id	sid	tid	token	lemma	pos
ejpam-6429	1	1	european	european	PROPN
ejpam-6429	1	2	journal	journal	PROPN
ejpam-6429	1	3	of	of	ADP
ejpam-6429	1	4	pure	pure	ADJ
ejpam-6429	1	5	and	and	CCONJ
ejpam-6429	1	6	applied	applied	ADJ
ejpam-6429	1	7	mathematics	mathematic	NOUN
ejpam-6429	1	8	2025	2025	NUM
ejpam-6429	1	9	,	,	PUNCT
ejpam-6429	1	10	vol	vol	NOUN
ejpam-6429	1	11	.	.	PROPN
ejpam-6429	1	12	18	18	NUM
ejpam-6429	1	13	,	,	PUNCT
ejpam-6429	1	14	issue	issue	NOUN
ejpam-6429	1	15	4	4	NUM
ejpam-6429	1	16	,	,	PUNCT
ejpam-6429	1	17	article	article	NOUN
ejpam-6429	1	18	number	number	NOUN
ejpam-6429	1	19	6429	6429	NUM
ejpam-6429	1	20	issn	issn	PROPN
ejpam-6429	1	21	1307	1307	NUM
ejpam-6429	1	22	-	-	SYM
ejpam-6429	1	23	5543	5543	NUM
ejpam-6429	1	24	–	–	PUNCT
ejpam-6429	1	25	ejpam.com	ejpam.com	X
ejpam-6429	1	26	published	publish	VERB
ejpam-6429	1	27	by	by	ADP
ejpam-6429	1	28	new	new	PROPN
ejpam-6429	1	29	york	york	PROPN
ejpam-6429	1	30	business	business	PROPN
ejpam-6429	1	31	global	global	ADJ
ejpam-6429	1	32	generous	generous	ADJ
ejpam-6429	1	33	roman	roman	ADJ
ejpam-6429	1	34	domination	domination	NOUN
ejpam-6429	1	35	subdivision	subdivision	NOUN
ejpam-6429	1	36	number	number	NOUN
ejpam-6429	1	37	in	in	ADP
ejpam-6429	1	38	graphs	graph	NOUN
ejpam-6429	1	39	jamil	jamil	PROPN
ejpam-6429	1	40	j.	j.	PROPN
ejpam-6429	1	41	hamja1,5	hamja1,5	PROPN
ejpam-6429	1	42	,	,	PUNCT
ejpam-6429	1	43	seyed	seyed	PROPN
ejpam-6429	1	44	mahmoud	mahmoud	PROPN
ejpam-6429	1	45	sheikholeslami2,∗	sheikholeslami2,∗	PROPN
ejpam-6429	1	46	,	,	PUNCT
ejpam-6429	1	47	mina	mina	PROPN
ejpam-6429	1	48	esmaeili2	esmaeili2	PROPN
ejpam-6429	1	49	,	,	PUNCT
ejpam-6429	1	50	lutz	lutz	PROPN
ejpam-6429	1	51	volkmann3	volkmann3	PROPN
ejpam-6429	1	52	,	,	PUNCT
ejpam-6429	1	53	imelda	imelda	PROPN
ejpam-6429	1	54	s.	s.	PROPN
ejpam-6429	1	55	aniversario4,5	aniversario4,5	PROPN
ejpam-6429	1	56	,	,	PUNCT
ejpam-6429	1	57	lucille	lucille	NOUN
ejpam-6429	1	58	m.	m.	NOUN
ejpam-6429	2	1	bugo4,5	bugo4,5	PROPN
ejpam-6429	2	2	1	1	NUM
ejpam-6429	2	3	department	department	NOUN
ejpam-6429	2	4	of	of	ADP
ejpam-6429	2	5	mathematics	mathematic	NOUN
ejpam-6429	2	6	,	,	PUNCT
ejpam-6429	2	7	college	college	NOUN
ejpam-6429	2	8	of	of	ADP
ejpam-6429	2	9	arts	art	NOUN
ejpam-6429	2	10	and	and	CCONJ
ejpam-6429	2	11	sciences	science	NOUN
ejpam-6429	2	12	,	,	PUNCT
ejpam-6429	2	13	msu	msu	PROPN
ejpam-6429	2	14	-	-	PUNCT
ejpam-6429	2	15	tawi	tawi	NOUN
ejpam-6429	2	16	-	-	PUNCT
ejpam-6429	2	17	tawi	tawi	NOUN
ejpam-6429	2	18	college	college	PROPN
ejpam-6429	2	19	of	of	ADP
ejpam-6429	2	20	technology	technology	NOUN
ejpam-6429	2	21	and	and	CCONJ
ejpam-6429	2	22	oceanography	oceanography	NOUN
ejpam-6429	2	23	,	,	PUNCT
ejpam-6429	2	24	7500	7500	NUM
ejpam-6429	2	25	tawi	tawi	NOUN
ejpam-6429	2	26	-	-	PUNCT
ejpam-6429	2	27	tawi	tawi	NOUN
ejpam-6429	2	28	,	,	PUNCT
ejpam-6429	2	29	philippines	philippines	PROPN
ejpam-6429	2	30	2	2	NUM
ejpam-6429	2	31	department	department	NOUN
ejpam-6429	2	32	of	of	ADP
ejpam-6429	2	33	mathematics	mathematics	PROPN
ejpam-6429	2	34	,	,	PUNCT
ejpam-6429	2	35	azarbaijan	azarbaijan	NOUN
ejpam-6429	2	36	shahid	shahid	PROPN
ejpam-6429	2	37	madani	madani	PROPN
ejpam-6429	2	38	university	university	PROPN
ejpam-6429	2	39	,	,	PUNCT
ejpam-6429	2	40	tabriz	tabriz	NOUN
ejpam-6429	2	41	,	,	PUNCT
ejpam-6429	2	42	i.r	i.r	PROPN
ejpam-6429	2	43	.	.	PROPN
ejpam-6429	2	44	iran	iran	PROPN
ejpam-6429	2	45	3	3	NUM
ejpam-6429	2	46	lehrstuhl	lehrstuhl	PROPN
ejpam-6429	2	47	ii	ii	PROPN
ejpam-6429	2	48	für	für	PROPN
ejpam-6429	2	49	mathematik	mathematik	PROPN
ejpam-6429	2	50	,	,	PUNCT
ejpam-6429	2	51	rwth	rwth	PROPN
ejpam-6429	2	52	aachen	aachen	PROPN
ejpam-6429	2	53	university	university	PROPN
ejpam-6429	2	54	,	,	PUNCT
ejpam-6429	2	55	52056	52056	NUM
ejpam-6429	2	56	aachen	aachen	PROPN
ejpam-6429	2	57	,	,	PUNCT
ejpam-6429	2	58	germany	germany	PROPN
ejpam-6429	2	59	4	4	NUM
ejpam-6429	2	60	department	department	NOUN
ejpam-6429	2	61	of	of	ADP
ejpam-6429	2	62	mathematics	mathematic	NOUN
ejpam-6429	2	63	and	and	CCONJ
ejpam-6429	2	64	statistics	statistic	NOUN
ejpam-6429	2	65	,	,	PUNCT
ejpam-6429	2	66	college	college	NOUN
ejpam-6429	2	67	of	of	ADP
ejpam-6429	2	68	science	science	NOUN
ejpam-6429	2	69	and	and	CCONJ
ejpam-6429	2	70	mathematics	mathematic	NOUN
ejpam-6429	2	71	,	,	PUNCT
ejpam-6429	2	72	msu	msu	PROPN
ejpam-6429	2	73	-	-	PUNCT
ejpam-6429	2	74	iligan	iligan	PROPN
ejpam-6429	2	75	institute	institute	PROPN
ejpam-6429	2	76	of	of	ADP
ejpam-6429	2	77	technology	technology	PROPN
ejpam-6429	2	78	,	,	PUNCT
ejpam-6429	2	79	9200	9200	NUM
ejpam-6429	2	80	iligan	iligan	ADJ
ejpam-6429	2	81	city	city	NOUN
ejpam-6429	2	82	,	,	PUNCT
ejpam-6429	2	83	philippines	philippine	NOUN
ejpam-6429	2	84	5	5	NUM
ejpam-6429	2	85	center	center	NOUN
ejpam-6429	2	86	for	for	ADP
ejpam-6429	2	87	mathematical	mathematical	ADJ
ejpam-6429	2	88	and	and	CCONJ
ejpam-6429	2	89	theoretical	theoretical	ADJ
ejpam-6429	2	90	physical	physical	ADJ
ejpam-6429	2	91	sciences	science	NOUN
ejpam-6429	2	92	,	,	PUNCT
ejpam-6429	2	93	premier	premier	PROPN
ejpam-6429	2	94	research	research	PROPN
ejpam-6429	2	95	institute	institute	PROPN
ejpam-6429	2	96	of	of	ADP
ejpam-6429	2	97	science	science	NOUN
ejpam-6429	2	98	and	and	CCONJ
ejpam-6429	2	99	mathematics	mathematics	PROPN
ejpam-6429	2	100	(	(	PUNCT
ejpam-6429	2	101	prism	prism	NOUN
ejpam-6429	2	102	)	)	PUNCT
ejpam-6429	2	103	,	,	PUNCT
ejpam-6429	2	104	msu	msu	PROPN
ejpam-6429	2	105	-	-	PUNCT
ejpam-6429	2	106	iligan	iligan	PROPN
ejpam-6429	2	107	institute	institute	PROPN
ejpam-6429	2	108	of	of	ADP
ejpam-6429	2	109	technology	technology	PROPN
ejpam-6429	2	110	,	,	PUNCT
ejpam-6429	2	111	9200	9200	NUM
ejpam-6429	2	112	iligan	iligan	ADJ
ejpam-6429	2	113	city	city	NOUN
ejpam-6429	2	114	,	,	PUNCT
ejpam-6429	2	115	philippines	philippine	NOUN
ejpam-6429	2	116	abstract	abstract	ADJ
ejpam-6429	2	117	.	.	PUNCT
ejpam-6429	3	1	let	let	VERB
ejpam-6429	3	2	g	g	PROPN
ejpam-6429	3	3	=	=	SYM
ejpam-6429	3	4	(	(	PUNCT
ejpam-6429	3	5	v	v	NOUN
ejpam-6429	3	6	,	,	PUNCT
ejpam-6429	3	7	e	e	NOUN
ejpam-6429	3	8	)	)	PUNCT
ejpam-6429	3	9	be	be	AUX
ejpam-6429	3	10	a	a	DET
ejpam-6429	3	11	simple	simple	ADJ
ejpam-6429	3	12	graph	graph	NOUN
ejpam-6429	3	13	,	,	PUNCT
ejpam-6429	3	14	and	and	CCONJ
ejpam-6429	3	15	let	let	VERB
ejpam-6429	3	16	f	f	NOUN
ejpam-6429	3	17	:	:	PUNCT
ejpam-6429	3	18	v	v	X
ejpam-6429	3	19	→	→	SYM
ejpam-6429	3	20	{	{	PUNCT
ejpam-6429	3	21	0	0	NUM
ejpam-6429	3	22	,	,	PUNCT
ejpam-6429	3	23	1	1	NUM
ejpam-6429	3	24	,	,	PUNCT
ejpam-6429	3	25	2	2	NUM
ejpam-6429	3	26	,	,	PUNCT
ejpam-6429	3	27	3	3	NUM
ejpam-6429	3	28	}	}	PUNCT
ejpam-6429	3	29	be	be	AUX
ejpam-6429	3	30	a	a	DET
ejpam-6429	3	31	function	function	NOUN
ejpam-6429	3	32	.	.	PUNCT
ejpam-6429	4	1	a	a	DET
ejpam-6429	4	2	vertex	vertex	NOUN
ejpam-6429	4	3	u	u	NOUN
ejpam-6429	4	4	is	be	AUX
ejpam-6429	4	5	considered	consider	VERB
ejpam-6429	4	6	an	an	DET
ejpam-6429	4	7	undefended	undefended	ADJ
ejpam-6429	4	8	vertex	vertex	NOUN
ejpam-6429	4	9	with	with	ADP
ejpam-6429	4	10	respect	respect	NOUN
ejpam-6429	4	11	to	to	ADP
ejpam-6429	4	12	f	f	PROPN
ejpam-6429	4	13	if	if	SCONJ
ejpam-6429	4	14	f(u	f(u	PROPN
ejpam-6429	4	15	)	)	PUNCT
ejpam-6429	5	1	=	=	SYM
ejpam-6429	5	2	0	0	PUNCT
ejpam-6429	6	1	and	and	CCONJ
ejpam-6429	6	2	there	there	PRON
ejpam-6429	6	3	is	be	VERB
ejpam-6429	6	4	no	no	DET
ejpam-6429	6	5	adjacent	adjacent	ADJ
ejpam-6429	6	6	vertex	vertex	NOUN
ejpam-6429	6	7	v	v	ADP
ejpam-6429	6	8	satisfying	satisfy	VERB
ejpam-6429	6	9	f(v	f(v	NOUN
ejpam-6429	6	10	)	)	PUNCT
ejpam-6429	6	11	≥	≥	NOUN
ejpam-6429	6	12	2	2	NUM
ejpam-6429	6	13	.	.	PUNCT
ejpam-6429	7	1	a	a	DET
ejpam-6429	7	2	function	function	NOUN
ejpam-6429	7	3	f	f	PROPN
ejpam-6429	7	4	is	be	AUX
ejpam-6429	7	5	termed	term	VERB
ejpam-6429	7	6	a	a	DET
ejpam-6429	7	7	generous	generous	ADJ
ejpam-6429	7	8	roman	roman	ADJ
ejpam-6429	7	9	dominating	dominating	NOUN
ejpam-6429	7	10	function	function	NOUN
ejpam-6429	7	11	(	(	PUNCT
ejpam-6429	7	12	grd	grd	NOUN
ejpam-6429	7	13	-	-	PUNCT
ejpam-6429	7	14	function	function	NOUN
ejpam-6429	7	15	)	)	PUNCT
ejpam-6429	7	16	if	if	SCONJ
ejpam-6429	7	17	,	,	PUNCT
ejpam-6429	7	18	for	for	ADP
ejpam-6429	7	19	every	every	DET
ejpam-6429	7	20	vertex	vertex	NOUN
ejpam-6429	7	21	u	u	NOUN
ejpam-6429	7	22	with	with	ADP
ejpam-6429	7	23	f(u	f(u	PROPN
ejpam-6429	7	24	)	)	PUNCT
ejpam-6429	7	25	=	=	SYM
ejpam-6429	7	26	0	0	NUM
ejpam-6429	7	27	,	,	PUNCT
ejpam-6429	7	28	there	there	PRON
ejpam-6429	7	29	exists	exist	VERB
ejpam-6429	7	30	at	at	ADP
ejpam-6429	7	31	least	least	ADV
ejpam-6429	7	32	one	one	NUM
ejpam-6429	7	33	adjacent	adjacent	ADJ
ejpam-6429	7	34	vertex	vertex	NOUN
ejpam-6429	7	35	v	v	ADP
ejpam-6429	7	36	such	such	ADJ
ejpam-6429	7	37	that	that	DET
ejpam-6429	7	38	f(v	f(v	NOUN
ejpam-6429	7	39	)	)	PUNCT
ejpam-6429	7	40	≥	≥	NOUN
ejpam-6429	7	41	2	2	NUM
ejpam-6429	7	42	and	and	CCONJ
ejpam-6429	7	43	the	the	DET
ejpam-6429	7	44	modified	modify	VERB
ejpam-6429	7	45	function	function	NOUN
ejpam-6429	7	46	f	f	NOUN
ejpam-6429	7	47	′	′	NUM
ejpam-6429	7	48	:	:	PUNCT
ejpam-6429	7	49	v	v	X
ejpam-6429	7	50	→	→	SYM
ejpam-6429	7	51	{	{	PUNCT
ejpam-6429	7	52	0	0	NUM
ejpam-6429	7	53	,	,	PUNCT
ejpam-6429	7	54	1	1	NUM
ejpam-6429	7	55	,	,	PUNCT
ejpam-6429	7	56	2	2	NUM
ejpam-6429	7	57	,	,	PUNCT
ejpam-6429	7	58	3	3	NUM
ejpam-6429	7	59	}	}	PUNCT
ejpam-6429	7	60	,	,	PUNCT
ejpam-6429	7	61	defined	define	VERB
ejpam-6429	7	62	as	as	ADP
ejpam-6429	7	63	f	f	PROPN
ejpam-6429	7	64	′(u	′(u	NOUN
ejpam-6429	7	65	)	)	PUNCT
ejpam-6429	7	66	=	=	SYM
ejpam-6429	8	1	α	α	PROPN
ejpam-6429	8	2	,	,	PUNCT
ejpam-6429	8	3	f	f	PROPN
ejpam-6429	8	4	′(v	′(v	PROPN
ejpam-6429	8	5	)	)	PUNCT
ejpam-6429	8	6	=	=	SYM
ejpam-6429	9	1	f(v)−	f(v)−	NOUN
ejpam-6429	9	2	α	α	NOUN
ejpam-6429	9	3	,	,	PUNCT
ejpam-6429	9	4	where	where	SCONJ
ejpam-6429	9	5	α	α	X
ejpam-6429	9	6	∈	∈	PROPN
ejpam-6429	9	7	{	{	PUNCT
ejpam-6429	9	8	1	1	NUM
ejpam-6429	9	9	,	,	PUNCT
ejpam-6429	9	10	2	2	NUM
ejpam-6429	9	11	}	}	PUNCT
ejpam-6429	9	12	,	,	PUNCT
ejpam-6429	9	13	and	and	CCONJ
ejpam-6429	9	14	f	f	PROPN
ejpam-6429	9	15	′(w	′(w	PROPN
ejpam-6429	9	16	)	)	PUNCT
ejpam-6429	9	17	=	=	SYM
ejpam-6429	9	18	f(w	f(w	PROPN
ejpam-6429	9	19	)	)	PUNCT
ejpam-6429	9	20	for	for	ADP
ejpam-6429	9	21	all	all	DET
ejpam-6429	9	22	w	w	PROPN
ejpam-6429	9	23	∈	∈	PROPN
ejpam-6429	9	24	v	v	ADP
ejpam-6429	9	25	\	\	NOUN
ejpam-6429	9	26	{	{	PUNCT
ejpam-6429	9	27	u	u	NOUN
ejpam-6429	9	28	,	,	PUNCT
ejpam-6429	9	29	v	v	NOUN
ejpam-6429	9	30	}	}	PUNCT
ejpam-6429	9	31	,	,	PUNCT
ejpam-6429	9	32	ensures	ensure	VERB
ejpam-6429	9	33	that	that	SCONJ
ejpam-6429	9	34	no	no	DET
ejpam-6429	9	35	vertex	vertex	NOUN
ejpam-6429	9	36	remains	remain	VERB
ejpam-6429	9	37	undefended	undefended	ADJ
ejpam-6429	9	38	.	.	PUNCT
ejpam-6429	10	1	the	the	DET
ejpam-6429	10	2	weight	weight	NOUN
ejpam-6429	10	3	of	of	ADP
ejpam-6429	10	4	a	a	DET
ejpam-6429	10	5	grd	grd	NOUN
ejpam-6429	10	6	-	-	PUNCT
ejpam-6429	10	7	function	function	NOUN
ejpam-6429	10	8	f	f	PROPN
ejpam-6429	10	9	is	be	AUX
ejpam-6429	10	10	defined	define	VERB
ejpam-6429	10	11	as	as	ADP
ejpam-6429	10	12	f(v	f(v	NOUN
ejpam-6429	10	13	)	)	PUNCT
ejpam-6429	11	1	=	=	SYM
ejpam-6429	11	2	∑	∑	PUNCT
ejpam-6429	11	3	u∈v	u∈v	NOUN
ejpam-6429	11	4	f(u	f(u	PROPN
ejpam-6429	11	5	)	)	PUNCT
ejpam-6429	11	6	.	.	PUNCT
ejpam-6429	12	1	the	the	DET
ejpam-6429	12	2	smallest	small	ADJ
ejpam-6429	12	3	possible	possible	ADJ
ejpam-6429	12	4	weight	weight	NOUN
ejpam-6429	12	5	of	of	ADP
ejpam-6429	12	6	a	a	DET
ejpam-6429	12	7	grd	grd	NOUN
ejpam-6429	12	8	-	-	PUNCT
ejpam-6429	12	9	function	function	NOUN
ejpam-6429	12	10	on	on	ADP
ejpam-6429	12	11	g	g	PROPN
ejpam-6429	12	12	is	be	AUX
ejpam-6429	12	13	known	know	VERB
ejpam-6429	12	14	as	as	ADP
ejpam-6429	12	15	the	the	DET
ejpam-6429	12	16	generous	generous	ADJ
ejpam-6429	12	17	roman	roman	ADJ
ejpam-6429	12	18	domination	domination	NOUN
ejpam-6429	12	19	number	number	NOUN
ejpam-6429	12	20	of	of	ADP
ejpam-6429	12	21	g	g	NOUN
ejpam-6429	12	22	,	,	PUNCT
ejpam-6429	12	23	denoted	denote	VERB
ejpam-6429	12	24	by	by	ADP
ejpam-6429	12	25	γgr(g	γgr(g	PROPN
ejpam-6429	12	26	)	)	PUNCT
ejpam-6429	12	27	.	.	PUNCT
ejpam-6429	13	1	the	the	DET
ejpam-6429	13	2	generous	generous	ADJ
ejpam-6429	13	3	roman	roman	ADJ
ejpam-6429	13	4	domination	domination	NOUN
ejpam-6429	13	5	subdivision	subdivision	NOUN
ejpam-6429	13	6	number	number	NOUN
ejpam-6429	13	7	,	,	PUNCT
ejpam-6429	13	8	denoted	denote	VERB
ejpam-6429	13	9	sdγgr(g	sdγgr(g	PROPN
ejpam-6429	13	10	)	)	PUNCT
ejpam-6429	13	11	,	,	PUNCT
ejpam-6429	13	12	is	be	AUX
ejpam-6429	13	13	the	the	DET
ejpam-6429	13	14	minimum	minimum	ADJ
ejpam-6429	13	15	number	number	NOUN
ejpam-6429	13	16	of	of	ADP
ejpam-6429	13	17	edges	edge	NOUN
ejpam-6429	13	18	that	that	PRON
ejpam-6429	13	19	must	must	AUX
ejpam-6429	13	20	be	be	AUX
ejpam-6429	13	21	subdivided	subdivide	VERB
ejpam-6429	13	22	(	(	PUNCT
ejpam-6429	13	23	each	each	PRON
ejpam-6429	13	24	at	at	ADV
ejpam-6429	13	25	most	most	ADV
ejpam-6429	13	26	once	once	ADV
ejpam-6429	13	27	)	)	PUNCT
ejpam-6429	13	28	to	to	PART
ejpam-6429	13	29	increase	increase	VERB
ejpam-6429	13	30	γgr(g	γgr(g	PROPN
ejpam-6429	13	31	)	)	PUNCT
ejpam-6429	13	32	.	.	PUNCT
ejpam-6429	14	1	in	in	ADP
ejpam-6429	14	2	this	this	DET
ejpam-6429	14	3	paper	paper	NOUN
ejpam-6429	14	4	,	,	PUNCT
ejpam-6429	14	5	we	we	PRON
ejpam-6429	14	6	establish	establish	VERB
ejpam-6429	14	7	upper	upper	ADJ
ejpam-6429	14	8	bounds	bound	NOUN
ejpam-6429	14	9	on	on	ADP
ejpam-6429	14	10	sdγgr(g	sdγgr(g	PROPN
ejpam-6429	14	11	)	)	PUNCT
ejpam-6429	14	12	,	,	PUNCT
ejpam-6429	14	13	and	and	CCONJ
ejpam-6429	14	14	determine	determine	VERB
ejpam-6429	14	15	its	its	PRON
ejpam-6429	14	16	exact	exact	ADJ
ejpam-6429	14	17	value	value	NOUN
ejpam-6429	14	18	for	for	ADP
ejpam-6429	14	19	certain	certain	ADJ
ejpam-6429	14	20	families	family	NOUN
ejpam-6429	14	21	of	of	ADP
ejpam-6429	14	22	graphs	graph	NOUN
ejpam-6429	14	23	,	,	PUNCT
ejpam-6429	14	24	including	include	VERB
ejpam-6429	14	25	paths	path	NOUN
ejpam-6429	14	26	,	,	PUNCT
ejpam-6429	14	27	cycles	cycle	NOUN
ejpam-6429	14	28	,	,	PUNCT
ejpam-6429	14	29	and	and	CCONJ
ejpam-6429	14	30	ladders	ladder	NOUN
ejpam-6429	14	31	.	.	PUNCT
ejpam-6429	15	1	furthermore	furthermore	ADV
ejpam-6429	15	2	,	,	PUNCT
ejpam-6429	15	3	we	we	PRON
ejpam-6429	15	4	provide	provide	VERB
ejpam-6429	15	5	sufficient	sufficient	ADJ
ejpam-6429	15	6	conditions	condition	NOUN
ejpam-6429	15	7	for	for	ADP
ejpam-6429	15	8	a	a	DET
ejpam-6429	15	9	graph	graph	NOUN
ejpam-6429	15	10	g	g	NOUN
ejpam-6429	15	11	to	to	PART
ejpam-6429	15	12	have	have	VERB
ejpam-6429	15	13	a	a	DET
ejpam-6429	15	14	small	small	ADJ
ejpam-6429	15	15	subdivision	subdivision	NOUN
ejpam-6429	15	16	number	number	NOUN
ejpam-6429	15	17	.	.	PUNCT
ejpam-6429	16	1	2020	2020	NUM
ejpam-6429	16	2	mathematics	mathematic	NOUN
ejpam-6429	16	3	subject	subject	NOUN
ejpam-6429	16	4	classifications	classification	NOUN
ejpam-6429	16	5	:	:	PUNCT
ejpam-6429	16	6	05c69	05c69	X
ejpam-6429	16	7	key	key	ADJ
ejpam-6429	16	8	words	word	NOUN
ejpam-6429	16	9	and	and	CCONJ
ejpam-6429	16	10	phrases	phrase	NOUN
ejpam-6429	16	11	:	:	PUNCT
ejpam-6429	16	12	generous	generous	ADJ
ejpam-6429	16	13	roman	roman	ADJ
ejpam-6429	16	14	domination	domination	NOUN
ejpam-6429	16	15	,	,	PUNCT
ejpam-6429	16	16	generous	generous	ADJ
ejpam-6429	16	17	roman	roman	ADJ
ejpam-6429	16	18	domination	domination	NOUN
ejpam-6429	16	19	subdivision	subdivision	NOUN
ejpam-6429	16	20	number	number	NOUN
ejpam-6429	16	21	∗corresponding	∗corresponde	VERB
ejpam-6429	16	22	author	author	NOUN
ejpam-6429	16	23	.	.	PUNCT
ejpam-6429	17	1	doi	doi	NOUN
ejpam-6429	17	2	:	:	PUNCT
ejpam-6429	17	3	https://doi.org/10.29020/nybg.ejpam.v18i4.6429	https://doi.org/10.29020/nybg.ejpam.v18i4.6429	ADJ
ejpam-6429	17	4	email	email	NOUN
ejpam-6429	17	5	addresses	address	VERB
ejpam-6429	17	6	:	:	PUNCT
ejpam-6429	17	7	jamilhamja@msutawi-tawi.edu.ph	jamilhamja@msutawi-tawi.edu.ph	PROPN
ejpam-6429	17	8	(	(	PUNCT
ejpam-6429	17	9	j.	j.	PROPN
ejpam-6429	17	10	j.	j.	PROPN
ejpam-6429	17	11	hamja	hamja	PROPN
ejpam-6429	17	12	)	)	PUNCT
ejpam-6429	17	13	,	,	PUNCT
ejpam-6429	17	14	s.m.sheikholeslami@azaruniv.ac.ir	s.m.sheikholeslami@azaruniv.ac.ir	PUNCT
ejpam-6429	17	15	(	(	PUNCT
ejpam-6429	17	16	s.	s.	PROPN
ejpam-6429	17	17	m.	m.	PROPN
ejpam-6429	17	18	sheikholeslami	sheikholeslami	PROPN
ejpam-6429	17	19	)	)	PUNCT
ejpam-6429	17	20	,	,	PUNCT
ejpam-6429	17	21	minaesmaeeli1999@gmail.com	minaesmaeeli1999@gmail.com	X
ejpam-6429	17	22	(	(	PUNCT
ejpam-6429	17	23	m.	m.	NOUN
ejpam-6429	17	24	esmaeili	esmaeili	NOUN
ejpam-6429	17	25	)	)	PUNCT
ejpam-6429	17	26	,	,	PUNCT
ejpam-6429	17	27	volkm@math2.rwth-aachen.de	volkm@math2.rwth-aachen.de	PROPN
ejpam-6429	17	28	(	(	PUNCT
ejpam-6429	17	29	l.	l.	PROPN
ejpam-6429	17	30	volkmann	volkmann	PROPN
ejpam-6429	17	31	)	)	PUNCT
ejpam-6429	17	32	,	,	PUNCT
ejpam-6429	17	33	imelda.aniversario@g.msuiit.edu.ph	imelda.aniversario@g.msuiit.edu.ph	PROPN
ejpam-6429	17	34	(	(	PUNCT
ejpam-6429	17	35	i.	i.	PROPN
ejpam-6429	17	36	s.	s.	PROPN
ejpam-6429	17	37	aniversario	aniversario	PROPN
ejpam-6429	17	38	)	)	PUNCT
ejpam-6429	17	39	,	,	PUNCT
ejpam-6429	17	40	lucille.bugo@g.msuiit.edu.ph	lucille.bugo@g.msuiit.edu.ph	PROPN
ejpam-6429	17	41	(	(	PUNCT
ejpam-6429	17	42	l.	l.	PROPN
ejpam-6429	17	43	m.	m.	PROPN
ejpam-6429	17	44	bugo	bugo	PROPN
ejpam-6429	17	45	)	)	PUNCT
ejpam-6429	17	46	https://www.ejpam.com	https://www.ejpam.com	X
ejpam-6429	17	47	1	1	NUM
ejpam-6429	17	48	copyright	copyright	NOUN
ejpam-6429	17	49	:	:	PUNCT
ejpam-6429	17	50	©	©	PROPN
ejpam-6429	17	51	2025	2025	NUM
ejpam-6429	17	52	the	the	DET
ejpam-6429	17	53	author(s	author(s	NOUN
ejpam-6429	17	54	)	)	PUNCT
ejpam-6429	17	55	.	.	PUNCT
ejpam-6429	18	1	(	(	PUNCT
ejpam-6429	18	2	cc	cc	NOUN
ejpam-6429	18	3	by	by	ADP
ejpam-6429	18	4	-	-	PUNCT
ejpam-6429	18	5	nc	nc	PROPN
ejpam-6429	18	6	4.0	4.0	NUM
ejpam-6429	18	7	)	)	PUNCT
ejpam-6429	18	8	j.	j.	PROPN
ejpam-6429	18	9	j.	j.	PROPN
ejpam-6429	18	10	hamja	hamja	PROPN
ejpam-6429	19	1	et	et	PROPN
ejpam-6429	19	2	al	al	PROPN
ejpam-6429	19	3	.	.	PUNCT
ejpam-6429	19	4	/	/	SYM
ejpam-6429	19	5	eur	eur	PROPN
ejpam-6429	19	6	.	.	PUNCT
ejpam-6429	20	1	j.	j.	PROPN
ejpam-6429	20	2	pure	pure	PROPN
ejpam-6429	20	3	appl	appl	PROPN
ejpam-6429	20	4	.	.	PROPN
ejpam-6429	20	5	math	math	PROPN
ejpam-6429	20	6	,	,	PUNCT
ejpam-6429	20	7	18	18	NUM
ejpam-6429	20	8	(	(	PUNCT
ejpam-6429	20	9	4	4	NUM
ejpam-6429	20	10	)	)	PUNCT
ejpam-6429	20	11	(	(	PUNCT
ejpam-6429	20	12	2025	2025	NUM
ejpam-6429	20	13	)	)	PUNCT
ejpam-6429	20	14	,	,	PUNCT
ejpam-6429	20	15	6429	6429	NUM
ejpam-6429	20	16	2	2	NUM
ejpam-6429	20	17	of	of	ADP
ejpam-6429	20	18	16	16	NUM
ejpam-6429	20	19	1	1	NUM
ejpam-6429	20	20	.	.	PUNCT
ejpam-6429	21	1	introduction	introduction	NOUN
ejpam-6429	21	2	motivated	motivate	VERB
ejpam-6429	21	3	by	by	ADP
ejpam-6429	21	4	resource	resource	NOUN
ejpam-6429	21	5	–	–	PUNCT
ejpam-6429	21	6	allocation	allocation	NOUN
ejpam-6429	21	7	strategies	strategy	NOUN
ejpam-6429	21	8	for	for	ADP
ejpam-6429	21	9	defending	defend	VERB
ejpam-6429	21	10	the	the	DET
ejpam-6429	21	11	roman	roman	ADJ
ejpam-6429	21	12	empire	empire	NOUN
ejpam-6429	21	13	,	,	PUNCT
ejpam-6429	21	14	where	where	SCONJ
ejpam-6429	21	15	lightly	lightly	ADV
ejpam-6429	21	16	defended	defend	VERB
ejpam-6429	21	17	regions	region	NOUN
ejpam-6429	21	18	must	must	AUX
ejpam-6429	21	19	be	be	AUX
ejpam-6429	21	20	able	able	ADJ
ejpam-6429	21	21	to	to	PART
ejpam-6429	21	22	call	call	VERB
ejpam-6429	21	23	in	in	ADP
ejpam-6429	21	24	reinforcements	reinforcement	NOUN
ejpam-6429	21	25	from	from	ADP
ejpam-6429	21	26	nearby	nearby	ADJ
ejpam-6429	21	27	strongholds	stronghold	NOUN
ejpam-6429	21	28	,	,	PUNCT
ejpam-6429	21	29	as	as	SCONJ
ejpam-6429	21	30	discussed	discuss	VERB
ejpam-6429	21	31	by	by	ADP
ejpam-6429	21	32	revelle	revelle	NOUN
ejpam-6429	21	33	and	and	CCONJ
ejpam-6429	21	34	rosing	rose	VERB
ejpam-6429	21	35	[	[	X
ejpam-6429	21	36	1	1	NUM
ejpam-6429	21	37	]	]	PUNCT
ejpam-6429	21	38	and	and	CCONJ
ejpam-6429	21	39	by	by	ADP
ejpam-6429	21	40	stewart	stewart	PROPN
ejpam-6429	22	1	[	[	X
ejpam-6429	22	2	2	2	NUM
ejpam-6429	22	3	]	]	PUNCT
ejpam-6429	22	4	,	,	PUNCT
ejpam-6429	22	5	cockayne	cockayne	PROPN
ejpam-6429	22	6	et	et	PROPN
ejpam-6429	22	7	al	al	PROPN
ejpam-6429	22	8	.	.	PROPN
ejpam-6429	22	9	introduced	introduce	VERB
ejpam-6429	22	10	roman	roman	ADJ
ejpam-6429	22	11	domination	domination	NOUN
ejpam-6429	22	12	in	in	ADP
ejpam-6429	22	13	graphs	graph	NOUN
ejpam-6429	22	14	in	in	ADP
ejpam-6429	22	15	2004	2004	NUM
ejpam-6429	22	16	[	[	X
ejpam-6429	22	17	3	3	NUM
ejpam-6429	22	18	]	]	PUNCT
ejpam-6429	22	19	.	.	PUNCT
ejpam-6429	23	1	since	since	SCONJ
ejpam-6429	23	2	its	its	PRON
ejpam-6429	23	3	introduction	introduction	NOUN
ejpam-6429	23	4	,	,	PUNCT
ejpam-6429	23	5	roman	roman	ADJ
ejpam-6429	23	6	domination	domination	NOUN
ejpam-6429	23	7	has	have	AUX
ejpam-6429	23	8	attracted	attract	VERB
ejpam-6429	23	9	wide	wide	ADV
ejpam-6429	23	10	and	and	CCONJ
ejpam-6429	23	11	sustained	sustained	ADJ
ejpam-6429	23	12	interest	interest	NOUN
ejpam-6429	23	13	,	,	PUNCT
ejpam-6429	23	14	spawning	spawn	VERB
ejpam-6429	23	15	a	a	DET
ejpam-6429	23	16	rich	rich	ADJ
ejpam-6429	23	17	family	family	NOUN
ejpam-6429	23	18	of	of	ADP
ejpam-6429	23	19	variants	variant	NOUN
ejpam-6429	23	20	and	and	CCONJ
ejpam-6429	23	21	extensions	extension	NOUN
ejpam-6429	23	22	,	,	PUNCT
ejpam-6429	23	23	e.g.	e.g.	ADV
ejpam-6429	23	24	,	,	PUNCT
ejpam-6429	23	25	double	double	ADJ
ejpam-6429	23	26	roman	roman	ADJ
ejpam-6429	23	27	domination	domination	NOUN
ejpam-6429	23	28	,	,	PUNCT
ejpam-6429	23	29	roman	roman	NOUN
ejpam-6429	23	30	{	{	PUNCT
ejpam-6429	23	31	2}-domination	2}-domination	NUM
ejpam-6429	23	32	,	,	PUNCT
ejpam-6429	23	33	independent	independent	ADJ
ejpam-6429	23	34	roman	roman	ADJ
ejpam-6429	23	35	domination	domination	NOUN
ejpam-6429	23	36	,	,	PUNCT
ejpam-6429	23	37	total	total	ADJ
ejpam-6429	23	38	roman	roman	ADJ
ejpam-6429	23	39	domination	domination	NOUN
ejpam-6429	23	40	,	,	PUNCT
ejpam-6429	23	41	bondage	bondage	NOUN
ejpam-6429	23	42	and	and	CCONJ
ejpam-6429	23	43	domatic	domatic	ADJ
ejpam-6429	23	44	parameters	parameter	NOUN
ejpam-6429	23	45	,	,	PUNCT
ejpam-6429	23	46	and	and	CCONJ
ejpam-6429	23	47	directed	direct	VERB
ejpam-6429	23	48	versions	version	NOUN
ejpam-6429	23	49	,	,	PUNCT
ejpam-6429	23	50	now	now	ADV
ejpam-6429	23	51	documented	document	VERB
ejpam-6429	23	52	in	in	ADP
ejpam-6429	23	53	well	well	ADV
ejpam-6429	23	54	over	over	ADP
ejpam-6429	23	55	two	two	NUM
ejpam-6429	23	56	hundred	hundred	NUM
ejpam-6429	23	57	publications	publication	NOUN
ejpam-6429	23	58	.	.	PUNCT
ejpam-6429	24	1	for	for	ADP
ejpam-6429	24	2	comprehensive	comprehensive	ADJ
ejpam-6429	24	3	accounts	account	NOUN
ejpam-6429	24	4	of	of	ADP
ejpam-6429	24	5	core	core	NOUN
ejpam-6429	24	6	results	result	NOUN
ejpam-6429	24	7	,	,	PUNCT
ejpam-6429	24	8	and	and	CCONJ
ejpam-6429	24	9	structural	structural	ADJ
ejpam-6429	24	10	characterizations	characterization	NOUN
ejpam-6429	24	11	,	,	PUNCT
ejpam-6429	24	12	we	we	PRON
ejpam-6429	24	13	refer	refer	VERB
ejpam-6429	24	14	to	to	ADP
ejpam-6429	24	15	the	the	DET
ejpam-6429	24	16	book	book	NOUN
ejpam-6429	24	17	chapters	chapter	NOUN
ejpam-6429	24	18	and	and	CCONJ
ejpam-6429	24	19	survey	survey	NOUN
ejpam-6429	24	20	papers	paper	NOUN
ejpam-6429	24	21	by	by	ADP
ejpam-6429	24	22	chellali	chellali	PROPN
ejpam-6429	24	23	,	,	PUNCT
ejpam-6429	24	24	jafari	jafari	PROPN
ejpam-6429	24	25	rad	rad	PROPN
ejpam-6429	24	26	,	,	PUNCT
ejpam-6429	24	27	sheikholeslami	sheikholeslami	NOUN
ejpam-6429	24	28	,	,	PUNCT
ejpam-6429	24	29	and	and	CCONJ
ejpam-6429	24	30	volkmann	volkmann	PROPN
ejpam-6429	25	1	[	[	X
ejpam-6429	25	2	4	4	NUM
ejpam-6429	25	3	,	,	PUNCT
ejpam-6429	25	4	5	5	NUM
ejpam-6429	25	5	]	]	PUNCT
ejpam-6429	25	6	.	.	PUNCT
ejpam-6429	26	1	focused	focus	VERB
ejpam-6429	26	2	overviews	overview	NOUN
ejpam-6429	26	3	of	of	ADP
ejpam-6429	26	4	specific	specific	ADJ
ejpam-6429	26	5	directions	direction	NOUN
ejpam-6429	26	6	include	include	VERB
ejpam-6429	26	7	surveys	survey	NOUN
ejpam-6429	26	8	on	on	ADP
ejpam-6429	26	9	varieties	variety	NOUN
ejpam-6429	26	10	of	of	ADP
ejpam-6429	26	11	roman	roman	ADJ
ejpam-6429	26	12	domination	domination	NOUN
ejpam-6429	27	1	[	[	X
ejpam-6429	27	2	6–8	6–8	X
ejpam-6429	27	3	]	]	X
ejpam-6429	27	4	,	,	PUNCT
ejpam-6429	27	5	roman	roman	ADJ
ejpam-6429	27	6	domatic	domatic	ADJ
ejpam-6429	27	7	problems	problem	NOUN
ejpam-6429	27	8	in	in	ADP
ejpam-6429	27	9	graphs	graph	NOUN
ejpam-6429	27	10	and	and	CCONJ
ejpam-6429	27	11	digraphs	digraph	VERB
ejpam-6429	27	12	[	[	X
ejpam-6429	27	13	9	9	NUM
ejpam-6429	27	14	]	]	PUNCT
ejpam-6429	27	15	,	,	PUNCT
ejpam-6429	27	16	and	and	CCONJ
ejpam-6429	27	17	roman	roman	ADJ
ejpam-6429	27	18	domination	domination	NOUN
ejpam-6429	27	19	parameters	parameter	NOUN
ejpam-6429	27	20	for	for	ADP
ejpam-6429	27	21	directed	direct	VERB
ejpam-6429	27	22	graphs	graph	NOUN
ejpam-6429	27	23	[	[	X
ejpam-6429	27	24	10	10	NUM
ejpam-6429	27	25	]	]	PUNCT
ejpam-6429	27	26	.	.	PUNCT
ejpam-6429	28	1	these	these	DET
ejpam-6429	28	2	sources	source	NOUN
ejpam-6429	28	3	together	together	ADV
ejpam-6429	28	4	chart	chart	VERB
ejpam-6429	28	5	the	the	DET
ejpam-6429	28	6	evolution	evolution	NOUN
ejpam-6429	28	7	of	of	ADP
ejpam-6429	28	8	roman	roman	ADJ
ejpam-6429	28	9	domination	domination	NOUN
ejpam-6429	28	10	from	from	ADP
ejpam-6429	28	11	its	its	PRON
ejpam-6429	28	12	facility	facility	NOUN
ejpam-6429	28	13	-	-	PUNCT
ejpam-6429	28	14	location	location	NOUN
ejpam-6429	28	15	and	and	CCONJ
ejpam-6429	28	16	defense	defense	NOUN
ejpam-6429	28	17	-	-	PUNCT
ejpam-6429	28	18	strategy	strategy	NOUN
ejpam-6429	28	19	origins	origin	NOUN
ejpam-6429	28	20	[	[	X
ejpam-6429	28	21	1	1	NUM
ejpam-6429	28	22	,	,	PUNCT
ejpam-6429	28	23	2	2	NUM
ejpam-6429	28	24	]	]	PUNCT
ejpam-6429	28	25	to	to	ADP
ejpam-6429	28	26	a	a	DET
ejpam-6429	28	27	mature	mature	ADJ
ejpam-6429	28	28	theory	theory	NOUN
ejpam-6429	28	29	with	with	ADP
ejpam-6429	28	30	broad	broad	ADJ
ejpam-6429	28	31	connections	connection	NOUN
ejpam-6429	28	32	across	across	ADP
ejpam-6429	28	33	domination	domination	NOUN
ejpam-6429	28	34	theory	theory	NOUN
ejpam-6429	28	35	and	and	CCONJ
ejpam-6429	28	36	algorithmic	algorithmic	ADJ
ejpam-6429	28	37	graph	graph	NOUN
ejpam-6429	28	38	problems	problem	NOUN
ejpam-6429	28	39	[	[	X
ejpam-6429	28	40	4–6	4–6	NOUN
ejpam-6429	28	41	,	,	PUNCT
ejpam-6429	28	42	9	9	NUM
ejpam-6429	28	43	,	,	PUNCT
ejpam-6429	28	44	10	10	NUM
ejpam-6429	28	45	]	]	PUNCT
ejpam-6429	28	46	.	.	PUNCT
ejpam-6429	29	1	among	among	ADP
ejpam-6429	29	2	the	the	DET
ejpam-6429	29	3	most	most	ADV
ejpam-6429	29	4	recent	recent	ADJ
ejpam-6429	29	5	contributions	contribution	NOUN
ejpam-6429	29	6	to	to	ADP
ejpam-6429	29	7	roman	roman	ADJ
ejpam-6429	29	8	domination	domination	NOUN
ejpam-6429	29	9	theory	theory	NOUN
ejpam-6429	29	10	is	be	AUX
ejpam-6429	29	11	the	the	DET
ejpam-6429	29	12	notion	notion	NOUN
ejpam-6429	29	13	of	of	ADP
ejpam-6429	29	14	generous	generous	ADJ
ejpam-6429	29	15	roman	roman	ADJ
ejpam-6429	29	16	domination	domination	NOUN
ejpam-6429	29	17	,	,	PUNCT
ejpam-6429	29	18	introduced	introduce	VERB
ejpam-6429	29	19	by	by	ADP
ejpam-6429	29	20	benatallah	benatallah	ADJ
ejpam-6429	29	21	,	,	PUNCT
ejpam-6429	29	22	blidia	blidia	NOUN
ejpam-6429	29	23	,	,	PUNCT
ejpam-6429	29	24	and	and	CCONJ
ejpam-6429	29	25	ouldrabah	ouldrabah	NOUN
ejpam-6429	29	26	in	in	ADP
ejpam-6429	29	27	2024	2024	NUM
ejpam-6429	29	28	[	[	X
ejpam-6429	29	29	11	11	NUM
ejpam-6429	29	30	]	]	PUNCT
ejpam-6429	29	31	.	.	PUNCT
ejpam-6429	30	1	this	this	DET
ejpam-6429	30	2	strengthening	strengthening	NOUN
ejpam-6429	30	3	reflects	reflect	VERB
ejpam-6429	30	4	scenarios	scenario	NOUN
ejpam-6429	30	5	where	where	SCONJ
ejpam-6429	30	6	“	"	PUNCT
ejpam-6429	30	7	reinforcements	reinforcement	NOUN
ejpam-6429	30	8	”	"	PUNCT
ejpam-6429	30	9	must	must	AUX
ejpam-6429	30	10	be	be	AUX
ejpam-6429	30	11	guaranteed	guarantee	VERB
ejpam-6429	30	12	from	from	ADP
ejpam-6429	30	13	more	more	ADJ
ejpam-6429	30	14	than	than	ADP
ejpam-6429	30	15	one	one	NUM
ejpam-6429	30	16	source	source	NOUN
ejpam-6429	30	17	,	,	PUNCT
ejpam-6429	30	18	thereby	thereby	ADV
ejpam-6429	30	19	increasing	increase	VERB
ejpam-6429	30	20	robustness	robustness	NOUN
ejpam-6429	30	21	compared	compare	VERB
ejpam-6429	30	22	to	to	ADP
ejpam-6429	30	23	classical	classical	ADJ
ejpam-6429	30	24	roman	roman	ADJ
ejpam-6429	30	25	domination	domination	NOUN
ejpam-6429	30	26	.	.	PUNCT
ejpam-6429	31	1	in	in	ADP
ejpam-6429	31	2	their	their	PRON
ejpam-6429	31	3	foundational	foundational	ADJ
ejpam-6429	31	4	paper	paper	NOUN
ejpam-6429	31	5	,	,	PUNCT
ejpam-6429	31	6	they	they	PRON
ejpam-6429	31	7	provided	provide	VERB
ejpam-6429	31	8	exact	exact	ADJ
ejpam-6429	31	9	values	value	NOUN
ejpam-6429	31	10	of	of	ADP
ejpam-6429	31	11	the	the	DET
ejpam-6429	31	12	generous	generous	ADJ
ejpam-6429	31	13	roman	roman	ADJ
ejpam-6429	31	14	domination	domination	NOUN
ejpam-6429	31	15	number	number	NOUN
ejpam-6429	31	16	for	for	ADP
ejpam-6429	31	17	paths	path	NOUN
ejpam-6429	31	18	and	and	CCONJ
ejpam-6429	31	19	cycles	cycle	NOUN
ejpam-6429	31	20	,	,	PUNCT
ejpam-6429	31	21	derived	derive	VERB
ejpam-6429	31	22	an	an	DET
ejpam-6429	31	23	upper	upper	ADJ
ejpam-6429	31	24	bound	bind	VERB
ejpam-6429	31	25	for	for	ADP
ejpam-6429	31	26	general	general	ADJ
ejpam-6429	31	27	graphs	graph	NOUN
ejpam-6429	31	28	,	,	PUNCT
ejpam-6429	31	29	and	and	CCONJ
ejpam-6429	31	30	characterized	characterize	VERB
ejpam-6429	31	31	cubic	cubic	ADJ
ejpam-6429	31	32	graphs	graph	NOUN
ejpam-6429	31	33	of	of	ADP
ejpam-6429	31	34	order	order	NOUN
ejpam-6429	31	35	n	n	PRON
ejpam-6429	31	36	with	with	ADP
ejpam-6429	31	37	respect	respect	NOUN
ejpam-6429	31	38	to	to	ADP
ejpam-6429	31	39	this	this	DET
ejpam-6429	31	40	parameter	parameter	NOUN
ejpam-6429	31	41	.	.	PUNCT
ejpam-6429	32	1	furthermore	furthermore	ADV
ejpam-6429	32	2	,	,	PUNCT
ejpam-6429	32	3	they	they	PRON
ejpam-6429	32	4	investigated	investigate	VERB
ejpam-6429	32	5	a	a	DET
ejpam-6429	32	6	nordhaus	nordhaus	NOUN
ejpam-6429	32	7	–	–	PUNCT
ejpam-6429	32	8	gaddum	gaddum	NOUN
ejpam-6429	32	9	type	type	NOUN
ejpam-6429	32	10	inequality	inequality	NOUN
ejpam-6429	32	11	for	for	ADP
ejpam-6429	32	12	the	the	DET
ejpam-6429	32	13	generous	generous	ADJ
ejpam-6429	32	14	roman	roman	ADJ
ejpam-6429	32	15	domination	domination	NOUN
ejpam-6429	32	16	number	number	NOUN
ejpam-6429	32	17	,	,	PUNCT
ejpam-6429	32	18	and	and	CCONJ
ejpam-6429	32	19	they	they	PRON
ejpam-6429	32	20	established	establish	VERB
ejpam-6429	32	21	results	result	NOUN
ejpam-6429	32	22	concerning	concern	VERB
ejpam-6429	32	23	its	its	PRON
ejpam-6429	32	24	computational	computational	ADJ
ejpam-6429	32	25	complexity	complexity	NOUN
ejpam-6429	32	26	,	,	PUNCT
ejpam-6429	32	27	demonstrating	demonstrate	VERB
ejpam-6429	32	28	that	that	SCONJ
ejpam-6429	32	29	the	the	DET
ejpam-6429	32	30	problem	problem	NOUN
ejpam-6429	32	31	of	of	ADP
ejpam-6429	32	32	determining	determine	VERB
ejpam-6429	32	33	this	this	DET
ejpam-6429	32	34	parameter	parameter	NOUN
ejpam-6429	32	35	remains	remain	VERB
ejpam-6429	32	36	np	np	NOUN
ejpam-6429	32	37	-	-	NOUN
ejpam-6429	32	38	complete	complete	ADJ
ejpam-6429	32	39	for	for	ADP
ejpam-6429	32	40	general	general	ADJ
ejpam-6429	32	41	graphs	graph	NOUN
ejpam-6429	32	42	.	.	PUNCT
ejpam-6429	33	1	building	build	VERB
ejpam-6429	33	2	on	on	ADP
ejpam-6429	33	3	this	this	DET
ejpam-6429	33	4	foundation	foundation	NOUN
ejpam-6429	33	5	,	,	PUNCT
ejpam-6429	33	6	sheikholeslami	sheikholeslami	NOUN
ejpam-6429	33	7	,	,	PUNCT
ejpam-6429	33	8	chellali	chellali	PROPN
ejpam-6429	33	9	,	,	PUNCT
ejpam-6429	33	10	and	and	CCONJ
ejpam-6429	33	11	kor	kor	VERB
ejpam-6429	34	1	[	[	X
ejpam-6429	34	2	12	12	NUM
ejpam-6429	34	3	]	]	X
ejpam-6429	34	4	advanced	advance	VERB
ejpam-6429	34	5	the	the	DET
ejpam-6429	34	6	study	study	NOUN
ejpam-6429	34	7	of	of	ADP
ejpam-6429	34	8	generous	generous	ADJ
ejpam-6429	34	9	roman	roman	ADJ
ejpam-6429	34	10	domination	domination	NOUN
ejpam-6429	34	11	in	in	ADP
ejpam-6429	34	12	2025	2025	NUM
ejpam-6429	34	13	by	by	ADP
ejpam-6429	34	14	determining	determine	VERB
ejpam-6429	34	15	its	its	PRON
ejpam-6429	34	16	exact	exact	ADJ
ejpam-6429	34	17	value	value	NOUN
ejpam-6429	34	18	for	for	ADP
ejpam-6429	34	19	ladder	ladder	NOUN
ejpam-6429	34	20	graphs	graph	NOUN
ejpam-6429	34	21	.	.	PUNCT
ejpam-6429	35	1	they	they	PRON
ejpam-6429	35	2	also	also	ADV
ejpam-6429	35	3	provided	provide	VERB
ejpam-6429	35	4	an	an	DET
ejpam-6429	35	5	upper	upper	ADJ
ejpam-6429	35	6	bound	bind	VERB
ejpam-6429	35	7	for	for	ADP
ejpam-6429	35	8	trees	tree	NOUN
ejpam-6429	35	9	in	in	ADP
ejpam-6429	35	10	terms	term	NOUN
ejpam-6429	35	11	of	of	ADP
ejpam-6429	35	12	the	the	DET
ejpam-6429	35	13	order	order	NOUN
ejpam-6429	35	14	,	,	PUNCT
ejpam-6429	35	15	the	the	DET
ejpam-6429	35	16	number	number	NOUN
ejpam-6429	35	17	of	of	ADP
ejpam-6429	35	18	leaves	leave	NOUN
ejpam-6429	35	19	,	,	PUNCT
ejpam-6429	35	20	and	and	CCONJ
ejpam-6429	35	21	the	the	DET
ejpam-6429	35	22	number	number	NOUN
ejpam-6429	35	23	of	of	ADP
ejpam-6429	35	24	stems	stem	NOUN
ejpam-6429	35	25	.	.	PUNCT
ejpam-6429	36	1	their	their	PRON
ejpam-6429	36	2	results	result	NOUN
ejpam-6429	36	3	highlighted	highlight	VERB
ejpam-6429	36	4	structural	structural	ADJ
ejpam-6429	36	5	dependencies	dependency	NOUN
ejpam-6429	36	6	of	of	ADP
ejpam-6429	36	7	this	this	DET
ejpam-6429	36	8	parameter	parameter	NOUN
ejpam-6429	36	9	within	within	ADP
ejpam-6429	36	10	tree	tree	NOUN
ejpam-6429	36	11	families	family	NOUN
ejpam-6429	36	12	,	,	PUNCT
ejpam-6429	36	13	leading	lead	VERB
ejpam-6429	36	14	to	to	ADP
ejpam-6429	36	15	a	a	DET
ejpam-6429	36	16	sharper	sharp	ADJ
ejpam-6429	36	17	understanding	understanding	NOUN
ejpam-6429	36	18	of	of	ADP
ejpam-6429	36	19	its	its	PRON
ejpam-6429	36	20	extremal	extremal	ADJ
ejpam-6429	36	21	behavior	behavior	NOUN
ejpam-6429	36	22	.	.	PUNCT
ejpam-6429	37	1	in	in	ADP
ejpam-6429	37	2	particular	particular	ADJ
ejpam-6429	37	3	,	,	PUNCT
ejpam-6429	37	4	they	they	PRON
ejpam-6429	37	5	proved	prove	VERB
ejpam-6429	37	6	that	that	SCONJ
ejpam-6429	37	7	for	for	ADP
ejpam-6429	37	8	every	every	DET
ejpam-6429	37	9	tree	tree	NOUN
ejpam-6429	37	10	t	t	X
ejpam-6429	37	11	on	on	ADP
ejpam-6429	37	12	at	at	ADV
ejpam-6429	37	13	least	least	ADV
ejpam-6429	37	14	three	three	NUM
ejpam-6429	37	15	vertices	vertex	NOUN
ejpam-6429	37	16	,	,	PUNCT
ejpam-6429	37	17	the	the	DET
ejpam-6429	37	18	generous	generous	ADJ
ejpam-6429	37	19	roman	roman	ADJ
ejpam-6429	37	20	domination	domination	NOUN
ejpam-6429	37	21	number	number	NOUN
ejpam-6429	37	22	satisfies	satisfy	VERB
ejpam-6429	37	23	γgr(t	γgr(t	PRON
ejpam-6429	37	24	)	)	PUNCT
ejpam-6429	37	25	≥	≥	NOUN
ejpam-6429	37	26	γ(t	γ(t	NOUN
ejpam-6429	37	27	)	)	PUNCT
ejpam-6429	38	1	+	+	CCONJ
ejpam-6429	38	2	2	2	NUM
ejpam-6429	38	3	,	,	PUNCT
ejpam-6429	38	4	where	where	SCONJ
ejpam-6429	38	5	γ(t	γ(t	NOUN
ejpam-6429	38	6	)	)	PUNCT
ejpam-6429	38	7	denotes	denote	VERB
ejpam-6429	38	8	the	the	DET
ejpam-6429	38	9	domination	domination	NOUN
ejpam-6429	38	10	number	number	NOUN
ejpam-6429	38	11	of	of	ADP
ejpam-6429	38	12	t	t	PROPN
ejpam-6429	38	13	,	,	PUNCT
ejpam-6429	38	14	and	and	CCONJ
ejpam-6429	38	15	they	they	PRON
ejpam-6429	38	16	completely	completely	ADV
ejpam-6429	38	17	characterized	characterize	VERB
ejpam-6429	38	18	the	the	DET
ejpam-6429	38	19	extremal	extremal	ADJ
ejpam-6429	38	20	trees	tree	NOUN
ejpam-6429	38	21	attaining	attain	VERB
ejpam-6429	38	22	this	this	DET
ejpam-6429	38	23	lower	lower	ADV
ejpam-6429	38	24	bound	bind	VERB
ejpam-6429	38	25	.	.	PUNCT
ejpam-6429	39	1	these	these	DET
ejpam-6429	39	2	findings	finding	NOUN
ejpam-6429	39	3	not	not	PART
ejpam-6429	39	4	only	only	ADV
ejpam-6429	39	5	extended	extend	VERB
ejpam-6429	39	6	the	the	DET
ejpam-6429	39	7	applicability	applicability	NOUN
ejpam-6429	39	8	of	of	ADP
ejpam-6429	39	9	generous	generous	ADJ
ejpam-6429	39	10	roman	roman	ADJ
ejpam-6429	39	11	domination	domination	NOUN
ejpam-6429	39	12	to	to	ADP
ejpam-6429	39	13	broader	broad	ADJ
ejpam-6429	39	14	classes	class	NOUN
ejpam-6429	39	15	of	of	ADP
ejpam-6429	39	16	graphs	graph	NOUN
ejpam-6429	39	17	,	,	PUNCT
ejpam-6429	39	18	but	but	CCONJ
ejpam-6429	39	19	also	also	ADV
ejpam-6429	39	20	established	establish	VERB
ejpam-6429	39	21	meaningful	meaningful	ADJ
ejpam-6429	39	22	links	link	NOUN
ejpam-6429	39	23	between	between	ADP
ejpam-6429	39	24	classical	classical	ADJ
ejpam-6429	39	25	domination	domination	NOUN
ejpam-6429	39	26	parameters	parameter	NOUN
ejpam-6429	39	27	and	and	CCONJ
ejpam-6429	39	28	this	this	DET
ejpam-6429	39	29	new	new	ADJ
ejpam-6429	39	30	variant	variant	NOUN
ejpam-6429	39	31	.	.	PUNCT
ejpam-6429	40	1	together	together	ADV
ejpam-6429	40	2	,	,	PUNCT
ejpam-6429	40	3	the	the	DET
ejpam-6429	40	4	works	work	NOUN
ejpam-6429	40	5	of	of	ADP
ejpam-6429	40	6	benatallah	benatallah	PROPN
ejpam-6429	40	7	et	et	PROPN
ejpam-6429	40	8	al	al	PROPN
ejpam-6429	40	9	.	.	PUNCT
ejpam-6429	41	1	[	[	X
ejpam-6429	41	2	11	11	NUM
ejpam-6429	41	3	]	]	PUNCT
ejpam-6429	41	4	and	and	CCONJ
ejpam-6429	41	5	sheikholeslami	sheikholeslami	NOUN
ejpam-6429	41	6	et	et	PROPN
ejpam-6429	41	7	al	al	PROPN
ejpam-6429	41	8	.	.	PUNCT
ejpam-6429	42	1	[	[	X
ejpam-6429	42	2	12	12	NUM
ejpam-6429	42	3	,	,	PUNCT
ejpam-6429	42	4	13	13	NUM
ejpam-6429	42	5	]	]	PUNCT
ejpam-6429	42	6	demonstrate	demonstrate	VERB
ejpam-6429	42	7	the	the	DET
ejpam-6429	42	8	rapid	rapid	ADJ
ejpam-6429	42	9	development	development	NOUN
ejpam-6429	42	10	of	of	ADP
ejpam-6429	42	11	generous	generous	ADJ
ejpam-6429	42	12	roman	roman	ADJ
ejpam-6429	42	13	domination	domination	NOUN
ejpam-6429	42	14	as	as	ADP
ejpam-6429	42	15	a	a	DET
ejpam-6429	42	16	promising	promising	ADJ
ejpam-6429	42	17	research	research	NOUN
ejpam-6429	42	18	direction	direction	NOUN
ejpam-6429	42	19	.	.	PUNCT
ejpam-6429	43	1	in	in	ADP
ejpam-6429	43	2	particular	particular	ADJ
ejpam-6429	43	3	,	,	PUNCT
ejpam-6429	43	4	they	they	PRON
ejpam-6429	43	5	show	show	VERB
ejpam-6429	43	6	how	how	SCONJ
ejpam-6429	43	7	the	the	DET
ejpam-6429	43	8	interplay	interplay	NOUN
ejpam-6429	43	9	between	between	ADP
ejpam-6429	43	10	structural	structural	ADJ
ejpam-6429	43	11	graph	graph	NOUN
ejpam-6429	43	12	properties	property	NOUN
ejpam-6429	43	13	and	and	CCONJ
ejpam-6429	43	14	domination	domination	NOUN
ejpam-6429	43	15	j.	j.	PROPN
ejpam-6429	43	16	j.	j.	PROPN
ejpam-6429	43	17	hamja	hamja	PROPN
ejpam-6429	43	18	et	et	PROPN
ejpam-6429	43	19	al	al	PROPN
ejpam-6429	43	20	.	.	PUNCT
ejpam-6429	43	21	/	/	SYM
ejpam-6429	43	22	eur	eur	PROPN
ejpam-6429	43	23	.	.	PUNCT
ejpam-6429	44	1	j.	j.	PROPN
ejpam-6429	44	2	pure	pure	PROPN
ejpam-6429	44	3	appl	appl	PROPN
ejpam-6429	44	4	.	.	PROPN
ejpam-6429	44	5	math	math	PROPN
ejpam-6429	44	6	,	,	PUNCT
ejpam-6429	44	7	18	18	NUM
ejpam-6429	44	8	(	(	PUNCT
ejpam-6429	44	9	4	4	NUM
ejpam-6429	44	10	)	)	PUNCT
ejpam-6429	44	11	(	(	PUNCT
ejpam-6429	44	12	2025	2025	NUM
ejpam-6429	44	13	)	)	PUNCT
ejpam-6429	44	14	,	,	PUNCT
ejpam-6429	44	15	6429	6429	NUM
ejpam-6429	44	16	3	3	NUM
ejpam-6429	44	17	of	of	ADP
ejpam-6429	44	18	16	16	NUM
ejpam-6429	44	19	parameters	parameter	NOUN
ejpam-6429	44	20	yields	yield	VERB
ejpam-6429	44	21	both	both	PRON
ejpam-6429	44	22	exact	exact	ADJ
ejpam-6429	44	23	values	value	NOUN
ejpam-6429	44	24	and	and	CCONJ
ejpam-6429	44	25	tight	tight	ADJ
ejpam-6429	44	26	bounds	bound	NOUN
ejpam-6429	44	27	,	,	PUNCT
ejpam-6429	44	28	while	while	SCONJ
ejpam-6429	44	29	also	also	ADV
ejpam-6429	44	30	raising	raise	VERB
ejpam-6429	44	31	new	new	ADJ
ejpam-6429	44	32	complexity	complexity	NOUN
ejpam-6429	44	33	and	and	CCONJ
ejpam-6429	44	34	extremal	extremal	ADJ
ejpam-6429	44	35	problems	problem	NOUN
ejpam-6429	44	36	.	.	PUNCT
ejpam-6429	45	1	this	this	PRON
ejpam-6429	45	2	suggests	suggest	VERB
ejpam-6429	45	3	that	that	SCONJ
ejpam-6429	45	4	generous	generous	ADJ
ejpam-6429	45	5	roman	roman	ADJ
ejpam-6429	45	6	domination	domination	NOUN
ejpam-6429	45	7	may	may	AUX
ejpam-6429	45	8	follow	follow	VERB
ejpam-6429	45	9	a	a	DET
ejpam-6429	45	10	trajectory	trajectory	NOUN
ejpam-6429	45	11	similar	similar	ADJ
ejpam-6429	45	12	to	to	ADP
ejpam-6429	45	13	classical	classical	ADJ
ejpam-6429	45	14	roman	roman	ADJ
ejpam-6429	45	15	domination	domination	NOUN
ejpam-6429	45	16	,	,	PUNCT
ejpam-6429	45	17	giving	give	VERB
ejpam-6429	45	18	rise	rise	NOUN
ejpam-6429	45	19	to	to	ADP
ejpam-6429	45	20	a	a	DET
ejpam-6429	45	21	wide	wide	ADJ
ejpam-6429	45	22	array	array	NOUN
ejpam-6429	45	23	of	of	ADP
ejpam-6429	45	24	variants	variant	NOUN
ejpam-6429	45	25	and	and	CCONJ
ejpam-6429	45	26	applications	application	NOUN
ejpam-6429	45	27	in	in	ADP
ejpam-6429	45	28	the	the	DET
ejpam-6429	45	29	years	year	NOUN
ejpam-6429	45	30	to	to	PART
ejpam-6429	45	31	come	come	VERB
ejpam-6429	45	32	.	.	PUNCT
ejpam-6429	46	1	given	give	VERB
ejpam-6429	46	2	that	that	DET
ejpam-6429	46	3	graphs	graph	NOUN
ejpam-6429	46	4	serve	serve	VERB
ejpam-6429	46	5	as	as	ADP
ejpam-6429	46	6	models	model	NOUN
ejpam-6429	46	7	for	for	ADP
ejpam-6429	46	8	numerous	numerous	ADJ
ejpam-6429	46	9	real	real	ADJ
ejpam-6429	46	10	-	-	PUNCT
ejpam-6429	46	11	world	world	NOUN
ejpam-6429	46	12	systems	system	NOUN
ejpam-6429	46	13	,	,	PUNCT
ejpam-6429	46	14	it	it	PRON
ejpam-6429	46	15	is	be	AUX
ejpam-6429	46	16	natural	natural	ADJ
ejpam-6429	46	17	to	to	PART
ejpam-6429	46	18	explore	explore	VERB
ejpam-6429	46	19	how	how	SCONJ
ejpam-6429	46	20	structural	structural	ADJ
ejpam-6429	46	21	changes	change	NOUN
ejpam-6429	46	22	such	such	ADJ
ejpam-6429	46	23	as	as	ADP
ejpam-6429	46	24	vertex	vertex	NOUN
ejpam-6429	46	25	or	or	CCONJ
ejpam-6429	46	26	edge	edge	NOUN
ejpam-6429	46	27	deletions	deletion	NOUN
ejpam-6429	46	28	,	,	PUNCT
ejpam-6429	46	29	edge	edge	NOUN
ejpam-6429	46	30	additions	addition	NOUN
ejpam-6429	46	31	,	,	PUNCT
ejpam-6429	46	32	or	or	CCONJ
ejpam-6429	46	33	edge	edge	NOUN
ejpam-6429	46	34	subdivisions	subdivision	NOUN
ejpam-6429	46	35	affect	affect	VERB
ejpam-6429	46	36	domination	domination	NOUN
ejpam-6429	46	37	-	-	PUNCT
ejpam-6429	46	38	related	relate	VERB
ejpam-6429	46	39	parameters	parameter	NOUN
ejpam-6429	46	40	.	.	PUNCT
ejpam-6429	47	1	in	in	ADP
ejpam-6429	47	2	this	this	DET
ejpam-6429	47	3	direction	direction	NOUN
ejpam-6429	47	4	,	,	PUNCT
ejpam-6429	47	5	velammal	velammal	ADJ
ejpam-6429	47	6	[	[	X
ejpam-6429	47	7	14	14	NUM
ejpam-6429	47	8	]	]	PUNCT
ejpam-6429	47	9	introduced	introduce	VERB
ejpam-6429	47	10	the	the	DET
ejpam-6429	47	11	domination	domination	NOUN
ejpam-6429	47	12	subdivision	subdivision	NOUN
ejpam-6429	47	13	number	number	NOUN
ejpam-6429	47	14	measuring	measure	VERB
ejpam-6429	47	15	the	the	DET
ejpam-6429	47	16	minimal	minimal	ADJ
ejpam-6429	47	17	number	number	NOUN
ejpam-6429	47	18	of	of	ADP
ejpam-6429	47	19	edge	edge	NOUN
ejpam-6429	47	20	subdivisions	subdivision	NOUN
ejpam-6429	47	21	needed	need	VERB
ejpam-6429	47	22	to	to	PART
ejpam-6429	47	23	increase	increase	VERB
ejpam-6429	47	24	a	a	DET
ejpam-6429	47	25	graph	graph	NOUN
ejpam-6429	47	26	’s	’s	PART
ejpam-6429	47	27	domination	domination	NOUN
ejpam-6429	47	28	number	number	NOUN
ejpam-6429	47	29	,	,	PUNCT
ejpam-6429	47	30	with	with	ADP
ejpam-6429	47	31	each	each	DET
ejpam-6429	47	32	edge	edge	NOUN
ejpam-6429	47	33	subdivided	subdivide	VERB
ejpam-6429	47	34	at	at	ADP
ejpam-6429	47	35	most	most	ADV
ejpam-6429	47	36	once	once	ADV
ejpam-6429	47	37	.	.	PUNCT
ejpam-6429	48	1	this	this	DET
ejpam-6429	48	2	line	line	NOUN
ejpam-6429	48	3	of	of	ADP
ejpam-6429	48	4	research	research	NOUN
ejpam-6429	48	5	has	have	AUX
ejpam-6429	48	6	since	since	ADV
ejpam-6429	48	7	been	be	AUX
ejpam-6429	48	8	extended	extend	VERB
ejpam-6429	48	9	to	to	ADP
ejpam-6429	48	10	various	various	ADJ
ejpam-6429	48	11	domination	domination	NOUN
ejpam-6429	48	12	parameters	parameter	NOUN
ejpam-6429	48	13	[	[	X
ejpam-6429	48	14	15–22	15–22	NUM
ejpam-6429	48	15	]	]	PUNCT
ejpam-6429	48	16	.	.	PUNCT
ejpam-6429	49	1	in	in	ADP
ejpam-6429	49	2	this	this	DET
ejpam-6429	49	3	paper	paper	NOUN
ejpam-6429	49	4	,	,	PUNCT
ejpam-6429	49	5	we	we	PRON
ejpam-6429	49	6	extend	extend	VERB
ejpam-6429	49	7	the	the	DET
ejpam-6429	49	8	study	study	NOUN
ejpam-6429	49	9	of	of	ADP
ejpam-6429	49	10	generous	generous	ADJ
ejpam-6429	49	11	roman	roman	ADJ
ejpam-6429	49	12	domination	domination	NOUN
ejpam-6429	49	13	by	by	ADP
ejpam-6429	49	14	examining	examine	VERB
ejpam-6429	49	15	its	its	PRON
ejpam-6429	49	16	behavior	behavior	NOUN
ejpam-6429	49	17	under	under	ADP
ejpam-6429	49	18	edge	edge	NOUN
ejpam-6429	49	19	subdivision	subdivision	NOUN
ejpam-6429	49	20	.	.	PUNCT
ejpam-6429	50	1	to	to	ADP
ejpam-6429	50	2	this	this	DET
ejpam-6429	50	3	end	end	NOUN
ejpam-6429	50	4	,	,	PUNCT
ejpam-6429	50	5	we	we	PRON
ejpam-6429	50	6	introduce	introduce	VERB
ejpam-6429	50	7	the	the	DET
ejpam-6429	50	8	generous	generous	ADJ
ejpam-6429	50	9	roman	roman	ADJ
ejpam-6429	50	10	domination	domination	NOUN
ejpam-6429	50	11	subdivision	subdivision	NOUN
ejpam-6429	50	12	number	number	NOUN
ejpam-6429	50	13	,	,	PUNCT
ejpam-6429	50	14	which	which	PRON
ejpam-6429	50	15	is	be	AUX
ejpam-6429	50	16	defined	define	VERB
ejpam-6429	50	17	as	as	ADP
ejpam-6429	50	18	the	the	DET
ejpam-6429	50	19	minimum	minimum	ADJ
ejpam-6429	50	20	number	number	NOUN
ejpam-6429	50	21	of	of	ADP
ejpam-6429	50	22	edges	edge	NOUN
ejpam-6429	50	23	in	in	ADP
ejpam-6429	50	24	a	a	DET
ejpam-6429	50	25	graph	graph	NOUN
ejpam-6429	50	26	g	g	NOUN
ejpam-6429	50	27	that	that	PRON
ejpam-6429	50	28	must	must	AUX
ejpam-6429	50	29	be	be	AUX
ejpam-6429	50	30	subdivided	subdivide	VERB
ejpam-6429	50	31	(	(	PUNCT
ejpam-6429	50	32	each	each	PRON
ejpam-6429	50	33	at	at	ADV
ejpam-6429	50	34	most	most	ADV
ejpam-6429	50	35	once	once	ADV
ejpam-6429	50	36	)	)	PUNCT
ejpam-6429	50	37	in	in	ADP
ejpam-6429	50	38	order	order	NOUN
ejpam-6429	50	39	to	to	PART
ejpam-6429	50	40	increase	increase	VERB
ejpam-6429	50	41	the	the	DET
ejpam-6429	50	42	generous	generous	ADJ
ejpam-6429	50	43	roman	roman	ADJ
ejpam-6429	50	44	domination	domination	NOUN
ejpam-6429	50	45	number	number	NOUN
ejpam-6429	50	46	of	of	ADP
ejpam-6429	50	47	g.	g.	NOUN
ejpam-6429	50	48	we	we	PRON
ejpam-6429	50	49	establish	establish	VERB
ejpam-6429	50	50	general	general	ADJ
ejpam-6429	50	51	upper	upper	ADJ
ejpam-6429	50	52	bounds	bound	NOUN
ejpam-6429	50	53	on	on	ADP
ejpam-6429	50	54	generous	generous	ADJ
ejpam-6429	50	55	roman	roman	ADJ
ejpam-6429	50	56	domination	domination	NOUN
ejpam-6429	50	57	subdivision	subdivision	NOUN
ejpam-6429	50	58	number	number	NOUN
ejpam-6429	50	59	and	and	CCONJ
