id	sid	tid	token	lemma	pos
ejpam-6713	1	1	european	european	PROPN
ejpam-6713	1	2	journal	journal	PROPN
ejpam-6713	1	3	of	of	ADP
ejpam-6713	1	4	pure	pure	ADJ
ejpam-6713	1	5	and	and	CCONJ
ejpam-6713	1	6	applied	applied	ADJ
ejpam-6713	1	7	mathematics	mathematic	NOUN
ejpam-6713	1	8	2025	2025	NUM
ejpam-6713	1	9	,	,	PUNCT
ejpam-6713	1	10	vol	vol	NOUN
ejpam-6713	1	11	.	.	PROPN
ejpam-6713	1	12	18	18	NUM
ejpam-6713	1	13	,	,	PUNCT
ejpam-6713	1	14	issue	issue	NOUN
ejpam-6713	1	15	4	4	NUM
ejpam-6713	1	16	,	,	PUNCT
ejpam-6713	1	17	article	article	NOUN
ejpam-6713	1	18	number	number	NOUN
ejpam-6713	1	19	6713	6713	NUM
ejpam-6713	1	20	issn	issn	VERB
ejpam-6713	1	21	1307	1307	NUM
ejpam-6713	1	22	-	-	SYM
ejpam-6713	1	23	5543	5543	NUM
ejpam-6713	1	24	–	–	PUNCT
ejpam-6713	1	25	ejpam.com	ejpam.com	X
ejpam-6713	1	26	published	publish	VERB
ejpam-6713	1	27	by	by	ADP
ejpam-6713	1	28	new	new	PROPN
ejpam-6713	1	29	york	york	PROPN
ejpam-6713	1	30	business	business	PROPN
ejpam-6713	1	31	global	global	ADJ
ejpam-6713	1	32	exploring	explore	VERB
ejpam-6713	1	33	certified	certify	VERB
ejpam-6713	1	34	domination	domination	NOUN
ejpam-6713	1	35	subdivision	subdivision	NOUN
ejpam-6713	1	36	numbers	number	NOUN
ejpam-6713	1	37	in	in	ADP
ejpam-6713	1	38	graph	graph	NOUN
ejpam-6713	1	39	theory	theory	NOUN
ejpam-6713	1	40	g.	g.	PROPN
ejpam-6713	1	41	navamani1	navamani1	PROPN
ejpam-6713	1	42	,	,	PUNCT
ejpam-6713	1	43	n.	n.	PROPN
ejpam-6713	1	44	sumathi1,∗	sumathi1,∗	NOUN
ejpam-6713	1	45	,	,	PUNCT
ejpam-6713	1	46	dharmaraj	dharmaraj	ADJ
ejpam-6713	1	47	mohankumar1	mohankumar1	PROPN
ejpam-6713	1	48	,	,	PUNCT
ejpam-6713	1	49	luminiţa	luminiţa	PROPN
ejpam-6713	1	50	-	-	PUNCT
ejpam-6713	1	51	ioana	ioana	PROPN
ejpam-6713	1	52	cotîrlă2	cotîrlă2	PROPN
ejpam-6713	1	53	,	,	PUNCT
ejpam-6713	1	54	daniel	daniel	PROPN
ejpam-6713	1	55	breaz3,∗	breaz3,∗	PROPN
ejpam-6713	1	56	1	1	NUM
ejpam-6713	1	57	department	department	NOUN
ejpam-6713	1	58	of	of	ADP
ejpam-6713	1	59	mathematics	mathematic	NOUN
ejpam-6713	1	60	,	,	PUNCT
ejpam-6713	1	61	saveetha	saveetha	PROPN
ejpam-6713	1	62	school	school	PROPN
ejpam-6713	1	63	of	of	ADP
ejpam-6713	1	64	engineering	engineering	PROPN
ejpam-6713	1	65	,	,	PUNCT
ejpam-6713	1	66	saveetha	saveetha	PROPN
ejpam-6713	1	67	institute	institute	PROPN
ejpam-6713	1	68	of	of	ADP
ejpam-6713	1	69	medical	medical	ADJ
ejpam-6713	1	70	and	and	CCONJ
ejpam-6713	1	71	technical	technical	ADJ
ejpam-6713	1	72	sciences	science	NOUN
ejpam-6713	1	73	,	,	PUNCT
ejpam-6713	1	74	chennai	chennai	NOUN
ejpam-6713	1	75	602105	602105	NUM
ejpam-6713	1	76	,	,	PUNCT
ejpam-6713	1	77	tamil	tamil	PROPN
ejpam-6713	1	78	nadu	nadu	PROPN
ejpam-6713	1	79	,	,	PUNCT
ejpam-6713	1	80	india	india	PROPN
ejpam-6713	1	81	2	2	NUM
ejpam-6713	1	82	department	department	NOUN
ejpam-6713	1	83	of	of	ADP
ejpam-6713	1	84	mathematics	mathematic	NOUN
ejpam-6713	1	85	,	,	PUNCT
ejpam-6713	1	86	technical	technical	ADJ
ejpam-6713	1	87	university	university	PROPN
ejpam-6713	1	88	of	of	ADP
ejpam-6713	1	89	cluj	cluj	PROPN
ejpam-6713	1	90	-	-	PUNCT
ejpam-6713	1	91	napoca	napoca	NOUN
ejpam-6713	1	92	,	,	PUNCT
ejpam-6713	1	93	400114	400114	NUM
ejpam-6713	1	94	cluj	cluj	PROPN
ejpam-6713	1	95	-	-	PUNCT
ejpam-6713	1	96	napoca	napoca	PROPN
ejpam-6713	1	97	,	,	PUNCT
ejpam-6713	1	98	romania	romania	PROPN
ejpam-6713	1	99	3	3	NUM
ejpam-6713	1	100	department	department	NOUN
ejpam-6713	1	101	of	of	ADP
ejpam-6713	1	102	mathematics	mathematic	NOUN
ejpam-6713	1	103	,	,	PUNCT
ejpam-6713	1	104	“	"	PUNCT
ejpam-6713	1	105	1	1	NUM
ejpam-6713	1	106	decembrie	decembrie	NOUN
ejpam-6713	1	107	1918	1918	NUM
ejpam-6713	1	108	”	"	PUNCT
ejpam-6713	1	109	university	university	PROPN
ejpam-6713	1	110	of	of	ADP
ejpam-6713	1	111	alba	alba	PROPN
ejpam-6713	1	112	iulia	iulia	PROPN
ejpam-6713	1	113	,	,	PUNCT
ejpam-6713	1	114	510009	510009	NUM
ejpam-6713	1	115	alba	alba	NOUN
ejpam-6713	1	116	iulia	iulia	PROPN
ejpam-6713	1	117	,	,	PUNCT
ejpam-6713	1	118	romania	romania	PROPN
ejpam-6713	1	119	abstract	abstract	NOUN
ejpam-6713	1	120	.	.	PUNCT
ejpam-6713	2	1	a	a	DET
ejpam-6713	2	2	certified	certify	VERB
ejpam-6713	2	3	dominating	dominating	NOUN
ejpam-6713	2	4	set	set	NOUN
ejpam-6713	2	5	s	s	VERB
ejpam-6713	2	6	is	be	AUX
ejpam-6713	2	7	a	a	DET
ejpam-6713	2	8	dominating	dominating	NOUN
ejpam-6713	2	9	set	set	NOUN
ejpam-6713	2	10	of	of	ADP
ejpam-6713	2	11	a	a	DET
ejpam-6713	2	12	graph	graph	NOUN
ejpam-6713	2	13	g	g	NOUN
ejpam-6713	2	14	,	,	PUNCT
ejpam-6713	2	15	if	if	SCONJ
ejpam-6713	2	16	every	every	DET
ejpam-6713	2	17	vertex	vertex	NOUN
ejpam-6713	2	18	in	in	ADP
ejpam-6713	2	19	s	s	PROPN
ejpam-6713	2	20	has	have	VERB
ejpam-6713	2	21	either	either	CCONJ
ejpam-6713	2	22	zero	zero	NUM
ejpam-6713	2	23	or	or	CCONJ
ejpam-6713	2	24	at	at	ADP
ejpam-6713	2	25	least	least	ADV
ejpam-6713	2	26	two	two	NUM
ejpam-6713	2	27	neighbours	neighbour	NOUN
ejpam-6713	2	28	in	in	ADP
ejpam-6713	2	29	v	v	NUM
ejpam-6713	2	30	\s	\s	NOUN
ejpam-6713	2	31	.	.	PUNCT
ejpam-6713	3	1	the	the	DET
ejpam-6713	3	2	minimum	minimum	ADJ
ejpam-6713	3	3	cardinality	cardinality	NOUN
ejpam-6713	3	4	of	of	ADP
ejpam-6713	3	5	certified	certify	VERB
ejpam-6713	3	6	dominating	dominating	NOUN
ejpam-6713	3	7	set	set	NOUN
ejpam-6713	3	8	of	of	ADP
ejpam-6713	3	9	g	g	PROPN
ejpam-6713	3	10	is	be	AUX
ejpam-6713	3	11	the	the	DET
ejpam-6713	3	12	certified	certify	VERB
ejpam-6713	3	13	domination	domination	NOUN
ejpam-6713	3	14	number	number	NOUN
ejpam-6713	3	15	of	of	ADP
ejpam-6713	3	16	g	g	PROPN
ejpam-6713	3	17	denoted	denote	VERB
ejpam-6713	3	18	by	by	ADP
ejpam-6713	3	19	γcer(g	γcer(g	PROPN
ejpam-6713	3	20	)	)	PUNCT
ejpam-6713	3	21	.	.	PUNCT
ejpam-6713	4	1	we	we	PRON
ejpam-6713	4	2	defined	define	VERB
ejpam-6713	4	3	certified	certify	VERB
ejpam-6713	4	4	domination	domination	NOUN
ejpam-6713	4	5	subdivision	subdivision	NOUN
ejpam-6713	4	6	number	number	NOUN
ejpam-6713	4	7	sd+γcer	sd+γcer	NOUN
ejpam-6713	4	8	(	(	PUNCT
ejpam-6713	4	9	g	g	NOUN
ejpam-6713	4	10	)	)	PUNCT
ejpam-6713	5	1	[	[	X
ejpam-6713	5	2	sd−γcer	sd−γcer	NOUN
ejpam-6713	5	3	(	(	PUNCT
ejpam-6713	5	4	g	g	NOUN
ejpam-6713	5	5	)	)	PUNCT
ejpam-6713	5	6	]	]	PUNCT
ejpam-6713	5	7	of	of	ADP
ejpam-6713	5	8	a	a	DET
ejpam-6713	5	9	graph	graph	NOUN
ejpam-6713	5	10	g	g	NOUN
ejpam-6713	5	11	to	to	PART
ejpam-6713	5	12	be	be	AUX
ejpam-6713	5	13	the	the	DET
ejpam-6713	5	14	minimum	minimum	ADJ
ejpam-6713	5	15	number	number	NOUN
ejpam-6713	5	16	of	of	ADP
ejpam-6713	5	17	edges	edge	NOUN
ejpam-6713	5	18	that	that	PRON
ejpam-6713	5	19	must	must	AUX
ejpam-6713	5	20	be	be	AUX
ejpam-6713	5	21	subdivided	subdivide	VERB
ejpam-6713	5	22	(	(	PUNCT
ejpam-6713	5	23	where	where	SCONJ
ejpam-6713	5	24	no	no	DET
ejpam-6713	5	25	edge	edge	NOUN
ejpam-6713	5	26	in	in	ADP
ejpam-6713	5	27	g	g	PROPN
ejpam-6713	5	28	can	can	AUX
ejpam-6713	5	29	be	be	AUX
ejpam-6713	5	30	subdivided	subdivide	VERB
ejpam-6713	5	31	more	more	ADV
ejpam-6713	5	32	than	than	ADP
ejpam-6713	5	33	once	once	ADV
ejpam-6713	5	34	)	)	PUNCT
ejpam-6713	5	35	in	in	ADP
ejpam-6713	5	36	order	order	NOUN
ejpam-6713	5	37	to	to	PART
ejpam-6713	5	38	construct	construct	VERB
ejpam-6713	5	39	a	a	DET
ejpam-6713	5	40	graph	graph	NOUN
ejpam-6713	5	41	with	with	ADP
ejpam-6713	5	42	a	a	DET
ejpam-6713	5	43	certified	certify	VERB
ejpam-6713	5	44	domination	domination	NOUN
ejpam-6713	5	45	number	number	NOUN
ejpam-6713	5	46	larger	large	ADJ
ejpam-6713	5	47	[	[	X
ejpam-6713	5	48	lesser	lesser	X
ejpam-6713	5	49	]	]	PUNCT
ejpam-6713	5	50	than	than	ADP
ejpam-6713	5	51	the	the	DET
ejpam-6713	5	52	certified	certify	VERB
ejpam-6713	5	53	domination	domination	NOUN
ejpam-6713	5	54	number	number	NOUN
ejpam-6713	5	55	of	of	ADP
ejpam-6713	5	56	g.	g.	PROPN
ejpam-6713	5	57	in	in	ADP
ejpam-6713	5	58	this	this	DET
ejpam-6713	5	59	paper	paper	NOUN
ejpam-6713	5	60	,	,	PUNCT
ejpam-6713	5	61	we	we	PRON
ejpam-6713	5	62	determine	determine	VERB
ejpam-6713	5	63	the	the	DET
ejpam-6713	5	64	values	value	NOUN
ejpam-6713	5	65	of	of	ADP
ejpam-6713	5	66	certified	certified	ADJ
ejpam-6713	5	67	domination	domination	NOUN
ejpam-6713	5	68	subdivision	subdivision	NOUN
ejpam-6713	5	69	number	number	NOUN
ejpam-6713	5	70	for	for	ADP
ejpam-6713	5	71	certain	certain	ADJ
ejpam-6713	5	72	classes	class	NOUN
ejpam-6713	5	73	of	of	ADP
ejpam-6713	5	74	graphs	graph	NOUN
ejpam-6713	5	75	including	include	VERB
ejpam-6713	5	76	circulant	circulant	ADJ
ejpam-6713	5	77	graphs	graph	NOUN
ejpam-6713	5	78	[	[	X
ejpam-6713	5	79	cn(1	cn(1	X
ejpam-6713	5	80	,	,	PUNCT
ejpam-6713	5	81	2	2	NUM
ejpam-6713	5	82	)	)	PUNCT
ejpam-6713	5	83	and	and	CCONJ
ejpam-6713	5	84	cn(1	cn(1	PROPN
ejpam-6713	5	85	,	,	PUNCT
ejpam-6713	5	86	3	3	NUM
ejpam-6713	5	87	)	)	PUNCT
ejpam-6713	5	88	]	]	PUNCT
ejpam-6713	5	89	and	and	CCONJ
ejpam-6713	5	90	petersen	petersen	NOUN
ejpam-6713	5	91	graphs	graph	VERB
ejpam-6713	5	92	[	[	X
ejpam-6713	5	93	p	p	X
ejpam-6713	5	94	(	(	PUNCT
ejpam-6713	5	95	n	n	CCONJ
ejpam-6713	5	96	,	,	PUNCT
ejpam-6713	5	97	1	1	NUM
ejpam-6713	5	98	)	)	PUNCT
ejpam-6713	5	99	and	and	CCONJ
ejpam-6713	5	100	p	p	X
ejpam-6713	5	101	(	(	PUNCT
ejpam-6713	5	102	n	n	CCONJ
ejpam-6713	5	103	,	,	PUNCT
ejpam-6713	5	104	2	2	NUM
ejpam-6713	5	105	)	)	PUNCT
ejpam-6713	5	106	]	]	PUNCT
ejpam-6713	5	107	.	.	PUNCT
ejpam-6713	6	1	2020	2020	NUM
ejpam-6713	6	2	mathematics	mathematic	NOUN
ejpam-6713	6	3	subject	subject	NOUN
ejpam-6713	6	4	classifications	classification	NOUN
ejpam-6713	6	5	:	:	PUNCT
ejpam-6713	6	6	05c38	05c38	NOUN
ejpam-6713	6	7	,	,	PUNCT
ejpam-6713	6	8	05c69	05c69	NUM
ejpam-6713	6	9	,	,	PUNCT
ejpam-6713	6	10	05c75	05c75	NUM
ejpam-6713	6	11	key	key	ADJ
ejpam-6713	6	12	words	word	NOUN
ejpam-6713	6	13	and	and	CCONJ
ejpam-6713	6	14	phrases	phrase	NOUN
ejpam-6713	6	15	:	:	PUNCT
ejpam-6713	6	16	domination	domination	NOUN
ejpam-6713	6	17	number	number	NOUN
ejpam-6713	6	18	,	,	PUNCT
ejpam-6713	6	19	certified	certify	VERB
ejpam-6713	6	20	domination	domination	NOUN
ejpam-6713	6	21	number	number	NOUN
ejpam-6713	6	22	,	,	PUNCT
ejpam-6713	6	23	subdivision	subdivision	NOUN
ejpam-6713	6	24	number	number	NOUN
ejpam-6713	6	25	,	,	PUNCT
ejpam-6713	6	26	certified	certify	VERB
ejpam-6713	6	27	domination	domination	NOUN
ejpam-6713	6	28	subdivision	subdivision	NOUN
ejpam-6713	6	29	number	number	NOUN
ejpam-6713	6	30	1	1	NUM
ejpam-6713	6	31	.	.	PUNCT
ejpam-6713	7	1	introduction	introduction	NOUN
ejpam-6713	7	2	haynes	hayne	NOUN
ejpam-6713	8	1	[	[	X
ejpam-6713	8	2	1	1	X
ejpam-6713	8	3	]	]	PUNCT
ejpam-6713	8	4	introduced	introduce	VERB
ejpam-6713	8	5	the	the	DET
ejpam-6713	8	6	most	most	ADV
ejpam-6713	8	7	fundamental	fundamental	ADJ
ejpam-6713	8	8	and	and	CCONJ
ejpam-6713	8	9	well	well	ADV
ejpam-6713	8	10	studied	study	VERB
ejpam-6713	8	11	concepts	concept	NOUN
ejpam-6713	8	12	in	in	ADP
ejpam-6713	8	13	graph	graph	NOUN
ejpam-6713	8	14	theory	theory	NOUN
ejpam-6713	8	15	called	call	VERB
ejpam-6713	8	16	domination	domination	NOUN
ejpam-6713	8	17	in	in	ADP
ejpam-6713	8	18	graphs	graph	NOUN
ejpam-6713	8	19	.	.	PUNCT
ejpam-6713	9	1	a	a	DET
ejpam-6713	9	2	dominating	dominating	NOUN
ejpam-6713	9	3	set	set	NOUN
ejpam-6713	9	4	of	of	ADP
ejpam-6713	9	5	a	a	DET
ejpam-6713	9	6	graph	graph	NOUN
ejpam-6713	9	7	g	g	NOUN
ejpam-6713	9	8	is	be	AUX
ejpam-6713	9	9	a	a	DET
ejpam-6713	9	10	set	set	NOUN
ejpam-6713	9	11	s	s	NOUN
ejpam-6713	9	12	⊆	⊆	NUM
ejpam-6713	9	13	v	v	NOUN
ejpam-6713	9	14	with	with	ADP
ejpam-6713	9	15	the	the	DET
ejpam-6713	9	16	property	property	NOUN
ejpam-6713	9	17	that	that	PRON
ejpam-6713	9	18	for	for	ADP
ejpam-6713	9	19	each	each	DET
ejpam-6713	9	20	vertex	vertex	NOUN
ejpam-6713	9	21	u	u	NOUN
ejpam-6713	9	22	∈	∈	PROPN
ejpam-6713	9	23	v	v	ADP
ejpam-6713	9	24	\	\	NOUN
ejpam-6713	9	25	s	s	PART
ejpam-6713	9	26	there	there	PRON
ejpam-6713	9	27	exists	exist	VERB
ejpam-6713	9	28	at	at	ADP
ejpam-6713	9	29	least	least	ADJ
ejpam-6713	9	30	a	a	DET
ejpam-6713	9	31	vertex	vertex	NOUN
ejpam-6713	9	32	x	x	PUNCT
ejpam-6713	9	33	∈	∈	NOUN
ejpam-6713	9	34	s	s	VERB
ejpam-6713	9	35	adjacent	adjacent	ADJ
ejpam-6713	9	36	to	to	PART
ejpam-6713	9	37	u.	u.	VERB
ejpam-6713	9	38	the	the	DET
ejpam-6713	9	39	minimum	minimum	ADJ
ejpam-6713	9	40	cardinality	cardinality	NOUN
ejpam-6713	9	41	amongst	amongst	ADP
ejpam-6713	9	42	all	all	DET
ejpam-6713	9	43	dominating	dominating	NOUN
ejpam-6713	9	44	sets	set	NOUN
ejpam-6713	9	45	of	of	ADP
ejpam-6713	9	46	g	g	PROPN
ejpam-6713	9	47	is	be	AUX
ejpam-6713	9	48	the	the	DET
ejpam-6713	9	49	domination	domination	NOUN
ejpam-6713	9	50	number	number	NOUN
ejpam-6713	9	51	γ(g	γ(g	PROPN
ejpam-6713	9	52	)	)	PUNCT
ejpam-6713	9	53	and	and	CCONJ
ejpam-6713	9	54	s	s	VERB
ejpam-6713	9	55	is	be	AUX
ejpam-6713	9	56	called	call	VERB
ejpam-6713	9	57	γ	γ	NOUN
ejpam-6713	9	58	-	-	PUNCT
ejpam-6713	9	59	set	set	NOUN
ejpam-6713	9	60	of	of	ADP
ejpam-6713	9	61	g	g	NOUN
ejpam-6713	9	62	,	,	PUNCT
ejpam-6713	9	63	if	if	SCONJ
ejpam-6713	9	64	s	s	VERB
ejpam-6713	9	65	is	be	AUX
ejpam-6713	9	66	minimum	minimum	ADJ
ejpam-6713	9	67	.	.	PUNCT
ejpam-6713	10	1	many	many	ADJ
ejpam-6713	10	2	advanced	advanced	ADJ
ejpam-6713	10	3	researches	research	NOUN
ejpam-6713	10	4	are	be	AUX
ejpam-6713	10	5	going	go	VERB
ejpam-6713	10	6	∗corresponding	∗corresponde	VERB
ejpam-6713	10	7	author	author	NOUN
ejpam-6713	10	8	.	.	PUNCT
ejpam-6713	11	1	∗corresponding	∗corresponde	VERB
ejpam-6713	11	2	author	author	NOUN
ejpam-6713	11	3	.	.	PUNCT
ejpam-6713	12	1	doi	doi	NOUN
ejpam-6713	12	2	:	:	PUNCT
ejpam-6713	12	3	https://doi.org/10.29020/nybg.ejpam.v18i4.6713	https://doi.org/10.29020/nybg.ejpam.v18i4.6713	PROPN
ejpam-6713	12	4	email	email	NOUN
ejpam-6713	12	5	addresses	address	NOUN
ejpam-6713	12	6	:	:	PUNCT
ejpam-6713	12	7	navamaniprakash69@gmail.com	navamaniprakash69@gmail.com	X
ejpam-6713	12	8	(	(	PUNCT
ejpam-6713	12	9	g.	g.	PROPN
ejpam-6713	12	10	navamani	navamani	PROPN
ejpam-6713	12	11	)	)	PUNCT
ejpam-6713	12	12	,	,	PUNCT
ejpam-6713	12	13	nsumathiphd2022@gmail.com	nsumathiphd2022@gmail.com	X
ejpam-6713	13	1	(	(	PUNCT
ejpam-6713	13	2	n.	n.	NOUN
ejpam-6713	13	3	sumathi	sumathi	PROPN
ejpam-6713	13	4	)	)	PUNCT
ejpam-6713	13	5	,	,	PUNCT
ejpam-6713	13	6	luminita.cotirla@math.utcluj.ro	luminita.cotirla@math.utcluj.ro	NOUN
ejpam-6713	13	7	(	(	PUNCT
ejpam-6713	13	8	l.	l.	PROPN
ejpam-6713	13	9	i.	i.	PROPN
ejpam-6713	13	10	cotîrlă	cotîrlă	PROPN
ejpam-6713	13	11	)	)	PUNCT
ejpam-6713	13	12	,	,	PUNCT
ejpam-6713	13	13	dbreaz@uab.ro	dbreaz@uab.ro	PROPN
ejpam-6713	13	14	(	(	PUNCT
ejpam-6713	13	15	d.	d.	NOUN
ejpam-6713	13	16	breaz	breaz	PROPN
ejpam-6713	13	17	)	)	PUNCT
ejpam-6713	13	18	https://www.ejpam.com	https://www.ejpam.com	X
ejpam-6713	14	1	1	1	NUM
ejpam-6713	14	2	copyright	copyright	NOUN
ejpam-6713	14	3	:	:	PUNCT
ejpam-6713	14	4	©	©	PROPN
ejpam-6713	14	5	2025	2025	NUM
ejpam-6713	14	6	the	the	DET
ejpam-6713	14	7	author(s	author(s	NOUN
ejpam-6713	14	8	)	)	PUNCT
ejpam-6713	14	9	.	.	PUNCT
ejpam-6713	15	1	(	(	PUNCT
ejpam-6713	15	2	cc	cc	NOUN
ejpam-6713	15	3	by	by	ADP
ejpam-6713	15	4	-	-	PUNCT
ejpam-6713	15	5	nc	nc	PROPN
ejpam-6713	15	6	4.0	4.0	NUM
ejpam-6713	15	7	)	)	PUNCT
ejpam-6713	15	8	g.	g.	NOUN
ejpam-6713	15	9	navamani	navamani	PROPN
ejpam-6713	15	10	et	et	PROPN
ejpam-6713	15	11	al	al	PROPN
ejpam-6713	15	12	.	.	PUNCT
ejpam-6713	15	13	/	/	SYM
ejpam-6713	15	14	eur	eur	PROPN
ejpam-6713	15	15	.	.	PUNCT
ejpam-6713	16	1	j.	j.	PROPN
ejpam-6713	16	2	pure	pure	PROPN
ejpam-6713	16	3	appl	appl	PROPN
ejpam-6713	16	4	.	.	PROPN
ejpam-6713	16	5	math	math	PROPN
ejpam-6713	16	6	,	,	PUNCT
ejpam-6713	16	7	18	18	NUM
ejpam-6713	16	8	(	(	PUNCT
ejpam-6713	16	9	4	4	NUM
ejpam-6713	16	10	)	)	PUNCT
ejpam-6713	16	11	(	(	PUNCT
ejpam-6713	16	12	2025	2025	NUM
ejpam-6713	16	13	)	)	PUNCT
ejpam-6713	16	14	,	,	PUNCT
ejpam-6713	16	15	6713	6713	NUM
ejpam-6713	16	16	2	2	NUM
ejpam-6713	16	17	of	of	ADP
ejpam-6713	16	18	15	15	NUM
ejpam-6713	16	19	on	on	ADV
ejpam-6713	16	20	in	in	ADP
ejpam-6713	16	21	the	the	DET
ejpam-6713	16	22	variety	variety	NOUN
ejpam-6713	16	23	of	of	ADP
ejpam-6713	16	24	domination	domination	NOUN
ejpam-6713	16	25	terminology	terminology	NOUN
ejpam-6713	16	26	[	[	X
ejpam-6713	16	27	2],[3	2],[3	NUM
ejpam-6713	16	28	]	]	PUNCT
ejpam-6713	16	29	.	.	PUNCT
ejpam-6713	17	1	one	one	NUM
ejpam-6713	17	2	latest	late	ADJ
ejpam-6713	17	3	among	among	ADP
ejpam-6713	17	4	these	these	DET
ejpam-6713	17	5	varieties	variety	NOUN
ejpam-6713	17	6	is	be	AUX
ejpam-6713	17	7	certified	certify	VERB
ejpam-6713	17	8	domination	domination	NOUN
ejpam-6713	17	9	which	which	PRON
ejpam-6713	17	10	was	be	AUX
ejpam-6713	17	11	introduced	introduce	VERB
ejpam-6713	17	12	by	by	ADP
ejpam-6713	17	13	magda	magda	PROPN
ejpam-6713	17	14	dettlaff	dettlaff	VERB
ejpam-6713	17	15	et	et	PROPN
ejpam-6713	17	16	al	al	PROPN
ejpam-6713	18	1	[	[	X
ejpam-6713	18	2	4	4	NUM
ejpam-6713	18	3	]	]	PUNCT
ejpam-6713	18	4	.	.	PUNCT
ejpam-6713	19	1	a	a	DET
ejpam-6713	19	2	certified	certify	VERB
ejpam-6713	19	3	dominating	dominating	NOUN
ejpam-6713	19	4	set	set	NOUN
ejpam-6713	19	5	is	be	AUX
ejpam-6713	19	6	defined	define	VERB
ejpam-6713	19	7	as	as	ADP
ejpam-6713	19	8	d	d	PROPN
ejpam-6713	19	9	⊆	⊆	PROPN
ejpam-6713	19	10	v	v	NOUN
ejpam-6713	19	11	is	be	AUX
ejpam-6713	19	12	a	a	DET
ejpam-6713	19	13	dominating	dominating	NOUN
ejpam-6713	19	14	set	set	NOUN
ejpam-6713	19	15	of	of	ADP
ejpam-6713	19	16	a	a	DET
ejpam-6713	19	17	graph	graph	NOUN
ejpam-6713	19	18	g	g	NOUN
ejpam-6713	19	19	and	and	CCONJ
ejpam-6713	19	20	every	every	DET
ejpam-6713	19	21	vertex	vertex	NOUN
ejpam-6713	19	22	in	in	ADP
ejpam-6713	19	23	d	d	PROPN
ejpam-6713	19	24	has	have	VERB
ejpam-6713	19	25	either	either	CCONJ
ejpam-6713	19	26	zero	zero	NUM
ejpam-6713	19	27	or	or	CCONJ
ejpam-6713	19	28	at	at	ADP
ejpam-6713	19	29	least	least	ADV
ejpam-6713	19	30	two	two	NUM
ejpam-6713	19	31	neighbours	neighbour	NOUN
ejpam-6713	19	32	in	in	ADP
ejpam-6713	19	33	v	v	PROPN
ejpam-6713	19	34	\d	\d	NOUN
ejpam-6713	19	35	.	.	PUNCT
ejpam-6713	20	1	γcer(g	γcer(g	NOUN
ejpam-6713	20	2	)	)	PUNCT
ejpam-6713	20	3	is	be	AUX
ejpam-6713	20	4	the	the	DET
ejpam-6713	20	5	certified	certify	VERB
ejpam-6713	20	6	domination	domination	NOUN
ejpam-6713	20	7	number	number	NOUN
ejpam-6713	20	8	of	of	ADP
ejpam-6713	20	9	g	g	NOUN
ejpam-6713	20	10	which	which	PRON
ejpam-6713	20	11	is	be	AUX
ejpam-6713	20	12	defined	define	VERB
ejpam-6713	20	13	as	as	ADP
ejpam-6713	20	14	the	the	DET
ejpam-6713	20	15	minimum	minimum	ADJ
ejpam-6713	20	16	cardinality	cardinality	NOUN
ejpam-6713	20	17	of	of	ADP
ejpam-6713	20	18	certified	certify	VERB
ejpam-6713	20	19	dominating	dominating	NOUN
ejpam-6713	20	20	set	set	NOUN
ejpam-6713	20	21	of	of	ADP
ejpam-6713	20	22	g	g	PROPN
ejpam-6713	20	23	and	and	CCONJ
ejpam-6713	20	24	d	d	PROPN
ejpam-6713	20	25	is	be	AUX
ejpam-6713	20	26	the	the	DET
ejpam-6713	20	27	γcer	γcer	NOUN
ejpam-6713	20	28	-	-	PUNCT
ejpam-6713	20	29	set	set	NOUN
ejpam-6713	20	30	of	of	ADP
ejpam-6713	20	31	g	g	NOUN
ejpam-6713	20	32	,	,	PUNCT
ejpam-6713	20	33	if	if	SCONJ
ejpam-6713	20	34	d	d	NOUN
ejpam-6713	20	35	is	be	AUX
ejpam-6713	20	36	minimum	minimum	ADJ
ejpam-6713	20	37	.	.	PUNCT
ejpam-6713	21	1	further	further	ADJ
ejpam-6713	21	2	results	result	NOUN
ejpam-6713	21	3	on	on	ADP
ejpam-6713	21	4	this	this	DET
ejpam-6713	21	5	parameter	parameter	NOUN
ejpam-6713	21	6	seen	see	VERB
ejpam-6713	21	7	in	in	ADP
ejpam-6713	21	8	[	[	X
ejpam-6713	21	9	5–9	5–9	X
ejpam-6713	21	10	]	]	X
ejpam-6713	21	11	.	.	PUNCT
ejpam-6713	22	1	an	an	DET
ejpam-6713	22	2	edge	edge	NOUN
ejpam-6713	22	3	uv	uv	PROPN
ejpam-6713	22	4	∈	∈	PROPN
ejpam-6713	22	5	e(g	e(g	PROPN
ejpam-6713	22	6	)	)	PUNCT
ejpam-6713	22	7	is	be	AUX
ejpam-6713	22	8	subdivided	subdivide	VERB
ejpam-6713	22	9	if	if	SCONJ
ejpam-6713	22	10	the	the	DET
ejpam-6713	22	11	edge	edge	NOUN
ejpam-6713	22	12	uv	uv	NOUN
ejpam-6713	22	13	is	be	AUX
ejpam-6713	22	14	deleted	delete	VERB
ejpam-6713	22	15	,	,	PUNCT
ejpam-6713	22	16	but	but	CCONJ
ejpam-6713	22	17	a	a	DET
ejpam-6713	22	18	new	new	ADJ
ejpam-6713	22	19	vertex	vertex	NOUN
ejpam-6713	22	20	called	call	VERB
ejpam-6713	22	21	subdivision	subdivision	NOUN
ejpam-6713	22	22	vertex	vertex	NOUN
ejpam-6713	22	23	w	w	NOUN
ejpam-6713	22	24	is	be	AUX
ejpam-6713	22	25	added	add	VERB
ejpam-6713	22	26	along	along	ADP
ejpam-6713	22	27	with	with	ADP
ejpam-6713	22	28	two	two	NUM
ejpam-6713	22	29	new	new	ADJ
ejpam-6713	22	30	edges	edge	NOUN
ejpam-6713	22	31	uw	uw	PROPN
ejpam-6713	22	32	and	and	CCONJ
ejpam-6713	22	33	vw	vw	PROPN
ejpam-6713	22	34	.	.	PUNCT
ejpam-6713	23	1	the	the	DET
ejpam-6713	23	2	domination	domination	NOUN
ejpam-6713	23	3	subdivision	subdivision	NOUN
ejpam-6713	23	4	number	number	NOUN
ejpam-6713	23	5	sd(g	sd(g	NUM
ejpam-6713	23	6	)	)	PUNCT
ejpam-6713	23	7	of	of	ADP
ejpam-6713	23	8	a	a	DET
ejpam-6713	23	9	graph	graph	NOUN
ejpam-6713	23	10	g	g	NOUN
ejpam-6713	23	11	is	be	AUX
ejpam-6713	23	12	the	the	DET
ejpam-6713	23	13	minimum	minimum	ADJ
ejpam-6713	23	14	number	number	NOUN
ejpam-6713	23	15	of	of	ADP
ejpam-6713	23	16	edges	edge	NOUN
ejpam-6713	23	17	which	which	PRON
ejpam-6713	23	18	must	must	AUX
ejpam-6713	23	19	be	be	AUX
ejpam-6713	23	20	subdivided	subdivide	VERB
ejpam-6713	23	21	(	(	PUNCT
ejpam-6713	23	22	where	where	SCONJ
ejpam-6713	23	23	each	each	DET
ejpam-6713	23	24	edge	edge	NOUN
ejpam-6713	23	25	can	can	AUX
ejpam-6713	23	26	be	be	AUX
ejpam-6713	23	27	subdivided	subdivide	VERB
ejpam-6713	23	28	at	at	ADP
ejpam-6713	23	29	most	most	ADV
ejpam-6713	23	30	once	once	ADV
ejpam-6713	23	31	)	)	PUNCT
ejpam-6713	23	32	in	in	ADP
ejpam-6713	23	33	order	order	NOUN
ejpam-6713	23	34	to	to	PART
ejpam-6713	23	35	increase	increase	VERB
ejpam-6713	23	36	the	the	DET
ejpam-6713	23	37	domination	domination	NOUN
ejpam-6713	23	38	number	number	NOUN
ejpam-6713	23	39	.	.	PUNCT
ejpam-6713	24	1	s.	s.	PROPN
ejpam-6713	24	2	arumugam	arumugam	PROPN
ejpam-6713	24	3	and	and	CCONJ
ejpam-6713	24	4	j.	j.	PROPN
ejpam-6713	24	5	paulraj	paulraj	PROPN
ejpam-6713	24	6	joseph	joseph	PROPN
ejpam-6713	25	1	[	[	X
ejpam-6713	25	2	10	10	NUM
ejpam-6713	25	3	]	]	PUNCT
ejpam-6713	25	4	first	first	ADV
ejpam-6713	25	5	defined	define	VERB
ejpam-6713	25	6	the	the	DET
ejpam-6713	25	7	domination	domination	NOUN
ejpam-6713	25	8	subdivision	subdivision	NOUN
ejpam-6713	25	9	number	number	NOUN
ejpam-6713	25	10	sd(g	sd(g	NUM
ejpam-6713	25	11	)	)	PUNCT
ejpam-6713	25	12	of	of	ADP
ejpam-6713	25	13	a	a	DET
ejpam-6713	25	14	graph	graph	NOUN
ejpam-6713	25	15	g	g	NOUN
ejpam-6713	25	16	and	and	CCONJ
ejpam-6713	25	17	showed	show	VERB
ejpam-6713	25	18	that	that	SCONJ
ejpam-6713	25	19	sd(t	sd(t	NUM
ejpam-6713	25	20	)	)	PUNCT
ejpam-6713	25	21	≤	≤	NUM
ejpam-6713	25	22	3	3	NUM
ejpam-6713	25	23	for	for	ADP
ejpam-6713	25	24	any	any	DET
ejpam-6713	25	25	tree	tree	NOUN
ejpam-6713	25	26	t	t	NOUN
ejpam-6713	25	27	with	with	ADP
ejpam-6713	25	28	at	at	ADV
ejpam-6713	25	29	least	least	ADV
ejpam-6713	25	30	three	three	NUM
ejpam-6713	25	31	vertices	vertex	NOUN
ejpam-6713	25	32	.	.	PUNCT
ejpam-6713	26	1	other	other	ADJ
ejpam-6713	26	2	results	result	NOUN
ejpam-6713	26	3	and	and	CCONJ
ejpam-6713	26	4	general	general	ADJ
ejpam-6713	26	5	bounds	bound	NOUN
ejpam-6713	26	6	of	of	ADP
ejpam-6713	26	7	domination	domination	NOUN
ejpam-6713	26	8	subdivision	subdivision	NOUN
ejpam-6713	26	9	number	number	NOUN
ejpam-6713	26	10	can	can	AUX
ejpam-6713	26	11	be	be	AUX
ejpam-6713	26	12	found	find	VERB
ejpam-6713	26	13	in	in	ADP
ejpam-6713	26	14	[	[	X
ejpam-6713	26	15	11–16	11–16	NUM
ejpam-6713	26	16	]	]	PUNCT
ejpam-6713	26	17	.	.	PUNCT
ejpam-6713	27	1	motivated	motivate	VERB
ejpam-6713	27	2	by	by	ADP
ejpam-6713	27	3	recent	recent	ADJ
ejpam-6713	27	4	researches	research	NOUN
ejpam-6713	27	5	focusing	focus	VERB
ejpam-6713	27	6	on	on	ADP
ejpam-6713	27	7	subdivision	subdivision	NOUN
ejpam-6713	27	8	number	number	NOUN
ejpam-6713	27	9	,	,	PUNCT
ejpam-6713	27	10	we	we	PRON
ejpam-6713	27	11	defined	define	VERB
ejpam-6713	27	12	certified	certify	VERB
ejpam-6713	27	13	domination	domination	NOUN
ejpam-6713	27	14	subdivision	subdivision	NOUN
ejpam-6713	27	15	number	number	NOUN
ejpam-6713	27	16	of	of	ADP
ejpam-6713	27	17	a	a	DET
ejpam-6713	27	18	graph	graph	NOUN
ejpam-6713	27	19	g	g	NOUN
ejpam-6713	27	20	denoted	denote	VERB
ejpam-6713	27	21	by	by	ADP
ejpam-6713	27	22	sd+γcer(g	sd+γcer(g	NOUN
ejpam-6713	27	23	)	)	PUNCT
ejpam-6713	28	1	[	[	X
ejpam-6713	28	2	sd−γcer(g	sd−γcer(g	NOUN
ejpam-6713	28	3	)	)	PUNCT
ejpam-6713	28	4	]	]	PUNCT
ejpam-6713	28	5	to	to	PART
ejpam-6713	28	6	be	be	AUX
ejpam-6713	28	7	the	the	DET
ejpam-6713	28	8	minimum	minimum	ADJ
ejpam-6713	28	9	number	number	NOUN
ejpam-6713	28	10	of	of	ADP
ejpam-6713	28	11	edges	edge	NOUN
ejpam-6713	28	12	which	which	PRON
ejpam-6713	28	13	must	must	AUX
ejpam-6713	28	14	be	be	AUX
ejpam-6713	28	15	subdivided	subdivide	VERB
ejpam-6713	28	16	(	(	PUNCT
ejpam-6713	28	17	where	where	SCONJ
ejpam-6713	28	18	each	each	DET
ejpam-6713	28	19	edge	edge	NOUN
ejpam-6713	28	20	can	can	AUX
ejpam-6713	28	21	be	be	AUX
ejpam-6713	28	22	subdivided	subdivide	VERB
ejpam-6713	28	23	at	at	ADP
ejpam-6713	28	24	most	most	ADV
ejpam-6713	28	25	once	once	ADV
ejpam-6713	28	26	)	)	PUNCT
ejpam-6713	28	27	in	in	ADP
ejpam-6713	28	28	order	order	NOUN
ejpam-6713	28	29	to	to	PART
ejpam-6713	28	30	increase	increase	VERB
ejpam-6713	28	31	[	[	X
ejpam-6713	28	32	decrease	decrease	NOUN
ejpam-6713	28	33	]	]	PUNCT
ejpam-6713	28	34	the	the	DET
ejpam-6713	28	35	certified	certify	VERB
ejpam-6713	28	36	domination	domination	NOUN
ejpam-6713	28	37	number	number	NOUN
ejpam-6713	28	38	of	of	ADP
ejpam-6713	28	39	g	g	PROPN
ejpam-6713	28	40	and	and	CCONJ
ejpam-6713	28	41	also	also	ADV
ejpam-6713	28	42	we	we	PRON
ejpam-6713	28	43	characterised	characterise	VERB
ejpam-6713	28	44	these	these	DET
ejpam-6713	28	45	parameters	parameter	NOUN
ejpam-6713	28	46	for	for	ADP
ejpam-6713	28	47	trees	tree	NOUN
ejpam-6713	28	48	in	in	ADP
ejpam-6713	28	49	[	[	X
ejpam-6713	28	50	17	17	NUM
ejpam-6713	28	51	]	]	PUNCT
ejpam-6713	28	52	.	.	PUNCT
ejpam-6713	29	1	domination	domination	NOUN
ejpam-6713	29	2	in	in	ADP
ejpam-6713	29	3	graphs	graph	NOUN
ejpam-6713	29	4	has	have	VERB
ejpam-6713	29	5	applications	application	NOUN
ejpam-6713	29	6	in	in	ADP
ejpam-6713	29	7	a	a	DET
ejpam-6713	29	8	variety	variety	NOUN
ejpam-6713	29	9	of	of	ADP
ejpam-6713	29	10	fields	field	NOUN
ejpam-6713	29	11	.	.	PUNCT
ejpam-6713	30	1	domination	domination	NOUN
ejpam-6713	30	2	occurs	occur	VERB
ejpam-6713	30	3	in	in	ADP
ejpam-6713	30	4	facility	facility	NOUN
ejpam-6713	30	5	location	location	NOUN
ejpam-6713	30	6	problems	problem	NOUN
ejpam-6713	30	7	in	in	ADP
ejpam-6713	30	8	which	which	PRON
ejpam-6713	30	9	the	the	DET
ejpam-6713	30	10	number	number	NOUN
ejpam-6713	30	11	of	of	ADP
ejpam-6713	30	12	facilities	facility	NOUN
ejpam-6713	30	13	(	(	PUNCT
ejpam-6713	30	14	e.g.	e.g.	ADV
ejpam-6713	30	15	,	,	PUNCT
ejpam-6713	30	16	health	health	NOUN
ejpam-6713	30	17	centres	centre	NOUN
ejpam-6713	30	18	,	,	PUNCT
ejpam-6713	30	19	police	police	NOUN
ejpam-6713	30	20	stations	station	NOUN
ejpam-6713	30	21	)	)	PUNCT
ejpam-6713	30	22	is	be	AUX
ejpam-6713	30	23	fixed	fix	VERB
ejpam-6713	30	24	and	and	CCONJ
ejpam-6713	30	25	an	an	DET
ejpam-6713	30	26	attempt	attempt	NOUN
ejpam-6713	30	27	is	be	AUX
ejpam-6713	30	28	made	make	VERB
ejpam-6713	30	29	to	to	PART
ejpam-6713	30	30	minimize	minimize	VERB
ejpam-6713	30	31	the	the	DET
ejpam-6713	30	32	distance	distance	NOUN
ejpam-6713	30	33	that	that	PRON
ejpam-6713	30	34	a	a	DET
ejpam-6713	30	35	person	person	NOUN
ejpam-6713	30	36	must	must	AUX
ejpam-6713	30	37	travel	travel	VERB
ejpam-6713	30	38	to	to	PART
ejpam-6713	30	39	reach	reach	VERB
ejpam-6713	30	40	the	the	DET
ejpam-6713	30	41	facility	facility	NOUN
ejpam-6713	30	42	.	.	PUNCT
ejpam-6713	31	1	certified	certify	VERB
ejpam-6713	31	2	domination	domination	NOUN
ejpam-6713	31	3	is	be	AUX
ejpam-6713	31	4	one	one	NUM
ejpam-6713	31	5	such	such	ADJ
ejpam-6713	31	6	latest	late	ADJ
ejpam-6713	31	7	parameter	parameter	NOUN
ejpam-6713	31	8	,	,	PUNCT
ejpam-6713	31	9	in	in	ADP
ejpam-6713	31	10	which	which	PRON
ejpam-6713	31	11	a	a	DET
ejpam-6713	31	12	set	set	NOUN
ejpam-6713	31	13	s	s	X
ejpam-6713	31	14	is	be	AUX
ejpam-6713	31	15	the	the	DET
ejpam-6713	31	16	facility	facility	NOUN
ejpam-6713	31	17	centres	centre	NOUN
ejpam-6713	31	18	and	and	CCONJ
ejpam-6713	31	19	set	set	VERB
ejpam-6713	31	20	t	t	PROPN
ejpam-6713	31	21	is	be	AUX
ejpam-6713	31	22	the	the	DET
ejpam-6713	31	23	area	area	NOUN
ejpam-6713	31	24	of	of	ADP
ejpam-6713	31	25	stakeholders	stakeholder	NOUN
ejpam-6713	31	26	.	.	PUNCT
ejpam-6713	32	1	for	for	ADP
ejpam-6713	32	2	each	each	DET
ejpam-6713	32	3	area	area	NOUN
ejpam-6713	32	4	x	x	SYM
ejpam-6713	32	5	∈	∈	PROPN
ejpam-6713	32	6	t	t	NOUN
ejpam-6713	32	7	,	,	PUNCT
ejpam-6713	32	8	there	there	PRON
ejpam-6713	32	9	must	must	AUX
ejpam-6713	32	10	be	be	AUX
ejpam-6713	32	11	a	a	DET
ejpam-6713	32	12	facility	facility	NOUN
ejpam-6713	32	13	centre	centre	NOUN
ejpam-6713	32	14	v	v	ADP
ejpam-6713	32	15	∈	∈	PROPN
ejpam-6713	32	16	s	s	NOUN
ejpam-6713	32	17	,	,	PUNCT
ejpam-6713	32	18	that	that	PRON
ejpam-6713	32	19	can	can	AUX
ejpam-6713	32	20	serve	serve	VERB
ejpam-6713	32	21	for	for	ADP
ejpam-6713	32	22	x	x	PUNCT
ejpam-6713	32	23	and	and	CCONJ
ejpam-6713	32	24	whenever	whenever	SCONJ
ejpam-6713	32	25	such	such	ADJ
ejpam-6713	32	26	v	v	NOUN
ejpam-6713	32	27	is	be	AUX
ejpam-6713	32	28	serving	serve	VERB
ejpam-6713	32	29	x	x	PRON
ejpam-6713	32	30	,	,	PUNCT
ejpam-6713	32	31	there	there	PRON
ejpam-6713	32	32	must	must	AUX
ejpam-6713	32	33	also	also	ADV
ejpam-6713	32	34	be	be	AUX
ejpam-6713	32	35	at	at	ADV
ejpam-6713	32	36	least	least	ADJ
ejpam-6713	32	37	one	one	NUM
ejpam-6713	32	38	neighbouring	neighbouring	NOUN
ejpam-6713	32	39	area	area	NOUN
ejpam-6713	32	40	y	y	PROPN
ejpam-6713	32	41	∈	∈	PROPN
ejpam-6713	32	42	t	t	PROPN
ejpam-6713	32	43	that	that	PRON
ejpam-6713	32	44	uses	use	VERB
ejpam-6713	32	45	the	the	DET
ejpam-6713	32	46	facility	facility	NOUN
ejpam-6713	32	47	centre	centre	NOUN
ejpam-6713	33	1	v.	v.	CCONJ
ejpam-6713	33	2	we	we	PRON
ejpam-6713	33	3	can	can	AUX
ejpam-6713	33	4	determine	determine	VERB
ejpam-6713	33	5	the	the	DET
ejpam-6713	33	6	minimum	minimum	ADJ
ejpam-6713	33	7	number	number	NOUN
ejpam-6713	33	8	of	of	ADP
ejpam-6713	33	9	facility	facility	NOUN
ejpam-6713	33	10	centres	centre	NOUN
ejpam-6713	33	11	either	either	ADV
ejpam-6713	33	12	to	to	PART
ejpam-6713	33	13	take	take	VERB
ejpam-6713	33	14	care	care	NOUN
ejpam-6713	33	15	of	of	ADP
ejpam-6713	33	16	its	its	PRON
ejpam-6713	33	17	area	area	NOUN
ejpam-6713	33	18	(	(	PUNCT
ejpam-6713	33	19	where	where	SCONJ
ejpam-6713	33	20	it	it	PRON
ejpam-6713	33	21	situated	situate	VERB
ejpam-6713	33	22	)	)	PUNCT
ejpam-6713	33	23	or	or	CCONJ
ejpam-6713	33	24	it	it	PRON
ejpam-6713	33	25	takes	take	VERB
ejpam-6713	33	26	care	care	NOUN
ejpam-6713	33	27	of	of	ADP
ejpam-6713	33	28	more	more	ADJ
ejpam-6713	33	29	than	than	ADP
ejpam-6713	33	30	one	one	NUM
ejpam-6713	33	31	neighbouring	neighbouring	ADJ
ejpam-6713	33	32	areas	area	NOUN
ejpam-6713	33	33	.	.	PUNCT
ejpam-6713	34	1	here	here	ADV
ejpam-6713	34	2	we	we	PRON
ejpam-6713	34	3	introduce	introduce	VERB
ejpam-6713	34	4	subdivision	subdivision	NOUN
ejpam-6713	34	5	certified	certify	VERB
ejpam-6713	34	6	domination	domination	NOUN
ejpam-6713	34	7	number	number	NOUN
ejpam-6713	34	8	,	,	PUNCT
ejpam-6713	34	9	which	which	PRON
ejpam-6713	34	10	is	be	AUX
ejpam-6713	34	11	to	to	PART
ejpam-6713	34	12	determine	determine	VERB
ejpam-6713	34	13	how	how	SCONJ
ejpam-6713	34	14	the	the	DET
ejpam-6713	34	15	areas	area	NOUN
ejpam-6713	34	16	are	be	AUX
ejpam-6713	34	17	subdivided	subdivide	VERB
ejpam-6713	34	18	according	accord	VERB
ejpam-6713	34	19	to	to	ADP
ejpam-6713	34	20	the	the	DET
ejpam-6713	34	21	convenience	convenience	NOUN
ejpam-6713	34	22	of	of	ADP
ejpam-6713	34	23	the	the	DET
ejpam-6713	34	24	stakeholders	stakeholder	NOUN
ejpam-6713	34	25	in	in	ADP
ejpam-6713	34	26	neighbour	neighbour	ADJ
ejpam-6713	34	27	areas	area	NOUN
ejpam-6713	34	28	so	so	SCONJ
ejpam-6713	34	29	as	as	SCONJ
ejpam-6713	34	30	to	to	PART
ejpam-6713	34	31	facilitate	facilitate	VERB
ejpam-6713	34	32	them	they	PRON
ejpam-6713	34	33	to	to	PART
ejpam-6713	34	34	save	save	VERB
ejpam-6713	34	35	time	time	NOUN
ejpam-6713	34	36	and	and	CCONJ
ejpam-6713	34	37	money	money	NOUN
ejpam-6713	34	38	without	without	ADP
