id	sid	tid	token	lemma	pos
cana-6235	1	1	communications	communication	NOUN
cana-6235	1	2	on	on	ADP
cana-6235	1	3	applied	apply	VERB
cana-6235	1	4	nonlinear	nonlinear	ADJ
cana-6235	1	5	analysis	analysis	NOUN
cana-6235	1	6	issn	issn	NOUN
cana-6235	1	7	:	:	PUNCT
cana-6235	1	8	1074	1074	NUM
cana-6235	1	9	-	-	PUNCT
cana-6235	1	10	133x	133x	NUM
cana-6235	1	11	vol	vol	NOUN
cana-6235	1	12	31	31	NUM
cana-6235	1	13	no	no	NOUN
cana-6235	1	14	.	.	NOUN
cana-6235	1	15	2	2	NUM
cana-6235	1	16	(	(	PUNCT
cana-6235	1	17	2024	2024	NUM
cana-6235	1	18	)	)	PUNCT
cana-6235	1	19	510	510	NUM
cana-6235	1	20	https://internationalpubls.com	https://internationalpubls.com	X
cana-6235	1	21	a	a	DET
cana-6235	1	22	study	study	NOUN
cana-6235	1	23	about	about	ADP
cana-6235	1	24	isolate	isolate	VERB
cana-6235	1	25	domination	domination	PROPN
cana-6235	1	26	&	&	CCONJ
cana-6235	1	27	isolate	isolate	PROPN
cana-6235	1	28	inclusive	inclusive	ADJ
cana-6235	1	29	set	set	NOUN
cana-6235	1	30	in	in	ADP
cana-6235	1	31	graphs	graph	NOUN
cana-6235	1	32	dr	dr	PROPN
cana-6235	1	33	.	.	PROPN
cana-6235	1	34	niketa	niketa	PROPN
cana-6235	1	35	j.	j.	PROPN
cana-6235	1	36	savaliya1	savaliya1	PROPN
cana-6235	1	37	,	,	PUNCT
cana-6235	1	38	dr	dr	PROPN
cana-6235	1	39	.	.	PROPN
cana-6235	1	40	mital	mital	ADJ
cana-6235	1	41	patel2	patel2	PROPN
cana-6235	1	42	1p	1p	NUM
cana-6235	1	43	.	.	PUNCT
cana-6235	2	1	p	p	PROPN
cana-6235	2	2	savani	savani	PROPN
cana-6235	2	3	university	university	PROPN
cana-6235	2	4	,	,	PUNCT
cana-6235	2	5	surat	surat	PROPN
cana-6235	2	6	,	,	PUNCT
cana-6235	2	7	gujarat	gujarat	PROPN
cana-6235	2	8	,	,	PUNCT
cana-6235	2	9	india	india	PROPN
cana-6235	2	10	2ahmedabad	2ahmedabad	PROPN
cana-6235	2	11	institute	institute	PROPN
cana-6235	2	12	of	of	ADP
cana-6235	2	13	technology	technology	PROPN
cana-6235	2	14	,	,	PUNCT
cana-6235	2	15	ahmedabad	ahmedabad	PROPN
cana-6235	2	16	,	,	PUNCT
cana-6235	2	17	gujarat	gujarat	PROPN
cana-6235	2	18	,	,	PUNCT
cana-6235	2	19	india	india	PROPN
cana-6235	2	20	nsavaliya24@gmail.com1	nsavaliya24@gmail.com1	PROPN
cana-6235	2	21	,	,	PUNCT
cana-6235	2	22	mital.kachhadia6611@gmail.com2	mital.kachhadia6611@gmail.com2	NOUN
cana-6235	2	23	article	article	NOUN
cana-6235	2	24	history	history	NOUN
cana-6235	2	25	:	:	PUNCT
cana-6235	2	26	received:04/10/2024	received:04/10/2024	PROPN
cana-6235	2	27	revised:03/11/2024	revised:03/11/2024	PROPN
cana-6235	2	28	accepted:05/12/2024	accepted:05/12/2024	VERB
cana-6235	2	29	abstract	abstract	NOUN
cana-6235	2	30	:	:	PUNCT
cana-6235	2	31	this	this	DET
cana-6235	2	32	paper	paper	NOUN
cana-6235	2	33	investigates	investigate	VERB
cana-6235	2	34	the	the	DET
cana-6235	2	35	concept	concept	NOUN
cana-6235	2	36	of	of	ADP
cana-6235	2	37	isolate	isolate	ADJ
cana-6235	2	38	domination	domination	NOUN
cana-6235	2	39	in	in	ADP
cana-6235	2	40	graphs	graph	NOUN
cana-6235	2	41	,	,	PUNCT
cana-6235	2	42	focusing	focus	VERB
cana-6235	2	43	on	on	ADP
cana-6235	2	44	how	how	SCONJ
cana-6235	2	45	structural	structural	ADJ
cana-6235	2	46	changes	change	NOUN
cana-6235	2	47	affect	affect	VERB
cana-6235	2	48	the	the	DET
cana-6235	2	49	isolate	isolate	ADJ
cana-6235	2	50	domination	domination	NOUN
cana-6235	2	51	number	number	NOUN
cana-6235	2	52	.	.	PUNCT
cana-6235	3	1	the	the	DET
cana-6235	3	2	impact	impact	NOUN
cana-6235	3	3	of	of	ADP
cana-6235	3	4	removing	remove	VERB
cana-6235	3	5	both	both	CCONJ
cana-6235	3	6	isolated	isolated	ADJ
cana-6235	3	7	and	and	CCONJ
cana-6235	3	8	non	non	ADJ
cana-6235	3	9	-	-	ADJ
cana-6235	3	10	isolated	isolated	ADJ
cana-6235	3	11	vertices	vertex	NOUN
cana-6235	3	12	is	be	AUX
cana-6235	3	13	examined	examine	VERB
cana-6235	3	14	,	,	PUNCT
cana-6235	3	15	and	and	CCONJ
cana-6235	3	16	necessary	necessary	ADJ
cana-6235	3	17	and	and	CCONJ
cana-6235	3	18	sufficient	sufficient	ADJ
cana-6235	3	19	conditions	condition	NOUN
cana-6235	3	20	are	be	AUX
cana-6235	3	21	established	establish	VERB
cana-6235	3	22	for	for	ADP
cana-6235	3	23	when	when	SCONJ
cana-6235	3	24	such	such	ADJ
cana-6235	3	25	removals	removal	NOUN
cana-6235	3	26	lead	lead	VERB
cana-6235	3	27	to	to	ADP
cana-6235	3	28	an	an	DET
cana-6235	3	29	increase	increase	NOUN
cana-6235	3	30	or	or	CCONJ
cana-6235	3	31	decrease	decrease	NOUN
cana-6235	3	32	in	in	ADP
cana-6235	3	33	the	the	DET
cana-6235	3	34	isolate	isolate	ADJ
cana-6235	3	35	domination	domination	NOUN
cana-6235	3	36	number	number	NOUN
cana-6235	3	37	.	.	PUNCT
cana-6235	4	1	it	it	PRON
cana-6235	4	2	is	be	AUX
cana-6235	4	3	shown	show	VERB
cana-6235	4	4	that	that	SCONJ
cana-6235	4	5	if	if	SCONJ
cana-6235	4	6	a	a	DET
cana-6235	4	7	graph	graph	NOUN
cana-6235	4	8	gcontains	gcontain	VERB
cana-6235	4	9	an	an	DET
cana-6235	4	10	isolated	isolated	ADJ
cana-6235	4	11	vertex	vertex	NOUN
cana-6235	4	12	v	v	ADP
cana-6235	4	13	such	such	ADJ
cana-6235	4	14	that	that	SCONJ
cana-6235	4	15	γ_0	γ_0	PROPN
cana-6235	4	16	(	(	PUNCT
cana-6235	4	17	g	g	NOUN
cana-6235	4	18	-	-	PUNCT
cana-6235	4	19	v)>γ_0	v)>γ_0	NOUN
cana-6235	4	20	(	(	PUNCT
cana-6235	4	21	g	g	NOUN
cana-6235	4	22	)	)	PUNCT
cana-6235	4	23	,	,	PUNCT
cana-6235	4	24	then	then	ADV
cana-6235	4	25	vmust	vmust	ADV
cana-6235	4	26	be	be	AUX
cana-6235	4	27	the	the	DET
cana-6235	4	28	only	only	ADJ
cana-6235	4	29	isolated	isolated	ADJ
cana-6235	4	30	vertex	vertex	NOUN
cana-6235	4	31	in	in	ADP
cana-6235	4	32	g.	g.	PROPN
cana-6235	4	33	the	the	DET
cana-6235	4	34	study	study	NOUN
cana-6235	4	35	further	far	ADV
cana-6235	4	36	explores	explore	VERB
cana-6235	4	37	the	the	DET
cana-6235	4	38	effect	effect	NOUN
cana-6235	4	39	of	of	ADP
cana-6235	4	40	edge	edge	NOUN
cana-6235	4	41	removal	removal	NOUN
cana-6235	4	42	,	,	PUNCT
cana-6235	4	43	identifying	identify	VERB
cana-6235	4	44	conditions	condition	NOUN
cana-6235	4	45	under	under	ADP
cana-6235	4	46	which	which	PRON
cana-6235	4	47	the	the	DET
cana-6235	4	48	isolate	isolate	ADJ
cana-6235	4	49	domination	domination	NOUN
cana-6235	4	50	number	number	NOUN
cana-6235	4	51	increases	increase	NOUN
cana-6235	4	52	.	.	PUNCT
cana-6235	5	1	additionally	additionally	ADV
cana-6235	5	2	,	,	PUNCT
cana-6235	5	3	the	the	DET
cana-6235	5	4	isolate	isolate	ADJ
cana-6235	5	5	inclusive	inclusive	ADJ
cana-6235	5	6	set	set	NOUN
cana-6235	5	7	number	number	NOUN
cana-6235	5	8	is	be	AUX
cana-6235	5	9	analyzed	analyze	VERB
cana-6235	5	10	,	,	PUNCT
cana-6235	5	11	with	with	ADP
cana-6235	5	12	criteria	criterion	NOUN
cana-6235	5	13	provided	provide	VERB
cana-6235	5	14	for	for	ADP
cana-6235	5	15	its	its	PRON
cana-6235	5	16	increase	increase	NOUN
cana-6235	5	17	following	follow	VERB
cana-6235	5	18	edge	edge	NOUN
cana-6235	5	19	deletion	deletion	NOUN
cana-6235	5	20	.	.	PUNCT
cana-6235	6	1	special	special	ADJ
cana-6235	6	2	attention	attention	NOUN
cana-6235	6	3	is	be	AUX
cana-6235	6	4	given	give	VERB
cana-6235	6	5	to	to	PART
cana-6235	6	6	graphs	graphs	VERB
cana-6235	6	7	where	where	SCONJ
cana-6235	6	8	the	the	DET
cana-6235	6	9	isolate	isolate	ADJ
cana-6235	6	10	domination	domination	NOUN
cana-6235	6	11	number	number	NOUN
cana-6235	6	12	equals	equal	VERB
cana-6235	6	13	one	one	NUM
cana-6235	6	14	or	or	CCONJ
cana-6235	6	15	two	two	NUM
cana-6235	6	16	,	,	PUNCT
cana-6235	6	17	offering	offer	VERB
cana-6235	6	18	characterizations	characterization	NOUN
cana-6235	6	19	that	that	PRON
cana-6235	6	20	enhance	enhance	VERB
cana-6235	6	21	understanding	understanding	NOUN
cana-6235	6	22	of	of	ADP
cana-6235	6	23	their	their	PRON
cana-6235	6	24	structural	structural	ADJ
cana-6235	6	25	properties	property	NOUN
cana-6235	6	26	.	.	PUNCT
cana-6235	7	1	these	these	DET
cana-6235	7	2	results	result	NOUN
cana-6235	7	3	contribute	contribute	VERB
cana-6235	7	4	to	to	ADP
cana-6235	7	5	the	the	DET
cana-6235	7	6	broader	broad	ADJ
cana-6235	7	7	theory	theory	NOUN
cana-6235	7	8	of	of	ADP
cana-6235	7	9	domination	domination	NOUN
cana-6235	7	10	in	in	ADP
cana-6235	7	11	graphs	graph	NOUN
cana-6235	7	12	by	by	ADP
cana-6235	7	13	clarifying	clarify	VERB
cana-6235	7	14	how	how	SCONJ
cana-6235	7	15	isolate	isolate	VERB
cana-6235	7	16	domination	domination	NOUN
cana-6235	7	17	responds	respond	VERB
cana-6235	7	18	to	to	ADP
cana-6235	7	19	local	local	ADJ
cana-6235	7	20	modifications	modification	NOUN
cana-6235	7	21	,	,	PUNCT
cana-6235	7	22	and	and	CCONJ
cana-6235	7	23	they	they	PRON
cana-6235	7	24	provide	provide	VERB
cana-6235	7	25	useful	useful	ADJ
cana-6235	7	26	tools	tool	NOUN
cana-6235	7	27	for	for	ADP
cana-6235	7	28	analyzing	analyze	VERB
cana-6235	7	29	graph	graph	NOUN
cana-6235	7	30	resilience	resilience	NOUN
cana-6235	7	31	and	and	CCONJ
cana-6235	7	32	optimization	optimization	NOUN
cana-6235	7	33	in	in	ADP
cana-6235	7	34	network	network	NOUN
cana-6235	7	35	structures	structure	NOUN
cana-6235	7	36	.	.	PUNCT
cana-6235	8	1	keywords	keyword	NOUN
cana-6235	8	2	:	:	PUNCT
cana-6235	8	3	isolate	isolate	VERB
cana-6235	8	4	dominating	dominating	NOUN
cana-6235	8	5	set	set	NOUN
cana-6235	8	6	,	,	PUNCT
cana-6235	8	7	minimal	minimal	ADJ
cana-6235	8	8	isolate	isolate	NOUN
cana-6235	8	9	dominating	dominating	NOUN
cana-6235	8	10	set	set	NOUN
cana-6235	8	11	,	,	PUNCT
cana-6235	8	12	minimum	minimum	NOUN
cana-6235	8	13	isolate	isolate	NOUN
cana-6235	8	14	dominating	dominating	NOUN
cana-6235	8	15	set	set	NOUN
cana-6235	8	16	,	,	PUNCT
cana-6235	8	17	isolate	isolate	VERB
cana-6235	8	18	domination	domination	NOUN
cana-6235	8	19	number	number	NOUN
cana-6235	8	20	,	,	PUNCT
cana-6235	8	21	isolate	isolate	VERB
cana-6235	8	22	inclusive	inclusive	ADJ
cana-6235	8	23	set	set	NOUN
cana-6235	8	24	,	,	PUNCT
cana-6235	8	25	1	1	NUM
cana-6235	8	26	-	-	PUNCT
cana-6235	8	27	maximal	maximal	ADJ
cana-6235	8	28	isolate	isolate	NOUN
cana-6235	8	29	inclusive	inclusive	ADJ
cana-6235	8	30	set	set	NOUN
cana-6235	8	31	,	,	PUNCT
cana-6235	8	32	private	private	ADJ
cana-6235	8	33	neighbourhood	neighbourhood	NOUN
cana-6235	8	34	.	.	PUNCT
cana-6235	9	1	1	1	X
cana-6235	9	2	.	.	X
cana-6235	9	3	introduction	introduction	NOUN
cana-6235	9	4	:	:	PUNCT
cana-6235	9	5	the	the	DET
cana-6235	9	6	concept	concept	NOUN
cana-6235	9	7	of	of	ADP
cana-6235	9	8	isolate	isolate	ADJ
cana-6235	9	9	domination	domination	NOUN
cana-6235	9	10	in	in	ADP
cana-6235	9	11	graphs	graph	NOUN
cana-6235	9	12	has	have	AUX
cana-6235	9	13	attracted	attract	VERB
cana-6235	9	14	considerable	considerable	ADJ
cana-6235	9	15	interest	interest	NOUN
cana-6235	9	16	due	due	ADP
cana-6235	9	17	to	to	ADP
cana-6235	9	18	its	its	PRON
cana-6235	9	19	relevance	relevance	NOUN
cana-6235	9	20	in	in	ADP
cana-6235	9	21	structural	structural	ADJ
cana-6235	9	22	graph	graph	NOUN
cana-6235	9	23	theory	theory	NOUN
cana-6235	9	24	and	and	CCONJ
cana-6235	9	25	its	its	PRON
cana-6235	9	26	applications	application	NOUN
cana-6235	9	27	in	in	ADP
cana-6235	9	28	network	network	NOUN
cana-6235	9	29	analysis	analysis	NOUN
cana-6235	9	30	.	.	PUNCT
cana-6235	10	1	an	an	DET
cana-6235	10	2	isolate	isolate	NOUN
cana-6235	10	3	dominating	dominating	NOUN
cana-6235	10	4	set	set	NOUN
cana-6235	10	5	is	be	AUX
cana-6235	10	6	a	a	DET
cana-6235	10	7	subset	subset	NOUN
cana-6235	10	8	of	of	ADP
cana-6235	10	9	vertices	vertex	NOUN
cana-6235	10	10	such	such	ADJ
cana-6235	10	11	that	that	SCONJ
cana-6235	10	12	every	every	DET
cana-6235	10	13	isolated	isolated	ADJ
cana-6235	10	14	vertex	vertex	NOUN
cana-6235	10	15	in	in	ADP
cana-6235	10	16	the	the	DET
cana-6235	10	17	graph	graph	NOUN
cana-6235	10	18	is	be	AUX
cana-6235	10	19	either	either	CCONJ
cana-6235	10	20	in	in	ADP
cana-6235	10	21	the	the	DET
cana-6235	10	22	set	set	NOUN
cana-6235	10	23	or	or	CCONJ
cana-6235	10	24	adjacent	adjacent	ADJ
cana-6235	10	25	to	to	ADP
cana-6235	10	26	a	a	DET
cana-6235	10	27	vertex	vertex	NOUN
cana-6235	10	28	in	in	ADP
cana-6235	10	29	the	the	DET
cana-6235	10	30	set	set	NOUN
cana-6235	10	31	.	.	PUNCT
cana-6235	11	1	building	build	VERB
cana-6235	11	2	on	on	ADP
cana-6235	11	3	this	this	DET
cana-6235	11	4	foundation	foundation	NOUN
cana-6235	11	5	,	,	PUNCT
cana-6235	11	6	the	the	DET
cana-6235	11	7	notion	notion	NOUN
cana-6235	11	8	of	of	ADP
cana-6235	11	9	an	an	DET
cana-6235	11	10	isolate	isolate	ADJ
cana-6235	11	11	inclusive	inclusive	ADJ
cana-6235	11	12	set	set	NOUN
cana-6235	11	13	was	be	AUX
cana-6235	11	14	introduced	introduce	VERB
cana-6235	11	15	to	to	PART
cana-6235	11	16	further	far	ADV
cana-6235	11	17	refine	refine	VERB
cana-6235	11	18	the	the	DET
cana-6235	11	19	understanding	understanding	NOUN
cana-6235	11	20	of	of	ADP
cana-6235	11	21	domination	domination	NOUN
cana-6235	11	22	in	in	ADP
cana-6235	11	23	graphs	graph	NOUN
cana-6235	11	24	.	.	PUNCT
cana-6235	12	1	it	it	PRON
cana-6235	12	2	was	be	AUX
cana-6235	12	3	established	establish	VERB
cana-6235	12	4	that	that	SCONJ
cana-6235	12	5	every	every	DET
cana-6235	12	6	1	1	NUM
cana-6235	12	7	-	-	PUNCT
cana-6235	12	8	maximal	maximal	ADJ
cana-6235	12	9	isolate	isolate	NOUN
cana-6235	12	10	inclusive	inclusive	ADJ
cana-6235	12	11	set	set	NOUN
cana-6235	12	12	qualifies	qualifie	NOUN
cana-6235	12	13	as	as	ADP
cana-6235	12	14	an	an	DET
cana-6235	12	15	isolate	isolate	NOUN
cana-6235	12	16	dominating	dominating	NOUN
cana-6235	12	17	set	set	NOUN
cana-6235	12	18	,	,	PUNCT
cana-6235	12	19	thereby	thereby	ADV
cana-6235	12	20	linking	link	VERB
cana-6235	12	21	these	these	DET
cana-6235	12	22	two	two	NUM
cana-6235	12	23	concepts	concept	NOUN
cana-6235	12	24	in	in	ADP
cana-6235	12	25	a	a	DET
cana-6235	12	26	meaningful	meaningful	ADJ
cana-6235	12	27	way	way	NOUN
cana-6235	12	28	.	.	PUNCT
cana-6235	13	1	this	this	DET
cana-6235	13	2	paper	paper	NOUN
cana-6235	13	3	extends	extend	VERB
cana-6235	13	4	the	the	DET
cana-6235	13	5	study	study	NOUN
cana-6235	13	6	of	of	ADP
cana-6235	13	7	isolate	isolate	ADJ
cana-6235	13	8	domination	domination	NOUN
cana-6235	13	9	by	by	ADP
cana-6235	13	10	examining	examine	VERB
cana-6235	13	11	how	how	SCONJ
cana-6235	13	12	various	various	ADJ
cana-6235	13	13	graph	graph	NOUN
cana-6235	13	14	operations	operation	NOUN
cana-6235	13	15	influence	influence	VERB
cana-6235	13	16	the	the	DET
cana-6235	13	17	isolate	isolate	ADJ
cana-6235	13	18	domination	domination	NOUN
cana-6235	13	19	number	number	NOUN
cana-6235	13	20	and	and	CCONJ
cana-6235	13	21	the	the	DET
cana-6235	13	22	isolate	isolate	ADJ
cana-6235	13	23	inclusive	inclusive	ADJ
cana-6235	13	24	set	set	VERB
cana-6235	13	25	number	number	NOUN
cana-6235	13	26	.	.	PUNCT
cana-6235	14	1	specifically	specifically	ADV
cana-6235	14	2	,	,	PUNCT
cana-6235	14	3	the	the	DET
cana-6235	14	4	effect	effect	NOUN
cana-6235	14	5	of	of	ADP
cana-6235	14	6	vertex	vertex	NOUN
cana-6235	14	7	removal	removal	NOUN
cana-6235	14	8	is	be	AUX
cana-6235	14	9	analyzed	analyze	VERB
cana-6235	14	10	in	in	ADP
cana-6235	14	11	detail	detail	NOUN
cana-6235	14	12	.	.	PUNCT
cana-6235	15	1	the	the	DET
cana-6235	15	2	removal	removal	NOUN
cana-6235	15	3	of	of	ADP
cana-6235	15	4	an	an	DET
cana-6235	15	5	isolated	isolated	ADJ
cana-6235	15	6	vertex	vertex	NOUN
cana-6235	15	7	is	be	AUX
cana-6235	15	8	considered	consider	VERB
cana-6235	15	9	first	first	ADJ
cana-6235	15	10	,	,	PUNCT
cana-6235	15	11	and	and	CCONJ
cana-6235	15	12	necessary	necessary	ADJ
cana-6235	15	13	and	and	CCONJ
cana-6235	15	14	sufficient	sufficient	ADJ
cana-6235	15	15	conditions	condition	NOUN
cana-6235	15	16	are	be	AUX
cana-6235	15	17	derived	derive	VERB
cana-6235	15	18	under	under	ADP
cana-6235	15	19	which	which	PRON
cana-6235	15	20	the	the	DET
cana-6235	15	21	isolate	isolate	ADJ
cana-6235	15	22	domination	domination	NOUN
cana-6235	15	23	number	number	NOUN
cana-6235	15	24	either	either	CCONJ
cana-6235	15	25	increases	increase	NOUN
cana-6235	15	26	or	or	CCONJ
cana-6235	15	27	decreases	decrease	NOUN
cana-6235	15	28	.	.	PUNCT
cana-6235	16	1	the	the	DET
cana-6235	16	2	analysis	analysis	NOUN
cana-6235	16	3	is	be	AUX
cana-6235	16	4	then	then	ADV
cana-6235	16	5	extended	extend	VERB
cana-6235	16	6	to	to	ADP
cana-6235	16	7	the	the	DET
cana-6235	16	8	removal	removal	NOUN
cana-6235	16	9	of	of	ADP
cana-6235	16	10	non	non	ADJ
cana-6235	16	11	-	-	ADJ
cana-6235	16	12	isolated	isolated	ADJ
cana-6235	16	13	vertices	vertex	NOUN
cana-6235	16	14	,	,	PUNCT
cana-6235	16	15	with	with	ADP
cana-6235	16	16	similar	similar	ADJ
cana-6235	16	17	conditions	condition	NOUN
cana-6235	16	18	established	establish	VERB
cana-6235	16	19	.	.	PUNCT
cana-6235	17	1	further	far	ADV
cana-6235	17	2	,	,	PUNCT
cana-6235	17	3	the	the	DET
cana-6235	17	4	paper	paper	NOUN
cana-6235	17	5	investigates	investigate	VERB
cana-6235	17	6	the	the	DET
cana-6235	17	7	behavior	behavior	NOUN
cana-6235	17	8	of	of	ADP
cana-6235	17	9	isolate	isolate	VERB
cana-6235	17	10	inclusive	inclusive	ADJ
cana-6235	17	11	sets	set	NOUN
cana-6235	17	12	under	under	ADP
cana-6235	17	13	vertex	vertex	NOUN
cana-6235	17	14	removal	removal	NOUN
cana-6235	17	15	,	,	PUNCT
cana-6235	17	16	offering	offer	VERB
cana-6235	17	17	insights	insight	NOUN
cana-6235	17	18	into	into	ADP
cana-6235	17	19	how	how	SCONJ
cana-6235	17	20	such	such	ADJ
cana-6235	17	21	operations	operation	NOUN
cana-6235	17	22	affect	affect	VERB
cana-6235	17	23	the	the	DET
cana-6235	17	24	isolate	isolate	ADJ
cana-6235	17	25	inclusive	inclusive	ADJ
cana-6235	17	26	set	set	VERB
cana-6235	17	27	number	number	NOUN
cana-6235	17	28	.	.	PUNCT
cana-6235	18	1	the	the	DET
cana-6235	18	2	study	study	NOUN
cana-6235	18	3	mailto:nsavaliya24@gmail.com	mailto:nsavaliya24@gmail.com	PROPN
cana-6235	18	4	mailto:mital.kachhadia6611@gmail.com	mailto:mital.kachhadia6611@gmail.com	PROPN
cana-6235	18	5	communications	communication	NOUN
cana-6235	18	6	on	on	ADP
cana-6235	18	7	applied	apply	VERB
cana-6235	18	8	nonlinear	nonlinear	ADJ
cana-6235	18	9	analysis	analysis	NOUN
cana-6235	18	10	issn	issn	NOUN
cana-6235	18	11	:	:	PUNCT
cana-6235	18	12	1074	1074	NUM
cana-6235	18	13	-	-	PUNCT
cana-6235	18	14	133x	133x	NUM
cana-6235	18	15	vol	vol	NOUN
cana-6235	18	16	31	31	NUM
cana-6235	18	17	no	no	NOUN
cana-6235	18	18	.	.	NOUN
cana-6235	18	19	2	2	NUM
cana-6235	18	20	(	(	PUNCT
cana-6235	18	21	2024	2024	NUM
cana-6235	18	22	)	)	PUNCT
cana-6235	18	23	511	511	NUM
cana-6235	18	24	https://internationalpubls.com	https://internationalpubls.com	X
cana-6235	18	25	also	also	ADV
cana-6235	18	26	explores	explore	VERB
cana-6235	18	27	the	the	DET
cana-6235	18	28	impact	impact	NOUN
cana-6235	18	29	of	of	ADP
cana-6235	18	30	edge	edge	NOUN
cana-6235	18	31	removal	removal	NOUN
cana-6235	18	32	on	on	ADP
cana-6235	18	33	both	both	CCONJ
cana-6235	18	34	the	the	DET
cana-6235	18	35	isolate	isolate	ADJ
cana-6235	18	36	domination	domination	NOUN
cana-6235	18	37	number	number	NOUN
cana-6235	18	38	and	and	CCONJ
cana-6235	18	39	the	the	DET
cana-6235	18	40	isolate	isolate	ADJ
cana-6235	18	41	inclusive	inclusive	ADJ
cana-6235	18	42	set	set	VERB
cana-6235	18	43	number	number	NOUN
cana-6235	18	44	.	.	PUNCT
cana-6235	19	1	conditions	condition	NOUN
cana-6235	19	2	are	be	AUX
cana-6235	19	3	provided	provide	VERB
cana-6235	19	4	to	to	PART
cana-6235	19	5	determine	determine	VERB
cana-6235	19	6	when	when	SCONJ
cana-6235	19	7	these	these	DET
cana-6235	19	8	parameters	parameter	NOUN
cana-6235	19	9	increase	increase	VERB
cana-6235	19	10	following	follow	VERB
cana-6235	19	11	the	the	DET
cana-6235	19	12	deletion	deletion	NOUN
cana-6235	19	13	of	of	ADP
cana-6235	19	14	an	an	DET
cana-6235	19	15	edge	edge	NOUN
cana-6235	19	16	.	.	PUNCT
cana-6235	20	1	finally	finally	ADV
cana-6235	20	2	,	,	PUNCT
cana-6235	20	3	graphs	graph	NOUN
cana-6235	20	4	with	with	ADP
cana-6235	20	5	isolate	isolate	NOUN
cana-6235	20	6	domination	domination	NOUN
cana-6235	20	7	numbers	number	NOUN
cana-6235	20	8	equal	equal	ADJ
cana-6235	20	9	to	to	ADP
cana-6235	20	10	one	one	NUM
cana-6235	20	11	or	or	CCONJ
cana-6235	20	12	two	two	NUM
cana-6235	20	13	are	be	AUX
cana-6235	20	14	characterized	characterize	VERB
cana-6235	20	15	,	,	PUNCT
cana-6235	20	16	contributing	contribute	VERB
cana-6235	20	17	to	to	ADP
cana-6235	20	18	a	a	DET
cana-6235	20	19	deeper	deep	ADJ
cana-6235	20	20	understanding	understanding	NOUN
cana-6235	20	21	of	of	ADP
cana-6235	20	22	their	their	PRON
cana-6235	20	23	structural	structural	ADJ
cana-6235	20	24	properties	property	NOUN
cana-6235	20	25	and	and	CCONJ
cana-6235	20	26	the	the	DET
cana-6235	20	27	role	role	NOUN
cana-6235	20	28	of	of	ADP
cana-6235	20	29	isolate	isolate	ADJ
cana-6235	20	30	domination	domination	NOUN
cana-6235	20	31	in	in	ADP
cana-6235	20	32	graph	graph	NOUN
cana-6235	20	33	classification	classification	NOUN
cana-6235	20	34	.	.	PUNCT
cana-6235	21	1	2	2	X
cana-6235	21	2	.	.	NUM
cana-6235	21	3	preliminaries	preliminary	NOUN
cana-6235	21	4	and	and	CCONJ
cana-6235	21	5	notations	notation	NOUN
cana-6235	21	6	let	let	VERB
cana-6235	21	7	𝐺be	𝐺be	PROPN
cana-6235	21	8	a	a	DET
cana-6235	21	9	graph	graph	NOUN
cana-6235	21	10	.	.	PUNCT
cana-6235	22	1	the	the	DET
cana-6235	22	2	vertex	vertex	NOUN
cana-6235	22	3	set	set	NOUN
cana-6235	22	4	of	of	ADP
cana-6235	22	5	𝐺is	𝐺is	PROPN
cana-6235	22	6	denoted	denote	VERB
cana-6235	22	7	by	by	ADP
cana-6235	22	8	𝑉(𝐺	𝑉(𝐺	NOUN
cana-6235	22	9	)	)	PUNCT
cana-6235	22	10	,	,	PUNCT
cana-6235	22	11	and	and	CCONJ
cana-6235	22	12	the	the	DET
cana-6235	22	13	edge	edge	NOUN
cana-6235	22	14	set	set	NOUN
cana-6235	22	15	is	be	AUX
cana-6235	22	16	denoted	denote	VERB
cana-6235	22	17	by	by	ADP
cana-6235	22	18	𝐸(𝐺	𝐸(𝐺	PROPN
cana-6235	22	19	)	)	PUNCT
cana-6235	22	20	.	.	PUNCT
cana-6235	23	1	for	for	ADP
cana-6235	23	2	an	an	DET
cana-6235	23	3	edge	edge	NOUN
cana-6235	23	4	𝑒	𝑒	PROPN
cana-6235	23	5	∈	∈	NOUN
cana-6235	23	6	𝐸(𝐺	𝐸(𝐺	NOUN
cana-6235	23	7	)	)	PUNCT
cana-6235	23	8	,	,	PUNCT
cana-6235	23	9	the	the	DET
cana-6235	23	10	graph	graph	NOUN
cana-6235	23	11	𝐺	𝐺	PROPN
cana-6235	23	12	−	−	PROPN
cana-6235	23	13	𝑒represents	𝑒represent	VERB
cana-6235	23	14	the	the	DET
cana-6235	23	15	subgraph	subgraph	NOUN
cana-6235	23	16	obtained	obtain	VERB
cana-6235	23	17	by	by	ADP
cana-6235	23	18	removing	remove	VERB
cana-6235	23	19	the	the	DET
cana-6235	23	20	edge	edge	NOUN
cana-6235	23	21	𝑒from	𝑒from	NOUN
cana-6235	23	22	𝐺.	𝐺.	PROPN
cana-6235	23	23	for	for	ADP
cana-6235	23	24	a	a	DET
cana-6235	23	25	vertex	vertex	NOUN
cana-6235	23	26	𝑣	𝑣	ADP
cana-6235	23	27	∈	∈	PROPN
cana-6235	23	28	𝑉(𝐺	𝑉(𝐺	NOUN
cana-6235	23	29	)	)	PUNCT
cana-6235	23	30	,	,	PUNCT
cana-6235	23	31	the	the	DET
cana-6235	23	32	graph	graph	NOUN
cana-6235	23	33	𝐺	𝐺	PROPN
cana-6235	23	34	−	−	PROPN
cana-6235	23	35	𝑣denotes	𝑣denote	VERB
cana-6235	23	36	the	the	DET
cana-6235	23	37	subgraph	subgraph	NOUN
cana-6235	23	38	induced	induce	VERB
cana-6235	23	39	by	by	ADP
cana-6235	23	40	all	all	DET
cana-6235	23	41	vertices	vertex	NOUN
cana-6235	23	42	in	in	ADP
cana-6235	23	43	𝑉(𝐺	𝑉(𝐺	NOUN
cana-6235	23	44	)	)	PUNCT
cana-6235	23	45	∖	∖	X
cana-6235	23	46	{	{	PUNCT
cana-6235	23	47	𝑣	𝑣	NOUN
cana-6235	23	48	}	}	PUNCT
cana-6235	23	49	.	.	PUNCT
cana-6235	24	1	if	if	SCONJ
cana-6235	24	2	𝑥	𝑥	PRON
cana-6235	24	3	∈	∈	PROPN
cana-6235	24	4	𝑉(𝐺	𝑉(𝐺	NOUN
cana-6235	24	5	)	)	PUNCT
cana-6235	24	6	,	,	PUNCT
cana-6235	24	7	then	then	ADV
cana-6235	24	8	𝑑(𝑥)denotes	𝑑(𝑥)denote	VERB
cana-6235	24	9	the	the	DET
cana-6235	24	10	degree	degree	NOUN
cana-6235	24	11	of	of	ADP
cana-6235	24	12	the	the	DET
cana-6235	24	13	vertex	vertex	NOUN
cana-6235	24	14	𝑥in	𝑥in	PROPN
cana-6235	24	15	𝐺	𝐺	PROPN
cana-6235	24	16	,	,	PUNCT
cana-6235	24	17	which	which	PRON
cana-6235	24	18	is	be	AUX
cana-6235	24	19	the	the	DET
cana-6235	24	20	number	number	NOUN
cana-6235	24	21	of	of	ADP
cana-6235	24	22	edges	edge	NOUN
cana-6235	24	23	incident	incident	NOUN
cana-6235	24	24	to	to	ADP
cana-6235	24	25	𝑥.	𝑥.	NOUN
cana-6235	24	26	throughout	throughout	ADP
cana-6235	24	27	this	this	DET
cana-6235	24	28	paper	paper	NOUN
cana-6235	24	29	,	,	PUNCT
cana-6235	24	30	only	only	ADV
cana-6235	24	31	simple	simple	ADJ
cana-6235	24	32	,	,	PUNCT
cana-6235	24	33	undirected	undirected	ADJ
cana-6235	24	34	graphs	graph	NOUN
cana-6235	24	35	with	with	ADP
cana-6235	24	36	finite	finite	ADJ
cana-6235	24	37	vertex	vertex	NOUN
cana-6235	24	38	sets	set	NOUN
cana-6235	24	39	are	be	AUX
cana-6235	24	40	considered	consider	VERB
cana-6235	24	41	.	.	PUNCT
cana-6235	25	1	3	3	X
cana-6235	25	2	.	.	X
cana-6235	25	3	definitions	definition	NOUN
cana-6235	25	4	and	and	CCONJ
cana-6235	25	5	examples	example	NOUN
cana-6235	25	6	definition	definition	NOUN
cana-6235	25	7	3.1	3.1	NUM
cana-6235	25	8	:	:	PUNCT
cana-6235	25	9	(	(	PUNCT
cana-6235	25	10	isolate	isolate	VERB
cana-6235	25	11	inclusive	inclusive	ADJ
cana-6235	25	12	set	set	NOUN
cana-6235	25	13	)	)	PUNCT
cana-6235	25	14	let	let	VERB
cana-6235	25	15	𝐺be	𝐺be	PROPN
cana-6235	25	16	a	a	DET
cana-6235	25	17	graph	graph	NOUN
cana-6235	25	18	and	and	CCONJ
cana-6235	25	19	𝑆a	𝑆a	PROPN
cana-6235	25	20	nonempty	nonempty	ADV
cana-6235	25	21	subset	subset	VERB
cana-6235	25	22	of	of	ADP
cana-6235	25	23	𝑉(𝐺	𝑉(𝐺	NOUN
cana-6235	25	24	)	)	PUNCT
cana-6235	25	25	.	.	PUNCT
cana-6235	26	1	the	the	DET
cana-6235	26	2	set	set	ADJ
cana-6235	26	3	𝑆is	𝑆is	PROPN
cana-6235	26	4	called	call	VERB
cana-6235	26	5	an	an	DET
cana-6235	26	6	isolate	isolate	ADJ
cana-6235	26	7	inclusive	inclusive	ADJ
cana-6235	26	8	set	set	NOUN
cana-6235	26	9	if	if	SCONJ
cana-6235	26	10	the	the	DET
cana-6235	26	11	subgraph	subgraph	NOUN
cana-6235	26	12	induced	induce	VERB
cana-6235	26	13	by	by	ADP
cana-6235	26	14	𝑆	𝑆	PROPN
cana-6235	26	15	,	,	PUNCT
cana-6235	26	16	denoted	denote	VERB
cana-6235	26	17	by	by	ADP
cana-6235	26	18	⟨𝑆⟩	⟨𝑆⟩	NOUN
cana-6235	26	19	,	,	PUNCT
cana-6235	26	20	contains	contain	VERB
cana-6235	26	21	at	at	ADV
cana-6235	26	22	least	least	ADV
cana-6235	26	23	one	one	NUM
cana-6235	26	24	isolated	isolated	ADJ
cana-6235	26	25	vertex	vertex	NOUN
cana-6235	26	26	.	.	PUNCT
cana-6235	27	1	an	an	DET
cana-6235	27	2	isolate	isolate	ADJ
cana-6235	27	3	inclusive	inclusive	ADJ
cana-6235	27	4	set	set	NOUN
cana-6235	27	5	of	of	ADP
cana-6235	27	6	maximum	maximum	ADJ
cana-6235	27	7	cardinality	cardinality	NOUN
cana-6235	27	8	is	be	AUX
cana-6235	27	9	referred	refer	VERB
cana-6235	27	10	to	to	ADP
cana-6235	27	11	as	as	ADP
cana-6235	27	12	a	a	DET
cana-6235	27	13	maximum	maximum	PROPN
cana-6235	27	14	isolate	isolate	NOUN
cana-6235	27	15	inclusive	inclusive	ADJ
cana-6235	27	16	set	set	NOUN
cana-6235	27	17	,	,	PUNCT
cana-6235	27	18	and	and	CCONJ
cana-6235	27	19	its	its	PRON
cana-6235	27	20	cardinality	cardinality	NOUN
cana-6235	27	21	is	be	AUX
cana-6235	27	22	denoted	denote	VERB
cana-6235	27	23	by	by	ADP
cana-6235	27	24	𝛽is(𝐺	𝛽is(𝐺	PROPN
cana-6235	27	25	)	)	PUNCT
cana-6235	27	26	.	.	PUNCT
cana-6235	28	1	definition	definition	NOUN
cana-6235	28	2	3.2	3.2	NUM
cana-6235	28	3	:	:	PUNCT
cana-6235	28	4	(	(	PUNCT
cana-6235	28	5	1	1	NUM
cana-6235	28	6	-	-	PUNCT
cana-6235	28	7	maximal	maximal	ADJ
cana-6235	28	8	isolate	isolate	NOUN
cana-6235	28	9	inclusive	inclusive	ADJ
cana-6235	28	10	set	set	NOUN
cana-6235	28	11	)	)	PUNCT
cana-6235	28	12	let	let	VERB
cana-6235	28	13	g	g	NOUN
cana-6235	28	14	be	be	AUX
cana-6235	28	15	a	a	DET
cana-6235	28	16	graph	graph	NOUN
cana-6235	28	17	and	and	CCONJ
cana-6235	28	18	s	s	AUX
cana-6235	28	19	be	be	AUX
cana-6235	28	20	a	a	DET
cana-6235	28	21	isolate	isolate	ADJ
cana-6235	28	22	inclusive	inclusive	ADJ
cana-6235	28	23	set	set	NOUN
cana-6235	28	24	of	of	ADP
cana-6235	28	25	g	g	PROPN
cana-6235	28	26	then	then	ADV
cana-6235	28	27	s	s	VERB
cana-6235	28	28	is	be	AUX
cana-6235	28	29	said	say	VERB
cana-6235	28	30	to	to	PART
cana-6235	28	31	be	be	AUX
cana-6235	28	32	a	a	DET
cana-6235	28	33	1	1	NUM
cana-6235	28	34	-	-	PUNCT
cana-6235	28	35	maximal	maximal	ADJ
cana-6235	28	36	isolate	isolate	NOUN
cana-6235	28	37	inclusive	inclusive	ADJ
cana-6235	28	38	set	set	NOUN
cana-6235	28	39	if	if	SCONJ
cana-6235	28	40	s	s	NOUN
cana-6235	28	41	∪	∪	X
cana-6235	28	42	{	{	PUNCT
cana-6235	28	43	v	v	NOUN
cana-6235	28	44	}	}	PUNCT
cana-6235	28	45	is	be	AUX
cana-6235	28	46	not	not	PART
cana-6235	28	47	an	an	DET
cana-6235	28	48	isolate	isolate	ADJ
