id	sid	tid	token	lemma	pos
ejpam-5648	1	1	european	european	PROPN
ejpam-5648	1	2	journal	journal	PROPN
ejpam-5648	1	3	of	of	ADP
ejpam-5648	1	4	pure	pure	ADJ
ejpam-5648	1	5	and	and	CCONJ
ejpam-5648	1	6	applied	applied	ADJ
ejpam-5648	1	7	mathematics	mathematic	NOUN
ejpam-5648	1	8	2025	2025	NUM
ejpam-5648	1	9	,	,	PUNCT
ejpam-5648	1	10	vol	vol	NOUN
ejpam-5648	1	11	.	.	PROPN
ejpam-5648	1	12	18	18	NUM
ejpam-5648	1	13	,	,	PUNCT
ejpam-5648	1	14	issue	issue	NOUN
ejpam-5648	1	15	1	1	NUM
ejpam-5648	1	16	,	,	PUNCT
ejpam-5648	1	17	article	article	NOUN
ejpam-5648	1	18	number	number	NOUN
ejpam-5648	1	19	5648	5648	NUM
ejpam-5648	1	20	issn	issn	PROPN
ejpam-5648	1	21	1307	1307	NUM
ejpam-5648	1	22	-	-	SYM
ejpam-5648	1	23	5543	5543	NUM
ejpam-5648	1	24	–	–	PUNCT
ejpam-5648	1	25	ejpam.com	ejpam.com	X
ejpam-5648	1	26	published	publish	VERB
ejpam-5648	1	27	by	by	ADP
ejpam-5648	1	28	new	new	PROPN
ejpam-5648	1	29	york	york	PROPN
ejpam-5648	1	30	business	business	PROPN
ejpam-5648	1	31	global	global	PROPN
ejpam-5648	1	32	connected	connect	VERB
ejpam-5648	1	33	hop	hop	NOUN
ejpam-5648	1	34	roman	roman	ADJ
ejpam-5648	1	35	dominating	dominating	NOUN
ejpam-5648	1	36	functions	function	NOUN
ejpam-5648	1	37	in	in	ADP
ejpam-5648	1	38	graphs	graph	NOUN
ejpam-5648	1	39	alkajim	alkajim	PROPN
ejpam-5648	1	40	a.	a.	NOUN
ejpam-5648	1	41	aradais3,∗	aradais3,∗	NOUN
ejpam-5648	1	42	,	,	PUNCT
ejpam-5648	1	43	jerry	jerry	NOUN
ejpam-5648	1	44	boy	boy	NOUN
ejpam-5648	1	45	g.	g.	PROPN
ejpam-5648	1	46	cariaga1	cariaga1	PROPN
ejpam-5648	1	47	,	,	PUNCT
ejpam-5648	1	48	sergio	sergio	PROPN
ejpam-5648	1	49	r.	r.	PROPN
ejpam-5648	1	50	canoy	canoy	PROPN
ejpam-5648	1	51	,	,	PUNCT
ejpam-5648	1	52	jr.1,2	jr.1,2	ADJ
ejpam-5648	1	53	1	1	NUM
ejpam-5648	1	54	department	department	NOUN
ejpam-5648	1	55	of	of	ADP
ejpam-5648	1	56	mathematics	mathematic	NOUN
ejpam-5648	1	57	and	and	CCONJ
ejpam-5648	1	58	statistics	statistic	NOUN
ejpam-5648	1	59	,	,	PUNCT
ejpam-5648	1	60	college	college	NOUN
ejpam-5648	1	61	of	of	ADP
ejpam-5648	1	62	science	science	NOUN
ejpam-5648	1	63	and	and	CCONJ
ejpam-5648	1	64	mathematics	mathematic	NOUN
ejpam-5648	1	65	,	,	PUNCT
ejpam-5648	1	66	msu	msu	PROPN
ejpam-5648	1	67	-	-	PUNCT
ejpam-5648	1	68	iligan	iligan	PROPN
ejpam-5648	1	69	institute	institute	PROPN
ejpam-5648	1	70	of	of	ADP
ejpam-5648	1	71	technology	technology	PROPN
ejpam-5648	1	72	,	,	PUNCT
ejpam-5648	1	73	9200	9200	NUM
ejpam-5648	1	74	iligan	iligan	ADJ
ejpam-5648	1	75	city	city	NOUN
ejpam-5648	1	76	,	,	PUNCT
ejpam-5648	1	77	philippines	philippine	NOUN
ejpam-5648	1	78	2	2	NUM
ejpam-5648	1	79	center	center	NOUN
ejpam-5648	1	80	of	of	ADP
ejpam-5648	1	81	mathematical	mathematical	ADJ
ejpam-5648	1	82	and	and	CCONJ
ejpam-5648	1	83	theoretical	theoretical	ADJ
ejpam-5648	1	84	physical	physical	ADJ
ejpam-5648	1	85	sciences	science	NOUN
ejpam-5648	1	86	,	,	PUNCT
ejpam-5648	1	87	premier	premier	PROPN
ejpam-5648	1	88	research	research	PROPN
ejpam-5648	1	89	institute	institute	PROPN
ejpam-5648	1	90	of	of	ADP
ejpam-5648	1	91	science	science	NOUN
ejpam-5648	1	92	and	and	CCONJ
ejpam-5648	1	93	mathematics	mathematic	NOUN
ejpam-5648	1	94	,	,	PUNCT
ejpam-5648	1	95	msu	msu	PROPN
ejpam-5648	1	96	-	-	PUNCT
ejpam-5648	1	97	iligan	iligan	PROPN
ejpam-5648	1	98	institute	institute	PROPN
ejpam-5648	1	99	of	of	ADP
ejpam-5648	1	100	technology	technology	PROPN
ejpam-5648	1	101	,	,	PUNCT
ejpam-5648	1	102	9200	9200	NUM
ejpam-5648	1	103	iligan	iligan	ADJ
ejpam-5648	1	104	city	city	NOUN
ejpam-5648	1	105	,	,	PUNCT
ejpam-5648	1	106	philippines	philippine	NOUN
ejpam-5648	1	107	3	3	NUM
ejpam-5648	1	108	integrated	integrate	VERB
ejpam-5648	1	109	laboratory	laboratory	NOUN
ejpam-5648	1	110	school	school	NOUN
ejpam-5648	1	111	,	,	PUNCT
ejpam-5648	1	112	college	college	NOUN
ejpam-5648	1	113	of	of	ADP
ejpam-5648	1	114	education	education	NOUN
ejpam-5648	1	115	,	,	PUNCT
ejpam-5648	1	116	msu	msu	PROPN
ejpam-5648	1	117	-	-	PUNCT
ejpam-5648	1	118	tawi	tawi	NOUN
ejpam-5648	1	119	-	-	PUNCT
ejpam-5648	1	120	tawi	tawi	NOUN
ejpam-5648	1	121	college	college	PROPN
ejpam-5648	1	122	of	of	ADP
ejpam-5648	1	123	technology	technology	NOUN
ejpam-5648	1	124	and	and	CCONJ
ejpam-5648	1	125	oceanography	oceanography	NOUN
ejpam-5648	1	126	,	,	PUNCT
ejpam-5648	1	127	7500	7500	NUM
ejpam-5648	1	128	bongao	bongao	NOUN
ejpam-5648	1	129	,	,	PUNCT
ejpam-5648	1	130	tawi	tawi	NOUN
ejpam-5648	1	131	-	-	PUNCT
ejpam-5648	1	132	tawi	tawi	NOUN
ejpam-5648	1	133	,	,	PUNCT
ejpam-5648	1	134	philippines	philippine	NOUN
ejpam-5648	1	135	abstract	abstract	ADJ
ejpam-5648	1	136	.	.	PUNCT
ejpam-5648	2	1	let	let	VERB
ejpam-5648	2	2	g	g	PRON
ejpam-5648	2	3	be	be	AUX
ejpam-5648	2	4	a	a	DET
ejpam-5648	2	5	connected	connected	ADJ
ejpam-5648	2	6	graph	graph	NOUN
ejpam-5648	2	7	.	.	PUNCT
ejpam-5648	3	1	a	a	DET
ejpam-5648	3	2	hop	hop	NOUN
ejpam-5648	3	3	roman	roman	ADJ
ejpam-5648	3	4	dominating	dominating	NOUN
ejpam-5648	3	5	function	function	NOUN
ejpam-5648	3	6	f	f	NOUN
ejpam-5648	3	7	:	:	PUNCT
ejpam-5648	3	8	v	v	X
ejpam-5648	3	9	(	(	PUNCT
ejpam-5648	3	10	g	g	NOUN
ejpam-5648	3	11	)	)	PUNCT
ejpam-5648	3	12	→	→	SYM
ejpam-5648	3	13	{	{	PUNCT
ejpam-5648	3	14	0	0	NUM
ejpam-5648	3	15	,	,	PUNCT
ejpam-5648	3	16	1	1	NUM
ejpam-5648	3	17	,	,	PUNCT
ejpam-5648	3	18	2	2	NUM
ejpam-5648	3	19	}	}	PUNCT
ejpam-5648	3	20	is	be	AUX
ejpam-5648	3	21	a	a	DET
ejpam-5648	3	22	connected	connected	ADJ
ejpam-5648	3	23	hop	hop	NOUN
ejpam-5648	3	24	roman	roman	ADJ
ejpam-5648	3	25	dominating	dominating	NOUN
ejpam-5648	3	26	function	function	NOUN
ejpam-5648	3	27	(	(	PUNCT
ejpam-5648	3	28	chrdf	chrdf	NOUN
ejpam-5648	3	29	)	)	PUNCT
ejpam-5648	3	30	on	on	ADP
ejpam-5648	3	31	g	g	PROPN
ejpam-5648	3	32	if	if	SCONJ
ejpam-5648	3	33	the	the	DET
ejpam-5648	3	34	set	set	NOUN
ejpam-5648	3	35	{	{	PUNCT
ejpam-5648	3	36	u	u	NOUN
ejpam-5648	3	37	∈	∈	PROPN
ejpam-5648	3	38	v	v	NOUN
ejpam-5648	3	39	(	(	PUNCT
ejpam-5648	3	40	g	g	NOUN
ejpam-5648	3	41	)	)	PUNCT
ejpam-5648	3	42	:	:	PUNCT
ejpam-5648	3	43	f(u	f(u	ADJ
ejpam-5648	3	44	)	)	PUNCT
ejpam-5648	3	45	̸=	̸=	PROPN
ejpam-5648	3	46	0	0	NUM
ejpam-5648	3	47	}	}	PUNCT
ejpam-5648	3	48	induces	induce	VERB
ejpam-5648	3	49	a	a	DET
ejpam-5648	3	50	connected	connected	ADJ
ejpam-5648	3	51	subgraph	subgraph	NOUN
ejpam-5648	3	52	of	of	ADP
ejpam-5648	3	53	g.	g.	PROPN
ejpam-5648	3	54	the	the	DET
ejpam-5648	3	55	weight	weight	NOUN
ejpam-5648	3	56	of	of	ADP
ejpam-5648	3	57	a	a	DET
ejpam-5648	3	58	chrdf	chrdf	NOUN
ejpam-5648	3	59	f	f	PROPN
ejpam-5648	3	60	is	be	AUX
ejpam-5648	3	61	given	give	VERB
ejpam-5648	3	62	by	by	ADP
ejpam-5648	3	63	ωcrh	ωcrh	ADJ
ejpam-5648	3	64	g	g	PROPN
ejpam-5648	3	65	(	(	PUNCT
ejpam-5648	3	66	f	f	X
ejpam-5648	3	67	)	)	PUNCT
ejpam-5648	3	68	=	=	SYM
ejpam-5648	4	1	∑	∑	PUNCT
ejpam-5648	4	2	v∈v	v∈v	PROPN
ejpam-5648	4	3	(	(	PUNCT
ejpam-5648	4	4	g	g	NOUN
ejpam-5648	4	5	)	)	PUNCT
ejpam-5648	4	6	f(v	f(v	NOUN
ejpam-5648	4	7	)	)	PUNCT
ejpam-5648	4	8	and	and	CCONJ
ejpam-5648	4	9	the	the	DET
ejpam-5648	4	10	minimum	minimum	ADJ
ejpam-5648	4	11	weight	weight	NOUN
ejpam-5648	4	12	among	among	ADP
ejpam-5648	4	13	all	all	DET
ejpam-5648	4	14	connected	connect	VERB
ejpam-5648	4	15	hop	hop	NOUN
ejpam-5648	4	16	roman	roman	ADJ
ejpam-5648	4	17	dominating	dominating	NOUN
ejpam-5648	4	18	functions	function	NOUN
ejpam-5648	4	19	on	on	ADP
ejpam-5648	4	20	g	g	NOUN
ejpam-5648	4	21	,	,	PUNCT
ejpam-5648	4	22	denoted	denote	VERB
ejpam-5648	4	23	γcrh(g	γcrh(g	NOUN
ejpam-5648	4	24	)	)	PUNCT
ejpam-5648	4	25	,	,	PUNCT
ejpam-5648	4	26	is	be	AUX
ejpam-5648	4	27	the	the	DET
ejpam-5648	4	28	connected	connect	VERB
ejpam-5648	4	29	hop	hop	NOUN
ejpam-5648	4	30	roman	roman	ADJ
ejpam-5648	4	31	domination	domination	NOUN
ejpam-5648	4	32	number	number	NOUN
ejpam-5648	4	33	of	of	ADP
ejpam-5648	4	34	g.	g.	PROPN
ejpam-5648	4	35	in	in	ADP
ejpam-5648	4	36	this	this	DET
ejpam-5648	4	37	paper	paper	NOUN
ejpam-5648	4	38	,	,	PUNCT
ejpam-5648	4	39	we	we	PRON
ejpam-5648	4	40	show	show	VERB
ejpam-5648	4	41	that	that	SCONJ
ejpam-5648	4	42	the	the	DET
ejpam-5648	4	43	parameter	parameter	NOUN
ejpam-5648	4	44	lies	lie	VERB
ejpam-5648	4	45	between	between	ADP
ejpam-5648	4	46	the	the	DET
ejpam-5648	4	47	connected	connect	VERB
ejpam-5648	4	48	hop	hop	NOUN
ejpam-5648	4	49	domination	domination	NOUN
ejpam-5648	4	50	number	number	NOUN
ejpam-5648	4	51	of	of	ADP
ejpam-5648	4	52	g	g	PROPN
ejpam-5648	4	53	and	and	CCONJ
ejpam-5648	4	54	twice	twice	DET
ejpam-5648	4	55	this	this	DET
ejpam-5648	4	56	number	number	NOUN
ejpam-5648	4	57	.	.	PUNCT
ejpam-5648	5	1	we	we	PRON
ejpam-5648	5	2	characterize	characterize	VERB
ejpam-5648	5	3	the	the	DET
ejpam-5648	5	4	graphs	graph	NOUN
ejpam-5648	5	5	that	that	PRON
ejpam-5648	5	6	attain	attain	VERB
ejpam-5648	5	7	small	small	ADJ
ejpam-5648	5	8	values	value	NOUN
ejpam-5648	5	9	of	of	ADP
ejpam-5648	5	10	the	the	DET
ejpam-5648	5	11	parameter	parameter	NOUN
ejpam-5648	5	12	and	and	CCONJ
ejpam-5648	5	13	determine	determine	VERB
ejpam-5648	5	14	the	the	DET
ejpam-5648	5	15	connected	connected	ADJ
ejpam-5648	5	16	hop	hop	NOUN
ejpam-5648	5	17	roman	roman	ADJ
ejpam-5648	5	18	domination	domination	NOUN
ejpam-5648	5	19	number	number	NOUN
ejpam-5648	5	20	of	of	ADP
ejpam-5648	5	21	some	some	DET
ejpam-5648	5	22	graphs	graph	NOUN
ejpam-5648	5	23	.	.	PUNCT
ejpam-5648	6	1	2020	2020	NUM
ejpam-5648	6	2	mathematics	mathematic	NOUN
ejpam-5648	6	3	subject	subject	NOUN
ejpam-5648	6	4	classifications	classification	NOUN
ejpam-5648	6	5	:	:	PUNCT
ejpam-5648	6	6	05c69	05c69	X
ejpam-5648	6	7	key	key	ADJ
ejpam-5648	6	8	words	word	NOUN
ejpam-5648	6	9	and	and	CCONJ
ejpam-5648	6	10	phrases	phrase	NOUN
ejpam-5648	6	11	:	:	PUNCT
ejpam-5648	6	12	connected	connect	VERB
ejpam-5648	6	13	,	,	PUNCT
ejpam-5648	6	14	hop	hop	NOUN
ejpam-5648	6	15	domination	domination	NOUN
ejpam-5648	6	16	,	,	PUNCT
ejpam-5648	6	17	hop	hop	NOUN
ejpam-5648	6	18	roman	roman	ADJ
ejpam-5648	6	19	dominating	dominating	NOUN
ejpam-5648	6	20	function	function	NOUN
ejpam-5648	6	21	,	,	PUNCT
ejpam-5648	6	22	connected	connect	VERB
ejpam-5648	6	23	hop	hop	NOUN
ejpam-5648	6	24	roman	roman	ADJ
ejpam-5648	6	25	domination	domination	NOUN
ejpam-5648	6	26	number	number	NOUN
ejpam-5648	6	27	1	1	NUM
ejpam-5648	6	28	.	.	PUNCT
ejpam-5648	7	1	introduction	introduction	NOUN
ejpam-5648	7	2	motivated	motivate	VERB
ejpam-5648	7	3	by	by	ADP
ejpam-5648	7	4	the	the	DET
ejpam-5648	7	5	ancient	ancient	ADJ
ejpam-5648	7	6	roman	roman	ADJ
ejpam-5648	7	7	empire	empire	NOUN
ejpam-5648	7	8	’s	’s	PART
ejpam-5648	7	9	military	military	ADJ
ejpam-5648	7	10	strategy	strategy	NOUN
ejpam-5648	7	11	,	,	PUNCT
ejpam-5648	7	12	cockayne	cockayne	NOUN
ejpam-5648	7	13	et	et	PROPN
ejpam-5648	7	14	al	al	PROPN
ejpam-5648	7	15	.	.	PUNCT
ejpam-5648	8	1	in	in	ADP
ejpam-5648	8	2	[	[	X
ejpam-5648	8	3	9	9	NUM
ejpam-5648	8	4	]	]	PUNCT
ejpam-5648	8	5	introduced	introduce	VERB
ejpam-5648	8	6	and	and	CCONJ
ejpam-5648	8	7	studied	study	VERB
ejpam-5648	8	8	the	the	DET
ejpam-5648	8	9	parameter	parameter	NOUN
ejpam-5648	8	10	called	call	VERB
ejpam-5648	8	11	roman	roman	ADJ
ejpam-5648	8	12	domination	domination	NOUN
ejpam-5648	8	13	.	.	PUNCT
ejpam-5648	9	1	in	in	ADP
ejpam-5648	9	2	a	a	DET
ejpam-5648	9	3	sense	sense	NOUN
ejpam-5648	9	4	,	,	PUNCT
ejpam-5648	9	5	the	the	DET
ejpam-5648	9	6	concept	concept	NOUN
ejpam-5648	9	7	is	be	AUX
ejpam-5648	9	8	one	one	NUM
ejpam-5648	9	9	of	of	ADP
ejpam-5648	9	10	the	the	DET
ejpam-5648	9	11	numerous	numerous	ADJ
ejpam-5648	9	12	variants	variant	NOUN
ejpam-5648	9	13	of	of	ADP
ejpam-5648	9	14	the	the	DET
ejpam-5648	9	15	standard	standard	ADJ
ejpam-5648	9	16	domination	domination	NOUN
ejpam-5648	9	17	concept	concept	NOUN
ejpam-5648	9	18	.	.	PUNCT
ejpam-5648	10	1	various	various	ADJ
ejpam-5648	10	2	studies	study	NOUN
ejpam-5648	10	3	have	have	AUX
ejpam-5648	10	4	been	be	AUX
ejpam-5648	10	5	done	do	VERB
ejpam-5648	10	6	since	since	SCONJ
ejpam-5648	10	7	the	the	DET
ejpam-5648	10	8	introduction	introduction	NOUN
ejpam-5648	10	9	of	of	ADP
ejpam-5648	10	10	roman	roman	ADJ
ejpam-5648	10	11	domination	domination	NOUN
ejpam-5648	10	12	.	.	PUNCT
ejpam-5648	11	1	in	in	ADP
ejpam-5648	11	2	particular	particular	ADJ
ejpam-5648	11	3	,	,	PUNCT
ejpam-5648	11	4	a	a	DET
ejpam-5648	11	5	significant	significant	ADJ
ejpam-5648	11	6	number	number	NOUN
ejpam-5648	11	7	of	of	ADP
ejpam-5648	11	8	variations	variation	NOUN
ejpam-5648	11	9	of	of	ADP
ejpam-5648	11	10	the	the	DET
ejpam-5648	11	11	parameter	parameter	NOUN
ejpam-5648	11	12	have	have	AUX
ejpam-5648	11	13	already	already	ADV
ejpam-5648	11	14	been	be	AUX
ejpam-5648	11	15	defined	define	VERB
ejpam-5648	11	16	and	and	CCONJ
ejpam-5648	11	17	investigated	investigate	VERB
ejpam-5648	11	18	(	(	PUNCT
ejpam-5648	11	19	see	see	VERB
ejpam-5648	11	20	[	[	X
ejpam-5648	11	21	1	1	NUM
ejpam-5648	11	22	]	]	PUNCT
ejpam-5648	11	23	,	,	PUNCT
ejpam-5648	11	24	[	[	X
ejpam-5648	11	25	3	3	NUM
ejpam-5648	11	26	]	]	PUNCT
ejpam-5648	11	27	,	,	PUNCT
ejpam-5648	11	28	[	[	X
ejpam-5648	11	29	4	4	NUM
ejpam-5648	11	30	]	]	PUNCT
ejpam-5648	11	31	,	,	PUNCT
ejpam-5648	11	32	[	[	X
ejpam-5648	11	33	7	7	NUM
ejpam-5648	11	34	]	]	PUNCT
ejpam-5648	11	35	,	,	PUNCT
ejpam-5648	12	1	[	[	X
ejpam-5648	12	2	8	8	NUM
ejpam-5648	12	3	]	]	PUNCT
ejpam-5648	12	4	,	,	PUNCT
ejpam-5648	12	5	[	[	X
ejpam-5648	12	6	10	10	NUM
ejpam-5648	12	7	]	]	PUNCT
ejpam-5648	12	8	,	,	PUNCT
ejpam-5648	13	1	[	[	X
ejpam-5648	13	2	11	11	NUM
ejpam-5648	13	3	]	]	PUNCT
ejpam-5648	13	4	,	,	PUNCT
ejpam-5648	13	5	[	[	X
ejpam-5648	13	6	12	12	NUM
ejpam-5648	13	7	]	]	PUNCT
ejpam-5648	13	8	,	,	PUNCT
ejpam-5648	13	9	[	[	X
ejpam-5648	13	10	16	16	NUM
ejpam-5648	13	11	]	]	PUNCT
ejpam-5648	13	12	,	,	PUNCT
ejpam-5648	13	13	[	[	X
ejpam-5648	13	14	20	20	NUM
ejpam-5648	13	15	]	]	PUNCT
ejpam-5648	13	16	,	,	PUNCT
ejpam-5648	14	1	[	[	X
ejpam-5648	14	2	21	21	NUM
ejpam-5648	14	3	]	]	PUNCT
ejpam-5648	14	4	,	,	PUNCT
ejpam-5648	14	5	[	[	X
ejpam-5648	14	6	24	24	NUM
ejpam-5648	14	7	]	]	PUNCT
ejpam-5648	14	8	)	)	PUNCT
ejpam-5648	14	9	.	.	PUNCT
ejpam-5648	15	1	the	the	DET
ejpam-5648	15	2	concept	concept	NOUN
ejpam-5648	15	3	of	of	ADP
ejpam-5648	15	4	hop	hop	PROPN
ejpam-5648	15	5	domination	domination	NOUN
ejpam-5648	15	6	,	,	PUNCT
ejpam-5648	15	7	one	one	NUM
ejpam-5648	15	8	that	that	PRON
ejpam-5648	15	9	utilizes	utilize	VERB
ejpam-5648	15	10	distance	distance	NOUN
ejpam-5648	15	11	two	two	NUM
ejpam-5648	15	12	rather	rather	ADV
ejpam-5648	15	13	than	than	ADP
ejpam-5648	15	14	unit	unit	NOUN
ejpam-5648	15	15	distance	distance	NOUN
ejpam-5648	15	16	,	,	PUNCT
ejpam-5648	15	17	has	have	AUX
ejpam-5648	15	18	also	also	ADV
ejpam-5648	15	19	been	be	AUX
ejpam-5648	15	20	widely	widely	ADV
ejpam-5648	15	21	studied	study	VERB
ejpam-5648	15	22	since	since	SCONJ
ejpam-5648	15	23	the	the	DET
ejpam-5648	15	24	time	time	NOUN
ejpam-5648	15	25	it	it	PRON
ejpam-5648	15	26	was	be	AUX
ejpam-5648	15	27	introduced	introduce	VERB
ejpam-5648	15	28	in	in	ADP
ejpam-5648	15	29	[	[	X
ejpam-5648	15	30	22	22	NUM
ejpam-5648	15	31	]	]	PUNCT
ejpam-5648	15	32	.	.	PUNCT
ejpam-5648	16	1	studies	study	NOUN
ejpam-5648	16	2	in	in	ADP
ejpam-5648	16	3	[	[	X
ejpam-5648	16	4	5	5	NUM
ejpam-5648	16	5	]	]	PUNCT
ejpam-5648	16	6	,	,	PUNCT
ejpam-5648	16	7	[	[	X
ejpam-5648	16	8	6	6	NUM
ejpam-5648	16	9	]	]	PUNCT
ejpam-5648	16	10	,	,	PUNCT
ejpam-5648	16	11	[	[	X
ejpam-5648	16	12	13	13	NUM
ejpam-5648	16	13	]	]	PUNCT
ejpam-5648	16	14	,	,	PUNCT
ejpam-5648	16	15	∗corresponding	∗corresponde	VERB
ejpam-5648	16	16	author	author	NOUN
ejpam-5648	16	17	.	.	PUNCT
ejpam-5648	17	1	doi	doi	NOUN
ejpam-5648	17	2	:	:	PUNCT
ejpam-5648	17	3	https://doi.org/10.29020/nybg.ejpam.v18i1.5648	https://doi.org/10.29020/nybg.ejpam.v18i1.5648	PROPN
ejpam-5648	17	4	email	email	NOUN
ejpam-5648	17	5	addresses	address	NOUN
ejpam-5648	17	6	:	:	PUNCT
ejpam-5648	17	7	alkajimaradais@msutawi-tawi.edu.ph	alkajimaradais@msutawi-tawi.edu.ph	PROPN
ejpam-5648	17	8	(	(	PUNCT
ejpam-5648	17	9	a.	a.	PROPN
ejpam-5648	17	10	aradais	aradais	PROPN
ejpam-5648	17	11	)	)	PUNCT
ejpam-5648	17	12	,	,	PUNCT
ejpam-5648	17	13	jerryboy.cariaga@g.msuiit.edu.ph	jerryboy.cariaga@g.msuiit.edu.ph	PROPN
ejpam-5648	17	14	(	(	PUNCT
ejpam-5648	17	15	j.	j.	PROPN
ejpam-5648	17	16	cariaga	cariaga	PROPN
ejpam-5648	17	17	)	)	PUNCT
ejpam-5648	17	18	,	,	PUNCT
ejpam-5648	17	19	sergio.canoy@g.msuiit.edu.ph	sergio.canoy@g.msuiit.edu.ph	PROPN
ejpam-5648	17	20	(	(	PUNCT
ejpam-5648	17	21	s.	s.	PROPN
ejpam-5648	17	22	canoy	canoy	PROPN
ejpam-5648	17	23	jr	jr	PROPN
ejpam-5648	17	24	.	.	PUNCT
ejpam-5648	17	25	)	)	PUNCT
ejpam-5648	17	26	https://www.ejpam.com	https://www.ejpam.com	X
ejpam-5648	18	1	1	1	NUM
ejpam-5648	18	2	copyright	copyright	NOUN
ejpam-5648	18	3	:	:	PUNCT
ejpam-5648	18	4	©	©	PROPN
ejpam-5648	18	5	2025	2025	NUM
ejpam-5648	18	6	the	the	DET
ejpam-5648	18	7	author(s	author(s	NOUN
ejpam-5648	18	8	)	)	PUNCT
ejpam-5648	18	9	.	.	PUNCT
ejpam-5648	19	1	(	(	PUNCT
ejpam-5648	19	2	cc	cc	NOUN
ejpam-5648	19	3	by	by	ADP
ejpam-5648	19	4	-	-	PUNCT
ejpam-5648	19	5	nc	nc	PROPN
ejpam-5648	19	6	4.0	4.0	NUM
ejpam-5648	19	7	)	)	PUNCT
ejpam-5648	19	8	a.	a.	NOUN
ejpam-5648	19	9	aradais	aradais	PROPN
ejpam-5648	19	10	,	,	PUNCT
ejpam-5648	19	11	j.	j.	PROPN
ejpam-5648	19	12	cariaga	cariaga	PROPN
ejpam-5648	19	13	,	,	PUNCT
ejpam-5648	19	14	s.	s.	PROPN
ejpam-5648	19	15	canoy	canoy	PROPN
ejpam-5648	19	16	jr	jr	PROPN
ejpam-5648	19	17	.	.	PROPN
ejpam-5648	19	18	/	/	SYM
ejpam-5648	19	19	eur	eur	PROPN
ejpam-5648	19	20	.	.	PUNCT
ejpam-5648	20	1	j.	j.	PROPN
ejpam-5648	20	2	pure	pure	PROPN
ejpam-5648	20	3	appl	appl	PROPN
ejpam-5648	20	4	.	.	PROPN
ejpam-5648	20	5	math	math	PROPN
ejpam-5648	20	6	,	,	PUNCT
ejpam-5648	20	7	18	18	NUM
ejpam-5648	20	8	(	(	PUNCT
ejpam-5648	20	9	1	1	NUM
ejpam-5648	20	10	)	)	PUNCT
ejpam-5648	20	11	(	(	PUNCT
ejpam-5648	20	12	2025	2025	NUM
ejpam-5648	20	13	)	)	PUNCT
ejpam-5648	20	14	,	,	PUNCT
ejpam-5648	20	15	5648	5648	NUM
ejpam-5648	20	16	2	2	NUM
ejpam-5648	20	17	of	of	ADP
ejpam-5648	20	18	13	13	NUM
ejpam-5648	20	19	[	[	X
ejpam-5648	20	20	14	14	NUM
ejpam-5648	20	21	]	]	PUNCT
ejpam-5648	20	22	,	,	PUNCT
ejpam-5648	20	23	[	[	X
ejpam-5648	20	24	15	15	NUM
ejpam-5648	20	25	]	]	PUNCT
ejpam-5648	20	26	,	,	PUNCT
ejpam-5648	21	1	[	[	X
ejpam-5648	21	2	17	17	NUM
ejpam-5648	21	3	]	]	PUNCT
ejpam-5648	21	4	,	,	PUNCT
ejpam-5648	22	1	[	[	X
ejpam-5648	22	2	19	19	NUM
ejpam-5648	22	3	]	]	PUNCT
ejpam-5648	22	4	,	,	PUNCT
ejpam-5648	22	5	[	[	X
ejpam-5648	22	6	18	18	NUM
ejpam-5648	22	7	]	]	PUNCT
ejpam-5648	22	8	,	,	PUNCT
ejpam-5648	22	9	[	[	X
ejpam-5648	22	10	23	23	NUM
ejpam-5648	22	11	]	]	PUNCT
ejpam-5648	22	12	,	,	PUNCT
ejpam-5648	23	1	[	[	X
ejpam-5648	23	2	26	26	NUM
ejpam-5648	23	3	]	]	PUNCT
ejpam-5648	23	4	,	,	PUNCT
ejpam-5648	23	5	and	and	CCONJ
ejpam-5648	23	6	[	[	X
ejpam-5648	23	7	27	27	NUM
ejpam-5648	23	8	]	]	PUNCT
ejpam-5648	23	9	considered	consider	VERB
ejpam-5648	23	10	further	far	ADV
ejpam-5648	23	11	the	the	DET
ejpam-5648	23	12	concept	concept	NOUN
ejpam-5648	23	13	and	and	CCONJ
ejpam-5648	23	14	some	some	PRON
ejpam-5648	23	15	of	of	ADP
ejpam-5648	23	16	its	its	PRON
ejpam-5648	23	17	constructs	construct	NOUN
ejpam-5648	23	18	or	or	CCONJ
ejpam-5648	23	19	variants	variant	NOUN
ejpam-5648	23	20	.	.	PUNCT
ejpam-5648	24	1	recently	recently	ADV
ejpam-5648	24	2	,	,	PUNCT
ejpam-5648	24	3	the	the	DET
ejpam-5648	24	4	parameter	parameter	NOUN
ejpam-5648	24	5	hop	hop	PROPN
ejpam-5648	24	6	roman	roman	PROPN
ejpam-5648	24	7	domination	domination	NOUN
ejpam-5648	24	8	was	be	AUX
ejpam-5648	24	9	introduced	introduce	VERB
ejpam-5648	24	10	and	and	CCONJ
ejpam-5648	24	11	,	,	PUNCT
ejpam-5648	24	12	just	just	ADV
ejpam-5648	24	13	like	like	ADP
ejpam-5648	24	14	roman	roman	ADJ
ejpam-5648	24	15	domination	domination	NOUN
ejpam-5648	24	16	,	,	PUNCT
ejpam-5648	24	17	various	various	ADJ
ejpam-5648	24	18	modifications	modification	NOUN
ejpam-5648	24	19	of	of	ADP
ejpam-5648	24	20	the	the	DET
ejpam-5648	24	21	concept	concept	NOUN
ejpam-5648	24	22	have	have	AUX
ejpam-5648	24	23	also	also	ADV
ejpam-5648	24	24	been	be	AUX
ejpam-5648	24	25	introduced	introduce	VERB
ejpam-5648	24	26	and	and	CCONJ
ejpam-5648	24	27	studied	study	VERB
ejpam-5648	24	28	(	(	PUNCT
ejpam-5648	24	29	see	see	VERB
ejpam-5648	24	30	[	[	X
ejpam-5648	24	31	2	2	NUM
ejpam-5648	24	32	]	]	PUNCT
ejpam-5648	24	33	,	,	PUNCT
ejpam-5648	25	1	[	[	X
ejpam-5648	25	2	25	25	NUM
ejpam-5648	25	3	]	]	PUNCT
ejpam-5648	25	4	,	,	PUNCT
ejpam-5648	25	5	and	and	CCONJ
ejpam-5648	25	6	[	[	X
ejpam-5648	25	7	28	28	NUM
ejpam-5648	25	8	]	]	NUM
ejpam-5648	25	9	)	)	PUNCT
ejpam-5648	25	10	.	.	PUNCT
ejpam-5648	26	1	this	this	DET
ejpam-5648	26	2	present	present	ADJ
ejpam-5648	26	3	study	study	NOUN
ejpam-5648	26	4	considers	consider	VERB
ejpam-5648	26	5	connected	connect	VERB
ejpam-5648	26	6	hop	hop	NOUN
ejpam-5648	26	7	roman	roman	ADJ
ejpam-5648	26	8	domination	domination	NOUN
ejpam-5648	26	9	.	.	PUNCT
ejpam-5648	27	1	since	since	SCONJ
ejpam-5648	27	2	hop	hop	NOUN
ejpam-5648	27	3	domination	domination	NOUN
ejpam-5648	27	4	and	and	CCONJ
ejpam-5648	27	5	roman	roman	ADJ
ejpam-5648	27	6	domination	domination	NOUN
ejpam-5648	27	7	have	have	VERB
ejpam-5648	27	8	both	both	DET
ejpam-5648	27	9	applications	application	NOUN
ejpam-5648	27	10	in	in	ADP
ejpam-5648	27	11	many	many	ADJ
ejpam-5648	27	12	networks	network	NOUN
ejpam-5648	27	13	(	(	PUNCT
ejpam-5648	27	14	for	for	ADP
ejpam-5648	27	15	example	example	NOUN
ejpam-5648	27	16	,	,	PUNCT
ejpam-5648	27	17	to	to	PART
ejpam-5648	27	18	model	model	VERB
ejpam-5648	27	19	defense	defense	NOUN
ejpam-5648	27	20	strategies	strategy	NOUN
ejpam-5648	27	21	,	,	PUNCT
ejpam-5648	27	22	communication	communication	NOUN
ejpam-5648	27	23	in	in	ADP
ejpam-5648	27	24	social	social	ADJ
ejpam-5648	27	25	networks	network	NOUN
ejpam-5648	27	26	,	,	PUNCT
ejpam-5648	27	27	and	and	CCONJ
ejpam-5648	27	28	management	management	NOUN
ejpam-5648	27	29	problems	problem	NOUN
ejpam-5648	27	30	)	)	PUNCT
ejpam-5648	27	31	,	,	PUNCT
ejpam-5648	27	32	this	this	DET
ejpam-5648	27	33	newly	newly	ADV
ejpam-5648	27	34	defined	define	VERB
ejpam-5648	27	35	parameter	parameter	NOUN
ejpam-5648	27	36	can	can	AUX
ejpam-5648	27	37	easily	easily	ADV
ejpam-5648	27	38	find	find	VERB
ejpam-5648	27	39	its	its	PRON
ejpam-5648	27	40	own	own	ADJ
ejpam-5648	27	41	similar	similar	ADJ
ejpam-5648	27	42	applications	application	NOUN
ejpam-5648	27	43	.	.	PUNCT
ejpam-5648	28	1	this	this	PRON
ejpam-5648	28	2	among	among	ADP
ejpam-5648	28	3	others	other	NOUN
ejpam-5648	28	4	gives	give	VERB
ejpam-5648	28	5	added	add	VERB
ejpam-5648	28	6	motivation	motivation	NOUN
ejpam-5648	28	7	for	for	ADP
ejpam-5648	28	8	introducing	introduce	VERB
ejpam-5648	28	9	and	and	CCONJ
ejpam-5648	28	10	studying	study	VERB
ejpam-5648	28	11	the	the	DET
ejpam-5648	28	12	said	say	VERB
ejpam-5648	28	13	parameter	parameter	NOUN
ejpam-5648	28	14	.	.	PUNCT
ejpam-5648	29	1	2	2	NUM
ejpam-5648	29	2	.	.	X
ejpam-5648	29	3	terminology	terminology	NOUN
ejpam-5648	29	4	and	and	CCONJ
ejpam-5648	29	5	notations	notation	NOUN
ejpam-5648	29	6	let	let	VERB
ejpam-5648	29	7	g	g	NOUN
ejpam-5648	29	8	=	=	SYM
ejpam-5648	29	9	(	(	PUNCT
ejpam-5648	29	10	v	v	NOUN
ejpam-5648	29	11	(	(	PUNCT
ejpam-5648	29	12	g	g	NOUN
ejpam-5648	29	13	)	)	PUNCT
ejpam-5648	29	14	,	,	PUNCT
ejpam-5648	29	15	e(g	e(g	PROPN
ejpam-5648	29	16	)	)	PUNCT
ejpam-5648	29	17	)	)	PUNCT
ejpam-5648	29	18	be	be	AUX
ejpam-5648	29	19	an	an	DET
ejpam-5648	29	20	undirected	undirected	ADJ
ejpam-5648	29	21	graph	graph	NOUN
ejpam-5648	29	22	.	.	PUNCT
ejpam-5648	30	1	the	the	DET
ejpam-5648	30	2	open	open	ADJ
ejpam-5648	30	3	neighborhood	neighborhood	NOUN
ejpam-5648	30	4	of	of	ADP
ejpam-5648	30	5	v	v	NUM
ejpam-5648	30	6	∈	∈	NOUN
ejpam-5648	30	7	v	v	NOUN
ejpam-5648	30	8	(	(	PUNCT
ejpam-5648	30	9	g	g	NOUN
ejpam-5648	30	10	)	)	PUNCT
ejpam-5648	30	11	is	be	AUX
ejpam-5648	30	12	the	the	DET
ejpam-5648	30	13	set	set	NOUN
ejpam-5648	30	14	ng(v	ng(v	PUNCT
ejpam-5648	30	15	)	)	PUNCT
ejpam-5648	30	16	=	=	SYM
ejpam-5648	31	1	{	{	PUNCT
ejpam-5648	31	2	u	u	NOUN
ejpam-5648	31	3	∈	∈	PROPN
ejpam-5648	31	4	v	v	NOUN
ejpam-5648	31	5	(	(	PUNCT
ejpam-5648	31	6	g	g	NOUN
ejpam-5648	31	7	)	)	PUNCT
ejpam-5648	31	8	:	:	PUNCT
ejpam-5648	31	9	uv	uv	PROPN
ejpam-5648	31	10	∈	∈	PROPN
ejpam-5648	31	11	e(g	e(g	PROPN
ejpam-5648	31	12	)	)	PUNCT
ejpam-5648	31	13	}	}	PUNCT
ejpam-5648	31	14	while	while	SCONJ
ejpam-5648	31	15	its	its	PRON
ejpam-5648	31	16	closed	closed	ADJ
ejpam-5648	31	17	neighborhood	neighborhood	NOUN
ejpam-5648	31	18	is	be	AUX
ejpam-5648	31	19	the	the	DET
ejpam-5648	31	20	set	set	NOUN
ejpam-5648	31	21	ng[v	ng[v	NOUN
ejpam-5648	31	22	]	]	X
ejpam-5648	31	23	=	=	SYM
ejpam-5648	31	24	{	{	PUNCT
ejpam-5648	31	25	v}∪ng(v	v}∪ng(v	PROPN
ejpam-5648	31	26	)	)	PUNCT
ejpam-5648	31	27	.	.	PUNCT
ejpam-5648	32	1	vertex	vertex	NOUN
ejpam-5648	32	2	v	v	NOUN
ejpam-5648	32	3	is	be	AUX
ejpam-5648	32	4	an	an	DET
ejpam-5648	32	5	isolated	isolated	ADJ
ejpam-5648	32	6	vertex	vertex	NOUN
ejpam-5648	32	7	if	if	SCONJ
ejpam-5648	32	8	ng(v	ng(v	NOUN
ejpam-5648	32	9	)	)	PUNCT
ejpam-5648	32	10	=	=	PUNCT
ejpam-5648	32	11	∅.	∅.	VERB
ejpam-5648	32	12	the	the	DET
ejpam-5648	32	13	open	open	ADJ
ejpam-5648	32	14	neighborhood	neighborhood	NOUN
ejpam-5648	32	15	and	and	CCONJ
ejpam-5648	32	16	closed	close	VERB
ejpam-5648	32	17	neighborhood	neighborhood	NOUN
ejpam-5648	32	18	of	of	ADP
ejpam-5648	32	19	set	set	NOUN
ejpam-5648	32	20	s	s	PROPN
ejpam-5648	32	21	⊆	⊆	NUM
ejpam-5648	32	22	v	v	NOUN
ejpam-5648	32	23	(	(	PUNCT
ejpam-5648	32	24	g	g	NOUN
ejpam-5648	32	25	)	)	PUNCT
ejpam-5648	32	26	are	be	AUX
ejpam-5648	32	27	the	the	DET
ejpam-5648	32	28	sets	set	NOUN
ejpam-5648	32	29	ng(s	ng(s	PRON
ejpam-5648	32	30	)	)	PUNCT
ejpam-5648	32	31	=	=	SYM
ejpam-5648	32	32	∪v∈sng(v	∪v∈sng(v	PROPN
ejpam-5648	32	33	)	)	PUNCT
ejpam-5648	32	34	and	and	CCONJ
ejpam-5648	32	35	ng[s	ng[s	PROPN
ejpam-5648	32	36	]	]	PUNCT
ejpam-5648	32	37	=	=	SYM
ejpam-5648	32	38	∪v∈sng[u	∪v∈sng[u	X
ejpam-5648	32	39	]	]	PUNCT
ejpam-5648	32	40	,	,	PUNCT
ejpam-5648	32	41	respectively	respectively	ADV
ejpam-5648	32	42	.	.	PUNCT
ejpam-5648	33	1	the	the	DET
ejpam-5648	33	2	degree	degree	NOUN
ejpam-5648	33	3	of	of	ADP
ejpam-5648	33	4	v	v	NOUN
ejpam-5648	33	5	,	,	PUNCT
ejpam-5648	33	6	denoted	denote	VERB
ejpam-5648	33	7	by	by	ADP
ejpam-5648	33	8	degg(v	degg(v	PROPN
ejpam-5648	33	9	)	)	PUNCT
ejpam-5648	33	10	,	,	PUNCT
ejpam-5648	33	11	is	be	AUX
ejpam-5648	33	12	equal	equal	ADJ
ejpam-5648	33	13	to	to	ADP
ejpam-5648	33	14	|ng(v)|	|ng(v)|	NOUN
ejpam-5648	33	15	.	.	PUNCT
ejpam-5648	34	1	any	any	DET
ejpam-5648	34	2	shortest	short	ADJ
ejpam-5648	34	3	path	path	NOUN
ejpam-5648	34	4	connecting	connect	VERB
ejpam-5648	34	5	two	two	NUM
ejpam-5648	34	6	vertices	vertex	NOUN
ejpam-5648	34	7	x	x	PUNCT
ejpam-5648	34	8	and	and	CCONJ
ejpam-5648	34	9	y	y	PROPN
ejpam-5648	34	10	of	of	ADP
ejpam-5648	34	11	g	g	PROPN
ejpam-5648	34	12	is	be	AUX
ejpam-5648	34	13	called	call	VERB
ejpam-5648	34	14	an	an	DET
ejpam-5648	34	15	x	x	NOUN
ejpam-5648	34	16	-	-	NOUN
ejpam-5648	34	17	y	y	ADJ
ejpam-5648	34	18	geodesic	geodesic	NOUN
ejpam-5648	34	19	and	and	CCONJ
ejpam-5648	34	20	the	the	DET
ejpam-5648	34	21	length	length	NOUN
ejpam-5648	34	22	of	of	ADP
ejpam-5648	34	23	an	an	DET
ejpam-5648	34	24	x	x	NOUN
ejpam-5648	34	25	-	-	NOUN
ejpam-5648	34	26	y	y	ADJ
ejpam-5648	34	27	geodesic	geodesic	NOUN
ejpam-5648	34	28	in	in	ADP
ejpam-5648	34	29	g	g	PROPN
ejpam-5648	34	30	is	be	AUX
ejpam-5648	34	31	the	the	DET
ejpam-5648	34	32	distance	distance	NOUN
ejpam-5648	34	33	dg(x	dg(x	PUNCT
ejpam-5648	34	34	,	,	PUNCT
ejpam-5648	34	35	y	y	NOUN
ejpam-5648	34	36	)	)	PUNCT
ejpam-5648	34	37	of	of	ADP
ejpam-5648	34	38	x	x	PUNCT
ejpam-5648	34	39	and	and	CCONJ
ejpam-5648	34	40	y.	y.	NOUN
ejpam-5648	34	41	the	the	DET
ejpam-5648	34	42	diameter	diameter	NOUN
ejpam-5648	34	43	of	of	ADP
ejpam-5648	34	44	g	g	PROPN
ejpam-5648	34	45	,	,	PUNCT
ejpam-5648	34	46	denoted	denote	VERB
ejpam-5648	34	47	diam(g	diam(g	PROPN
ejpam-5648	34	48	)	)	PUNCT
ejpam-5648	34	49	,	,	PUNCT
ejpam-5648	34	50	is	be	AUX
ejpam-5648	34	51	the	the	DET
ejpam-5648	34	52	maximum	maximum	ADJ
ejpam-5648	34	53	distance	distance	NOUN
ejpam-5648	34	54	between	between	ADP
ejpam-5648	34	55	the	the	DET
ejpam-5648	34	56	pair	pair	NOUN
ejpam-5648	34	57	of	of	ADP
ejpam-5648	34	58	vertices	vertex	NOUN
ejpam-5648	34	59	.	.	PUNCT
ejpam-5648	35	1	a	a	DET
ejpam-5648	35	2	vertex	vertex	NOUN
ejpam-5648	35	3	v	v	NOUN
ejpam-5648	35	4	of	of	ADP
ejpam-5648	35	5	g	g	PROPN
ejpam-5648	35	6	is	be	AUX
ejpam-5648	35	7	a	a	DET
ejpam-5648	35	8	leaf	leaf	NOUN
ejpam-5648	35	9	if	if	SCONJ
ejpam-5648	35	10	degg(v	degg(v	VERB
ejpam-5648	35	11	)	)	PUNCT
ejpam-5648	35	12	=	=	SYM
ejpam-5648	35	13	1	1	NUM
ejpam-5648	35	14	and	and	CCONJ
ejpam-5648	35	15	w	w	PROPN
ejpam-5648	35	16	∈	∈	PROPN
ejpam-5648	35	17	v	v	ADP
ejpam-5648	35	18	(	(	PUNCT
ejpam-5648	35	19	g	g	NOUN
ejpam-5648	35	20	)	)	PUNCT
ejpam-5648	35	21	is	be	AUX
ejpam-5648	35	22	a	a	DET
ejpam-5648	35	23	support	support	NOUN
ejpam-5648	35	24	vertex	vertex	NOUN
ejpam-5648	35	25	if	if	SCONJ
ejpam-5648	35	26	wz	wz	PROPN
ejpam-5648	35	27	∈	∈	PROPN
ejpam-5648	35	28	e(g	e(g	PROPN
ejpam-5648	35	29	)	)	PUNCT
ejpam-5648	35	30	for	for	ADP
ejpam-5648	35	31	some	some	DET
ejpam-5648	35	32	leaf	leaf	NOUN
ejpam-5648	35	33	z	z	NOUN
ejpam-5648	35	34	∈	∈	PROPN
ejpam-5648	35	35	v	v	ADP
ejpam-5648	35	36	(	(	PUNCT
ejpam-5648	35	37	g	g	NOUN
ejpam-5648	35	38	)	)	PUNCT
ejpam-5648	35	39	.	.	PUNCT
ejpam-5648	36	1	a	a	DET
ejpam-5648	36	2	set	set	NOUN
ejpam-5648	36	3	s	s	NOUN
ejpam-5648	36	4	⊆	⊆	NUM
ejpam-5648	36	5	v	v	NOUN
ejpam-5648	36	6	(	(	PUNCT
ejpam-5648	36	7	g	g	NOUN
ejpam-5648	36	8	)	)	PUNCT
ejpam-5648	36	9	is	be	AUX
ejpam-5648	36	10	said	say	VERB
ejpam-5648	36	11	to	to	PART
ejpam-5648	36	12	be	be	AUX
ejpam-5648	36	13	a	a	DET
ejpam-5648	36	14	dominating	dominating	NOUN
ejpam-5648	36	15	set	set	NOUN
ejpam-5648	36	16	of	of	ADP
ejpam-5648	36	17	g	g	PROPN
ejpam-5648	36	18	if	if	SCONJ
ejpam-5648	36	19	ng[s	ng[	NOUN
ejpam-5648	36	20	]	]	PUNCT
ejpam-5648	36	21	=	=	SYM
ejpam-5648	36	22	v	v	NOUN
ejpam-5648	36	23	(	(	PUNCT
ejpam-5648	36	24	g	g	NOUN
ejpam-5648	36	25	)	)	PUNCT
ejpam-5648	36	26	.	.	PUNCT
ejpam-5648	37	1	the	the	DET
ejpam-5648	37	2	minimum	minimum	ADJ
ejpam-5648	37	3	cardinality	cardinality	NOUN
ejpam-5648	37	4	of	of	ADP
ejpam-5648	37	5	a	a	DET
ejpam-5648	37	6	dominating	dominating	NOUN
ejpam-5648	37	7	set	set	NOUN
ejpam-5648	37	8	,	,	PUNCT
ejpam-5648	37	9	denoted	denote	VERB
ejpam-5648	37	10	by	by	ADP
ejpam-5648	37	11	γ(g	γ(g	PROPN
ejpam-5648	37	12	)	)	PUNCT
ejpam-5648	37	13	,	,	PUNCT
ejpam-5648	37	14	is	be	AUX
ejpam-5648	37	15	called	call	VERB
ejpam-5648	37	16	the	the	DET
ejpam-5648	37	17	domination	domination	NOUN
ejpam-5648	37	18	number	number	NOUN
ejpam-5648	37	19	of	of	ADP
ejpam-5648	37	20	g.	g.	PROPN
ejpam-5648	37	21	a	a	DET
ejpam-5648	37	22	vertex	vertex	NOUN
ejpam-5648	37	23	v	v	NOUN
ejpam-5648	37	24	is	be	AUX
ejpam-5648	37	25	a	a	DET
ejpam-5648	37	26	dominating	dominating	NOUN
ejpam-5648	37	27	vertex	vertex	NOUN
ejpam-5648	37	28	of	of	ADP
ejpam-5648	37	29	g	g	NOUN
ejpam-5648	37	30	if	if	SCONJ
ejpam-5648	37	31	ng[v	ng[v	NOUN
ejpam-5648	37	32	]	]	X
ejpam-5648	37	33	=	=	SYM
ejpam-5648	37	34	v	v	X
ejpam-5648	37	35	(	(	PUNCT
ejpam-5648	37	36	g	g	NOUN
ejpam-5648	37	37	)	)	PUNCT
ejpam-5648	37	38	.	.	PUNCT
ejpam-5648	38	1	any	any	DET
ejpam-5648	38	2	dominating	dominating	NOUN
ejpam-5648	38	3	set	set	NOUN
ejpam-5648	38	4	of	of	ADP
ejpam-5648	38	5	cardinality	cardinality	PROPN
ejpam-5648	38	6	γ(g	γ(g	PROPN
ejpam-5648	38	7	)	)	PUNCT
ejpam-5648	38	8	is	be	AUX
ejpam-5648	38	9	referred	refer	VERB
ejpam-5648	38	10	to	to	ADP
ejpam-5648	38	11	as	as	ADP
ejpam-5648	38	12	a	a	DET
ejpam-5648	38	13	γ	γ	NOUN
ejpam-5648	38	14	-	-	PUNCT
ejpam-5648	38	15	set	set	NOUN
ejpam-5648	38	16	of	of	ADP
ejpam-5648	38	17	g.	g.	PROPN
ejpam-5648	38	18	the	the	DET
ejpam-5648	38	19	open	open	ADJ
ejpam-5648	38	20	hop	hop	NOUN
ejpam-5648	38	21	neighborhood	neighborhood	NOUN
ejpam-5648	38	22	of	of	ADP
ejpam-5648	38	23	v	v	NUM
ejpam-5648	38	24	∈	∈	NOUN
ejpam-5648	38	25	v	v	NOUN
ejpam-5648	38	26	(	(	PUNCT
ejpam-5648	38	27	g	g	NOUN
ejpam-5648	38	28	)	)	PUNCT
ejpam-5648	38	29	is	be	AUX
ejpam-5648	38	30	the	the	DET
ejpam-5648	38	31	set	set	ADJ
ejpam-5648	38	32	n2	n2	ADJ
ejpam-5648	38	33	g(v	g(v	PROPN
ejpam-5648	38	34	)	)	PUNCT
ejpam-5648	39	1	=	=	PRON
ejpam-5648	39	2	{	{	PUNCT
ejpam-5648	39	3	u	u	NOUN
ejpam-5648	39	4	∈	∈	PROPN
ejpam-5648	39	5	v	v	NOUN
ejpam-5648	39	6	(	(	PUNCT
ejpam-5648	39	7	g	g	NOUN
ejpam-5648	39	8	)	)	PUNCT
ejpam-5648	39	9	:	:	PUNCT
ejpam-5648	39	10	dg(u	dg(u	X
ejpam-5648	39	11	,	,	PUNCT
ejpam-5648	39	12	v	v	NOUN
ejpam-5648	39	13	)	)	PUNCT
ejpam-5648	39	14	=	=	SYM
ejpam-5648	39	15	2	2	X
ejpam-5648	39	16	}	}	PUNCT
ejpam-5648	39	17	and	and	CCONJ
ejpam-5648	39	18	its	its	PRON
ejpam-5648	39	19	closed	closed	ADJ
ejpam-5648	39	20	hop	hop	NOUN
ejpam-5648	39	21	neighborhood	neighborhood	NOUN
ejpam-5648	39	22	is	be	AUX
ejpam-5648	39	23	n2	n2	ADJ
ejpam-5648	39	24	g[v	g[v	NOUN
ejpam-5648	39	25	]	]	X
ejpam-5648	39	26	=	=	SYM
ejpam-5648	39	27	{	{	PUNCT
ejpam-5648	39	28	v	v	NOUN
ejpam-5648	39	29	}	}	PUNCT
ejpam-5648	39	30	∪	∪	ADJ
ejpam-5648	39	31	n2	n2	ADJ
ejpam-5648	39	32	g(v	g(v	PROPN
ejpam-5648	39	33	)	)	PUNCT
ejpam-5648	39	34	.	.	PUNCT
ejpam-5648	40	1	the	the	DET
ejpam-5648	40	2	open	open	ADJ
ejpam-5648	40	3	hop	hop	NOUN
ejpam-5648	40	4	neighborhood	neighborhood	NOUN
ejpam-5648	40	5	and	and	CCONJ
ejpam-5648	40	6	closed	close	VERB
ejpam-5648	40	7	hop	hop	NOUN
ejpam-5648	40	8	neighborhood	neighborhood	NOUN
ejpam-5648	40	9	of	of	ADP
ejpam-5648	40	10	set	set	NOUN
ejpam-5648	40	11	s	s	PROPN
ejpam-5648	40	12	⊆	⊆	NUM
ejpam-5648	40	13	v	v	NOUN
ejpam-5648	40	14	(	(	PUNCT
ejpam-5648	40	15	g	g	NOUN
ejpam-5648	40	16	)	)	PUNCT
ejpam-5648	40	17	are	be	AUX
ejpam-5648	40	18	the	the	DET
ejpam-5648	40	19	sets	set	NOUN
ejpam-5648	40	20	n2	n2	ADJ
ejpam-5648	40	21	g(s	g(s	NOUN
ejpam-5648	40	22	)	)	PUNCT
ejpam-5648	40	23	=	=	PUNCT
ejpam-5648	41	1	∪v∈sn	∪v∈sn	VERB
ejpam-5648	41	2	2	2	NUM
ejpam-5648	41	3	g(v	g(v	NOUN
ejpam-5648	41	4	)	)	PUNCT
ejpam-5648	41	5	and	and	CCONJ
ejpam-5648	41	6	n2	n2	ADJ
ejpam-5648	41	7	g[s	g[s	PROPN
ejpam-5648	41	8	]	]	X
ejpam-5648	41	9	=	=	SYM
ejpam-5648	41	10	{	{	PUNCT
ejpam-5648	41	11	v	v	NOUN
ejpam-5648	41	12	}	}	PUNCT
ejpam-5648	41	13	∪n2	∪n2	NOUN
ejpam-5648	41	14	g(u	g(u	PROPN
ejpam-5648	41	15	)	)	PUNCT
ejpam-5648	41	16	,	,	PUNCT
ejpam-5648	41	17	respectively	respectively	ADV
ejpam-5648	41	18	.	.	PUNCT
ejpam-5648	42	1	a	a	DET
