id	sid	tid	token	lemma	pos
cana-4037	1	1	untitled	untitle	VERB
cana-4037	1	2	communications	communication	NOUN
cana-4037	1	3	on	on	ADP
cana-4037	1	4	applied	apply	VERB
cana-4037	1	5	nonlinear	nonlinear	ADJ
cana-4037	1	6	analysis	analysis	NOUN
cana-4037	1	7	issn	issn	NOUN
cana-4037	1	8	:	:	PUNCT
cana-4037	1	9	1074	1074	NUM
cana-4037	1	10	-	-	PUNCT
cana-4037	1	11	133x	133x	NUM
cana-4037	1	12	vol	vol	NOUN
cana-4037	1	13	32	32	NUM
cana-4037	1	14	no	no	NOUN
cana-4037	1	15	.	.	PUNCT
cana-4037	2	1	9s	9s	NUM
cana-4037	2	2	(	(	PUNCT
cana-4037	2	3	2025	2025	NUM
cana-4037	2	4	)	)	PUNCT
cana-4037	2	5	894	894	NUM
cana-4037	2	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-4037	2	7	1	1	NUM
cana-4037	2	8	t.	t.	NOUN
cana-4037	2	9	sindhuja	sindhuja	NOUN
cana-4037	2	10	,	,	PUNCT
cana-4037	2	11	2*v	2*v	NUM
cana-4037	2	12	.	.	X
cana-4037	2	13	maheswari	maheswari	PROPN
cana-4037	2	14	and	and	CCONJ
cana-4037	2	15	3v	3v	NUM
cana-4037	2	16	.	.	PUNCT
cana-4037	3	1	balaji	balaji	PROPN
cana-4037	3	2	1research	1research	NUM
cana-4037	3	3	scholar	scholar	NOUN
cana-4037	3	4	,	,	PUNCT
cana-4037	3	5	department	department	NOUN
cana-4037	3	6	of	of	ADP
cana-4037	3	7	mathematics	mathematics	PROPN
cana-4037	3	8	,	,	PUNCT
cana-4037	3	9	vels	vels	PROPN
cana-4037	3	10	institute	institute	PROPN
cana-4037	3	11	of	of	ADP
cana-4037	3	12	science	science	PROPN
cana-4037	3	13	technology	technology	NOUN
cana-4037	3	14	and	and	CCONJ
cana-4037	3	15	advanced	advanced	ADJ
cana-4037	3	16	studies	study	NOUN
cana-4037	3	17	,	,	PUNCT
cana-4037	3	18	chennai	chennai	PROPN
cana-4037	3	19	,	,	PUNCT
cana-4037	3	20	tamil	tamil	PROPN
cana-4037	3	21	nadu	nadu	PROPN
cana-4037	3	22	,	,	PUNCT
cana-4037	3	23	india	india	PROPN
cana-4037	3	24	.	.	PUNCT
cana-4037	3	25	email	email	NOUN
cana-4037	3	26	:	:	PUNCT
cana-4037	3	27	sindhutg86@gmail.com	sindhutg86@gmail.com	X
cana-4037	3	28	2,*research	2,*research	NUM
cana-4037	3	29	supervisor	supervisor	NOUN
cana-4037	3	30	,	,	PUNCT
cana-4037	3	31	professor	professor	NOUN
cana-4037	3	32	,	,	PUNCT
cana-4037	3	33	department	department	NOUN
cana-4037	3	34	of	of	ADP
cana-4037	3	35	mathematics	mathematics	PROPN
cana-4037	3	36	,	,	PUNCT
cana-4037	3	37	vels	vels	PROPN
cana-4037	3	38	institute	institute	PROPN
cana-4037	3	39	of	of	ADP
cana-4037	3	40	science	science	PROPN
cana-4037	3	41	technology	technology	NOUN
cana-4037	3	42	and	and	CCONJ
cana-4037	3	43	advanced	advanced	ADJ
cana-4037	3	44	studies	study	NOUN
cana-4037	3	45	,	,	PUNCT
cana-4037	3	46	chennai	chennai	PROPN
cana-4037	3	47	,	,	PUNCT
cana-4037	3	48	tamil	tamil	PROPN
cana-4037	3	49	nadu	nadu	PROPN
cana-4037	3	50	,	,	PUNCT
cana-4037	3	51	india	india	PROPN
cana-4037	3	52	.	.	PUNCT
cana-4037	4	1	corresponding	correspond	VERB
cana-4037	4	2	author	author	NOUN
cana-4037	4	3	:	:	PUNCT
cana-4037	4	4	3associate	3associate	NUM
cana-4037	4	5	professor	professor	NOUN
cana-4037	4	6	,	,	PUNCT
cana-4037	4	7	department	department	NOUN
cana-4037	4	8	of	of	ADP
cana-4037	4	9	mathematics	mathematic	NOUN
cana-4037	4	10	,	,	PUNCT
cana-4037	4	11	sacred	sacred	ADJ
cana-4037	4	12	heart	heart	NOUN
cana-4037	4	13	college	college	NOUN
cana-4037	4	14	,	,	PUNCT
cana-4037	4	15	tirupattur	tirupattur	PROPN
cana-4037	4	16	,	,	PUNCT
cana-4037	4	17	tamil	tamil	PROPN
cana-4037	4	18	nadu	nadu	PROPN
cana-4037	4	19	,	,	PUNCT
cana-4037	4	20	india	india	PROPN
cana-4037	4	21	.	.	PUNCT
cana-4037	4	22	email	email	NOUN
cana-4037	4	23	:	:	PUNCT
cana-4037	5	1	pulibala70@gmail.com	pulibala70@gmail.com	X
cana-4037	5	2	article	article	NOUN
cana-4037	5	3	history	history	NOUN
cana-4037	5	4	:	:	PUNCT
cana-4037	5	5	received	receive	VERB
cana-4037	5	6	:	:	PUNCT
cana-4037	5	7	12	12	NUM
cana-4037	5	8	-	-	SYM
cana-4037	5	9	11	11	NUM
cana-4037	5	10	-	-	PUNCT
cana-4037	5	11	2024	2024	NUM
cana-4037	5	12	revised	revise	VERB
cana-4037	5	13	:	:	PUNCT
cana-4037	5	14	17	17	NUM
cana-4037	5	15	-	-	SYM
cana-4037	5	16	12	12	NUM
cana-4037	5	17	-	-	PUNCT
cana-4037	5	18	2024	2024	NUM
cana-4037	5	19	accepted	accept	VERB
cana-4037	5	20	:	:	PUNCT
cana-4037	5	21	06	06	NUM
cana-4037	5	22	-	-	SYM
cana-4037	5	23	01	01	NUM
cana-4037	5	24	-	-	PUNCT
cana-4037	5	25	2025	2025	NUM
cana-4037	5	26	abstract	abstract	NOUN
cana-4037	5	27	:	:	PUNCT
cana-4037	5	28	let	let	VERB
cana-4037	5	29	d	d	PART
cana-4037	5	30	be	be	AUX
cana-4037	5	31	a	a	DET
cana-4037	5	32	subset	subset	NOUN
cana-4037	5	33	of	of	ADP
cana-4037	5	34	v	v	NOUN
cana-4037	5	35	and	and	CCONJ
cana-4037	5	36	𝑢	𝑢	PROPN
cana-4037	5	37	∈	∈	PROPN
cana-4037	5	38	𝐷	𝐷	PROPN
cana-4037	5	39	the	the	DET
cana-4037	5	40	out	out	ADJ
cana-4037	5	41	degree	degree	NOUN
cana-4037	5	42	of	of	ADP
cana-4037	5	43	u	u	NOUN
cana-4037	5	44	is	be	AUX
cana-4037	5	45	defined	define	VERB
cana-4037	5	46	by	by	ADP
cana-4037	5	47	|𝑁(𝑢	|𝑁(𝑢	ADJ
cana-4037	5	48	)	)	PUNCT
cana-4037	5	49	∩	∩	NOUN
cana-4037	5	50	(	(	PUNCT
cana-4037	5	51	𝑉	𝑉	PROPN
cana-4037	5	52	−	−	PROPN
cana-4037	5	53	𝐷)|	𝐷)|	NOUN
cana-4037	5	54	and	and	CCONJ
cana-4037	5	55	denoted	denote	VERB
cana-4037	5	56	by	by	ADP
cana-4037	5	57	𝑜𝑑𝐷(𝑢	𝑜𝑑𝐷(𝑢	PROPN
cana-4037	5	58	)	)	PUNCT
cana-4037	5	59	.	.	PUNCT
cana-4037	6	1	a	a	DET
cana-4037	6	2	dominating	dominating	NOUN
cana-4037	6	3	set	set	NOUN
cana-4037	6	4	d	d	NOUN
cana-4037	6	5	of	of	ADP
cana-4037	6	6	v	v	NOUN
cana-4037	6	7	is	be	AUX
cana-4037	6	8	said	say	VERB
cana-4037	6	9	to	to	PART
cana-4037	6	10	be	be	AUX
cana-4037	6	11	cototal	cototal	ADJ
cana-4037	6	12	dominating	dominating	NOUN
cana-4037	6	13	set	set	NOUN
cana-4037	6	14	if	if	SCONJ
cana-4037	6	15	the	the	DET
cana-4037	6	16	sub	sub	NOUN
cana-4037	6	17	graph	graph	NOUN
cana-4037	6	18	induced	induce	VERB
cana-4037	6	19	by	by	ADP
cana-4037	6	20	𝑉	𝑉	PROPN
cana-4037	6	21	−	−	PROPN
cana-4037	6	22	𝐷	𝐷	PROPN
cana-4037	6	23	has	have	VERB
cana-4037	6	24	no	no	DET
cana-4037	6	25	isolated	isolate	VERB
cana-4037	6	26	vertices	vertex	NOUN
cana-4037	6	27	based	base	VERB
cana-4037	6	28	on	on	ADP
cana-4037	6	29	the	the	DET
cana-4037	6	30	concepts	concept	NOUN
cana-4037	6	31	we	we	PRON
cana-4037	6	32	introduce	introduce	VERB
cana-4037	6	33	a	a	DET
cana-4037	6	34	new	new	ADJ
cana-4037	6	35	domination	domination	NOUN
cana-4037	6	36	called	call	VERB
cana-4037	6	37	co	co	NOUN
cana-4037	6	38	-	-	ADJ
cana-4037	6	39	total	total	ADJ
cana-4037	6	40	2	2	NUM
cana-4037	6	41	oded	oded	ADJ
cana-4037	6	42	number	number	NOUN
cana-4037	6	43	.	.	PUNCT
cana-4037	7	1	here	here	ADV
cana-4037	7	2	the	the	DET
cana-4037	7	3	proposed	propose	VERB
cana-4037	7	4	domination	domination	NOUN
cana-4037	7	5	number	number	NOUN
cana-4037	7	6	verified	verify	VERB
cana-4037	7	7	for	for	ADP
cana-4037	7	8	some	some	DET
cana-4037	7	9	general	general	ADJ
cana-4037	7	10	and	and	CCONJ
cana-4037	7	11	some	some	DET
cana-4037	7	12	named	name	VERB
cana-4037	7	13	graphs	graph	NOUN
cana-4037	7	14	.	.	PUNCT
cana-4037	8	1	keywords	keyword	NOUN
cana-4037	8	2	:	:	PUNCT
cana-4037	8	3	two	two	NUM
cana-4037	8	4	out	out	ADJ
cana-4037	8	5	degree	degree	NOUN
cana-4037	8	6	,	,	PUNCT
cana-4037	8	7	domination	domination	NOUN
cana-4037	8	8	number	number	NOUN
cana-4037	8	9	,	,	PUNCT
cana-4037	8	10	isolated	isolated	ADJ
cana-4037	8	11	vertices	vertex	NOUN
cana-4037	8	12	,	,	PUNCT
cana-4037	8	13	co	co	ADJ
cana-4037	8	14	-	-	NOUN
cana-4037	8	15	total	total	ADJ
cana-4037	8	16	1	1	NUM
cana-4037	8	17	.	.	PUNCT
cana-4037	9	1	the	the	DET
cana-4037	9	2	graphs	graph	NOUN
cana-4037	9	3	considered	consider	VERB
cana-4037	9	4	here	here	ADV
cana-4037	9	5	are	be	AUX
cana-4037	9	6	simple	simple	ADJ
cana-4037	9	7	and	and	CCONJ
cana-4037	9	8	without	without	ADP
cana-4037	9	9	isolated	isolated	ADJ
cana-4037	9	10	vertices	vertex	NOUN
cana-4037	9	11	defined	define	VERB
cana-4037	9	12	from	from	ADP
cana-4037	9	13	[	[	PUNCT
cana-4037	9	14	1	1	NUM
cana-4037	9	15	-	-	SYM
cana-4037	9	16	5	5	NUM
cana-4037	9	17	]	]	PUNCT
cana-4037	9	18	.	.	PUNCT
cana-4037	10	1	in	in	ADP
cana-4037	10	2	a	a	DET
cana-4037	10	3	graph	graph	NOUN
cana-4037	10	4	g=(v	g=(v	NOUN
cana-4037	10	5	,	,	PUNCT
cana-4037	10	6	e	e	NOUN
cana-4037	10	7	)	)	PUNCT
cana-4037	10	8	,	,	PUNCT
cana-4037	10	9	|v|=n	|v|=n	NOUN
cana-4037	10	10	and	and	CCONJ
cana-4037	10	11	|e|=m	|e|=m	NOUN
cana-4037	10	12	.	.	PUNCT
cana-4037	11	1	let	let	VERB
cana-4037	11	2	𝑢	𝑢	PRON
cana-4037	11	3	be	be	AUX
cana-4037	11	4	vertex	vertex	NOUN
cana-4037	11	5	and	and	CCONJ
cana-4037	11	6	n(u	n(u	PROPN
cana-4037	11	7	)	)	PUNCT
cana-4037	11	8	denotes	denote	VERB
cana-4037	11	9	the	the	DET
cana-4037	11	10	open	open	ADJ
cana-4037	11	11	neighborhood	neighborhood	NOUN
cana-4037	11	12	and	and	CCONJ
cana-4037	11	13	defined	define	VERB
cana-4037	11	14	by	by	ADP
cana-4037	11	15	𝑁(𝑢	𝑁(𝑢	NOUN
cana-4037	11	16	)	)	PUNCT
cana-4037	11	17	=	=	PRON
cana-4037	11	18	{	{	PUNCT
cana-4037	11	19	𝑣	𝑣	PRON
cana-4037	11	20	∈	∈	NOUN
cana-4037	11	21	𝑉(𝐺)/	𝑉(𝐺)/	SCONJ
cana-4037	11	22	𝑢𝑣	𝑢𝑣	NOUN
cana-4037	11	23	∈	∈	PROPN
cana-4037	11	24	𝐸(𝐺	𝐸(𝐺	NOUN
cana-4037	11	25	)	)	PUNCT
cana-4037	11	26	}	}	PUNCT
cana-4037	11	27	the	the	DET
cana-4037	11	28	degree	degree	NOUN
cana-4037	11	29	of	of	ADP
cana-4037	11	30	a	a	DET
cana-4037	11	31	vertex	vertex	NOUN
cana-4037	11	32	u	u	NOUN
cana-4037	11	33	is	be	AUX
cana-4037	11	34	denoted	denote	VERB
cana-4037	11	35	𝑑𝑒𝑔(𝑢	𝑑𝑒𝑔(𝑢	PROPN
cana-4037	11	36	)	)	PUNCT
cana-4037	11	37	and	and	CCONJ
cana-4037	11	38	defined	define	VERB
cana-4037	11	39	by	by	ADP
cana-4037	11	40	𝑑𝑒𝑔(𝑢	𝑑𝑒𝑔(𝑢	PROPN
cana-4037	11	41	)	)	PUNCT
cana-4037	11	42	=	=	SYM
cana-4037	12	1	|𝑁(𝑢)|	|𝑁(𝑢)|	PROPN
cana-4037	12	2	.	.	PUNCT
cana-4037	13	1	if	if	SCONJ
cana-4037	13	2	the	the	DET
cana-4037	13	3	vertex	vertex	NOUN
cana-4037	13	4	u	u	NOUN
cana-4037	13	5	of	of	ADP
cana-4037	13	6	a	a	DET
cana-4037	13	7	graph	graph	NOUN
cana-4037	13	8	is	be	AUX
cana-4037	13	9	isolated	isolate	VERB
cana-4037	13	10	vertex	vertex	NOUN
cana-4037	13	11	if	if	SCONJ
cana-4037	13	12	𝑑𝑒𝑔(𝑢	𝑑𝑒𝑔(𝑢	PROPN
cana-4037	13	13	)	)	PUNCT
cana-4037	13	14	=	=	SYM
cana-4037	14	1	0	0	X
cana-4037	14	2	.	.	PUNCT
cana-4037	14	3	o.	o.	PROPN
cana-4037	14	4	ore	ore	NOUN
cana-4037	15	1	[	[	X
cana-4037	15	2	10	10	NUM
cana-4037	15	3	]	]	PUNCT
cana-4037	15	4	and	and	CCONJ
cana-4037	15	5	c.	c.	PROPN
cana-4037	15	6	berge	berge	NOUN
cana-4037	15	7	[	[	X
cana-4037	15	8	4	4	X
cana-4037	15	9	]	]	PUNCT
cana-4037	15	10	introduced	introduce	VERB
cana-4037	15	11	the	the	DET
cana-4037	15	12	concept	concept	NOUN
cana-4037	15	13	of	of	ADP
cana-4037	15	14	domination	domination	NOUN
cana-4037	15	15	number	number	NOUN
cana-4037	15	16	.	.	PUNCT
cana-4037	16	1	a	a	DET
cana-4037	16	2	sub	sub	NOUN
cana-4037	16	3	set	set	NOUN
cana-4037	16	4	d	d	PROPN
cana-4037	16	5	of	of	ADP
cana-4037	16	6	v	v	NOUN
cana-4037	16	7	is	be	AUX
cana-4037	16	8	said	say	VERB
cana-4037	16	9	to	to	ADP
cana-4037	16	10	dominating	dominate	VERB
cana-4037	16	11	set	set	NOUN
cana-4037	16	12	if	if	SCONJ
cana-4037	16	13	each	each	DET
cana-4037	16	14	vertex	vertex	NOUN
cana-4037	16	15	of	of	ADP
cana-4037	16	16	v	v	NOUN
cana-4037	16	17	is	be	AUX
cana-4037	16	18	adjacent	adjacent	ADJ
cana-4037	16	19	to	to	ADP
cana-4037	16	20	some	some	DET
cana-4037	16	21	vertex	vertex	NOUN
cana-4037	16	22	in	in	ADP
cana-4037	16	23	d.	d.	PROPN
cana-4037	16	24	the	the	DET
cana-4037	16	25	domination	domination	NOUN
cana-4037	16	26	number	number	NOUN
cana-4037	16	27	𝛾(𝐺	𝛾(𝐺	PROPN
cana-4037	16	28	)	)	PUNCT
cana-4037	16	29	of	of	ADP
cana-4037	16	30	g	g	PROPN
cana-4037	16	31	is	be	AUX
cana-4037	16	32	the	the	DET
cana-4037	16	33	minimum	minimum	ADJ
cana-4037	16	34	cardinality	cardinality	NOUN
cana-4037	16	35	of	of	ADP
cana-4037	16	36	a	a	DET
cana-4037	16	37	dominating	dominating	NOUN
cana-4037	16	38	set	set	NOUN
cana-4037	16	39	.	.	PUNCT
cana-4037	17	1	kulli	kulli	PROPN
cana-4037	17	2	v.r	v.r	PROPN
cana-4037	17	3	.	.	PROPN
cana-4037	17	4	,	,	PUNCT
cana-4037	17	5	jankiram	jankiram	PROPN
cana-4037	17	6	b	b	PROPN
cana-4037	17	7	and	and	CCONJ
cana-4037	17	8	radha	radha	PROPN
cana-4037	17	9	r	r	NOUN
cana-4037	17	10	iyer	iyer	NOUN
cana-4037	18	1	[	[	X
cana-4037	18	2	6	6	NUM
cana-4037	18	3	]	]	PUNCT
cana-4037	18	4	introduce	introduce	VERB
cana-4037	18	5	co	co	ADJ
cana-4037	18	6	-	-	ADJ
cana-4037	18	7	total	total	ADJ
cana-4037	18	8	domination	domination	NOUN
cana-4037	18	9	number	number	NOUN
cana-4037	18	10	𝛾𝑐𝑡(𝐺	𝛾𝑐𝑡(𝐺	PROPN
cana-4037	18	11	)	)	PUNCT
cana-4037	18	12	.	.	PUNCT
cana-4037	19	1	dominating	dominate	VERB
cana-4037	19	2	set	set	NOUN
cana-4037	19	3	if	if	SCONJ
cana-4037	19	4	the	the	DET
cana-4037	19	5	sub	sub	NOUN
cana-4037	19	6	graph	graph	NOUN
cana-4037	19	7	induced	induce	VERB
cana-4037	19	8	by	by	ADP
cana-4037	19	9	v	v	NOUN
cana-4037	19	10	-	-	PUNCT
cana-4037	19	11	d	d	NOUN
cana-4037	19	12	has	have	VERB
cana-4037	19	13	no	no	DET
cana-4037	19	14	isolated	isolated	ADJ
cana-4037	19	15	vertices	vertex	NOUN
cana-4037	19	16	.	.	PUNCT
cana-4037	20	1	the	the	DET
cana-4037	20	2	minimum	minimum	ADJ
cana-4037	20	3	number	number	NOUN
cana-4037	20	4	of	of	ADP
cana-4037	20	5	vertices	vertex	NOUN
cana-4037	20	6	of	of	ADP
cana-4037	20	7	co	co	ADJ
cana-4037	20	8	-	-	ADJ
cana-4037	20	9	total	total	ADJ
cana-4037	20	10	dominating	dominating	NOUN
cana-4037	20	11	set	set	NOUN
cana-4037	20	12	is	be	AUX
cana-4037	20	13	called	call	VERB
cana-4037	20	14	co	co	ADJ
cana-4037	20	15	-	-	ADJ
cana-4037	20	16	total	total	ADJ
cana-4037	20	17	domination	domination	NOUN
cana-4037	20	18	number	number	NOUN
cana-4037	20	19	and	and	CCONJ
cana-4037	20	20	it	it	PRON
cana-4037	20	21	is	be	AUX
cana-4037	20	22	represented	represent	VERB
cana-4037	20	23	by	by	ADP
cana-4037	20	24	𝛾𝑐𝑡(𝐺	𝛾𝑐𝑡(𝐺	PROPN
cana-4037	20	25	)	)	PUNCT
cana-4037	20	26	.	.	PUNCT
cana-4037	21	1	sahal	sahal	NOUN
cana-4037	21	2	.	.	PUNCT
cana-4037	22	1	a	a	PRON
cana-4037	23	1	and	and	CCONJ
cana-4037	23	2	v.	v.	PROPN
cana-4037	23	3	mathad	mathad	VERB
cana-4037	23	4	[	[	X
cana-4037	23	5	11	11	NUM
cana-4037	23	6	]	]	PUNCT
cana-4037	23	7	introduce	introduce	VERB
cana-4037	23	8	the	the	DET
cana-4037	23	9	2oded	2oded	NUM
cana-4037	23	10	number	number	NOUN
cana-4037	23	11	.	.	PUNCT
cana-4037	24	1	a	a	DET
cana-4037	24	2	dominating	dominating	NOUN
cana-4037	24	3	set	set	NOUN
cana-4037	24	4	d	d	NOUN
cana-4037	24	5	of	of	ADP
cana-4037	24	6	v(g	v(g	PROPN
cana-4037	24	7	)	)	PUNCT
cana-4037	24	8	is	be	AUX
cana-4037	24	9	said	say	VERB
cana-4037	24	10	to	to	PART
cana-4037	24	11	be	be	AUX
cana-4037	24	12	2oded	2ode	VERB
cana-4037	24	13	set	set	VERB
cana-4037	24	14	for	for	ADP
cana-4037	24	15	any	any	DET
cana-4037	24	16	vertices	vertex	NOUN
cana-4037	24	17	𝑢	𝑢	PART
cana-4037	24	18	,	,	PUNCT
cana-4037	24	19	𝑣	𝑣	PRON
cana-4037	24	20	∈	∈	PROPN
cana-4037	24	21	𝐷	𝐷	NOUN
cana-4037	24	22	such	such	ADJ
cana-4037	24	23	that	that	DET
cana-4037	24	24	|𝑜𝑑𝐷(𝑢	|𝑜𝑑𝐷(𝑢	NOUN
cana-4037	24	25	)	)	PUNCT
cana-4037	25	1	−	−	ADP
cana-4037	25	2	𝑜𝑑𝐷(𝑣)|	𝑜𝑑𝐷(𝑣)|	NOUN
cana-4037	25	3	≤	≤	NOUN
cana-4037	25	4	2	2	NUM
cana-4037	25	5	where	where	SCONJ
cana-4037	25	6	𝑜𝑑𝐷(𝑢	𝑜𝑑𝐷(𝑢	VERB
cana-4037	25	7	)	)	PUNCT
cana-4037	25	8	=	=	SYM
cana-4037	25	9	|𝑁(𝑢	|𝑁(𝑢	ADJ
cana-4037	25	10	)	)	PUNCT
cana-4037	25	11	∩	∩	NOUN
cana-4037	25	12	(	(	PUNCT
cana-4037	25	13	𝑉	𝑉	PROPN
cana-4037	25	14	−	−	PROPN
cana-4037	25	15	𝐷)|	𝐷)|	PROPN
cana-4037	25	16	.	.	PUNCT
cana-4037	26	1	the	the	DET
cana-4037	26	2	minimum	minimum	NOUN
cana-4037	26	3	represented	represent	VERB
cana-4037	26	4	of	of	ADP
cana-4037	26	5	vertices	vertex	NOUN
cana-4037	26	6	in	in	ADP
cana-4037	26	7	2oded	2oded	NUM
cana-4037	26	8	set	set	NOUN
cana-4037	26	9	is	be	AUX
cana-4037	26	10	called	call	VERB
cana-4037	26	11	2oded	2ode	VERB
cana-4037	26	12	number	number	NOUN
cana-4037	26	13	and	and	CCONJ
cana-4037	26	14	it	it	PRON
cana-4037	26	15	is	be	AUX
cana-4037	26	16	denoted	denote	VERB
cana-4037	26	17	by	by	ADP
cana-4037	26	18	𝛾2𝑜𝑒(𝐺	𝛾2𝑜𝑒(𝐺	NOUN
cana-4037	26	19	)	)	PUNCT
cana-4037	26	20	.	.	PUNCT
cana-4037	27	1	based	base	VERB
cana-4037	27	2	on	on	ADP
cana-4037	27	3	above	above	ADP
cana-4037	27	4	domination	domination	NOUN
cana-4037	27	5	number	number	NOUN
cana-4037	27	6	,	,	PUNCT
cana-4037	27	7	here	here	ADV
cana-4037	27	8	we	we	PRON
cana-4037	27	9	introduce	introduce	VERB
cana-4037	27	10	a	a	DET
cana-4037	27	11	new	new	ADJ
cana-4037	27	12	domination	domination	NOUN
cana-4037	27	13	parameter	parameter	NOUN
cana-4037	27	14	called	call	VERB
cana-4037	27	15	co	co	NOUN
cana-4037	27	16	total	total	ADJ
cana-4037	27	17	2oded	2oded	NUM
cana-4037	27	18	(	(	PUNCT
cana-4037	27	19	𝛾𝑐𝑡2𝑜𝑒(𝐺	𝛾𝑐𝑡2𝑜𝑒(𝐺	X