cana-6235	28	49	inclusive	inclusive	ADJ
cana-6235	28	50	set	set	NOUN
cana-6235	28	51	,	,	PUNCT
cana-6235	28	52	for	for	ADP
cana-6235	28	53	every	every	DET
cana-6235	28	54	v	v	PRON
cana-6235	28	55	∈	∈	PROPN
cana-6235	28	56	v(g	v(g	NOUN
cana-6235	28	57	)	)	PUNCT
cana-6235	29	1	−	−	PROPN
cana-6235	29	2	s	s	PART
cana-6235	29	3	.	.	PUNCT
cana-6235	30	1	definition	definition	NOUN
cana-6235	30	2	3.3	3.3	NUM
cana-6235	30	3	:	:	PUNCT
cana-6235	30	4	(	(	PUNCT
cana-6235	30	5	minimal	minimal	ADJ
cana-6235	30	6	set	set	NOUN
cana-6235	30	7	)	)	PUNCT
cana-6235	30	8	a	a	DET
cana-6235	30	9	maximal	maximal	ADJ
cana-6235	30	10	isolate	isolate	NOUN
cana-6235	30	11	inclusive	inclusive	ADJ
cana-6235	30	12	set	set	VERB
cana-6235	30	13	with	with	ADP
cana-6235	30	14	minimum	minimum	ADJ
cana-6235	30	15	cardinality	cardinality	NOUN
cana-6235	30	16	is	be	AUX
cana-6235	30	17	called	call	VERB
cana-6235	30	18	a	a	DET
cana-6235	30	19	minimal	minimal	ADJ
cana-6235	30	20	set	set	NOUN
cana-6235	30	21	and	and	CCONJ
cana-6235	30	22	its	its	PRON
cana-6235	30	23	cardinality	cardinality	NOUN
cana-6235	30	24	is	be	AUX
cana-6235	30	25	denoted	denote	VERB
cana-6235	30	26	as	as	ADP
cana-6235	30	27	mis(g	mis(g	PROPN
cana-6235	30	28	)	)	PUNCT
cana-6235	31	1	and	and	CCONJ
cana-6235	31	2	it	it	PRON
cana-6235	31	3	’s	’s	AUX
cana-6235	31	4	called	call	VERB
cana-6235	31	5	the	the	DET
cana-6235	31	6	minimal	minimal	ADJ
cana-6235	31	7	number	number	NOUN
cana-6235	31	8	of	of	ADP
cana-6235	31	9	the	the	DET
cana-6235	31	10	graph	graph	NOUN
cana-6235	31	11	g.	g.	PROPN
cana-6235	31	12	let	let	VERB
cana-6235	31	13	g	g	NOUN
cana-6235	31	14	be	be	AUX
cana-6235	31	15	a	a	DET
cana-6235	31	16	graph	graph	NOUN
cana-6235	31	17	&	&	CCONJ
cana-6235	31	18	v	v	ADP
cana-6235	31	19	∈	∈	PROPN
cana-6235	31	20	v(g	v(g	NOUN
cana-6235	31	21	)	)	PUNCT
cana-6235	31	22	such	such	ADJ
cana-6235	31	23	that	that	SCONJ
cana-6235	31	24	d(v	d(v	PROPN
cana-6235	31	25	)	)	PUNCT
cana-6235	32	1	=	=	SYM
cana-6235	32	2	∇(g	∇(g	PROPN
cana-6235	32	3	)	)	PUNCT
cana-6235	32	4	.	.	PUNCT
cana-6235	33	1	now	now	ADV
cana-6235	33	2	v(g	v(g	ADJ
cana-6235	33	3	)	)	PUNCT
cana-6235	33	4	−	−	PROPN
cana-6235	33	5	n(v	n(v	PROPN
cana-6235	33	6	)	)	PUNCT
cana-6235	33	7	is	be	AUX
cana-6235	33	8	an	an	DET
cana-6235	33	9	isolate	isolate	ADJ
cana-6235	33	10	inclusive	inclusive	ADJ
cana-6235	33	11	set	set	NOUN
cana-6235	33	12	of	of	ADP
cana-6235	33	13	g	g	PROPN
cana-6235	34	1	but	but	CCONJ
cana-6235	34	2	it	it	PRON
cana-6235	34	3	did	do	AUX
cana-6235	34	4	not	not	PART
cana-6235	34	5	be	be	AUX
cana-6235	34	6	a	a	DET
cana-6235	34	7	1	1	NUM
cana-6235	34	8	-	-	PUNCT
cana-6235	34	9	maximal	maximal	ADJ
cana-6235	34	10	isolate	isolate	NOUN
cana-6235	34	11	inclusive	inclusive	ADJ
cana-6235	34	12	set	set	PROPN
cana-6235	34	13	of	of	ADP
cana-6235	34	14	g.	g.	PROPN
cana-6235	34	15	this	this	PRON
cana-6235	34	16	can	can	AUX
cana-6235	34	17	be	be	AUX
cana-6235	34	18	observed	observe	VERB
cana-6235	34	19	in	in	ADP
cana-6235	34	20	following	follow	VERB
cana-6235	34	21	example	example	NOUN
cana-6235	34	22	.	.	PUNCT
cana-6235	35	1	example	example	NOUN
cana-6235	35	2	1	1	NUM
cana-6235	35	3	:	:	PUNCT
cana-6235	35	4	consider	consider	VERB
cana-6235	35	5	the	the	DET
cana-6235	35	6	path	path	NOUN
cana-6235	35	7	graph	graph	NOUN
cana-6235	35	8	p5	p5	ADJ
cana-6235	35	9	with	with	ADP
cana-6235	35	10	5	5	NUM
cana-6235	35	11	vertices	vertex	NOUN
cana-6235	35	12	{	{	PUNCT
cana-6235	35	13	1	1	NUM
cana-6235	35	14	,	,	PUNCT
cana-6235	35	15	2	2	NUM
cana-6235	35	16	,	,	PUNCT
cana-6235	35	17	3	3	NUM
cana-6235	35	18	,	,	PUNCT
cana-6235	35	19	4	4	NUM
cana-6235	35	20	,	,	PUNCT
cana-6235	35	21	5	5	NUM
cana-6235	35	22	}	}	PUNCT
cana-6235	35	23	figure	figure	NOUN
cana-6235	35	24	1	1	NUM
cana-6235	35	25	.	.	PUNCT
cana-6235	35	26	path	path	NOUN
cana-6235	35	27	graph	graph	NOUN
cana-6235	35	28	consider	consider	VERB
cana-6235	35	29	the	the	DET
cana-6235	35	30	vertex	vertex	NOUN
cana-6235	35	31	3	3	NUM
cana-6235	35	32	.	.	PUNCT
cana-6235	36	1	d(3	d(3	NOUN
cana-6235	36	2	)	)	PUNCT
cana-6235	36	3	=	=	SYM
cana-6235	36	4	2	2	NUM
cana-6235	36	5	=	=	SYM
cana-6235	36	6	∇(g	∇(g	PROPN
cana-6235	36	7	)	)	PUNCT
cana-6235	36	8	.	.	PUNCT
cana-6235	37	1	communications	communication	NOUN
cana-6235	37	2	on	on	ADP
cana-6235	37	3	applied	apply	VERB
cana-6235	37	4	nonlinear	nonlinear	ADJ
cana-6235	37	5	analysis	analysis	NOUN
cana-6235	37	6	issn	issn	NOUN
cana-6235	37	7	:	:	PUNCT
cana-6235	37	8	1074	1074	NUM
cana-6235	37	9	-	-	PUNCT
cana-6235	37	10	133x	133x	NUM
cana-6235	37	11	vol	vol	NOUN
cana-6235	37	12	31	31	NUM
cana-6235	37	13	no	no	NOUN
cana-6235	37	14	.	.	NOUN
cana-6235	37	15	2	2	NUM
cana-6235	37	16	(	(	PUNCT
cana-6235	37	17	2024	2024	NUM
cana-6235	37	18	)	)	PUNCT
cana-6235	37	19	512	512	NUM
cana-6235	37	20	https://internationalpubls.com	https://internationalpubls.com	X
cana-6235	37	21	n(3	n(3	PROPN
cana-6235	37	22	)	)	PUNCT
cana-6235	37	23	=	=	PRON
cana-6235	38	1	{	{	PUNCT
cana-6235	38	2	2	2	NUM
cana-6235	38	3	,	,	PUNCT
cana-6235	38	4	4	4	NUM
cana-6235	38	5	}	}	PUNCT
cana-6235	38	6	&	&	CCONJ
cana-6235	38	7	v(g	v(g	ADJ
cana-6235	38	8	)	)	PUNCT
cana-6235	38	9	−	−	PROPN
cana-6235	38	10	n(3	n(3	PROPN
cana-6235	38	11	)	)	PUNCT
cana-6235	39	1	=	=	PRON
cana-6235	39	2	{	{	PUNCT
cana-6235	39	3	1	1	NUM
cana-6235	39	4	,	,	PUNCT
cana-6235	39	5	3	3	NUM
cana-6235	39	6	,	,	PUNCT
cana-6235	39	7	5	5	NUM
cana-6235	39	8	}	}	PUNCT
cana-6235	39	9	this	this	DET
cana-6235	39	10	set	set	NOUN
cana-6235	39	11	is	be	AUX
cana-6235	39	12	an	an	DET
cana-6235	39	13	isolate	isolate	ADJ
cana-6235	39	14	inclusive	inclusive	ADJ
cana-6235	39	15	set	set	NOUN
cana-6235	39	16	but	but	CCONJ
cana-6235	39	17	it	it	PRON
cana-6235	39	18	is	be	AUX
cana-6235	39	19	not	not	PART
cana-6235	39	20	1	1	NUM
cana-6235	39	21	-	-	PUNCT
cana-6235	39	22	maximal	maximal	ADJ
cana-6235	39	23	.	.	PUNCT
cana-6235	40	1	definition	definition	NOUN
cana-6235	40	2	3.4	3.4	NUM
cana-6235	40	3	:	:	PUNCT
cana-6235	40	4	(	(	PUNCT
cana-6235	40	5	isolate	isolate	VERB
cana-6235	40	6	dominating	dominating	NOUN
cana-6235	40	7	set	set	NOUN
cana-6235	40	8	)	)	PUNCT
cana-6235	40	9	let	let	VERB
cana-6235	40	10	g	g	NOUN
cana-6235	40	11	be	be	AUX
cana-6235	40	12	a	a	DET
cana-6235	40	13	graph	graph	NOUN
cana-6235	40	14	and	and	CCONJ
cana-6235	40	15	s	s	NOUN
cana-6235	40	16	⊂	⊂	X
cana-6235	40	17	v(g)then	v(g)then	X
cana-6235	40	18	s	s	NOUN
cana-6235	40	19	is	be	AUX
cana-6235	40	20	said	say	VERB
cana-6235	40	21	to	to	PART
cana-6235	40	22	be	be	AUX
cana-6235	40	23	an	an	DET
cana-6235	40	24	isolate	isolate	ADJ
cana-6235	40	25	dominating	dominating	NOUN
cana-6235	40	26	set	set	VERB
cana-6235	40	27	if	if	SCONJ
cana-6235	40	28	1	1	NUM
cana-6235	40	29	.	.	X
cana-6235	40	30	s	s	PART
cana-6235	40	31	is	be	AUX
cana-6235	40	32	a	a	DET
cana-6235	40	33	dominating	dominating	NOUN
cana-6235	40	34	set	set	NOUN
cana-6235	40	35	and	and	CCONJ
cana-6235	40	36	2	2	NUM
cana-6235	40	37	.	.	PUNCT
cana-6235	41	1	<	<	X
cana-6235	41	2	s	s	X
cana-6235	41	3	>	>	X
cana-6235	41	4	contains	contain	VERB
cana-6235	41	5	an	an	DET
cana-6235	41	6	isolated	isolated	ADJ
cana-6235	41	7	vertex	vertex	NOUN
cana-6235	41	8	.	.	PUNCT
cana-6235	42	1	an	an	DET
cana-6235	42	2	isolate	isolate	NOUN
cana-6235	42	3	dominating	dominating	NOUN
cana-6235	42	4	set	set	VERB
cana-6235	42	5	with	with	ADP
cana-6235	42	6	minimum	minimum	ADJ
cana-6235	42	7	cardinality	cardinality	NOUN
cana-6235	42	8	is	be	AUX
cana-6235	42	9	called	call	VERB
cana-6235	42	10	a	a	DET
cana-6235	42	11	minimum	minimum	NOUN
cana-6235	42	12	isolate	isolate	NOUN
cana-6235	42	13	dominating	dominating	NOUN
cana-6235	42	14	set	set	NOUN
cana-6235	42	15	.	.	PUNCT
cana-6235	43	1	the	the	DET
cana-6235	43	2	cardinality	cardinality	NOUN
cana-6235	43	3	of	of	ADP
cana-6235	43	4	a	a	DET
cana-6235	43	5	minimum	minimum	NOUN
cana-6235	43	6	isolate	isolate	NOUN
cana-6235	43	7	dominating	dominating	NOUN
cana-6235	43	8	set	set	NOUN
cana-6235	43	9	is	be	AUX
cana-6235	43	10	called	call	VERB
cana-6235	43	11	the	the	DET
cana-6235	43	12	isolate	isolate	ADJ
cana-6235	43	13	domination	domination	NOUN
cana-6235	43	14	number	number	NOUN
cana-6235	43	15	of	of	ADP
cana-6235	43	16	the	the	DET
cana-6235	43	17	graph	graph	NOUN
cana-6235	43	18	g	g	NOUN
cana-6235	43	19	and	and	CCONJ
cana-6235	43	20	it	it	PRON
cana-6235	43	21	is	be	AUX
cana-6235	43	22	denoted	denote	VERB
cana-6235	43	23	as	as	ADP
cana-6235	43	24	γ0(g	γ0(g	NOUN
cana-6235	43	25	)	)	PUNCT
cana-6235	43	26	.	.	PUNCT
cana-6235	44	1	obviously	obviously	ADV
cana-6235	44	2	for	for	ADP
cana-6235	44	3	any	any	DET
cana-6235	44	4	graph	graph	NOUN
cana-6235	44	5	g	g	PROPN
cana-6235	44	6	,	,	PUNCT
cana-6235	44	7	γ(g	γ(g	PROPN
cana-6235	44	8	)	)	PUNCT
cana-6235	44	9	≤	≤	PUNCT
cana-6235	45	1	γ0(g	γ0(g	NOUN
cana-6235	45	2	)	)	PUNCT
cana-6235	45	3	where	where	SCONJ
cana-6235	45	4	γ(g	γ(g	PROPN
cana-6235	45	5	)	)	PUNCT
cana-6235	45	6	denotes	denote	VERB
cana-6235	45	7	the	the	DET
cana-6235	45	8	domination	domination	NOUN
cana-6235	45	9	number	number	NOUN
cana-6235	45	10	of	of	ADP
cana-6235	45	11	the	the	DET
cana-6235	45	12	graph	graph	NOUN
cana-6235	45	13	g.	g.	NOUN
cana-6235	45	14	remark	remark	PROPN
cana-6235	45	15	:	:	PUNCT
cana-6235	45	16	note	note	VERB
cana-6235	45	17	that	that	SCONJ
cana-6235	45	18	every	every	DET
cana-6235	45	19	1	1	NUM
cana-6235	45	20	-	-	PUNCT
cana-6235	45	21	maximal	maximal	ADJ
cana-6235	45	22	isolate	isolate	NOUN
cana-6235	45	23	inclusive	inclusive	ADJ
cana-6235	45	24	set	set	NOUN
cana-6235	45	25	is	be	AUX
cana-6235	45	26	an	an	DET
cana-6235	45	27	isolate	isolate	ADJ
cana-6235	45	28	dominating	dominating	NOUN
cana-6235	45	29	set	set	NOUN
cana-6235	45	30	,	,	PUNCT
cana-6235	45	31	but	but	CCONJ
cana-6235	45	32	converse	converse	NOUN
cana-6235	45	33	is	be	AUX
cana-6235	45	34	not	not	PART
cana-6235	45	35	true	true	ADJ
cana-6235	45	36	.	.	PUNCT
cana-6235	46	1	we	we	PRON
cana-6235	46	2	introduce	introduce	VERB
cana-6235	46	3	the	the	DET
cana-6235	46	4	following	follow	VERB
cana-6235	46	5	symbols	symbol	NOUN
cana-6235	46	6	:	:	PUNCT
cana-6235	46	7	v0	v0	NOUN
cana-6235	46	8	+	+	NOUN
cana-6235	46	9	=	=	SYM
cana-6235	46	10	{	{	PUNCT
cana-6235	46	11	x	x	PUNCT
cana-6235	46	12	∈	∈	PROPN
cana-6235	46	13	v(g	v(g	ADJ
cana-6235	46	14	)	)	PUNCT
cana-6235	46	15	∋	∋	NOUN
cana-6235	47	1	γ0(g	γ0(g	NOUN
cana-6235	47	2	−	−	NOUN
cana-6235	47	3	x	x	NOUN
cana-6235	47	4	)	)	PUNCT
cana-6235	47	5	>	>	X
cana-6235	48	1	γ0(g	γ0(g	X
cana-6235	48	2	)	)	PUNCT
cana-6235	48	3	}	}	PUNCT
cana-6235	48	4	v0	v0	NOUN
cana-6235	48	5	−	−	NOUN
cana-6235	49	1	=	=	SYM
cana-6235	50	1	{	{	PUNCT
cana-6235	50	2	x	x	PUNCT
cana-6235	50	3	∈	∈	PROPN
cana-6235	50	4	v(g	v(g	ADJ
cana-6235	50	5	)	)	PUNCT
cana-6235	50	6	∋	∋	NOUN
cana-6235	51	1	γ0(g	γ0(g	NOUN
cana-6235	51	2	−	−	NOUN
cana-6235	51	3	x	x	X
cana-6235	51	4	)	)	PUNCT
cana-6235	51	5	<	<	X
cana-6235	51	6	γ0(g	γ0(g	X
cana-6235	51	7	)	)	PUNCT
cana-6235	51	8	}	}	PUNCT
cana-6235	51	9	v0	v0	NOUN
cana-6235	51	10	0	0	NUM
cana-6235	52	1	=	=	SYM
cana-6235	53	1	{	{	PUNCT
cana-6235	53	2	x	x	PUNCT
cana-6235	53	3	∈	∈	PROPN
cana-6235	53	4	v(g	v(g	ADJ
cana-6235	53	5	)	)	PUNCT
cana-6235	53	6	∋	∋	NOUN
cana-6235	54	1	γ0(g	γ0(g	NOUN
cana-6235	54	2	−	−	NOUN
cana-6235	54	3	x	x	X
cana-6235	54	4	)	)	PUNCT
cana-6235	54	5	=	=	PUNCT
cana-6235	54	6	γ0(g	γ0(g	NOUN
cana-6235	54	7	)	)	PUNCT
cana-6235	54	8	}	}	PUNCT
cana-6235	54	9	4	4	NUM
cana-6235	54	10	.	.	X
cana-6235	54	11	main	main	ADJ
cana-6235	54	12	result	result	NOUN
cana-6235	54	13	proposition	proposition	NOUN
cana-6235	54	14	4.1	4.1	NUM
cana-6235	54	15	:	:	PUNCT
cana-6235	54	16	let	let	VERB
cana-6235	54	17	g	g	PRON
cana-6235	54	18	be	be	AUX
cana-6235	54	19	a	a	DET
cana-6235	54	20	graph	graph	NOUN
cana-6235	54	21	&	&	CCONJ
cana-6235	54	22	s	s	PART
cana-6235	54	23	be	be	AUX
cana-6235	54	24	a	a	DET
cana-6235	54	25	1	1	NUM
cana-6235	54	26	-	-	PUNCT
cana-6235	54	27	maximal	maximal	ADJ
cana-6235	54	28	isolate	isolate	NOUN
cana-6235	54	29	inclusive	inclusive	ADJ
cana-6235	54	30	set	set	PROPN
cana-6235	54	31	of	of	ADP
cana-6235	54	32	g.	g.	PROPN
cana-6235	54	33	(	(	PUNCT
cana-6235	54	34	1	1	NUM
cana-6235	54	35	)	)	PUNCT
cana-6235	54	36	for	for	ADP
cana-6235	54	37	each	each	DET
cana-6235	54	38	isolated	isolate	VERB
cana-6235	54	39	vertex	vertex	NOUN
cana-6235	54	40	v	v	NOUN
cana-6235	54	41	of	of	ADP
cana-6235	54	42	s	s	PROPN
cana-6235	54	43	,	,	PUNCT
cana-6235	54	44	n(v	n(v	PROPN
cana-6235	54	45	)	)	PUNCT
cana-6235	54	46	=	=	SYM
cana-6235	54	47	v(g	v(g	ADJ
cana-6235	54	48	)	)	PUNCT
cana-6235	55	1	−	−	PROPN
cana-6235	55	2	s	s	PART
cana-6235	55	3	.	.	PUNCT
cana-6235	56	1	(	(	PUNCT
cana-6235	56	2	2	2	X
cana-6235	56	3	)	)	PUNCT
cana-6235	56	4	if	if	SCONJ
cana-6235	56	5	u	u	PROPN
cana-6235	56	6	&	&	CCONJ
cana-6235	56	7	v	v	PROPN
cana-6235	56	8	are	be	AUX
cana-6235	56	9	isolates	isolate	NOUN
cana-6235	56	10	of	of	ADP
cana-6235	56	11	s	s	PRON
cana-6235	56	12	then	then	ADV
cana-6235	56	13	d(u	d(u	PROPN
cana-6235	56	14	)	)	PUNCT
cana-6235	56	15	=	=	SYM
cana-6235	56	16	d(v	d(v	PROPN
cana-6235	56	17	)	)	PUNCT
cana-6235	56	18	.	.	PUNCT
cana-6235	57	1	proof	proof	NOUN
cana-6235	57	2	:	:	PUNCT
cana-6235	57	3	(	(	PUNCT
cana-6235	57	4	1	1	X
cana-6235	57	5	)	)	PUNCT
cana-6235	57	6	let	let	VERB
cana-6235	57	7	v	v	PART
cana-6235	57	8	be	be	AUX
cana-6235	57	9	an	an	DET
cana-6235	57	10	isolated	isolated	ADJ
cana-6235	57	11	vertex	vertex	NOUN
cana-6235	57	12	of	of	ADP
cana-6235	57	13	sthen	sthen	ADJ
cana-6235	57	14	n(v	n(v	PROPN
cana-6235	57	15	)	)	PUNCT
cana-6235	57	16	⊂	⊂	PROPN
cana-6235	57	17	v(g	v(g	PROPN
cana-6235	57	18	)	)	PUNCT
cana-6235	57	19	−	−	PROPN
cana-6235	58	1	s.	s.	PROPN
cana-6235	58	2	let	let	VERB
cana-6235	58	3	x	x	X
cana-6235	58	4	∈	∈	PROPN
cana-6235	58	5	v(g	v(g	ADJ
cana-6235	58	6	)	)	PUNCT
cana-6235	58	7	−	−	NOUN
cana-6235	58	8	s.since	s.since	NOUN
cana-6235	58	9	s	s	VERB
cana-6235	58	10	is	be	AUX
cana-6235	58	11	1	1	NUM
cana-6235	58	12	-	-	PUNCT
cana-6235	58	13	maximal	maximal	ADJ
cana-6235	58	14	,	,	PUNCT
cana-6235	58	15	x	x	X
cana-6235	58	16	is	be	AUX
cana-6235	58	17	adjacent	adjacent	ADJ
cana-6235	58	18	to	to	ADP
cana-6235	58	19	every	every	DET
cana-6235	58	20	isolated	isolate	VERB
cana-6235	58	21	vertex	vertex	NOUN
cana-6235	58	22	of	of	ADP
cana-6235	58	23	s	s	PRON
cana-6235	58	24	and	and	CCONJ
cana-6235	58	25	therefore	therefore	ADV
cana-6235	58	26	x	x	VERB
cana-6235	58	27	is	be	AUX
cana-6235	58	28	adjacent	adjacent	ADJ
cana-6235	58	29	to	to	ADP
cana-6235	58	30	v	v	ADP
cana-6235	58	31	which	which	PRON
cana-6235	58	32	implies	imply	VERB
cana-6235	58	33	that	that	SCONJ
cana-6235	58	34	x	x	SYM
cana-6235	58	35	∈	∈	PROPN
cana-6235	58	36	n(v	n(v	PROPN
cana-6235	58	37	)	)	PUNCT
cana-6235	58	38	.	.	PUNCT
cana-6235	59	1	thus	thus	ADV
cana-6235	59	2	,	,	PUNCT
cana-6235	59	3	(	(	PUNCT
cana-6235	59	4	v	v	NOUN
cana-6235	59	5	)	)	PUNCT
cana-6235	59	6	=	=	SYM
cana-6235	59	7	v(g	v(g	ADJ
cana-6235	59	8	)	)	PUNCT
cana-6235	60	1	−	−	PROPN
cana-6235	60	2	s	s	PART
cana-6235	60	3	.	.	PUNCT
cana-6235	61	1	(	(	PUNCT
cana-6235	61	2	2	2	X
cana-6235	61	3	)	)	PUNCT
cana-6235	61	4	let	let	VERB
cana-6235	61	5	u	u	PRON
cana-6235	61	6	&	&	CCONJ
cana-6235	61	7	v	v	NOUN
cana-6235	61	8	be	be	AUX
cana-6235	61	9	to	to	PART
cana-6235	61	10	isolate	isolate	VERB
cana-6235	61	11	sets	set	NOUN
cana-6235	61	12	of	of	ADP
cana-6235	61	13	sthen	sthen	ADJ
cana-6235	61	14	d(u	d(u	PROPN
cana-6235	61	15	)	)	PUNCT
cana-6235	62	1	=	=	PUNCT
cana-6235	62	2	|n(u)|	|n(u)|	NOUN
cana-6235	62	3	=	=	SYM
cana-6235	62	4	|v(g	|v(g	NOUN
cana-6235	62	5	)	)	PUNCT
cana-6235	62	6	−	−	PROPN
cana-6235	63	1	s|	s|	NOUN
cana-6235	63	2	=	=	SYM
cana-6235	64	1	|n(v)|	|n(v)|	NUM
cana-6235	64	2	=	=	SYM
cana-6235	64	3	d(v	d(v	PROPN
cana-6235	64	4	)	)	PUNCT
cana-6235	64	5	.	.	PUNCT
cana-6235	65	1	thus	thus	ADV
cana-6235	65	2	,	,	PUNCT
cana-6235	65	3	d(u	d(u	PROPN
cana-6235	65	4	)	)	PUNCT
cana-6235	65	5	=	=	SYM
cana-6235	65	6	d(v	d(v	PROPN
cana-6235	65	7	)	)	PUNCT
cana-6235	65	8	.	.	PUNCT
cana-6235	65	9	▐	▐	PROPN
cana-6235	65	10	proposition	proposition	NOUN
cana-6235	65	11	4.2	4.2	NUM
cana-6235	65	12	:	:	PUNCT
cana-6235	65	13	let	let	VERB
cana-6235	65	14	g	g	PRON
cana-6235	65	15	be	be	AUX
cana-6235	65	16	a	a	DET
cana-6235	65	17	graph	graph	NOUN
cana-6235	65	18	and	and	CCONJ
cana-6235	65	19	v	v	ADP
cana-6235	65	20	∈	∈	PROPN
cana-6235	65	21	v(g	v(g	NOUN
cana-6235	65	22	)	)	PUNCT
cana-6235	66	1	then	then	ADV
cana-6235	66	2	βis(g	βis(g	PRON
cana-6235	66	3	−	−	PROPN
cana-6235	66	4	v	v	NOUN
cana-6235	66	5	)	)	PUNCT
cana-6235	66	6	≤	≤	NOUN
cana-6235	66	7	βis(g	βis(g	PRON
cana-6235	66	8	)	)	PUNCT
cana-6235	66	9	.	.	PUNCT
cana-6235	67	1	communications	communication	NOUN
cana-6235	67	2	on	on	ADP
cana-6235	67	3	applied	apply	VERB
cana-6235	67	4	nonlinear	nonlinear	ADJ
cana-6235	67	5	analysis	analysis	NOUN
cana-6235	67	6	issn	issn	NOUN
cana-6235	67	7	:	:	PUNCT
cana-6235	67	8	1074	1074	NUM
cana-6235	67	9	-	-	PUNCT
cana-6235	67	10	133x	133x	NUM
cana-6235	67	11	vol	vol	NOUN
cana-6235	67	12	31	31	NUM
cana-6235	67	13	no	no	NOUN
cana-6235	67	14	.	.	NOUN
cana-6235	67	15	2	2	NUM
cana-6235	67	16	(	(	PUNCT
cana-6235	67	17	2024	2024	NUM
cana-6235	67	18	)	)	PUNCT
cana-6235	67	19	513	513	NUM
cana-6235	67	20	https://internationalpubls.com	https://internationalpubls.com	X
cana-6235	67	21	proof	proof	NOUN
cana-6235	67	22	:	:	PUNCT
cana-6235	67	23	let	let	VERB
cana-6235	67	24	m	m	PRON
cana-6235	67	25	be	be	AUX
cana-6235	67	26	a	a	DET
cana-6235	67	27	maximum	maximum	ADJ
cana-6235	67	28	isolate	isolate	NOUN
cana-6235	67	29	inclusive	inclusive	ADJ
cana-6235	67	30	set	set	NOUN
cana-6235	67	31	of	of	ADP
cana-6235	67	32	g	g	PROPN
cana-6235	67	33	−	−	PROPN
cana-6235	67	34	v	v	NOUN
cana-6235	67	35	.	.	PUNCT
cana-6235	68	1	by	by	ADP
cana-6235	68	2	the	the	DET
cana-6235	68	3	above	above	ADJ
cana-6235	68	4	proposition	proposition	NOUN
cana-6235	68	5	4.2	4.2	NUM
cana-6235	68	6	,	,	PUNCT
cana-6235	68	7	m	m	VERB
cana-6235	68	8	is	be	AUX
cana-6235	68	9	also	also	ADV
cana-6235	68	10	an	an	DET
cana-6235	68	11	isolate	isolate	ADJ
cana-6235	68	12	inclusive	inclusive	ADJ
cana-6235	68	13	set	set	NOUN
cana-6235	68	14	of	of	ADP
cana-6235	68	15	g	g	PROPN
cana-6235	68	16	.	.	PUNCT
cana-6235	69	1	therefore	therefore	ADV
cana-6235	69	2	,	,	PUNCT
cana-6235	69	3	βis(g	βis(g	PRON
cana-6235	69	4	−	−	PROPN
cana-6235	69	5	v	v	NOUN
cana-6235	69	6	)	)	PUNCT
cana-6235	69	7	≤	≤	NOUN
cana-6235	69	8	βis(g	βis(g	PRON
cana-6235	69	9	)	)	PUNCT
cana-6235	69	10	▐	▐	NUM
cana-6235	69	11	example	example	NOUN
cana-6235	69	12	2	2	NUM
cana-6235	69	13	:	:	PUNCT
cana-6235	69	14	consider	consider	VERB
cana-6235	69	15	the	the	DET
cana-6235	69	16	path	path	NOUN
cana-6235	69	17	graph	graph	NOUN
cana-6235	69	18	p5	p5	ADJ
cana-6235	69	19	with	with	ADP
cana-6235	69	20	vertices	vertex	NOUN
cana-6235	69	21	{	{	PUNCT
cana-6235	69	22	1	1	NUM
cana-6235	69	23	,	,	PUNCT
cana-6235	69	24	2	2	NUM
cana-6235	69	25	,	,	PUNCT
cana-6235	69	26	3	3	NUM
cana-6235	69	27	,	,	PUNCT
cana-6235	69	28	4	4	NUM
cana-6235	69	29	,	,	PUNCT
cana-6235	69	30	5	5	NUM
cana-6235	69	31	}	}	PUNCT
cana-6235	69	32	figure	figure	NOUN
cana-6235	69	33	2	2	NUM
cana-6235	69	34	.	.	NOUN
cana-6235	69	35	path	path	NOUN
cana-6235	69	36	graph	graph	NOUN
cana-6235	69	37	here	here	ADV
cana-6235	69	38	,	,	PUNCT
cana-6235	69	39	βis(g	βis(g	PRON
cana-6235	69	40	)	)	PUNCT
cana-6235	70	1	=	=	SYM
cana-6235	70	2	4	4	X
cana-6235	70	3	.	.	PUNCT
cana-6235	71	1	now	now	ADV
cana-6235	71	2	consider	consider	VERB
cana-6235	71	3	the	the	DET
cana-6235	71	4	subgraph	subgraph	NOUN
cana-6235	71	5	g	g	PROPN
cana-6235	71	6	−	−	PROPN
cana-6235	71	7	3	3	NUM
cana-6235	71	8	.	.	PUNCT
cana-6235	72	1	figure	figure	VERB
cana-6235	72	2	3	3	NUM
cana-6235	72	3	.	.	NOUN
cana-6235	72	4	path	path	NOUN
cana-6235	72	5	graph	graph	NOUN
cana-6235	72	6	here	here	ADV
cana-6235	72	7	,	,	PUNCT
cana-6235	72	8	βis(g	βis(g	PRON
cana-6235	72	9	−	−	NOUN
cana-6235	72	10	3	3	X
cana-6235	72	11	)	)	PUNCT
cana-6235	73	1	=	=	SYM
cana-6235	73	2	3	3	X
cana-6235	73	3	.	.	PUNCT
cana-6235	74	1	therefore	therefore	ADV
cana-6235	74	2	,	,	PUNCT
cana-6235	74	3	in	in	ADP
cana-6235	74	4	this	this	DET
cana-6235	74	5	example	example	NOUN
cana-6235	74	6	βis(g	βis(g	PUNCT
cana-6235	74	7	−	−	NOUN
cana-6235	74	8	3	3	NUM
cana-6235	74	9	)	)	PUNCT
cana-6235	74	10	<	<	X
cana-6235	74	11	βis(g	βis(g	PROPN
cana-6235	74	12	)	)	PUNCT
cana-6235	74	13	.	.	PUNCT
cana-6235	75	1	theorem	theorem	VERB
cana-6235	75	2	4.3	4.3	NUM
cana-6235	75	3	:	:	PUNCT
cana-6235	75	4	let	let	VERB
cana-6235	75	5	g	g	PRON
cana-6235	75	6	be	be	AUX
cana-6235	75	7	a	a	DET
cana-6235	75	8	graph	graph	NOUN
cana-6235	75	9	and	and	CCONJ
cana-6235	75	10	v	v	ADP
cana-6235	75	11	∈	∈	PROPN
cana-6235	75	12	v(g	v(g	NOUN
cana-6235	75	13	)	)	PUNCT
cana-6235	76	1	then	then	ADV
cana-6235	76	2	βis(g	βis(g	PRON
cana-6235	76	3	−	−	PROPN
cana-6235	76	4	v	v	NOUN
cana-6235	76	5	)	)	PUNCT
cana-6235	76	6	=	=	SYM
cana-6235	77	1	βis(g	βis(g	X
cana-6235	77	2	)	)	PUNCT
cana-6235	77	3	if	if	SCONJ
cana-6235	77	4	and	and	CCONJ
cana-6235	77	5	only	only	ADV
cana-6235	77	6	if	if	SCONJ
cana-6235	77	7	there	there	PRON
cana-6235	77	8	is	be	VERB
cana-6235	77	9	a	a	DET
cana-6235	77	10	maximum	maximum	PROPN
cana-6235	77	11	isolate	isolate	NOUN
cana-6235	77	12	inclusive	inclusive	ADJ
cana-6235	77	13	set	set	PROPN
cana-6235	77	14	m(g	m(g	PROPN
cana-6235	77	15	)	)	PUNCT
cana-6235	77	16	such	such	ADJ
cana-6235	77	17	that	that	DET
cana-6235	77	18	v	v	NUM
cana-6235	77	19	∉	∉	PROPN
cana-6235	77	20	m.	m.	NOUN
cana-6235	77	21	proof	proof	NOUN
cana-6235	77	22	:	:	PUNCT
cana-6235	77	23	first	first	ADV
cana-6235	77	24	suppose	suppose	VERB
cana-6235	77	25	that	that	SCONJ
cana-6235	77	26	βis(g	βis(g	PRON
cana-6235	77	27	−	−	PROPN
cana-6235	77	28	v	v	NOUN
cana-6235	77	29	)	)	PUNCT
cana-6235	77	30	=	=	SYM
cana-6235	77	31	βis(g	βis(g	PROPN
cana-6235	77	32	)	)	PUNCT
cana-6235	77	33	that	that	PRON
cana-6235	77	34	m	m	AUX
cana-6235	77	35	be	be	VERB
cana-6235	77	36	a	a	DET
cana-6235	77	37	maximum	maximum	ADJ
cana-6235	77	38	isolate	isolate	NOUN
cana-6235	77	39	inclusive	inclusive	ADJ
cana-6235	77	40	set	set	NOUN
cana-6235	77	41	of	of	ADP
cana-6235	77	42	g	g	PROPN
cana-6235	77	43	−	−	PROPN
cana-6235	77	44	v	v	NOUN
cana-6235	77	45	.	.	PUNCT
cana-6235	78	1	obviously	obviously	ADV
cana-6235	78	2	,	,	PUNCT
cana-6235	78	3	m	m	VERB
cana-6235	78	4	is	be	AUX
cana-6235	78	5	an	an	DET
cana-6235	78	6	isolate	isolate	ADJ
cana-6235	78	7	inclusive	inclusive	ADJ
cana-6235	78	8	set	set	NOUN
cana-6235	78	9	of	of	ADP
cana-6235	78	10	g	g	PROPN
cana-6235	78	11	.	.	PUNCT
cana-6235	79	1	since	since	SCONJ
cana-6235	79	2	βis(g	βis(g	PRON
cana-6235	79	3	−	−	PROPN
cana-6235	79	4	v	v	NOUN
cana-6235	79	5	)	)	PUNCT
cana-6235	79	6	=	=	SYM
cana-6235	79	7	βis(g	βis(g	PROPN
cana-6235	79	8	)	)	PUNCT
cana-6235	79	9	,	,	PUNCT
cana-6235	79	10	m	m	PROPN
cana-6235	79	11	must	must	AUX
cana-6235	79	12	be	be	AUX
cana-6235	79	13	a	a	DET
cana-6235	79	14	maximum	maximum	ADJ
cana-6235	79	15	isolate	isolate	NOUN
cana-6235	79	16	inclusive	inclusive	ADJ
cana-6235	79	17	set	set	NOUN
cana-6235	79	18	of	of	ADP
cana-6235	79	19	g	g	PROPN
cana-6235	79	20	.	.	PUNCT
cana-6235	80	1	note	note	VERB
cana-6235	80	2	that	that	SCONJ
cana-6235	80	3	v	v	NUM
cana-6235	80	4	∉	∉	PROPN
cana-6235	80	5	m	m	VERB
cana-6235	80	6	.	.	PUNCT
cana-6235	81	1	conversely	conversely	ADV
cana-6235	81	2	,	,	PUNCT
cana-6235	81	3	suppose	suppose	VERB
cana-6235	81	4	that	that	SCONJ
cana-6235	81	5	m	m	PROPN
cana-6235	81	6	is	be	AUX
cana-6235	81	7	a	a	DET
cana-6235	81	8	maximum	maximum	PROPN
cana-6235	81	9	isolate	isolate	NOUN
cana-6235	81	10	inclusive	inclusive	ADJ
cana-6235	81	11	set	set	NOUN
cana-6235	81	12	of	of	ADP
cana-6235	81	13	g	g	PROPN
cana-6235	81	14	such	such	ADJ
cana-6235	81	15	that	that	PRON
cana-6235	81	16	v	v	NOUN
cana-6235	81	17	∉	∉	PROPN
cana-6235	81	18	m	m	PROPN
cana-6235	81	19	.	.	PUNCT
cana-6235	82	1	now	now	ADV
cana-6235	82	2	m	m	VERB
cana-6235	82	3	is	be	AUX
cana-6235	82	4	a	a	DET
cana-6235	82	5	subset	subset	NOUN
cana-6235	82	6	of	of	ADP
cana-6235	82	7	g	g	PROPN
cana-6235	82	8	−	−	PROPN
cana-6235	82	9	v	v	PROPN
cana-6235	82	10	&	&	CCONJ
cana-6235	82	11	it	it	PRON
cana-6235	82	12	is	be	AUX
cana-6235	82	13	also	also	ADV
cana-6235	82	14	an	an	DET
cana-6235	82	15	isolate	isolate	ADJ
cana-6235	82	16	inclusive	inclusive	ADJ
cana-6235	82	17	set	set	NOUN
cana-6235	82	18	of	of	ADP
cana-6235	82	19	g	g	PROPN
cana-6235	83	1	−	−	PROPN
cana-6235	84	1	v.	v.	CCONJ
cana-6235	84	2	therefore	therefore	ADV
cana-6235	84	3	,	,	PUNCT
cana-6235	84	4	βis(g	βis(g	PRON
cana-6235	84	5	−	−	PROPN
cana-6235	84	6	v	v	NOUN
cana-6235	84	7	)	)	PUNCT
cana-6235	84	8	≥	≥	NOUN
cana-6235	84	9	|m|	|m|	VERB
cana-6235	84	10	=	=	PUNCT
cana-6235	84	11	βis(g	βis(g	PROPN
cana-6235	84	12	)	)	PUNCT
cana-6235	84	13	.	.	PUNCT
cana-6235	85	1	it	it	PRON
cana-6235	85	2	is	be	AUX
cana-6235	85	3	also	also	ADV
cana-6235	85	4	true	true	ADJ
cana-6235	85	5	that	that	SCONJ
cana-6235	85	6	βis(g	βis(g	PRON
cana-6235	85	7	−	−	PROPN
cana-6235	85	8	v	v	NOUN
cana-6235	85	9	)	)	PUNCT
cana-6235	85	10	≤	≤	NOUN
cana-6235	85	11	βis(g	βis(g	PRON
cana-6235	85	12	)	)	PUNCT
cana-6235	85	13	.	.	PUNCT
cana-6235	86	1	therefore	therefore	ADV
cana-6235	86	2	,	,	PUNCT
cana-6235	86	3	βis(g	βis(g	PRON
cana-6235	86	4	−	−	PROPN
cana-6235	86	5	v	v	NOUN
cana-6235	86	6	)	)	PUNCT
cana-6235	86	7	=	=	SYM
cana-6235	87	1	βis(g	βis(g	PROPN
cana-6235	87	2	)	)	PUNCT
cana-6235	87	3	▐	▐	NOUN
cana-6235	87	4	now	now	ADV
cana-6235	87	5	we	we	PRON
cana-6235	87	6	consider	consider	VERB
cana-6235	87	7	the	the	DET
cana-6235	87	8	effect	effect	NOUN
cana-6235	87	9	of	of	ADP
cana-6235	87	10	removing	remove	VERB
cana-6235	87	11	a	a	DET
cana-6235	87	12	vertex	vertex	NOUN
cana-6235	87	13	from	from	ADP
cana-6235	87	14	a	a	DET
cana-6235	87	15	graph	graph	NOUN
cana-6235	87	16	on	on	ADP
cana-6235	87	17	the	the	DET
cana-6235	87	18	isolate	isolate	ADJ
cana-6235	87	19	domination	domination	NOUN
cana-6235	87	20	number	number	NOUN
cana-6235	87	21	.	.	PUNCT
cana-6235	88	1	example	example	NOUN
cana-6235	88	2	3	3	NUM
cana-6235	88	3	:	:	PUNCT
cana-6235	88	4	consider	consider	VERB
cana-6235	88	5	the	the	DET
cana-6235	88	6	cycle	cycle	NOUN
cana-6235	88	7	graph	graph	NOUN
cana-6235	88	8	c7	c7	PROPN
cana-6235	88	9	with	with	ADP
cana-6235	88	10	7	7	NUM
cana-6235	88	11	vertices	vertex	NOUN
cana-6235	88	12	{	{	PUNCT
cana-6235	88	13	1	1	NUM
cana-6235	88	14	,	,	PUNCT
cana-6235	88	15	2	2	NUM
cana-6235	88	16	,	,	PUNCT
cana-6235	88	17	3	3	NUM
cana-6235	88	18	,	,	PUNCT
cana-6235	88	19	4	4	NUM
cana-6235	88	20	,	,	PUNCT
cana-6235	88	21	5	5	NUM
cana-6235	88	22	,	,	PUNCT
cana-6235	88	23	6	6	NUM
cana-6235	88	24	,	,	PUNCT
cana-6235	88	25	7	7	NUM
cana-6235	88	26	}	}	PUNCT
cana-6235	88	27	communications	communication	NOUN
cana-6235	88	28	on	on	ADP
cana-6235	88	29	applied	apply	VERB
cana-6235	88	30	nonlinear	nonlinear	ADJ
cana-6235	88	31	analysis	analysis	NOUN
cana-6235	88	32	issn	issn	NOUN
cana-6235	88	33	:	:	PUNCT
cana-6235	88	34	1074	1074	NUM
cana-6235	88	35	-	-	PUNCT
cana-6235	88	36	133x	133x	NUM
cana-6235	88	37	vol	vol	NOUN
cana-6235	88	38	31	31	NUM
cana-6235	88	39	no	no	NOUN
cana-6235	88	40	.	.	NOUN
cana-6235	88	41	2	2	NUM
cana-6235	88	42	(	(	PUNCT
cana-6235	88	43	2024	2024	NUM
cana-6235	88	44	)	)	PUNCT
cana-6235	88	45	514	514	NUM
cana-6235	88	46	https://internationalpubls.com	https://internationalpubls.com	X
cana-6235	88	47	figure	figure	NOUN
cana-6235	88	48	4	4	NUM
cana-6235	88	49	.	.	PUNCT
cana-6235	88	50	cycle	cycle	NOUN
cana-6235	88	51	graph	graph	NOUN
cana-6235	88	52	in	in	ADP
cana-6235	88	53	this	this	DET
cana-6235	88	54	graph	graph	NOUN
cana-6235	88	55	the	the	DET
cana-6235	88	56	set	set	NOUN
cana-6235	88	57	{	{	PUNCT
cana-6235	88	58	1	1	NUM
cana-6235	88	59	,	,	PUNCT
cana-6235	88	60	2	2	NUM
cana-6235	88	61	,	,	PUNCT
cana-6235	88	62	5	5	NUM
