id	sid	tid	token	lemma	pos
ejpam-6904	1	1	european	european	PROPN
ejpam-6904	1	2	journal	journal	PROPN
ejpam-6904	1	3	of	of	ADP
ejpam-6904	1	4	pure	pure	ADJ
ejpam-6904	1	5	and	and	CCONJ
ejpam-6904	1	6	applied	applied	ADJ
ejpam-6904	1	7	mathematics	mathematic	NOUN
ejpam-6904	1	8	2025	2025	NUM
ejpam-6904	1	9	,	,	PUNCT
ejpam-6904	1	10	vol	vol	NOUN
ejpam-6904	1	11	.	.	PROPN
ejpam-6904	1	12	18	18	NUM
ejpam-6904	1	13	,	,	PUNCT
ejpam-6904	1	14	issue	issue	NOUN
ejpam-6904	1	15	4	4	NUM
ejpam-6904	1	16	,	,	PUNCT
ejpam-6904	1	17	article	article	NOUN
ejpam-6904	1	18	number	number	NOUN
ejpam-6904	1	19	6904	6904	NUM
ejpam-6904	1	20	issn	issn	VERB
ejpam-6904	1	21	1307	1307	NUM
ejpam-6904	1	22	-	-	SYM
ejpam-6904	1	23	5543	5543	NUM
ejpam-6904	1	24	–	–	PUNCT
ejpam-6904	1	25	ejpam.com	ejpam.com	X
ejpam-6904	1	26	published	publish	VERB
ejpam-6904	1	27	by	by	ADP
ejpam-6904	1	28	new	new	PROPN
ejpam-6904	1	29	york	york	PROPN
ejpam-6904	1	30	business	business	PROPN
ejpam-6904	1	31	global	global	ADJ
ejpam-6904	1	32	2	2	NUM
ejpam-6904	1	33	-	-	PUNCT
ejpam-6904	1	34	path	path	NOUN
ejpam-6904	1	35	geodetic	geodetic	ADJ
ejpam-6904	1	36	vertex	vertex	NOUN
ejpam-6904	1	37	cover	cover	NOUN
ejpam-6904	1	38	of	of	ADP
ejpam-6904	1	39	graphs	graph	NOUN
ejpam-6904	1	40	aziz	aziz	PROPN
ejpam-6904	1	41	b.	b.	PROPN
ejpam-6904	1	42	tapeing1,2	tapeing1,2	PROPN
ejpam-6904	1	43	,	,	PUNCT
ejpam-6904	1	44	sergio	sergio	PROPN
ejpam-6904	1	45	r.	r.	PROPN
ejpam-6904	1	46	canoy	canoy	PROPN
ejpam-6904	1	47	,	,	PUNCT
ejpam-6904	1	48	jr.1,2,∗	jr.1,2,∗	PROPN
ejpam-6904	1	49	1	1	NUM
ejpam-6904	1	50	department	department	NOUN
ejpam-6904	1	51	of	of	ADP
ejpam-6904	1	52	mathematics	mathematic	NOUN
ejpam-6904	1	53	and	and	CCONJ
ejpam-6904	1	54	statistics	statistic	NOUN
ejpam-6904	1	55	,	,	PUNCT
ejpam-6904	1	56	college	college	NOUN
ejpam-6904	1	57	of	of	ADP
ejpam-6904	1	58	science	science	NOUN
ejpam-6904	1	59	and	and	CCONJ
ejpam-6904	1	60	mathematics	mathematic	NOUN
ejpam-6904	1	61	,	,	PUNCT
ejpam-6904	1	62	msu	msu	PROPN
ejpam-6904	1	63	-	-	PUNCT
ejpam-6904	1	64	iligan	iligan	PROPN
ejpam-6904	1	65	institute	institute	PROPN
ejpam-6904	1	66	of	of	ADP
ejpam-6904	1	67	technology	technology	PROPN
ejpam-6904	1	68	,	,	PUNCT
ejpam-6904	1	69	iligan	iligan	PROPN
ejpam-6904	1	70	city	city	PROPN
ejpam-6904	1	71	,	,	PUNCT
ejpam-6904	1	72	philippines	philippine	NOUN
ejpam-6904	1	73	2	2	NUM
ejpam-6904	1	74	center	center	NOUN
ejpam-6904	1	75	of	of	ADP
ejpam-6904	1	76	mathematical	mathematical	ADJ
ejpam-6904	1	77	and	and	CCONJ
ejpam-6904	1	78	theoretical	theoretical	ADJ
ejpam-6904	1	79	physical	physical	ADJ
ejpam-6904	1	80	sciences	science	NOUN
ejpam-6904	1	81	prism	prism	NOUN
ejpam-6904	1	82	,	,	PUNCT
ejpam-6904	1	83	msu	msu	PROPN
ejpam-6904	1	84	-	-	PUNCT
ejpam-6904	1	85	iligan	iligan	PROPN
ejpam-6904	1	86	institute	institute	PROPN
ejpam-6904	1	87	of	of	ADP
ejpam-6904	1	88	technology	technology	PROPN
ejpam-6904	1	89	,	,	PUNCT
ejpam-6904	1	90	iligan	iligan	PROPN
ejpam-6904	1	91	city	city	PROPN
ejpam-6904	1	92	,	,	PUNCT
ejpam-6904	1	93	philippines	philippine	NOUN
ejpam-6904	1	94	abstract	abstract	ADJ
ejpam-6904	1	95	.	.	PUNCT
ejpam-6904	2	1	a	a	DET
ejpam-6904	2	2	vertex	vertex	NOUN
ejpam-6904	2	3	cover	cover	NOUN
ejpam-6904	2	4	s	s	NOUN
ejpam-6904	2	5	⊆	⊆	NUM
ejpam-6904	2	6	v	v	NOUN
ejpam-6904	2	7	(	(	PUNCT
ejpam-6904	2	8	g	g	NOUN
ejpam-6904	2	9	)	)	PUNCT
ejpam-6904	2	10	is	be	AUX
ejpam-6904	2	11	called	call	VERB
ejpam-6904	2	12	a	a	DET
ejpam-6904	2	13	2	2	NUM
ejpam-6904	2	14	-	-	PUNCT
ejpam-6904	2	15	path	path	NOUN
ejpam-6904	2	16	geodetic	geodetic	ADJ
ejpam-6904	2	17	vertex	vertex	NOUN
ejpam-6904	2	18	cover	cover	NOUN
ejpam-6904	2	19	of	of	ADP
ejpam-6904	2	20	g	g	PROPN
ejpam-6904	2	21	if	if	SCONJ
ejpam-6904	2	22	for	for	ADP
ejpam-6904	2	23	every	every	PRON
ejpam-6904	2	24	v	v	NUM
ejpam-6904	2	25	∈	∈	NOUN
ejpam-6904	2	26	v	v	NOUN
ejpam-6904	2	27	(	(	PUNCT
ejpam-6904	2	28	g	g	NOUN
ejpam-6904	2	29	)	)	PUNCT
ejpam-6904	2	30	\	\	PROPN
ejpam-6904	3	1	s	s	X
ejpam-6904	3	2	,	,	PUNCT
ejpam-6904	3	3	there	there	PRON
ejpam-6904	3	4	exist	exist	VERB
ejpam-6904	3	5	vertices	vertex	NOUN
ejpam-6904	3	6	u	u	NOUN
ejpam-6904	3	7	,	,	PUNCT
ejpam-6904	3	8	w	w	PROPN
ejpam-6904	3	9	∈	∈	PROPN
ejpam-6904	3	10	s	s	VERB
ejpam-6904	3	11	such	such	ADJ
ejpam-6904	3	12	that	that	DET
ejpam-6904	3	13	dg(u	dg(u	ADJ
ejpam-6904	3	14	,	,	PUNCT
ejpam-6904	3	15	w	w	NOUN
ejpam-6904	3	16	)	)	PUNCT
ejpam-6904	3	17	=	=	SYM
ejpam-6904	3	18	2	2	NUM
ejpam-6904	3	19	and	and	CCONJ
ejpam-6904	3	20	v	v	ADP
ejpam-6904	3	21	∈	∈	PROPN
ejpam-6904	3	22	ig(u	ig(u	NOUN
ejpam-6904	3	23	,	,	PUNCT
ejpam-6904	3	24	w	w	NOUN
ejpam-6904	3	25	)	)	PUNCT
ejpam-6904	3	26	.	.	PUNCT
ejpam-6904	4	1	the	the	DET
ejpam-6904	4	2	2	2	NUM
ejpam-6904	4	3	-	-	PUNCT
ejpam-6904	4	4	path	path	NOUN
ejpam-6904	4	5	geodetic	geodetic	ADJ
ejpam-6904	4	6	vertex	vertex	NOUN
ejpam-6904	4	7	covering	cover	VERB
ejpam-6904	4	8	number	number	NOUN
ejpam-6904	4	9	of	of	ADP
ejpam-6904	4	10	g	g	NOUN
ejpam-6904	4	11	,	,	PUNCT
ejpam-6904	4	12	denoted	denote	VERB
ejpam-6904	4	13	β2pg(g	β2pg(g	PROPN
ejpam-6904	4	14	)	)	PUNCT
ejpam-6904	4	15	,	,	PUNCT
ejpam-6904	4	16	is	be	AUX
ejpam-6904	4	17	the	the	DET
ejpam-6904	4	18	minimum	minimum	ADJ
ejpam-6904	4	19	cardinality	cardinality	NOUN
ejpam-6904	4	20	of	of	ADP
ejpam-6904	4	21	a	a	DET
ejpam-6904	4	22	2	2	NUM
ejpam-6904	4	23	-	-	PUNCT
ejpam-6904	4	24	path	path	NOUN
ejpam-6904	4	25	geodetic	geodetic	ADJ
ejpam-6904	4	26	vertex	vertex	NOUN
ejpam-6904	4	27	covering	covering	NOUN
ejpam-6904	4	28	of	of	ADP
ejpam-6904	4	29	g.	g.	PROPN
ejpam-6904	4	30	in	in	ADP
ejpam-6904	4	31	this	this	DET
ejpam-6904	4	32	paper	paper	NOUN
ejpam-6904	4	33	,	,	PUNCT
ejpam-6904	4	34	we	we	PRON
ejpam-6904	4	35	show	show	VERB
ejpam-6904	4	36	that	that	SCONJ
ejpam-6904	4	37	given	give	VERB
ejpam-6904	4	38	two	two	NUM
ejpam-6904	4	39	positive	positive	ADJ
ejpam-6904	4	40	integers	integer	NOUN
ejpam-6904	4	41	a	a	DET
ejpam-6904	4	42	and	and	CCONJ
ejpam-6904	4	43	b	b	NOUN
ejpam-6904	4	44	such	such	ADJ
ejpam-6904	4	45	that	that	SCONJ
ejpam-6904	4	46	2	2	NUM
ejpam-6904	4	47	≤	≤	NOUN
ejpam-6904	4	48	a	a	DET
ejpam-6904	4	49	≤	≤	NUM
ejpam-6904	4	50	b	b	NOUN
ejpam-6904	4	51	,	,	PUNCT
ejpam-6904	4	52	there	there	PRON
ejpam-6904	4	53	exists	exist	VERB
ejpam-6904	4	54	a	a	DET
ejpam-6904	4	55	connected	connected	ADJ
ejpam-6904	4	56	graph	graph	NOUN
ejpam-6904	4	57	g	g	ADP
ejpam-6904	4	58	such	such	ADJ
ejpam-6904	4	59	that	that	PRON
ejpam-6904	4	60	β(g	β(g	PROPN
ejpam-6904	4	61	)	)	PUNCT
ejpam-6904	4	62	=	=	SYM
ejpam-6904	4	63	a	a	PRON
ejpam-6904	4	64	and	and	CCONJ
ejpam-6904	4	65	β2pg	β2pg	PUNCT
ejpam-6904	4	66	=	=	SYM
ejpam-6904	4	67	b.	b.	PROPN
ejpam-6904	4	68	as	as	ADP
ejpam-6904	4	69	a	a	DET
ejpam-6904	4	70	consequence	consequence	NOUN
ejpam-6904	4	71	,	,	PUNCT
ejpam-6904	4	72	the	the	DET
ejpam-6904	4	73	difference	difference	NOUN
ejpam-6904	4	74	between	between	ADP
ejpam-6904	4	75	the	the	DET
ejpam-6904	4	76	2	2	NUM
ejpam-6904	4	77	-	-	PUNCT
ejpam-6904	4	78	path	path	NOUN
ejpam-6904	4	79	geodetic	geodetic	ADJ
ejpam-6904	4	80	vertex	vertex	NOUN
ejpam-6904	4	81	covering	cover	VERB
ejpam-6904	4	82	number	number	NOUN
ejpam-6904	4	83	and	and	CCONJ
ejpam-6904	4	84	the	the	DET
ejpam-6904	4	85	classical	classical	ADJ
ejpam-6904	4	86	vertex	vertex	NOUN
ejpam-6904	4	87	covering	cover	VERB
ejpam-6904	4	88	number	number	NOUN
ejpam-6904	4	89	of	of	ADP
ejpam-6904	4	90	a	a	DET
ejpam-6904	4	91	graph	graph	NOUN
ejpam-6904	4	92	can	can	AUX
ejpam-6904	4	93	be	be	AUX
ejpam-6904	4	94	made	make	VERB
ejpam-6904	4	95	arbitrarily	arbitrarily	ADV
ejpam-6904	4	96	large	large	ADJ
ejpam-6904	4	97	.	.	PUNCT
ejpam-6904	5	1	we	we	PRON
ejpam-6904	5	2	characterize	characterize	VERB
ejpam-6904	5	3	graphs	graph	NOUN
ejpam-6904	5	4	with	with	ADP
ejpam-6904	5	5	small	small	ADJ
ejpam-6904	5	6	and	and	CCONJ
ejpam-6904	5	7	large	large	ADJ
ejpam-6904	5	8	values	value	NOUN
ejpam-6904	5	9	of	of	ADP
ejpam-6904	5	10	the	the	DET
ejpam-6904	5	11	2	2	NUM
ejpam-6904	5	12	-	-	PUNCT
ejpam-6904	5	13	path	path	NOUN
ejpam-6904	5	14	geodetic	geodetic	ADJ
ejpam-6904	5	15	vertex	vertex	NOUN
ejpam-6904	5	16	covering	cover	VERB
ejpam-6904	5	17	number	number	NOUN
ejpam-6904	5	18	.	.	PUNCT
ejpam-6904	6	1	furthermore	furthermore	ADV
ejpam-6904	6	2	,	,	PUNCT
ejpam-6904	6	3	we	we	PRON
ejpam-6904	6	4	provide	provide	VERB
ejpam-6904	6	5	necessary	necessary	ADJ
ejpam-6904	6	6	and	and	CCONJ
ejpam-6904	6	7	sufficient	sufficient	ADJ
ejpam-6904	6	8	conditions	condition	NOUN
ejpam-6904	6	9	for	for	ADP
ejpam-6904	6	10	the	the	DET
ejpam-6904	6	11	2	2	NUM
ejpam-6904	6	12	-	-	PUNCT
ejpam-6904	6	13	path	path	NOUN
ejpam-6904	6	14	geodetic	geodetic	ADJ
ejpam-6904	6	15	vertex	vertex	NOUN
ejpam-6904	6	16	covers	cover	NOUN
ejpam-6904	6	17	in	in	ADP
ejpam-6904	6	18	certain	certain	ADJ
ejpam-6904	6	19	graph	graph	NOUN
ejpam-6904	6	20	operations	operation	NOUN
ejpam-6904	6	21	.	.	PUNCT
ejpam-6904	7	1	the	the	DET
ejpam-6904	7	2	exact	exact	ADJ
ejpam-6904	7	3	values	value	NOUN
ejpam-6904	7	4	of	of	ADP
ejpam-6904	7	5	2	2	NUM
ejpam-6904	7	6	-	-	PUNCT
ejpam-6904	7	7	path	path	NOUN
ejpam-6904	7	8	geodetic	geodetic	ADJ
ejpam-6904	7	9	vertex	vertex	NOUN
ejpam-6904	7	10	cover	cover	NOUN
ejpam-6904	7	11	numbers	number	NOUN
ejpam-6904	7	12	of	of	ADP
ejpam-6904	7	13	these	these	DET
ejpam-6904	7	14	graphs	graph	NOUN
ejpam-6904	7	15	are	be	AUX
ejpam-6904	7	16	also	also	ADV
ejpam-6904	7	17	determined	determine	VERB
ejpam-6904	7	18	.	.	PUNCT
ejpam-6904	8	1	2020	2020	NUM
ejpam-6904	8	2	mathematics	mathematic	NOUN
ejpam-6904	8	3	subject	subject	NOUN
ejpam-6904	8	4	classifications	classification	NOUN
ejpam-6904	8	5	:	:	PUNCT
ejpam-6904	8	6	05c69	05c69	X
ejpam-6904	8	7	key	key	ADJ