cana-4037	27	20	)	)	PUNCT
cana-4037	27	21	−	−	PRON
cana-4037	27	22	𝑠𝑒𝑡	𝑠𝑒𝑡	NOUN
cana-4037	27	23	)	)	PUNCT
cana-4037	27	24	number	number	NOUN
cana-4037	27	25	.	.	PUNCT
cana-4037	28	1	2	2	X
cana-4037	28	2	.	.	X
cana-4037	28	3	co	co	NOUN
cana-4037	28	4	-	-	NOUN
cana-4037	28	5	total	total	ADJ
cana-4037	28	6	2	2	NUM
cana-4037	28	7	oded	oded	NOUN
cana-4037	28	8	number	number	NOUN
cana-4037	28	9	definition	definition	NOUN
cana-4037	28	10	2.1	2.1	NUM
cana-4037	28	11	a	a	DET
cana-4037	28	12	sub	sub	NOUN
cana-4037	28	13	set	set	NOUN
cana-4037	28	14	d	d	PROPN
cana-4037	28	15	vertices	vertice	VERB
cana-4037	28	16	𝑢	𝑢	PART
cana-4037	28	17	,	,	PUNCT
cana-4037	28	18	𝑣	𝑣	PRON
cana-4037	28	19	∈	∈	PROPN
cana-4037	28	20	𝐷	𝐷	NOUN
cana-4037	28	21	such	such	ADJ
cana-4037	28	22	that	that	DET
cana-4037	28	23	|𝑜𝑑𝐷(𝑢	|𝑜𝑑𝐷(𝑢	NOUN
cana-4037	28	24	)	)	PUNCT
cana-4037	29	1	−	−	ADP
cana-4037	29	2	𝑜𝑑𝐷(𝑣)|	𝑜𝑑𝐷(𝑣)|	NOUN
cana-4037	29	3	≤	≤	NOUN
cana-4037	29	4	2	2	NUM
cana-4037	29	5	where	where	SCONJ
cana-4037	29	6	𝑜𝑑𝐷(𝑢	𝑜𝑑𝐷(𝑢	VERB
cana-4037	29	7	)	)	PUNCT
cana-4037	29	8	=	=	SYM
cana-4037	29	9	|𝑁(𝑢	|𝑁(𝑢	NUM
cana-4037	29	10	)	)	PUNCT
cana-4037	29	11	∩	∩	NOUN
cana-4037	29	12	(	(	PUNCT
cana-4037	29	13	𝑉	𝑉	PROPN
cana-4037	29	14	−	−	PROPN
cana-4037	29	15	𝐷)|	𝐷)|	NOUN
cana-4037	29	16	and	and	CCONJ
cana-4037	29	17	the	the	DET
cana-4037	29	18	maheswari.sbs@vistas.ac.in	maheswari.sbs@vistas.ac.in	PROPN
cana-4037	29	19	co	co	NOUN
cana-4037	29	20	total	total	ADJ
cana-4037	29	21	2	2	NUM
cana-4037	29	22	oded	oded	ADJ
cana-4037	29	23	number	number	NOUN
cana-4037	29	24	in	in	ADP
cana-4037	29	25	graphs	graph	NOUN
cana-4037	29	26	introduction	introduction	NOUN
cana-4037	29	27	a	a	DET
cana-4037	29	28	dominating	dominating	NOUN
cana-4037	29	29	set	set	NOUN
cana-4037	29	30	d	d	NOUN
cana-4037	29	31	of	of	ADP
cana-4037	29	32	v	v	NOUN
cana-4037	29	33	is	be	AUX
cana-4037	29	34	said	say	VERB
cana-4037	29	35	to	to	PART
cana-4037	29	36	be	be	AUX
cana-4037	29	37	cototal	cototal	ADJ
cana-4037	29	38	of	of	ADP
cana-4037	29	39	v	v	NOUN
cana-4037	29	40	is	be	AUX
cana-4037	29	41	called	call	VERB
cana-4037	29	42	cototal	cototal	ADJ
cana-4037	29	43	2	2	NUM
cana-4037	29	44	oded	oded	NOUN
cana-4037	29	45	set	set	VERB
cana-4037	29	46	if	if	SCONJ
cana-4037	29	47	d	d	PROPN
cana-4037	29	48	is	be	AUX
cana-4037	29	49	dominating	dominate	VERB
cana-4037	29	50	set	set	NOUN
cana-4037	29	51	and	and	CCONJ
cana-4037	29	52	for	for	ADP
cana-4037	29	53	any	any	DET
cana-4037	29	54	mailto:sindhutg86@gmail.com	mailto:sindhutg86@gmail.com	PROPN
cana-4037	29	55	mailto:sasikala.sbs@velsuniv.ac.in	mailto:sasikala.sbs@velsuniv.ac.in	PROPN
cana-4037	29	56	mailto:pulibala70@gmail.com	mailto:pulibala70@gmail.com	PROPN
cana-4037	29	57	communications	communication	NOUN
cana-4037	29	58	on	on	ADP
cana-4037	29	59	applied	apply	VERB
cana-4037	29	60	nonlinear	nonlinear	ADJ
cana-4037	29	61	analysis	analysis	NOUN
cana-4037	29	62	issn	issn	NOUN
cana-4037	29	63	:	:	PUNCT
cana-4037	29	64	1074	1074	NUM
cana-4037	29	65	-	-	PUNCT
cana-4037	29	66	133x	133x	NUM
cana-4037	29	67	vol	vol	NOUN
cana-4037	29	68	32	32	NUM
cana-4037	29	69	no	no	NOUN
cana-4037	29	70	.	.	PUNCT
cana-4037	30	1	9s	9s	NUM
cana-4037	30	2	(	(	PUNCT
cana-4037	30	3	2025	2025	NUM
cana-4037	30	4	)	)	PUNCT
cana-4037	30	5	895	895	NUM
cana-4037	31	1	https://internationalpubls.com	https://internationalpubls.com	NUM
cana-4037	31	2	induced	induced	ADJ
cana-4037	31	3	sub	sub	NOUN
cana-4037	31	4	graph	graph	NOUN
cana-4037	31	5	<	<	X
cana-4037	31	6	𝑉	𝑉	PROPN
cana-4037	31	7	−	−	PROPN
cana-4037	31	8	𝐷	𝐷	PROPN
cana-4037	31	9	>	>	PUNCT
cana-4037	31	10	has	have	VERB
cana-4037	31	11	no	no	DET
cana-4037	31	12	vertices	vertex	NOUN
cana-4037	31	13	of	of	ADP
cana-4037	31	14	degree	degree	NOUN
cana-4037	31	15	zero	zero	NUM
cana-4037	31	16	.	.	PUNCT
cana-4037	32	1	the	the	DET
cana-4037	32	2	minimum	minimum	ADJ
cana-4037	32	3	cardinality	cardinality	NOUN
cana-4037	32	4	of	of	ADP
cana-4037	32	5	a	a	DET
cana-4037	32	6	co	co	NOUN
cana-4037	32	7	-	-	ADJ
cana-4037	32	8	total	total	ADJ
cana-4037	32	9	2	2	NUM
cana-4037	32	10	oded	oded	NOUN
cana-4037	32	11	set	set	NOUN
cana-4037	32	12	is	be	AUX
cana-4037	32	13	called	call	VERB
cana-4037	32	14	co	co	NOUN
cana-4037	32	15	-	-	ADJ
cana-4037	32	16	total	total	ADJ
cana-4037	32	17	2	2	NUM
cana-4037	32	18	oded	oded	NOUN
cana-4037	32	19	number	number	NOUN
cana-4037	32	20	and	and	CCONJ
cana-4037	32	21	it	it	PRON
cana-4037	32	22	is	be	AUX
cana-4037	32	23	represented	represent	VERB
cana-4037	32	24	by	by	ADP
cana-4037	32	25	𝛾𝑐𝑡2𝑜𝑒(g	𝛾𝑐𝑡2𝑜𝑒(g	NOUN
cana-4037	32	26	)	)	PUNCT
cana-4037	32	27	.	.	PUNCT
cana-4037	33	1	example	example	NOUN
cana-4037	33	2	2.2	2.2	NUM
cana-4037	33	3	figure	figure	NOUN
cana-4037	33	4	1	1	NUM
cana-4037	33	5	.	.	PUNCT
cana-4037	34	1	co	co	NOUN
cana-4037	34	2	-	-	NOUN
cana-4037	34	3	total	total	ADJ
cana-4037	34	4	2	2	NUM
cana-4037	34	5	oded	oded	NOUN
cana-4037	34	6	number	number	NOUN
cana-4037	34	7	take	take	VERB
cana-4037	34	8	d=	d=	NUM
cana-4037	34	9	{	{	PUNCT
cana-4037	34	10	4,8,9,10	4,8,9,10	NOUN
cana-4037	34	11	}	}	PUNCT
cana-4037	34	12	𝑎𝑛𝑑	𝑎𝑛𝑑	ADJ
cana-4037	34	13	𝑉	𝑉	PROPN
cana-4037	34	14	−	−	PROPN
cana-4037	34	15	𝐷	𝐷	NOUN
cana-4037	34	16	=	=	SYM
cana-4037	34	17	{	{	PUNCT
cana-4037	34	18	1,2,3,5,6,7	1,2,3,5,6,7	NOUN
cana-4037	34	19	}	}	PUNCT
cana-4037	34	20	𝑜𝑑𝑆(4	𝑜𝑑𝑆(4	NOUN
cana-4037	34	21	)	)	PUNCT
cana-4037	34	22	=	=	SYM
cana-4037	34	23	|𝑁(4	|𝑁(4	NOUN
cana-4037	34	24	)	)	PUNCT
cana-4037	34	25	∩	∩	NOUN
cana-4037	34	26	(	(	PUNCT
cana-4037	34	27	𝑉	𝑉	PROPN
cana-4037	34	28	−	−	PROPN
cana-4037	34	29	𝐷)|	𝐷)|	NOUN
cana-4037	34	30	=	=	SYM
cana-4037	34	31	|[1,2,9	|[1,2,9	NOUN
cana-4037	34	32	}	}	PUNCT
cana-4037	34	33	∩	∩	NOUN
cana-4037	34	34	{	{	PUNCT
cana-4037	34	35	1,2,3,5,6,7}|	1,2,3,5,6,7}|	PROPN
cana-4037	34	36	=	=	SYM
cana-4037	34	37	|{1,2}|	|{1,2}|	PROPN
cana-4037	34	38	=	=	SYM
cana-4037	34	39	2	2	NUM
cana-4037	34	40	𝑜𝑑𝑆(8	𝑜𝑑𝑆(8	PROPN
cana-4037	34	41	)	)	PUNCT
cana-4037	34	42	=	=	SYM
cana-4037	34	43	|𝑁(8	|𝑁(8	NOUN
cana-4037	34	44	)	)	PUNCT
cana-4037	34	45	∩	∩	NOUN
cana-4037	34	46	(	(	PUNCT
cana-4037	34	47	𝑉	𝑉	PROPN
cana-4037	34	48	−	−	PROPN
cana-4037	34	49	𝐷)|	𝐷)|	NOUN
cana-4037	34	50	=	=	NOUN
cana-4037	34	51	|[3,7,9	|[3,7,9	PROPN
cana-4037	34	52	}	}	PUNCT
cana-4037	34	53	∩	∩	NOUN
cana-4037	34	54	{	{	PUNCT
cana-4037	34	55	1,2,3,5,6,7}|	1,2,3,5,6,7}|	PROPN
cana-4037	34	56	=	=	SYM
cana-4037	34	57	|{3,7}|	|{3,7}|	PROPN
cana-4037	34	58	=	=	SYM
cana-4037	34	59	2	2	NUM
cana-4037	34	60	𝑜𝑑𝑆(9	𝑜𝑑𝑆(9	NOUN
cana-4037	34	61	)	)	PUNCT
cana-4037	34	62	=	=	SYM
cana-4037	34	63	|𝑁(4	|𝑁(4	NOUN
cana-4037	34	64	)	)	PUNCT
cana-4037	34	65	∩	∩	NOUN
cana-4037	34	66	(	(	PUNCT
cana-4037	34	67	𝑉	𝑉	PROPN
cana-4037	34	68	−	−	PROPN
cana-4037	34	69	𝐷)|	𝐷)|	NOUN
cana-4037	34	70	=	=	PUNCT
cana-4037	34	71	|[4,8,10	|[4,8,10	PROPN
cana-4037	34	72	}	}	PUNCT
cana-4037	34	73	∩	∩	NOUN
cana-4037	34	74	{	{	PUNCT
cana-4037	34	75	1,2,3,5,6,7}|	1,2,3,5,6,7}|	PROPN
cana-4037	34	76	=	=	SYM
cana-4037	34	77	|{∅}|	|{∅}|	NOUN
cana-4037	34	78	=	=	SYM
cana-4037	34	79	0	0	NUM
cana-4037	34	80	𝑜𝑑𝑆(10	𝑜𝑑𝑆(10	NOUN
cana-4037	34	81	)	)	PUNCT
cana-4037	35	1	=	=	SYM
cana-4037	35	2	|𝑁(10	|𝑁(10	ADJ
cana-4037	35	3	)	)	PUNCT
cana-4037	35	4	∩	∩	NOUN
cana-4037	35	5	(	(	PUNCT
cana-4037	35	6	𝑉	𝑉	PROPN
cana-4037	35	7	−	−	PROPN
cana-4037	35	8	𝐷)|	𝐷)|	NOUN
cana-4037	35	9	=	=	NOUN
cana-4037	35	10	|[5,6,9	|[5,6,9	X
cana-4037	35	11	}	}	PUNCT
cana-4037	35	12	∩	∩	NOUN
cana-4037	35	13	{	{	PUNCT
cana-4037	35	14	1,2,3,5,6,7}|	1,2,3,5,6,7}|	NUM
cana-4037	35	15	=	=	SYM
cana-4037	35	16	|{5,6}|	|{5,6}|	PROPN
cana-4037	35	17	=	=	SYM
cana-4037	35	18	2	2	NUM
cana-4037	35	19	then	then	ADV
cana-4037	35	20	any	any	DET
cana-4037	35	21	u	u	NOUN
cana-4037	35	22	,	,	PUNCT
cana-4037	35	23	v∈	v∈	PROPN
cana-4037	35	24	𝐷	𝐷	PROPN
cana-4037	35	25	,	,	PUNCT
cana-4037	35	26	|𝑜𝑑𝐷(𝑢	|𝑜𝑑𝐷(𝑢	ADJ
cana-4037	35	27	)	)	PUNCT
cana-4037	35	28	−	−	ADP
cana-4037	35	29	𝑜𝑑𝐷(𝑣)|	𝑜𝑑𝐷(𝑣)|	NOUN
cana-4037	35	30	≤	≤	NUM
cana-4037	35	31	2	2	NUM
cana-4037	35	32	so	so	ADV
cana-4037	35	33	d=	d=	NOUN
cana-4037	35	34	{	{	PUNCT
cana-4037	35	35	4,8,9,10}form	4,8,9,10}form	PROPN
cana-4037	35	36	2oded	2ode	VERB
cana-4037	35	37	set	set	NOUN
cana-4037	35	38	of	of	ADP
cana-4037	35	39	𝐺	𝐺	PROPN
cana-4037	35	40	and	and	CCONJ
cana-4037	35	41	the	the	DET
cana-4037	35	42	sub	sub	NOUN
cana-4037	35	43	graph	graph	NOUN
cana-4037	35	44	induced	induce	VERB
cana-4037	35	45	by	by	ADP
cana-4037	35	46	<	<	X
cana-4037	35	47	𝑉	𝑉	PROPN
cana-4037	35	48	−	−	PROPN
cana-4037	35	49	𝐷	𝐷	PROPN
cana-4037	35	50	>	>	X
cana-4037	35	51	has	have	VERB
cana-4037	35	52	no	no	DET
cana-4037	35	53	vertices	vertex	NOUN
cana-4037	35	54	of	of	ADP
cana-4037	35	55	degree	degree	NOUN
cana-4037	35	56	zero	zero	NUM
cana-4037	35	57	.	.	PUNCT
cana-4037	36	1	hence	hence	ADV
cana-4037	36	2	d	d	PROPN
cana-4037	36	3	is	be	AUX
cana-4037	36	4	a	a	DET
cana-4037	36	5	co	co	NOUN
cana-4037	36	6	-	-	ADJ
cana-4037	36	7	total	total	ADJ
cana-4037	36	8	2oded	2ode	VERB
cana-4037	36	9	set	set	NOUN
cana-4037	36	10	with	with	ADP
cana-4037	36	11	minimum	minimum	NOUN
cana-4037	36	12	cardinality.hence	cardinality.hence	NOUN
cana-4037	36	13	𝛾𝑐𝑡2𝑜𝑒(𝐺	𝛾𝑐𝑡2𝑜𝑒(𝐺	X
cana-4037	36	14	)	)	PUNCT
cana-4037	37	1	=	=	SYM
cana-4037	37	2	3	3	NUM
cana-4037	37	3	observation	observation	NOUN
cana-4037	37	4	2.3	2.3	NUM
cana-4037	37	5	a	a	DET
cana-4037	37	6	graph	graph	NOUN
cana-4037	37	7	g	g	NOUN
cana-4037	37	8	with	with	ADP
cana-4037	37	9	p	p	NOUN
cana-4037	37	10	vertices	vertex	NOUN
cana-4037	37	11	,	,	PUNCT
cana-4037	37	12	then	then	ADV
cana-4037	37	13	2≤	2≤	NUM
cana-4037	37	14	𝛾𝑐𝑡2𝑜𝑒(𝐺	𝛾𝑐𝑡2𝑜𝑒(𝐺	X
cana-4037	37	15	)	)	PUNCT
cana-4037	37	16	≤	≤	NUM
cana-4037	37	17	𝑛.	𝑛.	NOUN
cana-4037	37	18	observation	observation	NOUN
cana-4037	37	19	2.4	2.4	NUM
cana-4037	37	20	1	1	NUM
cana-4037	37	21	.	.	PUNCT
cana-4037	38	1	for	for	ADP
cana-4037	38	2	any	any	DET
cana-4037	38	3	complete	complete	ADJ
cana-4037	38	4	graph:𝛾𝑐𝑡2𝑜𝑒(𝐾𝑛	graph:𝛾𝑐𝑡2𝑜𝑒(𝐾𝑛	NOUN
cana-4037	38	5	)	)	PUNCT
cana-4037	38	6	=	=	SYM
cana-4037	38	7	2	2	NUM
cana-4037	38	8	2	2	NUM
cana-4037	38	9	.	.	PUNCT
cana-4037	38	10	for	for	ADP
cana-4037	38	11	any	any	DET
cana-4037	38	12	star	star	NOUN
cana-4037	38	13	𝐾1,𝑛	𝐾1,𝑛	PROPN
cana-4037	38	14	,	,	PUNCT
cana-4037	38	15	then	then	ADV
cana-4037	38	16	𝛾𝑐𝑡2𝑜𝑒(𝐾1,𝑛−1	𝛾𝑐𝑡2𝑜𝑒(𝐾1,𝑛−1	NOUN
cana-4037	38	17	)	)	PUNCT
cana-4037	38	18	=	=	SYM
cana-4037	38	19	𝑛	𝑛	DET
cana-4037	38	20	3	3	NUM
cana-4037	38	21	.	.	X
cana-4037	38	22	for	for	ADP
cana-4037	38	23	any	any	DET
cana-4037	38	24	complete	complete	ADJ
cana-4037	38	25	bipartite	bipartite	NOUN
cana-4037	38	26	graph	graph	NOUN
cana-4037	38	27	𝐾𝑝,𝑞	𝐾𝑝,𝑞	PROPN
cana-4037	38	28	,	,	PUNCT
cana-4037	38	29	is	be	AUX
cana-4037	38	30	𝛾𝑐𝑡2𝑜𝑒(𝐾𝑝,𝑞	𝛾𝑐𝑡2𝑜𝑒(𝐾𝑝,𝑞	NOUN
cana-4037	38	31	)	)	PUNCT
cana-4037	39	1	=	=	PRON
cana-4037	39	2	{	{	PUNCT
cana-4037	39	3	2	2	NUM
cana-4037	39	4	𝑖𝑓	𝑖𝑓	NOUN
cana-4037	39	5	|𝑝	|𝑝	NOUN
cana-4037	39	6	−	−	PROPN
cana-4037	39	7	𝑞|	𝑞|	PROPN
cana-4037	39	8	≤	≤	NUM
cana-4037	39	9	2𝑝	2𝑝	NOUN
cana-4037	39	10	+	+	CCONJ
cana-4037	39	11	𝑞	𝑞	X
cana-4037	39	12	−	−	PROPN
cana-4037	39	13	2	2	NUM
cana-4037	39	14	𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒	𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒	NOUN
cana-4037	39	15	4	4	NUM
cana-4037	39	16	.	.	X
cana-4037	40	1	for	for	ADP
cana-4037	40	2	any	any	DET
cana-4037	40	3	cycle	cycle	NOUN
cana-4037	40	4	𝐶𝑛	𝐶𝑛	PROPN
cana-4037	40	5	,	,	PUNCT
cana-4037	40	6	then	then	ADV
cana-4037	40	7	𝛾𝑐𝑡2𝑜𝑒(𝐶𝑛	𝛾𝑐𝑡2𝑜𝑒(𝐶𝑛	PROPN
cana-4037	40	8	)	)	PUNCT
cana-4037	40	9	=	=	SYM
cana-4037	40	10	𝑛	𝑛	DET
cana-4037	40	11	−	−	NUM
cana-4037	40	12	2	2	NUM
cana-4037	40	13	5	5	NUM
cana-4037	40	14	.	.	PUNCT
cana-4037	40	15	for	for	ADP
cana-4037	40	16	the	the	DET
cana-4037	40	17	path	path	NOUN
cana-4037	40	18	𝑃𝑛	𝑃𝑛	PROPN
cana-4037	40	19	,	,	PUNCT
cana-4037	40	20	𝛾𝑐𝑡2𝑜𝑒(𝑃𝑛	𝛾𝑐𝑡2𝑜𝑒(𝑃𝑛	PROPN
cana-4037	40	21	)	)	PUNCT
cana-4037	41	1	=	=	NOUN
cana-4037	41	2	𝑛	𝑛	PRON
cana-4037	41	3	−	−	NUM
cana-4037	41	4	2	2	NUM
cana-4037	41	5	,	,	PUNCT
cana-4037	41	6	n≥	n≥	NOUN
cana-4037	41	7	2	2	NUM
cana-4037	41	8	6	6	NUM
cana-4037	41	9	.	.	PUNCT
cana-4037	42	1	for	for	ADP
cana-4037	42	2	the	the	DET
cana-4037	42	3	double	double	ADJ
cana-4037	42	4	star	star	NOUN
cana-4037	42	5	𝑆𝑟,𝑡	𝑆𝑟,𝑡	PROPN
cana-4037	42	6	,	,	PUNCT
cana-4037	42	7	𝛾𝑐𝑡2𝑜𝑒(𝑆𝑟,𝑡)=	𝛾𝑐𝑡2𝑜𝑒(𝑆𝑟,𝑡)=	VERB
cana-4037	42	8	𝑟	𝑟	NOUN
cana-4037	42	9	+	+	CCONJ
cana-4037	42	10	𝑡	𝑡	PROPN
cana-4037	42	11	communications	communication	NOUN
cana-4037	42	12	on	on	ADP
cana-4037	42	13	applied	apply	VERB
cana-4037	42	14	nonlinear	nonlinear	ADJ
cana-4037	42	15	analysis	analysis	NOUN
cana-4037	42	16	issn	issn	NOUN
cana-4037	42	17	:	:	PUNCT
cana-4037	42	18	1074	1074	NUM
cana-4037	42	19	-	-	PUNCT
cana-4037	42	20	133x	133x	NUM
cana-4037	42	21	vol	vol	NOUN
cana-4037	42	22	32	32	NUM
cana-4037	43	1	no	no	NOUN
cana-4037	43	2	.	.	PUNCT
cana-4037	44	1	9s	9s	NUM
cana-4037	44	2	(	(	PUNCT
cana-4037	44	3	2025	2025	NUM
cana-4037	44	4	)	)	PUNCT
cana-4037	44	5	896	896	NUM
cana-4037	44	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-4037	44	7	3	3	X
cana-4037	44	8	.	.	PUNCT
cana-4037	45	1	co	co	NOUN
cana-4037	45	2	-	-	NOUN
cana-4037	45	3	total	total	ADJ
cana-4037	45	4	2	2	NUM
cana-4037	45	5	ode	ode	NOUN
cana-4037	45	6	d	d	NOUN
cana-4037	45	7	number	number	NOUN
cana-4037	45	8	for	for	ADP
cana-4037	45	9	different	different	ADJ
cana-4037	45	10	graphs	graph	NOUN
cana-4037	45	11	here	here	ADV
cana-4037	45	12	begin	begin	VERB
cana-4037	45	13	the	the	DET
cana-4037	45	14	investigation	investigation	NOUN
cana-4037	45	15	of	of	ADP
cana-4037	45	16	the	the	DET
cana-4037	45	17	co	co	NOUN
cana-4037	45	18	-	-	ADJ
cana-4037	45	19	total	total	ADJ
cana-4037	45	20	2oded	2ode	VERB
cana-4037	45	21	number	number	NOUN
cana-4037	45	22	by	by	ADP
cana-4037	45	23	computing	compute	VERB
cana-4037	45	24	its	its	PRON
cana-4037	45	25	values	value	NOUN
cana-4037	45	26	for	for	ADP
cana-4037	45	27	some	some	DET
cana-4037	45	28	general	general	NOUN
cana-4037	45	29	of	of	ADP
cana-4037	45	30	graphs	graph	NOUN
cana-4037	45	31	.	.	PUNCT
cana-4037	46	1	theorem	theorem	VERB
cana-4037	46	2	:	:	PUNCT
cana-4037	46	3	3.1	3.1	NUM
cana-4037	46	4	for	for	ADP
cana-4037	46	5	any	any	DET
cana-4037	46	6	combo	combo	NOUN
cana-4037	46	7	graph	graph	NOUN
cana-4037	46	8	𝛾𝑐𝑡2𝑜𝑒(𝑃𝑛+	𝛾𝑐𝑡2𝑜𝑒(𝑃𝑛+	NOUN
cana-4037	46	9	)	)	PUNCT
cana-4037	46	10	=	=	SYM
cana-4037	47	1	𝑛	𝑛	DET
cana-4037	47	2	proof	proof	NOUN
cana-4037	47	3	:	:	PUNCT
cana-4037	47	4	let	let	VERB
cana-4037	47	5	v(𝑃𝑛+	v(𝑃𝑛+	VERB
cana-4037	47	6	)	)	PUNCT
cana-4037	47	7	=	=	PRON
cana-4037	47	8	{	{	PUNCT
cana-4037	47	9	𝑣1	𝑣1	PROPN
cana-4037	47	10	,	,	PUNCT
cana-4037	47	11	𝑣2𝑣3	𝑣2𝑣3	PROPN
cana-4037	47	12	…	…	PUNCT
cana-4037	47	13	.	.	PUNCT
cana-4037	48	1	𝑣𝑛	𝑣𝑛	NOUN
cana-4037	48	2	,	,	PUNCT
cana-4037	48	3	𝑣𝑛+1	𝑣𝑛+1	NOUN
cana-4037	48	4	,	,	PUNCT
cana-4037	48	5	𝑣𝑛+2	𝑣𝑛+2	NUM
cana-4037	48	6	,	,	PUNCT
cana-4037	48	7	𝑣𝑛+3	𝑣𝑛+3	NOUN
cana-4037	48	8	,	,	PUNCT
cana-4037	48	9	…	…	PUNCT
cana-4037	48	10	.	.	PUNCT
cana-4037	49	1	𝑣2𝑛	𝑣2𝑛	NUM
cana-4037	49	2	}	}	PUNCT
cana-4037	49	3	.	.	PUNCT
cana-4037	50	1	here	here	ADV
cana-4037	50	2	{	{	PUNCT
cana-4037	50	3	𝑣1	𝑣1	PROPN
cana-4037	50	4	,	,	PUNCT
cana-4037	50	5	𝑣2𝑣3	𝑣2𝑣3	PROPN
cana-4037	50	6	…	…	PUNCT
cana-4037	50	7	.	.	PUNCT
cana-4037	51	1	𝑣𝑛	𝑣𝑛	X
cana-4037	51	2	}	}	PUNCT
cana-4037	51	3	be	be	VERB
cana-4037	51	4	the	the	DET
cana-4037	51	5	vertices	vertex	NOUN
cana-4037	51	6	of	of	ADP
cana-4037	51	7	the	the	DET
cana-4037	51	8	path	path	NOUN
cana-4037	51	9	{	{	PUNCT
cana-4037	51	10	𝑣𝑛+1	𝑣𝑛+1	PROPN
cana-4037	51	11	,	,	PUNCT
cana-4037	51	12	𝑣𝑛+2	𝑣𝑛+2	NUM
cana-4037	51	13	,	,	PUNCT
cana-4037	51	14	𝑣𝑛+3	𝑣𝑛+3	NOUN
cana-4037	51	15	,	,	PUNCT
cana-4037	51	16	…	…	PUNCT
cana-4037	51	17	.	.	PUNCT
cana-4037	52	1	𝑣2𝑛	𝑣2𝑛	X
cana-4037	52	2	}	}	PUNCT
cana-4037	52	3	be	be	AUX
cana-4037	52	4	the	the	DET
cana-4037	52	5	vertices	vertex	NOUN
cana-4037	52	6	of	of	ADP
cana-4037	52	7	degree	degree	NOUN
cana-4037	52	8	one	one	NOUN
cana-4037	52	9	is	be	AUX
cana-4037	52	10	incident	incident	NOUN
cana-4037	52	11	with	with	ADP
cana-4037	52	12	each	each	DET
cana-4037	52	13	vertex	vertex	NOUN
cana-4037	52	14	of	of	ADP
cana-4037	52	15	the	the	DET
cana-4037	52	16	path	path	NOUN
cana-4037	52	17	.	.	PUNCT
cana-4037	53	1	let	let	VERB
cana-4037	53	2	d=	d=	NUM
cana-4037	53	3	{	{	PUNCT
cana-4037	53	4	𝑣𝑛+1	𝑣𝑛+1	NOUN
cana-4037	53	5	,	,	PUNCT
cana-4037	53	6	𝑣𝑛+2	𝑣𝑛+2	NUM
cana-4037	53	7	,	,	PUNCT
