id	sid	tid	token	lemma	pos
ejpam-4912	1	1	european	european	PROPN
ejpam-4912	1	2	journal	journal	PROPN
ejpam-4912	1	3	of	of	ADP
ejpam-4912	1	4	pure	pure	ADJ
ejpam-4912	1	5	and	and	CCONJ
ejpam-4912	1	6	applied	apply	VERB
ejpam-4912	1	7	mathematics	mathematic	NOUN
ejpam-4912	1	8	vol	vol	NOUN
ejpam-4912	1	9	.	.	PUNCT
ejpam-4912	2	1	16	16	NUM
ejpam-4912	2	2	,	,	PUNCT
ejpam-4912	2	3	no	no	INTJ
ejpam-4912	2	4	.	.	NOUN
ejpam-4912	2	5	4	4	NUM
ejpam-4912	2	6	,	,	PUNCT
ejpam-4912	2	7	2023	2023	NUM
ejpam-4912	2	8	,	,	PUNCT
ejpam-4912	2	9	2106	2106	NUM
ejpam-4912	2	10	-	-	SYM
ejpam-4912	2	11	2117	2117	NUM
ejpam-4912	2	12	issn	issn	PROPN
ejpam-4912	2	13	1307	1307	NUM
ejpam-4912	2	14	-	-	SYM
ejpam-4912	2	15	5543	5543	NUM
ejpam-4912	2	16	–	–	PUNCT
ejpam-4912	2	17	ejpam.com	ejpam.com	X
ejpam-4912	2	18	published	publish	VERB
ejpam-4912	2	19	by	by	ADP
ejpam-4912	2	20	new	new	PROPN
ejpam-4912	2	21	york	york	PROPN
ejpam-4912	2	22	business	business	PROPN
ejpam-4912	2	23	global	global	ADJ
ejpam-4912	2	24	characterizations	characterization	NOUN
ejpam-4912	2	25	of	of	ADP
ejpam-4912	2	26	j	j	PROPN
ejpam-4912	2	27	-	-	ADJ
ejpam-4912	2	28	total	total	ADJ
ejpam-4912	2	29	dominating	dominating	NOUN
ejpam-4912	2	30	sets	set	NOUN
ejpam-4912	2	31	of	of	ADP
ejpam-4912	2	32	some	some	DET
ejpam-4912	2	33	graphs	graph	NOUN
ejpam-4912	2	34	javier	javier	PROPN
ejpam-4912	2	35	a.	a.	PROPN
ejpam-4912	2	36	hassan1	hassan1	PROPN
ejpam-4912	2	37	,	,	PUNCT
ejpam-4912	2	38	jahiri	jahiri	PROPN
ejpam-4912	2	39	u.	u.	PROPN
ejpam-4912	2	40	manditong1,∗	manditong1,∗	PROPN
ejpam-4912	2	41	,	,	PUNCT
ejpam-4912	2	42	alcyn	alcyn	NOUN
ejpam-4912	2	43	bakkang2	bakkang2	NOUN
ejpam-4912	2	44	,	,	PUNCT
ejpam-4912	2	45	sisteta	sisteta	PROPN
ejpam-4912	2	46	u.	u.	PROPN
ejpam-4912	2	47	kamdon1	kamdon1	PROPN
ejpam-4912	2	48	,	,	PUNCT
ejpam-4912	2	49	jeffrey	jeffrey	PROPN
ejpam-4912	2	50	imer	imer	PROPN
ejpam-4912	2	51	salim1	salim1	PROPN
ejpam-4912	3	1	1mathematics	1mathematics	NUM
ejpam-4912	3	2	and	and	CCONJ
ejpam-4912	3	3	sciences	sciences	PROPN
ejpam-4912	3	4	department	department	PROPN
ejpam-4912	3	5	,	,	PUNCT
ejpam-4912	3	6	college	college	NOUN
ejpam-4912	3	7	of	of	ADP
ejpam-4912	3	8	arts	art	NOUN
ejpam-4912	3	9	and	and	CCONJ
ejpam-4912	3	10	sciences	science	NOUN
ejpam-4912	3	11	,	,	PUNCT
ejpam-4912	3	12	msu	msu	PROPN
ejpam-4912	3	13	tawi	tawi	PROPN
ejpam-4912	3	14	-	-	PUNCT
ejpam-4912	3	15	tawi	tawi	PROPN
ejpam-4912	3	16	college	college	PROPN
ejpam-4912	3	17	of	of	ADP
ejpam-4912	3	18	technology	technology	NOUN
ejpam-4912	3	19	and	and	CCONJ
ejpam-4912	3	20	oceanography	oceanography	NOUN
ejpam-4912	3	21	,	,	PUNCT
ejpam-4912	3	22	bongao	bongao	NOUN
ejpam-4912	3	23	,	,	PUNCT
ejpam-4912	3	24	tawi	tawi	NOUN
ejpam-4912	3	25	-	-	PUNCT
ejpam-4912	3	26	tawi	tawi	NOUN
ejpam-4912	3	27	,	,	PUNCT
ejpam-4912	3	28	philippines	philippine	NOUN
ejpam-4912	3	29	2	2	NUM
ejpam-4912	3	30	secondary	secondary	ADJ
ejpam-4912	3	31	education	education	NOUN
ejpam-4912	3	32	department	department	NOUN
ejpam-4912	3	33	,	,	PUNCT
ejpam-4912	3	34	college	college	NOUN
ejpam-4912	3	35	of	of	ADP
ejpam-4912	3	36	education	education	NOUN
ejpam-4912	3	37	,	,	PUNCT
ejpam-4912	3	38	msu	msu	PROPN
ejpam-4912	3	39	tawi	tawi	PROPN
ejpam-4912	3	40	-	-	PUNCT
ejpam-4912	3	41	tawi	tawi	PROPN
ejpam-4912	3	42	college	college	PROPN
ejpam-4912	3	43	of	of	ADP
ejpam-4912	3	44	technology	technology	NOUN
ejpam-4912	3	45	and	and	CCONJ
ejpam-4912	3	46	oceanography	oceanography	NOUN
ejpam-4912	3	47	,	,	PUNCT
ejpam-4912	3	48	bongao	bongao	NOUN
ejpam-4912	3	49	,	,	PUNCT
ejpam-4912	3	50	tawi	tawi	NOUN
ejpam-4912	3	51	-	-	PUNCT
ejpam-4912	3	52	tawi	tawi	NOUN
ejpam-4912	3	53	,	,	PUNCT
ejpam-4912	3	54	philippines	philippine	NOUN
ejpam-4912	3	55	abstract	abstract	ADJ
ejpam-4912	3	56	.	.	PUNCT
ejpam-4912	4	1	let	let	VERB
ejpam-4912	4	2	g	g	PRON
ejpam-4912	4	3	be	be	AUX
ejpam-4912	4	4	a	a	DET
ejpam-4912	4	5	graph	graph	NOUN
ejpam-4912	4	6	with	with	ADP
ejpam-4912	4	7	no	no	DET
ejpam-4912	4	8	isolated	isolated	ADJ
ejpam-4912	4	9	vertex	vertex	NOUN
ejpam-4912	4	10	.	.	PUNCT
ejpam-4912	5	1	a	a	DET
ejpam-4912	5	2	subset	subset	NOUN
ejpam-4912	5	3	m	m	VERB
ejpam-4912	5	4	⊆	⊆	NUM
ejpam-4912	5	5	v	v	NOUN
ejpam-4912	5	6	(	(	PUNCT
ejpam-4912	5	7	g	g	NOUN
ejpam-4912	5	8	)	)	PUNCT
ejpam-4912	5	9	is	be	AUX
ejpam-4912	5	10	called	call	VERB
ejpam-4912	5	11	a	a	DET
ejpam-4912	5	12	j	j	NOUN
ejpam-4912	5	13	-	-	ADJ
ejpam-4912	5	14	open	open	ADJ
ejpam-4912	5	15	set	set	NOUN
ejpam-4912	5	16	if	if	SCONJ
ejpam-4912	5	17	ng(a)\ng(b	ng(a)\ng(b	ADJ
ejpam-4912	5	18	)	)	PUNCT
ejpam-4912	5	19	̸=	̸=	PROPN
ejpam-4912	5	20	∅	∅	NOUN
ejpam-4912	5	21	and	and	CCONJ
ejpam-4912	5	22	ng(b)\ng(a	ng(b)\ng(a	ADJ
ejpam-4912	5	23	)	)	PUNCT
ejpam-4912	5	24	̸=	̸=	PROPN
ejpam-4912	5	25	∅	∅	NOUN
ejpam-4912	5	26	∀	∀	X
ejpam-4912	5	27	a	a	PRON
ejpam-4912	5	28	,	,	PUNCT
ejpam-4912	5	29	b	b	PROPN
ejpam-4912	5	30	∈	∈	PROPN
ejpam-4912	5	31	m	m	NOUN
ejpam-4912	5	32	,	,	PUNCT
ejpam-4912	5	33	where	where	SCONJ
ejpam-4912	5	34	a	a	DET
ejpam-4912	5	35	̸=	̸=	PROPN
ejpam-4912	5	36	b.	b.	NOUN
ejpam-4912	5	37	if	if	SCONJ
ejpam-4912	5	38	in	in	ADP
ejpam-4912	5	39	addition	addition	NOUN
ejpam-4912	5	40	,	,	PUNCT
ejpam-4912	5	41	m	m	VERB
ejpam-4912	5	42	is	be	AUX
ejpam-4912	5	43	a	a	DET
ejpam-4912	5	44	total	total	ADJ
ejpam-4912	5	45	dominating	dominating	NOUN
ejpam-4912	5	46	in	in	ADP
ejpam-4912	5	47	g	g	PROPN
ejpam-4912	5	48	,	,	PUNCT
ejpam-4912	5	49	then	then	ADV
ejpam-4912	5	50	we	we	PRON
ejpam-4912	5	51	call	call	VERB
ejpam-4912	5	52	m	m	VERB
ejpam-4912	5	53	a	a	DET
ejpam-4912	5	54	j	j	PROPN
ejpam-4912	5	55	-	-	ADJ
ejpam-4912	5	56	total	total	ADJ
ejpam-4912	5	57	dominating	dominating	NOUN
ejpam-4912	5	58	set	set	VERB
ejpam-4912	5	59	in	in	ADP
ejpam-4912	5	60	g.	g.	PROPN
ejpam-4912	5	61	the	the	DET
ejpam-4912	5	62	maximum	maximum	PROPN
ejpam-4912	5	63	cardinality	cardinality	NOUN
ejpam-4912	5	64	among	among	ADP
ejpam-4912	5	65	all	all	DET
ejpam-4912	5	66	j	j	PROPN
ejpam-4912	5	67	-	-	ADJ
ejpam-4912	5	68	total	total	ADJ
ejpam-4912	5	69	dominating	dominating	NOUN
ejpam-4912	5	70	set	set	NOUN
ejpam-4912	5	71	in	in	ADP
ejpam-4912	5	72	g	g	NOUN
ejpam-4912	5	73	,	,	PUNCT
ejpam-4912	5	74	denoted	denote	VERB
ejpam-4912	5	75	by	by	ADP
ejpam-4912	5	76	γjt(g	γjt(g	PROPN
ejpam-4912	5	77	)	)	PUNCT
ejpam-4912	5	78	,	,	PUNCT
ejpam-4912	5	79	is	be	AUX
ejpam-4912	5	80	called	call	VERB
ejpam-4912	5	81	the	the	DET
ejpam-4912	5	82	j	j	PROPN
ejpam-4912	5	83	-	-	PUNCT
ejpam-4912	5	84	total	total	ADJ
ejpam-4912	5	85	domination	domination	NOUN
ejpam-4912	5	86	number	number	NOUN
ejpam-4912	5	87	of	of	ADP
ejpam-4912	5	88	g.	g.	PROPN
ejpam-4912	5	89	in	in	ADP
ejpam-4912	5	90	this	this	DET
ejpam-4912	5	91	paper	paper	NOUN
ejpam-4912	5	92	,	,	PUNCT
ejpam-4912	5	93	we	we	PRON
ejpam-4912	5	94	characterize	characterize	VERB
ejpam-4912	5	95	j	j	PROPN
ejpam-4912	5	96	-	-	ADJ
ejpam-4912	5	97	total	total	ADJ
ejpam-4912	5	98	dominating	dominating	NOUN
ejpam-4912	5	99	sets	set	NOUN
ejpam-4912	5	100	in	in	ADP
ejpam-4912	5	101	some	some	DET
ejpam-4912	5	102	special	special	ADJ
ejpam-4912	5	103	graphs	graph	NOUN
ejpam-4912	5	104	and	and	CCONJ
ejpam-4912	5	105	join	join	NOUN
ejpam-4912	5	106	of	of	ADP
ejpam-4912	5	107	two	two	NUM
ejpam-4912	5	108	graphs	graph	NOUN
ejpam-4912	5	109	,	,	PUNCT
ejpam-4912	5	110	and	and	CCONJ
ejpam-4912	5	111	we	we	PRON
ejpam-4912	5	112	use	use	VERB
ejpam-4912	5	113	these	these	DET
ejpam-4912	5	114	results	result	NOUN
ejpam-4912	5	115	to	to	PART
ejpam-4912	5	116	obtain	obtain	VERB
ejpam-4912	5	117	formulas	formula	NOUN
ejpam-4912	5	118	for	for	ADP
ejpam-4912	5	119	the	the	DET
ejpam-4912	5	120	parameters	parameter	NOUN
ejpam-4912	5	121	of	of	ADP
ejpam-4912	5	122	these	these	DET
ejpam-4912	5	123	graphs	graph	NOUN
ejpam-4912	5	124	.	.	PUNCT
ejpam-4912	6	1	moreover	moreover	ADV
ejpam-4912	6	2	,	,	PUNCT
ejpam-4912	6	3	we	we	PRON
ejpam-4912	6	4	determine	determine	VERB
ejpam-4912	6	5	its	its	PRON
ejpam-4912	6	6	relationships	relationship	NOUN
ejpam-4912	6	7	with	with	ADP
ejpam-4912	6	8	other	other	ADJ
ejpam-4912	6	9	known	know	VERB
ejpam-4912	6	10	parameters	parameter	NOUN
ejpam-4912	6	11	in	in	ADP
ejpam-4912	6	12	graph	graph	NOUN
ejpam-4912	6	13	theory	theory	NOUN
ejpam-4912	6	14	.	.	PUNCT
ejpam-4912	7	1	finally	finally	ADV
ejpam-4912	7	2	,	,	PUNCT
ejpam-4912	7	3	we	we	PRON
ejpam-4912	7	4	derive	derive	VERB
ejpam-4912	7	5	the	the	DET
ejpam-4912	7	6	lower	low	ADJ
ejpam-4912	7	7	bound	bind	VERB
ejpam-4912	7	8	of	of	ADP
ejpam-4912	7	9	the	the	DET
ejpam-4912	7	10	parameter	parameter	NOUN
ejpam-4912	7	11	for	for	ADP
ejpam-4912	7	12	the	the	DET
ejpam-4912	7	13	corona	corona	NOUN
ejpam-4912	7	14	of	of	ADP
ejpam-4912	7	15	two	two	NUM
ejpam-4912	7	16	graphs	graph	NOUN
ejpam-4912	7	17	.	.	PUNCT
ejpam-4912	8	1	2020	2020	NUM
ejpam-4912	8	2	mathematics	mathematic	NOUN
ejpam-4912	8	3	subject	subject	NOUN
ejpam-4912	8	4	classifications	classification	NOUN
ejpam-4912	8	5	:	:	PUNCT
ejpam-4912	8	6	05c69	05c69	X
ejpam-4912	8	7	key	key	ADJ
ejpam-4912	8	8	words	word	NOUN
ejpam-4912	8	9	and	and	CCONJ
ejpam-4912	8	10	phrases	phrase	NOUN
ejpam-4912	8	11	:	:	PUNCT
ejpam-4912	8	12	j	j	NOUN
ejpam-4912	8	13	-	-	ADJ
ejpam-4912	8	14	open	open	ADJ
ejpam-4912	8	15	set	set	NOUN
ejpam-4912	8	16	,	,	PUNCT
ejpam-4912	8	17	j	j	PROPN
ejpam-4912	8	18	-	-	ADJ
ejpam-4912	8	19	total	total	ADJ
ejpam-4912	8	20	dominating	dominating	NOUN
ejpam-4912	8	21	set	set	NOUN
ejpam-4912	8	22	,	,	PUNCT
ejpam-4912	8	23	j	j	PROPN
ejpam-4912	8	24	-	-	PUNCT
ejpam-4912	8	25	total	total	ADJ
ejpam-4912	8	26	domination	domination	NOUN
ejpam-4912	8	27	number	number	NOUN
ejpam-4912	8	28	1	1	NUM
ejpam-4912	8	29	.	.	PUNCT
ejpam-4912	8	30	introduction	introduction	NOUN
ejpam-4912	8	31	the	the	DET
ejpam-4912	8	32	study	study	NOUN
ejpam-4912	8	33	of	of	ADP
ejpam-4912	8	34	domination	domination	NOUN
ejpam-4912	8	35	in	in	ADP
ejpam-4912	8	36	graphs	graph	NOUN
ejpam-4912	8	37	came	come	VERB
ejpam-4912	8	38	about	about	ADV
ejpam-4912	8	39	partially	partially	ADV
ejpam-4912	8	40	as	as	ADP
ejpam-4912	8	41	a	a	DET
ejpam-4912	8	42	result	result	NOUN
ejpam-4912	8	43	of	of	ADP
ejpam-4912	8	44	the	the	DET
ejpam-4912	8	45	study	study	NOUN
ejpam-4912	8	46	of	of	ADP
ejpam-4912	8	47	games	game	NOUN
ejpam-4912	8	48	and	and	CCONJ
ejpam-4912	8	49	recreational	recreational	ADJ
ejpam-4912	8	50	mathematics	mathematic	NOUN
ejpam-4912	8	51	.	.	PUNCT
ejpam-4912	9	1	in	in	ADP
ejpam-4912	9	2	particular	particular	ADJ
ejpam-4912	9	3	,	,	PUNCT
ejpam-4912	9	4	mathematicians	mathematician	NOUN
ejpam-4912	9	5	studied	study	VERB
ejpam-4912	9	6	how	how	SCONJ
ejpam-4912	9	7	chess	chess	NOUN
ejpam-4912	9	8	pieces	piece	NOUN
ejpam-4912	9	9	of	of	ADP
ejpam-4912	9	10	a	a	DET
ejpam-4912	9	11	particular	particular	ADJ
ejpam-4912	9	12	type	type	NOUN
ejpam-4912	9	13	could	could	AUX
ejpam-4912	9	14	be	be	AUX
ejpam-4912	9	15	placed	place	VERB
ejpam-4912	9	16	on	on	ADP
ejpam-4912	9	17	a	a	DET
ejpam-4912	9	18	chessboard	chessboard	NOUN
ejpam-4912	9	19	in	in	ADP
ejpam-4912	9	20	such	such	DET
ejpam-4912	9	21	a	a	DET
ejpam-4912	9	22	way	way	NOUN
ejpam-4912	9	23	that	that	PRON
ejpam-4912	9	24	they	they	PRON
ejpam-4912	9	25	would	would	AUX
ejpam-4912	9	26	attack	attack	VERB
ejpam-4912	9	27	,	,	PUNCT
ejpam-4912	9	28	or	or	CCONJ
ejpam-4912	9	29	dominate	dominate	VERB
ejpam-4912	9	30	,	,	PUNCT
ejpam-4912	9	31	every	every	DET
ejpam-4912	9	32	square	square	NOUN
ejpam-4912	9	33	on	on	ADP
ejpam-4912	9	34	the	the	DET
ejpam-4912	9	35	board	board	NOUN
ejpam-4912	9	36	.	.	PUNCT
ejpam-4912	10	1	domination	domination	NOUN
ejpam-4912	10	2	in	in	ADP
ejpam-4912	10	3	a	a	DET
ejpam-4912	10	4	graph	graph	NOUN
ejpam-4912	10	5	was	be	AUX
ejpam-4912	10	6	introduced	introduce	VERB
ejpam-4912	10	7	by	by	ADP
ejpam-4912	10	8	oystein	oystein	ADJ
ejpam-4912	10	9	ore	ore	NOUN
ejpam-4912	10	10	in	in	ADP
ejpam-4912	10	11	1962	1962	NUM
ejpam-4912	10	12	in	in	ADP
ejpam-4912	10	13	his	his	PRON
ejpam-4912	10	14	book	book	NOUN
ejpam-4912	10	15	on	on	ADP
ejpam-4912	10	16	graph	graph	NOUN
ejpam-4912	10	17	theory	theory	NOUN
ejpam-4912	10	18	[	[	X
ejpam-4912	10	19	10	10	NUM
ejpam-4912	10	20	]	]	PUNCT
ejpam-4912	10	21	.	.	PUNCT
ejpam-4912	11	1	a	a	DET
ejpam-4912	11	2	subset	subset	NOUN
ejpam-4912	11	3	d	d	NOUN
ejpam-4912	11	4	of	of	ADP
ejpam-4912	11	5	vertices	vertex	NOUN
ejpam-4912	11	6	of	of	ADP
ejpam-4912	11	7	a	a	DET
ejpam-4912	11	8	graph	graph	NOUN
ejpam-4912	11	9	g	g	NOUN
ejpam-4912	11	10	is	be	AUX
ejpam-4912	11	11	called	call	VERB
ejpam-4912	11	12	a	a	DET
ejpam-4912	11	13	dominating	dominating	NOUN
ejpam-4912	11	14	of	of	ADP
ejpam-4912	11	15	g	g	PROPN
ejpam-4912	11	16	if	if	SCONJ
ejpam-4912	11	17	for	for	ADP
ejpam-4912	11	18	every	every	DET
ejpam-4912	11	19	x	x	SYM
ejpam-4912	11	20	∈	∈	PROPN
ejpam-4912	11	21	v	v	ADP
ejpam-4912	11	22	(	(	PUNCT
ejpam-4912	11	23	g	g	NOUN
ejpam-4912	11	24	)	)	PUNCT
ejpam-4912	11	25	\	\	PUNCT
ejpam-4912	12	1	d	d	X
ejpam-4912	12	2	,	,	PUNCT
ejpam-4912	12	3	there	there	PRON
ejpam-4912	12	4	exists	exist	VERB
ejpam-4912	12	5	y	y	PROPN
ejpam-4912	12	6	∈	∈	PROPN
ejpam-4912	13	1	d	d	X
ejpam-4912	13	2	such	such	ADJ
ejpam-4912	13	3	that	that	SCONJ
ejpam-4912	13	4	xy	xy	PROPN
ejpam-4912	13	5	∈	∈	PROPN
ejpam-4912	13	6	e(g	e(g	PROPN
ejpam-4912	13	7	)	)	PUNCT
ejpam-4912	13	8	,	,	PUNCT
ejpam-4912	13	9	that	that	ADV
ejpam-4912	13	10	is	is	ADV
ejpam-4912	13	11	,	,	PUNCT
ejpam-4912	13	12	ng[d	ng[d	PROPN
ejpam-4912	13	13	]	]	PUNCT
ejpam-4912	14	1	=	=	SYM
ejpam-4912	14	2	v	v	X
ejpam-4912	14	3	(	(	PUNCT
ejpam-4912	14	4	g	g	NOUN
ejpam-4912	14	5	)	)	PUNCT
ejpam-4912	14	6	.	.	PUNCT
ejpam-4912	15	1	the	the	DET
ejpam-4912	15	2	domination	domination	NOUN
ejpam-4912	15	3	number	number	NOUN
ejpam-4912	15	4	of	of	ADP
ejpam-4912	15	5	g	g	NOUN
ejpam-4912	15	6	,	,	PUNCT
ejpam-4912	15	7	denoted	denote	VERB
ejpam-4912	15	8	by	by	ADP
ejpam-4912	15	9	γ(g	γ(g	PROPN
ejpam-4912	15	10	)	)	PUNCT
ejpam-4912	15	11	,	,	PUNCT
ejpam-4912	15	12	is	be	AUX
ejpam-4912	15	13	the	the	DET
ejpam-4912	15	14	minimum	minimum	ADJ
ejpam-4912	15	15	cardinality	cardinality	NOUN
ejpam-4912	15	16	of	of	ADP
ejpam-4912	15	17	a	a	DET
ejpam-4912	15	18	dominating	dominating	NOUN
ejpam-4912	15	19	set	set	VERB
ejpam-4912	15	20	doi	doi	NOUN
ejpam-4912	15	21	:	:	PUNCT
ejpam-4912	15	22	https://doi.org/10.29020/nybg.ejpam.v16i4.4912	https://doi.org/10.29020/nybg.ejpam.v16i4.4912	PROPN
ejpam-4912	15	23	email	email	NOUN
ejpam-4912	15	24	addresses	address	NOUN
ejpam-4912	15	25	:	:	PUNCT
ejpam-4912	15	26	javierhassan@msutawi-tawi.edu.ph	javierhassan@msutawi-tawi.edu.ph	PROPN
ejpam-4912	15	27	(	(	PUNCT
ejpam-4912	15	28	j.	j.	PROPN
ejpam-4912	15	29	hassan	hassan	PROPN
ejpam-4912	15	30	)	)	PUNCT
ejpam-4912	15	31	,	,	PUNCT
ejpam-4912	15	32	jahirimanditong@msutawi-tawi.edu.ph	jahirimanditong@msutawi-tawi.edu.ph	PROPN
ejpam-4912	15	33	(	(	PUNCT
ejpam-4912	15	34	j.	j.	PROPN
ejpam-4912	15	35	manditong	manditong	PROPN
ejpam-4912	15	36	)	)	PUNCT
ejpam-4912	15	37	,	,	PUNCT
ejpam-4912	15	38	alcynbakkang@msutawi-tawi.edu.ph	alcynbakkang@msutawi-tawi.edu.ph	PROPN
ejpam-4912	15	39	(	(	PUNCT
ejpam-4912	15	40	a.	a.	PROPN
ejpam-4912	15	41	bakkang	bakkang	PROPN
ejpam-4912	15	42	)	)	PUNCT
ejpam-4912	15	43	,	,	PUNCT
ejpam-4912	15	44	sistetakamdon@msutawi-tawi.edu.ph	sistetakamdon@msutawi-tawi.edu.ph	PROPN
ejpam-4912	15	45	(	(	PUNCT
ejpam-4912	15	46	s.	s.	PROPN
ejpam-4912	15	47	kamdon	kamdon	PROPN
ejpam-4912	15	48	)	)	PUNCT
ejpam-4912	15	49	,	,	PUNCT
ejpam-4912	15	50	jeffreyimersalim@msutawi-tawi.edu.ph	jeffreyimersalim@msutawi-tawi.edu.ph	PROPN
ejpam-4912	15	51	(	(	PUNCT
ejpam-4912	15	52	j.	j.	PROPN
ejpam-4912	15	53	salim	salim	PROPN
ejpam-4912	15	54	)	)	PUNCT
ejpam-4912	15	55	https://www.ejpam.com	https://www.ejpam.com	X
ejpam-4912	15	56	2106	2106	NUM
ejpam-4912	16	1	©	©	ADP
ejpam-4912	16	2	2023	2023	NUM
ejpam-4912	16	3	ejpam	ejpam	NOUN
ejpam-4912	16	4	all	all	DET
ejpam-4912	16	5	rights	right	NOUN
ejpam-4912	16	6	reserved	reserve	VERB
ejpam-4912	16	7	.	.	PUNCT
ejpam-4912	17	1	j.a	j.a	PROPN
ejpam-4912	17	2	.	.	PROPN
ejpam-4912	17	3	hassan	hassan	PROPN
ejpam-4912	17	4	et	et	PROPN
ejpam-4912	17	5	al	al	PROPN
ejpam-4912	17	6	.	.	PUNCT
ejpam-4912	17	7	/	/	SYM
ejpam-4912	17	8	eur	eur	PROPN
ejpam-4912	17	9	.	.	PUNCT
ejpam-4912	18	1	j.	j.	PROPN
ejpam-4912	18	2	pure	pure	PROPN
ejpam-4912	18	3	appl	appl	PROPN
ejpam-4912	18	4	.	.	PROPN
ejpam-4912	18	5	math	math	PROPN
ejpam-4912	18	6	,	,	PUNCT
ejpam-4912	18	7	16	16	NUM
ejpam-4912	18	8	(	(	PUNCT
ejpam-4912	18	9	4	4	NUM
ejpam-4912	18	10	)	)	PUNCT
ejpam-4912	18	11	(	(	PUNCT
ejpam-4912	18	12	2023	2023	NUM
ejpam-4912	18	13	)	)	PUNCT
ejpam-4912	18	14	,	,	PUNCT
ejpam-4912	18	15	2106	2106	NUM
ejpam-4912	18	16	-	-	SYM
ejpam-4912	18	17	2117	2117	NUM
ejpam-4912	18	18	2107	2107	NUM
ejpam-4912	18	19	in	in	ADP
ejpam-4912	18	20	g.	g.	PROPN
ejpam-4912	18	21	a	a	DET
ejpam-4912	18	22	decade	decade	NOUN
ejpam-4912	18	23	later	later	ADV
ejpam-4912	18	24	,	,	PUNCT
ejpam-4912	18	25	cockayne	cockayne	NOUN
ejpam-4912	18	26	and	and	CCONJ
ejpam-4912	18	27	hedetniemi	hedetniemi	NOUN
ejpam-4912	19	1	[	[	X
ejpam-4912	19	2	1	1	X
ejpam-4912	19	3	]	]	PUNCT
ejpam-4912	19	4	published	publish	VERB
ejpam-4912	19	5	a	a	DET
ejpam-4912	19	6	survey	survey	NOUN
ejpam-4912	19	7	paper	paper	NOUN
ejpam-4912	19	8	,	,	PUNCT
ejpam-4912	19	9	in	in	ADP
ejpam-4912	19	10	which	which	PRON
ejpam-4912	19	11	the	the	DET
ejpam-4912	19	12	notation	notation	PROPN
ejpam-4912	19	13	γ(g	γ(g	PROPN
ejpam-4912	19	14	)	)	PUNCT
ejpam-4912	19	15	was	be	AUX
ejpam-4912	19	16	first	first	ADV
ejpam-4912	19	17	used	use	VERB
ejpam-4912	19	18	for	for	ADP
ejpam-4912	19	19	the	the	DET
ejpam-4912	19	20	domination	domination	NOUN
ejpam-4912	19	21	number	number	NOUN
ejpam-4912	19	22	of	of	ADP
ejpam-4912	19	23	a	a	DET
ejpam-4912	19	24	graph	graph	NOUN
ejpam-4912	19	25	g.	g.	NOUN
ejpam-4912	19	26	since	since	SCONJ
ejpam-4912	19	27	then	then	ADV
ejpam-4912	19	28	,	,	PUNCT
ejpam-4912	19	29	several	several	ADJ
ejpam-4912	19	30	mathematicians	mathematician	NOUN
ejpam-4912	19	31	had	have	AUX
ejpam-4912	19	32	studied	study	VERB
ejpam-4912	19	33	and	and	CCONJ
ejpam-4912	19	34	introduced	introduce	VERB
ejpam-4912	19	35	new	new	ADJ
ejpam-4912	19	36	domination	domination	NOUN
ejpam-4912	19	37	parameters	parameter	NOUN
ejpam-4912	19	38	in	in	ADP
ejpam-4912	19	39	graphs	graph	NOUN
ejpam-4912	19	40	.	.	PUNCT
ejpam-4912	20	1	some	some	DET
ejpam-4912	20	2	variants	variant	NOUN
ejpam-4912	20	3	of	of	ADP
ejpam-4912	20	4	domination	domination	NOUN
ejpam-4912	20	5	were	be	AUX
ejpam-4912	20	6	defined	define	VERB
ejpam-4912	20	7	and	and	CCONJ
ejpam-4912	20	8	further	far	ADV
ejpam-4912	20	9	studied	study	VERB
ejpam-4912	20	10	by	by	ADP
ejpam-4912	20	11	researchers	researcher	NOUN
ejpam-4912	20	12	in	in	ADP
ejpam-4912	20	13	[	[	X
ejpam-4912	20	14	2–9	2–9	NUM
ejpam-4912	20	15	,	,	PUNCT
ejpam-4912	20	16	11	11	NUM
ejpam-4912	20	17	,	,	PUNCT
ejpam-4912	20	18	12	12	NUM
ejpam-4912	20	19	]	]	PUNCT
ejpam-4912	20	20	.	.	PUNCT
ejpam-4912	21	1	in	in	ADP
ejpam-4912	21	2	this	this	DET
ejpam-4912	21	3	paper	paper	NOUN
ejpam-4912	21	4	,	,	PUNCT
ejpam-4912	21	5	new	new	ADJ
ejpam-4912	21	6	variant	variant	NOUN
ejpam-4912	21	7	of	of	ADP
ejpam-4912	21	8	domination	domination	NOUN
ejpam-4912	21	9	called	call	VERB
ejpam-4912	21	10	j	j	PROPN
ejpam-4912	21	11	-	-	ADJ
ejpam-4912	21	12	total	total	ADJ
ejpam-4912	21	13	domination	domination	NOUN
ejpam-4912	21	14	in	in	ADP
ejpam-4912	21	15	a	a	DET
ejpam-4912	21	16	graph	graph	NOUN
ejpam-4912	21	17	will	will	AUX
ejpam-4912	21	18	be	be	AUX
ejpam-4912	21	19	introduced	introduce	VERB
ejpam-4912	21	20	and	and	CCONJ
ejpam-4912	21	21	investigated	investigate	VERB
ejpam-4912	21	22	.	.	PUNCT
ejpam-4912	22	1	we	we	PRON
ejpam-4912	22	2	will	will	AUX
ejpam-4912	22	3	characterize	characterize	VERB
ejpam-4912	22	4	j	j	PROPN
ejpam-4912	22	5	-	-	ADJ
ejpam-4912	22	6	total	total	ADJ
ejpam-4912	22	7	dominating	dominating	NOUN
ejpam-4912	22	8	sets	set	NOUN
ejpam-4912	22	9	in	in	ADP
ejpam-4912	22	10	some	some	DET
ejpam-4912	22	11	classes	class	NOUN
ejpam-4912	22	12	of	of	ADP
ejpam-4912	22	13	graphs	graph	NOUN
ejpam-4912	22	14	and	and	CCONJ
ejpam-4912	22	15	the	the	DET
ejpam-4912	22	16	join	join	NOUN
ejpam-4912	22	17	of	of	ADP
ejpam-4912	22	18	two	two	NUM
ejpam-4912	22	19	graphs	graph	NOUN
ejpam-4912	22	20	,	,	PUNCT
ejpam-4912	22	21	and	and	CCONJ
ejpam-4912	22	22	we	we	PRON
ejpam-4912	22	23	will	will	AUX
ejpam-4912	22	24	use	use	VERB
ejpam-4912	22	25	these	these	DET
ejpam-4912	22	26	results	result	NOUN
ejpam-4912	22	27	to	to	PART
ejpam-4912	22	28	determine	determine	VERB
ejpam-4912	22	29	the	the	DET
ejpam-4912	22	30	exact	exact	ADJ
ejpam-4912	22	31	value	value	NOUN
ejpam-4912	22	32	of	of	ADP
ejpam-4912	22	33	each	each	PRON
ejpam-4912	22	34	of	of	ADP
ejpam-4912	22	35	these	these	DET
ejpam-4912	22	36	graphs	graph	NOUN
ejpam-4912	22	37	.	.	PUNCT
ejpam-4912	23	1	moreover	moreover	ADV
ejpam-4912	23	2	,	,	PUNCT
ejpam-4912	23	3	we	we	PRON
ejpam-4912	23	4	will	will	AUX
ejpam-4912	23	5	determine	determine	VERB
ejpam-4912	23	6	the	the	DET
ejpam-4912	23	7	bound	bind	VERB
ejpam-4912	23	8	of	of	ADP
ejpam-4912	23	9	the	the	DET
ejpam-4912	23	10	parameter	parameter	NOUN
ejpam-4912	23	11	for	for	ADP
ejpam-4912	23	12	the	the	DET
ejpam-4912	23	13	corona	corona	NOUN
ejpam-4912	23	14	of	of	ADP
ejpam-4912	23	15	two	two	NUM
ejpam-4912	23	16	graphs	graph	NOUN
ejpam-4912	23	17	.	.	PUNCT
ejpam-4912	24	1	we	we	PRON
ejpam-4912	24	2	believe	believe	VERB
ejpam-4912	24	3	that	that	SCONJ
ejpam-4912	24	4	this	this	DET
ejpam-4912	24	5	new	new	ADJ
ejpam-4912	24	6	parameter	parameter	NOUN
ejpam-4912	24	7	would	would	AUX
ejpam-4912	24	8	give	give	VERB
ejpam-4912	24	9	additional	additional	ADJ
ejpam-4912	24	10	insights	insight	NOUN
ejpam-4912	24	11	to	to	ADP
ejpam-4912	24	12	researchers	researcher	NOUN
ejpam-4912	24	13	in	in	ADP
ejpam-4912	24	14	the	the	DET
ejpam-4912	24	15	field	field	NOUN
ejpam-4912	24	16	and	and	CCONJ
ejpam-4912	24	17	would	would	AUX
ejpam-4912	24	18	help	help	VERB
ejpam-4912	24	19	them	they	PRON
ejpam-4912	24	20	for	for	ADP
ejpam-4912	24	21	more	more	ADJ
ejpam-4912	24	22	research	research	NOUN
ejpam-4912	24	23	directions	direction	NOUN
ejpam-4912	24	24	in	in	ADP
ejpam-4912	24	25	the	the	DET
ejpam-4912	24	26	future	future	NOUN
ejpam-4912	24	27	.	.	PUNCT
ejpam-4912	25	1	2	2	X
ejpam-4912	25	2	.	.	X
ejpam-4912	25	3	terminology	terminology	NOUN
ejpam-4912	25	4	and	and	CCONJ
ejpam-4912	25	5	notation	notation	NOUN
ejpam-4912	25	6	let	let	VERB
ejpam-4912	25	7	g	g	NOUN
ejpam-4912	25	8	=	=	SYM
ejpam-4912	25	9	(	(	PUNCT
ejpam-4912	25	10	v	v	NOUN
ejpam-4912	25	11	(	(	PUNCT
ejpam-4912	25	12	g	g	NOUN
ejpam-4912	25	13	)	)	PUNCT
ejpam-4912	25	14	,	,	PUNCT
ejpam-4912	25	15	e(g	e(g	PROPN
ejpam-4912	25	16	)	)	PUNCT
ejpam-4912	25	17	)	)	PUNCT
ejpam-4912	25	18	be	be	AUX
ejpam-4912	25	19	a	a	DET
ejpam-4912	25	20	graph	graph	NOUN
ejpam-4912	25	21	.	.	PUNCT
ejpam-4912	26	1	two	two	NUM
ejpam-4912	26	2	vertices	vertex	NOUN
ejpam-4912	26	3	a	a	PRON
ejpam-4912	26	4	,	,	PUNCT
ejpam-4912	26	5	b	b	NOUN
ejpam-4912	26	6	of	of	ADP
ejpam-4912	26	7	g	g	PROPN
ejpam-4912	26	8	are	be	AUX
ejpam-4912	26	9	said	say	VERB
ejpam-4912	26	10	to	to	PART
ejpam-4912	26	11	be	be	AUX
ejpam-4912	26	12	adjacent	adjacent	ADJ
ejpam-4912	26	13	,	,	PUNCT
ejpam-4912	26	14	or	or	CCONJ
ejpam-4912	26	15	neighbors	neighbor	NOUN
ejpam-4912	26	16	,	,	PUNCT
ejpam-4912	26	17	if	if	SCONJ
ejpam-4912	26	18	ab	ab	PROPN
ejpam-4912	26	19	is	be	AUX
ejpam-4912	26	20	an	an	DET
ejpam-4912	26	21	edge	edge	NOUN
ejpam-4912	26	22	of	of	ADP
ejpam-4912	26	23	g.	g.	PROPN
ejpam-4912	26	24	the	the	DET
ejpam-4912	26	25	open	open	ADJ
ejpam-4912	26	26	neighborhood	neighborhood	NOUN
ejpam-4912	26	27	of	of	ADP
ejpam-4912	26	28	x	x	PUNCT
ejpam-4912	26	29	in	in	ADP
ejpam-4912	26	30	g	g	PROPN
ejpam-4912	26	31	is	be	AUX
ejpam-4912	26	32	the	the	DET
ejpam-4912	26	33	set	set	NOUN
ejpam-4912	26	34	defined	define	VERB
ejpam-4912	26	35	by	by	ADP
ejpam-4912	26	36	ng(x	ng(x	NUM
ejpam-4912	26	37	)	)	PUNCT
ejpam-4912	27	1	=	=	PRON
ejpam-4912	27	2	{	{	PUNCT
ejpam-4912	27	3	y	y	PROPN
ejpam-4912	27	4	∈	∈	PROPN
ejpam-4912	27	5	v	v	NOUN
ejpam-4912	27	6	(	(	PUNCT
ejpam-4912	27	7	g	g	NOUN
ejpam-4912	27	8	)	)	PUNCT
ejpam-4912	27	9	:	:	PUNCT
ejpam-4912	27	10	xy	xy	PROPN
ejpam-4912	27	11	∈	∈	PROPN
ejpam-4912	27	12	e(g	e(g	PROPN
ejpam-4912	27	13	)	)	PUNCT
ejpam-4912	27	14	}	}	PUNCT
ejpam-4912	27	15	.	.	PUNCT
ejpam-4912	28	1	the	the	DET
ejpam-4912	28	2	closed	closed	ADJ
ejpam-4912	28	3	neighborhood	neighborhood	NOUN
ejpam-4912	28	4	of	of	ADP
ejpam-4912	28	5	x	x	PUNCT
ejpam-4912	28	6	in	in	ADP
ejpam-4912	28	7	g	g	PROPN
ejpam-4912	28	8	is	be	AUX
ejpam-4912	28	9	the	the	DET
ejpam-4912	28	10	set	set	NOUN
ejpam-4912	28	11	ng[x	ng[x	PROPN
ejpam-4912	28	12	]	]	X
ejpam-4912	28	13	=	=	PUNCT
ejpam-4912	28	14	ng(x	ng(x	X
ejpam-4912	28	15	)	)	PUNCT
ejpam-4912	28	16	∪	∪	ADP
ejpam-4912	28	17	{	{	PUNCT
ejpam-4912	28	18	x	x	NOUN
ejpam-4912	28	19	}	}	PUNCT
ejpam-4912	28	20	.	.	PUNCT
ejpam-4912	29	1	if	if	SCONJ
ejpam-4912	29	2	x	x	PROPN
ejpam-4912	29	3	⊆	⊆	NUM
ejpam-4912	29	4	v	v	X
ejpam-4912	29	5	(	(	PUNCT
ejpam-4912	29	6	g	g	NOUN
ejpam-4912	29	7	)	)	PUNCT
ejpam-4912	29	8	,	,	PUNCT
ejpam-4912	29	9	then	then	ADV
ejpam-4912	29	10	open	open	VERB
ejpam-4912	29	11	neighborhood	neighborhood	NOUN
ejpam-4912	29	12	of	of	ADP
ejpam-4912	29	13	x	x	PUNCT
ejpam-4912	29	14	in	in	ADP
ejpam-4912	29	15	g	g	PROPN
ejpam-4912	29	16	is	be	AUX
ejpam-4912	29	17	the	the	DET
ejpam-4912	29	18	set	set	NOUN
ejpam-4912	29	19	ng(x	ng(x	NUM
ejpam-4912	29	20	)	)	PUNCT
ejpam-4912	30	1	=	=	SYM
ejpam-4912	30	2	⋃	⋃	NOUN
ejpam-4912	30	3	x∈x	x∈x	NOUN
ejpam-4912	30	4	ng(x	ng(x	NUM
ejpam-4912	30	5	)	)	PUNCT
ejpam-4912	30	6	.	.	PUNCT
ejpam-4912	31	1	the	the	DET
ejpam-4912	31	2	closed	closed	ADJ
ejpam-4912	31	3	neighborhood	neighborhood	NOUN
ejpam-4912	31	4	of	of	ADP
ejpam-4912	31	5	x	x	PUNCT
ejpam-4912	31	6	in	in	ADP
ejpam-4912	31	7	g	g	PROPN
ejpam-4912	31	8	is	be	AUX
ejpam-4912	31	9	the	the	DET
ejpam-4912	31	10	set	set	NOUN
ejpam-4912	31	11	ng[x	ng[x	PROPN
ejpam-4912	31	12	]	]	X
ejpam-4912	31	13	=	=	SYM
ejpam-4912	31	14	ng(x)∪x	ng(x)∪x	PROPN
ejpam-4912	31	15	.	.	PUNCT
ejpam-4912	32	1	a	a	DET
ejpam-4912	32	2	path	path	NOUN
ejpam-4912	32	3	graph	graph	NOUN
ejpam-4912	32	4	is	be	AUX
ejpam-4912	32	5	a	a	DET
ejpam-4912	32	6	non	non	ADJ
ejpam-4912	32	7	-	-	ADJ
ejpam-4912	32	8	empty	empty	ADJ
ejpam-4912	32	9	graph	graph	NOUN
ejpam-4912	32	10	with	with	ADP
ejpam-4912	32	11	vertex	vertex	NOUN
ejpam-4912	32	12	-	-	PUNCT
ejpam-4912	32	13	set	set	VERB
ejpam-4912	32	14	{	{	PUNCT
ejpam-4912	32	15	x1	x1	PROPN
ejpam-4912	32	16	,	,	PUNCT
ejpam-4912	32	17	x2	x2	PROPN
ejpam-4912	32	18	,	,	PUNCT
ejpam-4912	32	19	.	.	PUNCT
ejpam-4912	32	20	.	.	PUNCT
ejpam-4912	33	1	.	.	PUNCT
ejpam-4912	34	1	,	,	PUNCT
ejpam-4912	34	2	xn	xn	X
ejpam-4912	34	3	}	}	PUNCT
ejpam-4912	34	4	and	and	CCONJ
ejpam-4912	34	5	edge	edge	NOUN
ejpam-4912	34	6	-	-	PUNCT
ejpam-4912	34	7	set	set	NOUN
ejpam-4912	34	8	{	{	PUNCT
ejpam-4912	34	9	x1x2	x1x2	NOUN
ejpam-4912	34	10	,	,	PUNCT
ejpam-4912	34	11	x2x3	x2x3	PROPN
ejpam-4912	34	12	,	,	PUNCT
ejpam-4912	34	13	.	.	PUNCT
ejpam-4912	34	14	.	.	PUNCT
ejpam-4912	35	1	.	.	PUNCT
ejpam-4912	36	1	,	,	PUNCT
ejpam-4912	36	2	xn−1xn	xn−1xn	PROPN
ejpam-4912	36	3	}	}	PUNCT
ejpam-4912	36	4	,	,	PUNCT
ejpam-4912	36	5	where	where	SCONJ
ejpam-4912	36	6	the	the	DET
ejpam-4912	36	7	x	x	NOUN
ejpam-4912	36	8	′	′	NOUN
ejpam-4912	36	9	is	be	AUX
ejpam-4912	36	10	are	be	AUX
ejpam-4912	36	11	all	all	ADV
ejpam-4912	36	12	distinct	distinct	ADJ
ejpam-4912	36	13	.	.	PUNCT
ejpam-4912	37	1	the	the	DET
ejpam-4912	37	2	path	path	NOUN
ejpam-4912	37	3	of	of	ADP
ejpam-4912	37	4	order	order	NOUN
ejpam-4912	37	5	n	n	NOUN
ejpam-4912	37	6	is	be	AUX
ejpam-4912	37	7	denoted	denote	VERB
ejpam-4912	37	8	by	by	ADP
ejpam-4912	37	9	pn	pn	PROPN