ejpam-5648	42	2	set	set	NOUN
ejpam-5648	42	3	s	s	NOUN
ejpam-5648	42	4	⊆	⊆	NUM
ejpam-5648	42	5	v	v	NOUN
ejpam-5648	42	6	(	(	PUNCT
ejpam-5648	42	7	g	g	NOUN
ejpam-5648	42	8	)	)	PUNCT
ejpam-5648	42	9	is	be	AUX
ejpam-5648	42	10	said	say	VERB
ejpam-5648	42	11	to	to	PART
ejpam-5648	42	12	be	be	AUX
ejpam-5648	42	13	a	a	DET
ejpam-5648	42	14	hop	hop	NOUN
ejpam-5648	42	15	dominating	dominating	NOUN
ejpam-5648	42	16	set	set	NOUN
ejpam-5648	42	17	of	of	ADP
ejpam-5648	42	18	g	g	PROPN
ejpam-5648	42	19	if	if	SCONJ
ejpam-5648	42	20	n2	n2	ADJ
ejpam-5648	42	21	g[s	g[s	PROPN
ejpam-5648	42	22	]	]	X
ejpam-5648	42	23	=	=	SYM
ejpam-5648	42	24	v	v	NOUN
ejpam-5648	42	25	(	(	PUNCT
ejpam-5648	42	26	g	g	NOUN
ejpam-5648	42	27	)	)	PUNCT
ejpam-5648	42	28	,	,	PUNCT
ejpam-5648	42	29	i.e.	i.e.	X
ejpam-5648	42	30	,	,	PUNCT
ejpam-5648	42	31	for	for	ADP
ejpam-5648	42	32	each	each	PRON
ejpam-5648	42	33	v	v	NUM
ejpam-5648	42	34	∈	∈	PROPN
ejpam-5648	42	35	v	v	NOUN
ejpam-5648	42	36	(	(	PUNCT
ejpam-5648	42	37	g	g	NOUN
ejpam-5648	42	38	)	)	PUNCT
ejpam-5648	42	39	\	\	PROPN
ejpam-5648	43	1	s	s	X
ejpam-5648	43	2	,	,	PUNCT
ejpam-5648	43	3	there	there	PRON
ejpam-5648	43	4	exists	exist	VERB
ejpam-5648	43	5	w	w	PROPN
ejpam-5648	43	6	∈	∈	PROPN
ejpam-5648	43	7	s	s	VERB
ejpam-5648	43	8	such	such	ADJ
ejpam-5648	43	9	that	that	PRON
ejpam-5648	43	10	dg(v	dg(v	ADJ
ejpam-5648	43	11	,	,	PUNCT
ejpam-5648	43	12	w	w	NOUN
ejpam-5648	43	13	)	)	PUNCT
ejpam-5648	43	14	=	=	SYM
ejpam-5648	43	15	2	2	X
ejpam-5648	43	16	.	.	X
ejpam-5648	43	17	a	a	DET
ejpam-5648	43	18	hop	hop	NOUN
ejpam-5648	43	19	dominating	dominating	NOUN
ejpam-5648	43	20	set	set	NOUN
ejpam-5648	43	21	s	s	VERB
ejpam-5648	43	22	is	be	AUX
ejpam-5648	43	23	connected	connect	VERB
ejpam-5648	43	24	hop	hop	NOUN
ejpam-5648	43	25	dominating	dominating	NOUN
ejpam-5648	43	26	if	if	SCONJ
ejpam-5648	43	27	the	the	DET
ejpam-5648	43	28	graph	graph	NOUN
ejpam-5648	43	29	⟨s⟩	⟨s⟩	VERB
ejpam-5648	43	30	induced	induce	VERB
ejpam-5648	43	31	by	by	ADP
ejpam-5648	43	32	s	s	PROPN
ejpam-5648	43	33	is	be	AUX
ejpam-5648	43	34	connected	connect	VERB
ejpam-5648	43	35	.	.	PUNCT
ejpam-5648	44	1	the	the	DET
ejpam-5648	44	2	minimum	minimum	ADJ
ejpam-5648	44	3	cardinality	cardinality	NOUN
ejpam-5648	44	4	among	among	ADP
ejpam-5648	44	5	all	all	DET
ejpam-5648	44	6	hop	hop	NOUN
ejpam-5648	44	7	dominating	dominating	NOUN
ejpam-5648	44	8	(	(	PUNCT
ejpam-5648	44	9	resp	resp	NOUN
ejpam-5648	44	10	.	.	PUNCT
ejpam-5648	45	1	connected	connect	VERB
ejpam-5648	45	2	hop	hop	NOUN
ejpam-5648	45	3	dominating	dominating	NOUN
ejpam-5648	45	4	)	)	PUNCT
ejpam-5648	45	5	sets	set	NOUN
ejpam-5648	45	6	in	in	ADP
ejpam-5648	45	7	g	g	PROPN
ejpam-5648	45	8	is	be	AUX
ejpam-5648	45	9	called	call	VERB
ejpam-5648	45	10	the	the	DET
ejpam-5648	45	11	hop	hop	NOUN
ejpam-5648	45	12	domination	domination	NOUN
ejpam-5648	45	13	number	number	NOUN
ejpam-5648	45	14	(	(	PUNCT
ejpam-5648	45	15	resp	resp	NOUN
ejpam-5648	45	16	.	.	PUNCT
ejpam-5648	46	1	connected	connect	VERB
ejpam-5648	46	2	hop	hop	NOUN
ejpam-5648	46	3	domination	domination	NOUN
ejpam-5648	46	4	number	number	NOUN
ejpam-5648	46	5	)	)	PUNCT
ejpam-5648	46	6	of	of	ADP
ejpam-5648	46	7	g	g	NOUN
ejpam-5648	46	8	,	,	PUNCT
ejpam-5648	46	9	and	and	CCONJ
ejpam-5648	46	10	is	be	AUX
ejpam-5648	46	11	denoted	denote	VERB
ejpam-5648	46	12	by	by	ADP
ejpam-5648	46	13	γh(g	γh(g	NOUN
ejpam-5648	46	14	)	)	PUNCT
ejpam-5648	46	15	(	(	PUNCT
ejpam-5648	46	16	resp	resp	NOUN
ejpam-5648	46	17	.	.	PUNCT
ejpam-5648	47	1	γch(g	γch(g	NOUN
ejpam-5648	47	2	)	)	PUNCT
ejpam-5648	47	3	.	.	PUNCT
ejpam-5648	48	1	any	any	DET
ejpam-5648	48	2	hop	hop	NOUN
ejpam-5648	48	3	dominating	dominating	NOUN
ejpam-5648	48	4	(	(	PUNCT
ejpam-5648	48	5	resp	resp	NOUN
ejpam-5648	48	6	.	.	PUNCT
ejpam-5648	49	1	connected	connect	VERB
ejpam-5648	49	2	hop	hop	NOUN
ejpam-5648	49	3	dominating	dominating	NOUN
ejpam-5648	49	4	)	)	PUNCT
ejpam-5648	49	5	set	set	NOUN
ejpam-5648	49	6	of	of	ADP
ejpam-5648	49	7	cardinality	cardinality	NOUN
ejpam-5648	49	8	γh(g	γh(g	NOUN
ejpam-5648	49	9	)	)	PUNCT
ejpam-5648	49	10	(	(	PUNCT
ejpam-5648	49	11	resp	resp	NOUN
ejpam-5648	49	12	.	.	PUNCT
ejpam-5648	49	13	γch(g	γch(g	NOUN
ejpam-5648	49	14	)	)	PUNCT
ejpam-5648	49	15	)	)	PUNCT
ejpam-5648	49	16	is	be	AUX
ejpam-5648	49	17	called	call	VERB
ejpam-5648	49	18	a	a	DET
ejpam-5648	49	19	γh	γh	ADV
ejpam-5648	49	20	-	-	PUNCT
ejpam-5648	49	21	set	set	VERB
ejpam-5648	49	22	(	(	PUNCT
ejpam-5648	49	23	resp	resp	NOUN
ejpam-5648	49	24	.	.	PUNCT
ejpam-5648	50	1	γch	γch	NOUN
ejpam-5648	50	2	-	-	PUNCT
ejpam-5648	50	3	set	set	NOUN
ejpam-5648	50	4	)	)	PUNCT
ejpam-5648	50	5	of	of	ADP
ejpam-5648	50	6	g.	g.	PROPN
ejpam-5648	50	7	a	a	DET
ejpam-5648	50	8	function	function	NOUN
ejpam-5648	51	1	f	f	NOUN
ejpam-5648	51	2	:	:	PUNCT
ejpam-5648	51	3	v	v	X
ejpam-5648	51	4	(	(	PUNCT
ejpam-5648	51	5	g	g	NOUN
ejpam-5648	51	6	)	)	PUNCT
ejpam-5648	51	7	→	→	SYM
ejpam-5648	51	8	{	{	PUNCT
ejpam-5648	51	9	0	0	NUM
ejpam-5648	51	10	,	,	PUNCT
ejpam-5648	51	11	1	1	NUM
ejpam-5648	51	12	,	,	PUNCT
ejpam-5648	51	13	2	2	NUM
ejpam-5648	51	14	}	}	PUNCT
ejpam-5648	51	15	is	be	AUX
ejpam-5648	51	16	a	a	DET
ejpam-5648	51	17	hop	hop	NOUN
ejpam-5648	51	18	roman	roman	ADJ
ejpam-5648	51	19	dominating	dominating	NOUN
ejpam-5648	51	20	function	function	NOUN
ejpam-5648	51	21	on	on	ADP
ejpam-5648	51	22	g	g	PROPN
ejpam-5648	51	23	if	if	SCONJ
ejpam-5648	51	24	for	for	ADP
ejpam-5648	51	25	each	each	DET
ejpam-5648	51	26	u	u	PROPN
ejpam-5648	51	27	∈	∈	PROPN
ejpam-5648	51	28	v	v	NOUN
ejpam-5648	51	29	(	(	PUNCT
ejpam-5648	51	30	g	g	NOUN
ejpam-5648	51	31	)	)	PUNCT
ejpam-5648	51	32	for	for	ADP
ejpam-5648	51	33	which	which	PRON
ejpam-5648	51	34	f(u	f(u	PROPN
ejpam-5648	51	35	)	)	PUNCT
ejpam-5648	52	1	=	=	SYM
ejpam-5648	52	2	0	0	NUM
ejpam-5648	52	3	,	,	PUNCT
ejpam-5648	52	4	there	there	PRON
ejpam-5648	52	5	exists	exist	VERB
ejpam-5648	52	6	v	v	ADP
ejpam-5648	52	7	∈	∈	PROPN
ejpam-5648	52	8	v	v	NOUN
ejpam-5648	52	9	(	(	PUNCT
ejpam-5648	52	10	g	g	NOUN
ejpam-5648	52	11	)	)	PUNCT
ejpam-5648	52	12	such	such	ADJ
ejpam-5648	52	13	that	that	SCONJ
ejpam-5648	52	14	f(v	f(v	NOUN
ejpam-5648	52	15	)	)	PUNCT
ejpam-5648	52	16	=	=	SYM
ejpam-5648	52	17	2	2	NUM
ejpam-5648	52	18	and	and	CCONJ
ejpam-5648	52	19	dg(u	dg(u	NOUN
ejpam-5648	52	20	,	,	PUNCT
ejpam-5648	52	21	v	v	NOUN
ejpam-5648	52	22	)	)	PUNCT
ejpam-5648	53	1	=	=	SYM
ejpam-5648	53	2	2	2	X
ejpam-5648	53	3	.	.	PUNCT
ejpam-5648	54	1	the	the	DET
ejpam-5648	54	2	weight	weight	NOUN
ejpam-5648	54	3	of	of	ADP
ejpam-5648	54	4	f	f	PROPN
ejpam-5648	54	5	is	be	AUX
ejpam-5648	54	6	given	give	VERB
ejpam-5648	54	7	by	by	ADP
ejpam-5648	54	8	ωrh	ωrh	NOUN
ejpam-5648	54	9	g	g	PROPN
ejpam-5648	54	10	(	(	PUNCT
ejpam-5648	54	11	f	f	X
ejpam-5648	54	12	)	)	PUNCT
ejpam-5648	54	13	=	=	SYM
ejpam-5648	54	14	∑	∑	PUNCT
ejpam-5648	54	15	v∈v	v∈v	PROPN
ejpam-5648	54	16	(	(	PUNCT
ejpam-5648	54	17	g	g	NOUN
ejpam-5648	54	18	)	)	PUNCT
ejpam-5648	54	19	f(v	f(v	NOUN
ejpam-5648	54	20	)	)	PUNCT
ejpam-5648	54	21	.	.	PUNCT
ejpam-5648	55	1	the	the	DET
ejpam-5648	55	2	hop	hop	PROPN
ejpam-5648	55	3	roman	roman	ADJ
ejpam-5648	55	4	domination	domination	NOUN
ejpam-5648	55	5	number	number	NOUN
ejpam-5648	55	6	of	of	ADP
ejpam-5648	55	7	g	g	NOUN
ejpam-5648	55	8	,	,	PUNCT
ejpam-5648	55	9	denoted	denote	VERB
ejpam-5648	55	10	by	by	ADP
ejpam-5648	55	11	γrh(g	γrh(g	NOUN
ejpam-5648	55	12	)	)	PUNCT
ejpam-5648	55	13	,	,	PUNCT
ejpam-5648	55	14	is	be	AUX
ejpam-5648	55	15	the	the	DET
ejpam-5648	55	16	minimum	minimum	ADJ
ejpam-5648	55	17	weight	weight	NOUN
ejpam-5648	55	18	of	of	ADP
ejpam-5648	55	19	a	a	DET
ejpam-5648	55	20	hop	hop	NOUN
ejpam-5648	55	21	roman	roman	ADJ
ejpam-5648	55	22	dominating	dominating	NOUN
ejpam-5648	55	23	function	function	NOUN
ejpam-5648	55	24	on	on	ADP
ejpam-5648	55	25	g.	g.	PROPN
ejpam-5648	55	26	a.	a.	PROPN
ejpam-5648	55	27	aradais	aradais	PROPN
ejpam-5648	55	28	,	,	PUNCT
ejpam-5648	55	29	j.	j.	PROPN
ejpam-5648	55	30	cariaga	cariaga	PROPN
ejpam-5648	55	31	,	,	PUNCT
ejpam-5648	55	32	s.	s.	PROPN
ejpam-5648	55	33	canoy	canoy	PROPN
ejpam-5648	55	34	jr	jr	PROPN
ejpam-5648	55	35	.	.	PROPN
ejpam-5648	55	36	/	/	SYM
ejpam-5648	55	37	eur	eur	PROPN
ejpam-5648	55	38	.	.	PUNCT
ejpam-5648	56	1	j.	j.	PROPN
ejpam-5648	56	2	pure	pure	PROPN
ejpam-5648	56	3	appl	appl	PROPN
ejpam-5648	56	4	.	.	PROPN
ejpam-5648	56	5	math	math	PROPN
ejpam-5648	56	6	,	,	PUNCT
ejpam-5648	56	7	18	18	NUM
ejpam-5648	56	8	(	(	PUNCT
ejpam-5648	56	9	1	1	NUM
ejpam-5648	56	10	)	)	PUNCT
ejpam-5648	56	11	(	(	PUNCT
ejpam-5648	56	12	2025	2025	NUM
ejpam-5648	56	13	)	)	PUNCT
ejpam-5648	56	14	,	,	PUNCT
ejpam-5648	56	15	5648	5648	NUM
ejpam-5648	56	16	3	3	NUM
ejpam-5648	56	17	of	of	ADP
ejpam-5648	56	18	13	13	NUM
ejpam-5648	56	19	let	let	VERB
ejpam-5648	56	20	g	g	NOUN
ejpam-5648	56	21	be	be	AUX
ejpam-5648	56	22	a	a	DET
ejpam-5648	56	23	connected	connected	ADJ
ejpam-5648	56	24	graph	graph	NOUN
ejpam-5648	56	25	.	.	PUNCT
ejpam-5648	57	1	a	a	DET
ejpam-5648	57	2	function	function	NOUN
ejpam-5648	57	3	f	f	NOUN
ejpam-5648	57	4	:	:	PUNCT
ejpam-5648	57	5	v	v	X
ejpam-5648	57	6	(	(	PUNCT
ejpam-5648	57	7	g	g	NOUN
ejpam-5648	57	8	)	)	PUNCT
ejpam-5648	57	9	−→	−→	NOUN
ejpam-5648	57	10	{	{	PUNCT
ejpam-5648	57	11	0	0	NUM
ejpam-5648	57	12	,	,	PUNCT
ejpam-5648	57	13	1	1	NUM
ejpam-5648	57	14	,	,	PUNCT
ejpam-5648	57	15	2	2	NUM
ejpam-5648	57	16	}	}	PUNCT
ejpam-5648	57	17	is	be	AUX
ejpam-5648	57	18	a	a	DET
ejpam-5648	57	19	connected	connected	ADJ
ejpam-5648	57	20	hop	hop	NOUN
ejpam-5648	57	21	roman	roman	ADJ
ejpam-5648	57	22	dominating	dominating	NOUN
ejpam-5648	57	23	function	function	NOUN
ejpam-5648	57	24	on	on	ADP
ejpam-5648	57	25	g	g	PROPN
ejpam-5648	57	26	provided	provide	VERB
ejpam-5648	57	27	that	that	SCONJ
ejpam-5648	57	28	it	it	PRON
ejpam-5648	57	29	satisfies	satisfy	VERB
ejpam-5648	57	30	the	the	DET
ejpam-5648	57	31	following	follow	VERB
ejpam-5648	57	32	properties	property	NOUN
ejpam-5648	57	33	:	:	PUNCT
ejpam-5648	57	34	(	(	PUNCT
ejpam-5648	57	35	p1	p1	NOUN
ejpam-5648	57	36	)	)	PUNCT
ejpam-5648	57	37	for	for	ADP
ejpam-5648	57	38	each	each	DET
ejpam-5648	57	39	v	v	NUM
ejpam-5648	57	40	∈	∈	PROPN
ejpam-5648	57	41	v	v	NOUN
ejpam-5648	57	42	(	(	PUNCT
ejpam-5648	57	43	g	g	NOUN
ejpam-5648	57	44	)	)	PUNCT
ejpam-5648	57	45	with	with	ADP
ejpam-5648	57	46	f(v	f(v	NOUN
ejpam-5648	57	47	)	)	PUNCT
ejpam-5648	58	1	=	=	SYM
ejpam-5648	58	2	0	0	NUM
ejpam-5648	58	3	,	,	PUNCT
ejpam-5648	58	4	there	there	PRON
ejpam-5648	58	5	exists	exist	VERB
ejpam-5648	58	6	w	w	PROPN
ejpam-5648	58	7	∈	∈	PROPN
ejpam-5648	58	8	v	v	ADP
ejpam-5648	58	9	(	(	PUNCT
ejpam-5648	58	10	g	g	NOUN
ejpam-5648	58	11	)	)	PUNCT
ejpam-5648	58	12	with	with	ADP
ejpam-5648	58	13	f(w	f(w	NOUN
ejpam-5648	58	14	)	)	PUNCT
ejpam-5648	58	15	=	=	SYM
ejpam-5648	58	16	2	2	NUM
ejpam-5648	58	17	and	and	CCONJ
ejpam-5648	58	18	dg(w	dg(w	NOUN
ejpam-5648	58	19	,	,	PUNCT
ejpam-5648	58	20	v	v	NOUN
ejpam-5648	58	21	)	)	PUNCT
ejpam-5648	58	22	=	=	SYM
ejpam-5648	58	23	2	2	NUM
ejpam-5648	58	24	(	(	PUNCT
ejpam-5648	58	25	i.e.	i.e.	X
ejpam-5648	58	26	,	,	PUNCT
ejpam-5648	58	27	f	f	PROPN
ejpam-5648	58	28	is	be	AUX
ejpam-5648	58	29	a	a	DET
ejpam-5648	58	30	hop	hop	NOUN
ejpam-5648	58	31	roman	roman	ADJ
ejpam-5648	58	32	dominating	dominating	NOUN
ejpam-5648	58	33	function	function	NOUN
ejpam-5648	58	34	on	on	ADP
ejpam-5648	58	35	g	g	NOUN
ejpam-5648	58	36	)	)	PUNCT
ejpam-5648	58	37	.	.	PUNCT
ejpam-5648	59	1	(	(	PUNCT
ejpam-5648	59	2	p2	p2	PROPN
ejpam-5648	59	3	)	)	PUNCT
ejpam-5648	59	4	the	the	DET
ejpam-5648	59	5	set	set	NOUN
ejpam-5648	59	6	{	{	PUNCT
ejpam-5648	59	7	u	u	NOUN
ejpam-5648	59	8	∈	∈	PROPN
ejpam-5648	59	9	v	v	NOUN
ejpam-5648	59	10	(	(	PUNCT
ejpam-5648	59	11	g	g	NOUN
ejpam-5648	59	12	)	)	PUNCT
ejpam-5648	59	13	:	:	PUNCT
ejpam-5648	59	14	f(u	f(u	ADJ
ejpam-5648	59	15	)	)	PUNCT
ejpam-5648	59	16	̸=	̸=	PROPN
ejpam-5648	59	17	0	0	NUM
ejpam-5648	59	18	}	}	PUNCT
ejpam-5648	59	19	induces	induce	VERB
ejpam-5648	59	20	a	a	DET
ejpam-5648	59	21	connected	connected	ADJ
ejpam-5648	59	22	subgraph	subgraph	NOUN
ejpam-5648	59	23	of	of	ADP
ejpam-5648	59	24	g.	g.	PROPN
ejpam-5648	59	25	the	the	DET
ejpam-5648	59	26	weight	weight	NOUN
ejpam-5648	59	27	of	of	ADP
ejpam-5648	59	28	f	f	PROPN
ejpam-5648	59	29	is	be	AUX
ejpam-5648	59	30	given	give	VERB
ejpam-5648	59	31	by	by	ADP
ejpam-5648	59	32	ωcrh	ωcrh	ADJ
ejpam-5648	59	33	g	g	PROPN
ejpam-5648	59	34	(	(	PUNCT
ejpam-5648	59	35	f	f	X
ejpam-5648	59	36	)	)	PUNCT
ejpam-5648	59	37	=	=	SYM
ejpam-5648	59	38	∑	∑	PUNCT
ejpam-5648	59	39	v∈v	v∈v	PROPN
ejpam-5648	59	40	(	(	PUNCT
ejpam-5648	59	41	g	g	NOUN
ejpam-5648	59	42	)	)	PUNCT
ejpam-5648	59	43	f(v	f(v	NOUN
ejpam-5648	59	44	)	)	PUNCT
ejpam-5648	59	45	.	.	PUNCT
ejpam-5648	60	1	the	the	DET
ejpam-5648	60	2	minimum	minimum	ADJ
ejpam-5648	60	3	weight	weight	NOUN
ejpam-5648	60	4	among	among	ADP
ejpam-5648	60	5	all	all	DET
ejpam-5648	60	6	connected	connect	VERB
ejpam-5648	60	7	hop	hop	NOUN
ejpam-5648	60	8	roman	roman	ADJ
ejpam-5648	60	9	dominating	dominating	NOUN
ejpam-5648	60	10	functions	function	NOUN
ejpam-5648	60	11	on	on	ADP
ejpam-5648	60	12	g	g	PROPN
ejpam-5648	60	13	is	be	AUX
ejpam-5648	60	14	the	the	DET
ejpam-5648	60	15	connected	connected	ADJ
ejpam-5648	60	16	hop	hop	NOUN
ejpam-5648	60	17	roman	roman	ADJ
ejpam-5648	60	18	domination	domination	NOUN
ejpam-5648	60	19	number	number	NOUN
ejpam-5648	60	20	γcrh(g	γcrh(g	PROPN
ejpam-5648	60	21	)	)	PUNCT
ejpam-5648	60	22	of	of	ADP
ejpam-5648	60	23	g.	g.	PROPN
ejpam-5648	60	24	if	if	SCONJ
ejpam-5648	60	25	f	f	PROPN
ejpam-5648	60	26	is	be	AUX
ejpam-5648	60	27	a	a	DET
ejpam-5648	60	28	connected	connected	ADJ
ejpam-5648	60	29	hop	hop	NOUN
ejpam-5648	60	30	roman	roman	ADJ
ejpam-5648	60	31	dominating	dominating	NOUN
ejpam-5648	60	32	function	function	NOUN
ejpam-5648	60	33	on	on	ADP
ejpam-5648	60	34	g	g	PROPN
ejpam-5648	60	35	and	and	CCONJ
ejpam-5648	60	36	ωcrh	ωcrh	ADJ
ejpam-5648	60	37	g	g	PROPN
ejpam-5648	60	38	(	(	PUNCT
ejpam-5648	60	39	f	f	X
ejpam-5648	60	40	)	)	PUNCT
ejpam-5648	60	41	=	=	SYM
ejpam-5648	60	42	γcrh(g	γcrh(g	NOUN
ejpam-5648	60	43	)	)	PUNCT
ejpam-5648	60	44	,	,	PUNCT
ejpam-5648	60	45	then	then	ADV
ejpam-5648	60	46	f	f	PROPN
ejpam-5648	60	47	is	be	AUX
ejpam-5648	60	48	called	call	VERB
ejpam-5648	60	49	a	a	DET
ejpam-5648	60	50	γcrh	γcrh	NOUN
ejpam-5648	60	51	-	-	PUNCT
ejpam-5648	60	52	function	function	NOUN
ejpam-5648	60	53	on	on	ADP
ejpam-5648	60	54	g.	g.	PROPN
ejpam-5648	60	55	if	if	SCONJ
ejpam-5648	60	56	f	f	PROPN
ejpam-5648	60	57	is	be	AUX
ejpam-5648	60	58	a	a	DET
ejpam-5648	60	59	(	(	PUNCT
ejpam-5648	60	60	connected	connected	ADJ
ejpam-5648	60	61	)	)	PUNCT
ejpam-5648	60	62	hop	hop	ADV
ejpam-5648	60	63	roman	roman	ADJ
ejpam-5648	60	64	dominating	dominating	NOUN
ejpam-5648	60	65	function	function	NOUN
ejpam-5648	60	66	on	on	ADP
ejpam-5648	60	67	g	g	PROPN
ejpam-5648	60	68	,	,	PUNCT
ejpam-5648	60	69	then	then	ADV
ejpam-5648	60	70	we	we	PRON
ejpam-5648	60	71	may	may	AUX
ejpam-5648	60	72	write	write	VERB
ejpam-5648	60	73	f	f	PROPN
ejpam-5648	60	74	=	=	SYM
ejpam-5648	60	75	(	(	PUNCT
ejpam-5648	60	76	v0	v0	PROPN
ejpam-5648	60	77	,	,	PUNCT
ejpam-5648	60	78	v1	v1	NOUN
ejpam-5648	60	79	,	,	PUNCT
ejpam-5648	60	80	v2	v2	PROPN
ejpam-5648	60	81	)	)	PUNCT
ejpam-5648	60	82	where	where	SCONJ
ejpam-5648	60	83	vj	vj	PROPN
ejpam-5648	60	84	=	=	PRON
ejpam-5648	60	85	{	{	PUNCT
ejpam-5648	60	86	x	x	PROPN
ejpam-5648	60	87	∈	∈	PROPN
ejpam-5648	60	88	v	v	NOUN
ejpam-5648	60	89	(	(	PUNCT
ejpam-5648	60	90	g	g	NOUN
ejpam-5648	60	91	)	)	PUNCT
ejpam-5648	60	92	:	:	PUNCT
ejpam-5648	60	93	f(x	f(x	PROPN
ejpam-5648	60	94	)	)	PUNCT
ejpam-5648	61	1	=	=	SYM
ejpam-5648	61	2	j	j	PROPN
ejpam-5648	61	3	}	}	PUNCT
ejpam-5648	61	4	for	for	ADP
ejpam-5648	61	5	j	j	PROPN
ejpam-5648	61	6	∈	∈	PROPN
ejpam-5648	61	7	{	{	PUNCT
ejpam-5648	61	8	0	0	NUM
ejpam-5648	61	9	,	,	PUNCT
ejpam-5648	61	10	1	1	NUM
ejpam-5648	61	11	,	,	PUNCT
ejpam-5648	61	12	2	2	NUM
ejpam-5648	61	13	}	}	PUNCT
ejpam-5648	61	14	.	.	PUNCT
ejpam-5648	62	1	consider	consider	VERB
ejpam-5648	62	2	graph	graph	NOUN
ejpam-5648	62	3	g	g	NOUN
ejpam-5648	62	4	in	in	ADP
ejpam-5648	62	5	figure	figure	NOUN
ejpam-5648	62	6	1	1	NUM
ejpam-5648	62	7	.	.	PUNCT
ejpam-5648	63	1	let	let	VERB
ejpam-5648	63	2	v0	v0	NOUN
ejpam-5648	63	3	=	=	SYM
ejpam-5648	63	4	{	{	PUNCT
ejpam-5648	63	5	a	a	PRON
ejpam-5648	63	6	,	,	PUNCT
ejpam-5648	63	7	b	b	NOUN
ejpam-5648	63	8	,	,	PUNCT
ejpam-5648	63	9	c	c	X
ejpam-5648	63	10	,	,	PUNCT
ejpam-5648	63	11	f	f	PROPN
ejpam-5648	63	12	,	,	PUNCT
ejpam-5648	63	13	g	g	PROPN
ejpam-5648	63	14	,	,	PUNCT
ejpam-5648	63	15	h	h	NOUN
ejpam-5648	63	16	}	}	PUNCT
ejpam-5648	63	17	,	,	PUNCT
ejpam-5648	63	18	v1	v1	NOUN
ejpam-5648	63	19	=	=	SYM
ejpam-5648	63	20	{	{	PUNCT
ejpam-5648	63	21	i	i	NOUN
ejpam-5648	63	22	}	}	PUNCT
ejpam-5648	63	23	,	,	PUNCT
ejpam-5648	63	24	and	and	CCONJ
ejpam-5648	63	25	v2	v2	NOUN
ejpam-5648	63	26	=	=	SYM
ejpam-5648	63	27	{	{	PUNCT
ejpam-5648	63	28	d	d	NOUN
ejpam-5648	63	29	,	,	PUNCT
ejpam-5648	63	30	e	e	NOUN
ejpam-5648	63	31	}	}	PUNCT
ejpam-5648	63	32	.	.	PUNCT
ejpam-5648	64	1	then	then	ADV
ejpam-5648	64	2	f	f	X
ejpam-5648	64	3	=	=	SYM
ejpam-5648	64	4	(	(	PUNCT
ejpam-5648	64	5	v0	v0	PROPN
ejpam-5648	64	6	,	,	PUNCT
ejpam-5648	64	7	v1	v1	NOUN
ejpam-5648	64	8	,	,	PUNCT
ejpam-5648	64	9	v2	v2	PROPN
ejpam-5648	64	10	)	)	PUNCT
ejpam-5648	64	11	is	be	AUX
ejpam-5648	64	12	a	a	DET
ejpam-5648	64	13	γrh	γrh	NOUN
ejpam-5648	64	14	-	-	PUNCT
ejpam-5648	64	15	function	function	NOUN
ejpam-5648	64	16	on	on	ADP
ejpam-5648	64	17	g.	g.	PROPN
ejpam-5648	64	18	on	on	ADP
ejpam-5648	64	19	the	the	DET
ejpam-5648	64	20	other	other	ADJ
ejpam-5648	64	21	hand	hand	NOUN
ejpam-5648	64	22	,	,	PUNCT
ejpam-5648	64	23	the	the	DET
ejpam-5648	64	24	function	function	NOUN
ejpam-5648	64	25	g	g	NOUN
ejpam-5648	64	26	=	=	SYM
ejpam-5648	64	27	(	(	PUNCT
ejpam-5648	64	28	v	v	NUM
ejpam-5648	64	29	′	′	NUM
ejpam-5648	64	30	0	0	NUM
ejpam-5648	64	31	,	,	PUNCT
ejpam-5648	64	32	v	v	NOUN
ejpam-5648	64	33	′	′	NUM
ejpam-5648	64	34	1	1	NUM
ejpam-5648	64	35	,	,	PUNCT
ejpam-5648	64	36	v	v	NOUN
ejpam-5648	64	37	′	′	NUM
ejpam-5648	64	38	2	2	NUM
ejpam-5648	64	39	)	)	PUNCT
ejpam-5648	64	40	,	,	PUNCT
ejpam-5648	64	41	where	where	SCONJ
ejpam-5648	64	42	v	v	X
ejpam-5648	64	43	′	′	NOUN
ejpam-5648	64	44	0	0	NUM
ejpam-5648	65	1	=	=	NOUN
ejpam-5648	65	2	{	{	PUNCT
ejpam-5648	65	3	a	a	PRON
ejpam-5648	65	4	,	,	PUNCT
ejpam-5648	65	5	b	b	NOUN
ejpam-5648	65	6	,	,	PUNCT
ejpam-5648	65	7	c	c	X
ejpam-5648	65	8	,	,	PUNCT
ejpam-5648	65	9	f	f	PROPN
ejpam-5648	65	10	,	,	PUNCT
ejpam-5648	65	11	g	g	PROPN
ejpam-5648	65	12	,	,	PUNCT
ejpam-5648	65	13	h	h	NOUN
ejpam-5648	65	14	,	,	PUNCT
ejpam-5648	65	15	i	i	PROPN
ejpam-5648	65	16	}	}	PUNCT
ejpam-5648	65	17	,	,	PUNCT
ejpam-5648	65	18	v	v	X
ejpam-5648	65	19	′	′	NOUN
ejpam-5648	65	20	1	1	NUM
ejpam-5648	65	21	=	=	NOUN
ejpam-5648	65	22	∅	∅	NOUN
ejpam-5648	65	23	,	,	PUNCT
ejpam-5648	65	24	and	and	CCONJ
ejpam-5648	65	25	v	v	X
ejpam-5648	65	26	′	′	NUM
ejpam-5648	65	27	2	2	NUM
ejpam-5648	65	28	=	=	SYM
ejpam-5648	65	29	{	{	PUNCT
ejpam-5648	65	30	d	d	NOUN
ejpam-5648	65	31	,	,	PUNCT
ejpam-5648	65	32	e	e	NOUN
ejpam-5648	65	33	,	,	PUNCT
ejpam-5648	65	34	f	f	PROPN
ejpam-5648	65	35	}	}	PUNCT
ejpam-5648	65	36	,	,	PUNCT
ejpam-5648	65	37	is	be	AUX
ejpam-5648	65	38	a	a	DET
ejpam-5648	65	39	γcrhfunction	γcrhfunction	NOUN
ejpam-5648	65	40	on	on	ADP
ejpam-5648	65	41	g.	g.	PROPN
ejpam-5648	65	42	therefore	therefore	ADV
ejpam-5648	65	43	,	,	PUNCT
ejpam-5648	65	44	γrh(g	γrh(g	PROPN
ejpam-5648	65	45	)	)	PUNCT
ejpam-5648	65	46	=	=	SYM
ejpam-5648	65	47	5	5	NUM
ejpam-5648	65	48	and	and	CCONJ
ejpam-5648	65	49	γcrh(g	γcrh(g	NOUN
ejpam-5648	65	50	)	)	PUNCT
ejpam-5648	65	51	=	=	SYM
ejpam-5648	65	52	6	6	NUM
ejpam-5648	65	53	.	.	PUNCT
ejpam-5648	65	54	...............	...............	PUNCT
ejpam-5648	65	55	..............	..............	PUNCT
ejpam-5648	66	1	..............	..............	PUNCT
ejpam-5648	66	2	..............	..............	PUNCT
ejpam-5648	67	1	..............	..............	PUNCT
ejpam-5648	67	2	..............	..............	PUNCT
ejpam-5648	67	3	........	........	PUNCT
ejpam-5648	68	1	....................................	....................................	PUNCT
ejpam-5648	68	2	.............................................................................................	.............................................................................................	PUNCT
ejpam-5648	69	1	....................................	....................................	PUNCT
ejpam-5648	69	2	.................................................................................................................................	.................................................................................................................................	PUNCT
ejpam-5648	70	1	....................................	....................................	PUNCT
ejpam-5648	70	2	...............	...............	PUNCT
ejpam-5648	71	1	..............	..............	PUNCT
ejpam-5648	71	2	..............	..............	PUNCT
ejpam-5648	72	1	..............	..............	PUNCT
ejpam-5648	72	2	..............	..............	PUNCT
ejpam-5648	73	1	..............	..............	PUNCT
ejpam-5648	73	2	........	........	PUNCT
ejpam-5648	74	1	....................................	....................................	PUNCT
ejpam-5648	74	2	....................................................................................................................................................	....................................................................................................................................................	PUNCT
ejpam-5648	75	1	...............	...............	PUNCT
ejpam-5648	75	2	..............	..............	PUNCT
ejpam-5648	76	1	..............	..............	PUNCT
ejpam-5648	76	2	..............	..............	PUNCT
ejpam-5648	77	1	..............	..............	PUNCT
ejpam-5648	77	2	..............	..............	PUNCT
ejpam-5648	77	3	........	........	PUNCT
ejpam-5648	78	1	....................................	....................................	PUNCT
ejpam-5648	78	2	.............................................................................................	.............................................................................................	PUNCT
ejpam-5648	79	1	....................................	....................................	PUNCT
ejpam-5648	79	2	.................................................................................................................................	.................................................................................................................................	PUNCT
ejpam-5648	80	1	....................................	....................................	PUNCT
ejpam-5648	80	2	...............	...............	PUNCT
ejpam-5648	81	1	..............	..............	PUNCT
ejpam-5648	81	2	..............	..............	PUNCT
ejpam-5648	82	1	..............	..............	PUNCT
ejpam-5648	82	2	..............	..............	PUNCT
ejpam-5648	83	1	..............	..............	PUNCT
ejpam-5648	83	2	........	........	PUNCT
ejpam-5648	84	1	....................................	....................................	PUNCT
ejpam-5648	84	2	....................................................................................................................................................	....................................................................................................................................................	PUNCT
ejpam-5648	85	1	....................................	....................................	PUNCT
ejpam-5648	85	2	a	a	DET
ejpam-5648	85	3	c	c	NOUN
ejpam-5648	85	4	b	b	PROPN
ejpam-5648	85	5	d	d	X
ejpam-5648	85	6	e	e	PROPN
ejpam-5648	85	7	h	h	NOUN
ejpam-5648	85	8	g	g	PROPN
ejpam-5648	86	1	f	f	PROPN
ejpam-5648	87	1	i	i	PRON
ejpam-5648	87	2	figure	figure	VERB
ejpam-5648	87	3	1	1	NUM
ejpam-5648	87	4	:	:	PUNCT
ejpam-5648	87	5	graph	graph	VERB
ejpam-5648	87	6	g	g	NOUN
ejpam-5648	87	7	with	with	ADP
ejpam-5648	87	8	γrh(g	γrh(g	NOUN
ejpam-5648	87	9	)	)	PUNCT
ejpam-5648	87	10	=	=	SYM
ejpam-5648	87	11	5	5	NUM
ejpam-5648	87	12	and	and	CCONJ
ejpam-5648	87	13	γcrh(g	γcrh(g	NOUN
ejpam-5648	87	14	)	)	PUNCT
ejpam-5648	87	15	=	=	PUNCT
ejpam-5648	87	16	6	6	NUM
ejpam-5648	87	17	henceforth	henceforth	ADV
ejpam-5648	87	18	,	,	PUNCT
ejpam-5648	87	19	the	the	DET
ejpam-5648	87	20	family	family	NOUN
ejpam-5648	87	21	chrdf	chrdf	VERB
ejpam-5648	87	22	(	(	PUNCT
ejpam-5648	87	23	g	g	NOUN
ejpam-5648	87	24	)	)	PUNCT
ejpam-5648	87	25	refers	refer	VERB
ejpam-5648	87	26	to	to	ADP
ejpam-5648	87	27	the	the	DET
ejpam-5648	87	28	set	set	NOUN
ejpam-5648	87	29	containing	contain	VERB
ejpam-5648	87	30	all	all	DET
ejpam-5648	87	31	the	the	DET
ejpam-5648	87	32	connected	connected	ADJ
ejpam-5648	87	33	hop	hop	NOUN
ejpam-5648	87	34	roman	roman	ADJ
ejpam-5648	87	35	dominating	dominating	NOUN
ejpam-5648	87	36	functions	function	NOUN
ejpam-5648	87	37	on	on	ADP
ejpam-5648	87	38	g.	g.	PROPN
ejpam-5648	87	39	3	3	NUM
ejpam-5648	87	40	.	.	PUNCT
ejpam-5648	88	1	results	result	NOUN
ejpam-5648	88	2	proposition	proposition	NOUN
ejpam-5648	88	3	1	1	X
ejpam-5648	88	4	.	.	PUNCT
ejpam-5648	89	1	let	let	VERB
ejpam-5648	89	2	g	g	PRON
ejpam-5648	89	3	be	be	AUX
ejpam-5648	89	4	a	a	DET
ejpam-5648	89	5	nontrivial	nontrivial	ADJ
ejpam-5648	89	6	connected	connect	VERB
ejpam-5648	89	7	graph	graph	NOUN
ejpam-5648	89	8	and	and	CCONJ
ejpam-5648	89	9	let	let	VERB
ejpam-5648	89	10	f	f	PROPN
ejpam-5648	89	11	=	=	SYM
ejpam-5648	89	12	(	(	PUNCT
ejpam-5648	89	13	v0	v0	PROPN
ejpam-5648	89	14	,	,	PUNCT
ejpam-5648	89	15	v1	v1	NOUN
ejpam-5648	89	16	,	,	PUNCT
ejpam-5648	89	17	v2	v2	PROPN
ejpam-5648	89	18	)	)	PUNCT
ejpam-5648	89	19	be	be	AUX
ejpam-5648	89	20	a	a	DET
ejpam-5648	89	21	γcrhfunction	γcrhfunction	NOUN
ejpam-5648	89	22	on	on	ADP
ejpam-5648	89	23	g.	g.	PROPN
ejpam-5648	90	1	then	then	ADV
ejpam-5648	90	2	each	each	PRON
ejpam-5648	90	3	of	of	ADP
ejpam-5648	90	4	the	the	DET
ejpam-5648	90	5	following	follow	VERB
ejpam-5648	90	6	holds	hold	VERB
ejpam-5648	90	7	:	:	PUNCT
ejpam-5648	90	8	(	(	PUNCT
ejpam-5648	90	9	i	i	NOUN
ejpam-5648	90	10	)	)	PUNCT
ejpam-5648	90	11	|v0|	|v0|	NOUN
ejpam-5648	90	12	=	=	SYM
ejpam-5648	90	13	0	0	PUNCT
ejpam-5648	91	1	if	if	SCONJ
ejpam-5648	91	2	and	and	CCONJ
ejpam-5648	91	3	only	only	ADV
ejpam-5648	91	4	if	if	SCONJ
ejpam-5648	91	5	|v2|	|v2|	ADV
ejpam-5648	91	6	=	=	SYM
ejpam-5648	91	7	0	0	X
ejpam-5648	91	8	.	.	PUNCT
ejpam-5648	91	9	(	(	PUNCT
ejpam-5648	91	10	ii	ii	NOUN
ejpam-5648	91	11	)	)	PUNCT
ejpam-5648	91	12	|v1|	|v1|	NOUN
ejpam-5648	91	13	=	=	SYM
ejpam-5648	91	14	0	0	PUNCT
ejpam-5648	92	1	if	if	SCONJ
ejpam-5648	92	2	and	and	CCONJ
ejpam-5648	92	3	only	only	ADV
ejpam-5648	92	4	if	if	SCONJ
ejpam-5648	92	5	v2	v2	PROPN
ejpam-5648	92	6	is	be	AUX
ejpam-5648	92	7	a	a	DET
ejpam-5648	92	8	γch	γch	NOUN
ejpam-5648	92	9	-	-	PUNCT
ejpam-5648	92	10	set	set	VERB
ejpam-5648	92	11	in	in	ADP
ejpam-5648	92	12	g.	g.	PROPN
ejpam-5648	92	13	proof	proof	NOUN
ejpam-5648	92	14	.	.	PUNCT
ejpam-5648	93	1	(	(	PUNCT
ejpam-5648	93	2	i	i	NOUN
ejpam-5648	93	3	)	)	PUNCT
ejpam-5648	93	4	the	the	DET
ejpam-5648	93	5	sufficiency	sufficiency	NOUN
ejpam-5648	93	6	part	part	NOUN
ejpam-5648	93	7	is	be	AUX
ejpam-5648	93	8	clear	clear	ADJ
ejpam-5648	93	9	by	by	ADP
ejpam-5648	93	10	property	property	NOUN
ejpam-5648	93	11	(	(	PUNCT
ejpam-5648	93	12	p1	p1	PROPN
ejpam-5648	93	13	)	)	PUNCT
ejpam-5648	93	14	.	.	PUNCT
ejpam-5648	94	1	suppose	suppose	VERB
ejpam-5648	94	2	|v0|	|v0|	NOUN
ejpam-5648	94	3	=	=	SYM
ejpam-5648	94	4	0	0	PUNCT
ejpam-5648	94	5	and	and	CCONJ
ejpam-5648	94	6	suppose	suppose	VERB
ejpam-5648	94	7	|v2|	|v2|	NOUN
ejpam-5648	94	8	=	=	NOUN
ejpam-5648	94	9	̸	̸	PUNCT
ejpam-5648	94	10	∅.	∅.	ADV
ejpam-5648	94	11	let	let	VERB
ejpam-5648	94	12	v	v	NOUN
ejpam-5648	94	13	′	′	NOUN
ejpam-5648	94	14	0	0	NUM
ejpam-5648	95	1	=	=	SYM
ejpam-5648	95	2	v0	v0	PROPN
ejpam-5648	95	3	,	,	PUNCT
ejpam-5648	95	4	v	v	NOUN
ejpam-5648	95	5	′	′	NUM
ejpam-5648	95	6	1	1	NUM
ejpam-5648	95	7	=	=	SYM
ejpam-5648	95	8	v1	v1	VERB
ejpam-5648	95	9	∪	∪	NOUN
ejpam-5648	95	10	v2	v2	NOUN
ejpam-5648	95	11	and	and	CCONJ
ejpam-5648	95	12	v	v	NOUN
ejpam-5648	95	13	′	′	NUM
ejpam-5648	95	14	2	2	NUM
ejpam-5648	95	15	=	=	PUNCT
ejpam-5648	95	16	∅.	∅.	NOUN
ejpam-5648	95	17	then	then	ADV
ejpam-5648	95	18	g	g	PROPN
ejpam-5648	95	19	=	=	PUNCT
ejpam-5648	95	20	(	(	PUNCT
ejpam-5648	95	21	v	v	NUM
ejpam-5648	95	22	′	′	NUM
ejpam-5648	95	23	0	0	NUM
ejpam-5648	95	24	,	,	PUNCT
ejpam-5648	95	25	v	v	NOUN
ejpam-5648	95	26	′	′	NUM
ejpam-5648	95	27	1	1	NUM
ejpam-5648	95	28	,	,	PUNCT
ejpam-5648	95	29	v	v	NOUN
ejpam-5648	95	30	′	′	NUM
ejpam-5648	95	31	2	2	NUM
ejpam-5648	95	32	)	)	PUNCT
ejpam-5648	95	33	∈	∈	PROPN
ejpam-5648	95	34	chrdf	chrdf	NOUN
ejpam-5648	95	35	(	(	PUNCT
ejpam-5648	95	36	g	g	NOUN
ejpam-5648	95	37	)	)	PUNCT
ejpam-5648	95	38	and	and	CCONJ
ejpam-5648	95	39	so	so	ADV
ejpam-5648	95	40	,	,	PUNCT
ejpam-5648	95	41	ωcrh	ωcrh	ADV
ejpam-5648	95	42	g	g	PROPN
ejpam-5648	95	43	(	(	PUNCT
ejpam-5648	95	44	g	g	NOUN
ejpam-5648	95	45	)	)	PUNCT
ejpam-5648	95	46	=	=	PUNCT
ejpam-5648	95	47	|v	|v	PROPN
ejpam-5648	95	48	′	′	NOUN
ejpam-5648	95	49	1	1	NUM
ejpam-5648	96	1	|	|	ADV
ejpam-5648	96	2	=	=	PUNCT
ejpam-5648	96	3	|v1	|v1	PRON
ejpam-5648	96	4	∪	∪	X
ejpam-5648	96	5	v2|	v2|	X
ejpam-5648	96	6	=	=	PUNCT
ejpam-5648	96	7	|v1|+	|v1|+	ADV
ejpam-5648	96	8	|v2|	|v2|	ADV
ejpam-5648	96	9	<	<	X
ejpam-5648	96	10	|v1|+	|v1|+	ADV
ejpam-5648	96	11	2|v2|	2|v2|	NUM
ejpam-5648	96	12	=	=	SYM
ejpam-5648	96	13	ωcrh	ωcrh	ADP
ejpam-5648	96	14	g	g	PROPN
ejpam-5648	96	15	(	(	PUNCT
ejpam-5648	96	16	f	f	PROPN
ejpam-5648	96	17	)	)	PUNCT
ejpam-5648	96	18	,	,	PUNCT
ejpam-5648	96	19	a.	a.	PROPN
ejpam-5648	96	20	aradais	aradais	PROPN
ejpam-5648	96	21	,	,	PUNCT
ejpam-5648	96	22	j.	j.	PROPN
ejpam-5648	96	23	cariaga	cariaga	PROPN
ejpam-5648	96	24	,	,	PUNCT
ejpam-5648	96	25	s.	s.	PROPN
ejpam-5648	96	26	canoy	canoy	PROPN
ejpam-5648	96	27	jr	jr	PROPN
ejpam-5648	96	28	.	.	PROPN
ejpam-5648	96	29	/	/	SYM
ejpam-5648	96	30	eur	eur	PROPN
ejpam-5648	96	31	.	.	PUNCT
ejpam-5648	97	1	j.	j.	PROPN
ejpam-5648	97	2	pure	pure	PROPN
ejpam-5648	97	3	appl	appl	PROPN
ejpam-5648	97	4	.	.	PROPN
ejpam-5648	97	5	math	math	PROPN
ejpam-5648	97	6	,	,	PUNCT
ejpam-5648	97	7	18	18	NUM
ejpam-5648	97	8	(	(	PUNCT
ejpam-5648	97	9	1	1	NUM
ejpam-5648	97	10	)	)	PUNCT
ejpam-5648	97	11	(	(	PUNCT
ejpam-5648	97	12	2025	2025	NUM
ejpam-5648	97	13	)	)	PUNCT
ejpam-5648	97	14	,	,	PUNCT
ejpam-5648	97	15	5648	5648	NUM
ejpam-5648	97	16	4	4	NUM
ejpam-5648	97	17	of	of	ADP
ejpam-5648	97	18	13	13	NUM
ejpam-5648	97	19	a	a	DET
ejpam-5648	97	20	contradiction	contradiction	NOUN
ejpam-5648	97	21	.	.	PUNCT
ejpam-5648	98	1	thus	thus	ADV
ejpam-5648	98	2	,	,	PUNCT
ejpam-5648	98	3	|v2|	|v2|	NOUN
ejpam-5648	98	4	=	=	SYM
ejpam-5648	98	5	∅.	∅.	PROPN
ejpam-5648	98	6	(	(	PUNCT
ejpam-5648	98	7	ii	ii	NOUN
ejpam-5648	98	8	)	)	PUNCT
ejpam-5648	98	9	suppose	suppose	VERB
ejpam-5648	98	10	|v1|	|v1|	NOUN
ejpam-5648	98	11	=	=	SYM
ejpam-5648	98	12	0	0	X
ejpam-5648	98	13	.	.	PUNCT
ejpam-5648	99	1	then	then	ADV
ejpam-5648	99	2	v2	v2	PROPN
ejpam-5648	99	3	is	be	AUX
ejpam-5648	99	4	a	a	DET
ejpam-5648	99	5	connected	connected	ADJ
ejpam-5648	99	6	hop	hop	NOUN
ejpam-5648	99	7	dominating	dominating	NOUN
ejpam-5648	99	8	set	set	VERB
ejpam-5648	99	9	in	in	ADP
ejpam-5648	99	10	g.	g.	PROPN
ejpam-5648	99	11	suppose	suppose	VERB
ejpam-5648	99	12	that	that	SCONJ
ejpam-5648	99	13	v2	v2	PROPN
ejpam-5648	99	14	is	be	AUX
ejpam-5648	99	15	not	not	PART
ejpam-5648	99	16	a	a	DET
ejpam-5648	99	17	γch	γch	NOUN
ejpam-5648	99	18	-	-	PUNCT
ejpam-5648	99	19	set	set	NOUN
ejpam-5648	99	20	in	in	ADP
ejpam-5648	99	21	g.	g.	PROPN
ejpam-5648	99	22	let	let	VERB
ejpam-5648	99	23	v	v	NOUN
ejpam-5648	99	24	′	′	NUM
ejpam-5648	99	25	2	2	NUM
ejpam-5648	99	26	⊆	⊆	NUM
ejpam-5648	99	27	v	v	NOUN
ejpam-5648	99	28	(	(	PUNCT
ejpam-5648	99	29	g	g	NOUN
ejpam-5648	99	30	)	)	PUNCT
ejpam-5648	99	31	be	be	AUX
ejpam-5648	99	32	a	a	DET
ejpam-5648	99	33	γch	γch	NOUN
ejpam-5648	99	34	-	-	PUNCT
ejpam-5648	99	35	set	set	NOUN
ejpam-5648	99	36	in	in	ADP
ejpam-5648	99	37	g.	g.	PROPN
ejpam-5648	99	38	then	then	ADV
ejpam-5648	99	39	|v	|v	PROPN
ejpam-5648	99	40	′	′	NOUN
ejpam-5648	99	41	2	2	NUM
ejpam-5648	100	1	|	|	ADV
ejpam-5648	100	2	<	<	X
ejpam-5648	100	3	|v2|	|v2|	NOUN
ejpam-5648	100	4	.	.	PUNCT
ejpam-5648	101	1	let	let	VERB
ejpam-5648	101	2	v	v	NUM
ejpam-5648	101	3	∗	∗	NOUN
ejpam-5648	101	4	2	2	NUM
ejpam-5648	101	5	=	=	SYM
ejpam-5648	101	6	v	v	NOUN
ejpam-5648	101	7	′	′	NUM
ejpam-5648	101	8	2	2	NUM
ejpam-5648	101	9	,	,	PUNCT
ejpam-5648	101	10	v	v	NOUN
ejpam-5648	101	11	∗	∗	NOUN
ejpam-5648	101	12	1	1	NUM
ejpam-5648	101	13	=	=	NOUN
ejpam-5648	101	14	∅	∅	NOUN
ejpam-5648	101	15	and	and	CCONJ
ejpam-5648	101	16	v	v	ADP
ejpam-5648	101	17	∗	∗	NOUN
ejpam-5648	101	18	0	0	NUM
ejpam-5648	102	1	=	=	SYM
ejpam-5648	102	2	v	v	NOUN
ejpam-5648	102	3	(	(	PUNCT
ejpam-5648	102	4	g	g	NOUN
ejpam-5648	102	5	)	)	PUNCT
ejpam-5648	102	6	\	\	PROPN
ejpam-5648	102	7	v	v	ADP
ejpam-5648	102	8	∗	∗	NOUN
ejpam-5648	102	9	2	2	NUM
ejpam-5648	102	10	.	.	PUNCT
ejpam-5648	103	1	then	then	ADV
ejpam-5648	103	2	h	h	NOUN
ejpam-5648	103	3	=	=	PUNCT
ejpam-5648	103	4	(	(	PUNCT
ejpam-5648	103	5	v	v	NOUN
ejpam-5648	103	6	∗	∗	NOUN
ejpam-5648	103	7	0	0	NUM
ejpam-5648	103	8	,	,	PUNCT
ejpam-5648	103	9	v	v	NOUN
ejpam-5648	103	10	∗	∗	NOUN
ejpam-5648	103	11	1	1	NUM
ejpam-5648	103	12	,	,	PUNCT
ejpam-5648	103	13	v	v	NOUN
ejpam-5648	103	14	∗	∗	X
ejpam-5648	103	15	2	2	NUM
ejpam-5648	103	16	)	)	PUNCT
ejpam-5648	103	17	∈	∈	NOUN
ejpam-5648	103	18	chrdf	chrdf	NOUN
ejpam-5648	103	19	(	(	PUNCT
ejpam-5648	103	20	g	g	NOUN
ejpam-5648	103	21	)	)	PUNCT
ejpam-5648	103	22	.	.	PUNCT
ejpam-5648	104	1	thus	thus	ADV
ejpam-5648	104	2	,	,	PUNCT
ejpam-5648	104	3	ωcrh	ωcrh	ADV
ejpam-5648	104	4	g	g	PROPN
ejpam-5648	104	5	(	(	PUNCT
ejpam-5648	104	6	h	h	NOUN
ejpam-5648	104	7	)	)	PUNCT
ejpam-5648	104	8	=	=	SYM
ejpam-5648	105	1	2|v	2|v	NUM
ejpam-5648	105	2	∗	∗	NOUN
ejpam-5648	105	3	2	2	NUM
ejpam-5648	105	4	|	|	NOUN
ejpam-5648	105	5	=	=	SYM
ejpam-5648	105	6	2|v	2|v	NUM
ejpam-5648	106	1	′	′	NUM
ejpam-5648	106	2	2	2	NUM
ejpam-5648	107	1	|	|	ADV
ejpam-5648	107	2	<	<	X
ejpam-5648	107	3	2|v2|	2|v2|	NUM
ejpam-5648	107	4	=	=	SYM
ejpam-5648	107	5	ωcrh	ωcrh	ADP
ejpam-5648	107	6	g	g	PROPN
ejpam-5648	107	7	(	(	PUNCT
ejpam-5648	107	8	f	f	PROPN