ejpam-6904	8	8	words	word	NOUN
ejpam-6904	8	9	and	and	CCONJ
ejpam-6904	8	10	phrases	phrase	NOUN
ejpam-6904	8	11	:	:	PUNCT
ejpam-6904	8	12	geodetic	geodetic	ADJ
ejpam-6904	8	13	,	,	PUNCT
ejpam-6904	8	14	vertex	vertex	NOUN
ejpam-6904	8	15	cover	cover	NOUN
ejpam-6904	8	16	,	,	PUNCT
ejpam-6904	8	17	vertex	vertex	NOUN
ejpam-6904	8	18	cover	cover	NOUN
ejpam-6904	8	19	number	number	NOUN
ejpam-6904	8	20	1	1	NUM
ejpam-6904	8	21	.	.	PUNCT
ejpam-6904	8	22	introduction	introduction	NOUN
ejpam-6904	8	23	the	the	DET
ejpam-6904	8	24	concept	concept	NOUN
ejpam-6904	8	25	of	of	ADP
ejpam-6904	8	26	vertex	vertex	NOUN
ejpam-6904	8	27	covering	cover	VERB
ejpam-6904	8	28	in	in	ADP
ejpam-6904	8	29	graphs	graph	NOUN
ejpam-6904	8	30	has	have	AUX
ejpam-6904	8	31	been	be	AUX
ejpam-6904	8	32	extensively	extensively	ADV
ejpam-6904	8	33	studied	study	VERB
ejpam-6904	8	34	(	(	PUNCT
ejpam-6904	8	35	see	see	VERB
ejpam-6904	8	36	,	,	PUNCT
ejpam-6904	8	37	for	for	ADP
ejpam-6904	8	38	instance	instance	NOUN
ejpam-6904	8	39	,	,	PUNCT
ejpam-6904	8	40	[	[	X
ejpam-6904	8	41	1	1	NUM
ejpam-6904	8	42	]	]	PUNCT
ejpam-6904	8	43	,	,	PUNCT
ejpam-6904	8	44	[	[	X
ejpam-6904	8	45	2	2	NUM
ejpam-6904	8	46	]	]	PUNCT
ejpam-6904	8	47	,	,	PUNCT
ejpam-6904	8	48	[	[	X
ejpam-6904	8	49	3	3	NUM
ejpam-6904	8	50	]	]	PUNCT
ejpam-6904	8	51	,	,	PUNCT
ejpam-6904	8	52	[	[	X
ejpam-6904	8	53	4	4	NUM
ejpam-6904	8	54	]	]	NUM
ejpam-6904	8	55	)	)	PUNCT
ejpam-6904	8	56	.	.	PUNCT
ejpam-6904	9	1	as	as	SCONJ
ejpam-6904	9	2	noted	note	VERB
ejpam-6904	9	3	by	by	ADP
ejpam-6904	9	4	angel	angel	NOUN
ejpam-6904	9	5	and	and	CCONJ
ejpam-6904	9	6	amutha	amutha	NOUN
ejpam-6904	9	7	[	[	X
ejpam-6904	9	8	5	5	NUM
ejpam-6904	9	9	]	]	PUNCT
ejpam-6904	9	10	,	,	PUNCT
ejpam-6904	9	11	this	this	DET
ejpam-6904	9	12	parameter	parameter	NOUN
ejpam-6904	9	13	has	have	VERB
ejpam-6904	9	14	practical	practical	ADJ
ejpam-6904	9	15	applications	application	NOUN
ejpam-6904	9	16	in	in	ADP
ejpam-6904	9	17	network	network	NOUN
ejpam-6904	9	18	security	security	NOUN
ejpam-6904	9	19	.	.	PUNCT
ejpam-6904	10	1	in	in	ADP
ejpam-6904	10	2	particular	particular	ADJ
ejpam-6904	10	3	,	,	PUNCT
ejpam-6904	10	4	their	their	PRON
ejpam-6904	10	5	study	study	NOUN
ejpam-6904	10	6	highlights	highlight	NOUN
ejpam-6904	10	7	that	that	SCONJ
ejpam-6904	10	8	in	in	ADP
ejpam-6904	10	9	computer	computer	NOUN
ejpam-6904	10	10	networks	network	NOUN
ejpam-6904	10	11	,	,	PUNCT
ejpam-6904	10	12	minimizing	minimize	VERB
ejpam-6904	10	13	the	the	DET
ejpam-6904	10	14	vertex	vertex	NOUN
ejpam-6904	10	15	cover	cover	NOUN
ejpam-6904	10	16	number	number	NOUN
ejpam-6904	10	17	provides	provide	VERB
ejpam-6904	10	18	an	an	DET
ejpam-6904	10	19	optimal	optimal	ADJ
ejpam-6904	10	20	strategy	strategy	NOUN
ejpam-6904	10	21	for	for	ADP
ejpam-6904	10	22	network	network	NOUN
ejpam-6904	10	23	defense	defense	NOUN
ejpam-6904	10	24	.	.	PUNCT
ejpam-6904	11	1	toregas	toregas	PROPN
ejpam-6904	11	2	et	et	PROPN
ejpam-6904	11	3	al	al	PROPN
ejpam-6904	11	4	.	.	PUNCT
ejpam-6904	12	1	[	[	X
ejpam-6904	12	2	6	6	NUM
ejpam-6904	12	3	]	]	PUNCT
ejpam-6904	12	4	further	far	ADV
ejpam-6904	12	5	demonstrated	demonstrate	VERB
ejpam-6904	12	6	that	that	SCONJ
ejpam-6904	12	7	this	this	DET
ejpam-6904	12	8	concept	concept	NOUN
ejpam-6904	12	9	is	be	AUX
ejpam-6904	12	10	utilized	utilize	VERB
ejpam-6904	12	11	in	in	ADP
ejpam-6904	12	12	determining	determine	VERB
ejpam-6904	12	13	the	the	DET
ejpam-6904	12	14	optimal	optimal	ADJ
ejpam-6904	12	15	placement	placement	NOUN
ejpam-6904	12	16	of	of	ADP
ejpam-6904	12	17	emergency	emergency	NOUN
ejpam-6904	12	18	facilities	facility	NOUN
ejpam-6904	12	19	within	within	ADP
ejpam-6904	12	20	telecommunication	telecommunication	NOUN
ejpam-6904	12	21	networks	network	NOUN
ejpam-6904	12	22	.	.	PUNCT
ejpam-6904	13	1	despite	despite	SCONJ
ejpam-6904	13	2	its	its	PRON
ejpam-6904	13	3	significance	significance	NOUN
ejpam-6904	13	4	,	,	PUNCT
ejpam-6904	13	5	the	the	DET
ejpam-6904	13	6	vertex	vertex	NOUN
ejpam-6904	13	7	cover	cover	NOUN
ejpam-6904	13	8	problem	problem	NOUN
ejpam-6904	13	9	is	be	AUX
ejpam-6904	13	10	classified	classify	VERB
ejpam-6904	13	11	as	as	ADP
ejpam-6904	13	12	an	an	DET
ejpam-6904	13	13	np	np	ADV
ejpam-6904	13	14	-	-	PUNCT
ejpam-6904	13	15	hard	hard	ADJ
ejpam-6904	13	16	optimization	optimization	NOUN
ejpam-6904	13	17	problem	problem	NOUN
ejpam-6904	13	18	.	.	PUNCT
ejpam-6904	14	1	specifically	specifically	ADV
ejpam-6904	14	2	,	,	PUNCT
ejpam-6904	14	3	karp	karp	PROPN
ejpam-6904	14	4	[	[	X
ejpam-6904	14	5	7	7	NUM
ejpam-6904	14	6	]	]	PUNCT
ejpam-6904	14	7	established	establish	VERB
ejpam-6904	14	8	its	its	PRON
ejpam-6904	14	9	np	np	NOUN
ejpam-6904	14	10	-	-	NOUN
ejpam-6904	14	11	completeness	completeness	NOUN
ejpam-6904	14	12	by	by	ADP
ejpam-6904	14	13	leveraging	leverage	VERB
ejpam-6904	14	14	the	the	DET
ejpam-6904	14	15	known	know	VERB
ejpam-6904	14	16	result	result	NOUN
ejpam-6904	14	17	that	that	SCONJ
ejpam-6904	14	18	the	the	DET
ejpam-6904	14	19	clique	clique	NOUN
ejpam-6904	14	20	problem	problem	NOUN
ejpam-6904	14	21	is	be	AUX
ejpam-6904	14	22	np	np	NOUN
ejpam-6904	14	23	-	-	PUNCT
ejpam-6904	14	24	complete	complete	ADJ
ejpam-6904	14	25	.	.	PUNCT
ejpam-6904	15	1	for	for	ADP
ejpam-6904	15	2	cubic	cubic	ADJ
ejpam-6904	15	3	and	and	CCONJ
ejpam-6904	15	4	planar	planar	ADJ
ejpam-6904	15	5	graphs	graph	NOUN
ejpam-6904	15	6	,	,	PUNCT
ejpam-6904	15	7	the	the	DET
ejpam-6904	15	8	∗corresponding	∗corresponde	VERB
ejpam-6904	15	9	author	author	NOUN
ejpam-6904	15	10	.	.	PUNCT
ejpam-6904	16	1	doi	doi	NOUN
ejpam-6904	16	2	:	:	PUNCT
ejpam-6904	16	3	https://doi.org/10.29020/nybg.ejpam.v18i4.6904	https://doi.org/10.29020/nybg.ejpam.v18i4.6904	NUM
ejpam-6904	16	4	email	email	NOUN
ejpam-6904	16	5	addresses	address	VERB
ejpam-6904	16	6	:	:	PUNCT
ejpam-6904	17	1	aziz.tapeing@g.msuiit.edu.ph	aziz.tapeing@g.msuiit.edu.ph	PROPN
ejpam-6904	17	2	(	(	PUNCT
ejpam-6904	17	3	a.	a.	PROPN
ejpam-6904	17	4	b.	b.	PROPN
ejpam-6904	17	5	tapeing	tapeing	PROPN
ejpam-6904	17	6	)	)	PUNCT
ejpam-6904	17	7	,	,	PUNCT
ejpam-6904	17	8	sergio.canoy@g.msuiit.edu.ph	sergio.canoy@g.msuiit.edu.ph	PROPN
ejpam-6904	17	9	(	(	PUNCT
ejpam-6904	17	10	s.	s.	PROPN
ejpam-6904	17	11	r.	r.	PROPN
ejpam-6904	17	12	canoy	canoy	PROPN
ejpam-6904	17	13	jr	jr	PROPN
ejpam-6904	17	14	.	.	PUNCT
ejpam-6904	17	15	)	)	PUNCT
ejpam-6904	17	16	https://www.ejpam.com	https://www.ejpam.com	X
ejpam-6904	18	1	1	1	NUM
ejpam-6904	18	2	copyright	copyright	NOUN
ejpam-6904	18	3	:	:	PUNCT
ejpam-6904	18	4	©	©	PROPN
ejpam-6904	18	5	2025	2025	NUM
ejpam-6904	18	6	the	the	DET
ejpam-6904	18	7	author(s	author(s	NOUN
ejpam-6904	18	8	)	)	PUNCT
ejpam-6904	18	9	.	.	PUNCT
ejpam-6904	19	1	(	(	PUNCT
ejpam-6904	19	2	cc	cc	NOUN
ejpam-6904	19	3	by	by	ADP
ejpam-6904	19	4	-	-	PUNCT
ejpam-6904	19	5	nc	nc	PROPN
ejpam-6904	19	6	4.0	4.0	NUM
ejpam-6904	19	7	)	)	PUNCT
ejpam-6904	19	8	a.	a.	PROPN
ejpam-6904	19	9	b.	b.	PROPN
ejpam-6904	19	10	tapeing	tapeing	PROPN
ejpam-6904	19	11	,	,	PUNCT
ejpam-6904	19	12	s.	s.	PROPN
ejpam-6904	19	13	r.	r.	PROPN
ejpam-6904	19	14	canoy	canoy	PROPN
ejpam-6904	19	15	/	/	SYM
ejpam-6904	19	16	eur	eur	PROPN
ejpam-6904	19	17	.	.	PUNCT
ejpam-6904	20	1	j.	j.	PROPN
ejpam-6904	20	2	pure	pure	PROPN
ejpam-6904	20	3	appl	appl	PROPN
ejpam-6904	20	4	.	.	PROPN
ejpam-6904	20	5	math	math	PROPN
ejpam-6904	20	6	,	,	PUNCT
ejpam-6904	20	7	18	18	NUM
ejpam-6904	20	8	(	(	PUNCT
ejpam-6904	20	9	4	4	NUM
ejpam-6904	20	10	)	)	PUNCT
ejpam-6904	20	11	(	(	PUNCT
ejpam-6904	20	12	2025	2025	NUM
ejpam-6904	20	13	)	)	PUNCT
ejpam-6904	20	14	,	,	PUNCT
ejpam-6904	20	15	6904	6904	NUM
ejpam-6904	20	16	2	2	NUM
ejpam-6904	20	17	of	of	ADP
ejpam-6904	20	18	14	14	NUM
ejpam-6904	20	19	vertex	vertex	NOUN
ejpam-6904	20	20	covering	covering	NOUN
ejpam-6904	20	21	problem	problem	NOUN
ejpam-6904	20	22	is	be	AUX
ejpam-6904	20	23	still	still	ADV
ejpam-6904	20	24	np	np	NOUN
ejpam-6904	20	25	-	-	PUNCT
ejpam-6904	20	26	complete	complete	ADJ
ejpam-6904	20	27	.	.	PUNCT
ejpam-6904	21	1	for	for	ADP
ejpam-6904	21	2	complete	complete	ADJ
ejpam-6904	21	3	details	detail	NOUN
ejpam-6904	21	4	,	,	PUNCT
ejpam-6904	21	5	one	one	PRON
ejpam-6904	21	6	may	may	AUX
ejpam-6904	21	7	refer	refer	VERB
ejpam-6904	21	8	to	to	ADP
ejpam-6904	21	9	[	[	X
ejpam-6904	21	10	8	8	NUM
ejpam-6904	21	11	]	]	PUNCT
ejpam-6904	21	12	and	and	CCONJ
ejpam-6904	21	13	[	[	X
ejpam-6904	21	14	9	9	NUM
ejpam-6904	21	15	]	]	PUNCT
ejpam-6904	21	16	.	.	PUNCT
ejpam-6904	22	1	the	the	DET
ejpam-6904	22	2	problem	problem	NOUN
ejpam-6904	22	3	of	of	ADP
ejpam-6904	22	4	determining	determine	VERB
ejpam-6904	22	5	bounds	bound	NOUN
ejpam-6904	22	6	and	and	CCONJ
ejpam-6904	22	7	exact	exact	ADJ
ejpam-6904	22	8	values	value	NOUN
ejpam-6904	22	9	for	for	ADP
ejpam-6904	22	10	the	the	DET
ejpam-6904	22	11	vertex	vertex	NOUN
ejpam-6904	22	12	cover	cover	NOUN
ejpam-6904	22	13	number	number	NOUN
ejpam-6904	22	14	of	of	ADP
ejpam-6904	22	15	specific	specific	ADJ
ejpam-6904	22	16	classes	class	NOUN
ejpam-6904	22	17	of	of	ADP
ejpam-6904	22	18	graphs	graph	NOUN
ejpam-6904	22	19	has	have	AUX
ejpam-6904	22	20	been	be	AUX
ejpam-6904	22	21	extensively	extensively	ADV
ejpam-6904	22	22	studied	study	VERB
ejpam-6904	22	23	(	(	PUNCT
ejpam-6904	22	24	see	see	VERB
ejpam-6904	22	25	[	[	X
ejpam-6904	22	26	10	10	NUM
ejpam-6904	22	27	]	]	PUNCT
ejpam-6904	22	28	,	,	PUNCT
ejpam-6904	22	29	[	[	X
ejpam-6904	22	30	11	11	NUM
ejpam-6904	22	31	]	]	NUM
ejpam-6904	22	32	)	)	PUNCT
ejpam-6904	22	33	.	.	PUNCT
ejpam-6904	23	1	in	in	ADP
ejpam-6904	23	2	recent	recent	ADJ
ejpam-6904	23	3	years	year	NOUN
ejpam-6904	23	4	,	,	PUNCT
ejpam-6904	23	5	several	several	ADJ
ejpam-6904	23	6	variations	variation	NOUN
ejpam-6904	23	7	of	of	ADP
ejpam-6904	23	8	the	the	DET
ejpam-6904	23	9	vertex	vertex	NOUN
ejpam-6904	23	10	cover	cover	NOUN
