id	sid	tid	token	lemma	pos
cana-577	1	1	communications	communication	NOUN
cana-577	1	2	on	on	ADP
cana-577	1	3	applied	apply	VERB
cana-577	1	4	nonlinear	nonlinear	ADJ
cana-577	1	5	analysis	analysis	NOUN
cana-577	1	6	issn	issn	NOUN
cana-577	1	7	:	:	PUNCT
cana-577	1	8	1074	1074	NUM
cana-577	1	9	-	-	PUNCT
cana-577	1	10	133x	133x	NUM
cana-577	1	11	vol	vol	NOUN
cana-577	1	12	31	31	NUM
cana-577	1	13	no	no	NOUN
cana-577	1	14	.	.	PUNCT
cana-577	2	1	1s	1s	NUM
cana-577	2	2	(	(	PUNCT
cana-577	2	3	2024	2024	NUM
cana-577	2	4	)	)	PUNCT
cana-577	2	5	180	180	NUM
cana-577	2	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-577	2	7	strong	strong	ADJ
cana-577	2	8	regular	regular	ADJ
cana-577	2	9	domination	domination	NOUN
cana-577	2	10	in	in	ADP
cana-577	2	11	litact	litact	NOUN
cana-577	2	12	graphs	graph	NOUN
cana-577	2	13	g.	g.	PROPN
cana-577	2	14	shankarajyothi1	shankarajyothi1	PROPN
cana-577	2	15	*	*	PROPN
cana-577	2	16	,	,	PUNCT
cana-577	2	17	g.	g.	PROPN
cana-577	2	18	upender	upender	PROPN
cana-577	2	19	reddy2	reddy2	PROPN
cana-577	3	1	1*research	1*research	NUM
cana-577	3	2	scholar	scholar	NOUN
cana-577	3	3	,	,	PUNCT
cana-577	3	4	department	department	NOUN
cana-577	3	5	of	of	ADP
cana-577	3	6	mathematics	mathematics	PROPN
cana-577	3	7	,	,	PUNCT
cana-577	3	8	osmania	osmania	PROPN
cana-577	3	9	university	university	PROPN
cana-577	3	10	,	,	PUNCT
cana-577	3	11	hyderabad	hyderabad	PROPN
cana-577	3	12	,	,	PUNCT
cana-577	3	13	india	india	PROPN
cana-577	3	14	.	.	PUNCT
cana-577	4	1	2department	2department	NUM
cana-577	4	2	of	of	ADP
cana-577	4	3	mathematics	mathematic	NOUN
cana-577	4	4	,	,	PUNCT
cana-577	4	5	nizam	nizam	PROPN
cana-577	4	6	college(a	college(a	PROPN
cana-577	4	7	)	)	PUNCT
cana-577	4	8	,	,	PUNCT
cana-577	4	9	osmania	osmania	PROPN
cana-577	4	10	university	university	PROPN
cana-577	4	11	,	,	PUNCT
cana-577	4	12	hyderabad	hyderabad	PROPN
cana-577	4	13	,	,	PUNCT
cana-577	4	14	india	india	PROPN
cana-577	4	15	.	.	PUNCT
cana-577	5	1	1e	1e	NOUN
cana-577	5	2	-	-	PUNCT
cana-577	5	3	mail	mail	NOUN
cana-577	5	4	:	:	PUNCT
cana-577	5	5	shankarajyothi.maths@gmail.com	shankarajyothi.maths@gmail.com	X
cana-577	5	6	*	*	PUNCT
cana-577	5	7	2e	2e	NOUN
cana-577	5	8	-	-	PUNCT
cana-577	5	9	mail	mail	NOUN
cana-577	5	10	:	:	PUNCT
cana-577	5	11	yuviganga@gmail.com	yuviganga@gmail.com	PROPN
cana-577	5	12	article	article	NOUN
cana-577	5	13	history	history	NOUN
cana-577	5	14	:	:	PUNCT
cana-577	5	15	received	receive	VERB
cana-577	5	16	:	:	PUNCT
cana-577	5	17	10	10	NUM
cana-577	5	18	-	-	PUNCT
cana-577	5	19	02	02	NUM
cana-577	5	20	-	-	PUNCT
cana-577	5	21	2024	2024	NUM
cana-577	5	22	revised	revise	VERB
cana-577	5	23	:	:	PUNCT
cana-577	5	24	12	12	NUM
cana-577	5	25	-	-	PUNCT
cana-577	5	26	04	04	NUM
cana-577	5	27	-	-	PUNCT
cana-577	5	28	2024	2024	NUM
cana-577	5	29	accepted	accept	VERB
cana-577	5	30	:	:	PUNCT
cana-577	5	31	26	26	NUM
cana-577	5	32	-	-	PUNCT
cana-577	5	33	04	04	NUM
cana-577	5	34	-	-	PUNCT
cana-577	5	35	2024	2024	NUM
cana-577	5	36	abstract	abstract	NOUN
cana-577	5	37	:	:	PUNCT
cana-577	5	38	strong	strong	ADJ
cana-577	5	39	regular	regular	ADJ
cana-577	5	40	domination	domination	NOUN
cana-577	5	41	in	in	ADP
cana-577	5	42	a	a	DET
cana-577	5	43	litact	litact	NOUN
cana-577	5	44	graph	graph	NOUN
cana-577	5	45	is	be	AUX
cana-577	5	46	a	a	DET
cana-577	5	47	novel	novel	ADJ
cana-577	5	48	domination	domination	NOUN
cana-577	5	49	parameter	parameter	NOUN
cana-577	5	50	that	that	PRON
cana-577	5	51	has	have	AUX
cana-577	5	52	been	be	AUX
cana-577	5	53	introduced	introduce	VERB
cana-577	5	54	in	in	ADP
cana-577	5	55	this	this	DET
cana-577	5	56	paper	paper	NOUN
cana-577	5	57	.	.	PUNCT
cana-577	6	1	a	a	DET
cana-577	6	2	dominating	dominating	NOUN
cana-577	6	3	set	set	VERB
cana-577	6	4	𝐷	𝐷	PROPN
cana-577	6	5	⊆	⊆	NUM
cana-577	6	6	𝑉(𝐺	𝑉(𝐺	NOUN
cana-577	6	7	)	)	PUNCT
cana-577	6	8	is	be	AUX
cana-577	6	9	known	know	VERB
cana-577	6	10	as	as	ADP
cana-577	6	11	strong	strong	ADJ
cana-577	6	12	regular	regular	ADJ
cana-577	6	13	dominating	dominating	NOUN
cana-577	6	14	set	set	NOUN
cana-577	6	15	of	of	ADP
cana-577	6	16	𝐺	𝐺	PROPN
cana-577	6	17	,	,	PUNCT
cana-577	6	18	if	if	SCONJ
cana-577	6	19	for	for	ADP
cana-577	6	20	each	each	DET
cana-577	6	21	point	point	NOUN
cana-577	6	22	𝑥	𝑥	DET
cana-577	6	23	∈	∈	PROPN
cana-577	6	24	𝑉(𝐺	𝑉(𝐺	NOUN
cana-577	6	25	)	)	PUNCT
cana-577	6	26	−	−	PROPN
cana-577	6	27	𝐷	𝐷	PROPN
cana-577	6	28	there	there	PRON
cana-577	6	29	is	be	VERB
cana-577	6	30	a	a	DET
cana-577	6	31	vertex	vertex	NOUN
cana-577	6	32	𝑦	𝑦	PROPN
cana-577	6	33	∈	∈	PROPN
cana-577	6	34	𝐷	𝐷	NOUN
cana-577	6	35	with	with	ADP
cana-577	6	36	an	an	DET
cana-577	6	37	edge	edge	NOUN
cana-577	7	1	𝑥𝑦	𝑥𝑦	NOUN
cana-577	7	2	∈	∈	PROPN
cana-577	7	3	𝐸(𝐺	𝐸(𝐺	NOUN
cana-577	7	4	)	)	PUNCT
cana-577	7	5	and	and	CCONJ
cana-577	7	6	deg⁡(𝑥	deg⁡(𝑥	NOUN
cana-577	7	7	)	)	PUNCT
cana-577	7	8	≤	≤	NOUN
cana-577	7	9	deg⁡(𝑦	deg⁡(𝑦	NUM
cana-577	7	10	)	)	PUNCT
cana-577	7	11	and	and	CCONJ
cana-577	7	12	all	all	DET
cana-577	7	13	vertices	vertex	NOUN
cana-577	7	14	of	of	ADP
cana-577	7	15	〈	〈	NOUN
cana-577	7	16	𝐷	𝐷	NOUN
cana-577	7	17	〉	〉	NOUN
cana-577	7	18	holds	hold	VERB
cana-577	7	19	the	the	DET
cana-577	7	20	equal	equal	ADJ
cana-577	7	21	degree	degree	NOUN
cana-577	7	22	.	.	PUNCT
cana-577	8	1	the	the	DET
cana-577	8	2	lowest	low	ADJ
cana-577	8	3	cardinality	cardinality	NOUN
cana-577	8	4	of	of	ADP
cana-577	8	5	such	such	ADJ
cana-577	8	6	vertices	vertex	NOUN
cana-577	8	7	of	of	ADP
cana-577	8	8	𝐷	𝐷	PROPN
cana-577	8	9	is	be	AUX
cana-577	8	10	known	know	VERB
cana-577	8	11	as	as	ADP
cana-577	8	12	strong	strong	ADJ
cana-577	8	13	regular	regular	ADJ
cana-577	8	14	domination	domination	NOUN
cana-577	8	15	number	number	NOUN
cana-577	8	16	of	of	ADP
cana-577	8	17	𝐺	𝐺	PROPN
cana-577	8	18	which	which	PRON
cana-577	8	19	is	be	AUX
cana-577	8	20	represented	represent	VERB
cana-577	8	21	by	by	ADP
cana-577	8	22	𝛾𝑠𝑡𝑟(𝐺	𝛾𝑠𝑡𝑟(𝐺	PROPN
cana-577	8	23	)	)	PUNCT
cana-577	8	24	.	.	PUNCT
cana-577	9	1	the	the	DET
cana-577	9	2	current	current	ADJ
cana-577	9	3	study	study	NOUN
cana-577	9	4	aims	aim	VERB
cana-577	9	5	by	by	ADP
cana-577	9	6	taking	take	VERB
cana-577	9	7	strong	strong	ADJ
cana-577	9	8	regular	regular	ADJ
cana-577	9	9	domination	domination	NOUN
cana-577	9	10	on	on	ADP
cana-577	9	11	a	a	DET
cana-577	9	12	litact	litact	NOUN
cana-577	9	13	graph	graph	NOUN
cana-577	9	14	𝑚(𝐺	𝑚(𝐺	NUM
cana-577	9	15	)	)	PUNCT
cana-577	9	16	denoted	denote	VERB
cana-577	9	17	by	by	ADP
cana-577	9	18	𝛾𝑠𝑡𝑟[𝑚(𝐺	𝛾𝑠𝑡𝑟[𝑚(𝐺	PROPN
cana-577	9	19	)	)	PUNCT
cana-577	9	20	]	]	PUNCT
cana-577	9	21	and	and	CCONJ
cana-577	9	22	to	to	PART
cana-577	9	23	obtain	obtain	VERB
cana-577	9	24	some	some	DET
cana-577	9	25	bounds	bound	NOUN
cana-577	9	26	on	on	ADP
cana-577	9	27	𝛾𝑠𝑡𝑟[𝑚(𝐺	𝛾𝑠𝑡𝑟[𝑚(𝐺	PROPN
cana-577	9	28	)	)	PUNCT
cana-577	9	29	]	]	PUNCT
cana-577	9	30	in	in	ADP
cana-577	9	31	terms	term	NOUN
cana-577	9	32	of	of	ADP
cana-577	9	33	various	various	ADJ
cana-577	9	34	parameters	parameter	NOUN
cana-577	9	35	of	of	ADP
cana-577	9	36	𝐺	𝐺	PROPN
cana-577	9	37	such	such	ADJ
cana-577	9	38	as	as	ADP
cana-577	9	39	vertices	vertex	NOUN
cana-577	9	40	,	,	PUNCT
cana-577	9	41	edges	edge	NOUN
cana-577	9	42	,	,	PUNCT
cana-577	9	43	maximum	maximum	ADJ
cana-577	9	44	degree	degree	NOUN
cana-577	9	45	,	,	PUNCT
cana-577	9	46	diameter	diameter	NOUN
cana-577	9	47	and	and	CCONJ
cana-577	9	48	so	so	ADV
cana-577	9	49	on	on	ADV
cana-577	9	50	and	and	CCONJ
cana-577	9	51	also	also	ADV
cana-577	9	52	in	in	ADP
cana-577	9	53	terms	term	NOUN
cana-577	9	54	of	of	ADP
cana-577	9	55	various	various	ADJ
cana-577	9	56	domination	domination	NOUN
cana-577	9	57	parameters	parameter	NOUN
cana-577	9	58	of	of	ADP
cana-577	9	59	𝐺	𝐺	PROPN
cana-577	9	60	such	such	ADJ
cana-577	9	61	as	as	ADP
cana-577	9	62	total	total	ADJ
cana-577	9	63	domination	domination	NOUN
cana-577	9	64	of	of	ADP
cana-577	9	65	𝐺	𝐺	PROPN
cana-577	9	66	,	,	PUNCT
cana-577	9	67	connected	connected	ADJ
cana-577	9	68	domination	domination	NOUN
cana-577	9	69	of	of	ADP
cana-577	9	70	𝐺	𝐺	PROPN
cana-577	9	71	and	and	CCONJ
cana-577	9	72	so	so	ADV
cana-577	9	73	on	on	ADV
cana-577	9	74	.	.	PUNCT
cana-577	10	1	furthermore	furthermore	ADV
cana-577	10	2	,	,	PUNCT
cana-577	10	3	outcomes	outcome	NOUN
cana-577	10	4	resembling	resemble	VERB
cana-577	10	5	those	those	PRON
cana-577	10	6	of	of	ADP
cana-577	10	7	nordhausgaddum	nordhausgaddum	NOUN
cana-577	10	8	were	be	AUX
cana-577	10	9	also	also	ADV
cana-577	10	10	obtained	obtain	VERB
cana-577	10	11	.	.	PUNCT
cana-577	11	1	keywords	keyword	NOUN
cana-577	11	2	:	:	PUNCT
cana-577	11	3	litact	litact	NOUN
cana-577	11	4	graph	graph	NOUN
cana-577	11	5	,	,	PUNCT
cana-577	11	6	strong	strong	ADJ
cana-577	11	7	domination	domination	NOUN
cana-577	11	8	number	number	NOUN
cana-577	11	9	,	,	PUNCT
cana-577	11	10	regular	regular	ADJ
cana-577	11	11	domination	domination	NOUN
cana-577	11	12	number	number	NOUN
cana-577	11	13	,	,	PUNCT
cana-577	11	14	strong	strong	ADJ
cana-577	11	15	regular	regular	ADJ
cana-577	11	16	domination	domination	NOUN
cana-577	11	17	number	number	NOUN
cana-577	11	18	.	.	PUNCT
cana-577	12	1	2020	2020	NUM
cana-577	12	2	ams	am	NOUN
cana-577	12	3	subject	subject	ADJ
cana-577	12	4	classification	classification	NOUN
cana-577	12	5	:	:	PUNCT
cana-577	12	6	05c72	05c72	NOUN
cana-577	12	7	.	.	PUNCT
cana-577	13	1	1	1	X
cana-577	13	2	.	.	X
cana-577	13	3	introduction	introduction	NOUN
cana-577	13	4	one	one	NUM
cana-577	13	5	area	area	NOUN
cana-577	13	6	of	of	ADP
cana-577	13	7	graph	graph	NOUN
cana-577	13	8	theory	theory	NOUN
cana-577	13	9	that	that	PRON
cana-577	13	10	has	have	AUX
cana-577	13	11	been	be	AUX
cana-577	13	12	studied	study	VERB
cana-577	13	13	in	in	ADP
cana-577	13	14	great	great	ADJ
cana-577	13	15	detail	detail	NOUN
cana-577	13	16	is	be	AUX
cana-577	13	17	domination	domination	NOUN
cana-577	13	18	.	.	PUNCT
cana-577	14	1	the	the	DET
cana-577	14	2	theory	theory	NOUN
cana-577	14	3	of	of	ADP
cana-577	14	4	domination	domination	NOUN
cana-577	14	5	has	have	VERB
cana-577	14	6	multiple	multiple	ADJ
cana-577	14	7	origins	origin	NOUN
cana-577	14	8	.	.	PUNCT
cana-577	15	1	according	accord	VERB
cana-577	15	2	to	to	ADP
cana-577	15	3	historical	historical	ADJ
cana-577	15	4	accounts	account	NOUN
cana-577	15	5	,	,	PUNCT
cana-577	15	6	the	the	DET
cana-577	15	7	first	first	ADJ
cana-577	15	8	domination	domination	NOUN
cana-577	15	9	-	-	PUNCT
cana-577	15	10	type	type	NOUN
cana-577	15	11	problem	problem	NOUN
cana-577	15	12	originated	originate	VERB
cana-577	15	13	with	with	ADP
cana-577	15	14	a	a	DET
cana-577	15	15	chess	chess	NOUN
cana-577	15	16	board	board	NOUN
cana-577	15	17	problem	problem	NOUN
cana-577	15	18	in	in	ADP
cana-577	15	19	1850	1850	NUM
cana-577	15	20	that	that	SCONJ
cana-577	15	21	c.f	c.f	PROPN
cana-577	15	22	.	.	PROPN
cana-577	15	23	de	de	PROPN
cana-577	15	24	jaenisch	jaenisch	PROPN
cana-577	15	25	mathematically	mathematically	ADV
cana-577	15	26	explained	explain	VERB
cana-577	15	27	in	in	ADP
cana-577	15	28	1862	1862	NUM
cana-577	15	29	.	.	PUNCT
cana-577	16	1	domination	domination	NOUN
cana-577	16	2	in	in	ADP
cana-577	16	3	graphs	graph	NOUN
cana-577	16	4	has	have	VERB
cana-577	16	5	implications	implication	NOUN
cana-577	16	6	beyond	beyond	ADP
cana-577	16	7	the	the	DET
cana-577	16	8	chess	chess	NOUN
cana-577	16	9	board	board	NOUN
cana-577	16	10	problem	problem	NOUN
cana-577	16	11	,	,	PUNCT
cana-577	16	12	including	include	VERB
cana-577	16	13	facility	facility	NOUN
cana-577	16	14	location	location	NOUN
cana-577	16	15	problems	problem	NOUN
cana-577	16	16	,	,	PUNCT
cana-577	16	17	electric	electric	ADJ
cana-577	16	18	networks	network	NOUN
cana-577	16	19	,	,	PUNCT
cana-577	16	20	power	power	NOUN
cana-577	16	21	grids	grid	NOUN
cana-577	16	22	,	,	PUNCT
cana-577	16	23	land	land	NOUN
cana-577	16	24	surveying	surveying	NOUN
cana-577	16	25	,	,	PUNCT
cana-577	16	26	and	and	CCONJ
cana-577	16	27	more	more	ADJ
cana-577	16	28	.	.	PUNCT
cana-577	17	1	the	the	DET
cana-577	17	2	topic	topic	NOUN
cana-577	17	3	of	of	ADP
cana-577	17	4	domination	domination	NOUN
cana-577	17	5	was	be	AUX
cana-577	17	6	first	first	ADV
cana-577	17	7	introduced	introduce	VERB
cana-577	17	8	by	by	ADP
cana-577	17	9	c.	c.	PROPN
cana-577	17	10	berge	berge	PROPN
cana-577	17	11	in	in	ADP
cana-577	17	12	‘	'	PUNCT
cana-577	17	13	the	the	DET
cana-577	17	14	theory	theory	NOUN
cana-577	17	15	of	of	ADP
cana-577	17	16	graphs	graph	NOUN
cana-577	17	17	and	and	CCONJ
cana-577	17	18	its	its	PRON
cana-577	17	19	application	application	NOUN
cana-577	17	20	’	'	PUNCT
cana-577	17	21	,	,	PUNCT
cana-577	17	22	and	and	CCONJ
cana-577	17	23	it	it	PRON
cana-577	17	24	was	be	AUX
cana-577	17	25	formalized	formalize	VERB
cana-577	17	26	mathematically	mathematically	ADV
cana-577	17	27	by	by	ADP
cana-577	17	28	ore	ore	NOUN
cana-577	17	29	in	in	ADP
cana-577	17	30	1962	1962	NUM
cana-577	17	31	.	.	PUNCT
cana-577	18	1	berge	berge	PROPN
cana-577	18	2	referred	refer	VERB
cana-577	18	3	to	to	ADP
cana-577	18	4	the	the	DET
cana-577	18	5	number	number	NOUN
cana-577	18	6	of	of	ADP
cana-577	18	7	domination	domination	NOUN
cana-577	18	8	as	as	ADP
cana-577	18	9	the	the	DET
cana-577	18	10	coefficient	coefficient	NOUN
cana-577	18	11	of	of	ADP
cana-577	18	12	external	external	ADJ
cana-577	18	13	stability	stability	NOUN
cana-577	18	14	and	and	CCONJ
cana-577	18	15	the	the	DET
cana-577	18	16	domination	domination	NOUN
cana-577	18	17	as	as	ADP
cana-577	18	18	the	the	DET
cana-577	18	19	external	external	ADJ
cana-577	18	20	stability	stability	NOUN
cana-577	18	21	.	.	PUNCT
cana-577	19	1	the	the	DET
cana-577	19	2	term	term	NOUN
cana-577	19	3	"	"	PUNCT
cana-577	19	4	domination	domination	NOUN
cana-577	19	5	"	"	PUNCT
cana-577	19	6	was	be	AUX
cana-577	19	7	first	first	ADV
cana-577	19	8	used	use	VERB
cana-577	19	9	by	by	ADP
cana-577	19	10	ore	ore	NOUN
cana-577	19	11	in	in	ADP
cana-577	19	12	his	his	PRON
cana-577	19	13	well	well	ADV
cana-577	19	14	-	-	PUNCT
cana-577	19	15	known	know	VERB
cana-577	19	16	1962	1962	NUM
cana-577	19	17	book	book	NOUN
cana-577	19	18	“	"	PUNCT
cana-577	19	19	theory	theory	NOUN
cana-577	19	20	of	of	ADP
cana-577	19	21	graphs	graph	NOUN
cana-577	19	22	”	"	PUNCT
cana-577	19	23	.	.	PUNCT
cana-577	20	1	certain	certain	ADJ
cana-577	20	2	real	real	ADJ
cana-577	20	3	-	-	PUNCT
cana-577	20	4	world	world	NOUN
cana-577	20	5	scenarios	scenario	NOUN
cana-577	20	6	naturally	naturally	ADV
cana-577	20	7	give	give	VERB
cana-577	20	8	rise	rise	NOUN
cana-577	20	9	to	to	ADP
cana-577	20	10	both	both	CCONJ
cana-577	20	11	strong	strong	ADJ
cana-577	20	12	and	and	CCONJ
cana-577	20	13	weak	weak	ADJ
cana-577	20	14	domination	domination	NOUN
cana-577	20	15	.	.	PUNCT
cana-577	21	1	take	take	VERB
cana-577	21	2	a	a	DET
cana-577	21	3	road	road	NOUN
cana-577	21	4	network	network	NOUN
cana-577	21	5	that	that	PRON
cana-577	21	6	connects	connect	VERB
cana-577	21	7	several	several	ADJ
cana-577	21	8	places	place	NOUN
cana-577	21	9	,	,	PUNCT
cana-577	21	10	for	for	ADP
cana-577	21	11	instance	instance	NOUN
cana-577	21	12	,	,	PUNCT
cana-577	21	13	the	the	DET
cana-577	21	14	degree	degree	NOUN
cana-577	21	15	of	of	ADP
cana-577	21	16	such	such	DET
cana-577	21	17	a	a	DET
cana-577	21	18	network	network	NOUN
cana-577	21	19	is	be	AUX
cana-577	21	20	determined	determine	VERB
cana-577	21	21	by	by	ADP
cana-577	21	22	‘	'	PUNCT
cana-577	21	23	the	the	DET
cana-577	21	24	number	number	NOUN
cana-577	21	25	of	of	ADP
cana-577	21	26	roads	road	NOUN
cana-577	21	27	that	that	PRON
cana-577	21	28	meet	meet	VERB
cana-577	21	29	at	at	ADP
cana-577	21	30	a	a	DET
cana-577	21	31	vertex	vertex	NOUN
cana-577	21	32	𝑣	𝑣	ADP
cana-577	21	33	’	'	PUNCT
cana-577	21	34	.	.	PUNCT
cana-577	22	1	let	let	VERB
cana-577	22	2	deg⁡(𝑢	deg⁡(𝑢	NUM
cana-577	22	3	)	)	PUNCT
cana-577	22	4	≥	≥	NOUN
cana-577	22	5	deg⁡(𝑣	deg⁡(𝑣	NOUN
cana-577	22	6	)	)	PUNCT
cana-577	22	7	.	.	PUNCT
cana-577	23	1	it	it	PRON
cana-577	23	2	seems	seem	VERB
cana-577	23	3	that	that	SCONJ
cana-577	23	4	,	,	PUNCT
cana-577	23	5	‘	'	PUNCT
cana-577	23	6	traffic	traffic	NOUN