ejpam-4912	37	10	.	.	PUNCT
ejpam-4912	38	1	if	if	SCONJ
ejpam-4912	38	2	g	g	PROPN
ejpam-4912	38	3	is	be	AUX
ejpam-4912	38	4	a	a	DET
ejpam-4912	38	5	graph	graph	NOUN
ejpam-4912	38	6	and	and	CCONJ
ejpam-4912	38	7	u	u	NOUN
ejpam-4912	38	8	and	and	CCONJ
ejpam-4912	38	9	v	v	NOUN
ejpam-4912	38	10	are	be	AUX
ejpam-4912	38	11	vertices	vertex	NOUN
ejpam-4912	38	12	of	of	ADP
ejpam-4912	38	13	g	g	NOUN
ejpam-4912	38	14	,	,	PUNCT
ejpam-4912	38	15	then	then	ADV
ejpam-4912	38	16	a	a	DET
ejpam-4912	38	17	path	path	NOUN
ejpam-4912	38	18	from	from	ADP
ejpam-4912	38	19	vertex	vertex	NOUN
ejpam-4912	38	20	u	u	NOUN
ejpam-4912	38	21	to	to	PART
ejpam-4912	38	22	vertex	vertex	NOUN
ejpam-4912	38	23	v	v	NOUN
ejpam-4912	38	24	is	be	AUX
ejpam-4912	38	25	sometimes	sometimes	ADV
ejpam-4912	38	26	called	call	VERB
ejpam-4912	38	27	a	a	DET
ejpam-4912	38	28	u	u	NOUN
ejpam-4912	38	29	-	-	NOUN
ejpam-4912	38	30	v	v	ADJ
ejpam-4912	38	31	path	path	NOUN
ejpam-4912	38	32	.	.	PUNCT
ejpam-4912	39	1	the	the	DET
ejpam-4912	39	2	cycle	cycle	NOUN
ejpam-4912	39	3	graph	graph	NOUN
ejpam-4912	39	4	is	be	AUX
ejpam-4912	39	5	the	the	DET
ejpam-4912	39	6	graph	graph	NOUN
ejpam-4912	39	7	of	of	ADP
ejpam-4912	39	8	order	order	NOUN
ejpam-4912	39	9	n	n	PRON
ejpam-4912	39	10	≥	≥	NOUN
ejpam-4912	39	11	3	3	NUM
ejpam-4912	39	12	with	with	ADP
ejpam-4912	39	13	vertex	vertex	NOUN
ejpam-4912	39	14	-	-	PUNCT
ejpam-4912	39	15	set	set	VERB
ejpam-4912	39	16	{	{	PUNCT
ejpam-4912	39	17	x1	x1	PROPN
ejpam-4912	39	18	,	,	PUNCT
ejpam-4912	39	19	x2	x2	PROPN
ejpam-4912	39	20	,	,	PUNCT
ejpam-4912	39	21	.	.	PUNCT
ejpam-4912	39	22	.	.	PUNCT
ejpam-4912	39	23	.	.	PUNCT
ejpam-4912	40	1	,	,	PUNCT
ejpam-4912	40	2	xn	xn	X
ejpam-4912	40	3	}	}	PUNCT
ejpam-4912	40	4	and	and	CCONJ
ejpam-4912	40	5	edge	edge	NOUN
ejpam-4912	40	6	-	-	PUNCT
ejpam-4912	40	7	set	set	NOUN
ejpam-4912	40	8	{	{	PUNCT
ejpam-4912	40	9	x1x2	x1x2	NOUN
ejpam-4912	40	10	,	,	PUNCT
ejpam-4912	40	11	x2x3	x2x3	PROPN
ejpam-4912	40	12	,	,	PUNCT
ejpam-4912	40	13	.	.	PUNCT
ejpam-4912	40	14	.	.	PUNCT
ejpam-4912	41	1	.	.	PUNCT
ejpam-4912	42	1	,	,	PUNCT
ejpam-4912	42	2	xn−1xn	xn−1xn	PROPN
ejpam-4912	42	3	,	,	PUNCT
ejpam-4912	42	4	xnx1	xnx1	PROPN
ejpam-4912	42	5	}	}	PUNCT
ejpam-4912	42	6	.	.	PUNCT
ejpam-4912	43	1	the	the	DET
ejpam-4912	43	2	cycle	cycle	NOUN
ejpam-4912	43	3	graph	graph	NOUN
ejpam-4912	43	4	of	of	ADP
ejpam-4912	43	5	order	order	NOUN
ejpam-4912	43	6	n	n	NOUN
ejpam-4912	43	7	is	be	AUX
ejpam-4912	43	8	denoted	denote	VERB
ejpam-4912	43	9	by	by	ADP
ejpam-4912	43	10	pn	pn	PROPN
ejpam-4912	43	11	.	.	PUNCT
ejpam-4912	43	12	a	a	DET
ejpam-4912	43	13	graph	graph	NOUN
ejpam-4912	43	14	g	g	NOUN
ejpam-4912	43	15	is	be	AUX
ejpam-4912	43	16	connected	connect	VERB
ejpam-4912	43	17	if	if	SCONJ
ejpam-4912	43	18	every	every	DET
ejpam-4912	43	19	pair	pair	NOUN
ejpam-4912	43	20	of	of	ADP
ejpam-4912	43	21	its	its	PRON
ejpam-4912	43	22	vertices	vertex	NOUN
ejpam-4912	43	23	can	can	AUX
ejpam-4912	43	24	be	be	AUX
ejpam-4912	43	25	joined	join	VERB
ejpam-4912	43	26	by	by	ADP
ejpam-4912	43	27	a	a	DET
ejpam-4912	43	28	path	path	NOUN
ejpam-4912	43	29	.	.	PUNCT
ejpam-4912	44	1	otherwise	otherwise	ADV
ejpam-4912	44	2	,	,	PUNCT
ejpam-4912	44	3	g	g	PROPN
ejpam-4912	44	4	is	be	AUX
ejpam-4912	44	5	disconnected	disconnect	VERB
ejpam-4912	44	6	.	.	PUNCT
ejpam-4912	45	1	a	a	DET
ejpam-4912	45	2	maximal	maximal	ADJ
ejpam-4912	45	3	connected	connected	ADJ
ejpam-4912	45	4	subgraph	subgraph	NOUN
ejpam-4912	45	5	(	(	PUNCT
ejpam-4912	45	6	not	not	PART
ejpam-4912	45	7	a	a	DET
ejpam-4912	45	8	subgraph	subgraph	NOUN
ejpam-4912	45	9	of	of	ADP
ejpam-4912	45	10	any	any	DET
ejpam-4912	45	11	connected	connected	ADJ
ejpam-4912	45	12	subgraph	subgraph	NOUN
ejpam-4912	45	13	)	)	PUNCT
ejpam-4912	45	14	of	of	ADP
ejpam-4912	45	15	g	g	PROPN
ejpam-4912	45	16	is	be	AUX
ejpam-4912	45	17	called	call	VERB
ejpam-4912	45	18	a	a	DET
ejpam-4912	45	19	component	component	NOUN
ejpam-4912	45	20	of	of	ADP
ejpam-4912	45	21	g.	g.	PROPN
ejpam-4912	45	22	the	the	DET
ejpam-4912	45	23	distance	distance	NOUN
ejpam-4912	45	24	dg(u	dg(u	X
ejpam-4912	45	25	,	,	PUNCT
ejpam-4912	45	26	v	v	NOUN
ejpam-4912	45	27	)	)	PUNCT
ejpam-4912	45	28	in	in	ADP
ejpam-4912	45	29	g	g	NOUN
ejpam-4912	45	30	of	of	ADP
ejpam-4912	45	31	two	two	NUM
ejpam-4912	45	32	vertices	vertex	NOUN
ejpam-4912	45	33	u	u	NOUN
ejpam-4912	45	34	,	,	PUNCT
ejpam-4912	45	35	v	v	PROPN
ejpam-4912	45	36	is	be	AUX
ejpam-4912	45	37	the	the	DET
ejpam-4912	45	38	length	length	NOUN
ejpam-4912	45	39	of	of	ADP
ejpam-4912	45	40	a	a	DET
ejpam-4912	45	41	shortest	short	ADJ
ejpam-4912	45	42	u	u	NOUN
ejpam-4912	45	43	-	-	NOUN
ejpam-4912	45	44	v	v	ADJ
ejpam-4912	45	45	path	path	NOUN
ejpam-4912	45	46	in	in	ADP
ejpam-4912	45	47	g.	g.	PROPN
ejpam-4912	45	48	the	the	DET
ejpam-4912	45	49	greatest	great	ADJ
ejpam-4912	45	50	distance	distance	NOUN
ejpam-4912	45	51	between	between	ADP
ejpam-4912	45	52	any	any	DET
ejpam-4912	45	53	two	two	NUM
ejpam-4912	45	54	vertices	vertex	NOUN
ejpam-4912	45	55	in	in	ADP
ejpam-4912	45	56	g	g	NOUN
ejpam-4912	45	57	,	,	PUNCT
ejpam-4912	45	58	denoted	denote	VERB
ejpam-4912	45	59	by	by	ADP
ejpam-4912	45	60	diam(g	diam(g	PROPN
ejpam-4912	45	61	)	)	PUNCT
ejpam-4912	45	62	,	,	PUNCT
ejpam-4912	45	63	is	be	AUX
ejpam-4912	45	64	called	call	VERB
ejpam-4912	45	65	the	the	DET
ejpam-4912	45	66	diameter	diameter	NOUN
ejpam-4912	45	67	of	of	ADP
ejpam-4912	45	68	g.	g.	PROPN
ejpam-4912	45	69	a	a	DET
ejpam-4912	45	70	subset	subset	NOUN
ejpam-4912	45	71	s	s	NOUN
ejpam-4912	45	72	of	of	ADP
ejpam-4912	45	73	v	v	NOUN
ejpam-4912	45	74	(	(	PUNCT
ejpam-4912	45	75	g	g	NOUN
ejpam-4912	45	76	)	)	PUNCT
ejpam-4912	45	77	is	be	AUX
ejpam-4912	45	78	called	call	VERB
ejpam-4912	45	79	a	a	DET
ejpam-4912	45	80	dominating	dominating	NOUN
ejpam-4912	45	81	of	of	ADP
ejpam-4912	45	82	g	g	PROPN
ejpam-4912	45	83	if	if	SCONJ
ejpam-4912	45	84	for	for	ADP
ejpam-4912	45	85	every	every	DET
ejpam-4912	45	86	x	x	SYM
ejpam-4912	45	87	∈	∈	PROPN
ejpam-4912	45	88	v	v	ADP
ejpam-4912	45	89	(	(	PUNCT
ejpam-4912	45	90	g	g	NOUN
ejpam-4912	45	91	)	)	PUNCT
ejpam-4912	45	92	\	\	PROPN
ejpam-4912	46	1	s	s	X
ejpam-4912	46	2	,	,	PUNCT
ejpam-4912	46	3	there	there	PRON
ejpam-4912	46	4	exists	exist	VERB
ejpam-4912	46	5	y	y	PROPN
ejpam-4912	46	6	∈	∈	PROPN
ejpam-4912	46	7	s	s	VERB
ejpam-4912	46	8	such	such	ADJ
ejpam-4912	46	9	that	that	SCONJ
ejpam-4912	46	10	xy	xy	PROPN
ejpam-4912	46	11	∈	∈	PROPN
ejpam-4912	46	12	e(g	e(g	PROPN
ejpam-4912	46	13	)	)	PUNCT
ejpam-4912	46	14	,	,	PUNCT
ejpam-4912	46	15	that	that	ADV
ejpam-4912	46	16	is	is	ADV
ejpam-4912	46	17	,	,	PUNCT
ejpam-4912	46	18	ng[s	ng[s	PROPN
ejpam-4912	46	19	]	]	PUNCT
ejpam-4912	46	20	=	=	SYM
ejpam-4912	46	21	v	v	X
ejpam-4912	46	22	(	(	PUNCT
ejpam-4912	46	23	g	g	NOUN
ejpam-4912	46	24	)	)	PUNCT
ejpam-4912	46	25	.	.	PUNCT
ejpam-4912	47	1	the	the	DET
ejpam-4912	47	2	domination	domination	NOUN
ejpam-4912	47	3	number	number	NOUN
ejpam-4912	47	4	of	of	ADP
ejpam-4912	47	5	g	g	NOUN
ejpam-4912	47	6	,	,	PUNCT
ejpam-4912	47	7	denoted	denote	VERB
ejpam-4912	47	8	by	by	ADP
ejpam-4912	47	9	γ(g	γ(g	PROPN
ejpam-4912	47	10	)	)	PUNCT
ejpam-4912	47	11	,	,	PUNCT
ejpam-4912	47	12	is	be	AUX
ejpam-4912	47	13	the	the	DET
ejpam-4912	47	14	minimum	minimum	ADJ
ejpam-4912	47	15	cardinality	cardinality	NOUN
ejpam-4912	47	16	of	of	ADP
ejpam-4912	47	17	a	a	DET
ejpam-4912	47	18	dominating	dominating	NOUN
ejpam-4912	47	19	set	set	VERB
ejpam-4912	47	20	in	in	ADP
ejpam-4912	47	21	g.	g.	PROPN
ejpam-4912	47	22	any	any	DET
ejpam-4912	47	23	dominating	dominating	NOUN
ejpam-4912	47	24	set	set	NOUN
ejpam-4912	47	25	s	s	NOUN
ejpam-4912	47	26	with	with	ADP
ejpam-4912	47	27	cardinality	cardinality	NOUN
ejpam-4912	47	28	equal	equal	ADJ
ejpam-4912	47	29	to	to	ADP
ejpam-4912	47	30	γ(g	γ(g	PROPN
ejpam-4912	47	31	)	)	PUNCT
ejpam-4912	47	32	is	be	AUX
ejpam-4912	47	33	called	call	VERB
ejpam-4912	47	34	a	a	DET
ejpam-4912	47	35	γ	γ	NOUN
ejpam-4912	47	36	-	-	PUNCT
ejpam-4912	47	37	set	set	NOUN
ejpam-4912	47	38	of	of	ADP
ejpam-4912	47	39	g.	g.	PROPN
ejpam-4912	47	40	a	a	DET
ejpam-4912	47	41	subset	subset	NOUN
ejpam-4912	47	42	t	t	NOUN
ejpam-4912	47	43	of	of	ADP
ejpam-4912	47	44	v	v	PROPN
ejpam-4912	47	45	(	(	PUNCT
ejpam-4912	47	46	g	g	NOUN
ejpam-4912	47	47	)	)	PUNCT
ejpam-4912	47	48	is	be	AUX
ejpam-4912	47	49	called	call	VERB
ejpam-4912	47	50	a	a	DET
ejpam-4912	47	51	total	total	ADJ
ejpam-4912	47	52	dominating	dominating	NOUN
ejpam-4912	47	53	of	of	ADP
ejpam-4912	47	54	g	g	PROPN
ejpam-4912	47	55	if	if	SCONJ
ejpam-4912	47	56	for	for	ADP
ejpam-4912	47	57	every	every	DET
ejpam-4912	47	58	x	x	SYM
ejpam-4912	47	59	∈	∈	PROPN
ejpam-4912	47	60	v	v	NOUN
ejpam-4912	47	61	(	(	PUNCT
ejpam-4912	47	62	g	g	NOUN
ejpam-4912	47	63	)	)	PUNCT
ejpam-4912	47	64	,	,	PUNCT
ejpam-4912	47	65	there	there	PRON
ejpam-4912	47	66	exists	exist	VERB
ejpam-4912	47	67	y	y	PROPN
ejpam-4912	47	68	∈	∈	PROPN
ejpam-4912	47	69	t	t	PROPN
ejpam-4912	47	70	such	such	ADJ
ejpam-4912	47	71	that	that	SCONJ
ejpam-4912	47	72	xy	xy	PROPN
ejpam-4912	47	73	∈	∈	PROPN
ejpam-4912	47	74	e(g	e(g	PROPN
ejpam-4912	47	75	)	)	PUNCT
ejpam-4912	47	76	,	,	PUNCT
ejpam-4912	47	77	that	that	ADV
ejpam-4912	47	78	is	is	ADV
ejpam-4912	47	79	,	,	PUNCT
ejpam-4912	47	80	ng(t	ng(t	PUNCT
ejpam-4912	47	81	)	)	PUNCT
ejpam-4912	48	1	=	=	SYM
ejpam-4912	48	2	v	v	X
ejpam-4912	48	3	(	(	PUNCT
ejpam-4912	48	4	g	g	NOUN
ejpam-4912	48	5	)	)	PUNCT
ejpam-4912	48	6	.	.	PUNCT
ejpam-4912	49	1	the	the	DET
ejpam-4912	49	2	total	total	ADJ
ejpam-4912	49	3	domination	domination	NOUN
ejpam-4912	49	4	number	number	NOUN
ejpam-4912	49	5	of	of	ADP
ejpam-4912	49	6	g	g	NOUN
ejpam-4912	49	7	,	,	PUNCT
ejpam-4912	49	8	denoted	denote	VERB
ejpam-4912	49	9	by	by	ADP
ejpam-4912	49	10	γt(g	γt(g	NOUN
ejpam-4912	49	11	)	)	PUNCT
ejpam-4912	49	12	,	,	PUNCT
ejpam-4912	49	13	is	be	AUX
ejpam-4912	49	14	the	the	DET
ejpam-4912	49	15	minimum	minimum	ADJ
ejpam-4912	49	16	cardinality	cardinality	NOUN
ejpam-4912	49	17	of	of	ADP
ejpam-4912	49	18	a	a	DET
ejpam-4912	49	19	total	total	ADJ
ejpam-4912	49	20	dominating	dominating	NOUN
ejpam-4912	49	21	set	set	VERB
ejpam-4912	49	22	in	in	ADP
ejpam-4912	49	23	g.	g.	PROPN
ejpam-4912	49	24	any	any	DET
ejpam-4912	49	25	total	total	ADJ
ejpam-4912	49	26	dominating	dominating	NOUN
ejpam-4912	49	27	set	set	VERB
ejpam-4912	49	28	t	t	PROPN
ejpam-4912	49	29	with	with	ADP
ejpam-4912	49	30	cardinality	cardinality	NOUN
ejpam-4912	49	31	equal	equal	ADJ
ejpam-4912	49	32	to	to	ADP
ejpam-4912	49	33	γt(g	γt(g	NOUN
ejpam-4912	49	34	)	)	PUNCT
ejpam-4912	49	35	is	be	AUX
ejpam-4912	49	36	called	call	VERB
ejpam-4912	49	37	a	a	DET
ejpam-4912	49	38	γt	γt	NOUN
ejpam-4912	49	39	-	-	NOUN
ejpam-4912	49	40	set	set	NOUN
ejpam-4912	49	41	of	of	ADP
ejpam-4912	49	42	g.	g.	PROPN
ejpam-4912	49	43	a	a	DET
ejpam-4912	49	44	graph	graph	NOUN
ejpam-4912	49	45	g	g	NOUN
ejpam-4912	49	46	is	be	AUX
ejpam-4912	49	47	complete	complete	ADJ
ejpam-4912	49	48	if	if	SCONJ
ejpam-4912	49	49	every	every	DET
ejpam-4912	49	50	pair	pair	NOUN
ejpam-4912	49	51	of	of	ADP
ejpam-4912	49	52	distinct	distinct	ADJ
ejpam-4912	49	53	vertices	vertex	NOUN
ejpam-4912	49	54	of	of	ADP
ejpam-4912	49	55	g	g	NOUN
ejpam-4912	49	56	are	be	AUX
ejpam-4912	49	57	adjacent	adjacent	ADJ
ejpam-4912	49	58	.	.	PUNCT
ejpam-4912	50	1	a	a	DET
ejpam-4912	50	2	complete	complete	ADJ
ejpam-4912	50	3	graph	graph	NOUN
ejpam-4912	50	4	of	of	ADP
ejpam-4912	50	5	order	order	NOUN
ejpam-4912	50	6	n	n	NOUN
ejpam-4912	50	7	is	be	AUX
ejpam-4912	50	8	denoted	denote	VERB
ejpam-4912	50	9	by	by	ADP
ejpam-4912	50	10	kn	kn	PROPN
ejpam-4912	50	11	.	.	PUNCT
ejpam-4912	51	1	a	a	DET
ejpam-4912	51	2	graph	graph	NOUN
ejpam-4912	51	3	g	g	NOUN
ejpam-4912	51	4	is	be	AUX
ejpam-4912	51	5	called	call	VERB
ejpam-4912	51	6	a	a	DET
ejpam-4912	51	7	bipartite	bipartite	NOUN
ejpam-4912	51	8	graph	graph	NOUN
ejpam-4912	51	9	if	if	SCONJ
ejpam-4912	51	10	its	its	PRON
ejpam-4912	51	11	vertex	vertex	NOUN
ejpam-4912	51	12	-	-	PUNCT
ejpam-4912	51	13	set	set	VERB
ejpam-4912	51	14	v	v	NOUN
ejpam-4912	51	15	(	(	PUNCT
ejpam-4912	51	16	g	g	NOUN
ejpam-4912	51	17	)	)	PUNCT
ejpam-4912	51	18	can	can	AUX
ejpam-4912	51	19	be	be	AUX
ejpam-4912	51	20	partitioned	partition	VERB
ejpam-4912	51	21	into	into	ADP
ejpam-4912	51	22	two	two	NUM
ejpam-4912	51	23	j.a	j.a	PROPN
ejpam-4912	51	24	.	.	PUNCT
ejpam-4912	52	1	hassan	hassan	PROPN
ejpam-4912	52	2	et	et	PROPN
ejpam-4912	52	3	al	al	PROPN
ejpam-4912	52	4	.	.	PUNCT
ejpam-4912	52	5	/	/	SYM
ejpam-4912	52	6	eur	eur	PROPN
ejpam-4912	52	7	.	.	PUNCT
ejpam-4912	53	1	j.	j.	PROPN
ejpam-4912	53	2	pure	pure	PROPN
ejpam-4912	53	3	appl	appl	PROPN
ejpam-4912	53	4	.	.	PROPN
ejpam-4912	53	5	math	math	PROPN
ejpam-4912	53	6	,	,	PUNCT
ejpam-4912	53	7	16	16	NUM
ejpam-4912	53	8	(	(	PUNCT
ejpam-4912	53	9	4	4	NUM
ejpam-4912	53	10	)	)	PUNCT
ejpam-4912	53	11	(	(	PUNCT
ejpam-4912	53	12	2023	2023	NUM
ejpam-4912	53	13	)	)	PUNCT
ejpam-4912	53	14	,	,	PUNCT
ejpam-4912	53	15	2106	2106	NUM
ejpam-4912	53	16	-	-	SYM
ejpam-4912	53	17	2117	2117	NUM
ejpam-4912	53	18	2108	2108	NUM
ejpam-4912	53	19	nonempty	nonempty	X
ejpam-4912	53	20	subsets	subset	NOUN
ejpam-4912	53	21	v1	v1	NOUN
ejpam-4912	53	22	and	and	CCONJ
ejpam-4912	53	23	v2	v2	VERB
ejpam-4912	53	24	such	such	ADJ
ejpam-4912	53	25	that	that	SCONJ
ejpam-4912	53	26	every	every	DET
ejpam-4912	53	27	edge	edge	NOUN
ejpam-4912	53	28	of	of	ADP
ejpam-4912	53	29	g	g	PROPN
ejpam-4912	53	30	has	have	VERB
ejpam-4912	53	31	one	one	NUM
ejpam-4912	53	32	end	end	NOUN
ejpam-4912	53	33	in	in	ADP
ejpam-4912	53	34	v1	v1	NOUN
ejpam-4912	53	35	and	and	CCONJ
ejpam-4912	53	36	one	one	NUM
ejpam-4912	53	37	end	end	NOUN
ejpam-4912	53	38	in	in	ADP
ejpam-4912	53	39	v2	v2	NOUN
ejpam-4912	53	40	.	.	PUNCT
ejpam-4912	54	1	the	the	DET
ejpam-4912	54	2	sets	set	NOUN
ejpam-4912	54	3	v1	v1	VERB
ejpam-4912	54	4	and	and	CCONJ
ejpam-4912	54	5	v2	v2	NOUN
ejpam-4912	54	6	are	be	AUX
ejpam-4912	54	7	called	call	VERB
ejpam-4912	54	8	the	the	DET
ejpam-4912	54	9	partite	partite	ADJ
ejpam-4912	54	10	sets	set	NOUN
ejpam-4912	54	11	of	of	ADP
ejpam-4912	54	12	g.	g.	PROPN
ejpam-4912	54	13	if	if	SCONJ
ejpam-4912	54	14	each	each	DET
ejpam-4912	54	15	vertex	vertex	NOUN
ejpam-4912	54	16	in	in	ADP
ejpam-4912	54	17	v1	v1	NOUN
ejpam-4912	54	18	is	be	AUX
ejpam-4912	54	19	adjacent	adjacent	ADJ
ejpam-4912	54	20	to	to	ADP
ejpam-4912	54	21	every	every	DET
ejpam-4912	54	22	vertex	vertex	NOUN
ejpam-4912	54	23	in	in	ADP
ejpam-4912	54	24	v2	v2	NOUN
ejpam-4912	54	25	,	,	PUNCT
ejpam-4912	54	26	then	then	ADV
ejpam-4912	54	27	g	g	PROPN
ejpam-4912	54	28	is	be	AUX
ejpam-4912	54	29	called	call	VERB
ejpam-4912	54	30	a	a	DET
ejpam-4912	54	31	complete	complete	ADJ
ejpam-4912	54	32	bipartite	bipartite	NOUN
ejpam-4912	54	33	graph	graph	NOUN
ejpam-4912	54	34	.	.	PUNCT
ejpam-4912	55	1	if	if	SCONJ
ejpam-4912	55	2	|v1|	|v1|	NUM
ejpam-4912	55	3	=	=	SYM
ejpam-4912	55	4	m	m	NOUN
ejpam-4912	55	5	and	and	CCONJ
ejpam-4912	55	6	|v2|	|v2|	NOUN
ejpam-4912	55	7	=	=	SYM
ejpam-4912	55	8	n	n	CCONJ
ejpam-4912	55	9	,	,	PUNCT
ejpam-4912	55	10	then	then	ADV
ejpam-4912	55	11	the	the	DET
ejpam-4912	55	12	complete	complete	ADJ
ejpam-4912	55	13	bipartite	bipartite	NOUN
ejpam-4912	55	14	graph	graph	NOUN
ejpam-4912	55	15	is	be	AUX
ejpam-4912	55	16	denoted	denote	VERB
ejpam-4912	55	17	by	by	ADP
ejpam-4912	55	18	km	km	PROPN
ejpam-4912	55	19	,	,	PUNCT
ejpam-4912	55	20	n.	n.	PROPN
ejpam-4912	55	21	a	a	DET
ejpam-4912	55	22	star	star	NOUN
ejpam-4912	55	23	graph	graph	NOUN
ejpam-4912	55	24	of	of	ADP
ejpam-4912	55	25	order	order	NOUN
ejpam-4912	55	26	n+	n+	ADP
ejpam-4912	55	27	1	1	NUM
ejpam-4912	55	28	is	be	AUX
ejpam-4912	55	29	the	the	DET
ejpam-4912	55	30	complete	complete	ADJ
ejpam-4912	55	31	bipartite	bipartite	PROPN
ejpam-4912	55	32	graph	graph	NOUN
ejpam-4912	55	33	k1,n	k1,n	PROPN
ejpam-4912	55	34	.	.	PUNCT
ejpam-4912	56	1	let	let	VERB
ejpam-4912	56	2	g	g	NOUN
ejpam-4912	56	3	and	and	CCONJ
ejpam-4912	56	4	h	h	NOUN
ejpam-4912	56	5	be	be	VERB
ejpam-4912	56	6	any	any	DET
ejpam-4912	56	7	two	two	NUM
ejpam-4912	56	8	graphs	graph	NOUN
ejpam-4912	56	9	.	.	PUNCT
ejpam-4912	57	1	the	the	DET
ejpam-4912	57	2	join	join	NOUN
ejpam-4912	57	3	of	of	ADP
ejpam-4912	57	4	g	g	PROPN
ejpam-4912	57	5	and	and	CCONJ
ejpam-4912	57	6	h	h	NOUN
ejpam-4912	57	7	,	,	PUNCT
ejpam-4912	57	8	denoted	denote	VERB
ejpam-4912	57	9	by	by	ADP
ejpam-4912	57	10	g+h	g+h	PROPN
ejpam-4912	57	11	is	be	AUX
ejpam-4912	57	12	the	the	DET
ejpam-4912	57	13	graph	graph	NOUN
ejpam-4912	57	14	with	with	ADP
ejpam-4912	57	15	vertex	vertex	NOUN
ejpam-4912	57	16	set	set	VERB
ejpam-4912	57	17	v	v	NOUN
ejpam-4912	57	18	(	(	PUNCT
ejpam-4912	57	19	g+h	g+h	NOUN
ejpam-4912	57	20	)	)	PUNCT
ejpam-4912	58	1	=	=	SYM
ejpam-4912	58	2	v	v	X
ejpam-4912	58	3	(	(	PUNCT
ejpam-4912	58	4	g	g	NOUN
ejpam-4912	58	5	)	)	PUNCT
ejpam-4912	58	6	∪	∪	NOUN
ejpam-4912	58	7	v	v	NOUN
ejpam-4912	58	8	(	(	PUNCT
ejpam-4912	58	9	h	h	NOUN
ejpam-4912	58	10	)	)	PUNCT
ejpam-4912	58	11	and	and	CCONJ
ejpam-4912	58	12	edge	edge	NOUN
ejpam-4912	58	13	set	set	VERB
ejpam-4912	58	14	e(g+h	e(g+h	NUM
ejpam-4912	58	15	)	)	PUNCT
ejpam-4912	58	16	=	=	SYM
ejpam-4912	58	17	e(g	e(g	NOUN
ejpam-4912	58	18	)	)	PUNCT
ejpam-4912	58	19	∪	∪	ADP
ejpam-4912	58	20	e(h	e(h	PROPN
ejpam-4912	58	21	)	)	PUNCT
ejpam-4912	58	22	∪	∪	NOUN
ejpam-4912	58	23	{	{	PUNCT
ejpam-4912	58	24	uv	uv	NOUN
ejpam-4912	58	25	:	:	PUNCT
ejpam-4912	58	26	u	u	PROPN
ejpam-4912	58	27	∈	∈	PROPN
ejpam-4912	58	28	v	v	ADP
ejpam-4912	58	29	(	(	PUNCT
ejpam-4912	58	30	g	g	NOUN
ejpam-4912	58	31	)	)	PUNCT
ejpam-4912	58	32	,	,	PUNCT
ejpam-4912	58	33	v	v	X
ejpam-4912	58	34	∈	∈	PROPN
ejpam-4912	58	35	v	v	NOUN
ejpam-4912	58	36	(	(	PUNCT
ejpam-4912	58	37	h	h	NOUN
ejpam-4912	58	38	)	)	PUNCT
ejpam-4912	58	39	}	}	PUNCT
ejpam-4912	58	40	.	.	PUNCT
ejpam-4912	59	1	the	the	DET
ejpam-4912	59	2	fan	fan	NOUN
ejpam-4912	59	3	fn	fn	PROPN
ejpam-4912	59	4	of	of	ADP
ejpam-4912	59	5	order	order	NOUN
ejpam-4912	59	6	n+	n+	PUNCT
ejpam-4912	59	7	1	1	NUM
ejpam-4912	59	8	,	,	PUNCT
ejpam-4912	59	9	where	where	SCONJ
ejpam-4912	59	10	n	n	PRON
ejpam-4912	59	11	≥	≥	NOUN
ejpam-4912	59	12	1	1	NUM
ejpam-4912	59	13	,	,	PUNCT
ejpam-4912	59	14	is	be	AUX
ejpam-4912	59	15	given	give	VERB
ejpam-4912	59	16	by	by	ADP
ejpam-4912	59	17	fn	fn	NOUN
ejpam-4912	59	18	=	=	PROPN
ejpam-4912	59	19	k1	k1	PROPN
ejpam-4912	59	20	+	+	CCONJ
ejpam-4912	59	21	pn	pn	PROPN
ejpam-4912	59	22	.	.	PUNCT
ejpam-4912	60	1	the	the	DET
ejpam-4912	60	2	corona	corona	NOUN
ejpam-4912	60	3	g	g	PROPN
ejpam-4912	60	4	and	and	CCONJ
ejpam-4912	60	5	h	h	NOUN
ejpam-4912	60	6	,	,	PUNCT
ejpam-4912	60	7	denoted	denote	VERB
ejpam-4912	60	8	by	by	ADP
ejpam-4912	60	9	g	g	PROPN
ejpam-4912	60	10	◦	◦	NOUN
ejpam-4912	60	11	h	h	NOUN
ejpam-4912	60	12	,	,	PUNCT
ejpam-4912	60	13	the	the	DET
ejpam-4912	60	14	graph	graph	NOUN
ejpam-4912	60	15	obtained	obtain	VERB
ejpam-4912	60	16	by	by	ADP
ejpam-4912	60	17	taking	take	VERB
ejpam-4912	60	18	one	one	NUM
ejpam-4912	60	19	copy	copy	NOUN
ejpam-4912	60	20	of	of	ADP
ejpam-4912	60	21	g	g	PROPN
ejpam-4912	60	22	and	and	CCONJ
ejpam-4912	60	23	|v	|v	PROPN
ejpam-4912	60	24	(	(	PUNCT
ejpam-4912	60	25	g)|	g)|	NOUN
ejpam-4912	60	26	copies	copy	NOUN
ejpam-4912	60	27	of	of	ADP
ejpam-4912	60	28	h	h	NOUN
ejpam-4912	60	29	,	,	PUNCT
ejpam-4912	60	30	and	and	CCONJ
ejpam-4912	60	31	then	then	ADV
ejpam-4912	60	32	joining	join	VERB
ejpam-4912	60	33	the	the	DET
ejpam-4912	60	34	ith	ith	PROPN
ejpam-4912	60	35	vertex	vertex	NOUN
ejpam-4912	60	36	of	of	ADP
ejpam-4912	60	37	g	g	NOUN
ejpam-4912	60	38	to	to	ADP
ejpam-4912	60	39	every	every	DET
ejpam-4912	60	40	vertex	vertex	NOUN
ejpam-4912	60	41	of	of	ADP
ejpam-4912	60	42	the	the	DET
ejpam-4912	60	43	ith	ith	PROPN
ejpam-4912	60	44	copy	copy	NOUN
ejpam-4912	60	45	of	of	ADP
ejpam-4912	60	46	h.	h.	PROPN
ejpam-4912	60	47	we	we	PRON
ejpam-4912	60	48	denote	denote	VERB
ejpam-4912	60	49	by	by	ADP
ejpam-4912	60	50	hv	hv	PROPN
ejpam-4912	61	1	the	the	DET
ejpam-4912	61	2	copy	copy	NOUN
ejpam-4912	61	3	of	of	ADP
ejpam-4912	61	4	h	h	NOUN
ejpam-4912	61	5	in	in	ADP
ejpam-4912	61	6	g	g	PROPN
ejpam-4912	61	7	◦	◦	NOUN
ejpam-4912	61	8	h	h	NOUN
ejpam-4912	61	9	corresponding	correspond	VERB
ejpam-4912	61	10	to	to	ADP
ejpam-4912	61	11	the	the	DET
ejpam-4912	61	12	vertex	vertex	NOUN
ejpam-4912	61	13	v	v	ADP
ejpam-4912	61	14	∈	∈	PROPN
ejpam-4912	61	15	g	g	NOUN
ejpam-4912	61	16	and	and	CCONJ
ejpam-4912	61	17	write	write	VERB
ejpam-4912	61	18	v	v	ADP
ejpam-4912	61	19	+	+	PROPN
ejpam-4912	61	20	hv	hv	NOUN
ejpam-4912	61	21	for	for	ADP
ejpam-4912	61	22	⟨{v}+hv⟩.	⟨{v}+hv⟩.	X
ejpam-4912	61	23	3	3	X
ejpam-4912	61	24	.	.	X
ejpam-4912	61	25	results	result	NOUN
ejpam-4912	61	26	we	we	PRON
ejpam-4912	61	27	begin	begin	VERB
ejpam-4912	61	28	this	this	DET
ejpam-4912	61	29	section	section	NOUN
ejpam-4912	61	30	by	by	ADP
ejpam-4912	61	31	introducing	introduce	VERB
ejpam-4912	61	32	the	the	DET
ejpam-4912	61	33	concept	concept	NOUN
ejpam-4912	61	34	of	of	ADP
ejpam-4912	61	35	j	j	PROPN
ejpam-4912	61	36	-	-	ADJ
ejpam-4912	61	37	total	total	ADJ
ejpam-4912	61	38	domination	domination	NOUN
ejpam-4912	61	39	in	in	ADP
ejpam-4912	61	40	a	a	DET
ejpam-4912	61	41	graph	graph	NOUN
ejpam-4912	61	42	.	.	PUNCT
ejpam-4912	62	1	definition	definition	NOUN
ejpam-4912	62	2	1	1	NUM
ejpam-4912	62	3	.	.	PUNCT
ejpam-4912	63	1	let	let	VERB
ejpam-4912	63	2	g	g	PRON
ejpam-4912	63	3	be	be	AUX
ejpam-4912	63	4	a	a	DET
ejpam-4912	63	5	graph	graph	NOUN
ejpam-4912	63	6	with	with	ADP
ejpam-4912	63	7	no	no	DET
ejpam-4912	63	8	isolated	isolated	ADJ
ejpam-4912	63	9	vertex	vertex	NOUN
ejpam-4912	63	10	.	.	PUNCT
ejpam-4912	64	1	a	a	DET
ejpam-4912	64	2	subset	subset	NOUN
ejpam-4912	64	3	m	m	VERB
ejpam-4912	64	4	⊆	⊆	NUM
ejpam-4912	64	5	v	v	NOUN
ejpam-4912	64	6	(	(	PUNCT
ejpam-4912	64	7	g	g	NOUN
ejpam-4912	64	8	)	)	PUNCT
ejpam-4912	64	9	is	be	AUX
ejpam-4912	64	10	called	call	VERB
ejpam-4912	64	11	a	a	DET
ejpam-4912	64	12	j	j	NOUN
ejpam-4912	64	13	-	-	ADJ
ejpam-4912	64	14	open	open	ADJ
ejpam-4912	64	15	set	set	NOUN
ejpam-4912	64	16	in	in	ADP
ejpam-4912	64	17	g	g	PROPN
ejpam-4912	64	18	if	if	SCONJ
ejpam-4912	64	19	ng(a	ng(a	NOUN
ejpam-4912	64	20	)	)	PUNCT
ejpam-4912	64	21	\	\	NOUN
ejpam-4912	65	1	ng(b	ng(b	NOUN
ejpam-4912	65	2	)	)	PUNCT
ejpam-4912	65	3	̸=	̸=	PROPN
ejpam-4912	65	4	∅	∅	NOUN
ejpam-4912	65	5	and	and	CCONJ
ejpam-4912	65	6	ng(b	ng(b	NOUN
ejpam-4912	65	7	)	)	PUNCT
ejpam-4912	65	8	\	\	NOUN
ejpam-4912	65	9	ng(a	ng(a	NOUN
ejpam-4912	65	10	)	)	PUNCT
ejpam-4912	65	11	̸=	̸=	PROPN
ejpam-4912	65	12	∅	∅	NOUN
ejpam-4912	65	13	∀	∀	X
ejpam-4912	65	14	a	a	PRON
ejpam-4912	65	15	,	,	PUNCT
ejpam-4912	65	16	b	b	PROPN
ejpam-4912	65	17	∈	∈	PROPN
ejpam-4912	65	18	m	m	PROPN
ejpam-4912	65	19	,	,	PUNCT
ejpam-4912	65	20	a	a	DET
ejpam-4912	65	21	̸=	̸=	PROPN
ejpam-4912	65	22	b.	b.	NOUN
ejpam-4912	65	23	if	if	SCONJ
ejpam-4912	65	24	in	in	ADP
ejpam-4912	65	25	addition	addition	NOUN
ejpam-4912	65	26	,	,	PUNCT
ejpam-4912	65	27	m	m	VERB
ejpam-4912	65	28	is	be	AUX
ejpam-4912	65	29	a	a	DET
ejpam-4912	65	30	total	total	ADJ
ejpam-4912	65	31	dominating	dominating	NOUN
ejpam-4912	65	32	in	in	ADP
ejpam-4912	65	33	g	g	PROPN
ejpam-4912	65	34	,	,	PUNCT
ejpam-4912	65	35	then	then	ADV
ejpam-4912	65	36	we	we	PRON
ejpam-4912	65	37	call	call	VERB
ejpam-4912	65	38	m	m	VERB
ejpam-4912	65	39	a	a	DET
ejpam-4912	65	40	j	j	PROPN
ejpam-4912	65	41	-	-	ADJ
ejpam-4912	65	42	total	total	ADJ
ejpam-4912	65	43	dominating	dominating	NOUN
ejpam-4912	65	44	set	set	VERB
ejpam-4912	65	45	in	in	ADP
ejpam-4912	65	46	g.	g.	PROPN
ejpam-4912	65	47	the	the	DET
ejpam-4912	65	48	maximun	maximun	PROPN
ejpam-4912	65	49	cardinality	cardinality	PROPN
ejpam-4912	65	50	among	among	ADP
ejpam-4912	65	51	all	all	DET
ejpam-4912	65	52	j	j	PROPN
ejpam-4912	65	53	-	-	ADJ
ejpam-4912	65	54	total	total	ADJ
ejpam-4912	65	55	dominating	dominating	NOUN
ejpam-4912	65	56	sets	set	NOUN
ejpam-4912	65	57	in	in	ADP
ejpam-4912	65	58	g	g	NOUN
ejpam-4912	65	59	,	,	PUNCT
ejpam-4912	65	60	denoted	denote	VERB
ejpam-4912	65	61	by	by	ADP
ejpam-4912	65	62	γjt(g	γjt(g	PROPN
ejpam-4912	65	63	)	)	PUNCT
ejpam-4912	65	64	,	,	PUNCT
ejpam-4912	65	65	is	be	AUX
ejpam-4912	65	66	called	call	VERB
ejpam-4912	65	67	the	the	DET
ejpam-4912	65	68	j	j	PROPN
ejpam-4912	65	69	-	-	PUNCT
ejpam-4912	65	70	total	total	ADJ
ejpam-4912	65	71	domination	domination	NOUN
ejpam-4912	65	72	number	number	NOUN
ejpam-4912	65	73	of	of	ADP
ejpam-4912	65	74	g.	g.	PROPN
ejpam-4912	65	75	any	any	DET
ejpam-4912	65	76	j	j	PROPN
ejpam-4912	65	77	-	-	ADJ
ejpam-4912	65	78	total	total	ADJ
ejpam-4912	65	79	dominating	dominating	NOUN
ejpam-4912	65	80	set	set	NOUN
ejpam-4912	65	81	m	m	NOUN
ejpam-4912	65	82	with	with	ADP
ejpam-4912	65	83	|	|	ADV
ejpam-4912	65	84	m	m	VERB
ejpam-4912	65	85	|=	|=	NOUN
ejpam-4912	65	86	γjt(g	γjt(g	NOUN
ejpam-4912	65	87	)	)	PUNCT
ejpam-4912	65	88	(	(	PUNCT
ejpam-4912	65	89	resp	resp	NOUN
ejpam-4912	65	90	.	.	PUNCT
ejpam-4912	66	1	|m	|m	NOUN
ejpam-4912	66	2	|	|	ADV
ejpam-4912	66	3	=	=	PRON
ejpam-4912	66	4	γt(g	γt(g	NUM
ejpam-4912	66	5	)	)	PUNCT
ejpam-4912	66	6	)	)	PUNCT
ejpam-4912	66	7	,	,	PUNCT
ejpam-4912	66	8	is	be	AUX
ejpam-4912	66	9	called	call	VERB
ejpam-4912	66	10	a	a	DET
ejpam-4912	66	11	γjt	γjt	NOUN
ejpam-4912	66	12	-	-	PUNCT
ejpam-4912	66	13	set	set	VERB
ejpam-4912	66	14	or	or	CCONJ
ejpam-4912	66	15	the	the	DET
ejpam-4912	66	16	maximum	maximum	ADJ
ejpam-4912	66	17	(	(	PUNCT
ejpam-4912	66	18	resp	resp	NOUN
ejpam-4912	66	19	.	.	PUNCT
ejpam-4912	67	1	minimum	minimum	ADJ
ejpam-4912	67	2	)	)	PUNCT
ejpam-4912	67	3	j	j	NOUN
ejpam-4912	67	4	-	-	PUNCT
ejpam-4912	67	5	total	total	ADJ
ejpam-4912	67	6	dominating	dominating	NOUN
ejpam-4912	67	7	set	set	VERB
ejpam-4912	67	8	in	in	ADP
ejpam-4912	67	9	g.	g.	PROPN
ejpam-4912	67	10	remark	remark	PROPN
ejpam-4912	67	11	1	1	NUM
ejpam-4912	67	12	.	.	PUNCT
ejpam-4912	68	1	let	let	VERB
ejpam-4912	68	2	g	g	PRON
ejpam-4912	68	3	be	be	AUX
ejpam-4912	68	4	a	a	DET
ejpam-4912	68	5	graph	graph	NOUN
ejpam-4912	68	6	with	with	ADP
ejpam-4912	68	7	no	no	DET
ejpam-4912	68	8	isolated	isolated	ADJ
ejpam-4912	68	9	vertex	vertex	NOUN
ejpam-4912	68	10	.	.	PUNCT
ejpam-4912	69	1	then	then	ADV
ejpam-4912	69	2	each	each	PRON
ejpam-4912	69	3	of	of	ADP
ejpam-4912	69	4	the	the	DET
ejpam-4912	69	5	following	following	NOUN
ejpam-4912	69	6	is	be	AUX
ejpam-4912	69	7	true	true	ADJ
ejpam-4912	69	8	.	.	PUNCT
ejpam-4912	70	1	(	(	PUNCT
ejpam-4912	70	2	i	i	NOUN
ejpam-4912	70	3	)	)	PUNCT
ejpam-4912	70	4	a	a	DET
ejpam-4912	70	5	total	total	ADJ
ejpam-4912	70	6	dominating	dominating	NOUN
ejpam-4912	70	7	set	set	VERB
ejpam-4912	70	8	t	t	PROPN
ejpam-4912	70	9	of	of	ADP
ejpam-4912	70	10	g	g	PROPN
ejpam-4912	70	11	may	may	AUX
ejpam-4912	70	12	not	not	PART
ejpam-4912	70	13	be	be	AUX
ejpam-4912	70	14	a	a	DET
ejpam-4912	70	15	j	j	NOUN
ejpam-4912	70	16	-	-	ADJ
ejpam-4912	70	17	open	open	ADJ
ejpam-4912	70	18	set	set	NOUN
ejpam-4912	70	19	in	in	ADP
ejpam-4912	70	20	g	g	PROPN
ejpam-4912	70	21	(	(	PUNCT
ejpam-4912	70	22	hence	hence	ADV
ejpam-4912	70	23	not	not	PART
ejpam-4912	70	24	a	a	DET
ejpam-4912	70	25	j	j	PROPN
ejpam-4912	70	26	-	-	ADJ
ejpam-4912	70	27	total	total	ADJ
ejpam-4912	70	28	dominating	dominating	NOUN
ejpam-4912	70	29	set	set	NOUN
ejpam-4912	70	30	)	)	PUNCT
ejpam-4912	70	31	.	.	PUNCT
ejpam-4912	71	1	(	(	PUNCT
ejpam-4912	71	2	ii	ii	NOUN
ejpam-4912	71	3	)	)	PUNCT
ejpam-4912	71	4	a	a	DET
ejpam-4912	71	5	j	j	NOUN
ejpam-4912	71	6	-	-	ADJ
ejpam-4912	71	7	open	open	ADJ
ejpam-4912	71	8	set	set	NOUN
ejpam-4912	71	9	q	q	NOUN
ejpam-4912	71	10	in	in	ADP
ejpam-4912	71	11	g	g	NOUN
ejpam-4912	71	12	may	may	AUX
ejpam-4912	71	13	not	not	PART
ejpam-4912	71	14	be	be	AUX
ejpam-4912	71	15	a	a	DET
ejpam-4912	71	16	total	total	ADJ
ejpam-4912	71	17	dominating	dominating	NOUN
ejpam-4912	71	18	set	set	NOUN
ejpam-4912	71	19	in	in	ADP
ejpam-4912	71	20	g	g	PROPN
ejpam-4912	71	21	(	(	PUNCT
ejpam-4912	71	22	hence	hence	ADV
ejpam-4912	71	23	not	not	PART
ejpam-4912	71	24	a	a	DET
ejpam-4912	71	25	j	j	PROPN