ejpam-6429	50	60	determine	determine	VERB
ejpam-6429	50	61	its	its	PRON
ejpam-6429	50	62	exact	exact	ADJ
ejpam-6429	50	63	value	value	NOUN
ejpam-6429	50	64	for	for	ADP
ejpam-6429	50	65	several	several	ADJ
ejpam-6429	50	66	families	family	NOUN
ejpam-6429	50	67	of	of	ADP
ejpam-6429	50	68	graphs	graph	NOUN
ejpam-6429	50	69	,	,	PUNCT
ejpam-6429	50	70	including	include	VERB
ejpam-6429	50	71	paths	path	NOUN
ejpam-6429	50	72	,	,	PUNCT
ejpam-6429	50	73	cycles	cycle	NOUN
ejpam-6429	50	74	,	,	PUNCT
ejpam-6429	50	75	and	and	CCONJ
ejpam-6429	50	76	ladders	ladder	NOUN
ejpam-6429	50	77	.	.	PUNCT
ejpam-6429	51	1	in	in	ADP
ejpam-6429	51	2	addition	addition	NOUN
ejpam-6429	51	3	,	,	PUNCT
ejpam-6429	51	4	we	we	PRON
ejpam-6429	51	5	provide	provide	VERB
ejpam-6429	51	6	sufficient	sufficient	ADJ
ejpam-6429	51	7	conditions	condition	NOUN
ejpam-6429	51	8	under	under	ADP
ejpam-6429	51	9	which	which	PRON
ejpam-6429	51	10	a	a	DET
ejpam-6429	51	11	graph	graph	NOUN
ejpam-6429	51	12	g	g	PROPN
ejpam-6429	51	13	admits	admit	VERB
ejpam-6429	51	14	a	a	DET
ejpam-6429	51	15	small	small	ADJ
ejpam-6429	51	16	generous	generous	ADJ
ejpam-6429	51	17	roman	roman	ADJ
ejpam-6429	51	18	domination	domination	NOUN
ejpam-6429	51	19	subdivision	subdivision	NOUN
ejpam-6429	51	20	number	number	NOUN
ejpam-6429	51	21	.	.	PUNCT
ejpam-6429	52	1	2	2	X
ejpam-6429	52	2	.	.	X
ejpam-6429	52	3	terminology	terminology	NOUN
ejpam-6429	52	4	and	and	CCONJ
ejpam-6429	52	5	notation	notation	NOUN
ejpam-6429	52	6	we	we	PRON
ejpam-6429	52	7	consider	consider	VERB
ejpam-6429	52	8	finite	finite	ADJ
ejpam-6429	52	9	,	,	PUNCT
ejpam-6429	52	10	undirected	undirected	ADJ
ejpam-6429	52	11	,	,	PUNCT
ejpam-6429	52	12	and	and	CCONJ
ejpam-6429	52	13	simple	simple	ADJ
ejpam-6429	52	14	graphs	graph	NOUN
ejpam-6429	52	15	g	g	NOUN
ejpam-6429	52	16	,	,	PUNCT
ejpam-6429	52	17	where	where	SCONJ
ejpam-6429	52	18	the	the	DET
ejpam-6429	52	19	vertex	vertex	NOUN
ejpam-6429	52	20	set	set	NOUN
ejpam-6429	52	21	is	be	AUX
ejpam-6429	52	22	denoted	denote	VERB
ejpam-6429	52	23	by	by	ADP
ejpam-6429	52	24	v	v	NOUN
ejpam-6429	52	25	=	=	SYM
ejpam-6429	52	26	v	v	NOUN
ejpam-6429	52	27	(	(	PUNCT
ejpam-6429	52	28	g	g	NOUN
ejpam-6429	52	29	)	)	PUNCT
ejpam-6429	52	30	and	and	CCONJ
ejpam-6429	52	31	the	the	DET
ejpam-6429	52	32	edge	edge	NOUN
ejpam-6429	52	33	set	set	VERB
ejpam-6429	52	34	by	by	ADP
ejpam-6429	52	35	e	e	PROPN
ejpam-6429	52	36	=	=	PROPN
ejpam-6429	52	37	e(g	e(g	PROPN
ejpam-6429	52	38	)	)	PUNCT
ejpam-6429	52	39	.	.	PUNCT
ejpam-6429	53	1	the	the	DET
ejpam-6429	53	2	order	order	NOUN
ejpam-6429	53	3	of	of	ADP
ejpam-6429	53	4	g	g	PROPN
ejpam-6429	53	5	is	be	AUX
ejpam-6429	53	6	given	give	VERB
ejpam-6429	53	7	by	by	ADP
ejpam-6429	53	8	|v	|v	PROPN
ejpam-6429	53	9	|	|	NOUN
ejpam-6429	53	10	=	=	SYM
ejpam-6429	53	11	n	n	CCONJ
ejpam-6429	53	12	,	,	PUNCT
ejpam-6429	53	13	while	while	SCONJ
ejpam-6429	53	14	the	the	DET
ejpam-6429	53	15	size	size	NOUN
ejpam-6429	53	16	of	of	ADP
ejpam-6429	53	17	g	g	PROPN
ejpam-6429	53	18	is	be	AUX
ejpam-6429	53	19	represented	represent	VERB
ejpam-6429	53	20	as	as	ADP
ejpam-6429	53	21	|e|	|e|	PROPN
ejpam-6429	53	22	=	=	PUNCT
ejpam-6429	53	23	m.	m.	NOUN
ejpam-6429	53	24	the	the	DET
ejpam-6429	53	25	open	open	ADJ
ejpam-6429	53	26	neighborhood	neighborhood	NOUN
ejpam-6429	53	27	of	of	ADP
ejpam-6429	53	28	a	a	DET
ejpam-6429	53	29	vertex	vertex	NOUN
ejpam-6429	53	30	v	v	ADP
ejpam-6429	53	31	∈	∈	NOUN
ejpam-6429	53	32	v	v	NOUN
ejpam-6429	53	33	is	be	AUX
ejpam-6429	53	34	defined	define	VERB
ejpam-6429	53	35	as	as	ADP
ejpam-6429	53	36	n(v	n(v	NOUN
ejpam-6429	53	37	)	)	PUNCT
ejpam-6429	53	38	=	=	PUNCT
ejpam-6429	53	39	ng(v	ng(v	X
ejpam-6429	53	40	)	)	PUNCT
ejpam-6429	53	41	=	=	PRON
ejpam-6429	53	42	{	{	PUNCT
ejpam-6429	53	43	u	u	NOUN
ejpam-6429	53	44	∈	∈	PROPN
ejpam-6429	53	45	v	v	ADP
ejpam-6429	53	46	|	|	ADV
ejpam-6429	53	47	uv	uv	NOUN
ejpam-6429	53	48	∈	∈	NOUN
ejpam-6429	53	49	e	e	NOUN
ejpam-6429	53	50	}	}	PUNCT
ejpam-6429	53	51	,	,	PUNCT
ejpam-6429	53	52	whereas	whereas	SCONJ
ejpam-6429	53	53	its	its	PRON
ejpam-6429	53	54	closed	closed	ADJ
ejpam-6429	53	55	neighborhood	neighborhood	NOUN
ejpam-6429	53	56	is	be	AUX
ejpam-6429	53	57	given	give	VERB
ejpam-6429	53	58	by	by	ADP
ejpam-6429	53	59	n	n	PRON
ejpam-6429	53	60	[	[	X
ejpam-6429	53	61	v	v	NOUN
ejpam-6429	53	62	]	]	X
ejpam-6429	53	63	=	=	PUNCT
ejpam-6429	53	64	n(v	n(v	PROPN
ejpam-6429	53	65	)	)	PUNCT
ejpam-6429	53	66	∪	∪	NOUN
ejpam-6429	53	67	{	{	PUNCT
ejpam-6429	53	68	v	v	NOUN
ejpam-6429	53	69	}	}	PUNCT
ejpam-6429	53	70	.	.	PUNCT
ejpam-6429	54	1	the	the	DET
ejpam-6429	54	2	degree	degree	NOUN
ejpam-6429	54	3	of	of	ADP
ejpam-6429	54	4	a	a	DET
ejpam-6429	54	5	vertex	vertex	NOUN
ejpam-6429	54	6	v	v	NOUN
ejpam-6429	54	7	,	,	PUNCT
ejpam-6429	54	8	denoted	denote	VERB
ejpam-6429	54	9	by	by	ADP
ejpam-6429	54	10	degg(v	degg(v	PROPN
ejpam-6429	54	11	)	)	PUNCT
ejpam-6429	54	12	=	=	SYM
ejpam-6429	54	13	deg(v	deg(v	PROPN
ejpam-6429	54	14	)	)	PUNCT
ejpam-6429	54	15	,	,	PUNCT
ejpam-6429	54	16	refers	refer	VERB
ejpam-6429	54	17	to	to	ADP
ejpam-6429	54	18	the	the	DET
ejpam-6429	54	19	number	number	NOUN
ejpam-6429	54	20	of	of	ADP
ejpam-6429	54	21	neighbors	neighbor	NOUN
ejpam-6429	54	22	of	of	ADP
ejpam-6429	54	23	v	v	NOUN
ejpam-6429	54	24	,	,	PUNCT
ejpam-6429	54	25	i.e.	i.e.	X
ejpam-6429	54	26	,	,	PUNCT
ejpam-6429	54	27	degg(v	degg(v	PROPN
ejpam-6429	54	28	)	)	PUNCT
ejpam-6429	54	29	=	=	SYM
ejpam-6429	54	30	|ng(v)|	|ng(v)|	NOUN
ejpam-6429	54	31	.	.	PUNCT
ejpam-6429	55	1	as	as	ADP
ejpam-6429	55	2	usual	usual	ADJ
ejpam-6429	55	3	,	,	PUNCT
ejpam-6429	55	4	a	a	DET
ejpam-6429	55	5	path	path	NOUN
ejpam-6429	55	6	,	,	PUNCT
ejpam-6429	55	7	cycle	cycle	NOUN
ejpam-6429	55	8	,	,	PUNCT
ejpam-6429	55	9	star	star	NOUN
ejpam-6429	55	10	,	,	PUNCT
ejpam-6429	55	11	and	and	CCONJ
ejpam-6429	55	12	complete	complete	ADJ
ejpam-6429	55	13	graph	graph	NOUN
ejpam-6429	55	14	with	with	ADP
ejpam-6429	55	15	n	n	DET
ejpam-6429	55	16	vertices	vertex	NOUN
ejpam-6429	55	17	are	be	AUX
ejpam-6429	55	18	denoted	denote	VERB
ejpam-6429	55	19	by	by	ADP
ejpam-6429	55	20	pn	pn	PROPN
ejpam-6429	55	21	,	,	PUNCT
ejpam-6429	55	22	cn	cn	PROPN
ejpam-6429	55	23	,	,	PUNCT
ejpam-6429	55	24	k1,n−1	k1,n−1	ADJ
ejpam-6429	55	25	,	,	PUNCT
ejpam-6429	55	26	and	and	CCONJ
ejpam-6429	55	27	kn	kn	PROPN
ejpam-6429	55	28	,	,	PUNCT
ejpam-6429	55	29	respectively	respectively	ADV
ejpam-6429	55	30	.	.	PUNCT
ejpam-6429	56	1	similarly	similarly	ADV
ejpam-6429	56	2	,	,	PUNCT
ejpam-6429	56	3	kn	kn	PROPN
ejpam-6429	56	4	,	,	PUNCT
ejpam-6429	56	5	m	m	VERB
ejpam-6429	56	6	denotes	denote	VERB
ejpam-6429	56	7	the	the	DET
ejpam-6429	56	8	complete	complete	ADJ
ejpam-6429	56	9	bipartite	bipartite	NOUN
ejpam-6429	56	10	graph	graph	NOUN
ejpam-6429	56	11	of	of	ADP
ejpam-6429	56	12	order	order	NOUN
ejpam-6429	56	13	m+	m+	NUM
ejpam-6429	56	14	n	n	CCONJ
ejpam-6429	56	15	,	,	PUNCT
ejpam-6429	56	16	while	while	SCONJ
ejpam-6429	56	17	dsp	dsp	NOUN
ejpam-6429	56	18	,	,	PUNCT
ejpam-6429	56	19	q	q	PROPN
ejpam-6429	56	20	denotes	denote	VERB
ejpam-6429	56	21	the	the	DET
ejpam-6429	56	22	double	double	ADJ
ejpam-6429	56	23	star	star	NOUN
ejpam-6429	56	24	of	of	ADP
ejpam-6429	56	25	order	order	NOUN
ejpam-6429	56	26	p	p	NOUN
ejpam-6429	57	1	+	+	NOUN
ejpam-6429	57	2	q	q	NOUN
ejpam-6429	57	3	+	+	NUM
ejpam-6429	57	4	2	2	X
ejpam-6429	57	5	.	.	X
ejpam-6429	57	6	a	a	DET
ejpam-6429	57	7	vertex	vertex	NOUN
ejpam-6429	57	8	with	with	ADP
ejpam-6429	57	9	degree	degree	NOUN
ejpam-6429	57	10	one	one	PRON
ejpam-6429	57	11	is	be	AUX
ejpam-6429	57	12	referred	refer	VERB
ejpam-6429	57	13	to	to	ADP
ejpam-6429	57	14	as	as	ADP
ejpam-6429	57	15	a	a	DET
ejpam-6429	57	16	leaf	leaf	NOUN
ejpam-6429	57	17	,	,	PUNCT
ejpam-6429	57	18	and	and	CCONJ
ejpam-6429	57	19	its	its	PRON
ejpam-6429	57	20	adjacent	adjacent	ADJ
ejpam-6429	57	21	vertex	vertex	NOUN
ejpam-6429	57	22	is	be	AUX
ejpam-6429	57	23	known	know	VERB
ejpam-6429	57	24	as	as	ADP
ejpam-6429	57	25	a	a	DET
ejpam-6429	57	26	support	support	NOUN
ejpam-6429	57	27	vertex	vertex	NOUN
ejpam-6429	57	28	.	.	PUNCT
ejpam-6429	58	1	a	a	DET
ejpam-6429	58	2	vertex	vertex	NOUN
ejpam-6429	58	3	connected	connect	VERB
ejpam-6429	58	4	to	to	ADP
ejpam-6429	58	5	at	at	ADV
ejpam-6429	58	6	least	least	ADV
ejpam-6429	58	7	two	two	NUM
ejpam-6429	58	8	leaves	leave	NOUN
ejpam-6429	58	9	is	be	AUX
ejpam-6429	58	10	called	call	VERB
ejpam-6429	58	11	a	a	DET
ejpam-6429	58	12	strong	strong	ADJ
ejpam-6429	58	13	support	support	NOUN
ejpam-6429	58	14	vertex	vertex	NOUN
ejpam-6429	58	15	.	.	PUNCT
ejpam-6429	59	1	the	the	DET
ejpam-6429	59	2	cartesian	cartesian	ADJ
ejpam-6429	59	3	product	product	NOUN
ejpam-6429	59	4	of	of	ADP
ejpam-6429	59	5	graphs	graph	NOUN
ejpam-6429	59	6	g	g	PROPN
ejpam-6429	59	7	and	and	CCONJ
ejpam-6429	59	8	h	h	NOUN
ejpam-6429	59	9	,	,	PUNCT
ejpam-6429	59	10	denoted	denote	VERB
ejpam-6429	59	11	by	by	ADP
ejpam-6429	59	12	g	g	PROPN
ejpam-6429	59	13	□	□	PROPN
ejpam-6429	59	14	h	h	NOUN
ejpam-6429	59	15	,	,	PUNCT
ejpam-6429	59	16	is	be	AUX
ejpam-6429	59	17	the	the	DET
ejpam-6429	59	18	graph	graph	NOUN
ejpam-6429	59	19	with	with	ADP
ejpam-6429	59	20	vertex	vertex	NOUN
ejpam-6429	59	21	set	set	VERB
ejpam-6429	59	22	v	v	NOUN
ejpam-6429	59	23	(	(	PUNCT
ejpam-6429	59	24	g	g	NOUN
ejpam-6429	59	25	□	□	NOUN
ejpam-6429	59	26	h	h	NOUN
ejpam-6429	59	27	)	)	PUNCT
ejpam-6429	59	28	=	=	NOUN
ejpam-6429	59	29	v	v	X
ejpam-6429	59	30	(	(	PUNCT
ejpam-6429	59	31	g)×v	g)×v	PROPN
ejpam-6429	59	32	(	(	PUNCT
ejpam-6429	59	33	h	h	NOUN
ejpam-6429	59	34	)	)	PUNCT
ejpam-6429	59	35	such	such	ADJ
ejpam-6429	59	36	that	that	SCONJ
ejpam-6429	59	37	two	two	NUM
ejpam-6429	59	38	vertices	vertex	NOUN
ejpam-6429	59	39	(	(	PUNCT
ejpam-6429	59	40	v	v	NOUN
ejpam-6429	59	41	,	,	PUNCT
ejpam-6429	59	42	p	p	NOUN
ejpam-6429	59	43	)	)	PUNCT
ejpam-6429	59	44	and	and	CCONJ
ejpam-6429	59	45	(	(	PUNCT
ejpam-6429	59	46	u	u	NOUN
ejpam-6429	59	47	,	,	PUNCT
ejpam-6429	59	48	q	q	NOUN
ejpam-6429	59	49	)	)	PUNCT
ejpam-6429	59	50	are	be	AUX
ejpam-6429	59	51	adjacent	adjacent	ADJ
ejpam-6429	59	52	in	in	ADP
ejpam-6429	59	53	g	g	PROPN
ejpam-6429	59	54	□	□	PROPN
ejpam-6429	59	55	h	h	NOUN
ejpam-6429	59	56	,	,	PUNCT
ejpam-6429	59	57	i.e.	i.e.	X
ejpam-6429	59	58	,	,	PUNCT
ejpam-6429	59	59	(	(	PUNCT
ejpam-6429	59	60	v	v	NOUN
ejpam-6429	59	61	,	,	PUNCT
ejpam-6429	59	62	p)(u	p)(u	ADJ
ejpam-6429	59	63	,	,	PUNCT
ejpam-6429	59	64	q	q	ADJ
ejpam-6429	59	65	)	)	PUNCT
ejpam-6429	59	66	∈	∈	PROPN
ejpam-6429	59	67	e(g	e(g	PROPN
ejpam-6429	59	68	□	□	PUNCT
ejpam-6429	59	69	h	h	NOUN
ejpam-6429	59	70	)	)	PUNCT
ejpam-6429	59	71	,	,	PUNCT
ejpam-6429	59	72	if	if	SCONJ
ejpam-6429	59	73	and	and	CCONJ
ejpam-6429	59	74	only	only	ADV
ejpam-6429	59	75	if	if	SCONJ
ejpam-6429	59	76	one	one	NUM
ejpam-6429	59	77	of	of	ADP
ejpam-6429	59	78	the	the	DET
ejpam-6429	59	79	following	follow	VERB
ejpam-6429	59	80	holds	hold	VERB
ejpam-6429	59	81	:	:	PUNCT
ejpam-6429	59	82	v	v	NOUN
ejpam-6429	59	83	=	=	SYM
ejpam-6429	59	84	u	u	NOUN
ejpam-6429	59	85	and	and	CCONJ
ejpam-6429	59	86	pq	pq	PROPN
ejpam-6429	59	87	∈	∈	PROPN
ejpam-6429	59	88	e(h	e(h	PROPN
ejpam-6429	59	89	)	)	PUNCT
ejpam-6429	59	90	,	,	PUNCT
ejpam-6429	59	91	or	or	CCONJ
ejpam-6429	59	92	p	p	NOUN
ejpam-6429	59	93	=	=	NOUN
ejpam-6429	59	94	q	q	NOUN
ejpam-6429	59	95	and	and	CCONJ
ejpam-6429	59	96	vu	vu	PROPN
ejpam-6429	59	97	∈	∈	PROPN
ejpam-6429	59	98	e(g	e(g	PROPN
ejpam-6429	59	99	)	)	PUNCT
ejpam-6429	59	100	.	.	PUNCT
ejpam-6429	60	1	for	for	ADP
ejpam-6429	60	2	any	any	DET
ejpam-6429	60	3	subset	subset	NOUN
ejpam-6429	60	4	a	a	PRON
ejpam-6429	60	5	⊆	⊆	NUM
ejpam-6429	60	6	v	v	NOUN
ejpam-6429	60	7	(	(	PUNCT
ejpam-6429	60	8	g	g	NOUN
ejpam-6429	60	9	)	)	PUNCT
ejpam-6429	60	10	and	and	CCONJ
ejpam-6429	60	11	a	a	DET
ejpam-6429	60	12	function	function	NOUN
ejpam-6429	60	13	f	f	NOUN
ejpam-6429	60	14	that	that	PRON
ejpam-6429	60	15	maps	map	VERB
ejpam-6429	60	16	v	v	ADP
ejpam-6429	60	17	(	(	PUNCT
ejpam-6429	60	18	g	g	NOUN
ejpam-6429	60	19	)	)	PUNCT
ejpam-6429	60	20	to	to	ADP
ejpam-6429	60	21	a	a	DET
ejpam-6429	60	22	numerical	numerical	ADJ
ejpam-6429	60	23	set	set	NOUN
ejpam-6429	60	24	,	,	PUNCT
ejpam-6429	60	25	the	the	DET
ejpam-6429	60	26	function	function	NOUN
ejpam-6429	60	27	sum	sum	NOUN
ejpam-6429	60	28	over	over	ADP
ejpam-6429	60	29	a	a	PRON
ejpam-6429	60	30	is	be	AUX
ejpam-6429	60	31	given	give	VERB
ejpam-6429	60	32	by	by	ADP
ejpam-6429	60	33	f(a	f(a	PROPN
ejpam-6429	60	34	)	)	PUNCT
ejpam-6429	61	1	=	=	SYM
ejpam-6429	61	2	∑	∑	PUNCT
ejpam-6429	61	3	x∈a	x∈a	VERB
ejpam-6429	61	4	f(x	f(x	PROPN
ejpam-6429	61	5	)	)	PUNCT
ejpam-6429	61	6	.	.	PUNCT
ejpam-6429	62	1	the	the	DET
ejpam-6429	62	2	total	total	ADJ
ejpam-6429	62	3	sum	sum	NOUN
ejpam-6429	62	4	over	over	ADP
ejpam-6429	62	5	all	all	DET
ejpam-6429	62	6	vertices	vertex	NOUN
ejpam-6429	62	7	,	,	PUNCT
ejpam-6429	62	8	f(v	f(v	PROPN
ejpam-6429	62	9	(	(	PUNCT
ejpam-6429	62	10	g	g	NOUN
ejpam-6429	62	11	)	)	PUNCT
ejpam-6429	62	12	)	)	PUNCT
ejpam-6429	62	13	,	,	PUNCT
ejpam-6429	62	14	is	be	AUX
ejpam-6429	62	15	referred	refer	VERB
ejpam-6429	62	16	to	to	ADP
ejpam-6429	62	17	as	as	ADP
ejpam-6429	62	18	the	the	DET
ejpam-6429	62	19	weight	weight	NOUN
ejpam-6429	62	20	of	of	ADP
ejpam-6429	62	21	f	f	PROPN
ejpam-6429	62	22	,	,	PUNCT
ejpam-6429	62	23	denoted	denote	VERB
ejpam-6429	62	24	by	by	ADP
ejpam-6429	62	25	ω(f	ω(f	NOUN
ejpam-6429	62	26	)	)	PUNCT
ejpam-6429	62	27	.	.	PUNCT
ejpam-6429	63	1	let	let	VERB
ejpam-6429	63	2	f	f	PRON
ejpam-6429	63	3	be	be	AUX
ejpam-6429	63	4	a	a	DET
ejpam-6429	63	5	function	function	NOUN
ejpam-6429	63	6	from	from	ADP
ejpam-6429	63	7	v	v	PRON
ejpam-6429	63	8	(	(	PUNCT
ejpam-6429	63	9	g	g	NOUN
ejpam-6429	63	10	)	)	PUNCT
ejpam-6429	63	11	to	to	ADP
ejpam-6429	63	12	{	{	PUNCT
ejpam-6429	63	13	0	0	NUM
ejpam-6429	63	14	,	,	PUNCT
ejpam-6429	63	15	1	1	NUM
ejpam-6429	63	16	,	,	PUNCT
ejpam-6429	63	17	2	2	NUM
ejpam-6429	63	18	,	,	PUNCT
ejpam-6429	63	19	3	3	NUM
ejpam-6429	63	20	}	}	PUNCT
ejpam-6429	63	21	.	.	PUNCT
ejpam-6429	64	1	a	a	DET
ejpam-6429	64	2	vertex	vertex	NOUN
ejpam-6429	64	3	u	u	NOUN
ejpam-6429	64	4	is	be	AUX
ejpam-6429	64	5	considered	consider	VERB
ejpam-6429	64	6	undefended	undefended	ADJ
ejpam-6429	64	7	with	with	ADP
ejpam-6429	64	8	respect	respect	NOUN
ejpam-6429	64	9	to	to	ADP
ejpam-6429	64	10	f	f	PROPN
ejpam-6429	64	11	if	if	SCONJ
ejpam-6429	64	12	f(u	f(u	PROPN
ejpam-6429	64	13	)	)	PUNCT
ejpam-6429	65	1	=	=	SYM
ejpam-6429	65	2	0	0	NUM
ejpam-6429	66	1	and	and	CCONJ
ejpam-6429	66	2	no	no	DET
ejpam-6429	66	3	adjacent	adjacent	ADJ
ejpam-6429	66	4	vertex	vertex	NOUN
ejpam-6429	66	5	v	v	ADP
ejpam-6429	66	6	satisfies	satisfie	NOUN
ejpam-6429	66	7	f(v	f(v	NOUN
ejpam-6429	66	8	)	)	PUNCT
ejpam-6429	66	9	≥	≥	NOUN
ejpam-6429	66	10	2	2	NUM
ejpam-6429	66	11	.	.	PUNCT
ejpam-6429	67	1	the	the	DET
ejpam-6429	67	2	function	function	NOUN
ejpam-6429	67	3	f	f	PROPN
ejpam-6429	67	4	is	be	AUX
ejpam-6429	67	5	a	a	DET
ejpam-6429	67	6	generous	generous	ADJ
ejpam-6429	67	7	roman	roman	ADJ
ejpam-6429	67	8	dominating	dominating	NOUN
ejpam-6429	67	9	function	function	NOUN
ejpam-6429	67	10	(	(	PUNCT
ejpam-6429	67	11	grd	grd	NOUN
ejpam-6429	67	12	-	-	PUNCT
ejpam-6429	67	13	function	function	NOUN
ejpam-6429	67	14	)	)	PUNCT
ejpam-6429	67	15	if	if	SCONJ
ejpam-6429	67	16	,	,	PUNCT
ejpam-6429	67	17	for	for	ADP
ejpam-6429	67	18	every	every	DET
ejpam-6429	67	19	vertex	vertex	NOUN
ejpam-6429	67	20	u	u	NOUN
ejpam-6429	67	21	with	with	ADP
ejpam-6429	67	22	j.	j.	PROPN
ejpam-6429	67	23	j.	j.	PROPN
ejpam-6429	67	24	hamja	hamja	PROPN
ejpam-6429	67	25	et	et	PROPN
ejpam-6429	67	26	al	al	PROPN
ejpam-6429	67	27	.	.	PUNCT
ejpam-6429	67	28	/	/	SYM
ejpam-6429	67	29	eur	eur	PROPN
ejpam-6429	67	30	.	.	PUNCT
ejpam-6429	68	1	j.	j.	PROPN
ejpam-6429	68	2	pure	pure	PROPN
ejpam-6429	68	3	appl	appl	PROPN
ejpam-6429	68	4	.	.	PROPN
ejpam-6429	68	5	math	math	PROPN
ejpam-6429	68	6	,	,	PUNCT
ejpam-6429	68	7	18	18	NUM
ejpam-6429	68	8	(	(	PUNCT
ejpam-6429	68	9	4	4	NUM
ejpam-6429	68	10	)	)	PUNCT
ejpam-6429	68	11	(	(	PUNCT
ejpam-6429	68	12	2025	2025	NUM
ejpam-6429	68	13	)	)	PUNCT
ejpam-6429	68	14	,	,	PUNCT
ejpam-6429	68	15	6429	6429	NUM
ejpam-6429	68	16	4	4	NUM
ejpam-6429	68	17	of	of	ADP
ejpam-6429	68	18	16	16	NUM
ejpam-6429	68	19	f(u	f(u	PROPN
ejpam-6429	68	20	)	)	PUNCT
ejpam-6429	69	1	=	=	SYM
ejpam-6429	69	2	0	0	NUM
ejpam-6429	69	3	,	,	PUNCT
ejpam-6429	69	4	there	there	PRON
ejpam-6429	69	5	exists	exist	VERB
ejpam-6429	69	6	at	at	ADP
ejpam-6429	69	7	least	least	ADV
ejpam-6429	69	8	one	one	NUM
ejpam-6429	69	9	adjacent	adjacent	ADJ
ejpam-6429	69	10	vertex	vertex	NOUN
ejpam-6429	69	11	v	v	NOUN
ejpam-6429	69	12	with	with	ADP
ejpam-6429	69	13	f(v	f(v	NOUN
ejpam-6429	69	14	)	)	PUNCT
ejpam-6429	69	15	≥	≥	NOUN
ejpam-6429	69	16	2	2	NUM
ejpam-6429	69	17	such	such	ADJ
ejpam-6429	69	18	that	that	SCONJ
ejpam-6429	69	19	the	the	DET
ejpam-6429	69	20	function	function	NOUN
ejpam-6429	69	21	g	g	NOUN
ejpam-6429	69	22	:	:	PUNCT
ejpam-6429	69	23	v	v	NOUN
ejpam-6429	69	24	→	→	SYM
ejpam-6429	69	25	{	{	PUNCT
ejpam-6429	69	26	0	0	NUM
ejpam-6429	69	27	,	,	PUNCT
ejpam-6429	69	28	1	1	NUM
ejpam-6429	69	29	,	,	PUNCT
ejpam-6429	69	30	2	2	NUM
ejpam-6429	69	31	,	,	PUNCT
ejpam-6429	69	32	3	3	NUM
ejpam-6429	69	33	}	}	PUNCT
ejpam-6429	69	34	,	,	PUNCT
ejpam-6429	69	35	given	give	VERB
ejpam-6429	69	36	by	by	ADP
ejpam-6429	69	37	g(u	g(u	PROPN
ejpam-6429	69	38	)	)	PUNCT
ejpam-6429	69	39	=	=	SYM
ejpam-6429	69	40	α	α	NOUN
ejpam-6429	69	41	,	,	PUNCT
ejpam-6429	69	42	g(v	g(v	PROPN
ejpam-6429	69	43	)	)	PUNCT
ejpam-6429	69	44	=	=	SYM
ejpam-6429	69	45	f(v	f(v	NOUN
ejpam-6429	69	46	)	)	PUNCT
ejpam-6429	70	1	−	−	PROPN
ejpam-6429	70	2	α	α	NOUN
ejpam-6429	70	3	,	,	PUNCT
ejpam-6429	70	4	where	where	SCONJ
ejpam-6429	70	5	α	α	X
ejpam-6429	70	6	∈	∈	PROPN
ejpam-6429	70	7	{	{	PUNCT
ejpam-6429	70	8	1	1	NUM
ejpam-6429	70	9	,	,	PUNCT
ejpam-6429	70	10	2	2	NUM
ejpam-6429	70	11	}	}	PUNCT
ejpam-6429	70	12	,	,	PUNCT
ejpam-6429	70	13	and	and	CCONJ
ejpam-6429	70	14	g(w	g(w	PROPN
ejpam-6429	70	15	)	)	PUNCT
ejpam-6429	70	16	=	=	SYM
ejpam-6429	70	17	f(w	f(w	PROPN
ejpam-6429	70	18	)	)	PUNCT
ejpam-6429	70	19	,	,	PUNCT
ejpam-6429	70	20	for	for	ADP
ejpam-6429	70	21	all	all	DET
ejpam-6429	70	22	w	w	NOUN
ejpam-6429	70	23	∈	∈	PROPN
ejpam-6429	70	24	v	v	ADP
ejpam-6429	70	25	\	\	NOUN
ejpam-6429	70	26	{	{	PUNCT
ejpam-6429	70	27	u	u	NOUN
ejpam-6429	70	28	,	,	PUNCT
ejpam-6429	70	29	v	v	NOUN
ejpam-6429	70	30	}	}	PUNCT
ejpam-6429	70	31	,	,	PUNCT
ejpam-6429	70	32	ensures	ensure	VERB
ejpam-6429	70	33	that	that	SCONJ
ejpam-6429	70	34	no	no	DET
ejpam-6429	70	35	vertex	vertex	NOUN
ejpam-6429	70	36	remains	remain	VERB
ejpam-6429	70	37	undefended	undefended	ADJ
ejpam-6429	70	38	.	.	PUNCT
ejpam-6429	71	1	the	the	DET
ejpam-6429	71	2	weight	weight	NOUN
ejpam-6429	71	3	of	of	ADP
ejpam-6429	71	4	a	a	DET
ejpam-6429	71	5	grd	grd	NOUN
ejpam-6429	71	6	-	-	PUNCT
ejpam-6429	71	7	function	function	NOUN
ejpam-6429	71	8	f	f	PROPN
ejpam-6429	71	9	is	be	AUX
ejpam-6429	71	10	defined	define	VERB
ejpam-6429	71	11	as	as	ADP
ejpam-6429	71	12	f(v	f(v	NOUN
ejpam-6429	71	13	)	)	PUNCT
ejpam-6429	72	1	=	=	SYM
ejpam-6429	72	2	∑	∑	PUNCT
ejpam-6429	72	3	u∈v	u∈v	NOUN
ejpam-6429	72	4	f(u	f(u	PROPN
ejpam-6429	72	5	)	)	PUNCT
ejpam-6429	72	6	.	.	PUNCT
ejpam-6429	73	1	the	the	DET
ejpam-6429	73	2	total	total	ADJ
ejpam-6429	73	3	sum	sum	NOUN
ejpam-6429	73	4	over	over	ADP
ejpam-6429	73	5	all	all	DET
ejpam-6429	73	6	vertices	vertex	NOUN
ejpam-6429	73	7	,	,	PUNCT
ejpam-6429	73	8	f(v	f(v	PROPN
ejpam-6429	73	9	)	)	PUNCT
ejpam-6429	73	10	,	,	PUNCT
ejpam-6429	73	11	is	be	AUX
ejpam-6429	73	12	referred	refer	VERB
ejpam-6429	73	13	to	to	ADP
ejpam-6429	73	14	as	as	ADP
ejpam-6429	73	15	the	the	DET
ejpam-6429	73	16	weight	weight	NOUN
ejpam-6429	73	17	of	of	ADP
ejpam-6429	73	18	f	f	PROPN
ejpam-6429	73	19	,	,	PUNCT
ejpam-6429	73	20	denoted	denote	VERB
ejpam-6429	73	21	by	by	ADP
ejpam-6429	73	22	ωg	ωg	NOUN
ejpam-6429	73	23	r(f	r(f	PROPN
ejpam-6429	73	24	)	)	PUNCT
ejpam-6429	73	25	,	,	PUNCT
ejpam-6429	73	26	and	and	CCONJ
ejpam-6429	73	27	the	the	DET
ejpam-6429	73	28	smallest	small	ADJ
ejpam-6429	73	29	possible	possible	ADJ
ejpam-6429	73	30	weight	weight	NOUN
ejpam-6429	73	31	of	of	ADP
ejpam-6429	73	32	a	a	DET
ejpam-6429	73	33	grd	grd	NOUN
ejpam-6429	73	34	-	-	PUNCT
ejpam-6429	73	35	function	function	NOUN
ejpam-6429	73	36	on	on	ADP
ejpam-6429	73	37	g	g	PROPN
ejpam-6429	73	38	is	be	AUX
ejpam-6429	73	39	referred	refer	VERB
ejpam-6429	73	40	to	to	ADP
ejpam-6429	73	41	as	as	ADP
ejpam-6429	73	42	the	the	DET
ejpam-6429	73	43	generous	generous	ADJ
ejpam-6429	73	44	roman	roman	ADJ
ejpam-6429	73	45	domination	domination	NOUN
ejpam-6429	73	46	number	number	NOUN
ejpam-6429	73	47	(	(	PUNCT
ejpam-6429	73	48	grd	grd	NOUN
ejpam-6429	73	49	-	-	PUNCT
ejpam-6429	73	50	number	number	NOUN
ejpam-6429	73	51	)	)	PUNCT
ejpam-6429	73	52	,	,	PUNCT
ejpam-6429	73	53	denoted	denote	VERB
ejpam-6429	73	54	by	by	ADP
ejpam-6429	73	55	γgr(g	γgr(g	PROPN
ejpam-6429	73	56	)	)	PUNCT
ejpam-6429	73	57	,	,	PUNCT
ejpam-6429	73	58	as	as	SCONJ
ejpam-6429	73	59	introduced	introduce	VERB
ejpam-6429	73	60	by	by	ADP
ejpam-6429	73	61	benatallah	benatallah	ADJ
ejpam-6429	73	62	,	,	PUNCT
ejpam-6429	73	63	blidia	blidia	NOUN
ejpam-6429	73	64	,	,	PUNCT
ejpam-6429	73	65	and	and	CCONJ
ejpam-6429	73	66	ouldrabah	ouldrabah	NOUN
ejpam-6429	74	1	[	[	X
ejpam-6429	74	2	11	11	NUM
ejpam-6429	74	3	]	]	PUNCT
ejpam-6429	74	4	.	.	PUNCT
ejpam-6429	75	1	for	for	ADP
ejpam-6429	75	2	any	any	DET
ejpam-6429	75	3	grd	grd	NOUN
ejpam-6429	75	4	-	-	PUNCT
ejpam-6429	75	5	function	function	NOUN
ejpam-6429	75	6	f	f	NOUN
ejpam-6429	75	7	of	of	ADP
ejpam-6429	75	8	g	g	PROPN
ejpam-6429	75	9	,	,	PUNCT
ejpam-6429	75	10	let	let	VERB
ejpam-6429	75	11	vi	vi	NOUN
ejpam-6429	75	12	=	=	PRON
ejpam-6429	75	13	{	{	PUNCT
ejpam-6429	75	14	v	v	NUM
ejpam-6429	75	15	∈	∈	NOUN
ejpam-6429	75	16	v	v	ADP
ejpam-6429	75	17	|	|	ADV
ejpam-6429	75	18	f(v	f(v	PRON
ejpam-6429	75	19	)	)	PUNCT
ejpam-6429	76	1	=	=	PUNCT
ejpam-6429	76	2	i	i	PROPN
ejpam-6429	76	3	}	}	PUNCT
ejpam-6429	76	4	,	,	PUNCT
ejpam-6429	76	5	where	where	SCONJ
ejpam-6429	76	6	i	i	PRON
ejpam-6429	76	7	∈	∈	PROPN
ejpam-6429	76	8	{	{	PUNCT
ejpam-6429	76	9	0	0	NUM
ejpam-6429	76	10	,	,	PUNCT
ejpam-6429	76	11	1	1	NUM
ejpam-6429	76	12	,	,	PUNCT
ejpam-6429	76	13	2	2	NUM
ejpam-6429	76	14	,	,	PUNCT
ejpam-6429	76	15	3	3	NUM
ejpam-6429	76	16	}	}	PUNCT
ejpam-6429	76	17	.	.	PUNCT
ejpam-6429	77	1	since	since	SCONJ
ejpam-6429	77	2	these	these	DET
ejpam-6429	77	3	four	four	NUM
ejpam-6429	77	4	sets	set	NOUN
ejpam-6429	77	5	uniquely	uniquely	ADV
ejpam-6429	77	6	define	define	VERB
ejpam-6429	77	7	f	f	PROPN
ejpam-6429	77	8	,	,	PUNCT
ejpam-6429	77	9	we	we	PRON
ejpam-6429	77	10	can	can	AUX
ejpam-6429	77	11	represent	represent	VERB
ejpam-6429	77	12	f	f	PROPN
ejpam-6429	77	13	as	as	ADP
ejpam-6429	77	14	(	(	PUNCT
ejpam-6429	77	15	v0	v0	NOUN
ejpam-6429	77	16	,	,	PUNCT
ejpam-6429	77	17	v1	v1	NOUN
ejpam-6429	77	18	,	,	PUNCT
ejpam-6429	77	19	v2	v2	PROPN
ejpam-6429	77	20	,	,	PUNCT
ejpam-6429	77	21	v3	v3	PROPN
ejpam-6429	77	22	)	)	PUNCT
ejpam-6429	77	23	.	.	PUNCT
ejpam-6429	78	1	furthermore	furthermore	ADV
ejpam-6429	78	2	,	,	PUNCT
ejpam-6429	78	3	a	a	DET
ejpam-6429	78	4	γgr(g)-function	γgr(g)-function	NOUN
ejpam-6429	78	5	is	be	AUX
ejpam-6429	78	6	a	a	DET
ejpam-6429	78	7	grd	grd	NOUN
ejpam-6429	78	8	-	-	PUNCT
ejpam-6429	78	9	function	function	NOUN
ejpam-6429	78	10	of	of	ADP
ejpam-6429	78	11	g	g	PROPN
ejpam-6429	78	12	if	if	SCONJ
ejpam-6429	78	13	ωg	ωg	PART
ejpam-6429	78	14	r(f	r(f	PROPN
ejpam-6429	78	15	)	)	PUNCT
ejpam-6429	78	16	=	=	SYM
ejpam-6429	78	17	γgr(g	γgr(g	PROPN
ejpam-6429	78	18	)	)	PUNCT
ejpam-6429	78	19	.	.	PUNCT
ejpam-6429	79	1	the	the	DET
ejpam-6429	79	2	generous	generous	ADJ
ejpam-6429	79	3	roman	roman	ADJ
ejpam-6429	79	4	domination	domination	NOUN
ejpam-6429	79	5	is	be	AUX
ejpam-6429	79	6	a	a	DET
ejpam-6429	79	7	variant	variant	NOUN
ejpam-6429	79	8	of	of	ADP
ejpam-6429	79	9	double	double	ADJ
ejpam-6429	79	10	roman	roman	ADJ
ejpam-6429	79	11	domination	domination	NOUN
ejpam-6429	79	12	with	with	ADP
ejpam-6429	79	13	less	less	ADJ
ejpam-6429	79	14	restriction	restriction	NOUN
ejpam-6429	79	15	.	.	PUNCT
ejpam-6429	80	1	importantly	importantly	ADV
ejpam-6429	80	2	,	,	PUNCT
ejpam-6429	80	3	if	if	SCONJ
ejpam-6429	80	4	g1	g1	NOUN
ejpam-6429	80	5	,	,	PUNCT
ejpam-6429	80	6	g2	g2	PROPN
ejpam-6429	80	7	,	,	PUNCT
ejpam-6429	80	8	.	.	PUNCT
ejpam-6429	80	9	.	.	PUNCT
ejpam-6429	81	1	.	.	PUNCT
ejpam-6429	82	1	,	,	PUNCT
ejpam-6429	82	2	gs	gs	PROPN
ejpam-6429	82	3	are	be	AUX
ejpam-6429	82	4	the	the	DET
ejpam-6429	82	5	components	component	NOUN
ejpam-6429	82	6	of	of	ADP
ejpam-6429	82	7	g	g	NOUN
ejpam-6429	82	8	,	,	PUNCT
ejpam-6429	82	9	then	then	ADV
ejpam-6429	82	10	γgr(g	γgr(g	PROPN
ejpam-6429	82	11	)	)	PUNCT
ejpam-6429	83	1	=	=	PUNCT
ejpam-6429	83	2	∑s	∑s	PROPN
ejpam-6429	83	3	i=1	i=1	PROPN
ejpam-6429	83	4	γgr(gi	γgr(gi	PROPN
ejpam-6429	83	5	)	)	PUNCT
ejpam-6429	83	6	.	.	PUNCT
ejpam-6429	84	1	furthermore	furthermore	ADV
ejpam-6429	84	2	,	,	PUNCT
ejpam-6429	84	3	if	if	SCONJ
ejpam-6429	84	4	g1	g1	NOUN
ejpam-6429	84	5	,	,	PUNCT
ejpam-6429	84	6	g2	g2	PROPN
ejpam-6429	84	7	,	,	PUNCT
ejpam-6429	84	8	.	.	PUNCT
ejpam-6429	84	9	.	.	PUNCT
ejpam-6429	85	1	.	.	PUNCT
ejpam-6429	86	1	,	,	PUNCT
ejpam-6429	86	2	gs	gs	PROPN
ejpam-6429	86	3	represent	represent	VERB
ejpam-6429	86	4	the	the	DET
ejpam-6429	86	5	components	component	NOUN
ejpam-6429	86	6	of	of	ADP
ejpam-6429	86	7	g	g	NOUN
ejpam-6429	86	8	with	with	ADP
ejpam-6429	86	9	order	order	NOUN
ejpam-6429	86	10	at	at	ADV
ejpam-6429	86	11	least	least	ADJ
ejpam-6429	86	12	2	2	NUM
ejpam-6429	86	13	,	,	PUNCT
ejpam-6429	86	14	then	then	ADV
ejpam-6429	86	15	sdγgr(g	sdγgr(g	PROPN
ejpam-6429	86	16	)	)	PUNCT
ejpam-6429	86	17	=	=	SYM
ejpam-6429	86	18	min{sdγgr(gi	min{sdγgr(gi	NOUN
ejpam-6429	86	19	)	)	PUNCT
ejpam-6429	87	1	|	|	ADV
ejpam-6429	87	2	1	1	NUM
ejpam-6429	87	3	≤	≤	NUM
ejpam-6429	87	4	i	i	PRON
ejpam-6429	87	5	≤	≤	PROPN
ejpam-6429	87	6	s	s	PART
ejpam-6429	87	7	}	}	PUNCT
ejpam-6429	87	8	.	.	PUNCT
ejpam-6429	88	1	consequently	consequently	ADV
ejpam-6429	88	2	,	,	PUNCT
ejpam-6429	88	3	we	we	PRON
ejpam-6429	88	4	restrict	restrict	VERB
ejpam-6429	88	5	our	our	PRON
ejpam-6429	88	6	study	study	NOUN
ejpam-6429	88	7	to	to	ADP
ejpam-6429	88	8	connected	connected	ADJ
ejpam-6429	88	9	graphs	graph	NOUN
ejpam-6429	88	10	of	of	ADP
ejpam-6429	88	11	order	order	NOUN
ejpam-6429	88	12	at	at	ADV
ejpam-6429	88	13	least	least	ADV
ejpam-6429	88	14	two	two	NUM
ejpam-6429	88	15	.	.	PUNCT
ejpam-6429	89	1	3	3	X
ejpam-6429	89	2	.	.	X
ejpam-6429	89	3	preliminary	preliminary	ADJ
ejpam-6429	89	4	results	result	NOUN
ejpam-6429	89	5	we	we	PRON
ejpam-6429	89	6	begin	begin	VERB
ejpam-6429	89	7	this	this	DET
ejpam-6429	89	8	section	section	NOUN
ejpam-6429	89	9	with	with	ADP
ejpam-6429	89	10	some	some	DET
ejpam-6429	89	11	results	result	NOUN
ejpam-6429	89	12	that	that	PRON
ejpam-6429	89	13	will	will	AUX
ejpam-6429	89	14	be	be	AUX
ejpam-6429	89	15	utilized	utilize	VERB
ejpam-6429	89	16	later	later	ADV
ejpam-6429	89	17	.	.	PUNCT
ejpam-6429	90	1	proposition	proposition	NOUN
ejpam-6429	90	2	1	1	NUM
ejpam-6429	90	3	.	.	PUNCT
ejpam-6429	91	1	let	let	VERB
ejpam-6429	91	2	g	g	PRON
ejpam-6429	91	3	be	be	AUX
ejpam-6429	91	4	a	a	DET
ejpam-6429	91	5	connected	connected	ADJ
ejpam-6429	91	6	graph	graph	NOUN
ejpam-6429	91	7	of	of	ADP
ejpam-6429	91	8	order	order	NOUN
ejpam-6429	91	9	n	n	PRON
ejpam-6429	91	10	≥	≥	NOUN
ejpam-6429	91	11	2	2	NUM
ejpam-6429	91	12	,	,	PUNCT
ejpam-6429	91	13	and	and	CCONJ
ejpam-6429	91	14	let	let	VERB
ejpam-6429	91	15	f	f	PROPN
ejpam-6429	91	16	⊆	⊆	NUM
ejpam-6429	91	17	e(g	e(g	PROPN
ejpam-6429	91	18	)	)	PUNCT
ejpam-6429	91	19	be	be	AUX
ejpam-6429	91	20	a	a	DET
ejpam-6429	91	21	subset	subset	NOUN
ejpam-6429	91	22	of	of	ADP
ejpam-6429	91	23	edges	edge	NOUN
ejpam-6429	91	24	in	in	ADP
ejpam-6429	91	25	g.	g.	PROPN
ejpam-6429	91	26	if	if	SCONJ
ejpam-6429	91	27	g′	g′	NOUN
ejpam-6429	91	28	is	be	AUX
ejpam-6429	91	29	obtained	obtain	VERB
ejpam-6429	91	30	by	by	ADP
ejpam-6429	91	31	subdividing	subdivide	VERB
ejpam-6429	91	32	each	each	DET
ejpam-6429	91	33	edge	edge	NOUN
ejpam-6429	91	34	in	in	ADP
ejpam-6429	91	35	f	f	PROPN
ejpam-6429	91	36	,	,	PUNCT
ejpam-6429	91	37	then	then	ADV
ejpam-6429	91	38	γgr(g	γgr(g	PROPN
ejpam-6429	91	39	′	′	NOUN
ejpam-6429	91	40	)	)	PUNCT
ejpam-6429	91	41	≥	≥	NOUN
ejpam-6429	91	42	γgr(g	γgr(g	PROPN
ejpam-6429	91	43	)	)	PUNCT
ejpam-6429	91	44	.	.	PUNCT
ejpam-6429	92	1	proof	proof	NOUN
ejpam-6429	92	2	.	.	PUNCT
ejpam-6429	93	1	the	the	DET
ejpam-6429	93	2	proof	proof	NOUN
ejpam-6429	93	3	is	be	AUX
ejpam-6429	93	4	carried	carry	VERB
ejpam-6429	93	5	out	out	ADP
ejpam-6429	93	6	using	use	VERB
ejpam-6429	93	7	induction	induction	NOUN
ejpam-6429	93	8	on	on	ADP
ejpam-6429	93	9	the	the	DET
ejpam-6429	93	10	size	size	NOUN
ejpam-6429	93	11	of	of	ADP
ejpam-6429	93	12	|f	|f	PROPN
ejpam-6429	93	13	|	|	PROPN
ejpam-6429	93	14	.	.	PUNCT
ejpam-6429	94	1	first	first	ADV
ejpam-6429	94	2	,	,	PUNCT
ejpam-6429	94	3	assume	assume	VERB
ejpam-6429	94	4	that	that	SCONJ
ejpam-6429	94	5	|f	|f	PROPN
ejpam-6429	95	1	|	|	ADV
ejpam-6429	95	2	=	=	SYM
ejpam-6429	95	3	1	1	NUM
ejpam-6429	95	4	,	,	PUNCT
ejpam-6429	95	5	and	and	CCONJ
ejpam-6429	95	6	let	let	VERB
ejpam-6429	95	7	e	e	NOUN
ejpam-6429	95	8	=	=	PRON
ejpam-6429	95	9	uv	uv	NOUN
ejpam-6429	95	10	be	be	AUX
ejpam-6429	95	11	an	an	DET
ejpam-6429	95	12	element	element	NOUN
ejpam-6429	95	13	of	of	ADP
ejpam-6429	95	14	f	f	PROPN
ejpam-6429	95	15	.	.	PUNCT
ejpam-6429	96	1	construct	construct	VERB
ejpam-6429	96	2	g′	g′	NOUN
ejpam-6429	96	3	from	from	ADP
ejpam-6429	96	4	g	g	NOUN
ejpam-6429	96	5	by	by	ADP
ejpam-6429	96	6	introducing	introduce	VERB
ejpam-6429	96	7	a	a	DET
ejpam-6429	96	8	new	new	ADJ
ejpam-6429	96	9	vertex	vertex	NOUN
ejpam-6429	96	10	x	x	VERB
ejpam-6429	96	11	to	to	PART
ejpam-6429	96	12	subdivide	subdivide	VERB
ejpam-6429	96	13	the	the	DET
ejpam-6429	96	14	edge	edge	NOUN
ejpam-6429	96	15	e.	e.	PROPN
ejpam-6429	96	16	let	let	VERB
ejpam-6429	96	17	f	f	PRON
ejpam-6429	96	18	be	be	AUX
ejpam-6429	96	19	a	a	DET
ejpam-6429	96	20	γgr(g	γgr(g	PROPN
ejpam-6429	96	21	′)-function	′)-function	NOUN
ejpam-6429	96	22	.	.	PUNCT
ejpam-6429	97	1	since	since	SCONJ
ejpam-6429	97	2	f	f	PROPN
ejpam-6429	97	3	is	be	AUX
ejpam-6429	97	4	a	a	DET
ejpam-6429	97	5	grd	grd	NOUN
ejpam-6429	97	6	-	-	PUNCT
ejpam-6429	97	7	function	function	NOUN
ejpam-6429	97	8	on	on	ADP
ejpam-6429	97	9	g′	g′	NOUN
ejpam-6429	97	10	,	,	PUNCT
ejpam-6429	97	11	it	it	PRON
ejpam-6429	97	12	follows	follow	VERB
ejpam-6429	97	13	that	that	SCONJ
ejpam-6429	97	14	f(u	f(u	PROPN
ejpam-6429	97	15	)	)	PUNCT
ejpam-6429	97	16	+	+	NUM
ejpam-6429	97	17	f(v	f(v	NOUN
ejpam-6429	97	18	)	)	PUNCT
ejpam-6429	98	1	+	+	CCONJ
ejpam-6429	98	2	f(x	f(x	PROPN
ejpam-6429	98	3	)	)	PUNCT
ejpam-6429	98	4	≥	≥	NOUN
ejpam-6429	98	5	1	1	NUM
ejpam-6429	98	6	.	.	PUNCT
ejpam-6429	99	1	let	let	VERB
ejpam-6429	99	2	g	g	NOUN
ejpam-6429	99	3	:	:	PUNCT
ejpam-6429	99	4	v	v	NOUN
ejpam-6429	99	5	(	(	PUNCT
ejpam-6429	99	6	g	g	NOUN
ejpam-6429	99	7	)	)	PUNCT
ejpam-6429	99	8	→	→	SYM
ejpam-6429	99	9	{	{	PUNCT
ejpam-6429	99	10	0	0	NUM
ejpam-6429	99	11	,	,	PUNCT
ejpam-6429	99	12	1	1	NUM
ejpam-6429	99	13	,	,	PUNCT
ejpam-6429	99	14	2	2	NUM
ejpam-6429	99	15	,	,	PUNCT
ejpam-6429	99	16	3	3	NUM
ejpam-6429	99	17	}	}	PUNCT
ejpam-6429	99	18	be	be	AUX
ejpam-6429	99	19	a	a	DET
ejpam-6429	99	20	function	function	NOUN
ejpam-6429	99	21	defined	define	VERB
ejpam-6429	99	22	by	by	ADP
ejpam-6429	99	23	g(u	g(u	PROPN
ejpam-6429	99	24	)	)	PUNCT
ejpam-6429	99	25	=	=	SYM
ejpam-6429	99	26	min{3	min{3	PROPN
ejpam-6429	99	27	,	,	PUNCT
ejpam-6429	99	28	f(u)+f(x	f(u)+f(x	PROPN
ejpam-6429	99	29	)	)	PUNCT
ejpam-6429	99	30	}	}	PUNCT
ejpam-6429	99	31	,	,	PUNCT
ejpam-6429	99	32	and	and	CCONJ
ejpam-6429	99	33	for	for	ADP
ejpam-6429	99	34	all	all	DET
ejpam-6429	99	35	y	y	PROPN
ejpam-6429	99	36	∈	∈	PROPN
ejpam-6429	99	37	v	v	NOUN
ejpam-6429	99	38	(	(	PUNCT
ejpam-6429	99	39	g)\{u	g)\{u	PROPN
ejpam-6429	99	40	}	}	PUNCT
ejpam-6429	99	41	,	,	PUNCT
ejpam-6429	99	42	by	by	ADP
ejpam-6429	99	43	g(y	g(y	NOUN
ejpam-6429	99	44	)	)	PUNCT
ejpam-6429	99	45	=	=	SYM
ejpam-6429	99	46	f(y	f(y	NOUN
ejpam-6429	99	47	)	)	PUNCT
ejpam-6429	99	48	.	.	PUNCT
ejpam-6429	100	1	it	it	PRON
ejpam-6429	100	2	is	be	AUX
ejpam-6429	100	3	clear	clear	ADJ
ejpam-6429	100	4	that	that	SCONJ
ejpam-6429	100	5	g	g	PROPN
ejpam-6429	100	6	is	be	AUX
ejpam-6429	100	7	a	a	DET
ejpam-6429	100	8	grd	grd	NOUN
ejpam-6429	100	9	-	-	PUNCT
ejpam-6429	100	10	function	function	NOUN
ejpam-6429	100	11	on	on	ADP
ejpam-6429	100	12	g	g	NOUN
ejpam-6429	100	13	,	,	PUNCT
ejpam-6429	100	14	and	and	CCONJ
ejpam-6429	100	15	ωg	ωg	ADP
ejpam-6429	100	16	r(g	r(g	NUM
ejpam-6429	100	17	)	)	PUNCT
ejpam-6429	100	18	≤	≤	NUM
ejpam-6429	100	19	ωg	ωg	ADP
ejpam-6429	100	20	r(f	r(f	PROPN
ejpam-6429	100	21	)	)	PUNCT
ejpam-6429	100	22	.	.	PUNCT
ejpam-6429	101	1	thus	thus	ADV
ejpam-6429	101	2	,	,	PUNCT
ejpam-6429	101	3	we	we	PRON
ejpam-6429	101	4	obtain	obtain	VERB
ejpam-6429	101	5	γgr(g	γgr(g	PROPN
ejpam-6429	101	6	′	′	NUM
ejpam-6429	101	7	)	)	PUNCT
ejpam-6429	101	8	≥	≥	PROPN
ejpam-6429	101	9	γgr(g	γgr(g	PROPN
ejpam-6429	101	10	)	)	PUNCT
ejpam-6429	101	11	,	,	PUNCT
ejpam-6429	101	12	which	which	PRON
ejpam-6429	101	13	establishes	establish	VERB
ejpam-6429	101	14	the	the	DET
ejpam-6429	101	15	base	base	NOUN
ejpam-6429	101	16	case	case	NOUN
ejpam-6429	101	17	.	.	PUNCT
ejpam-6429	102	1	suppose	suppose	VERB
ejpam-6429	102	2	the	the	DET
ejpam-6429	102	3	statement	statement	NOUN
ejpam-6429	102	4	holds	hold	VERB
ejpam-6429	102	5	for	for	ADP
ejpam-6429	102	6	any	any	DET
ejpam-6429	102	7	subset	subset	NOUN
ejpam-6429	102	8	f	f	PROPN
ejpam-6429	102	9	of	of	ADP
ejpam-6429	102	10	edges	edge	NOUN
ejpam-6429	102	11	with	with	ADP
ejpam-6429	102	12	1	1	NUM
ejpam-6429	102	13	≤	≤	NOUN
ejpam-6429	102	14	|f	|f	PUNCT
ejpam-6429	103	1	|	|	ADV
ejpam-6429	103	2	<	<	X
ejpam-6429	103	3	k.	k.	PROPN
ejpam-6429	103	4	let	let	VERB
ejpam-6429	103	5	f	f	PROPN
ejpam-6429	103	6	⊆	⊆	NUM
ejpam-6429	103	7	e(g	e(g	PROPN
ejpam-6429	103	8	)	)	PUNCT
ejpam-6429	103	9	be	be	AUX
ejpam-6429	103	10	a	a	DET
ejpam-6429	103	11	set	set	NOUN
ejpam-6429	103	12	of	of	ADP
ejpam-6429	103	13	edges	edge	NOUN
ejpam-6429	103	14	of	of	ADP
ejpam-6429	103	15	size	size	NOUN
ejpam-6429	103	16	k	k	PROPN
ejpam-6429	103	17	,	,	PUNCT
ejpam-6429	103	18	where	where	SCONJ
ejpam-6429	103	19	f	f	AUX
ejpam-6429	103	20	=	=	PRON
ejpam-6429	103	21	{	{	PUNCT
ejpam-6429	103	22	e1	e1	PROPN
ejpam-6429	103	23	,	,	PUNCT
ejpam-6429	103	24	e2	e2	PROPN
ejpam-6429	103	25	,	,	PUNCT
ejpam-6429	103	26	.	.	PUNCT
ejpam-6429	103	27	.	.	PUNCT
ejpam-6429	104	1	.	.	PUNCT
ejpam-6429	105	1	,	,	PUNCT
ejpam-6429	105	2	ek	ek	PROPN
ejpam-6429	105	3	}	}	PUNCT
ejpam-6429	105	4	,	,	PUNCT
ejpam-6429	105	5	k	k	X
ejpam-6429	105	6	≥	≥	NUM
ejpam-6429	105	7	2	2	X
ejpam-6429	105	8	.	.	PUNCT
ejpam-6429	105	9	let	let	VERB
ejpam-6429	105	10	g′	g′	NOUN
ejpam-6429	105	11	be	be	AUX
ejpam-6429	105	12	the	the	DET
ejpam-6429	105	13	graph	graph	NOUN
ejpam-6429	105	14	obtained	obtain	VERB
ejpam-6429	105	15	from	from	ADP
ejpam-6429	105	16	g	g	NOUN
ejpam-6429	105	17	by	by	ADP
ejpam-6429	105	18	subdividing	subdivide	VERB
ejpam-6429	105	19	all	all	DET
ejpam-6429	105	20	edges	edge	NOUN
ejpam-6429	105	21	in	in	ADP
ejpam-6429	105	22	f	f	PROPN
ejpam-6429	105	23	\{ek	\{ek	PROPN
ejpam-6429	105	24	}	}	PUNCT
ejpam-6429	105	25	.	.	PUNCT
ejpam-6429	106	1	then	then	ADV
ejpam-6429	106	2	,	,	PUNCT
ejpam-6429	106	3	let	let	VERB
ejpam-6429	106	4	g′′	g′′	PROPN
ejpam-6429	106	5	be	be	AUX
ejpam-6429	106	6	the	the	DET
ejpam-6429	106	7	graph	graph	NOUN
ejpam-6429	106	8	formed	form	VERB
ejpam-6429	106	9	by	by	ADP
ejpam-6429	106	10	further	further	ADJ
ejpam-6429	106	11	subdividing	subdivide	VERB
ejpam-6429	106	12	the	the	DET
ejpam-6429	106	13	edge	edge	NOUN
ejpam-6429	106	14	ek	ek	X
ejpam-6429	106	15	in	in	ADP
ejpam-6429	106	16	g′.	g′.	NOUN
ejpam-6429	106	17	by	by	ADP
ejpam-6429	106	18	the	the	DET
ejpam-6429	106	19	induction	induction	NOUN
ejpam-6429	106	20	hypothesis	hypothesis	NOUN
ejpam-6429	106	21	,	,	PUNCT
ejpam-6429	106	22	we	we	PRON
ejpam-6429	106	23	obtain	obtain	VERB
ejpam-6429	106	24	γgr(g	γgr(g	PROPN
ejpam-6429	106	25	′′	′′	PROPN
ejpam-6429	106	26	)	)	PUNCT
ejpam-6429	106	27	≥	≥	NOUN
ejpam-6429	106	28	γgr(g	γgr(g	PROPN
ejpam-6429	106	29	′	′	NUM
ejpam-6429	106	30	)	)	PUNCT
ejpam-6429	106	31	≥	≥	PROPN
ejpam-6429	106	32	γgr(g	γgr(g	PROPN
ejpam-6429	106	33	)	)	PUNCT
ejpam-6429	106	34	,	,	PUNCT
ejpam-6429	106	35	which	which	PRON
ejpam-6429	106	36	completes	complete	VERB
ejpam-6429	106	37	the	the	DET
ejpam-6429	106	38	proof	proof	NOUN
ejpam-6429	106	39	.	.	PUNCT
ejpam-6429	107	1	proposition	proposition	NOUN
ejpam-6429	107	2	2	2	NUM
ejpam-6429	107	3	.	.	PUNCT
ejpam-6429	108	1	let	let	VERB
ejpam-6429	108	2	g	g	PRON
ejpam-6429	108	3	be	be	AUX
ejpam-6429	108	4	a	a	DET
ejpam-6429	108	5	connected	connected	ADJ
ejpam-6429	108	6	graph	graph	NOUN
ejpam-6429	108	7	of	of	ADP
ejpam-6429	108	8	order	order	NOUN
ejpam-6429	108	9	n	n	PRON
ejpam-6429	108	10	≥	≥	NOUN
ejpam-6429	108	11	2	2	NUM
ejpam-6429	108	12	,	,	PUNCT
ejpam-6429	108	13	and	and	CCONJ
ejpam-6429	108	14	let	let	VERB
ejpam-6429	108	15	f	f	PROPN
ejpam-6429	108	16	⊆	⊆	NUM
ejpam-6429	108	17	e(g	e(g	PROPN
ejpam-6429	108	18	)	)	PUNCT
ejpam-6429	108	19	be	be	AUX
ejpam-6429	108	20	a	a	DET
ejpam-6429	108	21	set	set	NOUN
ejpam-6429	108	22	of	of	ADP
ejpam-6429	108	23	edges	edge	NOUN
ejpam-6429	108	24	in	in	ADP
ejpam-6429	108	25	g.	g.	PROPN
ejpam-6429	108	26	let	let	VERB
ejpam-6429	108	27	g′	g′	NOUN
ejpam-6429	108	28	be	be	AUX
ejpam-6429	108	29	the	the	DET
ejpam-6429	108	30	graph	graph	NOUN
ejpam-6429	108	31	obtained	obtain	VERB
ejpam-6429	108	32	from	from	ADP
ejpam-6429	108	33	g	g	NOUN
ejpam-6429	108	34	by	by	ADP
ejpam-6429	108	35	subdividing	subdivide	VERB
ejpam-6429	108	36	the	the	DET
ejpam-6429	108	37	edges	edge	NOUN
ejpam-6429	108	38	in	in	ADP
ejpam-6429	108	39	f	f	PROPN
ejpam-6429	108	40	.	.	PUNCT
ejpam-6429	109	1	if	if	SCONJ
ejpam-6429	109	2	there	there	PRON
ejpam-6429	109	3	exists	exist	VERB
ejpam-6429	109	4	a	a	DET
ejpam-6429	109	5	γgr(g	γgr(g	PROPN
ejpam-6429	109	6	′)-function	′)-function	NOUN
ejpam-6429	109	7	that	that	PRON
ejpam-6429	109	8	assigns	assign	VERB
ejpam-6429	109	9	a	a	DET
ejpam-6429	109	10	value	value	NOUN
ejpam-6429	109	11	of	of	ADP
ejpam-6429	109	12	1	1	NUM
ejpam-6429	109	13	or	or	CCONJ
ejpam-6429	109	14	3	3	NUM
ejpam-6429	109	15	to	to	ADP
ejpam-6429	109	16	at	at	ADV
ejpam-6429	109	17	least	least	ADV
ejpam-6429	109	18	one	one	NUM
ejpam-6429	109	19	subdivision	subdivision	NOUN
ejpam-6429	109	20	vertex	vertex	NOUN
ejpam-6429	109	21	,	,	PUNCT
ejpam-6429	109	22	then	then	ADV
ejpam-6429	109	23	γgr(g	γgr(g	PROPN
ejpam-6429	109	24	′	′	NUM
ejpam-6429	109	25	)	)	PUNCT
ejpam-6429	109	26	>	>	X
ejpam-6429	109	27	γgr(g	γgr(g	PROPN
ejpam-6429	109	28	)	)	PUNCT
ejpam-6429	109	29	.	.	PUNCT
ejpam-6429	110	1	proof	proof	NOUN
ejpam-6429	110	2	.	.	PUNCT
ejpam-6429	111	1	let	let	VERB
ejpam-6429	111	2	e	e	NOUN
ejpam-6429	111	3	=	=	NOUN
ejpam-6429	111	4	uv	uv	PROPN
ejpam-6429	111	5	∈	∈	PROPN
ejpam-6429	111	6	f	f	NOUN
ejpam-6429	111	7	and	and	CCONJ
ejpam-6429	111	8	suppose	suppose	VERB
ejpam-6429	111	9	that	that	SCONJ
ejpam-6429	111	10	e	e	PROPN
ejpam-6429	111	11	is	be	AUX
ejpam-6429	111	12	subdivided	subdivide	VERB
ejpam-6429	111	13	by	by	ADP
ejpam-6429	111	14	introducing	introduce	VERB
ejpam-6429	111	15	a	a	DET
ejpam-6429	111	16	new	new	ADJ
ejpam-6429	111	17	vertex	vertex	NOUN
ejpam-6429	111	18	x.	x.	NOUN
ejpam-6429	111	19	consider	consider	VERB
ejpam-6429	111	20	a	a	DET
ejpam-6429	111	21	function	function	NOUN
ejpam-6429	111	22	f	f	NOUN
ejpam-6429	111	23	that	that	PRON
ejpam-6429	111	24	is	be	AUX
ejpam-6429	111	25	a	a	DET
ejpam-6429	111	26	γgr(g	γgr(g	PROPN
ejpam-6429	111	27	′)-function	′)-function	NOUN
ejpam-6429	111	28	such	such	ADJ
ejpam-6429	111	29	that	that	SCONJ
ejpam-6429	111	30	f(x	f(x	PROPN
ejpam-6429	111	31	)	)	PUNCT
ejpam-6429	111	32	∈	∈	PROPN
ejpam-6429	111	33	{	{	PUNCT
ejpam-6429	111	34	1	1	NUM
ejpam-6429	111	35	,	,	PUNCT
ejpam-6429	111	36	3	3	NUM
ejpam-6429	111	37	}	}	PUNCT
ejpam-6429	111	38	.	.	PUNCT
ejpam-6429	112	1	construct	construct	VERB
ejpam-6429	112	2	g′′	g′′	PROPN
ejpam-6429	112	3	from	from	ADP
ejpam-6429	112	4	g	g	NOUN
ejpam-6429	112	5	by	by	ADP
ejpam-6429	112	6	subdividing	subdivide	VERB
ejpam-6429	112	7	all	all	DET
ejpam-6429	112	8	edges	edge	NOUN
ejpam-6429	112	9	in	in	ADP
ejpam-6429	112	10	f	f	PROPN
ejpam-6429	112	11	−	−	PROPN
ejpam-6429	112	12	{	{	PUNCT
ejpam-6429	112	13	e	e	NOUN
ejpam-6429	112	14	}	}	PUNCT
ejpam-6429	112	15	,	,	PUNCT
ejpam-6429	112	16	noting	note	VERB
ejpam-6429	112	17	that	that	SCONJ
ejpam-6429	112	18	if	if	SCONJ
ejpam-6429	112	19	f	f	PROPN
ejpam-6429	112	20	=	=	PRON
ejpam-6429	112	21	{	{	PUNCT
ejpam-6429	112	22	e	e	NOUN
ejpam-6429	112	23	}	}	PUNCT
ejpam-6429	112	24	,	,	PUNCT
ejpam-6429	112	25	then	then	ADV
ejpam-6429	112	26	g	g	PROPN
ejpam-6429	112	27	=	=	PROPN
ejpam-6429	112	28	g′′.	g′′.	PROPN
ejpam-6429	112	29	if	if	SCONJ
ejpam-6429	112	30	f(x	f(x	PROPN
ejpam-6429	112	31	)	)	PUNCT
ejpam-6429	112	32	=	=	SYM
ejpam-6429	113	1	1	1	NUM
ejpam-6429	113	2	,	,	PUNCT
ejpam-6429	113	3	then	then	ADV
ejpam-6429	113	4	the	the	DET
ejpam-6429	113	5	restriction	restriction	NOUN
ejpam-6429	113	6	of	of	ADP
ejpam-6429	113	7	f	f	PROPN
ejpam-6429	113	8	to	to	ADP
ejpam-6429	113	9	g′′	g′′	PROPN
ejpam-6429	113	10	acts	act	NOUN
ejpam-6429	113	11	as	as	ADP
ejpam-6429	113	12	a	a	DET
ejpam-6429	113	13	grd	grd	NOUN
ejpam-6429	113	14	-	-	PUNCT
ejpam-6429	113	15	function	function	NOUN
ejpam-6429	113	16	with	with	ADP
ejpam-6429	113	17	a	a	DET
ejpam-6429	113	18	weight	weight	NOUN
ejpam-6429	113	19	less	less	ADJ
ejpam-6429	113	20	j.	j.	PROPN
ejpam-6429	113	21	j.	j.	PROPN
ejpam-6429	113	22	hamja	hamja	PROPN
ejpam-6429	113	23	et	et	PROPN
ejpam-6429	113	24	al	al	PROPN
ejpam-6429	113	25	.	.	PUNCT
ejpam-6429	113	26	/	/	SYM
ejpam-6429	113	27	eur	eur	PROPN
ejpam-6429	113	28	.	.	PUNCT
ejpam-6429	114	1	j.	j.	PROPN
ejpam-6429	114	2	pure	pure	PROPN
ejpam-6429	114	3	appl	appl	PROPN
ejpam-6429	114	4	.	.	PROPN
ejpam-6429	114	5	math	math	PROPN
ejpam-6429	114	6	,	,	PUNCT
ejpam-6429	114	7	18	18	NUM
ejpam-6429	114	8	(	(	PUNCT
ejpam-6429	114	9	4	4	NUM
ejpam-6429	114	10	)	)	PUNCT
ejpam-6429	114	11	(	(	PUNCT
ejpam-6429	114	12	2025	2025	NUM
ejpam-6429	114	13	)	)	PUNCT
ejpam-6429	114	14	,	,	PUNCT
ejpam-6429	114	15	6429	6429	NUM
ejpam-6429	114	16	5	5	NUM
ejpam-6429	114	17	of	of	ADP
ejpam-6429	114	18	16	16	NUM
ejpam-6429	114	19	than	than	ADP
ejpam-6429	114	20	ωg	ωg	ADP
ejpam-6429	114	21	r(f	r(f	PROPN
ejpam-6429	114	22	)	)	PUNCT
ejpam-6429	114	23	,	,	PUNCT
ejpam-6429	114	24	which	which	PRON
ejpam-6429	114	25	leads	lead	VERB
ejpam-6429	114	26	to	to	ADP
ejpam-6429	114	27	the	the	DET
ejpam-6429	114	28	inequality	inequality	NOUN
ejpam-6429	114	29	γgr(g	γgr(g	PROPN
ejpam-6429	114	30	′	′	NUM
ejpam-6429	114	31	)	)	PUNCT
ejpam-6429	114	32	>	>	X
ejpam-6429	114	33	γgr(g	γgr(g	PROPN
ejpam-6429	114	34	′′	′′	PROPN
ejpam-6429	114	35	)	)	PUNCT
ejpam-6429	114	36	≥	≥	NOUN
ejpam-6429	114	37	γgr(g	γgr(g	PROPN
ejpam-6429	114	38	)	)	PUNCT
ejpam-6429	114	39	.	.	PUNCT
ejpam-6429	115	1	next	next	ADV
ejpam-6429	115	2	,	,	PUNCT
ejpam-6429	115	3	assume	assume	VERB
ejpam-6429	115	4	f(x	f(x	PROPN
ejpam-6429	115	5	)	)	PUNCT
ejpam-6429	115	6	=	=	SYM
ejpam-6429	116	1	3	3	X
ejpam-6429	116	2	.	.	PUNCT
ejpam-6429	116	3	define	define	VERB
ejpam-6429	116	4	a	a	DET
ejpam-6429	116	5	function	function	NOUN
ejpam-6429	116	6	g	g	NOUN
ejpam-6429	116	7	on	on	ADP
ejpam-6429	116	8	g′′	g′′	PROPN
ejpam-6429	116	9	by	by	ADP
ejpam-6429	116	10	g(u	g(u	PROPN
ejpam-6429	116	11	)	)	PUNCT
ejpam-6429	116	12	=	=	SYM
ejpam-6429	116	13	min{3	min{3	PROPN
ejpam-6429	116	14	,	,	PUNCT
ejpam-6429	116	15	f(u	f(u	PROPN
ejpam-6429	116	16	)	)	PUNCT
ejpam-6429	116	17	+	+	CCONJ
ejpam-6429	116	18	1	1	NUM
ejpam-6429	116	19	}	}	PUNCT
ejpam-6429	116	20	,	,	PUNCT
ejpam-6429	116	21	g(v	g(v	X
ejpam-6429	116	22	)	)	PUNCT
ejpam-6429	116	23	=	=	SYM
ejpam-6429	116	24	min{3	min{3	PROPN
ejpam-6429	116	25	,	,	PUNCT
ejpam-6429	116	26	f(v	f(v	PROPN
ejpam-6429	116	27	)	)	PUNCT
ejpam-6429	116	28	+	+	CCONJ
ejpam-6429	116	29	1	1	NUM
ejpam-6429	116	30	}	}	PUNCT
ejpam-6429	116	31	,	,	PUNCT
ejpam-6429	116	32	and	and	CCONJ
ejpam-6429	116	33	g(z	g(z	ADJ
ejpam-6429	116	34	)	)	PUNCT
ejpam-6429	116	35	=	=	SYM
ejpam-6429	116	36	f(z	f(z	PROPN
ejpam-6429	116	37	)	)	PUNCT
ejpam-6429	116	38	,	,	PUNCT
ejpam-6429	116	39	for	for	ADP
ejpam-6429	116	40	all	all	DET
ejpam-6429	116	41	z	z	NOUN
ejpam-6429	116	42	∈	∈	PROPN
ejpam-6429	116	43	v	v	NOUN
ejpam-6429	116	44	(	(	PUNCT
ejpam-6429	116	45	g′′)−	g′′)−	PROPN
ejpam-6429	116	46	{	{	PUNCT
ejpam-6429	116	47	u	u	NOUN
ejpam-6429	116	48	,	,	PUNCT
ejpam-6429	116	49	v	v	NOUN
ejpam-6429	116	50	}	}	PUNCT
ejpam-6429	116	51	.	.	PUNCT
ejpam-6429	117	1	then	then	ADV
ejpam-6429	117	2	,	,	PUNCT
ejpam-6429	117	3	g	g	PROPN
ejpam-6429	117	4	is	be	AUX
ejpam-6429	117	5	a	a	DET
ejpam-6429	117	6	grd	grd	NOUN
ejpam-6429	117	7	-	-	PUNCT
ejpam-6429	117	8	function	function	NOUN
ejpam-6429	117	9	with	with	ADP
ejpam-6429	117	10	a	a	DET
ejpam-6429	117	11	weight	weight	NOUN
ejpam-6429	117	12	less	less	ADJ
ejpam-6429	117	13	than	than	ADP
ejpam-6429	117	14	ωg	ωg	ADP
ejpam-6429	117	15	r(f	r(f	PROPN
ejpam-6429	117	16	)	)	PUNCT
ejpam-6429	117	17	,	,	PUNCT
ejpam-6429	117	18	leading	lead	VERB
ejpam-6429	117	19	to	to	ADP
ejpam-6429	117	20	the	the	DET
ejpam-6429	117	21	conclusion	conclusion	NOUN
ejpam-6429	117	22	that	that	SCONJ
ejpam-6429	117	23	γgr(g	γgr(g	PROPN
ejpam-6429	117	24	′	′	NUM
ejpam-6429	117	25	)	)	PUNCT
ejpam-6429	117	26	>	>	X
ejpam-6429	117	27	γgr(g	γgr(g	PROPN
ejpam-6429	117	28	′′	′′	PROPN
ejpam-6429	117	29	)	)	PUNCT
ejpam-6429	117	30	≥	≥	NOUN
ejpam-6429	117	31	γgr(g	γgr(g	PROPN
ejpam-6429	117	32	)	)	PUNCT
ejpam-6429	117	33	.	.	PUNCT
ejpam-6429	118	1	thus	thus	ADV
ejpam-6429	118	2	,	,	PUNCT
ejpam-6429	118	3	the	the	DET
ejpam-6429	118	4	proof	proof	NOUN
ejpam-6429	118	5	is	be	AUX
ejpam-6429	118	6	complete	complete	ADJ
ejpam-6429	118	7	.	.	PUNCT
ejpam-6429	119	1	4	4	X
ejpam-6429	119	2	.	.	X
ejpam-6429	119	3	exact	exact	ADJ
ejpam-6429	119	4	values	value	NOUN
ejpam-6429	119	5	the	the	DET
ejpam-6429	119	6	exact	exact	ADJ
ejpam-6429	119	7	values	value	NOUN
ejpam-6429	119	8	of	of	ADP
ejpam-6429	119	9	the	the	DET
ejpam-6429	119	10	grd	grd	NOUN
ejpam-6429	119	11	-	-	PUNCT
ejpam-6429	119	12	numbers	number	NOUN
ejpam-6429	119	13	of	of	ADP
ejpam-6429	119	14	paths	path	NOUN
ejpam-6429	119	15	and	and	CCONJ