cana-577	23	7	at	at	ADP
cana-577	23	8	𝑢	𝑢	PROPN
cana-577	23	9	appears	appear	VERB
cana-577	23	10	mailto:shankarajyothi.maths@gmail.com	mailto:shankarajyothi.maths@gmail.com	PROPN
cana-577	23	11	mailto:yuviganga@gmail.com	mailto:yuviganga@gmail.com	X
cana-577	23	12	communications	communication	NOUN
cana-577	23	13	on	on	ADP
cana-577	23	14	applied	apply	VERB
cana-577	23	15	nonlinear	nonlinear	ADJ
cana-577	23	16	analysis	analysis	NOUN
cana-577	23	17	issn	issn	NOUN
cana-577	23	18	:	:	PUNCT
cana-577	23	19	1074	1074	NUM
cana-577	23	20	-	-	PUNCT
cana-577	23	21	133x	133x	NUM
cana-577	23	22	vol	vol	NOUN
cana-577	23	23	31	31	NUM
cana-577	23	24	no	no	NOUN
cana-577	23	25	.	.	PUNCT
cana-577	24	1	1s	1s	NUM
cana-577	24	2	(	(	PUNCT
cana-577	24	3	2024	2024	NUM
cana-577	24	4	)	)	PUNCT
cana-577	24	5	181	181	NUM
cana-577	24	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-577	24	7	to	to	PART
cana-577	24	8	be	be	AUX
cana-577	24	9	heavier	heavy	ADJ
cana-577	24	10	than	than	ADP
cana-577	24	11	at	at	ADP
cana-577	24	12	𝑣	𝑣	NOUN
cana-577	24	13	’	'	PUNCT
cana-577	24	14	.	.	PUNCT
cana-577	25	1	vehicles	vehicle	NOUN
cana-577	25	2	travelling	travel	VERB
cana-577	25	3	from	from	ADP
cana-577	25	4	𝑢	𝑢	PRON
cana-577	25	5	to	to	ADP
cana-577	25	6	𝑣	𝑣	PRON
cana-577	25	7	should	should	AUX
cana-577	25	8	be	be	AUX
cana-577	25	9	given	give	VERB
cana-577	25	10	preference	preference	NOUN
cana-577	25	11	when	when	SCONJ
cana-577	25	12	it	it	PRON
cana-577	25	13	comes	come	VERB
cana-577	25	14	to	to	ADP
cana-577	25	15	traffic	traffic	NOUN
cana-577	25	16	between	between	ADP
cana-577	25	17	𝑢	𝑢	NOUN
cana-577	25	18	and	and	CCONJ
cana-577	25	19	𝑣.	𝑣.	PROPN
cana-577	25	20	therefore	therefore	ADV
cana-577	25	21	,	,	PUNCT
cana-577	25	22	‘	'	PUNCT
cana-577	25	23	𝑢	𝑢	PRON
cana-577	25	24	strongly	strongly	ADV
cana-577	25	25	dominates	dominate	VERB
cana-577	25	26	𝑣	𝑣	NOUN
cana-577	25	27	’	'	PUNCT
cana-577	25	28	and	and	CCONJ
cana-577	25	29	‘	'	PUNCT
cana-577	25	30	𝑣	𝑣	DET
cana-577	25	31	weakly	weakly	ADV
cana-577	25	32	dominates	dominate	VERB
cana-577	25	33	𝑢	𝑢	NOUN
cana-577	25	34	’	'	PUNCT
cana-577	25	35	in	in	ADP
cana-577	25	36	a	a	DET
cana-577	25	37	certain	certain	ADJ
cana-577	25	38	sense	sense	NOUN
cana-577	25	39	.	.	PUNCT
cana-577	26	1	in	in	ADP
cana-577	26	2	1996	1996	NUM
cana-577	26	3	,	,	PUNCT
cana-577	26	4	‘	'	PUNCT
cana-577	26	5	sampathkumar	sampathkumar	X
cana-577	26	6	and	and	CCONJ
cana-577	26	7	pushpa	pushpa	PROPN
cana-577	26	8	latha	latha	PROPN
cana-577	26	9	’	'	PUNCT
cana-577	26	10	presented	present	VERB
cana-577	26	11	the	the	DET
cana-577	26	12	ideas	idea	NOUN
cana-577	26	13	of	of	ADP
cana-577	26	14	‘	'	PUNCT
cana-577	26	15	strong	strong	ADJ
cana-577	26	16	and	and	CCONJ
cana-577	26	17	weak	weak	ADJ
cana-577	26	18	domination	domination	NOUN
cana-577	26	19	’	'	PUNCT
cana-577	26	20	.	.	PUNCT
cana-577	27	1	it	it	PRON
cana-577	27	2	was	be	AUX
cana-577	27	3	e.	e.	PROPN
cana-577	27	4	sampath	sampath	PROPN
cana-577	27	5	kumar	kumar	PROPN
cana-577	27	6	who	who	PRON
cana-577	27	7	first	first	ADV
cana-577	27	8	proposed	propose	VERB
cana-577	27	9	the	the	DET
cana-577	27	10	concept	concept	NOUN
cana-577	27	11	of	of	ADP
cana-577	27	12	regular	regular	ADJ
cana-577	27	13	domination	domination	NOUN
cana-577	27	14	number	number	NOUN
cana-577	27	15	.	.	PUNCT
cana-577	28	1	numerous	numerous	ADJ
cana-577	28	2	scholars	scholar	NOUN
cana-577	28	3	have	have	AUX
cana-577	28	4	investigated	investigate	VERB
cana-577	28	5	these	these	DET
cana-577	28	6	ideas	idea	NOUN
cana-577	28	7	,	,	PUNCT
cana-577	28	8	including	include	VERB
cana-577	28	9	(	(	PUNCT
cana-577	28	10	1,2,3,6,7	1,2,3,6,7	NUM
cana-577	28	11	)	)	PUNCT
cana-577	28	12	.	.	PUNCT
cana-577	29	1	2	2	X
cana-577	29	2	.	.	NUM
cana-577	29	3	notations	notation	NOUN
cana-577	29	4	and	and	CCONJ
cana-577	29	5	definitions	definition	NOUN
cana-577	29	6	definition-2.1	definition-2.1	NOUN
cana-577	29	7	:	:	PUNCT
cana-577	29	8	if	if	SCONJ
cana-577	29	9	each	each	DET
cana-577	29	10	point	point	NOUN
cana-577	29	11	in	in	ADP
cana-577	29	12	𝑉	𝑉	PROPN
cana-577	29	13	−	−	PROPN
cana-577	29	14	𝑋	𝑋	PROPN
cana-577	29	15	is	be	AUX
cana-577	29	16	adjacent	adjacent	ADJ
cana-577	29	17	to	to	ADP
cana-577	29	18	a	a	DET
cana-577	29	19	point	point	NOUN
cana-577	29	20	in	in	ADP
cana-577	29	21	𝑋	𝑋	PROPN
cana-577	29	22	,	,	PUNCT
cana-577	29	23	then	then	ADV
cana-577	29	24	𝑋	𝑋	PROPN
cana-577	29	25	is	be	AUX
cana-577	29	26	considered	consider	VERB
cana-577	29	27	a	a	DET
cana-577	29	28	‘	'	PUNCT
cana-577	29	29	dominating	dominating	NOUN
cana-577	29	30	set	set	NOUN
cana-577	29	31	’	'	PUNCT
cana-577	29	32	of	of	ADP
cana-577	29	33	𝐺(𝑉	𝐺(𝑉	NUM
cana-577	29	34	,	,	PUNCT
cana-577	29	35	𝐸	𝐸	PROPN
cana-577	29	36	)	)	PUNCT
cana-577	29	37	.	.	PUNCT
cana-577	30	1	the	the	DET
cana-577	30	2	domination	domination	NOUN
cana-577	30	3	number	number	NOUN
cana-577	30	4	of	of	ADP
cana-577	30	5	𝐺	𝐺	PROPN
cana-577	30	6	,	,	PUNCT
cana-577	30	7	represented	represent	VERB
cana-577	30	8	by	by	ADP
cana-577	30	9	𝛾(𝐺	𝛾(𝐺	PROPN
cana-577	30	10	)	)	PUNCT
cana-577	30	11	,	,	PUNCT
cana-577	30	12	is	be	AUX
cana-577	30	13	the	the	DET
cana-577	30	14	‘	'	PUNCT
cana-577	30	15	minimal	minimal	ADJ
cana-577	30	16	cardinality	cardinality	NOUN
cana-577	30	17	of	of	ADP
cana-577	30	18	such	such	DET
cana-577	30	19	a	a	DET
cana-577	30	20	set	set	NOUN
cana-577	30	21	’	'	PUNCT
cana-577	30	22	.	.	PUNCT
cana-577	31	1	definition-2.2	definition-2.2	NOUN
cana-577	31	2	:	:	PUNCT
cana-577	31	3	suppose	suppose	VERB
cana-577	31	4	𝐺	𝐺	PROPN
cana-577	31	5	is	be	AUX
cana-577	31	6	a	a	DET
cana-577	31	7	connected	connected	ADJ
cana-577	31	8	graph	graph	NOUN
cana-577	31	9	with	with	ADP
cana-577	31	10	an	an	DET
cana-577	31	11	edge	edge	NOUN
cana-577	31	12	𝑥𝑦𝜖𝐸(𝐺	𝑥𝑦𝜖𝐸(𝐺	NOUN
cana-577	31	13	)	)	PUNCT
cana-577	31	14	.	.	PUNCT
cana-577	32	1	then	then	ADV
cana-577	32	2	,	,	PUNCT
cana-577	32	3	‘	'	PUNCT
cana-577	32	4	𝑥	𝑥	PRON
cana-577	32	5	strongly	strongly	ADV
cana-577	32	6	dominates	dominate	VERB
cana-577	32	7	𝑦	𝑦	NOUN
cana-577	32	8	’	'	PUNCT
cana-577	32	9	(	(	PUNCT
cana-577	32	10	‘	'	PUNCT
cana-577	32	11	𝑦	𝑦	X
cana-577	32	12	weakly	weakly	ADV
cana-577	32	13	dominates	dominate	VERB
cana-577	32	14	𝑥	𝑥	NOUN
cana-577	32	15	’	'	PUNCT
cana-577	32	16	)	)	PUNCT
cana-577	32	17	if	if	SCONJ
cana-577	32	18	‘	'	PUNCT
cana-577	32	19	deg⁡(𝑥	deg⁡(𝑥	NOUN
cana-577	32	20	)	)	PUNCT
cana-577	32	21	≥	≥	NOUN
cana-577	32	22	deg⁡(𝑦	deg⁡(𝑦	NUM
cana-577	32	23	)	)	PUNCT
cana-577	32	24	’	'	PUNCT
cana-577	32	25	.	.	PUNCT
cana-577	33	1	it	it	PRON
cana-577	33	2	is	be	AUX
cana-577	33	3	clear	clear	ADJ
cana-577	33	4	that	that	SCONJ
cana-577	33	5	each	each	DET
cana-577	33	6	point	point	NOUN
cana-577	33	7	in	in	ADP
cana-577	33	8	𝑉(𝐺	𝑉(𝐺	NOUN
cana-577	33	9	)	)	PUNCT
cana-577	33	10	has	have	VERB
cana-577	33	11	the	the	DET
cana-577	33	12	ability	ability	NOUN
cana-577	33	13	to	to	PART
cana-577	33	14	strongly	strongly	ADV
cana-577	33	15	dominate	dominate	VERB
cana-577	33	16	itself	itself	PRON
cana-577	33	17	.	.	PUNCT
cana-577	34	1	a	a	DET
cana-577	34	2	set	set	VERB
cana-577	34	3	𝐷	𝐷	NOUN
cana-577	34	4	⊆	⊆	NUM
cana-577	34	5	𝑉(𝐺	𝑉(𝐺	NOUN
cana-577	34	6	)	)	PUNCT
cana-577	34	7	is	be	AUX
cana-577	34	8	a	a	DET
cana-577	34	9	‘	'	PUNCT
cana-577	34	10	strong	strong	ADJ
cana-577	34	11	dominating	dominating	NOUN
cana-577	34	12	set	set	NOUN
cana-577	34	13	’	'	PUNCT
cana-577	34	14	of	of	ADP
cana-577	34	15	g	g	PROPN
cana-577	34	16	,	,	PUNCT
cana-577	34	17	if	if	SCONJ
cana-577	34	18	for	for	ADP
cana-577	34	19	each	each	DET
cana-577	34	20	point	point	NOUN
cana-577	34	21	𝑦𝜖𝑉(𝐺	𝑦𝜖𝑉(𝐺	NOUN
cana-577	34	22	)	)	PUNCT
cana-577	34	23	−	−	PROPN
cana-577	34	24	𝐷	𝐷	PROPN
cana-577	34	25	there	there	PRON
cana-577	34	26	is	be	VERB
cana-577	34	27	another	another	DET
cana-577	34	28	point	point	NOUN
cana-577	34	29	𝑥𝜖𝐷	𝑥𝜖𝐷	NOUN
cana-577	34	30	such	such	ADJ
cana-577	34	31	that	that	SCONJ
cana-577	34	32	𝑥𝑦𝜖𝐸(𝐺	𝑥𝑦𝜖𝐸(𝐺	NOUN
cana-577	34	33	)	)	PUNCT
cana-577	34	34	and	and	CCONJ
cana-577	34	35	deg⁡(𝑥	deg⁡(𝑥	NOUN
cana-577	34	36	)	)	PUNCT
cana-577	34	37	≥	≥	NOUN
cana-577	34	38	deg⁡(𝑦	deg⁡(𝑦	NUM
cana-577	34	39	)	)	PUNCT
cana-577	34	40	.	.	PUNCT
cana-577	35	1	the	the	DET
cana-577	35	2	‘	'	PUNCT
cana-577	35	3	minimum	minimum	ADJ
cana-577	35	4	cardinality	cardinality	NOUN
cana-577	35	5	’	'	PUNCT
cana-577	35	6	of	of	ADP
cana-577	35	7	such	such	ADJ
cana-577	35	8	set	set	NOUN
cana-577	35	9	is	be	AUX
cana-577	35	10	known	know	VERB
cana-577	35	11	as	as	ADP
cana-577	35	12	‘	'	PUNCT
cana-577	35	13	strong	strong	ADJ
cana-577	35	14	domination	domination	NOUN
cana-577	35	15	number	number	NOUN
cana-577	35	16	’	'	PUNCT
cana-577	35	17	of	of	ADP
cana-577	35	18	g	g	PROPN
cana-577	35	19	and	and	CCONJ
cana-577	35	20	it	it	PRON
cana-577	35	21	is	be	AUX
cana-577	35	22	represented	represent	VERB
cana-577	35	23	by	by	ADP
cana-577	35	24	𝛾𝑠𝑡(𝐺	𝛾𝑠𝑡(𝐺	PROPN
cana-577	35	25	)	)	PUNCT
cana-577	35	26	.	.	PUNCT
cana-577	36	1	definition-2.3	definition-2.3	NOUN
cana-577	36	2	:	:	PUNCT
cana-577	36	3	a	a	DET
cana-577	36	4	‘	'	PUNCT
cana-577	36	5	graph	graph	NOUN
cana-577	36	6	’	'	PUNCT
cana-577	36	7	𝐺	𝐺	PROPN
cana-577	36	8	is	be	AUX
cana-577	36	9	called	call	VERB
cana-577	36	10	a	a	DET
cana-577	36	11	‘	'	PUNCT
cana-577	36	12	regular	regular	ADJ
cana-577	36	13	graph	graph	NOUN
cana-577	36	14	’	'	PUNCT
cana-577	36	15	if	if	SCONJ
cana-577	36	16	each	each	DET
cana-577	36	17	point	point	NOUN
cana-577	36	18	in	in	ADP
cana-577	36	19	it	it	PRON
cana-577	36	20	holds	hold	VERB
cana-577	36	21	the	the	DET
cana-577	36	22	equal	equal	ADJ
cana-577	36	23	degree	degree	NOUN
cana-577	36	24	.	.	PUNCT
cana-577	37	1	a	a	DET
cana-577	37	2	‘	'	PUNCT
cana-577	37	3	dominating	dominate	VERB
cana-577	37	4	set	set	VERB
cana-577	37	5	𝐷	𝐷	NOUN
cana-577	37	6	’	'	PUNCT
cana-577	37	7	is	be	AUX
cana-577	37	8	called	call	VERB
cana-577	37	9	a	a	DET
cana-577	37	10	‘	'	PUNCT
cana-577	37	11	regular	regular	ADJ
cana-577	37	12	dominating	dominating	NOUN
cana-577	37	13	set	set	NOUN
cana-577	37	14	’	'	PUNCT
cana-577	37	15	of	of	ADP
cana-577	37	16	𝐺	𝐺	PROPN
cana-577	37	17	if	if	SCONJ
cana-577	37	18	〈	〈	PROPN
cana-577	37	19	𝐷	𝐷	NOUN
cana-577	37	20	〉	〉	NOUN
cana-577	37	21	is	be	AUX
cana-577	37	22	regular	regular	ADJ
cana-577	37	23	.	.	PUNCT
cana-577	38	1	the	the	DET
cana-577	38	2	‘	'	PUNCT
cana-577	38	3	regular	regular	ADJ
cana-577	38	4	domination	domination	NOUN
cana-577	38	5	number	number	NOUN
cana-577	38	6	’	'	PUNCT
cana-577	38	7	of	of	ADP
cana-577	38	8	𝐺	𝐺	PROPN
cana-577	38	9	,	,	PUNCT
cana-577	38	10	represented	represent	VERB
cana-577	38	11	by	by	ADP
cana-577	38	12	𝛾𝑟(𝐺	𝛾𝑟(𝐺	PROPN
cana-577	38	13	)	)	PUNCT
cana-577	38	14	,	,	PUNCT
cana-577	38	15	is	be	AUX
cana-577	38	16	the	the	DET
cana-577	38	17	‘	'	PUNCT
cana-577	38	18	minimal	minimal	ADJ
cana-577	38	19	cardinality	cardinality	NOUN
cana-577	38	20	’	'	PUNCT
cana-577	38	21	of	of	ADP
cana-577	38	22	such	such	DET
cana-577	38	23	a	a	DET
cana-577	38	24	set	set	NOUN
cana-577	38	25	.	.	PUNCT
cana-577	39	1	definition-2.4	definition-2.4	ADP
cana-577	39	2	:	:	PUNCT
cana-577	39	3	a	a	DET
cana-577	39	4	‘	'	PUNCT
cana-577	39	5	dominating	dominating	NOUN
cana-577	39	6	set	set	NOUN
cana-577	39	7	d	d	NOUN
cana-577	39	8	’	'	PUNCT
cana-577	39	9	of	of	ADP
cana-577	39	10	a	a	DET
cana-577	39	11	‘	'	PUNCT
cana-577	39	12	graph	graph	NOUN
cana-577	39	13	g	g	NOUN
cana-577	39	14	’	'	PUNCT
cana-577	39	15	is	be	AUX
cana-577	39	16	known	know	VERB
cana-577	39	17	as	as	ADP
cana-577	39	18	‘	'	PUNCT
cana-577	39	19	strong	strong	ADJ
cana-577	39	20	regular	regular	ADJ
cana-577	39	21	dominating	dominating	NOUN
cana-577	39	22	set	set	NOUN
cana-577	39	23	’	'	PUNCT
cana-577	39	24	of	of	ADP
cana-577	39	25	g	g	PROPN
cana-577	39	26	if	if	SCONJ
cana-577	39	27	(	(	PUNCT
cana-577	39	28	i	i	NOUN
cana-577	39	29	)	)	PUNCT
cana-577	39	30	for	for	ADP
cana-577	39	31	each	each	DET
cana-577	39	32	point	point	NOUN
cana-577	39	33	𝑦𝜖𝑉(𝐺	𝑦𝜖𝑉(𝐺	NOUN
cana-577	39	34	)	)	PUNCT
cana-577	39	35	−	−	PROPN
cana-577	39	36	𝐷	𝐷	PROPN
cana-577	39	37	there	there	PRON
cana-577	39	38	is	be	VERB
cana-577	39	39	another	another	DET
cana-577	39	40	point	point	NOUN
cana-577	39	41	𝑥𝜖𝐷	𝑥𝜖𝐷	NOUN
cana-577	40	1	such	such	ADJ
cana-577	40	2	that	that	SCONJ
cana-577	40	3	𝑥𝑦𝜖𝐸(𝐺	𝑥𝑦𝜖𝐸(𝐺	NOUN
cana-577	40	4	)	)	PUNCT
cana-577	40	5	and	and	CCONJ
cana-577	40	6	deg⁡(𝑥	deg⁡(𝑥	NOUN
cana-577	40	7	)	)	PUNCT
cana-577	40	8	≥	≥	NOUN
cana-577	40	9	deg⁡(𝑦	deg⁡(𝑦	NUM
cana-577	40	10	)	)	PUNCT
cana-577	40	11	and	and	CCONJ
cana-577	40	12	(	(	PUNCT
cana-577	40	13	ii	ii	NOUN
cana-577	40	14	)	)	PUNCT
cana-577	40	15	an	an	DET
cana-577	40	16	induced	induced	ADJ
cana-577	40	17	subgraph	subgraph	NOUN
cana-577	40	18	〈	〈	PROPN
cana-577	40	19	𝐷	𝐷	NOUN
cana-577	40	20	〉	〉	NOUN
cana-577	40	21	is	be	AUX
cana-577	40	22	a	a	DET
cana-577	40	23	‘	'	PUNCT
cana-577	40	24	regular	regular	ADJ
cana-577	40	25	dominating	dominating	NOUN
cana-577	40	26	set	set	NOUN
cana-577	40	27	of	of	ADP
cana-577	40	28	g	g	NOUN
cana-577	40	29	’	'	PUNCT
cana-577	40	30	.	.	PUNCT
cana-577	41	1	the	the	DET
cana-577	41	2	smallest	small	ADJ
cana-577	41	3	cardinality	cardinality	NOUN
cana-577	41	4	of	of	ADP
cana-577	41	5	such	such	DET
cana-577	41	6	a	a	DET
cana-577	41	7	set	set	NOUN
cana-577	41	8	is	be	AUX
cana-577	41	9	known	know	VERB
cana-577	41	10	as	as	ADP
cana-577	41	11	‘	'	PUNCT
cana-577	41	12	strong	strong	ADJ
cana-577	41	13	regular	regular	ADJ
cana-577	41	14	domination	domination	NOUN
cana-577	41	15	number	number	NOUN
cana-577	41	16	’	'	PUNCT
cana-577	41	17	of	of	ADP
cana-577	41	18	g	g	PROPN
cana-577	41	19	and	and	CCONJ
cana-577	41	20	which	which	PRON
cana-577	41	21	is	be	AUX
cana-577	41	22	represented	represent	VERB
cana-577	41	23	by	by	ADP
cana-577	41	24	𝛾𝑠𝑡𝑟(𝐺	𝛾𝑠𝑡𝑟(𝐺	PROPN
cana-577	41	25	)	)	PUNCT
cana-577	41	26	.	.	PUNCT
cana-577	42	1	definition-2.5	definition-2.5	PROPN
cana-577	42	2	:	:	PUNCT
cana-577	42	3	the	the	DET
cana-577	42	4	point	point	NOUN
cana-577	42	5	set	set	NOUN
cana-577	42	6	of	of	ADP
cana-577	42	7	a	a	DET
cana-577	42	8	lit	light	VERB
cana-577	42	9	act	act	NOUN
cana-577	42	10	graph	graph	NOUN
cana-577	42	11	𝑚(𝐺	𝑚(𝐺	PROPN
cana-577	42	12	)	)	PUNCT
cana-577	42	13	is	be	AUX
cana-577	42	14	made	make	VERB
cana-577	42	15	up	up	ADP
cana-577	42	16	of	of	ADP
cana-577	42	17	the	the	DET
cana-577	42	18	edges	edge	NOUN
cana-577	42	19	and	and	CCONJ
cana-577	42	20	cut	cut	VERB
cana-577	42	21	vertices	vertex	NOUN
cana-577	42	22	of	of	ADP
cana-577	42	23	a	a	DET
cana-577	42	24	graph	graph	NOUN
cana-577	42	25	𝐺.	𝐺.	NOUN
cana-577	42	26	if	if	SCONJ
cana-577	42	27	edges	edge	NOUN
cana-577	42	28	and	and	CCONJ
cana-577	42	29	cut	cut	NOUN
cana-577	42	30	vertices	vertex	NOUN
cana-577	42	31	are	be	AUX
cana-577	42	32	incident	incident	NOUN
cana-577	42	33	or	or	CCONJ
cana-577	42	34	adjacent	adjacent	ADJ
cana-577	42	35	in	in	ADP
cana-577	42	36	𝐺	𝐺	PROPN
cana-577	42	37	,	,	PUNCT
cana-577	42	38	then	then	ADV
cana-577	42	39	the	the	DET
cana-577	42	40	two	two	NUM
cana-577	42	41	vertices	vertex	NOUN
cana-577	42	42	in	in	ADP
cana-577	42	43	𝑚(𝐺	𝑚(𝐺	NUM
cana-577	42	44	)	)	PUNCT
cana-577	42	45	are	be	AUX
cana-577	42	46	adjacent	adjacent	ADJ
cana-577	42	47	.	.	PUNCT
cana-577	43	1	example	example	NOUN
cana-577	43	2	.	.	PUNCT
cana-577	44	1	figure	figure	NOUN
cana-577	44	2	1	1	NUM
cana-577	44	3	below	below	ADV
cana-577	44	4	shows	show	VERB
cana-577	44	5	a	a	DET
cana-577	44	6	graph	graph	NOUN
cana-577	44	7	𝐺	𝐺	NOUN
cana-577	44	8	alongwith	alongwith	NOUN
cana-577	44	9	it	it	PRON
cana-577	44	10	’s	’	VERB
cana-577	44	11	litact	litact	ADJ
cana-577	44	12	graph	graph	NOUN
cana-577	44	13	𝑚(𝐺	𝑚(𝐺	PROPN
cana-577	44	14	)	)	PUNCT
cana-577	44	15	.	.	PUNCT
cana-577	45	1	fig	fig	NOUN
cana-577	45	2	.	.	PUNCT
cana-577	46	1	1	1	NUM
cana-577	46	2	:	:	PUNCT
cana-577	46	3	a	a	DET
cana-577	46	4	graph	graph	NOUN
cana-577	46	5	𝐺	𝐺	NOUN
cana-577	46	6	and	and	CCONJ
cana-577	46	7	it	it	PRON
cana-577	46	8	’s	’	VERB
cana-577	46	9	litact	litact	ADJ
cana-577	46	10	graph	graph	NOUN
cana-577	46	11	𝑚(𝐺	𝑚(𝐺	PROPN
cana-577	46	12	)	)	PUNCT