cana-4037	53	8	𝑣𝑛+3	𝑣𝑛+3	NOUN
cana-4037	53	9	,	,	PUNCT
cana-4037	53	10	…	…	PUNCT
cana-4037	53	11	.	.	PUNCT
cana-4037	54	1	𝑣2𝑛	𝑣2𝑛	X
cana-4037	54	2	}	}	PUNCT
cana-4037	54	3	be	be	AUX
cana-4037	54	4	minimal	minimal	ADJ
cana-4037	54	5	dominating	dominating	NOUN
cana-4037	54	6	set	set	NOUN
cana-4037	54	7	and	and	CCONJ
cana-4037	54	8	𝑉	𝑉	PROPN
cana-4037	54	9	−	−	PROPN
cana-4037	55	1	𝐷=	𝐷=	PROPN
cana-4037	55	2	{	{	PUNCT
cana-4037	55	3	𝑣1	𝑣1	PROPN
cana-4037	55	4	,	,	PUNCT
cana-4037	55	5	𝑣2𝑣3	𝑣2𝑣3	PROPN
cana-4037	55	6	…	…	PUNCT
cana-4037	55	7	.	.	PUNCT
cana-4037	56	1	𝑣𝑛	𝑣𝑛	X
cana-4037	56	2	}	}	PUNCT
cana-4037	56	3	.	.	PUNCT
cana-4037	57	1	each	each	DET
cana-4037	57	2	vertices	vertex	NOUN
cana-4037	57	3	d	d	NOUN
cana-4037	57	4	has	have	VERB
cana-4037	57	5	a	a	DET
cana-4037	57	6	neighborhood	neighborhood	NOUN
cana-4037	57	7	𝑉	𝑉	NOUN
cana-4037	57	8	−	−	PROPN
cana-4037	57	9	𝐷	𝐷	PROPN
cana-4037	57	10	and	and	CCONJ
cana-4037	57	11	𝑜𝑑𝐷(𝑣𝑛+𝑖	𝑜𝑑𝐷(𝑣𝑛+𝑖	NOUN
cana-4037	57	12	)	)	PUNCT
cana-4037	57	13	=	=	PUNCT
cana-4037	57	14	|𝑁(𝑣𝑛+𝑖)⋂(𝑉	|𝑁(𝑣𝑛+𝑖)⋂(𝑉	X
cana-4037	57	15	−𝐷)|=1	−𝐷)|=1	NUM
cana-4037	57	16	for	for	ADP
cana-4037	57	17	all	all	DET
cana-4037	57	18	i=1,2,3	i=1,2,3	NUM
cana-4037	57	19	…	…	SYM
cana-4037	57	20	…	…	PUNCT
cana-4037	57	21	n.	n.	NOUN
cana-4037	57	22	then	then	ADV
cana-4037	57	23	|𝑜𝑑𝐷(𝑣𝑛+𝑖	|𝑜𝑑𝐷(𝑣𝑛+𝑖	PROPN
cana-4037	57	24	)	)	PUNCT
cana-4037	57	25	−	−	PUNCT
cana-4037	58	1	𝑜𝑑𝐷(𝑣𝑛+𝑗)|	𝑜𝑑𝐷(𝑣𝑛+𝑗)|	SYM
cana-4037	59	1	≤	≤	ADV
cana-4037	59	2	2	2	NUM
cana-4037	59	3	.	.	PUNCT
cana-4037	60	1	then	then	ADV
cana-4037	60	2	d	d	X
cana-4037	60	3	is	be	AUX
cana-4037	60	4	2oded	2ode	VERB
cana-4037	60	5	set	set	VERB
cana-4037	60	6	and	and	CCONJ
cana-4037	60	7	<	<	X
cana-4037	60	8	𝑉	𝑉	PROPN
cana-4037	60	9	−𝐷	−𝐷	NOUN
cana-4037	60	10	>	>	X
cana-4037	60	11	has	have	VERB
cana-4037	60	12	no	no	DET
cana-4037	60	13	vertices	vertex	NOUN
cana-4037	60	14	of	of	ADP
cana-4037	60	15	degree	degree	NOUN
cana-4037	60	16	zero	zero	NUM
cana-4037	60	17	.	.	PUNCT
cana-4037	61	1	then	then	ADV
cana-4037	61	2	d	d	X
cana-4037	61	3	is	be	AUX
cana-4037	61	4	minimal	minimal	ADJ
cana-4037	61	5	co	co	ADJ
cana-4037	61	6	-	-	ADJ
cana-4037	61	7	total	total	ADJ
cana-4037	61	8	2oded	2ode	VERB
cana-4037	61	9	set	set	NOUN
cana-4037	61	10	.	.	PUNCT
cana-4037	62	1	then	then	ADV
cana-4037	62	2	𝛾𝑐𝑡2𝑜𝑒(𝑃𝑛+	𝛾𝑐𝑡2𝑜𝑒(𝑃𝑛+	NOUN
cana-4037	62	3	)	)	PUNCT
cana-4037	62	4	=	=	NOUN
cana-4037	62	5	|𝐷|	|𝐷|	NOUN
cana-4037	62	6	=	=	SYM
cana-4037	62	7	𝑛.	𝑛.	NOUN
cana-4037	62	8	theorem	theorem	VERB
cana-4037	62	9	3.2	3.2	NUM
cana-4037	62	10	for	for	ADP
cana-4037	62	11	any	any	DET
cana-4037	62	12	crown	crown	NOUN
cana-4037	62	13	graph	graph	NOUN
cana-4037	62	14	𝐶𝑛+	𝐶𝑛+	PROPN
cana-4037	62	15	,	,	PUNCT
cana-4037	62	16	𝛾𝑐𝑡2𝑜𝑒(𝐶𝑛+	𝛾𝑐𝑡2𝑜𝑒(𝐶𝑛+	PROPN
cana-4037	62	17	)	)	PUNCT
cana-4037	62	18	=	=	PUNCT
cana-4037	63	1	𝑛	𝑛	DET
cana-4037	63	2	proof	proof	NOUN
cana-4037	63	3	:	:	PUNCT
cana-4037	63	4	let	let	VERB
cana-4037	63	5	v(𝐶𝑛+	v(𝐶𝑛+	ADV
cana-4037	63	6	)	)	PUNCT
cana-4037	63	7	=	=	PRON
cana-4037	63	8	{	{	PUNCT
cana-4037	63	9	𝑣1	𝑣1	PROPN
cana-4037	63	10	,	,	PUNCT
cana-4037	63	11	𝑣2	𝑣2	PROPN
cana-4037	63	12	,	,	PUNCT
cana-4037	63	13	𝑣3	𝑣3	ADJ
cana-4037	63	14	…	…	PUNCT
cana-4037	63	15	.	.	PUNCT
cana-4037	64	1	𝑣𝑛	𝑣𝑛	NOUN
cana-4037	64	2	,	,	PUNCT
cana-4037	64	3	𝑣𝑛+1	𝑣𝑛+1	NOUN
cana-4037	64	4	,	,	PUNCT
cana-4037	64	5	𝑣𝑛+2	𝑣𝑛+2	NUM
cana-4037	64	6	…	…	PUNCT
cana-4037	64	7	.	.	PUNCT
cana-4037	65	1	𝑣2𝑛	𝑣2𝑛	ADJ
cana-4037	65	2	}	}	PUNCT
cana-4037	65	3	.	.	PUNCT
cana-4037	66	1	here	here	ADV
cana-4037	66	2	{	{	PUNCT
cana-4037	66	3	𝑣1	𝑣1	PROPN
cana-4037	66	4	,	,	PUNCT
cana-4037	66	5	𝑣2	𝑣2	PROPN
cana-4037	66	6	,	,	PUNCT
cana-4037	66	7	𝑣3	𝑣3	ADJ
cana-4037	66	8	…	…	PUNCT
cana-4037	66	9	.	.	PUNCT
cana-4037	67	1	𝑣𝑝	𝑣𝑝	X
cana-4037	67	2	}	}	PUNCT
cana-4037	67	3	be	be	AUX
cana-4037	67	4	the	the	DET
cana-4037	67	5	vertices	vertex	NOUN
cana-4037	67	6	of	of	ADP
cana-4037	67	7	cycle	cycle	NOUN
cana-4037	67	8	𝐶𝑝	𝐶𝑝	PROPN
cana-4037	67	9	𝑎𝑛𝑑	𝑎𝑛𝑑	ADJ
cana-4037	67	10	{	{	PUNCT
cana-4037	67	11	𝑣𝑛+1	𝑣𝑛+1	NOUN
cana-4037	67	12	,	,	PUNCT
cana-4037	67	13	𝑣𝑛+2	𝑣𝑛+2	NUM
cana-4037	67	14	,	,	PUNCT
cana-4037	67	15	…	…	PUNCT
cana-4037	67	16	.	.	PUNCT
cana-4037	68	1	,	,	PUNCT
cana-4037	68	2	𝑣2𝑛	𝑣2𝑛	PROPN
cana-4037	68	3	}	}	PUNCT
cana-4037	68	4	be	be	AUX
cana-4037	68	5	the	the	DET
cana-4037	68	6	vertices	vertex	NOUN
cana-4037	68	7	of	of	ADP
cana-4037	68	8	degree	degree	NOUN
cana-4037	68	9	zero	zero	NUM
cana-4037	68	10	which	which	PRON
cana-4037	68	11	adjacent	adjacent	ADJ
cana-4037	68	12	to	to	ADP
cana-4037	68	13	𝑣1	𝑣1	PROPN
cana-4037	68	14	,	,	PUNCT
cana-4037	68	15	𝑣2	𝑣2	PROPN
cana-4037	68	16	,	,	PUNCT
cana-4037	68	17	𝑣3	𝑣3	ADJ
cana-4037	68	18	…	…	PUNCT
cana-4037	68	19	.	.	PUNCT
cana-4037	69	1	𝑣𝑛	𝑣𝑛	ADP
cana-4037	69	2	respectively	respectively	ADV
cana-4037	69	3	.	.	PUNCT
cana-4037	70	1	let	let	VERB
cana-4037	70	2	us	we	PRON
cana-4037	70	3	take	take	VERB
cana-4037	70	4	d=	d=	NOUN
cana-4037	70	5	{	{	PUNCT
cana-4037	70	6	𝑣𝑛+1	𝑣𝑛+1	NOUN
cana-4037	70	7	,	,	PUNCT
cana-4037	70	8	𝑣𝑛+2	𝑣𝑛+2	NUM
cana-4037	70	9	,	,	PUNCT
cana-4037	70	10	…	…	PUNCT
cana-4037	70	11	.	.	PUNCT
cana-4037	71	1	,	,	PUNCT
cana-4037	71	2	𝑣2𝑛	𝑣2𝑛	AUX
cana-4037	71	3	}	}	PUNCT
cana-4037	71	4	be	be	AUX
cana-4037	71	5	dominating	dominate	VERB
cana-4037	71	6	set	set	VERB
cana-4037	71	7	with	with	ADP
cana-4037	71	8	minimal	minimal	ADJ
cana-4037	71	9	cardinality	cardinality	NOUN
cana-4037	71	10	and	and	CCONJ
cana-4037	71	11	𝑉	𝑉	PROPN
cana-4037	71	12	−	−	PROPN
cana-4037	71	13	𝐷	𝐷	NOUN
cana-4037	71	14	=	=	SYM
cana-4037	71	15	{	{	PUNCT
cana-4037	71	16	𝑣1	𝑣1	PROPN
cana-4037	71	17	,	,	PUNCT
cana-4037	71	18	𝑣2	𝑣2	PROPN
cana-4037	71	19	,	,	PUNCT
cana-4037	71	20	𝑣3	𝑣3	ADJ
cana-4037	71	21	…	…	PUNCT
cana-4037	71	22	.	.	PUNCT
cana-4037	72	1	𝑣𝑛	𝑣𝑛	NOUN
cana-4037	72	2	,	,	PUNCT
cana-4037	72	3	}	}	PUNCT
cana-4037	72	4	then	then	ADV
cana-4037	72	5	𝑜𝑑𝐷(𝑣𝑛+𝑖	𝑜𝑑𝐷(𝑣𝑛+𝑖	PROPN
cana-4037	72	6	)	)	PUNCT
cana-4037	72	7	=	=	SYM
cana-4037	73	1	|𝑁(𝑣𝑛+𝑖	|𝑁(𝑣𝑛+𝑖	NOUN
cana-4037	73	2	)	)	PUNCT
cana-4037	73	3	∩	∩	NOUN
cana-4037	73	4	(	(	PUNCT
cana-4037	73	5	𝑉	𝑉	PROPN
cana-4037	73	6	−	−	PROPN
cana-4037	73	7	𝐷	𝐷	PROPN
cana-4037	73	8	)	)	PUNCT
cana-4037	73	9	=	=	PUNCT
cana-4037	73	10	|{𝑣𝑖	|{𝑣𝑖	PROPN
cana-4037	73	11	}	}	PUNCT
cana-4037	73	12	∩	∩	NOUN
cana-4037	73	13	{	{	PUNCT
cana-4037	73	14	𝑣1	𝑣1	PROPN
cana-4037	73	15	,	,	PUNCT
cana-4037	73	16	𝑣2	𝑣2	PROPN
cana-4037	73	17	,	,	PUNCT
cana-4037	73	18	𝑣3	𝑣3	ADJ
cana-4037	73	19	…	…	PUNCT
cana-4037	73	20	.	.	PUNCT
cana-4037	74	1	𝑣𝑛}|	𝑣𝑛}|	X
cana-4037	74	2	=	=	PUNCT
cana-4037	74	3	|{𝑣𝑖}|=1	|{𝑣𝑖}|=1	NOUN
cana-4037	74	4	.	.	PUNCT
cana-4037	75	1	then	then	ADV
cana-4037	75	2	|𝑜𝑑𝐷(𝑣𝑖	|𝑜𝑑𝐷(𝑣𝑖	NOUN
cana-4037	75	3	)	)	PUNCT
cana-4037	76	1	−	−	NOUN
cana-4037	76	2	𝑜𝑑𝐷(𝑣𝑗)|	𝑜𝑑𝐷(𝑣𝑗)|	NOUN
cana-4037	76	3	=	=	NOUN
cana-4037	76	4	0	0	PUNCT
cana-4037	76	5	<	<	X
cana-4037	76	6	2	2	NUM
cana-4037	76	7	.	.	PUNCT
cana-4037	77	1	hence	hence	ADV
cana-4037	77	2	d	d	PROPN
cana-4037	77	3	is	be	AUX
cana-4037	77	4	a	a	DET
cana-4037	77	5	2	2	NUM
cana-4037	77	6	oded	oded	NOUN
cana-4037	77	7	set	set	NOUN
cana-4037	77	8	and	and	CCONJ
cana-4037	77	9	the	the	DET
cana-4037	77	10	sub	sub	NOUN
cana-4037	77	11	graph	graph	NOUN
cana-4037	77	12	induced	induce	VERB
cana-4037	77	13	by	by	ADP
cana-4037	77	14	𝑉	𝑉	PROPN
cana-4037	77	15	−	−	PROPN
cana-4037	77	16	𝐷	𝐷	PROPN
cana-4037	77	17	has	have	VERB
cana-4037	77	18	no	no	DET
cana-4037	77	19	vertices	vertex	NOUN
cana-4037	77	20	of	of	ADP
cana-4037	77	21	degree	degree	NOUN
cana-4037	77	22	zero	zero	NUM
cana-4037	77	23	.	.	PUNCT
cana-4037	78	1	then	then	ADV
cana-4037	78	2	𝛾𝑐𝑡2𝑜𝑒(𝐶𝑛+	𝛾𝑐𝑡2𝑜𝑒(𝐶𝑛+	PROPN
cana-4037	78	3	)	)	PUNCT
cana-4037	79	1	=	=	PUNCT
cana-4037	79	2	|𝐷|=	|𝐷|=	PROPN
cana-4037	79	3	𝑛.	𝑛.	NOUN
cana-4037	79	4	theorem:3.3	theorem:3.3	NOUN
cana-4037	79	5	for	for	ADP
cana-4037	79	6	any	any	DET
cana-4037	79	7	triangular	triangular	NOUN
cana-4037	79	8	snake	snake	NOUN
cana-4037	79	9	graph	graph	NOUN
cana-4037	79	10	𝛾𝑐𝑡2𝑜𝑒(𝑛𝐶3	𝛾𝑐𝑡2𝑜𝑒(𝑛𝐶3	NOUN
cana-4037	79	11	)	)	PUNCT
cana-4037	80	1	=	=	SYM
cana-4037	80	2	𝑛	𝑛	PRON
cana-4037	80	3	−	−	NOUN
cana-4037	80	4	1	1	NUM
cana-4037	80	5	proof	proof	NOUN
cana-4037	80	6	:	:	PUNCT
cana-4037	80	7	the	the	DET
cana-4037	80	8	graph	graph	NOUN
cana-4037	80	9	g	g	PROPN
cana-4037	80	10	contains	contain	VERB
cana-4037	80	11	2𝑛	2𝑛	PROPN
cana-4037	80	12	−	−	ADP
cana-4037	80	13	1	1	NUM
cana-4037	80	14	vertices	vertex	NOUN
cana-4037	80	15	and	and	CCONJ
cana-4037	80	16	𝑛	𝑛	DET
cana-4037	80	17	−	−	PROPN
cana-4037	80	18	1	1	NUM
cana-4037	80	19	triangles	triangle	NOUN
cana-4037	80	20	.	.	PUNCT
cana-4037	81	1	the	the	DET
cana-4037	81	2	upper	upper	ADJ
cana-4037	81	3	vertices	vertex	NOUN
cana-4037	81	4	labeled	label	VERB
cana-4037	81	5	from	from	ADP
cana-4037	81	6	𝑣1	𝑣1	PROPN
cana-4037	81	7	to	to	ADP
cana-4037	81	8	𝑣𝑛−1	𝑣𝑛−1	PROPN
cana-4037	81	9	and	and	CCONJ
cana-4037	81	10	the	the	DET
cana-4037	81	11	lower	low	ADJ
cana-4037	81	12	vertices	vertex	NOUN
cana-4037	81	13	are	be	AUX
cana-4037	81	14	labeled	label	VERB
cana-4037	81	15	form	form	NOUN
cana-4037	81	16	𝑣𝑛to	𝑣𝑛to	PROPN
cana-4037	81	17	𝑣2𝑛−1	𝑣2𝑛−1	PROPN
cana-4037	81	18	.	.	PUNCT
cana-4037	82	1	let	let	VERB
cana-4037	82	2	𝐷	𝐷	NOUN
cana-4037	82	3	=	=	SYM
cana-4037	82	4	{	{	PUNCT
cana-4037	82	5	𝑣1	𝑣1	PROPN
cana-4037	82	6	,	,	PUNCT
cana-4037	82	7	𝑣2	𝑣2	PROPN
cana-4037	82	8	,	,	PUNCT
cana-4037	82	9	…	…	PUNCT
cana-4037	82	10	…	…	PUNCT
cana-4037	82	11	.	.	PUNCT
cana-4037	83	1	𝑣𝑛−1}and	𝑣𝑛−1}and	VERB
cana-4037	83	2	v−𝐷	v−𝐷	NOUN
cana-4037	83	3	=	=	NOUN
cana-4037	83	4	{	{	PUNCT
cana-4037	83	5	𝑣𝑛	𝑣𝑛	NOUN
cana-4037	83	6	,	,	PUNCT
cana-4037	83	7	𝑣𝑛+1	𝑣𝑛+1	NOUN
cana-4037	83	8	,	,	PUNCT
cana-4037	83	9	𝑣𝑛+2	𝑣𝑛+2	NUM
cana-4037	83	10	,	,	PUNCT
cana-4037	83	11	𝑣𝑛+3	𝑣𝑛+3	NOUN
cana-4037	83	12	…	…	PUNCT
cana-4037	83	13	…	…	PUNCT
cana-4037	83	14	.	.	PUNCT
cana-4037	84	1	𝑣2𝑛−2	𝑣2𝑛−2	ADV
cana-4037	84	2	,	,	PUNCT
cana-4037	84	3	𝑣2𝑛−1	𝑣2𝑛−1	PROPN
cana-4037	84	4	}	}	PUNCT
cana-4037	84	5	.	.	PUNCT
cana-4037	85	1	the	the	DET
cana-4037	85	2	vertex	vertex	NOUN
cana-4037	85	3	𝑣1	𝑣1	NOUN
cana-4037	85	4	is	be	AUX
cana-4037	85	5	adjacent	adjacent	ADJ
cana-4037	85	6	to	to	ADP
cana-4037	85	7	𝑣𝑛	𝑣𝑛	PROPN
cana-4037	85	8	,	,	PUNCT
cana-4037	85	9	𝑣𝑛+1	𝑣𝑛+1	PROPN
cana-4037	85	10	.	.	PUNCT
cana-4037	86	1	the	the	DET
cana-4037	86	2	vertex	vertex	NOUN
cana-4037	86	3	𝑣2	𝑣2	NOUN
cana-4037	86	4	is	be	AUX
cana-4037	86	5	adjacent	adjacent	ADJ
cana-4037	86	6	to	to	ADP
cana-4037	86	7	𝑣𝑛+1	𝑣𝑛+1	NUM
cana-4037	86	8	,	,	PUNCT
cana-4037	86	9	𝑣𝑛+2	𝑣𝑛+2	NUM
cana-4037	86	10	.	.	PUNCT
cana-4037	87	1	the	the	DET
cana-4037	87	2	vertex	vertex	NOUN
cana-4037	87	3	𝑣𝑛−1	𝑣𝑛−1	PROPN
cana-4037	87	4	is	be	AUX
cana-4037	87	5	adjacent	adjacent	ADJ
cana-4037	87	6	to	to	ADP
cana-4037	87	7	𝑣2𝑛−2	𝑣2𝑛−2	PROPN
cana-4037	87	8	,	,	PUNCT
cana-4037	87	9	𝑣2𝑛−1	𝑣2𝑛−1	PROPN
cana-4037	87	10	.	.	PUNCT
cana-4037	88	1	hence	hence	ADV
cana-4037	88	2	the	the	DET
cana-4037	88	3	set	set	NOUN
cana-4037	88	4	{	{	PUNCT
cana-4037	88	5	𝑣𝑛+1	𝑣𝑛+1	NOUN
cana-4037	88	6	,	,	PUNCT
cana-4037	88	7	𝑣𝑛+3	𝑣𝑛+3	NOUN
cana-4037	88	8	…	…	PUNCT
cana-4037	88	9	…	…	PUNCT
cana-4037	88	10	.	.	PUNCT
cana-4037	88	11	.	.	PUNCT
cana-4037	89	1	𝑣2𝑛−2	𝑣2𝑛−2	ADV
cana-4037	89	2	}	}	PUNCT
cana-4037	89	3	is	be	AUX
cana-4037	89	4	minimum	minimum	ADJ
cana-4037	89	5	dominating	dominating	NOUN
cana-4037	89	6	set	set	NOUN
cana-4037	89	7	.	.	PUNCT
cana-4037	90	1	now	now	ADV
cana-4037	90	2	𝑜𝑑𝐷(𝑣1	𝑜𝑑𝐷(𝑣1	NUM
cana-4037	90	3	)	)	PUNCT
cana-4037	90	4	=	=	SYM
cana-4037	90	5	|𝑁(𝑣1	|𝑁(𝑣1	NUM
cana-4037	90	6	)	)	PUNCT
cana-4037	90	7	∩	∩	NOUN
cana-4037	90	8	(	(	PUNCT
cana-4037	90	9	𝑉	𝑉	PROPN
cana-4037	90	10	−	−	PROPN
cana-4037	90	11	𝐷)|	𝐷)|	NOUN
cana-4037	90	12	=	=	SYM
cana-4037	90	13	|{𝑣𝑛	|{𝑣𝑛	NOUN
cana-4037	90	14	,	,	PUNCT
cana-4037	90	15	𝑣𝑛+1}|	𝑣𝑛+1}|	X
cana-4037	90	16	=	=	SYM
cana-4037	90	17	2	2	NUM
cana-4037	90	18	𝑜𝑑𝐷(𝑣2	𝑜𝑑𝐷(𝑣2	ADV
cana-4037	90	19	)	)	PUNCT
cana-4037	90	20	=	=	SYM
cana-4037	90	21	|𝑁(𝑣2	|𝑁(𝑣2	PROPN
cana-4037	90	22	)	)	PUNCT
cana-4037	90	23	∩	∩	NOUN
cana-4037	90	24	(	(	PUNCT
cana-4037	90	25	𝑉	𝑉	PROPN
cana-4037	90	26	−	−	PROPN
cana-4037	90	27	𝐷)|	𝐷)|	NOUN
cana-4037	90	28	=	=	SYM
cana-4037	90	29	|{𝑣𝑛+1	|{𝑣𝑛+1	PROPN
cana-4037	90	30	,	,	PUNCT
cana-4037	90	31	𝑣𝑛+2}|	𝑣𝑛+2}|	NUM
cana-4037	90	32	=	=	SYM
cana-4037	90	33	2	2	NUM
cana-4037	90	34	and	and	CCONJ
cana-4037	90	35	𝑜𝑑𝐷(𝑣𝑛−1	𝑜𝑑𝐷(𝑣𝑛−1	NOUN
cana-4037	90	36	)	)	PUNCT
cana-4037	90	37	=	=	SYM
cana-4037	90	38	|𝑁(𝑣𝑛−1	|𝑁(𝑣𝑛−1	PROPN
cana-4037	90	39	)	)	PUNCT
cana-4037	90	40	∩	∩	NOUN
cana-4037	90	41	(	(	PUNCT
cana-4037	90	42	𝑉	𝑉	PROPN
cana-4037	90	43	−	−	PROPN
cana-4037	90	44	𝐷)|	𝐷)|	NOUN
cana-4037	90	45	=	=	SYM
cana-4037	90	46	|{𝑣2𝑛−2	|{𝑣2𝑛−2	PROPN
cana-4037	90	47	,	,	PUNCT
cana-4037	90	48	𝑣2𝑛−1}|	𝑣2𝑛−1}|	NOUN
cana-4037	90	49	=	=	SYM
cana-4037	90	50	2	2	NUM
cana-4037	90	51	.	.	PUNCT
cana-4037	90	52	hence	hence	ADV
cana-4037	90	53	|𝑜𝑑𝐷(𝑢	|𝑜𝑑𝐷(𝑢	VERB
cana-4037	90	54	)	)	PUNCT
cana-4037	91	1	−	−	ADP
cana-4037	91	2	𝑜𝑑𝐷(𝑣)|	𝑜𝑑𝐷(𝑣)|	NOUN
cana-4037	91	3	≤	≤	NOUN
cana-4037	91	4	2	2	NUM
cana-4037	91	5	𝑓𝑜𝑟	𝑓𝑜𝑟	NOUN
cana-4037	91	6	𝑎𝑙𝑙	𝑎𝑙𝑙	VERB
cana-4037	91	7	𝑢	𝑢	PROPN
cana-4037	91	8	,	,	PUNCT
cana-4037	91	9	𝑣	𝑣	PROPN
cana-4037	91	10	∈	∈	PROPN
cana-4037	91	11	𝐷	𝐷	PROPN
cana-4037	91	12	,	,	PUNCT
cana-4037	91	13	and	and	CCONJ
cana-4037	91	14	d	d	NOUN
cana-4037	91	15	is	be	AUX
cana-4037	91	16	minimum	minimum	ADJ
cana-4037	91	17	2	2	NUM
cana-4037	91	18	ode	ode	ADJ
cana-4037	91	19	dominating	dominating	NOUN
cana-4037	91	20	set	set	NOUN
cana-4037	91	21	and	and	CCONJ
cana-4037	91	22	clearly	clearly	ADV
cana-4037	91	23	<	<	X
cana-4037	91	24	𝑉	𝑉	PROPN
cana-4037	91	25	−	−	PROPN
cana-4037	91	26	𝐷	𝐷	PROPN
cana-4037	91	27	>	>	X
cana-4037	91	28	set	set	NOUN
cana-4037	91	29	.	.	PUNCT
cana-4037	92	1	hence	hence	ADV
cana-4037	92	2	𝛾𝑐𝑡2𝑜𝑒(𝐺	𝛾𝑐𝑡2𝑜𝑒(𝐺	X
cana-4037	92	3	)	)	PUNCT
cana-4037	93	1	=	=	SYM
cana-4037	93	2	𝑛	𝑛	PRON
cana-4037	93	3	−	−	NUM
cana-4037	93	4	1	1	NUM
cana-4037	93	5	.	.	PUNCT
cana-4037	93	6	theorem	theorem	VERB
cana-4037	93	7	3.4	3.4	NUM
cana-4037	93	8	for	for	ADP
cana-4037	93	9	any	any	DET
cana-4037	93	10	double	double	ADJ
cana-4037	93	11	triangular	triangular	NOUN
cana-4037	93	12	snake	snake	NOUN
cana-4037	93	13	graph	graph	NOUN
cana-4037	93	14	𝛾𝑐𝑡2𝑜𝑒(𝐷(𝑛𝐶3	𝛾𝑐𝑡2𝑜𝑒(𝐷(𝑛𝐶3	PROPN
cana-4037	93	15	)	)	PUNCT
cana-4037	93	16	)	)	PUNCT
cana-4037	94	1	=	=	SYM
cana-4037	94	2	𝑛	𝑛	PRON
cana-4037	94	3	+	+	NOUN
cana-4037	94	4	1	1	NUM
cana-4037	94	5	proof	proof	NOUN
cana-4037	94	6	:	:	PUNCT
cana-4037	94	7	let	let	VERB
cana-4037	94	8	v(𝐷(𝑛𝐶3))=	v(𝐷(𝑛𝐶3))=	PROPN
cana-4037	94	9	{	{	PUNCT
cana-4037	94	10	𝑢1	𝑢1	PROPN
cana-4037	94	11	,	,	PUNCT
cana-4037	94	12	𝑢2	𝑢2	PROPN
cana-4037	94	13	,	,	PUNCT
cana-4037	94	14	𝑢3	𝑢3	PROPN
cana-4037	94	15	…	…	PUNCT
cana-4037	94	16	…	…	PUNCT
cana-4037	94	17	𝑢𝑛+1	𝑢𝑛+1	NUM
cana-4037	94	18	,	,	PUNCT
cana-4037	94	19	𝑣1	𝑣1	PROPN
cana-4037	94	20	,	,	PUNCT
cana-4037	94	21	𝑣2	𝑣2	PROPN
cana-4037	94	22	,	,	PUNCT
cana-4037	94	23	𝑣3	𝑣3	ADJ
cana-4037	94	24	…	…	PUNCT
cana-4037	94	25	…	…	SYM
cana-4037	94	26	𝑣𝑛,𝑤1	𝑣𝑛,𝑤1	ADJ
cana-4037	94	27	,	,	PUNCT
cana-4037	94	28	𝑤2	𝑤2	NOUN
cana-4037	94	29	,	,	PUNCT
cana-4037	94	30	𝑤3	𝑤3	PROPN
cana-4037	94	31	…	…	PUNCT
cana-4037	94	32	…	…	PUNCT
cana-4037	94	33	𝑤𝑛	𝑤𝑛	NOUN
cana-4037	94	34	}	}	PUNCT
cana-4037	94	35	.	.	PUNCT
cana-4037	95	1	here	here	ADV
cana-4037	95	2	{	{	PUNCT
cana-4037	95	3	𝑢1	𝑢1	PROPN
cana-4037	95	4	,	,	PUNCT
cana-4037	95	5	𝑢2	𝑢2	PROPN
cana-4037	95	6	,	,	PUNCT
cana-4037	95	7	𝑢3	𝑢3	PROPN
cana-4037	95	8	…	…	PUNCT
cana-4037	95	9	…	…	PUNCT
cana-4037	95	10	.	.	PUNCT
cana-4037	95	11	.	.	PUNCT
cana-4037	96	1	𝑢𝑛+1	𝑢𝑛+1	X
cana-4037	96	2	}	}	PUNCT
cana-4037	96	3	be	be	AUX
cana-4037	96	4	the	the	DET
cana-4037	96	5	vertices	vertex	NOUN
cana-4037	96	6	of	of	ADP
cana-4037	96	7	path	path	NOUN
cana-4037	96	8	𝑃𝑛.	𝑃𝑛.	PROPN
cana-4037	96	9	from	from	ADP
cana-4037	96	10	path	path	NOUN
cana-4037	96	11	𝑃𝑛	𝑃𝑛	PROPN
cana-4037	96	12	join	join	VERB
cana-4037	96	13	𝑢𝑖	𝑢𝑖	PRON
cana-4037	96	14	and	and	CCONJ
cana-4037	96	15	𝑢𝑖+1	𝑢𝑖+1	NUM
cana-4037	96	16	to	to	ADP
cana-4037	96	17	a	a	DET
cana-4037	96	18	new	new	ADJ
cana-4037	96	19	edges𝑣𝑖	edges𝑣𝑖	NOUN
cana-4037	96	20	d	d	NOUN
cana-4037	96	21	is	be	AUX
cana-4037	96	22	minimal	minimal	ADJ
cana-4037	96	23	cototal	cototal	ADJ
cana-4037	96	24	2oded	2oded	NUM
cana-4037	96	25	set	set	NOUN
cana-4037	96	26	.	.	PUNCT
cana-4037	97	1	so	so	ADV
cana-4037	97	2	has	have	VERB
cana-4037	97	3	no	no	DET
cana-4037	97	4	vertices	vertex	NOUN
cana-4037	97	5	of	of	ADP