ejpam-6429	119	16	cycles	cycle	NOUN
ejpam-6429	119	17	are	be	AUX
ejpam-6429	119	18	determined	determine	VERB
ejpam-6429	119	19	in	in	ADP
ejpam-6429	119	20	[	[	PUNCT
ejpam-6429	119	21	11	11	NUM
ejpam-6429	119	22	]	]	PUNCT
ejpam-6429	119	23	.	.	PUNCT
ejpam-6429	120	1	proposition	proposition	NOUN
ejpam-6429	120	2	3	3	NUM
ejpam-6429	120	3	.	.	PUNCT
ejpam-6429	121	1	[	[	X
ejpam-6429	121	2	11	11	NUM
ejpam-6429	121	3	]	]	PUNCT
ejpam-6429	121	4	for	for	ADP
ejpam-6429	121	5	n	n	PRON
ejpam-6429	121	6	≥	≥	NUM
ejpam-6429	121	7	1	1	NUM
ejpam-6429	121	8	,	,	PUNCT
ejpam-6429	121	9	γgr(pn	γgr(pn	NUM
ejpam-6429	121	10	)	)	PUNCT
ejpam-6429	121	11	=	=	PUNCT
ejpam-6429	122	1	⌈	⌈	NOUN
ejpam-6429	122	2	6n	6n	NOUN
ejpam-6429	122	3	7	7	NUM
ejpam-6429	122	4	⌉	⌉	X
ejpam-6429	122	5	.	.	PUNCT
ejpam-6429	123	1	proposition	proposition	NOUN
ejpam-6429	123	2	4	4	NUM
ejpam-6429	123	3	.	.	PUNCT
ejpam-6429	124	1	[	[	X
ejpam-6429	124	2	11	11	NUM
ejpam-6429	124	3	]	]	PUNCT
ejpam-6429	124	4	for	for	ADP
ejpam-6429	124	5	n	n	X
ejpam-6429	124	6	≥	≥	NOUN
ejpam-6429	124	7	4	4	NUM
ejpam-6429	124	8	,	,	PUNCT
ejpam-6429	124	9	γgr(cn	γgr(cn	ADV
ejpam-6429	124	10	)	)	PUNCT
ejpam-6429	124	11	=	=	PUNCT
ejpam-6429	125	1	⌈	⌈	NOUN
ejpam-6429	125	2	6n	6n	NOUN
ejpam-6429	125	3	7	7	NUM
ejpam-6429	125	4	⌉	⌉	X
ejpam-6429	125	5	.	.	PUNCT
ejpam-6429	126	1	the	the	DET
ejpam-6429	126	2	following	follow	VERB
ejpam-6429	126	3	results	result	NOUN
ejpam-6429	126	4	directly	directly	ADV
ejpam-6429	126	5	follow	follow	VERB
ejpam-6429	126	6	from	from	ADP
ejpam-6429	126	7	propositions	proposition	NOUN
ejpam-6429	126	8	3	3	NUM
ejpam-6429	126	9	and	and	CCONJ
ejpam-6429	126	10	4	4	NUM
ejpam-6429	126	11	.	.	PUNCT
ejpam-6429	126	12	corollary	corollary	ADJ
ejpam-6429	126	13	1	1	NUM
ejpam-6429	126	14	.	.	PUNCT
ejpam-6429	126	15	for	for	ADP
ejpam-6429	126	16	n	n	PRON
ejpam-6429	126	17	≥	≥	NUM
ejpam-6429	126	18	2	2	NUM
ejpam-6429	126	19	,	,	PUNCT
ejpam-6429	126	20	sdγgr(pn	sdγgr(pn	NOUN
ejpam-6429	126	21	)	)	PUNCT
ejpam-6429	126	22	=	=	SYM
ejpam-6429	126	23	{	{	PUNCT
ejpam-6429	126	24	2	2	NUM
ejpam-6429	126	25	if	if	SCONJ
ejpam-6429	126	26	n	n	PRON
ejpam-6429	126	27	≡	≡	PROPN
ejpam-6429	126	28	6	6	NUM
ejpam-6429	126	29	(	(	PUNCT
ejpam-6429	126	30	mod	mod	PROPN
ejpam-6429	126	31	7	7	NUM
ejpam-6429	126	32	)	)	PUNCT
ejpam-6429	126	33	1	1	NUM
ejpam-6429	126	34	otherwise	otherwise	ADV
ejpam-6429	126	35	.	.	PUNCT
ejpam-6429	127	1	corollary	corollary	ADJ
ejpam-6429	127	2	2	2	NUM
ejpam-6429	127	3	.	.	PUNCT
ejpam-6429	127	4	for	for	ADP
ejpam-6429	127	5	n	n	X
ejpam-6429	127	6	≥	≥	NUM
ejpam-6429	127	7	4	4	NUM
ejpam-6429	127	8	,	,	PUNCT
ejpam-6429	127	9	sdγgr(cn	sdγgr(cn	NOUN
ejpam-6429	127	10	)	)	PUNCT
ejpam-6429	127	11	=	=	PRON
ejpam-6429	127	12	{	{	PUNCT
ejpam-6429	127	13	2	2	NUM
ejpam-6429	127	14	if	if	SCONJ
ejpam-6429	127	15	n	n	PRON
ejpam-6429	127	16	≡	≡	PROPN
ejpam-6429	127	17	6	6	NUM
ejpam-6429	127	18	(	(	PUNCT
ejpam-6429	127	19	mod	mod	PROPN
ejpam-6429	127	20	7	7	NUM
ejpam-6429	127	21	)	)	PUNCT
ejpam-6429	127	22	1	1	NUM
ejpam-6429	127	23	otherwise	otherwise	ADV
ejpam-6429	127	24	.	.	PUNCT
ejpam-6429	128	1	proposition	proposition	NOUN
ejpam-6429	128	2	5	5	NUM
ejpam-6429	128	3	.	.	PUNCT
ejpam-6429	129	1	[	[	X
ejpam-6429	129	2	11	11	NUM
ejpam-6429	129	3	]	]	PUNCT
ejpam-6429	129	4	γgr(g	γgr(g	PROPN
ejpam-6429	129	5	)	)	PUNCT
ejpam-6429	129	6	=	=	SYM
ejpam-6429	129	7	2	2	NUM
ejpam-6429	129	8	if	if	SCONJ
ejpam-6429	129	9	and	and	CCONJ
ejpam-6429	129	10	only	only	ADV
ejpam-6429	129	11	if	if	SCONJ
ejpam-6429	129	12	g	g	PROPN
ejpam-6429	129	13	=	=	VERB
ejpam-6429	129	14	kn	kn	PROPN
ejpam-6429	129	15	or	or	CCONJ
ejpam-6429	129	16	g	g	PROPN
ejpam-6429	129	17	=	=	SYM
ejpam-6429	129	18	k2	k2	PROPN
ejpam-6429	129	19	.	.	PUNCT
ejpam-6429	130	1	proposition	proposition	NOUN
ejpam-6429	130	2	6	6	NUM
ejpam-6429	130	3	.	.	PUNCT
ejpam-6429	131	1	[	[	X
ejpam-6429	131	2	13	13	NUM
ejpam-6429	131	3	]	]	PUNCT
ejpam-6429	131	4	let	let	VERB
ejpam-6429	131	5	g	g	PRON
ejpam-6429	131	6	be	be	AUX
ejpam-6429	131	7	a	a	DET
ejpam-6429	131	8	connected	connected	ADJ
ejpam-6429	131	9	graph	graph	NOUN
ejpam-6429	131	10	of	of	ADP
ejpam-6429	131	11	order	order	NOUN
ejpam-6429	131	12	n	n	PRON
ejpam-6429	131	13	≥	≥	NUM
ejpam-6429	131	14	3	3	NUM
ejpam-6429	131	15	different	different	ADV
ejpam-6429	131	16	from	from	ADP
ejpam-6429	131	17	kn	kn	PROPN
ejpam-6429	131	18	.	.	PUNCT
ejpam-6429	132	1	then	then	ADV
ejpam-6429	132	2	γgr(g	γgr(g	PROPN
ejpam-6429	132	3	)	)	PUNCT
ejpam-6429	132	4	=	=	SYM
ejpam-6429	132	5	3	3	NUM
ejpam-6429	132	6	if	if	SCONJ
ejpam-6429	132	7	and	and	CCONJ
ejpam-6429	132	8	only	only	ADV
ejpam-6429	132	9	if	if	SCONJ
ejpam-6429	132	10	∆(g	∆(g	NOUN
ejpam-6429	132	11	)	)	PUNCT
ejpam-6429	132	12	=	=	SYM
ejpam-6429	132	13	n−	n−	NOUN
ejpam-6429	132	14	1	1	NUM
ejpam-6429	132	15	.	.	PUNCT
ejpam-6429	133	1	as	as	ADP
ejpam-6429	133	2	a	a	DET
ejpam-6429	133	3	direct	direct	ADJ
ejpam-6429	133	4	consequence	consequence	NOUN
ejpam-6429	133	5	of	of	ADP
ejpam-6429	133	6	propositions	proposition	NOUN
ejpam-6429	133	7	5	5	NUM
ejpam-6429	133	8	and	and	CCONJ
ejpam-6429	133	9	6	6	NUM
ejpam-6429	133	10	,	,	PUNCT
ejpam-6429	133	11	we	we	PRON
ejpam-6429	133	12	obtain	obtain	VERB
ejpam-6429	133	13	:	:	PUNCT
ejpam-6429	133	14	corollary	corollary	ADJ
ejpam-6429	133	15	3	3	X
ejpam-6429	133	16	.	.	PUNCT
ejpam-6429	134	1	if	if	SCONJ
ejpam-6429	134	2	g	g	PROPN
ejpam-6429	134	3	is	be	AUX
ejpam-6429	134	4	a	a	DET
ejpam-6429	134	5	connected	connected	ADJ
ejpam-6429	134	6	graph	graph	NOUN
ejpam-6429	134	7	of	of	ADP
ejpam-6429	134	8	order	order	NOUN
ejpam-6429	134	9	n	n	PRON
ejpam-6429	134	10	≥	≥	NOUN
ejpam-6429	134	11	2	2	NUM
ejpam-6429	134	12	with	with	ADP
ejpam-6429	134	13	γgr(g	γgr(g	PROPN
ejpam-6429	134	14	)	)	PUNCT
ejpam-6429	134	15	∈	∈	PROPN
ejpam-6429	134	16	{	{	PUNCT
ejpam-6429	134	17	2	2	NUM
ejpam-6429	134	18	,	,	PUNCT
ejpam-6429	134	19	3	3	NUM
ejpam-6429	134	20	}	}	PUNCT
ejpam-6429	134	21	,	,	PUNCT
ejpam-6429	134	22	then	then	ADV
ejpam-6429	134	23	sdγgr(g	sdγgr(g	PROPN
ejpam-6429	134	24	)	)	PUNCT
ejpam-6429	134	25	=	=	SYM
ejpam-6429	134	26	1	1	X
ejpam-6429	134	27	.	.	PUNCT
ejpam-6429	134	28	proof	proof	NOUN
ejpam-6429	134	29	.	.	PUNCT
ejpam-6429	135	1	if	if	SCONJ
ejpam-6429	135	2	γgr(g	γgr(g	PROPN
ejpam-6429	135	3	)	)	PUNCT
ejpam-6429	135	4	=	=	SYM
ejpam-6429	135	5	2	2	NUM
ejpam-6429	135	6	,	,	PUNCT
ejpam-6429	135	7	then	then	ADV
ejpam-6429	135	8	according	accord	VERB
ejpam-6429	135	9	to	to	ADP
ejpam-6429	135	10	proposition	proposition	NOUN
ejpam-6429	135	11	5	5	NUM
ejpam-6429	135	12	,	,	PUNCT
ejpam-6429	135	13	we	we	PRON
ejpam-6429	135	14	have	have	VERB
ejpam-6429	135	15	g	g	PROPN
ejpam-6429	135	16	=	=	SYM
ejpam-6429	135	17	kn	kn	PROPN
ejpam-6429	135	18	.	.	PUNCT
ejpam-6429	136	1	furthermore	furthermore	ADV
ejpam-6429	136	2	,	,	PUNCT
ejpam-6429	136	3	proposition	proposition	NOUN
ejpam-6429	136	4	5	5	NUM
ejpam-6429	136	5	implies	imply	VERB
ejpam-6429	136	6	that	that	SCONJ
ejpam-6429	136	7	subdividing	subdivide	VERB
ejpam-6429	136	8	each	each	DET
ejpam-6429	136	9	edge	edge	NOUN
ejpam-6429	136	10	of	of	ADP
ejpam-6429	136	11	kn	kn	PROPN
ejpam-6429	136	12	increases	increase	VERB
ejpam-6429	136	13	the	the	DET
ejpam-6429	136	14	grd	grd	NOUN
ejpam-6429	136	15	-	-	PUNCT
ejpam-6429	136	16	number	number	NOUN
ejpam-6429	136	17	,	,	PUNCT
ejpam-6429	136	18	leading	lead	VERB
ejpam-6429	136	19	to	to	ADP
ejpam-6429	136	20	sdγgr(g	sdγgr(g	PROPN
ejpam-6429	136	21	)	)	PUNCT
ejpam-6429	136	22	=	=	SYM
ejpam-6429	137	1	1	1	X
ejpam-6429	137	2	.	.	PUNCT
ejpam-6429	137	3	now	now	ADV
ejpam-6429	137	4	,	,	PUNCT
ejpam-6429	137	5	suppose	suppose	VERB
ejpam-6429	137	6	γgr(g	γgr(g	PROPN
ejpam-6429	137	7	)	)	PUNCT
ejpam-6429	137	8	=	=	SYM
ejpam-6429	137	9	3	3	X
ejpam-6429	137	10	.	.	X
ejpam-6429	137	11	subdividing	subdivide	VERB
ejpam-6429	137	12	any	any	DET
ejpam-6429	137	13	edge	edge	NOUN
ejpam-6429	137	14	e	e	NOUN
ejpam-6429	137	15	of	of	ADP
ejpam-6429	137	16	g	g	PROPN
ejpam-6429	137	17	results	result	NOUN
ejpam-6429	137	18	in	in	ADP
ejpam-6429	137	19	a	a	DET
ejpam-6429	137	20	new	new	ADJ
ejpam-6429	137	21	graph	graph	NOUN
ejpam-6429	137	22	g′	g′	NOUN
ejpam-6429	137	23	with	with	ADP
ejpam-6429	137	24	order	order	NOUN
ejpam-6429	137	25	n	n	X
ejpam-6429	137	26	+	+	CCONJ
ejpam-6429	137	27	1	1	NUM
ejpam-6429	137	28	and	and	CCONJ
ejpam-6429	137	29	maximum	maximum	ADJ
ejpam-6429	137	30	degree	degree	NOUN
ejpam-6429	137	31	n	n	PRON
ejpam-6429	137	32	−	−	PROPN
ejpam-6429	137	33	1	1	NUM
ejpam-6429	137	34	.	.	PUNCT
ejpam-6429	137	35	by	by	ADP
ejpam-6429	137	36	proposition	proposition	NOUN
ejpam-6429	137	37	6	6	NUM
ejpam-6429	137	38	,	,	PUNCT
ejpam-6429	137	39	it	it	PRON
ejpam-6429	137	40	follows	follow	VERB
ejpam-6429	137	41	that	that	SCONJ
ejpam-6429	137	42	γgr(g	γgr(g	PROPN
ejpam-6429	137	43	)	)	PUNCT
ejpam-6429	137	44	>	>	X
ejpam-6429	137	45	3	3	NUM
ejpam-6429	137	46	,	,	PUNCT
ejpam-6429	137	47	which	which	PRON
ejpam-6429	137	48	implies	imply	VERB
ejpam-6429	137	49	sdγgr(g	sdγgr(g	PROPN
ejpam-6429	137	50	)	)	PUNCT
ejpam-6429	137	51	=	=	SYM
ejpam-6429	138	1	1	1	X
ejpam-6429	138	2	.	.	PUNCT
ejpam-6429	138	3	according	accord	VERB
ejpam-6429	138	4	to	to	ADP
ejpam-6429	138	5	corollary	corollary	ADJ
ejpam-6429	138	6	3	3	NUM
ejpam-6429	138	7	,	,	PUNCT
ejpam-6429	138	8	the	the	DET
ejpam-6429	138	9	following	following	ADJ
ejpam-6429	138	10	result	result	NOUN
ejpam-6429	138	11	is	be	AUX
ejpam-6429	138	12	obtained	obtain	VERB
ejpam-6429	138	13	.	.	PUNCT
ejpam-6429	139	1	corollary	corollary	ADJ
ejpam-6429	139	2	4	4	NUM
ejpam-6429	139	3	.	.	PUNCT
ejpam-6429	140	1	if	if	SCONJ
ejpam-6429	140	2	g	g	PROPN
ejpam-6429	140	3	is	be	AUX
ejpam-6429	140	4	a	a	DET
ejpam-6429	140	5	connected	connected	ADJ
ejpam-6429	140	6	graph	graph	NOUN
ejpam-6429	140	7	of	of	ADP
ejpam-6429	140	8	order	order	NOUN
ejpam-6429	140	9	n	n	PRON
ejpam-6429	140	10	≥	≥	NOUN
ejpam-6429	140	11	3	3	NUM
ejpam-6429	140	12	with	with	ADP
ejpam-6429	140	13	sdγgr(g	sdγgr(g	PROPN
ejpam-6429	140	14	)	)	PUNCT
ejpam-6429	140	15	≥	≥	NOUN
ejpam-6429	140	16	2	2	NUM
ejpam-6429	140	17	,	,	PUNCT
ejpam-6429	140	18	then	then	ADV
ejpam-6429	140	19	γgr(g	γgr(g	PROPN
ejpam-6429	140	20	)	)	PUNCT
ejpam-6429	140	21	≥	≥	NOUN
ejpam-6429	140	22	4	4	NUM
ejpam-6429	140	23	.	.	PUNCT
ejpam-6429	141	1	next	next	ADV
ejpam-6429	141	2	,	,	PUNCT
ejpam-6429	141	3	we	we	PRON
ejpam-6429	141	4	examine	examine	VERB
ejpam-6429	141	5	the	the	DET
ejpam-6429	141	6	ladder	ladder	NOUN
ejpam-6429	141	7	graphs	graph	NOUN
ejpam-6429	141	8	of	of	ADP
ejpam-6429	141	9	the	the	DET
ejpam-6429	141	10	form	form	NOUN
ejpam-6429	141	11	g	g	NOUN
ejpam-6429	141	12	=	=	NOUN
ejpam-6429	141	13	p2	p2	X
ejpam-6429	141	14	□	□	SYM
ejpam-6429	141	15	pn	pn	NOUN
ejpam-6429	141	16	and	and	CCONJ
ejpam-6429	141	17	demonstrate	demonstrate	VERB
ejpam-6429	141	18	that	that	SCONJ
ejpam-6429	141	19	sdγgr(p2	sdγgr(p2	ADJ
ejpam-6429	141	20	□	□	NUM
ejpam-6429	141	21	pn	pn	NOUN
ejpam-6429	141	22	)	)	PUNCT
ejpam-6429	141	23	=	=	SYM
ejpam-6429	141	24	1	1	NUM
ejpam-6429	141	25	when	when	SCONJ
ejpam-6429	141	26	n	n	X
ejpam-6429	141	27	is	be	AUX
ejpam-6429	141	28	odd	odd	ADJ
ejpam-6429	141	29	or	or	CCONJ
ejpam-6429	141	30	n	n	NOUN
ejpam-6429	141	31	=	=	NOUN
ejpam-6429	141	32	4	4	X
ejpam-6429	141	33	.	.	X
ejpam-6429	142	1	we	we	PRON
ejpam-6429	142	2	label	label	VERB
ejpam-6429	142	3	the	the	DET
ejpam-6429	142	4	vertices	vertex	NOUN
ejpam-6429	142	5	of	of	ADP
ejpam-6429	142	6	the	the	DET
ejpam-6429	142	7	i	i	PROPN
ejpam-6429	142	8	-	-	PUNCT
ejpam-6429	142	9	th	th	X
ejpam-6429	142	10	copy	copy	NOUN
ejpam-6429	142	11	of	of	ADP
ejpam-6429	142	12	p2	p2	PROPN
ejpam-6429	142	13	in	in	ADP
ejpam-6429	142	14	the	the	DET
ejpam-6429	142	15	ladder	ladder	NOUN
ejpam-6429	142	16	p2	p2	NOUN
ejpam-6429	142	17	□	□	SYM
ejpam-6429	142	18	pn	pn	NOUN
ejpam-6429	142	19	as	as	ADP
ejpam-6429	142	20	ui	ui	PROPN
ejpam-6429	142	21	and	and	CCONJ
ejpam-6429	142	22	vi	vi	NOUN
ejpam-6429	142	23	,	,	PUNCT
ejpam-6429	142	24	where	where	SCONJ
ejpam-6429	142	25	i	i	PRON
ejpam-6429	142	26	=	=	NOUN
ejpam-6429	142	27	1	1	NUM
ejpam-6429	142	28	,	,	PUNCT
ejpam-6429	142	29	2	2	NUM
ejpam-6429	142	30	,	,	PUNCT
ejpam-6429	142	31	.	.	PUNCT
ejpam-6429	142	32	.	.	PUNCT
ejpam-6429	143	1	.	.	PUNCT
ejpam-6429	144	1	,	,	PUNCT
ejpam-6429	144	2	n.	n.	VERB
ejpam-6429	144	3	the	the	DET
ejpam-6429	144	4	grd	grd	NOUN
ejpam-6429	144	5	-	-	PUNCT
ejpam-6429	144	6	number	number	NOUN
ejpam-6429	144	7	of	of	ADP
ejpam-6429	144	8	ladder	ladder	NOUN
ejpam-6429	144	9	graphs	graph	NOUN
ejpam-6429	144	10	is	be	AUX
ejpam-6429	144	11	established	establish	VERB
ejpam-6429	144	12	by	by	ADP
ejpam-6429	144	13	sheikholeslami	sheikholeslami	PROPN
ejpam-6429	144	14	et	et	PROPN
ejpam-6429	144	15	al	al	PROPN
ejpam-6429	144	16	.	.	PUNCT
ejpam-6429	145	1	in	in	ADP
ejpam-6429	145	2	[	[	X
ejpam-6429	145	3	12	12	NUM
ejpam-6429	145	4	]	]	PUNCT
ejpam-6429	145	5	.	.	PUNCT
ejpam-6429	146	1	theorem	theorem	NOUN
ejpam-6429	146	2	1	1	NUM
ejpam-6429	146	3	.	.	PUNCT
ejpam-6429	146	4	for	for	ADP
ejpam-6429	146	5	n	n	PRON
ejpam-6429	146	6	≥	≥	NUM
ejpam-6429	146	7	1	1	NUM
ejpam-6429	146	8	,	,	PUNCT
ejpam-6429	146	9	γgr(p2	γgr(p2	PROPN
ejpam-6429	146	10	□	□	SYM
ejpam-6429	146	11	pn	pn	NOUN
ejpam-6429	146	12	)	)	PUNCT
ejpam-6429	146	13	=	=	PUNCT
ejpam-6429	147	1	⌈3n+1	⌈3n+1	PROPN
ejpam-6429	147	2	2	2	NUM
ejpam-6429	147	3	⌉.	⌉.	ADV
ejpam-6429	147	4	j.	j.	PROPN
ejpam-6429	147	5	j.	j.	PROPN
ejpam-6429	147	6	hamja	hamja	PROPN
ejpam-6429	147	7	et	et	PROPN
ejpam-6429	147	8	al	al	PROPN
ejpam-6429	147	9	.	.	PUNCT
ejpam-6429	147	10	/	/	SYM
ejpam-6429	147	11	eur	eur	PROPN
ejpam-6429	147	12	.	.	PUNCT
ejpam-6429	148	1	j.	j.	PROPN
ejpam-6429	148	2	pure	pure	PROPN
ejpam-6429	148	3	appl	appl	PROPN
ejpam-6429	148	4	.	.	PROPN
ejpam-6429	148	5	math	math	PROPN
ejpam-6429	148	6	,	,	PUNCT
ejpam-6429	148	7	18	18	NUM
ejpam-6429	148	8	(	(	PUNCT
ejpam-6429	148	9	4	4	NUM
ejpam-6429	148	10	)	)	PUNCT
ejpam-6429	148	11	(	(	PUNCT
ejpam-6429	148	12	2025	2025	NUM
ejpam-6429	148	13	)	)	PUNCT
ejpam-6429	148	14	,	,	PUNCT
ejpam-6429	148	15	6429	6429	NUM
ejpam-6429	148	16	6	6	NUM
ejpam-6429	148	17	of	of	ADP
ejpam-6429	148	18	16	16	NUM
ejpam-6429	148	19	proposition	proposition	NOUN
ejpam-6429	148	20	7	7	NUM
ejpam-6429	148	21	.	.	PUNCT
ejpam-6429	148	22	sdγgr(p2	sdγgr(p2	ADJ
ejpam-6429	148	23	□	□	PUNCT
ejpam-6429	148	24	p4	p4	ADJ
ejpam-6429	148	25	)	)	PUNCT
ejpam-6429	148	26	=	=	SYM
ejpam-6429	149	1	1	1	X
ejpam-6429	149	2	.	.	PUNCT
ejpam-6429	149	3	proof	proof	NOUN
ejpam-6429	149	4	.	.	PUNCT
ejpam-6429	150	1	suppose	suppose	VERB
ejpam-6429	150	2	that	that	SCONJ
ejpam-6429	150	3	g′	g′	NOUN
ejpam-6429	150	4	is	be	AUX
ejpam-6429	150	5	obtained	obtain	VERB
ejpam-6429	150	6	from	from	ADP
ejpam-6429	150	7	g	g	NOUN
ejpam-6429	150	8	=	=	NOUN
ejpam-6429	150	9	p2	p2	X
ejpam-6429	150	10	□	□	NOUN
ejpam-6429	150	11	p4	p4	NOUN
ejpam-6429	150	12	by	by	ADP
ejpam-6429	150	13	introducing	introduce	VERB
ejpam-6429	150	14	a	a	DET
ejpam-6429	150	15	new	new	ADJ
ejpam-6429	150	16	vertex	vertex	NOUN
ejpam-6429	150	17	x	x	PUNCT
ejpam-6429	150	18	through	through	ADP
ejpam-6429	150	19	the	the	DET
ejpam-6429	150	20	subdivision	subdivision	NOUN
ejpam-6429	150	21	of	of	ADP
ejpam-6429	150	22	the	the	DET
ejpam-6429	150	23	edge	edge	NOUN
ejpam-6429	150	24	u2v2	u2v2	NOUN
ejpam-6429	150	25	.	.	PUNCT
ejpam-6429	151	1	let	let	VERB
ejpam-6429	151	2	f	f	PRON
ejpam-6429	151	3	be	be	AUX
ejpam-6429	151	4	a	a	DET
ejpam-6429	151	5	γgr(g	γgr(g	PROPN
ejpam-6429	151	6	′)-function	′)-function	NOUN
ejpam-6429	151	7	.	.	PUNCT
ejpam-6429	152	1	it	it	PRON
ejpam-6429	152	2	is	be	AUX
ejpam-6429	152	3	enough	enough	ADJ
ejpam-6429	152	4	to	to	PART
ejpam-6429	152	5	establish	establish	VERB
ejpam-6429	152	6	that	that	SCONJ
ejpam-6429	152	7	γgr(g′	γgr(g′	NOUN
ejpam-6429	152	8	)	)	PUNCT
ejpam-6429	152	9	>	>	X
ejpam-6429	152	10	γgr(g	γgr(g	PROPN
ejpam-6429	152	11	)	)	PUNCT
ejpam-6429	152	12	.	.	PUNCT
ejpam-6429	153	1	if	if	SCONJ
ejpam-6429	153	2	f(x	f(x	PROPN
ejpam-6429	153	3	)	)	PUNCT
ejpam-6429	153	4	∈	∈	PROPN
ejpam-6429	153	5	{	{	PUNCT
ejpam-6429	153	6	1	1	NUM
ejpam-6429	153	7	,	,	PUNCT
ejpam-6429	153	8	3	3	NUM
ejpam-6429	153	9	}	}	PUNCT
ejpam-6429	153	10	,	,	PUNCT
ejpam-6429	153	11	then	then	ADV
ejpam-6429	153	12	by	by	ADP
ejpam-6429	153	13	proposition	proposition	NOUN
ejpam-6429	153	14	2	2	NUM
ejpam-6429	153	15	,	,	PUNCT
ejpam-6429	153	16	we	we	PRON
ejpam-6429	153	17	conclude	conclude	VERB
ejpam-6429	153	18	that	that	SCONJ
ejpam-6429	153	19	γgr(g	γgr(g	PROPN
ejpam-6429	153	20	′	′	NUM
ejpam-6429	153	21	)	)	PUNCT
ejpam-6429	153	22	>	>	X
ejpam-6429	153	23	γgr(g	γgr(g	PROPN
ejpam-6429	153	24	)	)	PUNCT
ejpam-6429	153	25	,	,	PUNCT
ejpam-6429	153	26	as	as	SCONJ
ejpam-6429	153	27	required	require	VERB
ejpam-6429	153	28	.	.	PUNCT
ejpam-6429	154	1	we	we	PRON
ejpam-6429	154	2	now	now	ADV
ejpam-6429	154	3	analyze	analyze	VERB
ejpam-6429	154	4	two	two	NUM
ejpam-6429	154	5	cases	case	NOUN
ejpam-6429	154	6	.	.	PUNCT
ejpam-6429	155	1	case	case	NOUN
ejpam-6429	155	2	1	1	NUM
ejpam-6429	155	3	.	.	PUNCT
ejpam-6429	156	1	f(x	f(x	NOUN
ejpam-6429	156	2	)	)	PUNCT
ejpam-6429	157	1	=	=	PUNCT
ejpam-6429	158	1	2	2	X
ejpam-6429	158	2	.	.	X
ejpam-6429	158	3	if	if	SCONJ
ejpam-6429	158	4	f(u2	f(u2	PROPN
ejpam-6429	158	5	)	)	PUNCT
ejpam-6429	158	6	≥	≥	PROPN
ejpam-6429	158	7	1	1	NUM
ejpam-6429	158	8	(	(	PUNCT
ejpam-6429	158	9	a	a	DET
ejpam-6429	158	10	similar	similar	ADJ
ejpam-6429	158	11	argument	argument	NOUN
ejpam-6429	158	12	applies	apply	VERB
ejpam-6429	158	13	for	for	ADP
ejpam-6429	158	14	f(v2	f(v2	NOUN
ejpam-6429	158	15	)	)	PUNCT
ejpam-6429	158	16	≥	≥	NOUN
ejpam-6429	158	17	1	1	NUM
ejpam-6429	158	18	)	)	PUNCT
ejpam-6429	158	19	,	,	PUNCT
ejpam-6429	158	20	then	then	ADV
ejpam-6429	158	21	updating	update	VERB
ejpam-6429	158	22	v2	v2	NOUN
ejpam-6429	158	23	to	to	ADP
ejpam-6429	158	24	the	the	DET
ejpam-6429	158	25	value	value	NOUN
ejpam-6429	158	26	min{3	min{3	NOUN
ejpam-6429	158	27	,	,	PUNCT
ejpam-6429	158	28	f(v2	f(v2	NOUN
ejpam-6429	158	29	)	)	PUNCT
ejpam-6429	158	30	+	+	CCONJ
ejpam-6429	158	31	1	1	X
ejpam-6429	158	32	}	}	PUNCT
ejpam-6429	158	33	yields	yield	VERB
ejpam-6429	158	34	a	a	DET
ejpam-6429	158	35	grd	grd	NOUN
ejpam-6429	158	36	-	-	PUNCT
ejpam-6429	158	37	function	function	NOUN
ejpam-6429	158	38	for	for	ADP
ejpam-6429	158	39	g	g	NOUN
ejpam-6429	158	40	with	with	ADP
ejpam-6429	158	41	a	a	DET
ejpam-6429	158	42	weight	weight	NOUN
ejpam-6429	158	43	smaller	small	ADJ
ejpam-6429	158	44	than	than	ADP
ejpam-6429	158	45	ωg	ωg	ADP
ejpam-6429	158	46	r(f	r(f	PROPN
ejpam-6429	158	47	)	)	PUNCT
ejpam-6429	158	48	.	.	PUNCT
ejpam-6429	159	1	now	now	ADV
ejpam-6429	159	2	,	,	PUNCT
ejpam-6429	159	3	suppose	suppose	VERB
ejpam-6429	159	4	f(u2	f(u2	NOUN
ejpam-6429	159	5	)	)	PUNCT
ejpam-6429	159	6	=	=	SYM
ejpam-6429	159	7	f(v2	f(v2	NOUN
ejpam-6429	159	8	)	)	PUNCT
ejpam-6429	159	9	=	=	SYM
ejpam-6429	160	1	0	0	X
ejpam-6429	160	2	.	.	PUNCT
ejpam-6429	161	1	in	in	ADP
ejpam-6429	161	2	this	this	DET
ejpam-6429	161	3	case	case	NOUN
ejpam-6429	161	4	,	,	PUNCT
ejpam-6429	161	5	it	it	PRON
ejpam-6429	161	6	is	be	AUX
ejpam-6429	161	7	evident	evident	ADJ
ejpam-6429	161	8	that	that	SCONJ
ejpam-6429	161	9	f(u1)+f(v1	f(u1)+f(v1	NOUN
ejpam-6429	161	10	)	)	PUNCT
ejpam-6429	161	11	≥	≥	NOUN
ejpam-6429	161	12	2	2	NUM
ejpam-6429	161	13	.	.	PUNCT
ejpam-6429	161	14	additionally	additionally	ADV
ejpam-6429	161	15	,	,	PUNCT
ejpam-6429	161	16	the	the	DET
ejpam-6429	161	17	function	function	NOUN
ejpam-6429	161	18	f	f	PROPN
ejpam-6429	161	19	restricted	restrict	VERB
ejpam-6429	161	20	to	to	ADP
ejpam-6429	161	21	g	g	PROPN
ejpam-6429	161	22	\	\	PUNCT
ejpam-6429	161	23	{	{	PUNCT
ejpam-6429	161	24	u1	u1	NOUN
ejpam-6429	161	25	,	,	PUNCT
ejpam-6429	161	26	v1	v1	NOUN
ejpam-6429	161	27	,	,	PUNCT
ejpam-6429	161	28	u2	u2	NOUN
ejpam-6429	161	29	,	,	PUNCT
ejpam-6429	161	30	v2	v2	PROPN
ejpam-6429	161	31	}	}	PUNCT
ejpam-6429	161	32	serves	serve	VERB
ejpam-6429	161	33	as	as	ADP
ejpam-6429	161	34	a	a	DET
ejpam-6429	161	35	grd	grd	NOUN
ejpam-6429	161	36	-	-	PUNCT
ejpam-6429	161	37	function	function	NOUN
ejpam-6429	161	38	of	of	ADP
ejpam-6429	161	39	p2	p2	NOUN
ejpam-6429	161	40	□	□	SYM
ejpam-6429	161	41	p4	p4	ADJ
ejpam-6429	161	42	with	with	ADP
ejpam-6429	161	43	weight	weight	NOUN
ejpam-6429	161	44	at	at	ADP
ejpam-6429	161	45	most	most	ADV
ejpam-6429	161	46	ωg	ωg	ADP
ejpam-6429	161	47	r(f	r(f	PROPN
ejpam-6429	161	48	)	)	PUNCT
ejpam-6429	161	49	−	−	ADP
ejpam-6429	162	1	4	4	X
ejpam-6429	162	2	.	.	PUNCT
ejpam-6429	162	3	consequently	consequently	ADV
ejpam-6429	162	4	,	,	PUNCT
ejpam-6429	162	5	we	we	PRON
ejpam-6429	162	6	obtain	obtain	VERB
ejpam-6429	162	7	ωg	ωg	PART
ejpam-6429	162	8	r(f	r(f	PROPN
ejpam-6429	162	9	)	)	PUNCT
ejpam-6429	162	10	≥	≥	NOUN
ejpam-6429	162	11	8	8	NUM
ejpam-6429	162	12	>	>	SYM
ejpam-6429	162	13	7	7	NUM
ejpam-6429	162	14	=	=	SYM
ejpam-6429	162	15	γgr(g	γgr(g	PROPN
ejpam-6429	162	16	)	)	PUNCT
ejpam-6429	162	17	(	(	PUNCT
ejpam-6429	162	18	refer	refer	VERB
ejpam-6429	162	19	to	to	ADP
ejpam-6429	162	20	theorem	theorem	NOUN
ejpam-6429	162	21	1	1	NUM
ejpam-6429	162	22	)	)	PUNCT
ejpam-6429	162	23	.	.	PUNCT
ejpam-6429	163	1	case	case	NOUN
ejpam-6429	163	2	2	2	NUM
ejpam-6429	163	3	.	.	PUNCT
ejpam-6429	163	4	f(x	f(x	PROPN
ejpam-6429	163	5	)	)	PUNCT
ejpam-6429	164	1	=	=	PUNCT
ejpam-6429	165	1	0	0	X
ejpam-6429	165	2	.	.	PUNCT
ejpam-6429	166	1	without	without	ADP
ejpam-6429	166	2	loss	loss	NOUN
ejpam-6429	166	3	of	of	ADP
ejpam-6429	166	4	generality	generality	NOUN
ejpam-6429	166	5	,	,	PUNCT
ejpam-6429	166	6	we	we	PRON
ejpam-6429	166	7	assume	assume	VERB
ejpam-6429	166	8	that	that	SCONJ
ejpam-6429	166	9	u2	u2	PROPN
ejpam-6429	166	10	is	be	AUX
ejpam-6429	166	11	a	a	DET
ejpam-6429	166	12	moving	move	VERB
ejpam-6429	166	13	neighbor	neighbor	NOUN
ejpam-6429	166	14	of	of	ADP
ejpam-6429	166	15	x	x	PRON
ejpam-6429	166	16	,	,	PUNCT
ejpam-6429	166	17	implying	imply	VERB
ejpam-6429	166	18	that	that	DET
ejpam-6429	166	19	f(u2	f(u2	PROPN
ejpam-6429	166	20	)	)	PUNCT
ejpam-6429	166	21	≥	≥	NOUN
ejpam-6429	167	1	2	2	NUM
ejpam-6429	167	2	.	.	PUNCT
ejpam-6429	168	1	if	if	SCONJ
ejpam-6429	168	2	f(u2	f(u2	PROPN
ejpam-6429	168	3	)	)	PUNCT
ejpam-6429	169	1	=	=	SYM
ejpam-6429	169	2	3	3	X
ejpam-6429	169	3	,	,	PUNCT
ejpam-6429	169	4	it	it	PRON
ejpam-6429	169	5	is	be	AUX
ejpam-6429	169	6	easy	easy	ADJ
ejpam-6429	169	7	to	to	PART
ejpam-6429	169	8	verify	verify	VERB
ejpam-6429	169	9	that	that	SCONJ
ejpam-6429	169	10	in	in	ADP
ejpam-6429	169	11	order	order	NOUN
ejpam-6429	169	12	to	to	PART
ejpam-6429	169	13	protect	protect	VERB
ejpam-6429	169	14	the	the	DET
ejpam-6429	169	15	vertices	vertex	NOUN
ejpam-6429	169	16	u4	u4	PROPN
ejpam-6429	169	17	,	,	PUNCT
ejpam-6429	169	18	v1	v1	PROPN
ejpam-6429	169	19	,	,	PUNCT
ejpam-6429	169	20	v2	v2	PROPN
ejpam-6429	169	21	,	,	PUNCT
ejpam-6429	169	22	v3	v3	PROPN
ejpam-6429	169	23	,	,	PUNCT
ejpam-6429	169	24	v4	v4	PROPN
ejpam-6429	169	25	,	,	PUNCT
ejpam-6429	169	26	we	we	PRON
ejpam-6429	169	27	must	must	AUX
ejpam-6429	169	28	have	have	VERB
ejpam-6429	169	29	ωg	ωg	PART
ejpam-6429	169	30	r(f	r(f	PROPN
ejpam-6429	169	31	)	)	PUNCT
ejpam-6429	169	32	−	−	PROPN
ejpam-6429	169	33	f(u2	f(u2	PROPN
ejpam-6429	169	34	)	)	PUNCT
ejpam-6429	169	35	≥	≥	NOUN
ejpam-6429	169	36	5	5	NUM
ejpam-6429	169	37	,	,	PUNCT
ejpam-6429	169	38	which	which	PRON
ejpam-6429	169	39	implies	imply	VERB
ejpam-6429	169	40	ωg	ωg	PART
ejpam-6429	169	41	r(f	r(f	PROPN
ejpam-6429	169	42	)	)	PUNCT
ejpam-6429	169	43	≥	≥	NOUN
ejpam-6429	169	44	8	8	NUM
ejpam-6429	169	45	>	>	X
ejpam-6429	169	46	γgr(g	γgr(g	PROPN
ejpam-6429	169	47	)	)	PUNCT
ejpam-6429	169	48	.	.	PUNCT
ejpam-6429	170	1	let	let	VERB
ejpam-6429	170	2	f(u2	f(u2	NOUN
ejpam-6429	170	3	)	)	PUNCT
ejpam-6429	171	1	=	=	SYM
ejpam-6429	171	2	2	2	NUM
ejpam-6429	171	3	and	and	CCONJ
ejpam-6429	171	4	define	define	VERB
ejpam-6429	171	5	t	t	PROPN
ejpam-6429	171	6	=	=	SYM
ejpam-6429	171	7	f(u1)+	f(u1)+	PROPN
ejpam-6429	171	8	f(u2)+	f(u2)+	PROPN
ejpam-6429	171	9	f(v1)+	f(v1)+	PROPN
ejpam-6429	171	10	f(v2	f(v2	NOUN
ejpam-6429	171	11	)	)	PUNCT
ejpam-6429	171	12	.	.	PUNCT
ejpam-6429	172	1	since	since	SCONJ
ejpam-6429	172	2	u2	u2	PROPN
ejpam-6429	172	3	is	be	AUX
ejpam-6429	172	4	a	a	DET
ejpam-6429	172	5	moving	move	VERB
ejpam-6429	172	6	neighbor	neighbor	NOUN
ejpam-6429	172	7	of	of	ADP
ejpam-6429	172	8	x	x	SYM
ejpam-6429	172	9	,	,	PUNCT
ejpam-6429	172	10	we	we	PRON
ejpam-6429	172	11	must	must	AUX
ejpam-6429	172	12	have	have	VERB
ejpam-6429	172	13	either	either	CCONJ
ejpam-6429	172	14	f(u1	f(u1	NOUN
ejpam-6429	172	15	)	)	PUNCT
ejpam-6429	172	16	≥	≥	NOUN
ejpam-6429	172	17	1	1	NUM
ejpam-6429	172	18	or	or	CCONJ
ejpam-6429	172	19	f(v1	f(v1	ADJ
ejpam-6429	172	20	)	)	PUNCT
ejpam-6429	172	21	≥	≥	NOUN
ejpam-6429	172	22	2	2	NUM
ejpam-6429	172	23	.	.	PUNCT
ejpam-6429	173	1	first	first	ADV
ejpam-6429	173	2	,	,	PUNCT
ejpam-6429	173	3	assume	assume	VERB
ejpam-6429	173	4	that	that	SCONJ
ejpam-6429	173	5	f(v2	f(v2	NOUN
ejpam-6429	173	6	)	)	PUNCT
ejpam-6429	173	7	≥	≥	NOUN
ejpam-6429	174	1	2	2	NUM
ejpam-6429	174	2	.	.	PUNCT
ejpam-6429	174	3	if	if	SCONJ
ejpam-6429	174	4	f(v2	f(v2	NOUN
ejpam-6429	174	5	)	)	PUNCT
ejpam-6429	174	6	=	=	SYM
ejpam-6429	174	7	3	3	NUM
ejpam-6429	174	8	,	,	PUNCT
ejpam-6429	174	9	the	the	DET
ejpam-6429	174	10	result	result	NOUN
ejpam-6429	174	11	follows	follow	VERB
ejpam-6429	174	12	as	as	ADP
ejpam-6429	174	13	before	before	ADV
ejpam-6429	174	14	.	.	PUNCT
ejpam-6429	175	1	if	if	SCONJ
ejpam-6429	175	2	f(v2	f(v2	NOUN
ejpam-6429	175	3	)	)	PUNCT
ejpam-6429	175	4	=	=	SYM
ejpam-6429	175	5	2	2	NUM
ejpam-6429	175	6	,	,	PUNCT
ejpam-6429	175	7	then	then	ADV
ejpam-6429	175	8	we	we	PRON
ejpam-6429	175	9	have	have	VERB
ejpam-6429	175	10	t	t	PROPN
ejpam-6429	175	11	≥	≥	NUM
ejpam-6429	175	12	5	5	NUM
ejpam-6429	175	13	.	.	PUNCT
ejpam-6429	176	1	if	if	SCONJ
ejpam-6429	176	2	t	t	PROPN
ejpam-6429	176	3	≥	≥	NUM
ejpam-6429	176	4	6	6	NUM
ejpam-6429	176	5	,	,	PUNCT
ejpam-6429	176	6	in	in	ADP
ejpam-6429	176	7	order	order	NOUN
ejpam-6429	176	8	to	to	PART
ejpam-6429	176	9	protect	protect	VERB
ejpam-6429	176	10	other	other	ADJ
ejpam-6429	176	11	vertices	vertex	NOUN
ejpam-6429	176	12	,	,	PUNCT
ejpam-6429	176	13	we	we	PRON
ejpam-6429	176	14	must	must	AUX
ejpam-6429	176	15	have	have	VERB
ejpam-6429	176	16	f(u3	f(u3	NOUN
ejpam-6429	176	17	)	)	PUNCT
ejpam-6429	177	1	+	+	NUM
ejpam-6429	177	2	f(u4	f(u4	NOUN
ejpam-6429	177	3	)	)	PUNCT
ejpam-6429	178	1	+	+	SYM
ejpam-6429	178	2	f(v3	f(v3	X
ejpam-6429	178	3	)	)	PUNCT
ejpam-6429	178	4	+	+	CCONJ
ejpam-6429	178	5	f(v4	f(v4	PRON
ejpam-6429	178	6	)	)	PUNCT
ejpam-6429	178	7	≥	≥	NOUN
ejpam-6429	178	8	2	2	NUM
ejpam-6429	178	9	,	,	PUNCT
ejpam-6429	178	10	which	which	PRON
ejpam-6429	178	11	leads	lead	VERB
ejpam-6429	178	12	to	to	ADP
ejpam-6429	178	13	ωg	ωg	PART
ejpam-6429	178	14	r(f	r(f	PROPN
ejpam-6429	178	15	)	)	PUNCT
ejpam-6429	178	16	≥	≥	NOUN
ejpam-6429	178	17	8	8	NUM
ejpam-6429	178	18	>	>	X
ejpam-6429	178	19	γgr(g	γgr(g	PROPN
ejpam-6429	178	20	)	)	PUNCT
ejpam-6429	178	21	.	.	PUNCT
ejpam-6429	179	1	if	if	SCONJ
ejpam-6429	179	2	t	t	NOUN
ejpam-6429	179	3	=	=	SYM
ejpam-6429	179	4	5	5	NUM
ejpam-6429	179	5	,	,	PUNCT
ejpam-6429	179	6	then	then	ADV
ejpam-6429	179	7	f(u1	f(u1	NOUN
ejpam-6429	179	8	)	)	PUNCT
ejpam-6429	180	1	=	=	SYM
ejpam-6429	180	2	1	1	NUM
ejpam-6429	180	3	,	,	PUNCT
ejpam-6429	180	4	f(v1	f(v1	ADJ
ejpam-6429	180	5	)	)	PUNCT
ejpam-6429	180	6	=	=	SYM
ejpam-6429	180	7	0	0	NUM
ejpam-6429	180	8	,	,	PUNCT
ejpam-6429	180	9	and	and	CCONJ
ejpam-6429	180	10	v2	v2	NOUN
ejpam-6429	180	11	is	be	AUX
ejpam-6429	180	12	a	a	DET
ejpam-6429	180	13	moving	move	VERB
ejpam-6429	180	14	neighbor	neighbor	NOUN
ejpam-6429	180	15	only	only	ADV
ejpam-6429	180	16	for	for	ADP
ejpam-6429	180	17	v1	v1	NOUN
ejpam-6429	180	18	.	.	PUNCT
ejpam-6429	181	1	to	to	PART
ejpam-6429	181	2	protect	protect	VERB
ejpam-6429	181	3	the	the	DET
ejpam-6429	181	4	vertices	vertex	NOUN
ejpam-6429	181	5	u4	u4	PROPN
ejpam-6429	181	6	,	,	PUNCT
ejpam-6429	181	7	v3	v3	PROPN
ejpam-6429	181	8	,	,	PUNCT
ejpam-6429	181	9	v4	v4	PROPN
ejpam-6429	181	10	,	,	PUNCT
ejpam-6429	181	11	we	we	PRON
ejpam-6429	181	12	must	must	AUX
ejpam-6429	181	13	have	have	VERB
ejpam-6429	181	14	f(u3	f(u3	NOUN
ejpam-6429	181	15	)	)	PUNCT
ejpam-6429	182	1	+	+	NUM
ejpam-6429	182	2	f(u4	f(u4	NOUN
ejpam-6429	182	3	)	)	PUNCT
ejpam-6429	183	1	+	+	SYM
ejpam-6429	183	2	f(v3	f(v3	X
ejpam-6429	183	3	)	)	PUNCT
ejpam-6429	183	4	+	+	CCONJ
ejpam-6429	183	5	f(v4	f(v4	PRON
ejpam-6429	183	6	)	)	PUNCT
ejpam-6429	183	7	≥	≥	NOUN
ejpam-6429	183	8	3	3	NUM
ejpam-6429	183	9	,	,	PUNCT
ejpam-6429	183	10	so	so	ADV
ejpam-6429	183	11	ωg	ωg	ADP
ejpam-6429	183	12	r(f	r(f	PROPN
ejpam-6429	183	13	)	)	PUNCT
ejpam-6429	183	14	≥	≥	NOUN
ejpam-6429	183	15	8	8	NUM
ejpam-6429	183	16	>	>	X
ejpam-6429	183	17	γgr(g	γgr(g	PROPN
ejpam-6429	183	18	)	)	PUNCT
ejpam-6429	183	19	.	.	PUNCT
ejpam-6429	184	1	assume	assume	VERB
ejpam-6429	184	2	now	now	ADV
ejpam-6429	184	3	that	that	SCONJ
ejpam-6429	184	4	f(v2	f(v2	NOUN
ejpam-6429	184	5	)	)	PUNCT
ejpam-6429	184	6	≤	≤	NUM
ejpam-6429	184	7	1	1	NUM
ejpam-6429	184	8	.	.	PUNCT
ejpam-6429	185	1	in	in	ADP
ejpam-6429	185	2	this	this	DET
ejpam-6429	185	3	case	case	NOUN
ejpam-6429	185	4	,	,	PUNCT
ejpam-6429	185	5	u2	u2	PROPN
ejpam-6429	185	6	is	be	AUX
ejpam-6429	185	7	a	a	DET
ejpam-6429	185	8	moving	move	VERB
ejpam-6429	185	9	neighbor	neighbor	NOUN
ejpam-6429	185	10	only	only	ADV
ejpam-6429	185	11	for	for	ADP
ejpam-6429	185	12	x.	x.	NOUN
ejpam-6429	185	13	if	if	SCONJ
ejpam-6429	185	14	f(v2	f(v2	NOUN
ejpam-6429	185	15	)	)	PUNCT
ejpam-6429	185	16	=	=	SYM
ejpam-6429	185	17	1	1	NUM
ejpam-6429	185	18	,	,	PUNCT
ejpam-6429	185	19	then	then	ADV
ejpam-6429	185	20	to	to	PART
ejpam-6429	185	21	protect	protect	VERB
ejpam-6429	185	22	the	the	DET
ejpam-6429	185	23	vertices	vertex	NOUN
ejpam-6429	185	24	u1	u1	NOUN
ejpam-6429	185	25	and	and	CCONJ
ejpam-6429	185	26	v1	v1	NOUN
ejpam-6429	185	27	,	,	PUNCT
ejpam-6429	185	28	we	we	PRON
ejpam-6429	185	29	must	must	AUX
ejpam-6429	185	30	have	have	VERB
ejpam-6429	185	31	f(u1	f(u1	NOUN
ejpam-6429	185	32	)	)	PUNCT
ejpam-6429	186	1	+	+	NUM
ejpam-6429	186	2	f(v1	f(v1	NOUN
ejpam-6429	186	3	)	)	PUNCT
ejpam-6429	186	4	≥	≥	NOUN
ejpam-6429	186	5	2	2	NUM
ejpam-6429	186	6	.	.	PUNCT
ejpam-6429	186	7	reassigning	reassign	VERB
ejpam-6429	186	8	u1	u1	NOUN
ejpam-6429	186	9	and	and	CCONJ
ejpam-6429	186	10	v2	v2	VERB
ejpam-6429	186	11	the	the	DET
ejpam-6429	186	12	values	value	NOUN
ejpam-6429	186	13	0	0	NUM
ejpam-6429	186	14	and	and	CCONJ
ejpam-6429	186	15	2	2	NUM
ejpam-6429	186	16	,	,	PUNCT
ejpam-6429	186	17	respectively	respectively	ADV
ejpam-6429	186	18	,	,	PUNCT
ejpam-6429	186	19	yields	yield	VERB
ejpam-6429	186	20	a	a	DET
ejpam-6429	186	21	grd	grd	NOUN
ejpam-6429	186	22	-	-	PUNCT
ejpam-6429	186	23	function	function	NOUN
ejpam-6429	186	24	of	of	ADP
ejpam-6429	186	25	g	g	NOUN
ejpam-6429	186	26	with	with	ADP
ejpam-6429	186	27	a	a	DET
ejpam-6429	186	28	weight	weight	NOUN
ejpam-6429	186	29	less	less	ADJ
ejpam-6429	186	30	than	than	ADP
ejpam-6429	186	31	γgr(g1	γgr(g1	ADJ
ejpam-6429	186	32	)	)	PUNCT
ejpam-6429	186	33	,	,	PUNCT
ejpam-6429	186	34	as	as	SCONJ
ejpam-6429	186	35	desired	desire	VERB
ejpam-6429	186	36	(	(	PUNCT
ejpam-6429	186	37	note	note	VERB
ejpam-6429	186	38	that	that	SCONJ
ejpam-6429	186	39	u2	u2	NOUN
ejpam-6429	186	40	will	will	AUX
ejpam-6429	186	41	be	be	AUX
ejpam-6429	186	42	a	a	DET
ejpam-6429	186	43	moving	move	VERB
ejpam-6429	186	44	neighbor	neighbor	NOUN
ejpam-6429	186	45	for	for	ADP
ejpam-6429	186	46	u1	u1	NOUN
ejpam-6429	186	47	in	in	ADP
ejpam-6429	186	48	the	the	DET
ejpam-6429	186	49	new	new	ADJ
ejpam-6429	186	50	assignment	assignment	NOUN
ejpam-6429	186	51	)	)	PUNCT
ejpam-6429	186	52	.	.	PUNCT
ejpam-6429	187	1	thus	thus	ADV
ejpam-6429	187	2	,	,	PUNCT
ejpam-6429	187	3	we	we	PRON
ejpam-6429	187	4	assume	assume	VERB
ejpam-6429	187	5	that	that	SCONJ
ejpam-6429	187	6	f(v2	f(v2	NOUN
ejpam-6429	187	7	)	)	PUNCT
ejpam-6429	187	8	=	=	SYM
ejpam-6429	188	1	0	0	X
ejpam-6429	188	2	.	.	PUNCT
ejpam-6429	188	3	to	to	PART
ejpam-6429	188	4	protect	protect	VERB
ejpam-6429	188	5	u1	u1	NOUN
ejpam-6429	188	6	and	and	CCONJ
ejpam-6429	188	7	v1	v1	NOUN
ejpam-6429	188	8	,	,	PUNCT
ejpam-6429	188	9	we	we	PRON
ejpam-6429	188	10	must	must	AUX
ejpam-6429	188	11	have	have	VERB
ejpam-6429	188	12	f(u1)+	f(u1)+	PROPN
ejpam-6429	188	13	f(v1	f(v1	PROPN
ejpam-6429	188	14	)	)	PUNCT
ejpam-6429	188	15	≥	≥	NOUN
ejpam-6429	189	1	2	2	NUM
ejpam-6429	189	2	.	.	PUNCT
ejpam-6429	190	1	if	if	SCONJ
ejpam-6429	190	2	f(u1)+	f(u1)+	PROPN
ejpam-6429	190	3	f(v1	f(v1	NOUN
ejpam-6429	190	4	)	)	PUNCT
ejpam-6429	190	5	≥	≥	NOUN
ejpam-6429	190	6	3	3	NUM
ejpam-6429	190	7	,	,	PUNCT
ejpam-6429	190	8	as	as	ADP
ejpam-6429	190	9	before	before	ADV
ejpam-6429	190	10	,	,	PUNCT
ejpam-6429	190	11	we	we	PRON
ejpam-6429	190	12	observe	observe	VERB
ejpam-6429	190	13	that	that	SCONJ
ejpam-6429	190	14	γgr(g′	γgr(g′	NOUN
ejpam-6429	190	15	)	)	PUNCT
ejpam-6429	190	16	≥	≥	NOUN
ejpam-6429	190	17	8	8	NUM
ejpam-6429	190	18	>	>	X
ejpam-6429	190	19	γgr(g	γgr(g	PROPN
ejpam-6429	190	20	)	)	PUNCT
ejpam-6429	190	21	.	.	PUNCT
ejpam-6429	191	1	let	let	VERB
ejpam-6429	191	2	f(u1	f(u1	NOUN
ejpam-6429	191	3	)	)	PUNCT
ejpam-6429	192	1	+	+	NUM
ejpam-6429	192	2	f(v1	f(v1	NOUN
ejpam-6429	192	3	)	)	PUNCT
ejpam-6429	192	4	=	=	SYM
ejpam-6429	192	5	2	2	X
ejpam-6429	192	6	.	.	NOUN
ejpam-6429	192	7	without	without	ADP
ejpam-6429	192	8	loss	loss	NOUN
ejpam-6429	192	9	of	of	ADP
ejpam-6429	192	10	generality	generality	NOUN
ejpam-6429	192	11	,	,	PUNCT
ejpam-6429	192	12	assume	assume	VERB
ejpam-6429	192	13	that	that	SCONJ
ejpam-6429	192	14	f(v1	f(v1	VERB
ejpam-6429	192	15	)	)	PUNCT
ejpam-6429	192	16	=	=	SYM
ejpam-6429	192	17	2	2	NUM
ejpam-6429	192	18	and	and	CCONJ
ejpam-6429	192	19	f(u1	f(u1	NOUN
ejpam-6429	192	20	)	)	PUNCT
ejpam-6429	193	1	=	=	SYM
ejpam-6429	193	2	0	0	X
ejpam-6429	193	3	.	.	PUNCT
ejpam-6429	194	1	since	since	SCONJ
ejpam-6429	194	2	u2	u2	PROPN
ejpam-6429	194	3	is	be	AUX
ejpam-6429	194	4	a	a	DET
ejpam-6429	194	5	moving	move	VERB
ejpam-6429	194	6	neighbor	neighbor	NOUN
ejpam-6429	194	7	only	only	ADV
ejpam-6429	194	8	for	for	ADP
ejpam-6429	194	9	x	x	NOUN
ejpam-6429	194	10	,	,	PUNCT
ejpam-6429	194	11	v1	v1	NOUN
ejpam-6429	194	12	becomes	become	VERB
ejpam-6429	194	13	a	a	DET
ejpam-6429	194	14	moving	move	VERB
ejpam-6429	194	15	neighbor	neighbor	NOUN
ejpam-6429	194	16	of	of	ADP
ejpam-6429	194	17	u1	u1	NOUN
ejpam-6429	194	18	,	,	PUNCT
ejpam-6429	194	19	leading	lead	VERB
ejpam-6429	194	20	to	to	ADP
ejpam-6429	194	21	f(v3	f(v3	NUM
ejpam-6429	194	22	)	)	PUNCT
ejpam-6429	194	23	≥	≥	NOUN
ejpam-6429	195	1	2	2	NUM
ejpam-6429	195	2	.	.	X
ejpam-6429	196	1	if	if	SCONJ
ejpam-6429	196	2	f(v3	f(v3	NUM
ejpam-6429	196	3	)	)	PUNCT
ejpam-6429	197	1	=	=	SYM
ejpam-6429	197	2	3	3	NUM
ejpam-6429	197	3	,	,	PUNCT
ejpam-6429	197	4	the	the	DET
ejpam-6429	197	5	result	result	NOUN
ejpam-6429	197	6	follows	follow	VERB
ejpam-6429	197	7	as	as	ADP
ejpam-6429	197	8	before	before	ADV
ejpam-6429	197	9	.	.	PUNCT
ejpam-6429	198	1	suppose	suppose	VERB
ejpam-6429	198	2	f(v3	f(v3	NOUN
ejpam-6429	198	3	)	)	PUNCT
ejpam-6429	198	4	=	=	SYM
ejpam-6429	199	1	2	2	X
ejpam-6429	199	2	.	.	PUNCT
ejpam-6429	199	3	now	now	ADV
ejpam-6429	199	4	,	,	PUNCT
ejpam-6429	199	5	to	to	PART
ejpam-6429	199	6	protect	protect	VERB
ejpam-6429	199	7	u4	u4	PROPN
ejpam-6429	199	8	and	and	CCONJ
ejpam-6429	199	9	v4	v4	PROPN
ejpam-6429	199	10	,	,	PUNCT
ejpam-6429	199	11	we	we	PRON
ejpam-6429	199	12	must	must	AUX
ejpam-6429	199	13	have	have	VERB
ejpam-6429	199	14	f(u4)+f(v4)+f(u3	f(u4)+f(v4)+f(u3	NOUN
ejpam-6429	199	15	)	)	PUNCT
ejpam-6429	199	16	≥	≥	NOUN
ejpam-6429	199	17	2	2	NUM
ejpam-6429	199	18	,	,	PUNCT
ejpam-6429	199	19	which	which	PRON
ejpam-6429	199	20	again	again	ADV
ejpam-6429	199	21	gives	give	VERB
ejpam-6429	199	22	ωg	ωg	PART
ejpam-6429	199	23	r(f	r(f	PROPN
ejpam-6429	199	24	)	)	PUNCT
ejpam-6429	199	25	≥	≥	NOUN
ejpam-6429	199	26	8	8	NUM
ejpam-6429	199	27	>	>	X
ejpam-6429	199	28	γgr(g	γgr(g	PROPN
ejpam-6429	199	29	)	)	PUNCT
ejpam-6429	199	30	.	.	PUNCT
ejpam-6429	200	1	this	this	PRON
ejpam-6429	200	2	concludes	conclude	VERB
ejpam-6429	200	3	the	the	DET
ejpam-6429	200	4	proof	proof	NOUN
ejpam-6429	200	5	.	.	PUNCT
ejpam-6429	201	1	theorem	theorem	NOUN
ejpam-6429	201	2	2	2	NUM
ejpam-6429	201	3	.	.	PUNCT
ejpam-6429	202	1	if	if	SCONJ
ejpam-6429	202	2	n	n	NUM
ejpam-6429	202	3	≥	≥	NOUN
ejpam-6429	202	4	1	1	NUM
ejpam-6429	202	5	is	be	AUX
ejpam-6429	202	6	odd	odd	ADJ
ejpam-6429	202	7	,	,	PUNCT
ejpam-6429	202	8	then	then	ADV
ejpam-6429	202	9	sdγgr(p2	sdγgr(p2	ADJ
ejpam-6429	202	10	□	□	NUM
ejpam-6429	202	11	pn	pn	NOUN
ejpam-6429	202	12	)	)	PUNCT
ejpam-6429	202	13	=	=	SYM
ejpam-6429	202	14	1	1	X
ejpam-6429	202	15	.	.	PUNCT
ejpam-6429	203	1	proof	proof	NOUN
ejpam-6429	203	2	.	.	PUNCT
ejpam-6429	204	1	let	let	VERB
ejpam-6429	204	2	g	g	NOUN
ejpam-6429	204	3	=	=	VERB
ejpam-6429	204	4	p2	p2	X
ejpam-6429	204	5	□	□	SYM
ejpam-6429	204	6	pn	pn	NOUN
ejpam-6429	204	7	.	.	PUNCT
ejpam-6429	205	1	if	if	SCONJ
ejpam-6429	205	2	n	n	NUM
ejpam-6429	205	3	=	=	SYM
ejpam-6429	205	4	1	1	NUM
ejpam-6429	205	5	,	,	PUNCT
ejpam-6429	205	6	the	the	DET
ejpam-6429	205	7	result	result	NOUN
ejpam-6429	205	8	follows	follow	VERB
ejpam-6429	205	9	from	from	ADP
ejpam-6429	205	10	corollary	corollary	ADJ
ejpam-6429	205	11	1	1	NUM
ejpam-6429	205	12	.	.	PUNCT
ejpam-6429	205	13	assume	assume	VERB
ejpam-6429	205	14	n	n	PRON
ejpam-6429	205	15	≥	≥	NOUN
ejpam-6429	205	16	3	3	NUM
ejpam-6429	205	17	,	,	PUNCT
ejpam-6429	205	18	and	and	CCONJ
ejpam-6429	205	19	let	let	VERB
ejpam-6429	205	20	g′	g′	NOUN
ejpam-6429	205	21	be	be	AUX
ejpam-6429	205	22	the	the	DET
ejpam-6429	205	23	graph	graph	NOUN
ejpam-6429	205	24	derived	derive	VERB
ejpam-6429	205	25	from	from	ADP
ejpam-6429	205	26	g	g	NOUN
ejpam-6429	205	27	by	by	ADP
ejpam-6429	205	28	subdividing	subdivide	VERB
ejpam-6429	205	29	the	the	DET
ejpam-6429	205	30	edge	edge	NOUN
ejpam-6429	205	31	u1v1	u1v1	NOUN
ejpam-6429	205	32	with	with	ADP
ejpam-6429	205	33	a	a	DET
ejpam-6429	205	34	new	new	ADJ
ejpam-6429	205	35	vertex	vertex	NOUN
ejpam-6429	205	36	x.	x.	NOUN
ejpam-6429	205	37	let	let	VERB
ejpam-6429	205	38	f	f	PRON
ejpam-6429	205	39	be	be	AUX
ejpam-6429	205	40	a	a	DET
ejpam-6429	205	41	γgr(g	γgr(g	PROPN
ejpam-6429	205	42	′)-function	′)-function	NOUN
ejpam-6429	205	43	.	.	PUNCT
ejpam-6429	206	1	if	if	SCONJ
ejpam-6429	206	2	f(x	f(x	PROPN
ejpam-6429	206	3	)	)	PUNCT
ejpam-6429	206	4	∈	∈	PROPN
ejpam-6429	206	5	{	{	PUNCT
ejpam-6429	206	6	1	1	NUM
ejpam-6429	206	7	,	,	PUNCT
ejpam-6429	206	8	3	3	NUM
ejpam-6429	206	9	}	}	PUNCT
ejpam-6429	206	10	,	,	PUNCT
ejpam-6429	206	11	then	then	ADV
ejpam-6429	206	12	proposition	proposition	NOUN
ejpam-6429	206	13	2	2	NUM
ejpam-6429	206	14	implies	imply	VERB
ejpam-6429	206	15	that	that	SCONJ
ejpam-6429	206	16	γgr(g	γgr(g	PROPN
ejpam-6429	206	17	′	′	NUM
ejpam-6429	206	18	)	)	PUNCT
ejpam-6429	206	19	>	>	X
ejpam-6429	206	20	γgr(g	γgr(g	PROPN
ejpam-6429	206	21	)	)	PUNCT
ejpam-6429	206	22	,	,	PUNCT
ejpam-6429	206	23	so	so	SCONJ
ejpam-6429	206	24	we	we	PRON
ejpam-6429	206	25	have	have	VERB
ejpam-6429	206	26	sdγgr(p2	sdγgr(p2	ADJ
ejpam-6429	206	27	□	□	NUM
ejpam-6429	206	28	pn	pn	NOUN
ejpam-6429	206	29	)	)	PUNCT
ejpam-6429	206	30	=	=	SYM
ejpam-6429	207	1	1	1	X
ejpam-6429	207	2	.	.	PUNCT
ejpam-6429	208	1	next	next	ADV
ejpam-6429	208	2	,	,	PUNCT
ejpam-6429	208	3	we	we	PRON
ejpam-6429	208	4	consider	consider	VERB
ejpam-6429	208	5	two	two	NUM
ejpam-6429	208	6	cases	case	NOUN
ejpam-6429	208	7	.	.	PUNCT
ejpam-6429	209	1	j.	j.	PROPN
ejpam-6429	209	2	j.	j.	PROPN
ejpam-6429	209	3	hamja	hamja	PROPN
ejpam-6429	209	4	et	et	PROPN
ejpam-6429	209	5	al	al	PROPN
ejpam-6429	209	6	.	.	PUNCT
ejpam-6429	209	7	/	/	SYM
ejpam-6429	209	8	eur	eur	PROPN
ejpam-6429	209	9	.	.	PUNCT
ejpam-6429	210	1	j.	j.	PROPN
ejpam-6429	210	2	pure	pure	PROPN
ejpam-6429	210	3	appl	appl	PROPN
ejpam-6429	210	4	.	.	PROPN
ejpam-6429	210	5	math	math	PROPN
ejpam-6429	210	6	,	,	PUNCT
ejpam-6429	210	7	18	18	NUM
ejpam-6429	210	8	(	(	PUNCT
ejpam-6429	210	9	4	4	NUM
ejpam-6429	210	10	)	)	PUNCT
ejpam-6429	210	11	(	(	PUNCT
ejpam-6429	210	12	2025	2025	NUM
ejpam-6429	210	13	)	)	PUNCT
ejpam-6429	210	14	,	,	PUNCT
ejpam-6429	210	15	6429	6429	NUM
ejpam-6429	210	16	7	7	NUM
ejpam-6429	210	17	of	of	ADP
ejpam-6429	210	18	16	16	NUM
ejpam-6429	210	19	case	case	NOUN
ejpam-6429	210	20	1	1	NUM
ejpam-6429	210	21	.	.	PUNCT
ejpam-6429	211	1	f(x	f(x	NOUN
ejpam-6429	211	2	)	)	PUNCT
ejpam-6429	212	1	=	=	PUNCT
ejpam-6429	213	1	2	2	X
ejpam-6429	213	2	.	.	X
ejpam-6429	213	3	if	if	SCONJ
ejpam-6429	213	4	f(u1	f(u1	NOUN
ejpam-6429	213	5	)	)	PUNCT
ejpam-6429	213	6	≥	≥	NOUN
ejpam-6429	213	7	1	1	NUM
ejpam-6429	213	8	(	(	PUNCT
ejpam-6429	213	9	the	the	DET
ejpam-6429	213	10	case	case	NOUN
ejpam-6429	213	11	f(v1	f(v1	NOUN
ejpam-6429	213	12	)	)	PUNCT
ejpam-6429	213	13	≥	≥	NOUN
ejpam-6429	213	14	1	1	NUM
ejpam-6429	213	15	is	be	AUX
ejpam-6429	213	16	similar	similar	ADJ
ejpam-6429	213	17	)	)	PUNCT
ejpam-6429	213	18	,	,	PUNCT
ejpam-6429	213	19	then	then	ADV
ejpam-6429	213	20	by	by	ADP
ejpam-6429	213	21	reassigning	reassign	VERB
ejpam-6429	213	22	v1	v1	VERB
ejpam-6429	213	23	the	the	DET
ejpam-6429	213	24	value	value	NOUN
ejpam-6429	213	25	min{3	min{3	PROPN
ejpam-6429	213	26	,	,	PUNCT
ejpam-6429	213	27	f(v1)+	f(v1)+	NOUN
ejpam-6429	213	28	1	1	NUM
ejpam-6429	213	29	}	}	PUNCT
ejpam-6429	213	30	,	,	PUNCT
ejpam-6429	213	31	we	we	PRON
ejpam-6429	213	32	obtain	obtain	VERB
ejpam-6429	213	33	a	a	DET
ejpam-6429	213	34	grd	grd	NOUN
ejpam-6429	213	35	-	-	PUNCT
ejpam-6429	213	36	function	function	NOUN
ejpam-6429	213	37	for	for	ADP
ejpam-6429	213	38	g	g	NOUN
ejpam-6429	213	39	with	with	ADP
ejpam-6429	213	40	a	a	DET
ejpam-6429	213	41	weight	weight	NOUN
ejpam-6429	213	42	smaller	small	ADJ
ejpam-6429	213	43	than	than	ADP
ejpam-6429	213	44	ωg	ωg	ADP
ejpam-6429	213	45	r(f	r(f	PROPN
ejpam-6429	213	46	)	)	PUNCT
ejpam-6429	213	47	,	,	PUNCT
ejpam-6429	213	48	which	which	PRON
ejpam-6429	213	49	leads	lead	VERB
ejpam-6429	213	50	to	to	ADP
ejpam-6429	213	51	sdγgr(p2	sdγgr(p2	ADJ
ejpam-6429	213	52	□	□	NUM
ejpam-6429	213	53	pn	pn	NOUN
ejpam-6429	213	54	)	)	PUNCT
ejpam-6429	213	55	=	=	SYM
ejpam-6429	214	1	1	1	X
ejpam-6429	214	2	.	.	PUNCT
ejpam-6429	214	3	now	now	ADV
ejpam-6429	214	4	,	,	PUNCT
ejpam-6429	214	5	assume	assume	VERB
ejpam-6429	214	6	that	that	SCONJ
ejpam-6429	214	7	f(u1	f(u1	NOUN
ejpam-6429	214	8	)	)	PUNCT
ejpam-6429	214	9	=	=	SYM
ejpam-6429	214	10	f(v1	f(v1	NOUN
ejpam-6429	214	11	)	)	PUNCT
ejpam-6429	214	12	=	=	SYM
ejpam-6429	215	1	0	0	X
ejpam-6429	215	2	.	.	PUNCT
ejpam-6429	216	1	if	if	SCONJ
ejpam-6429	216	2	x	x	PRON
ejpam-6429	216	3	is	be	AUX
ejpam-6429	216	4	a	a	DET
ejpam-6429	216	5	moving	move	VERB
ejpam-6429	216	6	neighbor	neighbor	NOUN
ejpam-6429	216	7	of	of	ADP
ejpam-6429	216	8	neither	neither	CCONJ
ejpam-6429	216	9	u1	u1	NOUN
ejpam-6429	216	10	nor	nor	CCONJ
ejpam-6429	216	11	v1	v1	NOUN
ejpam-6429	216	12	,	,	PUNCT
ejpam-6429	216	13	or	or	CCONJ
ejpam-6429	216	14	if	if	SCONJ
ejpam-6429	216	15	x	x	PRON
ejpam-6429	216	16	is	be	AUX
ejpam-6429	216	17	a	a	DET
ejpam-6429	216	18	moving	move	VERB
ejpam-6429	216	19	neighbor	neighbor	NOUN
ejpam-6429	216	20	of	of	ADP
ejpam-6429	216	21	both	both	DET
ejpam-6429	216	22	u1	u1	NOUN
ejpam-6429	216	23	and	and	CCONJ
ejpam-6429	216	24	v1	v1	NOUN
ejpam-6429	216	25	,	,	PUNCT
ejpam-6429	216	26	then	then	ADV
ejpam-6429	216	27	we	we	PRON
ejpam-6429	216	28	must	must	AUX
ejpam-6429	216	29	have	have	VERB
ejpam-6429	216	30	min{f(u2	min{f(u2	ADV
ejpam-6429	216	31	)	)	PUNCT
ejpam-6429	216	32	,	,	PUNCT
ejpam-6429	216	33	f(v2	f(v2	NOUN
ejpam-6429	216	34	)	)	PUNCT
ejpam-6429	216	35	}	}	PUNCT
ejpam-6429	216	36	≥	≥	NOUN
ejpam-6429	216	37	2	2	NUM
ejpam-6429	216	38	.	.	PUNCT
ejpam-6429	216	39	reassigning	reassign	VERB
ejpam-6429	216	40	v1	v1	VERB
ejpam-6429	216	41	the	the	DET
ejpam-6429	216	42	value	value	NOUN
ejpam-6429	216	43	1	1	NUM
ejpam-6429	216	44	,	,	PUNCT
ejpam-6429	216	45	u2	u2	NOUN
ejpam-6429	216	46	the	the	DET
ejpam-6429	216	47	value	value	NOUN
ejpam-6429	216	48	2	2	NUM
ejpam-6429	216	49	,	,	PUNCT
ejpam-6429	216	50	v2	v2	VERB
ejpam-6429	216	51	the	the	DET
ejpam-6429	216	52	value	value	NOUN
ejpam-6429	216	53	0	0	PUNCT
ejpam-6429	216	54	and	and	CCONJ
ejpam-6429	216	55	v3	v3	VERB
ejpam-6429	216	56	the	the	DET
ejpam-6429	216	57	value	value	NOUN
ejpam-6429	216	58	min{3	min{3	PROPN
ejpam-6429	216	59	,	,	PUNCT
ejpam-6429	216	60	f(v3)+2	f(v3)+2	PROPN
ejpam-6429	216	61	}	}	PUNCT
ejpam-6429	216	62	gives	give	VERB
ejpam-6429	216	63	a	a	DET
ejpam-6429	216	64	grd	grd	NOUN
ejpam-6429	216	65	-	-	PUNCT
ejpam-6429	216	66	function	function	NOUN
ejpam-6429	216	67	for	for	ADP
ejpam-6429	216	68	g	g	NOUN
ejpam-6429	216	69	with	with	ADP
ejpam-6429	216	70	a	a	DET
ejpam-6429	216	71	weight	weight	NOUN
ejpam-6429	216	72	smaller	small	ADJ
ejpam-6429	216	73	than	than	ADP
ejpam-6429	216	74	ωg	ωg	ADP
ejpam-6429	216	75	r(f	r(f	PROPN
ejpam-6429	216	76	)	)	PUNCT
ejpam-6429	216	77	,	,	PUNCT
ejpam-6429	216	78	so	so	SCONJ
ejpam-6429	216	79	we	we	PRON
ejpam-6429	216	80	have	have	VERB
ejpam-6429	216	81	sdγgr(p2	sdγgr(p2	ADJ
ejpam-6429	216	82	□	□	NUM
ejpam-6429	216	83	pn	pn	NOUN
ejpam-6429	216	84	)	)	PUNCT
ejpam-6429	216	85	=	=	SYM
ejpam-6429	217	1	1	1	X
ejpam-6429	217	2	.	.	PUNCT
ejpam-6429	218	1	next	next	ADV
ejpam-6429	218	2	,	,	PUNCT
ejpam-6429	218	3	assume	assume	VERB
ejpam-6429	218	4	that	that	SCONJ
ejpam-6429	218	5	x	x	PRON
ejpam-6429	218	6	is	be	AUX
ejpam-6429	218	7	a	a	DET
ejpam-6429	218	8	moving	move	VERB
ejpam-6429	218	9	neighbor	neighbor	NOUN
ejpam-6429	218	10	of	of	ADP
ejpam-6429	218	11	u1	u1	NOUN
ejpam-6429	218	12	but	but	CCONJ
ejpam-6429	218	13	not	not	PART
ejpam-6429	218	14	of	of	ADP
ejpam-6429	218	15	v1	v1	NOUN
ejpam-6429	218	16	.	.	PUNCT
ejpam-6429	219	1	in	in	ADP
ejpam-6429	219	2	this	this	DET
ejpam-6429	219	3	case	case	NOUN
ejpam-6429	219	4	,	,	PUNCT
ejpam-6429	219	5	v2	v2	PROPN
ejpam-6429	219	6	must	must	AUX
ejpam-6429	219	7	be	be	AUX
ejpam-6429	219	8	a	a	DET
ejpam-6429	219	9	moving	move	VERB
ejpam-6429	219	10	neighbor	neighbor	NOUN
ejpam-6429	219	11	of	of	ADP
ejpam-6429	219	12	v1	v1	NOUN
ejpam-6429	219	13	,	,	PUNCT
ejpam-6429	219	14	so	so	SCONJ
ejpam-6429	219	15	f(v2	f(v2	NOUN
ejpam-6429	219	16	)	)	PUNCT
ejpam-6429	219	17	≥	≥	NOUN
ejpam-6429	220	1	2	2	NUM
ejpam-6429	220	2	.	.	PUNCT
ejpam-6429	221	1	it	it	PRON
ejpam-6429	221	2	is	be	AUX
ejpam-6429	221	3	straightforward	straightforward	ADJ
ejpam-6429	221	4	to	to	PART
ejpam-6429	221	5	observe	observe	VERB
ejpam-6429	221	6	that	that	SCONJ
ejpam-6429	221	7	the	the	DET
ejpam-6429	221	8	function	function	NOUN