cana-6235	88	63	}	}	PUNCT
cana-6235	88	64	is	be	AUX
cana-6235	88	65	a	a	DET
cana-6235	88	66	minimum	minimum	NOUN
cana-6235	88	67	isolate	isolate	NOUN
cana-6235	88	68	dominating	dominating	NOUN
cana-6235	88	69	set	set	NOUN
cana-6235	88	70	and	and	CCONJ
cana-6235	88	71	therefore	therefore	ADV
cana-6235	88	72	the	the	DET
cana-6235	88	73	isolate	isolate	ADJ
cana-6235	88	74	domination	domination	NOUN
cana-6235	88	75	number	number	NOUN
cana-6235	88	76	is	be	AUX
cana-6235	88	77	3	3	NUM
cana-6235	88	78	.	.	PUNCT
cana-6235	88	79	now	now	ADV
cana-6235	88	80	consider	consider	VERB
cana-6235	88	81	the	the	DET
cana-6235	88	82	graph	graph	NOUN
cana-6235	88	83	g	g	PROPN
cana-6235	88	84	−	−	PROPN
cana-6235	88	85	7	7	NUM
cana-6235	88	86	which	which	PRON
cana-6235	88	87	is	be	AUX
cana-6235	88	88	the	the	DET
cana-6235	88	89	path	path	NOUN
cana-6235	88	90	graph	graph	NOUN
cana-6235	88	91	with	with	ADP
cana-6235	88	92	6	6	NUM
cana-6235	88	93	vertices	vertex	NOUN
cana-6235	88	94	{	{	PUNCT
cana-6235	88	95	1	1	NUM
cana-6235	88	96	,	,	PUNCT
cana-6235	88	97	2	2	NUM
cana-6235	88	98	,	,	PUNCT
cana-6235	88	99	3	3	NUM
cana-6235	88	100	,	,	PUNCT
cana-6235	88	101	4	4	NUM
cana-6235	88	102	,	,	PUNCT
cana-6235	88	103	5	5	NUM
cana-6235	88	104	,	,	PUNCT
cana-6235	88	105	6	6	NUM
cana-6235	88	106	}	}	PUNCT
cana-6235	88	107	the	the	DET
cana-6235	88	108	isolate	isolate	ADJ
cana-6235	88	109	domination	domination	NOUN
cana-6235	88	110	number	number	NOUN
cana-6235	88	111	is	be	AUX
cana-6235	88	112	2	2	NUM
cana-6235	88	113	.	.	PUNCT
cana-6235	89	1	thus	thus	ADV
cana-6235	89	2	,	,	PUNCT
cana-6235	89	3	the	the	DET
cana-6235	89	4	isolate	isolate	ADJ
cana-6235	89	5	domination	domination	NOUN
cana-6235	89	6	number	number	NOUN
cana-6235	89	7	decreases	decrease	VERB
cana-6235	89	8	in	in	ADP
cana-6235	89	9	this	this	DET
cana-6235	89	10	graph	graph	NOUN
cana-6235	89	11	.	.	PUNCT
cana-6235	89	12	example	example	NOUN
cana-6235	89	13	4	4	NUM
cana-6235	89	14	:	:	PUNCT
cana-6235	89	15	consider	consider	VERB
cana-6235	89	16	the	the	DET
cana-6235	89	17	path	path	NOUN
cana-6235	89	18	graph	graph	NOUN
cana-6235	89	19	p5	p5	ADJ
cana-6235	89	20	with	with	ADP
cana-6235	89	21	5	5	NUM
cana-6235	89	22	vertices	vertex	NOUN
cana-6235	89	23	{	{	PUNCT
cana-6235	89	24	1	1	NUM
cana-6235	89	25	,	,	PUNCT
cana-6235	89	26	2	2	NUM
cana-6235	89	27	,	,	PUNCT
cana-6235	89	28	3	3	NUM
cana-6235	89	29	,	,	PUNCT
cana-6235	89	30	4	4	NUM
cana-6235	89	31	,	,	PUNCT
cana-6235	89	32	5	5	NUM
cana-6235	89	33	}	}	PUNCT
cana-6235	89	34	figure	figure	NOUN
cana-6235	89	35	5	5	NUM
cana-6235	89	36	.	.	PUNCT
cana-6235	89	37	path	path	NOUN
cana-6235	89	38	graph	graph	NOUN
cana-6235	89	39	the	the	DET
cana-6235	89	40	isolated	isolated	ADJ
cana-6235	89	41	domination	domination	NOUN
cana-6235	89	42	number	number	NOUN
cana-6235	89	43	of	of	ADP
cana-6235	89	44	this	this	DET
cana-6235	89	45	graph	graph	NOUN
cana-6235	89	46	is	be	AUX
cana-6235	89	47	2	2	NUM
cana-6235	89	48	.	.	PUNCT
cana-6235	90	1	if	if	SCONJ
cana-6235	90	2	we	we	PRON
cana-6235	90	3	remove	remove	VERB
cana-6235	90	4	any	any	DET
cana-6235	90	5	vertex	vertex	NOUN
cana-6235	90	6	from	from	ADP
cana-6235	90	7	the	the	DET
cana-6235	90	8	graph	graph	NOUN
cana-6235	90	9	the	the	DET
cana-6235	90	10	isolate	isolate	ADJ
cana-6235	90	11	domination	domination	NOUN
cana-6235	90	12	number	number	NOUN
cana-6235	90	13	of	of	ADP
cana-6235	90	14	the	the	DET
cana-6235	90	15	resulting	result	VERB
cana-6235	90	16	graph	graph	NOUN
cana-6235	90	17	remains	remain	VERB
cana-6235	90	18	unchanged	unchanged	ADJ
cana-6235	90	19	.	.	PUNCT
cana-6235	91	1	theorem	theorem	VERB
cana-6235	91	2	4.4	4.4	NUM
cana-6235	91	3	:	:	PUNCT
cana-6235	91	4	let	let	VERB
cana-6235	91	5	g	g	PRON
cana-6235	91	6	be	be	AUX
cana-6235	91	7	a	a	DET
cana-6235	91	8	graph	graph	NOUN
cana-6235	91	9	and	and	CCONJ
cana-6235	91	10	v	v	AUX
cana-6235	91	11	be	be	AUX
cana-6235	91	12	an	an	DET
cana-6235	91	13	isolated	isolated	ADJ
cana-6235	91	14	vertex	vertex	NOUN
cana-6235	91	15	in	in	ADP
cana-6235	91	16	g	g	PROPN
cana-6235	91	17	then	then	ADV
cana-6235	91	18	γ0(g	γ0(g	NOUN
cana-6235	91	19	)	)	PUNCT
cana-6235	91	20	<	<	X
cana-6235	92	1	γ0(g	γ0(g	X
cana-6235	93	1	−	−	X
cana-6235	93	2	v)if	v)if	PROPN
cana-6235	93	3	and	and	CCONJ
cana-6235	93	4	only	only	ADV
cana-6235	93	5	if	if	SCONJ
cana-6235	93	6	for	for	ADP
cana-6235	93	7	any	any	DET
cana-6235	93	8	minimum	minimum	NOUN
cana-6235	93	9	isolate	isolate	NOUN
cana-6235	93	10	dominating	dominating	NOUN
cana-6235	93	11	set	set	VERB
cana-6235	93	12	s.	s.	PROPN
cana-6235	93	13	the	the	DET
cana-6235	93	14	following	follow	VERB
cana-6235	93	15	two	two	NUM
cana-6235	93	16	conditions	condition	NOUN
cana-6235	93	17	are	be	AUX
cana-6235	93	18	satisfied	satisfied	ADJ
cana-6235	93	19	.	.	PUNCT
cana-6235	94	1	(	(	PUNCT
cana-6235	94	2	1	1	X
cana-6235	94	3	)	)	PUNCT
cana-6235	94	4	v	v	ADP
cana-6235	94	5	∈	∈	PROPN
cana-6235	94	6	s	s	X
cana-6235	94	7	(	(	PUNCT
cana-6235	94	8	2	2	NUM
cana-6235	94	9	)	)	PUNCT
cana-6235	94	10	v	v	NOUN
cana-6235	94	11	is	be	AUX
cana-6235	94	12	the	the	DET
cana-6235	94	13	only	only	ADJ
cana-6235	94	14	isolate	isolate	NOUN
cana-6235	94	15	in	in	ADP
cana-6235	94	16	the	the	DET
cana-6235	94	17	<	<	X
cana-6235	94	18	s	s	X
cana-6235	94	19	>	>	X
cana-6235	94	20	.	.	PUNCT
cana-6235	95	1	proof	proof	NOUN
cana-6235	95	2	:	:	PUNCT
cana-6235	95	3	suppose	suppose	VERB
cana-6235	95	4	γ0(g	γ0(g	PRON
cana-6235	95	5	−	−	PROPN
cana-6235	95	6	v	v	NOUN
cana-6235	95	7	)	)	PUNCT
cana-6235	95	8	>	>	X
cana-6235	96	1	γ0(g	γ0(g	PROPN
cana-6235	96	2	)	)	PUNCT
cana-6235	96	3	.	.	PUNCT
cana-6235	97	1	if	if	SCONJ
cana-6235	97	2	v	v	NUM
cana-6235	97	3	∉	∉	PROPN
cana-6235	97	4	s	s	PART
cana-6235	97	5	then	then	ADV
cana-6235	97	6	v	v	NOUN
cana-6235	97	7	is	be	AUX
cana-6235	97	8	not	not	PART
cana-6235	97	9	adjacent	adjacent	ADJ
cana-6235	97	10	to	to	ADP
cana-6235	97	11	any	any	DET
cana-6235	97	12	vertex	vertex	NOUN
cana-6235	97	13	of	of	ADP
cana-6235	97	14	s	s	PRON
cana-6235	97	15	which	which	PRON
cana-6235	97	16	implies	imply	VERB
cana-6235	97	17	that	that	SCONJ
cana-6235	97	18	s	s	VERB
cana-6235	97	19	is	be	AUX
cana-6235	97	20	not	not	PART
cana-6235	97	21	a	a	DET
cana-6235	97	22	dominating	dominating	NOUN
cana-6235	97	23	set	set	NOUN
cana-6235	97	24	.	.	PUNCT
cana-6235	98	1	which	which	PRON
cana-6235	98	2	is	be	AUX
cana-6235	98	3	a	a	DET
cana-6235	98	4	contradiction	contradiction	NOUN
cana-6235	98	5	.	.	PUNCT
cana-6235	99	1	thus	thus	ADV
cana-6235	99	2	,	,	PUNCT
cana-6235	99	3	v	v	X
cana-6235	99	4	∈	∈	PROPN
cana-6235	99	5	s.	s.	PROPN
cana-6235	99	6	hence	hence	ADV
cana-6235	99	7	condition	condition	NOUN
cana-6235	99	8	(	(	PUNCT
cana-6235	99	9	1	1	X
cana-6235	99	10	)	)	PUNCT
cana-6235	99	11	is	be	AUX
cana-6235	99	12	satisfied	satisfied	ADJ
cana-6235	99	13	.	.	PUNCT
cana-6235	100	1	suppose	suppose	VERB
cana-6235	100	2	u	u	NOUN
cana-6235	100	3	is	be	AUX
cana-6235	100	4	anther	anther	ADJ
cana-6235	100	5	vertex	vertex	NOUN
cana-6235	100	6	in	in	ADP
cana-6235	100	7	s.	s.	PROPN
cana-6235	100	8	which	which	PRON
cana-6235	100	9	is	be	AUX
cana-6235	100	10	an	an	DET
cana-6235	100	11	isolate	isolate	NOUN
cana-6235	100	12	in	in	ADP
cana-6235	100	13	the	the	DET
cana-6235	100	14	<	<	X
cana-6235	100	15	s	s	X
cana-6235	100	16	>	>	X
cana-6235	100	17	.	.	PUNCT
cana-6235	101	1	let	let	VERB
cana-6235	101	2	s1	s1	PROPN
cana-6235	101	3	=	=	PROPN
cana-6235	101	4	s	s	PART
cana-6235	101	5	−	−	NOUN
cana-6235	101	6	{	{	PUNCT
cana-6235	101	7	v	v	NOUN
cana-6235	101	8	}	}	PUNCT
cana-6235	101	9	.	.	PUNCT
cana-6235	102	1	note	note	VERB
cana-6235	102	2	that	that	SCONJ
cana-6235	102	3	u	u	PROPN
cana-6235	102	4	∈	∈	PROPN
cana-6235	102	5	s1	s1	NOUN
cana-6235	102	6	and	and	CCONJ
cana-6235	102	7	u	u	NOUN
cana-6235	102	8	is	be	AUX
cana-6235	102	9	an	an	DET
cana-6235	102	10	isolate	isolate	NOUN
cana-6235	102	11	in	in	ADP
cana-6235	102	12	<	<	X
cana-6235	102	13	s1	s1	PROPN
cana-6235	102	14	>	>	X
cana-6235	102	15	.	.	PUNCT
cana-6235	103	1	thus	thus	ADV
cana-6235	103	2	,	,	PUNCT
cana-6235	103	3	s1	s1	PROPN
cana-6235	103	4	is	be	AUX
cana-6235	103	5	an	an	DET
cana-6235	103	6	isolate	isolate	ADJ
cana-6235	103	7	dominating	dominating	NOUN
cana-6235	103	8	set	set	VERB
cana-6235	103	9	in	in	ADP
cana-6235	103	10	−v	−v	NOUN
cana-6235	103	11	.	.	PUNCT
cana-6235	104	1	therefore	therefore	ADV
cana-6235	104	2	,	,	PUNCT
cana-6235	104	3	γ0(g	γ0(g	NOUN
cana-6235	104	4	−	−	PROPN
cana-6235	104	5	v	v	NOUN
cana-6235	104	6	)	)	PUNCT
cana-6235	104	7	≤	≤	NOUN
cana-6235	104	8	|s1|	|s1|	NOUN
cana-6235	104	9	<	<	X
cana-6235	104	10	|s|	|s|	NOUN
cana-6235	104	11	=	=	SYM
cana-6235	104	12	γ0(g	γ0(g	NOUN
cana-6235	104	13	)	)	PUNCT
cana-6235	104	14	.	.	PUNCT
cana-6235	105	1	this	this	PRON
cana-6235	105	2	is	be	AUX
cana-6235	105	3	a	a	DET
cana-6235	105	4	contradiction	contradiction	NOUN
cana-6235	105	5	and	and	CCONJ
cana-6235	105	6	therefore	therefore	ADV
cana-6235	105	7	(	(	PUNCT
cana-6235	105	8	2	2	X
cana-6235	105	9	)	)	PUNCT
cana-6235	105	10	holds	hold	NOUN
cana-6235	105	11	.	.	PUNCT
cana-6235	106	1	communications	communication	NOUN
cana-6235	106	2	on	on	ADP
cana-6235	106	3	applied	apply	VERB
cana-6235	106	4	nonlinear	nonlinear	ADJ
cana-6235	106	5	analysis	analysis	NOUN
cana-6235	106	6	issn	issn	NOUN
cana-6235	106	7	:	:	PUNCT
cana-6235	106	8	1074	1074	NUM
cana-6235	106	9	-	-	PUNCT
cana-6235	106	10	133x	133x	NUM
cana-6235	106	11	vol	vol	NOUN
cana-6235	106	12	31	31	NUM
cana-6235	106	13	no	no	NOUN
cana-6235	106	14	.	.	NOUN
cana-6235	106	15	2	2	NUM
cana-6235	106	16	(	(	PUNCT
cana-6235	106	17	2024	2024	NUM
cana-6235	106	18	)	)	PUNCT
cana-6235	106	19	515	515	NUM
cana-6235	106	20	https://internationalpubls.com	https://internationalpubls.com	X
cana-6235	106	21	conversely	conversely	ADV
cana-6235	106	22	,	,	PUNCT
cana-6235	106	23	suppose	suppose	VERB
cana-6235	106	24	(	(	PUNCT
cana-6235	106	25	1	1	NUM
cana-6235	106	26	)	)	PUNCT
cana-6235	106	27	and	and	CCONJ
cana-6235	106	28	(	(	PUNCT
cana-6235	106	29	2	2	X
cana-6235	106	30	)	)	PUNCT
cana-6235	106	31	hold	hold	NOUN
cana-6235	106	32	.	.	PUNCT
cana-6235	107	1	let	let	AUX
cana-6235	107	2	t	t	NOUN
cana-6235	107	3	be	be	AUX
cana-6235	107	4	a	a	DET
cana-6235	107	5	set	set	NOUN
cana-6235	107	6	of	of	ADP
cana-6235	107	7	vertices	vertex	NOUN
cana-6235	107	8	of	of	ADP
cana-6235	107	9	g	g	PROPN
cana-6235	107	10	−	−	PROPN
cana-6235	107	11	v	v	ADP
cana-6235	107	12	such	such	DET
cana-6235	107	13	that	that	PRON
cana-6235	107	14	|t|	|t|	VERB
cana-6235	107	15	<	<	X
cana-6235	107	16	γ0(g	γ0(g	NOUN
cana-6235	107	17	)	)	PUNCT
cana-6235	107	18	and	and	CCONJ
cana-6235	107	19	<	<	X
cana-6235	107	20	t	t	X
cana-6235	107	21	>	>	X
cana-6235	107	22	contains	contain	VERB
cana-6235	107	23	an	an	DET
cana-6235	107	24	isolate	isolate	NOUN
cana-6235	107	25	.	.	PUNCT
cana-6235	108	1	if	if	SCONJ
cana-6235	108	2	t	t	PROPN
cana-6235	108	3	is	be	AUX
cana-6235	108	4	an	an	DET
cana-6235	108	5	isolate	isolate	ADJ
cana-6235	108	6	dominating	dominating	NOUN
cana-6235	108	7	set	set	VERB
cana-6235	108	8	in	in	ADP
cana-6235	108	9	g	g	PROPN
cana-6235	108	10	−	−	PROPN
cana-6235	108	11	v	v	ADP
cana-6235	108	12	then	then	ADV
cana-6235	108	13	t1	t1	PROPN
cana-6235	108	14	=	=	PUNCT
cana-6235	108	15	t	t	PROPN
cana-6235	108	16	∪	∪	X
cana-6235	108	17	{	{	PUNCT
cana-6235	108	18	v	v	NOUN
cana-6235	108	19	}	}	PUNCT
cana-6235	108	20	is	be	AUX
cana-6235	108	21	an	an	DET
cana-6235	108	22	isolate	isolate	NOUN
cana-6235	108	23	dominating	dominating	NOUN
cana-6235	108	24	set	set	NOUN
cana-6235	108	25	of	of	ADP
cana-6235	108	26	g.	g.	PROPN
cana-6235	108	27	note	note	VERB
cana-6235	108	28	that	that	SCONJ
cana-6235	108	29	t	t	PROPN
cana-6235	108	30	is	be	AUX
cana-6235	108	31	a	a	DET
cana-6235	108	32	minimum	minimum	NOUN
cana-6235	108	33	isolate	isolate	NOUN
cana-6235	108	34	dominating	dominating	NOUN
cana-6235	108	35	set	set	VERB
cana-6235	108	36	in	in	ADP
cana-6235	108	37	g	g	PROPN
cana-6235	108	38	−	−	PROPN
cana-6235	108	39	v	v	ADP
cana-6235	108	40	then	then	ADV
cana-6235	108	41	t1is	t1is	X
cana-6235	108	42	also	also	ADV
cana-6235	108	43	a	a	DET
cana-6235	108	44	minimum	minimum	NOUN
cana-6235	108	45	isolate	isolate	NOUN
cana-6235	108	46	dominating	dominating	NOUN
cana-6235	108	47	set	set	VERB
cana-6235	108	48	in	in	ADP
cana-6235	108	49	g.	g.	PROPN
cana-6235	108	50	then	then	ADV
cana-6235	108	51	t1	t1	PROPN
cana-6235	108	52	is	be	AUX
cana-6235	108	53	a	a	DET
cana-6235	108	54	minimum	minimum	NOUN
cana-6235	108	55	isolate	isolate	NOUN
cana-6235	108	56	dominating	dominating	NOUN
cana-6235	108	57	set	set	NOUN
cana-6235	108	58	of	of	ADP
cana-6235	108	59	g	g	NOUN
cana-6235	108	60	containing	contain	VERB
cana-6235	108	61	atleast	atleast	ADP
cana-6235	108	62	two	two	NUM
cana-6235	108	63	distinct	distinct	ADJ
cana-6235	108	64	isolate	isolate	NOUN
cana-6235	108	65	in	in	ADP
cana-6235	108	66	g	g	PROPN
cana-6235	108	67	.	.	PUNCT
cana-6235	109	1	this	this	PRON
cana-6235	109	2	contradicts	contradict	VERB
cana-6235	109	3	condition	condition	NOUN
cana-6235	109	4	(	(	PUNCT
cana-6235	109	5	2	2	NUM
cana-6235	109	6	)	)	PUNCT
cana-6235	109	7	.	.	PUNCT
cana-6235	110	1	therefore	therefore	ADV
cana-6235	110	2	,	,	PUNCT
cana-6235	110	3	any	any	DET
cana-6235	110	4	set	set	NOUN
cana-6235	110	5	t	t	PROPN
cana-6235	110	6	with	with	ADP
cana-6235	110	7	|t|	|t|	ADP
cana-6235	110	8	<	<	X
cana-6235	110	9	γ0(g	γ0(g	NOUN
cana-6235	110	10	)	)	PUNCT
cana-6235	110	11	can	can	AUX
cana-6235	110	12	not	not	PART
cana-6235	110	13	be	be	AUX
cana-6235	110	14	a	a	DET
cana-6235	110	15	minimum	minimum	NOUN
cana-6235	110	16	isolate	isolate	NOUN
cana-6235	110	17	dominating	dominating	NOUN
cana-6235	110	18	set	set	NOUN
cana-6235	110	19	of	of	ADP
cana-6235	110	20	g	g	PROPN
cana-6235	110	21	−	−	PROPN
cana-6235	110	22	v	v	NOUN
cana-6235	110	23	.	.	PUNCT
cana-6235	111	1	suppose	suppose	VERB
cana-6235	111	2	t	t	PROPN
cana-6235	111	3	is	be	AUX
cana-6235	111	4	a	a	DET
cana-6235	111	5	set	set	NOUN
cana-6235	111	6	of	of	ADP
cana-6235	111	7	vertices	vertex	NOUN
cana-6235	111	8	of	of	ADP
cana-6235	111	9	g	g	PROPN
cana-6235	111	10	−	−	PROPN
cana-6235	111	11	v	v	ADP
cana-6235	111	12	such	such	DET
cana-6235	111	13	that	that	DET
cana-6235	111	14	|t|	|t|	NOUN
cana-6235	111	15	=	=	SYM
cana-6235	111	16	γ0(g	γ0(g	NOUN
cana-6235	111	17	)	)	PUNCT
cana-6235	111	18	and	and	CCONJ
cana-6235	111	19	suppose	suppose	VERB
cana-6235	111	20	t	t	PROPN
cana-6235	111	21	is	be	AUX
cana-6235	111	22	a	a	DET
cana-6235	111	23	minimum	minimum	NOUN
cana-6235	111	24	isolate	isolate	NOUN
cana-6235	111	25	dominating	dominating	NOUN
cana-6235	111	26	set	set	NOUN
cana-6235	111	27	of	of	ADP
cana-6235	111	28	g	g	PROPN
cana-6235	111	29	−	−	PROPN
cana-6235	111	30	v	v	NOUN
cana-6235	111	31	.	.	PUNCT
cana-6235	112	1	now	now	ADV
cana-6235	112	2	t1	t1	NOUN
cana-6235	112	3	=	=	PUNCT
cana-6235	112	4	t	t	PROPN
cana-6235	112	5	∪	∪	X
cana-6235	112	6	{	{	PUNCT
cana-6235	112	7	v	v	NOUN
cana-6235	112	8	}	}	PUNCT
cana-6235	112	9	is	be	AUX
cana-6235	112	10	an	an	DET
cana-6235	112	11	isolate	isolate	NOUN
cana-6235	112	12	dominating	dominating	NOUN
cana-6235	112	13	set	set	NOUN
cana-6235	112	14	of	of	ADP
cana-6235	112	15	g	g	NOUN
cana-6235	112	16	with	with	ADP
cana-6235	112	17	|t1|	|t1|	NOUN
cana-6235	112	18	=	=	SYM
cana-6235	112	19	γ0(g	γ0(g	PROPN
cana-6235	112	20	)	)	PUNCT
cana-6235	113	1	+	+	NUM
cana-6235	113	2	1	1	NUM
cana-6235	113	3	.	.	PUNCT
cana-6235	113	4	also	also	ADV
cana-6235	113	5	t	t	PROPN
cana-6235	113	6	is	be	AUX
cana-6235	113	7	a	a	DET
cana-6235	113	8	proper	proper	ADJ
cana-6235	113	9	subset	subset	NOUN
cana-6235	113	10	of	of	ADP
cana-6235	113	11	t1	t1	NOUN
cana-6235	113	12	and	and	CCONJ
cana-6235	113	13	therefore	therefore	ADV
cana-6235	113	14	t1	t1	NOUN
cana-6235	113	15	is	be	AUX
cana-6235	113	16	not	not	PART
cana-6235	113	17	a	a	DET
cana-6235	113	18	minimal	minimal	ADJ
cana-6235	113	19	isolate	isolate	NOUN
cana-6235	113	20	dominating	dominate	VERB
cana-6235	113	21	set	set	NOUN
cana-6235	113	22	of	of	ADP
cana-6235	113	23	g.	g.	PROPN
cana-6235	113	24	hence	hence	ADV
cana-6235	113	25	there	there	PRON
cana-6235	113	26	is	be	VERB
cana-6235	113	27	a	a	DET
cana-6235	113	28	vertex	vertex	NOUN
cana-6235	113	29	u	u	NOUN
cana-6235	113	30	in	in	ADP
cana-6235	113	31	t1	t1	NOUN
cana-6235	114	1	such	such	ADJ
cana-6235	114	2	that	that	SCONJ
cana-6235	114	3	t1	t1	PROPN
cana-6235	114	4	−	−	PROPN
cana-6235	115	1	u	u	NOUN
cana-6235	115	2	is	be	AUX
cana-6235	115	3	an	an	DET
cana-6235	115	4	isolate	isolate	NOUN
cana-6235	115	5	dominating	dominating	NOUN
cana-6235	115	6	set	set	NOUN
cana-6235	115	7	of	of	ADP
cana-6235	115	8	g.	g.	PROPN
cana-6235	115	9	also	also	ADV
cana-6235	115	10	|t1	|t1	VERB
cana-6235	115	11	−	−	PRON
cana-6235	115	12	u|	u|	NOUN
cana-6235	115	13	=	=	PUNCT
cana-6235	115	14	|t|	|t|	NOUN
cana-6235	115	15	=	=	SYM
cana-6235	115	16	γ0(g	γ0(g	NOUN
cana-6235	115	17	)	)	PUNCT
cana-6235	115	18	.	.	PUNCT
cana-6235	116	1	further	far	ADV
cana-6235	116	2	note	note	VERB
cana-6235	116	3	that	that	SCONJ
cana-6235	116	4	u	u	PRON
cana-6235	116	5	can	can	AUX
cana-6235	116	6	not	not	PART
cana-6235	116	7	be	be	AUX
cana-6235	116	8	an	an	DET
cana-6235	116	9	isolate	isolate	NOUN
cana-6235	116	10	in	in	ADP
cana-6235	116	11	t1	t1	NOUN
cana-6235	116	12	because	because	SCONJ
cana-6235	116	13	otherwise	otherwise	ADV
cana-6235	116	14	t1	t1	NOUN
cana-6235	116	15	−	−	PROPN
cana-6235	117	1	u	u	NOUN
cana-6235	117	2	would	would	AUX
cana-6235	117	3	not	not	PART
cana-6235	117	4	be	be	AUX
cana-6235	117	5	a	a	DET
cana-6235	117	6	dominating	dominating	NOUN
cana-6235	117	7	set	set	NOUN
cana-6235	117	8	.	.	PUNCT
cana-6235	118	1	thus	thus	ADV
cana-6235	118	2	,	,	PUNCT
cana-6235	118	3	t1	t1	PROPN
cana-6235	118	4	−	−	PROPN
cana-6235	118	5	u	u	NOUN
cana-6235	118	6	is	be	AUX
cana-6235	118	7	a	a	DET
cana-6235	118	8	minimum	minimum	NOUN
cana-6235	118	9	isolate	isolate	NOUN
cana-6235	118	10	dominating	dominating	NOUN
cana-6235	118	11	set	set	NOUN
cana-6235	118	12	of	of	ADP
cana-6235	118	13	g	g	NOUN
cana-6235	118	14	containing	contain	VERB
cana-6235	118	15	at	at	ADV
cana-6235	118	16	least	least	ADV
cana-6235	118	17	two	two	NUM
cana-6235	118	18	isolate	isolate	ADJ
cana-6235	118	19	vertices	vertex	NOUN
cana-6235	118	20	.	.	PUNCT
cana-6235	119	1	which	which	PRON
cana-6235	119	2	is	be	AUX
cana-6235	119	3	a	a	DET
cana-6235	119	4	contradiction	contradiction	NOUN
cana-6235	119	5	.	.	PUNCT
cana-6235	120	1	thus	thus	ADV
cana-6235	120	2	,	,	PUNCT
cana-6235	120	3	there	there	PRON
cana-6235	120	4	is	be	VERB
cana-6235	120	5	no	no	DET
cana-6235	120	6	set	set	NOUN
cana-6235	120	7	of	of	ADP
cana-6235	120	8	t	t	NOUN
cana-6235	120	9	of	of	ADP
cana-6235	120	10	vertices	vertex	NOUN
cana-6235	120	11	of	of	ADP
cana-6235	120	12	g	g	PROPN
cana-6235	120	13	−	−	PROPN
cana-6235	120	14	v	v	ADP
cana-6235	120	15	such	such	DET
cana-6235	120	16	that	that	DET
cana-6235	120	17	|t|	|t|	NOUN
cana-6235	120	18	=	=	SYM
cana-6235	120	19	γ0(g	γ0(g	NOUN
cana-6235	120	20	)	)	PUNCT
cana-6235	120	21	and	and	CCONJ
cana-6235	120	22	t	t	PROPN
cana-6235	120	23	is	be	AUX
cana-6235	120	24	an	an	DET
cana-6235	120	25	isolate	isolate	NOUN
cana-6235	120	26	dominating	dominating	NOUN
cana-6235	120	27	set	set	NOUN
cana-6235	120	28	of	of	ADP
cana-6235	120	29	g	g	PROPN
cana-6235	120	30	−	−	PROPN
cana-6235	121	1	v.	v.	CCONJ
cana-6235	121	2	therefore	therefore	ADV
cana-6235	121	3	,	,	PUNCT
cana-6235	121	4	any	any	DET
cana-6235	121	5	isolate	isolate	NOUN
cana-6235	121	6	dominating	dominating	NOUN
cana-6235	121	7	set	set	NOUN
cana-6235	121	8	of	of	ADP
cana-6235	121	9	g	g	PROPN
cana-6235	121	10	−	−	PROPN
cana-6235	121	11	v	v	NOUN
cana-6235	121	12	must	must	AUX
cana-6235	121	13	have	have	VERB
cana-6235	121	14	cardinality	cardinality	NOUN
cana-6235	121	15	>	>	X
cana-6235	121	16	γ0(g	γ0(g	PROPN
cana-6235	121	17	)	)	PUNCT
cana-6235	121	18	.	.	PUNCT
cana-6235	122	1	therefore	therefore	ADV
cana-6235	122	2	,	,	PUNCT
cana-6235	122	3	γ0(g	γ0(g	NOUN
cana-6235	122	4	−	−	PROPN
cana-6235	122	5	v	v	NOUN
cana-6235	122	6	)	)	PUNCT
cana-6235	122	7	>	>	X
cana-6235	123	1	γ0(g	γ0(g	X
cana-6235	123	2	)	)	PUNCT
cana-6235	123	3	▐	▐	NOUN
cana-6235	123	4	theorem	theorem	VERB
cana-6235	123	5	4.5	4.5	NUM
cana-6235	123	6	:	:	PUNCT
cana-6235	123	7	let	let	VERB
cana-6235	123	8	g	g	PRON
cana-6235	123	9	be	be	AUX
cana-6235	123	10	a	a	DET
cana-6235	123	11	graph	graph	NOUN
cana-6235	123	12	and	and	CCONJ
cana-6235	123	13	v	v	AUX
cana-6235	123	14	be	be	AUX
cana-6235	123	15	an	an	DET
cana-6235	123	16	isolated	isolated	ADJ
cana-6235	123	17	vertex	vertex	NOUN
cana-6235	123	18	in	in	ADP
cana-6235	123	19	g	g	PROPN
cana-6235	123	20	then	then	ADV
cana-6235	123	21	γ0(g	γ0(g	INTJ
cana-6235	123	22	−	−	PROPN
cana-6235	123	23	v	v	NOUN
cana-6235	123	24	)	)	PUNCT
cana-6235	123	25	<	<	X
cana-6235	124	1	γ0(g	γ0(g	NOUN
cana-6235	124	2	)	)	PUNCT
cana-6235	124	3	if	if	SCONJ
cana-6235	124	4	and	and	CCONJ
cana-6235	124	5	only	only	ADV
cana-6235	124	6	if	if	SCONJ
cana-6235	124	7	there	there	PRON
cana-6235	124	8	is	be	VERB
cana-6235	124	9	a	a	DET
cana-6235	124	10	minimum	minimum	NOUN
cana-6235	124	11	isolate	isolate	NOUN
cana-6235	124	12	dominating	dominate	VERB
cana-6235	124	13	set	set	NOUN
cana-6235	124	14	s	s	PART
cana-6235	124	15	which	which	PRON
cana-6235	124	16	contains	contain	VERB
cana-6235	124	17	v	v	NOUN
cana-6235	124	18	and	and	CCONJ
cana-6235	124	19	it	it	PRON
cana-6235	124	20	also	also	ADV
cana-6235	124	21	contains	contain	VERB
cana-6235	124	22	some	some	DET
cana-6235	124	23	other	other	ADJ
cana-6235	124	24	isolate	isolate	NOUN
cana-6235	124	25	.	.	PUNCT
cana-6235	125	1	proof	proof	NOUN
cana-6235	125	2	:	:	PUNCT
cana-6235	125	3	suppose	suppose	VERB
cana-6235	125	4	γ0(g	γ0(g	PRON
cana-6235	125	5	−	−	NOUN
cana-6235	125	6	v	v	NOUN
cana-6235	125	7	)	)	PUNCT
cana-6235	125	8	<	<	X
cana-6235	125	9	γ0(g	γ0(g	NOUN
cana-6235	125	10	)	)	PUNCT
cana-6235	125	11	.	.	PUNCT
cana-6235	126	1	let	let	VERB
cana-6235	126	2	s1	s1	NOUN
cana-6235	126	3	be	be	AUX
cana-6235	126	4	a	a	DET
cana-6235	126	5	minimum	minimum	NOUN
cana-6235	126	6	isolate	isolate	NOUN
cana-6235	126	7	dominating	dominating	NOUN
cana-6235	126	8	set	set	NOUN
cana-6235	126	9	of	of	ADP
cana-6235	126	10	g	g	PROPN
cana-6235	126	11	−	−	PROPN
cana-6235	127	1	v	v	NOUN
cana-6235	127	2	then	then	ADV
cana-6235	127	3	s1	s1	PROPN
cana-6235	127	4	contain	contain	VERB
cana-6235	127	5	an	an	DET
cana-6235	127	6	isolate	isolate	NOUN
cana-6235	127	7	.	.	PUNCT
cana-6235	128	1	let	let	VERB
cana-6235	128	2	s1	s1	NOUN
cana-6235	128	3	=	=	VERB
cana-6235	128	4	s	s	PART
cana-6235	128	5	∪	∪	X
cana-6235	128	6	{	{	PUNCT
cana-6235	128	7	v	v	NOUN
cana-6235	128	8	}	}	PUNCT
cana-6235	128	9	.	.	PUNCT
cana-6235	129	1	then	then	ADV
cana-6235	129	2	s	s	VERB
cana-6235	129	3	is	be	AUX
cana-6235	129	4	a	a	DET
cana-6235	129	5	minimum	minimum	NOUN
cana-6235	129	6	isolate	isolate	NOUN
cana-6235	129	7	dominating	dominating	NOUN
cana-6235	129	8	set	set	NOUN
cana-6235	129	9	of	of	ADP
cana-6235	129	10	gand	gand	PROPN
cana-6235	129	11	v	v	ADP
cana-6235	129	12	∈	∈	PROPN
cana-6235	129	13	s	s	PART
cana-6235	129	14	.	.	PUNCT
cana-6235	130	1	further	far	ADV
cana-6235	130	2	,	,	PUNCT
cana-6235	130	3	s	s	NOUN
cana-6235	130	4	contains	contain	VERB
cana-6235	130	5	two	two	NUM
cana-6235	130	6	isolates	isolate	NOUN
cana-6235	130	7	one	one	NUM
cana-6235	130	8	of	of	ADP
cana-6235	130	9	them	they	PRON
cana-6235	130	10	is	be	AUX
cana-6235	130	11	v.	v.	ADP
cana-6235	130	12	conversely	conversely	ADV
cana-6235	130	13	,	,	PUNCT
cana-6235	130	14	suppose	suppose	VERB
cana-6235	130	15	that	that	SCONJ
cana-6235	130	16	condition	condition	NOUN
cana-6235	130	17	is	be	AUX
cana-6235	130	18	satisfied	satisfied	ADJ
cana-6235	130	19	.	.	PUNCT
cana-6235	131	1	let	let	VERB
cana-6235	131	2	s1	s1	PROPN
cana-6235	131	3	=	=	PROPN
cana-6235	131	4	s	s	PART
cana-6235	131	5	−	−	NOUN
cana-6235	131	6	{	{	PUNCT
cana-6235	131	7	v	v	NOUN
cana-6235	131	8	}	}	PUNCT
cana-6235	131	9	then	then	ADV
cana-6235	131	10	by	by	ADP
cana-6235	131	11	assumption	assumption	NOUN
cana-6235	131	12	s1	s1	NOUN
cana-6235	131	13	contains	contain	VERB
cana-6235	131	14	an	an	DET
cana-6235	131	15	isolate	isolate	NOUN
cana-6235	131	16	.	.	PUNCT
cana-6235	132	1	it	it	PRON
cana-6235	132	2	is	be	AUX
cana-6235	132	3	also	also	ADV
cana-6235	132	4	isolate	isolate	VERB
cana-6235	132	5	dominating	dominate	VERB
cana-6235	132	6	set	set	NOUN
cana-6235	132	7	of	of	ADP
cana-6235	132	8	g	g	PROPN
cana-6235	132	9	−	−	PROPN
cana-6235	132	10	v	v	NOUN
cana-6235	132	11	.	.	PUNCT
cana-6235	133	1	communications	communication	NOUN
cana-6235	133	2	on	on	ADP
cana-6235	133	3	applied	apply	VERB
cana-6235	133	4	nonlinear	nonlinear	ADJ
cana-6235	133	5	analysis	analysis	NOUN
cana-6235	133	6	issn	issn	NOUN
cana-6235	133	7	:	:	PUNCT
cana-6235	133	8	1074	1074	NUM
cana-6235	133	9	-	-	PUNCT
cana-6235	133	10	133x	133x	NUM
cana-6235	133	11	vol	vol	NOUN
cana-6235	133	12	31	31	NUM
cana-6235	133	13	no	no	NOUN
cana-6235	133	14	.	.	NOUN
cana-6235	133	15	2	2	NUM
cana-6235	133	16	(	(	PUNCT
cana-6235	133	17	2024	2024	NUM
cana-6235	133	18	)	)	PUNCT
cana-6235	133	19	516	516	NUM
cana-6235	133	20	https://internationalpubls.com	https://internationalpubls.com	X
cana-6235	133	21	therefore	therefore	ADV
cana-6235	133	22	,	,	PUNCT
cana-6235	133	23	γ0(g	γ0(g	PROPN
cana-6235	133	24	−	−	PROPN
cana-6235	133	25	v	v	NOUN
cana-6235	133	26	)	)	PUNCT
cana-6235	133	27	≤	≤	NOUN
cana-6235	133	28	|s1|	|s1|	NOUN
cana-6235	133	29	<	<	X
cana-6235	133	30	|s|	|s|	NOUN
cana-6235	133	31	=	=	SYM
cana-6235	133	32	γ0(g	γ0(g	PROPN
cana-6235	133	33	)	)	PUNCT
cana-6235	133	34	▐	▐	NOUN
cana-6235	133	35	theorem	theorem	VERB
cana-6235	133	36	4.6	4.6	NUM
cana-6235	133	37	:	:	PUNCT
cana-6235	133	38	let	let	VERB
cana-6235	133	39	g	g	PRON
cana-6235	133	40	be	be	AUX
cana-6235	133	41	a	a	DET
cana-6235	133	42	graph	graph	NOUN
cana-6235	133	43	and	and	CCONJ
cana-6235	133	44	v	v	AUX
cana-6235	133	45	be	be	AUX
cana-6235	133	46	a	a	DET
cana-6235	133	47	non	non	NOUN
cana-6235	133	48	isolated	isolate	VERB
cana-6235	133	49	vertex	vertex	NOUN
cana-6235	133	50	in	in	ADP
cana-6235	133	51	g	g	PROPN
cana-6235	133	52	then	then	ADV
cana-6235	133	53	γ0(g	γ0(g	VERB
cana-6235	133	54	−	−	PROPN
cana-6235	133	55	v	v	NOUN
cana-6235	133	56	)	)	PUNCT
cana-6235	133	57	>	>	X
cana-6235	134	1	γ0(g	γ0(g	X
cana-6235	134	2	)	)	PUNCT
cana-6235	134	3	if	if	SCONJ
cana-6235	134	4	and	and	CCONJ
cana-6235	134	5	only	only	ADV
cana-6235	134	6	if	if	SCONJ
cana-6235	134	7	the	the	DET
cana-6235	134	8	following	follow	VERB
cana-6235	134	9	two	two	NUM
cana-6235	134	10	conditions	condition	NOUN
cana-6235	134	11	are	be	AUX
cana-6235	134	12	satisfied	satisfied	ADJ
cana-6235	134	13	.	.	PUNCT
cana-6235	135	1	(	(	PUNCT
cana-6235	135	2	1	1	X
cana-6235	135	3	)	)	PUNCT
cana-6235	135	4	v	v	ADP
cana-6235	135	5	∈	∈	PROPN
cana-6235	135	6	s	s	NOUN
cana-6235	135	7	,	,	PUNCT
cana-6235	135	8	for	for	ADP
cana-6235	135	9	every	every	DET
cana-6235	135	10	minimum	minimum	NOUN
cana-6235	135	11	isolate	isolate	NOUN
cana-6235	135	12	dominating	dominate	VERB
cana-6235	135	13	set	set	NOUN
cana-6235	135	14	s	s	PROPN
cana-6235	135	15	of	of	ADP
cana-6235	135	16	g	g	PROPN
cana-6235	135	17	.	.	PUNCT
cana-6235	136	1	(	(	PUNCT
cana-6235	136	2	2	2	X
cana-6235	136	3	)	)	PUNCT
cana-6235	136	4	there	there	PRON
cana-6235	136	5	is	be	VERB
cana-6235	136	6	no	no	DET
cana-6235	136	7	subset	subset	NOUN
cana-6235	136	8	s	s	NOUN
cana-6235	136	9	of	of	ADP
cana-6235	136	10	g	g	NOUN
cana-6235	136	11	−	−	PROPN
cana-6235	136	12	v	v	ADP
cana-6235	136	13	such	such	ADJ
cana-6235	136	14	that	that	DET
cana-6235	136	15	|s|	|s|	NOUN
cana-6235	136	16	≤	≤	NUM
cana-6235	136	17	γ0(g	γ0(g	NOUN
cana-6235	136	18	)	)	PUNCT
cana-6235	136	19	,	,	PUNCT
cana-6235	136	20	s	s	VERB
cana-6235	136	21	is	be	AUX
cana-6235	136	22	a	a	DET
cana-6235	136	23	subset	subset	NOUN