cana-577	46	13	with	with	ADP
cana-577	46	14	𝛾𝑟(𝑚(𝐺	𝛾𝑟(𝑚(𝐺	PROPN
cana-577	46	15	)	)	PUNCT
cana-577	46	16	)	)	PUNCT
cana-577	47	1	=	=	SYM
cana-577	47	2	𝛾𝑠𝑡(𝑚(𝐺	𝛾𝑠𝑡(𝑚(𝐺	PROPN
cana-577	47	3	)	)	PUNCT
cana-577	47	4	)	)	PUNCT
cana-577	48	1	=	=	SYM
cana-577	48	2	𝛾𝑠𝑡𝑟(𝑚(𝐺	𝛾𝑠𝑡𝑟(𝑚(𝐺	X
cana-577	48	3	)	)	PUNCT
cana-577	48	4	)	)	PUNCT
cana-577	49	1	=	=	SYM
cana-577	49	2	2	2	NUM
cana-577	49	3	3	3	NUM
cana-577	49	4	.	.	PUNCT
cana-577	49	5	main	main	ADJ
cana-577	49	6	results	result	NOUN
cana-577	49	7	the	the	DET
cana-577	49	8	current	current	ADJ
cana-577	49	9	study	study	NOUN
cana-577	49	10	exclusively	exclusively	ADV
cana-577	49	11	considers	consider	VERB
cana-577	49	12	simple	simple	ADJ
cana-577	49	13	,	,	PUNCT
cana-577	49	14	finite	finite	ADJ
cana-577	49	15	,	,	PUNCT
cana-577	49	16	non	non	ADJ
cana-577	49	17	-	-	ADJ
cana-577	49	18	trivial	trivial	ADJ
cana-577	49	19	,	,	PUNCT
cana-577	49	20	connected	connected	ADJ
cana-577	49	21	and	and	CCONJ
cana-577	49	22	undirected	undirected	ADJ
cana-577	49	23	graphs	graph	NOUN
cana-577	49	24	.	.	PUNCT
cana-577	50	1	all	all	DET
cana-577	50	2	the	the	DET
cana-577	50	3	following	follow	VERB
cana-577	50	4	results	result	NOUN
cana-577	50	5	are	be	AUX
cana-577	50	6	hold	hold	ADJ
cana-577	50	7	for	for	ADP
cana-577	50	8	any	any	DET
cana-577	50	9	graph	graph	NOUN
cana-577	50	10	on	on	ADP
cana-577	50	11	which	which	PRON
cana-577	50	12	strong	strong	ADJ
cana-577	50	13	regular	regular	ADJ
cana-577	50	14	domination	domination	NOUN
cana-577	50	15	is	be	AUX
cana-577	50	16	found	find	VERB
cana-577	50	17	.	.	PUNCT
cana-577	51	1	𝐺	𝐺	PROPN
cana-577	51	2	𝑚(𝐺	𝑚(𝐺	NUM
cana-577	51	3	)	)	PUNCT
cana-577	51	4	communications	communication	NOUN
cana-577	51	5	on	on	ADP
cana-577	51	6	applied	apply	VERB
cana-577	51	7	nonlinear	nonlinear	ADJ
cana-577	51	8	analysis	analysis	NOUN
cana-577	51	9	issn	issn	NOUN
cana-577	51	10	:	:	PUNCT
cana-577	51	11	1074	1074	NUM
cana-577	51	12	-	-	PUNCT
cana-577	51	13	133x	133x	NUM
cana-577	51	14	vol	vol	NOUN
cana-577	51	15	31	31	NUM
cana-577	51	16	no	no	NOUN
cana-577	51	17	.	.	PUNCT
cana-577	52	1	1s	1s	NUM
cana-577	52	2	(	(	PUNCT
cana-577	52	3	2024	2024	NUM
cana-577	52	4	)	)	PUNCT
cana-577	52	5	182	182	NUM
cana-577	52	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-577	53	1	the	the	DET
cana-577	53	2	following	follow	VERB
cana-577	53	3	section	section	NOUN
cana-577	53	4	evaluates	evaluate	VERB
cana-577	53	5	a	a	DET
cana-577	53	6	graph	graph	NOUN
cana-577	53	7	g	g	NOUN
cana-577	53	8	’s	’s	PART
cana-577	54	1	and	and	CCONJ
cana-577	54	2	it	it	PRON
cana-577	54	3	’s	’	VERB
cana-577	54	4	litact	litact	ADJ
cana-577	54	5	graph	graph	NOUN
cana-577	54	6	𝑚(𝐺	𝑚(𝐺	NUM
cana-577	54	7	)	)	PUNCT
cana-577	54	8	’s	’	VERB
cana-577	54	9	strong	strong	ADJ
cana-577	54	10	regular	regular	ADJ
cana-577	54	11	domination	domination	NOUN
cana-577	54	12	number	number	NOUN
cana-577	54	13	for	for	ADP
cana-577	54	14	few	few	ADJ
cana-577	54	15	standard	standard	ADJ
cana-577	54	16	graphs	graph	NOUN
cana-577	54	17	,	,	PUNCT
cana-577	54	18	such	such	ADJ
cana-577	54	19	as	as	ADP
cana-577	54	20	a	a	DET
cana-577	54	21	‘	'	PUNCT
cana-577	54	22	cycle	cycle	NOUN
cana-577	54	23	graph	graph	NOUN
cana-577	54	24	’	'	PUNCT
cana-577	54	25	,	,	PUNCT
cana-577	54	26	‘	'	PUNCT
cana-577	54	27	star	star	NOUN
cana-577	54	28	graph	graph	NOUN
cana-577	54	29	’	'	PUNCT
cana-577	54	30	,	,	PUNCT
cana-577	54	31	‘	'	PUNCT
cana-577	54	32	wheel	wheel	NOUN
cana-577	54	33	graph	graph	NOUN
cana-577	54	34	’	'	PUNCT
cana-577	54	35	,	,	PUNCT
cana-577	54	36	‘	'	PUNCT
cana-577	54	37	complete	complete	ADJ
cana-577	54	38	graph	graph	NOUN
cana-577	54	39	’	'	PUNCT
cana-577	54	40	and	and	CCONJ
cana-577	54	41	so	so	ADV
cana-577	54	42	on	on	ADV
cana-577	54	43	.	.	PUNCT
cana-577	55	1	theorem-3.1	theorem-3.1	ADJ
cana-577	55	2	:	:	PUNCT
cana-577	55	3	for	for	ADP
cana-577	55	4	any	any	DET
cana-577	55	5	‘	'	PUNCT
cana-577	55	6	cycle	cycle	NOUN
cana-577	55	7	graph	graph	NOUN
cana-577	55	8	’	'	PUNCT
cana-577	55	9	𝐶𝑛⁡	𝐶𝑛⁡	PROPN
cana-577	55	10	,	,	PUNCT
cana-577	55	11	𝛾𝑠𝑡𝑟(𝐶𝑛⁡	𝛾𝑠𝑡𝑟(𝐶𝑛⁡	NOUN
cana-577	55	12	)	)	PUNCT
cana-577	56	1	=	=	SYM
cana-577	56	2	𝛾𝑠𝑡𝑟(𝑚(𝐶𝑛⁡	𝛾𝑠𝑡𝑟(𝑚(𝐶𝑛⁡	PROPN
cana-577	56	3	)	)	PUNCT
cana-577	56	4	)	)	PUNCT
cana-577	56	5	.	.	PUNCT
cana-577	57	1	theorem-3.2	theorem-3.2	NOUN
cana-577	57	2	:	:	PUNCT
cana-577	57	3	for	for	ADP
cana-577	57	4	any	any	DET
cana-577	57	5	‘	'	PUNCT
cana-577	57	6	wheel	wheel	NOUN
cana-577	57	7	graph	graph	NOUN
cana-577	57	8	’	'	PUNCT
cana-577	57	9	𝑊𝑛	𝑊𝑛	PROPN
cana-577	57	10	,	,	PUNCT
cana-577	57	11	⁡𝛾𝑠𝑡𝑟(𝑊𝑛	⁡𝛾𝑠𝑡𝑟(𝑊𝑛	PROPN
cana-577	57	12	)	)	PUNCT
cana-577	57	13	=	=	SYM
cana-577	57	14	1	1	X
cana-577	57	15	.	.	PUNCT
cana-577	58	1	theorem-3.3	theorem-3.3	NOUN
cana-577	58	2	:	:	PUNCT
cana-577	58	3	for	for	SCONJ
cana-577	58	4	any	any	DET
cana-577	58	5	‘	'	PUNCT
cana-577	58	6	complete	complete	ADJ
cana-577	58	7	graph	graph	NOUN
cana-577	58	8	’	'	PUNCT
cana-577	58	9	𝐾𝑛	𝐾𝑛	PROPN
cana-577	58	10	,	,	PUNCT
cana-577	58	11	⁡𝛾𝑠𝑡𝑟(𝐾𝑛	⁡𝛾𝑠𝑡𝑟(𝐾𝑛	VERB
cana-577	58	12	)	)	PUNCT
cana-577	58	13	=	=	SYM
cana-577	59	1	1	1	X
cana-577	59	2	.	.	X
cana-577	60	1	theorem-3.4	theorem-3.4	NOUN
cana-577	60	2	:	:	PUNCT
cana-577	60	3	for	for	ADP
cana-577	60	4	any	any	DET
cana-577	60	5	‘	'	PUNCT
cana-577	60	6	complete	complete	ADJ
cana-577	60	7	bipartite	bipartite	NOUN
cana-577	60	8	graph	graph	NOUN
cana-577	60	9	’	'	PUNCT
cana-577	60	10	𝐾𝑚,𝑛	𝐾𝑚,𝑛	PROPN
cana-577	60	11	,	,	PUNCT
cana-577	60	12	⁡𝛾𝑠𝑡𝑟(𝐾𝑚,𝑛	⁡𝛾𝑠𝑡𝑟(𝐾𝑚,𝑛	PUNCT
cana-577	60	13	)	)	PUNCT
cana-577	60	14	=	=	PUNCT
cana-577	60	15	𝑚	𝑚	X
cana-577	60	16	where	where	SCONJ
cana-577	60	17	𝑚	𝑚	ADP
cana-577	60	18	≤	≤	NUM
cana-577	60	19	𝑛.	𝑛.	NOUN
cana-577	60	20	theorem-3.5	theorem-3.5	PROPN
cana-577	60	21	:	:	PUNCT
cana-577	60	22	for	for	ADP
cana-577	60	23	any	any	DET
cana-577	60	24	‘	'	PUNCT
cana-577	60	25	star	star	NOUN
cana-577	60	26	graph	graph	NOUN
cana-577	60	27	’	'	PUNCT
cana-577	60	28	𝐾1,𝑛	𝐾1,𝑛	PROPN
cana-577	60	29	,	,	PUNCT
cana-577	60	30	⁡𝛾𝑠𝑡𝑟(𝐾1,𝑛	⁡𝛾𝑠𝑡𝑟(𝐾1,𝑛	NOUN
cana-577	60	31	)	)	PUNCT
cana-577	60	32	=	=	SYM
cana-577	60	33	⁡𝛾𝑠𝑡𝑟(𝑚(𝐾1,𝑛	⁡𝛾𝑠𝑡𝑟(𝑚(𝐾1,𝑛	NUM
cana-577	60	34	)	)	PUNCT
cana-577	60	35	)	)	PUNCT
cana-577	61	1	=	=	PUNCT
cana-577	61	2	1	1	X
cana-577	61	3	.	.	PUNCT
cana-577	61	4	an	an	DET
cana-577	61	5	upper	upper	ADJ
cana-577	61	6	bound	bind	VERB
cana-577	61	7	for	for	ADP
cana-577	61	8	𝛾𝑠𝑡𝑟(𝑚(𝐺	𝛾𝑠𝑡𝑟(𝑚(𝐺	NOUN
cana-577	61	9	)	)	PUNCT
cana-577	61	10	)	)	PUNCT
cana-577	61	11	in	in	ADP
cana-577	61	12	terms	term	NOUN
cana-577	61	13	of	of	ADP
cana-577	61	14	order	order	NOUN
cana-577	61	15	of	of	ADP
cana-577	61	16	g	g	PROPN
cana-577	61	17	has	have	AUX
cana-577	61	18	been	be	AUX
cana-577	61	19	determined	determine	VERB
cana-577	61	20	in	in	ADP
cana-577	61	21	the	the	DET
cana-577	61	22	following	following	NOUN
cana-577	61	23	theorem	theorem	VERB
cana-577	61	24	.	.	PUNCT
cana-577	62	1	theorem-3.6	theorem-3.6	X
cana-577	62	2	:	:	PUNCT
cana-577	62	3	if	if	SCONJ
cana-577	62	4	g	g	PROPN
cana-577	62	5	is	be	AUX
cana-577	62	6	any	any	DET
cana-577	62	7	‘	'	PUNCT
cana-577	62	8	graph	graph	NOUN
cana-577	62	9	g	g	NOUN
cana-577	62	10	’	'	PUNCT
cana-577	62	11	,	,	PUNCT
cana-577	62	12	then	then	ADV
cana-577	62	13	𝛾𝑠𝑡𝑟(𝑚(𝐺	𝛾𝑠𝑡𝑟(𝑚(𝐺	NUM
cana-577	62	14	)	)	PUNCT
cana-577	62	15	)	)	PUNCT
cana-577	62	16	<	<	X
cana-577	62	17	𝑝.	𝑝.	PROPN
cana-577	62	18	proof	proof	NOUN
cana-577	62	19	:	:	PUNCT
cana-577	62	20	suppose	suppose	VERB
cana-577	62	21	v	v	PART
cana-577	62	22	represent	represent	VERB
cana-577	62	23	g	g	PROPN
cana-577	62	24	's	's	PART
cana-577	62	25	vertex	vertex	NOUN
cana-577	62	26	set	set	NOUN
cana-577	62	27	,	,	PUNCT
cana-577	62	28	e	e	PRON
cana-577	62	29	represent	represent	VERB
cana-577	62	30	g	g	PROPN
cana-577	62	31	’s	’s	PART
cana-577	62	32	edge	edge	NOUN
cana-577	62	33	set	set	VERB
cana-577	62	34	and	and	CCONJ
cana-577	62	35	c	c	AUX
cana-577	62	36	represent	represent	VERB
cana-577	62	37	g	g	PROPN
cana-577	62	38	’s	’s	PART
cana-577	62	39	set	set	NOUN
cana-577	62	40	of	of	ADP
cana-577	62	41	cut	cut	NOUN
cana-577	62	42	vertices	vertex	NOUN
cana-577	62	43	.	.	PUNCT
cana-577	63	1	as	as	SCONJ
cana-577	63	2	stated	state	VERB
cana-577	63	3	in	in	ADP
cana-577	63	4	the	the	DET
cana-577	63	5	definition	definition	NOUN
cana-577	63	6	of	of	ADP
cana-577	63	7	a	a	DET
cana-577	63	8	litact	litact	NOUN
cana-577	63	9	graph	graph	NOUN
cana-577	63	10	𝑚(𝐺	𝑚(𝐺	NUM
cana-577	63	11	)	)	PUNCT
cana-577	63	12	,	,	PUNCT
cana-577	63	13	"	"	PUNCT
cana-577	63	14	𝑉(𝑚(𝐺	𝑉(𝑚(𝐺	NOUN
cana-577	63	15	)	)	PUNCT
cana-577	63	16	)	)	PUNCT
cana-577	64	1	=	=	SYM
cana-577	64	2	𝐸(𝐺	𝐸(𝐺	X
cana-577	64	3	)	)	PUNCT
cana-577	64	4	∪	∪	ADP
cana-577	64	5	𝐶(𝐺	𝐶(𝐺	PROPN
cana-577	64	6	)	)	PUNCT
cana-577	64	7	"	"	PUNCT
cana-577	64	8	.	.	PUNCT
cana-577	65	1	if	if	SCONJ
cana-577	65	2	𝑋	𝑋	PROPN
cana-577	65	3	⊆	⊆	NUM
cana-577	65	4	𝑉(𝑚(𝐺	𝑉(𝑚(𝐺	NOUN
cana-577	65	5	)	)	PUNCT
cana-577	65	6	)	)	PUNCT
cana-577	65	7	is	be	AUX
cana-577	65	8	the	the	DET
cana-577	65	9	smallest	small	ADJ
cana-577	65	10	dominating	dominating	NOUN
cana-577	65	11	set	set	NOUN
cana-577	65	12	of	of	ADP
cana-577	65	13	𝑚(𝐺	𝑚(𝐺	PROPN
cana-577	65	14	)	)	PUNCT
cana-577	65	15	and	and	CCONJ
cana-577	65	16	deg⁡(𝑥𝑖	deg⁡(𝑥𝑖	VERB
cana-577	65	17	)	)	PUNCT
cana-577	65	18	≤	≤	NOUN
cana-577	65	19	deg⁡(𝑥𝑗	deg⁡(𝑥𝑗	PROPN
cana-577	65	20	)	)	PUNCT
cana-577	66	1	where	where	SCONJ
cana-577	66	2	𝑥𝑖𝜖𝑉(𝑚(𝐺	𝑥𝑖𝜖𝑉(𝑚(𝐺	NUM
cana-577	66	3	)	)	PUNCT
cana-577	66	4	)	)	PUNCT
cana-577	67	1	−	−	PROPN
cana-577	67	2	𝑋	𝑋	PROPN
cana-577	67	3	and	and	CCONJ
cana-577	67	4	x	x	NOUN
cana-577	67	5	is	be	AUX
cana-577	67	6	k	k	NOUN
cana-577	67	7	-	-	ADJ
cana-577	67	8	regular	regular	ADJ
cana-577	67	9	then	then	ADV
cana-577	67	10	x	x	PRON
cana-577	67	11	itself	itself	PRON
cana-577	67	12	forms	form	VERB
cana-577	67	13	a	a	DET
cana-577	67	14	strong	strong	ADJ
cana-577	67	15	regular	regular	ADJ
cana-577	67	16	dominating	dominating	NOUN
cana-577	67	17	set	set	NOUN
cana-577	67	18	.	.	PUNCT
cana-577	68	1	then	then	ADV
cana-577	68	2	𝛾𝑠𝑡𝑟(𝑚(𝐺	𝛾𝑠𝑡𝑟(𝑚(𝐺	NUM
cana-577	68	3	)	)	PUNCT
cana-577	68	4	)	)	PUNCT
cana-577	69	1	=	=	PUNCT
cana-577	69	2	|𝑋|	|𝑋|	X
cana-577	69	3	.	.	PUNCT
cana-577	70	1	otherwise	otherwise	ADV
cana-577	70	2	,	,	PUNCT
cana-577	70	3	there	there	PRON
cana-577	70	4	exists	exist	VERB
cana-577	70	5	𝑥𝑖	𝑥𝑖	ADP
cana-577	70	6	∈	∈	PROPN
cana-577	70	7	𝑌	𝑌	PROPN
cana-577	70	8	⊆	⊆	NUM
cana-577	70	9	𝑉(𝑚(𝐺	𝑉(𝑚(𝐺	NOUN
cana-577	70	10	)	)	PUNCT
cana-577	70	11	)	)	PUNCT
cana-577	71	1	such	such	ADJ
cana-577	71	2	that	that	SCONJ
cana-577	71	3	〈	〈	NOUN
cana-577	71	4	𝑋	𝑋	PROPN
cana-577	71	5	∪	∪	ADP
cana-577	71	6	𝑌	𝑌	PROPN
cana-577	71	7	〉	〉	NOUN
cana-577	71	8	forms	form	VERB
cana-577	71	9	a	a	DET
cana-577	71	10	minimal	minimal	ADJ
cana-577	71	11	strong	strong	ADJ
cana-577	71	12	regular	regular	ADJ
cana-577	71	13	dominating	dominating	NOUN
cana-577	71	14	set	set	NOUN
cana-577	71	15	in	in	ADP
cana-577	71	16	𝑚(𝐺	𝑚(𝐺	NUM
cana-577	71	17	)	)	PUNCT
cana-577	71	18	.	.	PUNCT
cana-577	72	1	therefore	therefore	ADV
cana-577	72	2	,	,	PUNCT
cana-577	72	3	|𝑋	|𝑋	PROPN
cana-577	72	4	∪	∪	ADP
cana-577	72	5	𝑌|	𝑌|	PROPN
cana-577	72	6	=	=	SYM
cana-577	72	7	𝛾𝑠𝑡𝑟(𝑚(𝐺	𝛾𝑠𝑡𝑟(𝑚(𝐺	PROPN
cana-577	72	8	)	)	PUNCT
cana-577	72	9	)	)	PUNCT
cana-577	72	10	.	.	PUNCT
cana-577	73	1	then	then	ADV
cana-577	73	2	,	,	PUNCT
cana-577	73	3	𝑉(𝑚(𝐺	𝑉(𝑚(𝐺	PROPN
cana-577	73	4	)	)	PUNCT
cana-577	73	5	)	)	PUNCT
cana-577	74	1	<	<	X
cana-577	74	2	2𝑝.	2𝑝.	NUM
cana-577	74	3	also	also	ADV
cana-577	74	4	,	,	PUNCT
cana-577	74	5	|𝑋	|𝑋	PROPN
cana-577	74	6	∪	∪	ADP
cana-577	74	7	𝑌|	𝑌|	PROPN
cana-577	74	8	≤	≤	NUM
cana-577	74	9	𝑉(𝑚(𝐺	𝑉(𝑚(𝐺	PROPN
cana-577	74	10	)	)	PUNCT
cana-577	74	11	)	)	PUNCT
cana-577	75	1	−	−	PROPN
cana-577	76	1	|𝑋	|𝑋	PROPN
cana-577	76	2	∪	∪	ADP
cana-577	76	3	𝑌|	𝑌|	PROPN
cana-577	76	4	⟹	⟹	X
cana-577	76	5	|𝑋	|𝑋	NUM
cana-577	76	6	∪	∪	ADP
cana-577	76	7	𝑌|	𝑌|	PROPN
cana-577	76	8	+	+	CCONJ
cana-577	76	9	|𝑋	|𝑋	PROPN
cana-577	76	10	∪	∪	ADP
cana-577	76	11	𝑌|	𝑌|	PROPN
cana-577	76	12	≤	≤	NUM
cana-577	76	13	𝑉(𝑚(𝐺	𝑉(𝑚(𝐺	PROPN
cana-577	76	14	)	)	PUNCT
cana-577	76	15	)	)	PUNCT
cana-577	76	16	⟹	⟹	NUM
cana-577	77	1	2|𝑋	2|𝑋	NUM
cana-577	77	2	∪	∪	ADP
cana-577	77	3	𝑌|	𝑌|	PROPN
cana-577	77	4	≤	≤	NUM
cana-577	77	5	𝑉(𝑚(𝐺	𝑉(𝑚(𝐺	PROPN
cana-577	77	6	)	)	PUNCT
cana-577	77	7	)	)	PUNCT
cana-577	78	1	<	<	X
cana-577	78	2	2𝑝	2𝑝	NUM
cana-577	78	3	⟹	⟹	NUM
cana-577	78	4	2𝛾𝑠𝑡𝑟(𝑚(𝐺	2𝛾𝑠𝑡𝑟(𝑚(𝐺	NUM
cana-577	78	5	)	)	PUNCT
cana-577	78	6	)	)	PUNCT
cana-577	79	1	<	<	X
cana-577	79	2	2𝑝	2𝑝	NUM
cana-577	79	3	⟹	⟹	NUM
cana-577	79	4	𝛾𝑠𝑡𝑟(𝑚(𝐺	𝛾𝑠𝑡𝑟(𝑚(𝐺	PROPN
cana-577	79	5	)	)	PUNCT
cana-577	79	6	)	)	PUNCT
cana-577	80	1	<	<	X
cana-577	80	2	𝑝	𝑝	ADP
cana-577	80	3	the	the	DET
cana-577	80	4	relationship	relationship	NOUN
cana-577	80	5	between	between	ADP
cana-577	80	6	𝛾𝑠𝑡𝑟(𝑚(𝐺	𝛾𝑠𝑡𝑟(𝑚(𝐺	PROPN
cana-577	80	7	)	)	PUNCT
cana-577	80	8	)	)	PUNCT
cana-577	80	9	,	,	PUNCT
cana-577	80	10	⁡𝛾𝑠𝑡𝑟(𝐺	⁡𝛾𝑠𝑡𝑟(𝐺	NOUN
cana-577	80	11	)	)	PUNCT
cana-577	80	12	,	,	PUNCT
cana-577	80	13	deg(𝑥	deg(𝑥	PROPN
cana-577	80	14	)	)	PUNCT
cana-577	80	15	and	and	CCONJ
cana-577	80	16	deg⁡(𝑦	deg⁡(𝑦	NUM
cana-577	80	17	)	)	PUNCT
cana-577	80	18	is	be	AUX
cana-577	80	19	given	give	VERB
cana-577	80	20	below	below	ADV
cana-577	80	21	.	.	PUNCT
cana-577	81	1	theorem-3.7	theorem-3.7	NOUN
cana-577	81	2	:	:	PUNCT
cana-577	81	3	for	for	ADP
cana-577	81	4	any	any	DET
cana-577	81	5	‘	'	PUNCT
cana-577	81	6	graph	graph	NOUN
cana-577	81	7	g	g	NOUN
cana-577	81	8	’	'	PUNCT
cana-577	81	9	,	,	PUNCT
cana-577	81	10	𝛾𝑠𝑡𝑟(𝑚(𝐺	𝛾𝑠𝑡𝑟(𝑚(𝐺	PROPN
cana-577	81	11	)	)	PUNCT
cana-577	81	12	)	)	PUNCT
cana-577	81	13	≤	≤	NUM
cana-577	81	14	𝛾𝑠𝑡𝑟(𝐺	𝛾𝑠𝑡𝑟(𝐺	VERB
cana-577	81	15	)	)	PUNCT
cana-577	82	1	+	+	CCONJ
cana-577	82	2	deg(𝑥	deg(𝑥	PROPN
cana-577	82	3	)	)	PUNCT
cana-577	82	4	+	+	NUM
cana-577	82	5	deg⁡(𝑦	deg⁡(𝑦	NOUN
cana-577	82	6	)	)	PUNCT
cana-577	82	7	where	where	SCONJ
cana-577	82	8	𝑥	𝑥	X
cana-577	82	9	,	,	PUNCT
cana-577	82	10	𝑦	𝑦	PRON
cana-577	82	11	are	be	AUX
cana-577	82	12	the	the	DET
cana-577	82	13	elements	element	NOUN
cana-577	82	14	of	of	ADP
cana-577	82	15	a	a	DET
cana-577	82	16	‘	'	PUNCT
cana-577	82	17	strong	strong	ADJ
cana-577	82	18	regular	regular	ADJ
cana-577	82	19	dominating	dominating	NOUN
cana-577	82	20	set	set	NOUN
cana-577	82	21	of	of	ADP
cana-577	82	22	𝑚(𝐺	𝑚(𝐺	PROPN
cana-577	82	23	)	)	PUNCT
cana-577	82	24	’	'	PUNCT
cana-577	82	25	.	.	PUNCT
cana-577	83	1	proof	proof	NOUN
cana-577	83	2	:	:	PUNCT
cana-577	83	3	let	let	VERB
cana-577	83	4	𝑚(𝐺	𝑚(𝐺	NUM
cana-577	83	5	)	)	PUNCT
cana-577	83	6	be	be	AUX
cana-577	83	7	the	the	DET
cana-577	83	8	litact	litact	ADJ
cana-577	83	9	graph	graph	NOUN
cana-577	83	10	of	of	ADP
cana-577	83	11	g	g	NOUN
cana-577	83	12	with	with	ADP
cana-577	83	13	v	v	NOUN
cana-577	83	14	points	point	NOUN
cana-577	83	15	and	and	CCONJ
cana-577	83	16	e	e	NOUN
cana-577	83	17	edges	edge	NOUN
cana-577	83	18	.	.	PUNCT
cana-577	84	1	suppose	suppose	VERB
cana-577	84	2	d	d	X
cana-577	84	3	is	be	AUX
cana-577	84	4	a	a	DET
cana-577	84	5	‘	'	PUNCT
cana-577	84	6	strong	strong	ADJ
cana-577	84	7	regular	regular	ADJ
cana-577	84	8	dominating	dominating	NOUN
cana-577	84	9	set	set	NOUN
cana-577	84	10	’	'	PUNCT
cana-577	84	11	of	of	ADP
cana-577	84	12	𝑚(𝐺	𝑚(𝐺	NUM
cana-577	84	13	)	)	PUNCT
cana-577	84	14	.	.	PUNCT
cana-577	85	1	assume	assume	VERB
cana-577	85	2	that	that	SCONJ
cana-577	85	3	the	the	DET
cana-577	85	4	points	point	NOUN
cana-577	85	5	𝑥	𝑥	NOUN
cana-577	85	6	,	,	PUNCT
cana-577	85	7	𝑦	𝑦	PROPN
cana-577	85	8	∈	∈	PROPN
cana-577	85	9	𝐷	𝐷	NOUN
cana-577	85	10	and	and	CCONJ
cana-577	85	11	x	x	PRON
cana-577	85	12	has	have	VERB
cana-577	85	13	the	the	DET
cana-577	85	14	same	same	ADJ
cana-577	85	15	degree	degree	NOUN
cana-577	85	16	as	as	ADP
cana-577	85	17	some	some	PRON
cana-577	85	18	of	of	ADP
cana-577	85	19	its	its	PRON
cana-577	85	20	neighbours	neighbour	NOUN
cana-577	85	21	(	(	PUNCT
cana-577	85	22	except	except	SCONJ
cana-577	85	23	y	y	PROPN
cana-577	85	24	)	)	PUNCT
cana-577	85	25	and	and	CCONJ
cana-577	85	26	strongly	strongly	ADV
cana-577	85	27	dominates	dominate	VERB
cana-577	85	28	them	they	PRON
cana-577	85	29	,	,	PUNCT
cana-577	85	30	and	and	CCONJ
cana-577	85	31	that	that	SCONJ