ejpam-6904	23	11	concepts	concept	NOUN
ejpam-6904	23	12	have	have	AUX
ejpam-6904	23	13	been	be	AUX
ejpam-6904	23	14	introduced	introduce	VERB
ejpam-6904	23	15	and	and	CCONJ
ejpam-6904	23	16	investigated	investigate	VERB
ejpam-6904	23	17	(	(	PUNCT
ejpam-6904	23	18	see	see	VERB
ejpam-6904	23	19	[	[	X
ejpam-6904	23	20	5	5	NUM
ejpam-6904	23	21	]	]	PUNCT
ejpam-6904	23	22	,	,	PUNCT
ejpam-6904	23	23	[	[	X
ejpam-6904	23	24	12	12	NUM
ejpam-6904	23	25	]	]	PUNCT
ejpam-6904	23	26	,	,	PUNCT
ejpam-6904	23	27	[	[	X
ejpam-6904	23	28	13	13	NUM
ejpam-6904	23	29	]	]	PUNCT
ejpam-6904	23	30	,	,	PUNCT
ejpam-6904	23	31	[	[	X
ejpam-6904	23	32	14	14	NUM
ejpam-6904	23	33	]	]	PUNCT
ejpam-6904	23	34	,	,	PUNCT
ejpam-6904	23	35	[	[	X
ejpam-6904	23	36	15	15	NUM
ejpam-6904	23	37	]	]	PUNCT
ejpam-6904	23	38	,	,	PUNCT
ejpam-6904	23	39	[	[	X
ejpam-6904	23	40	1	1	NUM
ejpam-6904	23	41	]	]	PUNCT
ejpam-6904	23	42	,	,	PUNCT
ejpam-6904	23	43	[	[	X
ejpam-6904	23	44	16	16	NUM
ejpam-6904	23	45	]	]	PUNCT
ejpam-6904	23	46	,	,	PUNCT
ejpam-6904	23	47	[	[	X
ejpam-6904	23	48	17	17	NUM
ejpam-6904	23	49	]	]	PUNCT
ejpam-6904	23	50	,	,	PUNCT
ejpam-6904	23	51	and	and	CCONJ
ejpam-6904	23	52	[	[	X
ejpam-6904	23	53	18	18	NUM
ejpam-6904	23	54	]	]	NUM
ejpam-6904	23	55	)	)	PUNCT
ejpam-6904	23	56	.	.	PUNCT
ejpam-6904	24	1	motivated	motivate	VERB
ejpam-6904	24	2	by	by	ADP
ejpam-6904	24	3	the	the	DET
ejpam-6904	24	4	aforementioned	aforementioned	ADJ
ejpam-6904	24	5	studies	study	NOUN
ejpam-6904	24	6	,	,	PUNCT
ejpam-6904	24	7	we	we	PRON
ejpam-6904	24	8	introduce	introduce	VERB
ejpam-6904	24	9	and	and	CCONJ
ejpam-6904	24	10	initiate	initiate	VERB
ejpam-6904	24	11	the	the	DET
ejpam-6904	24	12	study	study	NOUN
ejpam-6904	24	13	2	2	NUM
ejpam-6904	24	14	-	-	PUNCT
ejpam-6904	24	15	path	path	NOUN
ejpam-6904	24	16	geodetic	geodetic	ADJ
ejpam-6904	24	17	vertex	vertex	NOUN
ejpam-6904	24	18	cover	cover	NOUN
ejpam-6904	24	19	of	of	ADP
ejpam-6904	24	20	a	a	DET
ejpam-6904	24	21	graph	graph	NOUN
ejpam-6904	24	22	.	.	PUNCT
ejpam-6904	25	1	this	this	DET
ejpam-6904	25	2	new	new	ADJ
ejpam-6904	25	3	parameter	parameter	NOUN
ejpam-6904	25	4	naturally	naturally	ADV
ejpam-6904	25	5	extends	extend	VERB
ejpam-6904	25	6	two	two	NUM
ejpam-6904	25	7	existing	exist	VERB
ejpam-6904	25	8	concepts	concept	NOUN
ejpam-6904	25	9	:	:	PUNCT
ejpam-6904	25	10	2	2	NUM
ejpam-6904	25	11	-	-	PUNCT
ejpam-6904	25	12	path	path	NOUN
ejpam-6904	25	13	geodetic	geodetic	ADJ
ejpam-6904	25	14	set	set	NOUN
ejpam-6904	25	15	and	and	CCONJ
ejpam-6904	25	16	vertex	vertex	NOUN
ejpam-6904	25	17	cover	cover	NOUN
ejpam-6904	25	18	of	of	ADP
ejpam-6904	25	19	a	a	DET
ejpam-6904	25	20	graph	graph	NOUN
ejpam-6904	25	21	.	.	PUNCT
ejpam-6904	26	1	for	for	ADP
ejpam-6904	26	2	some	some	DET
ejpam-6904	26	3	related	relate	VERB
ejpam-6904	26	4	works	work	NOUN
ejpam-6904	26	5	on	on	ADP
ejpam-6904	26	6	the	the	DET
ejpam-6904	26	7	concept	concept	NOUN
ejpam-6904	26	8	of	of	ADP
ejpam-6904	26	9	geodetic	geodetic	ADJ
ejpam-6904	26	10	and	and	CCONJ
ejpam-6904	26	11	2	2	NUM
ejpam-6904	26	12	-	-	PUNCT
ejpam-6904	26	13	path	path	NOUN
ejpam-6904	26	14	closure	closure	NOUN
ejpam-6904	26	15	absorbing	absorb	VERB
ejpam-6904	26	16	set	set	NOUN
ejpam-6904	26	17	,	,	PUNCT
ejpam-6904	26	18	readers	reader	NOUN
ejpam-6904	26	19	may	may	AUX
ejpam-6904	26	20	see	see	VERB
ejpam-6904	26	21	[	[	X
ejpam-6904	26	22	19	19	NUM
ejpam-6904	26	23	]	]	PUNCT
ejpam-6904	26	24	,	,	PUNCT
ejpam-6904	26	25	[	[	X
ejpam-6904	26	26	20	20	NUM
ejpam-6904	26	27	]	]	PUNCT
ejpam-6904	26	28	,	,	PUNCT
ejpam-6904	26	29	[	[	X
ejpam-6904	26	30	21	21	NUM
ejpam-6904	26	31	]	]	PUNCT
ejpam-6904	26	32	,	,	PUNCT
ejpam-6904	26	33	and	and	CCONJ
ejpam-6904	26	34	[	[	X
ejpam-6904	26	35	22	22	NUM
ejpam-6904	26	36	]	]	PUNCT
ejpam-6904	26	37	.	.	PUNCT
ejpam-6904	27	1	2	2	X
ejpam-6904	27	2	.	.	NOUN
ejpam-6904	27	3	terminologies	terminology	NOUN
ejpam-6904	27	4	and	and	CCONJ
ejpam-6904	27	5	notations	notation	NOUN
ejpam-6904	27	6	let	let	VERB
ejpam-6904	27	7	g	g	NOUN
ejpam-6904	27	8	=	=	SYM
ejpam-6904	27	9	(	(	PUNCT
ejpam-6904	27	10	v	v	NOUN
ejpam-6904	27	11	(	(	PUNCT
ejpam-6904	27	12	g	g	NOUN
ejpam-6904	27	13	)	)	PUNCT
ejpam-6904	27	14	,	,	PUNCT
ejpam-6904	27	15	e(g	e(g	PROPN
ejpam-6904	27	16	)	)	PUNCT
ejpam-6904	27	17	)	)	PUNCT
ejpam-6904	27	18	be	be	AUX
ejpam-6904	27	19	a	a	DET
ejpam-6904	27	20	simple	simple	ADJ
ejpam-6904	27	21	undirected	undirected	ADJ
ejpam-6904	27	22	graph	graph	NOUN
ejpam-6904	27	23	.	.	PUNCT
ejpam-6904	28	1	the	the	DET
ejpam-6904	28	2	open	open	ADJ
ejpam-6904	28	3	neighborhood	neighborhood	NOUN
ejpam-6904	28	4	of	of	ADP
ejpam-6904	28	5	a	a	DET
ejpam-6904	28	6	vertex	vertex	NOUN
ejpam-6904	28	7	v	v	NOUN
ejpam-6904	28	8	of	of	ADP
ejpam-6904	28	9	g	g	PROPN
ejpam-6904	28	10	is	be	AUX
ejpam-6904	28	11	the	the	DET
ejpam-6904	28	12	set	set	NOUN
ejpam-6904	28	13	ng(v	ng(v	PUNCT
ejpam-6904	28	14	)	)	PUNCT
ejpam-6904	28	15	=	=	SYM
ejpam-6904	29	1	{	{	PUNCT
ejpam-6904	29	2	u	u	NOUN
ejpam-6904	29	3	∈	∈	PROPN
ejpam-6904	29	4	v	v	NOUN
ejpam-6904	29	5	(	(	PUNCT
ejpam-6904	29	6	g	g	NOUN
ejpam-6904	29	7	)	)	PUNCT
ejpam-6904	29	8	:	:	PUNCT
ejpam-6904	29	9	uv	uv	PROPN
ejpam-6904	29	10	∈	∈	PROPN
ejpam-6904	29	11	e(g	e(g	PROPN
ejpam-6904	29	12	)	)	PUNCT
ejpam-6904	29	13	}	}	PUNCT
ejpam-6904	29	14	(	(	PUNCT
ejpam-6904	29	15	the	the	DET
ejpam-6904	29	16	set	set	NOUN
ejpam-6904	29	17	consisting	consist	VERB
ejpam-6904	29	18	of	of	ADP
ejpam-6904	29	19	all	all	DET
ejpam-6904	29	20	the	the	DET
ejpam-6904	29	21	neighbors	neighbor	NOUN
ejpam-6904	29	22	of	of	ADP
ejpam-6904	29	23	v	v	NOUN
ejpam-6904	29	24	)	)	PUNCT
ejpam-6904	29	25	,	,	PUNCT
ejpam-6904	29	26	while	while	SCONJ
ejpam-6904	29	27	its	its	PRON
ejpam-6904	29	28	closed	closed	ADJ
ejpam-6904	29	29	neighborhood	neighborhood	NOUN
ejpam-6904	29	30	is	be	AUX
ejpam-6904	29	31	the	the	DET
ejpam-6904	29	32	set	set	NOUN
ejpam-6904	29	33	ng[v	ng[v	NOUN
ejpam-6904	29	34	]	]	X
ejpam-6904	29	35	=	=	SYM
ejpam-6904	29	36	ng(v	ng(v	X
ejpam-6904	29	37	)	)	PUNCT
ejpam-6904	29	38	∪	∪	ADP
ejpam-6904	29	39	{	{	PUNCT
ejpam-6904	29	40	v	v	NOUN
ejpam-6904	29	41	}	}	PUNCT
ejpam-6904	29	42	.	.	PUNCT
ejpam-6904	30	1	the	the	DET
ejpam-6904	30	2	open	open	ADJ
ejpam-6904	30	3	neighborhood	neighborhood	NOUN
ejpam-6904	30	4	of	of	ADP
ejpam-6904	30	5	a	a	DET
ejpam-6904	30	6	set	set	NOUN
ejpam-6904	30	7	s	s	NOUN
ejpam-6904	30	8	⊆	⊆	NUM
ejpam-6904	30	9	v	v	NOUN
ejpam-6904	30	10	(	(	PUNCT
ejpam-6904	30	11	g	g	NOUN
ejpam-6904	30	12	)	)	PUNCT
ejpam-6904	30	13	is	be	AUX
ejpam-6904	30	14	the	the	DET
ejpam-6904	30	15	set	set	NOUN
ejpam-6904	30	16	ng(s	ng(s	NOUN
ejpam-6904	30	17	)	)	PUNCT
ejpam-6904	30	18	=	=	SYM
ejpam-6904	30	19	∪v∈sng(v	∪v∈sng(v	PROPN
ejpam-6904	30	20	)	)	PUNCT
ejpam-6904	30	21	and	and	CCONJ
ejpam-6904	30	22	its	its	PRON
ejpam-6904	30	23	closed	closed	ADJ
ejpam-6904	30	24	neighborhood	neighborhood	NOUN
ejpam-6904	30	25	is	be	AUX
ejpam-6904	30	26	the	the	DET
ejpam-6904	30	27	set	set	VERB
ejpam-6904	30	28	ng[s	ng[	NOUN
ejpam-6904	30	29	]	]	PUNCT
ejpam-6904	30	30	=	=	SYM
ejpam-6904	30	31	s∪ng(s	s∪ng(s	PROPN
ejpam-6904	30	32	)	)	PUNCT
ejpam-6904	30	33	.	.	PUNCT
ejpam-6904	31	1	any	any	DET
ejpam-6904	31	2	v	v	NUM
ejpam-6904	31	3	∈	∈	PROPN
ejpam-6904	31	4	v	v	NOUN
ejpam-6904	31	5	(	(	PUNCT
ejpam-6904	31	6	g	g	NOUN
ejpam-6904	31	7	)	)	PUNCT
ejpam-6904	31	8	with	with	ADP
ejpam-6904	31	9	|ng(v)|	|ng(v)|	NOUN
ejpam-6904	31	10	=	=	SYM
ejpam-6904	31	11	0	0	NUM
ejpam-6904	31	12	is	be	AUX
ejpam-6904	31	13	called	call	VERB
ejpam-6904	31	14	an	an	DET
ejpam-6904	31	15	isolated	isolated	ADJ
ejpam-6904	31	16	vertex	vertex	NOUN
ejpam-6904	31	17	.	.	PUNCT
ejpam-6904	32	1	vertex	vertex	NOUN
ejpam-6904	32	2	v	v	NOUN
ejpam-6904	32	3	is	be	AUX
ejpam-6904	32	4	a	a	DET
ejpam-6904	32	5	leaf	leaf	NOUN
ejpam-6904	32	6	or	or	CCONJ
ejpam-6904	32	7	an	an	DET
ejpam-6904	32	8	endvertex	endvertex	NOUN
ejpam-6904	32	9	if	if	SCONJ
ejpam-6904	32	10	|ng(v)|	|ng(v)|	VERB
ejpam-6904	32	11	=	=	SYM
ejpam-6904	32	12	1	1	X
ejpam-6904	32	13	.	.	PUNCT
ejpam-6904	33	1	a	a	DET
ejpam-6904	33	2	vertex	vertex	NOUN
ejpam-6904	33	3	w	w	NOUN
ejpam-6904	33	4	of	of	ADP
ejpam-6904	33	5	g	g	PROPN
ejpam-6904	33	6	is	be	AUX
ejpam-6904	33	7	a	a	DET
ejpam-6904	33	8	support	support	NOUN
ejpam-6904	33	9	vertex	vertex	NOUN
ejpam-6904	33	10	if	if	SCONJ
ejpam-6904	33	11	wv	wv	PROPN
ejpam-6904	33	12	∈	∈	PROPN
ejpam-6904	33	13	e(g	e(g	PROPN
ejpam-6904	33	14	)	)	PUNCT
ejpam-6904	33	15	for	for	ADP
ejpam-6904	33	16	some	some	DET
ejpam-6904	33	17	leaf	leaf	NOUN
ejpam-6904	33	18	v	v	NOUN
ejpam-6904	33	19	in	in	ADP
ejpam-6904	33	20	g.	g.	PROPN
ejpam-6904	33	21	a	a	DET
ejpam-6904	33	22	vertex	vertex	NOUN
ejpam-6904	33	23	w	w	NOUN
ejpam-6904	33	24	is	be	AUX
ejpam-6904	33	25	an	an	DET
ejpam-6904	33	26	extreme	extreme	ADJ
ejpam-6904	33	27	vertex	vertex	NOUN
ejpam-6904	33	28	in	in	ADP
ejpam-6904	33	29	g	g	PROPN
ejpam-6904	33	30	if	if	SCONJ
ejpam-6904	33	31	the	the	DET
ejpam-6904	33	32	induced	induced	ADJ
ejpam-6904	33	33	subgraph	subgraph	NOUN
ejpam-6904	33	34	⟨ng(v)⟩	⟨ng(v)⟩	NOUN
ejpam-6904	33	35	of	of	ADP
ejpam-6904	33	36	ng(v	ng(v	PUNCT
ejpam-6904	33	37	)	)	PUNCT
ejpam-6904	33	38	is	be	AUX
ejpam-6904	33	39	complete	complete	ADJ
ejpam-6904	33	40	.	.	PUNCT
ejpam-6904	34	1	the	the	DET
ejpam-6904	34	2	sets	set	NOUN
ejpam-6904	34	3	i(g	i(g	ADV
ejpam-6904	34	4	)	)	PUNCT
ejpam-6904	34	5	,	,	PUNCT
ejpam-6904	34	6	l(g	l(g	PROPN
ejpam-6904	34	7	)	)	PUNCT
ejpam-6904	34	8	,	,	PUNCT
ejpam-6904	34	9	s(g	s(g	PROPN
ejpam-6904	34	10	)	)	PUNCT
ejpam-6904	34	11	and	and	CCONJ
ejpam-6904	34	12	ext(g	ext(g	PROPN
ejpam-6904	34	13	)	)	PUNCT
ejpam-6904	34	14	denote	denote	VERB
ejpam-6904	34	15	the	the	DET
ejpam-6904	34	16	sets	set	NOUN
ejpam-6904	34	17	containing	contain	VERB