ejpam-6429	221	9	f	f	PROPN
ejpam-6429	221	10	restricted	restrict	VERB
ejpam-6429	221	11	to	to	ADP
ejpam-6429	221	12	g	g	NOUN
ejpam-6429	221	13	=	=	NOUN
ejpam-6429	221	14	p2	p2	X
ejpam-6429	221	15	□	□	SYM
ejpam-6429	221	16	pn	pn	PROPN
ejpam-6429	221	17	\	\	PROPN
ejpam-6429	221	18	{	{	PUNCT
ejpam-6429	221	19	u1	u1	NOUN
ejpam-6429	221	20	,	,	PUNCT
ejpam-6429	221	21	v1	v1	PROPN
ejpam-6429	221	22	}	}	PUNCT
ejpam-6429	221	23	is	be	AUX
ejpam-6429	221	24	a	a	DET
ejpam-6429	221	25	grd	grd	NOUN
ejpam-6429	221	26	-	-	PUNCT
ejpam-6429	221	27	function	function	NOUN
ejpam-6429	221	28	with	with	ADP
ejpam-6429	221	29	weight	weight	NOUN
ejpam-6429	221	30	ωg	ωg	PROPN
ejpam-6429	221	31	r(f)−	r(f)−	PROPN
ejpam-6429	221	32	2	2	NUM
ejpam-6429	221	33	.	.	PUNCT
ejpam-6429	221	34	by	by	ADP
ejpam-6429	221	35	theorem	theorem	NOUN
ejpam-6429	221	36	1	1	NUM
ejpam-6429	221	37	,	,	PUNCT
ejpam-6429	221	38	we	we	PRON
ejpam-6429	221	39	can	can	AUX
ejpam-6429	221	40	deduce	deduce	VERB
ejpam-6429	221	41	that	that	PRON
ejpam-6429	221	42	ωg	ωg	ADP
ejpam-6429	221	43	r(f	r(f	PROPN
ejpam-6429	221	44	)	)	PUNCT
ejpam-6429	221	45	≥	≥	PROPN
ejpam-6429	221	46	ωg	ωg	ADP
ejpam-6429	221	47	r(f	r(f	PROPN
ejpam-6429	221	48	|v	|v	PROPN
ejpam-6429	221	49	(	(	PUNCT
ejpam-6429	221	50	g	g	NOUN
ejpam-6429	221	51	)	)	PUNCT
ejpam-6429	221	52	)	)	PUNCT
ejpam-6429	222	1	+	+	CCONJ
ejpam-6429	222	2	2	2	NUM
ejpam-6429	222	3	>	>	PUNCT
ejpam-6429	222	4	⌈	⌈	NUM
ejpam-6429	222	5	3n+1	3n+1	PROPN
ejpam-6429	222	6	2	2	NUM
ejpam-6429	222	7	⌉	⌉	NOUN
ejpam-6429	222	8	,	,	PUNCT
ejpam-6429	222	9	which	which	PRON
ejpam-6429	222	10	results	result	VERB
ejpam-6429	222	11	in	in	ADP
ejpam-6429	222	12	sdγgr(p2	sdγgr(p2	ADJ
ejpam-6429	222	13	□	□	NUM
ejpam-6429	222	14	pn	pn	NOUN
ejpam-6429	222	15	)	)	PUNCT
ejpam-6429	222	16	=	=	SYM
ejpam-6429	222	17	1	1	X
ejpam-6429	222	18	.	.	PUNCT
ejpam-6429	222	19	case	case	NOUN
ejpam-6429	222	20	2	2	NUM
ejpam-6429	222	21	.	.	PUNCT
ejpam-6429	222	22	f(x	f(x	PROPN
ejpam-6429	222	23	)	)	PUNCT
ejpam-6429	223	1	=	=	PUNCT
ejpam-6429	223	2	0	0	X
ejpam-6429	223	3	.	.	PUNCT
ejpam-6429	223	4	to	to	PART
ejpam-6429	223	5	protect	protect	VERB
ejpam-6429	223	6	the	the	DET
ejpam-6429	223	7	vertex	vertex	NOUN
ejpam-6429	223	8	x	x	NOUN
ejpam-6429	223	9	,	,	PUNCT
ejpam-6429	223	10	we	we	PRON
ejpam-6429	223	11	may	may	AUX
ejpam-6429	223	12	assume	assume	VERB
ejpam-6429	223	13	that	that	SCONJ
ejpam-6429	223	14	f(u1	f(u1	NOUN
ejpam-6429	223	15	)	)	PUNCT
ejpam-6429	223	16	≥	≥	NOUN
ejpam-6429	224	1	2	2	NUM
ejpam-6429	224	2	.	.	PUNCT
ejpam-6429	225	1	if	if	SCONJ
ejpam-6429	225	2	f(u1	f(u1	NOUN
ejpam-6429	225	3	)	)	PUNCT
ejpam-6429	226	1	=	=	SYM
ejpam-6429	226	2	3	3	NUM
ejpam-6429	226	3	,	,	PUNCT
ejpam-6429	226	4	then	then	ADV
ejpam-6429	226	5	changing	change	VERB
ejpam-6429	226	6	the	the	DET
ejpam-6429	226	7	value	value	NOUN
ejpam-6429	226	8	of	of	ADP
ejpam-6429	226	9	u1	u1	NOUN
ejpam-6429	226	10	to	to	ADP
ejpam-6429	226	11	2	2	NUM
ejpam-6429	226	12	yields	yield	NOUN
ejpam-6429	226	13	a	a	DET
ejpam-6429	226	14	grd	grd	NOUN
ejpam-6429	226	15	-	-	PUNCT
ejpam-6429	226	16	function	function	NOUN
ejpam-6429	226	17	for	for	ADP
ejpam-6429	226	18	p2	p2	NOUN
ejpam-6429	226	19	□	□	SYM
ejpam-6429	226	20	pn	pn	NOUN
ejpam-6429	226	21	with	with	ADP
ejpam-6429	226	22	a	a	DET
ejpam-6429	226	23	weight	weight	NOUN
ejpam-6429	226	24	of	of	ADP
ejpam-6429	226	25	ωg	ωg	PART
ejpam-6429	226	26	r(f	r(f	PROPN
ejpam-6429	226	27	)	)	PUNCT
ejpam-6429	227	1	−	−	PROPN
ejpam-6429	227	2	1	1	NUM
ejpam-6429	227	3	,	,	PUNCT
ejpam-6429	227	4	which	which	PRON
ejpam-6429	227	5	implies	imply	VERB
ejpam-6429	227	6	that	that	SCONJ
ejpam-6429	227	7	sdγgr(p2	sdγgr(p2	ADJ
ejpam-6429	227	8	□	□	NUM
ejpam-6429	227	9	pn	pn	NOUN
ejpam-6429	227	10	)	)	PUNCT
ejpam-6429	227	11	=	=	SYM
ejpam-6429	227	12	1	1	X
ejpam-6429	227	13	.	.	PUNCT
ejpam-6429	227	14	therefore	therefore	ADV
ejpam-6429	227	15	,	,	PUNCT
ejpam-6429	227	16	we	we	PRON
ejpam-6429	227	17	assume	assume	VERB
ejpam-6429	227	18	that	that	SCONJ
ejpam-6429	227	19	f(u1	f(u1	NOUN
ejpam-6429	227	20	)	)	PUNCT
ejpam-6429	228	1	=	=	SYM
ejpam-6429	228	2	2	2	X
ejpam-6429	228	3	.	.	X
ejpam-6429	228	4	if	if	SCONJ
ejpam-6429	228	5	f(v1	f(v1	NOUN
ejpam-6429	228	6	)	)	PUNCT
ejpam-6429	228	7	≥	≥	NOUN
ejpam-6429	228	8	2	2	NUM
ejpam-6429	228	9	,	,	PUNCT
ejpam-6429	228	10	then	then	ADV
ejpam-6429	228	11	the	the	DET
ejpam-6429	228	12	function	function	NOUN
ejpam-6429	228	13	g	g	NOUN
ejpam-6429	228	14	,	,	PUNCT
ejpam-6429	228	15	defined	define	VERB
ejpam-6429	228	16	on	on	ADP
ejpam-6429	228	17	(	(	PUNCT
ejpam-6429	228	18	p2	p2	X
ejpam-6429	228	19	□	□	SYM
ejpam-6429	228	20	pn	pn	NOUN
ejpam-6429	228	21	)	)	PUNCT
ejpam-6429	228	22	\	\	NOUN
ejpam-6429	228	23	{	{	PUNCT
ejpam-6429	228	24	u1	u1	NOUN
ejpam-6429	228	25	,	,	PUNCT
ejpam-6429	228	26	v1	v1	NOUN
ejpam-6429	228	27	,	,	PUNCT
ejpam-6429	228	28	u2	u2	NOUN
ejpam-6429	228	29	,	,	PUNCT
ejpam-6429	228	30	v2	v2	PROPN
ejpam-6429	228	31	}	}	PUNCT
ejpam-6429	228	32	by	by	ADP
ejpam-6429	228	33	g(u3	g(u3	NOUN
ejpam-6429	228	34	)	)	PUNCT
ejpam-6429	228	35	=	=	SYM
ejpam-6429	228	36	min{3	min{3	PROPN
ejpam-6429	228	37	,	,	PUNCT
ejpam-6429	228	38	f(u3	f(u3	NOUN
ejpam-6429	228	39	)	)	PUNCT
ejpam-6429	228	40	+	+	NUM
ejpam-6429	228	41	f(u2	f(u2	NOUN
ejpam-6429	228	42	)	)	PUNCT
ejpam-6429	228	43	}	}	PUNCT
ejpam-6429	228	44	,	,	PUNCT
ejpam-6429	228	45	g(v3	g(v3	PROPN
ejpam-6429	228	46	)	)	PUNCT
ejpam-6429	228	47	=	=	SYM
ejpam-6429	228	48	min{3	min{3	PROPN
ejpam-6429	228	49	,	,	PUNCT
ejpam-6429	228	50	f(v3	f(v3	NOUN
ejpam-6429	228	51	)	)	PUNCT
ejpam-6429	228	52	+	+	SYM
ejpam-6429	228	53	f(v2	f(v2	NOUN
ejpam-6429	228	54	)	)	PUNCT
ejpam-6429	228	55	}	}	PUNCT
ejpam-6429	228	56	,	,	PUNCT
ejpam-6429	228	57	g(z	g(z	ADJ
ejpam-6429	228	58	)	)	PUNCT
ejpam-6429	228	59	=	=	SYM
ejpam-6429	228	60	f(z	f(z	PROPN
ejpam-6429	228	61	)	)	PUNCT
ejpam-6429	228	62	for	for	ADP
ejpam-6429	228	63	the	the	DET
ejpam-6429	228	64	remaining	remain	VERB
ejpam-6429	228	65	vertices	vertex	NOUN
ejpam-6429	228	66	z	z	NOUN
ejpam-6429	228	67	,	,	PUNCT
ejpam-6429	228	68	is	be	AUX
ejpam-6429	228	69	a	a	DET
ejpam-6429	228	70	grdfunction	grdfunction	NOUN
ejpam-6429	228	71	with	with	ADP
ejpam-6429	228	72	weight	weight	NOUN
ejpam-6429	228	73	at	at	ADP
ejpam-6429	228	74	most	most	ADV
ejpam-6429	228	75	ωg	ωg	ADP
ejpam-6429	228	76	r(f)−4	r(f)−4	PROPN
ejpam-6429	228	77	.	.	PUNCT
ejpam-6429	229	1	by	by	ADP
ejpam-6429	229	2	theorem	theorem	NOUN
ejpam-6429	229	3	1	1	NUM
ejpam-6429	229	4	,	,	PUNCT
ejpam-6429	229	5	it	it	PRON
ejpam-6429	229	6	follows	follow	VERB
ejpam-6429	229	7	that	that	SCONJ
ejpam-6429	229	8	ωg	ωg	ADP
ejpam-6429	229	9	r(f	r(f	PROPN
ejpam-6429	229	10	)	)	PUNCT
ejpam-6429	229	11	≥	≥	NUM
ejpam-6429	229	12	ωg	ωg	PART
ejpam-6429	229	13	r(g)+4	r(g)+4	X
ejpam-6429	229	14	≥⌈	≥⌈	NOUN
ejpam-6429	229	15	3(n−2)+1	3(n−2)+1	NUM
ejpam-6429	229	16	2	2	NUM
ejpam-6429	229	17	⌉	⌉	PRON
ejpam-6429	229	18	+	+	CCONJ
ejpam-6429	229	19	4	4	NUM
ejpam-6429	229	20	>	>	SYM
ejpam-6429	229	21	⌈	⌈	NUM
ejpam-6429	229	22	3n+1	3n+1	PROPN
ejpam-6429	229	23	2	2	NUM
ejpam-6429	229	24	⌉	⌉	X
ejpam-6429	229	25	.	.	PUNCT
ejpam-6429	230	1	if	if	SCONJ
ejpam-6429	230	2	f(v1	f(v1	ADJ
ejpam-6429	230	3	)	)	PUNCT
ejpam-6429	230	4	=	=	SYM
ejpam-6429	230	5	1	1	NUM
ejpam-6429	230	6	,	,	PUNCT
ejpam-6429	230	7	then	then	ADV
ejpam-6429	230	8	assigning	assign	VERB
ejpam-6429	230	9	the	the	DET
ejpam-6429	230	10	value	value	NOUN
ejpam-6429	230	11	0	0	NUM
ejpam-6429	230	12	to	to	PART
ejpam-6429	230	13	v1	v1	VERB
ejpam-6429	230	14	gives	give	VERB
ejpam-6429	230	15	a	a	DET
ejpam-6429	230	16	grdfunction	grdfunction	NOUN
ejpam-6429	230	17	for	for	ADP
ejpam-6429	230	18	p2	p2	NOUN
ejpam-6429	230	19	□	□	SYM
ejpam-6429	230	20	pn	pn	NOUN
ejpam-6429	230	21	with	with	ADP
ejpam-6429	230	22	weight	weight	NOUN
ejpam-6429	230	23	ωg	ωg	PART
ejpam-6429	230	24	r(f	r(f	PROPN
ejpam-6429	230	25	)	)	PUNCT
ejpam-6429	231	1	−	−	PROPN
ejpam-6429	231	2	1	1	NUM
ejpam-6429	231	3	,	,	PUNCT
ejpam-6429	231	4	leading	lead	VERB
ejpam-6429	231	5	to	to	ADP
ejpam-6429	231	6	sdγgr(p2	sdγgr(p2	ADJ
ejpam-6429	231	7	□	□	NUM
ejpam-6429	231	8	pn	pn	NOUN
ejpam-6429	231	9	)	)	PUNCT
ejpam-6429	231	10	=	=	SYM
ejpam-6429	232	1	1	1	X
ejpam-6429	232	2	.	.	PUNCT
ejpam-6429	233	1	hence	hence	ADV
ejpam-6429	233	2	,	,	PUNCT
ejpam-6429	233	3	we	we	PRON
ejpam-6429	233	4	assume	assume	VERB
ejpam-6429	233	5	f(v1	f(v1	VERB
ejpam-6429	233	6	)	)	PUNCT
ejpam-6429	233	7	=	=	SYM
ejpam-6429	234	1	0	0	X
ejpam-6429	234	2	.	.	PUNCT
ejpam-6429	235	1	in	in	ADP
ejpam-6429	235	2	this	this	DET
ejpam-6429	235	3	case	case	NOUN
ejpam-6429	235	4	,	,	PUNCT
ejpam-6429	235	5	v2	v2	PROPN
ejpam-6429	235	6	becomes	become	VERB
ejpam-6429	235	7	the	the	DET
ejpam-6429	235	8	moving	move	VERB
ejpam-6429	235	9	neighbor	neighbor	NOUN
ejpam-6429	235	10	for	for	ADP
ejpam-6429	235	11	v1	v1	NOUN
ejpam-6429	235	12	,	,	PUNCT
ejpam-6429	235	13	so	so	SCONJ
ejpam-6429	235	14	we	we	PRON
ejpam-6429	235	15	must	must	AUX
ejpam-6429	235	16	have	have	VERB
ejpam-6429	235	17	f(v2	f(v2	NOUN
ejpam-6429	235	18	)	)	PUNCT
ejpam-6429	235	19	≥	≥	NOUN
ejpam-6429	236	1	2	2	NUM
ejpam-6429	236	2	.	.	PUNCT
ejpam-6429	236	3	if	if	SCONJ
ejpam-6429	236	4	f(v2	f(v2	NOUN
ejpam-6429	236	5	)	)	PUNCT
ejpam-6429	236	6	=	=	SYM
ejpam-6429	236	7	3	3	NUM
ejpam-6429	236	8	or	or	CCONJ
ejpam-6429	236	9	if	if	SCONJ
ejpam-6429	236	10	f(v2	f(v2	NOUN
ejpam-6429	236	11	)	)	PUNCT
ejpam-6429	236	12	=	=	SYM
ejpam-6429	236	13	2	2	NUM
ejpam-6429	236	14	and	and	CCONJ
ejpam-6429	236	15	f(u2	f(u2	PROPN
ejpam-6429	236	16	)	)	PUNCT
ejpam-6429	236	17	≥	≥	NOUN
ejpam-6429	236	18	1	1	NUM
ejpam-6429	236	19	,	,	PUNCT
ejpam-6429	236	20	then	then	ADV
ejpam-6429	236	21	changing	change	VERB
ejpam-6429	236	22	the	the	DET
ejpam-6429	236	23	value	value	NOUN
ejpam-6429	236	24	of	of	ADP
ejpam-6429	236	25	u1	u1	NOUN
ejpam-6429	236	26	to	to	ADP
ejpam-6429	236	27	1	1	NUM
ejpam-6429	236	28	results	result	NOUN
ejpam-6429	236	29	in	in	ADP
ejpam-6429	236	30	a	a	DET
ejpam-6429	236	31	grd	grd	NOUN
ejpam-6429	236	32	-	-	PUNCT
ejpam-6429	236	33	function	function	NOUN
ejpam-6429	236	34	for	for	ADP
ejpam-6429	236	35	p2	p2	NOUN
ejpam-6429	236	36	□	□	SYM
ejpam-6429	236	37	pn	pn	NOUN
ejpam-6429	236	38	with	with	ADP
ejpam-6429	236	39	weight	weight	NOUN
ejpam-6429	236	40	ωg	ωg	NOUN
ejpam-6429	236	41	r(f)−1	r(f)−1	NOUN
ejpam-6429	236	42	,	,	PUNCT
ejpam-6429	236	43	and	and	CCONJ
ejpam-6429	236	44	again	again	ADV
ejpam-6429	236	45	sdγgr(p2	sdγgr(p2	ADJ
ejpam-6429	236	46	□	□	NUM
ejpam-6429	236	47	pn	pn	NOUN
ejpam-6429	236	48	)	)	PUNCT
ejpam-6429	236	49	=	=	SYM
ejpam-6429	236	50	1	1	X
ejpam-6429	236	51	.	.	X
ejpam-6429	236	52	let	let	VERB
ejpam-6429	236	53	f(v2	f(v2	NOUN
ejpam-6429	236	54	)	)	PUNCT
ejpam-6429	236	55	=	=	SYM
ejpam-6429	236	56	2	2	NUM
ejpam-6429	236	57	and	and	CCONJ
ejpam-6429	236	58	f(u2	f(u2	NOUN
ejpam-6429	236	59	)	)	PUNCT
ejpam-6429	237	1	=	=	PUNCT
ejpam-6429	238	1	0	0	X
ejpam-6429	238	2	.	.	PUNCT
ejpam-6429	238	3	from	from	ADP
ejpam-6429	238	4	f(x	f(x	PROPN
ejpam-6429	238	5	)	)	PUNCT
ejpam-6429	238	6	=	=	SYM
ejpam-6429	239	1	0	0	NUM
ejpam-6429	239	2	,	,	PUNCT
ejpam-6429	239	3	it	it	PRON
ejpam-6429	239	4	follows	follow	VERB
ejpam-6429	239	5	that	that	SCONJ
ejpam-6429	239	6	v2	v2	PROPN
ejpam-6429	239	7	is	be	AUX
ejpam-6429	239	8	only	only	ADV
ejpam-6429	239	9	a	a	DET
ejpam-6429	239	10	moving	move	VERB
ejpam-6429	239	11	neighbor	neighbor	NOUN
ejpam-6429	239	12	for	for	ADP
ejpam-6429	239	13	v1	v1	NOUN
ejpam-6429	239	14	and	and	CCONJ
ejpam-6429	239	15	that	that	SCONJ
ejpam-6429	239	16	u3	u3	NOUN
ejpam-6429	239	17	is	be	AUX
ejpam-6429	239	18	the	the	DET
ejpam-6429	239	19	only	only	ADV
ejpam-6429	239	20	moving	move	VERB
ejpam-6429	239	21	neighbor	neighbor	NOUN
ejpam-6429	239	22	of	of	ADP
ejpam-6429	239	23	u2	u2	NOUN
ejpam-6429	239	24	,	,	PUNCT
ejpam-6429	239	25	meaning	meaning	NOUN
ejpam-6429	239	26	f(u3	f(u3	NOUN
ejpam-6429	239	27	)	)	PUNCT
ejpam-6429	239	28	≥	≥	NOUN
ejpam-6429	240	1	2	2	NUM
ejpam-6429	240	2	.	.	PUNCT
ejpam-6429	241	1	if	if	SCONJ
ejpam-6429	241	2	n	n	NOUN
ejpam-6429	241	3	=	=	SYM
ejpam-6429	241	4	3	3	NUM
ejpam-6429	241	5	,	,	PUNCT
ejpam-6429	241	6	then	then	ADV
ejpam-6429	241	7	we	we	PRON
ejpam-6429	241	8	have	have	AUX
ejpam-6429	241	9	ωg	ωg	PART
ejpam-6429	241	10	r(f	r(f	PROPN
ejpam-6429	241	11	)	)	PUNCT
ejpam-6429	241	12	≥	≥	NOUN
ejpam-6429	241	13	6	6	NUM
ejpam-6429	241	14	>	>	PUNCT
ejpam-6429	241	15	⌈	⌈	NUM
ejpam-6429	241	16	3n+1	3n+1	PROPN
ejpam-6429	241	17	2	2	NUM
ejpam-6429	241	18	⌉	⌉	X
ejpam-6429	241	19	.	.	PUNCT
ejpam-6429	242	1	therefore	therefore	ADV
ejpam-6429	242	2	,	,	PUNCT
ejpam-6429	242	3	let	let	VERB
ejpam-6429	242	4	n	n	PRON
ejpam-6429	242	5	≥	≥	NOUN
ejpam-6429	242	6	5	5	NUM
ejpam-6429	242	7	.	.	PUNCT
ejpam-6429	243	1	if	if	SCONJ
ejpam-6429	243	2	f(u3	f(u3	NOUN
ejpam-6429	243	3	)	)	PUNCT
ejpam-6429	243	4	=	=	SYM
ejpam-6429	243	5	3	3	NUM
ejpam-6429	243	6	or	or	CCONJ
ejpam-6429	243	7	f(v3	f(v3	NUM
ejpam-6429	243	8	)	)	PUNCT
ejpam-6429	243	9	≥	≥	NOUN
ejpam-6429	243	10	1	1	NUM
ejpam-6429	243	11	,	,	PUNCT
ejpam-6429	243	12	then	then	ADV
ejpam-6429	243	13	the	the	DET
ejpam-6429	243	14	function	function	NOUN
ejpam-6429	243	15	f	f	PROPN
ejpam-6429	243	16	restricted	restrict	VERB
ejpam-6429	243	17	to	to	ADP
ejpam-6429	243	18	(	(	PUNCT
ejpam-6429	243	19	p2	p2	X
ejpam-6429	243	20	□	□	SYM
ejpam-6429	243	21	pn)−{v1	pn)−{v1	PROPN
ejpam-6429	243	22	,	,	PUNCT
ejpam-6429	243	23	v2	v2	NOUN
ejpam-6429	243	24	,	,	PUNCT
ejpam-6429	243	25	u1	u1	NOUN
ejpam-6429	243	26	,	,	PUNCT
ejpam-6429	243	27	u2	u2	PROPN
ejpam-6429	243	28	}	}	PUNCT
ejpam-6429	243	29	is	be	AUX
ejpam-6429	243	30	a	a	DET
ejpam-6429	243	31	grd	grd	NOUN
ejpam-6429	243	32	-	-	PUNCT
ejpam-6429	243	33	function	function	NOUN
ejpam-6429	243	34	with	with	ADP
ejpam-6429	243	35	weight	weight	NOUN
ejpam-6429	243	36	ωg	ωg	NOUN
ejpam-6429	243	37	r(f)−4	r(f)−4	PROPN
ejpam-6429	243	38	,	,	PUNCT
ejpam-6429	243	39	and	and	CCONJ
ejpam-6429	243	40	as	as	ADP
ejpam-6429	243	41	before	before	ADV
ejpam-6429	243	42	,	,	PUNCT
ejpam-6429	243	43	we	we	PRON
ejpam-6429	243	44	conclude	conclude	VERB
ejpam-6429	243	45	that	that	SCONJ
ejpam-6429	243	46	ωg	ωg	ADP
ejpam-6429	243	47	r(f	r(f	PROPN
ejpam-6429	243	48	)	)	PUNCT
ejpam-6429	243	49	>	>	X
ejpam-6429	244	1	⌈	⌈	X
ejpam-6429	244	2	3n+1	3n+1	NUM
ejpam-6429	244	3	2	2	NUM
ejpam-6429	244	4	⌉	⌉	NOUN
ejpam-6429	244	5	,	,	PUNCT
ejpam-6429	244	6	leading	lead	VERB
ejpam-6429	244	7	to	to	ADP
ejpam-6429	244	8	sdγgr(p2	sdγgr(p2	ADJ
ejpam-6429	244	9	□	□	NUM
ejpam-6429	244	10	pn	pn	NOUN
ejpam-6429	244	11	)	)	PUNCT
ejpam-6429	244	12	=	=	SYM
ejpam-6429	245	1	1	1	X
ejpam-6429	245	2	.	.	X
ejpam-6429	245	3	let	let	VERB
ejpam-6429	245	4	f(u3	f(u3	NOUN
ejpam-6429	245	5	)	)	PUNCT
ejpam-6429	245	6	=	=	SYM
ejpam-6429	245	7	2	2	NUM
ejpam-6429	245	8	and	and	CCONJ
ejpam-6429	245	9	f(v3	f(v3	NUM
ejpam-6429	245	10	)	)	PUNCT
ejpam-6429	245	11	=	=	SYM
ejpam-6429	246	1	0	0	X
ejpam-6429	246	2	.	.	PUNCT
ejpam-6429	247	1	if	if	SCONJ
ejpam-6429	247	2	f(u4	f(u4	NOUN
ejpam-6429	247	3	)	)	PUNCT
ejpam-6429	247	4	≥	≥	NOUN
ejpam-6429	247	5	1	1	NUM
ejpam-6429	247	6	or	or	CCONJ
ejpam-6429	247	7	f(v4	f(v4	NUM
ejpam-6429	247	8	)	)	PUNCT
ejpam-6429	247	9	≥	≥	NOUN
ejpam-6429	247	10	2	2	NUM
ejpam-6429	247	11	,	,	PUNCT
ejpam-6429	247	12	then	then	ADV
ejpam-6429	247	13	the	the	DET
ejpam-6429	247	14	function	function	NOUN
ejpam-6429	247	15	f	f	PROPN
ejpam-6429	247	16	restricted	restrict	VERB
ejpam-6429	247	17	to	to	ADP
ejpam-6429	247	18	(	(	PUNCT
ejpam-6429	247	19	p2	p2	X
ejpam-6429	247	20	□	□	SYM
ejpam-6429	247	21	pn	pn	NOUN
ejpam-6429	247	22	)	)	PUNCT
ejpam-6429	247	23	\	\	NOUN
ejpam-6429	247	24	{	{	PUNCT
ejpam-6429	247	25	v1	v1	NOUN
ejpam-6429	247	26	,	,	PUNCT
ejpam-6429	247	27	v2	v2	PROPN
ejpam-6429	247	28	,	,	PUNCT
ejpam-6429	247	29	u1	u1	NOUN
ejpam-6429	247	30	,	,	PUNCT
ejpam-6429	247	31	u2	u2	PROPN
ejpam-6429	247	32	}	}	PUNCT
ejpam-6429	247	33	is	be	AUX
ejpam-6429	247	34	a	a	DET
ejpam-6429	247	35	grd	grd	NOUN
ejpam-6429	247	36	-	-	PUNCT
ejpam-6429	247	37	function	function	NOUN
ejpam-6429	247	38	with	with	ADP
ejpam-6429	247	39	weight	weight	NOUN
ejpam-6429	247	40	ωg	ωg	PART
ejpam-6429	247	41	r(f	r(f	PROPN
ejpam-6429	247	42	)	)	PUNCT
ejpam-6429	248	1	−	−	PROPN
ejpam-6429	248	2	4	4	NUM
ejpam-6429	248	3	,	,	PUNCT
ejpam-6429	248	4	and	and	CCONJ
ejpam-6429	248	5	as	as	ADP
ejpam-6429	248	6	before	before	ADV
ejpam-6429	248	7	,	,	PUNCT
ejpam-6429	248	8	we	we	PRON
ejpam-6429	248	9	conclude	conclude	VERB
ejpam-6429	248	10	that	that	SCONJ
ejpam-6429	248	11	ωg	ωg	ADP
ejpam-6429	248	12	r(f	r(f	PROPN
ejpam-6429	248	13	)	)	PUNCT
ejpam-6429	248	14	>	>	X
ejpam-6429	249	1	⌈	⌈	X
ejpam-6429	249	2	3n+1	3n+1	NUM
ejpam-6429	249	3	2	2	NUM
ejpam-6429	249	4	⌉	⌉	X
ejpam-6429	249	5	,	,	PUNCT
ejpam-6429	249	6	showing	show	VERB
ejpam-6429	249	7	that	that	SCONJ
ejpam-6429	249	8	sdγgr(p2	sdγgr(p2	ADJ
ejpam-6429	249	9	□	□	NUM
ejpam-6429	249	10	pn	pn	NOUN
ejpam-6429	249	11	)	)	PUNCT
ejpam-6429	249	12	=	=	SYM
ejpam-6429	249	13	1	1	X
ejpam-6429	249	14	.	.	PUNCT
ejpam-6429	250	1	hence	hence	ADV
ejpam-6429	250	2	,	,	PUNCT
ejpam-6429	250	3	we	we	PRON
ejpam-6429	250	4	assume	assume	VERB
ejpam-6429	250	5	f(u4	f(u4	NOUN
ejpam-6429	250	6	)	)	PUNCT
ejpam-6429	250	7	=	=	SYM
ejpam-6429	250	8	0	0	NUM
ejpam-6429	250	9	and	and	CCONJ
ejpam-6429	250	10	f(v4	f(v4	NUM
ejpam-6429	250	11	)	)	PUNCT
ejpam-6429	250	12	≤	≤	NUM
ejpam-6429	251	1	1	1	NUM
ejpam-6429	251	2	.	.	PUNCT
ejpam-6429	252	1	if	if	SCONJ
ejpam-6429	252	2	f(v4	f(v4	PRON
ejpam-6429	252	3	)	)	PUNCT
ejpam-6429	252	4	=	=	SYM
ejpam-6429	252	5	1	1	NUM
ejpam-6429	252	6	,	,	PUNCT
ejpam-6429	252	7	then	then	ADV
ejpam-6429	252	8	the	the	DET
ejpam-6429	252	9	function	function	NOUN
ejpam-6429	252	10	f	f	PROPN
ejpam-6429	252	11	restricted	restrict	VERB
ejpam-6429	252	12	to	to	ADP
ejpam-6429	252	13	(	(	PUNCT
ejpam-6429	252	14	p2	p2	X
ejpam-6429	252	15	□	□	SYM
ejpam-6429	252	16	pn	pn	NOUN
ejpam-6429	252	17	)	)	PUNCT
ejpam-6429	252	18	\	\	PROPN
ejpam-6429	252	19	{	{	PUNCT
ejpam-6429	252	20	vi	vi	PROPN
ejpam-6429	252	21	,	,	PUNCT
ejpam-6429	252	22	ui	ui	NOUN
ejpam-6429	252	23	|	|	ADV
ejpam-6429	252	24	1	1	X
ejpam-6429	252	25	≤	≤	NUM
ejpam-6429	252	26	i	i	PRON
ejpam-6429	252	27	≤	≤	ADV
ejpam-6429	252	28	4	4	NUM
ejpam-6429	252	29	}	}	PUNCT
ejpam-6429	252	30	is	be	AUX
ejpam-6429	252	31	a	a	DET
ejpam-6429	252	32	grd	grd	NOUN
ejpam-6429	252	33	-	-	PUNCT
ejpam-6429	252	34	function	function	NOUN
ejpam-6429	252	35	with	with	ADP
ejpam-6429	252	36	weight	weight	NOUN
ejpam-6429	252	37	ωg	ωg	PART
ejpam-6429	252	38	r(f	r(f	PROPN
ejpam-6429	252	39	)	)	PUNCT
ejpam-6429	252	40	−	−	PROPN
ejpam-6429	252	41	7	7	NUM
ejpam-6429	252	42	,	,	PUNCT
ejpam-6429	252	43	and	and	CCONJ
ejpam-6429	252	44	using	use	VERB
ejpam-6429	252	45	theorem	theorem	NOUN
ejpam-6429	252	46	1	1	NUM
ejpam-6429	252	47	,	,	PUNCT
ejpam-6429	252	48	we	we	PRON
ejpam-6429	252	49	obtain	obtain	VERB
ejpam-6429	252	50	ωg	ωg	PART
ejpam-6429	252	51	r(f	r(f	PROPN
ejpam-6429	252	52	)	)	PUNCT
ejpam-6429	252	53	≥	≥	NOUN
ejpam-6429	252	54	7	7	NUM
ejpam-6429	252	55	+	+	CCONJ
ejpam-6429	252	56	⌈	⌈	SYM
ejpam-6429	252	57	3(n−4)+1	3(n−4)+1	NUM
ejpam-6429	252	58	2	2	NUM
ejpam-6429	252	59	⌉	⌉	X
ejpam-6429	252	60	>	>	X
ejpam-6429	252	61	⌈	⌈	SYM
ejpam-6429	252	62	3n+1	3n+1	PROPN
ejpam-6429	252	63	2	2	NUM
ejpam-6429	252	64	⌉	⌉	NOUN
ejpam-6429	252	65	,	,	PUNCT
ejpam-6429	252	66	thus	thus	ADV
ejpam-6429	252	67	sdγgr(p2	sdγgr(p2	ADJ
ejpam-6429	252	68	□	□	NUM
ejpam-6429	252	69	pn	pn	NOUN
ejpam-6429	252	70	)	)	PUNCT
ejpam-6429	252	71	=	=	SYM
ejpam-6429	253	1	1	1	X
ejpam-6429	253	2	.	.	PUNCT
ejpam-6429	253	3	thus	thus	ADV
ejpam-6429	253	4	,	,	PUNCT
ejpam-6429	253	5	let	let	VERB
ejpam-6429	253	6	f(v4	f(v4	PRON
ejpam-6429	253	7	)	)	PUNCT
ejpam-6429	253	8	=	=	SYM
ejpam-6429	254	1	0	0	X
ejpam-6429	254	2	.	.	PUNCT
ejpam-6429	254	3	to	to	PART
ejpam-6429	254	4	protect	protect	VERB
ejpam-6429	254	5	v4	v4	NOUN
ejpam-6429	254	6	,	,	PUNCT
ejpam-6429	254	7	we	we	PRON
ejpam-6429	254	8	must	must	AUX
ejpam-6429	254	9	have	have	VERB
ejpam-6429	254	10	f(v5	f(v5	NOUN
ejpam-6429	254	11	)	)	PUNCT
ejpam-6429	254	12	≥	≥	NOUN
ejpam-6429	255	1	2	2	NUM
ejpam-6429	255	2	.	.	PUNCT
ejpam-6429	255	3	on	on	ADP
ejpam-6429	255	4	the	the	DET
ejpam-6429	255	5	other	other	ADJ
ejpam-6429	255	6	hand	hand	NOUN
ejpam-6429	255	7	,	,	PUNCT
ejpam-6429	255	8	since	since	SCONJ
ejpam-6429	255	9	u3	u3	NOUN
ejpam-6429	255	10	is	be	AUX
ejpam-6429	255	11	the	the	DET
ejpam-6429	255	12	only	only	ADV
ejpam-6429	255	13	moving	move	VERB
ejpam-6429	255	14	neighbor	neighbor	NOUN
ejpam-6429	255	15	of	of	ADP
ejpam-6429	255	16	u2	u2	NOUN
ejpam-6429	255	17	,	,	PUNCT
ejpam-6429	255	18	we	we	PRON
ejpam-6429	255	19	have	have	VERB
ejpam-6429	255	20	f(u5	f(u5	NOUN
ejpam-6429	255	21	)	)	PUNCT
ejpam-6429	255	22	≥	≥	NOUN
ejpam-6429	256	1	2	2	NUM
ejpam-6429	256	2	.	.	PUNCT
ejpam-6429	257	1	if	if	SCONJ
ejpam-6429	257	2	n	n	NOUN
ejpam-6429	257	3	=	=	SYM
ejpam-6429	257	4	5	5	NUM
ejpam-6429	257	5	,	,	PUNCT
ejpam-6429	257	6	then	then	ADV
ejpam-6429	257	7	ωg	ωg	ADP
ejpam-6429	257	8	r(f	r(f	PROPN
ejpam-6429	257	9	)	)	PUNCT
ejpam-6429	257	10	≥	≥	NOUN
ejpam-6429	257	11	10	10	NUM
ejpam-6429	257	12	>	>	PUNCT
ejpam-6429	257	13	⌈	⌈	NUM
ejpam-6429	257	14	3n+1	3n+1	PROPN
ejpam-6429	257	15	2	2	NUM
ejpam-6429	257	16	⌉	⌉	X
ejpam-6429	257	17	.	.	PUNCT
ejpam-6429	258	1	hence	hence	ADV
ejpam-6429	258	2	,	,	PUNCT
ejpam-6429	258	3	let	let	VERB
ejpam-6429	258	4	n	n	PRON
ejpam-6429	258	5	≥	≥	NOUN
ejpam-6429	258	6	7	7	NUM
ejpam-6429	258	7	.	.	PUNCT
ejpam-6429	258	8	by	by	ADP
ejpam-6429	258	9	reassigning	reassign	VERB
ejpam-6429	258	10	u7	u7	PROPN
ejpam-6429	258	11	the	the	DET
ejpam-6429	258	12	value	value	NOUN
ejpam-6429	258	13	min{3	min{3	NOUN
ejpam-6429	258	14	,	,	PUNCT
ejpam-6429	258	15	f(u7	f(u7	NOUN
ejpam-6429	258	16	)	)	PUNCT
ejpam-6429	258	17	+	+	NUM
ejpam-6429	258	18	f(u6	f(u6	NOUN
ejpam-6429	258	19	)	)	PUNCT
ejpam-6429	258	20	}	}	PUNCT
ejpam-6429	258	21	and	and	CCONJ
ejpam-6429	258	22	v7	v7	VERB
ejpam-6429	258	23	the	the	DET
ejpam-6429	258	24	value	value	NOUN
ejpam-6429	258	25	min{3	min{3	NOUN
ejpam-6429	258	26	,	,	PUNCT
ejpam-6429	258	27	f(v7	f(v7	NOUN
ejpam-6429	258	28	)	)	PUNCT
ejpam-6429	259	1	+	+	SYM
ejpam-6429	259	2	f(v6	f(v6	NOUN
ejpam-6429	259	3	)	)	PUNCT
ejpam-6429	259	4	}	}	PUNCT
ejpam-6429	259	5	,	,	PUNCT
ejpam-6429	259	6	we	we	PRON
ejpam-6429	259	7	obtain	obtain	VERB
ejpam-6429	259	8	a	a	DET
ejpam-6429	259	9	grd	grd	NOUN
ejpam-6429	259	10	-	-	PUNCT
ejpam-6429	259	11	function	function	NOUN
ejpam-6429	259	12	on	on	ADP
ejpam-6429	259	13	(	(	PUNCT
ejpam-6429	259	14	p2	p2	X
ejpam-6429	259	15	□	□	SYM
ejpam-6429	259	16	pn	pn	NOUN
ejpam-6429	259	17	)	)	PUNCT
ejpam-6429	259	18	\	\	PROPN
ejpam-6429	259	19	{	{	PUNCT
ejpam-6429	259	20	ui	ui	PROPN
ejpam-6429	259	21	,	,	PUNCT
ejpam-6429	259	22	vi	vi	PROPN
ejpam-6429	260	1	|	|	ADV
ejpam-6429	260	2	1	1	NUM
ejpam-6429	260	3	≤	≤	NUM
ejpam-6429	260	4	i	i	PRON
ejpam-6429	260	5	≤	≤	ADV
ejpam-6429	260	6	6	6	NUM
ejpam-6429	260	7	}	}	PUNCT
ejpam-6429	260	8	with	with	ADP
ejpam-6429	260	9	weight	weight	NOUN
ejpam-6429	260	10	at	at	ADP
ejpam-6429	260	11	most	most	ADV
ejpam-6429	260	12	ωg	ωg	ADP
ejpam-6429	260	13	r(f)−	r(f)−	PROPN
ejpam-6429	260	14	10	10	NUM
ejpam-6429	260	15	.	.	PUNCT
ejpam-6429	261	1	by	by	ADP
ejpam-6429	261	2	theorem	theorem	NOUN
ejpam-6429	261	3	1	1	NUM
ejpam-6429	261	4	,	,	PUNCT
ejpam-6429	261	5	we	we	PRON
ejpam-6429	261	6	have	have	AUX
ejpam-6429	261	7	ωg	ωg	PART
ejpam-6429	261	8	r(f	r(f	PROPN
ejpam-6429	261	9	)	)	PUNCT
ejpam-6429	261	10	≥	≥	PROPN
ejpam-6429	261	11	ωg	ωg	X
ejpam-6429	261	12	r(g	r(g	NUM
ejpam-6429	261	13	)	)	PUNCT
ejpam-6429	262	1	+	+	CCONJ
ejpam-6429	262	2	10	10	NUM
ejpam-6429	262	3	≥	≥	NOUN
ejpam-6429	262	4	⌈	⌈	NOUN
ejpam-6429	262	5	3(n−6	3(n−6	PROPN
ejpam-6429	262	6	)	)	PUNCT
ejpam-6429	262	7	2	2	NUM
ejpam-6429	262	8	⌉	⌉	NOUN
ejpam-6429	262	9	+10	+10	NOUN
ejpam-6429	262	10	>	>	PUNCT
ejpam-6429	262	11	⌈	⌈	X
ejpam-6429	262	12	3n+1	3n+1	PROPN
ejpam-6429	262	13	2	2	NUM
ejpam-6429	262	14	⌉	⌉	X
ejpam-6429	262	15	.	.	PUNCT
ejpam-6429	263	1	in	in	ADP
ejpam-6429	263	2	conclusion	conclusion	NOUN
ejpam-6429	263	3	,	,	PUNCT
ejpam-6429	263	4	we	we	PRON
ejpam-6429	263	5	obtain	obtain	VERB
ejpam-6429	263	6	sdγgr(p2	sdγgr(p2	ADJ
ejpam-6429	263	7	□	□	NUM
ejpam-6429	263	8	pn	pn	NOUN
ejpam-6429	263	9	)	)	PUNCT
ejpam-6429	263	10	=	=	SYM
ejpam-6429	264	1	1	1	X
ejpam-6429	264	2	.	.	PUNCT
ejpam-6429	264	3	j.	j.	PROPN
ejpam-6429	264	4	j.	j.	PROPN
ejpam-6429	264	5	hamja	hamja	PROPN
ejpam-6429	264	6	et	et	PROPN
ejpam-6429	264	7	al	al	PROPN
ejpam-6429	264	8	.	.	PUNCT
ejpam-6429	264	9	/	/	SYM
ejpam-6429	264	10	eur	eur	PROPN
ejpam-6429	264	11	.	.	PUNCT
ejpam-6429	265	1	j.	j.	PROPN
ejpam-6429	265	2	pure	pure	PROPN
ejpam-6429	265	3	appl	appl	PROPN
ejpam-6429	265	4	.	.	PROPN
ejpam-6429	265	5	math	math	PROPN
ejpam-6429	265	6	,	,	PUNCT
ejpam-6429	265	7	18	18	NUM
ejpam-6429	265	8	(	(	PUNCT
ejpam-6429	265	9	4	4	NUM
ejpam-6429	265	10	)	)	PUNCT
ejpam-6429	265	11	(	(	PUNCT
ejpam-6429	265	12	2025	2025	NUM
ejpam-6429	265	13	)	)	PUNCT
ejpam-6429	265	14	,	,	PUNCT
ejpam-6429	265	15	6429	6429	NUM
ejpam-6429	265	16	8	8	NUM
ejpam-6429	265	17	of	of	ADP
ejpam-6429	265	18	16	16	NUM
ejpam-6429	265	19	5	5	NUM
ejpam-6429	265	20	.	.	PUNCT
ejpam-6429	265	21	sufficient	sufficient	ADJ
ejpam-6429	265	22	conditions	condition	NOUN
ejpam-6429	265	23	on	on	ADP
ejpam-6429	265	24	g	g	NOUN
ejpam-6429	265	25	having	have	VERB
ejpam-6429	265	26	small	small	ADJ
ejpam-6429	265	27	sdγgr(g	sdγgr(g	NOUN
ejpam-6429	265	28	)	)	PUNCT
ejpam-6429	265	29	in	in	ADP
ejpam-6429	265	30	this	this	DET
ejpam-6429	265	31	section	section	NOUN
ejpam-6429	265	32	,	,	PUNCT
ejpam-6429	265	33	we	we	PRON
ejpam-6429	265	34	present	present	VERB
ejpam-6429	265	35	several	several	ADJ
ejpam-6429	265	36	sufficient	sufficient	ADJ
ejpam-6429	265	37	conditions	condition	NOUN
ejpam-6429	265	38	for	for	ADP
ejpam-6429	265	39	a	a	DET
ejpam-6429	265	40	graph	graph	NOUN
ejpam-6429	265	41	g	g	NOUN
ejpam-6429	265	42	to	to	PART
ejpam-6429	265	43	have	have	VERB
ejpam-6429	265	44	a	a	DET
ejpam-6429	265	45	small	small	ADJ
ejpam-6429	265	46	value	value	NOUN
ejpam-6429	265	47	of	of	ADP
ejpam-6429	265	48	sdγgr(g	sdγgr(g	PROPN
ejpam-6429	265	49	)	)	PUNCT
ejpam-6429	265	50	.	.	PUNCT
ejpam-6429	266	1	proposition	proposition	NOUN
ejpam-6429	266	2	8	8	NUM
ejpam-6429	266	3	.	.	PUNCT
ejpam-6429	267	1	if	if	SCONJ
ejpam-6429	267	2	a	a	DET
ejpam-6429	267	3	connected	connected	ADJ
ejpam-6429	267	4	graph	graph	NOUN
ejpam-6429	267	5	g	g	PROPN
ejpam-6429	267	6	has	have	VERB
ejpam-6429	267	7	a	a	DET
ejpam-6429	267	8	support	support	NOUN
ejpam-6429	267	9	vertex	vertex	NOUN
ejpam-6429	267	10	with	with	ADP
ejpam-6429	267	11	at	at	ADV
ejpam-6429	267	12	least	least	ADV
ejpam-6429	267	13	three	three	NUM
ejpam-6429	267	14	leaf	leaf	NOUN
ejpam-6429	267	15	neighbors	neighbor	NOUN
ejpam-6429	267	16	,	,	PUNCT
ejpam-6429	267	17	then	then	ADV
ejpam-6429	267	18	sdγgr(g	sdγgr(g	PROPN
ejpam-6429	267	19	)	)	PUNCT
ejpam-6429	267	20	=	=	SYM
ejpam-6429	268	1	1	1	X
ejpam-6429	268	2	.	.	PUNCT
ejpam-6429	269	1	proof	proof	NOUN
ejpam-6429	269	2	.	.	PUNCT
ejpam-6429	270	1	if	if	SCONJ
ejpam-6429	270	2	g	g	PROPN
ejpam-6429	270	3	is	be	AUX
ejpam-6429	270	4	a	a	DET
ejpam-6429	270	5	star	star	NOUN
ejpam-6429	270	6	,	,	PUNCT
ejpam-6429	270	7	then	then	ADV
ejpam-6429	270	8	γgr(g	γgr(g	PROPN
ejpam-6429	270	9	)	)	PUNCT
ejpam-6429	270	10	=	=	SYM
ejpam-6429	270	11	3	3	NUM
ejpam-6429	270	12	,	,	PUNCT
ejpam-6429	270	13	and	and	CCONJ
ejpam-6429	270	14	by	by	ADP
ejpam-6429	270	15	corollary	corollary	ADJ
ejpam-6429	270	16	3	3	NUM
ejpam-6429	270	17	,	,	PUNCT
ejpam-6429	270	18	we	we	PRON
ejpam-6429	270	19	have	have	VERB
ejpam-6429	270	20	sdγgr(g	sdγgr(g	PROPN
ejpam-6429	270	21	)	)	PUNCT
ejpam-6429	271	1	=	=	SYM
ejpam-6429	271	2	1	1	X
ejpam-6429	271	3	.	.	X
ejpam-6429	271	4	assume	assume	VERB
ejpam-6429	271	5	now	now	ADV
ejpam-6429	271	6	that	that	SCONJ
ejpam-6429	271	7	g	g	PROPN
ejpam-6429	271	8	is	be	AUX
ejpam-6429	271	9	not	not	PART
ejpam-6429	271	10	a	a	DET
ejpam-6429	271	11	star	star	NOUN
ejpam-6429	271	12	.	.	PUNCT
ejpam-6429	272	1	let	let	VERB
ejpam-6429	272	2	u	u	NOUN
ejpam-6429	272	3	,	,	PUNCT
ejpam-6429	272	4	v	v	NOUN
ejpam-6429	272	5	and	and	CCONJ
ejpam-6429	272	6	y	y	PROPN
ejpam-6429	272	7	be	be	VERB
ejpam-6429	272	8	three	three	NUM
ejpam-6429	272	9	leaves	leave	NOUN
ejpam-6429	272	10	adjacent	adjacent	ADJ
ejpam-6429	272	11	to	to	ADP
ejpam-6429	272	12	a	a	DET
ejpam-6429	272	13	support	support	NOUN
ejpam-6429	272	14	vertex	vertex	NOUN
ejpam-6429	272	15	w	w	NOUN
ejpam-6429	272	16	,	,	PUNCT
ejpam-6429	272	17	and	and	CCONJ
ejpam-6429	272	18	let	let	VERB
ejpam-6429	272	19	z	z	PRON
ejpam-6429	272	20	be	be	AUX
ejpam-6429	272	21	a	a	DET
ejpam-6429	272	22	non	non	ADJ
ejpam-6429	272	23	-	-	ADJ
ejpam-6429	272	24	leaf	leaf	ADJ
ejpam-6429	272	25	neighbor	neighbor	NOUN
ejpam-6429	272	26	of	of	ADP
ejpam-6429	272	27	w.	w.	PROPN
ejpam-6429	272	28	let	let	VERB
ejpam-6429	272	29	g′	g′	NOUN
ejpam-6429	272	30	be	be	AUX
ejpam-6429	272	31	the	the	DET
ejpam-6429	272	32	graph	graph	NOUN
ejpam-6429	272	33	obtained	obtain	VERB
ejpam-6429	272	34	by	by	ADP
ejpam-6429	272	35	subdividing	subdivide	VERB
ejpam-6429	272	36	the	the	DET
ejpam-6429	272	37	edge	edge	NOUN
ejpam-6429	272	38	uw	uw	PROPN
ejpam-6429	272	39	with	with	ADP
ejpam-6429	272	40	a	a	DET
ejpam-6429	272	41	new	new	ADJ
ejpam-6429	272	42	vertex	vertex	NOUN
ejpam-6429	272	43	x	x	NOUN
ejpam-6429	272	44	,	,	PUNCT
ejpam-6429	272	45	and	and	CCONJ
ejpam-6429	272	46	let	let	VERB
ejpam-6429	272	47	f	f	PRON
ejpam-6429	272	48	be	be	AUX
ejpam-6429	272	49	a	a	DET
ejpam-6429	272	50	γgr(g	γgr(g	PROPN
ejpam-6429	272	51	′)-function	′)-function	NOUN
ejpam-6429	273	1	such	such	ADJ
ejpam-6429	273	2	that	that	SCONJ
ejpam-6429	273	3	f(w	f(w	PROPN
ejpam-6429	273	4	)	)	PUNCT
ejpam-6429	273	5	+	+	NUM
ejpam-6429	273	6	f(z	f(z	NOUN
ejpam-6429	273	7	)	)	PUNCT
ejpam-6429	273	8	is	be	AUX
ejpam-6429	273	9	maximized	maximize	VERB
ejpam-6429	273	10	.	.	PUNCT
ejpam-6429	274	1	if	if	SCONJ
ejpam-6429	274	2	f(w	f(w	PROPN
ejpam-6429	274	3	)	)	PUNCT
ejpam-6429	274	4	=	=	SYM
ejpam-6429	274	5	3	3	X
ejpam-6429	274	6	,	,	PUNCT
ejpam-6429	274	7	it	it	PRON
ejpam-6429	274	8	follows	follow	VERB
ejpam-6429	274	9	that	that	SCONJ
ejpam-6429	274	10	f(v	f(v	NOUN
ejpam-6429	274	11	)	)	PUNCT
ejpam-6429	274	12	=	=	SYM
ejpam-6429	274	13	f(x	f(x	PROPN
ejpam-6429	274	14	)	)	PUNCT
ejpam-6429	274	15	=	=	SYM
ejpam-6429	274	16	f(y	f(y	NOUN
ejpam-6429	274	17	)	)	PUNCT
ejpam-6429	274	18	=	=	SYM
ejpam-6429	274	19	0	0	NUM
ejpam-6429	274	20	,	,	PUNCT
ejpam-6429	274	21	and	and	CCONJ
ejpam-6429	274	22	thus	thus	ADV
ejpam-6429	274	23	f(u	f(u	PROPN
ejpam-6429	274	24	)	)	PUNCT
ejpam-6429	274	25	=	=	SYM
ejpam-6429	275	1	1	1	X
ejpam-6429	275	2	.	.	X
ejpam-6429	275	3	reassigning	reassign	VERB
ejpam-6429	275	4	the	the	DET
ejpam-6429	275	5	value	value	NOUN
ejpam-6429	275	6	0	0	NUM
ejpam-6429	275	7	to	to	ADP
ejpam-6429	275	8	u	u	PRON
ejpam-6429	275	9	results	result	NOUN
ejpam-6429	275	10	in	in	ADP
ejpam-6429	275	11	a	a	DET
ejpam-6429	275	12	grd	grd	NOUN
ejpam-6429	275	13	-	-	PUNCT
ejpam-6429	275	14	function	function	NOUN
ejpam-6429	275	15	for	for	ADP
ejpam-6429	275	16	g	g	NOUN
ejpam-6429	275	17	with	with	ADP
ejpam-6429	275	18	weight	weight	NOUN
ejpam-6429	275	19	smaller	small	ADJ
ejpam-6429	275	20	than	than	ADP
ejpam-6429	275	21	ωg	ωg	ADP
ejpam-6429	275	22	r(f	r(f	PROPN
ejpam-6429	275	23	)	)	PUNCT
ejpam-6429	275	24	;	;	PUNCT
ejpam-6429	275	25	therefore	therefore	ADV
ejpam-6429	275	26	,	,	PUNCT
ejpam-6429	275	27	sdγgr(g	sdγgr(g	PROPN
ejpam-6429	275	28	)	)	PUNCT
ejpam-6429	275	29	=	=	SYM
ejpam-6429	276	1	1	1	X
ejpam-6429	276	2	.	.	PUNCT
ejpam-6429	277	1	now	now	ADV
ejpam-6429	277	2	,	,	PUNCT
ejpam-6429	277	3	assume	assume	VERB
ejpam-6429	277	4	that	that	SCONJ
ejpam-6429	277	5	f(w	f(w	NOUN
ejpam-6429	277	6	)	)	PUNCT
ejpam-6429	277	7	=	=	SYM
ejpam-6429	278	1	2	2	X
ejpam-6429	278	2	.	.	X
ejpam-6429	278	3	if	if	SCONJ
ejpam-6429	278	4	f(v	f(v	NOUN
ejpam-6429	278	5	)	)	PUNCT
ejpam-6429	278	6	≥	≥	NOUN
ejpam-6429	278	7	1	1	NUM
ejpam-6429	278	8	or	or	CCONJ
ejpam-6429	278	9	f(y	f(y	NOUN
ejpam-6429	278	10	)	)	PUNCT
ejpam-6429	278	11	≥	≥	NOUN
ejpam-6429	278	12	1	1	NUM
ejpam-6429	278	13	,	,	PUNCT
ejpam-6429	278	14	then	then	ADV
ejpam-6429	278	15	by	by	ADP
ejpam-6429	278	16	reassigning	reassign	VERB
ejpam-6429	278	17	w	w	ADP
ejpam-6429	278	18	the	the	DET
ejpam-6429	278	19	value	value	NOUN
ejpam-6429	278	20	3	3	NUM
ejpam-6429	278	21	and	and	CCONJ
ejpam-6429	278	22	v	v	NOUN
ejpam-6429	278	23	and	and	CCONJ
ejpam-6429	278	24	y	y	PRON
ejpam-6429	278	25	the	the	DET
ejpam-6429	278	26	value	value	NOUN
ejpam-6429	278	27	0	0	NUM
ejpam-6429	278	28	,	,	PUNCT
ejpam-6429	278	29	we	we	PRON
ejpam-6429	278	30	obtain	obtain	VERB
ejpam-6429	278	31	a	a	DET
ejpam-6429	278	32	γgr(g	γgr(g	PROPN
ejpam-6429	278	33	′)-function	′)-function	NOUN
ejpam-6429	278	34	g	g	PROPN
ejpam-6429	278	35	such	such	ADJ
ejpam-6429	278	36	that	that	PRON
ejpam-6429	278	37	g(w	g(w	ADJ
ejpam-6429	278	38	)	)	PUNCT
ejpam-6429	279	1	+	+	PUNCT
ejpam-6429	279	2	g(z	g(z	ADJ
ejpam-6429	279	3	)	)	PUNCT
ejpam-6429	279	4	>	>	PUNCT
ejpam-6429	279	5	f(w	f(w	PROPN
ejpam-6429	279	6	)	)	PUNCT
ejpam-6429	280	1	+	+	NUM
ejpam-6429	281	1	f(z	f(z	NOUN
ejpam-6429	281	2	)	)	PUNCT
ejpam-6429	281	3	,	,	PUNCT
ejpam-6429	281	4	contradicting	contradict	VERB
ejpam-6429	281	5	the	the	DET
ejpam-6429	281	6	maximality	maximality	NOUN
ejpam-6429	281	7	of	of	ADP
ejpam-6429	281	8	f(w	f(w	PROPN
ejpam-6429	281	9	)	)	PUNCT
ejpam-6429	281	10	+	+	NUM
ejpam-6429	281	11	f(z	f(z	NOUN
ejpam-6429	281	12	)	)	PUNCT
ejpam-6429	281	13	.	.	PUNCT
ejpam-6429	282	1	therefore	therefore	ADV
ejpam-6429	282	2	,	,	PUNCT
ejpam-6429	282	3	we	we	PRON
ejpam-6429	282	4	must	must	AUX
ejpam-6429	282	5	have	have	VERB
ejpam-6429	282	6	f(v	f(v	NOUN
ejpam-6429	282	7	)	)	PUNCT
ejpam-6429	282	8	=	=	SYM
ejpam-6429	282	9	f(y	f(y	NOUN
ejpam-6429	282	10	)	)	PUNCT
ejpam-6429	282	11	=	=	SYM
ejpam-6429	283	1	0	0	X
ejpam-6429	283	2	.	.	PUNCT
ejpam-6429	284	1	however	however	ADV
ejpam-6429	284	2	,	,	PUNCT
ejpam-6429	284	3	in	in	ADP
ejpam-6429	284	4	this	this	DET
ejpam-6429	284	5	case	case	NOUN
ejpam-6429	284	6	,	,	PUNCT
ejpam-6429	284	7	the	the	DET
ejpam-6429	284	8	vertices	vertex	NOUN
ejpam-6429	284	9	v	v	NOUN
ejpam-6429	284	10	and	and	CCONJ
ejpam-6429	284	11	y	y	PROPN
ejpam-6429	284	12	can	can	AUX
ejpam-6429	284	13	not	not	PART
ejpam-6429	284	14	be	be	AUX
ejpam-6429	284	15	protected	protect	VERB
ejpam-6429	284	16	by	by	ADP
ejpam-6429	284	17	f	f	PROPN
ejpam-6429	284	18	,	,	PUNCT
ejpam-6429	284	19	which	which	PRON
ejpam-6429	284	20	is	be	AUX
ejpam-6429	284	21	a	a	DET
ejpam-6429	284	22	contradiction	contradiction	NOUN
ejpam-6429	284	23	.	.	PUNCT
ejpam-6429	285	1	if	if	SCONJ
ejpam-6429	285	2	f(w	f(w	PROPN
ejpam-6429	285	3	)	)	PUNCT
ejpam-6429	285	4	=	=	SYM
ejpam-6429	285	5	1	1	NUM
ejpam-6429	285	6	,	,	PUNCT
ejpam-6429	285	7	then	then	ADV
ejpam-6429	285	8	f(v	f(v	NOUN
ejpam-6429	285	9	)	)	PUNCT
ejpam-6429	285	10	=	=	SYM
ejpam-6429	285	11	1	1	NUM
ejpam-6429	285	12	,	,	PUNCT
ejpam-6429	285	13	and	and	CCONJ
ejpam-6429	285	14	by	by	ADP
ejpam-6429	285	15	reassigning	reassign	VERB
ejpam-6429	285	16	w	w	PROPN
ejpam-6429	285	17	and	and	CCONJ
ejpam-6429	285	18	v	v	ADP
ejpam-6429	285	19	the	the	DET
ejpam-6429	285	20	values	value	NOUN
ejpam-6429	285	21	2	2	NUM
ejpam-6429	285	22	and	and	CCONJ
ejpam-6429	285	23	0	0	NUM
ejpam-6429	285	24	,	,	PUNCT
ejpam-6429	285	25	respectively	respectively	ADV
ejpam-6429	285	26	,	,	PUNCT
ejpam-6429	285	27	we	we	PRON
ejpam-6429	285	28	get	get	VERB
ejpam-6429	285	29	a	a	DET
ejpam-6429	285	30	γgr(g)-function	γgr(g)-function	NOUN
ejpam-6429	285	31	g	g	ADP
ejpam-6429	285	32	such	such	ADJ
ejpam-6429	285	33	that	that	PRON
ejpam-6429	285	34	g(w	g(w	ADJ
ejpam-6429	285	35	)	)	PUNCT
ejpam-6429	286	1	+	+	PUNCT
ejpam-6429	286	2	g(z	g(z	ADJ
ejpam-6429	286	3	)	)	PUNCT
ejpam-6429	286	4	>	>	PUNCT
ejpam-6429	286	5	f(w	f(w	PROPN
ejpam-6429	286	6	)	)	PUNCT
ejpam-6429	287	1	+	+	NUM
ejpam-6429	288	1	f(z	f(z	NOUN
ejpam-6429	288	2	)	)	PUNCT
ejpam-6429	288	3	,	,	PUNCT
ejpam-6429	288	4	contradicting	contradict	VERB
ejpam-6429	288	5	the	the	DET
ejpam-6429	288	6	maximality	maximality	NOUN
ejpam-6429	288	7	of	of	ADP
ejpam-6429	288	8	f(w	f(w	PROPN
ejpam-6429	288	9	)	)	PUNCT
ejpam-6429	288	10	+	+	NUM
ejpam-6429	288	11	f(z	f(z	NOUN
ejpam-6429	288	12	)	)	PUNCT
ejpam-6429	288	13	.	.	PUNCT
ejpam-6429	289	1	thus	thus	ADV
ejpam-6429	289	2	,	,	PUNCT
ejpam-6429	289	3	we	we	PRON
ejpam-6429	289	4	assume	assume	VERB
ejpam-6429	289	5	that	that	SCONJ
ejpam-6429	289	6	f(w	f(w	NOUN
ejpam-6429	289	7	)	)	PUNCT
ejpam-6429	289	8	=	=	SYM
ejpam-6429	290	1	0	0	X
ejpam-6429	290	2	.	.	PUNCT
ejpam-6429	291	1	in	in	ADP
ejpam-6429	291	2	this	this	DET
ejpam-6429	291	3	case	case	NOUN
ejpam-6429	291	4	,	,	PUNCT
ejpam-6429	291	5	it	it	PRON
ejpam-6429	291	6	follows	follow	VERB
ejpam-6429	291	7	that	that	SCONJ
ejpam-6429	291	8	f(v	f(v	NOUN
ejpam-6429	291	9	)	)	PUNCT
ejpam-6429	291	10	=	=	SYM
ejpam-6429	291	11	f(y	f(y	NOUN
ejpam-6429	291	12	)	)	PUNCT
ejpam-6429	291	13	=	=	SYM
ejpam-6429	291	14	1	1	NUM
ejpam-6429	291	15	and	and	CCONJ
ejpam-6429	291	16	f(x	f(x	PROPN
ejpam-6429	291	17	)	)	PUNCT
ejpam-6429	292	1	+	+	CCONJ
ejpam-6429	292	2	f(u	f(u	NOUN
ejpam-6429	292	3	)	)	PUNCT
ejpam-6429	292	4	=	=	SYM
ejpam-6429	292	5	2	2	X
ejpam-6429	292	6	.	.	PUNCT
ejpam-6429	292	7	by	by	ADP
ejpam-6429	292	8	reassigning	reassign	VERB
ejpam-6429	292	9	the	the	DET
ejpam-6429	292	10	values	value	NOUN
ejpam-6429	292	11	of	of	ADP
ejpam-6429	292	12	w	w	PROPN
ejpam-6429	292	13	,	,	PUNCT
ejpam-6429	292	14	u	u	NOUN
ejpam-6429	292	15	,	,	PUNCT
ejpam-6429	292	16	v	v	NOUN
ejpam-6429	292	17	,	,	PUNCT
ejpam-6429	292	18	and	and	CCONJ
ejpam-6429	292	19	y	y	PROPN
ejpam-6429	292	20	to	to	ADP
ejpam-6429	292	21	3	3	NUM
ejpam-6429	292	22	,	,	PUNCT
ejpam-6429	292	23	1	1	NUM
ejpam-6429	292	24	,	,	PUNCT
ejpam-6429	292	25	0	0	NUM
ejpam-6429	292	26	,	,	PUNCT
ejpam-6429	292	27	and	and	CCONJ
ejpam-6429	292	28	0	0	NUM
ejpam-6429	292	29	,	,	PUNCT
ejpam-6429	292	30	respectively	respectively	ADV
ejpam-6429	292	31	,	,	PUNCT
ejpam-6429	292	32	we	we	PRON
ejpam-6429	292	33	obtain	obtain	VERB
ejpam-6429	292	34	a	a	DET
ejpam-6429	292	35	γgr(g)-function	γgr(g)-function	NOUN
ejpam-6429	292	36	g	g	ADP
ejpam-6429	292	37	such	such	ADJ
ejpam-6429	292	38	that	that	SCONJ
ejpam-6429	292	39	g(w)+	g(w)+	PROPN
ejpam-6429	292	40	g(z	g(z	PROPN
ejpam-6429	292	41	)	)	PUNCT
ejpam-6429	292	42	>	>	PUNCT
ejpam-6429	293	1	f(w)+	f(w)+	PROPN
ejpam-6429	293	2	f(z	f(z	PROPN
ejpam-6429	293	3	)	)	PUNCT
ejpam-6429	293	4	,	,	PUNCT
ejpam-6429	293	5	which	which	PRON
ejpam-6429	293	6	contradicts	contradict	VERB
ejpam-6429	293	7	the	the	DET
ejpam-6429	293	8	maximality	maximality	NOUN
ejpam-6429	293	9	of	of	ADP
ejpam-6429	293	10	f(w	f(w	PROPN
ejpam-6429	293	11	)	)	PUNCT
ejpam-6429	294	1	+	+	NUM
ejpam-6429	295	1	f(z	f(z	NOUN
ejpam-6429	295	2	)	)	PUNCT
ejpam-6429	295	3	.	.	PUNCT
ejpam-6429	296	1	therefore	therefore	ADV
ejpam-6429	296	2	,	,	PUNCT
ejpam-6429	296	3	we	we	PRON
ejpam-6429	296	4	conclude	conclude	VERB
ejpam-6429	296	5	that	that	PRON
ejpam-6429	296	6	sdγgr(g	sdγgr(g	PROPN
ejpam-6429	296	7	)	)	PUNCT
ejpam-6429	296	8	=	=	SYM
ejpam-6429	296	9	1	1	X
ejpam-6429	296	10	.	.	X
ejpam-6429	296	11	proposition	proposition	NOUN
ejpam-6429	296	12	9	9	NUM
ejpam-6429	296	13	.	.	PUNCT
ejpam-6429	297	1	if	if	SCONJ
ejpam-6429	297	2	g	g	PROPN
ejpam-6429	297	3	has	have	VERB
ejpam-6429	297	4	a	a	DET
ejpam-6429	297	5	support	support	NOUN
ejpam-6429	297	6	vertex	vertex	NOUN
ejpam-6429	297	7	w	w	NOUN
ejpam-6429	297	8	with	with	ADP
ejpam-6429	297	9	at	at	ADV
ejpam-6429	297	10	least	least	ADV
ejpam-6429	297	11	two	two	NUM
ejpam-6429	297	12	leaf	leaf	NOUN
ejpam-6429	297	13	neighbors	neighbor	NOUN
ejpam-6429	297	14	,	,	PUNCT
ejpam-6429	297	15	then	then	ADV
ejpam-6429	297	16	sdγgr(g	sdγgr(g	PROPN
ejpam-6429	297	17	)	)	PUNCT
ejpam-6429	297	18	≤	≤	NOUN
ejpam-6429	297	19	2	2	NUM
ejpam-6429	297	20	.	.	PUNCT
ejpam-6429	298	1	proof	proof	NOUN
ejpam-6429	298	2	.	.	PUNCT
ejpam-6429	299	1	if	if	SCONJ
ejpam-6429	299	2	g	g	PROPN
ejpam-6429	299	3	is	be	AUX
ejpam-6429	299	4	a	a	DET
ejpam-6429	299	5	star	star	NOUN
ejpam-6429	299	6	,	,	PUNCT
ejpam-6429	299	7	the	the	DET
ejpam-6429	299	8	result	result	NOUN
ejpam-6429	299	9	follows	follow	VERB
ejpam-6429	299	10	directly	directly	ADV
ejpam-6429	299	11	from	from	ADP
ejpam-6429	299	12	corollary	corollary	ADJ
ejpam-6429	299	13	3	3	NUM
ejpam-6429	299	14	,	,	PUNCT
ejpam-6429	299	15	and	and	CCONJ
ejpam-6429	299	16	if	if	SCONJ
ejpam-6429	299	17	w	w	NOUN
ejpam-6429	299	18	has	have	VERB
ejpam-6429	299	19	at	at	ADV
ejpam-6429	299	20	least	least	ADV
ejpam-6429	299	21	three	three	NUM
ejpam-6429	299	22	leaf	leaf	NOUN
ejpam-6429	299	23	neighbors	neighbor	NOUN
ejpam-6429	299	24	,	,	PUNCT
ejpam-6429	299	25	the	the	DET
ejpam-6429	299	26	conclusion	conclusion	NOUN
ejpam-6429	299	27	holds	hold	VERB
ejpam-6429	299	28	by	by	ADP
ejpam-6429	299	29	proposition	proposition	NOUN
ejpam-6429	299	30	8	8	NUM
ejpam-6429	299	31	.	.	PUNCT
ejpam-6429	300	1	thus	thus	ADV
ejpam-6429	300	2	,	,	PUNCT
ejpam-6429	300	3	assume	assume	VERB
ejpam-6429	300	4	that	that	SCONJ
ejpam-6429	300	5	g	g	PROPN
ejpam-6429	300	6	is	be	AUX
ejpam-6429	300	7	not	not	PART
ejpam-6429	300	8	a	a	DET
ejpam-6429	300	9	star	star	NOUN
ejpam-6429	300	10	and	and	CCONJ
ejpam-6429	300	11	that	that	SCONJ
ejpam-6429	300	12	w	w	NOUN
ejpam-6429	300	13	has	have	VERB
ejpam-6429	300	14	exactly	exactly	ADV
ejpam-6429	300	15	two	two	NUM
ejpam-6429	300	16	leaf	leaf	NOUN
ejpam-6429	300	17	neighbors	neighbor	NOUN
ejpam-6429	300	18	u	u	NOUN
ejpam-6429	300	19	and	and	CCONJ
ejpam-6429	300	20	v.	v.	CCONJ
ejpam-6429	300	21	let	let	VERB
ejpam-6429	300	22	z	z	PRON
ejpam-6429	300	23	be	be	AUX
ejpam-6429	300	24	a	a	DET
ejpam-6429	300	25	non	non	ADJ
ejpam-6429	300	26	-	-	ADJ
ejpam-6429	300	27	leaf	leaf	ADJ
ejpam-6429	300	28	neighbor	neighbor	NOUN
ejpam-6429	300	29	of	of	ADP
ejpam-6429	300	30	w	w	PROPN
ejpam-6429	300	31	and	and	CCONJ
ejpam-6429	300	32	let	let	VERB
ejpam-6429	300	33	g′	g′	NOUN
ejpam-6429	300	34	be	be	AUX
ejpam-6429	300	35	the	the	DET
ejpam-6429	300	36	graph	graph	NOUN
ejpam-6429	300	37	derived	derive	VERB
ejpam-6429	300	38	from	from	ADP
ejpam-6429	300	39	g	g	NOUN
ejpam-6429	300	40	by	by	ADP
ejpam-6429	300	41	inserting	insert	VERB
ejpam-6429	300	42	new	new	ADJ
ejpam-6429	300	43	vertices	vertex	NOUN
ejpam-6429	300	44	x	x	PUNCT
ejpam-6429	300	45	and	and	CCONJ
ejpam-6429	300	46	y	y	PROPN
ejpam-6429	300	47	to	to	PART
ejpam-6429	300	48	subdivide	subdivide	VERB
ejpam-6429	300	49	the	the	DET
ejpam-6429	300	50	edges	edge	NOUN
ejpam-6429	300	51	uw	uw	PROPN
ejpam-6429	300	52	and	and	CCONJ
ejpam-6429	300	53	vw	vw	PROPN
ejpam-6429	300	54	,	,	PUNCT
ejpam-6429	300	55	respectively	respectively	ADV
ejpam-6429	300	56	.	.	PUNCT
ejpam-6429	301	1	let	let	VERB
ejpam-6429	301	2	f	f	PRON
ejpam-6429	301	3	be	be	AUX
ejpam-6429	301	4	a	a	DET
ejpam-6429	301	5	γgr(g	γgr(g	PROPN
ejpam-6429	301	6	′)-function	′)-function	NOUN
ejpam-6429	301	7	that	that	PRON
ejpam-6429	301	8	maximizes	maximize	VERB
ejpam-6429	301	9	the	the	DET
ejpam-6429	301	10	sum	sum	NOUN
ejpam-6429	301	11	f(w	f(w	PROPN
ejpam-6429	301	12	)	)	PUNCT
ejpam-6429	302	1	+	+	NUM
ejpam-6429	303	1	f(z	f(z	NOUN
ejpam-6429	303	2	)	)	PUNCT
ejpam-6429	303	3	.	.	PUNCT
ejpam-6429	304	1	if	if	SCONJ
ejpam-6429	304	2	f(w	f(w	PROPN
ejpam-6429	304	3	)	)	PUNCT
ejpam-6429	304	4	=	=	SYM
ejpam-6429	304	5	3	3	NUM
ejpam-6429	304	6	,	,	PUNCT
ejpam-6429	304	7	then	then	ADV
ejpam-6429	304	8	we	we	PRON
ejpam-6429	304	9	have	have	VERB
ejpam-6429	304	10	f(u	f(u	PROPN
ejpam-6429	304	11	)	)	PUNCT
ejpam-6429	304	12	=	=	PUNCT
ejpam-6429	304	13	f(v	f(v	NOUN
ejpam-6429	304	14	)	)	PUNCT
ejpam-6429	304	15	=	=	SYM
ejpam-6429	304	16	1	1	NUM
ejpam-6429	304	17	,	,	PUNCT
ejpam-6429	304	18	and	and	CCONJ
ejpam-6429	304	19	by	by	ADP
ejpam-6429	304	20	reassigning	reassign	VERB
ejpam-6429	304	21	f(u	f(u	PROPN
ejpam-6429	304	22	)	)	PUNCT
ejpam-6429	304	23	=	=	PUNCT
ejpam-6429	304	24	f(v	f(v	NOUN
ejpam-6429	304	25	)	)	PUNCT
ejpam-6429	304	26	=	=	SYM
ejpam-6429	304	27	0	0	NUM
ejpam-6429	304	28	,	,	PUNCT
ejpam-6429	304	29	we	we	PRON
ejpam-6429	304	30	obtain	obtain	VERB
ejpam-6429	304	31	a	a	DET
ejpam-6429	304	32	grd	grd	NOUN
ejpam-6429	304	33	-	-	PUNCT
ejpam-6429	304	34	function	function	NOUN
ejpam-6429	304	35	for	for	ADP
ejpam-6429	304	36	g	g	NOUN
ejpam-6429	304	37	with	with	ADP
ejpam-6429	304	38	weight	weight	NOUN
ejpam-6429	304	39	less	less	ADJ
ejpam-6429	304	40	than	than	ADP
ejpam-6429	304	41	ωg	ωg	ADP
ejpam-6429	304	42	r(f	r(f	PROPN
ejpam-6429	304	43	)	)	PUNCT
ejpam-6429	304	44	,	,	PUNCT
ejpam-6429	304	45	thus	thus	ADV
ejpam-6429	304	46	implying	imply	VERB
ejpam-6429	304	47	that	that	SCONJ
ejpam-6429	304	48	sdγgr(g	sdγgr(g	NOUN
ejpam-6429	304	49	)	)	PUNCT
ejpam-6429	304	50	≤	≤	NOUN
ejpam-6429	304	51	2	2	NUM
ejpam-6429	304	52	.	.	X
ejpam-6429	304	53	assume	assume	VERB
ejpam-6429	304	54	now	now	ADV
ejpam-6429	304	55	that	that	SCONJ
ejpam-6429	304	56	f(w	f(w	NOUN
ejpam-6429	304	57	)	)	PUNCT
ejpam-6429	304	58	=	=	SYM
ejpam-6429	304	59	2	2	X
ejpam-6429	304	60	.	.	PUNCT
ejpam-6429	304	61	to	to	PART
ejpam-6429	304	62	protect	protect	VERB
ejpam-6429	304	63	the	the	DET
ejpam-6429	304	64	vertices	vertex	NOUN
ejpam-6429	304	65	u	u	NOUN
ejpam-6429	304	66	,	,	PUNCT
ejpam-6429	304	67	v	v	NOUN
ejpam-6429	304	68	,	,	PUNCT
ejpam-6429	304	69	x	x	PUNCT
ejpam-6429	304	70	and	and	CCONJ
ejpam-6429	304	71	y	y	PROPN
ejpam-6429	304	72	,	,	PUNCT
ejpam-6429	304	73	we	we	PRON
ejpam-6429	304	74	must	must	AUX
ejpam-6429	304	75	have	have	VERB
ejpam-6429	304	76	f(u	f(u	PROPN
ejpam-6429	304	77	)	)	PUNCT
ejpam-6429	304	78	+	+	NUM
ejpam-6429	304	79	f(v	f(v	NOUN
ejpam-6429	304	80	)	)	PUNCT
ejpam-6429	305	1	+	+	CCONJ
ejpam-6429	305	2	f(x	f(x	PROPN
ejpam-6429	305	3	)	)	PUNCT
ejpam-6429	306	1	+	+	SYM
ejpam-6429	306	2	f(y	f(y	NOUN
ejpam-6429	306	3	)	)	PUNCT
ejpam-6429	306	4	≥	≥	NOUN
ejpam-6429	306	5	3	3	NUM
ejpam-6429	306	6	.	.	PUNCT
ejpam-6429	307	1	by	by	ADP
ejpam-6429	307	2	reassigning	reassign	VERB
ejpam-6429	307	3	f(w	f(w	PROPN
ejpam-6429	307	4	)	)	PUNCT
ejpam-6429	307	5	=	=	SYM
ejpam-6429	307	6	3	3	NUM
ejpam-6429	307	7	,	,	PUNCT
ejpam-6429	307	8	f(u	f(u	PROPN
ejpam-6429	307	9	)	)	PUNCT
ejpam-6429	307	10	=	=	SYM
ejpam-6429	307	11	0	0	NUM
ejpam-6429	307	12	,	,	PUNCT
ejpam-6429	307	13	and	and	CCONJ
ejpam-6429	307	14	f(v	f(v	NOUN
ejpam-6429	307	15	)	)	PUNCT