cana-6235	136	24	of	of	ADP
cana-6235	136	25	v(g	v(g	NOUN
cana-6235	136	26	)	)	PUNCT
cana-6235	136	27	−	−	ADP
cana-6235	136	28	n[v	n[v	ADV
cana-6235	136	29	]	]	PUNCT
cana-6235	136	30	and	and	CCONJ
cana-6235	136	31	s	s	VERB
cana-6235	136	32	is	be	AUX
cana-6235	136	33	an	an	DET
cana-6235	136	34	isolate	isolate	NOUN
cana-6235	136	35	dominating	dominating	NOUN
cana-6235	136	36	set	set	NOUN
cana-6235	136	37	of	of	ADP
cana-6235	136	38	g	g	PROPN
cana-6235	136	39	−	−	PROPN
cana-6235	136	40	v	v	NOUN
cana-6235	136	41	.	.	PUNCT
cana-6235	137	1	proof	proof	NOUN
cana-6235	137	2	:	:	PUNCT
cana-6235	137	3	suppose	suppose	VERB
cana-6235	137	4	γ0(g	γ0(g	PRON
cana-6235	137	5	−	−	PROPN
cana-6235	137	6	v	v	NOUN
cana-6235	137	7	)	)	PUNCT
cana-6235	137	8	>	>	X
cana-6235	138	1	γ0(g	γ0(g	PROPN
cana-6235	138	2	)	)	PUNCT
cana-6235	138	3	.	.	PUNCT
cana-6235	139	1	(	(	PUNCT
cana-6235	139	2	1	1	X
cana-6235	139	3	)	)	PUNCT
cana-6235	139	4	suppose	suppose	VERB
cana-6235	139	5	there	there	PRON
cana-6235	139	6	is	be	VERB
cana-6235	139	7	a	a	DET
cana-6235	139	8	minimum	minimum	NOUN
cana-6235	139	9	isolate	isolate	NOUN
cana-6235	139	10	dominating	dominate	VERB
cana-6235	139	11	set	set	NOUN
cana-6235	139	12	s	s	PROPN
cana-6235	139	13	of	of	ADP
cana-6235	139	14	g	g	NOUN
cana-6235	139	15	such	such	ADJ
cana-6235	139	16	that	that	DET
cana-6235	139	17	v	v	NUM
cana-6235	139	18	∉	∉	PROPN
cana-6235	139	19	s	s	PART
cana-6235	139	20	then	then	ADV
cana-6235	139	21	s	s	VERB
cana-6235	139	22	is	be	AUX
cana-6235	139	23	an	an	DET
cana-6235	139	24	isolate	isolate	NOUN
cana-6235	139	25	dominating	dominating	NOUN
cana-6235	139	26	set	set	NOUN
cana-6235	139	27	of	of	ADP
cana-6235	139	28	g	g	PROPN
cana-6235	139	29	−	−	PROPN
cana-6235	139	30	v.	v.	CCONJ
cana-6235	139	31	therefore	therefore	ADV
cana-6235	139	32	,	,	PUNCT
cana-6235	139	33	γ0(g	γ0(g	NOUN
cana-6235	139	34	−	−	PROPN
cana-6235	139	35	v	v	NOUN
cana-6235	139	36	)	)	PUNCT
cana-6235	139	37	≤	≤	NUM
cana-6235	139	38	|s|	|s|	PROPN
cana-6235	139	39	=	=	SYM
cana-6235	139	40	γ0(g	γ0(g	NOUN
cana-6235	139	41	)	)	PUNCT
cana-6235	139	42	.	.	PUNCT
cana-6235	140	1	that	that	PRON
cana-6235	140	2	is	be	AUX
cana-6235	140	3	γ0(g	γ0(g	INTJ
cana-6235	140	4	−	−	PROPN
cana-6235	140	5	v	v	NOUN
cana-6235	140	6	)	)	PUNCT
cana-6235	140	7	≤	≤	NOUN
cana-6235	140	8	γ0(g	γ0(g	NOUN
cana-6235	140	9	)	)	PUNCT
cana-6235	140	10	which	which	PRON
cana-6235	140	11	is	be	AUX
cana-6235	140	12	a	a	DET
cana-6235	140	13	contradiction	contradiction	NOUN
cana-6235	140	14	.	.	PUNCT
cana-6235	141	1	therefore	therefore	ADV
cana-6235	141	2	,	,	PUNCT
cana-6235	141	3	v	v	X
cana-6235	141	4	∈	∈	PROPN
cana-6235	141	5	s	s	NOUN
cana-6235	141	6	,	,	PUNCT
cana-6235	141	7	for	for	ADP
cana-6235	141	8	every	every	DET
cana-6235	141	9	minimum	minimum	NOUN
cana-6235	141	10	isolate	isolate	NOUN
cana-6235	141	11	dominating	dominate	VERB
cana-6235	141	12	set	set	NOUN
cana-6235	141	13	s	s	PROPN
cana-6235	141	14	of	of	ADP
cana-6235	141	15	g	g	PROPN
cana-6235	141	16	.	.	PUNCT
cana-6235	142	1	(	(	PUNCT
cana-6235	142	2	2	2	X
cana-6235	142	3	)	)	PUNCT
cana-6235	142	4	suppose	suppose	VERB
cana-6235	142	5	there	there	PRON
cana-6235	142	6	is	be	VERB
cana-6235	142	7	a	a	DET
cana-6235	142	8	subset	subset	NOUN
cana-6235	142	9	s	s	NOUN
cana-6235	142	10	of	of	ADP
cana-6235	142	11	g	g	NOUN
cana-6235	142	12	−	−	PROPN
cana-6235	142	13	v	v	ADP
cana-6235	142	14	such	such	ADJ
cana-6235	142	15	that	that	DET
cana-6235	142	16	|s|	|s|	NOUN
cana-6235	142	17	≤	≤	NUM
cana-6235	142	18	γ0(g	γ0(g	NOUN
cana-6235	142	19	)	)	PUNCT
cana-6235	142	20	,	,	PUNCT
cana-6235	142	21	s	s	VERB
cana-6235	142	22	is	be	AUX
cana-6235	142	23	a	a	DET
cana-6235	142	24	subset	subset	NOUN
cana-6235	142	25	of	of	ADP
cana-6235	142	26	v(g	v(g	NOUN
cana-6235	142	27	)	)	PUNCT
cana-6235	142	28	−	−	ADP
cana-6235	142	29	n[v	n[v	ADV
cana-6235	142	30	]	]	PUNCT
cana-6235	142	31	and	and	CCONJ
cana-6235	142	32	s	s	VERB
cana-6235	142	33	is	be	AUX
cana-6235	142	34	an	an	DET
cana-6235	142	35	isolate	isolate	NOUN
cana-6235	142	36	dominating	dominating	NOUN
cana-6235	142	37	set	set	NOUN
cana-6235	142	38	of	of	ADP
cana-6235	142	39	g	g	PROPN
cana-6235	142	40	−	−	PROPN
cana-6235	142	41	v	v	NOUN
cana-6235	142	42	.	.	PUNCT
cana-6235	143	1	then	then	ADV
cana-6235	143	2	γ0(g	γ0(g	VERB
cana-6235	143	3	−	−	PROPN
cana-6235	143	4	v	v	NOUN
cana-6235	143	5	)	)	PUNCT
cana-6235	143	6	≤	≤	NUM
cana-6235	143	7	|s|	|s|	PROPN
cana-6235	143	8	=	=	SYM
cana-6235	143	9	γ0(g	γ0(g	PROPN
cana-6235	143	10	)	)	PUNCT
cana-6235	143	11	and	and	CCONJ
cana-6235	143	12	therefore	therefore	ADV
cana-6235	143	13	γ0(g	γ0(g	VERB
cana-6235	143	14	−	−	PROPN
cana-6235	143	15	v	v	NOUN
cana-6235	143	16	)	)	PUNCT
cana-6235	143	17	≤	≤	NOUN
cana-6235	144	1	γ0(g	γ0(g	NOUN
cana-6235	144	2	)	)	PUNCT
cana-6235	144	3	.	.	PUNCT
cana-6235	145	1	which	which	PRON
cana-6235	145	2	is	be	AUX
cana-6235	145	3	a	a	DET
cana-6235	145	4	contradiction	contradiction	NOUN
cana-6235	145	5	.	.	PUNCT
cana-6235	146	1	therefore	therefore	ADV
cana-6235	146	2	,	,	PUNCT
cana-6235	146	3	condition	condition	NOUN
cana-6235	146	4	(	(	PUNCT
cana-6235	146	5	2	2	X
cana-6235	146	6	)	)	PUNCT
cana-6235	146	7	is	be	AUX
cana-6235	146	8	also	also	ADV
cana-6235	146	9	satisfied	satisfied	ADJ
cana-6235	146	10	.	.	PUNCT
cana-6235	147	1	conversely	conversely	ADV
cana-6235	147	2	,	,	PUNCT
cana-6235	147	3	suppose	suppose	VERB
cana-6235	147	4	condition	condition	NOUN
cana-6235	147	5	(	(	PUNCT
cana-6235	147	6	1	1	NUM
cana-6235	147	7	)	)	PUNCT
cana-6235	147	8	and	and	CCONJ
cana-6235	147	9	(	(	PUNCT
cana-6235	147	10	2	2	X
cana-6235	147	11	)	)	PUNCT
cana-6235	147	12	are	be	AUX
cana-6235	147	13	satisfied	satisfied	ADJ
cana-6235	147	14	.	.	PUNCT
cana-6235	148	1	first	first	ADV
cana-6235	148	2	suppose	suppose	VERB
cana-6235	148	3	that	that	SCONJ
cana-6235	148	4	γ0(g	γ0(g	VERB
cana-6235	148	5	−	−	NOUN
cana-6235	148	6	v	v	NOUN
cana-6235	148	7	)	)	PUNCT
cana-6235	148	8	=	=	PUNCT
cana-6235	148	9	γ0(g	γ0(g	NOUN
cana-6235	148	10	)	)	PUNCT
cana-6235	148	11	.	.	PUNCT
cana-6235	149	1	let	let	VERB
cana-6235	149	2	s	s	PRON
cana-6235	149	3	be	be	AUX
cana-6235	149	4	a	a	DET
cana-6235	149	5	minimum	minimum	NOUN
cana-6235	149	6	isolate	isolate	NOUN
cana-6235	149	7	dominating	dominating	NOUN
cana-6235	149	8	set	set	NOUN
cana-6235	149	9	of	of	ADP
cana-6235	149	10	g	g	PROPN
cana-6235	149	11	−	−	PROPN
cana-6235	149	12	v	v	NOUN
cana-6235	149	13	then	then	ADV
cana-6235	149	14	|s|	|s|	PROPN
cana-6235	149	15	=	=	SYM
cana-6235	149	16	γ0(g	γ0(g	PROPN
cana-6235	149	17	−	−	NOUN
cana-6235	149	18	v	v	NOUN
cana-6235	149	19	)	)	PUNCT
cana-6235	149	20	=	=	PUNCT
cana-6235	150	1	γ0(g	γ0(g	NOUN
cana-6235	150	2	)	)	PUNCT
cana-6235	150	3	.	.	PUNCT
cana-6235	151	1	case	case	NOUN
cana-6235	151	2	(	(	PUNCT
cana-6235	151	3	1	1	NUM
cana-6235	151	4	):	):	PUNCT
cana-6235	151	5	suppose	suppose	VERB
cana-6235	151	6	v	v	NOUN
cana-6235	151	7	is	be	AUX
cana-6235	151	8	adjacent	adjacent	ADJ
cana-6235	151	9	to	to	ADP
cana-6235	151	10	some	some	DET
cana-6235	151	11	vertex	vertex	NOUN
cana-6235	151	12	of	of	ADP
cana-6235	151	13	s	s	PRON
cana-6235	151	14	then	then	ADV
cana-6235	151	15	s	s	VERB
cana-6235	151	16	is	be	AUX
cana-6235	151	17	a	a	DET
cana-6235	151	18	minimum	minimum	NOUN
cana-6235	151	19	isolate	isolate	NOUN
cana-6235	151	20	dominating	dominating	NOUN
cana-6235	151	21	set	set	NOUN
cana-6235	151	22	of	of	ADP
cana-6235	151	23	g	g	PROPN
cana-6235	151	24	not	not	PART
cana-6235	151	25	containing	contain	VERB
cana-6235	151	26	v.	v.	ADP
cana-6235	151	27	which	which	DET
cana-6235	151	28	contradiction	contradiction	NOUN
cana-6235	151	29	is	be	AUX
cana-6235	151	30	condition	condition	NOUN
cana-6235	151	31	(	(	PUNCT
cana-6235	151	32	1	1	NUM
cana-6235	151	33	)	)	PUNCT
cana-6235	151	34	.	.	PUNCT
cana-6235	152	1	case	case	NOUN
cana-6235	152	2	(	(	PUNCT
cana-6235	152	3	2	2	NUM
cana-6235	152	4	):	):	PUNCT
cana-6235	152	5	suppose	suppose	VERB
cana-6235	152	6	v	v	NOUN
cana-6235	152	7	is	be	AUX
cana-6235	152	8	not	not	PART
cana-6235	152	9	adjacent	adjacent	ADJ
cana-6235	152	10	any	any	DET
cana-6235	152	11	vertex	vertex	NOUN
cana-6235	152	12	of	of	ADP
cana-6235	152	13	s	s	PROPN
cana-6235	152	14	,	,	PUNCT
cana-6235	152	15	then	then	ADV
cana-6235	152	16	n[v	n[v	ADV
cana-6235	152	17	]	]	PUNCT
cana-6235	152	18	∩	∩	PROPN
cana-6235	152	19	s	s	PART
cana-6235	152	20	=	=	SYM
cana-6235	152	21	∅	∅	NOUN
cana-6235	152	22	,	,	PUNCT
cana-6235	152	23	which	which	PRON
cana-6235	152	24	is	be	AUX
cana-6235	152	25	equivalent	equivalent	ADJ
cana-6235	152	26	to	to	ADP
cana-6235	152	27	the	the	DET
cana-6235	152	28	fact	fact	NOUN
cana-6235	152	29	that	that	SCONJ
cana-6235	152	30	s	s	VERB
cana-6235	152	31	is	be	AUX
cana-6235	152	32	subset	subset	VERB
cana-6235	152	33	of	of	ADP
cana-6235	152	34	v(g	v(g	NOUN
cana-6235	152	35	)	)	PUNCT
cana-6235	152	36	−	−	ADP
cana-6235	152	37	n[v	n[v	SYM
cana-6235	152	38	]	]	PUNCT
cana-6235	152	39	also	also	ADV
cana-6235	152	40	|s|	|s|	VERB
cana-6235	152	41	≤	≤	NUM
cana-6235	152	42	γ0(g	γ0(g	NOUN
cana-6235	152	43	)	)	PUNCT
cana-6235	152	44	and	and	CCONJ
cana-6235	152	45	s	s	VERB
cana-6235	152	46	is	be	AUX
cana-6235	152	47	an	an	DET
cana-6235	152	48	isolate	isolate	NOUN
cana-6235	152	49	dominating	dominating	NOUN
cana-6235	152	50	set	set	NOUN
cana-6235	152	51	of	of	ADP
cana-6235	152	52	g	g	PROPN
cana-6235	152	53	−	−	PROPN
cana-6235	152	54	v	v	NOUN
cana-6235	152	55	.	.	PUNCT
cana-6235	153	1	this	this	PRON
cana-6235	153	2	is	be	AUX
cana-6235	153	3	again	again	ADV
cana-6235	153	4	contradiction	contradiction	NOUN
cana-6235	153	5	condition	condition	NOUN
cana-6235	153	6	(	(	PUNCT
cana-6235	153	7	2	2	NUM
cana-6235	153	8	)	)	PUNCT
cana-6235	153	9	.	.	PUNCT
cana-6235	154	1	from	from	ADP
cana-6235	154	2	case	case	NOUN
cana-6235	154	3	(	(	PUNCT
cana-6235	154	4	1	1	NUM
cana-6235	154	5	)	)	PUNCT
cana-6235	154	6	and	and	CCONJ
cana-6235	154	7	case	case	NOUN
cana-6235	154	8	(	(	PUNCT
cana-6235	154	9	2	2	X
cana-6235	154	10	)	)	PUNCT
cana-6235	154	11	it	it	PRON
cana-6235	154	12	follows	follow	VERB
cana-6235	154	13	that	that	SCONJ
cana-6235	154	14	γ0(g	γ0(g	VERB
cana-6235	154	15	−	−	NOUN
cana-6235	154	16	v	v	NOUN
cana-6235	154	17	)	)	PUNCT
cana-6235	154	18	=	=	PUNCT
cana-6235	155	1	γ0(g	γ0(g	NOUN
cana-6235	155	2	)	)	PUNCT
cana-6235	155	3	is	be	AUX
cana-6235	155	4	not	not	PART
cana-6235	155	5	possible	possible	ADJ
cana-6235	155	6	.	.	PUNCT
cana-6235	156	1	suppose	suppose	VERB
cana-6235	156	2	γ0(g	γ0(g	PRON
cana-6235	156	3	−	−	NOUN
cana-6235	156	4	v	v	NOUN
cana-6235	156	5	)	)	PUNCT
cana-6235	156	6	<	<	X
cana-6235	157	1	γ0(g	γ0(g	NOUN
cana-6235	157	2	)	)	PUNCT
cana-6235	157	3	.	.	PUNCT
cana-6235	158	1	let	let	VERB
cana-6235	158	2	s	s	PRON
cana-6235	158	3	be	be	AUX
cana-6235	158	4	a	a	DET
cana-6235	158	5	minimum	minimum	NOUN
cana-6235	158	6	isolate	isolate	NOUN
cana-6235	158	7	dominating	dominating	NOUN
cana-6235	158	8	set	set	NOUN
cana-6235	158	9	of	of	ADP
cana-6235	158	10	g	g	PROPN
cana-6235	158	11	−	−	PROPN
cana-6235	158	12	v	v	ADP
cana-6235	158	13	that	that	PRON
cana-6235	158	14	s	s	VERB
cana-6235	158	15	can	can	AUX
cana-6235	158	16	not	not	PART
cana-6235	158	17	be	be	AUX
cana-6235	158	18	isolate	isolate	VERB
cana-6235	158	19	dominating	dominate	VERB
cana-6235	158	20	set	set	NOUN
cana-6235	158	21	of	of	ADP
cana-6235	158	22	g	g	PROPN
cana-6235	158	23	because	because	SCONJ
cana-6235	158	24	|s|	|s|	NOUN
cana-6235	158	25	<	<	X
cana-6235	158	26	γ0(g	γ0(g	NOUN
cana-6235	158	27	)	)	PUNCT
cana-6235	158	28	this	this	PRON
cana-6235	158	29	means	mean	VERB
cana-6235	158	30	that	that	SCONJ
cana-6235	158	31	v	v	NOUN
cana-6235	158	32	is	be	AUX
cana-6235	158	33	not	not	PART
cana-6235	158	34	adjacent	adjacent	ADJ
cana-6235	158	35	to	to	ADP
cana-6235	158	36	any	any	DET
cana-6235	158	37	other	other	ADJ
cana-6235	158	38	vertex	vertex	NOUN
cana-6235	158	39	of	of	ADP
cana-6235	158	40	s	s	PRON
cana-6235	158	41	then	then	ADV
cana-6235	158	42	s	s	PART
cana-6235	158	43	is	be	AUX
cana-6235	158	44	subset	subset	VERB
cana-6235	158	45	of	of	ADP
cana-6235	158	46	v(g	v(g	NOUN
cana-6235	158	47	)	)	PUNCT
cana-6235	158	48	−	−	ADP
cana-6235	158	49	n[v	n[v	NUM
cana-6235	158	50	]	]	PUNCT
cana-6235	158	51	,	,	PUNCT
cana-6235	158	52	|s|	|s|	PROPN
cana-6235	158	53	≤	≤	NUM
cana-6235	158	54	γ0(g	γ0(g	NOUN
cana-6235	158	55	)	)	PUNCT
cana-6235	158	56	and	and	CCONJ
cana-6235	158	57	s	s	VERB
cana-6235	158	58	is	be	AUX
cana-6235	158	59	an	an	DET
cana-6235	158	60	isolate	isolate	NOUN
cana-6235	158	61	dominating	dominating	NOUN
cana-6235	158	62	set	set	NOUN
cana-6235	158	63	of	of	ADP
cana-6235	158	64	g	g	PROPN
cana-6235	158	65	−	−	PROPN
cana-6235	158	66	v	v	NOUN
cana-6235	158	67	.	.	PUNCT
cana-6235	159	1	this	this	PRON
cana-6235	159	2	is	be	AUX
cana-6235	159	3	again	again	ADV
cana-6235	159	4	contradiction	contradiction	NOUN
cana-6235	159	5	condition	condition	NOUN
cana-6235	159	6	(	(	PUNCT
cana-6235	159	7	2	2	NUM
cana-6235	159	8	)	)	PUNCT
cana-6235	159	9	.	.	PUNCT
cana-6235	160	1	therefore	therefore	ADV
cana-6235	160	2	,	,	PUNCT
cana-6235	160	3	γ0(g	γ0(g	NOUN
cana-6235	160	4	−	−	PROPN
cana-6235	160	5	v	v	NOUN
cana-6235	160	6	)	)	PUNCT
cana-6235	160	7	<	<	X
cana-6235	161	1	γ0(g	γ0(g	NOUN
cana-6235	161	2	)	)	PUNCT
cana-6235	161	3	is	be	AUX
cana-6235	161	4	also	also	ADV
cana-6235	161	5	not	not	PART
cana-6235	161	6	possible	possible	ADJ
cana-6235	161	7	.	.	PUNCT
cana-6235	162	1	communications	communication	NOUN
cana-6235	162	2	on	on	ADP
cana-6235	162	3	applied	apply	VERB
cana-6235	162	4	nonlinear	nonlinear	ADJ
cana-6235	162	5	analysis	analysis	NOUN
cana-6235	162	6	issn	issn	NOUN
cana-6235	162	7	:	:	PUNCT
cana-6235	162	8	1074	1074	NUM
cana-6235	162	9	-	-	PUNCT
cana-6235	162	10	133x	133x	NUM
cana-6235	162	11	vol	vol	NOUN
cana-6235	162	12	31	31	NUM
cana-6235	162	13	no	no	NOUN
cana-6235	162	14	.	.	NOUN
cana-6235	162	15	2	2	NUM
cana-6235	162	16	(	(	PUNCT
cana-6235	162	17	2024	2024	NUM
cana-6235	162	18	)	)	PUNCT
cana-6235	163	1	517	517	NUM
cana-6235	163	2	https://internationalpubls.com	https://internationalpubls.com	X
cana-6235	163	3	hence	hence	ADV
cana-6235	163	4	γ0(g	γ0(g	PART
cana-6235	163	5	−	−	PROPN
cana-6235	163	6	v	v	NOUN
cana-6235	163	7	)	)	PUNCT
cana-6235	163	8	>	>	X
cana-6235	164	1	γ0(g	γ0(g	X
cana-6235	164	2	)	)	PUNCT
cana-6235	164	3	▐	▐	NOUN
cana-6235	164	4	proposition	proposition	NOUN
cana-6235	164	5	4.7	4.7	NUM
cana-6235	164	6	:	:	PUNCT
cana-6235	164	7	let	let	VERB
cana-6235	164	8	g	g	PRON
cana-6235	164	9	be	be	AUX
cana-6235	164	10	a	a	DET
cana-6235	164	11	graph	graph	NOUN
cana-6235	164	12	and	and	CCONJ
cana-6235	164	13	v	v	AUX
cana-6235	164	14	be	be	AUX
cana-6235	164	15	a	a	DET
cana-6235	164	16	non	non	ADJ
cana-6235	164	17	isolated	isolate	VERB
cana-6235	164	18	vertex	vertex	NOUN
cana-6235	164	19	of	of	ADP
cana-6235	164	20	g	g	PROPN
cana-6235	164	21	if	if	SCONJ
cana-6235	164	22	γ0(g	γ0(g	NOUN
cana-6235	164	23	−	−	PROPN
cana-6235	164	24	v	v	NOUN
cana-6235	164	25	)	)	PUNCT
cana-6235	164	26	<	<	X
cana-6235	165	1	γ0(g	γ0(g	NOUN
cana-6235	165	2	)	)	PUNCT
cana-6235	165	3	then	then	ADV
cana-6235	165	4	γ0(g	γ0(g	VERB
cana-6235	165	5	−	−	PROPN
cana-6235	165	6	v	v	NOUN
cana-6235	165	7	)	)	PUNCT
cana-6235	166	1	=	=	PUNCT
cana-6235	166	2	γ0(g	γ0(g	NOUN
cana-6235	166	3	)	)	PUNCT
cana-6235	166	4	−	−	PROPN
cana-6235	166	5	1	1	NUM
cana-6235	166	6	.	.	PUNCT
cana-6235	167	1	proof	proof	NOUN
cana-6235	167	2	:	:	PUNCT
cana-6235	167	3	let	let	VERB
cana-6235	167	4	s1	s1	NOUN
cana-6235	167	5	be	be	AUX
cana-6235	167	6	a	a	DET
cana-6235	167	7	minimum	minimum	NOUN
cana-6235	167	8	isolate	isolate	NOUN
cana-6235	167	9	dominating	dominating	NOUN
cana-6235	167	10	set	set	NOUN
cana-6235	167	11	of	of	ADP
cana-6235	167	12	g	g	PROPN
cana-6235	167	13	−	−	PROPN
cana-6235	167	14	v	v	AUX
cana-6235	167	15	then	then	ADV
cana-6235	167	16	s1cannot	s1cannot	ADV
cana-6235	167	17	be	be	AUX
cana-6235	167	18	an	an	DET
cana-6235	167	19	isolate	isolate	NOUN
cana-6235	167	20	dominating	dominating	NOUN
cana-6235	167	21	set	set	NOUN
cana-6235	167	22	of	of	ADP
cana-6235	167	23	g	g	PROPN
cana-6235	167	24	because	because	SCONJ
cana-6235	167	25	|s1|	|s1|	NOUN
cana-6235	167	26	<	<	X
cana-6235	167	27	γ0(g	γ0(g	NOUN
cana-6235	167	28	)	)	PUNCT
cana-6235	167	29	.	.	PUNCT
cana-6235	168	1	let	let	VERB
cana-6235	168	2	s	s	PRON
cana-6235	168	3	=	=	VERB
cana-6235	168	4	s1	s1	PROPN
cana-6235	168	5	∪	∪	X
cana-6235	168	6	{	{	PUNCT
cana-6235	168	7	v	v	NOUN
cana-6235	168	8	}	}	PUNCT
cana-6235	168	9	then	then	ADV
cana-6235	168	10	s	s	VERB
cana-6235	168	11	is	be	AUX
cana-6235	168	12	an	an	DET
cana-6235	168	13	isolate	isolate	NOUN
cana-6235	168	14	dominating	dominating	NOUN
cana-6235	168	15	set	set	NOUN
cana-6235	168	16	of	of	ADP
cana-6235	168	17	g	g	PROPN
cana-6235	168	18	.	.	PUNCT
cana-6235	169	1	since	since	SCONJ
cana-6235	169	2	γ0(g	γ0(g	NOUN
cana-6235	169	3	−	−	PROPN
cana-6235	169	4	v	v	NOUN
cana-6235	169	5	)	)	PUNCT
cana-6235	169	6	<	<	X
cana-6235	169	7	γ0(g	γ0(g	NOUN
cana-6235	169	8	)	)	PUNCT
cana-6235	169	9	,	,	PUNCT
cana-6235	169	10	s	s	VERB
cana-6235	169	11	must	must	AUX
cana-6235	169	12	be	be	AUX
cana-6235	169	13	a	a	DET
cana-6235	169	14	minimum	minimum	NOUN
cana-6235	169	15	isolate	isolate	NOUN
cana-6235	169	16	dominating	dominating	NOUN
cana-6235	169	17	set	set	NOUN
cana-6235	169	18	of	of	ADP
cana-6235	169	19	g	g	PROPN
cana-6235	169	20	.	.	PUNCT
cana-6235	170	1	thus	thus	ADV
cana-6235	170	2	,	,	PUNCT
cana-6235	170	3	γ0(g	γ0(g	NOUN
cana-6235	170	4	)	)	PUNCT
cana-6235	170	5	=	=	SYM
cana-6235	170	6	|s|	|s|	NOUN
cana-6235	170	7	=	=	NOUN
cana-6235	170	8	|s1|	|s1|	NOUN
cana-6235	170	9	+	+	CCONJ
cana-6235	170	10	1	1	NUM
cana-6235	170	11	=	=	SYM
cana-6235	170	12	γ0(g	γ0(g	NOUN
cana-6235	170	13	−	−	NUM
cana-6235	170	14	v	v	NOUN
cana-6235	170	15	)	)	PUNCT
cana-6235	170	16	+	+	CCONJ
cana-6235	171	1	1	1	NUM
cana-6235	171	2	▐	▐	NUM
cana-6235	171	3	now	now	ADV
cana-6235	171	4	we	we	PRON
cana-6235	171	5	state	state	VERB
cana-6235	171	6	and	and	CCONJ
cana-6235	171	7	prove	prove	VERB
cana-6235	171	8	a	a	DET
cana-6235	171	9	necessary	necessary	ADJ
cana-6235	171	10	and	and	CCONJ
cana-6235	171	11	sufficient	sufficient	ADJ
cana-6235	171	12	condition	condition	NOUN
cana-6235	171	13	under	under	ADP
cana-6235	171	14	which	which	PRON
cana-6235	171	15	the	the	DET
cana-6235	171	16	isolate	isolate	ADJ
cana-6235	171	17	domination	domination	NOUN
cana-6235	171	18	number	number	NOUN
cana-6235	171	19	decreases	decrease	VERB
cana-6235	171	20	when	when	SCONJ
cana-6235	171	21	a	a	DET
cana-6235	171	22	non	non	ADJ
cana-6235	171	23	-	-	ADJ
cana-6235	171	24	isolated	isolated	ADJ
cana-6235	171	25	vertex	vertex	NOUN
cana-6235	171	26	is	be	AUX
cana-6235	171	27	removed	remove	VERB
cana-6235	171	28	from	from	ADP
cana-6235	171	29	the	the	DET
cana-6235	171	30	graph	graph	NOUN
cana-6235	171	31	.	.	PUNCT
cana-6235	172	1	theorem	theorem	NOUN
cana-6235	172	2	4.8	4.8	NUM
cana-6235	172	3	:	:	PUNCT
cana-6235	172	4	let	let	VERB
cana-6235	172	5	g	g	PRON
cana-6235	172	6	be	be	AUX
cana-6235	172	7	a	a	DET
cana-6235	172	8	graph	graph	NOUN
cana-6235	172	9	and	and	CCONJ
cana-6235	172	10	v	v	AUX
cana-6235	172	11	be	be	AUX
cana-6235	172	12	a	a	DET
cana-6235	172	13	non	non	NOUN
cana-6235	172	14	isolated	isolate	VERB
cana-6235	172	15	vertex	vertex	NOUN
cana-6235	172	16	in	in	ADP
cana-6235	172	17	g	g	PROPN
cana-6235	172	18	then	then	ADV
cana-6235	172	19	γ0(g	γ0(g	INTJ
cana-6235	172	20	−	−	PROPN
cana-6235	172	21	v	v	NOUN
cana-6235	172	22	)	)	PUNCT
cana-6235	172	23	<	<	X
cana-6235	173	1	γ0(g	γ0(g	NOUN
cana-6235	173	2	)	)	PUNCT
cana-6235	173	3	if	if	SCONJ
cana-6235	173	4	and	and	CCONJ
cana-6235	173	5	only	only	ADV
cana-6235	173	6	if	if	SCONJ
cana-6235	173	7	there	there	PRON
cana-6235	173	8	is	be	VERB
cana-6235	173	9	a	a	DET
cana-6235	173	10	minimum	minimum	NOUN
cana-6235	173	11	isolate	isolate	NOUN
cana-6235	173	12	dominating	dominate	VERB
cana-6235	173	13	set	set	NOUN
cana-6235	173	14	s	s	AUX
cana-6235	173	15	containing	contain	VERB
cana-6235	173	16	v	v	NOUN
cana-6235	173	17	and	and	CCONJ
cana-6235	173	18	some	some	DET
cana-6235	173	19	other	other	ADJ
cana-6235	173	20	isolate	isolate	VERB
cana-6235	173	21	such	such	ADJ
cana-6235	173	22	that	that	SCONJ
cana-6235	173	23	pn[v	pn[v	PROPN
cana-6235	173	24	,	,	PUNCT
cana-6235	173	25	s	s	X
cana-6235	173	26	]	]	X
cana-6235	173	27	=	=	PUNCT
cana-6235	173	28	{	{	PUNCT
cana-6235	173	29	v	v	NOUN
cana-6235	173	30	}	}	PUNCT
cana-6235	173	31	.	.	PUNCT
cana-6235	174	1	proof	proof	NOUN
cana-6235	174	2	:	:	PUNCT
cana-6235	174	3	suppose	suppose	VERB
cana-6235	174	4	γ0(g	γ0(g	PRON
cana-6235	174	5	−	−	NOUN
cana-6235	174	6	v	v	NOUN
cana-6235	174	7	)	)	PUNCT
cana-6235	174	8	<	<	X
cana-6235	174	9	γ0(g	γ0(g	NOUN
cana-6235	174	10	)	)	PUNCT
cana-6235	174	11	.	.	PUNCT
cana-6235	175	1	let	let	VERB
cana-6235	175	2	s1	s1	NOUN
cana-6235	175	3	be	be	AUX
cana-6235	175	4	a	a	DET
cana-6235	175	5	minimum	minimum	NOUN
cana-6235	175	6	isolate	isolate	NOUN
cana-6235	175	7	dominating	dominating	NOUN
cana-6235	175	8	set	set	NOUN
cana-6235	175	9	of	of	ADP
cana-6235	175	10	g	g	PROPN
cana-6235	175	11	−	−	PROPN
cana-6235	175	12	v	v	NOUN
cana-6235	175	13	.	.	PUNCT
cana-6235	176	1	then	then	ADV
cana-6235	176	2	s1	s1	PROPN
cana-6235	176	3	can	can	AUX
cana-6235	176	4	not	not	PART
cana-6235	176	5	be	be	AUX
cana-6235	176	6	an	an	DET
cana-6235	176	7	isolate	isolate	NOUN
cana-6235	176	8	dominating	dominating	NOUN
cana-6235	176	9	set	set	NOUN
cana-6235	176	10	of	of	ADP
cana-6235	176	11	g.	g.	PROPN
cana-6235	176	12	it	it	PRON
cana-6235	176	13	follows	follow	VERB
cana-6235	176	14	that	that	SCONJ
cana-6235	176	15	v	v	NOUN
cana-6235	176	16	can	can	AUX
cana-6235	176	17	not	not	PART
cana-6235	176	18	be	be	AUX
cana-6235	176	19	adjacent	adjacent	ADJ
cana-6235	176	20	to	to	ADP
cana-6235	176	21	any	any	DET
cana-6235	176	22	vertex	vertex	NOUN
cana-6235	176	23	of	of	ADP
cana-6235	176	24	s1	s1	PROPN
cana-6235	176	25	.	.	PUNCT
cana-6235	177	1	let	let	VERB
cana-6235	177	2	s	s	PRON
cana-6235	177	3	=	=	VERB
cana-6235	177	4	s1	s1	PROPN
cana-6235	177	5	∪	∪	X
cana-6235	177	6	{	{	PUNCT
cana-6235	177	7	v	v	NOUN
cana-6235	177	8	}	}	PUNCT
cana-6235	177	9	then	then	ADV
cana-6235	177	10	s	s	VERB
cana-6235	177	11	is	be	AUX
cana-6235	177	12	aminimum	aminimum	ADJ
cana-6235	177	13	isolate	isolate	NOUN
cana-6235	177	14	dominating	dominate	VERB
cana-6235	177	15	set	set	NOUN
cana-6235	177	16	of	of	ADP
cana-6235	177	17	gand	gand	PROPN
cana-6235	177	18	v	v	ADP
cana-6235	177	19	∈	∈	PROPN
cana-6235	177	20	s	s	PART
cana-6235	177	21	.	.	PUNCT
cana-6235	178	1	since	since	SCONJ
cana-6235	178	2	v	v	NOUN
cana-6235	178	3	is	be	AUX
cana-6235	178	4	not	not	PART
cana-6235	178	5	adjacent	adjacent	ADJ
cana-6235	178	6	to	to	ADP
cana-6235	178	7	any	any	DET
cana-6235	178	8	other	other	ADJ
cana-6235	178	9	vertex	vertex	NOUN
cana-6235	178	10	of	of	ADP
cana-6235	178	11	s	s	PROPN
cana-6235	178	12	,	,	PUNCT
cana-6235	178	13	v	v	NOUN
cana-6235	178	14	∈	∈	PROPN
cana-6235	178	15	pn[v	pn[v	PROPN
cana-6235	178	16	,	,	PUNCT
cana-6235	178	17	s	s	X
cana-6235	178	18	]	]	PUNCT
cana-6235	178	19	.	.	PUNCT
cana-6235	179	1	suppose	suppose	VERB
cana-6235	179	2	x	x	SYM
cana-6235	179	3	≠	≠	PROPN
cana-6235	179	4	v	v	PROPN
cana-6235	179	5	&	&	CCONJ
cana-6235	179	6	x	x	PRON
cana-6235	179	7	∈	∈	PROPN
cana-6235	179	8	pn[v	pn[v	PROPN
cana-6235	179	9	,	,	PUNCT
cana-6235	179	10	s	s	PROPN
cana-6235	179	11	]	]	X
cana-6235	179	12	then	then	ADV
cana-6235	179	13	x	x	SYM
cana-6235	179	14	∉	∉	PROPN
cana-6235	179	15	s1	s1	PROPN
cana-6235	179	16	.since	.since	PROPN
cana-6235	180	1	x	x	PRON
cana-6235	180	2	is	be	AUX
cana-6235	180	3	a	a	DET
cana-6235	180	4	vertex	vertex	NOUN
cana-6235	180	5	of	of	ADP
cana-6235	180	6	g	g	PROPN
cana-6235	180	7	−	−	PROPN
cana-6235	180	8	v	v	NOUN
cana-6235	180	9	,	,	PUNCT
cana-6235	180	10	x	x	PUNCT
cana-6235	180	11	is	be	AUX
cana-6235	180	12	adjacent	adjacent	ADJ
cana-6235	180	13	to	to	ADP
cana-6235	180	14	some	some	DET
cana-6235	180	15	vertex	vertex	NOUN
cana-6235	180	16	y	y	NOUN
cana-6235	180	17	of	of	ADP
cana-6235	180	18	s1	s1	PROPN
cana-6235	180	19	.	.	PUNCT
cana-6235	181	1	thus	thus	ADV
cana-6235	181	2	,	,	PUNCT
cana-6235	181	3	x	x	PRON
cana-6235	181	4	is	be	AUX
cana-6235	181	5	adjacent	adjacent	ADJ
cana-6235	181	6	to	to	ADP
cana-6235	181	7	v	v	NOUN
cana-6235	181	8	also	also	ADV
cana-6235	181	9	.	.	PUNCT
cana-6235	182	1	thus	thus	ADV
cana-6235	182	2	,	,	PUNCT
cana-6235	182	3	x	x	PRON
cana-6235	182	4	is	be	AUX
cana-6235	182	5	adjacent	adjacent	ADJ
cana-6235	182	6	to	to	ADP
cana-6235	182	7	two	two	NUM
cana-6235	182	8	distinct	distinct	ADJ
cana-6235	182	9	vertices	vertex	NOUN
cana-6235	182	10	of	of	ADP
cana-6235	182	11	g.	g.	NOUN
cana-6235	182	12	which	which	PRON
cana-6235	182	13	contradict	contradict	VERB
cana-6235	182	14	the	the	DET
cana-6235	182	15	fact	fact	NOUN
cana-6235	182	16	that	that	SCONJ
cana-6235	182	17	x	x	SYM
cana-6235	182	18	∈	∈	PROPN
cana-6235	182	19	pn[v	pn[v	PROPN
cana-6235	182	20	,	,	PUNCT
cana-6235	182	21	s	s	X
cana-6235	182	22	]	]	PUNCT
cana-6235	182	23	.	.	PUNCT
cana-6235	183	1	therefore	therefore	ADV
cana-6235	183	2	,	,	PUNCT
cana-6235	183	3	pn[v	pn[v	PROPN
cana-6235	183	4	,	,	PUNCT
cana-6235	183	5	s	s	X
cana-6235	183	6	]	]	X
cana-6235	183	7	=	=	PUNCT
cana-6235	183	8	{	{	PUNCT
cana-6235	183	9	v	v	NOUN
cana-6235	183	10	}	}	PUNCT
cana-6235	183	11	.	.	PUNCT
cana-6235	184	1	obviously	obviously	ADV
cana-6235	184	2	,	,	PUNCT
cana-6235	184	3	s	s	PROPN
cana-6235	184	4	contains	contain	VERB
cana-6235	184	5	atleast	atleast	VERB
cana-6235	184	6	two	two	NUM
cana-6235	184	7	isolates	isolate	NOUN
cana-6235	184	8	and	and	CCONJ
cana-6235	184	9	one	one	NUM
cana-6235	184	10	of	of	ADP
cana-6235	184	11	them	they	PRON
cana-6235	184	12	is	be	AUX
cana-6235	184	13	v	v	NOUN
cana-6235	184	14	.	.	PUNCT
cana-6235	185	1	conversely	conversely	ADV
cana-6235	185	2	,	,	PUNCT
cana-6235	185	3	suppose	suppose	VERB
cana-6235	185	4	that	that	SCONJ
cana-6235	185	5	there	there	PRON
cana-6235	185	6	is	be	VERB
cana-6235	185	7	a	a	DET
cana-6235	185	8	minimum	minimum	NOUN
cana-6235	185	9	isolate	isolate	NOUN
cana-6235	185	10	dominating	dominate	VERB
cana-6235	185	11	set	set	NOUN
cana-6235	185	12	s	s	PROPN
cana-6235	185	13	of	of	ADP
cana-6235	185	14	g	g	NOUN
cana-6235	185	15	such	such	DET
cana-6235	185	16	that	that	PRON
cana-6235	185	17	v	v	NUM
cana-6235	185	18	∈	∈	PROPN
cana-6235	185	19	s	s	NOUN
cana-6235	185	20	and	and	CCONJ
cana-6235	185	21	pn[v	pn[v	PROPN
cana-6235	185	22	,	,	PUNCT
cana-6235	185	23	s	s	X
cana-6235	185	24	]	]	X
cana-6235	185	25	=	=	SYM
cana-6235	185	26	{	{	PUNCT
cana-6235	185	27	v	v	NOUN
cana-6235	185	28	}	}	PUNCT
cana-6235	185	29	and	and	CCONJ
cana-6235	185	30	s	s	AUX
cana-6235	185	31	contains	contain	NOUN
cana-6235	185	32	atleast	atleast	VERB
cana-6235	185	33	two	two	NUM
cana-6235	185	34	isolates	isolate	NOUN
cana-6235	185	35	.	.	PUNCT
cana-6235	186	1	let	let	VERB
cana-6235	186	2	s1	s1	PROPN
cana-6235	186	3	=	=	PROPN
cana-6235	186	4	s	s	PART
cana-6235	186	5	−	−	NOUN
cana-6235	186	6	{	{	PUNCT
cana-6235	186	7	v	v	NOUN
cana-6235	186	8	}	}	PUNCT
cana-6235	186	9	then	then	ADV
cana-6235	186	10	s1	s1	NOUN
cana-6235	186	11	is	be	AUX
cana-6235	186	12	an	an	DET
cana-6235	186	13	isolate	isolate	NOUN
cana-6235	186	14	dominating	dominating	NOUN
cana-6235	186	15	set	set	NOUN
cana-6235	186	16	of	of	ADP
cana-6235	186	17	g	g	PROPN
cana-6235	186	18	−	−	PROPN
cana-6235	186	19	v	v	NOUN
cana-6235	186	20	.	.	PUNCT
cana-6235	187	1	therefore	therefore	ADV
cana-6235	187	2	,	,	PUNCT
cana-6235	187	3	γ0(g	γ0(g	NOUN
cana-6235	187	4	−	−	PROPN
cana-6235	187	5	v	v	NOUN
cana-6235	187	6	)	)	PUNCT
cana-6235	187	7	≤	≤	NOUN
cana-6235	187	8	|s1|	|s1|	NOUN
cana-6235	187	9	<	<	X
cana-6235	187	10	|s|	|s|	NOUN
cana-6235	187	11	=	=	SYM
cana-6235	187	12	γ0(g	γ0(g	PROPN
cana-6235	187	13	)	)	PUNCT
cana-6235	187	14	.	.	PUNCT