cana-4037	97	6	degree	degree	NOUN
cana-4037	97	7	zero	zero	NUM
cana-4037	97	8	.	.	PUNCT
cana-4037	98	1	hence	hence	ADV
cana-4037	98	2	d	d	PROPN
cana-4037	98	3	is	be	AUX
cana-4037	98	4	minimum	minimum	ADJ
cana-4037	98	5	cototal	cototal	ADJ
cana-4037	98	6	2ode	2ode	NUM
cana-4037	98	7	dominating	dominating	NOUN
cana-4037	98	8	communications	communication	NOUN
cana-4037	98	9	on	on	ADP
cana-4037	98	10	applied	apply	VERB
cana-4037	98	11	nonlinear	nonlinear	ADJ
cana-4037	98	12	analysis	analysis	NOUN
cana-4037	98	13	issn	issn	NOUN
cana-4037	98	14	:	:	PUNCT
cana-4037	98	15	1074	1074	NUM
cana-4037	98	16	-	-	PUNCT
cana-4037	98	17	133x	133x	NUM
cana-4037	98	18	vol	vol	NOUN
cana-4037	98	19	32	32	NUM
cana-4037	98	20	no	no	NOUN
cana-4037	98	21	.	.	PUNCT
cana-4037	99	1	9s	9s	NUM
cana-4037	99	2	(	(	PUNCT
cana-4037	99	3	2025	2025	NUM
cana-4037	99	4	)	)	PUNCT
cana-4037	99	5	897	897	NUM
cana-4037	99	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-4037	99	7	by	by	ADP
cana-4037	99	8	edges	edge	NOUN
cana-4037	99	9	𝑢𝑖𝑣𝑖	𝑢𝑖𝑣𝑖	NOUN
cana-4037	99	10	and	and	CCONJ
cana-4037	99	11	𝑢𝑖+1𝑣𝑖	𝑢𝑖+1𝑣𝑖	NOUN
cana-4037	99	12	,	,	PUNCT
cana-4037	99	13	for	for	ADP
cana-4037	99	14	i=1,2,3	i=1,2,3	NUM
cana-4037	99	15	…	…	PUNCT
cana-4037	99	16	…	…	PUNCT
cana-4037	99	17	..	..	PUNCT
cana-4037	99	18	n	n	CCONJ
cana-4037	99	19	and	and	CCONJ
cana-4037	99	20	join	join	VERB
cana-4037	99	21	𝑢𝑖	𝑢𝑖	NOUN
cana-4037	99	22	and	and	CCONJ
cana-4037	99	23	𝑢𝑖+1	𝑢𝑖+1	NUM
cana-4037	99	24	to	to	ADP
cana-4037	99	25	a	a	DET
cana-4037	99	26	new	new	ADJ
cana-4037	99	27	edges	edge	NOUN
cana-4037	99	28	𝑤𝑖	𝑤𝑖	NOUN
cana-4037	99	29	by	by	ADP
cana-4037	99	30	edges	edge	NOUN
cana-4037	99	31	𝑢𝑖𝑤𝑖	𝑢𝑖𝑤𝑖	NOUN
cana-4037	99	32	and	and	CCONJ
cana-4037	99	33	𝑢𝑖+1𝑤𝑖	𝑢𝑖+1𝑤𝑖	NOUN
cana-4037	99	34	for	for	ADP
cana-4037	99	35	i=1,2,3	i=1,2,3	NUM
cana-4037	99	36	…	…	PUNCT
cana-4037	99	37	…	…	PUNCT
cana-4037	99	38	..	..	PUNCT
cana-4037	100	1	n	n	PRON
cana-4037	100	2	take	take	VERB
cana-4037	100	3	the	the	DET
cana-4037	100	4	vertices	vertex	NOUN
cana-4037	100	5	of	of	ADP
cana-4037	100	6	path	path	NOUN
cana-4037	100	7	𝑃𝑛	𝑃𝑛	PROPN
cana-4037	100	8	,	,	PUNCT
cana-4037	100	9	d=	d=	NUM
cana-4037	100	10	{	{	PUNCT
cana-4037	100	11	𝑢1	𝑢1	PROPN
cana-4037	100	12	,	,	PUNCT
cana-4037	100	13	𝑢2	𝑢2	PROPN
cana-4037	100	14	,	,	PUNCT
cana-4037	100	15	𝑢3	𝑢3	PROPN
cana-4037	100	16	…	…	PUNCT
cana-4037	100	17	…	…	PUNCT
cana-4037	100	18	𝑢𝑛+1	𝑢𝑛+1	NUM
cana-4037	100	19	}	}	PUNCT
cana-4037	100	20	and	and	CCONJ
cana-4037	100	21	𝑉	𝑉	PROPN
cana-4037	100	22	−	−	PROPN
cana-4037	100	23	𝐷	𝐷	NOUN
cana-4037	100	24	=	=	NOUN
cana-4037	100	25	{	{	PUNCT
cana-4037	100	26	𝑣1	𝑣1	PROPN
cana-4037	100	27	,	,	PUNCT
cana-4037	100	28	𝑣2	𝑣2	PROPN
cana-4037	100	29	,	,	PUNCT
cana-4037	100	30	𝑣3	𝑣3	ADJ
cana-4037	100	31	…	…	PUNCT
cana-4037	100	32	…	…	PUNCT
cana-4037	100	33	𝑣𝑛	𝑣𝑛	NOUN
cana-4037	100	34	,	,	PUNCT
cana-4037	100	35	𝑤1	𝑤1	VERB
cana-4037	100	36	,	,	PUNCT
cana-4037	100	37	𝑤2	𝑤2	NOUN
cana-4037	100	38	,	,	PUNCT
cana-4037	100	39	𝑤3	𝑤3	PROPN
cana-4037	100	40	…	…	PUNCT
cana-4037	100	41	…	…	PUNCT
cana-4037	100	42	𝑤𝑛	𝑤𝑛	NOUN
cana-4037	100	43	}	}	PUNCT
cana-4037	100	44	.	.	PUNCT
cana-4037	101	1	clearly	clearly	ADV
cana-4037	101	2	d	d	X
cana-4037	101	3	is	be	AUX
cana-4037	101	4	a	a	DET
cana-4037	101	5	dominating	dominating	NOUN
cana-4037	101	6	𝑉	𝑉	PROPN
cana-4037	101	7	−	−	PROPN
cana-4037	101	8	𝐷.	𝐷.	PROPN
cana-4037	101	9	now	now	ADV
cana-4037	101	10	𝑜𝑑𝐷(𝑢𝑖	𝑜𝑑𝐷(𝑢𝑖	VERB
cana-4037	101	11	)	)	PUNCT
cana-4037	102	1	=	=	SYM
cana-4037	102	2	|𝑁(𝑢𝑖	|𝑁(𝑢𝑖	ADJ
cana-4037	102	3	)	)	PUNCT
cana-4037	102	4	∩	∩	NOUN
cana-4037	102	5	(	(	PUNCT
cana-4037	102	6	𝑉	𝑉	PROPN
cana-4037	102	7	−	−	PROPN
cana-4037	102	8	𝐷)|	𝐷)|	NOUN
cana-4037	102	9	=	=	SYM
cana-4037	102	10	|{𝑣𝑖	|{𝑣𝑖	PROPN
cana-4037	102	11	,	,	PUNCT
cana-4037	102	12	𝑤𝑖}|	𝑤𝑖}|	NOUN
cana-4037	102	13	=	=	SYM
cana-4037	102	14	2	2	NUM
cana-4037	102	15	for	for	ADP
cana-4037	102	16	𝑖	𝑖	NOUN
cana-4037	102	17	=	=	SYM
cana-4037	102	18	1	1	NUM
cana-4037	102	19	,	,	PUNCT
cana-4037	102	20	𝑛	𝑛	PRON
cana-4037	102	21	+	+	NUM
cana-4037	102	22	1	1	NUM
cana-4037	102	23	and	and	CCONJ
cana-4037	102	24	𝑜𝑑𝐷(𝑢𝑖	𝑜𝑑𝐷(𝑢𝑖	PROPN
cana-4037	102	25	)	)	PUNCT
cana-4037	102	26	=	=	SYM
cana-4037	102	27	|𝑁(𝑢𝑖	|𝑁(𝑢𝑖	ADJ
cana-4037	102	28	)	)	PUNCT
cana-4037	102	29	∩	∩	NOUN
cana-4037	102	30	(	(	PUNCT
cana-4037	102	31	𝑉	𝑉	PROPN
cana-4037	102	32	−	−	PROPN
cana-4037	102	33	𝐷)|	𝐷)|	NOUN
cana-4037	102	34	=	=	NOUN
cana-4037	102	35	|{𝑣𝑖−1	|{𝑣𝑖−1	NUM
cana-4037	102	36	,	,	PUNCT
cana-4037	102	37	𝑣𝑖	𝑣𝑖	ADV
cana-4037	102	38	,	,	PUNCT
cana-4037	102	39	𝑤𝑖−1	𝑤𝑖−1	PROPN
cana-4037	102	40	,	,	PUNCT
cana-4037	102	41	𝑤𝑖}|	𝑤𝑖}|	PRON
cana-4037	102	42	=	=	SYM
cana-4037	102	43	4	4	NUM
cana-4037	102	44	for	for	ADP
cana-4037	102	45	𝑖	𝑖	NOUN
cana-4037	102	46	=	=	SYM
cana-4037	102	47	2	2	NUM
cana-4037	102	48	…	…	PUNCT
cana-4037	102	49	.	.	PUNCT
cana-4037	102	50	.	.	PUNCT
cana-4037	103	1	𝑛.	𝑛.	NOUN
cana-4037	103	2	hence|𝑜𝑑𝐷(𝑢𝑖	hence|𝑜𝑑𝐷(𝑢𝑖	PROPN
cana-4037	103	3	)	)	PUNCT
cana-4037	103	4	−	−	ADP
cana-4037	104	1	𝑜𝑑𝐷(𝑢𝑗)|	𝑜𝑑𝐷(𝑢𝑗)|	NOUN
cana-4037	104	2	≤	≤	NOUN
cana-4037	104	3	2	2	NUM
cana-4037	104	4	for	for	ADP
cana-4037	104	5	all	all	DET
cana-4037	104	6	𝑢𝑖	𝑢𝑖	NOUN
cana-4037	104	7	,	,	PUNCT
cana-4037	104	8	𝑢𝑗	𝑢𝑗	NOUN
cana-4037	104	9	∈	∈	PROPN
cana-4037	104	10	𝐷	𝐷	PROPN
cana-4037	104	11	,	,	PUNCT
cana-4037	104	12	and	and	CCONJ
cana-4037	104	13	d	d	NOUN
cana-4037	104	14	is	be	AUX
cana-4037	104	15	minimum	minimum	ADJ
cana-4037	104	16	2	2	NUM
cana-4037	104	17	oded	oded	NOUN
cana-4037	104	18	set	set	NOUN
cana-4037	104	19	and	and	CCONJ
cana-4037	104	20	clearly	clearly	ADV
cana-4037	104	21	<	<	X
cana-4037	104	22	𝑉	𝑉	PROPN
cana-4037	104	23	−	−	PROPN
cana-4037	104	24	𝐷	𝐷	PROPN
cana-4037	104	25	>	>	PUNCT
cana-4037	104	26	has	have	VERB
cana-4037	104	27	no	no	DET
cana-4037	104	28	vertices	vertex	NOUN
cana-4037	104	29	of	of	ADP
cana-4037	104	30	degree	degree	NOUN
cana-4037	104	31	zero	zero	NUM
cana-4037	104	32	.	.	PUNCT
cana-4037	105	1	so	so	ADV
cana-4037	105	2	d	d	NOUN
cana-4037	105	3	is	be	AUX
cana-4037	105	4	minimum	minimum	ADJ
cana-4037	105	5	𝛾𝑐𝑡2𝑜𝑒(𝐷(𝑛𝐶3	𝛾𝑐𝑡2𝑜𝑒(𝐷(𝑛𝐶3	NOUN
cana-4037	105	6	)	)	PUNCT
cana-4037	105	7	)	)	PUNCT
cana-4037	106	1	=	=	SYM
cana-4037	106	2	𝑛	𝑛	PRON
cana-4037	106	3	+	+	SYM
cana-4037	106	4	1	1	NUM
cana-4037	106	5	definition	definition	NOUN
cana-4037	106	6	3.5	3.5	NUM
cana-4037	106	7	the	the	DET
cana-4037	106	8	square	square	NOUN
cana-4037	106	9	of	of	ADP
cana-4037	106	10	a	a	DET
cana-4037	106	11	given	give	VERB
cana-4037	106	12	graph	graph	NOUN
cana-4037	106	13	g	g	PROPN
cana-4037	106	14	denoted	denote	VERB
cana-4037	106	15	by	by	ADP
cana-4037	106	16	𝐺2	𝐺2	NOUN
cana-4037	106	17	has	have	VERB
cana-4037	106	18	the	the	DET
cana-4037	106	19	same	same	ADJ
cana-4037	106	20	number	number	NOUN
cana-4037	106	21	vertices	vertice	VERB
cana-4037	106	22	as	as	ADP
cana-4037	106	23	of	of	ADP
cana-4037	106	24	g	g	NOUN
cana-4037	106	25	and	and	CCONJ
cana-4037	106	26	has	have	VERB
cana-4037	106	27	a	a	DET
cana-4037	106	28	vertices	vertex	NOUN
cana-4037	106	29	are	be	AUX
cana-4037	106	30	adjacent	adjacent	ADJ
cana-4037	106	31	in	in	ADP
cana-4037	106	32	𝐺2	𝐺2	NOUN
cana-4037	106	33	if	if	SCONJ
cana-4037	106	34	they	they	PRON
cana-4037	106	35	are	be	AUX
cana-4037	106	36	at	at	ADP
cana-4037	106	37	distance	distance	NOUN
cana-4037	106	38	of	of	ADP
cana-4037	106	39	one	one	NUM
cana-4037	106	40	or	or	CCONJ
cana-4037	106	41	two	two	NUM
cana-4037	106	42	apart	apart	ADV
cana-4037	106	43	in	in	ADP
cana-4037	106	44	g.	g.	PROPN
cana-4037	106	45	theorem	theorem	VERB
cana-4037	106	46	3.6	3.6	NUM
cana-4037	106	47	for	for	ADP
cana-4037	106	48	any	any	DET
cana-4037	106	49	square	square	NOUN
cana-4037	106	50	of	of	ADP
cana-4037	106	51	bistar	bistar	PROPN
cana-4037	106	52	graph	graph	NOUN
cana-4037	106	53	𝛾𝑐𝑡2𝑜𝑒(𝐵𝑝,𝑞2	𝛾𝑐𝑡2𝑜𝑒(𝐵𝑝,𝑞2	PROPN
cana-4037	106	54	)	)	PUNCT
cana-4037	106	55	=	=	SYM
cana-4037	106	56	𝑝	𝑝	PROPN
cana-4037	107	1	+	+	CCONJ
cana-4037	107	2	𝑞	𝑞	PRON
cana-4037	107	3	proof	proof	NOUN
cana-4037	107	4	:	:	PUNCT
cana-4037	107	5	consider	consider	VERB
cana-4037	107	6	bistar𝐵𝑝,𝑞	bistar𝐵𝑝,𝑞	NOUN
cana-4037	107	7	,	,	PUNCT
cana-4037	107	8	with	with	ADP
cana-4037	107	9	vertices	vertex	NOUN
cana-4037	107	10	{	{	PUNCT
cana-4037	107	11	𝑢	𝑢	PROPN
cana-4037	107	12	,	,	PUNCT
cana-4037	107	13	𝑣	𝑣	NOUN
cana-4037	107	14	,	,	PUNCT
cana-4037	107	15	𝑢1	𝑢1	NOUN
cana-4037	107	16	,	,	PUNCT
cana-4037	107	17	𝑢2	𝑢2	PROPN
cana-4037	107	18	,	,	PUNCT
cana-4037	107	19	𝑢3	𝑢3	PROPN
cana-4037	107	20	…	…	PUNCT
cana-4037	107	21	…	…	PUNCT
cana-4037	107	22	𝑢𝑝	𝑢𝑝	NOUN
cana-4037	107	23	,	,	PUNCT
cana-4037	107	24	𝑣1	𝑣1	PROPN
cana-4037	107	25	,	,	PUNCT
cana-4037	107	26	𝑣2	𝑣2	PROPN
cana-4037	107	27	,	,	PUNCT
cana-4037	107	28	𝑣3	𝑣3	ADJ
cana-4037	107	29	…	…	PUNCT
cana-4037	107	30	…	…	PUNCT
cana-4037	107	31	𝑣𝑞	𝑣𝑞	NOUN
cana-4037	107	32	}	}	PUNCT
cana-4037	107	33	where	where	SCONJ
cana-4037	107	34	𝑢𝑖	𝑢𝑖	NOUN
cana-4037	107	35	,	,	PUNCT
cana-4037	107	36	𝑣𝑖are	𝑣𝑖are	ADJ
cana-4037	107	37	pendant	pendant	ADJ
cana-4037	107	38	vertices	vertex	NOUN
cana-4037	107	39	which	which	PRON
cana-4037	107	40	is	be	AUX
cana-4037	107	41	adjacent	adjacent	ADJ
cana-4037	107	42	u	u	NOUN
cana-4037	107	43	and	and	CCONJ
cana-4037	107	44	v	v	ADP
cana-4037	107	45	respectively	respectively	ADV
cana-4037	107	46	and	and	CCONJ
cana-4037	107	47	u	u	NOUN
cana-4037	107	48	and	and	CCONJ
cana-4037	107	49	v	v	NOUN
cana-4037	107	50	adjacent	adjacent	ADJ
cana-4037	107	51	.	.	PUNCT
cana-4037	108	1	take	take	VERB
cana-4037	108	2	d=	d=	NOUN
cana-4037	108	3	{	{	PUNCT
cana-4037	108	4	𝑢1	𝑢1	PROPN
cana-4037	108	5	,	,	PUNCT
cana-4037	108	6	𝑢2	𝑢2	PROPN
cana-4037	108	7	,	,	PUNCT
cana-4037	108	8	𝑢3	𝑢3	PROPN
cana-4037	108	9	…	…	PUNCT
cana-4037	108	10	…	…	PUNCT
cana-4037	108	11	𝑢𝑝	𝑢𝑝	NOUN
cana-4037	108	12	,	,	PUNCT
cana-4037	108	13	𝑣1	𝑣1	PROPN
cana-4037	108	14	,	,	PUNCT
cana-4037	108	15	𝑣2	𝑣2	PROPN
cana-4037	108	16	,	,	PUNCT
cana-4037	108	17	𝑣3	𝑣3	ADJ
cana-4037	108	18	…	…	PUNCT
cana-4037	108	19	…	…	PUNCT
cana-4037	108	20	𝑣𝑞	𝑣𝑞	NOUN
cana-4037	108	21	}	}	PUNCT
cana-4037	108	22	and	and	CCONJ
cana-4037	108	23	𝑉	𝑉	PROPN
cana-4037	108	24	−	−	PROPN
cana-4037	108	25	𝐷	𝐷	NOUN
cana-4037	108	26	=	=	SYM
cana-4037	108	27	{	{	PUNCT
cana-4037	108	28	𝑢	𝑢	X
cana-4037	108	29	,	,	PUNCT
cana-4037	108	30	𝑣	𝑣	NOUN
cana-4037	108	31	}	}	PUNCT
cana-4037	108	32	.	.	PUNCT
cana-4037	109	1	clearly	clearly	ADV
cana-4037	109	2	d	d	X
cana-4037	109	3	is	be	AUX
cana-4037	109	4	a	a	DET
cana-4037	109	5	dominating	dominating	NOUN
cana-4037	109	6	𝑉	𝑉	PROPN
cana-4037	109	7	−	−	PROPN
cana-4037	109	8	𝐷.	𝐷.	PROPN
cana-4037	109	9	now	now	ADV
cana-4037	109	10	𝑜𝑑𝐷(𝑢𝑖	𝑜𝑑𝐷(𝑢𝑖	VERB
cana-4037	109	11	)	)	PUNCT
cana-4037	109	12	=	=	SYM
cana-4037	109	13	|𝑁(𝑢𝑖	|𝑁(𝑢𝑖	ADJ
cana-4037	109	14	)	)	PUNCT
cana-4037	109	15	∩	∩	NOUN
cana-4037	109	16	(	(	PUNCT
cana-4037	109	17	𝑉	𝑉	PROPN
cana-4037	109	18	−	−	PROPN
cana-4037	109	19	𝐷)|	𝐷)|	NOUN
cana-4037	109	20	=	=	SYM
cana-4037	109	21	|𝑉	|𝑉	X
cana-4037	109	22	−	−	NOUN
cana-4037	109	23	𝐷|	𝐷|	NOUN
cana-4037	109	24	=	=	SYM
cana-4037	109	25	2	2	NUM
cana-4037	109	26	and	and	CCONJ
cana-4037	109	27	𝑜𝑑𝐷(𝑣𝑖	𝑜𝑑𝐷(𝑣𝑖	ADJ
cana-4037	109	28	)	)	PUNCT
cana-4037	109	29	=	=	SYM
cana-4037	109	30	|𝑁(𝑣𝑖	|𝑁(𝑣𝑖	PROPN
cana-4037	109	31	)	)	PUNCT
cana-4037	109	32	∩	∩	NOUN
cana-4037	109	33	(	(	PUNCT
cana-4037	109	34	𝑉	𝑉	PROPN
cana-4037	109	35	−	−	PROPN
cana-4037	109	36	𝐷)|	𝐷)|	NOUN
cana-4037	109	37	=	=	SYM
cana-4037	109	38	|𝑉	|𝑉	X
cana-4037	109	39	−	−	NOUN
cana-4037	109	40	𝐷|	𝐷|	PROPN
cana-4037	109	41	=	=	SYM
cana-4037	109	42	2	2	NUM
cana-4037	109	43	.	.	PUNCT
cana-4037	110	1	hence|𝑜𝑑𝐷(𝑢𝑖	hence|𝑜𝑑𝐷(𝑢𝑖	NOUN
cana-4037	110	2	)	)	PUNCT
cana-4037	110	3	−	−	PROPN
cana-4037	111	1	𝑜𝑑𝐷(𝑢𝑖)|	𝑜𝑑𝐷(𝑢𝑖)|	NOUN
cana-4037	111	2	=	=	SYM
cana-4037	111	3	0	0	NUM
cana-4037	111	4	≤	≤	NUM
cana-4037	111	5	2	2	NUM
cana-4037	111	6	and	and	CCONJ
cana-4037	111	7	d	d	NOUN
cana-4037	111	8	is	be	AUX
cana-4037	111	9	minimum	minimum	ADJ
cana-4037	111	10	co	co	NOUN
cana-4037	111	11	-	-	ADJ
cana-4037	111	12	total	total	ADJ
cana-4037	111	13	2	2	NUM
cana-4037	111	14	oded	oded	NOUN
cana-4037	111	15	set	set	NOUN
cana-4037	111	16	.	.	PUNCT
cana-4037	112	1	clearly	clearly	ADV
cana-4037	112	2	<	<	X
cana-4037	112	3	𝑉	𝑉	PROPN
cana-4037	112	4	−𝐷	−𝐷	NOUN
cana-4037	112	5	>	>	X
cana-4037	112	6	has	have	VERB
cana-4037	112	7	no	no	DET
cana-4037	112	8	vertices	vertex	NOUN
cana-4037	112	9	of	of	ADP
cana-4037	112	10	degree	degree	NOUN
cana-4037	112	11	zero	zero	NUM
cana-4037	112	12	.	.	PUNCT
cana-4037	113	1	so	so	ADV
cana-4037	113	2	d	d	PRON
cana-4037	113	3	is	be	AUX
cana-4037	113	4	a	a	DET
cana-4037	113	5	minimum	minimum	ADJ
cana-4037	113	6	cototal	cototal	ADJ
cana-4037	113	7	2	2	NUM
cana-4037	113	8	oded	oded	PROPN
cana-4037	113	9	set	set	NOUN
cana-4037	113	10	.	.	PUNCT
cana-4037	114	1	hence	hence	ADV
cana-4037	114	2	𝛾𝑐𝑡2𝑜𝑒(𝐵𝑝,𝑞2	𝛾𝑐𝑡2𝑜𝑒(𝐵𝑝,𝑞2	PROPN
cana-4037	114	3	)	)	PUNCT
cana-4037	115	1	=	=	SYM
cana-4037	115	2	2	2	X
cana-4037	115	3	.	.	PUNCT
cana-4037	115	4	theorem	theorem	VERB
cana-4037	115	5	3.7	3.7	NUM
cana-4037	115	6	for	for	ADP
cana-4037	115	7	any	any	DET
cana-4037	115	8	path	path	NOUN
cana-4037	115	9	𝑃𝑛	𝑃𝑛	PROPN
cana-4037	115	10	,	,	PUNCT
cana-4037	115	11	𝛾𝑐𝑡2𝑜𝑒(𝑃𝑛2	𝛾𝑐𝑡2𝑜𝑒(𝑃𝑛2	PROPN
cana-4037	115	12	)	)	PUNCT
cana-4037	115	13	=	=	PRON
cana-4037	116	1	{	{	PUNCT
cana-4037	116	2	2	2	NUM
cana-4037	116	3	⌈𝑛7⌉	⌈𝑛7⌉	NOUN
cana-4037	116	4	+	+	CCONJ
cana-4037	116	5	1	1	NUM
cana-4037	116	6	𝑖𝑓	𝑖𝑓	NOUN
cana-4037	116	7	𝑛	𝑛	DET
cana-4037	116	8	≡	≡	PROPN
cana-4037	116	9	0	0	NUM
cana-4037	116	10	𝑜𝑟	𝑜𝑟	ADP
cana-4037	116	11	6	6	NUM
cana-4037	116	12	(	(	PUNCT
cana-4037	116	13	𝑚𝑜𝑑	𝑚𝑜𝑑	NOUN
cana-4037	116	14	7)2	7)2	NUM
cana-4037	116	15	⌈𝑛7⌉	⌈𝑛7⌉	PROPN
cana-4037	116	16	𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒	𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒	NOUN
cana-4037	116	17	proof	proof	NOUN
cana-4037	116	18	:	:	PUNCT
cana-4037	116	19	let	let	VERB
cana-4037	116	20	𝑉(𝑃𝑛2	𝑉(𝑃𝑛2	VERB
cana-4037	116	21	)	)	PUNCT
cana-4037	116	22	=	=	PRON
cana-4037	116	23	{	{	PUNCT
cana-4037	116	24	𝑣1	𝑣1	PROPN
cana-4037	116	25	,	,	PUNCT
cana-4037	116	26	𝑣2	𝑣2	PROPN
cana-4037	116	27	,	,	PUNCT
cana-4037	116	28	𝑣3	𝑣3	ADJ
cana-4037	116	29	,	,	PUNCT
cana-4037	116	30	…	…	PUNCT
cana-4037	116	31	.	.	PUNCT
cana-4037	116	32	.	.	PUNCT
cana-4037	117	1	,	,	PUNCT
cana-4037	117	2	𝑣𝑛	𝑣𝑛	X
cana-4037	117	3	}	}	PUNCT
cana-4037	117	4	be	be	VERB
cana-4037	117	5	the	the	DET
cana-4037	117	6	vertex	vertex	NOUN
cana-4037	117	7	set	set	VERB
cana-4037	117	8	where	where	SCONJ
cana-4037	117	9	deg(𝑣1)=	deg(𝑣1)=	VERB
cana-4037	117	10	deg(𝑣𝑛)=2	deg(𝑣𝑛)=2	NOUN
cana-4037	117	11	,	,	PUNCT
cana-4037	117	12	deg(𝑣2)=	deg(𝑣2)=	VERB
cana-4037	117	13	deg(𝑣𝑛−1)=3	deg(𝑣𝑛−1)=3	NOUN
cana-4037	117	14	and	and	CCONJ
cana-4037	117	15	deg(𝑣𝑖)=4	deg(𝑣𝑖)=4	NUM
cana-4037	117	16	for	for	ADP
cana-4037	117	17	all	all	DET
cana-4037	117	18	i=1,2,3	i=1,2,3	NUM
cana-4037	117	19	…	…	PUNCT
cana-4037	117	20	…	…	PUNCT
cana-4037	117	21	n-2	n-2	PRON
cana-4037	117	22	case	case	NOUN
cana-4037	117	23	:	:	PUNCT
cana-4037	117	24	1	1	NUM
cana-4037	117	25	𝑛	𝑛	PRON
cana-4037	117	26	≡	≡	PROPN
cana-4037	117	27	0	0	NUM
cana-4037	118	1	𝑜𝑟	𝑜𝑟	ADP
cana-4037	118	2	6	6	NUM
cana-4037	118	3	(	(	PUNCT
cana-4037	118	4	𝑚𝑜𝑑	𝑚𝑜𝑑	NOUN
cana-4037	118	5	7	7	NUM
cana-4037	118	6	)	)	PUNCT
cana-4037	118	7	if	if	SCONJ
cana-4037	118	8	𝑛	𝑛	PRON
cana-4037	118	9	≡	≡	PROPN
cana-4037	118	10	0	0	PUNCT
cana-4037	119	1	(	(	PUNCT
cana-4037	119	2	𝑚𝑜𝑑	𝑚𝑜𝑑	NOUN
cana-4037	119	3	7	7	NUM
cana-4037	119	4	)	)	PUNCT
cana-4037	119	5	we	we	PRON
cana-4037	119	6	have	have	VERB
cana-4037	119	7	𝐷	𝐷	NOUN
cana-4037	119	8	=	=	SYM
cana-4037	119	9	{	{	PUNCT
cana-4037	119	10	(	(	PUNCT
cana-4037	119	11	𝑣7𝑖+2	𝑣7𝑖+2	PROPN
cana-4037	119	12	,	,	PUNCT
cana-4037	119	13	𝑣7𝑖+4	𝑣7𝑖+4	NOUN
cana-4037	119	14	)	)	PUNCT
cana-4037	119	15	∪	∪	NOUN
cana-4037	119	16	{	{	PUNCT
cana-4037	119	17	𝑣𝑛−1	𝑣𝑛−1	NOUN
cana-4037	119	18	}	}	PUNCT
cana-4037	119	19	}	}	PUNCT
cana-4037	119	20	for	for	ADP
cana-4037	119	21	0	0	NUM
cana-4037	119	22	≤	≤	NUM
cana-4037	119	23	𝑖	𝑖	SYM
cana-4037	119	24	≤	≤	NOUN
cana-4037	119	25	⌈𝑛7⌉	⌈𝑛7⌉	NOUN
cana-4037	119	26	−	−	PROPN
cana-4037	119	27	1	1	NUM
cana-4037	119	28	and	and	CCONJ
cana-4037	119	29	if	if	SCONJ
cana-4037	119	30	𝑛	𝑛	DET
cana-4037	119	31	≡	≡	PROPN
cana-4037	119	32	6	6	NUM
cana-4037	119	33	(	(	PUNCT
cana-4037	119	34	𝑚𝑜𝑑	𝑚𝑜𝑑	NOUN
cana-4037	119	35	7	7	NUM
cana-4037	119	36	)	)	PUNCT
cana-4037	119	37	we	we	PRON