ejpam-6429	307	16	=	=	SYM
ejpam-6429	307	17	0	0	NUM
ejpam-6429	307	18	,	,	PUNCT
ejpam-6429	307	19	we	we	PRON
ejpam-6429	307	20	obtain	obtain	VERB
ejpam-6429	307	21	a	a	DET
ejpam-6429	307	22	grd	grd	NOUN
ejpam-6429	307	23	-	-	PUNCT
ejpam-6429	307	24	function	function	NOUN
ejpam-6429	307	25	for	for	ADP
ejpam-6429	307	26	g	g	NOUN
ejpam-6429	307	27	with	with	ADP
ejpam-6429	307	28	weight	weight	NOUN
ejpam-6429	307	29	less	less	ADJ
ejpam-6429	307	30	than	than	ADP
ejpam-6429	307	31	ωg	ωg	ADP
ejpam-6429	307	32	r(f	r(f	PROPN
ejpam-6429	307	33	)	)	PUNCT
ejpam-6429	307	34	,	,	PUNCT
ejpam-6429	307	35	again	again	ADV
ejpam-6429	307	36	showing	show	VERB
ejpam-6429	307	37	that	that	SCONJ
ejpam-6429	307	38	sdγgr(g	sdγgr(g	NOUN
ejpam-6429	307	39	)	)	PUNCT
ejpam-6429	307	40	≤	≤	NOUN
ejpam-6429	307	41	2	2	NUM
ejpam-6429	307	42	.	.	PUNCT
ejpam-6429	308	1	finally	finally	ADV
ejpam-6429	308	2	,	,	PUNCT
ejpam-6429	308	3	if	if	SCONJ
ejpam-6429	308	4	f(w	f(w	NOUN
ejpam-6429	308	5	)	)	PUNCT
ejpam-6429	308	6	≤	≤	NUM
ejpam-6429	308	7	1	1	NUM
ejpam-6429	308	8	,	,	PUNCT
ejpam-6429	308	9	then	then	ADV
ejpam-6429	308	10	to	to	PART
ejpam-6429	308	11	protect	protect	VERB
ejpam-6429	308	12	the	the	DET
ejpam-6429	308	13	vertices	vertex	NOUN
ejpam-6429	308	14	u	u	NOUN
ejpam-6429	308	15	,	,	PUNCT
ejpam-6429	308	16	v	v	NOUN
ejpam-6429	308	17	,	,	PUNCT
ejpam-6429	308	18	x	x	PUNCT
ejpam-6429	308	19	and	and	CCONJ
ejpam-6429	308	20	y	y	PROPN
ejpam-6429	308	21	,	,	PUNCT
ejpam-6429	308	22	we	we	PRON
ejpam-6429	308	23	must	must	AUX
ejpam-6429	308	24	have	have	VERB
ejpam-6429	308	25	f(u)+f(v)+f(x)+f(y	f(u)+f(v)+f(x)+f(y	NOUN
ejpam-6429	308	26	)	)	PUNCT
ejpam-6429	308	27	≥	≥	NOUN
ejpam-6429	308	28	4	4	NUM
ejpam-6429	308	29	,	,	PUNCT
ejpam-6429	308	30	and	and	CCONJ
ejpam-6429	308	31	as	as	ADP
ejpam-6429	308	32	before	before	ADV
ejpam-6429	308	33	,	,	PUNCT
ejpam-6429	308	34	we	we	PRON
ejpam-6429	308	35	can	can	AUX
ejpam-6429	308	36	conclude	conclude	VERB
ejpam-6429	308	37	that	that	PRON
ejpam-6429	308	38	sdγgr(g	sdγgr(g	PROPN
ejpam-6429	308	39	)	)	PUNCT
ejpam-6429	308	40	≤	≤	NOUN
ejpam-6429	308	41	2	2	NUM
ejpam-6429	308	42	.	.	X
ejpam-6429	308	43	proposition	proposition	NOUN
ejpam-6429	308	44	10	10	NUM
ejpam-6429	308	45	.	.	PUNCT
ejpam-6429	309	1	for	for	ADP
ejpam-6429	309	2	every	every	DET
ejpam-6429	309	3	tree	tree	NOUN
ejpam-6429	309	4	t	t	NOUN
ejpam-6429	309	5	of	of	ADP
ejpam-6429	309	6	order	order	NOUN
ejpam-6429	309	7	n	n	PRON
ejpam-6429	309	8	≥	≥	NOUN
ejpam-6429	309	9	3	3	NUM
ejpam-6429	309	10	,	,	PUNCT
ejpam-6429	309	11	sdγgr(t	sdγgr(t	ADJ
ejpam-6429	309	12	)	)	PUNCT
ejpam-6429	309	13	≤	≤	NUM
ejpam-6429	309	14	2	2	NUM
ejpam-6429	309	15	.	.	PUNCT
ejpam-6429	310	1	j.	j.	PROPN
ejpam-6429	310	2	j.	j.	PROPN
ejpam-6429	310	3	hamja	hamja	PROPN
ejpam-6429	310	4	et	et	PROPN
ejpam-6429	310	5	al	al	PROPN
ejpam-6429	310	6	.	.	PUNCT
ejpam-6429	310	7	/	/	SYM
ejpam-6429	310	8	eur	eur	PROPN
ejpam-6429	310	9	.	.	PUNCT
ejpam-6429	311	1	j.	j.	PROPN
ejpam-6429	311	2	pure	pure	PROPN
ejpam-6429	311	3	appl	appl	PROPN
ejpam-6429	311	4	.	.	PROPN
ejpam-6429	311	5	math	math	PROPN
ejpam-6429	311	6	,	,	PUNCT
ejpam-6429	311	7	18	18	NUM
ejpam-6429	311	8	(	(	PUNCT
ejpam-6429	311	9	4	4	NUM
ejpam-6429	311	10	)	)	PUNCT
ejpam-6429	311	11	(	(	PUNCT
ejpam-6429	311	12	2025	2025	NUM
ejpam-6429	311	13	)	)	PUNCT
ejpam-6429	311	14	,	,	PUNCT
ejpam-6429	311	15	6429	6429	NUM
ejpam-6429	311	16	9	9	NUM
ejpam-6429	311	17	of	of	ADP
ejpam-6429	311	18	16	16	NUM
ejpam-6429	311	19	proof	proof	NOUN
ejpam-6429	311	20	.	.	PUNCT
ejpam-6429	312	1	if	if	SCONJ
ejpam-6429	312	2	t	t	PROPN
ejpam-6429	312	3	is	be	AUX
ejpam-6429	312	4	a	a	DET
ejpam-6429	312	5	star	star	NOUN
ejpam-6429	312	6	or	or	CCONJ
ejpam-6429	312	7	t	t	NOUN
ejpam-6429	312	8	contains	contain	VERB
ejpam-6429	312	9	a	a	DET
ejpam-6429	312	10	strong	strong	ADJ
ejpam-6429	312	11	support	support	NOUN
ejpam-6429	312	12	vertex	vertex	NOUN
ejpam-6429	312	13	,	,	PUNCT
ejpam-6429	312	14	then	then	ADV
ejpam-6429	312	15	the	the	DET
ejpam-6429	312	16	result	result	NOUN
ejpam-6429	312	17	follows	follow	VERB
ejpam-6429	312	18	directly	directly	ADV
ejpam-6429	312	19	from	from	ADP
ejpam-6429	312	20	corollary	corollary	ADJ
ejpam-6429	312	21	3	3	NUM
ejpam-6429	312	22	or	or	CCONJ
ejpam-6429	312	23	proposition	proposition	NOUN
ejpam-6429	312	24	9	9	NUM
ejpam-6429	312	25	.	.	PUNCT
ejpam-6429	313	1	now	now	ADV
ejpam-6429	313	2	,	,	PUNCT
ejpam-6429	313	3	assume	assume	VERB
ejpam-6429	313	4	that	that	SCONJ
ejpam-6429	313	5	t	t	PROPN
ejpam-6429	313	6	is	be	AUX
ejpam-6429	313	7	neither	neither	CCONJ
ejpam-6429	313	8	a	a	DET
ejpam-6429	313	9	star	star	NOUN
ejpam-6429	313	10	nor	nor	CCONJ
ejpam-6429	313	11	does	do	AUX
ejpam-6429	313	12	it	it	PRON
ejpam-6429	313	13	have	have	VERB
ejpam-6429	313	14	a	a	DET
ejpam-6429	313	15	strong	strong	ADJ
ejpam-6429	313	16	support	support	NOUN
ejpam-6429	313	17	vertex	vertex	NOUN
ejpam-6429	313	18	.	.	PUNCT
ejpam-6429	314	1	let	let	VERB
ejpam-6429	314	2	x1x2	x1x2	PUNCT
ejpam-6429	314	3	.	.	PUNCT
ejpam-6429	314	4	.	.	PUNCT
ejpam-6429	314	5	.	.	PUNCT
ejpam-6429	315	1	xk	xk	PROPN
ejpam-6429	315	2	represent	represent	VERB
ejpam-6429	315	3	a	a	DET
ejpam-6429	315	4	diametral	diametral	ADJ
ejpam-6429	315	5	path	path	NOUN
ejpam-6429	315	6	of	of	ADP
ejpam-6429	315	7	t	t	PROPN
ejpam-6429	315	8	,	,	PUNCT
ejpam-6429	315	9	and	and	CCONJ
ejpam-6429	315	10	root	root	PROPN
ejpam-6429	315	11	t	t	PROPN
ejpam-6429	315	12	at	at	ADP
ejpam-6429	315	13	vertex	vertex	PROPN
ejpam-6429	315	14	xk	xk	PROPN
ejpam-6429	315	15	.	.	PROPN
ejpam-6429	316	1	according	accord	VERB
ejpam-6429	316	2	to	to	ADP
ejpam-6429	316	3	our	our	PRON
ejpam-6429	316	4	previous	previous	ADJ
ejpam-6429	316	5	assumption	assumption	NOUN
ejpam-6429	316	6	,	,	PUNCT
ejpam-6429	316	7	we	we	PRON
ejpam-6429	316	8	have	have	VERB
ejpam-6429	316	9	deg(x2	deg(x2	NOUN
ejpam-6429	316	10	)	)	PUNCT
ejpam-6429	316	11	=	=	SYM
ejpam-6429	317	1	2	2	X
ejpam-6429	317	2	.	.	PUNCT
ejpam-6429	318	1	next	next	ADV
ejpam-6429	318	2	,	,	PUNCT
ejpam-6429	318	3	assume	assume	VERB
ejpam-6429	318	4	that	that	SCONJ
ejpam-6429	318	5	t	t	PROPN
ejpam-6429	318	6	′	′	NUM
ejpam-6429	318	7	is	be	AUX
ejpam-6429	318	8	obtained	obtain	VERB
ejpam-6429	318	9	by	by	ADP
ejpam-6429	318	10	subdividing	subdivide	VERB
ejpam-6429	318	11	the	the	DET
ejpam-6429	318	12	edges	edge	NOUN
ejpam-6429	318	13	x1x2	x1x2	PUNCT
ejpam-6429	318	14	and	and	CCONJ
ejpam-6429	318	15	x2x3	x2x3	X
ejpam-6429	318	16	with	with	ADP
ejpam-6429	318	17	new	new	ADJ
ejpam-6429	318	18	vertices	vertex	NOUN
ejpam-6429	318	19	x	x	PUNCT
ejpam-6429	318	20	and	and	CCONJ
ejpam-6429	318	21	y	y	PROPN
ejpam-6429	318	22	,	,	PUNCT
ejpam-6429	318	23	respectively	respectively	ADV
ejpam-6429	318	24	,	,	PUNCT
ejpam-6429	318	25	and	and	CCONJ
ejpam-6429	318	26	let	let	VERB
ejpam-6429	318	27	f	f	PRON
ejpam-6429	318	28	be	be	AUX
ejpam-6429	318	29	a	a	DET
ejpam-6429	318	30	γgr(t	γgr(t	PROPN
ejpam-6429	318	31	′)-function	′)-function	NOUN
ejpam-6429	318	32	.	.	PUNCT
ejpam-6429	319	1	if	if	SCONJ
ejpam-6429	319	2	f(x3	f(x3	VERB
ejpam-6429	319	3	)	)	PUNCT
ejpam-6429	319	4	≤	≤	NUM
ejpam-6429	319	5	1	1	NUM
ejpam-6429	319	6	,	,	PUNCT
ejpam-6429	319	7	then	then	ADV
ejpam-6429	319	8	we	we	PRON
ejpam-6429	319	9	must	must	AUX
ejpam-6429	319	10	have	have	VERB
ejpam-6429	319	11	f(x1	f(x1	ADJ
ejpam-6429	319	12	)	)	PUNCT
ejpam-6429	319	13	+	+	NUM
ejpam-6429	319	14	f(x2	f(x2	NOUN
ejpam-6429	319	15	)	)	PUNCT
ejpam-6429	320	1	+	+	CCONJ
ejpam-6429	320	2	f(x	f(x	PROPN
ejpam-6429	320	3	)	)	PUNCT
ejpam-6429	321	1	+	+	SYM
ejpam-6429	321	2	f(y	f(y	NOUN
ejpam-6429	321	3	)	)	PUNCT
ejpam-6429	321	4	≥	≥	NOUN
ejpam-6429	321	5	4	4	NUM
ejpam-6429	321	6	.	.	PUNCT
ejpam-6429	322	1	by	by	ADP
ejpam-6429	322	2	assigning	assign	VERB
ejpam-6429	322	3	the	the	DET
ejpam-6429	322	4	values	value	NOUN
ejpam-6429	322	5	0	0	NUM
ejpam-6429	322	6	and	and	CCONJ
ejpam-6429	322	7	2	2	NUM
ejpam-6429	322	8	to	to	ADP
ejpam-6429	322	9	x1	x1	PROPN
ejpam-6429	322	10	and	and	CCONJ
ejpam-6429	322	11	x2	x2	PROPN
ejpam-6429	322	12	,	,	PUNCT
ejpam-6429	322	13	respectively	respectively	ADV
ejpam-6429	322	14	,	,	PUNCT
ejpam-6429	322	15	we	we	PRON
ejpam-6429	322	16	obtain	obtain	VERB
ejpam-6429	322	17	a	a	DET
ejpam-6429	322	18	grd	grd	NOUN
ejpam-6429	322	19	-	-	PUNCT
ejpam-6429	322	20	function	function	NOUN
ejpam-6429	322	21	for	for	ADP
ejpam-6429	322	22	t	t	PROPN
ejpam-6429	322	23	with	with	ADP
ejpam-6429	322	24	weight	weight	NOUN
ejpam-6429	322	25	less	less	ADJ
ejpam-6429	322	26	than	than	ADP
ejpam-6429	322	27	ωg	ωg	ADP
ejpam-6429	322	28	r(f	r(f	PROPN
ejpam-6429	322	29	)	)	PUNCT
ejpam-6429	322	30	.	.	PUNCT
ejpam-6429	323	1	on	on	ADP
ejpam-6429	323	2	the	the	DET
ejpam-6429	323	3	other	other	ADJ
ejpam-6429	323	4	hand	hand	NOUN
ejpam-6429	323	5	,	,	PUNCT
ejpam-6429	323	6	if	if	SCONJ
ejpam-6429	323	7	f(x3	f(x3	VERB
ejpam-6429	323	8	)	)	PUNCT
ejpam-6429	323	9	≥	≥	NOUN
ejpam-6429	323	10	2	2	NUM
ejpam-6429	323	11	,	,	PUNCT
ejpam-6429	323	12	then	then	ADV
ejpam-6429	323	13	we	we	PRON
ejpam-6429	323	14	have	have	AUX
ejpam-6429	323	15	f(x1)+	f(x1)+	VERB
ejpam-6429	323	16	f(x2)+	f(x2)+	VERB
ejpam-6429	323	17	f(x)+	f(x)+	NOUN
ejpam-6429	323	18	f(y	f(y	NOUN
ejpam-6429	323	19	)	)	PUNCT
ejpam-6429	323	20	≥	≥	NOUN
ejpam-6429	323	21	3	3	NUM
ejpam-6429	323	22	,	,	PUNCT
ejpam-6429	323	23	and	and	CCONJ
ejpam-6429	323	24	assigning	assign	VERB
ejpam-6429	323	25	the	the	DET
ejpam-6429	323	26	values	value	NOUN
ejpam-6429	323	27	0	0	NUM
ejpam-6429	323	28	and	and	CCONJ
ejpam-6429	323	29	2	2	NUM
ejpam-6429	323	30	to	to	ADP
ejpam-6429	323	31	x1	x1	PROPN
ejpam-6429	323	32	and	and	CCONJ
ejpam-6429	323	33	x2	x2	PROPN
ejpam-6429	323	34	,	,	PUNCT
ejpam-6429	323	35	respectively	respectively	ADV
ejpam-6429	323	36	,	,	PUNCT
ejpam-6429	323	37	provides	provide	VERB
ejpam-6429	323	38	a	a	DET
ejpam-6429	323	39	grd	grd	NOUN
ejpam-6429	323	40	-	-	PUNCT
ejpam-6429	323	41	function	function	NOUN
ejpam-6429	323	42	for	for	ADP
ejpam-6429	323	43	t	t	PROPN
ejpam-6429	323	44	with	with	ADP
ejpam-6429	323	45	weight	weight	NOUN
ejpam-6429	323	46	less	less	ADJ
ejpam-6429	323	47	than	than	ADP
ejpam-6429	323	48	ωg	ωg	ADP
ejpam-6429	323	49	r(f	r(f	PROPN
ejpam-6429	323	50	)	)	PUNCT
ejpam-6429	323	51	.	.	PUNCT
ejpam-6429	324	1	therefore	therefore	ADV
ejpam-6429	324	2	,	,	PUNCT
ejpam-6429	324	3	we	we	PRON
ejpam-6429	324	4	conclude	conclude	VERB
ejpam-6429	324	5	that	that	PRON
ejpam-6429	324	6	sdγgr(g	sdγgr(g	PROPN
ejpam-6429	324	7	)	)	PUNCT
ejpam-6429	324	8	≤	≤	NOUN
ejpam-6429	324	9	2	2	NUM
ejpam-6429	324	10	.	.	X
ejpam-6429	324	11	proposition	proposition	NOUN
ejpam-6429	324	12	11	11	NUM
ejpam-6429	324	13	.	.	PUNCT
ejpam-6429	325	1	let	let	VERB
ejpam-6429	325	2	g	g	PRON
ejpam-6429	325	3	be	be	AUX
ejpam-6429	325	4	a	a	DET
ejpam-6429	325	5	connected	connected	ADJ
ejpam-6429	325	6	graph	graph	NOUN
ejpam-6429	325	7	of	of	ADP
ejpam-6429	325	8	order	order	NOUN
ejpam-6429	325	9	n	n	PRON
ejpam-6429	325	10	≥	≥	NOUN
ejpam-6429	325	11	3	3	NUM
ejpam-6429	325	12	.	.	PUNCT
ejpam-6429	326	1	if	if	SCONJ
ejpam-6429	326	2	g	g	PROPN
ejpam-6429	326	3	contains	contain	VERB
ejpam-6429	326	4	a	a	DET
ejpam-6429	326	5	vertex	vertex	NOUN
ejpam-6429	326	6	v	v	NOUN
ejpam-6429	326	7	that	that	PRON
ejpam-6429	326	8	lies	lie	VERB
ejpam-6429	326	9	in	in	ADP
ejpam-6429	326	10	a	a	DET
ejpam-6429	326	11	triangle	triangle	NOUN
ejpam-6429	326	12	uvwu	uvwu	ADJ
ejpam-6429	326	13	such	such	ADJ
ejpam-6429	326	14	that	that	DET
ejpam-6429	326	15	n(u	n(u	PROPN
ejpam-6429	326	16	)	)	PUNCT
ejpam-6429	326	17	∪n(w	∪n(w	ADV
ejpam-6429	326	18	)	)	PUNCT
ejpam-6429	327	1	⊆	⊆	NUM
ejpam-6429	327	2	n	n	NUM
ejpam-6429	327	3	[	[	X
ejpam-6429	327	4	v	v	NOUN
ejpam-6429	327	5	]	]	X
ejpam-6429	327	6	,	,	PUNCT
ejpam-6429	327	7	then	then	ADV
ejpam-6429	327	8	sdγgr(g	sdγgr(g	PROPN
ejpam-6429	327	9	)	)	PUNCT
ejpam-6429	327	10	≤	≤	NOUN
ejpam-6429	327	11	3	3	NUM
ejpam-6429	327	12	.	.	PUNCT
ejpam-6429	328	1	proof	proof	NOUN
ejpam-6429	328	2	.	.	PUNCT
ejpam-6429	329	1	let	let	VERB
ejpam-6429	329	2	g1	g1	PROPN
ejpam-6429	329	3	be	be	AUX
ejpam-6429	329	4	the	the	DET
ejpam-6429	329	5	graph	graph	NOUN
ejpam-6429	329	6	obtained	obtain	VERB
ejpam-6429	329	7	from	from	ADP
ejpam-6429	329	8	g	g	NOUN
ejpam-6429	329	9	by	by	ADP
ejpam-6429	329	10	subdividing	subdivide	VERB
ejpam-6429	329	11	the	the	DET
ejpam-6429	329	12	edges	edge	NOUN
ejpam-6429	329	13	vu	vu	PROPN
ejpam-6429	329	14	,	,	PUNCT
ejpam-6429	329	15	vw	vw	PROPN
ejpam-6429	329	16	,	,	PUNCT
ejpam-6429	329	17	and	and	CCONJ
ejpam-6429	329	18	uw	uw	NOUN
ejpam-6429	329	19	with	with	ADP
ejpam-6429	329	20	new	new	ADJ
ejpam-6429	329	21	vertices	vertex	NOUN
ejpam-6429	329	22	x1	x1	PROPN
ejpam-6429	329	23	,	,	PUNCT
ejpam-6429	329	24	x2	x2	PROPN
ejpam-6429	329	25	and	and	CCONJ
ejpam-6429	329	26	x3	x3	ADJ
ejpam-6429	329	27	,	,	PUNCT
ejpam-6429	329	28	respectively	respectively	ADV
ejpam-6429	329	29	.	.	PUNCT
ejpam-6429	330	1	it	it	PRON
ejpam-6429	330	2	suffices	suffice	VERB
ejpam-6429	330	3	to	to	PART
ejpam-6429	330	4	prove	prove	VERB
ejpam-6429	330	5	that	that	SCONJ
ejpam-6429	330	6	γgr(g1	γgr(g1	ADV
ejpam-6429	330	7	)	)	PUNCT
ejpam-6429	330	8	>	>	X
ejpam-6429	330	9	γgr(g	γgr(g	PROPN
ejpam-6429	330	10	)	)	PUNCT
ejpam-6429	330	11	.	.	PUNCT
ejpam-6429	331	1	let	let	VERB
ejpam-6429	331	2	g	g	PROPN
ejpam-6429	331	3	represent	represent	VERB
ejpam-6429	331	4	a	a	DET
ejpam-6429	331	5	minimum	minimum	ADJ
ejpam-6429	331	6	grd	grd	NOUN
ejpam-6429	331	7	-	-	PUNCT
ejpam-6429	331	8	function	function	NOUN
ejpam-6429	331	9	of	of	ADP
ejpam-6429	331	10	g1	g1	NOUN
ejpam-6429	331	11	.	.	PUNCT
ejpam-6429	332	1	by	by	ADP
ejpam-6429	332	2	proposition	proposition	NOUN
ejpam-6429	332	3	2	2	NUM
ejpam-6429	332	4	,	,	PUNCT
ejpam-6429	332	5	we	we	PRON
ejpam-6429	332	6	know	know	VERB
ejpam-6429	332	7	that	that	SCONJ
ejpam-6429	332	8	for	for	ADP
ejpam-6429	332	9	each	each	DET
ejpam-6429	332	10	i	i	PRON
ejpam-6429	332	11	∈	∈	PROPN
ejpam-6429	332	12	{	{	PUNCT
ejpam-6429	332	13	1	1	NUM
ejpam-6429	332	14	,	,	PUNCT
ejpam-6429	332	15	2	2	NUM
ejpam-6429	332	16	,	,	PUNCT
ejpam-6429	332	17	3	3	NUM
ejpam-6429	332	18	}	}	PUNCT
ejpam-6429	332	19	,	,	PUNCT
ejpam-6429	332	20	g(xi	g(xi	PROPN
ejpam-6429	332	21	)	)	PUNCT
ejpam-6429	332	22	/∈	/∈	PUNCT
ejpam-6429	333	1	{	{	PUNCT
ejpam-6429	333	2	1	1	NUM
ejpam-6429	333	3	,	,	PUNCT
ejpam-6429	333	4	3	3	NUM
ejpam-6429	333	5	}	}	PUNCT
ejpam-6429	333	6	.	.	PUNCT
ejpam-6429	334	1	if	if	SCONJ
ejpam-6429	334	2	two	two	NUM
ejpam-6429	334	3	of	of	ADP
ejpam-6429	334	4	the	the	DET
ejpam-6429	334	5	subdivision	subdivision	NOUN
ejpam-6429	334	6	vertices	vertex	NOUN
ejpam-6429	334	7	,	,	PUNCT
ejpam-6429	334	8	say	say	VERB
ejpam-6429	334	9	x1	x1	NUM
ejpam-6429	334	10	and	and	CCONJ
ejpam-6429	334	11	x2	x2	PROPN
ejpam-6429	334	12	,	,	PUNCT
ejpam-6429	334	13	have	have	VERB
ejpam-6429	334	14	positive	positive	ADJ
ejpam-6429	334	15	weights	weight	NOUN
ejpam-6429	334	16	under	under	ADP
ejpam-6429	334	17	g	g	NOUN
ejpam-6429	334	18	,	,	PUNCT
ejpam-6429	334	19	then	then	ADV
ejpam-6429	334	20	it	it	PRON
ejpam-6429	334	21	follows	follow	VERB
ejpam-6429	334	22	that	that	SCONJ
ejpam-6429	334	23	g(x1	g(x1	NOUN
ejpam-6429	334	24	)	)	PUNCT
ejpam-6429	334	25	+	+	NUM
ejpam-6429	334	26	g(x2	g(x2	NOUN
ejpam-6429	334	27	)	)	PUNCT
ejpam-6429	334	28	≥	≥	NOUN
ejpam-6429	334	29	4	4	NUM
ejpam-6429	334	30	,	,	PUNCT
ejpam-6429	334	31	and	and	CCONJ
ejpam-6429	334	32	by	by	ADP
ejpam-6429	334	33	reassigning	reassign	VERB
ejpam-6429	334	34	the	the	DET
ejpam-6429	334	35	value	value	NOUN
ejpam-6429	334	36	3	3	NUM
ejpam-6429	334	37	to	to	ADP
ejpam-6429	334	38	v	v	NUM
ejpam-6429	334	39	,	,	PUNCT
ejpam-6429	334	40	we	we	PRON
ejpam-6429	334	41	obtain	obtain	VERB
ejpam-6429	334	42	a	a	DET
ejpam-6429	334	43	grd	grd	NOUN
ejpam-6429	334	44	-	-	PUNCT
ejpam-6429	334	45	function	function	NOUN
ejpam-6429	334	46	on	on	ADP
ejpam-6429	334	47	g	g	PROPN
ejpam-6429	334	48	with	with	ADP
ejpam-6429	334	49	weight	weight	NOUN
ejpam-6429	334	50	less	less	ADJ
ejpam-6429	334	51	than	than	ADP
ejpam-6429	334	52	ωg	ωg	ADP
ejpam-6429	334	53	r(g	r(g	NUM
ejpam-6429	334	54	)	)	PUNCT
ejpam-6429	334	55	.	.	PUNCT
ejpam-6429	335	1	if	if	SCONJ
ejpam-6429	335	2	exactly	exactly	ADV
ejpam-6429	335	3	one	one	NUM
ejpam-6429	335	4	of	of	ADP
ejpam-6429	335	5	the	the	DET
ejpam-6429	335	6	subdivision	subdivision	NOUN
ejpam-6429	335	7	vertices	vertex	NOUN
ejpam-6429	335	8	,	,	PUNCT
ejpam-6429	335	9	say	say	VERB
ejpam-6429	335	10	x1	x1	PROPN
ejpam-6429	335	11	,	,	PUNCT
ejpam-6429	335	12	has	have	VERB
ejpam-6429	335	13	a	a	DET
ejpam-6429	335	14	positive	positive	ADJ
ejpam-6429	335	15	weight	weight	NOUN
ejpam-6429	335	16	under	under	ADP
ejpam-6429	335	17	g	g	NOUN
ejpam-6429	335	18	,	,	PUNCT
ejpam-6429	335	19	then	then	ADV
ejpam-6429	335	20	we	we	PRON
ejpam-6429	335	21	must	must	AUX
ejpam-6429	335	22	have	have	VERB
ejpam-6429	335	23	g(x2	g(x2	NOUN
ejpam-6429	335	24	)	)	PUNCT
ejpam-6429	335	25	=	=	SYM
ejpam-6429	335	26	g(x3	g(x3	NOUN
ejpam-6429	335	27	)	)	PUNCT
ejpam-6429	336	1	=	=	SYM
ejpam-6429	336	2	0	0	NUM
ejpam-6429	336	3	and	and	CCONJ
ejpam-6429	336	4	g(u	g(u	PROPN
ejpam-6429	336	5	)	)	PUNCT
ejpam-6429	336	6	+	+	CCONJ
ejpam-6429	336	7	g(v	g(v	X
ejpam-6429	336	8	)	)	PUNCT
ejpam-6429	336	9	+	+	SYM
ejpam-6429	336	10	g(w	g(w	PROPN
ejpam-6429	336	11	)	)	PUNCT
ejpam-6429	336	12	≥	≥	NOUN
ejpam-6429	336	13	2	2	NUM
ejpam-6429	336	14	.	.	PUNCT
ejpam-6429	336	15	again	again	ADV
ejpam-6429	336	16	,	,	PUNCT
ejpam-6429	336	17	by	by	ADP
ejpam-6429	336	18	reassigning	reassign	VERB
ejpam-6429	336	19	the	the	DET
ejpam-6429	336	20	value	value	NOUN
ejpam-6429	336	21	3	3	NUM
ejpam-6429	336	22	to	to	ADP
ejpam-6429	336	23	v	v	NOUN
ejpam-6429	336	24	,	,	PUNCT
ejpam-6429	336	25	we	we	PRON
ejpam-6429	336	26	get	get	VERB
ejpam-6429	336	27	a	a	DET
ejpam-6429	336	28	grd	grd	NOUN
ejpam-6429	336	29	-	-	PUNCT
ejpam-6429	336	30	function	function	NOUN
ejpam-6429	336	31	on	on	ADP
ejpam-6429	336	32	g	g	PROPN
ejpam-6429	336	33	with	with	ADP
ejpam-6429	336	34	weight	weight	NOUN
ejpam-6429	336	35	less	less	ADJ
ejpam-6429	336	36	than	than	ADP
ejpam-6429	336	37	ωg	ωg	ADP
ejpam-6429	336	38	r(g	r(g	NUM
ejpam-6429	336	39	)	)	PUNCT
ejpam-6429	336	40	,	,	PUNCT
ejpam-6429	336	41	as	as	SCONJ
ejpam-6429	336	42	the	the	DET
ejpam-6429	336	43	neighbors	neighbor	NOUN
ejpam-6429	336	44	of	of	ADP
ejpam-6429	336	45	u	u	PROPN
ejpam-6429	336	46	and	and	CCONJ
ejpam-6429	336	47	z	z	NOUN
ejpam-6429	336	48	are	be	AUX
ejpam-6429	336	49	also	also	ADV
ejpam-6429	336	50	neighbors	neighbor	NOUN
ejpam-6429	336	51	of	of	ADP
ejpam-6429	336	52	v.	v.	ADP
ejpam-6429	336	53	thus	thus	ADV
ejpam-6429	336	54	,	,	PUNCT
ejpam-6429	336	55	we	we	PRON
ejpam-6429	336	56	assume	assume	VERB
ejpam-6429	336	57	that	that	SCONJ
ejpam-6429	336	58	g(x1	g(x1	NOUN
ejpam-6429	336	59	)	)	PUNCT
ejpam-6429	336	60	=	=	SYM
ejpam-6429	336	61	g(x2	g(x2	NOUN
ejpam-6429	336	62	)	)	PUNCT
ejpam-6429	336	63	=	=	SYM
ejpam-6429	336	64	g(x3	g(x3	NOUN
ejpam-6429	336	65	)	)	PUNCT
ejpam-6429	337	1	=	=	SYM
ejpam-6429	337	2	0	0	X
ejpam-6429	337	3	.	.	PUNCT
ejpam-6429	337	4	to	to	PART
ejpam-6429	337	5	protect	protect	VERB
ejpam-6429	337	6	the	the	DET
ejpam-6429	337	7	subdivision	subdivision	NOUN
ejpam-6429	337	8	vertices	vertex	NOUN
ejpam-6429	337	9	,	,	PUNCT
ejpam-6429	337	10	we	we	PRON
ejpam-6429	337	11	must	must	AUX
ejpam-6429	337	12	have	have	VERB
ejpam-6429	337	13	g(v	g(v	NOUN
ejpam-6429	337	14	)	)	PUNCT
ejpam-6429	338	1	+	+	CCONJ
ejpam-6429	339	1	g(u	g(u	X
ejpam-6429	339	2	)	)	PUNCT
ejpam-6429	339	3	+	+	SYM
ejpam-6429	339	4	g(w	g(w	PROPN
ejpam-6429	339	5	)	)	PUNCT
ejpam-6429	339	6	≥	≥	NOUN
ejpam-6429	339	7	4	4	NUM
ejpam-6429	339	8	.	.	PUNCT
ejpam-6429	340	1	the	the	DET
ejpam-6429	340	2	function	function	NOUN
ejpam-6429	340	3	defined	define	VERB
ejpam-6429	340	4	above	above	ADV
ejpam-6429	340	5	is	be	AUX
ejpam-6429	340	6	a	a	DET
ejpam-6429	340	7	grd	grd	NOUN
ejpam-6429	340	8	-	-	PUNCT
ejpam-6429	340	9	function	function	NOUN
ejpam-6429	340	10	on	on	ADP
ejpam-6429	340	11	g	g	PROPN
ejpam-6429	340	12	with	with	ADP
ejpam-6429	340	13	weight	weight	NOUN
ejpam-6429	340	14	less	less	ADJ
ejpam-6429	340	15	than	than	ADP
ejpam-6429	340	16	ωg	ωg	ADP
ejpam-6429	340	17	r(g	r(g	NUM
ejpam-6429	340	18	)	)	PUNCT
ejpam-6429	340	19	.	.	PUNCT
ejpam-6429	341	1	this	this	PRON
ejpam-6429	341	2	completes	complete	VERB
ejpam-6429	341	3	the	the	DET
ejpam-6429	341	4	proof	proof	NOUN
ejpam-6429	341	5	.	.	PUNCT
ejpam-6429	342	1	6	6	X
ejpam-6429	342	2	.	.	NUM
ejpam-6429	342	3	bounds	bound	NOUN
ejpam-6429	342	4	in	in	ADP
ejpam-6429	342	5	this	this	DET
ejpam-6429	342	6	section	section	NOUN
ejpam-6429	342	7	,	,	PUNCT
ejpam-6429	342	8	we	we	PRON
ejpam-6429	342	9	derive	derive	VERB
ejpam-6429	342	10	several	several	ADJ
ejpam-6429	342	11	bounds	bound	NOUN
ejpam-6429	342	12	for	for	ADP
ejpam-6429	342	13	the	the	DET
ejpam-6429	342	14	generous	generous	ADJ
ejpam-6429	342	15	roman	roman	ADJ
ejpam-6429	342	16	domination	domination	NOUN
ejpam-6429	342	17	subdivision	subdivision	NOUN
ejpam-6429	342	18	number	number	NOUN
ejpam-6429	342	19	.	.	PUNCT
ejpam-6429	343	1	theorem	theorem	NOUN
ejpam-6429	343	2	3	3	X
ejpam-6429	343	3	.	.	PUNCT
ejpam-6429	344	1	let	let	VERB
ejpam-6429	344	2	g	g	PRON
ejpam-6429	344	3	be	be	AUX
ejpam-6429	344	4	a	a	DET
ejpam-6429	344	5	connected	connected	ADJ
ejpam-6429	344	6	graph	graph	NOUN
ejpam-6429	344	7	.	.	PUNCT
ejpam-6429	345	1	if	if	SCONJ
ejpam-6429	345	2	x	x	SYM
ejpam-6429	345	3	∈	∈	PROPN
ejpam-6429	345	4	v	v	X
ejpam-6429	345	5	(	(	PUNCT
ejpam-6429	345	6	g	g	NOUN
ejpam-6429	345	7	)	)	PUNCT
ejpam-6429	345	8	has	have	VERB
ejpam-6429	345	9	degree	degree	NOUN
ejpam-6429	345	10	at	at	ADV
ejpam-6429	345	11	least	least	ADV
ejpam-6429	345	12	two	two	NUM
ejpam-6429	345	13	,	,	PUNCT
ejpam-6429	345	14	then	then	ADV
ejpam-6429	345	15	sdγgr(g	sdγgr(g	PROPN
ejpam-6429	345	16	)	)	PUNCT
ejpam-6429	345	17	≤	≤	ADV
ejpam-6429	345	18	deg(x	deg(x	X
ejpam-6429	345	19	)	)	PUNCT
ejpam-6429	345	20	.	.	PUNCT
ejpam-6429	346	1	proof	proof	NOUN
ejpam-6429	346	2	.	.	PUNCT
ejpam-6429	347	1	let	let	VERB
ejpam-6429	347	2	s	s	VERB
ejpam-6429	347	3	=	=	SYM
ejpam-6429	347	4	deg(x	deg(x	X
ejpam-6429	347	5	)	)	PUNCT
ejpam-6429	347	6	and	and	CCONJ
ejpam-6429	347	7	n(x	n(x	X
ejpam-6429	347	8	)	)	PUNCT
ejpam-6429	347	9	=	=	PRON
ejpam-6429	347	10	{	{	PUNCT
ejpam-6429	347	11	x1	x1	PROPN
ejpam-6429	347	12	,	,	PUNCT
ejpam-6429	347	13	x2	x2	PROPN
ejpam-6429	347	14	,	,	PUNCT
ejpam-6429	347	15	.	.	PUNCT
ejpam-6429	347	16	.	.	PUNCT
ejpam-6429	348	1	.	.	PUNCT
ejpam-6429	349	1	,	,	PUNCT
ejpam-6429	349	2	xs	xs	PROPN
ejpam-6429	349	3	}	}	PUNCT
ejpam-6429	349	4	,	,	PUNCT
ejpam-6429	349	5	and	and	CCONJ
ejpam-6429	349	6	let	let	VERB
ejpam-6429	349	7	g1	g1	PROPN
ejpam-6429	349	8	be	be	AUX
ejpam-6429	349	9	the	the	DET
ejpam-6429	349	10	graph	graph	NOUN
ejpam-6429	349	11	obtained	obtain	VERB
ejpam-6429	349	12	from	from	ADP
ejpam-6429	349	13	g	g	NOUN
ejpam-6429	349	14	by	by	ADP
ejpam-6429	349	15	subdividing	subdivide	VERB
ejpam-6429	349	16	the	the	DET
ejpam-6429	349	17	edges	edge	NOUN
ejpam-6429	349	18	xx1	xx1	PROPN
ejpam-6429	349	19	,	,	PUNCT
ejpam-6429	349	20	xx2	xx2	PROPN
ejpam-6429	349	21	,	,	PUNCT
ejpam-6429	349	22	.	.	PUNCT
ejpam-6429	349	23	.	.	PUNCT
ejpam-6429	350	1	.	.	PUNCT
ejpam-6429	351	1	,	,	PUNCT
ejpam-6429	351	2	xxs	xxs	PROPN
ejpam-6429	351	3	with	with	ADP
ejpam-6429	351	4	new	new	ADJ
ejpam-6429	351	5	vertices	vertex	NOUN
ejpam-6429	351	6	y1	y1	NOUN
ejpam-6429	351	7	,	,	PUNCT
ejpam-6429	351	8	y2	y2	INTJ
ejpam-6429	351	9	,	,	PUNCT
ejpam-6429	351	10	.	.	PUNCT
ejpam-6429	351	11	.	.	PUNCT
ejpam-6429	352	1	.	.	PUNCT
ejpam-6429	353	1	,	,	PUNCT
ejpam-6429	353	2	ys	ys	INTJ
ejpam-6429	353	3	,	,	PUNCT
ejpam-6429	353	4	respectively	respectively	ADV
ejpam-6429	353	5	.	.	PUNCT
ejpam-6429	354	1	to	to	PART
ejpam-6429	354	2	prove	prove	VERB
ejpam-6429	354	3	the	the	DET
ejpam-6429	354	4	desired	desire	VERB
ejpam-6429	354	5	result	result	NOUN
ejpam-6429	354	6	,	,	PUNCT
ejpam-6429	354	7	it	it	PRON
ejpam-6429	354	8	is	be	AUX
ejpam-6429	354	9	sufficient	sufficient	ADJ
ejpam-6429	354	10	to	to	PART
ejpam-6429	354	11	show	show	VERB
ejpam-6429	354	12	that	that	SCONJ
ejpam-6429	354	13	γgr(g1	γgr(g1	ADV
ejpam-6429	354	14	)	)	PUNCT
ejpam-6429	354	15	>	>	X
ejpam-6429	354	16	γgr(g	γgr(g	PROPN
ejpam-6429	354	17	)	)	PUNCT
ejpam-6429	354	18	.	.	PUNCT
ejpam-6429	355	1	let	let	VERB
ejpam-6429	355	2	f	f	PRON
ejpam-6429	355	3	be	be	AUX
ejpam-6429	355	4	a	a	DET
ejpam-6429	355	5	γgr(g1)-function	γgr(g1)-function	PROPN
ejpam-6429	355	6	.	.	PUNCT
ejpam-6429	356	1	by	by	ADP
ejpam-6429	356	2	proposition	proposition	NOUN
ejpam-6429	356	3	2	2	NUM
ejpam-6429	356	4	,	,	PUNCT
ejpam-6429	356	5	we	we	PRON
ejpam-6429	356	6	may	may	AUX
ejpam-6429	356	7	assume	assume	VERB
ejpam-6429	356	8	that	that	SCONJ
ejpam-6429	356	9	f(yi	f(yi	PROPN
ejpam-6429	356	10	)	)	PUNCT
ejpam-6429	356	11	/∈	/∈	PUNCT
ejpam-6429	357	1	{	{	PUNCT
ejpam-6429	357	2	1	1	NUM
ejpam-6429	357	3	,	,	PUNCT
ejpam-6429	357	4	3	3	NUM
ejpam-6429	357	5	}	}	PUNCT
ejpam-6429	357	6	for	for	ADP
ejpam-6429	357	7	all	all	DET
ejpam-6429	357	8	1	1	NUM
ejpam-6429	357	9	≤	≤	NUM
ejpam-6429	357	10	i	i	PRON
ejpam-6429	357	11	≤	≤	VERB
ejpam-6429	357	12	s.	s.	PROPN
ejpam-6429	358	1	if	if	SCONJ
ejpam-6429	358	2	∑s	∑s	PROPN
ejpam-6429	358	3	i=1	i=1	PROPN
ejpam-6429	358	4	f(yi	f(yi	PROPN
ejpam-6429	358	5	)	)	PUNCT
ejpam-6429	359	1	+	+	CCONJ
ejpam-6429	359	2	f(x	f(x	PROPN
ejpam-6429	359	3	)	)	PUNCT
ejpam-6429	359	4	≥	≥	NOUN
ejpam-6429	359	5	4	4	NUM
ejpam-6429	359	6	,	,	PUNCT
ejpam-6429	359	7	then	then	ADV
ejpam-6429	359	8	assigning	assign	VERB
ejpam-6429	359	9	the	the	DET
ejpam-6429	359	10	value	value	NOUN
ejpam-6429	359	11	3	3	NUM
ejpam-6429	359	12	to	to	PART
ejpam-6429	359	13	x	x	NOUN
ejpam-6429	359	14	results	result	NOUN
ejpam-6429	359	15	in	in	ADP
ejpam-6429	359	16	a	a	DET
ejpam-6429	359	17	grd	grd	NOUN
ejpam-6429	359	18	-	-	PUNCT
ejpam-6429	359	19	function	function	NOUN
ejpam-6429	359	20	on	on	ADP
ejpam-6429	359	21	g	g	NOUN
ejpam-6429	359	22	with	with	ADP
ejpam-6429	359	23	a	a	DET
ejpam-6429	359	24	weight	weight	NOUN
ejpam-6429	359	25	smaller	small	ADJ
ejpam-6429	359	26	than	than	ADP
ejpam-6429	359	27	ωg	ωg	ADP
ejpam-6429	359	28	r(f	r(f	PROPN
ejpam-6429	359	29	)	)	PUNCT
ejpam-6429	359	30	,	,	PUNCT
ejpam-6429	359	31	as	as	SCONJ
ejpam-6429	359	32	desired	desire	VERB
ejpam-6429	359	33	.	.	PUNCT
ejpam-6429	360	1	therefore	therefore	ADV
ejpam-6429	360	2	,	,	PUNCT
ejpam-6429	360	3	we	we	PRON
ejpam-6429	360	4	assume	assume	VERB
ejpam-6429	360	5	that	that	SCONJ
ejpam-6429	360	6	∑s	∑s	PROPN
ejpam-6429	360	7	i=1	i=1	PROPN
ejpam-6429	360	8	f(yi	f(yi	PROPN
ejpam-6429	360	9	)	)	PUNCT
ejpam-6429	360	10	+	+	CCONJ
ejpam-6429	360	11	f(x	f(x	PROPN
ejpam-6429	360	12	)	)	PUNCT
ejpam-6429	360	13	≤	≤	NOUN
ejpam-6429	361	1	3	3	NUM
ejpam-6429	361	2	.	.	PUNCT
ejpam-6429	362	1	if	if	SCONJ
ejpam-6429	362	2	f(x	f(x	PROPN
ejpam-6429	362	3	)	)	PUNCT
ejpam-6429	362	4	∈	∈	PROPN
ejpam-6429	362	5	{	{	PUNCT
ejpam-6429	362	6	2	2	NUM
ejpam-6429	362	7	,	,	PUNCT
ejpam-6429	362	8	3	3	NUM
ejpam-6429	362	9	}	}	PUNCT
ejpam-6429	362	10	,	,	PUNCT
ejpam-6429	362	11	then	then	ADV
ejpam-6429	362	12	we	we	PRON
ejpam-6429	362	13	must	must	AUX
ejpam-6429	362	14	have	have	VERB
ejpam-6429	362	15	∑s	∑s	PROPN
ejpam-6429	362	16	i=1	i=1	PROPN
ejpam-6429	362	17	f(yi	f(yi	PROPN
ejpam-6429	362	18	)	)	PUNCT
ejpam-6429	363	1	=	=	SYM
ejpam-6429	363	2	0	0	X
ejpam-6429	364	1	j.	j.	PROPN
ejpam-6429	364	2	j.	j.	PROPN
ejpam-6429	364	3	hamja	hamja	PROPN
ejpam-6429	364	4	et	et	PROPN
ejpam-6429	364	5	al	al	PROPN
ejpam-6429	364	6	.	.	PUNCT
ejpam-6429	364	7	/	/	SYM
ejpam-6429	364	8	eur	eur	PROPN
ejpam-6429	364	9	.	.	PUNCT
ejpam-6429	365	1	j.	j.	PROPN
ejpam-6429	365	2	pure	pure	PROPN
ejpam-6429	365	3	appl	appl	PROPN
ejpam-6429	365	4	.	.	PROPN
ejpam-6429	365	5	math	math	PROPN
ejpam-6429	365	6	,	,	PUNCT
ejpam-6429	365	7	18	18	NUM
ejpam-6429	365	8	(	(	PUNCT
ejpam-6429	365	9	4	4	NUM
ejpam-6429	365	10	)	)	PUNCT
ejpam-6429	365	11	(	(	PUNCT
ejpam-6429	365	12	2025	2025	NUM
ejpam-6429	365	13	)	)	PUNCT
ejpam-6429	365	14	,	,	PUNCT
ejpam-6429	365	15	6429	6429	NUM
ejpam-6429	365	16	10	10	NUM
ejpam-6429	365	17	of	of	ADP
ejpam-6429	365	18	16	16	NUM
ejpam-6429	365	19	because	because	SCONJ
ejpam-6429	365	20	f(yi	f(yi	NUM
ejpam-6429	365	21	)	)	PUNCT
ejpam-6429	365	22	̸=	̸=	PROPN
ejpam-6429	365	23	1	1	NUM
ejpam-6429	365	24	for	for	ADP
ejpam-6429	365	25	each	each	DET
ejpam-6429	365	26	i.	i.	NOUN
ejpam-6429	365	27	in	in	ADP
ejpam-6429	365	28	this	this	DET
ejpam-6429	365	29	case	case	NOUN
ejpam-6429	365	30	,	,	PUNCT
ejpam-6429	365	31	reassigning	reassign	VERB
ejpam-6429	365	32	the	the	DET
ejpam-6429	365	33	value	value	NOUN
ejpam-6429	365	34	1	1	NUM
ejpam-6429	365	35	to	to	PART
ejpam-6429	365	36	x	x	NOUN
ejpam-6429	365	37	results	result	NOUN
ejpam-6429	365	38	in	in	ADP
ejpam-6429	365	39	a	a	DET
ejpam-6429	365	40	grdfunction	grdfunction	NOUN
ejpam-6429	365	41	on	on	ADP
ejpam-6429	365	42	g	g	NOUN
ejpam-6429	365	43	with	with	ADP
ejpam-6429	365	44	weight	weight	NOUN
ejpam-6429	365	45	less	less	ADJ
ejpam-6429	365	46	than	than	ADP
ejpam-6429	365	47	ωg	ωg	ADP
ejpam-6429	365	48	r(f	r(f	PROPN
ejpam-6429	365	49	)	)	PUNCT
ejpam-6429	365	50	,	,	PUNCT
ejpam-6429	365	51	as	as	SCONJ
ejpam-6429	365	52	desired	desire	VERB
ejpam-6429	365	53	.	.	PUNCT
ejpam-6429	366	1	if	if	SCONJ
ejpam-6429	366	2	f(x	f(x	PROPN
ejpam-6429	366	3	)	)	PUNCT
ejpam-6429	367	1	=	=	PUNCT
ejpam-6429	367	2	1	1	NUM
ejpam-6429	367	3	,	,	PUNCT
ejpam-6429	367	4	then	then	ADV
ejpam-6429	367	5	since	since	SCONJ
ejpam-6429	367	6	s	s	PRON
ejpam-6429	367	7	≥	≥	NUM
ejpam-6429	367	8	2	2	NUM
ejpam-6429	367	9	and∑s	and∑s	NOUN
ejpam-6429	367	10	i=1	i=1	PROPN
ejpam-6429	367	11	f(yi)+f(x	f(yi)+f(x	NOUN
ejpam-6429	367	12	)	)	PUNCT
ejpam-6429	367	13	≤	≤	NOUN
ejpam-6429	367	14	3	3	NUM
ejpam-6429	367	15	,	,	PUNCT
ejpam-6429	367	16	there	there	PRON
ejpam-6429	367	17	must	must	AUX
ejpam-6429	367	18	exist	exist	VERB
ejpam-6429	367	19	a	a	DET
ejpam-6429	367	20	vertex	vertex	NOUN
ejpam-6429	367	21	yi	yi	NOUN
ejpam-6429	367	22	,	,	PUNCT
ejpam-6429	367	23	say	say	VERB
ejpam-6429	367	24	y1	y1	NOUN
ejpam-6429	367	25	,	,	PUNCT
ejpam-6429	367	26	such	such	ADJ
ejpam-6429	367	27	that	that	DET
ejpam-6429	367	28	f(y1	f(y1	NOUN
ejpam-6429	367	29	)	)	PUNCT
ejpam-6429	367	30	=	=	SYM
ejpam-6429	368	1	0	0	X
ejpam-6429	368	2	.	.	PUNCT
ejpam-6429	369	1	therefore	therefore	ADV
ejpam-6429	369	2	,	,	PUNCT
ejpam-6429	369	3	x1	x1	PROPN
ejpam-6429	369	4	is	be	AUX
ejpam-6429	369	5	a	a	DET
ejpam-6429	369	6	moving	move	VERB
ejpam-6429	369	7	neighbor	neighbor	NOUN
ejpam-6429	369	8	of	of	ADP
ejpam-6429	369	9	y1	y1	PROPN
ejpam-6429	369	10	,	,	PUNCT
ejpam-6429	369	11	and	and	CCONJ
ejpam-6429	369	12	thus	thus	ADV
ejpam-6429	369	13	f(x1	f(x1	ADJ
ejpam-6429	369	14	)	)	PUNCT
ejpam-6429	369	15	≥	≥	NOUN
ejpam-6429	369	16	2	2	NUM
ejpam-6429	369	17	.	.	PUNCT
ejpam-6429	369	18	by	by	ADP
ejpam-6429	369	19	reassigning	reassign	VERB
ejpam-6429	369	20	the	the	DET
ejpam-6429	369	21	value	value	NOUN
ejpam-6429	369	22	0	0	NUM
ejpam-6429	369	23	to	to	ADP
ejpam-6429	369	24	x	x	SYM
ejpam-6429	369	25	,	,	PUNCT
ejpam-6429	369	26	we	we	PRON
ejpam-6429	369	27	obtain	obtain	VERB
ejpam-6429	369	28	a	a	DET
ejpam-6429	369	29	grd	grd	NOUN
ejpam-6429	369	30	-	-	PUNCT
ejpam-6429	369	31	function	function	NOUN
ejpam-6429	369	32	on	on	ADP
ejpam-6429	369	33	g	g	PROPN
ejpam-6429	369	34	with	with	ADP
ejpam-6429	369	35	weight	weight	NOUN
ejpam-6429	369	36	less	less	ADJ
ejpam-6429	369	37	than	than	ADP
ejpam-6429	369	38	ωg	ωg	ADP
ejpam-6429	369	39	r(f	r(f	PROPN
ejpam-6429	369	40	)	)	PUNCT
ejpam-6429	369	41	.	.	PUNCT
ejpam-6429	370	1	finally	finally	ADV
ejpam-6429	370	2	,	,	PUNCT
ejpam-6429	370	3	assume	assume	VERB
ejpam-6429	370	4	that	that	SCONJ
ejpam-6429	370	5	f(x	f(x	PROPN
ejpam-6429	370	6	)	)	PUNCT
ejpam-6429	370	7	=	=	PUNCT
ejpam-6429	371	1	0	0	X
ejpam-6429	371	2	.	.	PUNCT
ejpam-6429	372	1	in	in	ADP
ejpam-6429	372	2	this	this	DET
ejpam-6429	372	3	case	case	NOUN
ejpam-6429	372	4	,	,	PUNCT
ejpam-6429	372	5	one	one	NUM
ejpam-6429	372	6	of	of	ADP
ejpam-6429	372	7	the	the	DET
ejpam-6429	372	8	vertices	vertex	NOUN
ejpam-6429	372	9	yi	yi	PROPN
ejpam-6429	372	10	,	,	PUNCT
ejpam-6429	372	11	say	say	VERB
ejpam-6429	372	12	y1	y1	NOUN
ejpam-6429	372	13	,	,	PUNCT
ejpam-6429	372	14	must	must	AUX
ejpam-6429	372	15	be	be	AUX
ejpam-6429	372	16	a	a	DET
ejpam-6429	372	17	moving	move	VERB
ejpam-6429	372	18	neighbor	neighbor	NOUN
ejpam-6429	372	19	of	of	ADP
ejpam-6429	372	20	x	x	SYM
ejpam-6429	372	21	,	,	PUNCT
ejpam-6429	372	22	so	so	ADV
ejpam-6429	372	23	f(y1	f(y1	NOUN
ejpam-6429	372	24	)	)	PUNCT
ejpam-6429	372	25	≥	≥	NOUN
ejpam-6429	372	26	2	2	NUM
ejpam-6429	372	27	.	.	PUNCT
ejpam-6429	372	28	from	from	ADP
ejpam-6429	372	29	∑s	∑s	PROPN
ejpam-6429	372	30	i=1	i=1	PROPN
ejpam-6429	372	31	f(yi	f(yi	PROPN
ejpam-6429	372	32	)	)	PUNCT
ejpam-6429	373	1	+	+	CCONJ
ejpam-6429	373	2	f(x	f(x	PROPN
ejpam-6429	373	3	)	)	PUNCT
ejpam-6429	373	4	≤	≤	NOUN
ejpam-6429	373	5	3	3	NUM
ejpam-6429	373	6	and	and	CCONJ
ejpam-6429	373	7	the	the	DET
ejpam-6429	373	8	assumption	assumption	NOUN
ejpam-6429	373	9	that	that	SCONJ
ejpam-6429	373	10	f(yi	f(yi	AUX
ejpam-6429	373	11	)	)	PUNCT
ejpam-6429	373	12	̸=	̸=	PROPN
ejpam-6429	373	13	1	1	NUM
ejpam-6429	373	14	for	for	ADP
ejpam-6429	373	15	each	each	DET
ejpam-6429	373	16	i	i	PRON
ejpam-6429	373	17	,	,	PUNCT
ejpam-6429	373	18	we	we	PRON
ejpam-6429	373	19	conclude	conclude	VERB
ejpam-6429	373	20	that	that	PRON
ejpam-6429	373	21	f(yi	f(yi	NOUN
ejpam-6429	373	22	)	)	PUNCT
ejpam-6429	373	23	=	=	SYM
ejpam-6429	373	24	0	0	NUM
ejpam-6429	373	25	for	for	ADP
ejpam-6429	373	26	each	each	DET
ejpam-6429	373	27	i	i	PRON
ejpam-6429	373	28	∈	∈	PROPN
ejpam-6429	373	29	{	{	PUNCT
ejpam-6429	373	30	2	2	NUM
ejpam-6429	373	31	,	,	PUNCT
ejpam-6429	373	32	.	.	PUNCT
ejpam-6429	373	33	.	.	PUNCT
ejpam-6429	373	34	.	.	PUNCT
ejpam-6429	374	1	,	,	PUNCT
ejpam-6429	374	2	s	s	X
ejpam-6429	374	3	}	}	PUNCT
ejpam-6429	374	4	.	.	PUNCT
ejpam-6429	375	1	therefore	therefore	ADV
ejpam-6429	375	2	,	,	PUNCT
ejpam-6429	375	3	f(xi	f(xi	PROPN
ejpam-6429	375	4	)	)	PUNCT
ejpam-6429	375	5	≥	≥	NOUN
ejpam-6429	375	6	2	2	NUM
ejpam-6429	375	7	,	,	PUNCT
ejpam-6429	375	8	and	and	CCONJ
ejpam-6429	375	9	each	each	DET
ejpam-6429	375	10	xi	xi	ADP
ejpam-6429	375	11	is	be	AUX
ejpam-6429	375	12	a	a	DET
ejpam-6429	375	13	moving	move	VERB
ejpam-6429	375	14	neighbor	neighbor	NOUN
ejpam-6429	375	15	of	of	ADP
ejpam-6429	375	16	yi	yi	PROPN
ejpam-6429	375	17	for	for	ADP
ejpam-6429	375	18	i	i	PRON
ejpam-6429	375	19	∈	∈	PROPN
ejpam-6429	375	20	{	{	PUNCT
ejpam-6429	375	21	2	2	NUM
ejpam-6429	375	22	,	,	PUNCT
ejpam-6429	375	23	.	.	PUNCT
ejpam-6429	375	24	.	.	PUNCT
ejpam-6429	376	1	.	.	PUNCT
ejpam-6429	377	1	,	,	PUNCT
ejpam-6429	377	2	s	s	X
ejpam-6429	377	3	}	}	PUNCT
ejpam-6429	377	4	.	.	PUNCT
ejpam-6429	378	1	now	now	ADV
ejpam-6429	378	2	,	,	PUNCT
ejpam-6429	378	3	by	by	ADP
ejpam-6429	378	4	reassigning	reassign	VERB
ejpam-6429	378	5	the	the	DET
ejpam-6429	378	6	value	value	NOUN
ejpam-6429	378	7	min{3	min{3	PROPN
ejpam-6429	378	8	,	,	PUNCT
ejpam-6429	378	9	f(x1	f(x1	NOUN
ejpam-6429	378	10	)	)	PUNCT
ejpam-6429	378	11	+	+	CCONJ
ejpam-6429	378	12	1	1	X
ejpam-6429	378	13	}	}	PUNCT
ejpam-6429	378	14	to	to	ADP
ejpam-6429	378	15	x1	x1	PROPN
ejpam-6429	378	16	,	,	PUNCT
ejpam-6429	378	17	we	we	PRON
ejpam-6429	378	18	get	get	VERB
ejpam-6429	378	19	a	a	DET
ejpam-6429	378	20	grd	grd	NOUN
ejpam-6429	378	21	-	-	PUNCT
ejpam-6429	378	22	function	function	NOUN
ejpam-6429	378	23	on	on	ADP
ejpam-6429	378	24	g	g	PROPN
ejpam-6429	378	25	with	with	ADP
ejpam-6429	378	26	weight	weight	NOUN
ejpam-6429	378	27	less	less	ADJ
ejpam-6429	378	28	than	than	ADP
ejpam-6429	378	29	ωg	ωg	ADP
ejpam-6429	378	30	r(f	r(f	PROPN
ejpam-6429	378	31	)	)	PUNCT
ejpam-6429	378	32	,	,	PUNCT
ejpam-6429	378	33	completing	complete	VERB
ejpam-6429	378	34	the	the	DET
ejpam-6429	378	35	proof	proof	NOUN
ejpam-6429	378	36	.	.	PUNCT
ejpam-6429	379	1	as	as	ADP
ejpam-6429	379	2	a	a	DET
ejpam-6429	379	3	result	result	NOUN
ejpam-6429	379	4	of	of	ADP
ejpam-6429	379	5	corollary	corollary	ADJ
ejpam-6429	379	6	1	1	NUM
ejpam-6429	379	7	and	and	CCONJ
ejpam-6429	379	8	theorem	theorem	VERB
ejpam-6429	379	9	3	3	NUM
ejpam-6429	379	10	,	,	PUNCT
ejpam-6429	379	11	sdγgr(g	sdγgr(g	PROPN
ejpam-6429	379	12	)	)	PUNCT
ejpam-6429	379	13	is	be	AUX
ejpam-6429	379	14	well	well	ADV
ejpam-6429	379	15	-	-	PUNCT
ejpam-6429	379	16	defined	define	VERB
ejpam-6429	379	17	for	for	ADP
ejpam-6429	379	18	any	any	DET
ejpam-6429	379	19	connected	connected	ADJ
ejpam-6429	379	20	graph	graph	NOUN
ejpam-6429	379	21	g	g	NOUN
ejpam-6429	379	22	of	of	ADP
ejpam-6429	379	23	order	order	NOUN
ejpam-6429	379	24	n	n	PRON
ejpam-6429	379	25	≥	≥	NOUN
ejpam-6429	379	26	2	2	NUM
ejpam-6429	379	27	.	.	PUNCT
ejpam-6429	380	1	moreover	moreover	ADV
ejpam-6429	380	2	,	,	PUNCT
ejpam-6429	380	3	we	we	PRON
ejpam-6429	380	4	derive	derive	VERB
ejpam-6429	380	5	the	the	DET
ejpam-6429	380	6	following	follow	VERB
ejpam-6429	380	7	result	result	NOUN
ejpam-6429	380	8	.	.	PUNCT
ejpam-6429	381	1	corollary	corollary	ADJ
ejpam-6429	381	2	5	5	NUM
ejpam-6429	381	3	.	.	PUNCT
ejpam-6429	382	1	if	if	SCONJ
ejpam-6429	382	2	g	g	PROPN
ejpam-6429	382	3	is	be	AUX
ejpam-6429	382	4	a	a	DET
ejpam-6429	382	5	connected	connected	ADJ
ejpam-6429	382	6	graph	graph	NOUN
ejpam-6429	382	7	with	with	ADP
ejpam-6429	382	8	δ	δ	PROPN
ejpam-6429	382	9	≥	≥	NUM
ejpam-6429	382	10	2	2	NUM
ejpam-6429	382	11	,	,	PUNCT
ejpam-6429	382	12	then	then	ADV
ejpam-6429	382	13	sdγgr(g	sdγgr(g	PROPN
ejpam-6429	382	14	)	)	PUNCT
ejpam-6429	382	15	≤	≤	PROPN
ejpam-6429	382	16	δ	δ	PROPN
ejpam-6429	382	17	.	.	PUNCT
ejpam-6429	383	1	corollary	corollary	ADJ
ejpam-6429	383	2	6	6	NUM
ejpam-6429	383	3	.	.	PUNCT
ejpam-6429	384	1	if	if	SCONJ
ejpam-6429	384	2	g	g	PROPN
ejpam-6429	384	3	is	be	AUX
ejpam-6429	384	4	a	a	DET
ejpam-6429	384	5	connected	connected	ADJ
ejpam-6429	384	6	graph	graph	NOUN
ejpam-6429	384	7	with	with	ADP
ejpam-6429	384	8	δ	δ	PROPN
ejpam-6429	384	9	=	=	SYM
ejpam-6429	384	10	1	1	NUM
ejpam-6429	384	11	,	,	PUNCT
ejpam-6429	384	12	then	then	ADV
ejpam-6429	384	13	sdγgr(g	sdγgr(g	PROPN
ejpam-6429	384	14	)	)	PUNCT
ejpam-6429	384	15	≤	≤	NOUN
ejpam-6429	384	16	3	3	NUM
ejpam-6429	384	17	.	.	PUNCT
ejpam-6429	385	1	proof	proof	NOUN
ejpam-6429	385	2	.	.	PUNCT
ejpam-6429	386	1	if	if	SCONJ
ejpam-6429	386	2	g	g	PROPN
ejpam-6429	386	3	is	be	AUX
ejpam-6429	386	4	a	a	DET
ejpam-6429	386	5	star	star	NOUN
ejpam-6429	386	6	,	,	PUNCT
ejpam-6429	386	7	then	then	ADV
ejpam-6429	386	8	the	the	DET
ejpam-6429	386	9	result	result	NOUN
ejpam-6429	386	10	follows	follow	VERB
ejpam-6429	386	11	directly	directly	ADV
ejpam-6429	386	12	from	from	ADP
ejpam-6429	386	13	corollary	corollary	ADJ
ejpam-6429	386	14	3	3	NUM
ejpam-6429	386	15	.	.	PUNCT
ejpam-6429	386	16	suppose	suppose	VERB
ejpam-6429	386	17	that	that	SCONJ
ejpam-6429	386	18	g	g	PROPN
ejpam-6429	386	19	is	be	AUX
ejpam-6429	386	20	not	not	PART
ejpam-6429	386	21	a	a	DET
ejpam-6429	386	22	star	star	NOUN
ejpam-6429	386	23	.	.	PUNCT
ejpam-6429	387	1	let	let	VERB
ejpam-6429	387	2	v	v	NUM
ejpam-6429	387	3	∈	∈	PROPN
ejpam-6429	387	4	v	v	NOUN
ejpam-6429	387	5	(	(	PUNCT
ejpam-6429	387	6	g	g	NOUN
ejpam-6429	387	7	)	)	PUNCT
ejpam-6429	387	8	be	be	AUX
ejpam-6429	387	9	a	a	DET
ejpam-6429	387	10	support	support	NOUN
ejpam-6429	387	11	vertex	vertex	NOUN
ejpam-6429	387	12	,	,	PUNCT
ejpam-6429	387	13	and	and	CCONJ
ejpam-6429	387	14	let	let	VERB
ejpam-6429	387	15	v1	v1	NOUN
ejpam-6429	387	16	be	be	AUX
ejpam-6429	387	17	a	a	DET
ejpam-6429	387	18	leaf	leaf	NOUN
ejpam-6429	387	19	adjacent	adjacent	ADJ
ejpam-6429	387	20	to	to	ADP
ejpam-6429	387	21	v.	v.	VERB
ejpam-6429	387	22	if	if	SCONJ
ejpam-6429	387	23	deg(v	deg(v	PROPN
ejpam-6429	387	24	)	)	PUNCT
ejpam-6429	387	25	=	=	SYM
ejpam-6429	387	26	2	2	NUM
ejpam-6429	387	27	,	,	PUNCT
ejpam-6429	387	28	then	then	ADV
ejpam-6429	387	29	,	,	PUNCT
ejpam-6429	387	30	as	as	SCONJ
ejpam-6429	387	31	shown	show	VERB
ejpam-6429	387	32	in	in	ADP
ejpam-6429	387	33	the	the	DET
ejpam-6429	387	34	proof	proof	NOUN
ejpam-6429	387	35	of	of	ADP
ejpam-6429	387	36	proposition	proposition	NOUN
ejpam-6429	387	37	10	10	NUM
ejpam-6429	387	38	,	,	PUNCT
ejpam-6429	387	39	we	we	PRON
ejpam-6429	387	40	obtain	obtain	VERB
ejpam-6429	387	41	sdγgr(g	sdγgr(g	NOUN
ejpam-6429	387	42	)	)	PUNCT
ejpam-6429	387	43	≤	≤	NOUN
ejpam-6429	387	44	3	3	NUM
ejpam-6429	387	45	.	.	PUNCT
ejpam-6429	388	1	thus	thus	ADV
ejpam-6429	388	2	,	,	PUNCT
ejpam-6429	388	3	assume	assume	VERB
ejpam-6429	388	4	that	that	SCONJ
ejpam-6429	388	5	deg(v	deg(v	PROPN
ejpam-6429	388	6	)	)	PUNCT
ejpam-6429	388	7	≥	≥	NOUN
ejpam-6429	388	8	3	3	NUM
ejpam-6429	388	9	and	and	CCONJ
ejpam-6429	388	10	consider	consider	VERB
ejpam-6429	388	11	two	two	NUM
ejpam-6429	388	12	neighbors	neighbor	NOUN
ejpam-6429	388	13	v2	v2	PROPN
ejpam-6429	388	14	,	,	PUNCT
ejpam-6429	388	15	v3	v3	PROPN
ejpam-6429	388	16	∈	∈	PROPN
ejpam-6429	388	17	n(v	n(v	PROPN
ejpam-6429	388	18	)	)	PUNCT
ejpam-6429	388	19	\	\	PROPN
ejpam-6429	388	20	{	{	PUNCT
ejpam-6429	388	21	v1	v1	NOUN
ejpam-6429	388	22	}	}	PUNCT
ejpam-6429	388	23	.	.	PUNCT
ejpam-6429	389	1	let	let	VERB
ejpam-6429	389	2	g1	g1	PROPN
ejpam-6429	389	3	be	be	AUX
ejpam-6429	389	4	the	the	DET
ejpam-6429	389	5	graph	graph	NOUN
ejpam-6429	389	6	obtained	obtain	VERB
ejpam-6429	389	7	from	from	ADP
ejpam-6429	389	8	g	g	NOUN
ejpam-6429	389	9	by	by	ADP
ejpam-6429	389	10	subdividing	subdivide	VERB
ejpam-6429	389	11	the	the	DET
ejpam-6429	389	12	edges	edge	NOUN
ejpam-6429	389	13	vv1	vv1	NOUN
ejpam-6429	389	14	,	,	PUNCT
ejpam-6429	389	15	vv2	vv2	PROPN
ejpam-6429	389	16	,	,	PUNCT
ejpam-6429	389	17	vv3	vv3	ADP
ejpam-6429	389	18	with	with	ADP
ejpam-6429	389	19	new	new	ADJ
ejpam-6429	389	20	vertices	vertex	NOUN
ejpam-6429	389	21	x1	x1	PROPN
ejpam-6429	389	22	,	,	PUNCT
ejpam-6429	389	23	x2	x2	PROPN
ejpam-6429	389	24	,	,	PUNCT
ejpam-6429	389	25	x3	x3	ADJ
ejpam-6429	389	26	,	,	PUNCT
ejpam-6429	389	27	and	and	CCONJ
ejpam-6429	389	28	let	let	VERB
ejpam-6429	389	29	f	f	PRON
ejpam-6429	389	30	be	be	AUX
ejpam-6429	389	31	a	a	DET
ejpam-6429	389	32	γgr(g1)-function	γgr(g1)-function	NOUN
ejpam-6429	389	33	such	such	ADJ
ejpam-6429	389	34	that	that	SCONJ
ejpam-6429	389	35	f(v	f(v	NOUN
ejpam-6429	389	36	)	)	PUNCT
ejpam-6429	389	37	is	be	AUX
ejpam-6429	389	38	maximized	maximize	VERB
ejpam-6429	389	39	.	.	PUNCT
ejpam-6429	390	1	by	by	ADP
ejpam-6429	390	2	proposition	proposition	NOUN
ejpam-6429	390	3	2	2	NUM
ejpam-6429	390	4	,	,	PUNCT
ejpam-6429	390	5	we	we	PRON
ejpam-6429	390	6	may	may	AUX
ejpam-6429	390	7	assume	assume	VERB
ejpam-6429	390	8	that	that	SCONJ
ejpam-6429	390	9	f(xi	f(xi	PROPN
ejpam-6429	390	10	)	)	PUNCT
ejpam-6429	390	11	/∈	/∈	PUNCT
ejpam-6429	391	1	{	{	PUNCT
ejpam-6429	391	2	1	1	NUM
ejpam-6429	391	3	,	,	PUNCT
ejpam-6429	391	4	3	3	NUM
ejpam-6429	391	5	}	}	PUNCT
ejpam-6429	391	6	for	for	ADP
ejpam-6429	391	7	all	all	PRON
ejpam-6429	391	8	i	i	PRON
ejpam-6429	391	9	∈	∈	PROPN
ejpam-6429	391	10	{	{	PUNCT
ejpam-6429	391	11	1	1	NUM
ejpam-6429	391	12	,	,	PUNCT
ejpam-6429	391	13	2	2	NUM
ejpam-6429	391	14	,	,	PUNCT
ejpam-6429	391	15	3	3	NUM
ejpam-6429	391	16	}	}	PUNCT
ejpam-6429	391	17	.	.	PUNCT
ejpam-6429	392	1	if	if	SCONJ
ejpam-6429	392	2	f(v	f(v	NOUN
ejpam-6429	392	3	)	)	PUNCT
ejpam-6429	392	4	+	+	SYM
ejpam-6429	392	5	f(v1	f(v1	NOUN
ejpam-6429	392	6	)	)	PUNCT
ejpam-6429	392	7	+	+	CCONJ
ejpam-6429	392	8	∑3	∑3	PROPN
ejpam-6429	392	9	i=1	i=1	PROPN
ejpam-6429	392	10	f(xi	f(xi	PROPN
ejpam-6429	392	11	)	)	PUNCT
ejpam-6429	392	12	≥	≥	NOUN
ejpam-6429	392	13	4	4	NUM
ejpam-6429	392	14	,	,	PUNCT
ejpam-6429	392	15	then	then	ADV
ejpam-6429	392	16	assigning	assign	VERB
ejpam-6429	392	17	the	the	DET
ejpam-6429	392	18	value	value	NOUN
ejpam-6429	392	19	3	3	NUM
ejpam-6429	392	20	to	to	PART
ejpam-6429	392	21	v	v	NOUN
ejpam-6429	392	22	produces	produce	VERB
ejpam-6429	392	23	a	a	DET
ejpam-6429	392	24	grd	grd	NOUN
ejpam-6429	392	25	-	-	PUNCT
ejpam-6429	392	26	function	function	NOUN
ejpam-6429	392	27	of	of	ADP
ejpam-6429	392	28	g	g	NOUN
ejpam-6429	392	29	with	with	ADP
ejpam-6429	392	30	a	a	DET
ejpam-6429	392	31	weight	weight	NOUN
ejpam-6429	392	32	smaller	small	ADJ
ejpam-6429	392	33	than	than	ADP
ejpam-6429	392	34	ωg	ωg	ADP
ejpam-6429	392	35	r(f	r(f	PROPN
ejpam-6429	392	36	)	)	PUNCT
ejpam-6429	392	37	.	.	PUNCT
ejpam-6429	393	1	now	now	ADV
ejpam-6429	393	2	,	,	PUNCT
ejpam-6429	393	3	assume	assume	VERB
ejpam-6429	393	4	that	that	SCONJ
ejpam-6429	393	5	f(v	f(v	NOUN
ejpam-6429	393	6	)	)	PUNCT
ejpam-6429	394	1	+	+	SYM
ejpam-6429	394	2	f(v1	f(v1	NOUN
ejpam-6429	394	3	)	)	PUNCT
ejpam-6429	394	4	+	+	CCONJ
ejpam-6429	394	5	∑3	∑3	PROPN
ejpam-6429	394	6	i=1	i=1	PROPN
ejpam-6429	394	7	f(xi	f(xi	PROPN
ejpam-6429	394	8	)	)	PUNCT
ejpam-6429	394	9	≤	≤	NOUN
ejpam-6429	394	10	3	3	NUM
ejpam-6429	394	11	.	.	PUNCT
ejpam-6429	395	1	this	this	PRON
ejpam-6429	395	2	implies	imply	VERB
ejpam-6429	395	3	that	that	SCONJ
ejpam-6429	395	4	f(v	f(v	NOUN
ejpam-6429	395	5	)	)	PUNCT
ejpam-6429	395	6	∈	∈	PROPN
ejpam-6429	395	7	{	{	PUNCT
ejpam-6429	395	8	0	0	NUM
ejpam-6429	395	9	,	,	PUNCT
ejpam-6429	395	10	1	1	NUM
ejpam-6429	395	11	,	,	PUNCT
ejpam-6429	395	12	2	2	NUM
ejpam-6429	395	13	}	}	PUNCT
ejpam-6429	395	14	.	.	PUNCT
ejpam-6429	396	1	if	if	SCONJ
ejpam-6429	396	2	f(v	f(v	NOUN
ejpam-6429	396	3	)	)	PUNCT
ejpam-6429	396	4	≤	≤	NUM
ejpam-6429	396	5	1	1	NUM
ejpam-6429	396	6	,	,	PUNCT
ejpam-6429	396	7	then	then	ADV
ejpam-6429	396	8	we	we	PRON
ejpam-6429	396	9	must	must	AUX
ejpam-6429	396	10	have	have	VERB
ejpam-6429	396	11	f(x1	f(x1	ADJ
ejpam-6429	396	12	)	)	PUNCT
ejpam-6429	396	13	+	+	SYM
ejpam-6429	396	14	f(v1	f(v1	NOUN
ejpam-6429	396	15	)	)	PUNCT
ejpam-6429	396	16	=	=	SYM
ejpam-6429	396	17	2	2	NUM
ejpam-6429	396	18	and	and	CCONJ
ejpam-6429	396	19	f(x2	f(x2	NOUN
ejpam-6429	396	20	)	)	PUNCT
ejpam-6429	396	21	=	=	PUNCT
ejpam-6429	397	1	f(x3	f(x3	X
ejpam-6429	397	2	)	)	PUNCT
ejpam-6429	398	1	=	=	SYM
ejpam-6429	398	2	0	0	X
ejpam-6429	398	3	.	.	PUNCT
ejpam-6429	399	1	since	since	SCONJ
ejpam-6429	399	2	we	we	PRON
ejpam-6429	399	3	previously	previously	ADV
ejpam-6429	399	4	assumed	assume	VERB
ejpam-6429	399	5	that	that	SCONJ
ejpam-6429	399	6	f(yi	f(yi	NOUN
ejpam-6429	399	7	)	)	PUNCT
ejpam-6429	399	8	̸=	̸=	PROPN
ejpam-6429	399	9	1	1	NUM
ejpam-6429	399	10	for	for	ADP
ejpam-6429	399	11	all	all	DET
ejpam-6429	399	12	i	i	PRON
ejpam-6429	399	13	,	,	PUNCT
ejpam-6429	399	14	the	the	DET
ejpam-6429	399	15	function	function	NOUN
ejpam-6429	399	16	g	g	PROPN
ejpam-6429	399	17	defined	define	VERB
ejpam-6429	399	18	on	on	ADP
ejpam-6429	399	19	g	g	NOUN
