCT
ejpam-6326	38	31	denoted	denote	VERB
ejpam-6326	38	32	n(v	n(v	PROPN
ejpam-6326	38	33	)	)	PUNCT
ejpam-6326	38	34	,	,	PUNCT
ejpam-6326	38	35	is	be	AUX
ejpam-6326	38	36	the	the	DET
ejpam-6326	38	37	set	set	NOUN
ejpam-6326	38	38	of	of	ADP
ejpam-6326	38	39	all	all	DET
ejpam-6326	38	40	vertices	vertex	NOUN
ejpam-6326	38	41	adjacent	adjacent	ADJ
ejpam-6326	38	42	to	to	ADP
ejpam-6326	38	43	v	v	NOUN
ejpam-6326	38	44	,	,	PUNCT
ejpam-6326	38	45	that	that	ADV
ejpam-6326	38	46	is	is	ADV
ejpam-6326	38	47	,	,	PUNCT
ejpam-6326	38	48	n(v	n(v	PROPN
ejpam-6326	38	49	)	)	PUNCT
ejpam-6326	39	1	=	=	PRON
ejpam-6326	39	2	{	{	PUNCT
ejpam-6326	39	3	u	u	NOUN
ejpam-6326	39	4	∈	∈	PROPN
ejpam-6326	39	5	v	v	ADP
ejpam-6326	39	6	|	|	ADV
ejpam-6326	39	7	{	{	PUNCT
ejpam-6326	39	8	u	u	NOUN
ejpam-6326	39	9	,	,	PUNCT
ejpam-6326	39	10	v	v	NOUN
ejpam-6326	39	11	}	}	PUNCT
ejpam-6326	39	12	∈	∈	PROPN
ejpam-6326	39	13	e(g	e(g	PROPN
ejpam-6326	39	14	)	)	PUNCT
ejpam-6326	39	15	}	}	PUNCT
ejpam-6326	39	16	.	.	PUNCT
ejpam-6326	40	1	its	its	PRON
ejpam-6326	40	2	closed	closed	ADJ
ejpam-6326	40	3	neighborhood	neighborhood	NOUN
ejpam-6326	40	4	,	,	PUNCT
ejpam-6326	40	5	denoted	denote	VERB
ejpam-6326	40	6	n	n	PROPN
ejpam-6326	40	7	[	[	X
ejpam-6326	40	8	v	v	NOUN
ejpam-6326	40	9	]	]	X
ejpam-6326	40	10	,	,	PUNCT
ejpam-6326	40	11	is	be	AUX
ejpam-6326	40	12	obtained	obtain	VERB
ejpam-6326	40	13	by	by	ADP
ejpam-6326	40	14	adding	add	VERB
ejpam-6326	40	15	the	the	DET
ejpam-6326	40	16	vertex	vertex	NOUN
ejpam-6326	40	17	itself	itself	PRON
ejpam-6326	40	18	:	:	PUNCT
ejpam-6326	40	19	n	n	CCONJ
ejpam-6326	41	1	[	[	X
ejpam-6326	41	2	v	v	X
ejpam-6326	41	3	]	]	X
ejpam-6326	41	4	=	=	PUNCT
ejpam-6326	41	5	n(v	n(v	PROPN
ejpam-6326	41	6	)	)	PUNCT
ejpam-6326	41	7	∪	∪	NOUN
ejpam-6326	41	8	{	{	PUNCT
ejpam-6326	41	9	v	v	NOUN
ejpam-6326	41	10	}	}	PUNCT
ejpam-6326	41	11	.	.	PUNCT
ejpam-6326	42	1	more	more	ADV
ejpam-6326	42	2	generally	generally	ADV
ejpam-6326	42	3	,	,	PUNCT
ejpam-6326	42	4	for	for	ADP
ejpam-6326	42	5	any	any	DET
ejpam-6326	42	6	subset	subset	NOUN
ejpam-6326	42	7	s	s	VERB
ejpam-6326	42	8	⊆	⊆	NUM
ejpam-6326	42	9	v	v	NOUN
ejpam-6326	42	10	,	,	PUNCT
ejpam-6326	42	11	the	the	DET
ejpam-6326	42	12	open	open	ADJ
ejpam-6326	42	13	neighborhood	neighborhood	NOUN
ejpam-6326	42	14	of	of	ADP
ejpam-6326	42	15	s	s	NOUN
ejpam-6326	42	16	is	be	AUX
ejpam-6326	42	17	the	the	DET
ejpam-6326	42	18	union	union	NOUN
ejpam-6326	42	19	of	of	ADP
ejpam-6326	42	20	the	the	DET
ejpam-6326	42	21	open	open	ADJ
ejpam-6326	42	22	neighborhoods	neighborhood	NOUN
ejpam-6326	42	23	of	of	ADP
ejpam-6326	42	24	its	its	PRON
ejpam-6326	42	25	elements	element	NOUN
ejpam-6326	42	26	,	,	PUNCT
ejpam-6326	42	27	n(s	n(s	PROPN
ejpam-6326	42	28	)	)	PUNCT
ejpam-6326	42	29	=	=	SYM
ejpam-6326	42	30	⋃	⋃	ADP
ejpam-6326	42	31	v∈s	v∈s	ADJ
ejpam-6326	42	32	n(v	n(v	PROPN
ejpam-6326	42	33	)	)	PUNCT
ejpam-6326	42	34	,	,	PUNCT
ejpam-6326	42	35	and	and	CCONJ
ejpam-6326	42	36	the	the	DET
ejpam-6326	42	37	closed	closed	ADJ
ejpam-6326	42	38	neighborhood	neighborhood	NOUN
ejpam-6326	42	39	of	of	ADP
ejpam-6326	42	40	s	s	NOUN
ejpam-6326	42	41	is	be	AUX
ejpam-6326	42	42	the	the	DET
ejpam-6326	42	43	union	union	NOUN
ejpam-6326	42	44	of	of	ADP
ejpam-6326	42	45	their	their	PRON
ejpam-6326	42	46	closed	closed	ADJ
ejpam-6326	42	47	neighborhoods	neighborhood	NOUN
ejpam-6326	42	48	(	(	PUNCT
ejpam-6326	42	49	equivalently	equivalently	ADV
ejpam-6326	42	50	n(s	n(s	PROPN
ejpam-6326	42	51	)	)	PUNCT
ejpam-6326	42	52	∪	∪	ADP
ejpam-6326	42	53	s	s	NOUN
ejpam-6326	42	54	)	)	PUNCT
ejpam-6326	42	55	,	,	PUNCT
ejpam-6326	42	56	n	n	CCONJ
ejpam-6326	43	1	[	[	X
ejpam-6326	43	2	s	s	X
ejpam-6326	43	3	]	]	X
ejpam-6326	43	4	=	=	SYM
ejpam-6326	43	5	⋃	⋃	NOUN
ejpam-6326	43	6	v∈s	v∈s	ADJ
ejpam-6326	43	7	n	n	CCONJ
ejpam-6326	43	8	[	[	X
ejpam-6326	43	9	v	v	X
ejpam-6326	43	10	]	]	X
ejpam-6326	43	11	=	=	PUNCT
ejpam-6326	43	12	n(s	n(s	PROPN
ejpam-6326	43	13	)	)	PUNCT
ejpam-6326	43	14	∪	∪	ADP
ejpam-6326	43	15	s.	s.	PROPN
ejpam-6326	43	16	the	the	DET
ejpam-6326	43	17	complement	complement	NOUN
ejpam-6326	43	18	of	of	ADP
ejpam-6326	43	19	g	g	NOUN
ejpam-6326	43	20	,	,	PUNCT
ejpam-6326	43	21	denoted	denote	VERB
ejpam-6326	43	22	g	g	NOUN
ejpam-6326	43	23	,	,	PUNCT
ejpam-6326	43	24	is	be	AUX
ejpam-6326	43	25	the	the	DET
ejpam-6326	43	26	simple	simple	ADJ
ejpam-6326	43	27	graph	graph	NOUN
ejpam-6326	43	28	on	on	ADP
ejpam-6326	43	29	the	the	DET
ejpam-6326	43	30	same	same	ADJ
ejpam-6326	43	31	vertex	vertex	NOUN
ejpam-6326	43	32	set	set	VERB
ejpam-6326	43	33	v	v	ADP
ejpam-6326	43	34	whose	whose	DET
ejpam-6326	43	35	edge	edge	NOUN
ejpam-6326	43	36	set	set	NOUN
ejpam-6326	43	37	is	be	AUX
ejpam-6326	43	38	e(g	e(g	NOUN
ejpam-6326	43	39	)	)	PUNCT
ejpam-6326	44	1	=	=	PRON
ejpam-6326	44	2	{	{	PUNCT
ejpam-6326	44	3	{	{	PUNCT
ejpam-6326	44	4	u	u	NOUN
ejpam-6326	44	5	,	,	PUNCT
ejpam-6326	44	6	v	v	NOUN
ejpam-6326	44	7	}	}	PUNCT
ejpam-6326	44	8	⊆	⊆	NUM
ejpam-6326	44	9	v	v	NOUN
ejpam-6326	44	10	:	:	PUNCT
ejpam-6326	44	11	u	u	PROPN
ejpam-6326	44	12	̸=	̸=	PROPN
ejpam-6326	44	13	v	v	NOUN
ejpam-6326	44	14	and	and	CCONJ
ejpam-6326	44	15	{	{	PUNCT
ejpam-6326	44	16	u	u	NOUN
ejpam-6326	44	17	,	,	PUNCT
ejpam-6326	44	18	v	v	NOUN
ejpam-6326	44	19	}	}	PUNCT
ejpam-6326	44	20	/∈	/∈	PUNCT
ejpam-6326	44	21	e(g	e(g	NOUN
ejpam-6326	44	22	)	)	PUNCT
ejpam-6326	44	23	}	}	PUNCT
ejpam-6326	44	24	.	.	PUNCT
ejpam-6326	45	1	the	the	DET
ejpam-6326	45	2	degree	degree	NOUN
ejpam-6326	45	3	of	of	ADP
ejpam-6326	45	4	a	a	DET
ejpam-6326	45	5	vertex	vertex	NOUN
ejpam-6326	45	6	v	v	ADP
ejpam-6326	45	7	∈	∈	PROPN
ejpam-6326	45	8	v	v	NOUN
ejpam-6326	45	9	,	,	PUNCT
ejpam-6326	45	10	denoted	denote	VERB
ejpam-6326	45	11	deg(v	deg(v	NOUN
ejpam-6326	45	12	)	)	PUNCT
ejpam-6326	45	13	,	,	PUNCT
ejpam-6326	45	14	is	be	AUX
ejpam-6326	45	15	defined	define	VERB
ejpam-6326	45	16	to	to	PART
ejpam-6326	45	17	be	be	AUX
ejpam-6326	45	18	the	the	DET
ejpam-6326	45	19	number	number	NOUN
ejpam-6326	45	20	of	of	ADP
ejpam-6326	45	21	vertices	vertex	NOUN
ejpam-6326	45	22	adjacent	adjacent	ADJ
ejpam-6326	45	23	to	to	ADP
ejpam-6326	45	24	v.	v.	ADP
ejpam-6326	45	25	equivalently	equivalently	ADV
ejpam-6326	45	26	,	,	PUNCT
ejpam-6326	45	27	deg(v	deg(v	PROPN
ejpam-6326	45	28	)	)	PUNCT
ejpam-6326	45	29	=	=	SYM
ejpam-6326	45	30	∣∣n(v	∣∣n(v	PROPN
ejpam-6326	45	31	)	)	PUNCT
ejpam-6326	45	32	∣∣	∣∣	ADJ
ejpam-6326	45	33	,	,	PUNCT
ejpam-6326	45	34	n(v	n(v	NUM
ejpam-6326	45	35	)	)	PUNCT
ejpam-6326	46	1	=	=	PRON
ejpam-6326	46	2	{	{	PUNCT
ejpam-6326	46	3	u	u	NOUN
ejpam-6326	46	4	∈	∈	PROPN
ejpam-6326	46	5	v	v	ADP
ejpam-6326	46	6	|	|	ADV
ejpam-6326	46	7	{	{	PUNCT
ejpam-6326	46	8	u	u	NOUN
ejpam-6326	46	9	,	,	PUNCT
ejpam-6326	46	10	v	v	NOUN
ejpam-6326	46	11	}	}	PUNCT
ejpam-6326	46	12	∈	∈	PROPN
ejpam-6326	46	13	e(g	e(g	PROPN
ejpam-6326	46	14	)	)	PUNCT
ejpam-6326	46	15	}	}	PUNCT
ejpam-6326	46	16	.	.	PUNCT
ejpam-6326	47	1	a	a	DET
ejpam-6326	47	2	vertex	vertex	NOUN
ejpam-6326	47	3	v	v	NOUN
ejpam-6326	47	4	for	for	ADP
ejpam-6326	47	5	which	which	PRON
ejpam-6326	47	6	deg(v	deg(v	PROPN
ejpam-6326	47	7	)	)	PUNCT
ejpam-6326	47	8	=	=	SYM
ejpam-6326	47	9	1	1	NUM
ejpam-6326	47	10	is	be	AUX
ejpam-6326	47	11	called	call	VERB
ejpam-6326	47	12	a	a	DET
ejpam-6326	47	13	leaf	leaf	NOUN
ejpam-6326	47	14	.	.	PUNCT
ejpam-6326	48	1	the	the	DET
ejpam-6326	48	2	maximum	maximum	ADJ
ejpam-6326	48	3	degree	degree	NOUN
ejpam-6326	48	4	of	of	ADP
ejpam-6326	48	5	g	g	PROPN
ejpam-6326	48	6	is	be	AUX
ejpam-6326	48	7	∆(g	∆(g	NOUN
ejpam-6326	48	8	)	)	PUNCT
ejpam-6326	48	9	=	=	SYM
ejpam-6326	49	1	maxv∈v	maxv∈v	PROPN
ejpam-6326	49	2	deg(v	deg(v	PROPN
ejpam-6326	49	3	)	)	PUNCT
ejpam-6326	49	4	,	,	PUNCT
ejpam-6326	49	5	and	and	CCONJ
ejpam-6326	49	6	the	the	DET
ejpam-6326	49	7	minimum	minimum	NOUN
ejpam-6326	49	8	degree	degree	NOUN
ejpam-6326	49	9	of	of	ADP
ejpam-6326	49	10	g	g	PROPN
ejpam-6326	49	11	is	be	AUX
ejpam-6326	49	12	δ(g	δ(g	ADV
ejpam-6326	49	13	)	)	PUNCT
ejpam-6326	49	14	=	=	SYM
ejpam-6326	49	15	minv∈v	minv∈v	NOUN
ejpam-6326	49	16	deg(v	deg(v	PROPN
ejpam-6326	49	17	)	)	PUNCT
ejpam-6326	49	18	.	.	PUNCT
ejpam-6326	50	1	for	for	ADP
ejpam-6326	50	2	any	any	DET
ejpam-6326	50	3	subset	subset	NOUN
ejpam-6326	50	4	s	s	VERB
ejpam-6326	50	5	⊆	⊆	NUM
ejpam-6326	50	6	v	v	NOUN
ejpam-6326	50	7	,	,	PUNCT
ejpam-6326	50	8	the	the	DET
ejpam-6326	50	9	degree	degree	NOUN
ejpam-6326	50	10	of	of	ADP
ejpam-6326	50	11	v	v	NOUN
ejpam-6326	50	12	restricted	restrict	VERB
ejpam-6326	50	13	to	to	ADP
ejpam-6326	50	14	s	s	PROPN
ejpam-6326	50	15	is	be	AUX
ejpam-6326	50	16	degs(v	degs(v	NOUN
ejpam-6326	50	17	)	)	PUNCT
ejpam-6326	50	18	=	=	SYM
ejpam-6326	50	19	∣∣n(v	∣∣n(v	PROPN
ejpam-6326	50	20	)	)	PUNCT
ejpam-6326	50	21	∩	∩	NOUN
ejpam-6326	50	22	s	s	PART
ejpam-6326	50	23	∣∣.	∣∣.	PROPN
ejpam-6326	50	24	a	a	DET
ejpam-6326	50	25	subset	subset	NOUN
ejpam-6326	50	26	d	d	NOUN
ejpam-6326	50	27	⊆	⊆	NUM
ejpam-6326	50	28	v	v	NOUN
ejpam-6326	50	29	is	be	AUX
ejpam-6326	50	30	called	call	VERB
ejpam-6326	50	31	a	a	DET
ejpam-6326	50	32	dominating	dominating	NOUN
ejpam-6326	50	33	set	set	NOUN
ejpam-6326	50	34	if	if	SCONJ
ejpam-6326	50	35	every	every	DET
ejpam-6326	50	36	vertex	vertex	NOUN
ejpam-6326	50	37	not	not	PART
ejpam-6326	50	38	in	in	ADP
ejpam-6326	50	39	d	d	PROPN
ejpam-6326	50	40	is	be	AUX
ejpam-6326	50	41	adjacent	adjacent	ADJ
ejpam-6326	50	42	to	to	ADP
ejpam-6326	50	43	at	at	ADV
ejpam-6326	50	44	least	least	ADV
ejpam-6326	50	45	one	one	NUM
ejpam-6326	50	46	vertex	vertex	NOUN
ejpam-6326	50	47	in	in	ADP
ejpam-6326	50	48	d.	d.	PROPN
ejpam-6326	50	49	equivalently	equivalently	PROPN
ejpam-6326	50	50	,	,	PUNCT
ejpam-6326	50	51	the	the	DET
ejpam-6326	50	52	closed	closed	ADJ
ejpam-6326	50	53	neighborhood	neighborhood	NOUN
ejpam-6326	50	54	of	of	ADP
ejpam-6326	50	55	d	d	NOUN
ejpam-6326	50	56	,	,	PUNCT
ejpam-6326	50	57	defined	define	VERB
ejpam-6326	50	58	by	by	ADP
ejpam-6326	50	59	n	n	PRON
ejpam-6326	50	60	[	[	X
ejpam-6326	50	61	d	d	X
ejpam-6326	50	62	]	]	X
ejpam-6326	50	63	=	=	SYM
ejpam-6326	50	64	⋃	⋃	NOUN
ejpam-6326	50	65	v∈d	v∈d	NOUN
ejpam-6326	50	66	(	(	PUNCT
ejpam-6326	50	67	{	{	PUNCT
ejpam-6326	50	68	v	v	NOUN
ejpam-6326	50	69	}	}	PUNCT
ejpam-6326	50	70	∪n(v	∪n(v	PROPN
ejpam-6326	50	71	)	)	PUNCT
ejpam-6326	50	72	)	)	PUNCT
ejpam-6326	50	73	,	,	PUNCT
ejpam-6326	50	74	satisfies	satisfy	VERB
ejpam-6326	50	75	n	n	X
ejpam-6326	50	76	[	[	X
ejpam-6326	50	77	d	d	X
ejpam-6326	50	78	]	]	X
ejpam-6326	50	79	=	=	SYM
ejpam-6326	50	80	v	v	NOUN
ejpam-6326	50	81	,	,	PUNCT
ejpam-6326	50	82	so	so	SCONJ
ejpam-6326	50	83	that	that	SCONJ
ejpam-6326	50	84	every	every	DET
ejpam-6326	50	85	vertex	vertex	NOUN
ejpam-6326	50	86	of	of	ADP
ejpam-6326	50	87	g	g	PROPN
ejpam-6326	50	88	lies	lie	VERB
ejpam-6326	50	89	either	either	CCONJ
ejpam-6326	50	90	in	in	ADP
ejpam-6326	50	91	d	d	NOUN
ejpam-6326	50	92	or	or	CCONJ
ejpam-6326	50	93	has	have	VERB
ejpam-6326	50	94	a	a	DET
ejpam-6326	50	95	neighbor	neighbor	NOUN
ejpam-6326	50	96	in	in	ADP
ejpam-6326	50	97	d.	d.	PROPN
ejpam-6326	50	98	the	the	DET
ejpam-6326	50	99	domination	domination	NOUN
ejpam-6326	50	100	number	number	NOUN
ejpam-6326	50	101	of	of	ADP
ejpam-6326	50	102	g	g	NOUN
ejpam-6326	50	103	,	,	PUNCT
ejpam-6326	50	104	denoted	denote	VERB
ejpam-6326	50	105	γ(g	γ(g	PROPN
ejpam-6326	50	106	)	)	PUNCT
ejpam-6326	50	107	,	,	PUNCT
ejpam-6326	50	108	is	be	AUX
ejpam-6326	50	109	the	the	DET
ejpam-6326	50	110	smallest	small	ADJ
ejpam-6326	50	111	size	size	NOUN
ejpam-6326	50	112	of	of	ADP
ejpam-6326	50	113	any	any	DET
ejpam-6326	50	114	dominating	dominating	NOUN
ejpam-6326	50	115	set	set	VERB
ejpam-6326	50	116	in	in	ADP
ejpam-6326	50	117	g	g	PROPN
ejpam-6326	50	118	:	:	PUNCT
ejpam-6326	50	119	γ(g	γ(g	PROPN
ejpam-6326	50	120	)	)	PUNCT
ejpam-6326	51	1	=	=	SYM
ejpam-6326	51	2	min	min	PROPN
ejpam-6326	51	3	{	{	PUNCT
ejpam-6326	51	4	|d|	|d|	NOUN
ejpam-6326	51	5	:	:	PUNCT
ejpam-6326	52	1	d	d	PROPN
ejpam-6326	52	2	⊆	⊆	NUM
ejpam-6326	52	3	v	v	NOUN
ejpam-6326	52	4	,	,	PUNCT
ejpam-6326	52	5	n	n	X
ejpam-6326	52	6	[	[	X
ejpam-6326	52	7	d	d	X
ejpam-6326	52	8	]	]	X
ejpam-6326	52	9	=	=	SYM
ejpam-6326	52	10	v	v	NOUN
ejpam-6326	52	11	}	}	PUNCT
ejpam-6326	52	12	.	.	PUNCT
ejpam-6326	53	1	[	[	X
ejpam-6326	53	2	1	1	NUM
ejpam-6326	53	3	,	,	PUNCT
ejpam-6326	53	4	15	15	NUM
ejpam-6326	53	5	]	]	PUNCT
ejpam-6326	53	6	a	a	DET
ejpam-6326	53	7	subset	subset	NOUN
ejpam-6326	53	8	f	f	PROPN
ejpam-6326	53	9	⊆	⊆	NUM
ejpam-6326	53	10	v	v	NOUN
ejpam-6326	53	11	is	be	AUX
ejpam-6326	53	12	called	call	VERB
ejpam-6326	53	13	a	a	DET
ejpam-6326	53	14	friendly	friendly	ADJ
ejpam-6326	53	15	set	set	NOUN
ejpam-6326	53	16	if	if	SCONJ
ejpam-6326	53	17	each	each	DET
ejpam-6326	53	18	vertex	vertex	NOUN
ejpam-6326	53	19	outside	outside	ADP
ejpam-6326	53	20	f	f	PROPN
ejpam-6326	53	21	has	have	VERB
ejpam-6326	53	22	at	at	ADV
ejpam-6326	53	23	least	least	ADJ
ejpam-6326	53	24	as	as	ADP
ejpam-6326	53	25	many	many	ADJ
ejpam-6326	53	26	neighbors	neighbor	NOUN
ejpam-6326	53	27	outside	outside	ADP
ejpam-6326	53	28	f	f	PROPN
ejpam-6326	53	29	as	as	ADP
ejpam-6326	53	30	inside	inside	ADP
ejpam-6326	53	31	f	f	PROPN
ejpam-6326	53	32	.	.	PUNCT
ejpam-6326	54	1	equivalently	equivalently	ADV
ejpam-6326	54	2	,	,	PUNCT
ejpam-6326	54	3	for	for	ADP
ejpam-6326	54	4	every	every	PRON
ejpam-6326	54	5	v	v	NUM
ejpam-6326	54	6	∈	∈	PROPN
ejpam-6326	54	7	v	v	NOUN
ejpam-6326	54	8	(	(	PUNCT
ejpam-6326	54	9	g	g	NOUN
ejpam-6326	54	10	)	)	PUNCT
ejpam-6326	54	11	\	\	PROPN
ejpam-6326	54	12	f	f	PROPN
ejpam-6326	54	13	,	,	PUNCT
ejpam-6326	54	14	degf	degf	PROPN
ejpam-6326	54	15	(	(	PUNCT
ejpam-6326	54	16	v	v	NOUN
ejpam-6326	54	17	)	)	PUNCT
ejpam-6326	54	18	≤	≤	NOUN
ejpam-6326	54	19	degv	degv	NOUN
ejpam-6326	54	20	(	(	PUNCT
ejpam-6326	54	21	g)\f	g)\f	NOUN
ejpam-6326	54	22	(	(	PUNCT
ejpam-6326	54	23	v	v	NOUN
ejpam-6326	54	24	)	)	PUNCT
ejpam-6326	54	25	.	.	PUNCT
ejpam-6326	55	1	this	this	DET
ejpam-6326	55	2	notion	notion	NOUN
ejpam-6326	55	3	was	be	AUX
ejpam-6326	55	4	introduced	introduce	VERB
ejpam-6326	55	5	by	by	ADP
ejpam-6326	55	6	haynes	hayne	NOUN
ejpam-6326	55	7	,	,	PUNCT
ejpam-6326	55	8	hedetniemi	hedetniemi	ADV
ejpam-6326	55	9	,	,	PUNCT
ejpam-6326	55	10	and	and	CCONJ
ejpam-6326	55	11	henning	henning	NOUN
ejpam-6326	55	12	in	in	ADP
ejpam-6326	55	13	their	their	PRON
ejpam-6326	55	14	i.	i.	NOUN
ejpam-6326	55	15	s.	s.	PROPN
ejpam-6326	55	16	cabahug	cabahug	PROPN
ejpam-6326	55	17	,	,	PUNCT
ejpam-6326	55	18	jr	jr	PROPN
ejpam-6326	55	19	.	.	PROPN
ejpam-6326	55	20	,	,	PUNCT
ejpam-6326	55	21	r.	r.	PROPN
ejpam-6326	55	22	g.	g.	PROPN
ejpam-6326	55	23	eballe	eballe	PROPN
ejpam-6326	55	24	,	,	PUNCT
ejpam-6326	55	25	r.	r.	PROPN
ejpam-6326	55	26	t.	t.	PROPN
ejpam-6326	55	27	fernandez	fernandez	PROPN
ejpam-6326	55	28	/	/	SYM
ejpam-6326	55	29	eur	eur	PROPN
ejpam-6326	55	30	.	.	PUNCT
ejpam-6326	56	1	j.	j.	PROPN
ejpam-6326	56	2	pure	pure	PROPN
ejpam-6326	56	3	appl	appl	PROPN
ejpam-6326	56	4	.	.	PROPN
ejpam-6326	56	5	math	math	PROPN
ejpam-6326	56	6	,	,	PUNCT
ejpam-6326	56	7	18	18	NUM
ejpam-6326	56	8	(	(	PUNCT
ejpam-6326	56	9	4	4	NUM
ejpam-6326	56	10	)	)	PUNCT
ejpam-6326	56	11	(	(	PUNCT
ejpam-6326	56	12	2025	2025	NUM
ejpam-6326	56	13	)	)	PUNCT
ejpam-6326	56	14	,	,	PUNCT
ejpam-6326	56	15	6326	6326	NUM
ejpam-6326	56	16	3	3	NUM
ejpam-6326	56	17	of	of	ADP
ejpam-6326	56	18	15	15	NUM
ejpam-6326	56	19	study	study	NOUN
ejpam-6326	56	20	of	of	ADP
ejpam-6326	56	21	structures	structure	NOUN
ejpam-6326	56	22	of	of	ADP
ejpam-6326	56	23	domination	domination	NOUN
ejpam-6326	56	24	in	in	ADP
ejpam-6326	56	25	graphs	graph	NOUN
ejpam-6326	56	26	.	.	PUNCT
ejpam-6326	57	1	[	[	X
ejpam-6326	57	2	4	4	NUM
ejpam-6326	57	3	]	]	PUNCT
ejpam-6326	57	4	.	.	PUNCT
ejpam-6326	58	1	standard	standard	ADJ
ejpam-6326	58	2	graph	graph	NOUN
ejpam-6326	58	3	-	-	PUNCT
ejpam-6326	58	4	theoretic	theoretic	NOUN
ejpam-6326	58	5	terminology	terminology	NOUN
ejpam-6326	58	6	not	not	PART
ejpam-6326	58	7	defined	define	VERB
ejpam-6326	58	8	herein	herein	NOUN
ejpam-6326	58	9	may	may	AUX
ejpam-6326	58	10	be	be	AUX
ejpam-6326	58	11	found	find	VERB
ejpam-6326	58	12	in	in	ADP
ejpam-6326	58	13	classic	classic	ADJ
ejpam-6326	58	14	texts	text	NOUN
ejpam-6326	58	15	such	such	ADJ
ejpam-6326	58	16	as	as	ADP
ejpam-6326	58	17	[	[	X
ejpam-6326	58	18	1	1	NUM
ejpam-6326	58	19	,	,	PUNCT
ejpam-6326	58	20	2	2	NUM
ejpam-6326	58	21	,	,	PUNCT
ejpam-6326	58	22	15	15	NUM
ejpam-6326	58	23	]	]	PUNCT
ejpam-6326	58	24	.	.	PUNCT
ejpam-6326	59	1	for	for	ADP
ejpam-6326	59	2	more	more	ADV
ejpam-6326	59	3	advanced	advanced	ADJ
ejpam-6326	59	4	notions	notion	NOUN
ejpam-6326	59	5	of	of	ADP
ejpam-6326	59	6	domination	domination	NOUN
ejpam-6326	59	7	and	and	CCONJ
ejpam-6326	59	8	friendly	friendly	ADJ
ejpam-6326	59	9	sets	set	NOUN
ejpam-6326	59	10	,	,	PUNCT
ejpam-6326	59	11	the	the	DET
ejpam-6326	59	12	reader	reader	NOUN
ejpam-6326	59	13	is	be	AUX
ejpam-6326	59	14	referred	refer	VERB
ejpam-6326	59	15	to	to	ADP
ejpam-6326	59	16	haynes	haynes	PROPN
ejpam-6326	59	17	et	et	NOUN
ejpam-6326	59	18	al	al	PROPN
ejpam-6326	59	19	.	.	PUNCT
ejpam-6326	60	1	[	[	X
ejpam-6326	60	2	4	4	NUM
ejpam-6326	60	3	]	]	PUNCT
ejpam-6326	60	4	.	.	PUNCT
ejpam-6326	61	1	the	the	DET
ejpam-6326	61	2	join	join	NOUN
ejpam-6326	61	3	of	of	ADP
ejpam-6326	61	4	two	two	NUM
ejpam-6326	61	5	graphs	graph	NOUN
ejpam-6326	61	6	g	g	NOUN
ejpam-6326	61	7	and	and	CCONJ
ejpam-6326	61	8	h	h	NOUN
ejpam-6326	61	9	,	,	PUNCT
ejpam-6326	61	10	written	write	VERB
ejpam-6326	61	11	g	g	PROPN
ejpam-6326	61	12	∨	∨	NUM
ejpam-6326	61	13	h	h	NOUN
ejpam-6326	61	14	,	,	PUNCT
ejpam-6326	61	15	is	be	AUX
ejpam-6326	61	16	formed	form	VERB
ejpam-6326	61	17	by	by	ADP
ejpam-6326	61	18	first	first	ADV
ejpam-6326	61	19	taking	take	VERB
ejpam-6326	61	20	disjoint	disjoint	NOUN
ejpam-6326	61	21	copies	copy	NOUN
ejpam-6326	61	22	of	of	ADP
ejpam-6326	61	23	g	g	PROPN
ejpam-6326	61	24	and	and	CCONJ
ejpam-6326	61	25	h	h	NOUN
ejpam-6326	61	26	(	(	PUNCT
ejpam-6326	61	27	so	so	ADV
ejpam-6326	61	28	no	no	DET
ejpam-6326	61	29	vertex	vertex	NOUN
ejpam-6326	61	30	is	be	AUX
ejpam-6326	61	31	shared	share	VERB
ejpam-6326	61	32	and	and	CCONJ
ejpam-6326	61	33	no	no	DET
ejpam-6326	61	34	edge	edge	NOUN
ejpam-6326	61	35	connects	connect	VERB
ejpam-6326	61	36	the	the	DET
ejpam-6326	61	37	two	two	NUM
ejpam-6326	61	38	graphs	graph	NOUN
ejpam-6326	61	39	)	)	PUNCT
ejpam-6326	61	40	,	,	PUNCT
ejpam-6326	61	41	and	and	CCONJ
ejpam-6326	61	42	then	then	ADV
ejpam-6326	61	43	adding	add	VERB
ejpam-6326	61	44	every	every	DET
ejpam-6326	61	45	possible	possible	ADJ
ejpam-6326	61	46	edge	edge	NOUN
ejpam-6326	61	47	between	between	ADP
ejpam-6326	61	48	each	each	DET
ejpam-6326	61	49	vertex	vertex	NOUN
ejpam-6326	61	50	of	of	ADP
ejpam-6326	61	51	g	g	PROPN
ejpam-6326	61	52	and	and	CCONJ
ejpam-6326	61	53	each	each	DET
ejpam-6326	61	54	vertex	vertex	NOUN
ejpam-6326	61	55	of	of	ADP
ejpam-6326	61	56	h.	h.	PROPN
ejpam-6326	61	57	thus	thus	ADV
ejpam-6326	61	58	the	the	DET
ejpam-6326	61	59	vertex	vertex	NOUN
ejpam-6326	61	60	set	set	NOUN
ejpam-6326	61	61	of	of	ADP
ejpam-6326	61	62	g∨h	g∨h	PROPN
ejpam-6326	61	63	is	be	AUX
ejpam-6326	61	64	v	v	NOUN
ejpam-6326	61	65	(	(	PUNCT
ejpam-6326	61	66	g)∪v	g)∪v	X
ejpam-6326	61	67	(	(	PUNCT
ejpam-6326	61	68	h	h	NOUN
ejpam-6326	61	69	)	)	PUNCT
ejpam-6326	61	70	,	,	PUNCT
ejpam-6326	61	71	its	its	PRON
ejpam-6326	61	72	edge	edge	NOUN
ejpam-6326	61	73	set	set	NOUN
ejpam-6326	61	74	contains	contain	VERB
ejpam-6326	61	75	e(g	e(g	PROPN
ejpam-6326	61	76	)	)	PUNCT
ejpam-6326	61	77	and	and	CCONJ
ejpam-6326	61	78	e(h	e(h	PROPN
ejpam-6326	61	79	)	)	PUNCT
ejpam-6326	61	80	,	,	PUNCT
ejpam-6326	61	81	and	and	CCONJ
ejpam-6326	61	82	in	in	ADP
ejpam-6326	61	83	addition	addition	NOUN
ejpam-6326	61	84	it	it	PRON
ejpam-6326	61	85	contains	contain	VERB
ejpam-6326	61	86	the	the	DET
ejpam-6326	61	87	set	set	NOUN
ejpam-6326	61	88	of	of	ADP
ejpam-6326	61	89	edges	edge	NOUN
ejpam-6326	61	90	{	{	PUNCT
ejpam-6326	61	91	{	{	PUNCT
ejpam-6326	61	92	u	u	NOUN
ejpam-6326	61	93	,	,	PUNCT
ejpam-6326	61	94	v	v	NOUN
ejpam-6326	61	95	}	}	PUNCT
ejpam-6326	61	96	|	|	ADV
ejpam-6326	61	97	u	u	NOUN
ejpam-6326	61	98	∈	∈	PROPN
ejpam-6326	61	99	v	v	NOUN
ejpam-6326	61	100	(	(	PUNCT
ejpam-6326	61	101	g	g	NOUN
ejpam-6326	61	102	)	)	PUNCT
ejpam-6326	61	103	,	,	PUNCT
ejpam-6326	61	104	v	v	X
ejpam-6326	61	105	∈	∈	PROPN
ejpam-6326	61	106	v	v	NOUN
ejpam-6326	61	107	(	(	PUNCT
ejpam-6326	61	108	h	h	NOUN
ejpam-6326	61	109	)	)	PUNCT
ejpam-6326	61	110	}	}	PUNCT
ejpam-6326	61	111	.	.	PUNCT
ejpam-6326	62	1	intuitively	intuitively	ADV
ejpam-6326	62	2	,	,	PUNCT
ejpam-6326	62	3	the	the	DET
ejpam-6326	62	4	join	join	NOUN
ejpam-6326	62	5	“	"	PUNCT
ejpam-6326	62	6	glues	glue	NOUN
ejpam-6326	62	7	”	"	PUNCT
ejpam-6326	62	8	the	the	DET
ejpam-6326	62	9	two	two	NUM
ejpam-6326	62	10	graphs	graph	NOUN
ejpam-6326	62	11	together	together	ADV
ejpam-6326	62	12	by	by	ADP
ejpam-6326	62	13	making	make	VERB
ejpam-6326	62	14	every	every	DET
ejpam-6326	62	15	vertex	vertex	NOUN
ejpam-6326	62	16	of	of	ADP
ejpam-6326	62	17	one	one	NUM
ejpam-6326	62	18	graph	graph	NOUN
ejpam-6326	62	19	adjacent	adjacent	ADJ
ejpam-6326	62	20	to	to	ADP
ejpam-6326	62	21	every	every	DET
ejpam-6326	62	22	vertex	vertex	NOUN
ejpam-6326	62	23	of	of	ADP
ejpam-6326	62	24	the	the	DET
ejpam-6326	62	25	other	other	ADJ
ejpam-6326	62	26	.	.	PUNCT
ejpam-6326	63	1	[	[	X
ejpam-6326	63	2	1	1	NUM
ejpam-6326	63	3	,	,	PUNCT
ejpam-6326	63	4	15	15	NUM
ejpam-6326	63	5	]	]	PUNCT
ejpam-6326	63	6	the	the	DET
ejpam-6326	63	7	corona	corona	NOUN
ejpam-6326	63	8	of	of	ADP
ejpam-6326	63	9	two	two	NUM
ejpam-6326	63	10	graphs	graph	NOUN
ejpam-6326	63	11	g	g	NOUN
ejpam-6326	63	12	and	and	CCONJ
ejpam-6326	63	13	h	h	NOUN
ejpam-6326	63	14	,	,	PUNCT
ejpam-6326	63	15	written	write	VERB
ejpam-6326	63	16	g	g	PROPN
ejpam-6326	63	17	◦	◦	NOUN
ejpam-6326	63	18	h	h	NOUN
ejpam-6326	63	19	,	,	PUNCT
ejpam-6326	63	20	is	be	AUX
ejpam-6326	63	21	obtained	obtain	VERB
ejpam-6326	63	22	by	by	ADP
ejpam-6326	63	23	taking	take	VERB
ejpam-6326	63	24	one	one	NUM
ejpam-6326	63	25	copy	copy	NOUN
ejpam-6326	63	26	of	of	ADP
ejpam-6326	63	27	g	g	PROPN
ejpam-6326	63	28	and	and	CCONJ
ejpam-6326	63	29	,	,	PUNCT
ejpam-6326	63	30	for	for	ADP
ejpam-6326	63	31	every	every	DET
ejpam-6326	63	32	vertex	vertex	NOUN
ejpam-6326	63	33	x	x	X
ejpam-6326	63	34	of	of	ADP
ejpam-6326	63	35	g	g	NOUN
ejpam-6326	63	36	,	,	PUNCT
ejpam-6326	63	37	adjoining	adjoin	VERB
ejpam-6326	63	38	an	an	DET
ejpam-6326	63	39	entire	entire	ADJ
ejpam-6326	63	40	disjoint	disjoint	NOUN
ejpam-6326	63	41	copy	copy	NOUN
ejpam-6326	63	42	hx	hx	PROPN
ejpam-6326	63	43	of	of	ADP
ejpam-6326	63	44	h	h	PROPN
ejpam-6326	63	45	and	and	CCONJ
ejpam-6326	63	46	then	then	ADV
ejpam-6326	63	47	adding	add	VERB
ejpam-6326	63	48	edges	edge	NOUN
ejpam-6326	63	49	from	from	ADP
ejpam-6326	63	50	x	x	PUNCT
ejpam-6326	63	51	to	to	ADP
ejpam-6326	63	52	every	every	DET
ejpam-6326	63	53	vertex	vertex	NOUN
ejpam-6326	63	54	of	of	ADP
ejpam-6326	63	55	that	that	DET
ejpam-6326	63	56	copy	copy	NOUN
ejpam-6326	63	57	.	.	PUNCT
ejpam-6326	64	1	thus	thus	ADV
ejpam-6326	64	2	each	each	DET
ejpam-6326	64	3	vertex	vertex	NOUN
ejpam-6326	64	4	of	of	ADP
ejpam-6326	64	5	g	g	PROPN
ejpam-6326	64	6	becomes	become	VERB
ejpam-6326	64	7	the	the	DET
ejpam-6326	64	8	centre	centre	NOUN
ejpam-6326	64	9	of	of	ADP
ejpam-6326	64	10	a	a	DET
ejpam-6326	64	11	“	"	PUNCT
ejpam-6326	64	12	star	star	NOUN
ejpam-6326	64	13	”	"	PUNCT
ejpam-6326	64	14	whose	whose	DET
ejpam-6326	64	15	leaves	leave	NOUN
ejpam-6326	64	16	induce	induce	VERB
ejpam-6326	64	17	a	a	DET
ejpam-6326	64	18	copy	copy	NOUN
ejpam-6326	64	19	of	of	ADP
ejpam-6326	64	20	h	h	NOUN
ejpam-6326	64	21	,	,	PUNCT
ejpam-6326	64	22	while	while	SCONJ
ejpam-6326	64	23	all	all	DET
ejpam-6326	64	24	internal	internal	ADJ
ejpam-6326	64	25	adjacencies	adjacency	NOUN
ejpam-6326	64	26	of	of	ADP
ejpam-6326	64	27	g	g	PROPN
ejpam-6326	64	28	and	and	CCONJ
ejpam-6326	64	29	of	of	ADP
ejpam-6326	64	30	each	each	DET
ejpam-6326	64	31	hx	hx	PROPN
ejpam-6326	64	32	are	be	AUX
ejpam-6326	64	33	retained	retain	VERB
ejpam-6326	64	34	.	.	PUNCT
ejpam-6326	65	1	[	[	X
ejpam-6326	65	2	16	16	NUM
ejpam-6326	65	3	]	]	PUNCT
ejpam-6326	65	4	.	.	PUNCT
ejpam-6326	65	5	given	give	VERB
ejpam-6326	65	6	any	any	DET
ejpam-6326	65	7	statement	statement	NOUN
ejpam-6326	65	8	or	or	CCONJ
ejpam-6326	65	9	property	property	NOUN
ejpam-6326	65	10	a	a	PRON
ejpam-6326	65	11	,	,	PUNCT
ejpam-6326	65	12	the	the	DET
ejpam-6326	65	13	indicator	indicator	NOUN
ejpam-6326	65	14	(	(	PUNCT
ejpam-6326	65	15	or	or	CCONJ
ejpam-6326	65	16	characteristic	characteristic	ADJ
ejpam-6326	65	17	)	)	PUNCT
ejpam-6326	65	18	function	function	NOUN
ejpam-6326	65	19	1{a	1{a	NUM
ejpam-6326	65	20	}	}	PUNCT
ejpam-6326	65	21	is	be	AUX
ejpam-6326	65	22	defined	define	VERB
ejpam-6326	65	23	to	to	PART
ejpam-6326	65	24	be	be	AUX
ejpam-6326	65	25	1	1	NUM
ejpam-6326	65	26	if	if	SCONJ
ejpam-6326	65	27	the	the	DET
ejpam-6326	65	28	property	property	NOUN
ejpam-6326	65	29	a	a	PRON
ejpam-6326	65	30	is	be	AUX
ejpam-6326	65	31	true	true	ADJ
ejpam-6326	65	32	,	,	PUNCT
ejpam-6326	65	33	and	and	CCONJ
ejpam-6326	65	34	0	0	NUM
ejpam-6326	65	35	if	if	SCONJ
ejpam-6326	65	36	the	the	DET
ejpam-6326	65	37	property	property	NOUN
ejpam-6326	65	38	a	a	PRON
ejpam-6326	65	39	is	be	AUX
ejpam-6326	65	40	false	false	ADJ
ejpam-6326	65	41	.	.	PUNCT
ejpam-6326	66	1	in	in	ADP
ejpam-6326	66	2	particular	particular	ADJ
ejpam-6326	66	3	,	,	PUNCT
ejpam-6326	66	4	for	for	ADP
ejpam-6326	66	5	a	a	DET
ejpam-6326	66	6	subset	subset	NOUN
ejpam-6326	66	7	s	s	NOUN
ejpam-6326	66	8	of	of	ADP
ejpam-6326	66	9	vertices	vertex	NOUN
ejpam-6326	66	10	,	,	PUNCT
ejpam-6326	66	11	the	the	DET
ejpam-6326	66	12	indicator	indicator	NOUN
ejpam-6326	66	13	1{x∈s	1{x∈s	NUM
ejpam-6326	66	14	}	}	PUNCT
ejpam-6326	66	15	equals	equal	VERB
ejpam-6326	66	16	1	1	NUM
ejpam-6326	66	17	when	when	SCONJ
ejpam-6326	66	18	the	the	DET
ejpam-6326	66	19	vertex	vertex	NOUN
ejpam-6326	66	20	x	x	PUNCT
ejpam-6326	66	21	belongs	belong	VERB
ejpam-6326	66	22	to	to	ADP
ejpam-6326	66	23	s	s	PRON
ejpam-6326	66	24	and	and	CCONJ
ejpam-6326	66	25	0	0	NUM
ejpam-6326	66	26	otherwise	otherwise	ADV
ejpam-6326	66	27	,	,	PUNCT
ejpam-6326	66	28	while	while	SCONJ
ejpam-6326	66	29	1{x/∈s	1{x/∈s	NUM
ejpam-6326	66	30	}	}	PUNCT
ejpam-6326	66	31	equals	equal	VERB
ejpam-6326	66	32	1	1	NUM
ejpam-6326	66	33	when	when	SCONJ
ejpam-6326	66	34	x	x	PRON
ejpam-6326	66	35	does	do	AUX
ejpam-6326	66	36	not	not	PART
ejpam-6326	66	37	belong	belong	VERB
ejpam-6326	66	38	to	to	ADP
ejpam-6326	66	39	s	s	PRON
ejpam-6326	66	40	and	and	CCONJ
ejpam-6326	66	41	0	0	NUM
ejpam-6326	66	42	otherwise	otherwise	ADV
ejpam-6326	66	43	.	.	PUNCT
ejpam-6326	67	1	for	for	ADP
ejpam-6326	67	2	any	any	DET
ejpam-6326	67	3	fixed	fix	VERB
ejpam-6326	67	4	vertex	vertex	NOUN
ejpam-6326	67	5	x	x	NOUN
ejpam-6326	67	6	,	,	PUNCT
ejpam-6326	67	7	exactly	exactly	ADV
ejpam-6326	67	8	one	one	NUM
ejpam-6326	67	9	of	of	ADP
ejpam-6326	67	10	these	these	DET
ejpam-6326	67	11	two	two	NUM
ejpam-6326	67	12	indicators	indicator	NOUN
ejpam-6326	67	13	equals	equal	VERB
ejpam-6326	67	14	1	1	NUM
ejpam-6326	67	15	and	and	CCONJ
ejpam-6326	67	16	the	the	DET
ejpam-6326	67	17	other	other	ADJ
ejpam-6326	67	18	equals	equal	VERB
ejpam-6326	67	19	0	0	NUM
ejpam-6326	67	20	.	.	PUNCT
ejpam-6326	68	1	this	this	DET
ejpam-6326	68	2	convenient	convenient	ADJ
ejpam-6326	68	3	notation	notation	NOUN
ejpam-6326	68	4	allows	allow	VERB
ejpam-6326	68	5	case	case	NOUN
ejpam-6326	68	6	distinctions	distinction	NOUN
ejpam-6326	68	7	to	to	PART
ejpam-6326	68	8	be	be	AUX
ejpam-6326	68	9	written	write	VERB
ejpam-6326	68	10	compactly	compactly	ADV
ejpam-6326	68	11	inside	inside	ADP
ejpam-6326	68	12	a	a	DET
ejpam-6326	68	13	single	single	ADJ
ejpam-6326	68	14	unified	unified	ADJ
ejpam-6326	68	15	formula	formula	NOUN
ejpam-6326	68	16	,	,	PUNCT
ejpam-6326	68	17	simplifying	simplify	VERB
ejpam-6326	68	18	many	many	ADJ
ejpam-6326	68	19	expressions	expression	NOUN
ejpam-6326	68	20	and	and	CCONJ
ejpam-6326	68	21	arguments	argument	NOUN
ejpam-6326	68	22	.	.	PUNCT
ejpam-6326	69	1	for	for	ADP
ejpam-6326	69	2	non	non	ADJ
ejpam-6326	69	3	–	–	ADJ
ejpam-6326	69	4	negative	negative	ADJ
ejpam-6326	69	5	real	real	ADJ
ejpam-6326	69	6	numbers	number	NOUN
ejpam-6326	69	7	a1	a1	NOUN
ejpam-6326	69	8	,	,	PUNCT
ejpam-6326	69	9	a2	a2	PROPN
ejpam-6326	69	10	,	,	PUNCT
ejpam-6326	69	11	.	.	PUNCT
ejpam-6326	69	12	.	.	PUNCT
ejpam-6326	69	13	.	.	PUNCT
ejpam-6326	70	1	,	,	PUNCT
ejpam-6326	70	2	an	an	DET
ejpam-6326	70	3	the	the	DET
ejpam-6326	70	4	arithmetic	arithmetic	ADJ
ejpam-6326	70	5	–	–	PUNCT
ejpam-6326	70	6	geometric	geometric	ADJ
ejpam-6326	70	7	mean	mean	NOUN
ejpam-6326	70	8	inequality	inequality	NOUN
ejpam-6326	70	9	asserts	assert	VERB
ejpam-6326	70	10	that	that	SCONJ
ejpam-6326	70	11	the	the	DET
ejpam-6326	70	12	arithmetic	arithmetic	ADJ
ejpam-6326	70	13	mean	mean	NOUN
ejpam-6326	70	14	a1+···+an	a1+···+an	NOUN
ejpam-6326	70	15	n	n	NUM
ejpam-6326	70	16	is	be	AUX
ejpam-6326	70	17	never	never	ADV
ejpam-6326	70	18	smaller	small	ADJ
ejpam-6326	70	19	than	than	ADP
ejpam-6326	70	20	the	the	DET
ejpam-6326	70	21	geometric	geometric	ADJ
ejpam-6326	70	22	mean	mean	NOUN
ejpam-6326	70	23	(	(	PUNCT
ejpam-6326	70	24	a1	a1	NOUN
ejpam-6326	70	25	·	·	PUNCT
ejpam-6326	70	26	·	·	PUNCT
ejpam-6326	70	27	·	·	PUNCT
ejpam-6326	70	28	an)1	an)1	PROPN
ejpam-6326	70	29	/	/	SYM
ejpam-6326	70	30	n	n	CCONJ
ejpam-6326	70	31	;	;	PUNCT
ejpam-6326	70	32	in	in	ADP
ejpam-6326	70	33	symbols	symbol	NOUN
ejpam-6326	70	34	,	,	PUNCT
ejpam-6326	70	35	a1	a1	NOUN
ejpam-6326	70	36	+	+	CCONJ
ejpam-6326	70	37	a2	a2	PROPN
ejpam-6326	70	38	+	+	CCONJ
ejpam-6326	70	39	·	·	PUNCT
ejpam-6326	70	40	·	·	PUNCT
ejpam-6326	70	41	·	·	PUNCT
ejpam-6326	71	1	+	+	CCONJ
ejpam-6326	71	2	an	an	DET
ejpam-6326	71	3	n	n	NUM
ejpam-6326	71	4	≥	≥	NOUN
ejpam-6326	71	5	(	(	PUNCT
ejpam-6326	71	6	a1a2	a1a2	X
ejpam-6326	71	7	·	·	PUNCT
ejpam-6326	71	8	·	·	PUNCT
ejpam-6326	71	9	·	·	PUNCT
ejpam-6326	71	10	an)1	an)1	PROPN
ejpam-6326	71	11	/	/	SYM
ejpam-6326	71	12	n	n	CCONJ
ejpam-6326	71	13	,	,	PUNCT
ejpam-6326	71	14	with	with	ADP
ejpam-6326	71	15	equality	equality	NOUN
ejpam-6326	71	16	precisely	precisely	ADV
ejpam-6326	71	17	when	when	SCONJ
ejpam-6326	71	18	all	all	DET
ejpam-6326	71	19	the	the	DET
ejpam-6326	71	20	ai	ai	VERB
ejpam-6326	71	21	are	be	AUX
ejpam-6326	71	22	equal	equal	ADJ
ejpam-6326	71	23	[	[	X
ejpam-6326	71	24	17	17	NUM
ejpam-6326	71	25	]	]	SYM
ejpam-6326	71	26	.	.	PUNCT
ejpam-6326	72	1	3	3	X
ejpam-6326	72	2	.	.	X
ejpam-6326	72	3	results	result	VERB
ejpam-6326	72	4	definition	definition	NOUN
ejpam-6326	72	5	1	1	NUM
ejpam-6326	72	6	.	.	PUNCT
ejpam-6326	73	1	let	let	VERB
ejpam-6326	73	2	g	g	PRON
ejpam-6326	73	3	be	be	AUX
ejpam-6326	73	4	a	a	DET
ejpam-6326	73	5	nontrivial	nontrivial	ADJ
ejpam-6326	73	6	connected	connect	VERB
ejpam-6326	73	7	graph	graph	NOUN
ejpam-6326	73	8	.	.	PUNCT
ejpam-6326	74	1	a	a	DET
ejpam-6326	74	2	nonempty	nonempty	ADV
ejpam-6326	74	3	set	set	VERB
ejpam-6326	74	4	f	f	PROPN
ejpam-6326	74	5	⊆	⊆	NUM
ejpam-6326	74	6	v	v	NOUN
ejpam-6326	74	7	(	(	PUNCT
ejpam-6326	74	8	g	g	NOUN
ejpam-6326	74	9	)	)	PUNCT
ejpam-6326	74	10	is	be	AUX
ejpam-6326	74	11	a	a	DET
ejpam-6326	74	12	friendly	friendly	ADJ
ejpam-6326	74	13	dominating	dominating	NOUN
ejpam-6326	74	14	set	set	NOUN
ejpam-6326	74	15	if	if	SCONJ
ejpam-6326	74	16	n	n	PROPN
ejpam-6326	74	17	[	[	X
ejpam-6326	74	18	f	f	X
ejpam-6326	74	19	]	]	X
ejpam-6326	74	20	=	=	SYM
ejpam-6326	74	21	v	v	X
ejpam-6326	74	22	(	(	PUNCT
ejpam-6326	74	23	g	g	NOUN
ejpam-6326	74	24	)	)	PUNCT
ejpam-6326	74	25	and	and	CCONJ
ejpam-6326	74	26	for	for	ADP
ejpam-6326	74	27	every	every	PRON
ejpam-6326	74	28	v	v	NUM
ejpam-6326	74	29	∈	∈	PROPN
ejpam-6326	74	30	v	v	NOUN
ejpam-6326	74	31	(	(	PUNCT
ejpam-6326	74	32	g	g	NOUN
ejpam-6326	74	33	)	)	PUNCT
ejpam-6326	74	34	\	\	PROPN
ejpam-6326	75	1	f	f	PROPN
ejpam-6326	75	2	,	,	PUNCT
ejpam-6326	75	3	degf	degf	PROPN
ejpam-6326	75	4	(	(	PUNCT
ejpam-6326	75	5	v	v	NOUN
ejpam-6326	75	6	)	)	PUNCT
ejpam-6326	75	7	≤	≤	NOUN
ejpam-6326	75	8	degv	degv	NOUN
ejpam-6326	75	9	(	(	PUNCT
ejpam-6326	75	10	g)\f	g)\f	NOUN
ejpam-6326	75	11	(	(	PUNCT
ejpam-6326	75	12	v	v	NOUN
ejpam-6326	75	13	)	)	PUNCT
ejpam-6326	75	14	.	.	PUNCT
ejpam-6326	76	1	the	the	DET
ejpam-6326	76	2	minimum	minimum	ADJ
ejpam-6326	76	3	cardinality	cardinality	NOUN
ejpam-6326	76	4	of	of	ADP
ejpam-6326	76	5	a	a	DET
ejpam-6326	76	6	friendly	friendly	ADJ
ejpam-6326	76	7	dominating	dominating	NOUN
ejpam-6326	76	8	set	set	NOUN
ejpam-6326	76	9	in	in	ADP
ejpam-6326	76	10	g	g	PROPN
ejpam-6326	76	11	is	be	AUX
ejpam-6326	76	12	called	call	VERB
ejpam-6326	76	13	the	the	DET
ejpam-6326	76	14	friendly	friendly	ADJ
ejpam-6326	76	15	domination	domination	NOUN
ejpam-6326	76	16	number	number	NOUN
ejpam-6326	76	17	of	of	ADP
ejpam-6326	76	18	g	g	NOUN
ejpam-6326	76	19	,	,	PUNCT
ejpam-6326	76	20	denoted	denote	VERB
ejpam-6326	76	21	by	by	ADP
ejpam-6326	76	22	γf	γf	PROPN
ejpam-6326	76	23	(	(	PUNCT
ejpam-6326	76	24	g	g	NOUN
ejpam-6326	76	25	)	)	PUNCT
ejpam-6326	76	26	.	.	PUNCT
ejpam-6326	77	1	a	a	DET
ejpam-6326	77	2	friendly	friendly	ADJ
ejpam-6326	77	3	dominating	dominating	NOUN
ejpam-6326	77	4	set	set	NOUN
ejpam-6326	77	5	whose	whose	DET
ejpam-6326	77	6	cardinality	cardinality	NOUN
ejpam-6326	77	7	is	be	AUX
ejpam-6326	77	8	the	the	DET
ejpam-6326	77	9	γf	γf	ADJ
ejpam-6326	77	10	(	(	PUNCT
ejpam-6326	77	11	g	g	NOUN
ejpam-6326	77	12	)	)	PUNCT
ejpam-6326	77	13	,	,	PUNCT
ejpam-6326	77	14	is	be	AUX
ejpam-6326	77	15	called	call	VERB
ejpam-6326	77	16	the	the	DET
ejpam-6326	77	17	γf	γf	NOUN
ejpam-6326	77	18	-set	-set	PUNCT
ejpam-6326	77	19	of	of	ADP
ejpam-6326	77	20	g.	g.	PROPN
ejpam-6326	77	21	example	example	NOUN
ejpam-6326	77	22	1	1	X
ejpam-6326	77	23	.	.	X
ejpam-6326	77	24	consider	consider	VERB
ejpam-6326	77	25	the	the	DET
ejpam-6326	77	26	path	path	NOUN
ejpam-6326	77	27	graph	graph	NOUN
ejpam-6326	77	28	p4	p4	ADJ
ejpam-6326	77	29	with	with	ADP
ejpam-6326	77	30	vertices	vertex	NOUN
ejpam-6326	77	31	v1	v1	NOUN
ejpam-6326	77	32	,	,	PUNCT
ejpam-6326	77	33	v2	v2	PROPN
ejpam-6326	77	34	,	,	PUNCT
ejpam-6326	77	35	v3	v3	PROPN
ejpam-6326	77	36	,	,	PUNCT
ejpam-6326	77	37	v4	v4	VERB
ejpam-6326	77	38	in	in	ADP
ejpam-6326	77	39	order	order	NOUN
ejpam-6326	77	40	,	,	PUNCT
ejpam-6326	77	41	and	and	CCONJ
ejpam-6326	77	42	let	let	VERB
ejpam-6326	77	43	f	f	PROPN
ejpam-6326	77	44	=	=	X
ejpam-6326	77	45	{	{	PUNCT
ejpam-6326	77	46	v1	v1	PROPN
ejpam-6326	77	47	,	,	PUNCT
ejpam-6326	77	48	v4	v4	NOUN
ejpam-6326	77	49	}	}	PUNCT
ejpam-6326	77	50	.	.	PUNCT
ejpam-6326	78	1	i.	i.	PROPN
ejpam-6326	78	2	s.	s.	PROPN
ejpam-6326	78	3	cabahug	cabahug	PROPN
ejpam-6326	78	4	,	,	PUNCT
ejpam-6326	78	5	jr	jr	PROPN
ejpam-6326	78	6	.	.	PROPN
ejpam-6326	78	7	,	,	PUNCT
ejpam-6326	78	8	r.	r.	PROPN
ejpam-6326	78	9	g.	g.	PROPN
ejpam-6326	78	10	eballe	eballe	PROPN
ejpam-6326	78	11	,	,	PUNCT
ejpam-6326	78	12	r.	r.	PROPN
ejpam-6326	78	13	t.	t.	PROPN
ejpam-6326	78	14	fernandez	fernandez	PROPN
ejpam-6326	78	15	/	/	SYM
ejpam-6326	78	16	eur	eur	PROPN
ejpam-6326	78	17	.	.	PUNCT
ejpam-6326	79	1	j.	j.	PROPN
ejpam-6326	79	2	pure	pure	PROPN