cana-6235	188	1	thus	thus	ADV
cana-6235	188	2	,	,	PUNCT
cana-6235	188	3	γ0(g	γ0(g	NOUN
cana-6235	188	4	−	−	PROPN
cana-6235	188	5	v	v	NOUN
cana-6235	188	6	)	)	PUNCT
cana-6235	188	7	<	<	X
cana-6235	188	8	γ0(g	γ0(g	X
cana-6235	188	9	)	)	PUNCT
cana-6235	188	10	▐	▐	NOUN
cana-6235	188	11	theorem	theorem	VERB
cana-6235	188	12	4.9	4.9	NUM
cana-6235	188	13	:	:	PUNCT
cana-6235	188	14	let	let	VERB
cana-6235	188	15	g	g	PRON
cana-6235	188	16	be	be	AUX
cana-6235	188	17	a	a	DET
cana-6235	188	18	graph	graph	NOUN
cana-6235	188	19	and	and	CCONJ
cana-6235	188	20	v	v	AUX
cana-6235	188	21	be	be	AUX
cana-6235	188	22	a	a	DET
cana-6235	188	23	non	non	NOUN
cana-6235	188	24	isolated	isolate	VERB
cana-6235	188	25	vertex	vertex	NOUN
cana-6235	188	26	in	in	ADP
cana-6235	188	27	g	g	PROPN
cana-6235	188	28	then	then	ADV
cana-6235	188	29	γ0(g	γ0(g	VERB
cana-6235	188	30	−	−	PROPN
cana-6235	188	31	v	v	NOUN
cana-6235	188	32	)	)	PUNCT
cana-6235	188	33	>	>	X
cana-6235	189	1	γ0(g	γ0(g	NOUN
cana-6235	189	2	)	)	PUNCT
cana-6235	190	1	and	and	CCONJ
cana-6235	190	2	let	let	VERB
cana-6235	190	3	s	s	PRON
cana-6235	190	4	be	be	AUX
cana-6235	190	5	a	a	DET
cana-6235	190	6	minimum	minimum	NOUN
cana-6235	190	7	isolate	isolate	NOUN
cana-6235	190	8	dominating	dominating	NOUN
cana-6235	190	9	set	set	NOUN
cana-6235	190	10	of	of	ADP
cana-6235	190	11	g	g	PROPN
cana-6235	190	12	which	which	PRON
cana-6235	190	13	contains	contain	VERB
cana-6235	190	14	an	an	DET
cana-6235	190	15	isolate	isolate	NOUN
cana-6235	190	16	different	different	ADJ
cana-6235	190	17	from	from	ADP
cana-6235	190	18	v	v	PRON
cana-6235	190	19	then	then	ADV
cana-6235	190	20	v	v	ADP
cana-6235	190	21	∈	∈	PROPN
cana-6235	190	22	s	s	NOUN
cana-6235	190	23	and	and	CCONJ
cana-6235	190	24	pn[v	pn[v	PROPN
cana-6235	190	25	,	,	PUNCT
cana-6235	190	26	s	s	AUX
cana-6235	190	27	]	]	PUNCT
cana-6235	190	28	contains	contain	VERB
cana-6235	190	29	two	two	NUM
cana-6235	190	30	non	non	ADJ
cana-6235	190	31	-	-	ADJ
cana-6235	190	32	adjacent	adjacent	ADJ
cana-6235	190	33	vertices	vertex	NOUN
cana-6235	190	34	.	.	PUNCT
cana-6235	191	1	communications	communication	NOUN
cana-6235	191	2	on	on	ADP
cana-6235	191	3	applied	apply	VERB
cana-6235	191	4	nonlinear	nonlinear	ADJ
cana-6235	191	5	analysis	analysis	NOUN
cana-6235	191	6	issn	issn	NOUN
cana-6235	191	7	:	:	PUNCT
cana-6235	191	8	1074	1074	NUM
cana-6235	191	9	-	-	PUNCT
cana-6235	191	10	133x	133x	NUM
cana-6235	191	11	vol	vol	NOUN
cana-6235	191	12	31	31	NUM
cana-6235	191	13	no	no	NOUN
cana-6235	191	14	.	.	NOUN
cana-6235	191	15	2	2	NUM
cana-6235	191	16	(	(	PUNCT
cana-6235	191	17	2024	2024	NUM
cana-6235	191	18	)	)	PUNCT
cana-6235	191	19	518	518	NUM
cana-6235	191	20	https://internationalpubls.com	https://internationalpubls.com	X
cana-6235	191	21	proof	proof	NOUN
cana-6235	191	22	:	:	PUNCT
cana-6235	191	23	since	since	SCONJ
cana-6235	191	24	γ0(g	γ0(g	NOUN
cana-6235	191	25	−	−	PROPN
cana-6235	191	26	v	v	NOUN
cana-6235	191	27	)	)	PUNCT
cana-6235	191	28	>	>	X
cana-6235	192	1	γ0(g	γ0(g	PROPN
cana-6235	192	2	)	)	PUNCT
cana-6235	192	3	,	,	PUNCT
cana-6235	192	4	v	v	X
cana-6235	192	5	∈	∈	NOUN
cana-6235	192	6	s	s	X
cana-6235	192	7	since	since	SCONJ
cana-6235	192	8	s	s	NOUN
cana-6235	192	9	is	be	AUX
cana-6235	192	10	a	a	DET
cana-6235	192	11	minimal	minimal	ADJ
cana-6235	192	12	isolate	isolate	NOUN
cana-6235	192	13	dominating	dominating	NOUN
cana-6235	192	14	set	set	VERB
cana-6235	192	15	pn[v	pn[v	PROPN
cana-6235	192	16	,	,	PUNCT
cana-6235	192	17	s	s	X
cana-6235	192	18	]	]	PUNCT
cana-6235	192	19	≠	≠	PROPN
cana-6235	192	20	∅	∅	NOUN
cana-6235	192	21	.	.	PUNCT
cana-6235	193	1	if	if	SCONJ
cana-6235	193	2	pn[v	pn[v	PROPN
cana-6235	193	3	,	,	PUNCT
cana-6235	193	4	s	s	X
cana-6235	193	5	]	]	X
cana-6235	193	6	=	=	PUNCT
cana-6235	193	7	{	{	PUNCT
cana-6235	193	8	x	x	NOUN
cana-6235	193	9	}	}	PUNCT
cana-6235	193	10	,	,	PUNCT
cana-6235	193	11	then	then	ADV
cana-6235	193	12	γ0(g	γ0(g	VERB
cana-6235	193	13	−	−	PROPN
cana-6235	193	14	v	v	NOUN
cana-6235	193	15	)	)	PUNCT
cana-6235	193	16	<	<	X
cana-6235	194	1	γ0(g	γ0(g	NOUN
cana-6235	194	2	)	)	PUNCT
cana-6235	194	3	therefore	therefore	ADV
cana-6235	194	4	,	,	PUNCT
cana-6235	194	5	there	there	PRON
cana-6235	194	6	is	be	VERB
cana-6235	194	7	a	a	DET
cana-6235	194	8	vertex	vertex	NOUN
cana-6235	194	9	x	x	SYM
cana-6235	194	10	≠	≠	PROPN
cana-6235	194	11	v	v	NOUN
cana-6235	194	12	&	&	CCONJ
cana-6235	194	13	x	x	PRON
cana-6235	194	14	∈	∈	PROPN
cana-6235	194	15	pn[v	pn[v	PROPN
cana-6235	194	16	,	,	PUNCT
cana-6235	194	17	s	s	X
cana-6235	194	18	]	]	PUNCT
cana-6235	194	19	.	.	PUNCT
cana-6235	195	1	suppose	suppose	VERB
cana-6235	195	2	pn[v	pn[v	PROPN
cana-6235	195	3	,	,	PUNCT
cana-6235	195	4	s	s	X
cana-6235	195	5	]	]	X
cana-6235	195	6	=	=	PUNCT
cana-6235	195	7	{	{	PUNCT
cana-6235	195	8	x	x	NOUN
cana-6235	195	9	}	}	PUNCT
cana-6235	195	10	then	then	ADV
cana-6235	195	11	x	x	PROPN
cana-6235	195	12	∉	∉	PROPN
cana-6235	195	13	s	s	PART
cana-6235	195	14	.	.	PUNCT
cana-6235	196	1	let	let	VERB
cana-6235	196	2	s1	s1	PROPN
cana-6235	196	3	=	=	PROPN
cana-6235	196	4	s	s	PART
cana-6235	196	5	−	−	NOUN
cana-6235	196	6	{	{	PUNCT
cana-6235	196	7	v	v	NOUN
cana-6235	196	8	}	}	PUNCT
cana-6235	196	9	∪	∪	NOUN
cana-6235	196	10	{	{	PUNCT
cana-6235	196	11	x	x	NOUN
cana-6235	196	12	}	}	PUNCT
cana-6235	196	13	then	then	ADV
cana-6235	196	14	s1	s1	NOUN
cana-6235	196	15	is	be	AUX
cana-6235	196	16	a	a	DET
cana-6235	196	17	minimum	minimum	NOUN
cana-6235	196	18	isolate	isolate	NOUN
cana-6235	196	19	dominating	dominating	NOUN
cana-6235	196	20	set	set	NOUN
cana-6235	196	21	of	of	ADP
cana-6235	196	22	g	g	PROPN
cana-6235	196	23	not	not	PART
cana-6235	196	24	containing	contain	VERB
cana-6235	196	25	v	v	NOUN
cana-6235	196	26	.	.	PUNCT
cana-6235	197	1	this	this	PRON
cana-6235	197	2	is	be	AUX
cana-6235	197	3	a	a	DET
cana-6235	197	4	contradiction	contradiction	NOUN
cana-6235	197	5	.	.	PUNCT
cana-6235	198	1	suppose	suppose	VERB
cana-6235	198	2	pn[v	pn[v	PROPN
cana-6235	198	3	,	,	PUNCT
cana-6235	198	4	s	s	X
cana-6235	198	5	]	]	X
cana-6235	198	6	=	=	SYM
cana-6235	198	7	{	{	PUNCT
cana-6235	198	8	v	v	NOUN
cana-6235	198	9	,	,	PUNCT
cana-6235	198	10	y}for	y}for	PROPN
cana-6235	198	11	some	some	DET
cana-6235	198	12	vertex	vertex	NOUN
cana-6235	198	13	y≠	y≠	NOUN
cana-6235	198	14	v	v	NOUN
cana-6235	198	15	.	.	PUNCT
cana-6235	199	1	let	let	VERB
cana-6235	199	2	s1	s1	PROPN
cana-6235	199	3	=	=	PROPN
cana-6235	199	4	s	s	PART
cana-6235	199	5	−	−	NOUN
cana-6235	199	6	{	{	PUNCT
cana-6235	199	7	v	v	NOUN
cana-6235	199	8	}	}	PUNCT
cana-6235	199	9	∪	∪	ADJ
cana-6235	199	10	{	{	PUNCT
cana-6235	199	11	y	y	NOUN
cana-6235	199	12	}	}	PUNCT
cana-6235	199	13	then	then	ADV
cana-6235	199	14	s1	s1	NOUN
cana-6235	199	15	is	be	AUX
cana-6235	199	16	a	a	DET
cana-6235	199	17	minimum	minimum	NOUN
cana-6235	199	18	isolate	isolate	NOUN
cana-6235	199	19	dominating	dominating	NOUN
cana-6235	199	20	set	set	NOUN
cana-6235	199	21	of	of	ADP
cana-6235	199	22	g	g	PROPN
cana-6235	199	23	not	not	PART
cana-6235	199	24	containing	contain	VERB
cana-6235	199	25	v	v	NOUN
cana-6235	199	26	.	.	PUNCT
cana-6235	200	1	which	which	PRON
cana-6235	200	2	is	be	AUX
cana-6235	200	3	again	again	ADV
cana-6235	200	4	a	a	DET
cana-6235	200	5	contradiction	contradiction	NOUN
cana-6235	200	6	.	.	PUNCT
cana-6235	201	1	therefore	therefore	ADV
cana-6235	201	2	,	,	PUNCT
cana-6235	201	3	pn[v	pn[v	PROPN
cana-6235	201	4	,	,	PUNCT
cana-6235	201	5	s	s	AUX
cana-6235	201	6	]	]	PUNCT
cana-6235	201	7	contains	contain	VERB
cana-6235	201	8	atleast	atleast	VERB
cana-6235	201	9	two	two	NUM
cana-6235	201	10	distinct	distinct	ADJ
cana-6235	201	11	vertices	vertex	NOUN
cana-6235	201	12	different	different	ADJ
cana-6235	201	13	from	from	ADP
cana-6235	201	14	v	v	NUM
cana-6235	201	15	.	.	PUNCT
cana-6235	202	1	suppose	suppose	VERB
cana-6235	202	2	any	any	DET
cana-6235	202	3	two	two	NUM
cana-6235	202	4	vertices	vertex	NOUN
cana-6235	202	5	in	in	ADP
cana-6235	202	6	the	the	DET
cana-6235	202	7	pn[v	pn[v	PROPN
cana-6235	202	8	,	,	PUNCT
cana-6235	202	9	s	s	X
cana-6235	202	10	]	]	X
cana-6235	202	11	which	which	PRON
cana-6235	202	12	are	be	AUX
cana-6235	202	13	different	different	ADJ
cana-6235	202	14	from	from	ADP
cana-6235	202	15	v	v	NUM
cana-6235	202	16	are	be	AUX
cana-6235	202	17	adjacent	adjacent	ADJ
cana-6235	202	18	.	.	PUNCT
cana-6235	203	1	then	then	ADV
cana-6235	203	2	let	let	VERB
cana-6235	203	3	y1	y1	INTJ
cana-6235	203	4	,	,	PUNCT
cana-6235	203	5	y2	y2	PROPN
cana-6235	203	6	be	be	AUX
cana-6235	203	7	two	two	NUM
cana-6235	203	8	distinct	distinct	ADJ
cana-6235	203	9	vertices	vertex	NOUN
cana-6235	203	10	in	in	ADP
cana-6235	203	11	the	the	DET
cana-6235	203	12	pn[v	pn[v	PROPN
cana-6235	203	13	,	,	PUNCT
cana-6235	203	14	s	s	X
cana-6235	203	15	]	]	X
cana-6235	203	16	such	such	ADJ
cana-6235	203	17	that	that	SCONJ
cana-6235	203	18	y1	y1	PROPN
cana-6235	203	19	≠	≠	PROPN
cana-6235	203	20	v	v	NOUN
cana-6235	203	21	,	,	PUNCT
cana-6235	203	22	y2	y2	PROPN
cana-6235	203	23	≠	≠	PROPN
cana-6235	203	24	v	v	NOUN
cana-6235	203	25	.	.	PUNCT
cana-6235	204	1	now	now	ADV
cana-6235	204	2	y1	y1	INTJ
cana-6235	204	3	&	&	CCONJ
cana-6235	204	4	y2	y2	PROPN
cana-6235	204	5	are	be	AUX
cana-6235	204	6	adjacent	adjacent	ADJ
cana-6235	204	7	.	.	PUNCT
cana-6235	205	1	let	let	VERB
cana-6235	205	2	s1	s1	PROPN
cana-6235	205	3	=	=	PROPN
cana-6235	205	4	s	s	PART
cana-6235	205	5	−	−	NOUN
cana-6235	205	6	{	{	PUNCT
cana-6235	205	7	v	v	NOUN
cana-6235	205	8	}	}	PUNCT
cana-6235	205	9	∪	∪	ADJ
cana-6235	205	10	{	{	PUNCT
cana-6235	205	11	y1	y1	NOUN
cana-6235	205	12	}	}	PUNCT
cana-6235	205	13	then	then	ADV
cana-6235	205	14	s1	s1	PROPN
cana-6235	205	15	is	be	AUX
cana-6235	205	16	a	a	DET
cana-6235	205	17	minimum	minimum	NOUN
cana-6235	205	18	isolate	isolate	NOUN
cana-6235	205	19	dominating	dominating	NOUN
cana-6235	205	20	set	set	NOUN
cana-6235	205	21	of	of	ADP
cana-6235	205	22	g	g	PROPN
cana-6235	205	23	not	not	PART
cana-6235	205	24	containing	contain	VERB
cana-6235	205	25	v	v	NOUN
cana-6235	205	26	.	.	PUNCT
cana-6235	206	1	which	which	PRON
cana-6235	206	2	is	be	AUX
cana-6235	206	3	again	again	ADV
cana-6235	206	4	a	a	DET
cana-6235	206	5	contradiction	contradiction	NOUN
cana-6235	206	6	.	.	PUNCT
cana-6235	207	1	therefore	therefore	ADV
cana-6235	207	2	,	,	PUNCT
cana-6235	207	3	there	there	PRON
cana-6235	207	4	must	must	AUX
cana-6235	207	5	be	be	AUX
cana-6235	207	6	exist	exist	VERB
cana-6235	207	7	two	two	NUM
cana-6235	207	8	distinct	distinct	ADJ
cana-6235	207	9	vertices	vertex	NOUN
cana-6235	207	10	in	in	ADP
cana-6235	207	11	pn[v	pn[v	PROPN
cana-6235	207	12	,	,	PUNCT
cana-6235	207	13	s	s	X
cana-6235	207	14	]	]	X
cana-6235	207	15	which	which	PRON
cana-6235	207	16	are	be	AUX
cana-6235	207	17	non	non	ADJ
cana-6235	207	18	-	-	ADJ
cana-6235	207	19	adjacent	adjacent	ADJ
cana-6235	207	20	.	.	PUNCT
cana-6235	208	1	thus	thus	ADV
cana-6235	208	2	,	,	PUNCT
cana-6235	208	3	the	the	DET
cana-6235	208	4	theorem	theorem	NOUN
cana-6235	208	5	is	be	AUX
cana-6235	208	6	proved	prove	VERB
cana-6235	208	7	▐	▐	NOUN
cana-6235	208	8	proposition	proposition	NOUN
cana-6235	208	9	4.10	4.10	NUM
cana-6235	208	10	:	:	PUNCT
cana-6235	208	11	let	let	VERB
cana-6235	208	12	g	g	PRON
cana-6235	208	13	be	be	AUX
cana-6235	208	14	a	a	DET
cana-6235	208	15	graph	graph	NOUN
cana-6235	208	16	and	and	CCONJ
cana-6235	208	17	e	e	NOUN
cana-6235	208	18	be	be	AUX
cana-6235	208	19	an	an	DET
cana-6235	208	20	edge	edge	NOUN
cana-6235	208	21	of	of	ADP
cana-6235	208	22	g	g	NOUN
cana-6235	208	23	then	then	ADV
cana-6235	208	24	βis(g	βis(g	PRON
cana-6235	208	25	−	−	PROPN
cana-6235	208	26	e	e	NOUN
cana-6235	208	27	)	)	PUNCT
cana-6235	208	28	≥	≥	NOUN
cana-6235	208	29	βis(g	βis(g	PRON
cana-6235	208	30	)	)	PUNCT
cana-6235	208	31	.	.	PUNCT
cana-6235	209	1	proof	proof	NOUN
cana-6235	209	2	:	:	PUNCT
cana-6235	209	3	let	let	VERB
cana-6235	209	4	s	s	PRON
cana-6235	209	5	be	be	AUX
cana-6235	209	6	a	a	DET
cana-6235	209	7	maximum	maximum	ADJ
cana-6235	209	8	isolate	isolate	NOUN
cana-6235	209	9	inclusive	inclusive	ADJ
cana-6235	209	10	sets	set	NOUN
cana-6235	209	11	of	of	ADP
cana-6235	209	12	g.	g.	PROPN
cana-6235	210	1	then	then	ADV
cana-6235	210	2	s	s	VERB
cana-6235	210	3	is	be	AUX
cana-6235	210	4	also	also	ADV
cana-6235	210	5	isolate	isolate	VERB
cana-6235	210	6	inclusive	inclusive	ADJ
cana-6235	210	7	set	set	NOUN
cana-6235	210	8	of	of	ADP
cana-6235	210	9	g	g	PROPN
cana-6235	210	10	−	−	PROPN
cana-6235	210	11	e	e	NOUN
cana-6235	210	12	.	.	PUNCT
cana-6235	211	1	therefore	therefore	ADV
cana-6235	211	2	,	,	PUNCT
cana-6235	211	3	βis(g	βis(g	NOUN
cana-6235	211	4	−	−	PROPN
cana-6235	211	5	e	e	X
cana-6235	211	6	)	)	PUNCT
cana-6235	211	7	≥	≥	NOUN
cana-6235	211	8	|s|	|s|	NOUN
cana-6235	211	9	=	=	SYM
cana-6235	211	10	βis(g	βis(g	PROPN
cana-6235	211	11	)	)	PUNCT
cana-6235	211	12	.	.	PUNCT
cana-6235	212	1	thus	thus	ADV
cana-6235	212	2	,	,	PUNCT
cana-6235	212	3	βis(g	βis(g	PRON
cana-6235	212	4	−	−	PROPN
cana-6235	212	5	e	e	NOUN
cana-6235	212	6	)	)	PUNCT
cana-6235	212	7	≥	≥	NOUN
cana-6235	212	8	βis(g	βis(g	PRON
cana-6235	212	9	)	)	PUNCT
cana-6235	212	10	▐	▐	PUNCT
cana-6235	212	11	example	example	NOUN
cana-6235	212	12	5	5	NUM
cana-6235	212	13	:	:	PUNCT
cana-6235	212	14	consider	consider	VERB
cana-6235	212	15	the	the	DET
cana-6235	212	16	path	path	NOUN
cana-6235	212	17	graph	graph	NOUN
cana-6235	212	18	p5	p5	ADJ
cana-6235	212	19	with	with	ADP
cana-6235	212	20	5	5	NUM
cana-6235	212	21	vertices	vertex	NOUN
cana-6235	212	22	{	{	PUNCT
cana-6235	212	23	1	1	NUM
cana-6235	212	24	,	,	PUNCT
cana-6235	212	25	2	2	NUM
cana-6235	212	26	,	,	PUNCT
cana-6235	212	27	3	3	NUM
cana-6235	212	28	,	,	PUNCT
cana-6235	212	29	4	4	NUM
cana-6235	212	30	,	,	PUNCT
cana-6235	212	31	5	5	NUM
cana-6235	212	32	}	}	PUNCT
cana-6235	212	33	figure	figure	NOUN
cana-6235	212	34	6	6	NUM
cana-6235	212	35	.	.	PUNCT
cana-6235	212	36	path	path	NOUN
cana-6235	212	37	graph	graph	NOUN
cana-6235	212	38	here	here	ADV
cana-6235	212	39	,	,	PUNCT
cana-6235	212	40	βis(g	βis(g	PRON
cana-6235	212	41	)	)	PUNCT
cana-6235	213	1	=	=	SYM
cana-6235	213	2	4	4	X
cana-6235	213	3	.	.	PUNCT
cana-6235	214	1	now	now	ADV
cana-6235	214	2	consider	consider	VERB
cana-6235	214	3	the	the	DET
cana-6235	214	4	subgraph	subgraph	NOUN
cana-6235	214	5	g	g	PROPN
cana-6235	215	1	−	−	PROPN
cana-6235	215	2	e	e	NOUN
cana-6235	215	3	where	where	SCONJ
cana-6235	215	4	=	=	PRON
cana-6235	215	5	{	{	PUNCT
cana-6235	215	6	45	45	NUM
cana-6235	215	7	}	}	PUNCT
cana-6235	215	8	.	.	PUNCT
cana-6235	216	1	communications	communication	NOUN
cana-6235	216	2	on	on	ADP
cana-6235	216	3	applied	apply	VERB
cana-6235	216	4	nonlinear	nonlinear	ADJ
cana-6235	216	5	analysis	analysis	NOUN
cana-6235	216	6	issn	issn	NOUN
cana-6235	216	7	:	:	PUNCT
cana-6235	216	8	1074	1074	NUM
cana-6235	216	9	-	-	PUNCT
cana-6235	216	10	133x	133x	NUM
cana-6235	216	11	vol	vol	NOUN
cana-6235	216	12	31	31	NUM
cana-6235	216	13	no	no	NOUN
cana-6235	216	14	.	.	NOUN
cana-6235	216	15	2	2	NUM
cana-6235	216	16	(	(	PUNCT
cana-6235	216	17	2024	2024	NUM
cana-6235	216	18	)	)	PUNCT
cana-6235	216	19	519	519	NUM
cana-6235	216	20	https://internationalpubls.com	https://internationalpubls.com	X
cana-6235	216	21	figure	figure	NOUN
cana-6235	216	22	7	7	NUM
cana-6235	216	23	.	.	NOUN
cana-6235	216	24	path	path	NOUN
cana-6235	216	25	graph	graph	NOUN
cana-6235	216	26	here	here	ADV
cana-6235	216	27	,	,	PUNCT
cana-6235	216	28	βis(g	βis(g	PRON
cana-6235	216	29	−	−	PROPN
cana-6235	217	1	e	e	X
cana-6235	217	2	)	)	PUNCT
cana-6235	217	3	=	=	SYM
cana-6235	217	4	4	4	NUM
cana-6235	217	5	.	.	PUNCT
cana-6235	218	1	therefore	therefore	ADV
cana-6235	218	2	,	,	PUNCT
cana-6235	218	3	for	for	ADP
cana-6235	218	4	this	this	DET
cana-6235	218	5	graph	graph	NOUN
cana-6235	218	6	βis(g	βis(g	PRON
cana-6235	218	7	−	−	NOUN
cana-6235	218	8	e	e	NOUN
cana-6235	218	9	)	)	PUNCT
cana-6235	218	10	=	=	SYM
cana-6235	218	11	βis(g	βis(g	PROPN
cana-6235	218	12	)	)	PUNCT
cana-6235	218	13	.	.	PUNCT
cana-6235	219	1	example	example	NOUN
cana-6235	219	2	6	6	NUM
cana-6235	219	3	:	:	PUNCT
cana-6235	219	4	consider	consider	VERB
cana-6235	219	5	the	the	DET
cana-6235	219	6	cycle	cycle	NOUN
cana-6235	219	7	graph	graph	NOUN
cana-6235	219	8	c5	c5	PROPN
cana-6235	219	9	with	with	ADP
cana-6235	219	10	5	5	NUM
cana-6235	219	11	vertices	vertex	NOUN
cana-6235	219	12	{	{	PUNCT
cana-6235	219	13	1	1	NUM
cana-6235	219	14	,	,	PUNCT
cana-6235	219	15	2	2	NUM
cana-6235	219	16	,	,	PUNCT
cana-6235	219	17	3	3	NUM
cana-6235	219	18	,	,	PUNCT
cana-6235	219	19	4	4	NUM
cana-6235	219	20	,	,	PUNCT
cana-6235	219	21	5	5	NUM
cana-6235	219	22	}	}	PUNCT
cana-6235	219	23	figure	figure	NOUN
cana-6235	219	24	8	8	NUM
cana-6235	219	25	.	.	PUNCT
cana-6235	219	26	cycle	cycle	NOUN
cana-6235	219	27	graph	graph	NOUN
cana-6235	219	28	here	here	ADV
cana-6235	219	29	,	,	PUNCT
cana-6235	219	30	βis(g	βis(g	PRON
cana-6235	219	31	)	)	PUNCT
cana-6235	220	1	=	=	SYM
cana-6235	220	2	3	3	X
cana-6235	220	3	.	.	PUNCT
cana-6235	221	1	now	now	ADV
cana-6235	221	2	consider	consider	VERB
cana-6235	221	3	the	the	DET
cana-6235	221	4	subgraph	subgraph	NOUN
cana-6235	221	5	g	g	PROPN
cana-6235	222	1	−	−	PROPN
cana-6235	222	2	e	e	NOUN
cana-6235	222	3	where	where	SCONJ
cana-6235	222	4	=	=	PRON
cana-6235	222	5	{	{	PUNCT
cana-6235	222	6	15	15	NUM
cana-6235	222	7	}	}	PUNCT
cana-6235	222	8	.	.	PUNCT
cana-6235	223	1	figure	figure	NOUN
cana-6235	223	2	9	9	NUM
cana-6235	223	3	.	.	PUNCT
cana-6235	223	4	path	path	NOUN
cana-6235	223	5	graph	graph	NOUN
cana-6235	223	6	here	here	ADV
cana-6235	223	7	βis(g	βis(g	PUNCT
cana-6235	223	8	−	−	PROPN
cana-6235	224	1	e	e	X
cana-6235	224	2	)	)	PUNCT
cana-6235	224	3	=	=	SYM
cana-6235	224	4	4	4	NUM
cana-6235	224	5	.	.	PUNCT
cana-6235	225	1	therefore	therefore	ADV
cana-6235	225	2	,	,	PUNCT
cana-6235	225	3	for	for	ADP
cana-6235	225	4	this	this	DET
cana-6235	225	5	graph	graph	NOUN
cana-6235	225	6	βis(g	βis(g	PRON
cana-6235	225	7	−	−	NOUN
cana-6235	225	8	e	e	NOUN
cana-6235	225	9	)	)	PUNCT
cana-6235	225	10	>	>	X
cana-6235	225	11	βis(g	βis(g	PROPN
cana-6235	225	12	)	)	PUNCT
cana-6235	225	13	.	.	PUNCT
cana-6235	226	1	now	now	ADV
cana-6235	226	2	we	we	PRON
cana-6235	226	3	state	state	VERB
cana-6235	226	4	and	and	CCONJ
cana-6235	226	5	prove	prove	VERB
cana-6235	226	6	a	a	DET
cana-6235	226	7	necessary	necessary	ADJ
cana-6235	226	8	and	and	CCONJ
cana-6235	226	9	sufficient	sufficient	ADJ
cana-6235	226	10	condition	condition	NOUN
cana-6235	226	11	under	under	ADP
cana-6235	226	12	which	which	PRON
cana-6235	226	13	isolate	isolate	VERB
cana-6235	226	14	inclusive	inclusive	ADJ
cana-6235	226	15	set	set	NOUN
cana-6235	226	16	number	number	NOUN
cana-6235	226	17	of	of	ADP
cana-6235	226	18	a	a	DET
cana-6235	226	19	graph	graph	NOUN
cana-6235	226	20	increases	increase	NOUN
cana-6235	226	21	when	when	SCONJ
cana-6235	226	22	an	an	DET
cana-6235	226	23	edge	edge	NOUN
cana-6235	226	24	is	be	AUX
cana-6235	226	25	removed	remove	VERB
cana-6235	226	26	from	from	ADP
cana-6235	226	27	the	the	DET
cana-6235	226	28	graph	graph	NOUN
cana-6235	226	29	.	.	PUNCT
cana-6235	227	1	theorem	theorem	NOUN
cana-6235	227	2	4.11	4.11	NUM
cana-6235	227	3	:	:	PUNCT
cana-6235	227	4	let	let	VERB
cana-6235	227	5	g	g	PRON
cana-6235	227	6	be	be	AUX
cana-6235	227	7	a	a	DET
cana-6235	227	8	graph	graph	NOUN
cana-6235	227	9	and	and	CCONJ
cana-6235	227	10	e	e	NOUN
cana-6235	228	1	=	=	NOUN
cana-6235	228	2	{	{	PUNCT
cana-6235	228	3	uv	uv	NOUN
cana-6235	228	4	}	}	PUNCT
cana-6235	228	5	be	be	AUX
cana-6235	228	6	an	an	DET
cana-6235	228	7	edge	edge	NOUN
cana-6235	228	8	of	of	ADP
cana-6235	228	9	g	g	NOUN
cana-6235	228	10	then	then	ADV
cana-6235	228	11	βis(g	βis(g	PRON
cana-6235	228	12	−	−	PROPN
cana-6235	228	13	e	e	X
cana-6235	228	14	)	)	PUNCT
cana-6235	228	15	>	>	X
cana-6235	229	1	βis(g	βis(g	PROPN
cana-6235	229	2	)	)	PUNCT
cana-6235	229	3	if	if	SCONJ
cana-6235	229	4	and	and	CCONJ
cana-6235	229	5	only	only	ADV
cana-6235	229	6	if	if	SCONJ
cana-6235	229	7	there	there	PRON
cana-6235	229	8	is	be	VERB
cana-6235	229	9	a	a	DET
cana-6235	229	10	subset	subset	NOUN
cana-6235	229	11	s	s	NOUN
cana-6235	229	12	of	of	ADP
cana-6235	229	13	v(g	v(g	NUM
cana-6235	229	14	)	)	PUNCT
cana-6235	229	15	such	such	ADJ
cana-6235	229	16	that	that	SCONJ
cana-6235	229	17	s	s	NOUN
cana-6235	229	18	has	have	VERB
cana-6235	229	19	no	no	DET
cana-6235	229	20	isolated	isolated	ADJ
cana-6235	229	21	vertices	vertex	NOUN
cana-6235	229	22	,	,	PUNCT
cana-6235	229	23	|s|	|s|	PROPN
cana-6235	229	24	>	>	X
cana-6235	229	25	βis(g	βis(g	PROPN
cana-6235	229	26	)	)	PUNCT
cana-6235	229	27	,	,	PUNCT
cana-6235	229	28	u	u	NOUN
cana-6235	229	29	,	,	PUNCT
cana-6235	229	30	v	v	ADP
cana-6235	229	31	∈	∈	PROPN
cana-6235	229	32	s	s	PART
cana-6235	229	33	and	and	CCONJ
cana-6235	229	34	atleast	atleast	ADJ
cana-6235	229	35	one	one	NUM
cana-6235	229	36	of	of	ADP
cana-6235	229	37	u	u	PROPN
cana-6235	229	38	&	&	CCONJ
cana-6235	229	39	v	v	PROPN
cana-6235	229	40	is	be	AUX
cana-6235	229	41	a	a	DET
cana-6235	229	42	pendent	pendent	ADJ
cana-6235	229	43	vertex	vertex	NOUN
cana-6235	229	44	in	in	ADP
cana-6235	229	45	the	the	DET
cana-6235	229	46	<	<	X
cana-6235	229	47	s	s	X
cana-6235	229	48	>	>	X
cana-6235	229	49	.	.	PUNCT
cana-6235	230	1	proof	proof	NOUN
cana-6235	230	2	:	:	PUNCT
cana-6235	230	3	first	first	ADV
cana-6235	230	4	suppose	suppose	VERB
cana-6235	230	5	that	that	SCONJ
cana-6235	230	6	βis(g	βis(g	PRON
cana-6235	230	7	−	−	PROPN
cana-6235	230	8	e	e	X
cana-6235	230	9	)	)	PUNCT
cana-6235	230	10	>	>	X
cana-6235	230	11	βis(g	βis(g	PROPN
cana-6235	230	12	)	)	PUNCT
cana-6235	230	13	.	.	PUNCT
cana-6235	231	1	let	let	VERB
cana-6235	231	2	s	s	PRON
cana-6235	231	3	be	be	AUX
cana-6235	231	4	any	any	DET
cana-6235	231	5	maximum	maximum	ADJ
cana-6235	231	6	isolate	isolate	NOUN
cana-6235	231	7	inclusive	inclusive	ADJ
cana-6235	231	8	set	set	NOUN
cana-6235	231	9	of	of	ADP
cana-6235	231	10	g	g	PROPN
cana-6235	231	11	−	−	PROPN
cana-6235	231	12	e	e	NOUN
cana-6235	231	13	.	.	PUNCT
cana-6235	232	1	claim	claim	NOUN
cana-6235	232	2	:	:	PUNCT
cana-6235	232	3	u	u	PROPN
cana-6235	232	4	∈	∈	PROPN
cana-6235	232	5	s	s	PROPN
cana-6235	232	6	&	&	CCONJ
cana-6235	232	7	v	v	ADP
cana-6235	232	8	∈	∈	PROPN
cana-6235	232	9	s	s	VERB
cana-6235	232	10	now	now	ADV
cana-6235	232	11	consider	consider	VERB
cana-6235	232	12	the	the	DET
cana-6235	232	13	set	set	NOUN
cana-6235	232	14	s	s	X
cana-6235	232	15	in	in	ADP
cana-6235	232	16	the	the	DET
cana-6235	232	17	graph	graph	NOUN
cana-6235	232	18	g	g	PROPN
cana-6235	232	19	.	.	PUNCT
cana-6235	233	1	since	since	SCONJ
cana-6235	233	2	|s|	|s|	NOUN
cana-6235	233	3	>	>	X
cana-6235	233	4	βis(g	βis(g	PROPN
cana-6235	233	5	)	)	PUNCT
cana-6235	233	6	is	be	AUX
cana-6235	233	7	can	can	AUX
cana-6235	233	8	not	not	PART
cana-6235	233	9	be	be	AUX
cana-6235	233	10	isolate	isolate	ADJ
cana-6235	233	11	inclusive	inclusive	ADJ
cana-6235	233	12	set	set	NOUN
cana-6235	233	13	of	of	ADP
cana-6235	233	14	g	g	PROPN
cana-6235	233	15	but	but	CCONJ
cana-6235	233	16	s	s	NOUN
cana-6235	233	17	is	be	AUX
cana-6235	233	18	an	an	DET
cana-6235	233	19	isolate	isolate	ADJ
cana-6235	233	20	inclusive	inclusive	ADJ
cana-6235	233	21	set	set	NOUN
cana-6235	233	22	in	in	ADP
cana-6235	233	23	g	g	PROPN
cana-6235	233	24	−	−	PROPN
cana-6235	233	25	e	e	NOUN
cana-6235	233	26	.	.	PUNCT
cana-6235	234	1	communications	communication	NOUN
cana-6235	234	2	on	on	ADP
cana-6235	234	3	applied	apply	VERB
cana-6235	234	4	nonlinear	nonlinear	ADJ
cana-6235	234	5	analysis	analysis	NOUN
cana-6235	234	6	issn	issn	NOUN
cana-6235	234	7	:	:	PUNCT
cana-6235	234	8	1074	1074	NUM
cana-6235	234	9	-	-	PUNCT
cana-6235	234	10	133x	133x	NUM
cana-6235	234	11	vol	vol	NOUN
cana-6235	234	12	31	31	NUM
cana-6235	234	13	no	no	NOUN
cana-6235	234	14	.	.	NOUN
cana-6235	234	15	2	2	NUM
cana-6235	234	16	(	(	PUNCT
cana-6235	234	17	2024	2024	NUM
cana-6235	234	18	)	)	PUNCT
cana-6235	234	19	520	520	NUM
cana-6235	234	20	https://internationalpubls.com	https://internationalpubls.com	X
cana-6235	234	21	therefore	therefore	ADV
cana-6235	234	22	,	,	PUNCT
cana-6235	234	23	u	u	NOUN
cana-6235	234	24	or	or	CCONJ
cana-6235	234	25	v	v	NOUN
cana-6235	234	26	must	must	AUX
cana-6235	234	27	be	be	AUX
cana-6235	234	28	an	an	DET
cana-6235	234	29	isolated	isolated	ADJ
cana-6235	234	30	vertex	vertex	NOUN
cana-6235	234	31	in	in	ADP
cana-6235	234	32	s	s	PRON
cana-6235	234	33	when	when	SCONJ
cana-6235	234	34	s	s	NOUN
cana-6235	234	35	is	be	AUX
cana-6235	234	36	regard	regard	NOUN
cana-6235	234	37	as	as	ADP
cana-6235	234	38	a	a	DET
cana-6235	234	39	vertex	vertex	NOUN
cana-6235	234	40	set	set	NOUN
cana-6235	234	41	of	of	ADP
cana-6235	234	42	g	g	PROPN
cana-6235	234	43	−	−	PROPN
cana-6235	234	44	e	e	NOUN
cana-6235	234	45	.	.	PUNCT
cana-6235	235	1	therefore	therefore	ADV
cana-6235	235	2	,	,	PUNCT
cana-6235	235	3	u	u	NOUN
cana-6235	235	4	or	or	CCONJ
cana-6235	235	5	v	v	NOUN
cana-6235	235	6	must	must	AUX
cana-6235	235	7	be	be	AUX
cana-6235	235	8	a	a	DET
cana-6235	235	9	pendent	pendent	ADJ
cana-6235	235	10	vertex	vertex	NOUN
cana-6235	235	11	in	in	ADP
cana-6235	235	12	the	the	DET
cana-6235	235	13	<	<	X
cana-6235	235	14	s	s	X
cana-6235	235	15	>	>	X
cana-6235	235	16	when	when	SCONJ
cana-6235	235	17	s	s	NOUN
cana-6235	235	18	is	be	AUX
cana-6235	235	19	regard	regard	NOUN
cana-6235	235	20	as	as	ADP
cana-6235	235	21	a	a	DET
cana-6235	235	22	set	set	NOUN
cana-6235	235	23	of	of	ADP
cana-6235	235	24	vertices	vertex	NOUN
cana-6235	235	25	of	of	ADP
cana-6235	235	26	g	g	PROPN
cana-6235	235	27	.	.	PUNCT
cana-6235	236	1	since	since	SCONJ
cana-6235	236	2	s	s	NOUN
cana-6235	236	3	is	be	AUX
cana-6235	236	4	not	not	PART
cana-6235	236	5	isolate	isolate	VERB
cana-6235	236	6	inclusive	inclusive	ADJ
cana-6235	236	7	set	set	NOUN
cana-6235	236	8	of	of	ADP
cana-6235	236	9	g.	g.	PROPN
cana-6235	236	10	the	the	DET
cana-6235	236	11	<	<	X
cana-6235	236	12	s	s	X
cana-6235	236	13	>	>	X
cana-6235	236	14	does	do	AUX
cana-6235	236	15	not	not	PART
cana-6235	236	16	have	have	VERB
cana-6235	236	17	any	any	DET
cana-6235	236	18	isolated	isolated	ADJ
cana-6235	236	19	vertices	vertex	NOUN
cana-6235	236	20	.	.	PUNCT
cana-6235	237	1	conversely	conversely	ADV
cana-6235	237	2	,	,	PUNCT
cana-6235	237	3	suppose	suppose	VERB
cana-6235	237	4	condition	condition	NOUN
cana-6235	237	5	is	be	AUX
cana-6235	237	6	satisfied	satisfied	ADJ
cana-6235	237	7	.	.	PUNCT
cana-6235	238	1	let	let	VERB
cana-6235	238	2	s	s	PRON
cana-6235	238	3	be	be	AUX
cana-6235	238	4	a	a	DET
cana-6235	238	5	set	set	NOUN
cana-6235	238	6	of	of	ADP
cana-6235	238	7	vertices	vertex	NOUN
cana-6235	238	8	of	of	ADP
cana-6235	238	9	g	g	NOUN
cana-6235	238	10	such	such	ADJ
cana-6235	238	11	that	that	SCONJ
cana-6235	238	12	|s|	|s|	PROPN
cana-6235	238	13	>	>	X
cana-6235	238	14	βis(g	βis(g	PROPN
cana-6235	238	15	)	)	PUNCT
cana-6235	238	16	,	,	PUNCT
cana-6235	238	17	<	<	X
cana-6235	238	18	s	s	X
cana-6235	238	19	>	>	X
cana-6235	238	20	has	have	VERB
cana-6235	238	21	no	no	DET
cana-6235	238	22	isolated	isolated	ADJ
cana-6235	238	23	vertex	vertex	NOUN
cana-6235	238	24	and	and	CCONJ
cana-6235	238	25	u	u	NOUN
cana-6235	238	26	,	,	PUNCT
cana-6235	238	27	v	v	ADP
cana-6235	238	28	∈	∈	PROPN
cana-6235	238	29	s	s	PART
cana-6235	238	30	and	and	CCONJ
cana-6235	238	31	atleast	atleast	ADJ
cana-6235	238	32	one	one	NUM
cana-6235	238	33	of	of	ADP
cana-6235	238	34	u	u	PROPN
cana-6235	238	35	&	&	CCONJ
cana-6235	238	36	v	v	PROPN
cana-6235	238	37	is	be	AUX
cana-6235	238	38	a	a	DET
cana-6235	238	39	pendent	pendent	ADJ
cana-6235	238	40	vertex	vertex	NOUN
cana-6235	238	41	in	in	ADP
cana-6235	238	42	the	the	DET
cana-6235	238	43	<	<	X
cana-6235	238	44	s	s	X
cana-6235	238	45	>	>	X
cana-6235	238	46	.	.	PUNCT
cana-6235	239	1	suppose	suppose	VERB
cana-6235	239	2	u	u	PRON
cana-6235	239	3	is	be	AUX
cana-6235	239	4	a	a	DET
cana-6235	239	5	pendent	pendent	ADJ
cana-6235	239	6	vertex	vertex	NOUN
cana-6235	239	7	in	in	ADP
cana-6235	239	8	the	the	DET