ejpam-5648	107	9	)	)	PUNCT
ejpam-5648	107	10	.	.	PUNCT
ejpam-5648	108	1	this	this	PRON
ejpam-5648	108	2	is	be	AUX
ejpam-5648	108	3	a	a	DET
ejpam-5648	108	4	contradiction	contradiction	NOUN
ejpam-5648	108	5	to	to	ADP
ejpam-5648	108	6	the	the	DET
ejpam-5648	108	7	assumption	assumption	NOUN
ejpam-5648	108	8	that	that	SCONJ
ejpam-5648	108	9	f	f	PROPN
ejpam-5648	108	10	is	be	AUX
ejpam-5648	108	11	a	a	DET
ejpam-5648	108	12	γcrh	γcrh	NOUN
ejpam-5648	108	13	-	-	PUNCT
ejpam-5648	108	14	function	function	NOUN
ejpam-5648	108	15	on	on	ADP
ejpam-5648	108	16	g.	g.	PROPN
ejpam-5648	108	17	hence	hence	ADV
ejpam-5648	108	18	,	,	PUNCT
ejpam-5648	108	19	v2	v2	PROPN
ejpam-5648	108	20	is	be	AUX
ejpam-5648	108	21	a	a	DET
ejpam-5648	108	22	γch	γch	NOUN
ejpam-5648	108	23	-	-	PUNCT
ejpam-5648	108	24	set	set	NOUN
ejpam-5648	108	25	in	in	ADP
ejpam-5648	108	26	g.	g.	NOUN
ejpam-5648	108	27	conversely	conversely	ADV
ejpam-5648	108	28	,	,	PUNCT
ejpam-5648	108	29	suppose	suppose	VERB
ejpam-5648	108	30	that	that	SCONJ
ejpam-5648	108	31	v2	v2	PROPN
ejpam-5648	108	32	is	be	AUX
ejpam-5648	108	33	a	a	DET
ejpam-5648	108	34	γch	γch	NOUN
ejpam-5648	108	35	-	-	PUNCT
ejpam-5648	108	36	set	set	VERB
ejpam-5648	108	37	in	in	ADP
ejpam-5648	108	38	g.	g.	PROPN
ejpam-5648	108	39	suppose	suppose	VERB
ejpam-5648	108	40	further	far	ADV
ejpam-5648	108	41	that	that	SCONJ
ejpam-5648	108	42	|v1|	|v1|	NOUN
ejpam-5648	108	43	=	=	SYM
ejpam-5648	108	44	̸	̸	NUM
ejpam-5648	108	45	0	0	NUM
ejpam-5648	108	46	.	.	PUNCT
ejpam-5648	109	1	then	then	ADV
ejpam-5648	109	2	γcrh(g	γcrh(g	ADP
ejpam-5648	109	3	)	)	PUNCT
ejpam-5648	109	4	=	=	NOUN
ejpam-5648	109	5	|v1|	|v1|	NOUN
ejpam-5648	109	6	+	+	CCONJ
ejpam-5648	109	7	2|v2|	2|v2|	NUM
ejpam-5648	109	8	>	>	PUNCT
ejpam-5648	109	9	2|v2|	2|v2|	NUM
ejpam-5648	109	10	.	.	PUNCT
ejpam-5648	110	1	let	let	VERB
ejpam-5648	110	2	v	v	VERB
ejpam-5648	110	3	′′	′′	PROPN
ejpam-5648	110	4	0	0	PUNCT
ejpam-5648	111	1	=	=	SYM
ejpam-5648	111	2	v0	v0	NOUN
ejpam-5648	111	3	∪	∪	NOUN
ejpam-5648	111	4	v1	v1	PROPN
ejpam-5648	111	5	,	,	PUNCT
ejpam-5648	111	6	v	v	ADP
ejpam-5648	111	7	′′	′′	PROPN
ejpam-5648	111	8	1	1	NUM
ejpam-5648	111	9	=	=	NOUN
ejpam-5648	111	10	∅	∅	NOUN
ejpam-5648	111	11	and	and	CCONJ
ejpam-5648	111	12	v	v	ADP
ejpam-5648	111	13	′′	′′	PROPN
ejpam-5648	111	14	2	2	NUM
ejpam-5648	111	15	=	=	SYM
ejpam-5648	111	16	v2	v2	PROPN
ejpam-5648	111	17	.	.	PUNCT
ejpam-5648	112	1	then	then	ADV
ejpam-5648	112	2	v	v	ADP
ejpam-5648	112	3	′′	′′	PROPN
ejpam-5648	112	4	0	0	NUM
ejpam-5648	113	1	⊆	⊆	NUM
ejpam-5648	113	2	n2	n2	ADJ
ejpam-5648	113	3	g(v	g(v	PROPN
ejpam-5648	113	4	′′	′′	PROPN
ejpam-5648	113	5	2	2	NUM
ejpam-5648	113	6	)	)	PUNCT
ejpam-5648	113	7	and	and	CCONJ
ejpam-5648	113	8	v	v	ADP
ejpam-5648	113	9	′′	′′	PROPN
ejpam-5648	113	10	1	1	NUM
ejpam-5648	113	11	∪	∪	X
ejpam-5648	113	12	v	v	ADP
ejpam-5648	113	13	′′	′′	PROPN
ejpam-5648	113	14	2	2	NUM
ejpam-5648	113	15	=	=	SYM
ejpam-5648	113	16	v2	v2	NOUN
ejpam-5648	113	17	is	be	AUX
ejpam-5648	113	18	a	a	DET
ejpam-5648	113	19	connected	connected	ADJ
ejpam-5648	113	20	hop	hop	NOUN
ejpam-5648	113	21	dominating	dominating	NOUN
ejpam-5648	113	22	set	set	VERB
ejpam-5648	113	23	in	in	ADP
ejpam-5648	113	24	g.	g.	PROPN
ejpam-5648	113	25	thus	thus	ADV
ejpam-5648	113	26	,	,	PUNCT
ejpam-5648	113	27	h	h	NOUN
ejpam-5648	113	28	=	=	PRON
ejpam-5648	113	29	(	(	PUNCT
ejpam-5648	113	30	v	v	NUM
ejpam-5648	113	31	′′	′′	PROPN
ejpam-5648	113	32	0	0	NUM
ejpam-5648	113	33	,	,	PUNCT
ejpam-5648	113	34	v	v	ADP
ejpam-5648	113	35	′′	′′	PROPN
ejpam-5648	113	36	1	1	NUM
ejpam-5648	113	37	,	,	PUNCT
ejpam-5648	113	38	v	v	ADP
ejpam-5648	113	39	′′	′′	PROPN
ejpam-5648	113	40	2	2	NUM
ejpam-5648	113	41	)	)	PUNCT
ejpam-5648	113	42	∈	∈	PROPN
ejpam-5648	113	43	chrdf	chrdf	NOUN
ejpam-5648	113	44	(	(	PUNCT
ejpam-5648	113	45	g	g	NOUN
ejpam-5648	113	46	)	)	PUNCT
ejpam-5648	113	47	and	and	CCONJ
ejpam-5648	113	48	ωcrh	ωcrh	ADJ
ejpam-5648	113	49	g	g	PROPN
ejpam-5648	113	50	(	(	PUNCT
ejpam-5648	113	51	h	h	NOUN
ejpam-5648	113	52	)	)	PUNCT
ejpam-5648	113	53	=	=	SYM
ejpam-5648	114	1	2|v	2|v	X
ejpam-5648	115	1	′′	′′	NOUN
ejpam-5648	115	2	2	2	NUM
ejpam-5648	115	3	|	|	ADV
ejpam-5648	115	4	=	=	SYM
ejpam-5648	115	5	2|v2|	2|v2|	NUM
ejpam-5648	115	6	<	<	X
ejpam-5648	115	7	|v1|+	|v1|+	ADV
ejpam-5648	115	8	2|v2|	2|v2|	NUM
ejpam-5648	115	9	=	=	SYM
ejpam-5648	115	10	ωcrh	ωcrh	ADP
ejpam-5648	115	11	g	g	PROPN
ejpam-5648	115	12	(	(	PUNCT
ejpam-5648	115	13	f	f	PROPN
ejpam-5648	115	14	)	)	PUNCT
ejpam-5648	115	15	,	,	PUNCT
ejpam-5648	115	16	a	a	DET
ejpam-5648	115	17	contradiction	contradiction	NOUN
ejpam-5648	115	18	.	.	PUNCT
ejpam-5648	116	1	hence	hence	ADV
ejpam-5648	116	2	,	,	PUNCT
ejpam-5648	116	3	|v1|	|v1|	NOUN
ejpam-5648	116	4	=	=	SYM
ejpam-5648	116	5	0	0	X
ejpam-5648	116	6	.	.	PUNCT
ejpam-5648	116	7	proposition	proposition	NOUN
ejpam-5648	116	8	2	2	NUM
ejpam-5648	116	9	.	.	X
ejpam-5648	116	10	for	for	ADP
ejpam-5648	116	11	any	any	DET
ejpam-5648	116	12	graph	graph	NOUN
ejpam-5648	116	13	g	g	NOUN
ejpam-5648	116	14	of	of	ADP
ejpam-5648	116	15	order	order	NOUN
ejpam-5648	116	16	n	n	CCONJ
ejpam-5648	116	17	,	,	PUNCT
ejpam-5648	116	18	it	it	PRON
ejpam-5648	116	19	holds	hold	VERB
ejpam-5648	116	20	that	that	SCONJ
ejpam-5648	116	21	γch(g	γch(g	NOUN
ejpam-5648	116	22	)	)	PUNCT
ejpam-5648	116	23	≤	≤	NUM
ejpam-5648	116	24	γcrh(g	γcrh(g	NOUN
ejpam-5648	116	25	)	)	PUNCT
ejpam-5648	116	26	≤	≤	NUM
ejpam-5648	116	27	min{n	min{n	NOUN
ejpam-5648	116	28	,	,	PUNCT
ejpam-5648	116	29	2γch(g	2γch(g	NUM
ejpam-5648	116	30	)	)	PUNCT
ejpam-5648	116	31	}	}	PUNCT
ejpam-5648	116	32	.	.	PUNCT
ejpam-5648	117	1	proof	proof	NOUN
ejpam-5648	117	2	.	.	PUNCT
ejpam-5648	118	1	let	let	VERB
ejpam-5648	118	2	f	f	PROPN
ejpam-5648	118	3	=	=	SYM
ejpam-5648	118	4	(	(	PUNCT
ejpam-5648	118	5	v0	v0	PROPN
ejpam-5648	118	6	,	,	PUNCT
ejpam-5648	118	7	v1	v1	NOUN
ejpam-5648	118	8	,	,	PUNCT
ejpam-5648	118	9	v2	v2	PROPN
ejpam-5648	118	10	)	)	PUNCT
ejpam-5648	118	11	be	be	AUX
ejpam-5648	118	12	a	a	DET
ejpam-5648	118	13	γcrh	γcrh	NOUN
ejpam-5648	118	14	-	-	PUNCT
ejpam-5648	118	15	function	function	NOUN
ejpam-5648	118	16	of	of	ADP
ejpam-5648	118	17	g.	g.	PROPN
ejpam-5648	118	18	since	since	SCONJ
ejpam-5648	118	19	v1	v1	PROPN
ejpam-5648	118	20	∪	∪	NOUN
ejpam-5648	118	21	v2	v2	NOUN
ejpam-5648	118	22	is	be	AUX
ejpam-5648	118	23	a	a	DET
ejpam-5648	118	24	connected	connected	ADJ
ejpam-5648	118	25	hop	hop	NOUN
ejpam-5648	118	26	dominating	dominating	NOUN
ejpam-5648	118	27	set	set	NOUN
ejpam-5648	118	28	of	of	ADP
ejpam-5648	118	29	g	g	PROPN
ejpam-5648	118	30	,	,	PUNCT
ejpam-5648	118	31	we	we	PRON
ejpam-5648	118	32	have	have	VERB
ejpam-5648	118	33	γch(g	γch(g	NOUN
ejpam-5648	118	34	)	)	PUNCT
ejpam-5648	118	35	≤	≤	NOUN
ejpam-5648	118	36	|v1|+	|v1|+	PUNCT
ejpam-5648	118	37	|v2|	|v2|	ADV
ejpam-5648	118	38	≤	≤	NUM
ejpam-5648	118	39	|v1|+	|v1|+	CCONJ
ejpam-5648	118	40	2|v2|	2|v2|	NUM
ejpam-5648	118	41	=	=	SYM
ejpam-5648	118	42	γcrh(g	γcrh(g	NOUN
ejpam-5648	118	43	)	)	PUNCT
ejpam-5648	118	44	.	.	PUNCT
ejpam-5648	119	1	next	next	ADV
ejpam-5648	119	2	,	,	PUNCT
ejpam-5648	119	3	set	set	VERB
ejpam-5648	119	4	v	v	NUM
ejpam-5648	119	5	′	′	NOUN
ejpam-5648	119	6	0	0	NUM
ejpam-5648	120	1	=	=	SYM
ejpam-5648	120	2	v	v	NUM
ejpam-5648	120	3	′	′	NUM
ejpam-5648	120	4	2	2	NUM
ejpam-5648	120	5	=	=	NOUN
ejpam-5648	120	6	∅	∅	NOUN
ejpam-5648	120	7	and	and	CCONJ
ejpam-5648	120	8	v	v	NOUN
ejpam-5648	120	9	′	′	NUM
ejpam-5648	120	10	1	1	NUM
ejpam-5648	120	11	=	=	SYM
ejpam-5648	120	12	v	v	NOUN
ejpam-5648	120	13	(	(	PUNCT
ejpam-5648	120	14	g	g	NOUN
ejpam-5648	120	15	)	)	PUNCT
ejpam-5648	120	16	.	.	PUNCT
ejpam-5648	121	1	then	then	ADV
ejpam-5648	121	2	g1	g1	PROPN
ejpam-5648	121	3	=	=	PRON
ejpam-5648	121	4	(	(	PUNCT
ejpam-5648	121	5	v	v	NUM
ejpam-5648	121	6	′	′	NUM
ejpam-5648	121	7	0	0	NUM
ejpam-5648	121	8	,	,	PUNCT
ejpam-5648	121	9	v	v	NOUN
ejpam-5648	121	10	′	′	NUM
ejpam-5648	121	11	1	1	NUM
ejpam-5648	121	12	,	,	PUNCT
ejpam-5648	121	13	v	v	NOUN
ejpam-5648	121	14	′	′	NUM
ejpam-5648	121	15	2	2	NUM
ejpam-5648	121	16	)	)	PUNCT
ejpam-5648	121	17	∈	∈	PROPN
ejpam-5648	121	18	chrdf	chrdf	NOUN
ejpam-5648	121	19	(	(	PUNCT
ejpam-5648	121	20	g	g	NOUN
ejpam-5648	121	21	)	)	PUNCT
ejpam-5648	121	22	.	.	PUNCT
ejpam-5648	122	1	thus	thus	ADV
ejpam-5648	122	2	,	,	PUNCT
ejpam-5648	122	3	γcrh(g	γcrh(g	NOUN
ejpam-5648	122	4	)	)	PUNCT
ejpam-5648	122	5	≤	≤	NOUN
ejpam-5648	122	6	ωcrh	ωcrh	ADJ
ejpam-5648	122	7	g	g	PROPN
ejpam-5648	122	8	(	(	PUNCT
ejpam-5648	122	9	g1	g1	PROPN
ejpam-5648	122	10	)	)	PUNCT
ejpam-5648	122	11	=	=	SYM
ejpam-5648	123	1	|v	|v	ADJ
ejpam-5648	123	2	′	′	NOUN
ejpam-5648	123	3	1	1	NUM
ejpam-5648	124	1	|	|	ADV
ejpam-5648	124	2	=	=	SYM
ejpam-5648	124	3	|v1|	|v1|	NOUN
ejpam-5648	124	4	=	=	SYM
ejpam-5648	124	5	n.	n.	PROPN
ejpam-5648	124	6	finally	finally	ADV
ejpam-5648	124	7	,	,	PUNCT
ejpam-5648	124	8	let	let	VERB
ejpam-5648	124	9	s	s	PRON
ejpam-5648	124	10	be	be	AUX
ejpam-5648	124	11	a	a	DET
ejpam-5648	124	12	γch	γch	NOUN
ejpam-5648	124	13	-	-	PUNCT
ejpam-5648	124	14	set	set	NOUN
ejpam-5648	124	15	in	in	ADP
ejpam-5648	124	16	g	g	NOUN
ejpam-5648	124	17	and	and	CCONJ
ejpam-5648	124	18	let	let	VERB
ejpam-5648	124	19	v	v	VERB
ejpam-5648	124	20	′′	′′	PROPN
ejpam-5648	124	21	1	1	NUM
ejpam-5648	124	22	=	=	NOUN
ejpam-5648	124	23	∅	∅	NOUN
ejpam-5648	124	24	,	,	PUNCT
ejpam-5648	124	25	v	v	ADP
ejpam-5648	124	26	′′	′′	PROPN
ejpam-5648	124	27	0	0	NUM
ejpam-5648	125	1	=	=	SYM
ejpam-5648	125	2	v	v	NOUN
ejpam-5648	125	3	(	(	PUNCT
ejpam-5648	125	4	g	g	NOUN
ejpam-5648	125	5	)	)	PUNCT
ejpam-5648	125	6	\	\	PROPN
ejpam-5648	126	1	s	s	PROPN
ejpam-5648	126	2	,	,	PUNCT
ejpam-5648	126	3	and	and	CCONJ
ejpam-5648	126	4	v	v	ADP
ejpam-5648	126	5	′′	′′	PROPN
ejpam-5648	126	6	2	2	NUM
ejpam-5648	126	7	=	=	SYM
ejpam-5648	126	8	s.	s.	PROPN
ejpam-5648	126	9	then	then	ADV
ejpam-5648	126	10	g2	g2	PROPN
ejpam-5648	126	11	=	=	PRON
ejpam-5648	126	12	(	(	PUNCT
ejpam-5648	126	13	v	v	NUM
ejpam-5648	126	14	′′	′′	PROPN
ejpam-5648	126	15	0	0	NUM
ejpam-5648	126	16	,	,	PUNCT
ejpam-5648	126	17	v	v	ADP
ejpam-5648	126	18	′′	′′	PROPN
ejpam-5648	126	19	1	1	NUM
ejpam-5648	126	20	,	,	PUNCT
ejpam-5648	126	21	v	v	ADP
ejpam-5648	126	22	′′	′′	PROPN
ejpam-5648	126	23	2	2	NUM
ejpam-5648	126	24	)	)	PUNCT
ejpam-5648	126	25	∈	∈	PROPN
ejpam-5648	126	26	chrdf	chrdf	NOUN
ejpam-5648	126	27	(	(	PUNCT
ejpam-5648	126	28	g	g	NOUN
ejpam-5648	126	29	)	)	PUNCT
ejpam-5648	126	30	.	.	PUNCT
ejpam-5648	127	1	this	this	PRON
ejpam-5648	127	2	implies	imply	VERB
ejpam-5648	127	3	that	that	SCONJ
ejpam-5648	127	4	γcrh(g	γcrh(g	ADP
ejpam-5648	127	5	)	)	PUNCT
ejpam-5648	127	6	≤	≤	NOUN
ejpam-5648	127	7	ωcrh	ωcrh	ADJ
ejpam-5648	127	8	g	g	PROPN
ejpam-5648	127	9	(	(	PUNCT
ejpam-5648	127	10	g2	g2	PROPN
ejpam-5648	127	11	)	)	PUNCT
ejpam-5648	127	12	=	=	SYM
ejpam-5648	127	13	2|v	2|v	X
ejpam-5648	128	1	′′	′′	NOUN
ejpam-5648	128	2	2	2	NUM
ejpam-5648	128	3	|	|	NOUN
ejpam-5648	128	4	=	=	SYM
ejpam-5648	128	5	2γch(g	2γch(g	NOUN
ejpam-5648	128	6	)	)	PUNCT
ejpam-5648	128	7	.	.	PUNCT
ejpam-5648	129	1	hence	hence	ADV
ejpam-5648	129	2	,	,	PUNCT
ejpam-5648	129	3	γcrh(g	γcrh(g	NOUN
ejpam-5648	129	4	)	)	PUNCT
ejpam-5648	129	5	≤	≤	NUM
ejpam-5648	129	6	min{n	min{n	NOUN
ejpam-5648	129	7	,	,	PUNCT
ejpam-5648	129	8	2γch(g	2γch(g	NUM
ejpam-5648	129	9	)	)	PUNCT
ejpam-5648	129	10	}	}	PUNCT
ejpam-5648	129	11	.	.	PUNCT
ejpam-5648	130	1	therefore	therefore	ADV
ejpam-5648	130	2	,	,	PUNCT
ejpam-5648	130	3	γch(g	γch(g	NOUN
ejpam-5648	130	4	)	)	PUNCT
ejpam-5648	130	5	≤	≤	NUM
ejpam-5648	130	6	γcrh(g	γcrh(g	NOUN
ejpam-5648	130	7	)	)	PUNCT
ejpam-5648	130	8	≤	≤	NUM
ejpam-5648	130	9	min{n	min{n	NOUN
ejpam-5648	130	10	,	,	PUNCT
ejpam-5648	130	11	2γch(g	2γch(g	NUM
ejpam-5648	130	12	)	)	PUNCT
ejpam-5648	130	13	}	}	PUNCT
ejpam-5648	130	14	.	.	PUNCT
ejpam-5648	131	1	it	it	PRON
ejpam-5648	131	2	is	be	AUX
ejpam-5648	131	3	worth	worth	ADJ
ejpam-5648	131	4	noting	note	VERB
ejpam-5648	131	5	that	that	SCONJ
ejpam-5648	131	6	the	the	DET
ejpam-5648	131	7	bounds	bound	NOUN
ejpam-5648	131	8	given	give	VERB
ejpam-5648	131	9	in	in	ADP
ejpam-5648	131	10	proposition	proposition	NOUN
ejpam-5648	131	11	2	2	NUM
ejpam-5648	131	12	are	be	AUX
ejpam-5648	131	13	sharp	sharp	ADJ
ejpam-5648	131	14	.	.	PUNCT
ejpam-5648	132	1	indeed	indeed	ADV
ejpam-5648	132	2	,	,	PUNCT
ejpam-5648	132	3	it	it	PRON
ejpam-5648	132	4	can	can	AUX
ejpam-5648	132	5	be	be	AUX
ejpam-5648	132	6	verified	verify	VERB
ejpam-5648	132	7	easily	easily	ADV
ejpam-5648	132	8	that	that	DET
ejpam-5648	132	9	γcrh(k4	γcrh(k4	NOUN
ejpam-5648	132	10	)	)	PUNCT
ejpam-5648	133	1	=	=	SYM
ejpam-5648	133	2	4	4	NUM
ejpam-5648	133	3	=	=	SYM
ejpam-5648	133	4	γch(k4	γch(k4	PROPN
ejpam-5648	133	5	)	)	PUNCT
ejpam-5648	133	6	,	,	PUNCT
ejpam-5648	134	1	γch(p3	γch(p3	NOUN
ejpam-5648	134	2	)	)	PUNCT
ejpam-5648	135	1	=	=	SYM
ejpam-5648	135	2	2	2	NUM
ejpam-5648	135	3	<	<	SYM
ejpam-5648	135	4	3	3	NUM
ejpam-5648	135	5	=	=	SYM
ejpam-5648	135	6	|v	|v	X
ejpam-5648	135	7	(	(	PUNCT
ejpam-5648	135	8	p3)|	p3)|	NOUN
ejpam-5648	135	9	=	=	SYM
ejpam-5648	135	10	γcrh(p3	γcrh(p3	NOUN
ejpam-5648	135	11	)	)	PUNCT
ejpam-5648	135	12	,	,	PUNCT
ejpam-5648	135	13	and	and	CCONJ
ejpam-5648	135	14	γcrh(p5	γcrh(p5	NOUN
ejpam-5648	135	15	)	)	PUNCT
ejpam-5648	135	16	=	=	SYM
ejpam-5648	135	17	4	4	NUM
ejpam-5648	135	18	=	=	SYM
ejpam-5648	135	19	2γch(p5	2γch(p5	NUM
ejpam-5648	135	20	)	)	PUNCT
ejpam-5648	135	21	<	<	X
ejpam-5648	135	22	|v	|v	PROPN
ejpam-5648	135	23	(	(	PUNCT
ejpam-5648	135	24	p5)|	p5)|	PROPN
ejpam-5648	135	25	.	.	PUNCT
ejpam-5648	135	26	a.	a.	PROPN
ejpam-5648	135	27	aradais	aradais	PROPN
ejpam-5648	135	28	,	,	PUNCT
ejpam-5648	135	29	j.	j.	PROPN
ejpam-5648	135	30	cariaga	cariaga	PROPN
ejpam-5648	135	31	,	,	PUNCT
ejpam-5648	135	32	s.	s.	PROPN
ejpam-5648	135	33	canoy	canoy	PROPN
ejpam-5648	135	34	jr	jr	PROPN
ejpam-5648	135	35	.	.	PROPN
ejpam-5648	135	36	/	/	SYM
ejpam-5648	135	37	eur	eur	PROPN
ejpam-5648	135	38	.	.	PUNCT
ejpam-5648	136	1	j.	j.	PROPN
ejpam-5648	136	2	pure	pure	PROPN
ejpam-5648	136	3	appl	appl	PROPN
ejpam-5648	136	4	.	.	PROPN
ejpam-5648	136	5	math	math	PROPN
ejpam-5648	136	6	,	,	PUNCT
ejpam-5648	136	7	18	18	NUM
ejpam-5648	136	8	(	(	PUNCT
ejpam-5648	136	9	1	1	NUM
ejpam-5648	136	10	)	)	PUNCT
ejpam-5648	136	11	(	(	PUNCT
ejpam-5648	136	12	2025	2025	NUM
ejpam-5648	136	13	)	)	PUNCT
ejpam-5648	136	14	,	,	PUNCT
ejpam-5648	136	15	5648	5648	NUM
ejpam-5648	136	16	5	5	NUM
ejpam-5648	136	17	of	of	ADP
ejpam-5648	136	18	13	13	NUM
ejpam-5648	136	19	remark	remark	NOUN
ejpam-5648	136	20	1	1	NUM
ejpam-5648	136	21	.	.	PUNCT
ejpam-5648	137	1	let	let	VERB
ejpam-5648	137	2	g	g	PRON
ejpam-5648	137	3	be	be	AUX
ejpam-5648	137	4	a	a	DET
ejpam-5648	137	5	connected	connected	ADJ
ejpam-5648	137	6	graph	graph	NOUN
ejpam-5648	137	7	.	.	PUNCT
ejpam-5648	138	1	if	if	SCONJ
ejpam-5648	138	2	f	f	PROPN
ejpam-5648	138	3	=	=	SYM
ejpam-5648	138	4	(	(	PUNCT
ejpam-5648	138	5	v0	v0	PROPN
ejpam-5648	138	6	,	,	PUNCT
ejpam-5648	138	7	v1	v1	NOUN
ejpam-5648	138	8	,	,	PUNCT
ejpam-5648	138	9	v2	v2	PROPN
ejpam-5648	138	10	)	)	PUNCT
ejpam-5648	138	11	is	be	AUX
ejpam-5648	138	12	a	a	DET
ejpam-5648	138	13	γcrh	γcrh	NOUN
ejpam-5648	138	14	-	-	PUNCT
ejpam-5648	138	15	function	function	NOUN
ejpam-5648	138	16	in	in	ADP
ejpam-5648	138	17	g	g	NOUN
ejpam-5648	138	18	,	,	PUNCT
ejpam-5648	138	19	then	then	ADV
ejpam-5648	138	20	v1	v1	VERB
ejpam-5648	138	21	∪	∪	NOUN
ejpam-5648	138	22	v2	v2	NOUN
ejpam-5648	138	23	need	need	AUX
ejpam-5648	138	24	not	not	PART
ejpam-5648	138	25	be	be	AUX
ejpam-5648	138	26	a	a	DET
ejpam-5648	138	27	γch	γch	NOUN
ejpam-5648	138	28	-	-	PUNCT
ejpam-5648	138	29	set	set	VERB
ejpam-5648	138	30	in	in	ADP
ejpam-5648	138	31	g.	g.	PROPN
ejpam-5648	138	32	remark	remark	PROPN
ejpam-5648	138	33	2	2	NUM
ejpam-5648	138	34	.	.	PUNCT
ejpam-5648	139	1	let	let	VERB
ejpam-5648	139	2	g	g	PRON
ejpam-5648	139	3	be	be	AUX
ejpam-5648	139	4	a	a	DET
ejpam-5648	139	5	connected	connected	ADJ
ejpam-5648	139	6	graph	graph	NOUN
ejpam-5648	139	7	of	of	ADP
ejpam-5648	139	8	order	order	NOUN
ejpam-5648	139	9	n	n	NOUN
ejpam-5648	139	10	and	and	CCONJ
ejpam-5648	139	11	f	f	PROPN
ejpam-5648	139	12	=	=	SYM
ejpam-5648	139	13	(	(	PUNCT
ejpam-5648	139	14	v0	v0	PROPN
ejpam-5648	139	15	,	,	PUNCT
ejpam-5648	139	16	v1	v1	NOUN
ejpam-5648	139	17	,	,	PUNCT
ejpam-5648	139	18	v2	v2	PROPN
ejpam-5648	139	19	)	)	PUNCT
ejpam-5648	139	20	a	a	DET
ejpam-5648	139	21	γcrh	γcrh	NOUN
ejpam-5648	139	22	-	-	PUNCT
ejpam-5648	139	23	function	function	NOUN
ejpam-5648	139	24	on	on	ADP
ejpam-5648	139	25	g.	g.	PROPN
ejpam-5648	139	26	if	if	SCONJ
ejpam-5648	139	27	v	v	NUM
ejpam-5648	139	28	∈	∈	PROPN
ejpam-5648	139	29	v	v	NOUN
ejpam-5648	139	30	(	(	PUNCT
ejpam-5648	139	31	g	g	NOUN
ejpam-5648	139	32	)	)	PUNCT
ejpam-5648	139	33	and	and	CCONJ
ejpam-5648	139	34	degg(v	degg(v	PROPN
ejpam-5648	139	35	)	)	PUNCT
ejpam-5648	139	36	=	=	SYM
ejpam-5648	139	37	n	n	CCONJ
ejpam-5648	139	38	−	−	NUM
ejpam-5648	139	39	1	1	NUM
ejpam-5648	139	40	(	(	PUNCT
ejpam-5648	139	41	that	that	ADV
ejpam-5648	139	42	is	is	ADV
ejpam-5648	139	43	,	,	PUNCT
ejpam-5648	139	44	v	v	NOUN
ejpam-5648	139	45	is	be	AUX
ejpam-5648	139	46	a	a	DET
ejpam-5648	139	47	dominating	dominating	NOUN
ejpam-5648	139	48	vertex	vertex	NOUN
ejpam-5648	139	49	in	in	ADP
ejpam-5648	139	50	g	g	PROPN
ejpam-5648	139	51	)	)	PUNCT
ejpam-5648	139	52	,	,	PUNCT
ejpam-5648	139	53	then	then	ADV
ejpam-5648	139	54	v	v	ADP
ejpam-5648	139	55	∈	∈	PROPN
ejpam-5648	139	56	v1	v1	NOUN
ejpam-5648	139	57	.	.	PUNCT
ejpam-5648	140	1	proposition	proposition	NOUN
ejpam-5648	140	2	3	3	X
ejpam-5648	140	3	.	.	PUNCT
ejpam-5648	141	1	let	let	VERB
ejpam-5648	141	2	g	g	PRON
ejpam-5648	141	3	be	be	AUX
ejpam-5648	141	4	a	a	DET
ejpam-5648	141	5	connected	connected	ADJ
ejpam-5648	141	6	graph	graph	NOUN
ejpam-5648	141	7	of	of	ADP
ejpam-5648	141	8	order	order	NOUN
ejpam-5648	141	9	n.	n.	NOUN
ejpam-5648	141	10	then	then	ADV
ejpam-5648	141	11	each	each	DET
ejpam-5648	141	12	the	the	DET
ejpam-5648	141	13	following	follow	VERB
ejpam-5648	141	14	statements	statement	NOUN
ejpam-5648	141	15	holds	hold	VERB
ejpam-5648	141	16	:	:	PUNCT
ejpam-5648	141	17	(	(	PUNCT
ejpam-5648	141	18	i	i	NOUN
ejpam-5648	141	19	)	)	PUNCT
ejpam-5648	141	20	γcrh(g	γcrh(g	PROPN
ejpam-5648	141	21	)	)	PUNCT
ejpam-5648	141	22	=	=	SYM
ejpam-5648	141	23	1	1	NUM
ejpam-5648	142	1	if	if	SCONJ
ejpam-5648	142	2	and	and	CCONJ
ejpam-5648	142	3	only	only	ADV
ejpam-5648	142	4	if	if	SCONJ
ejpam-5648	142	5	g	g	PROPN
ejpam-5648	142	6	=	=	PROPN
ejpam-5648	142	7	k1	k1	PROPN
ejpam-5648	142	8	.	.	PUNCT
ejpam-5648	142	9	(	(	PUNCT
ejpam-5648	142	10	ii	ii	NOUN
ejpam-5648	142	11	)	)	PUNCT
ejpam-5648	142	12	γcrh(g	γcrh(g	PROPN
ejpam-5648	142	13	)	)	PUNCT
ejpam-5648	142	14	=	=	SYM
ejpam-5648	142	15	2	2	NUM
ejpam-5648	142	16	if	if	SCONJ
ejpam-5648	142	17	and	and	CCONJ
ejpam-5648	142	18	only	only	ADV
ejpam-5648	142	19	if	if	SCONJ
ejpam-5648	142	20	g	g	PROPN
ejpam-5648	142	21	=	=	SYM
ejpam-5648	142	22	k2	k2	PROPN
ejpam-5648	142	23	.	.	PUNCT
ejpam-5648	143	1	(	(	PUNCT
ejpam-5648	143	2	iii	iii	X
ejpam-5648	143	3	)	)	PUNCT
ejpam-5648	143	4	γcrh(g	γcrh(g	PROPN
ejpam-5648	143	5	)	)	PUNCT
ejpam-5648	143	6	=	=	SYM
ejpam-5648	143	7	3	3	NUM
ejpam-5648	144	1	if	if	SCONJ
ejpam-5648	144	2	and	and	CCONJ
ejpam-5648	144	3	only	only	ADV
ejpam-5648	144	4	if	if	SCONJ
ejpam-5648	144	5	n	n	NUM
ejpam-5648	144	6	≥	≥	NOUN
ejpam-5648	144	7	3	3	NUM
ejpam-5648	144	8	and	and	CCONJ
ejpam-5648	144	9	g	g	NOUN
ejpam-5648	144	10	=	=	PUNCT
ejpam-5648	144	11	k3	k3	X
ejpam-5648	144	12	or	or	CCONJ
ejpam-5648	144	13	g	g	NOUN
ejpam-5648	144	14	=	=	PROPN
ejpam-5648	144	15	k1	k1	PROPN
ejpam-5648	145	1	+	+	CCONJ
ejpam-5648	145	2	h	h	NOUN
ejpam-5648	145	3	,	,	PUNCT
ejpam-5648	145	4	where	where	SCONJ
ejpam-5648	145	5	h	h	NOUN
ejpam-5648	145	6	is	be	AUX
ejpam-5648	145	7	a	a	DET
ejpam-5648	145	8	(	(	PUNCT
ejpam-5648	145	9	disconnected	disconnected	ADJ
ejpam-5648	145	10	)	)	PUNCT
ejpam-5648	145	11	graph	graph	NOUN
ejpam-5648	145	12	of	of	ADP
ejpam-5648	145	13	order	order	NOUN
ejpam-5648	145	14	n−	n−	NOUN
ejpam-5648	145	15	1	1	NUM
ejpam-5648	145	16	and	and	CCONJ
ejpam-5648	145	17	has	have	VERB
ejpam-5648	145	18	at	at	ADV
ejpam-5648	145	19	least	least	ADJ
ejpam-5648	145	20	one	one	NUM
ejpam-5648	145	21	isolated	isolated	ADJ
ejpam-5648	145	22	vertex	vertex	NOUN
ejpam-5648	145	23	.	.	PUNCT
ejpam-5648	146	1	(	(	PUNCT
ejpam-5648	146	2	iv	iv	X
ejpam-5648	146	3	)	)	PUNCT
ejpam-5648	146	4	γcrh(g	γcrh(g	PROPN
ejpam-5648	146	5	)	)	PUNCT
ejpam-5648	146	6	=	=	SYM
ejpam-5648	146	7	4	4	NUM
ejpam-5648	147	1	if	if	SCONJ
ejpam-5648	147	2	and	and	CCONJ
ejpam-5648	147	3	only	only	ADV
ejpam-5648	147	4	if	if	SCONJ
ejpam-5648	147	5	n	n	NUM
ejpam-5648	147	6	≥	≥	NOUN
ejpam-5648	147	7	4	4	NUM
ejpam-5648	147	8	and	and	CCONJ
ejpam-5648	147	9	g	g	PROPN
ejpam-5648	147	10	∈	∈	PROPN
ejpam-5648	147	11	{	{	PUNCT
ejpam-5648	147	12	k4	k4	NOUN
ejpam-5648	147	13	,	,	PUNCT
ejpam-5648	147	14	p4	p4	ADJ
ejpam-5648	147	15	,	,	PUNCT
ejpam-5648	147	16	c4,k1	c4,k1	PROPN
ejpam-5648	147	17	+	+	CCONJ
ejpam-5648	147	18	p3	p3	PROPN
ejpam-5648	147	19	}	}	PUNCT
ejpam-5648	147	20	or	or	CCONJ
ejpam-5648	147	21	one	one	NUM
ejpam-5648	147	22	of	of	ADP
ejpam-5648	147	23	the	the	DET
ejpam-5648	147	24	following	follow	VERB
ejpam-5648	147	25	conditions	condition	NOUN
ejpam-5648	147	26	holds	hold	VERB
ejpam-5648	147	27	:	:	PUNCT
ejpam-5648	147	28	(	(	PUNCT
ejpam-5648	147	29	a	a	X
ejpam-5648	147	30	)	)	PUNCT
ejpam-5648	147	31	there	there	PRON
ejpam-5648	147	32	exists	exist	VERB
ejpam-5648	147	33	w	w	PROPN
ejpam-5648	147	34	∈	∈	PROPN
ejpam-5648	147	35	v	v	ADP
ejpam-5648	147	36	(	(	PUNCT
ejpam-5648	147	37	g	g	NOUN
ejpam-5648	147	38	)	)	PUNCT
ejpam-5648	147	39	such	such	ADJ
ejpam-5648	147	40	that	that	SCONJ
ejpam-5648	147	41	|ng(w)|	|ng(w)|	NOUN
ejpam-5648	147	42	=	=	SYM
ejpam-5648	147	43	2	2	NUM
ejpam-5648	147	44	,	,	PUNCT
ejpam-5648	147	45	n2	n2	NOUN
ejpam-5648	147	46	g(w	g(w	PROPN
ejpam-5648	147	47	)	)	PUNCT
ejpam-5648	147	48	=	=	SYM
ejpam-5648	147	49	v	v	X
ejpam-5648	147	50	(	(	PUNCT
ejpam-5648	147	51	g	g	NOUN
ejpam-5648	147	52	)	)	PUNCT
ejpam-5648	147	53	\ng[w	\ng[w	PROPN
ejpam-5648	147	54	]	]	PUNCT
ejpam-5648	147	55	,	,	PUNCT
ejpam-5648	147	56	and	and	CCONJ
ejpam-5648	147	57	x	x	X
ejpam-5648	147	58	is	be	AUX
ejpam-5648	147	59	not	not	PART
ejpam-5648	147	60	a	a	DET
ejpam-5648	147	61	support	support	NOUN
ejpam-5648	147	62	vertex	vertex	NOUN
ejpam-5648	147	63	whenever	whenever	SCONJ
ejpam-5648	147	64	x	x	SYM
ejpam-5648	147	65	∈	∈	NOUN
ejpam-5648	147	66	ng(w	ng(w	NOUN
ejpam-5648	147	67	)	)	PUNCT
ejpam-5648	147	68	and	and	CCONJ
ejpam-5648	147	69	is	be	AUX
ejpam-5648	147	70	a	a	DET
ejpam-5648	147	71	dominating	dominating	NOUN
ejpam-5648	147	72	vertex	vertex	NOUN
ejpam-5648	147	73	of	of	ADP
ejpam-5648	147	74	g.	g.	PROPN
ejpam-5648	147	75	(	(	PUNCT
ejpam-5648	147	76	b	b	X
ejpam-5648	147	77	)	)	PUNCT
ejpam-5648	147	78	there	there	PRON
ejpam-5648	147	79	exist	exist	VERB
ejpam-5648	147	80	adjacent	adjacent	ADJ
ejpam-5648	147	81	vertices	vertex	NOUN
ejpam-5648	147	82	u	u	NOUN
ejpam-5648	147	83	,	,	PUNCT
ejpam-5648	147	84	v	v	NOUN
ejpam-5648	147	85	∈	∈	PROPN
ejpam-5648	147	86	v	v	NOUN
ejpam-5648	147	87	(	(	PUNCT
ejpam-5648	147	88	g	g	NOUN
ejpam-5648	147	89	)	)	PUNCT
ejpam-5648	147	90	such	such	ADJ
ejpam-5648	147	91	that	that	DET
ejpam-5648	147	92	|ng(u)|	|ng(u)|	PROPN
ejpam-5648	147	93	≥	≥	NUM
ejpam-5648	147	94	2	2	NUM
ejpam-5648	147	95	,	,	PUNCT
ejpam-5648	147	96	|ng(v)|	|ng(v)|	NOUN
ejpam-5648	147	97	≥	≥	NOUN
ejpam-5648	147	98	2	2	NUM
ejpam-5648	147	99	,	,	PUNCT
ejpam-5648	147	100	n2	n2	ADJ
ejpam-5648	147	101	g(u	g(u	PROPN
ejpam-5648	147	102	)	)	PUNCT
ejpam-5648	147	103	∪	∪	ADP
ejpam-5648	147	104	n2	n2	ADJ
ejpam-5648	147	105	g(v	g(v	X
ejpam-5648	147	106	)	)	PUNCT
ejpam-5648	148	1	=	=	SYM
ejpam-5648	148	2	v	v	X
ejpam-5648	148	3	(	(	PUNCT
ejpam-5648	148	4	g	g	NOUN
ejpam-5648	148	5	)	)	PUNCT
ejpam-5648	148	6	\	\	NOUN
ejpam-5648	149	1	{	{	PUNCT
ejpam-5648	149	2	u	u	NOUN
ejpam-5648	149	3	,	,	PUNCT
ejpam-5648	149	4	v	v	NOUN
ejpam-5648	149	5	}	}	PUNCT
ejpam-5648	149	6	,	,	PUNCT
ejpam-5648	149	7	ng(u	ng(u	NOUN
ejpam-5648	149	8	)	)	PUNCT
ejpam-5648	149	9	∩	∩	NOUN
ejpam-5648	149	10	ng(v	ng(v	X
ejpam-5648	149	11	)	)	PUNCT
ejpam-5648	149	12	=	=	SYM
ejpam-5648	149	13	∅	∅	NOUN
ejpam-5648	149	14	,	,	PUNCT
ejpam-5648	149	15	and	and	CCONJ
ejpam-5648	149	16	{	{	PUNCT
ejpam-5648	149	17	u	u	NOUN
ejpam-5648	149	18	,	,	PUNCT
ejpam-5648	149	19	v	v	NOUN
ejpam-5648	149	20	}	}	PUNCT
ejpam-5648	149	21	is	be	AUX
ejpam-5648	149	22	a	a	DET
ejpam-5648	149	23	γch	γch	NOUN
ejpam-5648	149	24	-	-	PUNCT
ejpam-5648	149	25	set	set	VERB
ejpam-5648	149	26	in	in	ADP
ejpam-5648	149	27	g.	g.	PROPN
ejpam-5648	149	28	proof	proof	NOUN
ejpam-5648	149	29	.	.	PUNCT
ejpam-5648	150	1	(	(	PUNCT
ejpam-5648	150	2	i	i	NOUN
ejpam-5648	150	3	)	)	PUNCT
ejpam-5648	150	4	assume	assume	VERB
ejpam-5648	150	5	that	that	SCONJ
ejpam-5648	150	6	γcrh(g	γcrh(g	ADP
ejpam-5648	150	7	)	)	PUNCT
ejpam-5648	150	8	=	=	SYM
ejpam-5648	150	9	1	1	NUM
ejpam-5648	150	10	and	and	CCONJ
ejpam-5648	150	11	f	f	NOUN
ejpam-5648	150	12	=	=	SYM
ejpam-5648	150	13	(	(	PUNCT
ejpam-5648	150	14	v0	v0	PROPN
ejpam-5648	150	15	,	,	PUNCT
ejpam-5648	150	16	v1	v1	NOUN
ejpam-5648	150	17	,	,	PUNCT
ejpam-5648	150	18	v2	v2	PROPN
ejpam-5648	150	19	)	)	PUNCT
ejpam-5648	150	20	is	be	AUX
ejpam-5648	150	21	a	a	DET
ejpam-5648	150	22	γcrh	γcrh	NOUN
ejpam-5648	150	23	-	-	PUNCT
ejpam-5648	150	24	function	function	NOUN
ejpam-5648	150	25	on	on	ADP
ejpam-5648	150	26	g.	g.	PROPN
ejpam-5648	150	27	then	then	ADV
ejpam-5648	150	28	|v1|	|v1|	VERB
ejpam-5648	150	29	+	+	CCONJ
ejpam-5648	150	30	2|v2|	2|v2|	NUM
ejpam-5648	150	31	=	=	SYM
ejpam-5648	150	32	1	1	X
ejpam-5648	150	33	.	.	PUNCT
ejpam-5648	151	1	this	this	PRON
ejpam-5648	151	2	implies	imply	VERB
ejpam-5648	151	3	that	that	SCONJ
ejpam-5648	151	4	v2	v2	NOUN
ejpam-5648	151	5	=	=	PUNCT
ejpam-5648	151	6	∅.	∅.	NOUN
ejpam-5648	151	7	by	by	ADP
ejpam-5648	151	8	proposition	proposition	NOUN
ejpam-5648	151	9	1(i	1(i	NUM
ejpam-5648	151	10	)	)	PUNCT
ejpam-5648	151	11	,	,	PUNCT
ejpam-5648	151	12	v0	v0	NOUN
ejpam-5648	151	13	=	=	PUNCT
ejpam-5648	151	14	∅.	∅.	NOUN
ejpam-5648	151	15	thus	thus	ADV
ejpam-5648	151	16	,	,	PUNCT
ejpam-5648	151	17	|v	|v	PROPN
ejpam-5648	151	18	(	(	PUNCT
ejpam-5648	151	19	g)|	g)|	NOUN
ejpam-5648	151	20	=	=	PUNCT
ejpam-5648	151	21	|v1|	|v1|	NOUN
ejpam-5648	151	22	=	=	SYM
ejpam-5648	151	23	1	1	X
ejpam-5648	151	24	.	.	PUNCT
ejpam-5648	152	1	hence	hence	ADV
ejpam-5648	152	2	,	,	PUNCT
ejpam-5648	152	3	g	g	PROPN
ejpam-5648	152	4	=	=	PROPN
ejpam-5648	152	5	k1	k1	PROPN
ejpam-5648	152	6	.	.	PUNCT
ejpam-5648	153	1	the	the	DET
ejpam-5648	153	2	converse	converse	NOUN
ejpam-5648	153	3	is	be	AUX
ejpam-5648	153	4	clear	clear	ADJ
ejpam-5648	153	5	.	.	PUNCT
ejpam-5648	154	1	(	(	PUNCT
ejpam-5648	154	2	ii	ii	NOUN
ejpam-5648	154	3	)	)	PUNCT
ejpam-5648	154	4	suppose	suppose	VERB
ejpam-5648	154	5	γcrh(g	γcrh(g	NOUN
ejpam-5648	154	6	)	)	PUNCT
ejpam-5648	154	7	=	=	SYM
ejpam-5648	154	8	2	2	X
ejpam-5648	154	9	.	.	X
ejpam-5648	154	10	let	let	VERB
ejpam-5648	154	11	f	f	PROPN
ejpam-5648	154	12	=	=	SYM
ejpam-5648	154	13	(	(	PUNCT
ejpam-5648	154	14	v0	v0	PROPN
ejpam-5648	154	15	,	,	PUNCT
ejpam-5648	154	16	v1	v1	NOUN
ejpam-5648	154	17	,	,	PUNCT
ejpam-5648	154	18	v2	v2	PROPN
ejpam-5648	154	19	)	)	PUNCT
ejpam-5648	154	20	be	be	AUX
ejpam-5648	154	21	a	a	DET
ejpam-5648	154	22	γcrh	γcrh	NOUN
ejpam-5648	154	23	-	-	PUNCT
ejpam-5648	154	24	function	function	NOUN
ejpam-5648	154	25	in	in	ADP
ejpam-5648	154	26	g.	g.	PROPN
ejpam-5648	154	27	then	then	ADV
ejpam-5648	154	28	|v1|	|v1|	VERB
ejpam-5648	154	29	+	+	CCONJ
ejpam-5648	154	30	2|v2|	2|v2|	NUM
ejpam-5648	154	31	=	=	SYM
ejpam-5648	154	32	2	2	NUM
ejpam-5648	154	33	.	.	PUNCT
ejpam-5648	154	34	thus	thus	ADV
ejpam-5648	154	35	,	,	PUNCT
ejpam-5648	154	36	|v2|	|v2|	ADV
ejpam-5648	154	37	≤	≤	NOUN
ejpam-5648	154	38	1	1	NUM
ejpam-5648	154	39	.	.	PUNCT
ejpam-5648	154	40	assume	assume	VERB
ejpam-5648	154	41	that	that	SCONJ
ejpam-5648	154	42	|v2|	|v2|	NOUN
ejpam-5648	154	43	=	=	SYM
ejpam-5648	155	1	1	1	X
ejpam-5648	155	2	.	.	PUNCT
ejpam-5648	155	3	then	then	ADV
ejpam-5648	155	4	v1	v1	VERB
ejpam-5648	155	5	=	=	SYM
ejpam-5648	155	6	∅	∅	NOUN
ejpam-5648	155	7	and	and	CCONJ
ejpam-5648	155	8	,	,	PUNCT
ejpam-5648	155	9	by	by	ADP
ejpam-5648	155	10	proposition	proposition	NOUN
ejpam-5648	155	11	1(i	1(i	NUM
ejpam-5648	155	12	)	)	PUNCT
ejpam-5648	155	13	,	,	PUNCT
ejpam-5648	155	14	v0	v0	NOUN
ejpam-5648	155	15	̸=	̸=	PROPN
ejpam-5648	155	16	∅.	∅.	ADV
ejpam-5648	155	17	let	let	VERB
ejpam-5648	155	18	v2	v2	NOUN
ejpam-5648	155	19	=	=	SYM
ejpam-5648	155	20	{	{	PUNCT
ejpam-5648	155	21	x	x	NOUN
ejpam-5648	155	22	}	}	PUNCT
ejpam-5648	155	23	and	and	CCONJ
ejpam-5648	155	24	let	let	VERB
ejpam-5648	155	25	y	y	PROPN
ejpam-5648	155	26	∈	∈	PROPN
ejpam-5648	155	27	v0	v0	PROPN
ejpam-5648	155	28	.	.	PUNCT
ejpam-5648	156	1	then	then	ADV
ejpam-5648	156	2	dg(x	dg(x	NUM
ejpam-5648	156	3	,	,	PUNCT
ejpam-5648	156	4	y	y	NOUN
ejpam-5648	156	5	)	)	PUNCT
ejpam-5648	156	6	=	=	SYM
ejpam-5648	156	7	2	2	X
ejpam-5648	156	8	.	.	X
ejpam-5648	156	9	let	let	VERB
ejpam-5648	156	10	z	z	PROPN
ejpam-5648	156	11	∈	∈	PROPN
ejpam-5648	156	12	ng(x	ng(x	NUM
ejpam-5648	156	13	)	)	PUNCT
ejpam-5648	156	14	∩ng(y	∩ng(y	PROPN
ejpam-5648	156	15	)	)	PUNCT
ejpam-5648	156	16	.	.	PUNCT
ejpam-5648	157	1	then	then	ADV
ejpam-5648	157	2	z	z	PROPN
ejpam-5648	157	3	∈	∈	PROPN
ejpam-5648	157	4	v1	v1	NOUN
ejpam-5648	157	5	,	,	PUNCT
ejpam-5648	157	6	a	a	DET
ejpam-5648	157	7	contradiction	contradiction	NOUN
ejpam-5648	157	8	.	.	PUNCT
ejpam-5648	158	1	therefore	therefore	ADV
ejpam-5648	158	2	,	,	PUNCT
ejpam-5648	158	3	|v2|	|v2|	NOUN
ejpam-5648	158	4	=	=	SYM
ejpam-5648	158	5	0	0	X
ejpam-5648	158	6	.	.	PUNCT
ejpam-5648	159	1	it	it	PRON
ejpam-5648	159	2	follows	follow	VERB
ejpam-5648	159	3	that	that	SCONJ
ejpam-5648	159	4	|v1|	|v1|	NOUN
ejpam-5648	159	5	=	=	SYM
ejpam-5648	159	6	2	2	NUM
ejpam-5648	159	7	,	,	PUNCT
ejpam-5648	159	8	i.e.	i.e.	X
ejpam-5648	159	9	,	,	PUNCT
ejpam-5648	159	10	g	g	PROPN
ejpam-5648	159	11	=	=	SYM
ejpam-5648	159	12	k2	k2	PROPN
ejpam-5648	159	13	.	.	PUNCT
ejpam-5648	160	1	conversely	conversely	ADV
ejpam-5648	160	2	,	,	PUNCT
ejpam-5648	160	3	suppose	suppose	VERB
ejpam-5648	160	4	g	g	PROPN
ejpam-5648	160	5	=	=	PROPN
ejpam-5648	160	6	k2	k2	PROPN
ejpam-5648	160	7	=	=	PROPN
ejpam-5648	160	8	⟨{u	⟨{u	PROPN
ejpam-5648	160	9	,	,	PUNCT
ejpam-5648	160	10	v}⟩.	v}⟩.	NOUN
ejpam-5648	160	11	let	let	VERB
ejpam-5648	160	12	v1	v1	NOUN
ejpam-5648	160	13	=	=	SYM
ejpam-5648	160	14	{	{	PUNCT
ejpam-5648	160	15	u	u	NOUN
ejpam-5648	160	16	,	,	PUNCT
ejpam-5648	160	17	v	v	NOUN
ejpam-5648	160	18	}	}	PUNCT
ejpam-5648	160	19	and	and	CCONJ
ejpam-5648	160	20	v0	v0	NOUN
ejpam-5648	160	21	=	=	SYM
ejpam-5648	160	22	v2	v2	PROPN
ejpam-5648	160	23	=	=	PUNCT
ejpam-5648	160	24	∅.	∅.	NOUN
ejpam-5648	160	25	then	then	ADV
ejpam-5648	160	26	f	f	PROPN
ejpam-5648	160	27	=	=	SYM
ejpam-5648	160	28	(	(	PUNCT
ejpam-5648	160	29	v0	v0	PROPN
ejpam-5648	160	30	,	,	PUNCT
ejpam-5648	160	31	v1	v1	NOUN
ejpam-5648	160	32	,	,	PUNCT
ejpam-5648	160	33	v2	v2	NOUN
ejpam-5648	160	34	)	)	PUNCT
ejpam-5648	160	35	∈	∈	PROPN
ejpam-5648	160	36	chrdf	chrdf	NOUN
ejpam-5648	160	37	(	(	PUNCT
ejpam-5648	160	38	g	g	NOUN
ejpam-5648	160	39	)	)	PUNCT
ejpam-5648	160	40	.	.	PUNCT
ejpam-5648	161	1	thus	thus	ADV
ejpam-5648	161	2	,	,	PUNCT
ejpam-5648	161	3	γcrh(g	γcrh(g	NOUN
ejpam-5648	161	4	)	)	PUNCT
ejpam-5648	161	5	≤	≤	NOUN
ejpam-5648	161	6	ωcrh	ωcrh	ADV
ejpam-5648	161	7	g	g	PROPN
ejpam-5648	161	8	(	(	PUNCT
ejpam-5648	161	9	f	f	X
ejpam-5648	161	10	)	)	PUNCT
ejpam-5648	161	11	=	=	NOUN
ejpam-5648	161	12	|v1|	|v1|	NOUN
ejpam-5648	161	13	=	=	SYM
ejpam-5648	161	14	2	2	X
ejpam-5648	161	15	.	.	PUNCT
ejpam-5648	162	1	since	since	SCONJ
ejpam-5648	162	2	g	g	PROPN
ejpam-5648	162	3	̸=	̸=	PROPN
ejpam-5648	162	4	k1	k1	NOUN
ejpam-5648	162	5	,	,	PUNCT
ejpam-5648	162	6	γcrh(g	γcrh(g	NOUN
ejpam-5648	162	7	)	)	PUNCT
ejpam-5648	162	8	≥	≥	NOUN
ejpam-5648	162	9	2	2	NUM
ejpam-5648	162	10	.	.	PUNCT
ejpam-5648	162	11	hence	hence	ADV
ejpam-5648	162	12	,	,	PUNCT
ejpam-5648	162	13	γcrh(g	γcrh(g	NOUN
ejpam-5648	162	14	)	)	PUNCT
ejpam-5648	162	15	=	=	SYM
ejpam-5648	163	1	2	2	X
ejpam-5648	163	2	.	.	PUNCT
ejpam-5648	163	3	(	(	PUNCT
ejpam-5648	163	4	iii	iii	NOUN
ejpam-5648	163	5	)	)	PUNCT
ejpam-5648	163	6	suppose	suppose	VERB
ejpam-5648	163	7	γcrh(g	γcrh(g	NOUN
ejpam-5648	163	8	)	)	PUNCT
ejpam-5648	163	9	=	=	SYM
ejpam-5648	163	10	3	3	X
ejpam-5648	163	11	.	.	PUNCT
ejpam-5648	163	12	then	then	ADV
ejpam-5648	163	13	n	n	NUM
ejpam-5648	163	14	≥	≥	NOUN
ejpam-5648	163	15	3	3	NUM
ejpam-5648	163	16	by	by	ADP
ejpam-5648	163	17	(	(	PUNCT
ejpam-5648	163	18	i	i	NOUN
ejpam-5648	163	19	)	)	PUNCT
ejpam-5648	163	20	and	and	CCONJ
ejpam-5648	163	21	(	(	PUNCT
ejpam-5648	163	22	ii	ii	NOUN
ejpam-5648	163	23	)	)	PUNCT
ejpam-5648	163	24	.	.	PUNCT
ejpam-5648	164	1	let	let	VERB
ejpam-5648	164	2	f	f	PROPN
ejpam-5648	164	3	=	=	SYM
ejpam-5648	164	4	(	(	PUNCT
ejpam-5648	164	5	v0	v0	PROPN
ejpam-5648	164	6	,	,	PUNCT
ejpam-5648	164	7	v1	v1	NOUN
ejpam-5648	164	8	,	,	PUNCT
ejpam-5648	164	9	v2	v2	PROPN
ejpam-5648	164	10	)	)	PUNCT
ejpam-5648	164	11	be	be	AUX
ejpam-5648	164	12	a	a	DET
ejpam-5648	164	13	γch	γch	NOUN
ejpam-5648	164	14	-	-	PUNCT
ejpam-5648	164	15	function	function	NOUN
ejpam-5648	164	16	on	on	ADP
ejpam-5648	164	17	g.	g.	PROPN
ejpam-5648	164	18	then	then	ADV
ejpam-5648	164	19	|v1|+	|v1|+	ADV
ejpam-5648	164	20	2|v2|	2|v2|	NUM
ejpam-5648	164	21	=	=	SYM
ejpam-5648	164	22	3	3	X
ejpam-5648	164	23	.	.	PUNCT
ejpam-5648	164	24	thus	thus	ADV
ejpam-5648	164	25	,	,	PUNCT
ejpam-5648	164	26	|v2|	|v2|	ADV
ejpam-5648	164	27	≤	≤	NOUN
ejpam-5648	164	28	1	1	NUM
ejpam-5648	164	29	.	.	PUNCT
ejpam-5648	164	30	consider	consider	VERB
ejpam-5648	164	31	the	the	DET
ejpam-5648	164	32	following	follow	VERB
ejpam-5648	164	33	cases	case	NOUN
ejpam-5648	164	34	:	:	PUNCT
ejpam-5648	164	35	case	case	NOUN
ejpam-5648	164	36	1	1	NUM
ejpam-5648	164	37	.	.	NOUN
ejpam-5648	164	38	|v2|	|v2|	NOUN
ejpam-5648	164	39	=	=	SYM
ejpam-5648	165	1	0	0	X
ejpam-5648	165	2	.	.	PUNCT
ejpam-5648	165	3	then	then	ADV
ejpam-5648	165	4	|v0|	|v0|	NOUN
ejpam-5648	165	5	=	=	SYM
ejpam-5648	165	6	0	0	NUM
ejpam-5648	165	7	and	and	CCONJ
ejpam-5648	165	8	|v1|	|v1|	NOUN
ejpam-5648	165	9	=	=	SYM
ejpam-5648	165	10	|v	|v	PROPN
ejpam-5648	165	11	(	(	PUNCT
ejpam-5648	165	12	g)|	g)|	NOUN
ejpam-5648	165	13	=	=	SYM
ejpam-5648	165	14	3	3	X