ejpam-6326	79	3	appl	appl	PROPN
ejpam-6326	79	4	.	.	PROPN
ejpam-6326	79	5	math	math	PROPN
ejpam-6326	79	6	,	,	PUNCT
ejpam-6326	79	7	18	18	NUM
ejpam-6326	79	8	(	(	PUNCT
ejpam-6326	79	9	4	4	NUM
ejpam-6326	79	10	)	)	PUNCT
ejpam-6326	79	11	(	(	PUNCT
ejpam-6326	79	12	2025	2025	NUM
ejpam-6326	79	13	)	)	PUNCT
ejpam-6326	79	14	,	,	PUNCT
ejpam-6326	79	15	6326	6326	NUM
ejpam-6326	79	16	4	4	NUM
ejpam-6326	79	17	of	of	ADP
ejpam-6326	79	18	15	15	NUM
ejpam-6326	79	19	v1	v1	NOUN
ejpam-6326	79	20	v2	v2	PROPN
ejpam-6326	79	21	v3	v3	PROPN
ejpam-6326	79	22	v4	v4	PROPN
ejpam-6326	79	23	vertex	vertex	NOUN
ejpam-6326	79	24	in	in	ADP
ejpam-6326	79	25	f	f	PROPN
ejpam-6326	79	26	vertex	vertex	NOUN
ejpam-6326	79	27	in	in	ADP
ejpam-6326	79	28	v	v	NOUN
ejpam-6326	79	29	\	\	X
ejpam-6326	79	30	f	f	PROPN
ejpam-6326	79	31	clearly	clearly	ADV
ejpam-6326	79	32	n	n	PRON
ejpam-6326	80	1	[	[	X
ejpam-6326	80	2	f	f	X
ejpam-6326	80	3	]	]	X
ejpam-6326	80	4	=	=	SYM
ejpam-6326	80	5	v	v	X
ejpam-6326	80	6	(	(	PUNCT
ejpam-6326	80	7	p4	p4	ADJ
ejpam-6326	80	8	)	)	PUNCT
ejpam-6326	80	9	,	,	PUNCT
ejpam-6326	80	10	since	since	SCONJ
ejpam-6326	80	11	v1	v1	NOUN
ejpam-6326	80	12	dominates	dominate	VERB
ejpam-6326	80	13	v2	v2	PROPN
ejpam-6326	80	14	and	and	CCONJ
ejpam-6326	80	15	v4	v4	NOUN
ejpam-6326	80	16	dominates	dominate	VERB
ejpam-6326	80	17	v3	v3	PROPN
ejpam-6326	80	18	.	.	PUNCT
ejpam-6326	81	1	for	for	ADP
ejpam-6326	81	2	every	every	DET
ejpam-6326	81	3	vertex	vertex	NOUN
ejpam-6326	81	4	outside	outside	ADP
ejpam-6326	81	5	f	f	PROPN
ejpam-6326	81	6	we	we	PRON
ejpam-6326	81	7	have	have	VERB
ejpam-6326	81	8	degf	degf	NOUN
ejpam-6326	81	9	(	(	PUNCT
ejpam-6326	81	10	v2	v2	PROPN
ejpam-6326	81	11	)	)	PUNCT
ejpam-6326	81	12	=	=	SYM
ejpam-6326	81	13	1	1	NUM
ejpam-6326	81	14	≤	≤	NOUN
ejpam-6326	81	15	degv	degv	NOUN
ejpam-6326	81	16	\f	\f	X
ejpam-6326	81	17	(	(	PUNCT
ejpam-6326	81	18	v2	v2	NOUN
ejpam-6326	81	19	)	)	PUNCT
ejpam-6326	81	20	=	=	SYM
ejpam-6326	82	1	1	1	NUM
ejpam-6326	82	2	,	,	PUNCT
ejpam-6326	82	3	degf	degf	NOUN
ejpam-6326	82	4	(	(	PUNCT
ejpam-6326	82	5	v3	v3	PROPN
ejpam-6326	82	6	)	)	PUNCT
ejpam-6326	82	7	=	=	SYM
ejpam-6326	82	8	1	1	NUM
ejpam-6326	82	9	≤	≤	NOUN
ejpam-6326	82	10	degv	degv	NOUN
ejpam-6326	82	11	\f	\f	X
ejpam-6326	82	12	(	(	PUNCT
ejpam-6326	82	13	v3	v3	PROPN
ejpam-6326	82	14	)	)	PUNCT
ejpam-6326	82	15	=	=	PUNCT
ejpam-6326	83	1	1	1	X
ejpam-6326	83	2	.	.	PUNCT
ejpam-6326	83	3	thus	thus	ADV
ejpam-6326	83	4	f	f	PROPN
ejpam-6326	83	5	is	be	AUX
ejpam-6326	83	6	a	a	DET
ejpam-6326	83	7	friendly	friendly	ADJ
ejpam-6326	83	8	dominating	dominating	NOUN
ejpam-6326	83	9	set	set	NOUN
ejpam-6326	83	10	of	of	ADP
ejpam-6326	83	11	p4	p4	NOUN
ejpam-6326	83	12	.	.	PUNCT
ejpam-6326	84	1	in	in	ADP
ejpam-6326	84	2	particular	particular	ADJ
ejpam-6326	84	3	,	,	PUNCT
ejpam-6326	84	4	γf	γf	PROPN
ejpam-6326	84	5	(	(	PUNCT
ejpam-6326	84	6	p4	p4	ADJ
ejpam-6326	84	7	)	)	PUNCT
ejpam-6326	84	8	=	=	SYM
ejpam-6326	84	9	2	2	NUM
ejpam-6326	84	10	.	.	NOUN
ejpam-6326	84	11	3.1	3.1	NUM
ejpam-6326	84	12	.	.	PUNCT
ejpam-6326	85	1	some	some	DET
ejpam-6326	85	2	realizations	realization	NOUN
ejpam-6326	85	3	and	and	CCONJ
ejpam-6326	85	4	the	the	DET
ejpam-6326	85	5	friendly	friendly	ADJ
ejpam-6326	85	6	dominating	dominating	NOUN
ejpam-6326	85	7	sets	set	NOUN
ejpam-6326	85	8	of	of	ADP
ejpam-6326	85	9	some	some	DET
ejpam-6326	85	10	known	know	VERB
ejpam-6326	85	11	graphs	graph	NOUN
ejpam-6326	85	12	given	give	VERB
ejpam-6326	85	13	nontrivial	nontrivial	ADJ
ejpam-6326	85	14	connected	connect	VERB
ejpam-6326	85	15	graph	graph	NOUN
ejpam-6326	85	16	g	g	PROPN
ejpam-6326	85	17	=	=	PUNCT
ejpam-6326	85	18	(	(	PUNCT
ejpam-6326	85	19	v	v	NOUN
ejpam-6326	85	20	(	(	PUNCT
ejpam-6326	85	21	g	g	NOUN
ejpam-6326	85	22	)	)	PUNCT
ejpam-6326	85	23	,	,	PUNCT
ejpam-6326	85	24	e(g	e(g	PROPN
ejpam-6326	85	25	)	)	PUNCT
ejpam-6326	85	26	)	)	PUNCT
ejpam-6326	85	27	.	.	PUNCT
ejpam-6326	86	1	every	every	DET
ejpam-6326	86	2	friendly	friendly	ADJ
ejpam-6326	86	3	dominating	dominating	NOUN
ejpam-6326	86	4	set	set	NOUN
ejpam-6326	86	5	of	of	ADP
ejpam-6326	86	6	g	g	PROPN
ejpam-6326	86	7	is	be	AUX
ejpam-6326	86	8	a	a	DET
ejpam-6326	86	9	dominating	dominating	NOUN
ejpam-6326	86	10	set	set	NOUN
ejpam-6326	86	11	of	of	ADP
ejpam-6326	86	12	g.	g.	PROPN
ejpam-6326	86	13	thus	thus	ADV
ejpam-6326	86	14	we	we	PRON
ejpam-6326	86	15	have	have	VERB
ejpam-6326	86	16	the	the	DET
ejpam-6326	86	17	following	follow	VERB
ejpam-6326	86	18	remark	remark	NOUN
ejpam-6326	86	19	:	:	PUNCT
ejpam-6326	86	20	remark	remark	NOUN
ejpam-6326	86	21	1	1	NUM
ejpam-6326	86	22	.	.	PUNCT
ejpam-6326	87	1	let	let	VERB
ejpam-6326	87	2	g	g	PRON
ejpam-6326	87	3	be	be	AUX
ejpam-6326	87	4	a	a	DET
ejpam-6326	87	5	graph	graph	NOUN
ejpam-6326	87	6	.	.	PUNCT
ejpam-6326	88	1	then	then	ADV
ejpam-6326	88	2	,	,	PUNCT
ejpam-6326	88	3	γ(g	γ(g	PROPN
ejpam-6326	88	4	)	)	PUNCT
ejpam-6326	88	5	≤	≤	PUNCT
ejpam-6326	89	1	γf	γf	NOUN
ejpam-6326	89	2	(	(	PUNCT
ejpam-6326	89	3	g	g	NOUN
ejpam-6326	89	4	)	)	PUNCT
ejpam-6326	89	5	.	.	PUNCT
ejpam-6326	90	1	let	let	VERB
ejpam-6326	90	2	f	f	PRON
ejpam-6326	90	3	be	be	AUX
ejpam-6326	90	4	a	a	DET
ejpam-6326	90	5	nonempty	nonempty	ADJ
ejpam-6326	90	6	subset	subset	NOUN
ejpam-6326	90	7	of	of	ADP
ejpam-6326	90	8	v	v	NOUN
ejpam-6326	90	9	(	(	PUNCT
ejpam-6326	90	10	g	g	NOUN
ejpam-6326	90	11	)	)	PUNCT
ejpam-6326	90	12	.	.	PUNCT
ejpam-6326	91	1	if	if	SCONJ
ejpam-6326	91	2	f	f	PROPN
ejpam-6326	91	3	=	=	SYM
ejpam-6326	91	4	v	v	PROPN
ejpam-6326	91	5	(	(	PUNCT
ejpam-6326	91	6	g	g	NOUN
ejpam-6326	91	7	)	)	PUNCT
ejpam-6326	91	8	,	,	PUNCT
ejpam-6326	91	9	then	then	ADV
ejpam-6326	91	10	f	f	PROPN
ejpam-6326	91	11	is	be	AUX
ejpam-6326	91	12	trivially	trivially	ADV
ejpam-6326	91	13	a	a	DET
ejpam-6326	91	14	dominating	dominating	NOUN
ejpam-6326	91	15	set	set	NOUN
ejpam-6326	91	16	because	because	SCONJ
ejpam-6326	91	17	it	it	PRON
ejpam-6326	91	18	contains	contain	VERB
ejpam-6326	91	19	every	every	DET
ejpam-6326	91	20	vertex	vertex	NOUN
ejpam-6326	91	21	of	of	ADP
ejpam-6326	91	22	g.	g.	PROPN
ejpam-6326	91	23	moreover	moreover	ADV
ejpam-6326	91	24	,	,	PUNCT
ejpam-6326	91	25	the	the	DET
ejpam-6326	91	26	friendly	friendly	ADJ
ejpam-6326	91	27	condition	condition	NOUN
ejpam-6326	91	28	only	only	ADV
ejpam-6326	91	29	needs	need	VERB
ejpam-6326	91	30	to	to	PART
ejpam-6326	91	31	be	be	AUX
ejpam-6326	91	32	verified	verify	VERB
ejpam-6326	91	33	for	for	ADP
ejpam-6326	91	34	vertices	vertex	NOUN
ejpam-6326	91	35	in	in	ADP
ejpam-6326	91	36	v	v	ADP
ejpam-6326	91	37	(	(	PUNCT
ejpam-6326	91	38	g	g	NOUN
ejpam-6326	91	39	)	)	PUNCT
ejpam-6326	91	40	\	\	PROPN
ejpam-6326	92	1	f	f	PROPN
ejpam-6326	92	2	.	.	PUNCT
ejpam-6326	93	1	since	since	SCONJ
ejpam-6326	93	2	v	v	NOUN
ejpam-6326	93	3	(	(	PUNCT
ejpam-6326	93	4	g	g	NOUN
ejpam-6326	93	5	)	)	PUNCT
ejpam-6326	93	6	\	\	NOUN
ejpam-6326	93	7	f	f	NOUN
ejpam-6326	93	8	=	=	PUNCT
ejpam-6326	93	9	∅	∅	NOUN
ejpam-6326	93	10	,	,	PUNCT
ejpam-6326	93	11	there	there	PRON
ejpam-6326	93	12	are	be	VERB
ejpam-6326	93	13	no	no	DET
ejpam-6326	93	14	vertices	vertex	NOUN
ejpam-6326	93	15	to	to	PART
ejpam-6326	93	16	check	check	VERB
ejpam-6326	93	17	the	the	DET
ejpam-6326	93	18	inequality	inequality	NOUN
ejpam-6326	93	19	,	,	PUNCT
ejpam-6326	93	20	and	and	CCONJ
ejpam-6326	93	21	hence	hence	ADV
ejpam-6326	93	22	the	the	DET
ejpam-6326	93	23	condition	condition	NOUN
ejpam-6326	93	24	is	be	AUX
ejpam-6326	93	25	satisfied	satisfied	ADJ
ejpam-6326	93	26	vacuously	vacuously	ADV
ejpam-6326	93	27	.	.	PUNCT
ejpam-6326	94	1	remark	remark	PROPN
ejpam-6326	94	2	2	2	NUM
ejpam-6326	94	3	.	.	PUNCT
ejpam-6326	95	1	let	let	VERB
ejpam-6326	95	2	g	g	PRON
ejpam-6326	95	3	be	be	AUX
ejpam-6326	95	4	a	a	DET
ejpam-6326	95	5	graph	graph	NOUN
ejpam-6326	95	6	.	.	PUNCT
ejpam-6326	96	1	then	then	ADV
ejpam-6326	96	2	the	the	DET
ejpam-6326	96	3	vertex	vertex	NOUN
ejpam-6326	96	4	set	set	NOUN
ejpam-6326	96	5	,	,	PUNCT
ejpam-6326	96	6	v	v	NOUN
ejpam-6326	96	7	(	(	PUNCT
ejpam-6326	96	8	g	g	NOUN
ejpam-6326	96	9	)	)	PUNCT
ejpam-6326	96	10	,	,	PUNCT
ejpam-6326	96	11	is	be	AUX
ejpam-6326	96	12	a	a	DET
ejpam-6326	96	13	friendly	friendly	ADJ
ejpam-6326	96	14	dominating	dominating	NOUN
ejpam-6326	96	15	set	set	NOUN
ejpam-6326	96	16	of	of	ADP
ejpam-6326	96	17	g.	g.	PROPN
ejpam-6326	96	18	theorem	theorem	VERB
ejpam-6326	96	19	1	1	X
ejpam-6326	96	20	.	.	PUNCT
ejpam-6326	97	1	let	let	VERB
ejpam-6326	97	2	g	g	PRON
ejpam-6326	97	3	be	be	AUX
ejpam-6326	97	4	a	a	DET
ejpam-6326	97	5	nontrivial	nontrivial	ADJ
ejpam-6326	97	6	connected	connect	VERB
ejpam-6326	97	7	graph	graph	NOUN
ejpam-6326	97	8	and	and	CCONJ
ejpam-6326	97	9	let	let	VERB
ejpam-6326	97	10	f	f	PROPN
ejpam-6326	97	11	⊆	⊆	NUM
ejpam-6326	97	12	v	v	NOUN
ejpam-6326	97	13	(	(	PUNCT
ejpam-6326	97	14	g	g	NOUN
ejpam-6326	97	15	)	)	PUNCT
ejpam-6326	97	16	be	be	AUX
ejpam-6326	97	17	a	a	DET
ejpam-6326	97	18	friendly	friendly	ADJ
ejpam-6326	97	19	dominating	dominating	NOUN
ejpam-6326	97	20	set	set	NOUN
ejpam-6326	97	21	of	of	ADP
ejpam-6326	97	22	g.	g.	PROPN
ejpam-6326	97	23	if	if	SCONJ
ejpam-6326	97	24	g	g	PROPN
ejpam-6326	97	25	contains	contain	VERB
ejpam-6326	97	26	any	any	DET
ejpam-6326	97	27	leaf	leaf	NOUN
ejpam-6326	97	28	vertices	vertex	NOUN
ejpam-6326	97	29	,	,	PUNCT
ejpam-6326	97	30	then	then	ADV
ejpam-6326	97	31	every	every	DET
ejpam-6326	97	32	leaf	leaf	NOUN
ejpam-6326	97	33	vertex	vertex	NOUN
ejpam-6326	97	34	of	of	ADP
ejpam-6326	97	35	g	g	PROPN
ejpam-6326	97	36	must	must	AUX
ejpam-6326	97	37	belong	belong	VERB
ejpam-6326	97	38	to	to	ADP
ejpam-6326	97	39	f	f	PROPN
ejpam-6326	97	40	.	.	PUNCT
ejpam-6326	98	1	proof	proof	NOUN
ejpam-6326	98	2	.	.	PUNCT
ejpam-6326	99	1	let	let	VERB
ejpam-6326	99	2	g	g	PRON
ejpam-6326	99	3	be	be	AUX
ejpam-6326	99	4	a	a	DET
ejpam-6326	99	5	nontrivial	nontrivial	ADJ
ejpam-6326	99	6	connected	connect	VERB
ejpam-6326	99	7	graph	graph	NOUN
ejpam-6326	99	8	and	and	CCONJ
ejpam-6326	99	9	let	let	VERB
ejpam-6326	99	10	f	f	PROPN
ejpam-6326	99	11	⊆	⊆	NUM
ejpam-6326	99	12	v	v	NOUN
ejpam-6326	99	13	(	(	PUNCT
ejpam-6326	99	14	g	g	NOUN
ejpam-6326	99	15	)	)	PUNCT
ejpam-6326	99	16	be	be	AUX
ejpam-6326	99	17	a	a	DET
ejpam-6326	99	18	friendly	friendly	ADJ
ejpam-6326	99	19	dominating	dominating	NOUN
ejpam-6326	99	20	set	set	NOUN
ejpam-6326	99	21	of	of	ADP
ejpam-6326	99	22	g.	g.	PROPN
ejpam-6326	99	23	suppose	suppose	VERB
ejpam-6326	99	24	that	that	SCONJ
ejpam-6326	99	25	there	there	PRON
ejpam-6326	99	26	exists	exist	VERB
ejpam-6326	99	27	a	a	DET
ejpam-6326	99	28	leaf	leaf	NOUN
ejpam-6326	99	29	vertex	vertex	NOUN
ejpam-6326	99	30	u	u	NOUN
ejpam-6326	99	31	∈	∈	PROPN
ejpam-6326	99	32	v	v	ADP
ejpam-6326	99	33	(	(	PUNCT
ejpam-6326	99	34	g	g	NOUN
ejpam-6326	99	35	)	)	PUNCT
ejpam-6326	99	36	such	such	ADJ
ejpam-6326	99	37	that	that	DET
ejpam-6326	99	38	u	u	NOUN
ejpam-6326	99	39	/∈	/∈	PROPN
ejpam-6326	99	40	f	f	PROPN
ejpam-6326	99	41	.	.	PUNCT
ejpam-6326	100	1	since	since	SCONJ
ejpam-6326	100	2	f	f	PROPN
ejpam-6326	100	3	is	be	AUX
ejpam-6326	100	4	a	a	DET
ejpam-6326	100	5	dominating	dominating	NOUN
ejpam-6326	100	6	set	set	NOUN
ejpam-6326	100	7	,	,	PUNCT
ejpam-6326	100	8	u	u	PRON
ejpam-6326	100	9	must	must	AUX
ejpam-6326	100	10	be	be	AUX
ejpam-6326	100	11	adjacent	adjacent	ADJ
ejpam-6326	100	12	to	to	ADP
ejpam-6326	100	13	some	some	DET
ejpam-6326	100	14	vertex	vertex	NOUN
ejpam-6326	100	15	in	in	ADP
ejpam-6326	100	16	f	f	PROPN
ejpam-6326	100	17	.	.	PUNCT
ejpam-6326	101	1	however	however	ADV
ejpam-6326	101	2	,	,	PUNCT
ejpam-6326	101	3	as	as	SCONJ
ejpam-6326	101	4	u	u	NOUN
ejpam-6326	101	5	is	be	AUX
ejpam-6326	101	6	a	a	DET
ejpam-6326	101	7	leaf	leaf	NOUN
ejpam-6326	101	8	,	,	PUNCT
ejpam-6326	101	9	it	it	PRON
ejpam-6326	101	10	has	have	VERB
ejpam-6326	101	11	exactly	exactly	ADV
ejpam-6326	101	12	one	one	NUM
ejpam-6326	101	13	neighbor	neighbor	NOUN
ejpam-6326	101	14	in	in	ADP
ejpam-6326	101	15	g	g	PROPN
ejpam-6326	101	16	,	,	PUNCT
ejpam-6326	101	17	say	say	VERB
ejpam-6326	101	18	w.	w.	PROPN
ejpam-6326	101	19	thus	thus	ADV
ejpam-6326	101	20	,	,	PUNCT
ejpam-6326	101	21	w	w	PROPN
ejpam-6326	101	22	∈	∈	PROPN
ejpam-6326	101	23	f	f	NOUN
ejpam-6326	101	24	and	and	CCONJ
ejpam-6326	101	25	consequently	consequently	ADV
ejpam-6326	101	26	degf	degf	PROPN
ejpam-6326	101	27	(	(	PUNCT
ejpam-6326	101	28	u	u	NOUN
ejpam-6326	101	29	)	)	PUNCT
ejpam-6326	101	30	=	=	SYM
ejpam-6326	102	1	1	1	X
ejpam-6326	102	2	.	.	PUNCT
ejpam-6326	103	1	on	on	ADP
ejpam-6326	103	2	the	the	DET
ejpam-6326	103	3	other	other	ADJ
ejpam-6326	103	4	hand	hand	NOUN
ejpam-6326	103	5	,	,	PUNCT
ejpam-6326	103	6	u	u	NOUN
ejpam-6326	103	7	has	have	VERB
ejpam-6326	103	8	no	no	DET
ejpam-6326	103	9	other	other	ADJ
ejpam-6326	103	10	neighbors	neighbor	NOUN
ejpam-6326	103	11	,	,	PUNCT
ejpam-6326	103	12	so	so	ADV
ejpam-6326	103	13	degv	degv	NOUN
ejpam-6326	103	14	(	(	PUNCT
ejpam-6326	103	15	g)\f	g)\f	NOUN
ejpam-6326	103	16	(	(	PUNCT
ejpam-6326	103	17	u	u	NOUN
ejpam-6326	103	18	)	)	PUNCT
ejpam-6326	103	19	=	=	SYM
ejpam-6326	104	1	0	0	X
ejpam-6326	104	2	.	.	PUNCT
ejpam-6326	105	1	this	this	DET
ejpam-6326	105	2	yields	yield	NOUN
ejpam-6326	105	3	to	to	ADP
ejpam-6326	105	4	1	1	NUM
ejpam-6326	105	5	=	=	SYM
ejpam-6326	105	6	degf	degf	X
ejpam-6326	105	7	(	(	PUNCT
ejpam-6326	105	8	u	u	NOUN
ejpam-6326	105	9	)	)	PUNCT
ejpam-6326	105	10	>	>	X
ejpam-6326	106	1	degv	degv	PROPN
ejpam-6326	106	2	(	(	PUNCT
ejpam-6326	106	3	g)\f	g)\f	NOUN
ejpam-6326	106	4	(	(	PUNCT
ejpam-6326	106	5	u	u	NOUN
ejpam-6326	106	6	)	)	PUNCT
ejpam-6326	106	7	=	=	SYM
ejpam-6326	106	8	0	0	NUM
ejpam-6326	106	9	,	,	PUNCT
ejpam-6326	106	10	which	which	PRON
ejpam-6326	106	11	is	be	AUX
ejpam-6326	106	12	a	a	DET
ejpam-6326	106	13	contradiction	contradiction	NOUN
ejpam-6326	106	14	.	.	PUNCT
ejpam-6326	107	1	hence	hence	ADV
ejpam-6326	107	2	,	,	PUNCT
ejpam-6326	107	3	it	it	PRON
ejpam-6326	107	4	follows	follow	VERB
ejpam-6326	107	5	that	that	SCONJ
ejpam-6326	107	6	every	every	DET
ejpam-6326	107	7	leaf	leaf	NOUN
ejpam-6326	107	8	vertex	vertex	NOUN
ejpam-6326	107	9	of	of	ADP
ejpam-6326	107	10	g	g	PROPN
ejpam-6326	107	11	belongs	belong	VERB
ejpam-6326	107	12	to	to	ADP
ejpam-6326	107	13	f	f	PROPN
ejpam-6326	107	14	.	.	PUNCT
ejpam-6326	108	1	theorem	theorem	NOUN
ejpam-6326	108	2	2	2	NUM
ejpam-6326	108	3	.	.	PUNCT
ejpam-6326	109	1	let	let	VERB
ejpam-6326	109	2	g	g	PRON
ejpam-6326	109	3	be	be	AUX
ejpam-6326	109	4	a	a	DET
ejpam-6326	109	5	nontrivial	nontrivial	ADJ
ejpam-6326	109	6	connected	connect	VERB
ejpam-6326	109	7	graph	graph	NOUN
ejpam-6326	109	8	and	and	CCONJ
ejpam-6326	109	9	f	f	PROPN
ejpam-6326	109	10	⊆	⊆	NUM
ejpam-6326	109	11	v	v	ADP
ejpam-6326	109	12	(	(	PUNCT
ejpam-6326	109	13	g	g	NOUN
ejpam-6326	109	14	)	)	PUNCT
ejpam-6326	109	15	be	be	AUX
ejpam-6326	109	16	a	a	DET
ejpam-6326	109	17	friendly	friendly	ADJ
ejpam-6326	109	18	dominating	dominating	NOUN
ejpam-6326	109	19	set	set	NOUN
ejpam-6326	109	20	of	of	ADP
ejpam-6326	109	21	g.	g.	PROPN
ejpam-6326	109	22	then	then	ADV
ejpam-6326	109	23	the	the	DET
ejpam-6326	109	24	induced	induced	ADJ
ejpam-6326	109	25	subgraph	subgraph	NOUN
ejpam-6326	109	26	of	of	ADP
ejpam-6326	109	27	v	v	NOUN
ejpam-6326	109	28	(	(	PUNCT
ejpam-6326	109	29	g	g	NOUN
ejpam-6326	109	30	)	)	PUNCT
ejpam-6326	109	31	\	\	PROPN
ejpam-6326	110	1	f	f	PROPN
ejpam-6326	110	2	in	in	ADP
ejpam-6326	110	3	g	g	PROPN
ejpam-6326	110	4	,	,	PUNCT
ejpam-6326	110	5	has	have	VERB
ejpam-6326	110	6	no	no	DET
ejpam-6326	110	7	isolated	isolated	ADJ
ejpam-6326	110	8	vertices	vertex	NOUN
ejpam-6326	110	9	.	.	PUNCT
ejpam-6326	111	1	i.	i.	PROPN
ejpam-6326	111	2	s.	s.	PROPN
ejpam-6326	111	3	cabahug	cabahug	PROPN
ejpam-6326	111	4	,	,	PUNCT
ejpam-6326	111	5	jr	jr	PROPN
ejpam-6326	111	6	.	.	PROPN
ejpam-6326	111	7	,	,	PUNCT
ejpam-6326	111	8	r.	r.	PROPN
ejpam-6326	111	9	g.	g.	PROPN
ejpam-6326	111	10	eballe	eballe	PROPN
ejpam-6326	111	11	,	,	PUNCT
ejpam-6326	111	12	r.	r.	PROPN
ejpam-6326	111	13	t.	t.	PROPN
ejpam-6326	111	14	fernandez	fernandez	PROPN
ejpam-6326	111	15	/	/	SYM
ejpam-6326	111	16	eur	eur	PROPN
ejpam-6326	111	17	.	.	PUNCT
ejpam-6326	112	1	j.	j.	PROPN
ejpam-6326	112	2	pure	pure	PROPN
ejpam-6326	112	3	appl	appl	PROPN
ejpam-6326	112	4	.	.	PROPN
ejpam-6326	112	5	math	math	PROPN
ejpam-6326	112	6	,	,	PUNCT
ejpam-6326	112	7	18	18	NUM
ejpam-6326	112	8	(	(	PUNCT
ejpam-6326	112	9	4	4	NUM
ejpam-6326	112	10	)	)	PUNCT
ejpam-6326	112	11	(	(	PUNCT
ejpam-6326	112	12	2025	2025	NUM
ejpam-6326	112	13	)	)	PUNCT
ejpam-6326	112	14	,	,	PUNCT
ejpam-6326	112	15	6326	6326	NUM
ejpam-6326	112	16	5	5	NUM
ejpam-6326	112	17	of	of	ADP
ejpam-6326	112	18	15	15	NUM
ejpam-6326	112	19	proof	proof	NOUN
ejpam-6326	112	20	.	.	PUNCT
ejpam-6326	112	21	suppose	suppose	VERB
ejpam-6326	112	22	that	that	SCONJ
ejpam-6326	112	23	the	the	DET
ejpam-6326	112	24	induced	induced	ADJ
ejpam-6326	112	25	subgraph	subgraph	NOUN
ejpam-6326	112	26	g[v	g[v	PROPN
ejpam-6326	112	27	(	(	PUNCT
ejpam-6326	112	28	g	g	NOUN
ejpam-6326	112	29	)	)	PUNCT
ejpam-6326	112	30	\	\	X
ejpam-6326	113	1	f	f	PROPN
ejpam-6326	113	2	]	]	PUNCT
ejpam-6326	113	3	contains	contain	VERB
ejpam-6326	113	4	an	an	DET
ejpam-6326	113	5	isolated	isolated	ADJ
ejpam-6326	113	6	vertex	vertex	NOUN
ejpam-6326	113	7	w.	w.	NOUN
ejpam-6326	113	8	then	then	ADV
ejpam-6326	113	9	w	w	PROPN
ejpam-6326	113	10	∈	∈	PROPN
ejpam-6326	113	11	v	v	ADP
ejpam-6326	113	12	(	(	PUNCT
ejpam-6326	113	13	g	g	NOUN
ejpam-6326	113	14	)	)	PUNCT
ejpam-6326	113	15	\	\	PROPN
ejpam-6326	113	16	f	f	PROPN
ejpam-6326	113	17	and	and	CCONJ
ejpam-6326	113	18	degv	degv	PROPN
ejpam-6326	113	19	(	(	PUNCT
ejpam-6326	113	20	g)\f	g)\f	NOUN
ejpam-6326	113	21	(	(	PUNCT
ejpam-6326	113	22	w	w	NOUN
ejpam-6326	113	23	)	)	PUNCT
ejpam-6326	113	24	=	=	SYM
ejpam-6326	114	1	0	0	X
ejpam-6326	114	2	.	.	PUNCT
ejpam-6326	115	1	since	since	SCONJ
ejpam-6326	115	2	f	f	PROPN
ejpam-6326	115	3	is	be	AUX
ejpam-6326	115	4	a	a	DET
ejpam-6326	115	5	dominating	dominating	NOUN
ejpam-6326	115	6	set	set	NOUN
ejpam-6326	115	7	,	,	PUNCT
ejpam-6326	115	8	w	w	PROPN
ejpam-6326	115	9	must	must	AUX
ejpam-6326	115	10	be	be	AUX
ejpam-6326	115	11	adjacent	adjacent	ADJ
ejpam-6326	115	12	to	to	ADP
ejpam-6326	115	13	some	some	DET
ejpam-6326	115	14	vertex	vertex	NOUN
ejpam-6326	115	15	in	in	ADP
ejpam-6326	115	16	f	f	PROPN
ejpam-6326	115	17	,	,	PUNCT
ejpam-6326	115	18	implying	imply	VERB
ejpam-6326	115	19	that	that	SCONJ
ejpam-6326	115	20	degf	degf	PROPN
ejpam-6326	115	21	(	(	PUNCT
ejpam-6326	115	22	w	w	PROPN
ejpam-6326	115	23	)	)	PUNCT
ejpam-6326	115	24	≥	≥	NOUN
ejpam-6326	115	25	1	1	NUM
ejpam-6326	115	26	.	.	PUNCT
ejpam-6326	116	1	this	this	PRON
ejpam-6326	116	2	yields	yield	VERB
ejpam-6326	116	3	the	the	DET
ejpam-6326	116	4	inequality	inequality	NOUN
ejpam-6326	116	5	1	1	NUM
ejpam-6326	116	6	≤	≤	NUM
ejpam-6326	116	7	degf	degf	NOUN
ejpam-6326	116	8	(	(	PUNCT
ejpam-6326	116	9	w	w	NOUN
ejpam-6326	116	10	)	)	PUNCT
ejpam-6326	116	11	≤	≤	NOUN
ejpam-6326	116	12	degv	degv	NOUN
ejpam-6326	116	13	(	(	PUNCT
ejpam-6326	116	14	g)\f	g)\f	NOUN
ejpam-6326	116	15	(	(	PUNCT
ejpam-6326	116	16	w	w	NOUN
ejpam-6326	116	17	)	)	PUNCT
ejpam-6326	116	18	=	=	SYM
ejpam-6326	116	19	0	0	NUM
ejpam-6326	116	20	,	,	PUNCT
ejpam-6326	116	21	which	which	PRON
ejpam-6326	116	22	is	be	AUX
ejpam-6326	116	23	a	a	DET
ejpam-6326	116	24	contradiction	contradiction	NOUN
ejpam-6326	116	25	.	.	PUNCT
ejpam-6326	117	1	therefore	therefore	ADV
ejpam-6326	117	2	,	,	PUNCT
ejpam-6326	117	3	the	the	DET
ejpam-6326	117	4	induced	induced	ADJ
ejpam-6326	117	5	subgraph	subgraph	NOUN
ejpam-6326	117	6	g[v	g[v	PROPN
ejpam-6326	117	7	(	(	PUNCT
ejpam-6326	117	8	g	g	NOUN
ejpam-6326	117	9	)	)	PUNCT
ejpam-6326	117	10	\f	\f	PUNCT
ejpam-6326	117	11	]	]	PUNCT
ejpam-6326	117	12	can	can	AUX
ejpam-6326	117	13	not	not	PART
ejpam-6326	117	14	contain	contain	VERB
ejpam-6326	117	15	an	an	DET
ejpam-6326	117	16	isolated	isolated	ADJ
ejpam-6326	117	17	vertex	vertex	NOUN
ejpam-6326	117	18	.	.	PUNCT
ejpam-6326	118	1	theorem	theorem	NOUN
ejpam-6326	118	2	3	3	X
ejpam-6326	118	3	.	.	PUNCT
ejpam-6326	119	1	let	let	VERB
ejpam-6326	119	2	g	g	PRON
ejpam-6326	119	3	be	be	AUX
ejpam-6326	119	4	a	a	DET
ejpam-6326	119	5	nontrivial	nontrivial	ADJ
ejpam-6326	119	6	connected	connect	VERB
ejpam-6326	119	7	graph	graph	NOUN
ejpam-6326	119	8	of	of	ADP
ejpam-6326	119	9	order	order	NOUN
ejpam-6326	119	10	n	n	CCONJ
ejpam-6326	119	11	>	>	X
ejpam-6326	119	12	2	2	X
ejpam-6326	119	13	.	.	PUNCT
ejpam-6326	120	1	then	then	ADV
ejpam-6326	120	2	,	,	PUNCT
ejpam-6326	120	3	γf	γf	PROPN
ejpam-6326	120	4	(	(	PUNCT
ejpam-6326	120	5	g	g	NOUN
ejpam-6326	120	6	)	)	PUNCT
ejpam-6326	120	7	=	=	SYM
ejpam-6326	120	8	1	1	NUM
ejpam-6326	120	9	if	if	SCONJ
ejpam-6326	120	10	and	and	CCONJ
ejpam-6326	120	11	only	only	ADV
ejpam-6326	120	12	if	if	SCONJ
ejpam-6326	120	13	there	there	PRON
ejpam-6326	120	14	exist	exist	VERB
ejpam-6326	120	15	a	a	DET
ejpam-6326	120	16	vertex	vertex	NOUN
ejpam-6326	120	17	v	v	ADP
ejpam-6326	120	18	∈	∈	NOUN
ejpam-6326	120	19	v	v	NOUN
ejpam-6326	120	20	(	(	PUNCT
ejpam-6326	120	21	g	g	NOUN
ejpam-6326	120	22	)	)	PUNCT
ejpam-6326	120	23	such	such	ADJ
ejpam-6326	120	24	that	that	SCONJ
ejpam-6326	120	25	deg(v	deg(v	PROPN
ejpam-6326	120	26	)	)	PUNCT
ejpam-6326	120	27	=	=	SYM
ejpam-6326	120	28	n	n	CCONJ
ejpam-6326	120	29	−	−	PROPN
ejpam-6326	120	30	1	1	NUM
ejpam-6326	120	31	and	and	CCONJ
ejpam-6326	120	32	for	for	ADP
ejpam-6326	120	33	every	every	DET
ejpam-6326	120	34	other	other	ADJ
ejpam-6326	120	35	vertex	vertex	NOUN
ejpam-6326	120	36	u	u	NOUN
ejpam-6326	120	37	̸=	̸=	PROPN
ejpam-6326	120	38	v	v	NOUN
ejpam-6326	120	39	has	have	VERB
ejpam-6326	120	40	at	at	ADV
ejpam-6326	120	41	least	least	ADV
ejpam-6326	120	42	one	one	NUM
ejpam-6326	120	43	neighbor	neighbor	NOUN
ejpam-6326	120	44	besides	besides	ADV
ejpam-6326	120	45	v.	v.	ADP
ejpam-6326	120	46	proof	proof	NOUN
ejpam-6326	120	47	.	.	PUNCT
ejpam-6326	121	1	suppose	suppose	VERB
ejpam-6326	121	2	γf	γf	INTJ
ejpam-6326	121	3	(	(	PUNCT
ejpam-6326	121	4	g	g	NOUN
ejpam-6326	121	5	)	)	PUNCT
ejpam-6326	121	6	=	=	SYM
ejpam-6326	122	1	1	1	X
ejpam-6326	122	2	.	.	PUNCT
ejpam-6326	122	3	then	then	ADV
ejpam-6326	122	4	there	there	PRON
ejpam-6326	122	5	is	be	VERB
ejpam-6326	122	6	a	a	DET
ejpam-6326	122	7	friendly	friendly	ADJ
ejpam-6326	122	8	dominating	dominating	NOUN
ejpam-6326	122	9	set	set	NOUN
ejpam-6326	122	10	f	f	PROPN
ejpam-6326	122	11	⊆	⊆	NUM
ejpam-6326	122	12	v	v	NOUN
ejpam-6326	122	13	(	(	PUNCT
ejpam-6326	122	14	g	g	NOUN
ejpam-6326	122	15	)	)	PUNCT
ejpam-6326	122	16	with	with	ADP
ejpam-6326	122	17	|f	|f	PROPN
ejpam-6326	123	1	|	|	ADV
ejpam-6326	123	2	=	=	NOUN
ejpam-6326	123	3	1	1	X
ejpam-6326	123	4	.	.	PUNCT
ejpam-6326	124	1	let	let	VERB
ejpam-6326	124	2	f	f	NOUN
ejpam-6326	124	3	=	=	PRON
ejpam-6326	124	4	{	{	PUNCT
ejpam-6326	124	5	v	v	NOUN
ejpam-6326	124	6	}	}	PUNCT
ejpam-6326	124	7	.	.	PUNCT
ejpam-6326	125	1	since	since	SCONJ
ejpam-6326	125	2	f	f	PROPN
ejpam-6326	125	3	is	be	AUX
ejpam-6326	125	4	a	a	DET
ejpam-6326	125	5	dominating	dominating	NOUN
ejpam-6326	125	6	set	set	NOUN
ejpam-6326	125	7	,	,	PUNCT
ejpam-6326	125	8	v	v	X
ejpam-6326	125	9	must	must	AUX
ejpam-6326	125	10	be	be	AUX
ejpam-6326	125	11	adjacent	adjacent	ADJ
ejpam-6326	125	12	to	to	ADP
ejpam-6326	125	13	every	every	DET
ejpam-6326	125	14	other	other	ADJ
ejpam-6326	125	15	vertex	vertex	NOUN
ejpam-6326	125	16	.	.	PUNCT
ejpam-6326	126	1	hence	hence	ADV
ejpam-6326	126	2	,	,	PUNCT
ejpam-6326	126	3	deg(v	deg(v	PROPN
ejpam-6326	126	4	)	)	PUNCT
ejpam-6326	126	5	=	=	PUNCT
ejpam-6326	127	1	n−	n−	NOUN
ejpam-6326	127	2	1	1	NUM
ejpam-6326	127	3	.	.	PUNCT
ejpam-6326	128	1	now	now	ADV
ejpam-6326	128	2	,	,	PUNCT
ejpam-6326	128	3	for	for	ADP
ejpam-6326	128	4	each	each	DET
ejpam-6326	128	5	u	u	PROPN
ejpam-6326	128	6	∈	∈	PROPN
ejpam-6326	128	7	v	v	ADP
ejpam-6326	128	8	(	(	PUNCT
ejpam-6326	128	9	g	g	NOUN
ejpam-6326	128	10	)	)	PUNCT
ejpam-6326	128	11	\	\	NOUN
ejpam-6326	128	12	{	{	PUNCT
ejpam-6326	128	13	v	v	NOUN
ejpam-6326	128	14	}	}	PUNCT
ejpam-6326	128	15	,	,	PUNCT
ejpam-6326	128	16	we	we	PRON
ejpam-6326	128	17	have	have	VERB
ejpam-6326	128	18	degf	degf	PROPN
ejpam-6326	128	19	(	(	PUNCT
ejpam-6326	128	20	u	u	NOUN
ejpam-6326	128	21	)	)	PUNCT
ejpam-6326	128	22	=	=	SYM
ejpam-6326	129	1	1	1	X
ejpam-6326	129	2	.	.	PUNCT
ejpam-6326	129	3	since	since	SCONJ
ejpam-6326	129	4	f	f	PROPN
ejpam-6326	129	5	is	be	AUX
ejpam-6326	129	6	a	a	DET
ejpam-6326	129	7	friendly	friendly	ADJ
ejpam-6326	129	8	set	set	NOUN
ejpam-6326	129	9	,	,	PUNCT
ejpam-6326	129	10	degf	degf	PROPN
ejpam-6326	129	11	(	(	PUNCT
ejpam-6326	129	12	u	u	NOUN
ejpam-6326	129	13	)	)	PUNCT
ejpam-6326	129	14	≤	≤	NOUN
ejpam-6326	129	15	degv	degv	NOUN
ejpam-6326	129	16	(	(	PUNCT
ejpam-6326	129	17	g)\{v}(u	g)\{v}(u	NOUN
ejpam-6326	129	18	)	)	PUNCT
ejpam-6326	129	19	for	for	ADP
ejpam-6326	129	20	all	all	DET
ejpam-6326	129	21	u	u	NOUN
ejpam-6326	129	22	/∈	/∈	PROPN
ejpam-6326	129	23	f	f	PROPN
ejpam-6326	129	24	.	.	PUNCT
ejpam-6326	130	1	thus	thus	ADV
ejpam-6326	130	2	,	,	PUNCT
ejpam-6326	130	3	degv	degv	NOUN
ejpam-6326	130	4	(	(	PUNCT
ejpam-6326	130	5	g)\{v}(u	g)\{v}(u	NOUN
ejpam-6326	130	6	)	)	PUNCT
ejpam-6326	130	7	≥	≥	NOUN
ejpam-6326	130	8	1	1	NUM
ejpam-6326	130	9	,	,	PUNCT
ejpam-6326	130	10	meaning	mean	VERB
ejpam-6326	130	11	each	each	DET
ejpam-6326	130	12	u	u	NOUN
ejpam-6326	130	13	̸=	̸=	PROPN
ejpam-6326	130	14	v	v	NOUN
ejpam-6326	130	15	must	must	AUX
ejpam-6326	130	16	have	have	VERB
ejpam-6326	130	17	at	at	ADV
ejpam-6326	130	18	least	least	ADV
ejpam-6326	130	19	one	one	NUM
ejpam-6326	130	20	neighbor	neighbor	NOUN
ejpam-6326	130	21	other	other	ADJ
ejpam-6326	130	22	than	than	ADP
ejpam-6326	130	23	v.	v.	ADP
ejpam-6326	130	24	conversely	conversely	ADV
ejpam-6326	130	25	,	,	PUNCT
ejpam-6326	130	26	suppose	suppose	VERB
ejpam-6326	130	27	there	there	PRON
ejpam-6326	130	28	exists	exist	VERB
ejpam-6326	130	29	a	a	DET
ejpam-6326	130	30	vertex	vertex	NOUN
ejpam-6326	130	31	v	v	ADP
ejpam-6326	130	32	∈	∈	PROPN
ejpam-6326	130	33	v	v	NOUN
ejpam-6326	130	34	(	(	PUNCT
ejpam-6326	130	35	g	g	NOUN
ejpam-6326	130	36	)	)	PUNCT
ejpam-6326	130	37	with	with	ADP
ejpam-6326	130	38	deg(v	deg(v	NOUN
ejpam-6326	130	39	)	)	PUNCT
ejpam-6326	130	40	=	=	PUNCT
ejpam-6326	130	41	n−	n−	NOUN
ejpam-6326	130	42	1	1	NUM
ejpam-6326	130	43	and	and	CCONJ
ejpam-6326	130	44	for	for	ADP
ejpam-6326	130	45	every	every	DET
ejpam-6326	130	46	u	u	NOUN
ejpam-6326	130	47	̸=	̸=	PROPN
ejpam-6326	130	48	v	v	NOUN
ejpam-6326	130	49	,	,	PUNCT
ejpam-6326	130	50	degv	degv	NOUN
ejpam-6326	130	51	(	(	PUNCT
ejpam-6326	130	52	g)\f	g)\f	NOUN
ejpam-6326	130	53	(	(	PUNCT
ejpam-6326	130	54	u	u	NOUN
ejpam-6326	130	55	)	)	PUNCT
ejpam-6326	130	56	≥	≥	NOUN
ejpam-6326	130	57	1	1	NUM
ejpam-6326	130	58	.	.	PUNCT
ejpam-6326	130	59	consider	consider	VERB
ejpam-6326	130	60	f	f	NOUN
ejpam-6326	130	61	=	=	PRON
ejpam-6326	130	62	{	{	PUNCT
ejpam-6326	130	63	v	v	NOUN
ejpam-6326	130	64	}	}	PUNCT
ejpam-6326	130	65	.	.	PUNCT
ejpam-6326	131	1	then	then	ADV
ejpam-6326	131	2	,	,	PUNCT
ejpam-6326	131	3	f	f	PROPN
ejpam-6326	131	4	is	be	AUX
ejpam-6326	131	5	a	a	DET
ejpam-6326	131	6	dominating	dominating	NOUN
ejpam-6326	131	7	set	set	NOUN
ejpam-6326	131	8	of	of	ADP
ejpam-6326	131	9	g.	g.	PROPN
ejpam-6326	131	10	also	also	ADV
ejpam-6326	131	11	,	,	PUNCT
ejpam-6326	131	12	for	for	ADP
ejpam-6326	131	13	each	each	DET
ejpam-6326	131	14	u	u	NOUN
ejpam-6326	131	15	̸=	̸=	PROPN
ejpam-6326	131	16	v	v	NOUN
ejpam-6326	131	17	,	,	PUNCT
ejpam-6326	131	18	degf	degf	PROPN
ejpam-6326	131	19	(	(	PUNCT
ejpam-6326	131	20	u	u	NOUN
ejpam-6326	131	21	)	)	PUNCT
ejpam-6326	131	22	=	=	SYM
ejpam-6326	131	23	1	1	X
ejpam-6326	131	24	.	.	PUNCT
ejpam-6326	132	1	thus	thus	ADV
ejpam-6326	132	2	,	,	PUNCT
ejpam-6326	132	3	degf	degf	PROPN
ejpam-6326	132	4	(	(	PUNCT
ejpam-6326	132	5	u	u	NOUN
ejpam-6326	132	6	)	)	PUNCT
ejpam-6326	132	7	=	=	SYM
ejpam-6326	132	8	1	1	NUM
ejpam-6326	132	9	≤	≤	NOUN
ejpam-6326	132	10	degv	degv	NOUN
ejpam-6326	132	11	(	(	PUNCT
ejpam-6326	132	12	g)\f	g)\f	NOUN
ejpam-6326	132	13	(	(	PUNCT
ejpam-6326	132	14	u	u	NOUN
ejpam-6326	132	15	)	)	PUNCT
ejpam-6326	132	16	.	.	PUNCT
ejpam-6326	133	1	hence	hence	ADV
ejpam-6326	133	2	,	,	PUNCT
ejpam-6326	133	3	f	f	PROPN
ejpam-6326	133	4	is	be	AUX
ejpam-6326	133	5	a	a	DET
ejpam-6326	133	6	friendly	friendly	ADJ
ejpam-6326	133	7	set	set	NOUN
ejpam-6326	133	8	in	in	ADP
ejpam-6326	133	9	g.	g.	PROPN
ejpam-6326	133	10	therefore	therefore	ADV
ejpam-6326	133	11	,	,	PUNCT
ejpam-6326	133	12	f	f	PROPN
ejpam-6326	133	13	=	=	PRON
ejpam-6326	133	14	{	{	PUNCT
ejpam-6326	133	15	v	v	NOUN
ejpam-6326	133	16	}	}	PUNCT
ejpam-6326	133	17	is	be	AUX
ejpam-6326	133	18	a	a	DET
ejpam-6326	133	19	friendly	friendly	ADJ
ejpam-6326	133	20	dominating	dominating	NOUN
ejpam-6326	133	21	set	set	NOUN
ejpam-6326	133	22	of	of	ADP
ejpam-6326	133	23	g	g	PROPN
ejpam-6326	133	24	and	and	CCONJ
ejpam-6326	133	25	that	that	SCONJ
ejpam-6326	133	26	γf	γf	INTJ
ejpam-6326	133	27	(	(	PUNCT
ejpam-6326	133	28	g	g	NOUN
ejpam-6326	133	29	)	)	PUNCT
ejpam-6326	133	30	≤	≤	NUM
ejpam-6326	133	31	1	1	NUM
ejpam-6326	133	32	.	.	PUNCT
ejpam-6326	134	1	clearly	clearly	ADV
ejpam-6326	134	2	,	,	PUNCT
ejpam-6326	134	3	γf	γf	PROPN
ejpam-6326	134	4	(	(	PUNCT
ejpam-6326	134	5	g	g	NOUN
ejpam-6326	134	6	)	)	PUNCT
ejpam-6326	134	7	≥	≥	NOUN
ejpam-6326	134	8	1	1	NUM
ejpam-6326	134	9	.	.	PUNCT
ejpam-6326	135	1	thus	thus	ADV
ejpam-6326	135	2	,	,	PUNCT
ejpam-6326	135	3	γf	γf	INTJ
ejpam-6326	135	4	(	(	PUNCT
ejpam-6326	135	5	g	g	NOUN
ejpam-6326	135	6	)	)	PUNCT
ejpam-6326	135	7	=	=	SYM
ejpam-6326	135	8	1	1	X
ejpam-6326	135	9	.	.	PUNCT
ejpam-6326	135	10	corollary	corollary	ADJ
ejpam-6326	135	11	1	1	NUM
ejpam-6326	135	12	.	.	PUNCT
ejpam-6326	136	1	for	for	ADP
ejpam-6326	136	2	the	the	DET
ejpam-6326	136	3	complete	complete	ADJ
ejpam-6326	136	4	graph	graph	NOUN
ejpam-6326	136	5	kn	kn	PROPN
ejpam-6326	136	6	,	,	PUNCT
ejpam-6326	136	7	fan	fan	PROPN
ejpam-6326	136	8	graph	graph	NOUN
ejpam-6326	136	9	f1,n	f1,n	PROPN
ejpam-6326	136	10	,	,	PUNCT
ejpam-6326	136	11	and	and	CCONJ
ejpam-6326	136	12	wheel	wheel	NOUN
ejpam-6326	136	13	graph	graph	NOUN
ejpam-6326	136	14	w1,m	w1,m	PROPN
ejpam-6326	136	15	with	with	ADP
ejpam-6326	136	16	n	n	NOUN
ejpam-6326	136	17	>	>	SYM
ejpam-6326	136	18	1	1	NUM
ejpam-6326	136	19	and	and	CCONJ
ejpam-6326	136	20	m	m	VERB
ejpam-6326	136	21	>	>	X
ejpam-6326	136	22	2	2	NUM
ejpam-6326	136	23	,	,	PUNCT
ejpam-6326	136	24	γf	γf	ADJ
ejpam-6326	136	25	(	(	PUNCT
ejpam-6326	136	26	km	km	NOUN
ejpam-6326	136	27	)	)	PUNCT
ejpam-6326	136	28	=	=	SYM
ejpam-6326	136	29	γf	γf	PROPN
ejpam-6326	136	30	(	(	PUNCT
ejpam-6326	136	31	f1,n	f1,n	PROPN
ejpam-6326	136	32	)	)	PUNCT
ejpam-6326	136	33	=	=	SYM
ejpam-6326	136	34	γf	γf	PROPN
ejpam-6326	136	35	(	(	PUNCT
ejpam-6326	136	36	w1,m	w1,m	PROPN
ejpam-6326	136	37	)	)	PUNCT
ejpam-6326	136	38	=	=	SYM
ejpam-6326	136	39	1	1	X
ejpam-6326	136	40	.	.	PUNCT
ejpam-6326	136	41	proof	proof	NOUN
ejpam-6326	136	42	.	.	PUNCT
ejpam-6326	137	1	this	this	PRON
ejpam-6326	137	2	immediately	immediately	ADV
ejpam-6326	137	3	follows	follow	VERB
ejpam-6326	137	4	from	from	ADP
ejpam-6326	137	5	theorem	theorem	ADJ
ejpam-6326	137	6	3	3	NUM
ejpam-6326	137	7	.	.	PUNCT
ejpam-6326	137	8	theorem	theorem	NOUN
ejpam-6326	137	9	4	4	NUM
ejpam-6326	137	10	.	.	PUNCT
ejpam-6326	138	1	let	let	VERB
ejpam-6326	138	2	g	g	PRON
ejpam-6326	138	3	be	be	AUX
ejpam-6326	138	4	a	a	DET
ejpam-6326	138	5	non	non	ADJ
ejpam-6326	138	6	-	-	ADJ
ejpam-6326	138	7	trivial	trivial	ADJ
ejpam-6326	138	8	connected	connected	ADJ
ejpam-6326	138	9	graph	graph	NOUN
ejpam-6326	138	10	of	of	ADP
ejpam-6326	138	11	order	order	NOUN
ejpam-6326	138	12	n	n	PRON
ejpam-6326	138	13	≥	≥	NOUN
ejpam-6326	138	14	3	3	NUM
ejpam-6326	138	15	.	.	PUNCT
ejpam-6326	139	1	then	then	ADV
ejpam-6326	139	2	γf	γf	INTJ
ejpam-6326	139	3	(	(	PUNCT
ejpam-6326	139	4	g	g	NOUN
ejpam-6326	139	5	)	)	PUNCT
ejpam-6326	139	6	=	=	SYM
ejpam-6326	139	7	2	2	NUM
ejpam-6326	139	8	if	if	SCONJ
ejpam-6326	139	9	and	and	CCONJ
ejpam-6326	139	10	only	only	ADV
ejpam-6326	139	11	if	if	SCONJ
ejpam-6326	139	12	all	all	DET
ejpam-6326	139	13	three	three	NUM
ejpam-6326	139	14	conditions	condition	NOUN
ejpam-6326	139	15	hold	hold	VERB
ejpam-6326	139	16	.	.	PUNCT
ejpam-6326	140	1	(	(	PUNCT
ejpam-6326	140	2	i	i	NOUN
ejpam-6326	140	3	)	)	PUNCT
ejpam-6326	140	4	there	there	PRON
ejpam-6326	140	5	is	be	VERB
ejpam-6326	140	6	no	no	DET
ejpam-6326	140	7	vertex	vertex	NOUN
ejpam-6326	140	8	u	u	NOUN
ejpam-6326	140	9	∈	∈	PROPN
ejpam-6326	140	10	v	v	ADP
ejpam-6326	140	11	(	(	PUNCT
ejpam-6326	140	12	g	g	NOUN
ejpam-6326	140	13	)	)	PUNCT
ejpam-6326	140	14	such	such	ADJ
ejpam-6326	140	15	that	that	SCONJ
ejpam-6326	141	1	n	n	NUM
ejpam-6326	142	1	[	[	X
ejpam-6326	142	2	u	u	X
ejpam-6326	142	3	]	]	X
ejpam-6326	142	4	=	=	SYM
ejpam-6326	142	5	v	v	X
ejpam-6326	142	6	(	(	PUNCT
ejpam-6326	142	7	g	g	NOUN
ejpam-6326	142	8	)	)	PUNCT
ejpam-6326	142	9	and	and	CCONJ
ejpam-6326	142	10	the	the	DET
ejpam-6326	142	11	induced	induced	ADJ
ejpam-6326	142	12	subgraph	subgraph	NOUN
ejpam-6326	142	13	of	of	ADP
ejpam-6326	142	14	g−	g−	PROPN
ejpam-6326	142	15	{	{	PUNCT
ejpam-6326	142	16	u	u	NOUN
ejpam-6326	142	17	}	}	PUNCT
ejpam-6326	142	18	has	have	VERB
ejpam-6326	142	19	minimum	minimum	ADJ
ejpam-6326	142	20	degree	degree	NOUN
ejpam-6326	142	21	δ	δ	PROPN
ejpam-6326	142	22	(	(	PUNCT
ejpam-6326	142	23	g−	g−	PROPN
ejpam-6326	142	24	{	{	PUNCT
ejpam-6326	142	25	u	u	NOUN
ejpam-6326	142	26	}	}	PUNCT
ejpam-6326	142	27	)	)	PUNCT
ejpam-6326	142	28	≥	≥	NOUN
ejpam-6326	142	29	1	1	NUM
ejpam-6326	142	30	.	.	PUNCT
ejpam-6326	142	31	(	(	PUNCT
ejpam-6326	142	32	ii	ii	NOUN
ejpam-6326	142	33	)	)	PUNCT
ejpam-6326	142	34	there	there	PRON
ejpam-6326	142	35	exist	exist	VERB
ejpam-6326	142	36	two	two	NUM
ejpam-6326	142	37	distinct	distinct	ADJ
ejpam-6326	142	38	vertices	vertex	NOUN
ejpam-6326	142	39	x	x	X
ejpam-6326	142	40	,	,	PUNCT
ejpam-6326	142	41	y	y	PROPN
ejpam-6326	142	42	∈	∈	PROPN
ejpam-6326	142	43	v	v	ADP
ejpam-6326	142	44	(	(	PUNCT
ejpam-6326	142	45	g	g	NOUN
ejpam-6326	142	46	)	)	PUNCT
ejpam-6326	142	47	with	with	ADP
ejpam-6326	142	48	the	the	DET
ejpam-6326	142	49	property	property	NOUN
ejpam-6326	142	50	that	that	PRON
ejpam-6326	142	51	every	every	DET
ejpam-6326	142	52	other	other	ADJ
ejpam-6326	142	53	vertex	vertex	NOUN
ejpam-6326	142	54	v	v	ADP
ejpam-6326	142	55	∈	∈	PROPN
ejpam-6326	142	56	v	v	NOUN
ejpam-6326	142	57	(	(	PUNCT
ejpam-6326	142	58	g	g	NOUN
ejpam-6326	142	59	)	)	PUNCT
ejpam-6326	142	60	\	\	NOUN
ejpam-6326	142	61	{	{	PUNCT
ejpam-6326	142	62	x	x	NOUN
ejpam-6326	142	63	,	,	PUNCT
ejpam-6326	142	64	y	y	NOUN
ejpam-6326	142	65	}	}	PUNCT
ejpam-6326	142	66	is	be	AUX
ejpam-6326	142	67	adjacent	adjacent	ADJ
ejpam-6326	142	68	to	to	ADP
ejpam-6326	142	69	at	at	ADV
ejpam-6326	142	70	least	least	ADV
ejpam-6326	142	71	one	one	NUM
ejpam-6326	142	72	of	of	ADP
ejpam-6326	142	73	x	x	PUNCT
ejpam-6326	142	74	or	or	CCONJ
ejpam-6326	142	75	y.	y.	PROPN
ejpam-6326	142	76	(	(	PUNCT
ejpam-6326	142	77	iii	iii	NOUN
ejpam-6326	142	78	)	)	PUNCT
ejpam-6326	142	79	for	for	ADP
ejpam-6326	142	80	every	every	DET
ejpam-6326	142	81	vertex	vertex	NOUN
ejpam-6326	142	82	v	v	ADP
ejpam-6326	142	83	∈	∈	NOUN
ejpam-6326	142	84	v	v	NOUN