ejpam-4912	71	26	-	-	ADJ
ejpam-4912	71	27	total	total	ADJ
ejpam-4912	71	28	dominating	dominating	NOUN
ejpam-4912	71	29	set	set	NOUN
ejpam-4912	71	30	)	)	PUNCT
ejpam-4912	71	31	.	.	PUNCT
ejpam-4912	72	1	(	(	PUNCT
ejpam-4912	72	2	iii	iii	X
ejpam-4912	72	3	)	)	PUNCT
ejpam-4912	72	4	a	a	DET
ejpam-4912	72	5	vertex	vertex	NOUN
ejpam-4912	72	6	set	set	VERB
ejpam-4912	72	7	v	v	NOUN
ejpam-4912	72	8	(	(	PUNCT
ejpam-4912	72	9	g	g	NOUN
ejpam-4912	72	10	)	)	PUNCT
ejpam-4912	72	11	of	of	ADP
ejpam-4912	72	12	g	g	NOUN
ejpam-4912	72	13	may	may	AUX
ejpam-4912	72	14	not	not	PART
ejpam-4912	72	15	be	be	AUX
ejpam-4912	72	16	a	a	DET
ejpam-4912	72	17	j	j	PROPN
ejpam-4912	72	18	-	-	ADJ
ejpam-4912	72	19	total	total	ADJ
ejpam-4912	72	20	dominating	dominating	NOUN
ejpam-4912	72	21	set	set	VERB
ejpam-4912	72	22	in	in	ADP
ejpam-4912	72	23	g.	g.	PROPN
ejpam-4912	72	24	proposition	proposition	PROPN
ejpam-4912	72	25	1	1	X
ejpam-4912	72	26	.	.	PUNCT
ejpam-4912	73	1	let	let	VERB
ejpam-4912	73	2	g	g	PRON
ejpam-4912	73	3	be	be	AUX
ejpam-4912	73	4	a	a	DET
ejpam-4912	73	5	graph	graph	NOUN
ejpam-4912	73	6	with	with	ADP
ejpam-4912	73	7	no	no	DET
ejpam-4912	73	8	isolated	isolated	ADJ
ejpam-4912	73	9	vertex	vertex	NOUN
ejpam-4912	73	10	.	.	PUNCT
ejpam-4912	74	1	then	then	ADV
ejpam-4912	74	2	(	(	PUNCT
ejpam-4912	74	3	i	i	NOUN
ejpam-4912	74	4	)	)	PUNCT
ejpam-4912	74	5	γt(g	γt(g	PUNCT
ejpam-4912	74	6	)	)	PUNCT
ejpam-4912	74	7	≤	≤	NUM
ejpam-4912	74	8	γjt(g	γjt(g	PROPN
ejpam-4912	74	9	)	)	PUNCT
ejpam-4912	74	10	.	.	PUNCT
ejpam-4912	75	1	(	(	PUNCT
ejpam-4912	75	2	ii	ii	NOUN
ejpam-4912	75	3	)	)	PUNCT
ejpam-4912	75	4	2	2	NUM
ejpam-4912	75	5	≤	≤	NUM
ejpam-4912	75	6	γjt(g	γjt(g	NOUN
ejpam-4912	75	7	)	)	PUNCT
ejpam-4912	75	8	≤|	≤|	NOUN
ejpam-4912	75	9	v	v	ADP
ejpam-4912	75	10	(	(	PUNCT
ejpam-4912	75	11	g	g	NOUN
ejpam-4912	75	12	)	)	PUNCT
ejpam-4912	75	13	|	|	ADV
ejpam-4912	75	14	.	.	PUNCT
ejpam-4912	76	1	proof	proof	NOUN
ejpam-4912	76	2	.	.	PUNCT
ejpam-4912	77	1	(	(	PUNCT
ejpam-4912	77	2	i	i	NOUN
ejpam-4912	77	3	)	)	PUNCT
ejpam-4912	77	4	let	let	VERB
ejpam-4912	77	5	g	g	NOUN
ejpam-4912	77	6	be	be	AUX
ejpam-4912	77	7	a	a	DET
ejpam-4912	77	8	graph	graph	NOUN
ejpam-4912	77	9	with	with	ADP
ejpam-4912	77	10	no	no	DET
ejpam-4912	77	11	isolated	isolated	ADJ
ejpam-4912	77	12	vertex	vertex	NOUN
ejpam-4912	77	13	and	and	CCONJ
ejpam-4912	77	14	let	let	VERB
ejpam-4912	77	15	m	m	PRON
ejpam-4912	77	16	be	be	AUX
ejpam-4912	77	17	a	a	DET
ejpam-4912	77	18	maximum	maximum	ADJ
ejpam-4912	77	19	jtotal	jtotal	ADJ
ejpam-4912	77	20	dominating	dominating	NOUN
ejpam-4912	77	21	set	set	NOUN
ejpam-4912	77	22	of	of	ADP
ejpam-4912	77	23	g.	g.	PROPN
ejpam-4912	78	1	then	then	ADV
ejpam-4912	78	2	m	m	VERB
ejpam-4912	78	3	is	be	AUX
ejpam-4912	78	4	a	a	DET
ejpam-4912	78	5	total	total	ADJ
ejpam-4912	78	6	dominating	dominating	NOUN
ejpam-4912	78	7	set	set	NOUN
ejpam-4912	78	8	of	of	ADP
ejpam-4912	78	9	g	g	PROPN
ejpam-4912	78	10	(	(	PUNCT
ejpam-4912	78	11	by	by	ADP
ejpam-4912	78	12	defintion	defintion	NOUN
ejpam-4912	78	13	)	)	PUNCT
ejpam-4912	78	14	.	.	PUNCT
ejpam-4912	79	1	since	since	SCONJ
ejpam-4912	79	2	j.a	j.a	PROPN
ejpam-4912	79	3	.	.	PROPN
ejpam-4912	79	4	hassan	hassan	PROPN
ejpam-4912	79	5	et	et	PROPN
ejpam-4912	79	6	al	al	PROPN
ejpam-4912	79	7	.	.	PUNCT
ejpam-4912	79	8	/	/	SYM
ejpam-4912	79	9	eur	eur	PROPN
ejpam-4912	79	10	.	.	PUNCT
ejpam-4912	80	1	j.	j.	PROPN
ejpam-4912	80	2	pure	pure	PROPN
ejpam-4912	80	3	appl	appl	PROPN
ejpam-4912	80	4	.	.	PROPN
ejpam-4912	80	5	math	math	PROPN
ejpam-4912	80	6	,	,	PUNCT
ejpam-4912	80	7	16	16	NUM
ejpam-4912	80	8	(	(	PUNCT
ejpam-4912	80	9	4	4	NUM
ejpam-4912	80	10	)	)	PUNCT
ejpam-4912	80	11	(	(	PUNCT
ejpam-4912	80	12	2023	2023	NUM
ejpam-4912	80	13	)	)	PUNCT
ejpam-4912	80	14	,	,	PUNCT
ejpam-4912	80	15	2106	2106	NUM
ejpam-4912	80	16	-	-	SYM
ejpam-4912	80	17	2117	2117	NUM
ejpam-4912	80	18	2109	2109	NUM
ejpam-4912	80	19	γt(g	γt(g	PUNCT
ejpam-4912	80	20	)	)	PUNCT
ejpam-4912	80	21	is	be	AUX
ejpam-4912	80	22	the	the	DET
ejpam-4912	80	23	minimum	minimum	ADJ
ejpam-4912	80	24	cardinality	cardinality	NOUN
ejpam-4912	80	25	among	among	ADP
ejpam-4912	80	26	all	all	DET
ejpam-4912	80	27	total	total	ADJ
ejpam-4912	80	28	dominating	dominating	NOUN
ejpam-4912	80	29	sets	set	NOUN
ejpam-4912	80	30	in	in	ADP
ejpam-4912	80	31	g	g	NOUN
ejpam-4912	80	32	,	,	PUNCT
ejpam-4912	80	33	it	it	PRON
ejpam-4912	80	34	follows	follow	VERB
ejpam-4912	80	35	that	that	SCONJ
ejpam-4912	80	36	γjt(g	γjt(g	NOUN
ejpam-4912	80	37	)	)	PUNCT
ejpam-4912	80	38	=|	=|	NOUN
ejpam-4912	81	1	m	m	VERB
ejpam-4912	81	2	|≥	|≥	ADJ
ejpam-4912	81	3	γt(g	γt(g	PUNCT
ejpam-4912	81	4	)	)	PUNCT
ejpam-4912	81	5	.	.	PUNCT
ejpam-4912	82	1	(	(	PUNCT
ejpam-4912	82	2	ii	ii	NOUN
ejpam-4912	82	3	)	)	PUNCT
ejpam-4912	82	4	since	since	SCONJ
ejpam-4912	82	5	γt(g	γt(g	NOUN
ejpam-4912	82	6	)	)	PUNCT
ejpam-4912	82	7	≥	≥	NOUN
ejpam-4912	82	8	2	2	NUM
ejpam-4912	82	9	for	for	ADP
ejpam-4912	82	10	any	any	DET
ejpam-4912	82	11	graph	graph	NOUN
ejpam-4912	82	12	g	g	NOUN
ejpam-4912	82	13	with	with	ADP
ejpam-4912	82	14	no	no	DET
ejpam-4912	82	15	isolated	isolated	ADJ
ejpam-4912	82	16	vertex	vertex	NOUN
ejpam-4912	82	17	,	,	PUNCT
ejpam-4912	82	18	and	and	CCONJ
ejpam-4912	82	19	so	so	ADV
ejpam-4912	82	20	γjt(g	γjt(g	PROPN
ejpam-4912	82	21	)	)	PUNCT
ejpam-4912	82	22	≥	≥	NOUN
ejpam-4912	82	23	2	2	NUM
ejpam-4912	82	24	by	by	ADP
ejpam-4912	82	25	(	(	PUNCT
ejpam-4912	82	26	i	i	NOUN
ejpam-4912	82	27	)	)	PUNCT
ejpam-4912	82	28	.	.	PUNCT
ejpam-4912	83	1	since	since	SCONJ
ejpam-4912	83	2	any	any	DET
ejpam-4912	83	3	j	j	PROPN
ejpam-4912	83	4	-	-	ADJ
ejpam-4912	83	5	total	total	ADJ
ejpam-4912	83	6	dominating	dominating	NOUN
ejpam-4912	83	7	set	set	NOUN
ejpam-4912	83	8	m	m	VERB
ejpam-4912	83	9	is	be	AUX
ejpam-4912	83	10	always	always	ADV
ejpam-4912	83	11	a	a	DET
ejpam-4912	83	12	subset	subset	NOUN
ejpam-4912	83	13	of	of	ADP
ejpam-4912	83	14	a	a	DET
ejpam-4912	83	15	vertices	vertex	NOUN
ejpam-4912	83	16	v	v	X
ejpam-4912	83	17	(	(	PUNCT
ejpam-4912	83	18	g	g	NOUN
ejpam-4912	83	19	)	)	PUNCT
ejpam-4912	83	20	of	of	ADP
ejpam-4912	83	21	g	g	PROPN
ejpam-4912	83	22	,	,	PUNCT
ejpam-4912	83	23	it	it	PRON
ejpam-4912	83	24	follows	follow	VERB
ejpam-4912	83	25	that	that	SCONJ
ejpam-4912	83	26	γjt(g	γjt(g	NOUN
ejpam-4912	83	27	)	)	PUNCT
ejpam-4912	83	28	≤|	≤|	NOUN
ejpam-4912	83	29	v	v	ADP
ejpam-4912	83	30	(	(	PUNCT
ejpam-4912	83	31	g	g	NOUN
ejpam-4912	83	32	)	)	PUNCT
ejpam-4912	83	33	|	|	ADV
ejpam-4912	83	34	.	.	PUNCT
ejpam-4912	84	1	consequently	consequently	ADV
ejpam-4912	84	2	,	,	PUNCT
ejpam-4912	84	3	2	2	NUM
ejpam-4912	84	4	≤	≤	NUM
ejpam-4912	84	5	γjt(g	γjt(g	NOUN
ejpam-4912	84	6	)	)	PUNCT
ejpam-4912	84	7	≤|	≤|	NOUN
ejpam-4912	84	8	v	v	ADP
ejpam-4912	84	9	(	(	PUNCT
ejpam-4912	84	10	g	g	NOUN
ejpam-4912	84	11	)	)	PUNCT
ejpam-4912	84	12	|	|	ADV
ejpam-4912	84	13	.	.	PUNCT
ejpam-4912	85	1	remark	remark	PROPN
ejpam-4912	85	2	2	2	NUM
ejpam-4912	85	3	.	.	PUNCT
ejpam-4912	86	1	the	the	DET
ejpam-4912	86	2	bound	bind	VERB
ejpam-4912	86	3	given	give	VERB
ejpam-4912	86	4	in	in	ADP
ejpam-4912	86	5	proposition	proposition	NOUN
ejpam-4912	86	6	1	1	NUM
ejpam-4912	86	7	is	be	AUX
ejpam-4912	86	8	tight	tight	ADJ
ejpam-4912	86	9	.	.	PUNCT
ejpam-4912	87	1	moreover	moreover	ADV
ejpam-4912	87	2	,	,	PUNCT
ejpam-4912	87	3	strict	strict	ADJ
ejpam-4912	87	4	inequality	inequality	NOUN
ejpam-4912	87	5	is	be	AUX
ejpam-4912	87	6	attainable	attainable	ADJ
ejpam-4912	87	7	.	.	PUNCT
ejpam-4912	88	1	for	for	ADP
ejpam-4912	88	2	tightness	tightness	NOUN
ejpam-4912	88	3	,	,	PUNCT
ejpam-4912	88	4	consider	consider	VERB
ejpam-4912	88	5	the	the	DET
ejpam-4912	88	6	graph	graph	NOUN
ejpam-4912	88	7	g	g	NOUN
ejpam-4912	88	8	given	give	VERB
ejpam-4912	88	9	in	in	ADP
ejpam-4912	88	10	figure	figure	NOUN
ejpam-4912	88	11	1	1	NUM
ejpam-4912	88	12	below	below	ADV
ejpam-4912	88	13	.	.	PUNCT
ejpam-4912	89	1	g	g	NOUN
ejpam-4912	89	2	:	:	PUNCT
ejpam-4912	89	3	a	a	DET
ejpam-4912	89	4	b	b	X
ejpam-4912	89	5	c	c	NOUN
ejpam-4912	89	6	d	d	X
ejpam-4912	89	7	e	e	X
ejpam-4912	89	8	a′	a′	PROPN
ejpam-4912	89	9	b′	b′	NUM
ejpam-4912	89	10	c′	c′	NOUN
ejpam-4912	89	11	d′	d′	NOUN
ejpam-4912	89	12	e′	e′	X
ejpam-4912	89	13	figure	figure	NOUN
ejpam-4912	89	14	1	1	NUM
ejpam-4912	89	15	:	:	PUNCT
ejpam-4912	89	16	a	a	DET
ejpam-4912	89	17	graph	graph	NOUN
ejpam-4912	89	18	g	g	NOUN
ejpam-4912	89	19	with	with	ADP
ejpam-4912	89	20	γt(g	γt(g	NOUN
ejpam-4912	89	21	)	)	PUNCT
ejpam-4912	89	22	=	=	SYM
ejpam-4912	89	23	5	5	NUM
ejpam-4912	89	24	=	=	SYM
ejpam-4912	89	25	γjt(g	γjt(g	PROPN
ejpam-4912	89	26	)	)	PUNCT
ejpam-4912	89	27	let	let	VERB
ejpam-4912	89	28	s	s	PRON
ejpam-4912	89	29	=	=	X
ejpam-4912	89	30	{	{	PUNCT
ejpam-4912	89	31	a	a	PRON
ejpam-4912	89	32	,	,	PUNCT
ejpam-4912	89	33	b	b	NOUN
ejpam-4912	89	34	,	,	PUNCT
ejpam-4912	89	35	c	c	NOUN
ejpam-4912	89	36	,	,	PUNCT
ejpam-4912	89	37	d	d	NOUN
ejpam-4912	89	38	,	,	PUNCT
ejpam-4912	89	39	e	e	NOUN
ejpam-4912	89	40	}	}	PUNCT
ejpam-4912	89	41	.	.	PUNCT
ejpam-4912	90	1	clearly	clearly	ADV
ejpam-4912	90	2	,	,	PUNCT
ejpam-4912	90	3	s	s	VERB
ejpam-4912	90	4	is	be	AUX
ejpam-4912	90	5	the	the	DET
ejpam-4912	90	6	minimum	minimum	ADJ
ejpam-4912	90	7	total	total	ADJ
ejpam-4912	90	8	dominating	dominating	NOUN
ejpam-4912	90	9	set	set	NOUN
ejpam-4912	90	10	of	of	ADP
ejpam-4912	90	11	g.	g.	PROPN
ejpam-4912	90	12	thus	thus	ADV
ejpam-4912	90	13	,	,	PUNCT
ejpam-4912	90	14	γt(g	γt(g	PUNCT
ejpam-4912	90	15	)	)	PUNCT
ejpam-4912	90	16	=	=	SYM
ejpam-4912	90	17	5	5	X
ejpam-4912	90	18	.	.	X
ejpam-4912	90	19	observe	observe	VERB
ejpam-4912	90	20	that	that	SCONJ
ejpam-4912	90	21	x′	x′	PROPN
ejpam-4912	90	22	∈	∈	PROPN
ejpam-4912	90	23	ng(x	ng(x	NUM
ejpam-4912	90	24	)	)	PUNCT
ejpam-4912	90	25	\ng(y	\ng(y	NOUN
ejpam-4912	90	26	)	)	PUNCT
ejpam-4912	90	27	and	and	CCONJ
ejpam-4912	90	28	y′	y′	NOUN
ejpam-4912	90	29	∈	∈	PROPN
ejpam-4912	90	30	ng(y	ng(y	NOUN
ejpam-4912	90	31	)	)	PUNCT
ejpam-4912	90	32	\ng(x	\ng(x	NUM
ejpam-4912	90	33	)	)	PUNCT
ejpam-4912	90	34	for	for	ADP
ejpam-4912	90	35	every	every	DET
ejpam-4912	90	36	x	x	PROPN
ejpam-4912	90	37	,	,	PUNCT
ejpam-4912	90	38	y	y	PROPN
ejpam-4912	90	39	∈	∈	PROPN
ejpam-4912	90	40	s	s	PROPN
ejpam-4912	90	41	,	,	PUNCT
ejpam-4912	90	42	where	where	SCONJ
ejpam-4912	90	43	x	x	X
ejpam-4912	90	44	̸=	̸=	PROPN
ejpam-4912	90	45	y.	y.	NOUN
ejpam-4912	90	46	it	it	PRON
ejpam-4912	90	47	follows	follow	VERB
ejpam-4912	90	48	that	that	SCONJ
ejpam-4912	90	49	ng(x	ng(x	NUM
ejpam-4912	90	50	)	)	PUNCT
ejpam-4912	90	51	\	\	NOUN
ejpam-4912	90	52	ng(y	ng(y	NOUN
ejpam-4912	90	53	)	)	PUNCT
ejpam-4912	90	54	̸=	̸=	PROPN
ejpam-4912	90	55	∅	∅	NOUN
ejpam-4912	90	56	and	and	CCONJ
ejpam-4912	90	57	ng(y	ng(y	NOUN
ejpam-4912	90	58	)	)	PUNCT
ejpam-4912	90	59	\	\	NOUN
ejpam-4912	90	60	ng(x	ng(x	NUM
ejpam-4912	90	61	)	)	PUNCT
ejpam-4912	90	62	̸=	̸=	NOUN
ejpam-4912	90	63	∅	∅	NOUN
ejpam-4912	90	64	for	for	ADP
ejpam-4912	90	65	every	every	DET
ejpam-4912	90	66	x	x	NOUN
ejpam-4912	90	67	,	,	PUNCT
ejpam-4912	90	68	y	y	PROPN
ejpam-4912	90	69	∈	∈	PROPN
ejpam-4912	90	70	s	s	PROPN
ejpam-4912	90	71	,	,	PUNCT
ejpam-4912	90	72	x	x	PUNCT
ejpam-4912	90	73	̸=	̸=	PROPN
ejpam-4912	90	74	y.	y.	NOUN
ejpam-4912	90	75	hence	hence	ADV
ejpam-4912	90	76	,	,	PUNCT
ejpam-4912	90	77	s	s	VERB
ejpam-4912	90	78	is	be	AUX
ejpam-4912	90	79	a	a	DET
ejpam-4912	90	80	j	j	NOUN
ejpam-4912	90	81	-	-	ADJ
ejpam-4912	90	82	open	open	ADJ
ejpam-4912	90	83	set	set	NOUN
ejpam-4912	90	84	in	in	ADP
ejpam-4912	90	85	g	g	NOUN
ejpam-4912	90	86	,	,	PUNCT
ejpam-4912	90	87	showing	show	VERB
ejpam-4912	90	88	that	that	SCONJ
ejpam-4912	90	89	s	s	VERB
ejpam-4912	90	90	is	be	AUX
ejpam-4912	90	91	a	a	DET
ejpam-4912	90	92	j	j	PROPN
ejpam-4912	90	93	-	-	ADJ
ejpam-4912	90	94	total	total	ADJ
ejpam-4912	90	95	dominating	dominating	NOUN
ejpam-4912	90	96	set	set	NOUN
ejpam-4912	90	97	of	of	ADP
ejpam-4912	90	98	g.	g.	PROPN
ejpam-4912	90	99	notice	notice	VERB
ejpam-4912	90	100	that	that	SCONJ
ejpam-4912	90	101	ng(a	ng(a	NOUN
ejpam-4912	90	102	′	′	NUM
ejpam-4912	90	103	)	)	PUNCT
ejpam-4912	90	104	,	,	PUNCT
ejpam-4912	90	105	ng(c	ng(c	PROPN
ejpam-4912	90	106	′	′	NUM
ejpam-4912	90	107	)	)	PUNCT
ejpam-4912	90	108	⊆	⊆	NUM
ejpam-4912	90	109	ng(b	ng(b	X
ejpam-4912	90	110	)	)	PUNCT
ejpam-4912	90	111	,	,	PUNCT
ejpam-4912	90	112	ng(b	ng(b	X
ejpam-4912	90	113	′	′	NOUN
ejpam-4912	90	114	)	)	PUNCT
ejpam-4912	90	115	,	,	PUNCT
ejpam-4912	90	116	ng(d	ng(d	PRON
ejpam-4912	90	117	′	′	NOUN
ejpam-4912	90	118	)	)	PUNCT
ejpam-4912	90	119	⊆	⊆	NUM
ejpam-4912	90	120	ng(c	ng(c	NUM
ejpam-4912	90	121	)	)	PUNCT
ejpam-4912	90	122	and	and	CCONJ
ejpam-4912	90	123	ng(e	ng(e	ADV
ejpam-4912	90	124	′	′	NUM
ejpam-4912	90	125	)	)	PUNCT
ejpam-4912	90	126	⊆	⊆	NUM
ejpam-4912	90	127	ng(d	ng(d	NUM
ejpam-4912	90	128	)	)	PUNCT
ejpam-4912	90	129	and	and	CCONJ
ejpam-4912	90	130	a	a	DET
ejpam-4912	90	131	,	,	PUNCT
ejpam-4912	90	132	b	b	NOUN
ejpam-4912	90	133	,	,	PUNCT
ejpam-4912	90	134	c	c	NOUN
ejpam-4912	90	135	,	,	PUNCT
ejpam-4912	90	136	d	d	NOUN
ejpam-4912	90	137	,	,	PUNCT
ejpam-4912	90	138	e	e	NOUN
ejpam-4912	90	139	must	must	AUX
ejpam-4912	90	140	be	be	AUX
ejpam-4912	90	141	in	in	ADP
ejpam-4912	90	142	any	any	DET
ejpam-4912	90	143	total	total	ADJ
ejpam-4912	90	144	dominating	dominating	NOUN
ejpam-4912	90	145	set	set	NOUN
ejpam-4912	90	146	of	of	ADP
ejpam-4912	90	147	g.	g.	PROPN
ejpam-4912	90	148	consequently	consequently	ADV
ejpam-4912	90	149	,	,	PUNCT
ejpam-4912	90	150	s	s	VERB
ejpam-4912	90	151	is	be	AUX
ejpam-4912	90	152	the	the	DET
ejpam-4912	90	153	maximum	maximum	ADJ
ejpam-4912	90	154	j	j	PROPN
ejpam-4912	90	155	-	-	ADJ
ejpam-4912	90	156	total	total	ADJ
ejpam-4912	90	157	dominating	dominating	NOUN
ejpam-4912	90	158	set	set	NOUN
ejpam-4912	90	159	of	of	ADP
ejpam-4912	90	160	g	g	NOUN
ejpam-4912	90	161	,	,	PUNCT
ejpam-4912	90	162	and	and	CCONJ
ejpam-4912	90	163	so	so	ADV
ejpam-4912	90	164	γjt(g	γjt(g	PROPN
ejpam-4912	90	165	)	)	PUNCT
ejpam-4912	90	166	=	=	SYM
ejpam-4912	91	1	5	5	X
ejpam-4912	91	2	.	.	X
ejpam-4912	91	3	for	for	ADP
ejpam-4912	91	4	strict	strict	ADJ
ejpam-4912	91	5	inequality	inequality	NOUN
ejpam-4912	91	6	,	,	PUNCT
ejpam-4912	91	7	consider	consider	VERB
ejpam-4912	91	8	the	the	DET
ejpam-4912	91	9	graph	graph	NOUN
ejpam-4912	91	10	g′	g′	NOUN
ejpam-4912	91	11	given	give	VERB
ejpam-4912	91	12	in	in	ADP
ejpam-4912	91	13	figure	figure	NOUN
ejpam-4912	91	14	2	2	NUM
ejpam-4912	91	15	below	below	ADV
ejpam-4912	91	16	.	.	PUNCT
ejpam-4912	92	1	g′	g′	NOUN
ejpam-4912	92	2	:	:	PUNCT
ejpam-4912	92	3	u1	u1	PROPN
ejpam-4912	92	4	u2	u2	PROPN
ejpam-4912	92	5	u3	u3	PROPN
ejpam-4912	92	6	u4	u4	PROPN
ejpam-4912	92	7	u8	u8	PROPN
ejpam-4912	92	8	u5	u5	PROPN
ejpam-4912	92	9	u6	u6	PROPN
ejpam-4912	92	10	u7	u7	PROPN
ejpam-4912	92	11	figure	figure	NOUN
ejpam-4912	92	12	2	2	NUM
ejpam-4912	92	13	:	:	PUNCT
ejpam-4912	92	14	a	a	DET
ejpam-4912	92	15	graph	graph	NOUN
ejpam-4912	92	16	g′	g′	NOUN
ejpam-4912	92	17	with	with	ADP
ejpam-4912	92	18	γt(g′	γt(g′	NOUN
ejpam-4912	92	19	)	)	PUNCT
ejpam-4912	92	20	=	=	PUNCT
ejpam-4912	92	21	3	3	NUM
ejpam-4912	92	22	<	<	SYM
ejpam-4912	92	23	7	7	NUM
ejpam-4912	92	24	=	=	SYM
ejpam-4912	92	25	γjt(g	γjt(g	NOUN
ejpam-4912	92	26	′	′	NOUN
ejpam-4912	92	27	)	)	PUNCT
ejpam-4912	92	28	let	let	VERB
ejpam-4912	92	29	t1	t1	NOUN
ejpam-4912	92	30	=	=	PUNCT
ejpam-4912	92	31	{	{	PUNCT
ejpam-4912	92	32	u1	u1	NOUN
ejpam-4912	92	33	,	,	PUNCT
ejpam-4912	92	34	u2	u2	NOUN
ejpam-4912	92	35	,	,	PUNCT
ejpam-4912	92	36	u3	u3	PROPN
ejpam-4912	92	37	,	,	PUNCT
ejpam-4912	92	38	u4	u4	PROPN
ejpam-4912	92	39	,	,	PUNCT
ejpam-4912	92	40	u5	u5	PROPN
ejpam-4912	92	41	,	,	PUNCT
ejpam-4912	92	42	u6	u6	PROPN
ejpam-4912	92	43	,	,	PUNCT
ejpam-4912	92	44	u7	u7	PROPN
ejpam-4912	92	45	}	}	PUNCT
ejpam-4912	92	46	and	and	CCONJ
ejpam-4912	92	47	t2	t2	PROPN
ejpam-4912	92	48	=	=	SYM
ejpam-4912	92	49	{	{	PUNCT
ejpam-4912	92	50	u3	u3	PROPN
ejpam-4912	92	51	,	,	PUNCT
ejpam-4912	92	52	u4	u4	PROPN
ejpam-4912	92	53	,	,	PUNCT
ejpam-4912	92	54	u5	u5	PROPN
ejpam-4912	92	55	}	}	PUNCT
ejpam-4912	92	56	.	.	PUNCT
ejpam-4912	93	1	then	then	ADV
ejpam-4912	93	2	t2	t2	PROPN
ejpam-4912	93	3	is	be	AUX
ejpam-4912	93	4	a	a	DET
ejpam-4912	93	5	minimum	minimum	ADJ
ejpam-4912	93	6	total	total	ADJ
ejpam-4912	93	7	dominating	dominating	NOUN
ejpam-4912	93	8	set	set	NOUN
ejpam-4912	93	9	of	of	ADP
ejpam-4912	93	10	g′.	g′.	NOUN
ejpam-4912	93	11	hence	hence	ADV
ejpam-4912	93	12	,	,	PUNCT
ejpam-4912	93	13	γt(g	γt(g	NUM
ejpam-4912	93	14	′	′	NUM
ejpam-4912	93	15	)	)	PUNCT
ejpam-4912	93	16	=	=	SYM
ejpam-4912	94	1	3	3	X
ejpam-4912	94	2	.	.	PUNCT
ejpam-4912	94	3	since	since	SCONJ
ejpam-4912	94	4	t2	t2	PROPN
ejpam-4912	94	5	⊆	⊆	NUM
ejpam-4912	94	6	t1	t1	NOUN
ejpam-4912	94	7	,	,	PUNCT
ejpam-4912	94	8	it	it	PRON
ejpam-4912	94	9	follows	follow	VERB
ejpam-4912	94	10	that	that	SCONJ
ejpam-4912	94	11	t1	t1	PROPN
ejpam-4912	94	12	is	be	AUX
ejpam-4912	94	13	also	also	ADV
ejpam-4912	94	14	a	a	DET
ejpam-4912	94	15	total	total	ADJ
ejpam-4912	94	16	j.a	j.a	PROPN
ejpam-4912	94	17	.	.	PUNCT
ejpam-4912	95	1	hassan	hassan	PROPN
ejpam-4912	95	2	et	et	PROPN
ejpam-4912	95	3	al	al	PROPN
ejpam-4912	95	4	.	.	PUNCT
ejpam-4912	95	5	/	/	SYM
ejpam-4912	95	6	eur	eur	PROPN
ejpam-4912	95	7	.	.	PUNCT
ejpam-4912	96	1	j.	j.	PROPN
ejpam-4912	96	2	pure	pure	PROPN
ejpam-4912	96	3	appl	appl	PROPN
ejpam-4912	96	4	.	.	PROPN
ejpam-4912	96	5	math	math	PROPN
ejpam-4912	96	6	,	,	PUNCT
ejpam-4912	96	7	16	16	NUM
ejpam-4912	96	8	(	(	PUNCT
ejpam-4912	96	9	4	4	NUM
ejpam-4912	96	10	)	)	PUNCT
ejpam-4912	96	11	(	(	PUNCT
ejpam-4912	96	12	2023	2023	NUM
ejpam-4912	96	13	)	)	PUNCT
ejpam-4912	96	14	,	,	PUNCT
ejpam-4912	96	15	2106	2106	NUM
ejpam-4912	96	16	-	-	SYM
ejpam-4912	96	17	2117	2117	NUM
ejpam-4912	96	18	2110	2110	NUM
ejpam-4912	96	19	dominating	dominating	NOUN
ejpam-4912	96	20	set	set	NOUN
ejpam-4912	96	21	of	of	ADP
ejpam-4912	96	22	g′.	g′.	NOUN
ejpam-4912	96	23	observe	observe	VERB
ejpam-4912	96	24	that	that	SCONJ
ejpam-4912	96	25	u2	u2	PROPN
ejpam-4912	96	26	∈	∈	PROPN
ejpam-4912	96	27	ng′(u1)\ng′(ui	ng′(u1)\ng′(ui	PROPN
ejpam-4912	96	28	)	)	PUNCT
ejpam-4912	96	29	∀	∀	PUNCT
ejpam-4912	97	1	i	i	PRON
ejpam-4912	97	2	̸=	̸=	PROPN
ejpam-4912	97	3	3	3	NUM
ejpam-4912	97	4	,	,	PUNCT
ejpam-4912	97	5	u3	u3	NOUN
ejpam-4912	97	6	∈	∈	PROPN
ejpam-4912	97	7	ng′(u1)\ng′(u3	ng′(u1)\ng′(u3	NOUN
ejpam-4912	97	8	)	)	PUNCT
ejpam-4912	97	9	,	,	PUNCT
ejpam-4912	97	10	u1	u1	PROPN
ejpam-4912	97	11	∈	∈	PROPN
ejpam-4912	97	12	ng′(u2	ng′(u2	PROPN
ejpam-4912	97	13	)	)	PUNCT
ejpam-4912	97	14	\ng′(uj	\ng′(uj	PROPN
ejpam-4912	97	15	)	)	PUNCT
ejpam-4912	97	16	∀	∀	PUNCT
ejpam-4912	98	1	j	j	PROPN
ejpam-4912	98	2	̸=	̸=	PROPN
ejpam-4912	98	3	3	3	NUM
ejpam-4912	98	4	,	,	PUNCT
ejpam-4912	98	5	u3	u3	NOUN
ejpam-4912	98	6	∈	∈	PROPN
ejpam-4912	98	7	ng′(u2	ng′(u2	PROPN
ejpam-4912	98	8	)	)	PUNCT
ejpam-4912	98	9	\ng′(u3	\ng′(u3	NOUN
ejpam-4912	98	10	)	)	PUNCT
ejpam-4912	98	11	,	,	PUNCT
ejpam-4912	98	12	u1	u1	PROPN
ejpam-4912	98	13	∈	∈	PROPN
ejpam-4912	98	14	ng′(u3	ng′(u3	PROPN
ejpam-4912	98	15	)	)	PUNCT
ejpam-4912	98	16	\ng′(ur	\ng′(ur	NOUN
ejpam-4912	98	17	)	)	PUNCT
ejpam-4912	98	18	∀	∀	PUNCT
ejpam-4912	99	1	r	r	NOUN
ejpam-4912	99	2	̸=	̸=	PROPN
ejpam-4912	99	3	2	2	NUM
ejpam-4912	99	4	,	,	PUNCT
ejpam-4912	99	5	u2	u2	PROPN
ejpam-4912	99	6	∈	∈	PROPN
ejpam-4912	99	7	ng′(u3	ng′(u3	PROPN
ejpam-4912	99	8	)	)	PUNCT
ejpam-4912	99	9	\ng′(u2	\ng′(u2	NOUN
ejpam-4912	99	10	)	)	PUNCT
ejpam-4912	99	11	,	,	PUNCT
ejpam-4912	99	12	u8	u8	PROPN
ejpam-4912	99	13	∈	∈	PROPN
ejpam-4912	99	14	ng′(u4	ng′(u4	NOUN
ejpam-4912	99	15	)	)	PUNCT
ejpam-4912	99	16	\ng′(uq	\ng′(uq	NOUN
ejpam-4912	99	17	)	)	PUNCT
ejpam-4912	99	18	∀	∀	PUNCT
ejpam-4912	100	1	q	q	PROPN
ejpam-4912	100	2	̸=	̸=	PROPN
ejpam-4912	100	3	4	4	NUM
ejpam-4912	100	4	,	,	PUNCT
ejpam-4912	100	5	u7	u7	PROPN
ejpam-4912	100	6	∈	∈	PROPN
ejpam-4912	100	7	ng′(u5	ng′(u5	PROPN
ejpam-4912	100	8	)	)	PUNCT
ejpam-4912	100	9	\ng′(us	\ng′(us	NOUN
ejpam-4912	100	10	)	)	PUNCT
ejpam-4912	100	11	∀	∀	X
ejpam-4912	101	1	s	s	PART
ejpam-4912	101	2	̸=	̸=	PROPN
ejpam-4912	101	3	6	6	NUM
ejpam-4912	101	4	,	,	PUNCT
ejpam-4912	101	5	u6	u6	PROPN
ejpam-4912	101	6	∈	∈	PROPN
ejpam-4912	101	7	ng′(u5	ng′(u5	NOUN
ejpam-4912	101	8	)	)	PUNCT
ejpam-4912	101	9	\	\	NOUN
ejpam-4912	101	10	ng′(u6	ng′(u6	NOUN
ejpam-4912	101	11	)	)	PUNCT
ejpam-4912	101	12	,	,	PUNCT
ejpam-4912	101	13	u7	u7	PROPN
ejpam-4912	101	14	∈	∈	PROPN
ejpam-4912	101	15	ng′(u6	ng′(u6	NOUN
ejpam-4912	101	16	)	)	PUNCT
ejpam-4912	101	17	\	\	PROPN
ejpam-4912	101	18	ng′(ut	ng′(ut	NUM
ejpam-4912	101	19	)	)	PUNCT
ejpam-4912	101	20	∀	∀	PUNCT
ejpam-4912	101	21	t	t	NOUN
ejpam-4912	101	22	̸=	̸=	PROPN
ejpam-4912	101	23	5	5	NUM
ejpam-4912	101	24	,	,	PUNCT
ejpam-4912	101	25	u5	u5	PROPN
ejpam-4912	101	26	∈	∈	PROPN
ejpam-4912	101	27	ng′(u6	ng′(u6	NOUN
ejpam-4912	101	28	)	)	PUNCT
ejpam-4912	101	29	\	\	NOUN
ejpam-4912	101	30	ng′(u5	ng′(u5	PROPN
ejpam-4912	101	31	)	)	PUNCT
ejpam-4912	101	32	and	and	CCONJ
ejpam-4912	101	33	u6	u6	PROPN
ejpam-4912	101	34	∈	∈	PROPN
ejpam-4912	101	35	ng′(u7	ng′(u7	NOUN
ejpam-4912	101	36	)	)	PUNCT
ejpam-4912	101	37	\	\	PROPN
ejpam-4912	101	38	ng′(um	ng′(um	PROPN
ejpam-4912	101	39	)	)	PUNCT
ejpam-4912	101	40	∀	∀	X
ejpam-4912	102	1	m	m	VERB
ejpam-4912	102	2	̸=	̸=	PROPN
ejpam-4912	102	3	5	5	NUM
ejpam-4912	102	4	,	,	PUNCT
ejpam-4912	102	5	u5	u5	PROPN
ejpam-4912	102	6	∈	∈	PROPN
ejpam-4912	102	7	ng′(u7	ng′(u7	PROPN
ejpam-4912	102	8	)	)	PUNCT
ejpam-4912	102	9	\	\	PROPN
ejpam-4912	102	10	ng′(u5	ng′(u5	NOUN
ejpam-4912	102	11	)	)	PUNCT
ejpam-4912	102	12	.	.	PUNCT
ejpam-4912	103	1	thus	thus	ADV
ejpam-4912	103	2	,	,	PUNCT
ejpam-4912	103	3	t1	t1	PROPN
ejpam-4912	103	4	is	be	AUX
ejpam-4912	103	5	a	a	DET
ejpam-4912	103	6	j	j	NOUN
ejpam-4912	103	7	-	-	ADJ
ejpam-4912	103	8	open	open	ADJ
ejpam-4912	103	9	set	set	NOUN
ejpam-4912	103	10	of	of	ADP
ejpam-4912	103	11	g′	g′	NOUN
ejpam-4912	103	12	,	,	PUNCT
ejpam-4912	103	13	and	and	CCONJ
ejpam-4912	103	14	so	so	ADV
ejpam-4912	103	15	t1	t1	PROPN
ejpam-4912	103	16	is	be	AUX
ejpam-4912	103	17	a	a	DET
ejpam-4912	103	18	j	j	PROPN
ejpam-4912	103	19	-	-	ADJ
ejpam-4912	103	20	total	total	ADJ
ejpam-4912	103	21	dominating	dominating	NOUN
ejpam-4912	103	22	set	set	NOUN
ejpam-4912	103	23	of	of	ADP
ejpam-4912	103	24	g′.	g′.	NOUN
ejpam-4912	103	25	hence	hence	ADV
ejpam-4912	103	26	,	,	PUNCT
ejpam-4912	103	27	γjt(g	γjt(g	PROPN
ejpam-4912	103	28	′	′	NOUN
ejpam-4912	103	29	)	)	PUNCT
ejpam-4912	103	30	=	=	PUNCT
ejpam-4912	104	1	7	7	X
ejpam-4912	104	2	.	.	PUNCT
ejpam-4912	104	3	consequently	consequently	ADV
ejpam-4912	104	4	,	,	PUNCT
ejpam-4912	104	5	γjt(g	γjt(g	PROPN
ejpam-4912	104	6	′	′	NOUN
ejpam-4912	104	7	)	)	PUNCT
ejpam-4912	104	8	>	>	PUNCT
ejpam-4912	104	9	γt(g	γt(g	NUM
ejpam-4912	104	10	′	′	NUM
ejpam-4912	104	11	)	)	PUNCT
ejpam-4912	104	12	.	.	PUNCT
ejpam-4912	105	1	theorem	theorem	NOUN
ejpam-4912	105	2	1	1	X
ejpam-4912	105	3	.	.	PUNCT
ejpam-4912	106	1	let	let	VERB
ejpam-4912	106	2	kn	kn	PROPN
ejpam-4912	106	3	be	be	AUX
ejpam-4912	106	4	a	a	DET
ejpam-4912	106	5	complete	complete	ADJ
ejpam-4912	106	6	graph	graph	NOUN
ejpam-4912	106	7	of	of	ADP
ejpam-4912	106	8	order	order	NOUN
ejpam-4912	106	9	n	n	PRON
ejpam-4912	106	10	≥	≥	NOUN
ejpam-4912	106	11	2	2	NUM
ejpam-4912	106	12	.	.	PUNCT
ejpam-4912	107	1	then	then	ADV
ejpam-4912	107	2	m	m	VERB
ejpam-4912	107	3	⊆	⊆	NUM
ejpam-4912	107	4	v	v	NOUN
ejpam-4912	107	5	(	(	PUNCT
ejpam-4912	107	6	kn	kn	PROPN
ejpam-4912	107	7	)	)	PUNCT
ejpam-4912	107	8	is	be	AUX
ejpam-4912	107	9	a	a	DET
ejpam-4912	107	10	j	j	PROPN
ejpam-4912	107	11	-	-	ADJ
ejpam-4912	107	12	total	total	ADJ
ejpam-4912	107	13	dominating	dominating	NOUN
ejpam-4912	107	14	in	in	ADP
ejpam-4912	107	15	kn	kn	PROPN
ejpam-4912	107	16	if	if	SCONJ
ejpam-4912	107	17	and	and	CCONJ
ejpam-4912	107	18	only	only	ADV
ejpam-4912	107	19	if	if	SCONJ
ejpam-4912	107	20	|	|	ADV
ejpam-4912	107	21	m	m	VERB
ejpam-4912	107	22	|≥	|≥	ADJ
ejpam-4912	107	23	2	2	NUM
ejpam-4912	107	24	.	.	PUNCT
ejpam-4912	108	1	proof	proof	NOUN
ejpam-4912	108	2	.	.	PUNCT
ejpam-4912	109	1	let	let	VERB
ejpam-4912	109	2	m	m	PRON
ejpam-4912	109	3	⊆	⊆	NUM
ejpam-4912	109	4	v	v	NOUN
ejpam-4912	109	5	(	(	PUNCT
ejpam-4912	109	6	kn	kn	PROPN
ejpam-4912	109	7	)	)	PUNCT
ejpam-4912	109	8	be	be	AUX
ejpam-4912	109	9	a	a	DET
ejpam-4912	109	10	j	j	PROPN
ejpam-4912	109	11	-	-	ADJ
ejpam-4912	109	12	total	total	ADJ
ejpam-4912	109	13	dominating	dominating	NOUN
ejpam-4912	109	14	set	set	NOUN
ejpam-4912	109	15	in	in	ADP
ejpam-4912	109	16	kn	kn	PROPN
ejpam-4912	109	17	.	.	PUNCT
ejpam-4912	110	1	then	then	ADV
ejpam-4912	110	2	m	m	PROPN
ejpam-4912	110	3	is	be	AUX
ejpam-4912	110	4	a	a	DET
ejpam-4912	110	5	total	total	ADJ
ejpam-4912	110	6	dominating	dominating	NOUN
ejpam-4912	110	7	set	set	NOUN
ejpam-4912	110	8	in	in	ADP
ejpam-4912	110	9	kn	kn	PROPN
ejpam-4912	110	10	.	.	PUNCT
ejpam-4912	111	1	since	since	SCONJ
ejpam-4912	111	2	γt(kn	γt(kn	PROPN
ejpam-4912	111	3	)	)	PUNCT
ejpam-4912	111	4	=	=	SYM
ejpam-4912	111	5	2	2	NUM
ejpam-4912	111	6	for	for	ADP
ejpam-4912	111	7	all	all	DET
ejpam-4912	111	8	n	n	PRON
ejpam-4912	111	9	≥	≥	NOUN
ejpam-4912	111	10	2	2	NUM
ejpam-4912	111	11	,	,	PUNCT
ejpam-4912	111	12	it	it	PRON
ejpam-4912	111	13	follows	follow	VERB
ejpam-4912	111	14	that	that	SCONJ
ejpam-4912	112	1	|	|	ADV
ejpam-4912	112	2	m	m	VERB
ejpam-4912	112	3	|≥	|≥	ADJ
ejpam-4912	112	4	γt(kn	γt(kn	ADJ
ejpam-4912	112	5	)	)	PUNCT
ejpam-4912	113	1	=	=	SYM
ejpam-4912	113	2	2	2	X
ejpam-4912	113	3	.	.	X
ejpam-4912	113	4	conversely	conversely	ADV
ejpam-4912	113	5	,	,	PUNCT
ejpam-4912	113	6	suppose	suppose	VERB
ejpam-4912	113	7	thatm	thatm	VERB
ejpam-4912	113	8	⊆	⊆	NUM
ejpam-4912	113	9	v	v	NOUN
ejpam-4912	113	10	(	(	PUNCT
ejpam-4912	113	11	kn	kn	PROPN
ejpam-4912	113	12	)	)	PUNCT
ejpam-4912	113	13	with	with	ADP
ejpam-4912	113	14	|	|	ADV
ejpam-4912	113	15	m	m	VERB
ejpam-4912	113	16	|≥	|≥	ADJ
ejpam-4912	113	17	2	2	X
ejpam-4912	113	18	.	.	PUNCT
ejpam-4912	114	1	let	let	VERB
ejpam-4912	114	2	v	v	X
ejpam-4912	114	3	(	(	PUNCT
ejpam-4912	114	4	kn	kn	PROPN
ejpam-4912	114	5	)	)	PUNCT
ejpam-4912	114	6	=	=	SYM
ejpam-4912	114	7	{	{	PUNCT
ejpam-4912	114	8	v1	v1	PROPN
ejpam-4912	114	9	,	,	PUNCT
ejpam-4912	114	10	v2	v2	PROPN
ejpam-4912	114	11	,	,	PUNCT
ejpam-4912	114	12	.	.	PUNCT
ejpam-4912	114	13	.	.	PUNCT
ejpam-4912	115	1	.	.	PUNCT
ejpam-4912	116	1	,	,	PUNCT
ejpam-4912	116	2	vn	vn	NOUN
ejpam-4912	116	3	}	}	PUNCT
ejpam-4912	116	4	and	and	CCONJ
ejpam-4912	116	5	m	m	PROPN
ejpam-4912	116	6	=	=	SYM
ejpam-4912	116	7	{	{	PUNCT
ejpam-4912	116	8	v1	v1	PROPN
ejpam-4912	116	9	,	,	PUNCT
ejpam-4912	116	10	v2	v2	PROPN
ejpam-4912	116	11	,	,	PUNCT
ejpam-4912	116	12	.	.	PUNCT
ejpam-4912	116	13	.	.	PUNCT
ejpam-4912	117	1	.	.	PUNCT
ejpam-4912	118	1	,	,	PUNCT
ejpam-4912	118	2	vs	vs	ADP
ejpam-4912	118	3	}	}	PUNCT
ejpam-4912	118	4	⊆	⊆	NUM
ejpam-4912	118	5	v	v	NOUN
ejpam-4912	118	6	(	(	PUNCT
ejpam-4912	118	7	kn	kn	PROPN
ejpam-4912	118	8	)	)	PUNCT
ejpam-4912	118	9	where	where	SCONJ
ejpam-4912	118	10	s	s	VERB
ejpam-4912	118	11	∈	∈	PROPN
ejpam-4912	118	12	{	{	PUNCT
ejpam-4912	118	13	2	2	NUM
ejpam-4912	118	14	,	,	PUNCT
ejpam-4912	118	15	3	3	NUM
ejpam-4912	118	16	,	,	PUNCT
ejpam-4912	118	17	.	.	PUNCT
ejpam-4912	118	18	.	.	PUNCT
ejpam-4912	118	19	.	.	PUNCT
ejpam-4912	118	20	,	,	PUNCT
ejpam-4912	118	21	n	n	CCONJ
ejpam-4912	118	22	}	}	PUNCT
ejpam-4912	118	23	.	.	PUNCT
ejpam-4912	119	1	observe	observe	VERB
ejpam-4912	119	2	that	that	SCONJ
ejpam-4912	119	3	vi	vi	PROPN
ejpam-4912	119	4	∈	∈	PROPN
ejpam-4912	119	5	nkn(vj	nkn(vj	NOUN
ejpam-4912	119	6	)	)	PUNCT
ejpam-4912	119	7	\nkn(vi	\nkn(vi	PROPN
ejpam-4912	119	8	)	)	PUNCT
ejpam-4912	119	9	for	for	ADP
ejpam-4912	119	10	all	all	DET
ejpam-4912	119	11	i	i	PRON
ejpam-4912	119	12	̸=	̸=	PROPN
ejpam-4912	119	13	j	j	PROPN
ejpam-4912	119	14	,	,	PUNCT
ejpam-4912	119	15	i	i	PRON
ejpam-4912	119	16	,	,	PUNCT
ejpam-4912	119	17	j	j	PROPN
ejpam-4912	119	18	∈	∈	PROPN
ejpam-4912	119	19	{	{	PUNCT
ejpam-4912	119	20	1	1	NUM
ejpam-4912	119	21	,	,	PUNCT