cana-4037	119	38	have	have	VERB
cana-4037	119	39	𝐷	𝐷	NOUN
cana-4037	119	40	=	=	SYM
cana-4037	119	41	{	{	PUNCT
cana-4037	119	42	(	(	PUNCT
cana-4037	119	43	𝑣7𝑖+2	𝑣7𝑖+2	PROPN
cana-4037	119	44	,	,	PUNCT
cana-4037	119	45	𝑣7𝑖+4	𝑣7𝑖+4	NOUN
cana-4037	119	46	)	)	PUNCT
cana-4037	119	47	∪	∪	NOUN
cana-4037	119	48	{	{	PUNCT
cana-4037	119	49	𝑣𝑛	𝑣𝑛	NOUN
cana-4037	119	50	}	}	PUNCT
cana-4037	119	51	}	}	PUNCT
cana-4037	119	52	for	for	ADP
cana-4037	119	53	0	0	NUM
cana-4037	119	54	≤	≤	NUM
cana-4037	120	1	𝑖	𝑖	SYM
cana-4037	120	2	≤	≤	NOUN
cana-4037	120	3	⌈𝑛7⌉	⌈𝑛7⌉	NOUN
cana-4037	120	4	−	−	PROPN
cana-4037	120	5	1	1	NUM
cana-4037	120	6	now	now	ADV
cana-4037	120	7	𝑜𝑑𝐷(𝑣7𝑖+2	𝑜𝑑𝐷(𝑣7𝑖+2	ADJ
cana-4037	120	8	)	)	PUNCT
cana-4037	120	9	=	=	SYM
cana-4037	120	10	|𝑁(𝑣7𝑖+2	|𝑁(𝑣7𝑖+2	PROPN
cana-4037	120	11	)	)	PUNCT
cana-4037	120	12	∩	∩	NOUN
cana-4037	120	13	(	(	PUNCT
cana-4037	120	14	𝑉	𝑉	PROPN
cana-4037	120	15	−	−	PROPN
cana-4037	120	16	𝐷)|	𝐷)|	NOUN
cana-4037	120	17	=	=	NOUN
cana-4037	120	18	3	3	NUM
cana-4037	120	19	,	,	PUNCT
cana-4037	120	20	𝑜𝑑𝐷(𝑣7𝑖+4	𝑜𝑑𝐷(𝑣7𝑖+4	NOUN
cana-4037	120	21	)	)	PUNCT
cana-4037	120	22	=	=	SYM
cana-4037	120	23	|𝑁(𝑣7𝑖+4	|𝑁(𝑣7𝑖+4	PROPN
cana-4037	120	24	)	)	PUNCT
cana-4037	120	25	∩	∩	NOUN
cana-4037	120	26	(	(	PUNCT
cana-4037	120	27	𝑉	𝑉	PROPN
cana-4037	120	28	−	−	PROPN
cana-4037	120	29	𝐷)|	𝐷)|	NOUN
cana-4037	120	30	=	=	NOUN
cana-4037	120	31	4	4	NUM
cana-4037	120	32	,	,	PUNCT
cana-4037	120	33	𝑜𝑑𝐷(𝑣𝑛−1	𝑜𝑑𝐷(𝑣𝑛−1	NOUN
cana-4037	120	34	)	)	PUNCT
cana-4037	120	35	=	=	SYM
cana-4037	120	36	|𝑁(𝑣𝑛−1	|𝑁(𝑣𝑛−1	PROPN
cana-4037	120	37	)	)	PUNCT
cana-4037	120	38	∩	∩	NOUN
cana-4037	120	39	(	(	PUNCT
cana-4037	120	40	𝑉	𝑉	PROPN
cana-4037	120	41	−	−	PROPN
cana-4037	120	42	𝐷)|	𝐷)|	NOUN
cana-4037	120	43	=	=	NOUN
cana-4037	120	44	2	2	NUM
cana-4037	120	45	and	and	CCONJ
cana-4037	120	46	𝑜𝑑𝐷(𝑣𝑛	𝑜𝑑𝐷(𝑣𝑛	NUM
cana-4037	120	47	)	)	PUNCT
cana-4037	121	1	=	=	SYM
cana-4037	121	2	|𝑁(𝑣𝑛	|𝑁(𝑣𝑛	NOUN
cana-4037	121	3	)	)	PUNCT
cana-4037	121	4	∩	∩	NOUN
cana-4037	121	5	(	(	PUNCT
cana-4037	121	6	𝑉	𝑉	PROPN
cana-4037	121	7	−	−	PROPN
cana-4037	121	8	𝐷)|	𝐷)|	NOUN
cana-4037	121	9	=	=	NOUN
cana-4037	121	10	2	2	NUM
cana-4037	121	11	hence	hence	ADV
cana-4037	121	12	|𝑜𝑑𝐷(𝑢	|𝑜𝑑𝐷(𝑢	VERB
cana-4037	121	13	)	)	PUNCT
cana-4037	122	1	−	−	ADP
cana-4037	122	2	𝑜𝑑𝐷(𝑣)|	𝑜𝑑𝐷(𝑣)|	NOUN
cana-4037	122	3	≤	≤	NOUN
cana-4037	122	4	2	2	NUM
cana-4037	122	5	.	.	PUNCT
cana-4037	123	1	for	for	ADP
cana-4037	123	2	any	any	DET
cana-4037	123	3	𝑢	𝑢	NOUN
cana-4037	123	4	,	,	PUNCT
cana-4037	123	5	𝑣	𝑣	PRON
cana-4037	123	6	∈	∈	PROPN
cana-4037	123	7	𝐷.	𝐷.	PROPN
cana-4037	123	8	here	here	ADV
cana-4037	123	9	<	<	X
cana-4037	123	10	𝑉	𝑉	PROPN
cana-4037	123	11	−	−	PROPN
cana-4037	123	12	𝐷	𝐷	PROPN
cana-4037	123	13	>	>	PUNCT
cana-4037	123	14	has	have	VERB
cana-4037	123	15	no	no	DET
cana-4037	123	16	zero	zero	NUM
cana-4037	123	17	degree	degree	NOUN
cana-4037	123	18	vertices	vertex	NOUN
cana-4037	123	19	then	then	ADV
cana-4037	123	20	d	d	PROPN
cana-4037	123	21	is	be	AUX
cana-4037	123	22	co	co	ADJ
cana-4037	123	23	-	-	ADJ
cana-4037	123	24	total	total	ADJ
cana-4037	123	25	2	2	NUM
cana-4037	123	26	oded	oded	NOUN
cana-4037	123	27	set	set	NOUN
cana-4037	123	28	also	also	ADV
cana-4037	123	29	𝐷	𝐷	PROPN
cana-4037	123	30	−	−	PROPN
cana-4037	123	31	{	{	PUNCT
cana-4037	123	32	𝑢	𝑢	X
cana-4037	123	33	}	}	PUNCT
cana-4037	123	34	is	be	AUX
cana-4037	123	35	no	no	DET
cana-4037	123	36	zero	zero	NUM
cana-4037	123	37	degree	degree	NOUN
cana-4037	123	38	vertices	vertex	NOUN
cana-4037	123	39	then	then	ADV
cana-4037	123	40	d	d	PROPN
cana-4037	123	41	is	be	AUX
cana-4037	123	42	a	a	DET
cana-4037	123	43	minimal	minimal	ADJ
cana-4037	123	44	co	co	NOUN
cana-4037	123	45	-	-	ADJ
cana-4037	123	46	total	total	ADJ
cana-4037	123	47	2oded	2ode	VERB
cana-4037	123	48	set	set	NOUN
cana-4037	123	49	.	.	PUNCT
cana-4037	124	1	𝛾𝑡2𝑜𝑒(𝑃𝑛2	𝛾𝑡2𝑜𝑒(𝑃𝑛2	PROPN
cana-4037	124	2	)	)	PUNCT
cana-4037	125	1	=	=	SYM
cana-4037	125	2	2	2	NUM
cana-4037	125	3	⌈𝑛7⌉	⌈𝑛7⌉	NOUN
cana-4037	125	4	+	+	CCONJ
cana-4037	125	5	1	1	NUM
cana-4037	125	6	case	case	NOUN
cana-4037	125	7	:	:	PUNCT
cana-4037	125	8	2	2	NUM
cana-4037	125	9	𝑛	𝑛	ADP
cana-4037	125	10	≢	≢	NUM
cana-4037	125	11	0	0	NUM
cana-4037	125	12	𝑜𝑟	𝑜𝑟	ADP
cana-4037	125	13	6	6	NUM
cana-4037	125	14	(	(	PUNCT
cana-4037	125	15	𝑚𝑜𝑑	𝑚𝑜𝑑	PROPN
cana-4037	125	16	7	7	NUM
cana-4037	125	17	)	)	PUNCT
cana-4037	125	18	cototal	cototal	ADJ
cana-4037	125	19	2oded	2oded	NUM
cana-4037	125	20	set	set	NOUN
cana-4037	125	21	.	.	PUNCT
cana-4037	126	1	hence	hence	ADV
cana-4037	126	2	communications	communication	NOUN
cana-4037	126	3	on	on	ADP
cana-4037	126	4	applied	apply	VERB
cana-4037	126	5	nonlinear	nonlinear	ADJ
cana-4037	126	6	analysis	analysis	NOUN
cana-4037	126	7	issn	issn	NOUN
cana-4037	126	8	:	:	PUNCT
cana-4037	126	9	1074	1074	NUM
cana-4037	126	10	-	-	PUNCT
cana-4037	126	11	133x	133x	NUM
cana-4037	126	12	vol	vol	NOUN
cana-4037	126	13	32	32	NUM
cana-4037	126	14	no	no	NOUN
cana-4037	126	15	.	.	PUNCT
cana-4037	127	1	9s	9s	NUM
cana-4037	127	2	(	(	PUNCT
cana-4037	127	3	2025	2025	NUM
cana-4037	127	4	)	)	PUNCT
cana-4037	127	5	898	898	NUM
cana-4037	127	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-4037	128	1	if	if	SCONJ
cana-4037	128	2	𝑛	𝑛	PRON
cana-4037	128	3	≡	≡	PROPN
cana-4037	128	4	1	1	NUM
cana-4037	128	5	𝑜𝑟	𝑜𝑟	ADP
cana-4037	128	6	2	2	NUM
cana-4037	128	7	𝑜𝑟	𝑜𝑟	ADP
cana-4037	128	8	3	3	NUM
cana-4037	128	9	𝑜𝑟	𝑜𝑟	ADP
cana-4037	128	10	4	4	NUM
cana-4037	128	11	(	(	PUNCT
cana-4037	128	12	𝑚𝑜𝑑	𝑚𝑜𝑑	NOUN
cana-4037	128	13	7	7	NUM
cana-4037	128	14	)	)	PUNCT
cana-4037	128	15	we	we	PRON
cana-4037	128	16	have	have	VERB
cana-4037	128	17	𝐷	𝐷	NOUN
cana-4037	128	18	=	=	SYM
cana-4037	128	19	{	{	PUNCT
cana-4037	128	20	(	(	PUNCT
cana-4037	128	21	𝑣7𝑖+2	𝑣7𝑖+2	PROPN
cana-4037	128	22	,	,	PUNCT
cana-4037	128	23	𝑣7𝑖+4	𝑣7𝑖+4	NOUN
cana-4037	128	24	)	)	PUNCT
cana-4037	128	25	∪	∪	ADP
cana-4037	128	26	{	{	PUNCT
cana-4037	128	27	𝑣𝑛−2,𝑣𝑛	𝑣𝑛−2,𝑣𝑛	NUM
cana-4037	128	28	}	}	PUNCT
cana-4037	128	29	}	}	PUNCT
cana-4037	128	30	for	for	ADP
cana-4037	128	31	0	0	NUM
cana-4037	128	32	≤	≤	NUM
cana-4037	128	33	𝑖	𝑖	SYM
cana-4037	128	34	≤	≤	NOUN
cana-4037	128	35	⌊𝑛7⌋	⌊𝑛7⌋	X
cana-4037	129	1	−	−	NOUN
cana-4037	129	2	1	1	NUM
cana-4037	129	3	and	and	CCONJ
cana-4037	129	4	if	if	SCONJ
cana-4037	129	5	𝑛	𝑛	PRON
cana-4037	129	6	≡	≡	PROPN
cana-4037	129	7	5	5	NUM
cana-4037	129	8	(	(	PUNCT
cana-4037	129	9	𝑚𝑜𝑑	𝑚𝑜𝑑	NOUN
cana-4037	129	10	7	7	NUM
cana-4037	129	11	)	)	PUNCT
cana-4037	129	12	we	we	PRON
cana-4037	129	13	have	have	VERB
cana-4037	129	14	𝐷	𝐷	NOUN
cana-4037	129	15	=	=	SYM
cana-4037	129	16	{	{	PUNCT
cana-4037	129	17	(	(	PUNCT
cana-4037	129	18	𝑣7𝑖+2	𝑣7𝑖+2	PROPN
cana-4037	129	19	,	,	PUNCT
cana-4037	129	20	𝑣7𝑖+4	𝑣7𝑖+4	NOUN
cana-4037	129	21	)	)	PUNCT
cana-4037	129	22	}	}	PUNCT
cana-4037	129	23	for	for	ADP
cana-4037	129	24	0	0	NUM
cana-4037	129	25	≤	≤	NUM
cana-4037	129	26	𝑖	𝑖	SYM
cana-4037	129	27	≤	≤	NOUN
cana-4037	129	28	⌊𝑛7⌋	⌊𝑛7⌋	PUNCT
cana-4037	129	29	now	now	ADV
cana-4037	129	30	𝑜𝑑𝐷(𝑣7𝑖+2	𝑜𝑑𝐷(𝑣7𝑖+2	ADJ
cana-4037	129	31	)	)	PUNCT
cana-4037	129	32	=	=	SYM
cana-4037	129	33	|𝑁(𝑣7𝑖+2	|𝑁(𝑣7𝑖+2	PROPN
cana-4037	129	34	)	)	PUNCT
cana-4037	129	35	∩	∩	NOUN
cana-4037	129	36	(	(	PUNCT
cana-4037	129	37	𝑉	𝑉	PROPN
cana-4037	129	38	−	−	PROPN
cana-4037	129	39	𝐷)|	𝐷)|	NOUN
cana-4037	130	1	=	=	NOUN
cana-4037	130	2	3	3	NUM
cana-4037	130	3	,	,	PUNCT
cana-4037	130	4	𝑜𝑑𝐷(𝑣7𝑖+4	𝑜𝑑𝐷(𝑣7𝑖+4	NOUN
cana-4037	130	5	)	)	PUNCT
cana-4037	130	6	=	=	SYM
cana-4037	130	7	|𝑁(𝑣7𝑖+4	|𝑁(𝑣7𝑖+4	PROPN
cana-4037	130	8	)	)	PUNCT
cana-4037	130	9	∩	∩	NOUN
cana-4037	130	10	(	(	PUNCT
cana-4037	130	11	𝑉	𝑉	PROPN
cana-4037	130	12	−	−	PROPN
cana-4037	130	13	𝐷)|	𝐷)|	NOUN
cana-4037	130	14	=	=	NOUN
cana-4037	130	15	3	3	NUM
cana-4037	130	16	,	,	PUNCT
cana-4037	130	17	𝑜𝑑𝐷(𝑣𝑛	𝑜𝑑𝐷(𝑣𝑛	PROPN
cana-4037	130	18	)	)	PUNCT
cana-4037	130	19	=	=	SYM
cana-4037	130	20	|𝑁(𝑣𝑛	|𝑁(𝑣𝑛	X
cana-4037	130	21	)	)	PUNCT
cana-4037	130	22	∩	∩	NOUN
cana-4037	130	23	(	(	PUNCT
cana-4037	130	24	𝑉	𝑉	PROPN
cana-4037	130	25	−	−	PROPN
cana-4037	130	26	𝐷)|	𝐷)|	NOUN
cana-4037	130	27	=	=	NOUN
cana-4037	130	28	2	2	NUM
cana-4037	130	29	and	and	CCONJ
cana-4037	130	30	𝑜𝑑𝐷(𝑣𝑛−2	𝑜𝑑𝐷(𝑣𝑛−2	PROPN
cana-4037	130	31	)	)	PUNCT
cana-4037	131	1	=	=	SYM
cana-4037	131	2	|𝑁(𝑣𝑛−2	|𝑁(𝑣𝑛−2	NOUN
cana-4037	131	3	)	)	PUNCT
cana-4037	131	4	∩	∩	NOUN
cana-4037	131	5	(	(	PUNCT
cana-4037	131	6	𝑉	𝑉	PROPN
cana-4037	131	7	−	−	PROPN
cana-4037	131	8	𝐷)|	𝐷)|	NOUN
cana-4037	131	9	=	=	NOUN
cana-4037	131	10	2	2	NUM
cana-4037	131	11	hence	hence	ADV
cana-4037	131	12	|𝑜𝑑𝐷(𝑢	|𝑜𝑑𝐷(𝑢	VERB
cana-4037	131	13	)	)	PUNCT
cana-4037	131	14	−	−	ADP
cana-4037	131	15	𝑜𝑑𝐷(𝑣)|	𝑜𝑑𝐷(𝑣)|	NOUN
cana-4037	131	16	≤	≤	NOUN
cana-4037	131	17	2	2	NUM
cana-4037	131	18	.	.	PUNCT
cana-4037	132	1	for	for	ADP
cana-4037	132	2	any	any	DET
cana-4037	132	3	𝑢	𝑢	NOUN
cana-4037	132	4	,	,	PUNCT
cana-4037	132	5	𝑣	𝑣	PRON
cana-4037	132	6	∈	∈	PROPN
cana-4037	132	7	𝐷.	𝐷.	PROPN
cana-4037	132	8	here	here	ADV
cana-4037	132	9	<	<	X
cana-4037	132	10	𝑉	𝑉	PROPN
cana-4037	132	11	−	−	PROPN
cana-4037	132	12	𝐷	𝐷	PROPN
cana-4037	132	13	>	>	PUNCT
cana-4037	132	14	has	have	VERB
cana-4037	132	15	no	no	DET
cana-4037	132	16	vertices	vertex	NOUN
cana-4037	132	17	of	of	ADP
cana-4037	132	18	degree	degree	NOUN
cana-4037	132	19	zero	zero	NUM
cana-4037	132	20	then	then	ADV
cana-4037	132	21	d	d	X
cana-4037	132	22	is	be	AUX
cana-4037	132	23	a	a	DET
cana-4037	132	24	co	co	NOUN
cana-4037	132	25	-	-	ADJ
cana-4037	132	26	total	total	ADJ
cana-4037	132	27	2	2	NUM
cana-4037	132	28	oded	oded	NOUN
cana-4037	132	29	set	set	VERB
cana-4037	132	30	then	then	ADV
cana-4037	132	31	𝛾𝑡2𝑜𝑒(𝑃𝑛2	𝛾𝑡2𝑜𝑒(𝑃𝑛2	PROPN
cana-4037	132	32	)	)	PUNCT
cana-4037	133	1	=	=	NOUN
cana-4037	133	2	2	2	NUM
cana-4037	133	3	⌊𝑛7⌋.	⌊𝑛7⌋.	PUNCT
cana-4037	133	4	definition	definition	NOUN
cana-4037	133	5	3.8	3.8	NUM
cana-4037	133	6	the	the	DET
cana-4037	133	7	semi	semi	ADJ
cana-4037	133	8	total	total	ADJ
cana-4037	133	9	point	point	NOUN
cana-4037	133	10	graph	graph	NOUN
cana-4037	133	11	t2(g	t2(g	VERB
cana-4037	133	12	)	)	PUNCT
cana-4037	133	13	of	of	ADP
cana-4037	133	14	g	g	PROPN
cana-4037	133	15	is	be	AUX
cana-4037	133	16	the	the	DET
cana-4037	133	17	graph	graph	NOUN
cana-4037	133	18	whose	whose	DET
cana-4037	133	19	vertex	vertex	NOUN
cana-4037	133	20	set	set	NOUN
cana-4037	133	21	is	be	AUX
cana-4037	133	22	𝑉(𝐺	𝑉(𝐺	NOUN
cana-4037	133	23	)	)	PUNCT
cana-4037	133	24	∪	∪	ADP
cana-4037	133	25	𝐸(𝐺	𝐸(𝐺	PROPN
cana-4037	133	26	)	)	PUNCT
cana-4037	133	27	,	,	PUNCT
cana-4037	133	28	whose	whose	DET
cana-4037	133	29	vertices	vertex	NOUN
cana-4037	133	30	are	be	AUX
cana-4037	133	31	adjacent	adjacent	ADJ
cana-4037	133	32	if	if	SCONJ
cana-4037	133	33	they	they	PRON
cana-4037	133	34	adjacent	adjacent	ADJ
cana-4037	133	35	vertices	vertex	NOUN
cana-4037	133	36	of	of	ADP
cana-4037	133	37	g	g	NOUN
cana-4037	133	38	or	or	CCONJ
cana-4037	133	39	one	one	NUM
cana-4037	133	40	is	be	AUX
cana-4037	133	41	a	a	DET
cana-4037	133	42	vertex	vertex	NOUN
cana-4037	133	43	of	of	ADP
cana-4037	133	44	g	g	PROPN
cana-4037	133	45	and	and	CCONJ
cana-4037	133	46	another	another	DET
cana-4037	133	47	edge	edge	NOUN
cana-4037	133	48	of	of	ADP
cana-4037	133	49	g	g	PROPN
cana-4037	133	50	incident	incident	NOUN
cana-4037	133	51	with	with	ADP
cana-4037	133	52	it	it	PRON
cana-4037	133	53	.	.	PUNCT
cana-4037	134	1	theorem	theorem	VERB
cana-4037	134	2	3.9	3.9	NUM
cana-4037	134	3	for	for	ADP
cana-4037	134	4	any	any	DET
cana-4037	134	5	cycle	cycle	NOUN
cana-4037	134	6	cn	cn	PROPN
cana-4037	134	7	,	,	PUNCT
cana-4037	134	8	γct2oe	γct2oe	PROPN
cana-4037	134	9	t2(cn	t2(cn	PROPN
cana-4037	134	10	)	)	PUNCT
cana-4037	134	11	)	)	PUNCT
cana-4037	135	1	=	=	SYM
cana-4037	136	1	n	n	CCONJ
cana-4037	136	2	−	−	PROPN
cana-4037	136	3	1	1	NUM
cana-4037	136	4	forn	forn	X
cana-4037	136	5	≥	≥	NOUN
cana-4037	136	6	3	3	NUM
cana-4037	136	7	proof	proof	NOUN
cana-4037	136	8	:	:	PUNCT
cana-4037	136	9	let	let	VERB
cana-4037	136	10	vertex	vertex	NOUN
cana-4037	136	11	set	set	NOUN
cana-4037	136	12	of	of	ADP
cana-4037	136	13	v(c𝑛	v(c𝑛	NUM
cana-4037	136	14	)	)	PUNCT
cana-4037	137	1	=	=	PRON
cana-4037	137	2	{	{	PUNCT
cana-4037	137	3	𝑣1	𝑣1	PROPN
cana-4037	137	4	,	,	PUNCT
cana-4037	137	5	𝑣2	𝑣2	PROPN
cana-4037	137	6	,	,	PUNCT
cana-4037	137	7	…	…	PUNCT
cana-4037	137	8	,	,	PUNCT
cana-4037	137	9	𝑣𝑛	𝑣𝑛	NOUN
cana-4037	137	10	}	}	PUNCT
cana-4037	137	11	and	and	CCONJ
cana-4037	137	12	edge	edge	VERB
cana-4037	137	13	set	set	VERB
cana-4037	137	14	e(c𝑛	e(c𝑛	NOUN
cana-4037	137	15	)	)	PUNCT
cana-4037	137	16	=	=	PRON
cana-4037	137	17	{	{	PUNCT
cana-4037	137	18	𝑒1	𝑒1	NOUN
cana-4037	137	19	,	,	PUNCT
cana-4037	137	20	𝑒2	𝑒2	PROPN
cana-4037	137	21	,	,	PUNCT
cana-4037	137	22	…	…	PUNCT
cana-4037	137	23	,	,	PUNCT
cana-4037	137	24	𝑒𝑛	𝑒𝑛	NOUN
cana-4037	137	25	}	}	PUNCT
cana-4037	137	26	.	.	PUNCT
cana-4037	138	1	now	now	ADV
cana-4037	138	2	,	,	PUNCT
cana-4037	138	3	v(t2(c𝑛	v(t2(c𝑛	NOUN
cana-4037	138	4	)	)	PUNCT
cana-4037	138	5	)	)	PUNCT
cana-4037	139	1	=	=	PRON
cana-4037	139	2	{	{	PUNCT
cana-4037	139	3	𝑣1	𝑣1	PROPN
cana-4037	139	4	,	,	PUNCT
cana-4037	139	5	𝑣2	𝑣2	PROPN
cana-4037	139	6	,	,	PUNCT
cana-4037	139	7	…	…	PUNCT
cana-4037	139	8	,	,	PUNCT
cana-4037	139	9	𝑣𝑛	𝑣𝑛	NOUN
cana-4037	139	10	,	,	PUNCT
cana-4037	139	11	𝑒1	𝑒1	NOUN
cana-4037	139	12	,	,	PUNCT
cana-4037	139	13	𝑒2	𝑒2	PROPN
cana-4037	139	14	,	,	PUNCT
cana-4037	139	15	…	…	PUNCT
cana-4037	139	16	,	,	PUNCT
cana-4037	139	17	𝑒𝑛	𝑒𝑛	X
cana-4037	139	18	}	}	PUNCT
cana-4037	139	19	be	be	AUX
cana-4037	139	20	vertices	vertex	NOUN
cana-4037	139	21	of	of	ADP
cana-4037	139	22	t2(c𝑛	t2(c𝑛	PROPN
cana-4037	139	23	)	)	PUNCT
cana-4037	139	24	.	.	PUNCT
cana-4037	140	1	let	let	VERB
cana-4037	140	2	d	d	NOUN
cana-4037	140	3	=	=	PUNCT
cana-4037	140	4	{	{	PUNCT
cana-4037	140	5	𝑒1	𝑒1	NOUN
cana-4037	140	6	,	,	PUNCT
cana-4037	140	7	𝑒2	𝑒2	PROPN
cana-4037	140	8	,	,	PUNCT
cana-4037	140	9	…	…	PUNCT
cana-4037	140	10	,	,	PUNCT
cana-4037	140	11	𝑒𝑛−3	𝑒𝑛−3	PROPN
cana-4037	140	12	,	,	PUNCT
cana-4037	140	13	𝑒𝑛	𝑒𝑛	INTJ
cana-4037	140	14	,	,	PUNCT
cana-4037	140	15	𝑣𝑛−1	𝑣𝑛−1	PROPN
cana-4037	140	16	}	}	PUNCT
cana-4037	140	17	be	be	AUX
cana-4037	140	18	the	the	DET
cana-4037	140	19	minimal	minimal	ADJ
cana-4037	140	20	dominating	dominating	NOUN
cana-4037	140	21	set	set	NOUN
cana-4037	140	22	of	of	ADP
cana-4037	140	23	t2(c𝑛	t2(c𝑛	PROPN
cana-4037	140	24	)	)	PUNCT
cana-4037	140	25	then	then	ADV
cana-4037	140	26	v−d	v−d	VERB
cana-4037	140	27	=	=	SYM
cana-4037	140	28	{	{	PUNCT
cana-4037	140	29	𝑣1	𝑣1	PROPN
cana-4037	140	30	,	,	PUNCT
cana-4037	140	31	𝑣2	𝑣2	PROPN
cana-4037	140	32	,	,	PUNCT
cana-4037	140	33	…	…	PUNCT
cana-4037	140	34	,	,	PUNCT
cana-4037	140	35	𝑣𝑛−2	𝑣𝑛−2	PROPN
cana-4037	140	36	,	,	PUNCT
cana-4037	140	37	𝑣𝑛	𝑣𝑛	NOUN
cana-4037	140	38	,	,	PUNCT
cana-4037	140	39	𝑒𝑛−2	𝑒𝑛−2	PROPN
cana-4037	140	40	,	,	PUNCT
cana-4037	140	41	𝑒𝑛−1	𝑒𝑛−1	PROPN
cana-4037	140	42	}	}	PUNCT
cana-4037	140	43	.	.	PUNCT
cana-4037	141	1	clearly	clearly	ADV
cana-4037	141	2	,	,	PUNCT
cana-4037	141	3	the	the	DET
cana-4037	141	4	vertices	vertex	NOUN
cana-4037	141	5	𝑣1	𝑣1	PROPN
cana-4037	141	6	,	,	PUNCT
cana-4037	141	7	𝑣2	𝑣2	PROPN
cana-4037	141	8	,	,	PUNCT
cana-4037	141	9	…	…	PUNCT
cana-4037	141	10	,	,	PUNCT
cana-4037	141	11	𝑣𝑛−2	𝑣𝑛−2	PROPN
cana-4037	141	12	,	,	PUNCT
cana-4037	141	13	𝑣𝑛in	𝑣𝑛in	NOUN
cana-4037	141	14	v−d	v−d	NOUN
cana-4037	141	15	forms	form	NOUN
cana-4037	141	16	a	a	DET
cana-4037	141	17	path	path	NOUN
cana-4037	141	18	and	and	CCONJ
cana-4037	141	19	by	by	ADP
cana-4037	141	20	the	the	DET
cana-4037	141	21	definition	definition	NOUN
cana-4037	141	22	of	of	ADP
cana-4037	141	23	semi	semi	ADJ
cana-4037	141	24	total	total	ADJ
cana-4037	141	25	point	point	NOUN
cana-4037	141	26	graph	graph	NOUN
cana-4037	141	27	the	the	DET
cana-4037	141	28	vertices	vertex	NOUN
cana-4037	141	29	𝑒𝑛−2	𝑒𝑛−2	PROPN
cana-4037	141	30	,	,	PUNCT
cana-4037	141	31	𝑒𝑛−1is	𝑒𝑛−1i	NOUN
cana-4037	141	32	adjacent	adjacent	ADJ
cana-4037	141	33	to	to	ADP
cana-4037	141	34	𝑣𝑛−2	𝑣𝑛−2	PROPN
cana-4037	141	35	,	,	PUNCT
cana-4037	141	36	𝑣𝑛in	𝑣𝑛in	NOUN
cana-4037	141	37	v−d	v−d	PROPN
cana-4037	141	38	.	.	PUNCT
cana-4037	142	1	thus	thus	ADV
cana-4037	142	2	<	<	X
cana-4037	142	3	v−d	v−d	X
cana-4037	142	4	>	>	X
cana-4037	142	5	has	have	VERB
cana-4037	142	6	no	no	DET
cana-4037	142	7	isolated	isolated	ADJ
cana-4037	142	8	vertices	vertex	NOUN
cana-4037	142	9	.	.	PUNCT
cana-4037	143	1	except	except	SCONJ
cana-4037	143	2	𝑣𝑛−1all	𝑣𝑛−1all	ADV
cana-4037	143	3	other	other	ADJ
cana-4037	143	4	elements	element	NOUN
cana-4037	143	5	in	in	ADP