cana-577	85	32	y	y	PROPN
cana-577	85	33	does	do	VERB
cana-577	85	34	the	the	DET
cana-577	85	35	same	same	ADJ
cana-577	85	36	.	.	PUNCT
cana-577	86	1	let	let	VERB
cana-577	86	2	𝑣1	𝑣1	NOUN
cana-577	86	3	is	be	AUX
cana-577	86	4	adjacent	adjacent	ADJ
cana-577	86	5	to	to	ADP
cana-577	86	6	𝑥	𝑥	PRON
cana-577	86	7	,	,	PUNCT
cana-577	86	8	𝑣1	𝑣1	NOUN
cana-577	86	9	≠	≠	PROPN
cana-577	86	10	𝑦	𝑦	NOUN
cana-577	86	11	such	such	ADJ
cana-577	86	12	that	that	PRON
cana-577	86	13	deg(𝑥	deg(𝑥	PROPN
cana-577	86	14	)	)	PUNCT
cana-577	86	15	=	=	SYM
cana-577	86	16	deg⁡(𝑣1	deg⁡(𝑣1	NOUN
cana-577	86	17	)	)	PUNCT
cana-577	86	18	,	,	PUNCT
cana-577	86	19	and	and	CCONJ
cana-577	86	20	𝑣1	𝑣1	NOUN
cana-577	86	21	is	be	AUX
cana-577	86	22	strongly	strongly	ADV
cana-577	86	23	dominated	dominate	VERB
cana-577	86	24	only	only	ADV
cana-577	86	25	by	by	ADP
cana-577	86	26	x.	x.	PROPN
cana-577	86	27	hence	hence	ADV
cana-577	86	28	,	,	PUNCT
cana-577	86	29	‘	'	PUNCT
cana-577	86	30	d	d	X
cana-577	86	31	is	be	AUX
cana-577	86	32	a	a	DET
cana-577	86	33	strong	strong	ADJ
cana-577	86	34	regular	regular	ADJ
cana-577	86	35	dominating	dominating	NOUN
cana-577	86	36	set	set	NOUN
cana-577	86	37	’	'	PUNCT
cana-577	86	38	,	,	PUNCT
cana-577	86	39	so	so	SCONJ
cana-577	86	40	that	that	SCONJ
cana-577	86	41	|𝐷|	|𝐷|	NOUN
cana-577	86	42	=	=	SYM
cana-577	86	43	𝛾𝑠𝑡𝑟(𝑚(𝐺	𝛾𝑠𝑡𝑟(𝑚(𝐺	PROPN
cana-577	86	44	)	)	PUNCT
cana-577	86	45	)	)	PUNCT
cana-577	86	46	.	.	PUNCT
cana-577	87	1	let	let	VERB
cana-577	87	2	s	s	PRON
cana-577	87	3	be	be	AUX
cana-577	87	4	the	the	DET
cana-577	87	5	strong	strong	ADJ
cana-577	87	6	regular	regular	ADJ
cana-577	87	7	dominating	dominating	NOUN
cana-577	87	8	set	set	VERB
cana-577	87	9	in	in	ADP
cana-577	87	10	g	g	PROPN
cana-577	87	11	such	such	DET
cana-577	87	12	that	that	PRON
cana-577	87	13	|𝑆|	|𝑆|	VERB
cana-577	87	14	=	=	SYM
cana-577	87	15	𝛾𝑠𝑡𝑟(𝐺	𝛾𝑠𝑡𝑟(𝐺	PROPN
cana-577	87	16	)	)	PUNCT
cana-577	87	17	.	.	PUNCT
cana-577	88	1	from	from	ADP
cana-577	88	2	the	the	DET
cana-577	88	3	above	above	NOUN
cana-577	88	4	,	,	PUNCT
cana-577	88	5	it	it	PRON
cana-577	88	6	is	be	AUX
cana-577	88	7	easy	easy	ADJ
cana-577	88	8	to	to	PART
cana-577	88	9	verify	verify	VERB
cana-577	88	10	that	that	DET
cana-577	88	11	|𝐷|	|𝐷|	NOUN
cana-577	88	12	⊆	⊆	NUM
cana-577	88	13	|𝑆|	|𝑆|	VERB
cana-577	88	14	∪	∪	ADP
cana-577	88	15	|𝑁(𝑥)|	|𝑁(𝑥)|	PRON
cana-577	88	16	∪	∪	ADJ
cana-577	88	17	|𝑁(𝑦)|	|𝑁(𝑦)|	PROPN
cana-577	88	18	⟹	⟹	X
cana-577	88	19	𝛾𝑠𝑡𝑟(𝑚(𝐺	𝛾𝑠𝑡𝑟(𝑚(𝐺	PROPN
cana-577	88	20	)	)	PUNCT
cana-577	88	21	)	)	PUNCT
cana-577	88	22	≤	≤	NUM
cana-577	89	1	𝛾𝑠𝑡𝑟(𝐺	𝛾𝑠𝑡𝑟(𝐺	VERB
cana-577	89	2	)	)	PUNCT
cana-577	89	3	+	+	CCONJ
cana-577	89	4	deg(𝑥	deg(𝑥	PROPN
cana-577	89	5	)	)	PUNCT
cana-577	89	6	+	+	NUM
cana-577	90	1	deg⁡(𝑦	deg⁡(𝑦	NOUN
cana-577	90	2	)	)	PUNCT
cana-577	90	3	.	.	PUNCT
cana-577	91	1	the	the	DET
cana-577	91	2	relationship	relationship	NOUN
cana-577	91	3	between	between	ADP
cana-577	91	4	𝛾𝑠𝑡𝑟	𝛾𝑠𝑡𝑟	NOUN
cana-577	91	5	(	(	PUNCT
cana-577	91	6	𝑚(𝐻𝑝	𝑚(𝐻𝑝	PROPN
cana-577	91	7	)	)	PUNCT
cana-577	91	8	)	)	PUNCT
cana-577	91	9	,	,	PUNCT
cana-577	91	10	‘	'	PUNCT
cana-577	91	11	edges	edge	NOUN
cana-577	91	12	’	'	PUNCT
cana-577	91	13	and	and	CCONJ
cana-577	91	14	‘	'	PUNCT
cana-577	91	15	cut	cut	VERB
cana-577	91	16	vertices	vertex	NOUN
cana-577	91	17	’	'	PUNCT
cana-577	91	18	of	of	ADP
cana-577	91	19	the	the	DET
cana-577	91	20	‘	'	PUNCT
cana-577	91	21	helm	helm	ADJ
cana-577	91	22	graph	graph	NOUN
cana-577	91	23	’	'	PUNCT
cana-577	91	24	has	have	AUX
cana-577	91	25	been	be	AUX
cana-577	91	26	determined	determine	VERB
cana-577	91	27	in	in	ADP
cana-577	91	28	the	the	DET
cana-577	91	29	subsequent	subsequent	ADJ
cana-577	91	30	theorem	theorem	NOUN
cana-577	91	31	.	.	PUNCT
cana-577	92	1	communications	communication	NOUN
cana-577	92	2	on	on	ADP
cana-577	92	3	applied	apply	VERB
cana-577	92	4	nonlinear	nonlinear	ADJ
cana-577	92	5	analysis	analysis	NOUN
cana-577	92	6	issn	issn	NOUN
cana-577	92	7	:	:	PUNCT
cana-577	92	8	1074	1074	NUM
cana-577	92	9	-	-	PUNCT
cana-577	92	10	133x	133x	NUM
cana-577	92	11	vol	vol	NOUN
cana-577	92	12	31	31	NUM
cana-577	92	13	no	no	NOUN
cana-577	92	14	.	.	PUNCT
cana-577	93	1	1s	1s	NUM
cana-577	93	2	(	(	PUNCT
cana-577	93	3	2024	2024	NUM
cana-577	93	4	)	)	PUNCT
cana-577	93	5	183	183	NUM
cana-577	93	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-577	93	7	theorem-3.8	theorem-3.8	NUM
cana-577	93	8	:	:	PUNCT
cana-577	93	9	for	for	ADP
cana-577	93	10	any	any	DET
cana-577	93	11	(	(	PUNCT
cana-577	93	12	𝑝	𝑝	PROPN
cana-577	93	13	,	,	PUNCT
cana-577	93	14	𝑞	𝑞	NOUN
cana-577	93	15	)	)	PUNCT
cana-577	93	16	−helm	−helm	NOUN
cana-577	93	17	graph	graph	NOUN
cana-577	93	18	𝐻𝑝	𝐻𝑝	PROPN
cana-577	93	19	,	,	PUNCT
cana-577	93	20	𝛾𝑠𝑡𝑟	𝛾𝑠𝑡𝑟	NOUN
cana-577	93	21	(	(	PUNCT
cana-577	93	22	𝑚(𝐻𝑝	𝑚(𝐻𝑝	PROPN
cana-577	93	23	)	)	PUNCT
cana-577	93	24	)	)	PUNCT
cana-577	93	25	<	<	X
cana-577	94	1	𝑞	𝑞	X
cana-577	94	2	+	+	X
cana-577	94	3	𝑙	𝑙	X
cana-577	94	4	where	where	SCONJ
cana-577	94	5	𝑙	𝑙	PROPN
cana-577	94	6	is	be	AUX
cana-577	94	7	the	the	DET
cana-577	94	8	number	number	NOUN
cana-577	94	9	of	of	ADP
cana-577	94	10	cut	cut	VERB
cana-577	94	11	vertices	vertex	NOUN
cana-577	94	12	of	of	ADP
cana-577	94	13	𝐻𝑝.	𝐻𝑝.	PROPN
cana-577	94	14	proof	proof	NOUN
cana-577	94	15	:	:	PUNCT
cana-577	94	16	suppose	suppose	VERB
cana-577	94	17	𝐺(𝑉	𝐺(𝑉	NOUN
cana-577	94	18	,	,	PUNCT
cana-577	94	19	𝐸	𝐸	PROPN
cana-577	94	20	)	)	PUNCT
cana-577	94	21	be	be	VERB
cana-577	94	22	a	a	DET
cana-577	94	23	helm	helm	NOUN
cana-577	94	24	graph	graph	NOUN
cana-577	94	25	with	with	ADP
cana-577	94	26	𝑉	𝑉	PROPN
cana-577	94	27	=	=	PROPN
cana-577	94	28	{	{	PUNCT
cana-577	94	29	𝑣1	𝑣1	PROPN
cana-577	94	30	,	,	PUNCT
cana-577	94	31	𝑣2	𝑣2	PROPN
cana-577	94	32	,	,	PUNCT
cana-577	94	33	𝑣3	𝑣3	ADJ
cana-577	94	34	,	,	PUNCT
cana-577	94	35	…	…	PUNCT
cana-577	94	36	.	.	PUNCT
cana-577	95	1	,	,	PUNCT
cana-577	95	2	𝑣𝑝−1	𝑣𝑝−1	PROPN
cana-577	95	3	,	,	PUNCT
cana-577	95	4	𝑣1	𝑣1	NOUN
cana-577	95	5	′	′	NOUN
cana-577	95	6	,	,	PUNCT
cana-577	95	7	𝑣2	𝑣2	NOUN
cana-577	95	8	′	′	NUM
cana-577	95	9	,	,	PUNCT
cana-577	95	10	𝑣3	𝑣3	ADJ
cana-577	95	11	′	′	NUM
cana-577	95	12	,	,	PUNCT
cana-577	95	13	…	…	PUNCT
cana-577	95	14	.	.	PUNCT
cana-577	96	1	,	,	PUNCT
cana-577	96	2	𝑣𝑝−1	𝑣𝑝−1	PROPN
cana-577	96	3	′	′	NUM
cana-577	96	4	}	}	PUNCT
cana-577	96	5	vertices	vertex	NOUN
cana-577	96	6	and	and	CCONJ
cana-577	96	7	𝐸	𝐸	NOUN
cana-577	96	8	=	=	SYM
cana-577	96	9	{	{	PUNCT
cana-577	96	10	𝑣1𝑣𝑖	𝑣1𝑣𝑖	NOUN
cana-577	96	11	,	,	PUNCT
cana-577	96	12	2	2	NUM
cana-577	96	13	≤	≤	NOUN
cana-577	96	14	𝑖	𝑖	SYM
cana-577	96	15	≤	≤	PROPN
cana-577	96	16	𝑝	𝑝	NOUN
cana-577	96	17	}	}	PUNCT
cana-577	96	18	∪	∪	NOUN
cana-577	96	19	{	{	PUNCT
cana-577	96	20	𝑣𝑖𝑣𝑖+1	𝑣𝑖𝑣𝑖+1	PROPN
cana-577	96	21	,	,	PUNCT
cana-577	96	22	𝑣𝑖𝑣𝑖−1	𝑣𝑖𝑣𝑖−1	PRON
cana-577	96	23	′	′	NOUN
cana-577	96	24	,	,	PUNCT
cana-577	96	25	2	2	NUM
cana-577	96	26	≤	≤	NUM
cana-577	96	27	𝑖	𝑖	SYM
cana-577	96	28	≤	≤	NOUN
cana-577	96	29	𝑝	𝑝	ADP
cana-577	96	30	−	−	PROPN
cana-577	96	31	1	1	NUM
cana-577	96	32	}	}	PUNCT
cana-577	96	33	∪	∪	ADJ
cana-577	96	34	{	{	PUNCT
cana-577	96	35	𝑣2𝑣𝑝	𝑣2𝑣𝑝	NOUN
cana-577	96	36	}	}	PUNCT
cana-577	96	37	∪	∪	X
cana-577	96	38	{	{	PUNCT
cana-577	96	39	𝑣1𝑣𝑖	𝑣1𝑣𝑖	NUM
cana-577	96	40	′	′	NOUN
cana-577	96	41	,	,	PUNCT
cana-577	96	42	1	1	NUM
cana-577	96	43	≤	≤	NUM
cana-577	96	44	𝑖	𝑖	SYM
cana-577	96	45	≤	≤	PROPN
cana-577	96	46	𝑝	𝑝	ADP
cana-577	96	47	−	−	ADP
cana-577	96	48	1	1	NUM
cana-577	96	49	}	}	PUNCT
cana-577	96	50	edges	edge	NOUN
cana-577	96	51	.	.	PUNCT
cana-577	97	1	since	since	SCONJ
cana-577	97	2	the	the	DET
cana-577	97	3	helm	helm	NOUN
cana-577	97	4	graph	graph	NOUN
cana-577	97	5	is	be	AUX
cana-577	97	6	formed	form	VERB
cana-577	97	7	by	by	ADP
cana-577	97	8	attaching	attach	VERB
cana-577	97	9	one	one	NUM
cana-577	97	10	vertex	vertex	NOUN
cana-577	97	11	by	by	ADP
cana-577	97	12	an	an	DET
cana-577	97	13	edge	edge	NOUN
cana-577	97	14	to	to	ADP
cana-577	97	15	each	each	PRON
cana-577	97	16	of	of	ADP
cana-577	97	17	the	the	DET
cana-577	97	18	𝑝	𝑝	PROPN
cana-577	97	19	−	−	ADP
cana-577	97	20	1	1	NUM
cana-577	97	21	points	point	NOUN
cana-577	97	22	in	in	ADP
cana-577	97	23	the	the	DET
cana-577	97	24	wheel	wheel	NOUN
cana-577	97	25	graph	graph	NOUN
cana-577	97	26	's	's	PART
cana-577	97	27	‘	'	PUNCT
cana-577	97	28	outer	outer	ADJ
cana-577	97	29	circuit	circuit	NOUN
cana-577	97	30	’	'	PUNCT
cana-577	97	31	,	,	PUNCT
cana-577	97	32	the	the	DET
cana-577	97	33	vertex	vertex	NOUN
cana-577	97	34	𝑣1	𝑣1	NOUN
cana-577	97	35	comes	come	VERB
cana-577	97	36	after	after	ADP
cana-577	97	37	each	each	PRON
cana-577	97	38	of	of	ADP
cana-577	97	39	the	the	DET
cana-577	97	40	𝑝	𝑝	PROPN
cana-577	97	41	−	−	ADP
cana-577	97	42	1	1	NUM
cana-577	97	43	vertices	vertex	NOUN
cana-577	97	44	in	in	ADP
cana-577	97	45	the	the	DET
cana-577	97	46	wheel	wheel	NOUN
cana-577	97	47	graph	graph	NOUN
cana-577	97	48	.	.	PUNCT
cana-577	98	1	every	every	DET
cana-577	98	2	𝑝	𝑝	NOUN
cana-577	98	3	−	−	NUM
cana-577	98	4	1	1	NUM
cana-577	98	5	vertex	vertex	NOUN
cana-577	98	6	in	in	ADP
cana-577	98	7	the	the	DET
cana-577	98	8	wheel	wheel	NOUN
cana-577	98	9	network	network	NOUN
cana-577	98	10	is	be	AUX
cana-577	98	11	adjacent	adjacent	ADJ
cana-577	98	12	to	to	PART
cana-577	98	13	𝑣1	𝑣1	VERB
cana-577	98	14	.	.	PUNCT
cana-577	99	1	this	this	PRON
cana-577	99	2	indicates	indicate	VERB
cana-577	99	3	that	that	SCONJ
cana-577	99	4	the	the	DET
cana-577	99	5	strong	strong	ADJ
cana-577	99	6	regular	regular	ADJ
cana-577	99	7	domination	domination	NOUN
cana-577	99	8	for	for	ADP
cana-577	99	9	a	a	DET
cana-577	99	10	wheel	wheel	NOUN
cana-577	99	11	graph	graph	NOUN
cana-577	99	12	is	be	AUX
cana-577	99	13	1	1	NUM
cana-577	99	14	such	such	ADJ
cana-577	99	15	that	that	DET
cana-577	99	16	𝛾𝑠𝑡𝑟(𝑊𝑝	𝛾𝑠𝑡𝑟(𝑊𝑝	NOUN
cana-577	99	17	)	)	PUNCT
cana-577	99	18	=	=	PUNCT
cana-577	100	1	1	1	X
cana-577	100	2	.	.	PUNCT
cana-577	101	1	but	but	CCONJ
cana-577	101	2	the	the	DET
cana-577	101	3	vertex	vertex	NOUN
cana-577	101	4	𝑣1	𝑣1	NOUN
cana-577	101	5	strongly	strongly	ADV
cana-577	101	6	dominates	dominate	VERB
cana-577	101	7	only	only	ADV
cana-577	101	8	the	the	DET
cana-577	101	9	wheel	wheel	NOUN
cana-577	101	10	graph	graph	NOUN
cana-577	101	11	;	;	PUNCT
cana-577	101	12	the	the	DET
cana-577	101	13	helm	helm	NOUN
cana-577	101	14	graph	graph	NOUN
cana-577	101	15	is	be	AUX
cana-577	101	16	not	not	PART
cana-577	101	17	dominated	dominate	VERB
cana-577	101	18	by	by	ADP
cana-577	101	19	it	it	PRON
cana-577	101	20	.	.	PUNCT
cana-577	102	1	given	give	VERB
cana-577	102	2	that	that	SCONJ
cana-577	102	3	the	the	DET
cana-577	102	4	helm	helm	NOUN
cana-577	102	5	network	network	NOUN
cana-577	102	6	contains	contain	VERB
cana-577	102	7	𝑝	𝑝	PROPN
cana-577	102	8	−	−	PROPN
cana-577	102	9	1	1	NUM
cana-577	102	10	pendent	pendent	NOUN
cana-577	102	11	vertices	vertex	NOUN
cana-577	102	12	,	,	PUNCT
cana-577	102	13	and	and	CCONJ
cana-577	102	14	for	for	ADP
cana-577	102	15	each	each	DET
cana-577	102	16	pendant	pendant	ADJ
cana-577	102	17	vertex	vertex	NOUN
cana-577	102	18	𝑣𝑗	𝑣𝑗	ADP
cana-577	102	19	,	,	PUNCT
cana-577	102	20	2	2	NUM
cana-577	102	21	≤	≤	NUM
cana-577	102	22	𝑗	𝑗	PRON
cana-577	102	23	≤	≤	PROPN
cana-577	102	24	𝑝	𝑝	NOUN
cana-577	102	25	is	be	AUX
cana-577	102	26	connected	connect	VERB
cana-577	102	27	by	by	ADP
cana-577	102	28	an	an	DET
cana-577	102	29	edge	edge	NOUN
cana-577	102	30	.	.	PUNCT
cana-577	103	1	the	the	DET
cana-577	103	2	𝑝	𝑝	PROPN
cana-577	103	3	−	−	ADP
cana-577	103	4	1	1	NUM
cana-577	103	5	vertices	vertex	NOUN
cana-577	103	6	in	in	ADP
cana-577	103	7	the	the	DET
cana-577	103	8	outer	outer	ADJ
cana-577	103	9	circuit	circuit	NOUN
cana-577	103	10	are	be	AUX
cana-577	103	11	therefore	therefore	ADV
cana-577	103	12	superior	superior	ADJ
cana-577	103	13	to	to	ADP
cana-577	103	14	both	both	PRON
cana-577	103	15	of	of	ADP
cana-577	103	16	the	the	DET
cana-577	103	17	𝑝	𝑝	NOUN
cana-577	103	18	−	−	ADP
cana-577	103	19	1	1	NUM
cana-577	103	20	vertices	vertex	NOUN
cana-577	103	21	in	in	ADP
cana-577	103	22	the	the	DET
cana-577	103	23	helm	helm	NOUN
cana-577	103	24	graph	graph	NOUN
cana-577	103	25	.	.	PUNCT
cana-577	104	1	thus	thus	ADV
cana-577	104	2	,	,	PUNCT
cana-577	104	3	the	the	DET
cana-577	104	4	‘	'	PUNCT
cana-577	104	5	strong	strong	ADJ
cana-577	104	6	regular	regular	ADJ
cana-577	104	7	domination	domination	NOUN
cana-577	104	8	number	number	NOUN
cana-577	104	9	’	'	PUNCT
cana-577	104	10	of	of	ADP
cana-577	104	11	the	the	DET
cana-577	104	12	helm	helm	NOUN
cana-577	104	13	graph	graph	NOUN
cana-577	104	14	is	be	AUX
cana-577	104	15	𝑝	𝑝	ADJ
cana-577	104	16	−	−	PROPN
cana-577	104	17	1	1	NUM
cana-577	104	18	,	,	PUNCT
cana-577	104	19	that	that	PRON
cana-577	104	20	is	be	AUX
cana-577	104	21	t	t	PROPN
cana-577	104	22	𝛾𝑠𝑡𝑟(𝐶	𝛾𝑠𝑡𝑟(𝐶	PROPN
cana-577	104	23	)	)	PUNCT
cana-577	105	1	=	=	SYM
cana-577	105	2	𝑝	𝑝	ADJ
cana-577	105	3	−	−	NOUN
cana-577	105	4	1	1	NUM
cana-577	105	5	.	.	PUNCT
cana-577	106	1	since	since	SCONJ
cana-577	106	2	the	the	DET
cana-577	106	3	litact	litact	NOUN
cana-577	106	4	graph	graph	NOUN
cana-577	106	5	of	of	ADP
cana-577	106	6	helm	helm	NOUN
cana-577	106	7	graph	graph	NOUN
cana-577	106	8	is	be	AUX
cana-577	106	9	formed	form	VERB
cana-577	106	10	by	by	ADP
cana-577	106	11	the	the	DET
cana-577	106	12	cut	cut	NOUN
cana-577	106	13	vertices	vertex	NOUN
cana-577	106	14	and	and	CCONJ
cana-577	106	15	edges	edge	NOUN
cana-577	106	16	of	of	ADP
cana-577	106	17	the	the	DET
cana-577	106	18	helm	helm	NOUN
cana-577	106	19	graph	graph	NOUN
cana-577	106	20	,	,	PUNCT
cana-577	106	21	and	and	CCONJ
cana-577	106	22	it	it	PRON
cana-577	106	23	is	be	AUX
cana-577	106	24	clear	clear	ADJ
cana-577	106	25	that	that	SCONJ
cana-577	106	26	if	if	SCONJ
cana-577	106	27	d	d	PROPN
cana-577	106	28	is	be	AUX
cana-577	106	29	the	the	DET
cana-577	106	30	strong	strong	ADJ
cana-577	106	31	regular	regular	ADJ
cana-577	106	32	domination	domination	NOUN
cana-577	106	33	number	number	NOUN
cana-577	106	34	of	of	ADP
cana-577	106	35	the	the	DET
cana-577	106	36	helm	helm	NOUN
cana-577	106	37	graph	graph	NOUN
cana-577	106	38	,	,	PUNCT
cana-577	106	39	then	then	ADV
cana-577	106	40	|𝐷|	|𝐷|	VERB
cana-577	106	41	<	<	X
cana-577	106	42	𝑞(𝐻𝑝	𝑞(𝐻𝑝	X
cana-577	106	43	)	)	PUNCT
cana-577	106	44	+	+	NUM
cana-577	106	45	𝑙(𝐻𝑝	𝑙(𝐻𝑝	NOUN
cana-577	106	46	)	)	PUNCT
cana-577	106	47	.	.	PUNCT
cana-577	107	1	hence	hence	ADV
cana-577	107	2	,	,	PUNCT
cana-577	107	3	𝛾𝑠𝑡𝑟	𝛾𝑠𝑡𝑟	NOUN
cana-577	107	4	(	(	PUNCT
cana-577	107	5	𝑚(𝐻𝑝	𝑚(𝐻𝑝	PROPN
cana-577	107	6	)	)	PUNCT
cana-577	107	7	)	)	PUNCT
cana-577	107	8	<	<	X
cana-577	107	9	𝑞	𝑞	X
cana-577	108	1	+	+	X
cana-577	108	2	𝑙.	𝑙.	VERB
cana-577	108	3	the	the	DET
cana-577	108	4	next	next	ADJ
cana-577	108	5	theorem	theorem	NOUN
cana-577	108	6	connects	connect	VERB
cana-577	108	7	𝛾𝑠𝑡𝑟(𝑚(𝐺	𝛾𝑠𝑡𝑟(𝑚(𝐺	NOUN
cana-577	108	8	)	)	PUNCT
cana-577	108	9	)	)	PUNCT
cana-577	108	10	,	,	PUNCT
cana-577	108	11	𝛾(𝐺	𝛾(𝐺	PROPN
cana-577	108	12	)	)	PUNCT
cana-577	108	13	,	,	PUNCT
cana-577	108	14	𝑝(𝐺	𝑝(𝐺	NOUN
cana-577	108	15	)	)	PUNCT
cana-577	108	16	and	and	CCONJ
cana-577	108	17	𝛼0(𝐺	𝛼0(𝐺	PROPN
cana-577	108	18	)	)	PUNCT
cana-577	108	19	.	.	PUNCT
cana-577	109	1	theorem-3.9	theorem-3.9	NOUN
cana-577	109	2	:	:	PUNCT
cana-577	109	3	for	for	ADP
cana-577	109	4	any	any	DET
cana-577	109	5	connected	connected	ADJ
cana-577	109	6	(	(	PUNCT
cana-577	109	7	𝑝	𝑝	PROPN
cana-577	109	8	,	,	PUNCT
cana-577	109	9	𝑞	𝑞	NOUN
cana-577	109	10	)	)	PUNCT
cana-577	109	11	graph	graph	NOUN
cana-577	109	12	g	g	NOUN
cana-577	109	13	,	,	PUNCT
cana-577	109	14	𝛾𝑠𝑡𝑟(𝑚(𝐺	𝛾𝑠𝑡𝑟(𝑚(𝐺	PROPN
cana-577	109	15	)	)	PUNCT
cana-577	109	16	)	)	PUNCT
cana-577	110	1	+	+	CCONJ
cana-577	110	2	𝛾(𝐺	𝛾(𝐺	NOUN
cana-577	110	3	)	)	PUNCT
cana-577	110	4	≤	≤	NOUN
cana-577	110	5	𝑝	𝑝	ADP
cana-577	110	6	+	+	CCONJ
cana-577	110	7	⌈	⌈	NOUN
cana-577	110	8	𝛼0	𝛼0	PROPN
cana-577	110	9	2	2	NUM
cana-577	110	10	⌉.	⌉.	ADJ
cana-577	110	11	proof	proof	NOUN
cana-577	110	12	:	:	PUNCT
cana-577	110	13	let	let	VERB
cana-577	110	14	𝑉	𝑉	PROPN
cana-577	110	15	=	=	SYM
cana-577	110	16	{	{	PUNCT
cana-577	110	17	𝑣1	𝑣1	PROPN
cana-577	110	18	,	,	PUNCT