ejpam-4912	119	22	2	2	NUM
ejpam-4912	119	23	,	,	PUNCT
ejpam-4912	119	24	.	.	PUNCT
ejpam-4912	119	25	.	.	PUNCT
ejpam-4912	120	1	.	.	PUNCT
ejpam-4912	121	1	,	,	PUNCT
ejpam-4912	121	2	s	s	X
ejpam-4912	121	3	}	}	PUNCT
ejpam-4912	121	4	.	.	PUNCT
ejpam-4912	122	1	thus	thus	ADV
ejpam-4912	122	2	,	,	PUNCT
ejpam-4912	122	3	nkn(vj	nkn(vj	NOUN
ejpam-4912	122	4	)	)	PUNCT
ejpam-4912	122	5	\nkn(vi	\nkn(vi	ADJ
ejpam-4912	122	6	)	)	PUNCT
ejpam-4912	122	7	̸=	̸=	PROPN
ejpam-4912	122	8	∅	∅	NOUN
ejpam-4912	122	9	for	for	ADP
ejpam-4912	122	10	all	all	PRON
ejpam-4912	122	11	i	i	PRON
ejpam-4912	122	12	̸=	̸=	PROPN
ejpam-4912	122	13	j	j	PROPN
ejpam-4912	122	14	,	,	PUNCT
ejpam-4912	122	15	where	where	SCONJ
ejpam-4912	122	16	i	i	PRON
ejpam-4912	122	17	,	,	PUNCT
ejpam-4912	122	18	j	j	PROPN
ejpam-4912	122	19	∈	∈	PROPN
ejpam-4912	122	20	{	{	PUNCT
ejpam-4912	122	21	1	1	NUM
ejpam-4912	122	22	,	,	PUNCT
ejpam-4912	122	23	2	2	NUM
ejpam-4912	122	24	,	,	PUNCT
ejpam-4912	122	25	.	.	PUNCT
ejpam-4912	122	26	.	.	PUNCT
ejpam-4912	122	27	.	.	PUNCT
ejpam-4912	123	1	,	,	PUNCT
ejpam-4912	123	2	s	s	X
ejpam-4912	123	3	}	}	PUNCT
ejpam-4912	123	4	,	,	PUNCT
ejpam-4912	123	5	showing	show	VERB
ejpam-4912	123	6	that	that	SCONJ
ejpam-4912	123	7	m	m	PROPN
ejpam-4912	123	8	is	be	AUX
ejpam-4912	123	9	a	a	DET
ejpam-4912	123	10	j	j	NOUN
ejpam-4912	123	11	-	-	ADJ
ejpam-4912	123	12	open	open	ADJ
ejpam-4912	123	13	set	set	NOUN
ejpam-4912	123	14	in	in	ADP
ejpam-4912	123	15	kn	kn	PROPN
ejpam-4912	123	16	∀	∀	X
ejpam-4912	123	17	n	n	DET
ejpam-4912	123	18	≥	≥	NOUN
ejpam-4912	123	19	2	2	NUM
ejpam-4912	123	20	.	.	PUNCT
ejpam-4912	124	1	since	since	SCONJ
ejpam-4912	124	2	any	any	DET
ejpam-4912	124	3	set	set	NOUN
ejpam-4912	124	4	{	{	PUNCT
ejpam-4912	124	5	vi	vi	PROPN
ejpam-4912	124	6	,	,	PUNCT
ejpam-4912	124	7	vj	vj	ADJ
ejpam-4912	124	8	}	}	PUNCT
ejpam-4912	124	9	,	,	PUNCT
ejpam-4912	124	10	i	i	PROPN
ejpam-4912	124	11	̸=	̸=	PROPN
ejpam-4912	124	12	j	j	PROPN
ejpam-4912	124	13	,	,	PUNCT
ejpam-4912	124	14	is	be	AUX
ejpam-4912	124	15	a	a	DET
ejpam-4912	124	16	total	total	ADJ
ejpam-4912	124	17	dominating	dominating	NOUN
ejpam-4912	124	18	in	in	ADP
ejpam-4912	124	19	kn	kn	PROPN
ejpam-4912	124	20	,	,	PUNCT
ejpam-4912	124	21	it	it	PRON
ejpam-4912	124	22	follows	follow	VERB
ejpam-4912	124	23	that	that	SCONJ
ejpam-4912	124	24	m	m	PROPN
ejpam-4912	124	25	is	be	AUX
ejpam-4912	124	26	j	j	ADJ
ejpam-4912	124	27	-	-	ADJ
ejpam-4912	124	28	total	total	ADJ
ejpam-4912	124	29	dominating	dominating	NOUN
ejpam-4912	124	30	set	set	NOUN
ejpam-4912	124	31	in	in	ADP
ejpam-4912	124	32	kn	kn	PROPN
ejpam-4912	124	33	∀	∀	X
ejpam-4912	124	34	n	n	PRON
ejpam-4912	124	35	≥	≥	NOUN
ejpam-4912	124	36	2	2	NUM
ejpam-4912	124	37	.	.	PUNCT
ejpam-4912	124	38	corollary	corollary	ADJ
ejpam-4912	124	39	1	1	NUM
ejpam-4912	124	40	.	.	PUNCT
ejpam-4912	125	1	let	let	VERB
ejpam-4912	125	2	n	n	PRON
ejpam-4912	125	3	≥	≥	X
ejpam-4912	125	4	2	2	NUM
ejpam-4912	125	5	be	be	AUX
ejpam-4912	125	6	any	any	DET
ejpam-4912	125	7	positive	positive	ADJ
ejpam-4912	125	8	integer	integer	NOUN
ejpam-4912	125	9	.	.	PUNCT
ejpam-4912	126	1	then	then	ADV
ejpam-4912	126	2	γjt(kn	γjt(kn	ADV
ejpam-4912	126	3	)	)	PUNCT
ejpam-4912	126	4	=	=	SYM
ejpam-4912	126	5	n.	n.	NOUN
ejpam-4912	126	6	proof	proof	NOUN
ejpam-4912	126	7	.	.	PUNCT
ejpam-4912	127	1	letm	letm	NOUN
ejpam-4912	127	2	=	=	SYM
ejpam-4912	127	3	v	v	PROPN
ejpam-4912	127	4	(	(	PUNCT
ejpam-4912	127	5	kn	kn	PROPN
ejpam-4912	127	6	)	)	PUNCT
ejpam-4912	127	7	=	=	PRON
ejpam-4912	127	8	{	{	PUNCT
ejpam-4912	127	9	a1	a1	PROPN
ejpam-4912	127	10	,	,	PUNCT
ejpam-4912	127	11	a2	a2	PROPN
ejpam-4912	127	12	,	,	PUNCT
ejpam-4912	127	13	...	...	PUNCT
ejpam-4912	127	14	,	,	PUNCT
ejpam-4912	127	15	an	an	PRON
ejpam-4912	127	16	}	}	PUNCT
ejpam-4912	127	17	.	.	PUNCT
ejpam-4912	128	1	then	then	ADV
ejpam-4912	128	2	by	by	ADP
ejpam-4912	128	3	theorem	theorem	PROPN
ejpam-4912	128	4	1,m	1,m	PROPN
ejpam-4912	128	5	is	be	AUX
ejpam-4912	128	6	j	j	PROPN
ejpam-4912	128	7	-	-	ADJ
ejpam-4912	128	8	total	total	ADJ
ejpam-4912	128	9	dominating	dominating	NOUN
ejpam-4912	128	10	set	set	NOUN
ejpam-4912	128	11	in	in	ADP
ejpam-4912	128	12	kn	kn	PROPN
ejpam-4912	128	13	.	.	PUNCT
ejpam-4912	129	1	thus	thus	ADV
ejpam-4912	129	2	,	,	PUNCT
ejpam-4912	129	3	γjt(kn	γjt(kn	PROPN
ejpam-4912	129	4	)	)	PUNCT
ejpam-4912	129	5	≥	≥	X
ejpam-4912	129	6	n.	n.	VERB
ejpam-4912	129	7	by	by	ADP
ejpam-4912	129	8	proposition	proposition	NOUN
ejpam-4912	129	9	1	1	NUM
ejpam-4912	129	10	,	,	PUNCT
ejpam-4912	129	11	γjt(kn	γjt(kn	NUM
ejpam-4912	129	12	)	)	PUNCT
ejpam-4912	129	13	=	=	SYM
ejpam-4912	129	14	n.	n.	NOUN
ejpam-4912	129	15	theorem	theorem	NOUN
ejpam-4912	129	16	2	2	X
ejpam-4912	129	17	.	.	PUNCT
ejpam-4912	130	1	let	let	VERB
ejpam-4912	130	2	m	m	PRON
ejpam-4912	130	3	and	and	CCONJ
ejpam-4912	130	4	n	n	ADV
ejpam-4912	130	5	be	be	AUX
ejpam-4912	130	6	positive	positive	ADJ
ejpam-4912	130	7	integers	integer	NOUN
ejpam-4912	130	8	with	with	ADP
ejpam-4912	130	9	2	2	NUM
ejpam-4912	130	10	≤	≤	NUM
ejpam-4912	130	11	m	m	VERB
ejpam-4912	130	12	≤	≤	NOUN
ejpam-4912	131	1	n.	n.	NOUN
ejpam-4912	132	1	then	then	ADV
ejpam-4912	132	2	there	there	PRON
ejpam-4912	132	3	exists	exist	VERB
ejpam-4912	132	4	a	a	DET
ejpam-4912	132	5	connected	connected	ADJ
ejpam-4912	132	6	graph	graph	NOUN
ejpam-4912	132	7	h	h	NOUN
ejpam-4912	132	8	such	such	ADJ
ejpam-4912	132	9	that	that	PRON
ejpam-4912	132	10	γt(h	γt(h	NUM
ejpam-4912	132	11	)	)	PUNCT
ejpam-4912	132	12	=	=	SYM
ejpam-4912	132	13	m	m	PROPN
ejpam-4912	132	14	and	and	CCONJ
ejpam-4912	132	15	γjt(h	γjt(h	ADJ
ejpam-4912	132	16	)	)	PUNCT
ejpam-4912	132	17	=	=	VERB
ejpam-4912	133	1	n.	n.	NOUN
ejpam-4912	133	2	that	that	PRON
ejpam-4912	133	3	is	be	AUX
ejpam-4912	133	4	,	,	PUNCT
ejpam-4912	133	5	γjt(h)−	γjt(h)−	PROPN
ejpam-4912	133	6	γt(h	γt(h	NUM
ejpam-4912	133	7	)	)	PUNCT
ejpam-4912	133	8	can	can	AUX
ejpam-4912	133	9	be	be	AUX
ejpam-4912	133	10	made	make	VERB
ejpam-4912	133	11	arbitrarily	arbitrarily	ADV
ejpam-4912	133	12	large	large	ADJ
ejpam-4912	133	13	.	.	PUNCT
ejpam-4912	134	1	proof	proof	NOUN
ejpam-4912	134	2	.	.	PUNCT
ejpam-4912	135	1	for	for	ADP
ejpam-4912	135	2	m	m	PROPN
ejpam-4912	135	3	=	=	SYM
ejpam-4912	135	4	n	n	CCONJ
ejpam-4912	135	5	,	,	PUNCT
ejpam-4912	135	6	consider	consider	VERB
ejpam-4912	135	7	the	the	DET
ejpam-4912	135	8	graph	graph	NOUN
ejpam-4912	135	9	h	h	NOUN
ejpam-4912	135	10	in	in	ADP
ejpam-4912	135	11	figure	figure	NOUN
ejpam-4912	135	12	3	3	NUM
ejpam-4912	135	13	below	below	ADV
ejpam-4912	135	14	.	.	PUNCT
ejpam-4912	136	1	j.a	j.a	PROPN
ejpam-4912	136	2	.	.	PROPN
ejpam-4912	136	3	hassan	hassan	PROPN
ejpam-4912	136	4	et	et	PROPN
ejpam-4912	136	5	al	al	PROPN
ejpam-4912	136	6	.	.	PUNCT
ejpam-4912	136	7	/	/	SYM
ejpam-4912	136	8	eur	eur	PROPN
ejpam-4912	136	9	.	.	PUNCT
ejpam-4912	137	1	j.	j.	PROPN
ejpam-4912	137	2	pure	pure	PROPN
ejpam-4912	137	3	appl	appl	PROPN
ejpam-4912	137	4	.	.	PROPN
ejpam-4912	137	5	math	math	PROPN
ejpam-4912	137	6	,	,	PUNCT
ejpam-4912	137	7	16	16	NUM
ejpam-4912	137	8	(	(	PUNCT
ejpam-4912	137	9	4	4	NUM
ejpam-4912	137	10	)	)	PUNCT
ejpam-4912	137	11	(	(	PUNCT
ejpam-4912	137	12	2023	2023	NUM
ejpam-4912	137	13	)	)	PUNCT
ejpam-4912	137	14	,	,	PUNCT
ejpam-4912	137	15	2106	2106	NUM
ejpam-4912	137	16	-	-	SYM
ejpam-4912	137	17	2117	2117	NUM
ejpam-4912	137	18	2111	2111	NUM
ejpam-4912	137	19	b3b2b1	b3b2b1	X
ejpam-4912	137	20	c2c1	c2c1	PROPN
ejpam-4912	138	1	c3	c3	PROPN
ejpam-4912	138	2	a3a2a1	a3a2a1	PROPN
ejpam-4912	138	3	bm−1	bm−1	PROPN
ejpam-4912	138	4	bm	bm	PROPN
ejpam-4912	138	5	am−1	am−1	PROPN
ejpam-4912	138	6	am	be	AUX
ejpam-4912	138	7	ym−1	ym−1	PROPN
ejpam-4912	138	8	ym	ym	INTJ
ejpam-4912	138	9	.	.	PUNCT
ejpam-4912	138	10	.	.	PUNCT
ejpam-4912	139	1	.h	.h	NOUN
ejpam-4912	140	1	:	:	PUNCT
ejpam-4912	140	2	figure	figure	VERB
ejpam-4912	140	3	3	3	NUM
ejpam-4912	140	4	:	:	PUNCT
ejpam-4912	140	5	a	a	DET
ejpam-4912	140	6	graph	graph	NOUN
ejpam-4912	140	7	h	h	NOUN
ejpam-4912	140	8	with	with	ADP
ejpam-4912	140	9	γt(h	γt(h	NUM
ejpam-4912	140	10	)	)	PUNCT
ejpam-4912	141	1	=	=	SYM
ejpam-4912	141	2	m	m	PUNCT
ejpam-4912	141	3	=	=	SYM
ejpam-4912	141	4	γjt(h	γjt(h	PROPN
ejpam-4912	141	5	)	)	PUNCT
ejpam-4912	141	6	let	let	VERB
ejpam-4912	141	7	m	m	VERB
ejpam-4912	141	8	=	=	NOUN
ejpam-4912	141	9	{	{	PUNCT
ejpam-4912	141	10	a1	a1	PROPN
ejpam-4912	141	11	,	,	PUNCT
ejpam-4912	141	12	a2	a2	PROPN
ejpam-4912	141	13	,	,	PUNCT
ejpam-4912	141	14	...	...	PUNCT
ejpam-4912	141	15	,	,	PUNCT
ejpam-4912	141	16	am−1	am−1	PROPN
ejpam-4912	141	17	,	,	PUNCT
ejpam-4912	141	18	am	be	AUX
ejpam-4912	141	19	}	}	PUNCT
ejpam-4912	141	20	.	.	PUNCT
ejpam-4912	142	1	then	then	ADV
ejpam-4912	142	2	m	m	PROPN
ejpam-4912	142	3	is	be	AUX
ejpam-4912	142	4	a	a	DET
ejpam-4912	142	5	minimum	minimum	ADJ
ejpam-4912	142	6	total	total	ADJ
ejpam-4912	142	7	dominating	dominating	NOUN
ejpam-4912	142	8	set	set	NOUN
ejpam-4912	142	9	of	of	ADP
ejpam-4912	142	10	h	h	NOUN
ejpam-4912	142	11	,	,	PUNCT
ejpam-4912	142	12	and	and	CCONJ
ejpam-4912	142	13	so	so	ADV
ejpam-4912	142	14	γt(h	γt(h	NUM
ejpam-4912	142	15	)	)	PUNCT
ejpam-4912	142	16	=	=	SYM
ejpam-4912	142	17	m.	m.	NOUN
ejpam-4912	142	18	since	since	SCONJ
ejpam-4912	142	19	bi	bi	PROPN
ejpam-4912	142	20	,	,	PUNCT
ejpam-4912	142	21	ci	ci	PROPN
ejpam-4912	142	22	∈	∈	PROPN
ejpam-4912	142	23	nh(ai	nh(ai	PROPN
ejpam-4912	142	24	)	)	PUNCT
ejpam-4912	142	25	\nh(aj	\nh(aj	NOUN
ejpam-4912	142	26	)	)	PUNCT
ejpam-4912	142	27	for	for	ADP
ejpam-4912	142	28	every	every	DET
ejpam-4912	142	29	i	i	PROPN
ejpam-4912	142	30	̸=	̸=	PROPN
ejpam-4912	142	31	j	j	PROPN
ejpam-4912	142	32	,	,	PUNCT
ejpam-4912	142	33	i	i	PRON
ejpam-4912	142	34	,	,	PUNCT
ejpam-4912	142	35	j	j	PROPN
ejpam-4912	142	36	∈	∈	PROPN
ejpam-4912	142	37	{	{	PUNCT
ejpam-4912	142	38	1	1	NUM
ejpam-4912	142	39	,	,	PUNCT
ejpam-4912	142	40	2	2	NUM
ejpam-4912	142	41	,	,	PUNCT
ejpam-4912	142	42	.	.	PUNCT
ejpam-4912	142	43	.	.	PUNCT
ejpam-4912	142	44	.	.	PUNCT
ejpam-4912	143	1	,	,	PUNCT
ejpam-4912	143	2	m	m	VERB
ejpam-4912	143	3	}	}	PUNCT
ejpam-4912	143	4	,	,	PUNCT
ejpam-4912	143	5	it	it	PRON
ejpam-4912	143	6	follows	follow	VERB
ejpam-4912	143	7	that	that	SCONJ
ejpam-4912	143	8	m	m	PROPN
ejpam-4912	143	9	is	be	AUX
ejpam-4912	143	10	a	a	DET
ejpam-4912	143	11	j	j	NOUN
ejpam-4912	143	12	-	-	ADJ
ejpam-4912	143	13	open	open	ADJ
ejpam-4912	143	14	set	set	NOUN
ejpam-4912	143	15	in	in	ADP
ejpam-4912	143	16	h.	h.	PROPN
ejpam-4912	143	17	thus	thus	ADV
ejpam-4912	143	18	,	,	PUNCT
ejpam-4912	143	19	m	m	PROPN
ejpam-4912	143	20	is	be	AUX
ejpam-4912	143	21	a	a	DET
ejpam-4912	143	22	j	j	PROPN
ejpam-4912	143	23	-	-	ADJ
ejpam-4912	143	24	total	total	ADJ
ejpam-4912	143	25	dominating	dominating	NOUN
ejpam-4912	143	26	set	set	NOUN
ejpam-4912	143	27	of	of	ADP
ejpam-4912	143	28	h.	h.	PROPN
ejpam-4912	143	29	now	now	ADV
ejpam-4912	143	30	,	,	PUNCT
ejpam-4912	143	31	observe	observe	VERB
ejpam-4912	143	32	that	that	SCONJ
ejpam-4912	143	33	nh(bi	nh(bi	PROPN
ejpam-4912	143	34	)	)	PUNCT
ejpam-4912	143	35	,	,	PUNCT
ejpam-4912	143	36	nh(ci	nh(ci	NOUN
ejpam-4912	143	37	)	)	PUNCT
ejpam-4912	143	38	⊆	⊆	NUM
ejpam-4912	143	39	nh(ai+1	nh(ai+1	NOUN
ejpam-4912	143	40	)	)	PUNCT
ejpam-4912	143	41	,	,	PUNCT
ejpam-4912	143	42	∀	∀	PUNCT
ejpam-4912	144	1	i	i	PRON
ejpam-4912	144	2	∈	∈	PROPN
ejpam-4912	144	3	{	{	PUNCT
ejpam-4912	144	4	1	1	NUM
ejpam-4912	144	5	,	,	PUNCT
ejpam-4912	144	6	2	2	NUM
ejpam-4912	144	7	,	,	PUNCT
ejpam-4912	144	8	.	.	PUNCT
ejpam-4912	144	9	.	.	PUNCT
ejpam-4912	144	10	.	.	PUNCT
ejpam-4912	145	1	,	,	PUNCT
ejpam-4912	145	2	m	m	VERB
ejpam-4912	145	3	−	−	NOUN
ejpam-4912	145	4	1	1	NUM
ejpam-4912	145	5	}	}	PUNCT
ejpam-4912	145	6	,	,	PUNCT
ejpam-4912	145	7	nh(bm	nh(bm	PROPN
ejpam-4912	145	8	)	)	PUNCT
ejpam-4912	145	9	,	,	PUNCT
ejpam-4912	145	10	nh(cm	nh(cm	PROPN
ejpam-4912	145	11	)	)	PUNCT
ejpam-4912	145	12	⊆	⊆	NUM
ejpam-4912	145	13	nh(am−1	nh(am−1	NOUN
ejpam-4912	145	14	)	)	PUNCT
ejpam-4912	145	15	and	and	CCONJ
ejpam-4912	145	16	ai	ai	INTJ
ejpam-4912	145	17	must	must	AUX
ejpam-4912	145	18	be	be	AUX
ejpam-4912	145	19	in	in	ADP
ejpam-4912	145	20	any	any	DET
ejpam-4912	145	21	total	total	ADJ
ejpam-4912	145	22	dominating	dominating	NOUN
ejpam-4912	145	23	set	set	NOUN
ejpam-4912	145	24	of	of	ADP
ejpam-4912	145	25	h	h	NOUN
ejpam-4912	145	26	for	for	ADP
ejpam-4912	145	27	each	each	DET
ejpam-4912	145	28	i	i	PRON
ejpam-4912	145	29	∈	∈	PROPN
ejpam-4912	145	30	{	{	PUNCT
ejpam-4912	145	31	1	1	NUM
ejpam-4912	145	32	,	,	PUNCT
ejpam-4912	145	33	2	2	NUM
ejpam-4912	145	34	,	,	PUNCT
ejpam-4912	145	35	.	.	PUNCT
ejpam-4912	145	36	.	.	PUNCT
ejpam-4912	145	37	.	.	PUNCT
ejpam-4912	146	1	,	,	PUNCT
ejpam-4912	146	2	m	m	VERB
ejpam-4912	146	3	}	}	PUNCT
ejpam-4912	146	4	.	.	PUNCT
ejpam-4912	147	1	therefore	therefore	ADV
ejpam-4912	147	2	,	,	PUNCT
ejpam-4912	147	3	m	m	VERB
ejpam-4912	147	4	is	be	AUX
ejpam-4912	147	5	the	the	DET
ejpam-4912	147	6	maximum	maximum	ADJ
ejpam-4912	147	7	j	j	PROPN
ejpam-4912	147	8	-	-	ADJ
ejpam-4912	147	9	total	total	ADJ
ejpam-4912	147	10	dominating	dominating	NOUN
ejpam-4912	147	11	set	set	NOUN
ejpam-4912	147	12	of	of	ADP
ejpam-4912	147	13	h	h	NOUN
ejpam-4912	147	14	,	,	PUNCT
ejpam-4912	147	15	and	and	CCONJ
ejpam-4912	147	16	so	so	ADV
ejpam-4912	147	17	γjt(h	γjt(h	ADJ
ejpam-4912	147	18	)	)	PUNCT
ejpam-4912	148	1	=	=	SYM
ejpam-4912	148	2	m.	m.	NOUN
ejpam-4912	148	3	suppose	suppose	VERB
ejpam-4912	148	4	that	that	SCONJ
ejpam-4912	148	5	m	m	VERB
ejpam-4912	148	6	<	<	X
ejpam-4912	148	7	n.	n.	X
ejpam-4912	148	8	let	let	VERB
ejpam-4912	148	9	s	s	PRON
ejpam-4912	148	10	=	=	PUNCT
ejpam-4912	148	11	n	n	PRON
ejpam-4912	148	12	−	−	NOUN
ejpam-4912	148	13	m	m	NOUN
ejpam-4912	148	14	and	and	CCONJ
ejpam-4912	148	15	consider	consider	VERB
ejpam-4912	148	16	the	the	DET
ejpam-4912	148	17	graph	graph	NOUN
ejpam-4912	148	18	h	h	NOUN
ejpam-4912	148	19	′	′	NUM
ejpam-4912	149	1	in	in	ADP
ejpam-4912	149	2	figure	figure	NOUN
ejpam-4912	149	3	4	4	NUM
ejpam-4912	149	4	below	below	ADV
ejpam-4912	149	5	,	,	PUNCT
ejpam-4912	149	6	where	where	SCONJ
ejpam-4912	149	7	⟨{am	⟨{am	NOUN
ejpam-4912	149	8	,	,	PUNCT
ejpam-4912	149	9	b1	b1	NOUN
ejpam-4912	149	10	,	,	PUNCT
ejpam-4912	149	11	b2	b2	NOUN
ejpam-4912	149	12	,	,	PUNCT
ejpam-4912	149	13	.	.	PUNCT
ejpam-4912	149	14	.	.	PUNCT
ejpam-4912	149	15	.	.	PUNCT
ejpam-4912	150	1	,	,	PUNCT
ejpam-4912	150	2	bs}⟩	bs}⟩	PRON
ejpam-4912	150	3	induced	induce	VERB
ejpam-4912	150	4	a	a	DET
ejpam-4912	150	5	complete	complete	ADJ
ejpam-4912	150	6	graph	graph	NOUN
ejpam-4912	150	7	for	for	ADP
ejpam-4912	150	8	all	all	DET
ejpam-4912	150	9	positive	positive	ADJ
ejpam-4912	150	10	integer	integer	NOUN
ejpam-4912	150	11	s	s	PART
ejpam-4912	150	12	≥	≥	NOUN
ejpam-4912	150	13	1	1	NUM
ejpam-4912	150	14	.	.	PUNCT
ejpam-4912	151	1	x3x2x1	x3x2x1	NOUN
ejpam-4912	151	2	y2y1	y2y1	PROPN
ejpam-4912	151	3	y3	y3	PROPN
ejpam-4912	151	4	a3a2a1	a3a2a1	PROPN
ejpam-4912	151	5	xm−1	xm−1	PROPN
ejpam-4912	151	6	xm	xm	PROPN
ejpam-4912	151	7	am−1	am−1	PROPN
ejpam-4912	151	8	am	be	AUX
ejpam-4912	151	9	ym−1	ym−1	PROPN
ejpam-4912	151	10	ym	ym	PROPN
ejpam-4912	151	11	b1	b1	PROPN
ejpam-4912	151	12	b2	b2	NOUN
ejpam-4912	151	13	bs	bs	NOUN
ejpam-4912	151	14	.	.	PUNCT
ejpam-4912	151	15	.	.	PUNCT
ejpam-4912	151	16	.	.	PUNCT
ejpam-4912	151	17	.	.	PUNCT
ejpam-4912	151	18	.	.	PUNCT
ejpam-4912	152	1	.h	.h	NOUN
ejpam-4912	153	1	′	′	NUM
ejpam-4912	153	2	:	:	PUNCT
ejpam-4912	153	3	figure	figure	VERB
ejpam-4912	153	4	4	4	NUM
ejpam-4912	153	5	:	:	PUNCT
ejpam-4912	153	6	a	a	DET
ejpam-4912	153	7	graph	graph	NOUN
ejpam-4912	153	8	h′	h′	PROPN
ejpam-4912	153	9	with	with	ADP
ejpam-4912	153	10	γt(h′	γt(h′	PROPN
ejpam-4912	153	11	)	)	PUNCT
ejpam-4912	153	12	<	<	X
ejpam-4912	153	13	γjt(h	γjt(h	PROPN
ejpam-4912	153	14	′	′	NOUN
ejpam-4912	153	15	)	)	PUNCT
ejpam-4912	153	16	let	let	VERB
ejpam-4912	153	17	m1	m1	PROPN
ejpam-4912	153	18	=	=	SYM
ejpam-4912	153	19	{	{	PUNCT
ejpam-4912	153	20	a1	a1	PROPN
ejpam-4912	153	21	,	,	PUNCT
ejpam-4912	153	22	a2	a2	PROPN
ejpam-4912	153	23	,	,	PUNCT
ejpam-4912	153	24	...	...	PUNCT
ejpam-4912	153	25	,	,	PUNCT
ejpam-4912	153	26	am	be	AUX
ejpam-4912	153	27	}	}	PUNCT
ejpam-4912	153	28	and	and	CCONJ
ejpam-4912	153	29	m2	m2	PROPN
ejpam-4912	153	30	=	=	PROPN
ejpam-4912	153	31	{	{	PUNCT
ejpam-4912	153	32	a1	a1	PROPN
ejpam-4912	153	33	,	,	PUNCT
ejpam-4912	153	34	a2	a2	PROPN
ejpam-4912	153	35	,	,	PUNCT
ejpam-4912	153	36	.	.	PUNCT
ejpam-4912	153	37	.	.	PUNCT
ejpam-4912	154	1	.	.	PUNCT
ejpam-4912	155	1	,	,	PUNCT
ejpam-4912	155	2	am	be	AUX
ejpam-4912	155	3	,	,	PUNCT
ejpam-4912	155	4	b1	b1	NOUN
ejpam-4912	155	5	,	,	PUNCT
ejpam-4912	155	6	b2	b2	NOUN
ejpam-4912	155	7	,	,	PUNCT
ejpam-4912	155	8	.	.	PUNCT
ejpam-4912	155	9	.	.	PUNCT
ejpam-4912	156	1	.	.	PUNCT
ejpam-4912	157	1	,	,	PUNCT
ejpam-4912	157	2	bs	bs	NOUN
ejpam-4912	157	3	}	}	PUNCT
ejpam-4912	157	4	.	.	PUNCT
ejpam-4912	158	1	then	then	ADV
ejpam-4912	158	2	m1	m1	PROPN
ejpam-4912	158	3	is	be	AUX
ejpam-4912	158	4	the	the	DET
ejpam-4912	158	5	minimum	minimum	ADJ
ejpam-4912	158	6	total	total	ADJ
ejpam-4912	158	7	dominating	dominating	NOUN
ejpam-4912	158	8	set	set	NOUN
ejpam-4912	158	9	of	of	ADP
ejpam-4912	158	10	h	h	NOUN
ejpam-4912	158	11	′.	′.	PROPN
ejpam-4912	158	12	thus	thus	ADV
ejpam-4912	158	13	,	,	PUNCT
ejpam-4912	158	14	γt(h	γt(h	NUM
ejpam-4912	158	15	′	′	NUM
ejpam-4912	158	16	)	)	PUNCT
ejpam-4912	158	17	=	=	VERB
ejpam-4912	158	18	m.	m.	NOUN
ejpam-4912	158	19	since	since	SCONJ
ejpam-4912	158	20	m1	m1	PROPN
ejpam-4912	158	21	⊆	⊆	NUM
ejpam-4912	158	22	m2	m2	PROPN
ejpam-4912	158	23	,	,	PUNCT
ejpam-4912	158	24	m2	m2	PROPN
ejpam-4912	158	25	is	be	AUX
ejpam-4912	158	26	also	also	ADV
ejpam-4912	158	27	a	a	DET
ejpam-4912	158	28	total	total	ADJ
ejpam-4912	158	29	dominating	dominating	NOUN
ejpam-4912	158	30	set	set	NOUN
ejpam-4912	158	31	of	of	ADP
ejpam-4912	158	32	h	h	PROPN
ejpam-4912	158	33	′.	′.	PROPN
ejpam-4912	158	34	observe	observe	VERB
ejpam-4912	158	35	that	that	SCONJ
ejpam-4912	158	36	m2	m2	PROPN
ejpam-4912	158	37	is	be	AUX
ejpam-4912	158	38	a	a	DET
ejpam-4912	158	39	j	j	NOUN
ejpam-4912	158	40	-	-	ADJ
ejpam-4912	158	41	open	open	ADJ
ejpam-4912	158	42	set	set	NOUN
ejpam-4912	158	43	in	in	ADP
ejpam-4912	158	44	h	h	NOUN
ejpam-4912	158	45	′.	′.	NOUN
ejpam-4912	158	46	hence	hence	ADV
ejpam-4912	158	47	,	,	PUNCT
ejpam-4912	158	48	m2	m2	PROPN
ejpam-4912	158	49	is	be	AUX
ejpam-4912	158	50	a	a	DET
ejpam-4912	158	51	jtotal	jtotal	ADJ
ejpam-4912	158	52	dominating	dominating	NOUN
ejpam-4912	158	53	set	set	NOUN
ejpam-4912	158	54	of	of	ADP
ejpam-4912	158	55	h	h	NOUN
ejpam-4912	158	56	′.	′.	NOUN
ejpam-4912	158	57	applying	apply	VERB
ejpam-4912	158	58	the	the	DET
ejpam-4912	158	59	same	same	ADJ
ejpam-4912	158	60	argument	argument	NOUN
ejpam-4912	158	61	in	in	ADP
ejpam-4912	158	62	the	the	DET
ejpam-4912	158	63	equality	equality	NOUN
ejpam-4912	158	64	part	part	NOUN
ejpam-4912	158	65	and	and	CCONJ
ejpam-4912	158	66	v	v	NOUN
ejpam-4912	158	67	(	(	PUNCT
ejpam-4912	158	68	ks	ks	NOUN
ejpam-4912	158	69	)	)	PUNCT
ejpam-4912	158	70	is	be	AUX
ejpam-4912	158	71	a	a	DET
ejpam-4912	158	72	j	j	NOUN
ejpam-4912	158	73	-	-	ADJ
ejpam-4912	158	74	open	open	ADJ
ejpam-4912	158	75	set	set	NOUN
ejpam-4912	158	76	in	in	ADP
ejpam-4912	158	77	ks	ks	PROPN
ejpam-4912	158	78	,	,	PUNCT
ejpam-4912	158	79	s	s	PART
ejpam-4912	158	80	≥	≥	NOUN
ejpam-4912	158	81	2	2	NUM
ejpam-4912	158	82	by	by	ADP
ejpam-4912	158	83	theorem	theorem	NOUN
ejpam-4912	158	84	1	1	NUM
ejpam-4912	158	85	,	,	PUNCT
ejpam-4912	158	86	it	it	PRON
ejpam-4912	158	87	follows	follow	VERB
ejpam-4912	158	88	that	that	SCONJ
ejpam-4912	158	89	m2	m2	PROPN
ejpam-4912	158	90	is	be	AUX
ejpam-4912	158	91	the	the	DET
ejpam-4912	158	92	maximum	maximum	ADJ
ejpam-4912	158	93	j	j	PROPN
ejpam-4912	158	94	-	-	PROPN
ejpam-4912	158	95	total	total	ADJ
ejpam-4912	158	96	j.a	j.a	PROPN
ejpam-4912	158	97	.	.	PUNCT
ejpam-4912	159	1	hassan	hassan	PROPN
ejpam-4912	159	2	et	et	PROPN
ejpam-4912	159	3	al	al	PROPN
ejpam-4912	159	4	.	.	PUNCT
ejpam-4912	159	5	/	/	SYM
ejpam-4912	159	6	eur	eur	PROPN
ejpam-4912	159	7	.	.	PUNCT
ejpam-4912	160	1	j.	j.	PROPN
ejpam-4912	160	2	pure	pure	PROPN
ejpam-4912	160	3	appl	appl	PROPN
ejpam-4912	160	4	.	.	PROPN
ejpam-4912	160	5	math	math	PROPN
ejpam-4912	160	6	,	,	PUNCT
ejpam-4912	160	7	16	16	NUM
ejpam-4912	160	8	(	(	PUNCT
ejpam-4912	160	9	4	4	NUM
ejpam-4912	160	10	)	)	PUNCT
ejpam-4912	160	11	(	(	PUNCT
ejpam-4912	160	12	2023	2023	NUM
ejpam-4912	160	13	)	)	PUNCT
ejpam-4912	160	14	,	,	PUNCT
ejpam-4912	160	15	2106	2106	NUM
ejpam-4912	160	16	-	-	SYM
ejpam-4912	160	17	2117	2117	NUM
ejpam-4912	160	18	2112	2112	NUM
ejpam-4912	160	19	dominating	dominating	NOUN
ejpam-4912	160	20	set	set	NOUN
ejpam-4912	160	21	of	of	ADP
ejpam-4912	160	22	h	h	NOUN
ejpam-4912	160	23	′.	′.	PROPN
ejpam-4912	160	24	consequently	consequently	ADV
ejpam-4912	160	25	,	,	PUNCT
ejpam-4912	160	26	γjt(h	γjt(h	PROPN
ejpam-4912	160	27	′	′	NOUN
ejpam-4912	160	28	)	)	PUNCT
ejpam-4912	160	29	=	=	SYM
ejpam-4912	161	1	s+m	s+m	X
ejpam-4912	161	2	=	=	SYM
ejpam-4912	161	3	n.	n.	PROPN
ejpam-4912	161	4	theorem	theorem	VERB
ejpam-4912	161	5	3	3	X
ejpam-4912	161	6	.	.	PUNCT
ejpam-4912	162	1	let	let	VERB
ejpam-4912	162	2	g	g	PRON
ejpam-4912	162	3	be	be	AUX
ejpam-4912	162	4	a	a	DET
ejpam-4912	162	5	graph	graph	NOUN
ejpam-4912	162	6	with	with	ADP
ejpam-4912	162	7	no	no	DET
ejpam-4912	162	8	isolated	isolated	ADJ
ejpam-4912	162	9	vertex	vertex	NOUN
ejpam-4912	162	10	.	.	PUNCT
ejpam-4912	163	1	then	then	ADV
ejpam-4912	163	2	(	(	PUNCT
ejpam-4912	163	3	i	i	NOUN
ejpam-4912	163	4	)	)	PUNCT
ejpam-4912	163	5	m	m	VERB
ejpam-4912	163	6	is	be	AUX
ejpam-4912	163	7	a	a	DET
ejpam-4912	163	8	γt	γt	NOUN
ejpam-4912	163	9	-	-	ADJ
ejpam-4912	163	10	set	set	ADJ
ejpam-4912	163	11	in	in	ADP
ejpam-4912	163	12	g	g	PROPN
ejpam-4912	163	13	if	if	SCONJ
ejpam-4912	164	1	and	and	CCONJ
ejpam-4912	164	2	only	only	ADV
ejpam-4912	164	3	if	if	SCONJ
ejpam-4912	164	4	m	m	NOUN
ejpam-4912	164	5	is	be	AUX
ejpam-4912	164	6	a	a	DET
ejpam-4912	164	7	minimum	minimum	ADJ
ejpam-4912	164	8	j	j	ADJ
ejpam-4912	164	9	-	-	ADJ
ejpam-4912	164	10	total	total	ADJ
ejpam-4912	164	11	dominating	dominating	NOUN
ejpam-4912	164	12	set	set	VERB
ejpam-4912	164	13	in	in	ADP
ejpam-4912	164	14	g.	g.	PROPN
ejpam-4912	164	15	(	(	PUNCT
ejpam-4912	164	16	ii	ii	PROPN
ejpam-4912	164	17	)	)	PUNCT
ejpam-4912	164	18	if	if	SCONJ
ejpam-4912	164	19	every	every	DET
ejpam-4912	164	20	component	component	NOUN
ejpam-4912	164	21	of	of	ADP
ejpam-4912	164	22	g	g	PROPN
ejpam-4912	164	23	is	be	AUX
ejpam-4912	164	24	non	non	ADJ
ejpam-4912	164	25	-	-	ADJ
ejpam-4912	164	26	trivial	trivial	ADJ
ejpam-4912	164	27	complete	complete	ADJ
ejpam-4912	164	28	graph	graph	NOUN
ejpam-4912	164	29	,	,	PUNCT
ejpam-4912	164	30	then	then	ADV
ejpam-4912	164	31	γjt(g	γjt(g	PROPN
ejpam-4912	164	32	)	)	PUNCT
ejpam-4912	165	1	=	=	SYM
ejpam-4912	165	2	|v	|v	PROPN
ejpam-4912	165	3	(	(	PUNCT
ejpam-4912	165	4	g)|	g)|	NOUN
ejpam-4912	165	5	.	.	PUNCT
ejpam-4912	166	1	however	however	ADV
ejpam-4912	166	2	,	,	PUNCT
ejpam-4912	166	3	the	the	DET
ejpam-4912	166	4	converse	converse	NOUN
ejpam-4912	166	5	is	be	AUX
ejpam-4912	166	6	not	not	PART
ejpam-4912	166	7	true	true	ADJ
ejpam-4912	166	8	.	.	PUNCT
ejpam-4912	167	1	proof	proof	NOUN
ejpam-4912	167	2	.	.	PUNCT
ejpam-4912	168	1	(	(	PUNCT
ejpam-4912	168	2	i	i	NOUN
ejpam-4912	168	3	)	)	PUNCT
ejpam-4912	168	4	let	let	VERB
ejpam-4912	168	5	m	m	PRON
ejpam-4912	168	6	be	be	AUX
ejpam-4912	168	7	a	a	DET
ejpam-4912	168	8	γt	γt	NOUN
ejpam-4912	168	9	-	-	ADJ
ejpam-4912	168	10	set	set	ADJ
ejpam-4912	168	11	in	in	ADP
ejpam-4912	168	12	g.	g.	PROPN
ejpam-4912	168	13	then	then	ADV
ejpam-4912	168	14	m	m	PROPN
ejpam-4912	168	15	is	be	AUX
ejpam-4912	168	16	the	the	DET
ejpam-4912	168	17	minimum	minimum	ADJ
ejpam-4912	168	18	total	total	ADJ
ejpam-4912	168	19	dominating	dominating	NOUN
ejpam-4912	168	20	set	set	VERB
ejpam-4912	168	21	in	in	ADP
ejpam-4912	168	22	g.	g.	PROPN
ejpam-4912	168	23	suppose	suppose	VERB
ejpam-4912	168	24	that	that	SCONJ
ejpam-4912	168	25	m	m	PROPN
ejpam-4912	168	26	is	be	AUX
ejpam-4912	168	27	not	not	PART
ejpam-4912	168	28	a	a	DET
ejpam-4912	168	29	j	j	NOUN
ejpam-4912	168	30	-	-	ADJ
ejpam-4912	168	31	open	open	ADJ
ejpam-4912	168	32	set	set	NOUN
ejpam-4912	168	33	in	in	ADP
ejpam-4912	168	34	g.	g.	PROPN
ejpam-4912	168	35	then	then	ADV
ejpam-4912	168	36	there	there	PRON
ejpam-4912	168	37	exist	exist	VERB
ejpam-4912	168	38	a	a	DET
ejpam-4912	168	39	,	,	PUNCT
ejpam-4912	168	40	b	b	X
ejpam-4912	168	41	∈	∈	NOUN
ejpam-4912	168	42	m	m	VERB
ejpam-4912	168	43	such	such	ADJ
ejpam-4912	168	44	that	that	SCONJ
ejpam-4912	168	45	ng(a)\ng(b	ng(a)\ng(b	ADJ
ejpam-4912	168	46	)	)	PUNCT
ejpam-4912	168	47	=	=	PUNCT
ejpam-4912	168	48	∅	∅	NOUN
ejpam-4912	168	49	orng(b)\ng(a	orng(b)\ng(a	PROPN
ejpam-4912	168	50	)	)	PUNCT
ejpam-4912	169	1	=	=	VERB
ejpam-4912	169	2	∅.	∅.	NOUN
ejpam-4912	169	3	it	it	PRON
ejpam-4912	169	4	follows	follow	VERB
ejpam-4912	169	5	thatng(a	thatng(a	NUM
ejpam-4912	169	6	)	)	PUNCT
ejpam-4912	169	7	⊆	⊆	NUM
ejpam-4912	169	8	ng(b	ng(b	X
ejpam-4912	169	9	)	)	PUNCT
ejpam-4912	169	10	orng(b	orng(b	NOUN
ejpam-4912	169	11	)	)	PUNCT
ejpam-4912	169	12	⊆	⊆	NUM
ejpam-4912	169	13	ng(a	ng(a	NOUN
ejpam-4912	169	14	)	)	PUNCT
ejpam-4912	169	15	.	.	PUNCT
ejpam-4912	170	1	assume	assume	VERB
ejpam-4912	170	2	that	that	SCONJ
ejpam-4912	170	3	ng(a	ng(a	NOUN
ejpam-4912	170	4	)	)	PUNCT
ejpam-4912	170	5	⊆	⊆	NUM
ejpam-4912	170	6	ng(b	ng(b	X
ejpam-4912	170	7	)	)	PUNCT
ejpam-4912	170	8	,	,	PUNCT
ejpam-4912	170	9	then	then	ADV
ejpam-4912	170	10	m\{a	m\{a	NOUN
ejpam-4912	170	11	}	}	PUNCT
ejpam-4912	170	12	is	be	AUX
ejpam-4912	170	13	a	a	DET
ejpam-4912	170	14	total	total	ADJ
ejpam-4912	170	15	dominating	dominating	NOUN
ejpam-4912	170	16	set	set	VERB
ejpam-4912	170	17	in	in	ADP
ejpam-4912	170	18	g.	g.	PROPN
ejpam-4912	170	19	however	however	ADV
ejpam-4912	170	20	,	,	PUNCT
ejpam-4912	170	21	this	this	PRON
ejpam-4912	170	22	is	be	AUX
ejpam-4912	170	23	a	a	DET
ejpam-4912	170	24	contradiction	contradiction	NOUN
ejpam-4912	170	25	to	to	ADP
ejpam-4912	170	26	our	our	PRON
ejpam-4912	170	27	assumption	assumption	NOUN
ejpam-4912	170	28	that	that	SCONJ
ejpam-4912	170	29	m	m	NOUN
ejpam-4912	170	30	is	be	AUX
ejpam-4912	170	31	the	the	DET
ejpam-4912	170	32	minimum	minimum	ADJ
ejpam-4912	170	33	total	total	ADJ
ejpam-4912	170	34	dominating	dominating	NOUN
ejpam-4912	170	35	set	set	VERB
ejpam-4912	170	36	in	in	ADP
ejpam-4912	170	37	g.	g.	PROPN
ejpam-4912	170	38	hence	hence	ADV
ejpam-4912	170	39	,	,	PUNCT
ejpam-4912	170	40	m	m	PROPN
ejpam-4912	170	41	is	be	AUX
ejpam-4912	170	42	a	a	DET
ejpam-4912	170	43	j	j	NOUN
ejpam-4912	170	44	-	-	ADJ
ejpam-4912	170	45	open	open	ADJ
ejpam-4912	170	46	set	set	NOUN
ejpam-4912	170	47	in	in	ADP
ejpam-4912	170	48	g	g	NOUN
ejpam-4912	170	49	,	,	PUNCT
ejpam-4912	170	50	and	and	CCONJ
ejpam-4912	170	51	so	so	ADV
ejpam-4912	170	52	m	m	VERB
ejpam-4912	170	53	is	be	AUX
ejpam-4912	170	54	a	a	DET
ejpam-4912	170	55	minimum	minimum	ADJ
ejpam-4912	170	56	j	j	ADJ
ejpam-4912	170	57	-	-	ADJ
ejpam-4912	170	58	total	total	ADJ
ejpam-4912	170	59	dominating	dominating	NOUN
ejpam-4912	170	60	set	set	VERB
ejpam-4912	170	61	in	in	ADP
ejpam-4912	170	62	g.	g.	PROPN
ejpam-4912	170	63	the	the	DET
ejpam-4912	170	64	converse	converse	NOUN
ejpam-4912	170	65	is	be	AUX
ejpam-4912	170	66	clear	clear	ADJ
ejpam-4912	170	67	.	.	PUNCT
ejpam-4912	171	1	(	(	PUNCT
ejpam-4912	171	2	ii	ii	NOUN
ejpam-4912	171	3	)	)	PUNCT
ejpam-4912	171	4	suppose	suppose	VERB
ejpam-4912	171	5	that	that	SCONJ
ejpam-4912	171	6	every	every	DET
ejpam-4912	171	7	component	component	NOUN
ejpam-4912	171	8	h	h	NOUN
ejpam-4912	171	9	of	of	ADP
ejpam-4912	171	10	g	g	PROPN
ejpam-4912	171	11	is	be	AUX
ejpam-4912	171	12	a	a	DET
ejpam-4912	171	13	non	non	ADJ
ejpam-4912	171	14	-	-	ADJ
ejpam-4912	171	15	trivial	trivial	ADJ
ejpam-4912	171	16	complete	complete	ADJ
ejpam-4912	171	17	graph	graph	NOUN
