id	sid	tid	token	lemma	pos
cana-2085	1	1	communications	communication	NOUN
cana-2085	1	2	on	on	ADP
cana-2085	1	3	applied	apply	VERB
cana-2085	1	4	nonlinear	nonlinear	ADJ
cana-2085	1	5	analysis	analysis	NOUN
cana-2085	1	6	issn	issn	NOUN
cana-2085	1	7	:	:	PUNCT
cana-2085	1	8	1074	1074	NUM
cana-2085	1	9	-	-	PUNCT
cana-2085	1	10	133x	133x	NUM
cana-2085	1	11	vol	vol	NOUN
cana-2085	1	12	32	32	NUM
cana-2085	1	13	no	no	NOUN
cana-2085	1	14	.	.	PUNCT
cana-2085	2	1	1s	1s	NUM
cana-2085	2	2	(	(	PUNCT
cana-2085	2	3	2025	2025	NUM
cana-2085	2	4	)	)	PUNCT
cana-2085	2	5	17	17	NUM
cana-2085	2	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-2085	2	7	enhancing	enhance	VERB
cana-2085	2	8	network	network	NOUN
cana-2085	2	9	security	security	NOUN
cana-2085	2	10	in	in	ADP
cana-2085	2	11	distributed	distribute	VERB
cana-2085	2	12	systems	system	NOUN
cana-2085	2	13	using	use	VERB
cana-2085	2	14	middle	middle	ADJ
cana-2085	2	15	roman	roman	ADJ
cana-2085	2	16	dominating	dominating	NOUN
cana-2085	2	17	functions	function	NOUN
cana-2085	2	18	r.	r.	PROPN
cana-2085	2	19	vinodhini1	vinodhini1	PROPN
cana-2085	2	20	,	,	PUNCT
cana-2085	2	21	t.	t.	PROPN
cana-2085	2	22	n.	n.	PROPN
cana-2085	2	23	m.	m.	PROPN
cana-2085	2	24	malini	malini	PROPN
cana-2085	2	25	mai2	mai2	PROPN
cana-2085	2	26	1,2	1,2	NUM
cana-2085	2	27	simats	simat	NOUN
cana-2085	2	28	school	school	NOUN
cana-2085	2	29	of	of	ADP
cana-2085	2	30	engineering	engineering	NOUN
cana-2085	2	31	,	,	PUNCT
cana-2085	2	32	chennai	chennai	PROPN
cana-2085	2	33	–	–	PUNCT
cana-2085	2	34	600	600	NUM
cana-2085	2	35	097	097	NUM
cana-2085	2	36	,	,	PUNCT
cana-2085	2	37	tamil	tamil	PROPN
cana-2085	2	38	,	,	PUNCT
cana-2085	2	39	india	india	PROPN
cana-2085	2	40	vinodhinir41034.sse@saveetha.com1	vinodhinir41034.sse@saveetha.com1	PROPN
cana-2085	2	41	,	,	PUNCT
cana-2085	2	42	malinimait.sse@saveetha.com2	malinimait.sse@saveetha.com2	NOUN
cana-2085	2	43	article	article	NOUN
cana-2085	2	44	history	history	NOUN
cana-2085	2	45	:	:	PUNCT
cana-2085	2	46	received	receive	VERB
cana-2085	2	47	:	:	PUNCT
cana-2085	2	48	05	05	NUM
cana-2085	2	49	-	-	SYM
cana-2085	2	50	08	08	NUM
cana-2085	2	51	-	-	PUNCT
cana-2085	2	52	2024	2024	NUM
cana-2085	2	53	revised	revise	VERB
cana-2085	2	54	:	:	PUNCT
cana-2085	2	55	25	25	NUM
cana-2085	2	56	-	-	PUNCT
cana-2085	2	57	09	09	NUM
cana-2085	2	58	-	-	PUNCT
cana-2085	2	59	2024	2024	NUM
cana-2085	2	60	accepted	accept	VERB
cana-2085	2	61	:	:	PUNCT
cana-2085	2	62	07	07	NUM
cana-2085	2	63	-	-	SYM
cana-2085	2	64	10	10	NUM
cana-2085	2	65	-	-	PUNCT
cana-2085	2	66	2024	2024	NUM
cana-2085	2	67	abstract	abstract	NOUN
cana-2085	2	68	:	:	PUNCT
cana-2085	2	69	a	a	DET
cana-2085	2	70	middle	middle	ADJ
cana-2085	2	71	roman	roman	ADJ
cana-2085	2	72	dominating	dominating	NOUN
cana-2085	2	73	function	function	NOUN
cana-2085	2	74	(	(	PUNCT
cana-2085	2	75	mrdn	mrdn	NOUN
cana-2085	2	76	)	)	PUNCT
cana-2085	2	77	on	on	ADP
cana-2085	2	78	a	a	DET
cana-2085	2	79	graph	graph	NOUN
cana-2085	2	80	g	g	NOUN
cana-2085	2	81	=	=	PUNCT
cana-2085	2	82	(	(	PUNCT
cana-2085	2	83	v	v	NOUN
cana-2085	2	84	,	,	PUNCT
cana-2085	2	85	e	e	NOUN
cana-2085	2	86	)	)	PUNCT
cana-2085	2	87	is	be	AUX
cana-2085	2	88	a	a	DET
cana-2085	2	89	function	function	NOUN
cana-2085	2	90	f	f	NOUN
cana-2085	2	91	:	:	PUNCT
cana-2085	2	92	v→{0,1,2,3	v→{0,1,2,3	X
cana-2085	2	93	}	}	PUNCT
cana-2085	2	94	satisfying	satisfy	VERB
cana-2085	2	95	the	the	DET
cana-2085	2	96	condition	condition	NOUN
cana-2085	2	97	that	that	SCONJ
cana-2085	2	98	every	every	DET
cana-2085	2	99	vertex	vertex	NOUN
cana-2085	2	100	u	u	NOUN
cana-2085	2	101	with	with	ADP
cana-2085	2	102	f(u)=0	f(u)=0	NOUN
cana-2085	2	103	is	be	AUX
cana-2085	2	104	adjacent	adjacent	ADJ
cana-2085	2	105	to	to	ADP
cana-2085	2	106	at	at	ADP
cana-2085	2	107	most	most	ADV
cana-2085	2	108	one	one	NUM
cana-2085	2	109	vertex	vertex	NOUN
cana-2085	2	110	v	v	NOUN
cana-2085	2	111	with	with	ADP
cana-2085	2	112	f(v)=2	f(v)=2	NUM
cana-2085	2	113	or	or	CCONJ
cana-2085	2	114	3	3	NUM
cana-2085	2	115	.	.	PUNCT
cana-2085	2	116	further	far	ADV
cana-2085	2	117	if	if	SCONJ
cana-2085	2	118	a	a	DET
cana-2085	2	119	vertex	vertex	NOUN
cana-2085	2	120	is	be	AUX
cana-2085	2	121	assigned	assign	VERB
cana-2085	2	122	2	2	NUM
cana-2085	2	123	,	,	PUNCT
cana-2085	2	124	then	then	ADV
cana-2085	2	125	at	at	ADP
cana-2085	2	126	most	most	ADV
cana-2085	2	127	two	two	NUM
cana-2085	2	128	of	of	ADP
cana-2085	2	129	its	its	PRON
cana-2085	2	130	vertices	vertex	NOUN
cana-2085	2	131	can	can	AUX
cana-2085	2	132	be	be	AUX
cana-2085	2	133	assigned	assign	VERB
cana-2085	2	134	0	0	PUNCT
cana-2085	3	1	and	and	CCONJ
cana-2085	3	2	if	if	SCONJ
cana-2085	3	3	a	a	DET
cana-2085	3	4	vertex	vertex	NOUN
cana-2085	3	5	is	be	AUX
cana-2085	3	6	assigned	assign	VERB
cana-2085	3	7	3	3	NUM
cana-2085	3	8	,	,	PUNCT
cana-2085	3	9	then	then	ADV
cana-2085	3	10	all	all	DET
cana-2085	3	11	its	its	PRON
cana-2085	3	12	neighbours	neighbour	NOUN
cana-2085	3	13	can	can	AUX
cana-2085	3	14	be	be	AUX
cana-2085	3	15	assigned	assign	VERB
cana-2085	3	16	0	0	NUM
cana-2085	3	17	.	.	PUNCT
cana-2085	4	1	the	the	DET
cana-2085	4	2	weight	weight	NOUN
cana-2085	4	3	of	of	ADP
cana-2085	4	4	a	a	DET
cana-2085	4	5	mrdf	mrdf	NOUN
cana-2085	4	6	is	be	AUX
cana-2085	4	7	the	the	DET
cana-2085	4	8	value	value	NOUN
cana-2085	4	9	f(v(g))=∑_uϵv	f(v(g))=∑_uϵv	NOUN
cana-2085	4	10	▒	▒	PROPN
cana-2085	4	11	〖f(u〗	〖f(u〗	NOUN
cana-2085	4	12	)	)	PUNCT
cana-2085	4	13	.	.	PUNCT
cana-2085	5	1	the	the	DET
cana-2085	5	2	middle	middle	ADJ
cana-2085	5	3	roman	roman	ADJ
cana-2085	5	4	domination	domination	NOUN
cana-2085	5	5	number	number	NOUN
cana-2085	5	6	γ_mr	γ_mr	NOUN
cana-2085	5	7	(	(	PUNCT
cana-2085	5	8	g	g	NOUN
cana-2085	5	9	)	)	PUNCT
cana-2085	5	10	is	be	AUX
cana-2085	5	11	the	the	DET
cana-2085	5	12	minimum	minimum	ADJ
cana-2085	5	13	weight	weight	NOUN
cana-2085	5	14	of	of	ADP
cana-2085	5	15	a	a	DET
cana-2085	5	16	mrdf	mrdf	NOUN
cana-2085	5	17	on	on	ADP
cana-2085	5	18	g.	g.	PROPN
cana-2085	5	19	in	in	ADP
cana-2085	5	20	this	this	DET
cana-2085	5	21	paper	paper	NOUN
cana-2085	5	22	,	,	PUNCT
cana-2085	5	23	we	we	PRON
cana-2085	5	24	introduce	introduce	VERB
cana-2085	5	25	middle	middle	ADJ
cana-2085	5	26	roman	roman	ADJ
cana-2085	5	27	domination	domination	NOUN
cana-2085	5	28	number	number	NOUN
cana-2085	5	29	denoted	denote	VERB
cana-2085	5	30	as	as	ADP
cana-2085	5	31	γ_mr	γ_mr	PROPN
cana-2085	5	32	(	(	PUNCT
cana-2085	5	33	g	g	NOUN
cana-2085	5	34	)	)	PUNCT
cana-2085	5	35	,	,	PUNCT
cana-2085	5	36	study	study	VERB
cana-2085	5	37	the	the	DET
cana-2085	5	38	properties	property	NOUN
cana-2085	5	39	of	of	ADP
cana-2085	5	40	the	the	DET
cana-2085	5	41	function	function	NOUN
cana-2085	5	42	,	,	PUNCT
cana-2085	5	43	present	present	VERB
cana-2085	5	44	some	some	DET
cana-2085	5	45	characterization	characterization	NOUN
cana-2085	5	46	and	and	CCONJ
cana-2085	5	47	determine	determine	VERB
cana-2085	5	48	γ_mr	γ_mr	PROPN
cana-2085	5	49	(	(	PUNCT
cana-2085	5	50	g)-value	g)-value	NOUN
cana-2085	5	51	for	for	ADP
cana-2085	5	52	some	some	DET
cana-2085	5	53	graphs	graph	NOUN
cana-2085	5	54	.	.	PUNCT
cana-2085	6	1	middle	middle	ADJ
cana-2085	6	2	roman	roman	ADJ
cana-2085	6	3	dominating	dominating	NOUN
cana-2085	6	4	functions	function	NOUN
cana-2085	6	5	(	(	PUNCT
cana-2085	6	6	mrdfs	mrdfs	ADJ
cana-2085	6	7	)	)	PUNCT
cana-2085	6	8	present	present	VERB
cana-2085	6	9	a	a	DET
cana-2085	6	10	unique	unique	ADJ
cana-2085	6	11	approach	approach	NOUN
cana-2085	6	12	within	within	ADP
cana-2085	6	13	graph	graph	NOUN
cana-2085	6	14	theory	theory	NOUN
cana-2085	6	15	,	,	PUNCT
cana-2085	6	16	with	with	ADP
cana-2085	6	17	significant	significant	ADJ
cana-2085	6	18	implications	implication	NOUN
cana-2085	6	19	for	for	ADP
cana-2085	6	20	various	various	ADJ
cana-2085	6	21	fields	field	NOUN
cana-2085	6	22	in	in	ADP
cana-2085	6	23	computer	computer	NOUN
cana-2085	6	24	science	science	NOUN
cana-2085	6	25	.	.	PUNCT
cana-2085	7	1	by	by	ADP
cana-2085	7	2	assigning	assign	VERB
cana-2085	7	3	values	value	NOUN
cana-2085	7	4	to	to	PART
cana-2085	7	5	vertices	vertex	NOUN
cana-2085	7	6	under	under	ADP
cana-2085	7	7	specific	specific	ADJ
cana-2085	7	8	constraints	constraint	NOUN
cana-2085	7	9	,	,	PUNCT
cana-2085	7	10	mrdfs	mrdfs	ADV
cana-2085	7	11	enable	enable	VERB
cana-2085	7	12	the	the	DET
cana-2085	7	13	optimization	optimization	NOUN
cana-2085	7	14	of	of	ADP
cana-2085	7	15	resources	resource	NOUN
cana-2085	7	16	,	,	PUNCT
cana-2085	7	17	enhancing	enhance	VERB
cana-2085	7	18	network	network	NOUN
cana-2085	7	19	security	security	NOUN
cana-2085	7	20	,	,	PUNCT
cana-2085	7	21	load	load	NOUN
cana-2085	7	22	balancing	balancing	NOUN
cana-2085	7	23	,	,	PUNCT
cana-2085	7	24	and	and	CCONJ
cana-2085	7	25	energy	energy	NOUN
cana-2085	7	26	efficiency	efficiency	NOUN
cana-2085	7	27	in	in	ADP
cana-2085	7	28	distributed	distribute	VERB
cana-2085	7	29	systems	system	NOUN
cana-2085	7	30	.	.	PUNCT
cana-2085	8	1	this	this	DET
cana-2085	8	2	paper	paper	NOUN
cana-2085	8	3	explores	explore	VERB
cana-2085	8	4	the	the	DET
cana-2085	8	5	application	application	NOUN
cana-2085	8	6	of	of	ADP
cana-2085	8	7	mrdfs	mrdfs	ADJ
cana-2085	8	8	in	in	ADP
cana-2085	8	9	scenarios	scenario	NOUN
cana-2085	8	10	such	such	ADJ
cana-2085	8	11	as	as	ADP
cana-2085	8	12	intrusion	intrusion	NOUN
cana-2085	8	13	detection	detection	NOUN
cana-2085	8	14	,	,	PUNCT
cana-2085	8	15	resilient	resilient	ADJ
cana-2085	8	16	network	network	NOUN
cana-2085	8	17	design	design	NOUN
cana-2085	8	18	,	,	PUNCT
cana-2085	8	19	task	task	NOUN
cana-2085	8	20	scheduling	scheduling	NOUN
cana-2085	8	21	,	,	PUNCT
cana-2085	8	22	and	and	CCONJ
cana-2085	8	23	sensor	sensor	NOUN
cana-2085	8	24	activation	activation	NOUN
cana-2085	8	25	.	.	PUNCT
cana-2085	9	1	by	by	ADP
cana-2085	9	2	minimizing	minimize	VERB
cana-2085	9	3	the	the	DET
cana-2085	9	4	overall	overall	ADJ
cana-2085	9	5	weight	weight	NOUN
cana-2085	9	6	while	while	SCONJ
cana-2085	9	7	maintaining	maintain	VERB
cana-2085	9	8	functional	functional	ADJ
cana-2085	9	9	requirements	requirement	NOUN
cana-2085	9	10	,	,	PUNCT
cana-2085	9	11	mrdfs	mrdfs	ADJ
cana-2085	9	12	provide	provide	VERB
cana-2085	9	13	an	an	DET
cana-2085	9	14	effective	effective	ADJ
cana-2085	9	15	strategy	strategy	NOUN
cana-2085	9	16	for	for	ADP
cana-2085	9	17	addressing	address	VERB
cana-2085	9	18	challenges	challenge	NOUN
cana-2085	9	19	in	in	ADP
cana-2085	9	20	network	network	NOUN
cana-2085	9	21	topology	topology	NOUN
cana-2085	9	22	,	,	PUNCT
cana-2085	9	23	resource	resource	NOUN
cana-2085	9	24	allocation	allocation	NOUN
cana-2085	9	25	,	,	PUNCT
cana-2085	9	26	and	and	CCONJ
cana-2085	9	27	fault	fault	VERB
cana-2085	9	28	tolerance	tolerance	NOUN
cana-2085	9	29	.	.	PUNCT
cana-2085	10	1	the	the	DET
cana-2085	10	2	versatility	versatility	NOUN
cana-2085	10	3	of	of	ADP
cana-2085	10	4	mrdfs	mrdfs	ADJ
cana-2085	10	5	makes	make	VERB
cana-2085	10	6	them	they	PRON
cana-2085	10	7	a	a	DET
cana-2085	10	8	valuable	valuable	ADJ
cana-2085	10	9	tool	tool	NOUN
cana-2085	10	10	in	in	ADP
cana-2085	10	11	the	the	DET
cana-2085	10	12	development	development	NOUN
cana-2085	10	13	of	of	ADP
cana-2085	10	14	robust	robust	ADJ
cana-2085	10	15	and	and	CCONJ
cana-2085	10	16	efficient	efficient	ADJ
cana-2085	10	17	computing	computing	NOUN
cana-2085	10	18	systems	system	NOUN
cana-2085	10	19	.	.	PUNCT
cana-2085	11	1	keywords	keyword	NOUN
cana-2085	11	2	:	:	PUNCT
cana-2085	11	3	dominating	dominate	VERB
cana-2085	11	4	function	function	NOUN
cana-2085	11	5	,	,	PUNCT
cana-2085	11	6	roman	roman	ADJ
cana-2085	11	7	dominating	dominating	NOUN
cana-2085	11	8	function	function	NOUN
cana-2085	11	9	,	,	PUNCT
cana-2085	11	10	weak	weak	ADJ
cana-2085	11	11	roman	roman	ADJ
cana-2085	11	12	dominating	dominating	NOUN
cana-2085	11	13	function	function	NOUN
cana-2085	11	14	,	,	PUNCT
cana-2085	11	15	middle	middle	ADJ
cana-2085	11	16	roman	roman	ADJ
cana-2085	11	17	dominating	dominating	NOUN
cana-2085	11	18	function	function	NOUN
cana-2085	11	19	,	,	PUNCT
cana-2085	11	20	resource	resource	NOUN
cana-2085	11	21	allocation	allocation	NOUN
cana-2085	11	22	,	,	PUNCT
cana-2085	11	23	optimization	optimization	NOUN
cana-2085	11	24	,	,	PUNCT
cana-2085	11	25	network	network	NOUN
cana-2085	11	26	security	security	NOUN
cana-2085	11	27	.	.	PUNCT
cana-2085	12	1	mathematical	mathematical	ADJ
cana-2085	12	2	subject	subject	ADJ
cana-2085	12	3	classification	classification	NOUN
cana-2085	12	4	:	:	PUNCT
cana-2085	12	5	05c69	05c69	NUM
cana-2085	12	6	.	.	X
cana-2085	13	1	1	1	X
cana-2085	13	2	.	.	X
cana-2085	13	3	introduction	introduction	NOUN
cana-2085	13	4	a	a	DET
cana-2085	13	5	set	set	ADJ
cana-2085	13	6	𝐷	𝐷	NOUN
cana-2085	13	7	of	of	ADP
cana-2085	13	8	vertices	vertex	NOUN
cana-2085	13	9	in	in	ADP
cana-2085	13	10	a	a	DET
cana-2085	13	11	graph	graph	NOUN
cana-2085	13	12	g	g	NOUN
cana-2085	13	13	is	be	AUX
cana-2085	13	14	a	a	DET
cana-2085	13	15	dominating	dominating	NOUN
cana-2085	13	16	set	set	NOUN
cana-2085	13	17	if	if	SCONJ
cana-2085	13	18	every	every	DET
cana-2085	13	19	vertex	vertex	NOUN
cana-2085	13	20	in	in	ADP
cana-2085	13	21	𝑉	𝑉	PROPN
cana-2085	13	22	−	−	PROPN
cana-2085	13	23	𝐷	𝐷	PROPN
cana-2085	13	24	is	be	AUX
cana-2085	13	25	adjacent	adjacent	ADJ
cana-2085	13	26	to	to	ADP
cana-2085	13	27	some	some	DET
cana-2085	13	28	vertex	vertex	NOUN
cana-2085	13	29	in	in	ADP
cana-2085	13	30	𝐷.	𝐷.	PROPN
cana-2085	13	31	the	the	DET
cana-2085	13	32	domination	domination	NOUN
cana-2085	13	33	number	number	NOUN
cana-2085	13	34	𝛾(𝐺	𝛾(𝐺	PROPN
cana-2085	13	35	)	)	PUNCT
cana-2085	13	36	is	be	AUX
cana-2085	13	37	the	the	DET
cana-2085	13	38	minimum	minimum	ADJ
cana-2085	13	39	cardinality	cardinality	NOUN
cana-2085	13	40	of	of	ADP
cana-2085	13	41	the	the	DET
cana-2085	13	42	dominating	dominating	NOUN
cana-2085	13	43	set	set	NOUN
cana-2085	13	44	of	of	ADP
cana-2085	13	45	𝐺	𝐺	PROPN
cana-2085	13	46	.	.	PUNCT
cana-2085	14	1	in	in	ADP
cana-2085	14	2	2004	2004	NUM
cana-2085	14	3	,	,	PUNCT
cana-2085	14	4	cockayne	cockayne	NOUN
cana-2085	14	5	et	et	PROPN
cana-2085	14	6	al	al	PROPN
cana-2085	14	7	.	.	PROPN
cana-2085	14	8	published	publish	VERB
cana-2085	14	9	roman	roman	ADJ
cana-2085	14	10	domination	domination	NOUN
cana-2085	14	11	in	in	ADP
cana-2085	14	12	graphs	graph	NOUN
cana-2085	14	13	;	;	PUNCT
cana-2085	14	14	as	as	ADP
cana-2085	14	15	a	a	DET
cana-2085	14	16	result	result	NOUN
cana-2085	14	17	,	,	PUNCT
cana-2085	14	18	numerous	numerous	ADJ
cana-2085	14	19	roman	roman	ADJ
cana-2085	14	20	domination	domination	NOUN
cana-2085	14	21	parameters	parameter	NOUN
cana-2085	14	22	were	be	AUX
cana-2085	14	23	introduced	introduce	VERB
cana-2085	14	24	[	[	PUNCT
cana-2085	14	25	2	2	NUM
cana-2085	14	26	,	,	PUNCT
cana-2085	14	27	16	16	NUM
cana-2085	14	28	,	,	PUNCT
cana-2085	14	29	17	17	NUM
cana-2085	14	30	]	]	PUNCT
cana-2085	14	31	.	.	PUNCT
cana-2085	15	1	from	from	ADP
cana-2085	15	2	then	then	ADV
cana-2085	15	3	enormous	enormous	ADJ
cana-2085	15	4	work	work	NOUN
cana-2085	15	5	has	have	AUX
cana-2085	15	6	been	be	AUX
cana-2085	15	7	done	do	VERB
cana-2085	15	8	in	in	ADP
cana-2085	15	9	roman	roman	ADJ
cana-2085	15	10	domination	domination	NOUN
cana-2085	15	11	.	.	PUNCT
cana-2085	16	1	consider	consider	VERB
cana-2085	16	2	a	a	DET
cana-2085	16	3	𝐺	𝐺	NOUN
cana-2085	16	4	=	=	SYM
cana-2085	16	5	(	(	PUNCT
cana-2085	16	6	𝑉	𝑉	PROPN
cana-2085	16	7	,	,	PUNCT
cana-2085	16	8	𝐸	𝐸	PROPN
cana-2085	16	9	)	)	PUNCT
cana-2085	16	10	and	and	CCONJ
cana-2085	16	11	define	define	VERB
cana-2085	16	12	a	a	DET
cana-2085	16	13	function	function	NOUN
cana-2085	16	14	𝑓	𝑓	NOUN
cana-2085	16	15	:	:	PUNCT
cana-2085	16	16	𝑉	𝑉	PROPN
cana-2085	16	17	→	→	SYM
cana-2085	16	18	{	{	PUNCT
cana-2085	16	19	0	0	NUM
cana-2085	16	20	,	,	PUNCT
cana-2085	16	21	1	1	NUM
cana-2085	16	22	,	,	PUNCT
cana-2085	16	23	2	2	NUM
cana-2085	16	24	}	}	PUNCT
cana-2085	16	25	.	.	PUNCT
cana-2085	17	1	unguarded	unguarde	VERB
cana-2085	17	2	with	with	ADP
cana-2085	17	3	regard	regard	NOUN
cana-2085	17	4	to	to	ADP
cana-2085	17	5	f	f	PROPN
cana-2085	17	6	is	be	AUX
cana-2085	17	7	defined	define	VERB
cana-2085	17	8	as	as	ADP
cana-2085	17	9	a	a	DET
cana-2085	17	10	vertex	vertex	NOUN
cana-2085	17	11	u	u	NOUN
cana-2085	17	12	with	with	ADP
cana-2085	17	13	𝑓(𝑢	𝑓(𝑢	PROPN
cana-2085	17	14	)	)	PUNCT
cana-2085	18	1	=	=	SYM
cana-2085	18	2	0	0	PROPN
cana-2085	19	1	that	that	PRON
cana-2085	19	2	is	be	AUX
cana-2085	19	3	not	not	PART
cana-2085	19	4	next	next	ADJ
cana-2085	19	5	to	to	ADP
cana-2085	19	6	a	a	DET
cana-2085	19	7	vertex	vertex	NOUN
cana-2085	19	8	with	with	ADP
cana-2085	19	9	1	1	NUM
cana-2085	19	10	or	or	CCONJ
cana-2085	19	11	2	2	NUM
cana-2085	19	12	.	.	PUNCT
cana-2085	20	1	the	the	DET
cana-2085	20	2	function	function	NOUN
cana-2085	20	3	𝑓	𝑓	PROPN
cana-2085	20	4	:	:	PUNCT
cana-2085	20	5	𝑉	𝑉	PROPN
cana-2085	20	6	→	→	SYM
cana-2085	20	7	{	{	PUNCT
cana-2085	20	8	0	0	NUM
cana-2085	20	9	,	,	PUNCT
cana-2085	20	10	1	1	NUM
cana-2085	20	11	,	,	PUNCT
cana-2085	20	12	2	2	NUM
cana-2085	20	13	}	}	PUNCT
cana-2085	20	14	satisfying	satisfy	VERB
cana-2085	20	15	the	the	DET
cana-2085	20	16	condition	condition	NOUN
cana-2085	20	17	that	that	SCONJ
cana-2085	20	18	each	each	DET
cana-2085	20	19	vertex	vertex	NOUN
cana-2085	20	20	u	u	NOUN
cana-2085	20	21	for	for	ADP
cana-2085	20	22	which	which	PRON
cana-2085	20	23	𝑓(𝑢	𝑓(𝑢	ADV
cana-2085	20	24	)	)	PUNCT
cana-2085	20	25	=	=	SYM
cana-2085	20	26	0	0	NUM
cana-2085	20	27	is	be	AUX
cana-2085	20	28	adjacent	adjacent	ADJ
cana-2085	20	29	to	to	ADP
cana-2085	20	30	at	at	ADV
cana-2085	20	31	least	least	ADV
cana-2085	20	32	one	one	NUM
cana-2085	20	33	vertex	vertex	NOUN
cana-2085	20	34	v	v	NOUN
cana-2085	20	35	for	for	ADP
cana-2085	20	36	which	which	PRON
cana-2085	20	37	𝑓(𝑣	𝑓(𝑣	NOUN
cana-2085	20	38	)	)	PUNCT
cana-2085	20	39	=	=	SYM
cana-2085	20	40	2	2	NUM
cana-2085	20	41	,	,	PUNCT
cana-2085	20	42	is	be	AUX
cana-2085	20	43	referred	refer	VERB
cana-2085	20	44	to	to	PART
cana-2085	20	45	be	be	AUX
cana-2085	20	46	a	a	DET
cana-2085	20	47	roman	roman	ADJ
cana-2085	20	48	dominant	dominant	ADJ
cana-2085	20	49	function	function	NOUN
cana-2085	20	50	,	,	PUNCT
cana-2085	20	51	known	know	VERB
cana-2085	20	52	as	as	ADP
cana-2085	20	53	rdf	rdf	NOUN
cana-2085	20	54	of	of	ADP
cana-2085	20	55	a	a	DET
cana-2085	20	56	graph	graph	NOUN
cana-2085	20	57	𝐺	𝐺	NOUN
cana-2085	20	58	=	=	SYM
cana-2085	20	59	(	(	PUNCT
cana-2085	20	60	𝑉	𝑉	PROPN
cana-2085	20	61	,	,	PUNCT
cana-2085	20	62	𝐸	𝐸	PROPN
cana-2085	20	63	)	)	PUNCT
cana-2085	20	64	.	.	PUNCT
cana-2085	21	1	roman	roman	ADJ
cana-2085	21	2	domination	domination	NOUN
cana-2085	21	3	number	number	NOUN
cana-2085	21	4	(	(	PUNCT
cana-2085	21	5	rdn	rdn	NOUN
cana-2085	21	6	)	)	PUNCT
cana-2085	21	7	of	of	ADP
cana-2085	21	8	g	g	NOUN
cana-2085	21	9	,	,	PUNCT
cana-2085	21	10	which	which	PRON
cana-2085	21	11	is	be	AUX
cana-2085	21	12	represented	represent	VERB
cana-2085	21	13	by	by	ADP
cana-2085	21	14	𝛾𝑅(𝐺	𝛾𝑅(𝐺	NOUN
cana-2085	21	15	)	)	PUNCT
cana-2085	21	16	,	,	PUNCT
cana-2085	21	17	is	be	AUX
cana-2085	21	18	the	the	DET
cana-2085	21	19	bare	bare	ADJ
cana-2085	21	20	minimum	minimum	ADJ
cana-2085	21	21	number	number	NOUN
cana-2085	21	22	of	of	ADP
cana-2085	21	23	guards	guard	NOUN
cana-2085	21	24	that	that	PRON
cana-2085	21	25	must	must	AUX
cana-2085	21	26	be	be	AUX
cana-2085	21	27	employed	employ	VERB
cana-2085	21	28	in	in	ADP
cana-2085	21	29	any	any	DET
cana-2085	21	30	rdf	rdf	NOUN
cana-2085	21	31	.	.	PUNCT
cana-2085	22	1	let	let	VERB
cana-2085	22	2	𝐺	𝐺	PROPN
cana-2085	22	3	=	=	SYM
cana-2085	22	4	(	(	PUNCT
cana-2085	22	5	𝑉	𝑉	PROPN
cana-2085	22	6	,	,	PUNCT
cana-2085	22	7	𝐸	𝐸	PROPN
cana-2085	22	8	)	)	PUNCT
cana-2085	22	9	be	be	VERB
cana-2085	22	10	a	a	DET
cana-2085	22	11	graph	graph	NOUN
cana-2085	22	12	and	and	CCONJ
cana-2085	22	13	𝑓	𝑓	PRON
cana-2085	22	14	be	be	AUX
cana-2085	22	15	a	a	DET
cana-2085	22	16	function	function	NOUN
cana-2085	22	17	𝑓	𝑓	NOUN
cana-2085	22	18	:	:	PUNCT
cana-2085	22	19	𝑉	𝑉	PROPN
cana-2085	22	20	→	→	SYM
cana-2085	22	21	{	{	PUNCT
cana-2085	22	22	0,1,2	0,1,2	NOUN
cana-2085	22	23	}	}	PUNCT
cana-2085	22	24	.	.	PUNCT
cana-2085	23	1	the	the	DET
cana-2085	23	2	function	function	NOUN
cana-2085	23	3	𝑓	𝑓	PROPN
cana-2085	23	4	is	be	AUX
cana-2085	23	5	a	a	DET
cana-2085	23	6	weak	weak	ADJ
cana-2085	23	7	roman	roman	ADJ
cana-2085	23	8	dominating	dominating	NOUN
cana-2085	23	9	function	function	NOUN
cana-2085	23	10	(	(	PUNCT
cana-2085	23	11	wrdf	wrdf	PROPN
cana-2085	23	12	)	)	PUNCT
cana-2085	23	13	if	if	SCONJ
cana-2085	23	14	each	each	DET
cana-2085	23	15	vertex	vertex	NOUN
cana-2085	23	16	mailto:vinodhinir41034.sse@saveetha.com	mailto:vinodhinir41034.sse@saveetha.com	NOUN
cana-2085	23	17	mailto:malinimait.sse@saveetha.com	mailto:malinimait.sse@saveetha.com	PART
cana-2085	23	18	communications	communication	NOUN
cana-2085	23	19	on	on	ADP
cana-2085	23	20	applied	apply	VERB
cana-2085	23	21	nonlinear	nonlinear	ADJ
cana-2085	23	22	analysis	analysis	NOUN
cana-2085	23	23	issn	issn	NOUN
cana-2085	23	24	:	:	PUNCT
cana-2085	23	25	1074	1074	NUM
cana-2085	23	26	-	-	PUNCT
cana-2085	23	27	133x	133x	NUM
cana-2085	23	28	vol	vol	NOUN
cana-2085	23	29	32	32	NUM
cana-2085	23	30	no	no	NOUN
cana-2085	23	31	.	.	PUNCT
cana-2085	24	1	1s	1s	NUM
cana-2085	24	2	(	(	PUNCT
cana-2085	24	3	2025	2025	NUM
cana-2085	24	4	)	)	PUNCT
cana-2085	24	5	18	18	NUM
cana-2085	24	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-2085	24	7	𝑢	𝑢	NOUN
cana-2085	24	8	with	with	ADP
cana-2085	24	9	𝑓(𝑢	𝑓(𝑢	PROPN
cana-2085	24	10	)	)	PUNCT
cana-2085	25	1	=	=	SYM
cana-2085	25	2	0	0	NUM
cana-2085	25	3	is	be	AUX
cana-2085	25	4	adjacent	adjacent	ADJ
cana-2085	25	5	to	to	ADP
cana-2085	25	6	a	a	DET
cana-2085	25	7	vertex	vertex	NOUN
cana-2085	25	8	𝑉	𝑉	PROPN
cana-2085	25	9	with	with	ADP
cana-2085	25	10	𝑓(𝑣	𝑓(𝑣	NOUN
cana-2085	25	11	)	)	PUNCT
cana-2085	25	12	>	>	X
cana-2085	25	13	0	0	NUM
cana-2085	26	1	such	such	ADJ
cana-2085	26	2	that	that	SCONJ
cana-2085	26	3	𝑓	𝑓	DET
cana-2085	26	4	′	′	NOUN
cana-2085	26	5	:	:	PUNCT
cana-2085	26	6	𝑉	𝑉	PROPN
cana-2085	26	7	→	→	SYM
cana-2085	26	8	{	{	PUNCT
cana-2085	26	9	0,1,2	0,1,2	NOUN
cana-2085	26	10	}	}	PUNCT
cana-2085	26	11	defined	define	VERB
cana-2085	26	12	by	by	ADP
cana-2085	26	13	𝑓′(𝑢	𝑓′(𝑢	NOUN
cana-2085	26	14	)	)	PUNCT
cana-2085	26	15	=	=	SYM
cana-2085	26	16	1	1	NUM
cana-2085	26	17	,	,	PUNCT
cana-2085	26	18	𝑓′(𝑣	𝑓′(𝑣	ADJ
cana-2085	26	19	)	)	PUNCT
cana-2085	26	20	=	=	SYM
cana-2085	26	21	𝑓(𝑣	𝑓(𝑣	NOUN
cana-2085	26	22	)	)	PUNCT
cana-2085	26	23	−	−	PROPN
cana-2085	26	24	1	1	NUM
cana-2085	26	25	and	and	CCONJ
cana-2085	26	26	𝑓′(𝑤	𝑓′(𝑤	PRON
cana-2085	26	27	)	)	PUNCT
cana-2085	26	28	=	=	SYM
cana-2085	26	29	𝑓(𝑤	𝑓(𝑤	PROPN
cana-2085	26	30	)	)	PUNCT
cana-2085	26	31	if	if	SCONJ
cana-2085	26	32	𝑤𝜖	𝑤𝜖	ADP
cana-2085	26	33	𝑉	𝑉	PROPN
cana-2085	26	34	−	−	PUNCT
cana-2085	26	35	𝑢	𝑢	NOUN
cana-2085	26	36	,	,	PUNCT
cana-2085	26	37	𝑣	𝑣	PRON
cana-2085	26	38	has	have	VERB
cana-2085	26	39	no	no	DET
cana-2085	26	40	undefined	undefined	ADJ
cana-2085	26	41	vertex	vertex	NOUN
cana-2085	26	42	.	.	PUNCT
cana-2085	27	1	the	the	DET
cana-2085	27	2	weight	weight	NOUN
cana-2085	27	3	of	of	ADP
cana-2085	27	4	𝑓	𝑓	PRON
cana-2085	27	5	is	be	AUX
cana-2085	27	6	w(𝑓	w(𝑓	NOUN
cana-2085	27	7	)	)	PUNCT
cana-2085	27	8	=	=	SYM
cana-2085	28	1	𝞢𝑣𝜖𝑉𝑓(𝑣	𝞢𝑣𝜖𝑉𝑓(𝑣	NOUN
cana-2085	28	2	)	)	PUNCT
cana-2085	28	3	the	the	DET
cana-2085	28	4	weak	weak	ADJ
cana-2085	28	5	roman	roman	ADJ
cana-2085	28	6	domination	domination	NOUN
cana-2085	28	7	number	number	NOUN
cana-2085	28	8	denoted	denote	VERB
cana-2085	28	9	by	by	ADP
cana-2085	28	10	𝛾𝑟(𝐺	𝛾𝑟(𝐺	PROPN
cana-2085	28	11	)	)	PUNCT
cana-2085	28	12	is	be	AUX
cana-2085	28	13	the	the	DET
cana-2085	28	14	minimum	minimum	ADJ
cana-2085	28	15	weight	weight	NOUN
cana-2085	28	16	of	of	ADP
cana-2085	28	17	wrdf	wrdf	NOUN
cana-2085	28	18	in	in	ADP
cana-2085	28	19	g	g	PROPN
cana-2085	29	1	[	[	X
cana-2085	29	2	6	6	NUM
cana-2085	29	3	,	,	PUNCT
cana-2085	29	4	16	16	NUM
cana-2085	29	5	,	,	PUNCT
cana-2085	29	6	17	17	NUM
cana-2085	29	7	,	,	PUNCT
cana-2085	29	8	18	18	NUM
cana-2085	29	9	,	,	PUNCT
cana-2085	29	10	28	28	NUM
cana-2085	29	11	]	]	PUNCT
cana-2085	29	12	.	.	PUNCT
cana-2085	30	1	a	a	DET
cana-2085	30	2	middle	middle	ADJ
cana-2085	30	3	roman	roman	ADJ
cana-2085	30	4	dominating	dominating	NOUN
cana-2085	30	5	function	function	NOUN
cana-2085	30	6	(	(	PUNCT
cana-2085	30	7	mrdn	mrdn	NOUN
cana-2085	30	8	)	)	PUNCT
cana-2085	30	9	on	on	ADP
cana-2085	30	10	a	a	DET
cana-2085	30	11	graph	graph	NOUN
cana-2085	30	12	𝐺	𝐺	NOUN
cana-2085	30	13	=	=	SYM
cana-2085	30	14	(	(	PUNCT
cana-2085	30	15	𝑉	𝑉	PROPN
cana-2085	30	16	,	,	PUNCT
cana-2085	30	17	𝐸	𝐸	PROPN
cana-2085	30	18	)	)	PUNCT
cana-2085	30	19	is	be	AUX
cana-2085	30	20	a	a	DET
cana-2085	30	21	function	function	NOUN
cana-2085	30	22	𝑓	𝑓	NOUN
cana-2085	30	23	:	:	PUNCT
cana-2085	30	24	𝑉(𝐺	𝑉(𝐺	NOUN
cana-2085	30	25	)	)	PUNCT
cana-2085	30	26	→	→	SYM
cana-2085	30	27	{	{	PUNCT
cana-2085	30	28	0	0	NUM
cana-2085	30	29	,	,	PUNCT
cana-2085	30	30	1	1	NUM
cana-2085	30	31	,	,	PUNCT
cana-2085	30	32	2	2	NUM
cana-2085	30	33	,	,	PUNCT
cana-2085	30	34	3	3	NUM
cana-2085	30	35	}	}	PUNCT
cana-2085	30	36	satisfying	satisfy	VERB
cana-2085	30	37	the	the	DET
cana-2085	30	38	condition	condition	NOUN
cana-2085	30	39	for	for	ADP
cana-2085	30	40	every	every	DET
cana-2085	30	41	vertex	vertex	NOUN
cana-2085	30	42	𝑢	𝑢	PROPN
cana-2085	30	43	∈	∈	PROPN
cana-2085	30	44	𝑉	𝑉	PROPN
cana-2085	30	45	,	,	PUNCT
cana-2085	30	46	𝑓(𝑢	𝑓(𝑢	PROPN
cana-2085	30	47	)	)	PUNCT
cana-2085	31	1	=	=	SYM
cana-2085	31	2	0	0	NUM
cana-2085	31	3	,	,	PUNCT
cana-2085	31	4	then	then	ADV
cana-2085	31	5	vertex	vertex	NOUN
cana-2085	31	6	𝑢	𝑢	PROPN
cana-2085	31	7	has	have	VERB
cana-2085	31	8	at	at	ADV
cana-2085	31	9	least	least	ADV
cana-2085	31	10	one	one	NUM
cana-2085	31	11	neighbor	neighbor	NOUN
cana-2085	31	12	𝑣	𝑣	X
cana-2085	31	13	with	with	ADP
cana-2085	31	14	𝑓(𝑢	𝑓(𝑢	PROPN
cana-2085	31	15	)	)	PUNCT
cana-2085	31	16	≥	≥	NOUN
cana-2085	31	17	2	2	NUM
cana-2085	31	18	,	,	PUNCT
cana-2085	31	19	if	if	SCONJ
cana-2085	31	20	𝑓(𝑢	𝑓(𝑢	ADV
cana-2085	31	21	)	)	PUNCT
cana-2085	31	22	=	=	SYM
cana-2085	32	1	1	1	NUM
cana-2085	32	2	,	,	PUNCT
cana-2085	32	3	then	then	ADV
cana-2085	32	4	|𝑁(𝑢	|𝑁(𝑢	NUM
cana-2085	32	5	)	)	PUNCT
cana-2085	32	6	∩	∩	NOUN
cana-2085	32	7	𝑉0|	𝑉0|	X
cana-2085	32	8	=	=	SYM
cana-2085	32	9	0	0	PROPN
cana-2085	32	10	,	,	PUNCT
cana-2085	32	11	𝑓(𝑢	𝑓(𝑢	PROPN
cana-2085	32	12	)	)	PUNCT
cana-2085	32	13	=	=	SYM
cana-2085	32	14	2	2	NUM
cana-2085	32	15	,	,	PUNCT
cana-2085	32	16	then	then	ADV
cana-2085	32	17	|𝑁(𝑢	|𝑁(𝑢	NUM
cana-2085	32	18	)	)	PUNCT
cana-2085	32	19	∩	∩	NOUN
cana-2085	32	20	𝑉0|	𝑉0|	VERB
cana-2085	32	21	≤	≤	ADV
cana-2085	32	22	2	2	NUM
cana-2085	32	23	,	,	PUNCT
cana-2085	32	24	𝑓(𝑢	𝑓(𝑢	ADV
cana-2085	32	25	)	)	PUNCT
cana-2085	32	26	=	=	SYM
cana-2085	33	1	3	3	NUM
cana-2085	33	2	,	,	PUNCT
cana-2085	33	3	then	then	ADV
cana-2085	33	4	|𝑁(𝑢	|𝑁(𝑢	NUM
cana-2085	33	5	)	)	PUNCT
cana-2085	33	6	∩	∩	NOUN
cana-2085	33	7	𝑉0|	𝑉0|	NUM
cana-2085	33	8	≥	≥	PROPN
cana-2085	33	9	3	3	NUM
cana-2085	33	10	.	.	PUNCT
cana-2085	33	11	where	where	SCONJ
cana-2085	33	12	𝑉𝑖	𝑉𝑖	PROPN
cana-2085	33	13	,	,	PUNCT
cana-2085	33	14	𝑖	𝑖	NOUN
cana-2085	33	15	=	=	NOUN
cana-2085	33	16	1,2,3	1,2,3	NUM
cana-2085	33	17	.	.	PUNCT
cana-2085	33	18	.	.	PUNCT
cana-2085	33	19	.	.	PUNCT
cana-2085	34	1	,	,	PUNCT
cana-2085	34	2	𝑛.	𝑛.	VERB
cana-2085	34	3	the	the	DET
cana-2085	34	4	weight	weight	NOUN
cana-2085	34	5	of	of	ADP
cana-2085	34	6	a	a	DET
cana-2085	34	7	mrdf	mrdf	NOUN
cana-2085	34	8	is	be	AUX
cana-2085	34	9	the	the	DET
cana-2085	34	10	value	value	NOUN
cana-2085	34	11	𝑓(𝑉(𝐺	𝑓(𝑉(𝐺	NOUN
cana-2085	34	12	)	)	PUNCT
cana-2085	34	13	)	)	PUNCT
cana-2085	35	1	=	=	PUNCT
cana-2085	35	2	∑	∑	PUNCT
cana-2085	35	3	𝑓(𝑢𝑢𝜖𝑉(𝐺	𝑓(𝑢𝑢𝜖𝑉(𝐺	NOUN
cana-2085	35	4	)	)	PUNCT
cana-2085	35	5	)	)	PUNCT
cana-2085	35	6	.	.	PUNCT
cana-2085	36	1	the	the	DET
cana-2085	36	2	middle	middle	ADJ
cana-2085	36	3	roman	roman	ADJ
cana-2085	36	4	domination	domination	NOUN
cana-2085	36	5	number	number	NOUN
cana-2085	36	6	𝛾𝑀𝑅(𝐺	𝛾𝑀𝑅(𝐺	PROPN
cana-2085	36	7	)	)	PUNCT
cana-2085	36	8	is	be	AUX
cana-2085	36	9	the	the	DET
cana-2085	36	10	minimum	minimum	ADJ
cana-2085	36	11	weight	weight	NOUN
cana-2085	36	12	of	of	ADP
cana-2085	36	13	a	a	DET
cana-2085	36	14	mrdf	mrdf	NOUN
cana-2085	36	15	on	on	ADP
cana-2085	36	16	𝐺.	𝐺.	NOUN
cana-2085	36	17	we	we	PRON
cana-2085	36	18	define	define	VERB
cana-2085	36	19	the	the	DET
cana-2085	36	20	middle	middle	ADJ
cana-2085	36	21	roman	roman	ADJ
cana-2085	36	22	domination	domination	NOUN
cana-2085	36	23	number	number	NOUN
cana-2085	36	24	based	base	VERB
cana-2085	36	25	on	on	ADP
cana-2085	36	26	the	the	DET
cana-2085	36	27	military	military	ADJ
cana-2085	36	28	strategy	strategy	NOUN
cana-2085	36	29	that	that	SCONJ
cana-2085	36	30	if	if	SCONJ
cana-2085	36	31	any	any	DET
cana-2085	36	32	region	region	NOUN
cana-2085	36	33	is	be	AUX
cana-2085	36	34	un	un	PROPN
cana-2085	36	35	secured	secure	VERB
cana-2085	36	36	then	then	ADV
cana-2085	36	37	it	it	PRON
cana-2085	36	38	must	must	AUX
cana-2085	36	39	be	be	AUX
cana-2085	36	40	adjacent	adjacent	ADJ
cana-2085	36	41	to	to	ADP
cana-2085	36	42	a	a	DET
cana-2085	36	43	region	region	NOUN
cana-2085	36	44	with	with	ADP
cana-2085	36	45	either	either	CCONJ
cana-2085	36	46	2	2	NUM
cana-2085	36	47	or	or	CCONJ
cana-2085	36	48	3	3	NUM
cana-2085	36	49	legions	legion	NOUN
cana-2085	36	50	.	.	PUNCT
cana-2085	37	1	if	if	SCONJ
cana-2085	37	2	2	2	NUM
cana-2085	37	3	legions	legion	NOUN
cana-2085	37	4	are	be	AUX
cana-2085	37	5	placed	place	VERB
cana-2085	37	6	at	at	ADP
cana-2085	37	7	a	a	DET
cana-2085	37	8	region	region	NOUN
cana-2085	37	9	then	then	ADV
cana-2085	37	10	at	at	ADP
cana-2085	37	11	most	most	ADJ
cana-2085	37	12	two	two	NUM
cana-2085	37	13	neighboring	neighboring	NOUN
cana-2085	37	14	regions	region	NOUN
cana-2085	37	15	can	can	AUX
cana-2085	37	16	be	be	AUX
cana-2085	37	17	undefended	undefended	ADJ
cana-2085	37	18	.	.	PUNCT
cana-2085	38	1	if	if	SCONJ
cana-2085	38	2	3	3	NUM
cana-2085	38	3	legions	legion	NOUN
cana-2085	38	4	are	be	AUX
cana-2085	38	5	placed	place	VERB
cana-2085	38	6	at	at	ADP
cana-2085	38	7	a	a	DET
cana-2085	38	8	region	region	NOUN
cana-2085	38	9	then	then	ADV
cana-2085	38	10	any	any	DET
cana-2085	38	11	number	number	NOUN
cana-2085	38	12	of	of	ADP
cana-2085	38	13	neighboring	neighboring	NOUN
cana-2085	38	14	legions	legion	NOUN
cana-2085	38	15	can	can	AUX
cana-2085	38	16	be	be	AUX
cana-2085	38	17	undefended	undefended	ADJ
cana-2085	38	18	.	.	PUNCT
cana-2085	39	1	the	the	DET
cana-2085	39	2	idea	idea	NOUN
cana-2085	39	3	is	be	AUX
cana-2085	39	4	that	that	SCONJ
cana-2085	39	5	during	during	ADP
cana-2085	39	6	an	an	DET
cana-2085	39	7	attack	attack	NOUN
cana-2085	39	8	at	at	ADP
cana-2085	39	9	any	any	DET
cana-2085	39	10	undefended	undefended	ADJ
cana-2085	39	11	region	region	NOUN
cana-2085	39	12	,	,	PUNCT
cana-2085	39	13	the	the	DET
cana-2085	39	14	legions	legion	NOUN
cana-2085	39	15	placed	place	VERB
cana-2085	39	16	at	at	ADP
cana-2085	39	17	the	the	DET
cana-2085	39	18	neighboring	neighboring	NOUN
cana-2085	39	19	regions	region	NOUN
cana-2085	39	20	will	will	AUX
cana-2085	39	21	move	move	VERB
cana-2085	39	22	and	and	CCONJ
cana-2085	39	23	protect	protect	VERB
cana-2085	39	24	this	this	DET
cana-2085	39	25	undefended	undefended	ADJ
cana-2085	39	26	region	region	NOUN
cana-2085	39	27	.	.	PUNCT
cana-2085	40	1	which	which	PRON
cana-2085	40	2	provides	provide	VERB
cana-2085	40	3	a	a	DET
cana-2085	40	4	fruitful	fruitful	ADJ
cana-2085	40	5	level	level	NOUN
cana-2085	40	6	of	of	ADP
cana-2085	40	7	defence	defence	NOUN
cana-2085	40	8	at	at	ADP
cana-2085	40	9	a	a	DET
cana-2085	40	10	cheaper	cheap	ADJ
cana-2085	40	11	cost	cost	NOUN
cana-2085	40	12	.	.	PUNCT
cana-2085	41	1	in	in	ADP
cana-2085	41	2	some	some	DET
cana-2085	41	3	highly	highly	ADV
cana-2085	41	4	populated	populated	ADJ
cana-2085	41	5	areas	area	NOUN
cana-2085	41	6	of	of	ADP
cana-2085	41	7	the	the	DET
cana-2085	41	8	city	city	NOUN
cana-2085	41	9	where	where	SCONJ
cana-2085	41	10	emergency	emergency	NOUN
cana-2085	41	11	calls	call	VERB
cana-2085	41	12	for	for	ADP
cana-2085	41	13	police	police	NOUN
cana-2085	41	14	,	,	PUNCT
cana-2085	41	15	ambulance	ambulance	NOUN
cana-2085	41	16	,	,	PUNCT
cana-2085	41	17	fire	fire	NOUN
cana-2085	41	18	men	man	NOUN
cana-2085	41	19	service	service	NOUN
cana-2085	41	20	etc	etc	X
cana-2085	41	21	.	.	X
cana-2085	41	22	are	be	AUX
cana-2085	41	23	common	common	ADJ
cana-2085	41	24	.	.	PUNCT
cana-2085	42	1	three	three	NUM
cana-2085	42	2	units	unit	NOUN
cana-2085	42	3	of	of	ADP
cana-2085	42	4	servers	server	NOUN
cana-2085	42	5	can	can	AUX
cana-2085	42	6	be	be	AUX
cana-2085	42	7	placed	place	VERB
cana-2085	42	8	,	,	PUNCT
cana-2085	42	9	whereas	whereas	SCONJ
cana-2085	42	10	at	at	ADP
cana-2085	42	11	areas	area	NOUN
cana-2085	42	12	𝑉𝑖	𝑉𝑖	PROPN
cana-2085	42	13	,	,	PUNCT
cana-2085	42	14	𝑖	𝑖	NOUN
cana-2085	42	15	=	=	NOUN
cana-2085	42	16	1,2,3	1,2,3	NUM
cana-2085	42	17	.	.	PUNCT
cana-2085	42	18	.	.	PUNCT
cana-2085	42	19	.	.	PUNCT
cana-2085	43	1	,	,	PUNCT
cana-2085	43	2	𝑛	𝑛	PRON
cana-2085	43	3	where	where	SCONJ
cana-2085	43	4	only	only	ADV
cana-2085	43	5	two	two	NUM
cana-2085	43	6	neighbouring	neighbouring	ADJ
cana-2085	43	7	regions	region	NOUN
cana-2085	43	8	need	need	VERB
cana-2085	43	9	service	service	NOUN
cana-2085	43	10	,	,	PUNCT
cana-2085	43	11	exactly	exactly	ADV
cana-2085	43	12	two	two	NUM
cana-2085	43	13	units	unit	NOUN
cana-2085	43	14	of	of	ADP
cana-2085	43	15	servers	server	NOUN
cana-2085	43	16	can	can	AUX
cana-2085	43	17	be	be	AUX
cana-2085	43	18	placed	place	VERB
cana-2085	43	19	.	.	PUNCT
cana-2085	44	1	similarly	similarly	ADV
cana-2085	44	2	,	,	PUNCT
cana-2085	44	3	in	in	ADP
cana-2085	44	4	trust	trust	NOUN
cana-2085	44	5	worthy	worthy	ADJ
cana-2085	44	6	areas	area	NOUN
cana-2085	44	7	,	,	PUNCT
cana-2085	44	8	one	one	NUM
cana-2085	44	9	unit	unit	NOUN
cana-2085	44	10	of	of	ADP
cana-2085	44	11	server	server	NOUN
cana-2085	44	12	can	can	AUX
cana-2085	44	13	be	be	AUX
cana-2085	44	14	placed	place	VERB
cana-2085	44	15	.	.	PUNCT
cana-2085	45	1	such	such	ADJ
cana-2085	45	2	type	type	NOUN
cana-2085	45	3	of	of	ADP
cana-2085	45	4	arrangements	arrangement	NOUN
cana-2085	45	5	can	can	AUX
cana-2085	45	6	give	give	VERB
cana-2085	45	7	protection	protection	NOUN
cana-2085	45	8	with	with	ADP
cana-2085	45	9	minimum	minimum	ADJ
cana-2085	45	10	number	number	NOUN
cana-2085	45	11	of	of	ADP
cana-2085	45	12	servers	server	NOUN
cana-2085	45	13	which	which	PRON
cana-2085	45	14	also	also	ADV
cana-2085	45	15	optimize	optimize	VERB
cana-2085	45	16	the	the	DET
cana-2085	45	17	cost	cost	NOUN
cana-2085	45	18	.	.	PUNCT
cana-2085	46	1	server	server	NOUN
cana-2085	46	2	placements	placement	NOUN
cana-2085	46	3	will	will	AUX
cana-2085	46	4	maximize	maximize	VERB
cana-2085	46	5	the	the	DET
cana-2085	46	6	service	service	NOUN
cana-2085	46	7	of	of	ADP
cana-2085	46	8	the	the	DET
cana-2085	46	9	servers	server	NOUN
cana-2085	46	10	with	with	ADP
cana-2085	46	11	optimal	optimal	ADJ
cana-2085	46	12	cost	cost	NOUN
cana-2085	46	13	[	[	X
cana-2085	46	14	11,12,13,14	11,12,13,14	ADV
cana-2085	46	15	]	]	PUNCT
cana-2085	46	16	.	.	PUNCT
cana-2085	47	1	middle	middle	ADJ
cana-2085	47	2	roman	roman	ADJ
cana-2085	47	3	dominating	dominating	NOUN
cana-2085	47	4	functions	function	NOUN
cana-2085	47	5	(	(	PUNCT
cana-2085	47	6	mrdfs	mrdfs	ADJ
cana-2085	47	7	)	)	PUNCT
cana-2085	47	8	offer	offer	VERB
cana-2085	47	9	a	a	DET
cana-2085	47	10	fascinating	fascinating	ADJ
cana-2085	47	11	intersection	intersection	NOUN
cana-2085	47	12	of	of	ADP
cana-2085	47	13	graph	graph	NOUN
cana-2085	47	14	theory	theory	NOUN
cana-2085	47	15	and	and	CCONJ
cana-2085	47	16	computer	computer	NOUN
cana-2085	47	17	science	science	NOUN
cana-2085	47	18	,	,	PUNCT
cana-2085	47	19	providing	provide	VERB
cana-2085	47	20	powerful	powerful	ADJ
cana-2085	47	21	tools	tool	NOUN
cana-2085	47	22	for	for	ADP
cana-2085	47	23	addressing	address	VERB
cana-2085	47	24	complex	complex	ADJ
cana-2085	47	25	problems	problem	NOUN
cana-2085	47	26	in	in	ADP
cana-2085	47	27	network	network	NOUN
cana-2085	47	28	security	security	NOUN
cana-2085	47	29	,	,	PUNCT
cana-2085	47	30	resource	resource	NOUN
cana-2085	47	31	allocation	allocation	NOUN
cana-2085	47	32	,	,	PUNCT
cana-2085	47	33	and	and	CCONJ
cana-2085	47	34	optimization	optimization	NOUN
cana-2085	47	35	.	.	PUNCT
cana-2085	48	1	these	these	DET
cana-2085	48	2	functions	function	NOUN
cana-2085	48	3	,	,	PUNCT
cana-2085	48	4	defined	define	VERB
cana-2085	48	5	by	by	ADP
cana-2085	48	6	specific	specific	ADJ
cana-2085	48	7	conditions	condition	NOUN
cana-2085	48	8	on	on	ADP
cana-2085	48	9	vertex	vertex	NOUN
cana-2085	48	10	assignments	assignment	NOUN
cana-2085	48	11	within	within	ADP
cana-2085	48	12	a	a	DET
cana-2085	48	13	graph	graph	NOUN
cana-2085	48	14	,	,	PUNCT
cana-2085	48	15	allow	allow	VERB
cana-2085	48	16	for	for	ADP
cana-2085	48	17	strategic	strategic	ADJ
cana-2085	48	18	placement	placement	NOUN
cana-2085	48	19	and	and	CCONJ
cana-2085	48	20	allocation	allocation	NOUN
cana-2085	48	21	of	of	ADP
cana-2085	48	22	resources	resource	NOUN
cana-2085	48	23	,	,	PUNCT
cana-2085	48	24	ensuring	ensure	VERB
cana-2085	48	25	that	that	SCONJ
cana-2085	48	26	critical	critical	ADJ
cana-2085	48	27	nodes	node	NOUN
cana-2085	48	28	are	be	AUX
cana-2085	48	29	efficiently	efficiently	ADV
cana-2085	48	30	protected	protect	VERB
cana-2085	48	31	or	or	CCONJ
cana-2085	48	32	empowered	empower	VERB
cana-2085	48	33	.	.	PUNCT
cana-2085	49	1	by	by	ADP
cana-2085	49	2	minimizing	minimize	VERB
cana-2085	49	3	the	the	DET
cana-2085	49	4	overall	overall	ADJ
cana-2085	49	5	cost	cost	NOUN
cana-2085	49	6	or	or	CCONJ
cana-2085	49	7	weight	weight	NOUN
cana-2085	49	8	while	while	SCONJ
cana-2085	49	9	satisfying	satisfy	VERB
cana-2085	49	10	the	the	DET
cana-2085	49	11	constraints	constraint	NOUN
cana-2085	49	12	of	of	ADP
cana-2085	49	13	mrdfs	mrdfs	ADJ
cana-2085	49	14	,	,	PUNCT
cana-2085	49	15	computer	computer	NOUN
cana-2085	49	16	scientists	scientist	NOUN
cana-2085	49	17	can	can	AUX
cana-2085	49	18	design	design	VERB
cana-2085	49	19	robust	robust	ADJ
cana-2085	49	20	systems	system	NOUN
cana-2085	49	21	that	that	PRON
cana-2085	49	22	balance	balance	NOUN
cana-2085	49	23	efficiency	efficiency	NOUN
cana-2085	49	24	with	with	ADP
cana-2085	49	25	security	security	NOUN
cana-2085	49	26	.	.	PUNCT
cana-2085	50	1	this	this	DET
cana-2085	50	2	introduction	introduction	NOUN
cana-2085	50	3	explores	explore	VERB
cana-2085	50	4	the	the	DET
cana-2085	50	5	diverse	diverse	ADJ
cana-2085	50	6	applications	application	NOUN
cana-2085	50	7	of	of	ADP
cana-2085	50	8	mrdfs	mrdfs	ADJ
cana-2085	50	9	in	in	ADP
cana-2085	50	10	various	various	ADJ
cana-2085	50	11	domains	domain	NOUN
cana-2085	50	12	,	,	PUNCT
cana-2085	50	13	highlighting	highlight	VERB
cana-2085	50	14	their	their	PRON
cana-2085	50	15	significance	significance	NOUN
cana-2085	50	16	in	in	ADP
cana-2085	50	17	optimizing	optimize	VERB
cana-2085	50	18	network	network	NOUN
cana-2085	50	19	structures	structure	NOUN
cana-2085	50	20	,	,	PUNCT
cana-2085	50	21	managing	manage	VERB
cana-2085	50	22	distributed	distribute	VERB
cana-2085	50	23	systems	system	NOUN
cana-2085	50	24	,	,	PUNCT
cana-2085	50	25	and	and	CCONJ
cana-2085	50	26	enhancing	enhance	VERB
cana-2085	50	27	the	the	DET
cana-2085	50	28	resilience	resilience	NOUN
cana-2085	50	29	of	of	ADP
cana-2085	50	30	communication	communication	NOUN
cana-2085	50	31	networks	network	NOUN
cana-2085	50	32	.	.	PUNCT
cana-2085	51	1	here	here	ADV
cana-2085	51	2	we	we	PRON
cana-2085	51	3	stated	state	VERB
cana-2085	51	4	some	some	PRON
cana-2085	51	5	of	of	ADP
cana-2085	51	6	the	the	DET
cana-2085	51	7	characterization	characterization	NOUN
cana-2085	51	8	theorems	theorem	NOUN
cana-2085	51	9	of	of	ADP
cana-2085	51	10	mrdfs	mrdfs	PROPN
cana-2085	51	11	followed	follow	VERB
cana-2085	51	12	by	by	ADP
cana-2085	51	13	its	its	PRON
cana-2085	51	14	applications	application	NOUN
cana-2085	51	15	.	.	PUNCT
cana-2085	52	1	preposition	preposition	NOUN
cana-2085	52	2	1	1	NUM
cana-2085	52	3	:	:	PUNCT
cana-2085	52	4	if	if	SCONJ
cana-2085	52	5	𝐺	𝐺	PROPN
cana-2085	52	6	a	a	DET
cana-2085	52	7	graph	graph	NOUN
cana-2085	52	8	of	of	ADP
cana-2085	52	9	order	order	NOUN
cana-2085	52	10	𝑛	𝑛	PRON
cana-2085	52	11	≥	≥	NOUN
cana-2085	52	12	4	4	NUM
cana-2085	52	13	with	with	ADP
cana-2085	52	14	a	a	DET
cana-2085	52	15	vertex	vertex	NOUN
cana-2085	52	16	of	of	ADP
cana-2085	52	17	degree	degree	NOUN
cana-2085	52	18	𝑛	𝑛	PRON
cana-2085	52	19	−	−	PROPN
cana-2085	52	20	1	1	NUM
cana-2085	52	21	then	then	ADV
cana-2085	52	22	𝛾(𝐺	𝛾(𝐺	NOUN
cana-2085	52	23	)	)	PUNCT
cana-2085	53	1	=	=	SYM
cana-2085	53	2	1	1	NUM
cana-2085	53	3	and	and	CCONJ
cana-2085	53	4	𝛾𝑀𝑅(𝐺	𝛾𝑀𝑅(𝐺	PROPN
cana-2085	53	5	)	)	PUNCT
cana-2085	53	6	=	=	SYM
cana-2085	53	7	3	3	NUM
cana-2085	53	8	proposition	proposition	NOUN
cana-2085	53	9	2	2	NUM
cana-2085	53	10	:	:	PUNCT
cana-2085	53	11	let	let	VERB
cana-2085	53	12	𝑓	𝑓	PRON
cana-2085	53	13	=	=	X
cana-2085	53	14	(	(	PUNCT
cana-2085	53	15	𝑉0	𝑉0	NOUN
cana-2085	53	16	,	,	PUNCT
cana-2085	53	17	𝑉1	𝑉1	PROPN
cana-2085	53	18	,	,	PUNCT
cana-2085	53	19	𝑉2	𝑉2	NOUN
cana-2085	53	20	,	,	PUNCT
cana-2085	53	21	𝑉3	𝑉3	NOUN
cana-2085	53	22	)	)	PUNCT
cana-2085	53	23	be	be	AUX
cana-2085	53	24	a	a	DET
cana-2085	53	25	𝛾𝑀𝑅-function	𝛾𝑀𝑅-function	NOUN
cana-2085	53	26	then	then	ADV
cana-2085	53	27	,	,	PUNCT
cana-2085	53	28	a.	a.	NOUN
cana-2085	53	29	𝐺[𝑉1	𝐺[𝑉1	PROPN
cana-2085	53	30	]	]	PUNCT
cana-2085	53	31	the	the	DET
cana-2085	53	32	subgraph	subgraph	NOUN
cana-2085	53	33	induced	induce	VERB
cana-2085	53	34	by	by	ADP
cana-2085	53	35	𝑉1	𝑉1	PROPN
cana-2085	53	36	has	have	VERB
cana-2085	53	37	maximum	maximum	ADJ
cana-2085	53	38	degree	degree	NOUN
cana-2085	53	39	1	1	NUM
cana-2085	53	40	b.	b.	NOUN
cana-2085	53	41	no	no	DET
cana-2085	53	42	edge	edge	NOUN
cana-2085	53	43	of	of	ADP
cana-2085	53	44	𝐺	𝐺	PROPN
cana-2085	53	45	joins	join	VERB
cana-2085	53	46	𝑉1	𝑉1	PROPN
cana-2085	53	47	and	and	CCONJ
cana-2085	53	48	𝑉3	𝑉3	NOUN
cana-2085	53	49	.	.	PUNCT
cana-2085	54	1	c.	c.	NOUN
cana-2085	54	2	∀	∀	PUNCT
cana-2085	55	1	𝑣	𝑣	ADP
cana-2085	55	2	∈	∈	PROPN
cana-2085	55	3	𝑉0	𝑉0	NOUN
cana-2085	55	4	,	,	PUNCT
cana-2085	55	5	|𝑁(𝑣	|𝑁(𝑣	PROPN
cana-2085	55	6	)	)	PUNCT
cana-2085	55	7	∩	∩	NOUN
cana-2085	55	8	𝑉1|	𝑉1|	PROPN
cana-2085	55	9	≤	≤	ADV
cana-2085	55	10	1	1	NUM
cana-2085	55	11	=	=	SYM
cana-2085	55	12	{	{	PUNCT
cana-2085	55	13	𝑢	𝑢	PROPN
cana-2085	55	14	,	,	PUNCT
cana-2085	55	15	𝑣2	𝑣2	NOUN
cana-2085	55	16	}	}	PUNCT
cana-2085	55	17	∩	∩	NOUN
cana-2085	55	18	{	{	PUNCT
cana-2085	55	19	𝑣1	𝑣1	PROPN
cana-2085	55	20	,	,	PUNCT
cana-2085	55	21	𝑣2	𝑣2	PROPN
cana-2085	55	22	,	,	PUNCT
cana-2085	55	23	𝑣3	𝑣3	ADJ
cana-2085	55	24	}	}	PUNCT
cana-2085	55	25	=	=	SYM
cana-2085	55	26	{	{	PUNCT
cana-2085	55	27	𝑣2	𝑣2	NOUN
cana-2085	55	28	}	}	PUNCT
cana-2085	55	29	d.	d.	PROPN
cana-2085	55	30	𝑉2	𝑉2	PROPN
cana-2085	55	31	∪	∪	PROPN
cana-2085	55	32	𝑉3	𝑉3	NOUN
cana-2085	55	33	is	be	AUX
cana-2085	55	34	a	a	DET
cana-2085	55	35	𝛾set	𝛾set	NOUN
cana-2085	55	36	of	of	ADP
cana-2085	55	37	𝑉0	𝑉0	NOUN
cana-2085	55	38	.	.	PUNCT
cana-2085	56	1	𝑉2	𝑉2	PROPN
cana-2085	56	2	∪	∪	VERB
cana-2085	56	3	𝑉3	𝑉3	PROPN
cana-2085	56	4	≻	≻	PROPN
cana-2085	56	5	𝑉0	𝑉0	NOUN
cana-2085	56	6	theorem	theorem	ADJ
cana-2085	56	7	1	1	NUM
cana-2085	56	8	:	:	PUNCT
cana-2085	56	9	𝛾𝑀𝑅(𝑃𝑛	𝛾𝑀𝑅(𝑃𝑛	PROPN
cana-2085	56	10	)	)	PUNCT
cana-2085	56	11	=	=	SYM
cana-2085	56	12	𝛾𝑀𝑅(𝐶𝑛	𝛾𝑀𝑅(𝐶𝑛	PROPN