ejpam-5648	165	15	.	.	PUNCT
ejpam-5648	166	1	this	this	PRON
ejpam-5648	166	2	implies	imply	VERB
ejpam-5648	166	3	that	that	SCONJ
ejpam-5648	166	4	g	g	PROPN
ejpam-5648	166	5	∈	∈	PROPN
ejpam-5648	166	6	{	{	PUNCT
ejpam-5648	166	7	k3	k3	PROPN
ejpam-5648	166	8	,	,	PUNCT
ejpam-5648	166	9	p3	p3	PROPN
ejpam-5648	166	10	}	}	PUNCT
ejpam-5648	166	11	.	.	PUNCT
ejpam-5648	167	1	case	case	NOUN
ejpam-5648	167	2	2	2	NUM
ejpam-5648	167	3	.	.	NOUN
ejpam-5648	167	4	|v2|	|v2|	NOUN
ejpam-5648	167	5	=	=	NOUN
ejpam-5648	168	1	1	1	X
ejpam-5648	168	2	.	.	PUNCT
ejpam-5648	168	3	then	then	ADV
ejpam-5648	168	4	|v0|	|v0|	VERB
ejpam-5648	168	5	=	=	NOUN
ejpam-5648	168	6	̸	̸	ADJ
ejpam-5648	168	7	∅	∅	NOUN
ejpam-5648	168	8	and	and	CCONJ
ejpam-5648	168	9	|v1|	|v1|	NOUN
ejpam-5648	168	10	=	=	SYM
ejpam-5648	168	11	1	1	X
ejpam-5648	168	12	.	.	PUNCT
ejpam-5648	168	13	let	let	VERB
ejpam-5648	168	14	v1	v1	VERB
ejpam-5648	168	15	=	=	SYM
ejpam-5648	168	16	{	{	PUNCT
ejpam-5648	168	17	v	v	NOUN
ejpam-5648	168	18	}	}	PUNCT
ejpam-5648	168	19	and	and	CCONJ
ejpam-5648	168	20	v2	v2	NOUN
ejpam-5648	168	21	=	=	SYM
ejpam-5648	168	22	{	{	PUNCT
ejpam-5648	168	23	w	w	NOUN
ejpam-5648	168	24	}	}	PUNCT
ejpam-5648	168	25	.	.	PUNCT
ejpam-5648	169	1	since	since	SCONJ
ejpam-5648	169	2	⟨v1	⟨v1	PROPN
ejpam-5648	169	3	∪	∪	ADP
ejpam-5648	169	4	v2⟩	v2⟩	PROPN
ejpam-5648	169	5	is	be	AUX
ejpam-5648	169	6	connected	connect	VERB
ejpam-5648	169	7	,	,	PUNCT
ejpam-5648	169	8	uw	uw	PROPN
ejpam-5648	169	9	∈	∈	PROPN
ejpam-5648	169	10	e(g	e(g	PROPN
ejpam-5648	169	11	)	)	PUNCT
ejpam-5648	169	12	.	.	PUNCT
ejpam-5648	170	1	also	also	ADV
ejpam-5648	170	2	,	,	PUNCT
ejpam-5648	170	3	v0	v0	PROPN
ejpam-5648	170	4	=	=	SYM
ejpam-5648	170	5	v	v	PROPN
ejpam-5648	170	6	(	(	PUNCT
ejpam-5648	170	7	g)\{v	g)\{v	PROPN
ejpam-5648	170	8	,	,	PUNCT
ejpam-5648	170	9	w	w	NOUN
ejpam-5648	170	10	}	}	PUNCT
ejpam-5648	170	11	.	.	PUNCT
ejpam-5648	171	1	now	now	ADV
ejpam-5648	171	2	,	,	PUNCT
ejpam-5648	171	3	let	let	VERB
ejpam-5648	171	4	u	u	PRON
ejpam-5648	171	5	∈	∈	PROPN
ejpam-5648	171	6	v0	v0	NOUN
ejpam-5648	171	7	.	.	PUNCT
ejpam-5648	172	1	by	by	ADP
ejpam-5648	172	2	(	(	PUNCT
ejpam-5648	172	3	p1	p1	PROPN
ejpam-5648	172	4	)	)	PUNCT
ejpam-5648	172	5	,	,	PUNCT
ejpam-5648	172	6	u	u	PROPN
ejpam-5648	172	7	∈	∈	PROPN
ejpam-5648	172	8	n2	n2	NOUN
ejpam-5648	172	9	g(w	g(w	PROPN
ejpam-5648	172	10	)	)	PUNCT
ejpam-5648	172	11	.	.	PUNCT
ejpam-5648	173	1	let	let	VERB
ejpam-5648	173	2	[	[	X
ejpam-5648	173	3	u	u	NOUN
ejpam-5648	173	4	,	,	PUNCT
ejpam-5648	173	5	y	y	PROPN
ejpam-5648	173	6	,	,	PUNCT
ejpam-5648	173	7	w	w	PROPN
ejpam-5648	173	8	]	]	X
ejpam-5648	173	9	a.	a.	PROPN
ejpam-5648	173	10	aradais	aradais	PROPN
ejpam-5648	173	11	,	,	PUNCT
ejpam-5648	173	12	j.	j.	PROPN
ejpam-5648	173	13	cariaga	cariaga	PROPN
ejpam-5648	173	14	,	,	PUNCT
ejpam-5648	173	15	s.	s.	PROPN
ejpam-5648	173	16	canoy	canoy	PROPN
ejpam-5648	173	17	jr	jr	PROPN
ejpam-5648	173	18	.	.	PROPN
ejpam-5648	173	19	/	/	SYM
ejpam-5648	173	20	eur	eur	PROPN
ejpam-5648	173	21	.	.	PUNCT
ejpam-5648	174	1	j.	j.	PROPN
ejpam-5648	174	2	pure	pure	PROPN
ejpam-5648	174	3	appl	appl	PROPN
ejpam-5648	174	4	.	.	PROPN
ejpam-5648	174	5	math	math	PROPN
ejpam-5648	174	6	,	,	PUNCT
ejpam-5648	174	7	18	18	NUM
ejpam-5648	174	8	(	(	PUNCT
ejpam-5648	174	9	1	1	NUM
ejpam-5648	174	10	)	)	PUNCT
ejpam-5648	174	11	(	(	PUNCT
ejpam-5648	174	12	2025	2025	NUM
ejpam-5648	174	13	)	)	PUNCT
ejpam-5648	174	14	,	,	PUNCT
ejpam-5648	174	15	5648	5648	NUM
ejpam-5648	174	16	6	6	NUM
ejpam-5648	174	17	of	of	ADP
ejpam-5648	174	18	13	13	NUM
ejpam-5648	174	19	be	be	AUX
ejpam-5648	174	20	a	a	DET
ejpam-5648	174	21	u−	u−	PROPN
ejpam-5648	174	22	w	w	NOUN
ejpam-5648	174	23	geodesic	geodesic	NOUN
ejpam-5648	174	24	.	.	PUNCT
ejpam-5648	175	1	since	since	SCONJ
ejpam-5648	175	2	yw	yw	PROPN
ejpam-5648	175	3	∈	∈	PROPN
ejpam-5648	175	4	e(g	e(g	PROPN
ejpam-5648	175	5	)	)	PUNCT
ejpam-5648	175	6	,	,	PUNCT
ejpam-5648	175	7	y	y	PROPN
ejpam-5648	175	8	/∈	/∈	PUNCT
ejpam-5648	175	9	v0	v0	PROPN
ejpam-5648	175	10	.	.	PUNCT
ejpam-5648	176	1	hence	hence	ADV
ejpam-5648	176	2	,	,	PUNCT
ejpam-5648	176	3	y	y	PROPN
ejpam-5648	176	4	∈	∈	PROPN
ejpam-5648	176	5	v1	v1	NOUN
ejpam-5648	176	6	,	,	PUNCT
ejpam-5648	176	7	implying	imply	VERB
ejpam-5648	176	8	that	that	SCONJ
ejpam-5648	176	9	y	y	PROPN
ejpam-5648	176	10	=	=	PUNCT
ejpam-5648	177	1	v.	v.	CCONJ
ejpam-5648	177	2	let	let	VERB
ejpam-5648	177	3	h1	h1	VERB
ejpam-5648	177	4	=	=	NOUN
ejpam-5648	177	5	⟨v0⟩	⟨v0⟩	NOUN
ejpam-5648	177	6	and	and	CCONJ
ejpam-5648	177	7	h	h	NOUN
ejpam-5648	177	8	=	=	SYM
ejpam-5648	177	9	{	{	PUNCT
ejpam-5648	177	10	w	w	NOUN
ejpam-5648	177	11	}	}	PUNCT
ejpam-5648	177	12	∪h1	∪h1	NOUN
ejpam-5648	177	13	.	.	PUNCT
ejpam-5648	178	1	then	then	ADV
ejpam-5648	178	2	g	g	PROPN
ejpam-5648	178	3	=	=	PUNCT
ejpam-5648	178	4	{	{	PUNCT
ejpam-5648	178	5	v	v	NOUN
ejpam-5648	178	6	}	}	PUNCT
ejpam-5648	178	7	+	+	NOUN
ejpam-5648	178	8	h	h	NOUN
ejpam-5648	178	9	or	or	CCONJ
ejpam-5648	178	10	,	,	PUNCT
ejpam-5648	178	11	equivalently	equivalently	ADV
ejpam-5648	178	12	,	,	PUNCT
ejpam-5648	178	13	g	g	PROPN
ejpam-5648	178	14	=	=	PROPN
ejpam-5648	178	15	k1	k1	PROPN
ejpam-5648	179	1	+	+	NOUN
ejpam-5648	179	2	h	h	NOUN
ejpam-5648	179	3	,	,	PUNCT
ejpam-5648	179	4	where	where	SCONJ
ejpam-5648	179	5	h	h	NOUN
ejpam-5648	179	6	is	be	AUX
ejpam-5648	179	7	a	a	DET
ejpam-5648	179	8	disconnected	disconnected	ADJ
ejpam-5648	179	9	graph	graph	NOUN
ejpam-5648	179	10	with	with	ADP
ejpam-5648	179	11	isolated	isolated	ADJ
ejpam-5648	179	12	vertex	vertex	NOUN
ejpam-5648	179	13	{	{	PUNCT
ejpam-5648	179	14	w	w	NOUN
ejpam-5648	179	15	}	}	PUNCT
ejpam-5648	179	16	.	.	PUNCT
ejpam-5648	180	1	conversely	conversely	ADV
ejpam-5648	180	2	,	,	PUNCT
ejpam-5648	180	3	suppose	suppose	VERB
ejpam-5648	180	4	n	n	PRON
ejpam-5648	180	5	≥	≥	X
ejpam-5648	180	6	3	3	NUM
ejpam-5648	180	7	and	and	CCONJ
ejpam-5648	180	8	g	g	NOUN
ejpam-5648	180	9	satisfies	satisfy	VERB
ejpam-5648	180	10	the	the	DET
ejpam-5648	180	11	given	give	VERB
ejpam-5648	180	12	conditions	condition	NOUN
ejpam-5648	180	13	.	.	PUNCT
ejpam-5648	181	1	then	then	ADV
ejpam-5648	181	2	γcrh(g	γcrh(g	X
ejpam-5648	181	3	)	)	PUNCT
ejpam-5648	181	4	≥	≥	NOUN
ejpam-5648	181	5	3	3	NUM
ejpam-5648	181	6	by	by	ADP
ejpam-5648	181	7	(	(	PUNCT
ejpam-5648	181	8	i	i	NOUN
ejpam-5648	181	9	)	)	PUNCT
ejpam-5648	181	10	and	and	CCONJ
ejpam-5648	181	11	(	(	PUNCT
ejpam-5648	181	12	ii	ii	NOUN
ejpam-5648	181	13	)	)	PUNCT
ejpam-5648	181	14	.	.	PUNCT
ejpam-5648	182	1	clearly	clearly	ADV
ejpam-5648	182	2	,	,	PUNCT
ejpam-5648	182	3	γcrh(k3	γcrh(k3	NOUN
ejpam-5648	182	4	)	)	PUNCT
ejpam-5648	182	5	=	=	SYM
ejpam-5648	183	1	3	3	X
ejpam-5648	183	2	.	.	PUNCT
ejpam-5648	184	1	so	so	ADV
ejpam-5648	184	2	suppose	suppose	VERB
ejpam-5648	184	3	g	g	PROPN
ejpam-5648	184	4	=	=	PROPN
ejpam-5648	184	5	k1	k1	PROPN
ejpam-5648	185	1	+	+	NOUN
ejpam-5648	185	2	h	h	NOUN
ejpam-5648	185	3	and	and	CCONJ
ejpam-5648	185	4	u	u	NOUN
ejpam-5648	185	5	is	be	AUX
ejpam-5648	185	6	an	an	DET
ejpam-5648	185	7	isolated	isolated	ADJ
ejpam-5648	185	8	vertex	vertex	NOUN
ejpam-5648	185	9	of	of	ADP
ejpam-5648	185	10	h.	h.	PROPN
ejpam-5648	185	11	let	let	VERB
ejpam-5648	185	12	k1	k1	NOUN
ejpam-5648	185	13	=	=	SYM
ejpam-5648	185	14	{	{	PUNCT
ejpam-5648	185	15	v	v	NOUN
ejpam-5648	185	16	}	}	PUNCT
ejpam-5648	185	17	.	.	PUNCT
ejpam-5648	186	1	then	then	ADV
ejpam-5648	186	2	vu	vu	PROPN
ejpam-5648	186	3	∈	∈	PROPN
ejpam-5648	186	4	e(g	e(g	PROPN
ejpam-5648	186	5	)	)	PUNCT
ejpam-5648	186	6	.	.	PUNCT
ejpam-5648	187	1	since	since	SCONJ
ejpam-5648	187	2	u	u	NOUN
ejpam-5648	187	3	is	be	AUX
ejpam-5648	187	4	an	an	DET
ejpam-5648	187	5	isolated	isolated	ADJ
ejpam-5648	187	6	vertex	vertex	NOUN
ejpam-5648	187	7	of	of	ADP
ejpam-5648	187	8	h	h	NOUN
ejpam-5648	187	9	,	,	PUNCT
ejpam-5648	187	10	[	[	X
ejpam-5648	187	11	u	u	NOUN
ejpam-5648	187	12	,	,	PUNCT
ejpam-5648	187	13	v	v	NOUN
ejpam-5648	187	14	,	,	PUNCT
ejpam-5648	187	15	w	w	NOUN
ejpam-5648	187	16	]	]	X
ejpam-5648	187	17	is	be	AUX
ejpam-5648	187	18	a	a	DET
ejpam-5648	187	19	u−w	u−w	PROPN
ejpam-5648	187	20	geodesic	geodesic	NOUN
ejpam-5648	187	21	in	in	ADP
ejpam-5648	187	22	g	g	NOUN
ejpam-5648	187	23	for	for	ADP
ejpam-5648	187	24	every	every	DET
ejpam-5648	187	25	w	w	PROPN
ejpam-5648	187	26	∈	∈	PROPN
ejpam-5648	187	27	v	v	ADP
ejpam-5648	187	28	(	(	PUNCT
ejpam-5648	187	29	g	g	NOUN
ejpam-5648	187	30	)	)	PUNCT
ejpam-5648	187	31	\	\	NOUN
ejpam-5648	187	32	{	{	PUNCT
ejpam-5648	187	33	u	u	NOUN
ejpam-5648	187	34	,	,	PUNCT
ejpam-5648	187	35	v	v	NOUN
ejpam-5648	187	36	}	}	PUNCT
ejpam-5648	187	37	.	.	PUNCT
ejpam-5648	188	1	this	this	PRON
ejpam-5648	188	2	implies	imply	VERB
ejpam-5648	188	3	that	that	SCONJ
ejpam-5648	188	4	dg(u	dg(u	ADJ
ejpam-5648	188	5	,	,	PUNCT
ejpam-5648	188	6	w	w	NOUN
ejpam-5648	188	7	)	)	PUNCT
ejpam-5648	188	8	=	=	SYM
ejpam-5648	188	9	2	2	NUM
ejpam-5648	188	10	for	for	ADP
ejpam-5648	188	11	every	every	PRON
ejpam-5648	188	12	w	w	PROPN
ejpam-5648	188	13	∈	∈	PROPN
ejpam-5648	188	14	v	v	ADP
ejpam-5648	188	15	(	(	PUNCT
ejpam-5648	188	16	g	g	NOUN
ejpam-5648	188	17	)	)	PUNCT
ejpam-5648	188	18	\	\	NOUN
ejpam-5648	188	19	{	{	PUNCT
ejpam-5648	188	20	u	u	NOUN
ejpam-5648	188	21	,	,	PUNCT
ejpam-5648	188	22	v	v	NOUN
ejpam-5648	188	23	}	}	PUNCT
ejpam-5648	188	24	.	.	PUNCT
ejpam-5648	189	1	define	define	VERB
ejpam-5648	189	2	the	the	DET
ejpam-5648	189	3	function	function	NOUN
ejpam-5648	189	4	f	f	PROPN
ejpam-5648	189	5	=	=	SYM
ejpam-5648	189	6	(	(	PUNCT
ejpam-5648	189	7	v0	v0	PROPN
ejpam-5648	189	8	,	,	PUNCT
ejpam-5648	189	9	v1	v1	NOUN
ejpam-5648	189	10	,	,	PUNCT
ejpam-5648	189	11	v2	v2	NOUN
ejpam-5648	189	12	)	)	PUNCT
ejpam-5648	189	13	where	where	SCONJ
ejpam-5648	189	14	v2	v2	NOUN
ejpam-5648	189	15	=	=	SYM
ejpam-5648	189	16	{	{	PUNCT
ejpam-5648	189	17	u	u	NOUN
ejpam-5648	189	18	}	}	PUNCT
ejpam-5648	189	19	,	,	PUNCT
ejpam-5648	189	20	v1	v1	NOUN
ejpam-5648	189	21	=	=	SYM
ejpam-5648	189	22	{	{	PUNCT
ejpam-5648	189	23	v	v	NOUN
ejpam-5648	189	24	}	}	PUNCT
ejpam-5648	189	25	,	,	PUNCT
ejpam-5648	189	26	and	and	CCONJ
ejpam-5648	189	27	v0	v0	PROPN
ejpam-5648	189	28	=	=	SYM
ejpam-5648	189	29	v	v	PROPN
ejpam-5648	189	30	(	(	PUNCT
ejpam-5648	189	31	g	g	NOUN
ejpam-5648	189	32	)	)	PUNCT
ejpam-5648	189	33	\	\	NOUN
ejpam-5648	190	1	{	{	PUNCT
ejpam-5648	190	2	u	u	NOUN
ejpam-5648	190	3	,	,	PUNCT
ejpam-5648	190	4	v	v	NOUN
ejpam-5648	190	5	}	}	PUNCT
ejpam-5648	190	6	.	.	PUNCT
ejpam-5648	191	1	then	then	ADV
ejpam-5648	191	2	f	f	PROPN
ejpam-5648	191	3	∈	∈	PROPN
ejpam-5648	191	4	chrdf	chrdf	NOUN
ejpam-5648	191	5	(	(	PUNCT
ejpam-5648	191	6	g	g	NOUN
ejpam-5648	191	7	)	)	PUNCT
ejpam-5648	191	8	and	and	CCONJ
ejpam-5648	191	9	γcrh(g	γcrh(g	NOUN
ejpam-5648	191	10	)	)	PUNCT
ejpam-5648	191	11	≤	≤	NOUN
ejpam-5648	191	12	ωcrh	ωcrh	ADV
ejpam-5648	191	13	g	g	PROPN
ejpam-5648	191	14	(	(	PUNCT
ejpam-5648	191	15	f	f	X
ejpam-5648	191	16	)	)	PUNCT
ejpam-5648	191	17	=	=	PUNCT
ejpam-5648	191	18	|v1|+	|v1|+	PRON
ejpam-5648	191	19	2|v2|	2|v2|	NUM
ejpam-5648	191	20	=	=	SYM
ejpam-5648	191	21	3	3	X
ejpam-5648	191	22	.	.	X
ejpam-5648	191	23	therefore	therefore	ADV
ejpam-5648	191	24	,	,	PUNCT
ejpam-5648	191	25	γcrh(g	γcrh(g	NOUN
ejpam-5648	191	26	)	)	PUNCT
ejpam-5648	191	27	=	=	SYM
ejpam-5648	191	28	3	3	X
ejpam-5648	191	29	.	.	PUNCT
ejpam-5648	191	30	(	(	PUNCT
ejpam-5648	191	31	iv	iv	X
ejpam-5648	191	32	)	)	PUNCT
ejpam-5648	191	33	suppose	suppose	VERB
ejpam-5648	191	34	γcrh(g	γcrh(g	NOUN
ejpam-5648	191	35	)	)	PUNCT
ejpam-5648	191	36	=	=	SYM
ejpam-5648	191	37	4	4	X
ejpam-5648	191	38	.	.	PUNCT
ejpam-5648	191	39	then	then	ADV
ejpam-5648	191	40	n	n	CCONJ
ejpam-5648	191	41	≥	≥	NOUN
ejpam-5648	191	42	4	4	NUM
ejpam-5648	191	43	by	by	ADP
ejpam-5648	191	44	(	(	PUNCT
ejpam-5648	191	45	i	i	NOUN
ejpam-5648	191	46	)	)	PUNCT
ejpam-5648	191	47	,	,	PUNCT
ejpam-5648	191	48	(	(	PUNCT
ejpam-5648	191	49	ii	ii	NOUN
ejpam-5648	191	50	)	)	PUNCT
ejpam-5648	191	51	,	,	PUNCT
ejpam-5648	191	52	and	and	CCONJ
ejpam-5648	191	53	(	(	PUNCT
ejpam-5648	191	54	iii	iii	NOUN
ejpam-5648	191	55	)	)	PUNCT
ejpam-5648	191	56	.	.	PUNCT
ejpam-5648	192	1	let	let	VERB
ejpam-5648	192	2	f	f	PROPN
ejpam-5648	192	3	=	=	SYM
ejpam-5648	192	4	(	(	PUNCT
ejpam-5648	192	5	v0	v0	PROPN
ejpam-5648	192	6	,	,	PUNCT
ejpam-5648	192	7	v1	v1	NOUN
ejpam-5648	192	8	,	,	PUNCT
ejpam-5648	192	9	v2	v2	PROPN
ejpam-5648	192	10	)	)	PUNCT
ejpam-5648	192	11	be	be	AUX
ejpam-5648	192	12	a	a	DET
ejpam-5648	192	13	γcrh	γcrh	NOUN
ejpam-5648	192	14	-	-	PUNCT
ejpam-5648	192	15	function	function	NOUN
ejpam-5648	192	16	on	on	ADP
ejpam-5648	192	17	g.	g.	PROPN
ejpam-5648	192	18	then	then	ADV
ejpam-5648	192	19	|v1|+2|v2|	|v1|+2|v2|	NUM
ejpam-5648	193	1	=	=	SYM
ejpam-5648	193	2	4	4	X
ejpam-5648	193	3	.	.	PUNCT
ejpam-5648	194	1	it	it	PRON
ejpam-5648	194	2	follows	follow	VERB
ejpam-5648	194	3	that	that	DET
ejpam-5648	194	4	|v2|	|v2|	NOUN
ejpam-5648	194	5	≤	≤	NOUN
ejpam-5648	194	6	2	2	NUM
ejpam-5648	194	7	.	.	PUNCT
ejpam-5648	194	8	consider	consider	VERB
ejpam-5648	194	9	the	the	DET
ejpam-5648	194	10	following	follow	VERB
ejpam-5648	194	11	cases	case	NOUN
ejpam-5648	194	12	:	:	PUNCT
ejpam-5648	194	13	case	case	NOUN
ejpam-5648	194	14	1	1	NUM
ejpam-5648	194	15	.	.	NOUN
ejpam-5648	194	16	|v2|	|v2|	NOUN
ejpam-5648	194	17	=	=	SYM
ejpam-5648	195	1	0	0	X
ejpam-5648	195	2	.	.	PUNCT
ejpam-5648	196	1	then	then	ADV
ejpam-5648	196	2	v0	v0	NOUN
ejpam-5648	196	3	=	=	SYM
ejpam-5648	196	4	∅	∅	NOUN
ejpam-5648	196	5	and	and	CCONJ
ejpam-5648	196	6	|v1|	|v1|	NOUN
ejpam-5648	196	7	=	=	SYM
ejpam-5648	196	8	|v	|v	PROPN
ejpam-5648	196	9	(	(	PUNCT
ejpam-5648	196	10	g)|	g)|	NOUN
ejpam-5648	196	11	=	=	NOUN
ejpam-5648	196	12	4	4	NUM
ejpam-5648	196	13	.	.	PUNCT
ejpam-5648	197	1	therefore	therefore	ADV
ejpam-5648	197	2	,	,	PUNCT
ejpam-5648	197	3	with	with	ADP
ejpam-5648	197	4	reference	reference	NOUN
ejpam-5648	197	5	to	to	ADP
ejpam-5648	197	6	(	(	PUNCT
ejpam-5648	197	7	iii	iii	NOUN
ejpam-5648	197	8	)	)	PUNCT
ejpam-5648	197	9	,	,	PUNCT
ejpam-5648	197	10	we	we	PRON
ejpam-5648	197	11	must	must	AUX
ejpam-5648	197	12	have	have	VERB
ejpam-5648	197	13	g	g	PROPN
ejpam-5648	197	14	∈	∈	PROPN
ejpam-5648	197	15	{	{	PUNCT
ejpam-5648	197	16	k4	k4	NOUN
ejpam-5648	197	17	,	,	PUNCT
ejpam-5648	197	18	p4	p4	ADJ
ejpam-5648	197	19	,	,	PUNCT
ejpam-5648	197	20	c4,k1	c4,k1	PROPN
ejpam-5648	197	21	+	+	NUM
ejpam-5648	197	22	p3	p3	PROPN
ejpam-5648	197	23	}	}	PUNCT
ejpam-5648	197	24	.	.	PUNCT
ejpam-5648	198	1	case	case	NOUN
ejpam-5648	198	2	2	2	NUM
ejpam-5648	198	3	.	.	NOUN
ejpam-5648	198	4	|v2|	|v2|	NOUN
ejpam-5648	198	5	=	=	NOUN
ejpam-5648	199	1	1	1	X
ejpam-5648	199	2	.	.	PUNCT
ejpam-5648	199	3	then	then	ADV
ejpam-5648	199	4	|v1|	|v1|	NOUN
ejpam-5648	199	5	=	=	SYM
ejpam-5648	199	6	2	2	NUM
ejpam-5648	199	7	and	and	CCONJ
ejpam-5648	199	8	|v0|	|v0|	NOUN
ejpam-5648	199	9	=	=	NOUN
ejpam-5648	199	10	̸	̸	NUM
ejpam-5648	199	11	0	0	NUM
ejpam-5648	199	12	.	.	PUNCT
ejpam-5648	200	1	let	let	VERB
ejpam-5648	200	2	v1	v1	VERB
ejpam-5648	200	3	=	=	SYM
ejpam-5648	200	4	{	{	PUNCT
ejpam-5648	200	5	u	u	NOUN
ejpam-5648	200	6	,	,	PUNCT
ejpam-5648	200	7	v	v	NOUN
ejpam-5648	200	8	}	}	PUNCT
ejpam-5648	200	9	and	and	CCONJ
ejpam-5648	200	10	v2	v2	NOUN
ejpam-5648	200	11	=	=	SYM
ejpam-5648	200	12	{	{	PUNCT
ejpam-5648	200	13	w	w	NOUN
ejpam-5648	200	14	}	}	PUNCT
ejpam-5648	200	15	.	.	PUNCT
ejpam-5648	201	1	suppose	suppose	VERB
ejpam-5648	201	2	v	v	X
ejpam-5648	201	3	/∈	/∈	PUNCT
ejpam-5648	201	4	ng(w	ng(w	NUM
ejpam-5648	201	5	)	)	PUNCT
ejpam-5648	201	6	.	.	PUNCT
ejpam-5648	202	1	then	then	ADV
ejpam-5648	202	2	[	[	X
ejpam-5648	202	3	v	v	NOUN
ejpam-5648	202	4	,	,	PUNCT
ejpam-5648	202	5	u	u	NOUN
ejpam-5648	202	6	,	,	PUNCT
ejpam-5648	202	7	w	w	PROPN
ejpam-5648	202	8	]	]	X
ejpam-5648	202	9	is	be	AUX
ejpam-5648	202	10	v	v	NOUN
ejpam-5648	202	11	-	-	PUNCT
ejpam-5648	202	12	w	w	NOUN
ejpam-5648	202	13	geodesic	geodesic	NOUN
ejpam-5648	202	14	by	by	ADP
ejpam-5648	202	15	(	(	PUNCT
ejpam-5648	202	16	p2	p2	NOUN
ejpam-5648	202	17	)	)	PUNCT
ejpam-5648	202	18	.	.	PUNCT
ejpam-5648	203	1	this	this	PRON
ejpam-5648	203	2	and	and	CCONJ
ejpam-5648	203	3	(	(	PUNCT
ejpam-5648	203	4	p1	p1	NOUN
ejpam-5648	203	5	)	)	PUNCT
ejpam-5648	203	6	would	would	AUX
ejpam-5648	203	7	imply	imply	VERB
ejpam-5648	203	8	that	that	SCONJ
ejpam-5648	203	9	[	[	X
ejpam-5648	203	10	z	z	X
ejpam-5648	203	11	,	,	PUNCT
ejpam-5648	203	12	u	u	NOUN
ejpam-5648	203	13	,	,	PUNCT
ejpam-5648	203	14	w	w	PROPN
ejpam-5648	203	15	]	]	X
ejpam-5648	203	16	is	be	AUX
ejpam-5648	203	17	a	a	DET
ejpam-5648	203	18	z	z	PROPN
ejpam-5648	203	19	-	-	PUNCT
ejpam-5648	203	20	w	w	NOUN
ejpam-5648	203	21	geodesic	geodesic	NOUN
ejpam-5648	203	22	for	for	ADP
ejpam-5648	203	23	all	all	DET
ejpam-5648	203	24	z	z	NOUN
ejpam-5648	203	25	∈	∈	PROPN
ejpam-5648	203	26	v	v	ADP
ejpam-5648	203	27	(	(	PUNCT
ejpam-5648	203	28	g	g	NOUN
ejpam-5648	203	29	)	)	PUNCT
ejpam-5648	203	30	\	\	NOUN
ejpam-5648	203	31	{	{	PUNCT
ejpam-5648	203	32	u	u	NOUN
ejpam-5648	203	33	,	,	PUNCT
ejpam-5648	203	34	w	w	NOUN
ejpam-5648	203	35	}	}	PUNCT
ejpam-5648	203	36	.	.	PUNCT
ejpam-5648	204	1	let	let	VERB
ejpam-5648	204	2	v	v	X
ejpam-5648	204	3	′	′	NOUN
ejpam-5648	204	4	2	2	NUM
ejpam-5648	204	5	=	=	SYM
ejpam-5648	204	6	v2	v2	PROPN
ejpam-5648	204	7	,	,	PUNCT
ejpam-5648	204	8	v	v	NOUN
ejpam-5648	204	9	′	′	NOUN
ejpam-5648	204	10	1	1	NUM
ejpam-5648	204	11	=	=	SYM
ejpam-5648	204	12	{	{	PUNCT
ejpam-5648	204	13	u	u	NOUN
ejpam-5648	204	14	}	}	PUNCT
ejpam-5648	204	15	,	,	PUNCT
ejpam-5648	204	16	v	v	X
ejpam-5648	204	17	′	′	NOUN
ejpam-5648	204	18	0	0	NUM
ejpam-5648	205	1	=	=	SYM
ejpam-5648	205	2	v0	v0	NOUN
ejpam-5648	205	3	∪	∪	X
ejpam-5648	205	4	{	{	PUNCT
ejpam-5648	205	5	v	v	NOUN
ejpam-5648	205	6	}	}	PUNCT
ejpam-5648	205	7	=	=	SYM
ejpam-5648	205	8	v	v	NOUN
ejpam-5648	205	9	(	(	PUNCT
ejpam-5648	205	10	g	g	NOUN
ejpam-5648	205	11	)	)	PUNCT
ejpam-5648	205	12	\	\	NOUN
ejpam-5648	206	1	{	{	PUNCT
ejpam-5648	206	2	u	u	NOUN
ejpam-5648	206	3	,	,	PUNCT
ejpam-5648	206	4	w	w	NOUN
ejpam-5648	206	5	}	}	PUNCT
ejpam-5648	206	6	.	.	PUNCT
ejpam-5648	207	1	then	then	ADV
ejpam-5648	207	2	g	g	PROPN
ejpam-5648	207	3	=	=	PUNCT
ejpam-5648	207	4	(	(	PUNCT
ejpam-5648	207	5	v	v	NUM
ejpam-5648	207	6	′	′	NUM
ejpam-5648	207	7	0	0	NUM
ejpam-5648	207	8	,	,	PUNCT
ejpam-5648	207	9	v	v	NOUN
ejpam-5648	207	10	′	′	NUM
ejpam-5648	207	11	1	1	NUM
ejpam-5648	207	12	,	,	PUNCT
ejpam-5648	207	13	v	v	NOUN
ejpam-5648	207	14	′	′	NUM
ejpam-5648	207	15	2	2	NUM
ejpam-5648	207	16	)	)	PUNCT
ejpam-5648	207	17	∈	∈	PROPN
ejpam-5648	207	18	chrdf	chrdf	NOUN
ejpam-5648	207	19	(	(	PUNCT
ejpam-5648	207	20	g	g	NOUN
ejpam-5648	207	21	)	)	PUNCT
ejpam-5648	207	22	.	.	PUNCT
ejpam-5648	208	1	thus	thus	ADV
ejpam-5648	208	2	,	,	PUNCT
ejpam-5648	208	3	ωcrh	ωcrh	ADV
ejpam-5648	208	4	g	g	PROPN
ejpam-5648	208	5	(	(	PUNCT
ejpam-5648	208	6	g	g	NOUN
ejpam-5648	208	7	)	)	PUNCT
ejpam-5648	208	8	=	=	PUNCT
ejpam-5648	208	9	|v	|v	PROPN
ejpam-5648	208	10	′	′	NOUN
ejpam-5648	208	11	1	1	NUM
ejpam-5648	208	12	|+2|v	|+2|v	NOUN
ejpam-5648	209	1	′	′	NOUN
ejpam-5648	209	2	2	2	NUM
ejpam-5648	210	1	|	|	ADV
ejpam-5648	210	2	=	=	SYM
ejpam-5648	210	3	3	3	NUM
ejpam-5648	210	4	,	,	PUNCT
ejpam-5648	210	5	a	a	DET
ejpam-5648	210	6	contradiction	contradiction	NOUN
ejpam-5648	210	7	.	.	PUNCT
ejpam-5648	211	1	thus	thus	ADV
ejpam-5648	211	2	,	,	PUNCT
ejpam-5648	211	3	v	v	PROPN
ejpam-5648	211	4	∈	∈	PROPN
ejpam-5648	211	5	ng(w	ng(w	NOUN
ejpam-5648	211	6	)	)	PUNCT
ejpam-5648	211	7	.	.	PUNCT
ejpam-5648	212	1	similarly	similarly	ADV
ejpam-5648	212	2	,	,	PUNCT
ejpam-5648	212	3	u	u	PROPN
ejpam-5648	212	4	∈	∈	PROPN
ejpam-5648	212	5	ng(w	ng(w	NOUN
ejpam-5648	212	6	)	)	PUNCT
ejpam-5648	212	7	.	.	PUNCT
ejpam-5648	213	1	hence	hence	ADV
ejpam-5648	213	2	,	,	PUNCT
ejpam-5648	213	3	u	u	NOUN
ejpam-5648	213	4	,	,	PUNCT
ejpam-5648	213	5	v	v	PROPN
ejpam-5648	213	6	∈	∈	PROPN
ejpam-5648	213	7	ng(w	ng(w	NOUN
ejpam-5648	213	8	)	)	PUNCT
ejpam-5648	213	9	.	.	PUNCT
ejpam-5648	214	1	this	this	PRON
ejpam-5648	214	2	implies	imply	VERB
ejpam-5648	214	3	that	that	SCONJ
ejpam-5648	214	4	degg(w	degg(w	NOUN
ejpam-5648	214	5	)	)	PUNCT
ejpam-5648	214	6	=	=	SYM
ejpam-5648	214	7	2	2	NUM
ejpam-5648	214	8	,	,	PUNCT
ejpam-5648	214	9	that	that	ADV
ejpam-5648	214	10	is	is	ADV
ejpam-5648	214	11	,	,	PUNCT
ejpam-5648	214	12	|ng(w)|	|ng(w)|	X
ejpam-5648	214	13	=	=	NOUN
ejpam-5648	214	14	2	2	NUM
ejpam-5648	214	15	.	.	PUNCT
ejpam-5648	215	1	moreover	moreover	ADV
ejpam-5648	215	2	,	,	PUNCT
ejpam-5648	215	3	n2	n2	ADJ
ejpam-5648	215	4	g(w	g(w	PROPN
ejpam-5648	215	5	)	)	PUNCT
ejpam-5648	215	6	=	=	SYM
ejpam-5648	215	7	v0	v0	NOUN
ejpam-5648	215	8	=	=	SYM
ejpam-5648	215	9	v	v	PROPN
ejpam-5648	215	10	(	(	PUNCT
ejpam-5648	215	11	g	g	NOUN
ejpam-5648	215	12	)	)	PUNCT
ejpam-5648	215	13	\	\	PUNCT
ejpam-5648	216	1	ng[w	ng[w	PROPN
ejpam-5648	216	2	]	]	PUNCT
ejpam-5648	216	3	.	.	PUNCT
ejpam-5648	217	1	suppose	suppose	VERB
ejpam-5648	217	2	z	z	PROPN
ejpam-5648	217	3	∈	∈	PROPN
ejpam-5648	217	4	ng(w	ng(w	NOUN
ejpam-5648	217	5	)	)	PUNCT
ejpam-5648	217	6	is	be	AUX
ejpam-5648	217	7	a	a	DET
ejpam-5648	217	8	dominating	dominating	NOUN
ejpam-5648	217	9	vertex	vertex	NOUN
ejpam-5648	217	10	of	of	ADP
ejpam-5648	217	11	g.	g.	PROPN
ejpam-5648	217	12	suppose	suppose	VERB
ejpam-5648	217	13	further	far	ADV
ejpam-5648	217	14	that	that	SCONJ
ejpam-5648	217	15	z	z	PROPN
ejpam-5648	217	16	is	be	AUX
ejpam-5648	217	17	a	a	DET
ejpam-5648	217	18	support	support	NOUN
ejpam-5648	217	19	vertex	vertex	NOUN
ejpam-5648	217	20	of	of	ADP
ejpam-5648	217	21	some	some	DET
ejpam-5648	217	22	leaf	leaf	NOUN
ejpam-5648	218	1	p.	p.	NOUN
ejpam-5648	218	2	then	then	ADV
ejpam-5648	218	3	g	g	PROPN
ejpam-5648	218	4	=	=	SYM
ejpam-5648	218	5	⟨z⟩+h	⟨z⟩+h	PROPN
ejpam-5648	218	6	,	,	PUNCT
ejpam-5648	218	7	where	where	SCONJ
ejpam-5648	218	8	h	h	NOUN
ejpam-5648	218	9	=	=	SYM
ejpam-5648	218	10	⟨v	⟨v	X
ejpam-5648	218	11	(	(	PUNCT
ejpam-5648	218	12	g	g	NOUN
ejpam-5648	218	13	)	)	PUNCT
ejpam-5648	218	14	\	\	NOUN
ejpam-5648	219	1	{	{	PUNCT
ejpam-5648	219	2	z}⟩	z}⟩	PROPN
ejpam-5648	219	3	is	be	AUX
ejpam-5648	219	4	a	a	DET
ejpam-5648	219	5	graph	graph	NOUN
ejpam-5648	219	6	with	with	ADP
ejpam-5648	219	7	isolated	isolated	ADJ
ejpam-5648	219	8	vertex	vertex	NOUN
ejpam-5648	219	9	p.	p.	NOUN
ejpam-5648	219	10	by	by	ADP
ejpam-5648	219	11	part	part	NOUN
ejpam-5648	219	12	(	(	PUNCT
ejpam-5648	219	13	iii	iii	NOUN
ejpam-5648	219	14	)	)	PUNCT
ejpam-5648	219	15	,	,	PUNCT
ejpam-5648	219	16	it	it	PRON
ejpam-5648	219	17	follows	follow	VERB
ejpam-5648	219	18	that	that	SCONJ
ejpam-5648	219	19	γcrh(g	γcrh(g	NOUN
ejpam-5648	219	20	)	)	PUNCT
ejpam-5648	219	21	=	=	SYM
ejpam-5648	219	22	3	3	NUM
ejpam-5648	219	23	,	,	PUNCT
ejpam-5648	219	24	contrary	contrary	ADV
ejpam-5648	219	25	to	to	ADP
ejpam-5648	219	26	the	the	DET
ejpam-5648	219	27	assumption	assumption	NOUN
ejpam-5648	219	28	that	that	SCONJ
ejpam-5648	219	29	γcrh(g	γcrh(g	ADP
ejpam-5648	219	30	)	)	PUNCT
ejpam-5648	219	31	=	=	SYM
ejpam-5648	220	1	4	4	X
ejpam-5648	220	2	.	.	PUNCT
ejpam-5648	221	1	this	this	PRON
ejpam-5648	221	2	shows	show	VERB
ejpam-5648	221	3	that	that	SCONJ
ejpam-5648	221	4	(	(	PUNCT
ejpam-5648	221	5	a	a	X
ejpam-5648	221	6	)	)	PUNCT
ejpam-5648	221	7	holds	hold	NOUN
ejpam-5648	221	8	.	.	PUNCT
ejpam-5648	222	1	case	case	NOUN
ejpam-5648	222	2	3	3	NUM
ejpam-5648	222	3	.	.	NOUN
ejpam-5648	222	4	|v2|	|v2|	NOUN
ejpam-5648	222	5	=	=	SYM
ejpam-5648	223	1	2	2	X
ejpam-5648	223	2	.	.	PUNCT
ejpam-5648	223	3	then	then	ADV
ejpam-5648	223	4	|v1|	|v1|	NOUN
ejpam-5648	223	5	=	=	SYM
ejpam-5648	223	6	0	0	X
ejpam-5648	223	7	.	.	PUNCT
ejpam-5648	224	1	let	let	VERB
ejpam-5648	224	2	v2	v2	VERB
ejpam-5648	224	3	=	=	SYM
ejpam-5648	224	4	{	{	PUNCT
ejpam-5648	224	5	u	u	NOUN
ejpam-5648	224	6	,	,	PUNCT
ejpam-5648	224	7	v	v	NOUN
ejpam-5648	224	8	}	}	PUNCT
ejpam-5648	224	9	.	.	PUNCT
ejpam-5648	225	1	then	then	ADV
ejpam-5648	225	2	uv	uv	PROPN
ejpam-5648	225	3	∈	∈	PROPN
ejpam-5648	225	4	e(g	e(g	PROPN
ejpam-5648	225	5	)	)	PUNCT
ejpam-5648	225	6	by	by	ADP
ejpam-5648	225	7	(	(	PUNCT
ejpam-5648	225	8	p2	p2	PROPN
ejpam-5648	225	9	)	)	PUNCT
ejpam-5648	225	10	.	.	PUNCT
ejpam-5648	225	11	suppose	suppose	VERB
ejpam-5648	225	12	|ng(u)|	|ng(u)|	NOUN
ejpam-5648	225	13	=	=	SYM
ejpam-5648	225	14	1	1	X
ejpam-5648	225	15	.	.	PUNCT
ejpam-5648	225	16	then	then	ADV
ejpam-5648	225	17	ng(u	ng(u	NOUN
ejpam-5648	225	18	)	)	PUNCT
ejpam-5648	225	19	=	=	PRON
ejpam-5648	225	20	{	{	PUNCT
ejpam-5648	225	21	v	v	NOUN
ejpam-5648	225	22	}	}	PUNCT
ejpam-5648	225	23	.	.	PUNCT
ejpam-5648	226	1	since	since	SCONJ
ejpam-5648	226	2	g	g	PROPN
ejpam-5648	226	3	̸=	̸=	PROPN
ejpam-5648	226	4	k2	k2	NOUN
ejpam-5648	226	5	,	,	PUNCT
ejpam-5648	226	6	g	g	PROPN
ejpam-5648	226	7	=	=	SYM
ejpam-5648	226	8	⟨v⟩+h	⟨v⟩+h	NOUN
ejpam-5648	226	9	whereh	whereh	NOUN
ejpam-5648	226	10	=	=	SYM
ejpam-5648	226	11	⟨v	⟨v	X
ejpam-5648	226	12	(	(	PUNCT
ejpam-5648	226	13	g)\{v}⟩	g)\{v}⟩	VERB
ejpam-5648	226	14	is	be	AUX
ejpam-5648	226	15	a	a	DET
ejpam-5648	226	16	disconnected	disconnected	ADJ
ejpam-5648	226	17	graph	graph	NOUN
ejpam-5648	226	18	having	have	VERB
ejpam-5648	226	19	u	u	NOUN
ejpam-5648	226	20	as	as	ADP
ejpam-5648	226	21	an	an	DET
ejpam-5648	226	22	isolated	isolated	ADJ
ejpam-5648	226	23	vertex	vertex	NOUN
ejpam-5648	226	24	.	.	PUNCT
ejpam-5648	227	1	by	by	ADP
ejpam-5648	227	2	(	(	PUNCT
ejpam-5648	227	3	iii	iii	NOUN
ejpam-5648	227	4	)	)	PUNCT
ejpam-5648	227	5	,	,	PUNCT
ejpam-5648	227	6	γcrh(g	γcrh(g	NOUN
ejpam-5648	227	7	)	)	PUNCT
ejpam-5648	227	8	=	=	SYM
ejpam-5648	227	9	3	3	NUM
ejpam-5648	227	10	,	,	PUNCT
ejpam-5648	227	11	a	a	DET
ejpam-5648	227	12	contradiction	contradiction	NOUN
ejpam-5648	227	13	.	.	PUNCT
ejpam-5648	228	1	thus	thus	ADV
ejpam-5648	228	2	,	,	PUNCT
ejpam-5648	228	3	|ng(u)|	|ng(u)|	NOUN
ejpam-5648	228	4	≥	≥	NUM
ejpam-5648	228	5	2	2	NUM
ejpam-5648	228	6	.	.	PUNCT
ejpam-5648	228	7	similarly	similarly	ADV
ejpam-5648	228	8	,	,	PUNCT
ejpam-5648	228	9	|ng(v)|	|ng(v)|	NOUN
ejpam-5648	228	10	≥	≥	NOUN
ejpam-5648	228	11	2	2	NUM
ejpam-5648	228	12	.	.	PUNCT
ejpam-5648	228	13	suppose	suppose	VERB
ejpam-5648	228	14	z	z	PROPN
ejpam-5648	228	15	∈	∈	PROPN
ejpam-5648	228	16	ng(u	ng(u	PROPN
ejpam-5648	228	17	)	)	PUNCT
ejpam-5648	228	18	∩ng(v	∩ng(v	PROPN
ejpam-5648	228	19	)	)	PUNCT
ejpam-5648	228	20	.	.	PUNCT
ejpam-5648	229	1	then	then	ADV
ejpam-5648	229	2	z	z	PROPN
ejpam-5648	229	3	∈	∈	PROPN
ejpam-5648	229	4	v0	v0	NOUN
ejpam-5648	229	5	,	,	PUNCT
ejpam-5648	229	6	which	which	PRON
ejpam-5648	229	7	is	be	AUX
ejpam-5648	229	8	not	not	PART
ejpam-5648	229	9	possible	possible	ADJ
ejpam-5648	229	10	.	.	PUNCT
ejpam-5648	230	1	hence	hence	ADV
ejpam-5648	230	2	,	,	PUNCT
ejpam-5648	230	3	ng(u	ng(u	NOUN
ejpam-5648	230	4	)	)	PUNCT
ejpam-5648	230	5	∩	∩	NOUN
ejpam-5648	230	6	ng(v	ng(v	X
ejpam-5648	230	7	)	)	PUNCT
ejpam-5648	230	8	=	=	VERB
ejpam-5648	230	9	∅.	∅.	AUX
ejpam-5648	230	10	let	let	VERB
ejpam-5648	230	11	y	y	PROPN
ejpam-5648	230	12	∈	∈	PROPN
ejpam-5648	230	13	v	v	ADP
ejpam-5648	230	14	(	(	PUNCT
ejpam-5648	230	15	g	g	NOUN
ejpam-5648	230	16	)	)	PUNCT
ejpam-5648	230	17	\	\	PROPN
ejpam-5648	231	1	v2	v2	PROPN
ejpam-5648	231	2	=	=	SYM
ejpam-5648	231	3	v0	v0	NOUN
ejpam-5648	231	4	.	.	PUNCT
ejpam-5648	232	1	by	by	ADP
ejpam-5648	232	2	(	(	PUNCT
ejpam-5648	232	3	p1	p1	PROPN
ejpam-5648	232	4	)	)	PUNCT
ejpam-5648	232	5	,	,	PUNCT
ejpam-5648	232	6	y	y	PROPN
ejpam-5648	232	7	∈	∈	PROPN
ejpam-5648	232	8	n2	n2	NOUN
ejpam-5648	232	9	g(u	g(u	PROPN
ejpam-5648	232	10	)	)	PUNCT
ejpam-5648	232	11	∪	∪	ADP
ejpam-5648	232	12	n2	n2	ADJ
ejpam-5648	232	13	g(v	g(v	PROPN
ejpam-5648	232	14	)	)	PUNCT
ejpam-5648	232	15	.	.	PUNCT
ejpam-5648	233	1	hence	hence	ADV
ejpam-5648	233	2	,	,	PUNCT
ejpam-5648	233	3	n2	n2	ADJ
ejpam-5648	233	4	g(u	g(u	PROPN
ejpam-5648	233	5	)	)	PUNCT
ejpam-5648	233	6	∪n2	∪n2	NOUN
ejpam-5648	233	7	g(v	g(v	PROPN
ejpam-5648	233	8	)	)	PUNCT
ejpam-5648	233	9	=	=	SYM
ejpam-5648	233	10	v	v	X
ejpam-5648	233	11	(	(	PUNCT
ejpam-5648	233	12	g	g	NOUN
ejpam-5648	233	13	)	)	PUNCT
ejpam-5648	233	14	\	\	NOUN
ejpam-5648	233	15	{	{	PUNCT
ejpam-5648	233	16	u	u	NOUN
ejpam-5648	233	17	,	,	PUNCT
ejpam-5648	233	18	v	v	NOUN
ejpam-5648	233	19	}	}	PUNCT
ejpam-5648	233	20	.	.	PUNCT
ejpam-5648	234	1	cleary	cleary	PROPN
ejpam-5648	234	2	,	,	PUNCT
ejpam-5648	234	3	{	{	PUNCT
ejpam-5648	234	4	u	u	NOUN
ejpam-5648	234	5	,	,	PUNCT
ejpam-5648	234	6	v	v	NOUN
ejpam-5648	234	7	}	}	PUNCT
ejpam-5648	234	8	is	be	AUX
ejpam-5648	234	9	a	a	DET
ejpam-5648	234	10	γch	γch	NOUN
ejpam-5648	234	11	-	-	PUNCT
ejpam-5648	234	12	set	set	NOUN
ejpam-5648	234	13	in	in	ADP
ejpam-5648	234	14	g.	g.	PROPN
ejpam-5648	234	15	this	this	PRON
ejpam-5648	234	16	shows	show	VERB
ejpam-5648	234	17	that	that	SCONJ
ejpam-5648	234	18	(	(	PUNCT
ejpam-5648	234	19	b	b	X
ejpam-5648	234	20	)	)	PUNCT
ejpam-5648	234	21	holds	hold	VERB
ejpam-5648	234	22	.	.	PUNCT
ejpam-5648	235	1	conversely	conversely	ADV
ejpam-5648	235	2	,	,	PUNCT
ejpam-5648	235	3	suppose	suppose	VERB
ejpam-5648	235	4	n	n	PRON
ejpam-5648	235	5	≥	≥	X
ejpam-5648	235	6	4	4	NUM
ejpam-5648	235	7	and	and	CCONJ
ejpam-5648	235	8	suppose	suppose	VERB
ejpam-5648	235	9	g	g	PROPN
ejpam-5648	235	10	∈	∈	PROPN
ejpam-5648	235	11	{	{	PUNCT
ejpam-5648	235	12	k4	k4	NOUN
ejpam-5648	235	13	,	,	PUNCT
ejpam-5648	235	14	p4	p4	ADJ
ejpam-5648	235	15	,	,	PUNCT
ejpam-5648	235	16	c4,k1	c4,k1	PROPN
ejpam-5648	235	17	+	+	NUM
ejpam-5648	235	18	p3	p3	PROPN
ejpam-5648	235	19	.	.	PUNCT
ejpam-5648	236	1	let	let	VERB
ejpam-5648	236	2	v	v	X
ejpam-5648	236	3	(	(	PUNCT
ejpam-5648	236	4	g	g	NOUN
ejpam-5648	236	5	)	)	PUNCT
ejpam-5648	236	6	=	=	SYM
ejpam-5648	236	7	{	{	PUNCT
ejpam-5648	236	8	w	w	PROPN
ejpam-5648	236	9	,	,	PUNCT
ejpam-5648	236	10	x	x	NOUN
ejpam-5648	236	11	,	,	PUNCT
ejpam-5648	236	12	y	y	PROPN
ejpam-5648	236	13	,	,	PUNCT
ejpam-5648	236	14	z	z	NOUN
ejpam-5648	236	15	}	}	PUNCT
ejpam-5648	236	16	and	and	CCONJ
ejpam-5648	236	17	set	set	VERB
ejpam-5648	236	18	v1	v1	NOUN
ejpam-5648	236	19	=	=	SYM
ejpam-5648	236	20	{	{	PUNCT
ejpam-5648	236	21	w	w	PROPN
ejpam-5648	236	22	,	,	PUNCT
ejpam-5648	236	23	x	x	NOUN
ejpam-5648	236	24	,	,	PUNCT
ejpam-5648	236	25	y	y	PROPN
ejpam-5648	236	26	,	,	PUNCT
ejpam-5648	236	27	z	z	NOUN
ejpam-5648	236	28	}	}	PUNCT
ejpam-5648	236	29	and	and	CCONJ
ejpam-5648	236	30	v0	v0	NOUN
ejpam-5648	236	31	=	=	SYM
ejpam-5648	236	32	v2	v2	PROPN
ejpam-5648	236	33	=	=	PUNCT
ejpam-5648	236	34	∅.	∅.	NOUN
ejpam-5648	236	35	then	then	ADV
ejpam-5648	236	36	f	f	PROPN
ejpam-5648	237	1	=	=	SYM
ejpam-5648	238	1	(	(	PUNCT
ejpam-5648	238	2	v0	v0	PROPN
ejpam-5648	238	3	,	,	PUNCT
ejpam-5648	238	4	v1	v1	NOUN
ejpam-5648	238	5	,	,	PUNCT
ejpam-5648	238	6	v2	v2	NOUN
ejpam-5648	238	7	)	)	PUNCT
ejpam-5648	238	8	∈	∈	PROPN
ejpam-5648	238	9	a.	a.	NOUN
ejpam-5648	238	10	aradais	aradais	PROPN
ejpam-5648	238	11	,	,	PUNCT
ejpam-5648	238	12	j.	j.	PROPN
ejpam-5648	238	13	cariaga	cariaga	PROPN
ejpam-5648	238	14	,	,	PUNCT
ejpam-5648	238	15	s.	s.	PROPN
ejpam-5648	238	16	canoy	canoy	PROPN
ejpam-5648	238	17	jr	jr	PROPN
ejpam-5648	238	18	.	.	PROPN
ejpam-5648	238	19	/	/	SYM
ejpam-5648	238	20	eur	eur	PROPN
ejpam-5648	238	21	.	.	PUNCT
ejpam-5648	239	1	j.	j.	PROPN
ejpam-5648	239	2	pure	pure	PROPN
ejpam-5648	239	3	appl	appl	PROPN
ejpam-5648	239	4	.	.	PROPN
ejpam-5648	239	5	math	math	PROPN
ejpam-5648	239	6	,	,	PUNCT
ejpam-5648	239	7	18	18	NUM
ejpam-5648	239	8	(	(	PUNCT
ejpam-5648	239	9	1	1	NUM
ejpam-5648	239	10	)	)	PUNCT
ejpam-5648	239	11	(	(	PUNCT
ejpam-5648	239	12	2025	2025	NUM
ejpam-5648	239	13	)	)	PUNCT
ejpam-5648	239	14	,	,	PUNCT
ejpam-5648	239	15	5648	5648	NUM
ejpam-5648	239	16	7	7	NUM
ejpam-5648	239	17	of	of	ADP
ejpam-5648	239	18	13	13	NUM
ejpam-5648	239	19	chrdf	chrdf	NOUN
ejpam-5648	239	20	(	(	PUNCT
ejpam-5648	239	21	g	g	NOUN
ejpam-5648	239	22	)	)	PUNCT
ejpam-5648	239	23	.	.	PUNCT
ejpam-5648	240	1	thus	thus	ADV
ejpam-5648	240	2	,	,	PUNCT
ejpam-5648	240	3	γcrh(g	γcrh(g	NOUN
ejpam-5648	240	4	)	)	PUNCT
ejpam-5648	240	5	≤	≤	NOUN
ejpam-5648	240	6	ωcrh	ωcrh	ADV
ejpam-5648	240	7	g	g	PROPN
ejpam-5648	240	8	(	(	PUNCT
ejpam-5648	240	9	f	f	X
ejpam-5648	240	10	)	)	PUNCT
ejpam-5648	240	11	=	=	NOUN
ejpam-5648	240	12	|v1|	|v1|	NOUN
ejpam-5648	240	13	=	=	SYM
ejpam-5648	240	14	4	4	X
ejpam-5648	240	15	.	.	PUNCT
ejpam-5648	240	16	since	since	SCONJ
ejpam-5648	240	17	g	g	PROPN
ejpam-5648	240	18	̸=	̸=	PROPN
ejpam-5648	240	19	k1	k1	NOUN
ejpam-5648	240	20	+	+	PROPN
ejpam-5648	240	21	h	h	NOUN
ejpam-5648	240	22	for	for	ADP
ejpam-5648	240	23	any	any	DET
ejpam-5648	240	24	graph	graph	NOUN
ejpam-5648	240	25	h	h	NOUN
ejpam-5648	240	26	having	have	VERB
ejpam-5648	240	27	an	an	DET
ejpam-5648	240	28	isolated	isolated	ADJ
ejpam-5648	240	29	vertex	vertex	NOUN
ejpam-5648	240	30	,	,	PUNCT
ejpam-5648	240	31	it	it	PRON
ejpam-5648	240	32	follows	follow	VERB
ejpam-5648	240	33	from	from	ADP
ejpam-5648	240	34	(	(	PUNCT
ejpam-5648	240	35	iii	iii	NOUN
ejpam-5648	240	36	)	)	PUNCT
ejpam-5648	240	37	that	that	SCONJ
ejpam-5648	240	38	γcrh(g	γcrh(g	NOUN
ejpam-5648	240	39	)	)	PUNCT
ejpam-5648	240	40	≥	≥	NOUN
ejpam-5648	240	41	4	4	NUM
ejpam-5648	240	42	.	.	PUNCT
ejpam-5648	240	43	accordingly	accordingly	ADV
ejpam-5648	240	44	,	,	PUNCT
ejpam-5648	240	45	γcrh(g	γcrh(g	NOUN
ejpam-5648	240	46	)	)	PUNCT
ejpam-5648	240	47	=	=	SYM
ejpam-5648	240	48	4	4	X
ejpam-5648	240	49	.	.	PUNCT
ejpam-5648	241	1	now	now	ADV
ejpam-5648	241	2	,	,	PUNCT
ejpam-5648	241	3	suppose	suppose	VERB
ejpam-5648	241	4	(	(	PUNCT
ejpam-5648	241	5	a	a	PRON
ejpam-5648	241	6	)	)	PUNCT
ejpam-5648	241	7	holds	hold	NOUN
ejpam-5648	241	8	,	,	PUNCT
ejpam-5648	241	9	i.e.	i.e.	X
ejpam-5648	241	10	,	,	PUNCT
ejpam-5648	241	11	there	there	PRON
ejpam-5648	241	12	exists	exist	VERB
ejpam-5648	241	13	w	w	PROPN
ejpam-5648	241	14	∈	∈	PROPN
ejpam-5648	241	15	v	v	ADP
ejpam-5648	241	16	(	(	PUNCT
ejpam-5648	241	17	g	g	NOUN
ejpam-5648	241	18	)	)	PUNCT
ejpam-5648	241	19	such	such	ADJ
ejpam-5648	241	20	that	that	SCONJ
ejpam-5648	241	21	|ng(w)|	|ng(w)|	NOUN
ejpam-5648	241	22	=	=	SYM
ejpam-5648	241	23	2	2	NUM
ejpam-5648	241	24	,	,	PUNCT
ejpam-5648	241	25	n2	n2	NOUN
ejpam-5648	241	26	g(w	g(w	PROPN
ejpam-5648	241	27	)	)	PUNCT
ejpam-5648	241	28	=	=	SYM
ejpam-5648	241	29	v	v	X
ejpam-5648	241	30	(	(	PUNCT
ejpam-5648	241	31	g	g	NOUN
ejpam-5648	241	32	)	)	PUNCT
ejpam-5648	241	33	\	\	PUNCT
ejpam-5648	242	1	ng[w	ng[w	PROPN
ejpam-5648	242	2	]	]	PUNCT
ejpam-5648	242	3	,	,	PUNCT
ejpam-5648	242	4	and	and	CCONJ
ejpam-5648	242	5	x	x	X
ejpam-5648	242	6	is	be	AUX
ejpam-5648	242	7	not	not	PART
ejpam-5648	242	8	a	a	DET
ejpam-5648	242	9	support	support	NOUN
ejpam-5648	242	10	vertex	vertex	NOUN
ejpam-5648	242	11	whenever	whenever	SCONJ