cana-4037	143	6	d	d	PROPN
cana-4037	143	7	is	be	AUX
cana-4037	143	8	exactly	exactly	ADV
cana-4037	143	9	adjacent	adjacent	ADJ
cana-4037	143	10	to	to	ADP
cana-4037	143	11	two	two	NUM
cana-4037	143	12	vertices	vertex	NOUN
cana-4037	143	13	in	in	ADP
cana-4037	143	14	v−d	v−d	NOUN
cana-4037	143	15	,	,	PUNCT
cana-4037	143	16	so	so	SCONJ
cana-4037	143	17	the	the	DET
cana-4037	143	18	out	out	ADJ
cana-4037	143	19	degree	degree	NOUN
cana-4037	143	20	of	of	ADP
cana-4037	143	21	these	these	DET
cana-4037	143	22	elements	element	NOUN
cana-4037	143	23	in	in	ADP
cana-4037	143	24	d	d	PROPN
cana-4037	143	25	is	be	AUX
cana-4037	143	26	two	two	NUM
cana-4037	143	27	.	.	PUNCT
cana-4037	144	1	for	for	ADP
cana-4037	144	2	𝑣𝑛−1	𝑣𝑛−1	PROPN
cana-4037	144	3	in	in	ADP
cana-4037	144	4	d	d	PROPN
cana-4037	144	5	,	,	PUNCT
cana-4037	144	6	it	it	PRON
cana-4037	144	7	is	be	AUX
cana-4037	144	8	exactly	exactly	ADV
cana-4037	144	9	adjacent	adjacent	ADJ
cana-4037	144	10	with	with	ADP
cana-4037	144	11	four	four	NUM
cana-4037	144	12	vertices	vertex	NOUN
cana-4037	144	13	,	,	PUNCT
cana-4037	144	14	hence	hence	ADV
cana-4037	144	15	the	the	DET
cana-4037	144	16	out	out	ADJ
cana-4037	144	17	degree	degree	NOUN
cana-4037	144	18	is	be	AUX
cana-4037	144	19	four	four	NUM
cana-4037	144	20	.	.	PUNCT
cana-4037	145	1	thus	thus	ADV
cana-4037	145	2	,	,	PUNCT
cana-4037	145	3	for	for	ADP
cana-4037	145	4	any	any	DET
cana-4037	145	5	vertex	vertex	NOUN
cana-4037	145	6	𝑢	𝑢	NOUN
cana-4037	145	7	,	,	PUNCT
cana-4037	145	8	𝑣	𝑣	PRON
cana-4037	145	9	∈d	∈d	NOUN
cana-4037	145	10	,	,	PUNCT
cana-4037	145	11	|𝑜𝑑𝐷(𝑢	|𝑜𝑑𝐷(𝑢	ADJ
cana-4037	145	12	)	)	PUNCT
cana-4037	146	1	−	−	ADP
cana-4037	146	2	𝑜𝑑𝐷(𝑣)|	𝑜𝑑𝐷(𝑣)|	NOUN
cana-4037	146	3	≤	≤	NOUN
cana-4037	146	4	2	2	NUM
cana-4037	146	5	.	.	PUNCT
cana-4037	147	1	so	so	ADV
cana-4037	147	2	,	,	PUNCT
cana-4037	147	3	d	d	X
cana-4037	147	4	is	be	AUX
cana-4037	147	5	the	the	DET
cana-4037	147	6	minimum	minimum	ADJ
cana-4037	147	7	co	co	NOUN
cana-4037	147	8	-	-	ADJ
cana-4037	147	9	total	total	ADJ
cana-4037	147	10	2	2	NUM
cana-4037	147	11	oded	oded	NOUN
cana-4037	147	12	set	set	NOUN
cana-4037	147	13	.	.	PUNCT
cana-4037	148	1	now,|𝐷|	now,|𝐷|	PROPN
cana-4037	148	2	=	=	NOUN
cana-4037	148	3	𝑛	𝑛	NOUN
cana-4037	148	4	−	−	NUM
cana-4037	148	5	1	1	NUM
cana-4037	148	6	.	.	PUNCT
cana-4037	149	1	hence	hence	ADV
cana-4037	149	2	,	,	PUNCT
cana-4037	149	3	𝛾𝑐𝑡2𝑜𝑒(𝑇2(𝐶𝑛	𝛾𝑐𝑡2𝑜𝑒(𝑇2(𝐶𝑛	PROPN
cana-4037	149	4	)	)	PUNCT
cana-4037	149	5	)	)	PUNCT
cana-4037	150	1	=	=	SYM
cana-4037	150	2	𝑛	𝑛	DET
cana-4037	150	3	−	−	NUM
cana-4037	150	4	1	1	NUM
cana-4037	150	5	for	for	ADP
cana-4037	150	6	𝑛	𝑛	PRON
cana-4037	150	7	≥	≥	NUM
cana-4037	150	8	3	3	NUM
cana-4037	150	9	.	.	PUNCT
cana-4037	150	10	theorem	theorem	VERB
cana-4037	150	11	3.10	3.10	NUM
cana-4037	150	12	for	for	ADP
cana-4037	150	13	any	any	DET
cana-4037	150	14	path	path	NOUN
cana-4037	150	15	p𝑛	p𝑛	ADP
cana-4037	150	16	,	,	PUNCT
cana-4037	150	17	𝛾𝑛𝑠2𝑜𝑒(t2(p𝑛	𝛾𝑛𝑠2𝑜𝑒(t2(p𝑛	PROPN
cana-4037	150	18	)	)	PUNCT
cana-4037	150	19	)	)	PUNCT
cana-4037	151	1	=	=	SYM
cana-4037	151	2	𝑛	𝑛	DET
cana-4037	151	3	−	−	NUM
cana-4037	151	4	1	1	NUM
cana-4037	151	5	for	for	ADP
cana-4037	151	6	𝑛	𝑛	PRON
cana-4037	151	7	≥	≥	NUM
cana-4037	151	8	2	2	NUM
cana-4037	151	9	proof	proof	NOUN
cana-4037	151	10	:	:	PUNCT
cana-4037	151	11	let	let	VERB
cana-4037	151	12	p𝑛be	p𝑛be	NOUN
cana-4037	151	13	a	a	DET
cana-4037	151	14	path	path	NOUN
cana-4037	151	15	for	for	ADP
cana-4037	151	16	𝑛	𝑛	PRON
cana-4037	151	17	≥	≥	NUM
cana-4037	151	18	2	2	NUM
cana-4037	151	19	,	,	PUNCT
cana-4037	151	20	here	here	ADV
cana-4037	151	21	v(p𝑛	v(p𝑛	NUM
cana-4037	151	22	)	)	PUNCT
cana-4037	151	23	=	=	PRON
cana-4037	151	24	{	{	PUNCT
cana-4037	151	25	𝑣1	𝑣1	PROPN
cana-4037	151	26	,	,	PUNCT
cana-4037	151	27	𝑣2	𝑣2	PROPN
cana-4037	151	28	,	,	PUNCT
cana-4037	151	29	…	…	PUNCT
cana-4037	151	30	,	,	PUNCT
cana-4037	151	31	𝑣𝑛	𝑣𝑛	NOUN
cana-4037	151	32	}	}	PUNCT
cana-4037	151	33	and	and	CCONJ
cana-4037	151	34	e(p𝑛	e(p𝑛	NUM
cana-4037	151	35	)	)	PUNCT
cana-4037	151	36	=	=	PRON
cana-4037	151	37	{	{	PUNCT
cana-4037	151	38	𝑒1	𝑒1	NOUN
cana-4037	151	39	,	,	PUNCT
cana-4037	151	40	𝑒2	𝑒2	PROPN
cana-4037	151	41	,	,	PUNCT
cana-4037	151	42	…	…	PUNCT
cana-4037	151	43	,	,	PUNCT
cana-4037	151	44	𝑒𝑛−1	𝑒𝑛−1	PROPN
cana-4037	151	45	}	}	PUNCT
cana-4037	151	46	.	.	PUNCT
cana-4037	152	1	now	now	ADV
cana-4037	152	2	,	,	PUNCT
cana-4037	152	3	v(t2(p𝑛	v(t2(p𝑛	NOUN
cana-4037	152	4	)	)	PUNCT
cana-4037	152	5	)	)	PUNCT
cana-4037	153	1	=	=	PRON
cana-4037	153	2	{	{	PUNCT
cana-4037	153	3	𝑣1	𝑣1	PROPN
cana-4037	153	4	,	,	PUNCT
cana-4037	153	5	𝑣2	𝑣2	PROPN
cana-4037	153	6	,	,	PUNCT
cana-4037	153	7	…	…	PUNCT
cana-4037	153	8	,	,	PUNCT
cana-4037	153	9	𝑣𝑛	𝑣𝑛	NOUN
cana-4037	153	10	,	,	PUNCT
cana-4037	153	11	𝑒1	𝑒1	NOUN
cana-4037	153	12	,	,	PUNCT
cana-4037	153	13	𝑒2	𝑒2	PROPN
cana-4037	153	14	,	,	PUNCT
cana-4037	153	15	…	…	PUNCT
cana-4037	153	16	,	,	PUNCT
cana-4037	153	17	𝑒𝑛−1	𝑒𝑛−1	PROPN
cana-4037	153	18	}	}	PUNCT
cana-4037	153	19	be	be	AUX
cana-4037	153	20	vertices	vertex	NOUN
cana-4037	153	21	of	of	ADP
cana-4037	153	22	t2(p𝑛	t2(p𝑛	PROPN
cana-4037	153	23	)	)	PUNCT
cana-4037	153	24	.	.	PUNCT
cana-4037	154	1	let	let	VERB
cana-4037	154	2	d	d	NOUN
cana-4037	154	3	=	=	PUNCT
cana-4037	154	4	{	{	PUNCT
cana-4037	154	5	𝑒1	𝑒1	NOUN
cana-4037	154	6	,	,	PUNCT
cana-4037	154	7	𝑒2	𝑒2	PROPN
cana-4037	154	8	,	,	PUNCT
cana-4037	154	9	…	…	PUNCT
cana-4037	154	10	,	,	PUNCT
cana-4037	154	11	𝑒𝑛−1	𝑒𝑛−1	PROPN
cana-4037	154	12	}	}	PUNCT
cana-4037	154	13	be	be	VERB
cana-4037	154	14	the	the	DET
cana-4037	154	15	minimal	minimal	ADJ
cana-4037	154	16	dominating	dominating	NOUN
cana-4037	154	17	set	set	NOUN
cana-4037	154	18	of	of	ADP
cana-4037	154	19	t2(p𝑛	t2(p𝑛	PROPN
cana-4037	154	20	)	)	PUNCT
cana-4037	154	21	.	.	PUNCT
cana-4037	155	1	then	then	ADV
cana-4037	155	2	v−d	v−d	VERB
cana-4037	155	3	=	=	SYM
cana-4037	155	4	{	{	PUNCT
cana-4037	155	5	𝑣1	𝑣1	PROPN
cana-4037	155	6	,	,	PUNCT
cana-4037	155	7	𝑣2	𝑣2	PROPN
cana-4037	155	8	,	,	PUNCT
cana-4037	155	9	…	…	PUNCT
cana-4037	155	10	,	,	PUNCT
cana-4037	155	11	𝑣𝑛	𝑣𝑛	NOUN
cana-4037	155	12	}	}	PUNCT
cana-4037	155	13	.	.	PUNCT
cana-4037	156	1	the	the	DET
cana-4037	156	2	induced	induced	ADJ
cana-4037	156	3	subgraph	subgraph	NOUN
cana-4037	156	4	of	of	ADP
cana-4037	156	5	<	<	X
cana-4037	156	6	v−d	v−d	X
cana-4037	156	7	>	>	X
cana-4037	156	8	is	be	AUX
cana-4037	156	9	the	the	DET
cana-4037	156	10	given	give	VERB
cana-4037	156	11	p𝑛which	p𝑛which	NOUN
cana-4037	156	12	has	have	VERB
cana-4037	156	13	no	no	DET
cana-4037	156	14	isolated	isolate	VERB
cana-4037	156	15	vertices	vertex	NOUN
cana-4037	156	16	.hence	.hence	ADP
cana-4037	156	17	two	two	NUM
cana-4037	156	18	out	out	ADJ
cana-4037	156	19	degree	degree	NOUN
cana-4037	156	20	of	of	ADP
cana-4037	156	21	any	any	DET
cana-4037	156	22	vertex	vertex	NOUN
cana-4037	156	23	in	in	ADP
cana-4037	156	24	d	d	PROPN
cana-4037	156	25	is	be	AUX
cana-4037	156	26	2	2	NUM
cana-4037	156	27	.	.	PUNCT
cana-4037	156	28	clearly	clearly	ADV
cana-4037	156	29	for	for	ADP
cana-4037	156	30	any	any	DET
cana-4037	156	31	vertex	vertex	NOUN
cana-4037	156	32	𝑢	𝑢	NOUN
cana-4037	156	33	,	,	PUNCT
cana-4037	156	34	𝑣	𝑣	PRON
cana-4037	156	35	∈d	∈d	NOUN
cana-4037	156	36	,	,	PUNCT
cana-4037	156	37	|𝑜𝑑𝐷(𝑢	|𝑜𝑑𝐷(𝑢	ADJ
cana-4037	156	38	)	)	PUNCT
cana-4037	157	1	−	−	ADP
cana-4037	157	2	𝑜𝑑𝐷(𝑣)|	𝑜𝑑𝐷(𝑣)|	NOUN
cana-4037	157	3	≤	≤	NOUN
cana-4037	157	4	2	2	NUM
cana-4037	157	5	.	.	PUNCT
cana-4037	158	1	therefore	therefore	ADV
cana-4037	158	2	,	,	PUNCT
cana-4037	158	3	d	d	X
cana-4037	158	4	is	be	AUX
cana-4037	158	5	the	the	DET
cana-4037	158	6	minimum	minimum	ADJ
cana-4037	158	7	co	co	NOUN
cana-4037	158	8	-	-	ADJ
cana-4037	158	9	total	total	ADJ
cana-4037	158	10	2oded	2ode	VERB
cana-4037	158	11	set	set	NOUN
cana-4037	158	12	.	.	PUNCT
cana-4037	159	1	hence,𝛾𝑐𝑡2𝑜𝑒(t2(p𝑛	hence,𝛾𝑐𝑡2𝑜𝑒(t2(p𝑛	NOUN
cana-4037	159	2	)	)	PUNCT
cana-4037	159	3	)	)	PUNCT
cana-4037	160	1	=	=	SYM
cana-4037	160	2	𝑛	𝑛	DET
cana-4037	160	3	−	−	NUM
cana-4037	160	4	1	1	NUM
cana-4037	160	5	for	for	ADP
cana-4037	160	6	𝑛	𝑛	DET
cana-4037	160	7	≥	≥	NUM
cana-4037	160	8	2.\	2.\	NUM
cana-4037	160	9	theorem	theorem	VERB
cana-4037	160	10	3.11	3.11	NUM
cana-4037	160	11	for	for	ADP
cana-4037	160	12	all	all	DET
cana-4037	160	13	combo	combo	NOUN
cana-4037	160	14	graph	graph	NOUN
cana-4037	160	15	pn+	pn+	NOUN
cana-4037	160	16	,	,	PUNCT
cana-4037	160	17	γct2oe(t2(pn+	γct2oe(t2(pn+	NOUN
cana-4037	160	18	)	)	PUNCT
cana-4037	160	19	)	)	PUNCT
cana-4037	161	1	=	=	SYM
cana-4037	161	2	2n	2n	NUM
cana-4037	161	3	1	1	NUM
cana-4037	161	4	forn	forn	X
cana-4037	161	5	≥	≥	NOUN
cana-4037	161	6	2	2	NUM
cana-4037	161	7	proof	proof	NOUN
cana-4037	161	8	:	:	PUNCT
cana-4037	161	9	let	let	VERB
cana-4037	161	10	p𝑛+be	p𝑛+be	DET
cana-4037	161	11	a	a	DET
cana-4037	161	12	combo	combo	NOUN
cana-4037	161	13	graph	graph	NOUN
cana-4037	161	14	for	for	ADP
cana-4037	161	15	𝑛	𝑛	PRON
cana-4037	161	16	≥	≥	NUM
cana-4037	161	17	2	2	NUM
cana-4037	161	18	,	,	PUNCT
cana-4037	161	19	here	here	ADV
cana-4037	161	20	v(p𝑛+	v(p𝑛+	ADV
cana-4037	161	21	)	)	PUNCT
cana-4037	161	22	=	=	PRON
cana-4037	161	23	{	{	PUNCT
cana-4037	161	24	𝑣1	𝑣1	PROPN
cana-4037	161	25	,	,	PUNCT
cana-4037	161	26	𝑣2	𝑣2	PROPN
cana-4037	161	27	,	,	PUNCT
cana-4037	161	28	…	…	PUNCT
cana-4037	161	29	,	,	PUNCT
cana-4037	161	30	𝑣2𝑛	𝑣2𝑛	ADJ
cana-4037	161	31	}	}	PUNCT
cana-4037	161	32	and	and	CCONJ
cana-4037	161	33	e(p𝑛+	e(p𝑛+	NOUN
cana-4037	161	34	)	)	PUNCT
cana-4037	161	35	=	=	SYM
cana-4037	161	36	{	{	PUNCT
cana-4037	161	37	𝑒1	𝑒1	NOUN
cana-4037	161	38	,	,	PUNCT
cana-4037	161	39	𝑒2	𝑒2	PROPN
cana-4037	161	40	,	,	PUNCT
cana-4037	161	41	…	…	PUNCT
cana-4037	161	42	,	,	PUNCT
cana-4037	161	43	𝑒2𝑛−1	𝑒2𝑛−1	PROPN
cana-4037	161	44	}	}	PUNCT
cana-4037	161	45	.	.	PUNCT
cana-4037	162	1	now	now	ADV
cana-4037	162	2	,	,	PUNCT
cana-4037	162	3	v(t2(p𝑛+	v(t2(p𝑛+	NOUN
cana-4037	162	4	)	)	PUNCT
cana-4037	162	5	)	)	PUNCT
cana-4037	163	1	=	=	PRON
cana-4037	163	2	{	{	PUNCT
cana-4037	163	3	𝑣1	𝑣1	PROPN
cana-4037	163	4	,	,	PUNCT
cana-4037	163	5	𝑣2	𝑣2	PROPN
cana-4037	163	6	,	,	PUNCT
cana-4037	163	7	…	…	PUNCT
cana-4037	163	8	,	,	PUNCT
cana-4037	163	9	𝑣2𝑛	𝑣2𝑛	PROPN
cana-4037	163	10	,	,	PUNCT
cana-4037	163	11	𝑒1	𝑒1	NOUN
cana-4037	163	12	,	,	PUNCT
cana-4037	163	13	𝑒2	𝑒2	PROPN
cana-4037	163	14	,	,	PUNCT
cana-4037	163	15	…	…	PUNCT
cana-4037	163	16	,	,	PUNCT
cana-4037	163	17	𝑒2𝑛−1	𝑒2𝑛−1	PROPN
cana-4037	163	18	}	}	PUNCT
cana-4037	163	19	is	be	AUX
cana-4037	163	20	the	the	DET
cana-4037	163	21	vertex	vertex	NOUN
cana-4037	163	22	set	set	NOUN
cana-4037	163	23	of	of	ADP
cana-4037	163	24	t2(p𝑛+	t2(p𝑛+	NOUN
cana-4037	163	25	)	)	PUNCT
cana-4037	163	26	.	.	PUNCT
cana-4037	164	1	let	let	VERB
cana-4037	164	2	d	d	NOUN
cana-4037	164	3	=	=	PUNCT
cana-4037	164	4	{	{	PUNCT
cana-4037	164	5	𝑒1	𝑒1	NOUN
cana-4037	164	6	,	,	PUNCT
cana-4037	164	7	𝑒2	𝑒2	PROPN
cana-4037	164	8	,	,	PUNCT
cana-4037	164	9	…	…	PUNCT
cana-4037	164	10	,	,	PUNCT
cana-4037	164	11	𝑒2𝑛−1	𝑒2𝑛−1	PROPN
cana-4037	164	12	}	}	PUNCT
cana-4037	164	13	be	be	AUX
cana-4037	164	14	the	the	DET
cana-4037	164	15	minimal	minimal	ADJ
cana-4037	164	16	dominating	dominating	NOUN
cana-4037	164	17	set	set	NOUN
cana-4037	164	18	of	of	ADP
cana-4037	164	19	t2(p𝑛+	t2(p𝑛+	NOUN
cana-4037	164	20	)	)	PUNCT
cana-4037	164	21	.	.	PUNCT
cana-4037	165	1	then	then	ADV
cana-4037	165	2	v−d	v−d	VERB
cana-4037	165	3	=	=	SYM
cana-4037	165	4	{	{	PUNCT
cana-4037	165	5	𝑣1	𝑣1	PROPN
cana-4037	165	6	,	,	PUNCT
cana-4037	165	7	𝑣2	𝑣2	PROPN
cana-4037	165	8	,	,	PUNCT
cana-4037	165	9	…	…	PUNCT
cana-4037	165	10	,	,	PUNCT
cana-4037	165	11	𝑣2𝑛	𝑣2𝑛	PROPN
cana-4037	165	12	}	}	PUNCT
cana-4037	165	13	.	.	PUNCT
cana-4037	166	1	the	the	DET
cana-4037	166	2	subgraph	subgraph	NOUN
cana-4037	166	3	induced	induce	VERB
cana-4037	166	4	by	by	ADP
cana-4037	166	5	v−d	v−d	NOUN
cana-4037	166	6	is	be	AUX
cana-4037	166	7	the	the	DET
cana-4037	166	8	given	give	VERB
cana-4037	166	9	graph	graph	NOUN
cana-4037	166	10	p𝑛+which	p𝑛+which	NOUN
cana-4037	166	11	has	have	VERB
cana-4037	166	12	no	no	DET
cana-4037	166	13	isolated	isolate	VERB
cana-4037	166	14	vertices	vertex	NOUN
cana-4037	166	15	and	and	CCONJ
cana-4037	166	16	also	also	ADV
cana-4037	166	17	each	each	DET
cana-4037	166	18	vertex	vertex	NOUN
cana-4037	166	19	in	in	ADP
cana-4037	166	20	d	d	PROPN
cana-4037	166	21	is	be	AUX
cana-4037	166	22	exactly	exactly	ADV
cana-4037	166	23	adjacent	adjacent	ADJ
cana-4037	166	24	to	to	ADP
cana-4037	166	25	two	two	NUM
cana-4037	166	26	vertices	vertex	NOUN
cana-4037	166	27	in	in	ADP
cana-4037	166	28	v−d	v−d	NOUN
cana-4037	166	29	.	.	PUNCT
cana-4037	167	1	now	now	ADV
cana-4037	167	2	,	,	PUNCT
cana-4037	167	3	𝑜𝑑𝐷(𝑢	𝑜𝑑𝐷(𝑢	PROPN
cana-4037	167	4	)	)	PUNCT
cana-4037	167	5	=	=	SYM
cana-4037	167	6	2	2	NUM
cana-4037	167	7	for	for	ADP
cana-4037	167	8	any	any	DET
cana-4037	167	9	vertex𝑢	vertex𝑢	NOUN
cana-4037	167	10	,	,	PUNCT
cana-4037	167	11	𝑣	𝑣	PRON
cana-4037	167	12	∈d	∈d	NOUN
cana-4037	167	13	,	,	PUNCT
cana-4037	167	14	|𝑜𝑑𝐷(𝑢	|𝑜𝑑𝐷(𝑢	ADJ
cana-4037	167	15	)	)	PUNCT
cana-4037	168	1	−	−	ADP
cana-4037	168	2	𝑜𝑑𝐷(𝑣)|	𝑜𝑑𝐷(𝑣)|	NOUN
cana-4037	168	3	≤	≤	NOUN
cana-4037	168	4	2	2	NUM
cana-4037	168	5	.	.	PUNCT
cana-4037	169	1	therefore	therefore	ADV
cana-4037	169	2	,	,	PUNCT
cana-4037	169	3	d	d	X
cana-4037	169	4	is	be	AUX
cana-4037	169	5	the	the	DET
cana-4037	169	6	minimum	minimum	ADJ
cana-4037	169	7	co	co	NOUN
cana-4037	169	8	-	-	ADJ
cana-4037	169	9	total	total	ADJ
cana-4037	169	10	2	2	NUM
cana-4037	169	11	oded	oded	NOUN
cana-4037	169	12	set	set	NOUN
cana-4037	169	13	.	.	PUNCT
cana-4037	170	1	hence	hence	ADV
cana-4037	170	2	𝛾𝑐𝑡2𝑜𝑒(t2(p𝑛+	𝛾𝑐𝑡2𝑜𝑒(t2(p𝑛+	PROPN
cana-4037	170	3	)	)	PUNCT
cana-4037	170	4	)	)	PUNCT
cana-4037	171	1	=	=	SYM
cana-4037	171	2	2𝑛	2𝑛	PROPN
cana-4037	172	1	−	−	NOUN
cana-4037	172	2	1	1	NUM
cana-4037	172	3	for	for	ADP
cana-4037	172	4	𝑛	𝑛	PRON
cana-4037	172	5	≥	≥	NUM
cana-4037	172	6	2	2	NUM
cana-4037	172	7	.	.	PUNCT
cana-4037	172	8	communications	communication	NOUN
cana-4037	172	9	on	on	ADP
cana-4037	172	10	applied	apply	VERB
cana-4037	172	11	nonlinear	nonlinear	ADJ
cana-4037	172	12	analysis	analysis	NOUN
cana-4037	172	13	issn	issn	NOUN
cana-4037	172	14	:	:	PUNCT
cana-4037	172	15	1074	1074	NUM
cana-4037	172	16	-	-	PUNCT
cana-4037	172	17	133x	133x	NUM
cana-4037	172	18	vol	vol	NOUN
cana-4037	172	19	32	32	NUM
cana-4037	172	20	no	no	NOUN
cana-4037	172	21	.	.	PUNCT
cana-4037	173	1	9s	9s	NUM
cana-4037	173	2	(	(	PUNCT
cana-4037	173	3	2025	2025	NUM
cana-4037	173	4	)	)	PUNCT
cana-4037	173	5	899	899	NUM
cana-4037	174	1	https://internationalpubls.com	https://internationalpubls.com	X
cana-4037	174	2	theorem	theorem	VERB
cana-4037	174	3	3.12	3.12	NUM
cana-4037	174	4	for	for	ADP
cana-4037	174	5	all	all	DET
cana-4037	174	6	fan	fan	NOUN
cana-4037	174	7	graph	graph	NOUN
cana-4037	174	8	𝐹𝑛	𝐹𝑛	PROPN
cana-4037	174	9	,	,	PUNCT
cana-4037	174	10	γct2oe(t2(𝐹𝑛	γct2oe(t2(𝐹𝑛	PUNCT
cana-4037	174	11	)	)	PUNCT
cana-4037	174	12	)	)	PUNCT
cana-4037	175	1	=	=	SYM
cana-4037	175	2	2n	2n	NUM
cana-4037	175	3	1	1	NUM
cana-4037	175	4	forn	forn	X
cana-4037	175	5	≥	≥	NOUN
cana-4037	175	6	2	2	NUM
cana-4037	175	7	proof	proof	NOUN
cana-4037	175	8	:	:	PUNCT
cana-4037	175	9	let	let	VERB
cana-4037	175	10	vertex	vertex	NOUN
cana-4037	175	11	set	set	VERB
cana-4037	175	12	v(𝐹𝑛	v(𝐹𝑛	PROPN
cana-4037	175	13	)	)	PUNCT
cana-4037	175	14	=	=	PRON
cana-4037	175	15	{	{	PUNCT
cana-4037	175	16	𝑣	𝑣	NOUN
cana-4037	175	17	,	,	PUNCT
cana-4037	175	18	𝑣1	𝑣1	PROPN
cana-4037	175	19	,	,	PUNCT
cana-4037	175	20	𝑣2	𝑣2	PROPN
cana-4037	175	21	,	,	PUNCT
cana-4037	175	22	…	…	PUNCT
cana-4037	175	23	,	,	PUNCT
cana-4037	175	24	𝑣𝑛	𝑣𝑛	NOUN
cana-4037	175	25	}	}	PUNCT
cana-4037	175	26	and	and	CCONJ
cana-4037	175	27	edge	edge	VERB
cana-4037	175	28	set	set	ADJ
cana-4037	175	29	e(𝐹𝑛	e(𝐹𝑛	NOUN
cana-4037	175	30	)	)	PUNCT
cana-4037	175	31	=	=	SYM
cana-4037	175	32	{	{	PUNCT
cana-4037	175	33	𝑒1	𝑒1	NOUN
cana-4037	175	34	,	,	PUNCT
cana-4037	175	35	𝑒2	𝑒2	PROPN
cana-4037	175	36	,	,	PUNCT
cana-4037	175	37	…	…	PUNCT
cana-4037	175	38	,	,	PUNCT
cana-4037	175	39	𝑒2𝑛−1	𝑒2𝑛−1	PROPN
cana-4037	175	40	}	}	PUNCT
cana-4037	175	41	.	.	PUNCT
cana-4037	176	1	now	now	ADV
cana-4037	176	2	,	,	PUNCT
cana-4037	176	3	v(t2(𝐹𝑛	v(t2(𝐹𝑛	NOUN
cana-4037	176	4	)	)	PUNCT
cana-4037	176	5	)	)	PUNCT
cana-4037	177	1	=	=	PRON
cana-4037	177	2	{	{	PUNCT
cana-4037	177	3	𝑣	𝑣	NOUN
cana-4037	177	4	,	,	PUNCT
cana-4037	177	5	𝑣1	𝑣1	PROPN
cana-4037	177	6	,	,	PUNCT
cana-4037	177	7	𝑣2	𝑣2	PROPN