ejpam-6904	34	18	of	of	ADP
ejpam-6904	34	19	all	all	DET
ejpam-6904	34	20	the	the	DET
ejpam-6904	34	21	isolated	isolated	ADJ
ejpam-6904	34	22	vertices	vertex	NOUN
ejpam-6904	34	23	,	,	PUNCT
ejpam-6904	34	24	leaves	leave	NOUN
ejpam-6904	34	25	,	,	PUNCT
ejpam-6904	34	26	support	support	NOUN
ejpam-6904	34	27	vertices	vertex	NOUN
ejpam-6904	34	28	,	,	PUNCT
ejpam-6904	34	29	and	and	CCONJ
ejpam-6904	34	30	extreme	extreme	ADJ
ejpam-6904	34	31	vertices	vertex	NOUN
ejpam-6904	34	32	in	in	ADP
ejpam-6904	34	33	g	g	NOUN
ejpam-6904	34	34	,	,	PUNCT
ejpam-6904	34	35	respectively	respectively	ADV
ejpam-6904	34	36	.	.	PUNCT
ejpam-6904	35	1	a	a	DET
ejpam-6904	35	2	subset	subset	NOUN
ejpam-6904	35	3	a	a	PRON
ejpam-6904	35	4	of	of	ADP
ejpam-6904	35	5	v	v	NOUN
ejpam-6904	35	6	(	(	PUNCT
ejpam-6904	35	7	g	g	NOUN
ejpam-6904	35	8	)	)	PUNCT
ejpam-6904	35	9	is	be	AUX
ejpam-6904	35	10	an	an	DET
ejpam-6904	35	11	independent	independent	ADJ
ejpam-6904	35	12	set	set	NOUN
ejpam-6904	35	13	if	if	SCONJ
ejpam-6904	35	14	for	for	SCONJ
ejpam-6904	35	15	every	every	DET
ejpam-6904	35	16	pair	pair	NOUN
ejpam-6904	35	17	of	of	ADP
ejpam-6904	35	18	distinct	distinct	ADJ
ejpam-6904	35	19	vertices	vertex	NOUN
ejpam-6904	35	20	in	in	ADP
ejpam-6904	35	21	g	g	PROPN
ejpam-6904	35	22	do	do	AUX
ejpam-6904	35	23	not	not	PART
ejpam-6904	35	24	form	form	VERB
ejpam-6904	35	25	an	an	DET
ejpam-6904	35	26	edge	edge	NOUN
ejpam-6904	35	27	.	.	PUNCT
ejpam-6904	36	1	the	the	DET
ejpam-6904	36	2	maximum	maximum	ADJ
ejpam-6904	36	3	cardinality	cardinality	NOUN
ejpam-6904	36	4	of	of	ADP
ejpam-6904	36	5	an	an	DET
ejpam-6904	36	6	independent	independent	ADJ
ejpam-6904	36	7	set	set	NOUN
ejpam-6904	36	8	in	in	ADP
ejpam-6904	36	9	g	g	NOUN
ejpam-6904	36	10	,	,	PUNCT
ejpam-6904	36	11	denoted	denote	VERB
ejpam-6904	36	12	by	by	ADP
ejpam-6904	36	13	α(g	α(g	NOUN
ejpam-6904	36	14	)	)	PUNCT
ejpam-6904	36	15	,	,	PUNCT
ejpam-6904	36	16	is	be	AUX
ejpam-6904	36	17	called	call	VERB
ejpam-6904	36	18	the	the	DET
ejpam-6904	36	19	independence	independence	NOUN
ejpam-6904	36	20	number	number	NOUN
ejpam-6904	36	21	of	of	ADP
ejpam-6904	36	22	g.	g.	PROPN
ejpam-6904	36	23	any	any	DET
ejpam-6904	36	24	independent	independent	ADJ
ejpam-6904	36	25	set	set	NOUN
ejpam-6904	36	26	with	with	ADP
ejpam-6904	36	27	cardinality	cardinality	NOUN
ejpam-6904	36	28	equal	equal	ADJ
ejpam-6904	36	29	to	to	ADP
ejpam-6904	36	30	α(g	α(g	NUM
ejpam-6904	36	31	)	)	PUNCT
ejpam-6904	36	32	is	be	AUX
ejpam-6904	36	33	called	call	VERB
ejpam-6904	36	34	an	an	DET
ejpam-6904	36	35	α	α	NOUN
ejpam-6904	36	36	-	-	PUNCT
ejpam-6904	36	37	set	set	VERB
ejpam-6904	36	38	in	in	ADP
ejpam-6904	36	39	g.	g.	PROPN
ejpam-6904	36	40	a	a	DET
ejpam-6904	36	41	set	set	NOUN
ejpam-6904	36	42	s	s	PROPN
ejpam-6904	36	43	⊆	⊆	NUM
ejpam-6904	36	44	v	v	NOUN
ejpam-6904	36	45	(	(	PUNCT
ejpam-6904	36	46	g	g	NOUN
ejpam-6904	36	47	)	)	PUNCT
ejpam-6904	36	48	is	be	AUX
ejpam-6904	36	49	a	a	DET
ejpam-6904	36	50	dominating	dominating	NOUN
ejpam-6904	36	51	set	set	VERB
ejpam-6904	36	52	in	in	ADP
ejpam-6904	36	53	g	g	PROPN
ejpam-6904	36	54	if	if	SCONJ
ejpam-6904	36	55	ng[s	ng[	NOUN
ejpam-6904	36	56	]	]	PUNCT
ejpam-6904	36	57	=	=	SYM
ejpam-6904	36	58	v	v	NOUN
ejpam-6904	36	59	(	(	PUNCT
ejpam-6904	36	60	g	g	NOUN
ejpam-6904	36	61	)	)	PUNCT
ejpam-6904	36	62	.	.	PUNCT
ejpam-6904	37	1	it	it	PRON
ejpam-6904	37	2	is	be	AUX
ejpam-6904	37	3	a	a	DET
ejpam-6904	37	4	2	2	NUM
ejpam-6904	37	5	-	-	PUNCT
ejpam-6904	37	6	dominating	dominating	NOUN
ejpam-6904	37	7	set	set	NOUN
ejpam-6904	37	8	if	if	SCONJ
ejpam-6904	37	9	|ng(v	|ng(v	NOUN
ejpam-6904	37	10	)	)	PUNCT
ejpam-6904	37	11	∩	∩	NOUN
ejpam-6904	37	12	s|	s|	VERB
ejpam-6904	37	13	≥	≥	NUM
ejpam-6904	37	14	2	2	NUM
ejpam-6904	37	15	for	for	ADP
ejpam-6904	37	16	every	every	DET
ejpam-6904	37	17	v	v	NUM
ejpam-6904	37	18	∈	∈	NOUN
ejpam-6904	37	19	v	v	NOUN
ejpam-6904	37	20	(	(	PUNCT
ejpam-6904	37	21	g	g	NOUN
ejpam-6904	37	22	)	)	PUNCT
ejpam-6904	37	23	\	\	PUNCT
ejpam-6904	38	1	s.	s.	PROPN
ejpam-6904	38	2	the	the	DET
ejpam-6904	38	3	domination	domination	NOUN
ejpam-6904	38	4	number	number	NOUN
ejpam-6904	38	5	(	(	PUNCT
ejpam-6904	38	6	resp	resp	NOUN
ejpam-6904	38	7	.	.	PUNCT
ejpam-6904	39	1	2	2	NUM
ejpam-6904	39	2	-	-	PUNCT
ejpam-6904	39	3	domination	domination	NOUN
ejpam-6904	39	4	number	number	NOUN
ejpam-6904	39	5	)	)	PUNCT
ejpam-6904	39	6	of	of	ADP
ejpam-6904	39	7	g	g	NOUN
ejpam-6904	39	8	,	,	PUNCT
ejpam-6904	39	9	denoted	denote	VERB
ejpam-6904	39	10	γ(g	γ(g	PROPN
ejpam-6904	39	11	)	)	PUNCT
ejpam-6904	39	12	(	(	PUNCT
ejpam-6904	39	13	resp	resp	NOUN
ejpam-6904	39	14	.	.	PUNCT
ejpam-6904	40	1	γ2(g	γ2(g	VERB
ejpam-6904	40	2	)	)	PUNCT
ejpam-6904	40	3	)	)	PUNCT
ejpam-6904	40	4	is	be	AUX
ejpam-6904	40	5	the	the	DET
ejpam-6904	40	6	minimum	minimum	ADJ
ejpam-6904	40	7	cardinality	cardinality	NOUN
ejpam-6904	40	8	of	of	ADP
ejpam-6904	40	9	a	a	DET
ejpam-6904	40	10	dominating	dominating	NOUN
ejpam-6904	40	11	(	(	PUNCT
ejpam-6904	40	12	resp	resp	NOUN
ejpam-6904	40	13	.	.	PUNCT
ejpam-6904	41	1	2	2	NUM
ejpam-6904	41	2	-	-	PUNCT
ejpam-6904	41	3	dominating	dominating	NOUN
ejpam-6904	41	4	)	)	PUNCT
ejpam-6904	41	5	set	set	VERB
ejpam-6904	41	6	in	in	ADP
ejpam-6904	41	7	g.	g.	PROPN
ejpam-6904	41	8	any	any	DET
ejpam-6904	41	9	dominating	dominating	NOUN
ejpam-6904	41	10	set	set	NOUN
ejpam-6904	41	11	(	(	PUNCT
ejpam-6904	41	12	2	2	NUM
ejpam-6904	41	13	-	-	PUNCT
ejpam-6904	41	14	dominating	dominating	NOUN
ejpam-6904	41	15	set	set	NOUN
ejpam-6904	41	16	)	)	PUNCT
ejpam-6904	41	17	with	with	ADP
ejpam-6904	41	18	cardinality	cardinality	PROPN
ejpam-6904	41	19	γ(g	γ(g	PROPN
ejpam-6904	41	20	)	)	PUNCT
ejpam-6904	41	21	(	(	PUNCT
ejpam-6904	41	22	resp	resp	NOUN
ejpam-6904	41	23	.	.	PUNCT
ejpam-6904	42	1	γ2(g	γ2(g	VERB
ejpam-6904	42	2	)	)	PUNCT
ejpam-6904	42	3	)	)	PUNCT
ejpam-6904	42	4	is	be	AUX
ejpam-6904	42	5	called	call	VERB
ejpam-6904	42	6	a	a	DET
ejpam-6904	42	7	γ	γ	NOUN
ejpam-6904	42	8	-	-	PUNCT
ejpam-6904	42	9	set	set	ADJ
ejpam-6904	42	10	(	(	PUNCT
ejpam-6904	42	11	resp	resp	NOUN
ejpam-6904	42	12	.	.	PUNCT
ejpam-6904	43	1	γ2	γ2	NOUN
ejpam-6904	43	2	-	-	PUNCT
ejpam-6904	43	3	set	set	NOUN
ejpam-6904	43	4	)	)	PUNCT
ejpam-6904	43	5	in	in	ADP
ejpam-6904	43	6	g.	g.	PROPN
ejpam-6904	43	7	a	a	DET
ejpam-6904	43	8	subset	subset	NOUN
ejpam-6904	43	9	s	s	VERB
ejpam-6904	43	10	⊆	⊆	NUM
ejpam-6904	43	11	v	v	NOUN
ejpam-6904	43	12	(	(	PUNCT
ejpam-6904	43	13	g	g	NOUN
ejpam-6904	43	14	)	)	PUNCT
ejpam-6904	43	15	is	be	AUX
ejpam-6904	43	16	called	call	VERB
ejpam-6904	43	17	a	a	DET
ejpam-6904	43	18	geodetic	geodetic	ADJ
ejpam-6904	43	19	set	set	NOUN
ejpam-6904	43	20	of	of	ADP
ejpam-6904	43	21	a	a	DET
ejpam-6904	43	22	graph	graph	NOUN
ejpam-6904	43	23	g	g	NOUN
ejpam-6904	43	24	if	if	SCONJ
ejpam-6904	43	25	for	for	ADP
ejpam-6904	43	26	every	every	DET
ejpam-6904	43	27	vertex	vertex	NOUN
ejpam-6904	43	28	v	v	ADP
ejpam-6904	43	29	∈	∈	NOUN
ejpam-6904	43	30	v	v	NOUN
ejpam-6904	43	31	(	(	PUNCT
ejpam-6904	43	32	g	g	NOUN
ejpam-6904	43	33	)	)	PUNCT
ejpam-6904	43	34	,	,	PUNCT
ejpam-6904	43	35	there	there	PRON
ejpam-6904	43	36	exist	exist	VERB
ejpam-6904	43	37	vertices	vertex	NOUN
ejpam-6904	43	38	u	u	NOUN
ejpam-6904	43	39	,	,	PUNCT
ejpam-6904	43	40	w	w	PROPN
ejpam-6904	43	41	∈	∈	PROPN
ejpam-6904	43	42	s	s	VERB
ejpam-6904	43	43	such	such	ADJ
ejpam-6904	43	44	that	that	SCONJ
ejpam-6904	43	45	v	v	NUM
ejpam-6904	43	46	∈	∈	PROPN
ejpam-6904	43	47	ig[u	ig[u	PROPN
ejpam-6904	43	48	,	,	PUNCT
ejpam-6904	43	49	w	w	NOUN
ejpam-6904	43	50	]	]	X
ejpam-6904	43	51	,	,	PUNCT
ejpam-6904	43	52	where	where	SCONJ
ejpam-6904	43	53	ig[u	ig[u	PROPN
ejpam-6904	43	54	,	,	PUNCT
ejpam-6904	43	55	w	w	NOUN
ejpam-6904	43	56	]	]	X
ejpam-6904	43	57	denotes	denote	VERB
ejpam-6904	43	58	the	the	DET
ejpam-6904	43	59	set	set	NOUN
ejpam-6904	43	60	consisting	consisting	NOUN
ejpam-6904	43	61	of	of	ADP
ejpam-6904	43	62	u	u	NOUN
ejpam-6904	43	63	,	,	PUNCT
ejpam-6904	43	64	w	w	PROPN
ejpam-6904	43	65	,	,	PUNCT
ejpam-6904	43	66	and	and	CCONJ
ejpam-6904	43	67	all	all	DET
ejpam-6904	43	68	vertices	vertex	NOUN
ejpam-6904	43	69	that	that	PRON
ejpam-6904	43	70	lie	lie	VERB
ejpam-6904	43	71	on	on	ADP
ejpam-6904	43	72	some	some	DET
ejpam-6904	43	73	shortest	short	ADJ
ejpam-6904	43	74	path	path	NOUN
ejpam-6904	43	75	between	between	ADP
ejpam-6904	43	76	u	u	PROPN
ejpam-6904	43	77	and	and	CCONJ
ejpam-6904	43	78	w	w	NOUN
ejpam-6904	43	79	in	in	ADP
ejpam-6904	43	80	g.	g.	PROPN
ejpam-6904	43	81	this	this	DET
ejpam-6904	43	82	shortest	short	ADJ
ejpam-6904	43	83	path	path	NOUN
ejpam-6904	43	84	connecting	connect	VERB
ejpam-6904	43	85	u	u	NOUN
ejpam-6904	43	86	and	and	CCONJ
ejpam-6904	43	87	w	w	PROPN
ejpam-6904	43	88	is	be	AUX
ejpam-6904	43	89	called	call	VERB
ejpam-6904	43	90	a	a	DET
ejpam-6904	43	91	u	u	NOUN
ejpam-6904	43	92	-	-	PROPN
ejpam-6904	43	93	w	w	NOUN
ejpam-6904	43	94	geodesic	geodesic	NOUN
ejpam-6904	43	95	.	.	PUNCT
ejpam-6904	44	1	a	a	DET
ejpam-6904	44	2	set	set	NOUN
ejpam-6904	44	3	s	s	NOUN
ejpam-6904	44	4	⊆	⊆	NUM
ejpam-6904	44	5	v	v	NOUN
ejpam-6904	44	6	(	(	PUNCT
ejpam-6904	44	7	g	g	NOUN
ejpam-6904	44	8	)	)	PUNCT
ejpam-6904	44	9	is	be	AUX
ejpam-6904	44	10	a	a	DET
ejpam-6904	44	11	2	2	NUM
ejpam-6904	44	12	-	-	PUNCT
ejpam-6904	44	13	path	path	NOUN
ejpam-6904	44	14	geodetic	geodetic	ADJ
ejpam-6904	44	15	or	or	CCONJ
ejpam-6904	44	16	2	2	NUM
ejpam-6904	44	17	-	-	PUNCT
ejpam-6904	44	18	path	path	NOUN
ejpam-6904	44	19	closure	closure	NOUN
ejpam-6904	44	20	absorbing	absorb	VERB
ejpam-6904	44	21	set	set	NOUN
ejpam-6904	44	22	in	in	ADP
ejpam-6904	44	23	g	g	PROPN
ejpam-6904	44	24	if	if	SCONJ
ejpam-6904	44	25	for	for	ADP
ejpam-6904	44	26	each	each	DET
ejpam-6904	44	27	x	x	SYM
ejpam-6904	44	28	∈	∈	PROPN
ejpam-6904	44	29	v	v	ADP
ejpam-6904	44	30	(	(	PUNCT
ejpam-6904	44	31	g	g	NOUN