cana-2085	56	13	)	)	PUNCT
cana-2085	56	14	=	=	PUNCT
cana-2085	57	1	⌈	⌈	PROPN
cana-2085	57	2	2𝑛	2𝑛	PROPN
cana-2085	57	3	3	3	NUM
cana-2085	57	4	⌉	⌉	PRON
cana-2085	57	5	proof	proof	NOUN
cana-2085	57	6	is	be	AUX
cana-2085	57	7	obvious	obvious	ADJ
cana-2085	57	8	for	for	ADP
cana-2085	57	9	the	the	DET
cana-2085	57	10	above	above	ADJ
cana-2085	57	11	theorem	theorem	PROPN
cana-2085	57	12	.	.	PUNCT
cana-2085	57	13	theorem	theorem	NOUN
cana-2085	57	14	2	2	NUM
cana-2085	57	15	:	:	PUNCT
cana-2085	57	16	for	for	ADP
cana-2085	57	17	any	any	DET
cana-2085	57	18	graph	graph	NOUN
cana-2085	57	19	𝐺	𝐺	NOUN
cana-2085	57	20	,	,	PUNCT
cana-2085	57	21	𝛾𝑀𝑅(𝐺	𝛾𝑀𝑅(𝐺	PROPN
cana-2085	57	22	)	)	PUNCT
cana-2085	58	1	=	=	SYM
cana-2085	58	2	3𝛾(𝐺	3𝛾(𝐺	NUM
cana-2085	58	3	)	)	PUNCT
cana-2085	58	4	if	if	SCONJ
cana-2085	58	5	and	and	CCONJ
cana-2085	58	6	only	only	ADV
cana-2085	58	7	if	if	SCONJ
cana-2085	58	8	𝑉1	𝑉1	NOUN
cana-2085	58	9	=	=	SYM
cana-2085	58	10	∅	∅	NOUN
cana-2085	58	11	and	and	CCONJ
cana-2085	58	12	𝑉2	𝑉2	NOUN
cana-2085	58	13	=	=	SYM
cana-2085	58	14	∅	∅	NOUN
cana-2085	58	15	communications	communication	NOUN
cana-2085	58	16	on	on	ADP
cana-2085	58	17	applied	apply	VERB
cana-2085	58	18	nonlinear	nonlinear	ADJ
cana-2085	58	19	analysis	analysis	NOUN
cana-2085	58	20	issn	issn	NOUN
cana-2085	58	21	:	:	PUNCT
cana-2085	58	22	1074	1074	NUM
cana-2085	58	23	-	-	PUNCT
cana-2085	58	24	133x	133x	NUM
cana-2085	58	25	vol	vol	NOUN
cana-2085	58	26	32	32	NUM
cana-2085	58	27	no	no	NOUN
cana-2085	58	28	.	.	PUNCT
cana-2085	59	1	1s	1s	NUM
cana-2085	59	2	(	(	PUNCT
cana-2085	59	3	2025	2025	NUM
cana-2085	59	4	)	)	PUNCT
cana-2085	59	5	19	19	NUM
cana-2085	59	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-2085	59	7	proof	proof	NOUN
cana-2085	59	8	:	:	PUNCT
cana-2085	59	9	let	let	VERB
cana-2085	59	10	𝐺	𝐺	PRON
cana-2085	59	11	be	be	AUX
cana-2085	59	12	a	a	DET
cana-2085	59	13	middle	middle	ADJ
cana-2085	59	14	roman	roman	ADJ
cana-2085	59	15	graph	graph	NOUN
cana-2085	59	16	and	and	CCONJ
cana-2085	59	17	let	let	VERB
cana-2085	59	18	𝑓	𝑓	PRON
cana-2085	59	19	=	=	X
cana-2085	59	20	(	(	PUNCT
cana-2085	59	21	𝑉0	𝑉0	NOUN
cana-2085	59	22	,	,	PUNCT
cana-2085	59	23	𝑉1	𝑉1	PROPN
cana-2085	59	24	,	,	PUNCT
cana-2085	59	25	𝑉2	𝑉2	NOUN
cana-2085	59	26	,	,	PUNCT
cana-2085	59	27	𝑉3	𝑉3	NOUN
cana-2085	59	28	)	)	PUNCT
cana-2085	59	29	be	be	AUX
cana-2085	59	30	a	a	DET
cana-2085	59	31	𝛾𝑀𝑅-function	𝛾𝑀𝑅-function	NOUN
cana-2085	59	32	of	of	ADP
cana-2085	59	33	𝐺.	𝐺.	NOUN
cana-2085	59	34	from	from	ADP
cana-2085	59	35	the	the	DET
cana-2085	59	36	proposition	proposition	NOUN
cana-2085	59	37	3(d	3(d	NUM
cana-2085	59	38	)	)	PUNCT
cana-2085	59	39	𝑉2	𝑉2	NOUN
cana-2085	59	40	,	,	PUNCT
cana-2085	59	41	𝑉3	𝑉3	NOUN
cana-2085	59	42	≻	≻	PROPN
cana-2085	59	43	𝑉0	𝑉0	NOUN
cana-2085	59	44	and	and	CCONJ
cana-2085	59	45	𝑉1	𝑉1	PROPN
cana-2085	59	46	∪	∪	X
cana-2085	59	47	𝑉2	𝑉2	PROPN
cana-2085	59	48	∪	∪	ADP
cana-2085	59	49	𝑉3	𝑉3	PROPN
cana-2085	59	50	≻	≻	PROPN
cana-2085	59	51	𝑉0	𝑉0	NOUN
cana-2085	59	52	and	and	CCONJ
cana-2085	59	53	hence	hence	ADV
cana-2085	59	54	,	,	PUNCT
cana-2085	59	55	𝛾(𝐺	𝛾(𝐺	PROPN
cana-2085	59	56	)	)	PUNCT
cana-2085	59	57	≤	≤	NOUN
cana-2085	59	58	|𝑉1	|𝑉1	PUNCT
cana-2085	59	59	∪	∪	ADJ
cana-2085	59	60	𝑉2	𝑉2	NOUN
cana-2085	59	61	∪	∪	NOUN
cana-2085	59	62	𝑉3|	𝑉3|	X
cana-2085	59	63	=	=	SYM
cana-2085	59	64	|𝑉1|	|𝑉1|	PROPN
cana-2085	60	1	+	+	NUM
cana-2085	60	2	|𝑉2|	|𝑉2|	X
cana-2085	60	3	+	+	CCONJ
cana-2085	60	4	|𝑉3|	|𝑉3|	PRON
cana-2085	60	5	≤	≤	NUM
cana-2085	60	6	|𝑉1|	|𝑉1|	X
cana-2085	61	1	+	+	CCONJ
cana-2085	61	2	2|𝑉2|	2|𝑉2|	NUM
cana-2085	61	3	+	+	SYM
cana-2085	61	4	3|𝑉3|	3|𝑉3|	NUM
cana-2085	61	5	=	=	SYM
cana-2085	61	6	𝛾𝑀𝑅(𝐺	𝛾𝑀𝑅(𝐺	NOUN
cana-2085	61	7	)	)	PUNCT
cana-2085	61	8	.	.	PUNCT
cana-2085	62	1	but	but	CCONJ
cana-2085	62	2	since	since	ADV
cana-2085	62	3	,	,	PUNCT
cana-2085	62	4	𝐺	𝐺	PROPN
cana-2085	62	5	is	be	AUX
cana-2085	62	6	middle	middle	ADJ
cana-2085	62	7	roman	roman	ADJ
cana-2085	62	8	,	,	PUNCT
cana-2085	62	9	we	we	PRON
cana-2085	62	10	know	know	VERB
cana-2085	62	11	that	that	SCONJ
cana-2085	62	12	,	,	PUNCT
cana-2085	62	13	3𝛾(𝐺	3𝛾(𝐺	NUM
cana-2085	62	14	)	)	PUNCT
cana-2085	62	15	=	=	SYM
cana-2085	63	1	3|𝑉1|	3|𝑉1|	NUM
cana-2085	63	2	+	+	NUM
cana-2085	63	3	3|𝑉2|	3|𝑉2|	NUM
cana-2085	63	4	+	+	SYM
cana-2085	63	5	3|𝑉3|	3|𝑉3|	NUM
cana-2085	63	6	=	=	SYM
cana-2085	63	7	𝛾𝑀𝑅(𝐺	𝛾𝑀𝑅(𝐺	NOUN
cana-2085	63	8	)	)	PUNCT
cana-2085	63	9	=	=	SYM
cana-2085	64	1	|𝑉1|	|𝑉1|	PROPN
cana-2085	65	1	+	+	NUM
cana-2085	65	2	|𝑉2|	|𝑉2|	X
cana-2085	65	3	+	+	X
cana-2085	65	4	3|𝑉3|	3|𝑉3|	NUM
cana-2085	65	5	.	.	PUNCT
cana-2085	66	1	hence	hence	ADV
cana-2085	66	2	𝑛1	𝑛1	NOUN
cana-2085	66	3	=	=	PUNCT
cana-2085	66	4	|𝑉1|	|𝑉1|	PROPN
cana-2085	66	5	=	=	SYM
cana-2085	66	6	|𝑉2|	|𝑉2|	NOUN
cana-2085	66	7	=	=	SYM
cana-2085	66	8	0	0	X
cana-2085	66	9	.	.	PUNCT
cana-2085	67	1	conversely	conversely	ADV
cana-2085	67	2	,	,	PUNCT
cana-2085	67	3	let	let	VERB
cana-2085	67	4	𝑓	𝑓	PRON
cana-2085	67	5	=	=	X
cana-2085	67	6	(	(	PUNCT
cana-2085	67	7	𝑉0	𝑉0	NOUN
cana-2085	67	8	,	,	PUNCT
cana-2085	67	9	𝑉1	𝑉1	PROPN
cana-2085	67	10	,	,	PUNCT
cana-2085	67	11	𝑉2	𝑉2	NOUN
cana-2085	67	12	,	,	PUNCT
cana-2085	67	13	𝑉3	𝑉3	NOUN
cana-2085	67	14	)	)	PUNCT
cana-2085	67	15	be	be	AUX
cana-2085	67	16	a	a	DET
cana-2085	67	17	𝛾𝑀𝑅-function	𝛾𝑀𝑅-function	NOUN
cana-2085	67	18	of	of	ADP
cana-2085	67	19	𝐺	𝐺	PROPN
cana-2085	67	20	𝑛1	𝑛1	NOUN
cana-2085	67	21	=	=	PUNCT
cana-2085	67	22	|𝑉1|	|𝑉1|	PROPN
cana-2085	67	23	=	=	SYM
cana-2085	67	24	|𝑉2|	|𝑉2|	NOUN
cana-2085	67	25	=	=	SYM
cana-2085	67	26	0	0	X
cana-2085	67	27	.	.	PUNCT
cana-2085	68	1	therefore	therefore	ADV
cana-2085	68	2	𝛾𝑀𝑅(𝐺	𝛾𝑀𝑅(𝐺	PROPN
cana-2085	68	3	)	)	PUNCT
cana-2085	68	4	=	=	SYM
cana-2085	69	1	3|𝑉3|	3|𝑉3|	NUM
cana-2085	69	2	and	and	CCONJ
cana-2085	69	3	since	since	SCONJ
cana-2085	69	4	by	by	ADP
cana-2085	69	5	the	the	DET
cana-2085	69	6	definition	definition	NOUN
cana-2085	69	7	𝑉1	𝑉1	NOUN
cana-2085	69	8	∪	∪	ADP
cana-2085	69	9	𝑉2	𝑉2	PROPN
cana-2085	69	10	∪	∪	VERB
cana-2085	69	11	𝑉3	𝑉3	PROPN
cana-2085	69	12	≻	≻	PROPN
cana-2085	69	13	𝑉0	𝑉0	NOUN
cana-2085	69	14	it	it	PRON
cana-2085	69	15	follows	follow	VERB
cana-2085	69	16	that	that	SCONJ
cana-2085	69	17	𝑉3	𝑉3	NOUN
cana-2085	69	18	is	be	AUX
cana-2085	69	19	a	a	DET
cana-2085	69	20	dominating	dominating	NOUN
cana-2085	69	21	set	set	NOUN
cana-2085	69	22	of	of	ADP
cana-2085	69	23	𝐺.	𝐺.	NOUN
cana-2085	69	24	we	we	PRON
cana-2085	69	25	know	know	VERB
cana-2085	69	26	that	that	DET
cana-2085	69	27	𝑉3	𝑉3	NOUN
cana-2085	69	28	is	be	AUX
cana-2085	69	29	a	a	DET
cana-2085	69	30	𝛾set	𝛾set	NOUN
cana-2085	69	31	of	of	ADP
cana-2085	69	32	𝐺(𝑉2	𝐺(𝑉2	NOUN
cana-2085	69	33	∪	∪	ADJ
cana-2085	69	34	𝑉3	𝑉3	NOUN
cana-2085	69	35	)	)	PUNCT
cana-2085	69	36	,	,	PUNCT
cana-2085	70	1	i.e.	i.e.	X
cana-2085	70	2	,	,	PUNCT
cana-2085	70	3	|𝑉3|	|𝑉3|	X
cana-2085	70	4	=	=	SYM
cana-2085	70	5	𝛾(𝐺	𝛾(𝐺	PROPN
cana-2085	70	6	)	)	PUNCT
cana-2085	70	7	and	and	CCONJ
cana-2085	70	8	𝛾𝑀𝑅(𝐺	𝛾𝑀𝑅(𝐺	PROPN
cana-2085	70	9	)	)	PUNCT
cana-2085	70	10	=	=	SYM
cana-2085	71	1	3𝛾(𝐺	3𝛾(𝐺	NUM
cana-2085	71	2	)	)	PUNCT
cana-2085	71	3	i.e.	i.e.	X
cana-2085	71	4	,	,	PUNCT
cana-2085	71	5	is	be	AUX
cana-2085	71	6	a	a	DET
cana-2085	71	7	middle	middle	ADJ
cana-2085	71	8	roman	roman	ADJ
cana-2085	71	9	graph	graph	NOUN
cana-2085	71	10	.	.	PUNCT
cana-2085	72	1	the	the	DET
cana-2085	72	2	theorem	theorem	ADJ
cana-2085	72	3	states	state	NOUN
cana-2085	72	4	that	that	SCONJ
cana-2085	72	5	for	for	ADP
cana-2085	72	6	any	any	DET
cana-2085	72	7	graph	graph	NOUN
cana-2085	72	8	𝐺	𝐺	NOUN
cana-2085	72	9	,	,	PUNCT
cana-2085	72	10	the	the	DET
cana-2085	72	11	middle	middle	ADJ
cana-2085	72	12	roman	roman	ADJ
cana-2085	72	13	domination	domination	NOUN
cana-2085	72	14	number	number	NOUN
cana-2085	72	15	𝛾𝑀𝑅(𝐺	𝛾𝑀𝑅(𝐺	PROPN
cana-2085	72	16	)	)	PUNCT
cana-2085	72	17	equals	equal	VERB
cana-2085	72	18	three	three	NUM
cana-2085	72	19	times	time	NOUN
cana-2085	72	20	the	the	DET
cana-2085	72	21	domination	domination	NOUN
cana-2085	72	22	number	number	NOUN
cana-2085	72	23	𝛾(𝐺	𝛾(𝐺	PROPN
cana-2085	72	24	)	)	PUNCT
cana-2085	73	1	if	if	SCONJ
cana-2085	73	2	and	and	CCONJ
cana-2085	73	3	only	only	ADV
cana-2085	73	4	if	if	SCONJ
cana-2085	73	5	two	two	NUM
cana-2085	73	6	specific	specific	ADJ
cana-2085	73	7	vertex	vertex	NOUN
cana-2085	73	8	sets	set	NOUN
cana-2085	73	9	,	,	PUNCT
cana-2085	73	10	𝑉1	𝑉1	NOUN
cana-2085	73	11	and	and	CCONJ
cana-2085	73	12	𝑉2	𝑉2	NOUN
cana-2085	73	13	,	,	PUNCT
cana-2085	73	14	are	be	AUX
cana-2085	73	15	empty	empty	ADJ
cana-2085	73	16	.	.	PUNCT
cana-2085	74	1	these	these	DET
cana-2085	74	2	vertex	vertex	NOUN
cana-2085	74	3	sets	set	NOUN
cana-2085	74	4	likely	likely	ADV
cana-2085	74	5	represent	represent	VERB
cana-2085	74	6	particular	particular	ADJ
cana-2085	74	7	configurations	configuration	NOUN
cana-2085	74	8	or	or	CCONJ
cana-2085	74	9	substructures	substructure	NOUN
cana-2085	74	10	within	within	ADP
cana-2085	74	11	the	the	DET
cana-2085	74	12	graph	graph	NOUN
cana-2085	74	13	that	that	PRON
cana-2085	74	14	affect	affect	VERB
cana-2085	74	15	the	the	DET
cana-2085	74	16	domination	domination	NOUN
cana-2085	74	17	and	and	CCONJ
cana-2085	74	18	middle	middle	ADJ
cana-2085	74	19	roman	roman	ADJ
cana-2085	74	20	domination	domination	NOUN
cana-2085	74	21	properties	property	NOUN
cana-2085	74	22	and	and	CCONJ
cana-2085	74	23	the	the	DET
cana-2085	74	24	applications	application	NOUN
cana-2085	74	25	of	of	ADP
cana-2085	74	26	the	the	DET
cana-2085	74	27	theorem	theorem	NOUN
cana-2085	74	28	are	be	AUX
cana-2085	74	29	as	as	SCONJ
cana-2085	74	30	follows	follow	VERB
cana-2085	74	31	network	network	NOUN
cana-2085	74	32	design	design	NOUN
cana-2085	74	33	and	and	CCONJ
cana-2085	74	34	optimization	optimization	NOUN
cana-2085	74	35	-	-	PUNCT
cana-2085	74	36	simplified	simplify	VERB
cana-2085	74	37	network	network	NOUN
cana-2085	74	38	structures	structure	NOUN
cana-2085	74	39	:	:	PUNCT
cana-2085	74	40	in	in	ADP
cana-2085	74	41	the	the	DET
cana-2085	74	42	design	design	NOUN
cana-2085	74	43	and	and	CCONJ
cana-2085	74	44	optimization	optimization	NOUN
cana-2085	74	45	of	of	ADP
cana-2085	74	46	networks	network	NOUN
cana-2085	74	47	(	(	PUNCT
cana-2085	74	48	such	such	ADJ
cana-2085	74	49	as	as	ADP
cana-2085	74	50	communication	communication	NOUN
cana-2085	74	51	or	or	CCONJ
cana-2085	74	52	transportation	transportation	NOUN
cana-2085	74	53	networks	network	NOUN
cana-2085	74	54	)	)	PUNCT
cana-2085	74	55	,	,	PUNCT
cana-2085	74	56	this	this	DET
cana-2085	74	57	theorem	theorem	NOUN
cana-2085	74	58	implies	imply	VERB
cana-2085	74	59	that	that	SCONJ
cana-2085	74	60	achieving	achieve	VERB
cana-2085	74	61	a	a	DET
cana-2085	74	62	specific	specific	ADJ
cana-2085	74	63	domination	domination	NOUN
cana-2085	74	64	-	-	PUNCT
cana-2085	74	65	related	relate	VERB
cana-2085	74	66	property	property	NOUN
cana-2085	74	67	(	(	PUNCT
cana-2085	74	68	where	where	SCONJ
cana-2085	74	69	𝛾𝑀𝑅(𝐺	𝛾𝑀𝑅(𝐺	NOUN
cana-2085	74	70	)	)	PUNCT
cana-2085	74	71	=	=	PUNCT
cana-2085	74	72	3𝛾(𝐺	3𝛾(𝐺	NUM
cana-2085	74	73	)	)	PUNCT
cana-2085	74	74	)	)	PUNCT
cana-2085	74	75	requires	require	VERB
cana-2085	74	76	the	the	DET
cana-2085	74	77	absence	absence	NOUN
cana-2085	74	78	of	of	ADP
cana-2085	74	79	certain	certain	ADJ
cana-2085	74	80	substructures	substructure	NOUN
cana-2085	74	81	(	(	PUNCT
cana-2085	74	82	𝑉1	𝑉1	NOUN
cana-2085	74	83	and	and	CCONJ
cana-2085	74	84	𝑉2	𝑉2	NOUN
cana-2085	74	85	)	)	PUNCT
cana-2085	74	86	.	.	PUNCT
cana-2085	75	1	this	this	PRON
cana-2085	75	2	can	can	AUX
cana-2085	75	3	guide	guide	VERB
cana-2085	75	4	the	the	DET
cana-2085	75	5	design	design	NOUN
cana-2085	75	6	of	of	ADP
cana-2085	75	7	networks	network	NOUN
cana-2085	75	8	to	to	PART
cana-2085	75	9	ensure	ensure	VERB
cana-2085	75	10	they	they	PRON
cana-2085	75	11	meet	meet	VERB
cana-2085	75	12	specific	specific	ADJ
cana-2085	75	13	criteria	criterion	NOUN
cana-2085	75	14	,	,	PUNCT
cana-2085	75	15	such	such	ADJ
cana-2085	75	16	as	as	ADP
cana-2085	75	17	minimal	minimal	ADJ
cana-2085	75	18	redundancy	redundancy	NOUN
cana-2085	75	19	or	or	CCONJ
cana-2085	75	20	efficient	efficient	ADJ
cana-2085	75	21	coverage	coverage	NOUN
cana-2085	75	22	,	,	PUNCT
cana-2085	75	23	by	by	ADP
cana-2085	75	24	avoiding	avoid	VERB
cana-2085	75	25	these	these	DET
cana-2085	75	26	substructures	substructure	NOUN
cana-2085	75	27	[	[	X
cana-2085	75	28	20	20	NUM
cana-2085	75	29	,	,	PUNCT
cana-2085	75	30	21	21	NUM
cana-2085	75	31	]	]	PUNCT
cana-2085	75	32	.	.	PUNCT
cana-2085	76	1	fault	fault	NOUN
cana-2085	76	2	-	-	PUNCT
cana-2085	76	3	tolerant	tolerant	ADJ
cana-2085	76	4	systems	system	NOUN
cana-2085	76	5	-	-	PUNCT
cana-2085	76	6	design	design	NOUN
cana-2085	76	7	of	of	ADP
cana-2085	76	8	fault	fault	NOUN
cana-2085	76	9	-	-	PUNCT
cana-2085	76	10	tolerant	tolerant	ADJ
cana-2085	76	11	architectures	architecture	NOUN
cana-2085	76	12	:	:	PUNCT
cana-2085	76	13	in	in	ADP
cana-2085	76	14	systems	system	NOUN
cana-2085	76	15	that	that	PRON
cana-2085	76	16	require	require	VERB
cana-2085	76	17	fault	fault	NOUN
cana-2085	76	18	tolerance	tolerance	NOUN
cana-2085	76	19	,	,	PUNCT
cana-2085	76	20	ensuring	ensure	VERB
cana-2085	76	21	that	that	SCONJ
cana-2085	76	22	𝛾𝑀𝑅(𝐺	𝛾𝑀𝑅(𝐺	NOUN
cana-2085	76	23	)	)	PUNCT
cana-2085	76	24	=	=	SYM
cana-2085	77	1	3𝛾(𝐺	3𝛾(𝐺	NUM
cana-2085	77	2	)	)	PUNCT
cana-2085	77	3	can	can	AUX
cana-2085	77	4	lead	lead	VERB
cana-2085	77	5	to	to	ADP
cana-2085	77	6	designs	design	NOUN
cana-2085	77	7	where	where	SCONJ
cana-2085	77	8	the	the	DET
cana-2085	77	9	system	system	NOUN
cana-2085	77	10	's	's	PART
cana-2085	77	11	resilience	resilience	NOUN
cana-2085	77	12	is	be	AUX
cana-2085	77	13	maximized	maximize	VERB
cana-2085	77	14	under	under	ADP
cana-2085	77	15	specific	specific	ADJ
cana-2085	77	16	conditions	condition	NOUN
cana-2085	77	17	.	.	PUNCT
cana-2085	78	1	the	the	DET
cana-2085	78	2	absence	absence	NOUN
cana-2085	78	3	of	of	ADP
cana-2085	78	4	the	the	DET
cana-2085	78	5	vertex	vertex	NOUN
cana-2085	78	6	sets	set	VERB
cana-2085	78	7	𝑉1	𝑉1	NOUN
cana-2085	78	8	and	and	CCONJ
cana-2085	78	9	𝑉2	𝑉2	NOUN
cana-2085	78	10	may	may	AUX
cana-2085	78	11	correspond	correspond	VERB
cana-2085	78	12	to	to	ADP
cana-2085	78	13	configurations	configuration	NOUN
cana-2085	78	14	that	that	PRON
cana-2085	78	15	prevent	prevent	VERB
cana-2085	78	16	certain	certain	ADJ
cana-2085	78	17	types	type	NOUN
cana-2085	78	18	of	of	ADP
cana-2085	78	19	failures	failure	NOUN
cana-2085	78	20	or	or	CCONJ
cana-2085	78	21	ensure	ensure	VERB
cana-2085	78	22	that	that	SCONJ
cana-2085	78	23	every	every	DET
cana-2085	78	24	component	component	NOUN
cana-2085	78	25	is	be	AUX
cana-2085	78	26	adequately	adequately	ADV
cana-2085	78	27	backed	back	VERB
cana-2085	78	28	up	up	ADP
cana-2085	78	29	.	.	PUNCT
cana-2085	79	1	graph	graph	NOUN
cana-2085	79	2	-	-	PUNCT
cana-2085	79	3	based	base	VERB
cana-2085	79	4	modelling	modelling	NOUN
cana-2085	79	5	in	in	ADP
cana-2085	79	6	biological	biological	ADJ
cana-2085	79	7	networks	network	NOUN
cana-2085	79	8	-	-	PUNCT
cana-2085	79	9	stability	stability	NOUN
cana-2085	79	10	in	in	ADP
cana-2085	79	11	ecological	ecological	ADJ
cana-2085	79	12	or	or	CCONJ
cana-2085	79	13	biological	biological	ADJ
cana-2085	79	14	systems	system	NOUN
cana-2085	79	15	:	:	PUNCT
cana-2085	79	16	in	in	ADP
cana-2085	79	17	ecological	ecological	ADJ
cana-2085	79	18	networks	network	NOUN
cana-2085	79	19	or	or	CCONJ
cana-2085	79	20	biological	biological	ADJ
cana-2085	79	21	interaction	interaction	NOUN
cana-2085	79	22	networks	network	NOUN
cana-2085	79	23	,	,	PUNCT
cana-2085	79	24	this	this	DET
cana-2085	79	25	theorem	theorem	NOUN
cana-2085	79	26	can	can	AUX
cana-2085	79	27	be	be	AUX
cana-2085	79	28	used	use	VERB
cana-2085	79	29	to	to	PART
cana-2085	79	30	ensure	ensure	VERB
cana-2085	79	31	that	that	SCONJ
cana-2085	79	32	the	the	DET
cana-2085	79	33	network	network	NOUN
cana-2085	79	34	is	be	AUX
cana-2085	79	35	stable	stable	ADJ
cana-2085	79	36	and	and	CCONJ
cana-2085	79	37	robust	robust	ADJ
cana-2085	79	38	.	.	PUNCT
cana-2085	80	1	for	for	ADP
cana-2085	80	2	example	example	NOUN
cana-2085	80	3	,	,	PUNCT
cana-2085	80	4	if	if	SCONJ
cana-2085	80	5	𝛾𝑀𝑅(𝐺	𝛾𝑀𝑅(𝐺	PROPN
cana-2085	80	6	)	)	PUNCT
cana-2085	80	7	=	=	SYM
cana-2085	81	1	3𝛾(𝐺	3𝛾(𝐺	NUM
cana-2085	81	2	)	)	PUNCT
cana-2085	81	3	holds	hold	VERB
cana-2085	81	4	,	,	PUNCT
cana-2085	81	5	it	it	PRON
cana-2085	81	6	might	might	AUX
cana-2085	81	7	indicate	indicate	VERB
cana-2085	81	8	that	that	SCONJ
cana-2085	81	9	the	the	DET
cana-2085	81	10	system	system	NOUN
cana-2085	81	11	is	be	AUX
cana-2085	81	12	free	free	ADJ
cana-2085	81	13	from	from	ADP
cana-2085	81	14	certain	certain	ADJ
cana-2085	81	15	destabilizing	destabilizing	ADJ
cana-2085	81	16	interactions	interaction	NOUN
cana-2085	81	17	or	or	CCONJ
cana-2085	81	18	species	specie	NOUN
cana-2085	81	19	(	(	PUNCT
cana-2085	81	20	represented	represent	VERB
cana-2085	81	21	by	by	ADP
cana-2085	81	22	𝑉1	𝑉1	NOUN
cana-2085	81	23	and	and	CCONJ
cana-2085	81	24	𝑉2	𝑉2	NOUN
cana-2085	81	25	)	)	PUNCT
cana-2085	81	26	,	,	PUNCT
cana-2085	81	27	leading	lead	VERB
cana-2085	81	28	to	to	ADP
cana-2085	81	29	a	a	DET
cana-2085	81	30	more	more	ADV
cana-2085	81	31	stable	stable	ADJ
cana-2085	81	32	and	and	CCONJ
cana-2085	81	33	resilient	resilient	ADJ
cana-2085	81	34	ecosystem	ecosystem	NOUN
cana-2085	81	35	or	or	CCONJ
cana-2085	81	36	biological	biological	ADJ
cana-2085	81	37	network	network	NOUN
cana-2085	81	38	[	[	X
cana-2085	81	39	25	25	NUM
cana-2085	81	40	]	]	PUNCT
cana-2085	81	41	.	.	PUNCT
cana-2085	82	1	algorithm	algorithm	PROPN
cana-2085	82	2	design	design	NOUN
cana-2085	82	3	in	in	ADP
cana-2085	82	4	graph	graph	NOUN
cana-2085	82	5	theory	theory	NOUN
cana-2085	82	6	-	-	PUNCT
cana-2085	82	7	simplified	simplify	VERB
cana-2085	82	8	algorithms	algorithm	NOUN
cana-2085	82	9	for	for	ADP
cana-2085	82	10	specific	specific	ADJ
cana-2085	82	11	graph	graph	NOUN
cana-2085	82	12	classes	class	NOUN
cana-2085	82	13	:	:	PUNCT
cana-2085	82	14	in	in	ADP
cana-2085	82	15	algorithmic	algorithmic	ADJ
cana-2085	82	16	graph	graph	NOUN
cana-2085	82	17	theory	theory	NOUN
cana-2085	82	18	,	,	PUNCT
cana-2085	82	19	this	this	DET
cana-2085	82	20	theorem	theorem	NOUN
cana-2085	82	21	provides	provide	VERB
cana-2085	82	22	a	a	DET
cana-2085	82	23	characterization	characterization	NOUN
cana-2085	82	24	that	that	PRON
cana-2085	82	25	can	can	AUX
cana-2085	82	26	be	be	AUX
cana-2085	82	27	used	use	VERB
cana-2085	82	28	to	to	PART
cana-2085	82	29	design	design	VERB
cana-2085	82	30	simplified	simplified	ADJ
cana-2085	82	31	algorithms	algorithm	NOUN
cana-2085	82	32	for	for	ADP
cana-2085	82	33	specific	specific	ADJ
cana-2085	82	34	classes	class	NOUN
cana-2085	82	35	of	of	ADP
cana-2085	82	36	graphs	graph	NOUN
cana-2085	82	37	.	.	PUNCT
cana-2085	83	1	if	if	SCONJ
cana-2085	83	2	𝛾𝑀𝑅(𝐺	𝛾𝑀𝑅(𝐺	PROPN
cana-2085	83	3	)	)	PUNCT
cana-2085	83	4	=	=	PUNCT
cana-2085	84	1	3𝛾(𝐺	3𝛾(𝐺	NUM
cana-2085	84	2	)	)	PUNCT
cana-2085	84	3	and	and	CCONJ
cana-2085	84	4	𝑉1	𝑉1	NOUN
cana-2085	84	5	=	=	SYM
cana-2085	84	6	∅	∅	NOUN
cana-2085	84	7	and	and	CCONJ
cana-2085	84	8	𝑉2	𝑉2	NOUN
cana-2085	84	9	=	=	SYM
cana-2085	84	10	∅	∅	NOUN
cana-2085	84	11	,	,	PUNCT
cana-2085	84	12	algorithms	algorithm	NOUN
cana-2085	84	13	can	can	AUX
cana-2085	84	14	be	be	AUX
cana-2085	84	15	tailored	tailor	VERB
cana-2085	84	16	to	to	PART
cana-2085	84	17	efficiently	efficiently	ADV
cana-2085	84	18	handle	handle	VERB
cana-2085	84	19	these	these	DET
cana-2085	84	20	graphs	graph	NOUN
cana-2085	84	21	,	,	PUNCT
cana-2085	84	22	taking	take	VERB
cana-2085	84	23	advantage	advantage	NOUN
cana-2085	84	24	of	of	ADP
cana-2085	84	25	their	their	PRON
cana-2085	84	26	simplified	simplified	ADJ
cana-2085	84	27	structure	structure	NOUN
cana-2085	84	28	.	.	PUNCT
cana-2085	85	1	this	this	DET
cana-2085	85	2	theorem	theorem	NOUN
cana-2085	85	3	provides	provide	VERB
cana-2085	85	4	a	a	DET
cana-2085	85	5	valuable	valuable	ADJ
cana-2085	85	6	tool	tool	NOUN
cana-2085	85	7	for	for	ADP
cana-2085	85	8	understanding	understanding	NOUN
cana-2085	85	9	when	when	SCONJ
cana-2085	85	10	certain	certain	ADJ
cana-2085	85	11	domination	domination	NOUN
cana-2085	85	12	-	-	PUNCT
cana-2085	85	13	related	relate	VERB
cana-2085	85	14	properties	property	NOUN
cana-2085	85	15	hold	hold	VERB
cana-2085	85	16	in	in	ADP
cana-2085	85	17	a	a	DET
cana-2085	85	18	graph	graph	NOUN
cana-2085	85	19	,	,	PUNCT
cana-2085	85	20	with	with	ADP
cana-2085	85	21	wide	wide	ADV
cana-2085	85	22	-	-	PUNCT
cana-2085	85	23	ranging	range	VERB
cana-2085	85	24	applications	application	NOUN
cana-2085	85	25	in	in	ADP
cana-2085	85	26	network	network	NOUN
cana-2085	85	27	design	design	NOUN
cana-2085	85	28	,	,	PUNCT
cana-2085	85	29	optimization	optimization	NOUN
cana-2085	85	30	,	,	PUNCT
cana-2085	85	31	security	security	NOUN
cana-2085	85	32	,	,	PUNCT
cana-2085	85	33	biological	biological	ADJ
cana-2085	85	34	modelling	modelling	NOUN
cana-2085	85	35	,	,	PUNCT
cana-2085	85	36	and	and	CCONJ
cana-2085	85	37	more	more	ADJ
cana-2085	85	38	.	.	PUNCT
cana-2085	86	1	the	the	DET
cana-2085	86	2	conditions	condition	NOUN
cana-2085	86	3	𝑉1	𝑉1	NOUN
cana-2085	86	4	=	=	SYM
cana-2085	86	5	∅	∅	NOUN
cana-2085	86	6	and	and	CCONJ
cana-2085	86	7	𝑉2	𝑉2	NOUN
cana-2085	86	8	=	=	PUNCT
cana-2085	86	9	∅	∅	NOUN
cana-2085	86	10	highlight	highlight	VERB
cana-2085	86	11	specific	specific	ADJ
cana-2085	86	12	configurations	configuration	NOUN
cana-2085	86	13	that	that	PRON
cana-2085	86	14	need	need	VERB
cana-2085	86	15	to	to	PART
cana-2085	86	16	be	be	AUX
cana-2085	86	17	avoided	avoid	VERB
cana-2085	86	18	to	to	PART
cana-2085	86	19	achieve	achieve	VERB
cana-2085	86	20	these	these	DET
cana-2085	86	21	properties	property	NOUN
cana-2085	86	22	,	,	PUNCT
cana-2085	86	23	offering	offer	VERB
cana-2085	86	24	practical	practical	ADJ
cana-2085	86	25	guidance	guidance	NOUN
cana-2085	86	26	in	in	ADP
cana-2085	86	27	various	various	ADJ
cana-2085	86	28	fields	field	NOUN
cana-2085	86	29	.	.	PUNCT
cana-2085	87	1	characterization	characterization	NOUN
cana-2085	87	2	of	of	ADP
cana-2085	87	3	𝜸(𝑮	𝜸(𝑮	ADJ
cana-2085	87	4	)	)	PUNCT
cana-2085	87	5	=	=	SYM
cana-2085	87	6	𝜸𝑴𝑹(𝑮	𝜸𝑴𝑹(𝑮	VERB
cana-2085	87	7	)	)	PUNCT
cana-2085	87	8	theorem	theorem	VERB
cana-2085	87	9	3	3	NUM
cana-2085	87	10	:	:	PUNCT
cana-2085	87	11	for	for	ADP
cana-2085	87	12	any	any	DET
cana-2085	87	13	graph	graph	NOUN
cana-2085	87	14	𝐺	𝐺	NOUN
cana-2085	87	15	of	of	ADP
cana-2085	87	16	order	order	NOUN
cana-2085	87	17	𝑛	𝑛	NOUN
cana-2085	87	18	,	,	PUNCT
cana-2085	87	19	𝛾(𝐺	𝛾(𝐺	NUM
cana-2085	87	20	)	)	PUNCT
cana-2085	88	1	=	=	SYM
cana-2085	88	2	𝛾𝑀𝑅(𝐺	𝛾𝑀𝑅(𝐺	NOUN
cana-2085	88	3	)	)	PUNCT
cana-2085	89	1	if	if	SCONJ
cana-2085	89	2	and	and	CCONJ
cana-2085	89	3	only	only	ADV
cana-2085	89	4	if	if	SCONJ
cana-2085	89	5	𝐺	𝐺	PROPN
cana-2085	89	6	=	=	SYM
cana-2085	89	7	𝐾𝑛	𝐾𝑛	PROPN
cana-2085	89	8	̅̅̅̅	̅̅̅̅	ADJ
cana-2085	89	9	proof	proof	NOUN
cana-2085	89	10	:	:	PUNCT
cana-2085	89	11	it	it	PRON
cana-2085	89	12	is	be	AUX
cana-2085	89	13	obvious	obvious	ADJ
cana-2085	89	14	that	that	SCONJ
cana-2085	89	15	if	if	SCONJ
cana-2085	89	16	𝐺	𝐺	PROPN
cana-2085	89	17	=	=	SYM
cana-2085	89	18	𝐾𝑛	𝐾𝑛	PROPN
cana-2085	89	19	̅̅̅̅	̅̅̅̅	ADJ
cana-2085	89	20	then	then	ADV
cana-2085	89	21	𝛾(𝐺	𝛾(𝐺	PROPN
cana-2085	89	22	)	)	PUNCT
cana-2085	90	1	=	=	SYM
cana-2085	90	2	𝛾𝑀𝑅(𝐺	𝛾𝑀𝑅(𝐺	NOUN
cana-2085	90	3	)	)	PUNCT
cana-2085	90	4	.	.	PUNCT
cana-2085	91	1	let	let	VERB
cana-2085	91	2	𝑓	𝑓	PRON
cana-2085	91	3	=	=	X
cana-2085	91	4	(	(	PUNCT
cana-2085	91	5	𝑉0	𝑉0	NOUN
cana-2085	91	6	,	,	PUNCT
cana-2085	91	7	𝑉1	𝑉1	PROPN
cana-2085	91	8	,	,	PUNCT
cana-2085	91	9	𝑉2	𝑉2	NOUN
cana-2085	91	10	,	,	PUNCT
cana-2085	91	11	𝑉3	𝑉3	NOUN
cana-2085	91	12	)	)	PUNCT
cana-2085	91	13	be	be	AUX
cana-2085	91	14	a	a	DET
cana-2085	91	15	𝛾𝑀𝑅(𝐺)function	𝛾𝑀𝑅(𝐺)function	NOUN
cana-2085	91	16	.	.	PUNCT
cana-2085	92	1	the	the	DET
cana-2085	92	2	equality	equality	NOUN
cana-2085	92	3	𝛾(𝐺	𝛾(𝐺	NOUN
cana-2085	92	4	)	)	PUNCT
cana-2085	93	1	=	=	SYM
cana-2085	93	2	𝛾𝑀𝑅(𝐺	𝛾𝑀𝑅(𝐺	NOUN
cana-2085	93	3	)	)	PUNCT
cana-2085	93	4	implies	imply	VERB
cana-2085	93	5	that	that	SCONJ
cana-2085	93	6	we	we	PRON
cana-2085	93	7	have	have	VERB
cana-2085	93	8	equality	equality	NOUN
cana-2085	93	9	in	in	ADP
cana-2085	93	10	𝛾(𝐺	𝛾(𝐺	NOUN
cana-2085	93	11	)	)	PUNCT
cana-2085	93	12	≤	≤	NOUN
cana-2085	93	13	|𝑉1|	|𝑉1|	NOUN
cana-2085	94	1	+	+	NUM
cana-2085	94	2	|𝑉2|	|𝑉2|	X
cana-2085	94	3	+	+	CCONJ
cana-2085	94	4	|𝑉3|	|𝑉3|	PRON
cana-2085	94	5	=	=	SYM
cana-2085	94	6	|𝑉1|	|𝑉1|	PROPN
cana-2085	95	1	+	+	NUM
cana-2085	95	2	|𝑉2|	|𝑉2|	X
cana-2085	95	3	+	+	CCONJ
cana-2085	95	4	3|𝑉3|	3|𝑉3|	NUM
cana-2085	95	5	=	=	SYM
cana-2085	95	6	𝛾𝑀𝑅(𝐺	𝛾𝑀𝑅(𝐺	NOUN
cana-2085	95	7	)	)	PUNCT
cana-2085	95	8	.	.	PUNCT
cana-2085	96	1	hence	hence	ADV
cana-2085	96	2	|𝑉3|	|𝑉3|	PUNCT
cana-2085	96	3	=	=	SYM
cana-2085	96	4	0	0	NUM
cana-2085	96	5	,	,	PUNCT
cana-2085	96	6	which	which	PRON
cana-2085	96	7	implies	imply	VERB
cana-2085	96	8	that	that	SCONJ
cana-2085	96	9	𝑉0	𝑉0	NOUN
cana-2085	96	10	=	=	PUNCT
cana-2085	96	11	∅.	∅.	VERB
cana-2085	96	12	therefore	therefore	ADV
cana-2085	96	13	𝛾𝑀𝑅(𝐺	𝛾𝑀𝑅(𝐺	PROPN
cana-2085	96	14	)	)	PUNCT
cana-2085	96	15	=	=	SYM
cana-2085	97	1	|𝑉1|	|𝑉1|	PROPN
cana-2085	98	1	+	+	NUM
cana-2085	98	2	|𝑉2|	|𝑉2|	NOUN
cana-2085	98	3	=	=	SYM
cana-2085	98	4	|𝑉|	|𝑉|	NOUN
cana-2085	98	5	=	=	PUNCT
cana-2085	98	6	𝑛.	𝑛.	NOUN
cana-2085	98	7	this	this	PRON
cana-2085	98	8	implies	imply	VERB
cana-2085	98	9	that	that	SCONJ
cana-2085	98	10	𝐺	𝐺	PROPN
cana-2085	98	11	=	=	SYM
cana-2085	98	12	𝐾𝑛	𝐾𝑛	PROPN
cana-2085	98	13	̅̅̅̅	̅̅̅̅	VERB
cana-2085	98	14	the	the	DET
cana-2085	98	15	theorem	theorem	ADJ
cana-2085	98	16	states	state	NOUN
cana-2085	98	17	that	that	SCONJ
cana-2085	98	18	for	for	ADP
cana-2085	98	19	any	any	DET
cana-2085	98	20	graph	graph	NOUN
cana-2085	98	21	𝐺	𝐺	NOUN
cana-2085	98	22	of	of	ADP
cana-2085	98	23	order	order	NOUN
cana-2085	98	24	𝑛	𝑛	ADP
cana-2085	98	25	,	,	PUNCT
cana-2085	98	26	the	the	DET
cana-2085	98	27	equality	equality	NOUN
cana-2085	98	28	𝛾𝑀𝑅(𝐺	𝛾𝑀𝑅(𝐺	PROPN
cana-2085	98	29	)	)	PUNCT
cana-2085	98	30	holds	hold	VERB
cana-2085	98	31	if	if	SCONJ
cana-2085	98	32	and	and	CCONJ
cana-2085	98	33	only	only	ADV
cana-2085	98	34	if	if	SCONJ
cana-2085	98	35	𝐺	𝐺	PROPN
cana-2085	98	36	is	be	AUX
cana-2085	98	37	the	the	DET
cana-2085	98	38	complement	complement	NOUN
cana-2085	98	39	of	of	ADP
cana-2085	98	40	the	the	DET
cana-2085	98	41	complete	complete	ADJ
cana-2085	98	42	graph	graph	NOUN
cana-2085	98	43	𝐾𝑛.	𝐾𝑛.	PROPN
cana-2085	98	44	this	this	DET
cana-2085	98	45	result	result	NOUN
cana-2085	98	46	has	have	VERB
cana-2085	98	47	specific	specific	ADJ
cana-2085	98	48	implications	implication	NOUN
cana-2085	98	49	in	in	ADP
cana-2085	98	50	areas	area	NOUN
cana-2085	98	51	where	where	SCONJ
cana-2085	98	52	graph	graph	NOUN
cana-2085	98	53	communications	communication	NOUN
cana-2085	98	54	on	on	ADP
cana-2085	98	55	applied	apply	VERB
cana-2085	98	56	nonlinear	nonlinear	ADJ
cana-2085	98	57	analysis	analysis	NOUN
cana-2085	98	58	issn	issn	NOUN
cana-2085	98	59	:	:	PUNCT
cana-2085	98	60	1074	1074	NUM
cana-2085	98	61	-	-	PUNCT
cana-2085	98	62	133x	133x	NUM
cana-2085	98	63	vol	vol	NOUN
cana-2085	98	64	32	32	NUM
cana-2085	99	1	no	no	NOUN
cana-2085	99	2	.	.	PUNCT
cana-2085	100	1	1s	1s	NUM
cana-2085	100	2	(	(	PUNCT
cana-2085	100	3	2025	2025	NUM
cana-2085	100	4	)	)	PUNCT
cana-2085	100	5	20	20	NUM
cana-2085	100	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-2085	100	7	structure	structure	NOUN
cana-2085	100	8	plays	play	VERB
cana-2085	100	9	a	a	DET
cana-2085	100	10	crucial	crucial	ADJ
cana-2085	100	11	role	role	NOUN
cana-2085	100	12	and	and	CCONJ
cana-2085	100	13	the	the	DET
cana-2085	100	14	applications	application	NOUN
cana-2085	100	15	are	be	AUX
cana-2085	100	16	as	as	SCONJ
cana-2085	100	17	follows	follow	VERB
cana-2085	100	18	network	network	NOUN
cana-2085	100	19	security	security	NOUN
cana-2085	100	20	and	and	CCONJ
cana-2085	100	21	vulnerability	vulnerability	NOUN
cana-2085	100	22	analysis	analysis	NOUN
cana-2085	100	23	-	-	PUNCT
cana-2085	100	24	detection	detection	NOUN
cana-2085	100	25	of	of	ADP
cana-2085	100	26	isolated	isolated	ADJ
cana-2085	100	27	nodes	node	NOUN
cana-2085	100	28	:	:	PUNCT
cana-2085	100	29	in	in	ADP
cana-2085	100	30	network	network	NOUN
cana-2085	100	31	security	security	NOUN
cana-2085	100	32	,	,	PUNCT
cana-2085	100	33	this	this	DET
cana-2085	100	34	theorem	theorem	NOUN
cana-2085	100	35	can	can	AUX
cana-2085	100	36	be	be	AUX
cana-2085	100	37	used	use	VERB
cana-2085	100	38	to	to	PART
cana-2085	100	39	identify	identify	VERB
cana-2085	100	40	networks	network	NOUN
cana-2085	100	41	(	(	PUNCT
cana-2085	100	42	or	or	CCONJ
cana-2085	100	43	parts	part	NOUN
cana-2085	100	44	of	of	ADP
cana-2085	100	45	networks	network	NOUN
cana-2085	100	46	)	)	PUNCT
cana-2085	100	47	that	that	PRON
cana-2085	100	48	are	be	AUX
cana-2085	100	49	completely	completely	ADV
cana-2085	100	50	disconnected	disconnected	ADJ
cana-2085	100	51	.	.	PUNCT
cana-2085	101	1	the	the	DET
cana-2085	101	2	complement	complement	NOUN
cana-2085	101	3	of	of	ADP
cana-2085	101	4	a	a	DET
cana-2085	101	5	complete	complete	ADJ
cana-2085	101	6	graph	graph	NOUN
cana-2085	101	7	has	have	VERB
cana-2085	101	8	no	no	DET
cana-2085	101	9	edges	edge	NOUN
cana-2085	101	10	,	,	PUNCT
cana-2085	101	11	meaning	mean	VERB
cana-2085	101	12	each	each	DET
cana-2085	101	13	node	node	NOUN
cana-2085	101	14	is	be	AUX
cana-2085	101	15	isolated	isolate	VERB
cana-2085	101	16	.	.	PUNCT
cana-2085	102	1	if	if	SCONJ
cana-2085	102	2	𝛾(𝐺	𝛾(𝐺	NUM
cana-2085	102	3	)	)	PUNCT
cana-2085	103	1	=	=	SYM
cana-2085	103	2	𝛾𝑀𝑅(𝐺	𝛾𝑀𝑅(𝐺	NOUN
cana-2085	103	3	)	)	PUNCT
cana-2085	103	4	,	,	PUNCT
cana-2085	103	5	it	it	PRON
cana-2085	103	6	implies	imply	VERB
cana-2085	103	7	the	the	DET
cana-2085	103	8	network	network	NOUN
cana-2085	103	9	's	's	PART
cana-2085	103	10	vulnerability	vulnerability	NOUN
cana-2085	103	11	to	to	ADP
cana-2085	103	12	isolation	isolation	NOUN
cana-2085	103	13	,	,	PUNCT
cana-2085	103	14	which	which	PRON
cana-2085	103	15	could	could	AUX
cana-2085	103	16	be	be	AUX
cana-2085	103	17	critical	critical	ADJ
cana-2085	103	18	in	in	ADP
cana-2085	103	19	detecting	detect	VERB
cana-2085	103	20	potential	potential	ADJ
cana-2085	103	21	points	point	NOUN
cana-2085	103	22	of	of	ADP
cana-2085	103	23	failure	failure	NOUN
cana-2085	103	24	or	or	CCONJ
cana-2085	103	25	attack	attack	NOUN
cana-2085	103	26	.	.	PUNCT
cana-2085	104	1	social	social	ADJ
cana-2085	104	2	network	network	NOUN
cana-2085	104	3	analysis	analysis	NOUN
cana-2085	104	4	-	-	PUNCT
cana-2085	104	5	analysis	analysis	NOUN
cana-2085	104	6	of	of	ADP
cana-2085	104	7	isolated	isolated	ADJ
cana-2085	104	8	communities	community	NOUN
cana-2085	104	9	:	:	PUNCT
cana-2085	104	10	in	in	ADP
cana-2085	104	11	social	social	ADJ
cana-2085	104	12	networks	network	NOUN
cana-2085	104	13	,	,	PUNCT
cana-2085	104	14	where	where	SCONJ
cana-2085	104	15	nodes	node	NOUN
cana-2085	104	16	represent	represent	VERB
cana-2085	104	17	individuals	individual	NOUN
cana-2085	104	18	and	and	CCONJ
cana-2085	104	19	edges	edge	NOUN
cana-2085	104	20	represent	represent	VERB
cana-2085	104	21	interactions	interaction	NOUN
cana-2085	104	22	,	,	PUNCT
cana-2085	104	23	this	this	DET
cana-2085	104	24	theorem	theorem	NOUN
cana-2085	104	25	helps	helps	AUX
cana-2085	104	26	identify	identify	VERB
cana-2085	104	27	when	when	SCONJ
cana-2085	104	28	a	a	DET
cana-2085	104	29	community	community	NOUN
cana-2085	104	30	(	(	PUNCT
cana-2085	104	31	represented	represent	VERB
cana-2085	104	32	by	by	ADP
cana-2085	104	33	a	a	DET
cana-2085	104	34	graph	graph	NOUN
cana-2085	104	35	𝐺	𝐺	NOUN
cana-2085	104	36	)	)	PUNCT
cana-2085	104	37	has	have	VERB
cana-2085	104	38	no	no	DET
cana-2085	104	39	internal	internal	ADJ
cana-2085	104	40	interactions	interaction	NOUN
cana-2085	104	41	(	(	PUNCT
cana-2085	104	42	as	as	SCONJ
cana-2085	104	43	indicated	indicate	VERB
cana-2085	104	44	by	by	ADP
cana-2085	104	45	𝐺	𝐺	PROPN
cana-2085	104	46	=	=	SYM
cana-2085	104	47	𝐾𝑛	𝐾𝑛	PROPN
cana-2085	104	48	̅̅̅̅	̅̅̅̅	ADJ
cana-2085	104	49	)	)	PUNCT
cana-2085	104	50	.	.	PUNCT
cana-2085	105	1	this	this	PRON
cana-2085	105	2	could	could	AUX
cana-2085	105	3	be	be	AUX
cana-2085	105	4	useful	useful	ADJ
cana-2085	105	5	in	in	ADP
cana-2085	105	6	sociological	sociological	ADJ
cana-2085	105	7	studies	study	NOUN
cana-2085	105	8	to	to	PART
cana-2085	105	9	identify	identify	VERB
cana-2085	105	10	isolated	isolated	ADJ
cana-2085	105	11	or	or	CCONJ
cana-2085	105	12	inactive	inactive	ADJ
cana-2085	105	13	groups	group	NOUN
cana-2085	105	14	within	within	ADP
cana-2085	105	15	a	a	DET
cana-2085	105	16	larger	large	ADJ
cana-2085	105	17	social	social	ADJ
cana-2085	105	18	structure	structure	NOUN
cana-2085	105	19	.	.	PUNCT
cana-2085	106	1	biological	biological	ADJ
cana-2085	106	2	networksstudy	networksstudy	NOUN
cana-2085	106	3	of	of	ADP
cana-2085	106	4	non	non	ADJ
cana-2085	106	5	-	-	ADJ
cana-2085	106	6	interacting	interacting	ADJ
cana-2085	106	7	species	specie	NOUN
cana-2085	106	8	or	or	CCONJ
cana-2085	106	9	genes	gene	NOUN
cana-2085	106	10	:	:	PUNCT
cana-2085	106	11	in	in	ADP
cana-2085	106	12	biological	biological	ADJ
cana-2085	106	13	networks	network	NOUN
cana-2085	106	14	,	,	PUNCT
cana-2085	106	15	such	such	ADJ
cana-2085	106	16	as	as	ADP
cana-2085	106	17	gene	gene	NOUN
cana-2085	106	18	interaction	interaction	NOUN
cana-2085	106	19	or	or	CCONJ
cana-2085	106	20	ecological	ecological	ADJ
cana-2085	106	21	networks	network	NOUN
cana-2085	106	22	,	,	PUNCT
cana-2085	106	23	this	this	DET
cana-2085	106	24	theorem	theorem	NOUN
cana-2085	106	25	can	can	AUX
cana-2085	106	26	be	be	AUX
cana-2085	106	27	applied	apply	VERB
cana-2085	106	28	to	to	PART
cana-2085	106	29	understand	understand	VERB
cana-2085	106	30	when	when	SCONJ
cana-2085	106	31	a	a	DET
cana-2085	106	32	set	set	NOUN
cana-2085	106	33	of	of	ADP
cana-2085	106	34	species	specie	NOUN
cana-2085	106	35	or	or	CCONJ
cana-2085	106	36	genes	gene	NOUN
cana-2085	106	37	are	be	AUX
cana-2085	106	38	completely	completely	ADV
cana-2085	106	39	non	non	ADJ
cana-2085	106	40	-	-	ADJ
cana-2085	106	41	interacting	interacting	ADJ
cana-2085	106	42	.	.	PUNCT
cana-2085	107	1	if	if	SCONJ
cana-2085	107	2	a	a	DET
cana-2085	107	3	biological	biological	ADJ
cana-2085	107	4	network	network	NOUN
cana-2085	107	5	is	be	AUX
cana-2085	107	6	structured	structure	VERB
cana-2085	107	7	as	as	ADP
cana-2085	107	8	𝐾𝑛	𝐾𝑛	PROPN
cana-2085	107	9	̅̅̅̅	̅̅̅̅	ADJ
cana-2085	107	10	,	,	PUNCT
cana-2085	107	11	this	this	PRON
cana-2085	107	12	suggests	suggest	VERB
cana-2085	107	13	no	no	DET
cana-2085	107	14	direct	direct	ADJ
cana-2085	107	15	interaction	interaction	NOUN
cana-2085	107	16	between	between	ADP
cana-2085	107	17	the	the	DET
cana-2085	107	18	components	component	NOUN
cana-2085	107	19	,	,	PUNCT
cana-2085	107	20	which	which	PRON
cana-2085	107	21	could	could	AUX
cana-2085	107	22	have	have	VERB
cana-2085	107	23	implications	implication	NOUN
cana-2085	107	24	for	for	ADP
cana-2085	107	25	understanding	understand	VERB
cana-2085	107	26	certain	certain	ADJ
cana-2085	107	27	biological	biological	ADJ
cana-2085	107	28	phenomena	phenomenon	NOUN
cana-2085	107	29	or	or	CCONJ
cana-2085	107	30	for	for	ADP
cana-2085	107	31	the	the	DET
cana-2085	107	32	design	design	NOUN
cana-2085	107	33	of	of	ADP
cana-2085	107	34	experiments	experiment	NOUN
cana-2085	107	35	.	.	PUNCT
cana-2085	108	1	quantum	quantum	PROPN
cana-2085	108	2	computing	computing	NOUN
cana-2085	108	3	and	and	CCONJ
cana-2085	108	4	information	information	NOUN
cana-2085	108	5	theory	theory	NOUN
cana-2085	108	6	-	-	PUNCT
cana-2085	108	7	graph	graph	NOUN
cana-2085	108	8	-	-	PUNCT
cana-2085	108	9	based	base	VERB
cana-2085	108	10	quantum	quantum	ADJ
cana-2085	108	11	state	state	NOUN
cana-2085	108	12	analysis	analysis	NOUN
cana-2085	108	13	:	:	PUNCT
cana-2085	108	14	in	in	ADP
cana-2085	108	15	quantum	quantum	ADJ
cana-2085	108	16	computing	computing	NOUN
cana-2085	108	17	,	,	PUNCT
cana-2085	108	18	where	where	SCONJ
cana-2085	108	19	graph	graph	NOUN
cana-2085	108	20	structures	structure	NOUN
cana-2085	108	21	can	can	AUX
cana-2085	108	22	represent	represent	VERB
cana-2085	108	23	quantum	quantum	ADJ
cana-2085	108	24	states	state	NOUN
cana-2085	108	25	or	or	CCONJ
cana-2085	108	26	information	information	NOUN
cana-2085	108	27	flow	flow	NOUN
cana-2085	108	28	,	,	PUNCT
cana-2085	108	29	this	this	DET
cana-2085	108	30	theorem	theorem	NOUN
cana-2085	108	31	helps	helps	AUX
cana-2085	108	32	identify	identify	VERB
cana-2085	108	33	scenarios	scenario	NOUN
cana-2085	108	34	where	where	SCONJ
cana-2085	108	35	quantum	quantum	ADJ
cana-2085	108	36	states	state	NOUN
cana-2085	108	37	or	or	CCONJ
cana-2085	108	38	bits	bit	NOUN
cana-2085	108	39	are	be	AUX
cana-2085	108	40	completely	completely	ADV
cana-2085	108	41	independent	independent	ADJ
cana-2085	108	42	,	,	PUNCT
cana-2085	108	43	corresponding	correspond	VERB
cana-2085	108	44	to	to	ADP
cana-2085	108	45	the	the	DET
cana-2085	108	46	complement	complement	NOUN
cana-2085	108	47	of	of	ADP
cana-2085	108	48	a	a	DET
cana-2085	108	49	complete	complete	ADJ
cana-2085	108	50	graph	graph	NOUN
cana-2085	108	51	.	.	PUNCT
cana-2085	109	1	this	this	PRON
cana-2085	109	2	could	could	AUX
cana-2085	109	3	be	be	AUX
cana-2085	109	4	relevant	relevant	ADJ
cana-2085	109	5	in	in	ADP
cana-2085	109	6	designing	design	VERB
cana-2085	109	7	quantum	quantum	ADJ
cana-2085	109	8	algorithms	algorithm	NOUN
cana-2085	109	9	or	or	CCONJ
cana-2085	109	10	analyzing	analyze	VERB
cana-2085	109	11	quantum	quantum	ADJ
cana-2085	109	12	entanglement	entanglement	NOUN
cana-2085	109	13	properties	property	NOUN
cana-2085	109	14	.	.	PUNCT
cana-2085	110	1	optimization	optimization	NOUN
cana-2085	110	2	of	of	ADP
cana-2085	110	3	isolated	isolated	ADJ
cana-2085	110	4	systems	system	NOUN
cana-2085	110	5	-	-	PUNCT
cana-2085	110	6	design	design	NOUN
cana-2085	110	7	of	of	ADP
cana-2085	110	8	independent	independent	ADJ
cana-2085	110	9	subsystems	subsystem	NOUN
cana-2085	110	10	[	[	X
cana-2085	110	11	33	33	NUM
cana-2085	110	12	]	]	X
cana-2085	110	13	:	:	PUNCT
cana-2085	110	14	in	in	ADP
cana-2085	110	15	systems	system	NOUN
cana-2085	110	16	design	design	NOUN
cana-2085	110	17	,	,	PUNCT
cana-2085	110	18	particularly	particularly	ADV
cana-2085	110	19	where	where	SCONJ
cana-2085	110	20	subsystems	subsystem	NOUN
cana-2085	110	21	must	must	AUX
cana-2085	110	22	operate	operate	VERB
cana-2085	110	23	independently	independently	ADV
cana-2085	110	24	(	(	PUNCT
cana-2085	110	25	such	such	ADJ
cana-2085	110	26	as	as	ADP
cana-2085	110	27	in	in	ADP
cana-2085	110	28	modular	modular	ADJ
cana-2085	110	29	robotics	robotic	NOUN
cana-2085	110	30	or	or	CCONJ
cana-2085	110	31	independent	independent	ADJ
cana-2085	110	32	software	software	NOUN
cana-2085	110	33	components	component	NOUN
cana-2085	110	34	)	)	PUNCT
cana-2085	110	35	,	,	PUNCT
cana-2085	110	36	this	this	DET
cana-2085	110	37	theorem	theorem	NOUN
cana-2085	110	38	can	can	AUX
cana-2085	110	39	guide	guide	VERB
cana-2085	110	40	the	the	DET
cana-2085	110	41	design	design	NOUN
cana-2085	110	42	to	to	PART
cana-2085	110	43	ensure	ensure	VERB
cana-2085	110	44	that	that	SCONJ
cana-2085	110	45	no	no	DET
cana-2085	110	46	interaction	interaction	NOUN
cana-2085	110	47	occurs	occur	VERB
cana-2085	110	48	between	between	ADP
cana-2085	110	49	subsystems	subsystem	NOUN
cana-2085	110	50	,	,	PUNCT
cana-2085	110	51	modelled	model	VERB
cana-2085	110	52	by	by	ADP
cana-2085	110	53	𝐺	𝐺	PROPN
cana-2085	110	54	=	=	SYM
cana-2085	110	55	𝐾𝑛	𝐾𝑛	PROPN
cana-2085	110	56	̅̅̅̅	̅̅̅̅	ADJ
cana-2085	110	57	.	.	PUNCT
cana-2085	111	1	this	this	DET
cana-2085	111	2	theorem	theorem	NOUN
cana-2085	111	3	is	be	AUX
cana-2085	111	4	significant	significant	ADJ
cana-2085	111	5	in	in	ADP
cana-2085	111	6	identifying	identify	VERB
cana-2085	111	7	when	when	SCONJ
cana-2085	111	8	a	a	DET
cana-2085	111	9	graph	graph	NOUN
cana-2085	111	10	structure	structure	NOUN
cana-2085	111	11	corresponds	correspond	VERB
cana-2085	111	12	to	to	ADP
cana-2085	111	13	a	a	DET
cana-2085	111	14	completely	completely	ADV
cana-2085	111	15	independent	independent	ADJ
cana-2085	111	16	set	set	NOUN
cana-2085	111	17	of	of	ADP
cana-2085	111	18	vertices	vertex	NOUN
cana-2085	111	19	(	(	PUNCT
cana-2085	111	20	no	no	DET
cana-2085	111	21	edges	edge	NOUN
cana-2085	111	22	)	)	PUNCT
cana-2085	111	23	,	,	PUNCT
cana-2085	111	24	which	which	PRON
cana-2085	111	25	has	have	VERB
cana-2085	111	26	applications	application	NOUN
cana-2085	111	27	across	across	ADP
cana-2085	111	28	various	various	ADJ
cana-2085	111	29	fields	field	NOUN
cana-2085	111	30	where	where	SCONJ
cana-2085	111	31	isolation	isolation	NOUN
cana-2085	111	32	or	or	CCONJ
cana-2085	111	33	independence	independence	NOUN
cana-2085	111	34	of	of	ADP
cana-2085	111	35	components	component	NOUN
cana-2085	111	36	is	be	AUX
cana-2085	111	37	a	a	DET
cana-2085	111	38	key	key	ADJ
cana-2085	111	39	consideration	consideration	NOUN
cana-2085	111	40	.	.	PUNCT
cana-2085	112	1	theorem	theorem	ADJ
cana-2085	112	2	4	4	NUM
cana-2085	112	3	:	:	PUNCT
cana-2085	112	4	for	for	ADP
cana-2085	112	5	any	any	DET
cana-2085	112	6	complete	complete	ADJ
cana-2085	112	7	bipartite	bipartite	NOUN
cana-2085	112	8	graph	graph	NOUN
cana-2085	112	9	𝛾𝑀𝑅(𝐺	𝛾𝑀𝑅(𝐺	PROPN
cana-2085	112	10	)	)	PUNCT
cana-2085	112	11	=	=	PRON
cana-2085	112	12	{	{	PUNCT
cana-2085	112	13	4	4	NUM
cana-2085	112	14	,	,	PUNCT
cana-2085	112	15	min(|𝑋|	min(|𝑋|	NOUN
cana-2085	112	16	,	,	PUNCT
cana-2085	112	17	|𝑌|	|𝑌|	NOUN
cana-2085	112	18	)	)	PUNCT
cana-2085	112	19	=	=	SYM
cana-2085	112	20	2	2	NUM
cana-2085	112	21	5	5	NUM
cana-2085	112	22	,	,	PUNCT
cana-2085	112	23	min(|𝑋|	min(|𝑋|	ADJ
cana-2085	112	24	,	,	PUNCT
cana-2085	112	25	|𝑌|	|𝑌|	NOUN
cana-2085	112	26	)	)	PUNCT
cana-2085	112	27	=	=	SYM
cana-2085	112	28	3	3	NUM
cana-2085	112	29	6	6	NUM
cana-2085	112	30	,	,	PUNCT
cana-2085	112	31	min(|𝑋|	min(|𝑋|	NOUN
cana-2085	112	32	,	,	PUNCT
cana-2085	112	33	|𝑌|	|𝑌|	NOUN
cana-2085	112	34	)	)	PUNCT
cana-2085	112	35	=	=	SYM
cana-2085	112	36	4	4	NUM
cana-2085	112	37	the	the	DET
cana-2085	112	38	theorem	theorem	NOUN
cana-2085	112	39	describes	describe	VERB
cana-2085	112	40	the	the	DET
cana-2085	112	41	middle	middle	ADJ
cana-2085	112	42	roman	roman	ADJ
cana-2085	112	43	domination	domination	NOUN
cana-2085	112	44	number	number	NOUN
cana-2085	112	45	𝛾𝑀𝑅(𝐺	𝛾𝑀𝑅(𝐺	PROPN
cana-2085	112	46	)	)	PUNCT
cana-2085	112	47	for	for	ADP
cana-2085	112	48	complete	complete	ADJ
cana-2085	112	49	bipartite	bipartite	PROPN
cana-2085	112	50	graphs	graph	NOUN
cana-2085	112	51	𝐺	𝐺	PROPN
cana-2085	112	52	=	=	SYM
cana-2085	112	53	𝐾|𝑋||𝑌|	𝐾|𝑋||𝑌|	PROPN
cana-2085	112	54	,	,	PUNCT
cana-2085	112	55	where	where	SCONJ
cana-2085	112	56	the	the	DET
cana-2085	112	57	vertex	vertex	NOUN
cana-2085	112	58	set	set	NOUN
cana-2085	112	59	is	be	AUX
cana-2085	112	60	partitioned	partition	VERB
cana-2085	112	61	into	into	ADP
cana-2085	112	62	two	two	NUM
cana-2085	112	63	independent	independent	ADJ
cana-2085	112	64	sets	set	NOUN
cana-2085	112	65	𝑋	𝑋	PROPN
cana-2085	112	66	and	and	CCONJ
cana-2085	112	67	𝑌.	𝑌.	PROPN
cana-2085	112	68	the	the	DET
cana-2085	112	69	theorem	theorem	ADJ
cana-2085	112	70	states	state	NOUN
cana-2085	112	71	that	that	SCONJ
cana-2085	112	72	𝛾𝑀𝑅(𝐺	𝛾𝑀𝑅(𝐺	NOUN
cana-2085	112	73	)	)	PUNCT
cana-2085	112	74	takes	take	VERB
cana-2085	112	75	the	the	DET
cana-2085	112	76	following	follow	VERB
cana-2085	112	77	values	value	NOUN
cana-2085	112	78	based	base	VERB
cana-2085	112	79	on	on	ADP
cana-2085	112	80	the	the	DET