cana-577	110	19	𝑣2	𝑣2	PROPN
cana-577	110	20	,	,	PUNCT
cana-577	110	21	𝑣3	𝑣3	ADJ
cana-577	110	22	,	,	PUNCT
cana-577	110	23	…	…	PUNCT
cana-577	110	24	.	.	PUNCT
cana-577	110	25	,	,	PUNCT
cana-577	110	26	𝑣𝑖	𝑣𝑖	AUX
cana-577	110	27	}	}	PUNCT
cana-577	110	28	be	be	AUX
cana-577	110	29	the	the	DET
cana-577	110	30	smallest	small	ADJ
cana-577	110	31	set	set	NOUN
cana-577	110	32	of	of	ADP
cana-577	110	33	vertices	vertex	NOUN
cana-577	110	34	that	that	PRON
cana-577	110	35	encompasses	encompass	VERB
cana-577	110	36	every	every	DET
cana-577	110	37	edge	edge	NOUN
cana-577	110	38	in	in	ADP
cana-577	110	39	g	g	NOUN
cana-577	110	40	,	,	PUNCT
cana-577	110	41	ensuring	ensure	VERB
cana-577	110	42	that	that	SCONJ
cana-577	110	43	|𝑉|	|𝑉|	NOUN
cana-577	110	44	=	=	SYM
cana-577	110	45	𝛼0	𝛼0	PROPN
cana-577	110	46	.	.	PUNCT
cana-577	111	1	let	let	VERB
cana-577	111	2	d	d	PRON
cana-577	111	3	be	be	AUX
cana-577	111	4	the	the	DET
cana-577	111	5	‘	'	PUNCT
cana-577	111	6	dominating	dominating	NOUN
cana-577	111	7	set	set	NOUN
cana-577	111	8	of	of	ADP
cana-577	111	9	g	g	NOUN
cana-577	111	10	’	'	PUNCT
cana-577	111	11	going	go	VERB
cana-577	111	12	forward	forward	ADV
cana-577	111	13	so	so	SCONJ
cana-577	111	14	that	that	SCONJ
cana-577	111	15	|𝐷|	|𝐷|	NOUN
cana-577	111	16	=	=	SYM
cana-577	111	17	𝛾(𝐺	𝛾(𝐺	PROPN
cana-577	111	18	)	)	PUNCT
cana-577	111	19	.	.	PUNCT
cana-577	112	1	assume	assume	VERB
cana-577	112	2	that	that	SCONJ
cana-577	112	3	the	the	DET
cana-577	112	4	‘	'	PUNCT
cana-577	112	5	set	set	NOUN
cana-577	112	6	of	of	ADP
cana-577	112	7	edges	edge	NOUN
cana-577	112	8	’	'	PUNCT
cana-577	112	9	in	in	ADP
cana-577	112	10	g	g	PROPN
cana-577	112	11	that	that	PRON
cana-577	112	12	form	form	VERB
cana-577	112	13	a	a	DET
cana-577	112	14	minimal	minimal	ADJ
cana-577	112	15	connected	connect	VERB
cana-577	112	16	edge	edge	NOUN
cana-577	112	17	dominating	dominating	NOUN
cana-577	112	18	set	set	VERB
cana-577	112	19	in	in	ADP
cana-577	112	20	g	g	PROPN
cana-577	112	21	is	be	AUX
cana-577	112	22	represented	represent	VERB
cana-577	112	23	by	by	ADP
cana-577	112	24	the	the	DET
cana-577	112	25	set	set	NOUN
cana-577	112	26	𝐸	𝐸	PROPN
cana-577	112	27	=	=	SYM
cana-577	112	28	{	{	PUNCT
cana-577	112	29	𝑒1	𝑒1	NOUN
cana-577	112	30	,	,	PUNCT
cana-577	112	31	𝑒2	𝑒2	PROPN
cana-577	112	32	,	,	PUNCT
cana-577	112	33	𝑒3	𝑒3	PROPN
cana-577	112	34	,	,	PUNCT
cana-577	112	35	…	…	PUNCT
cana-577	112	36	.	.	PUNCT
cana-577	112	37	.	.	PUNCT
cana-577	113	1	,	,	PUNCT
cana-577	113	2	𝑒𝑘	𝑒𝑘	VERB
cana-577	113	3	}	}	PUNCT
cana-577	113	4	.	.	PUNCT
cana-577	114	1	let	let	VERB
cana-577	114	2	𝐶	𝐶	PROPN
cana-577	114	3	⊆	⊆	NUM
cana-577	114	4	𝑉(𝐺	𝑉(𝐺	NOUN
cana-577	114	5	)	)	PUNCT
cana-577	114	6	represent	represent	VERB
cana-577	114	7	g	g	PROPN
cana-577	114	8	's	's	PART
cana-577	114	9	set	set	NOUN
cana-577	114	10	of	of	ADP
cana-577	114	11	cut	cut	NOUN
cana-577	114	12	vertices	vertex	NOUN
cana-577	114	13	.	.	PUNCT
cana-577	115	1	the	the	DET
cana-577	115	2	minimal	minimal	ADJ
cana-577	115	3	strong	strong	ADJ
cana-577	115	4	dominating	dominating	NOUN
cana-577	115	5	set	set	NOUN
cana-577	115	6	in	in	ADP
cana-577	115	7	𝑚(𝐺	𝑚(𝐺	NUM
cana-577	115	8	)	)	PUNCT
cana-577	115	9	is	be	AUX
cana-577	115	10	then	then	ADV
cana-577	115	11	the	the	DET
cana-577	115	12	set	set	NOUN
cana-577	115	13	𝑋𝜖𝑉(𝑚(𝐺	𝑋𝜖𝑉(𝑚(𝐺	NOUN
cana-577	115	14	)	)	PUNCT
cana-577	115	15	)	)	PUNCT
cana-577	115	16	where	where	SCONJ
cana-577	115	17	𝑋	𝑋	PROPN
cana-577	115	18	⊆	⊆	NUM
cana-577	115	19	𝐹	𝐹	PROPN
cana-577	115	20	∪	∪	PROPN
cana-577	115	21	𝐶	𝐶	PROPN
cana-577	115	22	and	and	CCONJ
cana-577	115	23	𝑁(𝑋	𝑁(𝑋	NUM
cana-577	115	24	)	)	PUNCT
cana-577	115	25	=	=	SYM
cana-577	115	26	𝑉(𝑚(𝐺	𝑉(𝑚(𝐺	NOUN
cana-577	115	27	)	)	PUNCT
cana-577	115	28	)	)	PUNCT
cana-577	116	1	−	−	PROPN
cana-577	116	2	𝑋.	𝑋.	PROPN
cana-577	116	3	further	far	ADV
cana-577	116	4	,	,	PUNCT
cana-577	116	5	let	let	VERB
cana-577	116	6	𝑌	𝑌	PROPN
cana-577	116	7	⊆	⊆	NUM
cana-577	116	8	𝑉(𝑚(𝐺	𝑉(𝑚(𝐺	NOUN
cana-577	116	9	)	)	PUNCT
cana-577	116	10	)	)	PUNCT
cana-577	117	1	−	−	PROPN
cana-577	117	2	𝑋	𝑋	PROPN
cana-577	117	3	and	and	CCONJ
cana-577	117	4	𝑌	𝑌	PROPN
cana-577	117	5	∈	∈	PROPN
cana-577	117	6	𝑁(𝑋	𝑁(𝑋	NOUN
cana-577	117	7	)	)	PUNCT
cana-577	117	8	,	,	PUNCT
cana-577	117	9	then	then	ADV
cana-577	117	10	take	take	VERB
cana-577	117	11	a	a	DET
cana-577	117	12	set	set	NOUN
cana-577	117	13	𝑌′	𝑌′	NOUN
cana-577	117	14	⊂	⊂	PROPN
cana-577	117	15	𝑌	𝑌	PROPN
cana-577	117	16	so	so	SCONJ
cana-577	117	17	that	that	SCONJ
cana-577	117	18	〈	〈	ADP
cana-577	117	19	𝑋	𝑋	PROPN
cana-577	117	20	∪	∪	ADV
cana-577	117	21	𝑌′	𝑌′	NOUN
cana-577	117	22	〉	〉	NOUN
cana-577	117	23	is	be	AUX
cana-577	117	24	the	the	DET
cana-577	117	25	minimal	minimal	ADJ
cana-577	117	26	strong	strong	ADJ
cana-577	117	27	k	k	ADJ
cana-577	117	28	-	-	ADJ
cana-577	117	29	regular	regular	ADJ
cana-577	117	30	sub	sub	NOUN
cana-577	117	31	graph	graph	NOUN
cana-577	117	32	of	of	ADP
cana-577	117	33	𝑚(𝐺	𝑚(𝐺	NUM
cana-577	117	34	)	)	PUNCT
cana-577	117	35	,	,	PUNCT
cana-577	117	36	then	then	ADV
cana-577	117	37	𝑋	𝑋	PROPN
cana-577	117	38	∪	∪	NOUN
cana-577	117	39	𝑌′	𝑌′	PROPN
cana-577	117	40	is	be	AUX
cana-577	117	41	the	the	DET
cana-577	117	42	minimal	minimal	ADJ
cana-577	117	43	strong	strong	ADJ
cana-577	117	44	regular	regular	ADJ
cana-577	117	45	dominating	dominating	NOUN
cana-577	117	46	set	set	NOUN
cana-577	117	47	of	of	ADP
cana-577	117	48	𝑚(𝐺	𝑚(𝐺	PROPN
cana-577	117	49	)	)	PUNCT
cana-577	117	50	.	.	PUNCT
cana-577	118	1	then	then	ADV
cana-577	118	2	,	,	PUNCT
cana-577	118	3	|𝑋	|𝑋	PROPN
cana-577	118	4	∪	∪	ADP
cana-577	118	5	𝑌′|	𝑌′|	SYM
cana-577	118	6	∪	∪	ADJ
cana-577	118	7	|𝐷|	|𝐷|	X
cana-577	118	8	≤	≤	X
cana-577	118	9	𝑝	𝑝	PROPN
cana-577	118	10	+	+	CCONJ
cana-577	118	11	⌈	⌈	NOUN
cana-577	118	12	𝛼0	𝛼0	PROPN
cana-577	118	13	2	2	NUM
cana-577	118	14	⌉	⌉	X
cana-577	118	15	⟹	⟹	X
cana-577	118	16	𝛾𝑠𝑡𝑟(𝑚(𝐺	𝛾𝑠𝑡𝑟(𝑚(𝐺	NUM
cana-577	118	17	)	)	PUNCT
cana-577	118	18	)	)	PUNCT
cana-577	119	1	+	+	CCONJ
cana-577	119	2	𝛾(𝐺	𝛾(𝐺	NOUN
cana-577	119	3	)	)	PUNCT
cana-577	119	4	≤	≤	NOUN
cana-577	119	5	𝑝	𝑝	ADP
cana-577	119	6	+	+	CCONJ
cana-577	119	7	⌈	⌈	NOUN
cana-577	119	8	𝛼0	𝛼0	PROPN
cana-577	119	9	2	2	NUM
cana-577	119	10	⌉.	⌉.	ADV
cana-577	119	11	the	the	DET
cana-577	119	12	next	next	ADJ
cana-577	119	13	outcome	outcome	NOUN
cana-577	119	14	involving	involve	VERB
cana-577	119	15	𝛾𝑠𝑡𝑟(𝑚(𝐺	𝛾𝑠𝑡𝑟(𝑚(𝐺	NOUN
cana-577	119	16	)	)	PUNCT
cana-577	119	17	)	)	PUNCT
cana-577	119	18	,	,	PUNCT
cana-577	119	19	𝛾𝑐(𝐺	𝛾𝑐(𝐺	NOUN
cana-577	119	20	)	)	PUNCT
cana-577	119	21	,	,	PUNCT
cana-577	119	22	𝛽0(𝐺	𝛽0(𝐺	PROPN
cana-577	119	23	)	)	PUNCT
cana-577	119	24	and	and	CCONJ
cana-577	119	25	𝛾(𝐺	𝛾(𝐺	NUM
cana-577	119	26	)	)	PUNCT
cana-577	119	27	has	have	AUX
cana-577	119	28	been	be	AUX
cana-577	119	29	determined	determine	VERB
cana-577	119	30	.	.	PUNCT
cana-577	120	1	theorem-3.10	theorem-3.10	ADJ
cana-577	120	2	:	:	PUNCT
cana-577	120	3	for	for	ADP
cana-577	120	4	any	any	DET
cana-577	120	5	‘	'	PUNCT
cana-577	120	6	graph	graph	NOUN
cana-577	120	7	g	g	NOUN
cana-577	120	8	’	'	PUNCT
cana-577	120	9	,	,	PUNCT
cana-577	120	10	𝛾𝑠𝑡𝑟(𝑚(𝐺	𝛾𝑠𝑡𝑟(𝑚(𝐺	PROPN
cana-577	120	11	)	)	PUNCT
cana-577	120	12	)	)	PUNCT
cana-577	121	1	+	+	CCONJ
cana-577	121	2	𝛾𝑐(𝐺	𝛾𝑐(𝐺	NOUN
cana-577	121	3	)	)	PUNCT
cana-577	121	4	≤	≤	NOUN
cana-577	121	5	𝛽0(𝐺	𝛽0(𝐺	PROPN
cana-577	121	6	)	)	PUNCT
cana-577	122	1	+	+	NUM
cana-577	122	2	𝛾(𝐺	𝛾(𝐺	NOUN
cana-577	122	3	)	)	PUNCT
cana-577	122	4	.	.	PUNCT
cana-577	123	1	proof	proof	NOUN
cana-577	123	2	:	:	PUNCT
cana-577	123	3	let	let	VERB
cana-577	123	4	𝑋	𝑋	PROPN
cana-577	123	5	⊆	⊆	NUM
cana-577	123	6	𝑉(𝐺	𝑉(𝐺	NOUN
cana-577	123	7	)	)	PUNCT
cana-577	123	8	be	be	VERB
cana-577	123	9	the	the	DET
cana-577	123	10	set	set	NOUN
cana-577	123	11	of	of	ADP
cana-577	123	12	points	point	NOUN
cana-577	123	13	that	that	PRON
cana-577	123	14	cover	cover	VERB
cana-577	123	15	all	all	PRON
cana-577	123	16	of	of	ADP
cana-577	123	17	the	the	DET
cana-577	123	18	edges	edge	NOUN
cana-577	123	19	in	in	ADP
cana-577	123	20	g	g	NOUN
cana-577	123	21	at	at	ADP
cana-577	123	22	a	a	DET
cana-577	123	23	minimum	minimum	ADJ
cana-577	123	24	distance	distance	NOUN
cana-577	123	25	of	of	ADP
cana-577	123	26	two	two	NUM
cana-577	123	27	and	and	CCONJ
cana-577	123	28	have	have	AUX
cana-577	123	29	deg(𝑣𝑖	deg(𝑣𝑖	VERB
cana-577	123	30	)	)	PUNCT
cana-577	123	31	≥	≥	NOUN
cana-577	123	32	2	2	NUM
cana-577	123	33	,	,	PUNCT
cana-577	123	34	∀𝑣𝑖	∀𝑣𝑖	VERB
cana-577	123	35	∈	∈	PROPN
cana-577	123	36	𝑋	𝑋	PROPN
cana-577	123	37	,	,	PUNCT
cana-577	123	38	1	1	NUM
cana-577	123	39	≤	≤	NUM
cana-577	123	40	𝑖	𝑖	PRON
cana-577	123	41	≤	≤	PROPN
cana-577	123	42	𝑘.	𝑘.	NOUN
cana-577	123	43	additionally	additionally	ADV
cana-577	123	44	,	,	PUNCT
cana-577	123	45	if	if	SCONJ
cana-577	123	46	𝑁(𝑥	𝑁(𝑥	VERB
cana-577	123	47	)	)	PUNCT
cana-577	123	48	=	=	SYM
cana-577	123	49	𝑉(𝐺	𝑉(𝐺	NOUN
cana-577	123	50	)	)	PUNCT
cana-577	123	51	−	−	PROPN
cana-577	123	52	𝑋	𝑋	PROPN
cana-577	123	53	for	for	ADP
cana-577	123	54	every	every	DET
cana-577	123	55	vertex	vertex	NOUN
cana-577	123	56	𝑥	𝑥	PRON
cana-577	123	57	∈	∈	PROPN
cana-577	123	58	𝑋	𝑋	PROPN
cana-577	123	59	,	,	PUNCT
cana-577	123	60	then	then	ADV
cana-577	123	61	𝑋	𝑋	PROPN
cana-577	123	62	is	be	AUX
cana-577	123	63	an	an	DET
cana-577	123	64	independent	independent	ADJ
cana-577	123	65	vertex	vertex	NOUN
cana-577	123	66	collection	collection	NOUN
cana-577	123	67	.	.	PUNCT
cana-577	124	1	otherwise	otherwise	ADV
cana-577	124	2	,	,	PUNCT
cana-577	124	3	|𝑋1	|𝑋1	PUNCT
cana-577	124	4	∪	∪	VERB
cana-577	124	5	𝑋2|	𝑋2|	PROPN
cana-577	124	6	=	=	SYM
cana-577	124	7	𝛽0(𝐺	𝛽0(𝐺	PROPN
cana-577	124	8	)	)	PUNCT
cana-577	124	9	creates	create	VERB
cana-577	124	10	a	a	DET
cana-577	124	11	maximum	maximum	ADJ
cana-577	124	12	independent	independent	ADJ
cana-577	124	13	set	set	NOUN
cana-577	124	14	of	of	ADP
cana-577	124	15	g	g	NOUN
cana-577	124	16	where	where	SCONJ
cana-577	124	17	𝑋1	𝑋1	PROPN
cana-577	124	18	⊆	⊆	NUM
cana-577	124	19	𝑋	𝑋	PROPN
cana-577	124	20	and	and	CCONJ
cana-577	124	21	𝑋2	𝑋2	VERB
cana-577	124	22	⊆	⊆	NUM
cana-577	124	23	𝑉(𝐺	𝑉(𝐺	NOUN
cana-577	124	24	)	)	PUNCT
cana-577	124	25	−	−	PROPN
cana-577	124	26	𝑋.	𝑋.	PROPN
cana-577	124	27	let	let	VERB
cana-577	124	28	𝑌	𝑌	PROPN
cana-577	124	29	=	=	PUNCT
cana-577	124	30	𝑋′	𝑋′	PROPN
cana-577	124	31	∪	∪	ADJ
cana-577	124	32	𝑋	𝑋	PROPN
cana-577	124	33	"	"	PUNCT
cana-577	124	34	represent	represent	VERB
cana-577	124	35	the	the	DET
cana-577	124	36	minimal	minimal	ADJ
cana-577	124	37	set	set	NOUN
cana-577	124	38	of	of	ADP
cana-577	124	39	points	point	NOUN
cana-577	124	40	that	that	PRON
cana-577	124	41	covers	cover	VERB
cana-577	124	42	all	all	DET
cana-577	124	43	points	point	NOUN
cana-577	124	44	in	in	ADP
cana-577	124	45	g	g	NOUN
cana-577	124	46	,	,	PUNCT
cana-577	124	47	where	where	SCONJ
cana-577	124	48	𝑋′	𝑋′	PROPN
cana-577	124	49	⊆	⊆	NUM
cana-577	124	50	𝑋	𝑋	PROPN
cana-577	124	51	and	and	CCONJ
cana-577	124	52	𝑋	𝑋	PROPN
cana-577	124	53	"	"	PUNCT
cana-577	124	54	⊆	⊆	NUM
cana-577	124	55	𝑉(𝐺	𝑉(𝐺	NOUN
cana-577	124	56	)	)	PUNCT
cana-577	125	1	−	−	PROPN
cana-577	125	2	𝑋.	𝑋.	PROPN
cana-577	125	3	𝑌	𝑌	PROPN
cana-577	125	4	undoubtedly	undoubtedly	ADV
cana-577	125	5	makes	make	VERB
cana-577	125	6	up	up	ADP
cana-577	125	7	g	g	NOUN
cana-577	125	8	's	's	PART
cana-577	125	9	minimum	minimum	ADJ
cana-577	125	10	dominating	dominating	NOUN
cana-577	125	11	set	set	NOUN
cana-577	125	12	.	.	PUNCT
cana-577	126	1	assume	assume	VERB
cana-577	126	2	that	that	SCONJ
cana-577	126	3	there	there	PRON
cana-577	126	4	is	be	VERB
cana-577	126	5	just	just	ADV
cana-577	126	6	one	one	NUM
cana-577	126	7	element	element	NOUN
cana-577	126	8	in	in	ADP
cana-577	126	9	the	the	DET
cana-577	126	10	sub	sub	NOUN
cana-577	126	11	graph	graph	NOUN
cana-577	126	12	〈	〈	PROPN
cana-577	126	13	⁡𝑌	⁡𝑌	NUM
cana-577	126	14	〉	〉	NOUN
cana-577	126	15	.	.	PUNCT
cana-577	127	1	then	then	ADV
cana-577	127	2	,	,	PUNCT
cana-577	127	3	𝑌	𝑌	PROPN
cana-577	127	4	is	be	AUX
cana-577	127	5	a	a	DET
cana-577	127	6	‘	'	PUNCT
cana-577	127	7	connected	connected	ADJ
cana-577	127	8	dominating	dominating	NOUN
cana-577	127	9	set	set	NOUN
cana-577	127	10	of	of	ADP
cana-577	127	11	g	g	NOUN
cana-577	127	12	’	'	PUNCT
cana-577	127	13	on	on	ADP
cana-577	127	14	its	its	PRON
cana-577	127	15	own	own	ADJ
cana-577	127	16	.	.	PUNCT
cana-577	128	1	communications	communication	NOUN
cana-577	128	2	on	on	ADP
cana-577	128	3	applied	apply	VERB
cana-577	128	4	nonlinear	nonlinear	ADJ
cana-577	128	5	analysis	analysis	NOUN
cana-577	128	6	issn	issn	NOUN
cana-577	128	7	:	:	PUNCT
cana-577	128	8	1074	1074	NUM
cana-577	128	9	-	-	PUNCT
cana-577	128	10	133x	133x	NUM
cana-577	128	11	vol	vol	NOUN
cana-577	128	12	31	31	NUM
cana-577	128	13	no	no	NOUN
cana-577	128	14	.	.	PUNCT
cana-577	129	1	1s	1s	NUM
cana-577	129	2	(	(	PUNCT
cana-577	129	3	2024	2024	NUM
cana-577	129	4	)	)	PUNCT
cana-577	129	5	184	184	NUM
cana-577	129	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-577	129	7	otherwise	otherwise	ADV
cana-577	129	8	,	,	PUNCT
cana-577	129	9	attach	attach	VERB
cana-577	129	10	the	the	DET
cana-577	129	11	smallest	small	ADJ
cana-577	129	12	number	number	NOUN
cana-577	129	13	of	of	ADP
cana-577	129	14	vertices	vertex	NOUN
cana-577	129	15	{	{	PUNCT
cana-577	129	16	𝑢𝑗	𝑢𝑗	NOUN
cana-577	129	17	}	}	PUNCT
cana-577	129	18	∈	∈	PROPN
cana-577	129	19	𝑉(𝐺	𝑉(𝐺	NOUN
cana-577	129	20	)	)	PUNCT
cana-577	129	21	−	−	PROPN
cana-577	129	22	𝑌	𝑌	PROPN
cana-577	129	23	where	where	SCONJ
cana-577	129	24	deg⁡(𝑢𝑗	deg⁡(𝑢𝑗	NOUN
cana-577	129	25	)	)	PUNCT
cana-577	129	26	≥	≥	NOUN
cana-577	129	27	2	2	NUM
cana-577	129	28	that	that	PRON
cana-577	129	29	are	be	AUX
cana-577	129	30	between	between	ADP
cana-577	129	31	the	the	DET
cana-577	129	32	vertices	vertex	NOUN
cana-577	129	33	of	of	ADP
cana-577	129	34	𝑌	𝑌	PROPN
cana-577	129	35	such	such	ADJ
cana-577	129	36	that	that	PRON
cana-577	129	37	𝑌1	𝑌1	NOUN
cana-577	130	1	=	=	SYM
cana-577	130	2	𝑌	𝑌	PROPN
cana-577	130	3	∪	∪	ADJ
cana-577	130	4	{	{	PUNCT
cana-577	130	5	𝑢𝑗	𝑢𝑗	NOUN
cana-577	130	6	}	}	PUNCT
cana-577	130	7	forms	form	VERB
cana-577	130	8	precisely	precisely	ADV
cana-577	130	9	one	one	NUM
cana-577	130	10	component	component	NOUN
cana-577	130	11	in	in	ADP
cana-577	130	12	the	the	DET
cana-577	130	13	subgraph	subgraph	NOUN
cana-577	130	14	〈	〈	PROPN
cana-577	130	15	𝑌1	𝑌1	PROPN
cana-577	130	16	〉	〉	NOUN
cana-577	130	17	.	.	PUNCT
cana-577	131	1	𝑌1	𝑌1	PROPN
cana-577	131	2	undoubtedly	undoubtedly	ADV
cana-577	131	3	constitutes	constitute	VERB
cana-577	131	4	a	a	DET
cana-577	131	5	minimal	minimal	ADJ
cana-577	131	6	connected	connected	ADJ
cana-577	131	7	dominating	dominating	NOUN
cana-577	131	8	set	set	VERB
cana-577	131	9	within	within	ADP
cana-577	131	10	g.	g.	PROPN
cana-577	131	11	let	let	VERB
cana-577	131	12	𝐷	𝐷	PROPN
cana-577	131	13	⊆	⊆	NUM
cana-577	131	14	𝐶	𝐶	PROPN
cana-577	131	15	,	,	PUNCT
cana-577	131	16	where	where	SCONJ
cana-577	131	17	c	c	PROPN
cana-577	131	18	is	be	AUX
cana-577	131	19	the	the	DET
cana-577	131	20	set	set	NOUN
cana-577	131	21	of	of	ADP
cana-577	131	22	points	point	NOUN
cana-577	131	23	in	in	ADP
cana-577	131	24	g	g	PROPN
cana-577	131	25	that	that	PRON
cana-577	131	26	correspond	correspond	VERB
cana-577	131	27	to	to	ADP
cana-577	131	28	the	the	DET
cana-577	131	29	edges	edge	NOUN
cana-577	131	30	that	that	PRON
cana-577	131	31	meet	meet	VERB
cana-577	131	32	𝑌	𝑌	PROPN
cana-577	131	33	's	's	PART
cana-577	131	34	vertices	vertex	NOUN
cana-577	131	35	.	.	PUNCT
cana-577	132	1	a	a	DET
cana-577	132	2	minimal	minimal	ADJ
cana-577	132	3	set	set	NOUN
cana-577	132	4	of	of	ADP
cana-577	132	5	vertices	vertex	NOUN
cana-577	132	6	where	where	SCONJ
cana-577	132	7	∀𝑣𝑘	∀𝑣𝑘	PROPN
cana-577	132	8	∈	∈	PROPN
cana-577	132	9	𝐷	𝐷	PROPN
cana-577	132	10	and	and	CCONJ
cana-577	132	11	𝑁(𝐷	𝑁(𝐷	NOUN
cana-577	132	12	)	)	PUNCT
cana-577	132	13	=	=	SYM
cana-577	132	14	𝑉(𝑚(𝐺	𝑉(𝑚(𝐺	NOUN
cana-577	132	15	)	)	PUNCT
cana-577	132	16	)	)	PUNCT
cana-577	133	1	−	−	PROPN
cana-577	133	2	{	{	PUNCT
cana-577	133	3	𝑣𝑘	𝑣𝑘	NOUN
cana-577	133	4	}	}	PUNCT
cana-577	133	5	are	be	AUX
cana-577	133	6	found	find	VERB
cana-577	133	7	.	.	PUNCT