ejpam-6904	44	32	)	)	PUNCT
ejpam-6904	44	33	\	\	PROPN
ejpam-6904	45	1	s	s	X
ejpam-6904	45	2	,	,	PUNCT
ejpam-6904	45	3	there	there	PRON
ejpam-6904	45	4	exist	exist	VERB
ejpam-6904	45	5	p	p	PRON
ejpam-6904	45	6	,	,	PUNCT
ejpam-6904	45	7	q	q	PROPN
ejpam-6904	45	8	∈	∈	PROPN
ejpam-6904	45	9	s	s	VERB
ejpam-6904	45	10	such	such	ADJ
ejpam-6904	45	11	that	that	SCONJ
ejpam-6904	45	12	x	x	SYM
ejpam-6904	45	13	∈	∈	NOUN
ejpam-6904	45	14	ig(p	ig(p	NOUN
ejpam-6904	45	15	,	,	PUNCT
ejpam-6904	45	16	q	q	NOUN
ejpam-6904	45	17	)	)	PUNCT
ejpam-6904	45	18	and	and	CCONJ
ejpam-6904	45	19	dg(p	dg(p	NOUN
ejpam-6904	45	20	,	,	PUNCT
ejpam-6904	45	21	q	q	X
ejpam-6904	45	22	)	)	PUNCT
ejpam-6904	45	23	=	=	SYM
ejpam-6904	45	24	2	2	NUM
ejpam-6904	45	25	,	,	PUNCT
ejpam-6904	45	26	where	where	SCONJ
ejpam-6904	45	27	ig(p	ig(p	ADJ
ejpam-6904	45	28	,	,	PUNCT
ejpam-6904	45	29	q	q	NOUN
ejpam-6904	45	30	)	)	PUNCT
ejpam-6904	45	31	=	=	SYM
ejpam-6904	45	32	ig[p	ig[p	PROPN
ejpam-6904	45	33	,	,	PUNCT
ejpam-6904	45	34	q	q	X
ejpam-6904	45	35	]	]	X
ejpam-6904	45	36	\	\	NOUN
ejpam-6904	45	37	{	{	PUNCT
ejpam-6904	45	38	p	p	X
ejpam-6904	45	39	,	,	PUNCT
ejpam-6904	45	40	q	q	NOUN
ejpam-6904	45	41	}	}	PUNCT
ejpam-6904	45	42	.	.	PUNCT
ejpam-6904	46	1	the	the	DET
ejpam-6904	46	2	smallest	small	ADJ
ejpam-6904	46	3	cardinality	cardinality	NOUN
ejpam-6904	46	4	among	among	ADP
ejpam-6904	46	5	all	all	DET
ejpam-6904	46	6	2	2	NUM
ejpam-6904	46	7	-	-	PUNCT
ejpam-6904	46	8	path	path	NOUN
ejpam-6904	46	9	geodetic	geodetic	ADJ
ejpam-6904	46	10	sets	set	NOUN
ejpam-6904	46	11	in	in	ADP
ejpam-6904	46	12	g	g	NOUN
ejpam-6904	46	13	,	,	PUNCT
ejpam-6904	46	14	denoted	denote	VERB
ejpam-6904	46	15	g2p(g	g2p(g	PROPN
ejpam-6904	46	16	)	)	PUNCT
ejpam-6904	46	17	,	,	PUNCT
ejpam-6904	46	18	is	be	AUX
ejpam-6904	46	19	called	call	VERB
ejpam-6904	46	20	the	the	DET
ejpam-6904	46	21	2	2	NUM
ejpam-6904	46	22	-	-	PUNCT
ejpam-6904	46	23	path	path	NOUN
ejpam-6904	46	24	geodetic	geodetic	ADJ
ejpam-6904	46	25	number	number	NOUN
ejpam-6904	46	26	of	of	ADP
ejpam-6904	46	27	g.	g.	PROPN
ejpam-6904	46	28	a	a	DET
ejpam-6904	46	29	2	2	NUM
ejpam-6904	46	30	-	-	PUNCT
ejpam-6904	46	31	path	path	NOUN
ejpam-6904	46	32	geodetic	geodetic	ADJ
ejpam-6904	46	33	set	set	NOUN
ejpam-6904	46	34	s	s	PART
ejpam-6904	46	35	is	be	AUX
ejpam-6904	46	36	2	2	NUM
ejpam-6904	46	37	-	-	PUNCT
ejpam-6904	46	38	path	path	NOUN
ejpam-6904	46	39	geodetic	geodetic	ADJ
ejpam-6904	46	40	2	2	NUM
ejpam-6904	46	41	-	-	PUNCT
ejpam-6904	46	42	dominating	dominating	NOUN
ejpam-6904	46	43	if	if	SCONJ
ejpam-6904	46	44	s	s	NOUN
ejpam-6904	46	45	is	be	AUX
ejpam-6904	46	46	2	2	NUM
ejpam-6904	46	47	-	-	PUNCT
ejpam-6904	46	48	dominating	dominating	NOUN
ejpam-6904	46	49	in	in	ADP
ejpam-6904	46	50	g.	g.	PROPN
ejpam-6904	46	51	the	the	DET
ejpam-6904	46	52	smallest	small	ADJ
ejpam-6904	46	53	cardinality	cardinality	NOUN
ejpam-6904	46	54	of	of	ADP
ejpam-6904	46	55	a	a	DET
ejpam-6904	46	56	2	2	NUM
ejpam-6904	46	57	-	-	PUNCT
ejpam-6904	46	58	path	path	NOUN
ejpam-6904	46	59	geodetic	geodetic	ADJ
ejpam-6904	46	60	2	2	NUM
ejpam-6904	46	61	-	-	PUNCT
ejpam-6904	46	62	dominating	dominating	NOUN
ejpam-6904	46	63	set	set	NOUN
ejpam-6904	46	64	in	in	ADP
ejpam-6904	46	65	g	g	NOUN
ejpam-6904	46	66	,	,	PUNCT
ejpam-6904	46	67	denoted	denote	VERB
ejpam-6904	46	68	γ2pg2(g	γ2pg2(g	ADV
ejpam-6904	46	69	)	)	PUNCT
ejpam-6904	46	70	,	,	PUNCT
ejpam-6904	46	71	is	be	AUX
ejpam-6904	46	72	a.	a.	PROPN
ejpam-6904	46	73	b.	b.	PROPN
ejpam-6904	46	74	tapeing	tapeing	PROPN
ejpam-6904	46	75	,	,	PUNCT
ejpam-6904	46	76	s.	s.	PROPN
ejpam-6904	46	77	r.	r.	PROPN
ejpam-6904	46	78	canoy	canoy	PROPN
ejpam-6904	46	79	/	/	SYM
ejpam-6904	46	80	eur	eur	PROPN
ejpam-6904	46	81	.	.	PUNCT
ejpam-6904	47	1	j.	j.	PROPN
ejpam-6904	47	2	pure	pure	PROPN
ejpam-6904	47	3	appl	appl	PROPN
ejpam-6904	47	4	.	.	PROPN
ejpam-6904	47	5	math	math	PROPN
ejpam-6904	47	6	,	,	PUNCT
ejpam-6904	47	7	18	18	NUM
ejpam-6904	47	8	(	(	PUNCT
ejpam-6904	47	9	4	4	NUM
ejpam-6904	47	10	)	)	PUNCT
ejpam-6904	47	11	(	(	PUNCT
ejpam-6904	47	12	2025	2025	NUM
ejpam-6904	47	13	)	)	PUNCT
ejpam-6904	47	14	,	,	PUNCT
ejpam-6904	47	15	6904	6904	NUM
ejpam-6904	47	16	3	3	NUM
ejpam-6904	47	17	of	of	ADP
ejpam-6904	47	18	14	14	NUM
ejpam-6904	47	19	called	call	VERB
ejpam-6904	47	20	the	the	DET
ejpam-6904	47	21	2	2	NUM
ejpam-6904	47	22	-	-	PUNCT
ejpam-6904	47	23	path	path	NOUN
ejpam-6904	47	24	geodetic	geodetic	ADJ
ejpam-6904	47	25	2	2	NUM
ejpam-6904	47	26	-	-	PUNCT
ejpam-6904	47	27	domination	domination	NOUN
ejpam-6904	47	28	number	number	NOUN
ejpam-6904	47	29	of	of	ADP
ejpam-6904	47	30	g.	g.	PROPN
ejpam-6904	47	31	any	any	DET
ejpam-6904	47	32	2	2	NUM
ejpam-6904	47	33	-	-	PUNCT
ejpam-6904	47	34	path	path	NOUN
ejpam-6904	47	35	geodetic	geodetic	ADJ
ejpam-6904	47	36	2	2	NUM
ejpam-6904	47	37	-	-	PUNCT
ejpam-6904	47	38	dominating	dominating	NOUN
ejpam-6904	47	39	set	set	NOUN
ejpam-6904	47	40	with	with	ADP
ejpam-6904	47	41	cardinality	cardinality	NOUN
ejpam-6904	47	42	γ2pg2(g	γ2pg2(g	ADV
ejpam-6904	47	43	)	)	PUNCT
ejpam-6904	47	44	is	be	AUX
ejpam-6904	47	45	referred	refer	VERB
ejpam-6904	47	46	to	to	ADP
ejpam-6904	47	47	as	as	ADP
ejpam-6904	47	48	a	a	DET
ejpam-6904	47	49	γ2pg2	γ2pg2	ADV
ejpam-6904	47	50	-	-	PUNCT
ejpam-6904	47	51	set	set	NOUN
ejpam-6904	47	52	.	.	PUNCT
ejpam-6904	48	1	a	a	DET
ejpam-6904	48	2	subset	subset	ADJ
ejpam-6904	48	3	u	u	NOUN
ejpam-6904	48	4	of	of	ADP
ejpam-6904	48	5	vertices	vertex	NOUN
ejpam-6904	48	6	of	of	ADP
ejpam-6904	48	7	a	a	DET
ejpam-6904	48	8	graph	graph	NOUN
ejpam-6904	48	9	g	g	NOUN
ejpam-6904	48	10	is	be	AUX
ejpam-6904	48	11	called	call	VERB
ejpam-6904	48	12	a	a	DET
ejpam-6904	48	13	vertex	vertex	NOUN
ejpam-6904	48	14	cover	cover	NOUN
ejpam-6904	48	15	of	of	ADP
ejpam-6904	48	16	g	g	PROPN
ejpam-6904	48	17	if	if	SCONJ
ejpam-6904	48	18	for	for	ADP
ejpam-6904	48	19	every	every	DET
ejpam-6904	48	20	e	e	NOUN
ejpam-6904	48	21	=	=	PUNCT
ejpam-6904	48	22	uv	uv	PROPN
ejpam-6904	48	23	∈	∈	PROPN
ejpam-6904	48	24	e(g	e(g	PROPN
ejpam-6904	48	25	)	)	PUNCT
ejpam-6904	48	26	,	,	PUNCT
ejpam-6904	48	27	u	u	PROPN
ejpam-6904	48	28	∈	∈	PROPN
ejpam-6904	48	29	u	u	NOUN
ejpam-6904	48	30	or	or	CCONJ
ejpam-6904	48	31	v	v	ADP
ejpam-6904	48	32	∈	∈	PROPN
ejpam-6904	48	33	u	u	NOUN
ejpam-6904	48	34	.	.	PUNCT
ejpam-6904	49	1	the	the	DET
ejpam-6904	49	2	minimum	minimum	ADJ
ejpam-6904	49	3	cardinality	cardinality	NOUN
ejpam-6904	49	4	of	of	ADP
ejpam-6904	49	5	a	a	DET
ejpam-6904	49	6	vertex	vertex	NOUN
ejpam-6904	49	7	cover	cover	NOUN
ejpam-6904	49	8	of	of	ADP
ejpam-6904	49	9	g	g	NOUN
ejpam-6904	49	10	,	,	PUNCT
ejpam-6904	49	11	denoted	denote	VERB
ejpam-6904	49	12	β(g	β(g	PROPN
ejpam-6904	49	13	)	)	PUNCT
ejpam-6904	49	14	,	,	PUNCT
ejpam-6904	49	15	is	be	AUX
ejpam-6904	49	16	the	the	DET
ejpam-6904	49	17	vertex	vertex	NOUN
ejpam-6904	49	18	cover	cover	NOUN
ejpam-6904	49	19	number	number	NOUN
ejpam-6904	49	20	of	of	ADP
ejpam-6904	49	21	g.	g.	PROPN
ejpam-6904	49	22	any	any	DET
ejpam-6904	49	23	vertex	vertex	NOUN
ejpam-6904	49	24	cover	cover	NOUN
ejpam-6904	49	25	of	of	ADP
ejpam-6904	49	26	g	g	NOUN
ejpam-6904	49	27	with	with	ADP
ejpam-6904	49	28	cardinality	cardinality	PROPN
ejpam-6904	49	29	β(g	β(g	PROPN
ejpam-6904	49	30	)	)	PUNCT
ejpam-6904	49	31	is	be	AUX
ejpam-6904	49	32	called	call	VERB
ejpam-6904	49	33	a	a	DET
ejpam-6904	49	34	β	β	NOUN
ejpam-6904	49	35	-	-	NOUN
ejpam-6904	49	36	set	set	NOUN
ejpam-6904	49	37	.	.	PUNCT
ejpam-6904	50	1	a	a	DET
ejpam-6904	50	2	set	set	NOUN
ejpam-6904	50	3	s	s	NOUN
ejpam-6904	50	4	⊆	⊆	NUM
ejpam-6904	50	5	v	v	NOUN
ejpam-6904	50	6	(	(	PUNCT
ejpam-6904	50	7	g	g	NOUN
ejpam-6904	50	8	)	)	PUNCT
ejpam-6904	50	9	is	be	AUX
ejpam-6904	50	10	called	call	VERB
ejpam-6904	50	11	a	a	DET
ejpam-6904	50	12	2	2	NUM
ejpam-6904	50	13	-	-	PUNCT
ejpam-6904	50	14	path	path	NOUN
ejpam-6904	50	15	geodetic	geodetic	ADJ
ejpam-6904	50	16	vertex	vertex	NOUN
ejpam-6904	50	17	cover	cover	NOUN
ejpam-6904	50	18	of	of	ADP
ejpam-6904	50	19	g	g	PROPN
ejpam-6904	50	20	if	if	SCONJ
ejpam-6904	50	21	it	it	PRON
ejpam-6904	50	22	is	be	AUX
ejpam-6904	50	23	both	both	CCONJ
ejpam-6904	50	24	a	a	DET
ejpam-6904	50	25	vertex	vertex	NOUN
ejpam-6904	50	26	cover	cover	NOUN
ejpam-6904	50	27	and	and	CCONJ
ejpam-6904	50	28	a	a	DET
ejpam-6904	50	29	2	2	NUM
ejpam-6904	50	30	-	-	PUNCT
ejpam-6904	50	31	path	path	NOUN
ejpam-6904	50	32	geodetic	geodetic	ADJ
ejpam-6904	50	33	set	set	NOUN
ejpam-6904	50	34	in	in	ADP
ejpam-6904	50	35	g.	g.	PROPN
ejpam-6904	50	36	the	the	DET
ejpam-6904	50	37	smallest	small	ADJ
ejpam-6904	50	38	cardinality	cardinality	NOUN
ejpam-6904	50	39	of	of	ADP
ejpam-6904	50	40	a	a	DET
ejpam-6904	50	41	2	2	NUM
ejpam-6904	50	42	-	-	PUNCT
ejpam-6904	50	43	path	path	NOUN
ejpam-6904	50	44	geodetic	geodetic	ADJ
ejpam-6904	50	45	vertex	vertex	NOUN
ejpam-6904	50	46	cover	cover	NOUN
ejpam-6904	50	47	of	of	ADP
ejpam-6904	50	48	g	g	NOUN
ejpam-6904	50	49	,	,	PUNCT
ejpam-6904	50	50	denoted	denote	VERB
ejpam-6904	50	51	β2pg(g	β2pg(g	PROPN
ejpam-6904	50	52	)	)	PUNCT
ejpam-6904	50	53	,	,	PUNCT
ejpam-6904	50	54	is	be	AUX
ejpam-6904	50	55	called	call	VERB
ejpam-6904	50	56	the	the	DET
ejpam-6904	50	57	2	2	NUM
ejpam-6904	50	58	-	-	PUNCT
ejpam-6904	50	59	path	path	NOUN
ejpam-6904	50	60	geodetic	geodetic	ADJ
ejpam-6904	50	61	vertex	vertex	NOUN
ejpam-6904	50	62	cover	cover	NOUN
ejpam-6904	50	63	number	number	NOUN
ejpam-6904	50	64	of	of	ADP
ejpam-6904	50	65	g.	g.	PROPN
ejpam-6904	50	66	any	any	DET
ejpam-6904	50	67	2	2	NUM
ejpam-6904	50	68	-	-	PUNCT
ejpam-6904	50	69	path	path	NOUN
ejpam-6904	50	70	geodetic	geodetic	ADJ
ejpam-6904	50	71	vertex	vertex	NOUN
ejpam-6904	50	72	cover	cover	NOUN
ejpam-6904	50	73	of	of	ADP
ejpam-6904	50	74	g	g	NOUN
ejpam-6904	50	75	with	with	ADP
ejpam-6904	50	76	cardinality	cardinality	NOUN