cana-2085	112	81	sizes	size	NOUN
cana-2085	112	82	of	of	ADP
cana-2085	112	83	the	the	DET
cana-2085	112	84	partitions	partition	NOUN
cana-2085	112	85	:	:	PUNCT
cana-2085	112	86	•	•	NUM
cana-2085	112	87	𝛾𝑀𝑅(𝐺	𝛾𝑀𝑅(𝐺	NOUN
cana-2085	112	88	)	)	PUNCT
cana-2085	112	89	=	=	SYM
cana-2085	112	90	4	4	NUM
cana-2085	112	91	,	,	PUNCT
cana-2085	112	92	min(|𝑋|	min(|𝑋|	ADJ
cana-2085	112	93	,	,	PUNCT
cana-2085	112	94	|𝑌|	|𝑌|	NOUN
cana-2085	112	95	)	)	PUNCT
cana-2085	112	96	=	=	SYM
cana-2085	113	1	2	2	NUM
cana-2085	113	2	•	•	NUM
cana-2085	113	3	𝛾𝑀𝑅(𝐺	𝛾𝑀𝑅(𝐺	PROPN
cana-2085	113	4	)	)	PUNCT
cana-2085	113	5	=	=	SYM
cana-2085	113	6	5	5	NUM
cana-2085	113	7	,	,	PUNCT
cana-2085	113	8	min(|𝑋|	min(|𝑋|	ADJ
cana-2085	113	9	,	,	PUNCT
cana-2085	113	10	|𝑌|	|𝑌|	NOUN
cana-2085	113	11	)	)	PUNCT
cana-2085	113	12	=	=	SYM
cana-2085	113	13	3	3	NUM
cana-2085	113	14	,	,	PUNCT
cana-2085	113	15	•	•	X
cana-2085	113	16	𝛾𝑀𝑅(𝐺	𝛾𝑀𝑅(𝐺	NOUN
cana-2085	113	17	)	)	PUNCT
cana-2085	113	18	=	=	SYM
cana-2085	113	19	6	6	NUM
cana-2085	113	20	,	,	PUNCT
cana-2085	113	21	min(|𝑋|	min(|𝑋|	ADJ
cana-2085	113	22	,	,	PUNCT
cana-2085	113	23	|𝑌|	|𝑌|	NOUN
cana-2085	113	24	)	)	PUNCT
cana-2085	113	25	=	=	PUNCT
cana-2085	114	1	4	4	X
cana-2085	114	2	.	.	NOUN
cana-2085	114	3	which	which	PRON
cana-2085	114	4	can	can	AUX
cana-2085	114	5	provide	provide	VERB
cana-2085	114	6	the	the	DET
cana-2085	114	7	following	follow	VERB
cana-2085	114	8	applications	application	NOUN
cana-2085	114	9	bipartite	bipartite	PROPN
cana-2085	114	10	network	network	NOUN
cana-2085	114	11	design	design	NOUN
cana-2085	114	12	-	-	PUNCT
cana-2085	114	13	optimal	optimal	ADJ
cana-2085	114	14	resource	resource	NOUN
cana-2085	114	15	allocation	allocation	NOUN
cana-2085	114	16	:	:	PUNCT
cana-2085	114	17	in	in	ADP
cana-2085	114	18	designing	design	VERB
cana-2085	114	19	bipartite	bipartite	PROPN
cana-2085	114	20	networks	network	NOUN
cana-2085	114	21	,	,	PUNCT
cana-2085	114	22	such	such	ADJ
cana-2085	114	23	as	as	ADP
cana-2085	114	24	communication	communication	NOUN
cana-2085	114	25	networks	network	NOUN
cana-2085	114	26	or	or	CCONJ
cana-2085	114	27	supply	supply	NOUN
cana-2085	114	28	chains	chain	NOUN
cana-2085	114	29	,	,	PUNCT
cana-2085	114	30	this	this	DET
cana-2085	114	31	theorem	theorem	NOUN
cana-2085	114	32	provides	provide	VERB
cana-2085	114	33	a	a	DET
cana-2085	114	34	clear	clear	ADJ
cana-2085	114	35	guideline	guideline	NOUN
cana-2085	114	36	on	on	ADP
cana-2085	114	37	the	the	DET
cana-2085	114	38	minimal	minimal	ADJ
cana-2085	114	39	resources	resource	NOUN
cana-2085	114	40	required	require	VERB
cana-2085	114	41	to	to	PART
cana-2085	114	42	dominate	dominate	VERB
cana-2085	114	43	the	the	DET
cana-2085	114	44	entire	entire	ADJ
cana-2085	114	45	network	network	NOUN
cana-2085	114	46	.	.	PUNCT
cana-2085	115	1	for	for	ADP
cana-2085	115	2	example	example	NOUN
cana-2085	115	3	,	,	PUNCT
cana-2085	115	4	in	in	ADP
cana-2085	115	5	a	a	DET
cana-2085	115	6	communication	communication	NOUN
cana-2085	115	7	network	network	NOUN
cana-2085	115	8	modelled	model	VERB
cana-2085	115	9	as	as	ADP
cana-2085	115	10	a	a	DET
cana-2085	115	11	bipartite	bipartite	NOUN
cana-2085	115	12	graph	graph	NOUN
cana-2085	115	13	,	,	PUNCT
cana-2085	115	14	knowing	know	VERB
cana-2085	115	15	the	the	DET
cana-2085	115	16	middle	middle	ADJ
cana-2085	115	17	roman	roman	ADJ
cana-2085	115	18	domination	domination	NOUN
cana-2085	115	19	number	number	NOUN
cana-2085	115	20	helps	help	VERB
cana-2085	115	21	in	in	ADP
cana-2085	115	22	optimizing	optimize	VERB
cana-2085	115	23	the	the	DET
cana-2085	115	24	placement	placement	NOUN
cana-2085	115	25	of	of	ADP
cana-2085	115	26	communication	communication	NOUN
cana-2085	115	27	nodes	node	NOUN
cana-2085	115	28	to	to	PART
cana-2085	115	29	ensure	ensure	VERB
cana-2085	115	30	minimal	minimal	ADJ
cana-2085	115	31	resource	resource	NOUN
cana-2085	115	32	usage	usage	NOUN
cana-2085	115	33	while	while	SCONJ
cana-2085	115	34	maintaining	maintain	VERB
cana-2085	115	35	network	network	NOUN
cana-2085	115	36	coverage	coverage	NOUN
cana-2085	115	37	.	.	PUNCT
cana-2085	116	1	bi	bi	ADJ
cana-2085	116	2	-	-	ADJ
cana-2085	116	3	partitioned	partition	VERB
cana-2085	116	4	social	social	ADJ
cana-2085	116	5	networks	network	NOUN
cana-2085	116	6	-	-	PUNCT
cana-2085	116	7	influence	influence	NOUN
cana-2085	116	8	and	and	CCONJ
cana-2085	116	9	control	control	NOUN
cana-2085	116	10	:	:	PUNCT
cana-2085	116	11	in	in	ADP
cana-2085	116	12	social	social	ADJ
cana-2085	116	13	networks	network	NOUN
cana-2085	116	14	with	with	ADP
cana-2085	116	15	two	two	NUM
cana-2085	116	16	distinct	distinct	ADJ
cana-2085	116	17	groups	group	NOUN
cana-2085	116	18	(	(	PUNCT
cana-2085	116	19	e.g.	e.g.	ADV
cana-2085	116	20	,	,	PUNCT
cana-2085	116	21	buyers	buyer	NOUN
cana-2085	116	22	and	and	CCONJ
cana-2085	116	23	sellers	seller	NOUN
cana-2085	116	24	in	in	ADP
cana-2085	116	25	a	a	DET
cana-2085	116	26	marketplace	marketplace	NOUN
cana-2085	116	27	)	)	PUNCT
cana-2085	116	28	,	,	PUNCT
cana-2085	116	29	this	this	DET
cana-2085	116	30	theorem	theorem	NOUN
cana-2085	116	31	provides	provide	VERB
cana-2085	116	32	insights	insight	NOUN
cana-2085	116	33	into	into	ADP
cana-2085	116	34	the	the	DET
cana-2085	116	35	minimal	minimal	ADJ
cana-2085	116	36	influence	influence	NOUN
cana-2085	116	37	needed	need	VERB
cana-2085	116	38	to	to	PART
cana-2085	116	39	control	control	VERB
cana-2085	116	40	or	or	CCONJ
cana-2085	116	41	monitor	monitor	VERB
cana-2085	116	42	the	the	DET
cana-2085	116	43	interactions	interaction	NOUN
cana-2085	116	44	between	between	ADP
cana-2085	116	45	these	these	DET
cana-2085	116	46	groups	group	NOUN
cana-2085	116	47	.	.	PUNCT
cana-2085	117	1	it	it	PRON
cana-2085	117	2	can	can	AUX
cana-2085	117	3	be	be	AUX
cana-2085	117	4	useful	useful	ADJ
cana-2085	117	5	in	in	ADP
cana-2085	117	6	marketing	marketing	NOUN
cana-2085	117	7	strategies	strategy	NOUN
cana-2085	117	8	,	,	PUNCT
cana-2085	117	9	where	where	SCONJ
cana-2085	117	10	the	the	DET
cana-2085	117	11	goal	goal	NOUN
cana-2085	117	12	is	be	AUX
cana-2085	117	13	to	to	PART
cana-2085	117	14	dominate	dominate	VERB
cana-2085	117	15	a	a	DET
cana-2085	117	16	bipartite	bipartite	ADJ
cana-2085	117	17	network	network	NOUN
cana-2085	117	18	of	of	ADP
cana-2085	117	19	consumers	consumer	NOUN
cana-2085	117	20	and	and	CCONJ
cana-2085	117	21	products	product	NOUN
cana-2085	117	22	.	.	PUNCT
cana-2085	118	1	sensor	sensor	NOUN
cana-2085	118	2	networks	network	NOUN
cana-2085	118	3	-	-	PUNCT
cana-2085	118	4	energy	energy	NOUN
cana-2085	118	5	-	-	PUNCT
cana-2085	118	6	efficient	efficient	ADJ
cana-2085	118	7	sensor	sensor	NOUN
cana-2085	118	8	placement	placement	NOUN
cana-2085	118	9	:	:	PUNCT
cana-2085	118	10	in	in	ADP
cana-2085	118	11	wireless	wireless	ADJ
cana-2085	118	12	sensor	sensor	NOUN
cana-2085	118	13	networks	network	NOUN
cana-2085	118	14	where	where	SCONJ
cana-2085	118	15	sensors	sensor	NOUN
cana-2085	118	16	are	be	AUX
cana-2085	118	17	deployed	deploy	VERB
cana-2085	118	18	to	to	PART
cana-2085	118	19	monitor	monitor	VERB
cana-2085	118	20	two	two	NUM
cana-2085	118	21	different	different	ADJ
cana-2085	118	22	types	type	NOUN
cana-2085	118	23	of	of	ADP
cana-2085	118	24	areas	area	NOUN
cana-2085	118	25	(	(	PUNCT
cana-2085	118	26	e.g.	e.g.	ADV
cana-2085	118	27	,	,	PUNCT
cana-2085	118	28	communications	communication	NOUN
cana-2085	118	29	on	on	ADP
cana-2085	118	30	applied	apply	VERB
cana-2085	118	31	nonlinear	nonlinear	ADJ
cana-2085	118	32	analysis	analysis	NOUN
cana-2085	118	33	issn	issn	NOUN
cana-2085	118	34	:	:	PUNCT
cana-2085	118	35	1074	1074	NUM
cana-2085	118	36	-	-	PUNCT
cana-2085	118	37	133x	133x	NUM
cana-2085	118	38	vol	vol	NOUN
cana-2085	118	39	32	32	NUM
cana-2085	119	1	no	no	NOUN
cana-2085	119	2	.	.	PUNCT
cana-2085	120	1	1s	1s	NUM
cana-2085	120	2	(	(	PUNCT
cana-2085	120	3	2025	2025	NUM
cana-2085	120	4	)	)	PUNCT
cana-2085	120	5	21	21	NUM
cana-2085	120	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-2085	120	7	indoor	indoor	ADJ
cana-2085	120	8	vs.	vs.	X
cana-2085	120	9	outdoor	outdoor	NOUN
cana-2085	120	10	)	)	PUNCT
cana-2085	120	11	,	,	PUNCT
cana-2085	120	12	the	the	DET
cana-2085	120	13	theorem	theorem	NOUN
cana-2085	120	14	can	can	AUX
cana-2085	120	15	be	be	AUX
cana-2085	120	16	used	use	VERB
cana-2085	120	17	to	to	PART
cana-2085	120	18	determine	determine	VERB
cana-2085	120	19	the	the	DET
cana-2085	120	20	minimal	minimal	ADJ
cana-2085	120	21	number	number	NOUN
cana-2085	120	22	of	of	ADP
cana-2085	120	23	sensor	sensor	NOUN
cana-2085	120	24	nodes	node	NOUN
cana-2085	120	25	required	require	VERB
cana-2085	120	26	to	to	PART
cana-2085	120	27	ensure	ensure	VERB
cana-2085	120	28	full	full	ADJ
cana-2085	120	29	coverage	coverage	NOUN
cana-2085	120	30	and	and	CCONJ
cana-2085	120	31	minimal	minimal	ADJ
cana-2085	120	32	energy	energy	NOUN
cana-2085	120	33	consumption	consumption	NOUN
cana-2085	121	1	[	[	X
cana-2085	121	2	31	31	NUM
cana-2085	121	3	,	,	PUNCT
cana-2085	121	4	32	32	NUM
cana-2085	121	5	]	]	PUNCT
cana-2085	121	6	.	.	PUNCT
cana-2085	122	1	market	market	NOUN
cana-2085	122	2	research	research	NOUN
cana-2085	122	3	and	and	CCONJ
cana-2085	122	4	analysis	analysis	NOUN
cana-2085	122	5	:	:	PUNCT
cana-2085	122	6	consumer	consumer	NOUN
cana-2085	122	7	-	-	PUNCT
cana-2085	122	8	product	product	NOUN
cana-2085	122	9	matching	matching	NOUN
cana-2085	122	10	:	:	PUNCT
cana-2085	122	11	in	in	ADP
cana-2085	122	12	market	market	NOUN
cana-2085	122	13	analysis	analysis	NOUN
cana-2085	122	14	,	,	PUNCT
cana-2085	122	15	where	where	SCONJ
cana-2085	122	16	consumer	consumer	NOUN
cana-2085	122	17	preferences	preference	NOUN
cana-2085	122	18	and	and	CCONJ
cana-2085	122	19	products	product	NOUN
cana-2085	122	20	are	be	AUX
cana-2085	122	21	modelled	model	VERB
cana-2085	122	22	as	as	ADP
cana-2085	122	23	a	a	DET
cana-2085	122	24	bipartite	bipartite	NOUN
cana-2085	122	25	graph	graph	NOUN
cana-2085	122	26	,	,	PUNCT
cana-2085	122	27	the	the	DET
cana-2085	122	28	theorem	theorem	NOUN
cana-2085	122	29	assists	assist	NOUN
cana-2085	122	30	in	in	ADP
cana-2085	122	31	understanding	understand	VERB
cana-2085	122	32	the	the	DET
cana-2085	122	33	minimal	minimal	ADJ
cana-2085	122	34	marketing	marketing	NOUN
cana-2085	122	35	effort	effort	NOUN
cana-2085	122	36	needed	need	VERB
cana-2085	122	37	to	to	PART
cana-2085	122	38	ensure	ensure	VERB
cana-2085	122	39	that	that	SCONJ
cana-2085	122	40	all	all	DET
cana-2085	122	41	product	product	NOUN
cana-2085	122	42	categories	category	NOUN
cana-2085	122	43	are	be	AUX
cana-2085	122	44	effectively	effectively	ADV
cana-2085	122	45	promoted	promote	VERB
cana-2085	122	46	to	to	ADP
cana-2085	122	47	consumers	consumer	NOUN
cana-2085	122	48	.	.	PUNCT
cana-2085	123	1	this	this	DET
cana-2085	123	2	theorem	theorem	NOUN
cana-2085	123	3	is	be	AUX
cana-2085	123	4	particularly	particularly	ADV
cana-2085	123	5	useful	useful	ADJ
cana-2085	123	6	in	in	ADP
cana-2085	123	7	applications	application	NOUN
cana-2085	123	8	where	where	SCONJ
cana-2085	123	9	bipartite	bipartite	ADJ
cana-2085	123	10	graphs	graph	NOUN
cana-2085	123	11	are	be	AUX
cana-2085	123	12	prevalent	prevalent	ADJ
cana-2085	123	13	,	,	PUNCT
cana-2085	123	14	providing	provide	VERB
cana-2085	123	15	a	a	DET
cana-2085	123	16	direct	direct	ADJ
cana-2085	123	17	method	method	NOUN
cana-2085	123	18	to	to	PART
cana-2085	123	19	evaluate	evaluate	VERB
cana-2085	123	20	the	the	DET
cana-2085	123	21	minimal	minimal	ADJ
cana-2085	123	22	resources	resource	NOUN
cana-2085	123	23	or	or	CCONJ
cana-2085	123	24	influence	influence	NOUN
cana-2085	123	25	needed	need	VERB
cana-2085	123	26	to	to	PART
cana-2085	123	27	dominate	dominate	VERB
cana-2085	123	28	such	such	ADJ
cana-2085	123	29	graphs	graph	NOUN
cana-2085	123	30	efficiently	efficiently	ADV
cana-2085	123	31	.	.	PUNCT
cana-2085	124	1	theorem	theorem	VERB
cana-2085	124	2	5	5	NUM
cana-2085	124	3	:	:	PUNCT
cana-2085	124	4	for	for	ADP
cana-2085	124	5	any	any	DET
cana-2085	124	6	double	double	ADJ
cana-2085	124	7	star	star	NOUN
cana-2085	124	8	𝑇	𝑇	PROPN
cana-2085	124	9	,	,	PUNCT
cana-2085	124	10	𝛾𝑀𝑅(𝐺	𝛾𝑀𝑅(𝐺	PROPN
cana-2085	124	11	)	)	PUNCT
cana-2085	124	12	=	=	PRON
cana-2085	124	13	{	{	PUNCT
cana-2085	124	14	4	4	NUM
cana-2085	124	15	,	,	PUNCT
cana-2085	124	16	if	if	SCONJ
cana-2085	124	17	deg(𝑢1	deg(𝑢1	NOUN
cana-2085	124	18	)	)	PUNCT
cana-2085	125	1	=	=	SYM
cana-2085	125	2	deg(𝑢2	deg(𝑢2	PROPN
cana-2085	125	3	)	)	PUNCT
cana-2085	125	4	=	=	SYM
cana-2085	125	5	2	2	NUM
cana-2085	125	6	5	5	NUM
cana-2085	125	7	,	,	PUNCT
cana-2085	125	8	if	if	SCONJ
cana-2085	125	9	deg(𝑢1	deg(𝑢1	NOUN
cana-2085	125	10	)	)	PUNCT
cana-2085	125	11	=	=	SYM
cana-2085	125	12	3	3	NUM
cana-2085	125	13	,	,	PUNCT
cana-2085	125	14	deg(𝑢2	deg(𝑢2	PROPN
cana-2085	125	15	)	)	PUNCT
cana-2085	125	16	=	=	SYM
cana-2085	125	17	2	2	NUM
cana-2085	125	18	6	6	NUM
cana-2085	125	19	,	,	PUNCT
cana-2085	125	20	if	if	SCONJ
cana-2085	125	21	deg(𝑢1	deg(𝑢1	NOUN
cana-2085	125	22	)	)	PUNCT
cana-2085	125	23	=	=	SYM
cana-2085	125	24	deg(𝑢2	deg(𝑢2	PROPN
cana-2085	125	25	)	)	PUNCT
cana-2085	125	26	=	=	SYM
cana-2085	125	27	3	3	NUM
cana-2085	125	28	the	the	DET
cana-2085	125	29	theorem	theorem	ADJ
cana-2085	125	30	specifies	specifie	NOUN
cana-2085	125	31	the	the	DET
cana-2085	125	32	middle	middle	ADJ
cana-2085	125	33	roman	roman	ADJ
cana-2085	125	34	domination	domination	NOUN
cana-2085	125	35	number	number	NOUN
cana-2085	125	36	𝛾𝑀𝑅(𝐺	𝛾𝑀𝑅(𝐺	PROPN
cana-2085	125	37	)	)	PUNCT
cana-2085	125	38	for	for	ADP
cana-2085	125	39	a	a	DET
cana-2085	125	40	double	double	ADJ
cana-2085	125	41	star	star	NOUN
cana-2085	125	42	graph	graph	NOUN
cana-2085	125	43	t	t	PROPN
cana-2085	125	44	,	,	PUNCT
cana-2085	125	45	where	where	SCONJ
cana-2085	125	46	𝑇	𝑇	PROPN
cana-2085	125	47	consists	consist	VERB
cana-2085	125	48	of	of	ADP
cana-2085	125	49	two	two	NUM
cana-2085	125	50	central	central	ADJ
cana-2085	125	51	vertices	vertex	NOUN
cana-2085	125	52	𝑢1	𝑢1	PROPN
cana-2085	125	53	and	and	CCONJ
cana-2085	125	54	𝑢2	𝑢2	PROPN
cana-2085	125	55	with	with	ADP
cana-2085	125	56	specified	specified	ADJ
cana-2085	125	57	degrees	degree	NOUN
cana-2085	125	58	.	.	PUNCT
cana-2085	126	1	the	the	DET
cana-2085	126	2	values	value	NOUN
cana-2085	126	3	of	of	ADP
cana-2085	126	4	𝛾𝑀𝑅(𝐺	𝛾𝑀𝑅(𝐺	NOUN
cana-2085	126	5	)	)	PUNCT
cana-2085	126	6	depend	depend	VERB
cana-2085	126	7	on	on	ADP
cana-2085	126	8	the	the	DET
cana-2085	126	9	degrees	degree	NOUN
cana-2085	126	10	of	of	ADP
cana-2085	126	11	these	these	DET
cana-2085	126	12	central	central	ADJ
cana-2085	126	13	vertices	vertex	NOUN
cana-2085	126	14	:	:	PUNCT
cana-2085	126	15	•	•	NUM
cana-2085	126	16	𝛾𝑀𝑅(𝐺	𝛾𝑀𝑅(𝐺	NOUN
cana-2085	126	17	)	)	PUNCT
cana-2085	126	18	=	=	SYM
cana-2085	127	1	4	4	X
cana-2085	127	2	,	,	PUNCT
cana-2085	127	3	if	if	SCONJ
cana-2085	127	4	deg(𝑢1	deg(𝑢1	NOUN
cana-2085	127	5	)	)	PUNCT
cana-2085	127	6	=	=	SYM
cana-2085	127	7	deg(𝑢2	deg(𝑢2	PROPN
cana-2085	127	8	)	)	PUNCT
cana-2085	127	9	=	=	SYM
cana-2085	127	10	2	2	NUM
cana-2085	127	11	,	,	PUNCT
cana-2085	127	12	•	•	NUM
cana-2085	127	13	𝛾𝑀𝑅(𝐺	𝛾𝑀𝑅(𝐺	NOUN
cana-2085	127	14	)	)	PUNCT
cana-2085	127	15	=	=	SYM
cana-2085	128	1	5	5	NUM
cana-2085	128	2	,	,	PUNCT
cana-2085	128	3	if	if	SCONJ
cana-2085	128	4	deg(𝑢1	deg(𝑢1	NOUN
cana-2085	128	5	)	)	PUNCT
cana-2085	128	6	=	=	SYM
cana-2085	128	7	3	3	NUM
cana-2085	128	8	,	,	PUNCT
cana-2085	128	9	deg(𝑢2	deg(𝑢2	PROPN
cana-2085	128	10	)	)	PUNCT
cana-2085	128	11	=	=	SYM
cana-2085	128	12	2	2	NUM
cana-2085	128	13	,	,	PUNCT
cana-2085	128	14	•	•	NUM
cana-2085	128	15	𝛾𝑀𝑅(𝐺	𝛾𝑀𝑅(𝐺	NOUN
cana-2085	128	16	)	)	PUNCT
cana-2085	128	17	=	=	SYM
cana-2085	129	1	6	6	NUM
cana-2085	129	2	,	,	PUNCT
cana-2085	129	3	if	if	SCONJ
cana-2085	129	4	deg(𝑢1	deg(𝑢1	NOUN
cana-2085	129	5	)	)	PUNCT
cana-2085	129	6	=	=	SYM
cana-2085	129	7	deg(𝑢2	deg(𝑢2	PROPN
cana-2085	129	8	)	)	PUNCT
cana-2085	129	9	=	=	PUNCT
cana-2085	130	1	3	3	X
cana-2085	130	2	.	.	PUNCT
cana-2085	130	3	the	the	DET
cana-2085	130	4	above	above	ADJ
cana-2085	130	5	has	have	VERB
cana-2085	130	6	various	various	ADJ
cana-2085	130	7	applications	application	NOUN
cana-2085	130	8	as	as	SCONJ
cana-2085	130	9	follows	follow	VERB
cana-2085	130	10	network	network	NOUN
cana-2085	130	11	design	design	NOUN
cana-2085	130	12	and	and	CCONJ
cana-2085	130	13	reliability	reliability	NOUN
cana-2085	130	14	-	-	PUNCT
cana-2085	130	15	optimal	optimal	ADJ
cana-2085	130	16	node	node	ADJ
cana-2085	130	17	placement	placement	NOUN
cana-2085	130	18	:	:	PUNCT
cana-2085	130	19	in	in	ADP
cana-2085	130	20	network	network	NOUN
cana-2085	130	21	design	design	NOUN
cana-2085	130	22	,	,	PUNCT
cana-2085	130	23	where	where	SCONJ
cana-2085	130	24	nodes	node	NOUN
cana-2085	130	25	represent	represent	VERB
cana-2085	130	26	key	key	ADJ
cana-2085	130	27	components	component	NOUN
cana-2085	130	28	or	or	CCONJ
cana-2085	130	29	hubs	hub	NOUN
cana-2085	130	30	,	,	PUNCT
cana-2085	130	31	understanding	understand	VERB
cana-2085	130	32	the	the	DET
cana-2085	130	33	middle	middle	ADJ
cana-2085	130	34	roman	roman	ADJ
cana-2085	130	35	domination	domination	NOUN
cana-2085	130	36	number	number	NOUN
cana-2085	130	37	of	of	ADP
cana-2085	130	38	a	a	DET
cana-2085	130	39	double	double	ADJ
cana-2085	130	40	star	star	NOUN
cana-2085	130	41	graph	graph	NOUN
cana-2085	130	42	helps	help	VERB
cana-2085	130	43	in	in	ADP
cana-2085	130	44	determining	determine	VERB
cana-2085	130	45	the	the	DET
cana-2085	130	46	minimum	minimum	ADJ
cana-2085	130	47	number	number	NOUN
cana-2085	130	48	of	of	ADP
cana-2085	130	49	resources	resource	NOUN
cana-2085	130	50	required	require	VERB
cana-2085	130	51	to	to	PART
cana-2085	130	52	ensure	ensure	VERB
cana-2085	130	53	network	network	NOUN
cana-2085	130	54	reliability	reliability	NOUN
cana-2085	130	55	and	and	CCONJ
cana-2085	130	56	coverage	coverage	NOUN
cana-2085	130	57	,	,	PUNCT
cana-2085	130	58	particularly	particularly	ADV
cana-2085	130	59	in	in	ADP
cana-2085	130	60	hub	hub	NOUN
cana-2085	130	61	-	-	PUNCT
cana-2085	130	62	and	and	CCONJ
cana-2085	130	63	-	-	PUNCT
cana-2085	130	64	spoke	speak	VERB
cana-2085	130	65	models	model	NOUN
cana-2085	130	66	where	where	SCONJ
cana-2085	130	67	two	two	NUM
cana-2085	130	68	main	main	ADJ
cana-2085	130	69	hubs	hub	NOUN
cana-2085	130	70	are	be	AUX
cana-2085	130	71	critical	critical	ADJ
cana-2085	130	72	.	.	PUNCT
cana-2085	131	1	telecommunications	telecommunication	NOUN
cana-2085	131	2	-	-	PUNCT
cana-2085	131	3	minimal	minimal	ADJ
cana-2085	131	4	resource	resource	NOUN
cana-2085	131	5	deployment	deployment	NOUN
cana-2085	131	6	:	:	PUNCT
cana-2085	131	7	in	in	ADP
cana-2085	131	8	telecommunications	telecommunication	NOUN
cana-2085	131	9	,	,	PUNCT
cana-2085	131	10	where	where	SCONJ
cana-2085	131	11	double	double	ADJ
cana-2085	131	12	star	star	NOUN
cana-2085	131	13	structures	structure	NOUN
cana-2085	131	14	may	may	AUX
cana-2085	131	15	represent	represent	VERB
cana-2085	131	16	network	network	NOUN
cana-2085	131	17	topologies	topology	NOUN
cana-2085	131	18	with	with	ADP
cana-2085	131	19	two	two	NUM
cana-2085	131	20	central	central	ADJ
cana-2085	131	21	communication	communication	NOUN
cana-2085	131	22	nodes	node	NOUN
cana-2085	131	23	,	,	PUNCT
cana-2085	131	24	the	the	DET
cana-2085	131	25	theorem	theorem	NOUN
cana-2085	131	26	helps	help	VERB
cana-2085	131	27	in	in	ADP
cana-2085	131	28	deciding	decide	VERB
cana-2085	131	29	the	the	DET
cana-2085	131	30	minimal	minimal	ADJ
cana-2085	131	31	number	number	NOUN
cana-2085	131	32	of	of	ADP
cana-2085	131	33	backup	backup	ADJ
cana-2085	131	34	resources	resource	NOUN
cana-2085	131	35	(	(	PUNCT
cana-2085	131	36	e.g.	e.g.	ADV
cana-2085	131	37	,	,	PUNCT
cana-2085	131	38	routers	router	NOUN
cana-2085	131	39	,	,	PUNCT
cana-2085	131	40	servers	server	NOUN
cana-2085	131	41	)	)	PUNCT
cana-2085	131	42	needed	need	VERB
cana-2085	131	43	to	to	PART
cana-2085	131	44	maintain	maintain	VERB
cana-2085	131	45	network	network	NOUN
cana-2085	131	46	functionality	functionality	NOUN
cana-2085	131	47	in	in	ADP
cana-2085	131	48	case	case	NOUN
cana-2085	131	49	of	of	ADP
cana-2085	131	50	failures	failure	NOUN
cana-2085	131	51	.	.	PUNCT
cana-2085	132	1	biological	biological	ADJ
cana-2085	132	2	networksgene	networksgene	NOUN
cana-2085	132	3	interaction	interaction	NOUN
cana-2085	132	4	networks	network	NOUN
cana-2085	132	5	:	:	PUNCT
cana-2085	132	6	in	in	ADP
cana-2085	132	7	gene	gene	NOUN
cana-2085	132	8	interaction	interaction	NOUN
cana-2085	132	9	networks	network	NOUN
cana-2085	132	10	where	where	SCONJ
cana-2085	132	11	two	two	NUM
cana-2085	132	12	key	key	ADJ
cana-2085	132	13	genes	gene	NOUN
cana-2085	132	14	regulate	regulate	VERB
cana-2085	132	15	other	other	ADJ
cana-2085	132	16	genes	gene	NOUN
cana-2085	132	17	,	,	PUNCT
cana-2085	132	18	modelled	model	VERB
cana-2085	132	19	as	as	ADP
cana-2085	132	20	a	a	DET
cana-2085	132	21	double	double	ADJ
cana-2085	132	22	star	star	NOUN
cana-2085	132	23	graph	graph	NOUN
cana-2085	132	24	,	,	PUNCT
cana-2085	132	25	the	the	DET
cana-2085	132	26	theorem	theorem	NOUN
cana-2085	132	27	provides	provide	VERB
cana-2085	132	28	insights	insight	NOUN
cana-2085	132	29	into	into	ADP
cana-2085	132	30	the	the	DET
cana-2085	132	31	minimum	minimum	ADJ
cana-2085	132	32	interventions	intervention	NOUN
cana-2085	132	33	required	require	VERB
cana-2085	132	34	(	(	PUNCT
cana-2085	132	35	e.g.	e.g.	ADV
cana-2085	132	36	,	,	PUNCT
cana-2085	132	37	genetic	genetic	ADJ
cana-2085	132	38	modifications	modification	NOUN
cana-2085	132	39	or	or	CCONJ
cana-2085	132	40	treatments	treatment	NOUN
cana-2085	132	41	)	)	PUNCT
cana-2085	132	42	to	to	PART
cana-2085	132	43	influence	influence	VERB
cana-2085	132	44	or	or	CCONJ
cana-2085	132	45	control	control	VERB
cana-2085	132	46	the	the	DET
cana-2085	132	47	network	network	NOUN
cana-2085	132	48	effectively	effectively	ADV
cana-2085	132	49	.	.	PUNCT
cana-2085	133	1	data	datum	NOUN
cana-2085	133	2	centre	centre	PROPN
cana-2085	133	3	management	management	NOUN
cana-2085	133	4	-	-	PUNCT
cana-2085	133	5	redundancy	redundancy	NOUN
cana-2085	133	6	planning	planning	NOUN
cana-2085	133	7	:	:	PUNCT
cana-2085	133	8	in	in	ADP
cana-2085	133	9	data	data	NOUN
cana-2085	133	10	centres	centre	NOUN
cana-2085	133	11	organized	organize	VERB
cana-2085	133	12	with	with	ADP
cana-2085	133	13	two	two	NUM
cana-2085	133	14	main	main	ADJ
cana-2085	133	15	server	server	NOUN
cana-2085	133	16	clusters	cluster	NOUN
cana-2085	133	17	(	(	PUNCT
cana-2085	133	18	central	central	ADJ
cana-2085	133	19	vertices	vertex	NOUN
cana-2085	133	20	)	)	PUNCT
cana-2085	133	21	,	,	PUNCT
cana-2085	133	22	the	the	DET
cana-2085	133	23	theorem	theorem	ADJ
cana-2085	133	24	aids	aid	NOUN
cana-2085	133	25	in	in	ADP
cana-2085	133	26	determining	determine	VERB
cana-2085	133	27	the	the	DET
cana-2085	133	28	minimal	minimal	ADJ
cana-2085	133	29	redundancy	redundancy	NOUN
cana-2085	133	30	needed	need	VERB
cana-2085	133	31	to	to	PART
cana-2085	133	32	ensure	ensure	VERB
cana-2085	133	33	data	datum	NOUN
cana-2085	133	34	security	security	NOUN
cana-2085	133	35	and	and	CCONJ
cana-2085	133	36	availability	availability	NOUN
cana-2085	133	37	across	across	ADP
cana-2085	133	38	all	all	DET
cana-2085	133	39	connected	connected	ADJ
cana-2085	133	40	storage	storage	NOUN
cana-2085	133	41	units	unit	NOUN
cana-2085	133	42	or	or	CCONJ
cana-2085	133	43	servers	server	NOUN
cana-2085	133	44	.	.	PUNCT
cana-2085	134	1	transportation	transportation	NOUN
cana-2085	134	2	and	and	CCONJ
cana-2085	134	3	logistics	logistic	NOUN
cana-2085	134	4	-	-	PUNCT
cana-2085	134	5	airport	airport	NOUN
cana-2085	134	6	and	and	CCONJ
cana-2085	134	7	seaport	seaport	NOUN
cana-2085	134	8	operations	operation	NOUN
cana-2085	134	9	:	:	PUNCT
cana-2085	134	10	in	in	ADP
cana-2085	134	11	transportation	transportation	NOUN
cana-2085	134	12	logistics	logistic	NOUN
cana-2085	134	13	,	,	PUNCT
cana-2085	134	14	particularly	particularly	ADV
cana-2085	134	15	for	for	ADP
cana-2085	134	16	airports	airport	NOUN
cana-2085	134	17	or	or	CCONJ
cana-2085	134	18	seaports	seaport	NOUN
cana-2085	134	19	that	that	PRON
cana-2085	134	20	function	function	VERB
cana-2085	134	21	as	as	ADP
cana-2085	134	22	central	central	ADJ
cana-2085	134	23	hubs	hub	NOUN
cana-2085	134	24	,	,	PUNCT
cana-2085	134	25	this	this	DET
cana-2085	134	26	theorem	theorem	NOUN
cana-2085	134	27	helps	help	VERB
cana-2085	134	28	in	in	ADP
cana-2085	134	29	planning	plan	VERB
cana-2085	134	30	the	the	DET
cana-2085	134	31	minimal	minimal	ADJ
cana-2085	134	32	yet	yet	CCONJ
cana-2085	134	33	effective	effective	ADJ
cana-2085	134	34	allocation	allocation	NOUN
cana-2085	134	35	of	of	ADP
cana-2085	134	36	resources	resource	NOUN
cana-2085	134	37	like	like	ADP
cana-2085	134	38	security	security	NOUN
cana-2085	134	39	,	,	PUNCT
cana-2085	134	40	maintenance	maintenance	NOUN
cana-2085	134	41	,	,	PUNCT
cana-2085	134	42	or	or	CCONJ
cana-2085	134	43	logistics	logistic	NOUN
cana-2085	134	44	personnel	personnel	NOUN
cana-2085	134	45	to	to	PART
cana-2085	134	46	ensure	ensure	VERB
cana-2085	134	47	smooth	smooth	ADJ
cana-2085	134	48	operations	operation	NOUN
cana-2085	134	49	across	across	ADP
cana-2085	134	50	the	the	DET
cana-2085	134	51	network	network	NOUN
cana-2085	134	52	[	[	X
cana-2085	134	53	4	4	NUM
cana-2085	134	54	,	,	PUNCT
cana-2085	134	55	5	5	NUM
cana-2085	134	56	]	]	PUNCT
cana-2085	134	57	.	.	PUNCT
cana-2085	135	1	this	this	DET
cana-2085	135	2	theorem	theorem	NOUN
cana-2085	135	3	is	be	AUX
cana-2085	135	4	applicable	applicable	ADJ
cana-2085	135	5	in	in	ADP
cana-2085	135	6	various	various	ADJ
cana-2085	135	7	domains	domain	NOUN
cana-2085	135	8	where	where	SCONJ
cana-2085	135	9	the	the	DET
cana-2085	135	10	network	network	NOUN
cana-2085	135	11	or	or	CCONJ
cana-2085	135	12	structure	structure	NOUN
cana-2085	135	13	resembles	resemble	VERB
cana-2085	135	14	a	a	DET
cana-2085	135	15	double	double	ADJ
cana-2085	135	16	star	star	NOUN
cana-2085	135	17	graph	graph	NOUN
cana-2085	135	18	,	,	PUNCT
cana-2085	135	19	providing	provide	VERB
cana-2085	135	20	practical	practical	ADJ
cana-2085	135	21	insights	insight	NOUN
cana-2085	135	22	into	into	ADP
cana-2085	135	23	the	the	DET
cana-2085	135	24	minimal	minimal	ADJ
cana-2085	135	25	resource	resource	NOUN
cana-2085	135	26	allocation	allocation	NOUN
cana-2085	135	27	needed	need	VERB
cana-2085	135	28	to	to	PART
cana-2085	135	29	maintain	maintain	VERB
cana-2085	135	30	effective	effective	ADJ
cana-2085	135	31	operations	operation	NOUN
cana-2085	135	32	.	.	PUNCT
cana-2085	136	1	proof	proof	NOUN
cana-2085	136	2	is	be	AUX
cana-2085	136	3	obvious	obvious	ADJ
cana-2085	136	4	for	for	ADP
cana-2085	136	5	the	the	DET
cana-2085	136	6	above	above	ADJ
cana-2085	136	7	theorems	theorem	NOUN
cana-2085	136	8	.	.	PUNCT
cana-2085	137	1	theorem	theorem	VERB
cana-2085	137	2	6	6	NUM
cana-2085	137	3	:	:	PUNCT
cana-2085	137	4	for	for	ADP
cana-2085	137	5	any	any	DET
cana-2085	137	6	graph	graph	NOUN
cana-2085	137	7	𝐺	𝐺	NOUN
cana-2085	137	8	with	with	ADP
cana-2085	137	9	𝑛	𝑛	PROPN
cana-2085	137	10	vertices	vertex	NOUN
cana-2085	137	11	if	if	SCONJ
cana-2085	137	12	there	there	PRON
cana-2085	137	13	exists	exist	VERB
cana-2085	137	14	a	a	DET
cana-2085	137	15	vertex	vertex	NOUN
cana-2085	137	16	𝑛	𝑛	ADP
cana-2085	137	17	with	with	ADP
cana-2085	137	18	degree	degree	NOUN
cana-2085	137	19	𝑛	𝑛	PRON
cana-2085	137	20	−	−	PROPN
cana-2085	137	21	1	1	NUM
cana-2085	137	22	then	then	ADV
cana-2085	137	23	𝛾𝑀𝑅(𝐺	𝛾𝑀𝑅(𝐺	PROPN
cana-2085	137	24	)	)	PUNCT
cana-2085	137	25	=	=	SYM
cana-2085	137	26	2	2	NUM
cana-2085	137	27	𝑜𝑟	𝑜𝑟	PRON
cana-2085	137	28	3	3	NUM
cana-2085	137	29	.	.	PUNCT
cana-2085	138	1	proof	proof	NOUN
cana-2085	138	2	:	:	PUNCT
cana-2085	138	3	let	let	VERB
cana-2085	138	4	𝑣	𝑣	X
cana-2085	138	5	∈	∈	PROPN
cana-2085	138	6	𝑉(𝐺	𝑉(𝐺	NOUN
cana-2085	138	7	)	)	PUNCT
cana-2085	138	8	with	with	ADP
cana-2085	138	9	deg(𝑣	deg(𝑣	PROPN
cana-2085	138	10	)	)	PUNCT
cana-2085	138	11	=	=	SYM
cana-2085	139	1	𝑛	𝑛	PRON
cana-2085	139	2	−	−	NOUN
cana-2085	140	1	1	1	X
cana-2085	140	2	.	.	PUNCT
cana-2085	141	1	we	we	PRON
cana-2085	141	2	define	define	VERB
cana-2085	141	3	a	a	DET
cana-2085	141	4	function	function	NOUN
cana-2085	141	5	𝑓	𝑓	X
cana-2085	141	6	=	=	X
cana-2085	141	7	(	(	PUNCT
cana-2085	141	8	𝑉0	𝑉0	NOUN
cana-2085	141	9	,	,	PUNCT
cana-2085	141	10	𝑉1	𝑉1	PROPN
cana-2085	141	11	,	,	PUNCT
cana-2085	141	12	𝑉2	𝑉2	NOUN
cana-2085	141	13	,	,	PUNCT
cana-2085	141	14	𝑉3	𝑉3	NOUN
cana-2085	141	15	)	)	PUNCT
cana-2085	141	16	as	as	SCONJ
cana-2085	141	17	follows	follow	VERB
cana-2085	141	18	with	with	ADP
cana-2085	141	19	𝑛	𝑛	PROPN
cana-2085	141	20	=	=	SYM
cana-2085	141	21	3	3	NUM
cana-2085	141	22	,	,	PUNCT
cana-2085	141	23	𝑉1	𝑉1	NOUN
cana-2085	141	24	=	=	SYM
cana-2085	141	25	∅	∅	NOUN
cana-2085	141	26	,	,	PUNCT
cana-2085	141	27	𝑉2	𝑉2	NOUN
cana-2085	141	28	=	=	SYM
cana-2085	141	29	{	{	PUNCT
cana-2085	141	30	𝑣	𝑣	NOUN
cana-2085	141	31	}	}	PUNCT
cana-2085	141	32	,	,	PUNCT
cana-2085	141	33	𝑉3	𝑉3	NOUN
cana-2085	141	34	=	=	NOUN
cana-2085	141	35	∅	∅	NOUN
cana-2085	141	36	and	and	CCONJ
cana-2085	141	37	𝑉0	𝑉0	NOUN
cana-2085	141	38	=	=	SYM
cana-2085	141	39	𝑉	𝑉	PROPN
cana-2085	141	40	−	−	PROPN
cana-2085	141	41	𝑉2	𝑉2	NOUN
cana-2085	141	42	.	.	PUNCT
cana-2085	142	1	then	then	ADV
cana-2085	142	2	clearly	clearly	ADV
cana-2085	142	3	𝑓	𝑓	PRON
cana-2085	142	4	is	be	AUX
cana-2085	142	5	a	a	DET
cana-2085	142	6	mrdf	mrdf	NOUN
cana-2085	142	7	and	and	CCONJ
cana-2085	142	8	𝛾𝑀𝑅(𝐺	𝛾𝑀𝑅(𝐺	NOUN
cana-2085	142	9	)	)	PUNCT
cana-2085	142	10	=	=	SYM
cana-2085	143	1	2	2	X
cana-2085	143	2	.	.	PUNCT
cana-2085	143	3	when	when	SCONJ
cana-2085	143	4	𝑛	𝑛	PROPN
cana-2085	143	5	≥	≥	NOUN
cana-2085	143	6	4	4	NUM
cana-2085	143	7	,	,	PUNCT
cana-2085	143	8	𝑉1	𝑉1	NOUN
cana-2085	143	9	=	=	SYM
cana-2085	143	10	∅	∅	NOUN
cana-2085	143	11	,	,	PUNCT
cana-2085	143	12	𝑉2	𝑉2	NOUN
cana-2085	143	13	=	=	SYM
cana-2085	143	14	∅	∅	NOUN
cana-2085	143	15	,	,	PUNCT
cana-2085	143	16	𝑉3	𝑉3	NOUN
cana-2085	143	17	=	=	SYM
cana-2085	143	18	3	3	NUM
cana-2085	143	19	and	and	CCONJ
cana-2085	143	20	𝑉0	𝑉0	NOUN
cana-2085	143	21	=	=	SYM
cana-2085	143	22	𝑉	𝑉	PROPN
cana-2085	143	23	−	−	PROPN
cana-2085	143	24	𝑉3	𝑉3	NOUN
cana-2085	143	25	.	.	PUNCT
cana-2085	144	1	then	then	ADV
cana-2085	144	2	𝑓	𝑓	PRON
cana-2085	144	3	is	be	AUX
cana-2085	144	4	a	a	DET
cana-2085	144	5	mrdf	mrdf	NOUN
cana-2085	144	6	with	with	ADP
cana-2085	144	7	𝛾𝑀𝑅(𝐺	𝛾𝑀𝑅(𝐺	PROPN
cana-2085	144	8	)	)	PUNCT
cana-2085	144	9	=	=	PUNCT
cana-2085	145	1	3|𝑉3|	3|𝑉3|	NUM
cana-2085	145	2	=	=	SYM
cana-2085	145	3	3	3	X
cana-2085	145	4	.	.	X
cana-2085	145	5	communications	communication	NOUN
cana-2085	145	6	on	on	ADP
cana-2085	145	7	applied	apply	VERB
cana-2085	145	8	nonlinear	nonlinear	ADJ
cana-2085	145	9	analysis	analysis	NOUN
cana-2085	145	10	issn	issn	NOUN
cana-2085	145	11	:	:	PUNCT
cana-2085	145	12	1074	1074	NUM
cana-2085	145	13	-	-	PUNCT
cana-2085	145	14	133x	133x	NUM
cana-2085	145	15	vol	vol	NOUN
cana-2085	145	16	32	32	NUM
cana-2085	145	17	no	no	NOUN
cana-2085	145	18	.	.	PUNCT
cana-2085	146	1	1s	1s	NUM
cana-2085	146	2	(	(	PUNCT
cana-2085	146	3	2025	2025	NUM
cana-2085	146	4	)	)	PUNCT
cana-2085	146	5	22	22	NUM
cana-2085	146	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-2085	147	1	the	the	DET
cana-2085	147	2	theorem	theorem	ADJ
cana-2085	147	3	states	state	NOUN
cana-2085	147	4	that	that	SCONJ
cana-2085	147	5	for	for	ADP
cana-2085	147	6	any	any	DET
cana-2085	147	7	graph	graph	NOUN
cana-2085	147	8	𝐺	𝐺	NOUN
cana-2085	147	9	with	with	ADP
cana-2085	147	10	𝑛	𝑛	PROPN
cana-2085	147	11	vertices	vertex	NOUN
cana-2085	147	12	,	,	PUNCT
cana-2085	147	13	if	if	SCONJ
cana-2085	147	14	there	there	PRON
cana-2085	147	15	exists	exist	VERB
cana-2085	147	16	a	a	DET
cana-2085	147	17	vertex	vertex	NOUN
cana-2085	147	18	𝑣	𝑣	ADP
cana-2085	147	19	with	with	ADP
cana-2085	147	20	degree	degree	NOUN
cana-2085	147	21	𝑛	𝑛	PRON
cana-2085	147	22	−	−	PROPN
cana-2085	147	23	1(i.e	1(i.e	NUM
cana-2085	147	24	.	.	PUNCT
cana-2085	147	25	,	,	PUNCT
cana-2085	147	26	𝑣	𝑣	PRON
cana-2085	147	27	is	be	AUX
cana-2085	147	28	adjacent	adjacent	ADJ
cana-2085	147	29	to	to	ADP
cana-2085	147	30	every	every	DET
cana-2085	147	31	other	other	ADJ
cana-2085	147	32	vertex	vertex	NOUN
cana-2085	147	33	in	in	ADP
cana-2085	147	34	the	the	DET
cana-2085	147	35	graph	graph	NOUN
cana-2085	147	36	)	)	PUNCT
cana-2085	147	37	,	,	PUNCT
cana-2085	147	38	then	then	ADV
cana-2085	147	39	the	the	DET
cana-2085	147	40	middle	middle	ADJ
cana-2085	147	41	roman	roman	ADJ
cana-2085	147	42	domination	domination	NOUN
cana-2085	147	43	number	number	NOUN
cana-2085	147	44	𝛾𝑀𝑅(𝐺	𝛾𝑀𝑅(𝐺	PROPN
cana-2085	147	45	)	)	PUNCT
cana-2085	147	46	is	be	AUX
cana-2085	147	47	either	either	PRON
cana-2085	147	48	2	2	NUM
cana-2085	147	49	or	or	CCONJ
cana-2085	147	50	3	3	NUM
cana-2085	147	51	[	[	X
cana-2085	147	52	7	7	NUM
cana-2085	147	53	,	,	PUNCT
cana-2085	147	54	9	9	NUM
cana-2085	147	55	]	]	PUNCT
cana-2085	147	56	.	.	PUNCT
cana-2085	148	1	this	this	DET
cana-2085	148	2	situation	situation	NOUN
cana-2085	148	3	typically	typically	ADV
cana-2085	148	4	arises	arise	VERB
cana-2085	148	5	in	in	ADP
cana-2085	148	6	graphs	graph	NOUN
cana-2085	148	7	where	where	SCONJ
cana-2085	148	8	a	a	DET
cana-2085	148	9	single	single	ADJ
cana-2085	148	10	vertex	vertex	NOUN
cana-2085	148	11	has	have	VERB
cana-2085	148	12	a	a	DET
cana-2085	148	13	dominating	dominating	NOUN
cana-2085	148	14	influence	influence	NOUN
cana-2085	148	15	over	over	ADP
cana-2085	148	16	the	the	DET
cana-2085	148	17	rest	rest	NOUN
cana-2085	148	18	of	of	ADP
cana-2085	148	19	the	the	DET
cana-2085	148	20	graph	graph	NOUN
cana-2085	148	21	,	,	PUNCT
cana-2085	148	22	such	such	ADJ
cana-2085	148	23	as	as	ADP
cana-2085	148	24	in	in	ADP
cana-2085	148	25	complete	complete	ADJ
cana-2085	148	26	graphs	graph	NOUN
cana-2085	148	27	or	or	CCONJ
cana-2085	148	28	star	star	NOUN
cana-2085	148	29	graphs	graph	NOUN
cana-2085	148	30	.	.	PUNCT
cana-2085	149	1	here	here	ADV
cana-2085	149	2	are	be	AUX
cana-2085	149	3	some	some	DET
cana-2085	149	4	applications	application	NOUN
cana-2085	149	5	of	of	ADP
cana-2085	149	6	this	this	DET
cana-2085	149	7	theorem	theorem	NOUN
cana-2085	149	8	are	be	AUX
cana-2085	149	9	as	as	SCONJ
cana-2085	149	10	follows	follow	VERB
cana-2085	149	11	network	network	NOUN
cana-2085	149	12	hub	hub	NOUN
cana-2085	149	13	design	design	NOUN
cana-2085	149	14	-	-	PUNCT
cana-2085	149	15	optimal	optimal	ADJ
cana-2085	149	16	placement	placement	NOUN
cana-2085	149	17	of	of	ADP
cana-2085	149	18	central	central	ADJ
cana-2085	149	19	nodes	node	NOUN
cana-2085	149	20	:	:	PUNCT
cana-2085	149	21	in	in	ADP
cana-2085	149	22	network	network	NOUN
cana-2085	149	23	design	design	NOUN
cana-2085	149	24	,	,	PUNCT
cana-2085	149	25	especially	especially	ADV
cana-2085	149	26	in	in	ADP
cana-2085	149	27	star	star	NOUN
cana-2085	149	28	or	or	CCONJ
cana-2085	149	29	hub	hub	NOUN
cana-2085	149	30	-	-	PUNCT
cana-2085	149	31	and	and	CCONJ
cana-2085	149	32	-	-	PUNCT
cana-2085	149	33	spoke	speak	VERB
cana-2085	149	34	topologies	topology	NOUN
cana-2085	149	35	,	,	PUNCT
cana-2085	149	36	the	the	DET
cana-2085	149	37	theorem	theorem	NOUN
cana-2085	149	38	can	can	AUX
cana-2085	149	39	be	be	AUX
cana-2085	149	40	used	use	VERB
cana-2085	149	41	to	to	PART
cana-2085	149	42	determine	determine	VERB
cana-2085	149	43	the	the	DET
cana-2085	149	44	minimal	minimal	ADJ
cana-2085	149	45	effort	effort	NOUN
cana-2085	149	46	needed	need	VERB
cana-2085	149	47	to	to	PART
cana-2085	149	48	dominate	dominate	VERB
cana-2085	149	49	the	the	DET
cana-2085	149	50	entire	entire	ADJ
cana-2085	149	51	network	network	NOUN
cana-2085	149	52	.	.	PUNCT
cana-2085	150	1	if	if	SCONJ
cana-2085	150	2	a	a	DET
cana-2085	150	3	central	central	ADJ
cana-2085	150	4	hub	hub	NOUN
cana-2085	150	5	node	node	NOUN
cana-2085	150	6	is	be	AUX
cana-2085	150	7	connected	connect	VERB
cana-2085	150	8	to	to	ADP
cana-2085	150	9	all	all	DET
cana-2085	150	10	other	other	ADJ
cana-2085	150	11	nodes	node	NOUN
cana-2085	150	12	,	,	PUNCT
cana-2085	150	13	this	this	DET
cana-2085	150	14	configuration	configuration	NOUN
cana-2085	150	15	ensures	ensure	VERB
cana-2085	150	16	that	that	SCONJ
cana-2085	150	17	the	the	DET
cana-2085	150	18	middle	middle	ADJ
cana-2085	150	19	roman	roman	ADJ
cana-2085	150	20	domination	domination	NOUN
cana-2085	150	21	number	number	NOUN
cana-2085	150	22	is	be	AUX
cana-2085	150	23	minimized	minimize	VERB
cana-2085	150	24	(	(	PUNCT
cana-2085	150	25	either	either	CCONJ
cana-2085	150	26	2	2	NUM
cana-2085	150	27	or	or	CCONJ
cana-2085	150	28	3	3	NUM
cana-2085	150	29	)	)	PUNCT
cana-2085	150	30	,	,	PUNCT
cana-2085	150	31	leading	lead	VERB
cana-2085	150	32	to	to	ADP
cana-2085	150	33	efficient	efficient	ADJ
cana-2085	150	34	network	network	NOUN
cana-2085	150	35	design	design	NOUN
cana-2085	150	36	and	and	CCONJ
cana-2085	150	37	resource	resource	NOUN
cana-2085	150	38	allocation	allocation	NOUN
cana-2085	150	39	.	.	PUNCT
cana-2085	151	1	communication	communication	NOUN
cana-2085	151	2	networks	network	NOUN
cana-2085	151	3	-	-	PUNCT
cana-2085	151	4	designing	design	VERB
cana-2085	151	5	efficient	efficient	ADJ
cana-2085	151	6	broadcast	broadcast	NOUN
cana-2085	151	7	networks	network	NOUN
cana-2085	151	8	:	:	PUNCT
cana-2085	152	1	in	in	ADP
cana-2085	152	2	communication	communication	NOUN
cana-2085	152	3	networks	network	NOUN
cana-2085	152	4	,	,	PUNCT
cana-2085	152	5	where	where	SCONJ
cana-2085	152	6	one	one	NUM
cana-2085	152	7	central	central	ADJ
cana-2085	152	8	node	node	NOUN
cana-2085	152	9	needs	need	VERB
cana-2085	152	10	to	to	PART
cana-2085	152	11	broadcast	broadcast	VERB
cana-2085	152	12	to	to	ADP
cana-2085	152	13	all	all	DET
cana-2085	152	14	other	other	ADJ
cana-2085	152	15	nodes	node	NOUN
cana-2085	152	16	,	,	PUNCT
cana-2085	152	17	this	this	DET
cana-2085	152	18	theorem	theorem	NOUN
cana-2085	152	19	implies	imply	VERB
cana-2085	152	20	that	that	SCONJ
cana-2085	152	21	such	such	DET
cana-2085	152	22	a	a	DET
cana-2085	152	23	network	network	NOUN
cana-2085	152	24	is	be	AUX
cana-2085	152	25	optimally	optimally	ADV
cana-2085	152	26	dominated	dominate	VERB
cana-2085	152	27	with	with	ADP
cana-2085	152	28	minimal	minimal	ADJ
cana-2085	152	29	additional	additional	ADJ
cana-2085	152	30	resources	resource	NOUN
cana-2085	152	31	.	.	PUNCT
cana-2085	153	1	this	this	PRON
cana-2085	153	2	can	can	AUX
cana-2085	153	3	be	be	AUX
cana-2085	153	4	applied	apply	VERB
cana-2085	153	5	in	in	ADP
cana-2085	153	6	the	the	DET
cana-2085	153	7	design	design	NOUN
cana-2085	153	8	of	of	ADP
cana-2085	153	9	broadcasting	broadcasting	NOUN
cana-2085	153	10	stations	station	NOUN
cana-2085	153	11	,	,	PUNCT
cana-2085	153	12	wi	wi	PROPN
cana-2085	153	13	-	-	PUNCT
cana-2085	153	14	fi	fi	NOUN
cana-2085	153	15	networks	network	NOUN
cana-2085	153	16	,	,	PUNCT
cana-2085	153	17	or	or	CCONJ
cana-2085	153	18	cellular	cellular	ADJ
cana-2085	153	19	networks	network	NOUN
cana-2085	153	20	,	,	PUNCT
cana-2085	153	21	where	where	SCONJ
cana-2085	153	22	a	a	DET
cana-2085	153	23	central	central	ADJ
cana-2085	153	24	node	node	NOUN
cana-2085	153	25	serves	serve	VERB
cana-2085	153	26	as	as	ADP
cana-2085	153	27	a	a	DET
cana-2085	153	28	primary	primary	ADJ
cana-2085	153	29	transmitter	transmitter	NOUN
cana-2085	153	30	.	.	PUNCT
cana-2085	154	1	graph	graph	NOUN
cana-2085	154	2	theory	theory	NOUN
cana-2085	154	3	simplification	simplification	NOUN
cana-2085	154	4	-	-	PUNCT
cana-2085	154	5	characterization	characterization	NOUN
cana-2085	154	6	of	of	ADP
cana-2085	154	7	simple	simple	ADJ
cana-2085	154	8	graphs	graph	NOUN
cana-2085	155	1	:	:	PUNCT
cana-2085	155	2	this	this	DET
cana-2085	155	3	theorem	theorem	NOUN
cana-2085	155	4	provides	provide	VERB
cana-2085	155	5	a	a	DET
cana-2085	155	6	straightforward	straightforward	ADJ
cana-2085	155	7	criterion	criterion	NOUN
cana-2085	155	8	to	to	PART
cana-2085	155	9	identify	identify	VERB
cana-2085	155	10	graphs	graph	NOUN
cana-2085	155	11	with	with	ADP
cana-2085	155	12	low	low	ADJ
cana-2085	155	13	middle	middle	ADJ
cana-2085	155	14	roman	roman	ADJ
cana-2085	155	15	domination	domination	NOUN
cana-2085	155	16	numbers	number	NOUN
cana-2085	155	17	.	.	PUNCT
cana-2085	156	1	graphs	graph	NOUN
cana-2085	156	2	meeting	meet	VERB
cana-2085	156	3	the	the	DET
cana-2085	156	4	condition	condition	NOUN
cana-2085	156	5	can	can	AUX
cana-2085	156	6	be	be	AUX
cana-2085	156	7	easily	easily	ADV
cana-2085	156	8	classified	classify	VERB
cana-2085	156	9	and	and	CCONJ
cana-2085	156	10	analysed	analyse	VERB
cana-2085	156	11	,	,	PUNCT
cana-2085	156	12	simplifying	simplify	VERB
cana-2085	156	13	the	the	DET
cana-2085	156	14	study	study	NOUN
cana-2085	156	15	of	of	ADP
cana-2085	156	16	certain	certain	ADJ
cana-2085	156	17	classes	class	NOUN
cana-2085	156	18	of	of	ADP
cana-2085	156	19	graphs	graph	NOUN
cana-2085	156	20	,	,	PUNCT
cana-2085	156	21	such	such	ADJ
cana-2085	156	22	as	as	ADP
cana-2085	156	23	complete	complete	ADJ
cana-2085	156	24	or	or	CCONJ
cana-2085	156	25	nearly	nearly	ADV
cana-2085	156	26	complete	complete	ADJ
cana-2085	156	27	graphs	graph	NOUN
cana-2085	156	28	.	.	PUNCT
cana-2085	157	1	social	social	ADJ
cana-2085	157	2	network	network	NOUN
cana-2085	157	3	analysis	analysis	NOUN
cana-2085	157	4	-	-	PUNCT
cana-2085	157	5	influential	influential	ADJ
cana-2085	157	6	nodes	node	NOUN
cana-2085	157	7	identification	identification	NOUN
cana-2085	157	8	:	:	PUNCT
cana-2085	157	9	in	in	ADP
cana-2085	157	10	social	social	ADJ
cana-2085	157	11	networks	network	NOUN
cana-2085	157	12	,	,	PUNCT
cana-2085	157	13	a	a	DET
cana-2085	157	14	vertex	vertex	NOUN
cana-2085	157	15	with	with	ADP
cana-2085	157	16	degree	degree	NOUN
cana-2085	157	17	𝑛	𝑛	PRON
cana-2085	157	18	−	−	NOUN
cana-2085	157	19	1	1	NUM
cana-2085	157	20	represents	represent	VERB
cana-2085	157	21	an	an	DET
cana-2085	157	22	individual	individual	NOUN
cana-2085	157	23	connected	connect	VERB
cana-2085	157	24	to	to	ADP
cana-2085	157	25	every	every	DET
cana-2085	157	26	other	other	ADJ
cana-2085	157	27	person	person	NOUN
cana-2085	157	28	in	in	ADP
cana-2085	157	29	the	the	DET
cana-2085	157	30	network	network	NOUN
cana-2085	157	31	.	.	PUNCT
cana-2085	158	1	the	the	DET
cana-2085	158	2	theorem	theorem	NOUN
cana-2085	158	3	suggests	suggest	VERB
cana-2085	158	4	that	that	SCONJ
cana-2085	158	5	such	such	DET
cana-2085	158	6	a	a	DET
cana-2085	158	7	highly	highly	ADV
cana-2085	158	8	influential	influential	ADJ
cana-2085	158	9	individual	individual	NOUN
cana-2085	158	10	can	can	AUX
cana-2085	158	11	dominate	dominate	VERB
cana-2085	158	12	the	the	DET
cana-2085	158	13	network	network	NOUN
cana-2085	158	14	with	with	ADP
cana-2085	158	15	minimal	minimal	ADJ
cana-2085	158	16	additional	additional	ADJ
cana-2085	158	17	influence	influence	NOUN
cana-2085	158	18	𝛾𝑀𝑅(𝐺	𝛾𝑀𝑅(𝐺	NOUN
cana-2085	158	19	)	)	PUNCT
cana-2085	158	20	being	be	AUX
cana-2085	158	21	2	2	NUM
cana-2085	158	22	or	or	CCONJ
cana-2085	158	23	3	3	NUM
cana-2085	158	24	)	)	PUNCT
cana-2085	159	1	[	[	X
cana-2085	159	2	44	44	NUM
cana-2085	159	3	,	,	PUNCT
cana-2085	159	4	45	45	NUM
cana-2085	159	5	]	]	PUNCT
cana-2085	159	6	.	.	PUNCT
cana-2085	160	1	this	this	DET
cana-2085	160	2	insight	insight	NOUN
cana-2085	160	3	can	can	AUX
cana-2085	160	4	be	be	AUX
cana-2085	160	5	used	use	VERB
cana-2085	160	6	in	in	ADP
cana-2085	160	7	identifying	identify	VERB
cana-2085	160	8	key	key	ADJ
cana-2085	160	9	influencers	influencer	NOUN
cana-2085	160	10	or	or	CCONJ
cana-2085	160	11	leaders	leader	NOUN
cana-2085	160	12	in	in	ADP
cana-2085	160	13	social	social	ADJ
cana-2085	160	14	networks	network	NOUN