ejpam-6713	34	39	increasing	increase	VERB
ejpam-6713	34	40	the	the	DET
ejpam-6713	34	41	facility	facility	NOUN
ejpam-6713	34	42	centres	centre	NOUN
ejpam-6713	34	43	[	[	PUNCT
ejpam-6713	34	44	some	some	DET
ejpam-6713	34	45	times	time	NOUN
ejpam-6713	34	46	the	the	DET
ejpam-6713	34	47	number	number	NOUN
ejpam-6713	34	48	of	of	ADP
ejpam-6713	34	49	facility	facility	NOUN
ejpam-6713	34	50	centres	centre	NOUN
ejpam-6713	34	51	can	can	AUX
ejpam-6713	34	52	also	also	ADV
ejpam-6713	34	53	be	be	AUX
ejpam-6713	34	54	reduced	reduce	VERB
ejpam-6713	34	55	due	due	ADP
ejpam-6713	34	56	to	to	ADP
ejpam-6713	34	57	the	the	DET
ejpam-6713	34	58	subdivision	subdivision	NOUN
ejpam-6713	34	59	of	of	ADP
ejpam-6713	34	60	areas	area	NOUN
ejpam-6713	34	61	]	]	PUNCT
ejpam-6713	34	62	.	.	PUNCT
ejpam-6713	35	1	in	in	ADP
ejpam-6713	35	2	this	this	DET
ejpam-6713	35	3	paper	paper	NOUN
ejpam-6713	35	4	,	,	PUNCT
ejpam-6713	35	5	we	we	PRON
ejpam-6713	35	6	determine	determine	VERB
ejpam-6713	35	7	the	the	DET
ejpam-6713	35	8	values	value	NOUN
ejpam-6713	35	9	of	of	ADP
ejpam-6713	35	10	certified	certified	ADJ
ejpam-6713	35	11	domination	domination	NOUN
ejpam-6713	35	12	subdivision	subdivision	NOUN
ejpam-6713	35	13	number	number	NOUN
ejpam-6713	35	14	for	for	ADP
ejpam-6713	35	15	certain	certain	ADJ
ejpam-6713	35	16	classes	class	NOUN
ejpam-6713	35	17	of	of	ADP
ejpam-6713	35	18	graphs	graph	NOUN
ejpam-6713	35	19	including	include	VERB
ejpam-6713	35	20	circulant	circulant	ADJ
ejpam-6713	35	21	graphs	graph	NOUN
ejpam-6713	35	22	[	[	X
ejpam-6713	35	23	cn(1	cn(1	X
ejpam-6713	35	24	,	,	PUNCT
ejpam-6713	35	25	2	2	NUM
ejpam-6713	35	26	)	)	PUNCT
ejpam-6713	35	27	and	and	CCONJ
ejpam-6713	35	28	cn(1	cn(1	PROPN
ejpam-6713	35	29	,	,	PUNCT
ejpam-6713	35	30	3	3	NUM
ejpam-6713	35	31	)	)	PUNCT
ejpam-6713	35	32	]	]	PUNCT
ejpam-6713	35	33	and	and	CCONJ
ejpam-6713	35	34	petersen	petersen	NOUN
ejpam-6713	35	35	graphs	graph	VERB
ejpam-6713	35	36	[	[	X
ejpam-6713	35	37	p	p	X
ejpam-6713	35	38	(	(	PUNCT
ejpam-6713	35	39	n	n	CCONJ
ejpam-6713	35	40	,	,	PUNCT
ejpam-6713	35	41	1	1	NUM
ejpam-6713	35	42	)	)	PUNCT
ejpam-6713	35	43	and	and	CCONJ
ejpam-6713	35	44	p	p	X
ejpam-6713	35	45	(	(	PUNCT
ejpam-6713	35	46	n	n	CCONJ
ejpam-6713	35	47	,	,	PUNCT
ejpam-6713	35	48	2	2	NUM
ejpam-6713	35	49	)	)	PUNCT
ejpam-6713	35	50	]	]	PUNCT
ejpam-6713	35	51	.	.	PUNCT
ejpam-6713	36	1	2	2	X
ejpam-6713	36	2	.	.	X
ejpam-6713	36	3	notation	notation	NOUN
ejpam-6713	36	4	let	let	VERB
ejpam-6713	36	5	g	g	NOUN
ejpam-6713	36	6	=	=	SYM
ejpam-6713	36	7	(	(	PUNCT
ejpam-6713	36	8	v	v	NOUN
ejpam-6713	36	9	,	,	PUNCT
ejpam-6713	36	10	e	e	NOUN
ejpam-6713	36	11	)	)	PUNCT
ejpam-6713	36	12	be	be	AUX
ejpam-6713	36	13	a	a	DET
ejpam-6713	36	14	connected	connect	VERB
ejpam-6713	36	15	,	,	PUNCT
ejpam-6713	36	16	simple	simple	ADJ
ejpam-6713	36	17	graph	graph	NOUN
ejpam-6713	36	18	with	with	ADP
ejpam-6713	36	19	order	order	NOUN
ejpam-6713	36	20	|v	|v	VERB
ejpam-6713	37	1	|	|	ADV
ejpam-6713	37	2	=	=	PUNCT
ejpam-6713	37	3	n.	n.	NOUN
ejpam-6713	37	4	we	we	PRON
ejpam-6713	37	5	use	use	VERB
ejpam-6713	37	6	harary	harary	NOUN
ejpam-6713	37	7	’s	’s	PART
ejpam-6713	38	1	[	[	X
ejpam-6713	38	2	18	18	NUM
ejpam-6713	38	3	]	]	PUNCT
ejpam-6713	38	4	for	for	ADP
ejpam-6713	38	5	graph	graph	NOUN
ejpam-6713	38	6	theoretic	theoretic	ADJ
ejpam-6713	38	7	notation	notation	NOUN
ejpam-6713	38	8	.	.	PUNCT
ejpam-6713	39	1	for	for	ADP
ejpam-6713	39	2	any	any	DET
ejpam-6713	39	3	vertex	vertex	NOUN
ejpam-6713	39	4	v	v	ADP
ejpam-6713	39	5	∈	∈	PROPN
ejpam-6713	39	6	v	v	NOUN
ejpam-6713	39	7	,	,	PUNCT
ejpam-6713	39	8	the	the	DET
ejpam-6713	39	9	open	open	ADJ
ejpam-6713	39	10	neighbourhood	neighbourhood	NOUN
ejpam-6713	39	11	of	of	ADP
ejpam-6713	39	12	v	v	NOUN
ejpam-6713	39	13	is	be	AUX
ejpam-6713	39	14	the	the	DET
ejpam-6713	39	15	set	set	NOUN
ejpam-6713	39	16	n(v	n(v	PROPN
ejpam-6713	39	17	)	)	PUNCT
ejpam-6713	39	18	=	=	PRON
ejpam-6713	40	1	{	{	PUNCT
ejpam-6713	40	2	u	u	NOUN
ejpam-6713	40	3	∈	∈	PROPN
ejpam-6713	40	4	v	v	NOUN
ejpam-6713	40	5	:	:	PUNCT
ejpam-6713	40	6	uv	uv	NOUN
ejpam-6713	40	7	∈	∈	NOUN
ejpam-6713	40	8	e	e	NOUN
ejpam-6713	40	9	}	}	PUNCT
ejpam-6713	40	10	and	and	CCONJ
ejpam-6713	40	11	the	the	DET
ejpam-6713	40	12	closed	closed	ADJ
ejpam-6713	40	13	neighbourhood	neighbourhood	NOUN
ejpam-6713	40	14	is	be	AUX
ejpam-6713	40	15	the	the	DET
ejpam-6713	40	16	set	set	ADJ
ejpam-6713	40	17	n	n	PROPN
ejpam-6713	41	1	[	[	X
ejpam-6713	41	2	v	v	X
ejpam-6713	41	3	]	]	X
ejpam-6713	41	4	=	=	PUNCT
ejpam-6713	41	5	n(v	n(v	PROPN
ejpam-6713	41	6	)	)	PUNCT
ejpam-6713	41	7	∪	∪	NOUN
ejpam-6713	41	8	{	{	PUNCT
ejpam-6713	41	9	v	v	NOUN
ejpam-6713	41	10	}	}	PUNCT
ejpam-6713	41	11	.	.	PUNCT
ejpam-6713	42	1	for	for	ADP
ejpam-6713	42	2	g.	g.	PROPN
ejpam-6713	42	3	navamani	navamani	PROPN
ejpam-6713	42	4	et	et	PROPN
ejpam-6713	42	5	al	al	PROPN
ejpam-6713	42	6	.	.	PUNCT
ejpam-6713	42	7	/	/	SYM
ejpam-6713	42	8	eur	eur	PROPN
ejpam-6713	42	9	.	.	PUNCT
ejpam-6713	43	1	j.	j.	PROPN
ejpam-6713	43	2	pure	pure	PROPN
ejpam-6713	43	3	appl	appl	PROPN
ejpam-6713	43	4	.	.	PROPN
ejpam-6713	43	5	math	math	PROPN
ejpam-6713	43	6	,	,	PUNCT
ejpam-6713	43	7	18	18	NUM
ejpam-6713	43	8	(	(	PUNCT
ejpam-6713	43	9	4	4	NUM
ejpam-6713	43	10	)	)	PUNCT
ejpam-6713	43	11	(	(	PUNCT
ejpam-6713	43	12	2025	2025	NUM
ejpam-6713	43	13	)	)	PUNCT
ejpam-6713	43	14	,	,	PUNCT
ejpam-6713	43	15	6713	6713	NUM
ejpam-6713	43	16	3	3	NUM
ejpam-6713	43	17	of	of	ADP
ejpam-6713	43	18	15	15	NUM
ejpam-6713	43	19	a	a	DET
ejpam-6713	43	20	set	set	NOUN
ejpam-6713	43	21	s	s	NOUN
ejpam-6713	43	22	⊆	⊆	NUM
ejpam-6713	43	23	v	v	NOUN
ejpam-6713	43	24	,	,	PUNCT
ejpam-6713	43	25	the	the	DET
ejpam-6713	43	26	open	open	ADJ
ejpam-6713	43	27	neighbourhood	neighbourhood	NOUN
ejpam-6713	43	28	of	of	ADP
ejpam-6713	43	29	s	s	PROPN
ejpam-6713	43	30	is	be	AUX
ejpam-6713	43	31	n(s	n(s	PRON
ejpam-6713	43	32	)	)	PUNCT
ejpam-6713	43	33	=	=	SYM
ejpam-6713	44	1	⋃	⋃	ADP
ejpam-6713	44	2	v∈s	v∈s	ADJ
ejpam-6713	44	3	n(v	n(v	PROPN
ejpam-6713	44	4	)	)	PUNCT
ejpam-6713	44	5	,	,	PUNCT
ejpam-6713	44	6	the	the	DET
ejpam-6713	44	7	closed	closed	ADJ
ejpam-6713	44	8	neighbourhood	neighbourhood	NOUN
ejpam-6713	44	9	of	of	ADP
ejpam-6713	44	10	s	s	NOUN
ejpam-6713	44	11	is	be	AUX
ejpam-6713	44	12	n	n	PRON
ejpam-6713	44	13	[	[	X
ejpam-6713	44	14	s	s	X
ejpam-6713	44	15	]	]	X
ejpam-6713	44	16	=	=	PUNCT
ejpam-6713	44	17	n(s	n(s	PROPN
ejpam-6713	44	18	)	)	PUNCT
ejpam-6713	44	19	∪	∪	NOUN
ejpam-6713	44	20	s	s	NOUN
ejpam-6713	44	21	and	and	CCONJ
ejpam-6713	44	22	the	the	DET
ejpam-6713	44	23	private	private	ADJ
ejpam-6713	44	24	neighbourhood	neighbourhood	NOUN
ejpam-6713	44	25	pn(v	pn(v	NOUN
ejpam-6713	44	26	,	,	PUNCT
ejpam-6713	44	27	s	s	NOUN
ejpam-6713	44	28	)	)	PUNCT
ejpam-6713	44	29	of	of	ADP
ejpam-6713	44	30	a	a	DET
ejpam-6713	44	31	vertex	vertex	NOUN
ejpam-6713	44	32	u	u	NOUN
ejpam-6713	44	33	∈	∈	NOUN
ejpam-6713	44	34	s	s	VERB
ejpam-6713	44	35	is	be	AUX
ejpam-6713	44	36	defined	define	VERB
ejpam-6713	44	37	by	by	ADP
ejpam-6713	44	38	pn(v	pn(v	NOUN
ejpam-6713	44	39	,	,	PUNCT
ejpam-6713	44	40	s	s	X
ejpam-6713	44	41	)	)	PUNCT
ejpam-6713	44	42	=	=	SYM
ejpam-6713	44	43	{	{	PUNCT
ejpam-6713	44	44	u	u	NOUN
ejpam-6713	44	45	∈	∈	PROPN
ejpam-6713	44	46	v	v	ADP
ejpam-6713	44	47	−	−	PROPN
ejpam-6713	44	48	s	s	PART
ejpam-6713	44	49	:	:	PUNCT
ejpam-6713	44	50	n(u	n(u	PROPN
ejpam-6713	44	51	)	)	PUNCT
ejpam-6713	44	52	∩	∩	NOUN
ejpam-6713	44	53	s	s	PART
ejpam-6713	44	54	=	=	PUNCT
ejpam-6713	44	55	{	{	PUNCT
ejpam-6713	44	56	v	v	NOUN
ejpam-6713	44	57	}	}	PUNCT
ejpam-6713	44	58	}	}	PUNCT
ejpam-6713	44	59	.	.	PUNCT
ejpam-6713	45	1	a	a	DET
ejpam-6713	45	2	path	path	NOUN
ejpam-6713	45	3	is	be	AUX
ejpam-6713	45	4	a	a	DET
ejpam-6713	45	5	walk	walk	NOUN
ejpam-6713	45	6	with	with	ADP
ejpam-6713	45	7	no	no	DET
ejpam-6713	45	8	repeated	repeat	VERB
ejpam-6713	45	9	vertices	vertex	NOUN
ejpam-6713	45	10	.	.	PUNCT
ejpam-6713	46	1	a	a	DET
ejpam-6713	46	2	nontrivial	nontrivial	ADJ
ejpam-6713	46	3	closed	closed	ADJ
ejpam-6713	46	4	path	path	NOUN
ejpam-6713	46	5	is	be	AUX
ejpam-6713	46	6	called	call	VERB
ejpam-6713	46	7	a	a	DET
ejpam-6713	46	8	cycle	cycle	NOUN
ejpam-6713	46	9	.	.	PUNCT
ejpam-6713	47	1	a	a	DET
ejpam-6713	47	2	graph	graph	NOUN
ejpam-6713	47	3	g	g	PROPN
ejpam-6713	47	4	is	be	AUX
ejpam-6713	47	5	k	k	NOUN
ejpam-6713	47	6	-	-	ADJ
ejpam-6713	47	7	partite	partite	ADJ
ejpam-6713	47	8	,	,	PUNCT
ejpam-6713	47	9	k	k	X
ejpam-6713	47	10	≥	≥	NUM
ejpam-6713	47	11	1	1	NUM
ejpam-6713	47	12	if	if	SCONJ
ejpam-6713	47	13	it	it	PRON
ejpam-6713	47	14	is	be	AUX
ejpam-6713	47	15	possible	possible	ADJ
ejpam-6713	47	16	to	to	PART
ejpam-6713	47	17	partition	partition	VERB
ejpam-6713	47	18	v	v	NOUN
ejpam-6713	47	19	(	(	PUNCT
ejpam-6713	47	20	g	g	NOUN
ejpam-6713	47	21	)	)	PUNCT
ejpam-6713	47	22	into	into	ADP
ejpam-6713	47	23	k	k	PROPN
ejpam-6713	47	24	subsets	subset	NOUN
ejpam-6713	47	25	,	,	PUNCT
ejpam-6713	47	26	v1	v1	NOUN
ejpam-6713	47	27	,	,	PUNCT
ejpam-6713	47	28	v2	v2	PROPN
ejpam-6713	47	29	.	.	PUNCT
ejpam-6713	47	30	.	.	PUNCT
ejpam-6713	48	1	.	.	PUNCT
ejpam-6713	49	1	vk	vk	PROPN
ejpam-6713	49	2	(	(	PUNCT
ejpam-6713	49	3	called	call	VERB
ejpam-6713	49	4	partite	partite	ADJ
ejpam-6713	49	5	set	set	NOUN
ejpam-6713	49	6	)	)	PUNCT
ejpam-6713	49	7	such	such	ADJ
ejpam-6713	49	8	that	that	SCONJ
ejpam-6713	49	9	every	every	DET
ejpam-6713	49	10	element	element	NOUN
ejpam-6713	49	11	of	of	ADP
ejpam-6713	49	12	e(g	e(g	PROPN
ejpam-6713	49	13	)	)	PUNCT
ejpam-6713	49	14	joins	join	VERB
ejpam-6713	49	15	a	a	DET
ejpam-6713	49	16	vertex	vertex	NOUN
ejpam-6713	49	17	of	of	ADP
ejpam-6713	49	18	vi	vi	NOUN
ejpam-6713	49	19	to	to	ADP
ejpam-6713	49	20	a	a	DET
ejpam-6713	49	21	vertex	vertex	NOUN
ejpam-6713	49	22	of	of	ADP
ejpam-6713	49	23	vj	vj	PROPN
ejpam-6713	49	24	,	,	PUNCT
ejpam-6713	49	25	i	i	PROPN
ejpam-6713	49	26	6=	6=	PROPN
ejpam-6713	49	27	j.	j.	PROPN
ejpam-6713	49	28	if	if	SCONJ
ejpam-6713	49	29	g	g	PROPN
ejpam-6713	49	30	is	be	AUX
ejpam-6713	49	31	a	a	DET
ejpam-6713	49	32	1	1	NUM
ejpam-6713	49	33	-	-	PUNCT
ejpam-6713	49	34	partite	partite	ADJ
ejpam-6713	49	35	graph	graph	NOUN
ejpam-6713	49	36	of	of	ADP
ejpam-6713	49	37	order	order	NOUN
ejpam-6713	49	38	n	n	CCONJ
ejpam-6713	49	39	,	,	PUNCT
ejpam-6713	49	40	then	then	ADV
ejpam-6713	49	41	g	g	PROPN
ejpam-6713	49	42	=	=	PROPN
ejpam-6713	49	43	kn	kn	PROPN
ejpam-6713	49	44	.	.	PROPN
ejpam-6713	50	1	for	for	ADP
ejpam-6713	50	2	k	k	PROPN
ejpam-6713	50	3	=	=	SYM
ejpam-6713	50	4	2	2	NUM
ejpam-6713	50	5	,	,	PUNCT
ejpam-6713	50	6	such	such	ADJ
ejpam-6713	50	7	graphs	graph	NOUN
ejpam-6713	50	8	are	be	AUX
ejpam-6713	50	9	called	call	VERB
ejpam-6713	50	10	bipartite	bipartite	NOUN
ejpam-6713	50	11	graphs	graph	NOUN
ejpam-6713	50	12	.	.	PUNCT
ejpam-6713	51	1	a	a	DET
ejpam-6713	51	2	complete	complete	ADJ
ejpam-6713	51	3	bipartite	bipartite	NOUN
ejpam-6713	51	4	graph	graph	NOUN
ejpam-6713	51	5	is	be	AUX
ejpam-6713	51	6	a	a	DET
ejpam-6713	51	7	simple	simple	ADJ
ejpam-6713	51	8	bipartite	bipartite	NOUN
ejpam-6713	51	9	graph	graph	NOUN
ejpam-6713	51	10	such	such	ADJ
ejpam-6713	51	11	that	that	SCONJ
ejpam-6713	51	12	every	every	DET
ejpam-6713	51	13	vertex	vertex	NOUN
ejpam-6713	51	14	in	in	ADP
ejpam-6713	51	15	one	one	NUM
ejpam-6713	51	16	of	of	ADP
ejpam-6713	51	17	the	the	DET
ejpam-6713	51	18	bipartition	bipartition	NOUN
ejpam-6713	51	19	subsets	subset	NOUN
ejpam-6713	51	20	is	be	AUX
ejpam-6713	51	21	joined	join	VERB
ejpam-6713	51	22	to	to	ADP
ejpam-6713	51	23	every	every	DET
ejpam-6713	51	24	vertex	vertex	NOUN
ejpam-6713	51	25	in	in	ADP
ejpam-6713	51	26	the	the	DET
ejpam-6713	51	27	other	other	ADJ
ejpam-6713	51	28	bipartition	bipartition	NOUN
ejpam-6713	51	29	subset	subset	NOUN
ejpam-6713	51	30	.	.	PUNCT
ejpam-6713	52	1	any	any	DET
ejpam-6713	52	2	complete	complete	ADJ
ejpam-6713	52	3	bipartite	bipartite	NOUN
ejpam-6713	52	4	graph	graph	NOUN
ejpam-6713	52	5	that	that	PRON
ejpam-6713	52	6	has	have	AUX
ejpam-6713	52	7	m	m	NOUN
ejpam-6713	52	8	vertices	vertex	NOUN
ejpam-6713	52	9	in	in	ADP
ejpam-6713	52	10	one	one	NUM
ejpam-6713	52	11	of	of	ADP
ejpam-6713	52	12	its	its	PRON
ejpam-6713	52	13	bipartition	bipartition	NOUN
ejpam-6713	52	14	subsets	subset	NOUN
ejpam-6713	52	15	and	and	CCONJ
ejpam-6713	52	16	n	n	DET
ejpam-6713	52	17	vertices	vertex	NOUN
ejpam-6713	52	18	in	in	ADP
ejpam-6713	52	19	the	the	DET
ejpam-6713	52	20	other	other	ADJ
ejpam-6713	52	21	is	be	AUX
ejpam-6713	52	22	denoted	denote	VERB
ejpam-6713	52	23	km	km	NOUN
ejpam-6713	52	24	,	,	PUNCT
ejpam-6713	52	25	n.	n.	NOUN
ejpam-6713	52	26	a	a	DET
ejpam-6713	52	27	wheel	wheel	NOUN
ejpam-6713	52	28	graph	graph	NOUN
ejpam-6713	52	29	is	be	AUX
ejpam-6713	52	30	a	a	DET
ejpam-6713	52	31	graph	graph	NOUN
ejpam-6713	52	32	formed	form	VERB
ejpam-6713	52	33	by	by	ADP
ejpam-6713	52	34	connecting	connect	VERB
ejpam-6713	52	35	all	all	DET
ejpam-6713	52	36	vertices	vertex	NOUN
ejpam-6713	52	37	of	of	ADP
ejpam-6713	52	38	a	a	DET
ejpam-6713	52	39	cycle	cycle	NOUN
ejpam-6713	52	40	to	to	ADP
ejpam-6713	52	41	a	a	DET
ejpam-6713	52	42	single	single	ADJ
ejpam-6713	52	43	universal	universal	ADJ
ejpam-6713	52	44	vertex	vertex	NOUN
ejpam-6713	52	45	.	.	PUNCT
ejpam-6713	53	1	3	3	X
ejpam-6713	53	2	.	.	X
ejpam-6713	53	3	main	main	ADJ
ejpam-6713	53	4	results	result	NOUN
ejpam-6713	53	5	theorem	theorem	VERB
ejpam-6713	53	6	1	1	NUM
ejpam-6713	53	7	.	.	PUNCT
ejpam-6713	54	1	[	[	X
ejpam-6713	54	2	5	5	X
ejpam-6713	54	3	]	]	PUNCT
ejpam-6713	54	4	if	if	SCONJ
ejpam-6713	54	5	cn	cn	PROPN
ejpam-6713	54	6	is	be	AUX
ejpam-6713	54	7	an	an	DET
ejpam-6713	54	8	n	n	CCONJ
ejpam-6713	54	9	-	-	PUNCT
ejpam-6713	54	10	vertex	vertex	NOUN
ejpam-6713	54	11	cycle	cycle	NOUN
ejpam-6713	54	12	,	,	PUNCT
ejpam-6713	54	13	n	n	PRON
ejpam-6713	54	14	≥	≥	NOUN
ejpam-6713	54	15	3	3	NUM
ejpam-6713	54	16	,	,	PUNCT
ejpam-6713	54	17	then	then	ADV
ejpam-6713	54	18	γcer(cn	γcer(cn	NOUN
ejpam-6713	54	19	)	)	PUNCT
ejpam-6713	54	20	=	=	PUNCT
ejpam-6713	55	1	⌈	⌈	PROPN
ejpam-6713	55	2	n	n	CCONJ
ejpam-6713	55	3	3	3	NUM
ejpam-6713	55	4	⌉	⌉	X
ejpam-6713	55	5	theorem	theorem	VERB
ejpam-6713	55	6	2	2	NUM
ejpam-6713	55	7	.	.	X
ejpam-6713	55	8	for	for	ADP
ejpam-6713	55	9	any	any	DET
ejpam-6713	55	10	cycle	cycle	NOUN
ejpam-6713	55	11	cn	cn	PROPN
ejpam-6713	55	12	,	,	PUNCT
ejpam-6713	55	13	n	n	PRON
ejpam-6713	55	14	≥	≥	NOUN
ejpam-6713	55	15	4	4	NUM
ejpam-6713	55	16	sd+γcer(cn	sd+γcer(cn	NOUN
ejpam-6713	55	17	)	)	PUNCT
ejpam-6713	56	1	=	=	PUNCT
ejpam-6713	57	1			NOUN
ejpam-6713	57	2	1	1	NUM
ejpam-6713	57	3	if	if	SCONJ
ejpam-6713	57	4	n	n	PRON
ejpam-6713	57	5	≡	≡	PROPN
ejpam-6713	57	6	0	0	PUNCT
ejpam-6713	58	1	(	(	PUNCT
ejpam-6713	58	2	mod	mod	NOUN
ejpam-6713	58	3	3	3	NUM
ejpam-6713	58	4	)	)	PUNCT
ejpam-6713	58	5	2	2	NUM
ejpam-6713	58	6	if	if	SCONJ
ejpam-6713	58	7	n	n	PRON
ejpam-6713	58	8	≡	≡	PROPN
ejpam-6713	58	9	2	2	NUM
ejpam-6713	58	10	(	(	PUNCT
ejpam-6713	58	11	mod	mod	NOUN
ejpam-6713	58	12	3	3	NUM
ejpam-6713	58	13	)	)	PUNCT
ejpam-6713	58	14	3	3	NUM
ejpam-6713	58	15	if	if	SCONJ
ejpam-6713	58	16	n	n	PRON
ejpam-6713	58	17	≡	≡	PROPN
ejpam-6713	58	18	1	1	NUM
ejpam-6713	58	19	(	(	PUNCT
ejpam-6713	58	20	mod	mod	NOUN
ejpam-6713	58	21	3	3	NUM
ejpam-6713	58	22	)	)	PUNCT
ejpam-6713	58	23	proof	proof	NOUN
ejpam-6713	58	24	.	.	PUNCT
ejpam-6713	59	1	by	by	ADP
ejpam-6713	59	2	theorem	theorem	NOUN
ejpam-6713	59	3	1	1	NUM
ejpam-6713	59	4	,	,	PUNCT
ejpam-6713	59	5	γcer(cn	γcer(cn	NOUN
ejpam-6713	59	6	)	)	PUNCT
ejpam-6713	59	7	=	=	PUNCT
ejpam-6713	60	1	⌈	⌈	PROPN
ejpam-6713	60	2	n	n	CCONJ
ejpam-6713	60	3	3	3	NUM
ejpam-6713	60	4	⌉	⌉	NOUN
ejpam-6713	60	5	,	,	PUNCT
ejpam-6713	60	6	n	n	X
ejpam-6713	60	7	≥	≥	NOUN
ejpam-6713	60	8	4	4	NUM
ejpam-6713	60	9	.	.	PUNCT
ejpam-6713	61	1	let	let	AUX
ejpam-6713	61	2	d	d	PRON
ejpam-6713	61	3	be	be	AUX
ejpam-6713	61	4	a	a	DET
ejpam-6713	61	5	γcer	γcer	NOUN
ejpam-6713	61	6	-	-	PUNCT
ejpam-6713	61	7	set	set	NOUN
ejpam-6713	61	8	of	of	ADP
ejpam-6713	61	9	cn	cn	PROPN
ejpam-6713	61	10	.	.	PROPN
ejpam-6713	61	11	consider	consider	VERB
ejpam-6713	61	12	the	the	DET
ejpam-6713	61	13	following	follow	VERB
ejpam-6713	61	14	cases	case	NOUN
ejpam-6713	61	15	.	.	PUNCT
ejpam-6713	62	1	case	case	NOUN
ejpam-6713	62	2	(	(	PUNCT
ejpam-6713	62	3	i	i	NOUN
ejpam-6713	62	4	)	)	PUNCT
ejpam-6713	62	5	n	n	X
ejpam-6713	62	6	≡	≡	PROPN
ejpam-6713	62	7	0	0	PUNCT
ejpam-6713	63	1	(	(	PUNCT
ejpam-6713	63	2	mod	mod	NOUN
ejpam-6713	63	3	3	3	NUM
ejpam-6713	63	4	)	)	PUNCT
ejpam-6713	63	5	in	in	ADP
ejpam-6713	63	6	this	this	DET
ejpam-6713	63	7	case	case	NOUN
ejpam-6713	63	8	,	,	PUNCT
ejpam-6713	63	9	each	each	DET
ejpam-6713	63	10	vertex	vertex	NOUN
ejpam-6713	63	11	in	in	ADP
ejpam-6713	63	12	d	d	NOUN
ejpam-6713	63	13	dominates	dominate	VERB
ejpam-6713	63	14	exactly	exactly	ADV
ejpam-6713	63	15	3	3	NUM
ejpam-6713	63	16	vertices	vertex	NOUN
ejpam-6713	63	17	,	,	PUNCT
ejpam-6713	63	18	including	include	VERB
ejpam-6713	63	19	itself	itself	PRON
ejpam-6713	63	20	.	.	PUNCT
ejpam-6713	64	1	subdividing	subdivide	VERB
ejpam-6713	64	2	an	an	DET
ejpam-6713	64	3	edge	edge	NOUN
ejpam-6713	64	4	in	in	ADP
ejpam-6713	64	5	cn	cn	PROPN
ejpam-6713	64	6	results	result	NOUN
ejpam-6713	64	7	cn+1	cn+1	VERB
ejpam-6713	64	8	.	.	PUNCT
ejpam-6713	65	1	since	since	SCONJ
ejpam-6713	65	2	γcer(cn	γcer(cn	NOUN
ejpam-6713	65	3	)	)	PUNCT
ejpam-6713	65	4	=	=	PUNCT
ejpam-6713	65	5	⌈	⌈	PROPN
ejpam-6713	65	6	n	n	CCONJ
ejpam-6713	65	7	3	3	NUM
ejpam-6713	65	8	⌉	⌉	NOUN
ejpam-6713	65	9	,	,	PUNCT
ejpam-6713	65	10	for	for	ADP
ejpam-6713	65	11	n	n	X
ejpam-6713	65	12	≡	≡	PROPN
ejpam-6713	65	13	0	0	PUNCT
ejpam-6713	65	14	(	(	PUNCT
ejpam-6713	65	15	mod	mod	NOUN
ejpam-6713	65	16	3	3	NUM
ejpam-6713	65	17	)	)	PUNCT
ejpam-6713	65	18	.	.	PUNCT
ejpam-6713	66	1	so	so	ADV
ejpam-6713	66	2	we	we	PRON
ejpam-6713	66	3	need	need	VERB
ejpam-6713	66	4	to	to	PART
ejpam-6713	66	5	add	add	VERB
ejpam-6713	66	6	one	one	NUM
ejpam-6713	66	7	more	more	ADJ
ejpam-6713	66	8	vertex	vertex	NOUN
ejpam-6713	66	9	in	in	ADP
ejpam-6713	66	10	d	d	NOUN
ejpam-6713	66	11	to	to	PART
ejpam-6713	66	12	dominate	dominate	VERB
ejpam-6713	66	13	cn+1	cn+1	NUM
ejpam-6713	66	14	.	.	PUNCT
ejpam-6713	67	1	hence	hence	ADV
ejpam-6713	67	2	γcer	γcer	NOUN
ejpam-6713	67	3	(	(	PUNCT
ejpam-6713	67	4	cn+1	cn+1	NUM
ejpam-6713	67	5	)	)	PUNCT
ejpam-6713	67	6	>	>	X
ejpam-6713	67	7	γcer	γcer	X
ejpam-6713	67	8	(	(	PUNCT
ejpam-6713	67	9	cn	cn	PROPN
ejpam-6713	67	10	)	)	PUNCT
ejpam-6713	67	11	.	.	PUNCT
ejpam-6713	68	1	therefore	therefore	ADV
ejpam-6713	68	2	sd+γcer(cn	sd+γcer(cn	PROPN
ejpam-6713	68	3	)	)	PUNCT
ejpam-6713	68	4	=	=	SYM
ejpam-6713	69	1	1	1	X
ejpam-6713	69	2	.	.	X
ejpam-6713	69	3	case	case	NOUN
ejpam-6713	69	4	(	(	PUNCT
ejpam-6713	69	5	ii	ii	NOUN
ejpam-6713	69	6	)	)	PUNCT
ejpam-6713	69	7	n	n	CCONJ
ejpam-6713	69	8	≡	≡	PROPN
ejpam-6713	69	9	2	2	NUM
ejpam-6713	69	10	(	(	PUNCT
ejpam-6713	69	11	mod	mod	NOUN
ejpam-6713	69	12	3	3	NUM
ejpam-6713	69	13	)	)	PUNCT
ejpam-6713	69	14	in	in	ADP
ejpam-6713	69	15	this	this	DET
ejpam-6713	69	16	case	case	NOUN
ejpam-6713	69	17	,	,	PUNCT
ejpam-6713	69	18	each	each	DET
ejpam-6713	69	19	vertex	vertex	NOUN
ejpam-6713	69	20	in	in	ADP
ejpam-6713	69	21	d	d	NOUN
ejpam-6713	69	22	dominates	dominate	VERB
ejpam-6713	69	23	exactly	exactly	ADV
ejpam-6713	69	24	3	3	NUM
ejpam-6713	69	25	vertices	vertex	NOUN
ejpam-6713	69	26	including	include	VERB
ejpam-6713	69	27	itself	itself	PRON
ejpam-6713	69	28	except	except	SCONJ
ejpam-6713	69	29	one	one	NUM
ejpam-6713	69	30	which	which	PRON
ejpam-6713	69	31	dominates	dominate	VERB
ejpam-6713	69	32	2	2	NUM
ejpam-6713	69	33	vertices	vertex	NOUN
ejpam-6713	69	34	including	include	VERB
ejpam-6713	69	35	itself	itself	PRON
ejpam-6713	69	36	.	.	PUNCT
ejpam-6713	70	1	subdividing	subdivide	VERB
ejpam-6713	70	2	an	an	DET
ejpam-6713	70	3	edge	edge	NOUN
ejpam-6713	70	4	in	in	ADP
ejpam-6713	70	5	cn	cn	PROPN
ejpam-6713	70	6	results	result	NOUN
ejpam-6713	70	7	cn+1	cn+1	VERB
ejpam-6713	70	8	,	,	PUNCT
ejpam-6713	70	9	where	where	SCONJ
ejpam-6713	70	10	n	n	X
ejpam-6713	70	11	+	+	CCONJ
ejpam-6713	70	12	1	1	NUM
ejpam-6713	70	13	≡	≡	PROPN
ejpam-6713	70	14	0	0	PUNCT
ejpam-6713	70	15	(	(	PUNCT
ejpam-6713	70	16	mod	mod	NOUN
ejpam-6713	70	17	3	3	NUM
ejpam-6713	70	18	)	)	PUNCT
ejpam-6713	70	19	.	.	PUNCT
ejpam-6713	71	1	now	now	ADV
ejpam-6713	71	2	by	by	ADP
ejpam-6713	71	3	case	case	NOUN
ejpam-6713	71	4	(	(	PUNCT
ejpam-6713	71	5	i	i	NOUN
ejpam-6713	71	6	)	)	PUNCT
ejpam-6713	71	7	we	we	PRON
ejpam-6713	71	8	notice	notice	VERB
ejpam-6713	71	9	that	that	SCONJ
ejpam-6713	71	10	γcer(cn+1	γcer(cn+1	PROPN
ejpam-6713	71	11	)	)	PUNCT
ejpam-6713	72	1	=	=	SYM
ejpam-6713	72	2	γcer(cn	γcer(cn	PROPN
ejpam-6713	72	3	)	)	PUNCT
ejpam-6713	72	4	.	.	PUNCT
ejpam-6713	73	1	this	this	PRON
ejpam-6713	73	2	implies	imply	VERB
ejpam-6713	73	3	that	that	SCONJ
ejpam-6713	73	4	sd+γcer(g	sd+γcer(g	NOUN
ejpam-6713	73	5	)	)	PUNCT
ejpam-6713	73	6	>	>	X
ejpam-6713	74	1	1	1	X
ejpam-6713	74	2	.	.	PUNCT
ejpam-6713	74	3	by	by	ADP
ejpam-6713	74	4	case	case	NOUN
ejpam-6713	74	5	(	(	PUNCT
ejpam-6713	74	6	i	i	NOUN
ejpam-6713	74	7	)	)	PUNCT
ejpam-6713	74	8	we	we	PRON
ejpam-6713	74	9	need	need	VERB
ejpam-6713	74	10	to	to	PART
ejpam-6713	74	11	subdivide	subdivide	VERB
ejpam-6713	74	12	one	one	NUM
ejpam-6713	74	13	more	more	ADJ
ejpam-6713	74	14	edge	edge	NOUN
ejpam-6713	74	15	in	in	ADP
ejpam-6713	74	16	cn+1	cn+1	NUM
ejpam-6713	74	17	results	result	NOUN
ejpam-6713	74	18	cn+2	cn+2	PRON
ejpam-6713	74	19	.	.	PUNCT
ejpam-6713	75	1	hence	hence	ADV
ejpam-6713	75	2	γcer(cn+2	γcer(cn+2	PROPN
ejpam-6713	75	3	)	)	PUNCT
ejpam-6713	75	4	>	>	X
ejpam-6713	76	1	γcer(cn+1	γcer(cn+1	PROPN
ejpam-6713	76	2	)	)	PUNCT
ejpam-6713	76	3	.	.	PUNCT
ejpam-6713	77	1	therefore	therefore	ADV
ejpam-6713	77	2	sd+γcer(cn	sd+γcer(cn	PROPN
ejpam-6713	77	3	)	)	PUNCT
ejpam-6713	77	4	=	=	SYM
ejpam-6713	78	1	2	2	X
ejpam-6713	78	2	.	.	X
ejpam-6713	78	3	case	case	NOUN
ejpam-6713	78	4	(	(	PUNCT
ejpam-6713	78	5	iii	iii	NOUN
ejpam-6713	78	6	)	)	PUNCT
ejpam-6713	78	7	n	n	CCONJ
ejpam-6713	78	8	≡	≡	PROPN
ejpam-6713	78	9	1	1	NUM
ejpam-6713	78	10	(	(	PUNCT
ejpam-6713	78	11	mod	mod	NOUN
ejpam-6713	78	12	3	3	NUM
ejpam-6713	78	13	)	)	PUNCT
ejpam-6713	78	14	in	in	ADP
ejpam-6713	78	15	this	this	DET
ejpam-6713	78	16	case	case	NOUN
ejpam-6713	78	17	subdividing	subdivide	VERB
ejpam-6713	78	18	an	an	DET
ejpam-6713	78	19	edge	edge	NOUN
ejpam-6713	78	20	in	in	ADP
ejpam-6713	78	21	cn	cn	PROPN
ejpam-6713	78	22	results	result	NOUN
ejpam-6713	78	23	n	n	NUM
ejpam-6713	78	24	≡	≡	PROPN
ejpam-6713	78	25	2	2	NUM
ejpam-6713	78	26	(	(	PUNCT
ejpam-6713	78	27	mod	mod	NOUN
ejpam-6713	78	28	3	3	NUM
ejpam-6713	78	29	)	)	PUNCT
ejpam-6713	78	30	.	.	PUNCT
ejpam-6713	79	1	by	by	ADP
ejpam-6713	79	2	case	case	NOUN
ejpam-6713	79	3	(	(	PUNCT
ejpam-6713	79	4	ii	ii	NOUN
ejpam-6713	79	5	)	)	PUNCT
ejpam-6713	79	6	we	we	PRON
ejpam-6713	79	7	clearly	clearly	ADV
ejpam-6713	79	8	see	see	VERB
ejpam-6713	79	9	that	that	DET
ejpam-6713	79	10	sd+γcer(g	sd+γcer(g	NOUN
ejpam-6713	79	11	)	)	PUNCT
ejpam-6713	79	12	=	=	SYM
ejpam-6713	80	1	3	3	X
ejpam-6713	80	2	.	.	X
ejpam-6713	80	3	hence	hence	ADV
ejpam-6713	80	4	the	the	DET
ejpam-6713	80	5	proof	proof	NOUN
ejpam-6713	80	6	.	.	PUNCT
ejpam-6713	81	1	g.	g.	PROPN
ejpam-6713	81	2	navamani	navamani	PROPN
ejpam-6713	81	3	et	et	PROPN
ejpam-6713	81	4	al	al	PROPN
ejpam-6713	81	5	.	.	PUNCT
ejpam-6713	81	6	/	/	SYM
ejpam-6713	81	7	eur	eur	PROPN
ejpam-6713	81	8	.	.	PUNCT
ejpam-6713	82	1	j.	j.	PROPN
ejpam-6713	82	2	pure	pure	PROPN
ejpam-6713	82	3	appl	appl	PROPN
ejpam-6713	82	4	.	.	PROPN
ejpam-6713	82	5	math	math	PROPN
ejpam-6713	82	6	,	,	PUNCT
ejpam-6713	82	7	18	18	NUM
ejpam-6713	82	8	(	(	PUNCT
ejpam-6713	82	9	4	4	NUM
ejpam-6713	82	10	)	)	PUNCT
ejpam-6713	82	11	(	(	PUNCT
ejpam-6713	82	12	2025	2025	NUM
ejpam-6713	82	13	)	)	PUNCT
ejpam-6713	82	14	,	,	PUNCT
ejpam-6713	82	15	6713	6713	NUM
ejpam-6713	82	16	4	4	NUM
ejpam-6713	82	17	of	of	ADP
ejpam-6713	82	18	15	15	NUM
ejpam-6713	82	19	theorem	theorem	NOUN
ejpam-6713	82	20	3	3	NUM
ejpam-6713	82	21	.	.	PUNCT
ejpam-6713	83	1	[	[	X
ejpam-6713	83	2	4	4	X
ejpam-6713	83	3	]	]	PUNCT
ejpam-6713	83	4	let	let	VERB
ejpam-6713	83	5	g	g	PRON
ejpam-6713	83	6	be	be	AUX
ejpam-6713	83	7	a	a	DET
ejpam-6713	83	8	connected	connected	ADJ
ejpam-6713	83	9	graph	graph	NOUN
ejpam-6713	83	10	of	of	ADP
ejpam-6713	83	11	order	order	NOUN
ejpam-6713	83	12	at	at	ADV
ejpam-6713	83	13	least	least	ADV
ejpam-6713	83	14	three	three	NUM
ejpam-6713	83	15	vertices	vertex	NOUN
ejpam-6713	83	16	.	.	PUNCT
ejpam-6713	84	1	then	then	ADV
ejpam-6713	84	2	γ(g	γ(g	PROPN
ejpam-6713	84	3	)	)	PUNCT
ejpam-6713	85	1	=	=	SYM
ejpam-6713	85	2	γcer(g	γcer(g	NOUN
ejpam-6713	85	3	)	)	PUNCT
ejpam-6713	85	4	)	)	PUNCT
ejpam-6713	86	1	if	if	SCONJ
ejpam-6713	86	2	and	and	CCONJ
ejpam-6713	86	3	only	only	ADV
ejpam-6713	86	4	if	if	SCONJ
ejpam-6713	86	5	g	g	PROPN
ejpam-6713	86	6	has	have	VERB
ejpam-6713	86	7	a	a	DET
ejpam-6713	86	8	γ	γ	X
ejpam-6713	86	9	-	-	PUNCT
ejpam-6713	86	10	set	set	VERB
ejpam-6713	86	11	d	d	NOUN
ejpam-6713	86	12	such	such	ADJ
ejpam-6713	86	13	that	that	SCONJ
ejpam-6713	86	14	every	every	DET
ejpam-6713	86	15	vertex	vertex	NOUN
ejpam-6713	86	16	in	in	ADP
ejpam-6713	86	17	d	d	PROPN
ejpam-6713	86	18	has	have	VERB
ejpam-6713	86	19	at	at	ADV
ejpam-6713	86	20	least	least	ADV
ejpam-6713	86	21	two	two	NUM
ejpam-6713	86	22	neighbours	neighbour	NOUN
ejpam-6713	86	23	in	in	ADP
ejpam-6713	86	24	vg	vg	PROPN
ejpam-6713	86	25	−d	−d	PROPN
ejpam-6713	86	26	.	.	PUNCT
ejpam-6713	87	1	theorem	theorem	ADJ
ejpam-6713	87	2	4	4	NUM
ejpam-6713	87	3	.	.	PUNCT
ejpam-6713	88	1	[	[	X
ejpam-6713	88	2	5	5	X
ejpam-6713	88	3	]	]	PUNCT
ejpam-6713	88	4	if	if	SCONJ
ejpam-6713	88	5	km	km	PROPN
ejpam-6713	88	6	,	,	PUNCT
ejpam-6713	88	7	n	n	PRON
ejpam-6713	88	8	is	be	AUX
ejpam-6713	88	9	a	a	DET
ejpam-6713	88	10	complete	complete	ADJ
ejpam-6713	88	11	bipartite	bipartite	NOUN
ejpam-6713	88	12	graph	graph	NOUN
ejpam-6713	88	13	with	with	ADP
ejpam-6713	88	14	1	1	NUM
ejpam-6713	88	15	≤	≤	NUM
ejpam-6713	88	16	m	m	VERB
ejpam-6713	88	17	≤	≤	NOUN
ejpam-6713	88	18	n	n	CCONJ
ejpam-6713	88	19	,	,	PUNCT
ejpam-6713	88	20	then	then	ADV
ejpam-6713	88	21	γcer(km	γcer(km	PROPN
ejpam-6713	88	22	,	,	PUNCT
ejpam-6713	88	23	n	n	CCONJ
ejpam-6713	88	24	)	)	PUNCT
ejpam-6713	88	25	=	=	PRON
ejpam-6713	88	26	{	{	PUNCT
ejpam-6713	88	27	1	1	NUM
ejpam-6713	88	28	if	if	SCONJ
ejpam-6713	88	29	m	m	VERB
ejpam-6713	88	30	=	=	SYM
ejpam-6713	88	31	1	1	NUM
ejpam-6713	88	32	and	and	CCONJ
ejpam-6713	88	33	n	n	CCONJ
ejpam-6713	88	34	>	>	SYM
ejpam-6713	88	35	1	1	NUM
ejpam-6713	88	36	2	2	NUM
ejpam-6713	88	37	otherwise	otherwise	ADV
ejpam-6713	88	38	.	.	PUNCT
ejpam-6713	89	1	theorem	theorem	ADJ
ejpam-6713	89	2	5	5	NUM
ejpam-6713	89	3	.	.	PUNCT
ejpam-6713	90	1	for	for	ADP
ejpam-6713	90	2	complete	complete	ADJ
ejpam-6713	90	3	bipartite	bipartite	PROPN
ejpam-6713	90	4	graph	graph	NOUN
ejpam-6713	90	5	g	g	PROPN
ejpam-6713	90	6	=	=	SYM
ejpam-6713	90	7	km	km	PROPN
ejpam-6713	90	8	,	,	PUNCT
ejpam-6713	90	9	n	n	CCONJ
ejpam-6713	90	10	,	,	PUNCT
ejpam-6713	90	11	sd+γcer(g	sd+γcer(g	NOUN
ejpam-6713	90	12	)	)	PUNCT
ejpam-6713	91	1	=	=	PRON
ejpam-6713	91	2	{	{	PUNCT
ejpam-6713	91	3	3	3	NUM
ejpam-6713	91	4	if	if	SCONJ
ejpam-6713	91	5	m	m	VERB
ejpam-6713	91	6	=	=	SYM
ejpam-6713	91	7	2	2	NUM
ejpam-6713	91	8	and	and	CCONJ
ejpam-6713	91	9	n	n	PRON
ejpam-6713	91	10	≥	≥	NOUN
ejpam-6713	91	11	2	2	NUM
ejpam-6713	91	12	1	1	NUM
ejpam-6713	91	13	otherwise	otherwise	ADV
ejpam-6713	91	14	.	.	PUNCT
ejpam-6713	92	1	proof	proof	NOUN
ejpam-6713	92	2	.	.	PUNCT
ejpam-6713	93	1	let	let	VERB
ejpam-6713	93	2	g	g	PROPN
ejpam-6713	93	3	=	=	PROPN
ejpam-6713	93	4	km	km	PROPN
ejpam-6713	93	5	,	,	PUNCT
ejpam-6713	93	6	n	n	CCONJ
ejpam-6713	93	7	,	,	PUNCT
ejpam-6713	93	8	let	let	VERB
ejpam-6713	93	9	v1	v1	NOUN
ejpam-6713	93	10	,	,	PUNCT
ejpam-6713	93	11	v2	v2	PROPN
ejpam-6713	93	12	be	be	AUX
ejpam-6713	93	13	the	the	DET
ejpam-6713	93	14	vertex	vertex	NOUN
ejpam-6713	93	15	partition	partition	NOUN
ejpam-6713	93	16	of	of	ADP
ejpam-6713	93	17	g	g	PROPN
ejpam-6713	93	18	.	.	PUNCT
ejpam-6713	94	1	let	let	VERB
ejpam-6713	94	2	d	d	PRON
ejpam-6713	94	3	be	be	AUX
ejpam-6713	94	4	a	a	DET
ejpam-6713	94	5	γcer	γcer	NOUN
ejpam-6713	94	6	-	-	PUNCT
ejpam-6713	94	7	set	set	NOUN
ejpam-6713	94	8	of	of	ADP
ejpam-6713	94	9	g.	g.	PROPN
ejpam-6713	94	10	by	by	ADP
ejpam-6713	94	11	theorem	theorem	NOUN
ejpam-6713	94	12	3	3	NUM
ejpam-6713	94	13	,	,	PUNCT
ejpam-6713	94	14	for	for	ADP
ejpam-6713	94	15	all	all	DET
ejpam-6713	94	16	vertex	vertex	NOUN
ejpam-6713	94	17	v	v	ADP
ejpam-6713	94	18	∈	∈	PROPN
ejpam-6713	94	19	d	d	NOUN
ejpam-6713	94	20	,	,	PUNCT
ejpam-6713	94	21	|n(v)|	|n(v)|	PROPN
ejpam-6713	94	22	≥	≥	NOUN
ejpam-6713	94	23	2	2	NUM
ejpam-6713	94	24	,	,	PUNCT
ejpam-6713	94	25	we	we	PRON
ejpam-6713	94	26	clearly	clearly	ADV
ejpam-6713	94	27	see	see	VERB
ejpam-6713	94	28	that	that	PRON
ejpam-6713	94	29	γcer(g	γcer(g	NOUN
ejpam-6713	94	30	)	)	PUNCT
ejpam-6713	94	31	=	=	SYM
ejpam-6713	95	1	2	2	X
ejpam-6713	95	2	.	.	X
ejpam-6713	95	3	consider	consider	VERB
ejpam-6713	95	4	the	the	DET
ejpam-6713	95	5	following	follow	VERB
ejpam-6713	95	6	cases	case	NOUN
ejpam-6713	95	7	.	.	PUNCT
ejpam-6713	96	1	case	case	NOUN
ejpam-6713	96	2	(	(	PUNCT
ejpam-6713	96	3	i	i	NOUN
ejpam-6713	96	4	)	)	PUNCT
ejpam-6713	96	5	m	m	VERB
ejpam-6713	96	6	=	=	NOUN
ejpam-6713	96	7	2	2	NUM
ejpam-6713	96	8	in	in	ADP
ejpam-6713	96	9	this	this	DET
ejpam-6713	96	10	case	case	NOUN
ejpam-6713	96	11	,	,	PUNCT
ejpam-6713	96	12	we	we	PRON
ejpam-6713	96	13	have	have	VERB
ejpam-6713	96	14	the	the	DET
ejpam-6713	96	15	following	follow	VERB
ejpam-6713	96	16	subcases	subcase	NOUN
ejpam-6713	96	17	.	.	PUNCT
ejpam-6713	97	1	subcase	subcase	PROPN
ejpam-6713	97	2	(	(	PUNCT
ejpam-6713	97	3	i	i	NOUN
ejpam-6713	97	4	)	)	PUNCT
ejpam-6713	97	5	n	n	NOUN
ejpam-6713	97	6	=	=	SYM
ejpam-6713	97	7	2	2	NUM
ejpam-6713	97	8	here	here	ADV
ejpam-6713	97	9	k2,2	k2,2	PROPN
ejpam-6713	97	10	=	=	SYM
ejpam-6713	97	11	c4	c4	NOUN
ejpam-6713	97	12	,	,	PUNCT
ejpam-6713	97	13	by	by	ADP
ejpam-6713	97	14	theorem	theorem	NOUN
ejpam-6713	97	15	2	2	NUM
ejpam-6713	97	16	we	we	PRON
ejpam-6713	97	17	have	have	VERB
ejpam-6713	97	18	sd+γcer(c4	sd+γcer(c4	NOUN
ejpam-6713	97	19	)	)	PUNCT
ejpam-6713	97	20	=	=	SYM
ejpam-6713	98	1	3	3	X
ejpam-6713	98	2	.	.	PUNCT
ejpam-6713	98	3	hence	hence	ADV
ejpam-6713	98	4	sd+γcer(k2,2	sd+γcer(k2,2	NOUN
ejpam-6713	98	5	)	)	PUNCT
ejpam-6713	98	6	=	=	SYM
ejpam-6713	98	7	3	3	X
ejpam-6713	98	8	.	.	X
ejpam-6713	98	9	subcase	subcase	PROPN
ejpam-6713	98	10	(	(	PUNCT
ejpam-6713	98	11	ii	ii	NOUN
ejpam-6713	98	12	)	)	PUNCT
ejpam-6713	98	13	n	n	CCONJ
ejpam-6713	98	14	>	>	SYM
ejpam-6713	98	15	2	2	NUM
ejpam-6713	98	16	let	let	VERB
ejpam-6713	98	17	v1	v1	NOUN
ejpam-6713	98	18	,	,	PUNCT
ejpam-6713	98	19	v2	v2	PROPN
ejpam-6713	98	20	∈	∈	NOUN
ejpam-6713	98	21	v1	v1	NOUN
ejpam-6713	98	22	∩d	∩d	NOUN
ejpam-6713	98	23	and	and	CCONJ
ejpam-6713	98	24	u1	u1	NOUN
ejpam-6713	98	25	,	,	PUNCT
ejpam-6713	98	26	u2	u2	NOUN
ejpam-6713	98	27	,	,	PUNCT
ejpam-6713	98	28	.	.	PUNCT
ejpam-6713	98	29	.	.	PUNCT
ejpam-6713	98	30	.	.	PUNCT
ejpam-6713	99	1	un	un	PROPN
ejpam-6713	99	2	∈	∈	PROPN
ejpam-6713	99	3	v2	v2	PROPN
ejpam-6713	99	4	and	and	CCONJ
ejpam-6713	99	5	let	let	VERB
ejpam-6713	99	6	g′	g′	NOUN
ejpam-6713	99	7	be	be	AUX
ejpam-6713	99	8	a	a	DET
ejpam-6713	99	9	graph	graph	NOUN
ejpam-6713	99	10	derived	derive	VERB
ejpam-6713	99	11	from	from	ADP
ejpam-6713	99	12	g	g	PROPN
ejpam-6713	99	13	through	through	ADP
ejpam-6713	99	14	subdividing	subdivide	VERB
ejpam-6713	99	15	an	an	DET
ejpam-6713	99	16	edge	edge	NOUN
ejpam-6713	99	17	in	in	ADP
ejpam-6713	99	18	g	g	PROPN
ejpam-6713	99	19	say	say	VERB
ejpam-6713	99	20	e	e	X
ejpam-6713	99	21	=	=	SYM
ejpam-6713	99	22	v1ui	v1ui	PROPN
ejpam-6713	99	23	for	for	ADP
ejpam-6713	99	24	some	some	DET
ejpam-6713	99	25	i	i	PRON
ejpam-6713	99	26	by	by	ADP
ejpam-6713	99	27	a	a	DET
ejpam-6713	99	28	subdivision	subdivision	NOUN
ejpam-6713	99	29	vertex	vertex	NOUN
ejpam-6713	99	30	x1	x1	PRON
ejpam-6713	99	31	.	.	PUNCT
ejpam-6713	100	1	all	all	DET
ejpam-6713	100	2	vertices	vertex	NOUN
ejpam-6713	100	3	in	in	ADP
ejpam-6713	100	4	v2	v2	PROPN
ejpam-6713	100	5	are	be	AUX
ejpam-6713	100	6	dominated	dominate	VERB
ejpam-6713	100	7	by	by	ADP