ejpam-6429	399	20	by	by	ADP
ejpam-6429	399	21	g(v1	g(v1	NOUN
ejpam-6429	399	22	)	)	PUNCT
ejpam-6429	399	23	=	=	SYM
ejpam-6429	399	24	1	1	NUM
ejpam-6429	399	25	,	,	PUNCT
ejpam-6429	399	26	g(vi	g(vi	NUM
ejpam-6429	399	27	)	)	PUNCT
ejpam-6429	399	28	=	=	SYM
ejpam-6429	399	29	min{3	min{3	PROPN
ejpam-6429	399	30	,	,	PUNCT
ejpam-6429	399	31	f(vi	f(vi	PROPN
ejpam-6429	399	32	)	)	PUNCT
ejpam-6429	399	33	+	+	NUM
ejpam-6429	399	34	f(xi	f(xi	NUM
ejpam-6429	399	35	)	)	PUNCT
ejpam-6429	399	36	}	}	PUNCT
ejpam-6429	399	37	for	for	ADP
ejpam-6429	399	38	i	i	PROPN
ejpam-6429	399	39	=	=	SYM
ejpam-6429	399	40	2	2	NUM
ejpam-6429	399	41	,	,	PUNCT
ejpam-6429	399	42	3	3	NUM
ejpam-6429	399	43	,	,	PUNCT
ejpam-6429	399	44	and	and	CCONJ
ejpam-6429	399	45	g(x	g(x	NOUN
ejpam-6429	399	46	)	)	PUNCT
ejpam-6429	399	47	=	=	SYM
ejpam-6429	399	48	f(x	f(x	PROPN
ejpam-6429	399	49	)	)	PUNCT
ejpam-6429	399	50	for	for	ADP
ejpam-6429	399	51	all	all	DET
ejpam-6429	399	52	other	other	ADJ
ejpam-6429	399	53	vertices	vertex	NOUN
ejpam-6429	399	54	x	x	X
ejpam-6429	399	55	,	,	PUNCT
ejpam-6429	399	56	is	be	AUX
ejpam-6429	399	57	a	a	DET
ejpam-6429	399	58	grd	grd	NOUN
ejpam-6429	399	59	-	-	PUNCT
ejpam-6429	399	60	function	function	NOUN
ejpam-6429	399	61	of	of	ADP
ejpam-6429	399	62	g	g	NOUN
ejpam-6429	399	63	with	with	ADP
ejpam-6429	399	64	a	a	DET
ejpam-6429	399	65	weight	weight	NOUN
ejpam-6429	399	66	smaller	small	ADJ
ejpam-6429	399	67	than	than	ADP
ejpam-6429	399	68	ωg	ωg	ADP
ejpam-6429	399	69	r(f	r(f	PROPN
ejpam-6429	399	70	)	)	PUNCT
ejpam-6429	399	71	.	.	PUNCT
ejpam-6429	400	1	note	note	VERB
ejpam-6429	400	2	that	that	SCONJ
ejpam-6429	400	3	v2	v2	PROPN
ejpam-6429	400	4	is	be	AUX
ejpam-6429	400	5	a	a	DET
ejpam-6429	400	6	moving	move	VERB
ejpam-6429	400	7	neighbor	neighbor	NOUN
ejpam-6429	400	8	of	of	ADP
ejpam-6429	400	9	x2	x2	PROPN
ejpam-6429	400	10	under	under	ADP
ejpam-6429	400	11	f	f	PROPN
ejpam-6429	400	12	,	,	PUNCT
ejpam-6429	400	13	and	and	CCONJ
ejpam-6429	400	14	hence	hence	ADV
ejpam-6429	400	15	,	,	PUNCT
ejpam-6429	400	16	it	it	PRON
ejpam-6429	400	17	is	be	AUX
ejpam-6429	400	18	a	a	DET
ejpam-6429	400	19	moving	move	VERB
ejpam-6429	400	20	neighbor	neighbor	NOUN
ejpam-6429	400	21	of	of	ADP
ejpam-6429	400	22	v	v	NOUN
ejpam-6429	400	23	under	under	ADP
ejpam-6429	400	24	g.	g.	PROPN
ejpam-6429	400	25	next	next	ADV
ejpam-6429	400	26	,	,	PUNCT
ejpam-6429	400	27	assume	assume	VERB
ejpam-6429	400	28	that	that	SCONJ
ejpam-6429	400	29	f(v	f(v	NOUN
ejpam-6429	400	30	)	)	PUNCT
ejpam-6429	400	31	=	=	SYM
ejpam-6429	400	32	2	2	X
ejpam-6429	400	33	.	.	PUNCT
ejpam-6429	400	34	since	since	SCONJ
ejpam-6429	400	35	f(v)+f(v1)+	f(v)+f(v1)+	PROPN
ejpam-6429	400	36	∑3	∑3	PROPN
ejpam-6429	400	37	i=1	i=1	PROPN
ejpam-6429	400	38	f(xi	f(xi	PROPN
ejpam-6429	400	39	)	)	PUNCT
ejpam-6429	400	40	≤	≤	NOUN
ejpam-6429	400	41	3	3	NUM
ejpam-6429	400	42	,	,	PUNCT
ejpam-6429	400	43	it	it	PRON
ejpam-6429	400	44	follows	follow	VERB
ejpam-6429	400	45	that	that	SCONJ
ejpam-6429	400	46	f(v1	f(v1	VERB
ejpam-6429	400	47	)	)	PUNCT
ejpam-6429	400	48	=	=	SYM
ejpam-6429	400	49	1	1	NUM
ejpam-6429	400	50	and	and	CCONJ
ejpam-6429	400	51	f(xi	f(xi	PROPN
ejpam-6429	400	52	)	)	PUNCT
ejpam-6429	401	1	=	=	SYM
ejpam-6429	401	2	0	0	NUM
ejpam-6429	401	3	for	for	ADP
ejpam-6429	401	4	all	all	PRON
ejpam-6429	401	5	i	i	PRON
ejpam-6429	401	6	∈	∈	PROPN
ejpam-6429	401	7	{	{	PUNCT
ejpam-6429	401	8	1	1	NUM
ejpam-6429	401	9	,	,	PUNCT
ejpam-6429	401	10	2	2	NUM
ejpam-6429	401	11	,	,	PUNCT
ejpam-6429	401	12	3	3	NUM
ejpam-6429	401	13	}	}	PUNCT
ejpam-6429	401	14	.	.	PUNCT
ejpam-6429	402	1	in	in	ADP
ejpam-6429	402	2	this	this	DET
ejpam-6429	402	3	case	case	NOUN
ejpam-6429	402	4	,	,	PUNCT
ejpam-6429	402	5	v	v	NOUN
ejpam-6429	402	6	is	be	AUX
ejpam-6429	402	7	the	the	DET
ejpam-6429	402	8	moving	move	VERB
ejpam-6429	402	9	neighbor	neighbor	NOUN
ejpam-6429	402	10	only	only	ADV
ejpam-6429	402	11	of	of	ADP
ejpam-6429	402	12	x1	x1	PROPN
ejpam-6429	402	13	,	,	PUNCT
ejpam-6429	402	14	and	and	CCONJ
ejpam-6429	402	15	reassigning	reassign	VERB
ejpam-6429	402	16	v1	v1	VERB
ejpam-6429	402	17	the	the	DET
ejpam-6429	402	18	value	value	NOUN
ejpam-6429	402	19	0	0	NUM
ejpam-6429	402	20	produces	produce	VERB
ejpam-6429	402	21	a	a	DET
ejpam-6429	402	22	grd	grd	NOUN
ejpam-6429	402	23	-	-	PUNCT
ejpam-6429	402	24	function	function	NOUN
ejpam-6429	402	25	of	of	ADP
ejpam-6429	402	26	g	g	NOUN
ejpam-6429	402	27	with	with	ADP
ejpam-6429	402	28	a	a	DET
ejpam-6429	402	29	weight	weight	NOUN
ejpam-6429	402	30	smaller	small	ADJ
ejpam-6429	402	31	than	than	ADP
ejpam-6429	402	32	ωg	ωg	ADP
ejpam-6429	402	33	r(f	r(f	PROPN
ejpam-6429	402	34	)	)	PUNCT
ejpam-6429	402	35	.	.	PUNCT
ejpam-6429	403	1	thus	thus	ADV
ejpam-6429	403	2	,	,	PUNCT
ejpam-6429	403	3	we	we	PRON
ejpam-6429	403	4	conclude	conclude	VERB
ejpam-6429	403	5	that	that	PRON
ejpam-6429	403	6	sdγgr(g	sdγgr(g	PROPN
ejpam-6429	403	7	)	)	PUNCT
ejpam-6429	403	8	≤	≤	NOUN
ejpam-6429	403	9	3	3	NUM
ejpam-6429	403	10	,	,	PUNCT
ejpam-6429	403	11	completing	complete	VERB
ejpam-6429	403	12	the	the	DET
ejpam-6429	403	13	proof	proof	NOUN
ejpam-6429	403	14	.	.	PUNCT
ejpam-6429	404	1	since	since	SCONJ
ejpam-6429	404	2	every	every	DET
ejpam-6429	404	3	planar	planar	ADJ
ejpam-6429	404	4	graph	graph	NOUN
ejpam-6429	404	5	contains	contain	VERB
ejpam-6429	404	6	at	at	ADP
ejpam-6429	404	7	least	least	ADV
ejpam-6429	404	8	one	one	NUM
ejpam-6429	404	9	vertex	vertex	NOUN
ejpam-6429	404	10	of	of	ADP
ejpam-6429	404	11	degree	degree	NOUN
ejpam-6429	404	12	at	at	ADP
ejpam-6429	404	13	most	most	ADV
ejpam-6429	404	14	five	five	NUM
ejpam-6429	404	15	,	,	PUNCT
ejpam-6429	404	16	the	the	DET
ejpam-6429	404	17	following	following	ADJ
ejpam-6429	404	18	result	result	NOUN
ejpam-6429	404	19	is	be	AUX
ejpam-6429	404	20	an	an	DET
ejpam-6429	404	21	immediate	immediate	ADJ
ejpam-6429	404	22	consequence	consequence	NOUN
ejpam-6429	404	23	of	of	ADP
ejpam-6429	404	24	corollaries	corollary	NOUN
ejpam-6429	404	25	5	5	NUM
ejpam-6429	404	26	and	and	CCONJ
ejpam-6429	404	27	6	6	NUM
ejpam-6429	404	28	.	.	PUNCT
ejpam-6429	404	29	corollary	corollary	ADJ
ejpam-6429	404	30	7	7	NUM
ejpam-6429	404	31	.	.	PUNCT
ejpam-6429	405	1	if	if	SCONJ
ejpam-6429	405	2	g	g	PROPN
ejpam-6429	405	3	is	be	AUX
ejpam-6429	405	4	a	a	DET
ejpam-6429	405	5	planar	planar	ADJ
ejpam-6429	405	6	graph	graph	NOUN
ejpam-6429	405	7	,	,	PUNCT
ejpam-6429	405	8	then	then	ADV
ejpam-6429	405	9	sdγgr(g	sdγgr(g	PROPN
ejpam-6429	405	10	)	)	PUNCT
ejpam-6429	405	11	≤	≤	NOUN
ejpam-6429	405	12	5	5	NUM
ejpam-6429	405	13	.	.	PUNCT
ejpam-6429	406	1	in	in	ADP
ejpam-6429	406	2	the	the	DET
ejpam-6429	406	3	following	following	NOUN
ejpam-6429	406	4	,	,	PUNCT
ejpam-6429	406	5	we	we	PRON
ejpam-6429	406	6	establish	establish	VERB
ejpam-6429	406	7	an	an	DET
ejpam-6429	406	8	upper	upper	ADJ
ejpam-6429	406	9	bound	bind	VERB
ejpam-6429	406	10	on	on	ADP
ejpam-6429	406	11	the	the	DET
ejpam-6429	406	12	generous	generous	ADJ
ejpam-6429	406	13	roman	roman	ADJ
ejpam-6429	406	14	domination	domination	NOUN
ejpam-6429	406	15	number	number	NOUN
ejpam-6429	406	16	of	of	ADP
ejpam-6429	406	17	a	a	DET
ejpam-6429	406	18	graph	graph	NOUN
ejpam-6429	406	19	based	base	VERB
ejpam-6429	406	20	on	on	ADP
ejpam-6429	406	21	the	the	DET
ejpam-6429	406	22	number	number	NOUN
ejpam-6429	406	23	of	of	ADP
ejpam-6429	406	24	vertices	vertex	NOUN
ejpam-6429	406	25	that	that	PRON
ejpam-6429	406	26	are	be	AUX
ejpam-6429	406	27	at	at	ADP
ejpam-6429	406	28	a	a	DET
ejpam-6429	406	29	distance	distance	NOUN
ejpam-6429	406	30	of	of	ADP
ejpam-6429	406	31	2	2	NUM
ejpam-6429	406	32	from	from	ADP
ejpam-6429	406	33	a	a	DET
ejpam-6429	406	34	given	give	VERB
ejpam-6429	406	35	vertex	vertex	NOUN
ejpam-6429	406	36	.	.	PUNCT
ejpam-6429	407	1	for	for	ADP
ejpam-6429	407	2	a	a	DET
ejpam-6429	407	3	vertex	vertex	NOUN
ejpam-6429	407	4	x	x	SYM
ejpam-6429	407	5	∈	∈	NOUN
ejpam-6429	407	6	v	v	NOUN
ejpam-6429	407	7	(	(	PUNCT
ejpam-6429	407	8	g	g	NOUN
ejpam-6429	407	9	)	)	PUNCT
ejpam-6429	407	10	,	,	PUNCT
ejpam-6429	407	11	let	let	VERB
ejpam-6429	407	12	n2(x	n2(x	PRON
ejpam-6429	407	13	)	)	PUNCT
ejpam-6429	407	14	denote	denote	VERB
ejpam-6429	407	15	the	the	DET
ejpam-6429	407	16	set	set	NOUN
ejpam-6429	407	17	of	of	ADP
ejpam-6429	407	18	vertices	vertex	NOUN
ejpam-6429	407	19	in	in	ADP
ejpam-6429	407	20	g	g	PROPN
ejpam-6429	407	21	that	that	PRON
ejpam-6429	407	22	are	be	AUX
ejpam-6429	407	23	exactly	exactly	ADV
ejpam-6429	407	24	two	two	NUM
ejpam-6429	407	25	edges	edge	NOUN
ejpam-6429	407	26	away	away	ADV
ejpam-6429	407	27	from	from	ADP
ejpam-6429	407	28	x	x	PRON
ejpam-6429	407	29	,	,	PUNCT
ejpam-6429	407	30	and	and	CCONJ
ejpam-6429	407	31	define	define	VERB
ejpam-6429	407	32	d2(x	d2(x	NUM
ejpam-6429	408	1	)	)	PUNCT
ejpam-6429	408	2	=	=	SYM
ejpam-6429	408	3	|n2(x)|	|n2(x)|	PROPN
ejpam-6429	408	4	.	.	PUNCT
ejpam-6429	409	1	in	in	ADP
ejpam-6429	409	2	this	this	DET
ejpam-6429	409	3	setting	setting	NOUN
ejpam-6429	409	4	,	,	PUNCT
ejpam-6429	409	5	we	we	PRON
ejpam-6429	409	6	introduce	introduce	VERB
ejpam-6429	409	7	δ2(g	δ2(g	NOUN
ejpam-6429	409	8	)	)	PUNCT
ejpam-6429	409	9	=	=	SYM
ejpam-6429	409	10	min{d2(x	min{d2(x	NOUN
ejpam-6429	409	11	)	)	PUNCT
ejpam-6429	410	1	|	|	ADV
ejpam-6429	410	2	x	x	SYM
ejpam-6429	410	3	∈	∈	NOUN
ejpam-6429	410	4	v	v	ADP
ejpam-6429	410	5	(	(	PUNCT
ejpam-6429	410	6	g	g	NOUN
ejpam-6429	410	7	)	)	PUNCT
ejpam-6429	410	8	and	and	CCONJ
ejpam-6429	410	9	deg(x	deg(x	X
ejpam-6429	410	10	)	)	PUNCT
ejpam-6429	410	11	≥	≥	NOUN
ejpam-6429	410	12	2	2	NUM
ejpam-6429	410	13	}	}	PUNCT
ejpam-6429	410	14	.	.	PUNCT
ejpam-6429	411	1	j.	j.	PROPN
ejpam-6429	411	2	j.	j.	PROPN
ejpam-6429	411	3	hamja	hamja	PROPN
ejpam-6429	411	4	et	et	PROPN
ejpam-6429	411	5	al	al	PROPN
ejpam-6429	411	6	.	.	PUNCT
ejpam-6429	411	7	/	/	SYM
ejpam-6429	411	8	eur	eur	PROPN
ejpam-6429	411	9	.	.	PUNCT
ejpam-6429	412	1	j.	j.	PROPN
ejpam-6429	412	2	pure	pure	PROPN
ejpam-6429	412	3	appl	appl	PROPN
ejpam-6429	412	4	.	.	PROPN
ejpam-6429	412	5	math	math	PROPN
ejpam-6429	412	6	,	,	PUNCT
ejpam-6429	412	7	18	18	NUM
ejpam-6429	412	8	(	(	PUNCT
ejpam-6429	412	9	4	4	NUM
ejpam-6429	412	10	)	)	PUNCT
ejpam-6429	412	11	(	(	PUNCT
ejpam-6429	412	12	2025	2025	NUM
ejpam-6429	412	13	)	)	PUNCT
ejpam-6429	412	14	,	,	PUNCT
ejpam-6429	412	15	6429	6429	NUM
ejpam-6429	412	16	11	11	NUM
ejpam-6429	412	17	of	of	ADP
ejpam-6429	412	18	16	16	NUM
ejpam-6429	412	19	to	to	PART
ejpam-6429	412	20	establish	establish	VERB
ejpam-6429	412	21	this	this	DET
ejpam-6429	412	22	result	result	NOUN
ejpam-6429	412	23	,	,	PUNCT
ejpam-6429	412	24	we	we	PRON
ejpam-6429	412	25	begin	begin	VERB
ejpam-6429	412	26	with	with	ADP
ejpam-6429	412	27	a	a	DET
ejpam-6429	412	28	few	few	ADJ
ejpam-6429	412	29	lemmas	lemma	NOUN
ejpam-6429	412	30	.	.	PUNCT
ejpam-6429	413	1	the	the	DET
ejpam-6429	413	2	proof	proof	NOUN
ejpam-6429	413	3	of	of	ADP
ejpam-6429	413	4	the	the	DET
ejpam-6429	413	5	following	following	ADJ
ejpam-6429	413	6	lemmas	lemma	NOUN
ejpam-6429	413	7	are	be	AUX
ejpam-6429	413	8	essentially	essentially	ADV
ejpam-6429	413	9	similar	similar	ADJ
ejpam-6429	413	10	to	to	ADP
ejpam-6429	413	11	the	the	DET
ejpam-6429	413	12	proof	proof	NOUN
ejpam-6429	413	13	of	of	ADP
ejpam-6429	413	14	corresponding	corresponding	ADJ
ejpam-6429	413	15	lemmas	lemma	NOUN
ejpam-6429	413	16	in	in	ADP
ejpam-6429	413	17	[	[	X
ejpam-6429	413	18	17	17	NUM
ejpam-6429	413	19	]	]	PUNCT
ejpam-6429	413	20	.	.	PUNCT
ejpam-6429	414	1	lemma	lemma	PROPN
ejpam-6429	414	2	1	1	X
ejpam-6429	414	3	.	.	PUNCT
ejpam-6429	415	1	let	let	VERB
ejpam-6429	415	2	g	g	PRON
ejpam-6429	415	3	be	be	AUX
ejpam-6429	415	4	a	a	DET
ejpam-6429	415	5	connected	connected	ADJ
ejpam-6429	415	6	graph	graph	NOUN
ejpam-6429	415	7	of	of	ADP
ejpam-6429	415	8	order	order	NOUN
ejpam-6429	415	9	n	n	PRON
ejpam-6429	415	10	≥	≥	NOUN
ejpam-6429	415	11	3	3	NUM
ejpam-6429	415	12	.	.	PUNCT
ejpam-6429	416	1	if	if	SCONJ
ejpam-6429	416	2	g	g	PROPN
ejpam-6429	416	3	contains	contain	VERB
ejpam-6429	416	4	a	a	DET
ejpam-6429	416	5	vertex	vertex	NOUN
ejpam-6429	416	6	v1	v1	NOUN
ejpam-6429	416	7	that	that	PRON
ejpam-6429	416	8	is	be	AUX
ejpam-6429	416	9	part	part	NOUN
ejpam-6429	416	10	of	of	ADP
ejpam-6429	416	11	a	a	DET
ejpam-6429	416	12	triangle	triangle	NOUN
ejpam-6429	416	13	v1v2v3v1	v1v2v3v1	NOUN
ejpam-6429	416	14	such	such	ADJ
ejpam-6429	416	15	that	that	SCONJ
ejpam-6429	416	16	n(v2	n(v2	NOUN
ejpam-6429	416	17	)	)	PUNCT
ejpam-6429	416	18	⊆	⊆	NUM
ejpam-6429	416	19	n	n	NUM
ejpam-6429	416	20	[	[	X
ejpam-6429	416	21	v1	v1	NOUN
ejpam-6429	416	22	]	]	PUNCT
ejpam-6429	416	23	and	and	CCONJ
ejpam-6429	416	24	n(v3)−n	n(v3)−n	PROPN
ejpam-6429	417	1	[	[	X
ejpam-6429	417	2	v1	v1	X
ejpam-6429	417	3	]	]	X
ejpam-6429	417	4	̸=	̸=	PROPN
ejpam-6429	417	5	∅	∅	NOUN
ejpam-6429	417	6	,	,	PUNCT
ejpam-6429	417	7	then	then	ADV
ejpam-6429	417	8	sdγgr(g	sdγgr(g	PROPN
ejpam-6429	417	9	)	)	PUNCT
ejpam-6429	417	10	≤	≤	NOUN
ejpam-6429	417	11	3	3	NUM
ejpam-6429	417	12	+	+	SYM
ejpam-6429	417	13	|n(v3)−n	|n(v3)−n	X
ejpam-6429	417	14	[	[	X
ejpam-6429	417	15	v1]|	v1]|	PROPN
ejpam-6429	417	16	.	.	PUNCT
ejpam-6429	417	17	proof	proof	NOUN
ejpam-6429	417	18	.	.	PUNCT
ejpam-6429	418	1	let	let	VERB
ejpam-6429	418	2	w1	w1	NOUN
ejpam-6429	418	3	,	,	PUNCT
ejpam-6429	418	4	w2	w2	NOUN
ejpam-6429	418	5	,	,	PUNCT
ejpam-6429	418	6	.	.	PUNCT
ejpam-6429	418	7	.	.	PUNCT
ejpam-6429	419	1	.	.	PUNCT
ejpam-6429	420	1	,	,	PUNCT
ejpam-6429	420	2	wk	wk	X
ejpam-6429	420	3	be	be	AUX
ejpam-6429	420	4	the	the	DET
ejpam-6429	420	5	neighbors	neighbor	NOUN
ejpam-6429	420	6	of	of	ADP
ejpam-6429	420	7	v3	v3	PROPN
ejpam-6429	420	8	in	in	ADP
ejpam-6429	420	9	v	v	NUM
ejpam-6429	420	10	(	(	PUNCT
ejpam-6429	420	11	g)−n	g)−n	X
ejpam-6429	420	12	[	[	X
ejpam-6429	420	13	v1	v1	NOUN
ejpam-6429	420	14	]	]	PUNCT
ejpam-6429	420	15	,	,	PUNCT
ejpam-6429	420	16	and	and	CCONJ
ejpam-6429	420	17	let	let	VERB
ejpam-6429	420	18	g1	g1	PROPN
ejpam-6429	420	19	be	be	AUX
ejpam-6429	420	20	obtained	obtain	VERB
ejpam-6429	420	21	from	from	ADP
ejpam-6429	420	22	g	g	NOUN
ejpam-6429	420	23	by	by	ADP
ejpam-6429	420	24	subdividing	subdivide	VERB
ejpam-6429	420	25	the	the	DET
ejpam-6429	420	26	edges	edge	NOUN
ejpam-6429	420	27	v1v2	v1v2	VERB
ejpam-6429	420	28	,	,	PUNCT
ejpam-6429	420	29	v1v3	v1v3	NOUN
ejpam-6429	420	30	and	and	CCONJ
ejpam-6429	420	31	v2v3	v2v3	NUM
ejpam-6429	420	32	with	with	ADP
ejpam-6429	420	33	vertices	vertex	NOUN
ejpam-6429	420	34	x	x	X
ejpam-6429	420	35	,	,	PUNCT
ejpam-6429	420	36	y	y	PROPN
ejpam-6429	420	37	and	and	CCONJ
ejpam-6429	420	38	z	z	PROPN
ejpam-6429	420	39	,	,	PUNCT
ejpam-6429	420	40	respectively	respectively	ADV
ejpam-6429	420	41	,	,	PUNCT
ejpam-6429	420	42	and	and	CCONJ
ejpam-6429	420	43	the	the	DET
ejpam-6429	420	44	edge	edge	NOUN
ejpam-6429	420	45	v3wi	v3wi	NUM
ejpam-6429	420	46	with	with	ADP
ejpam-6429	420	47	vertex	vertex	NOUN
ejpam-6429	420	48	xi	xi	NUM
ejpam-6429	420	49	for	for	ADP
ejpam-6429	420	50	each	each	DET
ejpam-6429	420	51	1	1	NUM
ejpam-6429	420	52	≤	≤	NUM
ejpam-6429	421	1	i	i	PRON
ejpam-6429	421	2	≤	≤	PROPN
ejpam-6429	422	1	k.	k.	PROPN
ejpam-6429	422	2	assume	assume	VERB
ejpam-6429	422	3	that	that	SCONJ
ejpam-6429	422	4	f	f	PROPN
ejpam-6429	422	5	is	be	AUX
ejpam-6429	422	6	a	a	DET
ejpam-6429	422	7	γgr(g1)-function	γgr(g1)-function	PROPN
ejpam-6429	422	8	.	.	PUNCT
ejpam-6429	423	1	by	by	ADP
ejpam-6429	423	2	proposition	proposition	NOUN
ejpam-6429	423	3	2	2	NUM
ejpam-6429	423	4	,	,	PUNCT
ejpam-6429	423	5	we	we	PRON
ejpam-6429	423	6	may	may	AUX
ejpam-6429	423	7	assume	assume	VERB
ejpam-6429	423	8	that	that	SCONJ
ejpam-6429	423	9	{	{	PUNCT
ejpam-6429	423	10	1	1	NUM
ejpam-6429	423	11	,	,	PUNCT
ejpam-6429	423	12	3	3	NUM
ejpam-6429	423	13	}	}	PUNCT
ejpam-6429	423	14	∩	∩	NOUN
ejpam-6429	423	15	{	{	PUNCT
ejpam-6429	423	16	f(x	f(x	PROPN
ejpam-6429	423	17	)	)	PUNCT
ejpam-6429	423	18	,	,	PUNCT
ejpam-6429	423	19	f(y	f(y	NOUN
ejpam-6429	423	20	)	)	PUNCT
ejpam-6429	423	21	,	,	PUNCT
ejpam-6429	423	22	f(z	f(z	PROPN
ejpam-6429	423	23	)	)	PUNCT
ejpam-6429	423	24	,	,	PUNCT
ejpam-6429	423	25	f(x1	f(x1	NOUN
ejpam-6429	423	26	)	)	PUNCT
ejpam-6429	423	27	,	,	PUNCT
ejpam-6429	423	28	.	.	PUNCT
ejpam-6429	423	29	.	.	PUNCT
ejpam-6429	424	1	.	.	PUNCT
ejpam-6429	425	1	,	,	PUNCT
ejpam-6429	425	2	f(xk	f(xk	NOUN
ejpam-6429	425	3	)	)	PUNCT
ejpam-6429	425	4	}	}	PUNCT
ejpam-6429	426	1	=	=	PUNCT
ejpam-6429	426	2	∅.	∅.	VERB
ejpam-6429	426	3	similar	similar	ADJ
ejpam-6429	426	4	to	to	ADP
ejpam-6429	426	5	the	the	DET
ejpam-6429	426	6	proof	proof	NOUN
ejpam-6429	426	7	of	of	ADP
ejpam-6429	426	8	proposition	proposition	NOUN
ejpam-6429	426	9	11	11	NUM
ejpam-6429	426	10	,	,	PUNCT
ejpam-6429	426	11	we	we	PRON
ejpam-6429	426	12	can	can	AUX
ejpam-6429	426	13	observe	observe	VERB
ejpam-6429	426	14	that	that	SCONJ
ejpam-6429	426	15	f(x	f(x	PROPN
ejpam-6429	426	16	)	)	PUNCT
ejpam-6429	427	1	+	+	SYM
ejpam-6429	427	2	f(y	f(y	NOUN
ejpam-6429	427	3	)	)	PUNCT
ejpam-6429	428	1	+	+	NUM
ejpam-6429	428	2	f(z	f(z	NOUN
ejpam-6429	428	3	)	)	PUNCT
ejpam-6429	429	1	+	+	SYM
ejpam-6429	429	2	f(v1	f(v1	ADJ
ejpam-6429	429	3	)	)	PUNCT
ejpam-6429	429	4	+	+	SYM
ejpam-6429	429	5	f(v2	f(v2	NOUN
ejpam-6429	429	6	)	)	PUNCT
ejpam-6429	429	7	+	+	SYM
ejpam-6429	429	8	f(v3	f(v3	X
ejpam-6429	429	9	)	)	PUNCT
ejpam-6429	429	10	≥	≥	NOUN
ejpam-6429	429	11	4	4	NUM
ejpam-6429	429	12	.	.	PUNCT
ejpam-6429	429	13	define	define	VERB
ejpam-6429	429	14	a	a	DET
ejpam-6429	429	15	function	function	NOUN
ejpam-6429	429	16	g	g	NOUN
ejpam-6429	429	17	:	:	PUNCT
ejpam-6429	429	18	v	v	NOUN
ejpam-6429	429	19	(	(	PUNCT
ejpam-6429	429	20	g	g	NOUN
ejpam-6429	429	21	)	)	PUNCT
ejpam-6429	429	22	→	→	SYM
ejpam-6429	429	23	{	{	PUNCT
ejpam-6429	429	24	0	0	NUM
ejpam-6429	429	25	,	,	PUNCT
ejpam-6429	429	26	1	1	NUM
ejpam-6429	429	27	,	,	PUNCT
ejpam-6429	429	28	2	2	NUM
ejpam-6429	429	29	,	,	PUNCT
ejpam-6429	429	30	3	3	NUM
ejpam-6429	429	31	}	}	PUNCT
ejpam-6429	429	32	by	by	ADP
ejpam-6429	429	33	g(v1	g(v1	NOUN
ejpam-6429	429	34	)	)	PUNCT
ejpam-6429	429	35	=	=	SYM
ejpam-6429	429	36	3	3	NUM
ejpam-6429	429	37	,	,	PUNCT
ejpam-6429	429	38	g(v2	g(v2	NOUN
ejpam-6429	429	39	)	)	PUNCT
ejpam-6429	429	40	=	=	SYM
ejpam-6429	429	41	g(v3	g(v3	NOUN
ejpam-6429	429	42	)	)	PUNCT
ejpam-6429	429	43	=	=	SYM
ejpam-6429	429	44	0	0	NUM
ejpam-6429	429	45	,	,	PUNCT
ejpam-6429	429	46	g(wi	g(wi	ADJ
ejpam-6429	429	47	)	)	PUNCT
ejpam-6429	429	48	=	=	SYM
ejpam-6429	429	49	min{3	min{3	PROPN
ejpam-6429	429	50	,	,	PUNCT
ejpam-6429	429	51	f(wi	f(wi	NOUN
ejpam-6429	429	52	)	)	PUNCT
ejpam-6429	429	53	+	+	X
ejpam-6429	429	54	f(xi	f(xi	NUM
ejpam-6429	429	55	)	)	PUNCT
ejpam-6429	429	56	}	}	PUNCT
ejpam-6429	429	57	for	for	ADP
ejpam-6429	429	58	each	each	DET
ejpam-6429	429	59	1	1	NUM
ejpam-6429	429	60	≤	≤	NUM
ejpam-6429	429	61	i	i	NOUN
ejpam-6429	429	62	≤	≤	PUNCT
ejpam-6429	430	1	k	k	NOUN
ejpam-6429	430	2	,	,	PUNCT
ejpam-6429	430	3	and	and	CCONJ
ejpam-6429	430	4	g(t	g(t	PROPN
ejpam-6429	430	5	)	)	PUNCT
ejpam-6429	430	6	=	=	SYM
ejpam-6429	430	7	f(t	f(t	PROPN
ejpam-6429	430	8	)	)	PUNCT
ejpam-6429	430	9	for	for	ADP
ejpam-6429	430	10	all	all	DET
ejpam-6429	430	11	t	t	NOUN
ejpam-6429	430	12	∈	∈	PROPN
ejpam-6429	430	13	v	v	ADP
ejpam-6429	430	14	(	(	PUNCT
ejpam-6429	430	15	g	g	NOUN
ejpam-6429	430	16	)	)	PUNCT
ejpam-6429	430	17	\	\	NOUN
ejpam-6429	430	18	{	{	PUNCT
ejpam-6429	430	19	v1	v1	NOUN
ejpam-6429	430	20	,	,	PUNCT
ejpam-6429	430	21	v2	v2	PROPN
ejpam-6429	430	22	,	,	PUNCT
ejpam-6429	430	23	v3	v3	PROPN
ejpam-6429	430	24	,	,	PUNCT
ejpam-6429	430	25	wi	wi	PROPN
ejpam-6429	430	26	|	|	ADV
ejpam-6429	430	27	1	1	NUM
ejpam-6429	430	28	≤	≤	NUM
ejpam-6429	430	29	i	i	NOUN
ejpam-6429	430	30	≤	≤	PUNCT
ejpam-6429	431	1	k	k	X
ejpam-6429	431	2	}	}	PUNCT
ejpam-6429	431	3	.	.	PUNCT
ejpam-6429	432	1	it	it	PRON
ejpam-6429	432	2	is	be	AUX
ejpam-6429	432	3	straightforward	straightforward	ADJ
ejpam-6429	432	4	to	to	PART
ejpam-6429	432	5	verify	verify	VERB
ejpam-6429	432	6	that	that	SCONJ
ejpam-6429	432	7	g	g	PROPN
ejpam-6429	432	8	is	be	AUX
ejpam-6429	432	9	a	a	DET
ejpam-6429	432	10	grd	grd	NOUN
ejpam-6429	432	11	-	-	PUNCT
ejpam-6429	432	12	function	function	NOUN
ejpam-6429	432	13	of	of	ADP
ejpam-6429	432	14	g	g	NOUN
ejpam-6429	432	15	with	with	ADP
ejpam-6429	432	16	a	a	DET
ejpam-6429	432	17	weight	weight	NOUN
ejpam-6429	432	18	smaller	small	ADJ
ejpam-6429	432	19	than	than	ADP
ejpam-6429	432	20	γgr(g1	γgr(g1	ADJ
ejpam-6429	432	21	)	)	PUNCT
ejpam-6429	432	22	,	,	PUNCT
ejpam-6429	432	23	thereby	thereby	ADV
ejpam-6429	432	24	completing	complete	VERB
ejpam-6429	432	25	the	the	DET
ejpam-6429	432	26	proof	proof	NOUN
ejpam-6429	432	27	.	.	PUNCT
ejpam-6429	433	1	lemma	lemma	PROPN
ejpam-6429	433	2	2	2	X
ejpam-6429	433	3	.	.	PUNCT
ejpam-6429	434	1	let	let	VERB
ejpam-6429	434	2	g	g	PRON
ejpam-6429	434	3	be	be	AUX
ejpam-6429	434	4	a	a	DET
ejpam-6429	434	5	connected	connected	ADJ
ejpam-6429	434	6	graph	graph	NOUN
ejpam-6429	434	7	of	of	ADP
ejpam-6429	434	8	order	order	NOUN
ejpam-6429	434	9	n	n	PRON
ejpam-6429	434	10	≥	≥	NOUN
ejpam-6429	434	11	3	3	NUM
ejpam-6429	434	12	,	,	PUNCT
ejpam-6429	434	13	and	and	CCONJ
ejpam-6429	434	14	let	let	VERB
ejpam-6429	434	15	x	x	PRON
ejpam-6429	434	16	be	be	AUX
ejpam-6429	434	17	a	a	DET
ejpam-6429	434	18	vertex	vertex	NOUN
ejpam-6429	434	19	of	of	ADP
ejpam-6429	434	20	degree	degree	NOUN
ejpam-6429	434	21	at	at	ADV
ejpam-6429	434	22	least	least	ADJ
ejpam-6429	434	23	2	2	NUM
ejpam-6429	434	24	in	in	ADP
ejpam-6429	434	25	g	g	PROPN
ejpam-6429	434	26	that	that	PRON
ejpam-6429	434	27	satisfies	satisfy	VERB
ejpam-6429	434	28	the	the	DET
ejpam-6429	434	29	following	follow	VERB
ejpam-6429	434	30	conditions	condition	NOUN
ejpam-6429	434	31	:	:	PUNCT
ejpam-6429	434	32	(	(	PUNCT
ejpam-6429	434	33	i	i	NOUN
ejpam-6429	434	34	)	)	PUNCT
ejpam-6429	434	35	n(y	n(y	PROPN
ejpam-6429	434	36	)	)	PUNCT
ejpam-6429	434	37	\n	\n	PUNCT
ejpam-6429	435	1	[	[	X
ejpam-6429	435	2	x	x	X
ejpam-6429	435	3	]	]	X
ejpam-6429	435	4	̸=	̸=	NOUN
ejpam-6429	435	5	∅	∅	NOUN
ejpam-6429	435	6	for	for	ADP
ejpam-6429	435	7	each	each	DET
ejpam-6429	435	8	y	y	PROPN
ejpam-6429	435	9	∈	∈	PROPN
ejpam-6429	435	10	n(x	n(x	PROPN
ejpam-6429	435	11	)	)	PUNCT
ejpam-6429	435	12	,	,	PUNCT
ejpam-6429	435	13	(	(	PUNCT
ejpam-6429	435	14	ii	ii	NOUN
ejpam-6429	435	15	)	)	PUNCT
ejpam-6429	435	16	there	there	PRON
ejpam-6429	435	17	exist	exist	VERB
ejpam-6429	435	18	vertices	vertex	NOUN
ejpam-6429	435	19	a	a	PRON
ejpam-6429	435	20	,	,	PUNCT
ejpam-6429	435	21	b	b	NOUN
ejpam-6429	435	22	∈	∈	PROPN
ejpam-6429	435	23	n(x	n(x	PROPN
ejpam-6429	435	24	)	)	PUNCT
ejpam-6429	435	25	such	such	ADJ
ejpam-6429	435	26	that	that	SCONJ
ejpam-6429	435	27	(	(	PUNCT
ejpam-6429	435	28	n(a	n(a	NOUN
ejpam-6429	435	29	)	)	PUNCT
ejpam-6429	435	30	∩n(b	∩n(b	NOUN
ejpam-6429	435	31	)	)	PUNCT
ejpam-6429	435	32	)	)	PUNCT
ejpam-6429	435	33	\n	\n	PUNCT
ejpam-6429	436	1	[	[	X
ejpam-6429	436	2	x	x	X
ejpam-6429	436	3	]	]	X
ejpam-6429	436	4	=	=	PUNCT
ejpam-6429	436	5	∅.	∅.	NOUN
ejpam-6429	436	6	then	then	ADV
ejpam-6429	436	7	,	,	PUNCT
ejpam-6429	436	8	sdγgr(g	sdγgr(g	PROPN
ejpam-6429	436	9	)	)	PUNCT
ejpam-6429	436	10	≤	≤	NOUN
ejpam-6429	436	11	3	3	NUM
ejpam-6429	436	12	+	+	NUM
ejpam-6429	436	13	|n2(x)|	|n2(x)|	NOUN
ejpam-6429	436	14	.	.	PUNCT
ejpam-6429	437	1	proof	proof	NOUN
ejpam-6429	437	2	.	.	PUNCT
ejpam-6429	438	1	let	let	VERB
ejpam-6429	438	2	deg(x	deg(x	X
ejpam-6429	438	3	)	)	PUNCT
ejpam-6429	438	4	=	=	SYM
ejpam-6429	438	5	t	t	NOUN
ejpam-6429	438	6	and	and	CCONJ
ejpam-6429	438	7	n(x	n(x	PROPN
ejpam-6429	438	8	)	)	PUNCT
ejpam-6429	438	9	=	=	SYM
ejpam-6429	438	10	{	{	PUNCT
ejpam-6429	438	11	x1	x1	PROPN
ejpam-6429	438	12	,	,	PUNCT
ejpam-6429	438	13	x2	x2	PROPN
ejpam-6429	438	14	,	,	PUNCT
ejpam-6429	438	15	.	.	PUNCT
ejpam-6429	438	16	.	.	PUNCT
ejpam-6429	439	1	.	.	PUNCT
ejpam-6429	440	1	,	,	PUNCT
ejpam-6429	440	2	xt	xt	ADP
ejpam-6429	440	3	}	}	PUNCT
ejpam-6429	440	4	.	.	PUNCT
ejpam-6429	441	1	we	we	PRON
ejpam-6429	441	2	assume	assume	VERB
ejpam-6429	441	3	,	,	PUNCT
ejpam-6429	441	4	without	without	ADP
ejpam-6429	441	5	loss	loss	NOUN
ejpam-6429	441	6	of	of	ADP
ejpam-6429	441	7	generality	generality	NOUN
ejpam-6429	441	8	,	,	PUNCT
ejpam-6429	441	9	that	that	SCONJ
ejpam-6429	441	10	a	a	DET
ejpam-6429	441	11	=	=	SYM
ejpam-6429	441	12	x1	x1	PROPN
ejpam-6429	441	13	and	and	CCONJ
ejpam-6429	441	14	b	b	X
ejpam-6429	441	15	=	=	SYM
ejpam-6429	441	16	x2	x2	PROPN
ejpam-6429	441	17	.	.	PUNCT
ejpam-6429	442	1	additionally	additionally	ADV
ejpam-6429	442	2	,	,	PUNCT
ejpam-6429	442	3	we	we	PRON
ejpam-6429	442	4	assume	assume	VERB
ejpam-6429	442	5	that	that	SCONJ
ejpam-6429	442	6	the	the	DET
ejpam-6429	442	7	pair	pair	NOUN
ejpam-6429	442	8	a	a	PRON
ejpam-6429	442	9	,	,	PUNCT
ejpam-6429	442	10	b	b	PROPN
ejpam-6429	442	11	is	be	AUX
ejpam-6429	442	12	selected	select	VERB
ejpam-6429	442	13	first	first	ADV
ejpam-6429	442	14	among	among	ADP
ejpam-6429	442	15	the	the	DET
ejpam-6429	442	16	adjacent	adjacent	ADJ
ejpam-6429	442	17	vertices	vertex	NOUN
ejpam-6429	442	18	in	in	ADP
ejpam-6429	442	19	n(x	n(x	PROPN
ejpam-6429	442	20	)	)	PUNCT
ejpam-6429	442	21	.	.	PUNCT
ejpam-6429	443	1	therefore	therefore	ADV
ejpam-6429	443	2	,	,	PUNCT
ejpam-6429	443	3	if	if	SCONJ
ejpam-6429	443	4	ab	ab	PROPN
ejpam-6429	443	5	∈	∈	PROPN
ejpam-6429	443	6	e(g	e(g	PROPN
ejpam-6429	443	7	)	)	PUNCT
ejpam-6429	443	8	,	,	PUNCT
ejpam-6429	443	9	then	then	ADV
ejpam-6429	443	10	x	x	PRON
ejpam-6429	443	11	must	must	AUX
ejpam-6429	443	12	be	be	AUX
ejpam-6429	443	13	part	part	NOUN
ejpam-6429	443	14	of	of	ADP
ejpam-6429	443	15	the	the	DET
ejpam-6429	443	16	triangle	triangle	NOUN
ejpam-6429	443	17	xx1x2x	xx1x2x	PROPN
ejpam-6429	443	18	.	.	PUNCT
ejpam-6429	444	1	moreover	moreover	ADV
ejpam-6429	444	2	,	,	PUNCT
ejpam-6429	444	3	define	define	VERB
ejpam-6429	444	4	s	s	NOUN
ejpam-6429	444	5	=	=	PUNCT
ejpam-6429	444	6	{	{	PUNCT
ejpam-6429	444	7	x1	x1	PROPN
ejpam-6429	444	8	,	,	PUNCT
ejpam-6429	444	9	x2	x2	PROPN
ejpam-6429	444	10	,	,	PUNCT
ejpam-6429	444	11	.	.	PUNCT
ejpam-6429	444	12	.	.	PUNCT
ejpam-6429	445	1	.	.	PUNCT
ejpam-6429	446	1	,	,	PUNCT
ejpam-6429	446	2	xs	xs	PROPN
ejpam-6429	446	3	}	}	PUNCT
ejpam-6429	446	4	as	as	ADP
ejpam-6429	446	5	one	one	NUM
ejpam-6429	446	6	of	of	ADP
ejpam-6429	446	7	the	the	DET
ejpam-6429	446	8	largest	large	ADJ
ejpam-6429	446	9	subsets	subset	NOUN
ejpam-6429	446	10	of	of	ADP
ejpam-6429	446	11	n(x	n(x	NOUN
ejpam-6429	446	12	)	)	PUNCT
ejpam-6429	446	13	containing	contain	VERB
ejpam-6429	446	14	x1	x1	PROPN
ejpam-6429	446	15	and	and	CCONJ
ejpam-6429	446	16	x2	x2	PROPN
ejpam-6429	446	17	,	,	PUNCT
ejpam-6429	446	18	where	where	SCONJ
ejpam-6429	446	19	every	every	DET
ejpam-6429	446	20	pair	pair	NOUN
ejpam-6429	446	21	of	of	ADP
ejpam-6429	446	22	vertices	vertex	NOUN
ejpam-6429	446	23	a	a	PRON
ejpam-6429	446	24	,	,	PUNCT
ejpam-6429	446	25	b	b	NOUN
ejpam-6429	446	26	in	in	ADP
ejpam-6429	446	27	s	s	PART
ejpam-6429	446	28	satisfies	satisfie	NOUN
ejpam-6429	446	29	condition	condition	NOUN
ejpam-6429	446	30	(	(	PUNCT
ejpam-6429	446	31	ii	ii	NOUN
ejpam-6429	446	32	)	)	PUNCT
ejpam-6429	446	33	.	.	PUNCT
ejpam-6429	447	1	according	accord	VERB
ejpam-6429	447	2	to	to	ADP
ejpam-6429	447	3	item	item	NOUN
ejpam-6429	447	4	(	(	PUNCT
ejpam-6429	447	5	i	i	NOUN
ejpam-6429	447	6	)	)	PUNCT
ejpam-6429	447	7	,	,	PUNCT
ejpam-6429	447	8	for	for	ADP
ejpam-6429	447	9	each	each	DET
ejpam-6429	447	10	i	i	PRON
ejpam-6429	447	11	∈	∈	PROPN
ejpam-6429	447	12	{	{	PUNCT
ejpam-6429	447	13	1	1	NUM
ejpam-6429	447	14	,	,	PUNCT
ejpam-6429	447	15	2	2	NUM
ejpam-6429	447	16	,	,	PUNCT
ejpam-6429	447	17	.	.	PUNCT
ejpam-6429	447	18	.	.	PUNCT
ejpam-6429	447	19	.	.	PUNCT
ejpam-6429	448	1	,	,	PUNCT
ejpam-6429	448	2	s	s	X
ejpam-6429	448	3	}	}	PUNCT
ejpam-6429	448	4	,	,	PUNCT
ejpam-6429	448	5	we	we	PRON
ejpam-6429	448	6	define	define	VERB
ejpam-6429	448	7	n(xi)\n	n(xi)\n	PROPN
ejpam-6429	448	8	[	[	NOUN
ejpam-6429	448	9	x	x	X
ejpam-6429	448	10	]	]	X
ejpam-6429	448	11	=	=	X
ejpam-6429	448	12	{	{	PUNCT
ejpam-6429	448	13	xi1	xi1	PROPN
ejpam-6429	448	14	,	,	PUNCT
ejpam-6429	448	15	xi2	xi2	PROPN
ejpam-6429	448	16	,	,	PUNCT
ejpam-6429	448	17	.	.	PUNCT
ejpam-6429	448	18	.	.	PUNCT
ejpam-6429	448	19	.	.	PUNCT
ejpam-6429	449	1	,	,	PUNCT
ejpam-6429	449	2	xili	xili	NOUN
ejpam-6429	449	3	}	}	PUNCT
ejpam-6429	449	4	.	.	PUNCT
ejpam-6429	450	1	next	next	ADV
ejpam-6429	450	2	,	,	PUNCT
ejpam-6429	450	3	we	we	PRON
ejpam-6429	450	4	define	define	VERB
ejpam-6429	450	5	g1	g1	NOUN
ejpam-6429	450	6	as	as	ADP
ejpam-6429	450	7	the	the	DET
ejpam-6429	450	8	graph	graph	NOUN
ejpam-6429	450	9	obtained	obtain	VERB
ejpam-6429	450	10	from	from	ADP
ejpam-6429	450	11	g	g	NOUN
ejpam-6429	450	12	by	by	ADP
ejpam-6429	450	13	subdividing	subdivide	VERB
ejpam-6429	450	14	the	the	DET
ejpam-6429	450	15	edges	edge	NOUN
ejpam-6429	450	16	xx1	xx1	PROPN
ejpam-6429	450	17	and	and	CCONJ
ejpam-6429	450	18	xx2	xx2	NOUN
ejpam-6429	450	19	with	with	ADP
ejpam-6429	450	20	new	new	ADJ
ejpam-6429	450	21	vertices	vertex	NOUN
ejpam-6429	450	22	u1	u1	NOUN
ejpam-6429	450	23	and	and	CCONJ
ejpam-6429	450	24	u2	u2	NOUN
ejpam-6429	450	25	,	,	PUNCT
ejpam-6429	450	26	respectively	respectively	ADV
ejpam-6429	450	27	.	.	PUNCT
ejpam-6429	451	1	for	for	ADP
ejpam-6429	451	2	each	each	DET
ejpam-6429	451	3	i	i	PRON
ejpam-6429	451	4	∈	∈	PROPN
ejpam-6429	451	5	{	{	PUNCT
ejpam-6429	451	6	1	1	NUM
ejpam-6429	451	7	,	,	PUNCT
ejpam-6429	451	8	2	2	NUM
ejpam-6429	451	9	,	,	PUNCT
ejpam-6429	451	10	.	.	PUNCT
ejpam-6429	451	11	.	.	PUNCT
ejpam-6429	451	12	.	.	PUNCT
ejpam-6429	452	1	,	,	PUNCT
ejpam-6429	452	2	s	s	X
ejpam-6429	452	3	}	}	PUNCT
ejpam-6429	452	4	,	,	PUNCT
ejpam-6429	452	5	we	we	PRON
ejpam-6429	452	6	subdivide	subdivide	VERB
ejpam-6429	452	7	each	each	DET
ejpam-6429	452	8	edge	edge	NOUN
ejpam-6429	452	9	xixij	xixij	PROPN
ejpam-6429	452	10	,	,	PUNCT
ejpam-6429	452	11	where	where	SCONJ
ejpam-6429	452	12	1	1	NUM
ejpam-6429	452	13	≤	≤	NUM
ejpam-6429	452	14	j	j	PROPN
ejpam-6429	452	15	≤	≤	PROPN
ejpam-6429	452	16	li	li	PROPN
ejpam-6429	452	17	,	,	PUNCT
ejpam-6429	452	18	by	by	ADP
ejpam-6429	452	19	introducing	introduce	VERB
ejpam-6429	452	20	a	a	DET
ejpam-6429	452	21	new	new	ADJ
ejpam-6429	452	22	vertex	vertex	NOUN
ejpam-6429	452	23	xij	xij	PRON
ejpam-6429	452	24	.	.	PUNCT
ejpam-6429	453	1	we	we	PRON
ejpam-6429	453	2	define	define	VERB
ejpam-6429	453	3	wi	wi	PROPN
ejpam-6429	453	4	=	=	PROPN
ejpam-6429	453	5	{	{	PUNCT
ejpam-6429	453	6	xij	xij	PROPN
ejpam-6429	453	7	|	|	ADV
ejpam-6429	453	8	1	1	NUM
ejpam-6429	453	9	≤	≤	NUM
ejpam-6429	453	10	j	j	PROPN
ejpam-6429	453	11	≤	≤	PROPN
ejpam-6429	453	12	li	li	PROPN
ejpam-6429	453	13	}	}	PUNCT
ejpam-6429	453	14	and	and	CCONJ
ejpam-6429	453	15	w	w	NOUN
ejpam-6429	453	16	=	=	PUNCT
ejpam-6429	453	17	⋃	⋃	PROPN
ejpam-6429	453	18	1≤i≤swi	1≤i≤swi	NUM
ejpam-6429	453	19	.	.	PUNCT
ejpam-6429	454	1	furthermore	furthermore	ADV
ejpam-6429	454	2	,	,	PUNCT
ejpam-6429	454	3	if	if	SCONJ
ejpam-6429	454	4	x1	x1	PROPN
ejpam-6429	454	5	and	and	CCONJ
ejpam-6429	454	6	x2	x2	PROPN
ejpam-6429	454	7	are	be	AUX
ejpam-6429	454	8	adjacent	adjacent	ADJ
ejpam-6429	454	9	,	,	PUNCT
ejpam-6429	454	10	we	we	PRON
ejpam-6429	454	11	also	also	ADV
ejpam-6429	454	12	subdivide	subdivide	VERB
ejpam-6429	454	13	the	the	DET
ejpam-6429	454	14	edge	edge	NOUN
ejpam-6429	454	15	x1x2	x1x2	PUNCT
ejpam-6429	454	16	by	by	ADP
ejpam-6429	454	17	introducing	introduce	VERB
ejpam-6429	454	18	a	a	DET
ejpam-6429	454	19	new	new	ADJ
ejpam-6429	454	20	vertex	vertex	NOUN
ejpam-6429	454	21	u3	u3	NOUN
ejpam-6429	454	22	.	.	PUNCT
ejpam-6429	455	1	finally	finally	ADV
ejpam-6429	455	2	,	,	PUNCT
ejpam-6429	455	3	let	let	VERB
ejpam-6429	455	4	f	f	PRON
ejpam-6429	455	5	be	be	AUX
ejpam-6429	455	6	a	a	DET
ejpam-6429	455	7	γgr(g1)-function	γgr(g1)-function	PROPN
ejpam-6429	455	8	.	.	PUNCT
ejpam-6429	456	1	by	by	ADP
ejpam-6429	456	2	proposition	proposition	NOUN
ejpam-6429	456	3	2	2	NUM
ejpam-6429	456	4	,	,	PUNCT
ejpam-6429	456	5	we	we	PRON
ejpam-6429	456	6	can	can	AUX
ejpam-6429	456	7	assume	assume	VERB
ejpam-6429	456	8	that	that	SCONJ
ejpam-6429	456	9	no	no	DET
ejpam-6429	456	10	subdivision	subdivision	NOUN
ejpam-6429	456	11	vertex	vertex	NOUN
ejpam-6429	456	12	is	be	AUX
ejpam-6429	456	13	assigned	assign	VERB
ejpam-6429	456	14	the	the	DET
ejpam-6429	456	15	values	value	NOUN
ejpam-6429	456	16	1	1	NUM
ejpam-6429	456	17	or	or	CCONJ
ejpam-6429	456	18	3	3	NUM
ejpam-6429	456	19	under	under	ADP
ejpam-6429	456	20	f	f	PROPN
ejpam-6429	456	21	.	.	PUNCT
ejpam-6429	457	1	first	first	ADV
ejpam-6429	457	2	let	let	VERB
ejpam-6429	457	3	x1x2	x1x2	PROPN
ejpam-6429	457	4	∈	∈	PROPN
ejpam-6429	457	5	e(g	e(g	PROPN
ejpam-6429	457	6	)	)	PUNCT
ejpam-6429	457	7	.	.	PUNCT
ejpam-6429	458	1	similar	similar	ADJ
ejpam-6429	458	2	as	as	ADP
ejpam-6429	458	3	in	in	ADP
ejpam-6429	458	4	the	the	DET
ejpam-6429	458	5	proof	proof	NOUN
ejpam-6429	458	6	of	of	ADP
ejpam-6429	458	7	proposition	proposition	NOUN
ejpam-6429	458	8	11	11	NUM
ejpam-6429	458	9	,	,	PUNCT
ejpam-6429	458	10	we	we	PRON
ejpam-6429	458	11	can	can	AUX
ejpam-6429	458	12	see	see	VERB
ejpam-6429	458	13	that	that	SCONJ
ejpam-6429	458	14	f(x)+	f(x)+	NOUN
ejpam-6429	458	15	f(x1)+	f(x1)+	NOUN
ejpam-6429	458	16	f(x2)+	f(x2)+	VERB
ejpam-6429	458	17	f(u1)+	f(u1)+	PROPN
ejpam-6429	458	18	f(u2)+	f(u2)+	PROPN
ejpam-6429	458	19	f(u3	f(u3	PROPN
ejpam-6429	458	20	)	)	PUNCT
ejpam-6429	458	21	≥	≥	NOUN
ejpam-6429	458	22	4	4	NUM
ejpam-6429	458	23	.	.	PUNCT
ejpam-6429	459	1	by	by	ADP
ejpam-6429	459	2	reassigning	reassign	VERB
ejpam-6429	459	3	x	x	PUNCT
ejpam-6429	459	4	the	the	DET
ejpam-6429	459	5	value	value	NOUN
ejpam-6429	459	6	3	3	NUM
ejpam-6429	459	7	,	,	PUNCT
ejpam-6429	459	8	x1	x1	PROPN
ejpam-6429	459	9	,	,	PUNCT
ejpam-6429	459	10	x2	x2	PROPN
ejpam-6429	459	11	the	the	DET
ejpam-6429	459	12	value	value	NOUN
ejpam-6429	459	13	0	0	NUM
ejpam-6429	459	14	,	,	PUNCT
ejpam-6429	459	15	and	and	CCONJ
ejpam-6429	459	16	xij	xij	PRON
ejpam-6429	459	17	the	the	DET
ejpam-6429	459	18	value	value	NOUN
ejpam-6429	459	19	min{3	min{3	PROPN
ejpam-6429	459	20	,	,	PUNCT
ejpam-6429	459	21	f(xij	f(xij	PROPN
ejpam-6429	459	22	)	)	PUNCT
ejpam-6429	460	1	+	+	ADV
ejpam-6429	460	2	f(xij	f(xij	NOUN
ejpam-6429	460	3	)	)	PUNCT
ejpam-6429	460	4	}	}	PUNCT
ejpam-6429	460	5	for	for	ADP
ejpam-6429	460	6	all	all	DET
ejpam-6429	460	7	i	i	PROPN
ejpam-6429	460	8	and	and	CCONJ
ejpam-6429	460	9	j	j	PROPN
ejpam-6429	460	10	,	,	PUNCT
ejpam-6429	460	11	we	we	PRON
ejpam-6429	460	12	obtain	obtain	VERB
ejpam-6429	460	13	a	a	DET
ejpam-6429	460	14	grd	grd	NOUN
ejpam-6429	460	15	-	-	PUNCT
ejpam-6429	460	16	function	function	NOUN
ejpam-6429	460	17	of	of	ADP
ejpam-6429	460	18	g	g	NOUN
ejpam-6429	460	19	of	of	ADP
ejpam-6429	460	20	weight	weight	NOUN
ejpam-6429	460	21	less	less	ADV
ejpam-6429	460	22	than	than	ADP
ejpam-6429	460	23	γgr(g1	γgr(g1	ADJ
ejpam-6429	460	24	)	)	PUNCT
ejpam-6429	460	25	as	as	SCONJ
ejpam-6429	460	26	desired	desire	VERB
ejpam-6429	460	27	.	.	PUNCT
ejpam-6429	461	1	assume	assume	VERB
ejpam-6429	461	2	now	now	ADV
ejpam-6429	461	3	that	that	SCONJ
ejpam-6429	461	4	x1x2	x1x2	PUNCT
ejpam-6429	461	5	̸∈	̸∈	PROPN
ejpam-6429	461	6	e(g	e(g	PROPN
ejpam-6429	461	7	)	)	PUNCT
ejpam-6429	461	8	.	.	PUNCT
ejpam-6429	462	1	by	by	ADP
ejpam-6429	462	2	the	the	DET
ejpam-6429	462	3	choice	choice	NOUN
ejpam-6429	462	4	of	of	ADP
ejpam-6429	462	5	x1	x1	PROPN
ejpam-6429	462	6	,	,	PUNCT
ejpam-6429	462	7	x2	x2	PROPN
ejpam-6429	462	8	,	,	PUNCT
ejpam-6429	462	9	we	we	PRON
ejpam-6429	462	10	deduce	deduce	VERB
ejpam-6429	462	11	that	that	PRON
ejpam-6429	462	12	s	s	VERB
ejpam-6429	462	13	is	be	AUX
ejpam-6429	462	14	independent	independent	ADJ
ejpam-6429	462	15	.	.	PUNCT
ejpam-6429	463	1	to	to	PART
ejpam-6429	463	2	protect	protect	VERB
ejpam-6429	463	3	the	the	DET
ejpam-6429	463	4	vertices	vertex	NOUN
ejpam-6429	463	5	u1	u1	NOUN
ejpam-6429	463	6	and	and	CCONJ
ejpam-6429	463	7	u2	u2	NOUN
ejpam-6429	463	8	,	,	PUNCT
ejpam-6429	463	9	we	we	PRON
ejpam-6429	463	10	must	must	AUX
ejpam-6429	463	11	have	have	VERB
ejpam-6429	463	12	f(x)+f(x1)+f(x2)+f(u1)+f(u2	f(x)+f(x1)+f(x2)+f(u1)+f(u2	PROPN
ejpam-6429	463	13	)	)	PUNCT
ejpam-6429	463	14	≥	≥	PROPN
ejpam-6429	464	1	j.	j.	PROPN
ejpam-6429	464	2	j.	j.	PROPN
ejpam-6429	464	3	hamja	hamja	PROPN
ejpam-6429	464	4	et	et	PROPN
ejpam-6429	464	5	al	al	PROPN
ejpam-6429	464	6	.	.	PUNCT
ejpam-6429	464	7	/	/	SYM
ejpam-6429	464	8	eur	eur	PROPN
ejpam-6429	464	9	.	.	PUNCT
ejpam-6429	465	1	j.	j.	PROPN
ejpam-6429	465	2	pure	pure	PROPN
ejpam-6429	465	3	appl	appl	PROPN
ejpam-6429	465	4	.	.	PROPN
ejpam-6429	465	5	math	math	PROPN
ejpam-6429	465	6	,	,	PUNCT
ejpam-6429	465	7	18	18	NUM
ejpam-6429	465	8	(	(	PUNCT
ejpam-6429	465	9	4	4	NUM
ejpam-6429	465	10	)	)	PUNCT
ejpam-6429	465	11	(	(	PUNCT
ejpam-6429	465	12	2025	2025	NUM
ejpam-6429	465	13	)	)	PUNCT
ejpam-6429	465	14	,	,	PUNCT
ejpam-6429	465	15	6429	6429	NUM
ejpam-6429	465	16	12	12	NUM
ejpam-6429	465	17	of	of	ADP
ejpam-6429	465	18	16	16	NUM
ejpam-6429	465	19	3	3	NUM
ejpam-6429	465	20	.	.	PUNCT
ejpam-6429	466	1	if	if	SCONJ
ejpam-6429	466	2	f(x	f(x	PROPN
ejpam-6429	466	3	)	)	PUNCT
ejpam-6429	467	1	+	+	CCONJ
ejpam-6429	467	2	f(u1	f(u1	NOUN
ejpam-6429	467	3	)	)	PUNCT
ejpam-6429	468	1	+	+	NUM
ejpam-6429	468	2	f(u2	f(u2	NOUN
ejpam-6429	468	3	)	)	PUNCT
ejpam-6429	469	1	+	+	CCONJ
ejpam-6429	469	2	∑s	∑s	PROPN
ejpam-6429	469	3	i=1	i=1	PUNCT
ejpam-6429	469	4	f(xi	f(xi	PROPN
ejpam-6429	469	5	)	)	PUNCT
ejpam-6429	469	6	≥	≥	NOUN
ejpam-6429	469	7	4	4	NUM
ejpam-6429	469	8	,	,	PUNCT
ejpam-6429	469	9	then	then	ADV
ejpam-6429	469	10	reassigning	reassign	VERB
ejpam-6429	469	11	x	x	PUNCT
ejpam-6429	469	12	the	the	DET
ejpam-6429	469	13	value	value	NOUN
ejpam-6429	469	14	3	3	NUM
ejpam-6429	469	15	,	,	PUNCT
ejpam-6429	469	16	xi	xi	VERB
ejpam-6429	469	17	the	the	DET
ejpam-6429	469	18	value	value	NOUN
ejpam-6429	469	19	0	0	NUM
ejpam-6429	469	20	for	for	ADP
ejpam-6429	469	21	all	all	PRON
ejpam-6429	469	22	i	i	PRON
ejpam-6429	469	23	∈	∈	PROPN
ejpam-6429	469	24	{	{	PUNCT
ejpam-6429	469	25	1	1	NUM
ejpam-6429	469	26	,	,	PUNCT
ejpam-6429	469	27	2	2	NUM
ejpam-6429	469	28	,	,	PUNCT
ejpam-6429	469	29	.	.	PUNCT
ejpam-6429	469	30	.	.	PUNCT
ejpam-6429	469	31	.	.	PUNCT
ejpam-6429	470	1	,	,	PUNCT
ejpam-6429	470	2	s	s	X
ejpam-6429	470	3	}	}	PUNCT
ejpam-6429	470	4	,	,	PUNCT
ejpam-6429	470	5	and	and	CCONJ
ejpam-6429	470	6	xij	xij	PRON
ejpam-6429	470	7	the	the	DET
ejpam-6429	470	8	value	value	NOUN
ejpam-6429	470	9	min{3	min{3	PROPN
ejpam-6429	470	10	,	,	PUNCT
ejpam-6429	470	11	f(xij	f(xij	PROPN
ejpam-6429	470	12	)	)	PUNCT
ejpam-6429	471	1	+	+	CCONJ
ejpam-6429	471	2	f(xij	f(xij	PROPN
ejpam-6429	471	3	)	)	PUNCT
ejpam-6429	471	4	}	}	PUNCT
ejpam-6429	471	5	for	for	ADP
ejpam-6429	471	6	all	all	PRON
ejpam-6429	471	7	i	i	PROPN
ejpam-6429	471	8	,	,	PUNCT
ejpam-6429	471	9	j	j	PROPN
ejpam-6429	471	10	,	,	PUNCT
ejpam-6429	471	11	provides	provide	VERB
ejpam-6429	471	12	a	a	DET
ejpam-6429	471	13	grd	grd	NOUN
ejpam-6429	471	14	-	-	PUNCT
ejpam-6429	471	15	function	function	NOUN
ejpam-6429	471	16	of	of	ADP
ejpam-6429	471	17	g	g	NOUN
ejpam-6429	471	18	of	of	ADP
ejpam-6429	471	19	weight	weight	NOUN
ejpam-6429	471	20	less	less	ADV
ejpam-6429	471	21	than	than	ADP
ejpam-6429	471	22	γgr(g1	γgr(g1	ADJ
ejpam-6429	471	23	)	)	PUNCT
ejpam-6429	471	24	.	.	PUNCT
ejpam-6429	472	1	thus	thus	ADV
ejpam-6429	472	2	,	,	PUNCT
ejpam-6429	472	3	we	we	PRON
ejpam-6429	472	4	may	may	AUX
ejpam-6429	472	5	assume	assume	VERB
ejpam-6429	472	6	that	that	SCONJ
ejpam-6429	472	7	f(x	f(x	PROPN
ejpam-6429	472	8	)	)	PUNCT
ejpam-6429	473	1	+	+	CCONJ
ejpam-6429	473	2	f(u1	f(u1	NOUN
ejpam-6429	473	3	)	)	PUNCT
ejpam-6429	474	1	+	+	NUM
ejpam-6429	474	2	f(u2	f(u2	NOUN
ejpam-6429	474	3	)	)	PUNCT
ejpam-6429	475	1	+	+	CCONJ
ejpam-6429	475	2	∑s	∑s	PROPN
ejpam-6429	475	3	i=1	i=1	PUNCT
ejpam-6429	475	4	f(xi	f(xi	PROPN
ejpam-6429	475	5	)	)	PUNCT
ejpam-6429	475	6	≤	≤	NOUN
ejpam-6429	476	1	3	3	NUM
ejpam-6429	476	2	.	.	PUNCT
ejpam-6429	477	1	if	if	SCONJ
ejpam-6429	477	2	f(x	f(x	PROPN
ejpam-6429	477	3	)	)	PUNCT
ejpam-6429	477	4	∈	∈	PROPN
ejpam-6429	477	5	{	{	PUNCT
ejpam-6429	477	6	0	0	NUM
ejpam-6429	477	7	,	,	PUNCT
ejpam-6429	477	8	1	1	NUM
ejpam-6429	477	9	,	,	PUNCT
ejpam-6429	477	10	2	2	NUM
ejpam-6429	477	11	}	}	PUNCT
ejpam-6429	477	12	,	,	PUNCT
ejpam-6429	477	13	then	then	ADV
ejpam-6429	477	14	to	to	PART
ejpam-6429	477	15	protect	protect	VERB
ejpam-6429	477	16	the	the	DET
ejpam-6429	477	17	vertices	vertex	NOUN
ejpam-6429	477	18	u1	u1	NOUN
ejpam-6429	477	19	,	,	PUNCT
ejpam-6429	477	20	u2	u2	PROPN
ejpam-6429	477	21	,	,	PUNCT
ejpam-6429	477	22	we	we	PRON
ejpam-6429	477	23	must	must	AUX
ejpam-6429	477	24	have	have	VERB
ejpam-6429	477	25	f(x)+f(u1)+f(u2)+	f(x)+f(u1)+f(u2)+	PROPN
ejpam-6429	477	26	∑s	∑s	PROPN
ejpam-6429	477	27	i=1	i=1	PUNCT
ejpam-6429	477	28	f(xi	f(xi	PROPN
ejpam-6429	477	29	)	)	PUNCT
ejpam-6429	477	30	≥	≥	NOUN
ejpam-6429	477	31	f(x1)+f(x2)+f(u1)+f(u2	f(x1)+f(x2)+f(u1)+f(u2	X
ejpam-6429	477	32	)	)	PUNCT
ejpam-6429	477	33	≥	≥	NOUN
ejpam-6429	477	34	4	4	NUM
ejpam-6429	477	35	contradicting	contradict	VERB
ejpam-6429	477	36	our	our	PRON
ejpam-6429	477	37	assumption	assumption	NOUN
ejpam-6429	477	38	.	.	PUNCT
ejpam-6429	478	1	thus	thus	ADV
ejpam-6429	478	2	,	,	PUNCT
ejpam-6429	478	3	f(x	f(x	PROPN
ejpam-6429	478	4	)	)	PUNCT
ejpam-6429	478	5	=	=	SYM
ejpam-6429	479	1	3	3	NUM
ejpam-6429	479	2	and	and	CCONJ
ejpam-6429	479	3	so	so	ADV
ejpam-6429	479	4	f(u1	f(u1	NOUN
ejpam-6429	479	5	)	)	PUNCT
ejpam-6429	480	1	+	+	NUM
ejpam-6429	480	2	f(u2	f(u2	NOUN
ejpam-6429	480	3	)	)	PUNCT
ejpam-6429	481	1	+	+	CCONJ
ejpam-6429	481	2	∑s	∑s	PROPN
ejpam-6429	481	3	i=1	i=1	X
ejpam-6429	481	4	f(xi	f(xi	PROPN
ejpam-6429	481	5	)	)	PUNCT
ejpam-6429	481	6	=	=	SYM
ejpam-6429	482	1	0	0	X
ejpam-6429	482	2	.	.	PUNCT
ejpam-6429	483	1	if	if	SCONJ
ejpam-6429	483	2	∑li	∑li	NUM
ejpam-6429	483	3	j=1	j=1	PROPN
ejpam-6429	483	4	f(x	f(x	PROPN
ejpam-6429	483	5	ij	ij	INTJ
ejpam-6429	483	6	)	)	PUNCT
ejpam-6429	483	7	≥	≥	NOUN
ejpam-6429	483	8	4	4	NUM
ejpam-6429	483	9	for	for	ADP
ejpam-6429	483	10	some	some	DET
ejpam-6429	483	11	1	1	NUM
ejpam-6429	483	12	≤	≤	NUM
ejpam-6429	483	13	i	i	PRON
ejpam-6429	483	14	≤	≤	PROPN
ejpam-6429	483	15	s	s	X
ejpam-6429	483	16	,	,	PUNCT
ejpam-6429	483	17	say	say	VERB
ejpam-6429	483	18	i	i	NOUN
ejpam-6429	483	19	=	=	NOUN
ejpam-6429	483	20	1	1	NUM
ejpam-6429	483	21	,	,	PUNCT
ejpam-6429	483	22	then	then	ADV
ejpam-6429	483	23	reassigning	reassign	VERB
ejpam-6429	483	24	x1	x1	NOUN
ejpam-6429	483	25	the	the	DET
ejpam-6429	483	26	value	value	NOUN
ejpam-6429	483	27	3	3	NUM
ejpam-6429	483	28	and	and	CCONJ
ejpam-6429	483	29	xij	xij	PRON
ejpam-6429	483	30	the	the	DET
ejpam-6429	483	31	value	value	NOUN
ejpam-6429	483	32	min{3	min{3	PROPN
ejpam-6429	483	33	,	,	PUNCT
ejpam-6429	483	34	f(xij	f(xij	PROPN
ejpam-6429	483	35	)	)	PUNCT
ejpam-6429	484	1	+	+	CCONJ
ejpam-6429	484	2	f(xij	f(xij	PROPN
ejpam-6429	484	3	)	)	PUNCT
ejpam-6429	484	4	}	}	PUNCT
ejpam-6429	484	5	for	for	ADP
ejpam-6429	484	6	all	all	PRON
ejpam-6429	484	7	i	i	PRON
ejpam-6429	484	8	∈	∈	PROPN
ejpam-6429	484	9	{	{	PUNCT
ejpam-6429	484	10	2	2	NUM
ejpam-6429	484	11	,	,	PUNCT
ejpam-6429	484	12	.	.	PUNCT
ejpam-6429	484	13	.	.	PUNCT
ejpam-6429	485	1	.	.	PUNCT
ejpam-6429	486	1	,	,	PUNCT
ejpam-6429	486	2	s	s	X
ejpam-6429	486	3	}	}	PUNCT
ejpam-6429	486	4	and	and	CCONJ
ejpam-6429	486	5	all	all	DET
ejpam-6429	486	6	j	j	PROPN
ejpam-6429	486	7	∈	∈	PROPN
ejpam-6429	486	8	{	{	PUNCT
ejpam-6429	486	9	1	1	NUM
ejpam-6429	486	10	,	,	PUNCT
ejpam-6429	486	11	2	2	NUM
ejpam-6429	486	12	,	,	PUNCT
ejpam-6429	486	13	.	.	PUNCT
ejpam-6429	486	14	.	.	PUNCT
ejpam-6429	486	15	.	.	PUNCT
ejpam-6429	487	1	,	,	PUNCT
ejpam-6429	487	2	li	li	PROPN
ejpam-6429	487	3	}	}	PUNCT
ejpam-6429	487	4	,	,	PUNCT
ejpam-6429	487	5	provides	provide	VERB
ejpam-6429	487	6	a	a	DET
ejpam-6429	487	7	grd	grd	NOUN
ejpam-6429	487	8	-	-	PUNCT
ejpam-6429	487	9	function	function	NOUN
ejpam-6429	487	10	of	of	ADP
ejpam-6429	487	11	g	g	NOUN
ejpam-6429	487	12	of	of	ADP
ejpam-6429	487	13	weight	weight	NOUN
ejpam-6429	487	14	less	less	ADJ
ejpam-6429	487	15	than	than	ADP
ejpam-6429	487	16	ωg	ωg	ADP
ejpam-6429	487	17	r(f	r(f	PROPN
ejpam-6429	487	18	)	)	PUNCT
ejpam-6429	488	1	=	=	PUNCT
ejpam-6429	488	2	γgr(g1	γgr(g1	ADJ
ejpam-6429	488	3	)	)	PUNCT
ejpam-6429	488	4	.	.	PUNCT
ejpam-6429	489	1	hence	hence	ADV
ejpam-6429	489	2	,	,	PUNCT
ejpam-6429	489	3	suppose	suppose	VERB
ejpam-6429	489	4	that∑li	that∑li	PROPN
ejpam-6429	489	5	j=1	j=1	PROPN
ejpam-6429	489	6	f(x	f(x	PROPN
ejpam-6429	489	7	ij	ij	INTJ
ejpam-6429	489	8	)	)	PUNCT
ejpam-6429	489	9	≤	≤	NOUN
ejpam-6429	489	10	3	3	NUM
ejpam-6429	489	11	,	,	PUNCT
ejpam-6429	489	12	for	for	ADP
ejpam-6429	489	13	each	each	DET
ejpam-6429	489	14	i	i	PRON
ejpam-6429	489	15	∈	∈	PROPN
ejpam-6429	489	16	{	{	PUNCT
ejpam-6429	489	17	1	1	NUM
ejpam-6429	489	18	,	,	PUNCT
ejpam-6429	489	19	2	2	NUM
ejpam-6429	489	20	,	,	PUNCT
ejpam-6429	489	21	.	.	PUNCT
ejpam-6429	489	22	.	.	PUNCT
ejpam-6429	489	23	.	.	PUNCT
ejpam-6429	490	1	,	,	PUNCT
ejpam-6429	490	2	s	s	X
ejpam-6429	490	3	}	}	PUNCT
ejpam-6429	490	4	.	.	PUNCT
ejpam-6429	491	1	first	first	ADV
ejpam-6429	491	2	,	,	PUNCT
ejpam-6429	491	3	consider	consider	VERB
ejpam-6429	491	4	the	the	DET
ejpam-6429	491	5	case	case	NOUN
ejpam-6429	491	6	where	where	SCONJ
ejpam-6429	491	7	there	there	PRON
ejpam-6429	491	8	exist	exist	VERB
ejpam-6429	491	9	some	some	DET
ejpam-6429	491	10	i	i	PRON
ejpam-6429	491	11	∈	∈	PROPN
ejpam-6429	491	12	{	{	PUNCT
ejpam-6429	491	13	1	1	NUM
ejpam-6429	491	14	,	,	PUNCT
ejpam-6429	491	15	2	2	NUM
ejpam-6429	491	16	,	,	PUNCT
ejpam-6429	491	17	.	.	PUNCT
ejpam-6429	491	18	.	.	PUNCT
ejpam-6429	492	1	.	.	PUNCT
ejpam-6429	493	1	,	,	PUNCT
ejpam-6429	493	2	s	s	X
ejpam-6429	493	3	}	}	PUNCT
ejpam-6429	493	4	and	and	CCONJ
ejpam-6429	493	5	some	some	DET
ejpam-6429	493	6	j	j	PROPN
ejpam-6429	493	7	∈	∈	PROPN
ejpam-6429	493	8	{	{	PUNCT
ejpam-6429	493	9	1	1	NUM
ejpam-6429	493	10	,	,	PUNCT
ejpam-6429	493	11	2	2	NUM
ejpam-6429	493	12	,	,	PUNCT
ejpam-6429	493	13	.	.	PUNCT
ejpam-6429	493	14	.	.	PUNCT
ejpam-6429	493	15	.	.	PUNCT
ejpam-6429	494	1	,	,	PUNCT
ejpam-6429	494	2	li	li	PROPN
ejpam-6429	494	3	}	}	PUNCT
ejpam-6429	494	4	such	such	ADJ
ejpam-6429	494	5	that	that	SCONJ
ejpam-6429	494	6	f(xij	f(xij	PROPN
ejpam-6429	494	7	)	)	PUNCT
ejpam-6429	495	1	=	=	PUNCT
ejpam-6429	495	2	2	2	X
ejpam-6429	495	3	.	.	X
ejpam-6429	495	4	assume	assume	VERB
ejpam-6429	495	5	,	,	PUNCT
ejpam-6429	495	6	without	without	ADP
ejpam-6429	495	7	loss	loss	NOUN
ejpam-6429	495	8	of	of	ADP
ejpam-6429	495	9	generality	generality	NOUN
ejpam-6429	495	10	,	,	PUNCT
ejpam-6429	495	11	that	that	SCONJ
ejpam-6429	495	12	i	i	PRON
ejpam-6429	496	1	=	=	SYM
ejpam-6429	496	2	j	j	PROPN
ejpam-6429	496	3	=	=	SYM