ejpam-5648	242	12	x	x	SYM
ejpam-5648	242	13	∈	∈	NOUN
ejpam-5648	242	14	ng(w	ng(w	NOUN
ejpam-5648	242	15	)	)	PUNCT
ejpam-5648	242	16	and	and	CCONJ
ejpam-5648	242	17	is	be	AUX
ejpam-5648	242	18	a	a	DET
ejpam-5648	242	19	dominating	dominating	NOUN
ejpam-5648	242	20	vertex	vertex	NOUN
ejpam-5648	242	21	of	of	ADP
ejpam-5648	242	22	g.	g.	PROPN
ejpam-5648	242	23	let	let	VERB
ejpam-5648	242	24	v	v	NOUN
ejpam-5648	242	25	′	′	NOUN
ejpam-5648	242	26	0	0	NUM
ejpam-5648	243	1	=	=	SYM
ejpam-5648	243	2	v	v	X
ejpam-5648	243	3	(	(	PUNCT
ejpam-5648	243	4	g)\ng[w	g)\ng[w	PROPN
ejpam-5648	243	5	]	]	PUNCT
ejpam-5648	243	6	,	,	PUNCT
ejpam-5648	243	7	v	v	X
ejpam-5648	243	8	′	′	NOUN
ejpam-5648	243	9	1	1	NUM
ejpam-5648	243	10	=	=	NOUN
ejpam-5648	243	11	ng(w	ng(w	NOUN
ejpam-5648	243	12	)	)	PUNCT
ejpam-5648	243	13	,	,	PUNCT
ejpam-5648	243	14	and	and	CCONJ
ejpam-5648	243	15	v	v	X
ejpam-5648	243	16	′	′	NUM
ejpam-5648	243	17	2	2	NUM
ejpam-5648	243	18	=	=	SYM
ejpam-5648	243	19	{	{	PUNCT
ejpam-5648	243	20	w	w	NOUN
ejpam-5648	243	21	}	}	PUNCT
ejpam-5648	243	22	.	.	PUNCT
ejpam-5648	244	1	then	then	ADV
ejpam-5648	244	2	g	g	PROPN
ejpam-5648	244	3	=	=	PUNCT
ejpam-5648	244	4	(	(	PUNCT
ejpam-5648	244	5	v	v	NUM
ejpam-5648	244	6	′	′	NUM
ejpam-5648	244	7	0	0	NUM
ejpam-5648	244	8	,	,	PUNCT
ejpam-5648	244	9	v	v	NOUN
ejpam-5648	244	10	′	′	NUM
ejpam-5648	244	11	1	1	NUM
ejpam-5648	244	12	,	,	PUNCT
ejpam-5648	244	13	v	v	NOUN
ejpam-5648	244	14	′	′	NUM
ejpam-5648	244	15	2	2	NUM
ejpam-5648	244	16	)	)	PUNCT
ejpam-5648	244	17	∈	∈	PROPN
ejpam-5648	244	18	chrdf	chrdf	NOUN
ejpam-5648	244	19	(	(	PUNCT
ejpam-5648	244	20	g	g	NOUN
ejpam-5648	244	21	)	)	PUNCT
ejpam-5648	244	22	and	and	CCONJ
ejpam-5648	244	23	γcrh(g	γcrh(g	NOUN
ejpam-5648	244	24	)	)	PUNCT
ejpam-5648	244	25	≤	≤	NOUN
ejpam-5648	244	26	ωcrh	ωcrh	ADV
ejpam-5648	244	27	g	g	PROPN
ejpam-5648	244	28	(	(	PUNCT
ejpam-5648	244	29	g	g	NOUN
ejpam-5648	244	30	)	)	PUNCT
ejpam-5648	244	31	=	=	PUNCT
ejpam-5648	244	32	|v	|v	PROPN
ejpam-5648	244	33	′	′	NOUN
ejpam-5648	244	34	1	1	NUM
ejpam-5648	245	1	|	|	ADV
ejpam-5648	246	1	+	+	CCONJ
ejpam-5648	246	2	2|v	2|v	NUM
ejpam-5648	247	1	′	′	NUM
ejpam-5648	247	2	2	2	NUM
ejpam-5648	248	1	|	|	ADV
ejpam-5648	248	2	=	=	NOUN
ejpam-5648	248	3	4	4	X
ejpam-5648	248	4	.	.	PUNCT
ejpam-5648	249	1	the	the	DET
ejpam-5648	249	2	assumption	assumption	NOUN
ejpam-5648	249	3	that	that	SCONJ
ejpam-5648	249	4	x	x	PRON
ejpam-5648	249	5	is	be	AUX
ejpam-5648	249	6	not	not	PART
ejpam-5648	249	7	a	a	DET
ejpam-5648	249	8	support	support	NOUN
ejpam-5648	249	9	vertex	vertex	NOUN
ejpam-5648	249	10	whenever	whenever	SCONJ
ejpam-5648	249	11	x	x	SYM
ejpam-5648	249	12	∈	∈	NOUN
ejpam-5648	249	13	ng(w	ng(w	NOUN
ejpam-5648	249	14	)	)	PUNCT
ejpam-5648	249	15	and	and	CCONJ
ejpam-5648	249	16	is	be	AUX
ejpam-5648	249	17	a	a	DET
ejpam-5648	249	18	dominating	dominating	NOUN
ejpam-5648	249	19	vertex	vertex	NOUN
ejpam-5648	249	20	of	of	ADP
ejpam-5648	249	21	g	g	NOUN
ejpam-5648	249	22	,	,	PUNCT
ejpam-5648	249	23	implies	imply	VERB
ejpam-5648	249	24	that	that	SCONJ
ejpam-5648	249	25	g	g	PROPN
ejpam-5648	249	26	̸=	̸=	PROPN
ejpam-5648	249	27	k1	k1	NOUN
ejpam-5648	249	28	+	+	NOUN
ejpam-5648	249	29	h	h	NOUN
ejpam-5648	249	30	for	for	ADP
ejpam-5648	249	31	any	any	DET
ejpam-5648	249	32	graph	graph	NOUN
ejpam-5648	249	33	h	h	NOUN
ejpam-5648	249	34	having	have	VERB
ejpam-5648	249	35	an	an	DET
ejpam-5648	249	36	isolated	isolated	ADJ
ejpam-5648	249	37	vertex	vertex	NOUN
ejpam-5648	249	38	.	.	PUNCT
ejpam-5648	250	1	therefore	therefore	ADV
ejpam-5648	250	2	,	,	PUNCT
ejpam-5648	250	3	γcrh(g	γcrh(g	NOUN
ejpam-5648	250	4	)	)	PUNCT
ejpam-5648	250	5	=	=	SYM
ejpam-5648	251	1	4	4	X
ejpam-5648	251	2	.	.	PUNCT
ejpam-5648	251	3	lastly	lastly	ADV
ejpam-5648	251	4	,	,	PUNCT
ejpam-5648	251	5	suppose	suppose	VERB
ejpam-5648	251	6	there	there	PRON
ejpam-5648	251	7	exist	exist	VERB
ejpam-5648	251	8	adjacent	adjacent	ADJ
ejpam-5648	251	9	vertices	vertex	NOUN
ejpam-5648	251	10	u	u	NOUN
ejpam-5648	251	11	,	,	PUNCT
ejpam-5648	251	12	v	v	NOUN
ejpam-5648	251	13	∈	∈	PROPN
ejpam-5648	251	14	v	v	NOUN
ejpam-5648	251	15	(	(	PUNCT
ejpam-5648	251	16	g	g	NOUN
ejpam-5648	251	17	)	)	PUNCT
ejpam-5648	251	18	such	such	ADJ
ejpam-5648	251	19	that	that	DET
ejpam-5648	251	20	|ng(u)|	|ng(u)|	PROPN
ejpam-5648	251	21	≥	≥	NUM
ejpam-5648	251	22	2	2	NUM
ejpam-5648	251	23	and	and	CCONJ
ejpam-5648	251	24	|ng(v)|	|ng(v)|	PROPN
ejpam-5648	251	25	≥	≥	NOUN
ejpam-5648	251	26	2	2	NUM
ejpam-5648	251	27	,	,	PUNCT
ejpam-5648	251	28	n2	n2	ADJ
ejpam-5648	251	29	g(u	g(u	PROPN
ejpam-5648	251	30	)	)	PUNCT
ejpam-5648	251	31	∪n2	∪n2	NOUN
ejpam-5648	251	32	g(v	g(v	PROPN
ejpam-5648	251	33	)	)	PUNCT
ejpam-5648	252	1	=	=	SYM
ejpam-5648	252	2	v	v	X
ejpam-5648	252	3	(	(	PUNCT
ejpam-5648	252	4	g	g	NOUN
ejpam-5648	252	5	)	)	PUNCT
ejpam-5648	252	6	\	\	NOUN
ejpam-5648	253	1	{	{	PUNCT
ejpam-5648	253	2	u	u	NOUN
ejpam-5648	253	3	,	,	PUNCT
ejpam-5648	253	4	v	v	NOUN
ejpam-5648	253	5	}	}	PUNCT
ejpam-5648	253	6	,	,	PUNCT
ejpam-5648	253	7	ng(u	ng(u	PROPN
ejpam-5648	253	8	)	)	PUNCT
ejpam-5648	253	9	∩ng(v	∩ng(v	PROPN
ejpam-5648	253	10	)	)	PUNCT
ejpam-5648	253	11	=	=	NOUN
ejpam-5648	253	12	∅	∅	NOUN
ejpam-5648	253	13	and	and	CCONJ
ejpam-5648	253	14	{	{	PUNCT
ejpam-5648	253	15	u	u	NOUN
ejpam-5648	253	16	,	,	PUNCT
ejpam-5648	253	17	v	v	NOUN
ejpam-5648	253	18	}	}	PUNCT
ejpam-5648	253	19	is	be	AUX
ejpam-5648	253	20	a	a	DET
ejpam-5648	253	21	γch	γch	NOUN
ejpam-5648	253	22	-	-	PUNCT
ejpam-5648	253	23	set	set	NOUN
ejpam-5648	253	24	in	in	ADP
ejpam-5648	253	25	g.	g.	PROPN
ejpam-5648	253	26	define	define	VERB
ejpam-5648	253	27	a	a	DET
ejpam-5648	253	28	function	function	NOUN
ejpam-5648	253	29	h	h	NOUN
ejpam-5648	253	30	:	:	PUNCT
ejpam-5648	253	31	v	v	X
ejpam-5648	253	32	(	(	PUNCT
ejpam-5648	253	33	g	g	NOUN
ejpam-5648	253	34	)	)	PUNCT
ejpam-5648	253	35	→	→	SYM
ejpam-5648	253	36	{	{	PUNCT
ejpam-5648	253	37	0	0	NUM
ejpam-5648	253	38	,	,	PUNCT
ejpam-5648	253	39	1	1	NUM
ejpam-5648	253	40	,	,	PUNCT
ejpam-5648	253	41	2	2	NUM
ejpam-5648	253	42	}	}	PUNCT
ejpam-5648	253	43	by	by	ADP
ejpam-5648	253	44	h(x	h(x	PROPN
ejpam-5648	253	45	)	)	PUNCT
ejpam-5648	254	1	=	=	PRON
ejpam-5648	254	2	{	{	PUNCT
ejpam-5648	254	3	2	2	NUM
ejpam-5648	254	4	,	,	PUNCT
ejpam-5648	254	5	if	if	SCONJ
ejpam-5648	254	6	x	x	SYM
ejpam-5648	254	7	∈	∈	PROPN
ejpam-5648	254	8	{	{	PUNCT
ejpam-5648	254	9	u	u	NOUN
ejpam-5648	254	10	,	,	PUNCT
ejpam-5648	254	11	v	v	NOUN
ejpam-5648	254	12	}	}	PUNCT
ejpam-5648	254	13	,	,	PUNCT
ejpam-5648	254	14	0	0	NUM
ejpam-5648	254	15	,	,	PUNCT
ejpam-5648	254	16	if	if	SCONJ
ejpam-5648	254	17	x	x	PROPN
ejpam-5648	254	18	∈	∈	PROPN
ejpam-5648	254	19	v	v	X
ejpam-5648	254	20	(	(	PUNCT
ejpam-5648	254	21	g	g	NOUN
ejpam-5648	254	22	)	)	PUNCT
ejpam-5648	254	23	\	\	NOUN
ejpam-5648	254	24	{	{	PUNCT
ejpam-5648	254	25	u	u	NOUN
ejpam-5648	254	26	,	,	PUNCT
ejpam-5648	254	27	v	v	NOUN
ejpam-5648	254	28	}	}	PUNCT
ejpam-5648	254	29	.	.	PUNCT
ejpam-5648	255	1	then	then	ADV
ejpam-5648	255	2	h	h	PROPN
ejpam-5648	255	3	∈	∈	PROPN
ejpam-5648	255	4	chrdf	chrdf	NOUN
ejpam-5648	255	5	(	(	PUNCT
ejpam-5648	255	6	g	g	NOUN
ejpam-5648	255	7	)	)	PUNCT
ejpam-5648	255	8	and	and	CCONJ
ejpam-5648	255	9	γcrh(g)(h	γcrh(g)(h	NOUN
ejpam-5648	255	10	)	)	PUNCT
ejpam-5648	255	11	≤	≤	NOUN
ejpam-5648	255	12	ωcrh	ωcrh	ADV
ejpam-5648	255	13	g	g	PROPN
ejpam-5648	255	14	=	=	SYM
ejpam-5648	255	15	h(u	h(u	PROPN
ejpam-5648	255	16	)	)	PUNCT
ejpam-5648	256	1	+	+	CCONJ
ejpam-5648	256	2	h(v	h(v	NOUN
ejpam-5648	256	3	)	)	PUNCT
ejpam-5648	256	4	=	=	PUNCT
ejpam-5648	257	1	4	4	X
ejpam-5648	257	2	.	.	PUNCT
ejpam-5648	257	3	now	now	ADV
ejpam-5648	257	4	,	,	PUNCT
ejpam-5648	257	5	suppose	suppose	VERB
ejpam-5648	257	6	γ(g	γ(g	NOUN
ejpam-5648	257	7	)	)	PUNCT
ejpam-5648	257	8	=	=	SYM
ejpam-5648	258	1	1	1	X
ejpam-5648	258	2	,	,	PUNCT
ejpam-5648	258	3	say	say	VERB
ejpam-5648	258	4	q	q	NOUN
ejpam-5648	258	5	is	be	AUX
ejpam-5648	258	6	a	a	DET
ejpam-5648	258	7	dominating	dominating	NOUN
ejpam-5648	258	8	vertex	vertex	NOUN
ejpam-5648	258	9	of	of	ADP
ejpam-5648	258	10	g.	g.	PROPN
ejpam-5648	258	11	then	then	ADV
ejpam-5648	258	12	clearly	clearly	ADV
ejpam-5648	258	13	,	,	PUNCT
ejpam-5648	258	14	q	q	NOUN
ejpam-5648	258	15	/∈	/∈	PUNCT
ejpam-5648	259	1	v	v	INTJ
ejpam-5648	259	2	(	(	PUNCT
ejpam-5648	259	3	g	g	NOUN
ejpam-5648	259	4	)	)	PUNCT
ejpam-5648	259	5	\	\	NOUN
ejpam-5648	259	6	{	{	PUNCT
ejpam-5648	259	7	u	u	NOUN
ejpam-5648	259	8	,	,	PUNCT
ejpam-5648	259	9	v	v	NOUN
ejpam-5648	259	10	}	}	PUNCT
ejpam-5648	259	11	since	since	SCONJ
ejpam-5648	259	12	n2	n2	ADJ
ejpam-5648	259	13	g(u	g(u	PROPN
ejpam-5648	259	14	)	)	PUNCT
ejpam-5648	259	15	∪	∪	ADP
ejpam-5648	259	16	n2	n2	ADJ
ejpam-5648	259	17	g(v	g(v	X
ejpam-5648	259	18	)	)	PUNCT
ejpam-5648	259	19	=	=	SYM
ejpam-5648	259	20	v	v	X
ejpam-5648	259	21	(	(	PUNCT
ejpam-5648	259	22	g	g	NOUN
ejpam-5648	259	23	)	)	PUNCT
ejpam-5648	259	24	\	\	NOUN
ejpam-5648	259	25	{	{	PUNCT
ejpam-5648	259	26	u	u	NOUN
ejpam-5648	259	27	,	,	PUNCT
ejpam-5648	259	28	v	v	NOUN
ejpam-5648	259	29	}	}	PUNCT
ejpam-5648	259	30	.	.	PUNCT
ejpam-5648	260	1	this	this	PRON
ejpam-5648	260	2	implies	imply	VERB
ejpam-5648	260	3	that	that	SCONJ
ejpam-5648	260	4	q	q	PUNCT
ejpam-5648	260	5	∈	∈	PROPN
ejpam-5648	260	6	{	{	PUNCT
ejpam-5648	260	7	u	u	NOUN
ejpam-5648	260	8	,	,	PUNCT
ejpam-5648	260	9	v	v	NOUN
ejpam-5648	260	10	}	}	PUNCT
ejpam-5648	260	11	.	.	PUNCT
ejpam-5648	261	1	this	this	PRON
ejpam-5648	261	2	,	,	PUNCT
ejpam-5648	261	3	however	however	ADV
ejpam-5648	261	4	,	,	PUNCT
ejpam-5648	261	5	is	be	AUX
ejpam-5648	261	6	not	not	PART
ejpam-5648	261	7	possible	possible	ADJ
ejpam-5648	261	8	because	because	SCONJ
ejpam-5648	261	9	ng(u	ng(u	NOUN
ejpam-5648	261	10	)	)	PUNCT
ejpam-5648	261	11	∩	∩	NOUN
ejpam-5648	261	12	ng(v	ng(v	X
ejpam-5648	261	13	)	)	PUNCT
ejpam-5648	261	14	=	=	PUNCT
ejpam-5648	261	15	∅.	∅.	ADP
ejpam-5648	261	16	thus	thus	ADV
ejpam-5648	261	17	,	,	PUNCT
ejpam-5648	261	18	γ(g	γ(g	PROPN
ejpam-5648	261	19	)	)	PUNCT
ejpam-5648	261	20	̸=	̸=	PROPN
ejpam-5648	261	21	1	1	NUM
ejpam-5648	261	22	.	.	PUNCT
ejpam-5648	262	1	therefore	therefore	ADV
ejpam-5648	262	2	,	,	PUNCT
ejpam-5648	262	3	g	g	PROPN
ejpam-5648	262	4	̸=	̸=	PROPN
ejpam-5648	262	5	k1	k1	NOUN
ejpam-5648	262	6	+	+	NOUN
ejpam-5648	262	7	h	h	NOUN
ejpam-5648	262	8	for	for	ADP
ejpam-5648	262	9	any	any	DET
ejpam-5648	262	10	graph	graph	NOUN
ejpam-5648	262	11	h.	h.	PROPN
ejpam-5648	262	12	therefore	therefore	ADV
ejpam-5648	262	13	,	,	PUNCT
ejpam-5648	262	14	γcrh(g	γcrh(g	NOUN
ejpam-5648	262	15	)	)	PUNCT
ejpam-5648	262	16	=	=	SYM
ejpam-5648	262	17	4	4	X
ejpam-5648	262	18	.	.	PUNCT
ejpam-5648	263	1	the	the	DET
ejpam-5648	263	2	following	following	ADJ
ejpam-5648	263	3	result	result	NOUN
ejpam-5648	263	4	follows	follow	VERB
ejpam-5648	263	5	from	from	ADP
ejpam-5648	263	6	proposition	proposition	NOUN
ejpam-5648	263	7	3	3	NUM
ejpam-5648	263	8	.	.	PUNCT
ejpam-5648	263	9	corollary	corollary	ADJ
ejpam-5648	263	10	1	1	NUM
ejpam-5648	263	11	.	.	PUNCT
ejpam-5648	264	1	let	let	VERB
ejpam-5648	264	2	n	n	PRON
ejpam-5648	264	3	,	,	PUNCT
ejpam-5648	264	4	m	m	AUX
ejpam-5648	264	5	be	be	VERB
ejpam-5648	264	6	positive	positive	ADJ
ejpam-5648	264	7	integers	integer	NOUN
ejpam-5648	264	8	.	.	PUNCT
ejpam-5648	265	1	then	then	ADV
ejpam-5648	265	2	(	(	PUNCT
ejpam-5648	265	3	i	i	NOUN
ejpam-5648	265	4	)	)	PUNCT
ejpam-5648	265	5	γcrh(kn	γcrh(kn	NOUN
ejpam-5648	265	6	)	)	PUNCT
ejpam-5648	265	7	=	=	SYM
ejpam-5648	266	1	n	n	PROPN
ejpam-5648	266	2	for	for	ADP
ejpam-5648	266	3	all	all	DET
ejpam-5648	266	4	n	n	PRON
ejpam-5648	266	5	≥	≥	NOUN
ejpam-5648	266	6	1	1	NUM
ejpam-5648	266	7	;	;	PUNCT
ejpam-5648	266	8	(	(	PUNCT
ejpam-5648	266	9	ii	ii	NOUN
ejpam-5648	266	10	)	)	PUNCT
ejpam-5648	266	11	γcrh(k1,n	γcrh(k1,n	PROPN
ejpam-5648	266	12	)	)	PUNCT
ejpam-5648	266	13	=	=	SYM
ejpam-5648	266	14	3	3	NUM
ejpam-5648	266	15	for	for	ADP
ejpam-5648	266	16	all	all	DET
ejpam-5648	266	17	n	n	PRON
ejpam-5648	266	18	≥	≥	NOUN
ejpam-5648	266	19	2	2	NUM
ejpam-5648	266	20	;	;	PUNCT
ejpam-5648	266	21	(	(	PUNCT
ejpam-5648	266	22	iii	iii	X
ejpam-5648	266	23	)	)	PUNCT
ejpam-5648	266	24	γcrh(kn	γcrh(kn	NOUN
ejpam-5648	266	25	,	,	PUNCT
ejpam-5648	266	26	m	m	NOUN
ejpam-5648	266	27	)	)	PUNCT
ejpam-5648	266	28	=	=	SYM
ejpam-5648	266	29	4	4	NUM
ejpam-5648	266	30	for	for	ADP
ejpam-5648	266	31	all	all	DET
ejpam-5648	266	32	n	n	CCONJ
ejpam-5648	266	33	,	,	PUNCT
ejpam-5648	266	34	m	m	VERB
ejpam-5648	266	35	≥	≥	NOUN
ejpam-5648	266	36	2	2	NUM
ejpam-5648	266	37	;	;	PUNCT
ejpam-5648	266	38	(	(	PUNCT
ejpam-5648	266	39	iv	iv	X
ejpam-5648	266	40	)	)	PUNCT
ejpam-5648	266	41	γcrh(p	γcrh(p	PROPN
ejpam-5648	266	42	)	)	PUNCT
ejpam-5648	267	1	=	=	SYM
ejpam-5648	267	2	4	4	NUM
ejpam-5648	267	3	,	,	PUNCT
ejpam-5648	267	4	where	where	SCONJ
ejpam-5648	267	5	p	p	NOUN
ejpam-5648	267	6	is	be	AUX
ejpam-5648	267	7	the	the	DET
ejpam-5648	267	8	petersen	petersen	NOUN
ejpam-5648	267	9	graph	graph	NOUN
ejpam-5648	267	10	;	;	PUNCT
ejpam-5648	267	11	and	and	CCONJ
ejpam-5648	267	12	(	(	PUNCT
ejpam-5648	267	13	v	v	NOUN
ejpam-5648	267	14	)	)	PUNCT
ejpam-5648	267	15	γcrh(fn	γcrh(fn	NOUN
ejpam-5648	267	16	)	)	PUNCT
ejpam-5648	267	17	=	=	SYM
ejpam-5648	267	18	4	4	NUM
ejpam-5648	267	19	for	for	ADP
ejpam-5648	267	20	all	all	DET
ejpam-5648	267	21	n	n	PRON
ejpam-5648	267	22	≥	≥	NOUN
ejpam-5648	267	23	3	3	NUM
ejpam-5648	267	24	.	.	PUNCT
ejpam-5648	267	25	proposition	proposition	NOUN
ejpam-5648	267	26	4	4	NUM
ejpam-5648	267	27	.	.	X
ejpam-5648	268	1	for	for	ADP
ejpam-5648	268	2	any	any	DET
ejpam-5648	268	3	wheel	wheel	NOUN
ejpam-5648	268	4	graph	graph	NOUN
ejpam-5648	268	5	wn	wn	NOUN
ejpam-5648	268	6	with	with	ADP
ejpam-5648	268	7	n	n	PRON
ejpam-5648	268	8	≥	≥	NUM
ejpam-5648	268	9	4	4	NUM
ejpam-5648	268	10	,	,	PUNCT
ejpam-5648	268	11	γcrh(wn	γcrh(wn	NOUN
ejpam-5648	268	12	)	)	PUNCT
ejpam-5648	268	13	=	=	SYM
ejpam-5648	268	14	5	5	X
ejpam-5648	268	15	.	.	PUNCT
ejpam-5648	268	16	proof	proof	NOUN
ejpam-5648	268	17	.	.	PUNCT
ejpam-5648	269	1	clearly	clearly	ADV
ejpam-5648	269	2	,	,	PUNCT
ejpam-5648	269	3	γcrh(wn	γcrh(wn	PROPN
ejpam-5648	269	4	)	)	PUNCT
ejpam-5648	269	5	≥	≥	NOUN
ejpam-5648	269	6	4	4	NUM
ejpam-5648	269	7	.	.	PUNCT
ejpam-5648	270	1	let	let	VERB
ejpam-5648	270	2	v0	v0	NOUN
ejpam-5648	270	3	be	be	AUX
ejpam-5648	270	4	the	the	DET
ejpam-5648	270	5	hub	hub	NOUN
ejpam-5648	270	6	(	(	PUNCT
ejpam-5648	270	7	central	central	ADJ
ejpam-5648	270	8	)	)	PUNCT
ejpam-5648	270	9	vertex	vertex	NOUN
ejpam-5648	270	10	of	of	ADP
ejpam-5648	270	11	the	the	DET
ejpam-5648	270	12	wheel	wheel	NOUN
ejpam-5648	270	13	graph	graph	NOUN
ejpam-5648	270	14	wn	wn	PROPN
ejpam-5648	270	15	=	=	PROPN
ejpam-5648	270	16	k1	k1	PROPN
ejpam-5648	271	1	+	+	CCONJ
ejpam-5648	271	2	cn	cn	PROPN
ejpam-5648	271	3	and	and	CCONJ
ejpam-5648	271	4	let	let	VERB
ejpam-5648	271	5	v	v	X
ejpam-5648	271	6	(	(	PUNCT
ejpam-5648	271	7	wn	wn	PROPN
ejpam-5648	271	8	)	)	PUNCT
ejpam-5648	271	9	\	\	PROPN
ejpam-5648	271	10	{	{	PUNCT
ejpam-5648	271	11	v0	v0	NOUN
ejpam-5648	271	12	}	}	PUNCT
ejpam-5648	271	13	=	=	SYM
ejpam-5648	271	14	v	v	X
ejpam-5648	271	15	(	(	PUNCT
ejpam-5648	271	16	cn	cn	PROPN
ejpam-5648	271	17	)	)	PUNCT
ejpam-5648	271	18	=	=	SYM
ejpam-5648	271	19	{	{	PUNCT
ejpam-5648	271	20	v1	v1	PROPN
ejpam-5648	271	21	,	,	PUNCT
ejpam-5648	271	22	v2	v2	PROPN
ejpam-5648	271	23	,	,	PUNCT
ejpam-5648	271	24	.	.	PUNCT
ejpam-5648	271	25	.	.	PUNCT
ejpam-5648	272	1	.	.	PUNCT
ejpam-5648	273	1	,	,	PUNCT
ejpam-5648	273	2	vn	vn	PROPN
ejpam-5648	273	3	}	}	PUNCT
ejpam-5648	273	4	,	,	PUNCT
ejpam-5648	273	5	where	where	SCONJ
ejpam-5648	273	6	cn	cn	PROPN
ejpam-5648	273	7	=	=	PUNCT
ejpam-5648	273	8	[	[	X
ejpam-5648	273	9	v1	v1	NOUN
ejpam-5648	273	10	,	,	PUNCT
ejpam-5648	273	11	v2	v2	NOUN
ejpam-5648	273	12	,	,	PUNCT
ejpam-5648	273	13	.	.	PUNCT
ejpam-5648	273	14	.	.	PUNCT
ejpam-5648	274	1	.	.	PUNCT
ejpam-5648	275	1	,	,	PUNCT
ejpam-5648	275	2	vn	vn	X
ejpam-5648	275	3	,	,	PUNCT
ejpam-5648	275	4	v1	v1	PROPN
ejpam-5648	275	5	]	]	PUNCT
ejpam-5648	275	6	.	.	PUNCT
ejpam-5648	276	1	define	define	VERB
ejpam-5648	276	2	a	a	DET
ejpam-5648	276	3	function	function	NOUN
ejpam-5648	276	4	f	f	NOUN
ejpam-5648	276	5	:	:	PUNCT
ejpam-5648	276	6	v	v	PROPN
ejpam-5648	276	7	(	(	PUNCT
ejpam-5648	276	8	wn	wn	PROPN
ejpam-5648	276	9	)	)	PUNCT
ejpam-5648	276	10	→	→	SYM
ejpam-5648	276	11	{	{	PUNCT
ejpam-5648	276	12	0	0	NUM
ejpam-5648	276	13	,	,	PUNCT
ejpam-5648	276	14	1	1	NUM
ejpam-5648	276	15	,	,	PUNCT
ejpam-5648	276	16	2	2	NUM
ejpam-5648	276	17	}	}	PUNCT
ejpam-5648	276	18	by	by	ADP
ejpam-5648	276	19	f(v	f(v	NOUN
ejpam-5648	276	20	)	)	PUNCT
ejpam-5648	277	1	=	=	SYM
ejpam-5648	277	2			NOUN
ejpam-5648	277	3	2	2	NUM
ejpam-5648	277	4	,	,	PUNCT
ejpam-5648	277	5	if	if	SCONJ
ejpam-5648	277	6	v	v	NUM
ejpam-5648	277	7	∈	∈	PROPN
ejpam-5648	277	8	{	{	PUNCT
ejpam-5648	277	9	v1	v1	NOUN
ejpam-5648	277	10	,	,	PUNCT
ejpam-5648	277	11	v2	v2	NOUN
ejpam-5648	277	12	}	}	PUNCT
ejpam-5648	277	13	1	1	NUM
ejpam-5648	277	14	,	,	PUNCT
ejpam-5648	277	15	if	if	SCONJ
ejpam-5648	277	16	v	v	NOUN
ejpam-5648	277	17	=	=	SYM
ejpam-5648	277	18	v0	v0	NOUN
ejpam-5648	277	19	0	0	NUM
ejpam-5648	277	20	,	,	PUNCT
ejpam-5648	277	21	if	if	SCONJ
ejpam-5648	277	22	v	v	NUM
ejpam-5648	277	23	∈	∈	PROPN
ejpam-5648	277	24	v	v	NOUN
ejpam-5648	277	25	(	(	PUNCT
ejpam-5648	277	26	wn	wn	PROPN
ejpam-5648	277	27	)	)	PUNCT
ejpam-5648	277	28	\	\	PROPN
ejpam-5648	277	29	{	{	PUNCT
ejpam-5648	277	30	v0	v0	NOUN
ejpam-5648	277	31	,	,	PUNCT
ejpam-5648	277	32	v1	v1	NOUN
ejpam-5648	277	33	,	,	PUNCT
ejpam-5648	277	34	v2	v2	PROPN
ejpam-5648	277	35	}	}	PUNCT
ejpam-5648	277	36	.	.	PUNCT
ejpam-5648	278	1	a.	a.	PROPN
ejpam-5648	278	2	aradais	aradais	PROPN
ejpam-5648	278	3	,	,	PUNCT
ejpam-5648	278	4	j.	j.	PROPN
ejpam-5648	278	5	cariaga	cariaga	PROPN
ejpam-5648	278	6	,	,	PUNCT
ejpam-5648	278	7	s.	s.	PROPN
ejpam-5648	278	8	canoy	canoy	PROPN
ejpam-5648	278	9	jr	jr	PROPN
ejpam-5648	278	10	.	.	PROPN
ejpam-5648	278	11	/	/	SYM
ejpam-5648	278	12	eur	eur	PROPN
ejpam-5648	278	13	.	.	PUNCT
ejpam-5648	279	1	j.	j.	PROPN
ejpam-5648	279	2	pure	pure	PROPN
ejpam-5648	279	3	appl	appl	PROPN
ejpam-5648	279	4	.	.	PROPN
ejpam-5648	279	5	math	math	PROPN
ejpam-5648	279	6	,	,	PUNCT
ejpam-5648	279	7	18	18	NUM
ejpam-5648	279	8	(	(	PUNCT
ejpam-5648	279	9	1	1	NUM
ejpam-5648	279	10	)	)	PUNCT
ejpam-5648	279	11	(	(	PUNCT
ejpam-5648	279	12	2025	2025	NUM
ejpam-5648	279	13	)	)	PUNCT
ejpam-5648	279	14	,	,	PUNCT
ejpam-5648	279	15	5648	5648	NUM
ejpam-5648	279	16	8	8	NUM
ejpam-5648	279	17	of	of	ADP
ejpam-5648	279	18	13	13	NUM
ejpam-5648	279	19	then	then	ADV
ejpam-5648	279	20	f	f	PROPN
ejpam-5648	279	21	∈	∈	PROPN
ejpam-5648	279	22	chrdf	chrdf	NOUN
ejpam-5648	279	23	(	(	PUNCT
ejpam-5648	279	24	wn	wn	PROPN
ejpam-5648	279	25	)	)	PUNCT
ejpam-5648	279	26	and	and	CCONJ
ejpam-5648	279	27	γcrh(wn	γcrh(wn	NOUN
ejpam-5648	279	28	)	)	PUNCT
ejpam-5648	279	29	≤	≤	PROPN
ejpam-5648	279	30	ωcrh	ωcrh	ADJ
ejpam-5648	279	31	wn	wn	PROPN
ejpam-5648	279	32	(	(	PUNCT
ejpam-5648	279	33	f	f	X
ejpam-5648	279	34	)	)	PUNCT
ejpam-5648	279	35	=	=	SYM
ejpam-5648	279	36	f(v0	f(v0	ADJ
ejpam-5648	279	37	)	)	PUNCT
ejpam-5648	280	1	+	+	SYM
ejpam-5648	280	2	f(v1	f(v1	ADJ
ejpam-5648	280	3	)	)	PUNCT
ejpam-5648	280	4	+	+	NOUN
ejpam-5648	280	5	f(v2	f(v2	NOUN
ejpam-5648	280	6	)	)	PUNCT
ejpam-5648	280	7	=	=	SYM
ejpam-5648	281	1	5	5	X
ejpam-5648	281	2	.	.	PUNCT
ejpam-5648	281	3	since	since	SCONJ
ejpam-5648	281	4	γch(wn	γch(wn	NUM
ejpam-5648	281	5	)	)	PUNCT
ejpam-5648	281	6	=	=	SYM
ejpam-5648	281	7	3	3	NUM
ejpam-5648	281	8	and	and	CCONJ
ejpam-5648	281	9	|nwn(x)|	|nwn(x)|	NOUN
ejpam-5648	281	10	≥	≥	VERB
ejpam-5648	281	11	3	3	NUM
ejpam-5648	281	12	for	for	ADP
ejpam-5648	281	13	every	every	DET
ejpam-5648	281	14	x	x	SYM
ejpam-5648	281	15	∈	∈	PROPN
ejpam-5648	281	16	v	v	NOUN
ejpam-5648	281	17	(	(	PUNCT
ejpam-5648	281	18	wn	wn	PROPN
ejpam-5648	281	19	)	)	PUNCT
ejpam-5648	281	20	,	,	PUNCT
ejpam-5648	281	21	wn	wn	PROPN
ejpam-5648	281	22	does	do	AUX
ejpam-5648	281	23	not	not	PART
ejpam-5648	281	24	satisfy	satisfy	VERB
ejpam-5648	281	25	(	(	PUNCT
ejpam-5648	281	26	a	a	X
ejpam-5648	281	27	)	)	PUNCT
ejpam-5648	281	28	and	and	CCONJ
ejpam-5648	281	29	(	(	PUNCT
ejpam-5648	281	30	b	b	NOUN
ejpam-5648	281	31	)	)	PUNCT
ejpam-5648	281	32	of	of	ADP
ejpam-5648	281	33	proposition	proposition	NOUN
ejpam-5648	281	34	3(iv	3(iv	NUM
ejpam-5648	281	35	)	)	PUNCT
ejpam-5648	281	36	.	.	PUNCT
ejpam-5648	282	1	consequently	consequently	ADV
ejpam-5648	282	2	,	,	PUNCT
ejpam-5648	282	3	γcrh(wn	γcrh(wn	PROPN
ejpam-5648	282	4	)	)	PUNCT
ejpam-5648	282	5	=	=	SYM
ejpam-5648	283	1	5	5	X
ejpam-5648	283	2	.	.	X
ejpam-5648	283	3	proposition	proposition	NOUN
ejpam-5648	283	4	5	5	NUM
ejpam-5648	283	5	.	.	PUNCT
ejpam-5648	284	1	for	for	ADP
ejpam-5648	284	2	any	any	DET
ejpam-5648	284	3	path	path	NOUN
ejpam-5648	284	4	pn	pn	NOUN
ejpam-5648	284	5	of	of	ADP
ejpam-5648	284	6	order	order	NOUN
ejpam-5648	284	7	n	n	PRON
ejpam-5648	284	8	≥	≥	NOUN
ejpam-5648	284	9	1	1	NUM
ejpam-5648	284	10	,	,	PUNCT
ejpam-5648	284	11	γcrh(pn	γcrh(pn	NOUN
ejpam-5648	284	12	)	)	PUNCT
ejpam-5648	284	13	=	=	PUNCT
ejpam-5648	284	14			NOUN
ejpam-5648	284	15	4	4	NUM
ejpam-5648	284	16	,	,	PUNCT
ejpam-5648	284	17	if	if	SCONJ
ejpam-5648	284	18	n	n	NOUN
ejpam-5648	284	19	=	=	SYM
ejpam-5648	284	20	5	5	NUM
ejpam-5648	284	21	,	,	PUNCT
ejpam-5648	284	22	6	6	NUM
ejpam-5648	284	23	6	6	NUM
ejpam-5648	284	24	,	,	PUNCT
ejpam-5648	284	25	if	if	SCONJ
ejpam-5648	284	26	n	n	NOUN
ejpam-5648	284	27	=	=	SYM
ejpam-5648	284	28	7	7	NUM
ejpam-5648	284	29	n	n	CCONJ
ejpam-5648	284	30	,	,	PUNCT
ejpam-5648	284	31	if	if	SCONJ
ejpam-5648	284	32	n	n	PRON
ejpam-5648	284	33	̸=	̸=	PROPN
ejpam-5648	284	34	5	5	NUM
ejpam-5648	284	35	,	,	PUNCT
ejpam-5648	284	36	6	6	NUM
ejpam-5648	284	37	,	,	PUNCT
ejpam-5648	284	38	7	7	NUM
ejpam-5648	284	39	.	.	PUNCT
ejpam-5648	284	40	proof	proof	NOUN
ejpam-5648	284	41	.	.	PUNCT
ejpam-5648	285	1	for	for	ADP
ejpam-5648	285	2	n	n	NOUN
ejpam-5648	285	3	=	=	SYM
ejpam-5648	285	4	1	1	NUM
ejpam-5648	285	5	,	,	PUNCT
ejpam-5648	285	6	2	2	NUM
ejpam-5648	285	7	,	,	PUNCT
ejpam-5648	285	8	3	3	NUM
ejpam-5648	285	9	,	,	PUNCT
ejpam-5648	285	10	the	the	DET
ejpam-5648	285	11	result	result	NOUN
ejpam-5648	285	12	follows	follow	VERB
ejpam-5648	285	13	from	from	ADP
ejpam-5648	285	14	propositon	propositon	NOUN
ejpam-5648	285	15	3	3	NUM
ejpam-5648	285	16	(	(	PUNCT
ejpam-5648	285	17	i	i	NOUN
ejpam-5648	285	18	)	)	PUNCT
ejpam-5648	285	19	,	,	PUNCT
ejpam-5648	285	20	(	(	PUNCT
ejpam-5648	285	21	ii	ii	NOUN
ejpam-5648	285	22	)	)	PUNCT
ejpam-5648	285	23	,	,	PUNCT
ejpam-5648	285	24	(	(	PUNCT
ejpam-5648	285	25	iii	iii	NOUN
ejpam-5648	285	26	)	)	PUNCT
ejpam-5648	285	27	.	.	PUNCT
ejpam-5648	286	1	by	by	ADP
ejpam-5648	286	2	proposition	proposition	NOUN
ejpam-5648	286	3	3	3	NUM
ejpam-5648	286	4	(	(	PUNCT
ejpam-5648	286	5	iv	iv	NUM
ejpam-5648	286	6	)	)	PUNCT
ejpam-5648	286	7	,	,	PUNCT
ejpam-5648	286	8	γcrh(pn	γcrh(pn	NOUN
ejpam-5648	286	9	)	)	PUNCT
ejpam-5648	286	10	=	=	SYM
ejpam-5648	286	11	4	4	NUM
ejpam-5648	286	12	for	for	ADP
ejpam-5648	286	13	n	n	NOUN
ejpam-5648	286	14	=	=	SYM
ejpam-5648	286	15	4	4	NUM
ejpam-5648	286	16	,	,	PUNCT
ejpam-5648	286	17	5	5	NUM
ejpam-5648	286	18	,	,	PUNCT
ejpam-5648	286	19	6	6	NUM
ejpam-5648	286	20	.	.	PUNCT
ejpam-5648	286	21	let	let	VERB
ejpam-5648	286	22	n	n	PRON
ejpam-5648	286	23	≥	≥	X
ejpam-5648	286	24	7	7	NUM
ejpam-5648	286	25	and	and	CCONJ
ejpam-5648	286	26	let	let	VERB
ejpam-5648	286	27	pn	pn	VERB
ejpam-5648	286	28	=	=	PUNCT
ejpam-5648	287	1	[	[	X
ejpam-5648	287	2	x1	x1	PROPN
ejpam-5648	287	3	,	,	PUNCT
ejpam-5648	287	4	x2	x2	PROPN
ejpam-5648	287	5	,	,	PUNCT
ejpam-5648	287	6	...	...	PUNCT
ejpam-5648	287	7	,	,	PUNCT
ejpam-5648	287	8	xn	xn	PROPN
ejpam-5648	287	9	]	]	PUNCT
ejpam-5648	287	10	.	.	PUNCT
ejpam-5648	288	1	let	let	VERB
ejpam-5648	288	2	f	f	PROPN
ejpam-5648	288	3	=	=	SYM
ejpam-5648	288	4	(	(	PUNCT
ejpam-5648	288	5	v0	v0	PROPN
ejpam-5648	288	6	,	,	PUNCT
ejpam-5648	288	7	v1	v1	NOUN
ejpam-5648	288	8	,	,	PUNCT
ejpam-5648	288	9	v2	v2	PROPN
ejpam-5648	288	10	)	)	PUNCT
ejpam-5648	288	11	be	be	AUX
ejpam-5648	288	12	a	a	DET
ejpam-5648	288	13	γcrh	γcrh	NOUN
ejpam-5648	288	14	-	-	PUNCT
ejpam-5648	288	15	function	function	NOUN
ejpam-5648	288	16	on	on	ADP
ejpam-5648	288	17	pn	pn	PROPN
ejpam-5648	288	18	.	.	PROPN
ejpam-5648	289	1	since	since	SCONJ
ejpam-5648	289	2	v1	v1	NOUN
ejpam-5648	289	3	∪	∪	NOUN
ejpam-5648	289	4	v2	v2	NOUN
ejpam-5648	289	5	is	be	AUX
ejpam-5648	289	6	a	a	DET
ejpam-5648	289	7	connected	connected	ADJ
ejpam-5648	289	8	hop	hop	NOUN
ejpam-5648	289	9	dominating	dominating	NOUN
ejpam-5648	289	10	set	set	NOUN
ejpam-5648	289	11	,	,	PUNCT
ejpam-5648	289	12	v0	v0	PROPN
ejpam-5648	289	13	⊆	⊆	NUM
ejpam-5648	289	14	{	{	PUNCT
ejpam-5648	289	15	x1	x1	PROPN
ejpam-5648	289	16	,	,	PUNCT
ejpam-5648	289	17	x2	x2	PROPN
ejpam-5648	289	18	,	,	PUNCT
ejpam-5648	289	19	xn−1	xn−1	PROPN
ejpam-5648	289	20	,	,	PUNCT
ejpam-5648	289	21	xn	xn	PROPN
ejpam-5648	289	22	}	}	PUNCT
ejpam-5648	289	23	.	.	PUNCT
ejpam-5648	290	1	suppose	suppose	VERB
ejpam-5648	290	2	first	first	ADV
ejpam-5648	290	3	that	that	SCONJ
ejpam-5648	290	4	n	n	NOUN
ejpam-5648	290	5	=	=	SYM
ejpam-5648	290	6	7	7	X
ejpam-5648	290	7	.	.	PUNCT
ejpam-5648	291	1	since	since	SCONJ
ejpam-5648	291	2	the	the	DET
ejpam-5648	291	3	function	function	NOUN
ejpam-5648	291	4	g	g	NOUN
ejpam-5648	291	5	=	=	SYM
ejpam-5648	291	6	(	(	PUNCT
ejpam-5648	291	7	{	{	PUNCT
ejpam-5648	291	8	x1	x1	PROPN
ejpam-5648	291	9	,	,	PUNCT
ejpam-5648	291	10	x2	x2	PROPN
ejpam-5648	291	11	,	,	PUNCT
ejpam-5648	291	12	x6	x6	PROPN
ejpam-5648	291	13	,	,	PUNCT
ejpam-5648	291	14	x7},∅	x7},∅	PROPN
ejpam-5648	291	15	,	,	PUNCT
ejpam-5648	291	16	{	{	PUNCT
ejpam-5648	291	17	x3	x3	ADJ
ejpam-5648	291	18	,	,	PUNCT
ejpam-5648	291	19	x4	x4	PROPN
ejpam-5648	291	20	,	,	PUNCT
ejpam-5648	291	21	x5	x5	NOUN
ejpam-5648	291	22	}	}	PUNCT
ejpam-5648	291	23	)	)	PUNCT
ejpam-5648	291	24	is	be	AUX
ejpam-5648	291	25	a	a	DET
ejpam-5648	291	26	connected	connected	ADJ
ejpam-5648	291	27	hop	hop	NOUN
ejpam-5648	291	28	dominating	dominating	NOUN
ejpam-5648	291	29	function	function	NOUN
ejpam-5648	291	30	on	on	ADP
ejpam-5648	291	31	p7	p7	PROPN
ejpam-5648	291	32	,	,	PUNCT
ejpam-5648	291	33	and	and	CCONJ
ejpam-5648	291	34	p7	p7	PROPN
ejpam-5648	291	35	does	do	AUX
ejpam-5648	291	36	not	not	PART
ejpam-5648	291	37	satisfy	satisfy	VERB
ejpam-5648	291	38	any	any	PRON
ejpam-5648	291	39	of	of	ADP
ejpam-5648	291	40	the	the	DET
ejpam-5648	291	41	conditions	condition	NOUN
ejpam-5648	291	42	given	give	VERB
ejpam-5648	291	43	in	in	ADP
ejpam-5648	291	44	proposition	proposition	NOUN
ejpam-5648	291	45	3	3	NUM
ejpam-5648	291	46	,	,	PUNCT
ejpam-5648	291	47	it	it	PRON
ejpam-5648	291	48	follows	follow	VERB
ejpam-5648	291	49	that	that	SCONJ
ejpam-5648	291	50	5	5	NUM
ejpam-5648	291	51	≤	≤	PROPN
ejpam-5648	291	52	γcrh(p7	γcrh(p7	NOUN
ejpam-5648	291	53	)	)	PUNCT
ejpam-5648	291	54	≤	≤	NUM
ejpam-5648	291	55	6	6	NUM
ejpam-5648	291	56	.	.	PUNCT
ejpam-5648	292	1	now	now	ADV
ejpam-5648	292	2	,	,	PUNCT
ejpam-5648	292	3	suppose	suppose	VERB
ejpam-5648	292	4	that	that	SCONJ
ejpam-5648	292	5	γcrh(p5	γcrh(p5	NOUN
ejpam-5648	292	6	)	)	PUNCT
ejpam-5648	292	7	=	=	SYM
ejpam-5648	292	8	|v1|+2|v2|	|v1|+2|v2|	NUM
ejpam-5648	292	9	=	=	SYM
ejpam-5648	292	10	5	5	X
ejpam-5648	292	11	.	.	PUNCT
ejpam-5648	293	1	if	if	SCONJ
ejpam-5648	293	2	v0	v0	NOUN
ejpam-5648	293	3	=	=	SYM
ejpam-5648	293	4	∅	∅	NOUN
ejpam-5648	293	5	,	,	PUNCT
ejpam-5648	293	6	then	then	ADV
ejpam-5648	293	7	|v2|	|v2|	ADV
ejpam-5648	293	8	=	=	SYM
ejpam-5648	293	9	0	0	NUM
ejpam-5648	293	10	and	and	CCONJ
ejpam-5648	293	11	|v1|	|v1|	NOUN
ejpam-5648	293	12	=	=	SYM
ejpam-5648	293	13	7	7	X
ejpam-5648	293	14	.	.	PUNCT
ejpam-5648	294	1	this	this	PRON
ejpam-5648	294	2	implies	imply	VERB
ejpam-5648	294	3	that	that	SCONJ
ejpam-5648	294	4	γcrh(p7	γcrh(p7	NOUN
ejpam-5648	294	5	)	)	PUNCT
ejpam-5648	294	6	=	=	SYM
ejpam-5648	294	7	7	7	NUM
ejpam-5648	294	8	which	which	PRON
ejpam-5648	294	9	is	be	AUX
ejpam-5648	294	10	not	not	PART
ejpam-5648	294	11	possivle	possivle	ADJ
ejpam-5648	294	12	.	.	PUNCT
ejpam-5648	295	1	thus	thus	ADV
ejpam-5648	295	2	,	,	PUNCT
ejpam-5648	295	3	|v0|	|v0|	NOUN
ejpam-5648	295	4	=	=	SYM
ejpam-5648	295	5	̸	̸	NUM
ejpam-5648	295	6	0	0	NUM
ejpam-5648	295	7	and	and	CCONJ
ejpam-5648	295	8	1	1	NUM
ejpam-5648	295	9	≤	≤	NOUN
ejpam-5648	295	10	|v2|	|v2|	NOUN
ejpam-5648	295	11	≤	≤	NOUN
ejpam-5648	295	12	2	2	NUM
ejpam-5648	295	13	.	.	PUNCT
ejpam-5648	295	14	suppose	suppose	VERB
ejpam-5648	295	15	|v2|	|v2|	NOUN
ejpam-5648	295	16	=	=	SYM
ejpam-5648	295	17	1	1	X
ejpam-5648	295	18	.	.	X
ejpam-5648	295	19	then	then	ADV
ejpam-5648	295	20	|v0|	|v0|	NOUN
ejpam-5648	295	21	=	=	SYM
ejpam-5648	295	22	3	3	NUM
ejpam-5648	295	23	and	and	CCONJ
ejpam-5648	295	24	|v1|	|v1|	NOUN
ejpam-5648	295	25	=	=	SYM
ejpam-5648	295	26	3	3	X
ejpam-5648	295	27	.	.	X
ejpam-5648	296	1	if	if	SCONJ
ejpam-5648	296	2	x1	x1	PROPN
ejpam-5648	296	3	∈	∈	PROPN
ejpam-5648	296	4	v1	v1	NOUN
ejpam-5648	296	5	∪	∪	NOUN
ejpam-5648	296	6	v2	v2	NOUN
ejpam-5648	296	7	,	,	PUNCT
ejpam-5648	296	8	then	then	ADV
ejpam-5648	296	9	v1	v1	VERB
ejpam-5648	296	10	∪	∪	ADJ
ejpam-5648	296	11	v2	v2	NOUN
ejpam-5648	296	12	=	=	SYM
ejpam-5648	296	13	{	{	PUNCT
ejpam-5648	296	14	x1	x1	PROPN
ejpam-5648	296	15	,	,	PUNCT
ejpam-5648	296	16	x2	x2	PROPN
ejpam-5648	296	17	,	,	PUNCT
ejpam-5648	296	18	x3	x3	ADJ
ejpam-5648	296	19	,	,	PUNCT
ejpam-5648	296	20	x4	x4	PROPN
ejpam-5648	296	21	}	}	PUNCT
ejpam-5648	296	22	since	since	SCONJ
ejpam-5648	296	23	⟨v1	⟨v1	PROPN
ejpam-5648	296	24	∪	∪	ADP
ejpam-5648	296	25	v2⟩	v2⟩	PROPN
ejpam-5648	296	26	is	be	AUX
ejpam-5648	296	27	connected	connect	VERB
ejpam-5648	296	28	.	.	PUNCT
ejpam-5648	297	1	hence	hence	ADV
ejpam-5648	297	2	,	,	PUNCT
ejpam-5648	297	3	x7	x7	NOUN
ejpam-5648	297	4	∈	∈	NOUN
ejpam-5648	297	5	v0	v0	NOUN
ejpam-5648	297	6	and	and	CCONJ
ejpam-5648	297	7	dp7(x7	dp7(x7	PROPN
ejpam-5648	297	8	,	,	PUNCT
ejpam-5648	297	9	y	y	NOUN
ejpam-5648	297	10	)	)	PUNCT
ejpam-5648	297	11	̸=	̸=	PROPN
ejpam-5648	297	12	2	2	NUM
ejpam-5648	297	13	for	for	ADP
ejpam-5648	297	14	all	all	DET
ejpam-5648	297	15	y	y	PROPN
ejpam-5648	297	16	∈	∈	PROPN
ejpam-5648	297	17	v2	v2	PROPN
ejpam-5648	297	18	,	,	PUNCT
ejpam-5648	297	19	a	a	DET
ejpam-5648	297	20	contradiction	contradiction	NOUN
ejpam-5648	297	21	.	.	PUNCT
ejpam-5648	298	1	thus	thus	ADV
ejpam-5648	298	2	,	,	PUNCT
ejpam-5648	298	3	x1	x1	PROPN
ejpam-5648	298	4	∈	∈	PROPN
ejpam-5648	298	5	v0	v0	NOUN
ejpam-5648	298	6	.	.	PUNCT
ejpam-5648	299	1	similarly	similarly	ADV
ejpam-5648	299	2	,	,	PUNCT
ejpam-5648	299	3	x7	x7	NOUN
ejpam-5648	299	4	∈	∈	PROPN
ejpam-5648	299	5	v0	v0	NOUN
ejpam-5648	299	6	.	.	PUNCT
ejpam-5648	300	1	it	it	PRON
ejpam-5648	300	2	follows	follow	VERB
ejpam-5648	300	3	that	that	SCONJ
ejpam-5648	300	4	v1	v1	NOUN
ejpam-5648	300	5	∪	∪	NOUN
ejpam-5648	300	6	v2	v2	NOUN
ejpam-5648	300	7	is	be	AUX
ejpam-5648	300	8	{	{	PUNCT
ejpam-5648	300	9	x2	x2	PROPN
ejpam-5648	300	10	,	,	PUNCT
ejpam-5648	300	11	x3	x3	ADJ
ejpam-5648	300	12	,	,	PUNCT
ejpam-5648	300	13	x4	x4	PROPN
ejpam-5648	300	14	,	,	PUNCT
ejpam-5648	300	15	x5	x5	NOUN
ejpam-5648	300	16	}	}	PUNCT
ejpam-5648	300	17	or	or	CCONJ
ejpam-5648	300	18	{	{	PUNCT
ejpam-5648	300	19	x3	x3	ADJ
ejpam-5648	300	20	,	,	PUNCT
ejpam-5648	300	21	x4	x4	PROPN
ejpam-5648	300	22	,	,	PUNCT
ejpam-5648	300	23	x5	x5	PROPN
ejpam-5648	300	24	,	,	PUNCT
ejpam-5648	300	25	x6	x6	PROPN
ejpam-5648	300	26	}	}	PUNCT
ejpam-5648	300	27	.	.	PUNCT
ejpam-5648	301	1	this	this	PRON
ejpam-5648	301	2	implies	imply	VERB
ejpam-5648	301	3	that	that	SCONJ
ejpam-5648	301	4	x1	x1	PROPN
ejpam-5648	301	5	/∈	/∈	PUNCT
ejpam-5648	301	6	n2	n2	ADJ
ejpam-5648	301	7	p7	p7	PROPN
ejpam-5648	301	8	(	(	PUNCT
ejpam-5648	301	9	v2	v2	PROPN
ejpam-5648	301	10	)	)	PUNCT
ejpam-5648	301	11	or	or	CCONJ
ejpam-5648	301	12	x7	x7	NOUN
ejpam-5648	301	13	/∈	/∈	PUNCT
ejpam-5648	301	14	n2	n2	ADJ
ejpam-5648	301	15	p7	p7	PROPN
ejpam-5648	301	16	(	(	PUNCT
ejpam-5648	301	17	v2	v2	PROPN
ejpam-5648	301	18	)	)	PUNCT
ejpam-5648	301	19	,	,	PUNCT
ejpam-5648	301	20	a	a	DET
ejpam-5648	301	21	contradiction	contradiction	NOUN
ejpam-5648	301	22	.	.	PUNCT
ejpam-5648	302	1	this	this	DET
ejpam-5648	302	2	forces	force	NOUN
ejpam-5648	302	3	|v2|	|v2|	NOUN
ejpam-5648	302	4	=	=	SYM
ejpam-5648	302	5	2	2	NUM
ejpam-5648	302	6	and	and	CCONJ
ejpam-5648	302	7	|v1|	|v1|	NOUN
ejpam-5648	302	8	=	=	SYM
ejpam-5648	302	9	1	1	X
ejpam-5648	302	10	.	.	PUNCT
ejpam-5648	303	1	it	it	PRON
ejpam-5648	303	2	is	be	AUX
ejpam-5648	303	3	routine	routine	ADJ
ejpam-5648	303	4	to	to	PART
ejpam-5648	303	5	show	show	VERB
ejpam-5648	303	6	that	that	SCONJ
ejpam-5648	303	7	this	this	PRON
ejpam-5648	303	8	also	also	ADV
ejpam-5648	303	9	leads	lead	VERB
ejpam-5648	303	10	to	to	ADP
ejpam-5648	303	11	a	a	DET
ejpam-5648	303	12	contradiction	contradiction	NOUN
ejpam-5648	303	13	.	.	PUNCT
ejpam-5648	304	1	therefore	therefore	ADV
ejpam-5648	304	2	,	,	PUNCT
ejpam-5648	304	3	γcrh(p7	γcrh(p7	NOUN
ejpam-5648	304	4	)	)	PUNCT
ejpam-5648	304	5	=	=	PUNCT
ejpam-5648	305	1	6	6	X
ejpam-5648	305	2	.	.	PUNCT
ejpam-5648	306	1	next	next	ADV
ejpam-5648	306	2	,	,	PUNCT
ejpam-5648	306	3	suppose	suppose	VERB
ejpam-5648	306	4	that	that	SCONJ
ejpam-5648	306	5	n	n	PROPN
ejpam-5648	306	6	≥	≥	NUM
ejpam-5648	306	7	8	8	NUM
ejpam-5648	306	8	.	.	PUNCT
ejpam-5648	307	1	if	if	SCONJ
ejpam-5648	307	2	|v0|	|v0|	NOUN
ejpam-5648	307	3	=	=	SYM
ejpam-5648	307	4	0	0	NUM
ejpam-5648	307	5	,	,	PUNCT
ejpam-5648	307	6	then	then	ADV
ejpam-5648	307	7	|v2|	|v2|	ADV
ejpam-5648	307	8	=	=	SYM
ejpam-5648	307	9	0	0	NUM
ejpam-5648	307	10	and	and	CCONJ
ejpam-5648	307	11	|v1|	|v1|	NOUN
ejpam-5648	307	12	=	=	SYM
ejpam-5648	307	13	n.	n.	PROPN
ejpam-5648	307	14	hence	hence	ADV
ejpam-5648	307	15	,	,	PUNCT
ejpam-5648	307	16	γcrh(pn	γcrh(pn	NOUN
ejpam-5648	307	17	)	)	PUNCT
ejpam-5648	307	18	=	=	SYM
ejpam-5648	307	19	n.	n.	NOUN
ejpam-5648	307	20	suppose	suppose	VERB
ejpam-5648	307	21	|v0|	|v0|	NOUN
ejpam-5648	307	22	=	=	SYM
ejpam-5648	307	23	1	1	X
ejpam-5648	307	24	.	.	PUNCT
ejpam-5648	307	25	then	then	ADV
ejpam-5648	307	26	v0	v0	PROPN
ejpam-5648	307	27	=	=	SYM
ejpam-5648	307	28	{	{	PUNCT
ejpam-5648	307	29	x1	x1	PROPN
ejpam-5648	307	30	}	}	PUNCT
ejpam-5648	307	31	or	or	CCONJ
ejpam-5648	307	32	v0	v0	NOUN
ejpam-5648	307	33	=	=	SYM
ejpam-5648	307	34	{	{	PUNCT
ejpam-5648	307	35	xn	xn	NUM
ejpam-5648	307	36	}	}	PUNCT
ejpam-5648	307	37	.	.	PUNCT
ejpam-5648	308	1	assume	assume	VERB
ejpam-5648	308	2	that	that	SCONJ
ejpam-5648	308	3	v0	v0	NOUN