ejpam-6713	100	8	v2	v2	PROPN
ejpam-6713	100	9	and	and	CCONJ
ejpam-6713	100	10	x1	x1	PROPN
ejpam-6713	100	11	is	be	AUX
ejpam-6713	100	12	dominated	dominate	VERB
ejpam-6713	100	13	by	by	ADP
ejpam-6713	100	14	v1	v1	NOUN
ejpam-6713	100	15	.	.	PUNCT
ejpam-6713	101	1	we	we	PRON
ejpam-6713	101	2	clearly	clearly	ADV
ejpam-6713	101	3	see	see	VERB
ejpam-6713	101	4	that	that	SCONJ
ejpam-6713	101	5	d	d	NOUN
ejpam-6713	101	6	is	be	AUX
ejpam-6713	101	7	a	a	DET
ejpam-6713	101	8	γcer	γcer	NOUN
ejpam-6713	101	9	-	-	PUNCT
ejpam-6713	101	10	set	set	NOUN
ejpam-6713	101	11	of	of	ADP
ejpam-6713	101	12	g.	g.	PROPN
ejpam-6713	101	13	hence	hence	ADV
ejpam-6713	101	14	,	,	PUNCT
ejpam-6713	101	15	γcer(g	γcer(g	PROPN
ejpam-6713	101	16	)	)	PUNCT
ejpam-6713	101	17	=	=	PUNCT
ejpam-6713	101	18	γcer(g	γcer(g	PROPN
ejpam-6713	101	19	′	′	NUM
ejpam-6713	101	20	)	)	PUNCT
ejpam-6713	101	21	,	,	PUNCT
ejpam-6713	101	22	this	this	PRON
ejpam-6713	101	23	implies	imply	VERB
ejpam-6713	101	24	sd+γcer(g	sd+γcer(g	NOUN
ejpam-6713	101	25	)	)	PUNCT
ejpam-6713	101	26	>	>	X
ejpam-6713	102	1	1	1	X
ejpam-6713	102	2	.	.	PUNCT
ejpam-6713	102	3	let	let	VERB
ejpam-6713	102	4	g′′	g′′	PROPN
ejpam-6713	102	5	be	be	AUX
ejpam-6713	102	6	the	the	DET
ejpam-6713	102	7	graph	graph	NOUN
ejpam-6713	102	8	obtained	obtain	VERB
ejpam-6713	102	9	from	from	ADP
ejpam-6713	102	10	g′	g′	NOUN
ejpam-6713	102	11	,	,	PUNCT
ejpam-6713	102	12	by	by	ADP
ejpam-6713	102	13	subdividing	subdivide	VERB
ejpam-6713	102	14	an	an	DET
ejpam-6713	102	15	edge	edge	NOUN
ejpam-6713	102	16	in	in	ADP
ejpam-6713	102	17	g′	g′	NOUN
ejpam-6713	102	18	say	say	VERB
ejpam-6713	102	19	e	e	X
ejpam-6713	102	20	=	=	PUNCT
ejpam-6713	102	21	v2ui	v2ui	X
ejpam-6713	102	22	by	by	ADP
ejpam-6713	102	23	a	a	DET
ejpam-6713	102	24	subdivision	subdivision	NOUN
ejpam-6713	102	25	vertex	vertex	NOUN
ejpam-6713	102	26	x2	x2	PROPN
ejpam-6713	102	27	results	result	VERB
ejpam-6713	102	28	d1	d1	PROPN
ejpam-6713	102	29	=	=	PUNCT
ejpam-6713	103	1	{	{	PUNCT
ejpam-6713	103	2	d	d	X
ejpam-6713	103	3	−	−	PROPN
ejpam-6713	103	4	{	{	PUNCT
ejpam-6713	103	5	v2	v2	NOUN
ejpam-6713	103	6	}	}	PUNCT
ejpam-6713	103	7	∪	∪	NOUN
ejpam-6713	103	8	{	{	PUNCT
ejpam-6713	103	9	x2	x2	NOUN
ejpam-6713	103	10	}	}	PUNCT
ejpam-6713	103	11	}	}	PUNCT
ejpam-6713	103	12	is	be	AUX
ejpam-6713	103	13	a	a	DET
ejpam-6713	103	14	γcer	γcer	NOUN
ejpam-6713	103	15	-	-	PUNCT
ejpam-6713	103	16	set	set	NOUN
ejpam-6713	103	17	of	of	ADP
ejpam-6713	103	18	g′′.	g′′.	PROPN
ejpam-6713	103	19	hence	hence	ADV
ejpam-6713	103	20	,	,	PUNCT
ejpam-6713	103	21	sd+γcer(g	sd+γcer(g	NOUN
ejpam-6713	103	22	)	)	PUNCT
ejpam-6713	103	23	>	>	X
ejpam-6713	104	1	2	2	X
ejpam-6713	104	2	.	.	PUNCT
ejpam-6713	104	3	let	let	VERB
ejpam-6713	104	4	g′′′	g′′′	PROPN
ejpam-6713	104	5	be	be	AUX
ejpam-6713	104	6	the	the	DET
ejpam-6713	104	7	graph	graph	NOUN
ejpam-6713	104	8	obtained	obtain	VERB
ejpam-6713	104	9	from	from	ADP
ejpam-6713	104	10	g′′	g′′	PROPN
ejpam-6713	104	11	by	by	ADP
ejpam-6713	104	12	subdividing	subdivide	VERB
ejpam-6713	104	13	an	an	DET
ejpam-6713	104	14	edge	edge	NOUN
ejpam-6713	104	15	say	say	INTJ
ejpam-6713	105	1	e	e	NOUN
ejpam-6713	105	2	=	=	PUNCT
ejpam-6713	105	3	v2u2	v2u2	PROPN
ejpam-6713	105	4	in	in	ADP
ejpam-6713	105	5	g′′	g′′	NOUN
ejpam-6713	105	6	by	by	ADP
ejpam-6713	105	7	a	a	DET
ejpam-6713	105	8	subdivision	subdivision	NOUN
ejpam-6713	105	9	vertex	vertex	NOUN
ejpam-6713	105	10	x3	x3	PROPN
ejpam-6713	105	11	results	result	VERB
ejpam-6713	105	12	d2	d2	PROPN
ejpam-6713	105	13	=	=	SYM
ejpam-6713	105	14	d1∪{v2	d1∪{v2	NOUN
ejpam-6713	105	15	}	}	PUNCT
ejpam-6713	105	16	is	be	AUX
ejpam-6713	105	17	a	a	DET
ejpam-6713	105	18	γcer	γcer	NOUN
ejpam-6713	105	19	-	-	PUNCT
ejpam-6713	105	20	set	set	NOUN
ejpam-6713	105	21	of	of	ADP
ejpam-6713	105	22	g′′′.	g′′′.	PROPN
ejpam-6713	105	23	here	here	ADV
ejpam-6713	105	24	|d2|	|d2|	PROPN
ejpam-6713	105	25	>	>	X
ejpam-6713	105	26	|d1|	|d1|	NOUN
ejpam-6713	105	27	,	,	PUNCT
ejpam-6713	105	28	that	that	PRON
ejpam-6713	105	29	is	be	AUX
ejpam-6713	105	30	γcer(g	γcer(g	PROPN
ejpam-6713	105	31	)	)	PUNCT
ejpam-6713	105	32	<	<	X
ejpam-6713	105	33	γcer(g	γcer(g	PROPN
ejpam-6713	105	34	′′′	′′′	PROPN
ejpam-6713	105	35	)	)	PUNCT
ejpam-6713	105	36	.	.	PUNCT
ejpam-6713	106	1	hence	hence	ADV
ejpam-6713	106	2	sd+γcer(g	sd+γcer(g	NOUN
ejpam-6713	106	3	)	)	PUNCT
ejpam-6713	106	4	=	=	SYM
ejpam-6713	107	1	3	3	X
ejpam-6713	107	2	.	.	X
ejpam-6713	107	3	case	case	NOUN
ejpam-6713	107	4	(	(	PUNCT
ejpam-6713	107	5	ii	ii	NOUN
ejpam-6713	107	6	)	)	PUNCT
ejpam-6713	107	7	m	m	PROPN
ejpam-6713	107	8	=	=	SYM
ejpam-6713	107	9	1	1	NUM
ejpam-6713	107	10	and	and	CCONJ
ejpam-6713	107	11	n	n	PROPN
ejpam-6713	107	12	>	>	SYM
ejpam-6713	107	13	1	1	NUM
ejpam-6713	107	14	let	let	VERB
ejpam-6713	107	15	v1	v1	VERB
ejpam-6713	107	16	∈	∈	NOUN
ejpam-6713	107	17	v1	v1	NOUN
ejpam-6713	107	18	∩d	∩d	NOUN
ejpam-6713	107	19	and	and	CCONJ
ejpam-6713	107	20	u1	u1	NOUN
ejpam-6713	107	21	,	,	PUNCT
ejpam-6713	107	22	u2	u2	NOUN
ejpam-6713	107	23	,	,	PUNCT
ejpam-6713	107	24	u3	u3	NOUN
ejpam-6713	107	25	,	,	PUNCT
ejpam-6713	107	26	.	.	PUNCT
ejpam-6713	107	27	.	.	PUNCT
ejpam-6713	107	28	.	.	PUNCT
ejpam-6713	108	1	un	un	PROPN
ejpam-6713	108	2	∈	∈	PROPN
ejpam-6713	108	3	v2	v2	PROPN
ejpam-6713	108	4	.	.	PUNCT
ejpam-6713	109	1	we	we	PRON
ejpam-6713	109	2	know	know	VERB
ejpam-6713	109	3	that	that	PRON
ejpam-6713	109	4	γcer(g	γcer(g	NOUN
ejpam-6713	109	5	)	)	PUNCT
ejpam-6713	109	6	=	=	SYM
ejpam-6713	110	1	1	1	X
ejpam-6713	110	2	.	.	X
ejpam-6713	110	3	subdividing	subdivide	VERB
ejpam-6713	110	4	the	the	DET
ejpam-6713	110	5	edge	edge	NOUN
ejpam-6713	110	6	e	e	NOUN
ejpam-6713	110	7	=	=	PUNCT
ejpam-6713	110	8	v1ui	v1ui	PROPN
ejpam-6713	110	9	,	,	PUNCT
ejpam-6713	110	10	for	for	ADP
ejpam-6713	110	11	some	some	DET
ejpam-6713	110	12	1	1	NUM
ejpam-6713	110	13	<	<	X
ejpam-6713	110	14	i	i	PRON
ejpam-6713	110	15	<	<	X
ejpam-6713	110	16	n	n	X
ejpam-6713	110	17	by	by	ADP
ejpam-6713	110	18	subdivision	subdivision	NOUN
ejpam-6713	110	19	vertex	vertex	NOUN
ejpam-6713	110	20	x	x	PRON
ejpam-6713	110	21	results	result	VERB
ejpam-6713	110	22	d1	d1	NOUN
ejpam-6713	110	23	=	=	PUNCT
ejpam-6713	111	1	d	d	NOUN
ejpam-6713	111	2	−	−	PROPN
ejpam-6713	111	3	{	{	PUNCT
ejpam-6713	111	4	ui	ui	NOUN
ejpam-6713	111	5	}	}	PUNCT
ejpam-6713	111	6	.	.	PUNCT
ejpam-6713	112	1	hence	hence	ADV
ejpam-6713	112	2	sd+γcer(g	sd+γcer(g	NOUN
ejpam-6713	112	3	)	)	PUNCT
ejpam-6713	112	4	=	=	SYM
ejpam-6713	113	1	1	1	X
ejpam-6713	113	2	.	.	X
ejpam-6713	113	3	case	case	NOUN
ejpam-6713	113	4	(	(	PUNCT
ejpam-6713	113	5	iii	iii	X
ejpam-6713	113	6	)	)	PUNCT
ejpam-6713	113	7	m	m	VERB
ejpam-6713	113	8	>	>	X
ejpam-6713	113	9	2	2	NUM
ejpam-6713	113	10	and	and	CCONJ
ejpam-6713	113	11	n	n	NOUN
ejpam-6713	113	12	>	>	SYM
ejpam-6713	113	13	2	2	NUM
ejpam-6713	113	14	let	let	VERB
ejpam-6713	113	15	v1	v1	NOUN
ejpam-6713	113	16	,	,	PUNCT
ejpam-6713	113	17	v2	v2	PROPN
ejpam-6713	113	18	,	,	PUNCT
ejpam-6713	113	19	v3	v3	PROPN
ejpam-6713	113	20	,	,	PUNCT
ejpam-6713	113	21	.	.	PUNCT
ejpam-6713	113	22	.	.	PUNCT
ejpam-6713	113	23	.	.	PUNCT
ejpam-6713	114	1	vm	vm	PROPN
ejpam-6713	114	2	∈	∈	PROPN
ejpam-6713	114	3	v1	v1	PROPN
ejpam-6713	114	4	and	and	CCONJ
ejpam-6713	114	5	u1	u1	NOUN
ejpam-6713	114	6	,	,	PUNCT
ejpam-6713	114	7	u2	u2	NOUN
ejpam-6713	114	8	,	,	PUNCT
ejpam-6713	114	9	u3	u3	NOUN
ejpam-6713	114	10	,	,	PUNCT
ejpam-6713	114	11	.	.	PUNCT
ejpam-6713	114	12	.	.	PUNCT
ejpam-6713	114	13	.	.	PUNCT
ejpam-6713	115	1	un	un	PROPN
ejpam-6713	115	2	∈	∈	PROPN
ejpam-6713	115	3	v2	v2	PROPN
ejpam-6713	115	4	and	and	CCONJ
ejpam-6713	115	5	by	by	ADP
ejpam-6713	115	6	theorem	theorem	NOUN
ejpam-6713	115	7	[	[	X
ejpam-6713	115	8	4	4	NUM
ejpam-6713	115	9	]	]	PUNCT
ejpam-6713	115	10	,	,	PUNCT
ejpam-6713	115	11	γcer(g	γcer(g	NOUN
ejpam-6713	115	12	)	)	PUNCT
ejpam-6713	115	13	=	=	SYM
ejpam-6713	116	1	2	2	X
ejpam-6713	116	2	.	.	X
ejpam-6713	116	3	let	let	VERB
ejpam-6713	116	4	vi	vi	NOUN
ejpam-6713	116	5	,	,	PUNCT
ejpam-6713	116	6	uj	uj	PROPN
ejpam-6713	116	7	∈	∈	PROPN
ejpam-6713	116	8	d	d	PROPN
ejpam-6713	116	9	for	for	ADP
ejpam-6713	116	10	some	some	DET
ejpam-6713	116	11	1	1	NUM
ejpam-6713	116	12	<	<	X
ejpam-6713	116	13	i	i	PRON
ejpam-6713	116	14	<	<	X
ejpam-6713	116	15	m	m	VERB
ejpam-6713	116	16	and	and	CCONJ
ejpam-6713	116	17	1	1	NUM
ejpam-6713	116	18	<	<	X
ejpam-6713	116	19	j	j	X
ejpam-6713	116	20	<	<	X
ejpam-6713	116	21	n.	n.	PROPN
ejpam-6713	116	22	let	let	VERB
ejpam-6713	116	23	g′	g′	NOUN
ejpam-6713	116	24	be	be	AUX
ejpam-6713	116	25	a	a	DET
ejpam-6713	116	26	graph	graph	NOUN
ejpam-6713	116	27	obtained	obtain	VERB
ejpam-6713	116	28	by	by	ADP
ejpam-6713	116	29	subdividing	subdivide	VERB
ejpam-6713	116	30	an	an	DET
ejpam-6713	116	31	edge	edge	NOUN
ejpam-6713	116	32	e	e	NOUN
ejpam-6713	116	33	in	in	ADP
ejpam-6713	116	34	g	g	PROPN
ejpam-6713	116	35	say	say	VERB
ejpam-6713	116	36	e	e	X
ejpam-6713	116	37	=	=	PUNCT
ejpam-6713	116	38	v1u1	v1u1	PUNCT
ejpam-6713	116	39	by	by	ADP
ejpam-6713	116	40	a	a	DET
ejpam-6713	116	41	subdivision	subdivision	NOUN
ejpam-6713	116	42	vertex	vertex	NOUN
ejpam-6713	116	43	x1	x1	PROPN
ejpam-6713	116	44	,	,	PUNCT
ejpam-6713	116	45	results	result	VERB
ejpam-6713	116	46	a	a	DET
ejpam-6713	116	47	new	new	ADJ
ejpam-6713	116	48	configuration	configuration	NOUN
ejpam-6713	116	49	of	of	ADP
ejpam-6713	116	50	g.	g.	PROPN
ejpam-6713	116	51	navamani	navamani	PROPN
ejpam-6713	116	52	et	et	PROPN
ejpam-6713	116	53	al	al	PROPN
ejpam-6713	116	54	.	.	PUNCT
ejpam-6713	116	55	/	/	SYM
ejpam-6713	116	56	eur	eur	PROPN
ejpam-6713	116	57	.	.	PUNCT
ejpam-6713	117	1	j.	j.	PROPN
ejpam-6713	117	2	pure	pure	PROPN
ejpam-6713	117	3	appl	appl	PROPN
ejpam-6713	117	4	.	.	PROPN
ejpam-6713	117	5	math	math	PROPN
ejpam-6713	117	6	,	,	PUNCT
ejpam-6713	117	7	18	18	NUM
ejpam-6713	117	8	(	(	PUNCT
ejpam-6713	117	9	4	4	NUM
ejpam-6713	117	10	)	)	PUNCT
ejpam-6713	117	11	(	(	PUNCT
ejpam-6713	117	12	2025	2025	NUM
ejpam-6713	117	13	)	)	PUNCT
ejpam-6713	117	14	,	,	PUNCT
ejpam-6713	117	15	6713	6713	NUM
ejpam-6713	117	16	5	5	NUM
ejpam-6713	117	17	of	of	ADP
ejpam-6713	117	18	15	15	NUM
ejpam-6713	117	19	γcer	γcer	NOUN
ejpam-6713	117	20	-	-	PUNCT
ejpam-6713	117	21	set	set	NOUN
ejpam-6713	117	22	say	say	VERB
ejpam-6713	117	23	d1	d1	PROPN
ejpam-6713	117	24	=	=	SYM
ejpam-6713	117	25	{	{	PUNCT
ejpam-6713	117	26	v1	v1	NOUN
ejpam-6713	117	27	,	,	PUNCT
ejpam-6713	117	28	u1	u1	NOUN
ejpam-6713	117	29	}	}	PUNCT
ejpam-6713	117	30	which	which	PRON
ejpam-6713	117	31	implies	imply	VERB
ejpam-6713	117	32	|d|	|d|	PROPN
ejpam-6713	117	33	=	=	PUNCT
ejpam-6713	117	34	|d1|	|d1|	PROPN
ejpam-6713	117	35	.	.	PUNCT
ejpam-6713	118	1	therefore	therefore	ADV
ejpam-6713	118	2	sd+γcer(g	sd+γcer(g	NOUN
ejpam-6713	118	3	)	)	PUNCT
ejpam-6713	118	4	>	>	X
ejpam-6713	119	1	1	1	X
ejpam-6713	119	2	.	.	PUNCT
ejpam-6713	119	3	let	let	VERB
ejpam-6713	119	4	g′′	g′′	PROPN
ejpam-6713	119	5	be	be	AUX
ejpam-6713	119	6	a	a	DET
ejpam-6713	119	7	graph	graph	NOUN
ejpam-6713	119	8	obtained	obtain	VERB
ejpam-6713	119	9	from	from	ADP
ejpam-6713	119	10	g′	g′	NOUN
ejpam-6713	119	11	by	by	ADP
ejpam-6713	119	12	subdividing	subdivide	VERB
ejpam-6713	119	13	the	the	DET
ejpam-6713	119	14	edge	edge	NOUN
ejpam-6713	119	15	e	e	NOUN
ejpam-6713	119	16	=	=	PUNCT
ejpam-6713	119	17	u1v2	u1v2	AUX
ejpam-6713	119	18	by	by	ADP
ejpam-6713	119	19	a	a	DET
ejpam-6713	119	20	subdivision	subdivision	NOUN
ejpam-6713	119	21	vertices	vertice	VERB
ejpam-6713	119	22	x2	x2	PRON
ejpam-6713	119	23	.	.	PUNCT
ejpam-6713	120	1	here	here	ADV
ejpam-6713	120	2	x1	x1	PROPN
ejpam-6713	120	3	and	and	CCONJ
ejpam-6713	120	4	x2	x2	PROPN
ejpam-6713	120	5	are	be	AUX
ejpam-6713	120	6	dominated	dominate	VERB
ejpam-6713	120	7	by	by	ADP
ejpam-6713	120	8	u1	u1	NOUN
ejpam-6713	120	9	and	and	CCONJ
ejpam-6713	120	10	v2	v2	PROPN
ejpam-6713	120	11	/∈	/∈	PUNCT
ejpam-6713	121	1	n	n	CCONJ
ejpam-6713	122	1	[	[	X
ejpam-6713	122	2	v	v	X
ejpam-6713	122	3	]	]	X
ejpam-6713	122	4	for	for	ADP
ejpam-6713	122	5	all	all	PRON
ejpam-6713	122	6	v	v	NOUN
ejpam-6713	122	7	∈	∈	PROPN
ejpam-6713	122	8	d.	d.	NOUN
ejpam-6713	122	9	now	now	ADV
ejpam-6713	122	10	d2	d2	PROPN
ejpam-6713	122	11	=	=	SYM
ejpam-6713	122	12	d1	d1	PROPN
ejpam-6713	122	13	∪	∪	ADV
ejpam-6713	122	14	{	{	PUNCT
ejpam-6713	122	15	v2	v2	NOUN
ejpam-6713	122	16	}	}	PUNCT
ejpam-6713	122	17	.	.	PUNCT
ejpam-6713	123	1	hence	hence	ADV
ejpam-6713	123	2	|d2|	|d2|	VERB
ejpam-6713	123	3	>	>	X
ejpam-6713	123	4	|d1|	|d1|	NOUN
ejpam-6713	123	5	.	.	PUNCT
ejpam-6713	124	1	therefore	therefore	ADV
ejpam-6713	124	2	,	,	PUNCT
ejpam-6713	124	3	sd+γcer(g	sd+γcer(g	NOUN
ejpam-6713	124	4	)	)	PUNCT
ejpam-6713	124	5	=	=	SYM
ejpam-6713	125	1	2	2	X
ejpam-6713	125	2	.	.	X
ejpam-6713	126	1	we	we	PRON
ejpam-6713	126	2	observe	observe	VERB
ejpam-6713	126	3	that	that	SCONJ
ejpam-6713	126	4	,	,	PUNCT
ejpam-6713	126	5	for	for	ADP
ejpam-6713	126	6	all	all	DET
ejpam-6713	126	7	graphs	graph	NOUN
ejpam-6713	126	8	g	g	NOUN
ejpam-6713	126	9	which	which	PRON
ejpam-6713	126	10	is	be	AUX
ejpam-6713	126	11	isomorphic	isomorphic	ADJ
ejpam-6713	126	12	to	to	PART
ejpam-6713	126	13	complete	complete	VERB
ejpam-6713	126	14	graphs	graph	NOUN
ejpam-6713	126	15	,	,	PUNCT
ejpam-6713	126	16	wheel	wheel	NOUN
ejpam-6713	126	17	graphs	graph	NOUN
ejpam-6713	126	18	and	and	CCONJ
ejpam-6713	126	19	grid	grid	NOUN
ejpam-6713	126	20	graphs	graph	NOUN
ejpam-6713	126	21	(	(	PUNCT
ejpam-6713	126	22	p2	p2	PROPN
ejpam-6713	126	23	×	×	PROPN
ejpam-6713	126	24	pn	pn	PROPN
ejpam-6713	126	25	,	,	PUNCT
ejpam-6713	126	26	n	n	PRON
ejpam-6713	126	27	≥	≥	NOUN
ejpam-6713	126	28	2	2	NUM
ejpam-6713	126	29	)	)	PUNCT
ejpam-6713	126	30	,	,	PUNCT
ejpam-6713	126	31	sd+γcer(g	sd+γcer(g	NOUN
ejpam-6713	126	32	)	)	PUNCT
ejpam-6713	126	33	=	=	SYM
ejpam-6713	126	34	1	1	NUM
ejpam-6713	126	35	.	.	NOUN
ejpam-6713	126	36	4	4	NUM
ejpam-6713	126	37	.	.	X
ejpam-6713	126	38	circulant	circulant	ADJ
ejpam-6713	126	39	graphs	graph	NOUN
ejpam-6713	126	40	the	the	DET
ejpam-6713	126	41	circulant	circulant	ADJ
ejpam-6713	126	42	graph	graph	NOUN
ejpam-6713	126	43	cn(sc	cn(sc	NOUN
ejpam-6713	126	44	)	)	PUNCT
ejpam-6713	126	45	is	be	AUX
ejpam-6713	126	46	the	the	DET
ejpam-6713	126	47	graph	graph	NOUN
ejpam-6713	126	48	with	with	ADP
ejpam-6713	126	49	the	the	DET
ejpam-6713	126	50	vertex	vertex	NOUN
ejpam-6713	126	51	set	set	VERB
ejpam-6713	126	52	v	v	NOUN
ejpam-6713	126	53	(	(	PUNCT
ejpam-6713	126	54	cn(sc	cn(sc	NOUN
ejpam-6713	126	55	)	)	PUNCT
ejpam-6713	126	56	)	)	PUNCT
ejpam-6713	127	1	=	=	PRON
ejpam-6713	127	2	{	{	PUNCT
ejpam-6713	127	3	vi	vi	NOUN
ejpam-6713	127	4	:	:	SYM
ejpam-6713	127	5	0	0	NUM
ejpam-6713	127	6	≤	≤	NUM
ejpam-6713	128	1	i	i	PRON
ejpam-6713	128	2	≤	≤	ADJ
ejpam-6713	128	3	n−	n−	NOUN
ejpam-6713	128	4	1	1	NUM
ejpam-6713	128	5	}	}	PUNCT
ejpam-6713	128	6	and	and	CCONJ
ejpam-6713	128	7	the	the	DET
ejpam-6713	128	8	edge	edge	NOUN
ejpam-6713	128	9	set	set	VERB
ejpam-6713	128	10	e(cn(sc	e(cn(sc	ADV
ejpam-6713	128	11	)	)	PUNCT
ejpam-6713	128	12	)	)	PUNCT
ejpam-6713	129	1	=	=	PRON
ejpam-6713	129	2	{	{	PUNCT
ejpam-6713	129	3	vivj	vivj	NOUN
ejpam-6713	129	4	:	:	PUNCT
ejpam-6713	129	5	0	0	NUM
ejpam-6713	129	6	≤	≤	X
ejpam-6713	130	1	i	i	PRON
ejpam-6713	130	2	,	,	PUNCT
ejpam-6713	130	3	j	j	PROPN
ejpam-6713	130	4	≤	≤	PROPN
ejpam-6713	130	5	n−	n−	PROPN
ejpam-6713	130	6	1	1	NUM
ejpam-6713	130	7	,	,	PUNCT
ejpam-6713	130	8	(	(	PUNCT
ejpam-6713	130	9	i−	i−	PROPN
ejpam-6713	130	10	j)(mod	j)(mod	PROPN
ejpam-6713	130	11	n	n	CCONJ
ejpam-6713	130	12	)	)	PUNCT
ejpam-6713	130	13	∈	∈	PROPN
ejpam-6713	130	14	sc	sc	PROPN
ejpam-6713	130	15	}	}	PUNCT
ejpam-6713	130	16	.	.	PUNCT
ejpam-6713	131	1	here	here	ADV
ejpam-6713	131	2	sc	sc	PROPN
ejpam-6713	131	3	⊆	⊆	NUM
ejpam-6713	131	4	{	{	SYM
ejpam-6713	131	5	1	1	NUM
ejpam-6713	131	6	,	,	PUNCT
ejpam-6713	131	7	2	2	NUM
ejpam-6713	131	8	,	,	PUNCT
ejpam-6713	131	9	3	3	NUM
ejpam-6713	131	10	,	,	PUNCT
ejpam-6713	131	11	.	.	PUNCT
ejpam-6713	131	12	.	.	PUNCT
ejpam-6713	131	13	.	.	PUNCT
ejpam-6713	132	1	,	,	PUNCT
ejpam-6713	132	2	⌈	⌈	PROPN
ejpam-6713	132	3	n	n	CCONJ
ejpam-6713	132	4	2	2	NUM
ejpam-6713	132	5	⌉	⌉	NOUN
ejpam-6713	132	6	}	}	PUNCT
ejpam-6713	132	7	where	where	SCONJ
ejpam-6713	132	8	subscripts	subscript	NOUN
ejpam-6713	132	9	are	be	AUX
ejpam-6713	132	10	taken	take	VERB
ejpam-6713	132	11	modulo	modulo	ADJ
ejpam-6713	132	12	n.	n.	NOUN
ejpam-6713	132	13	in	in	ADP
ejpam-6713	132	14	this	this	DET
ejpam-6713	132	15	section	section	NOUN
ejpam-6713	132	16	we	we	PRON
ejpam-6713	132	17	find	find	VERB
ejpam-6713	132	18	the	the	DET
ejpam-6713	132	19	value	value	NOUN
ejpam-6713	132	20	of	of	ADP
ejpam-6713	132	21	sd+γcer(g	sd+γcer(g	NOUN
ejpam-6713	132	22	)	)	PUNCT
ejpam-6713	132	23	for	for	ADP
ejpam-6713	132	24	the	the	DET
ejpam-6713	132	25	circulant	circulant	ADJ
ejpam-6713	132	26	graphs	graph	NOUN
ejpam-6713	132	27	cn(1	cn(1	PROPN
ejpam-6713	132	28	,	,	PUNCT
ejpam-6713	132	29	2	2	NUM
ejpam-6713	132	30	)	)	PUNCT
ejpam-6713	132	31	and	and	CCONJ
ejpam-6713	132	32	cn(1	cn(1	PROPN
ejpam-6713	132	33	,	,	PUNCT
ejpam-6713	132	34	3	3	X
ejpam-6713	132	35	)	)	PUNCT
ejpam-6713	132	36	theorem	theorem	VERB
ejpam-6713	132	37	6	6	NUM
ejpam-6713	132	38	.	.	PUNCT
ejpam-6713	133	1	[	[	X
ejpam-6713	133	2	19	19	NUM
ejpam-6713	133	3	]	]	PUNCT
ejpam-6713	133	4	for	for	ADP
ejpam-6713	133	5	any	any	DET
ejpam-6713	133	6	integer	integer	NOUN
ejpam-6713	133	7	n	n	PRON
ejpam-6713	133	8	≥	≥	NOUN
ejpam-6713	133	9	5	5	NUM
ejpam-6713	133	10	,	,	PUNCT
ejpam-6713	133	11	γ(cn(1	γ(cn(1	PROPN
ejpam-6713	133	12	,	,	PUNCT
ejpam-6713	133	13	2	2	NUM
ejpam-6713	133	14	)	)	PUNCT
ejpam-6713	133	15	)	)	PUNCT
ejpam-6713	134	1	=	=	PUNCT
ejpam-6713	134	2	⌈	⌈	PROPN
ejpam-6713	134	3	n	n	CCONJ
ejpam-6713	134	4	5	5	NUM
ejpam-6713	134	5	⌉	⌉	X
ejpam-6713	134	6	theorem	theorem	VERB
ejpam-6713	134	7	7	7	NUM
ejpam-6713	134	8	.	.	X
ejpam-6713	134	9	for	for	ADP
ejpam-6713	134	10	any	any	DET
ejpam-6713	134	11	circulant	circulant	ADJ
ejpam-6713	134	12	graph	graph	NOUN
ejpam-6713	134	13	g	g	ADP
ejpam-6713	134	14	∼=	∼=	PROPN
ejpam-6713	134	15	cn(1	cn(1	NOUN
ejpam-6713	134	16	,	,	PUNCT
ejpam-6713	134	17	2	2	NUM
ejpam-6713	134	18	)	)	PUNCT
ejpam-6713	134	19	,	,	PUNCT
ejpam-6713	134	20	n	n	X
ejpam-6713	134	21	≥	≥	NOUN
ejpam-6713	134	22	6	6	NUM
ejpam-6713	134	23	,	,	PUNCT
ejpam-6713	134	24	sd+γcer(g	sd+γcer(g	NOUN
ejpam-6713	134	25	)	)	PUNCT
ejpam-6713	135	1	=	=	SYM
ejpam-6713	135	2			NOUN
ejpam-6713	135	3	1	1	NUM
ejpam-6713	135	4	if	if	SCONJ
ejpam-6713	135	5	n	n	PRON
ejpam-6713	135	6	≡	≡	PROPN
ejpam-6713	135	7	0	0	NUM
ejpam-6713	135	8	,	,	PUNCT
ejpam-6713	135	9	4	4	NUM
ejpam-6713	135	10	(	(	PUNCT
ejpam-6713	135	11	mod	mod	NOUN
ejpam-6713	135	12	5	5	NUM
ejpam-6713	135	13	)	)	PUNCT
ejpam-6713	135	14	2	2	NUM
ejpam-6713	135	15	if	if	SCONJ
ejpam-6713	135	16	n	n	PRON
ejpam-6713	135	17	≡	≡	PROPN
ejpam-6713	135	18	2	2	NUM
ejpam-6713	135	19	,	,	PUNCT
ejpam-6713	135	20	3	3	NUM
ejpam-6713	135	21	(	(	PUNCT
ejpam-6713	135	22	mod	mod	NOUN
ejpam-6713	135	23	5	5	NUM
ejpam-6713	135	24	)	)	PUNCT
ejpam-6713	135	25	3	3	NUM
ejpam-6713	135	26	if	if	SCONJ
ejpam-6713	135	27	n	n	PRON
ejpam-6713	135	28	≡	≡	PROPN
ejpam-6713	135	29	1	1	NUM
ejpam-6713	135	30	(	(	PUNCT
ejpam-6713	135	31	mod	mod	NOUN
ejpam-6713	135	32	5	5	NUM
ejpam-6713	135	33	)	)	PUNCT
ejpam-6713	135	34	proof	proof	NOUN
ejpam-6713	135	35	.	.	PUNCT
ejpam-6713	136	1	let	let	VERB
ejpam-6713	136	2	g	g	PROPN
ejpam-6713	136	3	∼=	∼=	PROPN
ejpam-6713	136	4	cn(1	cn(1	NOUN
ejpam-6713	136	5	,	,	PUNCT
ejpam-6713	136	6	2	2	NUM
ejpam-6713	136	7	)	)	PUNCT
ejpam-6713	136	8	.	.	PUNCT
ejpam-6713	137	1	let	let	VERB
ejpam-6713	137	2	v	v	X
ejpam-6713	137	3	(	(	PUNCT
ejpam-6713	137	4	g	g	NOUN
ejpam-6713	137	5	)	)	PUNCT
ejpam-6713	137	6	=	=	SYM
ejpam-6713	137	7	{	{	PUNCT
ejpam-6713	137	8	v1	v1	PROPN
ejpam-6713	137	9	,	,	PUNCT
ejpam-6713	137	10	v2	v2	PROPN
ejpam-6713	137	11	,	,	PUNCT
ejpam-6713	137	12	v3	v3	PROPN
ejpam-6713	137	13	,	,	PUNCT
ejpam-6713	137	14	.	.	PUNCT
ejpam-6713	137	15	.	.	PUNCT
ejpam-6713	137	16	.	.	PUNCT
ejpam-6713	138	1	vn	vn	AUX
ejpam-6713	138	2	}	}	PUNCT
ejpam-6713	138	3	be	be	AUX
ejpam-6713	138	4	the	the	DET
ejpam-6713	138	5	vertex	vertex	NOUN
ejpam-6713	138	6	set	set	NOUN
ejpam-6713	138	7	of	of	ADP
ejpam-6713	138	8	g	g	PROPN
ejpam-6713	138	9	and	and	CCONJ
ejpam-6713	138	10	d	d	NOUN
ejpam-6713	138	11	=	=	SYM
ejpam-6713	138	12	{	{	PUNCT
ejpam-6713	138	13	v5k−4	v5k−4	NOUN
ejpam-6713	138	14	:	:	PUNCT
ejpam-6713	138	15	1	1	NUM
ejpam-6713	138	16	≤	≤	NUM
ejpam-6713	138	17	k	k	X
ejpam-6713	138	18	≤	≤	PROPN
ejpam-6713	138	19	⌈	⌈	NUM
ejpam-6713	138	20	n	n	CCONJ
ejpam-6713	138	21	5	5	NUM
ejpam-6713	138	22	⌉	⌉	VERB
ejpam-6713	138	23	}	}	PUNCT
ejpam-6713	138	24	is	be	AUX
ejpam-6713	138	25	a	a	DET
ejpam-6713	138	26	γcer	γcer	NOUN
ejpam-6713	138	27	-	-	PUNCT
ejpam-6713	138	28	set	set	NOUN
ejpam-6713	138	29	of	of	ADP
ejpam-6713	138	30	g.	g.	NOUN
ejpam-6713	138	31	for	for	ADP
ejpam-6713	138	32	every	every	DET
ejpam-6713	138	33	vertex	vertex	NOUN
ejpam-6713	138	34	v	v	ADP
ejpam-6713	138	35	∈	∈	PROPN
ejpam-6713	138	36	d	d	NOUN
ejpam-6713	138	37	,	,	PUNCT
ejpam-6713	138	38	n(v	n(v	PROPN
ejpam-6713	138	39	)	)	PUNCT
ejpam-6713	138	40	≥	≥	NOUN
ejpam-6713	138	41	2	2	NUM
ejpam-6713	138	42	.	.	PUNCT
ejpam-6713	139	1	by	by	ADP
ejpam-6713	139	2	theorem	theorem	NOUN
ejpam-6713	139	3	6	6	NUM
ejpam-6713	139	4	and	and	CCONJ
ejpam-6713	139	5	theorem	theorem	VERB
ejpam-6713	139	6	3	3	NUM
ejpam-6713	139	7	,	,	PUNCT
ejpam-6713	139	8	we	we	PRON
ejpam-6713	139	9	clearly	clearly	ADV
ejpam-6713	139	10	see	see	VERB
ejpam-6713	139	11	that	that	PRON
ejpam-6713	139	12	γ(g	γ(g	PROPN
ejpam-6713	139	13	)	)	PUNCT
ejpam-6713	140	1	=	=	SYM
ejpam-6713	140	2	γcer(g	γcer(g	NOUN
ejpam-6713	140	3	)	)	PUNCT
ejpam-6713	140	4	=	=	PUNCT
ejpam-6713	141	1	⌈	⌈	PROPN
ejpam-6713	141	2	n	n	CCONJ
ejpam-6713	141	3	5	5	NUM
ejpam-6713	141	4	⌉	⌉	X
ejpam-6713	141	5	.	.	PUNCT
ejpam-6713	142	1	case	case	NOUN
ejpam-6713	142	2	(	(	PUNCT
ejpam-6713	142	3	i	i	NOUN
ejpam-6713	142	4	)	)	PUNCT
ejpam-6713	142	5	n	n	X
ejpam-6713	142	6	≡	≡	PROPN
ejpam-6713	142	7	0	0	NUM
ejpam-6713	142	8	,	,	PUNCT
ejpam-6713	142	9	4	4	NUM
ejpam-6713	142	10	(	(	PUNCT
ejpam-6713	142	11	mod	mod	NOUN
ejpam-6713	142	12	5	5	NUM
ejpam-6713	142	13	)	)	PUNCT
ejpam-6713	142	14	if	if	SCONJ
ejpam-6713	142	15	n	n	PRON
ejpam-6713	142	16	≡	≡	PROPN
ejpam-6713	142	17	0	0	PUNCT
ejpam-6713	142	18	(	(	PUNCT
ejpam-6713	142	19	mod	mod	NOUN
ejpam-6713	142	20	5	5	NUM
ejpam-6713	142	21	)	)	PUNCT
ejpam-6713	142	22	then	then	ADV
ejpam-6713	142	23	|pn(v	|pn(v	NUM
ejpam-6713	142	24	,	,	PUNCT
ejpam-6713	142	25	d)|	d)|	NOUN
ejpam-6713	142	26	=	=	NOUN
ejpam-6713	142	27	4	4	NUM
ejpam-6713	142	28	for	for	ADP
ejpam-6713	142	29	each	each	DET
ejpam-6713	142	30	vertex	vertex	NOUN
ejpam-6713	142	31	v	v	ADP
ejpam-6713	142	32	∈	∈	PROPN
ejpam-6713	142	33	d.	d.	NOUN
ejpam-6713	142	34	if	if	SCONJ
ejpam-6713	142	35	n	n	PRON
ejpam-6713	142	36	≡	≡	PROPN
ejpam-6713	142	37	4	4	NUM
ejpam-6713	142	38	(	(	PUNCT
ejpam-6713	142	39	mod	mod	NOUN
ejpam-6713	142	40	5	5	NUM
ejpam-6713	142	41	)	)	PUNCT
ejpam-6713	142	42	then	then	ADV
ejpam-6713	142	43	|pn(v	|pn(v	NUM
ejpam-6713	142	44	,	,	PUNCT
ejpam-6713	142	45	d)|	d)|	NOUN
ejpam-6713	142	46	=	=	NOUN
ejpam-6713	142	47	4	4	NUM
ejpam-6713	142	48	for	for	ADP
ejpam-6713	142	49	all	all	DET
ejpam-6713	142	50	vertex	vertex	NOUN
ejpam-6713	142	51	v	v	ADP
ejpam-6713	142	52	∈	∈	PROPN
ejpam-6713	142	53	d	d	NOUN
ejpam-6713	142	54	except	except	SCONJ
ejpam-6713	142	55	v1	v1	NOUN
ejpam-6713	142	56	and	and	CCONJ
ejpam-6713	142	57	vn−3	vn−3	PROPN
ejpam-6713	142	58	for	for	ADP
ejpam-6713	142	59	which	which	PRON
ejpam-6713	142	60	|pn(v1	|pn(v1	NOUN
ejpam-6713	142	61	,	,	PUNCT
ejpam-6713	142	62	d)|	d)|	NOUN
ejpam-6713	142	63	=	=	SYM
ejpam-6713	142	64	3	3	NUM
ejpam-6713	142	65	and	and	CCONJ
ejpam-6713	142	66	|pn(vn−3	|pn(vn−3	PROPN
ejpam-6713	142	67	,	,	PUNCT
ejpam-6713	142	68	d)|	d)|	NOUN
ejpam-6713	142	69	=	=	SYM
ejpam-6713	142	70	3	3	X
ejpam-6713	142	71	.	.	PUNCT
ejpam-6713	143	1	here	here	ADV
ejpam-6713	143	2	,	,	PUNCT
ejpam-6713	143	3	vn−1	vn−1	PROPN
ejpam-6713	143	4	∈	∈	PROPN
ejpam-6713	143	5	pn(v1	pn(v1	NOUN
ejpam-6713	143	6	,	,	PUNCT
ejpam-6713	143	7	d	d	NOUN
ejpam-6713	143	8	)	)	PUNCT
ejpam-6713	143	9	∩	∩	X
ejpam-6713	143	10	pn(vn−3	pn(vn−3	PROPN
ejpam-6713	143	11	,	,	PUNCT
ejpam-6713	143	12	d	d	NOUN
ejpam-6713	143	13	)	)	PUNCT
ejpam-6713	143	14	.	.	PUNCT
ejpam-6713	144	1	for	for	ADP
ejpam-6713	144	2	n	n	PRON
ejpam-6713	144	3	≡	≡	PROPN
ejpam-6713	144	4	0	0	NUM
ejpam-6713	144	5	,	,	PUNCT
ejpam-6713	144	6	4	4	NUM
ejpam-6713	144	7	(	(	PUNCT
ejpam-6713	144	8	mod	mod	NOUN
ejpam-6713	144	9	5	5	NUM
ejpam-6713	144	10	)	)	PUNCT
ejpam-6713	144	11	.	.	PUNCT
ejpam-6713	145	1	let	let	VERB
ejpam-6713	145	2	g′	g′	NOUN
ejpam-6713	145	3	be	be	AUX
ejpam-6713	145	4	a	a	DET
ejpam-6713	145	5	graph	graph	NOUN
ejpam-6713	145	6	obtained	obtain	VERB
ejpam-6713	145	7	from	from	ADP
ejpam-6713	145	8	g	g	NOUN
ejpam-6713	145	9	by	by	ADP
ejpam-6713	145	10	subdividing	subdivide	VERB
ejpam-6713	145	11	an	an	DET
ejpam-6713	145	12	edge	edge	NOUN
ejpam-6713	145	13	e	e	NOUN
ejpam-6713	145	14	=	=	SYM
ejpam-6713	145	15	v1vn	v1vn	PROPN
ejpam-6713	145	16	(	(	PUNCT
ejpam-6713	145	17	say	say	INTJ
ejpam-6713	145	18	)	)	PUNCT
ejpam-6713	145	19	by	by	ADP
ejpam-6713	145	20	a	a	DET
ejpam-6713	145	21	subdivision	subdivision	NOUN
ejpam-6713	145	22	vertex	vertex	NOUN
ejpam-6713	145	23	x.	x.	NOUN
ejpam-6713	145	24	here	here	ADV
ejpam-6713	145	25	,	,	PUNCT
ejpam-6713	145	26	we	we	PRON
ejpam-6713	145	27	clearly	clearly	ADV
ejpam-6713	145	28	see	see	VERB
ejpam-6713	145	29	that	that	DET
ejpam-6713	145	30	vn	vn	PROPN
ejpam-6713	145	31	/∈	/∈	PUNCT
ejpam-6713	146	1	ng′(d	ng′(d	NOUN
ejpam-6713	146	2	)	)	PUNCT
ejpam-6713	146	3	.	.	PUNCT
ejpam-6713	147	1	hence	hence	ADV
ejpam-6713	147	2	d′	d′	NOUN
ejpam-6713	147	3	=	=	PUNCT
ejpam-6713	147	4	d	d	X
ejpam-6713	147	5	∪	∪	X
ejpam-6713	147	6	{	{	PUNCT
ejpam-6713	147	7	vn	vn	NOUN
ejpam-6713	147	8	}	}	PUNCT
ejpam-6713	147	9	is	be	AUX
ejpam-6713	147	10	a	a	DET
ejpam-6713	147	11	γcer	γcer	NOUN
ejpam-6713	147	12	-	-	PUNCT
ejpam-6713	147	13	set	set	NOUN
ejpam-6713	147	14	of	of	ADP
ejpam-6713	147	15	g′.	g′.	X
ejpam-6713	147	16	therefore	therefore	ADV
ejpam-6713	147	17	,	,	PUNCT
ejpam-6713	147	18	sd+γcer(g	sd+γcer(g	NOUN
ejpam-6713	147	19	)	)	PUNCT
ejpam-6713	147	20	=	=	SYM
ejpam-6713	148	1	1	1	X
ejpam-6713	148	2	.	.	X
ejpam-6713	148	3	case	case	NOUN
ejpam-6713	148	4	(	(	PUNCT
ejpam-6713	148	5	ii	ii	NOUN
ejpam-6713	148	6	)	)	PUNCT
ejpam-6713	148	7	n	n	CCONJ
ejpam-6713	148	8	≡	≡	PROPN
ejpam-6713	148	9	2	2	NUM
ejpam-6713	148	10	,	,	PUNCT
ejpam-6713	148	11	3	3	NUM
ejpam-6713	148	12	(	(	PUNCT
ejpam-6713	148	13	mod	mod	NOUN
ejpam-6713	148	14	5	5	NUM
ejpam-6713	148	15	)	)	PUNCT
ejpam-6713	148	16	if	if	SCONJ
ejpam-6713	148	17	n	n	PRON
ejpam-6713	148	18	≡	≡	PROPN
ejpam-6713	148	19	2	2	NUM
ejpam-6713	148	20	(	(	PUNCT
ejpam-6713	148	21	mod	mod	NOUN
ejpam-6713	148	22	5	5	NUM
ejpam-6713	148	23	)	)	PUNCT
ejpam-6713	148	24	then	then	ADV
ejpam-6713	148	25	|pn(v	|pn(v	NUM
ejpam-6713	148	26	,	,	PUNCT
ejpam-6713	148	27	d)|	d)|	NOUN
ejpam-6713	148	28	=	=	NOUN
ejpam-6713	148	29	4	4	NUM
ejpam-6713	148	30	for	for	ADP
ejpam-6713	148	31	all	all	DET
ejpam-6713	148	32	vertex	vertex	NOUN
ejpam-6713	148	33	v	v	ADP
ejpam-6713	148	34	∈	∈	PROPN
ejpam-6713	148	35	d	d	NOUN
ejpam-6713	148	36	except	except	SCONJ
ejpam-6713	148	37	v1	v1	NOUN
ejpam-6713	148	38	and	and	CCONJ
ejpam-6713	148	39	vn−1	vn−1	ADJ
ejpam-6713	148	40	for	for	ADP
ejpam-6713	148	41	which	which	PRON
ejpam-6713	148	42	|pn(v1	|pn(v1	NOUN
ejpam-6713	148	43	,	,	PUNCT
ejpam-6713	148	44	d)|	d)|	NOUN
ejpam-6713	148	45	=	=	SYM
ejpam-6713	148	46	2	2	NUM
ejpam-6713	148	47	and	and	CCONJ
ejpam-6713	148	48	|pn(vn−1	|pn(vn−1	PROPN
ejpam-6713	148	49	,	,	PUNCT
ejpam-6713	148	50	d)|	d)|	NOUN
ejpam-6713	148	51	=	=	SYM
ejpam-6713	148	52	2	2	X
ejpam-6713	148	53	.	.	PUNCT
ejpam-6713	149	1	here	here	ADV
ejpam-6713	149	2	vn	vn	PROPN
ejpam-6713	149	3	is	be	AUX
ejpam-6713	149	4	a	a	DET
ejpam-6713	149	5	non	non	ADJ
ejpam-6713	149	6	-	-	ADJ
ejpam-6713	149	7	private	private	ADJ
ejpam-6713	149	8	neighbour	neighbour	NOUN
ejpam-6713	149	9	of	of	ADP
ejpam-6713	149	10	the	the	DET
ejpam-6713	149	11	vertices	vertex	NOUN
ejpam-6713	149	12	v1	v1	VERB
ejpam-6713	149	13	and	and	CCONJ
ejpam-6713	149	14	vn−1	vn−1	PROPN
ejpam-6713	149	15	.	.	PUNCT
ejpam-6713	150	1	if	if	SCONJ
ejpam-6713	150	2	n	n	NUM
ejpam-6713	150	3	≡	≡	PROPN
ejpam-6713	150	4	3	3	NUM
ejpam-6713	150	5	(	(	PUNCT
ejpam-6713	150	6	mod	mod	NOUN
ejpam-6713	150	7	5	5	NUM
ejpam-6713	150	8	)	)	PUNCT
ejpam-6713	150	9	then	then	ADV
ejpam-6713	150	10	|pn(v	|pn(v	NUM
ejpam-6713	150	11	,	,	PUNCT
ejpam-6713	150	12	d)|	d)|	NOUN
ejpam-6713	150	13	=	=	NOUN
ejpam-6713	150	14	4	4	NUM
ejpam-6713	150	15	for	for	ADP
ejpam-6713	150	16	all	all	DET
ejpam-6713	150	17	vertex	vertex	NOUN
ejpam-6713	150	18	v	v	ADP
ejpam-6713	150	19	∈	∈	PROPN
ejpam-6713	150	20	d	d	NOUN
ejpam-6713	150	21	except	except	SCONJ
ejpam-6713	150	22	v1	v1	NOUN
ejpam-6713	150	23	and	and	CCONJ
ejpam-6713	150	24	vn−2	vn−2	PROPN
ejpam-6713	150	25	for	for	ADP
ejpam-6713	150	26	which	which	DET
ejpam-6713	150	27	|pn(v1	|pn(v1	NOUN
ejpam-6713	150	28	,	,	PUNCT
ejpam-6713	150	29	d)|	d)|	NOUN
ejpam-6713	150	30	=	=	SYM
ejpam-6713	150	31	2	2	NUM
ejpam-6713	150	32	and	and	CCONJ
ejpam-6713	150	33	|pn(vn−2	|pn(vn−2	PROPN
ejpam-6713	150	34	,	,	PUNCT
ejpam-6713	150	35	d)|	d)|	NOUN
ejpam-6713	150	36	=	=	SYM
ejpam-6713	150	37	2	2	X
ejpam-6713	150	38	.	.	PUNCT
ejpam-6713	151	1	here	here	ADV
ejpam-6713	151	2	,	,	PUNCT
ejpam-6713	151	3	vn	vn	PROPN
ejpam-6713	151	4	/∈	/∈	PUNCT
ejpam-6713	152	1	pn(v1	pn(v1	PRON
ejpam-6713	152	2	,	,	PUNCT
ejpam-6713	152	3	d	d	NOUN
ejpam-6713	152	4	)	)	PUNCT
ejpam-6713	152	5	and	and	CCONJ
ejpam-6713	152	6	vn−1	vn−1	PROPN
ejpam-6713	152	7	/∈	/∈	PUNCT
ejpam-6713	153	1	pn(vn−2	pn(vn−2	PROPN
ejpam-6713	153	2	,	,	PUNCT
ejpam-6713	153	3	d	d	NOUN
ejpam-6713	153	4	)	)	PUNCT
ejpam-6713	153	5	.	.	PUNCT
ejpam-6713	154	1	let	let	VERB
ejpam-6713	154	2	g′	g′	NOUN
ejpam-6713	154	3	be	be	AUX
ejpam-6713	154	4	a	a	DET
ejpam-6713	154	5	graph	graph	NOUN
ejpam-6713	154	6	obtained	obtain	VERB
ejpam-6713	154	7	from	from	ADP
ejpam-6713	154	8	g	g	NOUN
ejpam-6713	154	9	by	by	ADP
ejpam-6713	154	10	subdividing	subdivide	VERB
ejpam-6713	154	11	an	an	DET
ejpam-6713	154	12	edge	edge	NOUN
ejpam-6713	154	13	e	e	NOUN
ejpam-6713	154	14	=	=	PUNCT
ejpam-6713	154	15	v1vn	v1vn	PROPN
ejpam-6713	155	1	[	[	X
ejpam-6713	155	2	or	or	CCONJ
ejpam-6713	155	3	v1vn−1	v1vn−1	PROPN
ejpam-6713	155	4	]	]	PUNCT
ejpam-6713	155	5	by	by	ADP
ejpam-6713	155	6	a	a	DET
ejpam-6713	155	7	subdivision	subdivision	NOUN
ejpam-6713	155	8	vertex	vertex	NOUN
ejpam-6713	155	9	x1	x1	PROPN
ejpam-6713	155	10	(	(	PUNCT
ejpam-6713	155	11	or	or	CCONJ
ejpam-6713	155	12	y1	y1	NOUN
ejpam-6713	155	13	)	)	PUNCT
ejpam-6713	155	14	.	.	PUNCT
ejpam-6713	156	1	here	here	ADV
ejpam-6713	156	2	,	,	PUNCT
ejpam-6713	156	3	d	d	PROPN
ejpam-6713	156	4	is	be	AUX
ejpam-6713	156	5	the	the	DET
ejpam-6713	156	6	γcer	γcer	NOUN
ejpam-6713	156	7	-	-	PUNCT
ejpam-6713	156	8	set	set	NOUN
ejpam-6713	156	9	of	of	ADP
ejpam-6713	156	10	g′.	g′.	X
ejpam-6713	156	11	therefore	therefore	ADV
ejpam-6713	156	12	,	,	PUNCT
ejpam-6713	156	13	sd+γcer(g	sd+γcer(g	NOUN
ejpam-6713	156	14	)	)	PUNCT
ejpam-6713	156	15	>	>	X
ejpam-6713	157	1	1	1	X
ejpam-6713	157	2	.	.	PUNCT
ejpam-6713	158	1	now	now	ADV
ejpam-6713	158	2	g′′	g′′	PROPN
ejpam-6713	158	3	be	be	AUX
ejpam-6713	158	4	a	a	DET
ejpam-6713	158	5	graph	graph	NOUN
ejpam-6713	158	6	obtained	obtain	VERB
ejpam-6713	158	7	from	from	ADP
ejpam-6713	158	8	g′	g′	NOUN
ejpam-6713	158	9	by	by	ADP
ejpam-6713	158	10	subdividing	subdivide	VERB
ejpam-6713	158	11	an	an	DET
ejpam-6713	158	12	edge	edge	NOUN
ejpam-6713	158	13	e	e	NOUN
ejpam-6713	158	14	=	=	SYM
ejpam-6713	159	1	vnvn−1	vnvn−1	PROPN
ejpam-6713	159	2	g.	g.	PROPN
ejpam-6713	159	3	navamani	navamani	PROPN
ejpam-6713	159	4	et	et	PROPN
ejpam-6713	159	5	al	al	PROPN
ejpam-6713	159	6	.	.	PUNCT
ejpam-6713	159	7	/	/	SYM
ejpam-6713	159	8	eur	eur	PROPN
ejpam-6713	159	9	.	.	PUNCT
ejpam-6713	160	1	j.	j.	PROPN
ejpam-6713	160	2	pure	pure	PROPN
ejpam-6713	160	3	appl	appl	PROPN