ejpam-6429	496	4	1	1	X
ejpam-6429	496	5	.	.	PUNCT
ejpam-6429	496	6	then	then	ADV
ejpam-6429	496	7	,	,	PUNCT
ejpam-6429	496	8	by	by	ADP
ejpam-6429	496	9	updating	update	VERB
ejpam-6429	496	10	x11	x11	PRON
ejpam-6429	496	11	to	to	PART
ejpam-6429	496	12	take	take	VERB
ejpam-6429	496	13	the	the	DET
ejpam-6429	496	14	value	value	NOUN
ejpam-6429	496	15	min{3	min{3	PROPN
ejpam-6429	496	16	,	,	PUNCT
ejpam-6429	496	17	1+f(x11	1+f(x11	NUM
ejpam-6429	496	18	)	)	PUNCT
ejpam-6429	496	19	}	}	PUNCT
ejpam-6429	496	20	and	and	CCONJ
ejpam-6429	496	21	redefining	redefine	VERB
ejpam-6429	496	22	xij	xij	PROPN
ejpam-6429	496	23	as	as	ADP
ejpam-6429	496	24	min{3	min{3	PROPN
ejpam-6429	496	25	,	,	PUNCT
ejpam-6429	496	26	f(xij	f(xij	PROPN
ejpam-6429	496	27	)	)	PUNCT
ejpam-6429	496	28	+	+	CCONJ
ejpam-6429	496	29	f(xij	f(xij	PROPN
ejpam-6429	496	30	)	)	PUNCT
ejpam-6429	496	31	}	}	PUNCT
ejpam-6429	496	32	,	,	PUNCT
ejpam-6429	496	33	for	for	ADP
ejpam-6429	496	34	ij	ij	NOUN
ejpam-6429	496	35	̸=	̸=	PROPN
ejpam-6429	496	36	11	11	NUM
ejpam-6429	496	37	,	,	PUNCT
ejpam-6429	496	38	we	we	PRON
ejpam-6429	496	39	obtain	obtain	VERB
ejpam-6429	496	40	a	a	DET
ejpam-6429	496	41	grd	grd	NOUN
ejpam-6429	496	42	-	-	PUNCT
ejpam-6429	496	43	function	function	NOUN
ejpam-6429	496	44	of	of	ADP
ejpam-6429	496	45	g	g	NOUN
ejpam-6429	496	46	with	with	ADP
ejpam-6429	496	47	weight	weight	NOUN
ejpam-6429	496	48	strictly	strictly	ADV
ejpam-6429	496	49	less	less	ADJ
ejpam-6429	496	50	than	than	ADP
ejpam-6429	496	51	γgr(g1	γgr(g1	ADJ
ejpam-6429	496	52	)	)	PUNCT
ejpam-6429	496	53	.	.	PUNCT
ejpam-6429	497	1	thus	thus	ADV
ejpam-6429	497	2	,	,	PUNCT
ejpam-6429	497	3	we	we	PRON
ejpam-6429	497	4	may	may	AUX
ejpam-6429	497	5	assume	assume	VERB
ejpam-6429	497	6	that	that	SCONJ
ejpam-6429	497	7	f(xij	f(xij	PROPN
ejpam-6429	497	8	)	)	PUNCT
ejpam-6429	498	1	=	=	PUNCT
ejpam-6429	498	2	0	0	NUM
ejpam-6429	499	1	for	for	ADP
ejpam-6429	499	2	all	all	DET
ejpam-6429	499	3	i	i	PRON
ejpam-6429	499	4	and	and	CCONJ
ejpam-6429	499	5	j.	j.	PROPN
ejpam-6429	499	6	this	this	PRON
ejpam-6429	499	7	directly	directly	ADV
ejpam-6429	499	8	implies	imply	VERB
ejpam-6429	499	9	that	that	SCONJ
ejpam-6429	499	10	f(xij	f(xij	PROPN
ejpam-6429	499	11	)	)	PUNCT
ejpam-6429	499	12	=	=	SYM
ejpam-6429	499	13	2	2	NUM
ejpam-6429	499	14	for	for	ADP
ejpam-6429	499	15	every	every	DET
ejpam-6429	499	16	i	i	PROPN
ejpam-6429	499	17	and	and	CCONJ
ejpam-6429	499	18	j.	j.	PROPN
ejpam-6429	499	19	clearly	clearly	ADV
ejpam-6429	499	20	,	,	PUNCT
ejpam-6429	499	21	reassigning	reassign	VERB
ejpam-6429	499	22	x	x	PUNCT
ejpam-6429	499	23	the	the	DET
ejpam-6429	499	24	value	value	NOUN
ejpam-6429	499	25	2	2	NUM
ejpam-6429	499	26	results	result	NOUN
ejpam-6429	499	27	in	in	ADP
ejpam-6429	499	28	a	a	DET
ejpam-6429	499	29	grd	grd	NOUN
ejpam-6429	499	30	-	-	PUNCT
ejpam-6429	499	31	function	function	NOUN
ejpam-6429	499	32	of	of	ADP
ejpam-6429	499	33	g	g	NOUN
ejpam-6429	499	34	with	with	ADP
ejpam-6429	499	35	weight	weight	NOUN
ejpam-6429	499	36	strictly	strictly	ADV
ejpam-6429	499	37	less	less	ADJ
ejpam-6429	499	38	than	than	ADP
ejpam-6429	499	39	γgr(g1	γgr(g1	ADJ
ejpam-6429	499	40	)	)	PUNCT
ejpam-6429	499	41	.	.	PUNCT
ejpam-6429	500	1	all	all	ADV
ejpam-6429	500	2	in	in	ADP
ejpam-6429	500	3	all	all	PRON
ejpam-6429	500	4	,	,	PUNCT
ejpam-6429	500	5	we	we	PRON
ejpam-6429	500	6	see	see	VERB
ejpam-6429	500	7	that	that	SCONJ
ejpam-6429	500	8	the	the	DET
ejpam-6429	500	9	graph	graph	NOUN
ejpam-6429	500	10	g	g	PROPN
ejpam-6429	500	11	has	have	VERB
ejpam-6429	500	12	a	a	DET
ejpam-6429	500	13	grd	grd	NOUN
ejpam-6429	500	14	-	-	PUNCT
ejpam-6429	500	15	function	function	NOUN
ejpam-6429	500	16	of	of	ADP
ejpam-6429	500	17	weight	weight	NOUN
ejpam-6429	500	18	less	less	ADJ
ejpam-6429	500	19	than	than	ADP
ejpam-6429	500	20	γgr(g1	γgr(g1	ADJ
ejpam-6429	500	21	)	)	PUNCT
ejpam-6429	500	22	.	.	PUNCT
ejpam-6429	501	1	moreover	moreover	ADV
ejpam-6429	501	2	,	,	PUNCT
ejpam-6429	501	3	since	since	SCONJ
ejpam-6429	501	4	g1	g1	PROPN
ejpam-6429	501	5	is	be	AUX
ejpam-6429	501	6	obtained	obtain	VERB
ejpam-6429	501	7	by	by	ADP
ejpam-6429	501	8	inserting	insert	VERB
ejpam-6429	501	9	at	at	ADP
ejpam-6429	501	10	most	most	ADJ
ejpam-6429	501	11	3	3	NUM
ejpam-6429	501	12	+	+	CCONJ
ejpam-6429	501	13	|w	|w	ADJ
ejpam-6429	501	14	|	|	ADV
ejpam-6429	501	15	≤	≤	NUM
ejpam-6429	501	16	3	3	NUM
ejpam-6429	501	17	+	+	CCONJ
ejpam-6429	501	18	|n2(x)|	|n2(x)|	X
ejpam-6429	501	19	new	new	ADJ
ejpam-6429	501	20	vertices	vertex	NOUN
ejpam-6429	501	21	,	,	PUNCT
ejpam-6429	501	22	we	we	PRON
ejpam-6429	501	23	obtain	obtain	VERB
ejpam-6429	501	24	sdγgr(g	sdγgr(g	NOUN
ejpam-6429	501	25	)	)	PUNCT
ejpam-6429	501	26	≤	≤	NOUN
ejpam-6429	501	27	3	3	NUM
ejpam-6429	501	28	+	+	NUM
ejpam-6429	501	29	|n2(x)|	|n2(x)|	NOUN
ejpam-6429	501	30	.	.	PUNCT
ejpam-6429	502	1	this	this	PRON
ejpam-6429	502	2	completes	complete	VERB
ejpam-6429	502	3	the	the	DET
ejpam-6429	502	4	proof	proof	NOUN
ejpam-6429	502	5	.	.	PUNCT
ejpam-6429	503	1	lemma	lemma	PROPN
ejpam-6429	503	2	3	3	X
ejpam-6429	503	3	.	.	PUNCT
ejpam-6429	504	1	let	let	VERB
ejpam-6429	504	2	g	g	PRON
ejpam-6429	504	3	be	be	AUX
ejpam-6429	504	4	a	a	DET
ejpam-6429	504	5	connected	connected	ADJ
ejpam-6429	504	6	graph	graph	NOUN
ejpam-6429	504	7	of	of	ADP
ejpam-6429	504	8	order	order	NOUN
ejpam-6429	504	9	n	n	PRON
ejpam-6429	504	10	≥	≥	NOUN
ejpam-6429	504	11	3	3	NUM
ejpam-6429	504	12	and	and	CCONJ
ejpam-6429	504	13	v	v	NOUN
ejpam-6429	504	14	be	be	AUX
ejpam-6429	504	15	a	a	DET
ejpam-6429	504	16	vertex	vertex	NOUN
ejpam-6429	504	17	of	of	ADP
ejpam-6429	504	18	degree	degree	NOUN
ejpam-6429	504	19	at	at	ADV
ejpam-6429	504	20	least	least	ADJ
ejpam-6429	504	21	2	2	NUM
ejpam-6429	504	22	of	of	ADP
ejpam-6429	504	23	g	g	NOUN
ejpam-6429	504	24	satisfying	satisfy	VERB
ejpam-6429	504	25	the	the	DET
ejpam-6429	504	26	following	follow	VERB
ejpam-6429	504	27	conditions	condition	NOUN
ejpam-6429	504	28	:	:	PUNCT
ejpam-6429	504	29	(	(	PUNCT
ejpam-6429	504	30	i	i	NOUN
ejpam-6429	504	31	)	)	PUNCT
ejpam-6429	504	32	n(y	n(y	PROPN
ejpam-6429	504	33	)	)	PUNCT
ejpam-6429	504	34	\n	\n	PUNCT
ejpam-6429	505	1	[	[	X
ejpam-6429	505	2	v	v	X
ejpam-6429	505	3	]	]	X
ejpam-6429	505	4	̸=	̸=	NOUN
ejpam-6429	505	5	∅	∅	NOUN
ejpam-6429	505	6	for	for	ADP
ejpam-6429	505	7	each	each	DET
ejpam-6429	505	8	y	y	PROPN
ejpam-6429	505	9	∈	∈	PROPN
ejpam-6429	505	10	n(v	n(v	PROPN
ejpam-6429	505	11	)	)	PUNCT
ejpam-6429	505	12	,	,	PUNCT
ejpam-6429	505	13	(	(	PUNCT
ejpam-6429	505	14	ii	ii	NOUN
ejpam-6429	505	15	)	)	PUNCT
ejpam-6429	505	16	for	for	ADP
ejpam-6429	505	17	every	every	DET
ejpam-6429	505	18	pair	pair	NOUN
ejpam-6429	505	19	of	of	ADP
ejpam-6429	505	20	vertices	vertex	NOUN
ejpam-6429	505	21	a	a	DET
ejpam-6429	505	22	,	,	PUNCT
ejpam-6429	505	23	b	b	NOUN
ejpam-6429	505	24	in	in	ADP
ejpam-6429	505	25	n(v	n(v	PROPN
ejpam-6429	505	26	)	)	PUNCT
ejpam-6429	505	27	,	,	PUNCT
ejpam-6429	505	28	(	(	PUNCT
ejpam-6429	505	29	n(a	n(a	X
ejpam-6429	505	30	)	)	PUNCT
ejpam-6429	505	31	∩n(b	∩n(b	NOUN
ejpam-6429	505	32	)	)	PUNCT
ejpam-6429	505	33	)	)	PUNCT
ejpam-6429	505	34	\n	\n	PUNCT
ejpam-6429	506	1	[	[	X
ejpam-6429	506	2	v	v	X
ejpam-6429	506	3	]	]	X
ejpam-6429	506	4	̸=	̸=	PROPN
ejpam-6429	506	5	∅.	∅.	VERB
ejpam-6429	506	6	then	then	ADV
ejpam-6429	506	7	sdγgr(g	sdγgr(g	PROPN
ejpam-6429	506	8	)	)	PUNCT
ejpam-6429	506	9	≤	≤	NOUN
ejpam-6429	506	10	3	3	NUM
ejpam-6429	506	11	+	+	CCONJ
ejpam-6429	506	12	|n2(v)|	|n2(v)|	ADJ
ejpam-6429	506	13	.	.	PUNCT
ejpam-6429	506	14	proof	proof	NOUN
ejpam-6429	506	15	.	.	PUNCT
ejpam-6429	507	1	if	if	SCONJ
ejpam-6429	507	2	deg(v	deg(v	PROPN
ejpam-6429	507	3	)	)	PUNCT
ejpam-6429	507	4	≤	≤	NOUN
ejpam-6429	507	5	3	3	NUM
ejpam-6429	507	6	+	+	NOUN
ejpam-6429	507	7	|n2(v)|	|n2(v)|	NUM
ejpam-6429	507	8	,	,	PUNCT
ejpam-6429	507	9	then	then	ADV
ejpam-6429	507	10	the	the	DET
ejpam-6429	507	11	result	result	NOUN
ejpam-6429	507	12	follows	follow	VERB
ejpam-6429	507	13	from	from	ADP
ejpam-6429	507	14	theorem	theorem	ADJ
ejpam-6429	507	15	3	3	NUM
ejpam-6429	507	16	.	.	PUNCT
ejpam-6429	508	1	henceforth	henceforth	ADV
ejpam-6429	508	2	,	,	PUNCT
ejpam-6429	508	3	we	we	PRON
ejpam-6429	508	4	assume	assume	VERB
ejpam-6429	508	5	that	that	SCONJ
ejpam-6429	508	6	deg(v	deg(v	PROPN
ejpam-6429	508	7	)	)	PUNCT
ejpam-6429	508	8	≥	≥	NOUN
ejpam-6429	508	9	4	4	NUM
ejpam-6429	508	10	+	+	CCONJ
ejpam-6429	508	11	|n2(v)|	|n2(v)|	NOUN
ejpam-6429	508	12	.	.	PUNCT
ejpam-6429	509	1	let	let	VERB
ejpam-6429	509	2	n(v	n(v	NOUN
ejpam-6429	509	3	)	)	PUNCT
ejpam-6429	509	4	=	=	PRON
ejpam-6429	509	5	{	{	PUNCT
ejpam-6429	509	6	v1	v1	PROPN
ejpam-6429	509	7	,	,	PUNCT
ejpam-6429	509	8	v2	v2	PROPN
ejpam-6429	509	9	,	,	PUNCT
ejpam-6429	509	10	.	.	PUNCT
ejpam-6429	509	11	.	.	PUNCT
ejpam-6429	510	1	.	.	PUNCT
ejpam-6429	511	1	,	,	PUNCT
ejpam-6429	511	2	vk	vk	VERB
ejpam-6429	511	3	}	}	PUNCT
ejpam-6429	511	4	and	and	CCONJ
ejpam-6429	511	5	k	k	NOUN
ejpam-6429	511	6	=	=	SYM
ejpam-6429	511	7	n(v1	n(v1	NOUN
ejpam-6429	511	8	)	)	PUNCT
ejpam-6429	511	9	\	\	NOUN
ejpam-6429	512	1	n	n	CCONJ
ejpam-6429	513	1	[	[	X
ejpam-6429	513	2	v	v	X
ejpam-6429	513	3	]	]	X
ejpam-6429	513	4	=	=	SYM
ejpam-6429	513	5	{	{	PUNCT
ejpam-6429	513	6	w1	w1	NOUN
ejpam-6429	513	7	,	,	PUNCT
ejpam-6429	513	8	w2	w2	NOUN
ejpam-6429	513	9	,	,	PUNCT
ejpam-6429	513	10	.	.	PUNCT
ejpam-6429	513	11	.	.	PUNCT
ejpam-6429	513	12	.	.	PUNCT
ejpam-6429	514	1	,	,	PUNCT
ejpam-6429	514	2	wp	wp	INTJ
ejpam-6429	514	3	}	}	PUNCT
ejpam-6429	514	4	.	.	PUNCT
ejpam-6429	515	1	by	by	ADP
ejpam-6429	515	2	(	(	PUNCT
ejpam-6429	515	3	ii	ii	NOUN
ejpam-6429	515	4	)	)	PUNCT
ejpam-6429	515	5	each	each	DET
ejpam-6429	515	6	vertex	vertex	NOUN
ejpam-6429	515	7	y	y	PROPN
ejpam-6429	515	8	∈	∈	PROPN
ejpam-6429	515	9	n(v	n(v	PROPN
ejpam-6429	515	10	)	)	PUNCT
ejpam-6429	515	11	\	\	PROPN
ejpam-6429	515	12	{	{	PUNCT
ejpam-6429	515	13	v1	v1	NOUN
ejpam-6429	515	14	}	}	PUNCT
ejpam-6429	515	15	has	have	VERB
ejpam-6429	515	16	a	a	DET
ejpam-6429	515	17	neighbor	neighbor	NOUN
ejpam-6429	515	18	in	in	ADP
ejpam-6429	515	19	k.	k.	PROPN
ejpam-6429	515	20	let	let	VERB
ejpam-6429	515	21	s	s	PRON
ejpam-6429	515	22	be	be	AUX
ejpam-6429	515	23	one	one	NUM
ejpam-6429	515	24	of	of	ADP
ejpam-6429	515	25	the	the	DET
ejpam-6429	515	26	largest	large	ADJ
ejpam-6429	515	27	subsets	subset	NOUN
ejpam-6429	515	28	of	of	ADP
ejpam-6429	515	29	n(v	n(v	PROPN
ejpam-6429	515	30	)	)	PUNCT
ejpam-6429	515	31	\	\	PROPN
ejpam-6429	515	32	{	{	PUNCT
ejpam-6429	515	33	v1	v1	NOUN
ejpam-6429	515	34	}	}	PUNCT
ejpam-6429	515	35	such	such	ADJ
ejpam-6429	515	36	that	that	PRON
ejpam-6429	515	37	for	for	ADP
ejpam-6429	515	38	every	every	DET
ejpam-6429	515	39	subset	subset	NOUN
ejpam-6429	515	40	s1	s1	NOUN
ejpam-6429	515	41	⊆	⊆	NUM
ejpam-6429	515	42	s	s	NOUN
ejpam-6429	515	43	,	,	PUNCT
ejpam-6429	515	44	the	the	DET
ejpam-6429	515	45	inequality	inequality	NOUN
ejpam-6429	515	46	|n(s1	|n(s1	ADJ
ejpam-6429	515	47	)	)	PUNCT
ejpam-6429	515	48	\	\	PUNCT
ejpam-6429	516	1	(	(	PUNCT
ejpam-6429	516	2	n	n	CCONJ
ejpam-6429	516	3	[	[	X
ejpam-6429	516	4	v	v	X
ejpam-6429	516	5	]	]	X
ejpam-6429	516	6	∪	∪	X
ejpam-6429	516	7	k)|	k)|	PROPN
ejpam-6429	516	8	≥	≥	NOUN
ejpam-6429	516	9	|s1|	|s1|	NOUN
ejpam-6429	516	10	holds	hold	NOUN
ejpam-6429	516	11	.	.	PUNCT
ejpam-6429	517	1	by	by	ADP
ejpam-6429	517	2	the	the	DET
ejpam-6429	517	3	choice	choice	NOUN
ejpam-6429	517	4	of	of	ADP
ejpam-6429	517	5	s	s	PROPN
ejpam-6429	517	6	,	,	PUNCT
ejpam-6429	517	7	we	we	PRON
ejpam-6429	517	8	have	have	VERB
ejpam-6429	517	9	|n2(v)|	|n2(v)|	NUM
ejpam-6429	517	10	≥	≥	NUM
ejpam-6429	517	11	|k|	|k|	PROPN
ejpam-6429	517	12	+	+	CCONJ
ejpam-6429	517	13	|s|	|s|	PROPN
ejpam-6429	517	14	.	.	PUNCT
ejpam-6429	518	1	furthermore	furthermore	ADV
ejpam-6429	518	2	,	,	PUNCT
ejpam-6429	518	3	every	every	DET
ejpam-6429	518	4	vertex	vertex	NOUN
ejpam-6429	518	5	u	u	NOUN
ejpam-6429	518	6	in	in	ADP
ejpam-6429	518	7	u	u	NOUN
ejpam-6429	518	8	=	=	PROPN
ejpam-6429	518	9	n(v)\(s∪{v1	n(v)\(s∪{v1	NOUN
ejpam-6429	518	10	}	}	PUNCT
ejpam-6429	518	11	)	)	PUNCT
ejpam-6429	518	12	has	have	VERB
ejpam-6429	518	13	at	at	ADV
ejpam-6429	518	14	least	least	ADV
ejpam-6429	518	15	one	one	NUM
ejpam-6429	518	16	neighbor	neighbor	NOUN
ejpam-6429	518	17	in	in	ADP
ejpam-6429	518	18	k	k	PROPN
ejpam-6429	518	19	,	,	PUNCT
ejpam-6429	518	20	and	and	CCONJ
ejpam-6429	518	21	the	the	DET
ejpam-6429	518	22	set	set	ADJ
ejpam-6429	518	23	n(u)\n	n(u)\n	ADJ
ejpam-6429	518	24	[	[	X
ejpam-6429	518	25	v	v	NOUN
ejpam-6429	518	26	]	]	PUNCT
ejpam-6429	518	27	satisfies	satisfy	VERB
ejpam-6429	518	28	n(u)\n	n(u)\n	NOUN
ejpam-6429	518	29	[	[	X
ejpam-6429	518	30	v	v	X
ejpam-6429	518	31	]	]	X
ejpam-6429	518	32	⊆	⊆	NUM
ejpam-6429	518	33	k∪n(s	k∪n(	NOUN
ejpam-6429	518	34	)	)	PUNCT
ejpam-6429	518	35	.	.	PUNCT
ejpam-6429	519	1	additionally	additionally	ADV
ejpam-6429	519	2	,	,	PUNCT
ejpam-6429	519	3	the	the	DET
ejpam-6429	519	4	set	set	NOUN
ejpam-6429	519	5	k	k	PROPN
ejpam-6429	519	6	dominates	dominate	VERB
ejpam-6429	519	7	n(v	n(v	PROPN
ejpam-6429	519	8	)	)	PUNCT
ejpam-6429	519	9	(	(	PUNCT
ejpam-6429	519	10	as	as	SCONJ
ejpam-6429	519	11	stated	state	VERB
ejpam-6429	519	12	in	in	ADP
ejpam-6429	519	13	item	item	NOUN
ejpam-6429	519	14	(	(	PUNCT
ejpam-6429	519	15	ii	ii	NOUN
ejpam-6429	519	16	)	)	PUNCT
ejpam-6429	519	17	)	)	PUNCT
ejpam-6429	519	18	.	.	PUNCT
ejpam-6429	520	1	from	from	ADP
ejpam-6429	520	2	the	the	DET
ejpam-6429	520	3	inequality	inequality	NOUN
ejpam-6429	520	4	4+|k|+|s|	4+|k|+|s|	VERB
ejpam-6429	520	5	≤	≤	NOUN
ejpam-6429	521	1	4+|n2(v)|	4+|n2(v)|	NUM
ejpam-6429	521	2	≤	≤	NUM
ejpam-6429	521	3	deg(v	deg(v	PROPN
ejpam-6429	521	4	)	)	PUNCT
ejpam-6429	521	5	=	=	SYM
ejpam-6429	521	6	|s|+1+|u	|s|+1+|u	NUM
ejpam-6429	521	7	|	|	ADV
ejpam-6429	521	8	,	,	PUNCT
ejpam-6429	521	9	we	we	PRON
ejpam-6429	521	10	conclude	conclude	VERB
ejpam-6429	521	11	that	that	SCONJ
ejpam-6429	521	12	|u	|u	ADJ
ejpam-6429	521	13	|	|	ADV
ejpam-6429	521	14	≥	≥	NOUN
ejpam-6429	521	15	4	4	NUM
ejpam-6429	521	16	.	.	PUNCT
ejpam-6429	522	1	if	if	SCONJ
ejpam-6429	522	2	s	s	VERB
ejpam-6429	522	3	̸=	̸=	PROPN
ejpam-6429	522	4	∅	∅	NOUN
ejpam-6429	522	5	,	,	PUNCT
ejpam-6429	522	6	then	then	ADV
ejpam-6429	522	7	,	,	PUNCT
ejpam-6429	522	8	without	without	ADP
ejpam-6429	522	9	loss	loss	NOUN
ejpam-6429	522	10	of	of	ADP
ejpam-6429	522	11	generality	generality	NOUN
ejpam-6429	522	12	,	,	PUNCT
ejpam-6429	522	13	we	we	PRON
ejpam-6429	522	14	may	may	AUX
ejpam-6429	522	15	assume	assume	VERB
ejpam-6429	522	16	s	s	X
ejpam-6429	522	17	=	=	PUNCT
ejpam-6429	522	18	{	{	PUNCT
ejpam-6429	522	19	v2	v2	PROPN
ejpam-6429	522	20	,	,	PUNCT
ejpam-6429	522	21	v3	v3	PROPN
ejpam-6429	522	22	,	,	PUNCT
ejpam-6429	522	23	.	.	PUNCT
ejpam-6429	522	24	.	.	PUNCT
ejpam-6429	523	1	.	.	PUNCT
ejpam-6429	524	1	,	,	PUNCT
ejpam-6429	524	2	vs	vs	ADP
ejpam-6429	524	3	}	}	PUNCT
ejpam-6429	524	4	.	.	PUNCT
ejpam-6429	524	5	assume	assume	VERB
ejpam-6429	524	6	that	that	SCONJ
ejpam-6429	524	7	g1	g1	PROPN
ejpam-6429	524	8	is	be	AUX
ejpam-6429	524	9	obtained	obtain	VERB
ejpam-6429	524	10	from	from	ADP
ejpam-6429	524	11	g	g	NOUN
ejpam-6429	524	12	by	by	ADP
ejpam-6429	524	13	subdividing	subdivide	VERB
ejpam-6429	524	14	the	the	DET
ejpam-6429	524	15	edges	edge	NOUN
ejpam-6429	524	16	v1wj	v1wj	PUNCT
ejpam-6429	524	17	with	with	ADP
ejpam-6429	524	18	new	new	ADJ
ejpam-6429	524	19	vertices	vertex	NOUN
ejpam-6429	524	20	yj	yj	PROPN
ejpam-6429	524	21	for	for	ADP
ejpam-6429	524	22	all	all	DET
ejpam-6429	524	23	j	j	PROPN
ejpam-6429	524	24	∈	∈	PROPN
ejpam-6429	524	25	{	{	PUNCT
ejpam-6429	524	26	1	1	NUM
ejpam-6429	524	27	,	,	PUNCT
ejpam-6429	524	28	.	.	PUNCT
ejpam-6429	524	29	.	.	PUNCT
ejpam-6429	525	1	.	.	PUNCT
ejpam-6429	526	1	,	,	PUNCT
ejpam-6429	526	2	p	p	X
ejpam-6429	526	3	}	}	PUNCT
ejpam-6429	526	4	,	,	PUNCT
ejpam-6429	526	5	and	and	CCONJ
ejpam-6429	526	6	subdividing	subdivide	VERB
ejpam-6429	526	7	the	the	DET
ejpam-6429	526	8	edges	edge	NOUN
ejpam-6429	526	9	vvi	vvi	NOUN
ejpam-6429	526	10	with	with	ADP
ejpam-6429	526	11	new	new	ADJ
ejpam-6429	526	12	vertices	vertex	NOUN
ejpam-6429	526	13	xi	xi	X
ejpam-6429	526	14	for	for	ADP
ejpam-6429	526	15	1	1	NUM
ejpam-6429	526	16	≤	≤	NUM
ejpam-6429	526	17	i	i	PRON
ejpam-6429	526	18	≤	≤	NOUN
ejpam-6429	526	19	s+2	s+2	NUM
ejpam-6429	526	20	when	when	SCONJ
ejpam-6429	526	21	s	s	VERB
ejpam-6429	526	22	̸=	̸=	PROPN
ejpam-6429	526	23	∅	∅	NOUN
ejpam-6429	526	24	,	,	PUNCT
ejpam-6429	526	25	or	or	CCONJ
ejpam-6429	526	26	for	for	ADP
ejpam-6429	526	27	1	1	NUM
ejpam-6429	526	28	≤	≤	NUM
ejpam-6429	526	29	i	i	PRON
ejpam-6429	526	30	≤	≤	NOUN
ejpam-6429	526	31	3	3	NUM
ejpam-6429	526	32	when	when	SCONJ
ejpam-6429	526	33	s	s	VERB
ejpam-6429	526	34	=	=	PUNCT
ejpam-6429	526	35	∅.	∅.	X
ejpam-6429	526	36	consequently	consequently	ADV
ejpam-6429	526	37	,	,	PUNCT
ejpam-6429	526	38	the	the	DET
ejpam-6429	526	39	number	number	NOUN
ejpam-6429	526	40	of	of	ADP
ejpam-6429	526	41	subdivided	subdivided	ADJ
ejpam-6429	526	42	edges	edge	NOUN
ejpam-6429	526	43	is	be	AUX
ejpam-6429	526	44	|k|	|k|	PROPN
ejpam-6429	526	45	+	+	CCONJ
ejpam-6429	526	46	|s|	|s|	PROPN
ejpam-6429	526	47	+	+	CCONJ
ejpam-6429	526	48	3	3	X
ejpam-6429	526	49	.	.	PUNCT
ejpam-6429	527	1	it	it	PRON
ejpam-6429	527	2	suffices	suffice	VERB
ejpam-6429	527	3	to	to	PART
ejpam-6429	527	4	demonstrate	demonstrate	VERB
ejpam-6429	527	5	that	that	SCONJ
ejpam-6429	527	6	γgr(g1	γgr(g1	ADV
ejpam-6429	527	7	)	)	PUNCT
ejpam-6429	527	8	>	>	X
ejpam-6429	527	9	γgr(g	γgr(g	PROPN
ejpam-6429	527	10	)	)	PUNCT
ejpam-6429	527	11	.	.	PUNCT
ejpam-6429	528	1	let	let	VERB
ejpam-6429	528	2	f	f	PRON
ejpam-6429	528	3	be	be	AUX
ejpam-6429	528	4	a	a	DET
ejpam-6429	528	5	γgr(g1)-function	γgr(g1)-function	PROPN
ejpam-6429	528	6	.	.	PUNCT
ejpam-6429	529	1	by	by	ADP
ejpam-6429	529	2	proposition	proposition	NOUN
ejpam-6429	529	3	3	3	NUM
ejpam-6429	529	4	,	,	PUNCT
ejpam-6429	529	5	we	we	PRON
ejpam-6429	529	6	can	can	AUX
ejpam-6429	529	7	assume	assume	VERB
ejpam-6429	529	8	that	that	SCONJ
ejpam-6429	529	9	f(z	f(z	NOUN
ejpam-6429	529	10	)	)	PUNCT
ejpam-6429	529	11	/∈	/∈	PUNCT
ejpam-6429	530	1	{	{	PUNCT
ejpam-6429	530	2	1	1	NUM
ejpam-6429	530	3	,	,	PUNCT
ejpam-6429	530	4	3	3	NUM
ejpam-6429	530	5	}	}	PUNCT
ejpam-6429	530	6	for	for	ADP
ejpam-6429	530	7	all	all	DET
ejpam-6429	530	8	subdivision	subdivision	NOUN
ejpam-6429	530	9	vertices	vertice	VERB
ejpam-6429	531	1	z.	z.	PROPN
ejpam-6429	531	2	j.	j.	PROPN
ejpam-6429	531	3	j.	j.	PROPN
ejpam-6429	531	4	hamja	hamja	PROPN
ejpam-6429	531	5	et	et	PROPN
ejpam-6429	531	6	al	al	PROPN
ejpam-6429	531	7	.	.	PUNCT
ejpam-6429	531	8	/	/	SYM
ejpam-6429	531	9	eur	eur	PROPN
ejpam-6429	531	10	.	.	PUNCT
ejpam-6429	532	1	j.	j.	PROPN
ejpam-6429	532	2	pure	pure	PROPN
ejpam-6429	532	3	appl	appl	PROPN
ejpam-6429	532	4	.	.	PROPN
ejpam-6429	532	5	math	math	PROPN
ejpam-6429	532	6	,	,	PUNCT
ejpam-6429	532	7	18	18	NUM
ejpam-6429	532	8	(	(	PUNCT
ejpam-6429	532	9	4	4	NUM
ejpam-6429	532	10	)	)	PUNCT
ejpam-6429	532	11	(	(	PUNCT
ejpam-6429	532	12	2025	2025	NUM
ejpam-6429	532	13	)	)	PUNCT
ejpam-6429	532	14	,	,	PUNCT
ejpam-6429	532	15	6429	6429	NUM
ejpam-6429	532	16	13	13	NUM
ejpam-6429	532	17	of	of	ADP
ejpam-6429	532	18	16	16	NUM
ejpam-6429	532	19	if	if	SCONJ
ejpam-6429	532	20	f(v)+	f(v)+	NOUN
ejpam-6429	532	21	f(v1)+	f(v1)+	PROPN
ejpam-6429	532	22	∑s+2	∑s+2	PROPN
ejpam-6429	532	23	i=1	i=1	PROPN
ejpam-6429	532	24	f(xi	f(xi	PROPN
ejpam-6429	532	25	)	)	PUNCT
ejpam-6429	532	26	≥	≥	NOUN
ejpam-6429	532	27	4	4	NUM
ejpam-6429	532	28	,	,	PUNCT
ejpam-6429	532	29	then	then	ADV
ejpam-6429	532	30	by	by	ADP
ejpam-6429	532	31	reassigning	reassign	VERB
ejpam-6429	532	32	v	v	ADP
ejpam-6429	532	33	the	the	DET
ejpam-6429	532	34	value	value	NOUN
ejpam-6429	532	35	3	3	NUM
ejpam-6429	532	36	,	,	PUNCT
ejpam-6429	532	37	v1	v1	VERB
ejpam-6429	532	38	the	the	DET
ejpam-6429	532	39	value	value	NOUN
ejpam-6429	532	40	0	0	NUM
ejpam-6429	532	41	,	,	PUNCT
ejpam-6429	532	42	and	and	CCONJ
ejpam-6429	532	43	wj	wj	VERB
ejpam-6429	532	44	the	the	DET
ejpam-6429	532	45	value	value	NOUN
ejpam-6429	532	46	min{3	min{3	PROPN
ejpam-6429	532	47	,	,	PUNCT
ejpam-6429	532	48	f(wj	f(wj	NUM
ejpam-6429	532	49	)	)	PUNCT
ejpam-6429	532	50	+	+	NUM
ejpam-6429	532	51	f(yj	f(yj	NOUN
ejpam-6429	532	52	)	)	PUNCT
ejpam-6429	532	53	}	}	PUNCT
ejpam-6429	532	54	for	for	ADP
ejpam-6429	532	55	all	all	DET
ejpam-6429	532	56	j	j	PROPN
ejpam-6429	532	57	∈	∈	PROPN
ejpam-6429	532	58	{	{	PUNCT
ejpam-6429	532	59	1	1	NUM
ejpam-6429	532	60	,	,	PUNCT
ejpam-6429	532	61	.	.	PUNCT
ejpam-6429	532	62	.	.	PUNCT
ejpam-6429	533	1	.	.	PUNCT
ejpam-6429	534	1	,	,	PUNCT
ejpam-6429	534	2	p	p	X
ejpam-6429	534	3	}	}	PUNCT
ejpam-6429	534	4	,	,	PUNCT
ejpam-6429	534	5	we	we	PRON
ejpam-6429	534	6	obtain	obtain	VERB
ejpam-6429	534	7	a	a	DET
ejpam-6429	534	8	grd	grd	NOUN
ejpam-6429	534	9	-	-	PUNCT
ejpam-6429	534	10	function	function	NOUN
ejpam-6429	534	11	of	of	ADP
ejpam-6429	534	12	g	g	NOUN
ejpam-6429	534	13	with	with	ADP
ejpam-6429	534	14	weight	weight	NOUN
ejpam-6429	534	15	less	less	ADJ
ejpam-6429	534	16	than	than	ADP
ejpam-6429	534	17	γgr(g1	γgr(g1	ADJ
ejpam-6429	534	18	)	)	PUNCT
ejpam-6429	534	19	.	.	PUNCT
ejpam-6429	535	1	thus	thus	ADV
ejpam-6429	535	2	,	,	PUNCT
ejpam-6429	535	3	we	we	PRON
ejpam-6429	535	4	assume	assume	VERB
ejpam-6429	535	5	that	that	SCONJ
ejpam-6429	535	6	f(v	f(v	NOUN
ejpam-6429	535	7	)	)	PUNCT
ejpam-6429	535	8	+	+	SYM
ejpam-6429	535	9	f(v1	f(v1	NOUN
ejpam-6429	535	10	)	)	PUNCT
ejpam-6429	535	11	+	+	CCONJ
ejpam-6429	535	12	s+2∑	s+2∑	PROPN
ejpam-6429	535	13	i=1	i=1	PROPN
ejpam-6429	535	14	f(xi	f(xi	PROPN
ejpam-6429	535	15	)	)	PUNCT
ejpam-6429	535	16	≤	≤	NOUN
ejpam-6429	535	17	3	3	NUM
ejpam-6429	535	18	.	.	PUNCT
ejpam-6429	536	1	(	(	PUNCT
ejpam-6429	536	2	1	1	X
ejpam-6429	536	3	)	)	PUNCT
ejpam-6429	536	4	we	we	PRON
ejpam-6429	536	5	now	now	ADV
ejpam-6429	536	6	distinguish	distinguish	VERB
ejpam-6429	536	7	four	four	NUM
ejpam-6429	536	8	different	different	ADJ
ejpam-6429	536	9	situations	situation	NOUN
ejpam-6429	536	10	.	.	PUNCT
ejpam-6429	537	1	case	case	NOUN
ejpam-6429	537	2	1	1	NUM
ejpam-6429	537	3	.	.	NUM
ejpam-6429	537	4	f(v	f(v	NOUN
ejpam-6429	537	5	)	)	PUNCT
ejpam-6429	538	1	=	=	SYM
ejpam-6429	538	2	3	3	X
ejpam-6429	538	3	.	.	NOUN
ejpam-6429	538	4	from	from	ADP
ejpam-6429	538	5	equation	equation	NOUN
ejpam-6429	538	6	(	(	PUNCT
ejpam-6429	538	7	1	1	NUM
ejpam-6429	538	8	)	)	PUNCT
ejpam-6429	538	9	,	,	PUNCT
ejpam-6429	538	10	we	we	PRON
ejpam-6429	538	11	have	have	AUX
ejpam-6429	538	12	f(v1	f(v1	NOUN
ejpam-6429	538	13	)	)	PUNCT
ejpam-6429	538	14	=	=	SYM
ejpam-6429	538	15	f(x1	f(x1	X
ejpam-6429	538	16	)	)	PUNCT
ejpam-6429	538	17	=	=	SYM
ejpam-6429	538	18	·	·	PUNCT
ejpam-6429	538	19	·	·	PUNCT
ejpam-6429	538	20	·	·	PUNCT
ejpam-6429	539	1	=	=	SYM
ejpam-6429	539	2	f(xs+2	f(xs+2	PROPN
ejpam-6429	539	3	)	)	PUNCT
ejpam-6429	539	4	=	=	SYM
ejpam-6429	540	1	0	0	X
ejpam-6429	540	2	.	.	PUNCT
ejpam-6429	541	1	if	if	SCONJ
ejpam-6429	541	2	∑p	∑p	ADJ
ejpam-6429	541	3	j=1	j=1	PROPN
ejpam-6429	541	4	f(yj	f(yj	NOUN
ejpam-6429	541	5	)	)	PUNCT
ejpam-6429	541	6	≥	≥	NOUN
ejpam-6429	541	7	4	4	NUM
ejpam-6429	541	8	,	,	PUNCT
ejpam-6429	541	9	then	then	ADV
ejpam-6429	541	10	by	by	ADP
ejpam-6429	541	11	assigning	assign	VERB
ejpam-6429	541	12	the	the	DET
ejpam-6429	541	13	value	value	NOUN
ejpam-6429	541	14	3	3	NUM
ejpam-6429	541	15	to	to	PART
ejpam-6429	541	16	v1	v1	VERB
ejpam-6429	541	17	,	,	PUNCT
ejpam-6429	541	18	we	we	PRON
ejpam-6429	541	19	can	can	AUX
ejpam-6429	541	20	obtain	obtain	VERB
ejpam-6429	541	21	a	a	DET
ejpam-6429	541	22	grd	grd	NOUN
ejpam-6429	541	23	-	-	PUNCT
ejpam-6429	541	24	function	function	NOUN
ejpam-6429	541	25	for	for	ADP
ejpam-6429	541	26	g	g	NOUN
ejpam-6429	541	27	with	with	ADP
ejpam-6429	541	28	weight	weight	NOUN
ejpam-6429	541	29	less	less	ADJ
ejpam-6429	541	30	than	than	ADP
ejpam-6429	541	31	γgr(g1	γgr(g1	ADJ
ejpam-6429	541	32	)	)	PUNCT
ejpam-6429	541	33	.	.	PUNCT
ejpam-6429	542	1	therefore	therefore	ADV
ejpam-6429	542	2	,	,	PUNCT
ejpam-6429	542	3	we	we	PRON
ejpam-6429	542	4	assume	assume	VERB
ejpam-6429	542	5	that	that	SCONJ
ejpam-6429	542	6	∑p	∑p	ADJ
ejpam-6429	542	7	j=1	j=1	ADJ
ejpam-6429	542	8	f(yj	f(yj	NOUN
ejpam-6429	542	9	)	)	PUNCT
ejpam-6429	542	10	≤	≤	NUM
ejpam-6429	542	11	3	3	NUM
ejpam-6429	542	12	.	.	PUNCT
ejpam-6429	542	13	based	base	VERB
ejpam-6429	542	14	on	on	ADP
ejpam-6429	542	15	our	our	PRON
ejpam-6429	542	16	earlier	early	ADJ
ejpam-6429	542	17	assumption	assumption	NOUN
ejpam-6429	542	18	,	,	PUNCT
ejpam-6429	542	19	we	we	PRON
ejpam-6429	542	20	also	also	ADV
ejpam-6429	542	21	have	have	VERB
ejpam-6429	542	22	∑p	∑p	PROPN
ejpam-6429	542	23	i=1	i=1	PRON
ejpam-6429	542	24	f(yi	f(yi	PROPN
ejpam-6429	542	25	)	)	PUNCT
ejpam-6429	542	26	≤	≤	NOUN
ejpam-6429	542	27	2	2	NUM
ejpam-6429	542	28	.	.	PUNCT
ejpam-6429	543	1	if	if	SCONJ
ejpam-6429	543	2	there	there	PRON
ejpam-6429	543	3	exists	exist	VERB
ejpam-6429	543	4	some	some	DET
ejpam-6429	543	5	j	j	PROPN
ejpam-6429	543	6	∈	∈	PROPN
ejpam-6429	543	7	{	{	PUNCT
ejpam-6429	543	8	1	1	NUM
ejpam-6429	543	9	,	,	PUNCT
ejpam-6429	543	10	2	2	NUM
ejpam-6429	543	11	,	,	PUNCT
ejpam-6429	543	12	.	.	PUNCT
ejpam-6429	543	13	.	.	PUNCT
ejpam-6429	544	1	.	.	PUNCT
ejpam-6429	545	1	,	,	PUNCT
ejpam-6429	545	2	p	p	X
ejpam-6429	545	3	}	}	PUNCT
ejpam-6429	545	4	,	,	PUNCT
ejpam-6429	545	5	say	say	VERB
ejpam-6429	545	6	j	j	PROPN
ejpam-6429	545	7	=	=	SYM
ejpam-6429	545	8	1	1	NUM
ejpam-6429	545	9	,	,	PUNCT
ejpam-6429	545	10	such	such	ADJ
ejpam-6429	545	11	that	that	DET
ejpam-6429	545	12	f(y1	f(y1	NOUN
ejpam-6429	545	13	)	)	PUNCT
ejpam-6429	545	14	=	=	SYM
ejpam-6429	545	15	2	2	NUM
ejpam-6429	545	16	,	,	PUNCT
ejpam-6429	545	17	then	then	ADV
ejpam-6429	545	18	it	it	PRON
ejpam-6429	545	19	follows	follow	VERB
ejpam-6429	545	20	that	that	SCONJ
ejpam-6429	545	21	f(yj	f(yj	NOUN
ejpam-6429	545	22	)	)	PUNCT
ejpam-6429	545	23	=	=	SYM
ejpam-6429	545	24	0	0	NUM
ejpam-6429	545	25	for	for	ADP
ejpam-6429	545	26	all	all	DET
ejpam-6429	545	27	j	j	PROPN
ejpam-6429	545	28	∈	∈	PROPN
ejpam-6429	545	29	{	{	PUNCT
ejpam-6429	545	30	2	2	NUM
ejpam-6429	545	31	,	,	PUNCT
ejpam-6429	545	32	.	.	PUNCT
ejpam-6429	545	33	.	.	PUNCT
ejpam-6429	546	1	.	.	PUNCT
ejpam-6429	547	1	,	,	PUNCT
ejpam-6429	547	2	p	p	X
ejpam-6429	547	3	}	}	PUNCT
ejpam-6429	547	4	.	.	PUNCT
ejpam-6429	548	1	in	in	ADP
ejpam-6429	548	2	this	this	DET
ejpam-6429	548	3	case	case	NOUN
ejpam-6429	548	4	,	,	PUNCT
ejpam-6429	548	5	reassigning	reassign	VERB
ejpam-6429	548	6	w1	w1	NOUN
ejpam-6429	548	7	the	the	DET
ejpam-6429	548	8	value	value	NOUN
ejpam-6429	548	9	min{3	min{3	PROPN
ejpam-6429	548	10	,	,	PUNCT
ejpam-6429	548	11	1+f(w1	1+f(w1	NUM
ejpam-6429	548	12	)	)	PUNCT
ejpam-6429	548	13	}	}	PUNCT
ejpam-6429	548	14	provides	provide	VERB
ejpam-6429	548	15	a	a	DET
ejpam-6429	548	16	grd	grd	NOUN
ejpam-6429	548	17	-	-	PUNCT
ejpam-6429	548	18	function	function	NOUN
ejpam-6429	548	19	for	for	ADP
ejpam-6429	548	20	g	g	NOUN
ejpam-6429	548	21	with	with	ADP
ejpam-6429	548	22	weight	weight	NOUN
ejpam-6429	548	23	less	less	ADJ
ejpam-6429	548	24	than	than	ADP
ejpam-6429	548	25	γgr(g1	γgr(g1	ADJ
ejpam-6429	548	26	)	)	PUNCT
ejpam-6429	548	27	.	.	PUNCT
ejpam-6429	549	1	hence	hence	ADV
ejpam-6429	549	2	,	,	PUNCT
ejpam-6429	549	3	we	we	PRON
ejpam-6429	549	4	assume	assume	VERB
ejpam-6429	549	5	that	that	SCONJ
ejpam-6429	549	6	∑p	∑p	ADJ
ejpam-6429	549	7	j=1	j=1	ADJ
ejpam-6429	549	8	f(yj	f(yj	NOUN
ejpam-6429	549	9	)	)	PUNCT
ejpam-6429	549	10	=	=	SYM
ejpam-6429	550	1	0	0	X
ejpam-6429	550	2	.	.	PUNCT
ejpam-6429	550	3	to	to	PART
ejpam-6429	550	4	protect	protect	VERB
ejpam-6429	550	5	the	the	DET
ejpam-6429	550	6	vertices	vertex	NOUN
ejpam-6429	550	7	yj	yj	PROPN
ejpam-6429	550	8	,	,	PUNCT
ejpam-6429	550	9	the	the	DET
ejpam-6429	550	10	vertex	vertex	NOUN
ejpam-6429	550	11	wj	wj	PROPN
ejpam-6429	550	12	must	must	AUX
ejpam-6429	550	13	be	be	AUX
ejpam-6429	550	14	the	the	DET
ejpam-6429	550	15	moving	move	VERB
ejpam-6429	550	16	neighbor	neighbor	NOUN
ejpam-6429	550	17	of	of	ADP
ejpam-6429	550	18	yj	yj	PROPN
ejpam-6429	550	19	,	,	PUNCT
ejpam-6429	550	20	implying	imply	VERB
ejpam-6429	550	21	that	that	SCONJ
ejpam-6429	550	22	f(wj	f(wj	NOUN
ejpam-6429	550	23	)	)	PUNCT
ejpam-6429	550	24	=	=	SYM
ejpam-6429	550	25	2	2	NUM
ejpam-6429	550	26	for	for	ADP
ejpam-6429	550	27	each	each	DET
ejpam-6429	550	28	j	j	PROPN
ejpam-6429	550	29	∈	∈	PROPN
ejpam-6429	550	30	{	{	PUNCT
ejpam-6429	550	31	1	1	NUM
ejpam-6429	550	32	,	,	PUNCT
ejpam-6429	550	33	2	2	NUM
ejpam-6429	550	34	,	,	PUNCT
ejpam-6429	550	35	.	.	PUNCT
ejpam-6429	550	36	.	.	PUNCT
ejpam-6429	551	1	.	.	PUNCT
ejpam-6429	552	1	,	,	PUNCT
ejpam-6429	552	2	p	p	X
ejpam-6429	552	3	}	}	PUNCT
ejpam-6429	552	4	.	.	PUNCT
ejpam-6429	553	1	then	then	ADV
ejpam-6429	553	2	,	,	PUNCT
ejpam-6429	553	3	by	by	ADP
ejpam-6429	553	4	reassigning	reassign	VERB
ejpam-6429	553	5	v	v	ADP
ejpam-6429	553	6	the	the	DET
ejpam-6429	553	7	value	value	NOUN
ejpam-6429	553	8	2	2	NUM
ejpam-6429	553	9	,	,	PUNCT
ejpam-6429	553	10	we	we	PRON
ejpam-6429	553	11	obtain	obtain	VERB
ejpam-6429	553	12	a	a	DET
ejpam-6429	553	13	grd	grd	NOUN
ejpam-6429	553	14	-	-	PUNCT
ejpam-6429	553	15	function	function	NOUN
ejpam-6429	553	16	for	for	ADP
ejpam-6429	553	17	g	g	NOUN
ejpam-6429	553	18	,	,	PUNCT
ejpam-6429	553	19	where	where	SCONJ
ejpam-6429	553	20	every	every	DET
ejpam-6429	553	21	vertex	vertex	NOUN
ejpam-6429	553	22	of	of	ADP
ejpam-6429	553	23	u	u	PROPN
ejpam-6429	553	24	has	have	VERB
ejpam-6429	553	25	a	a	DET
ejpam-6429	553	26	neighbor	neighbor	NOUN
ejpam-6429	553	27	in	in	ADP
ejpam-6429	553	28	k	k	PROPN
ejpam-6429	553	29	,	,	PUNCT
ejpam-6429	553	30	with	with	ADP
ejpam-6429	553	31	weight	weight	NOUN
ejpam-6429	553	32	less	less	ADJ
ejpam-6429	553	33	than	than	ADP
ejpam-6429	553	34	γgr(g1	γgr(g1	ADJ
ejpam-6429	553	35	)	)	PUNCT
ejpam-6429	553	36	.	.	PUNCT
ejpam-6429	554	1	case	case	NOUN
ejpam-6429	554	2	2	2	NUM
ejpam-6429	554	3	.	.	NUM
ejpam-6429	554	4	f(v	f(v	NOUN
ejpam-6429	554	5	)	)	PUNCT
ejpam-6429	555	1	=	=	SYM
ejpam-6429	555	2	2	2	X
ejpam-6429	555	3	.	.	NOUN
ejpam-6429	555	4	from	from	ADP
ejpam-6429	555	5	equation	equation	NOUN
ejpam-6429	555	6	(	(	PUNCT
ejpam-6429	555	7	1	1	NUM
ejpam-6429	555	8	)	)	PUNCT
ejpam-6429	555	9	and	and	CCONJ
ejpam-6429	555	10	our	our	PRON
ejpam-6429	555	11	previous	previous	ADJ
ejpam-6429	555	12	assumption	assumption	NOUN
ejpam-6429	555	13	,	,	PUNCT
ejpam-6429	555	14	it	it	PRON
ejpam-6429	555	15	follows	follow	VERB
ejpam-6429	555	16	that	that	SCONJ
ejpam-6429	555	17	f(x1	f(x1	NOUN
ejpam-6429	555	18	)	)	PUNCT
ejpam-6429	555	19	=	=	SYM
ejpam-6429	555	20	·	·	PUNCT
ejpam-6429	555	21	·	·	PUNCT
ejpam-6429	555	22	·	·	PUNCT
ejpam-6429	556	1	=	=	SYM
ejpam-6429	556	2	f(xs+2	f(xs+2	PROPN
ejpam-6429	556	3	)	)	PUNCT
ejpam-6429	556	4	=	=	SYM
ejpam-6429	557	1	0	0	X
ejpam-6429	557	2	.	.	PUNCT
ejpam-6429	558	1	if	if	SCONJ
ejpam-6429	558	2	f(v1	f(v1	ADJ
ejpam-6429	558	3	)	)	PUNCT
ejpam-6429	558	4	=	=	SYM
ejpam-6429	558	5	1	1	NUM
ejpam-6429	558	6	,	,	PUNCT
ejpam-6429	558	7	then	then	ADV
ejpam-6429	558	8	by	by	ADP
ejpam-6429	558	9	reassigning	reassign	VERB
ejpam-6429	558	10	the	the	DET
ejpam-6429	558	11	value	value	NOUN
ejpam-6429	558	12	0	0	NUM
ejpam-6429	558	13	to	to	PART
ejpam-6429	558	14	v1	v1	VERB
ejpam-6429	558	15	and	and	CCONJ
ejpam-6429	558	16	the	the	DET
ejpam-6429	558	17	value	value	NOUN
ejpam-6429	558	18	min{3	min{3	NOUN
ejpam-6429	558	19	,	,	PUNCT
ejpam-6429	558	20	f(wi	f(wi	NOUN
ejpam-6429	558	21	)	)	PUNCT
ejpam-6429	559	1	+	+	CCONJ
ejpam-6429	559	2	f(yi	f(yi	NUM
ejpam-6429	559	3	)	)	PUNCT
ejpam-6429	559	4	}	}	PUNCT
ejpam-6429	559	5	to	to	ADP
ejpam-6429	559	6	wi	wi	PROPN
ejpam-6429	559	7	for	for	ADP
ejpam-6429	559	8	all	all	PRON
ejpam-6429	559	9	i	i	PRON
ejpam-6429	559	10	∈	∈	PROPN
ejpam-6429	559	11	{	{	PUNCT
ejpam-6429	559	12	1	1	NUM
ejpam-6429	559	13	,	,	PUNCT
ejpam-6429	559	14	2	2	NUM
ejpam-6429	559	15	,	,	PUNCT
ejpam-6429	559	16	.	.	PUNCT
ejpam-6429	559	17	.	.	PUNCT
ejpam-6429	560	1	.	.	PUNCT
ejpam-6429	561	1	,	,	PUNCT
ejpam-6429	561	2	p	p	X
ejpam-6429	561	3	}	}	PUNCT
ejpam-6429	561	4	,	,	PUNCT
ejpam-6429	561	5	we	we	PRON
ejpam-6429	561	6	obtain	obtain	VERB
ejpam-6429	561	7	a	a	DET
ejpam-6429	561	8	grd	grd	NOUN
ejpam-6429	561	9	-	-	PUNCT
ejpam-6429	561	10	function	function	NOUN
ejpam-6429	561	11	for	for	ADP
ejpam-6429	561	12	g	g	NOUN
ejpam-6429	561	13	with	with	ADP
ejpam-6429	561	14	weight	weight	NOUN
ejpam-6429	561	15	less	less	ADJ
ejpam-6429	561	16	than	than	ADP
ejpam-6429	561	17	γgr(g1	γgr(g1	ADJ
ejpam-6429	561	18	)	)	PUNCT
ejpam-6429	561	19	(	(	PUNCT
ejpam-6429	561	20	note	note	VERB
ejpam-6429	561	21	that	that	SCONJ
ejpam-6429	561	22	v	v	NOUN
ejpam-6429	561	23	is	be	AUX
ejpam-6429	561	24	a	a	DET
ejpam-6429	561	25	moving	move	VERB
ejpam-6429	561	26	neighbor	neighbor	NOUN
ejpam-6429	561	27	of	of	ADP
ejpam-6429	561	28	v1	v1	NOUN
ejpam-6429	561	29	)	)	PUNCT
ejpam-6429	561	30	.	.	PUNCT
ejpam-6429	562	1	now	now	ADV
ejpam-6429	562	2	,	,	PUNCT
ejpam-6429	562	3	suppose	suppose	VERB
ejpam-6429	562	4	f(v1	f(v1	NOUN
ejpam-6429	562	5	)	)	PUNCT
ejpam-6429	562	6	=	=	SYM
ejpam-6429	563	1	0	0	X
ejpam-6429	563	2	.	.	PUNCT
ejpam-6429	564	1	this	this	PRON
ejpam-6429	564	2	implies	imply	VERB
ejpam-6429	564	3	that	that	SCONJ
ejpam-6429	564	4	v	v	NOUN
ejpam-6429	564	5	is	be	AUX
ejpam-6429	564	6	a	a	DET
ejpam-6429	564	7	moving	move	VERB
ejpam-6429	564	8	neighbor	neighbor	NOUN
ejpam-6429	564	9	only	only	ADV
ejpam-6429	564	10	for	for	ADP
ejpam-6429	564	11	x1	x1	PROPN
ejpam-6429	564	12	,	,	PUNCT
ejpam-6429	564	13	and	and	CCONJ
ejpam-6429	564	14	thus	thus	ADV
ejpam-6429	564	15	,	,	PUNCT
ejpam-6429	564	16	vi	vi	PROPN
ejpam-6429	564	17	is	be	AUX
ejpam-6429	564	18	the	the	DET
ejpam-6429	564	19	moving	move	VERB
ejpam-6429	564	20	neighbor	neighbor	NOUN
ejpam-6429	564	21	of	of	ADP
ejpam-6429	564	22	xi	xi	PROPN
ejpam-6429	564	23	in	in	ADP
ejpam-6429	564	24	order	order	NOUN
ejpam-6429	564	25	to	to	PART
ejpam-6429	564	26	protect	protect	VERB
ejpam-6429	564	27	xi	xi	PROPN
ejpam-6429	564	28	.	.	PUNCT
ejpam-6429	565	1	consequently	consequently	ADV
ejpam-6429	565	2	,	,	PUNCT
ejpam-6429	565	3	we	we	PRON
ejpam-6429	565	4	have	have	VERB
ejpam-6429	565	5	f(vi	f(vi	NOUN
ejpam-6429	565	6	)	)	PUNCT
ejpam-6429	565	7	≥	≥	NOUN
ejpam-6429	565	8	2	2	NUM
ejpam-6429	565	9	for	for	ADP
ejpam-6429	565	10	all	all	PRON
ejpam-6429	565	11	i	i	PRON
ejpam-6429	565	12	∈	∈	PROPN
ejpam-6429	565	13	{	{	PUNCT
ejpam-6429	565	14	2	2	NUM
ejpam-6429	565	15	,	,	PUNCT
ejpam-6429	565	16	3	3	NUM
ejpam-6429	565	17	,	,	PUNCT
ejpam-6429	565	18	.	.	PUNCT
ejpam-6429	565	19	.	.	PUNCT
ejpam-6429	565	20	.	.	PUNCT
ejpam-6429	566	1	,	,	PUNCT
ejpam-6429	566	2	s	s	PART
ejpam-6429	566	3	+	+	ADJ
ejpam-6429	566	4	2	2	NUM
ejpam-6429	566	5	}	}	PUNCT
ejpam-6429	566	6	.	.	PUNCT
ejpam-6429	567	1	if	if	SCONJ
ejpam-6429	567	2	∑p	∑p	ADJ
ejpam-6429	567	3	j=1	j=1	PROPN
ejpam-6429	567	4	f(yj	f(yj	NOUN
ejpam-6429	567	5	)	)	PUNCT
ejpam-6429	567	6	≥	≥	NOUN
ejpam-6429	567	7	4	4	NUM
ejpam-6429	567	8	,	,	PUNCT
ejpam-6429	567	9	then	then	ADV
ejpam-6429	567	10	reassigning	reassign	VERB
ejpam-6429	567	11	the	the	DET
ejpam-6429	567	12	value	value	NOUN
ejpam-6429	567	13	3	3	NUM
ejpam-6429	567	14	to	to	PART
ejpam-6429	567	15	v1	v1	VERB
ejpam-6429	567	16	provides	provide	VERB
ejpam-6429	567	17	a	a	DET
ejpam-6429	567	18	grd	grd	NOUN
ejpam-6429	567	19	-	-	PUNCT
ejpam-6429	567	20	function	function	NOUN
ejpam-6429	567	21	for	for	ADP
ejpam-6429	567	22	g	g	NOUN
ejpam-6429	567	23	with	with	ADP
ejpam-6429	567	24	weight	weight	NOUN
ejpam-6429	567	25	less	less	ADJ
ejpam-6429	567	26	than	than	ADP
ejpam-6429	567	27	γgr(g1	γgr(g1	ADJ
ejpam-6429	567	28	)	)	PUNCT
ejpam-6429	567	29	.	.	PUNCT
ejpam-6429	568	1	hence	hence	ADV
ejpam-6429	568	2	,	,	PUNCT
ejpam-6429	568	3	we	we	PRON
ejpam-6429	568	4	assume	assume	VERB
ejpam-6429	568	5	that	that	SCONJ
ejpam-6429	568	6	∑p	∑p	ADJ
ejpam-6429	568	7	j=1	j=1	ADJ
ejpam-6429	568	8	f(yj	f(yj	NOUN
ejpam-6429	568	9	)	)	PUNCT
ejpam-6429	568	10	≤	≤	NUM
ejpam-6429	568	11	3	3	NUM
ejpam-6429	568	12	.	.	PUNCT
ejpam-6429	568	13	from	from	ADP
ejpam-6429	568	14	our	our	PRON
ejpam-6429	568	15	earlier	early	ADJ
ejpam-6429	568	16	assumption	assumption	NOUN
ejpam-6429	568	17	,	,	PUNCT
ejpam-6429	568	18	it	it	PRON
ejpam-6429	568	19	follows	follow	VERB
ejpam-6429	568	20	that	that	SCONJ
ejpam-6429	568	21	either	either	CCONJ
ejpam-6429	568	22	f(yj	f(yj	NOUN
ejpam-6429	568	23	)	)	PUNCT
ejpam-6429	568	24	=	=	SYM
ejpam-6429	568	25	0	0	NUM
ejpam-6429	568	26	for	for	ADP
ejpam-6429	568	27	all	all	DET
ejpam-6429	568	28	j	j	PROPN
ejpam-6429	568	29	∈	∈	PROPN
ejpam-6429	568	30	{	{	PUNCT
ejpam-6429	568	31	1	1	NUM
ejpam-6429	568	32	,	,	PUNCT
ejpam-6429	568	33	2	2	NUM
ejpam-6429	568	34	,	,	PUNCT
ejpam-6429	568	35	.	.	PUNCT
ejpam-6429	568	36	.	.	PUNCT
ejpam-6429	568	37	.	.	PUNCT
ejpam-6429	569	1	,	,	PUNCT
ejpam-6429	569	2	p	p	X
ejpam-6429	569	3	}	}	PUNCT
ejpam-6429	569	4	or	or	CCONJ
ejpam-6429	569	5	f(yj	f(yj	NOUN
ejpam-6429	569	6	)	)	PUNCT
ejpam-6429	569	7	=	=	SYM
ejpam-6429	569	8	2	2	NUM
ejpam-6429	569	9	for	for	ADP
ejpam-6429	569	10	exactly	exactly	ADV
ejpam-6429	569	11	one	one	NUM
ejpam-6429	569	12	j.	j.	NOUN
ejpam-6429	569	13	in	in	ADP
ejpam-6429	569	14	the	the	DET
ejpam-6429	569	15	first	first	ADJ
ejpam-6429	569	16	case	case	NOUN
ejpam-6429	569	17	,	,	PUNCT
ejpam-6429	569	18	we	we	PRON
ejpam-6429	569	19	have	have	VERB
ejpam-6429	569	20	f(wj	f(wj	NOUN
ejpam-6429	569	21	)	)	PUNCT
ejpam-6429	569	22	≥	≥	NOUN
ejpam-6429	569	23	2	2	NUM
ejpam-6429	569	24	,	,	PUNCT
ejpam-6429	569	25	and	and	CCONJ
ejpam-6429	569	26	wj	wj	PROPN
ejpam-6429	569	27	is	be	AUX
ejpam-6429	569	28	the	the	DET
ejpam-6429	569	29	moving	move	VERB
ejpam-6429	569	30	neighbor	neighbor	NOUN
ejpam-6429	569	31	only	only	ADV
ejpam-6429	569	32	for	for	ADP
ejpam-6429	569	33	yj	yj	PROPN
ejpam-6429	569	34	if	if	SCONJ
ejpam-6429	569	35	f(wj	f(wj	NOUN
ejpam-6429	569	36	)	)	PUNCT
ejpam-6429	569	37	=	=	SYM
ejpam-6429	569	38	2	2	X
ejpam-6429	569	39	.	.	PUNCT
ejpam-6429	569	40	reassigning	reassign	VERB
ejpam-6429	569	41	the	the	DET
ejpam-6429	569	42	value	value	NOUN
ejpam-6429	569	43	1	1	NUM
ejpam-6429	569	44	to	to	PART
ejpam-6429	569	45	v	v	NOUN
ejpam-6429	569	46	then	then	ADV
ejpam-6429	569	47	provides	provide	VERB
ejpam-6429	569	48	a	a	DET
ejpam-6429	569	49	grd	grd	NOUN
ejpam-6429	569	50	-	-	PUNCT
ejpam-6429	569	51	function	function	NOUN
ejpam-6429	569	52	for	for	ADP
ejpam-6429	569	53	g	g	NOUN
ejpam-6429	569	54	with	with	ADP
ejpam-6429	569	55	weight	weight	NOUN
ejpam-6429	569	56	less	less	ADJ
ejpam-6429	569	57	than	than	ADP
ejpam-6429	569	58	γgr(g1	γgr(g1	ADJ
ejpam-6429	569	59	)	)	PUNCT
ejpam-6429	569	60	.	.	PUNCT
ejpam-6429	570	1	in	in	ADP
ejpam-6429	570	2	the	the	DET
ejpam-6429	570	3	second	second	ADJ
ejpam-6429	570	4	case	case	NOUN
ejpam-6429	570	5	,	,	PUNCT
ejpam-6429	570	6	reassigning	reassign	VERB
ejpam-6429	570	7	the	the	DET
ejpam-6429	570	8	value	value	NOUN
ejpam-6429	570	9	1	1	NUM
ejpam-6429	570	10	to	to	ADP
ejpam-6429	570	11	wj	wj	PROPN
ejpam-6429	570	12	and	and	CCONJ
ejpam-6429	570	13	the	the	DET
ejpam-6429	570	14	value	value	NOUN
ejpam-6429	570	15	min{3	min{3	PROPN
ejpam-6429	570	16	,	,	PUNCT
ejpam-6429	570	17	f(wi)+1	f(wi)+1	PROPN
ejpam-6429	570	18	}	}	PUNCT
ejpam-6429	570	19	to	to	ADP
ejpam-6429	570	20	wi	wi	PROPN
ejpam-6429	570	21	for	for	ADP
ejpam-6429	570	22	i	i	PRON
ejpam-6429	570	23	̸=	̸=	PROPN
ejpam-6429	570	24	j	j	PROPN
ejpam-6429	570	25	provides	provide	VERB
ejpam-6429	570	26	a	a	DET
ejpam-6429	570	27	grd	grd	NOUN
ejpam-6429	570	28	-	-	PUNCT
ejpam-6429	570	29	function	function	NOUN
ejpam-6429	570	30	for	for	ADP
ejpam-6429	570	31	g	g	NOUN
ejpam-6429	570	32	with	with	ADP
ejpam-6429	570	33	weight	weight	NOUN
ejpam-6429	570	34	less	less	ADJ
ejpam-6429	570	35	than	than	ADP
ejpam-6429	570	36	γgr(g1	γgr(g1	ADJ
ejpam-6429	570	37	)	)	PUNCT
ejpam-6429	570	38	.	.	PUNCT
ejpam-6429	571	1	case	case	NOUN
ejpam-6429	571	2	3	3	NUM
ejpam-6429	571	3	.	.	NUM
ejpam-6429	571	4	f(v	f(v	NOUN
ejpam-6429	571	5	)	)	PUNCT
ejpam-6429	572	1	=	=	SYM
ejpam-6429	572	2	1	1	X
ejpam-6429	572	3	.	.	PUNCT
ejpam-6429	572	4	to	to	PART
ejpam-6429	572	5	protect	protect	VERB
ejpam-6429	572	6	x1	x1	PROPN
ejpam-6429	572	7	,	,	PUNCT
ejpam-6429	572	8	it	it	PRON
ejpam-6429	572	9	is	be	AUX
ejpam-6429	572	10	required	require	VERB
ejpam-6429	572	11	that	that	SCONJ
ejpam-6429	572	12	f(x1	f(x1	NOUN
ejpam-6429	572	13	)	)	PUNCT
ejpam-6429	572	14	+	+	SYM
ejpam-6429	572	15	f(v1	f(v1	NOUN
ejpam-6429	572	16	)	)	PUNCT
ejpam-6429	572	17	≥	≥	NOUN
ejpam-6429	572	18	2	2	NUM
ejpam-6429	572	19	.	.	PUNCT
ejpam-6429	572	20	from	from	ADP
ejpam-6429	572	21	our	our	PRON
ejpam-6429	572	22	previous	previous	ADJ
ejpam-6429	572	23	assumption	assumption	NOUN
ejpam-6429	572	24	and	and	CCONJ
ejpam-6429	572	25	equation	equation	NOUN
ejpam-6429	572	26	(	(	PUNCT
ejpam-6429	572	27	1	1	NUM
ejpam-6429	572	28	)	)	PUNCT
ejpam-6429	572	29	,	,	PUNCT
ejpam-6429	572	30	it	it	PRON
ejpam-6429	572	31	follows	follow	VERB
ejpam-6429	572	32	that	that	SCONJ
ejpam-6429	572	33	∑s+2	∑s+2	PROPN
ejpam-6429	572	34	i=1	i=1	X
ejpam-6429	572	35	f(xi	f(xi	PROPN
ejpam-6429	572	36	)	)	PUNCT
ejpam-6429	572	37	=	=	SYM
ejpam-6429	573	1	0	0	X
ejpam-6429	573	2	.	.	PUNCT
ejpam-6429	574	1	if	if	SCONJ
ejpam-6429	574	2	f(v1	f(v1	ADJ
ejpam-6429	574	3	)	)	PUNCT
ejpam-6429	574	4	=	=	SYM
ejpam-6429	574	5	2	2	NUM
ejpam-6429	574	6	,	,	PUNCT
ejpam-6429	574	7	then	then	ADV
ejpam-6429	574	8	f(x1	f(x1	ADJ
ejpam-6429	574	9	)	)	PUNCT
ejpam-6429	575	1	=	=	SYM
ejpam-6429	575	2	0	0	NUM
ejpam-6429	575	3	,	,	PUNCT
ejpam-6429	575	4	which	which	PRON
ejpam-6429	575	5	means	mean	VERB
ejpam-6429	575	6	that	that	SCONJ
ejpam-6429	575	7	v1	v1	NOUN
ejpam-6429	575	8	is	be	AUX
ejpam-6429	575	9	a	a	DET
ejpam-6429	575	10	moving	move	VERB
ejpam-6429	575	11	neighbor	neighbor	NOUN
ejpam-6429	575	12	only	only	ADV
ejpam-6429	575	13	for	for	ADP
ejpam-6429	575	14	x1	x1	PROPN
ejpam-6429	575	15	.	.	PUNCT
ejpam-6429	576	1	by	by	ADP
ejpam-6429	576	2	assigning	assign	VERB
ejpam-6429	576	3	the	the	DET
ejpam-6429	576	4	value	value	NOUN
ejpam-6429	576	5	0	0	NUM
ejpam-6429	576	6	to	to	ADP
ejpam-6429	576	7	v	v	NOUN
ejpam-6429	576	8	and	and	CCONJ
ejpam-6429	576	9	the	the	DET
ejpam-6429	576	10	value	value	NOUN
ejpam-6429	576	11	min{3	min{3	NOUN
ejpam-6429	576	12	,	,	PUNCT
ejpam-6429	576	13	f(wi	f(wi	NOUN
ejpam-6429	576	14	)	)	PUNCT
ejpam-6429	576	15	+	+	CCONJ
ejpam-6429	576	16	f(yi	f(yi	NUM
ejpam-6429	576	17	)	)	PUNCT
ejpam-6429	576	18	}	}	PUNCT
ejpam-6429	576	19	to	to	ADP
ejpam-6429	576	20	wi	wi	PROPN
ejpam-6429	576	21	for	for	ADP
ejpam-6429	576	22	all	all	PRON
ejpam-6429	576	23	i	i	PRON
ejpam-6429	576	24	∈	∈	PROPN
ejpam-6429	576	25	{	{	PUNCT
ejpam-6429	576	26	1	1	NUM
ejpam-6429	576	27	,	,	PUNCT
ejpam-6429	576	28	2	2	NUM
ejpam-6429	576	29	,	,	PUNCT
ejpam-6429	576	30	.	.	PUNCT
ejpam-6429	576	31	.	.	PUNCT
ejpam-6429	576	32	.	.	PUNCT
ejpam-6429	577	1	,	,	PUNCT
ejpam-6429	577	2	p	p	X
ejpam-6429	577	3	}	}	PUNCT
ejpam-6429	577	4	,	,	PUNCT
ejpam-6429	577	5	we	we	PRON
ejpam-6429	577	6	obtain	obtain	VERB
ejpam-6429	577	7	a	a	DET
ejpam-6429	577	8	grd	grd	NOUN
ejpam-6429	577	9	-	-	PUNCT
ejpam-6429	577	10	function	function	NOUN
ejpam-6429	577	11	for	for	ADP
ejpam-6429	577	12	g	g	NOUN
ejpam-6429	577	13	with	with	ADP
ejpam-6429	577	14	weight	weight	NOUN
ejpam-6429	577	15	less	less	ADJ
ejpam-6429	577	16	than	than	ADP
ejpam-6429	577	17	γgr(g1	γgr(g1	ADJ
ejpam-6429	577	18	)	)	PUNCT
ejpam-6429	577	19	.	.	PUNCT
ejpam-6429	578	1	if	if	SCONJ
ejpam-6429	578	2	f(x1	f(x1	ADJ
ejpam-6429	578	3	)	)	PUNCT
ejpam-6429	578	4	=	=	SYM
ejpam-6429	578	5	2	2	NUM
ejpam-6429	578	6	,	,	PUNCT
ejpam-6429	578	7	then	then	ADV
ejpam-6429	578	8	by	by	ADP
ejpam-6429	578	9	reassigning	reassign	VERB
ejpam-6429	578	10	the	the	DET
ejpam-6429	578	11	value	value	NOUN
ejpam-6429	578	12	1	1	NUM
ejpam-6429	578	13	to	to	PART
ejpam-6429	578	14	v1	v1	VERB
ejpam-6429	578	15	and	and	CCONJ
ejpam-6429	578	16	the	the	DET
ejpam-6429	578	17	value	value	NOUN
ejpam-6429	578	18	min{3	min{3	NOUN
ejpam-6429	578	19	,	,	PUNCT
ejpam-6429	578	20	f(wi	f(wi	NOUN
ejpam-6429	578	21	)	)	PUNCT
ejpam-6429	578	22	+	+	CCONJ
ejpam-6429	578	23	f(yi	f(yi	NUM
ejpam-6429	578	24	)	)	PUNCT
ejpam-6429	578	25	}	}	PUNCT
ejpam-6429	578	26	to	to	ADP
ejpam-6429	578	27	wi	wi	PROPN
ejpam-6429	578	28	for	for	ADP
ejpam-6429	578	29	all	all	PRON
ejpam-6429	578	30	i	i	PRON
ejpam-6429	578	31	∈	∈	PROPN
ejpam-6429	578	32	{	{	PUNCT
ejpam-6429	578	33	1	1	NUM
ejpam-6429	578	34	,	,	PUNCT
ejpam-6429	578	35	2	2	NUM
ejpam-6429	578	36	,	,	PUNCT
ejpam-6429	578	37	.	.	PUNCT
ejpam-6429	578	38	.	.	PUNCT
ejpam-6429	579	1	.	.	PUNCT
ejpam-6429	580	1	,	,	PUNCT
ejpam-6429	580	2	p	p	X
ejpam-6429	580	3	}	}	PUNCT
ejpam-6429	580	4	,	,	PUNCT
ejpam-6429	580	5	we	we	PRON
ejpam-6429	580	6	obtain	obtain	VERB
ejpam-6429	580	7	a	a	DET
ejpam-6429	580	8	grd	grd	NOUN
ejpam-6429	580	9	-	-	PUNCT
ejpam-6429	580	10	function	function	NOUN
ejpam-6429	580	11	for	for	ADP
ejpam-6429	580	12	g	g	NOUN
ejpam-6429	580	13	with	with	ADP
ejpam-6429	580	14	weight	weight	NOUN
ejpam-6429	580	15	less	less	ADJ
ejpam-6429	580	16	than	than	ADP
ejpam-6429	580	17	γgr(g1	γgr(g1	ADJ
ejpam-6429	580	18	)	)	PUNCT
ejpam-6429	580	19	.	.	PUNCT
ejpam-6429	581	1	case	case	NOUN
ejpam-6429	581	2	4	4	NUM
ejpam-6429	581	3	.	.	NUM
ejpam-6429	581	4	f(v	f(v	NOUN
ejpam-6429	581	5	)	)	PUNCT
ejpam-6429	582	1	=	=	SYM
ejpam-6429	582	2	0	0	X
ejpam-6429	582	3	.	.	PUNCT
ejpam-6429	583	1	in	in	ADP
ejpam-6429	583	2	order	order	NOUN
ejpam-6429	583	3	to	to	PART
ejpam-6429	583	4	protect	protect	VERB
ejpam-6429	583	5	x1	x1	PROPN
ejpam-6429	583	6	,	,	PUNCT
ejpam-6429	583	7	we	we	PRON
ejpam-6429	583	8	require	require	VERB
ejpam-6429	583	9	that	that	SCONJ
ejpam-6429	583	10	f(x1	f(x1	NOUN
ejpam-6429	583	11	)	)	PUNCT
ejpam-6429	583	12	+	+	SYM
ejpam-6429	583	13	f(v1	f(v1	NOUN
ejpam-6429	583	14	)	)	PUNCT
ejpam-6429	583	15	≥	≥	NOUN
ejpam-6429	583	16	2	2	NUM
ejpam-6429	583	17	,	,	PUNCT
ejpam-6429	583	18	and	and	CCONJ
ejpam-6429	583	19	from	from	ADP
ejpam-6429	583	20	equation	equation	NOUN
ejpam-6429	583	21	(	(	PUNCT
ejpam-6429	583	22	1	1	NUM
ejpam-6429	583	23	)	)	PUNCT
ejpam-6429	583	24	,	,	PUNCT
ejpam-6429	583	25	we	we	PRON
ejpam-6429	583	26	know	know	VERB
ejpam-6429	583	27	j.	j.	PROPN
ejpam-6429	583	28	j.	j.	PROPN
ejpam-6429	583	29	hamja	hamja	PROPN
ejpam-6429	584	1	et	et	PROPN
ejpam-6429	584	2	al	al	PROPN
ejpam-6429	584	3	.	.	PUNCT
ejpam-6429	584	4	/	/	SYM
ejpam-6429	584	5	eur	eur	PROPN
ejpam-6429	584	6	.	.	PUNCT
ejpam-6429	585	1	j.	j.	PROPN
ejpam-6429	585	2	pure	pure	PROPN
ejpam-6429	585	3	appl	appl	PROPN
ejpam-6429	585	4	.	.	PROPN
ejpam-6429	585	5	math	math	PROPN
ejpam-6429	585	6	,	,	PUNCT