ejpam-6904	50	77	β2pg(g	β2pg(g	PUNCT
ejpam-6904	50	78	)	)	PUNCT
ejpam-6904	50	79	is	be	AUX
ejpam-6904	50	80	called	call	VERB
ejpam-6904	50	81	a	a	DET
ejpam-6904	50	82	β2pg	β2pg	PUNCT
ejpam-6904	50	83	-	-	VERB
ejpam-6904	50	84	set	set	NOUN
ejpam-6904	50	85	.	.	PUNCT
ejpam-6904	51	1	consider	consider	VERB
ejpam-6904	51	2	the	the	DET
ejpam-6904	51	3	graph	graph	NOUN
ejpam-6904	51	4	g	g	NOUN
ejpam-6904	51	5	in	in	ADP
ejpam-6904	51	6	figure	figure	NOUN
ejpam-6904	51	7	1	1	NUM
ejpam-6904	51	8	.	.	PUNCT
ejpam-6904	52	1	let	let	VERB
ejpam-6904	52	2	s	s	VERB
ejpam-6904	52	3	=	=	X
ejpam-6904	52	4	{	{	PUNCT
ejpam-6904	52	5	a	a	X
ejpam-6904	52	6	,	,	PUNCT
ejpam-6904	52	7	c	c	NOUN
ejpam-6904	52	8	,	,	PUNCT
ejpam-6904	52	9	f	f	NOUN
ejpam-6904	52	10	}	}	PUNCT
ejpam-6904	52	11	.	.	PUNCT
ejpam-6904	53	1	since	since	SCONJ
ejpam-6904	53	2	every	every	DET
ejpam-6904	53	3	edge	edge	NOUN
ejpam-6904	53	4	of	of	ADP
ejpam-6904	53	5	g	g	PROPN
ejpam-6904	53	6	is	be	AUX
ejpam-6904	53	7	incident	incident	NOUN
ejpam-6904	53	8	to	to	ADP
ejpam-6904	53	9	some	some	DET
ejpam-6904	53	10	vertex	vertex	NOUN
ejpam-6904	53	11	in	in	ADP
ejpam-6904	53	12	s	s	PROPN
ejpam-6904	53	13	,	,	PUNCT
ejpam-6904	53	14	it	it	PRON
ejpam-6904	53	15	follows	follow	VERB
ejpam-6904	53	16	that	that	SCONJ
ejpam-6904	53	17	s	s	VERB
ejpam-6904	53	18	is	be	AUX
ejpam-6904	53	19	a	a	DET
ejpam-6904	53	20	vertex	vertex	NOUN
ejpam-6904	53	21	cover	cover	NOUN
ejpam-6904	53	22	of	of	ADP
ejpam-6904	53	23	g.	g.	PROPN
ejpam-6904	53	24	moreover	moreover	ADV
ejpam-6904	53	25	,	,	PUNCT
ejpam-6904	53	26	b	b	X
ejpam-6904	53	27	,	,	PUNCT
ejpam-6904	53	28	d	d	PROPN
ejpam-6904	53	29	∈	∈	PROPN
ejpam-6904	53	30	ig(a	ig(a	NOUN
ejpam-6904	53	31	,	,	PUNCT
ejpam-6904	53	32	c	c	NOUN
ejpam-6904	53	33	)	)	PUNCT
ejpam-6904	53	34	and	and	CCONJ
ejpam-6904	53	35	e	e	X
ejpam-6904	53	36	∈	∈	PROPN
ejpam-6904	53	37	ig(c	ig(c	NOUN
ejpam-6904	53	38	,	,	PUNCT
ejpam-6904	53	39	f	f	NOUN
ejpam-6904	53	40	)	)	PUNCT
ejpam-6904	53	41	.	.	PUNCT
ejpam-6904	54	1	thus	thus	ADV
ejpam-6904	54	2	,	,	PUNCT
ejpam-6904	54	3	s	s	VERB
ejpam-6904	54	4	is	be	AUX
ejpam-6904	54	5	a	a	DET
ejpam-6904	54	6	2	2	NUM
ejpam-6904	54	7	-	-	PUNCT
ejpam-6904	54	8	path	path	NOUN
ejpam-6904	54	9	geodetic	geodetic	ADJ
ejpam-6904	54	10	vertex	vertex	NOUN
ejpam-6904	54	11	cover	cover	NOUN
ejpam-6904	54	12	of	of	ADP
ejpam-6904	54	13	g.	g.	PROPN
ejpam-6904	54	14	since	since	SCONJ
ejpam-6904	54	15	there	there	PRON
ejpam-6904	54	16	exists	exist	VERB
ejpam-6904	54	17	no	no	DET
ejpam-6904	54	18	2	2	NUM
ejpam-6904	54	19	-	-	PUNCT
ejpam-6904	54	20	path	path	NOUN
ejpam-6904	54	21	geodetic	geodetic	ADJ
ejpam-6904	54	22	vertex	vertex	NOUN
ejpam-6904	54	23	cover	cover	NOUN
ejpam-6904	54	24	with	with	ADP
ejpam-6904	54	25	cardinality	cardinality	NOUN
ejpam-6904	54	26	less	less	ADJ
ejpam-6904	54	27	than	than	ADP
ejpam-6904	54	28	3	3	NUM
ejpam-6904	54	29	,	,	PUNCT
ejpam-6904	54	30	it	it	PRON
ejpam-6904	54	31	follows	follow	VERB
ejpam-6904	54	32	that	that	PRON
ejpam-6904	54	33	β2pg(g	β2pg(g	PUNCT
ejpam-6904	54	34	)	)	PUNCT
ejpam-6904	54	35	=	=	SYM
ejpam-6904	54	36	|s|	|s|	NOUN
ejpam-6904	54	37	=	=	SYM
ejpam-6904	54	38	3	3	X
ejpam-6904	54	39	.	.	PUNCT
ejpam-6904	54	40	a	a	DET
ejpam-6904	54	41	b	b	PROPN
ejpam-6904	54	42	cd	cd	NOUN
ejpam-6904	54	43	e	e	NOUN
ejpam-6904	54	44	f	f	PROPN
ejpam-6904	54	45	figure	figure	VERB
ejpam-6904	54	46	1	1	NUM
ejpam-6904	54	47	:	:	PUNCT
ejpam-6904	54	48	a	a	DET
ejpam-6904	54	49	graph	graph	NOUN
ejpam-6904	54	50	g	g	NOUN
ejpam-6904	54	51	with	with	ADP
ejpam-6904	54	52	β2pg(g	β2pg(g	NOUN
ejpam-6904	54	53	)	)	PUNCT
ejpam-6904	54	54	=	=	SYM
ejpam-6904	54	55	3	3	NUM
ejpam-6904	54	56	let	let	VERB
ejpam-6904	54	57	g	g	NOUN
ejpam-6904	54	58	and	and	CCONJ
ejpam-6904	54	59	h	h	NOUN
ejpam-6904	54	60	be	be	VERB
ejpam-6904	54	61	any	any	DET
ejpam-6904	54	62	two	two	NUM
ejpam-6904	54	63	graphs	graph	NOUN
ejpam-6904	54	64	.	.	PUNCT
ejpam-6904	55	1	the	the	DET
ejpam-6904	55	2	join	join	NOUN
ejpam-6904	55	3	g	g	PROPN
ejpam-6904	55	4	+	+	CCONJ
ejpam-6904	55	5	h	h	NOUN
ejpam-6904	55	6	is	be	AUX
ejpam-6904	55	7	the	the	DET
ejpam-6904	55	8	graph	graph	NOUN
ejpam-6904	55	9	with	with	ADP
ejpam-6904	55	10	vertex	vertex	NOUN
ejpam-6904	55	11	set	set	VERB
ejpam-6904	55	12	v	v	NOUN
ejpam-6904	55	13	(	(	PUNCT
ejpam-6904	55	14	g+h	g+h	NOUN
ejpam-6904	55	15	)	)	PUNCT
ejpam-6904	55	16	=	=	SYM
ejpam-6904	55	17	v	v	NOUN
ejpam-6904	55	18	(	(	PUNCT
ejpam-6904	55	19	g)∪	g)∪	VERB
ejpam-6904	55	20	v	v	NUM
ejpam-6904	55	21	(	(	PUNCT
ejpam-6904	55	22	h	h	NOUN
ejpam-6904	55	23	)	)	PUNCT
ejpam-6904	55	24	and	and	CCONJ
ejpam-6904	55	25	edge	edge	NOUN
ejpam-6904	55	26	set	set	VERB
ejpam-6904	55	27	e(g+h	e(g+h	NUM
ejpam-6904	55	28	)	)	PUNCT
ejpam-6904	56	1	=	=	SYM
ejpam-6904	56	2	e(g)∪e(h)∪	e(g)∪e(h)∪	NOUN
ejpam-6904	56	3	{	{	PUNCT
ejpam-6904	56	4	uv	uv	NOUN
ejpam-6904	56	5	:	:	PUNCT
ejpam-6904	56	6	u	u	PROPN
ejpam-6904	56	7	∈	∈	PROPN
ejpam-6904	56	8	v	v	ADP
ejpam-6904	56	9	(	(	PUNCT
ejpam-6904	56	10	g	g	NOUN
ejpam-6904	56	11	)	)	PUNCT
ejpam-6904	56	12	,	,	PUNCT
ejpam-6904	56	13	v	v	X
ejpam-6904	56	14	∈	∈	PROPN
ejpam-6904	56	15	v	v	NOUN
ejpam-6904	56	16	(	(	PUNCT
ejpam-6904	56	17	h	h	NOUN
ejpam-6904	56	18	)	)	PUNCT
ejpam-6904	56	19	}	}	PUNCT
ejpam-6904	56	20	.	.	PUNCT
ejpam-6904	57	1	the	the	DET
ejpam-6904	57	2	shadow	shadow	NOUN
ejpam-6904	57	3	graph	graph	VERB
ejpam-6904	57	4	d2(g	d2(g	PROPN
ejpam-6904	57	5	)	)	PUNCT
ejpam-6904	57	6	of	of	ADP
ejpam-6904	57	7	graph	graph	NOUN
ejpam-6904	57	8	g	g	PROPN
ejpam-6904	57	9	is	be	AUX
ejpam-6904	57	10	constructed	construct	VERB
ejpam-6904	57	11	by	by	ADP
ejpam-6904	57	12	taking	take	VERB
ejpam-6904	57	13	two	two	NUM
ejpam-6904	57	14	copies	copy	NOUN
ejpam-6904	57	15	of	of	ADP
ejpam-6904	57	16	g	g	NOUN
ejpam-6904	57	17	,	,	PUNCT
ejpam-6904	57	18	say	say	VERB
ejpam-6904	57	19	g1	g1	PROPN
ejpam-6904	57	20	and	and	CCONJ
ejpam-6904	57	21	g2	g2	PROPN
ejpam-6904	57	22	,	,	PUNCT
ejpam-6904	57	23	and	and	CCONJ
ejpam-6904	57	24	then	then	ADV
ejpam-6904	57	25	joining	join	VERB
ejpam-6904	57	26	each	each	DET
ejpam-6904	57	27	vertex	vertex	NOUN
ejpam-6904	57	28	u	u	NOUN
ejpam-6904	57	29	∈	∈	PROPN
ejpam-6904	57	30	v	v	NOUN
ejpam-6904	57	31	(	(	PUNCT
ejpam-6904	57	32	g1	g1	PROPN
ejpam-6904	57	33	)	)	PUNCT
ejpam-6904	57	34	to	to	ADP
ejpam-6904	57	35	the	the	DET
ejpam-6904	57	36	neighbors	neighbor	NOUN
ejpam-6904	57	37	of	of	ADP
ejpam-6904	57	38	its	its	PRON
ejpam-6904	57	39	corresponding	correspond	VERB
ejpam-6904	57	40	vertex	vertex	NOUN
ejpam-6904	57	41	u′	u′	PROPN
ejpam-6904	57	42	∈	∈	PROPN
ejpam-6904	57	43	v	v	NOUN
ejpam-6904	57	44	(	(	PUNCT
ejpam-6904	57	45	g2	g2	PROPN
ejpam-6904	57	46	)	)	PUNCT
ejpam-6904	57	47	.	.	PUNCT
ejpam-6904	58	1	readers	reader	NOUN
ejpam-6904	58	2	are	be	AUX
ejpam-6904	58	3	referred	refer	VERB
ejpam-6904	58	4	to	to	ADP
ejpam-6904	58	5	[	[	X
ejpam-6904	58	6	23	23	NUM
ejpam-6904	58	7	]	]	PUNCT
ejpam-6904	58	8	for	for	ADP
ejpam-6904	58	9	other	other	ADJ
ejpam-6904	58	10	basic	basic	ADJ
ejpam-6904	58	11	definitions	definition	NOUN
ejpam-6904	58	12	that	that	PRON
ejpam-6904	58	13	are	be	AUX
ejpam-6904	58	14	not	not	PART
ejpam-6904	58	15	given	give	VERB
ejpam-6904	58	16	here	here	ADV
ejpam-6904	58	17	.	.	PUNCT
ejpam-6904	59	1	3	3	X
ejpam-6904	59	2	.	.	X
ejpam-6904	59	3	results	result	NOUN
ejpam-6904	59	4	since	since	SCONJ
ejpam-6904	59	5	v	v	NOUN
ejpam-6904	59	6	(	(	PUNCT
ejpam-6904	59	7	g	g	NOUN
ejpam-6904	59	8	)	)	PUNCT
ejpam-6904	59	9	is	be	AUX
ejpam-6904	59	10	a	a	DET
ejpam-6904	59	11	2	2	NUM
ejpam-6904	59	12	-	-	PUNCT
ejpam-6904	59	13	path	path	NOUN
ejpam-6904	59	14	geodetic	geodetic	ADJ
ejpam-6904	59	15	vertex	vertex	NOUN
ejpam-6904	59	16	cover	cover	NOUN
ejpam-6904	59	17	of	of	ADP
ejpam-6904	59	18	g	g	NOUN
ejpam-6904	59	19	,	,	PUNCT
ejpam-6904	59	20	it	it	PRON
ejpam-6904	59	21	follows	follow	VERB
ejpam-6904	59	22	that	that	SCONJ
ejpam-6904	59	23	any	any	DET
ejpam-6904	59	24	graph	graph	NOUN
ejpam-6904	59	25	g	g	PROPN
ejpam-6904	59	26	admits	admit	VERB
ejpam-6904	59	27	a	a	DET
ejpam-6904	59	28	2	2	NUM
ejpam-6904	59	29	-	-	PUNCT
ejpam-6904	59	30	path	path	NOUN
ejpam-6904	59	31	geodetic	geodetic	ADJ
ejpam-6904	59	32	vertex	vertex	NOUN
ejpam-6904	59	33	cover	cover	NOUN
ejpam-6904	59	34	.	.	PUNCT
ejpam-6904	60	1	remark	remark	NOUN
ejpam-6904	60	2	1	1	NUM
ejpam-6904	60	3	.	.	PUNCT
ejpam-6904	61	1	a	a	DET
ejpam-6904	61	2	2	2	NUM
ejpam-6904	61	3	-	-	PUNCT
ejpam-6904	61	4	path	path	NOUN
ejpam-6904	61	5	geodetic	geodetic	ADJ
ejpam-6904	61	6	set	set	NOUN
ejpam-6904	61	7	need	need	AUX
ejpam-6904	61	8	not	not	PART
ejpam-6904	61	9	be	be	AUX
ejpam-6904	61	10	a	a	DET
ejpam-6904	61	11	vertex	vertex	NOUN
ejpam-6904	61	12	cover	cover	NOUN
ejpam-6904	61	13	and	and	CCONJ
ejpam-6904	61	14	a	a	DET
ejpam-6904	61	15	vertex	vertex	NOUN
ejpam-6904	61	16	cover	cover	NOUN
ejpam-6904	61	17	need	need	AUX
ejpam-6904	61	18	not	not	PART
ejpam-6904	61	19	be	be	AUX
ejpam-6904	61	20	a	a	DET
ejpam-6904	61	21	2	2	NUM
ejpam-6904	61	22	-	-	PUNCT
ejpam-6904	61	23	path	path	NOUN
ejpam-6904	61	24	geodetic	geodetic	ADJ
ejpam-6904	61	25	set	set	NOUN
ejpam-6904	61	26	.	.	PUNCT
ejpam-6904	62	1	to	to	PART
ejpam-6904	62	2	see	see	VERB
ejpam-6904	62	3	this	this	PRON
ejpam-6904	62	4	,	,	PUNCT
ejpam-6904	62	5	consider	consider	VERB
ejpam-6904	62	6	graph	graph	NOUN
ejpam-6904	62	7	g	g	NOUN
ejpam-6904	62	8	in	in	ADP
ejpam-6904	62	9	figure	figure	NOUN
ejpam-6904	62	10	2	2	NUM
ejpam-6904	62	11	.	.	PUNCT
ejpam-6904	63	1	let	let	VERB
ejpam-6904	63	2	s1	s1	PROPN
ejpam-6904	63	3	=	=	SYM
ejpam-6904	63	4	{	{	PUNCT
ejpam-6904	63	5	v1	v1	PROPN
ejpam-6904	63	6	,	,	PUNCT
ejpam-6904	63	7	v3	v3	PROPN
ejpam-6904	63	8	,	,	PUNCT
ejpam-6904	63	9	w1	w1	NOUN
ejpam-6904	63	10	,	,	PUNCT
ejpam-6904	63	11	w3	w3	PROPN