cana-4037	177	8	,	,	PUNCT
cana-4037	177	9	…	…	PUNCT
cana-4037	177	10	,	,	PUNCT
cana-4037	177	11	𝑣𝑛	𝑣𝑛	NOUN
cana-4037	177	12	,	,	PUNCT
cana-4037	177	13	𝑒1	𝑒1	NOUN
cana-4037	177	14	,	,	PUNCT
cana-4037	177	15	𝑒2	𝑒2	PROPN
cana-4037	177	16	,	,	PUNCT
cana-4037	177	17	…	…	PUNCT
cana-4037	177	18	,	,	PUNCT
cana-4037	177	19	𝑒2𝑛−1	𝑒2𝑛−1	PROPN
cana-4037	177	20	}	}	PUNCT
cana-4037	177	21	is	be	AUX
cana-4037	177	22	the	the	DET
cana-4037	177	23	vertex	vertex	NOUN
cana-4037	177	24	set	set	NOUN
cana-4037	177	25	of	of	ADP
cana-4037	177	26	t2(𝐹𝑛	t2(𝐹𝑛	NUM
cana-4037	177	27	)	)	PUNCT
cana-4037	177	28	.	.	PUNCT
cana-4037	178	1	let	let	VERB
cana-4037	178	2	d	d	NOUN
cana-4037	178	3	=	=	PUNCT
cana-4037	178	4	{	{	PUNCT
cana-4037	178	5	𝑒1	𝑒1	NOUN
cana-4037	178	6	,	,	PUNCT
cana-4037	178	7	𝑒2	𝑒2	PROPN
cana-4037	178	8	,	,	PUNCT
cana-4037	178	9	…	…	PUNCT
cana-4037	178	10	,	,	PUNCT
cana-4037	178	11	𝑒2𝑛−1	𝑒2𝑛−1	PROPN
cana-4037	178	12	}	}	PUNCT
cana-4037	178	13	be	be	AUX
cana-4037	178	14	the	the	DET
cana-4037	178	15	minimal	minimal	ADJ
cana-4037	178	16	dominating	dominating	NOUN
cana-4037	178	17	set	set	NOUN
cana-4037	178	18	of	of	ADP
cana-4037	178	19	t2(𝐹𝑛	t2(𝐹𝑛	PROPN
cana-4037	178	20	)	)	PUNCT
cana-4037	178	21	.	.	PUNCT
cana-4037	178	22	.	.	PUNCT
cana-4037	179	1	then	then	ADV
cana-4037	179	2	v−d	v−d	VERB
cana-4037	179	3	=	=	SYM
cana-4037	179	4	{	{	PUNCT
cana-4037	179	5	𝑣	𝑣	NOUN
cana-4037	179	6	,	,	PUNCT
cana-4037	179	7	𝑣1	𝑣1	PROPN
cana-4037	179	8	,	,	PUNCT
cana-4037	179	9	𝑣2	𝑣2	PROPN
cana-4037	179	10	,	,	PUNCT
cana-4037	179	11	…	…	PUNCT
cana-4037	179	12	,	,	PUNCT
cana-4037	179	13	𝑣𝑛	𝑣𝑛	NOUN
cana-4037	179	14	}	}	PUNCT
cana-4037	179	15	.	.	PUNCT
cana-4037	180	1	the	the	DET
cana-4037	180	2	induced	induced	ADJ
cana-4037	180	3	subgraph	subgraph	NOUN
cana-4037	180	4	of	of	ADP
cana-4037	180	5	v−d	v−d	NOUN
cana-4037	180	6	is	be	AUX
cana-4037	180	7	the	the	DET
cana-4037	180	8	given	give	VERB
cana-4037	180	9	graph	graph	NOUN
cana-4037	180	10	𝐹𝑛	𝐹𝑛	PROPN
cana-4037	180	11	which	which	PRON
cana-4037	180	12	has	have	VERB
cana-4037	180	13	no	no	DET
cana-4037	180	14	isolated	isolate	VERB
cana-4037	180	15	vertices	vertex	NOUN
cana-4037	180	16	and	and	CCONJ
cana-4037	180	17	also	also	ADV
cana-4037	180	18	each	each	DET
cana-4037	180	19	vertex	vertex	NOUN
cana-4037	180	20	in	in	ADP
cana-4037	180	21	d	d	PROPN
cana-4037	180	22	is	be	AUX
cana-4037	180	23	exactly	exactly	ADV
cana-4037	180	24	adjacent	adjacent	ADJ
cana-4037	180	25	to	to	ADP
cana-4037	180	26	two	two	NUM
cana-4037	180	27	vertices	vertex	NOUN
cana-4037	180	28	in	in	ADP
cana-4037	180	29	v−d	v−d	NOUN
cana-4037	180	30	.	.	PUNCT
cana-4037	181	1	now	now	ADV
cana-4037	181	2	,	,	PUNCT
cana-4037	181	3	for	for	ADP
cana-4037	181	4	any	any	DET
cana-4037	181	5	vertex𝑢	vertex𝑢	NOUN
cana-4037	181	6	,	,	PUNCT
cana-4037	181	7	𝑣	𝑣	PRON
cana-4037	181	8	∈d	∈d	NOUN
cana-4037	181	9	,	,	PUNCT
cana-4037	181	10	|𝑜𝑑𝐷(𝑢	|𝑜𝑑𝐷(𝑢	ADJ
cana-4037	181	11	)	)	PUNCT
cana-4037	182	1	−	−	ADP
cana-4037	182	2	𝑜𝑑𝐷(𝑣)|	𝑜𝑑𝐷(𝑣)|	NOUN
cana-4037	182	3	≤	≤	NOUN
cana-4037	182	4	2	2	NUM
cana-4037	182	5	.	.	PUNCT
cana-4037	183	1	therefore	therefore	ADV
cana-4037	183	2	,	,	PUNCT
cana-4037	183	3	d	d	X
cana-4037	183	4	is	be	AUX
cana-4037	183	5	the	the	DET
cana-4037	183	6	minimum	minimum	ADJ
cana-4037	183	7	co	co	NOUN
cana-4037	183	8	-	-	ADJ
cana-4037	183	9	total	total	ADJ
cana-4037	183	10	2	2	NUM
cana-4037	183	11	oded	oded	NOUN
cana-4037	183	12	set	set	NOUN
cana-4037	183	13	.	.	PUNCT
cana-4037	184	1	hence	hence	ADV
cana-4037	184	2	𝛾𝑐𝑡2𝑜𝑒(t2(𝐹𝑛	𝛾𝑐𝑡2𝑜𝑒(t2(𝐹𝑛	PROPN
cana-4037	184	3	)	)	PUNCT
cana-4037	184	4	)	)	PUNCT
cana-4037	185	1	=	=	SYM
cana-4037	185	2	2𝑛	2𝑛	PROPN
cana-4037	186	1	−	−	NOUN
cana-4037	186	2	1	1	NUM
cana-4037	186	3	for	for	ADP
cana-4037	186	4	𝑛	𝑛	PRON
cana-4037	186	5	≥	≥	NUM
cana-4037	186	6	2	2	NUM
cana-4037	186	7	.	.	PUNCT
cana-4037	186	8	theorem	theorem	VERB
cana-4037	186	9	3.13	3.13	NUM
cana-4037	186	10	for	for	ADP
cana-4037	186	11	any	any	DET
cana-4037	186	12	star	star	NOUN
cana-4037	186	13	graph	graph	NOUN
cana-4037	186	14	k1,𝑛	k1,𝑛	PROPN
cana-4037	186	15	,	,	PUNCT
cana-4037	186	16	𝛾𝑛𝑠2𝑜𝑒(t2(k1,𝑛	𝛾𝑛𝑠2𝑜𝑒(t2(k1,𝑛	NOUN
cana-4037	186	17	)	)	PUNCT
cana-4037	186	18	)	)	PUNCT
cana-4037	187	1	=	=	SYM
cana-4037	187	2	𝑛	𝑛	PROPN
cana-4037	187	3	for	for	ADP
cana-4037	187	4	𝑛	𝑛	PRON
cana-4037	187	5	≥	≥	NUM
cana-4037	187	6	1	1	NUM
cana-4037	187	7	proof	proof	NOUN
cana-4037	187	8	:	:	PUNCT
cana-4037	187	9	let	let	VERB
cana-4037	187	10	k1,𝑛be	k1,𝑛be	PROPN
cana-4037	187	11	a	a	DET
cana-4037	187	12	star	star	NOUN
cana-4037	187	13	graph	graph	NOUN
cana-4037	187	14	for𝑛	for𝑛	PROPN
cana-4037	187	15	≥	≥	NUM
cana-4037	187	16	1with	1with	NUM
cana-4037	187	17	vertex	vertex	NOUN
cana-4037	187	18	set	set	NOUN
cana-4037	187	19	v(k1,𝑛	v(k1,𝑛	PROPN
cana-4037	187	20	)	)	PUNCT
cana-4037	187	21	=	=	SYM
cana-4037	187	22	{	{	PUNCT
cana-4037	187	23	𝑣1	𝑣1	PROPN
cana-4037	187	24	,	,	PUNCT
cana-4037	187	25	𝑣2	𝑣2	PROPN
cana-4037	187	26	,	,	PUNCT
cana-4037	187	27	…	…	PUNCT
cana-4037	187	28	,	,	PUNCT
cana-4037	187	29	𝑣𝑛+1	𝑣𝑛+1	NOUN
cana-4037	187	30	}	}	PUNCT
cana-4037	187	31	and	and	CCONJ
cana-4037	187	32	edge	edge	VERB
cana-4037	187	33	set	set	VERB
cana-4037	187	34	e(k1,𝑛	e(k1,𝑛	PROPN
cana-4037	187	35	)	)	PUNCT
cana-4037	188	1	=	=	PRON
cana-4037	188	2	{	{	PUNCT
cana-4037	188	3	𝑒1	𝑒1	NOUN
cana-4037	188	4	,	,	PUNCT
cana-4037	188	5	𝑒2	𝑒2	PROPN
cana-4037	188	6	,	,	PUNCT
cana-4037	188	7	…	…	PUNCT
cana-4037	188	8	,	,	PUNCT
cana-4037	188	9	𝑒𝑛	𝑒𝑛	NOUN
cana-4037	188	10	}	}	PUNCT
cana-4037	188	11	.	.	PUNCT
cana-4037	189	1	now	now	ADV
cana-4037	189	2	,	,	PUNCT
cana-4037	189	3	v(t2(k1,𝑛	v(t2(k1,𝑛	PROPN
cana-4037	189	4	)	)	PUNCT
cana-4037	189	5	)	)	PUNCT
cana-4037	190	1	=	=	PRON
cana-4037	190	2	{	{	PUNCT
cana-4037	190	3	𝑣1	𝑣1	PROPN
cana-4037	190	4	,	,	PUNCT
cana-4037	190	5	𝑣2	𝑣2	PROPN
cana-4037	190	6	,	,	PUNCT
cana-4037	190	7	…	…	PUNCT
cana-4037	190	8	,	,	PUNCT
cana-4037	190	9	𝑣𝑛+1	𝑣𝑛+1	NOUN
cana-4037	190	10	,	,	PUNCT
cana-4037	190	11	𝑒1	𝑒1	NOUN
cana-4037	190	12	,	,	PUNCT
cana-4037	190	13	𝑒2	𝑒2	PROPN
cana-4037	190	14	,	,	PUNCT
cana-4037	190	15	…	…	PUNCT
cana-4037	190	16	,	,	PUNCT
cana-4037	190	17	𝑒𝑛	𝑒𝑛	PROPN
cana-4037	190	18	}	}	PUNCT
cana-4037	190	19	be	be	VERB
cana-4037	190	20	the	the	DET
cana-4037	190	21	vertices	vertex	NOUN
cana-4037	190	22	of	of	ADP
cana-4037	190	23	t2(k1,𝑛	t2(k1,𝑛	NOUN
cana-4037	190	24	)	)	PUNCT
cana-4037	190	25	.	.	PUNCT
cana-4037	191	1	let	let	VERB
cana-4037	191	2	d	d	NOUN
cana-4037	191	3	=	=	PUNCT
cana-4037	191	4	{	{	PUNCT
cana-4037	191	5	𝑒1	𝑒1	NOUN
cana-4037	191	6	,	,	PUNCT
cana-4037	191	7	𝑒2	𝑒2	PROPN
cana-4037	191	8	,	,	PUNCT
cana-4037	191	9	…	…	PUNCT
cana-4037	191	10	,	,	PUNCT
cana-4037	191	11	𝑒𝑛	𝑒𝑛	X
cana-4037	191	12	}	}	PUNCT
cana-4037	191	13	then	then	ADV
cana-4037	191	14	v−d	v−d	VERB
cana-4037	191	15	=	=	SYM
cana-4037	191	16	{	{	PUNCT
cana-4037	191	17	𝑣1	𝑣1	PROPN
cana-4037	191	18	,	,	PUNCT
cana-4037	191	19	𝑣2	𝑣2	PROPN
cana-4037	191	20	,	,	PUNCT
cana-4037	191	21	…	…	PUNCT
cana-4037	191	22	,	,	PUNCT
cana-4037	191	23	𝑣𝑛+1	𝑣𝑛+1	NOUN
cana-4037	191	24	}	}	PUNCT
cana-4037	191	25	.	.	PUNCT
cana-4037	192	1	the	the	DET
cana-4037	192	2	induced	induced	ADJ
cana-4037	192	3	subgraph	subgraph	NOUN
cana-4037	192	4	of	of	ADP
cana-4037	192	5	v−d	v−d	NOUN
cana-4037	192	6	is	be	AUX
cana-4037	192	7	k1,𝑛	k1,𝑛	PROPN
cana-4037	192	8	,	,	PUNCT
cana-4037	192	9	hence	hence	ADV
cana-4037	192	10	it	it	PRON
cana-4037	192	11	has	have	VERB
cana-4037	192	12	no	no	DET
cana-4037	192	13	vertices	vertex	NOUN
cana-4037	192	14	of	of	ADP
cana-4037	192	15	degree	degree	NOUN
cana-4037	192	16	zero	zero	NUM
cana-4037	192	17	.	.	PUNCT
cana-4037	193	1	also	also	ADV
cana-4037	193	2	each	each	DET
cana-4037	193	3	vertices	vertex	NOUN
cana-4037	193	4	in	in	ADP
cana-4037	193	5	d	d	PROPN
cana-4037	193	6	is	be	AUX
cana-4037	193	7	exactly	exactly	ADV
cana-4037	193	8	adjacent	adjacent	ADJ
cana-4037	193	9	to	to	ADP
cana-4037	193	10	two	two	NUM
cana-4037	193	11	vertices	vertex	NOUN
cana-4037	193	12	in	in	ADP
cana-4037	193	13	v−d	v−d	NOUN
cana-4037	193	14	.	.	PUNCT
cana-4037	194	1	hence	hence	ADV
cana-4037	194	2	for	for	ADP
cana-4037	194	3	any	any	DET
cana-4037	194	4	vertex	vertex	NOUN
cana-4037	194	5	v	v	ADP
cana-4037	194	6	∈	∈	PROPN
cana-4037	194	7	𝐷	𝐷	PROPN
cana-4037	194	8	then	then	ADV
cana-4037	194	9	𝑜𝑑𝐷(𝑣	𝑜𝑑𝐷(𝑣	PROPN
cana-4037	194	10	)	)	PUNCT
cana-4037	194	11	=	=	SYM
cana-4037	195	1	2	2	X
cana-4037	195	2	.	.	PUNCT
cana-4037	195	3	clearly	clearly	ADV
cana-4037	195	4	,	,	PUNCT
cana-4037	195	5	for	for	ADP
cana-4037	195	6	any	any	DET
cana-4037	195	7	vertex	vertex	NOUN
cana-4037	195	8	𝑢	𝑢	NOUN
cana-4037	195	9	,	,	PUNCT
cana-4037	195	10	𝑣	𝑣	PRON
cana-4037	195	11	∈d	∈d	NOUN
cana-4037	195	12	,	,	PUNCT
cana-4037	195	13	|𝑜𝑑𝐷(𝑢	|𝑜𝑑𝐷(𝑢	ADJ
cana-4037	195	14	)	)	PUNCT
cana-4037	196	1	−	−	ADP
cana-4037	196	2	𝑜𝑑𝐷(𝑣)|	𝑜𝑑𝐷(𝑣)|	NOUN
cana-4037	196	3	≤	≤	NOUN
cana-4037	196	4	2	2	NUM
cana-4037	196	5	.	.	PUNCT
cana-4037	197	1	so	so	ADV
cana-4037	197	2	,	,	PUNCT
cana-4037	197	3	d	d	X
cana-4037	197	4	is	be	AUX
cana-4037	197	5	a	a	DET
cana-4037	197	6	minimum	minimum	ADJ
cana-4037	197	7	co	co	NOUN
cana-4037	197	8	-	-	ADJ
cana-4037	197	9	total	total	ADJ
cana-4037	197	10	2	2	NUM
cana-4037	197	11	oded	oded	NOUN
cana-4037	197	12	set	set	NOUN
cana-4037	197	13	.	.	PUNCT
cana-4037	198	1	hence	hence	ADV
cana-4037	198	2	𝛾𝑐𝑡2𝑜𝑒(t2(k1,𝑛	𝛾𝑐𝑡2𝑜𝑒(t2(k1,𝑛	NOUN
cana-4037	198	3	)	)	PUNCT
cana-4037	198	4	)	)	PUNCT
cana-4037	199	1	=	=	SYM
cana-4037	199	2	𝑛	𝑛	PROPN
cana-4037	199	3	for	for	ADP
cana-4037	199	4	𝑛	𝑛	PRON
cana-4037	199	5	≥	≥	NUM
cana-4037	199	6	1	1	NUM
cana-4037	199	7	.	.	PUNCT
cana-4037	199	8	theorem	theorem	VERB
cana-4037	199	9	3.14	3.14	NUM
cana-4037	199	10	for	for	ADP
cana-4037	199	11	any	any	DET
cana-4037	199	12	triangular	triangular	NOUN
cana-4037	199	13	snake	snake	NOUN
cana-4037	199	14	graph	graph	NOUN
cana-4037	199	15	nc3	nc3	PROPN
cana-4037	199	16	,	,	PUNCT
cana-4037	199	17	γns2oe(t2(nc3	γns2oe(t2(nc3	PROPN
cana-4037	199	18	)	)	PUNCT
cana-4037	199	19	)	)	PUNCT
cana-4037	200	1	=	=	SYM
cana-4037	200	2	2n	2n	NUM
cana-4037	200	3	for	for	ADP
cana-4037	200	4	n	n	PRON
cana-4037	200	5	≥	≥	NOUN
cana-4037	200	6	1	1	NUM
cana-4037	200	7	proof	proof	NOUN
cana-4037	200	8	:	:	PUNCT
cana-4037	200	9	let	let	VERB
cana-4037	200	10	nc3be	nc3be	PRON
cana-4037	200	11	a	a	DET
cana-4037	200	12	triangular	triangular	NOUN
cana-4037	200	13	snake	snake	NOUN
cana-4037	200	14	graph	graph	NOUN
cana-4037	200	15	for	for	ADP
cana-4037	200	16	𝑛	𝑛	PRON
cana-4037	200	17	≥	≥	NUM
cana-4037	200	18	1	1	NUM
cana-4037	200	19	,	,	PUNCT
cana-4037	200	20	with	with	ADP
cana-4037	200	21	vertex	vertex	NOUN
cana-4037	200	22	set	set	VERB
cana-4037	200	23	v(𝑛𝐶3	v(𝑛𝐶3	NUM
cana-4037	200	24	)	)	PUNCT
cana-4037	200	25	=	=	PRON
cana-4037	200	26	{	{	PUNCT
cana-4037	200	27	𝑣1	𝑣1	PROPN
cana-4037	200	28	,	,	PUNCT
cana-4037	200	29	𝑣2	𝑣2	PROPN
cana-4037	200	30	,	,	PUNCT
cana-4037	200	31	…	…	PUNCT
cana-4037	200	32	,	,	PUNCT
cana-4037	200	33	𝑣2𝑛+1	𝑣2𝑛+1	NOUN
cana-4037	200	34	}	}	PUNCT
cana-4037	200	35	and	and	CCONJ
cana-4037	200	36	edge	edge	VERB
cana-4037	200	37	set	set	VERB
cana-4037	200	38	e(nc3	e(nc3	NOUN
cana-4037	200	39	)	)	PUNCT
cana-4037	200	40	=	=	SYM
cana-4037	200	41	{	{	PUNCT
cana-4037	200	42	𝑒1	𝑒1	NOUN
cana-4037	200	43	,	,	PUNCT
cana-4037	200	44	𝑒2	𝑒2	PROPN
cana-4037	200	45	,	,	PUNCT
cana-4037	200	46	…	…	PUNCT
cana-4037	200	47	,	,	PUNCT
cana-4037	200	48	𝑒3𝑛	𝑒3𝑛	X
cana-4037	200	49	}	}	PUNCT
cana-4037	200	50	.	.	PUNCT
cana-4037	201	1	now	now	ADV
cana-4037	201	2	,	,	PUNCT
cana-4037	201	3	v(𝑇2(𝑛𝐶3	v(𝑇2(𝑛𝐶3	ADJ
cana-4037	201	4	)	)	PUNCT
cana-4037	201	5	)	)	PUNCT
cana-4037	202	1	=	=	PRON
cana-4037	202	2	{	{	PUNCT
cana-4037	202	3	𝑣1	𝑣1	PROPN
cana-4037	202	4	,	,	PUNCT
cana-4037	202	5	𝑣2	𝑣2	PROPN
cana-4037	202	6	,	,	PUNCT
cana-4037	202	7	…	…	PUNCT
cana-4037	202	8	,	,	PUNCT
cana-4037	202	9	𝑣2𝑛+1	𝑣2𝑛+1	NOUN
cana-4037	202	10	,	,	PUNCT
cana-4037	202	11	𝑒1	𝑒1	NOUN
cana-4037	202	12	,	,	PUNCT
cana-4037	202	13	𝑒2	𝑒2	PROPN
cana-4037	202	14	,	,	PUNCT
cana-4037	202	15	…	…	PUNCT
cana-4037	202	16	,	,	PUNCT
cana-4037	202	17	𝑒3𝑛	𝑒3𝑛	X
cana-4037	202	18	}	}	PUNCT
cana-4037	202	19	be	be	VERB
cana-4037	202	20	the	the	DET
cana-4037	202	21	vertex	vertex	NOUN
cana-4037	202	22	set	set	NOUN
cana-4037	202	23	of	of	ADP
cana-4037	202	24	t2(nc3	t2(nc3	NOUN
cana-4037	202	25	)	)	PUNCT
cana-4037	202	26	.	.	PUNCT
cana-4037	203	1	let	let	VERB
cana-4037	203	2	d	d	NOUN
cana-4037	203	3	=	=	PRON
cana-4037	203	4	{	{	PUNCT
cana-4037	203	5	𝑒2𝑛+1	𝑒2𝑛+1	PROPN
cana-4037	203	6	,	,	PUNCT
cana-4037	203	7	…	…	PUNCT
cana-4037	203	8	,	,	PUNCT
cana-4037	203	9	𝑒3𝑛	𝑒3𝑛	X
cana-4037	203	10	,	,	PUNCT
cana-4037	203	11	𝑣𝑛+2	𝑣𝑛+2	NUM
cana-4037	203	12	,	,	PUNCT
cana-4037	203	13	…	…	PUNCT
cana-4037	203	14	,	,	PUNCT
cana-4037	203	15	𝑣2𝑛+1	𝑣2𝑛+1	PROPN
cana-4037	203	16	}	}	PUNCT
cana-4037	203	17	be	be	VERB
cana-4037	203	18	the	the	DET
cana-4037	203	19	minimal	minimal	ADJ
cana-4037	203	20	dominating	dominating	NOUN
cana-4037	203	21	set	set	NOUN
cana-4037	203	22	of	of	ADP
cana-4037	203	23	t2(nc3	t2(nc3	NOUN
cana-4037	203	24	)	)	PUNCT
cana-4037	203	25	then	then	ADV
cana-4037	203	26	v−d	v−d	VERB
cana-4037	203	27	=	=	SYM
cana-4037	203	28	{	{	PUNCT
cana-4037	203	29	𝑣1	𝑣1	PROPN
cana-4037	203	30	,	,	PUNCT
cana-4037	203	31	𝑣2	𝑣2	PROPN
cana-4037	203	32	,	,	PUNCT
cana-4037	203	33	…	…	PUNCT
cana-4037	203	34	,	,	PUNCT
cana-4037	203	35	𝑣𝑛+1	𝑣𝑛+1	NOUN
cana-4037	203	36	,	,	PUNCT
cana-4037	203	37	𝑒1	𝑒1	NOUN
cana-4037	203	38	,	,	PUNCT
cana-4037	203	39	𝑒2	𝑒2	PROPN
cana-4037	203	40	,	,	PUNCT
cana-4037	203	41	…	…	PUNCT
cana-4037	203	42	,	,	PUNCT
cana-4037	203	43	𝑒2𝑛	𝑒2𝑛	VERB
cana-4037	203	44	}	}	PUNCT
cana-4037	203	45	.	.	PUNCT
cana-4037	204	1	clearly	clearly	ADV
cana-4037	204	2	,	,	PUNCT
cana-4037	204	3	the	the	DET
cana-4037	204	4	vertices	vertex	NOUN
cana-4037	204	5	𝑣1	𝑣1	PROPN
cana-4037	204	6	,	,	PUNCT
cana-4037	204	7	𝑣2	𝑣2	PROPN
cana-4037	204	8	,	,	PUNCT
cana-4037	204	9	…	…	PUNCT
cana-4037	204	10	,	,	PUNCT
cana-4037	204	11	𝑣𝑛+1	𝑣𝑛+1	NOUN
cana-4037	204	12	in	in	ADP
cana-4037	204	13	v−d	v−d	NOUN
cana-4037	204	14	forms	form	NOUN
cana-4037	204	15	a	a	DET
cana-4037	204	16	path	path	NOUN
cana-4037	204	17	and	and	CCONJ
cana-4037	204	18	by	by	ADP
cana-4037	204	19	the	the	DET
cana-4037	204	20	definition	definition	NOUN
cana-4037	204	21	of	of	ADP
cana-4037	204	22	semi	semi	ADJ
cana-4037	204	23	total	total	ADJ
cana-4037	204	24	point	point	NOUN
cana-4037	204	25	graph	graph	NOUN
cana-4037	204	26	the	the	DET
cana-4037	204	27	vertices	vertex	NOUN
cana-4037	204	28	𝑒1	𝑒1	NOUN
cana-4037	204	29	,	,	PUNCT
cana-4037	204	30	𝑒2	𝑒2	PROPN
cana-4037	204	31	,	,	PUNCT
cana-4037	204	32	…	…	PUNCT
cana-4037	204	33	,	,	PUNCT
cana-4037	204	34	𝑒2𝑛	𝑒2𝑛	PROPN
cana-4037	204	35	is	be	AUX
cana-4037	204	36	adjacent	adjacent	ADJ
cana-4037	204	37	to	to	PART
cana-4037	204	38	𝑣1	𝑣1	PROPN
cana-4037	204	39	,	,	PUNCT
cana-4037	204	40	𝑣2	𝑣2	PROPN
cana-4037	204	41	,	,	PUNCT
cana-4037	204	42	…	…	PUNCT
cana-4037	204	43	,	,	PUNCT
cana-4037	204	44	𝑣𝑛+1	𝑣𝑛+1	NOUN
cana-4037	204	45	in	in	ADP
cana-4037	204	46	v−d	v−d	NOUN
cana-4037	204	47	.	.	PUNCT
cana-4037	205	1	thus	thus	ADV
cana-4037	205	2	<	<	X
cana-4037	205	3	v−d	v−d	X
cana-4037	205	4	>	>	X
cana-4037	205	5	has	have	VERB
cana-4037	205	6	no	no	DET
cana-4037	205	7	isolated	isolated	ADJ
cana-4037	205	8	vertices	vertex	NOUN
cana-4037	205	9	.	.	PUNCT
cana-4037	206	1	the	the	DET
cana-4037	206	2	vertices	vertex	NOUN
cana-4037	206	3	𝑒2𝑛+1	𝑒2𝑛+1	PROPN
cana-4037	206	4	,	,	PUNCT
cana-4037	206	5	…	…	PUNCT
cana-4037	206	6	,	,	PUNCT
cana-4037	206	7	𝑒3𝑛	𝑒3𝑛	PROPN
cana-4037	206	8	in	in	ADP
cana-4037	206	9	d	d	PROPN
cana-4037	206	10	is	be	AUX
cana-4037	206	11	adjacent	adjacent	ADJ
cana-4037	206	12	exactly	exactly	ADV
cana-4037	206	13	two	two	NUM
cana-4037	206	14	vertices	vertex	NOUN
cana-4037	206	15	in	in	ADP
cana-4037	206	16	v−d	v−d	NOUN
cana-4037	206	17	,	,	PUNCT
cana-4037	206	18	so	so	SCONJ
cana-4037	206	19	the	the	DET
cana-4037	206	20	out	out	ADJ
cana-4037	206	21	degree	degree	NOUN
cana-4037	206	22	is	be	AUX
cana-4037	206	23	two	two	NUM
cana-4037	206	24	.	.	PUNCT
cana-4037	207	1	the	the	DET
cana-4037	207	2	remaining	remain	VERB
cana-4037	207	3	vertices	vertex	NOUN
cana-4037	207	4	𝑣𝑛+2	𝑣𝑛+2	NUM
cana-4037	207	5	,	,	PUNCT
cana-4037	207	6	…	…	PUNCT
cana-4037	207	7	,	,	PUNCT
cana-4037	207	8	𝑣2𝑛+1of	𝑣2𝑛+1of	NOUN
cana-4037	207	9	d	d	NOUN
cana-4037	207	10	is	be	AUX
cana-4037	207	11	adjacent	adjacent	ADJ