ejpam-6429	585	7	18	18	NUM
ejpam-6429	585	8	(	(	PUNCT
ejpam-6429	585	9	4	4	NUM
ejpam-6429	585	10	)	)	PUNCT
ejpam-6429	585	11	(	(	PUNCT
ejpam-6429	585	12	2025	2025	NUM
ejpam-6429	585	13	)	)	PUNCT
ejpam-6429	585	14	,	,	PUNCT
ejpam-6429	585	15	6429	6429	NUM
ejpam-6429	585	16	14	14	NUM
ejpam-6429	585	17	of	of	ADP
ejpam-6429	585	18	16	16	NUM
ejpam-6429	585	19	that	that	SCONJ
ejpam-6429	585	20	∑s+2	∑s+2	ADJ
ejpam-6429	585	21	i=2	i=2	PROPN
ejpam-6429	585	22	f(xi	f(xi	PROPN
ejpam-6429	585	23	)	)	PUNCT
ejpam-6429	585	24	=	=	SYM
ejpam-6429	586	1	0	0	X
ejpam-6429	586	2	.	.	PUNCT
ejpam-6429	587	1	this	this	PRON
ejpam-6429	587	2	implies	imply	VERB
ejpam-6429	587	3	that	that	SCONJ
ejpam-6429	587	4	f(vi	f(vi	NOUN
ejpam-6429	587	5	)	)	PUNCT
ejpam-6429	587	6	≥	≥	NOUN
ejpam-6429	587	7	2	2	NUM
ejpam-6429	587	8	,	,	PUNCT
ejpam-6429	587	9	and	and	CCONJ
ejpam-6429	587	10	that	that	DET
ejpam-6429	587	11	vi	vi	PROPN
ejpam-6429	587	12	is	be	AUX
ejpam-6429	587	13	a	a	DET
ejpam-6429	587	14	moving	move	VERB
ejpam-6429	587	15	neighbor	neighbor	NOUN
ejpam-6429	587	16	only	only	ADV
ejpam-6429	587	17	for	for	ADP
ejpam-6429	587	18	xi	xi	PROPN
ejpam-6429	587	19	for	for	ADP
ejpam-6429	587	20	all	all	PRON
ejpam-6429	587	21	i	i	PRON
ejpam-6429	587	22	∈	∈	PROPN
ejpam-6429	587	23	{	{	PUNCT
ejpam-6429	587	24	2	2	NUM
ejpam-6429	587	25	,	,	PUNCT
ejpam-6429	587	26	3	3	NUM
ejpam-6429	587	27	,	,	PUNCT
ejpam-6429	587	28	.	.	PUNCT
ejpam-6429	587	29	.	.	PUNCT
ejpam-6429	587	30	.	.	PUNCT
ejpam-6429	588	1	,	,	PUNCT
ejpam-6429	588	2	s	s	PART
ejpam-6429	588	3	+	+	ADJ
ejpam-6429	588	4	2	2	NUM
ejpam-6429	588	5	}	}	PUNCT
ejpam-6429	588	6	.	.	PUNCT
ejpam-6429	589	1	if	if	SCONJ
ejpam-6429	589	2	∑p	∑p	ADJ
ejpam-6429	589	3	j=1	j=1	PROPN
ejpam-6429	589	4	f(yj	f(yj	NOUN
ejpam-6429	589	5	)	)	PUNCT
ejpam-6429	589	6	≥	≥	NOUN
ejpam-6429	589	7	2	2	NUM
ejpam-6429	589	8	,	,	PUNCT
ejpam-6429	589	9	then	then	ADV
ejpam-6429	589	10	by	by	ADP
ejpam-6429	589	11	assigning	assign	VERB
ejpam-6429	589	12	the	the	DET
ejpam-6429	589	13	value	value	NOUN
ejpam-6429	589	14	3	3	NUM
ejpam-6429	589	15	to	to	PART
ejpam-6429	589	16	v1	v1	VERB
ejpam-6429	589	17	,	,	PUNCT
ejpam-6429	589	18	we	we	PRON
ejpam-6429	589	19	can	can	AUX
ejpam-6429	589	20	obtain	obtain	VERB
ejpam-6429	589	21	a	a	DET
ejpam-6429	589	22	grd	grd	NOUN
ejpam-6429	589	23	-	-	PUNCT
ejpam-6429	589	24	function	function	NOUN
ejpam-6429	589	25	for	for	ADP
ejpam-6429	589	26	g	g	NOUN
ejpam-6429	589	27	with	with	ADP
ejpam-6429	589	28	weight	weight	NOUN
ejpam-6429	589	29	less	less	ADJ
ejpam-6429	589	30	than	than	ADP
ejpam-6429	589	31	γgr(g1	γgr(g1	ADJ
ejpam-6429	589	32	)	)	PUNCT
ejpam-6429	589	33	.	.	PUNCT
ejpam-6429	590	1	therefore	therefore	ADV
ejpam-6429	590	2	,	,	PUNCT
ejpam-6429	590	3	we	we	PRON
ejpam-6429	590	4	assume	assume	VERB
ejpam-6429	590	5	that	that	SCONJ
ejpam-6429	590	6	∑p	∑p	ADJ
ejpam-6429	590	7	j=1	j=1	ADJ
ejpam-6429	590	8	f(yj	f(yj	NOUN
ejpam-6429	590	9	)	)	PUNCT
ejpam-6429	590	10	=	=	SYM
ejpam-6429	591	1	0	0	X
ejpam-6429	591	2	.	.	PUNCT
ejpam-6429	592	1	since	since	SCONJ
ejpam-6429	592	2	v1	v1	NOUN
ejpam-6429	592	3	is	be	AUX
ejpam-6429	592	4	a	a	DET
ejpam-6429	592	5	moving	move	VERB
ejpam-6429	592	6	neighbor	neighbor	NOUN
ejpam-6429	592	7	only	only	ADV
ejpam-6429	592	8	for	for	ADP
ejpam-6429	592	9	x1	x1	PROPN
ejpam-6429	592	10	,	,	PUNCT
ejpam-6429	592	11	in	in	ADP
ejpam-6429	592	12	order	order	NOUN
ejpam-6429	592	13	to	to	PART
ejpam-6429	592	14	protect	protect	VERB
ejpam-6429	592	15	yi	yi	PROPN
ejpam-6429	592	16	,	,	PUNCT
ejpam-6429	592	17	we	we	PRON
ejpam-6429	592	18	require	require	VERB
ejpam-6429	592	19	that	that	SCONJ
ejpam-6429	592	20	f(wi	f(wi	NOUN
ejpam-6429	592	21	)	)	PUNCT
ejpam-6429	592	22	≥	≥	NOUN
ejpam-6429	592	23	2	2	NUM
ejpam-6429	592	24	,	,	PUNCT
ejpam-6429	592	25	and	and	CCONJ
ejpam-6429	592	26	that	that	SCONJ
ejpam-6429	592	27	wi	wi	PROPN
ejpam-6429	592	28	is	be	AUX
ejpam-6429	592	29	a	a	DET
ejpam-6429	592	30	moving	move	VERB
ejpam-6429	592	31	neighbor	neighbor	NOUN
ejpam-6429	592	32	of	of	ADP
ejpam-6429	592	33	yi	yi	PROPN
ejpam-6429	592	34	.	.	PUNCT
ejpam-6429	593	1	finally	finally	ADV
ejpam-6429	593	2	,	,	PUNCT
ejpam-6429	593	3	by	by	ADP
ejpam-6429	593	4	assigning	assign	VERB
ejpam-6429	593	5	the	the	DET
ejpam-6429	593	6	value	value	NOUN
ejpam-6429	593	7	1	1	NUM
ejpam-6429	593	8	to	to	PART
ejpam-6429	593	9	v1	v1	VERB
ejpam-6429	593	10	,	,	PUNCT
ejpam-6429	593	11	we	we	PRON
ejpam-6429	593	12	obtain	obtain	VERB
ejpam-6429	593	13	a	a	DET
ejpam-6429	593	14	grd	grd	NOUN
ejpam-6429	593	15	-	-	PUNCT
ejpam-6429	593	16	function	function	NOUN
ejpam-6429	593	17	for	for	ADP
ejpam-6429	593	18	g	g	NOUN
ejpam-6429	593	19	with	with	ADP
ejpam-6429	593	20	weight	weight	NOUN
ejpam-6429	593	21	less	less	ADJ
ejpam-6429	593	22	than	than	ADP
ejpam-6429	593	23	γgr(g1	γgr(g1	ADJ
ejpam-6429	593	24	)	)	PUNCT
ejpam-6429	593	25	.	.	PUNCT
ejpam-6429	594	1	this	this	PRON
ejpam-6429	594	2	completes	complete	VERB
ejpam-6429	594	3	the	the	DET
ejpam-6429	594	4	proof	proof	NOUN
ejpam-6429	594	5	.	.	PUNCT
ejpam-6429	595	1	we	we	PRON
ejpam-6429	595	2	are	be	AUX
ejpam-6429	595	3	now	now	ADV
ejpam-6429	595	4	prepared	prepared	ADJ
ejpam-6429	595	5	to	to	PART
ejpam-6429	595	6	present	present	VERB
ejpam-6429	595	7	our	our	PRON
ejpam-6429	595	8	main	main	ADJ
ejpam-6429	595	9	result	result	NOUN
ejpam-6429	595	10	.	.	PUNCT
ejpam-6429	596	1	theorem	theorem	ADJ
ejpam-6429	596	2	4	4	NUM
ejpam-6429	596	3	.	.	PUNCT
ejpam-6429	597	1	let	let	VERB
ejpam-6429	597	2	g	g	PRON
ejpam-6429	597	3	be	be	AUX
ejpam-6429	597	4	a	a	DET
ejpam-6429	597	5	connected	connected	ADJ
ejpam-6429	597	6	graph	graph	NOUN
ejpam-6429	597	7	of	of	ADP
ejpam-6429	597	8	order	order	NOUN
ejpam-6429	597	9	n	n	PRON
ejpam-6429	597	10	≥	≥	NOUN
ejpam-6429	597	11	3	3	NUM
ejpam-6429	597	12	.	.	PUNCT
ejpam-6429	598	1	then	then	ADV
ejpam-6429	598	2	sdγgr(g	sdγgr(g	PROPN
ejpam-6429	598	3	)	)	PUNCT
ejpam-6429	598	4	≤	≤	NOUN
ejpam-6429	598	5	3	3	NUM
ejpam-6429	598	6	+	+	SYM
ejpam-6429	598	7	min{d2(x	min{d2(x	NOUN
ejpam-6429	598	8	)	)	PUNCT
ejpam-6429	599	1	|	|	ADV
ejpam-6429	599	2	x	x	SYM
ejpam-6429	599	3	∈	∈	PROPN
ejpam-6429	599	4	v	v	NOUN
ejpam-6429	599	5	and	and	CCONJ
ejpam-6429	599	6	deg(x	deg(x	NUM
ejpam-6429	599	7	)	)	PUNCT
ejpam-6429	599	8	≥	≥	NOUN
ejpam-6429	599	9	2	2	NUM
ejpam-6429	599	10	}	}	PUNCT
ejpam-6429	599	11	.	.	PUNCT
ejpam-6429	600	1	proof	proof	NOUN
ejpam-6429	600	2	.	.	PUNCT
ejpam-6429	601	1	if	if	SCONJ
ejpam-6429	601	2	g	g	PROPN
ejpam-6429	601	3	is	be	AUX
ejpam-6429	601	4	a	a	DET
ejpam-6429	601	5	star	star	NOUN
ejpam-6429	601	6	graph	graph	NOUN
ejpam-6429	601	7	k1,n−1	k1,n−1	PROPN
ejpam-6429	601	8	,	,	PUNCT
ejpam-6429	601	9	then	then	ADV
ejpam-6429	601	10	γgr(g	γgr(g	PROPN
ejpam-6429	601	11	)	)	PUNCT
ejpam-6429	601	12	=	=	SYM
ejpam-6429	602	1	3	3	X
ejpam-6429	602	2	.	.	PUNCT
ejpam-6429	602	3	moreover	moreover	ADV
ejpam-6429	602	4	,	,	PUNCT
ejpam-6429	602	5	by	by	ADP
ejpam-6429	602	6	corollary	corollary	ADJ
ejpam-6429	602	7	3	3	NUM
ejpam-6429	602	8	,	,	PUNCT
ejpam-6429	602	9	we	we	PRON
ejpam-6429	602	10	obtain	obtain	VERB
ejpam-6429	602	11	sdγgr(g	sdγgr(g	PRON
ejpam-6429	602	12	)	)	PUNCT
ejpam-6429	602	13	=	=	SYM
ejpam-6429	602	14	1	1	NUM
ejpam-6429	602	15	,	,	PUNCT
ejpam-6429	602	16	and	and	CCONJ
ejpam-6429	602	17	thus	thus	ADV
ejpam-6429	602	18	the	the	DET
ejpam-6429	602	19	result	result	NOUN
ejpam-6429	602	20	holds	hold	VERB
ejpam-6429	602	21	.	.	PUNCT
ejpam-6429	603	1	therefore	therefore	ADV
ejpam-6429	603	2	,	,	PUNCT
ejpam-6429	603	3	from	from	ADP
ejpam-6429	603	4	this	this	DET
ejpam-6429	603	5	point	point	NOUN
ejpam-6429	603	6	onward	onward	ADV
ejpam-6429	603	7	,	,	PUNCT
ejpam-6429	603	8	we	we	PRON
ejpam-6429	603	9	assume	assume	VERB
ejpam-6429	603	10	that	that	SCONJ
ejpam-6429	603	11	g	g	PROPN
ejpam-6429	603	12	̸=	̸=	PROPN
ejpam-6429	603	13	k1,n−1	k1,n−1	PROPN
ejpam-6429	603	14	.	.	PUNCT
ejpam-6429	604	1	if	if	SCONJ
ejpam-6429	604	2	g	g	PROPN
ejpam-6429	604	3	contains	contain	VERB
ejpam-6429	604	4	a	a	DET
ejpam-6429	604	5	leaf	leaf	NOUN
ejpam-6429	604	6	,	,	PUNCT
ejpam-6429	604	7	then	then	ADV
ejpam-6429	604	8	by	by	ADP
ejpam-6429	604	9	corollary	corollary	ADJ
ejpam-6429	604	10	6	6	NUM
ejpam-6429	604	11	,	,	PUNCT
ejpam-6429	604	12	the	the	DET
ejpam-6429	604	13	result	result	NOUN
ejpam-6429	604	14	also	also	ADV
ejpam-6429	604	15	holds	hold	VERB
ejpam-6429	604	16	.	.	PUNCT
ejpam-6429	605	1	next	next	ADV
ejpam-6429	605	2	,	,	PUNCT
ejpam-6429	605	3	let	let	VERB
ejpam-6429	605	4	g	g	PRON
ejpam-6429	605	5	be	be	AUX
ejpam-6429	605	6	a	a	DET
ejpam-6429	605	7	graph	graph	NOUN
ejpam-6429	605	8	such	such	ADJ
ejpam-6429	605	9	that	that	SCONJ
ejpam-6429	605	10	δ(g	δ(g	PROPN
ejpam-6429	605	11	)	)	PUNCT
ejpam-6429	605	12	≥	≥	NOUN
ejpam-6429	605	13	2	2	NUM
ejpam-6429	605	14	.	.	PUNCT
ejpam-6429	605	15	using	use	VERB
ejpam-6429	605	16	proposition	proposition	NOUN
ejpam-6429	605	17	11	11	NUM
ejpam-6429	605	18	and	and	CCONJ
ejpam-6429	605	19	lemmas	lemmas	PROPN
ejpam-6429	605	20	1	1	NUM
ejpam-6429	605	21	,	,	PUNCT
ejpam-6429	605	22	2	2	NUM
ejpam-6429	605	23	and	and	CCONJ
ejpam-6429	605	24	3	3	NUM
ejpam-6429	605	25	,	,	PUNCT
ejpam-6429	605	26	we	we	PRON
ejpam-6429	605	27	conclude	conclude	VERB
ejpam-6429	605	28	that	that	PRON
ejpam-6429	605	29	sdγgr(g	sdγgr(g	PROPN
ejpam-6429	605	30	)	)	PUNCT
ejpam-6429	605	31	≤	≤	NOUN
ejpam-6429	605	32	3	3	NUM
ejpam-6429	606	1	+	+	SYM
ejpam-6429	606	2	min{d2(x	min{d2(x	NOUN
ejpam-6429	606	3	)	)	PUNCT
ejpam-6429	607	1	|	|	ADV
ejpam-6429	607	2	x	x	SYM
ejpam-6429	607	3	∈	∈	PROPN
ejpam-6429	607	4	v	v	NOUN
ejpam-6429	607	5	and	and	CCONJ
ejpam-6429	607	6	deg(x	deg(x	NUM
ejpam-6429	607	7	)	)	PUNCT
ejpam-6429	607	8	≥	≥	NOUN
ejpam-6429	607	9	2	2	NUM
ejpam-6429	607	10	}	}	PUNCT
ejpam-6429	607	11	.	.	PUNCT
ejpam-6429	608	1	let	let	VERB
ejpam-6429	608	2	δ2(g	δ2(g	NUM
ejpam-6429	608	3	)	)	PUNCT
ejpam-6429	608	4	=	=	SYM
ejpam-6429	608	5	min{d2(v	min{d2(v	PROPN
ejpam-6429	608	6	)	)	PUNCT
ejpam-6429	608	7	|	|	ADV
ejpam-6429	608	8	v	v	ADP
ejpam-6429	608	9	∈	∈	PROPN
ejpam-6429	608	10	v	v	NOUN
ejpam-6429	608	11	(	(	PUNCT
ejpam-6429	608	12	g	g	NOUN
ejpam-6429	608	13	)	)	PUNCT
ejpam-6429	608	14	and	and	CCONJ
ejpam-6429	608	15	deg(v	deg(v	PROPN
ejpam-6429	608	16	)	)	PUNCT
ejpam-6429	608	17	≥	≥	NOUN
ejpam-6429	608	18	2	2	NUM
ejpam-6429	608	19	}	}	PUNCT
ejpam-6429	608	20	,	,	PUNCT
ejpam-6429	608	21	and	and	CCONJ
ejpam-6429	608	22	note	note	VERB
ejpam-6429	608	23	that	that	SCONJ
ejpam-6429	608	24	for	for	ADP
ejpam-6429	608	25	each	each	DET
ejpam-6429	608	26	vertex	vertex	NOUN
ejpam-6429	608	27	v	v	NOUN
ejpam-6429	608	28	with	with	ADP
ejpam-6429	608	29	degree	degree	NOUN
ejpam-6429	608	30	∆	∆	PROPN
ejpam-6429	608	31	,	,	PUNCT
ejpam-6429	608	32	we	we	PRON
ejpam-6429	608	33	have	have	VERB
ejpam-6429	608	34	δ2(g	δ2(g	NOUN
ejpam-6429	608	35	)	)	PUNCT
ejpam-6429	608	36	≤	≤	NOUN
ejpam-6429	608	37	|n2(v)|	|n2(v)|	NUM
ejpam-6429	608	38	≤	≤	NUM
ejpam-6429	608	39	n	n	CCONJ
ejpam-6429	608	40	−	−	PROPN
ejpam-6429	608	41	∆	∆	PROPN
ejpam-6429	608	42	−	−	PROPN
ejpam-6429	609	1	1	1	X
ejpam-6429	609	2	.	.	PUNCT
ejpam-6429	610	1	the	the	DET
ejpam-6429	610	2	next	next	ADJ
ejpam-6429	610	3	two	two	NUM
ejpam-6429	610	4	corollaries	corollary	NOUN
ejpam-6429	610	5	follow	follow	VERB
ejpam-6429	610	6	directly	directly	ADV
ejpam-6429	610	7	from	from	ADP
ejpam-6429	610	8	theorem	theorem	ADJ
ejpam-6429	610	9	4	4	NUM
ejpam-6429	610	10	.	.	PUNCT
ejpam-6429	610	11	corollary	corollary	ADJ
ejpam-6429	610	12	8	8	NUM
ejpam-6429	610	13	.	.	PUNCT
ejpam-6429	611	1	let	let	VERB
ejpam-6429	611	2	g	g	PRON
ejpam-6429	611	3	be	be	AUX
ejpam-6429	611	4	a	a	DET
ejpam-6429	611	5	connected	connected	ADJ
ejpam-6429	611	6	graph	graph	NOUN
ejpam-6429	611	7	with	with	ADP
ejpam-6429	611	8	δ(g	δ(g	PROPN
ejpam-6429	611	9	)	)	PUNCT
ejpam-6429	611	10	≥	≥	NOUN
ejpam-6429	611	11	2	2	NUM
ejpam-6429	611	12	,	,	PUNCT
ejpam-6429	611	13	sdγgr(g	sdγgr(g	PROPN
ejpam-6429	611	14	)	)	PUNCT
ejpam-6429	611	15	≤	≤	NOUN
ejpam-6429	611	16	3	3	NUM
ejpam-6429	611	17	+	+	CCONJ
ejpam-6429	611	18	δ2(g	δ2(g	NOUN
ejpam-6429	611	19	)	)	PUNCT
ejpam-6429	611	20	.	.	PUNCT
ejpam-6429	612	1	corollary	corollary	ADJ
ejpam-6429	612	2	9	9	NUM
ejpam-6429	612	3	.	.	PUNCT
ejpam-6429	613	1	let	let	VERB
ejpam-6429	613	2	g	g	PRON
ejpam-6429	613	3	be	be	AUX
ejpam-6429	613	4	a	a	DET
ejpam-6429	613	5	connected	connected	ADJ
ejpam-6429	613	6	graph	graph	NOUN
ejpam-6429	613	7	of	of	ADP
ejpam-6429	613	8	order	order	NOUN
ejpam-6429	613	9	n	n	PRON
ejpam-6429	613	10	≥	≥	NOUN
ejpam-6429	613	11	3	3	NUM
ejpam-6429	613	12	.	.	PUNCT
ejpam-6429	614	1	then	then	ADV
ejpam-6429	614	2	sdγgr(g	sdγgr(g	PROPN
ejpam-6429	614	3	)	)	PUNCT
ejpam-6429	614	4	≤	≤	NUM
ejpam-6429	614	5	n−∆+	n−∆+	NOUN
ejpam-6429	614	6	2	2	NUM
ejpam-6429	614	7	.	.	PUNCT
ejpam-6429	615	1	applying	apply	VERB
ejpam-6429	615	2	corollaries	corollary	NOUN
ejpam-6429	615	3	5	5	NUM
ejpam-6429	615	4	,	,	PUNCT
ejpam-6429	615	5	6	6	NUM
ejpam-6429	615	6	,	,	PUNCT
ejpam-6429	615	7	and	and	CCONJ
ejpam-6429	615	8	9	9	NUM
ejpam-6429	615	9	,	,	PUNCT
ejpam-6429	615	10	we	we	PRON
ejpam-6429	615	11	derive	derive	VERB
ejpam-6429	615	12	the	the	DET
ejpam-6429	615	13	following	follow	VERB
ejpam-6429	615	14	result	result	NOUN
ejpam-6429	615	15	.	.	PUNCT
ejpam-6429	616	1	proposition	proposition	NOUN
ejpam-6429	616	2	12	12	NUM
ejpam-6429	616	3	.	.	PUNCT
ejpam-6429	617	1	let	let	VERB
ejpam-6429	617	2	g	g	PRON
ejpam-6429	617	3	be	be	AUX
ejpam-6429	617	4	a	a	DET
ejpam-6429	617	5	connected	connected	ADJ
ejpam-6429	617	6	graph	graph	NOUN
ejpam-6429	617	7	of	of	ADP
ejpam-6429	617	8	order	order	NOUN
ejpam-6429	617	9	n	n	PRON
ejpam-6429	617	10	≥	≥	NOUN
ejpam-6429	617	11	3	3	NUM
ejpam-6429	617	12	.	.	PUNCT
ejpam-6429	618	1	then	then	ADV
ejpam-6429	618	2	sdγgr(g	sdγgr(g	PROPN
ejpam-6429	618	3	)	)	PUNCT
ejpam-6429	618	4	≤	≤	NOUN
ejpam-6429	618	5	n	n	PRON
ejpam-6429	618	6	2	2	NUM
ejpam-6429	618	7	+	+	NUM
ejpam-6429	618	8	1	1	NUM
ejpam-6429	618	9	.	.	X
ejpam-6429	619	1	conclusion	conclusion	NOUN
ejpam-6429	619	2	:	:	PUNCT
ejpam-6429	619	3	this	this	DET
ejpam-6429	619	4	study	study	NOUN
ejpam-6429	619	5	introduced	introduce	VERB
ejpam-6429	619	6	and	and	CCONJ
ejpam-6429	619	7	explored	explore	VERB
ejpam-6429	619	8	the	the	DET
ejpam-6429	619	9	concept	concept	NOUN
ejpam-6429	619	10	of	of	ADP
ejpam-6429	619	11	the	the	DET
ejpam-6429	619	12	generous	generous	ADJ
ejpam-6429	619	13	roman	roman	ADJ
ejpam-6429	619	14	domination	domination	NOUN
ejpam-6429	619	15	subdivision	subdivision	NOUN
ejpam-6429	619	16	number	number	NOUN
ejpam-6429	619	17	in	in	ADP
ejpam-6429	619	18	graphs	graph	NOUN
ejpam-6429	619	19	,	,	PUNCT
ejpam-6429	619	20	focusing	focus	VERB
ejpam-6429	619	21	on	on	ADP
ejpam-6429	619	22	its	its	PRON
ejpam-6429	619	23	effect	effect	NOUN
ejpam-6429	619	24	on	on	ADP
ejpam-6429	619	25	the	the	DET
ejpam-6429	619	26	generous	generous	ADJ
ejpam-6429	619	27	roman	roman	ADJ
ejpam-6429	619	28	domination	domination	NOUN
ejpam-6429	619	29	number	number	NOUN
ejpam-6429	619	30	.	.	PUNCT
ejpam-6429	620	1	by	by	ADP
ejpam-6429	620	2	analyzing	analyze	VERB
ejpam-6429	620	3	how	how	SCONJ
ejpam-6429	620	4	edge	edge	NOUN
ejpam-6429	620	5	subdivisions	subdivision	NOUN
ejpam-6429	620	6	influence	influence	VERB
ejpam-6429	620	7	this	this	DET
ejpam-6429	620	8	parameter	parameter	NOUN
ejpam-6429	620	9	,	,	PUNCT
ejpam-6429	620	10	we	we	PRON
ejpam-6429	620	11	established	establish	VERB
ejpam-6429	620	12	upper	upper	ADJ
ejpam-6429	620	13	bounds	bound	NOUN
ejpam-6429	620	14	and	and	CCONJ
ejpam-6429	620	15	exact	exact	ADJ
ejpam-6429	620	16	values	value	NOUN
ejpam-6429	620	17	for	for	ADP
ejpam-6429	620	18	specific	specific	ADJ
ejpam-6429	620	19	graph	graph	NOUN
ejpam-6429	620	20	families	family	NOUN
ejpam-6429	620	21	.	.	PUNCT
ejpam-6429	621	1	these	these	DET
ejpam-6429	621	2	findings	finding	NOUN
ejpam-6429	621	3	contributed	contribute	VERB
ejpam-6429	621	4	to	to	ADP
ejpam-6429	621	5	a	a	DET
ejpam-6429	621	6	deeper	deep	ADJ
ejpam-6429	621	7	understanding	understanding	NOUN
ejpam-6429	621	8	of	of	ADP
ejpam-6429	621	9	domination	domination	NOUN
ejpam-6429	621	10	parameters	parameter	NOUN
ejpam-6429	621	11	under	under	ADP
ejpam-6429	621	12	graph	graph	NOUN
ejpam-6429	621	13	modifications	modification	NOUN
ejpam-6429	621	14	.	.	PUNCT
ejpam-6429	622	1	future	future	ADJ
ejpam-6429	622	2	research	research	NOUN
ejpam-6429	622	3	may	may	AUX
ejpam-6429	622	4	focus	focus	VERB
ejpam-6429	622	5	on	on	ADP
ejpam-6429	622	6	characterizing	characterize	VERB
ejpam-6429	622	7	more	more	ADJ
ejpam-6429	622	8	graph	graph	NOUN
ejpam-6429	622	9	classes	class	NOUN
ejpam-6429	622	10	where	where	SCONJ
ejpam-6429	622	11	the	the	DET
ejpam-6429	622	12	exact	exact	ADJ
ejpam-6429	622	13	generous	generous	ADJ
ejpam-6429	622	14	roman	roman	ADJ
ejpam-6429	622	15	domination	domination	NOUN
ejpam-6429	622	16	subdivision	subdivision	NOUN
ejpam-6429	622	17	number	number	NOUN
ejpam-6429	622	18	can	can	AUX
ejpam-6429	622	19	be	be	AUX
ejpam-6429	622	20	determined	determine	VERB
ejpam-6429	622	21	.	.	PUNCT
ejpam-6429	623	1	additionally	additionally	ADV
ejpam-6429	623	2	,	,	PUNCT
ejpam-6429	623	3	algorithmic	algorithmic	ADJ
ejpam-6429	623	4	approaches	approach	NOUN
ejpam-6429	623	5	to	to	PART
ejpam-6429	623	6	compute	compute	VERB
ejpam-6429	623	7	this	this	DET
ejpam-6429	623	8	parameter	parameter	NOUN
ejpam-6429	623	9	efficiently	efficiently	ADV
ejpam-6429	623	10	in	in	ADP
ejpam-6429	623	11	general	general	ADJ
ejpam-6429	623	12	graphs	graph	NOUN
ejpam-6429	623	13	can	can	AUX
ejpam-6429	623	14	be	be	AUX
ejpam-6429	623	15	developed	develop	VERB
ejpam-6429	623	16	to	to	PART
ejpam-6429	623	17	support	support	VERB
ejpam-6429	623	18	applications	application	NOUN
ejpam-6429	623	19	in	in	ADP
ejpam-6429	623	20	network	network	NOUN
ejpam-6429	623	21	defense	defense	NOUN
ejpam-6429	623	22	and	and	CCONJ
ejpam-6429	623	23	resource	resource	NOUN
ejpam-6429	623	24	allocation	allocation	NOUN
ejpam-6429	623	25	.	.	PUNCT
ejpam-6429	624	1	j.	j.	PROPN
ejpam-6429	624	2	j.	j.	PROPN
ejpam-6429	624	3	hamja	hamja	PROPN
ejpam-6429	624	4	et	et	PROPN
ejpam-6429	624	5	al	al	PROPN
ejpam-6429	624	6	.	.	PUNCT
ejpam-6429	624	7	/	/	SYM
ejpam-6429	624	8	eur	eur	PROPN
ejpam-6429	624	9	.	.	PUNCT
ejpam-6429	625	1	j.	j.	PROPN
ejpam-6429	625	2	pure	pure	PROPN
ejpam-6429	625	3	appl	appl	PROPN
ejpam-6429	625	4	.	.	PROPN
ejpam-6429	625	5	math	math	PROPN
ejpam-6429	625	6	,	,	PUNCT
ejpam-6429	625	7	18	18	NUM
ejpam-6429	625	8	(	(	PUNCT
ejpam-6429	625	9	4	4	NUM
ejpam-6429	625	10	)	)	PUNCT
ejpam-6429	625	11	(	(	PUNCT
ejpam-6429	625	12	2025	2025	NUM
ejpam-6429	625	13	)	)	PUNCT
ejpam-6429	625	14	,	,	PUNCT
ejpam-6429	625	15	6429	6429	NUM
ejpam-6429	625	16	15	15	NUM
ejpam-6429	625	17	of	of	ADP
ejpam-6429	625	18	16	16	NUM
ejpam-6429	625	19	acknowledgements	acknowledgement	NOUN
ejpam-6429	625	20	we	we	PRON
ejpam-6429	625	21	gratefully	gratefully	ADV
ejpam-6429	625	22	acknowledge	acknowledge	VERB
ejpam-6429	625	23	the	the	DET
ejpam-6429	625	24	reviewers	reviewer	NOUN
ejpam-6429	625	25	for	for	ADP
ejpam-6429	625	26	their	their	PRON
ejpam-6429	625	27	insightful	insightful	ADJ
ejpam-6429	625	28	comments	comment	NOUN
ejpam-6429	625	29	and	and	CCONJ
ejpam-6429	625	30	suggestions	suggestion	NOUN
ejpam-6429	625	31	,	,	PUNCT
ejpam-6429	625	32	which	which	PRON
ejpam-6429	625	33	have	have	AUX
ejpam-6429	625	34	greatly	greatly	ADV
ejpam-6429	625	35	enhanced	enhance	VERB
ejpam-6429	625	36	the	the	DET
ejpam-6429	625	37	quality	quality	NOUN
ejpam-6429	625	38	of	of	ADP
ejpam-6429	625	39	this	this	DET
ejpam-6429	625	40	paper	paper	NOUN
ejpam-6429	625	41	.	.	PUNCT
ejpam-6429	626	1	we	we	PRON
ejpam-6429	626	2	also	also	ADV
ejpam-6429	626	3	extend	extend	VERB
ejpam-6429	626	4	our	our	PRON
ejpam-6429	626	5	sincere	sincere	ADJ
ejpam-6429	626	6	appreciation	appreciation	NOUN
ejpam-6429	626	7	to	to	ADP
ejpam-6429	626	8	mindanao	mindanao	PROPN
ejpam-6429	626	9	state	state	PROPN
ejpam-6429	626	10	university	university	PROPN
ejpam-6429	626	11	–	–	PUNCT
ejpam-6429	626	12	tawi	tawi	NOUN
ejpam-6429	626	13	-	-	PUNCT
ejpam-6429	626	14	tawi	tawi	NOUN
ejpam-6429	626	15	college	college	PROPN
ejpam-6429	626	16	of	of	ADP
ejpam-6429	626	17	technology	technology	NOUN
ejpam-6429	626	18	and	and	CCONJ
ejpam-6429	626	19	oceanography	oceanography	NOUN
ejpam-6429	626	20	(	(	PUNCT
ejpam-6429	626	21	msu	msu	PROPN
ejpam-6429	626	22	-	-	PUNCT
ejpam-6429	626	23	tcto	tcto	VERB
ejpam-6429	626	24	)	)	PUNCT
ejpam-6429	626	25	and	and	CCONJ
ejpam-6429	626	26	mindanao	mindanao	PROPN
ejpam-6429	626	27	state	state	PROPN
ejpam-6429	626	28	university	university	PROPN
ejpam-6429	626	29	–	–	PUNCT
ejpam-6429	626	30	iligan	iligan	PROPN
ejpam-6429	626	31	institute	institute	PROPN
ejpam-6429	626	32	of	of	ADP
ejpam-6429	626	33	technology	technology	PROPN
ejpam-6429	626	34	(	(	PUNCT
ejpam-6429	626	35	msu	msu	PROPN
ejpam-6429	626	36	-	-	PUNCT
ejpam-6429	626	37	iit	iit	NOUN
ejpam-6429	626	38	)	)	PUNCT
ejpam-6429	626	39	for	for	ADP
ejpam-6429	626	40	their	their	PRON
ejpam-6429	626	41	generous	generous	ADJ
ejpam-6429	626	42	financial	financial	ADJ
ejpam-6429	626	43	support	support	NOUN
ejpam-6429	626	44	of	of	ADP
ejpam-6429	626	45	this	this	DET
ejpam-6429	626	46	work	work	NOUN
ejpam-6429	626	47	.	.	PUNCT
ejpam-6429	627	1	references	reference	NOUN
ejpam-6429	627	2	[	[	X
ejpam-6429	627	3	1	1	NUM
ejpam-6429	627	4	]	]	PUNCT
ejpam-6429	627	5	c.	c.	PROPN
ejpam-6429	627	6	s.	s.	PROPN
ejpam-6429	627	7	revelle	revelle	PROPN
ejpam-6429	627	8	and	and	CCONJ
ejpam-6429	627	9	k.	k.	PROPN
ejpam-6429	627	10	e.	e.	PROPN
ejpam-6429	627	11	rosing	rosing	PROPN
ejpam-6429	627	12	.	.	PUNCT
ejpam-6429	628	1	defendens	defenden	VERB
ejpam-6429	628	2	imperium	imperium	NOUN
ejpam-6429	628	3	romanum	romanum	NOUN
ejpam-6429	628	4	:	:	PUNCT
ejpam-6429	628	5	a	a	DET
ejpam-6429	628	6	classical	classical	ADJ
ejpam-6429	628	7	problem	problem	NOUN
ejpam-6429	628	8	in	in	ADP
ejpam-6429	628	9	military	military	ADJ
ejpam-6429	628	10	strategy	strategy	NOUN
ejpam-6429	628	11	.	.	PUNCT
ejpam-6429	629	1	american	american	PROPN
ejpam-6429	629	2	mathematical	mathematical	PROPN
ejpam-6429	629	3	monthly	monthly	ADJ
ejpam-6429	629	4	,	,	PUNCT
ejpam-6429	629	5	107:585–594	107:585–594	NUM
ejpam-6429	629	6	,	,	PUNCT
ejpam-6429	629	7	2000	2000	NUM
ejpam-6429	629	8	.	.	PUNCT
ejpam-6429	630	1	[	[	X
ejpam-6429	630	2	2	2	NUM
ejpam-6429	630	3	]	]	PUNCT
ejpam-6429	630	4	i.	i.	PROPN
ejpam-6429	630	5	stewart	stewart	PROPN
ejpam-6429	630	6	.	.	PUNCT
ejpam-6429	631	1	defend	defend	VERB
ejpam-6429	631	2	the	the	DET
ejpam-6429	631	3	roman	roman	ADJ
ejpam-6429	631	4	empire	empire	NOUN
ejpam-6429	631	5	!	!	PUNCT
ejpam-6429	632	1	scientific	scientific	ADJ
ejpam-6429	632	2	american	american	PROPN
ejpam-6429	632	3	,	,	PUNCT
ejpam-6429	632	4	281:136–139	281:136–139	NUM
ejpam-6429	632	5	,	,	PUNCT
ejpam-6429	632	6	1999	1999	NUM
ejpam-6429	632	7	.	.	PUNCT
ejpam-6429	633	1	[	[	X
ejpam-6429	633	2	3	3	X
ejpam-6429	633	3	]	]	X
ejpam-6429	633	4	e.	e.	PROPN
ejpam-6429	633	5	j.	j.	PROPN
ejpam-6429	633	6	cockayne	cockayne	PROPN
ejpam-6429	633	7	,	,	PUNCT
ejpam-6429	633	8	jr	jr	PROPN
ejpam-6429	633	9	.	.	PUNCT
ejpam-6429	633	10	dreyer	dreyer	PROPN
ejpam-6429	633	11	,	,	PUNCT
ejpam-6429	633	12	p.	p.	PROPN
ejpam-6429	633	13	m.	m.	NOUN
ejpam-6429	633	14	,	,	PUNCT
ejpam-6429	633	15	s.	s.	PROPN
ejpam-6429	633	16	m.	m.	PROPN
ejpam-6429	633	17	hedetniemi	hedetniemi	ADV
ejpam-6429	633	18	,	,	PUNCT
ejpam-6429	633	19	and	and	CCONJ
ejpam-6429	633	20	s.	s.	PROPN
ejpam-6429	633	21	t.	t.	PROPN
ejpam-6429	633	22	hedetniemi	hedetniemi	PROPN
ejpam-6429	633	23	.	.	PUNCT
ejpam-6429	634	1	on	on	ADP
ejpam-6429	634	2	roman	roman	ADJ
ejpam-6429	634	3	domination	domination	NOUN
ejpam-6429	634	4	in	in	ADP
ejpam-6429	634	5	graphs	graph	NOUN
ejpam-6429	634	6	.	.	PUNCT
ejpam-6429	635	1	discrete	discrete	ADJ
ejpam-6429	635	2	mathematics	mathematic	NOUN
ejpam-6429	635	3	,	,	PUNCT
ejpam-6429	635	4	278:11–22	278:11–22	NUM
ejpam-6429	635	5	,	,	PUNCT
ejpam-6429	635	6	2004	2004	NUM
ejpam-6429	635	7	.	.	PUNCT
ejpam-6429	636	1	[	[	X
ejpam-6429	636	2	4	4	NUM
ejpam-6429	636	3	]	]	PUNCT
ejpam-6429	636	4	m.	m.	NOUN
ejpam-6429	636	5	chellali	chellali	PROPN
ejpam-6429	636	6	,	,	PUNCT
ejpam-6429	636	7	n.	n.	PROPN
ejpam-6429	636	8	jafari	jafari	PROPN
ejpam-6429	636	9	rad	rad	PROPN
ejpam-6429	636	10	,	,	PUNCT
ejpam-6429	636	11	s.	s.	PROPN
ejpam-6429	636	12	m.	m.	PROPN
ejpam-6429	636	13	sheikholeslami	sheikholeslami	PROPN
ejpam-6429	636	14	,	,	PUNCT
ejpam-6429	636	15	and	and	CCONJ
ejpam-6429	636	16	l.	l.	PROPN
ejpam-6429	636	17	volkmann	volkmann	PROPN
ejpam-6429	636	18	.	.	PUNCT
ejpam-6429	637	1	roman	roman	ADJ
ejpam-6429	637	2	domination	domination	NOUN
ejpam-6429	637	3	in	in	ADP
ejpam-6429	637	4	graphs	graph	NOUN
ejpam-6429	637	5	.	.	PUNCT
ejpam-6429	638	1	in	in	ADP
ejpam-6429	638	2	t.	t.	PROPN
ejpam-6429	638	3	w.	w.	PROPN
ejpam-6429	638	4	haynes	haynes	PROPN
ejpam-6429	638	5	,	,	PUNCT
ejpam-6429	638	6	s.	s.	PROPN
ejpam-6429	638	7	t.	t.	PROPN
ejpam-6429	638	8	hedetniemi	hedetniemi	PROPN
ejpam-6429	638	9	,	,	PUNCT
ejpam-6429	638	10	and	and	CCONJ
ejpam-6429	638	11	m.	m.	PROPN
ejpam-6429	638	12	a.	a.	PROPN
ejpam-6429	638	13	henning	henning	PROPN
ejpam-6429	638	14	,	,	PUNCT
ejpam-6429	638	15	editors	editor	NOUN
ejpam-6429	638	16	,	,	PUNCT
ejpam-6429	638	17	topics	topic	NOUN
ejpam-6429	638	18	in	in	ADP
ejpam-6429	638	19	domination	domination	NOUN
ejpam-6429	638	20	in	in	ADP
ejpam-6429	638	21	graphs	graph	NOUN
ejpam-6429	638	22	,	,	PUNCT
ejpam-6429	638	23	pages	page	NOUN
ejpam-6429	638	24	365–409	365–409	NUM
ejpam-6429	638	25	.	.	PUNCT
ejpam-6429	638	26	springer	springer	NOUN
ejpam-6429	638	27	,	,	PUNCT
ejpam-6429	638	28	berlin	berlin	PROPN
ejpam-6429	638	29	/	/	SYM
ejpam-6429	638	30	heidelberg	heidelberg	PROPN
ejpam-6429	638	31	,	,	PUNCT
ejpam-6429	638	32	2020	2020	NUM
ejpam-6429	638	33	.	.	PUNCT
ejpam-6429	639	1	[	[	X
ejpam-6429	639	2	5	5	NUM
ejpam-6429	639	3	]	]	PUNCT
ejpam-6429	639	4	m.	m.	NOUN
ejpam-6429	639	5	chellali	chellali	PROPN
ejpam-6429	639	6	,	,	PUNCT
ejpam-6429	639	7	n.	n.	PROPN
ejpam-6429	639	8	jafari	jafari	PROPN
ejpam-6429	639	9	rad	rad	PROPN
ejpam-6429	639	10	,	,	PUNCT
ejpam-6429	639	11	s.	s.	PROPN
ejpam-6429	639	12	m.	m.	PROPN
ejpam-6429	639	13	sheikholeslami	sheikholeslami	PROPN
ejpam-6429	639	14	,	,	PUNCT
ejpam-6429	639	15	and	and	CCONJ
ejpam-6429	639	16	l.	l.	PROPN
ejpam-6429	639	17	volkmann	volkmann	PROPN
ejpam-6429	639	18	.	.	PUNCT
ejpam-6429	640	1	varieties	variety	NOUN
ejpam-6429	640	2	of	of	ADP
ejpam-6429	640	3	roman	roman	ADJ
ejpam-6429	640	4	domination	domination	NOUN
ejpam-6429	640	5	.	.	PUNCT
ejpam-6429	641	1	in	in	ADP
ejpam-6429	641	2	t.	t.	PROPN
ejpam-6429	641	3	w.	w.	PROPN
ejpam-6429	641	4	haynes	haynes	PROPN
ejpam-6429	641	5	,	,	PUNCT
ejpam-6429	641	6	s.	s.	PROPN
ejpam-6429	641	7	t.	t.	PROPN
ejpam-6429	641	8	hedetniemi	hedetniemi	PROPN
ejpam-6429	641	9	,	,	PUNCT
ejpam-6429	641	10	and	and	CCONJ
ejpam-6429	641	11	m.	m.	PROPN
ejpam-6429	641	12	a.	a.	PROPN
ejpam-6429	641	13	henning	henning	PROPN
ejpam-6429	641	14	,	,	PUNCT
ejpam-6429	641	15	editors	editor	NOUN
ejpam-6429	641	16	,	,	PUNCT
ejpam-6429	641	17	structures	structure	NOUN
ejpam-6429	641	18	of	of	ADP
ejpam-6429	641	19	domination	domination	NOUN
ejpam-6429	641	20	in	in	ADP
ejpam-6429	641	21	graphs	graph	NOUN
ejpam-6429	641	22	,	,	PUNCT
ejpam-6429	641	23	pages	page	NOUN
ejpam-6429	641	24	273–307	273–307	NUM
ejpam-6429	641	25	.	.	PUNCT
ejpam-6429	641	26	springer	springer	NOUN
ejpam-6429	641	27	,	,	PUNCT
ejpam-6429	641	28	berlin	berlin	PROPN
ejpam-6429	641	29	/	/	SYM
ejpam-6429	641	30	heidelberg	heidelberg	PROPN
ejpam-6429	641	31	,	,	PUNCT
ejpam-6429	641	32	2021	2021	NUM
ejpam-6429	641	33	.	.	PUNCT
ejpam-6429	642	1	[	[	X
ejpam-6429	642	2	6	6	NUM
ejpam-6429	642	3	]	]	PUNCT
ejpam-6429	642	4	m.	m.	NOUN
ejpam-6429	642	5	chellali	chellali	PROPN
ejpam-6429	642	6	,	,	PUNCT
ejpam-6429	642	7	n.	n.	PROPN
ejpam-6429	642	8	jafari	jafari	PROPN
ejpam-6429	642	9	rad	rad	PROPN
ejpam-6429	642	10	,	,	PUNCT
ejpam-6429	642	11	s.	s.	PROPN
ejpam-6429	642	12	m.	m.	PROPN
ejpam-6429	642	13	sheikholeslami	sheikholeslami	PROPN
ejpam-6429	642	14	,	,	PUNCT
ejpam-6429	642	15	and	and	CCONJ
ejpam-6429	642	16	l.	l.	PROPN
ejpam-6429	642	17	volkmann	volkmann	PROPN
ejpam-6429	642	18	.	.	PUNCT
ejpam-6429	643	1	varieties	variety	NOUN
ejpam-6429	643	2	of	of	ADP
ejpam-6429	643	3	roman	roman	PROPN
ejpam-6429	643	4	domination	domination	PROPN
ejpam-6429	643	5	ii	ii	PROPN
ejpam-6429	643	6	.	.	PUNCT
ejpam-6429	644	1	akce	akce	PROPN
ejpam-6429	644	2	international	international	PROPN
ejpam-6429	644	3	journal	journal	NOUN
ejpam-6429	644	4	of	of	ADP
ejpam-6429	644	5	graphs	graph	NOUN
ejpam-6429	644	6	and	and	CCONJ
ejpam-6429	644	7	combinatorics	combinatoric	NOUN
ejpam-6429	644	8	,	,	PUNCT
ejpam-6429	644	9	17:966–984	17:966–984	NUM
ejpam-6429	644	10	,	,	PUNCT
ejpam-6429	644	11	2020	2020	NUM
ejpam-6429	644	12	.	.	PUNCT
ejpam-6429	645	1	[	[	X
ejpam-6429	645	2	7	7	X
ejpam-6429	645	3	]	]	X
ejpam-6429	645	4	m.	m.	NOUN
ejpam-6429	645	5	chellali	chellali	PROPN
ejpam-6429	645	6	,	,	PUNCT
ejpam-6429	645	7	n.	n.	PROPN
ejpam-6429	645	8	jafari	jafari	PROPN
ejpam-6429	645	9	rad	rad	PROPN
ejpam-6429	645	10	,	,	PUNCT
ejpam-6429	645	11	s.	s.	PROPN
ejpam-6429	645	12	m.	m.	PROPN
ejpam-6429	645	13	sheikholeslami	sheikholeslami	PROPN
ejpam-6429	645	14	,	,	PUNCT
ejpam-6429	645	15	and	and	CCONJ
ejpam-6429	645	16	l.	l.	PROPN
ejpam-6429	645	17	volkmann	volkmann	PROPN
ejpam-6429	645	18	.	.	PUNCT
ejpam-6429	646	1	varieties	variety	NOUN
ejpam-6429	646	2	of	of	ADP
ejpam-6429	646	3	roman	roman	ADJ
ejpam-6429	646	4	domination	domination	PROPN
ejpam-6429	646	5	iii	iii	PROPN
ejpam-6429	646	6	.	.	PROPN
ejpam-6429	646	7	submitted	submit	VERB
ejpam-6429	646	8	.	.	PUNCT
ejpam-6429	647	1	[	[	X
ejpam-6429	647	2	8	8	NUM
ejpam-6429	647	3	]	]	X
ejpam-6429	647	4	m.	m.	NOUN
ejpam-6429	647	5	chellali	chellali	PROPN
ejpam-6429	647	6	,	,	PUNCT
ejpam-6429	647	7	n.	n.	PROPN
ejpam-6429	647	8	jafari	jafari	PROPN
ejpam-6429	647	9	rad	rad	PROPN
ejpam-6429	647	10	,	,	PUNCT
ejpam-6429	647	11	s.	s.	PROPN
ejpam-6429	647	12	m.	m.	PROPN
ejpam-6429	647	13	sheikholeslami	sheikholeslami	PROPN
ejpam-6429	647	14	,	,	PUNCT
ejpam-6429	647	15	and	and	CCONJ
ejpam-6429	647	16	l.	l.	PROPN
ejpam-6429	647	17	volkmann	volkmann	PROPN
ejpam-6429	647	18	.	.	PUNCT
ejpam-6429	648	1	varieties	variety	NOUN
ejpam-6429	648	2	of	of	ADP
ejpam-6429	648	3	roman	roman	ADJ
ejpam-6429	648	4	domination	domination	NOUN
ejpam-6429	648	5	iv	iv	X
ejpam-6429	648	6	.	.	PUNCT
ejpam-6429	648	7	submitted	submit	VERB
ejpam-6429	648	8	.	.	PUNCT
ejpam-6429	649	1	[	[	X
ejpam-6429	649	2	9	9	NUM
ejpam-6429	649	3	]	]	PUNCT
ejpam-6429	649	4	m.	m.	NOUN
ejpam-6429	649	5	chellali	chellali	PROPN
ejpam-6429	649	6	,	,	PUNCT
ejpam-6429	649	7	n.	n.	PROPN
ejpam-6429	649	8	jafari	jafari	PROPN
ejpam-6429	649	9	rad	rad	PROPN
ejpam-6429	649	10	,	,	PUNCT
ejpam-6429	649	11	s.	s.	PROPN
ejpam-6429	649	12	m.	m.	PROPN
ejpam-6429	649	13	sheikholeslami	sheikholeslami	PROPN
ejpam-6429	649	14	,	,	PUNCT
ejpam-6429	649	15	and	and	CCONJ
ejpam-6429	649	16	l.	l.	PROPN
ejpam-6429	649	17	volkmann	volkmann	PROPN
ejpam-6429	649	18	.	.	PUNCT
ejpam-6429	650	1	the	the	DET
ejpam-6429	650	2	roman	roman	ADJ
ejpam-6429	650	3	domatic	domatic	ADJ
ejpam-6429	650	4	problem	problem	NOUN
ejpam-6429	650	5	in	in	ADP
ejpam-6429	650	6	graphs	graph	NOUN
ejpam-6429	650	7	and	and	CCONJ
ejpam-6429	650	8	digraphs	digraph	NOUN
ejpam-6429	650	9	:	:	PUNCT
ejpam-6429	650	10	a	a	DET
ejpam-6429	650	11	survey	survey	NOUN
ejpam-6429	650	12	.	.	PUNCT
ejpam-6429	651	1	discussiones	discussione	NOUN
ejpam-6429	651	2	mathematicae	mathematicae	PROPN
ejpam-6429	651	3	graph	graph	NOUN
ejpam-6429	651	4	theory	theory	NOUN
ejpam-6429	651	5	,	,	PUNCT
ejpam-6429	651	6	42:861–891	42:861–891	NUM
ejpam-6429	651	7	,	,	PUNCT
ejpam-6429	651	8	2022	2022	NUM
ejpam-6429	651	9	.	.	PUNCT
ejpam-6429	652	1	[	[	X
ejpam-6429	652	2	10	10	NUM
ejpam-6429	652	3	]	]	X
ejpam-6429	652	4	m.	m.	NOUN
ejpam-6429	652	5	chellali	chellali	PROPN
ejpam-6429	652	6	,	,	PUNCT
ejpam-6429	652	7	n.	n.	PROPN
ejpam-6429	652	8	jafari	jafari	PROPN
ejpam-6429	652	9	rad	rad	PROPN
ejpam-6429	652	10	,	,	PUNCT
ejpam-6429	652	11	s.	s.	PROPN
ejpam-6429	652	12	m.	m.	PROPN
ejpam-6429	652	13	sheikholeslami	sheikholeslami	PROPN
ejpam-6429	652	14	,	,	PUNCT
ejpam-6429	652	15	and	and	CCONJ
ejpam-6429	652	16	l.	l.	PROPN
ejpam-6429	652	17	volkmann	volkmann	PROPN
ejpam-6429	652	18	.	.	PUNCT
ejpam-6429	653	1	a	a	DET
ejpam-6429	653	2	survey	survey	NOUN
ejpam-6429	653	3	on	on	ADP
ejpam-6429	653	4	roman	roman	ADJ
ejpam-6429	653	5	domination	domination	NOUN
ejpam-6429	653	6	parameters	parameter	NOUN
ejpam-6429	653	7	in	in	ADP
ejpam-6429	653	8	directed	direct	VERB
ejpam-6429	653	9	graphs	graph	NOUN
ejpam-6429	653	10	.	.	PUNCT
ejpam-6429	654	1	journal	journal	NOUN
ejpam-6429	654	2	of	of	ADP
ejpam-6429	654	3	combinatorial	combinatorial	ADJ
ejpam-6429	654	4	mathematics	mathematic	NOUN
ejpam-6429	654	5	and	and	CCONJ
ejpam-6429	654	6	combinatorial	combinatorial	ADJ
ejpam-6429	654	7	computing	computing	NOUN
ejpam-6429	654	8	,	,	PUNCT
ejpam-6429	654	9	115:141–171	115:141–171	NUM
ejpam-6429	654	10	,	,	PUNCT
ejpam-6429	654	11	2020	2020	NUM
ejpam-6429	654	12	.	.	PUNCT
ejpam-6429	655	1	[	[	X
ejpam-6429	655	2	11	11	NUM
ejpam-6429	655	3	]	]	PUNCT
ejpam-6429	655	4	m.	m.	NOUN
ejpam-6429	655	5	benatallah	benatallah	PROPN
ejpam-6429	655	6	,	,	PUNCT
ejpam-6429	655	7	m.	m.	NOUN
ejpam-6429	655	8	blidia	blidia	PROPN
ejpam-6429	655	9	,	,	PUNCT
ejpam-6429	655	10	and	and	CCONJ
ejpam-6429	655	11	l.	l.	PROPN
ejpam-6429	655	12	ouldrabah	ouldrabah	PROPN
ejpam-6429	655	13	.	.	PUNCT
ejpam-6429	656	1	the	the	DET
ejpam-6429	656	2	generous	generous	ADJ
ejpam-6429	656	3	roman	roman	ADJ
ejpam-6429	656	4	domination	domination	NOUN
ejpam-6429	656	5	number	number	NOUN
ejpam-6429	656	6	.	.	PUNCT
ejpam-6429	657	1	transactions	transaction	NOUN
ejpam-6429	657	2	on	on	ADP
ejpam-6429	657	3	combinatorics	combinatoric	NOUN
ejpam-6429	657	4	,	,	PUNCT
ejpam-6429	657	5	13:179–196	13:179–196	NUM
ejpam-6429	657	6	,	,	PUNCT
ejpam-6429	657	7	2024	2024	NUM
ejpam-6429	657	8	.	.	PUNCT
ejpam-6429	658	1	[	[	X
ejpam-6429	658	2	12	12	NUM
ejpam-6429	658	3	]	]	X
ejpam-6429	658	4	s.	s.	PROPN
ejpam-6429	658	5	m.	m.	PROPN
ejpam-6429	658	6	sheikholeslami	sheikholeslami	PROPN
ejpam-6429	658	7	,	,	PUNCT
ejpam-6429	658	8	m.	m.	NOUN
ejpam-6429	658	9	chellali	chellali	PROPN
ejpam-6429	658	10	,	,	PUNCT
ejpam-6429	658	11	and	and	CCONJ
ejpam-6429	658	12	m.	m.	PROPN
ejpam-6429	658	13	kor	kor	PROPN
ejpam-6429	658	14	.	.	PUNCT
ejpam-6429	659	1	further	further	PROPN
ejpam-6429	659	2	results	result	NOUN
ejpam-6429	659	3	on	on	ADP
ejpam-6429	659	4	generous	generous	ADJ
ejpam-6429	659	5	roman	roman	ADJ
ejpam-6429	659	6	domination	domination	NOUN
ejpam-6429	659	7	.	.	PUNCT
ejpam-6429	660	1	mathematics	mathematic	NOUN
ejpam-6429	660	2	interdisciplinary	interdisciplinary	ADJ
ejpam-6429	660	3	research	research	NOUN
ejpam-6429	660	4	,	,	PUNCT
ejpam-6429	660	5	10:231–243	10:231–243	NUM
ejpam-6429	660	6	,	,	PUNCT
ejpam-6429	660	7	2025	2025	NUM
ejpam-6429	660	8	.	.	PUNCT
ejpam-6429	661	1	[	[	X
ejpam-6429	661	2	13	13	NUM
ejpam-6429	661	3	]	]	PUNCT
ejpam-6429	661	4	s.	s.	PROPN
ejpam-6429	661	5	m.	m.	PROPN
ejpam-6429	661	6	sheikholeslami	sheikholeslami	PROPN
ejpam-6429	661	7	,	,	PUNCT
ejpam-6429	661	8	m.	m.	NOUN
ejpam-6429	661	9	chellali	chellali	PROPN
ejpam-6429	661	10	,	,	PUNCT
ejpam-6429	661	11	and	and	CCONJ
ejpam-6429	661	12	m.	m.	PROPN
ejpam-6429	661	13	kor	kor	PROPN
ejpam-6429	661	14	.	.	PROPN
ejpam-6429	661	15	generous	generous	ADJ
ejpam-6429	661	16	roman	roman	ADJ
ejpam-6429	661	17	domination	domination	NOUN
ejpam-6429	661	18	stability	stability	NOUN
ejpam-6429	661	19	in	in	ADP
ejpam-6429	661	20	graphs	graph	NOUN
ejpam-6429	661	21	.	.	PUNCT
ejpam-6429	662	1	journal	journal	NOUN
ejpam-6429	662	2	of	of	ADP
ejpam-6429	662	3	discrete	discrete	ADJ
ejpam-6429	662	4	mathematical	mathematical	ADJ
ejpam-6429	662	5	applications	application	NOUN
ejpam-6429	662	6	,	,	PUNCT
ejpam-6429	662	7	10:231–243	10:231–243	NUM
ejpam-6429	662	8	,	,	PUNCT
ejpam-6429	662	9	2025	2025	NUM
ejpam-6429	662	10	.	.	PUNCT
ejpam-6429	663	1	to	to	PART
ejpam-6429	663	2	appear	appear	VERB
ejpam-6429	663	3	.	.	PUNCT
ejpam-6429	664	1	[	[	X
ejpam-6429	664	2	14	14	NUM
ejpam-6429	664	3	]	]	X
ejpam-6429	664	4	s.	s.	PROPN
ejpam-6429	664	5	velammal	velammal	PROPN
ejpam-6429	664	6	.	.	PUNCT
ejpam-6429	665	1	studies	study	NOUN
ejpam-6429	665	2	in	in	ADP
ejpam-6429	665	3	graph	graph	NOUN
ejpam-6429	665	4	theory	theory	NOUN
ejpam-6429	665	5	:	:	PUNCT
ejpam-6429	665	6	covering	covering	NOUN
ejpam-6429	665	7	,	,	PUNCT
ejpam-6429	665	8	independence	independence	NOUN
ejpam-6429	665	9	,	,	PUNCT
ejpam-6429	665	10	domination	domination	NOUN
ejpam-6429	665	11	and	and	CCONJ
ejpam-6429	665	12	j.	j.	PROPN
ejpam-6429	665	13	j.	j.	PROPN
ejpam-6429	665	14	hamja	hamja	PROPN
ejpam-6429	665	15	et	et	PROPN
ejpam-6429	665	16	al	al	PROPN
ejpam-6429	665	17	.	.	PUNCT
ejpam-6429	665	18	/	/	SYM
ejpam-6429	665	19	eur	eur	PROPN
ejpam-6429	665	20	.	.	PUNCT
ejpam-6429	666	1	j.	j.	PROPN
ejpam-6429	666	2	pure	pure	PROPN
ejpam-6429	666	3	appl	appl	PROPN
ejpam-6429	666	4	.	.	PROPN
ejpam-6429	666	5	math	math	PROPN
ejpam-6429	666	6	,	,	PUNCT
ejpam-6429	666	7	18	18	NUM
ejpam-6429	666	8	(	(	PUNCT
ejpam-6429	666	9	4	4	NUM
ejpam-6429	666	10	)	)	PUNCT
ejpam-6429	666	11	(	(	PUNCT
ejpam-6429	666	12	2025	2025	NUM
ejpam-6429	666	13	)	)	PUNCT
ejpam-6429	666	14	,	,	PUNCT
ejpam-6429	666	15	6429	6429	NUM
ejpam-6429	666	16	16	16	NUM
ejpam-6429	666	17	of	of	ADP
ejpam-6429	666	18	16	16	NUM
ejpam-6429	666	19	related	relate	VERB
ejpam-6429	666	20	topics	topic	NOUN
ejpam-6429	666	21	.	.	PUNCT
ejpam-6429	667	1	phd	phd	NOUN
ejpam-6429	667	2	thesis	thesis	PROPN
ejpam-6429	667	3	,	,	PUNCT
ejpam-6429	667	4	manonmaniam	manonmaniam	PROPN
ejpam-6429	667	5	sundaranar	sundaranar	PROPN
ejpam-6429	667	6	university	university	PROPN
ejpam-6429	667	7	,	,	PUNCT
ejpam-6429	667	8	tirunelveli	tirunelveli	PROPN
ejpam-6429	667	9	,	,	PUNCT
ejpam-6429	667	10	india	india	PROPN
ejpam-6429	667	11	,	,	PUNCT
ejpam-6429	667	12	1997	1997	NUM
ejpam-6429	667	13	.	.	PUNCT
ejpam-6429	668	1	[	[	X
ejpam-6429	668	2	15	15	NUM
ejpam-6429	668	3	]	]	X
ejpam-6429	668	4	j.	j.	PROPN
ejpam-6429	668	5	amjadi	amjadi	PROPN
ejpam-6429	668	6	.	.	PUNCT
ejpam-6429	669	1	total	total	ADJ
ejpam-6429	669	2	roman	roman	ADJ
ejpam-6429	669	3	domination	domination	NOUN
ejpam-6429	669	4	subdivision	subdivision	NOUN
ejpam-6429	669	5	number	number	NOUN
ejpam-6429	669	6	in	in	ADP
ejpam-6429	669	7	graphs	graph	NOUN
ejpam-6429	669	8	.	.	PUNCT
ejpam-6429	670	1	communications	communication	NOUN
ejpam-6429	670	2	in	in	ADP
ejpam-6429	670	3	combinatorics	combinatoric	NOUN
ejpam-6429	670	4	and	and	CCONJ
ejpam-6429	670	5	optimization	optimization	NOUN
ejpam-6429	670	6	,	,	PUNCT
ejpam-6429	670	7	5:157–168	5:157–168	NOUN
ejpam-6429	670	8	,	,	PUNCT
ejpam-6429	670	9	2020	2020	NUM
ejpam-6429	670	10	.	.	PUNCT
ejpam-6429	671	1	[	[	X
ejpam-6429	671	2	16	16	NUM
ejpam-6429	671	3	]	]	PUNCT
ejpam-6429	671	4	j.	j.	PROPN
ejpam-6429	671	5	amjadi	amjadi	PROPN
ejpam-6429	671	6	,	,	PUNCT
ejpam-6429	671	7	r.	r.	PROPN
ejpam-6429	671	8	khoeilar	khoeilar	PROPN
ejpam-6429	671	9	,	,	PUNCT
ejpam-6429	671	10	m.	m.	NOUN
ejpam-6429	671	11	chellali	chellali	PROPN
ejpam-6429	671	12	,	,	PUNCT
ejpam-6429	671	13	and	and	CCONJ
ejpam-6429	671	14	z.	z.	PROPN
ejpam-6429	671	15	shao	shao	PROPN
ejpam-6429	671	16	.	.	PUNCT
ejpam-6429	672	1	on	on	ADP
ejpam-6429	672	2	the	the	DET
ejpam-6429	672	3	roman	roman	ADJ
ejpam-6429	672	4	domination	domination	NOUN
ejpam-6429	672	5	subdivision	subdivision	NOUN
ejpam-6429	672	6	number	number	NOUN
ejpam-6429	672	7	of	of	ADP
ejpam-6429	672	8	a	a	DET
ejpam-6429	672	9	graph	graph	NOUN
ejpam-6429	672	10	.	.	PUNCT
ejpam-6429	672	11	journal	journal	NOUN
ejpam-6429	672	12	of	of	ADP
ejpam-6429	672	13	combinatorial	combinatorial	ADJ
ejpam-6429	672	14	optimization	optimization	NOUN
ejpam-6429	672	15	,	,	PUNCT
ejpam-6429	672	16	40:501–511	40:501–511	PROPN
ejpam-6429	672	17	,	,	PUNCT
ejpam-6429	672	18	2020	2020	NUM
ejpam-6429	672	19	.	.	PUNCT
ejpam-6429	673	1	[	[	X
ejpam-6429	673	2	17	17	NUM
ejpam-6429	673	3	]	]	PUNCT
ejpam-6429	673	4	j.	j.	PROPN
ejpam-6429	673	5	amjadi	amjadi	PROPN
ejpam-6429	673	6	and	and	CCONJ
ejpam-6429	673	7	h.	h.	PROPN
ejpam-6429	673	8	sadeghi	sadeghi	PROPN
ejpam-6429	673	9	.	.	PUNCT
ejpam-6429	674	1	double	double	ADJ
ejpam-6429	674	2	roman	roman	ADJ
ejpam-6429	674	3	domination	domination	NOUN
ejpam-6429	674	4	subdivision	subdivision	NOUN
ejpam-6429	674	5	number	number	NOUN
ejpam-6429	674	6	in	in	ADP
ejpam-6429	674	7	graphs	graph	NOUN
ejpam-6429	674	8	.	.	PUNCT
ejpam-6429	675	1	asian	asian	ADJ
ejpam-6429	675	2	-	-	PUNCT
ejpam-6429	675	3	european	european	ADJ
ejpam-6429	675	4	journal	journal	NOUN
ejpam-6429	675	5	of	of	ADP
ejpam-6429	675	6	mathematics	mathematic	NOUN
ejpam-6429	675	7	,	,	PUNCT
ejpam-6429	675	8	15:2250125	15:2250125	NUM
ejpam-6429	675	9	,	,	PUNCT
ejpam-6429	675	10	2020	2020	NUM
ejpam-6429	675	11	.	.	PUNCT
ejpam-6429	676	1	[	[	X
ejpam-6429	676	2	18	18	NUM
ejpam-6429	676	3	]	]	X
ejpam-6429	676	4	o.	o.	PROPN
ejpam-6429	676	5	favaron	favaron	PROPN
ejpam-6429	676	6	,	,	PUNCT
ejpam-6429	676	7	h.	h.	PROPN
ejpam-6429	676	8	karami	karami	PROPN
ejpam-6429	676	9	,	,	PUNCT
ejpam-6429	676	10	and	and	CCONJ
ejpam-6429	676	11	s.	s.	PROPN
ejpam-6429	676	12	m.	m.	PROPN
ejpam-6429	676	13	sheikholeslami	sheikholeslami	PROPN
ejpam-6429	676	14	.	.	PUNCT
ejpam-6429	677	1	paired	pair	VERB
ejpam-6429	677	2	-	-	PUNCT
ejpam-6429	677	3	domination	domination	NOUN
ejpam-6429	677	4	subdivision	subdivision	NOUN
ejpam-6429	677	5	numbers	number	NOUN
ejpam-6429	677	6	of	of	ADP
ejpam-6429	677	7	graphs	graph	NOUN
ejpam-6429	677	8	.	.	PUNCT
ejpam-6429	678	1	graphs	graph	NOUN
ejpam-6429	678	2	and	and	CCONJ
ejpam-6429	678	3	combinatorics	combinatoric	NOUN
ejpam-6429	678	4	,	,	PUNCT
ejpam-6429	678	5	25:503–512	25:503–512	PROPN
ejpam-6429	678	6	,	,	PUNCT
ejpam-6429	678	7	2009	2009	NUM
ejpam-6429	678	8	.	.	PUNCT
ejpam-6429	679	1	[	[	X
ejpam-6429	679	2	19	19	NUM
ejpam-6429	679	3	]	]	X
ejpam-6429	679	4	n.	n.	PROPN
ejpam-6429	679	5	meddah	meddah	PROPN
ejpam-6429	679	6	,	,	PUNCT
ejpam-6429	679	7	m.	m.	NOUN
ejpam-6429	679	8	blidia	blidia	PROPN
ejpam-6429	679	9	,	,	PUNCT
ejpam-6429	679	10	and	and	CCONJ
ejpam-6429	679	11	m.	m.	NOUN
ejpam-6429	679	12	chellali	chellali	PROPN
ejpam-6429	679	13	.	.	PUNCT
ejpam-6429	680	1	on	on	ADP
ejpam-6429	680	2	the	the	DET
ejpam-6429	680	3	2	2	NUM
ejpam-6429	680	4	-	-	PUNCT
ejpam-6429	680	5	independence	independence	NOUN
ejpam-6429	680	6	subdivision	subdivision	NOUN
ejpam-6429	680	7	number	number	NOUN
ejpam-6429	680	8	of	of	ADP
ejpam-6429	680	9	graphs	graph	NOUN
ejpam-6429	680	10	.	.	PUNCT
ejpam-6429	681	1	communications	communication	NOUN
ejpam-6429	681	2	in	in	ADP
ejpam-6429	681	3	combinatorics	combinatoric	NOUN
ejpam-6429	681	4	and	and	CCONJ
ejpam-6429	681	5	optimization	optimization	NOUN
ejpam-6429	681	6	,	,	PUNCT
ejpam-6429	681	7	7:105–112	7:105–112	PROPN
ejpam-6429	681	8	,	,	PUNCT
ejpam-6429	681	9	2022	2022	NUM
ejpam-6429	681	10	.	.	PUNCT
ejpam-6429	682	1	[	[	X
ejpam-6429	682	2	20	20	NUM
ejpam-6429	682	3	]	]	PUNCT
ejpam-6429	682	4	x.	x.	PROPN
ejpam-6429	682	5	qiang	qiang	PROPN
ejpam-6429	682	6	,	,	PUNCT
ejpam-6429	682	7	s.	s.	PROPN
ejpam-6429	682	8	kosari	kosari	PROPN
ejpam-6429	682	9	,	,	PUNCT
ejpam-6429	682	10	z.	z.	PROPN
ejpam-6429	682	11	shao	shao	PROPN
ejpam-6429	682	12	,	,	PUNCT
ejpam-6429	682	13	s.	s.	PROPN
ejpam-6429	682	14	m.	m.	PROPN
ejpam-6429	682	15	sheikholeslami	sheikholeslami	PROPN
ejpam-6429	682	16	,	,	PUNCT
ejpam-6429	682	17	m.	m.	NOUN
ejpam-6429	682	18	chellali	chellali	PROPN
ejpam-6429	682	19	,	,	PUNCT
ejpam-6429	682	20	and	and	CCONJ
ejpam-6429	682	21	h.	h.	PROPN
ejpam-6429	682	22	karami	karami	PROPN
ejpam-6429	682	23	.	.	PUNCT
ejpam-6429	683	1	a	a	DET
ejpam-6429	683	2	note	note	NOUN
ejpam-6429	683	3	on	on	ADP
ejpam-6429	683	4	the	the	DET
ejpam-6429	683	5	paired	pair	VERB
ejpam-6429	683	6	-	-	PUNCT
ejpam-6429	683	7	domination	domination	NOUN
ejpam-6429	683	8	subdivision	subdivision	NOUN
ejpam-6429	683	9	number	number	NOUN
ejpam-6429	683	10	of	of	ADP
ejpam-6429	683	11	trees	tree	NOUN
ejpam-6429	683	12	.	.	PUNCT
ejpam-6429	684	1	mathematics	mathematic	NOUN
ejpam-6429	684	2	,	,	PUNCT
ejpam-6429	684	3	9:181	9:181	NUM
ejpam-6429	684	4	,	,	PUNCT
ejpam-6429	684	5	2021	2021	NUM
ejpam-6429	684	6	.	.	PUNCT
ejpam-6429	685	1	[	[	X
ejpam-6429	685	2	21	21	NUM
ejpam-6429	685	3	]	]	X
ejpam-6429	685	4	p.	p.	PROPN
ejpam-6429	685	5	roushini	roushini	PROPN
ejpam-6429	685	6	leely	leely	ADV
ejpam-6429	685	7	pushpam	pushpam	VERB
ejpam-6429	685	8	and	and	CCONJ
ejpam-6429	685	9	k.	k.	PROPN
ejpam-6429	685	10	priya	priya	PROPN
ejpam-6429	685	11	bhanthavi	bhanthavi	VERB
ejpam-6429	685	12	.	.	PUNCT
ejpam-6429	686	1	independent	independent	ADJ
ejpam-6429	686	2	transversal	transversal	ADJ
ejpam-6429	686	3	domination	domination	NOUN
ejpam-6429	686	4	subdivision	subdivision	NOUN
ejpam-6429	686	5	number	number	NOUN
ejpam-6429	686	6	of	of	ADP
ejpam-6429	686	7	trees	tree	NOUN
ejpam-6429	686	8	.	.	PUNCT
ejpam-6429	687	1	communications	communication	NOUN
ejpam-6429	687	2	in	in	ADP
ejpam-6429	687	3	combinatorics	combinatoric	NOUN
ejpam-6429	687	4	and	and	CCONJ
ejpam-6429	687	5	optimization	optimization	NOUN
ejpam-6429	687	6	.	.	PUNCT
ejpam-6429	688	1	in	in	ADP
ejpam-6429	688	2	press	press	NOUN
ejpam-6429	688	3	.	.	PUNCT
ejpam-6429	689	1	[	[	X
ejpam-6429	689	2	22	22	NUM
ejpam-6429	689	3	]	]	PUNCT
ejpam-6429	689	4	p.	p.	NOUN
ejpam-6429	689	5	roushini	roushini	PROPN
ejpam-6429	689	6	leely	leely	ADV
ejpam-6429	689	7	pushpam	pushpam	VERB
ejpam-6429	689	8	and	and	CCONJ
ejpam-6429	689	9	n.	n.	NOUN
ejpam-6429	689	10	srilakshmi	srilakshmi	PROPN
ejpam-6429	689	11	.	.	PUNCT
ejpam-6429	690	1	weak	weak	ADJ
ejpam-6429	690	2	roman	roman	ADJ
ejpam-6429	690	3	subdivision	subdivision	NOUN
ejpam-6429	690	4	number	number	NOUN
ejpam-6429	690	5	of	of	ADP
ejpam-6429	690	6	graphs	graph	NOUN
ejpam-6429	690	7	.	.	PUNCT
ejpam-6429	691	1	discrete	discrete	ADJ
ejpam-6429	691	2	mathematics	mathematic	NOUN
ejpam-6429	691	3	,	,	PUNCT
ejpam-6429	691	4	algorithms	algorithm	NOUN
ejpam-6429	691	5	and	and	CCONJ
ejpam-6429	691	6	applications	application	NOUN
ejpam-6429	691	7	,	,	PUNCT
ejpam-6429	691	8	14:2150102	14:2150102	NUM
ejpam-6429	691	9	,	,	PUNCT
ejpam-6429	691	10	2022	2022	NUM
ejpam-6429	691	11	.	.	PUNCT