ejpam-5648	308	4	=	=	SYM
ejpam-5648	308	5	{	{	PUNCT
ejpam-5648	308	6	x1	x1	PROPN
ejpam-5648	308	7	}	}	PUNCT
ejpam-5648	308	8	.	.	PUNCT
ejpam-5648	309	1	then	then	ADV
ejpam-5648	309	2	v2	v2	VERB
ejpam-5648	309	3	=	=	SYM
ejpam-5648	309	4	{	{	PUNCT
ejpam-5648	309	5	x3	x3	ADJ
ejpam-5648	309	6	}	}	PUNCT
ejpam-5648	309	7	and	and	CCONJ
ejpam-5648	309	8	v1	v1	NOUN
ejpam-5648	309	9	=	=	SYM
ejpam-5648	309	10	v	v	NOUN
ejpam-5648	309	11	(	(	PUNCT
ejpam-5648	309	12	pn	pn	NOUN
ejpam-5648	309	13	)	)	PUNCT
ejpam-5648	309	14	\	\	NOUN
ejpam-5648	309	15	{	{	PUNCT
ejpam-5648	309	16	x1	x1	PROPN
ejpam-5648	309	17	,	,	PUNCT
ejpam-5648	309	18	x3	x3	ADJ
ejpam-5648	309	19	}	}	PUNCT
ejpam-5648	309	20	.	.	PUNCT
ejpam-5648	310	1	this	this	DET
ejpam-5648	310	2	yields	yield	NOUN
ejpam-5648	310	3	γcrh(pn	γcrh(pn	NOUN
ejpam-5648	310	4	)	)	PUNCT
ejpam-5648	310	5	=	=	PUNCT
ejpam-5648	310	6	ωcrh	ωcrh	PROPN
ejpam-5648	310	7	pn	pn	PROPN
ejpam-5648	310	8	(	(	PUNCT
ejpam-5648	310	9	f	f	X
ejpam-5648	310	10	)	)	PUNCT
ejpam-5648	310	11	=	=	PUNCT
ejpam-5648	311	1	|v1|+	|v1|+	ADV
ejpam-5648	311	2	2|v2	2|v2	NUM
ejpam-5648	312	1	=	=	PUNCT
ejpam-5648	313	1	(	(	PUNCT
ejpam-5648	313	2	n−	n−	NOUN
ejpam-5648	313	3	2	2	NUM
ejpam-5648	313	4	)	)	PUNCT
ejpam-5648	313	5	+	+	CCONJ
ejpam-5648	313	6	2	2	X
ejpam-5648	313	7	=	=	SYM
ejpam-5648	313	8	n.	n.	NOUN
ejpam-5648	313	9	if	if	SCONJ
ejpam-5648	313	10	|v0|	|v0|	NOUN
ejpam-5648	313	11	=	=	SYM
ejpam-5648	313	12	2	2	NUM
ejpam-5648	313	13	,	,	PUNCT
ejpam-5648	313	14	then	then	ADV
ejpam-5648	313	15	v0	v0	PROPN
ejpam-5648	313	16	=	=	SYM
ejpam-5648	313	17	{	{	PUNCT
ejpam-5648	313	18	x1	x1	PROPN
ejpam-5648	313	19	,	,	PUNCT
ejpam-5648	313	20	x2	x2	PROPN
ejpam-5648	313	21	}	}	PUNCT
ejpam-5648	313	22	or	or	CCONJ
ejpam-5648	313	23	v0	v0	NOUN
ejpam-5648	313	24	=	=	SYM
ejpam-5648	313	25	{	{	PUNCT
ejpam-5648	313	26	x1	x1	PROPN
ejpam-5648	313	27	,	,	PUNCT
ejpam-5648	313	28	xn	xn	PROPN
ejpam-5648	313	29	}	}	PUNCT
ejpam-5648	313	30	or	or	CCONJ
ejpam-5648	313	31	v0	v0	NOUN
ejpam-5648	313	32	=	=	SYM
ejpam-5648	313	33	{	{	PUNCT
ejpam-5648	313	34	xn−1	xn−1	PROPN
ejpam-5648	313	35	,	,	PUNCT
ejpam-5648	313	36	xn	xn	PROPN
ejpam-5648	313	37	}	}	PUNCT
ejpam-5648	313	38	.	.	PUNCT
ejpam-5648	314	1	it	it	PRON
ejpam-5648	314	2	follows	follow	VERB
ejpam-5648	314	3	that	that	SCONJ
ejpam-5648	314	4	v2	v2	PROPN
ejpam-5648	314	5	=	=	SYM
ejpam-5648	314	6	{	{	PUNCT
ejpam-5648	314	7	x3	x3	PROPN
ejpam-5648	314	8	,	,	PUNCT
ejpam-5648	314	9	x4	x4	ADJ
ejpam-5648	314	10	}	}	PUNCT
ejpam-5648	314	11	or	or	CCONJ
ejpam-5648	314	12	v2	v2	NOUN
ejpam-5648	314	13	=	=	SYM
ejpam-5648	314	14	{	{	PUNCT
ejpam-5648	314	15	x3	x3	PROPN
ejpam-5648	314	16	,	,	PUNCT
ejpam-5648	314	17	xn−2	xn−2	PROPN
ejpam-5648	314	18	}	}	PUNCT
ejpam-5648	314	19	or	or	CCONJ
ejpam-5648	314	20	v2	v2	NOUN
ejpam-5648	314	21	=	=	SYM
ejpam-5648	314	22	{	{	PUNCT
ejpam-5648	314	23	xn−3	xn−3	PROPN
ejpam-5648	314	24	,	,	PUNCT
ejpam-5648	314	25	xn−2	xn−2	PROPN
ejpam-5648	314	26	}	}	PUNCT
ejpam-5648	314	27	,	,	PUNCT
ejpam-5648	314	28	respectively	respectively	ADV
ejpam-5648	314	29	.	.	PUNCT
ejpam-5648	315	1	if	if	SCONJ
ejpam-5648	315	2	|v0|	|v0|	NOUN
ejpam-5648	315	3	=	=	SYM
ejpam-5648	315	4	3	3	NUM
ejpam-5648	315	5	,	,	PUNCT
ejpam-5648	315	6	then	then	ADV
ejpam-5648	315	7	v0	v0	PROPN
ejpam-5648	315	8	=	=	SYM
ejpam-5648	315	9	{	{	PUNCT
ejpam-5648	315	10	x1	x1	PROPN
ejpam-5648	315	11	,	,	PUNCT
ejpam-5648	315	12	x2	x2	PROPN
ejpam-5648	315	13	,	,	PUNCT
ejpam-5648	315	14	xn	xn	PRON
ejpam-5648	315	15	}	}	PUNCT
ejpam-5648	315	16	or	or	CCONJ
ejpam-5648	315	17	v0	v0	NOUN
ejpam-5648	315	18	=	=	SYM
ejpam-5648	315	19	{	{	PUNCT
ejpam-5648	315	20	x1	x1	PROPN
ejpam-5648	315	21	,	,	PUNCT
ejpam-5648	315	22	xn−1	xn−1	PROPN
ejpam-5648	315	23	,	,	PUNCT
ejpam-5648	315	24	xn	xn	PROPN
ejpam-5648	315	25	}	}	PUNCT
ejpam-5648	315	26	.	.	PUNCT
ejpam-5648	316	1	hence	hence	ADV
ejpam-5648	316	2	,	,	PUNCT
ejpam-5648	316	3	v2	v2	PROPN
ejpam-5648	316	4	=	=	SYM
ejpam-5648	316	5	{	{	PUNCT
ejpam-5648	316	6	x3	x3	PROPN
ejpam-5648	316	7	,	,	PUNCT
ejpam-5648	316	8	x4	x4	PROPN
ejpam-5648	316	9	,	,	PUNCT
ejpam-5648	316	10	xn−2	xn−2	PROPN
ejpam-5648	316	11	}	}	PUNCT
ejpam-5648	316	12	or	or	CCONJ
ejpam-5648	316	13	v2	v2	NOUN
ejpam-5648	316	14	=	=	SYM
ejpam-5648	316	15	{	{	PUNCT
ejpam-5648	316	16	x3	x3	ADJ
ejpam-5648	316	17	,	,	PUNCT
ejpam-5648	316	18	xn−3	xn−3	PROPN
ejpam-5648	316	19	,	,	PUNCT
ejpam-5648	316	20	xn−2	xn−2	PROPN
ejpam-5648	316	21	}	}	PUNCT
ejpam-5648	316	22	,	,	PUNCT
ejpam-5648	316	23	respectively	respectively	ADV
ejpam-5648	316	24	.	.	PUNCT
ejpam-5648	317	1	finally	finally	ADV
ejpam-5648	317	2	,	,	PUNCT
ejpam-5648	317	3	if	if	SCONJ
ejpam-5648	317	4	v0	v0	NOUN
ejpam-5648	317	5	=	=	SYM
ejpam-5648	317	6	{	{	PUNCT
ejpam-5648	317	7	x1	x1	PROPN
ejpam-5648	317	8	,	,	PUNCT
ejpam-5648	317	9	x2	x2	PROPN
ejpam-5648	317	10	,	,	PUNCT
ejpam-5648	317	11	xn−1	xn−1	PROPN
ejpam-5648	317	12	,	,	PUNCT
ejpam-5648	317	13	xn	xn	PROPN
ejpam-5648	317	14	}	}	PUNCT
ejpam-5648	317	15	,	,	PUNCT
ejpam-5648	317	16	then	then	ADV
ejpam-5648	317	17	v2	v2	VERB
ejpam-5648	317	18	=	=	SYM
ejpam-5648	317	19	{	{	PUNCT
ejpam-5648	317	20	x3	x3	PROPN
ejpam-5648	317	21	,	,	PUNCT
ejpam-5648	317	22	x4	x4	PROPN
ejpam-5648	317	23	,	,	PUNCT
ejpam-5648	317	24	xn−3	xn−3	PROPN
ejpam-5648	317	25	,	,	PUNCT
ejpam-5648	317	26	xn−2	xn−2	PROPN
ejpam-5648	317	27	}	}	PUNCT
ejpam-5648	317	28	.	.	PUNCT
ejpam-5648	318	1	it	it	PRON
ejpam-5648	318	2	can	can	AUX
ejpam-5648	318	3	easily	easily	ADV
ejpam-5648	318	4	be	be	AUX
ejpam-5648	318	5	shown	show	VERB
ejpam-5648	318	6	that	that	SCONJ
ejpam-5648	318	7	any	any	PRON
ejpam-5648	318	8	of	of	ADP
ejpam-5648	318	9	these	these	DET
ejpam-5648	318	10	cases	case	NOUN
ejpam-5648	318	11	will	will	AUX
ejpam-5648	318	12	imply	imply	VERB
ejpam-5648	318	13	that	that	DET
ejpam-5648	318	14	γcrh(pn	γcrh(pn	NOUN
ejpam-5648	318	15	)	)	PUNCT
ejpam-5648	318	16	=	=	PUNCT
ejpam-5648	318	17	ωcrh	ωcrh	PROPN
ejpam-5648	318	18	pn	pn	PROPN
ejpam-5648	318	19	(	(	PUNCT
ejpam-5648	318	20	f	f	X
ejpam-5648	318	21	)	)	PUNCT
ejpam-5648	318	22	=	=	PUNCT
ejpam-5648	318	23	|v1|+	|v1|+	ADV
ejpam-5648	318	24	2|v2	2|v2	NUM
ejpam-5648	318	25	=	=	PUNCT
ejpam-5648	318	26	(	(	PUNCT
ejpam-5648	318	27	n−	n−	NOUN
ejpam-5648	318	28	2	2	NUM
ejpam-5648	318	29	)	)	PUNCT
ejpam-5648	318	30	+	+	CCONJ
ejpam-5648	318	31	2	2	X
ejpam-5648	318	32	=	=	SYM
ejpam-5648	318	33	n.	n.	NOUN
ejpam-5648	318	34	this	this	PRON
ejpam-5648	318	35	proves	prove	VERB
ejpam-5648	318	36	the	the	DET
ejpam-5648	318	37	assertion	assertion	NOUN
ejpam-5648	318	38	.	.	PUNCT
ejpam-5648	319	1	a.	a.	PROPN
ejpam-5648	319	2	aradais	aradais	PROPN
ejpam-5648	319	3	,	,	PUNCT
ejpam-5648	319	4	j.	j.	PROPN
ejpam-5648	319	5	cariaga	cariaga	PROPN
ejpam-5648	319	6	,	,	PUNCT
ejpam-5648	319	7	s.	s.	PROPN
ejpam-5648	319	8	canoy	canoy	PROPN
ejpam-5648	319	9	jr	jr	PROPN
ejpam-5648	319	10	.	.	PROPN
ejpam-5648	319	11	/	/	SYM
ejpam-5648	319	12	eur	eur	PROPN
ejpam-5648	319	13	.	.	PUNCT
ejpam-5648	320	1	j.	j.	PROPN
ejpam-5648	320	2	pure	pure	PROPN
ejpam-5648	320	3	appl	appl	PROPN
ejpam-5648	320	4	.	.	PROPN
ejpam-5648	320	5	math	math	PROPN
ejpam-5648	320	6	,	,	PUNCT
ejpam-5648	320	7	18	18	NUM
ejpam-5648	320	8	(	(	PUNCT
ejpam-5648	320	9	1	1	NUM
ejpam-5648	320	10	)	)	PUNCT
ejpam-5648	320	11	(	(	PUNCT
ejpam-5648	320	12	2025	2025	NUM
ejpam-5648	320	13	)	)	PUNCT
ejpam-5648	320	14	,	,	PUNCT
ejpam-5648	320	15	5648	5648	NUM
ejpam-5648	320	16	9	9	NUM
ejpam-5648	320	17	of	of	ADP
ejpam-5648	320	18	13	13	NUM
ejpam-5648	320	19	proposition	proposition	NOUN
ejpam-5648	320	20	6	6	NUM
ejpam-5648	320	21	.	.	PUNCT
ejpam-5648	321	1	for	for	ADP
ejpam-5648	321	2	any	any	DET
ejpam-5648	321	3	cycle	cycle	NOUN
ejpam-5648	321	4	cn	cn	NOUN
ejpam-5648	321	5	of	of	ADP
ejpam-5648	321	6	order	order	NOUN
ejpam-5648	321	7	n	n	PRON
ejpam-5648	321	8	≥	≥	NOUN
ejpam-5648	321	9	3	3	NUM
ejpam-5648	321	10	,	,	PUNCT
ejpam-5648	321	11	γcrh(cn	γcrh(cn	NOUN
ejpam-5648	321	12	)	)	PUNCT
ejpam-5648	321	13	=	=	PUNCT
ejpam-5648	322	1			NOUN
ejpam-5648	322	2	4	4	NUM
ejpam-5648	322	3	,	,	PUNCT
ejpam-5648	322	4	if	if	SCONJ
ejpam-5648	322	5	n	n	NOUN
ejpam-5648	322	6	=	=	SYM
ejpam-5648	322	7	4	4	NUM
ejpam-5648	322	8	,	,	PUNCT
ejpam-5648	322	9	5	5	NUM
ejpam-5648	322	10	,	,	PUNCT
ejpam-5648	322	11	6	6	NUM
ejpam-5648	322	12	6	6	NUM
ejpam-5648	322	13	,	,	PUNCT
ejpam-5648	322	14	if	if	SCONJ
ejpam-5648	322	15	n	n	NOUN
ejpam-5648	322	16	=	=	SYM
ejpam-5648	322	17	7	7	NUM
ejpam-5648	322	18	n	n	CCONJ
ejpam-5648	322	19	,	,	PUNCT
ejpam-5648	322	20	if	if	SCONJ
ejpam-5648	322	21	n	n	PRON
ejpam-5648	322	22	̸=	̸=	PROPN
ejpam-5648	322	23	5	5	NUM
ejpam-5648	322	24	,	,	PUNCT
ejpam-5648	322	25	6	6	NUM
ejpam-5648	322	26	,	,	PUNCT
ejpam-5648	322	27	7	7	NUM
ejpam-5648	322	28	.	.	PUNCT
ejpam-5648	322	29	proof	proof	NOUN
ejpam-5648	322	30	.	.	PUNCT
ejpam-5648	323	1	by	by	ADP
ejpam-5648	323	2	proposition	proposition	NOUN
ejpam-5648	323	3	3(iii	3(iii	NUM
ejpam-5648	323	4	)	)	PUNCT
ejpam-5648	323	5	,	,	PUNCT
ejpam-5648	323	6	γcrh(c3	γcrh(c3	NOUN
ejpam-5648	323	7	)	)	PUNCT
ejpam-5648	323	8	=	=	SYM
ejpam-5648	323	9	3	3	NUM
ejpam-5648	323	10	and	and	CCONJ
ejpam-5648	323	11	by	by	ADP
ejpam-5648	323	12	proposition	proposition	NOUN
ejpam-5648	323	13	3(iv	3(iv	NUM
ejpam-5648	323	14	)	)	PUNCT
ejpam-5648	323	15	,	,	PUNCT
ejpam-5648	323	16	γcrh(cn	γcrh(cn	NOUN
ejpam-5648	323	17	)	)	PUNCT
ejpam-5648	323	18	=	=	SYM
ejpam-5648	323	19	4	4	NUM
ejpam-5648	323	20	for	for	ADP
ejpam-5648	323	21	n	n	NOUN
ejpam-5648	323	22	=	=	SYM
ejpam-5648	323	23	4	4	NUM
ejpam-5648	323	24	,	,	PUNCT
ejpam-5648	323	25	5	5	NUM
ejpam-5648	323	26	,	,	PUNCT
ejpam-5648	323	27	6	6	NUM
ejpam-5648	323	28	.	.	PUNCT
ejpam-5648	324	1	next	next	ADV
ejpam-5648	324	2	,	,	PUNCT
ejpam-5648	324	3	let	let	VERB
ejpam-5648	324	4	n	n	PRON
ejpam-5648	324	5	≥	≥	X
ejpam-5648	324	6	7	7	NUM
ejpam-5648	324	7	and	and	CCONJ
ejpam-5648	324	8	let	let	VERB
ejpam-5648	324	9	cn	cn	PROPN
ejpam-5648	324	10	=	=	PUNCT
ejpam-5648	325	1	[	[	X
ejpam-5648	325	2	v1	v1	NOUN
ejpam-5648	325	3	,	,	PUNCT
ejpam-5648	325	4	v2	v2	PROPN
ejpam-5648	325	5	,	,	PUNCT
ejpam-5648	325	6	...	...	PUNCT
ejpam-5648	325	7	,	,	PUNCT
ejpam-5648	325	8	vn	vn	X
ejpam-5648	325	9	,	,	PUNCT
ejpam-5648	325	10	v1	v1	PROPN
ejpam-5648	325	11	]	]	PUNCT
ejpam-5648	325	12	.	.	PUNCT
ejpam-5648	326	1	let	let	VERB
ejpam-5648	326	2	f	f	PROPN
ejpam-5648	326	3	=	=	SYM
ejpam-5648	326	4	(	(	PUNCT
ejpam-5648	326	5	v0	v0	PROPN
ejpam-5648	326	6	,	,	PUNCT
ejpam-5648	326	7	v1	v1	NOUN
ejpam-5648	326	8	,	,	PUNCT
ejpam-5648	326	9	v2	v2	PROPN
ejpam-5648	326	10	)	)	PUNCT
ejpam-5648	326	11	be	be	AUX
ejpam-5648	326	12	a	a	DET
ejpam-5648	326	13	γcrhfunction	γcrhfunction	NOUN
ejpam-5648	326	14	on	on	ADP
ejpam-5648	326	15	cn	cn	PROPN
ejpam-5648	326	16	.	.	PUNCT
ejpam-5648	327	1	since	since	SCONJ
ejpam-5648	327	2	v1	v1	NOUN
ejpam-5648	327	3	∪	∪	NOUN
ejpam-5648	327	4	v2	v2	NOUN
ejpam-5648	327	5	is	be	AUX
ejpam-5648	327	6	a	a	DET
ejpam-5648	327	7	connected	connected	ADJ
ejpam-5648	327	8	hop	hop	NOUN
ejpam-5648	327	9	dominating	dominating	NOUN
ejpam-5648	327	10	set	set	NOUN
ejpam-5648	327	11	,	,	PUNCT
ejpam-5648	327	12	⟨v0⟩	⟨v0⟩	NOUN
ejpam-5648	327	13	is	be	AUX
ejpam-5648	327	14	connected	connect	VERB
ejpam-5648	327	15	and	and	CCONJ
ejpam-5648	327	16	|v0|	|v0|	NOUN
ejpam-5648	327	17	≤	≤	NUM
ejpam-5648	327	18	4	4	NUM
ejpam-5648	327	19	.	.	PUNCT
ejpam-5648	327	20	suppose	suppose	VERB
ejpam-5648	327	21	first	first	ADV
ejpam-5648	327	22	that	that	SCONJ
ejpam-5648	327	23	n	n	NOUN
ejpam-5648	327	24	=	=	SYM
ejpam-5648	327	25	7	7	X
ejpam-5648	327	26	.	.	PUNCT
ejpam-5648	328	1	since	since	SCONJ
ejpam-5648	328	2	the	the	DET
ejpam-5648	328	3	function	function	NOUN
ejpam-5648	328	4	g	g	NOUN
ejpam-5648	328	5	=	=	SYM
ejpam-5648	328	6	(	(	PUNCT
ejpam-5648	328	7	{	{	PUNCT
ejpam-5648	328	8	v1	v1	NOUN
ejpam-5648	328	9	,	,	PUNCT
ejpam-5648	328	10	v2	v2	PROPN
ejpam-5648	328	11	,	,	PUNCT
ejpam-5648	328	12	v3	v3	PROPN
ejpam-5648	328	13	,	,	PUNCT
ejpam-5648	328	14	v4},∅	v4},∅	PROPN
ejpam-5648	328	15	,	,	PUNCT
ejpam-5648	328	16	{	{	PUNCT
ejpam-5648	328	17	v5	v5	NOUN
ejpam-5648	328	18	,	,	PUNCT
ejpam-5648	328	19	x6	x6	PROPN
ejpam-5648	328	20	,	,	PUNCT
ejpam-5648	328	21	v7	v7	NOUN
ejpam-5648	328	22	}	}	PUNCT
ejpam-5648	328	23	)	)	PUNCT
ejpam-5648	328	24	is	be	AUX
ejpam-5648	328	25	a	a	DET
ejpam-5648	328	26	connected	connected	ADJ
ejpam-5648	328	27	hop	hop	NOUN
ejpam-5648	328	28	dominating	dominating	NOUN
ejpam-5648	328	29	function	function	NOUN
ejpam-5648	328	30	on	on	ADP
ejpam-5648	328	31	c7	c7	PROPN
ejpam-5648	328	32	,	,	PUNCT
ejpam-5648	328	33	γcrh(c7	γcrh(c7	NOUN
ejpam-5648	328	34	)	)	PUNCT
ejpam-5648	328	35	≤	≤	NUM
ejpam-5648	328	36	6	6	NUM
ejpam-5648	328	37	.	.	PUNCT
ejpam-5648	329	1	if	if	SCONJ
ejpam-5648	329	2	|v0|	|v0|	NOUN
ejpam-5648	329	3	=	=	SYM
ejpam-5648	329	4	0	0	NUM
ejpam-5648	329	5	,	,	PUNCT
ejpam-5648	329	6	then	then	ADV
ejpam-5648	329	7	|v2|	|v2|	ADV
ejpam-5648	329	8	=	=	SYM
ejpam-5648	329	9	0	0	NUM
ejpam-5648	329	10	and	and	CCONJ
ejpam-5648	329	11	|v1|	|v1|	NOUN
ejpam-5648	329	12	=	=	SYM
ejpam-5648	329	13	7	7	X
ejpam-5648	329	14	.	.	X
ejpam-5648	329	15	if	if	SCONJ
ejpam-5648	329	16	|v0|	|v0|	NOUN
ejpam-5648	329	17	=	=	SYM
ejpam-5648	329	18	1	1	NUM
ejpam-5648	329	19	,	,	PUNCT
ejpam-5648	329	20	then	then	ADV
ejpam-5648	329	21	|v2|	|v2|	ADV
ejpam-5648	329	22	=	=	SYM
ejpam-5648	329	23	1	1	NUM
ejpam-5648	329	24	and	and	CCONJ
ejpam-5648	329	25	|v1|	|v1|	NOUN
ejpam-5648	329	26	=	=	SYM
ejpam-5648	329	27	5	5	X
ejpam-5648	329	28	.	.	PUNCT
ejpam-5648	330	1	if	if	SCONJ
ejpam-5648	330	2	|v0|	|v0|	NOUN
ejpam-5648	330	3	=	=	SYM
ejpam-5648	330	4	2	2	NUM
ejpam-5648	330	5	,	,	PUNCT
ejpam-5648	330	6	then	then	ADV
ejpam-5648	330	7	|v2|	|v2|	ADV
ejpam-5648	330	8	=	=	SYM
ejpam-5648	330	9	2	2	NUM
ejpam-5648	330	10	and	and	CCONJ
ejpam-5648	330	11	|v1|	|v1|	NOUN
ejpam-5648	330	12	=	=	SYM
ejpam-5648	330	13	3	3	NUM
ejpam-5648	330	14	and	and	CCONJ
ejpam-5648	330	15	if	if	SCONJ
ejpam-5648	330	16	|v0|	|v0|	NOUN
ejpam-5648	330	17	=	=	SYM
ejpam-5648	330	18	3	3	NUM
ejpam-5648	330	19	,	,	PUNCT
ejpam-5648	330	20	then	then	ADV
ejpam-5648	330	21	|v2|	|v2|	ADV
ejpam-5648	330	22	=	=	SYM
ejpam-5648	330	23	3	3	NUM
ejpam-5648	330	24	and	and	CCONJ
ejpam-5648	330	25	|v1|	|v1|	NOUN
ejpam-5648	330	26	=	=	SYM
ejpam-5648	330	27	1	1	X
ejpam-5648	330	28	.	.	X
ejpam-5648	331	1	any	any	PRON
ejpam-5648	331	2	of	of	ADP
ejpam-5648	331	3	these	these	DET
ejpam-5648	331	4	four	four	NUM
ejpam-5648	331	5	cases	case	NOUN
ejpam-5648	331	6	will	will	AUX
ejpam-5648	331	7	yield	yield	VERB
ejpam-5648	331	8	γcrh(c7	γcrh(c7	NOUN
ejpam-5648	331	9	)	)	PUNCT
ejpam-5648	332	1	=	=	SYM
ejpam-5648	332	2	7	7	NUM
ejpam-5648	332	3	which	which	PRON
ejpam-5648	332	4	is	be	AUX
ejpam-5648	332	5	not	not	PART
ejpam-5648	332	6	possible	possible	ADJ
ejpam-5648	332	7	.	.	PUNCT
ejpam-5648	333	1	therefore	therefore	ADV
ejpam-5648	333	2	,	,	PUNCT
ejpam-5648	333	3	|v0|	|v0|	NOUN
ejpam-5648	333	4	=	=	SYM
ejpam-5648	333	5	4	4	X
ejpam-5648	333	6	.	.	X
ejpam-5648	333	7	hence	hence	ADV
ejpam-5648	333	8	,	,	PUNCT
ejpam-5648	333	9	|v2|	|v2|	NOUN
ejpam-5648	333	10	=	=	SYM
ejpam-5648	333	11	3	3	NUM
ejpam-5648	333	12	and	and	CCONJ
ejpam-5648	333	13	|v1|	|v1|	NOUN
ejpam-5648	333	14	=	=	SYM
ejpam-5648	333	15	0	0	X
ejpam-5648	333	16	.	.	PUNCT
ejpam-5648	334	1	it	it	PRON
ejpam-5648	334	2	follows	follow	VERB
ejpam-5648	334	3	that	that	DET
ejpam-5648	334	4	γcrh(c7	γcrh(c7	NOUN
ejpam-5648	334	5	)	)	PUNCT
ejpam-5648	334	6	=	=	SYM
ejpam-5648	335	1	6	6	X
ejpam-5648	335	2	.	.	PUNCT
ejpam-5648	335	3	finally	finally	ADV
ejpam-5648	335	4	,	,	PUNCT
ejpam-5648	335	5	suppose	suppose	VERB
ejpam-5648	335	6	that	that	SCONJ
ejpam-5648	335	7	n	n	PROPN
ejpam-5648	335	8	≥	≥	NUM
ejpam-5648	335	9	8	8	NUM
ejpam-5648	335	10	.	.	PUNCT
ejpam-5648	336	1	clearly	clearly	ADV
ejpam-5648	336	2	,	,	PUNCT
ejpam-5648	336	3	if	if	SCONJ
ejpam-5648	336	4	|v0|	|v0|	NOUN
ejpam-5648	336	5	=	=	SYM
ejpam-5648	336	6	j	j	NOUN
ejpam-5648	336	7	,	,	PUNCT
ejpam-5648	336	8	then	then	ADV
ejpam-5648	336	9	|v2|	|v2|	ADV
ejpam-5648	336	10	=	=	SYM
ejpam-5648	336	11	j	j	PROPN
ejpam-5648	336	12	and	and	CCONJ
ejpam-5648	336	13	|v1|	|v1|	NOUN
ejpam-5648	336	14	=	=	PUNCT
ejpam-5648	336	15	n−	n−	NOUN
ejpam-5648	336	16	2j	2j	NOUN
ejpam-5648	336	17	.	.	PUNCT
ejpam-5648	337	1	therefore	therefore	ADV
ejpam-5648	337	2	,	,	PUNCT
ejpam-5648	337	3	γcrh(cn	γcrh(cn	NOUN
ejpam-5648	337	4	)	)	PUNCT
ejpam-5648	337	5	=	=	PUNCT
ejpam-5648	337	6	n−	n−	NOUN
ejpam-5648	337	7	2j	2j	NOUN
ejpam-5648	337	8	+	+	CCONJ
ejpam-5648	337	9	2j	2j	X
ejpam-5648	337	10	=	=	SYM
ejpam-5648	337	11	n.	n.	NOUN
ejpam-5648	337	12	proposition	proposition	NOUN
ejpam-5648	337	13	7	7	NUM
ejpam-5648	337	14	.	.	PUNCT
ejpam-5648	338	1	let	let	VERB
ejpam-5648	338	2	g	g	PROPN
ejpam-5648	338	3	=	=	SYM
ejpam-5648	338	4	kn1,n2,	kn1,n2,	PROPN
ejpam-5648	338	5	...	...	PUNCT
ejpam-5648	338	6	,nk	,nk	PUNCT
ejpam-5648	338	7	be	be	AUX
ejpam-5648	338	8	the	the	DET
ejpam-5648	338	9	complete	complete	ADJ
ejpam-5648	338	10	k	k	ADJ
ejpam-5648	338	11	-	-	ADJ
ejpam-5648	338	12	partite	partite	ADJ
ejpam-5648	338	13	graph	graph	NOUN
ejpam-5648	338	14	with	with	ADP
ejpam-5648	338	15	2	2	NUM
ejpam-5648	338	16	≤	≤	NUM
ejpam-5648	338	17	n1	n1	PROPN
ejpam-5648	338	18	≤	≤	NOUN
ejpam-5648	338	19	n2	n2	NOUN
ejpam-5648	338	20	≤	≤	NOUN
ejpam-5648	338	21	·	·	PUNCT
ejpam-5648	338	22	·	·	PUNCT
ejpam-5648	338	23	·	·	PUNCT
ejpam-5648	339	1	≤	≤	NUM
ejpam-5648	339	2	nk	nk	PROPN
ejpam-5648	339	3	,	,	PUNCT
ejpam-5648	339	4	where	where	SCONJ
ejpam-5648	339	5	k	k	PROPN
ejpam-5648	339	6	≥	≥	NUM
ejpam-5648	339	7	2	2	NUM
ejpam-5648	339	8	.	.	PUNCT
ejpam-5648	339	9	then	then	ADV
ejpam-5648	339	10	γcrh(g	γcrh(g	ADP
ejpam-5648	339	11	)	)	PUNCT
ejpam-5648	339	12	=	=	SYM
ejpam-5648	339	13	2k	2k	NUM
ejpam-5648	339	14	.	.	PUNCT
ejpam-5648	340	1	proof	proof	NOUN
ejpam-5648	340	2	.	.	PUNCT
ejpam-5648	341	1	let	let	VERB
ejpam-5648	341	2	sn1	sn1	PROPN
ejpam-5648	341	3	,	,	PUNCT
ejpam-5648	341	4	sn2	sn2	PROPN
ejpam-5648	341	5	,	,	PUNCT
ejpam-5648	341	6	.	.	PUNCT
ejpam-5648	341	7	.	.	PUNCT
ejpam-5648	342	1	.	.	PUNCT
ejpam-5648	343	1	,	,	PUNCT
ejpam-5648	343	2	snk	snk	PROPN
ejpam-5648	343	3	be	be	AUX
ejpam-5648	343	4	the	the	DET
ejpam-5648	343	5	partite	partite	ADJ
ejpam-5648	343	6	sets	set	NOUN
ejpam-5648	343	7	in	in	ADP
ejpam-5648	343	8	g.	g.	PROPN
ejpam-5648	343	9	pick	pick	VERB
ejpam-5648	343	10	vnj	vnj	PRON
ejpam-5648	343	11	∈	∈	PROPN
ejpam-5648	343	12	snj	snj	NOUN
ejpam-5648	343	13	for	for	ADP
ejpam-5648	343	14	each	each	DET
ejpam-5648	343	15	j	j	PROPN
ejpam-5648	343	16	∈	∈	PROPN
ejpam-5648	344	1	[	[	X
ejpam-5648	344	2	k	k	X
ejpam-5648	344	3	]	]	X
ejpam-5648	344	4	,	,	PUNCT
ejpam-5648	345	1	where	where	SCONJ
ejpam-5648	345	2	[	[	X
ejpam-5648	345	3	k	k	X
ejpam-5648	345	4	]	]	X
ejpam-5648	345	5	=	=	X
ejpam-5648	345	6	{	{	PUNCT
ejpam-5648	345	7	1	1	NUM
ejpam-5648	345	8	,	,	PUNCT
ejpam-5648	345	9	2	2	NUM
ejpam-5648	345	10	,	,	PUNCT
ejpam-5648	345	11	.	.	PUNCT
ejpam-5648	345	12	.	.	PUNCT
ejpam-5648	345	13	.	.	PUNCT
ejpam-5648	346	1	,	,	PUNCT
ejpam-5648	346	2	k	k	X
ejpam-5648	346	3	}	}	PUNCT
ejpam-5648	346	4	,	,	PUNCT
ejpam-5648	346	5	and	and	CCONJ
ejpam-5648	346	6	let	let	VERB
ejpam-5648	346	7	s	s	PRON
ejpam-5648	346	8	=	=	VERB
ejpam-5648	346	9	{	{	PUNCT
ejpam-5648	346	10	vnj	vnj	NOUN
ejpam-5648	346	11	:	:	PUNCT
ejpam-5648	346	12	j	j	PROPN
ejpam-5648	346	13	∈	∈	PROPN
ejpam-5648	347	1	[	[	X
ejpam-5648	347	2	k	k	X
ejpam-5648	347	3	]	]	X
ejpam-5648	347	4	}	}	PUNCT
ejpam-5648	347	5	.	.	PUNCT
ejpam-5648	348	1	set	set	VERB
ejpam-5648	348	2	v0	v0	PROPN
ejpam-5648	348	3	=	=	SYM
ejpam-5648	348	4	v	v	PROPN
ejpam-5648	348	5	(	(	PUNCT
ejpam-5648	348	6	g	g	NOUN
ejpam-5648	348	7	)	)	PUNCT
ejpam-5648	348	8	\	\	PROPN
ejpam-5648	349	1	s	s	X
ejpam-5648	349	2	,	,	PUNCT
ejpam-5648	349	3	v1	v1	NOUN
ejpam-5648	349	4	=	=	SYM
ejpam-5648	349	5	∅	∅	NOUN
ejpam-5648	349	6	,	,	PUNCT
ejpam-5648	349	7	and	and	CCONJ
ejpam-5648	349	8	v2	v2	X
ejpam-5648	349	9	=	=	SYM
ejpam-5648	349	10	s.	s.	PROPN
ejpam-5648	350	1	then	then	ADV
ejpam-5648	350	2	f	f	PROPN
ejpam-5648	350	3	=	=	SYM
ejpam-5648	350	4	(	(	PUNCT
ejpam-5648	350	5	v0	v0	PROPN
ejpam-5648	350	6	,	,	PUNCT
ejpam-5648	350	7	v1	v1	NOUN
ejpam-5648	350	8	,	,	PUNCT
ejpam-5648	350	9	v2	v2	NOUN
ejpam-5648	350	10	)	)	PUNCT
ejpam-5648	350	11	∈	∈	PROPN
ejpam-5648	350	12	chrdf	chrdf	NOUN
ejpam-5648	350	13	(	(	PUNCT
ejpam-5648	350	14	g	g	NOUN
ejpam-5648	350	15	)	)	PUNCT
ejpam-5648	350	16	and	and	CCONJ
ejpam-5648	350	17	γcrh(g	γcrh(g	X
ejpam-5648	350	18	)	)	PUNCT
ejpam-5648	350	19	≤	≤	NOUN
ejpam-5648	350	20	2|v2|	2|v2|	NUM
ejpam-5648	350	21	=	=	SYM
ejpam-5648	350	22	2k	2k	NUM
ejpam-5648	350	23	.	.	PUNCT
ejpam-5648	351	1	now	now	ADV
ejpam-5648	351	2	let	let	VERB
ejpam-5648	351	3	g	g	NOUN
ejpam-5648	351	4	=	=	SYM
ejpam-5648	351	5	(	(	PUNCT
ejpam-5648	351	6	w0,w1,w2	w0,w1,w2	ADV
ejpam-5648	351	7	)	)	PUNCT
ejpam-5648	351	8	be	be	AUX
ejpam-5648	351	9	a	a	DET
ejpam-5648	351	10	γcrh	γcrh	NOUN
ejpam-5648	351	11	-	-	PUNCT
ejpam-5648	351	12	function	function	NOUN
ejpam-5648	351	13	on	on	ADP
ejpam-5648	351	14	g.	g.	PROPN
ejpam-5648	351	15	let	let	VERB
ejpam-5648	351	16	m	m	VERB
ejpam-5648	351	17	=	=	PRON
ejpam-5648	351	18	{	{	PUNCT
ejpam-5648	351	19	j	j	PROPN
ejpam-5648	351	20	∈	∈	PROPN
ejpam-5648	352	1	[	[	X
ejpam-5648	352	2	k	k	X
ejpam-5648	352	3	]	]	X
ejpam-5648	352	4	:	:	PUNCT
ejpam-5648	352	5	nj	nj	PROPN
ejpam-5648	352	6	≥	≥	NUM
ejpam-5648	352	7	3	3	NUM
ejpam-5648	352	8	}	}	PUNCT
ejpam-5648	352	9	.	.	PUNCT
ejpam-5648	353	1	suppose	suppose	VERB
ejpam-5648	353	2	|w2|	|w2|	NOUN
ejpam-5648	353	3	=	=	PROPN
ejpam-5648	353	4	0	0	X
ejpam-5648	353	5	.	.	PUNCT
ejpam-5648	354	1	then	then	ADV
ejpam-5648	354	2	|w0|	|w0|	VERB
ejpam-5648	354	3	=	=	SYM
ejpam-5648	354	4	0	0	NUM
ejpam-5648	354	5	and	and	CCONJ
ejpam-5648	354	6	|w1|	|w1|	NOUN
ejpam-5648	354	7	=	=	SYM
ejpam-5648	354	8	v	v	NOUN
ejpam-5648	354	9	(	(	PUNCT
ejpam-5648	354	10	g	g	NOUN
ejpam-5648	354	11	)	)	PUNCT
ejpam-5648	354	12	.	.	PUNCT
ejpam-5648	355	1	it	it	PRON
ejpam-5648	355	2	follows	follow	VERB
ejpam-5648	355	3	that	that	SCONJ
ejpam-5648	355	4	γcrh(g	γcrh(g	NOUN
ejpam-5648	355	5	)	)	PUNCT
ejpam-5648	355	6	=	=	SYM
ejpam-5648	355	7	|v	|v	PROPN
ejpam-5648	355	8	(	(	PUNCT
ejpam-5648	355	9	g)|	g)|	PROPN
ejpam-5648	355	10	=	=	PROPN
ejpam-5648	355	11	∑k	∑k	PROPN
ejpam-5648	355	12	j=1	j=1	PROPN
ejpam-5648	355	13	nj	nj	PROPN
ejpam-5648	355	14	.	.	PUNCT
ejpam-5648	356	1	the	the	DET
ejpam-5648	356	2	value	value	NOUN
ejpam-5648	356	3	is	be	AUX
ejpam-5648	356	4	2k	2k	NUM
ejpam-5648	356	5	ifm	ifm	NOUN
ejpam-5648	356	6	=	=	NOUN
ejpam-5648	356	7	∅	∅	NOUN
ejpam-5648	356	8	and	and	CCONJ
ejpam-5648	356	9	strictly	strictly	ADV
ejpam-5648	356	10	greater	great	ADJ
ejpam-5648	356	11	than	than	ADP
ejpam-5648	356	12	2k	2k	NUM
ejpam-5648	356	13	ifm	ifm	NOUN
ejpam-5648	356	14	̸=	̸=	PROPN
ejpam-5648	356	15	∅.	∅.	PRON
ejpam-5648	356	16	hence	hence	ADV
ejpam-5648	356	17	,	,	PUNCT
ejpam-5648	356	18	|w2|	|w2|	NOUN
ejpam-5648	356	19	=	=	NOUN
ejpam-5648	356	20	̸	̸	ADV
ejpam-5648	356	21	0	0	PUNCT
ejpam-5648	357	1	if	if	SCONJ
ejpam-5648	357	2	m	m	VERB
ejpam-5648	357	3	̸=	̸=	PROPN
ejpam-5648	357	4	∅.	∅.	ADV
ejpam-5648	357	5	so	so	ADV
ejpam-5648	357	6	suppose	suppose	VERB
ejpam-5648	357	7	|w2|	|w2|	NOUN
ejpam-5648	357	8	̸=	̸=	PROPN
ejpam-5648	357	9	0	0	NUM
ejpam-5648	357	10	.	.	PUNCT
ejpam-5648	358	1	let	let	VERB
ejpam-5648	358	2	r	r	NOUN
ejpam-5648	358	3	=	=	PRON
ejpam-5648	358	4	{	{	PUNCT
ejpam-5648	358	5	j	j	PROPN
ejpam-5648	358	6	∈	∈	PROPN
ejpam-5648	359	1	[	[	X
ejpam-5648	359	2	k	k	X
ejpam-5648	359	3	]	]	X
ejpam-5648	359	4	:	:	PUNCT
ejpam-5648	359	5	w2	w2	NOUN
ejpam-5648	359	6	∩	∩	PROPN
ejpam-5648	359	7	snj	snj	VERB
ejpam-5648	359	8	̸=	̸=	PROPN
ejpam-5648	359	9	∅	∅	NOUN
ejpam-5648	359	10	}	}	PUNCT
ejpam-5648	359	11	.	.	PUNCT
ejpam-5648	360	1	since	since	SCONJ
ejpam-5648	360	2	g	g	PROPN
ejpam-5648	360	3	is	be	AUX
ejpam-5648	360	4	a	a	DET
ejpam-5648	360	5	roman	roman	ADJ
ejpam-5648	360	6	dominating	dominating	NOUN
ejpam-5648	360	7	function	function	NOUN
ejpam-5648	360	8	on	on	ADP
ejpam-5648	360	9	g	g	PROPN
ejpam-5648	360	10	,	,	PUNCT
ejpam-5648	360	11	it	it	PRON
ejpam-5648	360	12	follows	follow	VERB
ejpam-5648	360	13	that	that	DET
ejpam-5648	360	14	w0	w0	PROPN
ejpam-5648	360	15	=	=	PUNCT
ejpam-5648	360	16	∪j∈r[snj	∪j∈r[snj	PUNCT
ejpam-5648	360	17	\	\	X
ejpam-5648	361	1	[	[	X
ejpam-5648	361	2	(	(	PUNCT
ejpam-5648	361	3	w2	w2	NOUN
ejpam-5648	361	4	∩snj	∩snj	VERB
ejpam-5648	361	5	)	)	PUNCT
ejpam-5648	361	6	∪	∪	NOUN
ejpam-5648	361	7	(	(	PUNCT
ejpam-5648	361	8	w1	w1	NOUN
ejpam-5648	361	9	∩snj	∩snj	VERB
ejpam-5648	361	10	)	)	PUNCT
ejpam-5648	361	11	]	]	PUNCT
ejpam-5648	361	12	and	and	CCONJ
ejpam-5648	361	13	w1	w1	NOUN
ejpam-5648	361	14	=	=	PUNCT
ejpam-5648	362	1	[	[	X
ejpam-5648	362	2	∪j∈[k]\rsnj	∪j∈[k]\rsnj	X
ejpam-5648	362	3	]	]	PUNCT
ejpam-5648	362	4	∪	∪	ADP
ejpam-5648	362	5	[	[	X
ejpam-5648	362	6	∪j∈r(snj	∪j∈r(snj	NOUN
ejpam-5648	362	7	∩w1	∩w1	NOUN
ejpam-5648	362	8	)	)	PUNCT
ejpam-5648	362	9	]	]	PUNCT
ejpam-5648	362	10	.	.	PUNCT
ejpam-5648	363	1	it	it	PRON
ejpam-5648	363	2	follows	follow	VERB
ejpam-5648	363	3	that	that	SCONJ
ejpam-5648	363	4	γcrh(g	γcrh(g	ADP
ejpam-5648	363	5	)	)	PUNCT
ejpam-5648	363	6	=	=	SYM
ejpam-5648	363	7	∑	∑	PUNCT
ejpam-5648	363	8	j∈[k]\r	j∈[k]\r	PROPN
ejpam-5648	363	9	snj	snj	VERB
ejpam-5648	363	10	+	+	CCONJ
ejpam-5648	363	11	∑	∑	PROPN
ejpam-5648	363	12	j∈r	j∈r	NOUN
ejpam-5648	363	13	(	(	PUNCT
ejpam-5648	363	14	snj	snj	PROPN
ejpam-5648	363	15	∩w1	∩w1	NOUN
ejpam-5648	363	16	)	)	PUNCT
ejpam-5648	364	1	+	+	CCONJ
ejpam-5648	364	2	2	2	NUM
ejpam-5648	364	3	∑	∑	NOUN
ejpam-5648	364	4	j∈r	j∈r	NOUN
ejpam-5648	364	5	(	(	PUNCT
ejpam-5648	364	6	snj	snj	ADJ
ejpam-5648	364	7	∩w2	∩w2	PROPN
ejpam-5648	364	8	)	)	PUNCT
ejpam-5648	364	9	≥	≥	NOUN
ejpam-5648	364	10	2(k	2(k	NUM
ejpam-5648	364	11	−	−	PROPN
ejpam-5648	364	12	|r|	|r|	PROPN
ejpam-5648	364	13	)	)	PUNCT
ejpam-5648	364	14	+	+	NUM
ejpam-5648	364	15	2|r|	2|r|	NUM
ejpam-5648	364	16	=	=	SYM
ejpam-5648	364	17	2k	2k	NUM
ejpam-5648	364	18	.	.	PUNCT
ejpam-5648	365	1	this	this	PRON
ejpam-5648	365	2	establishes	establish	VERB
ejpam-5648	365	3	the	the	DET
ejpam-5648	365	4	desired	desire	VERB
ejpam-5648	365	5	equality	equality	NOUN
ejpam-5648	365	6	.	.	PUNCT
ejpam-5648	366	1	lemma	lemma	PROPN
ejpam-5648	366	2	1	1	X
ejpam-5648	366	3	.	.	PUNCT
ejpam-5648	367	1	let	let	VERB
ejpam-5648	367	2	g	g	PRON
ejpam-5648	367	3	be	be	AUX
ejpam-5648	367	4	a	a	DET
ejpam-5648	367	5	connected	connected	ADJ
ejpam-5648	367	6	graph	graph	NOUN
ejpam-5648	367	7	of	of	ADP
ejpam-5648	367	8	order	order	NOUN
ejpam-5648	367	9	n.	n.	NOUN
ejpam-5648	367	10	then	then	ADV
ejpam-5648	367	11	γch(g	γch(g	NOUN
ejpam-5648	367	12	)	)	PUNCT
ejpam-5648	367	13	=	=	SYM
ejpam-5648	368	1	n	n	NOUN
ejpam-5648	368	2	if	if	SCONJ
ejpam-5648	369	1	and	and	CCONJ
ejpam-5648	369	2	only	only	ADV
ejpam-5648	369	3	if	if	SCONJ
ejpam-5648	369	4	g	g	PROPN
ejpam-5648	369	5	=	=	PROPN
ejpam-5648	369	6	kn	kn	PROPN
ejpam-5648	369	7	.	.	PUNCT
ejpam-5648	369	8	proof	proof	PROPN
ejpam-5648	369	9	.	.	PUNCT
ejpam-5648	370	1	suppose	suppose	VERB
ejpam-5648	370	2	γch(g	γch(g	NOUN
ejpam-5648	370	3	)	)	PUNCT
ejpam-5648	370	4	=	=	SYM
ejpam-5648	370	5	n	n	NOUN
ejpam-5648	370	6	and	and	CCONJ
ejpam-5648	370	7	suppose	suppose	VERB
ejpam-5648	370	8	for	for	ADP
ejpam-5648	370	9	a	a	DET
ejpam-5648	370	10	contradiction	contradiction	NOUN
ejpam-5648	370	11	that	that	PRON
ejpam-5648	370	12	g	g	PROPN
ejpam-5648	370	13	̸=	̸=	PROPN
ejpam-5648	370	14	kn	kn	PROPN
ejpam-5648	370	15	.	.	PUNCT
ejpam-5648	371	1	then	then	ADV
ejpam-5648	371	2	diam(g	diam(g	PROPN
ejpam-5648	371	3	)	)	PUNCT
ejpam-5648	371	4	≥	≥	NOUN
ejpam-5648	371	5	2	2	NUM
ejpam-5648	371	6	.	.	PUNCT
ejpam-5648	371	7	pick	pick	VERB
ejpam-5648	371	8	any	any	DET
ejpam-5648	371	9	two	two	NUM
ejpam-5648	371	10	vertices	vertex	NOUN
ejpam-5648	371	11	x	x	PUNCT
ejpam-5648	371	12	and	and	CCONJ
ejpam-5648	371	13	y	y	PROPN
ejpam-5648	371	14	of	of	ADP
ejpam-5648	371	15	g	g	PROPN
ejpam-5648	371	16	such	such	ADJ
ejpam-5648	371	17	that	that	DET
ejpam-5648	371	18	dg(x	dg(x	NOUN
ejpam-5648	371	19	,	,	PUNCT
ejpam-5648	371	20	y	y	NOUN
ejpam-5648	371	21	)	)	PUNCT
ejpam-5648	371	22	=	=	SYM
ejpam-5648	371	23	diam(g	diam(g	NOUN
ejpam-5648	371	24	)	)	PUNCT
ejpam-5648	371	25	.	.	PUNCT
ejpam-5648	372	1	then	then	ADV
ejpam-5648	372	2	x	x	X
ejpam-5648	372	3	and	and	CCONJ
ejpam-5648	372	4	y	y	PROPN
ejpam-5648	372	5	are	be	AUX
ejpam-5648	372	6	non	non	ADJ
ejpam-5648	372	7	-	-	ADJ
ejpam-5648	372	8	cut	cut	ADJ
ejpam-5648	372	9	vertices	vertex	NOUN
ejpam-5648	372	10	of	of	ADP
ejpam-5648	372	11	g.	g.	PROPN
ejpam-5648	372	12	let	let	VERB
ejpam-5648	373	1	[	[	X
ejpam-5648	373	2	x1	x1	ADJ
ejpam-5648	373	3	,	,	PUNCT
ejpam-5648	373	4	x2	x2	PROPN
ejpam-5648	373	5	,	,	PUNCT
ejpam-5648	373	6	.	.	PUNCT
ejpam-5648	373	7	.	.	PUNCT
ejpam-5648	373	8	.	.	PUNCT
ejpam-5648	374	1	,	,	PUNCT
ejpam-5648	374	2	xk	xk	PROPN
ejpam-5648	374	3	]	]	X
ejpam-5648	374	4	,	,	PUNCT
ejpam-5648	374	5	where	where	SCONJ
ejpam-5648	374	6	x	x	ADP
ejpam-5648	374	7	=	=	SYM
ejpam-5648	374	8	x1	x1	PROPN
ejpam-5648	374	9	and	and	CCONJ
ejpam-5648	374	10	y	y	PROPN
ejpam-5648	374	11	=	=	SYM
ejpam-5648	374	12	xk	xk	PROPN
ejpam-5648	374	13	,	,	PUNCT
ejpam-5648	374	14	be	be	AUX
ejpam-5648	374	15	an	an	DET
ejpam-5648	374	16	x	x	NOUN
ejpam-5648	374	17	-	-	NOUN
ejpam-5648	374	18	y	y	ADJ
ejpam-5648	374	19	geodesic	geodesic	NOUN
ejpam-5648	374	20	in	in	ADP
ejpam-5648	374	21	g.	g.	PROPN
ejpam-5648	374	22	then	then	ADV
ejpam-5648	374	23	k	k	PROPN
ejpam-5648	374	24	≥	≥	NUM
ejpam-5648	374	25	3	3	NUM
ejpam-5648	374	26	and	and	CCONJ
ejpam-5648	374	27	dg(x1	dg(x1	NOUN
ejpam-5648	374	28	,	,	PUNCT
ejpam-5648	374	29	x3	x3	ADJ
ejpam-5648	374	30	)	)	PUNCT
ejpam-5648	375	1	=	=	SYM
ejpam-5648	375	2	2	2	X
ejpam-5648	375	3	.	.	X
ejpam-5648	375	4	put	put	VERB
ejpam-5648	375	5	s	s	PART
ejpam-5648	375	6	=	=	X
ejpam-5648	375	7	v	v	ADJ
ejpam-5648	375	8	(	(	PUNCT
ejpam-5648	375	9	g	g	NOUN
ejpam-5648	375	10	)	)	PUNCT
ejpam-5648	375	11	\	\	NOUN
ejpam-5648	375	12	{	{	PUNCT
ejpam-5648	375	13	x	x	NOUN
ejpam-5648	375	14	}	}	PUNCT
ejpam-5648	375	15	.	.	PUNCT
ejpam-5648	376	1	since	since	SCONJ
ejpam-5648	376	2	x	x	PRON
ejpam-5648	376	3	is	be	AUX
ejpam-5648	376	4	a	a	DET
ejpam-5648	376	5	non	non	ADJ
ejpam-5648	376	6	-	-	ADJ
ejpam-5648	376	7	cut	cut	ADJ
ejpam-5648	376	8	vertex	vertex	NOUN
ejpam-5648	376	9	,	,	PUNCT
ejpam-5648	376	10	it	it	PRON
ejpam-5648	376	11	follows	follow	VERB
ejpam-5648	376	12	that	that	SCONJ
ejpam-5648	376	13	⟨s⟩	⟨s⟩	PROPN
ejpam-5648	376	14	is	be	AUX
ejpam-5648	376	15	connected	connect	VERB
ejpam-5648	376	16	.	.	PUNCT
ejpam-5648	377	1	therefore	therefore	ADV
ejpam-5648	377	2	,	,	PUNCT
ejpam-5648	377	3	s	s	VERB
ejpam-5648	377	4	is	be	AUX
ejpam-5648	377	5	a	a	DET
ejpam-5648	377	6	connected	connected	ADJ
ejpam-5648	377	7	hop	hop	NOUN
ejpam-5648	377	8	dominating	dominating	NOUN
ejpam-5648	377	9	set	set	VERB
ejpam-5648	377	10	in	in	ADP
ejpam-5648	377	11	g	g	PROPN
ejpam-5648	377	12	and	and	CCONJ
ejpam-5648	377	13	γch(g	γch(g	NOUN
ejpam-5648	377	14	)	)	PUNCT
ejpam-5648	377	15	≤	≤	NUM
ejpam-5648	377	16	|s|	|s|	PROPN
ejpam-5648	377	17	=	=	SYM
ejpam-5648	377	18	n−	n−	NOUN
ejpam-5648	377	19	1	1	NUM
ejpam-5648	377	20	,	,	PUNCT
ejpam-5648	377	21	contrary	contrary	ADV
ejpam-5648	377	22	to	to	ADP
ejpam-5648	377	23	our	our	PRON
ejpam-5648	377	24	assumption	assumption	NOUN
ejpam-5648	377	25	.	.	PUNCT
ejpam-5648	378	1	thus	thus	ADV
ejpam-5648	378	2	,	,	PUNCT
ejpam-5648	378	3	g	g	PROPN
ejpam-5648	378	4	=	=	SYM
ejpam-5648	378	5	kn	kn	PROPN
ejpam-5648	378	6	.	.	PUNCT
ejpam-5648	378	7	a.	a.	PROPN
ejpam-5648	378	8	aradais	aradais	PROPN
ejpam-5648	378	9	,	,	PUNCT
ejpam-5648	378	10	j.	j.	PROPN
ejpam-5648	378	11	cariaga	cariaga	PROPN
ejpam-5648	378	12	,	,	PUNCT
ejpam-5648	378	13	s.	s.	PROPN
ejpam-5648	378	14	canoy	canoy	PROPN
ejpam-5648	378	15	jr	jr	PROPN
ejpam-5648	378	16	.	.	PROPN
ejpam-5648	378	17	/	/	SYM
ejpam-5648	378	18	eur	eur	PROPN
ejpam-5648	378	19	.	.	PUNCT
ejpam-5648	379	1	j.	j.	PROPN
ejpam-5648	379	2	pure	pure	PROPN
ejpam-5648	379	3	appl	appl	PROPN
ejpam-5648	379	4	.	.	PROPN
ejpam-5648	379	5	math	math	PROPN
ejpam-5648	379	6	,	,	PUNCT
ejpam-5648	379	7	18	18	NUM
ejpam-5648	379	8	(	(	PUNCT
ejpam-5648	379	9	1	1	NUM
ejpam-5648	379	10	)	)	PUNCT
ejpam-5648	379	11	(	(	PUNCT
ejpam-5648	379	12	2025	2025	NUM
ejpam-5648	379	13	)	)	PUNCT
ejpam-5648	379	14	,	,	PUNCT
ejpam-5648	379	15	5648	5648	NUM
ejpam-5648	379	16	10	10	NUM
ejpam-5648	379	17	of	of	ADP
ejpam-5648	379	18	13	13	NUM
ejpam-5648	379	19	for	for	ADP
ejpam-5648	379	20	the	the	DET
ejpam-5648	379	21	converse	converse	NOUN
ejpam-5648	379	22	,	,	PUNCT
ejpam-5648	379	23	suppose	suppose	VERB
ejpam-5648	379	24	g	g	PROPN
ejpam-5648	379	25	=	=	PROPN
ejpam-5648	379	26	kn	kn	PROPN
ejpam-5648	379	27	and	and	CCONJ
ejpam-5648	379	28	let	let	VERB
ejpam-5648	379	29	s	s	PRON
ejpam-5648	379	30	be	be	AUX
ejpam-5648	379	31	a	a	DET
ejpam-5648	379	32	gammach	gammach	NOUN
ejpam-5648	379	33	-	-	PUNCT
ejpam-5648	379	34	set	set	VERB
ejpam-5648	379	35	in	in	ADP
ejpam-5648	379	36	g.	g.	PROPN
ejpam-5648	379	37	since	since	SCONJ
ejpam-5648	379	38	s	s	PROPN
ejpam-5648	379	39	is	be	AUX
ejpam-5648	379	40	a	a	DET
ejpam-5648	379	41	hop	hop	NOUN
ejpam-5648	379	42	dominating	dominating	NOUN
ejpam-5648	379	43	set	set	NOUN
ejpam-5648	379	44	,	,	PUNCT
ejpam-5648	379	45	v	v	PROPN
ejpam-5648	379	46	(	(	PUNCT
ejpam-5648	379	47	kn	kn	PROPN
ejpam-5648	379	48	)	)	PUNCT
ejpam-5648	379	49	⊆	⊆	NUM
ejpam-5648	379	50	s.	s.	PROPN
ejpam-5648	379	51	therefore	therefore	ADV
ejpam-5648	379	52	,	,	PUNCT
ejpam-5648	379	53	s	s	NOUN
ejpam-5648	379	54	=	=	SYM
ejpam-5648	379	55	v	v	X
ejpam-5648	379	56	(	(	PUNCT
ejpam-5648	379	57	g	g	NOUN
ejpam-5648	379	58	)	)	PUNCT
ejpam-5648	379	59	and	and	CCONJ
ejpam-5648	379	60	γch(g	γch(g	NOUN
ejpam-5648	379	61	)	)	PUNCT
ejpam-5648	379	62	=	=	PUNCT
ejpam-5648	379	63	n.	n.	NOUN
ejpam-5648	379	64	proposition	proposition	NOUN
ejpam-5648	379	65	8	8	NUM
ejpam-5648	379	66	.	.	PUNCT
ejpam-5648	380	1	let	let	VERB
ejpam-5648	380	2	g	g	NOUN
ejpam-5648	380	3	be	be	AUX
ejpam-5648	380	4	any	any	DET
ejpam-5648	380	5	graph	graph	NOUN
ejpam-5648	380	6	of	of	ADP
ejpam-5648	380	7	order	order	NOUN
ejpam-5648	380	8	n.	n.	NOUN
ejpam-5648	380	9	then	then	ADV
ejpam-5648	380	10	γch(g	γch(g	NOUN
ejpam-5648	380	11	)	)	PUNCT
ejpam-5648	380	12	=	=	SYM