cana-6235	239	9	<	<	X
cana-6235	239	10	s	s	X
cana-6235	239	11	>	>	X
cana-6235	239	12	.	.	PUNCT
cana-6235	240	1	now	now	ADV
cana-6235	240	2	consider	consider	VERB
cana-6235	240	3	s	s	PRON
cana-6235	240	4	is	be	AUX
cana-6235	240	5	in	in	ADP
cana-6235	240	6	the	the	DET
cana-6235	240	7	graph	graph	NOUN
cana-6235	240	8	g	g	ADP
cana-6235	240	9	−	−	PROPN
cana-6235	240	10	e	e	NOUN
cana-6235	240	11	.	.	PUNCT
cana-6235	241	1	then	then	ADV
cana-6235	241	2	obviously	obviously	ADV
cana-6235	241	3	s	s	VERB
cana-6235	241	4	is	be	AUX
cana-6235	241	5	an	an	DET
cana-6235	241	6	isolate	isolate	ADJ
cana-6235	241	7	inclusive	inclusive	ADJ
cana-6235	241	8	set	set	NOUN
cana-6235	241	9	in	in	ADP
cana-6235	241	10	g	g	PROPN
cana-6235	241	11	−	−	PROPN
cana-6235	241	12	e	e	NOUN
cana-6235	241	13	.	.	PUNCT
cana-6235	242	1	then	then	ADV
cana-6235	242	2	βis(g	βis(g	PRON
cana-6235	242	3	−	−	PROPN
cana-6235	242	4	e	e	X
cana-6235	242	5	)	)	PUNCT
cana-6235	242	6	≥	≥	NOUN
cana-6235	242	7	|s|	|s|	VERB
cana-6235	242	8	>	>	X
cana-6235	242	9	βis(g	βis(g	PROPN
cana-6235	242	10	)	)	PUNCT
cana-6235	242	11	thus	thus	ADV
cana-6235	242	12	,	,	PUNCT
cana-6235	242	13	βis(g	βis(g	PRON
cana-6235	242	14	−	−	PROPN
cana-6235	242	15	e	e	X
cana-6235	242	16	)	)	PUNCT
cana-6235	242	17	>	>	X
cana-6235	242	18	βis(g	βis(g	PROPN
cana-6235	242	19	)	)	PUNCT
cana-6235	242	20	▐	▐	NOUN
cana-6235	242	21	theorem	theorem	VERB
cana-6235	242	22	4.12	4.12	NUM
cana-6235	242	23	:	:	PUNCT
cana-6235	242	24	let	let	VERB
cana-6235	242	25	g	g	PRON
cana-6235	242	26	be	be	AUX
cana-6235	242	27	a	a	DET
cana-6235	242	28	regular	regular	ADJ
cana-6235	242	29	graph	graph	NOUN
cana-6235	242	30	and	and	CCONJ
cana-6235	242	31	e	e	NOUN
cana-6235	242	32	be	be	AUX
cana-6235	242	33	any	any	DET
cana-6235	242	34	edge	edge	NOUN
cana-6235	242	35	of	of	ADP
cana-6235	242	36	g	g	NOUN
cana-6235	242	37	then	then	ADV
cana-6235	242	38	βis(g	βis(g	PRON
cana-6235	242	39	−	−	PROPN
cana-6235	242	40	e	e	X
cana-6235	242	41	)	)	PUNCT
cana-6235	242	42	>	>	X
cana-6235	242	43	βis(g	βis(g	PROPN
cana-6235	242	44	)	)	PUNCT
cana-6235	242	45	.	.	PUNCT
cana-6235	243	1	proof	proof	NOUN
cana-6235	243	2	:	:	PUNCT
cana-6235	243	3	suppose	suppose	VERB
cana-6235	243	4	g	g	PROPN
cana-6235	243	5	is	be	AUX
cana-6235	243	6	a	a	DET
cana-6235	243	7	k−regular	k−regular	X
cana-6235	243	8	graph	graph	NOUN
cana-6235	243	9	,	,	PUNCT
cana-6235	243	10	k	k	X
cana-6235	243	11	≥	≥	NUM
cana-6235	243	12	1	1	NUM
cana-6235	243	13	let	let	VERB
cana-6235	243	14	e	e	NOUN
cana-6235	243	15	=	=	PRON
cana-6235	243	16	{	{	PUNCT
cana-6235	243	17	uv	uv	NOUN
cana-6235	243	18	}	}	PUNCT
cana-6235	243	19	is	be	AUX
cana-6235	243	20	any	any	DET
cana-6235	243	21	edge	edge	NOUN
cana-6235	243	22	of	of	ADP
cana-6235	243	23	g.	g.	PROPN
cana-6235	243	24	now	now	ADV
cana-6235	243	25	(	(	PUNCT
cana-6235	243	26	v	v	NOUN
cana-6235	243	27	)	)	PUNCT
cana-6235	243	28	=	=	SYM
cana-6235	244	1	k	k	X
cana-6235	244	2	=	=	PUNCT
cana-6235	244	3	δ(g	δ(g	X
cana-6235	244	4	)	)	PUNCT
cana-6235	244	5	.	.	PUNCT
cana-6235	245	1	therefore	therefore	ADV
cana-6235	245	2	,	,	PUNCT
cana-6235	245	3	by	by	ADP
cana-6235	245	4	the	the	DET
cana-6235	245	5	above	above	ADJ
cana-6235	245	6	remark	remark	NOUN
cana-6235	245	7	,	,	PUNCT
cana-6235	245	8	βis(g	βis(g	PRON
cana-6235	245	9	−	−	PROPN
cana-6235	245	10	e	e	X
cana-6235	245	11	)	)	PUNCT
cana-6235	245	12	>	>	X
cana-6235	245	13	βis(g	βis(g	PROPN
cana-6235	245	14	)	)	PUNCT
cana-6235	245	15	▐	▐	NOUN
cana-6235	245	16	now	now	ADV
cana-6235	245	17	we	we	PRON
cana-6235	245	18	consider	consider	VERB
cana-6235	245	19	the	the	DET
cana-6235	245	20	operation	operation	NOUN
cana-6235	245	21	of	of	ADP
cana-6235	245	22	removing	remove	VERB
cana-6235	245	23	an	an	DET
cana-6235	245	24	edge	edge	NOUN
cana-6235	245	25	of	of	ADP
cana-6235	245	26	a	a	DET
cana-6235	245	27	graph	graph	NOUN
cana-6235	245	28	on	on	ADP
cana-6235	245	29	the	the	DET
cana-6235	245	30	isolate	isolate	ADJ
cana-6235	245	31	domination	domination	NOUN
cana-6235	245	32	number	number	NOUN
cana-6235	245	33	of	of	ADP
cana-6235	245	34	a	a	DET
cana-6235	245	35	graph	graph	NOUN
cana-6235	245	36	.	.	PUNCT
cana-6235	246	1	remark	remark	NOUN
cana-6235	246	2	:	:	PUNCT
cana-6235	246	3	let	let	VERB
cana-6235	246	4	g	g	PRON
cana-6235	246	5	be	be	AUX
cana-6235	246	6	a	a	DET
cana-6235	246	7	graph	graph	NOUN
cana-6235	246	8	and	and	CCONJ
cana-6235	246	9	e	e	NOUN
cana-6235	247	1	=	=	NOUN
cana-6235	247	2	{	{	PUNCT
cana-6235	247	3	uv	uv	NOUN
cana-6235	247	4	}	}	PUNCT
cana-6235	247	5	be	be	VERB
cana-6235	247	6	any	any	DET
cana-6235	247	7	edge	edge	NOUN
cana-6235	247	8	of	of	ADP
cana-6235	247	9	g	g	NOUN
cana-6235	247	10	then	then	ADV
cana-6235	247	11	any	any	PRON
cana-6235	247	12	of	of	ADP
cana-6235	247	13	the	the	DET
cana-6235	247	14	following	follow	VERB
cana-6235	247	15	three	three	NUM
cana-6235	247	16	possibilities	possibility	NOUN
cana-6235	247	17	exists	exist	VERB
cana-6235	247	18	(	(	PUNCT
cana-6235	247	19	i	i	NOUN
cana-6235	247	20	)	)	PUNCT
cana-6235	248	1	γ0(g	γ0(g	NOUN
cana-6235	249	1	−	−	NOUN
cana-6235	249	2	e	e	NOUN
cana-6235	249	3	)	)	PUNCT
cana-6235	249	4	=	=	SYM
cana-6235	249	5	γ0(g	γ0(g	NOUN
cana-6235	249	6	)	)	PUNCT
cana-6235	249	7	(	(	PUNCT
cana-6235	249	8	ii	ii	X
cana-6235	249	9	)	)	PUNCT
cana-6235	249	10	γ0(g	γ0(g	NOUN
cana-6235	249	11	−	−	NOUN
cana-6235	249	12	e	e	NOUN
cana-6235	249	13	)	)	PUNCT
cana-6235	249	14	<	<	X
cana-6235	250	1	γ0(g	γ0(g	NOUN
cana-6235	250	2	)	)	PUNCT
cana-6235	250	3	(	(	PUNCT
cana-6235	250	4	iii	iii	X
cana-6235	250	5	)	)	PUNCT
cana-6235	250	6	γ0(g	γ0(g	NOUN
cana-6235	251	1	−	−	NOUN
cana-6235	251	2	e	e	NOUN
cana-6235	251	3	)	)	PUNCT
cana-6235	251	4	>	>	X
cana-6235	252	1	γ0(g	γ0(g	PROPN
cana-6235	252	2	)	)	PUNCT
cana-6235	252	3	example	example	NOUN
cana-6235	252	4	7	7	NUM
cana-6235	252	5	:	:	PUNCT
cana-6235	252	6	let	let	VERB
cana-6235	252	7	g	g	PRON
cana-6235	252	8	be	be	AUX
cana-6235	252	9	a	a	DET
cana-6235	252	10	graph	graph	NOUN
cana-6235	252	11	with	with	ADP
cana-6235	252	12	4	4	NUM
cana-6235	252	13	vertices	vertex	NOUN
cana-6235	252	14	{	{	PUNCT
cana-6235	252	15	1	1	NUM
cana-6235	252	16	,	,	PUNCT
cana-6235	252	17	2	2	NUM
cana-6235	252	18	,	,	PUNCT
cana-6235	252	19	3	3	NUM
cana-6235	252	20	,	,	PUNCT
cana-6235	252	21	4	4	NUM
cana-6235	252	22	}	}	PUNCT
cana-6235	252	23	.	.	PUNCT
cana-6235	253	1	figure	figure	NOUN
cana-6235	253	2	10	10	NUM
cana-6235	253	3	.	.	PUNCT
cana-6235	254	1	graph	graph	VERB
cana-6235	254	2	g	g	PROPN
cana-6235	254	3	here	here	ADV
cana-6235	254	4	,	,	PUNCT
cana-6235	254	5	γ0(g	γ0(g	X
cana-6235	254	6	)	)	PUNCT
cana-6235	254	7	=	=	SYM
cana-6235	254	8	1	1	X
cana-6235	254	9	.	.	PUNCT
cana-6235	255	1	communications	communication	NOUN
cana-6235	255	2	on	on	ADP
cana-6235	255	3	applied	apply	VERB
cana-6235	255	4	nonlinear	nonlinear	ADJ
cana-6235	255	5	analysis	analysis	NOUN
cana-6235	255	6	issn	issn	NOUN
cana-6235	255	7	:	:	PUNCT
cana-6235	255	8	1074	1074	NUM
cana-6235	255	9	-	-	PUNCT
cana-6235	255	10	133x	133x	NUM
cana-6235	255	11	vol	vol	NOUN
cana-6235	255	12	31	31	NUM
cana-6235	255	13	no	no	NOUN
cana-6235	255	14	.	.	NOUN
cana-6235	255	15	2	2	NUM
cana-6235	255	16	(	(	PUNCT
cana-6235	255	17	2024	2024	NUM
cana-6235	255	18	)	)	PUNCT
cana-6235	255	19	521	521	NUM
cana-6235	255	20	https://internationalpubls.com	https://internationalpubls.com	X
cana-6235	255	21	now	now	ADV
cana-6235	255	22	we	we	PRON
cana-6235	255	23	state	state	VERB
cana-6235	255	24	and	and	CCONJ
cana-6235	255	25	prove	prove	VERB
cana-6235	255	26	a	a	DET
cana-6235	255	27	necessary	necessary	ADJ
cana-6235	255	28	and	and	CCONJ
cana-6235	255	29	sufficient	sufficient	ADJ
cana-6235	255	30	condition	condition	NOUN
cana-6235	255	31	under	under	ADP
cana-6235	255	32	which	which	PRON
cana-6235	255	33	the	the	DET
cana-6235	255	34	isolate	isolate	ADJ
cana-6235	255	35	domination	domination	NOUN
cana-6235	255	36	number	number	NOUN
cana-6235	255	37	of	of	ADP
cana-6235	255	38	a	a	DET
cana-6235	255	39	graph	graph	NOUN
cana-6235	255	40	increases	increase	NOUN
cana-6235	255	41	when	when	SCONJ
cana-6235	255	42	an	an	DET
cana-6235	255	43	edge	edge	NOUN
cana-6235	255	44	is	be	AUX
cana-6235	255	45	removing	remove	VERB
cana-6235	255	46	from	from	ADP
cana-6235	255	47	the	the	DET
cana-6235	255	48	graph	graph	NOUN
cana-6235	255	49	.	.	PUNCT
cana-6235	256	1	theorem	theorem	NOUN
cana-6235	256	2	4.13	4.13	NUM
cana-6235	256	3	:	:	PUNCT
cana-6235	256	4	let	let	VERB
cana-6235	256	5	g	g	PRON
cana-6235	256	6	be	be	AUX
cana-6235	256	7	a	a	DET
cana-6235	256	8	graph	graph	NOUN
cana-6235	256	9	with	with	ADP
cana-6235	256	10	γ0(g	γ0(g	NOUN
cana-6235	256	11	)	)	PUNCT
cana-6235	256	12	≥	≥	NOUN
cana-6235	256	13	2	2	NUM
cana-6235	256	14	and	and	CCONJ
cana-6235	256	15	e	e	NOUN
cana-6235	256	16	=	=	NOUN
cana-6235	256	17	{	{	PUNCT
cana-6235	256	18	uv	uv	NOUN
cana-6235	256	19	}	}	PUNCT
cana-6235	256	20	be	be	AUX
cana-6235	256	21	an	an	DET
cana-6235	256	22	edge	edge	NOUN
cana-6235	256	23	of	of	ADP
cana-6235	256	24	g	g	NOUN
cana-6235	256	25	then	then	ADV
cana-6235	256	26	the	the	DET
cana-6235	256	27	following	follow	VERB
cana-6235	256	28	statement	statement	NOUN
cana-6235	256	29	are	be	AUX
cana-6235	256	30	equivalent	equivalent	ADJ
cana-6235	256	31	.	.	PUNCT
cana-6235	257	1	(	(	PUNCT
cana-6235	257	2	1	1	X
cana-6235	257	3	)	)	PUNCT
cana-6235	257	4	γ0(g	γ0(g	NOUN
cana-6235	257	5	−	−	NOUN
cana-6235	257	6	e	e	NOUN
cana-6235	257	7	)	)	PUNCT
cana-6235	257	8	>	>	X
cana-6235	258	1	γ0(g	γ0(g	X
cana-6235	258	2	)	)	PUNCT
cana-6235	258	3	(	(	PUNCT
cana-6235	258	4	2	2	X
cana-6235	258	5	)	)	PUNCT
cana-6235	258	6	there	there	PRON
cana-6235	258	7	is	be	VERB
cana-6235	258	8	a	a	DET
cana-6235	258	9	minimum	minimum	NOUN
cana-6235	258	10	isolate	isolate	NOUN
cana-6235	258	11	dominating	dominate	VERB
cana-6235	258	12	set	set	NOUN
cana-6235	258	13	s	s	PROPN
cana-6235	258	14	of	of	ADP
cana-6235	258	15	g	g	NOUN
cana-6235	258	16	−	−	PROPN
cana-6235	258	17	e	e	NOUN
cana-6235	258	18	∋	∋	NOUN
cana-6235	258	19	u	u	NOUN
cana-6235	258	20	,	,	PUNCT
cana-6235	258	21	v	v	ADP
cana-6235	258	22	∈	∈	PROPN
cana-6235	258	23	s	s	NOUN
cana-6235	258	24	,	,	PUNCT
cana-6235	258	25	v	v	NOUN
cana-6235	258	26	is	be	AUX
cana-6235	258	27	an	an	DET
cana-6235	258	28	isolate	isolate	NOUN
cana-6235	258	29	in	in	ADP
cana-6235	258	30	s	s	PROPN
cana-6235	258	31	,	,	PUNCT
cana-6235	258	32	pextn[v	pextn[v	ADJ
cana-6235	258	33	,	,	PUNCT
cana-6235	258	34	s	s	AUX
cana-6235	258	35	]	]	X
cana-6235	258	36	is	be	AUX
cana-6235	258	37	empty	empty	ADJ
cana-6235	258	38	and	and	CCONJ
cana-6235	258	39	as	as	SCONJ
cana-6235	258	40	has	have	VERB
cana-6235	258	41	an	an	DET
cana-6235	258	42	isolate	isolate	NOUN
cana-6235	258	43	different	different	ADJ
cana-6235	258	44	from	from	ADP
cana-6235	258	45	v.	v.	PROPN
cana-6235	258	46	(	(	PUNCT
cana-6235	258	47	3	3	NUM
cana-6235	258	48	)	)	PUNCT
cana-6235	258	49	for	for	ADP
cana-6235	258	50	every	every	DET
cana-6235	258	51	minimum	minimum	NOUN
cana-6235	258	52	isolate	isolate	NOUN
cana-6235	258	53	dominating	dominate	VERB
cana-6235	258	54	set	set	VERB
cana-6235	258	55	t	t	PROPN
cana-6235	258	56	of	of	ADP
cana-6235	258	57	g	g	PROPN
cana-6235	258	58	,	,	PUNCT
cana-6235	258	59	u	u	PROPN
cana-6235	258	60	∈	∈	PROPN
cana-6235	258	61	t	t	PROPN
cana-6235	258	62	,	,	PUNCT
cana-6235	258	63	v	v	PROPN
cana-6235	258	64	∉	∉	PROPN
cana-6235	258	65	t	t	PROPN
cana-6235	258	66	&	&	CCONJ
cana-6235	258	67	v	v	ADP
cana-6235	258	68	∈	∈	PROPN
cana-6235	258	69	pextn[u	pextn[u	PROPN
cana-6235	258	70	,	,	PUNCT
cana-6235	258	71	t	t	PROPN
cana-6235	258	72	]	]	PUNCT
cana-6235	258	73	.	.	PUNCT
cana-6235	259	1	proof	proof	NOUN
cana-6235	259	2	:	:	PUNCT
cana-6235	259	3	(	(	PUNCT
cana-6235	259	4	1)	1)	NUM
cana-6235	259	5	(	(	PUNCT
cana-6235	259	6	3	3	NUM
cana-6235	259	7	)	)	PUNCT
cana-6235	259	8	let	let	VERB
cana-6235	259	9	t	t	NOUN
cana-6235	259	10	be	be	AUX
cana-6235	259	11	any	any	DET
cana-6235	259	12	minimum	minimum	NOUN
cana-6235	259	13	isolate	isolate	NOUN
cana-6235	259	14	dominating	dominate	VERB
cana-6235	259	15	set	set	NOUN
cana-6235	259	16	of	of	ADP
cana-6235	259	17	g.	g.	PROPN
cana-6235	259	18	if	if	SCONJ
cana-6235	259	19	u	u	PROPN
cana-6235	259	20	,	,	PUNCT
cana-6235	259	21	v	v	PROPN
cana-6235	259	22	∈	∈	PROPN
cana-6235	259	23	t	t	NOUN
cana-6235	259	24	or	or	CCONJ
cana-6235	259	25	u	u	NOUN
cana-6235	259	26	,	,	PUNCT
cana-6235	259	27	v	v	PROPN
cana-6235	259	28	∉	∉	PROPN
cana-6235	259	29	t	t	PROPN
cana-6235	259	30	then	then	ADV
cana-6235	259	31	obviously	obviously	ADV
cana-6235	259	32	t	t	PROPN
cana-6235	259	33	is	be	AUX
cana-6235	259	34	an	an	DET
cana-6235	259	35	isolate	isolate	ADJ
cana-6235	259	36	dominating	dominating	NOUN
cana-6235	259	37	set	set	VERB
cana-6235	259	38	in	in	ADP
cana-6235	259	39	g	g	PROPN
cana-6235	259	40	−	−	PROPN
cana-6235	259	41	e.	e.	PROPN
cana-6235	259	42	therefore	therefore	ADV
cana-6235	259	43	,	,	PUNCT
cana-6235	259	44	γ0(g	γ0(g	PROPN
cana-6235	259	45	−	−	NOUN
cana-6235	259	46	e	e	NOUN
cana-6235	259	47	)	)	PUNCT
cana-6235	259	48	≤	≤	NUM
cana-6235	259	49	|t|	|t|	NOUN
cana-6235	259	50	=	=	SYM
cana-6235	259	51	γ0(g	γ0(g	NOUN
cana-6235	259	52	)	)	PUNCT
cana-6235	259	53	.	.	PUNCT
cana-6235	260	1	which	which	PRON
cana-6235	260	2	is	be	AUX
cana-6235	260	3	a	a	DET
cana-6235	260	4	contradiction	contradiction	NOUN
cana-6235	260	5	.	.	PUNCT
cana-6235	261	1	therefore	therefore	ADV
cana-6235	261	2	,	,	PUNCT
cana-6235	261	3	u	u	PROPN
cana-6235	261	4	∈	∈	PROPN
cana-6235	261	5	t	t	PROPN
cana-6235	261	6	&	&	CCONJ
cana-6235	261	7	v	v	PROPN
cana-6235	261	8	∉	∉	PROPN
cana-6235	261	9	t	t	PROPN
cana-6235	261	10	or	or	CCONJ
cana-6235	261	11	v	v	ADP
cana-6235	261	12	∈	∈	PROPN
cana-6235	261	13	t	t	PROPN
cana-6235	261	14	&	&	CCONJ
cana-6235	261	15	u	u	PROPN
cana-6235	261	16	∉	∉	PROPN
cana-6235	261	17	t	t	PROPN
cana-6235	261	18	.	.	PUNCT
cana-6235	262	1	we	we	PRON
cana-6235	262	2	may	may	AUX
cana-6235	262	3	assume	assume	VERB
cana-6235	262	4	that	that	SCONJ
cana-6235	262	5	u	u	PROPN
cana-6235	262	6	∈	∈	PROPN
cana-6235	262	7	t	t	PROPN
cana-6235	262	8	&	&	CCONJ
cana-6235	262	9	v	v	X
cana-6235	262	10	∉	∉	PROPN
cana-6235	262	11	t.	t.	PROPN
cana-6235	262	12	now	now	ADV
cana-6235	262	13	|t|	|t|	VERB
cana-6235	262	14	=	=	SYM
cana-6235	262	15	γ0(g	γ0(g	NOUN
cana-6235	262	16	)	)	PUNCT
cana-6235	262	17	<	<	X
cana-6235	263	1	γ0(g	γ0(g	X
cana-6235	263	2	−	−	NOUN
cana-6235	263	3	e	e	NOUN
cana-6235	263	4	)	)	PUNCT
cana-6235	263	5	.	.	PUNCT
cana-6235	264	1	therefore	therefore	ADV
cana-6235	264	2	,	,	PUNCT
cana-6235	264	3	t	t	PROPN
cana-6235	264	4	can	can	AUX
cana-6235	264	5	not	not	PART
cana-6235	264	6	be	be	AUX
cana-6235	264	7	an	an	DET
cana-6235	264	8	isolate	isolate	NOUN
cana-6235	264	9	dominating	dominating	NOUN
cana-6235	264	10	set	set	VERB
cana-6235	264	11	in	in	ADP
cana-6235	264	12	−e	−e	NOUN
cana-6235	264	13	.	.	PUNCT
cana-6235	265	1	now	now	ADV
cana-6235	265	2	e	e	NOUN
cana-6235	265	3	has	have	AUX
cana-6235	265	4	isolated	isolate	VERB
cana-6235	265	5	vertices	vertex	NOUN
cana-6235	265	6	then	then	ADV
cana-6235	265	7	regarded	regard	VERB
cana-6235	265	8	as	as	ADP
cana-6235	265	9	a	a	DET
cana-6235	265	10	set	set	NOUN
cana-6235	265	11	of	of	ADP
cana-6235	265	12	vertices	vertex	NOUN
cana-6235	265	13	of	of	ADP
cana-6235	265	14	g	g	PROPN
cana-6235	265	15	−	−	PROPN
cana-6235	265	16	e.	e.	PROPN
cana-6235	265	17	therefore	therefore	ADV
cana-6235	265	18	,	,	PUNCT
cana-6235	265	19	t	t	PROPN
cana-6235	265	20	can	can	AUX
cana-6235	265	21	not	not	PART
cana-6235	265	22	be	be	AUX
cana-6235	265	23	dominating	dominate	VERB
cana-6235	265	24	set	set	NOUN
cana-6235	265	25	of	of	ADP
cana-6235	265	26	g	g	PROPN
cana-6235	265	27	−	−	PROPN
cana-6235	265	28	e.	e.	PROPN
cana-6235	265	29	therefore	therefore	ADV
cana-6235	265	30	,	,	PUNCT
cana-6235	265	31	v	v	NOUN
cana-6235	265	32	is	be	AUX
cana-6235	265	33	not	not	PART
cana-6235	265	34	adjacent	adjacent	ADJ
cana-6235	265	35	to	to	ADP
cana-6235	265	36	any	any	DET
cana-6235	265	37	vertex	vertex	NOUN
cana-6235	265	38	of	of	ADP
cana-6235	265	39	t	t	PROPN
cana-6235	265	40	in	in	ADP
cana-6235	265	41	g	g	PROPN
cana-6235	265	42	−	−	PROPN
cana-6235	265	43	v	v	NOUN
cana-6235	265	44	but	but	CCONJ
cana-6235	265	45	v	v	NOUN
cana-6235	265	46	is	be	AUX
cana-6235	265	47	adjacent	adjacent	ADJ
cana-6235	265	48	to	to	ADP
cana-6235	265	49	some	some	DET
cana-6235	265	50	vertex	vertex	NOUN
cana-6235	265	51	of	of	ADP
cana-6235	265	52	t	t	PROPN
cana-6235	265	53	in	in	ADP
cana-6235	265	54	g.	g.	PROPN
cana-6235	265	55	∴	∴	PROPN
cana-6235	265	56	v	v	ADP
cana-6235	265	57	∈	∈	PROPN
cana-6235	265	58	pn[u	pn[u	PROPN
cana-6235	265	59	,	,	PUNCT
cana-6235	265	60	t	t	PROPN
cana-6235	265	61	]	]	PUNCT
cana-6235	265	62	.	.	PUNCT
cana-6235	266	1	thus	thus	ADV
cana-6235	266	2	,	,	PUNCT
cana-6235	266	3	(	(	PUNCT
cana-6235	266	4	1)	1)	NUM
cana-6235	266	5	(	(	PUNCT
cana-6235	266	6	3	3	NUM
cana-6235	266	7	)	)	PUNCT
cana-6235	266	8	is	be	AUX
cana-6235	266	9	proved	prove	VERB
cana-6235	266	10	.	.	PUNCT
cana-6235	267	1	(	(	PUNCT
cana-6235	267	2	3)	3)	NUM
cana-6235	267	3	(	(	PUNCT
cana-6235	267	4	2	2	NUM
cana-6235	267	5	)	)	PUNCT
cana-6235	267	6	let	let	AUX
cana-6235	267	7	t	t	NOUN
cana-6235	267	8	be	be	AUX
cana-6235	267	9	any	any	DET
cana-6235	267	10	minimum	minimum	NOUN
cana-6235	267	11	isolate	isolate	NOUN
cana-6235	267	12	dominating	dominate	VERB
cana-6235	267	13	set	set	NOUN
cana-6235	267	14	of	of	ADP
cana-6235	267	15	g	g	PROPN
cana-6235	267	16	then	then	ADV
cana-6235	267	17	u	u	PROPN
cana-6235	267	18	∈	∈	PROPN
cana-6235	267	19	t	t	PROPN
cana-6235	267	20	,	,	PUNCT
cana-6235	267	21	v	v	PROPN
cana-6235	267	22	∉	∉	PROPN
cana-6235	267	23	t	t	PROPN
cana-6235	267	24	&	&	CCONJ
cana-6235	267	25	v	v	ADP
cana-6235	267	26	∈	∈	PROPN
cana-6235	267	27	pn[u	pn[u	PROPN
cana-6235	267	28	,	,	PUNCT
cana-6235	267	29	t	t	PROPN
cana-6235	267	30	]	]	PUNCT
cana-6235	267	31	in	in	ADP
cana-6235	267	32	g.	g.	PROPN
cana-6235	267	33	obviously	obviously	ADV
cana-6235	267	34	t	t	PROPN
cana-6235	267	35	can	can	AUX
cana-6235	267	36	not	not	PART
cana-6235	267	37	be	be	AUX
cana-6235	267	38	an	an	DET
cana-6235	267	39	isolate	isolate	NOUN
cana-6235	267	40	dominating	dominating	NOUN
cana-6235	267	41	set	set	VERB
cana-6235	267	42	in	in	ADP
cana-6235	267	43	g	g	PROPN
cana-6235	267	44	−	−	PROPN
cana-6235	267	45	e.	e.	PROPN
cana-6235	267	46	let	let	VERB
cana-6235	267	47	s	s	PRON
cana-6235	268	1	=	=	VERB
cana-6235	268	2	t	t	X
cana-6235	268	3	∪	∪	X
cana-6235	268	4	{	{	PUNCT
cana-6235	268	5	v	v	NOUN
cana-6235	268	6	}	}	PUNCT
cana-6235	268	7	.	.	PUNCT
cana-6235	269	1	then	then	ADV
cana-6235	269	2	v	v	X
cana-6235	269	3	∈	∈	PROPN
cana-6235	269	4	s	s	NOUN
cana-6235	269	5	&	&	CCONJ
cana-6235	269	6	u	u	PROPN
cana-6235	269	7	∈	∈	PROPN
cana-6235	269	8	s	s	PROPN
cana-6235	269	9	&	&	CCONJ
cana-6235	269	10	v	v	PROPN
cana-6235	269	11	is	be	AUX
cana-6235	269	12	an	an	DET
cana-6235	269	13	isolate	isolate	NOUN
cana-6235	269	14	in	in	ADP
cana-6235	269	15	s.	s.	PROPN
cana-6235	269	16	obviously	obviously	ADV
cana-6235	269	17	,	,	PUNCT
cana-6235	269	18	s	s	VERB
cana-6235	269	19	is	be	AUX
cana-6235	269	20	an	an	DET
cana-6235	269	21	isolate	isolate	NOUN
cana-6235	269	22	dominating	dominating	NOUN
cana-6235	269	23	set	set	NOUN
cana-6235	269	24	of	of	ADP
cana-6235	269	25	g	g	PROPN
cana-6235	269	26	−	−	PROPN
cana-6235	269	27	e	e	NOUN
cana-6235	269	28	.	.	PUNCT
cana-6235	270	1	if	if	SCONJ
cana-6235	270	2	there	there	PRON
cana-6235	270	3	is	be	VERB
cana-6235	270	4	no	no	DET
cana-6235	270	5	vertex	vertex	NOUN
cana-6235	270	6	w	w	NOUN
cana-6235	270	7	outside	outside	ADP
cana-6235	270	8	of	of	ADP
cana-6235	270	9	s	s	PRON
cana-6235	270	10	which	which	PRON
cana-6235	270	11	is	be	AUX
cana-6235	270	12	adjacent	adjacent	ADJ
cana-6235	270	13	to	to	ADP
cana-6235	270	14	v	v	PRON
cana-6235	270	15	then	then	ADV
cana-6235	270	16	pextn[v	pextn[v	ADV
cana-6235	270	17	,	,	PUNCT
cana-6235	270	18	s	s	X
cana-6235	270	19	]	]	X
cana-6235	270	20	=	=	PUNCT
cana-6235	270	21	∅.	∅.	NOUN
cana-6235	270	22	suppose	suppose	VERB
cana-6235	270	23	there	there	PRON
cana-6235	270	24	is	be	VERB
cana-6235	270	25	a	a	DET
cana-6235	270	26	vertex	vertex	NOUN
cana-6235	270	27	w	w	PROPN
cana-6235	270	28	∋	∋	NOUN
cana-6235	270	29	w	w	VERB
cana-6235	270	30	not	not	PART
cana-6235	270	31	in	in	ADP
cana-6235	270	32	g	g	PROPN
cana-6235	270	33	,	,	PUNCT
cana-6235	270	34	w	w	PROPN
cana-6235	270	35	is	be	AUX
cana-6235	270	36	adjacent	adjacent	ADJ
cana-6235	270	37	to	to	ADP
cana-6235	270	38	v	v	NOUN
cana-6235	270	39	in	in	ADP
cana-6235	270	40	g	g	NOUN
cana-6235	270	41	−	−	PROPN
cana-6235	270	42	e	e	NOUN
cana-6235	270	43	.	.	PUNCT
cana-6235	271	1	now	now	ADV
cana-6235	271	2	w	w	PROPN
cana-6235	271	3	∉	∉	PROPN
cana-6235	271	4	t	t	PROPN
cana-6235	271	5	&	&	CCONJ
cana-6235	271	6	t	t	PROPN
cana-6235	271	7	is	be	AUX
cana-6235	271	8	a	a	DET
cana-6235	271	9	dominating	dominating	NOUN
cana-6235	271	10	set	set	NOUN
cana-6235	271	11	of	of	ADP
cana-6235	271	12	g.	g.	PROPN
cana-6235	271	13	therefore	therefore	ADV
cana-6235	271	14	,	,	PUNCT
cana-6235	271	15	w	w	NOUN
cana-6235	271	16	is	be	AUX
cana-6235	271	17	adjacent	adjacent	ADJ
cana-6235	271	18	to	to	ADP
cana-6235	271	19	some	some	DET
cana-6235	271	20	vertex	vertex	NOUN
cana-6235	271	21	z	z	NOUN
cana-6235	271	22	of	of	ADP
cana-6235	271	23	g	g	PROPN
cana-6235	271	24	.	.	PUNCT
cana-6235	272	1	thus	thus	ADV
cana-6235	272	2	,	,	PUNCT
cana-6235	272	3	w	w	PROPN
cana-6235	272	4	is	be	AUX
cana-6235	272	5	adjacent	adjacent	ADJ
cana-6235	272	6	to	to	ADP
cana-6235	272	7	two	two	NUM
cana-6235	272	8	distinct	distinct	ADJ
cana-6235	272	9	vertices	vertex	NOUN
cana-6235	272	10	of	of	ADP
cana-6235	272	11	s	s	PRON
cana-6235	272	12	in	in	ADP
cana-6235	272	13	g	g	PROPN
cana-6235	272	14	−	−	PROPN
cana-6235	272	15	e.	e.	PROPN
cana-6235	272	16	thus	thus	ADV
cana-6235	272	17	,	,	PUNCT
cana-6235	272	18	pextn[v	pextn[v	ADV
cana-6235	272	19	,	,	PUNCT
cana-6235	272	20	s	s	X
cana-6235	272	21	]	]	X
cana-6235	272	22	=	=	SYM
cana-6235	272	23	∅	∅	NOUN
cana-6235	272	24	.	.	PUNCT
cana-6235	273	1	note	note	VERB
cana-6235	273	2	that	that	SCONJ
cana-6235	273	3	an	an	DET
cana-6235	273	4	isolate	isolate	NOUN
cana-6235	273	5	of	of	ADP
cana-6235	273	6	t	t	PROPN
cana-6235	273	7	in	in	ADP
cana-6235	273	8	g	g	PROPN
cana-6235	273	9	is	be	AUX
cana-6235	273	10	also	also	ADV
cana-6235	273	11	an	an	DET
cana-6235	273	12	isolate	isolate	NOUN
cana-6235	273	13	of	of	ADP
cana-6235	273	14	s	s	PRON
cana-6235	273	15	in	in	ADP
cana-6235	273	16	g	g	NOUN
cana-6235	273	17	−	−	NOUN
cana-6235	273	18	e	e	NOUN
cana-6235	274	1	and	and	CCONJ
cana-6235	274	2	it	it	PRON
cana-6235	274	3	is	be	AUX
cana-6235	274	4	different	different	ADJ
cana-6235	274	5	from	from	ADP
cana-6235	274	6	v.	v.	ADP
cana-6235	274	7	communications	communication	NOUN
cana-6235	274	8	on	on	ADP
cana-6235	274	9	applied	apply	VERB
cana-6235	274	10	nonlinear	nonlinear	ADJ
cana-6235	274	11	analysis	analysis	NOUN
cana-6235	274	12	issn	issn	NOUN
cana-6235	274	13	:	:	PUNCT
cana-6235	274	14	1074	1074	NUM
cana-6235	274	15	-	-	PUNCT
cana-6235	274	16	133x	133x	NUM
cana-6235	274	17	vol	vol	NOUN
cana-6235	274	18	31	31	NUM
cana-6235	274	19	no	no	NOUN
cana-6235	274	20	.	.	NOUN
cana-6235	274	21	2	2	NUM
cana-6235	274	22	(	(	PUNCT
cana-6235	274	23	2024	2024	NUM
cana-6235	274	24	)	)	PUNCT
cana-6235	274	25	522	522	NUM
cana-6235	274	26	https://internationalpubls.com	https://internationalpubls.com	X
cana-6235	274	27	thus	thus	ADV
cana-6235	274	28	,	,	PUNCT
cana-6235	274	29	(	(	PUNCT
cana-6235	274	30	3)	3)	NUM
cana-6235	274	31	(	(	PUNCT
cana-6235	274	32	2	2	NUM
cana-6235	274	33	)	)	PUNCT
cana-6235	274	34	is	be	AUX
cana-6235	274	35	proved	prove	VERB
cana-6235	274	36	.	.	PUNCT
cana-6235	275	1	(	(	PUNCT
cana-6235	275	2	2)	2)	NUM
cana-6235	275	3	(	(	PUNCT
cana-6235	275	4	1	1	NUM
cana-6235	275	5	)	)	PUNCT
cana-6235	275	6	let	let	VERB
cana-6235	275	7	s	s	PRON
cana-6235	275	8	be	be	AUX
cana-6235	275	9	minimum	minimum	NOUN
cana-6235	275	10	isolate	isolate	NOUN
cana-6235	275	11	dominating	dominate	VERB
cana-6235	275	12	set	set	NOUN
cana-6235	275	13	of	of	ADP
cana-6235	275	14	g	g	PROPN
cana-6235	275	15	−	−	PROPN
cana-6235	275	16	e	e	NOUN
cana-6235	275	17	such	such	ADJ
cana-6235	275	18	that	that	SCONJ
cana-6235	275	19	u	u	PROPN
cana-6235	275	20	∈	∈	PROPN
cana-6235	275	21	s	s	PROPN
cana-6235	275	22	&	&	CCONJ
cana-6235	275	23	v	v	ADP
cana-6235	275	24	∈	∈	PROPN
cana-6235	275	25	s	s	PROPN
cana-6235	275	26	&	&	CCONJ
cana-6235	275	27	v	v	PROPN
cana-6235	275	28	is	be	AUX
cana-6235	275	29	an	an	DET
cana-6235	275	30	isolate	isolate	NOUN
cana-6235	275	31	of	of	ADP
cana-6235	275	32	s	s	PROPN
cana-6235	275	33	,	,	PUNCT
cana-6235	275	34	pextn[v	pextn[v	ADJ
cana-6235	275	35	,	,	PUNCT
cana-6235	275	36	s	s	X
cana-6235	275	37	]	]	X
cana-6235	275	38	=	=	SYM
cana-6235	275	39	∅	∅	NOUN
cana-6235	275	40	and	and	CCONJ
cana-6235	275	41	suppose	suppose	VERB
cana-6235	275	42	s	s	X
cana-6235	275	43	has	have	VERB
cana-6235	275	44	an	an	DET
cana-6235	275	45	isolate	isolate	NOUN
cana-6235	275	46	different	different	ADJ
cana-6235	275	47	from	from	ADP
cana-6235	275	48	v.	v.	INTJ
cana-6235	275	49	let	let	VERB
cana-6235	275	50	t	t	NOUN
cana-6235	275	51	=	=	SYM
cana-6235	275	52	s	s	PART
cana-6235	275	53	−	−	NOUN
cana-6235	275	54	{	{	PUNCT
cana-6235	275	55	v	v	NOUN
cana-6235	275	56	}	}	PUNCT
cana-6235	275	57	.	.	PUNCT
cana-6235	276	1	let	let	VERB
cana-6235	276	2	z	z	NOUN
cana-6235	276	3	be	be	AUX
cana-6235	276	4	any	any	DET
cana-6235	276	5	vertex	vertex	NOUN
cana-6235	276	6	of	of	ADP
cana-6235	276	7	g	g	NOUN
cana-6235	276	8	which	which	PRON
cana-6235	276	9	is	be	AUX
cana-6235	276	10	not	not	PART
cana-6235	276	11	in	in	ADP
cana-6235	276	12	g	g	PROPN
cana-6235	276	13	−	−	PROPN
cana-6235	276	14	e	e	NOUN
cana-6235	276	15	.	.	PUNCT
cana-6235	277	1	if	if	SCONJ
cana-6235	277	2	z	z	NOUN
cana-6235	277	3	=	=	SYM
cana-6235	277	4	v	v	NOUN
cana-6235	277	5	then	then	ADV
cana-6235	277	6	z	z	PROPN
cana-6235	277	7	is	be	AUX
cana-6235	277	8	adjacent	adjacent	ADJ
cana-6235	277	9	to	to	ADP
cana-6235	277	10	u	u	NOUN
cana-6235	277	11	in	in	ADP
cana-6235	277	12	g.	g.	PROPN
cana-6235	277	13	if	if	SCONJ
cana-6235	277	14	z	z	PROPN
cana-6235	277	15	≠	≠	PROPN
cana-6235	277	16	v	v	X
cana-6235	277	17	then	then	ADV
cana-6235	277	18	∉	∉	PROPN
cana-6235	277	19	s	s	PART
cana-6235	277	20	.	.	PUNCT
cana-6235	278	1	suppose	suppose	VERB
cana-6235	278	2	z	z	NOUN
cana-6235	278	3	is	be	AUX
cana-6235	278	4	adjacent	adjacent	ADJ
cana-6235	278	5	to	to	ADP
cana-6235	278	6	v	v	NOUN
cana-6235	278	7	in	in	ADP
cana-6235	278	8	g	g	PROPN
cana-6235	278	9	−	−	PROPN
cana-6235	279	1	e	e	X
cana-6235	279	2	then	then	ADV
cana-6235	279	3	z	z	PROPN
cana-6235	279	4	must	must	AUX
cana-6235	279	5	be	be	AUX
cana-6235	279	6	adjacent	adjacent	ADJ
cana-6235	279	7	to	to	ADP
cana-6235	279	8	some	some	DET
cana-6235	279	9	other	other	ADJ
cana-6235	279	10	vertex	vertex	NOUN
cana-6235	279	11	v′	v′	NOUN
cana-6235	279	12	of	of	ADP
cana-6235	279	13	s	s	PRON
cana-6235	279	14	because	because	SCONJ
cana-6235	279	15	z	z	PROPN
cana-6235	279	16	∉	∉	PROPN
cana-6235	279	17	pextn[v	pextn[v	PROPN
cana-6235	279	18	,	,	PUNCT
cana-6235	279	19	s	s	PART
cana-6235	279	20	]	]	PUNCT
cana-6235	279	21	.	.	PUNCT
cana-6235	280	1	then	then	ADV
cana-6235	280	2	v′	v′	PROPN
cana-6235	280	3	∈	∈	PROPN
cana-6235	280	4	t	t	PROPN
cana-6235	280	5	&	&	CCONJ
cana-6235	280	6	z	z	PROPN
cana-6235	280	7	is	be	AUX
cana-6235	280	8	adjacent	adjacent	ADJ
cana-6235	280	9	to	to	ADP
cana-6235	280	10	v′	v′	NOUN
cana-6235	280	11	in	in	ADP
cana-6235	280	12	g.	g.	PROPN
cana-6235	280	13	therefore	therefore	ADV
cana-6235	280	14	,	,	PUNCT
cana-6235	280	15	t	t	PROPN
cana-6235	280	16	is	be	AUX
cana-6235	280	17	a	a	DET
cana-6235	280	18	dominating	dominating	NOUN
cana-6235	280	19	set	set	VERB
cana-6235	280	20	in	in	ADP
cana-6235	280	21	g.	g.	PROPN
cana-6235	280	22	note	note	PROPN