ejpam-6326	142	85	(	(	PUNCT
ejpam-6326	142	86	g	g	NOUN
ejpam-6326	142	87	)	)	PUNCT
ejpam-6326	142	88	\	\	NOUN
ejpam-6326	143	1	{	{	PUNCT
ejpam-6326	143	2	x	x	NOUN
ejpam-6326	143	3	,	,	PUNCT
ejpam-6326	143	4	y	y	PROPN
ejpam-6326	143	5	}	}	PUNCT
ejpam-6326	143	6	we	we	PRON
ejpam-6326	143	7	have	have	VERB
ejpam-6326	143	8	deg{x	deg{x	NOUN
ejpam-6326	143	9	,	,	PUNCT
ejpam-6326	143	10	y}(v	y}(v	PROPN
ejpam-6326	143	11	)	)	PUNCT
ejpam-6326	143	12	≤	≤	NUM
ejpam-6326	143	13	deg	deg	NOUN
ejpam-6326	143	14	v	v	X
ejpam-6326	143	15	(	(	PUNCT
ejpam-6326	143	16	g)\{x	g)\{x	PROPN
ejpam-6326	143	17	,	,	PUNCT
ejpam-6326	143	18	y}(v	y}(v	PROPN
ejpam-6326	143	19	)	)	PUNCT
ejpam-6326	143	20	.	.	PUNCT
ejpam-6326	144	1	proof	proof	NOUN
ejpam-6326	144	2	.	.	PUNCT
ejpam-6326	145	1	assume	assume	VERB
ejpam-6326	145	2	that	that	SCONJ
ejpam-6326	145	3	the	the	DET
ejpam-6326	145	4	friendly	friendly	ADJ
ejpam-6326	145	5	domination	domination	NOUN
ejpam-6326	145	6	number	number	NOUN
ejpam-6326	145	7	of	of	ADP
ejpam-6326	145	8	g	g	PROPN
ejpam-6326	145	9	is	be	AUX
ejpam-6326	145	10	γf	γf	ADJ
ejpam-6326	145	11	(	(	PUNCT
ejpam-6326	145	12	g	g	NOUN
ejpam-6326	145	13	)	)	PUNCT
ejpam-6326	145	14	=	=	SYM
ejpam-6326	146	1	2	2	X
ejpam-6326	146	2	.	.	PUNCT
ejpam-6326	146	3	then	then	ADV
ejpam-6326	146	4	there	there	PRON
ejpam-6326	146	5	exists	exist	VERB
ejpam-6326	146	6	a	a	DET
ejpam-6326	146	7	friendly	friendly	ADJ
ejpam-6326	146	8	dominating	dominating	NOUN
ejpam-6326	146	9	set	set	NOUN
ejpam-6326	146	10	f	f	NOUN
ejpam-6326	146	11	=	=	PRON
ejpam-6326	146	12	{	{	PUNCT
ejpam-6326	146	13	x	x	PROPN
ejpam-6326	146	14	,	,	PUNCT
ejpam-6326	146	15	y	y	PROPN
ejpam-6326	146	16	}	}	PUNCT
ejpam-6326	146	17	⊆	⊆	NUM
ejpam-6326	146	18	v	v	NOUN
ejpam-6326	146	19	(	(	PUNCT
ejpam-6326	146	20	g	g	NOUN
ejpam-6326	146	21	)	)	PUNCT
ejpam-6326	146	22	of	of	ADP
ejpam-6326	146	23	cardinality	cardinality	PROPN
ejpam-6326	146	24	two	two	NUM
ejpam-6326	146	25	.	.	PUNCT
ejpam-6326	147	1	since	since	SCONJ
ejpam-6326	147	2	f	f	PROPN
ejpam-6326	147	3	is	be	AUX
ejpam-6326	147	4	dominating	dominate	VERB
ejpam-6326	147	5	,	,	PUNCT
ejpam-6326	147	6	every	every	DET
ejpam-6326	147	7	vertex	vertex	NOUN
ejpam-6326	147	8	outside	outside	ADP
ejpam-6326	147	9	the	the	DET
ejpam-6326	147	10	pair	pair	NOUN
ejpam-6326	147	11	is	be	AUX
ejpam-6326	147	12	adjacent	adjacent	ADJ
ejpam-6326	147	13	to	to	ADP
ejpam-6326	147	14	at	at	ADV
ejpam-6326	147	15	least	least	ADV
ejpam-6326	147	16	one	one	NUM
ejpam-6326	147	17	of	of	ADP
ejpam-6326	147	18	x	x	X
ejpam-6326	147	19	or	or	CCONJ
ejpam-6326	147	20	y	y	PROPN
ejpam-6326	147	21	;	;	PUNCT
ejpam-6326	147	22	this	this	PRON
ejpam-6326	147	23	is	be	AUX
ejpam-6326	147	24	exactly	exactly	ADV
ejpam-6326	147	25	the	the	DET
ejpam-6326	147	26	content	content	NOUN
ejpam-6326	147	27	of	of	ADP
ejpam-6326	147	28	condition	condition	NOUN
ejpam-6326	147	29	(	(	PUNCT
ejpam-6326	147	30	ii	ii	NOUN
ejpam-6326	147	31	)	)	PUNCT
ejpam-6326	147	32	.	.	PUNCT
ejpam-6326	148	1	moreover	moreover	ADV
ejpam-6326	148	2	,	,	PUNCT
ejpam-6326	148	3	since	since	SCONJ
ejpam-6326	148	4	f	f	PROPN
ejpam-6326	148	5	is	be	AUX
ejpam-6326	148	6	friendly	friendly	ADJ
ejpam-6326	148	7	,	,	PUNCT
ejpam-6326	148	8	degf	degf	PROPN
ejpam-6326	148	9	(	(	PUNCT
ejpam-6326	148	10	v	v	NOUN
ejpam-6326	148	11	)	)	PUNCT
ejpam-6326	148	12	≤	≤	NOUN
ejpam-6326	148	13	degv	degv	NOUN
ejpam-6326	148	14	(	(	PUNCT
ejpam-6326	148	15	g)\f	g)\f	NOUN
ejpam-6326	148	16	(	(	PUNCT
ejpam-6326	148	17	v	v	NOUN
ejpam-6326	148	18	)	)	PUNCT
ejpam-6326	148	19	for	for	ADP
ejpam-6326	148	20	every	every	PRON
ejpam-6326	148	21	v	v	NUM
ejpam-6326	148	22	∈	∈	PROPN
ejpam-6326	148	23	v	v	NOUN
ejpam-6326	148	24	(	(	PUNCT
ejpam-6326	148	25	g	g	NOUN
ejpam-6326	148	26	)	)	PUNCT
ejpam-6326	148	27	\	\	PROPN
ejpam-6326	148	28	f	f	PROPN
ejpam-6326	148	29	,	,	PUNCT
ejpam-6326	148	30	is	be	AUX
ejpam-6326	148	31	identical	identical	ADJ
ejpam-6326	148	32	to	to	ADP
ejpam-6326	148	33	condition	condition	NOUN
ejpam-6326	148	34	(	(	PUNCT
ejpam-6326	148	35	iii	iii	NOUN
ejpam-6326	148	36	)	)	PUNCT
ejpam-6326	148	37	when	when	SCONJ
ejpam-6326	148	38	written	write	VERB
ejpam-6326	148	39	for	for	ADP
ejpam-6326	148	40	the	the	DET
ejpam-6326	148	41	set	set	NOUN
ejpam-6326	148	42	{	{	PUNCT
ejpam-6326	148	43	x	x	NOUN
ejpam-6326	148	44	,	,	PUNCT
ejpam-6326	148	45	y	y	PROPN
ejpam-6326	148	46	}	}	PUNCT
ejpam-6326	148	47	.	.	PUNCT
ejpam-6326	149	1	now	now	ADV
ejpam-6326	149	2	,	,	PUNCT
ejpam-6326	149	3	i.	i.	PROPN
ejpam-6326	149	4	s.	s.	PROPN
ejpam-6326	149	5	cabahug	cabahug	PROPN
ejpam-6326	149	6	,	,	PUNCT
ejpam-6326	149	7	jr	jr	PROPN
ejpam-6326	149	8	.	.	PROPN
ejpam-6326	149	9	,	,	PUNCT
ejpam-6326	149	10	r.	r.	PROPN
ejpam-6326	149	11	g.	g.	PROPN
ejpam-6326	149	12	eballe	eballe	PROPN
ejpam-6326	149	13	,	,	PUNCT
ejpam-6326	149	14	r.	r.	PROPN
ejpam-6326	149	15	t.	t.	PROPN
ejpam-6326	149	16	fernandez	fernandez	PROPN
ejpam-6326	149	17	/	/	SYM
ejpam-6326	149	18	eur	eur	PROPN
ejpam-6326	149	19	.	.	PUNCT
ejpam-6326	150	1	j.	j.	PROPN
ejpam-6326	150	2	pure	pure	PROPN
ejpam-6326	150	3	appl	appl	PROPN
ejpam-6326	150	4	.	.	PROPN
ejpam-6326	150	5	math	math	PROPN
ejpam-6326	150	6	,	,	PUNCT
ejpam-6326	150	7	18	18	NUM
ejpam-6326	150	8	(	(	PUNCT
ejpam-6326	150	9	4	4	NUM
ejpam-6326	150	10	)	)	PUNCT
ejpam-6326	150	11	(	(	PUNCT
ejpam-6326	150	12	2025	2025	NUM
ejpam-6326	150	13	)	)	PUNCT
ejpam-6326	150	14	,	,	PUNCT
ejpam-6326	150	15	6326	6326	NUM
ejpam-6326	150	16	6	6	NUM
ejpam-6326	150	17	of	of	ADP
ejpam-6326	150	18	15	15	NUM
ejpam-6326	150	19	suppose	suppose	VERB
ejpam-6326	150	20	(	(	PUNCT
ejpam-6326	150	21	i	i	NOUN
ejpam-6326	150	22	)	)	PUNCT
ejpam-6326	150	23	is	be	AUX
ejpam-6326	150	24	not	not	PART
ejpam-6326	150	25	true	true	ADJ
ejpam-6326	150	26	,	,	PUNCT
ejpam-6326	150	27	then	then	ADV
ejpam-6326	150	28	exist	exist	VERB
ejpam-6326	150	29	a	a	DET
ejpam-6326	150	30	single	single	ADJ
ejpam-6326	150	31	vertex	vertex	NOUN
ejpam-6326	150	32	u	u	NOUN
ejpam-6326	150	33	that	that	PRON
ejpam-6326	150	34	is	be	AUX
ejpam-6326	150	35	itself	itself	PRON
ejpam-6326	150	36	a	a	DET
ejpam-6326	150	37	friendly	friendly	ADJ
ejpam-6326	150	38	dominating	dominating	NOUN
ejpam-6326	150	39	set	set	NOUN
ejpam-6326	150	40	,	,	PUNCT
ejpam-6326	150	41	which	which	PRON
ejpam-6326	150	42	would	would	AUX
ejpam-6326	150	43	force	force	VERB
ejpam-6326	150	44	γf	γf	PROPN
ejpam-6326	150	45	(	(	PUNCT
ejpam-6326	150	46	g	g	NOUN
ejpam-6326	150	47	)	)	PUNCT
ejpam-6326	150	48	=	=	SYM
ejpam-6326	150	49	1	1	NUM
ejpam-6326	150	50	,	,	PUNCT
ejpam-6326	150	51	contradicting	contradict	VERB
ejpam-6326	150	52	our	our	PRON
ejpam-6326	150	53	assumption	assumption	NOUN
ejpam-6326	150	54	that	that	SCONJ
ejpam-6326	150	55	γf	γf	PROPN
ejpam-6326	150	56	(	(	PUNCT
ejpam-6326	150	57	g	g	NOUN
ejpam-6326	150	58	)	)	PUNCT
ejpam-6326	150	59	=	=	SYM
ejpam-6326	150	60	2	2	X
ejpam-6326	150	61	.	.	X
ejpam-6326	150	62	hence	hence	ADV
ejpam-6326	150	63	all	all	DET
ejpam-6326	150	64	three	three	NUM
ejpam-6326	150	65	conditions	condition	NOUN
ejpam-6326	150	66	(	(	PUNCT
ejpam-6326	150	67	i)−	i)−	PROPN
ejpam-6326	150	68	(	(	PUNCT
ejpam-6326	150	69	iii	iii	NOUN
ejpam-6326	150	70	)	)	PUNCT
ejpam-6326	150	71	must	must	AUX
ejpam-6326	150	72	hold	hold	VERB
ejpam-6326	150	73	.	.	PUNCT
ejpam-6326	151	1	conversely	conversely	ADV
ejpam-6326	151	2	,	,	PUNCT
ejpam-6326	151	3	suppose	suppose	VERB
ejpam-6326	151	4	that	that	SCONJ
ejpam-6326	151	5	conditions	condition	NOUN
ejpam-6326	151	6	(	(	PUNCT
ejpam-6326	151	7	i	i	NOUN
ejpam-6326	151	8	)	)	PUNCT
ejpam-6326	151	9	,	,	PUNCT
ejpam-6326	151	10	(	(	PUNCT
ejpam-6326	151	11	ii	ii	NOUN
ejpam-6326	151	12	)	)	PUNCT
ejpam-6326	151	13	and	and	CCONJ
ejpam-6326	151	14	(	(	PUNCT
ejpam-6326	151	15	iii	iii	X
ejpam-6326	151	16	)	)	PUNCT
ejpam-6326	151	17	are	be	AUX
ejpam-6326	151	18	satisfied	satisfied	ADJ
ejpam-6326	151	19	.	.	PUNCT
ejpam-6326	152	1	by	by	ADP
ejpam-6326	152	2	condition	condition	NOUN
ejpam-6326	152	3	(	(	PUNCT
ejpam-6326	152	4	ii	ii	NOUN
ejpam-6326	152	5	)	)	PUNCT
ejpam-6326	152	6	the	the	DET
ejpam-6326	152	7	pair	pair	NOUN
ejpam-6326	152	8	{	{	PUNCT
ejpam-6326	152	9	x	x	NOUN
ejpam-6326	152	10	,	,	PUNCT
ejpam-6326	152	11	y	y	PROPN
ejpam-6326	152	12	}	}	PUNCT
ejpam-6326	152	13	dominates	dominate	VERB
ejpam-6326	152	14	the	the	DET
ejpam-6326	152	15	graph	graph	NOUN
ejpam-6326	152	16	,	,	PUNCT
ejpam-6326	152	17	while	while	SCONJ
ejpam-6326	152	18	condition	condition	NOUN
ejpam-6326	152	19	(	(	PUNCT
ejpam-6326	152	20	iii	iii	NOUN
ejpam-6326	152	21	)	)	PUNCT
ejpam-6326	152	22	shows	show	VERB
ejpam-6326	152	23	that	that	SCONJ
ejpam-6326	152	24	the	the	DET
ejpam-6326	152	25	set	set	NOUN
ejpam-6326	152	26	{	{	PUNCT
ejpam-6326	152	27	x	x	NOUN
ejpam-6326	152	28	,	,	PUNCT
ejpam-6326	152	29	y	y	PRON
ejpam-6326	152	30	}	}	PUNCT
ejpam-6326	152	31	is	be	AUX
ejpam-6326	152	32	friendly	friendly	ADJ
ejpam-6326	152	33	.	.	PUNCT
ejpam-6326	153	1	thus	thus	ADV
ejpam-6326	153	2	{	{	PUNCT
ejpam-6326	153	3	x	x	X
ejpam-6326	153	4	,	,	PUNCT
ejpam-6326	153	5	y	y	PRON
ejpam-6326	153	6	}	}	PUNCT
ejpam-6326	153	7	is	be	AUX
ejpam-6326	153	8	a	a	DET
ejpam-6326	153	9	friendly	friendly	ADJ
ejpam-6326	153	10	dominating	dominating	NOUN
ejpam-6326	153	11	set	set	NOUN
ejpam-6326	153	12	,	,	PUNCT
ejpam-6326	153	13	so	so	ADV
ejpam-6326	153	14	γf	γf	PROPN
ejpam-6326	153	15	(	(	PUNCT
ejpam-6326	153	16	g	g	NOUN
ejpam-6326	153	17	)	)	PUNCT
ejpam-6326	153	18	≤	≤	NOUN
ejpam-6326	153	19	2	2	NUM
ejpam-6326	153	20	.	.	PUNCT
ejpam-6326	153	21	condition	condition	NOUN
ejpam-6326	153	22	(	(	PUNCT
ejpam-6326	153	23	i	i	NOUN
ejpam-6326	153	24	)	)	PUNCT
ejpam-6326	153	25	rules	rule	VERB
ejpam-6326	153	26	out	out	ADP
ejpam-6326	153	27	the	the	DET
ejpam-6326	153	28	possibility	possibility	NOUN
ejpam-6326	153	29	that	that	SCONJ
ejpam-6326	153	30	a	a	DET
ejpam-6326	153	31	single	single	ADJ
ejpam-6326	153	32	vertex	vertex	NOUN
ejpam-6326	153	33	can	can	AUX
ejpam-6326	153	34	act	act	VERB
ejpam-6326	153	35	as	as	ADP
ejpam-6326	153	36	a	a	DET
ejpam-6326	153	37	friendly	friendly	ADJ
ejpam-6326	153	38	dominating	dominating	NOUN
ejpam-6326	153	39	set	set	NOUN
ejpam-6326	153	40	.	.	PUNCT
ejpam-6326	154	1	thus	thus	ADV
ejpam-6326	154	2	,	,	PUNCT
ejpam-6326	154	3	γf	γf	INTJ
ejpam-6326	154	4	(	(	PUNCT
ejpam-6326	154	5	g	g	NOUN
ejpam-6326	154	6	)	)	PUNCT
ejpam-6326	154	7	>	>	X
ejpam-6326	154	8	1	1	NUM
ejpam-6326	154	9	,	,	PUNCT
ejpam-6326	154	10	hence	hence	ADV
ejpam-6326	154	11	,	,	PUNCT
ejpam-6326	154	12	γf	γf	PROPN
ejpam-6326	154	13	(	(	PUNCT
ejpam-6326	154	14	g	g	NOUN
ejpam-6326	154	15	)	)	PUNCT
ejpam-6326	154	16	≥	≥	NOUN
ejpam-6326	154	17	2	2	NUM
ejpam-6326	154	18	.	.	PUNCT
ejpam-6326	154	19	therefore	therefore	ADV
ejpam-6326	154	20	,	,	PUNCT
ejpam-6326	154	21	γf	γf	INTJ
ejpam-6326	154	22	(	(	PUNCT
ejpam-6326	154	23	g	g	NOUN
ejpam-6326	154	24	)	)	PUNCT
ejpam-6326	154	25	=	=	SYM
ejpam-6326	154	26	2	2	X
ejpam-6326	154	27	.	.	PUNCT
ejpam-6326	154	28	corollary	corollary	ADJ
ejpam-6326	154	29	2	2	NUM
ejpam-6326	154	30	.	.	PUNCT
ejpam-6326	155	1	let	let	VERB
ejpam-6326	155	2	g	g	NOUN
ejpam-6326	155	3	and	and	CCONJ
ejpam-6326	155	4	h	h	NOUN
ejpam-6326	155	5	be	be	AUX
ejpam-6326	155	6	graphs	graph	NOUN
ejpam-6326	155	7	with	with	ADP
ejpam-6326	155	8	|v	|v	PROPN
ejpam-6326	155	9	(	(	PUNCT
ejpam-6326	155	10	g)|	g)|	PROPN
ejpam-6326	155	11	,	,	PUNCT
ejpam-6326	155	12	|v	|v	PROPN
ejpam-6326	155	13	(	(	PUNCT
ejpam-6326	155	14	h)|	h)|	PROPN
ejpam-6326	155	15	≥	≥	NUM
ejpam-6326	155	16	2	2	NUM
ejpam-6326	155	17	such	such	ADJ
ejpam-6326	155	18	that	that	DET
ejpam-6326	155	19	∆(g	∆(g	NOUN
ejpam-6326	155	20	)	)	PUNCT
ejpam-6326	155	21	<	<	X
ejpam-6326	155	22	|v	|v	PROPN
ejpam-6326	155	23	(	(	PUNCT
ejpam-6326	155	24	g)|−1	g)|−1	NOUN
ejpam-6326	155	25	and	and	CCONJ
ejpam-6326	155	26	∆(h	∆(h	VERB
ejpam-6326	155	27	)	)	PUNCT
ejpam-6326	155	28	<	<	X
ejpam-6326	155	29	|v	|v	PROPN
ejpam-6326	155	30	(	(	PUNCT
ejpam-6326	155	31	h)|	h)|	NOUN
ejpam-6326	155	32	−	−	PROPN
ejpam-6326	155	33	1	1	NUM
ejpam-6326	155	34	.	.	PUNCT
ejpam-6326	156	1	then	then	ADV
ejpam-6326	156	2	,	,	PUNCT
ejpam-6326	156	3	γf	γf	PROPN
ejpam-6326	156	4	(	(	PUNCT
ejpam-6326	156	5	g∨h	g∨h	PROPN
ejpam-6326	156	6	)	)	PUNCT
ejpam-6326	156	7	=	=	SYM
ejpam-6326	157	1	2	2	X
ejpam-6326	157	2	.	.	X
ejpam-6326	157	3	in	in	ADP
ejpam-6326	157	4	particular	particular	ADJ
ejpam-6326	157	5	,	,	PUNCT
ejpam-6326	157	6	for	for	ADP
ejpam-6326	157	7	every	every	DET
ejpam-6326	157	8	complete	complete	ADJ
ejpam-6326	157	9	bipartite	bipartite	PROPN
ejpam-6326	157	10	graph	graph	NOUN
ejpam-6326	157	11	km	km	PROPN
ejpam-6326	157	12	,	,	PUNCT
ejpam-6326	157	13	n	n	CCONJ
ejpam-6326	157	14	with	with	ADP
ejpam-6326	157	15	m	m	PROPN
ejpam-6326	157	16	,	,	PUNCT
ejpam-6326	157	17	n	n	PRON
ejpam-6326	157	18	≥	≥	NOUN
ejpam-6326	157	19	2	2	NUM
ejpam-6326	157	20	we	we	PRON
ejpam-6326	157	21	have	have	VERB
ejpam-6326	157	22	γf	γf	PROPN
ejpam-6326	157	23	(	(	PUNCT
ejpam-6326	157	24	km	km	NOUN
ejpam-6326	157	25	,	,	PUNCT
ejpam-6326	157	26	n	n	CCONJ
ejpam-6326	157	27	)	)	PUNCT
ejpam-6326	157	28	=	=	SYM
ejpam-6326	158	1	2	2	X
ejpam-6326	158	2	.	.	PUNCT
ejpam-6326	158	3	proof	proof	NOUN
ejpam-6326	158	4	.	.	PUNCT
ejpam-6326	159	1	this	this	PRON
ejpam-6326	159	2	immediately	immediately	ADV
ejpam-6326	159	3	follows	follow	VERB
ejpam-6326	159	4	from	from	ADP
ejpam-6326	159	5	theorem	theorem	ADJ
ejpam-6326	159	6	4	4	NUM
ejpam-6326	159	7	.	.	PUNCT
ejpam-6326	159	8	theorem	theorem	NOUN
ejpam-6326	159	9	5	5	NUM
ejpam-6326	159	10	.	.	PUNCT
ejpam-6326	160	1	let	let	VERB
ejpam-6326	160	2	a	a	PRON
ejpam-6326	160	3	and	and	CCONJ
ejpam-6326	160	4	b	b	NOUN
ejpam-6326	160	5	be	be	AUX
ejpam-6326	160	6	positive	positive	ADJ
ejpam-6326	160	7	integers	integer	NOUN
ejpam-6326	160	8	such	such	ADJ
ejpam-6326	160	9	that	that	SCONJ
ejpam-6326	160	10	a	a	DET
ejpam-6326	160	11	≤	≤	PROPN
ejpam-6326	160	12	b.	b.	NOUN
ejpam-6326	160	13	then	then	ADV
ejpam-6326	160	14	there	there	PRON
ejpam-6326	160	15	exist	exist	VERB
ejpam-6326	160	16	a	a	DET
ejpam-6326	160	17	nontrivial	nontrivial	ADJ
ejpam-6326	160	18	connected	connect	VERB
ejpam-6326	160	19	graph	graph	NOUN
ejpam-6326	160	20	g	g	ADP
ejpam-6326	160	21	such	such	ADJ
ejpam-6326	160	22	that	that	DET
ejpam-6326	160	23	γ(g	γ(g	PROPN
ejpam-6326	160	24	)	)	PUNCT
ejpam-6326	161	1	=	=	SYM
ejpam-6326	162	1	a	a	PROPN
ejpam-6326	162	2	and	and	CCONJ
ejpam-6326	162	3	γf	γf	PROPN
ejpam-6326	162	4	(	(	PUNCT
ejpam-6326	162	5	g	g	NOUN
ejpam-6326	162	6	)	)	PUNCT
ejpam-6326	162	7	=	=	SYM
ejpam-6326	163	1	b.	b.	PROPN
ejpam-6326	163	2	proof	proof	NOUN
ejpam-6326	163	3	.	.	PUNCT
ejpam-6326	164	1	let	let	VERB
ejpam-6326	164	2	a	a	PRON
ejpam-6326	164	3	and	and	CCONJ
ejpam-6326	164	4	b	b	NOUN
ejpam-6326	164	5	be	be	AUX
ejpam-6326	164	6	positive	positive	ADJ
ejpam-6326	164	7	integers	integer	NOUN
ejpam-6326	164	8	with	with	ADP
ejpam-6326	164	9	a	a	DET
ejpam-6326	164	10	≤	≤	PROPN
ejpam-6326	164	11	b.	b.	NOUN
ejpam-6326	164	12	consider	consider	VERB
ejpam-6326	164	13	the	the	DET
ejpam-6326	164	14	following	follow	VERB
ejpam-6326	164	15	cases	case	NOUN
ejpam-6326	164	16	:	:	PUNCT
ejpam-6326	164	17	case	case	NOUN
ejpam-6326	164	18	1	1	NUM
ejpam-6326	164	19	:	:	PUNCT
ejpam-6326	164	20	a	a	DET
ejpam-6326	164	21	=	=	X
ejpam-6326	164	22	b	b	NOUN
ejpam-6326	164	23	consider	consider	VERB
ejpam-6326	164	24	the	the	DET
ejpam-6326	164	25	graph	graph	NOUN
ejpam-6326	164	26	g	g	NOUN
ejpam-6326	164	27	shown	show	VERB
ejpam-6326	164	28	in	in	ADP
ejpam-6326	164	29	figure	figure	NOUN
ejpam-6326	164	30	1	1	NUM
ejpam-6326	164	31	.	.	PUNCT
ejpam-6326	165	1	clearly	clearly	ADV
ejpam-6326	165	2	,	,	PUNCT
ejpam-6326	165	3	the	the	DET
ejpam-6326	165	4	set	set	NOUN
ejpam-6326	165	5	f	f	X
ejpam-6326	165	6	=	=	PRON
ejpam-6326	165	7	{	{	PUNCT
ejpam-6326	165	8	xi|i	xi|i	X
ejpam-6326	165	9	=	=	SYM
ejpam-6326	165	10	1	1	NUM
ejpam-6326	165	11	,	,	PUNCT
ejpam-6326	165	12	2	2	NUM
ejpam-6326	165	13	,	,	PUNCT
ejpam-6326	165	14	3	3	NUM
ejpam-6326	165	15	,	,	PUNCT
ejpam-6326	165	16	.	.	PUNCT
ejpam-6326	165	17	.	.	PUNCT
ejpam-6326	165	18	.	.	PUNCT
ejpam-6326	166	1	,	,	PUNCT
ejpam-6326	166	2	a	a	DET
ejpam-6326	166	3	−	−	PROPN
ejpam-6326	166	4	1	1	NUM
ejpam-6326	166	5	,	,	PUNCT
ejpam-6326	166	6	a	a	PRON
ejpam-6326	166	7	}	}	PUNCT
ejpam-6326	166	8	is	be	AUX
ejpam-6326	166	9	both	both	PRON
ejpam-6326	166	10	a	a	DET
ejpam-6326	166	11	γ	γ	X
ejpam-6326	166	12	-	-	PUNCT
ejpam-6326	166	13	set	set	VERB
ejpam-6326	166	14	and	and	CCONJ
ejpam-6326	166	15	a	a	DET
ejpam-6326	166	16	γf	γf	NOUN
ejpam-6326	166	17	-set	-set	PUNCT
ejpam-6326	166	18	of	of	ADP
ejpam-6326	166	19	g.	g.	PROPN
ejpam-6326	166	20	thus	thus	ADV
ejpam-6326	166	21	,	,	PUNCT
ejpam-6326	166	22	γ(g	γ(g	PROPN
ejpam-6326	166	23	)	)	PUNCT
ejpam-6326	167	1	=	=	SYM
ejpam-6326	167	2	γf	γf	ADJ
ejpam-6326	167	3	(	(	PUNCT
ejpam-6326	167	4	g	g	NOUN
ejpam-6326	167	5	)	)	PUNCT
ejpam-6326	167	6	=	=	SYM
ejpam-6326	167	7	|f	|f	PROPN
ejpam-6326	168	1	|	|	ADV
ejpam-6326	168	2	=	=	PUNCT
ejpam-6326	168	3	a	a	PROPN
ejpam-6326	168	4	=	=	X
ejpam-6326	168	5	b.	b.	PROPN
ejpam-6326	168	6	this	this	PRON
ejpam-6326	168	7	completes	complete	VERB
ejpam-6326	168	8	the	the	DET
ejpam-6326	168	9	proof	proof	NOUN
ejpam-6326	168	10	for	for	ADP
ejpam-6326	168	11	the	the	DET
ejpam-6326	168	12	case	case	NOUN
ejpam-6326	168	13	a	a	DET
ejpam-6326	168	14	=	=	X
ejpam-6326	168	15	b.	b.	NOUN
ejpam-6326	169	1	x1	x1	PROPN
ejpam-6326	170	1	x2	x2	NOUN
ejpam-6326	170	2	x3	x3	PROPN
ejpam-6326	170	3	·	·	PUNCT
ejpam-6326	170	4	·	·	PUNCT
ejpam-6326	170	5	·	·	PUNCT
ejpam-6326	170	6	xa−1	xa−1	PROPN
ejpam-6326	170	7	xa	xa	PROPN
ejpam-6326	170	8	figure	figure	VERB
ejpam-6326	170	9	1	1	NUM
ejpam-6326	170	10	:	:	PUNCT
ejpam-6326	170	11	a	a	DET
ejpam-6326	170	12	graph	graph	NOUN
ejpam-6326	170	13	with	with	ADP
ejpam-6326	170	14	γ(g	γ(g	PROPN
ejpam-6326	170	15	)	)	PUNCT
ejpam-6326	171	1	=	=	SYM
ejpam-6326	171	2	γf	γf	ADJ
ejpam-6326	171	3	(	(	PUNCT
ejpam-6326	171	4	g	g	NOUN
ejpam-6326	171	5	)	)	PUNCT
ejpam-6326	171	6	case	case	NOUN
ejpam-6326	171	7	2	2	NUM
ejpam-6326	171	8	:	:	PUNCT
ejpam-6326	171	9	a	a	DET
ejpam-6326	171	10	<	<	X
ejpam-6326	171	11	b	b	X
ejpam-6326	171	12	letg	letg	NOUN
ejpam-6326	171	13	be	be	VERB
ejpam-6326	171	14	the	the	DET
ejpam-6326	171	15	graph	graph	NOUN
ejpam-6326	171	16	shown	show	VERB
ejpam-6326	171	17	in	in	ADP
ejpam-6326	171	18	figure	figure	NOUN
ejpam-6326	171	19	2	2	NUM
ejpam-6326	171	20	.	.	X
ejpam-6326	171	21	observe	observe	VERB
ejpam-6326	171	22	that	that	SCONJ
ejpam-6326	171	23	the	the	DET
ejpam-6326	171	24	setd	setd	NOUN
ejpam-6326	171	25	=	=	PUNCT
ejpam-6326	171	26	{	{	PUNCT
ejpam-6326	171	27	x1	x1	PROPN
ejpam-6326	171	28	,	,	PUNCT
ejpam-6326	171	29	x2	x2	PROPN
ejpam-6326	171	30	,	,	PUNCT
ejpam-6326	171	31	x3	x3	ADJ
ejpam-6326	171	32	,	,	PUNCT
ejpam-6326	171	33	.	.	PUNCT
ejpam-6326	171	34	.	.	PUNCT
ejpam-6326	172	1	.	.	PUNCT
ejpam-6326	173	1	,	,	PUNCT
ejpam-6326	173	2	xa−1	xa−1	PROPN
ejpam-6326	173	3	,	,	PUNCT
ejpam-6326	173	4	xa	xa	PROPN
ejpam-6326	173	5	}	}	PUNCT
ejpam-6326	173	6	is	be	AUX
ejpam-6326	173	7	a	a	DET
ejpam-6326	173	8	γ	γ	NOUN
ejpam-6326	173	9	-	-	PUNCT
ejpam-6326	173	10	set	set	VERB
ejpam-6326	173	11	and	and	CCONJ
ejpam-6326	173	12	the	the	DET
ejpam-6326	173	13	set	set	NOUN
ejpam-6326	173	14	f	f	PROPN
ejpam-6326	173	15	=	=	SYM
ejpam-6326	173	16	d∪{y1	d∪{y1	PROPN
ejpam-6326	173	17	,	,	PUNCT
ejpam-6326	173	18	y2	y2	PROPN
ejpam-6326	173	19	,	,	PUNCT
ejpam-6326	173	20	y3	y3	PROPN
ejpam-6326	173	21	,	,	PUNCT
ejpam-6326	173	22	.	.	PUNCT
ejpam-6326	173	23	.	.	PUNCT
ejpam-6326	174	1	.	.	PUNCT
ejpam-6326	175	1	,	,	PUNCT
ejpam-6326	175	2	yb−a−1	yb−a−1	PROPN
ejpam-6326	175	3	,	,	PUNCT
ejpam-6326	175	4	yb−a	yb−a	PRON
ejpam-6326	175	5	}	}	PUNCT
ejpam-6326	175	6	is	be	AUX
ejpam-6326	175	7	a	a	DET
ejpam-6326	175	8	γf	γf	PROPN
ejpam-6326	175	9	-set	-set	NOUN
ejpam-6326	175	10	.	.	PUNCT
ejpam-6326	176	1	thus	thus	ADV
ejpam-6326	176	2	,	,	PUNCT
ejpam-6326	176	3	γ(g	γ(g	PROPN
ejpam-6326	176	4	)	)	PUNCT
ejpam-6326	177	1	=	=	SYM
ejpam-6326	177	2	|d|	|d|	PROPN
ejpam-6326	177	3	=	=	PUNCT
ejpam-6326	177	4	a	a	PROPN
ejpam-6326	177	5	and	and	CCONJ
ejpam-6326	177	6	γf	γf	PROPN
ejpam-6326	177	7	(	(	PUNCT
ejpam-6326	177	8	g	g	NOUN
ejpam-6326	177	9	)	)	PUNCT
ejpam-6326	177	10	=	=	SYM
ejpam-6326	177	11	|f	|f	PROPN
ejpam-6326	178	1	|	|	NOUN
ejpam-6326	178	2	=	=	PRON
ejpam-6326	178	3	|d|+	|d|+	VERB
ejpam-6326	178	4	|{y1	|{y1	NOUN
ejpam-6326	178	5	,	,	PUNCT
ejpam-6326	178	6	y2	y2	PROPN
ejpam-6326	178	7	,	,	PUNCT
ejpam-6326	178	8	y3	y3	PROPN
ejpam-6326	178	9	,	,	PUNCT
ejpam-6326	178	10	.	.	PUNCT
ejpam-6326	178	11	.	.	PUNCT
ejpam-6326	179	1	.	.	PUNCT
ejpam-6326	180	1	,	,	PUNCT
ejpam-6326	180	2	yb−a−1	yb−a−1	PROPN
ejpam-6326	180	3	,	,	PUNCT
ejpam-6326	180	4	yb−a}|	yb−a}|	PROPN
ejpam-6326	180	5	=	=	PUNCT
ejpam-6326	180	6	a+	a+	PUNCT
ejpam-6326	180	7	b−	b−	PROPN
ejpam-6326	180	8	a	a	DET
ejpam-6326	180	9	=	=	X
ejpam-6326	180	10	b.	b.	PROPN
ejpam-6326	181	1	this	this	PRON
ejpam-6326	181	2	proves	prove	VERB
ejpam-6326	181	3	the	the	DET
ejpam-6326	181	4	assertion	assertion	NOUN
ejpam-6326	181	5	.	.	PUNCT
ejpam-6326	182	1	corollary	corollary	ADJ
ejpam-6326	182	2	3	3	NUM
ejpam-6326	182	3	.	.	PUNCT
ejpam-6326	183	1	for	for	ADP
ejpam-6326	183	2	each	each	DET
ejpam-6326	183	3	positive	positive	ADJ
ejpam-6326	183	4	integer	integer	NOUN
ejpam-6326	183	5	n	n	CCONJ
ejpam-6326	183	6	,	,	PUNCT
ejpam-6326	183	7	there	there	PRON
ejpam-6326	183	8	exist	exist	VERB
ejpam-6326	183	9	a	a	DET
ejpam-6326	183	10	connected	connected	ADJ
ejpam-6326	183	11	graph	graph	NOUN
ejpam-6326	183	12	g	g	ADP
ejpam-6326	183	13	such	such	ADJ
ejpam-6326	183	14	that	that	DET
ejpam-6326	183	15	γf	γf	PROPN
ejpam-6326	183	16	(	(	PUNCT
ejpam-6326	183	17	g)−	g)−	PROPN
ejpam-6326	183	18	γ(g	γ(g	PROPN
ejpam-6326	183	19	)	)	PUNCT
ejpam-6326	183	20	=	=	SYM
ejpam-6326	184	1	n	n	CCONJ
ejpam-6326	184	2	,	,	PUNCT
ejpam-6326	184	3	that	that	ADV
ejpam-6326	184	4	is	is	ADV
ejpam-6326	184	5	,	,	PUNCT
ejpam-6326	184	6	the	the	DET
ejpam-6326	184	7	difference	difference	NOUN
ejpam-6326	184	8	between	between	ADP
ejpam-6326	184	9	γf	γf	PROPN
ejpam-6326	184	10	(	(	PUNCT
ejpam-6326	184	11	g	g	NOUN
ejpam-6326	184	12	)	)	PUNCT
ejpam-6326	184	13	and	and	CCONJ
ejpam-6326	184	14	γ(g	γ(g	PROPN
ejpam-6326	184	15	)	)	PUNCT
ejpam-6326	184	16	can	can	AUX
ejpam-6326	184	17	be	be	AUX
ejpam-6326	184	18	made	make	VERB
ejpam-6326	184	19	arbitrarily	arbitrarily	ADV
ejpam-6326	184	20	large	large	ADJ
ejpam-6326	184	21	.	.	PUNCT
ejpam-6326	185	1	proof	proof	NOUN
ejpam-6326	185	2	.	.	PUNCT
ejpam-6326	186	1	by	by	ADP
ejpam-6326	186	2	theorem	theorem	NOUN
ejpam-6326	186	3	5	5	NUM
ejpam-6326	186	4	,	,	PUNCT
ejpam-6326	186	5	for	for	ADP
ejpam-6326	186	6	any	any	DET
ejpam-6326	186	7	pair	pair	NOUN
ejpam-6326	186	8	of	of	ADP
ejpam-6326	186	9	positive	positive	ADJ
ejpam-6326	186	10	integers	integer	NOUN
ejpam-6326	186	11	a	a	PRON
ejpam-6326	186	12	and	and	CCONJ
ejpam-6326	186	13	b	b	NOUN
ejpam-6326	186	14	with	with	ADP
ejpam-6326	186	15	a	a	DET
ejpam-6326	186	16	<	<	X
ejpam-6326	186	17	b	b	NOUN
ejpam-6326	186	18	,	,	PUNCT
ejpam-6326	186	19	there	there	PRON
ejpam-6326	186	20	exists	exist	VERB
ejpam-6326	186	21	a	a	DET
ejpam-6326	186	22	connected	connected	ADJ
ejpam-6326	186	23	nontrivial	nontrivial	ADJ
ejpam-6326	186	24	graph	graph	NOUN
ejpam-6326	186	25	h	h	NOUN
ejpam-6326	186	26	such	such	ADJ
ejpam-6326	186	27	that	that	DET
ejpam-6326	186	28	γ(h	γ(h	NOUN
ejpam-6326	186	29	)	)	PUNCT
ejpam-6326	186	30	=	=	PUNCT
ejpam-6326	187	1	a	a	PROPN
ejpam-6326	187	2	and	and	CCONJ
ejpam-6326	187	3	γf	γf	PROPN
ejpam-6326	187	4	(	(	PUNCT
ejpam-6326	187	5	h	h	NOUN
ejpam-6326	187	6	)	)	PUNCT
ejpam-6326	187	7	=	=	SYM
ejpam-6326	187	8	b.	b.	PROPN
ejpam-6326	187	9	taking	take	VERB
ejpam-6326	187	10	a	a	DET
ejpam-6326	187	11	=	=	SYM
ejpam-6326	187	12	1	1	NUM
ejpam-6326	187	13	and	and	CCONJ
ejpam-6326	187	14	i.	i.	PROPN
ejpam-6326	187	15	s.	s.	PROPN
ejpam-6326	187	16	cabahug	cabahug	PROPN
ejpam-6326	187	17	,	,	PUNCT
ejpam-6326	187	18	jr	jr	PROPN
ejpam-6326	187	19	.	.	PROPN
ejpam-6326	187	20	,	,	PUNCT
ejpam-6326	187	21	r.	r.	PROPN
ejpam-6326	187	22	g.	g.	PROPN
ejpam-6326	187	23	eballe	eballe	PROPN
ejpam-6326	187	24	,	,	PUNCT
ejpam-6326	187	25	r.	r.	PROPN
ejpam-6326	187	26	t.	t.	PROPN
ejpam-6326	187	27	fernandez	fernandez	PROPN
ejpam-6326	187	28	/	/	SYM
ejpam-6326	187	29	eur	eur	PROPN
ejpam-6326	187	30	.	.	PUNCT
ejpam-6326	188	1	j.	j.	PROPN
ejpam-6326	188	2	pure	pure	PROPN
ejpam-6326	188	3	appl	appl	PROPN
ejpam-6326	188	4	.	.	PROPN
ejpam-6326	188	5	math	math	PROPN
ejpam-6326	188	6	,	,	PUNCT
ejpam-6326	188	7	18	18	NUM
ejpam-6326	188	8	(	(	PUNCT
ejpam-6326	188	9	4	4	NUM
ejpam-6326	188	10	)	)	PUNCT
ejpam-6326	188	11	(	(	PUNCT
ejpam-6326	188	12	2025	2025	NUM
ejpam-6326	188	13	)	)	PUNCT
ejpam-6326	188	14	,	,	PUNCT
ejpam-6326	188	15	6326	6326	NUM
ejpam-6326	188	16	7	7	NUM
ejpam-6326	188	17	of	of	ADP
ejpam-6326	188	18	15	15	NUM
ejpam-6326	189	1	x1	x1	NOUN
ejpam-6326	189	2	x2	x2	NOUN
ejpam-6326	189	3	x3	x3	PROPN
ejpam-6326	189	4	·	·	PUNCT
ejpam-6326	189	5	·	·	PUNCT
ejpam-6326	189	6	·	·	PUNCT
ejpam-6326	190	1	xa−1	xa−1	PROPN
ejpam-6326	190	2	xa	xa	PROPN
ejpam-6326	190	3	y1	y1	INTJ
ejpam-6326	190	4	y2	y2	PROPN
ejpam-6326	190	5	y3	y3	NOUN
ejpam-6326	190	6	...	...	PUNCT
ejpam-6326	190	7	yb−a−1	yb−a−1	PROPN
ejpam-6326	190	8	yb−a	yb−a	PROPN
ejpam-6326	190	9	figure	figure	NOUN
ejpam-6326	190	10	2	2	NUM
ejpam-6326	190	11	:	:	PUNCT
ejpam-6326	190	12	a	a	DET
ejpam-6326	190	13	graph	graph	NOUN
ejpam-6326	190	14	with	with	ADP
ejpam-6326	190	15	γ(g	γ(g	PROPN
ejpam-6326	190	16	)	)	PUNCT
ejpam-6326	190	17	<	<	X
ejpam-6326	191	1	γf	γf	PROPN
ejpam-6326	191	2	(	(	PUNCT
ejpam-6326	191	3	g	g	NOUN
ejpam-6326	191	4	)	)	PUNCT
ejpam-6326	191	5	b	b	NOUN
ejpam-6326	191	6	=	=	SYM
ejpam-6326	191	7	n	n	PROPN
ejpam-6326	191	8	+	+	NOUN
ejpam-6326	191	9	1	1	NUM
ejpam-6326	191	10	.	.	PUNCT
ejpam-6326	192	1	thus	thus	ADV
ejpam-6326	192	2	,	,	PUNCT
ejpam-6326	192	3	we	we	PRON
ejpam-6326	192	4	have	have	VERB
ejpam-6326	192	5	γf	γf	ADJ
ejpam-6326	192	6	(	(	PUNCT
ejpam-6326	192	7	g	g	NOUN
ejpam-6326	192	8	)	)	PUNCT
ejpam-6326	192	9	−	−	PROPN
ejpam-6326	192	10	γ(g	γ(g	PROPN
ejpam-6326	192	11	)	)	PUNCT
ejpam-6326	192	12	=	=	PUNCT
ejpam-6326	192	13	(	(	PUNCT
ejpam-6326	192	14	n	n	PROPN
ejpam-6326	192	15	+	+	NUM
ejpam-6326	192	16	1	1	NUM
ejpam-6326	192	17	)	)	PUNCT
ejpam-6326	193	1	−	−	PROPN
ejpam-6326	193	2	1	1	NUM
ejpam-6326	193	3	=	=	SYM
ejpam-6326	193	4	n.	n.	NOUN
ejpam-6326	193	5	since	since	SCONJ
ejpam-6326	193	6	n	n	PRON
ejpam-6326	193	7	is	be	AUX
ejpam-6326	193	8	arbitrary	arbitrary	ADJ
ejpam-6326	193	9	,	,	PUNCT
ejpam-6326	193	10	the	the	DET
ejpam-6326	193	11	difference	difference	NOUN
ejpam-6326	193	12	γf	γf	INTJ
ejpam-6326	193	13	(	(	PUNCT
ejpam-6326	193	14	g)−	g)−	PROPN
ejpam-6326	193	15	γ(g	γ(g	PROPN
ejpam-6326	193	16	)	)	PUNCT
ejpam-6326	193	17	can	can	AUX
ejpam-6326	193	18	be	be	AUX
ejpam-6326	193	19	made	make	VERB
ejpam-6326	193	20	as	as	ADV
ejpam-6326	193	21	large	large	ADJ
ejpam-6326	193	22	as	as	SCONJ
ejpam-6326	193	23	desired	desire	VERB
ejpam-6326	193	24	.	.	PUNCT
ejpam-6326	194	1	theorem	theorem	NOUN
ejpam-6326	194	2	6	6	NUM
ejpam-6326	194	3	.	.	PUNCT
ejpam-6326	195	1	let	let	VERB
ejpam-6326	195	2	g	g	PRON
ejpam-6326	195	3	be	be	AUX
ejpam-6326	195	4	a	a	DET
ejpam-6326	195	5	connected	connected	ADJ
ejpam-6326	195	6	nontrivial	nontrivial	ADJ
ejpam-6326	195	7	graph	graph	NOUN
ejpam-6326	195	8	with	with	ADP
ejpam-6326	195	9	maximum	maximum	ADJ
ejpam-6326	195	10	degree	degree	NOUN
ejpam-6326	195	11	at	at	ADP
ejpam-6326	195	12	most	most	ADV
ejpam-6326	195	13	2	2	NUM
ejpam-6326	195	14	,	,	PUNCT
ejpam-6326	195	15	and	and	CCONJ
ejpam-6326	195	16	let	let	VERB
ejpam-6326	195	17	f	f	PROPN
ejpam-6326	195	18	⊆	⊆	NUM
ejpam-6326	195	19	v	v	NOUN
ejpam-6326	195	20	(	(	PUNCT
ejpam-6326	195	21	g	g	NOUN
ejpam-6326	195	22	)	)	PUNCT
ejpam-6326	195	23	be	be	AUX
ejpam-6326	195	24	a	a	DET
ejpam-6326	195	25	nonempty	nonempty	NOUN
ejpam-6326	195	26	subset	subset	NOUN
ejpam-6326	195	27	.	.	PUNCT
ejpam-6326	196	1	then	then	ADV
ejpam-6326	196	2	f	f	PROPN
ejpam-6326	196	3	is	be	AUX
ejpam-6326	196	4	a	a	DET
ejpam-6326	196	5	friendly	friendly	ADJ
ejpam-6326	196	6	dominating	dominating	NOUN
ejpam-6326	196	7	set	set	NOUN
ejpam-6326	196	8	of	of	ADP
ejpam-6326	196	9	g	g	PROPN
ejpam-6326	196	10	if	if	SCONJ
ejpam-6326	197	1	and	and	CCONJ
ejpam-6326	197	2	only	only	ADV
ejpam-6326	197	3	if	if	SCONJ
ejpam-6326	197	4	the	the	DET
ejpam-6326	197	5	following	follow	VERB
ejpam-6326	197	6	conditions	condition	NOUN
ejpam-6326	197	7	hold	hold	VERB
ejpam-6326	197	8	:	:	PUNCT
ejpam-6326	197	9	(	(	PUNCT
ejpam-6326	197	10	i	i	NOUN
ejpam-6326	197	11	)	)	PUNCT
ejpam-6326	197	12	every	every	DET
ejpam-6326	197	13	vertex	vertex	NOUN
ejpam-6326	197	14	v	v	ADP
ejpam-6326	197	15	∈	∈	PROPN
ejpam-6326	197	16	v	v	NOUN
ejpam-6326	197	17	(	(	PUNCT
ejpam-6326	197	18	g	g	NOUN
ejpam-6326	197	19	)	)	PUNCT
ejpam-6326	197	20	\	\	PUNCT
ejpam-6326	198	1	f	f	PROPN
ejpam-6326	198	2	satisfies	satisfy	VERB
ejpam-6326	198	3	degf	degf	PROPN
ejpam-6326	198	4	(	(	PUNCT
ejpam-6326	198	5	v	v	NOUN
ejpam-6326	198	6	)	)	PUNCT
ejpam-6326	198	7	=	=	SYM
ejpam-6326	198	8	1	1	NUM
ejpam-6326	198	9	;	;	PUNCT
ejpam-6326	198	10	that	that	PRON
ejpam-6326	198	11	is	is	ADV
ejpam-6326	198	12	,	,	PUNCT
ejpam-6326	198	13	every	every	DET
ejpam-6326	198	14	vertex	vertex	NOUN
ejpam-6326	198	15	outside	outside	ADP
ejpam-6326	198	16	f	f	PROPN
ejpam-6326	198	17	is	be	AUX
ejpam-6326	198	18	adjacent	adjacent	ADJ
ejpam-6326	198	19	to	to	ADP
ejpam-6326	198	20	exactly	exactly	ADV
ejpam-6326	198	21	one	one	NUM
ejpam-6326	198	22	vertex	vertex	NOUN
ejpam-6326	198	23	in	in	ADP
ejpam-6326	198	24	f	f	PROPN
ejpam-6326	198	25	.	.	PUNCT
ejpam-6326	199	1	(	(	PUNCT
ejpam-6326	199	2	ii	ii	NOUN
ejpam-6326	199	3	)	)	PUNCT
ejpam-6326	199	4	every	every	DET
ejpam-6326	199	5	vertex	vertex	NOUN
ejpam-6326	199	6	v	v	ADP
ejpam-6326	199	7	∈	∈	PROPN
ejpam-6326	199	8	v	v	NOUN
ejpam-6326	199	9	(	(	PUNCT
ejpam-6326	199	10	g	g	NOUN
ejpam-6326	199	11	)	)	PUNCT
ejpam-6326	199	12	with	with	ADP
ejpam-6326	199	13	deg(v	deg(v	PROPN
ejpam-6326	199	14	)	)	PUNCT
ejpam-6326	199	15	=	=	SYM
ejpam-6326	199	16	1	1	NUM
ejpam-6326	199	17	belongs	belong	VERB
ejpam-6326	199	18	to	to	ADP
ejpam-6326	199	19	f	f	PROPN
ejpam-6326	199	20	.	.	PUNCT
ejpam-6326	200	1	proof	proof	NOUN
ejpam-6326	200	2	.	.	PUNCT
ejpam-6326	201	1	let	let	VERB
ejpam-6326	201	2	f	f	PROPN
ejpam-6326	201	3	⊆	⊆	NUM
ejpam-6326	201	4	v	v	X
ejpam-6326	201	5	(	(	PUNCT
ejpam-6326	201	6	g	g	NOUN
ejpam-6326	201	7	)	)	PUNCT
ejpam-6326	201	8	be	be	AUX
ejpam-6326	201	9	a	a	DET
ejpam-6326	201	10	friendly	friendly	ADJ
ejpam-6326	201	11	dominating	dominating	NOUN
ejpam-6326	201	12	set	set	NOUN
ejpam-6326	201	13	of	of	ADP
ejpam-6326	201	14	g.	g.	PROPN
ejpam-6326	201	15	by	by	ADP
ejpam-6326	201	16	theorem	theorem	NOUN
ejpam-6326	201	17	1	1	NUM
ejpam-6326	201	18	,	,	PUNCT
ejpam-6326	201	19	every	every	DET
ejpam-6326	201	20	vertex	vertex	NOUN
ejpam-6326	201	21	u	u	NOUN
ejpam-6326	201	22	∈	∈	PROPN
ejpam-6326	201	23	v	v	ADP
ejpam-6326	201	24	(	(	PUNCT
ejpam-6326	201	25	g	g	NOUN
ejpam-6326	201	26	)	)	PUNCT
ejpam-6326	201	27	with	with	ADP
ejpam-6326	201	28	deg(u	deg(u	PROPN
ejpam-6326	201	29	)	)	PUNCT
ejpam-6326	201	30	=	=	SYM
ejpam-6326	201	31	1	1	NUM
ejpam-6326	201	32	belongs	belong	VERB
ejpam-6326	201	33	to	to	ADP
ejpam-6326	201	34	f	f	PROPN
ejpam-6326	201	35	.	.	PUNCT
ejpam-6326	202	1	since	since	SCONJ
ejpam-6326	202	2	the	the	DET
ejpam-6326	202	3	maximum	maximum	ADJ
ejpam-6326	202	4	degree	degree	NOUN
ejpam-6326	202	5	in	in	ADP
ejpam-6326	202	6	g	g	PROPN
ejpam-6326	202	7	is	be	AUX
ejpam-6326	202	8	2	2	NUM
ejpam-6326	202	9	,	,	PUNCT
ejpam-6326	202	10	each	each	DET
ejpam-6326	202	11	vertex	vertex	NOUN
ejpam-6326	202	12	v	v	ADP
ejpam-6326	202	13	satisfies	satisfie	NOUN
ejpam-6326	202	14	deg(v	deg(v	PROPN
ejpam-6326	202	15	)	)	PUNCT
ejpam-6326	202	16	≤	≤	NOUN
ejpam-6326	202	17	2	2	NUM
ejpam-6326	202	18	.	.	PUNCT
ejpam-6326	203	1	if	if	SCONJ
ejpam-6326	203	2	a	a	DET
ejpam-6326	203	3	vertex	vertex	NOUN
ejpam-6326	203	4	v	v	NOUN
ejpam-6326	203	5	/∈	/∈	PUNCT
ejpam-6326	204	1	f	f	PROPN
ejpam-6326	204	2	were	be	AUX
ejpam-6326	204	3	adjacent	adjacent	ADJ
ejpam-6326	204	4	to	to	ADP
ejpam-6326	204	5	two	two	NUM
ejpam-6326	204	6	vertices	vertex	NOUN
ejpam-6326	204	7	in	in	ADP
ejpam-6326	204	8	f	f	PROPN
ejpam-6326	204	9	,	,	PUNCT
ejpam-6326	204	10	then	then	ADV
ejpam-6326	204	11	degf	degf	PROPN
ejpam-6326	204	12	(	(	PUNCT
ejpam-6326	204	13	v	v	NOUN
ejpam-6326	204	14	)	)	PUNCT
ejpam-6326	204	15	=	=	SYM
ejpam-6326	204	16	2	2	NUM
ejpam-6326	204	17	and	and	CCONJ
ejpam-6326	204	18	consequently	consequently	ADV
ejpam-6326	204	19	degv	degv	NOUN
ejpam-6326	204	20	(	(	PUNCT
ejpam-6326	204	21	g)\f	g)\f	NOUN
ejpam-6326	204	22	(	(	PUNCT
ejpam-6326	204	23	v	v	NOUN
ejpam-6326	204	24	)	)	PUNCT
ejpam-6326	204	25	=	=	SYM
ejpam-6326	204	26	0	0	NUM
ejpam-6326	204	27	,	,	PUNCT
ejpam-6326	204	28	which	which	PRON
ejpam-6326	204	29	is	be	AUX
ejpam-6326	204	30	a	a	DET
ejpam-6326	204	31	contradiction	contradiction	NOUN
ejpam-6326	204	32	to	to	ADP
ejpam-6326	204	33	the	the	DET
ejpam-6326	204	34	assumption	assumption	NOUN
ejpam-6326	204	35	of	of	ADP
ejpam-6326	204	36	f	f	PROPN
ejpam-6326	204	37	.	.	PUNCT
ejpam-6326	205	1	therefore	therefore	ADV
ejpam-6326	205	2	,	,	PUNCT
ejpam-6326	205	3	every	every	DET
ejpam-6326	205	4	vertex	vertex	NOUN
ejpam-6326	205	5	v	v	NOUN
ejpam-6326	205	6	/∈	/∈	PUNCT
ejpam-6326	206	1	f	f	PROPN
ejpam-6326	206	2	must	must	AUX
ejpam-6326	206	3	satisfy	satisfy	VERB
ejpam-6326	206	4	degf	degf	PROPN
ejpam-6326	206	5	(	(	PUNCT
ejpam-6326	206	6	v	v	NOUN
ejpam-6326	206	7	)	)	PUNCT
ejpam-6326	206	8	=	=	SYM
ejpam-6326	207	1	1	1	X
ejpam-6326	207	2	.	.	PUNCT
ejpam-6326	207	3	conversely	conversely	ADV
ejpam-6326	207	4	,	,	PUNCT
ejpam-6326	207	5	assume	assume	VERB
ejpam-6326	207	6	that	that	SCONJ
ejpam-6326	207	7	for	for	ADP
ejpam-6326	207	8	every	every	DET
ejpam-6326	207	9	vertex	vertex	NOUN
ejpam-6326	207	10	v	v	NOUN
ejpam-6326	207	11	/∈	/∈	PUNCT
ejpam-6326	208	1	f	f	AUX
ejpam-6326	208	2	we	we	PRON
ejpam-6326	208	3	have	have	VERB
ejpam-6326	208	4	degf	degf	PROPN
ejpam-6326	208	5	(	(	PUNCT
ejpam-6326	208	6	v	v	NOUN
ejpam-6326	208	7	)	)	PUNCT
ejpam-6326	208	8	=	=	SYM
ejpam-6326	208	9	1	1	NUM
ejpam-6326	208	10	and	and	CCONJ
ejpam-6326	208	11	that	that	SCONJ
ejpam-6326	208	12	every	every	DET
ejpam-6326	208	13	leaf	leaf	NOUN
ejpam-6326	208	14	of	of	ADP
ejpam-6326	208	15	g	g	PROPN
ejpam-6326	208	16	is	be	AUX
ejpam-6326	208	17	contained	contain	VERB
ejpam-6326	208	18	in	in	ADP
ejpam-6326	208	19	f	f	PROPN
ejpam-6326	208	20	.	.	PUNCT
ejpam-6326	209	1	then	then	ADV
ejpam-6326	209	2	f	f	PROPN
ejpam-6326	209	3	is	be	AUX
ejpam-6326	209	4	clearly	clearly	ADV
ejpam-6326	209	5	a	a	DET
ejpam-6326	209	6	dominating	dominating	NOUN
ejpam-6326	209	7	set	set	NOUN
ejpam-6326	209	8	since	since	SCONJ
ejpam-6326	209	9	every	every	DET
ejpam-6326	209	10	vertex	vertex	NOUN