ejpam-5648	380	13	γcrh(g	γcrh(g	PROPN
ejpam-5648	380	14	)	)	PUNCT
ejpam-5648	381	1	if	if	SCONJ
ejpam-5648	381	2	and	and	CCONJ
ejpam-5648	381	3	only	only	ADV
ejpam-5648	381	4	if	if	SCONJ
ejpam-5648	381	5	g	g	PROPN
ejpam-5648	381	6	=	=	PROPN
ejpam-5648	381	7	kn	kn	PROPN
ejpam-5648	381	8	.	.	PUNCT
ejpam-5648	381	9	proof	proof	PROPN
ejpam-5648	381	10	.	.	PUNCT
ejpam-5648	382	1	suppose	suppose	VERB
ejpam-5648	382	2	γch(g	γch(g	NOUN
ejpam-5648	382	3	)	)	PUNCT
ejpam-5648	382	4	=	=	SYM
ejpam-5648	382	5	γcrh(g	γcrh(g	PROPN
ejpam-5648	382	6	)	)	PUNCT
ejpam-5648	382	7	and	and	CCONJ
ejpam-5648	382	8	let	let	VERB
ejpam-5648	382	9	f	f	PROPN
ejpam-5648	382	10	=	=	SYM
ejpam-5648	382	11	(	(	PUNCT
ejpam-5648	382	12	v0	v0	PROPN
ejpam-5648	382	13	,	,	PUNCT
ejpam-5648	382	14	v1	v1	NOUN
ejpam-5648	382	15	,	,	PUNCT
ejpam-5648	382	16	v2	v2	PROPN
ejpam-5648	382	17	)	)	PUNCT
ejpam-5648	382	18	be	be	AUX
ejpam-5648	382	19	a	a	DET
ejpam-5648	382	20	γcrh	γcrh	NOUN
ejpam-5648	382	21	-	-	PUNCT
ejpam-5648	382	22	function	function	NOUN
ejpam-5648	382	23	in	in	ADP
ejpam-5648	382	24	g.	g.	PROPN
ejpam-5648	382	25	since	since	SCONJ
ejpam-5648	382	26	γch(g	γch(g	NOUN
ejpam-5648	382	27	)	)	PUNCT
ejpam-5648	382	28	≤	≤	NOUN
ejpam-5648	382	29	|v1|	|v1|	NOUN
ejpam-5648	382	30	+	+	CCONJ
ejpam-5648	382	31	|v2|	|v2|	NOUN
ejpam-5648	382	32	≤	≤	NOUN
ejpam-5648	382	33	|v1|	|v1|	NOUN
ejpam-5648	382	34	+	+	CCONJ
ejpam-5648	382	35	2|v2|	2|v2|	NUM
ejpam-5648	382	36	=	=	SYM
ejpam-5648	382	37	γcrh(g	γcrh(g	NOUN
ejpam-5648	382	38	)	)	PUNCT
ejpam-5648	382	39	and	and	CCONJ
ejpam-5648	382	40	γch(g	γch(g	NOUN
ejpam-5648	382	41	)	)	PUNCT
ejpam-5648	382	42	=	=	SYM
ejpam-5648	382	43	γcrh(g	γcrh(g	NOUN
ejpam-5648	382	44	)	)	PUNCT
ejpam-5648	382	45	,	,	PUNCT
ejpam-5648	382	46	|v1|	|v1|	NOUN
ejpam-5648	382	47	+	+	CCONJ
ejpam-5648	382	48	|v2|	|v2|	NOUN
ejpam-5648	382	49	=	=	SYM
ejpam-5648	382	50	|v1|	|v1|	NOUN
ejpam-5648	382	51	+	+	CCONJ
ejpam-5648	382	52	2|v2|	2|v2|	NUM
ejpam-5648	382	53	.	.	PUNCT
ejpam-5648	383	1	this	this	PRON
ejpam-5648	383	2	implies	imply	VERB
ejpam-5648	383	3	that	that	SCONJ
ejpam-5648	383	4	v2	v2	NOUN
ejpam-5648	383	5	=	=	NOUN
ejpam-5648	383	6	∅	∅	NOUN
ejpam-5648	383	7	and	and	CCONJ
ejpam-5648	383	8	thus	thus	ADV
ejpam-5648	383	9	,	,	PUNCT
ejpam-5648	383	10	v0	v0	NOUN
ejpam-5648	383	11	=	=	PUNCT
ejpam-5648	383	12	∅.	∅.	VERB
ejpam-5648	383	13	hence	hence	ADV
ejpam-5648	383	14	,	,	PUNCT
ejpam-5648	383	15	γcrh(g	γcrh(g	NOUN
ejpam-5648	383	16	)	)	PUNCT
ejpam-5648	383	17	=	=	NOUN
ejpam-5648	383	18	|v1|	|v1|	NOUN
ejpam-5648	383	19	=	=	SYM
ejpam-5648	383	20	n	n	NOUN
ejpam-5648	383	21	=	=	SYM
ejpam-5648	383	22	γch(g	γch(g	NOUN
ejpam-5648	383	23	)	)	PUNCT
ejpam-5648	383	24	.	.	PUNCT
ejpam-5648	384	1	by	by	ADP
ejpam-5648	384	2	lemma	lemma	PROPN
ejpam-5648	384	3	1	1	NUM
ejpam-5648	384	4	,	,	PUNCT
ejpam-5648	384	5	g	g	PROPN
ejpam-5648	384	6	=	=	SYM
ejpam-5648	384	7	kn	kn	PROPN
ejpam-5648	384	8	.	.	PROPN
ejpam-5648	385	1	for	for	ADP
ejpam-5648	385	2	the	the	DET
ejpam-5648	385	3	converse	converse	NOUN
ejpam-5648	385	4	,	,	PUNCT
ejpam-5648	385	5	suppose	suppose	VERB
ejpam-5648	385	6	that	that	SCONJ
ejpam-5648	385	7	g	g	PROPN
ejpam-5648	385	8	=	=	PROPN
ejpam-5648	385	9	kn	kn	PROPN
ejpam-5648	385	10	.	.	PUNCT
ejpam-5648	386	1	by	by	ADP
ejpam-5648	386	2	lemma	lemma	PROPN
ejpam-5648	386	3	1	1	NUM
ejpam-5648	386	4	,	,	PUNCT
ejpam-5648	386	5	γch(g	γch(g	NOUN
ejpam-5648	386	6	)	)	PUNCT
ejpam-5648	386	7	=	=	VERB
ejpam-5648	386	8	n.	n.	NOUN
ejpam-5648	386	9	by	by	ADP
ejpam-5648	386	10	proposition	proposition	NOUN
ejpam-5648	386	11	2	2	NUM
ejpam-5648	386	12	,	,	PUNCT
ejpam-5648	386	13	γcrh(g	γcrh(g	NOUN
ejpam-5648	386	14	)	)	PUNCT
ejpam-5648	386	15	=	=	VERB
ejpam-5648	386	16	n.	n.	NOUN
ejpam-5648	386	17	this	this	PRON
ejpam-5648	386	18	shows	show	VERB
ejpam-5648	386	19	that	that	SCONJ
ejpam-5648	386	20	γch(g	γch(g	NOUN
ejpam-5648	386	21	)	)	PUNCT
ejpam-5648	386	22	=	=	SYM
ejpam-5648	386	23	γcrh(g	γcrh(g	PROPN
ejpam-5648	386	24	)	)	PUNCT
ejpam-5648	386	25	.	.	PUNCT
ejpam-5648	387	1	the	the	DET
ejpam-5648	387	2	next	next	ADJ
ejpam-5648	387	3	result	result	NOUN
ejpam-5648	387	4	is	be	AUX
ejpam-5648	387	5	immediate	immediate	ADJ
ejpam-5648	387	6	from	from	ADP
ejpam-5648	387	7	proposition	proposition	NOUN
ejpam-5648	387	8	3	3	NUM
ejpam-5648	387	9	.	.	PUNCT
ejpam-5648	387	10	proposition	proposition	NOUN
ejpam-5648	387	11	9	9	NUM
ejpam-5648	387	12	.	.	PUNCT
ejpam-5648	388	1	let	let	VERB
ejpam-5648	388	2	g	g	NOUN
ejpam-5648	388	3	and	and	CCONJ
ejpam-5648	388	4	h	h	NOUN
ejpam-5648	388	5	be	be	AUX
ejpam-5648	388	6	connected	connect	VERB
ejpam-5648	388	7	graphs	graph	NOUN
ejpam-5648	388	8	.	.	PUNCT
ejpam-5648	389	1	then	then	ADV
ejpam-5648	389	2	γcrh(g	γcrh(g	ADP
ejpam-5648	389	3	+	+	NOUN
ejpam-5648	389	4	h	h	NOUN
ejpam-5648	389	5	)	)	PUNCT
ejpam-5648	389	6	≥	≥	NOUN
ejpam-5648	389	7	2	2	NUM
ejpam-5648	389	8	and	and	CCONJ
ejpam-5648	389	9	each	each	PRON
ejpam-5648	389	10	of	of	ADP
ejpam-5648	389	11	the	the	DET
ejpam-5648	389	12	following	follow	VERB
ejpam-5648	389	13	holds	hold	VERB
ejpam-5648	389	14	:	:	PUNCT
ejpam-5648	389	15	(	(	PUNCT
ejpam-5648	389	16	i	i	NOUN
ejpam-5648	389	17	)	)	PUNCT
ejpam-5648	389	18	γcrh(g+h	γcrh(g+h	PROPN
ejpam-5648	389	19	)	)	PUNCT
ejpam-5648	389	20	=	=	SYM
ejpam-5648	389	21	2	2	NUM
ejpam-5648	389	22	if	if	SCONJ
ejpam-5648	389	23	and	and	CCONJ
ejpam-5648	389	24	only	only	ADV
ejpam-5648	389	25	if	if	SCONJ
ejpam-5648	389	26	g	g	PROPN
ejpam-5648	389	27	and	and	CCONJ
ejpam-5648	389	28	h	h	NOUN
ejpam-5648	389	29	are	be	AUX
ejpam-5648	389	30	trivial	trivial	ADJ
ejpam-5648	389	31	graphs	graph	NOUN
ejpam-5648	389	32	.	.	PUNCT
ejpam-5648	390	1	(	(	PUNCT
ejpam-5648	390	2	ii	ii	NOUN
ejpam-5648	390	3	)	)	PUNCT
ejpam-5648	390	4	γcrh(g+h	γcrh(g+h	PROPN
ejpam-5648	390	5	)	)	PUNCT
ejpam-5648	391	1	=	=	SYM
ejpam-5648	391	2	3	3	NUM
ejpam-5648	391	3	if	if	SCONJ
ejpam-5648	391	4	and	and	CCONJ
ejpam-5648	391	5	only	only	ADV
ejpam-5648	391	6	if	if	SCONJ
ejpam-5648	391	7	g	g	PROPN
ejpam-5648	391	8	=	=	SYM
ejpam-5648	391	9	k1	k1	PROPN
ejpam-5648	391	10	and	and	CCONJ
ejpam-5648	391	11	h	h	NOUN
ejpam-5648	391	12	=	=	PROPN
ejpam-5648	391	13	k2	k2	X
ejpam-5648	391	14	(	(	PUNCT
ejpam-5648	391	15	or	or	CCONJ
ejpam-5648	391	16	h	h	NOUN
ejpam-5648	391	17	=	=	SYM
ejpam-5648	391	18	k1	k1	PROPN
ejpam-5648	391	19	and	and	CCONJ
ejpam-5648	391	20	g	g	NOUN
ejpam-5648	391	21	=	=	SYM
ejpam-5648	391	22	k2	k2	PROPN
ejpam-5648	391	23	)	)	PUNCT
ejpam-5648	391	24	or	or	CCONJ
ejpam-5648	391	25	g	g	NOUN
ejpam-5648	391	26	=	=	SYM
ejpam-5648	391	27	k1	k1	PROPN
ejpam-5648	391	28	and	and	CCONJ
ejpam-5648	391	29	h	h	NOUN
ejpam-5648	391	30	is	be	AUX
ejpam-5648	391	31	a	a	DET
ejpam-5648	391	32	graph	graph	NOUN
ejpam-5648	391	33	with	with	ADP
ejpam-5648	391	34	at	at	ADV
ejpam-5648	391	35	least	least	ADV
ejpam-5648	391	36	one	one	NUM
ejpam-5648	391	37	isolated	isolated	ADJ
ejpam-5648	391	38	vertex	vertex	NOUN
ejpam-5648	391	39	(	(	PUNCT
ejpam-5648	391	40	or	or	CCONJ
ejpam-5648	391	41	h	h	NOUN
ejpam-5648	391	42	=	=	SYM
ejpam-5648	391	43	k1	k1	PROPN
ejpam-5648	391	44	and	and	CCONJ
ejpam-5648	391	45	g	g	PROPN
ejpam-5648	391	46	is	be	AUX
ejpam-5648	391	47	a	a	DET
ejpam-5648	391	48	graph	graph	NOUN
ejpam-5648	391	49	with	with	ADP
ejpam-5648	391	50	at	at	ADV
ejpam-5648	391	51	least	least	ADV
ejpam-5648	391	52	one	one	NUM
ejpam-5648	391	53	isolated	isolated	ADJ
ejpam-5648	391	54	vertex	vertex	NOUN
ejpam-5648	391	55	)	)	PUNCT
ejpam-5648	391	56	.	.	PUNCT
ejpam-5648	392	1	(	(	PUNCT
ejpam-5648	392	2	iii	iii	X
ejpam-5648	392	3	)	)	PUNCT
ejpam-5648	392	4	if	if	SCONJ
ejpam-5648	392	5	g	g	PROPN
ejpam-5648	392	6	and	and	CCONJ
ejpam-5648	392	7	h	h	NOUN
ejpam-5648	392	8	are	be	AUX
ejpam-5648	392	9	non	non	ADJ
ejpam-5648	392	10	-	-	ADJ
ejpam-5648	392	11	trivial	trivial	ADJ
ejpam-5648	392	12	graphs	graph	NOUN
ejpam-5648	392	13	and	and	CCONJ
ejpam-5648	392	14	each	each	PRON
ejpam-5648	392	15	contains	contain	VERB
ejpam-5648	392	16	an	an	DET
ejpam-5648	392	17	isolated	isolated	ADJ
ejpam-5648	392	18	vertex	vertex	NOUN
ejpam-5648	392	19	,	,	PUNCT
ejpam-5648	392	20	then	then	ADV
ejpam-5648	392	21	γcrh(g+h	γcrh(g+h	VERB
ejpam-5648	392	22	)	)	PUNCT
ejpam-5648	392	23	=	=	SYM
ejpam-5648	392	24	4	4	X
ejpam-5648	392	25	.	.	X
ejpam-5648	392	26	proposition	proposition	NOUN
ejpam-5648	392	27	10	10	NUM
ejpam-5648	392	28	.	.	PUNCT
ejpam-5648	393	1	let	let	VERB
ejpam-5648	393	2	g	g	NOUN
ejpam-5648	393	3	and	and	CCONJ
ejpam-5648	393	4	h	h	NOUN
ejpam-5648	393	5	be	be	AUX
ejpam-5648	393	6	connected	connect	VERB
ejpam-5648	393	7	graphs	graph	NOUN
ejpam-5648	393	8	.	.	PUNCT
ejpam-5648	394	1	then	then	ADV
ejpam-5648	394	2	2	2	NUM
ejpam-5648	394	3	≤	≤	NUM
ejpam-5648	394	4	γcrh(g+h	γcrh(g+h	NOUN
ejpam-5648	394	5	)	)	PUNCT
ejpam-5648	394	6	≤	≤	NUM
ejpam-5648	394	7	γcrh(g	γcrh(g	NOUN
ejpam-5648	394	8	)	)	PUNCT
ejpam-5648	394	9	+	+	X
ejpam-5648	394	10	γcrh(h	γcrh(h	NOUN
ejpam-5648	394	11	)	)	PUNCT
ejpam-5648	394	12	.	.	PUNCT
ejpam-5648	395	1	proof	proof	NOUN
ejpam-5648	395	2	.	.	PUNCT
ejpam-5648	396	1	since	since	SCONJ
ejpam-5648	396	2	g	g	PROPN
ejpam-5648	396	3	+	+	NOUN
ejpam-5648	396	4	h	h	NOUN
ejpam-5648	396	5	is	be	AUX
ejpam-5648	396	6	nontrivial	nontrivial	ADJ
ejpam-5648	396	7	,	,	PUNCT
ejpam-5648	396	8	γcrh(g	γcrh(g	X
ejpam-5648	396	9	+	+	NOUN
ejpam-5648	396	10	h	h	NOUN
ejpam-5648	396	11	)	)	PUNCT
ejpam-5648	396	12	≥	≥	NOUN
ejpam-5648	397	1	2	2	NUM
ejpam-5648	397	2	.	.	PUNCT
ejpam-5648	397	3	let	let	VERB
ejpam-5648	397	4	f	f	PROPN
ejpam-5648	397	5	=	=	SYM
ejpam-5648	397	6	(	(	PUNCT
ejpam-5648	397	7	v0	v0	PROPN
ejpam-5648	397	8	,	,	PUNCT
ejpam-5648	397	9	v1	v1	NOUN
ejpam-5648	397	10	,	,	PUNCT
ejpam-5648	397	11	v2	v2	NOUN
ejpam-5648	397	12	)	)	PUNCT
ejpam-5648	397	13	and	and	CCONJ
ejpam-5648	397	14	g	g	NOUN
ejpam-5648	397	15	=	=	SYM
ejpam-5648	397	16	(	(	PUNCT
ejpam-5648	397	17	v	v	NUM
ejpam-5648	397	18	′	′	NUM
ejpam-5648	397	19	0	0	NUM
ejpam-5648	397	20	,	,	PUNCT
ejpam-5648	397	21	v	v	NOUN
ejpam-5648	397	22	′	′	NUM
ejpam-5648	397	23	1	1	NUM
ejpam-5648	397	24	,	,	PUNCT
ejpam-5648	397	25	v	v	NOUN
ejpam-5648	397	26	′	′	NUM
ejpam-5648	397	27	2	2	NUM
ejpam-5648	397	28	)	)	PUNCT
ejpam-5648	397	29	be	be	AUX
ejpam-5648	397	30	γcrh	γcrh	NOUN
ejpam-5648	397	31	-	-	PUNCT
ejpam-5648	397	32	functions	function	NOUN
ejpam-5648	397	33	ong	ong	NOUN
ejpam-5648	397	34	andh	andh	NOUN
ejpam-5648	397	35	,	,	PUNCT
ejpam-5648	397	36	respectively	respectively	ADV
ejpam-5648	397	37	.	.	PUNCT
ejpam-5648	398	1	define	define	VERB
ejpam-5648	398	2	a	a	DET
ejpam-5648	398	3	function	function	NOUN
ejpam-5648	398	4	h	h	NOUN
ejpam-5648	398	5	=	=	SYM
ejpam-5648	398	6	(	(	PUNCT
ejpam-5648	398	7	v	v	NUM
ejpam-5648	398	8	′′	′′	PROPN
ejpam-5648	398	9	0	0	NUM
ejpam-5648	398	10	,	,	PUNCT
ejpam-5648	398	11	v	v	ADP
ejpam-5648	398	12	′′	′′	PROPN
ejpam-5648	398	13	1	1	NUM
ejpam-5648	398	14	,	,	PUNCT
ejpam-5648	398	15	v	v	ADP
ejpam-5648	398	16	′′	′′	PROPN
ejpam-5648	398	17	2	2	NUM
ejpam-5648	398	18	)	)	PUNCT
ejpam-5648	398	19	for	for	ADP
ejpam-5648	398	20	which	which	PRON
ejpam-5648	398	21	v	v	ADP
ejpam-5648	398	22	′′	′′	PROPN
ejpam-5648	398	23	0	0	PUNCT
ejpam-5648	399	1	=	=	SYM
ejpam-5648	399	2	v0	v0	NOUN
ejpam-5648	399	3	∪v	∪v	NOUN
ejpam-5648	399	4	′	′	NOUN
ejpam-5648	399	5	0	0	NUM
ejpam-5648	399	6	,	,	PUNCT
ejpam-5648	399	7	v	v	ADP
ejpam-5648	399	8	′′	′′	PROPN
ejpam-5648	399	9	1	1	NUM
ejpam-5648	399	10	=	=	SYM
ejpam-5648	399	11	v1	v1	PROPN
ejpam-5648	399	12	∪v	∪v	NOUN
ejpam-5648	399	13	′	′	NOUN
ejpam-5648	399	14	1	1	NUM
ejpam-5648	399	15	and	and	CCONJ
ejpam-5648	399	16	v	v	ADP
ejpam-5648	399	17	′′	′′	PROPN
ejpam-5648	399	18	2	2	NUM
ejpam-5648	399	19	=	=	SYM
ejpam-5648	399	20	v2	v2	NOUN
ejpam-5648	399	21	∪v	∪v	NOUN
ejpam-5648	399	22	′	′	NOUN
ejpam-5648	399	23	2	2	NUM
ejpam-5648	399	24	.	.	PUNCT
ejpam-5648	400	1	then	then	ADV
ejpam-5648	400	2	h	h	PROPN
ejpam-5648	400	3	∈	∈	PROPN
ejpam-5648	400	4	chrdf	chrdf	NOUN
ejpam-5648	400	5	(	(	PUNCT
ejpam-5648	400	6	g+h	g+h	NOUN
ejpam-5648	400	7	)	)	PUNCT
ejpam-5648	400	8	and	and	CCONJ
ejpam-5648	400	9	γcrh(g+h	γcrh(g+h	PROPN
ejpam-5648	400	10	)	)	PUNCT
ejpam-5648	400	11	≤	≤	NOUN
ejpam-5648	400	12	ωcrh	ωcrh	ADJ
ejpam-5648	400	13	g+h	g+h	PUNCT
ejpam-5648	400	14	=	=	SYM
ejpam-5648	401	1	|v	|v	PROPN
ejpam-5648	401	2	′′	′′	PROPN
ejpam-5648	401	3	1	1	NUM
ejpam-5648	401	4	|+	|+	NOUN
ejpam-5648	401	5	2|v	2|v	NUM
ejpam-5648	402	1	′′	′′	NOUN
ejpam-5648	402	2	2	2	NUM
ejpam-5648	402	3	|	|	ADV
ejpam-5648	402	4	=	=	PUNCT
ejpam-5648	402	5	|v1	|v1	PROPN
ejpam-5648	402	6	∪	∪	PROPN
ejpam-5648	402	7	v	v	NOUN
ejpam-5648	402	8	′	′	NUM
ejpam-5648	402	9	1	1	NUM
ejpam-5648	402	10	|+	|+	NOUN
ejpam-5648	402	11	2|v2	2|v2	NUM
ejpam-5648	402	12	∪	∪	ADP
ejpam-5648	402	13	v	v	NUM
ejpam-5648	402	14	′	′	NUM
ejpam-5648	402	15	2	2	NUM
ejpam-5648	402	16	|	|	NOUN
ejpam-5648	402	17	=	=	SYM
ejpam-5648	402	18	|v1|+	|v1|+	ADV
ejpam-5648	402	19	|v	|v	ADJ
ejpam-5648	402	20	′	′	NUM
ejpam-5648	402	21	1	1	NUM
ejpam-5648	402	22	|+	|+	NOUN
ejpam-5648	402	23	2|v2|+	2|v2|+	NUM
ejpam-5648	402	24	2|v	2|v	NUM
ejpam-5648	403	1	′	′	NUM
ejpam-5648	403	2	2	2	NUM
ejpam-5648	403	3	|	|	NOUN
ejpam-5648	403	4	=	=	SYM
ejpam-5648	403	5	|v1|+	|v1|+	SYM
ejpam-5648	403	6	2|v2|+	2|v2|+	NUM
ejpam-5648	403	7	|v	|v	X
ejpam-5648	403	8	′	′	NOUN
ejpam-5648	403	9	1	1	NUM
ejpam-5648	403	10	|+	|+	NOUN
ejpam-5648	403	11	2|v	2|v	NOUN
ejpam-5648	404	1	′	′	NOUN
ejpam-5648	404	2	2	2	NUM
ejpam-5648	405	1	|	|	NOUN
ejpam-5648	405	2	=	=	SYM
ejpam-5648	405	3	γcrh(g	γcrh(g	PROPN
ejpam-5648	405	4	)	)	PUNCT
ejpam-5648	405	5	+	+	X
ejpam-5648	405	6	γcrh(h	γcrh(h	NOUN
ejpam-5648	405	7	)	)	PUNCT
ejpam-5648	405	8	.	.	PUNCT
ejpam-5648	406	1	(	(	PUNCT
ejpam-5648	406	2	1	1	X
ejpam-5648	406	3	)	)	PUNCT
ejpam-5648	406	4	remark	remark	NOUN
ejpam-5648	406	5	3	3	NUM
ejpam-5648	406	6	.	.	PUNCT
ejpam-5648	407	1	the	the	DET
ejpam-5648	407	2	upper	upper	ADJ
ejpam-5648	407	3	bound	bind	VERB
ejpam-5648	407	4	given	give	VERB
ejpam-5648	407	5	in	in	ADP
ejpam-5648	407	6	proposition	proposition	NOUN
ejpam-5648	407	7	10	10	NUM
ejpam-5648	407	8	is	be	AUX
ejpam-5648	407	9	tight	tight	ADJ
ejpam-5648	407	10	.	.	PUNCT
ejpam-5648	408	1	however	however	ADV
ejpam-5648	408	2	,	,	PUNCT
ejpam-5648	408	3	strict	strict	ADJ
ejpam-5648	408	4	inequality	inequality	NOUN
ejpam-5648	408	5	is	be	AUX
ejpam-5648	408	6	attainable	attainable	ADJ
ejpam-5648	408	7	.	.	PUNCT
ejpam-5648	409	1	to	to	PART
ejpam-5648	409	2	see	see	VERB
ejpam-5648	409	3	this	this	PRON
ejpam-5648	409	4	,	,	PUNCT
ejpam-5648	409	5	let	let	VERB
ejpam-5648	409	6	g	g	PROPN
ejpam-5648	409	7	=	=	PROPN
ejpam-5648	409	8	p3	p3	PROPN
ejpam-5648	409	9	=	=	PUNCT
ejpam-5648	410	1	[	[	X
ejpam-5648	410	2	a	a	PRON
ejpam-5648	410	3	,	,	PUNCT
ejpam-5648	410	4	b	b	NOUN
ejpam-5648	410	5	,	,	PUNCT
ejpam-5648	410	6	c	c	NOUN
ejpam-5648	410	7	]	]	PUNCT
ejpam-5648	410	8	and	and	CCONJ
ejpam-5648	410	9	h	h	NOUN
ejpam-5648	410	10	=	=	NOUN
ejpam-5648	410	11	p2	p2	PROPN
ejpam-5648	410	12	=	=	PUNCT
ejpam-5648	411	1	[	[	X
ejpam-5648	411	2	p	p	X
ejpam-5648	411	3	,	,	PUNCT
ejpam-5648	411	4	q	q	X
ejpam-5648	411	5	]	]	X
ejpam-5648	411	6	.	.	PUNCT
ejpam-5648	412	1	then	then	ADV
ejpam-5648	412	2	γcrh(g	γcrh(g	X
ejpam-5648	412	3	)	)	PUNCT
ejpam-5648	412	4	=	=	SYM
ejpam-5648	412	5	3	3	NUM
ejpam-5648	412	6	by	by	ADP
ejpam-5648	412	7	proposition	proposition	NOUN
ejpam-5648	412	8	3(iii	3(iii	NUM
ejpam-5648	412	9	)	)	PUNCT
ejpam-5648	412	10	and	and	CCONJ
ejpam-5648	412	11	γcrh(h	γcrh(h	NOUN
ejpam-5648	412	12	)	)	PUNCT
ejpam-5648	412	13	=	=	SYM
ejpam-5648	412	14	2	2	NUM
ejpam-5648	412	15	by	by	ADP
ejpam-5648	412	16	proposition	proposition	NOUN
ejpam-5648	412	17	3(ii	3(ii	NUM
ejpam-5648	412	18	)	)	PUNCT
ejpam-5648	412	19	.	.	PUNCT
ejpam-5648	413	1	let	let	VERB
ejpam-5648	413	2	f	f	PROPN
ejpam-5648	413	3	=	=	SYM
ejpam-5648	413	4	(	(	PUNCT
ejpam-5648	413	5	v0	v0	PROPN
ejpam-5648	413	6	,	,	PUNCT
ejpam-5648	413	7	v1	v1	NOUN
ejpam-5648	413	8	,	,	PUNCT
ejpam-5648	413	9	v2	v2	PROPN
ejpam-5648	413	10	)	)	PUNCT
ejpam-5648	413	11	be	be	AUX
ejpam-5648	413	12	a	a	DET
ejpam-5648	413	13	γcrh	γcrh	NOUN
ejpam-5648	413	14	-	-	PUNCT
ejpam-5648	413	15	function	function	NOUN
ejpam-5648	413	16	on	on	ADP
ejpam-5648	413	17	g	g	PROPN
ejpam-5648	413	18	+	+	PROPN
ejpam-5648	413	19	h.	h.	PROPN
ejpam-5648	413	20	since	since	SCONJ
ejpam-5648	413	21	b	b	PROPN
ejpam-5648	413	22	,	,	PUNCT
ejpam-5648	413	23	p	p	X
ejpam-5648	413	24	,	,	PUNCT
ejpam-5648	413	25	and	and	CCONJ
ejpam-5648	413	26	q	q	PROPN
ejpam-5648	413	27	are	be	AUX
ejpam-5648	413	28	dominating	dominate	VERB
ejpam-5648	413	29	vertices	vertex	NOUN
ejpam-5648	413	30	of	of	ADP
ejpam-5648	413	31	g	g	PROPN
ejpam-5648	413	32	+	+	CCONJ
ejpam-5648	413	33	h	h	NOUN
ejpam-5648	413	34	,	,	PUNCT
ejpam-5648	413	35	it	it	PRON
ejpam-5648	413	36	follows	follow	VERB
ejpam-5648	413	37	a.	a.	PROPN
ejpam-5648	413	38	aradais	aradais	PROPN
ejpam-5648	413	39	,	,	PUNCT
ejpam-5648	413	40	j.	j.	PROPN
ejpam-5648	413	41	cariaga	cariaga	PROPN
ejpam-5648	413	42	,	,	PUNCT
ejpam-5648	414	1	s.	s.	PROPN
ejpam-5648	414	2	canoy	canoy	PROPN
ejpam-5648	414	3	jr	jr	PROPN
ejpam-5648	414	4	.	.	PROPN
ejpam-5648	414	5	/	/	SYM
ejpam-5648	414	6	eur	eur	PROPN
ejpam-5648	414	7	.	.	PUNCT
ejpam-5648	415	1	j.	j.	PROPN
ejpam-5648	415	2	pure	pure	PROPN
ejpam-5648	415	3	appl	appl	PROPN
ejpam-5648	415	4	.	.	PROPN
ejpam-5648	415	5	math	math	PROPN
ejpam-5648	415	6	,	,	PUNCT
ejpam-5648	415	7	18	18	NUM
ejpam-5648	415	8	(	(	PUNCT
ejpam-5648	415	9	1	1	NUM
ejpam-5648	415	10	)	)	PUNCT
ejpam-5648	415	11	(	(	PUNCT
ejpam-5648	415	12	2025	2025	NUM
ejpam-5648	415	13	)	)	PUNCT
ejpam-5648	415	14	,	,	PUNCT
ejpam-5648	415	15	5648	5648	NUM
ejpam-5648	415	16	11	11	NUM
ejpam-5648	415	17	of	of	ADP
ejpam-5648	415	18	13	13	NUM
ejpam-5648	415	19	from	from	ADP
ejpam-5648	415	20	remark	remark	NOUN
ejpam-5648	415	21	1	1	NUM
ejpam-5648	415	22	that	that	PRON
ejpam-5648	415	23	b	b	X
ejpam-5648	415	24	,	,	PUNCT
ejpam-5648	415	25	p	p	X
ejpam-5648	415	26	,	,	PUNCT
ejpam-5648	415	27	q	q	PROPN
ejpam-5648	415	28	∈	∈	PROPN
ejpam-5648	415	29	v1	v1	NOUN
ejpam-5648	415	30	.	.	PUNCT
ejpam-5648	416	1	if	if	SCONJ
ejpam-5648	416	2	a	a	PRON
ejpam-5648	416	3	,	,	PUNCT
ejpam-5648	416	4	c	c	PROPN
ejpam-5648	416	5	∈	∈	PROPN
ejpam-5648	416	6	v1	v1	PROPN
ejpam-5648	416	7	∪	∪	NOUN
ejpam-5648	416	8	v2	v2	NOUN
ejpam-5648	416	9	,	,	PUNCT
ejpam-5648	416	10	then	then	ADV
ejpam-5648	416	11	v1	v1	VERB
ejpam-5648	416	12	=	=	SYM
ejpam-5648	416	13	{	{	PUNCT
ejpam-5648	416	14	a	a	PRON
ejpam-5648	416	15	,	,	PUNCT
ejpam-5648	416	16	b	b	NOUN
ejpam-5648	416	17	,	,	PUNCT
ejpam-5648	416	18	c	c	X
ejpam-5648	416	19	,	,	PUNCT
ejpam-5648	416	20	p	p	X
ejpam-5648	416	21	,	,	PUNCT
ejpam-5648	416	22	q	q	NOUN
ejpam-5648	416	23	}	}	PUNCT
ejpam-5648	416	24	.	.	PUNCT
ejpam-5648	417	1	hence	hence	ADV
ejpam-5648	417	2	,	,	PUNCT
ejpam-5648	417	3	γcrh(g	γcrh(g	X
ejpam-5648	417	4	+	+	NOUN
ejpam-5648	417	5	h	h	NOUN
ejpam-5648	417	6	)	)	PUNCT
ejpam-5648	417	7	=	=	SYM
ejpam-5648	417	8	5	5	X
ejpam-5648	417	9	.	.	PUNCT
ejpam-5648	417	10	suppose	suppose	VERB
ejpam-5648	417	11	one	one	NUM
ejpam-5648	417	12	of	of	ADP
ejpam-5648	417	13	a	a	PRON
ejpam-5648	417	14	and	and	CCONJ
ejpam-5648	417	15	c	c	NOUN
ejpam-5648	417	16	is	be	AUX
ejpam-5648	417	17	in	in	ADP
ejpam-5648	417	18	v0	v0	NOUN
ejpam-5648	417	19	,	,	PUNCT
ejpam-5648	417	20	say	say	VERB
ejpam-5648	417	21	v0	v0	NOUN
ejpam-5648	417	22	=	=	SYM
ejpam-5648	417	23	{	{	PUNCT
ejpam-5648	417	24	a	a	NOUN
ejpam-5648	417	25	}	}	PUNCT
ejpam-5648	417	26	.	.	PUNCT
ejpam-5648	418	1	then	then	ADV
ejpam-5648	418	2	necessarily	necessarily	ADV
ejpam-5648	418	3	,	,	PUNCT
ejpam-5648	418	4	c	c	PROPN
ejpam-5648	418	5	∈	∈	PROPN
ejpam-5648	418	6	v2	v2	PROPN
ejpam-5648	418	7	.	.	PUNCT
ejpam-5648	419	1	it	it	PRON
ejpam-5648	419	2	follows	follow	VERB
ejpam-5648	419	3	that	that	DET
ejpam-5648	419	4	v1	v1	NOUN
ejpam-5648	419	5	=	=	SYM
ejpam-5648	419	6	{	{	PUNCT
ejpam-5648	419	7	b	b	PROPN
ejpam-5648	419	8	,	,	PUNCT
ejpam-5648	419	9	p	p	X
ejpam-5648	419	10	,	,	PUNCT
ejpam-5648	419	11	q	q	ADJ
ejpam-5648	419	12	}	}	PUNCT
ejpam-5648	419	13	and	and	CCONJ
ejpam-5648	419	14	γcrh(g	γcrh(g	X
ejpam-5648	419	15	+	+	CCONJ
ejpam-5648	419	16	h	h	NOUN
ejpam-5648	419	17	)	)	PUNCT
ejpam-5648	419	18	=	=	NOUN
ejpam-5648	419	19	|v1|	|v1|	NOUN
ejpam-5648	419	20	+	+	CCONJ
ejpam-5648	419	21	2|v2|	2|v2|	NUM
ejpam-5648	419	22	=	=	SYM
ejpam-5648	419	23	5	5	NUM
ejpam-5648	419	24	.	.	PUNCT
ejpam-5648	420	1	therefore	therefore	ADV
ejpam-5648	420	2	,	,	PUNCT
ejpam-5648	420	3	γcrh(g+h	γcrh(g+h	PROPN
ejpam-5648	420	4	)	)	PUNCT
ejpam-5648	420	5	=	=	SYM
ejpam-5648	420	6	γcrh(g	γcrh(g	PROPN
ejpam-5648	420	7	)	)	PUNCT
ejpam-5648	420	8	+	+	X
ejpam-5648	421	1	γcrh(h	γcrh(h	NOUN
ejpam-5648	421	2	)	)	PUNCT
ejpam-5648	421	3	=	=	SYM
ejpam-5648	422	1	5	5	X
ejpam-5648	422	2	.	.	PUNCT
ejpam-5648	422	3	clearly	clearly	ADV
ejpam-5648	422	4	,	,	PUNCT
ejpam-5648	422	5	the	the	DET
ejpam-5648	422	6	upper	upper	ADJ
ejpam-5648	422	7	bound	bound	NOUN
ejpam-5648	422	8	is	be	AUX
ejpam-5648	422	9	also	also	ADV
ejpam-5648	422	10	attained	attain	VERB
ejpam-5648	422	11	when	when	SCONJ
ejpam-5648	422	12	g	g	PROPN
ejpam-5648	422	13	=	=	PUNCT
ejpam-5648	422	14	km	km	PROPN
ejpam-5648	422	15	and	and	CCONJ
ejpam-5648	422	16	h	h	NOUN
ejpam-5648	423	1	=	=	SYM
ejpam-5648	423	2	kn	kn	PROPN
ejpam-5648	423	3	.	.	PUNCT
ejpam-5648	424	1	next	next	ADV
ejpam-5648	424	2	,	,	PUNCT
ejpam-5648	424	3	to	to	PART
ejpam-5648	424	4	show	show	VERB
ejpam-5648	424	5	that	that	SCONJ
ejpam-5648	424	6	strict	strict	ADJ
ejpam-5648	424	7	inequality	inequality	NOUN
ejpam-5648	424	8	is	be	AUX
ejpam-5648	424	9	also	also	ADV
ejpam-5648	424	10	possible	possible	ADJ
ejpam-5648	424	11	,	,	PUNCT
ejpam-5648	424	12	consider	consider	VERB
ejpam-5648	424	13	g	g	NOUN
ejpam-5648	424	14	=	=	NOUN
ejpam-5648	424	15	p4	p4	NOUN
ejpam-5648	424	16	=	=	PUNCT
ejpam-5648	425	1	[	[	X
ejpam-5648	425	2	a	a	PRON
ejpam-5648	425	3	,	,	PUNCT
ejpam-5648	425	4	b	b	NOUN
ejpam-5648	425	5	,	,	PUNCT
ejpam-5648	425	6	c	c	NOUN
ejpam-5648	425	7	,	,	PUNCT
ejpam-5648	425	8	d	d	X
ejpam-5648	425	9	]	]	X
ejpam-5648	425	10	and	and	CCONJ
ejpam-5648	425	11	h	h	NOUN
ejpam-5648	426	1	=	=	NOUN
ejpam-5648	426	2	p4	p4	ADJ
ejpam-5648	426	3	=	=	PUNCT
ejpam-5648	427	1	[	[	X
ejpam-5648	427	2	x	x	X
ejpam-5648	427	3	,	,	PUNCT
ejpam-5648	427	4	y	y	PROPN
ejpam-5648	427	5	,	,	PUNCT
ejpam-5648	427	6	z	z	PROPN
ejpam-5648	427	7	,	,	PUNCT
ejpam-5648	427	8	w	w	PROPN
ejpam-5648	427	9	]	]	X
ejpam-5648	427	10	.	.	PUNCT
ejpam-5648	428	1	then	then	ADV
ejpam-5648	428	2	γcrh(g	γcrh(g	X
ejpam-5648	428	3	)	)	PUNCT
ejpam-5648	428	4	=	=	SYM
ejpam-5648	428	5	γcrh(h	γcrh(h	NOUN
ejpam-5648	428	6	)	)	PUNCT
ejpam-5648	428	7	=	=	SYM
ejpam-5648	428	8	4	4	NUM
ejpam-5648	428	9	by	by	ADP
ejpam-5648	428	10	proposition	proposition	NOUN
ejpam-5648	428	11	3(iv	3(iv	NUM
ejpam-5648	428	12	)	)	PUNCT
ejpam-5648	428	13	.	.	PUNCT
ejpam-5648	429	1	now	now	ADV
ejpam-5648	429	2	set	set	VERB
ejpam-5648	429	3	v0	v0	NOUN
ejpam-5648	429	4	=	=	SYM
ejpam-5648	429	5	{	{	PUNCT
ejpam-5648	429	6	a	a	PRON
ejpam-5648	429	7	,	,	PUNCT
ejpam-5648	429	8	b	b	NOUN
ejpam-5648	429	9	,	,	PUNCT
ejpam-5648	429	10	x	x	NOUN
ejpam-5648	429	11	,	,	PUNCT
ejpam-5648	429	12	y	y	NOUN
ejpam-5648	429	13	}	}	PUNCT
ejpam-5648	429	14	,	,	PUNCT
ejpam-5648	429	15	v1	v1	NOUN
ejpam-5648	429	16	=	=	SYM
ejpam-5648	429	17	{	{	PUNCT
ejpam-5648	429	18	c	c	NOUN
ejpam-5648	429	19	,	,	PUNCT
ejpam-5648	429	20	z	z	NOUN
ejpam-5648	429	21	}	}	PUNCT
ejpam-5648	429	22	,	,	PUNCT
ejpam-5648	429	23	and	and	CCONJ
ejpam-5648	429	24	v2	v2	NOUN
ejpam-5648	429	25	=	=	SYM
ejpam-5648	429	26	{	{	PUNCT
ejpam-5648	429	27	d	d	PROPN
ejpam-5648	429	28	,	,	PUNCT
ejpam-5648	429	29	w	w	NOUN
ejpam-5648	429	30	}	}	PUNCT
ejpam-5648	429	31	.	.	PUNCT
ejpam-5648	430	1	then	then	ADV
ejpam-5648	430	2	f	f	X
ejpam-5648	430	3	=	=	SYM
ejpam-5648	430	4	(	(	PUNCT
ejpam-5648	430	5	v0	v0	PROPN
ejpam-5648	430	6	,	,	PUNCT
ejpam-5648	430	7	v1	v1	NOUN
ejpam-5648	430	8	,	,	PUNCT
ejpam-5648	430	9	v2	v2	NOUN
ejpam-5648	430	10	)	)	PUNCT
ejpam-5648	430	11	∈	∈	PROPN
ejpam-5648	430	12	chrdf	chrdf	NOUN
ejpam-5648	430	13	(	(	PUNCT
ejpam-5648	430	14	g+h	g+h	NOUN
ejpam-5648	430	15	)	)	PUNCT
ejpam-5648	430	16	.	.	PUNCT
ejpam-5648	431	1	it	it	PRON
ejpam-5648	431	2	is	be	AUX
ejpam-5648	431	3	easy	easy	ADJ
ejpam-5648	431	4	to	to	PART
ejpam-5648	431	5	see	see	VERB
ejpam-5648	431	6	that	that	SCONJ
ejpam-5648	431	7	f	f	PROPN
ejpam-5648	431	8	is	be	AUX
ejpam-5648	431	9	a	a	DET
ejpam-5648	431	10	γcrh	γcrh	NOUN
ejpam-5648	431	11	-	-	PUNCT
ejpam-5648	431	12	function	function	NOUN
ejpam-5648	431	13	on	on	ADP
ejpam-5648	431	14	g	g	PROPN
ejpam-5648	431	15	+	+	PROPN
ejpam-5648	431	16	h.	h.	PROPN
ejpam-5648	431	17	therefore	therefore	ADV
ejpam-5648	431	18	,	,	PUNCT
ejpam-5648	431	19	γcrh(g	γcrh(g	ADP
ejpam-5648	431	20	+	+	NOUN
ejpam-5648	431	21	h	h	NOUN
ejpam-5648	431	22	)	)	PUNCT
ejpam-5648	431	23	=	=	PUNCT
ejpam-5648	431	24	6	6	NUM
ejpam-5648	431	25	<	<	SYM
ejpam-5648	431	26	8	8	NUM
ejpam-5648	431	27	=	=	SYM
ejpam-5648	431	28	γcrh(g	γcrh(g	PROPN
ejpam-5648	431	29	)	)	PUNCT
ejpam-5648	431	30	+	+	X
ejpam-5648	432	1	γcrh(h	γcrh(h	NOUN
ejpam-5648	432	2	)	)	PUNCT
ejpam-5648	432	3	.	.	PUNCT
ejpam-5648	433	1	proposition	proposition	NOUN
ejpam-5648	433	2	11	11	NUM
ejpam-5648	433	3	.	.	PUNCT
ejpam-5648	434	1	there	there	PRON
ejpam-5648	434	2	are	be	VERB
ejpam-5648	434	3	infinitely	infinitely	ADV
ejpam-5648	434	4	many	many	ADJ
ejpam-5648	434	5	graphs	graph	NOUN
ejpam-5648	434	6	g	g	NOUN
ejpam-5648	434	7	and	and	CCONJ
ejpam-5648	434	8	h	h	NOUN
ejpam-5648	434	9	such	such	ADJ
ejpam-5648	434	10	that	that	DET
ejpam-5648	434	11	γcrh(g+h	γcrh(g+h	NOUN
ejpam-5648	434	12	)	)	PUNCT
ejpam-5648	434	13	=	=	SYM
ejpam-5648	434	14	5	5	X
ejpam-5648	434	15	.	.	X
ejpam-5648	435	1	in	in	ADP
ejpam-5648	435	2	particular	particular	ADJ
ejpam-5648	435	3	,	,	PUNCT
ejpam-5648	435	4	if	if	SCONJ
ejpam-5648	435	5	g	g	PROPN
ejpam-5648	435	6	=	=	SYM
ejpam-5648	435	7	pn	pn	PROPN
ejpam-5648	435	8	,	,	PUNCT
ejpam-5648	435	9	where	where	SCONJ
ejpam-5648	435	10	n	n	PRON
ejpam-5648	435	11	≥	≥	NOUN
ejpam-5648	435	12	4	4	NUM
ejpam-5648	435	13	,	,	PUNCT
ejpam-5648	435	14	and	and	CCONJ
ejpam-5648	435	15	h	h	NOUN
ejpam-5648	435	16	is	be	AUX
ejpam-5648	435	17	any	any	DET
ejpam-5648	435	18	non	non	ADJ
ejpam-5648	435	19	-	-	ADJ
ejpam-5648	435	20	trivial	trivial	ADJ
ejpam-5648	435	21	graph	graph	NOUN
ejpam-5648	435	22	with	with	ADP
ejpam-5648	435	23	at	at	ADV
ejpam-5648	435	24	least	least	ADV
ejpam-5648	435	25	one	one	NUM
ejpam-5648	435	26	trivial	trivial	ADJ
ejpam-5648	435	27	component	component	NOUN
ejpam-5648	435	28	,	,	PUNCT
ejpam-5648	435	29	then	then	ADV
ejpam-5648	435	30	γcrh(g+h	γcrh(g+h	VERB
ejpam-5648	435	31	)	)	PUNCT
ejpam-5648	435	32	=	=	SYM
ejpam-5648	436	1	5	5	X
ejpam-5648	436	2	.	.	PUNCT
ejpam-5648	436	3	proof	proof	NOUN
ejpam-5648	436	4	.	.	PUNCT
ejpam-5648	437	1	let	let	VERB
ejpam-5648	437	2	n	n	PRON
ejpam-5648	437	3	≥	≥	X
ejpam-5648	437	4	4	4	NUM
ejpam-5648	437	5	and	and	CCONJ
ejpam-5648	437	6	let	let	VERB
ejpam-5648	437	7	g	g	PROPN
ejpam-5648	437	8	=	=	PUNCT
ejpam-5648	437	9	pn	pn	PROPN
ejpam-5648	437	10	=	=	PUNCT
ejpam-5648	438	1	[	[	X
ejpam-5648	438	2	x1	x1	PROPN
ejpam-5648	438	3	,	,	PUNCT
ejpam-5648	438	4	x2	x2	PROPN
ejpam-5648	438	5	,	,	PUNCT
ejpam-5648	438	6	.	.	PUNCT
ejpam-5648	438	7	.	.	PUNCT
ejpam-5648	439	1	.	.	PUNCT
ejpam-5648	440	1	,	,	PUNCT
ejpam-5648	440	2	xn	xn	PROPN
ejpam-5648	440	3	]	]	PUNCT
ejpam-5648	440	4	.	.	PUNCT
ejpam-5648	441	1	let	let	VERB
ejpam-5648	441	2	h1	h1	PROPN
ejpam-5648	441	3	,	,	PUNCT
ejpam-5648	441	4	h2	h2	PROPN
ejpam-5648	441	5	,	,	PUNCT
ejpam-5648	441	6	.	.	PUNCT
ejpam-5648	441	7	.	.	PUNCT
ejpam-5648	442	1	.	.	PUNCT
ejpam-5648	443	1	,	,	PUNCT
ejpam-5648	443	2	hk	hk	PROPN
ejpam-5648	443	3	be	be	AUX
ejpam-5648	443	4	the	the	DET
ejpam-5648	443	5	components	component	NOUN
ejpam-5648	443	6	of	of	ADP
ejpam-5648	443	7	h	h	NOUN
ejpam-5648	443	8	and	and	CCONJ
ejpam-5648	443	9	suppose	suppose	VERB
ejpam-5648	443	10	that	that	SCONJ
ejpam-5648	443	11	h1	h1	PROPN
ejpam-5648	443	12	=	=	SYM
ejpam-5648	443	13	⟨p⟩.	⟨p⟩.	PROPN
ejpam-5648	443	14	let	let	VERB
ejpam-5648	443	15	v0	v0	NOUN
ejpam-5648	443	16	=	=	SYM
ejpam-5648	443	17	(	(	PUNCT
ejpam-5648	443	18	v	v	NOUN
ejpam-5648	443	19	(	(	PUNCT
ejpam-5648	443	20	pn	pn	NOUN
ejpam-5648	443	21	)	)	PUNCT
ejpam-5648	443	22	\	\	NOUN
ejpam-5648	443	23	{	{	PUNCT
ejpam-5648	443	24	xn−1	xn−1	PROPN
ejpam-5648	443	25	,	,	PUNCT
ejpam-5648	443	26	xn	xn	PROPN
ejpam-5648	443	27	}	}	PUNCT
ejpam-5648	443	28	)	)	PUNCT
ejpam-5648	443	29	∪	∪	X
ejpam-5648	443	30	(	(	PUNCT
ejpam-5648	443	31	∪k	∪k	PROPN
ejpam-5648	443	32	j=2v	j=2v	PROPN
ejpam-5648	443	33	(	(	PUNCT
ejpam-5648	443	34	hj	hj	PROPN
ejpam-5648	443	35	)	)	PUNCT
ejpam-5648	443	36	)	)	PUNCT
ejpam-5648	443	37	,	,	PUNCT
ejpam-5648	443	38	v1	v1	NOUN
ejpam-5648	443	39	=	=	SYM
ejpam-5648	443	40	{	{	PUNCT
ejpam-5648	443	41	xn−1	xn−1	PROPN
ejpam-5648	443	42	}	}	PUNCT
ejpam-5648	443	43	and	and	CCONJ
ejpam-5648	443	44	v2	v2	NOUN
ejpam-5648	443	45	=	=	SYM
ejpam-5648	443	46	{	{	PUNCT
ejpam-5648	443	47	p	p	X
ejpam-5648	443	48	,	,	PUNCT
ejpam-5648	443	49	xn	xn	PROPN
ejpam-5648	443	50	}	}	PUNCT
ejpam-5648	443	51	.	.	PUNCT
ejpam-5648	444	1	then	then	ADV
ejpam-5648	444	2	g	g	PROPN
ejpam-5648	444	3	=	=	SYM
ejpam-5648	444	4	(	(	PUNCT
ejpam-5648	444	5	v0	v0	PROPN
ejpam-5648	444	6	,	,	PUNCT
ejpam-5648	444	7	v1	v1	NOUN
ejpam-5648	444	8	,	,	PUNCT
ejpam-5648	444	9	v2	v2	PROPN
ejpam-5648	444	10	)	)	PUNCT
ejpam-5648	444	11	is	be	AUX
ejpam-5648	444	12	a	a	DET
ejpam-5648	444	13	connected	connected	ADJ
ejpam-5648	444	14	hop	hop	NOUN
ejpam-5648	444	15	roman	roman	ADJ
ejpam-5648	444	16	dominating	dominating	NOUN
ejpam-5648	444	17	function	function	NOUN
ejpam-5648	444	18	on	on	ADP
ejpam-5648	444	19	g+h	g+h	PROPN
ejpam-5648	444	20	.	.	PUNCT
ejpam-5648	445	1	it	it	PRON
ejpam-5648	445	2	follows	follow	VERB
ejpam-5648	445	3	that	that	SCONJ
ejpam-5648	445	4	γcrh(g+h	γcrh(g+h	NOUN
ejpam-5648	445	5	)	)	PUNCT
ejpam-5648	445	6	≤	≤	NOUN
ejpam-5648	445	7	ωcrh	ωcrh	ADV
ejpam-5648	445	8	g	g	PROPN
ejpam-5648	445	9	(	(	PUNCT
ejpam-5648	445	10	f	f	X
ejpam-5648	445	11	)	)	PUNCT
ejpam-5648	445	12	=	=	SYM
ejpam-5648	445	13	5	5	X
ejpam-5648	445	14	.	.	PUNCT
ejpam-5648	446	1	since	since	SCONJ
ejpam-5648	446	2	g+h	g+h	PROPN
ejpam-5648	446	3	does	do	AUX
ejpam-5648	446	4	not	not	PART
ejpam-5648	446	5	satisfy	satisfy	VERB
ejpam-5648	446	6	any	any	PRON
ejpam-5648	446	7	of	of	ADP
ejpam-5648	446	8	the	the	DET
ejpam-5648	446	9	conditions	condition	NOUN
ejpam-5648	446	10	given	give	VERB
ejpam-5648	446	11	in	in	ADP
ejpam-5648	446	12	proposition	proposition	NOUN
ejpam-5648	446	13	3	3	NUM
ejpam-5648	446	14	,	,	PUNCT
ejpam-5648	446	15	it	it	PRON
ejpam-5648	446	16	follows	follow	VERB
ejpam-5648	446	17	that	that	SCONJ
ejpam-5648	446	18	γcrh(g+h	γcrh(g+h	NOUN
ejpam-5648	446	19	)	)	PUNCT
ejpam-5648	446	20	=	=	SYM
ejpam-5648	446	21	5	5	NUM
ejpam-5648	446	22	.	.	NOUN
ejpam-5648	446	23	4	4	NUM
ejpam-5648	446	24	.	.	X
ejpam-5648	446	25	conclusion	conclusion	NOUN
ejpam-5648	446	26	connected	connect	VERB
ejpam-5648	446	27	hop	hop	PROPN
ejpam-5648	446	28	roman	roman	ADJ
ejpam-5648	446	29	domination	domination	NOUN
ejpam-5648	446	30	has	have	AUX
ejpam-5648	446	31	been	be	AUX
ejpam-5648	446	32	defined	define	VERB
ejpam-5648	446	33	and	and	CCONJ
ejpam-5648	446	34	investigated	investigate	VERB
ejpam-5648	446	35	for	for	ADP
ejpam-5648	446	36	some	some	DET
ejpam-5648	446	37	graphs	graph	NOUN
ejpam-5648	446	38	.	.	PUNCT
ejpam-5648	447	1	there	there	PRON
ejpam-5648	447	2	are	be	VERB
ejpam-5648	447	3	still	still	ADV
ejpam-5648	447	4	a	a	DET
ejpam-5648	447	5	lot	lot	NOUN
ejpam-5648	447	6	of	of	ADP
ejpam-5648	447	7	aspects	aspect	NOUN
ejpam-5648	447	8	that	that	PRON
ejpam-5648	447	9	can	can	AUX
ejpam-5648	447	10	be	be	AUX
ejpam-5648	447	11	explored	explore	VERB
ejpam-5648	447	12	and	and	CCONJ
ejpam-5648	447	13	studied	study	VERB
ejpam-5648	447	14	for	for	ADP
ejpam-5648	447	15	this	this	DET
ejpam-5648	447	16	parameter	parameter	NOUN
ejpam-5648	447	17	.	.	PUNCT
ejpam-5648	448	1	for	for	ADP
ejpam-5648	448	2	any	any	DET
ejpam-5648	448	3	two	two	NUM
ejpam-5648	448	4	graphs	graph	NOUN
ejpam-5648	448	5	g	g	NOUN
ejpam-5648	448	6	and	and	CCONJ
ejpam-5648	448	7	h	h	NOUN
ejpam-5648	448	8	,	,	PUNCT
ejpam-5648	448	9	this	this	DET
ejpam-5648	448	10	initial	initial	ADJ
ejpam-5648	448	11	study	study	NOUN
ejpam-5648	448	12	has	have	AUX
ejpam-5648	448	13	only	only	ADV
ejpam-5648	448	14	provided	provide	VERB
ejpam-5648	448	15	sharp	sharp	ADJ
ejpam-5648	448	16	lower	low	ADJ
ejpam-5648	448	17	and	and	CCONJ
ejpam-5648	448	18	upper	upper	ADJ
ejpam-5648	448	19	bounds	bound	NOUN
ejpam-5648	448	20	for	for	ADP
ejpam-5648	448	21	the	the	DET
ejpam-5648	448	22	join	join	NOUN
ejpam-5648	448	23	g	g	PROPN
ejpam-5648	448	24	+	+	CCONJ
ejpam-5648	448	25	h.	h.	PROPN
ejpam-5648	448	26	it	it	PRON
ejpam-5648	448	27	was	be	AUX
ejpam-5648	448	28	,	,	PUNCT
ejpam-5648	448	29	however	however	ADV
ejpam-5648	448	30	,	,	PUNCT
ejpam-5648	448	31	shown	show	VERB
ejpam-5648	448	32	that	that	SCONJ
ejpam-5648	448	33	these	these	DET
ejpam-5648	448	34	bounds	bound	NOUN
ejpam-5648	448	35	may	may	AUX
ejpam-5648	448	36	not	not	PART
ejpam-5648	448	37	be	be	AUX
ejpam-5648	448	38	attained	attain	VERB
ejpam-5648	448	39	.	.	PUNCT
ejpam-5648	449	1	it	it	PRON
ejpam-5648	449	2	is	be	AUX
ejpam-5648	449	3	conjectured	conjecture	VERB
ejpam-5648	449	4	that	that	SCONJ
ejpam-5648	449	5	the	the	DET
ejpam-5648	449	6	exact	exact	ADJ
ejpam-5648	449	7	value	value	NOUN
ejpam-5648	449	8	of	of	ADP
ejpam-5648	449	9	the	the	DET
ejpam-5648	449	10	parameter	parameter	NOUN
ejpam-5648	449	11	for	for	ADP
ejpam-5648	449	12	the	the	DET
ejpam-5648	449	13	join	join	NOUN
ejpam-5648	449	14	g+h	g+h	PROPN
ejpam-5648	449	15	can	can	AUX
ejpam-5648	449	16	be	be	AUX
ejpam-5648	449	17	described	describe	VERB
ejpam-5648	449	18	by	by	ADP
ejpam-5648	449	19	defining	define	VERB
ejpam-5648	449	20	yet	yet	ADV
ejpam-5648	449	21	another	another	DET
ejpam-5648	449	22	parameter	parameter	NOUN
ejpam-5648	449	23	.	.	PUNCT
ejpam-5648	450	1	furthermore	furthermore	ADV
ejpam-5648	450	2	,	,	PUNCT
ejpam-5648	450	3	connected	connect	VERB
ejpam-5648	450	4	hop	hop	NOUN
ejpam-5648	450	5	roman	roman	ADJ
ejpam-5648	450	6	domination	domination	NOUN
ejpam-5648	450	7	can	can	AUX
ejpam-5648	450	8	also	also	ADV
ejpam-5648	450	9	be	be	AUX