cana-577	134	1	d	d	PROPN
cana-577	134	2	undoubtedly	undoubtedly	ADV
cana-577	134	3	creates	create	VERB
cana-577	134	4	a	a	DET
cana-577	134	5	strong	strong	ADJ
cana-577	134	6	regular	regular	ADJ
cana-577	134	7	dominating	dominating	NOUN
cana-577	134	8	set	set	NOUN
cana-577	134	9	in	in	ADP
cana-577	134	10	𝑚(𝐺	𝑚(𝐺	NUM
cana-577	134	11	)	)	PUNCT
cana-577	134	12	.	.	PUNCT
cana-577	135	1	therefore	therefore	ADV
cana-577	135	2	,	,	PUNCT
cana-577	135	3	|𝐷|	|𝐷|	NOUN
cana-577	135	4	∪	∪	ADV
cana-577	135	5	|𝑌1⁡|	|𝑌1⁡|	X
cana-577	135	6	≤	≤	NOUN
cana-577	135	7	|𝑋1	|𝑋1	VERB
cana-577	135	8	∪	∪	ADJ
cana-577	135	9	𝑋2|	𝑋2|	NOUN
cana-577	135	10	∪	∪	X
cana-577	135	11	|𝑌|	|𝑌|	X
cana-577	135	12	⟹	⟹	NUM
cana-577	135	13	𝛾𝑠𝑡𝑟(𝑚(𝐺	𝛾𝑠𝑡𝑟(𝑚(𝐺	PROPN
cana-577	135	14	)	)	PUNCT
cana-577	135	15	)	)	PUNCT
cana-577	136	1	+	+	CCONJ
cana-577	136	2	𝛾𝑐(𝐺	𝛾𝑐(𝐺	NOUN
cana-577	136	3	)	)	PUNCT
cana-577	136	4	≤	≤	NOUN
cana-577	137	1	𝛽0(𝐺	𝛽0(𝐺	PROPN
cana-577	137	2	)	)	PUNCT
cana-577	138	1	+	+	NUM
cana-577	138	2	𝛾(𝐺	𝛾(𝐺	NOUN
cana-577	138	3	)	)	PUNCT
cana-577	138	4	.	.	PUNCT
cana-577	139	1	hence	hence	ADV
cana-577	139	2	,	,	PUNCT
cana-577	139	3	the	the	DET
cana-577	139	4	proof	proof	NOUN
cana-577	139	5	.	.	PUNCT
cana-577	140	1	using	use	VERB
cana-577	140	2	𝛾𝑠𝑡𝑟(𝑚(𝐺	𝛾𝑠𝑡𝑟(𝑚(𝐺	NUM
cana-577	140	3	)	)	PUNCT
cana-577	140	4	in	in	ADP
cana-577	140	5	terms	term	NOUN
cana-577	140	6	of	of	ADP
cana-577	140	7	𝑞	𝑞	PROPN
cana-577	140	8	and	and	CCONJ
cana-577	140	9	∆′(𝐺	∆′(𝐺	NOUN
cana-577	140	10	)	)	PUNCT
cana-577	140	11	,	,	PUNCT
cana-577	140	12	the	the	DET
cana-577	140	13	following	follow	VERB
cana-577	140	14	theorem	theorem	NOUN
cana-577	140	15	has	have	AUX
cana-577	140	16	been	be	AUX
cana-577	140	17	discovered	discover	VERB
cana-577	140	18	.	.	PUNCT
cana-577	141	1	theorem-3.11	theorem-3.11	NOUN
cana-577	141	2	:	:	PUNCT
cana-577	141	3	for	for	ADP
cana-577	141	4	any	any	DET
cana-577	141	5	‘	'	PUNCT
cana-577	141	6	graph	graph	NOUN
cana-577	141	7	g	g	NOUN
cana-577	141	8	’	'	PUNCT
cana-577	141	9	,	,	PUNCT
cana-577	141	10	⌈	⌈	X
cana-577	141	11	𝑞	𝑞	X
cana-577	141	12	2∆′(𝐺)+1	2∆′(𝐺)+1	NUM
cana-577	141	13	⌉	⌉	X
cana-577	141	14	<	<	X
cana-577	141	15	𝛾𝑠𝑡𝑟(𝑚(𝐺	𝛾𝑠𝑡𝑟(𝑚(𝐺	X
cana-577	141	16	)	)	PUNCT
cana-577	141	17	+	+	NOUN
cana-577	141	18	1	1	X
cana-577	141	19	.	.	X
cana-577	141	20	proof	proof	NOUN
cana-577	141	21	:	:	PUNCT
cana-577	141	22	let	let	VERB
cana-577	141	23	‘	'	PUNCT
cana-577	141	24	g	g	NOUN
cana-577	141	25	be	be	AUX
cana-577	141	26	a	a	DET
cana-577	141	27	graph	graph	NOUN
cana-577	141	28	’	'	PUNCT
cana-577	141	29	with	with	ADP
cana-577	141	30	the	the	DET
cana-577	141	31	set	set	NOUN
cana-577	141	32	of	of	ADP
cana-577	141	33	edges	edge	NOUN
cana-577	141	34	𝐸	𝐸	NOUN
cana-577	141	35	=	=	SYM
cana-577	141	36	{	{	PUNCT
cana-577	141	37	𝑒1	𝑒1	NOUN
cana-577	141	38	,	,	PUNCT
cana-577	141	39	𝑒2	𝑒2	PROPN
cana-577	141	40	,	,	PUNCT
cana-577	141	41	𝑒3	𝑒3	PROPN
cana-577	141	42	,	,	PUNCT
cana-577	141	43	…	…	PUNCT
cana-577	141	44	…	…	PUNCT
cana-577	141	45	,	,	PUNCT
cana-577	141	46	𝑒𝑛	𝑒𝑛	PROPN
cana-577	141	47	}	}	PUNCT
cana-577	141	48	,	,	PUNCT
cana-577	141	49	such	such	ADJ
cana-577	141	50	that	that	PRON
cana-577	141	51	|𝐸(𝐺)|	|𝐸(𝐺)|	NOUN
cana-577	141	52	=	=	PUNCT
cana-577	141	53	𝑞	𝑞	PROPN
cana-577	141	54	and	and	CCONJ
cana-577	141	55	∆′(𝐺	∆′(𝐺	NOUN
cana-577	141	56	)	)	PUNCT
cana-577	141	57	be	be	VERB
cana-577	141	58	the	the	DET
cana-577	141	59	highest	high	ADJ
cana-577	141	60	degree	degree	NOUN
cana-577	141	61	of	of	ADP
cana-577	141	62	an	an	DET
cana-577	141	63	edge	edge	NOUN
cana-577	141	64	in	in	ADP
cana-577	141	65	g.	g.	PROPN
cana-577	141	66	suppose	suppose	VERB
cana-577	141	67	𝑋	𝑋	PROPN
cana-577	141	68	⊆	⊆	NUM
cana-577	141	69	𝑉(𝑚(𝐺	𝑉(𝑚(𝐺	NOUN
cana-577	141	70	)	)	PUNCT
cana-577	141	71	)	)	PUNCT
cana-577	141	72	is	be	AUX
cana-577	141	73	the	the	DET
cana-577	141	74	smallest	small	ADJ
cana-577	141	75	dominating	dominating	NOUN
cana-577	141	76	set	set	VERB
cana-577	141	77	in	in	ADP
cana-577	141	78	𝑚(𝐺	𝑚(𝐺	NUM
cana-577	141	79	)	)	PUNCT
cana-577	141	80	,	,	PUNCT
cana-577	141	81	then	then	ADV
cana-577	141	82	𝑋	𝑋	PROPN
cana-577	141	83	creates	create	VERB
cana-577	141	84	a	a	DET
cana-577	141	85	strong	strong	ADJ
cana-577	141	86	regular	regular	ADJ
cana-577	141	87	dominating	dominating	NOUN
cana-577	141	88	set	set	VERB
cana-577	141	89	by	by	ADP
cana-577	141	90	itself	itself	PRON
cana-577	141	91	.	.	PUNCT
cana-577	142	1	next	next	ADV
cana-577	142	2	,	,	PUNCT
cana-577	142	3	it	it	PRON
cana-577	142	4	is	be	AUX
cana-577	142	5	clear	clear	ADJ
cana-577	142	6	that	that	SCONJ
cana-577	142	7	,	,	PUNCT
cana-577	142	8	according	accord	VERB
cana-577	142	9	to	to	ADP
cana-577	142	10	the	the	DET
cana-577	142	11	litact	litact	NOUN
cana-577	142	12	graph	graph	NOUN
cana-577	142	13	definition	definition	NOUN
cana-577	142	14	𝑉(𝑚(𝐺	𝑉(𝑚(𝐺	PROPN
cana-577	142	15	)	)	PUNCT
cana-577	142	16	)	)	PUNCT
cana-577	143	1	=	=	SYM
cana-577	143	2	𝐸(𝐺	𝐸(𝐺	X
cana-577	143	3	)	)	PUNCT
cana-577	143	4	∪	∪	ADP
cana-577	143	5	𝐶(𝐺	𝐶(𝐺	PROPN
cana-577	143	6	)	)	PUNCT
cana-577	143	7	,	,	PUNCT
cana-577	143	8	we	we	PRON
cana-577	143	9	get	get	VERB
cana-577	143	10	,	,	PUNCT
cana-577	143	11	(	(	PUNCT
cana-577	143	12	2∆′(𝐺	2∆′(𝐺	NUM
cana-577	143	13	)	)	PUNCT
cana-577	143	14	+	+	NOUN
cana-577	143	15	1)|𝑋|	1)|𝑋|	NUM
cana-577	143	16	≥	≥	NOUN
cana-577	143	17	|𝐸(𝐺)|	|𝐸(𝐺)|	ADV
cana-577	143	18	⟹	⟹	X
cana-577	143	19	(	(	PUNCT
cana-577	143	20	2∆′(𝐺	2∆′(𝐺	NUM
cana-577	143	21	)	)	PUNCT
cana-577	143	22	+	+	NOUN
cana-577	144	1	1)|𝑋|	1)|𝑋|	NUM
cana-577	144	2	≥	≥	NOUN
cana-577	144	3	𝑞	𝑞	X
cana-577	144	4	⟹	⟹	PROPN
cana-577	144	5	|𝑋|	|𝑋|	PROPN
cana-577	144	6	≥	≥	PROPN
cana-577	144	7	𝑞	𝑞	X
cana-577	144	8	2∆′(𝐺	2∆′(𝐺	NUM
cana-577	144	9	)	)	PUNCT
cana-577	144	10	+	+	CCONJ
cana-577	144	11	1	1	NUM
cana-577	144	12	⟹	⟹	NUM
cana-577	144	13	⌈	⌈	NOUN
cana-577	144	14	𝑞	𝑞	X
cana-577	144	15	2∆′(𝐺	2∆′(𝐺	NUM
cana-577	144	16	)	)	PUNCT
cana-577	144	17	+	+	CCONJ
cana-577	144	18	1	1	NUM
cana-577	144	19	⌉	⌉	NOUN
cana-577	144	20	≤	≤	NOUN
cana-577	144	21	𝑞	𝑞	PRON
cana-577	144	22	2∆′(𝐺	2∆′(𝐺	NUM
cana-577	144	23	)	)	PUNCT
cana-577	144	24	+	+	CCONJ
cana-577	144	25	1	1	NUM
cana-577	144	26	<	<	X
cana-577	144	27	|𝑋|	|𝑋|	NOUN
cana-577	144	28	+	+	NUM
cana-577	144	29	1	1	NUM
cana-577	144	30	⟹	⟹	NOUN
cana-577	144	31	⌈	⌈	NOUN
cana-577	144	32	𝑞	𝑞	X
cana-577	144	33	2∆′(𝐺	2∆′(𝐺	NUM
cana-577	144	34	)	)	PUNCT
cana-577	144	35	+	+	CCONJ
cana-577	144	36	1	1	NUM
cana-577	144	37	⌉	⌉	X
cana-577	144	38	<	<	X
cana-577	144	39	𝛾𝑠𝑡𝑟(𝑚(𝐺	𝛾𝑠𝑡𝑟(𝑚(𝐺	X
cana-577	144	40	)	)	PUNCT
cana-577	144	41	+	+	CCONJ
cana-577	144	42	1	1	NUM
cana-577	144	43	we	we	PRON
cana-577	144	44	derive	derive	VERB
cana-577	144	45	the	the	DET
cana-577	144	46	lower	lower	ADV
cana-577	144	47	bound	bind	VERB
cana-577	144	48	for	for	ADP
cana-577	144	49	𝛾𝑠𝑡𝑟(𝑚(𝐺	𝛾𝑠𝑡𝑟(𝑚(𝐺	NOUN
cana-577	144	50	)	)	PUNCT
cana-577	144	51	)	)	PUNCT
cana-577	144	52	in	in	ADP
cana-577	144	53	terms	term	NOUN
cana-577	144	54	of	of	ADP
cana-577	144	55	diameter	diameter	NOUN
cana-577	144	56	of	of	ADP
cana-577	144	57	g	g	PROPN
cana-577	144	58	in	in	ADP
cana-577	144	59	the	the	DET
cana-577	144	60	subsequent	subsequent	ADJ
cana-577	144	61	theorem	theorem	NOUN
cana-577	144	62	.	.	PUNCT
cana-577	145	1	theorem-3.12	theorem-3.12	NOUN
cana-577	145	2	:	:	PUNCT
cana-577	145	3	for	for	ADP
cana-577	145	4	any	any	DET
cana-577	145	5	‘	'	PUNCT
cana-577	145	6	graph	graph	NOUN
cana-577	145	7	g	g	NOUN
cana-577	145	8	’	'	PUNCT
cana-577	145	9	,	,	PUNCT
cana-577	145	10	𝑑𝑖𝑎𝑚(𝐺	𝑑𝑖𝑎𝑚(𝐺	X
cana-577	145	11	)	)	PUNCT
cana-577	145	12	−	−	ADP
cana-577	145	13	1	1	NUM
cana-577	145	14	≤	≤	NOUN
cana-577	145	15	3𝛾𝑠𝑡𝑟(𝑚(𝐺	3𝛾𝑠𝑡𝑟(𝑚(𝐺	NUM
cana-577	145	16	)	)	PUNCT
cana-577	145	17	)	)	PUNCT
cana-577	145	18	.	.	PUNCT
cana-577	146	1	proof	proof	NOUN
cana-577	146	2	:	:	PUNCT
cana-577	146	3	let	let	VERB
cana-577	146	4	the	the	DET
cana-577	146	5	collection	collection	NOUN
cana-577	146	6	of	of	ADP
cana-577	146	7	edges	edge	NOUN
cana-577	146	8	in	in	ADP
cana-577	146	9	g	g	PROPN
cana-577	146	10	that	that	PRON
cana-577	146	11	make	make	VERB
cana-577	146	12	up	up	ADP
cana-577	146	13	the	the	DET
cana-577	146	14	diameteral	diameteral	ADJ
cana-577	146	15	path	path	NOUN
cana-577	146	16	in	in	ADP
cana-577	146	17	g	g	PROPN
cana-577	146	18	be	be	AUX
cana-577	146	19	denoted	denote	VERB
cana-577	146	20	by	by	ADP
cana-577	146	21	𝐸1	𝐸1	NOUN
cana-577	146	22	=	=	SYM
cana-577	146	23	{	{	PUNCT
cana-577	146	24	𝑒𝑖	𝑒𝑖	X
cana-577	146	25	;	;	PUNCT
cana-577	146	26	1	1	NUM
cana-577	146	27	≤	≤	NUM
cana-577	146	28	𝑖	𝑖	SYM
cana-577	146	29	≤	≤	NUM
cana-577	146	30	𝑛	𝑛	NOUN
cana-577	146	31	}	}	PUNCT
cana-577	146	32	.	.	PUNCT
cana-577	147	1	therefore	therefore	ADV
cana-577	147	2	,	,	PUNCT
cana-577	147	3	|𝐸1|	|𝐸1|	PROPN
cana-577	147	4	=	=	PUNCT
cana-577	147	5	𝑑𝑖𝑎𝑚(𝐺	𝑑𝑖𝑎𝑚(𝐺	X
cana-577	147	6	)	)	PUNCT
cana-577	147	7	.	.	PUNCT
cana-577	148	1	let	let	VERB
cana-577	148	2	𝐸2	𝐸2	VERB
cana-577	148	3	⊆	⊆	NUM
cana-577	148	4	𝐸(𝐺	𝐸(𝐺	NOUN
cana-577	148	5	)	)	PUNCT
cana-577	148	6	,	,	PUNCT
cana-577	148	7	∀𝑒𝑖	∀𝑒𝑖	PROPN
cana-577	148	8	∈	∈	PROPN
cana-577	148	9	𝐸2	𝐸2	PROPN
cana-577	148	10	has	have	AUX
cana-577	148	11	the	the	DET
cana-577	148	12	maximal	maximal	ADJ
cana-577	148	13	edge	edge	NOUN
cana-577	148	14	degree	degree	NOUN
cana-577	148	15	in	in	ADP
cana-577	148	16	g.	g.	PROPN
cana-577	148	17	𝐸2	𝐸2	PROPN
cana-577	148	18	becomes	become	VERB
cana-577	148	19	a	a	DET
cana-577	148	20	minimum	minimum	ADJ
cana-577	148	21	dominating	dominating	NOUN
cana-577	148	22	set	set	NOUN
cana-577	148	23	of	of	ADP
cana-577	148	24	𝑚(𝐺	𝑚(𝐺	PROPN
cana-577	148	25	)	)	PUNCT
cana-577	148	26	since	since	SCONJ
cana-577	148	27	𝐸2	𝐸2	ADJ
cana-577	148	28	⊆	⊆	NUM
cana-577	148	29	𝑉(𝑚(𝐺	𝑉(𝑚(𝐺	NOUN
cana-577	148	30	)	)	PUNCT
cana-577	148	31	)	)	PUNCT
cana-577	148	32	,	,	PUNCT
cana-577	148	33	it	it	PRON
cana-577	148	34	is	be	AUX
cana-577	148	35	the	the	DET
cana-577	148	36	smallest	small	ADJ
cana-577	148	37	set	set	NOUN
cana-577	148	38	of	of	ADP
cana-577	148	39	points	point	NOUN
cana-577	148	40	which	which	PRON
cana-577	148	41	covers	cover	VERB
cana-577	148	42	all	all	PRON
cana-577	148	43	of	of	ADP
cana-577	148	44	the	the	DET
cana-577	148	45	vertices	vertex	NOUN
cana-577	148	46	in	in	ADP
cana-577	148	47	𝑚(𝐺	𝑚(𝐺	NUM
cana-577	148	48	)	)	PUNCT
cana-577	148	49	.	.	PUNCT
cana-577	149	1	moreover	moreover	ADV
cana-577	149	2	,	,	PUNCT
cana-577	149	3	𝐸2	𝐸2	PROPN
cana-577	149	4	is	be	AUX
cana-577	149	5	a	a	DET
cana-577	149	6	smallest	small	ADJ
cana-577	149	7	strong	strong	ADJ
cana-577	149	8	regular	regular	ADJ
cana-577	149	9	dominating	dominating	NOUN
cana-577	149	10	set	set	NOUN
cana-577	149	11	in	in	ADP
cana-577	149	12	𝑚(𝐺	𝑚(𝐺	NUM
cana-577	149	13	)	)	PUNCT
cana-577	149	14	if	if	SCONJ
cana-577	149	15	deg(𝑥	deg(𝑥	PROPN
cana-577	149	16	)	)	PUNCT
cana-577	149	17	≤	≤	NUM
cana-577	149	18	deg(𝑦	deg(𝑦	PROPN
cana-577	149	19	)	)	PUNCT
cana-577	149	20	,	,	PUNCT
cana-577	149	21	∀𝑥𝜖𝑉(𝑚(𝐺	∀𝑥𝜖𝑉(𝑚(𝐺	PROPN
cana-577	149	22	)	)	PUNCT
cana-577	149	23	)	)	PUNCT
cana-577	150	1	−	−	PROPN
cana-577	150	2	𝐸2	𝐸2	ADJ
cana-577	150	3	and	and	CCONJ
cana-577	150	4	𝐸2	𝐸2	PROPN
cana-577	150	5	is	be	AUX
cana-577	150	6	k	k	NOUN
cana-577	150	7	-	-	ADJ
cana-577	150	8	regular	regular	ADJ
cana-577	150	9	.	.	PUNCT
cana-577	151	1	consequently	consequently	ADV
cana-577	151	2	,	,	PUNCT
cana-577	151	3	|𝐸1|	|𝐸1|	PROPN
cana-577	151	4	−	−	PROPN
cana-577	151	5	1	1	NUM
cana-577	151	6	≤	≤	NUM
cana-577	151	7	3|𝐸2|	3|𝐸2|	NUM
cana-577	151	8	⟹	⟹	NUM
cana-577	151	9	𝑑𝑖𝑎𝑚(𝐺	𝑑𝑖𝑎𝑚(𝐺	NOUN
cana-577	151	10	)	)	PUNCT
cana-577	151	11	−	−	ADP
cana-577	151	12	1	1	NUM
cana-577	151	13	≤	≤	NOUN
cana-577	151	14	3𝛾𝑠𝑡𝑟(𝑚(𝐺	3𝛾𝑠𝑡𝑟(𝑚(𝐺	NUM
cana-577	151	15	)	)	PUNCT
cana-577	151	16	.	.	PUNCT
cana-577	152	1	hence	hence	ADV
cana-577	152	2	,	,	PUNCT
cana-577	152	3	𝑑𝑖𝑎𝑚(𝐺	𝑑𝑖𝑎𝑚(𝐺	PUNCT
cana-577	152	4	)	)	PUNCT
cana-577	152	5	−	−	ADP
cana-577	152	6	1	1	NUM
cana-577	152	7	≤	≤	NOUN
cana-577	152	8	3𝛾𝑠𝑡𝑟(𝑚(𝐺	3𝛾𝑠𝑡𝑟(𝑚(𝐺	NUM
cana-577	152	9	)	)	PUNCT
cana-577	152	10	.	.	PUNCT
cana-577	153	1	the	the	DET
cana-577	153	2	subsequent	subsequent	ADJ
cana-577	153	3	theorem	theorem	NOUN
cana-577	153	4	relates	relate	VERB
cana-577	153	5	𝛾𝑠𝑡𝑟(𝑚(𝐺	𝛾𝑠𝑡𝑟(𝑚(𝐺	NUM
cana-577	153	6	)	)	PUNCT
cana-577	153	7	)	)	PUNCT
cana-577	153	8	,	,	PUNCT
cana-577	153	9	𝛾𝑐(𝐺	𝛾𝑐(𝐺	NOUN
cana-577	153	10	)	)	PUNCT
cana-577	153	11	,	,	PUNCT
cana-577	153	12	𝑑𝑖𝑎𝑚(𝐺	𝑑𝑖𝑎𝑚(𝐺	PROPN
cana-577	153	13	)	)	PUNCT
cana-577	153	14	,	,	PUNCT
cana-577	153	15	𝛼0(𝐺	𝛼0(𝐺	PROPN
cana-577	153	16	)	)	PUNCT
cana-577	153	17	and	and	CCONJ
cana-577	153	18	𝛾(𝐺	𝛾(𝐺	NUM
cana-577	153	19	)	)	PUNCT
cana-577	153	20	.	.	PUNCT
cana-577	154	1	theorem-3.13	theorem-3.13	NUM
cana-577	154	2	:	:	PUNCT
cana-577	154	3	for	for	ADP
cana-577	154	4	any	any	DET
cana-577	154	5	connected	connected	ADJ
cana-577	154	6	graph	graph	NOUN
cana-577	154	7	g	g	NOUN
cana-577	154	8	,	,	PUNCT
cana-577	154	9	𝛾𝑠𝑡𝑟(𝑚(𝐺	𝛾𝑠𝑡𝑟(𝑚(𝐺	PROPN
cana-577	154	10	)	)	PUNCT
cana-577	154	11	)	)	PUNCT
cana-577	155	1	+	+	CCONJ
cana-577	155	2	𝛾𝑐(𝐺	𝛾𝑐(𝐺	NOUN
cana-577	155	3	)	)	PUNCT
cana-577	155	4	<	<	X
cana-577	155	5	𝑑𝑖𝑎𝑚(𝐺	𝑑𝑖𝑎𝑚(𝐺	X
cana-577	155	6	)	)	PUNCT
cana-577	156	1	+	+	CCONJ
cana-577	156	2	𝛼0(𝐺	𝛼0(𝐺	PROPN
cana-577	156	3	)	)	PUNCT
cana-577	157	1	+	+	NUM
cana-577	157	2	𝛾(𝐺	𝛾(𝐺	NOUN
cana-577	157	3	)	)	PUNCT
cana-577	157	4	.	.	PUNCT
cana-577	158	1	proof	proof	NOUN
cana-577	158	2	:	:	PUNCT
cana-577	158	3	define	define	VERB
cana-577	158	4	𝐴	𝐴	PROPN
cana-577	158	5	⊆	⊆	NUM
cana-577	158	6	𝑉(𝐺	𝑉(𝐺	NOUN
cana-577	158	7	)	)	PUNCT
cana-577	158	8	as	as	ADP
cana-577	158	9	the	the	DET
cana-577	158	10	smallest	small	ADJ
cana-577	158	11	set	set	NOUN
cana-577	158	12	of	of	ADP
cana-577	158	13	vertices	vertex	NOUN
cana-577	158	14	that	that	PRON
cana-577	158	15	encompasses	encompass	VERB
cana-577	158	16	all	all	DET
cana-577	158	17	edges	edge	NOUN
cana-577	158	18	in	in	ADP
cana-577	158	19	g	g	NOUN
cana-577	158	20	,	,	PUNCT
cana-577	158	21	ensuring	ensure	VERB
cana-577	158	22	that	that	SCONJ
cana-577	158	23	|𝐴|	|𝐴|	NOUN
cana-577	158	24	equals	equal	VERB
cana-577	158	25	𝛼0(𝐺	𝛼0(𝐺	PROPN
cana-577	158	26	)	)	PUNCT
cana-577	158	27	.	.	PUNCT
cana-577	159	1	moreover	moreover	ADV
cana-577	159	2	,	,	PUNCT
cana-577	159	3	an	an	DET
cana-577	159	4	edge	edge	NOUN
cana-577	159	5	set	set	VERB
cana-577	159	6	𝐸	𝐸	PROPN
cana-577	159	7	⊆	⊆	NUM
cana-577	159	8	𝐸′	𝐸′	PROPN
cana-577	159	9	exists	exist	NOUN
cana-577	159	10	,	,	PUNCT
cana-577	159	11	where	where	SCONJ
cana-577	159	12	𝐸′	𝐸′	PROPN
cana-577	159	13	is	be	AUX
cana-577	159	14	the	the	DET
cana-577	159	15	collection	collection	NOUN
cana-577	159	16	of	of	ADP
cana-577	159	17	edges	edge	NOUN
cana-577	159	18	that	that	PRON
cana-577	159	19	coincide	coincide	VERB
cana-577	159	20	with	with	ADP
cana-577	159	21	the	the	DET
cana-577	159	22	vertices	vertex	NOUN
cana-577	159	23	of	of	ADP
cana-577	159	24	v	v	NOUN
cana-577	159	25	to	to	PART
cana-577	159	26	form	form	VERB
cana-577	159	27	the	the	DET
cana-577	159	28	longest	long	ADJ
cana-577	159	29	path	path	NOUN
cana-577	159	30	in	in	ADP
cana-577	159	31	g	g	NOUN
cana-577	159	32	,	,	PUNCT
cana-577	159	33	with	with	ADP
cana-577	159	34	|𝐸|	|𝐸|	NOUN
cana-577	159	35	equal	equal	ADJ
cana-577	159	36	to	to	ADP
cana-577	159	37	𝑑𝑖𝑎𝑚(𝐺	𝑑𝑖𝑎𝑚(𝐺	NOUN
cana-577	159	38	)	)	PUNCT
cana-577	159	39	.	.	PUNCT
cana-577	160	1	to	to	PART
cana-577	160	2	build	build	VERB
cana-577	160	3	a	a	DET
cana-577	160	4	‘	'	PUNCT
cana-577	160	5	minimal	minimal	ADJ
cana-577	160	6	dominating	dominating	NOUN
cana-577	160	7	set	set	NOUN
cana-577	160	8	of	of	ADP
cana-577	160	9	g	g	NOUN
cana-577	160	10	’	'	PUNCT
cana-577	160	11	,	,	PUNCT
cana-577	160	12	let	let	VERB
cana-577	160	13	𝑆	𝑆	PROPN
cana-577	160	14	=	=	PRON
cana-577	160	15	{	{	PUNCT
cana-577	160	16	𝑣𝑖	𝑣𝑖	ADP
cana-577	160	17	;	;	PUNCT
cana-577	160	18	1	1	NUM
cana-577	160	19	≤	≤	NUM
cana-577	160	20	𝑖	𝑖	SYM
cana-577	160	21	≤	≤	NUM
cana-577	160	22	𝑛	𝑛	PRON
cana-577	160	23	}	}	PUNCT
cana-577	160	24	⊆	⊆	NUM
cana-577	160	25	𝑉(𝐺	𝑉(𝐺	NOUN
cana-577	160	26	)	)	PUNCT
cana-577	160	27	.	.	PUNCT
cana-577	161	1	if	if	SCONJ
cana-577	161	2	〈	〈	PROPN
cana-577	161	3	𝑆′	𝑆′	ADJ
cana-577	161	4	〉	〉	NOUN