ejpam-6713	160	4	.	.	PROPN
ejpam-6713	160	5	math	math	PROPN
ejpam-6713	160	6	,	,	PUNCT
ejpam-6713	160	7	18	18	NUM
ejpam-6713	160	8	(	(	PUNCT
ejpam-6713	160	9	4	4	NUM
ejpam-6713	160	10	)	)	PUNCT
ejpam-6713	160	11	(	(	PUNCT
ejpam-6713	160	12	2025	2025	NUM
ejpam-6713	160	13	)	)	PUNCT
ejpam-6713	160	14	,	,	PUNCT
ejpam-6713	160	15	6713	6713	NUM
ejpam-6713	160	16	6	6	NUM
ejpam-6713	160	17	of	of	ADP
ejpam-6713	160	18	15	15	NUM
ejpam-6713	160	19	by	by	ADP
ejpam-6713	160	20	a	a	DET
ejpam-6713	160	21	subdivision	subdivision	NOUN
ejpam-6713	160	22	vertex	vertex	NOUN
ejpam-6713	160	23	x2	x2	PROPN
ejpam-6713	160	24	.	.	PUNCT
ejpam-6713	161	1	for	for	ADP
ejpam-6713	161	2	n	n	X
ejpam-6713	161	3	≡	≡	PROPN
ejpam-6713	161	4	2	2	NUM
ejpam-6713	161	5	(	(	PUNCT
ejpam-6713	161	6	mod	mod	NOUN
ejpam-6713	161	7	5	5	NUM
ejpam-6713	161	8	)	)	PUNCT
ejpam-6713	161	9	,	,	PUNCT
ejpam-6713	161	10	we	we	PRON
ejpam-6713	161	11	see	see	VERB
ejpam-6713	161	12	that	that	DET
ejpam-6713	161	13	vn	vn	PROPN
ejpam-6713	161	14	/∈	/∈	PUNCT
ejpam-6713	162	1	ng′′(d	ng′′(d	NUM
ejpam-6713	162	2	)	)	PUNCT
ejpam-6713	162	3	.	.	PUNCT
ejpam-6713	163	1	hence	hence	ADV
ejpam-6713	163	2	d′	d′	NOUN
ejpam-6713	163	3	=	=	PUNCT
ejpam-6713	163	4	d	d	X
ejpam-6713	163	5	∪	∪	X
ejpam-6713	163	6	{	{	PUNCT
ejpam-6713	163	7	vn	vn	NOUN
ejpam-6713	163	8	}	}	PUNCT
ejpam-6713	163	9	is	be	AUX
ejpam-6713	163	10	a	a	DET
ejpam-6713	163	11	γcer	γcer	NOUN
ejpam-6713	163	12	-	-	PUNCT
ejpam-6713	163	13	set	set	NOUN
ejpam-6713	163	14	of	of	ADP
ejpam-6713	163	15	g′′.	g′′.	PROPN
ejpam-6713	163	16	therefore	therefore	ADV
ejpam-6713	163	17	,	,	PUNCT
ejpam-6713	163	18	sd+γcer(g	sd+γcer(g	NOUN
ejpam-6713	163	19	)	)	PUNCT
ejpam-6713	163	20	=	=	SYM
ejpam-6713	163	21	2	2	X
ejpam-6713	163	22	.	.	PUNCT
ejpam-6713	164	1	now	now	ADV
ejpam-6713	164	2	for	for	ADP
ejpam-6713	164	3	n	n	X
ejpam-6713	164	4	≡	≡	PROPN
ejpam-6713	164	5	3	3	NUM
ejpam-6713	164	6	(	(	PUNCT
ejpam-6713	164	7	mod	mod	NOUN
ejpam-6713	164	8	5	5	NUM
ejpam-6713	164	9	)	)	PUNCT
ejpam-6713	164	10	x2	x2	PROPN
ejpam-6713	164	11	/∈	/∈	PUNCT
ejpam-6713	164	12	ng′′(d	ng′′(d	NOUN
ejpam-6713	164	13	)	)	PUNCT
ejpam-6713	164	14	.	.	PUNCT
ejpam-6713	165	1	hence	hence	ADV
ejpam-6713	165	2	d′	d′	NOUN
ejpam-6713	165	3	=	=	PUNCT
ejpam-6713	165	4	d	d	X
ejpam-6713	165	5	∪	∪	X
ejpam-6713	165	6	{	{	PUNCT
ejpam-6713	165	7	x2	x2	NOUN
ejpam-6713	165	8	}	}	PUNCT
ejpam-6713	165	9	is	be	AUX
ejpam-6713	165	10	a	a	DET
ejpam-6713	165	11	γcer	γcer	NOUN
ejpam-6713	165	12	-	-	PUNCT
ejpam-6713	165	13	set	set	NOUN
ejpam-6713	165	14	of	of	ADP
ejpam-6713	165	15	g′′.	g′′.	PROPN
ejpam-6713	165	16	therefore	therefore	ADV
ejpam-6713	165	17	,	,	PUNCT
ejpam-6713	165	18	sd+γcer(g	sd+γcer(g	NOUN
ejpam-6713	165	19	)	)	PUNCT
ejpam-6713	165	20	=	=	SYM
ejpam-6713	165	21	2	2	NUM
ejpam-6713	165	22	,	,	PUNCT
ejpam-6713	165	23	refer	refer	VERB
ejpam-6713	165	24	figure	figure	NOUN
ejpam-6713	165	25	1	1	NUM
ejpam-6713	165	26	.	.	PUNCT
ejpam-6713	166	1	figure	figure	VERB
ejpam-6713	166	2	1	1	NUM
ejpam-6713	166	3	:	:	PUNCT
ejpam-6713	166	4	a	a	DET
ejpam-6713	166	5	graph	graph	NOUN
ejpam-6713	166	6	illustrating	illustrate	VERB
ejpam-6713	166	7	case	case	NOUN
ejpam-6713	166	8	(	(	PUNCT
ejpam-6713	166	9	ii	ii	NOUN
ejpam-6713	166	10	)	)	PUNCT
ejpam-6713	166	11	of	of	ADP
ejpam-6713	166	12	theorem	theorem	ADJ
ejpam-6713	166	13	7	7	NUM
ejpam-6713	166	14	,	,	PUNCT
ejpam-6713	166	15	sd+γcer	sd+γcer	NOUN
ejpam-6713	166	16	(	(	PUNCT
ejpam-6713	166	17	g	g	NOUN
ejpam-6713	166	18	)	)	PUNCT
ejpam-6713	166	19	=	=	SYM
ejpam-6713	166	20	2	2	NUM
ejpam-6713	166	21	case	case	NOUN
ejpam-6713	166	22	(	(	PUNCT
ejpam-6713	166	23	iii	iii	NOUN
ejpam-6713	166	24	)	)	PUNCT
ejpam-6713	166	25	n	n	CCONJ
ejpam-6713	166	26	≡	≡	PROPN
ejpam-6713	166	27	1	1	NUM
ejpam-6713	166	28	(	(	PUNCT
ejpam-6713	166	29	mod	mod	NOUN
ejpam-6713	166	30	5	5	NUM
ejpam-6713	166	31	)	)	PUNCT
ejpam-6713	166	32	if	if	SCONJ
ejpam-6713	166	33	n	n	PRON
ejpam-6713	166	34	≡	≡	PROPN
ejpam-6713	166	35	1	1	NUM
ejpam-6713	166	36	(	(	PUNCT
ejpam-6713	166	37	mod	mod	NOUN
ejpam-6713	166	38	5	5	NUM
ejpam-6713	166	39	)	)	PUNCT
ejpam-6713	166	40	,	,	PUNCT
ejpam-6713	166	41	then	then	ADV
ejpam-6713	166	42	|pn(v	|pn(v	NUM
ejpam-6713	166	43	,	,	PUNCT
ejpam-6713	166	44	d)|	d)|	NOUN
ejpam-6713	166	45	=	=	NOUN
ejpam-6713	166	46	4	4	NUM
ejpam-6713	166	47	for	for	ADP
ejpam-6713	166	48	all	all	DET
ejpam-6713	166	49	vertex	vertex	NOUN
ejpam-6713	166	50	v	v	ADP
ejpam-6713	166	51	∈	∈	PROPN
ejpam-6713	166	52	d	d	NOUN
ejpam-6713	166	53	except	except	SCONJ
ejpam-6713	166	54	vn	vn	PROPN
ejpam-6713	166	55	and	and	CCONJ
ejpam-6713	166	56	v1	v1	VERB
ejpam-6713	166	57	for	for	ADP
ejpam-6713	166	58	which	which	DET
ejpam-6713	166	59	|pn(v1	|pn(v1	NOUN
ejpam-6713	166	60	,	,	PUNCT
ejpam-6713	166	61	d)|	d)|	NOUN
ejpam-6713	166	62	=	=	SYM
ejpam-6713	166	63	2	2	NUM
ejpam-6713	166	64	and	and	CCONJ
ejpam-6713	166	65	|pn(vn	|pn(vn	NOUN
ejpam-6713	166	66	,	,	PUNCT
ejpam-6713	166	67	d)|	d)|	NOUN
ejpam-6713	166	68	=	=	NOUN
ejpam-6713	166	69	1	1	X
ejpam-6713	166	70	.	.	PUNCT
ejpam-6713	167	1	let	let	VERB
ejpam-6713	167	2	g′	g′	NOUN
ejpam-6713	167	3	be	be	AUX
ejpam-6713	167	4	a	a	DET
ejpam-6713	167	5	graph	graph	NOUN
ejpam-6713	167	6	obtained	obtain	VERB
ejpam-6713	167	7	from	from	ADP
ejpam-6713	167	8	g	g	NOUN
ejpam-6713	167	9	by	by	ADP
ejpam-6713	167	10	subdividing	subdivide	VERB
ejpam-6713	167	11	an	an	DET
ejpam-6713	167	12	edge	edge	NOUN
ejpam-6713	167	13	e	e	NOUN
ejpam-6713	167	14	=	=	PUNCT
ejpam-6713	167	15	v1vn	v1vn	PROPN
ejpam-6713	168	1	[	[	X
ejpam-6713	168	2	or	or	CCONJ
ejpam-6713	168	3	v1vn−1	v1vn−1	PROPN
ejpam-6713	168	4	]	]	PUNCT
ejpam-6713	168	5	by	by	ADP
ejpam-6713	168	6	a	a	DET
ejpam-6713	168	7	subdivision	subdivision	NOUN
ejpam-6713	168	8	vertex	vertex	NOUN
ejpam-6713	168	9	y.	y.	PROPN
ejpam-6713	168	10	here	here	ADV
ejpam-6713	168	11	y	y	PROPN
ejpam-6713	168	12	is	be	AUX
ejpam-6713	168	13	dominated	dominate	VERB
ejpam-6713	168	14	by	by	ADP
ejpam-6713	168	15	v1	v1	NOUN
ejpam-6713	168	16	and	and	CCONJ
ejpam-6713	168	17	vn−1	vn−1	PROPN
ejpam-6713	168	18	is	be	AUX
ejpam-6713	168	19	dominated	dominate	VERB
ejpam-6713	168	20	by	by	ADP
ejpam-6713	168	21	vn	vn	PROPN
ejpam-6713	168	22	.	.	PUNCT
ejpam-6713	169	1	hence	hence	ADV
ejpam-6713	169	2	,	,	PUNCT
ejpam-6713	169	3	d	d	PROPN
ejpam-6713	169	4	is	be	AUX
ejpam-6713	169	5	the	the	DET
ejpam-6713	169	6	γcer	γcer	NOUN
ejpam-6713	169	7	-	-	PUNCT
ejpam-6713	169	8	set	set	NOUN
ejpam-6713	169	9	of	of	ADP
ejpam-6713	169	10	g′.	g′.	X
ejpam-6713	169	11	therefore	therefore	ADV
ejpam-6713	169	12	,	,	PUNCT
ejpam-6713	169	13	sd+γcer(g	sd+γcer(g	NOUN
ejpam-6713	169	14	)	)	PUNCT
ejpam-6713	169	15	>	>	X
ejpam-6713	170	1	1	1	X
ejpam-6713	170	2	.	.	PUNCT
ejpam-6713	170	3	now	now	ADV
ejpam-6713	170	4	let	let	VERB
ejpam-6713	170	5	g′′	g′′	PROPN
ejpam-6713	170	6	be	be	AUX
ejpam-6713	170	7	a	a	DET
ejpam-6713	170	8	graph	graph	NOUN
ejpam-6713	170	9	obtained	obtain	VERB
ejpam-6713	170	10	from	from	ADP
ejpam-6713	170	11	g	g	NOUN
ejpam-6713	170	12	by	by	ADP
ejpam-6713	170	13	subdividing	subdivide	VERB
ejpam-6713	170	14	the	the	DET
ejpam-6713	170	15	2	2	NUM
ejpam-6713	170	16	edges	edge	NOUN
ejpam-6713	170	17	e1	e1	NOUN
ejpam-6713	170	18	and	and	CCONJ
ejpam-6713	170	19	e2	e2	PROPN
ejpam-6713	170	20	by	by	ADP
ejpam-6713	170	21	subdivision	subdivision	NOUN
ejpam-6713	170	22	vertices	vertex	NOUN
ejpam-6713	170	23	x1	x1	PROPN
ejpam-6713	170	24	and	and	CCONJ
ejpam-6713	170	25	x2	x2	PROPN
ejpam-6713	170	26	respectively	respectively	ADV
ejpam-6713	170	27	.	.	PUNCT
ejpam-6713	171	1	now	now	ADV
ejpam-6713	171	2	we	we	PRON
ejpam-6713	171	3	have	have	VERB
ejpam-6713	171	4	the	the	DET
ejpam-6713	171	5	following	follow	VERB
ejpam-6713	171	6	subcases	subcase	NOUN
ejpam-6713	171	7	.	.	PUNCT
ejpam-6713	172	1	subcase	subcase	PROPN
ejpam-6713	172	2	(	(	PUNCT
ejpam-6713	172	3	a	a	NOUN
ejpam-6713	172	4	)	)	PUNCT
ejpam-6713	172	5	e1	e1	NOUN
ejpam-6713	172	6	=	=	SYM
ejpam-6713	172	7	v1vn	v1vn	PROPN
ejpam-6713	172	8	and	and	CCONJ
ejpam-6713	172	9	e2	e2	PROPN
ejpam-6713	172	10	=	=	PUNCT
ejpam-6713	172	11	v1v2	v1v2	PUNCT
ejpam-6713	173	1	[	[	X
ejpam-6713	173	2	adjacent	adjacent	ADJ
ejpam-6713	173	3	edges	edge	NOUN
ejpam-6713	173	4	]	]	PUNCT
ejpam-6713	173	5	.	.	PUNCT
ejpam-6713	174	1	in	in	ADP
ejpam-6713	174	2	this	this	DET
ejpam-6713	174	3	subcase	subcase	NOUN
ejpam-6713	174	4	,	,	PUNCT
ejpam-6713	174	5	x1	x1	PROPN
ejpam-6713	174	6	,	,	PUNCT
ejpam-6713	174	7	x2	x2	PROPN
ejpam-6713	174	8	∈	∈	PROPN
ejpam-6713	174	9	n(v1	n(v1	NOUN
ejpam-6713	174	10	)	)	PUNCT
ejpam-6713	174	11	and	and	CCONJ
ejpam-6713	174	12	v2	v2	PROPN
ejpam-6713	174	13	∈	∈	PROPN
ejpam-6713	174	14	n(vn	n(vn	PROPN
ejpam-6713	174	15	)	)	PUNCT
ejpam-6713	174	16	.	.	PUNCT
ejpam-6713	175	1	hence	hence	ADV
ejpam-6713	175	2	γcer(g	γcer(g	NUM
ejpam-6713	175	3	)	)	PUNCT
ejpam-6713	175	4	=	=	PUNCT
ejpam-6713	175	5	γcer(g	γcer(g	NUM
ejpam-6713	175	6	′′	′′	PROPN
ejpam-6713	175	7	)	)	PUNCT
ejpam-6713	175	8	.	.	PUNCT
ejpam-6713	176	1	subcase	subcase	NOUN
ejpam-6713	176	2	(	(	PUNCT
ejpam-6713	176	3	b	b	NOUN
ejpam-6713	176	4	)	)	PUNCT
ejpam-6713	176	5	e1	e1	NOUN
ejpam-6713	176	6	=	=	SYM
ejpam-6713	176	7	v1vn	v1vn	PROPN
ejpam-6713	176	8	and	and	CCONJ
ejpam-6713	176	9	e2	e2	PROPN
ejpam-6713	176	10	=	=	PUNCT
ejpam-6713	177	1	v6v7	v6v7	X
ejpam-6713	178	1	[	[	X
ejpam-6713	178	2	non	non	X
ejpam-6713	178	3	adjacent	adjacent	ADJ
ejpam-6713	178	4	edges	edge	NOUN
ejpam-6713	178	5	in	in	ADP
ejpam-6713	178	6	the	the	DET
ejpam-6713	178	7	outer	outer	ADJ
ejpam-6713	178	8	cycle	cycle	NOUN
ejpam-6713	178	9	]	]	PUNCT
ejpam-6713	178	10	here	here	ADV
ejpam-6713	178	11	x1	x1	PRON
ejpam-6713	178	12	∈	∈	NOUN
ejpam-6713	178	13	n(v1	n(v1	NOUN
ejpam-6713	178	14	)	)	PUNCT
ejpam-6713	178	15	and	and	CCONJ
ejpam-6713	178	16	x2	x2	PROPN
ejpam-6713	178	17	∈	∈	PROPN
ejpam-6713	178	18	n(v6	n(v6	NOUN
ejpam-6713	178	19	)	)	PUNCT
ejpam-6713	178	20	and	and	CCONJ
ejpam-6713	178	21	to	to	PART
ejpam-6713	178	22	dominate	dominate	VERB
ejpam-6713	178	23	v7	v7	NOUN
ejpam-6713	178	24	,	,	PUNCT
ejpam-6713	178	25	d1	d1	PROPN
ejpam-6713	178	26	=	=	PUNCT
ejpam-6713	178	27	{	{	PUNCT
ejpam-6713	178	28	d	d	X
ejpam-6713	178	29	−	−	PROPN
ejpam-6713	178	30	{	{	PUNCT
ejpam-6713	178	31	(	(	PUNCT
ejpam-6713	178	32	v5k+6	v5k+6	PROPN
ejpam-6713	178	33	)	)	PUNCT
ejpam-6713	178	34	:	:	PUNCT
ejpam-6713	178	35	1	1	NUM
ejpam-6713	178	36	≤	≤	NUM
ejpam-6713	178	37	k	k	X
ejpam-6713	178	38	≤	≤	NUM
ejpam-6713	178	39	n−6	n−6	PROPN
ejpam-6713	178	40	5	5	NUM
ejpam-6713	178	41	}	}	PUNCT
ejpam-6713	178	42	}	}	PUNCT
ejpam-6713	178	43	∪	∪	ADV
ejpam-6713	178	44	{	{	PUNCT
ejpam-6713	178	45	v5k+4	v5k+4	NOUN
ejpam-6713	178	46	:	:	PUNCT
ejpam-6713	178	47	1	1	NUM
ejpam-6713	178	48	≤	≤	NUM
ejpam-6713	178	49	k	k	X
ejpam-6713	178	50	≤	≤	NUM
ejpam-6713	178	51	n−6	n−6	PROPN
ejpam-6713	178	52	5	5	NUM
ejpam-6713	178	53	}	}	PUNCT
ejpam-6713	178	54	is	be	AUX
ejpam-6713	178	55	the	the	DET
ejpam-6713	178	56	new	new	ADJ
ejpam-6713	178	57	configuration	configuration	NOUN
ejpam-6713	178	58	of	of	ADP
ejpam-6713	178	59	the	the	DET
ejpam-6713	178	60	certified	certify	VERB
ejpam-6713	178	61	dominating	dominating	NOUN
ejpam-6713	178	62	set	set	NOUN
ejpam-6713	178	63	d	d	PROPN
ejpam-6713	178	64	g.	g.	PROPN
ejpam-6713	178	65	navamani	navamani	PROPN
ejpam-6713	178	66	et	et	PROPN
ejpam-6713	178	67	al	al	PROPN
ejpam-6713	178	68	.	.	PUNCT
ejpam-6713	178	69	/	/	SYM
ejpam-6713	178	70	eur	eur	PROPN
ejpam-6713	178	71	.	.	PUNCT
ejpam-6713	179	1	j.	j.	PROPN
ejpam-6713	179	2	pure	pure	PROPN
ejpam-6713	179	3	appl	appl	PROPN
ejpam-6713	179	4	.	.	PROPN
ejpam-6713	179	5	math	math	PROPN
ejpam-6713	179	6	,	,	PUNCT
ejpam-6713	179	7	18	18	NUM
ejpam-6713	179	8	(	(	PUNCT
ejpam-6713	179	9	4	4	NUM
ejpam-6713	179	10	)	)	PUNCT
ejpam-6713	179	11	(	(	PUNCT
ejpam-6713	179	12	2025	2025	NUM
ejpam-6713	179	13	)	)	PUNCT
ejpam-6713	179	14	,	,	PUNCT
ejpam-6713	179	15	6713	6713	NUM
ejpam-6713	179	16	7	7	NUM
ejpam-6713	179	17	of	of	ADP
ejpam-6713	179	18	15	15	NUM
ejpam-6713	179	19	and	and	CCONJ
ejpam-6713	179	20	|d|	|d|	PROPN
ejpam-6713	179	21	=	=	PUNCT
ejpam-6713	179	22	|d1|	|d1|	PROPN
ejpam-6713	179	23	.	.	PUNCT
ejpam-6713	180	1	hence	hence	ADV
ejpam-6713	180	2	γcer(g	γcer(g	NUM
ejpam-6713	180	3	)	)	PUNCT
ejpam-6713	180	4	=	=	PUNCT
ejpam-6713	180	5	γcer(g	γcer(g	NUM
ejpam-6713	180	6	′′	′′	PROPN
ejpam-6713	180	7	)	)	PUNCT
ejpam-6713	180	8	.	.	PUNCT
ejpam-6713	181	1	subcase	subcase	NOUN
ejpam-6713	181	2	(	(	PUNCT
ejpam-6713	181	3	c	c	NOUN
ejpam-6713	181	4	)	)	PUNCT
ejpam-6713	181	5	e1	e1	NOUN
ejpam-6713	181	6	=	=	SYM
ejpam-6713	181	7	v1vn	v1vn	PROPN
ejpam-6713	181	8	and	and	CCONJ
ejpam-6713	181	9	e2	e2	PROPN
ejpam-6713	181	10	=	=	SYM
ejpam-6713	181	11	v1vn−1	v1vn−1	PROPN
ejpam-6713	182	1	[	[	X
ejpam-6713	182	2	adjacent	adjacent	ADJ
ejpam-6713	182	3	edges	edge	NOUN
ejpam-6713	182	4	with	with	ADP
ejpam-6713	182	5	one	one	NUM
ejpam-6713	182	6	edge	edge	NOUN
ejpam-6713	182	7	in	in	ADP
ejpam-6713	182	8	the	the	DET
ejpam-6713	182	9	outer	outer	ADJ
ejpam-6713	182	10	cycle	cycle	NOUN
ejpam-6713	182	11	]	]	PUNCT
ejpam-6713	182	12	in	in	ADP
ejpam-6713	182	13	this	this	DET
ejpam-6713	182	14	subcase	subcase	NOUN
ejpam-6713	182	15	,	,	PUNCT
ejpam-6713	182	16	x1	x1	PROPN
ejpam-6713	182	17	,	,	PUNCT
ejpam-6713	182	18	x2	x2	PROPN
ejpam-6713	182	19	∈	∈	PROPN
ejpam-6713	182	20	n(v1	n(v1	NOUN
ejpam-6713	182	21	)	)	PUNCT
ejpam-6713	182	22	and	and	CCONJ
ejpam-6713	182	23	vn−1	vn−1	PROPN
ejpam-6713	182	24	∈	∈	PROPN
ejpam-6713	182	25	n(vn	n(vn	PROPN
ejpam-6713	182	26	)	)	PUNCT
ejpam-6713	182	27	.	.	PUNCT
ejpam-6713	183	1	hence	hence	ADV
ejpam-6713	183	2	γcer(g	γcer(g	NUM
ejpam-6713	183	3	)	)	PUNCT
ejpam-6713	183	4	=	=	PUNCT
ejpam-6713	183	5	γcer(g	γcer(g	NUM
ejpam-6713	183	6	′′	′′	PROPN
ejpam-6713	183	7	)	)	PUNCT
ejpam-6713	183	8	.	.	PUNCT
ejpam-6713	184	1	subcase	subcase	NOUN
ejpam-6713	184	2	(	(	PUNCT
ejpam-6713	184	3	d	d	NOUN
ejpam-6713	184	4	)	)	PUNCT
ejpam-6713	184	5	e1	e1	NOUN
ejpam-6713	184	6	=	=	SYM
ejpam-6713	184	7	v1vn	v1vn	PROPN
ejpam-6713	184	8	and	and	CCONJ
ejpam-6713	184	9	e2	e2	PROPN
ejpam-6713	184	10	=	=	PUNCT
ejpam-6713	185	1	v6v8	v6v8	X
ejpam-6713	186	1	[	[	X
ejpam-6713	186	2	non	non	X
ejpam-6713	186	3	adjacent	adjacent	ADJ
ejpam-6713	186	4	edges	edge	NOUN
ejpam-6713	186	5	with	with	ADP
ejpam-6713	186	6	one	one	NUM
ejpam-6713	186	7	edge	edge	NOUN
ejpam-6713	186	8	in	in	ADP
ejpam-6713	186	9	the	the	DET
ejpam-6713	186	10	outer	outer	ADJ
ejpam-6713	186	11	cycle	cycle	NOUN
ejpam-6713	186	12	]	]	PUNCT
ejpam-6713	186	13	in	in	ADP
ejpam-6713	186	14	this	this	DET
ejpam-6713	186	15	subcase	subcase	NOUN
ejpam-6713	186	16	,	,	PUNCT
ejpam-6713	186	17	x1	x1	PROPN
ejpam-6713	186	18	∈	∈	PROPN
ejpam-6713	186	19	n(v1	n(v1	NOUN
ejpam-6713	186	20	)	)	PUNCT
ejpam-6713	186	21	and	and	CCONJ
ejpam-6713	186	22	x2	x2	PROPN
ejpam-6713	186	23	∈	∈	PROPN
ejpam-6713	186	24	n(v6	n(v6	NOUN
ejpam-6713	186	25	)	)	PUNCT
ejpam-6713	186	26	as	as	ADP
ejpam-6713	186	27	in	in	ADP
ejpam-6713	186	28	subcase	subcase	NOUN
ejpam-6713	186	29	(	(	PUNCT
ejpam-6713	186	30	b	b	NOUN
ejpam-6713	186	31	)	)	PUNCT
ejpam-6713	186	32	,	,	PUNCT
ejpam-6713	186	33	d1	d1	PROPN
ejpam-6713	186	34	is	be	AUX
ejpam-6713	186	35	the	the	DET
ejpam-6713	186	36	certified	certify	VERB
ejpam-6713	186	37	dominating	dominating	NOUN
ejpam-6713	186	38	set	set	NOUN
ejpam-6713	186	39	of	of	ADP
ejpam-6713	186	40	g′′	g′′	PROPN
ejpam-6713	186	41	and	and	CCONJ
ejpam-6713	186	42	v8	v8	PROPN
ejpam-6713	186	43	∈	∈	PROPN
ejpam-6713	186	44	n(v9	n(v9	NUM
ejpam-6713	186	45	)	)	PUNCT
ejpam-6713	186	46	.	.	PUNCT
ejpam-6713	187	1	hence	hence	ADV
ejpam-6713	187	2	γcer(g	γcer(g	NUM
ejpam-6713	187	3	)	)	PUNCT
ejpam-6713	187	4	=	=	PUNCT
ejpam-6713	187	5	γcer(g	γcer(g	NUM
ejpam-6713	187	6	′′	′′	PROPN
ejpam-6713	187	7	)	)	PUNCT
ejpam-6713	187	8	.	.	PUNCT
ejpam-6713	188	1	subcase	subcase	NOUN
ejpam-6713	188	2	(	(	PUNCT
ejpam-6713	188	3	e	e	NOUN
ejpam-6713	188	4	)	)	PUNCT
ejpam-6713	188	5	e1	e1	NOUN
ejpam-6713	188	6	=	=	SYM
ejpam-6713	188	7	v1vn−1	v1vn−1	PROPN
ejpam-6713	188	8	and	and	CCONJ
ejpam-6713	188	9	e2	e2	PROPN
ejpam-6713	188	10	=	=	PUNCT
ejpam-6713	188	11	v1v3	v1v3	PROPN
ejpam-6713	189	1	[	[	X
ejpam-6713	189	2	adjacent	adjacent	ADJ
ejpam-6713	189	3	edges	edge	NOUN
ejpam-6713	189	4	,	,	PUNCT
ejpam-6713	189	5	which	which	PRON
ejpam-6713	189	6	are	be	AUX
ejpam-6713	189	7	not	not	PART
ejpam-6713	189	8	in	in	ADP
ejpam-6713	189	9	the	the	DET
ejpam-6713	189	10	outer	outer	ADJ
ejpam-6713	189	11	cycle	cycle	NOUN
ejpam-6713	189	12	]	]	PUNCT
ejpam-6713	189	13	here	here	ADV
ejpam-6713	189	14	,	,	PUNCT
ejpam-6713	189	15	x1	x1	PROPN
ejpam-6713	189	16	,	,	PUNCT
ejpam-6713	189	17	x2	x2	PROPN
ejpam-6713	189	18	∈	∈	PROPN
ejpam-6713	189	19	n(v1	n(v1	NOUN
ejpam-6713	189	20	)	)	PUNCT
ejpam-6713	189	21	.	.	PUNCT
ejpam-6713	190	1	in	in	ADP
ejpam-6713	190	2	order	order	NOUN
ejpam-6713	190	3	to	to	PART
ejpam-6713	190	4	dominate	dominate	VERB
ejpam-6713	190	5	v3	v3	PROPN
ejpam-6713	190	6	,	,	PUNCT
ejpam-6713	190	7	d1	d1	PROPN
ejpam-6713	190	8	=	=	PUNCT
ejpam-6713	190	9	{	{	PUNCT
ejpam-6713	190	10	v5k	v5k	NUM
ejpam-6713	190	11	:	:	PUNCT
ejpam-6713	190	12	1	1	NUM
ejpam-6713	190	13	≤	≤	NUM
ejpam-6713	190	14	k	k	X
ejpam-6713	190	15	≤	≤	NUM
ejpam-6713	190	16	n−1	n−1	PROPN
ejpam-6713	190	17	5	5	NUM
ejpam-6713	190	18	}	}	PUNCT
ejpam-6713	190	19	is	be	AUX
ejpam-6713	190	20	the	the	DET
ejpam-6713	190	21	new	new	ADJ
ejpam-6713	190	22	configuration	configuration	NOUN
ejpam-6713	190	23	of	of	ADP
ejpam-6713	190	24	the	the	DET
ejpam-6713	190	25	certified	certify	VERB
ejpam-6713	190	26	dominating	dominating	NOUN
ejpam-6713	190	27	set	set	VERB
ejpam-6713	190	28	d	d	PROPN
ejpam-6713	190	29	,	,	PUNCT
ejpam-6713	190	30	and	and	CCONJ
ejpam-6713	190	31	|d|	|d|	PROPN
ejpam-6713	190	32	=	=	PUNCT
ejpam-6713	190	33	|d1|	|d1|	PROPN
ejpam-6713	190	34	.	.	PUNCT
ejpam-6713	191	1	hence	hence	ADV
ejpam-6713	191	2	γcer(g	γcer(g	NUM
ejpam-6713	191	3	)	)	PUNCT
ejpam-6713	191	4	=	=	PUNCT
ejpam-6713	191	5	γcer(g	γcer(g	NUM
ejpam-6713	191	6	′′	′′	PROPN
ejpam-6713	191	7	)	)	PUNCT
ejpam-6713	191	8	.	.	PUNCT
ejpam-6713	192	1	subcase	subcase	NOUN
ejpam-6713	192	2	(	(	PUNCT
ejpam-6713	192	3	f	f	X
ejpam-6713	192	4	)	)	PUNCT
ejpam-6713	192	5	e1	e1	NOUN
ejpam-6713	192	6	=	=	SYM
ejpam-6713	192	7	v1v3	v1v3	NOUN
ejpam-6713	192	8	and	and	CCONJ
ejpam-6713	192	9	e2	e2	PROPN
ejpam-6713	192	10	=	=	SYM
ejpam-6713	192	11	v4v6	v4v6	X
ejpam-6713	192	12	[	[	PUNCT
ejpam-6713	192	13	non	non	X
ejpam-6713	192	14	adjacent	adjacent	ADJ
ejpam-6713	192	15	edges	edge	NOUN
ejpam-6713	192	16	,	,	PUNCT
ejpam-6713	192	17	which	which	PRON
ejpam-6713	192	18	are	be	AUX
ejpam-6713	192	19	not	not	PART
ejpam-6713	192	20	in	in	ADP
ejpam-6713	192	21	the	the	DET
ejpam-6713	192	22	outer	outer	ADJ
ejpam-6713	192	23	cycle	cycle	NOUN
ejpam-6713	192	24	]	]	PUNCT
ejpam-6713	192	25	in	in	ADP
ejpam-6713	192	26	this	this	DET
ejpam-6713	192	27	subcase	subcase	NOUN
ejpam-6713	192	28	,	,	PUNCT
ejpam-6713	192	29	x1	x1	PROPN
ejpam-6713	192	30	∈	∈	PROPN
ejpam-6713	192	31	n(v1	n(v1	NOUN
ejpam-6713	192	32	)	)	PUNCT
ejpam-6713	192	33	and	and	CCONJ
ejpam-6713	192	34	x2	x2	PROPN
ejpam-6713	192	35	∈	∈	PROPN
ejpam-6713	192	36	n(v4	n(v4	NOUN
ejpam-6713	192	37	)	)	PUNCT
ejpam-6713	192	38	.	.	PUNCT
ejpam-6713	193	1	hence	hence	ADV
ejpam-6713	193	2	γcer(g	γcer(g	NUM
ejpam-6713	193	3	)	)	PUNCT
ejpam-6713	193	4	=	=	PUNCT
ejpam-6713	193	5	γcer(g	γcer(g	NUM
ejpam-6713	193	6	′′	′′	PROPN
ejpam-6713	193	7	)	)	PUNCT
ejpam-6713	193	8	.	.	PUNCT
ejpam-6713	194	1	from	from	ADP
ejpam-6713	194	2	all	all	DET
ejpam-6713	194	3	the	the	DET
ejpam-6713	194	4	above	above	ADJ
ejpam-6713	194	5	subcases	subcase	NOUN
ejpam-6713	194	6	we	we	PRON
ejpam-6713	194	7	see	see	VERB
ejpam-6713	194	8	that	that	PRON
ejpam-6713	194	9	sd+γcer(g	sd+γcer(g	NOUN
ejpam-6713	194	10	)	)	PUNCT
ejpam-6713	194	11	>	>	X
ejpam-6713	195	1	2	2	X
ejpam-6713	195	2	.	.	PUNCT
ejpam-6713	195	3	let	let	VERB
ejpam-6713	195	4	g′′′	g′′′	PROPN
ejpam-6713	195	5	be	be	AUX
ejpam-6713	195	6	the	the	DET
ejpam-6713	195	7	graph	graph	NOUN
ejpam-6713	195	8	obtained	obtain	VERB
ejpam-6713	195	9	from	from	ADP
ejpam-6713	195	10	g	g	NOUN
ejpam-6713	195	11	by	by	ADP
ejpam-6713	195	12	subdividing	subdivide	VERB
ejpam-6713	195	13	3	3	NUM
ejpam-6713	195	14	edges	edge	NOUN
ejpam-6713	195	15	e1	e1	NOUN
ejpam-6713	195	16	=	=	SYM
ejpam-6713	195	17	v1vn	v1vn	PROPN
ejpam-6713	195	18	,	,	PUNCT
ejpam-6713	195	19	e2	e2	PROPN
ejpam-6713	195	20	=	=	PUNCT
ejpam-6713	195	21	v1v2	v1v2	X
ejpam-6713	195	22	and	and	CCONJ
ejpam-6713	195	23	e3	e3	PROPN
ejpam-6713	195	24	=	=	SYM
ejpam-6713	195	25	v2v3	v2v3	NUM
ejpam-6713	195	26	by	by	ADP
ejpam-6713	195	27	the	the	DET
ejpam-6713	195	28	subdivision	subdivision	NOUN
ejpam-6713	195	29	vertices	vertice	VERB
ejpam-6713	195	30	x1	x1	PROPN
ejpam-6713	195	31	,	,	PUNCT
ejpam-6713	195	32	x2	x2	PROPN
ejpam-6713	195	33	and	and	CCONJ
ejpam-6713	195	34	x3	x3	VERB
ejpam-6713	195	35	respectively	respectively	ADV
ejpam-6713	195	36	.	.	PUNCT
ejpam-6713	196	1	here	here	ADV
ejpam-6713	196	2	x3	x3	PROPN
ejpam-6713	196	3	/∈	/∈	PUNCT
ejpam-6713	197	1	n(v	n(v	PROPN
ejpam-6713	197	2	)	)	PUNCT
ejpam-6713	197	3	for	for	SCONJ
ejpam-6713	197	4	every	every	DET
ejpam-6713	197	5	vertex	vertex	NOUN
ejpam-6713	197	6	v	v	ADP
ejpam-6713	197	7	∈	∈	PROPN
ejpam-6713	197	8	d.	d.	NOUN
ejpam-6713	197	9	to	to	PART
ejpam-6713	197	10	dominate	dominate	VERB
ejpam-6713	197	11	x3	x3	PROPN
ejpam-6713	197	12	,	,	PUNCT
ejpam-6713	197	13	set	set	VERB
ejpam-6713	197	14	d1	d1	PROPN
ejpam-6713	197	15	=	=	SYM
ejpam-6713	197	16	{	{	PUNCT
ejpam-6713	197	17	v5k−2	v5k−2	PROPN
ejpam-6713	197	18	:	:	PUNCT
ejpam-6713	197	19	1	1	NUM
ejpam-6713	197	20	≤	≤	NUM
ejpam-6713	197	21	k	k	X
ejpam-6713	197	22	≤	≤	NUM
ejpam-6713	197	23	n−1	n−1	PROPN
ejpam-6713	197	24	5	5	NUM
ejpam-6713	197	25	}	}	PUNCT
ejpam-6713	197	26	∪	∪	ADJ
ejpam-6713	197	27	{	{	PUNCT
ejpam-6713	197	28	v1	v1	NOUN
ejpam-6713	197	29	}	}	PUNCT
ejpam-6713	197	30	which	which	PRON
ejpam-6713	197	31	is	be	AUX
ejpam-6713	197	32	the	the	DET
ejpam-6713	197	33	new	new	ADJ
ejpam-6713	197	34	configuration	configuration	NOUN
ejpam-6713	197	35	of	of	ADP
ejpam-6713	197	36	the	the	DET
ejpam-6713	197	37	certified	certify	VERB
ejpam-6713	197	38	dominating	dominating	NOUN
ejpam-6713	197	39	set	set	VERB
ejpam-6713	197	40	d.	d.	PROPN
ejpam-6713	197	41	but	but	CCONJ
ejpam-6713	197	42	{	{	PUNCT
ejpam-6713	197	43	vn	vn	X
ejpam-6713	197	44	,	,	PUNCT
ejpam-6713	197	45	v2	v2	PROPN
ejpam-6713	197	46	}	}	PUNCT
ejpam-6713	197	47	/∈	/∈	PUNCT
ejpam-6713	198	1	ng′′′(d1	ng′′′(d1	ADJ
ejpam-6713	198	2	)	)	PUNCT
ejpam-6713	198	3	.	.	PUNCT
ejpam-6713	199	1	so	so	ADV
ejpam-6713	199	2	d1	d1	PROPN
ejpam-6713	199	3	is	be	AUX
ejpam-6713	199	4	not	not	PART
ejpam-6713	199	5	a	a	DET
ejpam-6713	199	6	certified	certify	VERB
ejpam-6713	199	7	dominating	dominating	NOUN
ejpam-6713	199	8	set	set	NOUN
ejpam-6713	199	9	of	of	ADP
ejpam-6713	199	10	g′′′.	g′′′.	PROPN
ejpam-6713	199	11	hence	hence	ADV
ejpam-6713	199	12	d2	d2	PROPN
ejpam-6713	199	13	=	=	SYM
ejpam-6713	199	14	d1	d1	PROPN
ejpam-6713	199	15	∪	∪	ADV
ejpam-6713	199	16	{	{	PUNCT
ejpam-6713	199	17	vn	vn	NOUN
ejpam-6713	199	18	}	}	PUNCT
ejpam-6713	199	19	is	be	AUX
ejpam-6713	199	20	the	the	DET
ejpam-6713	199	21	certified	certify	VERB
ejpam-6713	199	22	dominating	dominating	NOUN
ejpam-6713	199	23	set	set	NOUN
ejpam-6713	199	24	of	of	ADP
ejpam-6713	199	25	g′′′.	g′′′.	PROPN
ejpam-6713	199	26	therefore	therefore	ADV
ejpam-6713	199	27	,	,	PUNCT
ejpam-6713	199	28	sd+γcer(g	sd+γcer(g	NOUN
ejpam-6713	199	29	)	)	PUNCT
ejpam-6713	199	30	=	=	SYM
ejpam-6713	200	1	3	3	X
ejpam-6713	200	2	.	.	X
ejpam-6713	200	3	theorem	theorem	VERB
ejpam-6713	200	4	8	8	NUM
ejpam-6713	200	5	.	.	PUNCT
ejpam-6713	201	1	[	[	X
ejpam-6713	201	2	19	19	NUM
ejpam-6713	201	3	]	]	PUNCT
ejpam-6713	201	4	for	for	ADP
ejpam-6713	201	5	any	any	DET
ejpam-6713	201	6	integer	integer	NOUN
ejpam-6713	201	7	n	n	PRON
ejpam-6713	201	8	≥	≥	NOUN
ejpam-6713	201	9	6	6	NUM
ejpam-6713	201	10	,	,	PUNCT
ejpam-6713	201	11	γ(cn(1	γ(cn(1	PROPN
ejpam-6713	201	12	,	,	PUNCT
ejpam-6713	201	13	3	3	NUM
ejpam-6713	201	14	)	)	PUNCT
ejpam-6713	201	15	)	)	PUNCT
ejpam-6713	202	1	=	=	PRON
ejpam-6713	202	2	{	{	PUNCT
ejpam-6713	202	3	⌈	⌈	PROPN
ejpam-6713	202	4	n	n	PRON
ejpam-6713	202	5	5	5	NUM
ejpam-6713	202	6	⌉	⌉	NOUN
ejpam-6713	202	7	,	,	PUNCT
ejpam-6713	202	8	n	n	PROPN
ejpam-6713	202	9	6∈	6∈	NOUN
ejpam-6713	202	10	4	4	NUM
ejpam-6713	202	11	(	(	PUNCT
ejpam-6713	202	12	mod	mod	PROPN
ejpam-6713	202	13	5)⌈	5)⌈	NUM
ejpam-6713	202	14	n	n	ADV
ejpam-6713	202	15	5	5	NUM
ejpam-6713	202	16	⌉	⌉	NOUN
ejpam-6713	202	17	+	+	CCONJ
ejpam-6713	202	18	1	1	NUM
ejpam-6713	202	19	,	,	PUNCT
ejpam-6713	202	20	n	n	PRON
ejpam-6713	202	21	≡	≡	PROPN
ejpam-6713	202	22	4	4	NUM
ejpam-6713	202	23	(	(	PUNCT
ejpam-6713	202	24	mod	mod	NOUN
ejpam-6713	202	25	5	5	NUM
ejpam-6713	202	26	)	)	PUNCT
ejpam-6713	202	27	theorem	theorem	NOUN
ejpam-6713	202	28	9	9	NUM
ejpam-6713	202	29	.	.	PUNCT
ejpam-6713	202	30	for	for	ADP
ejpam-6713	202	31	any	any	DET
ejpam-6713	202	32	circulant	circulant	ADJ
ejpam-6713	202	33	graph	graph	NOUN
ejpam-6713	202	34	g	g	ADP
ejpam-6713	202	35	∼=	∼=	PROPN
ejpam-6713	202	36	cn(1	cn(1	NOUN
ejpam-6713	202	37	,	,	PUNCT
ejpam-6713	202	38	3	3	NUM
ejpam-6713	202	39	)	)	PUNCT
ejpam-6713	202	40	,	,	PUNCT
ejpam-6713	202	41	n	n	X
ejpam-6713	202	42	≥	≥	NOUN
ejpam-6713	202	43	6	6	NUM
ejpam-6713	202	44	,	,	PUNCT
ejpam-6713	202	45	sd+γcer(g	sd+γcer(g	NOUN
ejpam-6713	202	46	)	)	PUNCT
ejpam-6713	202	47	=	=	SYM
ejpam-6713	202	48	{	{	PUNCT
ejpam-6713	202	49	1	1	NUM
ejpam-6713	202	50	if	if	SCONJ
ejpam-6713	202	51	n	n	PRON
ejpam-6713	202	52	≡	≡	PROPN
ejpam-6713	202	53	0	0	NUM
ejpam-6713	202	54	,	,	PUNCT
ejpam-6713	202	55	3(mod	3(mod	NUM
ejpam-6713	202	56	5	5	NUM
ejpam-6713	202	57	)	)	PUNCT
ejpam-6713	202	58	2	2	NUM
ejpam-6713	202	59	otherwise	otherwise	ADV
ejpam-6713	202	60	proof	proof	NOUN
ejpam-6713	202	61	.	.	PUNCT
ejpam-6713	203	1	let	let	VERB
ejpam-6713	203	2	g	g	PROPN
ejpam-6713	203	3	∼=	∼=	PROPN
ejpam-6713	203	4	cn(1	cn(1	NOUN
ejpam-6713	203	5	,	,	PUNCT
ejpam-6713	203	6	3	3	NUM
ejpam-6713	203	7	)	)	PUNCT
ejpam-6713	203	8	.	.	PUNCT
ejpam-6713	204	1	let	let	VERB
ejpam-6713	204	2	v	v	X
ejpam-6713	204	3	(	(	PUNCT
ejpam-6713	204	4	g	g	NOUN
ejpam-6713	204	5	)	)	PUNCT
ejpam-6713	204	6	=	=	SYM
ejpam-6713	204	7	{	{	PUNCT
ejpam-6713	204	8	v1	v1	PROPN
ejpam-6713	204	9	,	,	PUNCT
ejpam-6713	204	10	v2	v2	PROPN
ejpam-6713	204	11	,	,	PUNCT
ejpam-6713	204	12	v3	v3	PROPN
ejpam-6713	204	13	,	,	PUNCT
ejpam-6713	204	14	.	.	PUNCT
ejpam-6713	204	15	.	.	PUNCT
ejpam-6713	204	16	.	.	PUNCT
ejpam-6713	205	1	vn	vn	AUX
ejpam-6713	205	2	}	}	PUNCT
ejpam-6713	205	3	be	be	AUX
ejpam-6713	205	4	the	the	DET
ejpam-6713	205	5	vertex	vertex	NOUN
ejpam-6713	205	6	set	set	NOUN
ejpam-6713	205	7	of	of	ADP
ejpam-6713	205	8	g	g	PROPN
ejpam-6713	205	9	and	and	CCONJ
ejpam-6713	205	10	d	d	NOUN
ejpam-6713	205	11	=	=	SYM
ejpam-6713	205	12	{	{	PUNCT
ejpam-6713	205	13	v5k−4	v5k−4	NOUN
ejpam-6713	205	14	:	:	PUNCT
ejpam-6713	205	15	1	1	NUM
ejpam-6713	205	16	≤	≤	NUM
ejpam-6713	205	17	k	k	X
ejpam-6713	205	18	≤	≤	PROPN
ejpam-6713	205	19	⌈	⌈	NUM
ejpam-6713	205	20	n	n	CCONJ
ejpam-6713	205	21	5	5	NUM
ejpam-6713	205	22	⌉	⌉	AUX
ejpam-6713	205	23	}	}	PUNCT
ejpam-6713	205	24	be	be	AUX
ejpam-6713	205	25	a	a	DET
ejpam-6713	205	26	γcer	γcer	NOUN
ejpam-6713	205	27	-	-	PUNCT
ejpam-6713	205	28	set	set	NOUN
ejpam-6713	205	29	of	of	ADP
ejpam-6713	205	30	g.	g.	NOUN
ejpam-6713	205	31	for	for	ADP
ejpam-6713	205	32	every	every	DET
ejpam-6713	205	33	vertex	vertex	NOUN
ejpam-6713	205	34	v	v	ADP
ejpam-6713	205	35	∈	∈	PROPN
ejpam-6713	205	36	d	d	NOUN
ejpam-6713	205	37	,	,	PUNCT
ejpam-6713	205	38	n(v	n(v	PROPN
ejpam-6713	205	39	)	)	PUNCT
ejpam-6713	205	40	≥	≥	NOUN
ejpam-6713	205	41	2	2	NUM
ejpam-6713	205	42	.	.	PUNCT
ejpam-6713	206	1	by	by	ADP
ejpam-6713	206	2	theorem	theorem	ADJ
ejpam-6713	206	3	3	3	NUM
ejpam-6713	206	4	and	and	CCONJ
ejpam-6713	206	5	theorem	theorem	VERB
ejpam-6713	206	6	8	8	NUM
ejpam-6713	206	7	,	,	PUNCT
ejpam-6713	206	8	we	we	PRON
ejpam-6713	206	9	clearly	clearly	ADV
ejpam-6713	206	10	see	see	VERB
ejpam-6713	206	11	that	that	PRON
ejpam-6713	206	12	γ(g	γ(g	PROPN
ejpam-6713	206	13	)	)	PUNCT
ejpam-6713	207	1	=	=	SYM
ejpam-6713	207	2	γcer(g	γcer(g	NOUN
ejpam-6713	207	3	)	)	PUNCT
ejpam-6713	207	4	=	=	PUNCT
ejpam-6713	208	1	⌈	⌈	PROPN
ejpam-6713	208	2	n	n	CCONJ
ejpam-6713	208	3	5	5	NUM
ejpam-6713	208	4	⌉	⌉	X
ejpam-6713	208	5	.	.	PUNCT
ejpam-6713	209	1	case	case	NOUN
ejpam-6713	209	2	(	(	PUNCT
ejpam-6713	209	3	i	i	NOUN
ejpam-6713	209	4	)	)	PUNCT
ejpam-6713	209	5	n	n	X
ejpam-6713	209	6	≡	≡	PROPN
ejpam-6713	209	7	0	0	PUNCT
ejpam-6713	210	1	(	(	PUNCT
ejpam-6713	210	2	mod	mod	NOUN
ejpam-6713	210	3	5	5	NUM
ejpam-6713	210	4	)	)	PUNCT
ejpam-6713	210	5	here	here	ADV
ejpam-6713	210	6	|pn(v	|pn(v	NUM
ejpam-6713	210	7	,	,	PUNCT
ejpam-6713	210	8	d)|	d)|	NOUN
ejpam-6713	210	9	=	=	NOUN
ejpam-6713	210	10	4	4	NUM
ejpam-6713	210	11	for	for	ADP
ejpam-6713	210	12	each	each	DET
ejpam-6713	210	13	vertex	vertex	NOUN
ejpam-6713	210	14	v	v	ADP
ejpam-6713	210	15	∈	∈	PROPN
ejpam-6713	210	16	d.	d.	NOUN
ejpam-6713	210	17	let	let	VERB
ejpam-6713	210	18	g′	g′	NOUN
ejpam-6713	210	19	be	be	AUX
ejpam-6713	210	20	a	a	DET
ejpam-6713	210	21	graph	graph	NOUN
ejpam-6713	210	22	obtained	obtain	VERB
ejpam-6713	210	23	from	from	ADP
ejpam-6713	210	24	g	g	NOUN
ejpam-6713	210	25	by	by	ADP
ejpam-6713	210	26	subdividing	subdivide	VERB
ejpam-6713	210	27	an	an	DET
ejpam-6713	210	28	edge	edge	NOUN
ejpam-6713	210	29	e	e	NOUN
ejpam-6713	210	30	=	=	PUNCT
ejpam-6713	210	31	v1v2	v1v2	X
ejpam-6713	210	32	(	(	PUNCT
ejpam-6713	210	33	say	say	INTJ
ejpam-6713	210	34	)	)	PUNCT
ejpam-6713	210	35	by	by	ADP
ejpam-6713	210	36	a	a	DET
ejpam-6713	210	37	subdivision	subdivision	NOUN
ejpam-6713	210	38	vertex	vertex	NOUN
ejpam-6713	210	39	x	x	NOUN
ejpam-6713	210	40	,	,	PUNCT
ejpam-6713	210	41	here	here	ADV
ejpam-6713	210	42	x	x	PUNCT
ejpam-6713	210	43	is	be	AUX
ejpam-6713	210	44	dominated	dominate	VERB
ejpam-6713	210	45	by	by	ADP
ejpam-6713	210	46	v1	v1	NOUN
ejpam-6713	210	47	,	,	PUNCT
ejpam-6713	210	48	and	and	CCONJ
ejpam-6713	210	49	v2	v2	PROPN
ejpam-6713	210	50	/∈	/∈	PUNCT
ejpam-6713	211	1	n(v	n(v	PROPN
ejpam-6713	211	2	)	)	PUNCT
ejpam-6713	211	3	for	for	ADP
ejpam-6713	211	4	all	all	DET
ejpam-6713	211	5	vertex	vertex	NOUN
ejpam-6713	211	6	v	v	ADP
ejpam-6713	211	7	∈	∈	PROPN
ejpam-6713	211	8	d.	d.	NOUN
ejpam-6713	211	9	hence	hence	ADV
ejpam-6713	211	10	d1	d1	PROPN
ejpam-6713	211	11	=	=	PUNCT
ejpam-6713	212	1	d	d	X
ejpam-6713	212	2	∪	∪	X
ejpam-6713	212	3	{	{	PUNCT
ejpam-6713	212	4	v2	v2	NOUN
ejpam-6713	212	5	}	}	PUNCT
ejpam-6713	212	6	is	be	AUX
ejpam-6713	212	7	the	the	DET
ejpam-6713	212	8	γcer	γcer	NOUN
ejpam-6713	212	9	-	-	PUNCT
ejpam-6713	212	10	set	set	NOUN
ejpam-6713	212	11	of	of	ADP
ejpam-6713	212	12	g′.	g′.	X
ejpam-6713	212	13	therefore	therefore	ADV
ejpam-6713	212	14	,	,	PUNCT
ejpam-6713	212	15	sd+γcer(g	sd+γcer(g	NOUN
ejpam-6713	212	16	)	)	PUNCT
ejpam-6713	212	17	=	=	SYM
ejpam-6713	212	18	1	1	NUM
ejpam-6713	212	19	,	,	PUNCT
ejpam-6713	212	20	refer	refer	VERB
ejpam-6713	212	21	figure	figure	NOUN
ejpam-6713	212	22	2	2	NUM
ejpam-6713	212	23	.	.	NOUN
ejpam-6713	212	24	case	case	NOUN
ejpam-6713	212	25	(	(	PUNCT
ejpam-6713	212	26	ii	ii	NOUN
ejpam-6713	212	27	)	)	PUNCT
ejpam-6713	212	28	n	n	CCONJ
ejpam-6713	212	29	≡	≡	PROPN
ejpam-6713	212	30	3	3	NUM
ejpam-6713	212	31	(	(	PUNCT
ejpam-6713	212	32	mod	mod	NOUN
ejpam-6713	212	33	5	5	NUM
ejpam-6713	212	34	)	)	PUNCT
ejpam-6713	212	35	g.	g.	NOUN
ejpam-6713	212	36	navamani	navamani	PROPN
ejpam-6713	212	37	et	et	PROPN
ejpam-6713	212	38	al	al	PROPN
ejpam-6713	212	39	.	.	PUNCT
ejpam-6713	212	40	/	/	SYM
ejpam-6713	212	41	eur	eur	PROPN
ejpam-6713	212	42	.	.	PUNCT
ejpam-6713	213	1	j.	j.	PROPN
ejpam-6713	213	2	pure	pure	PROPN
ejpam-6713	213	3	appl	appl	PROPN
ejpam-6713	213	4	.	.	PROPN
ejpam-6713	213	5	math	math	PROPN
ejpam-6713	213	6	,	,	PUNCT
ejpam-6713	213	7	18	18	NUM
ejpam-6713	213	8	(	(	PUNCT
ejpam-6713	213	9	4	4	NUM
ejpam-6713	213	10	)	)	PUNCT
ejpam-6713	213	11	(	(	PUNCT
ejpam-6713	213	12	2025	2025	NUM
ejpam-6713	213	13	)	)	PUNCT