cana-6235	280	23	that	that	SCONJ
cana-6235	280	24	t	t	PROPN
cana-6235	280	25	has	have	VERB
cana-6235	280	26	an	an	DET
cana-6235	280	27	isolate	isolate	NOUN
cana-6235	280	28	because	because	SCONJ
cana-6235	280	29	s	s	PROPN
cana-6235	280	30	has	have	VERB
cana-6235	280	31	an	an	DET
cana-6235	280	32	isolate	isolate	NOUN
cana-6235	280	33	different	different	ADJ
cana-6235	280	34	from	from	ADP
cana-6235	280	35	v.	v.	ADP
cana-6235	280	36	thus	thus	ADV
cana-6235	280	37	,	,	PUNCT
cana-6235	280	38	t	t	PROPN
cana-6235	280	39	is	be	AUX
cana-6235	280	40	an	an	DET
cana-6235	280	41	isolate	isolate	NOUN
cana-6235	280	42	dominating	dominating	NOUN
cana-6235	280	43	set	set	NOUN
cana-6235	280	44	of	of	ADP
cana-6235	280	45	g.	g.	PROPN
cana-6235	280	46	therefore	therefore	ADV
cana-6235	280	47	,	,	PUNCT
cana-6235	280	48	γ0(g	γ0(g	PROPN
cana-6235	280	49	)	)	PUNCT
cana-6235	280	50	≤	≤	NUM
cana-6235	280	51	|t|	|t|	VERB
cana-6235	280	52	<	<	X
cana-6235	280	53	|s|	|s|	PROPN
cana-6235	280	54	=	=	SYM
cana-6235	280	55	γ0(g	γ0(g	PROPN
cana-6235	280	56	−	−	NOUN
cana-6235	280	57	e	e	NOUN
cana-6235	280	58	)	)	PUNCT
cana-6235	280	59	.	.	PUNCT
cana-6235	281	1	thus	thus	ADV
cana-6235	281	2	,	,	PUNCT
cana-6235	281	3	γ0(g	γ0(g	NOUN
cana-6235	281	4	−	−	NOUN
cana-6235	281	5	e	e	NOUN
cana-6235	281	6	)	)	PUNCT
cana-6235	281	7	>	>	X
cana-6235	281	8	γ0(g	γ0(g	X
cana-6235	281	9	)	)	PUNCT
cana-6235	281	10	▐	▐	NOUN
cana-6235	281	11	theorem	theorem	VERB
cana-6235	281	12	4.14	4.14	NUM
cana-6235	281	13	:	:	PUNCT
cana-6235	281	14	let	let	VERB
cana-6235	281	15	g	g	PRON
cana-6235	281	16	be	be	AUX
cana-6235	281	17	a	a	DET
cana-6235	281	18	graph	graph	NOUN
cana-6235	281	19	with	with	ADP
cana-6235	281	20	γ0(g	γ0(g	NOUN
cana-6235	281	21	)	)	PUNCT
cana-6235	281	22	=	=	SYM
cana-6235	282	1	1	1	X
cana-6235	282	2	.	.	PUNCT
cana-6235	282	3	let	let	VERB
cana-6235	282	4	e	e	NOUN
cana-6235	282	5	=	=	PRON
cana-6235	282	6	{	{	PUNCT
cana-6235	282	7	uv	uv	NOUN
cana-6235	282	8	}	}	PUNCT
cana-6235	282	9	be	be	VERB
cana-6235	282	10	any	any	DET
cana-6235	282	11	edge	edge	NOUN
cana-6235	282	12	of	of	ADP
cana-6235	282	13	g.	g.	NOUN
cana-6235	282	14	then	then	ADV
cana-6235	282	15	γ0(g	γ0(g	VERB
cana-6235	282	16	−	−	NOUN
cana-6235	282	17	e	e	NOUN
cana-6235	282	18	)	)	PUNCT
cana-6235	282	19	>	>	X
cana-6235	283	1	γ0(g	γ0(g	X
cana-6235	283	2	)	)	PUNCT
cana-6235	283	3	if	if	SCONJ
cana-6235	283	4	and	and	CCONJ
cana-6235	283	5	only	only	ADV
cana-6235	283	6	if	if	SCONJ
cana-6235	283	7	{	{	PUNCT
cana-6235	283	8	z	z	NOUN
cana-6235	283	9	}	}	PUNCT
cana-6235	283	10	is	be	AUX
cana-6235	283	11	an	an	DET
cana-6235	283	12	isolate	isolate	NOUN
cana-6235	283	13	dominating	dominating	NOUN
cana-6235	283	14	set	set	NOUN
cana-6235	283	15	of	of	ADP
cana-6235	283	16	g	g	NOUN
cana-6235	283	17	,	,	PUNCT
cana-6235	283	18	then	then	ADV
cana-6235	283	19	z	z	PROPN
cana-6235	283	20	∈	∈	PROPN
cana-6235	283	21	{	{	PUNCT
cana-6235	283	22	u	u	NOUN
cana-6235	283	23	,	,	PUNCT
cana-6235	283	24	v	v	NOUN
cana-6235	283	25	}	}	PUNCT
cana-6235	283	26	.	.	PUNCT
cana-6235	284	1	proof	proof	NOUN
cana-6235	284	2	:	:	PUNCT
cana-6235	284	3	suppose	suppose	VERB
cana-6235	284	4	condition	condition	NOUN
cana-6235	284	5	is	be	AUX
cana-6235	284	6	satisfied	satisfied	ADJ
cana-6235	284	7	.	.	PUNCT
cana-6235	285	1	if	if	SCONJ
cana-6235	285	2	γ0(g	γ0(g	PRON
cana-6235	285	3	−	−	NOUN
cana-6235	285	4	e	e	NOUN
cana-6235	285	5	)	)	PUNCT
cana-6235	285	6	=	=	SYM
cana-6235	286	1	γ0(g	γ0(g	NOUN
cana-6235	286	2	)	)	PUNCT
cana-6235	286	3	then	then	ADV
cana-6235	286	4	γ0(g	γ0(g	VERB
cana-6235	286	5	−	−	NOUN
cana-6235	286	6	e	e	NOUN
cana-6235	286	7	)	)	PUNCT
cana-6235	286	8	=	=	SYM
cana-6235	286	9	1	1	X
cana-6235	286	10	.	.	PUNCT
cana-6235	287	1	suppose	suppose	VERB
cana-6235	287	2	{	{	PUNCT
cana-6235	287	3	z	z	X
cana-6235	287	4	}	}	PUNCT
cana-6235	287	5	is	be	AUX
cana-6235	287	6	a	a	DET
cana-6235	287	7	minimum	minimum	NOUN
cana-6235	287	8	isolate	isolate	NOUN
cana-6235	287	9	dominating	dominating	NOUN
cana-6235	287	10	set	set	NOUN
cana-6235	287	11	of	of	ADP
cana-6235	287	12	g	g	PROPN
cana-6235	287	13	−	−	PROPN
cana-6235	287	14	e	e	NOUN
cana-6235	287	15	.	.	PUNCT
cana-6235	288	1	then	then	ADV
cana-6235	288	2	z	z	NOUN
cana-6235	288	3	≠	≠	PROPN
cana-6235	288	4	u	u	NOUN
cana-6235	288	5	because	because	SCONJ
cana-6235	288	6	v	v	NOUN
cana-6235	288	7	is	be	AUX
cana-6235	288	8	not	not	PART
cana-6235	288	9	adjacent	adjacent	ADJ
cana-6235	288	10	to	to	ADP
cana-6235	288	11	u	u	NOUN
cana-6235	288	12	in	in	ADP
cana-6235	288	13	g	g	PROPN
cana-6235	288	14	−	−	PROPN
cana-6235	288	15	e.	e.	PROPN
cana-6235	288	16	similarly	similarly	ADV
cana-6235	288	17	,	,	PUNCT
cana-6235	288	18	if	if	SCONJ
cana-6235	288	19	z	z	PROPN
cana-6235	288	20	≠	≠	PROPN
cana-6235	288	21	v	v	NOUN
cana-6235	288	22	.	.	PUNCT
cana-6235	289	1	this	this	PRON
cana-6235	289	2	contradicts	contradict	VERB
cana-6235	289	3	condition	condition	NOUN
cana-6235	289	4	c.	c.	PROPN
cana-6235	289	5	therefore	therefore	ADV
cana-6235	289	6	,	,	PUNCT
cana-6235	289	7	γ0(g	γ0(g	PROPN
cana-6235	289	8	−	−	NOUN
cana-6235	289	9	e	e	NOUN
cana-6235	289	10	)	)	PUNCT
cana-6235	289	11	>	>	X
cana-6235	290	1	γ0(g	γ0(g	X
cana-6235	290	2	)	)	PUNCT
cana-6235	290	3	(	(	PUNCT
cana-6235	290	4	γ0(g	γ0(g	NOUN
cana-6235	290	5	−	−	X
cana-6235	290	6	e	e	NOUN
cana-6235	290	7	)	)	PUNCT
cana-6235	290	8	<	<	X
cana-6235	290	9	γ0(g	γ0(g	NOUN
cana-6235	290	10	)	)	PUNCT
cana-6235	290	11	is	be	AUX
cana-6235	290	12	not	not	PART
cana-6235	290	13	possible	possible	ADJ
cana-6235	290	14	because	because	SCONJ
cana-6235	290	15	γ0(g	γ0(g	NOUN
cana-6235	290	16	)	)	PUNCT
cana-6235	290	17	=	=	SYM
cana-6235	290	18	1	1	NUM
cana-6235	290	19	)	)	PUNCT
cana-6235	290	20	.	.	PUNCT
cana-6235	291	1	conversely	conversely	ADV
cana-6235	291	2	,	,	PUNCT
cana-6235	291	3	suppose	suppose	VERB
cana-6235	291	4	γ0(g	γ0(g	PRON
cana-6235	291	5	−	−	NOUN
cana-6235	291	6	e	e	NOUN
cana-6235	291	7	)	)	PUNCT
cana-6235	291	8	>	>	X
cana-6235	292	1	γ0(g	γ0(g	PROPN
cana-6235	292	2	)	)	PUNCT
cana-6235	292	3	.	.	PUNCT
cana-6235	293	1	let	let	VERB
cana-6235	293	2	z	z	PRON
cana-6235	293	3	be	be	AUX
cana-6235	293	4	a	a	DET
cana-6235	293	5	vertex	vertex	NOUN
cana-6235	293	6	of	of	ADP
cana-6235	293	7	g	g	PROPN
cana-6235	293	8	∋	∋	PROPN
cana-6235	293	9	z	z	PROPN
cana-6235	293	10	∉	∉	PROPN
cana-6235	293	11	{	{	PUNCT
cana-6235	293	12	u	u	PROPN
cana-6235	293	13	,	,	PUNCT
cana-6235	293	14	v	v	NOUN
cana-6235	293	15	}	}	PUNCT
cana-6235	293	16	.	.	PUNCT
cana-6235	294	1	if	if	SCONJ
cana-6235	294	2	{	{	PUNCT
cana-6235	294	3	z	z	NOUN
cana-6235	294	4	}	}	PUNCT
cana-6235	294	5	is	be	AUX
cana-6235	294	6	a	a	DET
cana-6235	294	7	minimum	minimum	NOUN
cana-6235	294	8	isolate	isolate	NOUN
cana-6235	294	9	dominating	dominating	NOUN
cana-6235	294	10	set	set	NOUN
cana-6235	294	11	of	of	ADP
cana-6235	294	12	g	g	PROPN
cana-6235	294	13	then	then	ADV
cana-6235	294	14	{	{	PUNCT
cana-6235	294	15	z	z	NOUN
cana-6235	294	16	}	}	PUNCT
cana-6235	294	17	is	be	AUX
cana-6235	294	18	also	also	ADV
cana-6235	294	19	an	an	DET
cana-6235	294	20	isolate	isolate	ADJ
cana-6235	294	21	dominating	dominating	NOUN
cana-6235	294	22	set	set	NOUN
cana-6235	294	23	of	of	ADP
cana-6235	294	24	g	g	PROPN
cana-6235	294	25	−	−	PROPN
cana-6235	294	26	e.	e.	PROPN
cana-6235	294	27	⇒	⇒	PROPN
cana-6235	294	28	γ0(g	γ0(g	PRON
cana-6235	294	29	−	−	NOUN
cana-6235	294	30	e	e	NOUN
cana-6235	294	31	)	)	PUNCT
cana-6235	295	1	=	=	SYM
cana-6235	295	2	γ0(g	γ0(g	NOUN
cana-6235	295	3	)	)	PUNCT
cana-6235	295	4	.	.	PUNCT
cana-6235	296	1	which	which	PRON
cana-6235	296	2	is	be	AUX
cana-6235	296	3	a	a	DET
cana-6235	296	4	contradiction	contradiction	NOUN
cana-6235	296	5	.	.	PUNCT
cana-6235	297	1	therefore	therefore	ADV
cana-6235	297	2	,	,	PUNCT
cana-6235	297	3	z	z	PROPN
cana-6235	297	4	∈	∈	PROPN
cana-6235	297	5	{	{	PUNCT
cana-6235	297	6	u	u	NOUN
cana-6235	297	7	,	,	PUNCT
cana-6235	297	8	v	v	NOUN
cana-6235	297	9	}	}	PUNCT
cana-6235	297	10	▐	▐	NOUN
cana-6235	297	11	theorem	theorem	VERB
cana-6235	297	12	4.15	4.15	NUM
cana-6235	297	13	:	:	PUNCT
cana-6235	297	14	let	let	VERB
cana-6235	297	15	g	g	PRON
cana-6235	297	16	be	be	AUX
cana-6235	297	17	a	a	DET
cana-6235	297	18	graph	graph	NOUN
cana-6235	297	19	with	with	ADP
cana-6235	297	20	γ0(g	γ0(g	NOUN
cana-6235	297	21	)	)	PUNCT
cana-6235	298	1	=	=	SYM
cana-6235	298	2	2	2	NUM
cana-6235	298	3	and	and	CCONJ
cana-6235	298	4	e	e	NOUN
cana-6235	299	1	=	=	NOUN
cana-6235	299	2	uv	uv	NOUN
cana-6235	299	3	be	be	AUX
cana-6235	299	4	an	an	DET
cana-6235	299	5	edge	edge	NOUN
cana-6235	299	6	of	of	ADP
cana-6235	299	7	g.	g.	NOUN
cana-6235	299	8	then	then	ADV
cana-6235	299	9	γ0(g	γ0(g	VERB
cana-6235	299	10	−	−	NOUN
cana-6235	299	11	e	e	NOUN
cana-6235	299	12	)	)	PUNCT
cana-6235	299	13	>	>	X
cana-6235	300	1	γ0(g	γ0(g	X
cana-6235	300	2	)	)	PUNCT
cana-6235	300	3	if	if	SCONJ
cana-6235	300	4	and	and	CCONJ
cana-6235	300	5	only	only	ADV
cana-6235	300	6	if	if	SCONJ
cana-6235	300	7	for	for	ADP
cana-6235	300	8	every	every	DET
cana-6235	300	9	minimum	minimum	ADJ
cana-6235	300	10	dominating	dominating	NOUN
cana-6235	300	11	set	set	NOUN
cana-6235	300	12	s	s	PROPN
cana-6235	300	13	of	of	ADP
cana-6235	300	14	g.	g.	PROPN
cana-6235	300	15	the	the	DET
cana-6235	300	16	following	follow	VERB
cana-6235	300	17	conditions	condition	NOUN
cana-6235	300	18	holds	hold	VERB
cana-6235	300	19	:	:	PUNCT
cana-6235	300	20	communications	communication	NOUN
cana-6235	300	21	on	on	ADP
cana-6235	300	22	applied	apply	VERB
cana-6235	300	23	nonlinear	nonlinear	ADJ
cana-6235	300	24	analysis	analysis	NOUN
cana-6235	300	25	issn	issn	NOUN
cana-6235	300	26	:	:	PUNCT
cana-6235	300	27	1074	1074	NUM
cana-6235	300	28	-	-	PUNCT
cana-6235	300	29	133x	133x	NUM
cana-6235	300	30	vol	vol	NOUN
cana-6235	300	31	31	31	NUM
cana-6235	300	32	no	no	NOUN
cana-6235	300	33	.	.	NOUN
cana-6235	300	34	2	2	NUM
cana-6235	300	35	(	(	PUNCT
cana-6235	300	36	2024	2024	NUM
cana-6235	300	37	)	)	PUNCT
cana-6235	300	38	523	523	NUM
cana-6235	300	39	https://internationalpubls.com	https://internationalpubls.com	X
cana-6235	300	40	(	(	PUNCT
cana-6235	300	41	1	1	X
cana-6235	300	42	)	)	PUNCT
cana-6235	300	43	if	if	SCONJ
cana-6235	300	44	s	s	NOUN
cana-6235	300	45	is	be	AUX
cana-6235	300	46	an	an	DET
cana-6235	300	47	independent	independent	ADJ
cana-6235	300	48	dominating	dominating	NOUN
cana-6235	300	49	set	set	NOUN
cana-6235	300	50	then	then	ADV
cana-6235	300	51	u	u	PROPN
cana-6235	300	52	∈	∈	PROPN
cana-6235	300	53	s	s	PROPN
cana-6235	300	54	&	&	CCONJ
cana-6235	300	55	v	v	PROPN
cana-6235	300	56	∉	∉	PROPN
cana-6235	300	57	s	s	PROPN
cana-6235	300	58	&	&	CCONJ
cana-6235	300	59	v	v	ADP
cana-6235	300	60	∈	∈	PROPN
cana-6235	300	61	pextn[u	pextn[u	PROPN
cana-6235	300	62	,	,	PUNCT
cana-6235	300	63	s	s	X
cana-6235	300	64	]	]	PUNCT
cana-6235	300	65	or	or	CCONJ
cana-6235	300	66	v	v	ADP
cana-6235	300	67	∈	∈	PROPN
cana-6235	300	68	s	s	PROPN
cana-6235	300	69	&	&	CCONJ
cana-6235	300	70	u	u	PROPN
cana-6235	300	71	∉	∉	PROPN
cana-6235	300	72	s	s	PROPN
cana-6235	300	73	&	&	CCONJ
cana-6235	300	74	u	u	PROPN
cana-6235	300	75	∈	∈	PROPN
cana-6235	300	76	pextn[v	pextn[v	PROPN
cana-6235	300	77	,	,	PUNCT
cana-6235	300	78	s	s	PART
cana-6235	300	79	]	]	X
cana-6235	300	80	.	.	PUNCT
cana-6235	301	1	(	(	PUNCT
cana-6235	301	2	2	2	X
cana-6235	301	3	)	)	PUNCT
cana-6235	301	4	if	if	SCONJ
cana-6235	301	5	s	s	VERB
cana-6235	301	6	is	be	AUX
cana-6235	301	7	a	a	DET
cana-6235	301	8	total	total	ADJ
cana-6235	301	9	dominating	dominating	NOUN
cana-6235	301	10	set	set	NOUN
cana-6235	301	11	of	of	ADP
cana-6235	301	12	g	g	PROPN
cana-6235	301	13	then	then	ADV
cana-6235	301	14	u	u	PROPN
cana-6235	301	15	∉	∉	PROPN
cana-6235	301	16	s	s	PART
cana-6235	301	17	or	or	CCONJ
cana-6235	301	18	v	v	ADP
cana-6235	301	19	∉	∉	PROPN
cana-6235	301	20	s	s	PART
cana-6235	301	21	.	.	PUNCT
cana-6235	302	1	proof	proof	NOUN
cana-6235	302	2	:	:	PUNCT
cana-6235	302	3	suppose	suppose	VERB
cana-6235	302	4	γ0(g	γ0(g	PRON
cana-6235	302	5	−	−	NOUN
cana-6235	302	6	e	e	NOUN
cana-6235	302	7	)	)	PUNCT
cana-6235	302	8	>	>	X
cana-6235	303	1	γ0(g	γ0(g	PROPN
cana-6235	303	2	)	)	PUNCT
cana-6235	303	3	.	.	PUNCT
cana-6235	304	1	suppose	suppose	VERB
cana-6235	304	2	s	s	PRON
cana-6235	304	3	is	be	AUX
cana-6235	304	4	a	a	DET
cana-6235	304	5	minimum	minimum	ADJ
cana-6235	304	6	dominating	dominating	NOUN
cana-6235	304	7	set	set	NOUN
cana-6235	304	8	.	.	PUNCT
cana-6235	305	1	(	(	PUNCT
cana-6235	305	2	1	1	X
cana-6235	305	3	)	)	PUNCT
cana-6235	305	4	suppose	suppose	VERB
cana-6235	305	5	s	s	NOUN
cana-6235	305	6	is	be	AUX
cana-6235	305	7	an	an	DET
cana-6235	305	8	independent	independent	ADJ
cana-6235	305	9	set	set	NOUN
cana-6235	305	10	.	.	PUNCT
cana-6235	306	1	if	if	SCONJ
cana-6235	306	2	u	u	PROPN
cana-6235	306	3	∈	∈	PROPN
cana-6235	306	4	s	s	PROPN
cana-6235	306	5	&	&	CCONJ
cana-6235	306	6	v	v	ADP
cana-6235	306	7	∈	∈	PROPN
cana-6235	306	8	s	s	VERB
cana-6235	306	9	then	then	ADV
cana-6235	306	10	we	we	PRON
cana-6235	306	11	have	have	VERB
cana-6235	306	12	an	an	DET
cana-6235	306	13	obvious	obvious	ADJ
cana-6235	306	14	contradiction	contradiction	NOUN
cana-6235	306	15	because	because	SCONJ
cana-6235	306	16	s	s	VERB
cana-6235	306	17	is	be	AUX
cana-6235	306	18	an	an	DET
cana-6235	306	19	independent	independent	ADJ
cana-6235	306	20	set	set	NOUN
cana-6235	306	21	.	.	PUNCT
cana-6235	307	1	if	if	SCONJ
cana-6235	307	2	u	u	PROPN
cana-6235	307	3	∉	∉	PROPN
cana-6235	307	4	s	s	PROPN
cana-6235	307	5	&	&	CCONJ
cana-6235	307	6	v	v	PROPN
cana-6235	307	7	∉	∉	PROPN
cana-6235	307	8	s	s	PART
cana-6235	307	9	then	then	ADV
cana-6235	307	10	s	s	VERB
cana-6235	307	11	is	be	AUX
cana-6235	307	12	an	an	DET
cana-6235	307	13	isolate	isolate	ADJ
cana-6235	307	14	dominating	dominating	NOUN
cana-6235	307	15	set	set	VERB
cana-6235	307	16	in	in	ADP
cana-6235	307	17	g	g	PROPN
cana-6235	307	18	−	−	PROPN
cana-6235	308	1	e.	e.	PROPN
cana-6235	309	1	then	then	ADV
cana-6235	309	2	γ0(g	γ0(g	VERB
cana-6235	309	3	−	−	NOUN
cana-6235	309	4	e	e	NOUN
cana-6235	309	5	)	)	PUNCT
cana-6235	309	6	≤	≤	NUM
cana-6235	309	7	|s|	|s|	PROPN
cana-6235	309	8	=	=	SYM
cana-6235	309	9	γ0(g	γ0(g	PROPN
cana-6235	309	10	)	)	PUNCT
cana-6235	309	11	.	.	PUNCT
cana-6235	310	1	which	which	PRON
cana-6235	310	2	contradicts	contradict	VERB
cana-6235	310	3	the	the	DET
cana-6235	310	4	hypothesis	hypothesis	NOUN
cana-6235	310	5	.	.	PUNCT
cana-6235	311	1	therefore	therefore	ADV
cana-6235	311	2	,	,	PUNCT
cana-6235	311	3	u	u	PROPN
cana-6235	311	4	∈	∈	PROPN
cana-6235	311	5	s	s	PROPN
cana-6235	311	6	&	&	CCONJ
cana-6235	311	7	v	v	PROPN
cana-6235	311	8	∉	∉	PROPN
cana-6235	311	9	s	s	PART
cana-6235	311	10	or	or	CCONJ
cana-6235	311	11	∈	∈	PROPN
cana-6235	311	12	s	s	PROPN
cana-6235	311	13	&	&	CCONJ
cana-6235	311	14	u	u	PROPN
cana-6235	311	15	∉	∉	PROPN
cana-6235	311	16	s	s	PART
cana-6235	311	17	.	.	PUNCT
cana-6235	311	18	suppose	suppose	VERB
cana-6235	311	19	∈	∈	PROPN
cana-6235	311	20	s	s	PROPN
cana-6235	311	21	&	&	CCONJ
cana-6235	311	22	v	v	PROPN
cana-6235	311	23	∉	∉	PROPN
cana-6235	311	24	s	s	PART
cana-6235	311	25	.	.	PUNCT
cana-6235	312	1	now	now	ADV
cana-6235	312	2	suppose	suppose	VERB
cana-6235	312	3	v	v	ADP
cana-6235	312	4	∉	∉	PROPN
cana-6235	312	5	pextn[u	pextn[u	PROPN
cana-6235	312	6	,	,	PUNCT
cana-6235	312	7	s	s	X
cana-6235	312	8	]	]	X
cana-6235	312	9	then	then	ADV
cana-6235	312	10	s	s	VERB
cana-6235	312	11	is	be	AUX
cana-6235	312	12	an	an	DET
cana-6235	312	13	isolate	isolate	ADJ
cana-6235	312	14	dominating	dominating	NOUN
cana-6235	312	15	set	set	VERB
cana-6235	312	16	in	in	ADP
cana-6235	312	17	g	g	PROPN
cana-6235	312	18	−	−	PROPN
cana-6235	312	19	e	e	NOUN
cana-6235	312	20	,	,	PUNCT
cana-6235	312	21	which	which	PRON
cana-6235	312	22	implies	imply	VERB
cana-6235	312	23	that	that	SCONJ
cana-6235	312	24	γ0(g	γ0(g	NOUN
cana-6235	312	25	−	−	NOUN
cana-6235	312	26	e	e	NOUN
cana-6235	312	27	)	)	PUNCT
cana-6235	312	28	≤	≤	NOUN
cana-6235	312	29	γ0(g	γ0(g	NOUN
cana-6235	312	30	)	)	PUNCT
cana-6235	312	31	.	.	PUNCT
cana-6235	313	1	which	which	PRON
cana-6235	313	2	is	be	AUX
cana-6235	313	3	a	a	DET
cana-6235	313	4	contradiction	contradiction	NOUN
cana-6235	313	5	.	.	PUNCT
cana-6235	314	1	therefore	therefore	ADV
cana-6235	314	2	,	,	PUNCT
cana-6235	314	3	v	v	PROPN
cana-6235	314	4	∈	∈	PROPN
cana-6235	314	5	pextn[u	pextn[u	PROPN
cana-6235	314	6	,	,	PUNCT
cana-6235	314	7	s	s	PART
cana-6235	314	8	]	]	PUNCT
cana-6235	314	9	.	.	PUNCT
cana-6235	315	1	similarly	similarly	ADV
cana-6235	315	2	,	,	PUNCT
cana-6235	315	3	if	if	SCONJ
cana-6235	315	4	v	v	ADP
cana-6235	315	5	∈	∈	PROPN
cana-6235	315	6	s	s	PROPN
cana-6235	315	7	&	&	CCONJ
cana-6235	315	8	u	u	PROPN
cana-6235	315	9	∉	∉	PROPN
cana-6235	315	10	s	s	PART
cana-6235	315	11	then	then	ADV
cana-6235	315	12	u	u	NOUN
cana-6235	315	13	∈	∈	PROPN
cana-6235	315	14	pextn[v	pextn[v	PROPN
cana-6235	315	15	,	,	PUNCT
cana-6235	315	16	s	s	PART
cana-6235	315	17	]	]	X
cana-6235	315	18	.	.	PUNCT
cana-6235	316	1	(	(	PUNCT
cana-6235	316	2	2	2	X
cana-6235	316	3	)	)	PUNCT
cana-6235	316	4	suppose	suppose	VERB
cana-6235	316	5	s	s	NOUN
cana-6235	316	6	is	be	AUX
cana-6235	316	7	a	a	DET
cana-6235	316	8	total	total	ADJ
cana-6235	316	9	dominating	dominating	NOUN
cana-6235	316	10	set	set	NOUN
cana-6235	316	11	.	.	PUNCT
cana-6235	317	1	if	if	SCONJ
cana-6235	317	2	u	u	PROPN
cana-6235	317	3	∈	∈	PROPN
cana-6235	317	4	s	s	PROPN
cana-6235	317	5	&	&	CCONJ
cana-6235	317	6	v	v	ADP
cana-6235	317	7	∈	∈	PROPN
cana-6235	317	8	s	s	AUX
cana-6235	317	9	then	then	ADV
cana-6235	317	10	s	s	VERB
cana-6235	317	11	is	be	AUX
cana-6235	317	12	an	an	DET
cana-6235	317	13	isolate	isolate	ADJ
cana-6235	317	14	dominating	dominating	NOUN
cana-6235	317	15	set	set	VERB
cana-6235	317	16	in	in	ADP
cana-6235	317	17	−e	−e	NOUN
cana-6235	317	18	.	.	PUNCT
cana-6235	318	1	therefore	therefore	ADV
cana-6235	318	2	,	,	PUNCT
cana-6235	318	3	γ0(g	γ0(g	NOUN
cana-6235	318	4	−	−	NOUN
cana-6235	318	5	e	e	NOUN
cana-6235	318	6	)	)	PUNCT
cana-6235	318	7	≤	≤	NUM
cana-6235	318	8	|s|	|s|	NOUN
cana-6235	318	9	≤	≤	NUM
cana-6235	318	10	γ0(g	γ0(g	NOUN
cana-6235	318	11	)	)	PUNCT
cana-6235	318	12	.	.	PUNCT
cana-6235	319	1	which	which	PRON
cana-6235	319	2	is	be	AUX
cana-6235	319	3	a	a	DET
cana-6235	319	4	contradiction	contradiction	NOUN
cana-6235	319	5	.	.	PUNCT
cana-6235	320	1	therefore	therefore	ADV
cana-6235	320	2	,	,	PUNCT
cana-6235	320	3	u	u	PROPN
cana-6235	320	4	∉	∉	PROPN
cana-6235	320	5	s	s	PART
cana-6235	320	6	or	or	CCONJ
cana-6235	320	7	v	v	ADP
cana-6235	320	8	∉	∉	PROPN
cana-6235	320	9	s.	s.	PROPN
cana-6235	320	10	conversely	conversely	ADV
cana-6235	320	11	,	,	PUNCT
cana-6235	320	12	suppose	suppose	VERB
cana-6235	320	13	γ0(g	γ0(g	PRON
cana-6235	320	14	−	−	NOUN
cana-6235	320	15	e	e	NOUN
cana-6235	320	16	)	)	PUNCT
cana-6235	320	17	>	>	X
cana-6235	321	1	γ0(g	γ0(g	PROPN
cana-6235	321	2	)	)	PUNCT
cana-6235	321	3	.	.	PUNCT
cana-6235	322	1	let	let	VERB
cana-6235	322	2	t	t	PROPN
cana-6235	322	3	⊂	⊂	PRON
cana-6235	322	4	v(g	v(g	PROPN
cana-6235	322	5	)	)	PUNCT
cana-6235	322	6	be	be	AUX
cana-6235	322	7	such	such	ADJ
cana-6235	322	8	that	that	SCONJ
cana-6235	322	9	t	t	PROPN
cana-6235	322	10	≠	≠	PROPN
cana-6235	322	11	∅	∅	NOUN
cana-6235	322	12	&	&	CCONJ
cana-6235	322	13	|t|	|t|	PROPN
cana-6235	322	14	≤	≤	ADJ
cana-6235	323	1	γ0(g	γ0(g	NOUN
cana-6235	323	2	)	)	PUNCT
cana-6235	323	3	.	.	PUNCT
cana-6235	324	1	suppose	suppose	VERB
cana-6235	324	2	|t|	|t|	PROPN
cana-6235	324	3	=	=	NOUN
cana-6235	324	4	1	1	X
cana-6235	324	5	.	.	PUNCT
cana-6235	325	1	let	let	VERB
cana-6235	325	2	t	t	NOUN
cana-6235	325	3	=	=	PUNCT
cana-6235	325	4	{	{	PUNCT
cana-6235	325	5	z	z	NOUN
cana-6235	325	6	}	}	PUNCT
cana-6235	325	7	.	.	PUNCT
cana-6235	326	1	suppose	suppose	VERB
cana-6235	326	2	t	t	PROPN
cana-6235	326	3	is	be	AUX
cana-6235	326	4	an	an	DET
cana-6235	326	5	isolate	isolate	ADJ
cana-6235	326	6	dominating	dominating	NOUN
cana-6235	326	7	set	set	VERB
cana-6235	326	8	in	in	ADP
cana-6235	326	9	g	g	PROPN
cana-6235	326	10	−	−	PROPN
cana-6235	326	11	e	e	NOUN
cana-6235	326	12	then	then	ADV
cana-6235	326	13	z	z	PROPN
cana-6235	326	14	≠	≠	PROPN
cana-6235	326	15	u	u	PROPN
cana-6235	326	16	&	&	CCONJ
cana-6235	326	17	z	z	PROPN
cana-6235	326	18	≠	≠	PROPN
cana-6235	327	1	v.	v.	CCONJ
cana-6235	327	2	then	then	ADV
cana-6235	327	3	{	{	PUNCT
cana-6235	327	4	z	z	NOUN
cana-6235	327	5	}	}	PUNCT
cana-6235	327	6	is	be	AUX
cana-6235	327	7	an	an	DET
cana-6235	327	8	isolate	isolate	NOUN
cana-6235	327	9	dominating	dominating	NOUN
cana-6235	327	10	set	set	VERB
cana-6235	327	11	in	in	ADP
cana-6235	327	12	g	g	NOUN
cana-6235	327	13	which	which	PRON
cana-6235	327	14	implies	imply	VERB
cana-6235	327	15	that	that	SCONJ
cana-6235	328	1	γ0(g	γ0(g	NOUN
cana-6235	328	2	)	)	PUNCT
cana-6235	328	3	=	=	SYM
cana-6235	328	4	1	1	NUM
cana-6235	328	5	.	.	PUNCT
cana-6235	328	6	which	which	PRON
cana-6235	328	7	is	be	AUX
cana-6235	328	8	not	not	PART
cana-6235	328	9	true	true	ADJ
cana-6235	328	10	.	.	PUNCT
cana-6235	329	1	therefore	therefore	ADV
cana-6235	329	2	,	,	PUNCT
cana-6235	329	3	any	any	DET
cana-6235	329	4	set	set	NOUN
cana-6235	329	5	t	t	NOUN
cana-6235	329	6	with	with	ADP
cana-6235	329	7	|t|	|t|	NOUN
cana-6235	329	8	=	=	SYM
cana-6235	329	9	1	1	NUM
cana-6235	329	10	can	can	AUX
cana-6235	329	11	not	not	PART
cana-6235	329	12	be	be	AUX
cana-6235	329	13	an	an	DET
cana-6235	329	14	isolate	isolate	NOUN
cana-6235	329	15	dominating	dominating	NOUN
cana-6235	329	16	set	set	NOUN
cana-6235	329	17	of	of	ADP
cana-6235	329	18	g	g	PROPN
cana-6235	329	19	−	−	PROPN
cana-6235	329	20	e	e	PROPN
cana-6235	329	21	.	.	PUNCT
cana-6235	330	1	suppose	suppose	VERB
cana-6235	330	2	t	t	PROPN
cana-6235	330	3	⊂	⊂	PROPN
cana-6235	330	4	v(g	v(g	PROPN
cana-6235	330	5	)	)	PUNCT
cana-6235	330	6	be	be	AUX
cana-6235	330	7	such	such	ADJ
cana-6235	330	8	that	that	DET
cana-6235	330	9	|t|	|t|	NOUN
cana-6235	330	10	=	=	SYM
cana-6235	330	11	2	2	NUM
cana-6235	330	12	&	&	CCONJ
cana-6235	330	13	t	t	PROPN
cana-6235	330	14	is	be	AUX
cana-6235	330	15	an	an	DET
cana-6235	330	16	isolate	isolate	NOUN
cana-6235	330	17	dominating	dominating	NOUN
cana-6235	330	18	set	set	NOUN
cana-6235	330	19	of	of	ADP
cana-6235	330	20	g	g	PROPN
cana-6235	330	21	−	−	PROPN
cana-6235	330	22	e	e	NOUN
cana-6235	330	23	.	.	PUNCT
cana-6235	331	1	let	let	VERB
cana-6235	331	2	u	u	PRON
cana-6235	331	3	∈	∈	PROPN
cana-6235	331	4	t	t	PROPN
cana-6235	331	5	&	&	CCONJ
cana-6235	331	6	v	v	NOUN
cana-6235	331	7	∈	∈	PROPN
cana-6235	331	8	t.	t.	NOUN
cana-6235	331	9	then	then	ADV
cana-6235	331	10	t	t	PROPN
cana-6235	331	11	is	be	AUX
cana-6235	331	12	a	a	DET
cana-6235	331	13	minimum	minimum	ADJ
cana-6235	331	14	dominating	dominating	NOUN
cana-6235	331	15	set	set	NOUN
cana-6235	331	16	of	of	ADP
cana-6235	331	17	g.	g.	PROPN
cana-6235	331	18	which	which	PRON
cana-6235	331	19	is	be	AUX
cana-6235	331	20	a	a	DET
cana-6235	331	21	total	total	ADJ
cana-6235	331	22	dominating	dominating	NOUN
cana-6235	331	23	set	set	NOUN
cana-6235	331	24	and	and	CCONJ
cana-6235	331	25	u	u	NOUN
cana-6235	331	26	,	,	PUNCT
cana-6235	331	27	v	v	NOUN
cana-6235	331	28	∈	∈	NOUN
cana-6235	331	29	t.	t.	NOUN
cana-6235	331	30	this	this	PRON
cana-6235	331	31	contradicts	contradict	VERB
cana-6235	331	32	(	(	PUNCT
cana-6235	331	33	2	2	NUM
cana-6235	331	34	)	)	PUNCT
cana-6235	331	35	.	.	PUNCT
cana-6235	332	1	suppose	suppose	VERB
cana-6235	332	2	u	u	PROPN
cana-6235	332	3	∈	∈	PROPN
cana-6235	332	4	t	t	PROPN
cana-6235	332	5	&	&	CCONJ
cana-6235	332	6	v	v	X
cana-6235	332	7	∉	∉	PROPN
cana-6235	332	8	t.	t.	NOUN
cana-6235	332	9	communications	communication	NOUN
cana-6235	332	10	on	on	ADP
cana-6235	332	11	applied	apply	VERB
cana-6235	332	12	nonlinear	nonlinear	ADJ
cana-6235	332	13	analysis	analysis	NOUN
cana-6235	332	14	issn	issn	NOUN
cana-6235	332	15	:	:	PUNCT
cana-6235	332	16	1074	1074	NUM
cana-6235	332	17	-	-	PUNCT
cana-6235	332	18	133x	133x	NUM
cana-6235	332	19	vol	vol	NOUN
cana-6235	332	20	31	31	NUM
cana-6235	332	21	no	no	NOUN
cana-6235	332	22	.	.	NOUN
cana-6235	332	23	2	2	NUM
cana-6235	332	24	(	(	PUNCT
cana-6235	332	25	2024	2024	NUM
cana-6235	332	26	)	)	PUNCT
cana-6235	332	27	524	524	NUM
cana-6235	332	28	https://internationalpubls.com	https://internationalpubls.com	X
cana-6235	333	1	now	now	ADV
cana-6235	333	2	v	v	NOUN
cana-6235	333	3	is	be	AUX
cana-6235	333	4	an	an	DET
cana-6235	333	5	adjacent	adjacent	ADJ
cana-6235	333	6	to	to	ADP
cana-6235	333	7	some	some	DET
cana-6235	333	8	vertex	vertex	NOUN
cana-6235	333	9	x	x	PUNCT
cana-6235	333	10	of	of	ADP
cana-6235	333	11	t	t	PROPN
cana-6235	333	12	and	and	CCONJ
cana-6235	333	13	of	of	ADP
cana-6235	333	14	course	course	NOUN
cana-6235	333	15	x	x	SYM
cana-6235	333	16	≠	≠	PROPN
cana-6235	333	17	u	u	NOUN
cana-6235	333	18	because	because	SCONJ
cana-6235	333	19	u	u	PROPN
cana-6235	333	20	and	and	CCONJ
cana-6235	333	21	v	v	NOUN
cana-6235	333	22	are	be	AUX
cana-6235	333	23	not	not	PART
cana-6235	333	24	adjacent	adjacent	ADJ
cana-6235	333	25	in	in	ADP
cana-6235	333	26	g	g	PROPN
cana-6235	333	27	−	−	PROPN
cana-6235	334	1	e.	e.	PROPN
cana-6235	334	2	now	now	ADV
cana-6235	334	3	v	v	NOUN
cana-6235	334	4	is	be	AUX
cana-6235	334	5	an	an	DET
cana-6235	334	6	adjacent	adjacent	ADJ
cana-6235	334	7	two	two	NUM
cana-6235	334	8	distinct	distinct	ADJ
cana-6235	334	9	vertex	vertex	NOUN
cana-6235	334	10	u	u	NOUN
cana-6235	334	11	&	&	CCONJ
cana-6235	334	12	x	x	PROPN
cana-6235	334	13	of	of	ADP
cana-6235	334	14	t	t	PROPN
cana-6235	334	15	in	in	ADP
cana-6235	334	16	the	the	DET
cana-6235	334	17	graph	graph	NOUN
cana-6235	334	18	g.	g.	PROPN
cana-6235	334	19	therefore	therefore	ADV
cana-6235	334	20	,	,	PUNCT
cana-6235	334	21	v	v	PROPN
cana-6235	334	22	∉	∉	PROPN
cana-6235	334	23	pextn[u	pextn[u	PROPN
cana-6235	334	24	,	,	PUNCT
cana-6235	334	25	s	s	X
cana-6235	334	26	]	]	X
cana-6235	334	27	in	in	ADP
cana-6235	334	28	g.	g.	PROPN
cana-6235	334	29	similarly	similarly	ADV
cana-6235	334	30	,	,	PUNCT
cana-6235	334	31	if	if	SCONJ
cana-6235	334	32	v	v	ADP
cana-6235	334	33	∈	∈	PROPN
cana-6235	334	34	t	t	PROPN
cana-6235	334	35	&	&	CCONJ
cana-6235	334	36	u	u	PROPN
cana-6235	334	37	∉	∉	PROPN
cana-6235	334	38	t	t	PROPN
cana-6235	334	39	then	then	ADV
cana-6235	334	40	u	u	PROPN
cana-6235	334	41	∉	∉	PROPN
cana-6235	334	42	pextn[v	pextn[v	PROPN
cana-6235	334	43	,	,	PUNCT
cana-6235	334	44	t	t	PROPN
cana-6235	334	45	]	]	PUNCT
cana-6235	334	46	in	in	ADP
cana-6235	334	47	g.	g.	PROPN
cana-6235	334	48	thus	thus	ADV
cana-6235	334	49	,	,	PUNCT
cana-6235	334	50	if	if	SCONJ
cana-6235	334	51	u	u	PROPN
cana-6235	334	52	∈	∈	PROPN
cana-6235	334	53	t	t	PROPN
cana-6235	334	54	&	&	CCONJ
cana-6235	334	55	v	v	PROPN
cana-6235	334	56	∉	∉	PROPN
cana-6235	334	57	t	t	PROPN
cana-6235	334	58	or	or	CCONJ
cana-6235	334	59	v	v	ADP
cana-6235	334	60	∈	∈	PROPN
cana-6235	334	61	t	t	PROPN
cana-6235	334	62	&	&	CCONJ
cana-6235	334	63	u	u	PROPN
cana-6235	334	64	∉	∉	PROPN
cana-6235	334	65	t	t	PROPN
cana-6235	334	66	gives	give	VERB
cana-6235	334	67	rise	rise	NOUN
cana-6235	334	68	to	to	ADP
cana-6235	334	69	a	a	DET
cana-6235	334	70	contradiction	contradiction	NOUN
cana-6235	334	71	.	.	PUNCT
cana-6235	335	1	therefore	therefore	ADV
cana-6235	335	2	,	,	PUNCT
cana-6235	335	3	∉	∉	PROPN
cana-6235	335	4	t	t	PROPN
cana-6235	335	5	&	&	CCONJ
cana-6235	335	6	v	v	PROPN
cana-6235	335	7	∉	∉	PROPN
cana-6235	335	8	t	t	PROPN
cana-6235	335	9	.	.	PUNCT
cana-6235	336	1	therefore	therefore	ADV
cana-6235	336	2	,	,	PUNCT
cana-6235	336	3	t	t	PROPN
cana-6235	336	4	is	be	AUX
cana-6235	336	5	a	a	DET
cana-6235	336	6	minimum	minimum	NOUN
cana-6235	336	7	isolate	isolate	NOUN
cana-6235	336	8	dominating	dominating	NOUN
cana-6235	336	9	set	set	NOUN
cana-6235	336	10	of	of	ADP
cana-6235	336	11	g	g	PROPN
cana-6235	336	12	such	such	ADJ
cana-6235	336	13	that	that	SCONJ
cana-6235	336	14	u	u	PROPN
cana-6235	336	15	∉	∉	PROPN
cana-6235	336	16	t	t	PROPN
cana-6235	336	17	&	&	CCONJ
cana-6235	336	18	v	v	X
cana-6235	336	19	∉	∉	PROPN