cana-2085	160	15	for	for	ADP
cana-2085	160	16	marketing	marketing	NOUN
cana-2085	160	17	or	or	CCONJ
cana-2085	160	18	information	information	NOUN
cana-2085	160	19	dissemination	dissemination	NOUN
cana-2085	160	20	.	.	PUNCT
cana-2085	161	1	security	security	NOUN
cana-2085	161	2	and	and	CCONJ
cana-2085	161	3	defence	defence	NOUN
cana-2085	161	4	applications	application	NOUN
cana-2085	161	5	:	:	PUNCT
cana-2085	161	6	strategic	strategic	ADJ
cana-2085	161	7	positioning	positioning	NOUN
cana-2085	161	8	:	:	PUNCT
cana-2085	161	9	in	in	ADP
cana-2085	161	10	scenarios	scenario	NOUN
cana-2085	161	11	where	where	SCONJ
cana-2085	161	12	a	a	DET
cana-2085	161	13	central	central	ADJ
cana-2085	161	14	control	control	NOUN
cana-2085	161	15	or	or	CCONJ
cana-2085	161	16	command	command	VERB
cana-2085	161	17	node	node	NOUN
cana-2085	161	18	must	must	AUX
cana-2085	161	19	oversee	oversee	VERB
cana-2085	161	20	or	or	CCONJ
cana-2085	161	21	secure	secure	VERB
cana-2085	161	22	an	an	DET
cana-2085	161	23	entire	entire	ADJ
cana-2085	161	24	network	network	NOUN
cana-2085	161	25	,	,	PUNCT
cana-2085	161	26	this	this	DET
cana-2085	161	27	theorem	theorem	NOUN
cana-2085	161	28	indicates	indicate	VERB
cana-2085	161	29	that	that	SCONJ
cana-2085	161	30	minimal	minimal	ADJ
cana-2085	161	31	resources	resource	NOUN
cana-2085	161	32	are	be	AUX
cana-2085	161	33	needed	need	VERB
cana-2085	161	34	to	to	PART
cana-2085	161	35	ensure	ensure	VERB
cana-2085	161	36	complete	complete	ADJ
cana-2085	161	37	coverage	coverage	NOUN
cana-2085	161	38	and	and	CCONJ
cana-2085	161	39	control	control	NOUN
cana-2085	161	40	.	.	PUNCT
cana-2085	162	1	this	this	PRON
cana-2085	162	2	can	can	AUX
cana-2085	162	3	apply	apply	VERB
cana-2085	162	4	to	to	ADP
cana-2085	162	5	military	military	ADJ
cana-2085	162	6	networks	network	NOUN
cana-2085	162	7	,	,	PUNCT
cana-2085	162	8	cybersecurity	cybersecurity	NOUN
cana-2085	162	9	frameworks	framework	NOUN
cana-2085	162	10	,	,	PUNCT
cana-2085	162	11	or	or	CCONJ
cana-2085	162	12	surveillance	surveillance	NOUN
cana-2085	162	13	systems	system	NOUN
cana-2085	162	14	,	,	PUNCT
cana-2085	162	15	where	where	SCONJ
cana-2085	162	16	a	a	DET
cana-2085	162	17	central	central	ADJ
cana-2085	162	18	node	node	NOUN
cana-2085	162	19	needs	need	VERB
cana-2085	162	20	to	to	PART
cana-2085	162	21	ensure	ensure	VERB
cana-2085	162	22	the	the	DET
cana-2085	162	23	security	security	NOUN
cana-2085	162	24	of	of	ADP
cana-2085	162	25	the	the	DET
cana-2085	162	26	entire	entire	ADJ
cana-2085	162	27	system	system	NOUN
cana-2085	162	28	[	[	X
cana-2085	162	29	42	42	NUM
cana-2085	162	30	,	,	PUNCT
cana-2085	162	31	43	43	NUM
cana-2085	162	32	]	]	PUNCT
cana-2085	162	33	.	.	PUNCT
cana-2085	163	1	biological	biological	ADJ
cana-2085	163	2	networks	network	NOUN
cana-2085	163	3	-	-	PUNCT
cana-2085	163	4	central	central	ADJ
cana-2085	163	5	nodes	node	NOUN
cana-2085	163	6	in	in	ADP
cana-2085	163	7	biological	biological	ADJ
cana-2085	163	8	systems	system	NOUN
cana-2085	163	9	:	:	PUNCT
cana-2085	163	10	in	in	ADP
cana-2085	163	11	biological	biological	ADJ
cana-2085	163	12	networks	network	NOUN
cana-2085	163	13	,	,	PUNCT
cana-2085	163	14	such	such	ADJ
cana-2085	163	15	as	as	ADP
cana-2085	163	16	protein	protein	NOUN
cana-2085	163	17	interaction	interaction	NOUN
cana-2085	163	18	networks	network	NOUN
cana-2085	163	19	,	,	PUNCT
cana-2085	163	20	a	a	DET
cana-2085	163	21	central	central	ADJ
cana-2085	163	22	protein	protein	NOUN
cana-2085	163	23	(	(	PUNCT
cana-2085	163	24	node	node	NOUN
cana-2085	163	25	)	)	PUNCT
cana-2085	163	26	that	that	PRON
cana-2085	163	27	interacts	interact	VERB
cana-2085	163	28	with	with	ADP
cana-2085	163	29	nearly	nearly	ADV
cana-2085	163	30	all	all	PRON
cana-2085	163	31	others	other	NOUN
cana-2085	163	32	can	can	AUX
cana-2085	163	33	ensure	ensure	VERB
cana-2085	163	34	network	network	NOUN
cana-2085	163	35	stability	stability	NOUN
cana-2085	163	36	or	or	CCONJ
cana-2085	163	37	functionality	functionality	NOUN
cana-2085	163	38	with	with	ADP
cana-2085	163	39	minimal	minimal	ADJ
cana-2085	163	40	intervention	intervention	NOUN
cana-2085	163	41	.	.	PUNCT
cana-2085	164	1	the	the	DET
cana-2085	164	2	theorem	theorem	ADJ
cana-2085	164	3	highlights	highlight	NOUN
cana-2085	164	4	the	the	DET
cana-2085	164	5	significance	significance	NOUN
cana-2085	164	6	of	of	ADP
cana-2085	164	7	such	such	ADJ
cana-2085	164	8	central	central	ADJ
cana-2085	164	9	nodes	node	NOUN
cana-2085	164	10	in	in	ADP
cana-2085	164	11	maintaining	maintain	VERB
cana-2085	164	12	biological	biological	ADJ
cana-2085	164	13	processes	process	NOUN
cana-2085	164	14	.	.	PUNCT
cana-2085	165	1	urban	urban	ADJ
cana-2085	165	2	planning	planning	NOUN
cana-2085	165	3	and	and	CCONJ
cana-2085	165	4	infrastructure	infrastructure	NOUN
cana-2085	165	5	-	-	PUNCT
cana-2085	165	6	centralized	centralize	VERB
cana-2085	165	7	service	service	NOUN
cana-2085	165	8	locations	location	NOUN
cana-2085	165	9	:	:	PUNCT
cana-2085	165	10	in	in	ADP
cana-2085	165	11	urban	urban	ADJ
cana-2085	165	12	planning	planning	NOUN
cana-2085	165	13	,	,	PUNCT
cana-2085	165	14	where	where	SCONJ
cana-2085	165	15	a	a	DET
cana-2085	165	16	central	central	ADJ
cana-2085	165	17	service	service	NOUN
cana-2085	165	18	location	location	NOUN
cana-2085	165	19	(	(	PUNCT
cana-2085	165	20	e.g.	e.g.	ADV
cana-2085	165	21	,	,	PUNCT
cana-2085	165	22	a	a	DET
cana-2085	165	23	hospital	hospital	NOUN
cana-2085	165	24	or	or	CCONJ
cana-2085	165	25	fire	fire	NOUN
cana-2085	165	26	station	station	NOUN
cana-2085	165	27	)	)	PUNCT
cana-2085	165	28	serves	serve	VERB
cana-2085	165	29	an	an	DET
cana-2085	165	30	entire	entire	ADJ
cana-2085	165	31	community	community	NOUN
cana-2085	165	32	,	,	PUNCT
cana-2085	165	33	this	this	DET
cana-2085	165	34	theorem	theorem	NOUN
cana-2085	165	35	can	can	AUX
cana-2085	165	36	help	help	VERB
cana-2085	165	37	in	in	ADP
cana-2085	165	38	determining	determine	VERB
cana-2085	165	39	the	the	DET
cana-2085	165	40	minimal	minimal	ADJ
cana-2085	165	41	additional	additional	ADJ
cana-2085	165	42	infrastructure	infrastructure	NOUN
cana-2085	165	43	required	require	VERB
cana-2085	165	44	to	to	PART
cana-2085	165	45	ensure	ensure	VERB
cana-2085	165	46	full	full	ADJ
cana-2085	165	47	coverage	coverage	NOUN
cana-2085	165	48	.	.	PUNCT
cana-2085	166	1	this	this	PRON
cana-2085	166	2	leads	lead	VERB
cana-2085	166	3	to	to	ADP
cana-2085	166	4	costeffective	costeffective	ADJ
cana-2085	166	5	and	and	CCONJ
cana-2085	166	6	efficient	efficient	ADJ
cana-2085	166	7	service	service	NOUN
cana-2085	166	8	provision	provision	NOUN
cana-2085	166	9	in	in	ADP
cana-2085	166	10	cities	city	NOUN
cana-2085	166	11	.	.	PUNCT
cana-2085	167	1	data	datum	NOUN
cana-2085	167	2	centre	centre	PROPN
cana-2085	167	3	and	and	CCONJ
cana-2085	167	4	cloud	cloud	NOUN
cana-2085	167	5	computing	computing	NOUN
cana-2085	167	6	-	-	PUNCT
cana-2085	167	7	efficient	efficient	ADJ
cana-2085	167	8	resource	resource	NOUN
cana-2085	167	9	allocation	allocation	NOUN
cana-2085	167	10	:	:	PUNCT
cana-2085	168	1	in	in	ADP
cana-2085	168	2	data	data	NOUN
cana-2085	168	3	centres	centre	NOUN
cana-2085	168	4	or	or	CCONJ
cana-2085	168	5	cloud	cloud	NOUN
cana-2085	168	6	computing	compute	VERB
cana-2085	168	7	environments	environment	NOUN
cana-2085	168	8	,	,	PUNCT
cana-2085	168	9	where	where	SCONJ
cana-2085	168	10	one	one	NUM
cana-2085	168	11	node	node	NOUN
cana-2085	168	12	acts	act	VERB
cana-2085	168	13	as	as	ADP
cana-2085	168	14	a	a	DET
cana-2085	168	15	central	central	ADJ
cana-2085	168	16	coordinator	coordinator	NOUN
cana-2085	168	17	,	,	PUNCT
cana-2085	168	18	the	the	DET
cana-2085	168	19	theorem	theorem	NOUN
cana-2085	168	20	indicates	indicate	VERB
cana-2085	168	21	that	that	SCONJ
cana-2085	168	22	minimal	minimal	ADJ
cana-2085	168	23	resources	resource	NOUN
cana-2085	168	24	are	be	AUX
cana-2085	168	25	needed	need	VERB
cana-2085	168	26	to	to	PART
cana-2085	168	27	maintain	maintain	VERB
cana-2085	168	28	overall	overall	ADJ
cana-2085	168	29	system	system	NOUN
cana-2085	168	30	performance	performance	NOUN
cana-2085	168	31	and	and	CCONJ
cana-2085	168	32	redundancy	redundancy	NOUN
cana-2085	168	33	.	.	PUNCT
cana-2085	169	1	this	this	PRON
cana-2085	169	2	can	can	AUX
cana-2085	169	3	optimize	optimize	VERB
cana-2085	169	4	the	the	DET
cana-2085	169	5	design	design	NOUN
cana-2085	169	6	and	and	CCONJ
cana-2085	169	7	operation	operation	NOUN
cana-2085	169	8	of	of	ADP
cana-2085	169	9	cloud	cloud	PROPN
cana-2085	169	10	services	service	NOUN
cana-2085	169	11	.	.	PUNCT
cana-2085	170	1	strategic	strategic	ADJ
cana-2085	170	2	game	game	NOUN
cana-2085	170	3	design	design	NOUN
cana-2085	170	4	-	-	PUNCT
cana-2085	170	5	central	central	ADJ
cana-2085	170	6	control	control	NOUN
cana-2085	170	7	in	in	ADP
cana-2085	170	8	games	game	NOUN
cana-2085	170	9	:	:	PUNCT
cana-2085	170	10	in	in	ADP
cana-2085	170	11	strategic	strategic	ADJ
cana-2085	170	12	games	game	NOUN
cana-2085	170	13	modelled	model	VERB
cana-2085	170	14	by	by	ADP
cana-2085	170	15	graphs	graph	NOUN
cana-2085	170	16	,	,	PUNCT
cana-2085	170	17	where	where	SCONJ
cana-2085	170	18	one	one	NUM
cana-2085	170	19	player	player	NOUN
cana-2085	170	20	or	or	CCONJ
cana-2085	170	21	element	element	NOUN
cana-2085	170	22	controls	control	VERB
cana-2085	170	23	all	all	DET
cana-2085	170	24	others	other	NOUN
cana-2085	170	25	(	(	PUNCT
cana-2085	170	26	degree	degree	NOUN
cana-2085	170	27	𝑛	𝑛	PRON
cana-2085	170	28	−	−	NOUN
cana-2085	170	29	1	1	NUM
cana-2085	170	30	)	)	PUNCT
cana-2085	171	1	,	,	PUNCT
cana-2085	171	2	the	the	DET
cana-2085	171	3	theorem	theorem	NOUN
cana-2085	171	4	provides	provide	VERB
cana-2085	171	5	insights	insight	NOUN
cana-2085	171	6	into	into	ADP
cana-2085	171	7	minimal	minimal	ADJ
cana-2085	171	8	strategies	strategy	NOUN
cana-2085	171	9	needed	need	VERB
cana-2085	171	10	to	to	PART
cana-2085	171	11	dominate	dominate	VERB
cana-2085	171	12	the	the	DET
cana-2085	171	13	game	game	NOUN
cana-2085	171	14	,	,	PUNCT
cana-2085	171	15	making	make	VERB
cana-2085	171	16	it	it	PRON
cana-2085	171	17	useful	useful	ADJ
cana-2085	171	18	in	in	ADP
cana-2085	171	19	game	game	NOUN
cana-2085	171	20	theory	theory	NOUN
cana-2085	171	21	and	and	CCONJ
cana-2085	171	22	ai	ai	VERB
cana-2085	171	23	.	.	PUNCT
cana-2085	172	1	this	this	DET
cana-2085	172	2	theorem	theorem	NOUN
cana-2085	172	3	is	be	AUX
cana-2085	172	4	particularly	particularly	ADV
cana-2085	172	5	valuable	valuable	ADJ
cana-2085	172	6	in	in	ADP
cana-2085	172	7	scenarios	scenario	NOUN
cana-2085	172	8	where	where	SCONJ
cana-2085	172	9	a	a	DET
cana-2085	172	10	single	single	ADJ
cana-2085	172	11	,	,	PUNCT
cana-2085	172	12	highly	highly	ADV
cana-2085	172	13	connected	connect	VERB
cana-2085	172	14	node	node	NOUN
cana-2085	172	15	plays	play	VERB
cana-2085	172	16	a	a	DET
cana-2085	172	17	critical	critical	ADJ
cana-2085	172	18	role	role	NOUN
cana-2085	172	19	in	in	ADP
cana-2085	172	20	the	the	DET
cana-2085	172	21	structure	structure	NOUN
cana-2085	172	22	and	and	CCONJ
cana-2085	172	23	function	function	NOUN
cana-2085	172	24	of	of	ADP
cana-2085	172	25	a	a	DET
cana-2085	172	26	graph	graph	NOUN
cana-2085	172	27	.	.	PUNCT
cana-2085	173	1	it	it	PRON
cana-2085	173	2	provides	provide	VERB
cana-2085	173	3	a	a	DET
cana-2085	173	4	clear	clear	ADJ
cana-2085	173	5	understanding	understanding	NOUN
cana-2085	173	6	of	of	ADP
cana-2085	173	7	how	how	SCONJ
cana-2085	173	8	such	such	DET
cana-2085	173	9	a	a	DET
cana-2085	173	10	node	node	NOUN
cana-2085	173	11	can	can	AUX
cana-2085	173	12	influence	influence	VERB
cana-2085	173	13	the	the	DET
cana-2085	173	14	entire	entire	ADJ
cana-2085	173	15	system	system	NOUN
cana-2085	173	16	with	with	ADP
cana-2085	173	17	minimal	minimal	ADJ
cana-2085	173	18	additional	additional	ADJ
cana-2085	173	19	resources	resource	NOUN
cana-2085	173	20	or	or	CCONJ
cana-2085	173	21	effort	effort	NOUN
cana-2085	173	22	.	.	PUNCT
cana-2085	174	1	theorem	theorem	VERB
cana-2085	174	2	7	7	NUM
cana-2085	174	3	:	:	PUNCT
cana-2085	174	4	for	for	ADP
cana-2085	174	5	any	any	DET
cana-2085	174	6	connected	connected	ADJ
cana-2085	174	7	graph	graph	NOUN
cana-2085	174	8	𝐺	𝐺	PROPN
cana-2085	174	9	,	,	PUNCT
cana-2085	174	10	𝛾𝑀𝑅(𝐺	𝛾𝑀𝑅(𝐺	PROPN
cana-2085	174	11	)	)	PUNCT
cana-2085	174	12	=	=	PUNCT
cana-2085	175	1	𝛾	𝛾	ADP
cana-2085	175	2	+	+	ADV
cana-2085	175	3	1	1	NUM
cana-2085	175	4	if	if	SCONJ
cana-2085	175	5	and	and	CCONJ
cana-2085	175	6	only	only	ADV
cana-2085	175	7	if	if	SCONJ
cana-2085	175	8	the	the	DET
cana-2085	175	9	following	follow	VERB
cana-2085	175	10	conditions	condition	NOUN
cana-2085	175	11	holds	hold	VERB
cana-2085	175	12	(	(	PUNCT
cana-2085	175	13	i	i	NOUN
cana-2085	175	14	)	)	PUNCT
cana-2085	175	15	deg(𝑣	deg(𝑣	PROPN
cana-2085	175	16	)	)	PUNCT
cana-2085	175	17	≤	≤	NUM
cana-2085	175	18	2	2	NUM
cana-2085	175	19	∀	∀	NOUN
cana-2085	175	20	𝑣	𝑣	ADP
cana-2085	175	21	∈	∈	PROPN
cana-2085	175	22	𝑉	𝑉	PROPN
cana-2085	175	23	(	(	PUNCT
cana-2085	175	24	ii	ii	NOUN
cana-2085	175	25	)	)	PUNCT
cana-2085	176	1	𝑛	𝑛	DET
cana-2085	176	2	≤	≤	NUM
cana-2085	176	3	4	4	NUM
cana-2085	176	4	in	in	ADP
cana-2085	176	5	order	order	NOUN
cana-2085	176	6	for	for	ADP
cana-2085	176	7	a	a	DET
cana-2085	176	8	middle	middle	ADJ
cana-2085	176	9	roman	roman	ADJ
cana-2085	176	10	dominating	dominating	NOUN
cana-2085	176	11	function	function	NOUN
cana-2085	176	12	𝑓	𝑓	X
cana-2085	176	13	=	=	X
cana-2085	176	14	(	(	PUNCT
cana-2085	176	15	𝑉0	𝑉0	NOUN
cana-2085	176	16	,	,	PUNCT
cana-2085	176	17	𝑉1	𝑉1	PROPN
cana-2085	176	18	,	,	PUNCT
cana-2085	176	19	𝑉2	𝑉2	NOUN
cana-2085	176	20	,	,	PUNCT
cana-2085	176	21	𝑉3	𝑉3	NOUN
cana-2085	176	22	)	)	PUNCT
cana-2085	176	23	to	to	PART
cana-2085	176	24	have	have	VERB
cana-2085	176	25	𝛾𝑀𝑅(𝐺	𝛾𝑀𝑅(𝐺	PROPN
cana-2085	176	26	)	)	PUNCT
cana-2085	176	27	=	=	PUNCT
cana-2085	177	1	𝛾	𝛾	ADP
cana-2085	177	2	+	+	SYM
cana-2085	177	3	1	1	NUM
cana-2085	177	4	weight	weight	NOUN
cana-2085	177	5	𝛾(𝐺	𝛾(𝐺	NOUN
cana-2085	177	6	)	)	PUNCT
cana-2085	178	1	+	+	CCONJ
cana-2085	178	2	1	1	NUM
cana-2085	178	3	,	,	PUNCT
cana-2085	178	4	either	either	CCONJ
cana-2085	178	5	(	(	PUNCT
cana-2085	178	6	i	i	NOUN
cana-2085	178	7	)	)	PUNCT
cana-2085	178	8	|𝑉1|	|𝑉1|	PROPN
cana-2085	178	9	=	=	SYM
cana-2085	178	10	𝛾(𝐺	𝛾(𝐺	PROPN
cana-2085	178	11	)	)	PUNCT
cana-2085	179	1	+	+	CCONJ
cana-2085	179	2	1	1	NUM
cana-2085	179	3	,	,	PUNCT
cana-2085	179	4	|𝑉2|	|𝑉2|	PROPN
cana-2085	179	5	=	=	SYM
cana-2085	179	6	0	0	PROPN
cana-2085	179	7	and	and	CCONJ
cana-2085	179	8	|𝑉3|	|𝑉3|	PRON
cana-2085	179	9	=	=	SYM
cana-2085	179	10	0	0	NUM
cana-2085	179	11	or	or	CCONJ
cana-2085	179	12	(	(	PUNCT
cana-2085	179	13	ii	ii	NOUN
cana-2085	179	14	)	)	PUNCT
cana-2085	179	15	|𝑉1|	|𝑉1|	NOUN
cana-2085	180	1	=	=	SYM
cana-2085	180	2	𝛾(𝐺	𝛾(𝐺	PROPN
cana-2085	180	3	)	)	PUNCT
cana-2085	181	1	−	−	PROPN
cana-2085	181	2	1	1	NUM
cana-2085	181	3	,	,	PUNCT
cana-2085	181	4	|𝑉2|	|𝑉2|	PROPN
cana-2085	181	5	=	=	SYM
cana-2085	181	6	1	1	NUM
cana-2085	181	7	and	and	CCONJ
cana-2085	181	8	|𝑉3|	|𝑉3|	PRON
cana-2085	182	1	=	=	SYM
cana-2085	182	2	0	0	X
cana-2085	182	3	.	.	PUNCT
cana-2085	183	1	any	any	DET
cana-2085	183	2	other	other	ADJ
cana-2085	183	3	arrangement	arrangement	NOUN
cana-2085	183	4	of	of	ADP
cana-2085	183	5	weight	weight	NOUN
cana-2085	183	6	𝛾(𝐺	𝛾(𝐺	NOUN
cana-2085	183	7	)	)	PUNCT
cana-2085	184	1	+	+	CCONJ
cana-2085	184	2	1	1	NUM
cana-2085	184	3	would	would	AUX
cana-2085	184	4	have	have	VERB
cana-2085	184	5	|𝑉1|	|𝑉1|	PROPN
cana-2085	184	6	+	+	NUM
cana-2085	184	7	|𝑉2|	|𝑉2|	X
cana-2085	184	8	<	<	X
cana-2085	184	9	𝛾(𝐺	𝛾(𝐺	PROPN
cana-2085	184	10	)	)	PUNCT
cana-2085	184	11	.	.	PUNCT
cana-2085	185	1	communications	communication	NOUN
cana-2085	185	2	on	on	ADP
cana-2085	185	3	applied	apply	VERB
cana-2085	185	4	nonlinear	nonlinear	ADJ
cana-2085	185	5	analysis	analysis	NOUN
cana-2085	185	6	issn	issn	NOUN
cana-2085	185	7	:	:	PUNCT
cana-2085	185	8	1074	1074	NUM
cana-2085	185	9	-	-	PUNCT
cana-2085	185	10	133x	133x	NUM
cana-2085	185	11	vol	vol	NOUN
cana-2085	185	12	32	32	NUM
cana-2085	185	13	no	no	NOUN
cana-2085	185	14	.	.	PUNCT
cana-2085	186	1	1s	1s	NUM
cana-2085	186	2	(	(	PUNCT
cana-2085	186	3	2025	2025	NUM
cana-2085	186	4	)	)	PUNCT
cana-2085	186	5	23	23	NUM
cana-2085	186	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-2085	187	1	in	in	ADP
cana-2085	187	2	case	case	NOUN
cana-2085	187	3	(	(	PUNCT
cana-2085	187	4	i	i	NOUN
cana-2085	187	5	):	):	PUNCT
cana-2085	187	6	since	since	ADV
cana-2085	187	7	,	,	PUNCT
cana-2085	187	8	|𝑉2|	|𝑉2|	PROPN
cana-2085	187	9	=	=	SYM
cana-2085	187	10	0	0	PROPN
cana-2085	187	11	and	and	CCONJ
cana-2085	187	12	|𝑉3|	|𝑉3|	PRON
cana-2085	188	1	=	=	SYM
cana-2085	188	2	0	0	NUM
cana-2085	188	3	,	,	PUNCT
cana-2085	188	4	|𝑉1|	|𝑉1|	NOUN
cana-2085	188	5	=	=	SYM
cana-2085	188	6	𝑉.	𝑉.	PROPN
cana-2085	188	7	then	then	ADV
cana-2085	188	8	by	by	ADP
cana-2085	188	9	ore	ore	NOUN
cana-2085	188	10	’s	’s	PART
cana-2085	188	11	theorem	theorem	NOUN
cana-2085	188	12	,	,	PUNCT
cana-2085	188	13	for	for	ADP
cana-2085	188	14	any	any	DET
cana-2085	188	15	connected	connected	ADJ
cana-2085	188	16	graph	graph	NOUN
cana-2085	188	17	with	with	ADP
cana-2085	188	18	𝑛	𝑛	PROPN
cana-2085	188	19	vertices	vertex	NOUN
cana-2085	188	20	,	,	PUNCT
cana-2085	188	21	𝛾(𝐺	𝛾(𝐺	NUM
cana-2085	188	22	)	)	PUNCT
cana-2085	188	23	≤	≤	NUM
cana-2085	188	24	𝑛	𝑛	DET
cana-2085	188	25	2	2	NUM
cana-2085	188	26	.	.	PUNCT
cana-2085	189	1	thus	thus	ADV
cana-2085	189	2	𝑛	𝑛	PRON
cana-2085	189	3	=	=	SYM
cana-2085	189	4	𝛾(𝐺	𝛾(𝐺	PROPN
cana-2085	189	5	)	)	PUNCT
cana-2085	190	1	+	+	CCONJ
cana-2085	190	2	1	1	NUM
cana-2085	190	3	≤	≤	NUM
cana-2085	190	4	𝑛	𝑛	DET
cana-2085	190	5	2	2	NUM
cana-2085	190	6	+	+	NUM
cana-2085	190	7	1	1	NUM
cana-2085	190	8	.	.	X
cana-2085	190	9	hence	hence	ADV
cana-2085	190	10	𝑛	𝑛	VERB
cana-2085	190	11	=	=	SYM
cana-2085	190	12	2	2	X
cana-2085	190	13	.	.	PUNCT
cana-2085	190	14	let	let	VERB
cana-2085	190	15	𝑉1	𝑉1	NOUN
cana-2085	190	16	,	,	PUNCT
cana-2085	190	17	𝑉2	𝑉2	NOUN
cana-2085	190	18	∈	∈	PROPN
cana-2085	190	19	𝑉.	𝑉.	PROPN
cana-2085	190	20	if	if	SCONJ
cana-2085	190	21	𝑉1	𝑉1	NOUN
cana-2085	190	22	and	and	CCONJ
cana-2085	190	23	𝑉2	𝑉2	NOUN
cana-2085	190	24	are	be	AUX
cana-2085	190	25	not	not	PART
cana-2085	190	26	adjacent	adjacent	ADJ
cana-2085	190	27	,	,	PUNCT
cana-2085	190	28	then	then	ADV
cana-2085	190	29	𝛾(𝐺	𝛾(𝐺	NUM
cana-2085	190	30	)	)	PUNCT
cana-2085	191	1	=	=	SYM
cana-2085	191	2	𝛾𝑀𝑅(𝐺	𝛾𝑀𝑅(𝐺	NOUN
cana-2085	191	3	)	)	PUNCT
cana-2085	191	4	.	.	PUNCT
cana-2085	192	1	hence	hence	ADV
cana-2085	192	2	𝑉1	𝑉1	PROPN
cana-2085	192	3	and	and	CCONJ
cana-2085	192	4	𝑉2	𝑉2	NOUN
cana-2085	192	5	are	be	AUX
cana-2085	192	6	adjacent	adjacent	ADJ
cana-2085	192	7	and	and	CCONJ
cana-2085	192	8	hence	hence	ADV
cana-2085	192	9	𝐺	𝐺	PROPN
cana-2085	192	10	is	be	AUX
cana-2085	192	11	a	a	DET
cana-2085	192	12	𝑃2	𝑃2	NOUN
cana-2085	192	13	.	.	PUNCT
cana-2085	193	1	in	in	ADP
cana-2085	193	2	case	case	NOUN
cana-2085	193	3	(	(	PUNCT
cana-2085	193	4	ii	ii	NOUN
cana-2085	193	5	)	)	PUNCT
cana-2085	193	6	𝑓	𝑓	NOUN
cana-2085	193	7	=	=	X
cana-2085	193	8	(	(	PUNCT
cana-2085	193	9	𝑉0	𝑉0	NOUN
cana-2085	193	10	,	,	PUNCT
cana-2085	193	11	𝑉1	𝑉1	PROPN
cana-2085	193	12	,	,	PUNCT
cana-2085	193	13	𝑉2	𝑉2	NOUN
cana-2085	193	14	,	,	PUNCT
cana-2085	193	15	𝑉3	𝑉3	NOUN
cana-2085	193	16	)	)	PUNCT
cana-2085	193	17	be	be	AUX
cana-2085	193	18	a	a	DET
cana-2085	193	19	𝛾𝑀𝑅(𝐺	𝛾𝑀𝑅(𝐺	NOUN
cana-2085	193	20	)	)	PUNCT
cana-2085	193	21	function	function	NOUN
cana-2085	193	22	for	for	ADP
cana-2085	193	23	𝐺	𝐺	PROPN
cana-2085	193	24	of	of	ADP
cana-2085	193	25	weight	weight	NOUN
cana-2085	193	26	𝛾(𝐺	𝛾(𝐺	PROPN
cana-2085	193	27	)	)	PUNCT
cana-2085	194	1	+	+	CCONJ
cana-2085	194	2	1	1	NUM
cana-2085	194	3	with	with	ADP
cana-2085	194	4	|𝑉1|	|𝑉1|	NOUN
cana-2085	194	5	=	=	SYM
cana-2085	194	6	𝛾(𝐺	𝛾(𝐺	PROPN
cana-2085	194	7	)	)	PUNCT
cana-2085	194	8	−	−	PROPN
cana-2085	194	9	1	1	NUM
cana-2085	194	10	,	,	PUNCT
cana-2085	194	11	|𝑉2|	|𝑉2|	PROPN
cana-2085	194	12	=	=	SYM
cana-2085	194	13	1	1	NUM
cana-2085	194	14	and	and	CCONJ
cana-2085	194	15	|𝑉3|	|𝑉3|	PRON
cana-2085	195	1	=	=	SYM
cana-2085	195	2	0	0	X
cana-2085	195	3	.	.	PUNCT
cana-2085	196	1	let	let	VERB
cana-2085	196	2	𝑉2	𝑉2	NOUN
cana-2085	196	3	=	=	SYM
cana-2085	196	4	{	{	PUNCT
cana-2085	196	5	𝑢	𝑢	X
cana-2085	196	6	}	}	PUNCT
cana-2085	196	7	.	.	PUNCT
cana-2085	197	1	let	let	VERB
cana-2085	197	2	𝑣	𝑣	PRON
cana-2085	197	3	∈	∈	PROPN
cana-2085	197	4	𝑉1	𝑉1	NOUN
cana-2085	197	5	be	be	AUX
cana-2085	197	6	adjacent	adjacent	ADJ
cana-2085	197	7	to	to	AUX
cana-2085	197	8	𝑢.	𝑢.	VERB
cana-2085	197	9	then	then	ADV
cana-2085	197	10	{	{	PUNCT
cana-2085	197	11	𝑢	𝑢	X
cana-2085	197	12	}	}	PUNCT
cana-2085	197	13	is	be	AUX
cana-2085	197	14	the	the	DET
cana-2085	197	15	𝛾-set	𝛾-set	NOUN
cana-2085	197	16	of	of	ADP
cana-2085	197	17	𝐺	𝐺	PROPN
cana-2085	197	18	and	and	CCONJ
cana-2085	197	19	𝛾𝑀𝑅(𝐺	𝛾𝑀𝑅(𝐺	PROPN
cana-2085	197	20	)	)	PUNCT
cana-2085	197	21	=	=	SYM
cana-2085	198	1	|𝑉1|	|𝑉1|	PROPN
cana-2085	199	1	+	+	NUM
cana-2085	199	2	|𝑉2|	|𝑉2|	NOUN
cana-2085	199	3	=	=	SYM
cana-2085	199	4	𝛾	𝛾	PROPN
cana-2085	199	5	+	+	X
cana-2085	199	6	2	2	NUM
cana-2085	199	7	which	which	PRON
cana-2085	199	8	is	be	AUX
cana-2085	199	9	a	a	DET
cana-2085	199	10	contradiction	contradiction	NOUN
cana-2085	199	11	.	.	PUNCT
cana-2085	200	1	hence	hence	ADV
cana-2085	200	2	no	no	DET
cana-2085	200	3	edge	edge	NOUN
cana-2085	200	4	of	of	ADP
cana-2085	200	5	𝐺	𝐺	PROPN
cana-2085	200	6	joins	join	VERB
cana-2085	200	7	{	{	PUNCT
cana-2085	200	8	𝑢	𝑢	NOUN
cana-2085	200	9	}	}	PUNCT
cana-2085	200	10	and	and	CCONJ
cana-2085	200	11	𝑉1	𝑉1	NOUN
cana-2085	200	12	{	{	PUNCT
cana-2085	200	13	𝑢	𝑢	PROPN
cana-2085	200	14	}	}	PUNCT
cana-2085	200	15	≻	≻	NOUN
cana-2085	200	16	𝑉0	𝑉0	NOUN
cana-2085	200	17	,	,	PUNCT
cana-2085	200	18	deg(𝑢	deg(𝑢	PROPN
cana-2085	200	19	)	)	PUNCT
cana-2085	200	20	=	=	SYM
cana-2085	201	1	2	2	X
cana-2085	201	2	.	.	X
cana-2085	201	3	let	let	VERB
cana-2085	201	4	𝑢1	𝑢1	PROPN
cana-2085	201	5	and	and	CCONJ
cana-2085	201	6	𝑢2	𝑢2	PROPN
cana-2085	201	7	be	be	VERB
cana-2085	201	8	the	the	DET
cana-2085	201	9	vertices	vertex	NOUN
cana-2085	201	10	adjacent	adjacent	ADJ
cana-2085	201	11	to	to	ADP
cana-2085	201	12	𝑢.	𝑢.	PROPN
cana-2085	201	13	suppose	suppose	VERB
cana-2085	201	14	deg(𝑢1	deg(𝑢1	NOUN
cana-2085	201	15	)	)	PUNCT
cana-2085	201	16	=	=	SYM
cana-2085	201	17	deg(𝑢2	deg(𝑢2	PROPN
cana-2085	201	18	)	)	PUNCT
cana-2085	201	19	=	=	SYM
cana-2085	202	1	2	2	X
cana-2085	202	2	.	.	X
cana-2085	202	3	then	then	ADV
cana-2085	202	4	{	{	PUNCT
cana-2085	202	5	𝑢1	𝑢1	PROPN
cana-2085	202	6	,	,	PUNCT
cana-2085	202	7	𝑢2	𝑢2	PROPN
cana-2085	202	8	}	}	PUNCT
cana-2085	202	9	is	be	AUX
cana-2085	202	10	the	the	DET
cana-2085	202	11	𝛾-set	𝛾-set	NOUN
cana-2085	202	12	of	of	ADP
cana-2085	202	13	𝐺	𝐺	PROPN
cana-2085	202	14	,	,	PUNCT
cana-2085	202	15	𝑉2	𝑉2	NOUN
cana-2085	202	16	=	=	SYM
cana-2085	202	17	{	{	PUNCT
cana-2085	202	18	𝑢1	𝑢1	PROPN
cana-2085	202	19	,	,	PUNCT
cana-2085	202	20	𝑢2	𝑢2	PROPN
cana-2085	202	21	}	}	PUNCT
cana-2085	202	22	,	,	PUNCT
cana-2085	202	23	𝑉3	𝑉3	NOUN
cana-2085	202	24	=	=	NOUN
cana-2085	202	25	∅	∅	NOUN
cana-2085	202	26	and	and	CCONJ
cana-2085	202	27	𝑉1	𝑉1	NOUN
cana-2085	202	28	=	=	SYM
cana-2085	202	29	∅	∅	NOUN
cana-2085	202	30	,	,	PUNCT
cana-2085	202	31	which	which	PRON
cana-2085	202	32	implies	imply	VERB
cana-2085	202	33	that	that	SCONJ
cana-2085	202	34	𝛾𝑀𝑅(𝐺	𝛾𝑀𝑅(𝐺	NOUN
cana-2085	202	35	)	)	PUNCT
cana-2085	202	36	=	=	SYM
cana-2085	203	1	2|𝑉2|	2|𝑉2|	NUM
cana-2085	203	2	=	=	SYM
cana-2085	203	3	0	0	NUM
cana-2085	203	4	,	,	PUNCT
cana-2085	203	5	𝛾	𝛾	ADP
cana-2085	203	6	+	+	ADP
cana-2085	203	7	2	2	NUM
cana-2085	203	8	a	a	DET
cana-2085	203	9	contradiction	contradiction	NOUN
cana-2085	203	10	.	.	PUNCT
cana-2085	204	1	suppose	suppose	VERB
cana-2085	204	2	deg(𝑢1	deg(𝑢1	NOUN
cana-2085	204	3	)	)	PUNCT
cana-2085	204	4	=	=	SYM
cana-2085	204	5	2	2	NUM
cana-2085	204	6	and	and	CCONJ
cana-2085	204	7	deg(𝑢2	deg(𝑢2	PROPN
cana-2085	204	8	)	)	PUNCT
cana-2085	204	9	=	=	SYM
cana-2085	204	10	1	1	NUM
cana-2085	204	11	and	and	CCONJ
cana-2085	204	12	𝑢1	𝑢1	PROPN
cana-2085	204	13	,	,	PUNCT
cana-2085	204	14	𝑢2	𝑢2	PROPN
cana-2085	204	15	are	be	AUX
cana-2085	204	16	not	not	PART
cana-2085	204	17	adjacent	adjacent	ADJ
cana-2085	204	18	.	.	PUNCT
cana-2085	205	1	let	let	VERB
cana-2085	205	2	𝑤	𝑤	PART
cana-2085	205	3	be	be	AUX
cana-2085	205	4	the	the	DET
cana-2085	205	5	vertex	vertex	NOUN
cana-2085	205	6	adjacent	adjacent	ADJ
cana-2085	205	7	to	to	PART
cana-2085	205	8	𝑢1	𝑢1	VERB
cana-2085	205	9	.	.	PUNCT
cana-2085	206	1	then	then	ADV
cana-2085	206	2	{	{	PUNCT
cana-2085	206	3	𝑢1	𝑢1	PROPN
cana-2085	206	4	,	,	PUNCT
cana-2085	206	5	𝑢2	𝑢2	PROPN
cana-2085	206	6	}	}	PUNCT
cana-2085	206	7	is	be	AUX
cana-2085	206	8	the	the	DET
cana-2085	206	9	-set	-set	ADJ
cana-2085	206	10	of	of	ADP
cana-2085	206	11	𝐺	𝐺	PROPN
cana-2085	206	12	and	and	CCONJ
cana-2085	206	13	𝛾𝑀𝑅(𝐺	𝛾𝑀𝑅(𝐺	PROPN
cana-2085	206	14	)	)	PUNCT
cana-2085	206	15	=	=	SYM
cana-2085	207	1	|𝑉1|	|𝑉1|	PROPN
cana-2085	207	2	+	+	NUM
cana-2085	207	3	|𝑉2|	|𝑉2|	NOUN
cana-2085	207	4	=	=	SYM
cana-2085	207	5	𝛾(𝐺	𝛾(𝐺	PROPN
cana-2085	207	6	)	)	PUNCT
cana-2085	208	1	+	+	NUM
cana-2085	208	2	1	1	X
cana-2085	208	3	.	.	X
cana-2085	208	4	suppose	suppose	VERB
cana-2085	208	5	deg(𝑢1	deg(𝑢1	NOUN
cana-2085	208	6	)	)	PUNCT
cana-2085	208	7	=	=	SYM
cana-2085	209	1	3	3	NUM
cana-2085	209	2	,	,	PUNCT
cana-2085	209	3	deg(𝑢2	deg(𝑢2	PROPN
cana-2085	209	4	)	)	PUNCT
cana-2085	209	5	=	=	SYM
cana-2085	209	6	1	1	NUM
cana-2085	209	7	and	and	CCONJ
cana-2085	209	8	𝑢1	𝑢1	PROPN
cana-2085	209	9	,	,	PUNCT
cana-2085	209	10	𝑢2	𝑢2	PROPN
cana-2085	209	11	are	be	AUX
cana-2085	209	12	adjacent	adjacent	ADJ
cana-2085	209	13	.	.	PUNCT
cana-2085	210	1	then	then	ADV
cana-2085	210	2	𝑉3	𝑉3	NOUN
cana-2085	210	3	=	=	SYM
cana-2085	210	4	{	{	PUNCT
cana-2085	210	5	𝑢1	𝑢1	PROPN
cana-2085	210	6	}	}	PUNCT
cana-2085	210	7	,	,	PUNCT
cana-2085	210	8	i.e.	i.e.	X
cana-2085	210	9	,	,	PUNCT
cana-2085	210	10	|𝑉3|	|𝑉3|	X
cana-2085	210	11	=	=	SYM
cana-2085	210	12	1	1	NUM
cana-2085	210	13	which	which	PRON
cana-2085	210	14	is	be	AUX
cana-2085	210	15	a	a	DET
cana-2085	210	16	contradiction	contradiction	NOUN
cana-2085	210	17	.	.	PUNCT
cana-2085	211	1	hence	hence	ADV
cana-2085	211	2	𝑛	𝑛	VERB
cana-2085	211	3	=	=	SYM
cana-2085	211	4	4	4	NUM
cana-2085	211	5	and	and	CCONJ
cana-2085	211	6	deg(𝑣	deg(𝑣	PROPN
cana-2085	211	7	)	)	PUNCT
cana-2085	211	8	≤	≤	NUM
cana-2085	211	9	2	2	NUM
cana-2085	211	10	∀	∀	NOUN
cana-2085	211	11	𝑣	𝑣	ADP
cana-2085	211	12	∈	∈	PROPN
cana-2085	211	13	𝑉.	𝑉.	NOUN
cana-2085	211	14	conversely	conversely	ADV
cana-2085	211	15	,	,	PUNCT
cana-2085	211	16	assume	assume	VERB
cana-2085	211	17	that	that	SCONJ
cana-2085	211	18	for	for	ADP
cana-2085	211	19	every	every	DET
cana-2085	211	20	vertex	vertex	NOUN
cana-2085	211	21	of	of	ADP
cana-2085	211	22	𝐺	𝐺	PROPN
cana-2085	211	23	deg	deg	X
cana-2085	211	24	(	(	PUNCT
cana-2085	211	25	𝑣	𝑣	NOUN
cana-2085	211	26	)	)	PUNCT
cana-2085	211	27	≤	≤	NUM
cana-2085	211	28	2	2	NUM
cana-2085	211	29	and	and	CCONJ
cana-2085	211	30	𝑛	𝑛	DET
cana-2085	211	31	≤	≤	NOUN
cana-2085	211	32	4	4	NUM
cana-2085	211	33	.	.	PUNCT
cana-2085	212	1	if	if	SCONJ
cana-2085	212	2	𝑉2	𝑉2	NOUN
cana-2085	212	3	=	=	SYM
cana-2085	212	4	{	{	PUNCT
cana-2085	212	5	𝑢	𝑢	X
cana-2085	212	6	}	}	PUNCT
cana-2085	212	7	,	,	PUNCT
cana-2085	212	8	𝑉1	𝑉1	NOUN
cana-2085	212	9	=	=	SYM
cana-2085	212	10	𝑉	𝑉	PROPN
cana-2085	212	11	−	−	PROPN
cana-2085	212	12	𝑁[𝑢	𝑁[𝑢	NOUN
cana-2085	212	13	]	]	X
cana-2085	212	14	and	and	CCONJ
cana-2085	212	15	𝑉0	𝑉0	NOUN
cana-2085	212	16	=	=	SYM
cana-2085	212	17	𝑉	𝑉	PROPN
cana-2085	212	18	−	−	PROPN
cana-2085	212	19	𝑉1	𝑉1	NOUN
cana-2085	212	20	−	−	PROPN
cana-2085	212	21	𝑉2	𝑉2	NOUN
cana-2085	212	22	,	,	PUNCT
cana-2085	212	23	suppose	suppose	VERB
cana-2085	212	24	deg	deg	X
cana-2085	212	25	(	(	PUNCT
cana-2085	212	26	𝑣	𝑣	NOUN
cana-2085	212	27	)	)	PUNCT
cana-2085	212	28	≤	≤	NOUN
cana-2085	212	29	2	2	NUM
cana-2085	212	30	.	.	PUNCT
cana-2085	212	31	case	case	NOUN
cana-2085	212	32	(	(	PUNCT
cana-2085	212	33	ii	ii	NOUN
cana-2085	212	34	)	)	PUNCT
cana-2085	212	35	𝑛	𝑛	NOUN
cana-2085	212	36	=	=	SYM
cana-2085	212	37	2	2	NUM
cana-2085	212	38	,	,	PUNCT
cana-2085	212	39	then	then	ADV
cana-2085	212	40	since	since	ADV
cana-2085	212	41	,	,	PUNCT
cana-2085	212	42	𝐺	𝐺	PROPN
cana-2085	212	43	is	be	AUX
cana-2085	212	44	a	a	DET
cana-2085	212	45	connected	connect	VERB
cana-2085	212	46	deg(𝑣	deg(𝑣	PROPN
cana-2085	212	47	)	)	PUNCT
cana-2085	212	48	=	=	SYM
cana-2085	212	49	1	1	NUM
cana-2085	212	50	∀	∀	NOUN
cana-2085	212	51	𝑣	𝑣	ADP
cana-2085	212	52	∈	∈	PROPN
cana-2085	212	53	𝑉.	𝑉.	PROPN
cana-2085	212	54	hence	hence	ADV
cana-2085	212	55	𝐺	𝐺	PROPN
cana-2085	212	56	is	be	AUX
cana-2085	212	57	a	a	DET
cana-2085	212	58	𝑃2	𝑃2	NOUN
cana-2085	212	59	,	,	PUNCT
cana-2085	212	60	𝑉1	𝑉1	NOUN
cana-2085	212	61	=	=	SYM
cana-2085	212	62	{	{	PUNCT
cana-2085	212	63	𝑣1	𝑣1	PROPN
cana-2085	212	64	,	,	PUNCT
cana-2085	212	65	𝑣2	𝑣2	PROPN
cana-2085	212	66	}	}	PUNCT
cana-2085	212	67	and	and	CCONJ
cana-2085	212	68	{	{	PUNCT
cana-2085	212	69	𝑣1	𝑣1	NOUN
cana-2085	212	70	}	}	PUNCT
cana-2085	212	71	is	be	AUX
cana-2085	212	72	a	a	DET
cana-2085	212	73	𝛾set	𝛾set	NOUN
cana-2085	212	74	.	.	PUNCT
cana-2085	213	1	hence	hence	ADV
cana-2085	213	2	𝛾𝑀𝑅(𝐺	𝛾𝑀𝑅(𝐺	PROPN
cana-2085	213	3	)	)	PUNCT
cana-2085	213	4	=	=	PUNCT
cana-2085	214	1	𝛾	𝛾	ADP
cana-2085	214	2	+	+	NOUN
cana-2085	214	3	1	1	X
cana-2085	214	4	.	.	X
cana-2085	214	5	case	case	NOUN
cana-2085	214	6	(	(	PUNCT
cana-2085	214	7	ii	ii	NOUN
cana-2085	214	8	)	)	PUNCT
cana-2085	214	9	𝑛	𝑛	NOUN
cana-2085	214	10	=	=	SYM
cana-2085	214	11	2	2	NUM
cana-2085	214	12	let	let	VERB
cana-2085	214	13	𝑉1	𝑉1	NOUN
cana-2085	214	14	,	,	PUNCT
cana-2085	214	15	𝑉2	𝑉2	NOUN
cana-2085	214	16	,	,	PUNCT
cana-2085	214	17	𝑉3	𝑉3	NOUN
cana-2085	214	18	be	be	AUX
cana-2085	214	19	the	the	DET
cana-2085	214	20	vertices	vertex	NOUN
cana-2085	214	21	of	of	ADP
cana-2085	214	22	𝐺.	𝐺.	NOUN
cana-2085	214	23	if	if	SCONJ
cana-2085	214	24	deg(𝑣𝑖	deg(𝑣𝑖	VERB
cana-2085	214	25	)	)	PUNCT
cana-2085	215	1	=	=	SYM
cana-2085	215	2	2	2	NUM
cana-2085	215	3	∀	∀	NOUN
cana-2085	215	4	𝑖	𝑖	NOUN
cana-2085	215	5	=	=	SYM
cana-2085	215	6	1	1	NUM
cana-2085	215	7	,	,	PUNCT
cana-2085	215	8	2	2	NUM
cana-2085	215	9	,	,	PUNCT
cana-2085	215	10	3	3	NUM
cana-2085	215	11	…	…	PUNCT
cana-2085	215	12	,	,	PUNCT
cana-2085	215	13	then	then	ADV
cana-2085	215	14	𝐺	𝐺	PROPN
cana-2085	215	15	≅	≅	PROPN
cana-2085	215	16	𝐶3	𝐶3	PROPN
cana-2085	215	17	and	and	CCONJ
cana-2085	215	18	if	if	SCONJ
cana-2085	215	19	deg(𝑣𝑖	deg(𝑣𝑖	VERB
cana-2085	215	20	)	)	PUNCT
cana-2085	215	21	=	=	SYM
cana-2085	215	22	2	2	NUM
cana-2085	215	23	for	for	ADP
cana-2085	215	24	some	some	PRON
cana-2085	215	25	𝑖	𝑖	NOUN
cana-2085	215	26	=	=	SYM
cana-2085	215	27	1	1	NUM
cana-2085	215	28	say	say	VERB
cana-2085	215	29	,	,	PUNCT
cana-2085	215	30	then	then	ADV
cana-2085	215	31	𝐺	𝐺	PROPN
cana-2085	215	32	≅	≅	PROPN
cana-2085	215	33	𝑃3	𝑃3	PROPN
cana-2085	215	34	.	.	PUNCT
cana-2085	216	1	hence	hence	ADV
cana-2085	216	2	𝑉2	𝑉2	PROPN
cana-2085	216	3	=	=	SYM
cana-2085	216	4	{	{	PUNCT
cana-2085	216	5	𝑣1	𝑣1	PROPN
cana-2085	216	6	}	}	PUNCT
cana-2085	216	7	,	,	PUNCT
cana-2085	216	8	𝑉1	𝑉1	NOUN
cana-2085	216	9	=	=	SYM
cana-2085	216	10	∅	∅	NOUN
cana-2085	216	11	and	and	CCONJ
cana-2085	216	12	{	{	PUNCT
cana-2085	216	13	𝑣1	𝑣1	NOUN
cana-2085	216	14	}	}	PUNCT
cana-2085	216	15	is	be	AUX
cana-2085	216	16	the	the	DET
cana-2085	216	17	𝛾set	𝛾set	NOUN
cana-2085	216	18	.	.	PUNCT
cana-2085	217	1	therefore	therefore	ADV
cana-2085	217	2	𝛾𝑀𝑅(𝐺	𝛾𝑀𝑅(𝐺	PROPN
cana-2085	217	3	)	)	PUNCT
cana-2085	217	4	=	=	PUNCT
cana-2085	218	1	𝛾	𝛾	ADP
cana-2085	218	2	+	+	NOUN
cana-2085	218	3	1	1	X
cana-2085	218	4	.	.	PUNCT
cana-2085	219	1	the	the	DET
cana-2085	219	2	theorem	theorem	NOUN
cana-2085	219	3	provides	provide	VERB
cana-2085	219	4	a	a	DET
cana-2085	219	5	specific	specific	ADJ
cana-2085	219	6	condition	condition	NOUN
cana-2085	219	7	under	under	ADP
cana-2085	219	8	which	which	PRON
cana-2085	219	9	the	the	DET
cana-2085	219	10	middle	middle	ADJ
cana-2085	219	11	roman	roman	ADJ
cana-2085	219	12	domination	domination	NOUN
cana-2085	219	13	number	number	NOUN
cana-2085	219	14	𝛾𝑀𝑅(𝐺	𝛾𝑀𝑅(𝐺	PROPN
cana-2085	219	15	)	)	PUNCT
cana-2085	219	16	of	of	ADP
cana-2085	219	17	a	a	DET
cana-2085	219	18	connected	connected	ADJ
cana-2085	219	19	graph	graph	NOUN
cana-2085	219	20	𝐺	𝐺	NOUN
cana-2085	219	21	is	be	AUX
cana-2085	219	22	equal	equal	ADJ
cana-2085	219	23	to	to	ADP
cana-2085	219	24	𝛾	𝛾	PROPN
cana-2085	219	25	+	+	ADJ
cana-2085	219	26	1	1	NUM
cana-2085	219	27	,	,	PUNCT
cana-2085	219	28	where	where	SCONJ
cana-2085	219	29	𝛾	𝛾	NOUN
cana-2085	219	30	is	be	AUX
cana-2085	219	31	the	the	DET
cana-2085	219	32	domination	domination	NOUN
cana-2085	219	33	number	number	NOUN
cana-2085	219	34	of	of	ADP
cana-2085	219	35	the	the	DET
cana-2085	219	36	graph	graph	NOUN
cana-2085	219	37	.	.	PUNCT
cana-2085	220	1	the	the	DET
cana-2085	220	2	condition	condition	NOUN
cana-2085	220	3	given	give	VERB
cana-2085	220	4	is	be	AUX
cana-2085	220	5	:	:	PUNCT
cana-2085	220	6	1	1	X
cana-2085	220	7	.	.	X
cana-2085	221	1	deg(𝑣	deg(𝑣	PROPN
cana-2085	221	2	)	)	PUNCT
cana-2085	221	3	≤	≤	NUM
cana-2085	221	4	2	2	NUM
cana-2085	221	5	∀	∀	NOUN
cana-2085	221	6	𝑣	𝑣	ADP
cana-2085	221	7	∈	∈	PROPN
cana-2085	221	8	𝑉	𝑉	PROPN
cana-2085	221	9	(	(	PUNCT
cana-2085	221	10	i.e.	i.e.	X
cana-2085	221	11	,	,	PUNCT
cana-2085	221	12	each	each	DET
cana-2085	221	13	vertex	vertex	NOUN
cana-2085	221	14	has	have	VERB
cana-2085	221	15	a	a	DET
cana-2085	221	16	degree	degree	NOUN
cana-2085	221	17	of	of	ADP
cana-2085	221	18	at	at	ADV
cana-2085	221	19	most	most	ADJ
cana-2085	221	20	2	2	NUM
cana-2085	221	21	)	)	PUNCT
cana-2085	221	22	,	,	PUNCT
cana-2085	221	23	and	and	CCONJ
cana-2085	222	1	2	2	X
cana-2085	222	2	.	.	X
cana-2085	223	1	𝑛	𝑛	DET
cana-2085	223	2	≤	≤	NUM
cana-2085	223	3	4	4	NUM
cana-2085	223	4	(	(	PUNCT
cana-2085	223	5	the	the	DET
cana-2085	223	6	number	number	NOUN
cana-2085	223	7	of	of	ADP
cana-2085	223	8	vertices	vertex	NOUN
cana-2085	223	9	in	in	ADP
cana-2085	223	10	the	the	DET
cana-2085	223	11	graph	graph	NOUN
cana-2085	223	12	is	be	AUX
cana-2085	223	13	at	at	ADP
cana-2085	223	14	most	most	ADJ
cana-2085	223	15	4	4	NUM
cana-2085	223	16	)	)	PUNCT
cana-2085	223	17	.	.	PUNCT
cana-2085	224	1	this	this	DET
cana-2085	224	2	characterization	characterization	NOUN
cana-2085	224	3	has	have	VERB
cana-2085	224	4	several	several	ADJ
cana-2085	224	5	practical	practical	ADJ
cana-2085	224	6	applications	application	NOUN
cana-2085	224	7	in	in	ADP
cana-2085	224	8	various	various	ADJ
cana-2085	224	9	field	field	NOUN
cana-2085	224	10	such	such	ADJ
cana-2085	224	11	as	as	ADP
cana-2085	224	12	,	,	PUNCT
cana-2085	224	13	small	small	ADJ
cana-2085	224	14	-	-	PUNCT
cana-2085	224	15	scale	scale	NOUN
cana-2085	224	16	network	network	NOUN
cana-2085	224	17	design	design	NOUN
cana-2085	224	18	-	-	PUNCT
cana-2085	224	19	design	design	NOUN
cana-2085	224	20	of	of	ADP
cana-2085	224	21	simple	simple	ADJ
cana-2085	224	22	networks	network	NOUN
cana-2085	224	23	:	:	PUNCT
cana-2085	224	24	in	in	ADP
cana-2085	224	25	small	small	ADJ
cana-2085	224	26	networks	network	NOUN
cana-2085	224	27	(	(	PUNCT
cana-2085	224	28	with	with	ADP
cana-2085	224	29	up	up	ADP
cana-2085	224	30	to	to	PART
cana-2085	224	31	4	4	NUM
cana-2085	224	32	vertices	vertex	NOUN
cana-2085	224	33	)	)	PUNCT
cana-2085	224	34	,	,	PUNCT
cana-2085	224	35	this	this	DET
cana-2085	224	36	theorem	theorem	NOUN
cana-2085	224	37	can	can	AUX
cana-2085	224	38	simplify	simplify	VERB
cana-2085	224	39	the	the	DET
cana-2085	224	40	process	process	NOUN
cana-2085	224	41	of	of	ADP
cana-2085	224	42	determining	determine	VERB
cana-2085	224	43	optimal	optimal	ADJ
cana-2085	224	44	placements	placement	NOUN
cana-2085	224	45	of	of	ADP
cana-2085	224	46	resources	resource	NOUN
cana-2085	224	47	or	or	CCONJ
cana-2085	224	48	nodes	node	NOUN
cana-2085	224	49	to	to	PART
cana-2085	224	50	ensure	ensure	VERB
cana-2085	224	51	full	full	ADJ
cana-2085	224	52	coverage	coverage	NOUN
cana-2085	224	53	.	.	PUNCT
cana-2085	225	1	since	since	SCONJ
cana-2085	225	2	all	all	DET
cana-2085	225	3	nodes	node	NOUN
cana-2085	225	4	have	have	VERB
cana-2085	225	5	a	a	DET
cana-2085	225	6	maximum	maximum	ADJ
cana-2085	225	7	degree	degree	NOUN
cana-2085	225	8	of	of	ADP
cana-2085	225	9	2	2	NUM
cana-2085	225	10	,	,	PUNCT
cana-2085	225	11	the	the	DET
cana-2085	225	12	network	network	NOUN
cana-2085	225	13	structures	structure	NOUN
cana-2085	225	14	are	be	AUX
cana-2085	225	15	simple	simple	ADJ
cana-2085	225	16	(	(	PUNCT
cana-2085	225	17	such	such	ADJ
cana-2085	225	18	as	as	ADP
cana-2085	225	19	paths	path	NOUN
cana-2085	225	20	or	or	CCONJ
cana-2085	225	21	cycles	cycle	NOUN
cana-2085	225	22	)	)	PUNCT
cana-2085	225	23	,	,	PUNCT
cana-2085	225	24	which	which	PRON
cana-2085	225	25	can	can	AUX
cana-2085	225	26	be	be	AUX
cana-2085	225	27	used	use	VERB
cana-2085	225	28	to	to	PART
cana-2085	225	29	design	design	VERB
cana-2085	225	30	and	and	CCONJ
cana-2085	225	31	analyse	analyse	VERB
cana-2085	225	32	very	very	ADV
cana-2085	225	33	small	small	ADJ
cana-2085	225	34	and	and	CCONJ
cana-2085	225	35	specific	specific	ADJ
cana-2085	225	36	network	network	NOUN
cana-2085	225	37	configurations	configuration	NOUN
cana-2085	225	38	.	.	PUNCT
cana-2085	226	1	algorithmic	algorithmic	ADJ
cana-2085	226	2	efficiency	efficiency	NOUN
cana-2085	226	3	-	-	PUNCT
cana-2085	226	4	exact	exact	ADJ
cana-2085	226	5	computation	computation	NOUN
cana-2085	226	6	for	for	ADP
cana-2085	226	7	small	small	ADJ
cana-2085	226	8	graphs	graph	NOUN
cana-2085	226	9	:	:	PUNCT
cana-2085	226	10	for	for	ADP
cana-2085	226	11	graphs	graph	NOUN
cana-2085	226	12	with	with	ADP
cana-2085	226	13	up	up	ADP
cana-2085	226	14	to	to	PART
cana-2085	226	15	4	4	NUM
cana-2085	226	16	vertices	vertex	NOUN
cana-2085	226	17	,	,	PUNCT
cana-2085	226	18	the	the	DET
cana-2085	226	19	theorem	theorem	NOUN
cana-2085	226	20	provides	provide	VERB
cana-2085	226	21	an	an	DET
cana-2085	226	22	exact	exact	ADJ
cana-2085	226	23	and	and	CCONJ
cana-2085	226	24	straightforward	straightforward	ADJ
cana-2085	226	25	way	way	NOUN
cana-2085	226	26	to	to	PART
cana-2085	226	27	compute	compute	VERB
cana-2085	226	28	the	the	DET
cana-2085	226	29	middle	middle	ADJ
cana-2085	226	30	roman	roman	ADJ
cana-2085	226	31	domination	domination	NOUN
cana-2085	226	32	number	number	NOUN
cana-2085	226	33	[	[	X
cana-2085	226	34	34	34	NUM
cana-2085	226	35	,	,	PUNCT
cana-2085	226	36	35	35	NUM
cana-2085	226	37	]	]	PUNCT
cana-2085	226	38	.	.	PUNCT
cana-2085	227	1	this	this	PRON
cana-2085	227	2	can	can	AUX
cana-2085	227	3	be	be	AUX
cana-2085	227	4	useful	useful	ADJ
cana-2085	227	5	in	in	ADP
cana-2085	227	6	algorithm	algorithm	NOUN
cana-2085	227	7	design	design	NOUN
cana-2085	227	8	for	for	ADP
cana-2085	227	9	small	small	ADJ
cana-2085	227	10	-	-	PUNCT
cana-2085	227	11	scale	scale	NOUN
cana-2085	227	12	problems	problem	NOUN
cana-2085	227	13	where	where	SCONJ
cana-2085	227	14	brute	brute	ADJ
cana-2085	227	15	-	-	PUNCT
cana-2085	227	16	force	force	NOUN
cana-2085	227	17	methods	method	NOUN
cana-2085	227	18	or	or	CCONJ
cana-2085	227	19	exhaustive	exhaustive	ADJ
cana-2085	227	20	search	search	NOUN
cana-2085	227	21	can	can	AUX
cana-2085	227	22	be	be	AUX
cana-2085	227	23	feasible	feasible	ADJ
cana-2085	227	24	and	and	CCONJ
cana-2085	227	25	efficient	efficient	ADJ
cana-2085	227	26	[	[	X
cana-2085	227	27	41	41	NUM
cana-2085	227	28	]	]	PUNCT
cana-2085	227	29	.	.	PUNCT
cana-2085	228	1	graph	graph	NOUN
cana-2085	228	2	theory	theory	NOUN
cana-2085	228	3	education	education	NOUN
cana-2085	228	4	-	-	PUNCT
cana-2085	228	5	teaching	teaching	NOUN
cana-2085	228	6	and	and	CCONJ
cana-2085	228	7	learning	learning	NOUN
cana-2085	228	8	:	:	PUNCT
cana-2085	228	9	the	the	DET
cana-2085	228	10	theorem	theorem	NOUN
cana-2085	228	11	can	can	AUX
cana-2085	228	12	serve	serve	VERB
cana-2085	228	13	as	as	ADP
cana-2085	228	14	a	a	DET
cana-2085	228	15	pedagogical	pedagogical	ADJ
cana-2085	228	16	tool	tool	NOUN
cana-2085	228	17	to	to	PART
cana-2085	228	18	illustrate	illustrate	VERB
cana-2085	228	19	the	the	DET
cana-2085	228	20	concepts	concept	NOUN
cana-2085	228	21	of	of	ADP
cana-2085	228	22	domination	domination	NOUN
cana-2085	228	23	numbers	number	NOUN
cana-2085	228	24	and	and	CCONJ
cana-2085	228	25	middle	middle	ADJ
cana-2085	228	26	roman	roman	ADJ
cana-2085	228	27	domination	domination	NOUN
cana-2085	228	28	numbers	number	NOUN
cana-2085	228	29	in	in	ADP
cana-2085	228	30	graph	graph	NOUN
cana-2085	228	31	theory	theory	NOUN
cana-2085	228	32	.	.	PUNCT
cana-2085	229	1	its	its	PRON
cana-2085	229	2	simplicity	simplicity	NOUN
cana-2085	229	3	makes	make	VERB
cana-2085	229	4	it	it	PRON
cana-2085	229	5	an	an	DET
cana-2085	229	6	excellent	excellent	ADJ
cana-2085	229	7	example	example	NOUN
cana-2085	229	8	for	for	ADP
cana-2085	229	9	teaching	teach	VERB
cana-2085	229	10	these	these	DET
cana-2085	229	11	concepts	concept	NOUN
cana-2085	229	12	to	to	ADP
cana-2085	229	13	students	student	NOUN
cana-2085	229	14	in	in	ADP
cana-2085	229	15	an	an	DET
cana-2085	229	16	introductory	introductory	ADJ
cana-2085	229	17	graph	graph	NOUN
cana-2085	229	18	theory	theory	NOUN
cana-2085	229	19	course	course	NOUN
cana-2085	229	20	.	.	PUNCT
cana-2085	230	1	optimizing	optimize	VERB
cana-2085	230	2	small	small	ADJ
cana-2085	230	3	-	-	PUNCT
cana-2085	230	4	scale	scale	NOUN
cana-2085	230	5	graphs	graph	NOUN
cana-2085	230	6	in	in	ADP
cana-2085	230	7	research	research	NOUN
cana-2085	230	8	-	-	PUNCT
cana-2085	230	9	specialized	specialized	ADJ
cana-2085	230	10	applications	application	NOUN
cana-2085	230	11	:	:	PUNCT
cana-2085	230	12	in	in	ADP
cana-2085	230	13	research	research	NOUN
cana-2085	230	14	areas	area	NOUN
cana-2085	230	15	that	that	PRON
cana-2085	230	16	involve	involve	VERB
cana-2085	230	17	small	small	ADJ
cana-2085	230	18	,	,	PUNCT
cana-2085	230	19	specialized	specialized	ADJ
cana-2085	230	20	networks	network	NOUN
cana-2085	230	21	(	(	PUNCT
cana-2085	230	22	e.g.	e.g.	ADV
cana-2085	230	23	,	,	PUNCT
cana-2085	230	24	certain	certain	ADJ
cana-2085	230	25	types	type	NOUN
cana-2085	230	26	of	of	ADP
cana-2085	230	27	molecular	molecular	ADJ
cana-2085	230	28	networks	network	NOUN
cana-2085	230	29	,	,	PUNCT
cana-2085	230	30	small	small	ADJ
cana-2085	230	31	-	-	PUNCT
cana-2085	230	32	scale	scale	NOUN
cana-2085	230	33	computational	computational	ADJ
cana-2085	230	34	models	model	NOUN
cana-2085	230	35	)	)	PUNCT
cana-2085	230	36	,	,	PUNCT
cana-2085	230	37	the	the	DET
cana-2085	230	38	theorem	theorem	NOUN
cana-2085	230	39	can	can	AUX
cana-2085	230	40	be	be	AUX
cana-2085	230	41	used	use	VERB
cana-2085	230	42	to	to	PART
cana-2085	230	43	determine	determine	VERB
cana-2085	230	44	optimal	optimal	ADJ
cana-2085	230	45	configurations	configuration	NOUN
cana-2085	230	46	or	or	CCONJ
cana-2085	230	47	understand	understand	VERB
cana-2085	230	48	the	the	DET
cana-2085	230	49	properties	property	NOUN
cana-2085	230	50	of	of	ADP
cana-2085	230	51	these	these	DET
cana-2085	230	52	networks	network	NOUN
cana-2085	230	53	with	with	ADP
cana-2085	230	54	precision	precision	NOUN
cana-2085	230	55	.	.	PUNCT
cana-2085	231	1	resource	resource	NOUN
cana-2085	231	2	allocation	allocation	NOUN
cana-2085	231	3	in	in	ADP
cana-2085	231	4	simple	simple	ADJ
cana-2085	231	5	systems	system	NOUN
cana-2085	231	6	-	-	PUNCT
cana-2085	231	7	efficient	efficient	ADJ
cana-2085	231	8	resource	resource	NOUN
cana-2085	231	9	placement	placement	NOUN