ejpam-6713	213	14	,	,	PUNCT
ejpam-6713	213	15	6713	6713	NUM
ejpam-6713	213	16	8	8	NUM
ejpam-6713	213	17	of	of	ADP
ejpam-6713	213	18	15	15	NUM
ejpam-6713	213	19	figure	figure	NOUN
ejpam-6713	213	20	2	2	NUM
ejpam-6713	213	21	:	:	PUNCT
ejpam-6713	213	22	a	a	DET
ejpam-6713	213	23	graph	graph	NOUN
ejpam-6713	213	24	illustrating	illustrate	VERB
ejpam-6713	213	25	case	case	NOUN
ejpam-6713	213	26	(	(	PUNCT
ejpam-6713	213	27	i	i	NOUN
ejpam-6713	213	28	)	)	PUNCT
ejpam-6713	213	29	of	of	ADP
ejpam-6713	213	30	theorem	theorem	NOUN
ejpam-6713	213	31	9	9	NUM
ejpam-6713	213	32	,	,	PUNCT
ejpam-6713	213	33	sd+γcer	sd+γcer	NOUN
ejpam-6713	213	34	(	(	PUNCT
ejpam-6713	213	35	g	g	NOUN
ejpam-6713	213	36	)	)	PUNCT
ejpam-6713	213	37	=	=	SYM
ejpam-6713	214	1	1	1	NUM
ejpam-6713	214	2	here	here	ADV
ejpam-6713	214	3	|pn(v	|pn(v	NUM
ejpam-6713	214	4	,	,	PUNCT
ejpam-6713	214	5	d)|	d)|	NOUN
ejpam-6713	214	6	=	=	NOUN
ejpam-6713	214	7	4	4	NUM
ejpam-6713	214	8	for	for	ADP
ejpam-6713	214	9	all	all	DET
ejpam-6713	214	10	vertex	vertex	NOUN
ejpam-6713	214	11	v	v	ADP
ejpam-6713	214	12	∈	∈	PROPN
ejpam-6713	214	13	d	d	NOUN
ejpam-6713	214	14	except	except	SCONJ
ejpam-6713	214	15	the	the	DET
ejpam-6713	214	16	vertices	vertex	NOUN
ejpam-6713	214	17	v1	v1	VERB
ejpam-6713	214	18	and	and	CCONJ
ejpam-6713	214	19	vn−2	vn−2	PROPN
ejpam-6713	214	20	for	for	ADP
ejpam-6713	214	21	which	which	DET
ejpam-6713	214	22	|pn(v1	|pn(v1	NOUN
ejpam-6713	214	23	,	,	PUNCT
ejpam-6713	214	24	d)|	d)|	NOUN
ejpam-6713	214	25	=	=	SYM
ejpam-6713	214	26	3	3	NUM
ejpam-6713	214	27	and	and	CCONJ
ejpam-6713	214	28	|pn(vn−2	|pn(vn−2	PROPN
ejpam-6713	214	29	,	,	PUNCT
ejpam-6713	214	30	d)|	d)|	PROPN
ejpam-6713	214	31	=3	=3	VERB
ejpam-6713	214	32	.	.	PUNCT
ejpam-6713	215	1	let	let	VERB
ejpam-6713	215	2	g′	g′	NOUN
ejpam-6713	215	3	be	be	AUX
ejpam-6713	215	4	a	a	DET
ejpam-6713	215	5	graph	graph	NOUN
ejpam-6713	215	6	obtained	obtain	VERB
ejpam-6713	215	7	from	from	ADP
ejpam-6713	215	8	g	g	NOUN
ejpam-6713	215	9	by	by	ADP
ejpam-6713	215	10	subdividing	subdivide	VERB
ejpam-6713	215	11	an	an	DET
ejpam-6713	215	12	edge	edge	NOUN
ejpam-6713	215	13	e	e	NOUN
ejpam-6713	215	14	=	=	PUNCT
ejpam-6713	215	15	v1v2	v1v2	X
ejpam-6713	215	16	(	(	PUNCT
ejpam-6713	215	17	say	say	INTJ
ejpam-6713	215	18	)	)	PUNCT
ejpam-6713	215	19	by	by	ADP
ejpam-6713	215	20	a	a	DET
ejpam-6713	215	21	subdivision	subdivision	NOUN
ejpam-6713	215	22	vertex	vertex	NOUN
ejpam-6713	215	23	x	x	NOUN
ejpam-6713	215	24	,	,	PUNCT
ejpam-6713	215	25	here	here	ADV
ejpam-6713	215	26	x	x	PART
ejpam-6713	215	27	∈	∈	NOUN
ejpam-6713	215	28	n(v1	n(v1	NOUN
ejpam-6713	215	29	)	)	PUNCT
ejpam-6713	215	30	in	in	ADP
ejpam-6713	215	31	g′	g′	NOUN
ejpam-6713	215	32	and	and	CCONJ
ejpam-6713	215	33	v2	v2	PROPN
ejpam-6713	215	34	/∈	/∈	PUNCT
ejpam-6713	216	1	n(v	n(v	PROPN
ejpam-6713	216	2	)	)	PUNCT
ejpam-6713	216	3	,	,	PUNCT
ejpam-6713	216	4	for	for	ADP
ejpam-6713	216	5	all	all	PRON
ejpam-6713	216	6	v	v	NOUN
ejpam-6713	216	7	∈	∈	NOUN
ejpam-6713	216	8	d.	d.	NOUN
ejpam-6713	216	9	hence	hence	ADV
ejpam-6713	216	10	d1	d1	PROPN
ejpam-6713	216	11	=	=	PUNCT
ejpam-6713	217	1	d	d	X
ejpam-6713	217	2	∪	∪	X
ejpam-6713	217	3	{	{	PUNCT
ejpam-6713	217	4	v2	v2	NOUN
ejpam-6713	217	5	}	}	PUNCT
ejpam-6713	217	6	is	be	AUX
ejpam-6713	217	7	the	the	DET
ejpam-6713	217	8	γcer	γcer	NOUN
ejpam-6713	217	9	-	-	PUNCT
ejpam-6713	217	10	set	set	NOUN
ejpam-6713	217	11	of	of	ADP
ejpam-6713	217	12	g′.	g′.	X
ejpam-6713	217	13	therefore	therefore	ADV
ejpam-6713	217	14	,	,	PUNCT
ejpam-6713	217	15	sd+γcer(g	sd+γcer(g	NOUN
ejpam-6713	217	16	)	)	PUNCT
ejpam-6713	217	17	=	=	SYM
ejpam-6713	217	18	1	1	X
ejpam-6713	217	19	.	.	X
ejpam-6713	217	20	case	case	NOUN
ejpam-6713	217	21	(	(	PUNCT
ejpam-6713	217	22	iii	iii	NOUN
ejpam-6713	217	23	)	)	PUNCT
ejpam-6713	217	24	n	n	CCONJ
ejpam-6713	217	25	≡	≡	PROPN
ejpam-6713	217	26	1	1	NUM
ejpam-6713	217	27	(	(	PUNCT
ejpam-6713	217	28	mod	mod	NOUN
ejpam-6713	217	29	5	5	NUM
ejpam-6713	217	30	)	)	PUNCT
ejpam-6713	217	31	here	here	ADV
ejpam-6713	217	32	|pn(v	|pn(v	NUM
ejpam-6713	217	33	,	,	PUNCT
ejpam-6713	217	34	d)|	d)|	NOUN
ejpam-6713	217	35	=	=	NOUN
ejpam-6713	217	36	4	4	NUM
ejpam-6713	217	37	for	for	ADP
ejpam-6713	217	38	all	all	DET
ejpam-6713	217	39	vertex	vertex	NOUN
ejpam-6713	217	40	v	v	ADP
ejpam-6713	217	41	∈	∈	PROPN
ejpam-6713	217	42	d	d	NOUN
ejpam-6713	217	43	except	except	SCONJ
ejpam-6713	217	44	the	the	DET
ejpam-6713	217	45	vertices	vertex	NOUN
ejpam-6713	217	46	v1	v1	VERB
ejpam-6713	217	47	and	and	CCONJ
ejpam-6713	217	48	vn	vn	NOUN
ejpam-6713	217	49	for	for	ADP
ejpam-6713	217	50	which	which	PRON
ejpam-6713	217	51	|pn(v1	|pn(v1	NOUN
ejpam-6713	217	52	,	,	PUNCT
ejpam-6713	217	53	d)|	d)|	NOUN
ejpam-6713	217	54	=	=	SYM
ejpam-6713	217	55	2	2	NUM
ejpam-6713	217	56	and	and	CCONJ
ejpam-6713	217	57	|pn(vn	|pn(vn	NOUN
ejpam-6713	217	58	,	,	PUNCT
ejpam-6713	217	59	d)|	d)|	NOUN
ejpam-6713	217	60	=	=	SYM
ejpam-6713	217	61	2	2	X
ejpam-6713	217	62	.	.	PUNCT
ejpam-6713	218	1	let	let	VERB
ejpam-6713	218	2	g′	g′	NOUN
ejpam-6713	218	3	be	be	AUX
ejpam-6713	218	4	a	a	DET
ejpam-6713	218	5	graph	graph	NOUN
ejpam-6713	218	6	obtained	obtain	VERB
ejpam-6713	218	7	from	from	ADP
ejpam-6713	218	8	g	g	NOUN
ejpam-6713	218	9	by	by	ADP
ejpam-6713	218	10	subdividing	subdivide	VERB
ejpam-6713	218	11	an	an	DET
ejpam-6713	218	12	edge	edge	NOUN
ejpam-6713	218	13	e	e	NOUN
ejpam-6713	218	14	=	=	SYM
ejpam-6713	218	15	v1vn	v1vn	PROPN
ejpam-6713	218	16	(	(	PUNCT
ejpam-6713	218	17	or	or	CCONJ
ejpam-6713	218	18	e	e	X
ejpam-6713	218	19	=	=	PROPN
ejpam-6713	218	20	v1vn−2	v1vn−2	PROPN
ejpam-6713	218	21	)	)	PUNCT
ejpam-6713	218	22	by	by	ADP
ejpam-6713	218	23	a	a	DET
ejpam-6713	218	24	subdivision	subdivision	NOUN
ejpam-6713	218	25	vertex	vertex	NOUN
ejpam-6713	218	26	x.	x.	NOUN
ejpam-6713	219	1	here	here	ADV
ejpam-6713	219	2	,	,	PUNCT
ejpam-6713	219	3	x	x	X
ejpam-6713	219	4	∈	∈	NOUN
ejpam-6713	219	5	n(v1	n(v1	NOUN
ejpam-6713	219	6	)	)	PUNCT
ejpam-6713	219	7	,	,	PUNCT
ejpam-6713	219	8	vn−2	vn−2	PROPN
ejpam-6713	219	9	∈	∈	PROPN
ejpam-6713	219	10	n	n	CCONJ
ejpam-6713	219	11	[	[	X
ejpam-6713	219	12	vn−5	vn−5	NOUN
ejpam-6713	219	13	]	]	PUNCT
ejpam-6713	219	14	.	.	PUNCT
ejpam-6713	220	1	hence	hence	ADV
ejpam-6713	220	2	γcer(g	γcer(g	NUM
ejpam-6713	220	3	′	′	NUM
ejpam-6713	220	4	)	)	PUNCT
ejpam-6713	220	5	=	=	SYM
ejpam-6713	220	6	γcer(g	γcer(g	NOUN
ejpam-6713	220	7	)	)	PUNCT
ejpam-6713	220	8	.	.	PUNCT
ejpam-6713	221	1	therefore	therefore	ADV
ejpam-6713	221	2	sd+γcer(g	sd+γcer(g	NOUN
ejpam-6713	221	3	)	)	PUNCT
ejpam-6713	221	4	>	>	X
ejpam-6713	222	1	1	1	X
ejpam-6713	222	2	.	.	PUNCT
ejpam-6713	222	3	let	let	VERB
ejpam-6713	222	4	g′′	g′′	PROPN
ejpam-6713	222	5	be	be	AUX
ejpam-6713	222	6	a	a	DET
ejpam-6713	222	7	graph	graph	NOUN
ejpam-6713	222	8	obtained	obtain	VERB
ejpam-6713	222	9	from	from	ADP
ejpam-6713	222	10	g′	g′	NOUN
ejpam-6713	222	11	by	by	ADP
ejpam-6713	222	12	subdividing	subdivide	VERB
ejpam-6713	222	13	an	an	DET
ejpam-6713	222	14	edge	edge	NOUN
ejpam-6713	222	15	e	e	NOUN
ejpam-6713	222	16	=	=	PUNCT
ejpam-6713	222	17	v1v2	v1v2	X
ejpam-6713	222	18	(	(	PUNCT
ejpam-6713	222	19	say	say	INTJ
ejpam-6713	222	20	)	)	PUNCT
ejpam-6713	222	21	by	by	ADP
ejpam-6713	222	22	a	a	DET
ejpam-6713	222	23	subdivision	subdivision	NOUN
ejpam-6713	222	24	vertex	vertex	NOUN
ejpam-6713	222	25	y	y	PROPN
ejpam-6713	222	26	and	and	CCONJ
ejpam-6713	222	27	y	y	PROPN
ejpam-6713	222	28	∈	∈	PROPN
ejpam-6713	222	29	n(v1	n(v1	NOUN
ejpam-6713	222	30	)	)	PUNCT
ejpam-6713	222	31	.	.	PUNCT
ejpam-6713	223	1	now	now	ADV
ejpam-6713	223	2	v2	v2	VERB
ejpam-6713	223	3	/∈	/∈	PUNCT
ejpam-6713	223	4	ng′′(d	ng′′(d	NOUN
ejpam-6713	223	5	)	)	PUNCT
ejpam-6713	223	6	.	.	PUNCT
ejpam-6713	224	1	here	here	ADV
ejpam-6713	224	2	d1	d1	PROPN
ejpam-6713	224	3	=	=	PUNCT
ejpam-6713	225	1	d	d	X
ejpam-6713	225	2	∪	∪	X
ejpam-6713	225	3	{	{	PUNCT
ejpam-6713	225	4	v2	v2	NOUN
ejpam-6713	225	5	}	}	PUNCT
ejpam-6713	225	6	is	be	AUX
ejpam-6713	225	7	a	a	DET
ejpam-6713	225	8	γcer	γcer	NOUN
ejpam-6713	225	9	set	set	NOUN
ejpam-6713	225	10	of	of	ADP
ejpam-6713	225	11	g′′.	g′′.	PROPN
ejpam-6713	225	12	hence	hence	ADV
ejpam-6713	225	13	,	,	PUNCT
ejpam-6713	225	14	γcer(g	γcer(g	PROPN
ejpam-6713	225	15	′′	′′	PROPN
ejpam-6713	225	16	)	)	PUNCT
ejpam-6713	225	17	>	>	X
ejpam-6713	225	18	γcer(g	γcer(g	PROPN
ejpam-6713	225	19	′	′	NUM
ejpam-6713	225	20	)	)	PUNCT
ejpam-6713	225	21	.	.	PUNCT
ejpam-6713	226	1	therefore	therefore	ADV
ejpam-6713	226	2	sd+γcer(g	sd+γcer(g	NOUN
ejpam-6713	226	3	)	)	PUNCT
ejpam-6713	226	4	=	=	SYM
ejpam-6713	226	5	2	2	X
ejpam-6713	226	6	.	.	X
ejpam-6713	226	7	case	case	NOUN
ejpam-6713	226	8	(	(	PUNCT
ejpam-6713	226	9	iv	iv	X
ejpam-6713	226	10	)	)	PUNCT
ejpam-6713	226	11	n	n	CCONJ
ejpam-6713	226	12	≡	≡	PROPN
ejpam-6713	226	13	2	2	NUM
ejpam-6713	226	14	(	(	PUNCT
ejpam-6713	226	15	mod	mod	NOUN
ejpam-6713	226	16	5	5	NUM
ejpam-6713	226	17	)	)	PUNCT
ejpam-6713	226	18	here	here	ADV
ejpam-6713	226	19	|pn(v	|pn(v	NUM
ejpam-6713	226	20	,	,	PUNCT
ejpam-6713	226	21	d)|	d)|	NOUN
ejpam-6713	226	22	=	=	NOUN
ejpam-6713	226	23	4	4	NUM
ejpam-6713	226	24	for	for	ADP
ejpam-6713	226	25	all	all	DET
ejpam-6713	226	26	vertex	vertex	NOUN
ejpam-6713	226	27	v	v	ADP
ejpam-6713	226	28	∈	∈	PROPN
ejpam-6713	226	29	d	d	NOUN
ejpam-6713	226	30	except	except	SCONJ
ejpam-6713	226	31	the	the	DET
ejpam-6713	226	32	vertices	vertex	NOUN
ejpam-6713	226	33	v1	v1	VERB
ejpam-6713	226	34	and	and	CCONJ
ejpam-6713	226	35	vn−1	vn−1	ADJ
ejpam-6713	226	36	for	for	ADP
ejpam-6713	226	37	which	which	PRON
ejpam-6713	226	38	|pn(v1	|pn(v1	NOUN
ejpam-6713	226	39	,	,	PUNCT
ejpam-6713	226	40	d)|	d)|	NOUN
ejpam-6713	226	41	=	=	SYM
ejpam-6713	226	42	1	1	NUM
ejpam-6713	226	43	and	and	CCONJ
ejpam-6713	226	44	|pn(vn−1	|pn(vn−1	PROPN
ejpam-6713	226	45	,	,	PUNCT
ejpam-6713	226	46	d)|	d)|	NOUN
ejpam-6713	226	47	=	=	SYM
ejpam-6713	226	48	1	1	NUM
ejpam-6713	226	49	and	and	CCONJ
ejpam-6713	226	50	vn	vn	PROPN
ejpam-6713	226	51	is	be	AUX
ejpam-6713	226	52	a	a	DET
ejpam-6713	226	53	non	non	ADJ
ejpam-6713	226	54	-	-	ADJ
ejpam-6713	226	55	private	private	ADJ
ejpam-6713	226	56	neighbour	neighbour	NOUN
ejpam-6713	226	57	.	.	PUNCT
ejpam-6713	227	1	let	let	VERB
ejpam-6713	227	2	g′	g′	NOUN
ejpam-6713	227	3	be	be	AUX
ejpam-6713	227	4	a	a	DET
ejpam-6713	227	5	graph	graph	NOUN
ejpam-6713	227	6	obtained	obtain	VERB
ejpam-6713	227	7	from	from	ADP
ejpam-6713	227	8	g	g	NOUN
ejpam-6713	227	9	by	by	ADP
ejpam-6713	227	10	subdividing	subdivide	VERB
ejpam-6713	227	11	an	an	DET
ejpam-6713	227	12	edge	edge	NOUN
ejpam-6713	227	13	e	e	NOUN
ejpam-6713	227	14	=	=	PUNCT
ejpam-6713	227	15	v1vn	v1vn	PROPN
ejpam-6713	228	1	[	[	X
ejpam-6713	228	2	or	or	CCONJ
ejpam-6713	228	3	e	e	X
ejpam-6713	228	4	=	=	PROPN
ejpam-6713	228	5	v1vn−2	v1vn−2	PROPN
ejpam-6713	228	6	]	]	PUNCT
ejpam-6713	228	7	by	by	ADP
ejpam-6713	228	8	a	a	DET
ejpam-6713	228	9	subdivision	subdivision	NOUN
ejpam-6713	228	10	vertex	vertex	NOUN
ejpam-6713	228	11	x.	x.	NOUN
ejpam-6713	228	12	here	here	ADV
ejpam-6713	228	13	x	x	PUNCT
ejpam-6713	228	14	∈	∈	NOUN
ejpam-6713	228	15	n(v1	n(v1	NOUN
ejpam-6713	228	16	)	)	PUNCT
ejpam-6713	228	17	,	,	PUNCT
ejpam-6713	228	18	(	(	PUNCT
ejpam-6713	228	19	or	or	CCONJ
ejpam-6713	228	20	x	x	PROPN
ejpam-6713	228	21	∈	∈	PROPN
ejpam-6713	228	22	n(vn−1	n(vn−1	NOUN
ejpam-6713	228	23	)	)	PUNCT
ejpam-6713	228	24	)	)	PUNCT
ejpam-6713	228	25	.	.	PUNCT
ejpam-6713	229	1	hence	hence	ADV
ejpam-6713	229	2	γcer(g	γcer(g	NUM
ejpam-6713	229	3	′	′	NUM
ejpam-6713	229	4	)	)	PUNCT
ejpam-6713	229	5	=	=	SYM
ejpam-6713	229	6	γcer(g	γcer(g	NOUN
ejpam-6713	229	7	)	)	PUNCT
ejpam-6713	229	8	.	.	PUNCT
ejpam-6713	230	1	therefore	therefore	ADV
ejpam-6713	230	2	,	,	PUNCT
ejpam-6713	230	3	sd+γcer(g	sd+γcer(g	NOUN
ejpam-6713	230	4	)	)	PUNCT
ejpam-6713	230	5	>	>	X
ejpam-6713	231	1	1	1	X
ejpam-6713	231	2	.	.	PUNCT
ejpam-6713	231	3	let	let	VERB
ejpam-6713	231	4	g′′	g′′	PROPN
ejpam-6713	231	5	be	be	AUX
ejpam-6713	231	6	a	a	DET
ejpam-6713	231	7	graph	graph	NOUN
ejpam-6713	231	8	obtained	obtain	VERB
ejpam-6713	231	9	from	from	ADP
ejpam-6713	231	10	g	g	NOUN
ejpam-6713	231	11	by	by	ADP
ejpam-6713	231	12	subdividing	subdivide	VERB
ejpam-6713	231	13	the	the	DET
ejpam-6713	231	14	edges	edge	NOUN
ejpam-6713	231	15	e1	e1	NOUN
ejpam-6713	231	16	=	=	PUNCT
ejpam-6713	231	17	v1vn−2	v1vn−2	PROPN
ejpam-6713	231	18	and	and	CCONJ
ejpam-6713	231	19	e2	e2	PROPN
ejpam-6713	231	20	=	=	PUNCT
ejpam-6713	232	1	v1v4	v1v4	X
ejpam-6713	232	2	by	by	ADP
ejpam-6713	232	3	a	a	DET
ejpam-6713	232	4	subdivision	subdivision	NOUN
ejpam-6713	232	5	vertices	vertice	VERB
ejpam-6713	232	6	x	x	PUNCT
ejpam-6713	232	7	and	and	CCONJ
ejpam-6713	232	8	y	y	PROPN
ejpam-6713	232	9	respectively	respectively	ADV
ejpam-6713	232	10	,	,	PUNCT
ejpam-6713	232	11	and	and	CCONJ
ejpam-6713	232	12	x	x	X
ejpam-6713	232	13	,	,	PUNCT
ejpam-6713	232	14	y	y	PROPN
ejpam-6713	232	15	∈	∈	PROPN
ejpam-6713	232	16	n(v1	n(v1	NOUN
ejpam-6713	232	17	)	)	PUNCT
ejpam-6713	232	18	.	.	PUNCT
ejpam-6713	233	1	now	now	ADV
ejpam-6713	233	2	g.	g.	PROPN
ejpam-6713	233	3	navamani	navamani	PROPN
ejpam-6713	233	4	et	et	PROPN
ejpam-6713	233	5	al	al	PROPN
ejpam-6713	233	6	.	.	PUNCT
ejpam-6713	233	7	/	/	SYM
ejpam-6713	233	8	eur	eur	PROPN
ejpam-6713	233	9	.	.	PUNCT
ejpam-6713	234	1	j.	j.	PROPN
ejpam-6713	234	2	pure	pure	PROPN
ejpam-6713	234	3	appl	appl	PROPN
ejpam-6713	234	4	.	.	PROPN
ejpam-6713	234	5	math	math	PROPN
ejpam-6713	234	6	,	,	PUNCT
ejpam-6713	234	7	18	18	NUM
ejpam-6713	234	8	(	(	PUNCT
ejpam-6713	234	9	4	4	NUM
ejpam-6713	234	10	)	)	PUNCT
ejpam-6713	234	11	(	(	PUNCT
ejpam-6713	234	12	2025	2025	NUM
ejpam-6713	234	13	)	)	PUNCT
ejpam-6713	234	14	,	,	PUNCT
ejpam-6713	234	15	6713	6713	NUM
ejpam-6713	234	16	9	9	NUM
ejpam-6713	234	17	of	of	ADP
ejpam-6713	234	18	15	15	NUM
ejpam-6713	234	19	d1	d1	NOUN
ejpam-6713	234	20	=	=	PUNCT
ejpam-6713	235	1	d	d	X
ejpam-6713	235	2	∪	∪	X
ejpam-6713	235	3	{	{	PUNCT
ejpam-6713	235	4	v4	v4	NOUN
ejpam-6713	235	5	}	}	PUNCT
ejpam-6713	235	6	is	be	AUX
ejpam-6713	235	7	a	a	DET
ejpam-6713	235	8	γcer	γcer	NOUN
ejpam-6713	235	9	set	set	NOUN
ejpam-6713	235	10	of	of	ADP
ejpam-6713	235	11	g′′.	g′′.	PROPN
ejpam-6713	235	12	hence	hence	ADV
ejpam-6713	235	13	γcer(g	γcer(g	NUM
ejpam-6713	235	14	′′	′′	PROPN
ejpam-6713	235	15	)	)	PUNCT
ejpam-6713	235	16	>	>	X
ejpam-6713	235	17	γcer(g	γcer(g	PROPN
ejpam-6713	235	18	′	′	NUM
ejpam-6713	235	19	)	)	PUNCT
ejpam-6713	235	20	.	.	PUNCT
ejpam-6713	236	1	that	that	PRON
ejpam-6713	236	2	is	be	AUX
ejpam-6713	236	3	|d1|	|d1|	PROPN
ejpam-6713	236	4	>	>	X
ejpam-6713	236	5	|d|	|d|	PROPN
ejpam-6713	236	6	.	.	PUNCT
ejpam-6713	237	1	therefore	therefore	ADV
ejpam-6713	237	2	,	,	PUNCT
ejpam-6713	237	3	sd+γcer(g	sd+γcer(g	NOUN
ejpam-6713	237	4	)	)	PUNCT
ejpam-6713	237	5	=	=	SYM
ejpam-6713	237	6	2	2	X
ejpam-6713	237	7	.	.	X
ejpam-6713	237	8	case	case	NOUN
ejpam-6713	237	9	(	(	PUNCT
ejpam-6713	237	10	v	v	NOUN
ejpam-6713	237	11	)	)	PUNCT
ejpam-6713	237	12	n	n	CCONJ
ejpam-6713	237	13	≡	≡	PROPN
ejpam-6713	237	14	4	4	NUM
ejpam-6713	237	15	(	(	PUNCT
ejpam-6713	237	16	mod	mod	NOUN
ejpam-6713	237	17	5	5	NUM
ejpam-6713	237	18	)	)	PUNCT
ejpam-6713	237	19	in	in	ADP
ejpam-6713	237	20	this	this	DET
ejpam-6713	237	21	case	case	NOUN
ejpam-6713	237	22	d	d	X
ejpam-6713	237	23	∪	∪	X
ejpam-6713	237	24	{	{	PUNCT
ejpam-6713	237	25	vn−1	vn−1	PROPN
ejpam-6713	237	26	}	}	PUNCT
ejpam-6713	237	27	is	be	AUX
ejpam-6713	237	28	the	the	DET
ejpam-6713	237	29	γcer	γcer	NOUN
ejpam-6713	237	30	-	-	PUNCT
ejpam-6713	237	31	set	set	NOUN
ejpam-6713	237	32	of	of	ADP
ejpam-6713	237	33	g	g	PROPN
ejpam-6713	237	34	.	.	PUNCT
ejpam-6713	238	1	here	here	ADV
ejpam-6713	238	2	|pn(v	|pn(v	NUM
ejpam-6713	238	3	,	,	PUNCT
ejpam-6713	238	4	d)|	d)|	NOUN
ejpam-6713	238	5	=	=	NOUN
ejpam-6713	238	6	4	4	NUM
ejpam-6713	238	7	for	for	ADP
ejpam-6713	238	8	all	all	DET
ejpam-6713	238	9	vertex	vertex	NOUN
ejpam-6713	238	10	v	v	ADP
ejpam-6713	238	11	∈	∈	PROPN
ejpam-6713	238	12	d	d	NOUN
ejpam-6713	238	13	except	except	SCONJ
ejpam-6713	238	14	the	the	DET
ejpam-6713	238	15	vertices	vertex	NOUN
ejpam-6713	238	16	v1	v1	NOUN
ejpam-6713	238	17	,	,	PUNCT
ejpam-6713	238	18	vn−1	vn−1	ADJ
ejpam-6713	238	19	and	and	CCONJ
ejpam-6713	238	20	vn−3	vn−3	PROPN
ejpam-6713	238	21	for	for	ADP
ejpam-6713	238	22	which	which	PRON
ejpam-6713	238	23	|pn(v1	|pn(v1	NOUN
ejpam-6713	238	24	,	,	PUNCT
ejpam-6713	238	25	d)|	d)|	NOUN
ejpam-6713	238	26	=	=	SYM
ejpam-6713	238	27	2	2	NUM
ejpam-6713	238	28	,	,	PUNCT
ejpam-6713	238	29	|pn(vn−1	|pn(vn−1	PROPN
ejpam-6713	238	30	,	,	PUNCT
ejpam-6713	238	31	d)|	d)|	NOUN
ejpam-6713	238	32	=	=	SYM
ejpam-6713	238	33	0	0	NUM
ejpam-6713	238	34	and	and	CCONJ
ejpam-6713	238	35	|pn(vn−3	|pn(vn−3	PROPN
ejpam-6713	238	36	,	,	PUNCT
ejpam-6713	238	37	d)|	d)|	NOUN
ejpam-6713	238	38	=	=	SYM
ejpam-6713	238	39	2	2	X
ejpam-6713	238	40	.	.	X
ejpam-6713	239	1	here	here	ADV
ejpam-6713	239	2	vn	vn	PROPN
ejpam-6713	239	3	,	,	PUNCT
ejpam-6713	239	4	vn−2	vn−2	PROPN
ejpam-6713	239	5	/∈	/∈	PUNCT
ejpam-6713	239	6	pn(v	pn(v	PROPN
ejpam-6713	239	7	,	,	PUNCT
ejpam-6713	239	8	d	d	NOUN
ejpam-6713	239	9	)	)	PUNCT
ejpam-6713	239	10	for	for	ADP
ejpam-6713	239	11	all	all	DET
ejpam-6713	239	12	v	v	NOUN
ejpam-6713	239	13	∈	∈	PROPN
ejpam-6713	239	14	d.	d.	NOUN
ejpam-6713	239	15	let	let	VERB
ejpam-6713	239	16	g′	g′	NOUN
ejpam-6713	239	17	be	be	AUX
ejpam-6713	239	18	a	a	DET
ejpam-6713	239	19	graph	graph	NOUN
ejpam-6713	239	20	obtained	obtain	VERB
ejpam-6713	239	21	from	from	ADP
ejpam-6713	239	22	g	g	NOUN
ejpam-6713	239	23	by	by	ADP
ejpam-6713	239	24	subdividing	subdivide	VERB
ejpam-6713	239	25	an	an	DET
ejpam-6713	239	26	edge	edge	NOUN
ejpam-6713	239	27	e	e	NOUN
ejpam-6713	239	28	=	=	SYM
ejpam-6713	239	29	v1vn	v1vn	PROPN
ejpam-6713	239	30	(	(	PUNCT
ejpam-6713	239	31	or	or	CCONJ
ejpam-6713	239	32	e	e	X
ejpam-6713	239	33	=	=	PROPN
ejpam-6713	239	34	v1vn−2	v1vn−2	PROPN
ejpam-6713	239	35	)	)	PUNCT
ejpam-6713	239	36	by	by	ADP
ejpam-6713	239	37	a	a	DET
ejpam-6713	239	38	subdivision	subdivision	NOUN
ejpam-6713	239	39	vertex	vertex	NOUN
ejpam-6713	239	40	x.	x.	NOUN
ejpam-6713	239	41	here	here	ADV
ejpam-6713	239	42	x	x	PUNCT
ejpam-6713	239	43	∈	∈	NOUN
ejpam-6713	239	44	n(v1	n(v1	NOUN
ejpam-6713	239	45	)	)	PUNCT
ejpam-6713	239	46	,	,	PUNCT
ejpam-6713	239	47	vn	vn	PROPN
ejpam-6713	239	48	∈	∈	PROPN
ejpam-6713	239	49	n(vn−1	n(vn−1	NOUN
ejpam-6713	239	50	)	)	PUNCT
ejpam-6713	239	51	,	,	PUNCT
ejpam-6713	240	1	[	[	X
ejpam-6713	240	2	vn−2	vn−2	PROPN
ejpam-6713	240	3	∈	∈	PROPN
ejpam-6713	240	4	n(vn−3	n(vn−3	NUM
ejpam-6713	240	5	)	)	PUNCT
ejpam-6713	240	6	]	]	PUNCT
ejpam-6713	240	7	.	.	PUNCT
ejpam-6713	241	1	hence	hence	ADV
ejpam-6713	241	2	γcer(g	γcer(g	NUM
ejpam-6713	241	3	′	′	NUM
ejpam-6713	241	4	)	)	PUNCT
ejpam-6713	241	5	=	=	SYM
ejpam-6713	241	6	γcer(g	γcer(g	NOUN
ejpam-6713	241	7	)	)	PUNCT
ejpam-6713	241	8	.	.	PUNCT
ejpam-6713	242	1	therefore	therefore	ADV
ejpam-6713	242	2	sd+γcer(g	sd+γcer(g	NOUN
ejpam-6713	242	3	)	)	PUNCT
ejpam-6713	242	4	>	>	X
ejpam-6713	243	1	1	1	X
ejpam-6713	243	2	.	.	PUNCT
ejpam-6713	243	3	let	let	VERB
ejpam-6713	243	4	g′′	g′′	PROPN
ejpam-6713	243	5	be	be	AUX
ejpam-6713	243	6	a	a	DET
ejpam-6713	243	7	graph	graph	NOUN
ejpam-6713	243	8	obtained	obtain	VERB
ejpam-6713	243	9	from	from	ADP
ejpam-6713	243	10	g	g	NOUN
ejpam-6713	243	11	by	by	ADP
ejpam-6713	243	12	subdividing	subdivide	VERB
ejpam-6713	243	13	the	the	DET
ejpam-6713	243	14	edges	edge	NOUN
ejpam-6713	243	15	e1	e1	NOUN
ejpam-6713	243	16	=	=	SYM
ejpam-6713	243	17	v1vn	v1vn	PROPN
ejpam-6713	243	18	and	and	CCONJ
ejpam-6713	243	19	e2	e2	PROPN
ejpam-6713	243	20	=	=	PUNCT
ejpam-6713	244	1	v5v6	v5v6	X
ejpam-6713	244	2	by	by	ADP
ejpam-6713	244	3	a	a	DET
ejpam-6713	244	4	subdivision	subdivision	NOUN
ejpam-6713	244	5	vertices	vertice	VERB
ejpam-6713	244	6	x	x	PUNCT
ejpam-6713	244	7	and	and	CCONJ
ejpam-6713	244	8	y	y	PROPN
ejpam-6713	244	9	respectively	respectively	ADV
ejpam-6713	244	10	,	,	PUNCT
ejpam-6713	244	11	x	x	SYM
ejpam-6713	244	12	∈	∈	NOUN
ejpam-6713	244	13	n(v1	n(v1	NOUN
ejpam-6713	244	14	)	)	PUNCT
ejpam-6713	244	15	,	,	PUNCT
ejpam-6713	244	16	and	and	CCONJ
ejpam-6713	244	17	v4	v4	PROPN
ejpam-6713	244	18	/∈	/∈	PUNCT
ejpam-6713	245	1	n(v	n(v	PROPN
ejpam-6713	245	2	)	)	PUNCT
ejpam-6713	245	3	for	for	ADP
ejpam-6713	245	4	all	all	PRON
ejpam-6713	245	5	v	v	NOUN
ejpam-6713	245	6	∈	∈	NOUN
ejpam-6713	245	7	d.	d.	NOUN
ejpam-6713	245	8	hence	hence	ADV
ejpam-6713	245	9	we	we	PRON
ejpam-6713	245	10	have	have	VERB
ejpam-6713	245	11	d1	d1	NOUN
ejpam-6713	245	12	=	=	PUNCT
ejpam-6713	246	1	d	d	X
ejpam-6713	246	2	∪	∪	X
ejpam-6713	246	3	{	{	PUNCT
ejpam-6713	246	4	v4	v4	NOUN
ejpam-6713	246	5	}	}	PUNCT
ejpam-6713	246	6	is	be	AUX
ejpam-6713	246	7	a	a	DET
ejpam-6713	246	8	γcer	γcer	NOUN
ejpam-6713	246	9	set	set	NOUN
ejpam-6713	246	10	of	of	ADP
ejpam-6713	246	11	g′′.	g′′.	PROPN
ejpam-6713	246	12	so	so	ADV
ejpam-6713	246	13	γcer(g	γcer(g	PROPN
ejpam-6713	246	14	′′	′′	PROPN
ejpam-6713	246	15	)	)	PUNCT
ejpam-6713	246	16	>	>	X
ejpam-6713	246	17	γcer(g	γcer(g	PROPN
ejpam-6713	246	18	′	′	NUM
ejpam-6713	246	19	)	)	PUNCT
ejpam-6713	246	20	.	.	PUNCT
ejpam-6713	247	1	therefore	therefore	ADV
ejpam-6713	247	2	,	,	PUNCT
ejpam-6713	247	3	sd+γcer(g	sd+γcer(g	NOUN
ejpam-6713	247	4	)	)	PUNCT
ejpam-6713	247	5	=	=	SYM
ejpam-6713	248	1	2	2	NUM
ejpam-6713	248	2	.	.	NOUN
ejpam-6713	248	3	5	5	NUM
ejpam-6713	248	4	.	.	NUM
ejpam-6713	248	5	generalized	generalize	VERB
ejpam-6713	248	6	petersen	petersen	NOUN
ejpam-6713	248	7	graphs	graph	VERB
ejpam-6713	248	8	the	the	DET
ejpam-6713	248	9	generalized	generalized	ADJ
ejpam-6713	248	10	petersen	petersen	NOUN
ejpam-6713	248	11	graph	graph	NOUN
ejpam-6713	248	12	p	p	PROPN
ejpam-6713	248	13	(	(	PUNCT
ejpam-6713	248	14	n	n	X
ejpam-6713	248	15	,	,	PUNCT
ejpam-6713	248	16	k	k	NOUN
ejpam-6713	248	17	)	)	PUNCT
ejpam-6713	248	18	is	be	AUX
ejpam-6713	248	19	defined	define	VERB
ejpam-6713	248	20	to	to	PART
ejpam-6713	248	21	be	be	AUX
ejpam-6713	248	22	a	a	DET
ejpam-6713	248	23	graph	graph	NOUN
ejpam-6713	248	24	on	on	ADP
ejpam-6713	248	25	2n	2n	NUM
ejpam-6713	248	26	vertices	vertex	NOUN
ejpam-6713	248	27	with	with	ADP
ejpam-6713	248	28	v	v	NOUN
ejpam-6713	248	29	(	(	PUNCT
ejpam-6713	248	30	p	p	X
ejpam-6713	248	31	(	(	PUNCT
ejpam-6713	248	32	n	n	X
ejpam-6713	248	33	,	,	PUNCT
ejpam-6713	248	34	k	k	NOUN
ejpam-6713	248	35	)	)	PUNCT
ejpam-6713	248	36	)	)	PUNCT
ejpam-6713	249	1	=	=	PRON
ejpam-6713	249	2	{	{	PUNCT
ejpam-6713	249	3	vi	vi	PROPN
ejpam-6713	249	4	,	,	PUNCT
ejpam-6713	249	5	ui	ui	NOUN
ejpam-6713	249	6	:	:	PUNCT
ejpam-6713	249	7	0	0	NUM
ejpam-6713	249	8	≤	≤	NUM
ejpam-6713	249	9	i	i	PRON
ejpam-6713	249	10	≤	≤	ADJ
ejpam-6713	249	11	n	n	CCONJ
ejpam-6713	249	12	−	−	PROPN
ejpam-6713	249	13	1	1	NUM
ejpam-6713	249	14	}	}	PUNCT
ejpam-6713	249	15	and	and	CCONJ
ejpam-6713	249	16	e(p	e(p	PROPN
ejpam-6713	249	17	(	(	PUNCT
ejpam-6713	249	18	n	n	CCONJ
ejpam-6713	249	19	,	,	PUNCT
ejpam-6713	249	20	k	k	NOUN
ejpam-6713	249	21	)	)	PUNCT
ejpam-6713	249	22	)	)	PUNCT
ejpam-6713	250	1	=	=	PRON
ejpam-6713	250	2	{	{	PUNCT
ejpam-6713	250	3	vi	vi	PROPN
ejpam-6713	250	4	,	,	PUNCT
ejpam-6713	250	5	vi+1	vi+1	NOUN
ejpam-6713	250	6	,	,	PUNCT
ejpam-6713	250	7	,	,	PUNCT
ejpam-6713	250	8	vi	vi	PROPN
ejpam-6713	250	9	,	,	PUNCT
ejpam-6713	250	10	ui	ui	NOUN
ejpam-6713	250	11	,	,	PUNCT
ejpam-6713	250	12	uiui+k	uiui+k	PROPN
ejpam-6713	250	13	:	:	SYM
ejpam-6713	250	14	0	0	NUM
ejpam-6713	250	15	≤	≤	NUM
ejpam-6713	251	1	i	i	PRON
ejpam-6713	251	2	≤	≤	ADJ
ejpam-6713	251	3	n	n	CCONJ
ejpam-6713	251	4	−	−	PROPN
ejpam-6713	251	5	1	1	NUM
ejpam-6713	251	6	}	}	PUNCT
ejpam-6713	251	7	subscipts	subscipt	NOUN
ejpam-6713	251	8	taken	take	VERB
ejpam-6713	251	9	modulo	modulo	NOUN
ejpam-6713	251	10	n.	n.	NOUN
ejpam-6713	251	11	the	the	DET
ejpam-6713	251	12	edges	edge	NOUN
ejpam-6713	251	13	uivi	uivi	NOUN
ejpam-6713	251	14	for	for	ADP
ejpam-6713	251	15	0	0	NUM
ejpam-6713	251	16	≤	≤	NUM
ejpam-6713	251	17	i	i	PRON
ejpam-6713	252	1	≤	≤	NOUN
ejpam-6713	253	1	n	n	CCONJ
ejpam-6713	253	2	−	−	PROPN
ejpam-6713	253	3	1	1	NUM
ejpam-6713	253	4	are	be	AUX
ejpam-6713	253	5	called	call	VERB
ejpam-6713	253	6	the	the	DET
ejpam-6713	253	7	spokes	spoke	NOUN
ejpam-6713	253	8	.	.	PUNCT
ejpam-6713	254	1	in	in	ADP
ejpam-6713	254	2	this	this	DET
ejpam-6713	254	3	section	section	NOUN
ejpam-6713	254	4	we	we	PRON
ejpam-6713	254	5	find	find	VERB
ejpam-6713	254	6	the	the	DET
ejpam-6713	254	7	value	value	NOUN
ejpam-6713	254	8	of	of	ADP
ejpam-6713	254	9	sd+γcer(g	sd+γcer(g	NOUN
ejpam-6713	254	10	)	)	PUNCT
ejpam-6713	254	11	for	for	ADP
ejpam-6713	254	12	the	the	DET
ejpam-6713	254	13	generalised	generalise	VERB
ejpam-6713	254	14	petersen	petersen	NOUN
ejpam-6713	254	15	graphs	graph	NOUN
ejpam-6713	254	16	p	p	X
ejpam-6713	254	17	(	(	PUNCT
ejpam-6713	254	18	n	n	CCONJ
ejpam-6713	254	19	,	,	PUNCT
ejpam-6713	254	20	1	1	NUM
ejpam-6713	254	21	)	)	PUNCT
ejpam-6713	254	22	and	and	CCONJ
ejpam-6713	254	23	p	p	X
ejpam-6713	254	24	(	(	PUNCT
ejpam-6713	254	25	n	n	CCONJ
ejpam-6713	254	26	,	,	PUNCT
ejpam-6713	254	27	2	2	X
ejpam-6713	254	28	)	)	PUNCT
ejpam-6713	254	29	theorem	theorem	NOUN
ejpam-6713	254	30	10	10	NUM
ejpam-6713	254	31	.	.	PUNCT
ejpam-6713	255	1	[	[	X
ejpam-6713	255	2	20	20	NUM
ejpam-6713	255	3	]	]	PUNCT
ejpam-6713	255	4	for	for	ADP
ejpam-6713	255	5	n	n	PRON
ejpam-6713	255	6	≥	≥	NUM
ejpam-6713	255	7	3	3	NUM
ejpam-6713	255	8	,	,	PUNCT
ejpam-6713	255	9	γ(p	γ(p	PROPN
ejpam-6713	255	10	(	(	PUNCT
ejpam-6713	255	11	n	n	CCONJ
ejpam-6713	255	12	,	,	PUNCT
ejpam-6713	255	13	1	1	NUM
ejpam-6713	255	14	)	)	PUNCT
ejpam-6713	255	15	)	)	PUNCT
ejpam-6713	256	1	=	=	PRON
ejpam-6713	256	2	{	{	PUNCT
ejpam-6713	256	3	⌈	⌈	NOUN
ejpam-6713	256	4	n	n	CCONJ
ejpam-6713	256	5	2	2	NUM
ejpam-6713	256	6	⌉	⌉	NOUN
ejpam-6713	256	7	,	,	PUNCT
ejpam-6713	256	8	n	n	X
ejpam-6713	256	9	≡	≡	PROPN
ejpam-6713	256	10	0	0	NUM
ejpam-6713	256	11	,	,	PUNCT
ejpam-6713	256	12	1	1	NUM
ejpam-6713	256	13	,	,	PUNCT
ejpam-6713	256	14	3	3	NUM
ejpam-6713	256	15	(	(	PUNCT
ejpam-6713	256	16	mod	mod	PROPN
ejpam-6713	256	17	4)⌈	4)⌈	PROPN
ejpam-6713	257	1	n	n	PRON
ejpam-6713	257	2	2	2	NUM
ejpam-6713	257	3	⌉	⌉	NOUN
ejpam-6713	257	4	+	+	ADJ
ejpam-6713	257	5	1	1	NUM
ejpam-6713	257	6	,	,	PUNCT
ejpam-6713	257	7	n	n	PRON
ejpam-6713	257	8	≡	≡	PROPN
ejpam-6713	257	9	2	2	NUM
ejpam-6713	257	10	(	(	PUNCT
ejpam-6713	257	11	mod	mod	NOUN
ejpam-6713	257	12	4	4	NUM
ejpam-6713	257	13	)	)	PUNCT
ejpam-6713	257	14	theorem	theorem	NOUN
ejpam-6713	257	15	11	11	NUM
ejpam-6713	257	16	.	.	PUNCT
ejpam-6713	258	1	for	for	ADP
ejpam-6713	258	2	any	any	DET
ejpam-6713	258	3	petersen	petersen	NOUN
ejpam-6713	258	4	graph	graph	NOUN
ejpam-6713	258	5	g	g	ADP
ejpam-6713	258	6	∼=	∼=	PROPN
ejpam-6713	258	7	p	p	NOUN
ejpam-6713	258	8	(	(	PUNCT
ejpam-6713	258	9	n	n	CCONJ
ejpam-6713	258	10	,	,	PUNCT
ejpam-6713	258	11	1	1	NUM
ejpam-6713	258	12	)	)	PUNCT
ejpam-6713	258	13	,	,	PUNCT
ejpam-6713	258	14	n	n	X
ejpam-6713	258	15	≥	≥	NOUN
ejpam-6713	258	16	4	4	NUM
ejpam-6713	258	17	,	,	PUNCT
ejpam-6713	258	18	sd+γcer(g	sd+γcer(g	NOUN
ejpam-6713	258	19	)	)	PUNCT
ejpam-6713	259	1	=	=	SYM
ejpam-6713	259	2			NOUN
ejpam-6713	259	3	1	1	NUM
ejpam-6713	259	4	if	if	SCONJ
ejpam-6713	259	5	n	n	PRON
ejpam-6713	259	6	≡	≡	PROPN
ejpam-6713	259	7	0	0	NUM
ejpam-6713	259	8	,	,	PUNCT
ejpam-6713	259	9	1	1	NUM
ejpam-6713	259	10	(	(	PUNCT
ejpam-6713	259	11	mod	mod	NOUN
ejpam-6713	259	12	4	4	NUM
ejpam-6713	259	13	)	)	PUNCT
ejpam-6713	259	14	2	2	NUM
ejpam-6713	259	15	if	if	SCONJ
ejpam-6713	259	16	n	n	PRON
ejpam-6713	259	17	≡	≡	PROPN
ejpam-6713	259	18	3	3	NUM
ejpam-6713	259	19	(	(	PUNCT
ejpam-6713	259	20	mod	mod	NOUN
ejpam-6713	259	21	4	4	NUM
ejpam-6713	259	22	)	)	PUNCT
ejpam-6713	259	23	3	3	NUM
ejpam-6713	259	24	if	if	SCONJ
ejpam-6713	259	25	n	n	PRON
ejpam-6713	259	26	≡	≡	PROPN
ejpam-6713	259	27	2	2	NUM
ejpam-6713	259	28	(	(	PUNCT
ejpam-6713	259	29	mod	mod	NOUN
ejpam-6713	259	30	4	4	NUM
ejpam-6713	259	31	)	)	PUNCT
ejpam-6713	259	32	proof	proof	NOUN
ejpam-6713	259	33	.	.	PUNCT
ejpam-6713	260	1	let	let	VERB
ejpam-6713	260	2	g	g	PRON
ejpam-6713	260	3	∼=	∼=	PROPN
ejpam-6713	260	4	p	p	NOUN
ejpam-6713	260	5	(	(	PUNCT
ejpam-6713	260	6	n	n	CCONJ
ejpam-6713	260	7	,	,	PUNCT
ejpam-6713	260	8	1	1	NUM
ejpam-6713	260	9	)	)	PUNCT
ejpam-6713	260	10	.	.	PUNCT
ejpam-6713	261	1	let	let	VERB
ejpam-6713	261	2	c	c	NOUN
ejpam-6713	261	3	′	′	VERB
ejpam-6713	261	4	and	and	CCONJ
ejpam-6713	261	5	c	c	X
ejpam-6713	261	6	′′	′′	PROPN
ejpam-6713	261	7	be	be	VERB
ejpam-6713	261	8	the	the	DET
ejpam-6713	261	9	inner	inner	ADJ
ejpam-6713	261	10	and	and	CCONJ
ejpam-6713	261	11	outer	outer	ADJ
ejpam-6713	261	12	cycles	cycle	NOUN
ejpam-6713	261	13	of	of	ADP
ejpam-6713	261	14	g	g	NOUN
ejpam-6713	261	15	respectively	respectively	ADV
ejpam-6713	261	16	.	.	PUNCT
ejpam-6713	262	1	let	let	VERB
ejpam-6713	262	2	v	v	X
ejpam-6713	262	3	(	(	PUNCT
ejpam-6713	262	4	c	c	NOUN
ejpam-6713	262	5	′	′	NUM
ejpam-6713	262	6	)	)	PUNCT
ejpam-6713	262	7	=	=	PRON
ejpam-6713	262	8	{	{	PUNCT
ejpam-6713	262	9	v1	v1	PROPN
ejpam-6713	262	10	,	,	PUNCT
ejpam-6713	262	11	v2	v2	PROPN
ejpam-6713	262	12	,	,	PUNCT
ejpam-6713	262	13	v3	v3	PROPN
ejpam-6713	262	14	,	,	PUNCT
ejpam-6713	262	15	.	.	PUNCT
ejpam-6713	262	16	.	.	PUNCT
ejpam-6713	262	17	.	.	PUNCT
ejpam-6713	263	1	vn	vn	X
ejpam-6713	263	2	}	}	PUNCT
ejpam-6713	263	3	and	and	CCONJ
ejpam-6713	263	4	v	v	X
ejpam-6713	263	5	(	(	PUNCT
ejpam-6713	263	6	c	c	PROPN
ejpam-6713	263	7	′′	′′	PROPN
ejpam-6713	263	8	)	)	PUNCT
ejpam-6713	263	9	=	=	PRON
ejpam-6713	263	10	{	{	PUNCT
ejpam-6713	263	11	u1	u1	NOUN
ejpam-6713	263	12	,	,	PUNCT
ejpam-6713	263	13	u2	u2	NOUN
ejpam-6713	263	14	,	,	PUNCT
ejpam-6713	263	15	u3	u3	NOUN
ejpam-6713	263	16	,	,	PUNCT
ejpam-6713	263	17	.	.	PUNCT
ejpam-6713	263	18	.	.	PUNCT
ejpam-6713	263	19	.	.	PUNCT
ejpam-6713	264	1	un	un	PROPN
ejpam-6713	264	2	}	}	PUNCT
ejpam-6713	264	3	,	,	PUNCT
ejpam-6713	264	4	by	by	ADP
ejpam-6713	264	5	theorem	theorem	NOUN
ejpam-6713	264	6	3	3	NUM
ejpam-6713	264	7	and	and	CCONJ
ejpam-6713	264	8	theorem	theorem	VERB
ejpam-6713	264	9	10	10	NUM
ejpam-6713	264	10	,	,	PUNCT
ejpam-6713	264	11	hence	hence	ADV
ejpam-6713	264	12	,	,	PUNCT
ejpam-6713	264	13	γ(p	γ(p	PROPN
ejpam-6713	264	14	(	(	PUNCT
ejpam-6713	264	15	n	n	CCONJ
ejpam-6713	264	16	,	,	PUNCT
ejpam-6713	264	17	1	1	NUM
ejpam-6713	264	18	)	)	PUNCT
ejpam-6713	264	19	)	)	PUNCT
ejpam-6713	265	1	=	=	PUNCT
ejpam-6713	265	2	γcer(p	γcer(p	NOUN
ejpam-6713	265	3	(	(	PUNCT
ejpam-6713	265	4	n	n	CCONJ
ejpam-6713	265	5	,	,	PUNCT
ejpam-6713	265	6	1	1	NUM
ejpam-6713	265	7	)	)	PUNCT
ejpam-6713	265	8	)	)	PUNCT
ejpam-6713	265	9	.	.	PUNCT
ejpam-6713	266	1	let	let	VERB
ejpam-6713	266	2	d	d	PRON
ejpam-6713	266	3	be	be	AUX
ejpam-6713	266	4	a	a	DET
ejpam-6713	266	5	γcer	γcer	NOUN
ejpam-6713	266	6	-	-	PUNCT
ejpam-6713	266	7	set	set	NOUN
ejpam-6713	266	8	of	of	ADP
ejpam-6713	266	9	g	g	PROPN
ejpam-6713	266	10	and	and	CCONJ
ejpam-6713	266	11	d	d	NOUN
ejpam-6713	266	12	=	=	X
ejpam-6713	266	13	d′	d′	X
ejpam-6713	266	14	∪d′′	∪d′′	VERB
ejpam-6713	266	15	where	where	SCONJ
ejpam-6713	266	16	d′	d′	PRON
ejpam-6713	266	17	=	=	SYM
ejpam-6713	267	1	d	d	PROPN
ejpam-6713	267	2	∩	∩	ADJ
ejpam-6713	267	3	v	v	X
ejpam-6713	267	4	(	(	PUNCT
ejpam-6713	267	5	c	c	NOUN
ejpam-6713	267	6	′	′	NUM
ejpam-6713	267	7	)	)	PUNCT
ejpam-6713	267	8	and	and	CCONJ
ejpam-6713	267	9	d′′	d′′	NOUN
ejpam-6713	267	10	=	=	SYM
ejpam-6713	267	11	d	d	PROPN
ejpam-6713	267	12	∩	∩	X
ejpam-6713	267	13	v	v	X
ejpam-6713	267	14	(	(	PUNCT
ejpam-6713	267	15	c	c	PROPN
ejpam-6713	267	16	′′	′′	PROPN
ejpam-6713	267	17	)	)	PUNCT
ejpam-6713	267	18	.	.	PUNCT
ejpam-6713	268	1	let	let	VERB
ejpam-6713	268	2	d′	d′	PRON
ejpam-6713	268	3	=	=	PUNCT
ejpam-6713	268	4	{	{	PUNCT
ejpam-6713	268	5	v4k+1	v4k+1	X
ejpam-6713	268	6	:	:	PUNCT
ejpam-6713	268	7	0	0	NUM
ejpam-6713	268	8	≤	≤	NUM
ejpam-6713	268	9	k	k	X
ejpam-6713	268	10	≤	≤	NUM
ejpam-6713	268	11	⌊	⌊	VERB
ejpam-6713	268	12	n	n	ADV
ejpam-6713	268	13	4	4	NUM
ejpam-6713	268	14	⌋	⌋	NOUN
ejpam-6713	268	15	}	}	PUNCT
ejpam-6713	268	16	for	for	ADP
ejpam-6713	268	17	all	all	DET
ejpam-6713	268	18	n	n	NOUN
ejpam-6713	268	19	and	and	CCONJ
ejpam-6713	268	20	d′′	d′′	NOUN
ejpam-6713	268	21	=	=	PUNCT
ejpam-6713	268	22	{	{	PUNCT
ejpam-6713	268	23	u4k+3	u4k+3	ADV
ejpam-6713	268	24	,	,	PUNCT
ejpam-6713	268	25	0	0	NUM