ejpam-6326	209	11	not	not	PART
ejpam-6326	209	12	in	in	ADP
ejpam-6326	209	13	f	f	PROPN
ejpam-6326	209	14	has	have	VERB
ejpam-6326	209	15	exactly	exactly	ADV
ejpam-6326	209	16	one	one	NUM
ejpam-6326	209	17	neighbor	neighbor	NOUN
ejpam-6326	209	18	in	in	ADP
ejpam-6326	209	19	f	f	PROPN
ejpam-6326	209	20	.	.	PUNCT
ejpam-6326	210	1	moreover	moreover	ADV
ejpam-6326	210	2	,	,	PUNCT
ejpam-6326	210	3	for	for	ADP
ejpam-6326	210	4	every	every	DET
ejpam-6326	210	5	vertex	vertex	NOUN
ejpam-6326	210	6	v	v	ADP
ejpam-6326	210	7	/∈	/∈	PUNCT
ejpam-6326	210	8	f	f	PROPN
ejpam-6326	210	9	with	with	ADP
ejpam-6326	210	10	deg(v	deg(v	PROPN
ejpam-6326	210	11	)	)	PUNCT
ejpam-6326	210	12	≤	≤	NOUN
ejpam-6326	210	13	2	2	NUM
ejpam-6326	210	14	,	,	PUNCT
ejpam-6326	210	15	if	if	SCONJ
ejpam-6326	210	16	v	v	NOUN
ejpam-6326	210	17	is	be	AUX
ejpam-6326	210	18	not	not	PART
ejpam-6326	210	19	a	a	DET
ejpam-6326	210	20	leaf	leaf	NOUN
ejpam-6326	210	21	it	it	PRON
ejpam-6326	210	22	must	must	AUX
ejpam-6326	210	23	have	have	VERB
ejpam-6326	210	24	degree	degree	NOUN
ejpam-6326	210	25	2	2	NUM
ejpam-6326	210	26	,	,	PUNCT
ejpam-6326	210	27	and	and	CCONJ
ejpam-6326	210	28	the	the	DET
ejpam-6326	210	29	only	only	ADJ
ejpam-6326	210	30	possibility	possibility	NOUN
ejpam-6326	210	31	is	be	AUX
ejpam-6326	210	32	that	that	SCONJ
ejpam-6326	210	33	it	it	PRON
ejpam-6326	210	34	has	have	VERB
ejpam-6326	210	35	one	one	NUM
ejpam-6326	210	36	neighbor	neighbor	NOUN
ejpam-6326	210	37	in	in	ADP
ejpam-6326	210	38	f	f	PROPN
ejpam-6326	210	39	and	and	CCONJ
ejpam-6326	210	40	one	one	NUM
ejpam-6326	210	41	neighbor	neighbor	NOUN
ejpam-6326	210	42	in	in	ADP
ejpam-6326	210	43	v	v	PROPN
ejpam-6326	210	44	(	(	PUNCT
ejpam-6326	210	45	g	g	NOUN
ejpam-6326	210	46	)	)	PUNCT
ejpam-6326	210	47	\	\	PROPN
ejpam-6326	210	48	f	f	NOUN
ejpam-6326	210	49	;	;	PUNCT
ejpam-6326	210	50	hence	hence	ADV
ejpam-6326	210	51	degv	degv	NOUN
ejpam-6326	210	52	(	(	PUNCT
ejpam-6326	210	53	g)\f	g)\f	NOUN
ejpam-6326	210	54	(	(	PUNCT
ejpam-6326	210	55	v	v	NOUN
ejpam-6326	210	56	)	)	PUNCT
ejpam-6326	210	57	=	=	SYM
ejpam-6326	210	58	1	1	NUM
ejpam-6326	210	59	and	and	CCONJ
ejpam-6326	210	60	the	the	DET
ejpam-6326	210	61	inequality	inequality	NOUN
ejpam-6326	210	62	degf	degf	PROPN
ejpam-6326	210	63	(	(	PUNCT
ejpam-6326	210	64	v	v	NOUN
ejpam-6326	210	65	)	)	PUNCT
ejpam-6326	210	66	=	=	SYM
ejpam-6326	210	67	1	1	NUM
ejpam-6326	210	68	≤	≤	NUM
ejpam-6326	210	69	1	1	NUM
ejpam-6326	210	70	=	=	SYM
ejpam-6326	210	71	degv	degv	NOUN
ejpam-6326	210	72	(	(	PUNCT
ejpam-6326	210	73	g)\f	g)\f	NOUN
ejpam-6326	210	74	(	(	PUNCT
ejpam-6326	210	75	v	v	NOUN
ejpam-6326	210	76	)	)	PUNCT
ejpam-6326	210	77	holds	hold	VERB
ejpam-6326	210	78	.	.	PUNCT
ejpam-6326	211	1	thus	thus	ADV
ejpam-6326	211	2	,	,	PUNCT
ejpam-6326	211	3	f	f	PROPN
ejpam-6326	211	4	is	be	AUX
ejpam-6326	211	5	a	a	DET
ejpam-6326	211	6	friendly	friendly	ADJ
ejpam-6326	211	7	dominating	dominating	NOUN
ejpam-6326	211	8	set	set	NOUN
ejpam-6326	211	9	of	of	ADP
ejpam-6326	211	10	g.	g.	PROPN
ejpam-6326	211	11	corollary	corollary	PROPN
ejpam-6326	211	12	4	4	NUM
ejpam-6326	211	13	.	.	PUNCT
ejpam-6326	212	1	let	let	VERB
ejpam-6326	212	2	pn	pn	PART
ejpam-6326	212	3	be	be	AUX
ejpam-6326	212	4	the	the	DET
ejpam-6326	212	5	path	path	NOUN
ejpam-6326	212	6	graph	graph	NOUN
ejpam-6326	212	7	on	on	ADP
ejpam-6326	212	8	n	n	PRON
ejpam-6326	212	9	≥	≥	NUM
ejpam-6326	212	10	2	2	NUM
ejpam-6326	212	11	vertices	vertex	NOUN
ejpam-6326	212	12	.	.	PUNCT
ejpam-6326	213	1	then	then	ADV
ejpam-6326	213	2	the	the	DET
ejpam-6326	213	3	friendly	friendly	ADJ
ejpam-6326	213	4	domination	domination	NOUN
ejpam-6326	213	5	i.	i.	PROPN
ejpam-6326	213	6	s.	s.	PROPN
ejpam-6326	213	7	cabahug	cabahug	PROPN
ejpam-6326	213	8	,	,	PUNCT
ejpam-6326	213	9	jr	jr	PROPN
ejpam-6326	213	10	.	.	PROPN
ejpam-6326	213	11	,	,	PUNCT
ejpam-6326	213	12	r.	r.	PROPN
ejpam-6326	213	13	g.	g.	PROPN
ejpam-6326	213	14	eballe	eballe	PROPN
ejpam-6326	213	15	,	,	PUNCT
ejpam-6326	213	16	r.	r.	PROPN
ejpam-6326	213	17	t.	t.	PROPN
ejpam-6326	213	18	fernandez	fernandez	PROPN
ejpam-6326	213	19	/	/	SYM
ejpam-6326	213	20	eur	eur	PROPN
ejpam-6326	213	21	.	.	PUNCT
ejpam-6326	214	1	j.	j.	PROPN
ejpam-6326	214	2	pure	pure	PROPN
ejpam-6326	214	3	appl	appl	PROPN
ejpam-6326	214	4	.	.	PROPN
ejpam-6326	214	5	math	math	PROPN
ejpam-6326	214	6	,	,	PUNCT
ejpam-6326	214	7	18	18	NUM
ejpam-6326	214	8	(	(	PUNCT
ejpam-6326	214	9	4	4	NUM
ejpam-6326	214	10	)	)	PUNCT
ejpam-6326	214	11	(	(	PUNCT
ejpam-6326	214	12	2025	2025	NUM
ejpam-6326	214	13	)	)	PUNCT
ejpam-6326	214	14	,	,	PUNCT
ejpam-6326	214	15	6326	6326	NUM
ejpam-6326	214	16	8	8	NUM
ejpam-6326	214	17	of	of	ADP
ejpam-6326	214	18	15	15	NUM
ejpam-6326	214	19	number	number	NOUN
ejpam-6326	214	20	of	of	ADP
ejpam-6326	214	21	pn	pn	PROPN
ejpam-6326	214	22	,	,	PUNCT
ejpam-6326	214	23	is	be	AUX
ejpam-6326	214	24	given	give	VERB
ejpam-6326	214	25	by	by	ADP
ejpam-6326	214	26	γf	γf	PROPN
ejpam-6326	214	27	(	(	PUNCT
ejpam-6326	214	28	pn	pn	NOUN
ejpam-6326	214	29	)	)	PUNCT
ejpam-6326	214	30	=	=	PUNCT
ejpam-6326	215	1			NOUN
ejpam-6326	215	2	n+	n+	PUNCT
ejpam-6326	215	3	6	6	NUM
ejpam-6326	215	4	3	3	NUM
ejpam-6326	215	5	,	,	PUNCT
ejpam-6326	215	6	if	if	SCONJ
ejpam-6326	215	7	n	n	PRON
ejpam-6326	215	8	≡	≡	PROPN
ejpam-6326	215	9	0	0	PUNCT
ejpam-6326	215	10	(	(	PUNCT
ejpam-6326	215	11	mod	mod	NOUN
ejpam-6326	215	12	3	3	NUM
ejpam-6326	215	13	)	)	PUNCT
ejpam-6326	215	14	,	,	PUNCT
ejpam-6326	215	15	n+	n+	PUNCT
ejpam-6326	215	16	2	2	NUM
ejpam-6326	215	17	3	3	NUM
ejpam-6326	215	18	,	,	PUNCT
ejpam-6326	215	19	if	if	SCONJ
ejpam-6326	215	20	n	n	PRON
ejpam-6326	215	21	≡	≡	PROPN
ejpam-6326	215	22	1	1	NUM
ejpam-6326	215	23	(	(	PUNCT
ejpam-6326	215	24	mod	mod	NOUN
ejpam-6326	215	25	3	3	NUM
ejpam-6326	215	26	)	)	PUNCT
ejpam-6326	215	27	,	,	PUNCT
ejpam-6326	215	28	n+	n+	ADP
ejpam-6326	215	29	4	4	NUM
ejpam-6326	215	30	3	3	NUM
ejpam-6326	215	31	,	,	PUNCT
ejpam-6326	215	32	if	if	SCONJ
ejpam-6326	215	33	n	n	PRON
ejpam-6326	215	34	≡	≡	PROPN
ejpam-6326	215	35	2	2	NUM
ejpam-6326	215	36	(	(	PUNCT
ejpam-6326	215	37	mod	mod	NOUN
ejpam-6326	215	38	3	3	NUM
ejpam-6326	215	39	)	)	PUNCT
ejpam-6326	215	40	.	.	PUNCT
ejpam-6326	216	1	proof	proof	NOUN
ejpam-6326	216	2	.	.	PUNCT
ejpam-6326	217	1	let	let	VERB
ejpam-6326	217	2	pn	pn	VERB
ejpam-6326	217	3	=	=	PUNCT
ejpam-6326	218	1	[	[	X
ejpam-6326	218	2	v1	v1	NOUN
ejpam-6326	218	3	,	,	PUNCT
ejpam-6326	218	4	v2	v2	NOUN
ejpam-6326	218	5	,	,	PUNCT
ejpam-6326	218	6	.	.	PUNCT
ejpam-6326	218	7	.	.	PUNCT
ejpam-6326	219	1	.	.	PUNCT
ejpam-6326	220	1	,	,	PUNCT
ejpam-6326	220	2	vn−1	vn−1	PROPN
ejpam-6326	220	3	,	,	PUNCT
ejpam-6326	220	4	vn	vn	X
ejpam-6326	220	5	]	]	PUNCT
ejpam-6326	220	6	.	.	PUNCT
ejpam-6326	221	1	consider	consider	VERB
ejpam-6326	221	2	the	the	DET
ejpam-6326	221	3	following	follow	VERB
ejpam-6326	221	4	cases	case	NOUN
ejpam-6326	221	5	for	for	ADP
ejpam-6326	221	6	n	n	CCONJ
ejpam-6326	221	7	:	:	PUNCT
ejpam-6326	221	8	case	case	NOUN
ejpam-6326	221	9	1	1	NUM
ejpam-6326	221	10	:	:	PUNCT
ejpam-6326	221	11	n	n	NUM
ejpam-6326	221	12	≡	≡	PROPN
ejpam-6326	221	13	0	0	PUNCT
ejpam-6326	222	1	(	(	PUNCT
ejpam-6326	222	2	mod	mod	NOUN
ejpam-6326	222	3	3	3	X
ejpam-6326	222	4	)	)	PUNCT
ejpam-6326	222	5	if	if	SCONJ
ejpam-6326	222	6	n	n	NOUN
ejpam-6326	222	7	=	=	SYM
ejpam-6326	222	8	3	3	NUM
ejpam-6326	222	9	,	,	PUNCT
ejpam-6326	222	10	then	then	ADV
ejpam-6326	222	11	the	the	DET
ejpam-6326	222	12	γf	γf	PROPN
ejpam-6326	222	13	-set	-set	PUNCT
ejpam-6326	222	14	of	of	ADP
ejpam-6326	222	15	p3	p3	PROPN
ejpam-6326	222	16	is	be	AUX
ejpam-6326	222	17	v	v	NOUN
ejpam-6326	222	18	(	(	PUNCT
ejpam-6326	222	19	p3	p3	PROPN
ejpam-6326	222	20	)	)	PUNCT
ejpam-6326	222	21	.	.	PUNCT
ejpam-6326	223	1	thus	thus	ADV
ejpam-6326	223	2	,	,	PUNCT
ejpam-6326	223	3	γf	γf	PROPN
ejpam-6326	223	4	(	(	PUNCT
ejpam-6326	223	5	p3	p3	PROPN
ejpam-6326	223	6	)	)	PUNCT
ejpam-6326	223	7	=	=	SYM
ejpam-6326	223	8	n	n	PRON
ejpam-6326	223	9	3	3	NUM
ejpam-6326	223	10	+	+	SYM
ejpam-6326	223	11	2	2	NUM
ejpam-6326	223	12	=	=	SYM
ejpam-6326	223	13	n+	n+	NUM
ejpam-6326	223	14	6	6	NUM
ejpam-6326	223	15	3	3	NUM
ejpam-6326	223	16	=	=	SYM
ejpam-6326	223	17	3	3	NUM
ejpam-6326	223	18	+	+	CCONJ
ejpam-6326	223	19	6	6	NUM
ejpam-6326	223	20	3	3	NUM
ejpam-6326	223	21	=	=	SYM
ejpam-6326	223	22	3	3	NUM
ejpam-6326	223	23	=	=	SYM
ejpam-6326	223	24	v	v	PROPN
ejpam-6326	223	25	(	(	PUNCT
ejpam-6326	223	26	p3	p3	PROPN
ejpam-6326	223	27	)	)	PUNCT
ejpam-6326	223	28	.	.	PUNCT
ejpam-6326	224	1	now	now	ADV
ejpam-6326	224	2	,	,	PUNCT
ejpam-6326	224	3	for	for	ADP
ejpam-6326	224	4	n	n	X
ejpam-6326	224	5	>	>	X
ejpam-6326	224	6	3	3	X
ejpam-6326	224	7	.	.	PUNCT
ejpam-6326	224	8	choose	choose	VERB
ejpam-6326	224	9	f	f	X
ejpam-6326	224	10	=	=	NOUN
ejpam-6326	224	11	{	{	PUNCT
ejpam-6326	224	12	v1	v1	PROPN
ejpam-6326	224	13	,	,	PUNCT
ejpam-6326	224	14	vn	vn	PROPN
ejpam-6326	224	15	}	}	PUNCT
ejpam-6326	224	16	∪	∪	ADJ
ejpam-6326	224	17	{	{	PUNCT
ejpam-6326	224	18	v4	v4	NOUN
ejpam-6326	224	19	,	,	PUNCT
ejpam-6326	224	20	v7	v7	NOUN
ejpam-6326	224	21	,	,	PUNCT
ejpam-6326	224	22	.	.	PUNCT
ejpam-6326	224	23	.	.	PUNCT
ejpam-6326	224	24	.	.	PUNCT
ejpam-6326	225	1	,	,	PUNCT
ejpam-6326	225	2	vn−2	vn−2	PROPN
ejpam-6326	225	3	,	,	PUNCT
ejpam-6326	225	4	,	,	PUNCT
ejpam-6326	225	5	vn−1	vn−1	ADJ
ejpam-6326	225	6	}	}	PUNCT
ejpam-6326	225	7	,	,	PUNCT
ejpam-6326	225	8	then	then	ADV
ejpam-6326	225	9	|f	|f	PROPN
ejpam-6326	226	1	|	|	NOUN
ejpam-6326	226	2	=	=	SYM
ejpam-6326	226	3	2	2	NUM
ejpam-6326	226	4	+	+	CCONJ
ejpam-6326	226	5	n	n	PRON
ejpam-6326	226	6	3	3	NUM
ejpam-6326	226	7	=	=	SYM
ejpam-6326	226	8	n+	n+	ADP
ejpam-6326	226	9	6	6	NUM
ejpam-6326	226	10	3	3	NUM
ejpam-6326	226	11	.	.	PUNCT
ejpam-6326	227	1	by	by	ADP
ejpam-6326	227	2	theorem	theorem	NOUN
ejpam-6326	227	3	6	6	NUM
ejpam-6326	227	4	,	,	PUNCT
ejpam-6326	227	5	f	f	PROPN
ejpam-6326	227	6	is	be	AUX
ejpam-6326	227	7	a	a	DET
ejpam-6326	227	8	friendly	friendly	ADJ
ejpam-6326	227	9	dominating	dominating	NOUN
ejpam-6326	227	10	set	set	VERB
ejpam-6326	227	11	in	in	ADP
ejpam-6326	227	12	pn	pn	PROPN
ejpam-6326	227	13	.	.	PUNCT
ejpam-6326	228	1	thus	thus	ADV
ejpam-6326	228	2	,	,	PUNCT
ejpam-6326	228	3	γf	γf	INTJ
ejpam-6326	228	4	(	(	PUNCT
ejpam-6326	228	5	g	g	NOUN
ejpam-6326	228	6	)	)	PUNCT
ejpam-6326	228	7	≤	≤	NUM
ejpam-6326	228	8	n+	n+	PUNCT
ejpam-6326	228	9	6	6	NUM
ejpam-6326	228	10	3	3	NUM
ejpam-6326	228	11	.	.	PUNCT
ejpam-6326	229	1	since	since	SCONJ
ejpam-6326	229	2	there	there	PRON
ejpam-6326	229	3	can	can	AUX
ejpam-6326	229	4	be	be	AUX
ejpam-6326	229	5	no	no	DET
ejpam-6326	229	6	another	another	DET
ejpam-6326	229	7	friendly	friendly	ADJ
ejpam-6326	229	8	dominating	dominating	NOUN
ejpam-6326	229	9	sets	set	NOUN
ejpam-6326	229	10	whose	whose	DET
ejpam-6326	229	11	cardinality	cardinality	NOUN
ejpam-6326	229	12	is	be	AUX
ejpam-6326	229	13	strictly	strictly	ADV
ejpam-6326	229	14	less	less	ADJ
ejpam-6326	229	15	than	than	ADP
ejpam-6326	229	16	n+	n+	ADP
ejpam-6326	229	17	6	6	NUM
ejpam-6326	229	18	3	3	NUM
ejpam-6326	229	19	,	,	PUNCT
ejpam-6326	229	20	we	we	PRON
ejpam-6326	229	21	can	can	AUX
ejpam-6326	229	22	have	have	VERB
ejpam-6326	229	23	γf	γf	PROPN
ejpam-6326	229	24	(	(	PUNCT
ejpam-6326	229	25	pn	pn	NOUN
ejpam-6326	229	26	)	)	PUNCT
ejpam-6326	229	27	≥	≥	NOUN
ejpam-6326	229	28	n+	n+	PUNCT
ejpam-6326	229	29	6	6	NUM
ejpam-6326	229	30	3	3	NUM
ejpam-6326	229	31	.	.	PUNCT
ejpam-6326	230	1	thus	thus	ADV
ejpam-6326	230	2	,	,	PUNCT
ejpam-6326	230	3	γf	γf	PROPN
ejpam-6326	230	4	(	(	PUNCT
ejpam-6326	230	5	pn	pn	NOUN
ejpam-6326	230	6	)	)	PUNCT
ejpam-6326	230	7	=	=	PUNCT
ejpam-6326	230	8	n+	n+	ADP
ejpam-6326	230	9	6	6	NUM
ejpam-6326	230	10	3	3	NUM
ejpam-6326	230	11	.	.	PUNCT
ejpam-6326	231	1	case	case	NOUN
ejpam-6326	231	2	2	2	NUM
ejpam-6326	231	3	:	:	PUNCT
ejpam-6326	231	4	n	n	NUM
ejpam-6326	231	5	≡	≡	PROPN
ejpam-6326	231	6	1	1	NUM
ejpam-6326	231	7	(	(	PUNCT
ejpam-6326	231	8	mod	mod	NOUN
ejpam-6326	231	9	3	3	NUM
ejpam-6326	231	10	)	)	PUNCT
ejpam-6326	231	11	and	and	CCONJ
ejpam-6326	231	12	n	n	PRON
ejpam-6326	231	13	≡	≡	PROPN
ejpam-6326	231	14	2	2	NUM
ejpam-6326	231	15	(	(	PUNCT
ejpam-6326	231	16	mod	mod	NOUN
ejpam-6326	231	17	3	3	NUM
ejpam-6326	231	18	)	)	PUNCT
ejpam-6326	231	19	an	an	DET
ejpam-6326	231	20	entirely	entirely	ADV
ejpam-6326	231	21	analogous	analogous	ADJ
ejpam-6326	231	22	argument	argument	NOUN
ejpam-6326	231	23	as	as	ADP
ejpam-6326	231	24	case	case	NOUN
ejpam-6326	231	25	1	1	NUM
ejpam-6326	231	26	.	.	PUNCT
ejpam-6326	231	27	choosing	choose	VERB
ejpam-6326	231	28	f	f	PROPN
ejpam-6326	231	29	=	=	NOUN
ejpam-6326	231	30	{	{	PUNCT
ejpam-6326	231	31	v1	v1	PROPN
ejpam-6326	231	32	,	,	PUNCT
ejpam-6326	231	33	vn	vn	PROPN
ejpam-6326	231	34	}	}	PUNCT
ejpam-6326	231	35	∪	∪	ADJ
ejpam-6326	231	36	{	{	PUNCT
ejpam-6326	231	37	v4	v4	NOUN
ejpam-6326	231	38	,	,	PUNCT
ejpam-6326	231	39	v7	v7	NOUN
ejpam-6326	231	40	,	,	PUNCT
ejpam-6326	231	41	.	.	PUNCT
ejpam-6326	231	42	.	.	PUNCT
ejpam-6326	232	1	.	.	PUNCT
ejpam-6326	233	1	,	,	PUNCT
ejpam-6326	233	2	vn−3	vn−3	PROPN
ejpam-6326	233	3	}	}	PUNCT
ejpam-6326	233	4	with	with	ADP
ejpam-6326	233	5	|f	|f	PROPN
ejpam-6326	234	1	|	|	ADV
ejpam-6326	234	2	=	=	SYM
ejpam-6326	234	3	2	2	NUM
ejpam-6326	234	4	+	+	NUM
ejpam-6326	234	5	n−	n−	NOUN
ejpam-6326	234	6	4	4	NUM
ejpam-6326	234	7	3	3	NUM
ejpam-6326	234	8	=	=	SYM
ejpam-6326	234	9	n+	n+	ADP
ejpam-6326	234	10	2	2	NUM
ejpam-6326	234	11	3	3	NUM
ejpam-6326	234	12	for	for	ADP
ejpam-6326	234	13	n	n	X
ejpam-6326	234	14	≡	≡	PROPN
ejpam-6326	234	15	1	1	NUM
ejpam-6326	234	16	(	(	PUNCT
ejpam-6326	234	17	mod	mod	NOUN
ejpam-6326	234	18	3	3	NUM
ejpam-6326	234	19	)	)	PUNCT
ejpam-6326	234	20	and	and	CCONJ
ejpam-6326	234	21	f	f	X
ejpam-6326	234	22	=	=	NOUN
ejpam-6326	234	23	{	{	PUNCT
ejpam-6326	234	24	v1	v1	PROPN
ejpam-6326	234	25	,	,	PUNCT
ejpam-6326	234	26	vn	vn	PROPN
ejpam-6326	234	27	}	}	PUNCT
ejpam-6326	234	28	∪	∪	ADJ
ejpam-6326	234	29	{	{	PUNCT
ejpam-6326	234	30	v4	v4	NOUN
ejpam-6326	234	31	,	,	PUNCT
ejpam-6326	234	32	v7	v7	NOUN
ejpam-6326	234	33	,	,	PUNCT
ejpam-6326	234	34	.	.	PUNCT
ejpam-6326	234	35	.	.	PUNCT
ejpam-6326	235	1	.	.	PUNCT
ejpam-6326	236	1	,	,	PUNCT
ejpam-6326	236	2	vn−2	vn−2	PROPN
ejpam-6326	236	3	}	}	PUNCT
ejpam-6326	236	4	with	with	ADP
ejpam-6326	236	5	|f	|f	PROPN
ejpam-6326	237	1	|	|	ADV
ejpam-6326	237	2	=	=	SYM
ejpam-6326	237	3	2	2	NUM
ejpam-6326	237	4	+	+	NUM
ejpam-6326	237	5	n−	n−	NOUN
ejpam-6326	237	6	2	2	NUM
ejpam-6326	237	7	3	3	NUM
ejpam-6326	237	8	=	=	PUNCT
ejpam-6326	237	9	n+	n+	ADP
ejpam-6326	237	10	4	4	NUM
ejpam-6326	237	11	3	3	NUM
ejpam-6326	237	12	for	for	ADP
ejpam-6326	237	13	n	n	X
ejpam-6326	237	14	≡	≡	PROPN
ejpam-6326	237	15	2	2	NUM
ejpam-6326	237	16	(	(	PUNCT
ejpam-6326	237	17	mod	mod	NOUN
ejpam-6326	237	18	3	3	NUM
ejpam-6326	237	19	)	)	PUNCT
ejpam-6326	237	20	.	.	PUNCT
ejpam-6326	238	1	corollary	corollary	ADJ
ejpam-6326	238	2	5	5	NUM
ejpam-6326	238	3	.	.	PUNCT
ejpam-6326	239	1	let	let	VERB
ejpam-6326	239	2	cn	cn	PROPN
ejpam-6326	239	3	be	be	AUX
ejpam-6326	239	4	the	the	DET
ejpam-6326	239	5	cycle	cycle	NOUN
ejpam-6326	239	6	graph	graph	NOUN
ejpam-6326	239	7	on	on	ADP
ejpam-6326	239	8	n	n	PRON
ejpam-6326	239	9	≥	≥	NUM
ejpam-6326	239	10	3	3	NUM
ejpam-6326	239	11	vertices	vertex	NOUN
ejpam-6326	239	12	.	.	PUNCT
ejpam-6326	240	1	then	then	ADV
ejpam-6326	240	2	the	the	DET
ejpam-6326	240	3	friendly	friendly	ADJ
ejpam-6326	240	4	domination	domination	NOUN
ejpam-6326	240	5	number	number	NOUN
ejpam-6326	240	6	of	of	ADP
ejpam-6326	240	7	cn	cn	PROPN
ejpam-6326	240	8	is	be	AUX
ejpam-6326	240	9	γf	γf	PROPN
ejpam-6326	240	10	(	(	PUNCT
ejpam-6326	240	11	cn	cn	PROPN
ejpam-6326	240	12	)	)	PUNCT
ejpam-6326	240	13	=	=	PUNCT
ejpam-6326	240	14	⌈n	⌈n	NOUN
ejpam-6326	240	15	3	3	NUM
ejpam-6326	240	16	⌉	⌉	X
ejpam-6326	240	17	.	.	PUNCT
ejpam-6326	241	1	proof	proof	NOUN
ejpam-6326	241	2	.	.	PUNCT
ejpam-6326	242	1	by	by	ADP
ejpam-6326	242	2	theorem	theorem	NOUN
ejpam-6326	242	3	6	6	NUM
ejpam-6326	242	4	(	(	PUNCT
ejpam-6326	242	5	i	i	NOUN
ejpam-6326	242	6	)	)	PUNCT
ejpam-6326	242	7	,	,	PUNCT
ejpam-6326	242	8	finding	find	VERB
ejpam-6326	242	9	the	the	DET
ejpam-6326	242	10	friendly	friendly	ADJ
ejpam-6326	242	11	dominating	dominating	NOUN
ejpam-6326	242	12	set	set	NOUN
ejpam-6326	242	13	in	in	ADP
ejpam-6326	242	14	cn	cn	PROPN
ejpam-6326	242	15	is	be	AUX
ejpam-6326	242	16	equivalent	equivalent	ADJ
ejpam-6326	242	17	to	to	ADP
ejpam-6326	242	18	finding	find	VERB
ejpam-6326	242	19	a	a	DET
ejpam-6326	242	20	set	set	NOUN
ejpam-6326	242	21	f	f	PRON
ejpam-6326	242	22	such	such	ADJ
ejpam-6326	242	23	that	that	SCONJ
ejpam-6326	242	24	every	every	DET
ejpam-6326	242	25	vertex	vertex	NOUN
ejpam-6326	242	26	is	be	AUX
ejpam-6326	242	27	either	either	CCONJ
ejpam-6326	242	28	in	in	ADP
ejpam-6326	242	29	f	f	PROPN
ejpam-6326	242	30	or	or	CCONJ
ejpam-6326	242	31	adjacent	adjacent	ADJ
ejpam-6326	242	32	to	to	ADP
ejpam-6326	242	33	exactly	exactly	ADV
ejpam-6326	242	34	one	one	NUM
ejpam-6326	242	35	vertex	vertex	NOUN
ejpam-6326	242	36	of	of	ADP
ejpam-6326	242	37	f	f	PROPN
ejpam-6326	242	38	.	.	PUNCT
ejpam-6326	243	1	thus	thus	ADV
ejpam-6326	243	2	,	,	PUNCT
ejpam-6326	243	3	in	in	ADP
ejpam-6326	243	4	cn	cn	PROPN
ejpam-6326	243	5	,	,	PUNCT
ejpam-6326	243	6	the	the	DET
ejpam-6326	243	7	minimum	minimum	ADJ
ejpam-6326	243	8	number	number	NOUN
ejpam-6326	243	9	of	of	ADP
ejpam-6326	243	10	vertices	vertex	NOUN
ejpam-6326	243	11	needed	need	VERB
ejpam-6326	243	12	so	so	SCONJ
ejpam-6326	243	13	that	that	SCONJ
ejpam-6326	243	14	every	every	DET
ejpam-6326	243	15	vertex	vertex	NOUN
ejpam-6326	243	16	in	in	ADP
ejpam-6326	243	17	cn	cn	PROPN
ejpam-6326	243	18	is	be	AUX
ejpam-6326	243	19	dominated	dominate	VERB
ejpam-6326	243	20	by	by	ADP
ejpam-6326	243	21	exactly	exactly	ADV
ejpam-6326	243	22	one	one	NUM
ejpam-6326	243	23	member	member	NOUN
ejpam-6326	243	24	of	of	ADP
ejpam-6326	243	25	the	the	DET
ejpam-6326	243	26	set	set	NOUN
ejpam-6326	243	27	is	be	AUX
ejpam-6326	243	28	⌈n	⌈n	VERB
ejpam-6326	243	29	3	3	NUM
ejpam-6326	243	30	⌉	⌉	PUNCT
ejpam-6326	243	31	and	and	CCONJ
ejpam-6326	243	32	no	no	DET
ejpam-6326	243	33	smaller	small	ADJ
ejpam-6326	243	34	set	set	NOUN
ejpam-6326	243	35	can	can	AUX
ejpam-6326	243	36	satisfy	satisfy	VERB
ejpam-6326	243	37	this	this	DET
ejpam-6326	243	38	condition	condition	NOUN
ejpam-6326	243	39	in	in	ADP
ejpam-6326	243	40	cn	cn	PROPN
ejpam-6326	243	41	.	.	PUNCT
ejpam-6326	244	1	thus	thus	ADV
ejpam-6326	244	2	,	,	PUNCT
ejpam-6326	244	3	γf	γf	PROPN
ejpam-6326	244	4	(	(	PUNCT
ejpam-6326	244	5	cn	cn	PROPN
ejpam-6326	244	6	)	)	PUNCT
ejpam-6326	244	7	=	=	PUNCT
ejpam-6326	244	8	⌈n	⌈n	NOUN
ejpam-6326	244	9	3	3	NUM
ejpam-6326	244	10	⌉	⌉	NOUN
ejpam-6326	244	11	.	.	PUNCT
ejpam-6326	245	1	3.2	3.2	NUM
ejpam-6326	245	2	.	.	PUNCT
ejpam-6326	245	3	nordhaus	nordhaus	NOUN
ejpam-6326	245	4	-	-	PUNCT
ejpam-6326	245	5	gaddum	gaddum	PROPN
ejpam-6326	245	6	type	type	NOUN
ejpam-6326	245	7	inequalities	inequality	NOUN
ejpam-6326	245	8	for	for	ADP
ejpam-6326	245	9	γf	γf	PROPN
ejpam-6326	245	10	-set	-set	ADJ
ejpam-6326	245	11	.	.	PUNCT
ejpam-6326	246	1	theorem	theorem	VERB
ejpam-6326	246	2	7	7	NUM
ejpam-6326	246	3	.	.	PUNCT
ejpam-6326	247	1	let	let	VERB
ejpam-6326	247	2	g	g	PRON
ejpam-6326	247	3	be	be	AUX
ejpam-6326	247	4	a	a	DET
ejpam-6326	247	5	connected	connected	ADJ
ejpam-6326	247	6	nontrivial	nontrivial	ADJ
ejpam-6326	247	7	graph	graph	NOUN
ejpam-6326	247	8	on	on	ADP
ejpam-6326	247	9	n	n	DET
ejpam-6326	247	10	vertices	vertex	NOUN
ejpam-6326	247	11	.	.	PUNCT
ejpam-6326	248	1	then	then	ADV
ejpam-6326	248	2	,	,	PUNCT
ejpam-6326	248	3	0	0	NUM
ejpam-6326	248	4	≤	≤	NUM
ejpam-6326	248	5	∣∣γf	∣∣γf	VERB
ejpam-6326	248	6	(	(	PUNCT
ejpam-6326	248	7	g)−	g)−	PROPN
ejpam-6326	248	8	γf	γf	PROPN
ejpam-6326	248	9	(	(	PUNCT
ejpam-6326	248	10	g	g	NOUN
ejpam-6326	248	11	)	)	PUNCT
ejpam-6326	248	12	∣∣	∣∣	X
ejpam-6326	248	13	≤	≤	NUM
ejpam-6326	248	14	n−	n−	NOUN
ejpam-6326	248	15	1	1	NUM
ejpam-6326	248	16	.	.	PUNCT
ejpam-6326	249	1	proof	proof	NOUN
ejpam-6326	249	2	.	.	PUNCT
ejpam-6326	250	1	let	let	VERB
ejpam-6326	250	2	g	g	PRON
ejpam-6326	250	3	be	be	AUX
ejpam-6326	250	4	a	a	DET
ejpam-6326	250	5	connected	connected	ADJ
ejpam-6326	250	6	nontrivial	nontrivial	ADJ
ejpam-6326	250	7	graph	graph	NOUN
ejpam-6326	250	8	on	on	ADP
ejpam-6326	250	9	n	n	PRON
ejpam-6326	250	10	vertices	vertex	NOUN
ejpam-6326	250	11	,	,	PUNCT
ejpam-6326	250	12	and	and	CCONJ
ejpam-6326	250	13	let	let	VERB
ejpam-6326	250	14	γf	γf	VERB
ejpam-6326	250	15	(	(	PUNCT
ejpam-6326	250	16	g	g	NOUN
ejpam-6326	250	17	)	)	PUNCT
ejpam-6326	250	18	and	and	CCONJ
ejpam-6326	250	19	γf	γf	INTJ
ejpam-6326	250	20	(	(	PUNCT
ejpam-6326	250	21	g	g	NOUN
ejpam-6326	250	22	)	)	PUNCT
ejpam-6326	250	23	be	be	AUX
ejpam-6326	250	24	the	the	DET
ejpam-6326	250	25	friendly	friendly	ADJ
ejpam-6326	250	26	domination	domination	NOUN
ejpam-6326	250	27	numbers	number	NOUN
ejpam-6326	250	28	of	of	ADP
ejpam-6326	250	29	g	g	NOUN
ejpam-6326	250	30	and	and	CCONJ
ejpam-6326	250	31	its	its	PRON
ejpam-6326	250	32	complement	complement	NOUN
ejpam-6326	250	33	g	g	NOUN
ejpam-6326	250	34	,	,	PUNCT
ejpam-6326	250	35	respectively	respectively	ADV
ejpam-6326	250	36	.	.	PUNCT
ejpam-6326	251	1	now	now	ADV
ejpam-6326	251	2	,	,	PUNCT
ejpam-6326	251	3	γf	γf	PROPN
ejpam-6326	251	4	(	(	PUNCT
ejpam-6326	251	5	g	g	NOUN
ejpam-6326	251	6	)	)	PUNCT
ejpam-6326	251	7	i.	i.	PROPN
ejpam-6326	251	8	s.	s.	PROPN
ejpam-6326	251	9	cabahug	cabahug	PROPN
ejpam-6326	251	10	,	,	PUNCT
ejpam-6326	251	11	jr	jr	PROPN
ejpam-6326	251	12	.	.	PROPN
ejpam-6326	251	13	,	,	PUNCT
ejpam-6326	251	14	r.	r.	PROPN
ejpam-6326	251	15	g.	g.	PROPN
ejpam-6326	251	16	eballe	eballe	PROPN
ejpam-6326	251	17	,	,	PUNCT
ejpam-6326	251	18	r.	r.	PROPN
ejpam-6326	251	19	t.	t.	PROPN
ejpam-6326	251	20	fernandez	fernandez	PROPN
ejpam-6326	251	21	/	/	SYM
ejpam-6326	251	22	eur	eur	PROPN
ejpam-6326	251	23	.	.	PUNCT
ejpam-6326	252	1	j.	j.	PROPN
ejpam-6326	252	2	pure	pure	PROPN
ejpam-6326	252	3	appl	appl	PROPN
ejpam-6326	252	4	.	.	PROPN
ejpam-6326	252	5	math	math	PROPN
ejpam-6326	252	6	,	,	PUNCT
ejpam-6326	252	7	18	18	NUM
ejpam-6326	252	8	(	(	PUNCT
ejpam-6326	252	9	4	4	NUM
ejpam-6326	252	10	)	)	PUNCT
ejpam-6326	252	11	(	(	PUNCT
ejpam-6326	252	12	2025	2025	NUM
ejpam-6326	252	13	)	)	PUNCT
ejpam-6326	252	14	,	,	PUNCT
ejpam-6326	252	15	6326	6326	NUM
ejpam-6326	252	16	9	9	NUM
ejpam-6326	252	17	of	of	ADP
ejpam-6326	252	18	15	15	NUM
ejpam-6326	252	19	and	and	CCONJ
ejpam-6326	252	20	γf	γf	PROPN
ejpam-6326	252	21	(	(	PUNCT
ejpam-6326	252	22	g	g	NOUN
ejpam-6326	252	23	)	)	PUNCT
ejpam-6326	252	24	are	be	AUX
ejpam-6326	252	25	each	each	PRON
ejpam-6326	252	26	at	at	ADV
ejpam-6326	252	27	least	least	ADJ
ejpam-6326	252	28	1	1	NUM
ejpam-6326	252	29	since	since	SCONJ
ejpam-6326	252	30	a	a	DET
ejpam-6326	252	31	friendly	friendly	ADJ
ejpam-6326	252	32	dominating	dominating	NOUN
ejpam-6326	252	33	set	set	NOUN
ejpam-6326	252	34	must	must	AUX
ejpam-6326	252	35	be	be	AUX
ejpam-6326	252	36	nonempty	nonempty	ADJ
ejpam-6326	252	37	,	,	PUNCT
ejpam-6326	252	38	and	and	CCONJ
ejpam-6326	252	39	they	they	PRON
ejpam-6326	252	40	are	be	AUX
ejpam-6326	252	41	each	each	PRON
ejpam-6326	252	42	at	at	ADV
ejpam-6326	252	43	most	most	ADJ
ejpam-6326	252	44	n	n	CCONJ
ejpam-6326	252	45	since	since	SCONJ
ejpam-6326	252	46	taking	take	VERB
ejpam-6326	252	47	the	the	DET
ejpam-6326	252	48	entire	entire	ADJ
ejpam-6326	252	49	vertex	vertex	NOUN
ejpam-6326	252	50	set	set	VERB
ejpam-6326	252	51	v	v	NOUN
ejpam-6326	252	52	(	(	PUNCT
ejpam-6326	252	53	g	g	NOUN
ejpam-6326	252	54	)	)	PUNCT
ejpam-6326	252	55	and	and	CCONJ
ejpam-6326	252	56	v	v	NOUN
ejpam-6326	252	57	(	(	PUNCT
ejpam-6326	252	58	g	g	NOUN
ejpam-6326	252	59	)	)	PUNCT
ejpam-6326	252	60	always	always	ADV
ejpam-6326	252	61	yields	yield	VERB
ejpam-6326	252	62	a	a	DET
ejpam-6326	252	63	friendly	friendly	ADJ
ejpam-6326	252	64	dominating	dominating	NOUN
ejpam-6326	252	65	set	set	NOUN
ejpam-6326	252	66	.	.	PUNCT
ejpam-6326	253	1	it	it	PRON
ejpam-6326	253	2	follows	follow	VERB
ejpam-6326	253	3	that	that	SCONJ
ejpam-6326	253	4	γf	γf	INTJ
ejpam-6326	253	5	(	(	PUNCT
ejpam-6326	253	6	g	g	NOUN
ejpam-6326	253	7	)	)	PUNCT
ejpam-6326	253	8	and	and	CCONJ
ejpam-6326	253	9	γf	γf	INTJ
ejpam-6326	253	10	(	(	PUNCT
ejpam-6326	253	11	g	g	NOUN
ejpam-6326	253	12	)	)	PUNCT
ejpam-6326	253	13	both	both	PRON
ejpam-6326	253	14	lie	lie	VERB
ejpam-6326	253	15	in	in	ADP
ejpam-6326	253	16	the	the	DET
ejpam-6326	253	17	integer	integer	NOUN
ejpam-6326	253	18	interval	interval	NOUN
ejpam-6326	253	19	[	[	X
ejpam-6326	253	20	1	1	NUM
ejpam-6326	253	21	,	,	PUNCT
ejpam-6326	253	22	n	n	CCONJ
ejpam-6326	253	23	]	]	PUNCT
ejpam-6326	253	24	,	,	PUNCT
ejpam-6326	253	25	so	so	ADV
ejpam-6326	253	26	their	their	PRON
ejpam-6326	253	27	absolute	absolute	ADJ
ejpam-6326	253	28	difference	difference	NOUN
ejpam-6326	253	29	satisfies	satisfie	NOUN
ejpam-6326	253	30	0	0	NUM
ejpam-6326	253	31	≤	≤	NUM
ejpam-6326	253	32	∣∣γf	∣∣γf	VERB
ejpam-6326	253	33	(	(	PUNCT
ejpam-6326	253	34	g	g	NOUN
ejpam-6326	253	35	)	)	PUNCT
ejpam-6326	253	36	−	−	NOUN
ejpam-6326	254	1	γf	γf	INTJ
ejpam-6326	254	2	(	(	PUNCT
ejpam-6326	254	3	g	g	NOUN
ejpam-6326	254	4	)	)	PUNCT
ejpam-6326	254	5	∣∣	∣∣	X
ejpam-6326	254	6	≤	≤	NUM
ejpam-6326	254	7	n−	n−	NOUN
ejpam-6326	254	8	1	1	NUM
ejpam-6326	254	9	.	.	PUNCT
ejpam-6326	254	10	to	to	PART
ejpam-6326	254	11	see	see	VERB
ejpam-6326	254	12	that	that	SCONJ
ejpam-6326	254	13	these	these	DET
ejpam-6326	254	14	bounds	bound	NOUN
ejpam-6326	254	15	are	be	AUX
ejpam-6326	254	16	sharp	sharp	ADJ
ejpam-6326	254	17	,	,	PUNCT
ejpam-6326	254	18	consider	consider	VERB
ejpam-6326	254	19	the	the	DET
ejpam-6326	254	20	complete	complete	ADJ
ejpam-6326	254	21	graph	graph	NOUN
ejpam-6326	254	22	kn	kn	PROPN
ejpam-6326	254	23	.	.	PUNCT
ejpam-6326	255	1	by	by	ADP
ejpam-6326	255	2	corollary	corollary	ADJ
ejpam-6326	255	3	1	1	NUM
ejpam-6326	255	4	,	,	PUNCT
ejpam-6326	255	5	γf	γf	PRON
ejpam-6326	255	6	(	(	PUNCT
ejpam-6326	255	7	kn	kn	PROPN
ejpam-6326	255	8	)	)	PUNCT
ejpam-6326	255	9	=	=	PUNCT
ejpam-6326	256	1	1	1	X
ejpam-6326	256	2	.	.	PUNCT
ejpam-6326	256	3	however	however	ADV
ejpam-6326	256	4	,	,	PUNCT
ejpam-6326	256	5	the	the	DET
ejpam-6326	256	6	complement	complement	NOUN
ejpam-6326	256	7	of	of	ADP
ejpam-6326	256	8	kn	kn	PROPN
ejpam-6326	256	9	is	be	AUX
ejpam-6326	256	10	the	the	DET
ejpam-6326	256	11	empty	empty	ADJ
ejpam-6326	256	12	graph	graph	NOUN
ejpam-6326	256	13	on	on	ADP
ejpam-6326	256	14	n	n	DET
ejpam-6326	256	15	vertices	vertex	NOUN
ejpam-6326	256	16	,	,	PUNCT
ejpam-6326	256	17	kn	kn	PROPN
ejpam-6326	256	18	,	,	PUNCT
ejpam-6326	256	19	in	in	ADP
ejpam-6326	256	20	which	which	PRON
ejpam-6326	256	21	no	no	DET
ejpam-6326	256	22	vertices	vertex	NOUN
ejpam-6326	256	23	are	be	AUX
ejpam-6326	256	24	adjacent	adjacent	ADJ
ejpam-6326	256	25	.	.	PUNCT
ejpam-6326	257	1	in	in	ADP
ejpam-6326	257	2	kn	kn	PROPN
ejpam-6326	257	3	,	,	PUNCT
ejpam-6326	257	4	every	every	DET
ejpam-6326	257	5	vertex	vertex	NOUN
ejpam-6326	257	6	must	must	AUX
ejpam-6326	257	7	be	be	AUX
ejpam-6326	257	8	in	in	ADP
ejpam-6326	257	9	the	the	DET
ejpam-6326	257	10	friendly	friendly	ADJ
ejpam-6326	257	11	dominating	dominating	NOUN
ejpam-6326	257	12	set	set	NOUN
ejpam-6326	257	13	to	to	PART
ejpam-6326	257	14	dominate	dominate	VERB
ejpam-6326	257	15	itself	itself	PRON
ejpam-6326	257	16	,	,	PUNCT
ejpam-6326	257	17	so	so	ADV
ejpam-6326	257	18	γf	γf	PROPN
ejpam-6326	257	19	(	(	PUNCT
ejpam-6326	257	20	kn	kn	PROPN
ejpam-6326	257	21	)	)	PUNCT
ejpam-6326	257	22	=	=	VERB
ejpam-6326	257	23	n.	n.	NOUN
ejpam-6326	257	24	hence	hence	ADV
ejpam-6326	257	25	∣∣γf	∣∣γf	VERB
ejpam-6326	257	26	(	(	PUNCT
ejpam-6326	257	27	kn	kn	PROPN
ejpam-6326	257	28	)	)	PUNCT
ejpam-6326	257	29	−	−	PROPN
ejpam-6326	258	1	γf	γf	INTJ
ejpam-6326	258	2	(	(	PUNCT
ejpam-6326	258	3	kn	kn	PROPN
ejpam-6326	258	4	)	)	PUNCT
ejpam-6326	258	5	∣∣	∣∣	X
ejpam-6326	258	6	=	=	PUNCT
ejpam-6326	258	7	∣∣	∣∣	NUM
ejpam-6326	258	8	1	1	NUM
ejpam-6326	258	9	−	−	NOUN
ejpam-6326	258	10	n	n	PRON
ejpam-6326	258	11	∣∣	∣∣	NUM
ejpam-6326	258	12	=	=	PUNCT
ejpam-6326	258	13	n−	n−	NOUN
ejpam-6326	258	14	1	1	NUM
ejpam-6326	258	15	,	,	PUNCT
ejpam-6326	258	16	matching	match	VERB
ejpam-6326	258	17	the	the	DET
ejpam-6326	258	18	upper	upper	ADJ
ejpam-6326	258	19	bound	bind	VERB
ejpam-6326	258	20	exactly	exactly	ADV
ejpam-6326	258	21	.	.	PUNCT
ejpam-6326	259	1	on	on	ADP
ejpam-6326	259	2	the	the	DET
ejpam-6326	259	3	other	other	ADJ
ejpam-6326	259	4	hand	hand	NOUN
ejpam-6326	259	5	,	,	PUNCT
ejpam-6326	259	6	when	when	SCONJ
ejpam-6326	259	7	g	g	PROPN
ejpam-6326	259	8	is	be	AUX
ejpam-6326	259	9	self	self	NOUN
ejpam-6326	259	10	-	-	PUNCT
ejpam-6326	259	11	complementary	complementary	ADJ
ejpam-6326	259	12	,	,	PUNCT
ejpam-6326	259	13	γf	γf	ADJ
ejpam-6326	259	14	(	(	PUNCT
ejpam-6326	259	15	g	g	NOUN
ejpam-6326	259	16	)	)	PUNCT
ejpam-6326	259	17	=	=	SYM
ejpam-6326	259	18	γf	γf	ADJ
ejpam-6326	259	19	(	(	PUNCT
ejpam-6326	259	20	g	g	NOUN
ejpam-6326	259	21	)	)	PUNCT
ejpam-6326	259	22	,	,	PUNCT
ejpam-6326	259	23	implying	imply	VERB
ejpam-6326	259	24	that	that	SCONJ
ejpam-6326	259	25	,	,	PUNCT
ejpam-6326	259	26	∣∣γf	∣∣γf	VERB
ejpam-6326	259	27	(	(	PUNCT
ejpam-6326	259	28	g)−	g)−	PROPN
ejpam-6326	259	29	γf	γf	PROPN
ejpam-6326	259	30	(	(	PUNCT
ejpam-6326	259	31	g	g	NOUN
ejpam-6326	259	32	)	)	PUNCT
ejpam-6326	259	33	∣∣	∣∣	X
ejpam-6326	259	34	=	=	PUNCT
ejpam-6326	259	35	0	0	X
ejpam-6326	259	36	.	.	PUNCT
ejpam-6326	260	1	consequently	consequently	ADV
ejpam-6326	260	2	,	,	PUNCT
ejpam-6326	260	3	for	for	ADP
ejpam-6326	260	4	general	general	ADJ
ejpam-6326	260	5	connected	connect	VERB
ejpam-6326	260	6	nontrivial	nontrivial	ADJ
ejpam-6326	260	7	graphs	graph	NOUN
ejpam-6326	260	8	g	g	ADP
ejpam-6326	260	9	of	of	ADP
ejpam-6326	260	10	order	order	NOUN
ejpam-6326	260	11	n	n	CCONJ
ejpam-6326	260	12	,	,	PUNCT
ejpam-6326	260	13	the	the	DET
ejpam-6326	260	14	difference	difference	NOUN
ejpam-6326	260	15	∣∣γf	∣∣γf	VERB
ejpam-6326	260	16	(	(	PUNCT
ejpam-6326	260	17	g	g	NOUN
ejpam-6326	260	18	)	)	PUNCT
ejpam-6326	260	19	−	−	NOUN
ejpam-6326	261	1	γf	γf	INTJ
ejpam-6326	261	2	(	(	PUNCT
ejpam-6326	261	3	g	g	NOUN
ejpam-6326	261	4	)	)	PUNCT
ejpam-6326	261	5	∣∣	∣∣	NUM
ejpam-6326	261	6	is	be	AUX
ejpam-6326	261	7	always	always	ADV
ejpam-6326	261	8	between	between	ADP
ejpam-6326	261	9	0	0	NUM
ejpam-6326	261	10	and	and	CCONJ
ejpam-6326	261	11	n−	n−	NOUN
ejpam-6326	261	12	1	1	NUM
ejpam-6326	261	13	,	,	PUNCT
ejpam-6326	261	14	and	and	CCONJ
ejpam-6326	261	15	both	both	DET
ejpam-6326	261	16	extremes	extreme	NOUN
ejpam-6326	261	17	can	can	AUX
ejpam-6326	261	18	be	be	AUX
ejpam-6326	261	19	attained	attain	VERB
ejpam-6326	261	20	.	.	PUNCT
ejpam-6326	262	1	let	let	VERB
ejpam-6326	262	2	g	g	PRON
ejpam-6326	262	3	be	be	AUX
ejpam-6326	262	4	a	a	DET
ejpam-6326	262	5	graph	graph	NOUN
ejpam-6326	262	6	of	of	ADP
ejpam-6326	262	7	order	order	NOUN
ejpam-6326	262	8	n.	n.	NOUN
ejpam-6326	262	9	for	for	ADP
ejpam-6326	262	10	the	the	DET
ejpam-6326	262	11	empty	empty	ADJ
ejpam-6326	262	12	graph	graph	NOUN
ejpam-6326	262	13	we	we	PRON
ejpam-6326	262	14	clearly	clearly	ADV
ejpam-6326	262	15	have	have	VERB
ejpam-6326	262	16	both	both	CCONJ
ejpam-6326	262	17	the	the	DET
ejpam-6326	262	18	domination	domination	NOUN
ejpam-6326	262	19	number	number	NOUN
ejpam-6326	262	20	and	and	CCONJ
ejpam-6326	262	21	the	the	DET
ejpam-6326	262	22	friendly	friendly	ADJ
ejpam-6326	262	23	-	-	PUNCT
ejpam-6326	262	24	domination	domination	NOUN
ejpam-6326	262	25	number	number	NOUN
ejpam-6326	262	26	equal	equal	ADJ
ejpam-6326	262	27	to	to	ADP
ejpam-6326	262	28	n.	n.	NOUN
ejpam-6326	262	29	when	when	SCONJ
ejpam-6326	262	30	g	g	PROPN
ejpam-6326	262	31	is	be	AUX
ejpam-6326	262	32	connected	connect	VERB
ejpam-6326	262	33	and	and	CCONJ
ejpam-6326	262	34	nontrivial	nontrivial	ADJ
ejpam-6326	262	35	one	one	PRON
ejpam-6326	262	36	might	might	AUX
ejpam-6326	262	37	expect	expect	VERB
ejpam-6326	262	38	these	these	DET
ejpam-6326	262	39	parameters	parameter	NOUN
ejpam-6326	262	40	to	to	PART
ejpam-6326	262	41	drop	drop	VERB
ejpam-6326	262	42	,	,	PUNCT
ejpam-6326	262	43	and	and	CCONJ
ejpam-6326	262	44	indeed	indeed	ADV
ejpam-6326	262	45	the	the	DET
ejpam-6326	262	46	ordinary	ordinary	ADJ
ejpam-6326	262	47	domination	domination	NOUN
ejpam-6326	262	48	number	number	NOUN
ejpam-6326	262	49	always	always	ADV
ejpam-6326	262	50	does	do	VERB
ejpam-6326	262	51	.	.	PUNCT
ejpam-6326	263	1	in	in	ADP
ejpam-6326	263	2	contrast	contrast	NOUN
ejpam-6326	263	3	,	,	PUNCT
ejpam-6326	263	4	the	the	DET
ejpam-6326	263	5	friendly	friendly	ADJ
ejpam-6326	263	6	-	-	PUNCT
ejpam-6326	263	7	domination	domination	NOUN
ejpam-6326	263	8	number	number	NOUN
ejpam-6326	263	9	can	can	AUX
ejpam-6326	263	10	remain	remain	VERB
ejpam-6326	263	11	as	as	ADV
ejpam-6326	263	12	large	large	ADJ
ejpam-6326	263	13	as	as	ADP
ejpam-6326	263	14	n	n	CCONJ
ejpam-6326	263	15	,	,	PUNCT
ejpam-6326	263	16	even	even	ADV
ejpam-6326	263	17	in	in	ADP
ejpam-6326	263	18	connected	connected	ADJ
ejpam-6326	263	19	graphs	graph	NOUN
ejpam-6326	263	20	,	,	PUNCT
ejpam-6326	263	21	as	as	ADP
ejpam-6326	263	22	the	the	DET
ejpam-6326	263	23	next	next	ADJ
ejpam-6326	263	24	lemma	lemma	PROPN
ejpam-6326	263	25	shows	show	VERB
ejpam-6326	263	26	.	.	PUNCT
ejpam-6326	264	1	lemma	lemma	PROPN
ejpam-6326	264	2	1	1	X
ejpam-6326	264	3	.	.	PUNCT
ejpam-6326	265	1	let	let	VERB
ejpam-6326	265	2	g	g	PRON
ejpam-6326	265	3	be	be	AUX
ejpam-6326	265	4	a	a	DET
ejpam-6326	265	5	connected	connected	ADJ
ejpam-6326	265	6	graph	graph	NOUN
ejpam-6326	265	7	on	on	ADP
ejpam-6326	265	8	at	at	ADV
ejpam-6326	265	9	least	least	ADV
ejpam-6326	265	10	two	two	NUM
ejpam-6326	265	11	vertices	vertex	NOUN
ejpam-6326	265	12	.	.	PUNCT
ejpam-6326	266	1	then	then	ADV
ejpam-6326	266	2	γf	γf	INTJ
ejpam-6326	266	3	(	(	PUNCT
ejpam-6326	266	4	g	g	NOUN
ejpam-6326	266	5	)	)	PUNCT
ejpam-6326	266	6	≤	≤	NOUN
ejpam-6326	266	7	n	n	CCONJ
ejpam-6326	266	8	,	,	PUNCT
ejpam-6326	266	9	and	and	CCONJ
ejpam-6326	266	10	the	the	DET
ejpam-6326	266	11	bound	bind	VERB
ejpam-6326	266	12	is	be	AUX
ejpam-6326	266	13	best	well	ADV
ejpam-6326	266	14	possible	possible	ADJ
ejpam-6326	266	15	.	.	PUNCT
ejpam-6326	267	1	proof	proof	NOUN
ejpam-6326	267	2	.	.	PUNCT
ejpam-6326	268	1	since	since	SCONJ
ejpam-6326	268	2	g	g	PROPN
ejpam-6326	268	3	is	be	AUX
ejpam-6326	268	4	connected	connect	VERB
ejpam-6326	268	5	,	,	PUNCT
ejpam-6326	268	6	by	by	ADP
ejpam-6326	268	7	remark	remark	NOUN
ejpam-6326	268	8	2	2	NUM
ejpam-6326	268	9	,	,	PUNCT
ejpam-6326	268	10	the	the	DET
ejpam-6326	268	11	whole	whole	ADJ
ejpam-6326	268	12	vertex	vertex	NOUN
ejpam-6326	268	13	set	set	VERB
ejpam-6326	268	14	v	v	NOUN
ejpam-6326	268	15	(	(	PUNCT
ejpam-6326	268	16	g	g	NOUN
ejpam-6326	268	17	)	)	PUNCT
ejpam-6326	268	18	is	be	AUX
ejpam-6326	268	19	a	a	DET
ejpam-6326	268	20	non	non	ADJ
ejpam-6326	268	21	-	-	ADJ
ejpam-6326	268	22	empty	empty	ADJ
ejpam-6326	268	23	friendly	friendly	ADJ
ejpam-6326	268	24	dominating	dominating	NOUN
ejpam-6326	268	25	set	set	NOUN
ejpam-6326	268	26	.	.	PUNCT
ejpam-6326	269	1	hence	hence	ADV
ejpam-6326	269	2	γf	γf	PROPN
ejpam-6326	269	3	(	(	PUNCT