cana-2085	231	10	:	:	PUNCT
cana-2085	231	11	for	for	ADP
cana-2085	231	12	systems	system	NOUN
cana-2085	231	13	or	or	CCONJ
cana-2085	231	14	models	model	NOUN
cana-2085	231	15	with	with	ADP
cana-2085	231	16	up	up	ADP
cana-2085	231	17	to	to	PART
cana-2085	231	18	4	4	NUM
cana-2085	231	19	nodes	node	NOUN
cana-2085	231	20	,	,	PUNCT
cana-2085	231	21	where	where	SCONJ
cana-2085	231	22	resources	resource	NOUN
cana-2085	231	23	need	need	VERB
cana-2085	231	24	to	to	PART
cana-2085	231	25	be	be	AUX
cana-2085	231	26	allocated	allocate	VERB
cana-2085	231	27	or	or	CCONJ
cana-2085	231	28	placed	place	VERB
cana-2085	231	29	optimally	optimally	ADV
cana-2085	231	30	,	,	PUNCT
cana-2085	231	31	the	the	DET
cana-2085	231	32	theorem	theorem	NOUN
cana-2085	231	33	provides	provide	VERB
cana-2085	231	34	a	a	DET
cana-2085	231	35	clear	clear	ADJ
cana-2085	231	36	understanding	understanding	NOUN
cana-2085	231	37	of	of	ADP
cana-2085	231	38	the	the	DET
cana-2085	231	39	relationship	relationship	NOUN
cana-2085	231	40	between	between	ADP
cana-2085	231	41	the	the	DET
cana-2085	231	42	domination	domination	NOUN
cana-2085	231	43	number	number	NOUN
cana-2085	231	44	and	and	CCONJ
cana-2085	231	45	the	the	DET
cana-2085	231	46	middle	middle	ADJ
cana-2085	231	47	roman	roman	ADJ
cana-2085	231	48	domination	domination	NOUN
cana-2085	231	49	number	number	NOUN
cana-2085	231	50	.	.	PUNCT
cana-2085	232	1	hence	hence	ADV
cana-2085	232	2	,	,	PUNCT
cana-2085	232	3	the	the	DET
cana-2085	232	4	theorem	theorem	NOUN
cana-2085	232	5	provides	provide	VERB
cana-2085	232	6	a	a	DET
cana-2085	232	7	clear	clear	ADJ
cana-2085	232	8	and	and	CCONJ
cana-2085	232	9	specific	specific	ADJ
cana-2085	232	10	framework	framework	NOUN
cana-2085	232	11	for	for	ADP
cana-2085	232	12	understanding	understanding	NOUN
cana-2085	232	13	and	and	CCONJ
cana-2085	232	14	solving	solve	VERB
cana-2085	232	15	problems	problem	NOUN
cana-2085	232	16	related	relate	VERB
cana-2085	232	17	to	to	ADP
cana-2085	232	18	middle	middle	ADJ
cana-2085	232	19	roman	roman	ADJ
cana-2085	232	20	domination	domination	NOUN
cana-2085	232	21	in	in	ADP
cana-2085	232	22	very	very	ADV
cana-2085	232	23	simple	simple	ADJ
cana-2085	232	24	or	or	CCONJ
cana-2085	232	25	small	small	ADJ
cana-2085	232	26	communications	communication	NOUN
cana-2085	232	27	on	on	ADP
cana-2085	232	28	applied	apply	VERB
cana-2085	232	29	nonlinear	nonlinear	ADJ
cana-2085	232	30	analysis	analysis	NOUN
cana-2085	232	31	issn	issn	NOUN
cana-2085	232	32	:	:	PUNCT
cana-2085	232	33	1074	1074	NUM
cana-2085	232	34	-	-	PUNCT
cana-2085	232	35	133x	133x	NUM
cana-2085	232	36	vol	vol	NOUN
cana-2085	232	37	32	32	NUM
cana-2085	232	38	no	no	NOUN
cana-2085	232	39	.	.	PUNCT
cana-2085	233	1	1s	1s	NUM
cana-2085	233	2	(	(	PUNCT
cana-2085	233	3	2025	2025	NUM
cana-2085	233	4	)	)	PUNCT
cana-2085	233	5	24	24	NUM
cana-2085	233	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-2085	233	7	graphs	graph	NOUN
cana-2085	233	8	,	,	PUNCT
cana-2085	233	9	making	make	VERB
cana-2085	233	10	it	it	PRON
cana-2085	233	11	valuable	valuable	ADJ
cana-2085	233	12	in	in	ADP
cana-2085	233	13	theoretical	theoretical	ADJ
cana-2085	233	14	studies	study	NOUN
cana-2085	233	15	,	,	PUNCT
cana-2085	233	16	educational	educational	ADJ
cana-2085	233	17	contexts	contexts	NOUN
cana-2085	233	18	,	,	PUNCT
cana-2085	233	19	and	and	CCONJ
cana-2085	233	20	practical	practical	ADJ
cana-2085	233	21	applications	application	NOUN
cana-2085	233	22	involving	involve	VERB
cana-2085	233	23	small	small	ADJ
cana-2085	233	24	-	-	PUNCT
cana-2085	233	25	scale	scale	NOUN
cana-2085	233	26	systems	system	NOUN
cana-2085	233	27	.	.	PUNCT
cana-2085	234	1	theorem	theorem	VERB
cana-2085	234	2	8	8	NUM
cana-2085	234	3	:	:	PUNCT
cana-2085	234	4	for	for	ADP
cana-2085	234	5	any	any	DET
cana-2085	234	6	connected	connected	ADJ
cana-2085	234	7	graph	graph	NOUN
cana-2085	234	8	𝐺	𝐺	PROPN
cana-2085	234	9	,	,	PUNCT
cana-2085	234	10	𝛾𝑀𝑅(𝐺	𝛾𝑀𝑅(𝐺	PROPN
cana-2085	234	11	)	)	PUNCT
cana-2085	234	12	=	=	PUNCT
cana-2085	235	1	𝛾	𝛾	ADP
cana-2085	235	2	+	+	ADV
cana-2085	235	3	2	2	NUM
cana-2085	235	4	if	if	SCONJ
cana-2085	235	5	and	and	CCONJ
cana-2085	235	6	only	only	ADV
cana-2085	235	7	if	if	SCONJ
cana-2085	235	8	there	there	PRON
cana-2085	235	9	exist	exist	VERB
cana-2085	235	10	a	a	DET
cana-2085	235	11	minimal	minimal	ADJ
cana-2085	235	12	dominating	dominating	NOUN
cana-2085	235	13	set	set	VERB
cana-2085	235	14	𝑆	𝑆	PROPN
cana-2085	235	15	satisfying	satisfy	VERB
cana-2085	235	16	one	one	NUM
cana-2085	235	17	of	of	ADP
cana-2085	235	18	the	the	DET
cana-2085	235	19	following	following	ADJ
cana-2085	235	20	conditions	condition	NOUN
cana-2085	235	21	:	:	PUNCT
cana-2085	235	22	(	(	PUNCT
cana-2085	235	23	i	i	NOUN
cana-2085	235	24	)	)	PUNCT
cana-2085	235	25	there	there	PRON
cana-2085	235	26	exist	exist	VERB
cana-2085	235	27	a	a	DET
cana-2085	235	28	vertex	vertex	NOUN
cana-2085	235	29	𝑣𝜖𝑆	𝑣𝜖𝑆	PROPN
cana-2085	235	30	such	such	ADJ
cana-2085	235	31	that	that	SCONJ
cana-2085	235	32	𝑣	𝑣	PRON
cana-2085	235	33	is	be	AUX
cana-2085	235	34	a	a	DET
cana-2085	235	35	support	support	NOUN
cana-2085	235	36	and	and	CCONJ
cana-2085	235	37	𝑣	𝑣	ADP
cana-2085	235	38	⊆	⊆	NUM
cana-2085	235	39	𝑁[𝑃𝑛𝑠(𝑣	𝑁[𝑃𝑛𝑠(𝑣	NUM
cana-2085	235	40	,	,	PUNCT
cana-2085	235	41	𝑠	𝑠	PROPN
cana-2085	235	42	)	)	PUNCT
cana-2085	235	43	]	]	PUNCT
cana-2085	236	1	(	(	PUNCT
cana-2085	236	2	ii	ii	NOUN
cana-2085	236	3	)	)	PUNCT
cana-2085	236	4	there	there	PRON
cana-2085	236	5	exist	exist	VERB
cana-2085	236	6	two	two	NUM
cana-2085	236	7	vertices	vertex	NOUN
cana-2085	236	8	𝑢	𝑢	NOUN
cana-2085	236	9	and	and	CCONJ
cana-2085	236	10	𝑣	𝑣	X
cana-2085	236	11	in	in	ADP
cana-2085	236	12	𝑆	𝑆	PROPN
cana-2085	236	13	such	such	ADJ
cana-2085	236	14	that	that	SCONJ
cana-2085	236	15	𝑣	𝑣	ADP
cana-2085	236	16	⊆	⊆	NUM
cana-2085	236	17	𝑁[𝑃𝑛𝑠(𝑢	𝑁[𝑃𝑛𝑠(𝑢	NUM
cana-2085	236	18	,	,	PUNCT
cana-2085	236	19	𝑆	𝑆	PROPN
cana-2085	236	20	)	)	PUNCT
cana-2085	236	21	]	]	PUNCT
cana-2085	236	22	∪	∪	ADP
cana-2085	236	23	𝑁[𝑃𝑛𝑠(𝑣	𝑁[𝑃𝑛𝑠(𝑣	PROPN
cana-2085	236	24	,	,	PUNCT
cana-2085	236	25	𝑆	𝑆	PROPN
cana-2085	236	26	)	)	PUNCT
cana-2085	236	27	]	]	PUNCT
cana-2085	236	28	𝐺	𝐺	PROPN
cana-2085	236	29	has	have	VERB
cana-2085	236	30	a	a	DET
cana-2085	236	31	vertex	vertex	NOUN
cana-2085	236	32	𝑣	𝑣	ADP
cana-2085	236	33	of	of	ADP
cana-2085	236	34	degree	degree	NOUN
cana-2085	236	35	𝑛	𝑛	DET
cana-2085	236	36	−	−	PROPN
cana-2085	236	37	𝛾	𝛾	NOUN
cana-2085	236	38	or	or	CCONJ
cana-2085	236	39	𝐺	𝐺	PROPN
cana-2085	236	40	has	have	VERB
cana-2085	236	41	two	two	NUM
cana-2085	236	42	vertices	vertex	NOUN
cana-2085	236	43	𝑣	𝑣	ADP
cana-2085	236	44	and	and	CCONJ
cana-2085	236	45	𝑤	𝑤	ADP
cana-2085	236	46	such	such	ADJ
cana-2085	236	47	that	that	PRON
cana-2085	236	48	|𝑉[𝑣	|𝑉[𝑣	VERB
cana-2085	236	49	]	]	PUNCT
cana-2085	236	50	∪	∪	X
cana-2085	236	51	𝑁[𝑤]|	𝑁[𝑤]|	PUNCT
cana-2085	236	52	=	=	SYM
cana-2085	236	53	𝑛	𝑛	DET
cana-2085	236	54	−	−	PROPN
cana-2085	236	55	𝛾(𝐺	𝛾(𝐺	NOUN
cana-2085	236	56	)	)	PUNCT
cana-2085	237	1	+	+	CCONJ
cana-2085	237	2	2	2	NUM
cana-2085	237	3	proof	proof	NOUN
cana-2085	237	4	:	:	PUNCT
cana-2085	237	5	if	if	SCONJ
cana-2085	237	6	𝐺	𝐺	PROPN
cana-2085	237	7	has	have	VERB
cana-2085	237	8	a	a	DET
cana-2085	237	9	vertex	vertex	NOUN
cana-2085	237	10	𝑣	𝑣	ADP
cana-2085	237	11	of	of	ADP
cana-2085	237	12	degree	degree	NOUN
cana-2085	237	13	𝑛	𝑛	DET
cana-2085	237	14	−	−	PROPN
cana-2085	237	15	𝛾	𝛾	ADP
cana-2085	237	16	≥	≥	NOUN
cana-2085	237	17	3	3	NUM
cana-2085	237	18	,	,	PUNCT
cana-2085	237	19	we	we	PRON
cana-2085	237	20	define	define	VERB
cana-2085	237	21	𝑉0	𝑉0	NOUN
cana-2085	237	22	=	=	SYM
cana-2085	237	23	𝑁(𝑣	𝑁(𝑣	PROPN
cana-2085	237	24	)	)	PUNCT
cana-2085	237	25	,	,	PUNCT
cana-2085	237	26	𝑉1	𝑉1	NOUN
cana-2085	237	27	=	=	SYM
cana-2085	237	28	𝑉	𝑉	PROPN
cana-2085	237	29	−	−	PROPN
cana-2085	237	30	𝑁[𝑣	𝑁[𝑣	PROPN
cana-2085	237	31	]	]	X
cana-2085	237	32	,	,	PUNCT
cana-2085	237	33	𝑉2	𝑉2	NOUN
cana-2085	237	34	=	=	SYM
cana-2085	237	35	∅	∅	NOUN
cana-2085	237	36	and	and	CCONJ
cana-2085	237	37	𝑉3	𝑉3	NOUN
cana-2085	237	38	=	=	SYM
cana-2085	237	39	{	{	PUNCT
cana-2085	237	40	𝑣	𝑣	NOUN
cana-2085	237	41	}	}	PUNCT
cana-2085	237	42	,	,	PUNCT
cana-2085	237	43	then	then	ADV
cana-2085	237	44	𝑓	𝑓	X
cana-2085	237	45	=	=	X
cana-2085	237	46	(	(	PUNCT
cana-2085	237	47	𝑉0	𝑉0	NOUN
cana-2085	237	48	,	,	PUNCT
cana-2085	237	49	𝑉1	𝑉1	PROPN
cana-2085	237	50	,	,	PUNCT
cana-2085	237	51	𝑉2	𝑉2	NOUN
cana-2085	237	52	,	,	PUNCT
cana-2085	237	53	𝑉3	𝑉3	NOUN
cana-2085	237	54	)	)	PUNCT
cana-2085	237	55	is	be	AUX
cana-2085	237	56	a	a	DET
cana-2085	237	57	mrdf	mrdf	NOUN
cana-2085	237	58	with	with	ADP
cana-2085	237	59	𝑓(𝑣	𝑓(𝑣	NOUN
cana-2085	237	60	)	)	PUNCT
cana-2085	237	61	=	=	SYM
cana-2085	237	62	𝛾(𝐺	𝛾(𝐺	NOUN
cana-2085	237	63	)	)	PUNCT
cana-2085	238	1	+	+	CCONJ
cana-2085	238	2	2	2	NUM
cana-2085	238	3	and	and	CCONJ
cana-2085	238	4	hence	hence	ADV
cana-2085	238	5	is	be	AUX
cana-2085	238	6	a	a	DET
cana-2085	238	7	𝛾𝑀𝑅function	𝛾𝑀𝑅function	NOUN
cana-2085	238	8	with	with	ADP
cana-2085	238	9	𝑓(𝑣	𝑓(𝑣	NOUN
cana-2085	238	10	)	)	PUNCT
cana-2085	238	11	=	=	SYM
cana-2085	238	12	𝛾(𝐺	𝛾(𝐺	NOUN
cana-2085	238	13	)	)	PUNCT
cana-2085	239	1	+	+	CCONJ
cana-2085	239	2	2	2	X
cana-2085	239	3	.	.	PUNCT
cana-2085	240	1	if	if	SCONJ
cana-2085	240	2	there	there	PRON
cana-2085	240	3	are	be	VERB
cana-2085	240	4	two	two	NUM
cana-2085	240	5	vertices	vertex	NOUN
cana-2085	240	6	𝑢	𝑢	NOUN
cana-2085	240	7	and	and	CCONJ
cana-2085	240	8	𝑣	𝑣	ADP
cana-2085	240	9	such	such	ADJ
cana-2085	240	10	that	that	DET
cana-2085	240	11	𝑁[𝑢	𝑁[𝑢	NOUN
cana-2085	240	12	]	]	PUNCT
cana-2085	240	13	∪	∪	ADP
cana-2085	240	14	𝑁[𝑣	𝑁[𝑣	PROPN
cana-2085	240	15	]	]	X
cana-2085	240	16	=	=	SYM
cana-2085	240	17	𝑛	𝑛	PRON
cana-2085	240	18	−	−	PROPN
cana-2085	240	19	𝛾(𝐺	𝛾(𝐺	NOUN
cana-2085	240	20	)	)	PUNCT
cana-2085	240	21	+	+	CCONJ
cana-2085	240	22	1	1	NUM
cana-2085	240	23	,	,	PUNCT
cana-2085	240	24	𝑉3	𝑉3	NOUN
cana-2085	240	25	=	=	NOUN
cana-2085	240	26	∅	∅	NOUN
cana-2085	240	27	we	we	PRON
cana-2085	240	28	define	define	VERB
cana-2085	240	29	𝑉2	𝑉2	NOUN
cana-2085	240	30	=	=	SYM
cana-2085	240	31	{	{	PUNCT
cana-2085	240	32	𝑢	𝑢	X
cana-2085	240	33	,	,	PUNCT
cana-2085	240	34	𝑣	𝑣	NOUN
cana-2085	240	35	}	}	PUNCT
cana-2085	240	36	,	,	PUNCT
cana-2085	240	37	𝑉0	𝑉0	NOUN
cana-2085	240	38	=	=	SYM
cana-2085	240	39	𝑁[𝑢	𝑁[𝑢	NOUN
cana-2085	240	40	]	]	PUNCT
cana-2085	240	41	∪	∪	ADP
cana-2085	240	42	𝑁[𝑣	𝑁[𝑣	PROPN
cana-2085	240	43	]	]	X
cana-2085	240	44	−	−	PROPN
cana-2085	240	45	{	{	PUNCT
cana-2085	240	46	𝑢	𝑢	X
cana-2085	240	47	,	,	PUNCT
cana-2085	240	48	𝑣	𝑣	NOUN
cana-2085	240	49	}	}	PUNCT
cana-2085	240	50	,	,	PUNCT
cana-2085	240	51	𝑉1	𝑉1	NOUN
cana-2085	240	52	=	=	SYM
cana-2085	240	53	𝑉	𝑉	PROPN
cana-2085	240	54	−	−	PROPN
cana-2085	240	55	𝑁[𝑢	𝑁[𝑢	NOUN
cana-2085	240	56	]	]	PUNCT
cana-2085	240	57	∪	∪	ADP
cana-2085	240	58	𝑁[𝑣	𝑁[𝑣	PROPN
cana-2085	240	59	]	]	PUNCT
cana-2085	240	60	.	.	PUNCT
cana-2085	241	1	then	then	ADV
cana-2085	241	2	𝑓	𝑓	X
cana-2085	241	3	=	=	X
cana-2085	241	4	(	(	PUNCT
cana-2085	241	5	𝑉0	𝑉0	NOUN
cana-2085	241	6	,	,	PUNCT
cana-2085	241	7	𝑉1	𝑉1	PROPN
cana-2085	241	8	,	,	PUNCT
cana-2085	241	9	𝑉2	𝑉2	NOUN
cana-2085	241	10	,	,	PUNCT
cana-2085	241	11	𝑉3	𝑉3	NOUN
cana-2085	241	12	)	)	PUNCT
cana-2085	241	13	is	be	AUX
cana-2085	241	14	a	a	DET
cana-2085	241	15	mrdf	mrdf	NOUN
cana-2085	241	16	with	with	ADP
cana-2085	241	17	𝑓(𝑣	𝑓(𝑣	NOUN
cana-2085	241	18	)	)	PUNCT
cana-2085	241	19	=	=	SYM
cana-2085	241	20	𝛾(𝐺	𝛾(𝐺	NOUN
cana-2085	241	21	)	)	PUNCT
cana-2085	242	1	+	+	CCONJ
cana-2085	242	2	2	2	NUM
cana-2085	242	3	and	and	CCONJ
cana-2085	242	4	hence	hence	ADV
cana-2085	242	5	is	be	AUX
cana-2085	242	6	a	a	DET
cana-2085	242	7	𝛾𝑀𝑅function	𝛾𝑀𝑅function	NOUN
cana-2085	242	8	.	.	PUNCT
cana-2085	243	1	in	in	ADP
cana-2085	243	2	order	order	NOUN
cana-2085	243	3	for	for	ADP
cana-2085	243	4	a	a	DET
cana-2085	243	5	mrdf	mrdf	NOUN
cana-2085	243	6	𝑓	𝑓	X
cana-2085	243	7	=	=	X
cana-2085	243	8	(	(	PUNCT
cana-2085	243	9	𝑉0	𝑉0	NOUN
cana-2085	243	10	,	,	PUNCT
cana-2085	243	11	𝑉1	𝑉1	PROPN
cana-2085	243	12	,	,	PUNCT
cana-2085	243	13	𝑉2	𝑉2	NOUN
cana-2085	243	14	,	,	PUNCT
cana-2085	243	15	𝑉3	𝑉3	NOUN
cana-2085	243	16	)	)	PUNCT
cana-2085	243	17	to	to	PART
cana-2085	243	18	have	have	AUX
cana-2085	243	19	weight	weight	NOUN
cana-2085	243	20	𝛾(𝐺	𝛾(𝐺	NOUN
cana-2085	243	21	)	)	PUNCT
cana-2085	243	22	+	+	CCONJ
cana-2085	243	23	2	2	NUM
cana-2085	243	24	either	either	CCONJ
cana-2085	243	25	(	(	PUNCT
cana-2085	243	26	i	i	NOUN
cana-2085	243	27	)	)	PUNCT
cana-2085	243	28	|𝑉3|	|𝑉3|	X
cana-2085	243	29	=	=	SYM
cana-2085	243	30	1	1	NUM
cana-2085	243	31	,	,	PUNCT
cana-2085	243	32	|𝑉2|	|𝑉2|	PROPN
cana-2085	243	33	=	=	SYM
cana-2085	243	34	0	0	NUM
cana-2085	243	35	and	and	CCONJ
cana-2085	243	36	|𝑉1|	|𝑉1|	NOUN
cana-2085	243	37	=	=	SYM
cana-2085	243	38	𝛾(𝐺	𝛾(𝐺	PROPN
cana-2085	243	39	)	)	PUNCT
cana-2085	243	40	−	−	NOUN
cana-2085	243	41	1	1	NUM
cana-2085	243	42	or	or	CCONJ
cana-2085	243	43	(	(	PUNCT
cana-2085	243	44	ii	ii	NOUN
cana-2085	243	45	)	)	PUNCT
cana-2085	243	46	|𝑉3|	|𝑉3|	X
cana-2085	243	47	=	=	SYM
cana-2085	243	48	0	0	NUM
cana-2085	243	49	,	,	PUNCT
cana-2085	243	50	|𝑉2|	|𝑉2|	PROPN
cana-2085	243	51	=	=	SYM
cana-2085	243	52	2	2	NUM
cana-2085	243	53	and	and	CCONJ
cana-2085	243	54	|𝑉1|	|𝑉1|	NOUN
cana-2085	243	55	=	=	SYM
cana-2085	243	56	𝛾(𝐺	𝛾(𝐺	PROPN
cana-2085	243	57	)	)	PUNCT
cana-2085	243	58	−	−	PROPN
cana-2085	243	59	2	2	X
cana-2085	243	60	.	.	PUNCT
cana-2085	243	61	in	in	ADP
cana-2085	243	62	case	case	NOUN
cana-2085	243	63	(	(	PUNCT
cana-2085	243	64	i	i	NOUN
cana-2085	243	65	)	)	PUNCT
cana-2085	243	66	let	let	VERB
cana-2085	243	67	𝑓	𝑓	PRON
cana-2085	243	68	=	=	X
cana-2085	243	69	(	(	PUNCT
cana-2085	243	70	𝑉0	𝑉0	NOUN
cana-2085	243	71	,	,	PUNCT
cana-2085	243	72	𝑉1	𝑉1	PROPN
cana-2085	243	73	,	,	PUNCT
cana-2085	243	74	𝑉2	𝑉2	NOUN
cana-2085	243	75	,	,	PUNCT
cana-2085	243	76	𝑉3	𝑉3	NOUN
cana-2085	243	77	)	)	PUNCT
cana-2085	243	78	be	be	AUX
cana-2085	243	79	a	a	DET
cana-2085	243	80	𝛾𝑀𝑅function	𝛾𝑀𝑅function	NOUN
cana-2085	243	81	for	for	ADP
cana-2085	243	82	𝐺	𝐺	PROPN
cana-2085	243	83	of	of	ADP
cana-2085	243	84	weight	weight	NOUN
cana-2085	243	85	𝛾(𝐺	𝛾(𝐺	PROPN
cana-2085	243	86	)	)	PUNCT
cana-2085	244	1	+	+	CCONJ
cana-2085	244	2	2	2	NUM
cana-2085	244	3	with	with	ADP
cana-2085	244	4	|𝑉3|	|𝑉3|	X
cana-2085	244	5	=	=	SYM
cana-2085	244	6	1	1	NUM
cana-2085	244	7	,	,	PUNCT
cana-2085	244	8	|𝑉1|	|𝑉1|	NOUN
cana-2085	244	9	=	=	SYM
cana-2085	244	10	𝛾(𝐺	𝛾(𝐺	PROPN
cana-2085	244	11	)	)	PUNCT
cana-2085	244	12	−	−	PROPN
cana-2085	245	1	1	1	X
cana-2085	245	2	.	.	PUNCT
cana-2085	245	3	let	let	VERB
cana-2085	245	4	𝑉3	𝑉3	NOUN
cana-2085	245	5	=	=	SYM
cana-2085	245	6	{	{	PUNCT
cana-2085	245	7	𝑣	𝑣	NOUN
cana-2085	245	8	}	}	PUNCT
cana-2085	245	9	.	.	PUNCT
cana-2085	246	1	since	since	SCONJ
cana-2085	246	2	,	,	PUNCT
cana-2085	246	3	no	no	DET
cana-2085	246	4	edge	edge	NOUN
cana-2085	246	5	of	of	ADP
cana-2085	246	6	𝐺	𝐺	PROPN
cana-2085	246	7	join	join	VERB
cana-2085	246	8	𝑉1	𝑉1	NOUN
cana-2085	246	9	and	and	CCONJ
cana-2085	246	10	𝑣	𝑣	ADP
cana-2085	246	11	,	,	PUNCT
cana-2085	246	12	𝑣	𝑣	PRON
cana-2085	246	13	≻	≻	NOUN
cana-2085	246	14	𝑉0	𝑉0	NOUN
cana-2085	246	15	,	,	PUNCT
cana-2085	246	16	it	it	PRON
cana-2085	246	17	follows	follow	VERB
cana-2085	246	18	thatdeg(𝑣	thatdeg(𝑣	NOUN
cana-2085	246	19	)	)	PUNCT
cana-2085	246	20	=	=	PUNCT
cana-2085	246	21	|𝑉0|	|𝑉0|	X
cana-2085	247	1	=	=	PUNCT
cana-2085	247	2	𝑛	𝑛	DET
cana-2085	247	3	−	−	PROPN
cana-2085	247	4	|𝑉1|	|𝑉1|	NOUN
cana-2085	247	5	−	−	NOUN
cana-2085	247	6	|𝑉3|	|𝑉3|	NOUN
cana-2085	247	7	=	=	PUNCT
cana-2085	247	8	𝑛	𝑛	PRON
cana-2085	247	9	−	−	PROPN
cana-2085	247	10	(	(	PUNCT
cana-2085	247	11	𝛾(𝐺	𝛾(𝐺	PROPN
cana-2085	247	12	)	)	PUNCT
cana-2085	248	1	−	−	PROPN
cana-2085	248	2	1	1	NUM
cana-2085	248	3	)	)	PUNCT
cana-2085	248	4	−	−	PROPN
cana-2085	248	5	1	1	NUM
cana-2085	248	6	=	=	SYM
cana-2085	248	7	𝑛	𝑛	DET
cana-2085	248	8	−	−	PROPN
cana-2085	248	9	𝛾(𝐺	𝛾(𝐺	NOUN
cana-2085	248	10	)	)	PUNCT
cana-2085	248	11	.	.	PUNCT
cana-2085	249	1	in	in	ADP
cana-2085	249	2	case	case	NOUN
cana-2085	249	3	(	(	PUNCT
cana-2085	249	4	ii	ii	NOUN
cana-2085	249	5	)	)	PUNCT
cana-2085	249	6	let	let	VERB
cana-2085	249	7	𝑓	𝑓	PRON
cana-2085	249	8	=	=	X
cana-2085	249	9	(	(	PUNCT
cana-2085	249	10	𝑉0	𝑉0	NOUN
cana-2085	249	11	,	,	PUNCT
cana-2085	249	12	𝑉1	𝑉1	PROPN
cana-2085	249	13	,	,	PUNCT
cana-2085	249	14	𝑉2	𝑉2	NOUN
cana-2085	249	15	,	,	PUNCT
cana-2085	249	16	𝑉3	𝑉3	NOUN
cana-2085	249	17	)	)	PUNCT
cana-2085	249	18	be	be	AUX
cana-2085	249	19	a	a	DET
cana-2085	249	20	𝛾𝑀𝑅function	𝛾𝑀𝑅function	NOUN
cana-2085	249	21	for	for	ADP
cana-2085	249	22	𝐺	𝐺	PROPN
cana-2085	249	23	of	of	ADP
cana-2085	249	24	weight	weight	NOUN
cana-2085	249	25	𝛾(𝐺	𝛾(𝐺	PROPN
cana-2085	249	26	)	)	PUNCT
cana-2085	250	1	+	+	CCONJ
cana-2085	250	2	2	2	NUM
cana-2085	250	3	with	with	ADP
cana-2085	250	4	|𝑉3|	|𝑉3|	X
cana-2085	250	5	=	=	SYM
cana-2085	250	6	0	0	NUM
cana-2085	250	7	,	,	PUNCT
cana-2085	250	8	|𝑉2|	|𝑉2|	PROPN
cana-2085	250	9	=	=	SYM
cana-2085	250	10	2	2	NUM
cana-2085	250	11	and	and	CCONJ
cana-2085	250	12	|𝑉1|	|𝑉1|	NOUN
cana-2085	250	13	=	=	SYM
cana-2085	250	14	𝛾(𝐺	𝛾(𝐺	PROPN
cana-2085	250	15	)	)	PUNCT
cana-2085	251	1	−	−	PROPN
cana-2085	251	2	2	2	X
cana-2085	251	3	.	.	PUNCT
cana-2085	252	1	let	let	VERB
cana-2085	252	2	𝑉2	𝑉2	NOUN
cana-2085	252	3	=	=	SYM
cana-2085	252	4	{	{	PUNCT
cana-2085	252	5	𝑢	𝑢	X
cana-2085	252	6	,	,	PUNCT
cana-2085	252	7	𝑣	𝑣	NOUN
cana-2085	252	8	}	}	PUNCT
cana-2085	252	9	.	.	PUNCT
cana-2085	253	1	since	since	SCONJ
cana-2085	253	2	,	,	PUNCT
cana-2085	253	3	no	no	DET
cana-2085	253	4	edge	edge	NOUN
cana-2085	253	5	of	of	ADP
cana-2085	253	6	𝐺	𝐺	PROPN
cana-2085	253	7	join	join	VERB
cana-2085	253	8	𝑉1	𝑉1	NOUN
cana-2085	253	9	and	and	CCONJ
cana-2085	253	10	to	to	ADP
cana-2085	253	11	𝑢	𝑢	PRON
cana-2085	253	12	or	or	CCONJ
cana-2085	253	13	𝑣	𝑣	ADP
cana-2085	253	14	,	,	PUNCT
cana-2085	253	15	{	{	PUNCT
cana-2085	253	16	𝑢	𝑢	X
cana-2085	253	17	,	,	PUNCT
cana-2085	253	18	𝑣	𝑣	NOUN
cana-2085	253	19	}	}	PUNCT
cana-2085	253	20	≻	≻	VERB
cana-2085	253	21	𝑉0	𝑉0	NOUN
cana-2085	253	22	.	.	PUNCT
cana-2085	254	1	it	it	PRON
cana-2085	254	2	follows	follow	VERB
cana-2085	254	3	that	that	SCONJ
cana-2085	254	4	|𝑁[𝑢	|𝑁[𝑢	NOUN
cana-2085	254	5	]	]	PUNCT
cana-2085	254	6	∪	∪	NOUN
cana-2085	254	7	𝑁[𝑣]|	𝑁[𝑣]|	PROPN
cana-2085	254	8	=	=	SYM
cana-2085	254	9	𝑛	𝑛	DET
cana-2085	254	10	−	−	NOUN
cana-2085	254	11	|𝑉1|	|𝑉1|	NOUN
cana-2085	254	12	=	=	SYM
cana-2085	254	13	𝑛[𝛾(𝐺	𝑛[𝛾(𝐺	NUM
cana-2085	254	14	)	)	PUNCT
cana-2085	254	15	−	−	PROPN
cana-2085	254	16	2	2	NUM
cana-2085	254	17	]	]	X
cana-2085	254	18	=	=	PUNCT
cana-2085	254	19	𝑛	𝑛	PRON
cana-2085	254	20	−	−	PROPN
cana-2085	254	21	𝛾(𝐺	𝛾(𝐺	NOUN
cana-2085	254	22	)	)	PUNCT
cana-2085	255	1	+	+	CCONJ
cana-2085	255	2	2	2	X
cana-2085	255	3	.	.	X
cana-2085	255	4	the	the	DET
cana-2085	255	5	theorem	theorem	NOUN
cana-2085	255	6	provides	provide	VERB
cana-2085	255	7	a	a	DET
cana-2085	255	8	characterization	characterization	NOUN
cana-2085	255	9	of	of	ADP
cana-2085	255	10	the	the	DET
cana-2085	255	11	middle	middle	ADJ
cana-2085	255	12	roman	roman	ADJ
cana-2085	255	13	domination	domination	NOUN
cana-2085	255	14	number	number	NOUN
cana-2085	255	15	𝛾𝑀𝑅(𝐺	𝛾𝑀𝑅(𝐺	PROPN
cana-2085	255	16	)	)	PUNCT
cana-2085	255	17	for	for	ADP
cana-2085	255	18	a	a	DET
cana-2085	255	19	connected	connected	ADJ
cana-2085	255	20	graph	graph	NOUN
cana-2085	255	21	𝐺	𝐺	PROPN
cana-2085	255	22	in	in	ADP
cana-2085	255	23	terms	term	NOUN
cana-2085	255	24	of	of	ADP
cana-2085	255	25	its	its	PRON
cana-2085	255	26	minimal	minimal	ADJ
cana-2085	255	27	dominating	dominating	NOUN
cana-2085	255	28	sets	set	NOUN
cana-2085	255	29	and	and	CCONJ
cana-2085	255	30	specific	specific	ADJ
cana-2085	255	31	conditions	condition	NOUN
cana-2085	255	32	involving	involve	VERB
cana-2085	255	33	vertex	vertex	NOUN
cana-2085	255	34	degrees	degree	NOUN
cana-2085	255	35	and	and	CCONJ
cana-2085	255	36	neighborhoods	neighborhood	NOUN
cana-2085	255	37	.	.	PUNCT
cana-2085	256	1	this	this	DET
cana-2085	256	2	result	result	NOUN
cana-2085	256	3	has	have	VERB
cana-2085	256	4	several	several	ADJ
cana-2085	256	5	applications	application	NOUN
cana-2085	256	6	in	in	ADP
cana-2085	256	7	various	various	ADJ
cana-2085	256	8	domains	domain	NOUN
cana-2085	256	9	,	,	PUNCT
cana-2085	256	10	particularly	particularly	ADV
cana-2085	256	11	where	where	SCONJ
cana-2085	256	12	network	network	NOUN
cana-2085	256	13	design	design	NOUN
cana-2085	256	14	,	,	PUNCT
cana-2085	256	15	analysis	analysis	NOUN
cana-2085	256	16	,	,	PUNCT
cana-2085	256	17	and	and	CCONJ
cana-2085	256	18	optimization	optimization	NOUN
cana-2085	256	19	are	be	AUX
cana-2085	256	20	critical	critical	ADJ
cana-2085	256	21	.	.	PUNCT
cana-2085	257	1	here	here	ADV
cana-2085	257	2	are	be	AUX
cana-2085	257	3	some	some	DET
cana-2085	257	4	potential	potential	ADJ
cana-2085	257	5	applications	application	NOUN
cana-2085	257	6	,	,	PUNCT
cana-2085	257	7	network	network	NOUN
cana-2085	257	8	design	design	NOUN
cana-2085	257	9	and	and	CCONJ
cana-2085	257	10	optimization	optimization	NOUN
cana-2085	257	11	-	-	PUNCT
cana-2085	257	12	optimal	optimal	ADJ
cana-2085	257	13	resource	resource	NOUN
cana-2085	257	14	placement	placement	NOUN
cana-2085	257	15	:	:	PUNCT
cana-2085	257	16	the	the	DET
cana-2085	257	17	theorem	theorem	NOUN
cana-2085	257	18	helps	helps	AUX
cana-2085	257	19	identify	identify	VERB
cana-2085	257	20	minimal	minimal	ADJ
cana-2085	257	21	dominating	dominating	NOUN
cana-2085	257	22	sets	set	NOUN
cana-2085	257	23	with	with	ADP
cana-2085	257	24	specific	specific	ADJ
cana-2085	257	25	properties	property	NOUN
cana-2085	257	26	that	that	PRON
cana-2085	257	27	can	can	AUX
cana-2085	257	28	be	be	AUX
cana-2085	257	29	used	use	VERB
cana-2085	257	30	to	to	PART
cana-2085	257	31	optimize	optimize	VERB
cana-2085	257	32	the	the	DET
cana-2085	257	33	placement	placement	NOUN
cana-2085	257	34	of	of	ADP
cana-2085	257	35	resources	resource	NOUN
cana-2085	257	36	or	or	CCONJ
cana-2085	257	37	infrastructure	infrastructure	NOUN
cana-2085	257	38	within	within	ADP
cana-2085	257	39	a	a	DET
cana-2085	257	40	network	network	NOUN
cana-2085	257	41	[	[	X
cana-2085	257	42	23	23	NUM
cana-2085	257	43	,	,	PUNCT
cana-2085	257	44	24	24	NUM
cana-2085	257	45	]	]	PUNCT
cana-2085	257	46	.	.	PUNCT
cana-2085	258	1	for	for	ADP
cana-2085	258	2	example	example	NOUN
cana-2085	258	3	,	,	PUNCT
cana-2085	258	4	if	if	SCONJ
cana-2085	258	5	a	a	DET
cana-2085	258	6	network	network	NOUN
cana-2085	258	7	(	(	PUNCT
cana-2085	258	8	graph	graph	NOUN
cana-2085	258	9	)	)	PUNCT
cana-2085	258	10	needs	need	VERB
cana-2085	258	11	to	to	PART
cana-2085	258	12	ensure	ensure	VERB
cana-2085	258	13	full	full	ADJ
cana-2085	258	14	coverage	coverage	NOUN
cana-2085	258	15	with	with	ADP
cana-2085	258	16	minimal	minimal	ADJ
cana-2085	258	17	redundancy	redundancy	NOUN
cana-2085	258	18	,	,	PUNCT
cana-2085	258	19	the	the	DET
cana-2085	258	20	theorem	theorem	NOUN
cana-2085	258	21	can	can	AUX
cana-2085	258	22	guide	guide	VERB
cana-2085	258	23	the	the	DET
cana-2085	258	24	placement	placement	NOUN
cana-2085	258	25	of	of	ADP
cana-2085	258	26	resources	resource	NOUN
cana-2085	258	27	such	such	ADJ
cana-2085	258	28	that	that	SCONJ
cana-2085	258	29	the	the	DET
cana-2085	258	30	network	network	NOUN
cana-2085	258	31	is	be	AUX
cana-2085	258	32	efficiently	efficiently	ADV
cana-2085	258	33	dominated	dominate	VERB
cana-2085	258	34	.	.	PUNCT
cana-2085	259	1	robustness	robustness	NOUN
cana-2085	259	2	and	and	CCONJ
cana-2085	259	3	fault	fault	VERB
cana-2085	259	4	tolerance	tolerance	NOUN
cana-2085	259	5	-	-	PUNCT
cana-2085	259	6	identifying	identify	VERB
cana-2085	259	7	critical	critical	ADJ
cana-2085	259	8	nodes	node	NOUN
cana-2085	259	9	:	:	PUNCT
cana-2085	259	10	the	the	DET
cana-2085	259	11	conditions	condition	NOUN
cana-2085	259	12	specified	specify	VERB
cana-2085	259	13	in	in	ADP
cana-2085	259	14	the	the	DET
cana-2085	259	15	theorem	theorem	ADJ
cana-2085	259	16	help	help	NOUN
cana-2085	259	17	in	in	ADP
cana-2085	259	18	identifying	identify	VERB
cana-2085	259	19	critical	critical	ADJ
cana-2085	259	20	nodes	node	NOUN
cana-2085	259	21	that	that	PRON
cana-2085	259	22	can	can	AUX
cana-2085	259	23	be	be	AUX
cana-2085	259	24	used	use	VERB
cana-2085	259	25	to	to	PART
cana-2085	259	26	enhance	enhance	VERB
cana-2085	259	27	the	the	DET
cana-2085	259	28	robustness	robustness	NOUN
cana-2085	259	29	of	of	ADP
cana-2085	259	30	a	a	DET
cana-2085	259	31	network	network	NOUN
cana-2085	259	32	[	[	X
cana-2085	259	33	14	14	NUM
cana-2085	259	34	,	,	PUNCT
cana-2085	259	35	24	24	NUM
cana-2085	259	36	,	,	PUNCT
cana-2085	259	37	42	42	NUM
cana-2085	259	38	]	]	PUNCT
cana-2085	259	39	.	.	PUNCT
cana-2085	260	1	if	if	SCONJ
cana-2085	260	2	a	a	DET
cana-2085	260	3	minimal	minimal	ADJ
cana-2085	260	4	dominating	dominating	NOUN
cana-2085	260	5	set	set	NOUN
cana-2085	260	6	satisfies	satisfy	VERB
cana-2085	260	7	one	one	NUM
cana-2085	260	8	of	of	ADP
cana-2085	260	9	the	the	DET
cana-2085	260	10	conditions	condition	NOUN
cana-2085	260	11	,	,	PUNCT
cana-2085	260	12	it	it	PRON
cana-2085	260	13	suggests	suggest	VERB
cana-2085	260	14	that	that	SCONJ
cana-2085	260	15	certain	certain	ADJ
cana-2085	260	16	nodes	node	NOUN
cana-2085	260	17	are	be	AUX
cana-2085	260	18	crucial	crucial	ADJ
cana-2085	260	19	for	for	ADP
cana-2085	260	20	maintaining	maintain	VERB
cana-2085	260	21	network	network	NOUN
cana-2085	260	22	coverage	coverage	NOUN
cana-2085	260	23	,	,	PUNCT
cana-2085	260	24	which	which	PRON
cana-2085	260	25	can	can	AUX
cana-2085	260	26	be	be	AUX
cana-2085	260	27	useful	useful	ADJ
cana-2085	260	28	for	for	ADP
cana-2085	260	29	designing	design	VERB
cana-2085	260	30	fault	fault	NOUN
cana-2085	260	31	-	-	PUNCT
cana-2085	260	32	tolerant	tolerant	ADJ
cana-2085	260	33	networks	network	NOUN
cana-2085	260	34	.	.	PUNCT
cana-2085	261	1	wireless	wireless	ADJ
cana-2085	261	2	sensor	sensor	NOUN
cana-2085	261	3	networks	network	NOUN
cana-2085	261	4	-	-	PUNCT
cana-2085	261	5	coverage	coverage	NOUN
cana-2085	261	6	and	and	CCONJ
cana-2085	261	7	connectivity	connectivity	NOUN
cana-2085	261	8	:	:	PUNCT
cana-2085	261	9	in	in	ADP
cana-2085	261	10	wireless	wireless	ADJ
cana-2085	261	11	sensor	sensor	NOUN
cana-2085	261	12	networks	network	NOUN
cana-2085	261	13	,	,	PUNCT
cana-2085	261	14	ensuring	ensure	VERB
cana-2085	261	15	coverage	coverage	NOUN
cana-2085	261	16	and	and	CCONJ
cana-2085	261	17	connectivity	connectivity	NOUN
cana-2085	261	18	with	with	ADP
cana-2085	261	19	minimal	minimal	ADJ
cana-2085	261	20	nodes	node	NOUN
cana-2085	261	21	is	be	AUX
cana-2085	261	22	crucial	crucial	ADJ
cana-2085	261	23	.	.	PUNCT
cana-2085	262	1	the	the	DET
cana-2085	262	2	theorem	theorem	NOUN
cana-2085	262	3	can	can	AUX
cana-2085	262	4	be	be	AUX
cana-2085	262	5	applied	apply	VERB
cana-2085	262	6	to	to	PART
cana-2085	262	7	identify	identify	VERB
cana-2085	262	8	configurations	configuration	NOUN
cana-2085	262	9	of	of	ADP
cana-2085	262	10	sensor	sensor	NOUN
cana-2085	262	11	nodes	node	NOUN
cana-2085	262	12	that	that	PRON
cana-2085	262	13	provide	provide	VERB
cana-2085	262	14	optimal	optimal	ADJ
cana-2085	262	15	coverage	coverage	NOUN
cana-2085	262	16	and	and	CCONJ
cana-2085	262	17	redundancy	redundancy	NOUN
cana-2085	262	18	,	,	PUNCT
cana-2085	262	19	which	which	PRON
cana-2085	262	20	is	be	AUX
cana-2085	262	21	important	important	ADJ
cana-2085	262	22	for	for	ADP
cana-2085	262	23	maintaining	maintain	VERB
cana-2085	262	24	network	network	NOUN
cana-2085	262	25	performance	performance	NOUN
cana-2085	262	26	and	and	CCONJ
cana-2085	262	27	reliability	reliability	NOUN
cana-2085	262	28	[	[	X
cana-2085	262	29	39	39	NUM
cana-2085	262	30	,	,	PUNCT
cana-2085	262	31	40	40	NUM
cana-2085	262	32	]	]	PUNCT
cana-2085	262	33	.	.	PUNCT
cana-2085	263	1	biological	biological	ADJ
cana-2085	263	2	network	network	NOUN
cana-2085	263	3	analysis	analysis	NOUN
cana-2085	263	4	-	-	PUNCT
cana-2085	263	5	understanding	understand	VERB
cana-2085	263	6	biological	biological	ADJ
cana-2085	263	7	interactions	interaction	NOUN
cana-2085	263	8	:	:	PUNCT
cana-2085	263	9	in	in	ADP
cana-2085	263	10	biological	biological	ADJ
cana-2085	263	11	networks	network	NOUN
cana-2085	263	12	,	,	PUNCT
cana-2085	263	13	such	such	ADJ
cana-2085	263	14	as	as	ADP
cana-2085	263	15	protein	protein	NOUN
cana-2085	263	16	-	-	PUNCT
cana-2085	263	17	protein	protein	NOUN
cana-2085	263	18	interaction	interaction	NOUN
cana-2085	263	19	networks	network	NOUN
cana-2085	263	20	,	,	PUNCT
cana-2085	263	21	identifying	identify	VERB
cana-2085	263	22	minimal	minimal	ADJ
cana-2085	263	23	sets	set	NOUN
cana-2085	263	24	of	of	ADP
cana-2085	263	25	proteins	protein	NOUN
cana-2085	263	26	(	(	PUNCT
cana-2085	263	27	nodes	node	NOUN
cana-2085	263	28	)	)	PUNCT
cana-2085	263	29	that	that	PRON
cana-2085	263	30	dominate	dominate	VERB
cana-2085	263	31	the	the	DET
cana-2085	263	32	network	network	NOUN
cana-2085	263	33	can	can	AUX
cana-2085	263	34	help	help	VERB
cana-2085	263	35	in	in	ADP
cana-2085	263	36	understanding	understand	VERB
cana-2085	263	37	key	key	ADJ
cana-2085	263	38	interactions	interaction	NOUN
cana-2085	263	39	and	and	CCONJ
cana-2085	263	40	functional	functional	ADJ
cana-2085	263	41	modules	module	NOUN
cana-2085	263	42	[	[	X
cana-2085	263	43	26	26	NUM
cana-2085	263	44	,	,	PUNCT
cana-2085	263	45	27	27	NUM
cana-2085	263	46	,	,	PUNCT
cana-2085	263	47	29	29	NUM
cana-2085	263	48	]	]	PUNCT
cana-2085	263	49	.	.	PUNCT
cana-2085	264	1	the	the	DET
cana-2085	264	2	theorem	theorem	NOUN
cana-2085	264	3	provides	provide	VERB
cana-2085	264	4	a	a	DET
cana-2085	264	5	framework	framework	NOUN
cana-2085	264	6	for	for	ADP
cana-2085	264	7	analyzing	analyze	VERB
cana-2085	264	8	and	and	CCONJ
cana-2085	264	9	optimizing	optimize	VERB
cana-2085	264	10	these	these	DET
cana-2085	264	11	networks	network	NOUN
cana-2085	264	12	.	.	PUNCT
cana-2085	265	1	social	social	ADJ
cana-2085	265	2	network	network	NOUN
cana-2085	265	3	analysisinfluence	analysisinfluence	NOUN
cana-2085	265	4	and	and	CCONJ
cana-2085	265	5	control	control	NOUN
cana-2085	265	6	:	:	PUNCT
cana-2085	265	7	in	in	ADP
cana-2085	265	8	social	social	ADJ
cana-2085	265	9	networks	network	NOUN
cana-2085	265	10	,	,	PUNCT
cana-2085	265	11	identifying	identify	VERB
cana-2085	265	12	key	key	ADJ
cana-2085	265	13	individuals	individual	NOUN
cana-2085	265	14	or	or	CCONJ
cana-2085	265	15	groups	group	NOUN
cana-2085	265	16	that	that	PRON
cana-2085	265	17	can	can	AUX
cana-2085	265	18	effectively	effectively	ADV
cana-2085	265	19	influence	influence	VERB
cana-2085	265	20	or	or	CCONJ
cana-2085	265	21	control	control	VERB
cana-2085	265	22	the	the	DET
cana-2085	265	23	network	network	NOUN
cana-2085	265	24	is	be	AUX
cana-2085	265	25	crucial	crucial	ADJ
cana-2085	265	26	[	[	X
cana-2085	265	27	1	1	NUM
cana-2085	265	28	,	,	PUNCT
cana-2085	265	29	3	3	NUM
cana-2085	265	30	]	]	PUNCT
cana-2085	265	31	.	.	PUNCT
cana-2085	266	1	the	the	DET
cana-2085	266	2	theorem	theorem	NOUN
cana-2085	266	3	can	can	AUX
cana-2085	266	4	be	be	AUX
cana-2085	266	5	used	use	VERB
cana-2085	266	6	to	to	PART
cana-2085	266	7	find	find	VERB
cana-2085	266	8	minimal	minimal	ADJ
cana-2085	266	9	sets	set	NOUN
cana-2085	266	10	of	of	ADP
cana-2085	266	11	influential	influential	ADJ
cana-2085	266	12	nodes	node	NOUN
cana-2085	266	13	that	that	PRON
cana-2085	266	14	meet	meet	VERB
cana-2085	266	15	certain	certain	ADJ
cana-2085	266	16	criteria	criterion	NOUN
cana-2085	266	17	,	,	PUNCT
cana-2085	266	18	aiding	aid	VERB
cana-2085	266	19	in	in	ADP
cana-2085	266	20	strategic	strategic	ADJ
cana-2085	266	21	decision	decision	NOUN
cana-2085	266	22	-	-	PUNCT
cana-2085	266	23	making	making	NOUN
cana-2085	266	24	and	and	CCONJ
cana-2085	266	25	intervention	intervention	NOUN
cana-2085	266	26	planning	planning	NOUN
cana-2085	266	27	.	.	PUNCT
cana-2085	267	1	security	security	NOUN
cana-2085	267	2	and	and	CCONJ
cana-2085	267	3	surveillance	surveillance	NOUN
cana-2085	267	4	-	-	PUNCT
cana-2085	267	5	optimizing	optimize	VERB
cana-2085	267	6	coverage	coverage	NOUN
cana-2085	267	7	:	:	PUNCT
cana-2085	267	8	in	in	ADP
cana-2085	267	9	security	security	NOUN
cana-2085	267	10	and	and	CCONJ
cana-2085	267	11	surveillance	surveillance	NOUN
cana-2085	267	12	systems	system	NOUN
cana-2085	267	13	,	,	PUNCT
cana-2085	267	14	the	the	DET
cana-2085	267	15	communications	communication	NOUN
cana-2085	267	16	on	on	ADP
cana-2085	267	17	applied	apply	VERB
cana-2085	267	18	nonlinear	nonlinear	ADJ
cana-2085	267	19	analysis	analysis	NOUN
cana-2085	267	20	issn	issn	NOUN
cana-2085	267	21	:	:	PUNCT
cana-2085	267	22	1074	1074	NUM
cana-2085	267	23	-	-	PUNCT
cana-2085	267	24	133x	133x	NUM
cana-2085	267	25	vol	vol	NOUN
cana-2085	267	26	32	32	NUM
cana-2085	268	1	no	no	NOUN
cana-2085	268	2	.	.	PUNCT
cana-2085	269	1	1s	1s	NUM
cana-2085	269	2	(	(	PUNCT
cana-2085	269	3	2025	2025	NUM
cana-2085	269	4	)	)	PUNCT
cana-2085	269	5	25	25	NUM
cana-2085	269	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-2085	269	7	theorem	theorem	NOUN
cana-2085	269	8	can	can	AUX
cana-2085	269	9	be	be	AUX
cana-2085	269	10	used	use	VERB
cana-2085	269	11	to	to	PART
cana-2085	269	12	determine	determine	VERB
cana-2085	269	13	the	the	DET
cana-2085	269	14	optimal	optimal	ADJ
cana-2085	269	15	placement	placement	NOUN
cana-2085	269	16	of	of	ADP
cana-2085	269	17	cameras	camera	NOUN
cana-2085	269	18	or	or	CCONJ
cana-2085	269	19	sensors	sensor	NOUN
cana-2085	269	20	to	to	PART
cana-2085	269	21	ensure	ensure	VERB
cana-2085	269	22	full	full	ADJ
cana-2085	269	23	coverage	coverage	NOUN
cana-2085	269	24	of	of	ADP
cana-2085	269	25	an	an	DET
cana-2085	269	26	area	area	NOUN
cana-2085	269	27	with	with	ADP
cana-2085	269	28	minimal	minimal	ADJ
cana-2085	269	29	equipment	equipment	NOUN
cana-2085	269	30	.	.	PUNCT
cana-2085	270	1	the	the	DET
cana-2085	270	2	conditions	condition	NOUN
cana-2085	270	3	help	help	VERB
cana-2085	270	4	in	in	ADP
cana-2085	270	5	selecting	select	VERB
cana-2085	270	6	strategic	strategic	ADJ
cana-2085	270	7	locations	location	NOUN
cana-2085	270	8	for	for	ADP
cana-2085	270	9	surveillance	surveillance	NOUN
cana-2085	270	10	that	that	PRON
cana-2085	270	11	provide	provide	VERB
cana-2085	270	12	comprehensive	comprehensive	ADJ
cana-2085	270	13	coverage	coverage	NOUN
cana-2085	271	1	[	[	X
cana-2085	271	2	36,37	36,37	X
cana-2085	271	3	]	]	X
cana-2085	271	4	.	.	PUNCT
cana-2085	272	1	overall	overall	ADV
cana-2085	272	2	,	,	PUNCT
cana-2085	272	3	the	the	DET
cana-2085	272	4	theorem	theorem	NOUN
cana-2085	272	5	provides	provide	VERB
cana-2085	272	6	a	a	DET
cana-2085	272	7	theoretical	theoretical	ADJ
cana-2085	272	8	foundation	foundation	NOUN
cana-2085	272	9	for	for	ADP
cana-2085	272	10	understanding	understanding	NOUN
cana-2085	272	11	and	and	CCONJ
cana-2085	272	12	optimizing	optimize	VERB
cana-2085	272	13	various	various	ADJ
cana-2085	272	14	network	network	NOUN
cana-2085	272	15	structures	structure	NOUN
cana-2085	272	16	and	and	CCONJ
cana-2085	272	17	configurations	configuration	NOUN
cana-2085	272	18	.	.	PUNCT
cana-2085	273	1	it	it	PRON
cana-2085	273	2	helps	help	VERB
cana-2085	273	3	in	in	ADP
cana-2085	273	4	solving	solve	VERB
cana-2085	273	5	practical	practical	ADJ
cana-2085	273	6	problems	problem	NOUN
cana-2085	273	7	related	relate	VERB
cana-2085	273	8	to	to	ADP
cana-2085	273	9	network	network	NOUN
cana-2085	273	10	coverage	coverage	NOUN
cana-2085	273	11	,	,	PUNCT
cana-2085	273	12	resource	resource	NOUN
cana-2085	273	13	allocation	allocation	NOUN
cana-2085	273	14	,	,	PUNCT
cana-2085	273	15	and	and	CCONJ
cana-2085	273	16	system	system	NOUN
cana-2085	273	17	robustness	robustness	NOUN
cana-2085	273	18	across	across	ADP
cana-2085	273	19	different	different	ADJ
cana-2085	273	20	applications	application	NOUN
cana-2085	273	21	.	.	PUNCT
cana-2085	274	1	determination	determination	NOUN
cana-2085	274	2	of	of	ADP
cana-2085	274	3	𝜸𝑴𝑹	𝜸𝑴𝑹	NOUN
cana-2085	274	4	value	value	NOUN
cana-2085	274	5	of	of	ADP
cana-2085	274	6	some	some	DET
cana-2085	274	7	graphs	graph	NOUN
cana-2085	274	8	:	:	PUNCT
cana-2085	274	9	in	in	ADP
cana-2085	274	10	this	this	DET
cana-2085	274	11	section	section	NOUN
cana-2085	274	12	we	we	PRON
cana-2085	274	13	first	first	ADV
cana-2085	274	14	determine	determine	VERB
cana-2085	274	15	the	the	DET
cana-2085	274	16	value	value	NOUN
cana-2085	274	17	of	of	ADP
cana-2085	274	18	𝛾𝑀𝑅	𝛾𝑀𝑅	NOUN
cana-2085	274	19	for	for	ADP
cana-2085	274	20	a	a	DET
cana-2085	274	21	caterpillar	caterpillar	ADJ
cana-2085	274	22	𝑇.	𝑇.	PROPN
cana-2085	274	23	for	for	ADP
cana-2085	274	24	this	this	DET
cana-2085	274	25	purpose	purpose	NOUN
cana-2085	274	26	,	,	PUNCT
cana-2085	274	27	we	we	PRON
cana-2085	274	28	discuss	discuss	VERB
cana-2085	274	29	as	as	SCONJ
cana-2085	274	30	follows	follow	VERB
cana-2085	274	31	:	:	PUNCT
cana-2085	274	32	let	let	VERB
cana-2085	274	33	𝑣1	𝑣1	PROPN
cana-2085	274	34	,	,	PUNCT
cana-2085	274	35	𝑣2	𝑣2	PROPN
cana-2085	274	36	,	,	PUNCT
cana-2085	274	37	…	…	PUNCT
cana-2085	274	38	,	,	PUNCT
cana-2085	275	1	𝑣𝑛	𝑣𝑛	X
cana-2085	275	2	be	be	AUX
cana-2085	275	3	the	the	DET
cana-2085	275	4	consecutive	consecutive	ADJ
cana-2085	275	5	super	super	ADV
cana-2085	275	6	strong	strong	ADJ
cana-2085	275	7	support	support	NOUN
cana-2085	275	8	of	of	ADP
cana-2085	275	9	𝑇.	𝑇.	PROPN
cana-2085	275	10	we	we	PRON
cana-2085	275	11	call	call	VERB
cana-2085	275	12	the	the	DET
cana-2085	275	13	graph	graph	NOUN
cana-2085	275	14	induced	induce	VERB
cana-2085	275	15	by	by	ADP
cana-2085	275	16	𝑁[𝑣𝑖	𝑁[𝑣𝑖	NOUN
cana-2085	275	17	]	]	PUNCT
cana-2085	275	18	as	as	ADP
cana-2085	275	19	super	super	ADV
cana-2085	275	20	strong	strong	ADJ
cana-2085	275	21	support	support	NOUN
cana-2085	275	22	chain	chain	NOUN
cana-2085	275	23	[	[	X
cana-2085	275	24	sss	sss	NOUN
cana-2085	275	25	chain	chain	NOUN
cana-2085	275	26	]	]	PUNCT
cana-2085	275	27	for	for	ADP
cana-2085	275	28	convenience	convenience	NOUN
cana-2085	275	29	.	.	PUNCT
cana-2085	276	1	next	next	ADV
cana-2085	276	2	,	,	PUNCT
cana-2085	276	3	we	we	PRON
cana-2085	276	4	consider	consider	VERB
cana-2085	276	5	a	a	DET
cana-2085	276	6	moderate	moderate	ADJ
cana-2085	276	7	support	support	NOUN
cana-2085	276	8	𝑢𝑟	𝑢𝑟	ADP
cana-2085	276	9	as	as	ADP
cana-2085	276	10	an	an	DET
cana-2085	276	11	artificial	artificial	ADJ
cana-2085	276	12	sss	sss	NOUN
cana-2085	276	13	of	of	ADP
cana-2085	276	14	the	the	DET
cana-2085	276	15	following	follow	VERB
cana-2085	276	16	conditions	condition	NOUN
cana-2085	276	17	hold	hold	VERB
cana-2085	276	18	.	.	PUNCT
cana-2085	277	1	let	let	VERB
cana-2085	277	2	𝑢𝑟−1	𝑢𝑟−1	NOUN
cana-2085	277	3	and	and	CCONJ
cana-2085	277	4	𝑢𝑟+1	𝑢𝑟+1	PROPN
cana-2085	277	5	be	be	AUX
cana-2085	277	6	the	the	DET
cana-2085	277	7	preceding	precede	VERB
cana-2085	277	8	and	and	CCONJ
cana-2085	277	9	succeeding	succeed	VERB
cana-2085	277	10	vertices	vertex	NOUN
cana-2085	277	11	of	of	ADP
cana-2085	277	12	a	a	DET
cana-2085	277	13	moderate	moderate	ADJ
cana-2085	277	14	support	support	NOUN
cana-2085	277	15	𝑢𝑟.	𝑢𝑟.	AUX
cana-2085	277	16	let	let	VERB
cana-2085	277	17	𝑛𝑖	𝑛𝑖	PRON
cana-2085	277	18	and	and	CCONJ
cana-2085	277	19	𝑛𝑗	𝑛𝑗	AUX
cana-2085	277	20	be	be	AUX
cana-2085	277	21	the	the	DET
cana-2085	277	22	number	number	NOUN
cana-2085	277	23	of	of	ADP
cana-2085	277	24	internal	internal	ADJ
cana-2085	277	25	vertices	vertex	NOUN
cana-2085	277	26	of	of	ADP
cana-2085	277	27	the	the	DET
cana-2085	277	28	(	(	PUNCT
cana-2085	277	29	𝑢𝑟−1	𝑢𝑟−1	NOUN
cana-2085	277	30	,	,	PUNCT
cana-2085	277	31	𝑢𝑟	𝑢𝑟	ADV
cana-2085	277	32	)	)	PUNCT
cana-2085	277	33	path	path	NOUN
cana-2085	277	34	and	and	CCONJ
cana-2085	277	35	(	(	PUNCT
cana-2085	277	36	𝑢𝑟	𝑢𝑟	ADV
cana-2085	277	37	,	,	PUNCT
cana-2085	277	38	𝑢𝑟+1	𝑢𝑟+1	NOUN
cana-2085	277	39	)	)	PUNCT
cana-2085	277	40	respectively	respectively	ADV
cana-2085	277	41	.	.	PUNCT
cana-2085	278	1	then	then	ADV
cana-2085	278	2	𝑢𝑟	𝑢𝑟	ADV
cana-2085	278	3	is	be	AUX
cana-2085	278	4	said	say	VERB
cana-2085	278	5	to	to	PART
cana-2085	278	6	be	be	AUX
cana-2085	278	7	an	an	DET
cana-2085	278	8	artificial	artificial	ADJ
cana-2085	278	9	super	super	ADV
cana-2085	278	10	strong	strong	ADJ
cana-2085	278	11	support	support	NOUN
cana-2085	278	12	if	if	SCONJ
cana-2085	278	13	one	one	NUM
cana-2085	278	14	of	of	ADP
cana-2085	278	15	the	the	DET
cana-2085	278	16	following	follow	VERB
cana-2085	278	17	conditions	condition	NOUN
cana-2085	278	18	hold	hold	VERB
cana-2085	278	19	.	.	PUNCT
cana-2085	279	1	(	(	PUNCT
cana-2085	279	2	i	i	NOUN
cana-2085	279	3	)	)	PUNCT
cana-2085	279	4	if	if	SCONJ
cana-2085	279	5	both	both	DET
cana-2085	279	6	𝑢𝑟−1	𝑢𝑟−1	NOUN
cana-2085	279	7	and	and	CCONJ
cana-2085	279	8	𝑢𝑟+1	𝑢𝑟+1	PROPN
cana-2085	279	9	are	be	AUX
cana-2085	279	10	weak	weak	ADJ
cana-2085	279	11	support	support	NOUN
cana-2085	279	12	then	then	ADV
cana-2085	279	13	𝑛𝑖	𝑛𝑖	PROPN
cana-2085	279	14	≡	≡	PROPN
cana-2085	279	15	0	0	NUM
cana-2085	279	16	,	,	PUNCT
cana-2085	279	17	2(𝑚𝑜𝑑	2(𝑚𝑜𝑑	NOUN
cana-2085	279	18	3	3	NUM
cana-2085	279	19	)	)	PUNCT
cana-2085	279	20	and	and	CCONJ
cana-2085	279	21	𝑛𝑗	𝑛𝑗	ADP
cana-2085	279	22	≡	≡	PROPN
cana-2085	279	23	0	0	NUM
cana-2085	279	24	,	,	PUNCT
cana-2085	279	25	2(𝑚𝑜𝑑	2(𝑚𝑜𝑑	NOUN
cana-2085	279	26	3	3	NUM
cana-2085	279	27	)	)	PUNCT
cana-2085	279	28	.	.	PUNCT
cana-2085	280	1	(	(	PUNCT
cana-2085	280	2	ii	ii	NOUN
cana-2085	280	3	)	)	PUNCT
cana-2085	280	4	if	if	SCONJ
cana-2085	280	5	𝑢𝑟−1	𝑢𝑟−1	NOUN
cana-2085	280	6	is	be	AUX
cana-2085	280	7	a	a	DET
cana-2085	280	8	moderate	moderate	ADJ
cana-2085	280	9	support	support	NOUN
cana-2085	280	10	and	and	CCONJ
cana-2085	280	11	𝑢𝑟+1	𝑢𝑟+1	PROPN
cana-2085	280	12	is	be	AUX
cana-2085	280	13	a	a	DET
cana-2085	280	14	weak	weak	ADJ
cana-2085	280	15	support	support	NOUN
cana-2085	280	16	then	then	ADV
cana-2085	280	17	𝑛𝑖	𝑛𝑖	PROPN
cana-2085	280	18	≡	≡	PROPN
cana-2085	280	19	1	1	NUM
cana-2085	280	20	,	,	PUNCT
cana-2085	280	21	2(𝑚𝑜𝑑	2(𝑚𝑜𝑑	NOUN
cana-2085	280	22	3	3	NUM
cana-2085	280	23	)	)	PUNCT
cana-2085	280	24	and	and	CCONJ
cana-2085	280	25	𝑛𝑗	𝑛𝑗	ADP
cana-2085	280	26	≡	≡	PROPN
cana-2085	280	27	0	0	NUM
cana-2085	280	28	,	,	PUNCT
cana-2085	280	29	2(𝑚𝑜𝑑	2(𝑚𝑜𝑑	NOUN
cana-2085	280	30	3	3	NUM
cana-2085	280	31	)	)	PUNCT
cana-2085	280	32	and	and	CCONJ
cana-2085	280	33	vice	vice	ADV
cana-2085	280	34	versa	versa	ADV
cana-2085	280	35	.	.	PUNCT
cana-2085	281	1	(	(	PUNCT
cana-2085	281	2	iii	iii	X
cana-2085	281	3	)	)	PUNCT
cana-2085	281	4	if	if	SCONJ
cana-2085	281	5	both	both	DET
cana-2085	281	6	𝑢𝑟−1	𝑢𝑟−1	NOUN
cana-2085	281	7	and	and	CCONJ
cana-2085	281	8	𝑢𝑟+1	𝑢𝑟+1	PROPN
cana-2085	281	9	are	be	AUX
cana-2085	281	10	moderate	moderate	ADJ
cana-2085	281	11	supports	support	NOUN
cana-2085	281	12	then	then	ADV
cana-2085	281	13	𝑛𝑖	𝑛𝑖	PROPN
cana-2085	281	14	≡	≡	PROPN
cana-2085	281	15	1	1	NUM
cana-2085	281	16	,	,	PUNCT
cana-2085	281	17	2(𝑚𝑜𝑑	2(𝑚𝑜𝑑	NOUN
cana-2085	281	18	3	3	NUM
cana-2085	281	19	)	)	PUNCT
cana-2085	281	20	and	and	CCONJ
cana-2085	281	21	𝑛𝑗	𝑛𝑗	ADP
cana-2085	281	22	≡	≡	PROPN
cana-2085	281	23	1	1	NUM
cana-2085	281	24	,	,	PUNCT
cana-2085	281	25	2(𝑚𝑜𝑑	2(𝑚𝑜𝑑	NOUN
cana-2085	281	26	3	3	NUM
cana-2085	281	27	)	)	PUNCT
cana-2085	281	28	.	.	PUNCT
cana-2085	282	1	let	let	VERB
cana-2085	282	2	𝑢𝑠	𝑢𝑠	PRON
cana-2085	282	3	be	be	AUX
cana-2085	282	4	the	the	DET
cana-2085	282	5	weak	weak	ADJ
cana-2085	282	6	support	support	NOUN
cana-2085	282	7	on	on	ADP
cana-2085	282	8	the	the	DET
cana-2085	282	9	spine	spine	NOUN
cana-2085	282	10	of	of	ADP
cana-2085	282	11	𝑇.	𝑇.	PROPN
cana-2085	282	12	then	then	ADV
cana-2085	282	13	it	it	PRON
cana-2085	282	14	will	will	AUX
cana-2085	282	15	be	be	AUX
cana-2085	282	16	considered	consider	VERB