ejpam-6713	269	1	≤	≤	PUNCT
ejpam-6713	270	1	k	k	X
ejpam-6713	270	2	<	<	X
ejpam-6713	270	3	⌈	⌈	X
ejpam-6713	270	4	n	n	PRON
ejpam-6713	270	5	4	4	NUM
ejpam-6713	270	6	⌉	⌉	NOUN
ejpam-6713	270	7	∪	∪	X
ejpam-6713	270	8	{	{	PUNCT
ejpam-6713	270	9	un	un	PROPN
ejpam-6713	270	10	}	}	PUNCT
ejpam-6713	270	11	for	for	ADP
ejpam-6713	270	12	n	n	X
ejpam-6713	270	13	≡	≡	PROPN
ejpam-6713	270	14	2	2	NUM
ejpam-6713	270	15	(	(	PUNCT
ejpam-6713	270	16	mod	mod	NOUN
ejpam-6713	270	17	4	4	NUM
ejpam-6713	270	18	)	)	PUNCT
ejpam-6713	270	19	u4k+3	u4k+3	ADP
ejpam-6713	270	20	,	,	PUNCT
ejpam-6713	270	21	0	0	NUM
ejpam-6713	270	22	≤	≤	PUNCT
ejpam-6713	270	23	k	k	X
ejpam-6713	270	24	<	<	X
ejpam-6713	270	25	⌈	⌈	X
ejpam-6713	270	26	n	n	CCONJ
ejpam-6713	270	27	4	4	NUM
ejpam-6713	270	28	⌉	⌉	VERB
ejpam-6713	270	29	otherwise	otherwise	ADV
ejpam-6713	270	30	now	now	ADV
ejpam-6713	270	31	consider	consider	VERB
ejpam-6713	270	32	the	the	DET
ejpam-6713	270	33	following	follow	VERB
ejpam-6713	270	34	cases	case	NOUN
ejpam-6713	270	35	g.	g.	PROPN
ejpam-6713	270	36	navamani	navamani	PROPN
ejpam-6713	270	37	et	et	PROPN
ejpam-6713	270	38	al	al	PROPN
ejpam-6713	270	39	.	.	PUNCT
ejpam-6713	270	40	/	/	SYM
ejpam-6713	270	41	eur	eur	PROPN
ejpam-6713	270	42	.	.	PUNCT
ejpam-6713	271	1	j.	j.	PROPN
ejpam-6713	271	2	pure	pure	PROPN
ejpam-6713	271	3	appl	appl	PROPN
ejpam-6713	271	4	.	.	PROPN
ejpam-6713	271	5	math	math	PROPN
ejpam-6713	271	6	,	,	PUNCT
ejpam-6713	271	7	18	18	NUM
ejpam-6713	271	8	(	(	PUNCT
ejpam-6713	271	9	4	4	NUM
ejpam-6713	271	10	)	)	PUNCT
ejpam-6713	271	11	(	(	PUNCT
ejpam-6713	271	12	2025	2025	NUM
ejpam-6713	271	13	)	)	PUNCT
ejpam-6713	271	14	,	,	PUNCT
ejpam-6713	271	15	6713	6713	NUM
ejpam-6713	271	16	10	10	NUM
ejpam-6713	271	17	of	of	ADP
ejpam-6713	271	18	15	15	NUM
ejpam-6713	271	19	case	case	NOUN
ejpam-6713	271	20	(	(	PUNCT
ejpam-6713	271	21	i	i	NOUN
ejpam-6713	271	22	)	)	PUNCT
ejpam-6713	271	23	n	n	X
ejpam-6713	271	24	≡	≡	PROPN
ejpam-6713	271	25	0	0	PUNCT
ejpam-6713	272	1	(	(	PUNCT
ejpam-6713	272	2	mod	mod	NOUN
ejpam-6713	272	3	4	4	NUM
ejpam-6713	272	4	)	)	PUNCT
ejpam-6713	272	5	in	in	ADP
ejpam-6713	272	6	this	this	DET
ejpam-6713	272	7	case	case	NOUN
ejpam-6713	272	8	each	each	DET
ejpam-6713	272	9	vertex	vertex	NOUN
ejpam-6713	272	10	in	in	ADP
ejpam-6713	272	11	d	d	NOUN
ejpam-6713	272	12	dominates	dominate	VERB
ejpam-6713	272	13	exactly	exactly	ADV
ejpam-6713	272	14	4	4	NUM
ejpam-6713	272	15	vertices	vertex	NOUN
ejpam-6713	272	16	including	include	VERB
ejpam-6713	272	17	itself	itself	PRON
ejpam-6713	272	18	.	.	PUNCT
ejpam-6713	273	1	let	let	VERB
ejpam-6713	273	2	g′	g′	NOUN
ejpam-6713	273	3	be	be	AUX
ejpam-6713	273	4	a	a	DET
ejpam-6713	273	5	graph	graph	NOUN
ejpam-6713	273	6	obtained	obtain	VERB
ejpam-6713	273	7	from	from	ADP
ejpam-6713	273	8	g	g	NOUN
ejpam-6713	273	9	by	by	ADP
ejpam-6713	273	10	subdividing	subdivide	VERB
ejpam-6713	273	11	an	an	DET
ejpam-6713	273	12	edge	edge	NOUN
ejpam-6713	273	13	e	e	NOUN
ejpam-6713	273	14	=	=	PUNCT
ejpam-6713	273	15	u1un	u1un	PUNCT
ejpam-6713	273	16	by	by	ADP
ejpam-6713	273	17	a	a	DET
ejpam-6713	273	18	subdivision	subdivision	NOUN
ejpam-6713	273	19	vertex	vertex	NOUN
ejpam-6713	273	20	x	x	NOUN
ejpam-6713	273	21	,	,	PUNCT
ejpam-6713	273	22	we	we	PRON
ejpam-6713	273	23	clearly	clearly	ADV
ejpam-6713	273	24	see	see	VERB
ejpam-6713	273	25	that	that	PRON
ejpam-6713	273	26	x	x	PUNCT
ejpam-6713	273	27	/∈	/∈	PUNCT
ejpam-6713	273	28	n(v	n(v	PROPN
ejpam-6713	273	29	)	)	PUNCT
ejpam-6713	273	30	for	for	ADP
ejpam-6713	273	31	all	all	DET
ejpam-6713	273	32	v	v	NOUN
ejpam-6713	273	33	∈	∈	NOUN
ejpam-6713	273	34	d.	d.	NOUN
ejpam-6713	273	35	hence	hence	ADV
ejpam-6713	273	36	d1	d1	PROPN
ejpam-6713	273	37	=	=	PUNCT
ejpam-6713	274	1	d	d	X
ejpam-6713	274	2	∪	∪	X
ejpam-6713	274	3	{	{	PUNCT
ejpam-6713	274	4	x	x	NOUN
ejpam-6713	274	5	}	}	PUNCT
ejpam-6713	274	6	is	be	AUX
ejpam-6713	274	7	a	a	DET
ejpam-6713	274	8	γcerset	γcerset	NOUN
ejpam-6713	274	9	of	of	ADP
ejpam-6713	274	10	g′.	g′.	ADP
ejpam-6713	274	11	this	this	PRON
ejpam-6713	274	12	implies	imply	VERB
ejpam-6713	274	13	that	that	SCONJ
ejpam-6713	274	14	γcer(g	γcer(g	PROPN
ejpam-6713	274	15	′	′	NUM
ejpam-6713	274	16	)	)	PUNCT
ejpam-6713	274	17	>	>	X
ejpam-6713	275	1	γcer(g	γcer(g	NOUN
ejpam-6713	275	2	)	)	PUNCT
ejpam-6713	275	3	.	.	PUNCT
ejpam-6713	276	1	hence	hence	ADV
ejpam-6713	276	2	sd+γcer(g	sd+γcer(g	NOUN
ejpam-6713	276	3	)	)	PUNCT
ejpam-6713	276	4	=	=	SYM
ejpam-6713	277	1	1	1	X
ejpam-6713	277	2	.	.	X
ejpam-6713	277	3	case	case	NOUN
ejpam-6713	277	4	(	(	PUNCT
ejpam-6713	277	5	ii	ii	NOUN
ejpam-6713	277	6	)	)	PUNCT
ejpam-6713	277	7	n	n	CCONJ
ejpam-6713	277	8	≡	≡	PROPN
ejpam-6713	277	9	1	1	NUM
ejpam-6713	277	10	(	(	PUNCT
ejpam-6713	277	11	mod	mod	NOUN
ejpam-6713	277	12	4	4	X
ejpam-6713	277	13	)	)	PUNCT
ejpam-6713	277	14	let	let	VERB
ejpam-6713	277	15	g′	g′	NOUN
ejpam-6713	277	16	be	be	AUX
ejpam-6713	277	17	a	a	DET
ejpam-6713	277	18	graph	graph	NOUN
ejpam-6713	277	19	obtained	obtain	VERB
ejpam-6713	277	20	from	from	ADP
ejpam-6713	277	21	g	g	NOUN
ejpam-6713	277	22	by	by	ADP
ejpam-6713	277	23	subdividing	subdivide	VERB
ejpam-6713	277	24	an	an	DET
ejpam-6713	277	25	edge	edge	NOUN
ejpam-6713	277	26	e	e	NOUN
ejpam-6713	277	27	=	=	PUNCT
ejpam-6713	277	28	u1v1	u1v1	ADJ
ejpam-6713	277	29	by	by	ADP
ejpam-6713	277	30	a	a	DET
ejpam-6713	277	31	subdivision	subdivision	NOUN
ejpam-6713	277	32	vertex	vertex	NOUN
ejpam-6713	277	33	x	x	NOUN
ejpam-6713	277	34	,	,	PUNCT
ejpam-6713	277	35	we	we	PRON
ejpam-6713	277	36	clearly	clearly	ADV
ejpam-6713	277	37	see	see	VERB
ejpam-6713	277	38	that	that	SCONJ
ejpam-6713	277	39	x	x	PUNCT
ejpam-6713	277	40	∈	∈	NOUN
ejpam-6713	277	41	n(v1	n(v1	NOUN
ejpam-6713	277	42	)	)	PUNCT
ejpam-6713	277	43	.	.	PUNCT
ejpam-6713	278	1	in	in	ADP
ejpam-6713	278	2	order	order	NOUN
ejpam-6713	278	3	to	to	PART
ejpam-6713	278	4	dominate	dominate	VERB
ejpam-6713	278	5	u1	u1	NOUN
ejpam-6713	278	6	,	,	PUNCT
ejpam-6713	278	7	the	the	DET
ejpam-6713	278	8	position	position	NOUN
ejpam-6713	278	9	of	of	ADP
ejpam-6713	278	10	the	the	DET
ejpam-6713	278	11	d	d	NOUN
ejpam-6713	278	12	will	will	AUX
ejpam-6713	278	13	be	be	AUX
ejpam-6713	278	14	changed	change	VERB
ejpam-6713	278	15	to	to	ADP
ejpam-6713	278	16	d′	d′	PROPN
ejpam-6713	278	17	where	where	SCONJ
ejpam-6713	278	18	the	the	DET
ejpam-6713	278	19	dissimilar	dissimilar	ADJ
ejpam-6713	278	20	sets	set	NOUN
ejpam-6713	278	21	are	be	AUX
ejpam-6713	278	22	d′	d′	X
ejpam-6713	278	23	=	=	SYM
ejpam-6713	278	24	{	{	PUNCT
ejpam-6713	278	25	x	x	NOUN
ejpam-6713	278	26	}	}	PUNCT
ejpam-6713	278	27	∪	∪	ADJ
ejpam-6713	278	28	{	{	PUNCT
ejpam-6713	278	29	v4k−2	v4k−2	PROPN
ejpam-6713	278	30	,	,	PUNCT
ejpam-6713	278	31	1	1	NUM
ejpam-6713	278	32	≤	≤	NUM
ejpam-6713	279	1	k	k	X
ejpam-6713	279	2	<	<	X
ejpam-6713	279	3	n+2	n+2	PROPN
ejpam-6713	279	4	4	4	NUM
ejpam-6713	279	5	}	}	PUNCT
ejpam-6713	279	6	∪	∪	ADJ
ejpam-6713	279	7	{	{	PUNCT
ejpam-6713	279	8	u4k	u4k	NOUN
ejpam-6713	279	9	,	,	PUNCT
ejpam-6713	279	10	1	1	NUM
ejpam-6713	279	11	≤	≤	NUM
ejpam-6713	280	1	k	k	X
ejpam-6713	280	2	≤	≤	NUM
ejpam-6713	280	3	n−1	n−1	PROPN
ejpam-6713	280	4	4	4	NUM
ejpam-6713	280	5	}	}	PUNCT
ejpam-6713	280	6	}	}	PUNCT
ejpam-6713	280	7	and	and	CCONJ
ejpam-6713	280	8	d′′	d′′	NOUN
ejpam-6713	280	9	=	=	PUNCT
ejpam-6713	280	10	{	{	PUNCT
ejpam-6713	280	11	{	{	PUNCT
ejpam-6713	280	12	u3k−2	u3k−2	NOUN
ejpam-6713	280	13	,	,	PUNCT
ejpam-6713	280	14	1	1	NUM
ejpam-6713	280	15	≤	≤	NUM
ejpam-6713	280	16	k	k	X
ejpam-6713	280	17	≤	≤	NUM
ejpam-6713	280	18	n+2	n+2	ADV
ejpam-6713	280	19	4	4	NUM
ejpam-6713	280	20	}	}	PUNCT
ejpam-6713	280	21	∪	∪	ADJ
ejpam-6713	280	22	{	{	PUNCT
ejpam-6713	280	23	u4k−2	u4k−2	PROPN
ejpam-6713	280	24	,	,	PUNCT
ejpam-6713	280	25	1	1	NUM
ejpam-6713	280	26	≤	≤	NUM
ejpam-6713	280	27	k	k	X
ejpam-6713	280	28	≤	≤	NUM
ejpam-6713	280	29	n+1	n+1	NUM
ejpam-6713	280	30	4	4	NUM
ejpam-6713	280	31	}	}	PUNCT
ejpam-6713	280	32	}	}	PUNCT
ejpam-6713	280	33	.	.	PUNCT
ejpam-6713	281	1	hence	hence	ADV
ejpam-6713	281	2	γcer(g	γcer(g	PROPN
ejpam-6713	281	3	′	′	NUM
ejpam-6713	281	4	)	)	PUNCT
ejpam-6713	281	5	>	>	X
ejpam-6713	282	1	γcer(g	γcer(g	PROPN
ejpam-6713	282	2	)	)	PUNCT
ejpam-6713	282	3	.	.	PUNCT
ejpam-6713	283	1	therefore	therefore	ADV
ejpam-6713	283	2	,	,	PUNCT
ejpam-6713	283	3	sd+γcer(g	sd+γcer(g	NOUN
ejpam-6713	283	4	)	)	PUNCT
ejpam-6713	283	5	=	=	SYM
ejpam-6713	284	1	1	1	X
ejpam-6713	284	2	.	.	X
ejpam-6713	284	3	case	case	NOUN
ejpam-6713	284	4	(	(	PUNCT
ejpam-6713	284	5	iii	iii	NOUN
ejpam-6713	284	6	)	)	PUNCT
ejpam-6713	284	7	n	n	NUM
ejpam-6713	284	8	≡	≡	PROPN
ejpam-6713	284	9	3	3	NUM
ejpam-6713	284	10	(	(	PUNCT
ejpam-6713	284	11	mod	mod	NOUN
ejpam-6713	284	12	4	4	NUM
ejpam-6713	284	13	)	)	PUNCT
ejpam-6713	284	14	figure	figure	NOUN
ejpam-6713	284	15	3	3	NUM
ejpam-6713	284	16	:	:	PUNCT
ejpam-6713	284	17	a	a	DET
ejpam-6713	284	18	graph	graph	NOUN
ejpam-6713	284	19	illustrating	illustrate	VERB
ejpam-6713	284	20	case	case	NOUN
ejpam-6713	284	21	(	(	PUNCT
ejpam-6713	284	22	iii	iii	NOUN
ejpam-6713	284	23	)	)	PUNCT
ejpam-6713	284	24	of	of	ADP
ejpam-6713	284	25	theorem	theorem	ADJ
ejpam-6713	284	26	11	11	NUM
ejpam-6713	284	27	,	,	PUNCT
ejpam-6713	284	28	sd+γcer	sd+γcer	NOUN
ejpam-6713	284	29	(	(	PUNCT
ejpam-6713	284	30	g	g	NOUN
ejpam-6713	284	31	)	)	PUNCT
ejpam-6713	284	32	=	=	SYM
ejpam-6713	284	33	2	2	NUM
ejpam-6713	284	34	let	let	VERB
ejpam-6713	284	35	g′	g′	NOUN
ejpam-6713	284	36	be	be	AUX
ejpam-6713	284	37	a	a	DET
ejpam-6713	284	38	graph	graph	NOUN
ejpam-6713	284	39	obtained	obtain	VERB
ejpam-6713	284	40	from	from	ADP
ejpam-6713	284	41	g	g	NOUN
ejpam-6713	284	42	by	by	ADP
ejpam-6713	284	43	subdividing	subdivide	VERB
ejpam-6713	284	44	an	an	DET
ejpam-6713	284	45	edge	edge	NOUN
ejpam-6713	284	46	e	e	NOUN
ejpam-6713	284	47	=	=	PRON
ejpam-6713	284	48	u1un	u1un	PUNCT
ejpam-6713	285	1	[	[	X
ejpam-6713	285	2	or	or	CCONJ
ejpam-6713	285	3	e	e	NOUN
ejpam-6713	285	4	=	=	NOUN
ejpam-6713	285	5	unvn	unvn	PROPN
ejpam-6713	285	6	,	,	PUNCT
ejpam-6713	285	7	or	or	CCONJ
ejpam-6713	285	8	e	e	X
ejpam-6713	285	9	=	=	SYM
ejpam-6713	285	10	v1vn	v1vn	PROPN
ejpam-6713	285	11	]	]	PUNCT
ejpam-6713	285	12	by	by	ADP
ejpam-6713	285	13	a	a	DET
ejpam-6713	285	14	subdivision	subdivision	NOUN
ejpam-6713	285	15	vertex	vertex	NOUN
ejpam-6713	285	16	x	x	NOUN
ejpam-6713	285	17	,	,	PUNCT
ejpam-6713	285	18	here	here	ADV
ejpam-6713	285	19	x	x	PART
ejpam-6713	285	20	∈	∈	PROPN
ejpam-6713	285	21	n(un	n(un	PROPN
ejpam-6713	285	22	)	)	PUNCT
ejpam-6713	286	1	[	[	X
ejpam-6713	286	2	or	or	CCONJ
ejpam-6713	286	3	n(un	n(un	NUM
ejpam-6713	286	4	)	)	PUNCT
ejpam-6713	286	5	or	or	CCONJ
ejpam-6713	286	6	n(v1	n(v1	NOUN
ejpam-6713	286	7	)	)	PUNCT
ejpam-6713	286	8	]	]	PUNCT
ejpam-6713	286	9	.	.	PUNCT
ejpam-6713	287	1	hence	hence	ADV
ejpam-6713	287	2	γcer(g	γcer(g	NUM
ejpam-6713	287	3	′	′	NUM
ejpam-6713	287	4	)	)	PUNCT
ejpam-6713	287	5	=	=	SYM
ejpam-6713	287	6	γcer(g	γcer(g	NOUN
ejpam-6713	287	7	)	)	PUNCT
ejpam-6713	287	8	.	.	PUNCT
ejpam-6713	288	1	therefore	therefore	ADV
ejpam-6713	288	2	,	,	PUNCT
ejpam-6713	288	3	sd+γcer(g	sd+γcer(g	NOUN
ejpam-6713	288	4	)	)	PUNCT
ejpam-6713	288	5	>	>	X
ejpam-6713	289	1	1	1	X
ejpam-6713	289	2	.	.	PUNCT
ejpam-6713	289	3	let	let	VERB
ejpam-6713	289	4	g′′	g′′	PROPN
ejpam-6713	289	5	be	be	AUX
ejpam-6713	289	6	a	a	DET
ejpam-6713	289	7	graph	graph	NOUN
ejpam-6713	289	8	obtained	obtain	VERB
ejpam-6713	289	9	from	from	ADP
ejpam-6713	289	10	g	g	NOUN
ejpam-6713	289	11	by	by	ADP
ejpam-6713	289	12	g.	g.	PROPN
ejpam-6713	289	13	navamani	navamani	PROPN
ejpam-6713	289	14	et	et	PROPN
ejpam-6713	289	15	al	al	PROPN
ejpam-6713	289	16	.	.	PUNCT
ejpam-6713	289	17	/	/	SYM
ejpam-6713	289	18	eur	eur	PROPN
ejpam-6713	289	19	.	.	PUNCT
ejpam-6713	290	1	j.	j.	PROPN
ejpam-6713	290	2	pure	pure	PROPN
ejpam-6713	290	3	appl	appl	PROPN
ejpam-6713	290	4	.	.	PROPN
ejpam-6713	290	5	math	math	PROPN
ejpam-6713	290	6	,	,	PUNCT
ejpam-6713	290	7	18	18	NUM
ejpam-6713	290	8	(	(	PUNCT
ejpam-6713	290	9	4	4	NUM
ejpam-6713	290	10	)	)	PUNCT
ejpam-6713	290	11	(	(	PUNCT
ejpam-6713	290	12	2025	2025	NUM
ejpam-6713	290	13	)	)	PUNCT
ejpam-6713	290	14	,	,	PUNCT
ejpam-6713	290	15	6713	6713	NUM
ejpam-6713	290	16	11	11	NUM
ejpam-6713	290	17	of	of	ADP
ejpam-6713	290	18	15	15	NUM
ejpam-6713	290	19	subdividing	subdivide	VERB
ejpam-6713	290	20	two	two	NUM
ejpam-6713	290	21	edges	edge	NOUN
ejpam-6713	290	22	(	(	PUNCT
ejpam-6713	290	23	say	say	INTJ
ejpam-6713	290	24	)	)	PUNCT
ejpam-6713	290	25	e1	e1	PROPN
ejpam-6713	290	26	=	=	SYM
ejpam-6713	290	27	u1un	u1un	X
ejpam-6713	290	28	and	and	CCONJ
ejpam-6713	290	29	e2	e2	PROPN
ejpam-6713	290	30	=	=	SYM
ejpam-6713	291	1	u1u2	u1u2	PUNCT
ejpam-6713	291	2	by	by	ADP
ejpam-6713	291	3	a	a	DET
ejpam-6713	291	4	subdivision	subdivision	NOUN
ejpam-6713	291	5	vertices	vertice	VERB
ejpam-6713	291	6	x	x	PUNCT
ejpam-6713	291	7	and	and	CCONJ
ejpam-6713	291	8	y	y	PROPN
ejpam-6713	291	9	respectively	respectively	ADV
ejpam-6713	291	10	,	,	PUNCT
ejpam-6713	291	11	x	x	PROPN
ejpam-6713	291	12	∈	∈	PROPN
ejpam-6713	291	13	n(un	n(un	PROPN
ejpam-6713	291	14	)	)	PUNCT
ejpam-6713	291	15	here	here	ADV
ejpam-6713	291	16	y	y	PROPN
ejpam-6713	291	17	/∈	/∈	PUNCT
ejpam-6713	292	1	n	n	CCONJ
ejpam-6713	293	1	[	[	X
ejpam-6713	293	2	d	d	X
ejpam-6713	293	3	]	]	PUNCT
ejpam-6713	293	4	.	.	PUNCT
ejpam-6713	294	1	now	now	ADV
ejpam-6713	294	2	d′	d′	X
ejpam-6713	295	1	=	=	SYM
ejpam-6713	295	2	d	d	X
ejpam-6713	295	3	∪	∪	X
ejpam-6713	295	4	{	{	PUNCT
ejpam-6713	295	5	y	y	NOUN
ejpam-6713	295	6	}	}	PUNCT
ejpam-6713	295	7	is	be	AUX
ejpam-6713	295	8	a	a	DET
ejpam-6713	295	9	γcer	γcer	NOUN
ejpam-6713	295	10	-	-	PUNCT
ejpam-6713	295	11	set	set	NOUN
ejpam-6713	295	12	of	of	ADP
ejpam-6713	295	13	g′′.	g′′.	PROPN
ejpam-6713	295	14	hence	hence	ADV
ejpam-6713	295	15	,	,	PUNCT
ejpam-6713	295	16	sd+γcer(g	sd+γcer(g	NOUN
ejpam-6713	295	17	)	)	PUNCT
ejpam-6713	295	18	=	=	SYM
ejpam-6713	295	19	2	2	NUM
ejpam-6713	295	20	,	,	PUNCT
ejpam-6713	295	21	refer	refer	VERB
ejpam-6713	295	22	figure	figure	NOUN
ejpam-6713	295	23	3	3	NUM
ejpam-6713	295	24	.	.	NOUN
ejpam-6713	295	25	case	case	NOUN
ejpam-6713	295	26	(	(	PUNCT
ejpam-6713	295	27	iv	iv	X
ejpam-6713	295	28	)	)	PUNCT
ejpam-6713	295	29	n	n	CCONJ
ejpam-6713	295	30	≡	≡	PROPN
ejpam-6713	295	31	2	2	NUM
ejpam-6713	295	32	(	(	PUNCT
ejpam-6713	295	33	mod	mod	NOUN
ejpam-6713	295	34	4	4	NUM
ejpam-6713	295	35	)	)	PUNCT
ejpam-6713	295	36	let	let	VERB
ejpam-6713	295	37	g′	g′	NOUN
ejpam-6713	295	38	be	be	AUX
ejpam-6713	295	39	a	a	DET
ejpam-6713	295	40	graph	graph	NOUN
ejpam-6713	295	41	obtained	obtain	VERB
ejpam-6713	295	42	from	from	ADP
ejpam-6713	295	43	g	g	NOUN
ejpam-6713	295	44	by	by	ADP
ejpam-6713	295	45	subdividing	subdivide	VERB
ejpam-6713	295	46	an	an	DET
ejpam-6713	295	47	edge	edge	NOUN
ejpam-6713	295	48	e	e	NOUN
ejpam-6713	295	49	=	=	PRON
ejpam-6713	295	50	u1un	u1un	PUNCT
ejpam-6713	296	1	[	[	X
ejpam-6713	296	2	or	or	CCONJ
ejpam-6713	296	3	e	e	NOUN
ejpam-6713	296	4	=	=	NOUN
ejpam-6713	296	5	unvn	unvn	ADJ
ejpam-6713	296	6	or	or	CCONJ
ejpam-6713	296	7	e	e	NOUN
ejpam-6713	296	8	=	=	SYM
ejpam-6713	296	9	v1vn	v1vn	PROPN
ejpam-6713	296	10	]	]	PUNCT
ejpam-6713	296	11	by	by	ADP
ejpam-6713	296	12	a	a	DET
ejpam-6713	296	13	subdivision	subdivision	NOUN
ejpam-6713	296	14	vertex	vertex	NOUN
ejpam-6713	296	15	x	x	NOUN
ejpam-6713	296	16	,	,	PUNCT
ejpam-6713	296	17	here	here	ADV
ejpam-6713	296	18	x	x	PART
ejpam-6713	296	19	∈	∈	PROPN
ejpam-6713	296	20	n(un	n(un	PROPN
ejpam-6713	296	21	)	)	PUNCT
ejpam-6713	296	22	and	and	CCONJ
ejpam-6713	296	23	u1	u1	NOUN
ejpam-6713	296	24	∈	∈	NOUN
ejpam-6713	296	25	n(v1	n(v1	NOUN
ejpam-6713	296	26	)	)	PUNCT
ejpam-6713	297	1	[	[	X
ejpam-6713	297	2	or	or	CCONJ
ejpam-6713	297	3	n(un	n(un	NUM
ejpam-6713	297	4	)	)	PUNCT
ejpam-6713	297	5	or	or	CCONJ
ejpam-6713	297	6	n(v1	n(v1	NOUN
ejpam-6713	297	7	)	)	PUNCT
ejpam-6713	297	8	]	]	PUNCT
ejpam-6713	297	9	respectively	respectively	ADV
ejpam-6713	297	10	.	.	PUNCT
ejpam-6713	298	1	hence	hence	ADV
ejpam-6713	298	2	γcer(g	γcer(g	NUM
ejpam-6713	298	3	′	′	NUM
ejpam-6713	298	4	)	)	PUNCT
ejpam-6713	298	5	=	=	SYM
ejpam-6713	298	6	γcer(g	γcer(g	NOUN
ejpam-6713	298	7	)	)	PUNCT
ejpam-6713	298	8	.	.	PUNCT
ejpam-6713	299	1	therefore	therefore	ADV
ejpam-6713	299	2	sd+γcer(g	sd+γcer(g	NOUN
ejpam-6713	299	3	)	)	PUNCT
ejpam-6713	299	4	>	>	X
ejpam-6713	300	1	1	1	X
ejpam-6713	300	2	.	.	PUNCT
ejpam-6713	300	3	let	let	VERB
ejpam-6713	300	4	g′′	g′′	PROPN
ejpam-6713	300	5	be	be	AUX
ejpam-6713	300	6	a	a	DET
ejpam-6713	300	7	graph	graph	NOUN
ejpam-6713	300	8	obtained	obtain	VERB
ejpam-6713	300	9	from	from	ADP
ejpam-6713	300	10	g	g	NOUN
ejpam-6713	300	11	by	by	ADP
ejpam-6713	300	12	subdividing	subdivide	VERB
ejpam-6713	300	13	two	two	NUM
ejpam-6713	300	14	edges	edge	NOUN
ejpam-6713	300	15	(	(	PUNCT
ejpam-6713	300	16	say	say	INTJ
ejpam-6713	300	17	)	)	PUNCT
ejpam-6713	300	18	e1	e1	PROPN
ejpam-6713	300	19	and	and	CCONJ
ejpam-6713	300	20	e2	e2	NOUN
ejpam-6713	300	21	by	by	ADP
ejpam-6713	300	22	a	a	DET
ejpam-6713	300	23	subdivision	subdivision	NOUN
ejpam-6713	300	24	vertices	vertice	VERB
ejpam-6713	300	25	x	x	PUNCT
ejpam-6713	300	26	and	and	CCONJ
ejpam-6713	300	27	y	y	PROPN
ejpam-6713	300	28	respectively	respectively	ADV
ejpam-6713	300	29	.	.	PUNCT
ejpam-6713	301	1	we	we	PRON
ejpam-6713	301	2	have	have	VERB
ejpam-6713	301	3	the	the	DET
ejpam-6713	301	4	following	follow	VERB
ejpam-6713	301	5	subcases	subcase	NOUN
ejpam-6713	301	6	.	.	PUNCT
ejpam-6713	302	1	subcase	subcase	PROPN
ejpam-6713	302	2	(	(	PUNCT
ejpam-6713	302	3	a	a	NOUN
ejpam-6713	302	4	)	)	PUNCT
ejpam-6713	302	5	e1	e1	NOUN
ejpam-6713	302	6	=	=	SYM
ejpam-6713	302	7	u1un	u1un	X
ejpam-6713	302	8	and	and	CCONJ
ejpam-6713	302	9	e2	e2	PROPN
ejpam-6713	302	10	=	=	SYM
ejpam-6713	302	11	unun−1	unun−1	PROPN
ejpam-6713	302	12	(	(	PUNCT
ejpam-6713	302	13	adjacent	adjacent	ADJ
ejpam-6713	302	14	edges	edge	NOUN
ejpam-6713	302	15	in	in	ADP
ejpam-6713	302	16	the	the	DET
ejpam-6713	302	17	outer	outer	ADJ
ejpam-6713	302	18	cycle	cycle	NOUN
ejpam-6713	302	19	)	)	PUNCT
ejpam-6713	302	20	in	in	ADP
ejpam-6713	302	21	this	this	DET
ejpam-6713	302	22	subcase	subcase	NOUN
ejpam-6713	302	23	,	,	PUNCT
ejpam-6713	302	24	x	x	PRON
ejpam-6713	302	25	,	,	PUNCT
ejpam-6713	302	26	y	y	PROPN
ejpam-6713	302	27	∈	∈	PROPN
ejpam-6713	302	28	n(un	n(un	PROPN
ejpam-6713	302	29	)	)	PUNCT
ejpam-6713	302	30	and	and	CCONJ
ejpam-6713	302	31	un−1	un−1	PROPN
ejpam-6713	302	32	∈	∈	PROPN
ejpam-6713	302	33	n(vn−1	n(vn−1	NOUN
ejpam-6713	302	34	)	)	PUNCT
ejpam-6713	302	35	and	and	CCONJ
ejpam-6713	302	36	u1	u1	NOUN
ejpam-6713	302	37	∈	∈	NOUN
ejpam-6713	302	38	n(v1	n(v1	NOUN
ejpam-6713	302	39	)	)	PUNCT
ejpam-6713	302	40	,	,	PUNCT
ejpam-6713	302	41	hence	hence	ADV
ejpam-6713	302	42	γcer(g	γcer(g	NUM
ejpam-6713	302	43	′′	′′	PROPN
ejpam-6713	302	44	)	)	PUNCT
ejpam-6713	302	45	=	=	SYM
ejpam-6713	302	46	γcer(g	γcer(g	NOUN
ejpam-6713	302	47	)	)	PUNCT
ejpam-6713	302	48	.	.	PUNCT
ejpam-6713	303	1	subcase	subcase	NOUN
ejpam-6713	303	2	(	(	PUNCT
ejpam-6713	303	3	b	b	NOUN
ejpam-6713	303	4	)	)	PUNCT
ejpam-6713	303	5	e1	e1	NOUN
ejpam-6713	303	6	=	=	SYM
ejpam-6713	303	7	unun−1	unun−1	ADJ
ejpam-6713	303	8	and	and	CCONJ
ejpam-6713	303	9	e2	e2	PROPN
ejpam-6713	303	10	=	=	SYM
ejpam-6713	303	11	vnvn−1	vnvn−1	PROPN
ejpam-6713	303	12	(	(	PUNCT
ejpam-6713	303	13	an	an	DET
ejpam-6713	303	14	edge	edge	NOUN
ejpam-6713	303	15	in	in	ADP
ejpam-6713	303	16	the	the	DET
ejpam-6713	303	17	outer	outer	ADJ
ejpam-6713	303	18	cycle	cycle	NOUN
ejpam-6713	303	19	and	and	CCONJ
ejpam-6713	303	20	an	an	DET
ejpam-6713	303	21	edge	edge	NOUN
ejpam-6713	303	22	in	in	ADP
ejpam-6713	303	23	the	the	DET
ejpam-6713	303	24	inner	inner	ADJ
ejpam-6713	303	25	cycle	cycle	NOUN
ejpam-6713	303	26	)	)	PUNCT
ejpam-6713	303	27	in	in	ADP
ejpam-6713	303	28	this	this	DET
ejpam-6713	303	29	subcase	subcase	NOUN
ejpam-6713	303	30	,	,	PUNCT
ejpam-6713	303	31	x	x	PROPN
ejpam-6713	303	32	∈	∈	PROPN
ejpam-6713	303	33	n(un	n(un	PROPN
ejpam-6713	303	34	)	)	PUNCT
ejpam-6713	303	35	and	and	CCONJ
ejpam-6713	303	36	y	y	PROPN
ejpam-6713	303	37	∈	∈	PROPN
ejpam-6713	303	38	n(vn−1	n(vn−1	NOUN
ejpam-6713	303	39	)	)	PUNCT
ejpam-6713	303	40	and	and	CCONJ
ejpam-6713	303	41	vn−1	vn−1	PROPN
ejpam-6713	303	42	∈	∈	PROPN
ejpam-6713	303	43	n(un−1	n(un−1	PRON
ejpam-6713	303	44	)	)	PUNCT
ejpam-6713	303	45	,	,	PUNCT
ejpam-6713	303	46	hence	hence	ADV
ejpam-6713	303	47	γcer(g	γcer(g	NUM
ejpam-6713	303	48	′′	′′	PROPN
ejpam-6713	303	49	)	)	PUNCT
ejpam-6713	303	50	=	=	SYM
ejpam-6713	303	51	γcer(g	γcer(g	NOUN
ejpam-6713	303	52	)	)	PUNCT
ejpam-6713	303	53	.	.	PUNCT
ejpam-6713	304	1	subcase	subcase	NOUN
ejpam-6713	304	2	(	(	PUNCT
ejpam-6713	304	3	c	c	NOUN
ejpam-6713	304	4	)	)	PUNCT
ejpam-6713	304	5	e1	e1	NOUN
ejpam-6713	304	6	=	=	SYM
ejpam-6713	304	7	u1v1	u1v1	PROPN
ejpam-6713	304	8	and	and	CCONJ
ejpam-6713	304	9	e2	e2	PROPN
ejpam-6713	304	10	=	=	SYM
ejpam-6713	304	11	unvn	unvn	PROPN
ejpam-6713	304	12	(	(	PUNCT
ejpam-6713	304	13	edges	edge	NOUN
ejpam-6713	304	14	in	in	ADP
ejpam-6713	304	15	the	the	DET
ejpam-6713	304	16	spokes	spoke	NOUN
ejpam-6713	304	17	)	)	PUNCT
ejpam-6713	304	18	in	in	ADP
ejpam-6713	304	19	this	this	DET
ejpam-6713	304	20	subcase	subcase	NOUN
ejpam-6713	304	21	,	,	PUNCT
ejpam-6713	304	22	x	x	SYM
ejpam-6713	304	23	∈	∈	NOUN
ejpam-6713	304	24	n(v1	n(v1	NOUN
ejpam-6713	304	25	)	)	PUNCT
ejpam-6713	304	26	and	and	CCONJ
ejpam-6713	304	27	y	y	PROPN
ejpam-6713	304	28	∈	∈	PROPN
ejpam-6713	304	29	n(un	n(un	PROPN
ejpam-6713	304	30	)	)	PUNCT
ejpam-6713	304	31	,	,	PUNCT
ejpam-6713	304	32	u1	u1	PROPN
ejpam-6713	304	33	∈	∈	PROPN
ejpam-6713	304	34	n(un	n(un	PROPN
ejpam-6713	304	35	)	)	PUNCT
ejpam-6713	304	36	and	and	CCONJ
ejpam-6713	304	37	vn	vn	PROPN
ejpam-6713	304	38	∈	∈	PROPN
ejpam-6713	304	39	n(v1	n(v1	NOUN
ejpam-6713	304	40	)	)	PUNCT
ejpam-6713	304	41	.	.	PUNCT
ejpam-6713	305	1	hence	hence	ADV
ejpam-6713	305	2	γcer(g	γcer(g	NUM
ejpam-6713	305	3	′′	′′	PROPN
ejpam-6713	305	4	)	)	PUNCT
ejpam-6713	305	5	=	=	SYM
ejpam-6713	305	6	γcer(g	γcer(g	NOUN
ejpam-6713	305	7	)	)	PUNCT
ejpam-6713	305	8	.	.	PUNCT
ejpam-6713	306	1	subcase	subcase	NOUN
ejpam-6713	306	2	(	(	PUNCT
ejpam-6713	306	3	d	d	NOUN
ejpam-6713	306	4	)	)	PUNCT
ejpam-6713	306	5	e1	e1	NOUN
ejpam-6713	306	6	=	=	SYM
ejpam-6713	306	7	v1vn	v1vn	PROPN
ejpam-6713	306	8	and	and	CCONJ
ejpam-6713	306	9	e2	e2	PROPN
ejpam-6713	306	10	=	=	SYM
ejpam-6713	306	11	vnvn−1	vnvn−1	PROPN
ejpam-6713	306	12	(	(	PUNCT
ejpam-6713	306	13	adjacent	adjacent	ADJ
ejpam-6713	306	14	edges	edge	NOUN
ejpam-6713	306	15	in	in	ADP
ejpam-6713	306	16	the	the	DET
ejpam-6713	306	17	inner	inner	ADJ
ejpam-6713	306	18	cycles	cycle	NOUN
ejpam-6713	306	19	)	)	PUNCT
ejpam-6713	306	20	in	in	ADP
ejpam-6713	306	21	this	this	DET
ejpam-6713	306	22	subcase	subcase	NOUN
ejpam-6713	306	23	,	,	PUNCT
ejpam-6713	306	24	x	x	SYM
ejpam-6713	306	25	∈	∈	NOUN
ejpam-6713	306	26	n(v1	n(v1	NOUN
ejpam-6713	306	27	)	)	PUNCT
ejpam-6713	306	28	and	and	CCONJ
ejpam-6713	306	29	y	y	PROPN
ejpam-6713	306	30	∈	∈	PROPN
ejpam-6713	306	31	n(vn−1	n(vn−1	NOUN
ejpam-6713	306	32	)	)	PUNCT
ejpam-6713	306	33	,	,	PUNCT
ejpam-6713	306	34	vn	vn	PROPN
ejpam-6713	306	35	∈	∈	PROPN
ejpam-6713	306	36	n(un	n(un	PROPN
ejpam-6713	306	37	)	)	PUNCT
ejpam-6713	306	38	.	.	PUNCT
ejpam-6713	307	1	hence	hence	ADV
ejpam-6713	307	2	γcer(g	γcer(g	NUM
ejpam-6713	307	3	′′	′′	PROPN
ejpam-6713	307	4	)	)	PUNCT
ejpam-6713	307	5	=	=	SYM
ejpam-6713	307	6	γcer(g	γcer(g	NOUN
ejpam-6713	307	7	)	)	PUNCT
ejpam-6713	307	8	.	.	PUNCT
ejpam-6713	308	1	subcase	subcase	NOUN
ejpam-6713	308	2	(	(	PUNCT
ejpam-6713	308	3	e	e	NOUN
ejpam-6713	308	4	)	)	PUNCT
ejpam-6713	308	5	e1	e1	NOUN
ejpam-6713	308	6	=	=	SYM
ejpam-6713	308	7	u1un	u1un	X
ejpam-6713	308	8	and	and	CCONJ
ejpam-6713	308	9	e2	e2	PROPN
ejpam-6713	308	10	=	=	SYM
ejpam-6713	308	11	un−1vn−1	un−1vn−1	PROPN
ejpam-6713	308	12	(	(	PUNCT
ejpam-6713	308	13	non	non	X
ejpam-6713	308	14	adjacent	adjacent	ADJ
ejpam-6713	308	15	edges	edge	NOUN
ejpam-6713	308	16	with	with	ADP
ejpam-6713	308	17	one	one	NUM
ejpam-6713	308	18	edge	edge	NOUN
ejpam-6713	308	19	in	in	ADP
ejpam-6713	308	20	the	the	DET
ejpam-6713	308	21	outer	outer	ADJ
ejpam-6713	308	22	cycle	cycle	NOUN
ejpam-6713	308	23	and	and	CCONJ
ejpam-6713	308	24	another	another	DET
ejpam-6713	308	25	edge	edge	NOUN
ejpam-6713	308	26	in	in	ADP
ejpam-6713	308	27	the	the	DET
ejpam-6713	308	28	spoke	spoke	NOUN
ejpam-6713	308	29	)	)	PUNCT
ejpam-6713	308	30	in	in	ADP
ejpam-6713	308	31	this	this	DET
ejpam-6713	308	32	subcase	subcase	NOUN
ejpam-6713	308	33	,	,	PUNCT
ejpam-6713	308	34	x	x	PROPN
ejpam-6713	308	35	∈	∈	PROPN
ejpam-6713	308	36	n(un	n(un	PROPN
ejpam-6713	308	37	)	)	PUNCT
ejpam-6713	308	38	and	and	CCONJ
ejpam-6713	308	39	y	y	PROPN
ejpam-6713	308	40	∈	∈	PROPN
ejpam-6713	308	41	n(vn−1	n(vn−1	NOUN
ejpam-6713	308	42	)	)	PUNCT
ejpam-6713	308	43	,	,	PUNCT
ejpam-6713	308	44	un−1	un−1	PROPN
ejpam-6713	308	45	∈	∈	PROPN
ejpam-6713	308	46	n(un	n(un	PROPN
ejpam-6713	308	47	)	)	PUNCT
ejpam-6713	308	48	,	,	PUNCT
ejpam-6713	308	49	u1	u1	NOUN
ejpam-6713	308	50	∈	∈	NOUN
ejpam-6713	308	51	n(v1	n(v1	NOUN
ejpam-6713	308	52	)	)	PUNCT
ejpam-6713	308	53	.	.	PUNCT
ejpam-6713	309	1	hence	hence	ADV
ejpam-6713	309	2	γcer(g	γcer(g	NUM
ejpam-6713	309	3	′′	′′	PROPN
ejpam-6713	309	4	)	)	PUNCT
ejpam-6713	309	5	=	=	SYM
ejpam-6713	309	6	γcer(g	γcer(g	NOUN
ejpam-6713	309	7	)	)	PUNCT
ejpam-6713	309	8	.	.	PUNCT
ejpam-6713	310	1	subcase	subcase	NOUN
ejpam-6713	310	2	(	(	PUNCT
ejpam-6713	310	3	f	f	X
ejpam-6713	310	4	)	)	PUNCT
ejpam-6713	310	5	e1	e1	NOUN
ejpam-6713	310	6	=	=	SYM
ejpam-6713	310	7	unvn	unvn	NOUN
ejpam-6713	310	8	and	and	CCONJ
ejpam-6713	310	9	e2	e2	PROPN
ejpam-6713	310	10	=	=	PUNCT
ejpam-6713	310	11	vn−1vn	vn−1vn	NUM
ejpam-6713	310	12	(	(	PUNCT
ejpam-6713	310	13	an	an	DET
ejpam-6713	310	14	edge	edge	NOUN
ejpam-6713	310	15	in	in	ADP
ejpam-6713	310	16	spoke	speak	VERB
ejpam-6713	310	17	and	and	CCONJ
ejpam-6713	310	18	an	an	DET
ejpam-6713	310	19	edge	edge	NOUN
ejpam-6713	310	20	in	in	ADP
ejpam-6713	310	21	inner	inner	ADJ
ejpam-6713	310	22	cycle	cycle	NOUN
ejpam-6713	310	23	)	)	PUNCT
ejpam-6713	310	24	in	in	ADP
ejpam-6713	310	25	this	this	DET
ejpam-6713	310	26	subcase	subcase	NOUN
ejpam-6713	310	27	x	x	X
ejpam-6713	310	28	∈	∈	PROPN
ejpam-6713	310	29	n(un	n(un	PROPN
ejpam-6713	310	30	)	)	PUNCT
ejpam-6713	310	31	and	and	CCONJ
ejpam-6713	310	32	y	y	PROPN
ejpam-6713	310	33	∈	∈	PROPN
ejpam-6713	310	34	n(vn−1	n(vn−1	NOUN
ejpam-6713	310	35	)	)	PUNCT
ejpam-6713	310	36	,	,	PUNCT
ejpam-6713	310	37	un−1	un−1	PROPN
ejpam-6713	310	38	∈	∈	PROPN
ejpam-6713	310	39	n(un	n(un	PROPN
ejpam-6713	310	40	)	)	PUNCT
ejpam-6713	310	41	,	,	PUNCT
ejpam-6713	310	42	vn	vn	PROPN
ejpam-6713	310	43	∈	∈	PROPN
ejpam-6713	310	44	n(v1	n(v1	NOUN
ejpam-6713	310	45	)	)	PUNCT
ejpam-6713	310	46	.	.	PUNCT
ejpam-6713	311	1	hence	hence	ADV
ejpam-6713	311	2	,	,	PUNCT
ejpam-6713	311	3	γcer(g	γcer(g	PROPN
ejpam-6713	311	4	′′	′′	PROPN
ejpam-6713	311	5	)	)	PUNCT
ejpam-6713	311	6	=	=	SYM
ejpam-6713	311	7	γcer(g	γcer(g	PROPN
ejpam-6713	311	8	)	)	PUNCT
ejpam-6713	311	9	.	.	PUNCT
ejpam-6713	312	1	from	from	ADP
ejpam-6713	312	2	all	all	DET
ejpam-6713	312	3	the	the	DET
ejpam-6713	312	4	cases	case	NOUN
ejpam-6713	312	5	mentioned	mention	VERB
ejpam-6713	312	6	above	above	ADP
ejpam-6713	312	7	clearly	clearly	ADV
ejpam-6713	312	8	,	,	PUNCT
ejpam-6713	312	9	we	we	PRON
ejpam-6713	312	10	see	see	VERB
ejpam-6713	312	11	that	that	PRON
ejpam-6713	312	12	sd+γcer(g	sd+γcer(g	NOUN
ejpam-6713	312	13	)	)	PUNCT
ejpam-6713	312	14	>	>	X
ejpam-6713	313	1	2	2	X
ejpam-6713	313	2	.	.	PUNCT
ejpam-6713	313	3	let	let	VERB
ejpam-6713	313	4	g′′′	g′′′	PROPN
ejpam-6713	313	5	be	be	AUX
ejpam-6713	313	6	a	a	DET
ejpam-6713	313	7	graph	graph	NOUN
ejpam-6713	313	8	obtained	obtain	VERB
ejpam-6713	313	9	from	from	ADP
ejpam-6713	313	10	g	g	NOUN
ejpam-6713	313	11	by	by	ADP
ejpam-6713	313	12	subdividing	subdivide	VERB
ejpam-6713	313	13	the	the	DET
ejpam-6713	313	14	edges	edge	NOUN
ejpam-6713	313	15	e1	e1	NOUN
ejpam-6713	313	16	=	=	SYM
ejpam-6713	313	17	unu1	unu1	PROPN
ejpam-6713	313	18	,	,	PUNCT
ejpam-6713	313	19	e2	e2	PROPN
ejpam-6713	313	20	=	=	PUNCT
ejpam-6713	313	21	unun−1	unun−1	ADJ
ejpam-6713	313	22	and	and	CCONJ
ejpam-6713	313	23	e3	e3	VERB
ejpam-6713	313	24	=	=	SYM
ejpam-6713	313	25	u1u2	u1u2	NOUN
ejpam-6713	313	26	by	by	ADP
ejpam-6713	313	27	the	the	DET
ejpam-6713	313	28	subdivision	subdivision	NOUN
ejpam-6713	313	29	vertices	vertice	VERB
ejpam-6713	313	30	x	x	PRON
ejpam-6713	313	31	,	,	PUNCT
ejpam-6713	313	32	y	y	PROPN
ejpam-6713	313	33	and	and	CCONJ
ejpam-6713	313	34	z	z	NOUN
ejpam-6713	313	35	respectively	respectively	ADV
ejpam-6713	313	36	,	,	PUNCT
ejpam-6713	313	37	where	where	SCONJ
ejpam-6713	313	38	x	x	X
ejpam-6713	313	39	,	,	PUNCT
ejpam-6713	313	40	y	y	PROPN
ejpam-6713	313	41	∈	∈	PROPN
ejpam-6713	313	42	n(un	n(un	PROPN
ejpam-6713	313	43	)	)	PUNCT
ejpam-6713	313	44	and	and	CCONJ
ejpam-6713	313	45	z	z	NOUN
ejpam-6713	313	46	/∈	/∈	PUNCT
ejpam-6713	314	1	n(v	n(v	PROPN
ejpam-6713	314	2	)	)	PUNCT
ejpam-6713	314	3	.	.	PUNCT
ejpam-6713	315	1	hence	hence	ADV
ejpam-6713	315	2	,	,	PUNCT
ejpam-6713	315	3	d′	d′	X
ejpam-6713	315	4	=	=	PUNCT
ejpam-6713	315	5	d	d	X
ejpam-6713	315	6	∪	∪	X
ejpam-6713	315	7	{	{	PUNCT
ejpam-6713	315	8	z	z	NOUN
ejpam-6713	315	9	}	}	PUNCT
ejpam-6713	315	10	is	be	AUX
ejpam-6713	315	11	a	a	DET
ejpam-6713	315	12	γcer	γcer	NOUN
ejpam-6713	315	13	-	-	PUNCT
ejpam-6713	315	14	set	set	NOUN
ejpam-6713	315	15	of	of	ADP
ejpam-6713	315	16	g′′′.	g′′′.	PROPN
ejpam-6713	315	17	therefore	therefore	ADV
ejpam-6713	315	18	,	,	PUNCT
ejpam-6713	315	19	sd+γcer(g	sd+γcer(g	NOUN
ejpam-6713	315	20	)	)	PUNCT
ejpam-6713	315	21	=	=	SYM
ejpam-6713	316	1	3	3	X
ejpam-6713	316	2	.	.	X
ejpam-6713	316	3	theorem	theorem	NOUN
ejpam-6713	316	4	12	12	NUM
ejpam-6713	316	5	.	.	PUNCT
ejpam-6713	317	1	[	[	X
ejpam-6713	317	2	20	20	NUM
ejpam-6713	317	3	]	]	PUNCT
ejpam-6713	317	4	for	for	ADP
ejpam-6713	317	5	n	n	X
ejpam-6713	317	6	≥	≥	NUM
ejpam-6713	317	7	5	5	NUM
ejpam-6713	317	8	,	,	PUNCT
ejpam-6713	317	9	we	we	PRON
ejpam-6713	317	10	have	have	VERB
ejpam-6713	317	11	γ(p	γ(p	PROPN
ejpam-6713	317	12	(	(	PUNCT
ejpam-6713	317	13	n	n	CCONJ
ejpam-6713	317	14	,	,	PUNCT
ejpam-6713	317	15	2	2	NUM
ejpam-6713	317	16	)	)	PUNCT
ejpam-6713	317	17	)	)	PUNCT
ejpam-6713	318	1	=	=	PUNCT
ejpam-6713	319	1	⌈	⌈	NUM
ejpam-6713	319	2	3n	3n	NUM
ejpam-6713	319	3	5	5	NUM
ejpam-6713	319	4	⌉	⌉	SCONJ
ejpam-6713	319	5	g.	g.	PROPN
ejpam-6713	319	6	navamani	navamani	PROPN
ejpam-6713	319	7	et	et	PROPN
ejpam-6713	319	8	al	al	PROPN
ejpam-6713	319	9	.	.	PUNCT
ejpam-6713	319	10	/	/	SYM
ejpam-6713	319	11	eur	eur	PROPN
ejpam-6713	319	12	.	.	PUNCT
ejpam-6713	320	1	j.	j.	PROPN
ejpam-6713	320	2	pure	pure	PROPN
ejpam-6713	320	3	appl	appl	PROPN
ejpam-6713	320	4	.	.	PROPN
ejpam-6713	320	5	math	math	PROPN
ejpam-6713	320	6	,	,	PUNCT
ejpam-6713	320	7	18	18	NUM
ejpam-6713	320	8	(	(	PUNCT
ejpam-6713	320	9	4	4	NUM
ejpam-6713	320	10	)	)	PUNCT
ejpam-6713	320	11	(	(	PUNCT
ejpam-6713	320	12	2025	2025	NUM
ejpam-6713	320	13	)	)	PUNCT
ejpam-6713	320	14	,	,	PUNCT
ejpam-6713	320	15	6713	6713	NUM
ejpam-6713	320	16	12	12	NUM
ejpam-6713	320	17	of	of	ADP
ejpam-6713	320	18	15	15	NUM
ejpam-6713	320	19	theorem	theorem	VERB
ejpam-6713	320	20	13	13	NUM
ejpam-6713	320	21	.	.	PUNCT
ejpam-6713	321	1	for	for	ADP
ejpam-6713	321	2	any	any	DET
ejpam-6713	321	3	petersen	petersen	NOUN
ejpam-6713	321	4	graph	graph	NOUN
ejpam-6713	321	5	g	g	ADP
ejpam-6713	321	6	∼=	∼=	PROPN
ejpam-6713	321	7	p	p	NOUN
ejpam-6713	321	8	(	(	PUNCT
ejpam-6713	321	9	n	n	CCONJ
ejpam-6713	321	10	,	,	PUNCT
ejpam-6713	321	11	2	2	NUM
ejpam-6713	321	12	)	)	PUNCT
ejpam-6713	321	13	,	,	PUNCT
ejpam-6713	321	14	n	n	X
ejpam-6713	321	15	≥	≥	NOUN
ejpam-6713	321	16	5	5	NUM
ejpam-6713	321	17	,	,	PUNCT
ejpam-6713	321	18	sd+γcer(g	sd+γcer(g	NOUN
ejpam-6713	321	19	)	)	PUNCT
ejpam-6713	321	20	=	=	SYM
ejpam-6713	321	21	{	{	PUNCT
ejpam-6713	321	22	1	1	NUM
ejpam-6713	321	23	if	if	SCONJ
ejpam-6713	321	24	n	n	PRON
ejpam-6713	321	25	≡	≡	PROPN
ejpam-6713	321	26	0	0	NUM
ejpam-6713	321	27	,	,	PUNCT
ejpam-6713	321	28	3	3	NUM
ejpam-6713	321	29	(	(	PUNCT
ejpam-6713	321	30	mod	mod	NOUN
ejpam-6713	321	31	5	5	NUM
ejpam-6713	321	32	)	)	PUNCT