ejpam-4912	171	18	.	.	PUNCT
ejpam-4912	172	1	let	let	VERB
ejpam-4912	172	2	h1	h1	VERB
ejpam-4912	172	3	,	,	PUNCT
ejpam-4912	172	4	.	.	PUNCT
ejpam-4912	172	5	.	.	PUNCT
ejpam-4912	173	1	.	.	PUNCT
ejpam-4912	174	1	,	,	PUNCT
ejpam-4912	174	2	hk	hk	PROPN
ejpam-4912	174	3	,	,	PUNCT
ejpam-4912	174	4	k	k	PROPN
ejpam-4912	174	5	≥	≥	NUM
ejpam-4912	174	6	2	2	NUM
ejpam-4912	174	7	be	be	AUX
ejpam-4912	174	8	components	component	NOUN
ejpam-4912	174	9	of	of	ADP
ejpam-4912	174	10	g.	g.	PROPN
ejpam-4912	174	11	then	then	ADV
ejpam-4912	174	12	by	by	ADP
ejpam-4912	174	13	corollary	corollary	ADJ
ejpam-4912	174	14	1	1	NUM
ejpam-4912	174	15	,	,	PUNCT
ejpam-4912	174	16	γjt(hi	γjt(hi	NOUN
ejpam-4912	174	17	)	)	PUNCT
ejpam-4912	174	18	=|	=|	NOUN
ejpam-4912	174	19	v	v	NOUN
ejpam-4912	174	20	(	(	PUNCT
ejpam-4912	174	21	hi	hi	INTJ
ejpam-4912	174	22	)	)	PUNCT
ejpam-4912	174	23	|	|	ADV
ejpam-4912	174	24	.	.	PUNCT
ejpam-4912	175	1	it	it	PRON
ejpam-4912	175	2	follows	follow	VERB
ejpam-4912	175	3	that	that	SCONJ
ejpam-4912	175	4	γjt(g	γjt(g	NOUN
ejpam-4912	175	5	)	)	PUNCT
ejpam-4912	175	6	=	=	SYM
ejpam-4912	175	7	γjt(hi	γjt(hi	NOUN
ejpam-4912	175	8	)	)	PUNCT
ejpam-4912	176	1	+	+	CCONJ
ejpam-4912	176	2	·	·	PUNCT
ejpam-4912	176	3	·	·	PUNCT
ejpam-4912	176	4	·	·	PUNCT
ejpam-4912	176	5	+	+	NUM
ejpam-4912	176	6	γjt(hk	γjt(hk	NOUN
ejpam-4912	176	7	)	)	PUNCT
ejpam-4912	176	8	=|	=|	NOUN
ejpam-4912	176	9	v	v	NOUN
ejpam-4912	176	10	(	(	PUNCT
ejpam-4912	176	11	hi	hi	INTJ
ejpam-4912	176	12	)	)	PUNCT
ejpam-4912	176	13	|	|	ADV
ejpam-4912	176	14	+	+	CCONJ
ejpam-4912	176	15	·	·	PUNCT
ejpam-4912	176	16	·	·	PUNCT
ejpam-4912	176	17	·	·	PUNCT
ejpam-4912	176	18	+	+	CCONJ
ejpam-4912	176	19	|	|	ADV
ejpam-4912	176	20	v	v	ADP
ejpam-4912	176	21	(	(	PUNCT
ejpam-4912	176	22	hk	hk	PROPN
ejpam-4912	176	23	)	)	PUNCT
ejpam-4912	176	24	=|	=|	NOUN
ejpam-4912	176	25	v	v	NOUN
ejpam-4912	176	26	(	(	PUNCT
ejpam-4912	176	27	g	g	NOUN
ejpam-4912	176	28	)	)	PUNCT
ejpam-4912	176	29	|	|	ADV
ejpam-4912	176	30	.	.	PUNCT
ejpam-4912	177	1	to	to	PART
ejpam-4912	177	2	see	see	VERB
ejpam-4912	177	3	that	that	SCONJ
ejpam-4912	177	4	the	the	DET
ejpam-4912	177	5	converse	converse	NOUN
ejpam-4912	177	6	is	be	AUX
ejpam-4912	177	7	not	not	PART
ejpam-4912	177	8	true	true	ADJ
ejpam-4912	177	9	,	,	PUNCT
ejpam-4912	177	10	consider	consider	VERB
ejpam-4912	177	11	g	g	NOUN
ejpam-4912	177	12	=	=	PROPN
ejpam-4912	177	13	c5	c5	PROPN
ejpam-4912	177	14	below	below	ADV
ejpam-4912	177	15	.	.	PUNCT
ejpam-4912	178	1	c5	c5	PROPN
ejpam-4912	178	2	:	:	PUNCT
ejpam-4912	178	3	a5	a5	PROPN
ejpam-4912	178	4	a3	a3	PROPN
ejpam-4912	178	5	a2	a2	PROPN
ejpam-4912	178	6	a1	a1	NOUN
ejpam-4912	178	7	a4	a4	NOUN
ejpam-4912	178	8	figure	figure	NOUN
ejpam-4912	178	9	5	5	NUM
ejpam-4912	178	10	:	:	PUNCT
ejpam-4912	178	11	a	a	DET
ejpam-4912	178	12	graph	graph	NOUN
ejpam-4912	178	13	c5	c5	PROPN
ejpam-4912	178	14	with	with	ADP
ejpam-4912	178	15	γjt(c5	γjt(c5	PROPN
ejpam-4912	178	16	)	)	PUNCT
ejpam-4912	178	17	=	=	SYM
ejpam-4912	178	18	5	5	NUM
ejpam-4912	178	19	let	let	VERB
ejpam-4912	178	20	m	m	VERB
ejpam-4912	178	21	=	=	NOUN
ejpam-4912	178	22	{	{	PUNCT
ejpam-4912	178	23	a1	a1	PROPN
ejpam-4912	178	24	,	,	PUNCT
ejpam-4912	178	25	a2	a2	PROPN
ejpam-4912	178	26	,	,	PUNCT
ejpam-4912	178	27	...	...	PUNCT
ejpam-4912	178	28	,	,	PUNCT
ejpam-4912	178	29	a5	a5	PROPN
ejpam-4912	178	30	}	}	PUNCT
ejpam-4912	178	31	=	=	SYM
ejpam-4912	178	32	v	v	NOUN
ejpam-4912	178	33	(	(	PUNCT
ejpam-4912	178	34	g	g	NOUN
ejpam-4912	178	35	)	)	PUNCT
ejpam-4912	178	36	.	.	PUNCT
ejpam-4912	179	1	observe	observe	VERB
ejpam-4912	179	2	that	that	SCONJ
ejpam-4912	179	3	a2	a2	PROPN
ejpam-4912	179	4	∈	∈	PROPN
ejpam-4912	179	5	ng(a1	ng(a1	NOUN
ejpam-4912	179	6	)	)	PUNCT
ejpam-4912	179	7	\ng(ai	\ng(ai	NOUN
ejpam-4912	179	8	)	)	PUNCT
ejpam-4912	179	9	∀	∀	NOUN
ejpam-4912	180	1	i	i	PRON
ejpam-4912	180	2	̸=	̸=	PROPN
ejpam-4912	180	3	3	3	NUM
ejpam-4912	180	4	,	,	PUNCT
ejpam-4912	180	5	a5	a5	PROPN
ejpam-4912	180	6	∈	∈	PROPN
ejpam-4912	180	7	ng(a1	ng(a1	NUM
ejpam-4912	180	8	)	)	PUNCT
ejpam-4912	180	9	\ng(aj	\ng(aj	NOUN
ejpam-4912	180	10	)	)	PUNCT
ejpam-4912	180	11	∀	∀	NOUN
ejpam-4912	181	1	j	j	PROPN
ejpam-4912	181	2	̸=	̸=	PROPN
ejpam-4912	181	3	4	4	NUM
ejpam-4912	181	4	,	,	PUNCT
ejpam-4912	181	5	a1	a1	NOUN
ejpam-4912	181	6	∈	∈	PROPN
ejpam-4912	181	7	ng(a2	ng(a2	NOUN
ejpam-4912	181	8	)	)	PUNCT
ejpam-4912	181	9	\ng(ak	\ng(ak	NOUN
ejpam-4912	181	10	)	)	PUNCT
ejpam-4912	181	11	∀	∀	PUNCT
ejpam-4912	182	1	k	k	X
ejpam-4912	182	2	̸=	̸=	PROPN
ejpam-4912	182	3	5	5	NUM
ejpam-4912	182	4	,	,	PUNCT
ejpam-4912	182	5	a3	a3	NOUN
ejpam-4912	182	6	∈	∈	PROPN
ejpam-4912	182	7	ng(a2	ng(a2	NOUN
ejpam-4912	182	8	)	)	PUNCT
ejpam-4912	182	9	\ng(ae	\ng(ae	NOUN
ejpam-4912	182	10	)	)	PUNCT
ejpam-4912	182	11	∀	∀	X
ejpam-4912	182	12	e	e	X
ejpam-4912	182	13	̸=	̸=	PROPN
ejpam-4912	182	14	4	4	NUM
ejpam-4912	182	15	,	,	PUNCT
ejpam-4912	182	16	a2	a2	PROPN
ejpam-4912	182	17	∈	∈	PROPN
ejpam-4912	182	18	ng(a3	ng(a3	PROPN
ejpam-4912	182	19	)	)	PUNCT
ejpam-4912	182	20	\ng(as	\ng(as	NOUN
ejpam-4912	182	21	)	)	PUNCT
ejpam-4912	182	22	∀	∀	X
ejpam-4912	183	1	s	s	PART
ejpam-4912	183	2	̸=	̸=	PROPN
ejpam-4912	183	3	1	1	NUM
ejpam-4912	183	4	,	,	PUNCT
ejpam-4912	183	5	j.a	j.a	PROPN
ejpam-4912	183	6	.	.	PROPN
ejpam-4912	183	7	hassan	hassan	PROPN
ejpam-4912	183	8	et	et	PROPN
ejpam-4912	183	9	al	al	PROPN
ejpam-4912	183	10	.	.	PUNCT
ejpam-4912	183	11	/	/	SYM
ejpam-4912	183	12	eur	eur	PROPN
ejpam-4912	183	13	.	.	PUNCT
ejpam-4912	184	1	j.	j.	PROPN
ejpam-4912	184	2	pure	pure	PROPN
ejpam-4912	184	3	appl	appl	PROPN
ejpam-4912	184	4	.	.	PROPN
ejpam-4912	184	5	math	math	PROPN
ejpam-4912	184	6	,	,	PUNCT
ejpam-4912	184	7	16	16	NUM
ejpam-4912	184	8	(	(	PUNCT
ejpam-4912	184	9	4	4	NUM
ejpam-4912	184	10	)	)	PUNCT
ejpam-4912	184	11	(	(	PUNCT
ejpam-4912	184	12	2023	2023	NUM
ejpam-4912	184	13	)	)	PUNCT
ejpam-4912	184	14	,	,	PUNCT
ejpam-4912	184	15	2106	2106	NUM
ejpam-4912	184	16	-	-	SYM
ejpam-4912	184	17	2117	2117	NUM
ejpam-4912	184	18	2113	2113	NUM
ejpam-4912	184	19	a4	a4	NOUN
ejpam-4912	184	20	∈	∈	PROPN
ejpam-4912	184	21	ng(a3	ng(a3	NOUN
ejpam-4912	184	22	)	)	PUNCT
ejpam-4912	184	23	\ng(at	\ng(at	NOUN
ejpam-4912	184	24	)	)	PUNCT
ejpam-4912	184	25	∀	∀	X
ejpam-4912	185	1	t	t	X
ejpam-4912	185	2	̸=	̸=	PROPN
ejpam-4912	185	3	5	5	NUM
ejpam-4912	185	4	,	,	PUNCT
ejpam-4912	185	5	a3	a3	NOUN
ejpam-4912	185	6	∈	∈	PROPN
ejpam-4912	185	7	ng(a4	ng(a4	X
ejpam-4912	185	8	)	)	PUNCT
ejpam-4912	185	9	\ng(ar	\ng(ar	NOUN
ejpam-4912	185	10	)	)	PUNCT
ejpam-4912	185	11	∀	∀	X
ejpam-4912	186	1	r	r	NOUN
ejpam-4912	186	2	̸=	̸=	PROPN
ejpam-4912	186	3	2	2	NUM
ejpam-4912	186	4	,	,	PUNCT
ejpam-4912	186	5	a5	a5	PROPN
ejpam-4912	186	6	∈	∈	PROPN
ejpam-4912	186	7	ng(a4	ng(a4	X
ejpam-4912	186	8	)	)	PUNCT
ejpam-4912	186	9	\ng(aq	\ng(aq	PROPN
ejpam-4912	186	10	)	)	PUNCT
ejpam-4912	186	11	∀	∀	X
ejpam-4912	187	1	q	q	PROPN
ejpam-4912	187	2	̸=	̸=	PROPN
ejpam-4912	187	3	1	1	NUM
ejpam-4912	187	4	,	,	PUNCT
ejpam-4912	187	5	a1	a1	NOUN
ejpam-4912	187	6	∈	∈	PROPN
ejpam-4912	187	7	ng(a5	ng(a5	ADV
ejpam-4912	187	8	)	)	PUNCT
ejpam-4912	187	9	\ng(am	\ng(am	NOUN
ejpam-4912	187	10	)	)	PUNCT
ejpam-4912	187	11	∀	∀	X
ejpam-4912	188	1	m	m	VERB
ejpam-4912	188	2	̸=	̸=	PROPN
ejpam-4912	188	3	2	2	NUM
ejpam-4912	188	4	,	,	PUNCT
ejpam-4912	188	5	and	and	CCONJ
ejpam-4912	188	6	a4	a4	NOUN
ejpam-4912	188	7	∈	∈	NOUN
ejpam-4912	188	8	ng(a5	ng(a5	ADV
ejpam-4912	188	9	)	)	PUNCT
ejpam-4912	188	10	\ng(an	\ng(an	NOUN
ejpam-4912	188	11	)	)	PUNCT
ejpam-4912	188	12	∀	∀	X
ejpam-4912	189	1	n	n	CCONJ
ejpam-4912	189	2	̸=	̸=	PROPN
ejpam-4912	189	3	3	3	NUM
ejpam-4912	189	4	.	.	PUNCT
ejpam-4912	190	1	thus	thus	ADV
ejpam-4912	190	2	,	,	PUNCT
ejpam-4912	190	3	m	m	VERB
ejpam-4912	190	4	=	=	ADJ
ejpam-4912	190	5	v	v	ADJ
ejpam-4912	190	6	(	(	PUNCT
ejpam-4912	190	7	g	g	NOUN
ejpam-4912	190	8	)	)	PUNCT
ejpam-4912	190	9	is	be	AUX
ejpam-4912	190	10	a	a	DET
ejpam-4912	190	11	j	j	NOUN
ejpam-4912	190	12	-	-	ADJ
ejpam-4912	190	13	open	open	ADJ
ejpam-4912	190	14	set	set	NOUN
ejpam-4912	190	15	in	in	ADP
ejpam-4912	190	16	g.	g.	PROPN
ejpam-4912	190	17	since	since	SCONJ
ejpam-4912	190	18	ng(m	ng(m	NOUN
ejpam-4912	190	19	)	)	PUNCT
ejpam-4912	190	20	=	=	SYM
ejpam-4912	190	21	v	v	X
ejpam-4912	190	22	(	(	PUNCT
ejpam-4912	190	23	g	g	NOUN
ejpam-4912	190	24	)	)	PUNCT
ejpam-4912	190	25	,	,	PUNCT
ejpam-4912	190	26	it	it	PRON
ejpam-4912	190	27	follows	follow	VERB
ejpam-4912	190	28	that	that	SCONJ
ejpam-4912	190	29	m	m	PROPN
ejpam-4912	190	30	is	be	AUX
ejpam-4912	190	31	a	a	DET
ejpam-4912	190	32	j	j	PROPN
ejpam-4912	190	33	-	-	ADJ
ejpam-4912	190	34	total	total	ADJ
ejpam-4912	190	35	dominating	dominating	NOUN
ejpam-4912	190	36	set	set	VERB
ejpam-4912	190	37	in	in	ADP
ejpam-4912	190	38	g.	g.	PROPN
ejpam-4912	190	39	hence	hence	ADV
ejpam-4912	190	40	,	,	PUNCT
ejpam-4912	190	41	γjt(g	γjt(g	PROPN
ejpam-4912	190	42	)	)	PUNCT
ejpam-4912	190	43	=	=	SYM
ejpam-4912	191	1	5	5	NUM
ejpam-4912	191	2	=|	=|	SYM
ejpam-4912	191	3	v	v	NOUN
ejpam-4912	191	4	(	(	PUNCT
ejpam-4912	191	5	g	g	NOUN
ejpam-4912	191	6	)	)	PUNCT
ejpam-4912	191	7	|	|	ADV
ejpam-4912	191	8	.	.	PUNCT
ejpam-4912	192	1	theorem	theorem	ADJ
ejpam-4912	192	2	4	4	NUM
ejpam-4912	192	3	.	.	PUNCT
ejpam-4912	193	1	let	let	VERB
ejpam-4912	193	2	km	km	PROPN
ejpam-4912	193	3	,	,	PUNCT
ejpam-4912	193	4	n	n	PRON
ejpam-4912	193	5	be	be	VERB
ejpam-4912	193	6	a	a	DET
ejpam-4912	193	7	complete	complete	ADJ
ejpam-4912	193	8	bipartite	bipartite	NOUN
ejpam-4912	193	9	graph	graph	NOUN
ejpam-4912	193	10	where	where	SCONJ
ejpam-4912	193	11	m	m	VERB
ejpam-4912	193	12	,	,	PUNCT
ejpam-4912	193	13	n	n	PRON
ejpam-4912	193	14	≥	≥	NOUN
ejpam-4912	193	15	1	1	NUM
ejpam-4912	193	16	.	.	PUNCT
ejpam-4912	194	1	then	then	ADV
ejpam-4912	194	2	n	n	PROPN
ejpam-4912	194	3	⊆	⊆	NUM
ejpam-4912	194	4	v	v	NOUN
ejpam-4912	194	5	(	(	PUNCT
ejpam-4912	194	6	km	km	PROPN
ejpam-4912	194	7	,	,	PUNCT
ejpam-4912	194	8	n	n	CCONJ
ejpam-4912	194	9	)	)	PUNCT
ejpam-4912	194	10	is	be	AUX
ejpam-4912	194	11	a	a	DET
ejpam-4912	194	12	j	j	PROPN
ejpam-4912	194	13	-	-	ADJ
ejpam-4912	194	14	total	total	ADJ
ejpam-4912	194	15	dominating	dominating	NOUN
ejpam-4912	194	16	in	in	ADP
ejpam-4912	194	17	km	km	PROPN
ejpam-4912	194	18	,	,	PUNCT
ejpam-4912	194	19	n	n	CCONJ
ejpam-4912	194	20	if	if	SCONJ
ejpam-4912	194	21	and	and	CCONJ
ejpam-4912	194	22	only	only	ADV
ejpam-4912	194	23	if	if	SCONJ
ejpam-4912	194	24	n	n	ADV
ejpam-4912	194	25	=	=	X
ejpam-4912	194	26	{	{	PUNCT
ejpam-4912	194	27	a	a	DET
ejpam-4912	194	28	,	,	PUNCT
ejpam-4912	194	29	b	b	NOUN
ejpam-4912	194	30	}	}	PUNCT
ejpam-4912	194	31	for	for	ADP
ejpam-4912	194	32	some	some	PRON
ejpam-4912	194	33	a	a	DET
ejpam-4912	194	34	∈	∈	PROPN
ejpam-4912	194	35	v	v	NOUN
ejpam-4912	194	36	(	(	PUNCT
ejpam-4912	194	37	km	km	PROPN
ejpam-4912	194	38	)	)	PUNCT
ejpam-4912	194	39	and	and	CCONJ
ejpam-4912	194	40	b	b	X
ejpam-4912	194	41	∈	∈	PROPN
ejpam-4912	194	42	v	v	NOUN
ejpam-4912	194	43	(	(	PUNCT
ejpam-4912	194	44	kn	kn	PROPN
ejpam-4912	194	45	)	)	PUNCT
ejpam-4912	194	46	.	.	PUNCT
ejpam-4912	195	1	proof	proof	NOUN
ejpam-4912	195	2	.	.	PUNCT
ejpam-4912	196	1	let	let	VERB
ejpam-4912	196	2	n	n	PRON
ejpam-4912	196	3	⊆	⊆	NUM
ejpam-4912	196	4	v	v	NOUN
ejpam-4912	196	5	(	(	PUNCT
ejpam-4912	196	6	km	km	PROPN
ejpam-4912	196	7	,	,	PUNCT
ejpam-4912	196	8	n	n	CCONJ
ejpam-4912	196	9	)	)	PUNCT
ejpam-4912	196	10	be	be	AUX
ejpam-4912	196	11	a	a	DET
ejpam-4912	196	12	j	j	PROPN
ejpam-4912	196	13	-	-	ADJ
ejpam-4912	196	14	total	total	ADJ
ejpam-4912	196	15	dominating	dominating	NOUN
ejpam-4912	196	16	in	in	ADP
ejpam-4912	196	17	km	km	PROPN
ejpam-4912	196	18	,	,	PUNCT
ejpam-4912	196	19	n	n	CCONJ
ejpam-4912	196	20	,	,	PUNCT
ejpam-4912	196	21	m	m	PROPN
ejpam-4912	196	22	,	,	PUNCT
ejpam-4912	196	23	n	n	PRON
ejpam-4912	196	24	≥	≥	NOUN
ejpam-4912	196	25	1	1	NUM
ejpam-4912	196	26	.	.	PUNCT
ejpam-4912	197	1	then	then	ADV
ejpam-4912	197	2	n	n	PRON
ejpam-4912	197	3	is	be	AUX
ejpam-4912	197	4	a	a	DET
ejpam-4912	197	5	total	total	ADJ
ejpam-4912	197	6	dominating	dominating	NOUN
ejpam-4912	197	7	set	set	NOUN
ejpam-4912	197	8	in	in	ADP
ejpam-4912	197	9	km	km	PROPN
ejpam-4912	197	10	,	,	PUNCT
ejpam-4912	197	11	n.	n.	PROPN
ejpam-4912	197	12	thus	thus	ADV
ejpam-4912	197	13	,	,	PUNCT
ejpam-4912	197	14	|n	|n	NOUN
ejpam-4912	197	15	|	|	ADV
ejpam-4912	197	16	≥	≥	NOUN
ejpam-4912	197	17	2	2	NUM
ejpam-4912	197	18	.	.	PUNCT
ejpam-4912	198	1	let	let	VERB
ejpam-4912	198	2	v	v	NOUN
ejpam-4912	198	3	(	(	PUNCT
ejpam-4912	198	4	km	km	NOUN
ejpam-4912	198	5	)	)	PUNCT
ejpam-4912	198	6	=	=	PRON
ejpam-4912	198	7	{	{	PUNCT
ejpam-4912	198	8	u1	u1	NOUN
ejpam-4912	198	9	,	,	PUNCT
ejpam-4912	198	10	u2	u2	NOUN
ejpam-4912	198	11	,	,	PUNCT
ejpam-4912	198	12	.	.	PUNCT
ejpam-4912	198	13	.	.	PUNCT
ejpam-4912	199	1	.	.	PUNCT
ejpam-4912	200	1	,	,	PUNCT
ejpam-4912	200	2	um	um	INTJ
ejpam-4912	200	3	}	}	PUNCT
ejpam-4912	200	4	and	and	CCONJ
ejpam-4912	200	5	v	v	INTJ
ejpam-4912	200	6	(	(	PUNCT
ejpam-4912	200	7	kn	kn	PROPN
ejpam-4912	200	8	)	)	PUNCT
ejpam-4912	200	9	=	=	SYM
ejpam-4912	200	10	{	{	PUNCT
ejpam-4912	200	11	v1	v1	PROPN
ejpam-4912	200	12	,	,	PUNCT
ejpam-4912	200	13	v2	v2	PROPN
ejpam-4912	200	14	,	,	PUNCT
ejpam-4912	200	15	.	.	PUNCT
ejpam-4912	200	16	.	.	PUNCT
ejpam-4912	201	1	.	.	PUNCT
ejpam-4912	202	1	,	,	PUNCT
ejpam-4912	202	2	vn	vn	PROPN
ejpam-4912	202	3	}	}	PUNCT
ejpam-4912	202	4	.	.	PUNCT
ejpam-4912	203	1	observe	observe	VERB
ejpam-4912	203	2	that	that	SCONJ
ejpam-4912	203	3	nkm	nkm	NOUN
ejpam-4912	203	4	,	,	PUNCT
ejpam-4912	203	5	n(ui	n(ui	PROPN
ejpam-4912	203	6	)	)	PUNCT
ejpam-4912	203	7	=	=	SYM
ejpam-4912	203	8	nkm	nkm	PROPN
ejpam-4912	203	9	,	,	PUNCT
ejpam-4912	203	10	n(uj	n(uj	NUM
ejpam-4912	203	11	)	)	PUNCT
ejpam-4912	203	12	∀	∀	VERB
ejpam-4912	204	1	i	i	PRON
ejpam-4912	204	2	̸=	̸=	PROPN
ejpam-4912	204	3	j	j	PROPN
ejpam-4912	204	4	,	,	PUNCT
ejpam-4912	204	5	i	i	PRON
ejpam-4912	204	6	,	,	PUNCT
ejpam-4912	204	7	j	j	PROPN
ejpam-4912	204	8	∈	∈	PROPN
ejpam-4912	204	9	{	{	PUNCT
ejpam-4912	204	10	1	1	NUM
ejpam-4912	204	11	,	,	PUNCT
ejpam-4912	204	12	2	2	NUM
ejpam-4912	204	13	,	,	PUNCT
ejpam-4912	204	14	.	.	PUNCT
ejpam-4912	204	15	.	.	PUNCT
ejpam-4912	204	16	.	.	PUNCT
ejpam-4912	205	1	,	,	PUNCT
ejpam-4912	205	2	m	m	VERB
ejpam-4912	205	3	}	}	PUNCT
ejpam-4912	205	4	and	and	CCONJ
ejpam-4912	205	5	nkm	nkm	PROPN
ejpam-4912	205	6	,	,	PUNCT
ejpam-4912	205	7	n(vr	n(vr	PROPN
ejpam-4912	205	8	)	)	PUNCT
ejpam-4912	205	9	=	=	SYM
ejpam-4912	205	10	nkm	nkm	NOUN
ejpam-4912	205	11	,	,	PUNCT
ejpam-4912	205	12	n(vq	n(vq	NOUN
ejpam-4912	205	13	)	)	PUNCT
ejpam-4912	205	14	∀r	∀r	X
ejpam-4912	205	15	̸=	̸=	PROPN
ejpam-4912	205	16	q	q	NOUN
ejpam-4912	205	17	,	,	PUNCT
ejpam-4912	205	18	r	r	NOUN
ejpam-4912	205	19	,	,	PUNCT
ejpam-4912	205	20	q	q	NOUN
ejpam-4912	205	21	∈	∈	PROPN
ejpam-4912	205	22	{	{	PUNCT
ejpam-4912	205	23	1	1	NUM
ejpam-4912	205	24	,	,	PUNCT
ejpam-4912	205	25	2	2	NUM
ejpam-4912	205	26	,	,	PUNCT
ejpam-4912	205	27	.	.	PUNCT
ejpam-4912	205	28	.	.	PUNCT
ejpam-4912	206	1	.	.	PUNCT
ejpam-4912	206	2	,	,	PUNCT
ejpam-4912	206	3	n	n	CCONJ
ejpam-4912	206	4	}	}	PUNCT
ejpam-4912	206	5	.	.	PUNCT
ejpam-4912	207	1	this	this	PRON
ejpam-4912	207	2	means	mean	VERB
ejpam-4912	207	3	that	that	SCONJ
ejpam-4912	207	4	there	there	PRON
ejpam-4912	207	5	are	be	VERB
ejpam-4912	207	6	m	m	VERB
ejpam-4912	207	7	−	−	NOUN
ejpam-4912	207	8	1	1	NUM
ejpam-4912	207	9	and	and	CCONJ
ejpam-4912	207	10	n	n	CCONJ
ejpam-4912	207	11	−	−	NOUN
ejpam-4912	207	12	1	1	NUM
ejpam-4912	207	13	vertices	vertex	NOUN
ejpam-4912	207	14	of	of	ADP
ejpam-4912	207	15	km	km	PROPN
ejpam-4912	207	16	and	and	CCONJ
ejpam-4912	207	17	kn	kn	PROPN
ejpam-4912	207	18	,	,	PUNCT
ejpam-4912	207	19	respectively	respectively	ADV
ejpam-4912	207	20	,	,	PUNCT
ejpam-4912	207	21	can	can	AUX
ejpam-4912	207	22	not	not	PART
ejpam-4912	207	23	be	be	AUX
ejpam-4912	207	24	in	in	ADP
ejpam-4912	207	25	any	any	DET
ejpam-4912	207	26	j	j	PROPN
ejpam-4912	207	27	-	-	ADJ
ejpam-4912	207	28	total	total	ADJ
ejpam-4912	207	29	dominating	dominating	NOUN
ejpam-4912	207	30	set	set	NOUN
ejpam-4912	207	31	of	of	ADP
ejpam-4912	207	32	km	km	PROPN
ejpam-4912	207	33	,	,	PUNCT
ejpam-4912	207	34	n.	n.	PROPN
ejpam-4912	207	35	hence	hence	ADV
ejpam-4912	207	36	,	,	PUNCT
ejpam-4912	207	37	|	|	ADV
ejpam-4912	207	38	n	n	CCONJ
ejpam-4912	207	39	|≤	|≤	PROPN
ejpam-4912	207	40	2	2	NUM
ejpam-4912	207	41	,	,	PUNCT
ejpam-4912	207	42	and	and	CCONJ
ejpam-4912	207	43	so	so	ADV
ejpam-4912	207	44	|	|	ADV
ejpam-4912	207	45	n	n	CCONJ
ejpam-4912	207	46	|=	|=	X
ejpam-4912	207	47	2	2	NUM
ejpam-4912	207	48	.	.	PUNCT
ejpam-4912	208	1	thus	thus	ADV
ejpam-4912	208	2	,	,	PUNCT
ejpam-4912	208	3	n	n	PROPN
ejpam-4912	208	4	=	=	X
ejpam-4912	208	5	{	{	PUNCT
ejpam-4912	208	6	a	a	DET
ejpam-4912	208	7	,	,	PUNCT
ejpam-4912	208	8	b	b	NOUN
ejpam-4912	208	9	}	}	PUNCT
ejpam-4912	208	10	for	for	ADP
ejpam-4912	208	11	some	some	PRON
ejpam-4912	208	12	a	a	DET
ejpam-4912	208	13	∈	∈	PROPN
ejpam-4912	208	14	v	v	NOUN
ejpam-4912	208	15	(	(	PUNCT
ejpam-4912	208	16	km	km	PROPN
ejpam-4912	208	17	)	)	PUNCT
ejpam-4912	208	18	and	and	CCONJ
ejpam-4912	208	19	b	b	X
ejpam-4912	208	20	∈	∈	PROPN
ejpam-4912	208	21	v	v	NOUN
ejpam-4912	208	22	(	(	PUNCT
ejpam-4912	208	23	kn	kn	PROPN
ejpam-4912	208	24	)	)	PUNCT
ejpam-4912	208	25	.	.	PUNCT
ejpam-4912	209	1	conversely	conversely	ADV
ejpam-4912	209	2	,	,	PUNCT
ejpam-4912	209	3	let	let	VERB
ejpam-4912	209	4	n	n	X
ejpam-4912	209	5	=	=	PRON
ejpam-4912	209	6	{	{	PUNCT
ejpam-4912	209	7	a	a	DET
ejpam-4912	209	8	,	,	PUNCT
ejpam-4912	209	9	b	b	NOUN
ejpam-4912	209	10	}	}	PUNCT
ejpam-4912	209	11	for	for	ADP
ejpam-4912	209	12	some	some	PRON
ejpam-4912	209	13	a	a	DET
ejpam-4912	209	14	∈	∈	PROPN
ejpam-4912	209	15	v	v	NOUN
ejpam-4912	209	16	(	(	PUNCT
ejpam-4912	209	17	km	km	PROPN
ejpam-4912	209	18	)	)	PUNCT
ejpam-4912	209	19	and	and	CCONJ
ejpam-4912	209	20	b	b	X
ejpam-4912	209	21	∈	∈	PROPN
ejpam-4912	209	22	v	v	NOUN
ejpam-4912	209	23	(	(	PUNCT
ejpam-4912	209	24	kn	kn	PROPN
ejpam-4912	209	25	)	)	PUNCT
ejpam-4912	209	26	.	.	PUNCT
ejpam-4912	210	1	then	then	ADV
ejpam-4912	210	2	nkm	nkm	VERB
ejpam-4912	210	3	,	,	PUNCT
ejpam-4912	210	4	n(a	n(a	X
ejpam-4912	210	5	)	)	PUNCT
ejpam-4912	211	1	=	=	SYM
ejpam-4912	211	2	v	v	X
ejpam-4912	211	3	(	(	PUNCT
ejpam-4912	211	4	kn	kn	PROPN
ejpam-4912	211	5	)	)	PUNCT
ejpam-4912	211	6	and	and	CCONJ
ejpam-4912	211	7	nkm	nkm	PROPN
ejpam-4912	211	8	,	,	PUNCT
ejpam-4912	211	9	n(b	n(b	PROPN
ejpam-4912	211	10	)	)	PUNCT
ejpam-4912	211	11	=	=	SYM
ejpam-4912	211	12	v	v	X
ejpam-4912	211	13	(	(	PUNCT
ejpam-4912	211	14	km	km	PROPN
ejpam-4912	211	15	)	)	PUNCT
ejpam-4912	211	16	.	.	PUNCT
ejpam-4912	212	1	hence	hence	ADV
ejpam-4912	212	2	,	,	PUNCT
ejpam-4912	212	3	nkm	nkm	PROPN
ejpam-4912	212	4	,	,	PUNCT
ejpam-4912	212	5	n(n	n(n	X
ejpam-4912	212	6	)	)	PUNCT
ejpam-4912	212	7	=	=	SYM
ejpam-4912	212	8	v	v	X
ejpam-4912	212	9	(	(	PUNCT
ejpam-4912	212	10	km	km	PROPN
ejpam-4912	212	11	,	,	PUNCT
ejpam-4912	212	12	n	n	CCONJ
ejpam-4912	212	13	)	)	PUNCT
ejpam-4912	212	14	,	,	PUNCT
ejpam-4912	212	15	and	and	CCONJ
ejpam-4912	212	16	nkm	nkm	PROPN
ejpam-4912	212	17	,	,	PUNCT
ejpam-4912	212	18	n(a)\nkm	n(a)\nkm	NOUN
ejpam-4912	212	19	,	,	PUNCT
ejpam-4912	212	20	n(b	n(b	X
ejpam-4912	212	21	)	)	PUNCT
ejpam-4912	212	22	=	=	SYM
ejpam-4912	212	23	v	v	X
ejpam-4912	212	24	(	(	PUNCT
ejpam-4912	212	25	kn	kn	PROPN
ejpam-4912	212	26	)	)	PUNCT
ejpam-4912	212	27	̸=	̸=	PROPN
ejpam-4912	212	28	∅	∅	NOUN
ejpam-4912	212	29	and	and	CCONJ
ejpam-4912	212	30	nkm	nkm	NOUN
ejpam-4912	212	31	,	,	PUNCT
ejpam-4912	212	32	n(b)\nkm	n(b)\nkm	VERB
ejpam-4912	212	33	,	,	PUNCT
ejpam-4912	212	34	n(a	n(a	PRON
ejpam-4912	212	35	)	)	PUNCT
ejpam-4912	212	36	=	=	SYM
ejpam-4912	212	37	v	v	X
ejpam-4912	212	38	(	(	PUNCT
ejpam-4912	212	39	km	km	NOUN
ejpam-4912	212	40	)	)	PUNCT
ejpam-4912	212	41	̸=	̸=	PROPN
ejpam-4912	212	42	∅.	∅.	ADP
ejpam-4912	212	43	consequently	consequently	ADV
ejpam-4912	212	44	,	,	PUNCT
ejpam-4912	212	45	n	n	PROPN
ejpam-4912	212	46	=	=	PRON
ejpam-4912	212	47	{	{	PUNCT
ejpam-4912	212	48	a	a	DET
ejpam-4912	212	49	,	,	PUNCT
ejpam-4912	212	50	b	b	NOUN
ejpam-4912	212	51	}	}	PUNCT
ejpam-4912	212	52	is	be	AUX
ejpam-4912	212	53	a	a	DET
ejpam-4912	212	54	j	j	PROPN
ejpam-4912	212	55	-	-	ADJ
ejpam-4912	212	56	total	total	ADJ
ejpam-4912	212	57	dominating	dominating	NOUN
ejpam-4912	212	58	set	set	NOUN
ejpam-4912	212	59	in	in	ADP
ejpam-4912	212	60	km	km	PROPN
ejpam-4912	212	61	,	,	PUNCT
ejpam-4912	212	62	n.	n.	NOUN
ejpam-4912	212	63	the	the	DET
ejpam-4912	212	64	following	follow	VERB
ejpam-4912	212	65	result	result	NOUN
ejpam-4912	212	66	follows	follow	VERB
ejpam-4912	212	67	immediately	immediately	ADV
ejpam-4912	212	68	for	for	ADP
ejpam-4912	212	69	theorem	theorem	ADJ
ejpam-4912	212	70	4	4	NUM
ejpam-4912	212	71	.	.	PUNCT
ejpam-4912	212	72	corollary	corollary	ADJ
ejpam-4912	212	73	2	2	NUM
ejpam-4912	212	74	.	.	PUNCT
ejpam-4912	213	1	let	let	VERB
ejpam-4912	213	2	m	m	PRON
ejpam-4912	213	3	,	,	PUNCT
ejpam-4912	213	4	n	n	PRON
ejpam-4912	213	5	≥	≥	NOUN
ejpam-4912	213	6	1	1	NUM
ejpam-4912	213	7	be	be	AUX
ejpam-4912	213	8	positive	positive	ADJ
ejpam-4912	213	9	integers	integer	NOUN
ejpam-4912	213	10	.	.	PUNCT
ejpam-4912	214	1	then	then	ADV
ejpam-4912	214	2	γjt(km	γjt(km	NOUN
ejpam-4912	214	3	,	,	PUNCT
ejpam-4912	214	4	n	n	CCONJ
ejpam-4912	214	5	)	)	PUNCT
ejpam-4912	214	6	=	=	SYM
ejpam-4912	214	7	2	2	X
ejpam-4912	214	8	.	.	X
ejpam-4912	214	9	theorem	theorem	NOUN
ejpam-4912	214	10	5	5	NUM
ejpam-4912	214	11	.	.	PUNCT
ejpam-4912	215	1	let	let	VERB
ejpam-4912	215	2	m	m	PRON
ejpam-4912	215	3	≥	≥	NOUN
ejpam-4912	215	4	2	2	NUM
ejpam-4912	215	5	be	be	AUX
ejpam-4912	215	6	positive	positive	ADJ
ejpam-4912	215	7	integer	integer	NOUN
ejpam-4912	215	8	.	.	PUNCT
ejpam-4912	216	1	then	then	ADV
ejpam-4912	216	2	γjt(pm	γjt(pm	VERB
ejpam-4912	216	3	)	)	PUNCT
ejpam-4912	217	1	=	=	PUNCT
ejpam-4912	217	2			NOUN
ejpam-4912	217	3	2	2	NUM
ejpam-4912	217	4	if	if	SCONJ
ejpam-4912	217	5	m	m	VERB
ejpam-4912	217	6	=	=	SYM
ejpam-4912	217	7	2	2	NUM
ejpam-4912	217	8	,	,	PUNCT
ejpam-4912	217	9	3	3	NUM
ejpam-4912	217	10	,	,	PUNCT
ejpam-4912	217	11	4	4	NUM
ejpam-4912	217	12	4	4	NUM
ejpam-4912	217	13	if	if	SCONJ
ejpam-4912	217	14	m	m	VERB
ejpam-4912	217	15	=	=	SYM
ejpam-4912	217	16	5	5	NUM
ejpam-4912	217	17	m−	m−	PROPN
ejpam-4912	217	18	2	2	NUM
ejpam-4912	217	19	if	if	SCONJ
ejpam-4912	217	20	m	m	PROPN
ejpam-4912	217	21	≥	≥	VERB
ejpam-4912	217	22	6	6	NUM
ejpam-4912	217	23	.	.	PUNCT
ejpam-4912	218	1	proof	proof	NOUN
ejpam-4912	218	2	.	.	PUNCT
ejpam-4912	219	1	clearly	clearly	ADV
ejpam-4912	219	2	,	,	PUNCT
ejpam-4912	219	3	γjt(p2	γjt(p2	PROPN
ejpam-4912	219	4	)	)	PUNCT
ejpam-4912	219	5	=	=	SYM
ejpam-4912	219	6	2	2	X
ejpam-4912	219	7	.	.	X
ejpam-4912	219	8	for	for	ADP
ejpam-4912	219	9	m	m	PROPN
ejpam-4912	219	10	=	=	SYM
ejpam-4912	219	11	3	3	NUM
ejpam-4912	219	12	,	,	PUNCT
ejpam-4912	219	13	let	let	VERB
ejpam-4912	219	14	v	v	NOUN
ejpam-4912	219	15	(	(	PUNCT
ejpam-4912	219	16	p3	p3	PROPN
ejpam-4912	219	17	)	)	PUNCT
ejpam-4912	219	18	=	=	PRON
ejpam-4912	219	19	{	{	PUNCT
ejpam-4912	219	20	v1	v1	PROPN
ejpam-4912	219	21	,	,	PUNCT
ejpam-4912	219	22	v2	v2	PROPN
ejpam-4912	219	23	,	,	PUNCT
ejpam-4912	219	24	v3	v3	PROPN
ejpam-4912	219	25	}	}	PUNCT
ejpam-4912	219	26	and	and	CCONJ
ejpam-4912	219	27	let	let	VERB
ejpam-4912	219	28	s	s	AUX
ejpam-4912	219	29	=	=	NOUN
ejpam-4912	219	30	{	{	PUNCT
ejpam-4912	219	31	v1	v1	PROPN
ejpam-4912	219	32	,	,	PUNCT
ejpam-4912	219	33	v2	v2	PROPN
ejpam-4912	219	34	}	}	PUNCT
ejpam-4912	219	35	.	.	PUNCT
ejpam-4912	220	1	then	then	ADV
ejpam-4912	220	2	v2	v2	PROPN
ejpam-4912	220	3	∈	∈	PROPN
ejpam-4912	220	4	np3(v1)\np3(v2	np3(v1)\np3(v2	NOUN
ejpam-4912	220	5	)	)	PUNCT
ejpam-4912	220	6	and	and	CCONJ
ejpam-4912	220	7	v1	v1	NOUN
ejpam-4912	220	8	∈	∈	NOUN
ejpam-4912	220	9	np3(v2)\np3(v1	np3(v2)\np3(v1	NOUN
ejpam-4912	220	10	)	)	PUNCT
ejpam-4912	220	11	.	.	PUNCT
ejpam-4912	221	1	thus	thus	ADV
ejpam-4912	221	2	,	,	PUNCT
ejpam-4912	221	3	s	s	VERB
ejpam-4912	221	4	is	be	AUX
ejpam-4912	221	5	a	a	DET
ejpam-4912	221	6	j	j	NOUN
ejpam-4912	221	7	-	-	ADJ
ejpam-4912	221	8	open	open	ADJ
ejpam-4912	221	9	set	set	NOUN
ejpam-4912	221	10	in	in	ADP
ejpam-4912	221	11	p3	p3	PROPN
ejpam-4912	221	12	.	.	PUNCT
ejpam-4912	222	1	since	since	SCONJ
ejpam-4912	222	2	np3(s	np3(s	PROPN
ejpam-4912	222	3	)	)	PUNCT
ejpam-4912	222	4	=	=	SYM
ejpam-4912	222	5	v	v	X
ejpam-4912	222	6	(	(	PUNCT
ejpam-4912	222	7	p3	p3	PROPN
ejpam-4912	222	8	)	)	PUNCT
ejpam-4912	222	9	,	,	PUNCT
ejpam-4912	222	10	it	it	PRON
ejpam-4912	222	11	follows	follow	VERB
ejpam-4912	222	12	that	that	SCONJ
ejpam-4912	222	13	s	s	VERB
ejpam-4912	222	14	is	be	AUX
ejpam-4912	222	15	a	a	DET
ejpam-4912	222	16	j	j	PROPN
ejpam-4912	222	17	-	-	ADJ
ejpam-4912	222	18	total	total	ADJ
ejpam-4912	222	19	dominating	dominating	NOUN
ejpam-4912	222	20	set	set	NOUN
ejpam-4912	222	21	of	of	ADP
ejpam-4912	222	22	p3	p3	PROPN
ejpam-4912	222	23	.	.	PUNCT
ejpam-4912	223	1	notice	notice	VERB
ejpam-4912	223	2	that	that	SCONJ
ejpam-4912	223	3	np3(v1	np3(v1	NOUN
ejpam-4912	223	4	)	)	PUNCT
ejpam-4912	223	5	=	=	SYM
ejpam-4912	223	6	np3(v3	np3(v3	NOUN
ejpam-4912	223	7	)	)	PUNCT
ejpam-4912	223	8	.	.	PUNCT
ejpam-4912	224	1	hence	hence	ADV
ejpam-4912	224	2	,	,	PUNCT
ejpam-4912	224	3	v1	v1	PROPN
ejpam-4912	224	4	and	and	CCONJ
ejpam-4912	224	5	v3	v3	PROPN
ejpam-4912	224	6	can	can	AUX
ejpam-4912	224	7	not	not	PART
ejpam-4912	224	8	be	be	AUX
ejpam-4912	224	9	both	both	PRON
ejpam-4912	224	10	in	in	ADP
ejpam-4912	224	11	any	any	DET
ejpam-4912	224	12	j	j	NOUN
ejpam-4912	224	13	-	-	ADJ
ejpam-4912	224	14	open	open	ADJ
ejpam-4912	224	15	set	set	NOUN
ejpam-4912	224	16	of	of	ADP
ejpam-4912	224	17	p3	p3	PROPN
ejpam-4912	224	18	.	.	PUNCT
ejpam-4912	225	1	therefore	therefore	ADV
ejpam-4912	225	2	,	,	PUNCT
ejpam-4912	225	3	s	s	VERB
ejpam-4912	225	4	=	=	NOUN
ejpam-4912	225	5	{	{	PUNCT
ejpam-4912	225	6	v1	v1	PROPN
ejpam-4912	225	7	,	,	PUNCT
ejpam-4912	225	8	v2	v2	PROPN
ejpam-4912	225	9	}	}	PUNCT
ejpam-4912	225	10	is	be	AUX
ejpam-4912	225	11	a	a	DET
ejpam-4912	225	12	maximum	maximum	ADJ
ejpam-4912	225	13	j	j	ADJ
ejpam-4912	225	14	-	-	ADJ
ejpam-4912	225	15	total	total	ADJ
ejpam-4912	225	16	dominating	dominating	NOUN
ejpam-4912	225	17	set	set	NOUN
ejpam-4912	225	18	in	in	ADP
ejpam-4912	225	19	p3	p3	PROPN
ejpam-4912	225	20	,	,	PUNCT
ejpam-4912	225	21	showing	show	VERB
ejpam-4912	225	22	that	that	SCONJ
ejpam-4912	225	23	γjt(p3	γjt(p3	PROPN
ejpam-4912	225	24	)	)	PUNCT
ejpam-4912	225	25	=	=	SYM
ejpam-4912	225	26	2	2	X
ejpam-4912	225	27	.	.	X
ejpam-4912	225	28	for	for	ADP
ejpam-4912	225	29	m	m	PROPN
ejpam-4912	225	30	=	=	SYM
ejpam-4912	225	31	4	4	NUM
ejpam-4912	225	32	,	,	PUNCT
ejpam-4912	225	33	let	let	VERB
ejpam-4912	225	34	v	v	NOUN
ejpam-4912	225	35	(	(	PUNCT
ejpam-4912	225	36	p4	p4	ADJ
ejpam-4912	225	37	)	)	PUNCT
ejpam-4912	225	38	=	=	SYM
ejpam-4912	225	39	{	{	PUNCT
ejpam-4912	225	40	a1	a1	PROPN
ejpam-4912	225	41	,	,	PUNCT
ejpam-4912	225	42	a2	a2	PROPN
ejpam-4912	225	43	,	,	PUNCT
ejpam-4912	225	44	a3	a3	NOUN
ejpam-4912	225	45	,	,	PUNCT
ejpam-4912	225	46	a4	a4	NOUN
ejpam-4912	225	47	}	}	PUNCT
ejpam-4912	225	48	and	and	CCONJ
ejpam-4912	225	49	s′	s′	ADJ
ejpam-4912	225	50	=	=	PUNCT
ejpam-4912	225	51	{	{	PUNCT
ejpam-4912	225	52	a2	a2	PROPN
ejpam-4912	225	53	,	,	PUNCT
ejpam-4912	225	54	a3	a3	NOUN
ejpam-4912	225	55	}	}	PUNCT
ejpam-4912	225	56	.	.	PUNCT
ejpam-4912	226	1	then	then	ADV
ejpam-4912	226	2	a1	a1	NOUN
ejpam-4912	226	3	,	,	PUNCT
ejpam-4912	226	4	a3	a3	NOUN
ejpam-4912	226	5	∈	∈	PROPN
ejpam-4912	226	6	np4(a2)\np4(a3	np4(a2)\np4(a3	NOUN
ejpam-4912	226	7	)	)	PUNCT
ejpam-4912	226	8	and	and	CCONJ
ejpam-4912	226	9	a2	a2	PROPN
ejpam-4912	226	10	,	,	PUNCT
ejpam-4912	226	11	a4	a4	NOUN
ejpam-4912	226	12	∈	∈	PROPN
ejpam-4912	226	13	np4(a3)\np4(a2	np4(a3)\np4(a2	PROPN
ejpam-4912	226	14	)	)	PUNCT
ejpam-4912	226	15	.	.	PUNCT
ejpam-4912	227	1	j.a	j.a	PROPN
ejpam-4912	227	2	.	.	PROPN