ejpam-6904	63	12	}	}	PUNCT
ejpam-6904	63	13	and	and	CCONJ
ejpam-6904	63	14	s2	s2	VERB
ejpam-6904	63	15	=	=	SYM
ejpam-6904	63	16	{	{	PUNCT
ejpam-6904	63	17	v2	v2	PROPN
ejpam-6904	63	18	,	,	PUNCT
ejpam-6904	63	19	w2	w2	NOUN
ejpam-6904	63	20	}	}	PUNCT
ejpam-6904	63	21	.	.	PUNCT
ejpam-6904	64	1	clearly	clearly	ADV
ejpam-6904	64	2	,	,	PUNCT
ejpam-6904	64	3	s1	s1	PROPN
ejpam-6904	64	4	and	and	CCONJ
ejpam-6904	64	5	s2	s2	PROPN
ejpam-6904	64	6	are	be	AUX
ejpam-6904	64	7	2	2	NUM
ejpam-6904	64	8	-	-	PUNCT
ejpam-6904	64	9	path	path	NOUN
ejpam-6904	64	10	geodetic	geodetic	ADJ
ejpam-6904	64	11	set	set	NOUN
ejpam-6904	64	12	and	and	CCONJ
ejpam-6904	64	13	vertex	vertex	NOUN
ejpam-6904	64	14	cover	cover	NOUN
ejpam-6904	64	15	of	of	ADP
ejpam-6904	64	16	g	g	NOUN
ejpam-6904	64	17	,	,	PUNCT
ejpam-6904	64	18	respectively	respectively	ADV
ejpam-6904	64	19	.	.	PUNCT
ejpam-6904	65	1	since	since	SCONJ
ejpam-6904	65	2	v2w2	v2w2	PROPN
ejpam-6904	65	3	∈	∈	PROPN
ejpam-6904	65	4	e(g	e(g	PROPN
ejpam-6904	65	5	)	)	PUNCT
ejpam-6904	65	6	and	and	CCONJ
ejpam-6904	65	7	v2	v2	PROPN
ejpam-6904	65	8	,	,	PUNCT
ejpam-6904	65	9	w2	w2	NOUN
ejpam-6904	65	10	/∈	/∈	PUNCT
ejpam-6904	65	11	s1	s1	PROPN
ejpam-6904	65	12	,	,	PUNCT
ejpam-6904	65	13	it	it	PRON
ejpam-6904	65	14	follows	follow	VERB
ejpam-6904	65	15	that	that	SCONJ
ejpam-6904	65	16	s1	s1	NOUN
ejpam-6904	65	17	is	be	AUX
ejpam-6904	65	18	not	not	PART
ejpam-6904	65	19	a	a	DET
ejpam-6904	65	20	vertex	vertex	NOUN
ejpam-6904	65	21	cover	cover	NOUN
ejpam-6904	65	22	of	of	ADP
ejpam-6904	65	23	g.	g.	NOUN
ejpam-6904	65	24	by	by	ADP
ejpam-6904	65	25	definition	definition	NOUN
ejpam-6904	65	26	,	,	PUNCT
ejpam-6904	65	27	s2	s2	PROPN
ejpam-6904	65	28	is	be	AUX
ejpam-6904	65	29	not	not	PART
ejpam-6904	65	30	a	a	DET
ejpam-6904	65	31	2	2	NUM
ejpam-6904	65	32	-	-	PUNCT
ejpam-6904	65	33	path	path	NOUN
ejpam-6904	65	34	geodetic	geodetic	ADJ
ejpam-6904	65	35	set	set	NOUN
ejpam-6904	65	36	in	in	ADP
ejpam-6904	65	37	g.	g.	PROPN
ejpam-6904	65	38	it	it	PRON
ejpam-6904	65	39	can	can	AUX
ejpam-6904	65	40	easily	easily	ADV
ejpam-6904	65	41	be	be	AUX
ejpam-6904	65	42	verified	verify	VERB
ejpam-6904	65	43	that	that	SCONJ
ejpam-6904	65	44	g2p(g	g2p(g	NOUN
ejpam-6904	65	45	)	)	PUNCT
ejpam-6904	66	1	=	=	SYM
ejpam-6904	66	2	|s1|	|s1|	NOUN
ejpam-6904	66	3	=	=	SYM
ejpam-6904	66	4	4	4	NUM
ejpam-6904	66	5	and	and	CCONJ
ejpam-6904	66	6	a.	a.	PROPN
ejpam-6904	66	7	b.	b.	PROPN
ejpam-6904	66	8	tapeing	tapeing	PROPN
ejpam-6904	66	9	,	,	PUNCT
ejpam-6904	66	10	s.	s.	PROPN
ejpam-6904	66	11	r.	r.	PROPN
ejpam-6904	66	12	canoy	canoy	PROPN
ejpam-6904	66	13	/	/	SYM
ejpam-6904	66	14	eur	eur	PROPN
ejpam-6904	66	15	.	.	PUNCT
ejpam-6904	67	1	j.	j.	PROPN
ejpam-6904	67	2	pure	pure	PROPN
ejpam-6904	67	3	appl	appl	PROPN
ejpam-6904	67	4	.	.	PROPN
ejpam-6904	67	5	math	math	PROPN
ejpam-6904	67	6	,	,	PUNCT
ejpam-6904	67	7	18	18	NUM
ejpam-6904	67	8	(	(	PUNCT
ejpam-6904	67	9	4	4	NUM
ejpam-6904	67	10	)	)	PUNCT
ejpam-6904	67	11	(	(	PUNCT
ejpam-6904	67	12	2025	2025	NUM
ejpam-6904	67	13	)	)	PUNCT
ejpam-6904	67	14	,	,	PUNCT
ejpam-6904	67	15	6904	6904	NUM
ejpam-6904	67	16	4	4	NUM
ejpam-6904	67	17	of	of	ADP
ejpam-6904	67	18	14	14	NUM
ejpam-6904	67	19	β(g	β(g	NUM
ejpam-6904	67	20	)	)	PUNCT
ejpam-6904	67	21	=	=	SYM
ejpam-6904	68	1	2	2	X
ejpam-6904	68	2	.	.	PUNCT
ejpam-6904	68	3	................................................................................................................	................................................................................................................	PUNCT
ejpam-6904	68	4	................................................................................................................	................................................................................................................	PUNCT
ejpam-6904	68	5	....................................	....................................	PUNCT
ejpam-6904	68	6	................................................................................................................	................................................................................................................	PUNCT
ejpam-6904	68	7	................................................................................................................	................................................................................................................	PUNCT
ejpam-6904	68	8	....................................	....................................	PUNCT
ejpam-6904	68	9	.........	.........	PUNCT
ejpam-6904	68	10	........	........	PUNCT
ejpam-6904	68	11	........	........	PUNCT
ejpam-6904	68	12	........	........	PUNCT
ejpam-6904	68	13	........	........	PUNCT
ejpam-6904	68	14	........	........	PUNCT
ejpam-6904	68	15	........	........	PUNCT
ejpam-6904	68	16	........	........	PUNCT
ejpam-6904	68	17	........	........	PUNCT
ejpam-6904	68	18	...	...	PUNCT
ejpam-6904	68	19	....................................	....................................	PUNCT
ejpam-6904	68	20	....................................	....................................	PUNCT
ejpam-6904	69	1	v1	v1	VERB
ejpam-6904	69	2	v2	v2	PROPN
ejpam-6904	69	3	v3	v3	PROPN
ejpam-6904	69	4	w1	w1	PROPN
ejpam-6904	69	5	w2	w2	PROPN
ejpam-6904	69	6	w3	w3	PROPN
ejpam-6904	69	7	g	g	PROPN
ejpam-6904	69	8	figure	figure	NOUN
ejpam-6904	69	9	2	2	NUM
ejpam-6904	69	10	:	:	PUNCT
ejpam-6904	69	11	graph	graph	VERB
ejpam-6904	69	12	g	g	NOUN
ejpam-6904	69	13	with	with	ADP
ejpam-6904	69	14	β(g	β(g	PROPN
ejpam-6904	69	15	)	)	PUNCT
ejpam-6904	69	16	=	=	SYM
ejpam-6904	69	17	2	2	NUM
ejpam-6904	69	18	and	and	CCONJ
ejpam-6904	69	19	g2p(g	g2p(g	NOUN
ejpam-6904	69	20	)	)	PUNCT
ejpam-6904	69	21	=	=	SYM
ejpam-6904	70	1	4	4	NUM
ejpam-6904	70	2	remark	remark	NOUN
ejpam-6904	70	3	2	2	NUM
ejpam-6904	70	4	.	.	PUNCT
ejpam-6904	71	1	let	let	VERB
ejpam-6904	71	2	g	g	PRON
ejpam-6904	71	3	be	be	AUX
ejpam-6904	71	4	a	a	DET
ejpam-6904	71	5	graph	graph	NOUN
ejpam-6904	71	6	and	and	CCONJ
ejpam-6904	71	7	let	let	VERB
ejpam-6904	71	8	s	s	PRON
ejpam-6904	71	9	be	be	AUX
ejpam-6904	71	10	a	a	DET
ejpam-6904	71	11	vertex	vertex	NOUN
ejpam-6904	71	12	cover	cover	NOUN
ejpam-6904	71	13	of	of	ADP
ejpam-6904	71	14	g.	g.	PROPN
ejpam-6904	72	1	then	then	ADV
ejpam-6904	72	2	v	v	X
ejpam-6904	72	3	(	(	PUNCT
ejpam-6904	72	4	g	g	NOUN
ejpam-6904	72	5	)	)	PUNCT
ejpam-6904	72	6	\	\	PROPN
ejpam-6904	73	1	s	s	PART
ejpam-6904	73	2	is	be	AUX
ejpam-6904	73	3	an	an	DET
ejpam-6904	73	4	independent	independent	ADJ
ejpam-6904	73	5	set	set	NOUN
ejpam-6904	73	6	.	.	PUNCT
ejpam-6904	74	1	moreover	moreover	ADV
ejpam-6904	74	2	,	,	PUNCT
ejpam-6904	74	3	α(g	α(g	NUM
ejpam-6904	74	4	)	)	PUNCT
ejpam-6904	74	5	+	+	NUM
ejpam-6904	74	6	β(g	β(g	PROPN
ejpam-6904	74	7	)	)	PUNCT
ejpam-6904	74	8	=	=	SYM
ejpam-6904	74	9	|v	|v	PROPN
ejpam-6904	74	10	(	(	PUNCT
ejpam-6904	74	11	g)|	g)|	NOUN
ejpam-6904	74	12	.	.	PUNCT
ejpam-6904	75	1	proposition	proposition	NOUN
ejpam-6904	75	2	1	1	NUM
ejpam-6904	75	3	.	.	PUNCT
ejpam-6904	76	1	let	let	VERB
ejpam-6904	76	2	g	g	NOUN
ejpam-6904	76	3	be	be	AUX
ejpam-6904	76	4	any	any	DET
ejpam-6904	76	5	graph	graph	NOUN
ejpam-6904	76	6	and	and	CCONJ
ejpam-6904	76	7	let	let	VERB
ejpam-6904	76	8	s	s	PRON
ejpam-6904	76	9	be	be	AUX
ejpam-6904	76	10	a	a	DET
ejpam-6904	76	11	2	2	NUM
ejpam-6904	76	12	-	-	PUNCT
ejpam-6904	76	13	path	path	NOUN
ejpam-6904	76	14	geodetic	geodetic	ADJ
ejpam-6904	76	15	set	set	NOUN
ejpam-6904	76	16	in	in	ADP
ejpam-6904	76	17	g.	g.	PROPN
ejpam-6904	76	18	then	then	ADV
ejpam-6904	76	19	each	each	PRON
ejpam-6904	76	20	of	of	ADP
ejpam-6904	76	21	the	the	DET
ejpam-6904	76	22	following	follow	VERB
ejpam-6904	76	23	holds	hold	VERB
ejpam-6904	76	24	:	:	PUNCT
ejpam-6904	76	25	(	(	PUNCT
ejpam-6904	76	26	i	i	NOUN
ejpam-6904	76	27	)	)	PUNCT
ejpam-6904	76	28	ext(g	ext(g	PROPN
ejpam-6904	76	29	)	)	PUNCT
ejpam-6904	76	30	∪	∪	ADP
ejpam-6904	76	31	i(g	i(g	NOUN
ejpam-6904	76	32	)	)	PUNCT
ejpam-6904	77	1	⊆	⊆	NUM
ejpam-6904	77	2	s	s	NOUN
ejpam-6904	77	3	and	and	CCONJ
ejpam-6904	77	4	|ext(g)|+	|ext(g)|+	NOUN
ejpam-6904	77	5	|i(g)|	|i(g)|	PROPN
ejpam-6904	77	6	≤	≤	ADJ
ejpam-6904	77	7	g2p(g	g2p(g	NOUN
ejpam-6904	77	8	)	)	PUNCT
ejpam-6904	77	9	.	.	PUNCT
ejpam-6904	78	1	(	(	PUNCT
ejpam-6904	78	2	ii	ii	X
ejpam-6904	78	3	)	)	PUNCT
ejpam-6904	78	4	s	s	VERB
ejpam-6904	78	5	is	be	AUX
ejpam-6904	78	6	a	a	DET
ejpam-6904	78	7	2	2	NUM
ejpam-6904	78	8	-	-	PUNCT
ejpam-6904	78	9	dominating	dominating	NOUN
ejpam-6904	78	10	set	set	NOUN
ejpam-6904	78	11	and	and	CCONJ
ejpam-6904	78	12	γ2(g	γ2(g	NUM
ejpam-6904	78	13	)	)	PUNCT
ejpam-6904	78	14	≤	≤	ADJ
ejpam-6904	78	15	g2p(g	g2p(g	NOUN
ejpam-6904	78	16	)	)	PUNCT
ejpam-6904	78	17	.	.	PUNCT
ejpam-6904	79	1	proof	proof	NOUN
ejpam-6904	79	2	.	.	PUNCT
ejpam-6904	80	1	clearly	clearly	ADV
ejpam-6904	80	2	,	,	PUNCT
ejpam-6904	80	3	every	every	DET
ejpam-6904	80	4	2	2	NUM
ejpam-6904	80	5	-	-	PUNCT
ejpam-6904	80	6	path	path	NOUN
ejpam-6904	80	7	geodetic	geodetic	ADJ
ejpam-6904	80	8	set	set	NOUN
ejpam-6904	80	9	is	be	AUX
ejpam-6904	80	10	a	a	DET
ejpam-6904	80	11	geodetic	geodetic	ADJ
ejpam-6904	80	12	set	set	NOUN
ejpam-6904	80	13	.	.	PUNCT
ejpam-6904	81	1	now	now	ADV
ejpam-6904	81	2	,	,	PUNCT
ejpam-6904	81	3	since	since	SCONJ
ejpam-6904	81	4	every	every	DET
ejpam-6904	81	5	geodetic	geodetic	ADJ
ejpam-6904	81	6	set	set	NOUN
ejpam-6904	81	7	in	in	ADP
ejpam-6904	81	8	a	a	DET
ejpam-6904	81	9	graph	graph	NOUN
ejpam-6904	81	10	contains	contain	VERB
ejpam-6904	81	11	all	all	DET
ejpam-6904	81	12	the	the	DET
ejpam-6904	81	13	extreme	extreme	ADJ
ejpam-6904	81	14	and	and	CCONJ
ejpam-6904	81	15	isolated	isolated	ADJ
ejpam-6904	81	16	vertices	vertex	NOUN
ejpam-6904	81	17	,	,	PUNCT
ejpam-6904	81	18	it	it	PRON
ejpam-6904	81	19	follows	follow	VERB