cana-577	161	5	is	be	AUX
cana-577	161	6	a	a	DET
cana-577	161	7	connected	connected	ADJ
cana-577	161	8	sub	sub	NOUN
cana-577	161	9	graph	graph	NOUN
cana-577	161	10	of	of	ADP
cana-577	161	11	communications	communication	NOUN
cana-577	161	12	on	on	ADP
cana-577	161	13	applied	apply	VERB
cana-577	161	14	nonlinear	nonlinear	ADJ
cana-577	161	15	analysis	analysis	NOUN
cana-577	161	16	issn	issn	NOUN
cana-577	161	17	:	:	PUNCT
cana-577	161	18	1074	1074	NUM
cana-577	161	19	-	-	PUNCT
cana-577	161	20	133x	133x	NUM
cana-577	161	21	vol	vol	NOUN
cana-577	161	22	31	31	NUM
cana-577	161	23	no	no	NOUN
cana-577	161	24	.	.	PUNCT
cana-577	162	1	1s	1s	NUM
cana-577	162	2	(	(	PUNCT
cana-577	162	3	2024	2024	NUM
cana-577	162	4	)	)	PUNCT
cana-577	162	5	185	185	NUM
cana-577	162	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-577	162	7	s	s	X
cana-577	162	8	,	,	PUNCT
cana-577	162	9	then	then	ADV
cana-577	162	10	‘s	‘	VERB
cana-577	162	11	is	be	AUX
cana-577	162	12	a	a	DET
cana-577	162	13	connected	connected	ADJ
cana-577	162	14	dominating	dominating	NOUN
cana-577	162	15	set	set	NOUN
cana-577	162	16	’	'	PUNCT
cana-577	162	17	.	.	PUNCT
cana-577	163	1	in	in	ADP
cana-577	163	2	the	the	DET
cana-577	163	3	absence	absence	NOUN
cana-577	163	4	of	of	ADP
cana-577	163	5	this	this	PRON
cana-577	163	6	,	,	PUNCT
cana-577	163	7	a	a	DET
cana-577	163	8	‘	'	PUNCT
cana-577	163	9	minimal	minimal	ADJ
cana-577	163	10	connected	connected	ADJ
cana-577	163	11	dominating	dominating	NOUN
cana-577	163	12	set	set	NOUN
cana-577	163	13	of	of	ADP
cana-577	163	14	g	g	NOUN
cana-577	163	15	’	'	PUNCT
cana-577	163	16	is	be	AUX
cana-577	163	17	formed	form	VERB
cana-577	163	18	by	by	ADP
cana-577	163	19	at	at	ADV
cana-577	163	20	least	least	ADV
cana-577	163	21	one	one	NUM
cana-577	163	22	vertex	vertex	NOUN
cana-577	163	23	,	,	PUNCT
cana-577	163	24	𝑥	𝑥	DET
cana-577	163	25	∈	∈	PROPN
cana-577	163	26	𝑉(𝐺	𝑉(𝐺	NOUN
cana-577	163	27	)	)	PUNCT
cana-577	163	28	−	−	PROPN
cana-577	163	29	𝑆′	𝑆′	PROPN
cana-577	163	30	and	and	CCONJ
cana-577	163	31	𝑆	𝑆	PROPN
cana-577	163	32	"	"	PUNCT
cana-577	163	33	=	=	PROPN
cana-577	163	34	𝑆′	𝑆′	PROPN
cana-577	163	35	∪	∪	X
cana-577	163	36	{	{	PUNCT
cana-577	163	37	𝑥	𝑥	NOUN
cana-577	163	38	}	}	PUNCT
cana-577	163	39	.	.	PUNCT
cana-577	164	1	now	now	ADV
cana-577	164	2	,	,	PUNCT
cana-577	164	3	in	in	ADP
cana-577	164	4	𝑚(𝐺	𝑚(𝐺	NUM
cana-577	164	5	)	)	PUNCT
cana-577	164	6	,	,	PUNCT
cana-577	164	7	let	let	VERB
cana-577	164	8	𝑈	𝑈	PROPN
cana-577	164	9	=	=	PRON
cana-577	164	10	{	{	PUNCT
cana-577	164	11	𝑢𝑗	𝑢𝑗	NOUN
cana-577	164	12	;	;	PUNCT
cana-577	164	13	1	1	NUM
cana-577	164	14	≤	≤	NUM
cana-577	164	15	𝑗	𝑗	PRON
cana-577	164	16	≤	≤	NUM
cana-577	164	17	𝑛	𝑛	NOUN
cana-577	164	18	}	}	PUNCT
cana-577	164	19	⊆	⊆	NUM
cana-577	164	20	𝑉(𝑚(𝐺	𝑉(𝑚(𝐺	NOUN
cana-577	164	21	)	)	PUNCT
cana-577	164	22	)	)	PUNCT
cana-577	165	1	such	such	ADJ
cana-577	165	2	that	that	SCONJ
cana-577	165	3	{	{	PUNCT
cana-577	165	4	𝑢𝑗	𝑢𝑗	NOUN
cana-577	165	5	}	}	PUNCT
cana-577	165	6	=	=	SYM
cana-577	165	7	{	{	PUNCT
cana-577	165	8	𝑒𝑗	𝑒𝑗	NOUN
cana-577	165	9	}	}	PUNCT
cana-577	165	10	∈	∈	PROPN
cana-577	165	11	𝐸(𝐺	𝐸(𝐺	NOUN
cana-577	165	12	)	)	PUNCT
cana-577	165	13	,	,	PUNCT
cana-577	165	14	1	1	NUM
cana-577	165	15	≤	≤	NUM
cana-577	165	16	𝑗	𝑗	PRON
cana-577	165	17	≤	≤	NUM
cana-577	165	18	𝑛	𝑛	NOUN
cana-577	165	19	,	,	PUNCT
cana-577	165	20	where	where	SCONJ
cana-577	165	21	{	{	PUNCT
cana-577	165	22	𝑒𝑗	𝑒𝑗	NOUN
cana-577	165	23	}	}	PUNCT
cana-577	165	24	are	be	AUX
cana-577	165	25	incident	incident	NOUN
cana-577	165	26	with	with	ADP
cana-577	165	27	the	the	DET
cana-577	165	28	vertices	vertex	NOUN
cana-577	165	29	of	of	ADP
cana-577	165	30	𝑆′.	𝑆′.	NOUN
cana-577	165	31	additionally	additionally	ADV
cana-577	165	32	,	,	PUNCT
cana-577	165	33	assume	assume	VERB
cana-577	165	34	that	that	SCONJ
cana-577	165	35	d	d	NOUN
cana-577	165	36	represents	represent	VERB
cana-577	165	37	the	the	DET
cana-577	165	38	dominant	dominant	ADJ
cana-577	165	39	set	set	NOUN
cana-577	165	40	of	of	ADP
cana-577	165	41	𝑚(𝐺	𝑚(𝐺	PROPN
cana-577	165	42	)	)	PUNCT
cana-577	165	43	and	and	CCONJ
cana-577	165	44	that	that	SCONJ
cana-577	165	45	∀𝑢𝑘	∀𝑢𝑘	PRON
cana-577	165	46	∈	∈	PROPN
cana-577	165	47	〈	〈	NOUN
cana-577	165	48	𝑉(𝑚(𝐺	𝑉(𝑚(𝐺	NOUN
cana-577	165	49	)	)	PUNCT
cana-577	165	50	)	)	PUNCT
cana-577	166	1	−	−	PROPN
cana-577	166	2	𝐷	𝐷	NOUN
cana-577	166	3	〉	〉	NOUN
cana-577	166	4	,	,	PUNCT
cana-577	166	5	deg(𝑢𝑘	deg(𝑢𝑘	NOUN
cana-577	166	6	)	)	PUNCT
cana-577	166	7	≤	≤	NUM
cana-577	166	8	deg(𝑢𝑗	deg(𝑢𝑗	NOUN
cana-577	166	9	)	)	PUNCT
cana-577	166	10	,	,	PUNCT
cana-577	166	11	∀𝑢𝑗	∀𝑢𝑗	PROPN
cana-577	166	12	∈	∈	PROPN
cana-577	166	13	𝐷	𝐷	PROPN
cana-577	166	14	and	and	CCONJ
cana-577	166	15	d	d	PROPN
cana-577	166	16	is	be	AUX
cana-577	166	17	k	k	NOUN
cana-577	166	18	-	-	ADJ
cana-577	166	19	regular	regular	ADJ
cana-577	166	20	.	.	PUNCT
cana-577	167	1	so	so	ADV
cana-577	167	2	,	,	PUNCT
cana-577	167	3	d	d	PRON
cana-577	167	4	creates	create	VERB
cana-577	167	5	a	a	DET
cana-577	167	6	strong	strong	ADJ
cana-577	167	7	regular	regular	ADJ
cana-577	167	8	dominating	dominating	NOUN
cana-577	167	9	set	set	NOUN
cana-577	167	10	in	in	ADP
cana-577	167	11	𝑚(𝐺	𝑚(𝐺	NUM
cana-577	167	12	)	)	PUNCT
cana-577	167	13	.	.	PUNCT
cana-577	168	1	otherwise	otherwise	ADV
cana-577	168	2	,	,	PUNCT
cana-577	168	3	deg(𝑢𝑘	deg(𝑢𝑘	NOUN
cana-577	168	4	)	)	PUNCT
cana-577	168	5	>	>	X
cana-577	168	6	deg(𝑢𝑗	deg(𝑢𝑗	NOUN
cana-577	168	7	)	)	PUNCT
cana-577	168	8	,	,	PUNCT
cana-577	168	9	∀𝑢𝑗	∀𝑢𝑗	PROPN
cana-577	168	10	∈	∈	PROPN
cana-577	168	11	𝐷	𝐷	PROPN
cana-577	168	12	,	,	PUNCT
cana-577	168	13	exists	exist	VERB
cana-577	168	14	for	for	ADP
cana-577	168	15	at	at	ADV
cana-577	168	16	least	least	ADV
cana-577	168	17	one	one	NUM
cana-577	168	18	vertex	vertex	NOUN
cana-577	168	19	{	{	PUNCT
cana-577	168	20	𝑢	𝑢	X
cana-577	168	21	}	}	PUNCT
cana-577	168	22	∈	∈	PROPN
cana-577	168	23	𝑉(𝑚(𝐺	𝑉(𝑚(𝐺	NOUN
cana-577	168	24	)	)	PUNCT
cana-577	168	25	)	)	PUNCT
cana-577	169	1	−	−	ADP
cana-577	169	2	𝐷.	𝐷.	PROPN
cana-577	169	3	it	it	PRON
cana-577	169	4	is	be	AUX
cana-577	169	5	evident	evident	ADJ
cana-577	169	6	that	that	SCONJ
cana-577	169	7	𝐷	𝐷	NOUN
cana-577	169	8	∪	∪	NOUN
cana-577	169	9	{	{	PUNCT
cana-577	169	10	𝑢	𝑢	NOUN
cana-577	169	11	}	}	PUNCT
cana-577	169	12	constitutes	constitute	VERB
cana-577	169	13	a	a	DET
cana-577	169	14	minimal	minimal	ADJ
cana-577	169	15	strong	strong	ADJ
cana-577	169	16	regular	regular	ADJ
cana-577	169	17	dominating	dominating	NOUN
cana-577	169	18	set	set	NOUN
cana-577	169	19	within	within	ADP
cana-577	169	20	𝑚(𝐺	𝑚(𝐺	PROPN
cana-577	169	21	)	)	PUNCT
cana-577	169	22	.	.	PUNCT
cana-577	170	1	consequently	consequently	ADV
cana-577	170	2	,	,	PUNCT
cana-577	170	3	|𝐷	|𝐷	NOUN
cana-577	170	4	∪	∪	X
cana-577	170	5	{	{	PUNCT
cana-577	170	6	𝑢}|	𝑢}|	NOUN
cana-577	170	7	∪	∪	ADP
cana-577	170	8	|𝑆"|	|𝑆"|	PROPN
cana-577	170	9	<	<	X
cana-577	170	10	|𝐸|	|𝐸|	NOUN
cana-577	170	11	+	+	CCONJ
cana-577	170	12	|𝑆′|	|𝑆′|	NOUN
cana-577	170	13	+	+	CCONJ
cana-577	170	14	|𝐴|	|𝐴|	VERB
cana-577	170	15	⟹	⟹	X
cana-577	170	16	𝛾𝑠𝑡𝑟(𝑚(𝐺	𝛾𝑠𝑡𝑟(𝑚(𝐺	PROPN
cana-577	170	17	)	)	PUNCT
cana-577	170	18	+	+	NUM
cana-577	170	19	𝛾𝑐(𝐺	𝛾𝑐(𝐺	NOUN
cana-577	170	20	)	)	PUNCT
cana-577	170	21	<	<	X
cana-577	170	22	𝑑𝑖𝑎𝑚(𝐺	𝑑𝑖𝑎𝑚(𝐺	X
cana-577	170	23	)	)	PUNCT
cana-577	171	1	+	+	CCONJ
cana-577	171	2	𝛼0(𝐺	𝛼0(𝐺	PROPN
cana-577	171	3	)	)	PUNCT
cana-577	172	1	+	+	NUM
cana-577	172	2	𝛾(𝐺	𝛾(𝐺	NOUN
cana-577	172	3	)	)	PUNCT
cana-577	172	4	the	the	DET
cana-577	172	5	following	follow	VERB
cana-577	172	6	outcome	outcome	NOUN
cana-577	172	7	has	have	AUX
cana-577	172	8	been	be	AUX
cana-577	172	9	ascertained	ascertain	VERB
cana-577	172	10	.	.	PUNCT
cana-577	173	1	𝛾𝑠𝑡𝑟(𝑚(𝑇	𝛾𝑠𝑡𝑟(𝑚(𝑇	PROPN
cana-577	173	2	)	)	PUNCT
cana-577	173	3	)	)	PUNCT
cana-577	173	4	concerning	concern	VERB
cana-577	173	5	𝛾(𝑇	𝛾(𝑇	PROPN
cana-577	173	6	)	)	PUNCT
cana-577	173	7	and	and	CCONJ
cana-577	173	8	𝛾𝑐(𝑇	𝛾𝑐(𝑇	ADV
cana-577	173	9	)	)	PUNCT
cana-577	173	10	.	.	PUNCT
cana-577	174	1	theorem-3.14	theorem-3.14	NOUN
cana-577	174	2	:	:	PUNCT
cana-577	174	3	for	for	ADP
cana-577	174	4	any	any	DET
cana-577	174	5	tree	tree	NOUN
cana-577	174	6	t	t	PROPN
cana-577	174	7	,	,	PUNCT
cana-577	174	8	𝛾𝑠𝑡𝑟(𝑚(𝑇	𝛾𝑠𝑡𝑟(𝑚(𝑇	PROPN
cana-577	174	9	)	)	PUNCT
cana-577	174	10	)	)	PUNCT
cana-577	174	11	≤	≤	NUM
cana-577	174	12	𝛾(𝑇	𝛾(𝑇	PROPN
cana-577	174	13	)	)	PUNCT
cana-577	174	14	+	+	CCONJ
cana-577	174	15	𝛾𝑐(𝑇	𝛾𝑐(𝑇	ADJ
cana-577	174	16	)	)	PUNCT
cana-577	174	17	.	.	PUNCT
cana-577	175	1	proof	proof	NOUN
cana-577	175	2	:	:	PUNCT
cana-577	175	3	assume	assume	VERB
cana-577	175	4	that	that	SCONJ
cana-577	175	5	𝑋	𝑋	PROPN
cana-577	175	6	=	=	SYM
cana-577	175	7	{	{	PUNCT
cana-577	175	8	𝑣𝑖	𝑣𝑖	NOUN
cana-577	175	9	;	;	PUNCT
cana-577	175	10	1	1	NUM
cana-577	175	11	≤	≤	NUM
cana-577	175	12	𝑖	𝑖	SYM
cana-577	175	13	≤	≤	NUM
cana-577	175	14	𝑛	𝑛	PRON
cana-577	175	15	}	}	PUNCT
cana-577	175	16	⊆	⊆	NUM
cana-577	175	17	𝑉(𝑇	𝑉(𝑇	NOUN
cana-577	175	18	)	)	PUNCT
cana-577	175	19	is	be	AUX
cana-577	175	20	the	the	DET
cana-577	175	21	smallest	small	ADJ
cana-577	175	22	set	set	NOUN
cana-577	175	23	of	of	ADP
cana-577	175	24	vertices	vertex	NOUN
cana-577	175	25	that	that	PRON
cana-577	175	26	encompasses	encompass	VERB
cana-577	175	27	every	every	DET
cana-577	175	28	edge	edge	NOUN
cana-577	175	29	in	in	ADP
cana-577	175	30	t.	t.	NOUN
cana-577	175	31	it	it	PRON
cana-577	175	32	is	be	AUX
cana-577	175	33	evident	evident	ADJ
cana-577	175	34	that	that	SCONJ
cana-577	175	35	𝑋	𝑋	PROPN
cana-577	175	36	makes	make	VERB
cana-577	175	37	up	up	ADP
cana-577	175	38	t	t	PROPN
cana-577	175	39	's	's	PART
cana-577	175	40	minimal	minimal	ADJ
cana-577	175	41	dominating	dominating	NOUN
cana-577	175	42	set	set	NOUN
cana-577	175	43	,	,	PUNCT
cana-577	175	44	so	so	SCONJ
cana-577	175	45	that	that	SCONJ
cana-577	175	46	|𝑋|	|𝑋|	ADV
cana-577	175	47	=	=	PUNCT
cana-577	175	48	𝛾(𝑇	𝛾(𝑇	PROPN
cana-577	175	49	)	)	PUNCT
cana-577	175	50	.	.	PUNCT
cana-577	176	1	assume	assume	VERB
cana-577	176	2	that	that	SCONJ
cana-577	176	3	〈	〈	NOUN
cana-577	176	4	𝑋′	𝑋′	X
cana-577	176	5	〉	〉	NOUN
cana-577	176	6	is	be	AUX
cana-577	176	7	a	a	DET
cana-577	176	8	connected	connected	ADJ
cana-577	176	9	sub	sub	NOUN
cana-577	176	10	graph	graph	NOUN
cana-577	176	11	of	of	ADP
cana-577	176	12	𝑋.	𝑋.	PROPN
cana-577	176	13	then	then	ADV
cana-577	176	14	𝑋′	𝑋′	PROPN
cana-577	176	15	is	be	AUX
cana-577	176	16	a	a	DET
cana-577	176	17	connected	connected	ADJ
cana-577	176	18	dominating	dominating	NOUN
cana-577	176	19	set	set	VERB
cana-577	176	20	in	in	ADP
cana-577	176	21	and	and	CCONJ
cana-577	176	22	of	of	ADP
cana-577	176	23	itself	itself	PRON
cana-577	176	24	,	,	PUNCT
cana-577	176	25	so	so	SCONJ
cana-577	176	26	that	that	SCONJ
cana-577	176	27	|𝑋′|	|𝑋′|	NOUN
cana-577	176	28	=	=	PUNCT
cana-577	176	29	𝛾𝑐(𝑇	𝛾𝑐(𝑇	ADJ
cana-577	176	30	)	)	PUNCT
cana-577	176	31	.	.	PUNCT
cana-577	177	1	in	in	ADP
cana-577	177	2	the	the	DET
cana-577	177	3	absence	absence	NOUN
cana-577	177	4	of	of	ADP
cana-577	177	5	this	this	PRON
cana-577	177	6	,	,	PUNCT
cana-577	177	7	𝑋	𝑋	PROPN
cana-577	177	8	"	"	PUNCT
cana-577	177	9	=	=	SYM
cana-577	177	10	𝑋′	𝑋′	X
cana-577	177	11	∪	∪	X
cana-577	177	12	{	{	PUNCT
cana-577	177	13	𝑥	𝑥	NOUN
cana-577	177	14	}	}	PUNCT
cana-577	177	15	forms	form	NOUN
cana-577	177	16	a	a	DET
cana-577	177	17	‘	'	PUNCT
cana-577	177	18	minimal	minimal	ADJ
cana-577	177	19	connected	connected	ADJ
cana-577	177	20	dominating	dominating	NOUN
cana-577	177	21	set	set	NOUN
cana-577	177	22	’	'	PUNCT
cana-577	177	23	of	of	ADP
cana-577	177	24	t	t	PROPN
cana-577	177	25	,	,	PUNCT
cana-577	177	26	and	and	CCONJ
cana-577	177	27	there	there	PRON
cana-577	177	28	exists	exist	VERB
cana-577	177	29	at	at	ADP
cana-577	177	30	least	least	ADV
cana-577	177	31	one	one	NUM
cana-577	177	32	vertex	vertex	NOUN
cana-577	177	33	𝑥	𝑥	PRON
cana-577	177	34	∈	∈	NOUN
cana-577	177	35	𝑉(𝑇	𝑉(𝑇	NOUN
cana-577	177	36	)	)	PUNCT
cana-577	178	1	−	−	PROPN
cana-577	178	2	𝑋′.	𝑋′.	NOUN
cana-577	178	3	according	accord	VERB
cana-577	178	4	to	to	ADP
cana-577	178	5	this	this	PRON
cana-577	178	6	,	,	PUNCT
cana-577	178	7	d	d	PRON
cana-577	178	8	creates	create	VERB
cana-577	178	9	a	a	DET
cana-577	178	10	strong	strong	ADJ
cana-577	178	11	regular	regular	ADJ
cana-577	178	12	dominating	dominating	NOUN
cana-577	178	13	set	set	NOUN
cana-577	178	14	in	in	ADP
cana-577	178	15	𝑚(𝑇	𝑚(𝑇	PROPN
cana-577	178	16	)	)	PUNCT
cana-577	178	17	,	,	PUNCT
cana-577	178	18	so	so	SCONJ
cana-577	178	19	that	that	SCONJ
cana-577	178	20	|𝐷|	|𝐷|	NOUN
cana-577	178	21	=	=	SYM
cana-577	178	22	𝛾𝑠𝑡𝑟(𝑚(𝑇	𝛾𝑠𝑡𝑟(𝑚(𝑇	PROPN
cana-577	178	23	)	)	PUNCT
cana-577	178	24	)	)	PUNCT
cana-577	178	25	.	.	PUNCT
cana-577	179	1	consequently	consequently	ADV
cana-577	179	2	,	,	PUNCT
cana-577	179	3	|𝐷|	|𝐷|	VERB
cana-577	179	4	⊆	⊆	NUM
cana-577	179	5	|𝑋|	|𝑋|	NOUN
cana-577	179	6	∪	∪	ADP
cana-577	179	7	|𝑋′|	|𝑋′|	PROPN
cana-577	179	8	⟹	⟹	NUM
cana-577	179	9	𝛾𝑠𝑡𝑟(𝑚(𝑇	𝛾𝑠𝑡𝑟(𝑚(𝑇	PROPN
cana-577	179	10	)	)	PUNCT
cana-577	179	11	)	)	PUNCT
cana-577	179	12	≤	≤	NUM
cana-577	179	13	𝛾(𝑇	𝛾(𝑇	PROPN
cana-577	179	14	)	)	PUNCT
cana-577	179	15	+	+	CCONJ
cana-577	179	16	𝛾𝑐(𝑇	𝛾𝑐(𝑇	ADJ
cana-577	179	17	)	)	PUNCT
cana-577	179	18	.	.	PUNCT
cana-577	180	1	we	we	PRON
cana-577	180	2	require	require	VERB
cana-577	180	3	the	the	DET
cana-577	180	4	following	follow	VERB
cana-577	180	5	theorems	theorem	NOUN
cana-577	180	6	in	in	ADP
cana-577	180	7	order	order	NOUN
cana-577	180	8	to	to	PART
cana-577	180	9	demonstrate	demonstrate	VERB
cana-577	180	10	our	our	PRON
cana-577	180	11	subsequent	subsequent	ADJ
cana-577	180	12	findings	finding	NOUN
cana-577	180	13	.	.	PUNCT
cana-577	181	1	theorem	theorem	VERB
cana-577	181	2	-	-	PUNCT
cana-577	181	3	a[4	a[4	PROPN
cana-577	181	4	]	]	X
cana-577	181	5	:	:	PUNCT
cana-577	181	6	for	for	ADP
cana-577	181	7	each	each	DET
cana-577	181	8	graph	graph	NOUN
cana-577	181	9	g	g	PROPN
cana-577	181	10	,	,	PUNCT
cana-577	181	11	𝛾𝑚	𝛾𝑚	ADP
cana-577	181	12	−1(𝐺	−1(𝐺	NOUN
cana-577	181	13	)	)	PUNCT
cana-577	181	14	<	<	X
cana-577	181	15	𝛾𝑐(𝐺	𝛾𝑐(𝐺	NOUN
cana-577	181	16	)	)	PUNCT
cana-577	181	17	+	+	CCONJ
cana-577	181	18	𝛾𝑡(𝐺	𝛾𝑡(𝐺	PROPN
cana-577	181	19	)	)	PUNCT
cana-577	181	20	.	.	PUNCT
cana-577	182	1	theorem	theorem	PROPN
cana-577	182	2	-	-	PROPN
cana-577	182	3	e[5	e[5	PROPN
cana-577	182	4	]	]	X
cana-577	182	5	:	:	PUNCT
cana-577	182	6	in	in	ADP
cana-577	182	7	each	each	DET
cana-577	182	8	graph	graph	NOUN
cana-577	182	9	g	g	NOUN
cana-577	182	10	,	,	PUNCT
cana-577	182	11	𝛾𝑚	𝛾𝑚	NOUN
cana-577	182	12	′	′	NUM
cana-577	182	13	(	(	PUNCT
cana-577	182	14	𝐺	𝐺	NOUN
cana-577	182	15	)	)	PUNCT
cana-577	182	16	+	+	CCONJ
cana-577	182	17	𝑑𝑖𝑎𝑚(𝐺	𝑑𝑖𝑎𝑚(𝐺	NOUN
cana-577	182	18	)	)	PUNCT
cana-577	182	19	≤	≤	NOUN
cana-577	182	20	𝑝	𝑝	ADP
cana-577	182	21	+	+	NUM
cana-577	182	22	𝛾(𝐺	𝛾(𝐺	NOUN
cana-577	182	23	)	)	PUNCT
cana-577	182	24	.	.	PUNCT
cana-577	183	1	corollary-3.15	corollary-3.15	PROPN
cana-577	183	2	:	:	PUNCT
cana-577	183	3	for	for	ADP
cana-577	183	4	each	each	DET
cana-577	183	5	graph	graph	NOUN
cana-577	183	6	g	g	NOUN
cana-577	183	7	,	,	PUNCT
cana-577	183	8	𝛾𝑠𝑡𝑟(𝑚(𝐺	𝛾𝑠𝑡𝑟(𝑚(𝐺	PROPN
cana-577	183	9	)	)	PUNCT
cana-577	183	10	)	)	PUNCT
cana-577	184	1	+	+	CCONJ
cana-577	184	2	𝛾𝑚	𝛾𝑚	NUM
cana-577	184	3	−1(𝐺	−1(𝐺	NOUN
cana-577	184	4	)	)	PUNCT
cana-577	184	5	<	<	X
cana-577	184	6	𝑝	𝑝	PROPN
cana-577	184	7	+	+	CCONJ
cana-577	184	8	𝛾𝑐(𝐺	𝛾𝑐(𝐺	NOUN
cana-577	184	9	)	)	PUNCT
cana-577	184	10	+	+	CCONJ
cana-577	184	11	𝛾𝑡(𝐺	𝛾𝑡(𝐺	PROPN
cana-577	184	12	)	)	PUNCT
cana-577	184	13	.	.	PUNCT
cana-577	185	1	proof	proof	NOUN
cana-577	185	2	:	:	PUNCT
cana-577	185	3	theorems	theorem	VERB
cana-577	185	4	3.6	3.6	NUM
cana-577	185	5	and	and	CCONJ
cana-577	185	6	a	a	DET
cana-577	185	7	make	make	NOUN
cana-577	185	8	it	it	PRON
cana-577	185	9	simple	simple	ADJ
cana-577	185	10	to	to	PART
cana-577	185	11	demonstrate	demonstrate	VERB
cana-577	185	12	the	the	DET
cana-577	185	13	aforementioned	aforementioned	ADJ
cana-577	185	14	result	result	NOUN
cana-577	185	15	.	.	PUNCT
cana-577	186	1	corollary-3.16	corollary-3.16	NOUN
cana-577	186	2	:	:	PUNCT
cana-577	186	3	for	for	ADP
cana-577	186	4	any	any	DET
cana-577	186	5	‘	'	PUNCT