ejpam-4912	227	3	hassan	hassan	PROPN
ejpam-4912	227	4	et	et	PROPN
ejpam-4912	227	5	al	al	PROPN
ejpam-4912	227	6	.	.	PUNCT
ejpam-4912	227	7	/	/	SYM
ejpam-4912	227	8	eur	eur	PROPN
ejpam-4912	227	9	.	.	PUNCT
ejpam-4912	228	1	j.	j.	PROPN
ejpam-4912	228	2	pure	pure	PROPN
ejpam-4912	228	3	appl	appl	PROPN
ejpam-4912	228	4	.	.	PROPN
ejpam-4912	228	5	math	math	PROPN
ejpam-4912	228	6	,	,	PUNCT
ejpam-4912	228	7	16	16	NUM
ejpam-4912	228	8	(	(	PUNCT
ejpam-4912	228	9	4	4	NUM
ejpam-4912	228	10	)	)	PUNCT
ejpam-4912	228	11	(	(	PUNCT
ejpam-4912	228	12	2023	2023	NUM
ejpam-4912	228	13	)	)	PUNCT
ejpam-4912	228	14	,	,	PUNCT
ejpam-4912	228	15	2106	2106	NUM
ejpam-4912	228	16	-	-	SYM
ejpam-4912	228	17	2117	2117	NUM
ejpam-4912	228	18	2114	2114	NUM
ejpam-4912	228	19	thus	thus	ADV
ejpam-4912	228	20	,	,	PUNCT
ejpam-4912	228	21	s′	s′	PROPN
ejpam-4912	228	22	is	be	AUX
ejpam-4912	228	23	a	a	DET
ejpam-4912	228	24	j	j	NOUN
ejpam-4912	228	25	-	-	ADJ
ejpam-4912	228	26	open	open	ADJ
ejpam-4912	228	27	set	set	NOUN
ejpam-4912	228	28	in	in	ADP
ejpam-4912	228	29	p4	p4	ADJ
ejpam-4912	228	30	.	.	PUNCT
ejpam-4912	229	1	observe	observe	VERB
ejpam-4912	229	2	that	that	SCONJ
ejpam-4912	229	3	np4(s	np4(s	PROPN
ejpam-4912	229	4	′	′	NOUN
ejpam-4912	229	5	)	)	PUNCT
ejpam-4912	230	1	=	=	SYM
ejpam-4912	230	2	v	v	X
ejpam-4912	230	3	(	(	PUNCT
ejpam-4912	230	4	p4	p4	ADJ
ejpam-4912	230	5	)	)	PUNCT
ejpam-4912	230	6	.	.	PUNCT
ejpam-4912	231	1	therefore	therefore	ADV
ejpam-4912	231	2	,	,	PUNCT
ejpam-4912	231	3	s	s	VERB
ejpam-4912	231	4	′	′	NOUN
ejpam-4912	231	5	is	be	AUX
ejpam-4912	231	6	a	a	DET
ejpam-4912	231	7	j	j	PROPN
ejpam-4912	231	8	-	-	ADJ
ejpam-4912	231	9	total	total	ADJ
ejpam-4912	231	10	dominating	dominating	NOUN
ejpam-4912	231	11	set	set	NOUN
ejpam-4912	231	12	in	in	ADP
ejpam-4912	231	13	p4	p4	ADJ
ejpam-4912	231	14	.	.	PUNCT
ejpam-4912	232	1	notice	notice	VERB
ejpam-4912	232	2	that	that	SCONJ
ejpam-4912	232	3	np4(a1	np4(a1	NOUN
ejpam-4912	232	4	)	)	PUNCT
ejpam-4912	232	5	⊆	⊆	NUM
ejpam-4912	232	6	np4(a3	np4(a3	NUM
ejpam-4912	232	7	)	)	PUNCT
ejpam-4912	232	8	and	and	CCONJ
ejpam-4912	232	9	np4(a4	np4(a4	NUM
ejpam-4912	232	10	)	)	PUNCT
ejpam-4912	232	11	⊆	⊆	NUM
ejpam-4912	232	12	np4(a2	np4(a2	NUM
ejpam-4912	232	13	)	)	PUNCT
ejpam-4912	232	14	.	.	PUNCT
ejpam-4912	233	1	this	this	PRON
ejpam-4912	233	2	means	mean	VERB
ejpam-4912	233	3	that	that	SCONJ
ejpam-4912	233	4	a1	a1	NOUN
ejpam-4912	233	5	and	and	CCONJ
ejpam-4912	233	6	a3	a3	NOUN
ejpam-4912	233	7	(	(	PUNCT
ejpam-4912	233	8	resp	resp	NOUN
ejpam-4912	233	9	.	.	PUNCT
ejpam-4912	234	1	a2	a2	PROPN
ejpam-4912	234	2	and	and	CCONJ
ejpam-4912	234	3	a4	a4	NOUN
ejpam-4912	234	4	)	)	PUNCT
ejpam-4912	234	5	can	can	AUX
ejpam-4912	234	6	not	not	PART
ejpam-4912	234	7	be	be	AUX
ejpam-4912	234	8	both	both	PRON
ejpam-4912	234	9	in	in	ADP
ejpam-4912	234	10	any	any	DET
ejpam-4912	234	11	j	j	NOUN
ejpam-4912	234	12	-	-	ADJ
ejpam-4912	234	13	open	open	ADJ
ejpam-4912	234	14	set	set	NOUN
ejpam-4912	234	15	of	of	ADP
ejpam-4912	234	16	p4	p4	NOUN
ejpam-4912	234	17	.	.	PUNCT
ejpam-4912	235	1	consequently	consequently	ADV
ejpam-4912	235	2	,	,	PUNCT
ejpam-4912	235	3	s′	s′	ADJ
ejpam-4912	235	4	=	=	PUNCT
ejpam-4912	235	5	{	{	PUNCT
ejpam-4912	235	6	v2	v2	PROPN
ejpam-4912	235	7	,	,	PUNCT
ejpam-4912	235	8	v3	v3	PROPN
ejpam-4912	235	9	}	}	PUNCT
ejpam-4912	235	10	is	be	AUX
ejpam-4912	235	11	a	a	DET
ejpam-4912	235	12	maximum	maximum	ADJ
ejpam-4912	235	13	j	j	ADJ
ejpam-4912	235	14	-	-	ADJ
ejpam-4912	235	15	total	total	ADJ
ejpam-4912	235	16	dominating	dominating	NOUN
ejpam-4912	235	17	set	set	NOUN
ejpam-4912	235	18	of	of	ADP
ejpam-4912	235	19	p4	p4	ADJ
ejpam-4912	235	20	,	,	PUNCT
ejpam-4912	235	21	and	and	CCONJ
ejpam-4912	235	22	so	so	ADV
ejpam-4912	235	23	γjt(p4	γjt(p4	NOUN
ejpam-4912	235	24	)	)	PUNCT
ejpam-4912	236	1	=	=	SYM
ejpam-4912	236	2	2	2	X
ejpam-4912	236	3	.	.	X
ejpam-4912	236	4	for	for	ADP
ejpam-4912	236	5	m	m	PROPN
ejpam-4912	236	6	=	=	SYM
ejpam-4912	236	7	5	5	NUM
ejpam-4912	236	8	,	,	PUNCT
ejpam-4912	236	9	let	let	VERB
ejpam-4912	236	10	v	v	NOUN
ejpam-4912	236	11	(	(	PUNCT
ejpam-4912	236	12	p5	p5	ADJ
ejpam-4912	236	13	)	)	PUNCT
ejpam-4912	236	14	=	=	SYM
ejpam-4912	236	15	{	{	PUNCT
ejpam-4912	236	16	u1	u1	NOUN
ejpam-4912	236	17	,	,	PUNCT
ejpam-4912	236	18	u2	u2	NOUN
ejpam-4912	236	19	,	,	PUNCT
ejpam-4912	236	20	u3	u3	PROPN
ejpam-4912	236	21	,	,	PUNCT
ejpam-4912	236	22	u4	u4	PROPN
ejpam-4912	236	23	,	,	PUNCT
ejpam-4912	236	24	u5	u5	PROPN
ejpam-4912	236	25	}	}	PUNCT
ejpam-4912	236	26	and	and	CCONJ
ejpam-4912	236	27	consider	consider	VERB
ejpam-4912	236	28	c	c	NOUN
ejpam-4912	236	29	=	=	SYM
ejpam-4912	236	30	{	{	PUNCT
ejpam-4912	236	31	u1	u1	NOUN
ejpam-4912	236	32	,	,	PUNCT
ejpam-4912	236	33	u2	u2	PROPN
ejpam-4912	236	34	,	,	PUNCT
ejpam-4912	236	35	u4	u4	PROPN
ejpam-4912	236	36	,	,	PUNCT
ejpam-4912	236	37	u5	u5	PROPN
ejpam-4912	236	38	}	}	PUNCT
ejpam-4912	236	39	.	.	PUNCT
ejpam-4912	237	1	then	then	ADV
ejpam-4912	237	2	u1	u1	PROPN
ejpam-4912	237	3	∈	∈	PROPN
ejpam-4912	237	4	np5(u2	np5(u2	PROPN
ejpam-4912	237	5	)	)	PUNCT
ejpam-4912	237	6	\np5(ui	\np5(ui	PROPN
ejpam-4912	237	7	)	)	PUNCT
ejpam-4912	237	8	∀	∀	PUNCT
ejpam-4912	238	1	i	i	PRON
ejpam-4912	238	2	̸=	̸=	PROPN
ejpam-4912	238	3	2	2	NUM
ejpam-4912	238	4	,	,	PUNCT
ejpam-4912	238	5	u2	u2	PROPN
ejpam-4912	238	6	∈	∈	PROPN
ejpam-4912	238	7	np5(u1)\np5(uj	np5(u1)\np5(uj	PROPN
ejpam-4912	238	8	)	)	PUNCT
ejpam-4912	238	9	∀j	∀j	PROPN
ejpam-4912	238	10	̸=	̸=	PROPN
ejpam-4912	238	11	1	1	NUM
ejpam-4912	238	12	,	,	PUNCT
ejpam-4912	238	13	u4	u4	PROPN
ejpam-4912	238	14	∈	∈	PROPN
ejpam-4912	238	15	np5(u5)\np5(ur	np5(u5)\np5(ur	PROPN
ejpam-4912	238	16	)	)	PUNCT
ejpam-4912	238	17	∀	∀	X
ejpam-4912	239	1	r	r	NOUN
ejpam-4912	239	2	̸=	̸=	PROPN
ejpam-4912	239	3	5	5	NUM
ejpam-4912	239	4	and	and	CCONJ
ejpam-4912	239	5	u5	u5	PROPN
ejpam-4912	239	6	∈	∈	PROPN
ejpam-4912	239	7	np5(u4)\np5(uq	np5(u4)\np5(uq	PROPN
ejpam-4912	239	8	)	)	PUNCT
ejpam-4912	239	9	∀	∀	X
ejpam-4912	240	1	q	q	PROPN
ejpam-4912	240	2	̸=	̸=	PROPN
ejpam-4912	240	3	4	4	NUM
ejpam-4912	240	4	.	.	PUNCT
ejpam-4912	241	1	hence	hence	ADV
ejpam-4912	241	2	,	,	PUNCT
ejpam-4912	241	3	c	c	PROPN
ejpam-4912	241	4	is	be	AUX
ejpam-4912	241	5	a	a	DET
ejpam-4912	241	6	j	j	NOUN
ejpam-4912	241	7	-	-	ADJ
ejpam-4912	241	8	open	open	ADJ
ejpam-4912	241	9	set	set	NOUN
ejpam-4912	241	10	in	in	ADP
ejpam-4912	241	11	p5	p5	PROPN
ejpam-4912	241	12	.	.	PUNCT
ejpam-4912	242	1	since	since	SCONJ
ejpam-4912	242	2	np5(c	np5(c	NOUN
ejpam-4912	242	3	)	)	PUNCT
ejpam-4912	242	4	=	=	NOUN
ejpam-4912	242	5	v	v	NOUN
ejpam-4912	242	6	(	(	PUNCT
ejpam-4912	242	7	5	5	NUM
ejpam-4912	242	8	)	)	PUNCT
ejpam-4912	242	9	,	,	PUNCT
ejpam-4912	242	10	it	it	PRON
ejpam-4912	242	11	follows	follow	VERB
ejpam-4912	242	12	that	that	SCONJ
ejpam-4912	242	13	c	c	PROPN
ejpam-4912	242	14	is	be	AUX
ejpam-4912	242	15	a	a	DET
ejpam-4912	242	16	j	j	PROPN
ejpam-4912	242	17	-	-	ADJ
ejpam-4912	242	18	total	total	ADJ
ejpam-4912	242	19	dominating	dominating	NOUN
ejpam-4912	242	20	set	set	NOUN
ejpam-4912	242	21	of	of	ADP
ejpam-4912	242	22	p5	p5	PROPN
ejpam-4912	242	23	.	.	PUNCT
ejpam-4912	243	1	notice	notice	VERB
ejpam-4912	243	2	that	that	SCONJ
ejpam-4912	243	3	np5(u1	np5(u1	NUM
ejpam-4912	243	4	)	)	PUNCT
ejpam-4912	243	5	⊆	⊆	NUM
ejpam-4912	243	6	np5(u3	np5(u3	NOUN
ejpam-4912	243	7	)	)	PUNCT
ejpam-4912	243	8	.	.	PUNCT
ejpam-4912	244	1	thus	thus	ADV
ejpam-4912	244	2	,	,	PUNCT
ejpam-4912	244	3	u1	u1	NOUN
ejpam-4912	244	4	and	and	CCONJ
ejpam-4912	244	5	u3	u3	NOUN
ejpam-4912	244	6	can	can	AUX
ejpam-4912	244	7	not	not	PART
ejpam-4912	244	8	be	be	AUX
ejpam-4912	244	9	both	both	PRON
ejpam-4912	244	10	in	in	ADP
ejpam-4912	244	11	any	any	DET
ejpam-4912	244	12	j	j	NOUN
ejpam-4912	244	13	-	-	ADJ
ejpam-4912	244	14	open	open	ADJ
ejpam-4912	244	15	set	set	NOUN
ejpam-4912	244	16	of	of	ADP
ejpam-4912	244	17	p5	p5	PROPN
ejpam-4912	244	18	.	.	PUNCT
ejpam-4912	245	1	consequently	consequently	ADV
ejpam-4912	245	2	,	,	PUNCT
ejpam-4912	245	3	c	c	PROPN
ejpam-4912	245	4	is	be	AUX
ejpam-4912	245	5	a	a	DET
ejpam-4912	245	6	maximum	maximum	ADJ
ejpam-4912	245	7	j	j	ADJ
ejpam-4912	245	8	-	-	ADJ
ejpam-4912	245	9	total	total	ADJ
ejpam-4912	245	10	domianting	domianting	NOUN
ejpam-4912	245	11	set	set	NOUN
ejpam-4912	245	12	of	of	ADP
ejpam-4912	245	13	p5	p5	ADJ
ejpam-4912	245	14	,	,	PUNCT
ejpam-4912	245	15	and	and	CCONJ
ejpam-4912	245	16	so	so	ADV
ejpam-4912	245	17	γjt(p5	γjt(p5	ADJ
ejpam-4912	245	18	)	)	PUNCT
ejpam-4912	245	19	=	=	PUNCT
ejpam-4912	246	1	4	4	X
ejpam-4912	246	2	.	.	PUNCT
ejpam-4912	247	1	next	next	ADV
ejpam-4912	247	2	,	,	PUNCT
ejpam-4912	247	3	suppose	suppose	VERB
ejpam-4912	247	4	that	that	SCONJ
ejpam-4912	247	5	m	m	PROPN
ejpam-4912	247	6	≥	≥	NUM
ejpam-4912	247	7	6	6	NUM
ejpam-4912	247	8	.	.	PUNCT
ejpam-4912	248	1	let	let	VERB
ejpam-4912	248	2	v	v	NOUN
ejpam-4912	248	3	(	(	PUNCT
ejpam-4912	248	4	pm	pm	NOUN
ejpam-4912	248	5	)	)	PUNCT
ejpam-4912	248	6	=	=	SYM
ejpam-4912	248	7	{	{	PUNCT
ejpam-4912	248	8	w1	w1	NOUN
ejpam-4912	248	9	,	,	PUNCT
ejpam-4912	248	10	w2	w2	NOUN
ejpam-4912	248	11	,	,	PUNCT
ejpam-4912	248	12	.	.	PUNCT
ejpam-4912	248	13	.	.	PUNCT
ejpam-4912	249	1	.	.	PUNCT
ejpam-4912	250	1	,	,	PUNCT
ejpam-4912	250	2	wm	wm	X
ejpam-4912	250	3	}	}	PUNCT
ejpam-4912	250	4	and	and	CCONJ
ejpam-4912	250	5	consider	consider	VERB
ejpam-4912	250	6	c	c	NOUN
ejpam-4912	250	7	′	′	NOUN
ejpam-4912	250	8	=	=	SYM
ejpam-4912	250	9	{	{	PUNCT
ejpam-4912	250	10	w2	w2	NOUN
ejpam-4912	250	11	,	,	PUNCT
ejpam-4912	250	12	w3	w3	PROPN
ejpam-4912	250	13	,	,	PUNCT
ejpam-4912	250	14	...	...	PUNCT
ejpam-4912	250	15	,	,	PUNCT
ejpam-4912	250	16	wm−2	wm−2	NOUN
ejpam-4912	250	17	,	,	PUNCT
ejpam-4912	250	18	wm−1	wm−1	NOUN
ejpam-4912	250	19	}	}	PUNCT
ejpam-4912	250	20	.	.	PUNCT
ejpam-4912	251	1	notice	notice	VERB
ejpam-4912	251	2	that	that	SCONJ
ejpam-4912	251	3	wi−1	wi−1	PROPN
ejpam-4912	251	4	∈	∈	PROPN
ejpam-4912	251	5	npm(wi	npm(wi	ADV
ejpam-4912	251	6	)	)	PUNCT
ejpam-4912	251	7	\	\	NOUN
ejpam-4912	251	8	npm(wj	npm(wj	NOUN
ejpam-4912	251	9	)	)	PUNCT
ejpam-4912	251	10	and	and	CCONJ
ejpam-4912	251	11	wj+1	wj+1	PROPN
ejpam-4912	251	12	∈	∈	PROPN
ejpam-4912	251	13	npm(wj	npm(wj	NOUN
ejpam-4912	251	14	)	)	PUNCT
ejpam-4912	251	15	\npm(wi	\npm(wi	NOUN
ejpam-4912	251	16	)	)	PUNCT
ejpam-4912	251	17	∀i	∀i	NOUN
ejpam-4912	251	18	<	<	X
ejpam-4912	251	19	j	j	PROPN
ejpam-4912	251	20	,	,	PUNCT
ejpam-4912	251	21	i	i	PRON
ejpam-4912	251	22	,	,	PUNCT
ejpam-4912	251	23	j	j	PROPN
ejpam-4912	251	24	∈	∈	PROPN
ejpam-4912	251	25	{	{	PUNCT
ejpam-4912	251	26	2	2	NUM
ejpam-4912	251	27	,	,	PUNCT
ejpam-4912	251	28	3	3	NUM
ejpam-4912	251	29	,	,	PUNCT
ejpam-4912	251	30	.	.	PUNCT
ejpam-4912	251	31	.	.	PUNCT
ejpam-4912	251	32	.	.	PUNCT
ejpam-4912	252	1	,	,	PUNCT
ejpam-4912	252	2	m−	m−	PROPN
ejpam-4912	252	3	1	1	NUM
ejpam-4912	252	4	}	}	PUNCT
ejpam-4912	252	5	.	.	PUNCT
ejpam-4912	253	1	it	it	PRON
ejpam-4912	253	2	follows	follow	VERB
ejpam-4912	253	3	that	that	SCONJ
ejpam-4912	253	4	npm(wi	npm(wi	ADV
ejpam-4912	253	5	)	)	PUNCT
ejpam-4912	253	6	\npm(wj	\npm(wj	ADJ
ejpam-4912	253	7	)	)	PUNCT
ejpam-4912	253	8	̸=	̸=	PROPN
ejpam-4912	253	9	∅	∅	NOUN
ejpam-4912	253	10	∀	∀	NOUN
ejpam-4912	254	1	i	i	PRON
ejpam-4912	254	2	̸=	̸=	PROPN
ejpam-4912	254	3	j	j	PROPN
ejpam-4912	254	4	,	,	PUNCT
ejpam-4912	254	5	i	i	PRON
ejpam-4912	254	6	,	,	PUNCT
ejpam-4912	254	7	j	j	PROPN
ejpam-4912	254	8	∈	∈	PROPN
ejpam-4912	254	9	{	{	PUNCT
ejpam-4912	254	10	2	2	NUM
ejpam-4912	254	11	,	,	PUNCT
ejpam-4912	254	12	3	3	NUM
ejpam-4912	254	13	,	,	PUNCT
ejpam-4912	254	14	...	...	PUNCT
ejpam-4912	254	15	,	,	PUNCT
ejpam-4912	254	16	m−	m−	PROPN
ejpam-4912	254	17	1	1	NUM
ejpam-4912	254	18	}	}	PUNCT
ejpam-4912	254	19	.	.	PUNCT
ejpam-4912	255	1	thus	thus	ADV
ejpam-4912	255	2	,	,	PUNCT
ejpam-4912	255	3	c	c	NOUN
ejpam-4912	255	4	′	′	PROPN
ejpam-4912	255	5	is	be	AUX
ejpam-4912	255	6	a	a	DET
ejpam-4912	255	7	j	j	NOUN
ejpam-4912	255	8	-	-	ADJ
ejpam-4912	255	9	open	open	ADJ
ejpam-4912	255	10	set	set	NOUN
ejpam-4912	255	11	in	in	ADP
ejpam-4912	255	12	pm	pm	NOUN
ejpam-4912	255	13	for	for	ADP
ejpam-4912	255	14	allm	allm	NOUN
ejpam-4912	255	15	≥	≥	NOUN
ejpam-4912	255	16	6	6	NUM
ejpam-4912	255	17	.	.	PUNCT
ejpam-4912	256	1	sincenpm(c	sincenpm(c	NUM
ejpam-4912	256	2	′	′	NOUN
ejpam-4912	256	3	)	)	PUNCT
ejpam-4912	256	4	=	=	NOUN
ejpam-4912	256	5	v	v	X
ejpam-4912	256	6	(	(	PUNCT
ejpam-4912	256	7	pm	pm	NOUN
ejpam-4912	256	8	)	)	PUNCT
ejpam-4912	256	9	,	,	PUNCT
ejpam-4912	256	10	c	c	NOUN
ejpam-4912	256	11	′	′	PROPN
ejpam-4912	256	12	is	be	AUX
ejpam-4912	256	13	a	a	DET
ejpam-4912	256	14	j	j	PROPN
ejpam-4912	256	15	-	-	ADJ
ejpam-4912	256	16	total	total	ADJ
ejpam-4912	256	17	dominating	dominating	NOUN
ejpam-4912	256	18	set	set	NOUN
ejpam-4912	256	19	of	of	ADP
ejpam-4912	256	20	pm	pm	NOUN
ejpam-4912	256	21	.	.	PUNCT
ejpam-4912	257	1	now	now	ADV
ejpam-4912	257	2	,	,	PUNCT
ejpam-4912	257	3	observe	observe	VERB
ejpam-4912	257	4	that	that	SCONJ
ejpam-4912	257	5	npm(w1	npm(w1	NOUN
ejpam-4912	257	6	)	)	PUNCT
ejpam-4912	257	7	⊆	⊆	NUM
ejpam-4912	257	8	npm(w3	npm(w3	NOUN
ejpam-4912	257	9	)	)	PUNCT
ejpam-4912	257	10	and	and	CCONJ
ejpam-4912	257	11	npm(wm	npm(wm	NUM
ejpam-4912	257	12	)	)	PUNCT
ejpam-4912	257	13	⊆	⊆	NUM
ejpam-4912	257	14	npm(wm−2	npm(wm−2	NOUN
ejpam-4912	257	15	)	)	PUNCT
ejpam-4912	257	16	.	.	PUNCT
ejpam-4912	258	1	hence	hence	ADV
ejpam-4912	258	2	,	,	PUNCT
ejpam-4912	258	3	w1	w1	NOUN
ejpam-4912	258	4	and	and	CCONJ
ejpam-4912	258	5	w3	w3	PROPN
ejpam-4912	258	6	(	(	PUNCT
ejpam-4912	258	7	resp	resp	PROPN
ejpam-4912	258	8	.	.	PUNCT
ejpam-4912	259	1	wm−2	wm−2	NOUN
ejpam-4912	259	2	and	and	CCONJ
ejpam-4912	259	3	wm	wm	PROPN
ejpam-4912	259	4	)	)	PUNCT
ejpam-4912	259	5	can	can	AUX
ejpam-4912	259	6	not	not	PART
ejpam-4912	259	7	be	be	AUX
ejpam-4912	259	8	both	both	PRON
ejpam-4912	259	9	in	in	ADP
ejpam-4912	259	10	any	any	DET
ejpam-4912	259	11	j	j	NOUN
ejpam-4912	259	12	-	-	ADJ
ejpam-4912	259	13	open	open	ADJ
ejpam-4912	259	14	set	set	NOUN
ejpam-4912	259	15	of	of	ADP
ejpam-4912	259	16	pm	pm	NOUN
ejpam-4912	259	17	.	.	PUNCT
ejpam-4912	260	1	therefore	therefore	ADV
ejpam-4912	260	2	,	,	PUNCT
ejpam-4912	260	3	c	c	NOUN
ejpam-4912	260	4	′	′	NOUN
ejpam-4912	260	5	is	be	AUX
ejpam-4912	260	6	a	a	DET
ejpam-4912	260	7	maximum	maximum	ADJ
ejpam-4912	260	8	j	j	ADJ
ejpam-4912	260	9	-	-	ADJ
ejpam-4912	260	10	total	total	ADJ
ejpam-4912	260	11	dominating	dominating	NOUN
ejpam-4912	260	12	set	set	NOUN
ejpam-4912	260	13	of	of	ADP
ejpam-4912	260	14	pm	pm	NOUN
ejpam-4912	260	15	,	,	PUNCT
ejpam-4912	260	16	and	and	CCONJ
ejpam-4912	260	17	so	so	ADV
ejpam-4912	260	18	γjt(pm	γjt(pm	X
ejpam-4912	260	19	)	)	PUNCT
ejpam-4912	261	1	=	=	SYM
ejpam-4912	261	2	m−	m−	PROPN
ejpam-4912	261	3	2	2	NUM
ejpam-4912	261	4	for	for	ADP
ejpam-4912	261	5	all	all	DET
ejpam-4912	261	6	m	m	PROPN
ejpam-4912	261	7	≥	≥	NOUN
ejpam-4912	261	8	6	6	NUM
ejpam-4912	261	9	.	.	PUNCT
ejpam-4912	261	10	theorem	theorem	VERB
ejpam-4912	261	11	6	6	NUM
ejpam-4912	261	12	.	.	PUNCT
ejpam-4912	262	1	let	let	VERB
ejpam-4912	262	2	n	n	PRON
ejpam-4912	262	3	be	be	AUX
ejpam-4912	262	4	any	any	DET
ejpam-4912	262	5	positive	positive	ADJ
ejpam-4912	262	6	integer	integer	NOUN
ejpam-4912	262	7	.	.	PUNCT
ejpam-4912	263	1	then	then	ADV
ejpam-4912	263	2	γjt(fn	γjt(fn	X
ejpam-4912	263	3	)	)	PUNCT
ejpam-4912	263	4	=	=	PUNCT
ejpam-4912	263	5			NOUN
ejpam-4912	263	6	2	2	NUM
ejpam-4912	263	7	if	if	SCONJ
ejpam-4912	263	8	n	n	NOUN
ejpam-4912	263	9	=	=	SYM
ejpam-4912	263	10	1	1	NUM
ejpam-4912	263	11	3	3	NUM
ejpam-4912	263	12	if	if	SCONJ
ejpam-4912	263	13	n	n	NOUN
ejpam-4912	263	14	=	=	SYM
ejpam-4912	263	15	2	2	NUM
ejpam-4912	263	16	,	,	PUNCT
ejpam-4912	263	17	3	3	NUM
ejpam-4912	263	18	,	,	PUNCT
ejpam-4912	263	19	4	4	NUM
ejpam-4912	263	20	5	5	NUM
ejpam-4912	263	21	if	if	SCONJ
ejpam-4912	263	22	n	n	NOUN
ejpam-4912	263	23	=	=	SYM
ejpam-4912	263	24	5	5	NUM
ejpam-4912	263	25	n−	n−	NOUN
ejpam-4912	263	26	1	1	NUM
ejpam-4912	263	27	if	if	SCONJ
ejpam-4912	263	28	n	n	PRON
ejpam-4912	263	29	≥	≥	VERB
ejpam-4912	263	30	6	6	NUM
ejpam-4912	263	31	proof	proof	NOUN
ejpam-4912	263	32	.	.	PUNCT
ejpam-4912	264	1	since	since	SCONJ
ejpam-4912	264	2	f1	f1	NOUN
ejpam-4912	264	3	and	and	CCONJ
ejpam-4912	264	4	f2	f2	PROPN
ejpam-4912	264	5	are	be	AUX
ejpam-4912	264	6	complete	complete	ADJ
ejpam-4912	264	7	graphs	graph	NOUN
ejpam-4912	264	8	,	,	PUNCT
ejpam-4912	264	9	γjt(f1	γjt(f1	NOUN
ejpam-4912	264	10	)	)	PUNCT
ejpam-4912	264	11	=	=	SYM
ejpam-4912	264	12	2	2	NUM
ejpam-4912	264	13	and	and	CCONJ
ejpam-4912	264	14	γjt(f2	γjt(f2	VERB
ejpam-4912	264	15	)	)	PUNCT
ejpam-4912	264	16	=	=	SYM
ejpam-4912	264	17	3	3	NUM
ejpam-4912	264	18	by	by	ADP
ejpam-4912	264	19	corollary	corollary	ADJ
ejpam-4912	264	20	1	1	NUM
ejpam-4912	264	21	.	.	PUNCT
ejpam-4912	265	1	for	for	ADP
ejpam-4912	265	2	n	n	NOUN
ejpam-4912	265	3	=	=	SYM
ejpam-4912	265	4	3	3	NUM
ejpam-4912	265	5	,	,	PUNCT
ejpam-4912	265	6	let	let	VERB
ejpam-4912	265	7	v	v	NOUN
ejpam-4912	265	8	(	(	PUNCT
ejpam-4912	265	9	f3	f3	ADJ
ejpam-4912	265	10	)	)	PUNCT
ejpam-4912	265	11	=	=	SYM
ejpam-4912	265	12	{	{	PUNCT
ejpam-4912	265	13	v0	v0	NOUN
ejpam-4912	265	14	,	,	PUNCT
ejpam-4912	265	15	v1	v1	NOUN
ejpam-4912	265	16	,	,	PUNCT
ejpam-4912	265	17	v2	v2	PROPN
ejpam-4912	265	18	,	,	PUNCT
ejpam-4912	265	19	v3	v3	PROPN
ejpam-4912	265	20	}	}	PUNCT
ejpam-4912	265	21	,	,	PUNCT
ejpam-4912	265	22	where	where	SCONJ
ejpam-4912	265	23	v0	v0	NOUN
ejpam-4912	265	24	is	be	AUX
ejpam-4912	265	25	the	the	DET
ejpam-4912	265	26	dominating	dominating	NOUN
ejpam-4912	265	27	vertex	vertex	NOUN
ejpam-4912	265	28	of	of	ADP
ejpam-4912	265	29	f3	f3	PROPN
ejpam-4912	265	30	.	.	PUNCT
ejpam-4912	266	1	consider	consider	VERB
ejpam-4912	266	2	m	m	NOUN
ejpam-4912	266	3	=	=	PUNCT
ejpam-4912	266	4	{	{	PUNCT
ejpam-4912	266	5	v0	v0	NOUN
ejpam-4912	266	6	,	,	PUNCT
ejpam-4912	266	7	v1	v1	NOUN
ejpam-4912	266	8	,	,	PUNCT
ejpam-4912	266	9	v2	v2	PROPN
ejpam-4912	266	10	}	}	PUNCT
ejpam-4912	266	11	.	.	PUNCT
ejpam-4912	267	1	then	then	ADV
ejpam-4912	267	2	vi	vi	PROPN
ejpam-4912	267	3	∈	∈	PROPN
ejpam-4912	267	4	nf3(v0	nf3(v0	NUM
ejpam-4912	267	5	)	)	PUNCT
ejpam-4912	267	6	\	\	NOUN
ejpam-4912	267	7	nf3(vi	nf3(vi	NOUN
ejpam-4912	267	8	)	)	PUNCT
ejpam-4912	267	9	and	and	CCONJ
ejpam-4912	267	10	v0	v0	PROPN
ejpam-4912	267	11	∈	∈	PROPN
ejpam-4912	267	12	nf3(vi	nf3(vi	NOUN
ejpam-4912	267	13	)	)	PUNCT
ejpam-4912	267	14	\	\	NOUN
ejpam-4912	267	15	nf3(v0	nf3(v0	NUM
ejpam-4912	267	16	)	)	PUNCT
ejpam-4912	267	17	∀	∀	NOUN
ejpam-4912	268	1	i	i	PRON
ejpam-4912	268	2	̸=	̸=	PROPN
ejpam-4912	268	3	0	0	NUM
ejpam-4912	268	4	,	,	PUNCT
ejpam-4912	268	5	and	and	CCONJ
ejpam-4912	268	6	v2	v2	PROPN
ejpam-4912	268	7	∈	∈	PROPN
ejpam-4912	268	8	nf3(v1)\nf3(v2	nf3(v1)\nf3(v2	NOUN
ejpam-4912	268	9	)	)	PUNCT
ejpam-4912	268	10	and	and	CCONJ
ejpam-4912	268	11	v1	v1	NOUN
ejpam-4912	268	12	∈	∈	NOUN
ejpam-4912	268	13	nf3(v2)\nf3(v1	nf3(v2)\nf3(v1	NOUN
ejpam-4912	268	14	)	)	PUNCT
ejpam-4912	268	15	.	.	PUNCT
ejpam-4912	269	1	thus	thus	ADV
ejpam-4912	269	2	,	,	PUNCT
ejpam-4912	269	3	m	m	VERB
ejpam-4912	269	4	is	be	AUX
ejpam-4912	269	5	a	a	DET
ejpam-4912	269	6	j	j	NOUN
ejpam-4912	269	7	-	-	ADJ
ejpam-4912	269	8	open	open	ADJ
ejpam-4912	269	9	set	set	NOUN
ejpam-4912	269	10	in	in	ADP
ejpam-4912	269	11	f3	f3	PROPN
ejpam-4912	269	12	.	.	PUNCT
ejpam-4912	270	1	since	since	SCONJ
ejpam-4912	270	2	nfn(m	nfn(m	NUM
ejpam-4912	270	3	)	)	PUNCT
ejpam-4912	270	4	=	=	SYM
ejpam-4912	270	5	v	v	X
ejpam-4912	270	6	(	(	PUNCT
ejpam-4912	270	7	f3	f3	NOUN
ejpam-4912	270	8	)	)	PUNCT
ejpam-4912	270	9	,	,	PUNCT
ejpam-4912	270	10	it	it	PRON
ejpam-4912	270	11	follows	follow	VERB
ejpam-4912	270	12	that	that	SCONJ
ejpam-4912	270	13	m	m	PROPN
ejpam-4912	270	14	is	be	AUX
ejpam-4912	270	15	a	a	DET
ejpam-4912	270	16	j	j	PROPN
ejpam-4912	270	17	-	-	ADJ
ejpam-4912	270	18	total	total	ADJ
ejpam-4912	270	19	dominating	dominating	NOUN
ejpam-4912	270	20	set	set	NOUN
ejpam-4912	270	21	of	of	ADP
ejpam-4912	270	22	f3	f3	PROPN
ejpam-4912	270	23	.	.	PUNCT
ejpam-4912	271	1	since	since	SCONJ
ejpam-4912	271	2	nf3(v1	nf3(v1	NOUN
ejpam-4912	271	3	)	)	PUNCT
ejpam-4912	271	4	=	=	PUNCT
ejpam-4912	271	5	nf3(v3	nf3(v3	NUM
ejpam-4912	271	6	)	)	PUNCT
ejpam-4912	271	7	,	,	PUNCT
ejpam-4912	271	8	v1	v1	NOUN
ejpam-4912	271	9	and	and	CCONJ
ejpam-4912	271	10	v3	v3	PROPN
ejpam-4912	271	11	can	can	AUX
ejpam-4912	271	12	not	not	PART
ejpam-4912	271	13	be	be	AUX
ejpam-4912	271	14	both	both	PRON
ejpam-4912	271	15	in	in	ADP
ejpam-4912	271	16	any	any	DET
ejpam-4912	271	17	j	j	NOUN
ejpam-4912	271	18	-	-	ADJ
ejpam-4912	271	19	open	open	ADJ
ejpam-4912	271	20	set	set	NOUN
ejpam-4912	271	21	of	of	ADP
ejpam-4912	271	22	f3	f3	PROPN
ejpam-4912	271	23	.	.	PUNCT
ejpam-4912	272	1	therefore	therefore	ADV
ejpam-4912	272	2	,	,	PUNCT
ejpam-4912	272	3	m	m	VERB
ejpam-4912	272	4	is	be	AUX
ejpam-4912	272	5	a	a	DET
ejpam-4912	272	6	maximum	maximum	ADJ
ejpam-4912	272	7	j	j	ADJ
ejpam-4912	272	8	-	-	ADJ
ejpam-4912	272	9	total	total	ADJ
ejpam-4912	272	10	dominating	dominating	NOUN
ejpam-4912	272	11	set	set	NOUN
ejpam-4912	272	12	of	of	ADP
ejpam-4912	272	13	f3	f3	ADJ
ejpam-4912	272	14	,	,	PUNCT
ejpam-4912	272	15	and	and	CCONJ
ejpam-4912	272	16	so	so	ADV
ejpam-4912	272	17	γjt(f3	γjt(f3	PROPN
ejpam-4912	272	18	)	)	PUNCT
ejpam-4912	273	1	=	=	SYM
ejpam-4912	274	1	3	3	X
ejpam-4912	274	2	.	.	X
ejpam-4912	274	3	similarly	similarly	ADV
ejpam-4912	274	4	,	,	PUNCT
ejpam-4912	274	5	if	if	SCONJ
ejpam-4912	274	6	n	n	CCONJ
ejpam-4912	274	7	=	=	SYM
ejpam-4912	274	8	4	4	NUM
ejpam-4912	274	9	,	,	PUNCT
ejpam-4912	274	10	then	then	ADV
ejpam-4912	274	11	γjt(f4	γjt(f4	ADJ
ejpam-4912	274	12	)	)	PUNCT
ejpam-4912	274	13	=	=	SYM
ejpam-4912	275	1	3	3	X
ejpam-4912	275	2	.	.	X
ejpam-4912	275	3	for	for	ADP
ejpam-4912	275	4	n	n	NOUN
ejpam-4912	275	5	=	=	SYM
ejpam-4912	275	6	5	5	NUM
ejpam-4912	275	7	,	,	PUNCT
ejpam-4912	275	8	let	let	VERB
ejpam-4912	275	9	v	v	NOUN
ejpam-4912	275	10	(	(	PUNCT
ejpam-4912	275	11	f5	f5	PROPN
ejpam-4912	275	12	)	)	PUNCT
ejpam-4912	275	13	=	=	SYM
ejpam-4912	275	14	{	{	PUNCT
ejpam-4912	275	15	v0	v0	NOUN
ejpam-4912	275	16	,	,	PUNCT
ejpam-4912	275	17	v1	v1	NOUN
ejpam-4912	275	18	,	,	PUNCT
ejpam-4912	275	19	v2	v2	PROPN
ejpam-4912	275	20	,	,	PUNCT
ejpam-4912	275	21	v3	v3	PROPN
ejpam-4912	275	22	,	,	PUNCT
ejpam-4912	275	23	v4	v4	PROPN
ejpam-4912	275	24	,	,	PUNCT
ejpam-4912	275	25	v5	v5	PROPN
ejpam-4912	275	26	}	}	PUNCT
ejpam-4912	275	27	,	,	PUNCT
ejpam-4912	275	28	where	where	SCONJ
ejpam-4912	275	29	v0	v0	NOUN
ejpam-4912	275	30	is	be	AUX
ejpam-4912	275	31	the	the	DET
ejpam-4912	275	32	dominating	dominating	NOUN
ejpam-4912	275	33	vertex	vertex	NOUN
ejpam-4912	275	34	of	of	ADP
ejpam-4912	275	35	f5	f5	NOUN
ejpam-4912	275	36	.	.	PUNCT
ejpam-4912	276	1	let	let	VERB
ejpam-4912	276	2	m	m	PRON
ejpam-4912	276	3	′	′	NUM
ejpam-4912	276	4	=	=	PUNCT
ejpam-4912	276	5	{	{	PUNCT
ejpam-4912	276	6	v0	v0	NOUN
ejpam-4912	276	7	,	,	PUNCT
ejpam-4912	276	8	v1	v1	NOUN
ejpam-4912	276	9	,	,	PUNCT
ejpam-4912	276	10	v2	v2	PROPN
ejpam-4912	276	11	,	,	PUNCT
ejpam-4912	276	12	,	,	PUNCT
ejpam-4912	276	13	v4	v4	PROPN
ejpam-4912	276	14	,	,	PUNCT
ejpam-4912	276	15	v5	v5	PROPN
ejpam-4912	276	16	}	}	PUNCT
ejpam-4912	276	17	.	.	PUNCT
ejpam-4912	277	1	then	then	ADV
ejpam-4912	277	2	vj	vj	PROPN
ejpam-4912	277	3	∈	∈	PROPN
ejpam-4912	277	4	nf5(v0	nf5(v0	PROPN
ejpam-4912	277	5	)	)	PUNCT
ejpam-4912	277	6	\	\	NOUN
ejpam-4912	277	7	nf5(vj	nf5(vj	NOUN
ejpam-4912	277	8	)	)	PUNCT
ejpam-4912	277	9	and	and	CCONJ
ejpam-4912	277	10	v0	v0	PROPN
ejpam-4912	277	11	∈	∈	PROPN
ejpam-4912	277	12	nf5(vj	nf5(vj	NOUN
ejpam-4912	277	13	)	)	PUNCT
ejpam-4912	277	14	\	\	NOUN
ejpam-4912	278	1	nf5(v0	nf5(v0	NUM
ejpam-4912	278	2	)	)	PUNCT
ejpam-4912	278	3	∀	∀	PUNCT
ejpam-4912	279	1	j	j	PROPN
ejpam-4912	279	2	̸=	̸=	PROPN
ejpam-4912	279	3	0	0	NUM
ejpam-4912	279	4	.	.	PUNCT
ejpam-4912	280	1	since	since	SCONJ
ejpam-4912	280	2	{	{	PUNCT
ejpam-4912	280	3	v1	v1	NOUN
ejpam-4912	280	4	,	,	PUNCT
ejpam-4912	280	5	v2	v2	PROPN
ejpam-4912	280	6	,	,	PUNCT
ejpam-4912	280	7	v4	v4	NOUN
ejpam-4912	280	8	,	,	PUNCT
ejpam-4912	280	9	v5	v5	PROPN
ejpam-4912	280	10	}	}	PUNCT
ejpam-4912	280	11	is	be	AUX
ejpam-4912	280	12	a	a	DET
ejpam-4912	280	13	j	j	NOUN
ejpam-4912	280	14	-	-	ADJ
ejpam-4912	280	15	open	open	ADJ
ejpam-4912	280	16	set	set	NOUN
ejpam-4912	280	17	in	in	ADP
ejpam-4912	280	18	p5	p5	ADJ
ejpam-4912	280	19	by	by	ADP
ejpam-4912	280	20	theorem	theorem	NOUN
ejpam-4912	280	21	5	5	NUM
ejpam-4912	280	22	,	,	PUNCT
ejpam-4912	280	23	it	it	PRON
ejpam-4912	280	24	follows	follow	VERB
ejpam-4912	280	25	that	that	SCONJ
ejpam-4912	280	26	m	m	VERB
ejpam-4912	280	27	′	′	ADJ
ejpam-4912	280	28	is	be	AUX
ejpam-4912	280	29	a	a	DET
ejpam-4912	280	30	j	j	NOUN
ejpam-4912	280	31	-	-	ADJ
ejpam-4912	280	32	open	open	ADJ
ejpam-4912	280	33	set	set	NOUN
ejpam-4912	280	34	in	in	ADP
ejpam-4912	280	35	f5	f5	NOUN
ejpam-4912	280	36	.	.	PUNCT
ejpam-4912	281	1	notice	notice	VERB
ejpam-4912	281	2	that	that	SCONJ
ejpam-4912	281	3	nf5(m	nf5(m	PROPN
ejpam-4912	281	4	′	′	NOUN
ejpam-4912	281	5	)	)	PUNCT
ejpam-4912	281	6	=	=	SYM
ejpam-4912	281	7	v	v	X
ejpam-4912	281	8	(	(	PUNCT
ejpam-4912	281	9	f5	f5	PROPN
ejpam-4912	281	10	)	)	PUNCT
ejpam-4912	281	11	.	.	PUNCT
ejpam-4912	282	1	thus	thus	ADV
ejpam-4912	282	2	,	,	PUNCT
ejpam-4912	282	3	m	m	VERB
ejpam-4912	282	4	′	′	ADJ
ejpam-4912	282	5	is	be	AUX
ejpam-4912	282	6	a	a	DET
ejpam-4912	282	7	j	j	PROPN
ejpam-4912	282	8	-	-	ADJ
ejpam-4912	282	9	total	total	ADJ
ejpam-4912	282	10	dominating	dominating	NOUN
ejpam-4912	282	11	set	set	NOUN
ejpam-4912	282	12	of	of	ADP
ejpam-4912	282	13	f5	f5	NOUN
ejpam-4912	282	14	.	.	PUNCT
ejpam-4912	283	1	since	since	SCONJ
ejpam-4912	283	2	nf5(v1	nf5(v1	NOUN
ejpam-4912	283	3	)	)	PUNCT
ejpam-4912	283	4	⊆	⊆	NUM
ejpam-4912	283	5	nf5(v3	nf5(v3	NOUN
ejpam-4912	283	6	)	)	PUNCT
ejpam-4912	283	7	,	,	PUNCT
ejpam-4912	283	8	it	it	PRON
ejpam-4912	283	9	follows	follow	VERB
ejpam-4912	283	10	that	that	SCONJ
ejpam-4912	283	11	v1	v1	NOUN
ejpam-4912	283	12	and	and	CCONJ
ejpam-4912	283	13	v3	v3	PROPN
ejpam-4912	283	14	can	can	AUX
ejpam-4912	283	15	not	not	PART
ejpam-4912	283	16	be	be	AUX
ejpam-4912	283	17	both	both	PRON
ejpam-4912	283	18	in	in	ADP
ejpam-4912	283	19	any	any	DET
ejpam-4912	283	20	j	j	NOUN
ejpam-4912	283	21	-	-	ADJ
ejpam-4912	283	22	open	open	ADJ
ejpam-4912	283	23	set	set	NOUN
ejpam-4912	283	24	of	of	ADP
ejpam-4912	283	25	f5	f5	NOUN
ejpam-4912	283	26	.	.	PUNCT
ejpam-4912	284	1	therefore	therefore	ADV
ejpam-4912	284	2	,	,	PUNCT
ejpam-4912	284	3	m	m	VERB
ejpam-4912	284	4	′	′	ADJ
ejpam-4912	284	5	is	be	AUX
ejpam-4912	284	6	a	a	DET
ejpam-4912	284	7	maximum	maximum	ADJ
ejpam-4912	284	8	j	j	ADJ
ejpam-4912	284	9	-	-	ADJ
ejpam-4912	284	10	total	total	ADJ
ejpam-4912	284	11	dominating	dominating	NOUN
ejpam-4912	284	12	set	set	NOUN
ejpam-4912	284	13	of	of	ADP
ejpam-4912	284	14	f5	f5	NOUN
ejpam-4912	284	15	,	,	PUNCT
ejpam-4912	284	16	and	and	CCONJ
ejpam-4912	284	17	so	so	ADV
ejpam-4912	284	18	γjt(f5	γjt(f5	ADJ
ejpam-4912	284	19	)	)	PUNCT
ejpam-4912	284	20	=	=	SYM
ejpam-4912	284	21	5	5	X
ejpam-4912	284	22	.	.	PUNCT
ejpam-4912	285	1	j.a	j.a	PROPN
ejpam-4912	285	2	.	.	PROPN
ejpam-4912	285	3	hassan	hassan	PROPN
ejpam-4912	285	4	et	et	PROPN
ejpam-4912	285	5	al	al	PROPN
ejpam-4912	285	6	.	.	PUNCT
ejpam-4912	285	7	/	/	SYM
ejpam-4912	285	8	eur	eur	PROPN
ejpam-4912	285	9	.	.	PUNCT
ejpam-4912	286	1	j.	j.	PROPN
ejpam-4912	286	2	pure	pure	PROPN
ejpam-4912	286	3	appl	appl	PROPN
ejpam-4912	286	4	.	.	PROPN
ejpam-4912	286	5	math	math	PROPN
ejpam-4912	286	6	,	,	PUNCT
ejpam-4912	286	7	16	16	NUM
ejpam-4912	286	8	(	(	PUNCT
ejpam-4912	286	9	4	4	NUM
ejpam-4912	286	10	)	)	PUNCT
ejpam-4912	286	11	(	(	PUNCT
ejpam-4912	286	12	2023	2023	NUM
ejpam-4912	286	13	)	)	PUNCT
ejpam-4912	286	14	,	,	PUNCT
ejpam-4912	286	15	2106	2106	NUM
ejpam-4912	286	16	-	-	SYM
ejpam-4912	286	17	2117	2117	NUM
ejpam-4912	286	18	2115	2115	NUM
ejpam-4912	286	19	next	next	ADV