ejpam-5648	450	10	investigated	investigate	VERB
ejpam-5648	450	11	for	for	ADP
ejpam-5648	450	12	other	other	ADJ
ejpam-5648	450	13	graphs	graph	NOUN
ejpam-5648	450	14	under	under	ADP
ejpam-5648	450	15	binary	binary	ADJ
ejpam-5648	450	16	operations	operation	NOUN
ejpam-5648	450	17	and	and	CCONJ
ejpam-5648	450	18	for	for	ADP
ejpam-5648	450	19	the	the	DET
ejpam-5648	450	20	complexity	complexity	NOUN
ejpam-5648	450	21	of	of	ADP
ejpam-5648	450	22	the	the	DET
ejpam-5648	450	23	connected	connect	VERB
ejpam-5648	450	24	hop	hop	NOUN
ejpam-5648	450	25	roman	roman	ADJ
ejpam-5648	450	26	dominating	dominating	NOUN
ejpam-5648	450	27	function	function	NOUN
ejpam-5648	450	28	problem	problem	NOUN
ejpam-5648	450	29	.	.	PUNCT
ejpam-5648	451	1	acknowledgements	acknowledgement	NOUN
ejpam-5648	451	2	the	the	DET
ejpam-5648	451	3	authors	author	NOUN
ejpam-5648	451	4	would	would	AUX
ejpam-5648	451	5	like	like	VERB
ejpam-5648	451	6	to	to	PART
ejpam-5648	451	7	thank	thank	VERB
ejpam-5648	451	8	the	the	DET
ejpam-5648	451	9	referees	referee	NOUN
ejpam-5648	451	10	for	for	ADP
ejpam-5648	451	11	the	the	DET
ejpam-5648	451	12	their	their	PRON
ejpam-5648	451	13	comments	comment	NOUN
ejpam-5648	451	14	and	and	CCONJ
ejpam-5648	451	15	suggestions	suggestion	NOUN
ejpam-5648	451	16	which	which	PRON
ejpam-5648	451	17	led	lead	VERB
ejpam-5648	451	18	to	to	ADP
ejpam-5648	451	19	the	the	DET
ejpam-5648	451	20	improvement	improvement	NOUN
ejpam-5648	451	21	of	of	ADP
ejpam-5648	451	22	the	the	DET
ejpam-5648	451	23	paper	paper	NOUN
ejpam-5648	451	24	.	.	PUNCT
ejpam-5648	452	1	also	also	ADV
ejpam-5648	452	2	,	,	PUNCT
ejpam-5648	452	3	the	the	DET
ejpam-5648	452	4	authors	author	NOUN
ejpam-5648	452	5	would	would	AUX
ejpam-5648	452	6	like	like	VERB
ejpam-5648	452	7	to	to	PART
ejpam-5648	452	8	thank	thank	VERB
ejpam-5648	452	9	the	the	DET
ejpam-5648	452	10	department	department	NOUN
ejpam-5648	452	11	of	of	ADP
ejpam-5648	452	12	science	science	NOUN
ejpam-5648	452	13	and	and	CCONJ
ejpam-5648	452	14	technology	technology	NOUN
ejpam-5648	452	15	accelerated	accelerate	VERB
ejpam-5648	452	16	science	science	NOUN
ejpam-5648	452	17	and	and	CCONJ
ejpam-5648	452	18	technology	technology	NOUN
ejpam-5648	452	19	human	human	ADJ
ejpam-5648	452	20	resource	resource	NOUN
ejpam-5648	452	21	development	development	NOUN
ejpam-5648	452	22	program	program	NOUN
ejpam-5648	452	23	(	(	PUNCT
ejpam-5648	452	24	dost	dost	NOUN
ejpam-5648	452	25	-	-	PUNCT
ejpam-5648	452	26	asthrdp)-philippines	asthrdp)-philippine	NOUN
ejpam-5648	452	27	,	,	PUNCT
ejpam-5648	452	28	and	and	CCONJ
ejpam-5648	452	29	msu	msu	PROPN
ejpam-5648	452	30	-	-	PUNCT
ejpam-5648	452	31	iligan	iligan	PROPN
ejpam-5648	452	32	institute	institute	PROPN
ejpam-5648	452	33	of	of	ADP
ejpam-5648	452	34	technology	technology	PROPN
ejpam-5648	452	35	(	(	PUNCT
ejpam-5648	452	36	iligan	iligan	ADJ
ejpam-5648	452	37	city	city	NOUN
ejpam-5648	452	38	,	,	PUNCT
ejpam-5648	452	39	philippines	philippine	NOUN
ejpam-5648	452	40	)	)	PUNCT
ejpam-5648	452	41	for	for	ADP
ejpam-5648	452	42	funding	fund	VERB
ejpam-5648	452	43	this	this	DET
ejpam-5648	452	44	research	research	NOUN
ejpam-5648	452	45	.	.	PUNCT
ejpam-5648	453	1	a.	a.	PROPN
ejpam-5648	453	2	aradais	aradais	PROPN
ejpam-5648	453	3	,	,	PUNCT
ejpam-5648	453	4	j.	j.	PROPN
ejpam-5648	453	5	cariaga	cariaga	PROPN
ejpam-5648	453	6	,	,	PUNCT
ejpam-5648	453	7	s.	s.	PROPN
ejpam-5648	453	8	canoy	canoy	PROPN
ejpam-5648	453	9	jr	jr	PROPN
ejpam-5648	453	10	.	.	PROPN
ejpam-5648	453	11	/	/	SYM
ejpam-5648	453	12	eur	eur	PROPN
ejpam-5648	453	13	.	.	PUNCT
ejpam-5648	454	1	j.	j.	PROPN
ejpam-5648	454	2	pure	pure	PROPN
ejpam-5648	454	3	appl	appl	PROPN
ejpam-5648	454	4	.	.	PROPN
ejpam-5648	454	5	math	math	PROPN
ejpam-5648	454	6	,	,	PUNCT
ejpam-5648	454	7	18	18	NUM
ejpam-5648	454	8	(	(	PUNCT
ejpam-5648	454	9	1	1	NUM
ejpam-5648	454	10	)	)	PUNCT
ejpam-5648	454	11	(	(	PUNCT
ejpam-5648	454	12	2025	2025	NUM
ejpam-5648	454	13	)	)	PUNCT
ejpam-5648	454	14	,	,	PUNCT
ejpam-5648	454	15	5648	5648	NUM
ejpam-5648	454	16	12	12	NUM
ejpam-5648	454	17	of	of	ADP
ejpam-5648	454	18	13	13	NUM
ejpam-5648	454	19	references	reference	NOUN
ejpam-5648	454	20	[	[	X
ejpam-5648	454	21	1	1	NUM
ejpam-5648	454	22	]	]	PUNCT
ejpam-5648	454	23	m.	m.	NOUN
ejpam-5648	454	24	adabi	adabi	PROPN
ejpam-5648	454	25	,	,	PUNCT
ejpam-5648	454	26	e.	e.	PROPN
ejpam-5648	454	27	ebrahimi	ebrahimi	PROPN
ejpam-5648	454	28	targhi	targhi	PROPN
ejpam-5648	454	29	,	,	PUNCT
ejpam-5648	454	30	n.	n.	PROPN
ejpam-5648	454	31	jafari	jafari	PROPN
ejpam-5648	454	32	rad	rad	PROPN
ejpam-5648	454	33	,	,	PUNCT
ejpam-5648	454	34	and	and	CCONJ
ejpam-5648	454	35	m.	m.	NOUN
ejpam-5648	454	36	saied	saie	VERB
ejpam-5648	454	37	moradi	moradi	NOUN
ejpam-5648	454	38	.	.	PUNCT
ejpam-5648	455	1	properties	property	NOUN
ejpam-5648	455	2	of	of	ADP
ejpam-5648	455	3	independent	independent	ADJ
ejpam-5648	455	4	roman	roman	ADJ
ejpam-5648	455	5	domination	domination	NOUN
ejpam-5648	455	6	in	in	ADP
ejpam-5648	455	7	graphs	graph	NOUN
ejpam-5648	455	8	.	.	PUNCT
ejpam-5648	456	1	australasian	australasian	ADJ
ejpam-5648	456	2	journal	journal	NOUN
ejpam-5648	456	3	of	of	ADP
ejpam-5648	456	4	combinatorics	combinatoric	NOUN
ejpam-5648	456	5	,	,	PUNCT
ejpam-5648	456	6	52:11–18	52:11–18	NUM
ejpam-5648	456	7	,	,	PUNCT
ejpam-5648	456	8	2012	2012	NUM
ejpam-5648	456	9	.	.	PUNCT
ejpam-5648	457	1	[	[	X
ejpam-5648	457	2	2	2	X
ejpam-5648	457	3	]	]	PUNCT
ejpam-5648	457	4	h.	h.	NOUN
ejpam-5648	457	5	ahangar	ahangar	PROPN
ejpam-5648	457	6	,	,	PUNCT
ejpam-5648	457	7	m.	m.	NOUN
ejpam-5648	457	8	cellali	cellali	PROPN
ejpam-5648	457	9	,	,	PUNCT
ejpam-5648	457	10	s.	s.	PROPN
ejpam-5648	457	11	sheikholeslami	sheikholeslami	PROPN
ejpam-5648	457	12	,	,	PUNCT
ejpam-5648	457	13	and	and	CCONJ
ejpam-5648	457	14	m.	m.	NOUN
ejpam-5648	457	15	soroudi	soroudi	PROPN
ejpam-5648	457	16	.	.	PUNCT
ejpam-5648	458	1	hop	hop	PROPN
ejpam-5648	458	2	total	total	ADJ
ejpam-5648	458	3	roman	roman	ADJ
ejpam-5648	458	4	domination	domination	NOUN
ejpam-5648	458	5	in	in	ADP
ejpam-5648	458	6	graphs	graph	NOUN
ejpam-5648	458	7	.	.	PUNCT
ejpam-5648	459	1	akce	akce	PROPN
ejpam-5648	459	2	international	international	PROPN
ejpam-5648	459	3	journal	journal	NOUN
ejpam-5648	459	4	of	of	ADP
ejpam-5648	459	5	graphs	graph	NOUN
ejpam-5648	459	6	and	and	CCONJ
ejpam-5648	459	7	combinatorics	combinatoric	NOUN
ejpam-5648	459	8	,	,	PUNCT
ejpam-5648	459	9	20(1):73	20(1):73	NUM
ejpam-5648	459	10	–	–	PUNCT
ejpam-5648	459	11	78	78	NUM
ejpam-5648	459	12	,	,	PUNCT
ejpam-5648	459	13	2023	2023	NUM
ejpam-5648	459	14	.	.	PUNCT
ejpam-5648	460	1	[	[	X
ejpam-5648	460	2	3	3	NUM
ejpam-5648	460	3	]	]	X
ejpam-5648	460	4	h.a	h.a	PROPN
ejpam-5648	460	5	.	.	PROPN
ejpam-5648	460	6	ahangar	ahangar	PROPN
ejpam-5648	460	7	,	,	PUNCT
ejpam-5648	460	8	m.a	m.a	PROPN
ejpam-5648	460	9	.	.	PROPN
ejpam-5648	460	10	henning	henning	PROPN
ejpam-5648	460	11	,	,	PUNCT
ejpam-5648	460	12	v.	v.	ADP
ejpam-5648	460	13	samodivkin	samodivkin	NOUN
ejpam-5648	460	14	,	,	PUNCT
ejpam-5648	460	15	and	and	CCONJ
ejpam-5648	460	16	i.g	i.g	PROPN
ejpam-5648	460	17	.	.	PROPN
ejpam-5648	460	18	yero	yero	PROPN
ejpam-5648	460	19	.	.	PUNCT
ejpam-5648	461	1	total	total	ADJ
ejpam-5648	461	2	roman	roman	ADJ
ejpam-5648	461	3	domination	domination	NOUN
ejpam-5648	461	4	in	in	ADP
ejpam-5648	461	5	graphs	graph	NOUN
ejpam-5648	461	6	.	.	PUNCT
ejpam-5648	462	1	applicable	applicable	ADJ
ejpam-5648	462	2	analysis	analysis	NOUN
ejpam-5648	462	3	and	and	CCONJ
ejpam-5648	462	4	discrete	discrete	ADJ
ejpam-5648	462	5	mathematics	mathematic	NOUN
ejpam-5648	462	6	,	,	PUNCT
ejpam-5648	462	7	10(2):501–517	10(2):501–517	PROPN
ejpam-5648	462	8	,	,	PUNCT
ejpam-5648	462	9	2016	2016	NUM
ejpam-5648	462	10	.	.	PUNCT
ejpam-5648	463	1	[	[	X
ejpam-5648	463	2	4	4	NUM
ejpam-5648	463	3	]	]	X
ejpam-5648	463	4	m.p	m.p	PROPN
ejpam-5648	463	5	.	.	PROPN
ejpam-5648	463	6	alvarez	alvarez	PROPN
ejpam-5648	463	7	-	-	PUNCT
ejpam-5648	463	8	ruiz	ruiz	PROPN
ejpam-5648	463	9	,	,	PUNCT
ejpam-5648	463	10	t.	t.	PROPN
ejpam-5648	463	11	mediavilla	mediavilla	PROPN
ejpam-5648	463	12	-	-	PUNCT
ejpam-5648	463	13	gradolph	gradolph	NOUN
ejpam-5648	463	14	,	,	PUNCT
ejpam-5648	463	15	s.m	s.m	PROPN
ejpam-5648	463	16	.	.	PROPN
ejpam-5648	463	17	sheikholeslami	sheikholeslami	PROPN
ejpam-5648	463	18	,	,	PUNCT
ejpam-5648	463	19	j.c	j.c	PROPN
ejpam-5648	463	20	.	.	PROPN
ejpam-5648	463	21	valenzuelatripodoro	valenzuelatripodoro	PROPN
ejpam-5648	463	22	,	,	PUNCT
ejpam-5648	463	23	and	and	CCONJ
ejpam-5648	463	24	i.g	i.g	PROPN
ejpam-5648	463	25	.	.	PROPN
ejpam-5648	463	26	yero	yero	PROPN
ejpam-5648	463	27	.	.	PUNCT
ejpam-5648	464	1	on	on	ADP
ejpam-5648	464	2	the	the	DET
ejpam-5648	464	3	strong	strong	ADJ
ejpam-5648	464	4	roman	roman	ADJ
ejpam-5648	464	5	domination	domination	NOUN
ejpam-5648	464	6	number	number	NOUN
ejpam-5648	464	7	of	of	ADP
ejpam-5648	464	8	graphs	graph	NOUN
ejpam-5648	464	9	.	.	PUNCT
ejpam-5648	465	1	discrete	discrete	ADJ
ejpam-5648	465	2	applied	apply	VERB
ejpam-5648	465	3	mathematics	mathematic	NOUN
ejpam-5648	465	4	,	,	PUNCT
ejpam-5648	465	5	231:44–59	231:44–59	NUM
ejpam-5648	465	6	,	,	PUNCT
ejpam-5648	465	7	2017	2017	NUM
ejpam-5648	465	8	.	.	PUNCT
ejpam-5648	466	1	[	[	X
ejpam-5648	466	2	5	5	X
ejpam-5648	466	3	]	]	PUNCT
ejpam-5648	466	4	s.	s.	PROPN
ejpam-5648	466	5	ayyaswamy	ayyaswamy	PROPN
ejpam-5648	466	6	,	,	PUNCT
ejpam-5648	466	7	b.	b.	PROPN
ejpam-5648	466	8	krishnakumari	krishnakumari	PROPN
ejpam-5648	466	9	,	,	PUNCT
ejpam-5648	466	10	b.	b.	PROPN
ejpam-5648	466	11	natarjan	natarjan	PROPN
ejpam-5648	466	12	,	,	PUNCT
ejpam-5648	466	13	and	and	CCONJ
ejpam-5648	466	14	y.	y.	PROPN
ejpam-5648	466	15	venkatakrishnan	venkatakrishnan	PROPN
ejpam-5648	466	16	.	.	PUNCT
ejpam-5648	467	1	bounds	bound	NOUN
ejpam-5648	467	2	on	on	ADP
ejpam-5648	467	3	the	the	DET
ejpam-5648	467	4	hop	hop	NOUN
ejpam-5648	467	5	domination	domination	NOUN
ejpam-5648	467	6	number	number	NOUN
ejpam-5648	467	7	of	of	ADP
ejpam-5648	467	8	a	a	DET
ejpam-5648	467	9	tree	tree	NOUN
ejpam-5648	467	10	.	.	PUNCT
ejpam-5648	468	1	proceedings	proceeding	NOUN
ejpam-5648	468	2	-	-	PUNCT
ejpam-5648	468	3	mathematical	mathematical	ADJ
ejpam-5648	468	4	sciences	science	NOUN
ejpam-5648	468	5	.	.	PUNCT
ejpam-5648	468	6	,	,	PUNCT
ejpam-5648	468	7	125(4):449–455	125(4):449–455	ADP
ejpam-5648	468	8	,	,	PUNCT
ejpam-5648	468	9	2015	2015	NUM
ejpam-5648	468	10	.	.	PUNCT
ejpam-5648	469	1	[	[	X
ejpam-5648	469	2	6	6	NUM
ejpam-5648	469	3	]	]	PUNCT
ejpam-5648	469	4	s.	s.	PROPN
ejpam-5648	469	5	ayyaswamy	ayyaswamy	PROPN
ejpam-5648	469	6	,	,	PUNCT
ejpam-5648	469	7	c.	c.	PROPN
ejpam-5648	469	8	natarajan	natarajan	PROPN
ejpam-5648	469	9	,	,	PUNCT
ejpam-5648	469	10	and	and	CCONJ
ejpam-5648	469	11	g.	g.	PROPN
ejpam-5648	469	12	sathiamoorphy	sathiamoorphy	PROPN
ejpam-5648	469	13	.	.	PUNCT
ejpam-5648	470	1	a	a	DET
ejpam-5648	470	2	note	note	NOUN
ejpam-5648	470	3	on	on	ADP
ejpam-5648	470	4	hop	hop	NOUN
ejpam-5648	470	5	domination	domination	NOUN
ejpam-5648	470	6	number	number	NOUN
ejpam-5648	470	7	of	of	ADP
ejpam-5648	470	8	some	some	DET
ejpam-5648	470	9	special	special	ADJ
ejpam-5648	470	10	families	family	NOUN
ejpam-5648	470	11	of	of	ADP
ejpam-5648	470	12	graphs	graph	NOUN
ejpam-5648	470	13	.	.	PUNCT
ejpam-5648	471	1	international	international	ADJ
ejpam-5648	471	2	journal	journal	NOUN
ejpam-5648	471	3	of	of	ADP
ejpam-5648	471	4	pure	pure	ADJ
ejpam-5648	471	5	and	and	CCONJ
ejpam-5648	471	6	applied	applied	ADJ
ejpam-5648	471	7	mathematics	mathematic	NOUN
ejpam-5648	471	8	.	.	PUNCT
ejpam-5648	471	9	,	,	PUNCT
ejpam-5648	471	10	119(12):11465–14171	119(12):11465–14171	NUM
ejpam-5648	471	11	,	,	PUNCT
ejpam-5648	471	12	2018	2018	NUM
ejpam-5648	471	13	.	.	PUNCT
ejpam-5648	472	1	[	[	X
ejpam-5648	472	2	7	7	X
ejpam-5648	472	3	]	]	X
ejpam-5648	472	4	s.	s.	PROPN
ejpam-5648	472	5	banerjee	banerjee	PROPN
ejpam-5648	472	6	,	,	PUNCT
ejpam-5648	472	7	j.m	j.m	PROPN
ejpam-5648	472	8	.	.	PROPN
ejpam-5648	472	9	keil	keil	PROPN
ejpam-5648	472	10	,	,	PUNCT
ejpam-5648	472	11	and	and	CCONJ
ejpam-5648	472	12	d.	d.	PROPN
ejpam-5648	472	13	pradhan	pradhan	PROPN
ejpam-5648	472	14	.	.	PUNCT
ejpam-5648	473	1	perfect	perfect	ADJ
ejpam-5648	473	2	roman	roman	ADJ
ejpam-5648	473	3	domination	domination	NOUN
ejpam-5648	473	4	in	in	ADP
ejpam-5648	473	5	graphs	graph	NOUN
ejpam-5648	473	6	.	.	PUNCT
ejpam-5648	474	1	theoretical	theoretical	ADJ
ejpam-5648	474	2	computer	computer	NOUN
ejpam-5648	474	3	science	science	NOUN
ejpam-5648	474	4	,	,	PUNCT
ejpam-5648	474	5	796:1–21	796:1–21	NUM
ejpam-5648	474	6	,	,	PUNCT
ejpam-5648	474	7	2019	2019	NUM
ejpam-5648	474	8	.	.	PUNCT
ejpam-5648	475	1	[	[	X
ejpam-5648	475	2	8	8	NUM
ejpam-5648	475	3	]	]	X
ejpam-5648	475	4	m.	m.	NOUN
ejpam-5648	475	5	chellali	chellali	PROPN
ejpam-5648	475	6	,	,	PUNCT
ejpam-5648	475	7	t.w	t.w	PROPN
ejpam-5648	475	8	.	.	PROPN
ejpam-5648	475	9	haynes	haynes	PROPN
ejpam-5648	475	10	,	,	PUNCT
ejpam-5648	475	11	and	and	CCONJ
ejpam-5648	475	12	s.t	s.t	PROPN
ejpam-5648	475	13	.	.	PROPN
ejpam-5648	475	14	hedetnieme	hedetnieme	PROPN
ejpam-5648	475	15	.	.	PUNCT
ejpam-5648	476	1	roman	roman	NOUN
ejpam-5648	476	2	{	{	PUNCT
ejpam-5648	476	3	2	2	NUM
ejpam-5648	476	4	}	}	PUNCT
ejpam-5648	476	5	domination	domination	NOUN
ejpam-5648	476	6	.	.	PUNCT
ejpam-5648	477	1	discrete	discrete	ADJ
ejpam-5648	477	2	applied	apply	VERB
ejpam-5648	477	3	math	math	NOUN
ejpam-5648	477	4	,	,	PUNCT
ejpam-5648	477	5	204:22–28	204:22–28	NUM
ejpam-5648	477	6	,	,	PUNCT
ejpam-5648	477	7	2016	2016	NUM
ejpam-5648	477	8	.	.	PUNCT
ejpam-5648	478	1	[	[	X
ejpam-5648	478	2	9	9	NUM
ejpam-5648	478	3	]	]	SYM
ejpam-5648	478	4	e.j	e.j	PROPN
ejpam-5648	478	5	.	.	PROPN
ejpam-5648	478	6	cockayne	cockayne	PROPN
ejpam-5648	478	7	,	,	PUNCT
ejpam-5648	478	8	p.a	p.a	PROPN
ejpam-5648	478	9	.	.	PROPN
ejpam-5648	478	10	deryer	deryer	PROPN
ejpam-5648	478	11	,	,	PUNCT
ejpam-5648	478	12	s.m	s.m	PROPN
ejpam-5648	478	13	.	.	PROPN
ejpam-5648	478	14	hedetnieme	hedetnieme	PROPN
ejpam-5648	478	15	,	,	PUNCT
ejpam-5648	478	16	and	and	CCONJ
ejpam-5648	478	17	s.t	s.t	PROPN
ejpam-5648	478	18	.	.	PROPN
ejpam-5648	478	19	hedetnieme	hedetnieme	PROPN
ejpam-5648	478	20	.	.	PUNCT
ejpam-5648	479	1	roman	roman	ADJ
ejpam-5648	479	2	domination	domination	NOUN
ejpam-5648	479	3	in	in	ADP
ejpam-5648	479	4	graphs	graph	NOUN
ejpam-5648	479	5	.	.	PUNCT
ejpam-5648	480	1	discrete	discrete	ADJ
ejpam-5648	480	2	mathematics	mathematic	NOUN
ejpam-5648	480	3	,	,	PUNCT
ejpam-5648	480	4	278(13):11–22	278(13):11–22	NUM
ejpam-5648	480	5	,	,	PUNCT
ejpam-5648	480	6	2004	2004	NUM
ejpam-5648	480	7	.	.	PUNCT
ejpam-5648	481	1	[	[	X
ejpam-5648	481	2	10	10	NUM
ejpam-5648	481	3	]	]	X
ejpam-5648	481	4	r.j	r.j	PROPN
ejpam-5648	481	5	.	.	PROPN
ejpam-5648	481	6	fortosa	fortosa	PROPN
ejpam-5648	481	7	,	,	PUNCT
ejpam-5648	481	8	f.	f.	PROPN
ejpam-5648	481	9	jamil	jamil	PROPN
ejpam-5648	481	10	,	,	PUNCT
ejpam-5648	481	11	and	and	CCONJ
ejpam-5648	481	12	s.	s.	PROPN
ejpam-5648	481	13	canoy	canoy	PROPN
ejpam-5648	481	14	jr	jr	PROPN
ejpam-5648	481	15	.	.	PROPN
ejpam-5648	481	16	convex	convex	PROPN
ejpam-5648	481	17	roman	roman	ADJ
ejpam-5648	481	18	dominating	dominating	NOUN
ejpam-5648	481	19	functions	function	NOUN
ejpam-5648	481	20	on	on	ADP
ejpam-5648	481	21	graphs	graph	NOUN
ejpam-5648	481	22	under	under	ADP
ejpam-5648	481	23	some	some	DET
ejpam-5648	481	24	binary	binary	ADJ
ejpam-5648	481	25	operations	operation	NOUN
ejpam-5648	481	26	.	.	PUNCT
ejpam-5648	482	1	european	european	ADJ
ejpam-5648	482	2	journal	journal	PROPN
ejpam-5648	482	3	of	of	ADP
ejpam-5648	482	4	pure	pure	ADJ
ejpam-5648	482	5	and	and	CCONJ
ejpam-5648	482	6	applied	applied	ADJ
ejpam-5648	482	7	mathematics	mathematic	NOUN
ejpam-5648	482	8	,	,	PUNCT
ejpam-5648	482	9	17(2):1335–1351	17(2):1335–1351	NUM
ejpam-5648	482	10	,	,	PUNCT
ejpam-5648	482	11	2024	2024	NUM
ejpam-5648	482	12	.	.	PUNCT
ejpam-5648	483	1	[	[	X
ejpam-5648	483	2	11	11	NUM
ejpam-5648	483	3	]	]	X
ejpam-5648	483	4	r.j	r.j	PROPN
ejpam-5648	483	5	.	.	PROPN
ejpam-5648	483	6	fortosa	fortosa	PROPN
ejpam-5648	483	7	and	and	CCONJ
ejpam-5648	483	8	s.	s.	PROPN
ejpam-5648	483	9	canoy	canoy	PROPN
ejpam-5648	483	10	jr	jr	PROPN
ejpam-5648	483	11	.	.	PROPN
ejpam-5648	483	12	geodetic	geodetic	ADJ
ejpam-5648	483	13	roman	roman	ADJ
ejpam-5648	483	14	dominating	dominating	NOUN
ejpam-5648	483	15	functions	function	NOUN
ejpam-5648	483	16	in	in	ADP
ejpam-5648	483	17	a	a	DET
ejpam-5648	483	18	graph	graph	NOUN
ejpam-5648	483	19	.	.	PUNCT
ejpam-5648	484	1	european	european	ADJ
ejpam-5648	484	2	journal	journal	PROPN
ejpam-5648	484	3	of	of	ADP
ejpam-5648	484	4	pure	pure	ADJ
ejpam-5648	484	5	and	and	CCONJ
ejpam-5648	484	6	applied	applied	ADJ
ejpam-5648	484	7	mathematics	mathematic	NOUN
ejpam-5648	484	8	,	,	PUNCT
ejpam-5648	484	9	16(4):2368–2383	16(4):2368–2383	NUM
ejpam-5648	484	10	,	,	PUNCT
ejpam-5648	484	11	2023	2023	NUM
ejpam-5648	484	12	.	.	PUNCT
ejpam-5648	485	1	[	[	X
ejpam-5648	485	2	12	12	NUM
ejpam-5648	485	3	]	]	X
ejpam-5648	485	4	s.m	s.m	PROPN
ejpam-5648	485	5	.	.	PROPN
ejpam-5648	485	6	sheikholeslami	sheikholeslami	PROPN
ejpam-5648	485	7	h.	h.	PROPN
ejpam-5648	485	8	abdollahzadeh	abdollahzadeh	PROPN
ejpam-5648	485	9	ahangar	ahangar	NOUN
ejpam-5648	485	10	,	,	PUNCT
ejpam-5648	485	11	m.	m.	NOUN
ejpam-5648	485	12	chellali	chellali	PROPN
ejpam-5648	485	13	.	.	PUNCT
ejpam-5648	486	1	on	on	ADP
ejpam-5648	486	2	the	the	DET
ejpam-5648	486	3	double	double	ADJ
ejpam-5648	486	4	roman	roman	ADJ
ejpam-5648	486	5	domination	domination	NOUN
ejpam-5648	486	6	in	in	ADP
ejpam-5648	486	7	graphs	graph	NOUN
ejpam-5648	486	8	.	.	PUNCT
ejpam-5648	487	1	discrete	discrete	ADJ
ejpam-5648	487	2	appl	appl	PROPN
ejpam-5648	487	3	.	.	PUNCT
ejpam-5648	487	4	math	math	PROPN
ejpam-5648	487	5	.	.	PUNCT
ejpam-5648	488	1	,	,	PUNCT
ejpam-5648	488	2	pages	page	NOUN
ejpam-5648	488	3	1–7	1–7	NUM
ejpam-5648	488	4	,	,	PUNCT
ejpam-5648	488	5	2017	2017	NUM
ejpam-5648	488	6	.	.	PUNCT
ejpam-5648	489	1	[	[	X
ejpam-5648	489	2	13	13	NUM
ejpam-5648	489	3	]	]	PUNCT
ejpam-5648	489	4	j.	j.	PROPN
ejpam-5648	489	5	hassan	hassan	PROPN
ejpam-5648	489	6	and	and	CCONJ
ejpam-5648	489	7	s.	s.	PROPN
ejpam-5648	489	8	canoy	canoy	PROPN
ejpam-5648	489	9	jr	jr	PROPN
ejpam-5648	489	10	.	.	PROPN
ejpam-5648	489	11	hop	hop	PROPN
ejpam-5648	489	12	independent	independent	ADJ
ejpam-5648	489	13	hop	hop	NOUN
ejpam-5648	489	14	domination	domination	NOUN
ejpam-5648	489	15	in	in	ADP
ejpam-5648	489	16	graphs	graph	NOUN
ejpam-5648	489	17	.	.	PUNCT
ejpam-5648	490	1	eur	eur	PROPN
ejpam-5648	490	2	.	.	PUNCT
ejpam-5648	491	1	j.	j.	PROPN
ejpam-5648	491	2	pure	pure	PROPN
ejpam-5648	491	3	appl	appl	PROPN
ejpam-5648	491	4	.	.	PUNCT
ejpam-5648	491	5	math	math	PROPN
ejpam-5648	491	6	.	.	PUNCT
ejpam-5648	491	7	,	,	PUNCT
ejpam-5648	491	8	15(4):1783–1796	15(4):1783–1796	NUM
ejpam-5648	491	9	,	,	PUNCT
ejpam-5648	491	10	2022	2022	NUM
ejpam-5648	491	11	.	.	PUNCT
ejpam-5648	492	1	[	[	X
ejpam-5648	492	2	14	14	NUM
ejpam-5648	492	3	]	]	PUNCT
ejpam-5648	492	4	j.	j.	PROPN
ejpam-5648	492	5	hassan	hassan	PROPN
ejpam-5648	492	6	,	,	PUNCT
ejpam-5648	492	7	s.	s.	PROPN
ejpam-5648	492	8	canoy	canoy	PROPN
ejpam-5648	492	9	jr	jr	PROPN
ejpam-5648	492	10	.	.	PROPN
ejpam-5648	492	11	,	,	PUNCT
ejpam-5648	492	12	and	and	CCONJ
ejpam-5648	492	13	a.	a.	PROPN
ejpam-5648	492	14	aradais	aradais	PROPN
ejpam-5648	492	15	.	.	PUNCT
ejpam-5648	493	1	hop	hop	PROPN
ejpam-5648	493	2	independent	independent	ADJ
ejpam-5648	493	3	sets	set	NOUN
ejpam-5648	493	4	in	in	ADP
ejpam-5648	493	5	graphs	graph	NOUN
ejpam-5648	493	6	.	.	PUNCT
ejpam-5648	494	1	eur	eur	PROPN
ejpam-5648	494	2	.	.	PUNCT
ejpam-5648	495	1	j.	j.	PROPN
ejpam-5648	495	2	pure	pure	PROPN
ejpam-5648	495	3	appl	appl	PROPN
ejpam-5648	495	4	.	.	PUNCT
ejpam-5648	495	5	math	math	PROPN
ejpam-5648	495	6	.	.	PUNCT
ejpam-5648	495	7	,	,	PUNCT
ejpam-5648	495	8	15(2):467–477	15(2):467–477	PROPN
ejpam-5648	495	9	,	,	PUNCT
ejpam-5648	495	10	2022	2022	NUM
ejpam-5648	495	11	.	.	PUNCT
ejpam-5648	496	1	[	[	X
ejpam-5648	496	2	15	15	NUM
ejpam-5648	496	3	]	]	X
ejpam-5648	496	4	m.	m.	NOUN
ejpam-5648	496	5	henning	henning	PROPN
ejpam-5648	496	6	and	and	CCONJ
ejpam-5648	496	7	n.	n.	PROPN
ejpam-5648	496	8	rad	rad	PROPN
ejpam-5648	496	9	.	.	PROPN
ejpam-5648	497	1	on	on	ADP
ejpam-5648	497	2	2	2	NUM
ejpam-5648	497	3	-	-	PUNCT
ejpam-5648	497	4	step	step	NOUN
ejpam-5648	497	5	and	and	CCONJ
ejpam-5648	497	6	hop	hop	NOUN
ejpam-5648	497	7	dominating	dominating	NOUN
ejpam-5648	497	8	sets	set	NOUN
ejpam-5648	497	9	in	in	ADP
ejpam-5648	497	10	graphs	graph	NOUN
ejpam-5648	497	11	.	.	PUNCT
ejpam-5648	498	1	graphs	graph	NOUN
ejpam-5648	498	2	and	and	CCONJ
ejpam-5648	498	3	combinatorics	combinatoric	NOUN
ejpam-5648	498	4	.	.	PUNCT
ejpam-5648	498	5	,	,	PUNCT
ejpam-5648	498	6	33(4):913–927	33(4):913–927	PROPN
ejpam-5648	498	7	,	,	PUNCT
ejpam-5648	498	8	2017	2017	NUM
ejpam-5648	498	9	.	.	PUNCT
ejpam-5648	499	1	[	[	X
ejpam-5648	499	2	16	16	NUM
ejpam-5648	499	3	]	]	X
ejpam-5648	499	4	m.a	m.a	PROPN
ejpam-5648	499	5	.	.	PROPN
ejpam-5648	499	6	henning	henning	PROPN
ejpam-5648	499	7	,	,	PUNCT
ejpam-5648	499	8	w.f	w.f	PROPN
ejpam-5648	499	9	.	.	PROPN
ejpam-5648	499	10	klostermeyer	klostermeyer	PROPN
ejpam-5648	499	11	,	,	PUNCT
ejpam-5648	499	12	and	and	CCONJ
ejpam-5648	499	13	g.	g.	PROPN
ejpam-5648	499	14	macgillivray	macgillivray	PROPN
ejpam-5648	499	15	.	.	PUNCT
ejpam-5648	500	1	perfect	perfect	ADJ
ejpam-5648	500	2	roman	roman	ADJ
ejpam-5648	500	3	domination	domination	NOUN
ejpam-5648	500	4	in	in	ADP
ejpam-5648	500	5	trees	tree	NOUN
ejpam-5648	500	6	.	.	PUNCT
ejpam-5648	501	1	discrete	discrete	ADJ
ejpam-5648	501	2	applied	apply	VERB
ejpam-5648	501	3	mathematics	mathematic	NOUN
ejpam-5648	501	4	,	,	PUNCT
ejpam-5648	501	5	236:235–245	236:235–245	NUM
ejpam-5648	501	6	,	,	PUNCT
ejpam-5648	501	7	2018	2018	NUM
ejpam-5648	501	8	.	.	PUNCT
ejpam-5648	502	1	[	[	X
ejpam-5648	502	2	17	17	NUM
ejpam-5648	502	3	]	]	X
ejpam-5648	502	4	s.	s.	PROPN
ejpam-5648	502	5	canoy	canoy	PROPN
ejpam-5648	502	6	jr	jr	PROPN
ejpam-5648	502	7	.	.	PROPN
ejpam-5648	502	8	,	,	PUNCT
ejpam-5648	502	9	r.	r.	PROPN
ejpam-5648	502	10	mollejon	mollejon	NOUN
ejpam-5648	502	11	,	,	PUNCT
ejpam-5648	502	12	and	and	CCONJ
ejpam-5648	502	13	j.	j.	PROPN
ejpam-5648	502	14	g.	g.	PROPN
ejpam-5648	502	15	canoy	canoy	PROPN
ejpam-5648	502	16	.	.	PUNCT
ejpam-5648	503	1	hop	hop	PROPN
ejpam-5648	503	2	dominating	dominating	NOUN
ejpam-5648	503	3	sets	set	NOUN
ejpam-5648	503	4	in	in	ADP
ejpam-5648	503	5	graphs	graph	NOUN
ejpam-5648	503	6	under	under	ADP
ejpam-5648	503	7	binary	binary	ADJ
ejpam-5648	503	8	operations	operation	NOUN
ejpam-5648	503	9	.	.	PUNCT
ejpam-5648	504	1	eur	eur	PROPN
ejpam-5648	504	2	.	.	PUNCT
ejpam-5648	505	1	j.	j.	PROPN
ejpam-5648	505	2	pure	pure	PROPN
ejpam-5648	505	3	appl	appl	PROPN
ejpam-5648	505	4	.	.	PUNCT
ejpam-5648	505	5	math	math	PROPN
ejpam-5648	505	6	.	.	PUNCT
ejpam-5648	505	7	,	,	PUNCT
ejpam-5648	506	1	12(4):1455–1463	12(4):1455–1463	NUM
ejpam-5648	506	2	,	,	PUNCT
ejpam-5648	506	3	2019	2019	NUM
ejpam-5648	506	4	.	.	PUNCT
ejpam-5648	507	1	[	[	X
ejpam-5648	507	2	18	18	NUM
ejpam-5648	507	3	]	]	X
ejpam-5648	507	4	s.	s.	PROPN
ejpam-5648	507	5	canoy	canoy	PROPN
ejpam-5648	507	6	jr	jr	PROPN
ejpam-5648	507	7	.	.	PROPN
ejpam-5648	507	8	and	and	CCONJ
ejpam-5648	507	9	g.	g.	PROPN
ejpam-5648	507	10	salasalan	salasalan	NOUN
ejpam-5648	507	11	.	.	PUNCT
ejpam-5648	508	1	revisiting	revisit	VERB
ejpam-5648	508	2	domination	domination	NOUN
ejpam-5648	508	3	,	,	PUNCT
ejpam-5648	508	4	hop	hop	NOUN
ejpam-5648	508	5	domination	domination	NOUN
ejpam-5648	508	6	,	,	PUNCT
ejpam-5648	508	7	and	and	CCONJ
ejpam-5648	508	8	global	global	ADJ
ejpam-5648	508	9	a.	a.	PROPN
ejpam-5648	508	10	aradais	aradais	PROPN
ejpam-5648	508	11	,	,	PUNCT
ejpam-5648	508	12	j.	j.	PROPN
ejpam-5648	508	13	cariaga	cariaga	PROPN
ejpam-5648	508	14	,	,	PUNCT
ejpam-5648	508	15	s.	s.	PROPN
ejpam-5648	508	16	canoy	canoy	PROPN
ejpam-5648	508	17	jr	jr	PROPN
ejpam-5648	508	18	.	.	PROPN
ejpam-5648	508	19	/	/	SYM
ejpam-5648	508	20	eur	eur	PROPN
ejpam-5648	508	21	.	.	PUNCT
ejpam-5648	509	1	j.	j.	PROPN
ejpam-5648	509	2	pure	pure	PROPN
ejpam-5648	509	3	appl	appl	PROPN
ejpam-5648	509	4	.	.	PROPN
ejpam-5648	509	5	math	math	PROPN
ejpam-5648	509	6	,	,	PUNCT
ejpam-5648	509	7	18	18	NUM
ejpam-5648	509	8	(	(	PUNCT
ejpam-5648	509	9	1	1	NUM
ejpam-5648	509	10	)	)	PUNCT
ejpam-5648	509	11	(	(	PUNCT
ejpam-5648	509	12	2025	2025	NUM
ejpam-5648	509	13	)	)	PUNCT
ejpam-5648	509	14	,	,	PUNCT
ejpam-5648	509	15	5648	5648	NUM
ejpam-5648	509	16	13	13	NUM
ejpam-5648	509	17	of	of	ADP
ejpam-5648	509	18	13	13	NUM
ejpam-5648	509	19	hop	hop	NOUN
ejpam-5648	509	20	domination	domination	NOUN
ejpam-5648	509	21	in	in	ADP
ejpam-5648	509	22	graphs	graph	NOUN
ejpam-5648	509	23	.	.	PUNCT
ejpam-5648	510	1	eur	eur	PROPN
ejpam-5648	510	2	.	.	PUNCT
ejpam-5648	511	1	j.	j.	PROPN
ejpam-5648	511	2	pure	pure	PROPN
ejpam-5648	511	3	appl	appl	PROPN
ejpam-5648	511	4	.	.	PUNCT
ejpam-5648	511	5	math	math	PROPN
ejpam-5648	511	6	.	.	PUNCT
ejpam-5648	511	7	,	,	PUNCT
ejpam-5648	511	8	14:1415–1428	14:1415–1428	NUM
ejpam-5648	511	9	,	,	PUNCT
ejpam-5648	511	10	2021	2021	NUM
ejpam-5648	511	11	.	.	PUNCT
ejpam-5648	512	1	[	[	X
ejpam-5648	512	2	19	19	NUM
ejpam-5648	512	3	]	]	X
ejpam-5648	512	4	s.	s.	PROPN
ejpam-5648	512	5	canoy	canoy	PROPN
ejpam-5648	512	6	jr	jr	PROPN
ejpam-5648	512	7	.	.	PROPN
ejpam-5648	512	8	and	and	CCONJ
ejpam-5648	512	9	g.	g.	PROPN
ejpam-5648	512	10	salasalan	salasalan	NOUN
ejpam-5648	512	11	.	.	PUNCT
ejpam-5648	513	1	locating	locate	VERB
ejpam-5648	513	2	-	-	PUNCT
ejpam-5648	513	3	hop	hop	NOUN
ejpam-5648	513	4	domination	domination	NOUN
ejpam-5648	513	5	in	in	ADP
ejpam-5648	513	6	graphs	graph	NOUN
ejpam-5648	513	7	.	.	PUNCT
ejpam-5648	514	1	kyungpook	kyungpook	PROPN
ejpam-5648	514	2	mathematical	mathematical	PROPN
ejpam-5648	514	3	journal	journal	PROPN
ejpam-5648	514	4	.	.	PUNCT
ejpam-5648	514	5	,	,	PUNCT
ejpam-5648	514	6	62:193–204	62:193–204	NUM
ejpam-5648	514	7	,	,	PUNCT
ejpam-5648	514	8	2022	2022	NUM
ejpam-5648	514	9	.	.	PUNCT
ejpam-5648	515	1	[	[	X
ejpam-5648	515	2	20	20	NUM
ejpam-5648	515	3	]	]	PUNCT
ejpam-5648	515	4	k.	k.	PROPN
ejpam-5648	515	5	kammerling	kammerling	PROPN
ejpam-5648	515	6	and	and	CCONJ
ejpam-5648	515	7	l.	l.	PROPN
ejpam-5648	515	8	volkman	volkman	PROPN
ejpam-5648	515	9	.	.	PUNCT
ejpam-5648	516	1	roman	roman	ADJ
ejpam-5648	516	2	k	k	NOUN
ejpam-5648	516	3	-	-	PUNCT
ejpam-5648	516	4	domination	domination	NOUN
ejpam-5648	516	5	in	in	ADP
ejpam-5648	516	6	graphs	graph	NOUN
ejpam-5648	516	7	.	.	PUNCT
ejpam-5648	517	1	journal	journal	NOUN
ejpam-5648	517	2	of	of	ADP
ejpam-5648	517	3	the	the	DET
ejpam-5648	517	4	korean	korean	PROPN
ejpam-5648	517	5	mathematical	mathematical	ADJ
ejpam-5648	517	6	society	society	NOUN
ejpam-5648	517	7	,	,	PUNCT
ejpam-5648	517	8	46(6):1309–1318	46(6):1309–1318	PROPN
ejpam-5648	517	9	,	,	PUNCT
ejpam-5648	517	10	2009	2009	NUM
ejpam-5648	517	11	.	.	PUNCT
ejpam-5648	518	1	[	[	X
ejpam-5648	518	2	21	21	NUM
ejpam-5648	518	3	]	]	X
ejpam-5648	518	4	m.h	m.h	PROPN
ejpam-5648	518	5	.	.	PROPN
ejpam-5648	518	6	muddebihal	muddebihal	PROPN
ejpam-5648	518	7	and	and	CCONJ
ejpam-5648	518	8	sumangaladevi	sumangaladevi	ADJ
ejpam-5648	518	9	.	.	PUNCT
ejpam-5648	519	1	connected	connect	VERB
ejpam-5648	519	2	roman	roman	ADJ
ejpam-5648	519	3	domination	domination	NOUN
ejpam-5648	519	4	in	in	ADP
ejpam-5648	519	5	graphs	graph	NOUN
ejpam-5648	519	6	.	.	PUNCT
ejpam-5648	520	1	international	international	ADJ
ejpam-5648	520	2	journal	journal	PROPN
ejpam-5648	520	3	of	of	ADP
ejpam-5648	520	4	research	research	NOUN
ejpam-5648	520	5	and	and	CCONJ
ejpam-5648	520	6	engineering	engineering	NOUN
ejpam-5648	520	7	technology	technology	NOUN
ejpam-5648	520	8	,	,	PUNCT
ejpam-5648	520	9	2(10):333–340	2(10):333–340	NUM
ejpam-5648	520	10	,	,	PUNCT
ejpam-5648	520	11	2013	2013	NUM
ejpam-5648	520	12	.	.	PUNCT
ejpam-5648	521	1	[	[	X
ejpam-5648	521	2	22	22	NUM
ejpam-5648	521	3	]	]	X
ejpam-5648	521	4	c.	c.	PROPN
ejpam-5648	521	5	natarajan	natarajan	PROPN
ejpam-5648	521	6	and	and	CCONJ
ejpam-5648	521	7	s.	s.	PROPN
ejpam-5648	521	8	ayyaswamy	ayyaswamy	PROPN
ejpam-5648	521	9	.	.	PUNCT
ejpam-5648	522	1	hop	hop	PROPN
ejpam-5648	522	2	domination	domination	NOUN
ejpam-5648	522	3	in	in	ADP
ejpam-5648	522	4	graphs	graphs	PROPN
ejpam-5648	522	5	ii	ii	PROPN
ejpam-5648	522	6	.	.	PUNCT
ejpam-5648	522	7	versita	versita	PROPN
ejpam-5648	522	8	,	,	PUNCT
ejpam-5648	522	9	23(2):187	23(2):187	NUM
ejpam-5648	522	10	–	–	PUNCT
ejpam-5648	522	11	199	199	NUM
ejpam-5648	522	12	,	,	PUNCT
ejpam-5648	522	13	2015	2015	NUM
ejpam-5648	522	14	.	.	PUNCT
ejpam-5648	523	1	[	[	X
ejpam-5648	523	2	23	23	NUM
ejpam-5648	523	3	]	]	X
ejpam-5648	523	4	y.	y.	PROPN
ejpam-5648	523	5	pabilona	pabilona	PROPN
ejpam-5648	523	6	and	and	CCONJ
ejpam-5648	523	7	h.	h.	PROPN
ejpam-5648	523	8	rara	rara	PROPN
ejpam-5648	523	9	.	.	PUNCT
ejpam-5648	524	1	connected	connect	VERB
ejpam-5648	524	2	hop	hop	NOUN
ejpam-5648	524	3	domination	domination	NOUN
ejpam-5648	524	4	in	in	ADP
ejpam-5648	524	5	graphs	graph	NOUN
ejpam-5648	524	6	under	under	ADP
ejpam-5648	524	7	some	some	DET
ejpam-5648	524	8	binary	binary	ADJ
ejpam-5648	524	9	operations	operation	NOUN
ejpam-5648	524	10	.	.	PUNCT
ejpam-5648	525	1	asian	asian	ADJ
ejpam-5648	525	2	-	-	PUNCT
ejpam-5648	525	3	eur	eur	NOUN
ejpam-5648	525	4	.	.	PUNCT
ejpam-5648	526	1	j.	j.	PROPN
ejpam-5648	526	2	math	math	PROPN
ejpam-5648	526	3	.	.	PROPN
ejpam-5648	526	4	,	,	PUNCT
ejpam-5648	526	5	11(5):1850075–1–1850075–11	11(5):1850075–1–1850075–11	NUM
ejpam-5648	526	6	,	,	PUNCT
ejpam-5648	526	7	2018	2018	NUM
ejpam-5648	526	8	.	.	PUNCT
ejpam-5648	527	1	[	[	X
ejpam-5648	527	2	24	24	NUM
ejpam-5648	527	3	]	]	X
ejpam-5648	527	4	l.	l.	PROPN
ejpam-5648	527	5	paleta	paleta	PROPN
ejpam-5648	527	6	and	and	CCONJ
ejpam-5648	527	7	f.	f.	PROPN
ejpam-5648	527	8	jamil	jamil	PROPN
ejpam-5648	527	9	.	.	PUNCT
ejpam-5648	528	1	more	more	ADJ
ejpam-5648	528	2	on	on	ADP
ejpam-5648	528	3	perfect	perfect	ADJ
ejpam-5648	528	4	roman	roman	ADJ
ejpam-5648	528	5	domination	domination	NOUN
ejpam-5648	528	6	in	in	ADP
ejpam-5648	528	7	graphs	graph	NOUN
ejpam-5648	528	8	.	.	PUNCT
ejpam-5648	529	1	european	european	ADJ
ejpam-5648	529	2	journal	journal	PROPN
ejpam-5648	529	3	of	of	ADP
ejpam-5648	529	4	pure	pure	ADJ
ejpam-5648	529	5	and	and	CCONJ
ejpam-5648	529	6	applied	applied	ADJ
ejpam-5648	529	7	mathematics	mathematic	NOUN
ejpam-5648	529	8	.	.	PUNCT
ejpam-5648	530	1	,	,	PUNCT
ejpam-5648	530	2	13(3):529–548	13(3):529–548	NOUN
ejpam-5648	530	3	,	,	PUNCT
ejpam-5648	530	4	2020	2020	NUM
ejpam-5648	530	5	.	.	PUNCT
ejpam-5648	531	1	[	[	X
ejpam-5648	531	2	25	25	NUM
ejpam-5648	531	3	]	]	X
ejpam-5648	531	4	n.j	n.j	PROPN
ejpam-5648	531	5	.	.	PROPN
ejpam-5648	531	6	rad	rad	PROPN
ejpam-5648	531	7	and	and	CCONJ
ejpam-5648	531	8	a.	a.	NOUN
ejpam-5648	531	9	poureidi	poureidi	PROPN
ejpam-5648	531	10	.	.	PUNCT
ejpam-5648	532	1	on	on	ADP
ejpam-5648	532	2	hop	hop	PROPN
ejpam-5648	532	3	roman	roman	ADJ
ejpam-5648	532	4	domination	domination	NOUN
ejpam-5648	532	5	in	in	ADP
ejpam-5648	532	6	trees	tree	NOUN
ejpam-5648	532	7	.	.	PUNCT
ejpam-5648	533	1	communications	communication	NOUN
ejpam-5648	533	2	in	in	ADP
ejpam-5648	533	3	combinatorics	combinatoric	NOUN
ejpam-5648	533	4	and	and	CCONJ
ejpam-5648	533	5	optimization	optimization	NOUN
ejpam-5648	533	6	,	,	PUNCT
ejpam-5648	533	7	4(2):201–208	4(2):201–208	NOUN
ejpam-5648	533	8	,	,	PUNCT
ejpam-5648	533	9	2019	2019	NUM
ejpam-5648	533	10	.	.	PUNCT
ejpam-5648	534	1	[	[	X
ejpam-5648	534	2	26	26	NUM
ejpam-5648	534	3	]	]	PUNCT
ejpam-5648	534	4	r.	r.	PROPN
ejpam-5648	534	5	rakim	rakim	PROPN
ejpam-5648	534	6	,	,	PUNCT
ejpam-5648	534	7	h.	h.	PROPN
ejpam-5648	534	8	rara	rara	PROPN
ejpam-5648	534	9	,	,	PUNCT
ejpam-5648	534	10	and	and	CCONJ
ejpam-5648	534	11	c.j	c.j	PROPN
ejpam-5648	534	12	.	.	PROPN
ejpam-5648	534	13	saromines	saromine	NOUN
ejpam-5648	534	14	.	.	PUNCT
ejpam-5648	535	1	perfect	perfect	ADJ
ejpam-5648	535	2	hop	hop	NOUN
ejpam-5648	535	3	domination	domination	NOUN
ejpam-5648	535	4	in	in	ADP
ejpam-5648	535	5	graphs	graph	NOUN
ejpam-5648	535	6	.	.	PUNCT
ejpam-5648	536	1	applied	apply	VERB
ejpam-5648	536	2	mathematical	mathematical	ADJ
ejpam-5648	536	3	sciences	sciences	PROPN
ejpam-5648	536	4	,	,	PUNCT
ejpam-5648	536	5	12(13):635–649	12(13):635–649	NUM
ejpam-5648	536	6	,	,	PUNCT
ejpam-5648	536	7	2018	2018	NUM
ejpam-5648	536	8	.	.	PUNCT
ejpam-5648	537	1	[	[	X
ejpam-5648	537	2	27	27	NUM
ejpam-5648	537	3	]	]	X
ejpam-5648	537	4	g.	g.	NOUN
ejpam-5648	537	5	salasalan	salasalan	NOUN
ejpam-5648	537	6	and	and	CCONJ
ejpam-5648	537	7	s.	s.	PROPN
ejpam-5648	537	8	canoy	canoy	PROPN
ejpam-5648	537	9	jr	jr	PROPN
ejpam-5648	537	10	.	.	PROPN
ejpam-5648	537	11	global	global	PROPN
ejpam-5648	537	12	hop	hop	PROPN
ejpam-5648	537	13	domination	domination	PROPN
ejpam-5648	537	14	numbers	number	NOUN
ejpam-5648	537	15	of	of	ADP
ejpam-5648	537	16	graphs	graph	NOUN
ejpam-5648	537	17	.	.	PUNCT
ejpam-5648	538	1	eur	eur	PROPN
ejpam-5648	538	2	.	.	PUNCT
ejpam-5648	539	1	j.	j.	PROPN
ejpam-5648	539	2	pure	pure	PROPN
ejpam-5648	539	3	appl	appl	PROPN
ejpam-5648	539	4	.	.	PUNCT
ejpam-5648	539	5	math	math	PROPN
ejpam-5648	539	6	.	.	PUNCT
ejpam-5648	539	7	,	,	PUNCT
ejpam-5648	539	8	14(1):112–125	14(1):112–125	NUM
ejpam-5648	539	9	,	,	PUNCT
ejpam-5648	539	10	2021	2021	NUM
ejpam-5648	539	11	.	.	PUNCT
ejpam-5648	540	1	[	[	X
ejpam-5648	540	2	28	28	NUM
ejpam-5648	540	3	]	]	X
ejpam-5648	540	4	e.	e.	PROPN
ejpam-5648	540	5	shabani	shabani	PROPN
ejpam-5648	540	6	,	,	PUNCT
ejpam-5648	540	7	n.j	n.j	PROPN
ejpam-5648	540	8	.	.	PROPN
ejpam-5648	540	9	rad	rad	PROPN
ejpam-5648	540	10	,	,	PUNCT
ejpam-5648	540	11	and	and	CCONJ
ejpam-5648	540	12	a.	a.	NOUN
ejpam-5648	540	13	poureidi	poureidi	PROPN
ejpam-5648	540	14	.	.	PUNCT
ejpam-5648	541	1	graphs	graph	NOUN
ejpam-5648	541	2	with	with	ADP
ejpam-5648	541	3	large	large	ADJ
ejpam-5648	541	4	hop	hop	NOUN
ejpam-5648	541	5	roman	roman	ADJ
ejpam-5648	541	6	domination	domination	NOUN
ejpam-5648	541	7	numbers	number	NOUN
ejpam-5648	541	8	.	.	PUNCT
ejpam-5648	542	1	computer	computer	NOUN
ejpam-5648	542	2	science	science	PROPN
ejpam-5648	542	3	journal	journal	PROPN
ejpam-5648	542	4	of	of	ADP
ejpam-5648	542	5	modeva	modeva	PROPN
ejpam-5648	542	6	,	,	PUNCT
ejpam-5648	542	7	27(1):3–22	27(1):3–22	NUM
ejpam-5648	542	8	,	,	PUNCT
ejpam-5648	542	9	2019	2019	NUM
ejpam-5648	542	10	.	.	PUNCT