ejpam-6326	269	4	g	g	NOUN
ejpam-6326	269	5	)	)	PUNCT
ejpam-6326	269	6	≤	≤	NOUN
ejpam-6326	269	7	|v	|v	X
ejpam-6326	269	8	(	(	PUNCT
ejpam-6326	269	9	g)|	g)|	NOUN
ejpam-6326	269	10	=	=	PUNCT
ejpam-6326	269	11	n.	n.	NOUN
ejpam-6326	269	12	to	to	PART
ejpam-6326	269	13	show	show	VERB
ejpam-6326	269	14	that	that	SCONJ
ejpam-6326	269	15	the	the	DET
ejpam-6326	269	16	bound	bind	VERB
ejpam-6326	269	17	is	be	AUX
ejpam-6326	269	18	tight	tight	ADJ
ejpam-6326	269	19	.	.	PUNCT
ejpam-6326	270	1	consider	consider	VERB
ejpam-6326	270	2	the	the	DET
ejpam-6326	270	3	star	star	NOUN
ejpam-6326	270	4	k1,n−1	k1,n−1	ADJ
ejpam-6326	270	5	with	with	ADP
ejpam-6326	270	6	center	center	NOUN
ejpam-6326	270	7	c	c	PROPN
ejpam-6326	270	8	and	and	CCONJ
ejpam-6326	270	9	leaves	leave	VERB
ejpam-6326	270	10	l	l	NOUN
ejpam-6326	270	11	=	=	SYM
ejpam-6326	270	12	{	{	PUNCT
ejpam-6326	270	13	ℓ1	ℓ1	NOUN
ejpam-6326	270	14	,	,	PUNCT
ejpam-6326	270	15	.	.	PUNCT
ejpam-6326	270	16	.	.	PUNCT
ejpam-6326	271	1	.	.	PUNCT
ejpam-6326	272	1	,	,	PUNCT
ejpam-6326	272	2	ℓn−1	ℓn−1	PROPN
ejpam-6326	272	3	}	}	PUNCT
ejpam-6326	272	4	.	.	PUNCT
ejpam-6326	273	1	let	let	VERB
ejpam-6326	273	2	f	f	PROPN
ejpam-6326	273	3	⊆	⊆	NUM
ejpam-6326	273	4	v	v	PROPN
ejpam-6326	273	5	(	(	PUNCT
ejpam-6326	273	6	k1,n−1	k1,n−1	ADJ
ejpam-6326	273	7	)	)	PUNCT
ejpam-6326	273	8	be	be	AUX
ejpam-6326	273	9	a	a	DET
ejpam-6326	273	10	friendly	friendly	ADJ
ejpam-6326	273	11	dominating	dominating	NOUN
ejpam-6326	273	12	set	set	VERB
ejpam-6326	273	13	in	in	ADP
ejpam-6326	273	14	k1,n−1	k1,n−1	PROPN
ejpam-6326	273	15	.	.	PUNCT
ejpam-6326	274	1	if	if	SCONJ
ejpam-6326	274	2	c	c	PROPN
ejpam-6326	274	3	/∈	/∈	PUNCT
ejpam-6326	275	1	f	f	PROPN
ejpam-6326	275	2	,	,	PUNCT
ejpam-6326	275	3	then	then	ADV
ejpam-6326	275	4	c	c	PROPN
ejpam-6326	275	5	must	must	AUX
ejpam-6326	275	6	be	be	AUX
ejpam-6326	275	7	dominated	dominate	VERB
ejpam-6326	275	8	by	by	ADP
ejpam-6326	275	9	some	some	DET
ejpam-6326	275	10	leaf	leaf	NOUN
ejpam-6326	275	11	ℓi	ℓi	PROPN
ejpam-6326	275	12	∈	∈	PROPN
ejpam-6326	275	13	f	f	X
ejpam-6326	275	14	.	.	PUNCT
ejpam-6326	276	1	vertex	vertex	PROPN
ejpam-6326	276	2	c	c	PROPN
ejpam-6326	276	3	has	have	VERB
ejpam-6326	276	4	no	no	DET
ejpam-6326	276	5	neighbor	neighbor	NOUN
ejpam-6326	276	6	outside	outside	ADP
ejpam-6326	276	7	f	f	PROPN
ejpam-6326	276	8	while	while	SCONJ
ejpam-6326	276	9	degf	degf	PROPN
ejpam-6326	276	10	(	(	PUNCT
ejpam-6326	276	11	c	c	NOUN
ejpam-6326	276	12	)	)	PUNCT
ejpam-6326	276	13	≥	≥	NOUN
ejpam-6326	276	14	1	1	NUM
ejpam-6326	276	15	,	,	PUNCT
ejpam-6326	276	16	contradicting	contradict	VERB
ejpam-6326	276	17	degf	degf	NOUN
ejpam-6326	276	18	(	(	PUNCT
ejpam-6326	276	19	c	c	NOUN
ejpam-6326	276	20	)	)	PUNCT
ejpam-6326	276	21	≤	≤	NOUN
ejpam-6326	276	22	degv	degv	NOUN
ejpam-6326	276	23	(	(	PUNCT
ejpam-6326	276	24	k1,n−1)\f	k1,n−1)\f	PROPN
ejpam-6326	276	25	(	(	PUNCT
ejpam-6326	276	26	c	c	NOUN
ejpam-6326	276	27	)	)	PUNCT
ejpam-6326	276	28	.	.	PUNCT
ejpam-6326	277	1	if	if	SCONJ
ejpam-6326	277	2	c	c	PROPN
ejpam-6326	277	3	∈	∈	PROPN
ejpam-6326	277	4	f	f	PROPN
ejpam-6326	277	5	but	but	CCONJ
ejpam-6326	277	6	some	some	DET
ejpam-6326	277	7	leaf	leaf	NOUN
ejpam-6326	277	8	ℓ	ℓ	NOUN
ejpam-6326	277	9	/∈	/∈	PUNCT
ejpam-6326	278	1	f	f	PROPN
ejpam-6326	278	2	,	,	PUNCT
ejpam-6326	278	3	then	then	ADV
ejpam-6326	278	4	degf	degf	PROPN
ejpam-6326	278	5	(	(	PUNCT
ejpam-6326	278	6	ℓ	ℓ	NOUN
ejpam-6326	278	7	)	)	PUNCT
ejpam-6326	278	8	=	=	SYM
ejpam-6326	278	9	1	1	NUM
ejpam-6326	278	10	(	(	PUNCT
ejpam-6326	278	11	its	its	PRON
ejpam-6326	278	12	neighbour	neighbour	NOUN
ejpam-6326	278	13	c	c	NOUN
ejpam-6326	278	14	)	)	PUNCT
ejpam-6326	278	15	and	and	CCONJ
ejpam-6326	278	16	degv	degv	NOUN
ejpam-6326	278	17	(	(	PUNCT
ejpam-6326	278	18	k1,n−1)\f	k1,n−1)\f	PROPN
ejpam-6326	278	19	(	(	PUNCT
ejpam-6326	278	20	ℓ	ℓ	NOUN
ejpam-6326	278	21	)	)	PUNCT
ejpam-6326	278	22	=	=	SYM
ejpam-6326	278	23	0	0	NUM
ejpam-6326	278	24	,	,	PUNCT
ejpam-6326	278	25	again	again	ADV
ejpam-6326	278	26	violating	violate	VERB
ejpam-6326	278	27	the	the	DET
ejpam-6326	278	28	friendly	friendly	ADJ
ejpam-6326	278	29	condition	condition	NOUN
ejpam-6326	278	30	.	.	PUNCT
ejpam-6326	279	1	therefore	therefore	ADV
ejpam-6326	279	2	every	every	DET
ejpam-6326	279	3	friendly	friendly	ADJ
ejpam-6326	279	4	dominating	dominating	NOUN
ejpam-6326	279	5	set	set	NOUN
ejpam-6326	279	6	of	of	ADP
ejpam-6326	279	7	the	the	DET
ejpam-6326	279	8	star	star	NOUN
ejpam-6326	279	9	must	must	AUX
ejpam-6326	279	10	be	be	AUX
ejpam-6326	279	11	the	the	DET
ejpam-6326	279	12	entire	entire	ADJ
ejpam-6326	279	13	vertex	vertex	NOUN
ejpam-6326	279	14	set	set	NOUN
ejpam-6326	279	15	,	,	PUNCT
ejpam-6326	279	16	giving	give	VERB
ejpam-6326	279	17	γf	γf	PRON
ejpam-6326	279	18	(	(	PUNCT
ejpam-6326	279	19	k1,n−1	k1,n−1	ADJ
ejpam-6326	279	20	)	)	PUNCT
ejpam-6326	279	21	=	=	SYM
ejpam-6326	279	22	n.	n.	PROPN
ejpam-6326	279	23	thus	thus	ADV
ejpam-6326	279	24	,	,	PUNCT
ejpam-6326	279	25	the	the	DET
ejpam-6326	279	26	upper	upper	ADJ
ejpam-6326	279	27	bound	bind	VERB
ejpam-6326	279	28	n	n	NOUN
ejpam-6326	279	29	is	be	AUX
ejpam-6326	279	30	tight	tight	ADJ
ejpam-6326	279	31	.	.	PUNCT
ejpam-6326	280	1	theorem	theorem	ADJ
ejpam-6326	280	2	8	8	NUM
ejpam-6326	280	3	.	.	PUNCT
ejpam-6326	281	1	let	let	VERB
ejpam-6326	281	2	g	g	PRON
ejpam-6326	281	3	be	be	AUX
ejpam-6326	281	4	a	a	DET
ejpam-6326	281	5	graph	graph	NOUN
ejpam-6326	281	6	on	on	ADP
ejpam-6326	281	7	n	n	PRON
ejpam-6326	281	8	≥	≥	NUM
ejpam-6326	281	9	4	4	NUM
ejpam-6326	281	10	vertices	vertex	NOUN
ejpam-6326	281	11	such	such	ADJ
ejpam-6326	281	12	that	that	SCONJ
ejpam-6326	281	13	both	both	CCONJ
ejpam-6326	281	14	g	g	PROPN
ejpam-6326	281	15	and	and	CCONJ
ejpam-6326	281	16	its	its	PRON
ejpam-6326	281	17	complement	complement	NOUN
ejpam-6326	281	18	g	g	NOUN
ejpam-6326	281	19	are	be	AUX
ejpam-6326	281	20	connected	connect	VERB
ejpam-6326	281	21	.	.	PUNCT
ejpam-6326	282	1	then	then	ADV
ejpam-6326	282	2	γf	γf	INTJ
ejpam-6326	282	3	(	(	PUNCT
ejpam-6326	282	4	g	g	NOUN
ejpam-6326	282	5	)	)	PUNCT
ejpam-6326	282	6	+	+	CCONJ
ejpam-6326	282	7	γf	γf	ADJ
ejpam-6326	282	8	(	(	PUNCT
ejpam-6326	282	9	g	g	NOUN
ejpam-6326	282	10	)	)	PUNCT
ejpam-6326	282	11	≤	≤	NOUN
ejpam-6326	282	12	2n	2n	NUM
ejpam-6326	282	13	.	.	PUNCT
ejpam-6326	283	1	proof	proof	NOUN
ejpam-6326	283	2	.	.	PUNCT
ejpam-6326	284	1	by	by	ADP
ejpam-6326	284	2	lemma	lemma	PROPN
ejpam-6326	284	3	1	1	NUM
ejpam-6326	284	4	,	,	PUNCT
ejpam-6326	284	5	γf	γf	ADJ
ejpam-6326	284	6	(	(	PUNCT
ejpam-6326	284	7	g	g	NOUN
ejpam-6326	284	8	)	)	PUNCT
ejpam-6326	284	9	≤	≤	NOUN
ejpam-6326	284	10	n.	n.	NOUN
ejpam-6326	284	11	thus	thus	ADV
ejpam-6326	284	12	,	,	PUNCT
ejpam-6326	284	13	γf(g	γf(g	NUM
ejpam-6326	284	14	)	)	PUNCT
ejpam-6326	284	15	+	+	CCONJ
ejpam-6326	284	16	γf	γf	ADJ
ejpam-6326	284	17	(	(	PUNCT
ejpam-6326	284	18	g	g	NOUN
ejpam-6326	284	19	)	)	PUNCT
ejpam-6326	284	20	≤	≤	NUM
ejpam-6326	284	21	n+	n+	PUNCT
ejpam-6326	284	22	n	n	NOUN
ejpam-6326	284	23	=	=	SYM
ejpam-6326	284	24	2n	2n	NUM
ejpam-6326	284	25	.	.	PUNCT
ejpam-6326	285	1	corollary	corollary	ADJ
ejpam-6326	285	2	6	6	NUM
ejpam-6326	285	3	.	.	PUNCT
ejpam-6326	286	1	for	for	ADP
ejpam-6326	286	2	nontrivial	nontrivial	ADJ
ejpam-6326	286	3	connected	connect	VERB
ejpam-6326	286	4	graph	graph	NOUN
ejpam-6326	286	5	g	g	PROPN
ejpam-6326	286	6	on	on	ADP
ejpam-6326	286	7	n	n	PRON
ejpam-6326	286	8	≥	≥	NUM
ejpam-6326	286	9	4	4	NUM
ejpam-6326	286	10	vertices	vertex	NOUN
ejpam-6326	286	11	,	,	PUNCT
ejpam-6326	286	12	we	we	PRON
ejpam-6326	286	13	have	have	VERB
ejpam-6326	286	14	γf	γf	ADJ
ejpam-6326	286	15	(	(	PUNCT
ejpam-6326	286	16	g	g	NOUN
ejpam-6326	286	17	)	)	PUNCT
ejpam-6326	286	18	γf	γf	PROPN
ejpam-6326	286	19	(	(	PUNCT
ejpam-6326	286	20	g	g	NOUN
ejpam-6326	286	21	)	)	PUNCT
ejpam-6326	286	22	≤	≤	NUM
ejpam-6326	286	23	n2	n2	NOUN
ejpam-6326	286	24	.	.	PUNCT
ejpam-6326	287	1	i.	i.	PROPN
ejpam-6326	287	2	s.	s.	PROPN
ejpam-6326	287	3	cabahug	cabahug	PROPN
ejpam-6326	287	4	,	,	PUNCT
ejpam-6326	287	5	jr	jr	PROPN
ejpam-6326	287	6	.	.	PROPN
ejpam-6326	287	7	,	,	PUNCT
ejpam-6326	287	8	r.	r.	PROPN
ejpam-6326	287	9	g.	g.	PROPN
ejpam-6326	287	10	eballe	eballe	PROPN
ejpam-6326	287	11	,	,	PUNCT
ejpam-6326	287	12	r.	r.	PROPN
ejpam-6326	287	13	t.	t.	PROPN
ejpam-6326	287	14	fernandez	fernandez	PROPN
ejpam-6326	287	15	/	/	SYM
ejpam-6326	287	16	eur	eur	PROPN
ejpam-6326	287	17	.	.	PUNCT
ejpam-6326	288	1	j.	j.	PROPN
ejpam-6326	288	2	pure	pure	PROPN
ejpam-6326	288	3	appl	appl	PROPN
ejpam-6326	288	4	.	.	PROPN
ejpam-6326	288	5	math	math	PROPN
ejpam-6326	288	6	,	,	PUNCT
ejpam-6326	288	7	18	18	NUM
ejpam-6326	288	8	(	(	PUNCT
ejpam-6326	288	9	4	4	NUM
ejpam-6326	288	10	)	)	PUNCT
ejpam-6326	288	11	(	(	PUNCT
ejpam-6326	288	12	2025	2025	NUM
ejpam-6326	288	13	)	)	PUNCT
ejpam-6326	288	14	,	,	PUNCT
ejpam-6326	288	15	6326	6326	NUM
ejpam-6326	288	16	10	10	NUM
ejpam-6326	288	17	of	of	ADP
ejpam-6326	288	18	15	15	NUM
ejpam-6326	288	19	proof	proof	NOUN
ejpam-6326	288	20	.	.	PUNCT
ejpam-6326	289	1	by	by	ADP
ejpam-6326	289	2	theorem	theorem	NOUN
ejpam-6326	289	3	8	8	NUM
ejpam-6326	289	4	,	,	PUNCT
ejpam-6326	289	5	γf	γf	ADJ
ejpam-6326	289	6	(	(	PUNCT
ejpam-6326	289	7	g	g	NOUN
ejpam-6326	289	8	)	)	PUNCT
ejpam-6326	289	9	+	+	CCONJ
ejpam-6326	289	10	γf	γf	PRON
ejpam-6326	289	11	(	(	PUNCT
ejpam-6326	289	12	g	g	PROPN
ejpam-6326	289	13	)	)	PUNCT
ejpam-6326	289	14	≤	≤	NUM
ejpam-6326	289	15	2n	2n	NUM
ejpam-6326	289	16	.	.	PUNCT
ejpam-6326	290	1	since	since	SCONJ
ejpam-6326	290	2	both	both	CCONJ
ejpam-6326	290	3	γf	γf	PROPN
ejpam-6326	290	4	(	(	PUNCT
ejpam-6326	290	5	g	g	NOUN
ejpam-6326	290	6	)	)	PUNCT
ejpam-6326	290	7	and	and	CCONJ
ejpam-6326	290	8	γf	γf	ADJ
ejpam-6326	290	9	(	(	PUNCT
ejpam-6326	290	10	g	g	NOUN
ejpam-6326	290	11	)	)	PUNCT
ejpam-6326	290	12	are	be	AUX
ejpam-6326	290	13	nonnegative	nonnegative	ADJ
ejpam-6326	290	14	numbers	number	NOUN
ejpam-6326	290	15	,	,	PUNCT
ejpam-6326	290	16	the	the	DET
ejpam-6326	290	17	arithmetic	arithmetic	ADJ
ejpam-6326	290	18	mean	mean	ADJ
ejpam-6326	290	19	–	–	PUNCT
ejpam-6326	290	20	geometric	geometric	ADJ
ejpam-6326	290	21	mean	mean	NOUN
ejpam-6326	290	22	(	(	PUNCT
ejpam-6326	290	23	am	am	PROPN
ejpam-6326	290	24	–	–	PUNCT
ejpam-6326	290	25	gm	gm	ADJ
ejpam-6326	290	26	)	)	PUNCT
ejpam-6326	290	27	inequality	inequality	NOUN
ejpam-6326	290	28	yields	yield	VERB
ejpam-6326	290	29	γf	γf	INTJ
ejpam-6326	290	30	(	(	PUNCT
ejpam-6326	290	31	g	g	NOUN
ejpam-6326	290	32	)	)	PUNCT
ejpam-6326	291	1	γf	γf	PROPN
ejpam-6326	292	1	(	(	PUNCT
ejpam-6326	292	2	g	g	NOUN
ejpam-6326	292	3	)	)	PUNCT
ejpam-6326	292	4	≤	≤	NOUN
ejpam-6326	292	5	(	(	PUNCT
ejpam-6326	292	6	γf	γf	INTJ
ejpam-6326	292	7	(	(	PUNCT
ejpam-6326	292	8	g	g	NOUN
ejpam-6326	292	9	)	)	PUNCT
ejpam-6326	293	1	+	+	CCONJ
ejpam-6326	293	2	γf	γf	PRON
ejpam-6326	293	3	(	(	PUNCT
ejpam-6326	293	4	g	g	NOUN
ejpam-6326	293	5	)	)	PUNCT
ejpam-6326	293	6	2	2	NUM
ejpam-6326	293	7	)	)	PUNCT
ejpam-6326	293	8	2	2	NUM
ejpam-6326	293	9	.	.	PUNCT
ejpam-6326	294	1	substituting	substitute	VERB
ejpam-6326	294	2	the	the	DET
ejpam-6326	294	3	upper	upper	ADJ
ejpam-6326	294	4	bound	bind	VERB
ejpam-6326	294	5	on	on	ADP
ejpam-6326	294	6	the	the	DET
ejpam-6326	294	7	sum	sum	NOUN
ejpam-6326	294	8	,	,	PUNCT
ejpam-6326	294	9	we	we	PRON
ejpam-6326	294	10	obtain	obtain	VERB
ejpam-6326	294	11	,	,	PUNCT
ejpam-6326	294	12	γf	γf	PROPN
ejpam-6326	294	13	(	(	PUNCT
ejpam-6326	294	14	g	g	NOUN
ejpam-6326	294	15	)	)	PUNCT
ejpam-6326	295	1	γf	γf	PROPN
ejpam-6326	296	1	(	(	PUNCT
ejpam-6326	296	2	g	g	NOUN
ejpam-6326	296	3	)	)	PUNCT
ejpam-6326	296	4	≤	≤	NOUN
ejpam-6326	296	5	(	(	PUNCT
ejpam-6326	296	6	2n	2n	NUM
ejpam-6326	296	7	2	2	NUM
ejpam-6326	296	8	)	)	SYM
ejpam-6326	296	9	2	2	NUM
ejpam-6326	296	10	=	=	SYM
ejpam-6326	296	11	n2	n2	NOUN
ejpam-6326	296	12	.	.	PUNCT
ejpam-6326	297	1	3.3	3.3	NUM
ejpam-6326	297	2	.	.	PUNCT
ejpam-6326	298	1	structure	structure	NOUN
ejpam-6326	298	2	of	of	ADP
ejpam-6326	298	3	friendly	friendly	ADJ
ejpam-6326	298	4	dominating	dominating	NOUN
ejpam-6326	298	5	sets	set	NOUN
ejpam-6326	298	6	in	in	ADP
ejpam-6326	298	7	the	the	DET
ejpam-6326	298	8	join	join	NOUN
ejpam-6326	298	9	of	of	ADP
ejpam-6326	298	10	graphs	graph	NOUN
ejpam-6326	298	11	remark	remark	VERB
ejpam-6326	298	12	3	3	NUM
ejpam-6326	298	13	.	.	PUNCT
ejpam-6326	299	1	let	let	VERB
ejpam-6326	299	2	g	g	NOUN
ejpam-6326	299	3	and	and	CCONJ
ejpam-6326	299	4	h	h	NOUN
ejpam-6326	299	5	be	be	AUX
ejpam-6326	299	6	empty	empty	ADJ
ejpam-6326	299	7	graphs	graph	NOUN
ejpam-6326	299	8	and	and	CCONJ
ejpam-6326	299	9	let	let	VERB
ejpam-6326	299	10	f	f	PROPN
ejpam-6326	299	11	⊆	⊆	NUM
ejpam-6326	299	12	v	v	NOUN
ejpam-6326	299	13	(	(	PUNCT
ejpam-6326	299	14	g	g	NOUN
ejpam-6326	299	15	∨h	∨h	PROPN
ejpam-6326	299	16	)	)	PUNCT
ejpam-6326	299	17	be	be	VERB
ejpam-6326	299	18	a	a	DET
ejpam-6326	299	19	nonempty	nonempty	ADV
ejpam-6326	299	20	set	set	VERB
ejpam-6326	299	21	.	.	PUNCT
ejpam-6326	300	1	if	if	SCONJ
ejpam-6326	300	2	f	f	PROPN
ejpam-6326	300	3	⊆	⊆	NUM
ejpam-6326	300	4	v	v	ADP
ejpam-6326	300	5	(	(	PUNCT
ejpam-6326	300	6	g	g	NOUN
ejpam-6326	300	7	)	)	PUNCT
ejpam-6326	300	8	or	or	CCONJ
ejpam-6326	300	9	f	f	PROPN
ejpam-6326	300	10	⊆	⊆	NUM
ejpam-6326	300	11	v	v	NOUN
ejpam-6326	300	12	(	(	PUNCT
ejpam-6326	300	13	h	h	NOUN
ejpam-6326	300	14	)	)	PUNCT
ejpam-6326	300	15	,	,	PUNCT
ejpam-6326	300	16	then	then	ADV
ejpam-6326	300	17	f	f	PROPN
ejpam-6326	300	18	can	can	AUX
ejpam-6326	300	19	not	not	PART
ejpam-6326	300	20	be	be	AUX
ejpam-6326	300	21	a	a	DET
ejpam-6326	300	22	friendly	friendly	ADJ
ejpam-6326	300	23	dominating	dominating	NOUN
ejpam-6326	300	24	set	set	VERB
ejpam-6326	300	25	in	in	ADP
ejpam-6326	300	26	g	g	PROPN
ejpam-6326	300	27	∨h	∨h	NOUN
ejpam-6326	300	28	.	.	PUNCT
ejpam-6326	301	1	theorem	theorem	NOUN
ejpam-6326	301	2	9	9	NUM
ejpam-6326	301	3	.	.	PUNCT
ejpam-6326	302	1	let	let	VERB
ejpam-6326	302	2	g	g	NOUN
ejpam-6326	302	3	and	and	CCONJ
ejpam-6326	302	4	h	h	NOUN
ejpam-6326	302	5	be	be	AUX
ejpam-6326	302	6	empty	empty	ADJ
ejpam-6326	302	7	graphs	graph	NOUN
ejpam-6326	302	8	with	with	ADP
ejpam-6326	302	9	|v	|v	PROPN
ejpam-6326	302	10	(	(	PUNCT
ejpam-6326	302	11	g)|	g)|	NOUN
ejpam-6326	302	12	=	=	PUNCT
ejpam-6326	302	13	n1	n1	PROPN
ejpam-6326	302	14	and	and	CCONJ
ejpam-6326	302	15	|v	|v	PROPN
ejpam-6326	302	16	(	(	PUNCT
ejpam-6326	302	17	h)|	h)|	NOUN
ejpam-6326	302	18	=	=	SYM
ejpam-6326	302	19	n2	n2	PROPN
ejpam-6326	302	20	.	.	PUNCT
ejpam-6326	303	1	let	let	VERB
ejpam-6326	303	2	f	f	PROPN
ejpam-6326	303	3	=	=	PROPN
ejpam-6326	303	4	f1	f1	PROPN
ejpam-6326	303	5	∪f2	∪f2	NOUN
ejpam-6326	303	6	,	,	PUNCT
ejpam-6326	303	7	where	where	SCONJ
ejpam-6326	303	8	f1	f1	PROPN
ejpam-6326	303	9	⊆	⊆	NUM
ejpam-6326	303	10	v	v	NOUN
ejpam-6326	303	11	(	(	PUNCT
ejpam-6326	303	12	g	g	NOUN
ejpam-6326	303	13	)	)	PUNCT
ejpam-6326	303	14	and	and	CCONJ
ejpam-6326	303	15	f2	f2	VERB
ejpam-6326	303	16	⊆	⊆	NUM
ejpam-6326	303	17	v	v	NOUN
ejpam-6326	303	18	(	(	PUNCT
ejpam-6326	303	19	h	h	NOUN
ejpam-6326	303	20	)	)	PUNCT
ejpam-6326	303	21	are	be	AUX
ejpam-6326	303	22	both	both	PRON
ejpam-6326	303	23	nonempty	nonempty	ADJ
ejpam-6326	303	24	and	and	CCONJ
ejpam-6326	303	25	assume	assume	VERB
ejpam-6326	303	26	f	f	PROPN
ejpam-6326	303	27	̸=	̸=	PROPN
ejpam-6326	303	28	v	v	PROPN
ejpam-6326	303	29	(	(	PUNCT
ejpam-6326	303	30	g∨h	g∨h	PROPN
ejpam-6326	303	31	)	)	PUNCT
ejpam-6326	303	32	(	(	PUNCT
ejpam-6326	303	33	equivalently	equivalently	ADV
ejpam-6326	303	34	,	,	PUNCT
ejpam-6326	303	35	v	v	INTJ
ejpam-6326	303	36	(	(	PUNCT
ejpam-6326	303	37	g∨h	g∨h	PROPN
ejpam-6326	303	38	)	)	PUNCT
ejpam-6326	303	39	\f	\f	PUNCT
ejpam-6326	304	1	̸=	̸=	PROPN
ejpam-6326	304	2	∅	∅	NOUN
ejpam-6326	304	3	)	)	PUNCT
ejpam-6326	304	4	.	.	PUNCT
ejpam-6326	305	1	then	then	ADV
ejpam-6326	305	2	f	f	PROPN
ejpam-6326	305	3	is	be	AUX
ejpam-6326	305	4	a	a	DET
ejpam-6326	305	5	friendly	friendly	ADJ
ejpam-6326	305	6	dominating	dominating	NOUN
ejpam-6326	305	7	set	set	NOUN
ejpam-6326	305	8	in	in	ADP
ejpam-6326	305	9	the	the	DET
ejpam-6326	305	10	join	join	NOUN
ejpam-6326	305	11	g∨h	g∨h	PROPN
ejpam-6326	306	1	if	if	SCONJ
ejpam-6326	306	2	and	and	CCONJ
ejpam-6326	306	3	only	only	ADV
ejpam-6326	306	4	if	if	SCONJ
ejpam-6326	306	5	|f1|	|f1|	NOUN
ejpam-6326	306	6	≤	≤	NUM
ejpam-6326	306	7	⌊n1	⌊n1	NOUN
ejpam-6326	306	8	2	2	NUM
ejpam-6326	306	9	⌋	⌋	NOUN
ejpam-6326	306	10	and	and	CCONJ
ejpam-6326	306	11	|f2|	|f2|	VERB
ejpam-6326	306	12	≤	≤	NUM
ejpam-6326	306	13	⌊n2	⌊n2	X
ejpam-6326	306	14	2	2	NUM
ejpam-6326	306	15	⌋	⌋	NOUN
ejpam-6326	306	16	.	.	PUNCT
ejpam-6326	307	1	proof	proof	NOUN
ejpam-6326	307	2	.	.	PUNCT
ejpam-6326	308	1	assume	assume	VERB
ejpam-6326	308	2	that	that	SCONJ
ejpam-6326	308	3	f	f	PROPN
ejpam-6326	308	4	is	be	AUX
ejpam-6326	308	5	a	a	DET
ejpam-6326	308	6	friendly	friendly	ADJ
ejpam-6326	308	7	dominating	dominating	NOUN
ejpam-6326	308	8	set	set	VERB
ejpam-6326	308	9	with	with	ADP
ejpam-6326	308	10	f	f	PROPN
ejpam-6326	308	11	̸=	̸=	PROPN
ejpam-6326	308	12	v	v	NOUN
ejpam-6326	308	13	(	(	PUNCT
ejpam-6326	308	14	g	g	NOUN
ejpam-6326	308	15	∨h	∨h	PROPN
ejpam-6326	308	16	)	)	PUNCT
ejpam-6326	308	17	.	.	PUNCT
ejpam-6326	309	1	because	because	SCONJ
ejpam-6326	309	2	f1	f1	PROPN
ejpam-6326	309	3	and	and	CCONJ
ejpam-6326	309	4	f2	f2	PROPN
ejpam-6326	309	5	are	be	AUX
ejpam-6326	309	6	both	both	PRON
ejpam-6326	309	7	nonempty	nonempty	ADJ
ejpam-6326	309	8	,	,	PUNCT
ejpam-6326	309	9	every	every	DET
ejpam-6326	309	10	vertex	vertex	NOUN
ejpam-6326	309	11	is	be	AUX
ejpam-6326	309	12	either	either	CCONJ
ejpam-6326	309	13	in	in	ADP
ejpam-6326	309	14	f	f	PROPN
ejpam-6326	309	15	or	or	CCONJ
ejpam-6326	309	16	adjacent	adjacent	ADJ
ejpam-6326	309	17	to	to	ADP
ejpam-6326	309	18	some	some	DET
ejpam-6326	309	19	vertex	vertex	NOUN
ejpam-6326	309	20	of	of	ADP
ejpam-6326	309	21	f	f	PROPN
ejpam-6326	309	22	.	.	PUNCT
ejpam-6326	310	1	hence	hence	ADV
ejpam-6326	310	2	,	,	PUNCT
ejpam-6326	310	3	f	f	PROPN
ejpam-6326	310	4	is	be	AUX
ejpam-6326	310	5	a	a	DET
ejpam-6326	310	6	dominating	dominating	NOUN
ejpam-6326	310	7	set	set	NOUN
ejpam-6326	310	8	.	.	PUNCT
ejpam-6326	311	1	now	now	ADV
ejpam-6326	311	2	,	,	PUNCT
ejpam-6326	311	3	pick	pick	VERB
ejpam-6326	311	4	any	any	DET
ejpam-6326	311	5	vertex	vertex	NOUN
ejpam-6326	311	6	y	y	PROPN
ejpam-6326	311	7	∈	∈	PROPN
ejpam-6326	311	8	v	v	PROPN
ejpam-6326	311	9	(	(	PUNCT
ejpam-6326	311	10	h)\f2	h)\f2	NOUN
ejpam-6326	311	11	.	.	PROPN
ejpam-6326	311	12	all	all	PRON
ejpam-6326	311	13	of	of	ADP
ejpam-6326	311	14	y	y	PROPN
ejpam-6326	311	15	’s	’s	PART
ejpam-6326	311	16	neighbors	neighbor	NOUN
ejpam-6326	311	17	lie	lie	VERB
ejpam-6326	311	18	in	in	ADP
ejpam-6326	311	19	v	v	NOUN
ejpam-6326	311	20	(	(	PUNCT
ejpam-6326	311	21	g	g	NOUN
ejpam-6326	311	22	)	)	PUNCT
ejpam-6326	311	23	;	;	PUNCT
ejpam-6326	311	24	specifically	specifically	ADV
ejpam-6326	311	25	,	,	PUNCT
ejpam-6326	311	26	degf	degf	PROPN
ejpam-6326	311	27	(	(	PUNCT
ejpam-6326	311	28	y	y	NOUN
ejpam-6326	311	29	)	)	PUNCT
ejpam-6326	311	30	=	=	NOUN
ejpam-6326	311	31	|f1|	|f1|	NOUN
ejpam-6326	311	32	and	and	CCONJ
ejpam-6326	311	33	degv	degv	NOUN
ejpam-6326	311	34	(	(	PUNCT
ejpam-6326	311	35	g∨h)\f	g∨h)\f	PROPN
ejpam-6326	311	36	(	(	PUNCT
ejpam-6326	311	37	y	y	NOUN
ejpam-6326	311	38	)	)	PUNCT
ejpam-6326	311	39	=	=	SYM
ejpam-6326	311	40	n1	n1	ADJ
ejpam-6326	311	41	−	−	NOUN
ejpam-6326	311	42	|f1|	|f1|	NOUN
ejpam-6326	311	43	.	.	PUNCT
ejpam-6326	312	1	since	since	SCONJ
ejpam-6326	312	2	f	f	PROPN
ejpam-6326	312	3	is	be	AUX
ejpam-6326	312	4	friendly	friendly	ADJ
ejpam-6326	312	5	,	,	PUNCT
ejpam-6326	312	6	we	we	PRON
ejpam-6326	312	7	have	have	AUX
ejpam-6326	312	8	,	,	PUNCT
ejpam-6326	312	9	|f1|	|f1|	VERB
ejpam-6326	312	10	≤	≤	ADJ
ejpam-6326	312	11	n1−|f1|	n1−|f1|	NOUN
ejpam-6326	312	12	.	.	PUNCT
ejpam-6326	313	1	thus	thus	ADV
ejpam-6326	313	2	,	,	PUNCT
ejpam-6326	313	3	|f1|	|f1|	VERB
ejpam-6326	313	4	≤	≤	NUM
ejpam-6326	313	5	⌊n1	⌊n1	NOUN
ejpam-6326	313	6	2	2	NUM
ejpam-6326	313	7	⌋	⌋	NOUN
ejpam-6326	313	8	.	.	PUNCT
ejpam-6326	314	1	repeating	repeat	VERB
ejpam-6326	314	2	the	the	DET
ejpam-6326	314	3	same	same	ADJ
ejpam-6326	314	4	argument	argument	NOUN
ejpam-6326	314	5	with	with	ADP
ejpam-6326	314	6	a	a	DET
ejpam-6326	314	7	vertex	vertex	NOUN
ejpam-6326	314	8	x	x	SYM
ejpam-6326	314	9	∈	∈	NOUN
ejpam-6326	314	10	v	v	ADP
ejpam-6326	314	11	(	(	PUNCT
ejpam-6326	314	12	g	g	NOUN
ejpam-6326	314	13	)	)	PUNCT
ejpam-6326	314	14	\	\	PROPN
ejpam-6326	314	15	f1	f1	NOUN
ejpam-6326	314	16	gives	give	VERB
ejpam-6326	314	17	|f2|	|f2|	VERB
ejpam-6326	314	18	≤	≤	NOUN
ejpam-6326	314	19	⌊n2	⌊n2	X
ejpam-6326	314	20	2	2	NUM
ejpam-6326	314	21	⌋	⌋	NOUN
ejpam-6326	314	22	.	.	PUNCT
ejpam-6326	315	1	conversely	conversely	ADV
ejpam-6326	315	2	,	,	PUNCT
ejpam-6326	315	3	suppose	suppose	VERB
ejpam-6326	315	4	that	that	SCONJ
ejpam-6326	315	5	|f1|	|f1|	VERB
ejpam-6326	315	6	≤	≤	NUM
ejpam-6326	315	7	⌊n1	⌊n1	NOUN
ejpam-6326	315	8	2	2	NUM
ejpam-6326	315	9	⌋	⌋	NOUN
ejpam-6326	315	10	and	and	CCONJ
ejpam-6326	315	11	|f2|	|f2|	VERB
ejpam-6326	315	12	≤	≤	NUM
ejpam-6326	315	13	⌊n2	⌊n2	X
ejpam-6326	315	14	2	2	NUM
ejpam-6326	315	15	⌋	⌋	NOUN
ejpam-6326	315	16	with	with	ADP
ejpam-6326	315	17	f1	f1	PROPN
ejpam-6326	315	18	and	and	CCONJ
ejpam-6326	315	19	f2	f2	PROPN
ejpam-6326	315	20	both	both	PRON
ejpam-6326	315	21	nonempty	nonempty	ADJ
ejpam-6326	315	22	.	.	PUNCT
ejpam-6326	316	1	because	because	SCONJ
ejpam-6326	316	2	f1	f1	PROPN
ejpam-6326	316	3	̸=	̸=	PROPN
ejpam-6326	316	4	∅	∅	NOUN
ejpam-6326	316	5	,	,	PUNCT
ejpam-6326	316	6	every	every	DET
ejpam-6326	316	7	vertex	vertex	NOUN
ejpam-6326	316	8	in	in	ADP
ejpam-6326	316	9	v	v	NUM
ejpam-6326	316	10	(	(	PUNCT
ejpam-6326	316	11	h	h	NOUN
ejpam-6326	316	12	)	)	PUNCT
ejpam-6326	316	13	is	be	AUX
ejpam-6326	316	14	adjacent	adjacent	ADJ
ejpam-6326	316	15	to	to	ADP
ejpam-6326	316	16	some	some	DET
ejpam-6326	316	17	vertex	vertex	NOUN
ejpam-6326	316	18	of	of	ADP
ejpam-6326	316	19	f	f	PROPN
ejpam-6326	316	20	;	;	PUNCT
ejpam-6326	316	21	similarly	similarly	ADV
ejpam-6326	316	22	,	,	PUNCT
ejpam-6326	316	23	f2	f2	PROPN
ejpam-6326	316	24	̸=	̸=	PROPN
ejpam-6326	316	25	∅	∅	NOUN
ejpam-6326	316	26	guarantees	guarantee	VERB
ejpam-6326	316	27	that	that	SCONJ
ejpam-6326	316	28	every	every	DET
ejpam-6326	316	29	vertex	vertex	NOUN
ejpam-6326	316	30	in	in	ADP
ejpam-6326	316	31	v	v	NOUN
ejpam-6326	316	32	(	(	PUNCT
ejpam-6326	316	33	g	g	NOUN
ejpam-6326	316	34	)	)	PUNCT
ejpam-6326	316	35	is	be	AUX
ejpam-6326	316	36	adjacent	adjacent	ADJ
ejpam-6326	316	37	to	to	ADP
ejpam-6326	316	38	some	some	DET
ejpam-6326	316	39	vertex	vertex	NOUN
ejpam-6326	316	40	of	of	ADP
ejpam-6326	316	41	f	f	PROPN
ejpam-6326	316	42	.	.	PUNCT
ejpam-6326	317	1	hence	hence	ADV
ejpam-6326	317	2	,	,	PUNCT
ejpam-6326	317	3	n	n	PROPN
ejpam-6326	318	1	[	[	X
ejpam-6326	318	2	f	f	X
ejpam-6326	318	3	]	]	X
ejpam-6326	318	4	=	=	SYM
ejpam-6326	318	5	v	v	X
ejpam-6326	318	6	(	(	PUNCT
ejpam-6326	318	7	g∨h	g∨h	PROPN
ejpam-6326	318	8	)	)	PUNCT
ejpam-6326	318	9	.	.	PUNCT
ejpam-6326	319	1	thus	thus	ADV
ejpam-6326	319	2	,	,	PUNCT
ejpam-6326	319	3	f	f	PROPN
ejpam-6326	319	4	is	be	AUX
ejpam-6326	319	5	a	a	DET
ejpam-6326	319	6	dominating	dominating	NOUN
ejpam-6326	319	7	set	set	VERB
ejpam-6326	319	8	in	in	ADP
ejpam-6326	319	9	g∨h	g∨h	PROPN
ejpam-6326	319	10	.	.	PUNCT
ejpam-6326	320	1	now	now	ADV
ejpam-6326	320	2	,	,	PUNCT
ejpam-6326	320	3	let	let	VERB
ejpam-6326	320	4	v	v	NUM
ejpam-6326	320	5	∈	∈	NOUN
ejpam-6326	320	6	v	v	NOUN
ejpam-6326	320	7	(	(	PUNCT
ejpam-6326	320	8	g)\f1	g)\f1	NOUN
ejpam-6326	320	9	.	.	PUNCT
ejpam-6326	321	1	its	its	PRON
ejpam-6326	321	2	neighbors	neighbor	NOUN
ejpam-6326	321	3	are	be	AUX
ejpam-6326	321	4	precisely	precisely	ADV
ejpam-6326	321	5	the	the	DET
ejpam-6326	321	6	n2	n2	ADJ
ejpam-6326	321	7	vertices	vertex	NOUN
ejpam-6326	321	8	of	of	ADP
ejpam-6326	321	9	v	v	NOUN
ejpam-6326	321	10	(	(	PUNCT
ejpam-6326	321	11	h	h	NOUN
ejpam-6326	321	12	)	)	PUNCT
ejpam-6326	321	13	,	,	PUNCT
ejpam-6326	321	14	of	of	ADP
ejpam-6326	321	15	which	which	PRON
ejpam-6326	321	16	|f2|	|f2|	VERB
ejpam-6326	321	17	lie	lie	VERB
ejpam-6326	321	18	in	in	ADP
ejpam-6326	321	19	f	f	PROPN
ejpam-6326	321	20	and	and	CCONJ
ejpam-6326	321	21	n2−|f2|	n2−|f2|	PROPN
ejpam-6326	321	22	lie	lie	VERB
ejpam-6326	321	23	outside	outside	ADP
ejpam-6326	321	24	f	f	PROPN
ejpam-6326	321	25	.	.	PUNCT
ejpam-6326	322	1	hence	hence	ADV
ejpam-6326	322	2	,	,	PUNCT
ejpam-6326	322	3	degf	degf	PROPN
ejpam-6326	322	4	(	(	PUNCT
ejpam-6326	322	5	v	v	NOUN
ejpam-6326	322	6	)	)	PUNCT
ejpam-6326	322	7	=	=	SYM
ejpam-6326	322	8	|f2|	|f2|	NOUN
ejpam-6326	322	9	and	and	CCONJ
ejpam-6326	322	10	degv	degv	NOUN
ejpam-6326	322	11	(	(	PUNCT
ejpam-6326	322	12	g∨h)\f	g∨h)\f	PROPN
ejpam-6326	322	13	(	(	PUNCT
ejpam-6326	322	14	v	v	NOUN
ejpam-6326	322	15	)	)	PUNCT
ejpam-6326	322	16	=	=	SYM
ejpam-6326	322	17	n2	n2	NOUN
ejpam-6326	322	18	−	−	PROPN
ejpam-6326	322	19	|f2|	|f2|	NOUN
ejpam-6326	322	20	.	.	PUNCT
ejpam-6326	323	1	by	by	ADP
ejpam-6326	323	2	assumption	assumption	NOUN
ejpam-6326	323	3	,	,	PUNCT
ejpam-6326	323	4	|f2|	|f2|	ADJ
ejpam-6326	323	5	≤	≤	NUM
ejpam-6326	323	6	⌊n2	⌊n2	X
ejpam-6326	323	7	2	2	NUM
ejpam-6326	323	8	⌋	⌋	NOUN
ejpam-6326	323	9	,	,	PUNCT
ejpam-6326	323	10	implying	imply	VERB
ejpam-6326	323	11	that,|f2|	that,|f2|	NOUN
ejpam-6326	323	12	≤	≤	NUM
ejpam-6326	323	13	n2−	n2−	NUM
ejpam-6326	323	14	|f2|	|f2|	NOUN
ejpam-6326	323	15	.	.	PUNCT
ejpam-6326	324	1	thus	thus	ADV
ejpam-6326	324	2	,	,	PUNCT
ejpam-6326	324	3	degf	degf	PROPN
ejpam-6326	324	4	(	(	PUNCT
ejpam-6326	324	5	v	v	NOUN
ejpam-6326	324	6	)	)	PUNCT
ejpam-6326	324	7	≤	≤	NOUN
ejpam-6326	324	8	degv	degv	NOUN
ejpam-6326	324	9	(	(	PUNCT
ejpam-6326	324	10	g∨h)\f	g∨h)\f	PROPN
ejpam-6326	324	11	(	(	PUNCT
ejpam-6326	324	12	v	v	NOUN
ejpam-6326	324	13	)	)	PUNCT
ejpam-6326	324	14	.	.	PUNCT
ejpam-6326	325	1	an	an	DET
ejpam-6326	325	2	analogous	analogous	ADJ
ejpam-6326	325	3	calculation	calculation	NOUN
ejpam-6326	325	4	applies	apply	VERB
ejpam-6326	325	5	for	for	ADP
ejpam-6326	325	6	w	w	PROPN
ejpam-6326	325	7	∈	∈	PROPN
ejpam-6326	325	8	v	v	ADP
ejpam-6326	325	9	(	(	PUNCT
ejpam-6326	325	10	h	h	NOUN
ejpam-6326	325	11	)	)	PUNCT
ejpam-6326	325	12	\	\	NOUN
ejpam-6326	325	13	f2	f2	PROPN
ejpam-6326	325	14	,	,	PUNCT
ejpam-6326	325	15	using	use	VERB
ejpam-6326	325	16	|f1|	|f1|	NOUN
ejpam-6326	325	17	≤	≤	ADJ
ejpam-6326	325	18	n1	n1	PROPN
ejpam-6326	325	19	−	−	PROPN
ejpam-6326	325	20	|f1|	|f1|	NOUN
ejpam-6326	325	21	gives	give	VERB
ejpam-6326	325	22	degf	degf	PROPN
ejpam-6326	325	23	(	(	PUNCT
ejpam-6326	325	24	w	w	NOUN
ejpam-6326	325	25	)	)	PUNCT
ejpam-6326	325	26	≤	≤	NOUN
ejpam-6326	325	27	degv	degv	NOUN
ejpam-6326	325	28	(	(	PUNCT
ejpam-6326	325	29	g∨h)\f	g∨h)\f	PROPN
ejpam-6326	325	30	(	(	PUNCT
ejpam-6326	325	31	w	w	NOUN
ejpam-6326	325	32	)	)	PUNCT
ejpam-6326	325	33	.	.	PUNCT
ejpam-6326	326	1	thus	thus	ADV
ejpam-6326	326	2	,	,	PUNCT
ejpam-6326	326	3	f	f	PROPN
ejpam-6326	326	4	is	be	AUX
ejpam-6326	326	5	friendly	friendly	ADJ
ejpam-6326	326	6	.	.	PUNCT
ejpam-6326	327	1	therefore	therefore	ADV
ejpam-6326	327	2	,	,	PUNCT
ejpam-6326	327	3	f	f	PROPN
ejpam-6326	327	4	is	be	AUX
ejpam-6326	327	5	a	a	DET
ejpam-6326	327	6	friendly	friendly	ADJ
ejpam-6326	327	7	dominating	dominating	NOUN
ejpam-6326	327	8	set	set	VERB
ejpam-6326	327	9	in	in	ADP
ejpam-6326	327	10	g	g	PROPN
ejpam-6326	327	11	∨h	∨h	NOUN
ejpam-6326	327	12	.	.	PUNCT
ejpam-6326	328	1	theorem	theorem	ADJ
ejpam-6326	328	2	10	10	NUM
ejpam-6326	328	3	.	.	PUNCT
ejpam-6326	329	1	let	let	VERB
ejpam-6326	329	2	g	g	NOUN
ejpam-6326	329	3	and	and	CCONJ
ejpam-6326	329	4	h	h	PROPN
ejpam-6326	329	5	be	be	VERB
ejpam-6326	329	6	non	non	ADJ
ejpam-6326	329	7	-	-	ADJ
ejpam-6326	329	8	trivial	trivial	ADJ
ejpam-6326	329	9	connected	connected	ADJ
ejpam-6326	329	10	graphs	graph	NOUN
ejpam-6326	329	11	and	and	CCONJ
ejpam-6326	329	12	let	let	VERB
ejpam-6326	329	13	f	f	PROPN
ejpam-6326	329	14	⊆	⊆	NUM
ejpam-6326	329	15	v	v	NOUN
ejpam-6326	329	16	(	(	PUNCT
ejpam-6326	329	17	g	g	NOUN
ejpam-6326	329	18	)	)	PUNCT
ejpam-6326	329	19	.	.	PUNCT
ejpam-6326	330	1	then	then	ADV
ejpam-6326	330	2	f	f	PROPN
ejpam-6326	330	3	is	be	AUX
ejpam-6326	330	4	a	a	DET
ejpam-6326	330	5	friendly	friendly	ADJ
ejpam-6326	330	6	dominating	dominating	NOUN
ejpam-6326	330	7	set	set	NOUN
ejpam-6326	330	8	of	of	ADP
ejpam-6326	330	9	g	g	PROPN
ejpam-6326	330	10	∨h	∨h	NOUN
ejpam-6326	330	11	if	if	SCONJ
ejpam-6326	330	12	and	and	CCONJ
ejpam-6326	330	13	only	only	ADV
ejpam-6326	330	14	if	if	SCONJ
ejpam-6326	330	15	the	the	DET
ejpam-6326	330	16	following	follow	VERB
ejpam-6326	330	17	three	three	NUM
ejpam-6326	330	18	conditions	condition	NOUN
ejpam-6326	330	19	hold	hold	VERB
ejpam-6326	330	20	:	:	PUNCT
ejpam-6326	330	21	(	(	PUNCT
ejpam-6326	330	22	i	i	NOUN
ejpam-6326	330	23	)	)	PUNCT
ejpam-6326	330	24	f	f	PROPN
ejpam-6326	330	25	dominates	dominate	VERB
ejpam-6326	330	26	g	g	NOUN
ejpam-6326	330	27	;	;	PUNCT
ejpam-6326	330	28	(	(	PUNCT
ejpam-6326	330	29	ii	ii	NOUN
ejpam-6326	330	30	)	)	PUNCT
ejpam-6326	330	31	|f	|f	PUNCT
ejpam-6326	331	1	|	|	ADV
ejpam-6326	331	2	≤	≤	NUM
ejpam-6326	331	3	|v	|v	X
ejpam-6326	331	4	(	(	PUNCT
ejpam-6326	331	5	g	g	NOUN
ejpam-6326	331	6	)	)	PUNCT
ejpam-6326	331	7	\	\	PROPN
ejpam-6326	332	1	f	f	PROPN
ejpam-6326	332	2	|+	|+	X
ejpam-6326	332	3	δ(h	δ(h	PROPN
ejpam-6326	332	4	)	)	PUNCT
ejpam-6326	332	5	;	;	PUNCT
ejpam-6326	332	6	i.	i.	PROPN
ejpam-6326	332	7	s.	s.	PROPN
ejpam-6326	332	8	cabahug	cabahug	PROPN
ejpam-6326	332	9	,	,	PUNCT
ejpam-6326	332	10	jr	jr	PROPN
ejpam-6326	332	11	.	.	PROPN
ejpam-6326	332	12	,	,	PUNCT
ejpam-6326	332	13	r.	r.	PROPN
ejpam-6326	332	14	g.	g.	PROPN
ejpam-6326	332	15	eballe	eballe	PROPN
ejpam-6326	332	16	,	,	PUNCT
ejpam-6326	332	17	r.	r.	PROPN
ejpam-6326	332	18	t.	t.	PROPN
ejpam-6326	332	19	fernandez	fernandez	PROPN
ejpam-6326	332	20	/	/	SYM
ejpam-6326	332	21	eur	eur	PROPN
ejpam-6326	332	22	.	.	PUNCT
ejpam-6326	333	1	j.	j.	PROPN
ejpam-6326	333	2	pure	pure	PROPN
ejpam-6326	333	3	appl	appl	PROPN
ejpam-6326	333	4	.	.	PROPN
ejpam-6326	333	5	math	math	PROPN
ejpam-6326	333	6	,	,	PUNCT
ejpam-6326	333	7	18	18	NUM
ejpam-6326	333	8	(	(	PUNCT
ejpam-6326	333	9	4	4	NUM
ejpam-6326	333	10	)	)	PUNCT
ejpam-6326	333	11	(	(	PUNCT
ejpam-6326	333	12	2025	2025	NUM
ejpam-6326	333	13	)	)	PUNCT
ejpam-6326	333	14	,	,	PUNCT
ejpam-6326	333	15	6326	6326	NUM
ejpam-6326	333	16	11	11	NUM
ejpam-6326	333	17	of	of	ADP
ejpam-6326	333	18	15	15	NUM
ejpam-6326	333	19	(	(	PUNCT
ejpam-6326	333	20	iii	iii	NOUN
ejpam-6326	333	21	)	)	PUNCT
ejpam-6326	333	22	deggf	deggf	NOUN
ejpam-6326	333	23	(	(	PUNCT
ejpam-6326	333	24	x	x	X
ejpam-6326	333	25	)	)	PUNCT
ejpam-6326	333	26	≤	≤	NOUN
ejpam-6326	333	27	|v	|v	X
ejpam-6326	333	28	(	(	PUNCT
ejpam-6326	333	29	h)|+	h)|+	ADJ
ejpam-6326	333	30	deggv	deggv	NOUN
ejpam-6326	333	31	(	(	PUNCT
ejpam-6326	333	32	g)\f	g)\f	NOUN
ejpam-6326	333	33	(	(	PUNCT
ejpam-6326	333	34	x	x	NOUN
ejpam-6326	333	35	)	)	PUNCT
ejpam-6326	333	36	for	for	ADP
ejpam-6326	333	37	every	every	PRON
ejpam-6326	333	38	x	x	SYM
ejpam-6326	333	39	∈	∈	PROPN
ejpam-6326	333	40	v	v	ADP
ejpam-6326	333	41	(	(	PUNCT
ejpam-6326	333	42	g	g	NOUN
ejpam-6326	333	43	)	)	PUNCT
ejpam-6326	333	44	\	\	NOUN
ejpam-6326	334	1	f	f	PROPN
ejpam-6326	334	2	proof	proof	NOUN
ejpam-6326	334	3	.	.	PUNCT
ejpam-6326	335	1	assume	assume	VERB
ejpam-6326	335	2	f	f	PROPN
ejpam-6326	335	3	is	be	AUX
ejpam-6326	335	4	a	a	DET
ejpam-6326	335	5	friendly	friendly	ADJ
ejpam-6326	335	6	dominating	dominating	NOUN
ejpam-6326	335	7	set	set	NOUN
ejpam-6326	335	8	of	of	ADP
ejpam-6326	335	9	g	g	PROPN
ejpam-6326	335	10	∨	∨	PROPN
ejpam-6326	335	11	h.	h.	PROPN
ejpam-6326	335	12	since	since	SCONJ
ejpam-6326	335	13	every	every	DET
ejpam-6326	335	14	vertex	vertex	NOUN
ejpam-6326	335	15	of	of	ADP
ejpam-6326	335	16	h	h	NOUN
ejpam-6326	335	17	is	be	AUX
ejpam-6326	335	18	adjacent	adjacent	ADJ
ejpam-6326	335	19	to	to	ADP
ejpam-6326	335	20	all	all	DET
ejpam-6326	335	21	vertices	vertex	NOUN
ejpam-6326	335	22	of	of	ADP
ejpam-6326	335	23	g	g	NOUN
ejpam-6326	335	24	in	in	ADP
ejpam-6326	335	25	the	the	DET
ejpam-6326	335	26	join	join	NOUN
ejpam-6326	335	27	,	,	PUNCT
ejpam-6326	335	28	f	f	PROPN
ejpam-6326	335	29	already	already	ADV
ejpam-6326	335	30	dominates	dominate	VERB
ejpam-6326	335	31	g	g	PROPN
ejpam-6326	335	32	∨h	∨h	NOUN
ejpam-6326	335	33	,	,	PUNCT
ejpam-6326	335	34	hence	hence	ADV
ejpam-6326	335	35	in	in	ADP
ejpam-6326	335	36	particular	particular	ADJ
ejpam-6326	335	37	g	g	NOUN
ejpam-6326	335	38	;	;	PUNCT
ejpam-6326	335	39	this	this	PRON
ejpam-6326	335	40	is	be	AUX
ejpam-6326	335	41	(	(	PUNCT
ejpam-6326	335	42	i	i	NOUN
ejpam-6326	335	43	)	)	PUNCT
ejpam-6326	335	44	.	.	PUNCT
ejpam-6326	336	1	choose	choose	VERB
ejpam-6326	336	2	y	y	PROPN
ejpam-6326	336	3	∈	∈	PROPN
ejpam-6326	336	4	v	v	PROPN
ejpam-6326	336	5	(	(	PUNCT
ejpam-6326	336	6	h	h	NOUN
ejpam-6326	336	7	)	)	PUNCT
ejpam-6326	336	8	.	.	PUNCT
ejpam-6326	337	1	then	then	ADV
ejpam-6326	337	2	degf	degf	PROPN
ejpam-6326	337	3	(	(	PUNCT
ejpam-6326	337	4	y	y	NOUN
ejpam-6326	337	5	)	)	PUNCT
ejpam-6326	337	6	=	=	PRON
ejpam-6326	337	7	|f	|f	PROPN
ejpam-6326	338	1	|	|	ADV
ejpam-6326	338	2	and	and	CCONJ
ejpam-6326	338	3	degv	degv	NOUN
ejpam-6326	338	4	(	(	PUNCT
ejpam-6326	338	5	g∨h)\f	g∨h)\f	PROPN
ejpam-6326	338	6	(	(	PUNCT
ejpam-6326	338	7	y	y	NOUN
ejpam-6326	338	8	)	)	PUNCT
ejpam-6326	338	9	=	=	SYM
ejpam-6326	338	10	|v	|v	PROPN
ejpam-6326	338	11	(	(	PUNCT
ejpam-6326	338	12	g)\f	g)\f	NOUN
ejpam-6326	338	13	|+	|+	NOUN
ejpam-6326	338	14	degh(y	degh(y	PROPN
ejpam-6326	338	15	)	)	PUNCT
ejpam-6326	338	16	.	.	PUNCT
ejpam-6326	339	1	since	since	SCONJ
ejpam-6326	339	2	f	f	PROPN
ejpam-6326	339	3	is	be	AUX
ejpam-6326	339	4	friendly	friendly	ADJ
ejpam-6326	339	5	and	and	CCONJ
ejpam-6326	339	6	degh(y	degh(y	ADJ
ejpam-6326	339	7	)	)	PUNCT
ejpam-6326	339	8	≥	≥	PROPN
ejpam-6326	339	9	δ(h	δ(h	PROPN
ejpam-6326	339	10	)	)	PUNCT
ejpam-6326	339	11	,	,	PUNCT
ejpam-6326	339	12	we	we	PRON
ejpam-6326	339	13	have	have	VERB
ejpam-6326	339	14	,	,	PUNCT
ejpam-6326	339	15	|f	|f	PROPN
ejpam-6326	340	1	|	|	ADV
ejpam-6326	340	2	≤	≤	NUM
ejpam-6326	340	3	|v	|v	X
ejpam-6326	340	4	(	(	PUNCT
ejpam-6326	340	5	g	g	NOUN
ejpam-6326	340	6	)	)	PUNCT
ejpam-6326	340	7	\	\	NOUN
ejpam-6326	340	8	f	f	PROPN
ejpam-6326	341	1	|	|	ADV
ejpam-6326	341	2	+	+	CCONJ
ejpam-6326	341	3	δ(h	δ(h	PROPN
ejpam-6326	341	4	)	)	PUNCT
ejpam-6326	341	5	.	.	PUNCT
ejpam-6326	342	1	thus	thus	ADV
ejpam-6326	342	2	(	(	PUNCT
ejpam-6326	342	3	ii	ii	NOUN
ejpam-6326	342	4	)	)	PUNCT
ejpam-6326	342	5	holds	hold	VERB
ejpam-6326	342	6	.	.	PUNCT
ejpam-6326	343	1	now	now	ADV
ejpam-6326	343	2	take	take	VERB
ejpam-6326	343	3	x	x	PUNCT
ejpam-6326	343	4	∈	∈	PROPN
ejpam-6326	343	5	v	v	NOUN
ejpam-6326	343	6	(	(	PUNCT
ejpam-6326	343	7	g)\f	g)\f	PROPN
ejpam-6326	343	8	.	.	PUNCT
ejpam-6326	344	1	then	then	ADV
ejpam-6326	344	2	,	,	PUNCT
ejpam-6326	344	3	degf	degf	PROPN
ejpam-6326	344	4	(	(	PUNCT
ejpam-6326	344	5	x	x	NOUN
ejpam-6326	344	6	)	)	PUNCT
ejpam-6326	344	7	=	=	SYM
ejpam-6326	344	8	deggf	deggf	ADJ
ejpam-6326	344	9	(	(	PUNCT
ejpam-6326	344	10	x	x	NOUN
ejpam-6326	344	11	)	)	PUNCT
ejpam-6326	344	12	and	and	CCONJ