ejpam-4912	286	20	,	,	PUNCT
ejpam-4912	286	21	suppose	suppose	VERB
ejpam-4912	286	22	that	that	SCONJ
ejpam-4912	286	23	n	n	PROPN
ejpam-4912	286	24	≥	≥	NUM
ejpam-4912	286	25	6	6	NUM
ejpam-4912	286	26	.	.	PUNCT
ejpam-4912	287	1	let	let	VERB
ejpam-4912	287	2	v	v	NOUN
ejpam-4912	287	3	(	(	PUNCT
ejpam-4912	287	4	fn	fn	NOUN
ejpam-4912	287	5	)	)	PUNCT
ejpam-4912	287	6	=	=	SYM
ejpam-4912	287	7	{	{	PUNCT
ejpam-4912	287	8	u0	u0	ADJ
ejpam-4912	287	9	,	,	PUNCT
ejpam-4912	287	10	u1	u1	NOUN
ejpam-4912	287	11	,	,	PUNCT
ejpam-4912	287	12	.	.	PUNCT
ejpam-4912	287	13	.	.	PUNCT
ejpam-4912	288	1	.	.	PUNCT
ejpam-4912	289	1	,	,	PUNCT
ejpam-4912	289	2	un	un	PROPN
ejpam-4912	289	3	}	}	PUNCT
ejpam-4912	289	4	,	,	PUNCT
ejpam-4912	289	5	where	where	SCONJ
ejpam-4912	289	6	u0	u0	ADJ
ejpam-4912	289	7	is	be	AUX
ejpam-4912	289	8	the	the	DET
ejpam-4912	289	9	dominating	dominating	NOUN
ejpam-4912	289	10	vertex	vertex	NOUN
ejpam-4912	289	11	of	of	ADP
ejpam-4912	289	12	fn	fn	NOUN
ejpam-4912	289	13	.	.	PUNCT
ejpam-4912	290	1	let	let	VERB
ejpam-4912	290	2	c	c	NOUN
ejpam-4912	290	3	=	=	PUNCT
ejpam-4912	290	4	{	{	PUNCT
ejpam-4912	290	5	u0	u0	PROPN
ejpam-4912	290	6	,	,	PUNCT
ejpam-4912	290	7	u2	u2	PROPN
ejpam-4912	290	8	,	,	PUNCT
ejpam-4912	290	9	u3	u3	NOUN
ejpam-4912	290	10	,	,	PUNCT
ejpam-4912	290	11	.	.	PUNCT
ejpam-4912	290	12	.	.	PUNCT
ejpam-4912	291	1	.	.	PUNCT
ejpam-4912	292	1	,	,	PUNCT
ejpam-4912	292	2	un−1	un−1	ADJ
ejpam-4912	292	3	}	}	PUNCT
ejpam-4912	292	4	.	.	PUNCT
ejpam-4912	293	1	observe	observe	VERB
ejpam-4912	293	2	that	that	SCONJ
ejpam-4912	293	3	ur	ur	PROPN
ejpam-4912	293	4	∈	∈	PROPN
ejpam-4912	293	5	nfn(uo	nfn(uo	NOUN
ejpam-4912	293	6	)	)	PUNCT
ejpam-4912	293	7	\	\	NOUN
ejpam-4912	293	8	nfn(ur	nfn(ur	NOUN
ejpam-4912	293	9	)	)	PUNCT
ejpam-4912	293	10	and	and	CCONJ
ejpam-4912	293	11	u0	u0	PROPN
ejpam-4912	293	12	∈	∈	PROPN
ejpam-4912	293	13	nfn(ur	nfn(ur	NUM
ejpam-4912	293	14	)	)	PUNCT
ejpam-4912	293	15	\nfn(uo	\nfn(uo	NOUN
ejpam-4912	293	16	)	)	PUNCT
ejpam-4912	293	17	∀	∀	PUNCT
ejpam-4912	294	1	r	r	NOUN
ejpam-4912	294	2	̸=	̸=	PROPN
ejpam-4912	294	3	0	0	NUM
ejpam-4912	294	4	.	.	PUNCT
ejpam-4912	295	1	since	since	SCONJ
ejpam-4912	295	2	{	{	PUNCT
ejpam-4912	295	3	u2	u2	NOUN
ejpam-4912	295	4	,	,	PUNCT
ejpam-4912	295	5	u3	u3	NOUN
ejpam-4912	295	6	,	,	PUNCT
ejpam-4912	295	7	...	...	PUNCT
ejpam-4912	295	8	,	,	PUNCT
ejpam-4912	295	9	un−1	un−1	ADJ
ejpam-4912	295	10	}	}	PUNCT
ejpam-4912	295	11	is	be	AUX
ejpam-4912	295	12	a	a	DET
ejpam-4912	295	13	j	j	NOUN
ejpam-4912	295	14	-	-	ADJ
ejpam-4912	295	15	open	open	ADJ
ejpam-4912	295	16	set	set	NOUN
ejpam-4912	295	17	in	in	ADP
ejpam-4912	295	18	pn	pn	PROPN
ejpam-4912	295	19	by	by	ADP
ejpam-4912	295	20	theorem	theorem	NOUN
ejpam-4912	295	21	5	5	NUM
ejpam-4912	295	22	,	,	PUNCT
ejpam-4912	295	23	it	it	PRON
ejpam-4912	295	24	follows	follow	VERB
ejpam-4912	295	25	that	that	SCONJ
ejpam-4912	295	26	c	c	PROPN
ejpam-4912	295	27	is	be	AUX
ejpam-4912	295	28	a	a	DET
ejpam-4912	295	29	j	j	NOUN
ejpam-4912	295	30	-	-	ADJ
ejpam-4912	295	31	open	open	ADJ
ejpam-4912	295	32	set	set	NOUN
ejpam-4912	295	33	in	in	ADP
ejpam-4912	295	34	fn	fn	PROPN
ejpam-4912	295	35	.	.	PUNCT
ejpam-4912	296	1	observe	observe	VERB
ejpam-4912	296	2	further	far	ADV
ejpam-4912	296	3	that	that	SCONJ
ejpam-4912	296	4	nfn(c	nfn(c	NOUN
ejpam-4912	296	5	)	)	PUNCT
ejpam-4912	296	6	=	=	SYM
ejpam-4912	296	7	v	v	X
ejpam-4912	296	8	(	(	PUNCT
ejpam-4912	296	9	fn	fn	NOUN
ejpam-4912	296	10	)	)	PUNCT
ejpam-4912	296	11	.	.	PUNCT
ejpam-4912	297	1	hence	hence	ADV
ejpam-4912	297	2	,	,	PUNCT
ejpam-4912	297	3	c	c	PROPN
ejpam-4912	297	4	is	be	AUX
ejpam-4912	297	5	a	a	DET
ejpam-4912	297	6	j	j	PROPN
ejpam-4912	297	7	-	-	ADJ
ejpam-4912	297	8	total	total	ADJ
ejpam-4912	297	9	dominating	dominating	NOUN
ejpam-4912	297	10	set	set	NOUN
ejpam-4912	297	11	of	of	ADP
ejpam-4912	297	12	fn	fn	PROPN
ejpam-4912	297	13	.	.	PUNCT
ejpam-4912	298	1	since	since	SCONJ
ejpam-4912	298	2	nfn(u1	nfn(u1	NUM
ejpam-4912	298	3	)	)	PUNCT
ejpam-4912	298	4	⊆	⊆	NUM
ejpam-4912	298	5	nfn(u3	nfn(u3	NOUN
ejpam-4912	298	6	)	)	PUNCT
ejpam-4912	298	7	and	and	CCONJ
ejpam-4912	298	8	nfn(un	nfn(un	NOUN
ejpam-4912	298	9	)	)	PUNCT
ejpam-4912	298	10	⊆	⊆	NUM
ejpam-4912	298	11	nfn(un−2	nfn(un−2	PROPN
ejpam-4912	298	12	)	)	PUNCT
ejpam-4912	298	13	,	,	PUNCT
ejpam-4912	298	14	u1	u1	NOUN
ejpam-4912	298	15	and	and	CCONJ
ejpam-4912	298	16	u3	u3	PROPN
ejpam-4912	298	17	(	(	PUNCT
ejpam-4912	298	18	resp	resp	NOUN
ejpam-4912	298	19	.	.	PUNCT
ejpam-4912	298	20	un−2	un−2	VERB
ejpam-4912	298	21	and	and	CCONJ
ejpam-4912	298	22	un	un	ADJ
ejpam-4912	298	23	)	)	PUNCT
ejpam-4912	298	24	can	can	AUX
ejpam-4912	298	25	not	not	PART
ejpam-4912	298	26	be	be	AUX
ejpam-4912	298	27	both	both	PRON
ejpam-4912	298	28	in	in	ADP
ejpam-4912	298	29	any	any	DET
ejpam-4912	298	30	j	j	NOUN
ejpam-4912	298	31	-	-	ADJ
ejpam-4912	298	32	open	open	ADJ
ejpam-4912	298	33	set	set	NOUN
ejpam-4912	298	34	of	of	ADP
ejpam-4912	298	35	fn	fn	PROPN
ejpam-4912	298	36	.	.	PUNCT
ejpam-4912	299	1	therefore	therefore	ADV
ejpam-4912	299	2	,	,	PUNCT
ejpam-4912	299	3	c	c	PROPN
ejpam-4912	299	4	is	be	AUX
ejpam-4912	299	5	a	a	DET
ejpam-4912	299	6	maximum	maximum	ADJ
ejpam-4912	299	7	j	j	ADJ
ejpam-4912	299	8	-	-	ADJ
ejpam-4912	299	9	total	total	ADJ
ejpam-4912	299	10	dominating	dominating	NOUN
ejpam-4912	299	11	set	set	NOUN
ejpam-4912	299	12	of	of	ADP
ejpam-4912	299	13	fn	fn	NOUN
ejpam-4912	299	14	,	,	PUNCT
ejpam-4912	299	15	showing	show	VERB
ejpam-4912	299	16	that	that	SCONJ
ejpam-4912	299	17	γjt(fn	γjt(fn	NOUN
ejpam-4912	299	18	)	)	PUNCT
ejpam-4912	299	19	=	=	SYM
ejpam-4912	299	20	n−	n−	NOUN
ejpam-4912	299	21	1	1	NUM
ejpam-4912	299	22	for	for	ADP
ejpam-4912	299	23	all	all	DET
ejpam-4912	299	24	n	n	PRON
ejpam-4912	299	25	≥	≥	NUM
ejpam-4912	299	26	6	6	NUM
ejpam-4912	299	27	.	.	PUNCT
ejpam-4912	299	28	theorem	theorem	VERB
ejpam-4912	299	29	7	7	NUM
ejpam-4912	299	30	.	.	PUNCT
ejpam-4912	300	1	let	let	VERB
ejpam-4912	300	2	g	g	NOUN
ejpam-4912	300	3	and	and	CCONJ
ejpam-4912	300	4	h	h	NOUN
ejpam-4912	300	5	be	be	VERB
ejpam-4912	300	6	two	two	NUM
ejpam-4912	300	7	graphs	graph	NOUN
ejpam-4912	300	8	with	with	ADP
ejpam-4912	300	9	no	no	DET
ejpam-4912	300	10	isolated	isolated	ADJ
ejpam-4912	300	11	vertices	vertex	NOUN
ejpam-4912	300	12	.	.	PUNCT
ejpam-4912	301	1	a	a	DET
ejpam-4912	301	2	subset	subset	NOUN
ejpam-4912	301	3	m	m	VERB
ejpam-4912	301	4	of	of	ADP
ejpam-4912	301	5	vertices	vertex	NOUN
ejpam-4912	301	6	of	of	ADP
ejpam-4912	301	7	g+h	g+h	PROPN
ejpam-4912	301	8	is	be	AUX
ejpam-4912	301	9	a	a	DET
ejpam-4912	301	10	j	j	PROPN
ejpam-4912	301	11	-	-	ADJ
ejpam-4912	301	12	total	total	ADJ
ejpam-4912	301	13	dominating	dominating	NOUN
ejpam-4912	301	14	set	set	NOUN
ejpam-4912	301	15	of	of	ADP
ejpam-4912	301	16	g+h	g+h	PROPN
ejpam-4912	302	1	if	if	SCONJ
ejpam-4912	302	2	and	and	CCONJ
ejpam-4912	302	3	only	only	ADV
ejpam-4912	302	4	if	if	SCONJ
ejpam-4912	302	5	one	one	NUM
ejpam-4912	302	6	of	of	ADP
ejpam-4912	302	7	the	the	DET
ejpam-4912	302	8	following	follow	VERB
ejpam-4912	302	9	conditions	condition	NOUN
ejpam-4912	302	10	holds	hold	VERB
ejpam-4912	302	11	:	:	PUNCT
ejpam-4912	302	12	(	(	PUNCT
ejpam-4912	302	13	i	i	NOUN
ejpam-4912	302	14	)	)	PUNCT
ejpam-4912	302	15	m	m	VERB
ejpam-4912	302	16	is	be	AUX
ejpam-4912	302	17	a	a	DET
ejpam-4912	302	18	j	j	PROPN
ejpam-4912	302	19	-	-	ADJ
ejpam-4912	302	20	total	total	ADJ
ejpam-4912	302	21	dominating	dominating	NOUN
ejpam-4912	302	22	set	set	NOUN
ejpam-4912	302	23	of	of	ADP
ejpam-4912	302	24	g.	g.	PROPN
ejpam-4912	302	25	(	(	PUNCT
ejpam-4912	302	26	ii	ii	PROPN
ejpam-4912	302	27	)	)	PUNCT
ejpam-4912	302	28	m	m	VERB
ejpam-4912	302	29	is	be	AUX
ejpam-4912	302	30	a	a	DET
ejpam-4912	302	31	j	j	PROPN
ejpam-4912	302	32	-	-	ADJ
ejpam-4912	302	33	total	total	ADJ
ejpam-4912	302	34	dominating	dominating	NOUN
ejpam-4912	302	35	set	set	NOUN
ejpam-4912	302	36	of	of	ADP
ejpam-4912	302	37	h.	h.	PROPN
ejpam-4912	302	38	(	(	PUNCT
ejpam-4912	302	39	iii	iii	X
ejpam-4912	302	40	)	)	PUNCT
ejpam-4912	302	41	m	m	PROPN
ejpam-4912	302	42	=	=	SYM
ejpam-4912	302	43	mg	mg	PROPN
ejpam-4912	302	44	∪mh	∪mh	NOUN
ejpam-4912	302	45	,	,	PUNCT
ejpam-4912	302	46	where	where	SCONJ
ejpam-4912	302	47	mg	mg	PROPN
ejpam-4912	302	48	and	and	CCONJ
ejpam-4912	302	49	mh	mh	PROPN
ejpam-4912	302	50	are	be	AUX
ejpam-4912	302	51	j	j	NOUN
ejpam-4912	302	52	-	-	ADJ
ejpam-4912	302	53	open	open	ADJ
ejpam-4912	302	54	sets	set	NOUN
ejpam-4912	302	55	in	in	ADP
ejpam-4912	302	56	g	g	PROPN
ejpam-4912	302	57	and	and	CCONJ
ejpam-4912	302	58	h	h	NOUN
ejpam-4912	302	59	,	,	PUNCT
ejpam-4912	302	60	respectively	respectively	ADV
ejpam-4912	302	61	.	.	PUNCT
ejpam-4912	303	1	proof	proof	NOUN
ejpam-4912	303	2	.	.	PUNCT
ejpam-4912	304	1	let	let	VERB
ejpam-4912	304	2	m	m	PRON
ejpam-4912	304	3	be	be	AUX
ejpam-4912	304	4	a	a	DET
ejpam-4912	304	5	j	j	PROPN
ejpam-4912	304	6	-	-	ADJ
ejpam-4912	304	7	total	total	ADJ
ejpam-4912	304	8	dominating	dominating	NOUN
ejpam-4912	304	9	set	set	NOUN
ejpam-4912	304	10	of	of	ADP
ejpam-4912	304	11	g	g	PROPN
ejpam-4912	304	12	+	+	CCONJ
ejpam-4912	304	13	h.	h.	PROPN
ejpam-4912	305	1	if	if	SCONJ
ejpam-4912	305	2	mh	mh	PROPN
ejpam-4912	305	3	=	=	SYM
ejpam-4912	305	4	∅	∅	NOUN
ejpam-4912	305	5	,	,	PUNCT
ejpam-4912	305	6	then	then	ADV
ejpam-4912	305	7	m	m	VERB
ejpam-4912	305	8	=	=	ADJ
ejpam-4912	305	9	mg	mg	PROPN
ejpam-4912	305	10	is	be	AUX
ejpam-4912	305	11	a	a	DET
ejpam-4912	305	12	j	j	PROPN
ejpam-4912	305	13	-	-	ADJ
ejpam-4912	305	14	total	total	ADJ
ejpam-4912	305	15	dominating	dominating	NOUN
ejpam-4912	305	16	set	set	VERB
ejpam-4912	305	17	in	in	ADP
ejpam-4912	305	18	g.	g.	PROPN
ejpam-4912	305	19	thus	thus	ADV
ejpam-4912	305	20	,	,	PUNCT
ejpam-4912	305	21	(	(	PUNCT
ejpam-4912	305	22	i	i	NOUN
ejpam-4912	305	23	)	)	PUNCT
ejpam-4912	305	24	holds	hold	VERB
ejpam-4912	305	25	.	.	PUNCT
ejpam-4912	306	1	if	if	SCONJ
ejpam-4912	306	2	mg	mg	NOUN
ejpam-4912	306	3	=	=	SYM
ejpam-4912	306	4	∅	∅	NOUN
ejpam-4912	306	5	,	,	PUNCT
ejpam-4912	306	6	then	then	ADV
ejpam-4912	306	7	m	m	VERB
ejpam-4912	306	8	=	=	ADJ
ejpam-4912	306	9	mh	mh	PROPN
ejpam-4912	306	10	is	be	AUX
ejpam-4912	306	11	a	a	DET
ejpam-4912	306	12	j	j	PROPN
ejpam-4912	306	13	-	-	ADJ
ejpam-4912	306	14	total	total	ADJ
ejpam-4912	306	15	dominating	dominating	NOUN
ejpam-4912	306	16	set	set	NOUN
ejpam-4912	306	17	in	in	ADP
ejpam-4912	306	18	h	h	NOUN
ejpam-4912	306	19	,	,	PUNCT
ejpam-4912	306	20	and	and	CCONJ
ejpam-4912	306	21	hence	hence	ADV
ejpam-4912	306	22	(	(	PUNCT
ejpam-4912	306	23	ii	ii	NOUN
ejpam-4912	306	24	)	)	PUNCT
ejpam-4912	306	25	holds	hold	VERB
ejpam-4912	306	26	.	.	PUNCT
ejpam-4912	307	1	next	next	ADV
ejpam-4912	307	2	,	,	PUNCT
ejpam-4912	307	3	assume	assume	VERB
ejpam-4912	307	4	that	that	SCONJ
ejpam-4912	307	5	mg	mg	PROPN
ejpam-4912	307	6	and	and	CCONJ
ejpam-4912	307	7	mh	mh	PROPN
ejpam-4912	307	8	are	be	AUX
ejpam-4912	307	9	both	both	PRON
ejpam-4912	307	10	non	non	ADJ
ejpam-4912	307	11	-	-	ADJ
ejpam-4912	307	12	empty	empty	ADJ
ejpam-4912	307	13	.	.	PUNCT
ejpam-4912	308	1	suppose	suppose	VERB
ejpam-4912	308	2	on	on	ADP
ejpam-4912	308	3	the	the	DET
ejpam-4912	308	4	contrary	contrary	NOUN
ejpam-4912	308	5	that	that	PRON
ejpam-4912	308	6	mg	mg	PROPN
ejpam-4912	308	7	is	be	AUX
ejpam-4912	308	8	not	not	PART
ejpam-4912	308	9	a	a	DET
ejpam-4912	308	10	j	j	NOUN
ejpam-4912	308	11	-	-	ADJ
ejpam-4912	308	12	open	open	ADJ
ejpam-4912	308	13	set	set	NOUN
ejpam-4912	308	14	in	in	ADP
ejpam-4912	308	15	g.	g.	PROPN
ejpam-4912	308	16	then	then	ADV
ejpam-4912	308	17	there	there	PRON
ejpam-4912	308	18	exist	exist	VERB
ejpam-4912	308	19	a	a	DET
ejpam-4912	308	20	,	,	PUNCT
ejpam-4912	308	21	b	b	X
ejpam-4912	308	22	∈	∈	NOUN
ejpam-4912	308	23	mg	mg	PROPN
ejpam-4912	308	24	⊆	⊆	NUM
ejpam-4912	308	25	m	m	NOUN
ejpam-4912	308	26	such	such	ADJ
ejpam-4912	308	27	that	that	SCONJ
ejpam-4912	308	28	either	either	ADV
ejpam-4912	308	29	ng(a	ng(a	NOUN
ejpam-4912	308	30	)	)	PUNCT
ejpam-4912	308	31	\	\	NOUN
ejpam-4912	308	32	ng(b	ng(b	PUNCT
ejpam-4912	308	33	)	)	PUNCT
ejpam-4912	308	34	=	=	SYM
ejpam-4912	308	35	∅	∅	NOUN
ejpam-4912	308	36	or	or	CCONJ
ejpam-4912	308	37	ng(b	ng(b	NOUN
ejpam-4912	308	38	)	)	PUNCT
ejpam-4912	308	39	\	\	NOUN
ejpam-4912	308	40	ng(a	ng(a	NOUN
ejpam-4912	308	41	)	)	PUNCT
ejpam-4912	308	42	=	=	PUNCT
ejpam-4912	308	43	∅.	∅.	ADP
ejpam-4912	308	44	thus	thus	ADV
ejpam-4912	308	45	,	,	PUNCT
ejpam-4912	308	46	ng+h(a	ng+h(a	NOUN
ejpam-4912	308	47	)	)	PUNCT
ejpam-4912	308	48	\ng+h(b	\ng+h(b	PROPN
ejpam-4912	308	49	)	)	PUNCT
ejpam-4912	308	50	=	=	NOUN
ejpam-4912	308	51	∅	∅	NOUN
ejpam-4912	308	52	or	or	CCONJ
ejpam-4912	308	53	ng+h(a	ng+h(a	NUM
ejpam-4912	308	54	)	)	PUNCT
ejpam-4912	308	55	\ng+h(b	\ng+h(b	PROPN
ejpam-4912	308	56	)	)	PUNCT
ejpam-4912	308	57	=	=	NOUN
ejpam-4912	308	58	∅	∅	NOUN
ejpam-4912	308	59	,	,	PUNCT
ejpam-4912	308	60	a	a	DET
ejpam-4912	308	61	contradiction	contradiction	NOUN
ejpam-4912	308	62	to	to	ADP
ejpam-4912	308	63	the	the	DET
ejpam-4912	308	64	fact	fact	NOUN
ejpam-4912	308	65	that	that	SCONJ
ejpam-4912	308	66	m	m	NOUN
ejpam-4912	308	67	is	be	AUX
ejpam-4912	308	68	a	a	DET
ejpam-4912	308	69	j	j	NOUN
ejpam-4912	308	70	-	-	ADJ
ejpam-4912	308	71	open	open	ADJ
ejpam-4912	308	72	set	set	NOUN
ejpam-4912	308	73	in	in	ADP
ejpam-4912	308	74	g+h	g+h	PROPN
ejpam-4912	308	75	.	.	PUNCT
ejpam-4912	309	1	therefore	therefore	ADV
ejpam-4912	309	2	,	,	PUNCT
ejpam-4912	309	3	dg	dg	PROPN
ejpam-4912	309	4	is	be	AUX
ejpam-4912	309	5	a	a	DET
ejpam-4912	309	6	j	j	NOUN
ejpam-4912	309	7	-	-	ADJ
ejpam-4912	309	8	open	open	ADJ
ejpam-4912	309	9	set	set	NOUN
ejpam-4912	309	10	in	in	ADP
ejpam-4912	309	11	g.	g.	NOUN
ejpam-4912	309	12	similarly	similarly	ADV
ejpam-4912	309	13	,	,	PUNCT
ejpam-4912	309	14	mh	mh	PROPN
ejpam-4912	309	15	is	be	AUX
ejpam-4912	309	16	a	a	DET
ejpam-4912	309	17	j	j	NOUN
ejpam-4912	309	18	-	-	ADJ
ejpam-4912	309	19	open	open	ADJ
ejpam-4912	309	20	set	set	NOUN
ejpam-4912	309	21	in	in	ADP
ejpam-4912	309	22	h.	h.	PROPN
ejpam-4912	309	23	consequently	consequently	ADV
ejpam-4912	309	24	,	,	PUNCT
ejpam-4912	309	25	(	(	PUNCT
ejpam-4912	309	26	iii	iii	NOUN
ejpam-4912	309	27	)	)	PUNCT
ejpam-4912	309	28	holds	hold	VERB
ejpam-4912	309	29	.	.	PUNCT
ejpam-4912	310	1	conversely	conversely	ADV
ejpam-4912	310	2	,	,	PUNCT
ejpam-4912	310	3	if	if	SCONJ
ejpam-4912	310	4	(	(	PUNCT
ejpam-4912	310	5	i	i	NOUN
ejpam-4912	310	6	)	)	PUNCT
ejpam-4912	310	7	or	or	CCONJ
ejpam-4912	310	8	(	(	PUNCT
ejpam-4912	310	9	ii	ii	NOUN
ejpam-4912	310	10	)	)	PUNCT
ejpam-4912	310	11	holds	hold	VERB
ejpam-4912	310	12	,	,	PUNCT
ejpam-4912	310	13	then	then	ADV
ejpam-4912	310	14	the	the	DET
ejpam-4912	310	15	assertion	assertion	NOUN
ejpam-4912	310	16	follows	follow	VERB
ejpam-4912	310	17	.	.	PUNCT
ejpam-4912	311	1	next	next	ADV
ejpam-4912	311	2	,	,	PUNCT
ejpam-4912	311	3	suppose	suppose	VERB
ejpam-4912	311	4	that	that	SCONJ
ejpam-4912	311	5	(	(	PUNCT
ejpam-4912	311	6	iii	iii	NOUN
ejpam-4912	311	7	)	)	PUNCT
ejpam-4912	311	8	holds	hold	VERB
ejpam-4912	311	9	.	.	PUNCT
ejpam-4912	312	1	sincemg	sincemg	PROPN
ejpam-4912	312	2	andmh	andmh	PROPN
ejpam-4912	312	3	are	be	AUX
ejpam-4912	312	4	both	both	PRON
ejpam-4912	312	5	non	non	ADJ
ejpam-4912	312	6	-	-	ADJ
ejpam-4912	312	7	empty	empty	ADJ
ejpam-4912	312	8	,	,	PUNCT
ejpam-4912	312	9	it	it	PRON
ejpam-4912	312	10	follows	follow	VERB
ejpam-4912	312	11	thatm	thatm	NOUN
ejpam-4912	312	12	is	be	AUX
ejpam-4912	312	13	a	a	DET
ejpam-4912	312	14	total	total	ADJ
ejpam-4912	312	15	dominating	dominating	NOUN
ejpam-4912	312	16	set	set	NOUN
ejpam-4912	312	17	ing+h	ing+h	PRON
ejpam-4912	312	18	.	.	PUNCT
ejpam-4912	313	1	let	let	VERB
ejpam-4912	313	2	a	a	DET
ejpam-4912	313	3	,	,	PUNCT
ejpam-4912	313	4	b	b	X
ejpam-4912	313	5	∈	∈	ADV
ejpam-4912	313	6	m	m	VERB
ejpam-4912	313	7	.	.	PUNCT
ejpam-4912	314	1	suppose	suppose	VERB
ejpam-4912	314	2	that	that	SCONJ
ejpam-4912	314	3	a	a	DET
ejpam-4912	314	4	,	,	PUNCT
ejpam-4912	314	5	b	b	X
ejpam-4912	314	6	∈	∈	NOUN
ejpam-4912	314	7	mg	mg	PROPN
ejpam-4912	314	8	⊆	⊆	NUM
ejpam-4912	314	9	m	m	NOUN
ejpam-4912	314	10	.	.	PUNCT
ejpam-4912	315	1	since	since	SCONJ
ejpam-4912	315	2	mg	mg	PROPN
ejpam-4912	315	3	is	be	AUX
ejpam-4912	315	4	a	a	DET
ejpam-4912	315	5	j	j	NOUN
ejpam-4912	315	6	-	-	ADJ
ejpam-4912	315	7	open	open	ADJ
ejpam-4912	315	8	set	set	NOUN
ejpam-4912	315	9	in	in	ADP
ejpam-4912	315	10	g	g	NOUN
ejpam-4912	315	11	,	,	PUNCT
ejpam-4912	315	12	we	we	PRON
ejpam-4912	315	13	have	have	VERB
ejpam-4912	315	14	ng(a	ng(a	NOUN
ejpam-4912	315	15	)	)	PUNCT
ejpam-4912	315	16	\ng(b	\ng(b	NOUN
ejpam-4912	315	17	)	)	PUNCT
ejpam-4912	315	18	̸=	̸=	PROPN
ejpam-4912	315	19	∅	∅	NOUN
ejpam-4912	315	20	and	and	CCONJ
ejpam-4912	315	21	ng(b	ng(b	X
ejpam-4912	315	22	)	)	PUNCT
ejpam-4912	315	23	\ng(a	\ng(a	X
ejpam-4912	315	24	)	)	PUNCT
ejpam-4912	315	25	̸=	̸=	PROPN
ejpam-4912	315	26	∅.	∅.	NOUN
ejpam-4912	315	27	it	it	PRON
ejpam-4912	315	28	follows	follow	VERB
ejpam-4912	315	29	that	that	SCONJ
ejpam-4912	315	30	ng+h(a	ng+h(a	NOUN
ejpam-4912	315	31	)	)	PUNCT
ejpam-4912	315	32	\ng+h(b	\ng+h(b	PROPN
ejpam-4912	315	33	)	)	PUNCT
ejpam-4912	315	34	̸=	̸=	PROPN
ejpam-4912	315	35	∅	∅	NOUN
ejpam-4912	315	36	and	and	CCONJ
ejpam-4912	315	37	ng+h(b)\ng+h(a	ng+h(b)\ng+h(a	NUM
ejpam-4912	315	38	)	)	PUNCT
ejpam-4912	315	39	̸=	̸=	PROPN
ejpam-4912	315	40	∅.	∅.	PRON
ejpam-4912	315	41	hence	hence	ADV
ejpam-4912	315	42	m	m	VERB
ejpam-4912	315	43	is	be	AUX
ejpam-4912	315	44	a	a	DET
ejpam-4912	315	45	j	j	NOUN
ejpam-4912	315	46	-	-	ADJ
ejpam-4912	315	47	open	open	ADJ
ejpam-4912	315	48	set	set	NOUN
ejpam-4912	315	49	in	in	ADP
ejpam-4912	315	50	g+h	g+h	PROPN
ejpam-4912	315	51	.	.	PUNCT
ejpam-4912	316	1	similarly	similarly	ADV
ejpam-4912	316	2	,	,	PUNCT
ejpam-4912	316	3	if	if	SCONJ
ejpam-4912	316	4	a	a	DET
ejpam-4912	316	5	,	,	PUNCT
ejpam-4912	316	6	b	b	PROPN
ejpam-4912	316	7	∈	∈	PROPN
ejpam-4912	316	8	mh	mh	PROPN
ejpam-4912	316	9	⊆	⊆	NUM
ejpam-4912	316	10	m	m	PROPN
ejpam-4912	316	11	,	,	PUNCT
ejpam-4912	316	12	then	then	ADV
ejpam-4912	316	13	m	m	VERB
ejpam-4912	316	14	is	be	AUX
ejpam-4912	316	15	a	a	DET
ejpam-4912	316	16	j	j	NOUN
ejpam-4912	316	17	-	-	ADJ
ejpam-4912	316	18	open	open	ADJ
ejpam-4912	316	19	set	set	NOUN
ejpam-4912	316	20	in	in	ADP
ejpam-4912	316	21	g	g	PROPN
ejpam-4912	316	22	+	+	CCONJ
ejpam-4912	316	23	h.	h.	PROPN
ejpam-4912	316	24	now	now	ADV
ejpam-4912	316	25	,	,	PUNCT
ejpam-4912	316	26	assume	assume	VERB
ejpam-4912	316	27	that	that	SCONJ
ejpam-4912	316	28	a	a	DET
ejpam-4912	316	29	∈	∈	PROPN
ejpam-4912	316	30	mg	mg	PROPN
ejpam-4912	316	31	and	and	CCONJ
ejpam-4912	316	32	b	b	PROPN
ejpam-4912	316	33	∈	∈	PROPN
ejpam-4912	316	34	mh	mh	PROPN
ejpam-4912	316	35	.	.	PUNCT
ejpam-4912	317	1	if	if	SCONJ
ejpam-4912	317	2	a	a	PRON
ejpam-4912	317	3	is	be	AUX
ejpam-4912	317	4	a	a	DET
ejpam-4912	317	5	dominating	dominating	NOUN
ejpam-4912	317	6	vertex	vertex	NOUN
ejpam-4912	317	7	of	of	ADP
ejpam-4912	317	8	g	g	PROPN
ejpam-4912	317	9	,	,	PUNCT
ejpam-4912	317	10	then	then	ADV
ejpam-4912	317	11	we	we	PRON
ejpam-4912	317	12	are	be	AUX
ejpam-4912	317	13	done	do	VERB
ejpam-4912	317	14	.	.	PUNCT
ejpam-4912	318	1	similarly	similarly	ADV
ejpam-4912	318	2	,	,	PUNCT
ejpam-4912	318	3	if	if	SCONJ
ejpam-4912	318	4	b	b	PROPN
ejpam-4912	318	5	a	a	DET
ejpam-4912	318	6	dominating	dominating	NOUN
ejpam-4912	318	7	vertex	vertex	NOUN
ejpam-4912	318	8	of	of	ADP
ejpam-4912	318	9	h.	h.	PROPN
ejpam-4912	318	10	suppose	suppose	VERB
ejpam-4912	318	11	that	that	SCONJ
ejpam-4912	318	12	a	a	PRON
ejpam-4912	318	13	and	and	CCONJ
ejpam-4912	318	14	b	b	NOUN
ejpam-4912	318	15	are	be	AUX
ejpam-4912	318	16	not	not	PART
ejpam-4912	318	17	dominating	dominate	VERB
ejpam-4912	318	18	vertices	vertex	NOUN
ejpam-4912	318	19	of	of	ADP
ejpam-4912	318	20	g	g	PROPN
ejpam-4912	318	21	and	and	CCONJ
ejpam-4912	318	22	h	h	NOUN
ejpam-4912	318	23	,	,	PUNCT
ejpam-4912	318	24	respectively	respectively	ADV
ejpam-4912	318	25	.	.	PUNCT
ejpam-4912	319	1	let	let	VERB
ejpam-4912	319	2	x	x	SYM
ejpam-4912	319	3	∈	∈	PROPN
ejpam-4912	319	4	v	v	X
ejpam-4912	319	5	(	(	PUNCT
ejpam-4912	319	6	g	g	NOUN
ejpam-4912	319	7	)	)	PUNCT
ejpam-4912	319	8	and	and	CCONJ
ejpam-4912	319	9	y	y	PROPN
ejpam-4912	319	10	∈	∈	PROPN
ejpam-4912	319	11	v	v	ADP
ejpam-4912	319	12	(	(	PUNCT
ejpam-4912	319	13	h	h	NOUN
ejpam-4912	319	14	)	)	PUNCT
ejpam-4912	319	15	,	,	PUNCT
ejpam-4912	319	16	where	where	SCONJ
ejpam-4912	319	17	x	x	X
ejpam-4912	319	18	/∈	/∈	PUNCT
ejpam-4912	319	19	ng(a	ng(a	NUM
ejpam-4912	319	20	)	)	PUNCT
ejpam-4912	319	21	and	and	CCONJ
ejpam-4912	319	22	y	y	PROPN
ejpam-4912	319	23	/∈	/∈	PUNCT
ejpam-4912	319	24	nh(b	nh(b	NUM
ejpam-4912	319	25	)	)	PUNCT
ejpam-4912	319	26	.	.	PUNCT
ejpam-4912	320	1	then	then	ADV
ejpam-4912	320	2	y	y	PROPN
ejpam-4912	320	3	∈	∈	PROPN
ejpam-4912	320	4	ng+h(a	ng+h(a	PROPN
ejpam-4912	320	5	)	)	PUNCT
ejpam-4912	320	6	\	\	NOUN
ejpam-4912	321	1	ng+h(b	ng+h(b	PROPN
ejpam-4912	321	2	)	)	PUNCT
ejpam-4912	322	1	and	and	CCONJ
ejpam-4912	322	2	x	x	PUNCT
ejpam-4912	322	3	∈	∈	PROPN
ejpam-4912	322	4	ng+h(b)\ng+h(a	ng+h(b)\ng+h(a	NOUN
ejpam-4912	322	5	)	)	PUNCT
ejpam-4912	322	6	.	.	PUNCT
ejpam-4912	323	1	thus	thus	ADV
ejpam-4912	323	2	,	,	PUNCT
ejpam-4912	323	3	m	m	VERB
ejpam-4912	323	4	is	be	AUX
ejpam-4912	323	5	a	a	DET
ejpam-4912	323	6	j	j	NOUN
ejpam-4912	323	7	-	-	ADJ
ejpam-4912	323	8	open	open	ADJ
ejpam-4912	323	9	set	set	NOUN
ejpam-4912	323	10	in	in	ADP
ejpam-4912	323	11	g+h	g+h	PROPN
ejpam-4912	323	12	.	.	PUNCT
ejpam-4912	324	1	consequently	consequently	ADV
ejpam-4912	324	2	,	,	PUNCT
ejpam-4912	324	3	d	d	PROPN
ejpam-4912	324	4	is	be	AUX
ejpam-4912	324	5	a	a	DET
ejpam-4912	324	6	j	j	PROPN
ejpam-4912	324	7	-	-	ADJ
ejpam-4912	324	8	total	total	ADJ
ejpam-4912	324	9	dominating	dominating	NOUN
ejpam-4912	324	10	set	set	VERB
ejpam-4912	324	11	in	in	ADP
ejpam-4912	324	12	g+h	g+h	PROPN
ejpam-4912	324	13	.	.	PUNCT
ejpam-4912	325	1	the	the	DET
ejpam-4912	325	2	following	following	ADJ
ejpam-4912	325	3	result	result	NOUN
ejpam-4912	325	4	follows	follow	VERB
ejpam-4912	325	5	immediately	immediately	ADV
ejpam-4912	325	6	from	from	ADP
ejpam-4912	325	7	theorem	theorem	ADJ
ejpam-4912	325	8	7	7	NUM
ejpam-4912	325	9	.	.	PUNCT
ejpam-4912	325	10	corollary	corollary	ADJ
ejpam-4912	325	11	3	3	X
ejpam-4912	325	12	.	.	PUNCT
ejpam-4912	326	1	let	let	VERB
ejpam-4912	326	2	g	g	NOUN
ejpam-4912	326	3	and	and	CCONJ
ejpam-4912	326	4	h	h	NOUN
ejpam-4912	326	5	be	be	VERB
ejpam-4912	326	6	two	two	NUM
ejpam-4912	326	7	graphs	graph	NOUN
ejpam-4912	326	8	with	with	ADP
ejpam-4912	326	9	no	no	DET
ejpam-4912	326	10	isolated	isolated	ADJ
ejpam-4912	326	11	vertices	vertex	NOUN
ejpam-4912	326	12	.	.	PUNCT
ejpam-4912	327	1	then	then	ADV
ejpam-4912	327	2	γjt(g+h	γjt(g+h	NOUN
ejpam-4912	327	3	)	)	PUNCT
ejpam-4912	327	4	=	=	SYM
ejpam-4912	327	5	γjt(g	γjt(g	PROPN
ejpam-4912	327	6	)	)	PUNCT
ejpam-4912	327	7	+	+	CCONJ
ejpam-4912	327	8	γjt(h	γjt(h	PROPN
ejpam-4912	327	9	)	)	PUNCT
ejpam-4912	327	10	.	.	PUNCT
ejpam-4912	328	1	j.a	j.a	PROPN
ejpam-4912	328	2	.	.	PROPN
ejpam-4912	328	3	hassan	hassan	PROPN
ejpam-4912	328	4	et	et	PROPN
ejpam-4912	328	5	al	al	PROPN
ejpam-4912	328	6	.	.	PUNCT
ejpam-4912	328	7	/	/	SYM
ejpam-4912	328	8	eur	eur	PROPN
ejpam-4912	328	9	.	.	PUNCT
ejpam-4912	329	1	j.	j.	PROPN
ejpam-4912	329	2	pure	pure	PROPN
ejpam-4912	329	3	appl	appl	PROPN
ejpam-4912	329	4	.	.	PROPN
ejpam-4912	329	5	math	math	PROPN
ejpam-4912	329	6	,	,	PUNCT
ejpam-4912	329	7	16	16	NUM
ejpam-4912	329	8	(	(	PUNCT
ejpam-4912	329	9	4	4	NUM
ejpam-4912	329	10	)	)	PUNCT
ejpam-4912	329	11	(	(	PUNCT
ejpam-4912	329	12	2023	2023	NUM
ejpam-4912	329	13	)	)	PUNCT
ejpam-4912	329	14	,	,	PUNCT
ejpam-4912	329	15	2106	2106	NUM
ejpam-4912	329	16	-	-	SYM
ejpam-4912	329	17	2117	2117	NUM
ejpam-4912	329	18	2116	2116	NUM
ejpam-4912	329	19	theorem	theorem	VERB
ejpam-4912	329	20	8	8	NUM
ejpam-4912	329	21	.	.	PUNCT
ejpam-4912	330	1	let	let	VERB
ejpam-4912	330	2	g	g	PRON
ejpam-4912	330	3	be	be	AUX
ejpam-4912	330	4	a	a	DET
ejpam-4912	330	5	connected	connected	ADJ
ejpam-4912	330	6	non	non	ADJ
ejpam-4912	330	7	-	-	ADJ
ejpam-4912	330	8	trivial	trivial	ADJ
ejpam-4912	330	9	graph	graph	NOUN
ejpam-4912	330	10	and	and	CCONJ
ejpam-4912	330	11	h	h	NOUN
ejpam-4912	330	12	be	be	AUX
ejpam-4912	330	13	a	a	DET
ejpam-4912	330	14	graph	graph	NOUN
ejpam-4912	330	15	with	with	ADP
ejpam-4912	330	16	no	no	DET
ejpam-4912	330	17	isolated	isolated	ADJ
ejpam-4912	330	18	vertex	vertex	NOUN
ejpam-4912	330	19	.	.	PUNCT
ejpam-4912	331	1	if	if	SCONJ
ejpam-4912	331	2	m	m	PROPN
ejpam-4912	331	3	=	=	SYM
ejpam-4912	331	4	v	v	ADJ
ejpam-4912	331	5	(	(	PUNCT
ejpam-4912	331	6	g	g	NOUN
ejpam-4912	331	7	)	)	PUNCT
ejpam-4912	331	8	∪	∪	NOUN
ejpam-4912	331	9	(	(	PUNCT
ejpam-4912	331	10	⋃	⋃	ADJ
ejpam-4912	331	11	v∈v	v∈v	NOUN
ejpam-4912	331	12	(	(	PUNCT
ejpam-4912	331	13	g)mv	g)mv	PROPN
ejpam-4912	331	14	)	)	PUNCT
ejpam-4912	331	15	,	,	PUNCT
ejpam-4912	331	16	where	where	SCONJ
ejpam-4912	331	17	mv	mv	PROPN
ejpam-4912	331	18	is	be	AUX
ejpam-4912	331	19	a	a	DET
ejpam-4912	331	20	j	j	PROPN
ejpam-4912	331	21	-	-	ADJ
ejpam-4912	331	22	total	total	ADJ
ejpam-4912	331	23	dominating	dominating	NOUN
ejpam-4912	331	24	set	set	VERB
ejpam-4912	331	25	in	in	ADP
ejpam-4912	331	26	hv	hv	PROPN
ejpam-4912	331	27	for	for	ADP
ejpam-4912	331	28	each	each	DET
ejpam-4912	331	29	v	v	NUM
ejpam-4912	331	30	∈	∈	PROPN
ejpam-4912	331	31	v	v	NOUN
ejpam-4912	331	32	(	(	PUNCT
ejpam-4912	331	33	g	g	NOUN
ejpam-4912	331	34	)	)	PUNCT
ejpam-4912	331	35	,	,	PUNCT
ejpam-4912	331	36	then	then	ADV
ejpam-4912	331	37	m	m	PROPN
ejpam-4912	331	38	is	be	AUX
ejpam-4912	331	39	a	a	DET
ejpam-4912	331	40	j	j	PROPN
ejpam-4912	331	41	-	-	ADJ
ejpam-4912	331	42	total	total	ADJ
ejpam-4912	331	43	dominating	dominating	NOUN
ejpam-4912	331	44	set	set	VERB
ejpam-4912	331	45	in	in	ADP
ejpam-4912	331	46	g	g	PROPN
ejpam-4912	331	47	◦	◦	NOUN
ejpam-4912	331	48	h.	h.	NOUN
ejpam-4912	331	49	moreover	moreover	ADV
ejpam-4912	331	50	,	,	PUNCT
ejpam-4912	331	51	γjt(g	γjt(g	PROPN
ejpam-4912	331	52	◦	◦	NOUN
ejpam-4912	331	53	h	h	NOUN
ejpam-4912	331	54	)	)	PUNCT
ejpam-4912	331	55	≥	≥	NOUN
ejpam-4912	331	56	|v	|v	PROPN
ejpam-4912	331	57	(	(	PUNCT
ejpam-4912	331	58	g)|+	g)|+	PROPN
ejpam-4912	331	59	γjt(h	γjt(h	PROPN
ejpam-4912	331	60	)	)	PUNCT
ejpam-4912	331	61	·	·	PUNCT
ejpam-4912	332	1	|v	|v	PROPN
ejpam-4912	332	2	(	(	PUNCT
ejpam-4912	332	3	g)|	g)|	NOUN
ejpam-4912	332	4	.	.	PUNCT
ejpam-4912	332	5	proof	proof	NOUN
ejpam-4912	332	6	.	.	PUNCT
ejpam-4912	333	1	let	let	VERB
ejpam-4912	333	2	m	m	PROPN
ejpam-4912	333	3	=	=	VERB
ejpam-4912	333	4	v	v	ADJ
ejpam-4912	333	5	(	(	PUNCT
ejpam-4912	333	6	g	g	NOUN
ejpam-4912	333	7	)	)	PUNCT
ejpam-4912	333	8	∪	∪	NOUN
ejpam-4912	333	9	(	(	PUNCT
ejpam-4912	333	10	⋃	⋃	ADJ
ejpam-4912	333	11	v∈v	v∈v	NOUN
ejpam-4912	333	12	(	(	PUNCT
ejpam-4912	333	13	g)mv	g)mv	PROPN
ejpam-4912	333	14	)	)	PUNCT
ejpam-4912	333	15	,	,	PUNCT
ejpam-4912	333	16	where	where	SCONJ
ejpam-4912	333	17	mv	mv	PROPN
ejpam-4912	333	18	is	be	AUX
ejpam-4912	333	19	a	a	DET
ejpam-4912	333	20	j	j	PROPN
ejpam-4912	333	21	-	-	ADJ
ejpam-4912	333	22	total	total	ADJ
ejpam-4912	333	23	dominating	dominating	NOUN
ejpam-4912	333	24	set	set	VERB
ejpam-4912	333	25	in	in	ADP
ejpam-4912	333	26	hv	hv	PROPN
ejpam-4912	333	27	for	for	ADP
ejpam-4912	333	28	each	each	DET
ejpam-4912	333	29	v	v	NUM
ejpam-4912	333	30	∈	∈	PROPN
ejpam-4912	333	31	v	v	NOUN
ejpam-4912	333	32	(	(	PUNCT
ejpam-4912	333	33	g	g	NOUN
ejpam-4912	333	34	)	)	PUNCT
ejpam-4912	333	35	.	.	PUNCT
ejpam-4912	334	1	since	since	SCONJ
ejpam-4912	334	2	g	g	PROPN
ejpam-4912	334	3	is	be	AUX
ejpam-4912	334	4	connected	connect	VERB
ejpam-4912	334	5	,	,	PUNCT
ejpam-4912	334	6	it	it	PRON
ejpam-4912	334	7	follows	follow	VERB
ejpam-4912	334	8	that	that	SCONJ
ejpam-4912	334	9	v	v	X
ejpam-4912	334	10	(	(	PUNCT
ejpam-4912	334	11	g	g	NOUN
ejpam-4912	334	12	)	)	PUNCT
ejpam-4912	334	13	is	be	AUX
ejpam-4912	334	14	a	a	DET
ejpam-4912	334	15	total	total	ADJ