cana-4037	207	12	exactly	exactly	ADV
cana-4037	207	13	four	four	NUM
cana-4037	207	14	vertices	vertex	NOUN
cana-4037	207	15	in	in	ADP
cana-4037	207	16	v−d	v−d	NOUN
cana-4037	207	17	,	,	PUNCT
cana-4037	207	18	hence	hence	ADV
cana-4037	207	19	the	the	DET
cana-4037	207	20	out	out	ADJ
cana-4037	207	21	degree	degree	NOUN
cana-4037	207	22	is	be	AUX
cana-4037	207	23	four	four	NUM
cana-4037	207	24	.	.	PUNCT
cana-4037	208	1	thus	thus	ADV
cana-4037	208	2	,	,	PUNCT
cana-4037	208	3	for	for	ADP
cana-4037	208	4	any	any	DET
cana-4037	208	5	vertex	vertex	NOUN
cana-4037	208	6	𝑢	𝑢	NOUN
cana-4037	208	7	,	,	PUNCT
cana-4037	208	8	𝑣	𝑣	PRON
cana-4037	208	9	∈d	∈d	NOUN
cana-4037	208	10	,	,	PUNCT
cana-4037	208	11	|𝑜𝑑𝐷(𝑢	|𝑜𝑑𝐷(𝑢	ADJ
cana-4037	208	12	)	)	PUNCT
cana-4037	209	1	−	−	ADP
cana-4037	209	2	𝑜𝑑𝐷(𝑣)|	𝑜𝑑𝐷(𝑣)|	NOUN
cana-4037	209	3	≤	≤	NOUN
cana-4037	209	4	2	2	NUM
cana-4037	209	5	.	.	PUNCT
cana-4037	210	1	so	so	ADV
cana-4037	210	2	,	,	PUNCT
cana-4037	210	3	d	d	X
cana-4037	210	4	is	be	AUX
cana-4037	210	5	the	the	DET
cana-4037	210	6	minimum	minimum	ADJ
cana-4037	210	7	co	co	NOUN
cana-4037	210	8	-	-	ADJ
cana-4037	210	9	total	total	ADJ
cana-4037	210	10	2	2	NUM
cana-4037	210	11	oded	oded	NOUN
cana-4037	210	12	set	set	NOUN
cana-4037	210	13	.	.	PUNCT
cana-4037	211	1	now	now	ADV
cana-4037	211	2	,	,	PUNCT
cana-4037	211	3	|𝐷|	|𝐷|	X
cana-4037	211	4	=	=	PUNCT
cana-4037	211	5	2𝑛.	2𝑛.	NOUN
cana-4037	211	6	hence	hence	ADV
cana-4037	211	7	𝛾𝑐𝑡2𝑜𝑒(t2(nc3	𝛾𝑐𝑡2𝑜𝑒(t2(nc3	NOUN
cana-4037	211	8	)	)	PUNCT
cana-4037	211	9	)	)	PUNCT
cana-4037	212	1	=	=	SYM
cana-4037	212	2	2𝑛	2𝑛	PROPN
cana-4037	212	3	for	for	ADP
cana-4037	212	4	𝑛	𝑛	PRON
cana-4037	212	5	≥1	≥1	NUM
cana-4037	212	6	4	4	NUM
cana-4037	212	7	.	.	PUNCT
cana-4037	212	8	conclusion	conclusion	NOUN
cana-4037	212	9	in	in	ADP
cana-4037	212	10	the	the	DET
cana-4037	212	11	next	next	ADJ
cana-4037	212	12	paper	paper	NOUN
cana-4037	212	13	we	we	PRON
cana-4037	212	14	study	study	VERB
cana-4037	212	15	the	the	DET
cana-4037	212	16	bounds	bound	NOUN
cana-4037	212	17	of	of	ADP
cana-4037	212	18	co	co	NOUN
cana-4037	212	19	-	-	ADJ
cana-4037	212	20	total	total	ADJ
cana-4037	212	21	2oded	2ode	VERB
cana-4037	212	22	number	number	NOUN
cana-4037	212	23	and	and	CCONJ
cana-4037	212	24	find	find	VERB
cana-4037	212	25	the	the	DET
cana-4037	212	26	above	above	ADJ
cana-4037	212	27	number	number	NOUN
cana-4037	212	28	for	for	ADP
cana-4037	212	29	some	some	DET
cana-4037	212	30	new	new	ADJ
cana-4037	212	31	family	family	NOUN
cana-4037	212	32	of	of	ADP
cana-4037	212	33	graphs	graph	NOUN
cana-4037	212	34	.	.	PUNCT
cana-4037	213	1	also	also	ADV
cana-4037	213	2	we	we	PRON
cana-4037	213	3	like	like	VERB
cana-4037	213	4	to	to	PART
cana-4037	213	5	extend	extend	VERB
cana-4037	213	6	the	the	DET
cana-4037	213	7	study	study	NOUN
cana-4037	213	8	to	to	PART
cana-4037	213	9	find	find	VERB
cana-4037	213	10	the	the	DET
cana-4037	213	11	limitations	limitation	NOUN
cana-4037	213	12	and	and	CCONJ
cana-4037	213	13	applications	application	NOUN
cana-4037	213	14	of	of	ADP
cana-4037	213	15	the	the	DET
cana-4037	213	16	co	co	NOUN
cana-4037	213	17	-	-	ADJ
cana-4037	213	18	total	total	ADJ
cana-4037	213	19	2oded	2ode	VERB
cana-4037	213	20	number	number	NOUN
cana-4037	213	21	.	.	PUNCT
cana-4037	214	1	references	reference	NOUN
cana-4037	214	2	1	1	NUM
cana-4037	214	3	.	.	PUNCT
cana-4037	215	1	basavanagoud	basavanagoud	PROPN
cana-4037	215	2	,	,	PUNCT
cana-4037	215	3	b.	b.	PROPN
cana-4037	215	4	,	,	PUNCT
cana-4037	215	5	&	&	CCONJ
cana-4037	215	6	tali	tali	PROPN
cana-4037	215	7	,	,	PUNCT
cana-4037	215	8	v.	v.	PROPN
cana-4037	216	1	v.	v.	PROPN
cana-4037	216	2	(	(	PUNCT
cana-4037	216	3	2014	2014	NUM
cana-4037	216	4	)	)	PUNCT
cana-4037	216	5	.	.	PUNCT
cana-4037	217	1	equitable	equitable	ADJ
cana-4037	217	2	co	co	ADJ
cana-4037	217	3	-	-	ADJ
cana-4037	217	4	total	total	ADJ
cana-4037	217	5	domination	domination	NOUN
cana-4037	217	6	number	number	NOUN
cana-4037	217	7	in	in	ADP
cana-4037	217	8	graphs	graph	NOUN
cana-4037	217	9	.	.	PUNCT
cana-4037	218	1	international	international	ADJ
cana-4037	218	2	journal	journal	NOUN
cana-4037	218	3	of	of	ADP
cana-4037	218	4	scientific	scientific	ADJ
cana-4037	218	5	research	research	NOUN
cana-4037	218	6	,	,	PUNCT
cana-4037	218	7	3(7	3(7	NUM
cana-4037	218	8	)	)	PUNCT
cana-4037	218	9	,	,	PUNCT
cana-4037	218	10	301	301	NUM
cana-4037	218	11	-	-	SYM
cana-4037	218	12	305	305	NUM
cana-4037	218	13	.	.	PUNCT
cana-4037	219	1	2	2	NUM
cana-4037	219	2	.	.	X
cana-4037	219	3	basavanagoud	basavanagoud	NOUN
cana-4037	219	4	,	,	PUNCT
cana-4037	219	5	b.	b.	PROPN
cana-4037	219	6	,	,	PUNCT
cana-4037	219	7	&	&	CCONJ
cana-4037	219	8	hosamani	hosamani	PROPN
cana-4037	219	9	,	,	PUNCT
cana-4037	219	10	s.	s.	PROPN
cana-4037	219	11	m.	m.	PROPN
cana-4037	219	12	(	(	PUNCT
cana-4037	219	13	2011	2011	NUM
cana-4037	219	14	)	)	PUNCT
cana-4037	219	15	.	.	PUNCT
cana-4037	220	1	connected	connect	VERB
cana-4037	220	2	semi	semi	ADJ
cana-4037	220	3	-	-	ADJ
cana-4037	220	4	total	total	ADJ
cana-4037	220	5	point	point	NOUN
cana-4037	220	6	domination	domination	NOUN
cana-4037	220	7	in	in	ADP
cana-4037	220	8	graphs	graph	NOUN
cana-4037	220	9	.	.	PUNCT
cana-4037	221	1	international	international	ADJ
cana-4037	221	2	journal	journal	PROPN
cana-4037	221	3	of	of	ADP
cana-4037	221	4	science	science	NOUN
cana-4037	221	5	and	and	CCONJ
cana-4037	221	6	technology	technology	NOUN
cana-4037	221	7	,	,	PUNCT
cana-4037	221	8	2(4	2(4	NUM
cana-4037	221	9	)	)	PUNCT
cana-4037	221	10	,	,	PUNCT
cana-4037	221	11	116	116	NUM
cana-4037	221	12	-	-	SYM
cana-4037	221	13	125	125	NUM
cana-4037	221	14	.	.	PUNCT
cana-4037	222	1	3	3	NUM
cana-4037	222	2	.	.	X
cana-4037	222	3	basavanagoud	basavanagoud	PROPN
cana-4037	222	4	,	,	PUNCT
cana-4037	222	5	b.	b.	PROPN
cana-4037	222	6	,	,	PUNCT
cana-4037	222	7	&	&	CCONJ
cana-4037	222	8	malgham	malgham	PROPN
cana-4037	222	9	,	,	PUNCT
cana-4037	222	10	s.	s.	PROPN
cana-4037	222	11	h.	h.	PROPN
cana-4037	222	12	(	(	PUNCT
cana-4037	222	13	2010	2010	NUM
cana-4037	222	14	)	)	PUNCT
cana-4037	222	15	.	.	PUNCT
cana-4037	223	1	domination	domination	NOUN
cana-4037	223	2	in	in	ADP
cana-4037	223	3	semi	semi	ADJ
cana-4037	223	4	-	-	ADJ
cana-4037	223	5	total	total	ADJ
cana-4037	223	6	point	point	NOUN
cana-4037	223	7	graph	graph	NOUN
cana-4037	223	8	.	.	PUNCT
cana-4037	224	1	journal	journal	PROPN
cana-4037	224	2	of	of	ADP
cana-4037	224	3	computer	computer	NOUN
cana-4037	224	4	and	and	CCONJ
cana-4037	224	5	mathematical	mathematical	ADJ
cana-4037	224	6	sciences	science	NOUN
cana-4037	224	7	,	,	PUNCT
cana-4037	224	8	5	5	NUM
cana-4037	224	9	,	,	PUNCT
cana-4037	224	10	598	598	NUM
cana-4037	224	11	-	-	SYM
cana-4037	224	12	605	605	NUM
cana-4037	224	13	.	.	PUNCT
cana-4037	225	1	communications	communication	NOUN
cana-4037	225	2	on	on	ADP
cana-4037	225	3	applied	apply	VERB
cana-4037	225	4	nonlinear	nonlinear	ADJ
cana-4037	225	5	analysis	analysis	NOUN
cana-4037	225	6	issn	issn	NOUN
cana-4037	225	7	:	:	PUNCT
cana-4037	225	8	1074	1074	NUM
cana-4037	225	9	-	-	PUNCT
cana-4037	225	10	133x	133x	NUM
cana-4037	225	11	vol	vol	NOUN
cana-4037	225	12	32	32	NUM
cana-4037	225	13	no	no	NOUN
cana-4037	225	14	.	.	PUNCT
cana-4037	226	1	9s	9s	NUM
cana-4037	226	2	(	(	PUNCT
cana-4037	226	3	2025	2025	NUM
cana-4037	226	4	)	)	PUNCT
cana-4037	226	5	900	900	NUM
cana-4037	226	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-4037	226	7	4	4	NUM
cana-4037	226	8	.	.	X
cana-4037	227	1	berge	berge	NOUN
cana-4037	227	2	,	,	PUNCT
cana-4037	227	3	c.	c.	PROPN
cana-4037	227	4	(	(	PUNCT
cana-4037	227	5	1962	1962	NUM
cana-4037	227	6	)	)	PUNCT
cana-4037	227	7	.	.	PUNCT
cana-4037	228	1	theory	theory	NOUN
cana-4037	228	2	of	of	ADP
cana-4037	228	3	graphs	graph	NOUN
cana-4037	228	4	and	and	CCONJ
cana-4037	228	5	its	its	PRON
cana-4037	228	6	applications	application	NOUN
cana-4037	228	7	.	.	PUNCT
cana-4037	229	1	methuen	methuen	PROPN
cana-4037	229	2	,	,	PUNCT
cana-4037	229	3	london	london	PROPN
cana-4037	229	4	.	.	PUNCT
cana-4037	230	1	5	5	NUM
cana-4037	230	2	.	.	X
cana-4037	230	3	harary	harary	PROPN
cana-4037	230	4	,	,	PUNCT
cana-4037	230	5	f.	f.	PROPN
cana-4037	230	6	(	(	PUNCT
cana-4037	230	7	1969	1969	NUM
cana-4037	230	8	)	)	PUNCT
cana-4037	230	9	.	.	PUNCT
cana-4037	231	1	graph	graph	NOUN
cana-4037	231	2	theory	theory	NOUN
cana-4037	231	3	.	.	PUNCT
cana-4037	232	1	addison	addison	PROPN
cana-4037	232	2	-	-	PUNCT
cana-4037	232	3	wesley	wesley	PROPN
cana-4037	232	4	,	,	PUNCT
cana-4037	232	5	reading	reading	NOUN
cana-4037	232	6	,	,	PUNCT
cana-4037	232	7	ma	ma	PROPN
cana-4037	232	8	.	.	PROPN
cana-4037	232	9	6	6	NUM
cana-4037	232	10	.	.	X
cana-4037	233	1	kulli	kulli	PROPN
cana-4037	233	2	,	,	PUNCT
cana-4037	233	3	v.	v.	PROPN
cana-4037	233	4	r.	r.	PROPN
cana-4037	233	5	,	,	PUNCT
cana-4037	233	6	jankiram	jankiram	PROPN
cana-4037	233	7	,	,	PUNCT
cana-4037	233	8	b.	b.	PROPN
cana-4037	233	9	,	,	PUNCT
cana-4037	233	10	&	&	CCONJ
cana-4037	233	11	radha	radha	PROPN
cana-4037	233	12	r.	r.	PROPN
cana-4037	233	13	iyer	iyer	PROPN
cana-4037	233	14	.	.	PUNCT
cana-4037	234	1	(	(	PUNCT
cana-4037	234	2	1999	1999	NUM
cana-4037	234	3	)	)	PUNCT
cana-4037	234	4	.	.	PUNCT
cana-4037	235	1	the	the	DET
cana-4037	235	2	co	co	ADJ
cana-4037	235	3	-	-	ADJ
cana-4037	235	4	total	total	ADJ
cana-4037	235	5	domination	domination	NOUN
cana-4037	235	6	number	number	NOUN
cana-4037	235	7	of	of	ADP
cana-4037	235	8	a	a	DET
cana-4037	235	9	graph	graph	NOUN
cana-4037	235	10	.	.	PUNCT
cana-4037	236	1	journal	journal	NOUN
cana-4037	236	2	of	of	ADP
cana-4037	236	3	discrete	discrete	ADJ
cana-4037	236	4	mathematical	mathematical	ADJ
cana-4037	236	5	sciences	sciences	PROPN
cana-4037	236	6	&	&	CCONJ
cana-4037	236	7	cryptography	cryptography	NOUN
cana-4037	236	8	,	,	PUNCT
cana-4037	236	9	2(2	2(2	NUM
cana-4037	236	10	-	-	SYM
cana-4037	236	11	3	3	NUM
cana-4037	236	12	)	)	PUNCT
cana-4037	236	13	,	,	PUNCT
cana-4037	236	14	179	179	NUM
cana-4037	236	15	-	-	SYM
cana-4037	236	16	184	184	NUM
cana-4037	236	17	.	.	PUNCT
cana-4037	237	1	7	7	X
cana-4037	237	2	.	.	X
cana-4037	237	3	mahesh	mahesh	PROPN
cana-4037	237	4	,	,	PUNCT
cana-4037	237	5	m.	m.	PROPN
cana-4037	237	6	s.	s.	PROPN
cana-4037	237	7	,	,	PUNCT
cana-4037	237	8	&	&	CCONJ
cana-4037	237	9	namasivayam	namasivayam	PROPN
cana-4037	237	10	,	,	PUNCT
cana-4037	237	11	p.	p.	NOUN
cana-4037	237	12	(	(	PUNCT
cana-4037	237	13	2017	2017	NUM
cana-4037	237	14	)	)	PUNCT
cana-4037	237	15	.	.	PUNCT
cana-4037	238	1	connected	connect	VERB
cana-4037	238	2	two	two	NUM
cana-4037	238	3	out	out	ADJ
cana-4037	238	4	-	-	PUNCT
cana-4037	238	5	degree	degree	NOUN
cana-4037	238	6	equitable	equitable	ADJ
cana-4037	238	7	domination	domination	NOUN
cana-4037	238	8	of	of	ADP
cana-4037	238	9	semi	semi	ADJ
cana-4037	238	10	-	-	ADJ
cana-4037	238	11	total	total	ADJ
cana-4037	238	12	point	point	NOUN
cana-4037	238	13	graphs	graph	NOUN
cana-4037	238	14	.	.	PUNCT
cana-4037	239	1	journal	journal	NOUN
cana-4037	239	2	of	of	ADP
cana-4037	239	3	computer	computer	NOUN
cana-4037	239	4	and	and	CCONJ
cana-4037	239	5	mathematical	mathematical	ADJ
cana-4037	239	6	sciences	science	NOUN
cana-4037	239	7	,	,	PUNCT
cana-4037	239	8	8(4	8(4	NUM
cana-4037	239	9	)	)	PUNCT
cana-4037	239	10	,	,	PUNCT
cana-4037	239	11	133	133	NUM
cana-4037	239	12	-	-	SYM
cana-4037	239	13	138	138	NUM
cana-4037	239	14	.	.	NOUN
cana-4037	239	15	8	8	NUM
cana-4037	239	16	.	.	PUNCT
cana-4037	239	17	mathevan	mathevan	PROPN
cana-4037	239	18	pillai	pillai	PROPN
cana-4037	239	19	,	,	PUNCT
cana-4037	239	20	k.	k.	PROPN
cana-4037	239	21	,	,	PUNCT
cana-4037	239	22	mahesh	mahesh	PROPN
cana-4037	239	23	,	,	PUNCT
cana-4037	239	24	m.	m.	PROPN
cana-4037	239	25	s.	s.	PROPN
cana-4037	239	26	,	,	PUNCT
cana-4037	239	27	santiago	santiago	PROPN
cana-4037	239	28	stephan	stephan	PROPN
cana-4037	239	29	,	,	PUNCT
cana-4037	239	30	a.	a.	PROPN
cana-4037	239	31	,	,	PUNCT
cana-4037	239	32	&	&	CCONJ
cana-4037	239	33	selvam	selvam	PROPN
cana-4037	239	34	,	,	PUNCT
cana-4037	239	35	g.	g.	PROPN
cana-4037	239	36	(	(	PUNCT
cana-4037	239	37	2018	2018	NUM
cana-4037	239	38	)	)	PUNCT
cana-4037	239	39	.	.	PUNCT
cana-4037	240	1	some	some	DET
cana-4037	240	2	more	more	ADJ
cana-4037	240	3	results	result	NOUN
cana-4037	240	4	on	on	ADP
cana-4037	240	5	non	non	ADJ
cana-4037	240	6	-	-	ADJ
cana-4037	240	7	split	split	ADJ
cana-4037	240	8	two	two	NUM
cana-4037	240	9	out	out	ADJ
cana-4037	240	10	-	-	PUNCT
cana-4037	240	11	degree	degree	NOUN
cana-4037	240	12	equitable	equitable	ADJ
cana-4037	240	13	domination	domination	NOUN
cana-4037	240	14	number	number	NOUN
cana-4037	240	15	.	.	PUNCT
cana-4037	241	1	international	international	ADJ
cana-4037	241	2	journal	journal	PROPN
cana-4037	241	3	of	of	ADP
cana-4037	241	4	mechanical	mechanical	ADJ
cana-4037	241	5	and	and	CCONJ
cana-4037	241	6	production	production	NOUN
cana-4037	241	7	engineering	engineering	NOUN
cana-4037	241	8	research	research	NOUN
cana-4037	241	9	and	and	CCONJ
cana-4037	241	10	development	development	NOUN
cana-4037	241	11	,	,	PUNCT
cana-4037	241	12	8(2	8(2	NUM
cana-4037	241	13	)	)	PUNCT
cana-4037	241	14	,	,	PUNCT
cana-4037	241	15	361	361	NUM
cana-4037	241	16	-	-	SYM
cana-4037	241	17	369	369	NUM
cana-4037	241	18	.	.	PUNCT
cana-4037	242	1	9	9	X
cana-4037	242	2	.	.	X
cana-4037	242	3	mathevan	mathevan	PROPN
cana-4037	242	4	pillai	pillai	PROPN
cana-4037	242	5	,	,	PUNCT
cana-4037	242	6	k.	k.	PROPN
cana-4037	242	7	,	,	PUNCT
cana-4037	242	8	mahesh	mahesh	PROPN
cana-4037	242	9	,	,	PUNCT
cana-4037	242	10	m.	m.	PROPN
cana-4037	242	11	s.	s.	PROPN
cana-4037	242	12	,	,	PUNCT
cana-4037	242	13	&	&	CCONJ
cana-4037	242	14	selvam	selvam	PROPN
cana-4037	242	15	,	,	PUNCT
cana-4037	242	16	g.	g.	PROPN
cana-4037	242	17	(	(	PUNCT
cana-4037	242	18	2016	2016	NUM
cana-4037	242	19	)	)	PUNCT
cana-4037	242	20	.	.	PUNCT
cana-4037	243	1	non	non	ADJ
cana-4037	243	2	-	-	ADJ
cana-4037	243	3	split	split	ADJ
cana-4037	243	4	two	two	NUM
cana-4037	243	5	out	out	ADJ
cana-4037	243	6	-	-	PUNCT
cana-4037	243	7	degree	degree	NOUN
cana-4037	243	8	equitable	equitable	ADJ
cana-4037	243	9	domination	domination	NOUN
cana-4037	243	10	number	number	NOUN
cana-4037	243	11	in	in	ADP
cana-4037	243	12	graphs	graph	NOUN
cana-4037	243	13	.	.	PUNCT
cana-4037	244	1	journal	journal	NOUN
cana-4037	244	2	of	of	ADP
cana-4037	244	3	chemical	chemical	PROPN
cana-4037	244	4	and	and	CCONJ
cana-4037	244	5	pharmaceutical	pharmaceutical	NOUN
cana-4037	244	6	sciences	science	NOUN
cana-4037	244	7	,	,	PUNCT
cana-4037	244	8	9(4	9(4	NOUN
cana-4037	244	9	)	)	PUNCT
cana-4037	244	10	,	,	PUNCT
cana-4037	244	11	22662270	22662270	NUM
cana-4037	244	12	.	.	PUNCT
cana-4037	245	1	10	10	NUM
cana-4037	245	2	.	.	X
cana-4037	245	3	ore	ore	NOUN
cana-4037	245	4	,	,	PUNCT
cana-4037	245	5	o.	o.	PROPN
cana-4037	245	6	(	(	PUNCT
cana-4037	245	7	1962	1962	NUM
cana-4037	245	8	)	)	PUNCT
cana-4037	245	9	.	.	PUNCT
cana-4037	246	1	theory	theory	NOUN
cana-4037	246	2	of	of	ADP
cana-4037	246	3	graphs	graph	NOUN
cana-4037	246	4	.	.	PUNCT
cana-4037	247	1	american	american	PROPN
cana-4037	247	2	mathematical	mathematical	PROPN
cana-4037	247	3	society	society	NOUN
cana-4037	247	4	colloquium	colloquium	NOUN
cana-4037	247	5	publications	publication	NOUN
cana-4037	247	6	,	,	PUNCT
cana-4037	247	7	38	38	NUM
cana-4037	247	8	.	.	PUNCT
cana-4037	248	1	providence	providence	NOUN
cana-4037	248	2	,	,	PUNCT
cana-4037	248	3	ri	ri	PROPN
cana-4037	248	4	.	.	PROPN
cana-4037	248	5	11	11	NUM
cana-4037	248	6	.	.	X
cana-4037	249	1	sahal	sahal	ADJ
cana-4037	249	2	,	,	PUNCT
cana-4037	249	3	a.	a.	NOUN
cana-4037	249	4	,	,	PUNCT
cana-4037	249	5	&	&	CCONJ
cana-4037	249	6	mathad	mathad	VERB
cana-4037	249	7	,	,	PUNCT
cana-4037	249	8	v.	v.	PROPN
cana-4037	249	9	(	(	PUNCT
cana-4037	249	10	2013	2013	NUM
cana-4037	249	11	)	)	PUNCT
cana-4037	249	12	.	.	PUNCT
cana-4037	250	1	two	two	NUM
cana-4037	250	2	-	-	PUNCT
cana-4037	250	3	out	out	ADP
cana-4037	250	4	degree	degree	NOUN
cana-4037	250	5	equitable	equitable	ADJ
cana-4037	250	6	domination	domination	NOUN
cana-4037	250	7	in	in	ADP
cana-4037	250	8	graphs	graph	NOUN
cana-4037	250	9	.	.	PUNCT
cana-4037	251	1	transactions	transaction	NOUN
cana-4037	251	2	on	on	ADP
cana-4037	251	3	combinatorics	combinatoric	NOUN
cana-4037	251	4	,	,	PUNCT
cana-4037	251	5	2(3	2(3	NUM
cana-4037	251	6	)	)	PUNCT
cana-4037	251	7	,	,	PUNCT
cana-4037	251	8	13	13	NUM
cana-4037	251	9	-	-	SYM
cana-4037	251	10	19	19	NUM
cana-4037	251	11	.	.	PUNCT