ejpam-6326	344	13	degv	degv	NOUN
ejpam-6326	344	14	(	(	PUNCT
ejpam-6326	344	15	g∨h)\f	g∨h)\f	PROPN
ejpam-6326	344	16	(	(	PUNCT
ejpam-6326	344	17	x	x	NOUN
ejpam-6326	344	18	)	)	PUNCT
ejpam-6326	344	19	=	=	SYM
ejpam-6326	344	20	|v	|v	PROPN
ejpam-6326	344	21	(	(	PUNCT
ejpam-6326	344	22	h)|	h)|	NOUN
ejpam-6326	344	23	+	+	CCONJ
ejpam-6326	344	24	deggv	deggv	PROPN
ejpam-6326	344	25	(	(	PUNCT
ejpam-6326	344	26	g)\f	g)\f	PROPN
ejpam-6326	344	27	(	(	PUNCT
ejpam-6326	344	28	x	x	NOUN
ejpam-6326	344	29	)	)	PUNCT
ejpam-6326	344	30	.	.	PUNCT
ejpam-6326	345	1	again	again	ADV
ejpam-6326	345	2	,	,	PUNCT
ejpam-6326	345	3	since	since	SCONJ
ejpam-6326	345	4	f	f	PROPN
ejpam-6326	345	5	is	be	AUX
ejpam-6326	345	6	friendly	friendly	ADJ
ejpam-6326	345	7	,	,	PUNCT
ejpam-6326	345	8	deggf	deggf	ADJ
ejpam-6326	345	9	(	(	PUNCT
ejpam-6326	345	10	x	x	NOUN
ejpam-6326	345	11	)	)	PUNCT
ejpam-6326	345	12	≤	≤	NOUN
ejpam-6326	345	13	|v	|v	X
ejpam-6326	345	14	(	(	PUNCT
ejpam-6326	345	15	h)|	h)|	PROPN
ejpam-6326	345	16	+	+	CCONJ
ejpam-6326	345	17	deggv	deggv	PROPN
ejpam-6326	345	18	(	(	PUNCT
ejpam-6326	345	19	g)\f	g)\f	PROPN
ejpam-6326	345	20	(	(	PUNCT
ejpam-6326	345	21	x	x	NOUN
ejpam-6326	345	22	)	)	PUNCT
ejpam-6326	345	23	.	.	PUNCT
ejpam-6326	346	1	thus	thus	ADV
ejpam-6326	346	2	,	,	PUNCT
ejpam-6326	346	3	(	(	PUNCT
ejpam-6326	346	4	iii	iii	NOUN
ejpam-6326	346	5	)	)	PUNCT
ejpam-6326	346	6	holds	hold	VERB
ejpam-6326	346	7	.	.	PUNCT
ejpam-6326	347	1	conversely	conversely	ADV
ejpam-6326	347	2	,	,	PUNCT
ejpam-6326	347	3	suppose	suppose	VERB
ejpam-6326	347	4	(	(	PUNCT
ejpam-6326	347	5	i)–(iii	i)–(iii	NOUN
ejpam-6326	347	6	)	)	PUNCT
ejpam-6326	347	7	hold	hold	NOUN
ejpam-6326	347	8	.	.	PUNCT
ejpam-6326	348	1	because	because	SCONJ
ejpam-6326	348	2	f	f	PROPN
ejpam-6326	348	3	dominates	dominate	VERB
ejpam-6326	348	4	g	g	NOUN
ejpam-6326	348	5	and	and	CCONJ
ejpam-6326	348	6	every	every	DET
ejpam-6326	348	7	vertex	vertex	NOUN
ejpam-6326	348	8	of	of	ADP
ejpam-6326	348	9	h	h	NOUN
ejpam-6326	348	10	is	be	AUX
ejpam-6326	348	11	adjacent	adjacent	ADJ
ejpam-6326	348	12	to	to	ADP
ejpam-6326	348	13	all	all	DET
ejpam-6326	348	14	vertices	vertex	NOUN
ejpam-6326	348	15	of	of	ADP
ejpam-6326	348	16	g	g	NOUN
ejpam-6326	348	17	in	in	ADP
ejpam-6326	348	18	the	the	DET
ejpam-6326	348	19	join	join	NOUN
ejpam-6326	348	20	,	,	PUNCT
ejpam-6326	348	21	f	f	PROPN
ejpam-6326	348	22	also	also	ADV
ejpam-6326	348	23	dominates	dominate	VERB
ejpam-6326	348	24	h	h	NOUN
ejpam-6326	348	25	;	;	PUNCT
ejpam-6326	348	26	hence	hence	ADV
ejpam-6326	348	27	it	it	PRON
ejpam-6326	348	28	dominates	dominate	VERB
ejpam-6326	348	29	g	g	ADP
ejpam-6326	348	30	∨h	∨h	NOUN
ejpam-6326	348	31	.	.	PUNCT
ejpam-6326	349	1	to	to	PART
ejpam-6326	349	2	show	show	VERB
ejpam-6326	349	3	that	that	SCONJ
ejpam-6326	349	4	f	f	PROPN
ejpam-6326	349	5	is	be	AUX
ejpam-6326	349	6	friendly	friendly	ADJ
ejpam-6326	349	7	,	,	PUNCT
ejpam-6326	349	8	we	we	PRON
ejpam-6326	349	9	consider	consider	VERB
ejpam-6326	349	10	two	two	NUM
ejpam-6326	349	11	cases	case	NOUN
ejpam-6326	349	12	.	.	PUNCT
ejpam-6326	350	1	case	case	NOUN
ejpam-6326	350	2	1	1	NUM
ejpam-6326	350	3	:	:	PUNCT
ejpam-6326	350	4	let	let	VERB
ejpam-6326	350	5	y	y	PROPN
ejpam-6326	350	6	∈	∈	PROPN
ejpam-6326	350	7	v	v	PROPN
ejpam-6326	350	8	(	(	PUNCT
ejpam-6326	350	9	h	h	NOUN
ejpam-6326	350	10	)	)	PUNCT
ejpam-6326	350	11	.	.	PUNCT
ejpam-6326	351	1	we	we	PRON
ejpam-6326	351	2	have	have	VERB
ejpam-6326	351	3	degf	degf	PROPN
ejpam-6326	351	4	(	(	PUNCT
ejpam-6326	351	5	y	y	NOUN
ejpam-6326	351	6	)	)	PUNCT
ejpam-6326	351	7	=	=	PRON
ejpam-6326	351	8	|f	|f	PROPN
ejpam-6326	352	1	|	|	ADV
ejpam-6326	352	2	and	and	CCONJ
ejpam-6326	352	3	degv	degv	NOUN
ejpam-6326	352	4	(	(	PUNCT
ejpam-6326	352	5	g∨h)\f	g∨h)\f	PROPN
ejpam-6326	352	6	(	(	PUNCT
ejpam-6326	352	7	y	y	NOUN
ejpam-6326	352	8	)	)	PUNCT
ejpam-6326	352	9	=	=	SYM
ejpam-6326	352	10	|v	|v	X
ejpam-6326	352	11	(	(	PUNCT
ejpam-6326	352	12	g	g	NOUN
ejpam-6326	352	13	)	)	PUNCT
ejpam-6326	352	14	\	\	NOUN
ejpam-6326	353	1	f	f	PROPN
ejpam-6326	354	1	|	|	ADV
ejpam-6326	354	2	+	+	CCONJ
ejpam-6326	354	3	degh(y	degh(y	ADJ
ejpam-6326	354	4	)	)	PUNCT
ejpam-6326	354	5	.	.	PUNCT
ejpam-6326	355	1	with	with	ADP
ejpam-6326	355	2	degh(y	degh(y	PROPN
ejpam-6326	355	3	)	)	PUNCT
ejpam-6326	355	4	≥	≥	PROPN
ejpam-6326	355	5	δ(h	δ(h	PROPN
ejpam-6326	355	6	)	)	PUNCT
ejpam-6326	355	7	and	and	CCONJ
ejpam-6326	355	8	(	(	PUNCT
ejpam-6326	355	9	ii	ii	NOUN
ejpam-6326	355	10	)	)	PUNCT
ejpam-6326	355	11	we	we	PRON
ejpam-6326	355	12	get	get	VERB
ejpam-6326	355	13	degf	degf	PROPN
ejpam-6326	355	14	(	(	PUNCT
ejpam-6326	355	15	y	y	NOUN
ejpam-6326	355	16	)	)	PUNCT
ejpam-6326	355	17	≤	≤	NOUN
ejpam-6326	355	18	degv	degv	NOUN
ejpam-6326	355	19	(	(	PUNCT
ejpam-6326	355	20	g∨h)\f	g∨h)\f	PROPN
ejpam-6326	355	21	(	(	PUNCT
ejpam-6326	355	22	y	y	NOUN
ejpam-6326	355	23	)	)	PUNCT
ejpam-6326	355	24	.	.	PUNCT
ejpam-6326	356	1	case	case	NOUN
ejpam-6326	356	2	2	2	NUM
ejpam-6326	356	3	:	:	PUNCT
ejpam-6326	356	4	let	let	VERB
ejpam-6326	356	5	x	x	X
ejpam-6326	356	6	∈	∈	PROPN
ejpam-6326	356	7	v	v	X
ejpam-6326	356	8	(	(	PUNCT
ejpam-6326	356	9	g	g	NOUN
ejpam-6326	356	10	)	)	PUNCT
ejpam-6326	356	11	\	\	PROPN
ejpam-6326	357	1	f	f	X
ejpam-6326	357	2	.	.	PUNCT
ejpam-6326	358	1	here	here	ADV
ejpam-6326	358	2	degf	degf	PROPN
ejpam-6326	358	3	(	(	PUNCT
ejpam-6326	358	4	x	x	NOUN
ejpam-6326	358	5	)	)	PUNCT
ejpam-6326	358	6	=	=	SYM
ejpam-6326	358	7	deggf	deggf	ADJ
ejpam-6326	358	8	(	(	PUNCT
ejpam-6326	358	9	x	x	NOUN
ejpam-6326	358	10	)	)	PUNCT
ejpam-6326	358	11	and	and	CCONJ
ejpam-6326	358	12	degv	degv	NOUN
ejpam-6326	358	13	(	(	PUNCT
ejpam-6326	358	14	g∨h)\f	g∨h)\f	PROPN
ejpam-6326	358	15	(	(	PUNCT
ejpam-6326	358	16	x	x	NOUN
ejpam-6326	358	17	)	)	PUNCT
ejpam-6326	358	18	=	=	SYM
ejpam-6326	358	19	|v	|v	X
ejpam-6326	358	20	(	(	PUNCT
ejpam-6326	358	21	h)|+	h)|+	ADJ
ejpam-6326	358	22	deggv	deggv	NOUN
ejpam-6326	358	23	(	(	PUNCT
ejpam-6326	358	24	g)\f	g)\f	NOUN
ejpam-6326	358	25	(	(	PUNCT
ejpam-6326	358	26	x	x	NOUN
ejpam-6326	358	27	)	)	PUNCT
ejpam-6326	358	28	.	.	PUNCT
ejpam-6326	359	1	by	by	ADP
ejpam-6326	359	2	condition	condition	NOUN
ejpam-6326	359	3	(	(	PUNCT
ejpam-6326	359	4	iii	iii	NOUN
ejpam-6326	359	5	)	)	PUNCT
ejpam-6326	359	6	,	,	PUNCT
ejpam-6326	359	7	again	again	ADV
ejpam-6326	359	8	,	,	PUNCT
ejpam-6326	359	9	degf	degf	PROPN
ejpam-6326	359	10	(	(	PUNCT
ejpam-6326	359	11	y	y	NOUN
ejpam-6326	359	12	)	)	PUNCT
ejpam-6326	359	13	≤	≤	NOUN
ejpam-6326	359	14	degv	degv	NOUN
ejpam-6326	359	15	(	(	PUNCT
ejpam-6326	359	16	g∨h)\f	g∨h)\f	PROPN
ejpam-6326	359	17	(	(	PUNCT
ejpam-6326	359	18	y	y	NOUN
ejpam-6326	359	19	)	)	PUNCT
ejpam-6326	359	20	.	.	PUNCT
ejpam-6326	360	1	thus	thus	ADV
ejpam-6326	360	2	f	f	PROPN
ejpam-6326	360	3	is	be	AUX
ejpam-6326	360	4	friendly	friendly	ADJ
ejpam-6326	360	5	.	.	PUNCT
ejpam-6326	361	1	hence	hence	ADV
ejpam-6326	361	2	,	,	PUNCT
ejpam-6326	361	3	f	f	PROPN
ejpam-6326	361	4	is	be	AUX
ejpam-6326	361	5	a	a	DET
ejpam-6326	361	6	friendly	friendly	ADJ
ejpam-6326	361	7	dominating	dominating	NOUN
ejpam-6326	361	8	set	set	VERB
ejpam-6326	361	9	in	in	ADP
ejpam-6326	361	10	g	g	PROPN
ejpam-6326	361	11	∨h	∨h	NOUN
ejpam-6326	361	12	.	.	PUNCT
ejpam-6326	362	1	theorem	theorem	NOUN
ejpam-6326	362	2	11	11	NUM
ejpam-6326	362	3	.	.	PUNCT
ejpam-6326	363	1	let	let	VERB
ejpam-6326	363	2	g	g	NOUN
ejpam-6326	363	3	and	and	CCONJ
ejpam-6326	363	4	h	h	PROPN
ejpam-6326	363	5	be	be	VERB
ejpam-6326	363	6	non	non	ADJ
ejpam-6326	363	7	-	-	ADJ
ejpam-6326	363	8	trivial	trivial	ADJ
ejpam-6326	363	9	connected	connected	ADJ
ejpam-6326	363	10	graphs	graph	NOUN
ejpam-6326	363	11	and	and	CCONJ
ejpam-6326	363	12	let	let	VERB
ejpam-6326	363	13	f	f	PROPN
ejpam-6326	363	14	⊆	⊆	NUM
ejpam-6326	363	15	v	v	PROPN
ejpam-6326	363	16	(	(	PUNCT
ejpam-6326	363	17	h	h	NOUN
ejpam-6326	363	18	)	)	PUNCT
ejpam-6326	363	19	.	.	PUNCT
ejpam-6326	364	1	then	then	ADV
ejpam-6326	364	2	f	f	PROPN
ejpam-6326	364	3	is	be	AUX
ejpam-6326	364	4	a	a	DET
ejpam-6326	364	5	friendly	friendly	ADJ
ejpam-6326	364	6	dominating	dominating	NOUN
ejpam-6326	364	7	set	set	NOUN
ejpam-6326	364	8	of	of	ADP
ejpam-6326	364	9	g	g	PROPN
ejpam-6326	364	10	∨h	∨h	NOUN
ejpam-6326	364	11	if	if	SCONJ
ejpam-6326	364	12	and	and	CCONJ
ejpam-6326	364	13	only	only	ADV
ejpam-6326	364	14	if	if	SCONJ
ejpam-6326	364	15	the	the	DET
ejpam-6326	364	16	following	follow	VERB
ejpam-6326	364	17	three	three	NUM
ejpam-6326	364	18	conditions	condition	NOUN
ejpam-6326	364	19	hold	hold	VERB
ejpam-6326	364	20	:	:	PUNCT
ejpam-6326	364	21	(	(	PUNCT
ejpam-6326	364	22	i	i	NOUN
ejpam-6326	364	23	)	)	PUNCT
ejpam-6326	364	24	f	f	PROPN
ejpam-6326	364	25	dominates	dominate	VERB
ejpam-6326	364	26	h	h	NOUN
ejpam-6326	364	27	;	;	PUNCT
ejpam-6326	364	28	(	(	PUNCT
ejpam-6326	364	29	ii	ii	NOUN
ejpam-6326	364	30	)	)	PUNCT
ejpam-6326	364	31	|f	|f	PUNCT
ejpam-6326	365	1	|	|	ADV
ejpam-6326	365	2	≤	≤	NUM
ejpam-6326	365	3	|v	|v	X
ejpam-6326	365	4	(	(	PUNCT
ejpam-6326	365	5	h	h	NOUN
ejpam-6326	365	6	)	)	PUNCT
ejpam-6326	365	7	\	\	PROPN
ejpam-6326	365	8	f	f	PROPN
ejpam-6326	365	9	|+	|+	X
ejpam-6326	365	10	δ(g	δ(g	PROPN
ejpam-6326	365	11	)	)	PUNCT
ejpam-6326	365	12	;	;	PUNCT
ejpam-6326	365	13	(	(	PUNCT
ejpam-6326	365	14	iii	iii	X
ejpam-6326	365	15	)	)	PUNCT
ejpam-6326	365	16	deghf	deghf	NOUN
ejpam-6326	365	17	(	(	PUNCT
ejpam-6326	365	18	x	x	NOUN
ejpam-6326	365	19	)	)	PUNCT
ejpam-6326	365	20	≤	≤	NOUN
ejpam-6326	365	21	|v	|v	X
ejpam-6326	365	22	(	(	PUNCT
ejpam-6326	365	23	g)|+	g)|+	NOUN
ejpam-6326	365	24	deghv	deghv	NOUN
ejpam-6326	365	25	(	(	PUNCT
ejpam-6326	365	26	h)\f	h)\f	PROPN
ejpam-6326	365	27	(	(	PUNCT
ejpam-6326	365	28	x	x	NOUN
ejpam-6326	365	29	)	)	PUNCT
ejpam-6326	365	30	for	for	ADP
ejpam-6326	365	31	every	every	PRON
ejpam-6326	365	32	x	x	SYM
ejpam-6326	365	33	∈	∈	PROPN
ejpam-6326	365	34	v	v	ADP
ejpam-6326	365	35	(	(	PUNCT
ejpam-6326	365	36	h	h	NOUN
ejpam-6326	365	37	)	)	PUNCT
ejpam-6326	365	38	\	\	NOUN
ejpam-6326	366	1	f	f	PROPN
ejpam-6326	366	2	proof	proof	NOUN
ejpam-6326	366	3	.	.	PUNCT
ejpam-6326	367	1	the	the	DET
ejpam-6326	367	2	same	same	ADJ
ejpam-6326	367	3	analogous	analogous	ADJ
ejpam-6326	367	4	proof	proof	NOUN
ejpam-6326	367	5	as	as	ADP
ejpam-6326	367	6	theorem	theorem	ADJ
ejpam-6326	367	7	10	10	NUM
ejpam-6326	367	8	.	.	PUNCT
ejpam-6326	367	9	theorem	theorem	NOUN
ejpam-6326	367	10	12	12	NUM
ejpam-6326	367	11	.	.	PUNCT
ejpam-6326	368	1	let	let	VERB
ejpam-6326	368	2	g	g	NOUN
ejpam-6326	368	3	and	and	CCONJ
ejpam-6326	368	4	h	h	PROPN
ejpam-6326	368	5	be	be	VERB
ejpam-6326	368	6	non	non	ADJ
ejpam-6326	368	7	-	-	ADJ
ejpam-6326	368	8	trivial	trivial	ADJ
ejpam-6326	368	9	connected	connected	ADJ
ejpam-6326	368	10	graphs	graph	NOUN
ejpam-6326	368	11	on	on	ADP
ejpam-6326	368	12	|v	|v	PROPN
ejpam-6326	368	13	(	(	PUNCT
ejpam-6326	368	14	g)|	g)|	NOUN
ejpam-6326	368	15	=	=	PUNCT
ejpam-6326	368	16	n1	n1	PROPN
ejpam-6326	368	17	and	and	CCONJ
ejpam-6326	368	18	|v	|v	PROPN
ejpam-6326	368	19	(	(	PUNCT
ejpam-6326	368	20	h)|	h)|	NOUN
ejpam-6326	368	21	=	=	SYM
ejpam-6326	368	22	n2	n2	ADJ
ejpam-6326	368	23	vertices	vertex	NOUN
ejpam-6326	368	24	,	,	PUNCT
ejpam-6326	368	25	respectively	respectively	ADV
ejpam-6326	368	26	.	.	PUNCT
ejpam-6326	369	1	let	let	VERB
ejpam-6326	369	2	f1	f1	PROPN
ejpam-6326	369	3	⊆	⊆	NUM
ejpam-6326	369	4	v	v	NOUN
ejpam-6326	369	5	(	(	PUNCT
ejpam-6326	369	6	g	g	NOUN
ejpam-6326	369	7	)	)	PUNCT
ejpam-6326	369	8	and	and	CCONJ
ejpam-6326	369	9	f2	f2	VERB
ejpam-6326	369	10	⊆	⊆	NUM
ejpam-6326	369	11	v	v	NOUN
ejpam-6326	369	12	(	(	PUNCT
ejpam-6326	369	13	h	h	NOUN
ejpam-6326	369	14	)	)	PUNCT
ejpam-6326	369	15	and	and	CCONJ
ejpam-6326	369	16	put	put	VERB
ejpam-6326	369	17	f	f	NOUN
ejpam-6326	369	18	=	=	SYM
ejpam-6326	369	19	f1	f1	PROPN
ejpam-6326	369	20	∪	∪	X
ejpam-6326	369	21	f2	f2	PROPN
ejpam-6326	369	22	.	.	PUNCT
ejpam-6326	370	1	then	then	ADV
ejpam-6326	370	2	f	f	PROPN
ejpam-6326	370	3	is	be	AUX
ejpam-6326	370	4	a	a	DET
ejpam-6326	370	5	friendly	friendly	ADJ
ejpam-6326	370	6	dominating	dominating	NOUN
ejpam-6326	370	7	set	set	NOUN
ejpam-6326	370	8	of	of	ADP
ejpam-6326	370	9	g	g	PROPN
ejpam-6326	370	10	∨	∨	NUM
ejpam-6326	370	11	h	h	NOUN
ejpam-6326	370	12	if	if	SCONJ
ejpam-6326	371	1	and	and	CCONJ
ejpam-6326	371	2	only	only	ADV
ejpam-6326	371	3	if	if	SCONJ
ejpam-6326	371	4	the	the	DET
ejpam-6326	371	5	following	follow	VERB
ejpam-6326	371	6	two	two	NUM
ejpam-6326	371	7	inequalities	inequality	NOUN
ejpam-6326	371	8	are	be	AUX
ejpam-6326	371	9	satisfied	satisfied	ADJ
ejpam-6326	371	10	:	:	PUNCT
ejpam-6326	371	11	(	(	PUNCT
ejpam-6326	371	12	i	i	NOUN
ejpam-6326	371	13	)	)	PUNCT
ejpam-6326	371	14	deggf1	deggf1	VERB
ejpam-6326	372	1	(	(	PUNCT
ejpam-6326	372	2	x	x	X
ejpam-6326	372	3	)	)	PUNCT
ejpam-6326	372	4	≤	≤	ADJ
ejpam-6326	372	5	deggv	deggv	NOUN
ejpam-6326	372	6	(	(	PUNCT
ejpam-6326	372	7	g)\f1	g)\f1	X
ejpam-6326	372	8	(	(	PUNCT
ejpam-6326	372	9	x	x	NOUN
ejpam-6326	372	10	)	)	PUNCT
ejpam-6326	372	11	+	+	CCONJ
ejpam-6326	372	12	n2	n2	ADJ
ejpam-6326	372	13	−	−	PROPN
ejpam-6326	372	14	2|f2|	2|f2|	NUM
ejpam-6326	372	15	,	,	PUNCT
ejpam-6326	372	16	for	for	ADP
ejpam-6326	372	17	every	every	DET
ejpam-6326	372	18	x	x	SYM
ejpam-6326	372	19	∈	∈	PROPN
ejpam-6326	372	20	v	v	ADP
ejpam-6326	372	21	(	(	PUNCT
ejpam-6326	372	22	g	g	NOUN
ejpam-6326	372	23	)	)	PUNCT
ejpam-6326	372	24	\	\	NOUN
ejpam-6326	372	25	f1	f1	NOUN
ejpam-6326	372	26	,	,	PUNCT
ejpam-6326	372	27	(	(	PUNCT
ejpam-6326	372	28	ii	ii	NOUN
ejpam-6326	372	29	)	)	PUNCT
ejpam-6326	372	30	deghf2	deghf2	PROPN
ejpam-6326	372	31	(	(	PUNCT
ejpam-6326	372	32	y	y	NOUN
ejpam-6326	372	33	)	)	PUNCT
ejpam-6326	372	34	≤	≤	NOUN
ejpam-6326	372	35	deghv	deghv	NOUN
ejpam-6326	372	36	(	(	PUNCT
ejpam-6326	372	37	h)\f2	h)\f2	NOUN
ejpam-6326	372	38	(	(	PUNCT
ejpam-6326	372	39	y	y	NOUN
ejpam-6326	372	40	)	)	PUNCT
ejpam-6326	373	1	+	+	CCONJ
ejpam-6326	373	2	n1	n1	PROPN
ejpam-6326	373	3	−	−	PROPN
ejpam-6326	373	4	2|f1|	2|f1|	NUM
ejpam-6326	374	1	,	,	PUNCT
ejpam-6326	374	2	for	for	ADP
ejpam-6326	374	3	every	every	DET
ejpam-6326	374	4	y	y	PROPN
ejpam-6326	374	5	∈	∈	PROPN
ejpam-6326	374	6	v	v	ADP
ejpam-6326	374	7	(	(	PUNCT
ejpam-6326	374	8	h	h	NOUN
ejpam-6326	374	9	)	)	PUNCT
ejpam-6326	374	10	\	\	PROPN
ejpam-6326	374	11	f2	f2	PROPN
ejpam-6326	374	12	.	.	PUNCT
ejpam-6326	375	1	proof	proof	NOUN
ejpam-6326	375	2	.	.	PUNCT
ejpam-6326	376	1	assume	assume	VERB
ejpam-6326	376	2	f	f	PROPN
ejpam-6326	376	3	is	be	AUX
ejpam-6326	376	4	friendly	friendly	ADJ
ejpam-6326	376	5	dominating	dominating	NOUN
ejpam-6326	376	6	set	set	VERB
ejpam-6326	376	7	in	in	ADP
ejpam-6326	376	8	g	g	PROPN
ejpam-6326	376	9	∨h	∨h	NOUN
ejpam-6326	376	10	.	.	PUNCT
ejpam-6326	377	1	pick	pick	VERB
ejpam-6326	377	2	x	x	SYM
ejpam-6326	377	3	∈	∈	PROPN
ejpam-6326	377	4	v	v	ADP
ejpam-6326	377	5	(	(	PUNCT
ejpam-6326	377	6	g	g	NOUN
ejpam-6326	377	7	)	)	PUNCT
ejpam-6326	377	8	\	\	NOUN
ejpam-6326	377	9	f1	f1	NOUN
ejpam-6326	377	10	and	and	CCONJ
ejpam-6326	377	11	count	count	VERB
ejpam-6326	377	12	separately	separately	ADV
ejpam-6326	377	13	the	the	DET
ejpam-6326	377	14	neighbors	neighbor	NOUN
ejpam-6326	377	15	of	of	ADP
ejpam-6326	377	16	x	x	SYM
ejpam-6326	377	17	inside	inside	ADP
ejpam-6326	377	18	and	and	CCONJ
ejpam-6326	377	19	outside	outside	ADP
ejpam-6326	377	20	f	f	PROPN
ejpam-6326	377	21	.	.	PUNCT
ejpam-6326	378	1	then	then	ADV
ejpam-6326	378	2	,	,	PUNCT
ejpam-6326	378	3	degf	degf	PROPN
ejpam-6326	378	4	(	(	PUNCT
ejpam-6326	378	5	x	x	NOUN
ejpam-6326	378	6	)	)	PUNCT
ejpam-6326	378	7	=	=	SYM
ejpam-6326	378	8	deggf1	deggf1	X
ejpam-6326	378	9	(	(	PUNCT
ejpam-6326	378	10	x	x	X
ejpam-6326	378	11	)	)	PUNCT
ejpam-6326	378	12	+	+	CCONJ
ejpam-6326	378	13	|f2|	|f2|	NOUN
ejpam-6326	378	14	and	and	CCONJ
ejpam-6326	378	15	degv	degv	NOUN
ejpam-6326	378	16	(	(	PUNCT
ejpam-6326	378	17	g∨h)\f	g∨h)\f	PROPN
ejpam-6326	378	18	(	(	PUNCT
ejpam-6326	378	19	x	x	NOUN
ejpam-6326	378	20	)	)	PUNCT
ejpam-6326	378	21	=	=	SYM
ejpam-6326	378	22	deggv	deggv	NOUN
ejpam-6326	378	23	(	(	PUNCT
ejpam-6326	378	24	g)\f1	g)\f1	X
ejpam-6326	378	25	(	(	PUNCT
ejpam-6326	378	26	x	x	NOUN
ejpam-6326	378	27	)	)	PUNCT
ejpam-6326	378	28	+	+	CCONJ
ejpam-6326	378	29	|v	|v	X
ejpam-6326	378	30	(	(	PUNCT
ejpam-6326	378	31	h	h	NOUN
ejpam-6326	378	32	)	)	PUNCT
ejpam-6326	378	33	\	\	PUNCT
ejpam-6326	379	1	f2|	f2|	NOUN
ejpam-6326	379	2	=	=	SYM
ejpam-6326	379	3	deggv	deggv	NOUN
ejpam-6326	379	4	(	(	PUNCT
ejpam-6326	379	5	g)\f1	g)\f1	X
ejpam-6326	379	6	(	(	PUNCT
ejpam-6326	379	7	x	x	NOUN
ejpam-6326	379	8	)	)	PUNCT
ejpam-6326	379	9	+	+	CCONJ
ejpam-6326	379	10	n2	n2	ADJ
ejpam-6326	379	11	−	−	PROPN
ejpam-6326	379	12	|f2|	|f2|	NOUN
ejpam-6326	379	13	.	.	PUNCT
ejpam-6326	380	1	since	since	SCONJ
ejpam-6326	380	2	f	f	PROPN
ejpam-6326	380	3	is	be	AUX
ejpam-6326	380	4	friendly	friendly	ADJ
ejpam-6326	380	5	,	,	PUNCT
ejpam-6326	380	6	we	we	PRON
ejpam-6326	380	7	have	have	VERB
ejpam-6326	380	8	,	,	PUNCT
ejpam-6326	380	9	deggf1	deggf1	VERB
ejpam-6326	380	10	(	(	PUNCT
ejpam-6326	380	11	x	x	X
ejpam-6326	380	12	)	)	PUNCT
ejpam-6326	380	13	≤	≤	ADJ
ejpam-6326	380	14	deggv	deggv	NOUN
ejpam-6326	380	15	(	(	PUNCT
ejpam-6326	380	16	g)\f1	g)\f1	X
ejpam-6326	380	17	(	(	PUNCT
ejpam-6326	380	18	x	x	NOUN
ejpam-6326	380	19	)	)	PUNCT
ejpam-6326	380	20	+	+	NOUN
ejpam-6326	380	21	n2	n2	ADJ
ejpam-6326	380	22	−	−	PROPN
ejpam-6326	380	23	2|f2|	2|f2|	NUM
ejpam-6326	380	24	.	.	PUNCT
ejpam-6326	381	1	an	an	DET
ejpam-6326	381	2	analogous	analogous	ADJ
ejpam-6326	381	3	statement	statement	NOUN
ejpam-6326	381	4	for	for	ADP
ejpam-6326	381	5	y	y	PROPN
ejpam-6326	381	6	∈	∈	PROPN
ejpam-6326	381	7	v	v	ADP
ejpam-6326	381	8	(	(	PUNCT
ejpam-6326	381	9	g	g	NOUN
ejpam-6326	381	10	)	)	PUNCT
ejpam-6326	381	11	\	\	PUNCT
ejpam-6326	381	12	f2	f2	PROPN
ejpam-6326	381	13	gives	give	VERB
ejpam-6326	381	14	deghf2	deghf2	PROPN
ejpam-6326	381	15	(	(	PUNCT
ejpam-6326	381	16	y	y	NOUN
ejpam-6326	381	17	)	)	PUNCT
ejpam-6326	381	18	≤	≤	NOUN
ejpam-6326	381	19	deghv	deghv	NOUN
ejpam-6326	381	20	(	(	PUNCT
ejpam-6326	381	21	h)\f2	h)\f2	NOUN
ejpam-6326	381	22	(	(	PUNCT
ejpam-6326	381	23	y	y	NOUN
ejpam-6326	381	24	)	)	PUNCT
ejpam-6326	381	25	+	+	CCONJ
ejpam-6326	382	1	n1	n1	ADJ
ejpam-6326	382	2	−	−	PROPN
ejpam-6326	382	3	2|f1|	2|f1|	NUM
ejpam-6326	382	4	.	.	PUNCT
ejpam-6326	383	1	conversely	conversely	ADV
ejpam-6326	383	2	,	,	PUNCT
ejpam-6326	383	3	suppose	suppose	VERB
ejpam-6326	383	4	(	(	PUNCT
ejpam-6326	383	5	i	i	NOUN
ejpam-6326	383	6	)	)	PUNCT
ejpam-6326	383	7	and	and	CCONJ
ejpam-6326	383	8	(	(	PUNCT
ejpam-6326	383	9	ii	ii	NOUN
ejpam-6326	383	10	)	)	PUNCT
ejpam-6326	383	11	hold	hold	VERB
ejpam-6326	383	12	.	.	PUNCT
ejpam-6326	384	1	since	since	SCONJ
ejpam-6326	384	2	f1	f1	PROPN
ejpam-6326	384	3	(	(	PUNCT
ejpam-6326	384	4	resp	resp	NOUN
ejpam-6326	384	5	.	.	PUNCT
ejpam-6326	385	1	f2	f2	PROPN
ejpam-6326	385	2	)	)	PUNCT
ejpam-6326	385	3	contains	contain	VERB
ejpam-6326	385	4	at	at	ADP
ejpam-6326	385	5	least	least	ADV
ejpam-6326	385	6	one	one	NUM
ejpam-6326	385	7	vertex	vertex	NOUN
ejpam-6326	385	8	of	of	ADP
ejpam-6326	385	9	g	g	PROPN
ejpam-6326	385	10	(	(	PUNCT
ejpam-6326	385	11	resp	resp	NOUN
ejpam-6326	385	12	.	.	PUNCT
ejpam-6326	386	1	h	h	X
ejpam-6326	386	2	)	)	PUNCT
ejpam-6326	386	3	,	,	PUNCT
ejpam-6326	386	4	every	every	DET
ejpam-6326	386	5	vertex	vertex	NOUN
ejpam-6326	386	6	of	of	ADP
ejpam-6326	386	7	h	h	PROPN
ejpam-6326	386	8	(	(	PUNCT
ejpam-6326	386	9	resp	resp	NOUN
ejpam-6326	386	10	.	.	PUNCT
ejpam-6326	387	1	g	g	NOUN
ejpam-6326	387	2	)	)	PUNCT
ejpam-6326	387	3	is	be	AUX
ejpam-6326	387	4	adjacent	adjacent	ADJ
ejpam-6326	387	5	(	(	PUNCT
ejpam-6326	387	6	through	through	ADP
ejpam-6326	387	7	the	the	DET
ejpam-6326	387	8	join	join	NOUN
ejpam-6326	387	9	)	)	PUNCT
ejpam-6326	387	10	to	to	ADP
ejpam-6326	387	11	a	a	DET
ejpam-6326	387	12	vertex	vertex	NOUN
ejpam-6326	387	13	of	of	ADP
ejpam-6326	387	14	f	f	PROPN
ejpam-6326	387	15	;	;	PUNCT
ejpam-6326	387	16	i.	i.	PROPN
ejpam-6326	387	17	s.	s.	PROPN
ejpam-6326	387	18	cabahug	cabahug	PROPN
ejpam-6326	387	19	,	,	PUNCT
ejpam-6326	387	20	jr	jr	PROPN
ejpam-6326	387	21	.	.	PROPN
ejpam-6326	387	22	,	,	PUNCT
ejpam-6326	387	23	r.	r.	PROPN
ejpam-6326	387	24	g.	g.	PROPN
ejpam-6326	387	25	eballe	eballe	PROPN
ejpam-6326	387	26	,	,	PUNCT
ejpam-6326	387	27	r.	r.	PROPN
ejpam-6326	387	28	t.	t.	PROPN
ejpam-6326	387	29	fernandez	fernandez	PROPN
ejpam-6326	387	30	/	/	SYM
ejpam-6326	387	31	eur	eur	PROPN
ejpam-6326	387	32	.	.	PUNCT
ejpam-6326	388	1	j.	j.	PROPN
ejpam-6326	388	2	pure	pure	PROPN
ejpam-6326	388	3	appl	appl	PROPN
ejpam-6326	388	4	.	.	PROPN
ejpam-6326	388	5	math	math	PROPN
ejpam-6326	388	6	,	,	PUNCT
ejpam-6326	388	7	18	18	NUM
ejpam-6326	388	8	(	(	PUNCT
ejpam-6326	388	9	4	4	NUM
ejpam-6326	388	10	)	)	PUNCT
ejpam-6326	388	11	(	(	PUNCT
ejpam-6326	388	12	2025	2025	NUM
ejpam-6326	388	13	)	)	PUNCT
ejpam-6326	388	14	,	,	PUNCT
ejpam-6326	388	15	6326	6326	NUM
ejpam-6326	388	16	12	12	NUM
ejpam-6326	388	17	of	of	ADP
ejpam-6326	388	18	15	15	NUM
ejpam-6326	389	1	hence	hence	ADV
ejpam-6326	389	2	f	f	PROPN
ejpam-6326	389	3	is	be	AUX
ejpam-6326	389	4	a	a	DET
ejpam-6326	389	5	dominating	dominating	NOUN
ejpam-6326	389	6	set	set	VERB
ejpam-6326	389	7	in	in	ADP
ejpam-6326	389	8	g∨h	g∨h	PROPN
ejpam-6326	389	9	.	.	PUNCT
ejpam-6326	390	1	to	to	PART
ejpam-6326	390	2	show	show	VERB
ejpam-6326	390	3	that	that	SCONJ
ejpam-6326	390	4	f	f	PROPN
ejpam-6326	390	5	is	be	AUX
ejpam-6326	390	6	friendly	friendly	ADJ
ejpam-6326	390	7	.	.	PUNCT
ejpam-6326	391	1	let	let	VERB
ejpam-6326	391	2	x	x	SYM
ejpam-6326	391	3	∈	∈	PROPN
ejpam-6326	391	4	v	v	NOUN
ejpam-6326	391	5	(	(	PUNCT
ejpam-6326	391	6	g)\f1	g)\f1	NOUN
ejpam-6326	391	7	.	.	PUNCT
ejpam-6326	391	8	by	by	ADP
ejpam-6326	391	9	(	(	PUNCT
ejpam-6326	391	10	i	i	NOUN
ejpam-6326	391	11	)	)	PUNCT
ejpam-6326	391	12	,	,	PUNCT
ejpam-6326	391	13	deggf1	deggf1	VERB
ejpam-6326	391	14	(	(	PUNCT
ejpam-6326	391	15	x	x	X
ejpam-6326	391	16	)	)	PUNCT
ejpam-6326	391	17	≤	≤	ADJ
ejpam-6326	391	18	deggv	deggv	NOUN
ejpam-6326	391	19	(	(	PUNCT
ejpam-6326	391	20	g)\f1	g)\f1	X
ejpam-6326	391	21	(	(	PUNCT
ejpam-6326	391	22	x)+n2−2|f2|	x)+n2−2|f2|	PROPN
ejpam-6326	391	23	,	,	PUNCT
ejpam-6326	391	24	.	.	PUNCT
ejpam-6326	392	1	this	this	PRON
ejpam-6326	392	2	implies	imply	VERB
ejpam-6326	392	3	that	that	SCONJ
ejpam-6326	392	4	,	,	PUNCT
ejpam-6326	392	5	deggf1	deggf1	PROPN
ejpam-6326	392	6	(	(	PUNCT
ejpam-6326	392	7	x)+|f2|	x)+|f2|	PROPN
ejpam-6326	392	8	≤	≤	X
ejpam-6326	392	9	deggv	deggv	PROPN
ejpam-6326	392	10	(	(	PUNCT
ejpam-6326	392	11	g)\f1	g)\f1	PROPN
ejpam-6326	392	12	(	(	PUNCT
ejpam-6326	392	13	x)+	x)+	PROPN
ejpam-6326	392	14	n2	n2	PROPN
ejpam-6326	392	15	−	−	PROPN
ejpam-6326	392	16	|f2|	|f2|	NOUN
ejpam-6326	392	17	.	.	PUNCT
ejpam-6326	393	1	now	now	ADV
ejpam-6326	393	2	,	,	PUNCT
ejpam-6326	393	3	degf	degf	PROPN
ejpam-6326	393	4	(	(	PUNCT
ejpam-6326	393	5	x	x	NOUN
ejpam-6326	393	6	)	)	PUNCT
ejpam-6326	393	7	=	=	SYM
ejpam-6326	393	8	deggf1	deggf1	X
ejpam-6326	393	9	(	(	PUNCT
ejpam-6326	393	10	x	x	X
ejpam-6326	393	11	)	)	PUNCT
ejpam-6326	393	12	+	+	CCONJ
ejpam-6326	393	13	|f2|	|f2|	NOUN
ejpam-6326	393	14	and	and	CCONJ
ejpam-6326	393	15	degv	degv	NOUN
ejpam-6326	393	16	(	(	PUNCT
ejpam-6326	393	17	g∨h)\f	g∨h)\f	PROPN
ejpam-6326	393	18	(	(	PUNCT
ejpam-6326	393	19	x	x	NOUN
ejpam-6326	393	20	)	)	PUNCT
ejpam-6326	393	21	=	=	SYM
ejpam-6326	393	22	deggv	deggv	NOUN
ejpam-6326	393	23	(	(	PUNCT
ejpam-6326	393	24	g)\f1	g)\f1	X
ejpam-6326	393	25	(	(	PUNCT
ejpam-6326	393	26	x	x	NOUN
ejpam-6326	393	27	)	)	PUNCT
ejpam-6326	393	28	+	+	CCONJ
ejpam-6326	393	29	n2	n2	ADJ
ejpam-6326	393	30	−	−	PROPN
ejpam-6326	393	31	|f2|	|f2|	NOUN
ejpam-6326	393	32	.	.	PUNCT
ejpam-6326	394	1	thus	thus	ADV
ejpam-6326	394	2	,	,	PUNCT
ejpam-6326	394	3	degf	degf	PROPN
ejpam-6326	394	4	(	(	PUNCT
ejpam-6326	394	5	x	x	NOUN
ejpam-6326	394	6	)	)	PUNCT
ejpam-6326	394	7	≤	≤	ADJ
ejpam-6326	394	8	degv	degv	NOUN
ejpam-6326	394	9	(	(	PUNCT
ejpam-6326	394	10	g∨h)\f	g∨h)\f	PROPN
ejpam-6326	394	11	(	(	PUNCT
ejpam-6326	394	12	x	x	NOUN
ejpam-6326	394	13	)	)	PUNCT
ejpam-6326	394	14	.	.	PUNCT
ejpam-6326	395	1	similarly	similarly	ADV
ejpam-6326	395	2	for	for	ADP
ejpam-6326	395	3	y	y	PROPN
ejpam-6326	395	4	∈	∈	PROPN
ejpam-6326	395	5	v	v	ADP
ejpam-6326	395	6	(	(	PUNCT
ejpam-6326	395	7	h	h	NOUN
ejpam-6326	395	8	)	)	PUNCT
ejpam-6326	395	9	\	\	NOUN
ejpam-6326	395	10	f2	f2	PROPN
ejpam-6326	395	11	and	and	CCONJ
ejpam-6326	395	12	by	by	ADP
ejpam-6326	395	13	(	(	PUNCT
ejpam-6326	395	14	ii	ii	NOUN
ejpam-6326	395	15	)	)	PUNCT
ejpam-6326	395	16	gives	give	VERB
ejpam-6326	395	17	,	,	PUNCT
ejpam-6326	395	18	degf	degf	PROPN
ejpam-6326	395	19	(	(	PUNCT
ejpam-6326	395	20	y	y	NOUN
ejpam-6326	395	21	)	)	PUNCT
ejpam-6326	395	22	≤	≤	NOUN
ejpam-6326	395	23	degv	degv	NOUN
ejpam-6326	395	24	(	(	PUNCT
ejpam-6326	395	25	g∨h)\f	g∨h)\f	PROPN
ejpam-6326	395	26	(	(	PUNCT
ejpam-6326	395	27	y	y	NOUN
ejpam-6326	395	28	)	)	PUNCT
ejpam-6326	395	29	.	.	PUNCT
ejpam-6326	396	1	hence	hence	ADV
ejpam-6326	396	2	,	,	PUNCT
ejpam-6326	396	3	f	f	PROPN
ejpam-6326	396	4	is	be	AUX
ejpam-6326	396	5	friendly	friendly	ADJ
ejpam-6326	396	6	.	.	PUNCT
ejpam-6326	397	1	therefore	therefore	ADV
ejpam-6326	397	2	,	,	PUNCT
ejpam-6326	397	3	f	f	PROPN
ejpam-6326	397	4	is	be	AUX
ejpam-6326	397	5	a	a	DET
ejpam-6326	397	6	friendly	friendly	ADJ
ejpam-6326	397	7	dominating	dominating	NOUN
ejpam-6326	397	8	set	set	VERB
ejpam-6326	397	9	in	in	ADP
ejpam-6326	397	10	g	g	PROPN
ejpam-6326	397	11	∨h	∨h	NOUN
ejpam-6326	397	12	.	.	PUNCT
ejpam-6326	398	1	3.4	3.4	NUM
ejpam-6326	398	2	.	.	PUNCT
ejpam-6326	398	3	friendly	friendly	ADJ
ejpam-6326	398	4	dominating	dominating	NOUN
ejpam-6326	398	5	sets	set	NOUN
ejpam-6326	398	6	in	in	ADP
ejpam-6326	398	7	the	the	DET
ejpam-6326	398	8	corona	corona	NOUN
ejpam-6326	398	9	of	of	ADP
ejpam-6326	398	10	graphs	graph	NOUN
ejpam-6326	398	11	theorem	theorem	VERB
ejpam-6326	398	12	13	13	NUM
ejpam-6326	398	13	.	.	PUNCT
ejpam-6326	399	1	let	let	VERB
ejpam-6326	399	2	g	g	NOUN
ejpam-6326	399	3	and	and	CCONJ
ejpam-6326	399	4	h	h	NOUN
ejpam-6326	399	5	be	be	AUX
ejpam-6326	399	6	connected	connect	VERB
ejpam-6326	399	7	non	non	ADJ
ejpam-6326	399	8	-	-	ADJ
ejpam-6326	399	9	trivial	trivial	ADJ
ejpam-6326	399	10	graphs	graph	NOUN
ejpam-6326	399	11	.	.	PUNCT
ejpam-6326	400	1	suppose	suppose	VERB
ejpam-6326	400	2	f	f	PROPN
ejpam-6326	400	3	⊆	⊆	NUM
ejpam-6326	400	4	v	v	NOUN
ejpam-6326	400	5	(	(	PUNCT
ejpam-6326	400	6	g	g	PROPN
ejpam-6326	400	7	◦	◦	NOUN
ejpam-6326	400	8	h	h	NOUN
ejpam-6326	400	9	)	)	PUNCT
ejpam-6326	400	10	for	for	ADP
ejpam-6326	400	11	which	which	PRON
ejpam-6326	400	12	f	f	PROPN
ejpam-6326	400	13	⊆	⊆	NUM
ejpam-6326	400	14	v	v	NOUN
ejpam-6326	400	15	(	(	PUNCT
ejpam-6326	400	16	g	g	NOUN
ejpam-6326	400	17	)	)	PUNCT
ejpam-6326	400	18	.	.	PUNCT
ejpam-6326	401	1	then	then	ADV
ejpam-6326	401	2	f	f	PROPN
ejpam-6326	401	3	is	be	AUX
ejpam-6326	401	4	a	a	DET
ejpam-6326	401	5	friendly	friendly	ADJ
ejpam-6326	401	6	dominating	dominating	NOUN
ejpam-6326	401	7	set	set	NOUN
ejpam-6326	401	8	of	of	ADP
ejpam-6326	401	9	g	g	PROPN
ejpam-6326	401	10	◦	◦	NOUN
ejpam-6326	401	11	h	h	NOUN
ejpam-6326	401	12	if	if	SCONJ
ejpam-6326	402	1	and	and	CCONJ
ejpam-6326	402	2	only	only	ADV
ejpam-6326	402	3	if	if	SCONJ
ejpam-6326	402	4	f	f	PROPN
ejpam-6326	402	5	=	=	SYM
ejpam-6326	402	6	v	v	PROPN
ejpam-6326	402	7	(	(	PUNCT
ejpam-6326	402	8	g	g	NOUN
ejpam-6326	402	9	)	)	PUNCT
ejpam-6326	402	10	.	.	PUNCT
ejpam-6326	403	1	proof	proof	NOUN
ejpam-6326	403	2	.	.	PUNCT
ejpam-6326	404	1	assume	assume	VERB
ejpam-6326	404	2	first	first	ADV
ejpam-6326	404	3	that	that	SCONJ
ejpam-6326	404	4	f	f	PROPN
ejpam-6326	404	5	is	be	AUX
ejpam-6326	404	6	a	a	DET
ejpam-6326	404	7	friendly	friendly	ADJ
ejpam-6326	404	8	dominating	dominating	NOUN
ejpam-6326	404	9	set	set	NOUN
ejpam-6326	404	10	of	of	ADP
ejpam-6326	404	11	g	g	PROPN
ejpam-6326	404	12	◦	◦	NOUN
ejpam-6326	404	13	h.	h.	NOUN
ejpam-6326	404	14	if	if	SCONJ
ejpam-6326	404	15	f	f	PROPN
ejpam-6326	404	16	̸=	̸=	PROPN
ejpam-6326	404	17	v	v	ADP
ejpam-6326	404	18	(	(	PUNCT
ejpam-6326	404	19	g	g	NOUN
ejpam-6326	404	20	)	)	PUNCT
ejpam-6326	404	21	choose	choose	VERB
ejpam-6326	404	22	a	a	DET
ejpam-6326	404	23	vertex	vertex	NOUN
ejpam-6326	404	24	y	y	PROPN
ejpam-6326	404	25	∈	∈	PROPN
ejpam-6326	404	26	v	v	PROPN
ejpam-6326	404	27	(	(	PUNCT
ejpam-6326	404	28	g)\f	g)\f	PROPN
ejpam-6326	404	29	.	.	PUNCT
ejpam-6326	405	1	since	since	SCONJ
ejpam-6326	405	2	f	f	PROPN
ejpam-6326	405	3	contains	contain	VERB
ejpam-6326	405	4	no	no	DET
ejpam-6326	405	5	vertices	vertex	NOUN
ejpam-6326	405	6	of	of	ADP
ejpam-6326	405	7	the	the	DET
ejpam-6326	405	8	copy	copy	NOUN
ejpam-6326	405	9	hy	hy	PROPN
ejpam-6326	405	10	,	,	PUNCT
ejpam-6326	405	11	every	every	DET
ejpam-6326	405	12	vertex	vertex	NOUN
ejpam-6326	405	13	h	h	NOUN
ejpam-6326	405	14	∈	∈	PROPN
ejpam-6326	405	15	v	v	ADP
ejpam-6326	405	16	(	(	PUNCT
ejpam-6326	405	17	hy	hy	NOUN
ejpam-6326	405	18	)	)	PUNCT
ejpam-6326	405	19	is	be	AUX
ejpam-6326	405	20	adjacent	adjacent	ADJ
ejpam-6326	405	21	only	only	ADV
ejpam-6326	405	22	to	to	ADP
ejpam-6326	405	23	its	its	PRON
ejpam-6326	405	24	root	root	NOUN
ejpam-6326	405	25	y	y	PROPN
ejpam-6326	405	26	and	and	CCONJ
ejpam-6326	405	27	to	to	ADP
ejpam-6326	405	28	vertices	vertex	NOUN
ejpam-6326	405	29	of	of	ADP
ejpam-6326	405	30	hy	hy	NOUN
ejpam-6326	405	31	itself	itself	PRON
ejpam-6326	405	32	.	.	PUNCT
ejpam-6326	406	1	since	since	SCONJ
ejpam-6326	406	2	neither	neither	CCONJ
ejpam-6326	406	3	y	y	PROPN
ejpam-6326	406	4	nor	nor	CCONJ
ejpam-6326	406	5	any	any	DET
ejpam-6326	406	6	vertex	vertex	NOUN
ejpam-6326	406	7	of	of	ADP
ejpam-6326	406	8	hy	hy	NOUN
ejpam-6326	406	9	lies	lie	VERB
ejpam-6326	406	10	in	in	ADP
ejpam-6326	406	11	f	f	PROPN
ejpam-6326	406	12	,	,	PUNCT
ejpam-6326	406	13	the	the	DET
ejpam-6326	406	14	vertex	vertex	NOUN
ejpam-6326	406	15	h	h	NOUN
ejpam-6326	406	16	has	have	VERB
ejpam-6326	406	17	no	no	DET
ejpam-6326	406	18	neighbor	neighbor	NOUN
ejpam-6326	406	19	in	in	ADP
ejpam-6326	406	20	f	f	PROPN
ejpam-6326	406	21	;	;	PUNCT
ejpam-6326	406	22	hence	hence	ADV
ejpam-6326	406	23	h	h	NOUN
ejpam-6326	406	24	/∈	/∈	PUNCT
ejpam-6326	407	1	n	n	PROPN
ejpam-6326	408	1	[	[	X
ejpam-6326	408	2	f	f	X
ejpam-6326	408	3	]	]	X
ejpam-6326	408	4	,	,	PUNCT
ejpam-6326	408	5	contradicting	contradict	VERB
ejpam-6326	408	6	the	the	DET
ejpam-6326	408	7	assumption	assumption	NOUN
ejpam-6326	408	8	that	that	SCONJ
ejpam-6326	408	9	f	f	PROPN
ejpam-6326	408	10	is	be	AUX
ejpam-6326	408	11	dominating	dominate	VERB
ejpam-6326	408	12	.	.	PUNCT
ejpam-6326	409	1	therefore	therefore	ADV
ejpam-6326	409	2	domination	domination	NOUN
ejpam-6326	409	3	forces	force	NOUN
ejpam-6326	409	4	f	f	PROPN
ejpam-6326	409	5	=	=	SYM
ejpam-6326	409	6	v	v	PROPN
ejpam-6326	409	7	(	(	PUNCT
ejpam-6326	409	8	g	g	NOUN
ejpam-6326	409	9	)	)	PUNCT
ejpam-6326	409	10	.	.	PUNCT
ejpam-6326	410	1	conversely	conversely	ADV
ejpam-6326	410	2	,	,	PUNCT
ejpam-6326	410	3	suppose	suppose	VERB
ejpam-6326	410	4	f	f	PROPN
ejpam-6326	410	5	=	=	SYM
ejpam-6326	410	6	v	v	PROPN
ejpam-6326	410	7	(	(	PUNCT
ejpam-6326	410	8	g	g	NOUN
ejpam-6326	410	9	)	)	PUNCT
ejpam-6326	410	10	.	.	PUNCT
ejpam-6326	411	1	clearly	clearly	ADV
ejpam-6326	411	2	,	,	PUNCT
ejpam-6326	411	3	f	f	PROPN
ejpam-6326	411	4	is	be	AUX
ejpam-6326	411	5	a	a	DET
ejpam-6326	411	6	dominating	dominating	NOUN
ejpam-6326	411	7	set	set	VERB
ejpam-6326	411	8	in	in	ADP
ejpam-6326	411	9	g	g	PROPN
ejpam-6326	411	10	◦	◦	NOUN
ejpam-6326	411	11	h.	h.	NOUN
ejpam-6326	411	12	fix	fix	NOUN
ejpam-6326	411	13	x	x	X
ejpam-6326	411	14	∈	∈	PROPN
ejpam-6326	411	15	v	v	NOUN
ejpam-6326	411	16	(	(	PUNCT
ejpam-6326	411	17	g	g	NOUN
ejpam-6326	411	18	)	)	PUNCT
ejpam-6326	411	19	and	and	CCONJ
ejpam-6326	411	20	let	let	VERB
ejpam-6326	411	21	h	h	PRON
ejpam-6326	411	22	∈	∈	PROPN
ejpam-6326	411	23	hx	hx	PROPN
ejpam-6326	411	24	\	\	PROPN
ejpam-6326	411	25	f	f	PROPN
ejpam-6326	411	26	.	.	PUNCT
ejpam-6326	412	1	exactly	exactly	ADV
ejpam-6326	412	2	one	one	NUM
ejpam-6326	412	3	edge	edge	NOUN
ejpam-6326	412	4	,	,	PUNCT
ejpam-6326	412	5	namely	namely	ADV
ejpam-6326	412	6	{	{	PUNCT
ejpam-6326	412	7	h	h	NOUN
ejpam-6326	412	8	,	,	PUNCT
ejpam-6326	412	9	x	x	NOUN
ejpam-6326	412	10	}	}	PUNCT
ejpam-6326	412	11	,	,	PUNCT
ejpam-6326	412	12	joins	join	VERB
ejpam-6326	412	13	h	h	NOUN
ejpam-6326	412	14	to	to	ADP
ejpam-6326	412	15	f	f	PROPN
ejpam-6326	412	16	,	,	PUNCT
ejpam-6326	412	17	so	so	ADV
ejpam-6326	412	18	degf	degf	PROPN
ejpam-6326	412	19	(	(	PUNCT
ejpam-6326	412	20	h	h	NOUN
ejpam-6326	412	21	)	)	PUNCT
ejpam-6326	412	22	=	=	SYM
ejpam-6326	413	1	1	1	X
ejpam-6326	413	2	.	.	PUNCT
ejpam-6326	414	1	on	on	ADP
ejpam-6326	414	2	the	the	DET
ejpam-6326	414	3	other	other	ADJ
ejpam-6326	414	4	hand	hand	NOUN
ejpam-6326	414	5	,	,	PUNCT
ejpam-6326	414	6	h	h	NOUN
ejpam-6326	414	7	is	be	AUX
ejpam-6326	414	8	adjacent	adjacent	ADJ
ejpam-6326	414	9	inside	inside	ADP
ejpam-6326	414	10	hx	hx	PROPN
ejpam-6326	414	11	to	to	ADP
ejpam-6326	414	12	degh(h	degh(h	VERB
ejpam-6326	414	13	)	)	PUNCT
ejpam-6326	414	14	vertices	vertex	NOUN
ejpam-6326	414	15	,	,	PUNCT
ejpam-6326	414	16	all	all	PRON
ejpam-6326	414	17	of	of	ADP
ejpam-6326	414	18	which	which	PRON
ejpam-6326	414	19	lie	lie	VERB
ejpam-6326	414	20	in	in	ADP
ejpam-6326	414	21	v	v	NOUN
ejpam-6326	414	22	(	(	PUNCT
ejpam-6326	414	23	g	g	PROPN
ejpam-6326	414	24	◦	◦	NOUN
ejpam-6326	414	25	h	h	NOUN
ejpam-6326	414	26	)	)	PUNCT
ejpam-6326	414	27	\	\	PROPN
ejpam-6326	415	1	f	f	PROPN
ejpam-6326	415	2	.	.	PUNCT
ejpam-6326	416	1	since	since	SCONJ
ejpam-6326	416	2	h	h	NOUN
ejpam-6326	416	3	is	be	AUX
ejpam-6326	416	4	connected	connect	VERB
ejpam-6326	416	5	and	and	CCONJ
ejpam-6326	416	6	non	non	ADJ
ejpam-6326	416	7	–	–	NOUN
ejpam-6326	416	8	trivial	trivial	ADJ
ejpam-6326	416	9	,	,	PUNCT
ejpam-6326	416	10	degh(h	degh(h	NOUN
ejpam-6326	416	11	)	)	PUNCT
ejpam-6326	416	12	≥	≥	NOUN
ejpam-6326	416	13	1	1	NUM
ejpam-6326	416	14	,	,	PUNCT
ejpam-6326	416	15	whence	whence	NOUN
ejpam-6326	416	16	,	,	PUNCT
ejpam-6326	416	17	degf	degf	PROPN
ejpam-6326	416	18	(	(	PUNCT
ejpam-6326	416	19	h	h	NOUN
ejpam-6326	416	20	)	)	PUNCT
ejpam-6326	416	21	=	=	SYM
ejpam-6326	416	22	1	1	NUM
ejpam-6326	416	23	≤	≤	NOUN
ejpam-6326	416	24	degv	degv	NOUN
ejpam-6326	416	25	(	(	PUNCT
ejpam-6326	416	26	g	g	NOUN
ejpam-6326	416	27	◦	◦	NOUN
ejpam-6326	416	28	h)\f	h)\f	ADJ
ejpam-6326	416	29	(	(	PUNCT
ejpam-6326	416	30	h	h	NOUN
ejpam-6326	416	31	)	)	PUNCT
ejpam-6326	416	32	=	=	SYM
ejpam-6326	416	33	degh(h	degh(h	NOUN
ejpam-6326	416	34	)	)	PUNCT
ejpam-6326	416	35	.	.	PUNCT
ejpam-6326	417	1	thus	thus	ADV
ejpam-6326	417	2	the	the	DET
ejpam-6326	417	3	inequality	inequality	NOUN
ejpam-6326	417	4	holds	hold	VERB
ejpam-6326	417	5	for	for	ADP
ejpam-6326	417	6	every	every	DET
ejpam-6326	417	7	vertex	vertex	NOUN
ejpam-6326	417	8	outside	outside	ADP
ejpam-6326	417	9	f	f	PROPN
ejpam-6326	417	10	,	,	PUNCT
ejpam-6326	417	11	so	so	CCONJ
ejpam-6326	417	12	f	f	PROPN
ejpam-6326	417	13	is	be	AUX
ejpam-6326	417	14	friendly	friendly	ADJ
ejpam-6326	417	15	.	.	PUNCT
ejpam-6326	418	1	therefore	therefore	ADV
ejpam-6326	418	2	f	f	PROPN
ejpam-6326	418	3	=	=	SYM
ejpam-6326	418	4	v	v	PROPN