cana-2085	282	17	as	as	ADP
cana-2085	282	18	an	an	DET
cana-2085	282	19	artificial	artificial	ADJ
cana-2085	282	20	super	super	ADV
cana-2085	282	21	strong	strong	ADJ
cana-2085	282	22	support	support	NOUN
cana-2085	282	23	if	if	SCONJ
cana-2085	282	24	one	one	NUM
cana-2085	282	25	of	of	ADP
cana-2085	282	26	the	the	DET
cana-2085	282	27	following	follow	VERB
cana-2085	282	28	conditions	condition	NOUN
cana-2085	282	29	hold	hold	VERB
cana-2085	282	30	.	.	PUNCT
cana-2085	283	1	(	(	PUNCT
cana-2085	283	2	i	i	NOUN
cana-2085	283	3	)	)	PUNCT
cana-2085	283	4	if	if	SCONJ
cana-2085	283	5	both	both	DET
cana-2085	283	6	𝑢𝑟−1	𝑢𝑟−1	NOUN
cana-2085	283	7	and	and	CCONJ
cana-2085	283	8	𝑢𝑟+1	𝑢𝑟+1	PROPN
cana-2085	283	9	are	be	AUX
cana-2085	283	10	weak	weak	ADJ
cana-2085	283	11	support	support	NOUN
cana-2085	283	12	then	then	ADV
cana-2085	283	13	𝑛𝑖	𝑛𝑖	PROPN
cana-2085	283	14	≡	≡	PROPN
cana-2085	283	15	0	0	NUM
cana-2085	283	16	,	,	PUNCT
cana-2085	283	17	2(𝑚𝑜𝑑	2(𝑚𝑜𝑑	NOUN
cana-2085	283	18	3	3	NUM
cana-2085	283	19	)	)	PUNCT
cana-2085	283	20	and	and	CCONJ
cana-2085	283	21	𝑛𝑗	𝑛𝑗	ADP
cana-2085	283	22	≡	≡	PROPN
cana-2085	283	23	0	0	NUM
cana-2085	283	24	,	,	PUNCT
cana-2085	283	25	2(𝑚𝑜𝑑	2(𝑚𝑜𝑑	NOUN
cana-2085	283	26	3	3	NUM
cana-2085	283	27	)	)	PUNCT
cana-2085	283	28	.	.	PUNCT
cana-2085	284	1	(	(	PUNCT
cana-2085	284	2	ii	ii	NOUN
cana-2085	284	3	)	)	PUNCT
cana-2085	284	4	if	if	SCONJ
cana-2085	284	5	𝑢𝑟−1	𝑢𝑟−1	NOUN
cana-2085	284	6	is	be	AUX
cana-2085	284	7	a	a	DET
cana-2085	284	8	moderate	moderate	ADJ
cana-2085	284	9	support	support	NOUN
cana-2085	284	10	and	and	CCONJ
cana-2085	284	11	𝑢𝑟+1	𝑢𝑟+1	PROPN
cana-2085	284	12	is	be	AUX
cana-2085	284	13	a	a	DET
cana-2085	284	14	weak	weak	ADJ
cana-2085	284	15	support	support	NOUN
cana-2085	284	16	then	then	ADV
cana-2085	284	17	𝑛𝑖	𝑛𝑖	PROPN
cana-2085	284	18	≡	≡	PROPN
cana-2085	284	19	1	1	NUM
cana-2085	284	20	,	,	PUNCT
cana-2085	284	21	2(𝑚𝑜𝑑	2(𝑚𝑜𝑑	NOUN
cana-2085	284	22	3	3	NUM
cana-2085	284	23	)	)	PUNCT
cana-2085	284	24	and	and	CCONJ
cana-2085	284	25	𝑛𝑗	𝑛𝑗	ADP
cana-2085	284	26	≡	≡	PROPN
cana-2085	284	27	0	0	NUM
cana-2085	284	28	,	,	PUNCT
cana-2085	284	29	2(𝑚𝑜𝑑	2(𝑚𝑜𝑑	NOUN
cana-2085	284	30	3	3	NUM
cana-2085	284	31	)	)	PUNCT
cana-2085	284	32	.	.	PUNCT
cana-2085	285	1	(	(	PUNCT
cana-2085	285	2	iii	iii	X
cana-2085	285	3	)	)	PUNCT
cana-2085	285	4	if	if	SCONJ
cana-2085	285	5	both	both	DET
cana-2085	285	6	𝑢𝑟−1	𝑢𝑟−1	NOUN
cana-2085	285	7	and	and	CCONJ
cana-2085	285	8	𝑢𝑟+1	𝑢𝑟+1	PROPN
cana-2085	285	9	are	be	AUX
cana-2085	285	10	moderate	moderate	ADJ
cana-2085	285	11	supports	support	NOUN
cana-2085	285	12	then	then	ADV
cana-2085	285	13	𝑛𝑖	𝑛𝑖	PROPN
cana-2085	285	14	≡	≡	PROPN
cana-2085	285	15	1	1	NUM
cana-2085	285	16	,	,	PUNCT
cana-2085	285	17	2(𝑚𝑜𝑑	2(𝑚𝑜𝑑	NOUN
cana-2085	285	18	3	3	NUM
cana-2085	285	19	)	)	PUNCT
cana-2085	285	20	and	and	CCONJ
cana-2085	285	21	𝑛𝑗	𝑛𝑗	ADP
cana-2085	285	22	≡	≡	PROPN
cana-2085	285	23	1	1	NUM
cana-2085	285	24	,	,	PUNCT
cana-2085	285	25	2(𝑚𝑜𝑑	2(𝑚𝑜𝑑	NOUN
cana-2085	285	26	3	3	NUM
cana-2085	285	27	)	)	PUNCT
cana-2085	285	28	.	.	PUNCT
cana-2085	286	1	next	next	ADV
cana-2085	286	2	,	,	PUNCT
cana-2085	286	3	identify	identify	VERB
cana-2085	286	4	the	the	DET
cana-2085	286	5	following	follow	VERB
cana-2085	286	6	(	(	PUNCT
cana-2085	286	7	i	i	NOUN
cana-2085	286	8	)	)	PUNCT
cana-2085	286	9	all	all	DET
cana-2085	286	10	consecutive	consecutive	ADJ
cana-2085	286	11	sss	sss	NOUN
cana-2085	286	12	say	say	VERB
cana-2085	286	13	𝑠𝑛1	𝑠𝑛1	NOUN
cana-2085	286	14	,	,	PUNCT
cana-2085	286	15	𝑠𝑛2	𝑠𝑛2	PROPN
cana-2085	286	16	,	,	PUNCT
cana-2085	286	17	𝑠𝑛3	𝑠𝑛3	ADJ
cana-2085	286	18	,	,	PUNCT
cana-2085	286	19	…	…	PUNCT
cana-2085	286	20	,	,	PUNCT
cana-2085	286	21	𝑠𝑛𝑘	𝑠𝑛𝑘	VERB
cana-2085	286	22	where	where	SCONJ
cana-2085	286	23	𝑠𝑛𝑖	𝑠𝑛𝑖	NOUN
cana-2085	286	24	=	=	SYM
cana-2085	286	25	{	{	PUNCT
cana-2085	286	26	𝑠1	𝑠1	PROPN
cana-2085	286	27	,	,	PUNCT
cana-2085	286	28	𝑠2	𝑠2	NOUN
cana-2085	286	29	,	,	PUNCT
cana-2085	286	30	𝑠3	𝑠3	NOUN
cana-2085	286	31	,	,	PUNCT
cana-2085	286	32	…	…	PUNCT
cana-2085	286	33	,	,	PUNCT
cana-2085	286	34	𝑠𝑛𝑖	𝑠𝑛𝑖	ADV
cana-2085	286	35	}	}	PUNCT
cana-2085	286	36	such	such	ADJ
cana-2085	286	37	that	that	SCONJ
cana-2085	286	38	each	each	DET
cana-2085	286	39	vertex	vertex	NOUN
cana-2085	286	40	of	of	ADP
cana-2085	286	41	𝑠𝑛𝑖	𝑠𝑛𝑖	NOUN
cana-2085	286	42	is	be	AUX
cana-2085	286	43	a	a	DET
cana-2085	286	44	sss	sss	NOUN
cana-2085	286	45	.	.	PUNCT
cana-2085	286	46	(	(	PUNCT
cana-2085	286	47	ii	ii	NOUN
cana-2085	286	48	)	)	PUNCT
cana-2085	286	49	consecutive	consecutive	ADJ
cana-2085	286	50	weak	weak	ADJ
cana-2085	286	51	supports	support	NOUN
cana-2085	286	52	𝑤𝑛1	𝑤𝑛1	NOUN
cana-2085	286	53	,	,	PUNCT
cana-2085	286	54	𝑤𝑛2	𝑤𝑛2	NOUN
cana-2085	286	55	,	,	PUNCT
cana-2085	286	56	𝑤𝑛3	𝑤𝑛3	NOUN
cana-2085	286	57	,	,	PUNCT
cana-2085	286	58	…	…	PUNCT
cana-2085	286	59	,	,	PUNCT
cana-2085	286	60	𝑤𝑛𝑟	𝑤𝑛𝑟	ADJ
cana-2085	286	61	where	where	SCONJ
cana-2085	286	62	each	each	DET
cana-2085	286	63	𝑤𝑛𝑖	𝑤𝑛𝑖	NOUN
cana-2085	286	64	=	=	SYM
cana-2085	286	65	{	{	PUNCT
cana-2085	286	66	𝑤1	𝑤1	VERB
cana-2085	286	67	,	,	PUNCT
cana-2085	286	68	𝑤2	𝑤2	NOUN
cana-2085	286	69	,	,	PUNCT
cana-2085	286	70	𝑤3	𝑤3	PROPN
cana-2085	286	71	,	,	PUNCT
cana-2085	286	72	…	…	PUNCT
cana-2085	286	73	,	,	PUNCT
cana-2085	286	74	𝑤𝑛𝑖	𝑤𝑛𝑖	X
cana-2085	286	75	}	}	PUNCT
cana-2085	286	76	such	such	ADJ
cana-2085	286	77	that	that	SCONJ
cana-2085	286	78	each	each	DET
cana-2085	286	79	vertex	vertex	NOUN
cana-2085	286	80	of	of	ADP
cana-2085	286	81	𝑤𝑛𝑖	𝑤𝑛𝑖	NOUN
cana-2085	286	82	is	be	AUX
cana-2085	286	83	a	a	DET
cana-2085	286	84	weak	weak	ADJ
cana-2085	286	85	support	support	NOUN
cana-2085	286	86	.	.	PUNCT
cana-2085	287	1	now	now	ADV
cana-2085	287	2	we	we	PRON
cana-2085	287	3	determine	determine	VERB
cana-2085	287	4	the	the	DET
cana-2085	287	5	value	value	NOUN
cana-2085	287	6	of	of	ADP
cana-2085	287	7	𝛾𝑀𝑅	𝛾𝑀𝑅	NOUN
cana-2085	287	8	for	for	ADP
cana-2085	287	9	a	a	DET
cana-2085	287	10	caterpillar	caterpillar	NOUN
cana-2085	287	11	in	in	ADP
cana-2085	287	12	the	the	DET
cana-2085	287	13	next	next	ADJ
cana-2085	287	14	theorem	theorem	NOUN
cana-2085	287	15	theorem	theorem	VERB
cana-2085	287	16	9	9	NUM
cana-2085	287	17	:	:	PUNCT
cana-2085	287	18	let	let	VERB
cana-2085	287	19	𝑇	𝑇	PROPN
cana-2085	287	20	be	be	AUX
cana-2085	287	21	any	any	DET
cana-2085	287	22	caterpillar	caterpillar	NOUN
cana-2085	287	23	.	.	PUNCT
cana-2085	288	1	let	let	VERB
cana-2085	288	2	𝑆	𝑆	PROPN
cana-2085	288	3	=	=	SYM
cana-2085	288	4	{	{	PUNCT
cana-2085	288	5	𝑆𝑛𝑖	𝑆𝑛𝑖	PROPN
cana-2085	288	6	𝑖	𝑖	PROPN
cana-2085	288	7	=	=	SYM
cana-2085	288	8	1	1	NUM
cana-2085	288	9	,	,	PUNCT
cana-2085	288	10	2	2	NUM
cana-2085	288	11	,	,	PUNCT
cana-2085	288	12	…	…	PUNCT
cana-2085	288	13	,	,	PUNCT
cana-2085	288	14	𝑘	𝑘	X
cana-2085	288	15	𝑆𝑖	𝑆𝑖	PROPN
cana-2085	288	16	𝑖𝑠	𝑖𝑠	NOUN
cana-2085	288	17	𝑎	𝑎	DET
cana-2085	288	18	𝑆𝑆𝑆	𝑆𝑆𝑆	PROPN
cana-2085	288	19	𝑐ℎ𝑎𝑖𝑛	𝑐ℎ𝑎𝑖𝑛	ADV
cana-2085	288	20	}	}	PUNCT
cana-2085	288	21	,	,	PUNCT
cana-2085	288	22	𝑆1	𝑆1	NOUN
cana-2085	288	23	=	=	SYM
cana-2085	288	24	{	{	PUNCT
cana-2085	288	25	𝑠/	𝑠/	X
cana-2085	288	26	𝑒𝑎𝑐ℎ	𝑒𝑎𝑐ℎ	PROPN
cana-2085	288	27	𝑣𝑒𝑟𝑡𝑒𝑥	𝑣𝑒𝑟𝑡𝑒𝑥	NOUN
cana-2085	289	1	𝑠	𝑠	PROPN
cana-2085	290	1	𝑖𝑠	𝑖𝑠	INTJ
cana-2085	290	2	𝑒𝑖𝑡ℎ𝑒𝑟	𝑒𝑖𝑡ℎ𝑒𝑟	VERB
cana-2085	290	3	𝑎	𝑎	PRON
cana-2085	290	4	𝑆𝑆𝑆	𝑆𝑆𝑆	PROPN
cana-2085	290	5	𝑜𝑟	𝑜𝑟	PRON
cana-2085	290	6	𝐴𝑆𝑆𝑆	𝐴𝑆𝑆𝑆	PROPN
cana-2085	290	7	}	}	PUNCT
cana-2085	290	8	,	,	PUNCT
cana-2085	290	9	𝑆2	𝑆2	PROPN
cana-2085	290	10	=	=	SYM
cana-2085	290	11	{	{	PUNCT
cana-2085	290	12	𝑚/	𝑚/	NOUN
cana-2085	290	13	𝑒𝑎𝑐ℎ	𝑒𝑎𝑐ℎ	NOUN
cana-2085	290	14	𝑚	𝑚	PROPN
cana-2085	291	1	𝑖𝑠	𝑖𝑠	PROPN
cana-2085	291	2	𝑒𝑖𝑡ℎ𝑒𝑟	𝑒𝑖𝑡ℎ𝑒𝑟	PROPN
cana-2085	291	3	𝑎	𝑎	DET
cana-2085	291	4	𝑀𝑆	𝑀𝑆	PROPN
cana-2085	291	5	𝑜𝑟	𝑜𝑟	ADP
cana-2085	291	6	𝐴𝑀𝑆	𝐴𝑀𝑆	PROPN
cana-2085	291	7	}	}	PUNCT
cana-2085	291	8	,	,	PUNCT
cana-2085	291	9	𝑊	𝑊	PROPN
cana-2085	291	10	=	=	SYM
cana-2085	291	11	{	{	PUNCT
cana-2085	291	12	𝑊𝑛𝑗	𝑊𝑛𝑗	PROPN
cana-2085	291	13	,	,	PUNCT
cana-2085	291	14	𝐽	𝐽	PROPN
cana-2085	291	15	=	=	SYM
cana-2085	291	16	1	1	NUM
cana-2085	291	17	,	,	PUNCT
cana-2085	291	18	2	2	NUM
cana-2085	291	19	,	,	PUNCT
cana-2085	291	20	…	…	PUNCT
cana-2085	291	21	,	,	PUNCT
cana-2085	291	22	𝑙	𝑙	X
cana-2085	291	23	/	/	SYM
cana-2085	291	24	𝑒𝑎𝑐ℎ	𝑒𝑎𝑐ℎ	NOUN
cana-2085	291	25	𝑊𝑛𝑗	𝑊𝑛𝑗	PROPN
cana-2085	291	26	𝑖𝑠	𝑖𝑠	PROPN
cana-2085	292	1	𝑎	𝑎	DET
cana-2085	292	2	𝑐𝑜𝑚𝑏	𝑐𝑜𝑚𝑏	ADJ
cana-2085	292	3	𝑔𝑟𝑎𝑝ℎ	𝑔𝑟𝑎𝑝ℎ	NOUN
cana-2085	292	4	𝑤𝑖𝑡ℎ	𝑤𝑖𝑡ℎ	PROPN
cana-2085	292	5	𝑛𝑗	𝑛𝑗	ADP
cana-2085	292	6	𝑤𝑒𝑎𝑘	𝑤𝑒𝑎𝑘	NOUN
cana-2085	292	7	𝑠𝑢𝑝𝑝𝑜𝑟𝑡𝑠	𝑠𝑢𝑝𝑝𝑜𝑟𝑡𝑠	ADJ
cana-2085	292	8	}	}	PUNCT
cana-2085	292	9	.	.	PUNCT
cana-2085	293	1	let	let	VERB
cana-2085	293	2	𝑇1	𝑇1	NOUN
cana-2085	293	3	=	=	PUNCT
cana-2085	294	1	[	[	X
cana-2085	294	2	𝑁[𝑆	𝑁[𝑆	X
cana-2085	294	3	]	]	PUNCT
cana-2085	294	4	∪	∪	X
cana-2085	294	5	𝑁[𝑆1	𝑁[𝑆1	PROPN
cana-2085	294	6	]	]	PUNCT
cana-2085	294	7	∪	∪	ADP
cana-2085	294	8	𝑁[𝑆2	𝑁[𝑆2	NOUN
cana-2085	294	9	]	]	PUNCT
cana-2085	294	10	∪	∪	ADP
cana-2085	294	11	𝑁[𝑊	𝑁[𝑊	NOUN
cana-2085	294	12	]	]	X
cana-2085	294	13	]	]	PUNCT
cana-2085	294	14	.	.	PUNCT
cana-2085	295	1	let	let	VERB
cana-2085	295	2	𝑇2	𝑇2	NOUN
cana-2085	295	3	=	=	SYM
cana-2085	295	4	𝑇	𝑇	PROPN
cana-2085	295	5	−	−	PROPN
cana-2085	295	6	𝑇1	𝑇1	NOUN
cana-2085	295	7	and	and	CCONJ
cana-2085	295	8	𝑟	𝑟	NOUN
cana-2085	295	9	=	=	SYM
cana-2085	295	10	1	1	NUM
cana-2085	295	11	,	,	PUNCT
cana-2085	295	12	2	2	NUM
cana-2085	295	13	,	,	PUNCT
cana-2085	295	14	…	…	PUNCT
cana-2085	295	15	,	,	PUNCT
cana-2085	295	16	𝑡.	𝑡.	PROPN
cana-2085	295	17	𝑅1	𝑅1	PROPN
cana-2085	295	18	,	,	PUNCT
cana-2085	295	19	𝑅2	𝑅2	ADP
cana-2085	295	20	,	,	PUNCT
cana-2085	295	21	…	…	PUNCT
cana-2085	295	22	,	,	PUNCT
cana-2085	295	23	𝑅𝑡	𝑅𝑡	PROPN
cana-2085	295	24	be	be	VERB
cana-2085	295	25	the	the	DET
cana-2085	295	26	components	component	NOUN
cana-2085	295	27	of	of	ADP
cana-2085	295	28	𝑇2	𝑇2	PROPN
cana-2085	295	29	.	.	PUNCT
cana-2085	296	1	then	then	ADV
cana-2085	296	2	𝛾𝑀𝑅	𝛾𝑀𝑅	NOUN
cana-2085	296	3	=	=	SYM
cana-2085	296	4	3[(𝑥	3[(𝑥	NUM
cana-2085	296	5	+	+	ADJ
cana-2085	296	6	𝑎)𝑛	𝑎)𝑛	NOUN
cana-2085	296	7	]	]	X
cana-2085	297	1	+	+	CCONJ
cana-2085	297	2	[	[	X
cana-2085	297	3	∑	∑	PUNCT
cana-2085	297	4	𝑛𝑖	𝑛𝑖	PROPN
cana-2085	297	5	+	+	PROPN
cana-2085	297	6	𝑘	𝑘	PRON
cana-2085	297	7	𝑖=1	𝑖=1	PROPN
cana-2085	297	8	|𝑆1|	|𝑆1|	PROPN
cana-2085	297	9	]	]	PUNCT
cana-2085	298	1	+	+	CCONJ
cana-2085	298	2	2|𝑆2|	2|𝑆2|	NUM
cana-2085	298	3	+	+	CCONJ
cana-2085	298	4	3	3	NUM
cana-2085	298	5	4	4	NUM
cana-2085	298	6	∑	∑	ADP
cana-2085	298	7	𝑇1	𝑇1	NOUN
cana-2085	298	8	𝑙	𝑙	PRON
cana-2085	298	9	𝑗=1	𝑗=1	PROPN
cana-2085	298	10	+	+	CCONJ
cana-2085	298	11	2|𝑇2|	2|𝑇2|	NUM
cana-2085	298	12	proof	proof	NOUN
cana-2085	298	13	:	:	PUNCT
cana-2085	298	14	let	let	VERB
cana-2085	298	15	𝑇	𝑇	PROPN
cana-2085	298	16	be	be	AUX
cana-2085	298	17	any	any	DET
cana-2085	298	18	caterpillar	caterpillar	NOUN
cana-2085	298	19	.	.	PUNCT
cana-2085	299	1	identify	identify	VERB
cana-2085	299	2	the	the	DET
cana-2085	299	3	sss	sss	NOUN
cana-2085	299	4	chain	chain	NOUN
cana-2085	299	5	,	,	PUNCT
cana-2085	299	6	sss	sss	PROPN
cana-2085	299	7	,	,	PUNCT
cana-2085	299	8	asss	asss	NOUN
cana-2085	299	9	,	,	PUNCT
cana-2085	299	10	mss	mss	PROPN
cana-2085	299	11	,	,	PUNCT
cana-2085	299	12	amss	amss	PROPN
cana-2085	299	13	and	and	CCONJ
cana-2085	299	14	comb	comb	VERB
cana-2085	299	15	graphs	graph	NOUN
cana-2085	299	16	using	use	VERB
cana-2085	299	17	the	the	DET
cana-2085	299	18	above	above	ADJ
cana-2085	299	19	procedure	procedure	NOUN
cana-2085	299	20	.	.	PUNCT
cana-2085	300	1	let	let	VERB
cana-2085	300	2	𝑆	𝑆	PROPN
cana-2085	300	3	,	,	PUNCT
cana-2085	300	4	𝑆1	𝑆1	PROPN
cana-2085	300	5	,	,	PUNCT
cana-2085	300	6	𝑆2	𝑆2	PROPN
cana-2085	300	7	,	,	PUNCT
cana-2085	300	8	𝑊	𝑊	PROPN
cana-2085	300	9	be	be	VERB
cana-2085	300	10	as	as	ADV
cana-2085	300	11	defined	define	VERB
cana-2085	300	12	in	in	ADP
cana-2085	300	13	the	the	DET
cana-2085	300	14	theorem	theorem	NOUN
cana-2085	300	15	.	.	PUNCT
cana-2085	301	1	let	let	VERB
cana-2085	301	2	𝑣	𝑣	PART
cana-2085	301	3	be	be	AUX
cana-2085	301	4	an	an	DET
cana-2085	301	5	artificial	artificial	ADJ
cana-2085	301	6	super	super	ADV
cana-2085	301	7	strong	strong	ADJ
cana-2085	301	8	support	support	NOUN
cana-2085	301	9	.	.	PUNCT
cana-2085	302	1	let	let	VERB
cana-2085	302	2	𝑣1	𝑣1	NOUN
cana-2085	302	3	and	and	CCONJ
cana-2085	302	4	𝑣2	𝑣2	PROPN
cana-2085	302	5	be	be	AUX
cana-2085	302	6	the	the	DET
cana-2085	302	7	supports	support	NOUN
cana-2085	302	8	that	that	PRON
cana-2085	302	9	precede	precede	VERB
cana-2085	302	10	and	and	CCONJ
cana-2085	302	11	succeed	succeed	VERB
cana-2085	302	12	𝑣	𝑣	ADP
cana-2085	302	13	on	on	ADP
cana-2085	302	14	the	the	DET
cana-2085	302	15	spine	spine	NOUN
cana-2085	302	16	.	.	PUNCT
cana-2085	303	1	let	let	VERB
cana-2085	303	2	𝑃	𝑃	PRON
cana-2085	303	3	be	be	AUX
cana-2085	303	4	the	the	DET
cana-2085	303	5	(	(	PUNCT
cana-2085	303	6	𝑣1	𝑣1	PROPN
cana-2085	303	7	,	,	PUNCT
cana-2085	303	8	𝑣2	𝑣2	NOUN
cana-2085	303	9	)	)	PUNCT
cana-2085	303	10	path	path	NOUN
cana-2085	303	11	.	.	PUNCT
cana-2085	304	1	let	let	VERB
cana-2085	304	2	𝑢1	𝑢1	PROPN
cana-2085	304	3	,	,	PUNCT
cana-2085	304	4	𝑢2	𝑢2	PROPN
cana-2085	304	5	,	,	PUNCT
cana-2085	304	6	…	…	PUNCT
cana-2085	304	7	,	,	PUNCT
cana-2085	304	8	𝑢𝑥	𝑢𝑥	AUX
cana-2085	304	9	be	be	AUX
cana-2085	304	10	the	the	DET
cana-2085	304	11	internal	internal	ADJ
cana-2085	304	12	vertices	vertex	NOUN
cana-2085	304	13	of	of	ADP
cana-2085	304	14	(	(	PUNCT
cana-2085	304	15	𝑣1	𝑣1	NOUN
cana-2085	304	16	,	,	PUNCT
cana-2085	304	17	𝑣)path	𝑣)path	NUM
cana-2085	304	18	and	and	CCONJ
cana-2085	304	19	𝑤1	𝑤1	VERB
cana-2085	304	20	,	,	PUNCT
cana-2085	304	21	𝑤2	𝑤2	NOUN
cana-2085	304	22	,	,	PUNCT
cana-2085	304	23	…	…	PUNCT
cana-2085	304	24	,	,	PUNCT
cana-2085	304	25	𝑤𝑦	𝑤𝑦	PART
cana-2085	304	26	be	be	AUX
cana-2085	304	27	the	the	DET
cana-2085	304	28	internal	internal	ADJ
cana-2085	304	29	vertices	vertex	NOUN
cana-2085	304	30	of	of	ADP
cana-2085	304	31	(	(	PUNCT
cana-2085	304	32	𝑣	𝑣	NOUN
cana-2085	304	33	,	,	PUNCT
cana-2085	304	34	𝑣2)path	𝑣2)path	NOUN
cana-2085	304	35	.	.	PUNCT
cana-2085	305	1	communications	communication	NOUN
cana-2085	305	2	on	on	ADP
cana-2085	305	3	applied	apply	VERB
cana-2085	305	4	nonlinear	nonlinear	ADJ
cana-2085	305	5	analysis	analysis	NOUN
cana-2085	305	6	issn	issn	NOUN
cana-2085	305	7	:	:	PUNCT
cana-2085	305	8	1074	1074	NUM
cana-2085	305	9	-	-	PUNCT
cana-2085	305	10	133x	133x	NUM
cana-2085	305	11	vol	vol	NOUN
cana-2085	305	12	32	32	NUM
cana-2085	305	13	no	no	NOUN
cana-2085	305	14	.	.	PUNCT
cana-2085	306	1	1s	1s	NUM
cana-2085	306	2	(	(	PUNCT
cana-2085	306	3	2025	2025	NUM
cana-2085	306	4	)	)	PUNCT
cana-2085	306	5	26	26	NUM
cana-2085	306	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-2085	306	7	case	case	NOUN
cana-2085	306	8	(	(	PUNCT
cana-2085	306	9	i	i	NOUN
cana-2085	306	10	)	)	PUNCT
cana-2085	306	11	𝑣1	𝑣1	NOUN
cana-2085	306	12	and	and	CCONJ
cana-2085	306	13	𝑣2are	𝑣2are	VERB
cana-2085	306	14	the	the	DET
cana-2085	306	15	weak	weak	ADJ
cana-2085	306	16	supports	support	NOUN
cana-2085	306	17	.	.	PUNCT
cana-2085	307	1	if	if	SCONJ
cana-2085	307	2	two	two	NUM
cana-2085	307	3	legions	legion	NOUN
cana-2085	307	4	are	be	AUX
cana-2085	307	5	posted	post	VERB
cana-2085	307	6	at	at	ADP
cana-2085	307	7	𝑣	𝑣	PRON
cana-2085	307	8	then	then	ADV
cana-2085	307	9	⌈	⌈	NOUN
cana-2085	307	10	2(𝑥−1	2(𝑥−1	NOUN
cana-2085	307	11	)	)	PUNCT
cana-2085	307	12	3	3	NUM
cana-2085	307	13	⌉	⌉	NOUN
cana-2085	307	14	+	+	CCONJ
cana-2085	307	15	⌈	⌈	SYM
cana-2085	307	16	2(𝑦−1	2(𝑦−1	NUM
cana-2085	307	17	)	)	PUNCT
cana-2085	307	18	3	3	NUM
cana-2085	307	19	⌉	⌉	NOUN
cana-2085	307	20	+	+	CCONJ
cana-2085	307	21	2	2	NUM
cana-2085	307	22	=	=	NOUN
cana-2085	307	23	𝐾1	𝐾1	PROPN
cana-2085	307	24	legions	legion	NOUN
cana-2085	307	25	are	be	AUX
cana-2085	307	26	needed	need	VERB
cana-2085	307	27	to	to	PART
cana-2085	307	28	safeguard	safeguard	VERB
cana-2085	307	29	the	the	DET
cana-2085	307	30	vertices	vertex	NOUN
cana-2085	307	31	on	on	ADP
cana-2085	307	32	the	the	DET
cana-2085	307	33	path	path	NOUN
cana-2085	307	34	𝑃.	𝑃.	PROPN
cana-2085	307	35	but	but	CCONJ
cana-2085	307	36	,	,	PUNCT
cana-2085	307	37	on	on	ADP
cana-2085	307	38	the	the	DET
cana-2085	307	39	other	other	ADJ
cana-2085	307	40	hand	hand	NOUN
cana-2085	307	41	if	if	SCONJ
cana-2085	307	42	three	three	NUM
cana-2085	307	43	legions	legion	NOUN
cana-2085	307	44	are	be	AUX
cana-2085	307	45	posted	post	VERB
cana-2085	307	46	at	at	ADP
cana-2085	307	47	𝑣1	𝑣1	NOUN
cana-2085	307	48	then	then	ADV
cana-2085	307	49	⌈	⌈	NOUN
cana-2085	307	50	2(𝑥−2	2(𝑥−2	NUM
cana-2085	307	51	)	)	PUNCT
cana-2085	307	52	3	3	NUM
cana-2085	308	1	⌉	⌉	NOUN
cana-2085	308	2	+	+	CCONJ
cana-2085	308	3	⌈	⌈	NUM
cana-2085	308	4	2(𝑦−2	2(𝑦−2	NOUN
cana-2085	308	5	)	)	PUNCT
cana-2085	308	6	3	3	NUM
cana-2085	308	7	⌉	⌉	NOUN
cana-2085	308	8	+	+	SYM
cana-2085	308	9	3	3	NUM
cana-2085	308	10	legions	legion	NOUN
cana-2085	308	11	are	be	AUX
cana-2085	308	12	required	require	VERB
cana-2085	308	13	to	to	PART
cana-2085	308	14	safeguard	safeguard	VERB
cana-2085	308	15	the	the	DET
cana-2085	308	16	path	path	NOUN
cana-2085	308	17	𝑃	𝑃	NOUN
cana-2085	308	18	,	,	PUNCT
cana-2085	308	19	which	which	PRON
cana-2085	308	20	is	be	AUX
cana-2085	308	21	less	less	ADJ
cana-2085	308	22	than	than	ADP
cana-2085	308	23	or	or	CCONJ
cana-2085	308	24	equal	equal	ADJ
cana-2085	308	25	to	to	ADP
cana-2085	308	26	𝑀1	𝑀1	PROPN
cana-2085	308	27	.	.	PUNCT
cana-2085	309	1	hence	hence	ADV
cana-2085	309	2	,	,	PUNCT
cana-2085	309	3	we	we	PRON
cana-2085	309	4	assign	assign	VERB
cana-2085	309	5	three	three	NUM
cana-2085	309	6	legions	legion	NOUN
cana-2085	309	7	at	at	ADP
cana-2085	309	8	𝑣	𝑣	PRON
cana-2085	309	9	to	to	PART
cana-2085	309	10	safeguard	safeguard	VERB
cana-2085	309	11	𝑁[𝑣	𝑁[𝑣	PROPN
cana-2085	309	12	]	]	PUNCT
cana-2085	309	13	.	.	PUNCT
cana-2085	310	1	case	case	NOUN
cana-2085	310	2	(	(	PUNCT
cana-2085	310	3	iii	iii	NOUN
cana-2085	310	4	)	)	PUNCT
cana-2085	310	5	𝑣1	𝑣1	NOUN
cana-2085	310	6	is	be	AUX
cana-2085	310	7	a	a	DET
cana-2085	310	8	weak	weak	ADJ
cana-2085	310	9	support	support	NOUN
cana-2085	310	10	and	and	CCONJ
cana-2085	310	11	𝑣2	𝑣2	NOUN
cana-2085	310	12	is	be	AUX
cana-2085	310	13	a	a	DET
cana-2085	310	14	moderate	moderate	ADJ
cana-2085	310	15	support	support	NOUN
cana-2085	310	16	.	.	PUNCT
cana-2085	311	1	if	if	SCONJ
cana-2085	311	2	the	the	DET
cana-2085	311	3	legions	legion	NOUN
cana-2085	311	4	are	be	AUX
cana-2085	311	5	posted	post	VERB
cana-2085	311	6	at	at	ADP
cana-2085	311	7	𝑣	𝑣	PRON
cana-2085	311	8	then	then	ADV
cana-2085	311	9	⌈	⌈	NOUN
cana-2085	311	10	2(𝑥−1	2(𝑥−1	NOUN
cana-2085	311	11	)	)	PUNCT
cana-2085	311	12	3	3	NUM
cana-2085	311	13	⌉	⌉	NOUN
cana-2085	311	14	+	+	PUNCT
cana-2085	311	15	⌈	⌈	SYM
cana-2085	311	16	2𝑦	2𝑦	NUM
cana-2085	311	17	3	3	NUM
cana-2085	311	18	⌉	⌉	NOUN
cana-2085	311	19	+	+	CCONJ
cana-2085	311	20	2	2	NUM
cana-2085	311	21	=	=	SYM
cana-2085	311	22	𝐾2	𝐾2	NOUN
cana-2085	311	23	legions	legion	NOUN
cana-2085	311	24	are	be	AUX
cana-2085	311	25	required	require	VERB
cana-2085	311	26	to	to	PART
cana-2085	311	27	safeguard	safeguard	VERB
cana-2085	311	28	the	the	DET
cana-2085	311	29	vertices	vertex	NOUN
cana-2085	311	30	on	on	ADP
cana-2085	311	31	the	the	DET
cana-2085	311	32	path	path	NOUN
cana-2085	311	33	𝑃.	𝑃.	PROPN
cana-2085	311	34	but	but	CCONJ
cana-2085	311	35	,	,	PUNCT
cana-2085	311	36	on	on	ADP
cana-2085	311	37	the	the	DET
cana-2085	311	38	other	other	ADJ
cana-2085	311	39	hand	hand	NOUN
cana-2085	311	40	,	,	PUNCT
cana-2085	311	41	if	if	SCONJ
cana-2085	311	42	three	three	NUM
cana-2085	311	43	legions	legion	NOUN
cana-2085	311	44	are	be	AUX
cana-2085	311	45	posted	post	VERB
cana-2085	311	46	at	at	ADP
cana-2085	311	47	𝑣	𝑣	ADP
cana-2085	311	48	,	,	PUNCT
cana-2085	311	49	then	then	ADV
cana-2085	311	50	⌈	⌈	NOUN
cana-2085	311	51	2(𝑥−2	2(𝑥−2	NUM
cana-2085	311	52	)	)	PUNCT
cana-2085	311	53	3	3	NUM
cana-2085	311	54	⌉	⌉	NOUN
cana-2085	311	55	+	+	CCONJ
cana-2085	311	56	⌈	⌈	SYM
cana-2085	311	57	2(𝑦−1	2(𝑦−1	NUM
cana-2085	311	58	)	)	PUNCT
cana-2085	311	59	3	3	NUM
cana-2085	311	60	⌉	⌉	X
cana-2085	311	61	+	+	SYM
cana-2085	311	62	3	3	NUM
cana-2085	311	63	legions	legion	NOUN
cana-2085	311	64	are	be	AUX
cana-2085	311	65	required	require	VERB
cana-2085	311	66	to	to	PART
cana-2085	311	67	safeguard	safeguard	VERB
cana-2085	311	68	the	the	DET
cana-2085	311	69	path	path	NOUN
cana-2085	311	70	𝑃	𝑃	NOUN
cana-2085	311	71	which	which	PRON
cana-2085	311	72	is	be	AUX
cana-2085	311	73	less	less	ADJ
cana-2085	311	74	than	than	ADP
cana-2085	311	75	𝐾2	𝐾2	NOUN
cana-2085	311	76	.	.	PUNCT
cana-2085	312	1	hence	hence	ADV
cana-2085	312	2	,	,	PUNCT
cana-2085	312	3	we	we	PRON
cana-2085	312	4	assign	assign	VERB
cana-2085	312	5	three	three	NUM
cana-2085	312	6	legions	legion	NOUN
cana-2085	312	7	at	at	ADP
cana-2085	312	8	𝑣.	𝑣.	NOUN
cana-2085	312	9	case	case	NOUN
cana-2085	312	10	(	(	PUNCT
cana-2085	312	11	iv	iv	X
cana-2085	312	12	)	)	PUNCT
cana-2085	312	13	𝑣1	𝑣1	NOUN
cana-2085	312	14	and	and	CCONJ
cana-2085	312	15	𝑣2	𝑣2	NOUN
cana-2085	312	16	are	be	AUX
cana-2085	312	17	moderate	moderate	ADJ
cana-2085	312	18	supports	support	NOUN
cana-2085	312	19	.	.	PUNCT
cana-2085	313	1	if	if	SCONJ
cana-2085	313	2	legions	legion	NOUN
cana-2085	313	3	are	be	AUX
cana-2085	313	4	posted	post	VERB
cana-2085	313	5	at	at	ADP
cana-2085	313	6	𝑣1	𝑣1	NOUN
cana-2085	313	7	then	then	ADV
cana-2085	313	8	⌈	⌈	PROPN
cana-2085	313	9	2𝑥	2𝑥	NOUN
cana-2085	313	10	3	3	NUM
cana-2085	313	11	⌉	⌉	NOUN
cana-2085	313	12	+	+	PUNCT
cana-2085	313	13	⌈	⌈	SYM
cana-2085	313	14	2𝑦	2𝑦	NUM
cana-2085	313	15	3	3	NUM
cana-2085	313	16	⌉	⌉	NOUN
cana-2085	313	17	+	+	CCONJ
cana-2085	313	18	2	2	NUM
cana-2085	313	19	=	=	NUM
cana-2085	313	20	𝐾3	𝐾3	NOUN
cana-2085	313	21	legions	legion	NOUN
cana-2085	313	22	are	be	AUX
cana-2085	313	23	required	require	VERB
cana-2085	313	24	to	to	PART
cana-2085	313	25	safeguard	safeguard	VERB
cana-2085	313	26	the	the	DET
cana-2085	313	27	legions	legion	NOUN
cana-2085	313	28	on	on	ADP
cana-2085	313	29	the	the	DET
cana-2085	313	30	path	path	NOUN
cana-2085	313	31	.	.	PUNCT
cana-2085	314	1	but	but	CCONJ
cana-2085	314	2	,	,	PUNCT
cana-2085	314	3	on	on	ADP
cana-2085	314	4	the	the	DET
cana-2085	314	5	other	other	ADJ
cana-2085	314	6	hand	hand	NOUN
cana-2085	314	7	if	if	SCONJ
cana-2085	314	8	three	three	NUM
cana-2085	314	9	legions	legion	NOUN
cana-2085	314	10	are	be	AUX
cana-2085	314	11	posted	post	VERB
cana-2085	314	12	at	at	ADP
cana-2085	314	13	𝑣	𝑣	ADP
cana-2085	314	14	,	,	PUNCT
cana-2085	314	15	then	then	ADV
cana-2085	314	16	⌈	⌈	PUNCT
cana-2085	314	17	2(𝑥−1	2(𝑥−1	NOUN
cana-2085	314	18	)	)	PUNCT
cana-2085	314	19	3	3	NUM
cana-2085	314	20	⌉	⌉	NOUN
cana-2085	314	21	+	+	CCONJ
cana-2085	314	22	⌈	⌈	SYM
cana-2085	314	23	2𝑦−1	2𝑦−1	NUM
cana-2085	314	24	3	3	NUM
cana-2085	314	25	⌉	⌉	PRON
cana-2085	314	26	+	+	SYM
cana-2085	314	27	3	3	NUM
cana-2085	314	28	legions	legion	NOUN
cana-2085	314	29	are	be	AUX
cana-2085	314	30	required	require	VERB
cana-2085	314	31	to	to	PART
cana-2085	314	32	safeguard	safeguard	VERB
cana-2085	314	33	the	the	DET
cana-2085	314	34	path	path	NOUN
cana-2085	314	35	𝑃	𝑃	NOUN
cana-2085	314	36	,	,	PUNCT
cana-2085	314	37	which	which	PRON
cana-2085	314	38	is	be	AUX
cana-2085	314	39	less	less	ADJ
cana-2085	314	40	than	than	ADP
cana-2085	314	41	or	or	CCONJ
cana-2085	314	42	equal	equal	ADJ
cana-2085	314	43	to	to	PART
cana-2085	314	44	𝐾3	𝐾3	VERB
cana-2085	314	45	.	.	PUNCT
cana-2085	315	1	hence	hence	ADV
cana-2085	315	2	,	,	PUNCT
cana-2085	315	3	we	we	PRON
cana-2085	315	4	assign	assign	VERB
cana-2085	315	5	three	three	NUM
cana-2085	315	6	legions	legion	NOUN
cana-2085	315	7	at	at	ADP
cana-2085	315	8	𝑣	𝑣	PRON
cana-2085	315	9	to	to	PART
cana-2085	315	10	safeguard	safeguard	VERB
cana-2085	315	11	𝑁[𝑣	𝑁[𝑣	PROPN
cana-2085	315	12	]	]	PUNCT
cana-2085	315	13	.	.	PUNCT
cana-2085	316	1	let	let	VERB
cana-2085	316	2	𝑤	𝑤	PART
cana-2085	316	3	be	be	AUX
cana-2085	316	4	an	an	DET
cana-2085	316	5	artificial	artificial	ADJ
cana-2085	316	6	strong	strong	ADJ
cana-2085	316	7	support	support	NOUN
cana-2085	316	8	.	.	PUNCT
cana-2085	317	1	let	let	VERB
cana-2085	317	2	𝑤1	𝑤1	VERB
cana-2085	317	3	and	and	CCONJ
cana-2085	317	4	𝑤2	𝑤2	NOUN
cana-2085	317	5	be	be	VERB
cana-2085	317	6	the	the	DET
cana-2085	317	7	supports	support	NOUN
cana-2085	317	8	that	that	PRON
cana-2085	317	9	precede	precede	VERB
cana-2085	317	10	and	and	CCONJ
cana-2085	317	11	succeed	succeed	VERB
cana-2085	317	12	𝑤	𝑤	ADP
cana-2085	317	13	on	on	ADP
cana-2085	317	14	the	the	DET
cana-2085	317	15	spine	spine	NOUN
cana-2085	317	16	.	.	PUNCT
cana-2085	318	1	let	let	VERB
cana-2085	318	2	𝑃	𝑃	PRON
cana-2085	318	3	be	be	AUX
cana-2085	318	4	the	the	DET
cana-2085	318	5	(	(	PUNCT
cana-2085	318	6	𝑤1	𝑤1	ADJ
cana-2085	318	7	,	,	PUNCT
cana-2085	318	8	𝑤2	𝑤2	NOUN
cana-2085	318	9	)	)	PUNCT
cana-2085	318	10	path	path	NOUN
cana-2085	318	11	.	.	PUNCT
cana-2085	319	1	let	let	VERB
cana-2085	319	2	𝑥1	𝑥1	NOUN
cana-2085	319	3	,	,	PUNCT
cana-2085	319	4	𝑥2	𝑥2	NOUN
cana-2085	319	5	,	,	PUNCT
cana-2085	319	6	…	…	PUNCT
cana-2085	319	7	,	,	PUNCT
cana-2085	319	8	𝑥𝑝	𝑥𝑝	NOUN
cana-2085	319	9	and	and	CCONJ
cana-2085	319	10	𝑦1	𝑦1	PROPN
cana-2085	319	11	,	,	PUNCT
cana-2085	319	12	𝑦2	𝑦2	NOUN
cana-2085	319	13	,	,	PUNCT
cana-2085	319	14	…	…	PUNCT
cana-2085	319	15	,	,	PUNCT
cana-2085	319	16	𝑦𝑞	𝑦𝑞	PROPN
cana-2085	319	17	be	be	AUX
cana-2085	319	18	the	the	DET
cana-2085	319	19	internal	internal	ADJ
cana-2085	319	20	vertices	vertex	NOUN
cana-2085	319	21	of	of	ADP
cana-2085	319	22	(	(	PUNCT
cana-2085	319	23	𝑤1	𝑤1	VERB
cana-2085	319	24	,	,	PUNCT
cana-2085	319	25	𝑤	𝑤	NOUN
cana-2085	319	26	)	)	PUNCT
cana-2085	319	27	path	path	NOUN
cana-2085	319	28	and	and	CCONJ
cana-2085	319	29	(	(	PUNCT
cana-2085	319	30	𝑤	𝑤	ADP
cana-2085	319	31	,	,	PUNCT
cana-2085	319	32	𝑤2	𝑤2	NOUN
cana-2085	319	33	)	)	PUNCT
cana-2085	319	34	path	path	NOUN
cana-2085	319	35	respectively	respectively	ADV
cana-2085	319	36	.	.	PUNCT
cana-2085	320	1	case	case	NOUN
cana-2085	320	2	(	(	PUNCT
cana-2085	320	3	v	v	NOUN
cana-2085	320	4	)	)	PUNCT
cana-2085	320	5	𝑤1	𝑤1	NOUN
cana-2085	320	6	and	and	CCONJ
cana-2085	320	7	𝑤2	𝑤2	NOUN
cana-2085	320	8	are	be	AUX
cana-2085	320	9	weak	weak	ADJ
cana-2085	320	10	supports	support	NOUN
cana-2085	320	11	.	.	PUNCT
cana-2085	321	1	if	if	SCONJ
cana-2085	321	2	two	two	NUM
cana-2085	321	3	legions	legion	NOUN
cana-2085	321	4	are	be	AUX
cana-2085	321	5	posted	post	VERB
cana-2085	321	6	at	at	ADP
cana-2085	321	7	𝑤	𝑤	ADP
cana-2085	321	8	then	then	ADV
cana-2085	321	9	,	,	PUNCT
cana-2085	321	10	⌈	⌈	NOUN
cana-2085	321	11	2(𝑝−1	2(𝑝−1	NUM
cana-2085	321	12	)	)	PUNCT
cana-2085	321	13	3	3	NUM
cana-2085	321	14	⌉	⌉	NOUN
cana-2085	321	15	+	+	ADJ
cana-2085	321	16	2	2	NUM
cana-2085	321	17	=	=	SYM
cana-2085	321	18	𝐿1	𝐿1	PROPN
cana-2085	321	19	legions	legion	NOUN
cana-2085	321	20	are	be	AUX
cana-2085	321	21	required	require	VERB
cana-2085	321	22	to	to	PART
cana-2085	321	23	safeguard	safeguard	VERB
cana-2085	321	24	the	the	DET
cana-2085	321	25	internal	internal	ADJ
cana-2085	321	26	vertices	vertex	NOUN
cana-2085	321	27	on	on	ADP
cana-2085	321	28	the	the	DET
cana-2085	321	29	path	path	NOUN
cana-2085	321	30	(	(	PUNCT
cana-2085	321	31	𝑤1	𝑤1	VERB
cana-2085	321	32	,	,	PUNCT
cana-2085	321	33	𝑤	𝑤	ADP
cana-2085	321	34	)	)	PUNCT
cana-2085	321	35	.	.	PUNCT
cana-2085	322	1	but	but	CCONJ
cana-2085	322	2	on	on	ADP
cana-2085	322	3	the	the	DET
cana-2085	322	4	other	other	ADJ
cana-2085	322	5	hand	hand	NOUN
cana-2085	322	6	,	,	PUNCT
cana-2085	322	7	if	if	SCONJ
cana-2085	322	8	three	three	NUM
cana-2085	322	9	legions	legion	NOUN
cana-2085	322	10	are	be	AUX
cana-2085	322	11	posted	post	VERB
cana-2085	322	12	at	at	ADP
cana-2085	322	13	𝑤	𝑤	ADP
cana-2085	322	14	,	,	PUNCT
cana-2085	322	15	then	then	ADV
cana-2085	322	16	⌈	⌈	NOUN
cana-2085	322	17	2(𝑝−2	2(𝑝−2	NOUN
cana-2085	322	18	)	)	PUNCT
cana-2085	322	19	3	3	NUM
cana-2085	322	20	⌉	⌉	NOUN
cana-2085	322	21	+	+	CCONJ
cana-2085	322	22	⌈	⌈	SYM
cana-2085	322	23	2𝑞−2	2𝑞−2	NUM
cana-2085	322	24	3	3	NUM
cana-2085	322	25	⌉	⌉	NOUN
cana-2085	322	26	+	+	SYM
cana-2085	322	27	3	3	NUM
cana-2085	322	28	legions	legion	NOUN
cana-2085	322	29	are	be	AUX
cana-2085	322	30	required	require	VERB
cana-2085	322	31	to	to	PART
cana-2085	322	32	safeguard	safeguard	VERB
cana-2085	322	33	the	the	DET
cana-2085	322	34	path	path	NOUN
cana-2085	322	35	𝑃	𝑃	NOUN
cana-2085	322	36	,	,	PUNCT
cana-2085	322	37	which	which	PRON
cana-2085	322	38	is	be	AUX
cana-2085	322	39	less	less	ADJ
cana-2085	322	40	than	than	ADP
cana-2085	322	41	𝐿1	𝐿1	PROPN
cana-2085	322	42	.	.	PUNCT
cana-2085	323	1	hence	hence	ADV
cana-2085	323	2	,	,	PUNCT
cana-2085	323	3	we	we	PRON
cana-2085	323	4	assign	assign	VERB
cana-2085	323	5	three	three	NUM
cana-2085	323	6	legions	legion	NOUN
cana-2085	323	7	at	at	ADP
cana-2085	323	8	𝑤	𝑤	ADP
cana-2085	323	9	to	to	PART
cana-2085	323	10	safeguard	safeguard	VERB
cana-2085	323	11	𝑁[𝑤	𝑁[𝑤	ADJ
cana-2085	323	12	]	]	PUNCT
cana-2085	323	13	.	.	PUNCT
cana-2085	324	1	case	case	NOUN
cana-2085	324	2	(	(	PUNCT
cana-2085	324	3	vi	vi	NOUN
cana-2085	324	4	)	)	PUNCT
cana-2085	324	5	𝑤1	𝑤1	VERB
cana-2085	324	6	is	be	AUX
cana-2085	324	7	a	a	DET
cana-2085	324	8	moderate	moderate	ADJ
cana-2085	324	9	support	support	NOUN
cana-2085	324	10	and	and	CCONJ
cana-2085	324	11	𝑤2	𝑤2	NOUN
cana-2085	324	12	is	be	AUX
cana-2085	324	13	a	a	DET
cana-2085	324	14	weak	weak	ADJ
cana-2085	324	15	support	support	NOUN
cana-2085	324	16	.	.	PUNCT
cana-2085	325	1	if	if	SCONJ
cana-2085	325	2	two	two	NUM
cana-2085	325	3	legions	legion	NOUN
cana-2085	325	4	are	be	AUX
cana-2085	325	5	posted	post	VERB
cana-2085	325	6	at	at	ADP
cana-2085	325	7	𝑤	𝑤	ADP
cana-2085	325	8	then	then	ADV
cana-2085	325	9	⌈	⌈	CCONJ
cana-2085	325	10	2𝑝	2𝑝	PROPN
cana-2085	325	11	3	3	NUM
cana-2085	325	12	⌉	⌉	NOUN
cana-2085	325	13	+	+	CCONJ
cana-2085	325	14	⌈	⌈	SYM
cana-2085	325	15	2(𝑞−1	2(𝑞−1	NUM
cana-2085	325	16	)	)	PUNCT
cana-2085	325	17	3	3	NUM
cana-2085	325	18	⌉	⌉	NOUN
cana-2085	325	19	+	+	ADJ
cana-2085	325	20	2	2	NUM
cana-2085	325	21	=	=	SYM
cana-2085	325	22	𝐿2	𝐿2	PROPN
cana-2085	325	23	legions	legion	NOUN
cana-2085	325	24	are	be	AUX
cana-2085	325	25	required	require	VERB
cana-2085	325	26	to	to	PART
cana-2085	325	27	safeguard	safeguard	VERB
cana-2085	325	28	the	the	DET
cana-2085	325	29	vertices	vertex	NOUN
cana-2085	325	30	on	on	ADP
cana-2085	325	31	the	the	DET
cana-2085	325	32	path	path	NOUN
cana-2085	325	33	𝑃.	𝑃.	PROPN
cana-2085	325	34	but	but	CCONJ
cana-2085	325	35	on	on	ADP
cana-2085	325	36	the	the	DET
cana-2085	325	37	other	other	ADJ
cana-2085	325	38	hand	hand	NOUN
cana-2085	325	39	,	,	PUNCT
cana-2085	325	40	if	if	SCONJ
cana-2085	325	41	three	three	NUM
cana-2085	325	42	legions	legion	NOUN
cana-2085	325	43	are	be	AUX
cana-2085	325	44	posted	post	VERB
cana-2085	325	45	at	at	ADP
cana-2085	325	46	𝑤	𝑤	ADP
cana-2085	325	47	,	,	PUNCT
cana-2085	325	48	then	then	ADV
cana-2085	325	49	⌈	⌈	NOUN
cana-2085	325	50	2(𝑝−1	2(𝑝−1	NUM
cana-2085	325	51	)	)	PUNCT
cana-2085	325	52	3	3	NUM
cana-2085	325	53	⌉	⌉	NOUN
cana-2085	325	54	+	+	CCONJ
cana-2085	325	55	⌈	⌈	NUM
cana-2085	325	56	2(𝑞−2	2(𝑞−2	NUM
cana-2085	325	57	)	)	PUNCT
cana-2085	325	58	3	3	NUM
cana-2085	325	59	⌉	⌉	X
cana-2085	325	60	+	+	SYM
cana-2085	325	61	3	3	NUM
cana-2085	325	62	legions	legion	NOUN
cana-2085	325	63	are	be	AUX
cana-2085	325	64	required	require	VERB
cana-2085	325	65	which	which	PRON
cana-2085	325	66	is	be	AUX
cana-2085	325	67	less	less	ADJ
cana-2085	325	68	than	than	ADP
cana-2085	325	69	𝐿2	𝐿2	PROPN
cana-2085	325	70	.	.	PUNCT
cana-2085	326	1	hence	hence	ADV
cana-2085	326	2	,	,	PUNCT
cana-2085	326	3	we	we	PRON
cana-2085	326	4	assign	assign	VERB
cana-2085	326	5	three	three	NUM
cana-2085	326	6	at	at	ADP
cana-2085	326	7	𝑤	𝑤	ADP
cana-2085	326	8	to	to	PART
cana-2085	326	9	safeguard	safeguard	VERB
cana-2085	326	10	𝑁[𝑤	𝑁[𝑤	ADJ
cana-2085	326	11	]	]	PUNCT
cana-2085	326	12	.	.	PUNCT
cana-2085	327	1	case	case	NOUN
cana-2085	327	2	(	(	PUNCT
cana-2085	327	3	ii	ii	NOUN
cana-2085	327	4	)	)	PUNCT
cana-2085	327	5	both	both	PRON
cana-2085	327	6	,	,	PUNCT
cana-2085	327	7	𝑤𝑟−1	𝑤𝑟−1	PROPN
cana-2085	327	8	and	and	CCONJ
cana-2085	327	9	𝑤𝑟+1	𝑤𝑟+1	PRON
cana-2085	327	10	are	be	AUX
cana-2085	327	11	moderate	moderate	ADJ
cana-2085	327	12	supports	support	NOUN
cana-2085	327	13	.	.	PUNCT
cana-2085	328	1	if	if	SCONJ
cana-2085	328	2	two	two	NUM
cana-2085	328	3	legions	legion	NOUN
cana-2085	328	4	are	be	AUX
cana-2085	328	5	posted	post	VERB
cana-2085	328	6	at	at	ADP
cana-2085	328	7	𝑤	𝑤	ADP
cana-2085	328	8	,	,	PUNCT
cana-2085	328	9	then	then	ADV
cana-2085	328	10	⌈	⌈	X
cana-2085	328	11	2𝑝	2𝑝	PROPN
cana-2085	328	12	3	3	NUM
cana-2085	328	13	⌉	⌉	NOUN
cana-2085	328	14	+	+	CCONJ
cana-2085	328	15	⌈	⌈	NUM
cana-2085	328	16	2𝑞	2𝑞	NUM
cana-2085	328	17	3	3	NUM
cana-2085	328	18	⌉	⌉	NOUN
cana-2085	328	19	+	+	ADJ
cana-2085	328	20	2	2	NUM
cana-2085	328	21	=	=	SYM
cana-2085	328	22	𝐿3	𝐿3	NOUN
cana-2085	328	23	legions	legion	NOUN
cana-2085	328	24	are	be	AUX
cana-2085	328	25	required	require	VERB
cana-2085	328	26	to	to	PART
cana-2085	328	27	safeguard	safeguard	VERB
cana-2085	328	28	the	the	DET
cana-2085	328	29	path	path	NOUN
cana-2085	328	30	𝑃.	𝑃.	PROPN
cana-2085	329	1	but	but	CCONJ
cana-2085	329	2	,	,	PUNCT
cana-2085	329	3	on	on	ADP
cana-2085	329	4	the	the	DET
cana-2085	329	5	other	other	ADJ
cana-2085	329	6	hand	hand	NOUN
cana-2085	329	7	if	if	SCONJ
cana-2085	329	8	three	three	NUM
cana-2085	329	9	legions	legion	NOUN
cana-2085	329	10	are	be	AUX
cana-2085	329	11	placed	place	VERB
cana-2085	329	12	at	at	ADP
cana-2085	329	13	𝑤	𝑤	ADP
cana-2085	329	14	then	then	ADV
cana-2085	329	15	⌈	⌈	NOUN
cana-2085	329	16	2(𝑝−1	2(𝑝−1	NUM
cana-2085	329	17	)	)	PUNCT
cana-2085	329	18	3	3	NUM
cana-2085	329	19	⌉	⌉	NOUN
cana-2085	329	20	+	+	CCONJ
cana-2085	329	21	⌈	⌈	NOUN
cana-2085	329	22	2(𝑞−1	2(𝑞−1	NUM
cana-2085	329	23	)	)	PUNCT
cana-2085	329	24	3	3	NUM
cana-2085	329	25	⌉	⌉	PRON
cana-2085	329	26	+	+	SYM
cana-2085	329	27	3	3	NUM
cana-2085	329	28	legions	legion	NOUN
cana-2085	329	29	are	be	AUX
cana-2085	329	30	required	require	VERB
cana-2085	329	31	to	to	PART
cana-2085	329	32	safeguard	safeguard	VERB
cana-2085	329	33	the	the	DET
cana-2085	329	34	vertices	vertex	NOUN
cana-2085	329	35	on	on	ADP
cana-2085	329	36	the	the	DET
cana-2085	329	37	path	path	NOUN
cana-2085	329	38	(	(	PUNCT
cana-2085	329	39	𝑤1	𝑤1	VERB
cana-2085	329	40	,	,	PUNCT
cana-2085	329	41	𝑤2	𝑤2	NOUN
cana-2085	329	42	)	)	PUNCT
cana-2085	329	43	.	.	PUNCT
cana-2085	330	1	which	which	PRON
cana-2085	330	2	is	be	AUX
cana-2085	330	3	less	less	ADJ
cana-2085	330	4	than	than	ADP
cana-2085	330	5	𝐿3	𝐿3	NOUN
cana-2085	330	6	.	.	PUNCT
cana-2085	331	1	hence	hence	ADV
cana-2085	331	2	,	,	PUNCT
cana-2085	331	3	we	we	PRON
cana-2085	331	4	assign	assign	VERB
cana-2085	331	5	three	three	NUM
cana-2085	331	6	legions	legion	NOUN
cana-2085	331	7	at	at	ADP
cana-2085	331	8	𝑤	𝑤	ADP
cana-2085	331	9	to	to	PART
cana-2085	331	10	safeguard	safeguard	VERB
cana-2085	331	11	𝑁[𝑤	𝑁[𝑤	ADJ
cana-2085	331	12	]	]	PUNCT
cana-2085	331	13	.	.	PUNCT
cana-2085	332	1	hence	hence	ADV
cana-2085	332	2	,	,	PUNCT
cana-2085	332	3	in	in	ADP
cana-2085	332	4	all	all	DET
cana-2085	332	5	the	the	DET
cana-2085	332	6	cases	case	NOUN
cana-2085	332	7	we	we	PRON
cana-2085	332	8	see	see	VERB
cana-2085	332	9	that	that	SCONJ
cana-2085	332	10	three	three	NUM
cana-2085	332	11	legions	legion	NOUN
cana-2085	332	12	are	be	AUX
cana-2085	332	13	required	require	VERB
cana-2085	332	14	to	to	PART
cana-2085	332	15	safeguard	safeguard	VERB
cana-2085	332	16	𝑁[𝑤	𝑁[𝑤	ADJ
cana-2085	332	17	]	]	PUNCT
cana-2085	332	18	.	.	PUNCT
cana-2085	333	1	next	next	ADV
cana-2085	333	2	,	,	PUNCT
cana-2085	333	3	we	we	PRON
cana-2085	333	4	consider	consider	VERB
cana-2085	333	5	a	a	DET
cana-2085	333	6	weak	weak	ADJ
cana-2085	333	7	support	support	NOUN
cana-2085	333	8	𝑢𝑟	𝑢𝑟	ADP
cana-2085	333	9	as	as	ADP
cana-2085	333	10	an	an	DET
cana-2085	333	11	artificial	artificial	ADJ
cana-2085	333	12	moderate	moderate	ADJ
cana-2085	333	13	support	support	NOUN
cana-2085	333	14	,	,	PUNCT
cana-2085	333	15	if	if	SCONJ
cana-2085	333	16	any	any	DET
cana-2085	333	17	one	one	NUM
cana-2085	333	18	of	of	ADP
cana-2085	333	19	the	the	DET
cana-2085	333	20	following	follow	VERB
cana-2085	333	21	conditions	condition	NOUN
cana-2085	333	22	hold	hold	VERB
cana-2085	333	23	.	.	PUNCT
cana-2085	334	1	communications	communication	NOUN
cana-2085	334	2	on	on	ADP
cana-2085	334	3	applied	apply	VERB
cana-2085	334	4	nonlinear	nonlinear	ADJ
cana-2085	334	5	analysis	analysis	NOUN
cana-2085	334	6	issn	issn	NOUN
cana-2085	334	7	:	:	PUNCT
cana-2085	334	8	1074	1074	NUM
cana-2085	334	9	-	-	PUNCT
cana-2085	334	10	133x	133x	NUM
cana-2085	334	11	vol	vol	NOUN
cana-2085	334	12	32	32	NUM
cana-2085	334	13	no	no	NOUN
cana-2085	334	14	.	.	PUNCT
cana-2085	335	1	1s	1s	NUM
cana-2085	335	2	(	(	PUNCT
cana-2085	335	3	2025	2025	NUM
cana-2085	335	4	)	)	PUNCT
cana-2085	335	5	27	27	NUM
cana-2085	335	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-2085	335	7	(	(	PUNCT
cana-2085	335	8	i	i	NOUN
cana-2085	335	9	)	)	PUNCT
cana-2085	335	10	if	if	SCONJ
cana-2085	335	11	both	both	DET
cana-2085	335	12	𝑢𝑟−1	𝑢𝑟−1	NOUN
cana-2085	335	13	and	and	CCONJ
cana-2085	335	14	𝑢𝑟+1	𝑢𝑟+1	PROPN
cana-2085	335	15	are	be	AUX
cana-2085	335	16	weak	weak	ADJ
cana-2085	335	17	supports	support	NOUN
cana-2085	335	18	then	then	ADV
cana-2085	335	19	𝑛𝑖	𝑛𝑖	PROPN
cana-2085	335	20	≡	≡	PROPN
cana-2085	335	21	0	0	NUM
cana-2085	335	22	,	,	PUNCT
cana-2085	335	23	2(𝑚𝑜𝑑	2(𝑚𝑜𝑑	NOUN
cana-2085	335	24	3	3	NUM
cana-2085	335	25	)	)	PUNCT
cana-2085	335	26	and	and	CCONJ
cana-2085	335	27	𝑛𝑗	𝑛𝑗	ADP
cana-2085	335	28	≡	≡	PROPN
cana-2085	335	29	1(𝑚𝑜𝑑	1(𝑚𝑜𝑑	ADJ
cana-2085	335	30	3	3	NUM
cana-2085	335	31	)	)	PUNCT
cana-2085	335	32	or	or	CCONJ
cana-2085	335	33	vice	vice	NOUN
cana-2085	335	34	versa	versa	ADV
cana-2085	335	35	.	.	PUNCT
cana-2085	336	1	(	(	PUNCT
cana-2085	336	2	ii	ii	NOUN
cana-2085	336	3	)	)	PUNCT
cana-2085	336	4	if	if	SCONJ
cana-2085	336	5	both	both	DET
cana-2085	336	6	𝑢𝑟−1	𝑢𝑟−1	NOUN
cana-2085	336	7	and	and	CCONJ
cana-2085	336	8	𝑢𝑟+1	𝑢𝑟+1	PROPN
cana-2085	336	9	are	be	AUX
cana-2085	336	10	moderate	moderate	ADJ
cana-2085	336	11	supports	support	NOUN
cana-2085	336	12	then	then	ADV
cana-2085	336	13	𝑛𝑖	𝑛𝑖	PROPN
cana-2085	336	14	≡	≡	PROPN
cana-2085	336	15	1	1	NUM
cana-2085	336	16	,	,	PUNCT
cana-2085	336	17	2(𝑚𝑜𝑑	2(𝑚𝑜𝑑	NOUN
cana-2085	336	18	3	3	NUM
cana-2085	336	19	)	)	PUNCT
cana-2085	336	20	and	and	CCONJ
cana-2085	336	21	𝑛𝑗	𝑛𝑗	ADP
cana-2085	336	22	≡	≡	PROPN
cana-2085	336	23	0(𝑚𝑜𝑑	0(𝑚𝑜𝑑	ADJ
cana-2085	336	24	3	3	NUM
cana-2085	336	25	)	)	PUNCT
cana-2085	336	26	or	or	CCONJ
cana-2085	336	27	vice	vice	NOUN
cana-2085	336	28	versa	versa	ADV
cana-2085	336	29	.	.	PUNCT
cana-2085	337	1	(	(	PUNCT
cana-2085	337	2	iii	iii	X
cana-2085	337	3	)	)	PUNCT
cana-2085	337	4	it	it	PRON
cana-2085	337	5	either	either	CCONJ
cana-2085	337	6	𝑢𝑟−1	𝑢𝑟−1	NOUN
cana-2085	337	7	is	be	AUX
cana-2085	337	8	moderate	moderate	ADJ
cana-2085	337	9	support	support	NOUN
cana-2085	337	10	and	and	CCONJ
cana-2085	337	11	𝑢𝑟+1	𝑢𝑟+1	PROPN
cana-2085	337	12	is	be	AUX
cana-2085	337	13	a	a	DET
cana-2085	337	14	weak	weak	ADJ
cana-2085	337	15	support	support	NOUN
cana-2085	337	16	then	then	ADV
cana-2085	337	17	𝑛𝑖	𝑛𝑖	PROPN
cana-2085	337	18	≡	≡	PROPN
cana-2085	337	19	1	1	NUM
cana-2085	337	20	,	,	PUNCT
cana-2085	337	21	2(𝑚𝑜𝑑	2(𝑚𝑜𝑑	NOUN
cana-2085	337	22	3	3	NUM
cana-2085	337	23	)	)	PUNCT
cana-2085	337	24	and	and	CCONJ
cana-2085	337	25	𝑛𝑗	𝑛𝑗	ADP
cana-2085	337	26	≡	≡	PROPN
cana-2085	337	27	0(𝑚𝑜𝑑	0(𝑚𝑜𝑑	X
cana-2085	338	1	3	3	X
cana-2085	338	2	)	)	PUNCT
cana-2085	338	3	let	let	VERB
cana-2085	338	4	𝑢	𝑢	PRON
cana-2085	338	5	be	be	AUX
cana-2085	338	6	an	an	DET
cana-2085	338	7	artificial	artificial	ADJ
cana-2085	338	8	moderate	moderate	ADJ
cana-2085	338	9	support	support	NOUN
cana-2085	338	10	.	.	PUNCT
cana-2085	339	1	let	let	VERB
cana-2085	339	2	𝑢1	𝑢1	PROPN
cana-2085	339	3	and	and	CCONJ
cana-2085	339	4	𝑢2	𝑢2	PROPN
cana-2085	339	5	be	be	VERB
cana-2085	339	6	the	the	DET
cana-2085	339	7	supports	support	NOUN
cana-2085	339	8	that	that	PRON
cana-2085	339	9	precede	precede	VERB
cana-2085	339	10	and	and	CCONJ
cana-2085	339	11	succeed	succeed	VERB
cana-2085	339	12	𝑢	𝑢	NOUN
cana-2085	339	13	on	on	ADP
cana-2085	339	14	the	the	DET
cana-2085	339	15	spine	spine	NOUN
cana-2085	339	16	.	.	PUNCT
cana-2085	340	1	let	let	VERB
cana-2085	340	2	𝑃	𝑃	PRON
cana-2085	340	3	be	be	AUX
cana-2085	340	4	the	the	DET
cana-2085	340	5	(	(	PUNCT
cana-2085	340	6	𝑢1	𝑢1	PROPN
cana-2085	340	7	,	,	PUNCT
cana-2085	340	8	𝑢2)path	𝑢2)path	NOUN
cana-2085	340	9	.	.	PUNCT
cana-2085	341	1	let	let	VERB
cana-2085	341	2	𝑡1	𝑡1	NOUN
cana-2085	341	3	,	,	PUNCT
cana-2085	341	4	𝑡2	𝑡2	PROPN
cana-2085	341	5	,	,	PUNCT
cana-2085	341	6	…	…	PUNCT
cana-2085	341	7	,	,	PUNCT
cana-2085	341	8	𝑡𝑚	𝑡𝑚	NOUN
cana-2085	341	9	and	and	CCONJ
cana-2085	341	10	𝑙1	𝑙1	PROPN
cana-2085	341	11	,	,	PUNCT
cana-2085	341	12	𝑙2	𝑙2	PROPN
cana-2085	341	13	,	,	PUNCT
cana-2085	341	14	…	…	PUNCT
cana-2085	341	15	,	,	PUNCT
cana-2085	341	16	𝑙𝑠	𝑙𝑠	PROPN
cana-2085	341	17	be	be	AUX
cana-2085	341	18	the	the	DET
cana-2085	341	19	internal	internal	ADJ
cana-2085	341	20	vertices	vertex	NOUN
cana-2085	341	21	of	of	ADP
cana-2085	341	22	(	(	PUNCT
cana-2085	341	23	𝑢1	𝑢1	PROPN
cana-2085	341	24	,	,	PUNCT
cana-2085	341	25	𝑢	𝑢	NOUN
cana-2085	341	26	)	)	PUNCT
cana-2085	341	27	path	path	NOUN
cana-2085	341	28	and	and	CCONJ
cana-2085	341	29	(	(	PUNCT
cana-2085	341	30	𝑢	𝑢	X
cana-2085	341	31	,	,	PUNCT
cana-2085	341	32	𝑢2)-path	𝑢2)-path	ADV
cana-2085	341	33	respectively	respectively	ADV
cana-2085	341	34	.	.	PUNCT
cana-2085	342	1	case	case	NOUN
cana-2085	342	2	(	(	PUNCT
cana-2085	342	3	i	i	NOUN
cana-2085	342	4	)	)	PUNCT
cana-2085	342	5	both	both	PRON
cana-2085	342	6	,	,	PUNCT
cana-2085	342	7	𝑢𝑟−1	𝑢𝑟−1	NOUN
cana-2085	342	8	and	and	CCONJ
cana-2085	342	9	𝑢𝑟+1	𝑢𝑟+1	PROPN
cana-2085	342	10	are	be	AUX
cana-2085	342	11	weak	weak	ADJ
cana-2085	342	12	supports	support	NOUN
cana-2085	342	13	.	.	PUNCT
cana-2085	343	1	claim	claim	NOUN
cana-2085	343	2	:	:	PUNCT
cana-2085	343	3	𝑛𝑖	𝑛𝑖	PROPN
cana-2085	343	4	≡	≡	PROPN
cana-2085	343	5	0	0	NUM
cana-2085	343	6	,	,	PUNCT
cana-2085	343	7	2(𝑚𝑜𝑑	2(𝑚𝑜𝑑	NOUN
cana-2085	343	8	3	3	NUM
cana-2085	343	9	)	)	PUNCT
cana-2085	343	10	and	and	CCONJ
cana-2085	343	11	𝑛𝑗	𝑛𝑗	ADP
cana-2085	343	12	≡	≡	PROPN
cana-2085	343	13	1(𝑚𝑜𝑑	1(𝑚𝑜𝑑	ADJ
cana-2085	343	14	3	3	X
cana-2085	343	15	)	)	PUNCT
cana-2085	343	16	sub	sub	NOUN
cana-2085	343	17	case	case	NOUN
cana-2085	343	18	(	(	PUNCT
cana-2085	343	19	i	i	NOUN
cana-2085	343	20	):	):	PUNCT
cana-2085	343	21	𝑛𝑖	𝑛𝑖	PROPN
cana-2085	343	22	≡	≡	PROPN
cana-2085	343	23	0	0	NUM
cana-2085	343	24	,	,	PUNCT
cana-2085	343	25	2(𝑚𝑜𝑑	2(𝑚𝑜𝑑	NOUN
cana-2085	343	26	3	3	NUM
cana-2085	343	27	)	)	PUNCT
cana-2085	343	28	suppose	suppose	VERB
cana-2085	343	29	,	,	PUNCT
cana-2085	343	30	𝑛𝑗	𝑛𝑗	ADP
cana-2085	343	31	≢	≢	NUM
cana-2085	343	32	1(𝑚𝑜𝑑	1(𝑚𝑜𝑑	NOUN
cana-2085	343	33	3	3	NUM
cana-2085	343	34	)	)	PUNCT
cana-2085	343	35	.	.	PUNCT
cana-2085	344	1	then	then	ADV
cana-2085	344	2	either	either	CCONJ
cana-2085	344	3	𝑛𝑗	𝑛𝑗	ADP
cana-2085	344	4	≡	≡	PROPN
cana-2085	344	5	0	0	NUM
cana-2085	344	6	,	,	PUNCT
cana-2085	344	7	(	(	PUNCT
cana-2085	344	8	𝑚𝑜𝑑	𝑚𝑜𝑑	NOUN
cana-2085	344	9	3	3	NUM
cana-2085	344	10	)	)	PUNCT
cana-2085	344	11	or	or	CCONJ
cana-2085	344	12	𝑛𝑗	𝑛𝑗	ADP
cana-2085	344	13	≡	≡	PROPN
cana-2085	344	14	2	2	NUM
cana-2085	344	15	,	,	PUNCT
cana-2085	344	16	(	(	PUNCT
cana-2085	344	17	𝑚𝑜𝑑	𝑚𝑜𝑑	NOUN
cana-2085	344	18	3	3	NUM
cana-2085	344	19	)	)	PUNCT
cana-2085	344	20	.	.	PUNCT
cana-2085	345	1	in	in	ADP
cana-2085	345	2	both	both	DET