ejpam-6713	321	33	2	2	NUM
ejpam-6713	321	34	otherwise	otherwise	ADV
ejpam-6713	321	35	proof	proof	NOUN
ejpam-6713	321	36	.	.	PUNCT
ejpam-6713	322	1	let	let	VERB
ejpam-6713	322	2	g	g	PRON
ejpam-6713	322	3	∼=	∼=	PROPN
ejpam-6713	322	4	p	p	NOUN
ejpam-6713	322	5	(	(	PUNCT
ejpam-6713	322	6	n	n	CCONJ
ejpam-6713	322	7	,	,	PUNCT
ejpam-6713	322	8	2	2	NUM
ejpam-6713	322	9	)	)	PUNCT
ejpam-6713	322	10	,	,	PUNCT
ejpam-6713	322	11	c	c	NOUN
ejpam-6713	322	12	′	′	NOUN
ejpam-6713	322	13	and	and	CCONJ
ejpam-6713	322	14	c	c	X
ejpam-6713	322	15	′′	′′	PROPN
ejpam-6713	322	16	be	be	VERB
ejpam-6713	322	17	the	the	DET
ejpam-6713	322	18	inner	inner	ADJ
ejpam-6713	322	19	and	and	CCONJ
ejpam-6713	322	20	outer	outer	ADJ
ejpam-6713	322	21	cycles	cycle	NOUN
ejpam-6713	322	22	of	of	ADP
ejpam-6713	322	23	g.	g.	PROPN
ejpam-6713	322	24	let	let	VERB
ejpam-6713	322	25	v	v	X
ejpam-6713	322	26	(	(	PUNCT
ejpam-6713	322	27	c	c	NOUN
ejpam-6713	322	28	′	′	NUM
ejpam-6713	322	29	)	)	PUNCT
ejpam-6713	323	1	=	=	PRON
ejpam-6713	323	2	{	{	PUNCT
ejpam-6713	323	3	v1	v1	PROPN
ejpam-6713	323	4	,	,	PUNCT
ejpam-6713	323	5	v2	v2	PROPN
ejpam-6713	323	6	,	,	PUNCT
ejpam-6713	323	7	v3	v3	PROPN
ejpam-6713	323	8	,	,	PUNCT
ejpam-6713	323	9	.	.	PUNCT
ejpam-6713	323	10	.	.	PUNCT
ejpam-6713	323	11	.	.	PUNCT
ejpam-6713	324	1	vn	vn	X
ejpam-6713	324	2	}	}	PUNCT
ejpam-6713	324	3	and	and	CCONJ
ejpam-6713	324	4	v	v	X
ejpam-6713	324	5	(	(	PUNCT
ejpam-6713	324	6	c	c	PROPN
ejpam-6713	324	7	′′	′′	PROPN
ejpam-6713	324	8	)	)	PUNCT
ejpam-6713	324	9	=	=	PRON
ejpam-6713	324	10	{	{	PUNCT
ejpam-6713	324	11	u1	u1	NOUN
ejpam-6713	324	12	,	,	PUNCT
ejpam-6713	324	13	u2	u2	NOUN
ejpam-6713	324	14	,	,	PUNCT
ejpam-6713	324	15	u3	u3	NOUN
ejpam-6713	324	16	,	,	PUNCT
ejpam-6713	324	17	.	.	PUNCT
ejpam-6713	324	18	.	.	PUNCT
ejpam-6713	324	19	.	.	PUNCT
ejpam-6713	325	1	un	un	PROPN
ejpam-6713	325	2	}	}	PUNCT
ejpam-6713	325	3	by	by	ADP
ejpam-6713	325	4	theorem	theorem	ADJ
ejpam-6713	325	5	3	3	NUM
ejpam-6713	325	6	and	and	CCONJ
ejpam-6713	325	7	theorem	theorem	VERB
ejpam-6713	325	8	12	12	NUM
ejpam-6713	325	9	,	,	PUNCT
ejpam-6713	325	10	γ(p	γ(p	PROPN
ejpam-6713	325	11	(	(	PUNCT
ejpam-6713	325	12	n	n	CCONJ
ejpam-6713	325	13	,	,	PUNCT
ejpam-6713	325	14	2	2	NUM
ejpam-6713	325	15	)	)	PUNCT
ejpam-6713	325	16	)	)	PUNCT
ejpam-6713	326	1	=	=	PUNCT
ejpam-6713	326	2	γcer(p	γcer(p	NOUN
ejpam-6713	326	3	(	(	PUNCT
ejpam-6713	326	4	n	n	CCONJ
ejpam-6713	326	5	,	,	PUNCT
ejpam-6713	326	6	2	2	NUM
ejpam-6713	326	7	)	)	PUNCT
ejpam-6713	326	8	)	)	PUNCT
ejpam-6713	326	9	.	.	PUNCT
ejpam-6713	327	1	let	let	VERB
ejpam-6713	327	2	d	d	PRON
ejpam-6713	327	3	be	be	AUX
ejpam-6713	327	4	a	a	DET
ejpam-6713	327	5	γcer	γcer	NOUN
ejpam-6713	327	6	-	-	PUNCT
ejpam-6713	327	7	set	set	NOUN
ejpam-6713	327	8	of	of	ADP
ejpam-6713	327	9	g	g	PROPN
ejpam-6713	327	10	and	and	CCONJ
ejpam-6713	327	11	d	d	NOUN
ejpam-6713	327	12	=	=	X
ejpam-6713	327	13	d′	d′	X
ejpam-6713	327	14	∪	∪	ADP
ejpam-6713	327	15	d′′	d′′	PROPN
ejpam-6713	327	16	,	,	PUNCT
ejpam-6713	327	17	where	where	SCONJ
ejpam-6713	327	18	d′	d′	PRON
ejpam-6713	327	19	=	=	SYM
ejpam-6713	328	1	d	d	PROPN
ejpam-6713	328	2	∩	∩	ADJ
ejpam-6713	328	3	v	v	X
ejpam-6713	328	4	(	(	PUNCT
ejpam-6713	328	5	c	c	NOUN
ejpam-6713	328	6	′	′	NUM
ejpam-6713	328	7	)	)	PUNCT
ejpam-6713	328	8	and	and	CCONJ
ejpam-6713	328	9	d′′	d′′	NOUN
ejpam-6713	328	10	=	=	SYM
ejpam-6713	328	11	d	d	PROPN
ejpam-6713	328	12	∩	∩	X
ejpam-6713	328	13	v	v	X
ejpam-6713	328	14	(	(	PUNCT
ejpam-6713	328	15	c	c	PROPN
ejpam-6713	328	16	′′	′′	PROPN
ejpam-6713	328	17	)	)	PUNCT
ejpam-6713	328	18	.	.	PUNCT
ejpam-6713	329	1	now	now	ADV
ejpam-6713	329	2	consider	consider	VERB
ejpam-6713	329	3	the	the	DET
ejpam-6713	329	4	following	follow	VERB
ejpam-6713	329	5	cases	case	NOUN
ejpam-6713	329	6	case	case	NOUN
ejpam-6713	329	7	(	(	PUNCT
ejpam-6713	329	8	i	i	NOUN
ejpam-6713	329	9	)	)	PUNCT
ejpam-6713	329	10	n	n	X
ejpam-6713	329	11	≡	≡	PROPN
ejpam-6713	329	12	0	0	PUNCT
ejpam-6713	330	1	(	(	PUNCT
ejpam-6713	330	2	mod	mod	NOUN
ejpam-6713	330	3	5	5	NUM
ejpam-6713	330	4	)	)	PUNCT
ejpam-6713	330	5	let	let	VERB
ejpam-6713	330	6	d′	d′	X
ejpam-6713	330	7	=	=	PUNCT
ejpam-6713	330	8	{	{	PUNCT
ejpam-6713	330	9	v5k−4	v5k−4	NOUN
ejpam-6713	330	10	:	:	PUNCT
ejpam-6713	330	11	1	1	NUM
ejpam-6713	330	12	≤	≤	NUM
ejpam-6713	330	13	k	k	X
ejpam-6713	330	14	≤	≤	NUM
ejpam-6713	330	15	n	n	PRON
ejpam-6713	330	16	5	5	NUM
ejpam-6713	330	17	}	}	PUNCT
ejpam-6713	330	18	∪	∪	ADJ
ejpam-6713	330	19	{	{	PUNCT
ejpam-6713	330	20	v5k−3	v5k−3	NOUN
ejpam-6713	330	21	:	:	PUNCT
ejpam-6713	330	22	1	1	NUM
ejpam-6713	330	23	≤	≤	NUM
ejpam-6713	330	24	k	k	X
ejpam-6713	330	25	≤	≤	NUM
ejpam-6713	330	26	n	n	PRON
ejpam-6713	330	27	5	5	NUM
ejpam-6713	330	28	}	}	PUNCT
ejpam-6713	330	29	and	and	CCONJ
ejpam-6713	330	30	d′′	d′′	NOUN
ejpam-6713	330	31	=	=	PUNCT
ejpam-6713	330	32	{	{	PUNCT
ejpam-6713	330	33	u5k−1	u5k−1	INTJ
ejpam-6713	330	34	:	:	PUNCT
ejpam-6713	330	35	1	1	NUM
ejpam-6713	330	36	≤	≤	NUM
ejpam-6713	330	37	k	k	X
ejpam-6713	330	38	≤	≤	NUM
ejpam-6713	330	39	n	n	PRON
ejpam-6713	330	40	5	5	NUM
ejpam-6713	330	41	}	}	PUNCT
ejpam-6713	330	42	and	and	CCONJ
ejpam-6713	330	43	g′	g′	NOUN
ejpam-6713	330	44	be	be	AUX
ejpam-6713	330	45	a	a	DET
ejpam-6713	330	46	graph	graph	NOUN
ejpam-6713	330	47	obtained	obtain	VERB
ejpam-6713	330	48	from	from	ADP
ejpam-6713	330	49	g	g	NOUN
ejpam-6713	330	50	by	by	ADP
ejpam-6713	330	51	subdividing	subdivide	VERB
ejpam-6713	330	52	an	an	DET
ejpam-6713	330	53	edge	edge	NOUN
ejpam-6713	330	54	e	e	NOUN
ejpam-6713	330	55	=	=	NOUN
ejpam-6713	330	56	u1u2	u1u2	PUNCT
ejpam-6713	330	57	by	by	ADP
ejpam-6713	330	58	a	a	DET
ejpam-6713	330	59	subdivision	subdivision	NOUN
ejpam-6713	330	60	vertex	vertex	NOUN
ejpam-6713	330	61	x	x	NOUN
ejpam-6713	330	62	,	,	PUNCT
ejpam-6713	330	63	where	where	SCONJ
ejpam-6713	330	64	x	x	X
ejpam-6713	330	65	/∈	/∈	PUNCT
ejpam-6713	330	66	n(v	n(v	PROPN
ejpam-6713	330	67	)	)	PUNCT
ejpam-6713	330	68	for	for	ADP
ejpam-6713	330	69	all	all	DET
ejpam-6713	330	70	v	v	NOUN
ejpam-6713	330	71	∈	∈	PRON
ejpam-6713	330	72	d.	d.	NOUN
ejpam-6713	330	73	now	now	ADV
ejpam-6713	330	74	d1	d1	PROPN
ejpam-6713	330	75	=	=	PUNCT
ejpam-6713	331	1	d	d	X
ejpam-6713	331	2	∪	∪	X
ejpam-6713	331	3	{	{	PUNCT
ejpam-6713	331	4	x	x	NOUN
ejpam-6713	331	5	}	}	PUNCT
ejpam-6713	331	6	is	be	AUX
ejpam-6713	331	7	the	the	DET
ejpam-6713	331	8	γcer	γcer	NOUN
ejpam-6713	331	9	-	-	PUNCT
ejpam-6713	331	10	set	set	NOUN
ejpam-6713	331	11	of	of	ADP
ejpam-6713	331	12	g′.	g′.	X
ejpam-6713	331	13	we	we	PRON
ejpam-6713	331	14	clearly	clearly	ADV
ejpam-6713	331	15	see	see	VERB
ejpam-6713	331	16	that	that	SCONJ
ejpam-6713	331	17	|d1|	|d1|	NOUN
ejpam-6713	331	18	>	>	X
ejpam-6713	331	19	|d|	|d|	PROPN
ejpam-6713	331	20	,	,	PUNCT
ejpam-6713	331	21	hence	hence	ADV
ejpam-6713	331	22	γcer(g	γcer(g	PROPN
ejpam-6713	331	23	′	′	NUM
ejpam-6713	331	24	)	)	PUNCT
ejpam-6713	331	25	>	>	X
ejpam-6713	332	1	γcer(g	γcer(g	PROPN
ejpam-6713	332	2	)	)	PUNCT
ejpam-6713	332	3	.	.	PUNCT
ejpam-6713	333	1	therefore	therefore	ADV
ejpam-6713	333	2	,	,	PUNCT
ejpam-6713	333	3	sd+γcer(g	sd+γcer(g	NOUN
ejpam-6713	333	4	)	)	PUNCT
ejpam-6713	333	5	=	=	SYM
ejpam-6713	334	1	1	1	X
ejpam-6713	334	2	.	.	X
ejpam-6713	334	3	case	case	NOUN
ejpam-6713	334	4	(	(	PUNCT
ejpam-6713	334	5	ii	ii	NOUN
ejpam-6713	334	6	)	)	PUNCT
ejpam-6713	334	7	n	n	CCONJ
ejpam-6713	334	8	≡	≡	PROPN
ejpam-6713	334	9	1	1	NUM
ejpam-6713	334	10	(	(	PUNCT
ejpam-6713	334	11	mod	mod	NOUN
ejpam-6713	334	12	5	5	NUM
ejpam-6713	334	13	)	)	PUNCT
ejpam-6713	334	14	let	let	VERB
ejpam-6713	334	15	d′	d′	X
ejpam-6713	334	16	=	=	PUNCT
ejpam-6713	334	17	{	{	PUNCT
ejpam-6713	334	18	v5k−4	v5k−4	NOUN
ejpam-6713	334	19	:	:	PUNCT
ejpam-6713	334	20	1	1	NUM
ejpam-6713	334	21	≤	≤	NUM
ejpam-6713	334	22	k	k	X
ejpam-6713	334	23	≤	≤	PROPN
ejpam-6713	334	24	⌈	⌈	NUM
ejpam-6713	334	25	n	n	CCONJ
ejpam-6713	334	26	5	5	NUM
ejpam-6713	334	27	⌉	⌉	NOUN
ejpam-6713	334	28	}	}	PUNCT
ejpam-6713	334	29	∪	∪	ADJ
ejpam-6713	334	30	{	{	PUNCT
ejpam-6713	334	31	v5k−3	v5k−3	NOUN
ejpam-6713	334	32	:	:	PUNCT
ejpam-6713	334	33	1	1	NUM
ejpam-6713	334	34	≤	≤	NUM
ejpam-6713	334	35	k	k	X
ejpam-6713	334	36	≤	≤	NUM
ejpam-6713	334	37	⌊	⌊	VERB
ejpam-6713	334	38	n	n	ADV
ejpam-6713	334	39	5	5	NUM
ejpam-6713	334	40	⌋	⌋	NOUN
ejpam-6713	334	41	}	}	PUNCT
ejpam-6713	334	42	and	and	CCONJ
ejpam-6713	334	43	d′′	d′′	NOUN
ejpam-6713	334	44	=	=	PUNCT
ejpam-6713	334	45	{	{	PUNCT
ejpam-6713	334	46	u5k−1	u5k−1	INTJ
ejpam-6713	334	47	:	:	PUNCT
ejpam-6713	334	48	1	1	NUM
ejpam-6713	334	49	≤	≤	NUM
ejpam-6713	334	50	k	k	X
ejpam-6713	334	51	≤	≤	NUM
ejpam-6713	334	52	⌊	⌊	VERB
ejpam-6713	334	53	n	n	ADV
ejpam-6713	334	54	5	5	NUM
ejpam-6713	334	55	⌋	⌋	NOUN
ejpam-6713	334	56	}	}	PUNCT
ejpam-6713	334	57	and	and	CCONJ
ejpam-6713	334	58	g′	g′	NOUN
ejpam-6713	334	59	be	be	AUX
ejpam-6713	334	60	a	a	DET
ejpam-6713	334	61	graph	graph	NOUN
ejpam-6713	334	62	obtained	obtain	VERB
ejpam-6713	334	63	from	from	ADP
ejpam-6713	334	64	g	g	NOUN
ejpam-6713	334	65	by	by	ADP
ejpam-6713	334	66	subdividing	subdivide	VERB
ejpam-6713	334	67	an	an	DET
ejpam-6713	334	68	edge	edge	NOUN
ejpam-6713	334	69	e	e	NOUN
ejpam-6713	334	70	=	=	SYM
ejpam-6713	334	71	u3u4	u3u4	X
ejpam-6713	334	72	(	(	PUNCT
ejpam-6713	334	73	or	or	CCONJ
ejpam-6713	334	74	e	e	X
ejpam-6713	334	75	=	=	NOUN
ejpam-6713	334	76	v1v3	v1v3	NOUN
ejpam-6713	334	77	)	)	PUNCT
ejpam-6713	334	78	,	,	PUNCT
ejpam-6713	334	79	(	(	PUNCT
ejpam-6713	334	80	or	or	CCONJ
ejpam-6713	334	81	e	e	X
ejpam-6713	334	82	=	=	PUNCT
ejpam-6713	334	83	u4v4	u4v4	PROPN
ejpam-6713	334	84	)	)	PUNCT
ejpam-6713	334	85	by	by	ADP
ejpam-6713	334	86	a	a	DET
ejpam-6713	334	87	subdivision	subdivision	NOUN
ejpam-6713	334	88	vertex	vertex	NOUN
ejpam-6713	334	89	x	x	NOUN
ejpam-6713	334	90	,	,	PUNCT
ejpam-6713	334	91	here	here	ADV
ejpam-6713	334	92	x	x	PART
ejpam-6713	334	93	∈	∈	PROPN
ejpam-6713	334	94	n(u4	n(u4	PRON
ejpam-6713	334	95	)	)	PUNCT
ejpam-6713	334	96	or	or	CCONJ
ejpam-6713	334	97	n(v1	n(v1	NOUN
ejpam-6713	334	98	)	)	PUNCT
ejpam-6713	334	99	or	or	CCONJ
ejpam-6713	334	100	n(v6	n(v6	NOUN
ejpam-6713	334	101	)	)	PUNCT
ejpam-6713	334	102	respectively	respectively	ADV
ejpam-6713	334	103	.	.	PUNCT
ejpam-6713	335	1	for	for	ADP
ejpam-6713	335	2	the	the	DET
ejpam-6713	335	3	first	first	ADJ
ejpam-6713	335	4	two	two	NUM
ejpam-6713	335	5	category	category	NOUN
ejpam-6713	335	6	in	in	ADP
ejpam-6713	335	7	order	order	NOUN
ejpam-6713	335	8	to	to	PART
ejpam-6713	335	9	dominate	dominate	VERB
ejpam-6713	335	10	u3	u3	NOUN
ejpam-6713	335	11	(	(	PUNCT
ejpam-6713	335	12	or	or	CCONJ
ejpam-6713	335	13	v3	v3	PROPN
ejpam-6713	335	14	)	)	PUNCT
ejpam-6713	335	15	the	the	DET
ejpam-6713	335	16	configuration	configuration	NOUN
ejpam-6713	335	17	of	of	ADP
ejpam-6713	335	18	d	d	PROPN
ejpam-6713	335	19	has	have	VERB
ejpam-6713	335	20	to	to	PART
ejpam-6713	335	21	be	be	AUX
ejpam-6713	335	22	changed	change	VERB
ejpam-6713	335	23	to	to	ADP
ejpam-6713	335	24	d1	d1	PROPN
ejpam-6713	335	25	=	=	PUNCT
ejpam-6713	336	1	d	d	X
ejpam-6713	336	2	−	−	PROPN
ejpam-6713	336	3	{	{	PUNCT
ejpam-6713	336	4	v2	v2	NOUN
ejpam-6713	336	5	}	}	PUNCT
ejpam-6713	336	6	∪	∪	NOUN
ejpam-6713	336	7	{	{	PUNCT
ejpam-6713	336	8	u3	u3	NOUN
ejpam-6713	336	9	}	}	PUNCT
ejpam-6713	336	10	again	again	ADV
ejpam-6713	336	11	|d1|	|d1|	NOUN
ejpam-6713	336	12	=	=	SYM
ejpam-6713	336	13	|d|	|d|	PROPN
ejpam-6713	336	14	.	.	PUNCT
ejpam-6713	337	1	hence	hence	ADV
ejpam-6713	337	2	γcer(g	γcer(g	PROPN
ejpam-6713	337	3	′	′	NOUN
ejpam-6713	337	4	)	)	PUNCT
ejpam-6713	337	5	=	=	SYM
ejpam-6713	337	6	γcer(g	γcer(g	NOUN
ejpam-6713	337	7	)	)	PUNCT
ejpam-6713	337	8	.	.	PUNCT
ejpam-6713	338	1	therefore	therefore	ADV
ejpam-6713	338	2	,	,	PUNCT
ejpam-6713	338	3	sd+γcer(g	sd+γcer(g	NOUN
ejpam-6713	338	4	)	)	PUNCT
ejpam-6713	338	5	>	>	X
ejpam-6713	339	1	1	1	X
ejpam-6713	339	2	.	.	PUNCT
ejpam-6713	339	3	let	let	VERB
ejpam-6713	339	4	g′′	g′′	PROPN
ejpam-6713	339	5	be	be	AUX
ejpam-6713	339	6	a	a	DET
ejpam-6713	339	7	graph	graph	NOUN
ejpam-6713	339	8	obtained	obtain	VERB
ejpam-6713	339	9	from	from	ADP
ejpam-6713	339	10	g	g	NOUN
ejpam-6713	339	11	by	by	ADP
ejpam-6713	339	12	subdividing	subdivide	VERB
ejpam-6713	339	13	the	the	DET
ejpam-6713	339	14	two	two	NUM
ejpam-6713	339	15	edges	edge	NOUN
ejpam-6713	339	16	(	(	PUNCT
ejpam-6713	339	17	say	say	INTJ
ejpam-6713	339	18	)	)	PUNCT
ejpam-6713	339	19	e1	e1	PROPN
ejpam-6713	339	20	=	=	SYM
ejpam-6713	339	21	u3u4	u3u4	PROPN
ejpam-6713	339	22	and	and	CCONJ
ejpam-6713	339	23	e2	e2	PROPN
ejpam-6713	339	24	=	=	SYM
ejpam-6713	339	25	u4u5	u4u5	PROPN
ejpam-6713	339	26	by	by	ADP
ejpam-6713	339	27	a	a	DET
ejpam-6713	339	28	subdivision	subdivision	NOUN
ejpam-6713	339	29	vertices	vertice	VERB
ejpam-6713	339	30	x	x	PUNCT
ejpam-6713	339	31	and	and	CCONJ
ejpam-6713	339	32	y	y	PROPN
ejpam-6713	339	33	respectively	respectively	ADV
ejpam-6713	339	34	.	.	PUNCT
ejpam-6713	340	1	here	here	ADV
ejpam-6713	340	2	,	,	PUNCT
ejpam-6713	340	3	x	x	PROPN
ejpam-6713	340	4	∈	∈	PROPN
ejpam-6713	340	5	n(u3	n(u3	PROPN
ejpam-6713	340	6	)	)	PUNCT
ejpam-6713	340	7	,	,	PUNCT
ejpam-6713	340	8	y	y	PROPN
ejpam-6713	340	9	∈	∈	PROPN
ejpam-6713	340	10	n(u4	n(u4	PRON
ejpam-6713	340	11	)	)	PUNCT
ejpam-6713	340	12	.	.	PUNCT
ejpam-6713	341	1	u5	u5	PROPN
ejpam-6713	341	2	/∈	/∈	PUNCT
ejpam-6713	342	1	n(v	n(v	PROPN
ejpam-6713	342	2	)	)	PUNCT
ejpam-6713	342	3	.	.	PUNCT
ejpam-6713	343	1	now	now	ADV
ejpam-6713	343	2	d′	d′	PRON
ejpam-6713	343	3	=	=	SYM
ejpam-6713	343	4	d	d	X
ejpam-6713	343	5	∪	∪	X
ejpam-6713	343	6	{	{	PUNCT
ejpam-6713	343	7	u5	u5	NOUN
ejpam-6713	343	8	}	}	PUNCT
ejpam-6713	343	9	is	be	AUX
ejpam-6713	343	10	the	the	DET
ejpam-6713	343	11	γcer	γcer	NOUN
ejpam-6713	343	12	-	-	PUNCT
ejpam-6713	343	13	set	set	NOUN
ejpam-6713	343	14	of	of	ADP
ejpam-6713	343	15	g′′.	g′′.	PROPN
ejpam-6713	343	16	hence	hence	ADV
ejpam-6713	343	17	γcer(g	γcer(g	NUM
ejpam-6713	343	18	′′	′′	PROPN
ejpam-6713	343	19	)	)	PUNCT
ejpam-6713	343	20	>	>	X
ejpam-6713	343	21	γcer(g	γcer(g	PROPN
ejpam-6713	343	22	)	)	PUNCT
ejpam-6713	343	23	.	.	PUNCT
ejpam-6713	344	1	therefore	therefore	ADV
ejpam-6713	344	2	,	,	PUNCT
ejpam-6713	344	3	sd+γcer(g	sd+γcer(g	NOUN
ejpam-6713	344	4	)	)	PUNCT
ejpam-6713	344	5	=	=	SYM
ejpam-6713	344	6	2	2	X
ejpam-6713	344	7	.	.	X
ejpam-6713	344	8	case	case	NOUN
ejpam-6713	344	9	(	(	PUNCT
ejpam-6713	344	10	iii	iii	NOUN
ejpam-6713	344	11	)	)	PUNCT
ejpam-6713	344	12	n	n	CCONJ
ejpam-6713	344	13	≡	≡	PROPN
ejpam-6713	344	14	2	2	NUM
ejpam-6713	344	15	(	(	PUNCT
ejpam-6713	344	16	mod	mod	NOUN
ejpam-6713	344	17	5	5	NUM
ejpam-6713	344	18	)	)	PUNCT
ejpam-6713	344	19	let	let	VERB
ejpam-6713	344	20	d′	d′	X
ejpam-6713	344	21	=	=	PUNCT
ejpam-6713	344	22	{	{	PUNCT
ejpam-6713	344	23	v5k−4	v5k−4	NOUN
ejpam-6713	344	24	:	:	PUNCT
ejpam-6713	344	25	1	1	NUM
ejpam-6713	344	26	≤	≤	NUM
ejpam-6713	344	27	k	k	X
ejpam-6713	344	28	≤	≤	PROPN
ejpam-6713	344	29	⌈	⌈	NUM
ejpam-6713	344	30	n	n	CCONJ
ejpam-6713	344	31	5	5	NUM
ejpam-6713	344	32	⌉	⌉	NOUN
ejpam-6713	344	33	}	}	PUNCT
ejpam-6713	344	34	∪	∪	ADJ
ejpam-6713	344	35	{	{	PUNCT
ejpam-6713	344	36	v5k−3	v5k−3	NOUN
ejpam-6713	344	37	:	:	PUNCT
ejpam-6713	344	38	1	1	NUM
ejpam-6713	344	39	≤	≤	NUM
ejpam-6713	344	40	k	k	X
ejpam-6713	344	41	≤	≤	PROPN
ejpam-6713	344	42	⌈	⌈	NUM
ejpam-6713	344	43	n	n	CCONJ
ejpam-6713	344	44	5	5	NUM
ejpam-6713	344	45	⌉	⌉	NOUN
ejpam-6713	344	46	}	}	PUNCT
ejpam-6713	344	47	and	and	CCONJ
ejpam-6713	344	48	d′′	d′′	NOUN
ejpam-6713	344	49	=	=	PUNCT
ejpam-6713	344	50	{	{	PUNCT
ejpam-6713	344	51	u5k−1	u5k−1	INTJ
ejpam-6713	344	52	:	:	PUNCT
ejpam-6713	344	53	1	1	NUM
ejpam-6713	344	54	≤	≤	NUM
ejpam-6713	344	55	k	k	X
ejpam-6713	344	56	≤	≤	NUM
ejpam-6713	344	57	⌊	⌊	VERB
ejpam-6713	344	58	n	n	ADV
ejpam-6713	344	59	5	5	NUM
ejpam-6713	344	60	⌋	⌋	NOUN
ejpam-6713	344	61	}	}	PUNCT
ejpam-6713	344	62	and	and	CCONJ
ejpam-6713	344	63	g′	g′	NOUN
ejpam-6713	344	64	be	be	AUX
ejpam-6713	344	65	a	a	DET
ejpam-6713	344	66	graph	graph	NOUN
ejpam-6713	344	67	obtained	obtain	VERB
ejpam-6713	344	68	from	from	ADP
ejpam-6713	344	69	g	g	NOUN
ejpam-6713	344	70	by	by	ADP
ejpam-6713	344	71	subdividing	subdivide	VERB
ejpam-6713	344	72	an	an	DET
ejpam-6713	344	73	edge	edge	NOUN
ejpam-6713	344	74	e	e	NOUN
ejpam-6713	345	1	=	=	SYM
ejpam-6713	345	2	u3u4	u3u4	X
ejpam-6713	345	3	(	(	PUNCT
ejpam-6713	345	4	or	or	CCONJ
ejpam-6713	345	5	e	e	X
ejpam-6713	345	6	=	=	SYM
ejpam-6713	345	7	u4v4	u4v4	PROPN
ejpam-6713	345	8	,	,	PUNCT
ejpam-6713	345	9	or	or	CCONJ
ejpam-6713	345	10	e	e	X
ejpam-6713	345	11	=	=	PUNCT
ejpam-6713	345	12	v2v4	v2v4	NOUN
ejpam-6713	345	13	)	)	PUNCT
ejpam-6713	345	14	by	by	ADP
ejpam-6713	345	15	a	a	DET
ejpam-6713	345	16	subdivision	subdivision	NOUN
ejpam-6713	345	17	vertex	vertex	NOUN
ejpam-6713	345	18	x.	x.	NOUN
ejpam-6713	345	19	here	here	ADV
ejpam-6713	345	20	x	x	X
ejpam-6713	345	21	∈	∈	PROPN
ejpam-6713	345	22	n(u4	n(u4	PRON
ejpam-6713	345	23	)	)	PUNCT
ejpam-6713	345	24	and	and	CCONJ
ejpam-6713	345	25	in	in	ADP
ejpam-6713	345	26	order	order	NOUN
ejpam-6713	345	27	to	to	PART
ejpam-6713	345	28	dominate	dominate	VERB
ejpam-6713	345	29	u3	u3	NOUN
ejpam-6713	345	30	the	the	DET
ejpam-6713	345	31	position	position	NOUN
ejpam-6713	345	32	of	of	ADP
ejpam-6713	345	33	the	the	DET
ejpam-6713	345	34	d	d	NOUN
ejpam-6713	345	35	is	be	AUX
ejpam-6713	345	36	changed	change	VERB
ejpam-6713	345	37	to	to	ADP
ejpam-6713	345	38	d1	d1	NOUN
ejpam-6713	345	39	,	,	PUNCT
ejpam-6713	345	40	where	where	SCONJ
ejpam-6713	345	41	d1	d1	PROPN
ejpam-6713	345	42	=	=	PUNCT
ejpam-6713	346	1	(	(	PUNCT
ejpam-6713	346	2	d	d	X
ejpam-6713	346	3	−	−	PROPN
ejpam-6713	346	4	{	{	PUNCT
ejpam-6713	346	5	v2	v2	NOUN
ejpam-6713	346	6	}	}	PUNCT
ejpam-6713	346	7	)	)	PUNCT
ejpam-6713	346	8	∪	∪	ADP
ejpam-6713	346	9	{	{	PUNCT
ejpam-6713	346	10	u2	u2	NOUN
ejpam-6713	346	11	}	}	PUNCT
ejpam-6713	346	12	,	,	PUNCT
ejpam-6713	346	13	(	(	PUNCT
ejpam-6713	346	14	or	or	CCONJ
ejpam-6713	346	15	x	x	PROPN
ejpam-6713	346	16	∈	∈	PROPN
ejpam-6713	346	17	n(u4	n(u4	PRON
ejpam-6713	346	18	)	)	PUNCT
ejpam-6713	346	19	or	or	CCONJ
ejpam-6713	346	20	x	x	PUNCT
ejpam-6713	346	21	∈	∈	NOUN
ejpam-6713	346	22	n(v2	n(v2	NOUN
ejpam-6713	346	23	)	)	PUNCT
ejpam-6713	346	24	)	)	PUNCT
ejpam-6713	346	25	.	.	PUNCT
ejpam-6713	347	1	hence	hence	ADV
ejpam-6713	347	2	γcer(g	γcer(g	NUM
ejpam-6713	347	3	′	′	NUM
ejpam-6713	347	4	)	)	PUNCT
ejpam-6713	347	5	=	=	SYM
ejpam-6713	347	6	γcer(g	γcer(g	NOUN
ejpam-6713	347	7	)	)	PUNCT
ejpam-6713	347	8	.	.	PUNCT
ejpam-6713	348	1	therefore	therefore	ADV
ejpam-6713	348	2	,	,	PUNCT
ejpam-6713	348	3	sd+γcer(g	sd+γcer(g	NOUN
ejpam-6713	348	4	)	)	PUNCT
ejpam-6713	348	5	>	>	X
ejpam-6713	349	1	1	1	X
ejpam-6713	349	2	.	.	PUNCT
ejpam-6713	349	3	let	let	VERB
ejpam-6713	349	4	g′′	g′′	PROPN
ejpam-6713	349	5	be	be	AUX
ejpam-6713	349	6	a	a	DET
ejpam-6713	349	7	graph	graph	NOUN
ejpam-6713	349	8	obtained	obtain	VERB
ejpam-6713	349	9	from	from	ADP
ejpam-6713	349	10	g	g	NOUN
ejpam-6713	349	11	by	by	ADP
ejpam-6713	349	12	subdividing	subdivide	VERB
ejpam-6713	349	13	two	two	NUM
ejpam-6713	349	14	edges	edge	NOUN
ejpam-6713	349	15	e1	e1	NOUN
ejpam-6713	349	16	=	=	SYM
ejpam-6713	349	17	u1v1	u1v1	PROPN
ejpam-6713	349	18	and	and	CCONJ
ejpam-6713	349	19	e2	e2	PROPN
ejpam-6713	349	20	=	=	SYM
ejpam-6713	349	21	unvn	unvn	ADJ
ejpam-6713	349	22	by	by	ADP
ejpam-6713	349	23	a	a	DET
ejpam-6713	349	24	subdivision	subdivision	NOUN
ejpam-6713	349	25	vertices	vertice	VERB
ejpam-6713	349	26	x	x	PUNCT
ejpam-6713	349	27	and	and	CCONJ
ejpam-6713	349	28	y	y	PROPN
ejpam-6713	349	29	respectively	respectively	ADV
ejpam-6713	349	30	.	.	PUNCT
ejpam-6713	350	1	here	here	ADV
ejpam-6713	350	2	x	x	PUNCT
ejpam-6713	350	3	∈	∈	NOUN
ejpam-6713	350	4	n(v1	n(v1	NOUN
ejpam-6713	350	5	)	)	PUNCT
ejpam-6713	350	6	and	and	CCONJ
ejpam-6713	350	7	y	y	PROPN
ejpam-6713	350	8	∈	∈	PROPN
ejpam-6713	350	9	n(vn	n(vn	PROPN
ejpam-6713	350	10	)	)	PUNCT
ejpam-6713	350	11	.	.	PUNCT
ejpam-6713	351	1	now	now	ADV
ejpam-6713	351	2	u1	u1	PROPN
ejpam-6713	351	3	,	,	PUNCT
ejpam-6713	351	4	un	un	PROPN
ejpam-6713	351	5	/∈	/∈	PROPN
ejpam-6713	351	6	n	n	PROPN
ejpam-6713	352	1	[	[	X
ejpam-6713	352	2	d1	d1	NOUN
ejpam-6713	352	3	]	]	PUNCT
ejpam-6713	352	4	.	.	PUNCT
ejpam-6713	353	1	now	now	ADV
ejpam-6713	353	2	d2	d2	PROPN
ejpam-6713	353	3	=	=	SYM
ejpam-6713	353	4	d1	d1	PROPN
ejpam-6713	353	5	∪	∪	ADV
ejpam-6713	353	6	{	{	PUNCT
ejpam-6713	353	7	un	un	PROPN
ejpam-6713	353	8	}	}	PUNCT
ejpam-6713	353	9	.	.	PUNCT
ejpam-6713	354	1	hence	hence	ADV
ejpam-6713	354	2	γcer(g	γcer(g	PROPN
ejpam-6713	354	3	′′	′′	PROPN
ejpam-6713	354	4	)	)	PUNCT
ejpam-6713	354	5	>	>	X
ejpam-6713	355	1	γcer(g	γcer(g	PROPN
ejpam-6713	355	2	)	)	PUNCT
ejpam-6713	355	3	.	.	PUNCT
ejpam-6713	356	1	therefore	therefore	ADV
ejpam-6713	356	2	,	,	PUNCT
ejpam-6713	356	3	sd+γcer(g	sd+γcer(g	NOUN
ejpam-6713	356	4	)	)	PUNCT
ejpam-6713	356	5	=	=	SYM
ejpam-6713	356	6	2	2	X
ejpam-6713	356	7	.	.	X
ejpam-6713	356	8	case	case	NOUN
ejpam-6713	356	9	(	(	PUNCT
ejpam-6713	356	10	iv	iv	X
ejpam-6713	356	11	)	)	PUNCT
ejpam-6713	356	12	n	n	CCONJ
ejpam-6713	356	13	≡	≡	PROPN
ejpam-6713	356	14	4	4	NUM
ejpam-6713	356	15	(	(	PUNCT
ejpam-6713	356	16	mod	mod	NOUN
ejpam-6713	356	17	5	5	NUM
ejpam-6713	356	18	)	)	PUNCT
ejpam-6713	356	19	let	let	VERB
ejpam-6713	356	20	d′	d′	X
ejpam-6713	356	21	=	=	PUNCT
ejpam-6713	356	22	{	{	PUNCT
ejpam-6713	356	23	v5k−4	v5k−4	NOUN
ejpam-6713	356	24	:	:	PUNCT
ejpam-6713	356	25	1	1	NUM
ejpam-6713	356	26	≤	≤	NUM
ejpam-6713	356	27	k	k	X
ejpam-6713	356	28	≤	≤	PROPN
ejpam-6713	356	29	⌈	⌈	NUM
ejpam-6713	356	30	n	n	CCONJ
ejpam-6713	356	31	5	5	NUM
ejpam-6713	356	32	⌉	⌉	NOUN
ejpam-6713	356	33	}	}	PUNCT
ejpam-6713	356	34	∪	∪	ADJ
ejpam-6713	356	35	{	{	PUNCT
ejpam-6713	356	36	v5k−3	v5k−3	NOUN
ejpam-6713	356	37	:	:	PUNCT
ejpam-6713	356	38	1	1	NUM
ejpam-6713	356	39	≤	≤	NUM
ejpam-6713	356	40	k	k	X
ejpam-6713	356	41	≤	≤	PROPN
ejpam-6713	356	42	⌈	⌈	NUM
ejpam-6713	356	43	n	n	CCONJ
ejpam-6713	356	44	5	5	NUM
ejpam-6713	356	45	⌉	⌉	NOUN
ejpam-6713	356	46	}	}	PUNCT
ejpam-6713	356	47	and	and	CCONJ
ejpam-6713	356	48	d′′	d′′	NOUN
ejpam-6713	356	49	=	=	PUNCT
ejpam-6713	356	50	{	{	PUNCT
ejpam-6713	356	51	u5k−1	u5k−1	INTJ
ejpam-6713	356	52	:	:	PUNCT
ejpam-6713	356	53	1	1	NUM
ejpam-6713	356	54	≤	≤	NUM
ejpam-6713	356	55	k	k	X
ejpam-6713	356	56	≤	≤	PROPN
ejpam-6713	356	57	⌈	⌈	NUM
ejpam-6713	356	58	n	n	CCONJ
ejpam-6713	356	59	5	5	NUM
ejpam-6713	356	60	⌉	⌉	NOUN
ejpam-6713	356	61	}	}	PUNCT
ejpam-6713	356	62	and	and	CCONJ
ejpam-6713	356	63	g′	g′	NOUN
ejpam-6713	356	64	be	be	AUX
ejpam-6713	356	65	a	a	DET
ejpam-6713	356	66	graph	graph	NOUN
ejpam-6713	356	67	obtained	obtain	VERB
ejpam-6713	356	68	from	from	ADP
ejpam-6713	356	69	g	g	NOUN
ejpam-6713	356	70	by	by	ADP
ejpam-6713	356	71	subdividing	subdivide	VERB
ejpam-6713	356	72	an	an	DET
ejpam-6713	356	73	edge	edge	NOUN
ejpam-6713	356	74	e	e	NOUN
ejpam-6713	356	75	=	=	PUNCT
ejpam-6713	356	76	u3u4	u3u4	X
ejpam-6713	357	1	[	[	X
ejpam-6713	357	2	or	or	CCONJ
ejpam-6713	357	3	e	e	X
ejpam-6713	357	4	=	=	SYM
ejpam-6713	357	5	u4v4	u4v4	PROPN
ejpam-6713	357	6	,	,	PUNCT
ejpam-6713	357	7	or	or	CCONJ
ejpam-6713	357	8	e	e	NOUN
ejpam-6713	357	9	=	=	PUNCT
ejpam-6713	357	10	v2v4	v2v4	PROPN
ejpam-6713	357	11	]	]	PUNCT
ejpam-6713	357	12	by	by	ADP
ejpam-6713	357	13	a	a	DET
ejpam-6713	357	14	subdivision	subdivision	NOUN
ejpam-6713	357	15	vertex	vertex	NOUN
ejpam-6713	357	16	x.	x.	NOUN
ejpam-6713	357	17	here	here	ADV
ejpam-6713	357	18	,	,	PUNCT
ejpam-6713	357	19	x	x	PROPN
ejpam-6713	357	20	∈	∈	PROPN
ejpam-6713	357	21	n(u4	n(u4	PRON
ejpam-6713	357	22	)	)	PUNCT
ejpam-6713	357	23	and	and	CCONJ
ejpam-6713	357	24	in	in	ADP
ejpam-6713	357	25	order	order	NOUN
ejpam-6713	357	26	to	to	PART
ejpam-6713	357	27	dominate	dominate	VERB
ejpam-6713	357	28	u3	u3	NOUN
ejpam-6713	357	29	the	the	DET
ejpam-6713	357	30	position	position	NOUN
ejpam-6713	357	31	of	of	ADP
ejpam-6713	357	32	the	the	DET
ejpam-6713	357	33	d	d	NOUN
ejpam-6713	357	34	is	be	AUX
ejpam-6713	357	35	changed	change	VERB
ejpam-6713	357	36	to	to	ADP
ejpam-6713	357	37	d1	d1	NOUN
ejpam-6713	357	38	,	,	PUNCT
ejpam-6713	357	39	where	where	SCONJ
ejpam-6713	357	40	d1	d1	PROPN
ejpam-6713	357	41	=	=	PUNCT
ejpam-6713	358	1	(	(	PUNCT
ejpam-6713	358	2	d	d	X
ejpam-6713	358	3	−	−	PROPN
ejpam-6713	358	4	{	{	PUNCT
ejpam-6713	358	5	v2	v2	NOUN
ejpam-6713	358	6	}	}	PUNCT
ejpam-6713	358	7	)	)	PUNCT
ejpam-6713	358	8	∪	∪	ADP
ejpam-6713	358	9	{	{	PUNCT
ejpam-6713	358	10	u2	u2	NOUN
ejpam-6713	358	11	}	}	PUNCT
ejpam-6713	358	12	,	,	PUNCT
ejpam-6713	358	13	[	[	PUNCT
ejpam-6713	358	14	or	or	CCONJ
ejpam-6713	358	15	x	x	PROPN
ejpam-6713	358	16	∈	∈	PROPN
ejpam-6713	358	17	n(u4	n(u4	PRON
ejpam-6713	358	18	)	)	PUNCT
ejpam-6713	358	19	or	or	CCONJ
ejpam-6713	358	20	x	x	PUNCT
ejpam-6713	358	21	∈	∈	NOUN
ejpam-6713	358	22	n(v2	n(v2	NOUN
ejpam-6713	358	23	)	)	PUNCT
ejpam-6713	358	24	]	]	PUNCT
ejpam-6713	358	25	.	.	PUNCT
ejpam-6713	359	1	hence	hence	ADV
ejpam-6713	359	2	γcer(g	γcer(g	NUM
ejpam-6713	359	3	′	′	NUM
ejpam-6713	359	4	)	)	PUNCT
ejpam-6713	359	5	=	=	SYM
ejpam-6713	359	6	γcer(g	γcer(g	NOUN
ejpam-6713	359	7	)	)	PUNCT
ejpam-6713	359	8	.	.	PUNCT
ejpam-6713	360	1	therefore	therefore	ADV
ejpam-6713	360	2	sd+γcer(g	sd+γcer(g	NOUN
ejpam-6713	360	3	)	)	PUNCT
ejpam-6713	360	4	>	>	X
ejpam-6713	361	1	1	1	X
ejpam-6713	361	2	.	.	PUNCT
ejpam-6713	361	3	let	let	VERB
ejpam-6713	361	4	g′′	g′′	PROPN
ejpam-6713	361	5	be	be	AUX
ejpam-6713	361	6	a	a	DET
ejpam-6713	361	7	graph	graph	NOUN
ejpam-6713	361	8	g.	g.	NOUN
ejpam-6713	361	9	navamani	navamani	PROPN
ejpam-6713	361	10	et	et	PROPN
ejpam-6713	361	11	al	al	PROPN
ejpam-6713	361	12	.	.	PUNCT
ejpam-6713	361	13	/	/	SYM
ejpam-6713	361	14	eur	eur	PROPN
ejpam-6713	361	15	.	.	PUNCT
ejpam-6713	362	1	j.	j.	PROPN
ejpam-6713	362	2	pure	pure	PROPN
ejpam-6713	362	3	appl	appl	PROPN
ejpam-6713	362	4	.	.	PROPN
ejpam-6713	362	5	math	math	PROPN
ejpam-6713	362	6	,	,	PUNCT
ejpam-6713	362	7	18	18	NUM
ejpam-6713	362	8	(	(	PUNCT
ejpam-6713	362	9	4	4	NUM
ejpam-6713	362	10	)	)	PUNCT
ejpam-6713	362	11	(	(	PUNCT
ejpam-6713	362	12	2025	2025	NUM
ejpam-6713	362	13	)	)	PUNCT
ejpam-6713	362	14	,	,	PUNCT
ejpam-6713	362	15	6713	6713	NUM
ejpam-6713	362	16	13	13	NUM
ejpam-6713	362	17	of	of	ADP
ejpam-6713	362	18	15	15	NUM
ejpam-6713	362	19	obtained	obtain	VERB
ejpam-6713	362	20	from	from	ADP
ejpam-6713	362	21	g	g	NOUN
ejpam-6713	362	22	by	by	ADP
ejpam-6713	362	23	subdividing	subdivide	VERB
ejpam-6713	362	24	two	two	NUM
ejpam-6713	362	25	edges	edge	NOUN
ejpam-6713	362	26	e1	e1	NOUN
ejpam-6713	362	27	=	=	SYM
ejpam-6713	362	28	u2u3	u2u3	PROPN
ejpam-6713	362	29	,	,	PUNCT
ejpam-6713	362	30	e2	e2	PROPN
ejpam-6713	362	31	=	=	PUNCT
ejpam-6713	363	1	u3u4	u3u4	X
ejpam-6713	363	2	by	by	ADP
ejpam-6713	363	3	a	a	DET
ejpam-6713	363	4	subdivision	subdivision	NOUN
ejpam-6713	363	5	vertices	vertice	VERB
ejpam-6713	363	6	x	x	PUNCT
ejpam-6713	363	7	and	and	CCONJ
ejpam-6713	363	8	y	y	PROPN
ejpam-6713	363	9	respectively	respectively	ADV
ejpam-6713	363	10	.	.	PUNCT
ejpam-6713	364	1	here	here	ADV
ejpam-6713	364	2	,	,	PUNCT
ejpam-6713	364	3	x	x	PROPN
ejpam-6713	364	4	∈	∈	PROPN
ejpam-6713	364	5	n(u2	n(u2	PROPN
ejpam-6713	364	6	)	)	PUNCT
ejpam-6713	364	7	and	and	CCONJ
ejpam-6713	364	8	y	y	PROPN
ejpam-6713	364	9	∈	∈	PROPN
ejpam-6713	364	10	n(u4	n(u4	PRON
ejpam-6713	364	11	)	)	PUNCT
ejpam-6713	364	12	.	.	PUNCT
ejpam-6713	365	1	now	now	ADV
ejpam-6713	365	2	u3	u3	PROPN
ejpam-6713	365	3	/∈	/∈	PUNCT
ejpam-6713	365	4	n	n	CCONJ
ejpam-6713	365	5	[	[	X
ejpam-6713	365	6	d1	d1	NOUN
ejpam-6713	365	7	]	]	PUNCT
ejpam-6713	365	8	.	.	PUNCT
ejpam-6713	366	1	now	now	ADV
ejpam-6713	366	2	d2	d2	PROPN
ejpam-6713	366	3	=	=	SYM
ejpam-6713	366	4	d1	d1	PROPN
ejpam-6713	366	5	∪	∪	ADP
ejpam-6713	366	6	{	{	PUNCT
ejpam-6713	366	7	u3	u3	NOUN
ejpam-6713	366	8	}	}	PUNCT
ejpam-6713	366	9	is	be	AUX
ejpam-6713	366	10	a	a	DET
ejpam-6713	366	11	γcer	γcer	NOUN
ejpam-6713	366	12	-	-	PUNCT
ejpam-6713	366	13	set	set	NOUN
ejpam-6713	366	14	of	of	ADP
ejpam-6713	366	15	g′.	g′.	NOUN
ejpam-6713	366	16	hence	hence	ADV
ejpam-6713	366	17	γcer(g	γcer(g	NUM
ejpam-6713	366	18	′′	′′	PROPN
ejpam-6713	366	19	)	)	PUNCT
ejpam-6713	366	20	>	>	X
ejpam-6713	366	21	γcer(g	γcer(g	PROPN
ejpam-6713	366	22	)	)	PUNCT
ejpam-6713	366	23	.	.	PUNCT
ejpam-6713	367	1	therefore	therefore	ADV
ejpam-6713	367	2	,	,	PUNCT
ejpam-6713	367	3	sd+γcer(g	sd+γcer(g	NOUN
ejpam-6713	367	4	)	)	PUNCT
ejpam-6713	367	5	=	=	SYM
ejpam-6713	367	6	2	2	X
ejpam-6713	367	7	.	.	X
ejpam-6713	367	8	case	case	NOUN
ejpam-6713	367	9	(	(	PUNCT
ejpam-6713	367	10	v	v	NOUN
ejpam-6713	367	11	)	)	PUNCT
ejpam-6713	367	12	n	n	CCONJ
ejpam-6713	367	13	≡	≡	PROPN
ejpam-6713	367	14	3	3	NUM
ejpam-6713	367	15	(	(	PUNCT
ejpam-6713	367	16	mod	mod	NOUN
ejpam-6713	367	17	5	5	NUM
ejpam-6713	367	18	)	)	PUNCT
ejpam-6713	367	19	let	let	VERB
ejpam-6713	367	20	d	d	PRON
ejpam-6713	367	21	be	be	AUX
ejpam-6713	367	22	a	a	DET
ejpam-6713	367	23	γcer	γcer	NOUN
ejpam-6713	367	24	-	-	PUNCT
ejpam-6713	367	25	set	set	NOUN
ejpam-6713	367	26	of	of	ADP
ejpam-6713	367	27	g.	g.	PROPN
ejpam-6713	367	28	d	d	PROPN
ejpam-6713	367	29	=	=	PUNCT
ejpam-6713	367	30	d′∪d′′	d′∪d′′	PROPN
ejpam-6713	367	31	and	and	CCONJ
ejpam-6713	367	32	|d|	|d|	PROPN
ejpam-6713	367	33	=	=	PUNCT
ejpam-6713	368	1	⌈	⌈	NOUN
ejpam-6713	368	2	3n	3n	NUM
ejpam-6713	368	3	5	5	NUM
ejpam-6713	368	4	⌉	⌉	NOUN
ejpam-6713	368	5	,	,	PUNCT
ejpam-6713	368	6	where	where	SCONJ
ejpam-6713	368	7	d′	d′	PRON
ejpam-6713	368	8	=	=	SYM
ejpam-6713	368	9	{	{	PUNCT
ejpam-6713	368	10	v5k−4	v5k−4	PROPN
ejpam-6713	368	11	,	,	PUNCT
ejpam-6713	368	12	1	1	NUM
ejpam-6713	368	13	≤	≤	NUM
ejpam-6713	368	14	k	k	X
ejpam-6713	368	15	≤	≤	PROPN
ejpam-6713	368	16	⌈	⌈	NUM
ejpam-6713	368	17	n	n	CCONJ
ejpam-6713	368	18	5	5	NUM
ejpam-6713	368	19	⌉	⌉	PRON
ejpam-6713	368	20	}	}	PUNCT
ejpam-6713	368	21	figure	figure	VERB
ejpam-6713	368	22	4	4	NUM
ejpam-6713	368	23	:	:	PUNCT
ejpam-6713	368	24	a	a	DET
ejpam-6713	368	25	graph	graph	NOUN
ejpam-6713	368	26	illustrating	illustrate	VERB
ejpam-6713	368	27	case	case	NOUN
ejpam-6713	368	28	(	(	PUNCT
ejpam-6713	368	29	v	v	NOUN
ejpam-6713	368	30	)	)	PUNCT
ejpam-6713	368	31	of	of	ADP
ejpam-6713	368	32	theorem	theorem	ADJ
ejpam-6713	368	33	13	13	NUM
ejpam-6713	368	34	,	,	PUNCT
ejpam-6713	368	35	sd+γcer	sd+γcer	NOUN
ejpam-6713	368	36	(	(	PUNCT
ejpam-6713	368	37	g	g	NOUN
ejpam-6713	368	38	)	)	PUNCT
ejpam-6713	368	39	=	=	SYM
ejpam-6713	368	40	1	1	NUM
ejpam-6713	368	41	and	and	CCONJ
ejpam-6713	368	42	d′′	d′′	NOUN
ejpam-6713	368	43	=	=	SYM
ejpam-6713	368	44	{	{	PUNCT
ejpam-6713	368	45	u5i−3	u5i−3	PROPN
ejpam-6713	368	46	,	,	PUNCT
ejpam-6713	368	47	1	1	NUM
ejpam-6713	368	48	≤	≤	NUM
ejpam-6713	368	49	i	i	PRON
ejpam-6713	368	50	≤	≤	NOUN
ejpam-6713	368	51	⌈	⌈	NUM
ejpam-6713	368	52	n	n	CCONJ
ejpam-6713	368	53	5	5	NUM
ejpam-6713	368	54	⌉	⌉	NOUN
ejpam-6713	368	55	}	}	PUNCT
ejpam-6713	368	56	∪	∪	X
ejpam-6713	368	57	{	{	PUNCT
ejpam-6713	368	58	u5j	u5j	NOUN
ejpam-6713	368	59	,	,	PUNCT
ejpam-6713	368	60	1	1	NUM
ejpam-6713	368	61	≤	≤	NUM
ejpam-6713	368	62	j	j	PROPN
ejpam-6713	368	63	≤	≤	NUM
ejpam-6713	368	64	⌊	⌊	VERB
ejpam-6713	368	65	n	n	ADV
ejpam-6713	368	66	5	5	NUM
ejpam-6713	368	67	⌋	⌋	NOUN
ejpam-6713	368	68	}	}	PUNCT
ejpam-6713	368	69	.	.	PUNCT
ejpam-6713	369	1	let	let	VERB
ejpam-6713	369	2	g′	g′	NOUN
ejpam-6713	369	3	be	be	AUX
ejpam-6713	369	4	a	a	DET
ejpam-6713	369	5	graph	graph	NOUN
ejpam-6713	369	6	obtained	obtain	VERB
ejpam-6713	369	7	from	from	ADP
ejpam-6713	369	8	g	g	NOUN
ejpam-6713	369	9	by	by	ADP
ejpam-6713	369	10	subdividing	subdivide	VERB
ejpam-6713	369	11	an	an	DET
ejpam-6713	369	12	edge	edge	NOUN
ejpam-6713	369	13	e	e	NOUN
ejpam-6713	369	14	=	=	NOUN
ejpam-6713	369	15	u2u3	u2u3	X
ejpam-6713	369	16	by	by	ADP
ejpam-6713	369	17	a	a	DET
ejpam-6713	369	18	subdivision	subdivision	NOUN
ejpam-6713	369	19	vertex	vertex	NOUN
ejpam-6713	369	20	x	x	INTJ
ejpam-6713	369	21	.	.	PUNCT
ejpam-6713	370	1	here	here	ADV
ejpam-6713	370	2	,	,	PUNCT
ejpam-6713	370	3	x	x	PROPN
ejpam-6713	370	4	∈	∈	PROPN
ejpam-6713	370	5	n(u2	n(u2	PROPN
ejpam-6713	370	6	)	)	PUNCT
ejpam-6713	370	7	,	,	PUNCT
ejpam-6713	370	8	u3	u3	NOUN
ejpam-6713	370	9	/∈	/∈	PUNCT
ejpam-6713	370	10	n	n	CCONJ
ejpam-6713	371	1	[	[	X
ejpam-6713	371	2	d	d	X
ejpam-6713	371	3	]	]	PUNCT
ejpam-6713	371	4	.	.	PUNCT
ejpam-6713	372	1	hence	hence	ADV