ejpam-6326	418	5	(	(	PUNCT
ejpam-6326	418	6	g	g	NOUN
ejpam-6326	418	7	)	)	PUNCT
ejpam-6326	418	8	is	be	AUX
ejpam-6326	418	9	indeed	indeed	ADV
ejpam-6326	418	10	a	a	DET
ejpam-6326	418	11	friendly	friendly	ADJ
ejpam-6326	418	12	dominating	dominating	NOUN
ejpam-6326	418	13	set	set	NOUN
ejpam-6326	418	14	.	.	PUNCT
ejpam-6326	419	1	theorem	theorem	VERB
ejpam-6326	419	2	14	14	NUM
ejpam-6326	419	3	.	.	PUNCT
ejpam-6326	420	1	let	let	VERB
ejpam-6326	420	2	g	g	NOUN
ejpam-6326	420	3	and	and	CCONJ
ejpam-6326	420	4	h	h	NOUN
ejpam-6326	420	5	be	be	AUX
ejpam-6326	420	6	connected	connect	VERB
ejpam-6326	420	7	non	non	ADJ
ejpam-6326	420	8	–	–	ADJ
ejpam-6326	420	9	trivial	trivial	ADJ
ejpam-6326	420	10	graphs	graph	NOUN
ejpam-6326	420	11	and	and	CCONJ
ejpam-6326	420	12	let	let	VERB
ejpam-6326	420	13	f	f	PROPN
ejpam-6326	420	14	⊆	⊆	NUM
ejpam-6326	420	15	v	v	NOUN
ejpam-6326	420	16	(	(	PUNCT
ejpam-6326	420	17	g	g	PROPN
ejpam-6326	420	18	◦	◦	NOUN
ejpam-6326	420	19	h	h	NOUN
ejpam-6326	420	20	)	)	PUNCT
ejpam-6326	420	21	.	.	PUNCT
ejpam-6326	421	1	suppose	suppose	VERB
ejpam-6326	421	2	f	f	PROPN
ejpam-6326	421	3	⊆	⊆	NUM
ejpam-6326	421	4	⋃	⋃	PROPN
ejpam-6326	421	5	x∈v	x∈v	PROPN
ejpam-6326	421	6	(	(	PUNCT
ejpam-6326	421	7	g	g	NOUN
ejpam-6326	421	8	)	)	PUNCT
ejpam-6326	421	9	v	v	NOUN
ejpam-6326	421	10	(	(	PUNCT
ejpam-6326	421	11	hx	hx	PROPN
ejpam-6326	421	12	)	)	PUNCT
ejpam-6326	421	13	and	and	CCONJ
ejpam-6326	421	14	px	px	X
ejpam-6326	421	15	=	=	SYM
ejpam-6326	421	16	f	f	PROPN
ejpam-6326	421	17	∩	∩	PROPN
ejpam-6326	421	18	v	v	PROPN
ejpam-6326	421	19	(	(	PUNCT
ejpam-6326	421	20	hx	hx	PROPN
ejpam-6326	421	21	)	)	PUNCT
ejpam-6326	421	22	̸=	̸=	PROPN
ejpam-6326	421	23	∅	∅	NOUN
ejpam-6326	421	24	for	for	ADP
ejpam-6326	421	25	every	every	DET
ejpam-6326	421	26	x	x	SYM
ejpam-6326	421	27	∈	∈	PROPN
ejpam-6326	421	28	v	v	NOUN
ejpam-6326	421	29	(	(	PUNCT
ejpam-6326	421	30	g	g	NOUN
ejpam-6326	421	31	)	)	PUNCT
ejpam-6326	421	32	.	.	PUNCT
ejpam-6326	422	1	then	then	ADV
ejpam-6326	422	2	f	f	PROPN
ejpam-6326	422	3	is	be	AUX
ejpam-6326	422	4	a	a	DET
ejpam-6326	422	5	friendly	friendly	ADJ
ejpam-6326	422	6	dominating	dominating	NOUN
ejpam-6326	422	7	set	set	NOUN
ejpam-6326	422	8	of	of	ADP
ejpam-6326	422	9	g	g	PROPN
ejpam-6326	422	10	◦	◦	NOUN
ejpam-6326	422	11	h	h	NOUN
ejpam-6326	422	12	if	if	SCONJ
ejpam-6326	423	1	and	and	CCONJ
ejpam-6326	423	2	only	only	ADV
ejpam-6326	423	3	if	if	SCONJ
ejpam-6326	423	4	the	the	DET
ejpam-6326	423	5	following	follow	VERB
ejpam-6326	423	6	hold	hold	NOUN
ejpam-6326	423	7	for	for	ADP
ejpam-6326	423	8	every	every	DET
ejpam-6326	423	9	x	x	SYM
ejpam-6326	423	10	∈	∈	PROPN
ejpam-6326	423	11	v	v	NOUN
ejpam-6326	423	12	(	(	PUNCT
ejpam-6326	423	13	g	g	NOUN
ejpam-6326	423	14	):	):	PUNCT
ejpam-6326	423	15	(	(	PUNCT
ejpam-6326	423	16	i	i	NOUN
ejpam-6326	423	17	)	)	PUNCT
ejpam-6326	423	18	px	px	PROPN
ejpam-6326	423	19	is	be	AUX
ejpam-6326	423	20	a	a	DET
ejpam-6326	423	21	dominating	dominating	NOUN
ejpam-6326	423	22	set	set	NOUN
ejpam-6326	423	23	of	of	ADP
ejpam-6326	423	24	hx	hx	PROPN
ejpam-6326	423	25	;	;	PUNCT
ejpam-6326	423	26	(	(	PUNCT
ejpam-6326	423	27	ii	ii	NOUN
ejpam-6326	423	28	)	)	PUNCT
ejpam-6326	423	29	|px|	|px|	VERB
ejpam-6326	423	30	≤	≤	NUM
ejpam-6326	423	31	⌊	⌊	AUX
ejpam-6326	423	32	|v	|v	PROPN
ejpam-6326	423	33	(	(	PUNCT
ejpam-6326	423	34	h)|+	h)|+	ADJ
ejpam-6326	423	35	degg(x	degg(x	NOUN
ejpam-6326	423	36	)	)	PUNCT
ejpam-6326	423	37	2	2	NUM
ejpam-6326	423	38	⌋	⌋	NOUN
ejpam-6326	423	39	.	.	PUNCT
ejpam-6326	424	1	proof	proof	NOUN
ejpam-6326	424	2	.	.	PUNCT
ejpam-6326	425	1	assume	assume	VERB
ejpam-6326	425	2	f	f	PROPN
ejpam-6326	425	3	is	be	AUX
ejpam-6326	425	4	a	a	DET
ejpam-6326	425	5	friendly	friendly	ADJ
ejpam-6326	425	6	dominating	dominating	NOUN
ejpam-6326	425	7	set	set	NOUN
ejpam-6326	425	8	.	.	PUNCT
ejpam-6326	426	1	since	since	SCONJ
ejpam-6326	426	2	x	x	PROPN
ejpam-6326	426	3	/∈	/∈	PROPN
ejpam-6326	426	4	f	f	PROPN
ejpam-6326	426	5	for	for	ADP
ejpam-6326	426	6	each	each	DET
ejpam-6326	426	7	x	x	SYM
ejpam-6326	426	8	∈	∈	PROPN
ejpam-6326	426	9	v	v	NOUN
ejpam-6326	426	10	(	(	PUNCT
ejpam-6326	426	11	g	g	NOUN
ejpam-6326	426	12	)	)	PUNCT
ejpam-6326	426	13	,	,	PUNCT
ejpam-6326	426	14	domination	domination	NOUN
ejpam-6326	426	15	of	of	ADP
ejpam-6326	426	16	xmust	xmust	PROPN
ejpam-6326	426	17	occur	occur	VERB
ejpam-6326	426	18	via	via	ADP
ejpam-6326	426	19	an	an	DET
ejpam-6326	426	20	edge	edge	NOUN
ejpam-6326	426	21	xp	xp	INTJ
ejpam-6326	426	22	with	with	ADP
ejpam-6326	426	23	p	p	PROPN
ejpam-6326	426	24	∈	∈	PROPN
ejpam-6326	426	25	px	px	NOUN
ejpam-6326	426	26	,	,	PUNCT
ejpam-6326	426	27	so	so	SCONJ
ejpam-6326	426	28	px	px	PROPN
ejpam-6326	426	29	̸=	̸=	PROPN
ejpam-6326	426	30	∅.	∅.	VERB
ejpam-6326	426	31	if	if	SCONJ
ejpam-6326	426	32	some	some	DET
ejpam-6326	426	33	h	h	NOUN
ejpam-6326	426	34	∈	∈	PROPN
ejpam-6326	426	35	hx\px	hx\px	PROPN
ejpam-6326	426	36	had	have	VERB
ejpam-6326	426	37	no	no	DET
ejpam-6326	426	38	neighbour	neighbour	NOUN
ejpam-6326	426	39	in	in	ADP
ejpam-6326	426	40	px	px	PROPN
ejpam-6326	426	41	,	,	PUNCT
ejpam-6326	426	42	it	it	PRON
ejpam-6326	426	43	would	would	AUX
ejpam-6326	426	44	not	not	PART
ejpam-6326	426	45	be	be	AUX
ejpam-6326	426	46	dominated	dominate	VERB
ejpam-6326	426	47	;	;	PUNCT
ejpam-6326	426	48	hence	hence	ADV
ejpam-6326	426	49	px	px	PROPN
ejpam-6326	426	50	dominates	dominate	VERB
ejpam-6326	426	51	hx	hx	PROPN
ejpam-6326	426	52	,	,	PUNCT
ejpam-6326	426	53	establishing	establish	VERB
ejpam-6326	426	54	(	(	PUNCT
ejpam-6326	426	55	i	i	NOUN
ejpam-6326	426	56	)	)	PUNCT
ejpam-6326	426	57	.	.	PUNCT
ejpam-6326	427	1	for	for	ADP
ejpam-6326	427	2	(	(	PUNCT
ejpam-6326	427	3	ii	ii	NOUN
ejpam-6326	427	4	)	)	PUNCT
ejpam-6326	427	5	,	,	PUNCT
ejpam-6326	427	6	observe	observe	VERB
ejpam-6326	427	7	that	that	SCONJ
ejpam-6326	427	8	,	,	PUNCT
ejpam-6326	427	9	degf	degf	PROPN
ejpam-6326	427	10	(	(	PUNCT
ejpam-6326	427	11	x	x	NOUN
ejpam-6326	427	12	)	)	PUNCT
ejpam-6326	427	13	=	=	SYM
ejpam-6326	427	14	|px|	|px|	NOUN
ejpam-6326	427	15	and	and	CCONJ
ejpam-6326	427	16	degv	degv	NOUN
ejpam-6326	427	17	(	(	PUNCT
ejpam-6326	427	18	g	g	NOUN
ejpam-6326	427	19	◦	◦	NOUN
ejpam-6326	427	20	h)\f	h)\f	ADJ
ejpam-6326	427	21	(	(	PUNCT
ejpam-6326	427	22	x	x	NOUN
ejpam-6326	427	23	)	)	PUNCT
ejpam-6326	427	24	=	=	SYM
ejpam-6326	427	25	degg(x)+|v	degg(x)+|v	NOUN
ejpam-6326	427	26	(	(	PUNCT
ejpam-6326	427	27	h)|−|px|	h)|−|px|	PROPN
ejpam-6326	427	28	,	,	PUNCT
ejpam-6326	427	29	so	so	CCONJ
ejpam-6326	427	30	the	the	DET
ejpam-6326	427	31	friendly	friendly	ADJ
ejpam-6326	427	32	inequality	inequality	NOUN
ejpam-6326	427	33	degf	degf	NOUN
ejpam-6326	427	34	(	(	PUNCT
ejpam-6326	427	35	x	x	NOUN
ejpam-6326	427	36	)	)	PUNCT
ejpam-6326	427	37	≤	≤	ADJ
ejpam-6326	427	38	degv	degv	NOUN
ejpam-6326	427	39	(	(	PUNCT
ejpam-6326	427	40	g	g	NOUN
ejpam-6326	427	41	◦	◦	NOUN
ejpam-6326	427	42	h)\f	h)\f	ADJ
ejpam-6326	427	43	(	(	PUNCT
ejpam-6326	427	44	x	x	X
ejpam-6326	427	45	)	)	PUNCT
ejpam-6326	427	46	is	be	AUX
ejpam-6326	427	47	equivalent	equivalent	ADJ
ejpam-6326	427	48	to	to	PART
ejpam-6326	427	49	|px|	|px|	VERB
ejpam-6326	427	50	≤	≤	NUM
ejpam-6326	427	51	|v	|v	X
ejpam-6326	427	52	(	(	PUNCT
ejpam-6326	427	53	h)|+	h)|+	ADJ
ejpam-6326	427	54	degg(x	degg(x	NOUN
ejpam-6326	427	55	)	)	PUNCT
ejpam-6326	427	56	2	2	NUM
ejpam-6326	427	57	.	.	PUNCT
ejpam-6326	428	1	conversely	conversely	ADV
ejpam-6326	428	2	,	,	PUNCT
ejpam-6326	428	3	suppose	suppose	VERB
ejpam-6326	428	4	that	that	SCONJ
ejpam-6326	428	5	(	(	PUNCT
ejpam-6326	428	6	i	i	NOUN
ejpam-6326	428	7	)	)	PUNCT
ejpam-6326	428	8	and	and	CCONJ
ejpam-6326	428	9	(	(	PUNCT
ejpam-6326	428	10	ii	ii	NOUN
ejpam-6326	428	11	)	)	PUNCT
ejpam-6326	428	12	hold	hold	VERB
ejpam-6326	428	13	.	.	PUNCT
ejpam-6326	429	1	every	every	DET
ejpam-6326	429	2	x	x	PROPN
ejpam-6326	429	3	∈	∈	PROPN
ejpam-6326	429	4	v	v	NOUN
ejpam-6326	429	5	(	(	PUNCT
ejpam-6326	429	6	g	g	NOUN
ejpam-6326	429	7	)	)	PUNCT
ejpam-6326	429	8	is	be	AUX
ejpam-6326	429	9	adjacent	adjacent	ADJ
ejpam-6326	429	10	to	to	ADP
ejpam-6326	429	11	px	px	PROPN
ejpam-6326	429	12	̸=	̸=	PROPN
ejpam-6326	429	13	∅	∅	NOUN
ejpam-6326	429	14	,	,	PUNCT
ejpam-6326	429	15	and	and	CCONJ
ejpam-6326	429	16	(	(	PUNCT
ejpam-6326	429	17	i	i	NOUN
ejpam-6326	429	18	)	)	PUNCT
ejpam-6326	429	19	ensures	ensure	VERB
ejpam-6326	429	20	px	px	PROPN
ejpam-6326	429	21	dominates	dominate	VERB
ejpam-6326	429	22	hx	hx	NOUN
ejpam-6326	429	23	;	;	PUNCT
ejpam-6326	430	1	hence	hence	ADV
ejpam-6326	430	2	ng	ng	PROPN
ejpam-6326	430	3	◦	◦	NOUN
ejpam-6326	430	4	h	h	NOUN
ejpam-6326	431	1	[	[	X
ejpam-6326	431	2	f	f	X
ejpam-6326	431	3	]	]	X
ejpam-6326	431	4	=	=	SYM
ejpam-6326	431	5	v	v	X
ejpam-6326	431	6	(	(	PUNCT
ejpam-6326	431	7	g	g	PROPN
ejpam-6326	431	8	◦	◦	NOUN
ejpam-6326	431	9	h	h	NOUN
ejpam-6326	431	10	)	)	PUNCT
ejpam-6326	431	11	.	.	PUNCT
ejpam-6326	432	1	to	to	PART
ejpam-6326	432	2	show	show	VERB
ejpam-6326	432	3	that	that	SCONJ
ejpam-6326	432	4	f	f	PROPN
ejpam-6326	432	5	is	be	AUX
ejpam-6326	432	6	friendly	friendly	ADJ
ejpam-6326	432	7	.	.	PUNCT
ejpam-6326	433	1	let	let	VERB
ejpam-6326	433	2	v	v	NUM
ejpam-6326	433	3	∈	∈	PROPN
ejpam-6326	433	4	v	v	NOUN
ejpam-6326	433	5	(	(	PUNCT
ejpam-6326	433	6	g	g	PROPN
ejpam-6326	433	7	◦	◦	NOUN
ejpam-6326	433	8	h	h	NOUN
ejpam-6326	433	9	)	)	PUNCT
ejpam-6326	433	10	\	\	PROPN
ejpam-6326	434	1	f	f	X
ejpam-6326	434	2	.	.	PUNCT
ejpam-6326	435	1	if	if	SCONJ
ejpam-6326	435	2	v	v	NOUN
ejpam-6326	435	3	=	=	PUNCT
ejpam-6326	435	4	x	x	SYM
ejpam-6326	435	5	∈	∈	PROPN
ejpam-6326	435	6	v	v	NOUN
ejpam-6326	435	7	(	(	PUNCT
ejpam-6326	435	8	g	g	NOUN
ejpam-6326	435	9	)	)	PUNCT
ejpam-6326	435	10	,	,	PUNCT
ejpam-6326	435	11	then	then	ADV
ejpam-6326	435	12	degf	degf	PROPN
ejpam-6326	435	13	(	(	PUNCT
ejpam-6326	435	14	v	v	NOUN
ejpam-6326	435	15	)	)	PUNCT
ejpam-6326	435	16	=	=	PUNCT
ejpam-6326	435	17	|px|	|px|	VERB
ejpam-6326	435	18	≤	≤	NUM
ejpam-6326	435	19	|v	|v	X
ejpam-6326	435	20	(	(	PUNCT
ejpam-6326	435	21	h)|+degg(x	h)|+degg(x	NOUN
ejpam-6326	435	22	)	)	PUNCT
ejpam-6326	435	23	2	2	NUM
ejpam-6326	435	24	=	=	SYM
ejpam-6326	435	25	degv	degv	NOUN
ejpam-6326	435	26	(	(	PUNCT
ejpam-6326	435	27	g	g	NOUN
ejpam-6326	435	28	◦	◦	NOUN
ejpam-6326	435	29	h)\f	h)\f	ADJ
ejpam-6326	435	30	(	(	PUNCT
ejpam-6326	435	31	v	v	NOUN
ejpam-6326	435	32	)	)	PUNCT
ejpam-6326	435	33	by	by	ADP
ejpam-6326	435	34	(	(	PUNCT
ejpam-6326	435	35	ii	ii	NOUN
ejpam-6326	435	36	)	)	PUNCT
ejpam-6326	435	37	.	.	PUNCT
ejpam-6326	436	1	now	now	ADV
ejpam-6326	436	2	,	,	PUNCT
ejpam-6326	436	3	if	if	SCONJ
ejpam-6326	436	4	v	v	NOUN
ejpam-6326	436	5	=	=	NOUN
ejpam-6326	436	6	h	h	NOUN
ejpam-6326	436	7	∈	∈	PROPN
ejpam-6326	436	8	hx	hx	PROPN
ejpam-6326	436	9	\	\	PROPN
ejpam-6326	436	10	px	px	PROPN
ejpam-6326	436	11	,	,	PUNCT
ejpam-6326	436	12	put	put	VERB
ejpam-6326	436	13	a	a	DET
ejpam-6326	436	14	=	=	NOUN
ejpam-6326	436	15	degpx	degpx	NOUN
ejpam-6326	436	16	(	(	PUNCT
ejpam-6326	436	17	h	h	NOUN
ejpam-6326	436	18	)	)	PUNCT
ejpam-6326	436	19	.	.	PUNCT
ejpam-6326	437	1	then	then	ADV
ejpam-6326	437	2	degf	degf	PROPN
ejpam-6326	437	3	(	(	PUNCT
ejpam-6326	437	4	h	h	NOUN
ejpam-6326	437	5	)	)	PUNCT
ejpam-6326	437	6	=	=	SYM
ejpam-6326	437	7	a	a	PRON
ejpam-6326	437	8	and	and	CCONJ
ejpam-6326	437	9	degv	degv	NOUN
ejpam-6326	437	10	(	(	PUNCT
ejpam-6326	437	11	g	g	NOUN
ejpam-6326	437	12	◦	◦	NOUN
ejpam-6326	437	13	h)\f	h)\f	ADJ
ejpam-6326	437	14	(	(	PUNCT
ejpam-6326	437	15	h	h	NOUN
ejpam-6326	437	16	)	)	PUNCT
ejpam-6326	437	17	=	=	SYM
ejpam-6326	437	18	degh(h)−	degh(h)−	PROPN
ejpam-6326	437	19	a+	a+	PUNCT
ejpam-6326	437	20	1	1	NUM
ejpam-6326	437	21	.	.	PUNCT
ejpam-6326	437	22	because	because	SCONJ
ejpam-6326	437	23	a	a	DET
ejpam-6326	437	24	≤	≤	NUM
ejpam-6326	437	25	|px|	|px|	NOUN
ejpam-6326	437	26	and	and	CCONJ
ejpam-6326	437	27	2|px|	2|px|	PROPN
ejpam-6326	437	28	≤	≤	NOUN
ejpam-6326	437	29	|v	|v	X
ejpam-6326	437	30	(	(	PUNCT
ejpam-6326	437	31	h)|+	h)|+	ADJ
ejpam-6326	437	32	degg(x	degg(x	NOUN
ejpam-6326	437	33	)	)	PUNCT
ejpam-6326	437	34	by	by	ADP
ejpam-6326	437	35	(	(	PUNCT
ejpam-6326	437	36	ii	ii	NOUN
ejpam-6326	437	37	)	)	PUNCT
ejpam-6326	437	38	,	,	PUNCT
ejpam-6326	437	39	we	we	PRON
ejpam-6326	437	40	have	have	VERB
ejpam-6326	437	41	2a	2a	NUM
ejpam-6326	437	42	≤	≤	NUM
ejpam-6326	437	43	|v	|v	X
ejpam-6326	437	44	(	(	PUNCT
ejpam-6326	437	45	h)|+degg(x	h)|+degg(x	NOUN
ejpam-6326	437	46	)	)	PUNCT
ejpam-6326	437	47	.	.	PUNCT
ejpam-6326	438	1	since	since	SCONJ
ejpam-6326	438	2	degh(h)+1	degh(h)+1	NOUN
ejpam-6326	438	3	≤	≤	PUNCT
ejpam-6326	438	4	|v	|v	X
ejpam-6326	438	5	(	(	PUNCT
ejpam-6326	438	6	h)|+degg(x	h)|+degg(x	NOUN
ejpam-6326	438	7	)	)	PUNCT
ejpam-6326	438	8	,	,	PUNCT
ejpam-6326	438	9	it	it	PRON
ejpam-6326	438	10	follows	follow	VERB
ejpam-6326	438	11	i.	i.	PROPN
ejpam-6326	438	12	s.	s.	PROPN
ejpam-6326	438	13	cabahug	cabahug	PROPN
ejpam-6326	438	14	,	,	PUNCT
ejpam-6326	438	15	jr	jr	PROPN
ejpam-6326	438	16	.	.	PROPN
ejpam-6326	438	17	,	,	PUNCT
ejpam-6326	438	18	r.	r.	PROPN
ejpam-6326	438	19	g.	g.	PROPN
ejpam-6326	438	20	eballe	eballe	PROPN
ejpam-6326	438	21	,	,	PUNCT
ejpam-6326	438	22	r.	r.	PROPN
ejpam-6326	438	23	t.	t.	PROPN
ejpam-6326	438	24	fernandez	fernandez	PROPN
ejpam-6326	438	25	/	/	SYM
ejpam-6326	438	26	eur	eur	PROPN
ejpam-6326	438	27	.	.	PUNCT
ejpam-6326	439	1	j.	j.	PROPN
ejpam-6326	439	2	pure	pure	PROPN
ejpam-6326	439	3	appl	appl	PROPN
ejpam-6326	439	4	.	.	PROPN
ejpam-6326	439	5	math	math	PROPN
ejpam-6326	439	6	,	,	PUNCT
ejpam-6326	439	7	18	18	NUM
ejpam-6326	439	8	(	(	PUNCT
ejpam-6326	439	9	4	4	NUM
ejpam-6326	439	10	)	)	PUNCT
ejpam-6326	439	11	(	(	PUNCT
ejpam-6326	439	12	2025	2025	NUM
ejpam-6326	439	13	)	)	PUNCT
ejpam-6326	439	14	,	,	PUNCT
ejpam-6326	439	15	6326	6326	NUM
ejpam-6326	439	16	13	13	NUM
ejpam-6326	439	17	of	of	ADP
ejpam-6326	439	18	15	15	NUM
ejpam-6326	439	19	that	that	SCONJ
ejpam-6326	439	20	2	2	NUM
ejpam-6326	439	21	degf	degf	NOUN
ejpam-6326	439	22	(	(	PUNCT
ejpam-6326	439	23	h	h	NOUN
ejpam-6326	439	24	)	)	PUNCT
ejpam-6326	439	25	≤	≤	NUM
ejpam-6326	439	26	degh(h	degh(h	NOUN
ejpam-6326	439	27	)	)	PUNCT
ejpam-6326	439	28	+	+	CCONJ
ejpam-6326	439	29	1	1	NUM
ejpam-6326	439	30	=	=	SYM
ejpam-6326	439	31	degg	degg	NOUN
ejpam-6326	439	32	◦	◦	NOUN
ejpam-6326	439	33	h(h	h(h	NOUN
ejpam-6326	439	34	)	)	PUNCT
ejpam-6326	439	35	,	,	PUNCT
ejpam-6326	439	36	i.e.	i.e.	X
ejpam-6326	439	37	degf	degf	X
ejpam-6326	439	38	(	(	PUNCT
ejpam-6326	439	39	h	h	NOUN
ejpam-6326	439	40	)	)	PUNCT
ejpam-6326	439	41	≤	≤	NOUN
ejpam-6326	439	42	degv	degv	NOUN
ejpam-6326	439	43	(	(	PUNCT
ejpam-6326	439	44	g	g	NOUN
ejpam-6326	439	45	◦	◦	NOUN
ejpam-6326	439	46	h)\f	h)\f	ADJ
ejpam-6326	439	47	(	(	PUNCT
ejpam-6326	439	48	h	h	NOUN
ejpam-6326	439	49	)	)	PUNCT
ejpam-6326	439	50	.	.	PUNCT
ejpam-6326	440	1	hence	hence	ADV
ejpam-6326	440	2	,	,	PUNCT
ejpam-6326	440	3	f	f	PROPN
ejpam-6326	440	4	is	be	AUX
ejpam-6326	440	5	friendly	friendly	ADJ
ejpam-6326	440	6	.	.	PUNCT
ejpam-6326	441	1	therefore	therefore	ADV
ejpam-6326	441	2	,	,	PUNCT
ejpam-6326	441	3	f	f	PROPN
ejpam-6326	441	4	is	be	AUX
ejpam-6326	441	5	a	a	DET
ejpam-6326	441	6	friendly	friendly	ADJ
ejpam-6326	441	7	dominating	dominating	NOUN
ejpam-6326	441	8	set	set	NOUN
ejpam-6326	441	9	of	of	ADP
ejpam-6326	441	10	g	g	PROPN
ejpam-6326	441	11	◦	◦	PROPN
ejpam-6326	441	12	h.	h.	NOUN
ejpam-6326	441	13	theorem	theorem	NOUN
ejpam-6326	441	14	15	15	NUM
ejpam-6326	441	15	.	.	PUNCT
ejpam-6326	442	1	let	let	VERB
ejpam-6326	442	2	g	g	NOUN
ejpam-6326	442	3	and	and	CCONJ
ejpam-6326	442	4	h	h	NOUN
ejpam-6326	442	5	be	be	AUX
ejpam-6326	442	6	connected	connect	VERB
ejpam-6326	442	7	non	non	ADJ
ejpam-6326	442	8	-	-	ADJ
ejpam-6326	442	9	trivial	trivial	ADJ
ejpam-6326	442	10	graphs	graph	NOUN
ejpam-6326	442	11	and	and	CCONJ
ejpam-6326	442	12	let	let	VERB
ejpam-6326	442	13	f	f	PROPN
ejpam-6326	442	14	⊆	⊆	NUM
ejpam-6326	442	15	v	v	NOUN
ejpam-6326	442	16	(	(	PUNCT
ejpam-6326	442	17	g	g	PROPN
ejpam-6326	442	18	◦	◦	NOUN
ejpam-6326	442	19	h	h	NOUN
ejpam-6326	442	20	)	)	PUNCT
ejpam-6326	442	21	.	.	PUNCT
ejpam-6326	443	1	choose	choose	VERB
ejpam-6326	443	2	a	a	DET
ejpam-6326	443	3	non	non	ADJ
ejpam-6326	443	4	-	-	ADJ
ejpam-6326	443	5	empty	empty	ADJ
ejpam-6326	443	6	subset	subset	NOUN
ejpam-6326	443	7	f0	f0	PROPN
ejpam-6326	443	8	⊆	⊆	NUM
ejpam-6326	443	9	v	v	NOUN
ejpam-6326	443	10	(	(	PUNCT
ejpam-6326	443	11	g	g	NOUN
ejpam-6326	443	12	)	)	PUNCT
ejpam-6326	443	13	and	and	CCONJ
ejpam-6326	443	14	,	,	PUNCT
ejpam-6326	443	15	for	for	ADP
ejpam-6326	443	16	every	every	DET
ejpam-6326	443	17	x	x	SYM
ejpam-6326	443	18	∈	∈	PROPN
ejpam-6326	443	19	v	v	NOUN
ejpam-6326	443	20	(	(	PUNCT
ejpam-6326	443	21	g	g	NOUN
ejpam-6326	443	22	)	)	PUNCT
ejpam-6326	443	23	,	,	PUNCT
ejpam-6326	443	24	a	a	DET
ejpam-6326	443	25	subset	subset	NOUN
ejpam-6326	443	26	px	px	PROPN
ejpam-6326	443	27	⊆	⊆	NUM
ejpam-6326	443	28	v	v	NOUN
ejpam-6326	443	29	(	(	PUNCT
ejpam-6326	443	30	hx	hx	PROPN
ejpam-6326	443	31	)	)	PUNCT
ejpam-6326	443	32	such	such	ADJ
ejpam-6326	443	33	that	that	SCONJ
ejpam-6326	443	34	px	px	PROPN
ejpam-6326	443	35	̸=	̸=	PROPN
ejpam-6326	443	36	∅	∅	NOUN
ejpam-6326	443	37	for	for	ADP
ejpam-6326	443	38	at	at	ADV
ejpam-6326	443	39	least	least	ADV
ejpam-6326	443	40	one	one	NUM
ejpam-6326	443	41	vertex	vertex	NOUN
ejpam-6326	443	42	x.	x.	NOUN
ejpam-6326	443	43	suppose	suppose	VERB
ejpam-6326	443	44	f	f	X
ejpam-6326	443	45	=	=	SYM
ejpam-6326	443	46	f0	f0	PROPN
ejpam-6326	443	47	∪	∪	X
ejpam-6326	443	48	(	(	PUNCT
ejpam-6326	443	49	⋃	⋃	PROPN
ejpam-6326	443	50	x∈v	x∈v	PROPN
ejpam-6326	443	51	(	(	PUNCT
ejpam-6326	443	52	g	g	NOUN
ejpam-6326	443	53	)	)	PUNCT
ejpam-6326	443	54	px	px	PROPN
ejpam-6326	443	55	)	)	PUNCT
ejpam-6326	443	56	.	.	PUNCT
ejpam-6326	444	1	then	then	ADV
ejpam-6326	444	2	the	the	DET
ejpam-6326	444	3	set	set	NOUN
ejpam-6326	444	4	f	f	PROPN
ejpam-6326	444	5	is	be	AUX
ejpam-6326	444	6	a	a	DET
ejpam-6326	444	7	friendly	friendly	ADJ
ejpam-6326	444	8	dominating	dominating	NOUN
ejpam-6326	444	9	set	set	NOUN
ejpam-6326	444	10	of	of	ADP
ejpam-6326	444	11	g	g	PROPN
ejpam-6326	444	12	◦	◦	NOUN
ejpam-6326	444	13	h	h	NOUN
ejpam-6326	444	14	if	if	SCONJ
ejpam-6326	445	1	and	and	CCONJ
ejpam-6326	445	2	only	only	ADV
ejpam-6326	445	3	if	if	SCONJ
ejpam-6326	445	4	the	the	DET
ejpam-6326	445	5	following	follow	VERB
ejpam-6326	445	6	conditions	condition	NOUN
ejpam-6326	445	7	are	be	AUX
ejpam-6326	445	8	satisfied	satisfied	ADJ
ejpam-6326	445	9	.	.	PUNCT
ejpam-6326	446	1	(	(	PUNCT
ejpam-6326	446	2	i	i	NOUN
ejpam-6326	446	3	)	)	PUNCT
ejpam-6326	446	4	for	for	ADP
ejpam-6326	446	5	every	every	PRON
ejpam-6326	446	6	x	x	SYM
ejpam-6326	446	7	∈	∈	PROPN
ejpam-6326	446	8	v	v	NOUN
ejpam-6326	446	9	(	(	PUNCT
ejpam-6326	446	10	g	g	NOUN
ejpam-6326	446	11	):	):	PUNCT
ejpam-6326	446	12	(	(	PUNCT
ejpam-6326	446	13	a	a	X
ejpam-6326	446	14	)	)	PUNCT
ejpam-6326	446	15	x	x	SYM
ejpam-6326	446	16	∈	∈	PROPN
ejpam-6326	446	17	f0	f0	PROPN
ejpam-6326	446	18	,	,	PUNCT
ejpam-6326	446	19	or	or	CCONJ
ejpam-6326	446	20	ng(x	ng(x	NUM
ejpam-6326	446	21	)	)	PUNCT
ejpam-6326	446	22	∩	∩	PROPN
ejpam-6326	446	23	f0	f0	PROPN
ejpam-6326	446	24	̸=	̸=	PROPN
ejpam-6326	446	25	∅	∅	NOUN
ejpam-6326	446	26	,	,	PUNCT
ejpam-6326	446	27	or	or	CCONJ
ejpam-6326	446	28	px	px	PROPN
ejpam-6326	446	29	̸=	̸=	PROPN
ejpam-6326	446	30	∅	∅	NOUN
ejpam-6326	446	31	;	;	PUNCT
ejpam-6326	446	32	(	(	PUNCT
ejpam-6326	446	33	b	b	X
ejpam-6326	446	34	)	)	PUNCT
ejpam-6326	446	35	if	if	SCONJ
ejpam-6326	446	36	x	x	PROPN
ejpam-6326	446	37	/∈	/∈	PUNCT
ejpam-6326	447	1	f0	f0	PROPN
ejpam-6326	447	2	then	then	ADV
ejpam-6326	447	3	every	every	DET
ejpam-6326	447	4	vertex	vertex	NOUN
ejpam-6326	447	5	of	of	ADP
ejpam-6326	447	6	hx	hx	PROPN
ejpam-6326	447	7	is	be	AUX
ejpam-6326	447	8	adjacent	adjacent	ADJ
ejpam-6326	447	9	to	to	ADP
ejpam-6326	447	10	a	a	DET
ejpam-6326	447	11	vertex	vertex	NOUN
ejpam-6326	447	12	of	of	ADP
ejpam-6326	447	13	px	px	PROPN
ejpam-6326	447	14	;	;	PUNCT
ejpam-6326	447	15	equivalently	equivalently	ADV
ejpam-6326	447	16	nhx	nhx	PROPN
ejpam-6326	448	1	[	[	X
ejpam-6326	448	2	px	px	X
ejpam-6326	448	3	]	]	X
ejpam-6326	448	4	=	=	SYM
ejpam-6326	448	5	v	v	X
ejpam-6326	448	6	(	(	PUNCT
ejpam-6326	448	7	hx	hx	PROPN
ejpam-6326	448	8	)	)	PUNCT
ejpam-6326	448	9	.	.	PUNCT
ejpam-6326	449	1	(	(	PUNCT
ejpam-6326	449	2	ii	ii	NOUN
ejpam-6326	449	3	)	)	PUNCT
ejpam-6326	449	4	for	for	ADP
ejpam-6326	449	5	every	every	DET
ejpam-6326	449	6	vertex	vertex	NOUN
ejpam-6326	449	7	x	x	SYM
ejpam-6326	449	8	∈	∈	NOUN
ejpam-6326	449	9	v	v	ADP
ejpam-6326	449	10	(	(	PUNCT
ejpam-6326	449	11	g	g	NOUN
ejpam-6326	449	12	)	)	PUNCT
ejpam-6326	449	13	\	\	PROPN
ejpam-6326	449	14	f0	f0	PROPN
ejpam-6326	449	15	,	,	PUNCT
ejpam-6326	449	16	|px|+	|px|+	NOUN
ejpam-6326	449	17	degf0	degf0	NOUN
ejpam-6326	449	18	(	(	PUNCT
ejpam-6326	449	19	x	x	NOUN
ejpam-6326	449	20	)	)	PUNCT
ejpam-6326	449	21	≤	≤	NOUN
ejpam-6326	449	22	(	(	PUNCT
ejpam-6326	449	23	|v	|v	PROPN
ejpam-6326	449	24	(	(	PUNCT
ejpam-6326	449	25	h)|	h)|	NOUN
ejpam-6326	449	26	−	−	PROPN
ejpam-6326	449	27	|px|	|px|	PROPN
ejpam-6326	449	28	)	)	PUNCT
ejpam-6326	450	1	+	+	CCONJ
ejpam-6326	450	2	(	(	PUNCT
ejpam-6326	450	3	degg(x)−	degg(x)−	PROPN
ejpam-6326	450	4	degf0	degf0	X
ejpam-6326	450	5	(	(	PUNCT
ejpam-6326	450	6	x	x	NOUN
ejpam-6326	450	7	)	)	PUNCT
ejpam-6326	450	8	)	)	PUNCT
ejpam-6326	450	9	.	.	PUNCT
ejpam-6326	451	1	(	(	PUNCT
ejpam-6326	451	2	iii	iii	X
ejpam-6326	451	3	)	)	PUNCT
ejpam-6326	451	4	for	for	ADP
ejpam-6326	451	5	every	every	DET
ejpam-6326	451	6	vertex	vertex	NOUN
ejpam-6326	451	7	h	h	NOUN
ejpam-6326	451	8	∈	∈	PROPN
ejpam-6326	451	9	v	v	PROPN
ejpam-6326	451	10	(	(	PUNCT
ejpam-6326	451	11	hx	hx	PROPN
ejpam-6326	451	12	)	)	PUNCT
ejpam-6326	451	13	\	\	PROPN
ejpam-6326	452	1	px	px	PROPN
ejpam-6326	452	2	,	,	PUNCT
ejpam-6326	452	3	degpx	degpx	NOUN
ejpam-6326	452	4	(	(	PUNCT
ejpam-6326	452	5	h	h	NOUN
ejpam-6326	452	6	)	)	PUNCT
ejpam-6326	453	1	+	+	CCONJ
ejpam-6326	453	2	1{x∈f0	1{x∈f0	NUM
ejpam-6326	453	3	}	}	PUNCT
ejpam-6326	453	4	≤	≤	NUM
ejpam-6326	453	5	(	(	PUNCT
ejpam-6326	453	6	degh(h)−	degh(h)−	PROPN
ejpam-6326	453	7	degpx	degpx	NOUN
ejpam-6326	453	8	(	(	PUNCT
ejpam-6326	453	9	h	h	NOUN
ejpam-6326	453	10	)	)	PUNCT
ejpam-6326	453	11	)	)	PUNCT
ejpam-6326	454	1	+	+	CCONJ
ejpam-6326	454	2	1{x/∈f0	1{x/∈f0	NUM
ejpam-6326	454	3	}	}	PUNCT
ejpam-6326	454	4	.	.	PUNCT
ejpam-6326	455	1	proof	proof	NOUN
ejpam-6326	455	2	.	.	PUNCT
ejpam-6326	456	1	assume	assume	VERB
ejpam-6326	456	2	that	that	SCONJ
ejpam-6326	456	3	f	f	PROPN
ejpam-6326	456	4	is	be	AUX
ejpam-6326	456	5	a	a	DET
ejpam-6326	456	6	friendly	friendly	ADJ
ejpam-6326	456	7	dominating	dominating	NOUN
ejpam-6326	456	8	set	set	NOUN
ejpam-6326	456	9	of	of	ADP
ejpam-6326	456	10	g	g	PROPN
ejpam-6326	456	11	◦	◦	NOUN
ejpam-6326	456	12	h.	h.	NOUN
ejpam-6326	456	13	let	let	VERB
ejpam-6326	456	14	x	x	SYM
ejpam-6326	456	15	∈	∈	PROPN
ejpam-6326	456	16	v	v	X
ejpam-6326	456	17	(	(	PUNCT
ejpam-6326	456	18	g	g	NOUN
ejpam-6326	456	19	)	)	PUNCT
ejpam-6326	456	20	.	.	PUNCT
ejpam-6326	457	1	if	if	SCONJ
ejpam-6326	457	2	x	x	SYM
ejpam-6326	457	3	∈	∈	PROPN
ejpam-6326	457	4	f0	f0	PROPN
ejpam-6326	457	5	,	,	PUNCT
ejpam-6326	457	6	then	then	ADV
ejpam-6326	457	7	we	we	PRON
ejpam-6326	457	8	are	be	AUX
ejpam-6326	457	9	done	do	VERB
ejpam-6326	457	10	.	.	PUNCT
ejpam-6326	458	1	otherwise	otherwise	ADV
ejpam-6326	458	2	x	x	X
ejpam-6326	458	3	/∈	/∈	PROPN
ejpam-6326	459	1	f	f	PROPN
ejpam-6326	459	2	,	,	PUNCT
ejpam-6326	459	3	so	so	ADV
ejpam-6326	459	4	x	x	PRON
ejpam-6326	459	5	must	must	AUX
ejpam-6326	459	6	be	be	AUX
ejpam-6326	459	7	dominated	dominate	VERB
ejpam-6326	459	8	by	by	ADP
ejpam-6326	459	9	a	a	DET
ejpam-6326	459	10	neighbor	neighbor	NOUN
ejpam-6326	459	11	belonging	belong	VERB
ejpam-6326	459	12	to	to	ADP
ejpam-6326	459	13	f	f	PROPN
ejpam-6326	459	14	.	.	PUNCT
ejpam-6326	460	1	its	its	PRON
ejpam-6326	460	2	neighbors	neighbor	NOUN
ejpam-6326	460	3	are	be	AUX
ejpam-6326	460	4	exactly	exactly	ADV
ejpam-6326	460	5	the	the	DET
ejpam-6326	460	6	vertices	vertex	NOUN
ejpam-6326	460	7	of	of	ADP
ejpam-6326	460	8	ng(x)∪hx	ng(x)∪hx	ADJ
ejpam-6326	460	9	;	;	PUNCT
ejpam-6326	460	10	hence	hence	ADV
ejpam-6326	460	11	either	either	CCONJ
ejpam-6326	460	12	ng(x)∩f0	ng(x)∩f0	PROPN
ejpam-6326	460	13	̸=	̸=	PROPN
ejpam-6326	460	14	∅	∅	NOUN
ejpam-6326	460	15	or	or	CCONJ
ejpam-6326	460	16	px	px	ADP
ejpam-6326	460	17	̸=	̸=	PROPN
ejpam-6326	460	18	∅	∅	NOUN
ejpam-6326	460	19	,	,	PUNCT
ejpam-6326	460	20	which	which	PRON
ejpam-6326	460	21	is	be	AUX
ejpam-6326	460	22	condition	condition	NOUN
ejpam-6326	460	23	(	(	PUNCT
ejpam-6326	460	24	i.a	i.a	PROPN
ejpam-6326	460	25	)	)	PUNCT
ejpam-6326	460	26	.	.	PUNCT
ejpam-6326	461	1	for	for	ADP
ejpam-6326	461	2	condition	condition	NOUN
ejpam-6326	461	3	(	(	PUNCT
ejpam-6326	461	4	i.b	i.b	PROPN
ejpam-6326	461	5	)	)	PUNCT
ejpam-6326	461	6	keep	keep	VERB
ejpam-6326	461	7	x	x	PROPN
ejpam-6326	461	8	/∈	/∈	PUNCT
ejpam-6326	461	9	f0	f0	PROPN
ejpam-6326	461	10	and	and	CCONJ
ejpam-6326	461	11	choose	choose	VERB
ejpam-6326	461	12	an	an	DET
ejpam-6326	461	13	arbitrary	arbitrary	ADJ
ejpam-6326	461	14	h	h	NOUN
ejpam-6326	461	15	∈	∈	PROPN
ejpam-6326	461	16	hx	hx	PROPN
ejpam-6326	461	17	\	\	PROPN
ejpam-6326	461	18	px	px	PROPN
ejpam-6326	461	19	.	.	PROPN
ejpam-6326	462	1	since	since	SCONJ
ejpam-6326	462	2	x	x	PROPN
ejpam-6326	462	3	/∈	/∈	PROPN
ejpam-6326	462	4	f	f	PROPN
ejpam-6326	462	5	,	,	PUNCT
ejpam-6326	462	6	the	the	DET
ejpam-6326	462	7	vertex	vertex	NOUN
ejpam-6326	462	8	h	h	NOUN
ejpam-6326	462	9	can	can	AUX
ejpam-6326	462	10	be	be	AUX
ejpam-6326	462	11	dominated	dominate	VERB
ejpam-6326	462	12	only	only	ADV
ejpam-6326	462	13	through	through	ADP
ejpam-6326	462	14	a	a	DET
ejpam-6326	462	15	neighbor	neighbor	NOUN
ejpam-6326	462	16	in	in	ADP
ejpam-6326	462	17	px	px	PROPN
ejpam-6326	462	18	,	,	PUNCT
ejpam-6326	462	19	so	so	ADV
ejpam-6326	462	20	nhx(h	nhx(h	PROPN
ejpam-6326	462	21	)	)	PUNCT
ejpam-6326	462	22	∩	∩	NOUN
ejpam-6326	462	23	px	px	ADP
ejpam-6326	462	24	̸=	̸=	PROPN
ejpam-6326	462	25	∅.	∅.	ADV
ejpam-6326	462	26	thus	thus	ADV
ejpam-6326	462	27	,	,	PUNCT
ejpam-6326	462	28	px	px	PROPN
ejpam-6326	462	29	dominates	dominate	VERB
ejpam-6326	462	30	hx	hx	PROPN
ejpam-6326	462	31	.	.	PUNCT
ejpam-6326	463	1	now	now	ADV
ejpam-6326	463	2	take	take	VERB
ejpam-6326	463	3	x	x	PUNCT
ejpam-6326	463	4	∈	∈	PROPN
ejpam-6326	463	5	v	v	NOUN
ejpam-6326	463	6	(	(	PUNCT
ejpam-6326	463	7	g	g	NOUN
ejpam-6326	463	8	)	)	PUNCT
ejpam-6326	463	9	\	\	PROPN
ejpam-6326	463	10	f0	f0	PROPN
ejpam-6326	463	11	.	.	PUNCT
ejpam-6326	464	1	the	the	DET
ejpam-6326	464	2	number	number	NOUN
ejpam-6326	464	3	of	of	ADP
ejpam-6326	464	4	neighbors	neighbor	NOUN
ejpam-6326	464	5	of	of	ADP
ejpam-6326	464	6	x	x	PRON
ejpam-6326	464	7	that	that	DET
ejpam-6326	464	8	lie	lie	VERB
ejpam-6326	464	9	in	in	ADP
ejpam-6326	464	10	f	f	PROPN
ejpam-6326	464	11	equals	equal	VERB
ejpam-6326	464	12	degf	degf	PROPN
ejpam-6326	464	13	(	(	PUNCT
ejpam-6326	464	14	x	x	NOUN
ejpam-6326	464	15	)	)	PUNCT
ejpam-6326	464	16	=	=	SYM
ejpam-6326	465	1	degf0	degf0	NOUN
ejpam-6326	465	2	(	(	PUNCT
ejpam-6326	465	3	x	x	X
ejpam-6326	465	4	)	)	PUNCT
ejpam-6326	466	1	+	+	CCONJ
ejpam-6326	466	2	|px|	|px|	VERB
ejpam-6326	466	3	,	,	PUNCT
ejpam-6326	466	4	while	while	SCONJ
ejpam-6326	466	5	the	the	DET
ejpam-6326	466	6	number	number	NOUN
ejpam-6326	466	7	that	that	PRON
ejpam-6326	466	8	lie	lie	VERB
ejpam-6326	466	9	outside	outside	ADP
ejpam-6326	466	10	f	f	PROPN
ejpam-6326	466	11	is	be	AUX
ejpam-6326	466	12	degv	degv	NOUN
ejpam-6326	466	13	(	(	PUNCT
ejpam-6326	466	14	g	g	NOUN
ejpam-6326	466	15	◦	◦	NOUN
ejpam-6326	466	16	h)\f	h)\f	ADJ
ejpam-6326	466	17	(	(	PUNCT
ejpam-6326	466	18	x	x	NOUN
ejpam-6326	466	19	)	)	PUNCT
ejpam-6326	466	20	=	=	SYM
ejpam-6326	466	21	(	(	PUNCT
ejpam-6326	466	22	|v	|v	X
ejpam-6326	466	23	(	(	PUNCT
ejpam-6326	466	24	h)|−|px|	h)|−|px|	PROPN
ejpam-6326	466	25	)	)	PUNCT
ejpam-6326	467	1	+	+	CCONJ
ejpam-6326	467	2	(	(	PUNCT
ejpam-6326	467	3	degg(x)−degf0	degg(x)−degf0	PROPN
ejpam-6326	467	4	(	(	PUNCT
ejpam-6326	467	5	x	x	NOUN
ejpam-6326	467	6	)	)	PUNCT
ejpam-6326	467	7	)	)	PUNCT
ejpam-6326	467	8	.	.	PUNCT
ejpam-6326	468	1	hence	hence	ADV
ejpam-6326	468	2	the	the	DET
ejpam-6326	468	3	inequality	inequality	NOUN
ejpam-6326	468	4	degf	degf	NOUN
ejpam-6326	468	5	(	(	PUNCT
ejpam-6326	468	6	x	x	NOUN
ejpam-6326	468	7	)	)	PUNCT
ejpam-6326	468	8	≤	≤	ADJ
ejpam-6326	468	9	degv	degv	NOUN
ejpam-6326	468	10	(	(	PUNCT
ejpam-6326	468	11	g	g	NOUN
ejpam-6326	468	12	◦	◦	NOUN
ejpam-6326	468	13	h)\f	h)\f	ADJ
ejpam-6326	468	14	(	(	PUNCT
ejpam-6326	468	15	x	x	NOUN
ejpam-6326	468	16	)	)	PUNCT
ejpam-6326	468	17	reproduces	reproduce	VERB
ejpam-6326	468	18	condition	condition	NOUN
ejpam-6326	468	19	(	(	PUNCT
ejpam-6326	468	20	ii	ii	NOUN
ejpam-6326	468	21	)	)	PUNCT
ejpam-6326	468	22	.	.	PUNCT
ejpam-6326	469	1	finally	finally	ADV
ejpam-6326	469	2	,	,	PUNCT
ejpam-6326	469	3	fix	fix	VERB
ejpam-6326	469	4	h	h	NOUN
ejpam-6326	469	5	∈	∈	PROPN
ejpam-6326	469	6	hx	hx	PROPN
ejpam-6326	469	7	\	\	PROPN
ejpam-6326	469	8	px	px	PROPN
ejpam-6326	469	9	and	and	CCONJ
ejpam-6326	469	10	write	write	VERB
ejpam-6326	469	11	a	a	DET
ejpam-6326	469	12	=	=	NOUN
ejpam-6326	469	13	degpx	degpx	NOUN
ejpam-6326	469	14	(	(	PUNCT
ejpam-6326	469	15	h	h	NOUN
ejpam-6326	469	16	)	)	PUNCT
ejpam-6326	469	17	.	.	PUNCT
ejpam-6326	470	1	then	then	ADV
ejpam-6326	470	2	,	,	PUNCT
ejpam-6326	470	3	degf	degf	PROPN
ejpam-6326	470	4	(	(	PUNCT
ejpam-6326	470	5	h	h	NOUN
ejpam-6326	470	6	)	)	PUNCT
ejpam-6326	470	7	=	=	SYM
ejpam-6326	470	8	a+	a+	PUNCT
ejpam-6326	470	9	1{x∈f0	1{x∈f0	NUM
ejpam-6326	470	10	}	}	PUNCT
ejpam-6326	470	11	,	,	PUNCT
ejpam-6326	470	12	degv	degv	NOUN
ejpam-6326	470	13	(	(	PUNCT
ejpam-6326	470	14	g	g	NOUN
ejpam-6326	470	15	◦	◦	NOUN
ejpam-6326	470	16	h)\f	h)\f	ADJ
ejpam-6326	470	17	(	(	PUNCT
ejpam-6326	470	18	h	h	NOUN
ejpam-6326	470	19	)	)	PUNCT
ejpam-6326	470	20	=	=	SYM
ejpam-6326	470	21	(	(	PUNCT
ejpam-6326	470	22	degh(h)−	degh(h)−	PROPN
ejpam-6326	470	23	a	a	PRON
ejpam-6326	470	24	)	)	PUNCT
ejpam-6326	470	25	+	+	CCONJ
ejpam-6326	470	26	1{x/∈f0	1{x/∈f0	NUM
ejpam-6326	470	27	}	}	PUNCT
ejpam-6326	470	28	,	,	PUNCT
ejpam-6326	470	29	so	so	CCONJ
ejpam-6326	470	30	the	the	DET
ejpam-6326	470	31	friendly	friendly	ADJ
ejpam-6326	470	32	inequality	inequality	NOUN
ejpam-6326	470	33	for	for	ADP
ejpam-6326	470	34	h	h	NOUN
ejpam-6326	470	35	is	be	AUX
ejpam-6326	470	36	exactly	exactly	ADV
ejpam-6326	470	37	condition	condition	NOUN
ejpam-6326	470	38	(	(	PUNCT
ejpam-6326	470	39	iii	iii	NOUN
ejpam-6326	470	40	)	)	PUNCT
ejpam-6326	470	41	.	.	PUNCT
ejpam-6326	471	1	conversely	conversely	ADV
ejpam-6326	471	2	,	,	PUNCT
ejpam-6326	471	3	suppose	suppose	VERB
ejpam-6326	471	4	that	that	SCONJ
ejpam-6326	471	5	conditions	condition	NOUN
ejpam-6326	471	6	(	(	PUNCT
ejpam-6326	471	7	i	i	NOUN
ejpam-6326	471	8	)	)	PUNCT
ejpam-6326	471	9	−	−	PROPN
ejpam-6326	471	10	(	(	PUNCT
ejpam-6326	471	11	iii	iii	NOUN
ejpam-6326	471	12	)	)	PUNCT
ejpam-6326	471	13	hold	hold	NOUN
ejpam-6326	471	14	.	.	PUNCT
ejpam-6326	472	1	for	for	ADP
ejpam-6326	472	2	every	every	DET
ejpam-6326	472	3	x	x	SYM
ejpam-6326	472	4	∈	∈	PROPN
ejpam-6326	472	5	v	v	NOUN
ejpam-6326	472	6	(	(	PUNCT
ejpam-6326	472	7	g	g	NOUN
ejpam-6326	472	8	)	)	PUNCT
ejpam-6326	472	9	,	,	PUNCT
ejpam-6326	472	10	by	by	ADP
ejpam-6326	472	11	(	(	PUNCT
ejpam-6326	472	12	i.a	i.a	PROPN
ejpam-6326	472	13	)	)	PUNCT
ejpam-6326	472	14	,	,	PUNCT
ejpam-6326	472	15	guarantees	guarantee	VERB
ejpam-6326	472	16	a	a	DET
ejpam-6326	472	17	neighbor	neighbor	NOUN
ejpam-6326	472	18	of	of	ADP
ejpam-6326	472	19	x	x	DET
ejpam-6326	472	20	that	that	PRON
ejpam-6326	472	21	lies	lie	VERB
ejpam-6326	472	22	in	in	ADP
ejpam-6326	472	23	f	f	PROPN
ejpam-6326	472	24	;	;	PUNCT
ejpam-6326	472	25	hence	hence	ADV
ejpam-6326	472	26	x	x	VERB
ejpam-6326	472	27	is	be	AUX
ejpam-6326	472	28	dominated	dominate	VERB
ejpam-6326	472	29	.	.	PUNCT
ejpam-6326	473	1	now	now	ADV
ejpam-6326	473	2	fix	fix	VERB
ejpam-6326	473	3	h	h	NOUN
ejpam-6326	473	4	∈	∈	PROPN
ejpam-6326	473	5	hx	hx	PROPN
ejpam-6326	473	6	.	.	PUNCT
ejpam-6326	474	1	if	if	SCONJ
ejpam-6326	474	2	x	x	SYM
ejpam-6326	474	3	∈	∈	PROPN
ejpam-6326	474	4	f0	f0	PROPN
ejpam-6326	474	5	,	,	PUNCT
ejpam-6326	474	6	the	the	DET
ejpam-6326	474	7	edge	edge	NOUN
ejpam-6326	474	8	xh	xh	PROPN
ejpam-6326	474	9	dominates	dominate	VERB
ejpam-6326	474	10	h.	h.	PROPN
ejpam-6326	475	1	if	if	SCONJ
ejpam-6326	475	2	x	x	PROPN
ejpam-6326	475	3	/∈	/∈	PUNCT
ejpam-6326	475	4	f0	f0	PROPN
ejpam-6326	475	5	,	,	PUNCT
ejpam-6326	475	6	by	by	ADP
ejpam-6326	475	7	(	(	PUNCT
ejpam-6326	475	8	i.b	i.b	PROPN
ejpam-6326	475	9	)	)	PUNCT
ejpam-6326	475	10	,	,	PUNCT
ejpam-6326	475	11	guarantees	guarantee	VERB
ejpam-6326	475	12	a	a	DET
ejpam-6326	475	13	neighbor	neighbor	NOUN
ejpam-6326	475	14	of	of	ADP
ejpam-6326	475	15	h	h	NOUN
ejpam-6326	475	16	in	in	ADP
ejpam-6326	475	17	px	px	PROPN
ejpam-6326	475	18	⊆	⊆	NUM
ejpam-6326	475	19	f	f	NOUN
ejpam-6326	475	20	.	.	PUNCT
ejpam-6326	476	1	thus	thus	ADV
ejpam-6326	476	2	every	every	DET
ejpam-6326	476	3	vertex	vertex	NOUN
ejpam-6326	476	4	of	of	ADP
ejpam-6326	476	5	hx	hx	PROPN
ejpam-6326	476	6	is	be	AUX
ejpam-6326	476	7	dominated	dominate	VERB
ejpam-6326	476	8	,	,	PUNCT
ejpam-6326	476	9	and	and	CCONJ
ejpam-6326	476	10	we	we	PRON
ejpam-6326	476	11	obtain	obtain	VERB
ejpam-6326	476	12	ng	ng	PROPN
ejpam-6326	476	13	◦	◦	NOUN
ejpam-6326	476	14	h	h	NOUN
ejpam-6326	477	1	[	[	X
ejpam-6326	477	2	f	f	X
ejpam-6326	477	3	]	]	X
ejpam-6326	477	4	=	=	SYM
ejpam-6326	477	5	v	v	X
ejpam-6326	477	6	(	(	PUNCT
ejpam-6326	477	7	g	g	PROPN
ejpam-6326	477	8	◦	◦	NOUN
ejpam-6326	477	9	h	h	NOUN
ejpam-6326	477	10	)	)	PUNCT
ejpam-6326	477	11	.	.	PUNCT
ejpam-6326	478	1	thus	thus	ADV
ejpam-6326	478	2	,	,	PUNCT
ejpam-6326	478	3	f	f	PROPN
ejpam-6326	478	4	is	be	AUX
ejpam-6326	478	5	a	a	DET
ejpam-6326	478	6	dominating	dominating	NOUN
ejpam-6326	478	7	set	set	NOUN
ejpam-6326	478	8	.	.	PUNCT
ejpam-6326	479	1	to	to	PART
ejpam-6326	479	2	show	show	VERB
ejpam-6326	479	3	that	that	SCONJ
ejpam-6326	479	4	f	f	PROPN
ejpam-6326	479	5	is	be	AUX
ejpam-6326	479	6	friendly	friendly	ADJ
ejpam-6326	479	7	.	.	PUNCT
ejpam-6326	480	1	let	let	VERB
ejpam-6326	480	2	v	v	X
ejpam-6326	480	3	/∈	/∈	PUNCT
ejpam-6326	481	1	f	f	PROPN
ejpam-6326	481	2	.	.	PUNCT
ejpam-6326	482	1	if	if	SCONJ
ejpam-6326	482	2	v	v	NOUN
ejpam-6326	482	3	=	=	PUNCT
ejpam-6326	482	4	x	x	SYM
ejpam-6326	482	5	∈	∈	PROPN
ejpam-6326	482	6	v	v	ADP
ejpam-6326	482	7	(	(	PUNCT
ejpam-6326	482	8	g)\f0	g)\f0	PROPN
ejpam-6326	482	9	,	,	PUNCT
ejpam-6326	482	10	by	by	ADP
ejpam-6326	482	11	(	(	PUNCT
ejpam-6326	482	12	ii	ii	NOUN
ejpam-6326	482	13	)	)	PUNCT
ejpam-6326	482	14	,	,	PUNCT
ejpam-6326	482	15	we	we	PRON
ejpam-6326	482	16	have	have	VERB
ejpam-6326	482	17	degf	degf	PROPN
ejpam-6326	482	18	(	(	PUNCT
ejpam-6326	482	19	v	v	NOUN
ejpam-6326	482	20	)	)	PUNCT
ejpam-6326	482	21	≤	≤	NOUN
ejpam-6326	482	22	degv	degv	NOUN
ejpam-6326	482	23	(	(	PUNCT
ejpam-6326	482	24	g	g	NOUN
ejpam-6326	482	25	◦	◦	NOUN
ejpam-6326	482	26	h)\f	h)\f	ADJ
ejpam-6326	482	27	(	(	PUNCT
ejpam-6326	482	28	v	v	NOUN
ejpam-6326	482	29	)	)	PUNCT