cana-577	186	6	graph	graph	NOUN
cana-577	186	7	g	g	NOUN
cana-577	186	8	’	'	PUNCT
cana-577	186	9	,	,	PUNCT
cana-577	186	10	𝛾𝑠𝑡𝑟(𝑚(𝐺	𝛾𝑠𝑡𝑟(𝑚(𝐺	PROPN
cana-577	186	11	)	)	PUNCT
cana-577	186	12	)	)	PUNCT
cana-577	187	1	+	+	CCONJ
cana-577	187	2	𝛾𝑐(𝐺	𝛾𝑐(𝐺	NOUN
cana-577	187	3	)	)	PUNCT
cana-577	188	1	+	+	NUM
cana-577	188	2	𝛾𝑚	𝛾𝑚	X
cana-577	188	3	′	′	NUM
cana-577	188	4	(	(	PUNCT
cana-577	188	5	𝐺	𝐺	NOUN
cana-577	188	6	)	)	PUNCT
cana-577	188	7	<	<	X
cana-577	188	8	2𝛼0(𝐺	2𝛼0(𝐺	NUM
cana-577	188	9	)	)	PUNCT
cana-577	189	1	+	+	CCONJ
cana-577	189	2	𝛽0(𝐺	𝛽0(𝐺	ADJ
cana-577	189	3	)	)	PUNCT
cana-577	189	4	+	+	NUM
cana-577	189	5	2𝛾(𝐺	2𝛾(𝐺	NUM
cana-577	189	6	)	)	PUNCT
cana-577	189	7	.	.	PUNCT
cana-577	190	1	proof	proof	NOUN
cana-577	190	2	:	:	PUNCT
cana-577	190	3	since	since	SCONJ
cana-577	190	4	𝑝(𝐺	𝑝(𝐺	NOUN
cana-577	190	5	)	)	PUNCT
cana-577	191	1	=	=	SYM
cana-577	191	2	𝛼0(𝐺	𝛼0(𝐺	PROPN
cana-577	191	3	)	)	PUNCT
cana-577	192	1	+	+	CCONJ
cana-577	192	2	𝛽0(𝐺	𝛽0(𝐺	PROPN
cana-577	192	3	)	)	PUNCT
cana-577	192	4	,	,	PUNCT
cana-577	192	5	theorem	theorem	VERB
cana-577	192	6	3.13	3.13	NUM
cana-577	192	7	and	and	CCONJ
cana-577	192	8	theorem	theorem	VERB
cana-577	192	9	e	e	NOUN
cana-577	192	10	lead	lead	NOUN
cana-577	192	11	to	to	ADP
cana-577	192	12	the	the	DET
cana-577	192	13	above	above	ADJ
cana-577	192	14	outcome	outcome	NOUN
cana-577	192	15	.	.	PUNCT
cana-577	193	1	finally	finally	ADV
cana-577	193	2	,	,	PUNCT
cana-577	193	3	nordhaus	nordhaus	NOUN
cana-577	193	4	-	-	PUNCT
cana-577	193	5	gaddum	gaddum	NOUN
cana-577	193	6	type	type	NOUN
cana-577	193	7	outcomes	outcome	NOUN
cana-577	193	8	are	be	AUX
cana-577	193	9	found	find	VERB
cana-577	193	10	at	at	ADP
cana-577	193	11	the	the	DET
cana-577	193	12	end	end	NOUN
cana-577	193	13	.	.	PUNCT
cana-577	194	1	theorem-3.17	theorem-3.17	ADV
cana-577	194	2	:	:	PUNCT
cana-577	194	3	for	for	ADP
cana-577	194	4	any	any	DET
cana-577	194	5	(	(	PUNCT
cana-577	194	6	𝑝	𝑝	NOUN
cana-577	194	7	,	,	PUNCT
cana-577	194	8	𝑞)⁡connected	𝑞)⁡connecte	VERB
cana-577	194	9	graphs	graph	NOUN
cana-577	194	10	𝐺	𝐺	PROPN
cana-577	194	11	and	and	CCONJ
cana-577	194	12	�	�	PROPN
cana-577	194	13	̅	̅	NOUN
cana-577	194	14	�	�	NOUN
cana-577	194	15	,	,	PUNCT
cana-577	194	16	i	i	NOUN
cana-577	194	17	)	)	PUNCT
cana-577	194	18	𝛾𝑠𝑡𝑟(𝑚(𝐺	𝛾𝑠𝑡𝑟(𝑚(𝐺	PROPN
cana-577	194	19	)	)	PUNCT
cana-577	194	20	)	)	PUNCT
cana-577	195	1	+	+	CCONJ
cana-577	195	2	𝛾𝑠𝑡𝑟(𝑚(	𝛾𝑠𝑡𝑟(𝑚(	X
cana-577	195	3	�	�	NOUN
cana-577	195	4	̅	̅	NOUN
cana-577	195	5	�	�	NOUN
cana-577	195	6	)	)	PUNCT
cana-577	195	7	)	)	PUNCT
cana-577	195	8	≤	≤	NUM
cana-577	195	9	2𝑝	2𝑝	NUM
cana-577	195	10	ii	ii	PROPN
cana-577	195	11	)	)	PUNCT
cana-577	195	12	𝛾𝑠𝑡𝑟(𝑚(𝐺	𝛾𝑠𝑡𝑟(𝑚(𝐺	PROPN
cana-577	195	13	)	)	PUNCT
cana-577	195	14	)	)	PUNCT
cana-577	195	15	.	.	PUNCT
cana-577	196	1	𝛾𝑠𝑡𝑟(𝑚(	𝛾𝑠𝑡𝑟(𝑚(	X
cana-577	196	2	�	�	SYM
cana-577	196	3	̅	̅	NOUN
cana-577	196	4	�	�	NOUN
cana-577	196	5	)	)	PUNCT
cana-577	196	6	)	)	PUNCT
cana-577	196	7	≤	≤	NUM
cana-577	196	8	𝑝𝑞	𝑝𝑞	VERB
cana-577	196	9	4	4	NUM
cana-577	196	10	.	.	PUNCT
cana-577	196	11	conclusion	conclusion	NOUN
cana-577	196	12	in	in	ADP
cana-577	196	13	the	the	DET
cana-577	196	14	current	current	ADJ
cana-577	196	15	study	study	NOUN
cana-577	196	16	some	some	DET
cana-577	196	17	results	result	NOUN
cana-577	196	18	have	have	AUX
cana-577	196	19	been	be	AUX
cana-577	196	20	obtained	obtain	VERB
cana-577	196	21	on	on	ADP
cana-577	196	22	𝛾𝑠𝑡𝑟(𝑚(𝐺	𝛾𝑠𝑡𝑟(𝑚(𝐺	NUM
cana-577	196	23	)	)	PUNCT
cana-577	196	24	)	)	PUNCT
cana-577	196	25	in	in	ADP
cana-577	196	26	terms	term	NOUN
cana-577	196	27	of	of	ADP
cana-577	196	28	several	several	ADJ
cana-577	196	29	parameters	parameter	NOUN
cana-577	196	30	of	of	ADP
cana-577	196	31	g	g	NOUN
cana-577	196	32	,	,	PUNCT
cana-577	196	33	such	such	ADJ
cana-577	196	34	as	as	ADP
cana-577	196	35	its	its	PRON
cana-577	196	36	vertices	vertex	NOUN
cana-577	196	37	,	,	PUNCT
cana-577	196	38	edges	edge	NOUN
cana-577	196	39	,	,	PUNCT
cana-577	196	40	diameter	diameter	NOUN
cana-577	196	41	and	and	CCONJ
cana-577	196	42	so	so	ADV
cana-577	196	43	on	on	ADV
cana-577	196	44	,	,	PUNCT
cana-577	196	45	as	as	ADV
cana-577	196	46	well	well	ADV
cana-577	196	47	as	as	ADP
cana-577	196	48	several	several	ADJ
cana-577	196	49	domination	domination	NOUN
cana-577	196	50	parameters	parameter	NOUN
cana-577	196	51	of	of	ADP
cana-577	196	52	g	g	NOUN
cana-577	196	53	,	,	PUNCT
cana-577	196	54	such	such	ADJ
cana-577	196	55	as	as	ADP
cana-577	196	56	edge	edge	NOUN
cana-577	196	57	domination	domination	NOUN
cana-577	196	58	,	,	PUNCT
cana-577	196	59	connected	connected	ADJ
cana-577	196	60	domination	domination	NOUN
cana-577	196	61	and	and	CCONJ
cana-577	196	62	total	total	ADJ
cana-577	196	63	domination	domination	NOUN
cana-577	196	64	and	and	CCONJ
cana-577	196	65	many	many	ADJ
cana-577	196	66	more	more	ADJ
cana-577	196	67	.	.	PUNCT
cana-577	197	1	communications	communication	NOUN
cana-577	197	2	on	on	ADP
cana-577	197	3	applied	apply	VERB
cana-577	197	4	nonlinear	nonlinear	ADJ
cana-577	197	5	analysis	analysis	NOUN
cana-577	197	6	issn	issn	NOUN
cana-577	197	7	:	:	PUNCT
cana-577	197	8	1074	1074	NUM
cana-577	197	9	-	-	PUNCT
cana-577	197	10	133x	133x	NUM
cana-577	197	11	vol	vol	NOUN
cana-577	197	12	31	31	NUM
cana-577	197	13	no	no	NOUN
cana-577	197	14	.	.	PUNCT
cana-577	198	1	1s	1s	NUM
cana-577	198	2	(	(	PUNCT
cana-577	198	3	2024	2024	NUM
cana-577	198	4	)	)	PUNCT
cana-577	198	5	186	186	NUM
cana-577	198	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-577	198	7	references	reference	NOUN
cana-577	198	8	[	[	X
cana-577	198	9	1	1	NUM
cana-577	198	10	]	]	X
cana-577	198	11	sri	sri	PROPN
cana-577	198	12	,	,	PUNCT
cana-577	198	13	p.a	p.a	PROPN
cana-577	198	14	.	.	PROPN
cana-577	198	15	,	,	PUNCT
cana-577	198	16	thamaraikannan	thamaraikannan	NOUN
cana-577	198	17	,	,	PUNCT
cana-577	198	18	n.	n.	PROPN
cana-577	198	19	,	,	PUNCT
cana-577	198	20	loganathan	loganathan	PROPN
cana-577	198	21	,	,	PUNCT
cana-577	198	22	k.	k.	PROPN
cana-577	198	23	,	,	PUNCT
cana-577	198	24	&	&	CCONJ
cana-577	198	25	chaudhary	chaudhary	PROPN
cana-577	198	26	,	,	PUNCT
cana-577	198	27	d.k	d.k	PROPN
cana-577	198	28	.	.	PROPN
cana-577	198	29	(	(	PUNCT
cana-577	198	30	2022	2022	NUM
cana-577	198	31	)	)	PUNCT
cana-577	198	32	.	.	PUNCT
cana-577	199	1	double	double	ADJ
cana-577	199	2	domination	domination	NOUN
cana-577	199	3	and	and	CCONJ
cana-577	199	4	regular	regular	ADJ
cana-577	199	5	domination	domination	NOUN
cana-577	199	6	in	in	ADP
cana-577	199	7	intuitionistic	intuitionistic	ADJ
cana-577	199	8	fuzzy	fuzzy	ADJ
cana-577	199	9	hypergraph	hypergraph	NOUN
cana-577	199	10	.	.	PUNCT
cana-577	200	1	journal	journal	NOUN
cana-577	200	2	of	of	ADP
cana-577	200	3	mathematics	mathematic	NOUN
cana-577	200	4	,	,	PUNCT
cana-577	200	5	1436194	1436194	NUM
cana-577	200	6	.	.	PUNCT
cana-577	201	1	https://doi.org/10.1155/2022/1436194	https://doi.org/10.1155/2022/1436194	PROPN
cana-577	201	2	.	.	PUNCT
cana-577	202	1	[	[	X
cana-577	202	2	2	2	NUM
cana-577	202	3	]	]	PUNCT
cana-577	202	4	agustin	agustin	PROPN
cana-577	202	5	,	,	PUNCT
cana-577	202	6	i.h	i.h	PROPN
cana-577	202	7	.	.	PROPN
cana-577	202	8	,	,	PUNCT
cana-577	202	9	retnowardani	retnowardani	PROPN
cana-577	202	10	,	,	PUNCT
cana-577	202	11	d.a	d.a	PROPN
cana-577	202	12	.	.	PROPN
cana-577	202	13	,	,	PUNCT
cana-577	202	14	&	&	CCONJ
cana-577	202	15	kurniawati	kurniawati	PROPN
cana-577	202	16	,	,	PUNCT
cana-577	202	17	e.y	e.y	PROPN
cana-577	202	18	.	.	PROPN
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cana-577	202	21	)	)	PUNCT
cana-577	202	22	.	.	PUNCT
cana-577	203	1	on	on	ADP
cana-577	203	2	the	the	DET
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cana-577	203	4	strong	strong	ADJ
cana-577	203	5	domination	domination	NOUN
cana-577	203	6	number	number	NOUN
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cana-577	203	8	graphs	graph	NOUN
cana-577	203	9	:	:	PUNCT
cana-577	203	10	a	a	DET
cana-577	203	11	new	new	ADJ
cana-577	203	12	notion	notion	NOUN
cana-577	203	13	.	.	PUNCT
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cana-577	204	2	journal	journal	PROPN
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cana-577	204	4	physics	physics	PROPN
cana-577	204	5	:	:	PUNCT
cana-577	204	6	conference	conference	NOUN
cana-577	204	7	series	series	NOUN
cana-577	204	8	,	,	PUNCT
cana-577	204	9	1836(1	1836(1	NUM
cana-577	204	10	)	)	PUNCT
cana-577	204	11	.	.	PUNCT
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cana-577	205	2	.	.	PUNCT
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cana-577	206	2	3	3	NUM
cana-577	206	3	]	]	X
cana-577	206	4	elavarasan	elavarasan	PROPN
cana-577	206	5	,	,	PUNCT
cana-577	206	6	k.	k.	PROPN
cana-577	206	7	,	,	PUNCT
cana-577	206	8	gunasekar	gunasekar	PROPN
cana-577	206	9	,	,	PUNCT
cana-577	206	10	t.	t.	PROPN
cana-577	206	11	,	,	PUNCT
cana-577	206	12	cepova	cepova	PROPN
cana-577	206	13	,	,	PUNCT
cana-577	206	14	l.	l.	PROPN
cana-577	206	15	,	,	PUNCT
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cana-577	206	18	,	,	PUNCT
cana-577	206	19	r.	r.	PROPN
cana-577	206	20	(	(	PUNCT
cana-577	206	21	2022	2022	NUM
cana-577	206	22	)	)	PUNCT
cana-577	206	23	.	.	PUNCT
cana-577	207	1	study	study	NOUN
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cana-577	207	3	a	a	DET
cana-577	207	4	strong	strong	ADJ
cana-577	207	5	and	and	CCONJ
cana-577	207	6	weak	weak	ADJ
cana-577	207	7	n	n	CCONJ
cana-577	207	8	-	-	PUNCT
cana-577	207	9	connected	connect	VERB
cana-577	207	10	total	total	ADJ
cana-577	207	11	perfect	perfect	ADJ
cana-577	207	12	k	k	ADJ
cana-577	207	13	-	-	PUNCT
cana-577	207	14	dominating	dominating	NOUN
cana-577	207	15	set	set	NOUN
cana-577	207	16	in	in	ADP
cana-577	207	17	fuzzy	fuzzy	ADJ
cana-577	207	18	graphs	graph	NOUN
cana-577	207	19	.	.	PUNCT
cana-577	208	1	mathematics	mathematic	NOUN
cana-577	208	2	,	,	PUNCT
cana-577	208	3	10(17	10(17	NUM
cana-577	208	4	)	)	PUNCT
cana-577	208	5	,	,	PUNCT
cana-577	208	6	3178	3178	NUM
cana-577	208	7	.	.	PUNCT
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cana-577	209	2	.	.	PUNCT
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cana-577	210	2	4	4	NUM
cana-577	210	3	]	]	PUNCT
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cana-577	210	5	,	,	PUNCT
cana-577	210	6	a.r	a.r	PROPN
cana-577	210	7	.	.	PROPN
cana-577	210	8	,	,	PUNCT
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cana-577	210	10	,	,	PUNCT
cana-577	210	11	m.	m.	NOUN
cana-577	210	12	,	,	PUNCT
cana-577	210	13	&	&	CCONJ
cana-577	210	14	majeed	majeed	PROPN
cana-577	210	15	,	,	PUNCT
cana-577	210	16	a.	a.	NOUN
cana-577	210	17	(	(	PUNCT
cana-577	210	18	2022	2022	NUM
cana-577	210	19	)	)	PUNCT
cana-577	210	20	.	.	PUNCT
cana-577	211	1	inverse	inverse	PROPN
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cana-577	211	3	domination	domination	NOUN
cana-577	211	4	in	in	ADP
cana-577	211	5	graphs	graph	NOUN
cana-577	211	6	.	.	PUNCT
cana-577	212	1	materials	material	NOUN
cana-577	212	2	today	today	NOUN
cana-577	212	3	:	:	PUNCT
cana-577	212	4	proceedings	proceeding	NOUN
cana-577	212	5	,	,	PUNCT
cana-577	212	6	65(8	65(8	NOUN
cana-577	212	7	)	)	PUNCT
cana-577	212	8	,	,	PUNCT
cana-577	212	9	3552	3552	NUM
cana-577	212	10	-	-	SYM
cana-577	212	11	3557	3557	NUM
cana-577	212	12	.	.	PUNCT
cana-577	213	1	https://doi.org/10.1016/j.matpr.2022.06.147	https://doi.org/10.1016/j.matpr.2022.06.147	NOUN
cana-577	213	2	.	.	PUNCT
cana-577	214	1	[	[	X
cana-577	214	2	5	5	NUM
cana-577	214	3	]	]	X
cana-577	214	4	vani	vani	NOUN
cana-577	214	5	,	,	PUNCT
cana-577	214	6	m.	m.	NOUN
cana-577	214	7	,	,	PUNCT
cana-577	214	8	abdul	abdul	PROPN
cana-577	214	9	,	,	PUNCT
cana-577	214	10	m.	m.	NOUN
cana-577	214	11	,	,	PUNCT
cana-577	214	12	&	&	CCONJ
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cana-577	214	14	,	,	PUNCT
cana-577	214	15	d.j	d.j	PROPN
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cana-577	214	17	(	(	PUNCT
cana-577	214	18	2020	2020	NUM
cana-577	214	19	)	)	PUNCT
cana-577	214	20	.	.	PUNCT
cana-577	215	1	edge	edge	PROPN
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cana-577	215	3	domination	domination	NOUN
cana-577	215	4	in	in	ADP
cana-577	215	5	graphs	graph	NOUN
cana-577	215	6	.	.	PUNCT
cana-577	216	1	international	international	ADJ
cana-577	216	2	journal	journal	NOUN
cana-577	216	3	of	of	ADP
cana-577	216	4	future	future	ADJ
cana-577	216	5	generation	generation	NOUN
cana-577	216	6	communication	communication	NOUN
cana-577	216	7	and	and	CCONJ
cana-577	216	8	networking	networking	NOUN
cana-577	216	9	,	,	PUNCT
cana-577	216	10	13(3	13(3	NUM
cana-577	216	11	)	)	PUNCT
cana-577	216	12	,	,	PUNCT
cana-577	216	13	3636	3636	NUM
cana-577	216	14	-	-	SYM
cana-577	216	15	3641	3641	NUM
cana-577	216	16	.	.	PUNCT
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cana-577	217	2	.	.	PUNCT
cana-577	218	1	[	[	X
cana-577	218	2	6	6	NUM
cana-577	218	3	]	]	X
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cana-577	218	5	,	,	PUNCT
cana-577	218	6	x.	x.	PROPN
cana-577	218	7	,	,	PUNCT
cana-577	218	8	akhoundi	akhoundi	ADJ
cana-577	218	9	,	,	PUNCT
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cana-577	218	11	,	,	PUNCT
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cana-577	218	13	,	,	PUNCT
cana-577	218	14	a.a	a.a	PROPN
cana-577	218	15	.	.	PROPN
cana-577	218	16	,	,	PUNCT
cana-577	218	17	&	&	CCONJ
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cana-577	218	19	,	,	PUNCT
cana-577	218	20	m.	m.	NOUN
cana-577	218	21	(	(	PUNCT
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cana-577	218	23	)	)	PUNCT
cana-577	218	24	.	.	PUNCT
cana-577	219	1	a	a	DET
cana-577	219	2	study	study	NOUN
cana-577	219	3	on	on	ADP
cana-577	219	4	regular	regular	ADJ
cana-577	219	5	domination	domination	NOUN
cana-577	219	6	in	in	ADP
cana-577	219	7	vague	vague	ADJ
cana-577	219	8	graphs	graph	NOUN
cana-577	219	9	with	with	ADP
cana-577	219	10	application	application	NOUN
cana-577	219	11	.	.	PUNCT
cana-577	220	1	advances	advance	NOUN
cana-577	220	2	in	in	ADP
cana-577	220	3	mathematical	mathematical	ADJ
cana-577	220	4	physics	physics	NOUN
cana-577	220	5	,	,	PUNCT
cana-577	220	6	2023	2023	NUM
cana-577	220	7	.	.	PUNCT
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cana-577	222	1	[	[	X
cana-577	222	2	7	7	NUM
cana-577	222	3	]	]	SYM
cana-577	222	4	zaherifar	zaherifar	PROPN
cana-577	222	5	,	,	PUNCT
cana-577	222	6	h.	h.	PROPN
cana-577	222	7	,	,	PUNCT
cana-577	222	8	alikhani	alikhani	PROPN
cana-577	222	9	,	,	PUNCT
cana-577	222	10	s.	s.	PROPN
cana-577	222	11	,	,	PUNCT
cana-577	222	12	&	&	CCONJ
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cana-577	222	14	,	,	PUNCT
cana-577	222	15	n.	n.	PROPN
cana-577	222	16	(	(	PUNCT
cana-577	222	17	2023	2023	NUM
cana-577	222	18	)	)	PUNCT
cana-577	222	19	.	.	PUNCT
cana-577	223	1	on	on	ADP
cana-577	223	2	the	the	DET
cana-577	223	3	strong	strong	ADJ
cana-577	223	4	dominating	dominating	NOUN
cana-577	223	5	sets	set	NOUN
cana-577	223	6	of	of	ADP
cana-577	223	7	graphs	graph	NOUN
cana-577	223	8	.	.	PUNCT
cana-577	224	1	journal	journal	NOUN
cana-577	224	2	of	of	ADP
cana-577	224	3	algebraic	algebraic	PROPN
cana-577	224	4	systems	system	NOUN
cana-577	224	5	,	,	PUNCT
cana-577	224	6	11(1	11(1	NUM
cana-577	224	7	)	)	PUNCT
cana-577	224	8	,	,	PUNCT
cana-577	224	9	65	65	NUM
cana-577	224	10	-	-	SYM
cana-577	224	11	76	76	NUM
cana-577	224	12	.	.	PUNCT
cana-577	225	1	https://doi.org/10.22044/jas.2022.11646.1595	https://doi.org/10.22044/jas.2022.11646.1595	NOUN
cana-577	225	2	.	.	PUNCT
cana-577	226	1	https://doi.org/10.1155/2022/1436194	https://doi.org/10.1155/2022/1436194	PROPN
cana-577	226	2	https://doi.org/10.3390/math10173178	https://doi.org/10.3390/math10173178	PROPN
cana-577	226	3	https://doi.org/10.1016/j.matpr.2022.06.147	https://doi.org/10.1016/j.matpr.2022.06.147	PROPN
cana-577	226	4	http://sersc.org/journals/index.php/ijfgcn/article/view/30753	http://sersc.org/journals/index.php/ijfgcn/article/view/30753	X
cana-577	226	5	https://doi.org/10.1155/2023/7098134	https://doi.org/10.1155/2023/7098134	VERB