ejpam-4912	334	16	dominating	dominating	NOUN
ejpam-4912	334	17	set	set	VERB
ejpam-4912	334	18	in	in	ADP
ejpam-4912	334	19	g.	g.	PROPN
ejpam-4912	334	20	moreover	moreover	ADV
ejpam-4912	334	21	,	,	PUNCT
ejpam-4912	334	22	since	since	SCONJ
ejpam-4912	334	23	mv	mv	PROPN
ejpam-4912	334	24	is	be	AUX
ejpam-4912	334	25	a	a	DET
ejpam-4912	334	26	total	total	ADJ
ejpam-4912	334	27	dominating	dominating	NOUN
ejpam-4912	334	28	set	set	NOUN
ejpam-4912	334	29	in	in	ADP
ejpam-4912	334	30	hv	hv	PROPN
ejpam-4912	334	31	for	for	ADP
ejpam-4912	334	32	each	each	DET
ejpam-4912	334	33	v	v	NUM
ejpam-4912	334	34	∈	∈	PROPN
ejpam-4912	334	35	v	v	NOUN
ejpam-4912	334	36	(	(	PUNCT
ejpam-4912	334	37	g	g	NOUN
ejpam-4912	334	38	)	)	PUNCT
ejpam-4912	334	39	,	,	PUNCT
ejpam-4912	334	40	m	m	VERB
ejpam-4912	334	41	is	be	AUX
ejpam-4912	334	42	a	a	DET
ejpam-4912	334	43	total	total	ADJ
ejpam-4912	334	44	dominating	dominating	NOUN
ejpam-4912	334	45	set	set	VERB
ejpam-4912	334	46	in	in	ADP
ejpam-4912	334	47	g	g	PROPN
ejpam-4912	334	48	◦	◦	NOUN
ejpam-4912	334	49	h.	h.	PROPN
ejpam-4912	334	50	now	now	ADV
ejpam-4912	334	51	,	,	PUNCT
ejpam-4912	334	52	let	let	VERB
ejpam-4912	334	53	a	a	DET
ejpam-4912	334	54	,	,	PUNCT
ejpam-4912	334	55	b	b	X
ejpam-4912	334	56	∈	∈	ADV
ejpam-4912	334	57	m	m	VERB
ejpam-4912	334	58	.	.	PUNCT
ejpam-4912	335	1	if	if	SCONJ
ejpam-4912	335	2	a	a	PRON
ejpam-4912	335	3	,	,	PUNCT
ejpam-4912	335	4	b	b	X
ejpam-4912	335	5	∈	∈	NOUN
ejpam-4912	335	6	mu	mu	NOUN
ejpam-4912	335	7	for	for	ADP
ejpam-4912	335	8	some	some	PRON
ejpam-4912	335	9	u	u	NOUN
ejpam-4912	335	10	∈	∈	PROPN
ejpam-4912	335	11	v	v	NOUN
ejpam-4912	335	12	(	(	PUNCT
ejpam-4912	335	13	g	g	NOUN
ejpam-4912	335	14	)	)	PUNCT
ejpam-4912	335	15	,	,	PUNCT
ejpam-4912	335	16	then	then	ADV
ejpam-4912	335	17	nh(a	nh(a	NUM
ejpam-4912	335	18	)	)	PUNCT
ejpam-4912	335	19	\nh(b	\nh(b	NUM
ejpam-4912	335	20	)	)	PUNCT
ejpam-4912	335	21	̸=	̸=	PROPN
ejpam-4912	335	22	∅	∅	NOUN
ejpam-4912	335	23	and	and	CCONJ
ejpam-4912	335	24	nh(b	nh(b	NUM
ejpam-4912	335	25	)	)	PUNCT
ejpam-4912	335	26	\nh(a	\nh(a	X
ejpam-4912	335	27	)	)	PUNCT
ejpam-4912	335	28	̸=	̸=	PROPN
ejpam-4912	335	29	∅.	∅.	NOUN
ejpam-4912	335	30	it	it	PRON
ejpam-4912	335	31	follows	follow	VERB
ejpam-4912	335	32	that	that	SCONJ
ejpam-4912	335	33	ng	ng	PROPN
ejpam-4912	335	34	◦	◦	PROPN
ejpam-4912	335	35	h(a	h(a	PROPN
ejpam-4912	335	36	)	)	PUNCT
ejpam-4912	335	37	\ng	\ng	PROPN
ejpam-4912	335	38	◦	◦	NOUN
ejpam-4912	335	39	h(b	h(b	NOUN
ejpam-4912	335	40	)	)	PUNCT
ejpam-4912	335	41	̸=	̸=	PROPN
ejpam-4912	335	42	∅	∅	NOUN
ejpam-4912	335	43	and	and	CCONJ
ejpam-4912	335	44	ng	ng	PROPN
ejpam-4912	335	45	◦	◦	NOUN
ejpam-4912	335	46	h(b)\ng	h(b)\ng	PROPN
ejpam-4912	335	47	◦	◦	NOUN
ejpam-4912	335	48	h(a	h(a	PROPN
ejpam-4912	335	49	)	)	PUNCT
ejpam-4912	335	50	̸=	̸=	PROPN
ejpam-4912	335	51	∅.	∅.	ADP
ejpam-4912	335	52	thus	thus	ADV
ejpam-4912	335	53	,	,	PUNCT
ejpam-4912	335	54	m	m	PROPN
ejpam-4912	335	55	is	be	AUX
ejpam-4912	335	56	a	a	DET
ejpam-4912	335	57	j	j	NOUN
ejpam-4912	335	58	-	-	ADJ
ejpam-4912	335	59	open	open	ADJ
ejpam-4912	335	60	set	set	NOUN
ejpam-4912	335	61	in	in	ADP
ejpam-4912	335	62	g	g	PROPN
ejpam-4912	335	63	◦	◦	NOUN
ejpam-4912	335	64	h.	h.	NOUN
ejpam-4912	335	65	similarly	similarly	ADV
ejpam-4912	335	66	,	,	PUNCT
ejpam-4912	335	67	if	if	SCONJ
ejpam-4912	335	68	a	a	DET
ejpam-4912	335	69	,	,	PUNCT
ejpam-4912	335	70	b	b	PROPN
ejpam-4912	335	71	∈	∈	PROPN
ejpam-4912	335	72	v	v	NOUN
ejpam-4912	335	73	(	(	PUNCT
ejpam-4912	335	74	g	g	NOUN
ejpam-4912	335	75	)	)	PUNCT
ejpam-4912	335	76	,	,	PUNCT
ejpam-4912	335	77	then	then	ADV
ejpam-4912	335	78	m	m	PROPN
ejpam-4912	335	79	is	be	AUX
ejpam-4912	335	80	a	a	DET
ejpam-4912	335	81	j	j	NOUN
ejpam-4912	335	82	-	-	ADJ
ejpam-4912	335	83	open	open	ADJ
ejpam-4912	335	84	set	set	NOUN
ejpam-4912	335	85	in	in	ADP
ejpam-4912	335	86	g	g	PROPN
ejpam-4912	335	87	◦	◦	PROPN
ejpam-4912	335	88	h.	h.	NOUN
ejpam-4912	335	89	assume	assume	VERB
ejpam-4912	335	90	that	that	SCONJ
ejpam-4912	335	91	a	a	DET
ejpam-4912	335	92	∈	∈	PROPN
ejpam-4912	335	93	ms	ms	NOUN
ejpam-4912	335	94	and	and	CCONJ
ejpam-4912	335	95	b	b	PROPN
ejpam-4912	335	96	∈	∈	PROPN
ejpam-4912	335	97	mt	mt	PROPN
ejpam-4912	335	98	for	for	ADP
ejpam-4912	335	99	some	some	DET
ejpam-4912	335	100	s	s	PROPN
ejpam-4912	335	101	,	,	PUNCT
ejpam-4912	335	102	t	t	PROPN
ejpam-4912	335	103	∈	∈	PROPN
ejpam-4912	335	104	v	v	ADP
ejpam-4912	335	105	(	(	PUNCT
ejpam-4912	335	106	g	g	NOUN
ejpam-4912	335	107	)	)	PUNCT
ejpam-4912	335	108	,	,	PUNCT
ejpam-4912	335	109	s	s	VERB
ejpam-4912	335	110	̸=	̸=	PROPN
ejpam-4912	335	111	t.	t.	NOUN
ejpam-4912	335	112	then	then	ADV
ejpam-4912	335	113	s	s	VERB
ejpam-4912	335	114	∈	∈	PROPN
ejpam-4912	335	115	ng	ng	PROPN
ejpam-4912	335	116	◦	◦	PROPN
ejpam-4912	335	117	h(a	h(a	PROPN
ejpam-4912	335	118	)	)	PUNCT
ejpam-4912	335	119	\	\	PROPN
ejpam-4912	336	1	ng	ng	PROPN
ejpam-4912	336	2	◦	◦	PROPN
ejpam-4912	336	3	h(b	h(b	PROPN
ejpam-4912	336	4	)	)	PUNCT
ejpam-4912	337	1	and	and	CCONJ
ejpam-4912	337	2	t	t	PROPN
ejpam-4912	337	3	∈	∈	PROPN
ejpam-4912	337	4	ng	ng	PROPN
ejpam-4912	337	5	◦	◦	PROPN
ejpam-4912	337	6	h(b	h(b	PROPN
ejpam-4912	337	7	)	)	PUNCT
ejpam-4912	337	8	\	\	PROPN
ejpam-4912	337	9	ng	ng	PROPN
ejpam-4912	337	10	◦	◦	PROPN
ejpam-4912	337	11	h(a	h(a	PROPN
ejpam-4912	337	12	)	)	PUNCT
ejpam-4912	337	13	,	,	PUNCT
ejpam-4912	337	14	hence	hence	ADV
ejpam-4912	337	15	we	we	PRON
ejpam-4912	337	16	are	be	AUX
ejpam-4912	337	17	done	do	VERB
ejpam-4912	337	18	.	.	PUNCT
ejpam-4912	338	1	suppose	suppose	VERB
ejpam-4912	338	2	that	that	SCONJ
ejpam-4912	338	3	a	a	DET
ejpam-4912	338	4	∈	∈	NOUN
ejpam-4912	338	5	mw	mw	NOUN
ejpam-4912	338	6	for	for	ADP
ejpam-4912	338	7	some	some	DET
ejpam-4912	338	8	w	w	PROPN
ejpam-4912	338	9	∈	∈	PROPN
ejpam-4912	338	10	v	v	ADP
ejpam-4912	338	11	(	(	PUNCT
ejpam-4912	338	12	g	g	NOUN
ejpam-4912	338	13	)	)	PUNCT
ejpam-4912	338	14	and	and	CCONJ
ejpam-4912	338	15	b	b	X
ejpam-4912	338	16	∈	∈	NOUN
ejpam-4912	338	17	v	v	NOUN
ejpam-4912	338	18	(	(	PUNCT
ejpam-4912	338	19	g	g	NOUN
ejpam-4912	338	20	)	)	PUNCT
ejpam-4912	338	21	.	.	PUNCT
ejpam-4912	339	1	if	if	SCONJ
ejpam-4912	339	2	w	w	PROPN
ejpam-4912	339	3	=	=	SYM
ejpam-4912	339	4	b	b	PROPN
ejpam-4912	339	5	,	,	PUNCT
ejpam-4912	339	6	then	then	ADV
ejpam-4912	339	7	b	b	PROPN
ejpam-4912	339	8	∈	∈	PROPN
ejpam-4912	339	9	ng	ng	PROPN
ejpam-4912	339	10	◦	◦	PROPN
ejpam-4912	339	11	h(a	h(a	PROPN
ejpam-4912	339	12	)	)	PUNCT
ejpam-4912	339	13	\	\	PROPN
ejpam-4912	340	1	ng	ng	PROPN
ejpam-4912	340	2	◦	◦	PROPN
ejpam-4912	340	3	h(b	h(b	PROPN
ejpam-4912	340	4	)	)	PUNCT
ejpam-4912	340	5	and	and	CCONJ
ejpam-4912	340	6	a	a	DET
ejpam-4912	340	7	∈	∈	PROPN
ejpam-4912	340	8	ng	ng	PROPN
ejpam-4912	340	9	◦	◦	NOUN
ejpam-4912	340	10	h(b	h(b	PROPN
ejpam-4912	340	11	)	)	PUNCT
ejpam-4912	340	12	\	\	PROPN
ejpam-4912	341	1	ng	ng	PROPN
ejpam-4912	341	2	◦	◦	PROPN
ejpam-4912	341	3	h(a	h(a	PROPN
ejpam-4912	341	4	)	)	PUNCT
ejpam-4912	342	1	,	,	PUNCT
ejpam-4912	342	2	and	and	CCONJ
ejpam-4912	342	3	we	we	PRON
ejpam-4912	342	4	are	be	AUX
ejpam-4912	342	5	done	do	VERB
ejpam-4912	342	6	.	.	PUNCT
ejpam-4912	343	1	suppose	suppose	VERB
ejpam-4912	343	2	w	w	ADP
ejpam-4912	343	3	̸=	̸=	PROPN
ejpam-4912	343	4	b.	b.	NOUN
ejpam-4912	343	5	since	since	SCONJ
ejpam-4912	343	6	h	h	PROPN
ejpam-4912	343	7	is	be	AUX
ejpam-4912	343	8	graph	graph	VERB
ejpam-4912	343	9	with	with	ADP
ejpam-4912	343	10	no	no	DET
ejpam-4912	343	11	isolated	isolated	ADJ
ejpam-4912	343	12	vertex	vertex	NOUN
ejpam-4912	343	13	,	,	PUNCT
ejpam-4912	343	14	there	there	PRON
ejpam-4912	343	15	exists	exist	VERB
ejpam-4912	343	16	q	q	PROPN
ejpam-4912	343	17	∈	∈	PROPN
ejpam-4912	343	18	hw	hw	VERB
ejpam-4912	343	19	such	such	ADJ
ejpam-4912	343	20	that	that	SCONJ
ejpam-4912	343	21	q	q	PROPN
ejpam-4912	343	22	∈	∈	PROPN
ejpam-4912	343	23	ng	ng	PROPN
ejpam-4912	343	24	◦	◦	PROPN
ejpam-4912	343	25	h(a	h(a	PROPN
ejpam-4912	343	26	)	)	PUNCT
ejpam-4912	343	27	\	\	PROPN
ejpam-4912	343	28	ng	ng	PROPN
ejpam-4912	343	29	◦	◦	PROPN
ejpam-4912	343	30	h(b	h(b	PROPN
ejpam-4912	343	31	)	)	PUNCT
ejpam-4912	343	32	.	.	PUNCT
ejpam-4912	344	1	clearly	clearly	ADV
ejpam-4912	344	2	,	,	PUNCT
ejpam-4912	344	3	ng	ng	PROPN
ejpam-4912	344	4	◦	◦	PROPN
ejpam-4912	344	5	h(b	h(b	PROPN
ejpam-4912	344	6	)	)	PUNCT
ejpam-4912	344	7	\ng	\ng	PROPN
ejpam-4912	344	8	◦	◦	NOUN
ejpam-4912	344	9	h(a	h(a	PROPN
ejpam-4912	344	10	)	)	PUNCT
ejpam-4912	345	1	=	=	PUNCT
ejpam-4912	346	1	mb	mb	ADP
ejpam-4912	346	2	̸=	̸=	PROPN
ejpam-4912	346	3	∅.	∅.	VERB
ejpam-4912	346	4	therefore	therefore	ADV
ejpam-4912	346	5	,	,	PUNCT
ejpam-4912	346	6	m	m	PROPN
ejpam-4912	346	7	is	be	AUX
ejpam-4912	346	8	a	a	DET
ejpam-4912	346	9	j	j	NOUN
ejpam-4912	346	10	-	-	ADJ
ejpam-4912	346	11	open	open	ADJ
ejpam-4912	346	12	set	set	NOUN
ejpam-4912	346	13	in	in	ADP
ejpam-4912	346	14	g	g	PROPN
ejpam-4912	346	15	◦	◦	NOUN
ejpam-4912	346	16	h	h	NOUN
ejpam-4912	346	17	,	,	PUNCT
ejpam-4912	346	18	showing	show	VERB
ejpam-4912	346	19	that	that	SCONJ
ejpam-4912	346	20	m	m	NOUN
ejpam-4912	346	21	is	be	AUX
ejpam-4912	346	22	a	a	DET
ejpam-4912	346	23	j	j	PROPN
ejpam-4912	346	24	-	-	ADJ
ejpam-4912	346	25	total	total	ADJ
ejpam-4912	346	26	dominating	dominating	NOUN
ejpam-4912	346	27	set	set	VERB
ejpam-4912	346	28	in	in	ADP
ejpam-4912	346	29	g	g	PROPN
ejpam-4912	346	30	◦	◦	NOUN
ejpam-4912	346	31	h.	h.	PROPN
ejpam-4912	346	32	consequently	consequently	ADV
ejpam-4912	346	33	,	,	PUNCT
ejpam-4912	346	34	γjt(g	γjt(g	PROPN
ejpam-4912	346	35	◦	◦	NOUN
ejpam-4912	346	36	h	h	NOUN
ejpam-4912	346	37	)	)	PUNCT
ejpam-4912	346	38	≥	≥	NOUN
ejpam-4912	346	39	|v	|v	PROPN
ejpam-4912	346	40	(	(	PUNCT
ejpam-4912	346	41	g)|+	g)|+	PROPN
ejpam-4912	346	42	γjt(h	γjt(h	PROPN
ejpam-4912	346	43	)	)	PUNCT
ejpam-4912	346	44	·	·	PUNCT
ejpam-4912	347	1	|v	|v	PROPN
ejpam-4912	347	2	(	(	PUNCT
ejpam-4912	347	3	g)|	g)|	NOUN
ejpam-4912	347	4	.	.	PUNCT
ejpam-4912	348	1	4	4	X
ejpam-4912	348	2	.	.	X
ejpam-4912	348	3	conclusion	conclusion	VERB
ejpam-4912	348	4	the	the	DET
ejpam-4912	348	5	concept	concept	NOUN
ejpam-4912	348	6	of	of	ADP
ejpam-4912	348	7	j	j	PROPN
ejpam-4912	348	8	-	-	ADJ
ejpam-4912	348	9	total	total	ADJ
ejpam-4912	348	10	domination	domination	NOUN
ejpam-4912	348	11	has	have	AUX
ejpam-4912	348	12	been	be	AUX
ejpam-4912	348	13	introduced	introduce	VERB
ejpam-4912	348	14	and	and	CCONJ
ejpam-4912	348	15	investigated	investigate	VERB
ejpam-4912	348	16	in	in	ADP
ejpam-4912	348	17	this	this	DET
ejpam-4912	348	18	study	study	NOUN
ejpam-4912	348	19	.	.	PUNCT
ejpam-4912	349	1	characterizations	characterization	NOUN
ejpam-4912	349	2	of	of	ADP
ejpam-4912	349	3	j	j	PROPN
ejpam-4912	349	4	-	-	ADJ
ejpam-4912	349	5	total	total	ADJ
ejpam-4912	349	6	dominating	dominating	NOUN
ejpam-4912	349	7	sets	set	NOUN
ejpam-4912	349	8	in	in	ADP
ejpam-4912	349	9	some	some	DET
ejpam-4912	349	10	graphs	graph	NOUN
ejpam-4912	349	11	and	and	CCONJ
ejpam-4912	349	12	join	join	NOUN
ejpam-4912	349	13	of	of	ADP
ejpam-4912	349	14	two	two	NUM
ejpam-4912	349	15	graphs	graph	NOUN
ejpam-4912	349	16	are	be	AUX
ejpam-4912	349	17	formulated	formulate	VERB
ejpam-4912	349	18	and	and	CCONJ
ejpam-4912	349	19	were	be	AUX
ejpam-4912	349	20	used	use	VERB
ejpam-4912	349	21	to	to	PART
ejpam-4912	349	22	solve	solve	VERB
ejpam-4912	349	23	exact	exact	ADJ
ejpam-4912	349	24	values	value	NOUN
ejpam-4912	349	25	of	of	ADP
ejpam-4912	349	26	the	the	DET
ejpam-4912	349	27	parameters	parameter	NOUN
ejpam-4912	349	28	of	of	ADP
ejpam-4912	349	29	these	these	DET
ejpam-4912	349	30	graphs	graph	NOUN
ejpam-4912	349	31	.	.	PUNCT
ejpam-4912	350	1	some	some	DET
ejpam-4912	350	2	bounds	bound	NOUN
ejpam-4912	350	3	and	and	CCONJ
ejpam-4912	350	4	relationships	relationship	NOUN
ejpam-4912	350	5	of	of	ADP
ejpam-4912	350	6	this	this	DET
ejpam-4912	350	7	newly	newly	ADV
ejpam-4912	350	8	defined	define	VERB
ejpam-4912	350	9	parameter	parameter	NOUN
ejpam-4912	350	10	have	have	AUX
ejpam-4912	350	11	been	be	AUX
ejpam-4912	350	12	established	establish	VERB
ejpam-4912	350	13	.	.	PUNCT
ejpam-4912	351	1	other	other	ADJ
ejpam-4912	351	2	graphs	graph	NOUN
ejpam-4912	351	3	that	that	PRON
ejpam-4912	351	4	were	be	AUX
ejpam-4912	351	5	not	not	PART
ejpam-4912	351	6	considered	consider	VERB
ejpam-4912	351	7	in	in	ADP
ejpam-4912	351	8	this	this	DET
ejpam-4912	351	9	study	study	NOUN
ejpam-4912	351	10	could	could	AUX
ejpam-4912	351	11	be	be	AUX
ejpam-4912	351	12	an	an	DET
ejpam-4912	351	13	interesting	interesting	ADJ
ejpam-4912	351	14	topic	topic	NOUN
ejpam-4912	351	15	to	to	PART
ejpam-4912	351	16	consider	consider	VERB
ejpam-4912	351	17	by	by	ADP
ejpam-4912	351	18	researchers	researcher	NOUN
ejpam-4912	351	19	for	for	ADP
ejpam-4912	351	20	further	further	ADJ
ejpam-4912	351	21	investigation	investigation	NOUN
ejpam-4912	351	22	of	of	ADP
ejpam-4912	351	23	the	the	DET
ejpam-4912	351	24	concept	concept	NOUN
ejpam-4912	351	25	.	.	PUNCT
ejpam-4912	352	1	they	they	PRON
ejpam-4912	352	2	may	may	AUX
ejpam-4912	352	3	also	also	ADV
ejpam-4912	352	4	consider	consider	VERB
ejpam-4912	352	5	the	the	DET
ejpam-4912	352	6	bounds	bound	NOUN
ejpam-4912	352	7	of	of	ADP
ejpam-4912	352	8	the	the	DET
ejpam-4912	352	9	parameter	parameter	NOUN
ejpam-4912	352	10	with	with	ADP
ejpam-4912	352	11	respect	respect	NOUN
ejpam-4912	352	12	to	to	ADP
ejpam-4912	352	13	other	other	ADJ
ejpam-4912	352	14	well	well	ADV
ejpam-4912	352	15	known	know	VERB
ejpam-4912	352	16	parameters	parameter	NOUN
ejpam-4912	352	17	in	in	ADP
ejpam-4912	352	18	graph	graph	NOUN
ejpam-4912	352	19	theory	theory	NOUN
ejpam-4912	352	20	.	.	PUNCT
ejpam-4912	353	1	acknowledgements	acknowledgement	NOUN
ejpam-4912	353	2	the	the	DET
ejpam-4912	353	3	authors	author	NOUN
ejpam-4912	353	4	would	would	AUX
ejpam-4912	353	5	like	like	VERB
ejpam-4912	353	6	to	to	PART
ejpam-4912	353	7	thank	thank	VERB
ejpam-4912	353	8	mindanao	mindanao	PROPN
ejpam-4912	353	9	state	state	PROPN
ejpam-4912	353	10	universitytawi	universitytawi	PROPN
ejpam-4912	353	11	-	-	PUNCT
ejpam-4912	353	12	tawi	tawi	NOUN
ejpam-4912	353	13	college	college	PROPN
ejpam-4912	353	14	of	of	ADP
ejpam-4912	353	15	technology	technology	NOUN
ejpam-4912	353	16	and	and	CCONJ
ejpam-4912	353	17	oceanography	oceanography	NOUN
ejpam-4912	353	18	for	for	ADP
ejpam-4912	353	19	funding	fund	VERB
ejpam-4912	353	20	this	this	DET
ejpam-4912	353	21	research	research	NOUN
ejpam-4912	353	22	.	.	PUNCT
ejpam-4912	354	1	moreover	moreover	ADV
ejpam-4912	354	2	,	,	PUNCT
ejpam-4912	354	3	the	the	DET
ejpam-4912	354	4	authors	author	NOUN
ejpam-4912	354	5	would	would	AUX
ejpam-4912	354	6	like	like	VERB
ejpam-4912	354	7	to	to	PART
ejpam-4912	354	8	thank	thank	VERB
ejpam-4912	354	9	the	the	DET
ejpam-4912	354	10	referees	referee	NOUN
ejpam-4912	354	11	for	for	ADP
ejpam-4912	354	12	their	their	PRON
ejpam-4912	354	13	invaluable	invaluable	ADJ
ejpam-4912	354	14	comments	comment	NOUN
ejpam-4912	354	15	and	and	CCONJ
ejpam-4912	354	16	suggestions	suggestion	NOUN
ejpam-4912	354	17	that	that	PRON
ejpam-4912	354	18	contributed	contribute	VERB
ejpam-4912	354	19	a	a	DET
ejpam-4912	354	20	lot	lot	NOUN
ejpam-4912	354	21	for	for	ADP
ejpam-4912	354	22	the	the	DET
ejpam-4912	354	23	improvement	improvement	NOUN
ejpam-4912	354	24	of	of	ADP
ejpam-4912	354	25	this	this	DET
ejpam-4912	354	26	paper	paper	NOUN
ejpam-4912	354	27	.	.	PUNCT
ejpam-4912	355	1	references	reference	NOUN
ejpam-4912	355	2	2117	2117	NUM
ejpam-4912	355	3	references	reference	NOUN
ejpam-4912	355	4	[	[	X
ejpam-4912	355	5	1	1	NUM
ejpam-4912	355	6	]	]	X
ejpam-4912	355	7	e.j	e.j	PROPN
ejpam-4912	355	8	.	.	PROPN
ejpam-4912	355	9	cockayne	cockayne	PROPN
ejpam-4912	355	10	and	and	CCONJ
ejpam-4912	355	11	s.t	s.t	PROPN
ejpam-4912	355	12	.	.	PROPN
ejpam-4912	355	13	hedetniemi	hedetniemi	PROPN
ejpam-4912	355	14	.	.	PUNCT
ejpam-4912	356	1	towards	towards	ADP
ejpam-4912	356	2	a	a	DET
ejpam-4912	356	3	theory	theory	NOUN
ejpam-4912	356	4	of	of	ADP
ejpam-4912	356	5	domination	domination	NOUN
ejpam-4912	356	6	in	in	ADP
ejpam-4912	356	7	graphs	graph	NOUN
ejpam-4912	356	8	.	.	PUNCT
ejpam-4912	357	1	networks	network	NOUN
ejpam-4912	357	2	,	,	PUNCT
ejpam-4912	357	3	,	,	PUNCT
ejpam-4912	357	4	7:247–261	7:247–261	NUM
ejpam-4912	357	5	,	,	PUNCT
ejpam-4912	357	6	1977	1977	NUM
ejpam-4912	357	7	.	.	PUNCT
ejpam-4912	358	1	[	[	X
ejpam-4912	358	2	2	2	NUM
ejpam-4912	358	3	]	]	PUNCT
ejpam-4912	358	4	b.	b.	PROPN
ejpam-4912	358	5	gayathri	gayathri	PROPN
ejpam-4912	358	6	and	and	CCONJ
ejpam-4912	358	7	s.	s.	PROPN
ejpam-4912	358	8	kaspar	kaspar	PROPN
ejpam-4912	358	9	.	.	PUNCT
ejpam-4912	359	1	connected	connect	VERB
ejpam-4912	359	2	co	co	ADJ
ejpam-4912	359	3	-	-	ADJ
ejpam-4912	359	4	independent	independent	ADJ
ejpam-4912	359	5	domination	domination	NOUN
ejpam-4912	359	6	of	of	ADP
ejpam-4912	359	7	a	a	DET
ejpam-4912	359	8	graph	graph	NOUN
ejpam-4912	359	9	.	.	PUNCT
ejpam-4912	359	10	intl	intl	PROPN
ejpam-4912	359	11	.	.	PUNCT
ejpam-4912	360	1	j.	j.	PROPN
ejpam-4912	360	2	contemp	contemp	PROPN
ejpam-4912	360	3	.	.	PUNCT
ejpam-4912	361	1	math	math	NOUN
ejpam-4912	361	2	.	.	PUNCT
ejpam-4912	362	1	sciences	science	NOUN
ejpam-4912	362	2	,	,	PUNCT
ejpam-4912	362	3	9(6):423–429	9(6):423–429	NUM
ejpam-4912	362	4	,	,	PUNCT
ejpam-4912	362	5	2011	2011	NUM
ejpam-4912	362	6	.	.	PUNCT
ejpam-4912	363	1	[	[	X
ejpam-4912	363	2	3	3	X
ejpam-4912	363	3	]	]	X
ejpam-4912	363	4	j.	j.	PROPN
ejpam-4912	363	5	hassan	hassan	PROPN
ejpam-4912	363	6	and	and	CCONJ
ejpam-4912	363	7	s.	s.	PROPN
ejpam-4912	363	8	canoy	canoy	PROPN
ejpam-4912	363	9	jr	jr	PROPN
ejpam-4912	363	10	.	.	PROPN
ejpam-4912	363	11	connected	connect	VERB
ejpam-4912	363	12	grundy	grundy	PROPN
ejpam-4912	363	13	hop	hop	NOUN
ejpam-4912	363	14	dominating	dominate	VERB
ejpam-4912	363	15	sequences	sequence	NOUN
ejpam-4912	363	16	in	in	ADP
ejpam-4912	363	17	graphs	graph	NOUN
ejpam-4912	363	18	.	.	PUNCT
ejpam-4912	364	1	eur	eur	PROPN
ejpam-4912	364	2	.	.	PUNCT
ejpam-4912	365	1	j.	j.	PROPN
ejpam-4912	365	2	pure	pure	PROPN
ejpam-4912	365	3	appl	appl	PROPN
ejpam-4912	365	4	.	.	PUNCT
ejpam-4912	365	5	math	math	PROPN
ejpam-4912	365	6	.	.	PUNCT
ejpam-4912	365	7	,	,	PUNCT
ejpam-4912	366	1	16(2):1212–1227	16(2):1212–1227	NUM
ejpam-4912	366	2	,	,	PUNCT
ejpam-4912	366	3	2023	2023	NUM
ejpam-4912	366	4	.	.	PUNCT
ejpam-4912	367	1	[	[	X
ejpam-4912	367	2	4	4	X
ejpam-4912	367	3	]	]	PUNCT
ejpam-4912	367	4	j.	j.	PROPN
ejpam-4912	367	5	hassan	hassan	PROPN
ejpam-4912	367	6	,	,	PUNCT
ejpam-4912	367	7	s.	s.	PROPN
ejpam-4912	367	8	canoy	canoy	PROPN
ejpam-4912	367	9	jr	jr	PROPN
ejpam-4912	367	10	.	.	PROPN
ejpam-4912	367	11	,	,	PUNCT
ejpam-4912	367	12	and	and	CCONJ
ejpam-4912	367	13	chrisley	chrisley	PROPN
ejpam-4912	367	14	jade	jade	NOUN
ejpam-4912	367	15	saromines	saromine	NOUN
ejpam-4912	367	16	.	.	PUNCT
ejpam-4912	368	1	convex	convex	VERB
ejpam-4912	368	2	hop	hop	NOUN
ejpam-4912	368	3	domination	domination	NOUN
ejpam-4912	368	4	in	in	ADP
ejpam-4912	368	5	graphs	graph	NOUN
ejpam-4912	368	6	.	.	PUNCT
ejpam-4912	369	1	eur	eur	PROPN
ejpam-4912	369	2	.	.	PUNCT
ejpam-4912	370	1	j.	j.	PROPN
ejpam-4912	370	2	pure	pure	PROPN
ejpam-4912	370	3	appl	appl	PROPN
ejpam-4912	370	4	.	.	PUNCT
ejpam-4912	370	5	math	math	PROPN
ejpam-4912	370	6	.	.	PUNCT
ejpam-4912	370	7	,	,	PUNCT
ejpam-4912	370	8	16(1):319–335	16(1):319–335	NOUN
ejpam-4912	370	9	,	,	PUNCT
ejpam-4912	370	10	2023	2023	NUM
ejpam-4912	370	11	.	.	PUNCT
ejpam-4912	371	1	[	[	X
ejpam-4912	371	2	5	5	X
ejpam-4912	371	3	]	]	PUNCT
ejpam-4912	371	4	j.	j.	PROPN
ejpam-4912	371	5	hassan	hassan	PROPN
ejpam-4912	371	6	,	,	PUNCT
ejpam-4912	371	7	a.	a.	PROPN
ejpam-4912	371	8	lintasan	lintasan	PROPN
ejpam-4912	371	9	,	,	PUNCT
ejpam-4912	371	10	and	and	CCONJ
ejpam-4912	371	11	n.h	n.h	PROPN
ejpam-4912	371	12	.	.	PUNCT
ejpam-4912	372	1	mohammad	mohammad	PROPN
ejpam-4912	372	2	.	.	PUNCT
ejpam-4912	373	1	some	some	DET
ejpam-4912	373	2	properties	property	NOUN
ejpam-4912	373	3	and	and	CCONJ
ejpam-4912	373	4	realization	realization	NOUN
ejpam-4912	373	5	problems	problem	NOUN
ejpam-4912	373	6	involving	involve	VERB
ejpam-4912	373	7	connected	connected	ADJ
ejpam-4912	373	8	outer	outer	ADJ
ejpam-4912	373	9	-	-	PUNCT
ejpam-4912	373	10	hop	hop	NOUN
ejpam-4912	373	11	independent	independent	ADJ
ejpam-4912	373	12	hop	hop	NOUN
ejpam-4912	373	13	domination	domination	NOUN
ejpam-4912	373	14	in	in	ADP
ejpam-4912	373	15	graphs	graph	NOUN
ejpam-4912	373	16	.	.	PUNCT
ejpam-4912	374	1	eur	eur	PROPN
ejpam-4912	374	2	.	.	PUNCT
ejpam-4912	375	1	j.	j.	PROPN
ejpam-4912	375	2	pure	pure	PROPN
ejpam-4912	375	3	appl	appl	PROPN
ejpam-4912	375	4	.	.	PUNCT
ejpam-4912	375	5	math	math	PROPN
ejpam-4912	375	6	.	.	PUNCT
ejpam-4912	375	7	,	,	PUNCT
ejpam-4912	376	1	6(3):1848–1861	6(3):1848–1861	NOUN
ejpam-4912	376	2	,	,	PUNCT
ejpam-4912	376	3	2023	2023	NUM
ejpam-4912	376	4	.	.	PUNCT
ejpam-4912	377	1	[	[	X
ejpam-4912	377	2	6	6	NUM
ejpam-4912	377	3	]	]	PUNCT
ejpam-4912	377	4	m.	m.	NOUN
ejpam-4912	377	5	henning	henning	PROPN
ejpam-4912	377	6	and	and	CCONJ
ejpam-4912	377	7	a.	a.	PROPN
ejpam-4912	377	8	yeo	yeo	PROPN
ejpam-4912	377	9	.	.	PROPN
ejpam-4912	378	1	total	total	ADJ
ejpam-4912	378	2	domination	domination	NOUN
ejpam-4912	378	3	in	in	ADP
ejpam-4912	378	4	graphs	graph	NOUN
ejpam-4912	378	5	.	.	PUNCT
ejpam-4912	379	1	in	in	ADP
ejpam-4912	379	2	springer	springer	NOUN
ejpam-4912	379	3	monographs	monograph	NOUN
ejpam-4912	379	4	in	in	ADP
ejpam-4912	379	5	mathematics	mathematic	NOUN
ejpam-4912	379	6	;	;	PUNCT
ejpam-4912	379	7	springer	springer	PROPN
ejpam-4912	379	8	new	new	PROPN
ejpam-4912	379	9	york	york	PROPN
ejpam-4912	379	10	,	,	PUNCT
ejpam-4912	379	11	ny	ny	PROPN
ejpam-4912	379	12	,	,	PUNCT
ejpam-4912	379	13	usa	usa	PROPN
ejpam-4912	379	14	,	,	PUNCT
ejpam-4912	379	15	,	,	PUNCT
ejpam-4912	379	16	2013	2013	NUM
ejpam-4912	379	17	.	.	PUNCT
ejpam-4912	380	1	[	[	X
ejpam-4912	380	2	7	7	X
ejpam-4912	380	3	]	]	X
ejpam-4912	380	4	s.	s.	PROPN
ejpam-4912	380	5	canoy	canoy	PROPN
ejpam-4912	380	6	jr	jr	PROPN
ejpam-4912	380	7	and	and	CCONJ
ejpam-4912	380	8	j.	j.	PROPN
ejpam-4912	380	9	hassan	hassan	PROPN
ejpam-4912	380	10	.	.	PUNCT
ejpam-4912	381	1	weakly	weakly	ADJ
ejpam-4912	381	2	convex	convex	VERB
ejpam-4912	381	3	hop	hop	NOUN
ejpam-4912	381	4	dominating	dominating	NOUN
ejpam-4912	381	5	sets	set	NOUN
ejpam-4912	381	6	in	in	ADP
ejpam-4912	381	7	graphs	graph	NOUN
ejpam-4912	381	8	.	.	PUNCT
ejpam-4912	382	1	eur	eur	PROPN
ejpam-4912	382	2	.	.	PUNCT
ejpam-4912	383	1	j.	j.	PROPN
ejpam-4912	383	2	pure	pure	PROPN
ejpam-4912	383	3	appl	appl	PROPN
ejpam-4912	383	4	.	.	PUNCT
ejpam-4912	383	5	math	math	PROPN
ejpam-4912	383	6	.	.	PUNCT
ejpam-4912	383	7	,	,	PUNCT
ejpam-4912	383	8	16(2):1196–1211	16(2):1196–1211	NUM
ejpam-4912	383	9	,	,	PUNCT
ejpam-4912	383	10	2023	2023	NUM
ejpam-4912	383	11	.	.	PUNCT
ejpam-4912	384	1	[	[	X
ejpam-4912	384	2	8	8	NUM
ejpam-4912	384	3	]	]	PUNCT
ejpam-4912	384	4	a.	a.	NOUN
ejpam-4912	384	5	klobucar	klobucar	NOUN
ejpam-4912	384	6	.	.	PUNCT
ejpam-4912	385	1	total	total	ADJ
ejpam-4912	385	2	domination	domination	NOUN
ejpam-4912	385	3	numbers	number	NOUN
ejpam-4912	385	4	of	of	ADP
ejpam-4912	385	5	cartesian	cartesian	ADJ
ejpam-4912	385	6	products	product	NOUN
ejpam-4912	385	7	.	.	PUNCT
ejpam-4912	386	1	math	math	NOUN
ejpam-4912	386	2	.	.	PUNCT
ejpam-4912	387	1	commun	commun	PROPN
ejpam-4912	387	2	.	.	PROPN
ejpam-4912	387	3	,	,	PUNCT
ejpam-4912	387	4	1:35–44	1:35–44	NUM
ejpam-4912	387	5	,	,	PUNCT
ejpam-4912	387	6	2014	2014	NUM
ejpam-4912	387	7	.	.	PUNCT
ejpam-4912	388	1	[	[	X
ejpam-4912	388	2	9	9	X
ejpam-4912	388	3	]	]	PUNCT
ejpam-4912	388	4	j.	j.	PROPN
ejpam-4912	388	5	manditong	manditong	PROPN
ejpam-4912	388	6	,	,	PUNCT
ejpam-4912	388	7	j.	j.	PROPN
ejpam-4912	388	8	hassan	hassan	PROPN
ejpam-4912	388	9	,	,	PUNCT
ejpam-4912	388	10	l.	l.	PROPN
ejpam-4912	388	11	laja	laja	PROPN
ejpam-4912	388	12	,	,	PUNCT
ejpam-4912	388	13	a.	a.	NOUN
ejpam-4912	388	14	laja	laja	PROPN
ejpam-4912	388	15	,	,	PUNCT
ejpam-4912	388	16	n.h	n.h	PROPN
ejpam-4912	388	17	.	.	PUNCT
ejpam-4912	388	18	mohammad	mohammad	PROPN
ejpam-4912	388	19	,	,	PUNCT
ejpam-4912	388	20	and	and	CCONJ
ejpam-4912	388	21	s.	s.	PROPN
ejpam-4912	388	22	kamdon	kamdon	PROPN
ejpam-4912	388	23	.	.	PUNCT
ejpam-4912	389	1	connected	connected	ADJ
ejpam-4912	389	2	outer	outer	ADJ
ejpam-4912	389	3	-	-	PUNCT
ejpam-4912	389	4	hop	hop	NOUN
ejpam-4912	389	5	independent	independent	ADJ
ejpam-4912	389	6	dominating	dominating	NOUN
ejpam-4912	389	7	sets	set	NOUN
ejpam-4912	389	8	in	in	ADP
ejpam-4912	389	9	graphs	graph	NOUN
ejpam-4912	389	10	under	under	ADP
ejpam-4912	389	11	some	some	DET
ejpam-4912	389	12	binary	binary	ADJ
ejpam-4912	389	13	operations	operation	NOUN
ejpam-4912	389	14	,	,	PUNCT
ejpam-4912	389	15	.	.	PUNCT
ejpam-4912	390	1	eur	eur	PROPN
ejpam-4912	390	2	.	.	PUNCT
ejpam-4912	391	1	j.	j.	PROPN
ejpam-4912	391	2	pure	pure	PROPN
ejpam-4912	391	3	appl	appl	PROPN
ejpam-4912	391	4	.	.	PUNCT
ejpam-4912	391	5	math	math	PROPN
ejpam-4912	391	6	.	.	PUNCT
ejpam-4912	391	7	,	,	PUNCT
ejpam-4912	392	1	16(3):1817–1829	16(3):1817–1829	NUM
ejpam-4912	392	2	,	,	PUNCT
ejpam-4912	392	3	2023	2023	NUM
ejpam-4912	392	4	.	.	PUNCT
ejpam-4912	393	1	[	[	X
ejpam-4912	393	2	10	10	NUM
ejpam-4912	393	3	]	]	X
ejpam-4912	393	4	o.	o.	NOUN
ejpam-4912	393	5	ore	ore	PROPN
ejpam-4912	393	6	.	.	PUNCT
ejpam-4912	394	1	theory	theory	NOUN
ejpam-4912	394	2	of	of	ADP
ejpam-4912	394	3	graphs	graph	NOUN
ejpam-4912	394	4	.	.	PUNCT
ejpam-4912	395	1	amer	amer	PROPN
ejpam-4912	395	2	math	math	PROPN
ejpam-4912	395	3	.	.	PUNCT
ejpam-4912	396	1	soc	soc	PROPN
ejpam-4912	396	2	.	.	PUNCT
ejpam-4912	397	1	colloq	colloq	PROPN
ejpam-4912	397	2	.	.	PUNCT
ejpam-4912	398	1	publ	publ	PROPN
ejpam-4912	398	2	.	.	PUNCT
ejpam-4912	398	3	,	,	PUNCT
ejpam-4912	398	4	38	38	NUM
ejpam-4912	398	5	(	(	PUNCT
ejpam-4912	398	6	amer	amer	PROPN
ejpam-4912	398	7	.	.	PROPN
ejpam-4912	398	8	math	math	PROPN
ejpam-4912	398	9	.	.	PUNCT
ejpam-4912	399	1	soc	soc	PROPN
ejpam-4912	399	2	.	.	PUNCT
ejpam-4912	399	3	,	,	PUNCT
ejpam-4912	399	4	providence	providence	NOUN
ejpam-4912	399	5	,	,	PUNCT
ejpam-4912	399	6	ri	ri	PROPN
ejpam-4912	399	7	)	)	PUNCT
ejpam-4912	399	8	,	,	PUNCT
ejpam-4912	399	9	1962	1962	NUM
ejpam-4912	399	10	.	.	PUNCT
ejpam-4912	400	1	[	[	X
ejpam-4912	400	2	11	11	NUM
ejpam-4912	400	3	]	]	PUNCT
ejpam-4912	400	4	j.	j.	PROPN
ejpam-4912	400	5	sigarreta	sigarreta	PROPN
ejpam-4912	400	6	.	.	PUNCT
ejpam-4912	401	1	total	total	ADJ
ejpam-4912	401	2	domination	domination	NOUN
ejpam-4912	401	3	on	on	ADP
ejpam-4912	401	4	some	some	DET
ejpam-4912	401	5	graph	graph	NOUN
ejpam-4912	401	6	operators	operator	NOUN
ejpam-4912	401	7	.	.	PUNCT
ejpam-4912	402	1	mathematics	mathematic	NOUN
ejpam-4912	402	2	,	,	PUNCT
ejpam-4912	402	3	9(241):1–9	9(241):1–9	NUM
ejpam-4912	402	4	,	,	PUNCT
ejpam-4912	402	5	2021	2021	NUM
ejpam-4912	402	6	.	.	PUNCT
ejpam-4912	403	1	[	[	X
ejpam-4912	403	2	12	12	NUM
ejpam-4912	403	3	]	]	PUNCT
ejpam-4912	403	4	a.	a.	NOUN
ejpam-4912	403	5	sugumaran	sugumaran	NOUN
ejpam-4912	403	6	and	and	CCONJ
ejpam-4912	403	7	e.	e.	PROPN
ejpam-4912	403	8	jayachandran	jayachandran	PROPN
ejpam-4912	403	9	.	.	PUNCT
ejpam-4912	404	1	domination	domination	NOUN
ejpam-4912	404	2	number	number	NOUN
ejpam-4912	404	3	of	of	ADP
ejpam-4912	404	4	some	some	DET
ejpam-4912	404	5	graphs	graph	NOUN
ejpam-4912	404	6	.	.	PUNCT
ejpam-4912	405	1	intl	intl	PROPN
ejpam-4912	405	2	.	.	PUNCT
ejpam-4912	406	1	jour	jour	PROPN
ejpam-4912	406	2	.	.	PROPN
ejpam-4912	406	3	of	of	ADP
ejpam-4912	406	4	scientific	scientific	ADJ
ejpam-4912	406	5	,	,	PUNCT
ejpam-4912	406	6	development	development	NOUN
ejpam-4912	406	7	and	and	CCONJ
ejpam-4912	406	8	research	research	NOUN
ejpam-4912	406	9	(	(	PUNCT
ejpam-4912	406	10	ijsdr	ijsdr	PROPN
ejpam-4912	406	11	)	)	PUNCT
ejpam-4912	406	12	,	,	PUNCT
ejpam-4912	406	13	3(11):2455–2631	3(11):2455–2631	NUM
ejpam-4912	406	14	,	,	PUNCT
ejpam-4912	406	15	2018	2018	NUM
ejpam-4912	406	16	.	.	PUNCT