ejpam-6326	482	30	.	.	PUNCT
ejpam-6326	483	1	if	if	SCONJ
ejpam-6326	483	2	v	v	NOUN
ejpam-6326	483	3	=	=	SYM
ejpam-6326	483	4	h	h	NOUN
ejpam-6326	483	5	∈	∈	PROPN
ejpam-6326	483	6	hx	hx	PROPN
ejpam-6326	483	7	\px	\px	PROPN
ejpam-6326	483	8	,	,	PUNCT
ejpam-6326	483	9	by	by	ADP
ejpam-6326	483	10	(	(	PUNCT
ejpam-6326	483	11	iii	iii	NOUN
ejpam-6326	483	12	)	)	PUNCT
ejpam-6326	483	13	,	,	PUNCT
ejpam-6326	483	14	again	again	ADV
ejpam-6326	483	15	,	,	PUNCT
ejpam-6326	483	16	degf	degf	PROPN
ejpam-6326	483	17	(	(	PUNCT
ejpam-6326	483	18	v	v	NOUN
ejpam-6326	483	19	)	)	PUNCT
ejpam-6326	483	20	≤	≤	NOUN
ejpam-6326	483	21	degv	degv	NOUN
ejpam-6326	483	22	(	(	PUNCT
ejpam-6326	483	23	g	g	NOUN
ejpam-6326	483	24	◦	◦	NOUN
ejpam-6326	483	25	h)\f	h)\f	ADJ
ejpam-6326	483	26	(	(	PUNCT
ejpam-6326	483	27	v	v	NOUN
ejpam-6326	483	28	)	)	PUNCT
ejpam-6326	483	29	.	.	PUNCT
ejpam-6326	484	1	hence	hence	ADV
ejpam-6326	484	2	,	,	PUNCT
ejpam-6326	484	3	f	f	PROPN
ejpam-6326	484	4	is	be	AUX
ejpam-6326	484	5	friendly	friendly	ADJ
ejpam-6326	484	6	.	.	PUNCT
ejpam-6326	485	1	therefore	therefore	ADV
ejpam-6326	485	2	,	,	PUNCT
ejpam-6326	485	3	f	f	PROPN
ejpam-6326	485	4	is	be	AUX
ejpam-6326	485	5	a	a	DET
ejpam-6326	485	6	friendly	friendly	ADJ
ejpam-6326	485	7	dominating	dominating	NOUN
ejpam-6326	485	8	set	set	NOUN
ejpam-6326	485	9	of	of	ADP
ejpam-6326	485	10	g	g	PROPN
ejpam-6326	485	11	◦	◦	NOUN
ejpam-6326	485	12	h.	h.	NOUN
ejpam-6326	485	13	corollary	corollary	ADJ
ejpam-6326	485	14	7	7	PROPN
ejpam-6326	485	15	.	.	PUNCT
ejpam-6326	486	1	let	let	VERB
ejpam-6326	486	2	g	g	NOUN
ejpam-6326	486	3	and	and	CCONJ
ejpam-6326	486	4	h	h	NOUN
ejpam-6326	486	5	be	be	AUX
ejpam-6326	486	6	connected	connect	VERB
ejpam-6326	486	7	,	,	PUNCT
ejpam-6326	486	8	non	non	ADJ
ejpam-6326	486	9	-	-	ADJ
ejpam-6326	486	10	trivial	trivial	ADJ
ejpam-6326	486	11	graphs	graph	NOUN
ejpam-6326	486	12	and	and	CCONJ
ejpam-6326	486	13	let	let	VERB
ejpam-6326	486	14	g	g	NOUN
ejpam-6326	486	15	◦	◦	NOUN
ejpam-6326	486	16	h	h	NOUN
ejpam-6326	486	17	denote	denote	VERB
ejpam-6326	486	18	their	their	PRON
ejpam-6326	486	19	corona	corona	NOUN
ejpam-6326	486	20	.	.	PUNCT
ejpam-6326	487	1	then	then	ADV
ejpam-6326	487	2	γf	γf	INTJ
ejpam-6326	487	3	(	(	PUNCT
ejpam-6326	487	4	g	g	PROPN
ejpam-6326	487	5	◦	◦	NOUN
ejpam-6326	487	6	h	h	NOUN
ejpam-6326	487	7	)	)	PUNCT
ejpam-6326	487	8	=	=	SYM
ejpam-6326	487	9	|v	|v	PROPN
ejpam-6326	487	10	(	(	PUNCT
ejpam-6326	487	11	g)|	g)|	PROPN
ejpam-6326	487	12	.	.	PUNCT
ejpam-6326	487	13	i.	i.	PROPN
ejpam-6326	487	14	s.	s.	PROPN
ejpam-6326	487	15	cabahug	cabahug	PROPN
ejpam-6326	487	16	,	,	PUNCT
ejpam-6326	487	17	jr	jr	PROPN
ejpam-6326	487	18	.	.	PROPN
ejpam-6326	487	19	,	,	PUNCT
ejpam-6326	487	20	r.	r.	PROPN
ejpam-6326	487	21	g.	g.	PROPN
ejpam-6326	487	22	eballe	eballe	PROPN
ejpam-6326	487	23	,	,	PUNCT
ejpam-6326	487	24	r.	r.	PROPN
ejpam-6326	487	25	t.	t.	PROPN
ejpam-6326	487	26	fernandez	fernandez	PROPN
ejpam-6326	487	27	/	/	SYM
ejpam-6326	487	28	eur	eur	PROPN
ejpam-6326	487	29	.	.	PUNCT
ejpam-6326	488	1	j.	j.	PROPN
ejpam-6326	488	2	pure	pure	PROPN
ejpam-6326	488	3	appl	appl	PROPN
ejpam-6326	488	4	.	.	PROPN
ejpam-6326	488	5	math	math	PROPN
ejpam-6326	488	6	,	,	PUNCT
ejpam-6326	488	7	18	18	NUM
ejpam-6326	488	8	(	(	PUNCT
ejpam-6326	488	9	4	4	NUM
ejpam-6326	488	10	)	)	PUNCT
ejpam-6326	488	11	(	(	PUNCT
ejpam-6326	488	12	2025	2025	NUM
ejpam-6326	488	13	)	)	PUNCT
ejpam-6326	488	14	,	,	PUNCT
ejpam-6326	488	15	6326	6326	NUM
ejpam-6326	488	16	14	14	NUM
ejpam-6326	488	17	of	of	ADP
ejpam-6326	488	18	15	15	NUM
ejpam-6326	488	19	proof	proof	NOUN
ejpam-6326	488	20	.	.	PUNCT
ejpam-6326	489	1	theorem	theorem	VERB
ejpam-6326	489	2	13	13	NUM
ejpam-6326	489	3	shows	show	VERB
ejpam-6326	489	4	that	that	SCONJ
ejpam-6326	489	5	the	the	DET
ejpam-6326	489	6	set	set	NOUN
ejpam-6326	489	7	f	f	PROPN
ejpam-6326	489	8	=	=	SYM
ejpam-6326	489	9	v	v	PROPN
ejpam-6326	489	10	(	(	PUNCT
ejpam-6326	489	11	g	g	NOUN
ejpam-6326	489	12	)	)	PUNCT
ejpam-6326	489	13	is	be	AUX
ejpam-6326	489	14	a	a	DET
ejpam-6326	489	15	friendly	friendly	ADJ
ejpam-6326	489	16	dominating	dominating	NOUN
ejpam-6326	489	17	set	set	NOUN
ejpam-6326	489	18	of	of	ADP
ejpam-6326	489	19	g	g	PROPN
ejpam-6326	489	20	◦	◦	NOUN
ejpam-6326	489	21	h	h	NOUN
ejpam-6326	489	22	,	,	PUNCT
ejpam-6326	490	1	so	so	ADV
ejpam-6326	490	2	γf	γf	PROPN
ejpam-6326	490	3	(	(	PUNCT
ejpam-6326	490	4	g	g	PROPN
ejpam-6326	490	5	◦	◦	NOUN
ejpam-6326	490	6	h	h	NOUN
ejpam-6326	490	7	)	)	PUNCT
ejpam-6326	490	8	≤	≤	NOUN
ejpam-6326	490	9	|v	|v	X
ejpam-6326	490	10	(	(	PUNCT
ejpam-6326	490	11	g)|	g)|	NOUN
ejpam-6326	490	12	.	.	PUNCT
ejpam-6326	491	1	now	now	ADV
ejpam-6326	491	2	,	,	PUNCT
ejpam-6326	491	3	theorem	theorem	VERB
ejpam-6326	491	4	14	14	NUM
ejpam-6326	491	5	and	and	CCONJ
ejpam-6326	491	6	theorem	theorem	VERB
ejpam-6326	491	7	15	15	NUM
ejpam-6326	491	8	together	together	ADV
ejpam-6326	491	9	implies	imply	VERB
ejpam-6326	491	10	that	that	SCONJ
ejpam-6326	491	11	every	every	DET
ejpam-6326	491	12	friendly	friendly	ADJ
ejpam-6326	491	13	dominating	dominating	NOUN
ejpam-6326	491	14	set	set	NOUN
ejpam-6326	491	15	which	which	PRON
ejpam-6326	491	16	omits	omit	VERB
ejpam-6326	491	17	at	at	ADV
ejpam-6326	491	18	least	least	ADV
ejpam-6326	491	19	one	one	NUM
ejpam-6326	491	20	vertex	vertex	NOUN
ejpam-6326	491	21	of	of	ADP
ejpam-6326	491	22	v	v	NOUN
ejpam-6326	491	23	(	(	PUNCT
ejpam-6326	491	24	g	g	NOUN
ejpam-6326	491	25	)	)	PUNCT
ejpam-6326	491	26	must	must	AUX
ejpam-6326	491	27	contain	contain	VERB
ejpam-6326	491	28	strictly	strictly	ADV
ejpam-6326	491	29	more	more	ADJ
ejpam-6326	491	30	than	than	ADP
ejpam-6326	491	31	|v	|v	PROPN
ejpam-6326	491	32	(	(	PUNCT
ejpam-6326	491	33	g)|	g)|	NOUN
ejpam-6326	491	34	vertices	vertex	NOUN
ejpam-6326	491	35	.	.	PUNCT
ejpam-6326	492	1	hence	hence	ADV
ejpam-6326	492	2	no	no	DET
ejpam-6326	492	3	friendly	friendly	ADJ
ejpam-6326	492	4	dominating	dominating	NOUN
ejpam-6326	492	5	set	set	NOUN
ejpam-6326	492	6	of	of	ADP
ejpam-6326	492	7	smaller	small	ADJ
ejpam-6326	492	8	size	size	NOUN
ejpam-6326	492	9	than	than	ADP
ejpam-6326	492	10	|v	|v	PROPN
ejpam-6326	492	11	(	(	PUNCT
ejpam-6326	492	12	g)|	g)|	NOUN
ejpam-6326	492	13	exists	exist	VERB
ejpam-6326	492	14	,	,	PUNCT
ejpam-6326	492	15	so	so	ADV
ejpam-6326	492	16	γf	γf	PROPN
ejpam-6326	492	17	(	(	PUNCT
ejpam-6326	492	18	g	g	PROPN
ejpam-6326	492	19	◦	◦	NOUN
ejpam-6326	492	20	h	h	NOUN
ejpam-6326	492	21	)	)	PUNCT
ejpam-6326	492	22	≥	≥	NOUN
ejpam-6326	492	23	|v	|v	PROPN
ejpam-6326	492	24	(	(	PUNCT
ejpam-6326	492	25	g)|	g)|	PROPN
ejpam-6326	492	26	.	.	PUNCT
ejpam-6326	493	1	therefore	therefore	ADV
ejpam-6326	493	2	,	,	PUNCT
ejpam-6326	493	3	γf	γf	INTJ
ejpam-6326	493	4	(	(	PUNCT
ejpam-6326	493	5	g	g	PROPN
ejpam-6326	493	6	◦	◦	NOUN
ejpam-6326	493	7	h	h	NOUN
ejpam-6326	493	8	)	)	PUNCT
ejpam-6326	493	9	=	=	SYM
ejpam-6326	493	10	|v	|v	PROPN
ejpam-6326	493	11	(	(	PUNCT
ejpam-6326	493	12	g)|	g)|	NOUN
ejpam-6326	493	13	.	.	PUNCT
ejpam-6326	494	1	4	4	X
ejpam-6326	494	2	.	.	X
ejpam-6326	494	3	conclusion	conclusion	NOUN
ejpam-6326	494	4	this	this	DET
ejpam-6326	494	5	paper	paper	NOUN
ejpam-6326	494	6	demonstrated	demonstrate	VERB
ejpam-6326	494	7	that	that	SCONJ
ejpam-6326	494	8	the	the	DET
ejpam-6326	494	9	friendly	friendly	ADJ
ejpam-6326	494	10	domination	domination	NOUN
ejpam-6326	494	11	number	number	NOUN
ejpam-6326	494	12	γf	γf	PROPN
ejpam-6326	494	13	,	,	PUNCT
ejpam-6326	494	14	the	the	DET
ejpam-6326	494	15	smallest	small	ADJ
ejpam-6326	494	16	size	size	NOUN
ejpam-6326	494	17	of	of	ADP
ejpam-6326	494	18	a	a	DET
ejpam-6326	494	19	dominating	dominating	NOUN
ejpam-6326	494	20	set	set	NOUN
ejpam-6326	494	21	that	that	PRON
ejpam-6326	494	22	is	be	AUX
ejpam-6326	494	23	also	also	ADV
ejpam-6326	494	24	friendly	friendly	ADJ
ejpam-6326	494	25	—	—	PUNCT
ejpam-6326	494	26	always	always	ADV
ejpam-6326	494	27	lies	lie	VERB
ejpam-6326	494	28	above	above	ADP
ejpam-6326	494	29	or	or	CCONJ
ejpam-6326	494	30	on	on	ADP
ejpam-6326	494	31	the	the	DET
ejpam-6326	494	32	ordinary	ordinary	ADJ
ejpam-6326	494	33	domination	domination	NOUN
ejpam-6326	494	34	number	number	NOUN
ejpam-6326	494	35	γ	γ	PROPN
ejpam-6326	494	36	.	.	PUNCT
ejpam-6326	495	1	we	we	PRON
ejpam-6326	495	2	gave	give	VERB
ejpam-6326	495	3	complete	complete	ADJ
ejpam-6326	495	4	characterizations	characterization	NOUN
ejpam-6326	495	5	of	of	ADP
ejpam-6326	495	6	graphs	graph	NOUN
ejpam-6326	495	7	with	with	ADP
ejpam-6326	495	8	γf	γf	PROPN
ejpam-6326	495	9	=	=	SYM
ejpam-6326	495	10	1	1	NUM
ejpam-6326	495	11	or	or	CCONJ
ejpam-6326	495	12	2	2	NUM
ejpam-6326	495	13	,	,	PUNCT
ejpam-6326	495	14	construct	construct	VERB
ejpam-6326	495	15	families	family	NOUN
ejpam-6326	495	16	in	in	ADP
ejpam-6326	495	17	which	which	PRON
ejpam-6326	495	18	the	the	DET
ejpam-6326	495	19	gap	gap	NOUN
ejpam-6326	495	20	γf	γf	ADJ
ejpam-6326	495	21	−	−	NOUN
ejpam-6326	495	22	γ	γ	PROPN
ejpam-6326	495	23	is	be	AUX
ejpam-6326	495	24	arbitrarily	arbitrarily	ADV
ejpam-6326	495	25	large	large	ADJ
ejpam-6326	495	26	,	,	PUNCT
ejpam-6326	495	27	and	and	CCONJ
ejpam-6326	495	28	derive	derive	VERB
ejpam-6326	495	29	exact	exact	ADJ
ejpam-6326	495	30	expression	expression	NOUN
ejpam-6326	495	31	for	for	ADP
ejpam-6326	495	32	paths	path	NOUN
ejpam-6326	495	33	and	and	CCONJ
ejpam-6326	495	34	cycles	cycle	NOUN
ejpam-6326	495	35	.	.	PUNCT
ejpam-6326	496	1	moreover	moreover	ADV
ejpam-6326	496	2	,	,	PUNCT
ejpam-6326	496	3	we	we	PRON
ejpam-6326	496	4	established	establish	VERB
ejpam-6326	496	5	sharp	sharp	ADJ
ejpam-6326	496	6	nordhaus	nordhaus	NOUN
ejpam-6326	496	7	–	–	PUNCT
ejpam-6326	496	8	gaddum	gaddum	NOUN
ejpam-6326	496	9	bounds	bound	VERB
ejpam-6326	496	10	for	for	ADP
ejpam-6326	496	11	the	the	DET
ejpam-6326	496	12	difference	difference	NOUN
ejpam-6326	496	13	,	,	PUNCT
ejpam-6326	496	14	the	the	DET
ejpam-6326	496	15	sum	sum	NOUN
ejpam-6326	496	16	,	,	PUNCT
ejpam-6326	496	17	and	and	CCONJ
ejpam-6326	496	18	the	the	DET
ejpam-6326	496	19	product	product	NOUN
ejpam-6326	496	20	of	of	ADP
ejpam-6326	496	21	γf	γf	PRON
ejpam-6326	496	22	of	of	ADP
ejpam-6326	496	23	a	a	DET
ejpam-6326	496	24	graph	graph	NOUN
ejpam-6326	496	25	and	and	CCONJ
ejpam-6326	496	26	its	its	PRON
ejpam-6326	496	27	complement	complement	NOUN
ejpam-6326	496	28	,	,	PUNCT
ejpam-6326	496	29	and	and	CCONJ
ejpam-6326	496	30	we	we	PRON
ejpam-6326	496	31	identified	identify	VERB
ejpam-6326	496	32	all	all	DET
ejpam-6326	496	33	friendly	friendly	ADJ
ejpam-6326	496	34	dominating	dominating	NOUN
ejpam-6326	496	35	sets	set	NOUN
ejpam-6326	496	36	in	in	ADP
ejpam-6326	496	37	the	the	DET
ejpam-6326	496	38	join	join	NOUN
ejpam-6326	496	39	and	and	CCONJ
ejpam-6326	496	40	the	the	DET
ejpam-6326	496	41	corona	corona	NOUN
ejpam-6326	496	42	of	of	ADP
ejpam-6326	496	43	two	two	NUM
ejpam-6326	496	44	graphs	graph	NOUN
ejpam-6326	496	45	,	,	PUNCT
ejpam-6326	496	46	proving	prove	VERB
ejpam-6326	496	47	that	that	SCONJ
ejpam-6326	496	48	in	in	ADP
ejpam-6326	496	49	every	every	DET
ejpam-6326	496	50	corona	corona	NOUN
ejpam-6326	496	51	γf	γf	NOUN
ejpam-6326	496	52	equals	equal	VERB
ejpam-6326	496	53	the	the	DET
ejpam-6326	496	54	order	order	NOUN
ejpam-6326	496	55	of	of	ADP
ejpam-6326	496	56	the	the	DET
ejpam-6326	496	57	host	host	NOUN
ejpam-6326	496	58	graph	graph	NOUN
ejpam-6326	496	59	.	.	PUNCT
ejpam-6326	497	1	the	the	DET
ejpam-6326	497	2	present	present	ADJ
ejpam-6326	497	3	work	work	NOUN
ejpam-6326	497	4	is	be	AUX
ejpam-6326	497	5	restricted	restrict	VERB
ejpam-6326	497	6	to	to	ADP
ejpam-6326	497	7	simple	simple	ADJ
ejpam-6326	497	8	,	,	PUNCT
ejpam-6326	497	9	undirected	undirected	ADJ
ejpam-6326	497	10	graphs	graph	NOUN
ejpam-6326	497	11	and	and	CCONJ
ejpam-6326	497	12	to	to	ADP
ejpam-6326	497	13	two	two	NUM
ejpam-6326	497	14	basic	basic	ADJ
ejpam-6326	497	15	graph	graph	NOUN
ejpam-6326	497	16	operations	operation	NOUN
ejpam-6326	497	17	.	.	PUNCT
ejpam-6326	498	1	it	it	PRON
ejpam-6326	498	2	does	do	AUX
ejpam-6326	498	3	not	not	PART
ejpam-6326	498	4	treat	treat	VERB
ejpam-6326	498	5	weighted	weight	VERB
ejpam-6326	498	6	or	or	CCONJ
ejpam-6326	498	7	directed	direct	VERB
ejpam-6326	498	8	variants	variant	NOUN
ejpam-6326	498	9	,	,	PUNCT
ejpam-6326	498	10	computational	computational	ADJ
ejpam-6326	498	11	complexity	complexity	NOUN
ejpam-6326	498	12	,	,	PUNCT
ejpam-6326	498	13	or	or	CCONJ
ejpam-6326	498	14	behaviour	behaviour	NOUN
ejpam-6326	498	15	under	under	ADP
ejpam-6326	498	16	richer	rich	ADJ
ejpam-6326	498	17	products	product	NOUN
ejpam-6326	498	18	such	such	ADJ
ejpam-6326	498	19	as	as	ADP
ejpam-6326	498	20	cartesian	cartesian	ADJ
ejpam-6326	498	21	or	or	CCONJ
ejpam-6326	498	22	strong	strong	ADJ
ejpam-6326	498	23	products	product	NOUN
ejpam-6326	498	24	.	.	PUNCT
ejpam-6326	499	1	future	future	ADJ
ejpam-6326	499	2	research	research	NOUN
ejpam-6326	499	3	should	should	AUX
ejpam-6326	499	4	therefore	therefore	ADV
ejpam-6326	499	5	address	address	VERB
ejpam-6326	499	6	several	several	ADJ
ejpam-6326	499	7	open	open	ADJ
ejpam-6326	499	8	directions	direction	NOUN
ejpam-6326	499	9	:	:	PUNCT
ejpam-6326	499	10	determining	determine	VERB
ejpam-6326	499	11	γf	γf	NOUN
ejpam-6326	499	12	for	for	ADP
ejpam-6326	499	13	additional	additional	ADJ
ejpam-6326	499	14	graph	graph	NOUN
ejpam-6326	499	15	families	family	NOUN
ejpam-6326	499	16	and	and	CCONJ
ejpam-6326	499	17	products	product	NOUN
ejpam-6326	499	18	;	;	PUNCT
ejpam-6326	499	19	analysing	analyse	VERB
ejpam-6326	499	20	the	the	DET
ejpam-6326	499	21	complexity	complexity	NOUN
ejpam-6326	499	22	of	of	ADP
ejpam-6326	499	23	computing	compute	VERB
ejpam-6326	499	24	γf	γf	PROPN
ejpam-6326	499	25	and	and	CCONJ
ejpam-6326	499	26	designing	design	VERB
ejpam-6326	499	27	efficient	efficient	ADJ
ejpam-6326	499	28	approximation	approximation	NOUN
ejpam-6326	499	29	schemes	scheme	NOUN
ejpam-6326	499	30	;	;	PUNCT
ejpam-6326	499	31	extending	extend	VERB
ejpam-6326	499	32	the	the	DET
ejpam-6326	499	33	concept	concept	NOUN
ejpam-6326	499	34	to	to	PART
ejpam-6326	499	35	weighted	weighted	VERB
ejpam-6326	499	36	,	,	PUNCT
ejpam-6326	499	37	directed	direct	VERB
ejpam-6326	499	38	,	,	PUNCT
ejpam-6326	499	39	or	or	CCONJ
ejpam-6326	499	40	temporal	temporal	ADJ
ejpam-6326	499	41	graphs	graph	NOUN
ejpam-6326	499	42	;	;	PUNCT
ejpam-6326	499	43	and	and	CCONJ
ejpam-6326	499	44	studying	study	VERB
ejpam-6326	499	45	probabilistic	probabilistic	ADJ
ejpam-6326	499	46	thresholds	threshold	NOUN
ejpam-6326	499	47	for	for	ADP
ejpam-6326	499	48	friendly	friendly	ADJ
ejpam-6326	499	49	domination	domination	NOUN
ejpam-6326	499	50	in	in	ADP
ejpam-6326	499	51	random	random	ADJ
ejpam-6326	499	52	graph	graph	NOUN
ejpam-6326	499	53	models	model	NOUN
ejpam-6326	499	54	.	.	PUNCT
ejpam-6326	500	1	progress	progress	NOUN
ejpam-6326	500	2	on	on	ADP
ejpam-6326	500	3	these	these	DET
ejpam-6326	500	4	questions	question	NOUN
ejpam-6326	500	5	will	will	AUX
ejpam-6326	500	6	deepen	deepen	VERB
ejpam-6326	500	7	our	our	PRON
ejpam-6326	500	8	understanding	understanding	NOUN
ejpam-6326	500	9	of	of	ADP
ejpam-6326	500	10	how	how	SCONJ
ejpam-6326	500	11	equitable	equitable	ADJ
ejpam-6326	500	12	influence	influence	NOUN
ejpam-6326	500	13	can	can	AUX
ejpam-6326	500	14	be	be	AUX
ejpam-6326	500	15	maintained	maintain	VERB
ejpam-6326	500	16	in	in	ADP
ejpam-6326	500	17	increasingly	increasingly	ADV
ejpam-6326	500	18	complex	complex	ADJ
ejpam-6326	500	19	networks	network	NOUN
ejpam-6326	500	20	.	.	PUNCT
ejpam-6326	501	1	acknowledgements	acknowledgement	NOUN
ejpam-6326	501	2	the	the	DET
ejpam-6326	501	3	authors	author	NOUN
ejpam-6326	501	4	would	would	AUX
ejpam-6326	501	5	like	like	VERB
ejpam-6326	501	6	to	to	PART
ejpam-6326	501	7	thank	thank	VERB
ejpam-6326	501	8	central	central	PROPN
ejpam-6326	501	9	mindanao	mindanao	PROPN
ejpam-6326	501	10	university	university	PROPN
ejpam-6326	501	11	for	for	ADP
ejpam-6326	501	12	providing	provide	VERB
ejpam-6326	501	13	financial	financial	ADJ
ejpam-6326	501	14	support	support	NOUN
ejpam-6326	501	15	for	for	ADP
ejpam-6326	501	16	this	this	DET
ejpam-6326	501	17	research	research	NOUN
ejpam-6326	501	18	.	.	PUNCT
ejpam-6326	502	1	they	they	PRON
ejpam-6326	502	2	also	also	ADV
ejpam-6326	502	3	express	express	VERB
ejpam-6326	502	4	their	their	PRON
ejpam-6326	502	5	sincere	sincere	ADJ
ejpam-6326	502	6	gratitude	gratitude	NOUN
ejpam-6326	502	7	to	to	ADP
ejpam-6326	502	8	the	the	DET
ejpam-6326	502	9	referee	referee	NOUN
ejpam-6326	502	10	for	for	ADP
ejpam-6326	502	11	the	the	DET
ejpam-6326	502	12	helpful	helpful	ADJ
ejpam-6326	502	13	comments	comment	NOUN
ejpam-6326	502	14	and	and	CCONJ
ejpam-6326	502	15	valuable	valuable	ADJ
ejpam-6326	502	16	suggestions	suggestion	NOUN
ejpam-6326	502	17	that	that	PRON
ejpam-6326	502	18	greatly	greatly	ADV
ejpam-6326	502	19	improved	improve	VERB
ejpam-6326	502	20	the	the	DET
ejpam-6326	502	21	quality	quality	NOUN
ejpam-6326	502	22	of	of	ADP
ejpam-6326	502	23	this	this	DET
ejpam-6326	502	24	paper	paper	NOUN
ejpam-6326	502	25	.	.	PUNCT
ejpam-6326	503	1	references	reference	NOUN
ejpam-6326	503	2	[	[	X
ejpam-6326	503	3	1	1	X
ejpam-6326	503	4	]	]	PUNCT
ejpam-6326	503	5	j.	j.	PROPN
ejpam-6326	503	6	a.	a.	PROPN
ejpam-6326	503	7	bondy	bondy	PROPN
ejpam-6326	503	8	and	and	CCONJ
ejpam-6326	503	9	u.	u.	PROPN
ejpam-6326	503	10	s.	s.	PROPN
ejpam-6326	503	11	r.	r.	PROPN
ejpam-6326	503	12	murty	murty	PROPN
ejpam-6326	503	13	.	.	PUNCT
ejpam-6326	504	1	graph	graph	NOUN
ejpam-6326	504	2	theory	theory	NOUN
ejpam-6326	504	3	.	.	PUNCT
ejpam-6326	505	1	springer	springer	PROPN
ejpam-6326	505	2	,	,	PUNCT
ejpam-6326	505	3	berlin	berlin	PROPN
ejpam-6326	505	4	,	,	PUNCT
ejpam-6326	505	5	germany	germany	PROPN
ejpam-6326	505	6	,	,	PUNCT
ejpam-6326	505	7	2008	2008	NUM
ejpam-6326	505	8	.	.	PUNCT
ejpam-6326	506	1	[	[	X
ejpam-6326	506	2	2	2	X
ejpam-6326	506	3	]	]	PUNCT
ejpam-6326	506	4	d.	d.	PROPN
ejpam-6326	506	5	b.	b.	PROPN
ejpam-6326	506	6	west	west	PROPN
ejpam-6326	506	7	.	.	PUNCT
ejpam-6326	507	1	introduction	introduction	NOUN
ejpam-6326	507	2	to	to	AUX
ejpam-6326	507	3	graph	graph	NOUN
ejpam-6326	507	4	theory	theory	NOUN
ejpam-6326	507	5	.	.	PUNCT
ejpam-6326	508	1	prentice	prentice	PROPN
ejpam-6326	508	2	hall	hall	PROPN
ejpam-6326	508	3	,	,	PUNCT
ejpam-6326	508	4	upper	upper	ADJ
ejpam-6326	508	5	saddle	saddle	NOUN
ejpam-6326	508	6	river	river	PROPN
ejpam-6326	508	7	,	,	PUNCT
ejpam-6326	508	8	nj	nj	PROPN
ejpam-6326	508	9	,	,	PUNCT
ejpam-6326	508	10	2001	2001	NUM
ejpam-6326	508	11	.	.	PUNCT
ejpam-6326	509	1	[	[	X
ejpam-6326	509	2	3	3	X
ejpam-6326	509	3	]	]	X
ejpam-6326	509	4	m.	m.	NOUN
ejpam-6326	509	5	a.	a.	PROPN
ejpam-6326	509	6	henning	henning	PROPN
ejpam-6326	509	7	and	and	CCONJ
ejpam-6326	509	8	a.	a.	PROPN
ejpam-6326	509	9	yeo	yeo	PROPN
ejpam-6326	509	10	.	.	PROPN
ejpam-6326	510	1	total	total	ADJ
ejpam-6326	510	2	domination	domination	NOUN
ejpam-6326	510	3	in	in	ADP
ejpam-6326	510	4	graphs	graph	NOUN
ejpam-6326	510	5	.	.	PUNCT
ejpam-6326	511	1	springer	springer	NOUN
ejpam-6326	511	2	,	,	PUNCT
ejpam-6326	511	3	new	new	PROPN
ejpam-6326	511	4	york	york	PROPN
ejpam-6326	511	5	,	,	PUNCT
ejpam-6326	511	6	usa	usa	PROPN
ejpam-6326	511	7	,	,	PUNCT
ejpam-6326	511	8	2013	2013	NUM
ejpam-6326	511	9	.	.	PUNCT
ejpam-6326	512	1	[	[	X
ejpam-6326	512	2	4	4	X
ejpam-6326	512	3	]	]	PUNCT
ejpam-6326	512	4	t.	t.	PROPN
ejpam-6326	512	5	w.	w.	PROPN
ejpam-6326	512	6	haynes	haynes	PROPN
ejpam-6326	512	7	,	,	PUNCT
ejpam-6326	512	8	s.	s.	PROPN
ejpam-6326	512	9	t.	t.	PROPN
ejpam-6326	512	10	hedetniemi	hedetniemi	PROPN
ejpam-6326	512	11	,	,	PUNCT
ejpam-6326	512	12	and	and	CCONJ
ejpam-6326	512	13	m.	m.	PROPN
ejpam-6326	512	14	a.	a.	PROPN
ejpam-6326	512	15	henning	henning	PROPN
ejpam-6326	512	16	.	.	PUNCT
ejpam-6326	513	1	structures	structure	NOUN
ejpam-6326	513	2	of	of	ADP
ejpam-6326	513	3	domination	domination	NOUN
ejpam-6326	513	4	in	in	ADP
ejpam-6326	513	5	graphs	graph	NOUN
ejpam-6326	513	6	.	.	PUNCT
ejpam-6326	514	1	springer	springer	NOUN
ejpam-6326	514	2	,	,	PUNCT
ejpam-6326	514	3	cham	cham	PROPN
ejpam-6326	514	4	,	,	PUNCT
ejpam-6326	514	5	switzerland	switzerland	PROPN
ejpam-6326	514	6	,	,	PUNCT
ejpam-6326	514	7	2021	2021	NUM
ejpam-6326	514	8	.	.	PUNCT
ejpam-6326	515	1	[	[	X
ejpam-6326	515	2	5	5	X
ejpam-6326	515	3	]	]	PUNCT
ejpam-6326	515	4	j.	j.	PROPN
ejpam-6326	515	5	m.	m.	PROPN
ejpam-6326	515	6	sigarreta	sigarreta	PROPN
ejpam-6326	515	7	and	and	CCONJ
ejpam-6326	515	8	j.	j.	PROPN
ejpam-6326	515	9	a.	a.	PROPN
ejpam-6326	515	10	rodŕıguez	rodŕıguez	PROPN
ejpam-6326	515	11	-	-	NOUN
ejpam-6326	515	12	velázquez	velázquez	NOUN
ejpam-6326	515	13	.	.	PUNCT
ejpam-6326	516	1	on	on	ADP
ejpam-6326	516	2	the	the	DET
ejpam-6326	516	3	global	global	ADJ
ejpam-6326	516	4	offensive	offensive	PROPN
ejpam-6326	516	5	alliance	alliance	NOUN
ejpam-6326	516	6	number	number	NOUN
ejpam-6326	516	7	of	of	ADP
ejpam-6326	516	8	a	a	DET
ejpam-6326	516	9	graph	graph	NOUN
ejpam-6326	516	10	.	.	PUNCT
ejpam-6326	517	1	discrete	discrete	ADJ
ejpam-6326	517	2	applied	apply	VERB
ejpam-6326	517	3	mathematics	mathematic	NOUN
ejpam-6326	517	4	,	,	PUNCT
ejpam-6326	517	5	157(2):219–226	157(2):219–226	NUM
ejpam-6326	517	6	,	,	PUNCT
ejpam-6326	517	7	2009	2009	NUM
ejpam-6326	517	8	.	.	PUNCT
ejpam-6326	518	1	i.	i.	PROPN
ejpam-6326	518	2	s.	s.	PROPN
ejpam-6326	518	3	cabahug	cabahug	PROPN
ejpam-6326	518	4	,	,	PUNCT
ejpam-6326	518	5	jr	jr	PROPN
ejpam-6326	518	6	.	.	PROPN
ejpam-6326	518	7	,	,	PUNCT
ejpam-6326	518	8	r.	r.	PROPN
ejpam-6326	518	9	g.	g.	PROPN
ejpam-6326	518	10	eballe	eballe	PROPN
ejpam-6326	518	11	,	,	PUNCT
ejpam-6326	518	12	r.	r.	PROPN
ejpam-6326	518	13	t.	t.	PROPN
ejpam-6326	518	14	fernandez	fernandez	PROPN
ejpam-6326	518	15	/	/	SYM
ejpam-6326	518	16	eur	eur	PROPN
ejpam-6326	518	17	.	.	PUNCT
ejpam-6326	519	1	j.	j.	PROPN
ejpam-6326	519	2	pure	pure	PROPN
ejpam-6326	519	3	appl	appl	PROPN
ejpam-6326	519	4	.	.	PROPN
ejpam-6326	519	5	math	math	PROPN
ejpam-6326	519	6	,	,	PUNCT
ejpam-6326	519	7	18	18	NUM
ejpam-6326	519	8	(	(	PUNCT
ejpam-6326	519	9	4	4	NUM
ejpam-6326	519	10	)	)	PUNCT
ejpam-6326	519	11	(	(	PUNCT
ejpam-6326	519	12	2025	2025	NUM
ejpam-6326	519	13	)	)	PUNCT
ejpam-6326	519	14	,	,	PUNCT
ejpam-6326	519	15	6326	6326	NUM
ejpam-6326	519	16	15	15	NUM
ejpam-6326	519	17	of	of	ADP
ejpam-6326	519	18	15	15	NUM
ejpam-6326	519	19	[	[	SYM
ejpam-6326	519	20	6	6	NUM
ejpam-6326	519	21	]	]	PUNCT
ejpam-6326	519	22	o.	o.	PROPN
ejpam-6326	519	23	favaron	favaron	PROPN
ejpam-6326	519	24	,	,	PUNCT
ejpam-6326	519	25	g.	g.	PROPN
ejpam-6326	519	26	fricke	fricke	PROPN
ejpam-6326	519	27	,	,	PUNCT
ejpam-6326	519	28	w.	w.	PROPN
ejpam-6326	519	29	goddard	goddard	PROPN
ejpam-6326	519	30	,	,	PUNCT
ejpam-6326	519	31	s.	s.	PROPN
ejpam-6326	519	32	m.	m.	PROPN
ejpam-6326	519	33	hedetniemi	hedetniemi	PROPN
ejpam-6326	519	34	,	,	PUNCT
ejpam-6326	519	35	s.	s.	PROPN
ejpam-6326	519	36	t.	t.	PROPN
ejpam-6326	519	37	hedetniemi	hedetniemi	PROPN
ejpam-6326	519	38	,	,	PUNCT
ejpam-6326	519	39	p.	p.	PROPN
ejpam-6326	519	40	kristiansen	kristiansen	PROPN
ejpam-6326	519	41	,	,	PUNCT
ejpam-6326	519	42	r.	r.	PROPN
ejpam-6326	519	43	c.	c.	PROPN
ejpam-6326	519	44	laskar	laskar	PROPN
ejpam-6326	519	45	,	,	PUNCT
ejpam-6326	519	46	and	and	CCONJ
ejpam-6326	519	47	r.	r.	PROPN
ejpam-6326	519	48	d.	d.	PROPN
ejpam-6326	519	49	skaggs	skaggs	PROPN
ejpam-6326	519	50	.	.	PUNCT
ejpam-6326	520	1	offensive	offensive	ADJ
ejpam-6326	520	2	alliances	alliance	NOUN
ejpam-6326	520	3	in	in	ADP
ejpam-6326	520	4	graphs	graph	NOUN
ejpam-6326	520	5	.	.	PUNCT
ejpam-6326	521	1	discussiones	discussione	NOUN
ejpam-6326	521	2	mathematicae	mathematicae	PROPN
ejpam-6326	521	3	graph	graph	NOUN
ejpam-6326	521	4	theory	theory	NOUN
ejpam-6326	521	5	,	,	PUNCT
ejpam-6326	521	6	24(2):263–275	24(2):263–275	NUM
ejpam-6326	521	7	,	,	PUNCT
ejpam-6326	521	8	2004	2004	NUM
ejpam-6326	521	9	.	.	PUNCT
ejpam-6326	522	1	[	[	X
ejpam-6326	522	2	7	7	X
ejpam-6326	522	3	]	]	PUNCT
ejpam-6326	522	4	m.	m.	NOUN
ejpam-6326	522	5	s.	s.	PROPN
ejpam-6326	522	6	hablo	hablo	PROPN
ejpam-6326	522	7	and	and	CCONJ
ejpam-6326	522	8	i.	i.	PROPN
ejpam-6326	522	9	s.	s.	PROPN
ejpam-6326	522	10	cabahug	cabahug	PROPN
ejpam-6326	522	11	jr	jr	PROPN
ejpam-6326	522	12	.	.	PROPN
ejpam-6326	522	13	total	total	ADJ
ejpam-6326	522	14	offensive	offensive	ADJ
ejpam-6326	522	15	alliance	alliance	NOUN
ejpam-6326	522	16	of	of	ADP
ejpam-6326	522	17	total	total	ADJ
ejpam-6326	522	18	graphs	graph	NOUN
ejpam-6326	522	19	generated	generate	VERB
ejpam-6326	522	20	from	from	ADP
ejpam-6326	522	21	graphs	graph	NOUN
ejpam-6326	522	22	with	with	ADP
ejpam-6326	522	23	maximum	maximum	ADJ
ejpam-6326	522	24	degree	degree	NOUN
ejpam-6326	522	25	2	2	NUM
ejpam-6326	522	26	.	.	NOUN
ejpam-6326	522	27	advances	advance	NOUN
ejpam-6326	522	28	and	and	CCONJ
ejpam-6326	522	29	applications	application	NOUN
ejpam-6326	522	30	in	in	ADP
ejpam-6326	522	31	discrete	discrete	ADJ
ejpam-6326	522	32	mathematics	mathematic	NOUN
ejpam-6326	522	33	,	,	PUNCT
ejpam-6326	522	34	42(4):391–399	42(4):391–399	PROPN
ejpam-6326	522	35	,	,	PUNCT
ejpam-6326	522	36	2025	2025	NUM
ejpam-6326	522	37	.	.	PUNCT
ejpam-6326	523	1	[	[	X
ejpam-6326	523	2	8	8	NUM
ejpam-6326	523	3	]	]	PUNCT
ejpam-6326	523	4	k.	k.	PROPN
ejpam-6326	523	5	s.	s.	PROPN
ejpam-6326	523	6	r.	r.	PROPN
ejpam-6326	523	7	tan	tan	PROPN
ejpam-6326	523	8	and	and	CCONJ
ejpam-6326	523	9	i.	i.	PROPN
ejpam-6326	523	10	s.	s.	PROPN
ejpam-6326	523	11	cabahug	cabahug	PROPN
ejpam-6326	523	12	jr	jr	PROPN
ejpam-6326	523	13	.	.	PUNCT
ejpam-6326	524	1	restrained	restrained	ADJ
ejpam-6326	524	2	global	global	ADJ
ejpam-6326	524	3	offensive	offensive	ADJ
ejpam-6326	524	4	alliances	alliance	NOUN
ejpam-6326	524	5	in	in	ADP
ejpam-6326	524	6	some	some	DET
ejpam-6326	524	7	graphs	graph	NOUN
ejpam-6326	524	8	.	.	PUNCT
ejpam-6326	525	1	advances	advance	NOUN
ejpam-6326	525	2	and	and	CCONJ
ejpam-6326	525	3	applications	application	NOUN
ejpam-6326	525	4	in	in	ADP
ejpam-6326	525	5	discrete	discrete	ADJ
ejpam-6326	525	6	mathematics	mathematic	NOUN
ejpam-6326	525	7	,	,	PUNCT
ejpam-6326	525	8	42(3):191–204	42(3):191–204	NOUN
ejpam-6326	525	9	,	,	PUNCT
ejpam-6326	525	10	2025	2025	NUM
ejpam-6326	525	11	.	.	PUNCT
ejpam-6326	526	1	[	[	X
ejpam-6326	526	2	9	9	NUM
ejpam-6326	526	3	]	]	PUNCT
ejpam-6326	526	4	t.	t.	PROPN
ejpam-6326	526	5	w.	w.	PROPN
ejpam-6326	526	6	haynes	haynes	PROPN
ejpam-6326	526	7	,	,	PUNCT
ejpam-6326	526	8	s.	s.	PROPN
ejpam-6326	526	9	t.	t.	PROPN
ejpam-6326	526	10	hedetniemi	hedetniemi	PROPN
ejpam-6326	526	11	,	,	PUNCT
ejpam-6326	526	12	and	and	CCONJ
ejpam-6326	526	13	m.	m.	PROPN
ejpam-6326	526	14	a.	a.	PROPN
ejpam-6326	526	15	henning	henning	PROPN
ejpam-6326	526	16	.	.	PUNCT
ejpam-6326	527	1	global	global	ADJ
ejpam-6326	527	2	defensive	defensive	ADJ
ejpam-6326	527	3	alliances	alliance	NOUN
ejpam-6326	527	4	in	in	ADP
ejpam-6326	527	5	graphs	graph	NOUN
ejpam-6326	527	6	.	.	PUNCT
ejpam-6326	528	1	electronic	electronic	ADJ
ejpam-6326	528	2	journal	journal	NOUN
ejpam-6326	528	3	of	of	ADP
ejpam-6326	528	4	combinatorics	combinatoric	NOUN
ejpam-6326	528	5	,	,	PUNCT
ejpam-6326	528	6	10(1):r47	10(1):r47	NUM
ejpam-6326	528	7	,	,	PUNCT
ejpam-6326	528	8	2003	2003	NUM
ejpam-6326	528	9	.	.	PUNCT
ejpam-6326	529	1	[	[	X
ejpam-6326	529	2	10	10	NUM
ejpam-6326	529	3	]	]	X
ejpam-6326	529	4	j.	j.	PROPN
ejpam-6326	529	5	m.	m.	PROPN
ejpam-6326	529	6	sigarreta	sigarreta	PROPN
ejpam-6326	529	7	and	and	CCONJ
ejpam-6326	529	8	j.	j.	PROPN
ejpam-6326	529	9	a.	a.	PROPN
ejpam-6326	529	10	rodŕıguez	rodŕıguez	PROPN
ejpam-6326	529	11	-	-	NOUN
ejpam-6326	529	12	velázquez	velázquez	NOUN
ejpam-6326	529	13	.	.	PUNCT
ejpam-6326	530	1	on	on	ADP
ejpam-6326	530	2	defensive	defensive	ADJ
ejpam-6326	530	3	alliances	alliance	NOUN
ejpam-6326	530	4	and	and	CCONJ
ejpam-6326	530	5	line	line	NOUN
ejpam-6326	530	6	graphs	graph	NOUN
ejpam-6326	530	7	.	.	PUNCT
ejpam-6326	531	1	applied	apply	VERB
ejpam-6326	531	2	mathematics	mathematics	NOUN
ejpam-6326	531	3	letters	letter	NOUN
ejpam-6326	531	4	,	,	PUNCT
ejpam-6326	531	5	19(12):1345–1350	19(12):1345–1350	NUM
ejpam-6326	531	6	,	,	PUNCT
ejpam-6326	531	7	2006	2006	NUM
ejpam-6326	531	8	.	.	PUNCT
ejpam-6326	532	1	[	[	X
ejpam-6326	532	2	11	11	NUM
ejpam-6326	532	3	]	]	PUNCT
ejpam-6326	532	4	t.	t.	PROPN
ejpam-6326	532	5	w.	w.	PROPN
ejpam-6326	532	6	haynes	haynes	PROPN
ejpam-6326	532	7	,	,	PUNCT
ejpam-6326	532	8	s.	s.	PROPN
ejpam-6326	532	9	t.	t.	PROPN
ejpam-6326	532	10	hedetniemi	hedetniemi	PROPN
ejpam-6326	532	11	,	,	PUNCT
ejpam-6326	532	12	and	and	CCONJ
ejpam-6326	532	13	m.	m.	PROPN
ejpam-6326	532	14	a.	a.	PROPN
ejpam-6326	532	15	henning	henning	PROPN
ejpam-6326	532	16	.	.	PUNCT
ejpam-6326	533	1	global	global	ADJ
ejpam-6326	533	2	defensive	defensive	ADJ
ejpam-6326	533	3	alliances	alliance	NOUN
ejpam-6326	533	4	.	.	PUNCT
ejpam-6326	534	1	in	in	ADP
ejpam-6326	534	2	proceedings	proceeding	NOUN
ejpam-6326	534	3	of	of	ADP
ejpam-6326	534	4	the	the	DET
ejpam-6326	534	5	17th	17th	ADJ
ejpam-6326	534	6	international	international	ADJ
ejpam-6326	534	7	symposium	symposium	NOUN
ejpam-6326	534	8	on	on	ADP
ejpam-6326	534	9	computer	computer	NOUN
ejpam-6326	534	10	and	and	CCONJ
ejpam-6326	534	11	information	information	NOUN
ejpam-6326	534	12	sciences	sciences	PROPN
ejpam-6326	534	13	(	(	PUNCT
ejpam-6326	534	14	iscis	iscis	PROPN
ejpam-6326	534	15	)	)	PUNCT
ejpam-6326	534	16	,	,	PUNCT
ejpam-6326	534	17	pages	page	NOUN
ejpam-6326	534	18	303–307	303–307	NUM
ejpam-6326	534	19	.	.	PUNCT
ejpam-6326	535	1	crc	crc	PROPN
ejpam-6326	535	2	press	press	PROPN
ejpam-6326	535	3	/	/	SYM
ejpam-6326	535	4	taylor	taylor	PROPN
ejpam-6326	535	5	&	&	CCONJ
ejpam-6326	535	6	francis	francis	PROPN
ejpam-6326	535	7	,	,	PUNCT
ejpam-6326	535	8	2022	2022	NUM
ejpam-6326	535	9	.	.	PUNCT
ejpam-6326	536	1	[	[	X
ejpam-6326	536	2	12	12	NUM
ejpam-6326	536	3	]	]	PUNCT
ejpam-6326	536	4	l.	l.	PROPN
ejpam-6326	536	5	f.	f.	PROPN
ejpam-6326	536	6	consistente	consistente	PROPN
ejpam-6326	536	7	and	and	CCONJ
ejpam-6326	536	8	i.s	i.s	PROPN
ejpam-6326	536	9	.	.	PROPN
ejpam-6326	536	10	cabahug	cabahug	PROPN
ejpam-6326	536	11	jr	jr	PROPN
ejpam-6326	536	12	.	.	PROPN
ejpam-6326	536	13	restrained	restrain	VERB
ejpam-6326	536	14	global	global	ADJ
ejpam-6326	536	15	defensive	defensive	ADJ
ejpam-6326	536	16	alliances	alliance	NOUN
ejpam-6326	536	17	in	in	ADP
ejpam-6326	536	18	graphs	graph	NOUN
ejpam-6326	536	19	.	.	PUNCT
ejpam-6326	537	1	european	european	ADJ
ejpam-6326	537	2	journal	journal	PROPN
ejpam-6326	537	3	of	of	ADP
ejpam-6326	537	4	pure	pure	ADJ
ejpam-6326	537	5	and	and	CCONJ
ejpam-6326	537	6	applied	applied	ADJ
ejpam-6326	537	7	mathematics	mathematic	NOUN
ejpam-6326	537	8	,	,	PUNCT
ejpam-6326	537	9	17(3):2196–2209	17(3):2196–2209	NUM
ejpam-6326	537	10	,	,	PUNCT
ejpam-6326	537	11	2024	2024	NUM
ejpam-6326	537	12	.	.	PUNCT
ejpam-6326	538	1	[	[	X
ejpam-6326	538	2	13	13	NUM
ejpam-6326	538	3	]	]	X
ejpam-6326	538	4	y.	y.	PROPN
ejpam-6326	538	5	caro	caro	PROPN
ejpam-6326	538	6	,	,	PUNCT
ejpam-6326	538	7	a.	a.	NOUN
ejpam-6326	538	8	hansberg	hansberg	PROPN
ejpam-6326	538	9	,	,	PUNCT
ejpam-6326	538	10	and	and	CCONJ
ejpam-6326	538	11	m.	m.	PROPN
ejpam-6326	538	12	a.	a.	PROPN
ejpam-6326	538	13	henning	henning	PROPN
ejpam-6326	538	14	.	.	PUNCT
ejpam-6326	539	1	fair	fair	ADJ
ejpam-6326	539	2	domination	domination	NOUN
ejpam-6326	539	3	in	in	ADP
ejpam-6326	539	4	graphs	graph	NOUN
ejpam-6326	539	5	.	.	PUNCT
ejpam-6326	540	1	discrete	discrete	ADJ
ejpam-6326	540	2	mathematics	mathematic	NOUN
ejpam-6326	540	3	,	,	PUNCT
ejpam-6326	540	4	312:2905–2914	312:2905–2914	NOUN
ejpam-6326	540	5	,	,	PUNCT
ejpam-6326	540	6	2012	2012	NUM
ejpam-6326	540	7	.	.	PUNCT
ejpam-6326	541	1	[	[	X
ejpam-6326	541	2	14	14	NUM
ejpam-6326	541	3	]	]	X
ejpam-6326	541	4	f.	f.	PROPN
ejpam-6326	541	5	harary	harary	PROPN
ejpam-6326	541	6	and	and	CCONJ
ejpam-6326	541	7	t.	t.	PROPN
ejpam-6326	541	8	w.	w.	PROPN
ejpam-6326	541	9	haynes	haynes	PROPN
ejpam-6326	541	10	.	.	PUNCT
ejpam-6326	542	1	nordhaus	nordhaus	PROPN
ejpam-6326	542	2	–	–	PUNCT
ejpam-6326	542	3	gaddum	gaddum	PROPN
ejpam-6326	542	4	inequalities	inequality	NOUN
ejpam-6326	542	5	for	for	ADP
ejpam-6326	542	6	domination	domination	NOUN
ejpam-6326	542	7	in	in	ADP
ejpam-6326	542	8	graphs	graph	NOUN
ejpam-6326	542	9	.	.	PUNCT
ejpam-6326	543	1	discrete	discrete	ADJ
ejpam-6326	543	2	mathematics	mathematic	NOUN
ejpam-6326	543	3	,	,	PUNCT
ejpam-6326	543	4	155(1	155(1	NUM
ejpam-6326	543	5	-	-	SYM
ejpam-6326	543	6	3):99–105	3):99–105	NUM
ejpam-6326	543	7	,	,	PUNCT
ejpam-6326	543	8	1996	1996	NUM
ejpam-6326	543	9	.	.	PUNCT
ejpam-6326	544	1	[	[	X
ejpam-6326	544	2	15	15	NUM
ejpam-6326	544	3	]	]	X
ejpam-6326	544	4	g.	g.	PROPN
ejpam-6326	544	5	chartrand	chartrand	PROPN
ejpam-6326	544	6	and	and	CCONJ
ejpam-6326	544	7	l.	l.	PROPN
ejpam-6326	544	8	lesniak	lesniak	PROPN
ejpam-6326	544	9	.	.	PUNCT
ejpam-6326	544	10	graphs	graph	NOUN
ejpam-6326	544	11	&	&	CCONJ
ejpam-6326	544	12	digraphs	digraph	NOUN
ejpam-6326	544	13	.	.	PUNCT
ejpam-6326	545	1	crc	crc	PROPN
ejpam-6326	545	2	press	press	PROPN
ejpam-6326	545	3	,	,	PUNCT
ejpam-6326	545	4	boca	boca	PROPN
ejpam-6326	545	5	raton	raton	PROPN
ejpam-6326	545	6	,	,	PUNCT
ejpam-6326	545	7	fl	fl	PROPN
ejpam-6326	545	8	,	,	PUNCT
ejpam-6326	545	9	usa	usa	PROPN
ejpam-6326	545	10	,	,	PUNCT
ejpam-6326	545	11	6th	6th	ADJ
ejpam-6326	545	12	edition	edition	NOUN
ejpam-6326	545	13	,	,	PUNCT
ejpam-6326	545	14	2016	2016	NUM
ejpam-6326	545	15	.	.	PUNCT
ejpam-6326	546	1	[	[	X
ejpam-6326	546	2	16	16	NUM
ejpam-6326	546	3	]	]	X
ejpam-6326	546	4	r.	r.	PROPN
ejpam-6326	546	5	frucht	frucht	PROPN
ejpam-6326	546	6	and	and	CCONJ
ejpam-6326	546	7	f.	f.	PROPN
ejpam-6326	546	8	harary	harary	PROPN
ejpam-6326	546	9	.	.	PUNCT
ejpam-6326	547	1	on	on	ADP
ejpam-6326	547	2	the	the	DET
ejpam-6326	547	3	corona	corona	NOUN
ejpam-6326	547	4	of	of	ADP
ejpam-6326	547	5	two	two	NUM
ejpam-6326	547	6	graphs	graph	NOUN
ejpam-6326	547	7	.	.	PUNCT
ejpam-6326	548	1	aequationes	aequatione	NOUN
ejpam-6326	548	2	mathematicae	mathematicae	PROPN
ejpam-6326	548	3	,	,	PUNCT
ejpam-6326	548	4	4:322–325	4:322–325	PROPN
ejpam-6326	548	5	,	,	PUNCT
ejpam-6326	548	6	1970	1970	NUM
ejpam-6326	548	7	.	.	PUNCT
ejpam-6326	549	1	[	[	X
ejpam-6326	549	2	17	17	NUM
ejpam-6326	549	3	]	]	X
ejpam-6326	549	4	g.	g.	PROPN
ejpam-6326	549	5	h.	h.	PROPN
ejpam-6326	549	6	hardy	hardy	PROPN
ejpam-6326	549	7	,	,	PUNCT
ejpam-6326	549	8	j.	j.	PROPN
ejpam-6326	549	9	e.	e.	PROPN
ejpam-6326	549	10	littlewood	littlewood	PROPN
ejpam-6326	549	11	,	,	PUNCT
ejpam-6326	549	12	and	and	CCONJ
ejpam-6326	549	13	g.	g.	PROPN
ejpam-6326	549	14	pólya	pólya	PROPN
ejpam-6326	549	15	.	.	PUNCT
ejpam-6326	550	1	inequalities	inequality	NOUN
ejpam-6326	550	2	.	.	PUNCT
ejpam-6326	551	1	cambridge	cambridge	PROPN
ejpam-6326	551	2	university	university	PROPN
ejpam-6326	551	3	press	press	PROPN
ejpam-6326	551	4	,	,	PUNCT
ejpam-6326	551	5	cambridge	cambridge	PROPN
ejpam-6326	551	6	,	,	PUNCT
ejpam-6326	551	7	uk	uk	PROPN
ejpam-6326	551	8	,	,	PUNCT
ejpam-6326	551	9	2nd	2nd	ADJ
ejpam-6326	551	10	edition	edition	NOUN
ejpam-6326	551	11	,	,	PUNCT
ejpam-6326	551	12	1952	1952	NUM
ejpam-6326	551	13	.	.	PUNCT