ejpam-6713	372	2	d1	d1	PROPN
ejpam-6713	372	3	=	=	PUNCT
ejpam-6713	373	1	d	d	X
ejpam-6713	373	2	∪	∪	X
ejpam-6713	373	3	{	{	PUNCT
ejpam-6713	373	4	u3	u3	NOUN
ejpam-6713	373	5	}	}	PUNCT
ejpam-6713	373	6	is	be	AUX
ejpam-6713	373	7	the	the	DET
ejpam-6713	373	8	γcer	γcer	NOUN
ejpam-6713	373	9	-	-	PUNCT
ejpam-6713	373	10	set	set	NOUN
ejpam-6713	373	11	of	of	ADP
ejpam-6713	373	12	g′.	g′.	NOUN
ejpam-6713	373	13	hence	hence	ADV
ejpam-6713	373	14	γcer(g	γcer(g	NUM
ejpam-6713	373	15	′	′	NUM
ejpam-6713	373	16	)	)	PUNCT
ejpam-6713	373	17	>	>	X
ejpam-6713	374	1	γcer(g	γcer(g	PROPN
ejpam-6713	374	2	)	)	PUNCT
ejpam-6713	374	3	.	.	PUNCT
ejpam-6713	375	1	therefore	therefore	ADV
ejpam-6713	375	2	,	,	PUNCT
ejpam-6713	375	3	sd+γcer(g	sd+γcer(g	NOUN
ejpam-6713	375	4	)	)	PUNCT
ejpam-6713	375	5	=	=	SYM
ejpam-6713	376	1	1	1	X
ejpam-6713	376	2	.	.	PUNCT
ejpam-6713	376	3	acknowledgements	acknowledgement	NOUN
ejpam-6713	376	4	the	the	DET
ejpam-6713	376	5	authors	author	NOUN
ejpam-6713	376	6	thank	thank	VERB
ejpam-6713	376	7	the	the	DET
ejpam-6713	376	8	readers	reader	NOUN
ejpam-6713	376	9	of	of	ADP
ejpam-6713	376	10	european	european	PROPN
ejpam-6713	376	11	journal	journal	PROPN
ejpam-6713	376	12	of	of	ADP
ejpam-6713	376	13	pure	pure	ADJ
ejpam-6713	376	14	and	and	CCONJ
ejpam-6713	376	15	applied	applied	ADJ
ejpam-6713	376	16	mathematics	mathematic	NOUN
ejpam-6713	376	17	,	,	PUNCT
ejpam-6713	376	18	for	for	ADP
ejpam-6713	376	19	making	make	VERB
ejpam-6713	376	20	our	our	PRON
ejpam-6713	376	21	journal	journal	NOUN
ejpam-6713	376	22	successful	successful	ADJ
ejpam-6713	376	23	.	.	PUNCT
ejpam-6713	377	1	g.	g.	PROPN
ejpam-6713	377	2	navamani	navamani	PROPN
ejpam-6713	377	3	et	et	PROPN
ejpam-6713	377	4	al	al	PROPN
ejpam-6713	377	5	.	.	PUNCT
ejpam-6713	377	6	/	/	SYM
ejpam-6713	377	7	eur	eur	PROPN
ejpam-6713	377	8	.	.	PUNCT
ejpam-6713	378	1	j.	j.	PROPN
ejpam-6713	378	2	pure	pure	PROPN
ejpam-6713	378	3	appl	appl	PROPN
ejpam-6713	378	4	.	.	PROPN
ejpam-6713	378	5	math	math	PROPN
ejpam-6713	378	6	,	,	PUNCT
ejpam-6713	378	7	18	18	NUM
ejpam-6713	378	8	(	(	PUNCT
ejpam-6713	378	9	4	4	NUM
ejpam-6713	378	10	)	)	PUNCT
ejpam-6713	378	11	(	(	PUNCT
ejpam-6713	378	12	2025	2025	NUM
ejpam-6713	378	13	)	)	PUNCT
ejpam-6713	378	14	,	,	PUNCT
ejpam-6713	378	15	6713	6713	NUM
ejpam-6713	378	16	14	14	NUM
ejpam-6713	378	17	of	of	ADP
ejpam-6713	378	18	15	15	NUM
ejpam-6713	378	19	6	6	NUM
ejpam-6713	378	20	.	.	PUNCT
ejpam-6713	379	1	conclusion	conclusion	NOUN
ejpam-6713	379	2	in	in	ADP
ejpam-6713	379	3	conclusion	conclusion	NOUN
ejpam-6713	379	4	,	,	PUNCT
ejpam-6713	379	5	this	this	DET
ejpam-6713	379	6	study	study	NOUN
ejpam-6713	379	7	establishes	establish	VERB
ejpam-6713	379	8	the	the	DET
ejpam-6713	379	9	certified	certify	VERB
ejpam-6713	379	10	domination	domination	NOUN
ejpam-6713	379	11	subdivision	subdivision	NOUN
ejpam-6713	379	12	number	number	NOUN
ejpam-6713	379	13	as	as	ADP
ejpam-6713	379	14	a	a	DET
ejpam-6713	379	15	measure	measure	NOUN
ejpam-6713	379	16	of	of	ADP
ejpam-6713	379	17	how	how	SCONJ
ejpam-6713	379	18	the	the	DET
ejpam-6713	379	19	certified	certified	ADJ
ejpam-6713	379	20	domination	domination	NOUN
ejpam-6713	379	21	number	number	NOUN
ejpam-6713	379	22	of	of	ADP
ejpam-6713	379	23	a	a	DET
ejpam-6713	379	24	graph	graph	NOUN
ejpam-6713	379	25	responds	respond	VERB
ejpam-6713	379	26	to	to	PART
ejpam-6713	379	27	edge	edge	VERB
ejpam-6713	379	28	subdivisions	subdivision	NOUN
ejpam-6713	379	29	,	,	PUNCT
ejpam-6713	379	30	revealing	reveal	VERB
ejpam-6713	379	31	distinct	distinct	ADJ
ejpam-6713	379	32	behaviors	behavior	NOUN
ejpam-6713	379	33	across	across	ADP
ejpam-6713	379	34	different	different	ADJ
ejpam-6713	379	35	graph	graph	NOUN
ejpam-6713	379	36	classes	class	NOUN
ejpam-6713	379	37	.	.	PUNCT
ejpam-6713	380	1	for	for	ADP
ejpam-6713	380	2	circulant	circulant	ADJ
ejpam-6713	380	3	graphs	graph	NOUN
ejpam-6713	380	4	cn(1	cn(1	PROPN
ejpam-6713	380	5	,	,	PUNCT
ejpam-6713	380	6	2	2	NUM
ejpam-6713	380	7	)	)	PUNCT
ejpam-6713	380	8	and	and	CCONJ
ejpam-6713	380	9	cn(1	cn(1	PROPN
ejpam-6713	380	10	,	,	PUNCT
ejpam-6713	380	11	3	3	NUM
ejpam-6713	380	12	)	)	PUNCT
ejpam-6713	380	13	,	,	PUNCT
ejpam-6713	380	14	as	as	ADV
ejpam-6713	380	15	well	well	ADV
ejpam-6713	380	16	as	as	ADP
ejpam-6713	380	17	petersen	petersen	NOUN
ejpam-6713	380	18	graphs	graph	NOUN
ejpam-6713	380	19	p	p	X
ejpam-6713	380	20	(	(	PUNCT
ejpam-6713	380	21	n	n	CCONJ
ejpam-6713	380	22	,	,	PUNCT
ejpam-6713	380	23	1	1	NUM
ejpam-6713	380	24	)	)	PUNCT
ejpam-6713	380	25	and	and	CCONJ
ejpam-6713	380	26	p	p	X
ejpam-6713	380	27	(	(	PUNCT
ejpam-6713	380	28	n	n	CCONJ
ejpam-6713	380	29	,	,	PUNCT
ejpam-6713	380	30	2	2	NUM
ejpam-6713	380	31	)	)	PUNCT
ejpam-6713	380	32	,	,	PUNCT
ejpam-6713	380	33	we	we	PRON
ejpam-6713	380	34	identify	identify	VERB
ejpam-6713	380	35	the	the	DET
ejpam-6713	380	36	precise	precise	ADJ
ejpam-6713	380	37	number	number	NOUN
ejpam-6713	380	38	of	of	ADP
ejpam-6713	380	39	subdivisions	subdivision	NOUN
ejpam-6713	380	40	required	require	VERB
ejpam-6713	380	41	to	to	PART
ejpam-6713	380	42	alter	alter	VERB
ejpam-6713	380	43	the	the	DET
ejpam-6713	380	44	parameter	parameter	NOUN
ejpam-6713	380	45	,	,	PUNCT
ejpam-6713	380	46	thereby	thereby	ADV
ejpam-6713	380	47	offering	offer	VERB
ejpam-6713	380	48	new	new	ADJ
ejpam-6713	380	49	insights	insight	NOUN
ejpam-6713	380	50	into	into	ADP
ejpam-6713	380	51	structural	structural	ADJ
ejpam-6713	380	52	properties	property	NOUN
ejpam-6713	380	53	of	of	ADP
ejpam-6713	380	54	these	these	DET
ejpam-6713	380	55	graphs	graph	NOUN
ejpam-6713	380	56	with	with	ADP
ejpam-6713	380	57	potential	potential	ADJ
ejpam-6713	380	58	applications	application	NOUN
ejpam-6713	380	59	in	in	ADP
ejpam-6713	380	60	network	network	NOUN
ejpam-6713	380	61	design	design	NOUN
ejpam-6713	380	62	and	and	CCONJ
ejpam-6713	380	63	optimization	optimization	NOUN
ejpam-6713	380	64	.	.	PUNCT
ejpam-6713	381	1	the	the	DET
ejpam-6713	381	2	findings	finding	NOUN
ejpam-6713	381	3	further	far	ADV
ejpam-6713	381	4	suggest	suggest	VERB
ejpam-6713	381	5	that	that	SCONJ
ejpam-6713	381	6	strategic	strategic	ADJ
ejpam-6713	381	7	edge	edge	NOUN
ejpam-6713	381	8	subdivisions	subdivision	NOUN
ejpam-6713	381	9	can	can	AUX
ejpam-6713	381	10	reduce	reduce	VERB
ejpam-6713	381	11	the	the	DET
ejpam-6713	381	12	number	number	NOUN
ejpam-6713	381	13	of	of	ADP
ejpam-6713	381	14	facility	facility	NOUN
ejpam-6713	381	15	centers	center	NOUN
ejpam-6713	381	16	without	without	ADP
ejpam-6713	381	17	compromising	compromise	VERB
ejpam-6713	381	18	service	service	NOUN
ejpam-6713	381	19	quality	quality	NOUN
ejpam-6713	381	20	,	,	PUNCT
ejpam-6713	381	21	highlighting	highlight	VERB
ejpam-6713	381	22	their	their	PRON
ejpam-6713	381	23	relevance	relevance	NOUN
ejpam-6713	381	24	to	to	ADP
ejpam-6713	381	25	urban	urban	ADJ
ejpam-6713	381	26	planning	planning	NOUN
ejpam-6713	381	27	and	and	CCONJ
ejpam-6713	381	28	resource	resource	NOUN
ejpam-6713	381	29	management	management	NOUN
ejpam-6713	381	30	.	.	PUNCT
ejpam-6713	382	1	beyond	beyond	ADP
ejpam-6713	382	2	these	these	DET
ejpam-6713	382	3	results	result	NOUN
ejpam-6713	382	4	,	,	PUNCT
ejpam-6713	382	5	the	the	DET
ejpam-6713	382	6	concept	concept	NOUN
ejpam-6713	382	7	of	of	ADP
ejpam-6713	382	8	certified	certified	ADJ
ejpam-6713	382	9	domination	domination	NOUN
ejpam-6713	382	10	subdivision	subdivision	NOUN
ejpam-6713	382	11	numbers	number	NOUN
ejpam-6713	382	12	opens	open	VERB
ejpam-6713	382	13	promising	promise	VERB
ejpam-6713	382	14	avenues	avenue	NOUN
ejpam-6713	382	15	for	for	ADP
ejpam-6713	382	16	future	future	ADJ
ejpam-6713	382	17	research	research	NOUN
ejpam-6713	382	18	,	,	PUNCT
ejpam-6713	382	19	including	include	VERB
ejpam-6713	382	20	extensions	extension	NOUN
ejpam-6713	382	21	to	to	ADP
ejpam-6713	382	22	bipartite	bipartite	VERB
ejpam-6713	382	23	graphs	graph	NOUN
ejpam-6713	382	24	,	,	PUNCT
ejpam-6713	382	25	hypercubes	hypercube	NOUN
ejpam-6713	382	26	,	,	PUNCT
ejpam-6713	382	27	grid	grid	NOUN
ejpam-6713	382	28	graphs	graph	NOUN
ejpam-6713	382	29	,	,	PUNCT
ejpam-6713	382	30	chordal	chordal	NOUN
ejpam-6713	382	31	graphs	graph	NOUN
ejpam-6713	382	32	,	,	PUNCT
ejpam-6713	382	33	and	and	CCONJ
ejpam-6713	382	34	random	random	ADJ
ejpam-6713	382	35	graphs	graph	NOUN
ejpam-6713	382	36	,	,	PUNCT
ejpam-6713	382	37	as	as	ADV
ejpam-6713	382	38	well	well	ADV
ejpam-6713	382	39	as	as	ADP
ejpam-6713	382	40	the	the	DET
ejpam-6713	382	41	development	development	NOUN
ejpam-6713	382	42	of	of	ADP
ejpam-6713	382	43	efficient	efficient	ADJ
ejpam-6713	382	44	algorithms	algorithm	NOUN
ejpam-6713	382	45	for	for	ADP
ejpam-6713	382	46	computing	compute	VERB
ejpam-6713	382	47	sd+γcer(g	sd+γcer(g	NOUN
ejpam-6713	382	48	)	)	PUNCT
ejpam-6713	382	49	and	and	CCONJ
ejpam-6713	382	50	sd−γcer(g	sd−γcer(g	NOUN
ejpam-6713	382	51	)	)	PUNCT
ejpam-6713	382	52	in	in	ADP
ejpam-6713	382	53	large	large	ADJ
ejpam-6713	382	54	-	-	PUNCT
ejpam-6713	382	55	scale	scale	NOUN
ejpam-6713	382	56	networks	network	NOUN
ejpam-6713	382	57	.	.	PUNCT
ejpam-6713	383	1	comparative	comparative	ADJ
ejpam-6713	383	2	analyses	analysis	NOUN
ejpam-6713	383	3	with	with	ADP
ejpam-6713	383	4	related	related	ADJ
ejpam-6713	383	5	parameters	parameter	NOUN
ejpam-6713	383	6	such	such	ADJ
ejpam-6713	383	7	as	as	ADP
ejpam-6713	383	8	bondage	bondage	NOUN
ejpam-6713	383	9	,	,	PUNCT
ejpam-6713	383	10	reinforcement	reinforcement	NOUN
ejpam-6713	383	11	,	,	PUNCT
ejpam-6713	383	12	and	and	CCONJ
ejpam-6713	383	13	classical	classical	ADJ
ejpam-6713	383	14	subdivision	subdivision	NOUN
ejpam-6713	383	15	numbers	number	NOUN
ejpam-6713	383	16	,	,	PUNCT
ejpam-6713	383	17	along	along	ADP
ejpam-6713	383	18	with	with	ADP
ejpam-6713	383	19	empirical	empirical	ADJ
ejpam-6713	383	20	validation	validation	NOUN
ejpam-6713	383	21	on	on	ADP
ejpam-6713	383	22	synthetic	synthetic	ADJ
ejpam-6713	383	23	and	and	CCONJ
ejpam-6713	383	24	real	real	ADJ
ejpam-6713	383	25	-	-	PUNCT
ejpam-6713	383	26	world	world	NOUN
ejpam-6713	383	27	networks	network	NOUN
ejpam-6713	383	28	,	,	PUNCT
ejpam-6713	383	29	may	may	AUX
ejpam-6713	383	30	yield	yield	VERB
ejpam-6713	383	31	deeper	deep	ADJ
ejpam-6713	383	32	theoretical	theoretical	ADJ
ejpam-6713	383	33	insights	insight	NOUN
ejpam-6713	383	34	and	and	CCONJ
ejpam-6713	383	35	strengthen	strengthen	VERB
ejpam-6713	383	36	the	the	DET
ejpam-6713	383	37	practical	practical	ADJ
ejpam-6713	383	38	significance	significance	NOUN
ejpam-6713	383	39	of	of	ADP
ejpam-6713	383	40	these	these	DET
ejpam-6713	383	41	concepts	concept	NOUN
ejpam-6713	383	42	in	in	ADP
ejpam-6713	383	43	dynamic	dynamic	ADJ
ejpam-6713	383	44	and	and	CCONJ
ejpam-6713	383	45	weighted	weight	VERB
ejpam-6713	383	46	network	network	NOUN
ejpam-6713	383	47	models	model	NOUN
ejpam-6713	383	48	.	.	PUNCT
ejpam-6713	384	1	references	reference	NOUN
ejpam-6713	384	2	[	[	X
ejpam-6713	384	3	1	1	X
ejpam-6713	384	4	]	]	X
ejpam-6713	384	5	teresa	teresa	PROPN
ejpam-6713	384	6	w	w	PROPN
ejpam-6713	384	7	haynes	haynes	PROPN
ejpam-6713	384	8	,	,	PUNCT
ejpam-6713	384	9	stephen	stephen	PROPN
ejpam-6713	384	10	hedetniemi	hedetniemi	PROPN
ejpam-6713	384	11	,	,	PUNCT
ejpam-6713	384	12	and	and	CCONJ
ejpam-6713	384	13	peter	peter	PROPN
ejpam-6713	384	14	slater	slater	PROPN
ejpam-6713	384	15	.	.	PUNCT
ejpam-6713	385	1	fundamentals	fundamental	NOUN
ejpam-6713	385	2	of	of	ADP
ejpam-6713	385	3	domination	domination	NOUN
ejpam-6713	385	4	in	in	ADP
ejpam-6713	385	5	graphs	graph	NOUN
ejpam-6713	385	6	.	.	PUNCT
ejpam-6713	386	1	crc	crc	PROPN
ejpam-6713	386	2	press	press	PROPN
ejpam-6713	386	3	,	,	PUNCT
ejpam-6713	386	4	2013	2013	NUM
ejpam-6713	386	5	.	.	PUNCT
ejpam-6713	387	1	[	[	X
ejpam-6713	387	2	2	2	NUM
ejpam-6713	387	3	]	]	PUNCT
ejpam-6713	387	4	enrico	enrico	PROPN
ejpam-6713	387	5	enriquez	enriquez	PROPN
ejpam-6713	387	6	,	,	PUNCT
ejpam-6713	387	7	grace	grace	PROPN
ejpam-6713	387	8	estrada	estrada	PROPN
ejpam-6713	387	9	,	,	PUNCT
ejpam-6713	387	10	carmelita	carmelita	PROPN
ejpam-6713	387	11	loquias	loquias	PROPN
ejpam-6713	387	12	,	,	PUNCT
ejpam-6713	387	13	reuella	reuella	PROPN
ejpam-6713	387	14	j	j	PROPN
ejpam-6713	387	15	bacalso	bacalso	NOUN
ejpam-6713	387	16	,	,	PUNCT
ejpam-6713	387	17	and	and	CCONJ
ejpam-6713	387	18	lanndon	lanndon	PROPN
ejpam-6713	387	19	ocampo	ocampo	PROPN
ejpam-6713	387	20	.	.	PUNCT
ejpam-6713	388	1	domination	domination	NOUN
ejpam-6713	388	2	in	in	ADP
ejpam-6713	388	3	fuzzy	fuzzy	ADJ
ejpam-6713	388	4	directed	direct	VERB
ejpam-6713	388	5	graphs	graph	NOUN
ejpam-6713	388	6	.	.	PUNCT
ejpam-6713	389	1	mathematics	mathematic	NOUN
ejpam-6713	389	2	,	,	PUNCT
ejpam-6713	389	3	9(17):2143	9(17):2143	NOUN
ejpam-6713	389	4	,	,	PUNCT
ejpam-6713	389	5	2021	2021	NUM
ejpam-6713	389	6	.	.	PUNCT
ejpam-6713	390	1	[	[	X
ejpam-6713	390	2	3	3	X
ejpam-6713	390	3	]	]	PUNCT
ejpam-6713	390	4	aldwin	aldwin	NOUN
ejpam-6713	390	5	t	t	PROPN
ejpam-6713	390	6	miranda	miranda	PROPN
ejpam-6713	390	7	and	and	CCONJ
ejpam-6713	390	8	rolito	rolito	ADJ
ejpam-6713	390	9	g	g	PROPN
ejpam-6713	390	10	eballe	eballe	NOUN
ejpam-6713	390	11	.	.	PUNCT
ejpam-6713	391	1	domination	domination	NOUN
ejpam-6713	391	2	defect	defect	NOUN
ejpam-6713	391	3	for	for	ADP
ejpam-6713	391	4	the	the	DET
ejpam-6713	391	5	join	join	NOUN
ejpam-6713	391	6	and	and	CCONJ
ejpam-6713	391	7	corona	corona	NOUN
ejpam-6713	391	8	of	of	ADP
ejpam-6713	391	9	graphs	graph	NOUN
ejpam-6713	391	10	.	.	PUNCT
ejpam-6713	392	1	applied	apply	VERB
ejpam-6713	392	2	mathematical	mathematical	ADJ
ejpam-6713	392	3	sciences	science	NOUN
ejpam-6713	392	4	,	,	PUNCT
ejpam-6713	392	5	15(12):615–623	15(12):615–623	NUM
ejpam-6713	392	6	,	,	PUNCT
ejpam-6713	392	7	2021	2021	NUM
ejpam-6713	392	8	.	.	PUNCT
ejpam-6713	393	1	[	[	X
ejpam-6713	393	2	4	4	NUM
ejpam-6713	393	3	]	]	X
ejpam-6713	393	4	magda	magda	PROPN
ejpam-6713	393	5	dettlaff	dettlaff	PROPN
ejpam-6713	393	6	,	,	PUNCT
ejpam-6713	393	7	magdalena	magdalena	PROPN
ejpam-6713	393	8	lemańska	lemańska	PROPN
ejpam-6713	393	9	,	,	PUNCT
ejpam-6713	393	10	mateusz	mateusz	NOUN
ejpam-6713	393	11	miotk	miotk	NOUN
ejpam-6713	393	12	,	,	PUNCT
ejpam-6713	393	13	jerzy	jerzy	PROPN
ejpam-6713	393	14	topp	topp	PROPN
ejpam-6713	393	15	,	,	PUNCT
ejpam-6713	393	16	radosław	radosław	PROPN
ejpam-6713	393	17	ziemann	ziemann	PROPN
ejpam-6713	393	18	,	,	PUNCT
ejpam-6713	393	19	and	and	CCONJ
ejpam-6713	393	20	paweł	paweł	VERB
ejpam-6713	393	21	żyliński	żyliński	NOUN
ejpam-6713	393	22	.	.	PUNCT
ejpam-6713	394	1	graphs	graph	NOUN
ejpam-6713	394	2	with	with	ADP
ejpam-6713	394	3	equal	equal	ADJ
ejpam-6713	394	4	domination	domination	NOUN
ejpam-6713	394	5	and	and	CCONJ
ejpam-6713	394	6	certified	certified	ADJ
ejpam-6713	394	7	domination	domination	NOUN
ejpam-6713	394	8	numbers	number	NOUN
ejpam-6713	394	9	.	.	PUNCT
ejpam-6713	395	1	arxiv	arxiv	PROPN
ejpam-6713	395	2	preprint	preprint	PROPN
ejpam-6713	395	3	arxiv:1710.02059	arxiv:1710.02059	PROPN
ejpam-6713	395	4	,	,	PUNCT
ejpam-6713	395	5	2017	2017	NUM
ejpam-6713	395	6	.	.	PUNCT
ejpam-6713	396	1	[	[	X
ejpam-6713	396	2	5	5	NUM
ejpam-6713	396	3	]	]	X
ejpam-6713	396	4	magda	magda	PROPN
ejpam-6713	396	5	dettlaff	dettlaff	PROPN
ejpam-6713	396	6	,	,	PUNCT
ejpam-6713	396	7	magdalena	magdalena	PROPN
ejpam-6713	396	8	lemańska	lemańska	PROPN
ejpam-6713	396	9	,	,	PUNCT
ejpam-6713	396	10	jerzy	jerzy	PROPN
ejpam-6713	396	11	topp	topp	PROPN
ejpam-6713	396	12	,	,	PUNCT
ejpam-6713	396	13	radosław	radosław	PROPN
ejpam-6713	396	14	ziemann	ziemann	PROPN
ejpam-6713	396	15	,	,	PUNCT
ejpam-6713	396	16	and	and	CCONJ
ejpam-6713	396	17	paweł	paweł	VERB
ejpam-6713	396	18	żyliński	żyliński	PROPN
ejpam-6713	396	19	.	.	PUNCT
ejpam-6713	397	1	certified	certified	ADJ
ejpam-6713	397	2	domination	domination	NOUN
ejpam-6713	397	3	.	.	PUNCT
ejpam-6713	398	1	akce	akce	PROPN
ejpam-6713	398	2	international	international	PROPN
ejpam-6713	398	3	journal	journal	NOUN
ejpam-6713	398	4	of	of	ADP
ejpam-6713	398	5	graphs	graph	NOUN
ejpam-6713	398	6	and	and	CCONJ
ejpam-6713	398	7	combinatorics	combinatoric	NOUN
ejpam-6713	398	8	,	,	PUNCT
ejpam-6713	398	9	17(1):86–97	17(1):86–97	NUM
ejpam-6713	398	10	,	,	PUNCT
ejpam-6713	398	11	2020	2020	NUM
ejpam-6713	398	12	.	.	PUNCT
ejpam-6713	399	1	[	[	X
ejpam-6713	399	2	6	6	NUM
ejpam-6713	399	3	]	]	SYM
ejpam-6713	399	4	vishwajeet	vishwajeet	PROPN
ejpam-6713	399	5	s	s	PROPN
ejpam-6713	399	6	goswami	goswami	NOUN
ejpam-6713	399	7	,	,	PUNCT
ejpam-6713	399	8	azham	azham	PROPN
ejpam-6713	399	9	ilyass	ilyass	NOUN
ejpam-6713	399	10	,	,	PUNCT
ejpam-6713	399	11	and	and	CCONJ
ejpam-6713	399	12	kamaljit	kamaljit	PROPN
ejpam-6713	399	13	kaur	kaur	PROPN
ejpam-6713	399	14	bhagwat	bhagwat	PROPN
ejpam-6713	399	15	.	.	PUNCT
ejpam-6713	400	1	certified	certify	VERB
ejpam-6713	400	2	domination	domination	NOUN
ejpam-6713	400	3	number	number	NOUN
ejpam-6713	400	4	of	of	ADP
ejpam-6713	400	5	some	some	DET
ejpam-6713	400	6	graphs	graph	NOUN
ejpam-6713	400	7	.	.	PUNCT
ejpam-6713	401	1	in	in	ADP
ejpam-6713	401	2	aip	aip	PROPN
ejpam-6713	401	3	conference	conference	NOUN
ejpam-6713	401	4	proceedings	proceeding	NOUN
ejpam-6713	401	5	,	,	PUNCT
ejpam-6713	401	6	volume	volume	NOUN
ejpam-6713	401	7	2735	2735	NUM
ejpam-6713	401	8	,	,	PUNCT
ejpam-6713	401	9	page	page	NOUN
ejpam-6713	401	10	040029	040029	NUM
ejpam-6713	401	11	.	.	PUNCT
ejpam-6713	402	1	aip	aip	PROPN
ejpam-6713	402	2	publishing	publishing	PROPN
ejpam-6713	402	3	llc	llc	PROPN
ejpam-6713	402	4	,	,	PUNCT
ejpam-6713	402	5	2023	2023	NUM
ejpam-6713	402	6	.	.	PUNCT
ejpam-6713	403	1	[	[	X
ejpam-6713	403	2	7	7	NUM
ejpam-6713	403	3	]	]	SYM
ejpam-6713	403	4	mateusz	mateusz	NOUN
ejpam-6713	403	5	miotk	miotk	NOUN
ejpam-6713	403	6	.	.	PUNCT
ejpam-6713	404	1	on	on	ADP
ejpam-6713	404	2	a	a	DET
ejpam-6713	404	3	class	class	NOUN
ejpam-6713	404	4	of	of	ADP
ejpam-6713	404	5	graphs	graph	NOUN
ejpam-6713	404	6	with	with	ADP
ejpam-6713	404	7	equal	equal	ADJ
ejpam-6713	404	8	domination	domination	NOUN
ejpam-6713	404	9	and	and	CCONJ
ejpam-6713	404	10	certified	certified	ADJ
ejpam-6713	404	11	domination	domination	NOUN
ejpam-6713	404	12	numbers	number	NOUN
ejpam-6713	404	13	.	.	PUNCT
ejpam-6713	405	1	discrete	discrete	ADJ
ejpam-6713	405	2	mathematics	mathematic	NOUN
ejpam-6713	405	3	letters	letter	NOUN
ejpam-6713	405	4	,	,	PUNCT
ejpam-6713	405	5	16	16	NUM
ejpam-6713	405	6	,	,	PUNCT
ejpam-6713	405	7	2025	2025	NUM
ejpam-6713	405	8	.	.	PUNCT
ejpam-6713	406	1	[	[	X
ejpam-6713	406	2	8	8	NUM
ejpam-6713	406	3	]	]	PUNCT
ejpam-6713	406	4	s	s	PART
ejpam-6713	406	5	durai	durai	PROPN
ejpam-6713	406	6	raj	raj	PROPN
ejpam-6713	406	7	and	and	CCONJ
ejpam-6713	406	8	sg	sg	ADP
ejpam-6713	406	9	shiji	shiji	PROPN
ejpam-6713	406	10	kumari	kumari	PROPN
ejpam-6713	406	11	.	.	PUNCT
ejpam-6713	407	1	certified	certify	VERB
ejpam-6713	407	2	domination	domination	NOUN
ejpam-6713	407	3	number	number	NOUN
ejpam-6713	407	4	in	in	ADP
ejpam-6713	407	5	product	product	NOUN
ejpam-6713	407	6	of	of	ADP
ejpam-6713	407	7	graphs	graph	NOUN
ejpam-6713	407	8	.	.	PUNCT
ejpam-6713	408	1	turkish	turkish	ADJ
ejpam-6713	408	2	journal	journal	NOUN
ejpam-6713	408	3	of	of	ADP
ejpam-6713	408	4	computer	computer	NOUN
ejpam-6713	408	5	and	and	CCONJ
ejpam-6713	408	6	mathematics	mathematics	PROPN
ejpam-6713	408	7	education	education	NOUN
ejpam-6713	408	8	(	(	PUNCT
ejpam-6713	408	9	turcomat	turcomat	NOUN
ejpam-6713	408	10	)	)	PUNCT
ejpam-6713	408	11	,	,	PUNCT
ejpam-6713	408	12	11(3):1166–1170	11(3):1166–1170	NUM
ejpam-6713	408	13	,	,	PUNCT
ejpam-6713	408	14	2020	2020	NUM
ejpam-6713	408	15	.	.	PUNCT
ejpam-6713	409	1	[	[	X
ejpam-6713	409	2	9	9	NUM
ejpam-6713	409	3	]	]	X
ejpam-6713	409	4	p	p	X
ejpam-6713	409	5	roushini	roushini	NOUN
ejpam-6713	409	6	leely	leely	ADV
ejpam-6713	409	7	pushpam	pushpam	VERB
ejpam-6713	409	8	,	,	PUNCT
ejpam-6713	409	9	m	m	VERB
ejpam-6713	409	10	kamalam	kamalam	ADJ
ejpam-6713	409	11	,	,	PUNCT
ejpam-6713	409	12	and	and	CCONJ
ejpam-6713	409	13	b	b	PROPN
ejpam-6713	409	14	mahavir	mahavir	PROPN
ejpam-6713	409	15	.	.	PUNCT
ejpam-6713	410	1	stability	stability	NOUN
ejpam-6713	410	2	of	of	ADP
ejpam-6713	410	3	certified	certify	VERB
ejpam-6713	410	4	g.	g.	PROPN
ejpam-6713	410	5	navamani	navamani	PROPN
ejpam-6713	410	6	et	et	PROPN
ejpam-6713	410	7	al	al	PROPN
ejpam-6713	410	8	.	.	PUNCT
ejpam-6713	410	9	/	/	SYM
ejpam-6713	410	10	eur	eur	PROPN
ejpam-6713	410	11	.	.	PUNCT
ejpam-6713	411	1	j.	j.	PROPN
ejpam-6713	411	2	pure	pure	PROPN
ejpam-6713	411	3	appl	appl	PROPN
ejpam-6713	411	4	.	.	PROPN
ejpam-6713	411	5	math	math	PROPN
ejpam-6713	411	6	,	,	PUNCT
ejpam-6713	411	7	18	18	NUM
ejpam-6713	411	8	(	(	PUNCT
ejpam-6713	411	9	4	4	NUM
ejpam-6713	411	10	)	)	PUNCT
ejpam-6713	411	11	(	(	PUNCT
ejpam-6713	411	12	2025	2025	NUM
ejpam-6713	411	13	)	)	PUNCT
ejpam-6713	411	14	,	,	PUNCT
ejpam-6713	411	15	6713	6713	NUM
ejpam-6713	411	16	15	15	NUM
ejpam-6713	411	17	of	of	ADP
ejpam-6713	411	18	15	15	NUM
ejpam-6713	411	19	domination	domination	NOUN
ejpam-6713	411	20	upon	upon	SCONJ
ejpam-6713	411	21	edge	edge	NOUN
ejpam-6713	411	22	addition	addition	NOUN
ejpam-6713	411	23	.	.	PUNCT
ejpam-6713	412	1	discrete	discrete	ADJ
ejpam-6713	412	2	mathematics	mathematic	NOUN
ejpam-6713	412	3	,	,	PUNCT
ejpam-6713	412	4	algorithms	algorithm	NOUN
ejpam-6713	412	5	and	and	CCONJ
ejpam-6713	412	6	applications	application	NOUN
ejpam-6713	412	7	,	,	PUNCT
ejpam-6713	412	8	16(06):2350068	16(06):2350068	NUM
ejpam-6713	412	9	,	,	PUNCT
ejpam-6713	412	10	2024	2024	NUM
ejpam-6713	412	11	.	.	PUNCT
ejpam-6713	413	1	[	[	X
ejpam-6713	413	2	10	10	NUM
ejpam-6713	413	3	]	]	X
ejpam-6713	413	4	s	s	VERB
ejpam-6713	413	5	arumugam	arumugam	ADJ
ejpam-6713	413	6	and	and	CCONJ
ejpam-6713	413	7	j	j	PROPN
ejpam-6713	413	8	paulraj	paulraj	PROPN
ejpam-6713	413	9	joseph	joseph	PROPN
ejpam-6713	413	10	.	.	PUNCT
ejpam-6713	414	1	domination	domination	NOUN
ejpam-6713	414	2	in	in	ADP
ejpam-6713	414	3	subdivision	subdivision	NOUN
ejpam-6713	414	4	graphs	graph	NOUN
ejpam-6713	414	5	.	.	PUNCT
ejpam-6713	415	1	j.	j.	PROPN
ejpam-6713	415	2	indian	indian	PROPN
ejpam-6713	415	3	math	math	PROPN
ejpam-6713	415	4	.	.	PUNCT
ejpam-6713	416	1	soc.(ns	soc.(ns	ADJ
ejpam-6713	416	2	)	)	PUNCT
ejpam-6713	416	3	,	,	PUNCT
ejpam-6713	416	4	62(1	62(1	PROPN
ejpam-6713	416	5	-	-	NOUN
ejpam-6713	416	6	4):274–282	4):274–282	NUM
ejpam-6713	416	7	,	,	PUNCT
ejpam-6713	416	8	1996	1996	NUM
ejpam-6713	416	9	.	.	PUNCT
ejpam-6713	417	1	[	[	X
ejpam-6713	417	2	11	11	NUM
ejpam-6713	417	3	]	]	SYM
ejpam-6713	417	4	hamideh	hamideh	PROPN
ejpam-6713	417	5	aram	aram	PROPN
ejpam-6713	417	6	,	,	PUNCT
ejpam-6713	417	7	seyed	seyed	PROPN
ejpam-6713	417	8	mahmoud	mahmoud	PROPN
ejpam-6713	417	9	sheikholeslami	sheikholeslami	PROPN
ejpam-6713	417	10	,	,	PUNCT
ejpam-6713	417	11	and	and	CCONJ
ejpam-6713	417	12	odile	odile	PROPN
ejpam-6713	417	13	favaron	favaron	PROPN
ejpam-6713	417	14	.	.	PUNCT
ejpam-6713	418	1	domination	domination	NOUN
ejpam-6713	418	2	subdivision	subdivision	NOUN
ejpam-6713	418	3	numbers	number	NOUN
ejpam-6713	418	4	of	of	ADP
ejpam-6713	418	5	trees	tree	NOUN
ejpam-6713	418	6	.	.	PUNCT
ejpam-6713	419	1	discrete	discrete	ADJ
ejpam-6713	419	2	mathematics	mathematic	NOUN
ejpam-6713	419	3	,	,	PUNCT
ejpam-6713	419	4	309(4):622–628	309(4):622–628	NUM
ejpam-6713	419	5	,	,	PUNCT
ejpam-6713	419	6	2009	2009	NUM
ejpam-6713	419	7	.	.	PUNCT
ejpam-6713	420	1	[	[	X
ejpam-6713	420	2	12	12	NUM
ejpam-6713	420	3	]	]	PUNCT
ejpam-6713	420	4	ammar	ammar	PROPN
ejpam-6713	420	5	babikir	babikir	PROPN
ejpam-6713	420	6	,	,	PUNCT
ejpam-6713	420	7	magda	magda	PROPN
ejpam-6713	420	8	dettlaff	dettlaff	PROPN
ejpam-6713	420	9	,	,	PUNCT
ejpam-6713	420	10	michael	michael	PROPN
ejpam-6713	420	11	a	a	DET
ejpam-6713	420	12	henning	henning	NOUN
ejpam-6713	420	13	,	,	PUNCT
ejpam-6713	420	14	and	and	CCONJ
ejpam-6713	420	15	magdalena	magdalena	PROPN
ejpam-6713	420	16	lemańska	lemańska	PROPN
ejpam-6713	420	17	.	.	PUNCT
ejpam-6713	421	1	independent	independent	ADJ
ejpam-6713	421	2	domination	domination	NOUN
ejpam-6713	421	3	subdivision	subdivision	NOUN
ejpam-6713	421	4	in	in	ADP
ejpam-6713	421	5	graphs	graph	NOUN
ejpam-6713	421	6	.	.	PUNCT
ejpam-6713	422	1	graphs	graph	NOUN
ejpam-6713	422	2	and	and	CCONJ
ejpam-6713	422	3	combinatorics	combinatoric	NOUN
ejpam-6713	422	4	,	,	PUNCT
ejpam-6713	422	5	37(3):691–709	37(3):691–709	PROPN
ejpam-6713	422	6	,	,	PUNCT
ejpam-6713	422	7	2021	2021	NUM
ejpam-6713	422	8	.	.	PUNCT
ejpam-6713	423	1	[	[	X
ejpam-6713	423	2	13	13	NUM
ejpam-6713	423	3	]	]	PUNCT
ejpam-6713	423	4	amitava	amitava	PROPN
ejpam-6713	423	5	bhattacharya	bhattacharya	PROPN
ejpam-6713	423	6	and	and	CCONJ
ejpam-6713	423	7	gurusamy	gurusamy	PROPN
ejpam-6713	423	8	vijayakumar	vijayakumar	PROPN
ejpam-6713	423	9	.	.	PROPN
ejpam-6713	423	10	effect	effect	NOUN
ejpam-6713	423	11	of	of	ADP
ejpam-6713	423	12	edge	edge	NOUN
ejpam-6713	423	13	-	-	PUNCT
ejpam-6713	423	14	subdivision	subdivision	NOUN
ejpam-6713	423	15	on	on	ADP
ejpam-6713	423	16	vertex	vertex	NOUN
ejpam-6713	423	17	-	-	PUNCT
ejpam-6713	423	18	domination	domination	NOUN
ejpam-6713	423	19	in	in	ADP
ejpam-6713	423	20	a	a	DET
ejpam-6713	423	21	graph	graph	NOUN
ejpam-6713	423	22	.	.	PUNCT
ejpam-6713	424	1	discussiones	discussione	NOUN
ejpam-6713	424	2	mathematicae	mathematicae	PROPN
ejpam-6713	424	3	graph	graph	NOUN
ejpam-6713	424	4	theory	theory	NOUN
ejpam-6713	424	5	,	,	PUNCT
ejpam-6713	424	6	22(2):335–347	22(2):335–347	PROPN
ejpam-6713	424	7	,	,	PUNCT
ejpam-6713	424	8	2002	2002	NUM
ejpam-6713	424	9	.	.	PUNCT
ejpam-6713	425	1	[	[	X
ejpam-6713	425	2	14	14	NUM
ejpam-6713	425	3	]	]	X
ejpam-6713	425	4	odile	odile	PROPN
ejpam-6713	425	5	favaron	favaron	PROPN
ejpam-6713	425	6	,	,	PUNCT
ejpam-6713	425	7	teresa	teresa	PROPN
ejpam-6713	425	8	w	w	PROPN
ejpam-6713	425	9	haynes	hayne	NOUN
ejpam-6713	425	10	,	,	PUNCT
ejpam-6713	425	11	and	and	CCONJ
ejpam-6713	425	12	stephen	stephen	PROPN
ejpam-6713	425	13	t	t	PROPN
ejpam-6713	425	14	hedetniemi	hedetniemi	PROPN
ejpam-6713	425	15	.	.	PUNCT
ejpam-6713	426	1	domination	domination	NOUN
ejpam-6713	426	2	subdivision	subdivision	NOUN
ejpam-6713	426	3	numbers	number	NOUN
ejpam-6713	426	4	in	in	ADP
ejpam-6713	426	5	graphs	graph	NOUN
ejpam-6713	426	6	.	.	PUNCT
ejpam-6713	427	1	utilitas	utilitas	PROPN
ejpam-6713	427	2	mathematica	mathematica	PROPN
ejpam-6713	427	3	,	,	PUNCT
ejpam-6713	427	4	66:195–209	66:195–209	PROPN
ejpam-6713	427	5	,	,	PUNCT
ejpam-6713	427	6	2004	2004	NUM
ejpam-6713	427	7	.	.	PUNCT
ejpam-6713	428	1	[	[	X
ejpam-6713	428	2	15	15	NUM
ejpam-6713	428	3	]	]	X
ejpam-6713	428	4	odile	odile	PROPN
ejpam-6713	428	5	favaron	favaron	PROPN
ejpam-6713	428	6	,	,	PUNCT
ejpam-6713	428	7	hossein	hossein	PROPN
ejpam-6713	428	8	karami	karami	PROPN
ejpam-6713	428	9	,	,	PUNCT
ejpam-6713	428	10	r	r	NOUN
ejpam-6713	428	11	khoeilar	khoeilar	NOUN
ejpam-6713	428	12	,	,	PUNCT
ejpam-6713	428	13	and	and	CCONJ
ejpam-6713	428	14	seyed	seyed	PROPN
ejpam-6713	428	15	mahmoud	mahmoud	PROPN
ejpam-6713	428	16	sheikholeslami	sheikholeslami	PROPN
ejpam-6713	428	17	.	.	PUNCT
ejpam-6713	429	1	a	a	DET
ejpam-6713	429	2	new	new	ADJ
ejpam-6713	429	3	bound	bind	VERB
ejpam-6713	429	4	on	on	ADP
ejpam-6713	429	5	the	the	DET
ejpam-6713	429	6	total	total	ADJ
ejpam-6713	429	7	domination	domination	NOUN
ejpam-6713	429	8	subdivision	subdivision	NOUN
ejpam-6713	429	9	number	number	NOUN
ejpam-6713	429	10	.	.	PUNCT
ejpam-6713	430	1	graphs	graph	NOUN
ejpam-6713	430	2	and	and	CCONJ
ejpam-6713	430	3	combinatorics	combinatoric	NOUN
ejpam-6713	430	4	,	,	PUNCT
ejpam-6713	430	5	25(1):41–47	25(1):41–47	NUM
ejpam-6713	430	6	,	,	PUNCT
ejpam-6713	430	7	2009	2009	NUM
ejpam-6713	430	8	.	.	PUNCT
ejpam-6713	431	1	[	[	X
ejpam-6713	431	2	16	16	NUM
ejpam-6713	431	3	]	]	X
ejpam-6713	431	4	teresa	teresa	PROPN
ejpam-6713	431	5	haynes	haynes	PROPN
ejpam-6713	431	6	,	,	PUNCT
ejpam-6713	431	7	sandra	sandra	PROPN
ejpam-6713	431	8	hedetniemi	hedetniemi	PROPN
ejpam-6713	431	9	,	,	PUNCT
ejpam-6713	431	10	stephen	stephen	PROPN
ejpam-6713	431	11	hedetniemi	hedetniemi	PROPN
ejpam-6713	431	12	,	,	PUNCT
ejpam-6713	431	13	david	david	PROPN
ejpam-6713	431	14	jacobs	jacobs	PROPN
ejpam-6713	431	15	,	,	PUNCT
ejpam-6713	431	16	james	james	PROPN
ejpam-6713	431	17	knisely	knisely	ADV
ejpam-6713	431	18	,	,	PUNCT
ejpam-6713	431	19	and	and	CCONJ
ejpam-6713	431	20	lucas	lucas	PROPN
ejpam-6713	431	21	van	van	PROPN
ejpam-6713	431	22	der	der	PROPN
ejpam-6713	431	23	merwe	merwe	PROPN
ejpam-6713	431	24	.	.	PUNCT
ejpam-6713	431	25	domination	domination	NOUN
ejpam-6713	431	26	subdivision	subdivision	NOUN
ejpam-6713	431	27	numbers	number	NOUN
ejpam-6713	431	28	.	.	PUNCT
ejpam-6713	432	1	discussiones	discussione	NOUN
ejpam-6713	432	2	mathematicae	mathematicae	PROPN
ejpam-6713	432	3	graph	graph	NOUN
ejpam-6713	432	4	theory	theory	NOUN
ejpam-6713	432	5	,	,	PUNCT
ejpam-6713	432	6	21(2):239–253	21(2):239–253	NUM
ejpam-6713	432	7	,	,	PUNCT
ejpam-6713	432	8	2001	2001	NUM
ejpam-6713	432	9	.	.	PUNCT
ejpam-6713	433	1	[	[	X
ejpam-6713	433	2	17	17	NUM
ejpam-6713	433	3	]	]	X
ejpam-6713	433	4	g	g	NOUN
ejpam-6713	433	5	navamani	navamani	NOUN
ejpam-6713	433	6	and	and	CCONJ
ejpam-6713	433	7	n	n	PRON
ejpam-6713	433	8	sumathi	sumathi	ADV
ejpam-6713	433	9	.	.	PUNCT
ejpam-6713	434	1	certified	certify	VERB
ejpam-6713	434	2	domination	domination	NOUN
ejpam-6713	434	3	subdivision	subdivision	NOUN
ejpam-6713	434	4	number	number	NOUN
ejpam-6713	434	5	of	of	ADP
ejpam-6713	434	6	trees	tree	NOUN
ejpam-6713	434	7	.	.	PUNCT
ejpam-6713	435	1	neuroquantology	neuroquantology	NOUN
ejpam-6713	435	2	,	,	PUNCT
ejpam-6713	435	3	20(11):6161	20(11):6161	NUM
ejpam-6713	435	4	,	,	PUNCT
ejpam-6713	435	5	2022	2022	NUM
ejpam-6713	435	6	.	.	PUNCT
ejpam-6713	436	1	[	[	X
ejpam-6713	436	2	18	18	NUM
ejpam-6713	436	3	]	]	X
ejpam-6713	436	4	f	f	PROPN
ejpam-6713	436	5	harary	harary	NOUN
ejpam-6713	436	6	.	.	PUNCT
ejpam-6713	437	1	graph	graph	NOUN
ejpam-6713	437	2	theory	theory	NOUN
ejpam-6713	437	3	.	.	PUNCT
ejpam-6713	438	1	addison	addison	PROPN
ejpam-6713	438	2	wesley	wesley	PROPN
ejpam-6713	438	3	publishing	publishing	PROPN
ejpam-6713	438	4	company	company	NOUN
ejpam-6713	438	5	.	.	PUNCT
ejpam-6713	439	1	reading	reading	PROPN
ejpam-6713	439	2	,	,	PUNCT
ejpam-6713	439	3	massachusetts	massachusetts	PROPN
ejpam-6713	439	4	,	,	PUNCT
ejpam-6713	439	5	1969	1969	NUM
ejpam-6713	439	6	.	.	PUNCT
ejpam-6713	440	1	[	[	X
ejpam-6713	440	2	19	19	NUM
ejpam-6713	440	3	]	]	X
ejpam-6713	440	4	nader	nader	PROPN
ejpam-6713	440	5	jafari	jafari	PROPN
ejpam-6713	440	6	rad	rad	PROPN
ejpam-6713	440	7	.	.	PROPN
ejpam-6713	440	8	domination	domination	NOUN
ejpam-6713	440	9	in	in	ADP
ejpam-6713	440	10	circulant	circulant	ADJ
ejpam-6713	440	11	graphs	graph	NOUN
ejpam-6713	440	12	.	.	PUNCT
ejpam-6713	441	1	analele	analele	ADP
ejpam-6713	441	2	stiintifice	stiintifice	PROPN
ejpam-6713	441	3	ale	ale	PROPN
ejpam-6713	441	4	universitatii	universitatii	PROPN
ejpam-6713	441	5	ovidius	ovidius	PROPN
ejpam-6713	441	6	constanta	constanta	PROPN
ejpam-6713	441	7	,	,	PUNCT
ejpam-6713	441	8	seria	seria	PROPN
ejpam-6713	441	9	matematica	matematica	PROPN
ejpam-6713	441	10	,	,	PUNCT
ejpam-6713	441	11	17	17	NUM
ejpam-6713	441	12	,	,	PUNCT
ejpam-6713	441	13	01	01	NUM
ejpam-6713	441	14	2009	2009	NUM
ejpam-6713	441	15	.	.	PUNCT
ejpam-6713	442	1	[	[	X
ejpam-6713	442	2	20	20	NUM
ejpam-6713	442	3	]	]	SYM
ejpam-6713	442	4	b	b	PROPN
ejpam-6713	442	5	javad	javad	PROPN
ejpam-6713	442	6	ebrahimi	ebrahimi	PROPN
ejpam-6713	442	7	,	,	PUNCT
ejpam-6713	442	8	nafiseh	nafiseh	ADJ
ejpam-6713	442	9	jahanbakht	jahanbakht	PROPN
ejpam-6713	442	10	,	,	PUNCT
ejpam-6713	442	11	and	and	CCONJ
ejpam-6713	442	12	ebadollah	ebadollah	ADP
ejpam-6713	442	13	s	s	PROPN
ejpam-6713	442	14	mahmoodian	mahmoodian	PROPN
ejpam-6713	442	15	.	.	PUNCT
ejpam-6713	443	1	vertex	vertex	NOUN
ejpam-6713	443	2	domination	domination	NOUN
ejpam-6713	443	3	of	of	ADP
ejpam-6713	443	4	generalized	generalized	ADJ
ejpam-6713	443	5	petersen	petersen	NOUN
ejpam-6713	443	6	graphs	graph	NOUN
ejpam-6713	443	7	.	.	PUNCT
ejpam-6713	444	1	discrete	discrete	ADJ
ejpam-6713	444	2	mathematics	mathematic	NOUN
ejpam-6713	444	3	,	,	PUNCT
ejpam-6713	444	4	309(13):4355–4361	309(13):4355–4361	NOUN
ejpam-6713	444	5	,	,	PUNCT
ejpam-6713	444	6	2009	2009	NUM
ejpam-6713	444	7	.	.	PUNCT