cana-2085	345	3	cases	case	NOUN
cana-2085	345	4	𝑢	𝑢	NOUN
cana-2085	345	5	will	will	AUX
cana-2085	345	6	become	become	VERB
cana-2085	345	7	an	an	DET
cana-2085	345	8	artificial	artificial	ADJ
cana-2085	345	9	super	super	ADV
cana-2085	345	10	strong	strong	ADJ
cana-2085	345	11	support	support	NOUN
cana-2085	345	12	which	which	PRON
cana-2085	345	13	is	be	AUX
cana-2085	345	14	a	a	DET
cana-2085	345	15	contradiction	contradiction	NOUN
cana-2085	345	16	to	to	ADP
cana-2085	345	17	the	the	DET
cana-2085	345	18	artificial	artificial	ADJ
cana-2085	345	19	moderate	moderate	ADJ
cana-2085	345	20	support	support	NOUN
cana-2085	345	21	𝑢𝑟.	𝑢𝑟.	NOUN
cana-2085	345	22	sub	sub	NOUN
cana-2085	345	23	case	case	NOUN
cana-2085	345	24	(	(	PUNCT
cana-2085	345	25	ii	ii	NOUN
cana-2085	345	26	):	):	PUNCT
cana-2085	345	27	𝑛𝑗	𝑛𝑗	ADP
cana-2085	345	28	≡	≡	PROPN
cana-2085	345	29	1(𝑚𝑜𝑑	1(𝑚𝑜𝑑	NOUN
cana-2085	345	30	3	3	X
cana-2085	345	31	)	)	PUNCT
cana-2085	345	32	suppose	suppose	VERB
cana-2085	345	33	𝑛𝑖	𝑛𝑖	PROPN
cana-2085	345	34	≢	≢	NOUN
cana-2085	345	35	0	0	NUM
cana-2085	345	36	,	,	PUNCT
cana-2085	345	37	2(𝑚𝑜𝑑	2(𝑚𝑜𝑑	NOUN
cana-2085	345	38	3	3	NUM
cana-2085	345	39	)	)	PUNCT
cana-2085	345	40	.	.	PUNCT
cana-2085	346	1	then	then	ADV
cana-2085	346	2	𝑛𝑖	𝑛𝑖	PROPN
cana-2085	346	3	≡	≡	PROPN
cana-2085	346	4	1(𝑚𝑜𝑑	1(𝑚𝑜𝑑	ADJ
cana-2085	346	5	3	3	X
cana-2085	346	6	)	)	PUNCT
cana-2085	346	7	legions	legion	NOUN
cana-2085	346	8	are	be	AUX
cana-2085	346	9	posted	post	VERB
cana-2085	346	10	at	at	ADP
cana-2085	346	11	𝑢	𝑢	X
cana-2085	346	12	,	,	PUNCT
cana-2085	346	13	it	it	PRON
cana-2085	346	14	can	can	AUX
cana-2085	346	15	safeguard	safeguard	VERB
cana-2085	346	16	only	only	ADV
cana-2085	346	17	two	two	NUM
cana-2085	346	18	vertices	vertex	NOUN
cana-2085	346	19	𝑢	𝑢	NOUN
cana-2085	346	20	and	and	CCONJ
cana-2085	346	21	the	the	DET
cana-2085	346	22	corresponding	corresponding	ADJ
cana-2085	346	23	leaf	leaf	NOUN
cana-2085	346	24	vertices	vertex	NOUN
cana-2085	346	25	,	,	PUNCT
cana-2085	346	26	where	where	SCONJ
cana-2085	346	27	as	as	SCONJ
cana-2085	346	28	by	by	ADP
cana-2085	346	29	definition	definition	NOUN
cana-2085	346	30	two	two	NUM
cana-2085	346	31	legions	legion	NOUN
cana-2085	346	32	placed	place	VERB
cana-2085	346	33	at	at	ADP
cana-2085	346	34	a	a	DET
cana-2085	346	35	region	region	NOUN
cana-2085	346	36	can	can	AUX
cana-2085	346	37	safeguard	safeguard	VERB
cana-2085	346	38	at	at	ADP
cana-2085	346	39	most	most	ADV
cana-2085	346	40	two	two	NUM
cana-2085	346	41	adjacent	adjacent	ADJ
cana-2085	346	42	regions	region	NOUN
cana-2085	346	43	.	.	PUNCT
cana-2085	347	1	hence	hence	ADV
cana-2085	347	2	the	the	DET
cana-2085	347	3	middle	middle	ADJ
cana-2085	347	4	roman	roman	ADJ
cana-2085	347	5	domination	domination	NOUN
cana-2085	347	6	number	number	NOUN
cana-2085	347	7	in	in	ADP
cana-2085	347	8	this	this	DET
cana-2085	347	9	case	case	NOUN
cana-2085	347	10	may	may	AUX
cana-2085	347	11	not	not	PART
cana-2085	347	12	be	be	AUX
cana-2085	347	13	minimum	minimum	ADJ
cana-2085	347	14	.	.	PUNCT
cana-2085	348	1	hence	hence	ADV
cana-2085	348	2	,	,	PUNCT
cana-2085	348	3	𝑛𝑖	𝑛𝑖	PROPN
cana-2085	348	4	≡	≡	PROPN
cana-2085	348	5	0,2(𝑚𝑜𝑑	0,2(𝑚𝑜𝑑	PUNCT
cana-2085	348	6	3	3	NUM
cana-2085	348	7	)	)	PUNCT
cana-2085	348	8	.	.	PUNCT
cana-2085	349	1	case	case	NOUN
cana-2085	349	2	(	(	PUNCT
cana-2085	349	3	ii	ii	NOUN
cana-2085	349	4	)	)	PUNCT
cana-2085	349	5	both	both	PRON
cana-2085	349	6	,	,	PUNCT
cana-2085	349	7	𝑢𝑟−1	𝑢𝑟−1	NOUN
cana-2085	349	8	and	and	CCONJ
cana-2085	349	9	𝑢𝑟+1	𝑢𝑟+1	PROPN
cana-2085	349	10	are	be	AUX
cana-2085	349	11	moderate	moderate	ADJ
cana-2085	349	12	supports	support	NOUN
cana-2085	349	13	claim	claim	NOUN
cana-2085	349	14	:	:	PUNCT
cana-2085	349	15	𝑛𝑖	𝑛𝑖	PROPN
cana-2085	349	16	≡	≡	PROPN
cana-2085	349	17	1,2(𝑚𝑜𝑑	1,2(𝑚𝑜𝑑	NUM
cana-2085	349	18	3	3	NUM
cana-2085	349	19	)	)	PUNCT
cana-2085	349	20	and	and	CCONJ
cana-2085	349	21	𝑛𝑗	𝑛𝑗	ADP
cana-2085	349	22	≡	≡	PROPN
cana-2085	349	23	0(𝑚𝑜𝑑	0(𝑚𝑜𝑑	ADJ
cana-2085	349	24	3	3	NUM
cana-2085	349	25	)	)	PUNCT
cana-2085	349	26	or	or	CCONJ
cana-2085	349	27	vice	vice	NOUN
cana-2085	349	28	versa	versa	ADV
cana-2085	349	29	.	.	PUNCT
cana-2085	350	1	subcase	subcase	NOUN
cana-2085	350	2	(	(	PUNCT
cana-2085	350	3	a	a	X
cana-2085	350	4	):	):	PUNCT
cana-2085	350	5	suppose	suppose	PROPN
cana-2085	350	6	𝑛𝑖	𝑛𝑖	PROPN
cana-2085	350	7	≡	≡	PROPN
cana-2085	350	8	1,2(𝑚𝑜𝑑	1,2(𝑚𝑜𝑑	NUM
cana-2085	350	9	3	3	NUM
cana-2085	350	10	)	)	PUNCT
cana-2085	350	11	and	and	CCONJ
cana-2085	350	12	𝑛𝑗	𝑛𝑗	ADP
cana-2085	350	13	≢	≢	NUM
cana-2085	350	14	0(𝑚𝑜𝑑	0(𝑚𝑜𝑑	ADJ
cana-2085	350	15	3	3	NUM
cana-2085	350	16	)	)	PUNCT
cana-2085	350	17	then	then	ADV
cana-2085	350	18	𝑛𝑗	𝑛𝑗	ADP
cana-2085	350	19	≡	≡	PROPN
cana-2085	350	20	1	1	NUM
cana-2085	350	21	,	,	PUNCT
cana-2085	350	22	2(𝑚𝑜𝑑	2(𝑚𝑜𝑑	NOUN
cana-2085	350	23	3	3	NUM
cana-2085	350	24	)	)	PUNCT
cana-2085	350	25	.	.	PUNCT
cana-2085	351	1	then	then	ADV
cana-2085	351	2	in	in	ADP
cana-2085	351	3	both	both	DET
cana-2085	351	4	cases	case	NOUN
cana-2085	351	5	𝑢	𝑢	NOUN
cana-2085	351	6	will	will	AUX
cana-2085	351	7	become	become	VERB
cana-2085	351	8	artificial	artificial	ADJ
cana-2085	351	9	sss	sss	NOUN
cana-2085	351	10	which	which	PRON
cana-2085	351	11	is	be	AUX
cana-2085	351	12	a	a	DET
cana-2085	351	13	contradiction	contradiction	NOUN
cana-2085	351	14	to	to	ADP
cana-2085	351	15	the	the	DET
cana-2085	351	16	artificial	artificial	ADJ
cana-2085	351	17	moderate	moderate	ADJ
cana-2085	351	18	support	support	NOUN
cana-2085	351	19	𝑢𝑟.	𝑢𝑟.	ADP
cana-2085	351	20	hence	hence	ADV
cana-2085	351	21	,	,	PUNCT
cana-2085	351	22	the	the	DET
cana-2085	351	23	claim	claim	NOUN
cana-2085	351	24	.	.	PUNCT
cana-2085	352	1	subcase	subcase	PROPN
cana-2085	352	2	(	(	PUNCT
cana-2085	352	3	iii	iii	NOUN
cana-2085	352	4	):	):	PUNCT
cana-2085	352	5	𝑢𝑟−1	𝑢𝑟−1	NOUN
cana-2085	352	6	is	be	AUX
cana-2085	352	7	a	a	DET
cana-2085	352	8	moderate	moderate	ADJ
cana-2085	352	9	support	support	NOUN
cana-2085	352	10	and	and	CCONJ
cana-2085	352	11	𝑢𝑟+1	𝑢𝑟+1	PROPN
cana-2085	352	12	is	be	AUX
cana-2085	352	13	a	a	DET
cana-2085	352	14	weak	weak	ADJ
cana-2085	352	15	support	support	NOUN
cana-2085	352	16	.	.	PUNCT
cana-2085	353	1	claim	claim	NOUN
cana-2085	353	2	:	:	PUNCT
cana-2085	353	3	𝑛𝑖	𝑛𝑖	PROPN
cana-2085	353	4	≡	≡	PROPN
cana-2085	353	5	1,2(𝑚𝑜𝑑	1,2(𝑚𝑜𝑑	NUM
cana-2085	353	6	3	3	NUM
cana-2085	353	7	)	)	PUNCT
cana-2085	353	8	and	and	CCONJ
cana-2085	353	9	𝑛𝑗	𝑛𝑗	ADP
cana-2085	353	10	≢	≢	NUM
cana-2085	353	11	1(𝑚𝑜𝑑	1(𝑚𝑜𝑑	NOUN
cana-2085	353	12	3	3	NUM
cana-2085	353	13	)	)	PUNCT
cana-2085	353	14	.	.	PUNCT
cana-2085	354	1	then	then	ADV
cana-2085	354	2	𝑛𝑗	𝑛𝑗	ADP
cana-2085	354	3	≡	≡	PROPN
cana-2085	354	4	0(𝑚𝑜𝑑	0(𝑚𝑜𝑑	ADJ
cana-2085	354	5	3	3	NUM
cana-2085	354	6	)	)	PUNCT
cana-2085	354	7	or	or	CCONJ
cana-2085	354	8	𝑛𝑗	𝑛𝑗	ADP
cana-2085	354	9	≡	≡	PROPN
cana-2085	354	10	2(𝑚𝑜𝑑	2(𝑚𝑜𝑑	NOUN
cana-2085	354	11	3	3	NUM
cana-2085	354	12	)	)	PUNCT
cana-2085	354	13	.	.	PUNCT
cana-2085	355	1	in	in	ADP
cana-2085	355	2	both	both	CCONJ
cana-2085	355	3	the	the	DET
cana-2085	355	4	cases	case	NOUN
cana-2085	355	5	,	,	PUNCT
cana-2085	355	6	𝑢𝑟	𝑢𝑟	ADV
cana-2085	355	7	will	will	AUX
cana-2085	355	8	become	become	VERB
cana-2085	355	9	artificial	artificial	ADJ
cana-2085	355	10	sss	sss	NOUN
cana-2085	355	11	a	a	DET
cana-2085	355	12	contradiction	contradiction	NOUN
cana-2085	355	13	to	to	ADP
cana-2085	355	14	the	the	DET
cana-2085	355	15	artificial	artificial	ADJ
cana-2085	355	16	moderate	moderate	ADJ
cana-2085	355	17	support	support	NOUN
cana-2085	355	18	𝑢𝑟.	𝑢𝑟.	VERB
cana-2085	355	19	𝑛𝑗	𝑛𝑗	ADP
cana-2085	355	20	≡	≡	PROPN
cana-2085	355	21	2(𝑚𝑜𝑑	2(𝑚𝑜𝑑	NOUN
cana-2085	355	22	3	3	NUM
cana-2085	355	23	)	)	PUNCT
cana-2085	355	24	,	,	PUNCT
cana-2085	355	25	subcase	subcase	NOUN
cana-2085	355	26	(	(	PUNCT
cana-2085	355	27	b	b	NOUN
cana-2085	355	28	):	):	PUNCT
cana-2085	355	29	𝑛𝑗	𝑛𝑗	ADP
cana-2085	355	30	≡	≡	PROPN
cana-2085	355	31	1(𝑚𝑜𝑑	1(𝑚𝑜𝑑	ADJ
cana-2085	355	32	3	3	NUM
cana-2085	355	33	)	)	PUNCT
cana-2085	355	34	and	and	CCONJ
cana-2085	355	35	𝑛𝑖	𝑛𝑖	PROPN
cana-2085	355	36	≢	≢	NUM
cana-2085	355	37	1,2(𝑚𝑜𝑑	1,2(𝑚𝑜𝑑	NUM
cana-2085	355	38	3	3	NUM
cana-2085	355	39	)	)	PUNCT
cana-2085	355	40	suppose	suppose	VERB
cana-2085	355	41	not	not	PART
cana-2085	355	42	.	.	PUNCT
cana-2085	356	1	let	let	VERB
cana-2085	356	2	𝑛𝑗	𝑛𝑗	ADP
cana-2085	356	3	≡	≡	PROPN
cana-2085	356	4	1(𝑚𝑜𝑑	1(𝑚𝑜𝑑	ADJ
cana-2085	356	5	3	3	NUM
cana-2085	356	6	)	)	PUNCT
cana-2085	356	7	and	and	CCONJ
cana-2085	356	8	𝑛𝑖	𝑛𝑖	PRON
cana-2085	356	9	≡	≡	PROPN
cana-2085	356	10	0(𝑚𝑜𝑑	0(𝑚𝑜𝑑	X
cana-2085	356	11	3	3	NUM
cana-2085	356	12	)	)	PUNCT
cana-2085	356	13	.	.	PUNCT
cana-2085	357	1	then	then	ADV
cana-2085	357	2	the	the	DET
cana-2085	357	3	two	two	NUM
cana-2085	357	4	legions	legion	NOUN
cana-2085	357	5	placed	place	VERB
cana-2085	357	6	at	at	ADP
cana-2085	357	7	𝑢𝑟	𝑢𝑟	ADP
cana-2085	357	8	protect	protect	VERB
cana-2085	357	9	only	only	ADV
cana-2085	357	10	two	two	NUM
cana-2085	357	11	regions	region	NOUN
cana-2085	357	12	whereas	whereas	SCONJ
cana-2085	357	13	,	,	PUNCT
cana-2085	357	14	by	by	ADP
cana-2085	357	15	definition	definition	NOUN
cana-2085	357	16	two	two	NUM
cana-2085	357	17	legions	legion	NOUN
cana-2085	357	18	can	can	AUX
cana-2085	357	19	protect	protect	VERB
cana-2085	357	20	at	at	ADP
cana-2085	357	21	most	most	ADV
cana-2085	357	22	three	three	NUM
cana-2085	357	23	regions	region	NOUN
cana-2085	357	24	which	which	PRON
cana-2085	357	25	may	may	AUX
cana-2085	357	26	or	or	CCONJ
cana-2085	357	27	may	may	AUX
cana-2085	357	28	not	not	PART
cana-2085	357	29	minimize	minimize	VERB
cana-2085	357	30	the	the	DET
cana-2085	357	31	middle	middle	ADJ
cana-2085	357	32	roman	roman	ADJ
cana-2085	357	33	domination	domination	NOUN
cana-2085	357	34	number	number	NOUN
cana-2085	357	35	.	.	PUNCT
cana-2085	358	1	hence	hence	ADV
cana-2085	358	2	,	,	PUNCT
cana-2085	358	3	𝑛𝑖	𝑛𝑖	PROPN
cana-2085	358	4	≡	≡	PROPN
cana-2085	358	5	0(𝑚𝑜𝑑	0(𝑚𝑜𝑑	X
cana-2085	358	6	3	3	NUM
cana-2085	358	7	)	)	PUNCT
cana-2085	358	8	.	.	PUNCT
cana-2085	359	1	similarly	similarly	ADV
cana-2085	359	2	,	,	PUNCT
cana-2085	359	3	we	we	PRON
cana-2085	359	4	can	can	AUX
cana-2085	359	5	bring	bring	VERB
cana-2085	359	6	𝑛𝑖	𝑛𝑖	PRON
cana-2085	359	7	≡	≡	PROPN
cana-2085	359	8	1	1	NUM
cana-2085	359	9	,	,	PUNCT
cana-2085	359	10	2(𝑚𝑜𝑑	2(𝑚𝑜𝑑	NOUN
cana-2085	359	11	3	3	NUM
cana-2085	359	12	)	)	PUNCT
cana-2085	359	13	and	and	CCONJ
cana-2085	359	14	𝑛𝑗	𝑛𝑗	ADP
cana-2085	359	15	≢	≢	NUM
cana-2085	359	16	1	1	NUM
cana-2085	359	17	(	(	PUNCT
cana-2085	359	18	𝑚𝑜𝑑	𝑚𝑜𝑑	NOUN
cana-2085	359	19	3	3	NUM
cana-2085	359	20	)	)	PUNCT
cana-2085	359	21	.	.	PUNCT
cana-2085	360	1	hence	hence	ADV
cana-2085	360	2	the	the	DET
cana-2085	360	3	claim	claim	NOUN
cana-2085	360	4	.	.	PUNCT
cana-2085	361	1	let	let	VERB
cana-2085	361	2	𝐶1	𝐶1	PRON
cana-2085	361	3	,	,	PUNCT
cana-2085	361	4	𝐶2	𝐶2	ADJ
cana-2085	361	5	,	,	PUNCT
cana-2085	361	6	…	…	PUNCT
cana-2085	361	7	𝐶𝑛𝑖	𝐶𝑛𝑖	PROPN
cana-2085	361	8	be	be	VERB
cana-2085	361	9	the	the	DET
cana-2085	361	10	support	support	NOUN
cana-2085	361	11	vertices	vertex	NOUN
cana-2085	361	12	of	of	ADP
cana-2085	361	13	ƈ𝑛𝑖.	ƈ𝑛𝑖.	X
cana-2085	361	14	if	if	SCONJ
cana-2085	361	15	𝑛𝑖	𝑛𝑖	PROPN
cana-2085	361	16	is	be	AUX
cana-2085	361	17	odd	odd	ADJ
cana-2085	361	18	,	,	PUNCT
cana-2085	361	19	then	then	ADV
cana-2085	361	20	omit	omit	VERB
cana-2085	361	21	𝐶1	𝐶1	PRON
cana-2085	361	22	or	or	CCONJ
cana-2085	361	23	𝐶𝑗	𝐶𝑗	PROPN
cana-2085	361	24	according	accord	VERB
cana-2085	361	25	to	to	ADP
cana-2085	361	26	the	the	DET
cana-2085	361	27	following	follow	VERB
cana-2085	361	28	(	(	PUNCT
cana-2085	361	29	i	i	NOUN
cana-2085	361	30	)	)	PUNCT
cana-2085	361	31	if	if	SCONJ
cana-2085	361	32	𝑢𝑟−1	𝑢𝑟−1	NOUN
cana-2085	361	33	is	be	AUX
cana-2085	361	34	a	a	DET
cana-2085	361	35	moderate	moderate	ADJ
cana-2085	361	36	support	support	NOUN
cana-2085	361	37	,	,	PUNCT
cana-2085	361	38	then	then	ADV
cana-2085	361	39	𝑛𝑖	𝑛𝑖	PROPN
cana-2085	361	40	≡	≡	PROPN
cana-2085	361	41	1,2(𝑚𝑜𝑑	1,2(𝑚𝑜𝑑	NUM
cana-2085	361	42	3	3	NUM
cana-2085	361	43	)	)	PUNCT
cana-2085	361	44	communications	communication	NOUN
cana-2085	361	45	on	on	ADP
cana-2085	361	46	applied	apply	VERB
cana-2085	361	47	nonlinear	nonlinear	ADJ
cana-2085	361	48	analysis	analysis	NOUN
cana-2085	361	49	issn	issn	NOUN
cana-2085	361	50	:	:	PUNCT
cana-2085	361	51	1074	1074	NUM
cana-2085	361	52	-	-	PUNCT
cana-2085	361	53	133x	133x	NUM
cana-2085	361	54	vol	vol	NOUN
cana-2085	361	55	32	32	NUM
cana-2085	361	56	no	no	NOUN
cana-2085	361	57	.	.	PUNCT
cana-2085	362	1	1s	1s	NUM
cana-2085	362	2	(	(	PUNCT
cana-2085	362	3	2025	2025	NUM
cana-2085	362	4	)	)	PUNCT
cana-2085	362	5	28	28	NUM
cana-2085	363	1	https://internationalpubls.com	https://internationalpubls.com	X
cana-2085	363	2	(	(	PUNCT
cana-2085	363	3	ii	ii	NOUN
cana-2085	363	4	)	)	PUNCT
cana-2085	363	5	if	if	SCONJ
cana-2085	363	6	𝑢𝑟−1	𝑢𝑟−1	NOUN
cana-2085	363	7	is	be	AUX
cana-2085	363	8	a	a	DET
cana-2085	363	9	weak	weak	ADJ
cana-2085	363	10	support	support	NOUN
cana-2085	363	11	,	,	PUNCT
cana-2085	363	12	then	then	ADV
cana-2085	363	13	𝑛𝑖	𝑛𝑖	PROPN
cana-2085	363	14	≡	≡	PROPN
cana-2085	363	15	0,2(𝑚𝑜𝑑	0,2(𝑚𝑜𝑑	PUNCT
cana-2085	363	16	3	3	X
cana-2085	363	17	)	)	PUNCT
cana-2085	363	18	(	(	PUNCT
cana-2085	363	19	iii	iii	X
cana-2085	363	20	)	)	PUNCT
cana-2085	363	21	if	if	SCONJ
cana-2085	363	22	𝑢𝑟+1	𝑢𝑟+1	PROPN
cana-2085	363	23	is	be	AUX
cana-2085	363	24	a	a	DET
cana-2085	363	25	moderate	moderate	ADJ
cana-2085	363	26	support	support	NOUN
cana-2085	363	27	,	,	PUNCT
cana-2085	363	28	then	then	ADV
cana-2085	363	29	omit	omit	VERB
cana-2085	363	30	𝐶𝑛	𝐶𝑛	PROPN
cana-2085	363	31	if	if	SCONJ
cana-2085	363	32	𝑛𝑗	𝑛𝑗	ADP
cana-2085	363	33	≡	≡	PROPN
cana-2085	363	34	1,2(𝑚𝑜𝑑	1,2(𝑚𝑜𝑑	NUM
cana-2085	363	35	3	3	NUM
cana-2085	363	36	)	)	PUNCT
cana-2085	363	37	(	(	PUNCT
cana-2085	363	38	iv	iv	X
cana-2085	363	39	)	)	PUNCT
cana-2085	363	40	if	if	SCONJ
cana-2085	363	41	𝑢𝑟+1	𝑢𝑟+1	PROPN
cana-2085	363	42	is	be	AUX
cana-2085	363	43	a	a	DET
cana-2085	363	44	weak	weak	ADJ
cana-2085	363	45	support	support	NOUN
cana-2085	363	46	,	,	PUNCT
cana-2085	363	47	then	then	ADV
cana-2085	363	48	omit	omit	VERB
cana-2085	363	49	𝐶𝑛	𝐶𝑛	PROPN
cana-2085	363	50	𝑛𝑗	𝑛𝑗	ADP
cana-2085	363	51	≡	≡	PROPN
cana-2085	363	52	0,2(𝑚𝑜𝑑	0,2(𝑚𝑜𝑑	PUNCT
cana-2085	364	1	3	3	X
cana-2085	364	2	)	)	PUNCT
cana-2085	364	3	let	let	VERB
cana-2085	364	4	ʂ	ʂ	X
cana-2085	364	5	=	=	NOUN
cana-2085	364	6	{	{	PUNCT
cana-2085	364	7	𝑆𝑚1	𝑆𝑚1	NOUN
cana-2085	364	8	,	,	PUNCT
cana-2085	364	9	𝑆𝑚1	𝑆𝑚1	NOUN
cana-2085	364	10	,	,	PUNCT
cana-2085	364	11	…	…	PUNCT
cana-2085	364	12	,	,	PUNCT
cana-2085	364	13	𝑆𝑚𝑟	𝑆𝑚𝑟	PROPN
cana-2085	364	14	}	}	PUNCT
cana-2085	364	15	be	be	VERB
cana-2085	364	16	the	the	DET
cana-2085	364	17	sss	sss	NOUN
cana-2085	364	18	chain	chain	NOUN
cana-2085	364	19	in	in	ADP
cana-2085	364	20	𝑇	𝑇	PROPN
cana-2085	364	21	and	and	CCONJ
cana-2085	364	22	with	with	ADP
cana-2085	364	23	each	each	DET
cana-2085	364	24	chain	chain	NOUN
cana-2085	364	25	𝑆𝑚𝑖	𝑆𝑚𝑖	PROPN
cana-2085	364	26	,	,	PUNCT
cana-2085	364	27	𝑖	𝑖	NOUN
cana-2085	364	28	=	=	SYM
cana-2085	364	29	1	1	NUM
cana-2085	364	30	,	,	PUNCT
cana-2085	364	31	2	2	NUM
cana-2085	364	32	,	,	PUNCT
cana-2085	364	33	3	3	NUM
cana-2085	364	34	,	,	PUNCT
cana-2085	364	35	…	…	PUNCT
cana-2085	364	36	,	,	PUNCT
cana-2085	364	37	𝑚𝑟	𝑚𝑟	PROPN
cana-2085	364	38	has	have	VERB
cana-2085	364	39	𝑚𝑖	𝑚𝑖	PART
cana-2085	364	40	sss	sss	VERB
cana-2085	364	41	.	.	PUNCT
cana-2085	365	1	let	let	VERB
cana-2085	365	2	𝑥	𝑥	DET
cana-2085	365	3	∈	∈	VERB
cana-2085	365	4	𝑁(𝑆𝑚1	𝑁(𝑆𝑚1	NOUN
cana-2085	365	5	)	)	PUNCT
cana-2085	365	6	and	and	CCONJ
cana-2085	365	7	𝑦	𝑦	NOUN
cana-2085	365	8	∈	∈	PROPN
cana-2085	365	9	𝑁(𝑆𝑚𝑟	𝑁(𝑆𝑚𝑟	PROPN
cana-2085	365	10	)	)	PUNCT
cana-2085	365	11	,	,	PUNCT
cana-2085	365	12	𝑆	𝑆	PROPN
cana-2085	365	13	=	=	SYM
cana-2085	365	14	{	{	PUNCT
cana-2085	365	15	𝑠1	𝑠1	PROPN
cana-2085	365	16	,	,	PUNCT
cana-2085	365	17	𝑠2	𝑠2	NOUN
cana-2085	365	18	,	,	PUNCT
cana-2085	365	19	𝑠3	𝑠3	NOUN
cana-2085	365	20	,	,	PUNCT
cana-2085	365	21	…	…	PUNCT
cana-2085	365	22	,	,	PUNCT
cana-2085	365	23	𝑠𝑛	𝑠𝑛	NOUN
cana-2085	365	24	}	}	PUNCT
cana-2085	365	25	be	be	AUX
cana-2085	365	26	the	the	DET
cana-2085	365	27	sss	sss	NOUN
cana-2085	365	28	or	or	CCONJ
cana-2085	365	29	asss	asss	NOUN
cana-2085	365	30	of	of	ADP
cana-2085	365	31	𝑇	𝑇	PROPN
cana-2085	365	32	,	,	PUNCT
cana-2085	365	33	ƈ	ƈ	PROPN
cana-2085	365	34	=	=	NOUN
cana-2085	365	35	{	{	PUNCT
cana-2085	365	36	ƈ𝑛1,ƈ𝑛2,ƈ𝑛3	ƈ𝑛1,ƈ𝑛2,ƈ𝑛3	NOUN
cana-2085	365	37	,	,	PUNCT
cana-2085	365	38	…	…	PUNCT
cana-2085	365	39	,	,	PUNCT
cana-2085	365	40	ƈ𝑛𝑝	ƈ𝑛𝑝	VERB
cana-2085	365	41	}	}	PUNCT
cana-2085	365	42	be	be	VERB
cana-2085	365	43	the	the	DET
cana-2085	365	44	comb	comb	NOUN
cana-2085	365	45	graph	graph	NOUN
cana-2085	365	46	in	in	ADP
cana-2085	365	47	𝑇	𝑇	PROPN
cana-2085	365	48	,	,	PUNCT
cana-2085	365	49	where	where	SCONJ
cana-2085	365	50	𝑛𝑗	𝑛𝑗	ADV
cana-2085	365	51	,	,	PUNCT
cana-2085	365	52	𝑗	𝑗	NOUN
cana-2085	365	53	=	=	SYM
cana-2085	365	54	1	1	NUM
cana-2085	365	55	,	,	PUNCT
cana-2085	365	56	2	2	NUM
cana-2085	365	57	,	,	PUNCT
cana-2085	365	58	3	3	NUM
cana-2085	365	59	,	,	PUNCT
cana-2085	365	60	…	…	PUNCT
cana-2085	365	61	,	,	PUNCT
cana-2085	365	62	𝑝	𝑝	NOUN
cana-2085	365	63	are	be	AUX
cana-2085	365	64	the	the	DET
cana-2085	365	65	even	even	ADJ
cana-2085	365	66	number	number	NOUN
cana-2085	365	67	of	of	ADP
cana-2085	365	68	supports	support	NOUN
cana-2085	365	69	in	in	ADP
cana-2085	365	70	ƈ𝑗.	ƈ𝑗.	NOUN
cana-2085	365	71	let	let	VERB
cana-2085	365	72	𝑇1	𝑇1	NOUN
cana-2085	365	73	=	=	SYM
cana-2085	365	74	𝑇	𝑇	PROPN
cana-2085	365	75	−	−	PROPN
cana-2085	366	1	[	[	X
cana-2085	366	2	𝑁[𝑆	𝑁[𝑆	X
cana-2085	366	3	]	]	PUNCT
cana-2085	366	4	∪	∪	ADP
cana-2085	366	5	𝑁[𝑠	𝑁[𝑠	X
cana-2085	366	6	]	]	PUNCT
cana-2085	366	7	∪	∪	ADP
cana-2085	366	8	𝑁[ƈ	𝑁[ƈ	NOUN
cana-2085	366	9	]	]	X
cana-2085	366	10	−	−	PROPN
cana-2085	366	11	{	{	PUNCT
cana-2085	366	12	𝑥	𝑥	NOUN
cana-2085	366	13	,	,	PUNCT
cana-2085	366	14	𝑦	𝑦	NOUN
cana-2085	366	15	𝑥	𝑥	X
cana-2085	366	16	∈	∈	PROPN
cana-2085	366	17	𝑁(𝑐1	𝑁(𝑐1	PROPN
cana-2085	366	18	)	)	PUNCT
cana-2085	366	19	,	,	PUNCT
cana-2085	366	20	𝑦	𝑦	NOUN
cana-2085	366	21	∈	∈	NOUN
cana-2085	366	22	𝑁(𝑐𝑛	𝑁(𝑐𝑛	NOUN
cana-2085	366	23	)	)	PUNCT
cana-2085	366	24	}	}	PUNCT
cana-2085	366	25	]	]	PUNCT
cana-2085	366	26	∪	∪	ADP
cana-2085	366	27	𝐿	𝐿	PROPN
cana-2085	366	28	∪	∪	NOUN
cana-2085	366	29	𝐴.	𝐴.	PROPN
cana-2085	366	30	where	where	SCONJ
cana-2085	366	31	,	,	PUNCT
cana-2085	366	32	𝐿	𝐿	PROPN
cana-2085	366	33	=	=	SYM
cana-2085	366	34	{	{	PUNCT
cana-2085	366	35	𝑙	𝑙	NOUN
cana-2085	366	36	𝑙	𝑙	X
cana-2085	366	37	𝑖𝑠	𝑖𝑠	NUM
cana-2085	366	38	𝑎	𝑎	PROPN
cana-2085	366	39	𝑙𝑒𝑎𝑓	𝑙𝑒𝑎𝑓	NOUN
cana-2085	366	40	𝑣𝑒𝑟𝑡𝑒𝑥	𝑣𝑒𝑟𝑡𝑒𝑥	NOUN
cana-2085	366	41	𝑖𝑛	𝑖𝑛	NOUN
cana-2085	366	42	𝑁(𝑚𝑘	𝑁(𝑚𝑘	PROPN
cana-2085	366	43	)	)	PUNCT
cana-2085	366	44	,	,	PUNCT
cana-2085	366	45	𝑘	𝑘	X
cana-2085	366	46	=	=	SYM
cana-2085	366	47	1	1	NUM
cana-2085	366	48	,	,	PUNCT
cana-2085	366	49	2	2	NUM
cana-2085	366	50	,	,	PUNCT
cana-2085	366	51	3	3	NUM
cana-2085	366	52	,	,	PUNCT
cana-2085	366	53	…	…	PUNCT
cana-2085	366	54	,	,	PUNCT
cana-2085	366	55	𝑝	𝑝	NOUN
cana-2085	366	56	}	}	PUNCT
cana-2085	366	57	,	,	PUNCT
cana-2085	366	58	𝐴	𝐴	PROPN
cana-2085	366	59	=	=	SYM
cana-2085	366	60	{	{	PUNCT
cana-2085	366	61	𝑁[𝑎	𝑁[𝑎	ADJ
cana-2085	366	62	]	]	X
cana-2085	366	63	−	−	X
cana-2085	366	64	{	{	PUNCT
cana-2085	366	65	ℎ	ℎ	X
cana-2085	366	66	}	}	PUNCT
cana-2085	366	67	,	,	PUNCT
cana-2085	366	68	where	where	SCONJ
cana-2085	366	69	a	a	PRON
cana-2085	366	70	is	be	AUX
cana-2085	366	71	an	an	DET
cana-2085	366	72	artificial	artificial	ADJ
cana-2085	366	73	sss	sss	NOUN
cana-2085	366	74	or	or	CCONJ
cana-2085	366	75	h	h	NOUN
cana-2085	366	76	is	be	AUX
cana-2085	366	77	the	the	DET
cana-2085	366	78	vertex	vertex	NOUN
cana-2085	366	79	on	on	ADP
cana-2085	366	80	the	the	DET
cana-2085	366	81	internal	internal	ADJ
cana-2085	366	82	path	path	NOUN
cana-2085	366	83	(	(	PUNCT
cana-2085	366	84	𝑢𝑟−1	𝑢𝑟−1	NOUN
cana-2085	366	85	,	,	PUNCT
cana-2085	366	86	𝑎	𝑎	NOUN
cana-2085	366	87	)	)	PUNCT
cana-2085	366	88	𝑜𝑟	𝑜𝑟	NOUN
cana-2085	366	89	(	(	PUNCT
cana-2085	366	90	𝑎	𝑎	X
cana-2085	366	91	,	,	PUNCT
cana-2085	366	92	𝑢𝑟−1	𝑢𝑟−1	NOUN
cana-2085	366	93	)	)	PUNCT
cana-2085	366	94	𝑤𝑖𝑡ℎ	𝑤𝑖𝑡ℎ	NOUN
cana-2085	366	95	{	{	PUNCT
cana-2085	366	96	𝑛𝑗	𝑛𝑗	ADP
cana-2085	366	97	≢	≢	NUM
cana-2085	366	98	0	0	NUM
cana-2085	366	99	,	,	PUNCT
cana-2085	366	100	2	2	NUM
cana-2085	366	101	𝑚𝑜𝑑	𝑚𝑜𝑑	NOUN
cana-2085	366	102	3	3	NUM
cana-2085	366	103	}	}	PUNCT
cana-2085	366	104	hence	hence	ADV
cana-2085	366	105	,	,	PUNCT
cana-2085	366	106	𝛾𝑀𝑅	𝛾𝑀𝑅	NOUN
cana-2085	366	107	=	=	SYM
cana-2085	366	108	3[(𝑥	3[(𝑥	NUM
cana-2085	366	109	+	+	ADJ
cana-2085	366	110	𝑎)𝑛	𝑎)𝑛	NOUN
cana-2085	366	111	]	]	X
cana-2085	367	1	+	+	CCONJ
cana-2085	367	2	[	[	X
cana-2085	367	3	∑	∑	PUNCT
cana-2085	367	4	𝑛𝑖	𝑛𝑖	PROPN
cana-2085	367	5	+	+	PROPN
cana-2085	367	6	𝑘	𝑘	PRON
cana-2085	367	7	𝑖=1	𝑖=1	PROPN
cana-2085	367	8	|𝑆1|	|𝑆1|	PROPN
cana-2085	367	9	]	]	PUNCT
cana-2085	368	1	+	+	CCONJ
cana-2085	368	2	2|𝑆2|	2|𝑆2|	NUM
cana-2085	368	3	+	+	CCONJ
cana-2085	368	4	3	3	NUM
cana-2085	368	5	4	4	NUM
cana-2085	368	6	∑	∑	ADP
cana-2085	368	7	𝑇1	𝑇1	NOUN
cana-2085	368	8	𝑙	𝑙	PRON
cana-2085	368	9	𝑗=1	𝑗=1	PROPN
cana-2085	368	10	+	+	PROPN
cana-2085	369	1	2|𝑇2|	2|𝑇2|	NUM
cana-2085	369	2	.	.	PUNCT
cana-2085	370	1	the	the	DET
cana-2085	370	2	above	above	ADJ
cana-2085	370	3	theorem	theorem	NOUN
cana-2085	370	4	has	have	VERB
cana-2085	370	5	various	various	ADJ
cana-2085	370	6	applications	application	NOUN
cana-2085	370	7	in	in	ADP
cana-2085	370	8	various	various	ADJ
cana-2085	370	9	applications	application	NOUN
cana-2085	370	10	including	include	VERB
cana-2085	370	11	network	network	NOUN
cana-2085	370	12	design	design	NOUN
cana-2085	370	13	and	and	CCONJ
cana-2085	370	14	optimization	optimization	NOUN
cana-2085	370	15	:	:	PUNCT
cana-2085	370	16	in	in	ADP
cana-2085	370	17	telecommunications	telecommunications	NOUN
cana-2085	370	18	and	and	CCONJ
cana-2085	370	19	computer	computer	NOUN
cana-2085	370	20	networks	network	NOUN
cana-2085	370	21	,	,	PUNCT
cana-2085	370	22	designing	design	VERB
cana-2085	370	23	efficient	efficient	ADJ
cana-2085	370	24	network	network	NOUN
cana-2085	370	25	topologies	topology	NOUN
cana-2085	370	26	and	and	CCONJ
cana-2085	370	27	optimizing	optimize	VERB
cana-2085	370	28	resource	resource	NOUN
cana-2085	370	29	allocation	allocation	NOUN
cana-2085	370	30	often	often	ADV
cana-2085	370	31	involves	involve	VERB
cana-2085	370	32	understanding	understand	VERB
cana-2085	370	33	the	the	DET
cana-2085	370	34	domination	domination	NOUN
cana-2085	370	35	properties	property	NOUN
cana-2085	370	36	of	of	ADP
cana-2085	370	37	the	the	DET
cana-2085	370	38	network	network	NOUN
cana-2085	370	39	.	.	PUNCT
cana-2085	371	1	the	the	DET
cana-2085	371	2	theorem	theorem	NOUN
cana-2085	371	3	can	can	AUX
cana-2085	371	4	be	be	AUX
cana-2085	371	5	used	use	VERB
cana-2085	371	6	to	to	PART
cana-2085	371	7	determine	determine	VERB
cana-2085	371	8	the	the	DET
cana-2085	371	9	optimal	optimal	ADJ
cana-2085	371	10	placement	placement	NOUN
cana-2085	371	11	of	of	ADP
cana-2085	371	12	network	network	NOUN
cana-2085	371	13	nodes	node	NOUN
cana-2085	371	14	or	or	CCONJ
cana-2085	371	15	resources	resource	NOUN
cana-2085	371	16	to	to	PART
cana-2085	371	17	achieve	achieve	VERB
cana-2085	371	18	desired	desire	VERB
cana-2085	371	19	network	network	NOUN
cana-2085	371	20	performance	performance	NOUN
cana-2085	371	21	or	or	CCONJ
cana-2085	371	22	reliability	reliability	NOUN
cana-2085	371	23	.	.	PUNCT
cana-2085	372	1	vlsi	vlsi	PROPN
cana-2085	372	2	circuit	circuit	PROPN
cana-2085	372	3	design	design	PROPN
cana-2085	372	4	:	:	PUNCT
cana-2085	372	5	in	in	ADP
cana-2085	372	6	very	very	ADV
cana-2085	372	7	-	-	PUNCT
cana-2085	372	8	largescale	largescale	ADJ
cana-2085	372	9	integration	integration	NOUN
cana-2085	372	10	(	(	PUNCT
cana-2085	372	11	vlsi	vlsi	PROPN
cana-2085	372	12	)	)	PUNCT
cana-2085	372	13	circuit	circuit	NOUN
cana-2085	372	14	design	design	NOUN
cana-2085	372	15	,	,	PUNCT
cana-2085	372	16	the	the	DET
cana-2085	372	17	problem	problem	NOUN
cana-2085	372	18	of	of	ADP
cana-2085	372	19	minimizing	minimize	VERB
cana-2085	372	20	the	the	DET
cana-2085	372	21	number	number	NOUN
cana-2085	372	22	of	of	ADP
cana-2085	372	23	components	component	NOUN
cana-2085	372	24	(	(	PUNCT
cana-2085	372	25	e.g.	e.g.	ADV
cana-2085	372	26	,	,	PUNCT
cana-2085	372	27	transistors	transistor	NOUN
cana-2085	372	28	,	,	PUNCT
cana-2085	372	29	gates	gate	NOUN
cana-2085	372	30	)	)	PUNCT
cana-2085	372	31	while	while	SCONJ
cana-2085	372	32	ensuring	ensure	VERB
cana-2085	372	33	that	that	SCONJ
cana-2085	372	34	all	all	DET
cana-2085	372	35	necessary	necessary	ADJ
cana-2085	372	36	connections	connection	NOUN
cana-2085	372	37	and	and	CCONJ
cana-2085	372	38	functionalities	functionality	NOUN
cana-2085	372	39	are	be	AUX
cana-2085	372	40	present	present	ADJ
cana-2085	372	41	can	can	AUX
cana-2085	372	42	be	be	AUX
cana-2085	372	43	modelled	model	VERB
cana-2085	372	44	using	use	VERB
cana-2085	372	45	graph	graph	NOUN
cana-2085	372	46	theory	theory	NOUN
cana-2085	372	47	.	.	PUNCT
cana-2085	373	1	the	the	DET
cana-2085	373	2	middle	middle	ADJ
cana-2085	373	3	roman	roman	ADJ
cana-2085	373	4	domination	domination	NOUN
cana-2085	373	5	number	number	NOUN
cana-2085	373	6	can	can	AUX
cana-2085	373	7	help	help	VERB
cana-2085	373	8	in	in	ADP
cana-2085	373	9	optimizing	optimize	VERB
cana-2085	373	10	circuit	circuit	NOUN
cana-2085	373	11	layouts	layout	NOUN
cana-2085	373	12	to	to	PART
cana-2085	373	13	minimize	minimize	VERB
cana-2085	373	14	power	power	NOUN
cana-2085	373	15	consumption	consumption	NOUN
cana-2085	373	16	or	or	CCONJ
cana-2085	373	17	improve	improve	VERB
cana-2085	373	18	performance	performance	NOUN
cana-2085	373	19	.	.	PUNCT
cana-2085	374	1	bioinformatics	bioinformatic	NOUN
cana-2085	374	2	:	:	PUNCT
cana-2085	374	3	in	in	ADP
cana-2085	374	4	bioinformatics	bioinformatics	NOUN
cana-2085	374	5	,	,	PUNCT
cana-2085	374	6	especially	especially	ADV
cana-2085	374	7	in	in	ADP
cana-2085	374	8	the	the	DET
cana-2085	374	9	study	study	NOUN
cana-2085	374	10	of	of	ADP
cana-2085	374	11	protein	protein	NOUN
cana-2085	374	12	-	-	PUNCT
cana-2085	374	13	protein	protein	NOUN
cana-2085	374	14	interaction	interaction	NOUN
cana-2085	374	15	networks	network	NOUN
cana-2085	374	16	or	or	CCONJ
cana-2085	374	17	gene	gene	NOUN
cana-2085	374	18	regulatory	regulatory	ADJ
cana-2085	374	19	networks	network	NOUN
cana-2085	374	20	,	,	PUNCT
cana-2085	374	21	graphtheoretic	graphtheoretic	ADJ
cana-2085	374	22	models	model	NOUN
cana-2085	374	23	are	be	AUX
cana-2085	374	24	used	use	VERB
cana-2085	374	25	to	to	PART
cana-2085	374	26	understand	understand	VERB
cana-2085	374	27	the	the	DET
cana-2085	374	28	structure	structure	NOUN
cana-2085	374	29	and	and	CCONJ
cana-2085	374	30	function	function	NOUN
cana-2085	374	31	of	of	ADP
cana-2085	374	32	biological	biological	ADJ
cana-2085	374	33	networks	network	NOUN
cana-2085	374	34	.	.	PUNCT
cana-2085	375	1	the	the	DET
cana-2085	375	2	theorem	theorem	NOUN
cana-2085	375	3	can	can	AUX
cana-2085	375	4	be	be	AUX
cana-2085	375	5	used	use	VERB
cana-2085	375	6	to	to	PART
cana-2085	375	7	analyse	analyse	VERB
cana-2085	375	8	and	and	CCONJ
cana-2085	375	9	optimize	optimize	VERB
cana-2085	375	10	these	these	DET
cana-2085	375	11	networks	network	NOUN
cana-2085	375	12	by	by	ADP
cana-2085	375	13	providing	provide	VERB
cana-2085	375	14	insights	insight	NOUN
cana-2085	375	15	into	into	ADP
cana-2085	375	16	their	their	PRON
cana-2085	375	17	dominating	dominating	NOUN
cana-2085	375	18	sets	set	NOUN
cana-2085	375	19	and	and	CCONJ
cana-2085	375	20	efficient	efficient	ADJ
cana-2085	375	21	network	network	NOUN
cana-2085	375	22	configurations	configuration	NOUN
cana-2085	375	23	.	.	PUNCT
cana-2085	376	1	urban	urban	ADJ
cana-2085	376	2	planning	planning	NOUN
cana-2085	376	3	and	and	CCONJ
cana-2085	376	4	infrastructure	infrastructure	NOUN
cana-2085	376	5	:	:	PUNCT
cana-2085	376	6	for	for	ADP
cana-2085	376	7	urban	urban	ADJ
cana-2085	376	8	planning	planning	NOUN
cana-2085	376	9	and	and	CCONJ
cana-2085	376	10	the	the	DET
cana-2085	376	11	design	design	NOUN
cana-2085	376	12	of	of	ADP
cana-2085	376	13	infrastructure	infrastructure	NOUN
cana-2085	376	14	networks	network	NOUN
cana-2085	376	15	(	(	PUNCT
cana-2085	376	16	such	such	ADJ
cana-2085	376	17	as	as	ADP
cana-2085	376	18	roads	road	NOUN
cana-2085	376	19	,	,	PUNCT
cana-2085	376	20	utilities	utility	NOUN
cana-2085	376	21	,	,	PUNCT
cana-2085	376	22	and	and	CCONJ
cana-2085	376	23	public	public	ADJ
cana-2085	376	24	services	service	NOUN
cana-2085	376	25	)	)	PUNCT
cana-2085	376	26	,	,	PUNCT
cana-2085	376	27	the	the	DET
cana-2085	376	28	theorem	theorem	NOUN
cana-2085	376	29	can	can	AUX
cana-2085	376	30	assist	assist	VERB
cana-2085	376	31	in	in	ADP
cana-2085	376	32	optimizing	optimize	VERB
cana-2085	376	33	the	the	DET
cana-2085	376	34	placement	placement	NOUN
cana-2085	376	35	of	of	ADP
cana-2085	376	36	facilities	facility	NOUN
cana-2085	376	37	and	and	CCONJ
cana-2085	376	38	services	service	NOUN
cana-2085	376	39	to	to	PART
cana-2085	376	40	ensure	ensure	VERB
cana-2085	376	41	maximum	maximum	ADJ
cana-2085	376	42	coverage	coverage	NOUN
cana-2085	376	43	and	and	CCONJ
cana-2085	376	44	efficiency	efficiency	NOUN
cana-2085	376	45	.	.	PUNCT
cana-2085	377	1	this	this	PRON
cana-2085	377	2	can	can	AUX
cana-2085	377	3	help	help	VERB
cana-2085	377	4	in	in	ADP
cana-2085	377	5	minimizing	minimize	VERB
cana-2085	377	6	costs	cost	NOUN
cana-2085	377	7	while	while	SCONJ
cana-2085	377	8	ensuring	ensure	VERB
cana-2085	377	9	adequate	adequate	ADJ
cana-2085	377	10	service	service	NOUN
cana-2085	377	11	provision	provision	NOUN
cana-2085	377	12	.	.	PUNCT
cana-2085	378	1	social	social	ADJ
cana-2085	378	2	network	network	NOUN
cana-2085	378	3	analysis	analysis	NOUN
cana-2085	378	4	:	:	PUNCT
cana-2085	378	5	in	in	ADP
cana-2085	378	6	social	social	ADJ
cana-2085	378	7	network	network	NOUN
cana-2085	378	8	analysis	analysis	NOUN
cana-2085	378	9	,	,	PUNCT
cana-2085	378	10	the	the	DET
cana-2085	378	11	middle	middle	ADJ
cana-2085	378	12	roman	roman	ADJ
cana-2085	378	13	domination	domination	NOUN
cana-2085	378	14	number	number	NOUN
cana-2085	378	15	can	can	AUX
cana-2085	378	16	be	be	AUX
cana-2085	378	17	used	use	VERB
cana-2085	378	18	to	to	PART
cana-2085	378	19	identify	identify	VERB
cana-2085	378	20	key	key	ADJ
cana-2085	378	21	individuals	individual	NOUN
cana-2085	378	22	or	or	CCONJ
cana-2085	378	23	groups	group	NOUN
cana-2085	378	24	that	that	PRON
cana-2085	378	25	influence	influence	VERB
cana-2085	378	26	the	the	DET
cana-2085	378	27	entire	entire	ADJ
cana-2085	378	28	network	network	NOUN
cana-2085	378	29	.	.	PUNCT
cana-2085	379	1	this	this	PRON
cana-2085	379	2	can	can	AUX
cana-2085	379	3	be	be	AUX
cana-2085	379	4	useful	useful	ADJ
cana-2085	379	5	in	in	ADP
cana-2085	379	6	marketing	marketing	NOUN
cana-2085	379	7	,	,	PUNCT
cana-2085	379	8	spreading	spread	VERB
cana-2085	379	9	information	information	NOUN
cana-2085	379	10	,	,	PUNCT
cana-2085	379	11	or	or	CCONJ
cana-2085	379	12	managing	manage	VERB
cana-2085	379	13	social	social	ADJ
cana-2085	379	14	dynamics	dynamic	NOUN
cana-2085	379	15	[	[	X
cana-2085	379	16	46	46	NUM
cana-2085	379	17	,	,	PUNCT
cana-2085	379	18	47	47	NUM
cana-2085	379	19	]	]	PUNCT
cana-2085	379	20	.	.	PUNCT
cana-2085	380	1	algorithm	algorithm	PROPN
cana-2085	380	2	design	design	PROPN
cana-2085	380	3	:	:	PUNCT
cana-2085	380	4	the	the	DET
cana-2085	380	5	theorem	theorem	NOUN
cana-2085	380	6	provides	provide	VERB
cana-2085	380	7	a	a	DET
cana-2085	380	8	formula	formula	NOUN
cana-2085	380	9	that	that	PRON
cana-2085	380	10	can	can	AUX
cana-2085	380	11	be	be	AUX
cana-2085	380	12	used	use	VERB
cana-2085	380	13	in	in	ADP
cana-2085	380	14	the	the	DET
cana-2085	380	15	development	development	NOUN
cana-2085	380	16	of	of	ADP
cana-2085	380	17	algorithms	algorithm	NOUN
cana-2085	380	18	for	for	ADP
cana-2085	380	19	graph	graph	NOUN
cana-2085	380	20	-	-	PUNCT
cana-2085	380	21	based	base	VERB
cana-2085	380	22	problems	problem	NOUN
cana-2085	380	23	.	.	PUNCT
cana-2085	381	1	it	it	PRON
cana-2085	381	2	can	can	AUX
cana-2085	381	3	be	be	AUX
cana-2085	381	4	applied	apply	VERB
cana-2085	381	5	to	to	PART
cana-2085	381	6	design	design	VERB
cana-2085	381	7	algorithms	algorithm	NOUN
cana-2085	381	8	for	for	ADP
cana-2085	381	9	computing	compute	VERB
cana-2085	381	10	the	the	DET
cana-2085	381	11	middle	middle	ADJ
cana-2085	381	12	roman	roman	ADJ
cana-2085	381	13	domination	domination	NOUN
cana-2085	381	14	number	number	NOUN
cana-2085	381	15	in	in	ADP
cana-2085	381	16	practical	practical	ADJ
cana-2085	381	17	scenarios	scenario	NOUN
cana-2085	381	18	where	where	SCONJ
cana-2085	381	19	such	such	ADJ
cana-2085	381	20	graphs	graph	NOUN
cana-2085	381	21	are	be	AUX
cana-2085	381	22	encountered	encounter	VERB
cana-2085	381	23	[	[	PUNCT
cana-2085	381	24	38	38	NUM
cana-2085	381	25	]	]	PUNCT
cana-2085	381	26	.	.	PUNCT
cana-2085	382	1	conclusion	conclusion	NOUN
cana-2085	382	2	:	:	PUNCT
cana-2085	382	3	the	the	DET
cana-2085	382	4	concept	concept	NOUN
cana-2085	382	5	of	of	ADP
cana-2085	382	6	a	a	DET
cana-2085	382	7	middle	middle	ADJ
cana-2085	382	8	roman	roman	ADJ
cana-2085	382	9	dominating	dominating	NOUN
cana-2085	382	10	function	function	NOUN
cana-2085	382	11	(	(	PUNCT
cana-2085	382	12	mrdf	mrdf	NOUN
cana-2085	382	13	)	)	PUNCT
cana-2085	382	14	in	in	ADP
cana-2085	382	15	graph	graph	NOUN
cana-2085	382	16	theory	theory	NOUN
cana-2085	382	17	has	have	VERB
cana-2085	382	18	wide	wide	ADV
cana-2085	382	19	-	-	PUNCT
cana-2085	382	20	ranging	range	VERB
cana-2085	382	21	applications	application	NOUN
cana-2085	382	22	in	in	ADP
cana-2085	382	23	computer	computer	NOUN
cana-2085	382	24	science	science	NOUN
cana-2085	382	25	,	,	PUNCT
cana-2085	382	26	particularly	particularly	ADV
cana-2085	382	27	in	in	ADP
cana-2085	382	28	optimizing	optimize	VERB
cana-2085	382	29	resource	resource	NOUN
cana-2085	382	30	allocation	allocation	NOUN
cana-2085	382	31	,	,	PUNCT
cana-2085	382	32	enhancing	enhance	VERB
cana-2085	382	33	network	network	NOUN
cana-2085	382	34	security	security	NOUN
cana-2085	382	35	,	,	PUNCT
cana-2085	382	36	and	and	CCONJ
cana-2085	382	37	improving	improve	VERB
cana-2085	382	38	efficiency	efficiency	NOUN
cana-2085	382	39	in	in	ADP
cana-2085	382	40	distributed	distribute	VERB
cana-2085	382	41	systems	system	NOUN
cana-2085	382	42	.	.	PUNCT
cana-2085	383	1	by	by	ADP
cana-2085	383	2	strategically	strategically	ADV
cana-2085	383	3	assigning	assign	VERB
cana-2085	383	4	values	value	NOUN
cana-2085	383	5	to	to	ADP
cana-2085	383	6	nodes	node	NOUN
cana-2085	383	7	,	,	PUNCT
cana-2085	383	8	mrdf	mrdf	NOUN
cana-2085	383	9	enables	enable	VERB
cana-2085	383	10	efficient	efficient	ADJ
cana-2085	383	11	task	task	NOUN
cana-2085	383	12	scheduling	scheduling	NOUN
cana-2085	383	13	,	,	PUNCT
cana-2085	383	14	load	load	NOUN
cana-2085	383	15	balancing	balancing	NOUN
cana-2085	383	16	,	,	PUNCT
cana-2085	383	17	and	and	CCONJ
cana-2085	383	18	energy	energy	NOUN
cana-2085	383	19	conservation	conservation	NOUN
cana-2085	383	20	in	in	ADP
cana-2085	383	21	sensor	sensor	NOUN
cana-2085	383	22	networks	network	NOUN
cana-2085	383	23	.	.	PUNCT
cana-2085	384	1	it	it	PRON
cana-2085	384	2	also	also	ADV
cana-2085	384	3	supports	support	VERB
cana-2085	384	4	resilient	resilient	ADJ
cana-2085	384	5	network	network	NOUN
cana-2085	384	6	design	design	NOUN
cana-2085	384	7	and	and	CCONJ
cana-2085	384	8	fault	fault	VERB
cana-2085	384	9	tolerance	tolerance	NOUN
cana-2085	384	10	in	in	ADP
cana-2085	384	11	communication	communication	NOUN
cana-2085	384	12	networks	network	NOUN
cana-2085	384	13	.	.	PUNCT
cana-2085	385	1	furthermore	furthermore	ADV
cana-2085	385	2	,	,	PUNCT
cana-2085	385	3	mrdf	mrdf	NOUN
cana-2085	385	4	can	can	AUX
cana-2085	385	5	be	be	AUX
cana-2085	385	6	leveraged	leverage	VERB
cana-2085	385	7	in	in	ADP
cana-2085	385	8	robotics	robotic	NOUN
cana-2085	385	9	for	for	ADP
cana-2085	385	10	path	path	NOUN
cana-2085	385	11	planning	planning	NOUN
cana-2085	385	12	and	and	CCONJ
cana-2085	385	13	resource	resource	NOUN
cana-2085	385	14	deployment	deployment	NOUN
cana-2085	385	15	,	,	PUNCT
cana-2085	385	16	and	and	CCONJ
cana-2085	385	17	in	in	ADP
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cana-2085	385	19	theory	theory	NOUN
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cana-2085	385	23	.	.	PUNCT
cana-2085	386	1	overall	overall	ADV
cana-2085	386	2	,	,	PUNCT
cana-2085	386	3	mrdf	mrdf	NOUN
cana-2085	386	4	provides	provide	VERB
cana-2085	386	5	powerful	powerful	ADJ
cana-2085	386	6	solutions	solution	NOUN
cana-2085	386	7	for	for	ADP
cana-2085	386	8	balancing	balance	VERB
cana-2085	386	9	resource	resource	NOUN
cana-2085	386	10	utilization	utilization	NOUN
cana-2085	386	11	,	,	PUNCT
cana-2085	386	12	security	security	NOUN
cana-2085	386	13	,	,	PUNCT
cana-2085	386	14	and	and	CCONJ
cana-2085	386	15	efficiency	efficiency	NOUN
cana-2085	386	16	across	across	ADP
cana-2085	386	17	various	various	ADJ
cana-2085	386	18	fields	field	NOUN
cana-2085	386	19	.	.	PUNCT
cana-2085	387	1	reference	reference	NOUN
cana-2085	387	2	:	:	PUNCT
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cana-2085	388	2	1	1	NUM
cana-2085	388	3	]	]	X
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cana-2085	388	5	,	,	PUNCT
cana-2085	388	6	i.f	i.f	PROPN
cana-2085	388	7	.	.	PROPN
cana-2085	388	8	,	,	PUNCT
cana-2085	388	9	&	&	CCONJ
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cana-2085	388	11	,	,	PUNCT
cana-2085	388	12	i.h	i.h	PROPN
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cana-2085	388	16	)	)	PUNCT
cana-2085	388	17	.	.	PUNCT
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cana-2085	388	19	sensor	sensor	NOUN
cana-2085	388	20	networks	network	NOUN
cana-2085	388	21	(	(	PUNCT
cana-2085	388	22	advanced	advanced	ADJ
cana-2085	388	23	texts	text	NOUN
cana-2085	388	24	in	in	ADP
cana-2085	388	25	communications	communication	NOUN
cana-2085	388	26	)	)	PUNCT
cana-2085	388	27	.	.	PUNCT
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cana-2085	389	2	.	.	PUNCT
cana-2085	390	1	communications	communication	NOUN
cana-2085	390	2	on	on	ADP
cana-2085	390	3	applied	apply	VERB
cana-2085	390	4	nonlinear	nonlinear	ADJ
cana-2085	390	5	analysis	analysis	NOUN
cana-2085	390	6	issn	issn	NOUN
cana-2085	390	7	:	:	PUNCT
cana-2085	390	8	1074	1074	NUM
cana-2085	390	9	-	-	PUNCT
cana-2085	390	10	133x	133x	NUM
cana-2085	390	11	vol	vol	NOUN
cana-2085	390	12	32	32	NUM
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cana-2085	390	14	.	.	PUNCT
cana-2085	391	1	1s	1s	NUM
cana-2085	391	2	(	(	PUNCT
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cana-2085	391	4	)	)	PUNCT
cana-2085	391	5	29	29	NUM
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cana-2085	392	2	2	2	NUM
cana-2085	392	3	]	]	PUNCT
cana-2085	392	4	aldred	aldre	VERB
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cana-2085	392	7	.	.	PUNCT
cana-2085	392	8	,	,	PUNCT
cana-2085	393	1	dyer	dyer	PROPN
cana-2085	393	2	,	,	PUNCT
cana-2085	393	3	d.m	d.m	PROPN
cana-2085	393	4	.	.	PROPN
cana-2085	393	5	,	,	PUNCT
cana-2085	393	6	&	&	CCONJ
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cana-2085	393	8	,	,	PUNCT
cana-2085	393	9	k.b	k.b	PROPN
cana-2085	393	10	.	.	PUNCT
cana-2085	393	11	(	(	PUNCT
cana-2085	393	12	2002	2002	NUM
cana-2085	393	13	)	)	PUNCT
cana-2085	393	14	.	.	PUNCT
cana-2085	394	1	"	"	PUNCT
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cana-2085	394	7	strategies	strategy	NOUN
cana-2085	394	8	.	.	PUNCT
cana-2085	394	9	"	"	PUNCT
cana-2085	394	10	discrete	discrete	ADJ
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cana-2085	394	12	,	,	PUNCT
cana-2085	394	13	254(1	254(1	NUM
cana-2085	394	14	-	-	SYM
cana-2085	394	15	3	3	NUM
cana-2085	394	16	)	)	PUNCT
cana-2085	394	17	,	,	PUNCT
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cana-2085	394	19	-	-	SYM
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cana-2085	394	21	.	.	PUNCT
cana-2085	395	1	[	[	X
cana-2085	395	2	3	3	NUM
cana-2085	395	3	]	]	X
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cana-2085	395	14	)	)	PUNCT
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cana-2085	396	1	"	"	PUNCT
cana-2085	396	2	network	network	NOUN
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cana-2085	396	5	:	:	PUNCT
cana-2085	396	6	a	a	DET
cana-2085	396	7	survey	survey	NOUN
cana-2085	396	8	and	and	CCONJ
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cana-2085	397	7	,	,	PUNCT
cana-2085	397	8	32(4	32(4	NUM
cana-2085	397	9	)	)	PUNCT
cana-2085	397	10	,	,	PUNCT
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cana-2085	397	12	-	-	SYM
cana-2085	397	13	414	414	NUM
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cana-2085	398	2	4	4	NUM
cana-2085	398	3	]	]	SYM
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cana-2085	398	5	,	,	PUNCT
cana-2085	398	6	n.	n.	NOUN
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cana-2085	398	8	&	&	CCONJ
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cana-2085	398	11	b.	b.	PROPN
cana-2085	398	12	(	(	PUNCT
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cana-2085	398	14	)	)	PUNCT
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cana-2085	399	1	"	"	PUNCT
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cana-2085	399	3	problems	problem	NOUN
cana-2085	399	4	in	in	ADP
cana-2085	399	5	graph	graph	NOUN
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cana-2085	399	8	"	"	PUNCT
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cana-2085	400	5	,	,	PUNCT
cana-2085	400	6	95(3	95(3	NUM
cana-2085	400	7	)	)	PUNCT
cana-2085	400	8	,	,	PUNCT
cana-2085	400	9	391	391	NUM
cana-2085	400	10	-	-	SYM
cana-2085	400	11	403	403	NUM
cana-2085	400	12	.	.	PUNCT
cana-2085	401	1	[	[	X
cana-2085	401	2	5	5	NUM
cana-2085	401	3	]	]	X
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cana-2085	401	5	,	,	PUNCT
cana-2085	401	6	h.h	h.h	PROPN
cana-2085	401	7	.	.	PROPN
cana-2085	401	8	,	,	PUNCT
cana-2085	401	9	&	&	CCONJ
cana-2085	401	10	dowdy	dowdy	ADJ
cana-2085	401	11	,	,	PUNCT
cana-2085	401	12	l.w	l.w	PROPN
cana-2085	401	13	.	.	PROPN
cana-2085	401	14	(	(	PUNCT
cana-2085	401	15	1995	1995	NUM
cana-2085	401	16	)	)	PUNCT
cana-2085	401	17	.	.	PUNCT
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cana-2085	402	4	prediction	prediction	NOUN
cana-2085	402	5	,	,	PUNCT
cana-2085	402	6	and	and	CCONJ
cana-2085	402	7	visualization	visualization	NOUN
cana-2085	402	8	of	of	ADP
cana-2085	402	9	parallel	parallel	ADJ
cana-2085	402	10	systems	system	NOUN
cana-2085	402	11	.	.	PUNCT
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cana-2085	403	2	-	-	PUNCT
cana-2085	403	3	ieee	ieee	PROPN
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cana-2085	403	5	.	.	PUNCT
cana-2085	404	1	[	[	X
cana-2085	404	2	6	6	NUM
cana-2085	404	3	]	]	X
cana-2085	404	4	bang	bang	PROPN
cana-2085	404	5	-	-	PUNCT
cana-2085	404	6	jensen	jensen	PROPN
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cana-2085	404	10	gutin	gutin	PROPN
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cana-2085	404	12	g.	g.	PROPN
cana-2085	404	13	,	,	PUNCT
cana-2085	404	14	&	&	CCONJ
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cana-2085	404	20	)	)	PUNCT
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cana-2085	405	1	"	"	PUNCT
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cana-2085	405	18	)	)	PUNCT