cana-6235	336	20	t.	t.	NOUN
cana-6235	336	21	which	which	PRON
cana-6235	336	22	is	be	AUX
cana-6235	336	23	contradict	contradict	VERB
cana-6235	336	24	(	(	PUNCT
cana-6235	336	25	1	1	NUM
cana-6235	336	26	)	)	PUNCT
cana-6235	336	27	.	.	PUNCT
cana-6235	337	1	therefore	therefore	ADV
cana-6235	337	2	,	,	PUNCT
cana-6235	337	3	any	any	DET
cana-6235	337	4	set	set	NOUN
cana-6235	337	5	t	t	NOUN
cana-6235	337	6	with	with	ADP
cana-6235	337	7	|t|	|t|	NOUN
cana-6235	337	8	=	=	SYM
cana-6235	337	9	2	2	NUM
cana-6235	337	10	can	can	AUX
cana-6235	337	11	not	not	PART
cana-6235	337	12	be	be	AUX
cana-6235	337	13	an	an	DET
cana-6235	337	14	isolate	isolate	NOUN
cana-6235	337	15	dominating	dominating	NOUN
cana-6235	337	16	set	set	NOUN
cana-6235	337	17	of	of	ADP
cana-6235	337	18	g	g	PROPN
cana-6235	337	19	−	−	PROPN
cana-6235	337	20	e.	e.	PROPN
cana-6235	337	21	thus	thus	ADV
cana-6235	337	22	,	,	PUNCT
cana-6235	337	23	we	we	PRON
cana-6235	337	24	conclude	conclude	VERB
cana-6235	337	25	that	that	SCONJ
cana-6235	337	26	any	any	DET
cana-6235	337	27	set	set	NOUN
cana-6235	337	28	t	t	NOUN
cana-6235	337	29	of	of	ADP
cana-6235	337	30	vertices	vertex	NOUN
cana-6235	337	31	of	of	ADP
cana-6235	337	32	g	g	NOUN
cana-6235	337	33	−	−	PROPN
cana-6235	337	34	e	e	NOUN
cana-6235	337	35	with	with	ADP
cana-6235	337	36	|t|	|t|	ADJ
cana-6235	337	37	≤	≤	ADJ
cana-6235	337	38	2	2	NUM
cana-6235	337	39	can	can	AUX
cana-6235	337	40	not	not	PART
cana-6235	337	41	be	be	AUX
cana-6235	337	42	an	an	DET
cana-6235	337	43	isolate	isolate	NOUN
cana-6235	337	44	dominating	dominating	NOUN
cana-6235	337	45	set	set	NOUN
cana-6235	337	46	of	of	ADP
cana-6235	337	47	g	g	PROPN
cana-6235	337	48	−	−	PROPN
cana-6235	338	1	e.	e.	PROPN
cana-6235	339	1	therefore	therefore	ADV
cana-6235	339	2	γ0(g	γ0(g	X
cana-6235	339	3	−	−	PROPN
cana-6235	339	4	e	e	NOUN
cana-6235	339	5	)	)	PUNCT
cana-6235	339	6	>	>	X
cana-6235	339	7	2	2	X
cana-6235	339	8	=	=	SYM
cana-6235	339	9	γ0(g	γ0(g	NOUN
cana-6235	339	10	)	)	PUNCT
cana-6235	339	11	.	.	PUNCT
cana-6235	340	1	▐	▐	NOUN
cana-6235	340	2	conclusion	conclusion	NOUN
cana-6235	340	3	:	:	PUNCT
cana-6235	340	4	if	if	SCONJ
cana-6235	340	5	g	g	PROPN
cana-6235	340	6	is	be	AUX
cana-6235	340	7	a	a	DET
cana-6235	340	8	graph	graph	NOUN
cana-6235	340	9	,	,	PUNCT
cana-6235	340	10	v	v	NOUN
cana-6235	340	11	∈	∈	PROPN
cana-6235	340	12	v(g	v(g	NOUN
cana-6235	340	13	)	)	PUNCT
cana-6235	340	14	and	and	CCONJ
cana-6235	340	15	suppose	suppose	VERB
cana-6235	340	16	g	g	PROPN
cana-6235	340	17	has	have	VERB
cana-6235	340	18	a	a	DET
cana-6235	340	19	minimum	minimum	NOUN
cana-6235	340	20	isolate	isolate	NOUN
cana-6235	340	21	dominating	dominate	VERB
cana-6235	340	22	set	set	NOUN
cana-6235	340	23	s	s	PRON
cana-6235	340	24	which	which	PRON
cana-6235	340	25	has	have	VERB
cana-6235	340	26	an	an	DET
cana-6235	340	27	isolate	isolate	NOUN
cana-6235	340	28	different	different	ADJ
cana-6235	340	29	from	from	ADP
cana-6235	340	30	v	v	PRON
cana-6235	340	31	then	then	ADV
cana-6235	340	32	v	v	NOUN
cana-6235	340	33	gives	give	VERB
cana-6235	340	34	rises	rise	NOUN
cana-6235	340	35	to	to	ADP
cana-6235	340	36	two	two	NUM
cana-6235	340	37	distinct	distinct	ADJ
cana-6235	340	38	vertices	vertex	NOUN
cana-6235	340	39	v1	v1	PROPN
cana-6235	340	40	&	&	CCONJ
cana-6235	340	41	v2	v2	PROPN
cana-6235	340	42	∋	∋	NOUN
cana-6235	341	1	γ0(g	γ0(g	NOUN
cana-6235	341	2	−	−	PROPN
cana-6235	341	3	vi	vi	NOUN
cana-6235	341	4	)	)	PUNCT
cana-6235	341	5	=	=	SYM
cana-6235	342	1	γ0(g	γ0(g	NOUN
cana-6235	342	2	)	)	PUNCT
cana-6235	342	3	or	or	CCONJ
cana-6235	342	4	i	i	NOUN
cana-6235	342	5	=	=	NOUN
cana-6235	342	6	1,2	1,2	NUM
cana-6235	342	7	.	.	PUNCT
cana-6235	343	1	thus	thus	ADV
cana-6235	343	2	,	,	PUNCT
cana-6235	343	3	it	it	PRON
cana-6235	343	4	follows	follow	VERB
cana-6235	343	5	that	that	SCONJ
cana-6235	343	6	for	for	ADP
cana-6235	343	7	any	any	DET
cana-6235	343	8	graph	graph	NOUN
cana-6235	343	9	g	g	NOUN
cana-6235	343	10	which	which	PRON
cana-6235	343	11	contains	contain	VERB
cana-6235	343	12	a	a	DET
cana-6235	343	13	minimum	minimum	NOUN
cana-6235	343	14	isolate	isolate	NOUN
cana-6235	343	15	dominating	dominating	NOUN
cana-6235	343	16	set	set	NOUN
cana-6235	343	17	of	of	ADP
cana-6235	343	18	s	s	PRON
cana-6235	343	19	with	with	ADP
cana-6235	343	20	isolate	isolate	NOUN
cana-6235	343	21	u	u	NOUN
cana-6235	343	22	and	and	CCONJ
cana-6235	343	23	if	if	SCONJ
cana-6235	343	24	there	there	PRON
cana-6235	343	25	is	be	VERB
cana-6235	343	26	a	a	DET
cana-6235	343	27	vertex	vertex	NOUN
cana-6235	343	28	v	v	ADP
cana-6235	343	29	inv0	inv0	NOUN
cana-6235	343	30	+	+	CCONJ
cana-6235	343	31	then	then	ADV
cana-6235	343	32	v0	v0	PROPN
cana-6235	343	33	0	0	NUM
cana-6235	343	34	≥	≥	NOUN
cana-6235	343	35	2|v0	2|v0	NOUN
cana-6235	344	1	+	+	NOUN
cana-6235	344	2	|	|	ADV
cana-6235	344	3	.	.	PUNCT
cana-6235	345	1	references	reference	NOUN
cana-6235	345	2	1	1	NUM
cana-6235	345	3	.	.	PUNCT
cana-6235	345	4	b.	b.	PROPN
cana-6235	345	5	h.	h.	PROPN
cana-6235	345	6	arriola	arriola	PROPN
cana-6235	345	7	,	,	PUNCT
cana-6235	345	8	isolate	isolate	VERB
cana-6235	345	9	domination	domination	NOUN
cana-6235	345	10	in	in	ADP
cana-6235	345	11	the	the	DET
cana-6235	345	12	join	join	NOUN
cana-6235	345	13	and	and	CCONJ
cana-6235	345	14	corona	corona	NOUN
cana-6235	345	15	of	of	ADP
cana-6235	345	16	graphs	graph	NOUN
cana-6235	345	17	,	,	PUNCT
cana-6235	345	18	appl	appl	PROPN
cana-6235	345	19	.	.	PROPN
cana-6235	345	20	math	math	PROPN
cana-6235	345	21	.	.	PUNCT
cana-6235	346	1	sci	sci	PROPN
cana-6235	346	2	.	.	PROPN
cana-6235	346	3	9	9	NUM
cana-6235	346	4	,	,	PUNCT
cana-6235	346	5	(	(	PUNCT
cana-6235	346	6	2015	2015	NUM
cana-6235	346	7	)	)	PUNCT
cana-6235	346	8	,	,	PUNCT
cana-6235	346	9	1543	1543	NUM
cana-6235	346	10	-	-	SYM
cana-6235	346	11	1549	1549	NUM
cana-6235	346	12	.	.	PUNCT
cana-6235	347	1	2	2	X
cana-6235	347	2	.	.	X
cana-6235	347	3	c.	c.	PROPN
cana-6235	347	4	berge	berge	PROPN
cana-6235	347	5	,	,	PUNCT
cana-6235	347	6	theory	theory	NOUN
cana-6235	347	7	of	of	ADP
cana-6235	347	8	graphs	graph	NOUN
cana-6235	347	9	and	and	CCONJ
cana-6235	347	10	its	its	PRON
cana-6235	347	11	applications	application	NOUN
cana-6235	347	12	,	,	PUNCT
cana-6235	347	13	methuen	methuen	PROPN
cana-6235	347	14	,	,	PUNCT
cana-6235	347	15	london	london	PROPN
cana-6235	347	16	(	(	PUNCT
cana-6235	347	17	1962	1962	NUM
cana-6235	347	18	)	)	PUNCT
cana-6235	347	19	.	.	PUNCT
cana-6235	348	1	3	3	X
cana-6235	348	2	.	.	X
cana-6235	348	3	d.	d.	PROPN
cana-6235	348	4	b.	b.	PROPN
cana-6235	348	5	west	west	PROPN
cana-6235	348	6	,	,	PUNCT
cana-6235	348	7	introduction	introduction	NOUN
cana-6235	348	8	to	to	AUX
cana-6235	348	9	graph	graph	NOUN
cana-6235	348	10	theory	theory	NOUN
cana-6235	348	11	,	,	PUNCT
cana-6235	348	12	prentice	prentice	NOUN
cana-6235	348	13	hall	hall	NOUN
cana-6235	348	14	of	of	ADP
cana-6235	348	15	india	india	PROPN
cana-6235	348	16	,	,	PUNCT
cana-6235	348	17	new	new	ADJ
cana-6235	348	18	delhi	delhi	PROPN
cana-6235	348	19	(	(	PUNCT
cana-6235	348	20	2006	2006	NUM
cana-6235	348	21	)	)	PUNCT
cana-6235	348	22	.	.	PUNCT
cana-6235	349	1	4	4	X
cana-6235	349	2	.	.	X
cana-6235	349	3	d.	d.	PROPN
cana-6235	349	4	k.	k.	PROPN
cana-6235	349	5	thakkar	thakkar	PROPN
cana-6235	349	6	and	and	CCONJ
cana-6235	349	7	j.	j.	PROPN
cana-6235	349	8	c.	c.	PROPN
cana-6235	349	9	bosamiya	bosamiya	PROPN
cana-6235	349	10	,	,	PUNCT
cana-6235	349	11	graph	graph	VERB
cana-6235	349	12	critical	critical	ADJ
cana-6235	349	13	with	with	ADP
cana-6235	349	14	respect	respect	NOUN
cana-6235	349	15	to	to	ADP
cana-6235	349	16	independent	independent	ADJ
cana-6235	349	17	domination	domination	NOUN
cana-6235	349	18	,	,	PUNCT
cana-6235	349	19	journal	journal	NOUN
cana-6235	349	20	of	of	ADP
cana-6235	349	21	discrete	discrete	ADJ
cana-6235	349	22	mathematical	mathematical	ADJ
cana-6235	349	23	sciences	science	NOUN
cana-6235	349	24	and	and	CCONJ
cana-6235	349	25	cryptography	cryptography	NOUN
cana-6235	349	26	,	,	PUNCT
cana-6235	349	27	16	16	NUM
cana-6235	349	28	,	,	PUNCT
cana-6235	349	29	(	(	PUNCT
cana-6235	349	30	2013	2013	NUM
cana-6235	349	31	)	)	PUNCT
cana-6235	349	32	,	,	PUNCT
cana-6235	349	33	179	179	NUM
cana-6235	349	34	-	-	SYM
cana-6235	349	35	186	186	NUM
cana-6235	349	36	.	.	PUNCT
cana-6235	350	1	5	5	NUM
cana-6235	350	2	.	.	PUNCT
cana-6235	350	3	e.	e.	PROPN
cana-6235	350	4	j.	j.	PROPN
cana-6235	350	5	cockayne	cockayne	PROPN
cana-6235	350	6	,	,	PUNCT
cana-6235	350	7	b.	b.	PROPN
cana-6235	350	8	gamble	gamble	PROPN
cana-6235	350	9	and	and	CCONJ
cana-6235	350	10	b.	b.	PROPN
cana-6235	350	11	shepherd	shepherd	NOUN
cana-6235	350	12	,	,	PUNCT
cana-6235	350	13	an	an	DET
cana-6235	350	14	upper	upper	ADJ
cana-6235	350	15	bound	bind	VERB
cana-6235	350	16	for	for	ADP
cana-6235	350	17	the	the	DET
cana-6235	350	18	k	k	ADJ
cana-6235	350	19	-	-	PUNCT
cana-6235	350	20	domination	domination	NOUN
cana-6235	350	21	number	number	NOUN
cana-6235	350	22	of	of	ADP
cana-6235	350	23	a	a	DET
cana-6235	350	24	graph	graph	NOUN
cana-6235	350	25	,	,	PUNCT
cana-6235	350	26	j.	j.	PROPN
cana-6235	350	27	graph	graph	PROPN
cana-6235	350	28	theory	theory	NOUN
cana-6235	350	29	,	,	PUNCT
cana-6235	350	30	9	9	NUM
cana-6235	350	31	,	,	PUNCT
cana-6235	350	32	(	(	PUNCT
cana-6235	350	33	1985	1985	NUM
cana-6235	350	34	)	)	PUNCT
cana-6235	350	35	,	,	PUNCT
cana-6235	350	36	533	533	NUM
cana-6235	350	37	-	-	SYM
cana-6235	350	38	534	534	NUM
cana-6235	350	39	.	.	NOUN
cana-6235	351	1	6	6	NUM
cana-6235	351	2	.	.	PUNCT
cana-6235	351	3	e.	e.	PROPN
cana-6235	351	4	j.	j.	PROPN
cana-6235	351	5	cockayne	cockayne	PROPN
cana-6235	351	6	,	,	PUNCT
cana-6235	351	7	r.	r.	PROPN
cana-6235	351	8	m.	m.	PROPN
cana-6235	351	9	dawes	dawes	PROPN
cana-6235	351	10	and	and	CCONJ
cana-6235	351	11	s.	s.	PROPN
cana-6235	351	12	t.	t.	PROPN
cana-6235	351	13	hedetniemi	hedetniemi	PROPN
cana-6235	351	14	,	,	PUNCT
cana-6235	351	15	total	total	ADJ
cana-6235	351	16	domination	domination	NOUN
cana-6235	351	17	in	in	ADP
cana-6235	351	18	graphs	graph	NOUN
cana-6235	351	19	,	,	PUNCT
cana-6235	351	20	networks	network	NOUN
cana-6235	351	21	,	,	PUNCT
cana-6235	351	22	10	10	NUM
cana-6235	351	23	,	,	PUNCT
cana-6235	351	24	(	(	PUNCT
cana-6235	351	25	1980	1980	NUM
cana-6235	351	26	)	)	PUNCT
cana-6235	351	27	,	,	PUNCT
cana-6235	351	28	211	211	NUM
cana-6235	351	29	-	-	SYM
cana-6235	351	30	219	219	NUM
cana-6235	351	31	.	.	PUNCT
cana-6235	352	1	7	7	X
cana-6235	352	2	.	.	PUNCT
cana-6235	352	3	f.	f.	PROPN
cana-6235	352	4	harary	harary	PROPN
cana-6235	352	5	,	,	PUNCT
cana-6235	352	6	graph	graph	NOUN
cana-6235	352	7	theory	theory	NOUN
cana-6235	352	8	,	,	PUNCT
cana-6235	352	9	addison	addison	PROPN
cana-6235	352	10	wesley	wesley	PROPN
cana-6235	352	11	,	,	PUNCT
cana-6235	352	12	reading	read	VERB
cana-6235	352	13	mass	mass	PROPN
cana-6235	352	14	.	.	PUNCT
cana-6235	352	15	,	,	PUNCT
cana-6235	352	16	(	(	PUNCT
cana-6235	352	17	1969	1969	NUM
cana-6235	352	18	)	)	PUNCT
cana-6235	352	19	.	.	PUNCT
cana-6235	353	1	8	8	X
cana-6235	353	2	.	.	PUNCT
cana-6235	353	3	g.	g.	PROPN
cana-6235	353	4	chartrand	chartrand	PROPN
cana-6235	353	5	and	and	CCONJ
cana-6235	353	6	l.	l.	PROPN
cana-6235	353	7	lesniak	lesniak	PROPN
cana-6235	353	8	,	,	PUNCT
cana-6235	353	9	graphs	graph	NOUN
cana-6235	353	10	and	and	CCONJ
cana-6235	353	11	digraphs	digraph	NOUN
cana-6235	353	12	,	,	PUNCT
cana-6235	353	13	4th	4th	ADJ
cana-6235	353	14	ed	ed	NOUN
cana-6235	353	15	.	.	PROPN
cana-6235	353	16	,	,	PUNCT
cana-6235	353	17	chapman	chapman	NOUN
cana-6235	353	18	and	and	CCONJ
cana-6235	353	19	hall	hall	PROPN
cana-6235	353	20	/	/	SYM
cana-6235	353	21	crc	crc	NOUN
cana-6235	353	22	(	(	PUNCT
cana-6235	353	23	2005	2005	NUM
cana-6235	353	24	)	)	PUNCT
cana-6235	353	25	9	9	NUM
cana-6235	353	26	.	.	PUNCT
cana-6235	353	27	i.	i.	PROPN
cana-6235	353	28	s.	s.	PROPN
cana-6235	353	29	hamid	hamid	PROPN
cana-6235	353	30	,	,	PUNCT
cana-6235	353	31	s.	s.	PROPN
cana-6235	353	32	balamurugan	balamurugan	VERB
cana-6235	353	33	and	and	CCONJ
cana-6235	353	34	a.	a.	NOUN
cana-6235	353	35	navaneethakrishnan	navaneethakrishnan	PROPN
cana-6235	353	36	,	,	PUNCT
cana-6235	353	37	electronic	electronic	ADJ
cana-6235	353	38	journal	journal	NOUN
cana-6235	353	39	of	of	ADP
cana-6235	353	40	graph	graph	NOUN
cana-6235	353	41	theory	theory	NOUN
cana-6235	353	42	and	and	CCONJ
cana-6235	353	43	applications	application	NOUN
cana-6235	353	44	4(1	4(1	NOUN
cana-6235	353	45	)	)	PUNCT
cana-6235	353	46	,	,	PUNCT
cana-6235	353	47	(	(	PUNCT
cana-6235	353	48	2016	2016	NUM
cana-6235	353	49	)	)	PUNCT
cana-6235	353	50	,	,	PUNCT
cana-6235	353	51	94	94	NUM
cana-6235	353	52	-	-	SYM
cana-6235	353	53	100	100	NUM
cana-6235	353	54	.	.	PUNCT
cana-6235	354	1	communications	communication	NOUN
cana-6235	354	2	on	on	ADP
cana-6235	354	3	applied	apply	VERB
cana-6235	354	4	nonlinear	nonlinear	ADJ
cana-6235	354	5	analysis	analysis	NOUN
cana-6235	354	6	issn	issn	NOUN
cana-6235	354	7	:	:	PUNCT
cana-6235	354	8	1074	1074	NUM
cana-6235	354	9	-	-	PUNCT
cana-6235	354	10	133x	133x	NUM
cana-6235	354	11	vol	vol	NOUN
cana-6235	354	12	31	31	NUM
cana-6235	354	13	no	no	NOUN
cana-6235	354	14	.	.	NOUN
cana-6235	354	15	2	2	NUM
cana-6235	354	16	(	(	PUNCT
cana-6235	354	17	2024	2024	NUM
cana-6235	354	18	)	)	PUNCT
cana-6235	354	19	525	525	NUM
cana-6235	354	20	https://internationalpubls.com	https://internationalpubls.com	X
cana-6235	354	21	10	10	NUM
cana-6235	354	22	.	.	PUNCT
cana-6235	355	1	i.	i.	PROPN
cana-6235	355	2	s.	s.	PROPN
cana-6235	355	3	hamid	hamid	PROPN
cana-6235	355	4	and	and	CCONJ
cana-6235	355	5	s.	s.	PROPN
cana-6235	355	6	balamurugan	balamurugan	PROPN
cana-6235	355	7	,	,	PUNCT
cana-6235	355	8	isolate	isolate	VERB
cana-6235	355	9	domination	domination	NOUN
cana-6235	355	10	in	in	ADP
cana-6235	355	11	unicyclic	unicyclic	ADJ
cana-6235	355	12	graphs	graph	NOUN
cana-6235	355	13	,	,	PUNCT
cana-6235	355	14	international	international	ADJ
cana-6235	355	15	journal	journal	NOUN
cana-6235	355	16	of	of	ADP
cana-6235	355	17	mathematics	mathematic	NOUN
cana-6235	355	18	and	and	CCONJ
cana-6235	355	19	soft	soft	ADJ
cana-6235	355	20	computing	computing	NOUN
cana-6235	355	21	vol	vol	NOUN
cana-6235	355	22	3	3	NUM
cana-6235	355	23	,	,	PUNCT
cana-6235	355	24	no.3	no.3	VERB
cana-6235	355	25	,	,	PUNCT
cana-6235	355	26	(	(	PUNCT
cana-6235	355	27	2013	2013	NUM
cana-6235	355	28	)	)	PUNCT
cana-6235	355	29	,	,	PUNCT
cana-6235	355	30	79	79	NUM
cana-6235	355	31	-	-	SYM
cana-6235	355	32	83	83	NUM
cana-6235	355	33	.	.	PUNCT
cana-6235	356	1	11	11	NUM
cana-6235	356	2	.	.	PUNCT
cana-6235	356	3	i.	i.	PROPN
cana-6235	356	4	sahul	sahul	PROPN
cana-6235	356	5	hamid	hamid	PROPN
cana-6235	356	6	and	and	CCONJ
cana-6235	356	7	s.	s.	PROPN
cana-6235	356	8	balamurugan	balamurugan	VERB
cana-6235	356	9	,	,	PUNCT
cana-6235	356	10	”	"	PUNCT
cana-6235	356	11	isolate	isolate	VERB
cana-6235	356	12	domination	domination	NOUN
cana-6235	356	13	number	number	NOUN
cana-6235	356	14	and	and	CCONJ
cana-6235	356	15	maximum	maximum	ADJ
cana-6235	356	16	degree	degree	NOUN
cana-6235	356	17	”	"	PUNCT
cana-6235	356	18	bulletin	bulletin	NOUN
cana-6235	356	19	of	of	ADP
cana-6235	356	20	the	the	DET
cana-6235	356	21	international	international	ADJ
cana-6235	356	22	mathematical	mathematical	ADJ
cana-6235	356	23	virtual	virtual	PROPN
cana-6235	356	24	institute	institute	PROPN
cana-6235	356	25	vol.3(2013),127	vol.3(2013),127	PROPN
cana-6235	356	26	-	-	PUNCT
cana-6235	356	27	133	133	NUM
cana-6235	356	28	.	.	PUNCT
cana-6235	357	1	12	12	NUM
cana-6235	357	2	.	.	PUNCT
cana-6235	357	3	i.	i.	PROPN
cana-6235	357	4	sahul	sahul	PROPN
cana-6235	357	5	hamid	hamid	PROPN
cana-6235	357	6	and	and	CCONJ
cana-6235	357	7	s.	s.	PROPN
cana-6235	357	8	balamurugan	balamurugan	VERB
cana-6235	357	9	,	,	PUNCT
cana-6235	357	10	“	"	PUNCT
cana-6235	357	11	isolate	isolate	VERB
cana-6235	357	12	domination	domination	NOUN
cana-6235	357	13	in	in	ADP
cana-6235	357	14	graphs	graph	NOUN
cana-6235	357	15	”	"	PUNCT
cana-6235	357	16	,	,	PUNCT
cana-6235	357	17	arab	arab	ADJ
cana-6235	357	18	journal	journal	PROPN
cana-6235	357	19	of	of	ADP
cana-6235	357	20	mathematical	mathematical	PROPN
cana-6235	357	21	sciences	sciences	PROPN
cana-6235	357	22	math	math	PROPN
cana-6235	357	23	sci	sci	PROPN
cana-6235	357	24	22(2016	22(2016	NUM
cana-6235	357	25	)	)	PUNCT
cana-6235	357	26	,	,	PUNCT
cana-6235	357	27	232	232	NUM
cana-6235	357	28	-	-	SYM
cana-6235	357	29	241	241	NUM
cana-6235	357	30	.	.	PUNCT
cana-6235	357	31	13	13	NUM
cana-6235	357	32	.	.	PUNCT
cana-6235	358	1	j.	j.	PROPN
cana-6235	358	2	a.	a.	PROPN
cana-6235	358	3	bondy	bondy	PROPN
cana-6235	358	4	and	and	CCONJ
cana-6235	358	5	u.	u.	PROPN
cana-6235	358	6	s.	s.	PROPN
cana-6235	358	7	r.	r.	PROPN
cana-6235	358	8	murty	murty	PROPN
cana-6235	358	9	,	,	PUNCT
cana-6235	358	10	graph	graph	NOUN
cana-6235	358	11	theory	theory	NOUN
cana-6235	358	12	,	,	PUNCT
cana-6235	358	13	springer	springer	NOUN
cana-6235	358	14	(	(	PUNCT
cana-6235	358	15	2008	2008	NUM
cana-6235	358	16	)	)	PUNCT
cana-6235	358	17	.	.	PUNCT
cana-6235	359	1	14	14	NUM
cana-6235	359	2	.	.	PUNCT
cana-6235	360	1	j.	j.	PROPN
cana-6235	360	2	clerk	clerk	PROPN
cana-6235	360	3	and	and	CCONJ
cana-6235	360	4	d.	d.	PROPN
cana-6235	360	5	a.	a.	PROPN
cana-6235	360	6	holton	holton	PROPN
cana-6235	360	7	,	,	PUNCT
cana-6235	360	8	a	a	DET
cana-6235	360	9	first	first	ADJ
cana-6235	360	10	look	look	NOUN
cana-6235	360	11	at	at	ADP
cana-6235	360	12	graph	graph	NOUN
cana-6235	360	13	theory	theory	NOUN
cana-6235	360	14	,	,	PUNCT
cana-6235	360	15	world	world	NOUN
cana-6235	360	16	scientic	scientic	NOUN
cana-6235	360	17	(	(	PUNCT
cana-6235	360	18	1991	1991	NUM
cana-6235	360	19	)	)	PUNCT
cana-6235	360	20	.	.	PUNCT
cana-6235	361	1	15	15	NUM
cana-6235	361	2	.	.	PUNCT
cana-6235	361	3	j.	j.	PROPN
cana-6235	361	4	f.	f.	PROPN
cana-6235	361	5	fink	fink	PROPN
cana-6235	361	6	and	and	CCONJ
cana-6235	361	7	m.	m.	PROPN
cana-6235	361	8	s.	s.	PROPN
cana-6235	361	9	jacobson	jacobson	PROPN
cana-6235	361	10	,	,	PUNCT
cana-6235	361	11	n	n	CCONJ
cana-6235	361	12	-	-	PUNCT
cana-6235	361	13	domination	domination	NOUN
cana-6235	361	14	in	in	ADP
cana-6235	361	15	graphs	graph	NOUN
cana-6235	361	16	,	,	PUNCT
cana-6235	361	17	in	in	ADP
cana-6235	361	18	y.	y.	PROPN
cana-6235	361	19	alavi	alavi	PROPN
cana-6235	361	20	and	and	CCONJ
cana-6235	361	21	a.	a.	NOUN
cana-6235	361	22	j.	j.	PROPN
cana-6235	362	1	schwenk	schwenk	PROPN
cana-6235	362	2	editiors	editiors	PROPN
cana-6235	362	3	,	,	PUNCT
cana-6235	362	4	graph	graph	NOUN
cana-6235	362	5	theory	theory	NOUN
cana-6235	362	6	with	with	ADP
cana-6235	362	7	applications	application	NOUN
cana-6235	362	8	to	to	ADP
cana-6235	362	9	algorithms	algorithm	NOUN
cana-6235	362	10	and	and	CCONJ
cana-6235	362	11	computer	computer	NOUN
cana-6235	362	12	science	science	NOUN
cana-6235	362	13	,	,	PUNCT
cana-6235	362	14	pages	page	NOUN
cana-6235	362	15	23	23	NUM
cana-6235	362	16	-	-	SYM
cana-6235	362	17	300	300	NUM
cana-6235	362	18	,	,	PUNCT
cana-6235	362	19	(	(	PUNCT
cana-6235	362	20	kalamazoo	kalamazoo	PROPN
cana-6235	362	21	,	,	PUNCT
cana-6235	362	22	mi	mi	PROPN
cana-6235	362	23	1984	1984	NUM
cana-6235	362	24	)	)	PUNCT
cana-6235	362	25	,	,	PUNCT
cana-6235	362	26	(	(	PUNCT
cana-6235	362	27	1985	1985	NUM
cana-6235	362	28	)	)	PUNCT
cana-6235	362	29	wiley	wiley	NOUN
cana-6235	362	30	.	.	PUNCT
cana-6235	363	1	16	16	NUM
cana-6235	363	2	.	.	PUNCT
cana-6235	364	1	j.	j.	PROPN
cana-6235	364	2	l.	l.	PROPN
cana-6235	364	3	gross	gross	PROPN
cana-6235	364	4	and	and	CCONJ
cana-6235	364	5	j.	j.	PROPN
cana-6235	364	6	yellen	yellen	PROPN
cana-6235	364	7	,	,	PUNCT
cana-6235	364	8	graph	graph	NOUN
cana-6235	364	9	theory	theory	NOUN
cana-6235	364	10	and	and	CCONJ
cana-6235	364	11	its	its	PRON
cana-6235	364	12	applications	application	NOUN
cana-6235	364	13	,	,	PUNCT
cana-6235	364	14	crc	crc	NOUN
cana-6235	364	15	press	press	NOUN
cana-6235	364	16	(	(	PUNCT
cana-6235	364	17	1998	1998	NUM
cana-6235	364	18	)	)	PUNCT
cana-6235	364	19	.	.	PUNCT
cana-6235	365	1	17	17	NUM
cana-6235	365	2	.	.	PUNCT
cana-6235	366	1	michael	michael	PROPN
cana-6235	366	2	a.	a.	PROPN
cana-6235	366	3	henning	henning	PROPN
cana-6235	366	4	.	.	PROPN
cana-6235	366	5	,	,	PUNCT
cana-6235	366	6	anders	anders	PROPN
cana-6235	366	7	yeo	yeo	PROPN
cana-6235	367	1	“	"	PUNCT
cana-6235	367	2	total	total	ADJ
cana-6235	367	3	domination	domination	NOUN
cana-6235	367	4	in	in	ADP
cana-6235	367	5	graphs	graph	NOUN
cana-6235	367	6	”	"	PUNCT
cana-6235	367	7	,	,	PUNCT
cana-6235	367	8	springer	springer	NOUN
cana-6235	367	9	,	,	PUNCT
cana-6235	367	10	new	new	PROPN
cana-6235	367	11	york	york	PROPN
cana-6235	367	12	,	,	PUNCT
cana-6235	367	13	(	(	PUNCT
cana-6235	367	14	2013	2013	NUM
cana-6235	367	15	)	)	PUNCT
cana-6235	367	16	.	.	PUNCT
cana-6235	368	1	18	18	NUM
cana-6235	368	2	.	.	PUNCT
cana-6235	368	3	m.	m.	NOUN
cana-6235	368	4	a.	a.	PROPN
cana-6235	368	5	henning	henning	PROPN
cana-6235	368	6	and	and	CCONJ
cana-6235	368	7	a.	a.	PROPN
cana-6235	368	8	yeo	yeo	PROPN
cana-6235	368	9	,	,	PUNCT
cana-6235	368	10	total	total	ADJ
cana-6235	368	11	domination	domination	NOUN
cana-6235	368	12	in	in	ADP
cana-6235	368	13	graphs	graph	NOUN
cana-6235	368	14	,	,	PUNCT
cana-6235	368	15	springer	springer	NOUN
cana-6235	368	16	,	,	PUNCT
cana-6235	368	17	new	new	PROPN
cana-6235	368	18	york	york	PROPN
cana-6235	368	19	(	(	PUNCT
cana-6235	368	20	2013	2013	NUM
cana-6235	368	21	)	)	PUNCT
cana-6235	368	22	.	.	PUNCT
cana-6235	369	1	19	19	NUM
cana-6235	369	2	.	.	X
cana-6235	369	3	m.	m.	PROPN
cana-6235	369	4	a.	a.	PROPN
cana-6235	369	5	henning	henning	PROPN
cana-6235	369	6	,	,	PUNCT
cana-6235	369	7	o.	o.	PROPN
cana-6235	369	8	r.	r.	PROPN
cana-6235	369	9	oellermann	oellermann	PROPN
cana-6235	369	10	and	and	CCONJ
cana-6235	369	11	h.	h.	PROPN
cana-6235	369	12	c.	c.	PROPN
cana-6235	369	13	swart	swart	PROPN
cana-6235	369	14	,	,	PUNCT
cana-6235	369	15	bounds	bound	VERB
cana-6235	369	16	on	on	ADP
cana-6235	369	17	distance	distance	NOUN
cana-6235	369	18	domination	domination	NOUN
cana-6235	369	19	parameters	parameter	NOUN
cana-6235	369	20	,	,	PUNCT
cana-6235	369	21	j.	j.	PROPN
cana-6235	369	22	combin	combin	PROPN
cana-6235	369	23	inform	inform	NOUN
cana-6235	369	24	.	.	PUNCT
cana-6235	370	1	system	system	NOUN
cana-6235	370	2	sci	sci	PROPN
cana-6235	370	3	.	.	PROPN
cana-6235	370	4	,	,	PUNCT
cana-6235	370	5	16	16	NUM
cana-6235	370	6	,	,	PUNCT
cana-6235	370	7	(	(	PUNCT
cana-6235	370	8	1991	1991	NUM
cana-6235	370	9	)	)	PUNCT
cana-6235	370	10	,	,	PUNCT
cana-6235	370	11	11	11	NUM
cana-6235	370	12	-	-	SYM
cana-6235	370	13	18	18	NUM
cana-6235	370	14	.	.	NOUN
cana-6235	370	15	20	20	NUM
cana-6235	370	16	.	.	PUNCT
cana-6235	370	17	r.	r.	PROPN
cana-6235	370	18	balakrishnan	balakrishnan	PROPN
cana-6235	370	19	and	and	CCONJ
cana-6235	370	20	k.	k.	PROPN
cana-6235	370	21	ranganathan	ranganathan	PROPN
cana-6235	370	22	,	,	PUNCT
cana-6235	370	23	a	a	DET
cana-6235	370	24	textbook	textbook	NOUN
cana-6235	370	25	of	of	ADP
cana-6235	370	26	graph	graph	NOUN
cana-6235	370	27	theory	theory	NOUN
cana-6235	370	28	,	,	PUNCT
cana-6235	370	29	springer	springer	NOUN
cana-6235	370	30	,	,	PUNCT
cana-6235	370	31	new	new	PROPN
cana-6235	370	32	york	york	PROPN
cana-6235	370	33	(	(	PUNCT
cana-6235	370	34	2012	2012	NUM
cana-6235	370	35	)	)	PUNCT
cana-6235	370	36	.	.	PUNCT
cana-6235	371	1	21	21	NUM
cana-6235	371	2	.	.	PUNCT
cana-6235	371	3	r.	r.	PROPN
cana-6235	371	4	j.	j.	PROPN
cana-6235	371	5	wilson	wilson	PROPN
cana-6235	371	6	and	and	CCONJ
cana-6235	371	7	j.	j.	PROPN
cana-6235	371	8	j.	j.	PROPN
cana-6235	371	9	watkins	watkins	PROPN
cana-6235	371	10	,	,	PUNCT
cana-6235	371	11	an	an	DET
cana-6235	371	12	introductory	introductory	ADJ
cana-6235	371	13	approach	approach	NOUN
cana-6235	371	14	,	,	PUNCT
cana-6235	371	15	john	john	PROPN
cana-6235	371	16	wiley	wiley	PROPN
cana-6235	371	17	&	&	CCONJ
cana-6235	371	18	sons	sons	PROPN
cana-6235	371	19	,	,	PUNCT
cana-6235	371	20	inc	inc	PROPN
cana-6235	371	21	.	.	PROPN
cana-6235	371	22	new	new	PROPN
cana-6235	371	23	york	york	PROPN
cana-6235	371	24	(	(	PUNCT
cana-6235	371	25	1990	1990	NUM
cana-6235	371	26	)	)	PUNCT
cana-6235	371	27	.	.	PUNCT
cana-6235	372	1	22	22	NUM
cana-6235	372	2	.	.	PUNCT
cana-6235	373	1	s.	s.	PROPN
cana-6235	373	2	balamurugan	balamurugan	VERB
cana-6235	373	3	,	,	PUNCT
cana-6235	373	4	changing	change	VERB
cana-6235	373	5	and	and	CCONJ
cana-6235	373	6	unchanging	unchanging	ADJ
cana-6235	373	7	isolate	isolate	NOUN
cana-6235	373	8	domination	domination	NOUN
cana-6235	373	9	:	:	PUNCT
cana-6235	373	10	edge	edge	NOUN
cana-6235	373	11	removal	removal	NOUN
cana-6235	373	12	discrete	discrete	ADJ
cana-6235	373	13	mathematics	mathematic	NOUN
cana-6235	373	14	,	,	PUNCT
cana-6235	373	15	algorithms	algorithm	NOUN
cana-6235	373	16	and	and	CCONJ
cana-6235	373	17	applications	application	NOUN
cana-6235	373	18	vol	vol	NOUN
cana-6235	373	19	.	.	PUNCT
cana-6235	373	20	9	9	NUM
cana-6235	373	21	,	,	PUNCT
cana-6235	373	22	no.1	no.1	NUM
cana-6235	373	23	,	,	PUNCT
cana-6235	373	24	(	(	PUNCT
cana-6235	373	25	2017	2017	NUM
cana-6235	373	26	)	)	PUNCT
cana-6235	373	27	.	.	PUNCT
cana-6235	374	1	23	23	NUM
cana-6235	374	2	.	.	PUNCT
cana-6235	375	1	s.	s.	PROPN
cana-6235	375	2	t.	t.	PROPN
cana-6235	375	3	hedetniemi	hedetniemi	PROPN
cana-6235	375	4	and	and	CCONJ
cana-6235	375	5	r.	r.	PROPN
cana-6235	375	6	laskar	laskar	PROPN
cana-6235	375	7	,	,	PUNCT
cana-6235	375	8	eds	ed	NOUN
cana-6235	375	9	.	.	PUNCT
cana-6235	376	1	topics	topic	NOUN
cana-6235	376	2	in	in	ADP
cana-6235	376	3	domination	domination	NOUN
cana-6235	376	4	in	in	ADP
cana-6235	376	5	graphs	graph	NOUN
cana-6235	376	6	,	,	PUNCT
cana-6235	376	7	discrete	discrete	ADJ
cana-6235	376	8	mathematics	mathematic	NOUN
cana-6235	376	9	,	,	PUNCT
cana-6235	376	10	86	86	NUM
cana-6235	376	11	(	(	PUNCT
cana-6235	376	12	1990	1990	NUM
cana-6235	376	13	)	)	PUNCT
cana-6235	376	14	.	.	PUNCT
cana-6235	377	1	24	24	NUM
cana-6235	377	2	.	.	PUNCT
cana-6235	378	1	t.	t.	PROPN
cana-6235	378	2	w.	w.	PROPN
cana-6235	378	3	haynes	haynes	PROPN
cana-6235	378	4	,	,	PUNCT
cana-6235	378	5	s.	s.	PROPN
cana-6235	378	6	t.	t.	PROPN
cana-6235	378	7	hedetniemi	hedetniemi	PROPN
cana-6235	378	8	and	and	CCONJ
cana-6235	378	9	p.	p.	PROPN
cana-6235	378	10	j.	j.	PROPN
cana-6235	378	11	slater	slater	PROPN
cana-6235	378	12	,	,	PUNCT
cana-6235	378	13	domination	domination	NOUN
cana-6235	378	14	in	in	ADP
cana-6235	378	15	graphs	graph	NOUN
cana-6235	378	16	advanced	advanced	ADJ
cana-6235	378	17	topics	topic	NOUN
cana-6235	378	18	,	,	PUNCT
cana-6235	378	19	marcel	marcel	PROPN
cana-6235	378	20	dekker	dekker	PROPN
cana-6235	378	21	,	,	PUNCT
cana-6235	378	22	inc	inc	PROPN
cana-6235	378	23	.	.	PROPN
cana-6235	378	24	,	,	PUNCT
cana-6235	378	25	new	new	PROPN
cana-6235	378	26	york	york	PROPN
cana-6235	378	27	(	(	PUNCT
cana-6235	378	28	1998	1998	NUM
cana-6235	378	29	)	)	PUNCT
cana-6235	378	30	.	.	PUNCT
cana-6235	379	1	25	25	NUM
cana-6235	379	2	.	.	PUNCT
cana-6235	380	1	t.	t.	PROPN
cana-6235	380	2	w.	w.	PROPN
cana-6235	380	3	haynes	haynes	PROPN
cana-6235	380	4	,	,	PUNCT
cana-6235	380	5	s.	s.	PROPN
cana-6235	380	6	t.	t.	PROPN
cana-6235	380	7	hedetniemi	hedetniemi	PROPN
cana-6235	380	8	,	,	PUNCT
cana-6235	380	9	p.	p.	PROPN
cana-6235	380	10	j.	j.	PROPN
cana-6235	380	11	slater	slater	PROPN
cana-6235	380	12	,	,	PUNCT
cana-6235	380	13	“	"	PUNCT
cana-6235	380	14	fundamental	fundamental	ADJ
cana-6235	380	15	of	of	ADP
cana-6235	380	16	domination	domination	NOUN
cana-6235	380	17	in	in	ADP
cana-6235	380	18	graphs	graph	NOUN
cana-6235	380	19	”	"	PUNCT
cana-6235	380	20	,	,	PUNCT
cana-6235	380	21	marcel	marcel	PROPN
cana-6235	380	22	dekker	dekker	PROPN
cana-6235	380	23	,	,	PUNCT
cana-6235	380	24	new	new	PROPN
cana-6235	380	25	york	york	PROPN
cana-6235	380	26	,	,	PUNCT
cana-6235	380	27	(	(	PUNCT
cana-6235	380	28	1998	1998	NUM
cana-6235	380	29	)	)	PUNCT
cana-6235	380	30	.	.	PUNCT