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cana-2085	407	1	"	"	PUNCT
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cana-2085	407	10	"	"	PUNCT
cana-2085	408	1	journal	journal	NOUN
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cana-2085	408	7	)	)	PUNCT
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cana-2085	409	2	8	8	NUM
cana-2085	409	3	]	]	PUNCT
cana-2085	409	4	bertsekas	bertsekas	PROPN
cana-2085	409	5	,	,	PUNCT
cana-2085	409	6	d.p	d.p	PROPN
cana-2085	409	7	.	.	PUNCT
cana-2085	409	8	(	(	PUNCT
cana-2085	409	9	1999	1999	NUM
cana-2085	409	10	)	)	PUNCT
cana-2085	409	11	.	.	PUNCT
cana-2085	410	1	nonlinear	nonlinear	ADJ
cana-2085	410	2	programming	programming	NOUN
cana-2085	410	3	(	(	PUNCT
cana-2085	410	4	2nd	2nd	ADJ
cana-2085	410	5	ed	ed	NOUN
cana-2085	410	6	.	.	PUNCT
cana-2085	410	7	)	)	PUNCT
cana-2085	410	8	.	.	PUNCT
cana-2085	411	1	athena	athena	PROPN
cana-2085	411	2	scientific	scientific	PROPN
cana-2085	411	3	.	.	PUNCT
cana-2085	412	1	[	[	X
cana-2085	412	2	9	9	NUM
cana-2085	412	3	]	]	X
cana-2085	412	4	bollobás	bollobás	PROPN
cana-2085	412	5	,	,	PUNCT
cana-2085	412	6	b.	b.	PROPN
cana-2085	412	7	(	(	PUNCT
cana-2085	412	8	1998	1998	NUM
cana-2085	412	9	)	)	PUNCT
cana-2085	412	10	.	.	PUNCT
cana-2085	413	1	"	"	PUNCT
cana-2085	413	2	modern	modern	ADJ
cana-2085	413	3	graph	graph	NOUN
cana-2085	413	4	theory	theory	NOUN
cana-2085	413	5	.	.	PUNCT
cana-2085	413	6	"	"	PUNCT
cana-2085	413	7	springer	springer	NOUN
cana-2085	413	8	graduate	graduate	NOUN
cana-2085	413	9	texts	text	NOUN
cana-2085	413	10	in	in	ADP
cana-2085	413	11	mathematics	mathematic	NOUN
cana-2085	413	12	,	,	PUNCT
cana-2085	413	13	vol	vol	NOUN
cana-2085	413	14	.	.	PUNCT
cana-2085	413	15	184	184	NUM
cana-2085	413	16	.	.	PUNCT
cana-2085	414	1	[	[	X
cana-2085	414	2	10	10	NUM
cana-2085	414	3	]	]	X
cana-2085	414	4	bondy	bondy	PROPN
cana-2085	414	5	,	,	PUNCT
cana-2085	414	6	j.a	j.a	PROPN
cana-2085	414	7	.	.	PROPN
cana-2085	414	8	,	,	PUNCT
cana-2085	414	9	&	&	CCONJ
cana-2085	414	10	murty	murty	NOUN
cana-2085	414	11	,	,	PUNCT
cana-2085	414	12	u.s.r	u.s.r	NOUN
cana-2085	414	13	.	.	PUNCT
cana-2085	414	14	(	(	PUNCT
cana-2085	414	15	2008	2008	NUM
cana-2085	414	16	)	)	PUNCT
cana-2085	414	17	.	.	PUNCT
cana-2085	415	1	graph	graph	NOUN
cana-2085	415	2	theory	theory	NOUN
cana-2085	415	3	(	(	PUNCT
cana-2085	415	4	graduate	graduate	NOUN
cana-2085	415	5	texts	text	NOUN
cana-2085	415	6	in	in	ADP
cana-2085	415	7	mathematics	mathematic	NOUN
cana-2085	415	8	)	)	PUNCT
cana-2085	415	9	.	.	PUNCT
cana-2085	416	1	springer	springer	NOUN
cana-2085	416	2	.	.	PUNCT
cana-2085	417	1	[	[	X
cana-2085	417	2	11	11	NUM
cana-2085	417	3	]	]	X
cana-2085	417	4	bondy	bondy	PROPN
cana-2085	417	5	,	,	PUNCT
cana-2085	417	6	j.a	j.a	PROPN
cana-2085	417	7	.	.	PROPN
cana-2085	417	8	,	,	PUNCT
cana-2085	417	9	&	&	CCONJ
cana-2085	417	10	murty	murty	NOUN
cana-2085	417	11	,	,	PUNCT
cana-2085	417	12	u.s.r	u.s.r	NOUN
cana-2085	417	13	.	.	PUNCT
cana-2085	417	14	(	(	PUNCT
cana-2085	417	15	2021	2021	NUM
cana-2085	417	16	)	)	PUNCT
cana-2085	417	17	.	.	PUNCT
cana-2085	417	18	graph	graph	NOUN
cana-2085	417	19	theory	theory	NOUN
cana-2085	417	20	(	(	PUNCT
cana-2085	417	21	graduate	graduate	NOUN
cana-2085	417	22	texts	text	NOUN
cana-2085	417	23	in	in	ADP
cana-2085	417	24	mathematics	mathematic	NOUN
cana-2085	417	25	)	)	PUNCT
cana-2085	417	26	.	.	PUNCT
cana-2085	418	1	springer	springer	NOUN
cana-2085	418	2	.	.	PUNCT
cana-2085	419	1	(	(	PUNCT
cana-2085	419	2	updated	update	VERB
cana-2085	419	3	edition	edition	NOUN
cana-2085	419	4	)	)	PUNCT
cana-2085	420	1	[	[	X
cana-2085	420	2	12	12	NUM
cana-2085	420	3	]	]	PUNCT
cana-2085	420	4	brešar	brešar	PROPN
cana-2085	420	5	,	,	PUNCT
cana-2085	420	6	b.	b.	PROPN
cana-2085	420	7	,	,	PUNCT
cana-2085	420	8	klavžar	klavžar	PROPN
cana-2085	420	9	,	,	PUNCT
cana-2085	420	10	s.	s.	PROPN
cana-2085	420	11	,	,	PUNCT
cana-2085	420	12	&	&	CCONJ
cana-2085	420	13	rall	rall	PROPN
cana-2085	420	14	,	,	PUNCT
cana-2085	420	15	d.f	d.f	PROPN
cana-2085	420	16	.	.	PROPN
cana-2085	420	17	(	(	PUNCT
cana-2085	420	18	1998	1998	NUM
cana-2085	420	19	)	)	PUNCT
cana-2085	420	20	.	.	PUNCT
cana-2085	421	1	"	"	PUNCT
cana-2085	421	2	domination	domination	NOUN
cana-2085	421	3	game	game	NOUN
cana-2085	421	4	and	and	CCONJ
cana-2085	421	5	an	an	DET
cana-2085	421	6	algorithm	algorithm	NOUN
cana-2085	421	7	for	for	ADP
cana-2085	421	8	the	the	DET
cana-2085	421	9	domination	domination	NOUN
cana-2085	421	10	number	number	NOUN
cana-2085	421	11	.	.	PUNCT
cana-2085	421	12	"	"	PUNCT
cana-2085	422	1	siam	siam	ADJ
cana-2085	422	2	journal	journal	NOUN
cana-2085	422	3	on	on	ADP
cana-2085	422	4	discrete	discrete	ADJ
cana-2085	422	5	mathematics	mathematic	NOUN
cana-2085	422	6	,	,	PUNCT
cana-2085	422	7	12(4	12(4	NUM
cana-2085	422	8	)	)	PUNCT
cana-2085	422	9	,	,	PUNCT
cana-2085	422	10	536	536	NUM
cana-2085	422	11	-	-	SYM
cana-2085	422	12	547	547	NUM
cana-2085	422	13	.	.	PUNCT
cana-2085	423	1	[	[	X
cana-2085	423	2	13	13	NUM
cana-2085	423	3	]	]	SYM
cana-2085	423	4	chen	chen	PROPN
cana-2085	423	5	,	,	PUNCT
cana-2085	423	6	l.	l.	PROPN
cana-2085	423	7	,	,	PUNCT
cana-2085	423	8	&	&	CCONJ
cana-2085	423	9	wang	wang	PROPN
cana-2085	423	10	,	,	PUNCT
cana-2085	423	11	r.	r.	PROPN
cana-2085	423	12	(	(	PUNCT
cana-2085	423	13	2023	2023	NUM
cana-2085	423	14	)	)	PUNCT
cana-2085	423	15	.	.	PUNCT
cana-2085	423	16	"	"	PUNCT
cana-2085	423	17	graph	graph	NOUN
cana-2085	423	18	-	-	PUNCT
cana-2085	423	19	based	base	VERB
cana-2085	423	20	routing	routing	NOUN
cana-2085	423	21	protocols	protocol	NOUN
cana-2085	423	22	for	for	ADP
cana-2085	423	23	wireless	wireless	ADJ
cana-2085	423	24	sensor	sensor	NOUN
cana-2085	423	25	networks	network	NOUN
cana-2085	423	26	in	in	ADP
cana-2085	423	27	smart	smart	ADJ
cana-2085	423	28	cities	city	NOUN
cana-2085	423	29	.	.	PUNCT
cana-2085	423	30	"	"	PUNCT
cana-2085	424	1	ieee	ieee	NOUN
cana-2085	424	2	communications	communication	NOUN
cana-2085	424	3	surveys	survey	NOUN
cana-2085	424	4	&	&	CCONJ
cana-2085	424	5	tutorials	tutorial	NOUN
cana-2085	424	6	,	,	PUNCT
cana-2085	424	7	25(1	25(1	NUM
cana-2085	424	8	)	)	PUNCT
cana-2085	424	9	,	,	PUNCT
cana-2085	424	10	1	1	NUM
cana-2085	424	11	-	-	SYM
cana-2085	424	12	25	25	NUM
cana-2085	424	13	.	.	PUNCT
cana-2085	425	1	[	[	X
cana-2085	425	2	14	14	NUM
cana-2085	425	3	]	]	X
cana-2085	425	4	chiu	chiu	PROPN
cana-2085	425	5	,	,	PUNCT
cana-2085	425	6	c.m	c.m	PROPN
cana-2085	425	7	.	.	PROPN
cana-2085	425	8	,	,	PUNCT
cana-2085	425	9	wang	wang	PROPN
cana-2085	425	10	,	,	PUNCT
cana-2085	425	11	e.t.g	e.t.g	PROPN
cana-2085	425	12	.	.	PROPN
cana-2085	425	13	,	,	PUNCT
cana-2085	425	14	&	&	CCONJ
cana-2085	425	15	fan	fan	PROPN
cana-2085	425	16	,	,	PUNCT
cana-2085	425	17	y.w	y.w	PROPN
cana-2085	425	18	.	.	PROPN
cana-2085	425	19	(	(	PUNCT
cana-2085	425	20	2007	2007	NUM
cana-2085	425	21	)	)	PUNCT
cana-2085	425	22	.	.	PUNCT
cana-2085	426	1	"	"	PUNCT
cana-2085	426	2	service	service	NOUN
cana-2085	426	3	optimization	optimization	NOUN
cana-2085	426	4	and	and	CCONJ
cana-2085	426	5	fault	fault	VERB
cana-2085	426	6	tolerance	tolerance	NOUN
cana-2085	426	7	in	in	ADP
cana-2085	426	8	network	network	NOUN
cana-2085	426	9	systems	system	NOUN
cana-2085	426	10	.	.	PUNCT
cana-2085	426	11	"	"	PUNCT
cana-2085	427	1	ieee	ieee	NOUN
cana-2085	427	2	transactions	transaction	NOUN
cana-2085	427	3	on	on	ADP
cana-2085	427	4	network	network	NOUN
cana-2085	427	5	and	and	CCONJ
cana-2085	427	6	service	service	NOUN
cana-2085	427	7	management	management	NOUN
cana-2085	427	8	,	,	PUNCT
cana-2085	427	9	4(2	4(2	NUM
cana-2085	427	10	)	)	PUNCT
cana-2085	427	11	,	,	PUNCT
cana-2085	427	12	143	143	NUM
cana-2085	427	13	-	-	SYM
cana-2085	427	14	155	155	NUM
cana-2085	427	15	.	.	PUNCT
cana-2085	428	1	[	[	X
cana-2085	428	2	15	15	NUM
cana-2085	428	3	]	]	X
cana-2085	428	4	chudnovsky	chudnovsky	NOUN
cana-2085	428	5	,	,	PUNCT
cana-2085	428	6	m.	m.	NOUN
cana-2085	428	7	,	,	PUNCT
cana-2085	428	8	scott	scott	PROPN
cana-2085	428	9	,	,	PUNCT
cana-2085	428	10	a.	a.	PROPN
cana-2085	428	11	,	,	PUNCT
cana-2085	428	12	seymour	seymour	ADJ
cana-2085	428	13	,	,	PUNCT
cana-2085	428	14	p.	p.	PROPN
cana-2085	428	15	,	,	PUNCT
cana-2085	428	16	&	&	CCONJ
cana-2085	428	17	spirkl	spirkl	PROPN
cana-2085	428	18	,	,	PUNCT
cana-2085	428	19	s.	s.	PROPN
cana-2085	428	20	(	(	PUNCT
cana-2085	428	21	2022	2022	NUM
cana-2085	428	22	)	)	PUNCT
cana-2085	428	23	.	.	PUNCT
cana-2085	429	1	"	"	PUNCT
cana-2085	429	2	erdős	erdős	NOUN
cana-2085	429	3	-	-	PUNCT
cana-2085	429	4	hajnal	hajnal	ADJ
cana-2085	429	5	conjecture	conjecture	NOUN
cana-2085	429	6	for	for	ADP
cana-2085	429	7	paths	path	NOUN
cana-2085	429	8	and	and	CCONJ
cana-2085	429	9	antipaths	antipath	NOUN
cana-2085	429	10	.	.	PUNCT
cana-2085	429	11	"	"	PUNCT
cana-2085	430	1	journal	journal	NOUN
cana-2085	430	2	of	of	ADP
cana-2085	430	3	combinatorial	combinatorial	ADJ
cana-2085	430	4	theory	theory	NOUN
cana-2085	430	5	,	,	PUNCT
cana-2085	430	6	series	series	PROPN
cana-2085	430	7	b	b	PROPN
cana-2085	430	8	,	,	PUNCT
cana-2085	430	9	153	153	NUM
cana-2085	430	10	,	,	PUNCT
cana-2085	430	11	1	1	NUM
cana-2085	430	12	-	-	SYM
cana-2085	430	13	23	23	NUM
cana-2085	430	14	.	.	PUNCT
cana-2085	431	1	[	[	X
cana-2085	431	2	16	16	NUM
cana-2085	431	3	]	]	X
cana-2085	431	4	cockayne	cockayne	PROPN
cana-2085	431	5	,	,	PUNCT
cana-2085	431	6	e.j	e.j	PROPN
cana-2085	431	7	.	.	PROPN
cana-2085	431	8	,	,	PUNCT
cana-2085	431	9	&	&	CCONJ
cana-2085	431	10	hedetniemi	hedetniemi	PROPN
cana-2085	431	11	,	,	PUNCT
cana-2085	431	12	s.t	s.t	PROPN
cana-2085	431	13	.	.	PROPN
cana-2085	431	14	(	(	PUNCT
cana-2085	431	15	1975	1975	NUM
cana-2085	431	16	)	)	PUNCT
cana-2085	431	17	.	.	PUNCT
cana-2085	432	1	"	"	PUNCT
cana-2085	432	2	on	on	ADP
cana-2085	432	3	the	the	DET
cana-2085	432	4	dominating	dominating	NOUN
cana-2085	432	5	numbers	number	NOUN
cana-2085	432	6	of	of	ADP
cana-2085	432	7	a	a	DET
cana-2085	432	8	graph	graph	NOUN
cana-2085	432	9	.	.	PUNCT
cana-2085	432	10	"	"	PUNCT
cana-2085	432	11	proceedings	proceeding	NOUN
cana-2085	432	12	of	of	ADP
cana-2085	432	13	the	the	DET
cana-2085	432	14	american	american	PROPN
cana-2085	432	15	mathematical	mathematical	PROPN
cana-2085	432	16	society	society	NOUN
cana-2085	432	17	,	,	PUNCT
cana-2085	432	18	32(2	32(2	NUM
cana-2085	432	19	)	)	PUNCT
cana-2085	432	20	,	,	PUNCT
cana-2085	432	21	357	357	NUM
cana-2085	432	22	-	-	SYM
cana-2085	432	23	362	362	NUM
cana-2085	432	24	.	.	PUNCT
cana-2085	433	1	[	[	X
cana-2085	433	2	17	17	NUM
cana-2085	433	3	]	]	X
cana-2085	433	4	cockayne	cockayne	NOUN
cana-2085	433	5	,	,	PUNCT
cana-2085	433	6	e.j	e.j	PROPN
cana-2085	433	7	.	.	PROPN
cana-2085	433	8	,	,	PUNCT
cana-2085	433	9	&	&	CCONJ
cana-2085	433	10	hedetniemi	hedetniemi	PROPN
cana-2085	433	11	,	,	PUNCT
cana-2085	433	12	s.t	s.t	PROPN
cana-2085	433	13	.	.	PROPN
cana-2085	433	14	(	(	PUNCT
cana-2085	433	15	1977	1977	NUM
cana-2085	433	16	)	)	PUNCT
cana-2085	433	17	.	.	PUNCT
cana-2085	434	1	"	"	PUNCT
cana-2085	434	2	towards	towards	ADP
cana-2085	434	3	a	a	DET
cana-2085	434	4	theory	theory	NOUN
cana-2085	434	5	of	of	ADP
cana-2085	434	6	domination	domination	NOUN
cana-2085	434	7	in	in	ADP
cana-2085	434	8	graphs	graph	NOUN
cana-2085	434	9	.	.	PUNCT
cana-2085	434	10	"	"	PUNCT
cana-2085	434	11	networks	network	NOUN
cana-2085	434	12	,	,	PUNCT
cana-2085	434	13	7(3	7(3	NUM
cana-2085	434	14	)	)	PUNCT
cana-2085	434	15	,	,	PUNCT
cana-2085	434	16	247261	247261	NUM
cana-2085	434	17	.	.	PUNCT
cana-2085	435	1	[	[	X
cana-2085	435	2	18	18	NUM
cana-2085	435	3	]	]	PUNCT
cana-2085	435	4	diestel	diestel	PROPN
cana-2085	435	5	,	,	PUNCT
cana-2085	435	6	r.	r.	PROPN
cana-2085	435	7	(	(	PUNCT
cana-2085	435	8	2017	2017	NUM
cana-2085	435	9	)	)	PUNCT
cana-2085	435	10	.	.	PUNCT
cana-2085	436	1	graph	graph	NOUN
cana-2085	436	2	theory	theory	NOUN
cana-2085	436	3	(	(	PUNCT
cana-2085	436	4	5th	5th	ADJ
cana-2085	436	5	ed	ed	NOUN
cana-2085	436	6	.	.	PUNCT
cana-2085	436	7	)	)	PUNCT
cana-2085	436	8	.	.	PUNCT
cana-2085	437	1	springer	springer	NOUN
cana-2085	437	2	.	.	PUNCT
cana-2085	438	1	[	[	X
cana-2085	438	2	19	19	NUM
cana-2085	438	3	]	]	X
cana-2085	438	4	gao	gao	PROPN
cana-2085	438	5	,	,	PUNCT
cana-2085	438	6	j.	j.	PROPN
cana-2085	438	7	,	,	PUNCT
cana-2085	438	8	hu	hu	PROPN
cana-2085	438	9	,	,	PUNCT
cana-2085	438	10	y.	y.	PROPN
cana-2085	438	11	,	,	PUNCT
cana-2085	438	12	&	&	CCONJ
cana-2085	438	13	zhou	zhou	PROPN
cana-2085	438	14	,	,	PUNCT
cana-2085	438	15	b.	b.	PROPN
cana-2085	438	16	(	(	PUNCT
cana-2085	438	17	2022	2022	NUM
cana-2085	438	18	)	)	PUNCT
cana-2085	438	19	.	.	PUNCT
cana-2085	439	1	"	"	PUNCT
cana-2085	439	2	graph	graph	VERB
cana-2085	439	3	neural	neural	ADJ
cana-2085	439	4	networks	network	NOUN
cana-2085	439	5	for	for	ADP
cana-2085	439	6	data	datum	NOUN
cana-2085	439	7	mining	mining	NOUN
cana-2085	439	8	:	:	PUNCT
cana-2085	439	9	a	a	DET
cana-2085	439	10	survey	survey	NOUN
cana-2085	439	11	.	.	PUNCT
cana-2085	439	12	"	"	PUNCT
cana-2085	440	1	acm	acm	PROPN
cana-2085	440	2	computing	computing	NOUN
cana-2085	440	3	surveys	survey	NOUN
cana-2085	440	4	,	,	PUNCT
cana-2085	440	5	55(1	55(1	NOUN
cana-2085	440	6	)	)	PUNCT
cana-2085	440	7	,	,	PUNCT
cana-2085	440	8	1	1	NUM
cana-2085	440	9	-	-	SYM
cana-2085	440	10	35	35	NUM
cana-2085	440	11	.	.	PUNCT
cana-2085	441	1	[	[	X
cana-2085	441	2	20	20	NUM
cana-2085	441	3	]	]	X
cana-2085	441	4	ghosh	ghosh	PROPN
cana-2085	441	5	,	,	PUNCT
cana-2085	441	6	s.	s.	PROPN
cana-2085	441	7	(	(	PUNCT
cana-2085	441	8	2007	2007	NUM
cana-2085	441	9	)	)	PUNCT
cana-2085	441	10	.	.	PUNCT
cana-2085	442	1	distributed	distribute	VERB
cana-2085	442	2	systems	system	NOUN
cana-2085	442	3	:	:	PUNCT
cana-2085	442	4	an	an	DET
cana-2085	442	5	algorithmic	algorithmic	ADJ
cana-2085	442	6	approach	approach	NOUN
cana-2085	442	7	.	.	PUNCT
cana-2085	443	1	crc	crc	PROPN
cana-2085	443	2	press	press	PROPN
cana-2085	443	3	.	.	PUNCT
cana-2085	444	1	[	[	X
cana-2085	444	2	21	21	NUM
cana-2085	444	3	]	]	X
cana-2085	444	4	gupta	gupta	PROPN
cana-2085	444	5	,	,	PUNCT
cana-2085	444	6	i.	i.	PROPN
cana-2085	444	7	,	,	PUNCT
cana-2085	444	8	&	&	CCONJ
cana-2085	444	9	chandra	chandra	PROPN
cana-2085	444	10	,	,	PUNCT
cana-2085	444	11	a.	a.	NOUN
cana-2085	444	12	(	(	PUNCT
cana-2085	444	13	2003	2003	NUM
cana-2085	444	14	)	)	PUNCT
cana-2085	444	15	.	.	PUNCT
cana-2085	445	1	"	"	PUNCT
cana-2085	445	2	load	load	NOUN
cana-2085	445	3	balancing	balance	VERB
cana-2085	445	4	in	in	ADP
cana-2085	445	5	distributed	distributed	ADJ
cana-2085	445	6	systems	system	NOUN
cana-2085	445	7	:	:	PUNCT
cana-2085	445	8	a	a	DET
cana-2085	445	9	perspective	perspective	NOUN
cana-2085	445	10	.	.	PUNCT
cana-2085	445	11	"	"	PUNCT
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cana-2085	445	13	science	science	NOUN
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cana-2085	445	15	,	,	PUNCT
cana-2085	445	16	2(2	2(2	NUM
cana-2085	445	17	)	)	PUNCT
cana-2085	445	18	,	,	PUNCT
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cana-2085	445	20	-	-	SYM
cana-2085	445	21	108	108	NUM
cana-2085	445	22	.	.	PUNCT
cana-2085	446	1	[	[	X
cana-2085	446	2	22	22	NUM
cana-2085	446	3	]	]	X
cana-2085	446	4	haynes	hayne	NOUN
cana-2085	446	5	,	,	PUNCT
cana-2085	446	6	t.w	t.w	PROPN
cana-2085	446	7	.	.	PROPN
cana-2085	446	8	,	,	PUNCT
cana-2085	446	9	hedetniemi	hedetniemi	PROPN
cana-2085	446	10	,	,	PUNCT
cana-2085	446	11	s.t	s.t	PROPN
cana-2085	446	12	.	.	PROPN
cana-2085	446	13	,	,	PUNCT
cana-2085	446	14	&	&	CCONJ
cana-2085	446	15	slater	slater	PROPN
cana-2085	446	16	,	,	PUNCT
cana-2085	446	17	p.j	p.j	PROPN
cana-2085	446	18	.	.	PROPN
cana-2085	446	19	(	(	PUNCT
cana-2085	446	20	1998	1998	NUM
cana-2085	446	21	)	)	PUNCT
cana-2085	446	22	.	.	PUNCT
cana-2085	447	1	fundamentals	fundamental	NOUN
cana-2085	447	2	of	of	ADP
cana-2085	447	3	domination	domination	NOUN
cana-2085	447	4	in	in	ADP
cana-2085	447	5	graphs	graph	NOUN
cana-2085	447	6	.	.	PUNCT
cana-2085	448	1	marcel	marcel	PROPN
cana-2085	448	2	dekker	dekker	PROPN
cana-2085	448	3	.	.	PUNCT
cana-2085	449	1	[	[	X
cana-2085	449	2	23	23	NUM
cana-2085	449	3	]	]	X
cana-2085	449	4	heinzelman	heinzelman	NOUN
cana-2085	449	5	,	,	PUNCT
cana-2085	449	6	w.b	w.b	PROPN
cana-2085	449	7	.	.	PROPN
cana-2085	449	8	,	,	PUNCT
cana-2085	449	9	chandrakasan	chandrakasan	ADJ
cana-2085	449	10	,	,	PUNCT
cana-2085	449	11	a.	a.	NOUN
cana-2085	449	12	,	,	PUNCT
cana-2085	449	13	&	&	CCONJ
cana-2085	449	14	balakrishnan	balakrishnan	PROPN
cana-2085	449	15	,	,	PUNCT
cana-2085	449	16	h.	h.	PROPN
cana-2085	449	17	(	(	PUNCT
cana-2085	449	18	2000	2000	NUM
cana-2085	449	19	)	)	PUNCT
cana-2085	449	20	.	.	PUNCT
cana-2085	450	1	"	"	PUNCT
cana-2085	450	2	energy	energy	NOUN
cana-2085	450	3	-	-	PUNCT
cana-2085	450	4	efficient	efficient	ADJ
cana-2085	450	5	communication	communication	NOUN
cana-2085	450	6	protocol	protocol	NOUN
cana-2085	450	7	for	for	ADP
cana-2085	450	8	wireless	wireless	ADJ
cana-2085	450	9	microsensor	microsensor	NOUN
cana-2085	450	10	networks	network	NOUN
cana-2085	450	11	.	.	PUNCT
cana-2085	450	12	"	"	PUNCT
cana-2085	451	1	proceedings	proceeding	NOUN
cana-2085	451	2	of	of	ADP
cana-2085	451	3	the	the	DET
cana-2085	451	4	33rd	33rd	ADJ
cana-2085	451	5	annual	annual	ADJ
cana-2085	451	6	hawaii	hawaii	PROPN
cana-2085	451	7	international	international	PROPN
cana-2085	451	8	conference	conference	NOUN
cana-2085	451	9	on	on	ADP
cana-2085	451	10	system	system	NOUN
cana-2085	451	11	sciences	science	NOUN
cana-2085	451	12	,	,	PUNCT
cana-2085	451	13	vol	vol	NOUN
cana-2085	451	14	.	.	PROPN
cana-2085	452	1	2	2	NUM
cana-2085	452	2	.	.	PUNCT
cana-2085	453	1	[	[	X
cana-2085	453	2	24	24	NUM
cana-2085	453	3	]	]	X
cana-2085	453	4	henning	henning	PROPN
cana-2085	453	5	,	,	PUNCT
cana-2085	453	6	m.a	m.a	PROPN
cana-2085	453	7	.	.	PROPN
cana-2085	453	8	,	,	PUNCT
cana-2085	453	9	&	&	CCONJ
cana-2085	453	10	plummer	plummer	PROPN
cana-2085	453	11	,	,	PUNCT
cana-2085	453	12	m.d	m.d	PROPN
cana-2085	453	13	.	.	PROPN
cana-2085	453	14	(	(	PUNCT
cana-2085	453	15	2009	2009	NUM
cana-2085	453	16	)	)	PUNCT
cana-2085	453	17	.	.	PUNCT
cana-2085	454	1	"	"	PUNCT
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cana-2085	454	3	in	in	ADP
cana-2085	454	4	graphs	graph	NOUN
cana-2085	454	5	applied	apply	VERB
cana-2085	454	6	to	to	ADP
cana-2085	454	7	network	network	NOUN
cana-2085	454	8	stability	stability	NOUN
cana-2085	454	9	.	.	PUNCT
cana-2085	454	10	"	"	PUNCT
cana-2085	455	1	journal	journal	NOUN
cana-2085	455	2	of	of	ADP
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cana-2085	455	4	optimization	optimization	NOUN
cana-2085	455	5	,	,	PUNCT
cana-2085	455	6	18(3	18(3	NUM
cana-2085	455	7	)	)	PUNCT
cana-2085	455	8	,	,	PUNCT
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cana-2085	455	10	-	-	SYM
cana-2085	455	11	231	231	NUM
cana-2085	455	12	.	.	PUNCT
cana-2085	456	1	[	[	X
cana-2085	456	2	25	25	NUM
cana-2085	456	3	]	]	X
cana-2085	456	4	henning	henning	PROPN
cana-2085	456	5	,	,	PUNCT
cana-2085	456	6	m.a	m.a	PROPN
cana-2085	456	7	.	.	PROPN
cana-2085	456	8	,	,	PUNCT
cana-2085	456	9	&	&	CCONJ
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cana-2085	456	11	,	,	PUNCT
cana-2085	456	12	a.	a.	NOUN
cana-2085	456	13	(	(	PUNCT
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cana-2085	456	15	)	)	PUNCT
cana-2085	456	16	.	.	PUNCT
cana-2085	457	1	"	"	PUNCT
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cana-2085	457	4	in	in	ADP
cana-2085	457	5	graphs	graph	NOUN
cana-2085	457	6	.	.	PUNCT
cana-2085	457	7	"	"	PUNCT
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cana-2085	458	2	monographs	monograph	NOUN
cana-2085	458	3	in	in	ADP
cana-2085	458	4	mathematics	mathematic	NOUN
cana-2085	458	5	.	.	PUNCT
cana-2085	459	1	[	[	X
cana-2085	459	2	26	26	NUM
cana-2085	459	3	]	]	X
cana-2085	459	4	javad	javad	NOUN
cana-2085	459	5	,	,	PUNCT
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cana-2085	459	7	(	(	PUNCT
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cana-2085	459	9	)	)	PUNCT
cana-2085	459	10	.	.	PUNCT
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cana-2085	460	5	.	.	PUNCT
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cana-2085	461	2	.	.	PUNCT
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cana-2085	462	3	]	]	X
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cana-2085	462	5	,	,	PUNCT
cana-2085	462	6	m.	m.	NOUN
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cana-2085	462	8	&	&	CCONJ
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cana-2085	462	10	,	,	PUNCT
cana-2085	462	11	b.	b.	PROPN
cana-2085	462	12	(	(	PUNCT
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cana-2085	462	14	)	)	PUNCT
cana-2085	462	15	.	.	PUNCT
cana-2085	463	1	"	"	PUNCT
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cana-2085	463	8	.	.	PUNCT
cana-2085	463	9	"	"	PUNCT
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cana-2085	464	2	journal	journal	PROPN
cana-2085	464	3	of	of	ADP
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cana-2085	464	5	,	,	PUNCT
cana-2085	464	6	93	93	NUM
cana-2085	464	7	,	,	PUNCT
cana-2085	464	8	103273	103273	NUM
cana-2085	464	9	.	.	PUNCT
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cana-2085	465	3	]	]	X
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cana-2085	465	5	,	,	PUNCT
cana-2085	465	6	j.	j.	PROPN
cana-2085	465	7	,	,	PUNCT
cana-2085	465	8	&	&	CCONJ
cana-2085	465	9	scapellato	scapellato	PROPN
cana-2085	465	10	,	,	PUNCT
cana-2085	465	11	r.	r.	PROPN
cana-2085	465	12	(	(	PUNCT
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cana-2085	466	12	its	its	PRON
cana-2085	466	13	applications	application	NOUN
cana-2085	466	14	)	)	PUNCT
cana-2085	466	15	.	.	PUNCT
cana-2085	467	1	cambridge	cambridge	PROPN
cana-2085	467	2	university	university	PROPN
cana-2085	467	3	press	press	NOUN
cana-2085	467	4	.	.	PUNCT
cana-2085	468	1	[	[	X
cana-2085	468	2	29	29	NUM
cana-2085	468	3	]	]	SYM
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cana-2085	468	5	,	,	PUNCT
cana-2085	468	6	d.	d.	PROPN
cana-2085	468	7	,	,	PUNCT
cana-2085	468	8	&	&	CCONJ
cana-2085	468	9	liao	liao	PROPN
cana-2085	468	10	,	,	PUNCT
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cana-2085	468	12	(	(	PUNCT
cana-2085	468	13	2005	2005	NUM
cana-2085	468	14	)	)	PUNCT
cana-2085	468	15	.	.	PUNCT
cana-2085	469	1	"	"	PUNCT
cana-2085	469	2	design	design	NOUN
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cana-2085	469	7	-	-	PUNCT
cana-2085	469	8	tolerant	tolerant	ADJ
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cana-2085	469	10	.	.	PUNCT
cana-2085	469	11	"	"	PUNCT
cana-2085	470	1	ieee	ieee	NOUN
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cana-2085	470	4	parallel	parallel	ADJ
cana-2085	470	5	and	and	CCONJ
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cana-2085	470	7	systems	system	NOUN
cana-2085	470	8	,	,	PUNCT
cana-2085	470	9	16(10	16(10	NOUN
cana-2085	470	10	)	)	PUNCT
cana-2085	470	11	,	,	PUNCT
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cana-2085	470	13	-	-	SYM
cana-2085	470	14	982	982	NUM
cana-2085	470	15	.	.	PUNCT
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cana-2085	471	2	30	30	NUM
cana-2085	471	3	]	]	X
cana-2085	471	4	li	li	PROPN
cana-2085	471	5	,	,	PUNCT
cana-2085	471	6	h.	h.	PROPN
cana-2085	471	7	,	,	PUNCT
cana-2085	471	8	li	li	PROPN
cana-2085	471	9	,	,	PUNCT
cana-2085	471	10	j.	j.	PROPN
cana-2085	471	11	,	,	PUNCT
cana-2085	471	12	&	&	CCONJ
cana-2085	471	13	zhang	zhang	PROPN
cana-2085	471	14	,	,	PUNCT
cana-2085	471	15	y.	y.	PROPN
cana-2085	471	16	(	(	PUNCT
cana-2085	471	17	2023	2023	NUM
cana-2085	471	18	)	)	PUNCT
cana-2085	471	19	.	.	PUNCT
cana-2085	472	1	"	"	PUNCT
cana-2085	472	2	graph	graph	NOUN
cana-2085	472	3	-	-	PUNCT
cana-2085	472	4	based	base	VERB
cana-2085	472	5	pathfinding	pathfinding	NOUN
cana-2085	472	6	algorithms	algorithm	NOUN
cana-2085	472	7	for	for	ADP
cana-2085	472	8	autonomous	autonomous	ADJ
cana-2085	472	9	vehicles	vehicle	NOUN
cana-2085	472	10	in	in	ADP
cana-2085	472	11	dynamic	dynamic	ADJ
cana-2085	472	12	environments	environment	NOUN
cana-2085	472	13	.	.	PUNCT
cana-2085	472	14	"	"	PUNCT
cana-2085	473	1	acm	acm	PROPN
cana-2085	473	2	transactions	transaction	NOUN
cana-2085	473	3	on	on	ADP
cana-2085	473	4	intelligent	intelligent	ADJ
cana-2085	473	5	systems	system	NOUN
cana-2085	473	6	and	and	CCONJ
cana-2085	473	7	technology	technology	NOUN
cana-2085	473	8	,	,	PUNCT
cana-2085	473	9	14(3	14(3	NUM
cana-2085	473	10	)	)	PUNCT
cana-2085	473	11	,	,	PUNCT
cana-2085	473	12	32	32	NUM
cana-2085	473	13	-	-	SYM
cana-2085	473	14	44	44	NUM
cana-2085	473	15	.	.	PUNCT
cana-2085	474	1	[	[	X
cana-2085	474	2	31	31	NUM
cana-2085	474	3	]	]	X
cana-2085	474	4	liu	liu	PROPN
cana-2085	474	5	,	,	PUNCT
cana-2085	474	6	z.	z.	PROPN
cana-2085	474	7	,	,	PUNCT
cana-2085	474	8	&	&	CCONJ
cana-2085	474	9	zhang	zhang	PROPN
cana-2085	474	10	,	,	PUNCT
cana-2085	474	11	y.	y.	PROPN
cana-2085	474	12	(	(	PUNCT
cana-2085	474	13	2022	2022	NUM
cana-2085	474	14	)	)	PUNCT
cana-2085	474	15	.	.	PUNCT
cana-2085	475	1	"	"	PUNCT
cana-2085	475	2	graph	graph	NOUN
cana-2085	475	3	-	-	PUNCT
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cana-2085	475	7	protein	protein	NOUN
cana-2085	475	8	-	-	PUNCT
cana-2085	475	9	protein	protein	NOUN
cana-2085	475	10	interaction	interaction	NOUN
cana-2085	475	11	network	network	NOUN
cana-2085	475	12	analysis	analysis	NOUN
cana-2085	475	13	.	.	PUNCT
cana-2085	475	14	"	"	PUNCT
cana-2085	476	1	bioinformatics	bioinformatics	NOUN
cana-2085	476	2	,	,	PUNCT
cana-2085	476	3	38(14	38(14	NUM
cana-2085	476	4	)	)	PUNCT
cana-2085	476	5	,	,	PUNCT
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cana-2085	476	7	-	-	SYM
cana-2085	476	8	3543	3543	NUM
cana-2085	476	9	.	.	PUNCT
cana-2085	477	1	[	[	X
cana-2085	477	2	32	32	NUM
cana-2085	477	3	]	]	SYM
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cana-2085	477	5	,	,	PUNCT
cana-2085	477	6	l.	l.	PROPN
cana-2085	477	7	,	,	PUNCT
cana-2085	477	8	&	&	CCONJ
cana-2085	477	9	vesztergombi	vesztergombi	PROPN
cana-2085	477	10	,	,	PUNCT
cana-2085	477	11	k.	k.	PROPN
cana-2085	477	12	(	(	PUNCT
cana-2085	477	13	2021	2021	NUM
cana-2085	477	14	)	)	PUNCT
cana-2085	477	15	.	.	PUNCT
cana-2085	478	1	"	"	PUNCT
cana-2085	478	2	graph	graph	NOUN
cana-2085	478	3	theory	theory	NOUN
cana-2085	478	4	:	:	PUNCT
cana-2085	478	5	from	from	ADP
cana-2085	478	6	königsberg	königsberg	PROPN
cana-2085	478	7	to	to	ADP
cana-2085	478	8	the	the	DET
cana-2085	478	9	internet	internet	NOUN
cana-2085	478	10	.	.	PUNCT
cana-2085	478	11	"	"	PUNCT
cana-2085	479	1	cambridge	cambridge	PROPN
cana-2085	479	2	university	university	PROPN
cana-2085	479	3	press	press	NOUN
cana-2085	479	4	.	.	PUNCT
cana-2085	480	1	[	[	X
cana-2085	480	2	33	33	NUM
cana-2085	480	3	]	]	X
cana-2085	480	4	mishra	mishra	PROPN
cana-2085	480	5	,	,	PUNCT
cana-2085	480	6	b.	b.	PROPN
cana-2085	480	7	,	,	PUNCT
cana-2085	480	8	&	&	CCONJ
cana-2085	480	9	yadav	yadav	PROPN
cana-2085	480	10	,	,	PUNCT
cana-2085	480	11	p.	p.	NOUN
cana-2085	480	12	(	(	PUNCT
cana-2085	480	13	2022	2022	NUM
cana-2085	480	14	)	)	PUNCT
cana-2085	480	15	.	.	PUNCT
cana-2085	481	1	"	"	PUNCT
cana-2085	481	2	graph	graph	NOUN
cana-2085	481	3	-	-	PUNCT
cana-2085	481	4	theoretic	theoretic	NOUN
cana-2085	481	5	approaches	approach	NOUN
cana-2085	481	6	for	for	ADP
cana-2085	481	7	optimization	optimization	NOUN
cana-2085	481	8	in	in	ADP
cana-2085	481	9	distributed	distribute	VERB
cana-2085	481	10	systems	system	NOUN
cana-2085	481	11	:	:	PUNCT
cana-2085	481	12	a	a	DET
cana-2085	481	13	survey	survey	NOUN
cana-2085	481	14	.	.	PUNCT
cana-2085	481	15	"	"	PUNCT
cana-2085	482	1	ieee	ieee	NOUN
cana-2085	482	2	transactions	transaction	NOUN
cana-2085	482	3	on	on	ADP
cana-2085	482	4	parallel	parallel	ADJ
cana-2085	482	5	and	and	CCONJ
cana-2085	482	6	distributed	distributed	ADJ
cana-2085	482	7	systems	system	NOUN
cana-2085	482	8	,	,	PUNCT
cana-2085	482	9	33(12	33(12	NUM
cana-2085	482	10	)	)	PUNCT
cana-2085	482	11	,	,	PUNCT
cana-2085	482	12	4101	4101	NUM
cana-2085	482	13	-	-	SYM
cana-2085	482	14	4114	4114	NUM
cana-2085	482	15	.	.	PUNCT
cana-2085	483	1	[	[	X
cana-2085	483	2	34	34	NUM
cana-2085	483	3	]	]	X
cana-2085	483	4	mitra	mitra	PROPN
cana-2085	483	5	,	,	PUNCT
cana-2085	483	6	d.	d.	PROPN
cana-2085	483	7	,	,	PUNCT
cana-2085	483	8	&	&	CCONJ
cana-2085	483	9	morrison	morrison	PROPN
cana-2085	483	10	,	,	PUNCT
cana-2085	483	11	j.a	j.a	PROPN
cana-2085	483	12	.	.	PROPN
cana-2085	483	13	(	(	PUNCT
cana-2085	483	14	1997	1997	NUM
cana-2085	483	15	)	)	PUNCT
cana-2085	483	16	.	.	PUNCT
cana-2085	484	1	"	"	PUNCT
cana-2085	484	2	optimal	optimal	ADJ
cana-2085	484	3	design	design	NOUN
cana-2085	484	4	of	of	ADP
cana-2085	484	5	distributed	distribute	VERB
cana-2085	484	6	call	call	NOUN
cana-2085	484	7	centers	center	NOUN
cana-2085	484	8	.	.	PUNCT
cana-2085	484	9	"	"	PUNCT
cana-2085	484	10	ieee	ieee	PROPN
cana-2085	484	11	/	/	SYM
cana-2085	484	12	acm	acm	NOUN
cana-2085	484	13	transactions	transaction	NOUN
cana-2085	484	14	on	on	ADP
cana-2085	484	15	networking	networking	NOUN
cana-2085	484	16	,	,	PUNCT
cana-2085	484	17	5(2	5(2	NUM
cana-2085	484	18	)	)	PUNCT
cana-2085	484	19	,	,	PUNCT
cana-2085	484	20	254	254	NUM
cana-2085	484	21	-	-	SYM
cana-2085	484	22	266	266	NUM
cana-2085	484	23	.	.	PUNCT
cana-2085	485	1	[	[	X
cana-2085	485	2	35	35	NUM
cana-2085	485	3	]	]	X
cana-2085	485	4	mitrani	mitrani	PROPN
cana-2085	485	5	,	,	PUNCT
cana-2085	485	6	i.	i.	NOUN
cana-2085	485	7	(	(	PUNCT
cana-2085	485	8	1991	1991	NUM
cana-2085	485	9	)	)	PUNCT
cana-2085	485	10	.	.	PUNCT
cana-2085	486	1	probabilistic	probabilistic	ADJ
cana-2085	486	2	modelling	modelling	NOUN
cana-2085	486	3	of	of	ADP
cana-2085	486	4	computer	computer	NOUN
cana-2085	486	5	systems	system	NOUN
cana-2085	486	6	.	.	PUNCT
cana-2085	487	1	cambridge	cambridge	PROPN
cana-2085	487	2	university	university	PROPN
cana-2085	487	3	press	press	NOUN
cana-2085	487	4	.	.	PUNCT
cana-2085	488	1	communications	communication	NOUN
cana-2085	488	2	on	on	ADP
cana-2085	488	3	applied	apply	VERB
cana-2085	488	4	nonlinear	nonlinear	ADJ
cana-2085	488	5	analysis	analysis	NOUN
cana-2085	488	6	issn	issn	NOUN
cana-2085	488	7	:	:	PUNCT
cana-2085	488	8	1074	1074	NUM
cana-2085	488	9	-	-	PUNCT
cana-2085	488	10	133x	133x	NUM
cana-2085	488	11	vol	vol	NOUN
cana-2085	488	12	32	32	NUM
cana-2085	488	13	no	no	NOUN
cana-2085	488	14	.	.	PUNCT
cana-2085	489	1	1s	1s	NUM
cana-2085	489	2	(	(	PUNCT
cana-2085	489	3	2025	2025	NUM
cana-2085	489	4	)	)	PUNCT
cana-2085	489	5	30	30	NUM
cana-2085	489	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-2085	490	1	[	[	X
cana-2085	490	2	36	36	NUM
cana-2085	490	3	]	]	X
cana-2085	490	4	molloy	molloy	PROPN
cana-2085	490	5	,	,	PUNCT
cana-2085	490	6	m.	m.	NOUN
cana-2085	490	7	,	,	PUNCT
cana-2085	490	8	&	&	CCONJ
cana-2085	490	9	reed	reed	PROPN
cana-2085	490	10	,	,	PUNCT
cana-2085	490	11	b.	b.	PROPN
cana-2085	490	12	(	(	PUNCT
cana-2085	490	13	2020	2020	NUM
cana-2085	490	14	)	)	PUNCT
cana-2085	490	15	.	.	PUNCT
cana-2085	491	1	"	"	PUNCT
cana-2085	491	2	graph	graph	NOUN
cana-2085	491	3	colouring	colouring	NOUN
cana-2085	491	4	and	and	CCONJ
cana-2085	491	5	the	the	DET
cana-2085	491	6	probabilistic	probabilistic	ADJ
cana-2085	491	7	method	method	NOUN
cana-2085	491	8	:	:	PUNCT
cana-2085	491	9	recent	recent	ADJ
cana-2085	491	10	progress	progress	NOUN
cana-2085	491	11	.	.	PUNCT
cana-2085	491	12	"	"	PUNCT
cana-2085	491	13	journal	journal	NOUN
cana-2085	491	14	of	of	ADP
cana-2085	491	15	combinatorial	combinatorial	ADJ
cana-2085	491	16	theory	theory	NOUN
cana-2085	491	17	,	,	PUNCT
cana-2085	491	18	series	series	PROPN
cana-2085	491	19	b	b	PROPN
cana-2085	491	20	,	,	PUNCT
cana-2085	491	21	142	142	NUM
cana-2085	491	22	,	,	PUNCT
cana-2085	491	23	138	138	NUM
cana-2085	491	24	-	-	SYM
cana-2085	491	25	163	163	NUM
cana-2085	491	26	.	.	PUNCT
cana-2085	492	1	[	[	X
cana-2085	492	2	37	37	NUM
cana-2085	492	3	]	]	PUNCT
cana-2085	492	4	m.	m.	NOUN
cana-2085	492	5	salmerón	salmerón	PROPN
cana-2085	492	6	,	,	PUNCT
cana-2085	492	7	k.	k.	PROPN
cana-2085	492	8	wood	wood	PROPN
cana-2085	492	9	,	,	PUNCT
cana-2085	492	10	&	&	CCONJ
cana-2085	492	11	r.	r.	PROPN
cana-2085	492	12	baldick	baldick	PROPN
cana-2085	492	13	(	(	PUNCT
cana-2085	492	14	2004	2004	NUM
cana-2085	492	15	)	)	PUNCT
cana-2085	492	16	.	.	PUNCT
cana-2085	493	1	"	"	PUNCT
cana-2085	493	2	analysis	analysis	NOUN
cana-2085	493	3	of	of	ADP
cana-2085	493	4	electric	electric	ADJ
cana-2085	493	5	grid	grid	NOUN
cana-2085	493	6	security	security	NOUN
cana-2085	493	7	under	under	ADP
cana-2085	493	8	terrorist	terrorist	ADJ
cana-2085	493	9	threat	threat	NOUN
cana-2085	493	10	.	.	PUNCT
cana-2085	493	11	"	"	PUNCT
cana-2085	494	1	ieee	ieee	NOUN
cana-2085	494	2	transactions	transaction	NOUN
cana-2085	494	3	on	on	ADP
cana-2085	494	4	power	power	NOUN
cana-2085	494	5	systems	system	NOUN
cana-2085	494	6	,	,	PUNCT
cana-2085	494	7	19(2	19(2	NUM
cana-2085	494	8	)	)	PUNCT
cana-2085	494	9	,	,	PUNCT
cana-2085	494	10	905	905	NUM
cana-2085	494	11	-	-	SYM
cana-2085	494	12	912	912	NUM
cana-2085	494	13	.	.	PUNCT
cana-2085	495	1	[	[	X
cana-2085	495	2	38	38	NUM
cana-2085	495	3	]	]	SYM
cana-2085	495	4	natarajan	natarajan	PROPN
cana-2085	495	5	,	,	PUNCT
cana-2085	495	6	p.	p.	PROPN
cana-2085	495	7	,	,	PUNCT
cana-2085	495	8	&	&	CCONJ
cana-2085	495	9	premkumar	premkumar	PROPN
cana-2085	495	10	,	,	PUNCT
cana-2085	495	11	p.b	p.b	PROPN
cana-2085	495	12	.	.	PROPN
cana-2085	495	13	(	(	PUNCT
cana-2085	495	14	2015	2015	NUM
cana-2085	495	15	)	)	PUNCT
cana-2085	495	16	.	.	PUNCT
cana-2085	496	1	"	"	PUNCT
cana-2085	496	2	energy	energy	NOUN
cana-2085	496	3	-	-	PUNCT
cana-2085	496	4	efficient	efficient	ADJ
cana-2085	496	5	algorithms	algorithm	NOUN
cana-2085	496	6	for	for	ADP
cana-2085	496	7	wireless	wireless	ADJ
cana-2085	496	8	sensor	sensor	NOUN
cana-2085	496	9	networks	network	NOUN
cana-2085	496	10	using	use	VERB
cana-2085	496	11	graph	graph	NOUN
cana-2085	496	12	domination	domination	NOUN
cana-2085	496	13	.	.	PUNCT
cana-2085	496	14	"	"	PUNCT
cana-2085	497	1	international	international	ADJ
cana-2085	497	2	journal	journal	NOUN
cana-2085	497	3	of	of	ADP
cana-2085	497	4	ad	ad	X
cana-2085	497	5	hoc	hoc	X
cana-2085	497	6	and	and	CCONJ
cana-2085	497	7	ubiquitous	ubiquitous	ADJ
cana-2085	497	8	computing	computing	NOUN
cana-2085	497	9	,	,	PUNCT
cana-2085	497	10	19(4	19(4	PROPN
cana-2085	497	11	)	)	PUNCT
cana-2085	497	12	,	,	PUNCT
cana-2085	497	13	263	263	NUM
cana-2085	497	14	-	-	SYM
cana-2085	497	15	271	271	NUM
cana-2085	497	16	.	.	PUNCT
cana-2085	498	1	[	[	X
cana-2085	498	2	39	39	NUM
cana-2085	498	3	]	]	PUNCT
cana-2085	498	4	papadimitriou	papadimitriou	NOUN
cana-2085	498	5	,	,	PUNCT
cana-2085	498	6	c.h	c.h	PROPN
cana-2085	498	7	.	.	PROPN
cana-2085	498	8	,	,	PUNCT
cana-2085	498	9	&	&	CCONJ
cana-2085	498	10	steiglitz	steiglitz	PROPN
cana-2085	498	11	,	,	PUNCT
cana-2085	498	12	k.	k.	PROPN
cana-2085	498	13	(	(	PUNCT
cana-2085	498	14	1998	1998	NUM
cana-2085	498	15	)	)	PUNCT
cana-2085	498	16	.	.	PUNCT
cana-2085	499	1	combinatorial	combinatorial	ADJ
cana-2085	499	2	optimization	optimization	NOUN
cana-2085	499	3	:	:	PUNCT
cana-2085	499	4	algorithms	algorithm	NOUN
cana-2085	499	5	and	and	CCONJ
cana-2085	499	6	complexity	complexity	NOUN
cana-2085	499	7	.	.	PUNCT
cana-2085	500	1	dover	dover	PROPN
cana-2085	500	2	publications	publication	NOUN
cana-2085	500	3	.	.	PUNCT
cana-2085	501	1	[	[	X
cana-2085	501	2	40	40	NUM
cana-2085	501	3	]	]	PUNCT
cana-2085	501	4	prabhu	prabhu	NOUN
cana-2085	501	5	,	,	PUNCT
cana-2085	501	6	r.	r.	PROPN
cana-2085	501	7	,	,	PUNCT
cana-2085	501	8	&	&	CCONJ
cana-2085	501	9	manimaran	manimaran	PROPN
cana-2085	501	10	,	,	PUNCT
cana-2085	501	11	g.	g.	PROPN
cana-2085	501	12	(	(	PUNCT
cana-2085	501	13	2005	2005	NUM
cana-2085	501	14	)	)	PUNCT
cana-2085	501	15	.	.	PUNCT
cana-2085	502	1	"	"	PUNCT
cana-2085	502	2	network	network	NOUN
cana-2085	502	3	security	security	NOUN
cana-2085	502	4	through	through	ADP
cana-2085	502	5	optimal	optimal	ADJ
cana-2085	502	6	resource	resource	NOUN
cana-2085	502	7	allocation	allocation	NOUN
cana-2085	502	8	.	.	PUNCT
cana-2085	502	9	"	"	PUNCT
cana-2085	503	1	ieee	ieee	NOUN
cana-2085	503	2	transactions	transaction	NOUN
cana-2085	503	3	on	on	ADP
cana-2085	503	4	dependable	dependable	ADJ
cana-2085	503	5	and	and	CCONJ
cana-2085	503	6	secure	secure	ADJ
cana-2085	503	7	computing	computing	NOUN
cana-2085	503	8	,	,	PUNCT
cana-2085	503	9	2(3	2(3	NUM
cana-2085	503	10	)	)	PUNCT
cana-2085	503	11	,	,	PUNCT
cana-2085	503	12	180	180	NUM
cana-2085	503	13	-	-	SYM
cana-2085	503	14	189	189	NUM
cana-2085	503	15	.	.	PUNCT
cana-2085	504	1	[	[	X
cana-2085	504	2	41	41	NUM
cana-2085	504	3	]	]	PUNCT
cana-2085	504	4	quisquater	quisquater	NOUN
cana-2085	504	5	,	,	PUNCT
cana-2085	504	6	j.j	j.j	PROPN
cana-2085	504	7	.	.	PROPN
cana-2085	504	8	,	,	PUNCT
cana-2085	504	9	&	&	CCONJ
cana-2085	504	10	schneier	schneier	PROPN
cana-2085	504	11	,	,	PUNCT
cana-2085	504	12	b.	b.	PROPN
cana-2085	504	13	(	(	PUNCT
cana-2085	504	14	1999	1999	NUM
cana-2085	504	15	)	)	PUNCT
cana-2085	504	16	.	.	PUNCT
cana-2085	505	1	applied	apply	VERB
cana-2085	505	2	cryptography	cryptography	NOUN
cana-2085	505	3	:	:	PUNCT
cana-2085	505	4	protocols	protocol	NOUN
cana-2085	505	5	,	,	PUNCT
cana-2085	505	6	algorithms	algorithm	NOUN
cana-2085	505	7	,	,	PUNCT
cana-2085	505	8	and	and	CCONJ
cana-2085	505	9	source	source	NOUN
cana-2085	505	10	code	code	NOUN
cana-2085	505	11	in	in	ADP
cana-2085	505	12	c	c	PROPN
cana-2085	505	13	(	(	PUNCT
cana-2085	505	14	2nd	2nd	ADJ
cana-2085	505	15	ed	ed	NOUN
cana-2085	505	16	.	.	PUNCT
cana-2085	505	17	)	)	PUNCT
cana-2085	505	18	.	.	PUNCT
cana-2085	506	1	wiley	wiley	PROPN
cana-2085	506	2	.	.	PUNCT
cana-2085	507	1	[	[	X
cana-2085	507	2	42	42	NUM
cana-2085	507	3	]	]	PUNCT
cana-2085	507	4	revelle	revelle	NOUN
cana-2085	507	5	,	,	PUNCT
cana-2085	507	6	c.s	c.s	PROPN
cana-2085	507	7	.	.	PROPN
cana-2085	507	8	,	,	PUNCT
cana-2085	507	9	&	&	CCONJ
cana-2085	507	10	rosing	rosing	PROPN
cana-2085	507	11	,	,	PUNCT
cana-2085	507	12	k.e	k.e	PROPN
cana-2085	507	13	.	.	PROPN
cana-2085	507	14	(	(	PUNCT
cana-2085	507	15	2000	2000	NUM
cana-2085	507	16	)	)	PUNCT
cana-2085	507	17	.	.	PUNCT
cana-2085	508	1	"	"	PUNCT
cana-2085	508	2	defendens	defenden	VERB
cana-2085	508	3	imperium	imperium	NOUN
cana-2085	508	4	romanum	romanum	NOUN
cana-2085	508	5	:	:	PUNCT
cana-2085	508	6	a	a	DET
cana-2085	508	7	classical	classical	ADJ
cana-2085	508	8	problem	problem	NOUN
cana-2085	508	9	in	in	ADP
cana-2085	508	10	military	military	ADJ
cana-2085	508	11	strategy	strategy	NOUN
cana-2085	508	12	.	.	PUNCT
cana-2085	508	13	"	"	PUNCT
cana-2085	509	1	american	american	PROPN
cana-2085	509	2	mathematical	mathematical	PROPN
cana-2085	509	3	monthly	monthly	ADV
cana-2085	509	4	,	,	PUNCT
cana-2085	509	5	107(7	107(7	NUM
cana-2085	509	6	)	)	PUNCT
cana-2085	509	7	,	,	PUNCT
cana-2085	509	8	585	585	NUM
cana-2085	509	9	-	-	SYM
cana-2085	509	10	594	594	NUM
cana-2085	509	11	.	.	PUNCT
cana-2085	510	1	[	[	X
cana-2085	510	2	43	43	NUM
cana-2085	510	3	]	]	X
cana-2085	510	4	schneier	schneier	PROPN
cana-2085	510	5	,	,	PUNCT
cana-2085	510	6	b.	b.	PROPN
cana-2085	510	7	(	(	PUNCT
cana-2085	510	8	1999	1999	NUM
cana-2085	510	9	)	)	PUNCT
cana-2085	510	10	.	.	PUNCT
cana-2085	511	1	"	"	PUNCT
cana-2085	511	2	a	a	DET
cana-2085	511	3	taxonomy	taxonomy	NOUN
cana-2085	511	4	of	of	ADP
cana-2085	511	5	security	security	NOUN
cana-2085	511	6	considerations	consideration	NOUN
cana-2085	511	7	in	in	ADP
cana-2085	511	8	computer	computer	NOUN
cana-2085	511	9	networks	network	NOUN
cana-2085	511	10	.	.	PUNCT
cana-2085	511	11	"	"	PUNCT
cana-2085	512	1	communications	communication	NOUN
cana-2085	512	2	of	of	ADP
cana-2085	512	3	the	the	DET
cana-2085	512	4	acm	acm	PROPN
cana-2085	512	5	,	,	PUNCT
cana-2085	512	6	40(6	40(6	PROPN
cana-2085	512	7	)	)	PUNCT
cana-2085	512	8	,	,	PUNCT
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cana-2085	512	10	-	-	SYM
cana-2085	512	11	102	102	NUM
cana-2085	512	12	.	.	PUNCT
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cana-2085	513	3	]	]	X
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cana-2085	513	5	,	,	PUNCT
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cana-2085	513	7	,	,	PUNCT
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cana-2085	513	9	,	,	PUNCT
cana-2085	513	10	v.	v.	PROPN
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cana-2085	513	18	)	)	PUNCT
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cana-2085	514	3	-	-	PUNCT
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cana-2085	514	5	anomaly	anomaly	NOUN
cana-2085	514	6	detection	detection	NOUN
cana-2085	514	7	in	in	ADP
cana-2085	514	8	complex	complex	ADJ
cana-2085	514	9	networks	network	NOUN
cana-2085	514	10	using	use	VERB
cana-2085	514	11	deep	deep	ADJ
cana-2085	514	12	learning	learning	NOUN
cana-2085	514	13	.	.	PUNCT
cana-2085	514	14	"	"	PUNCT
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cana-2085	515	2	transactions	transaction	NOUN
cana-2085	515	3	on	on	ADP
cana-2085	515	4	network	network	NOUN
cana-2085	515	5	and	and	CCONJ
cana-2085	515	6	service	service	NOUN
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cana-2085	515	8	,	,	PUNCT
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cana-2085	515	10	)	)	PUNCT
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cana-2085	515	13	-	-	SYM
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cana-2085	516	3	]	]	X
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cana-2085	516	10	)	)	PUNCT
cana-2085	516	11	.	.	PUNCT
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cana-2085	517	4	location	location	NOUN
cana-2085	517	5	in	in	ADP
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cana-2085	517	7	.	.	PUNCT
cana-2085	517	8	"	"	PUNCT
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cana-2085	517	10	,	,	PUNCT
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cana-2085	517	17	.	.	PUNCT
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cana-2085	518	2	46	46	NUM
cana-2085	518	3	]	]	X
cana-2085	518	4	stallings	stallings	PROPN
cana-2085	518	5	,	,	PUNCT
cana-2085	518	6	w.	w.	PROPN
cana-2085	518	7	(	(	PUNCT
cana-2085	518	8	2017	2017	NUM
cana-2085	518	9	)	)	PUNCT
cana-2085	518	10	.	.	PUNCT
cana-2085	519	1	network	network	NOUN
cana-2085	519	2	security	security	NOUN
cana-2085	519	3	essentials	essential	NOUN
cana-2085	519	4	:	:	PUNCT
cana-2085	519	5	applications	application	NOUN
cana-2085	519	6	and	and	CCONJ
cana-2085	519	7	standards	standard	NOUN
cana-2085	519	8	(	(	PUNCT
cana-2085	519	9	6th	6th	ADJ
cana-2085	519	10	ed	ed	NOUN
cana-2085	519	11	.	.	PUNCT
cana-2085	519	12	)	)	PUNCT
cana-2085	519	13	.	.	PUNCT
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cana-2085	520	2	.	.	PUNCT
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cana-2085	521	2	47	47	NUM
cana-2085	521	3	]	]	X
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cana-2085	521	5	,	,	PUNCT
cana-2085	521	6	s.	s.	PROPN
cana-2085	521	7	,	,	PUNCT
cana-2085	521	8	&	&	CCONJ
cana-2085	521	9	thiyagarajan	thiyagarajan	PROPN
cana-2085	521	10	,	,	PUNCT
cana-2085	521	11	r.	r.	PROPN
cana-2085	521	12	(	(	PUNCT
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cana-2085	521	14	)	)	PUNCT
cana-2085	521	15	.	.	PUNCT
cana-2085	522	1	"	"	PUNCT
cana-2085	522	2	graph	graph	NOUN
cana-2085	522	3	-	-	PUNCT
cana-2085	522	4	based	base	VERB
cana-2085	522	5	trust	trust	NOUN
cana-2085	522	6	management	management	NOUN