ejpam-6904	81	20	that	that	SCONJ
ejpam-6904	81	21	ext(g)∪i(g	ext(g)∪i(g	PROPN
ejpam-6904	81	22	)	)	PUNCT
ejpam-6904	81	23	⊆	⊆	NUM
ejpam-6904	81	24	s	s	NOUN
ejpam-6904	81	25	and	and	CCONJ
ejpam-6904	81	26	the	the	DET
ejpam-6904	81	27	given	give	VERB
ejpam-6904	81	28	inequality	inequality	NOUN
ejpam-6904	81	29	in	in	ADP
ejpam-6904	81	30	(	(	PUNCT
ejpam-6904	81	31	i	i	NOUN
ejpam-6904	81	32	)	)	PUNCT
ejpam-6904	81	33	holds	hold	VERB
ejpam-6904	81	34	.	.	PUNCT
ejpam-6904	82	1	next	next	ADV
ejpam-6904	82	2	,	,	PUNCT
ejpam-6904	82	3	let	let	VERB
ejpam-6904	82	4	v	v	NUM
ejpam-6904	82	5	∈	∈	PROPN
ejpam-6904	82	6	v	v	NOUN
ejpam-6904	82	7	(	(	PUNCT
ejpam-6904	82	8	g	g	NOUN
ejpam-6904	82	9	)	)	PUNCT
ejpam-6904	82	10	\s	\s	NOUN
ejpam-6904	82	11	.	.	PUNCT
ejpam-6904	83	1	since	since	SCONJ
ejpam-6904	83	2	s	s	PROPN
ejpam-6904	83	3	is	be	AUX
ejpam-6904	83	4	a	a	DET
ejpam-6904	83	5	2	2	NUM
ejpam-6904	83	6	-	-	PUNCT
ejpam-6904	83	7	path	path	NOUN
ejpam-6904	83	8	geodetic	geodetic	ADJ
ejpam-6904	83	9	set	set	NOUN
ejpam-6904	83	10	in	in	ADP
ejpam-6904	83	11	g	g	NOUN
ejpam-6904	83	12	,	,	PUNCT
ejpam-6904	83	13	there	there	PRON
ejpam-6904	83	14	exist	exist	VERB
ejpam-6904	83	15	p	p	PRON
ejpam-6904	83	16	,	,	PUNCT
ejpam-6904	83	17	q	q	PROPN
ejpam-6904	83	18	∈	∈	PROPN
ejpam-6904	83	19	s	s	VERB
ejpam-6904	83	20	such	such	ADJ
ejpam-6904	83	21	that	that	SCONJ
ejpam-6904	83	22	dg(p	dg(p	NOUN
ejpam-6904	83	23	,	,	PUNCT
ejpam-6904	83	24	q	q	X
ejpam-6904	83	25	)	)	PUNCT
ejpam-6904	83	26	=	=	SYM
ejpam-6904	83	27	2	2	NUM
ejpam-6904	83	28	and	and	CCONJ
ejpam-6904	83	29	v	v	ADP
ejpam-6904	83	30	∈	∈	NOUN
ejpam-6904	83	31	ig(p	ig(p	NOUN
ejpam-6904	83	32	,	,	PUNCT
ejpam-6904	83	33	q	q	NOUN
ejpam-6904	83	34	)	)	PUNCT
ejpam-6904	83	35	.	.	PUNCT
ejpam-6904	84	1	it	it	PRON
ejpam-6904	84	2	follows	follow	VERB
ejpam-6904	84	3	that	that	SCONJ
ejpam-6904	84	4	s	s	VERB
ejpam-6904	84	5	is	be	AUX
ejpam-6904	84	6	a	a	DET
ejpam-6904	84	7	2	2	NUM
ejpam-6904	84	8	-	-	PUNCT
ejpam-6904	84	9	dominating	dominating	NOUN
ejpam-6904	84	10	set	set	NOUN
ejpam-6904	84	11	in	in	ADP
ejpam-6904	84	12	g.	g.	PROPN
ejpam-6904	84	13	thus	thus	ADV
ejpam-6904	84	14	,	,	PUNCT
ejpam-6904	84	15	γ2(g	γ2(g	ADP
ejpam-6904	84	16	)	)	PUNCT
ejpam-6904	84	17	≤	≤	NOUN
ejpam-6904	84	18	g2(g	g2(g	NOUN
ejpam-6904	84	19	)	)	PUNCT
ejpam-6904	84	20	.	.	PUNCT
ejpam-6904	85	1	the	the	DET
ejpam-6904	85	2	next	next	ADJ
ejpam-6904	85	3	result	result	NOUN
ejpam-6904	85	4	is	be	AUX
ejpam-6904	85	5	a	a	DET
ejpam-6904	85	6	direct	direct	ADJ
ejpam-6904	85	7	consequence	consequence	NOUN
ejpam-6904	85	8	of	of	ADP
ejpam-6904	85	9	proposition	proposition	NOUN
ejpam-6904	85	10	1(i	1(i	NUM
ejpam-6904	85	11	)	)	PUNCT
ejpam-6904	85	12	.	.	PUNCT
ejpam-6904	86	1	corollary	corollary	ADJ
ejpam-6904	86	2	1	1	NUM
ejpam-6904	86	3	.	.	PUNCT
ejpam-6904	87	1	let	let	VERB
ejpam-6904	87	2	n	n	PRON
ejpam-6904	87	3	be	be	AUX
ejpam-6904	87	4	a	a	DET
ejpam-6904	87	5	positive	positive	ADJ
ejpam-6904	87	6	integer	integer	NOUN
ejpam-6904	87	7	.	.	PUNCT
ejpam-6904	88	1	then	then	ADV
ejpam-6904	88	2	β2pg(kn	β2pg(kn	PUNCT
ejpam-6904	88	3	)	)	PUNCT
ejpam-6904	88	4	=	=	SYM
ejpam-6904	89	1	β2pg(kn	β2pg(kn	PUNCT
ejpam-6904	89	2	)	)	PUNCT
ejpam-6904	89	3	=	=	SYM
ejpam-6904	89	4	n.	n.	NOUN
ejpam-6904	89	5	theorem	theorem	NOUN
ejpam-6904	89	6	1	1	X
ejpam-6904	89	7	.	.	PUNCT
ejpam-6904	90	1	let	let	VERB
ejpam-6904	90	2	g1	g1	PROPN
ejpam-6904	90	3	,	,	PUNCT
ejpam-6904	90	4	g2	g2	PROPN
ejpam-6904	90	5	,	,	PUNCT
ejpam-6904	90	6	..	..	PUNCT
ejpam-6904	90	7	,	,	PUNCT
ejpam-6904	90	8	gk	gk	PROPN
ejpam-6904	90	9	,	,	PUNCT
ejpam-6904	90	10	where	where	SCONJ
ejpam-6904	90	11	k	k	PROPN
ejpam-6904	90	12	≥	≥	PROPN
ejpam-6904	90	13	1	1	NUM
ejpam-6904	90	14	,	,	PUNCT
ejpam-6904	90	15	be	be	AUX
ejpam-6904	90	16	the	the	DET
ejpam-6904	90	17	components	component	NOUN
ejpam-6904	90	18	of	of	ADP
ejpam-6904	90	19	g.	g.	PROPN
ejpam-6904	90	20	then	then	ADV
ejpam-6904	90	21	s	s	VERB
ejpam-6904	90	22	is	be	AUX
ejpam-6904	90	23	a	a	DET
ejpam-6904	90	24	2	2	NUM
ejpam-6904	90	25	-	-	PUNCT
ejpam-6904	90	26	path	path	NOUN
ejpam-6904	90	27	geodetic	geodetic	ADJ
ejpam-6904	90	28	vertex	vertex	NOUN
ejpam-6904	90	29	cover	cover	NOUN
ejpam-6904	90	30	of	of	ADP
ejpam-6904	90	31	g	g	PROPN
ejpam-6904	91	1	if	if	SCONJ
ejpam-6904	91	2	and	and	CCONJ
ejpam-6904	91	3	only	only	ADV
ejpam-6904	91	4	if	if	SCONJ
ejpam-6904	91	5	s	s	NOUN
ejpam-6904	91	6	=	=	SYM
ejpam-6904	91	7	∪	∪	ADP
ejpam-6904	91	8	j∈[k	j∈[k	PROPN
ejpam-6904	91	9	]	]	PUNCT
ejpam-6904	91	10	sj	sj	NOUN
ejpam-6904	91	11	,	,	PUNCT
ejpam-6904	91	12	where	where	SCONJ
ejpam-6904	91	13	sj	sj	PROPN
ejpam-6904	91	14	is	be	AUX
ejpam-6904	91	15	a	a	DET
ejpam-6904	91	16	2	2	NUM
ejpam-6904	91	17	-	-	PUNCT
ejpam-6904	91	18	path	path	NOUN
ejpam-6904	91	19	geodetic	geodetic	ADJ
ejpam-6904	91	20	vertex	vertex	NOUN
ejpam-6904	91	21	cover	cover	NOUN
ejpam-6904	91	22	of	of	ADP
ejpam-6904	91	23	gj	gj	NOUN
ejpam-6904	91	24	for	for	ADP
ejpam-6904	91	25	every	every	DET
ejpam-6904	91	26	j	j	PROPN
ejpam-6904	91	27	∈	∈	PROPN
ejpam-6904	92	1	[	[	X
ejpam-6904	92	2	k	k	X
ejpam-6904	92	3	]	]	X
ejpam-6904	92	4	=	=	X
ejpam-6904	92	5	{	{	PUNCT
ejpam-6904	92	6	1	1	NUM
ejpam-6904	92	7	,	,	PUNCT
ejpam-6904	92	8	2	2	NUM
ejpam-6904	92	9	,	,	PUNCT
ejpam-6904	92	10	..	..	PUNCT
ejpam-6904	92	11	,	,	PUNCT
ejpam-6904	92	12	k	k	NOUN
ejpam-6904	92	13	}	}	PUNCT
ejpam-6904	92	14	.	.	PUNCT
ejpam-6904	93	1	moreover	moreover	ADV
ejpam-6904	93	2	,	,	PUNCT
ejpam-6904	93	3	β2pg(g	β2pg(g	PUNCT
ejpam-6904	93	4	)	)	PUNCT
ejpam-6904	93	5	=	=	SYM
ejpam-6904	93	6	∑	∑	PUNCT
ejpam-6904	93	7	j∈[k	j∈[k	PROPN
ejpam-6904	93	8	]	]	PUNCT
ejpam-6904	93	9	β2pg(gj	β2pg(gj	PUNCT
ejpam-6904	93	10	)	)	PUNCT
ejpam-6904	93	11	.	.	PUNCT
ejpam-6904	94	1	proof	proof	NOUN
ejpam-6904	94	2	.	.	PUNCT
ejpam-6904	95	1	suppose	suppose	VERB
ejpam-6904	95	2	s	s	NOUN
ejpam-6904	95	3	is	be	AUX
ejpam-6904	95	4	a	a	DET
ejpam-6904	95	5	2	2	NUM
ejpam-6904	95	6	-	-	PUNCT
ejpam-6904	95	7	path	path	NOUN
ejpam-6904	95	8	geodetic	geodetic	ADJ
ejpam-6904	95	9	vertex	vertex	NOUN
ejpam-6904	95	10	cover	cover	NOUN
ejpam-6904	95	11	of	of	ADP
ejpam-6904	95	12	g.	g.	PROPN
ejpam-6904	95	13	let	let	VERB
ejpam-6904	95	14	sj	sj	INTJ
ejpam-6904	95	15	=	=	NOUN
ejpam-6904	95	16	s	s	PART
ejpam-6904	95	17	∩	∩	ADJ
ejpam-6904	95	18	v	v	NOUN
ejpam-6904	95	19	(	(	PUNCT
ejpam-6904	95	20	gj	gj	NOUN
ejpam-6904	95	21	)	)	PUNCT
ejpam-6904	95	22	for	for	ADP
ejpam-6904	95	23	each	each	DET
ejpam-6904	95	24	j	j	PROPN
ejpam-6904	95	25	∈	∈	PROPN
ejpam-6904	96	1	[	[	X
ejpam-6904	96	2	k	k	X
ejpam-6904	96	3	]	]	X
ejpam-6904	96	4	=	=	X
ejpam-6904	96	5	{	{	PUNCT
ejpam-6904	96	6	1	1	NUM
ejpam-6904	96	7	,	,	PUNCT
ejpam-6904	96	8	2	2	NUM
ejpam-6904	96	9	,	,	PUNCT
ejpam-6904	96	10	..	..	PUNCT
ejpam-6904	96	11	,	,	PUNCT
ejpam-6904	96	12	k	k	NOUN
ejpam-6904	96	13	}	}	PUNCT
ejpam-6904	96	14	.	.	PUNCT
ejpam-6904	97	1	then	then	ADV
ejpam-6904	97	2	s	s	VERB
ejpam-6904	97	3	=	=	SYM
ejpam-6904	97	4	∪j∈[k]sj	∪j∈[k]sj	PROPN
ejpam-6904	97	5	.	.	PUNCT
ejpam-6904	98	1	since	since	SCONJ
ejpam-6904	98	2	s	s	PROPN
ejpam-6904	98	3	is	be	AUX
ejpam-6904	98	4	a	a	DET
ejpam-6904	98	5	2	2	NUM
ejpam-6904	98	6	-	-	PUNCT
ejpam-6904	98	7	path	path	NOUN
ejpam-6904	98	8	geodetic	geodetic	ADJ
ejpam-6904	98	9	set	set	NOUN
ejpam-6904	98	10	in	in	ADP
ejpam-6904	98	11	g	g	NOUN
ejpam-6904	98	12	,	,	PUNCT
ejpam-6904	98	13	sj	sj	PROPN
ejpam-6904	98	14	̸=	̸=	PROPN
ejpam-6904	98	15	∅	∅	NOUN
ejpam-6904	98	16	for	for	ADP
ejpam-6904	98	17	all	all	DET
ejpam-6904	98	18	j	j	PROPN
ejpam-6904	98	19	∈	∈	PROPN
ejpam-6904	99	1	[	[	X
ejpam-6904	99	2	k	k	X
ejpam-6904	99	3	]	]	X
ejpam-6904	99	4	.	.	PUNCT
ejpam-6904	100	1	next	next	ADJ
ejpam-6904	100	2	,	,	PUNCT
ejpam-6904	100	3	let	let	VERB
ejpam-6904	100	4	j	j	PROPN
ejpam-6904	100	5	∈	∈	PROPN
ejpam-6904	101	1	[	[	X
ejpam-6904	101	2	k	k	X
ejpam-6904	101	3	]	]	PUNCT
ejpam-6904	101	4	and	and	CCONJ
ejpam-6904	101	5	let	let	VERB
ejpam-6904	101	6	ab	ab	PROPN
ejpam-6904	101	7	∈	∈	PROPN
ejpam-6904	101	8	e(gj	e(gj	NUM
ejpam-6904	101	9	)	)	PUNCT
ejpam-6904	101	10	.	.	PUNCT
ejpam-6904	102	1	since	since	SCONJ
ejpam-6904	102	2	s	s	PROPN
ejpam-6904	102	3	is	be	AUX
ejpam-6904	102	4	a	a	DET
ejpam-6904	102	5	vertex	vertex	NOUN
ejpam-6904	102	6	cover	cover	NOUN
ejpam-6904	102	7	in	in	ADP
ejpam-6904	102	8	g	g	PROPN
ejpam-6904	102	9	,	,	PUNCT
ejpam-6904	102	10	it	it	PRON
ejpam-6904	102	11	follows	follow	VERB
ejpam-6904	102	12	that	that	SCONJ
ejpam-6904	102	13	a	a	DET
ejpam-6904	102	14	∈	∈	PROPN
ejpam-6904	102	15	s	s	NOUN
ejpam-6904	102	16	or	or	CCONJ
ejpam-6904	102	17	b	b	PROPN
ejpam-6904	102	18	∈	∈	PROPN
ejpam-6904	102	19	s.	s.	PROPN
ejpam-6904	102	20	hence	hence	ADV
ejpam-6904	102	21	,	,	PUNCT
ejpam-6904	102	22	a	a	DET
ejpam-6904	102	23	∈	∈	NOUN
ejpam-6904	102	24	sj	sj	NOUN
ejpam-6904	102	25	or	or	CCONJ
ejpam-6904	102	26	b	b	X
ejpam-6904	102	27	∈	∈	NOUN
ejpam-6904	102	28	sj	sj	INTJ
ejpam-6904	102	29	,	,	PUNCT
ejpam-6904	102	30	showing	show	VERB
ejpam-6904	102	31	that	that	SCONJ
ejpam-6904	102	32	sj	sj	PROPN
ejpam-6904	102	33	is	be	AUX
ejpam-6904	102	34	a	a	DET
ejpam-6904	102	35	vertex	vertex	NOUN
ejpam-6904	102	36	cover	cover	NOUN
ejpam-6904	102	37	of	of	ADP
ejpam-6904	102	38	gj	gj	NOUN
ejpam-6904	102	39	.	.	PUNCT
ejpam-6904	103	1	now	now	ADV
ejpam-6904	103	2	let	let	VERB
ejpam-6904	103	3	v	v	NUM
ejpam-6904	103	4	∈	∈	PROPN
ejpam-6904	103	5	v	v	NOUN
ejpam-6904	103	6	(	(	PUNCT
ejpam-6904	103	7	gj	gj	NOUN
ejpam-6904	103	8	)	)	PUNCT
ejpam-6904	103	9	\	\	PROPN
ejpam-6904	104	1	s.	s.	PROPN
ejpam-6904	104	2	since	since	SCONJ
ejpam-6904	104	3	s	s	PROPN
ejpam-6904	104	4	is	be	AUX
ejpam-6904	104	5	a	a	DET
ejpam-6904	104	6	2	2	NUM
ejpam-6904	104	7	-	-	PUNCT
ejpam-6904	104	8	path	path	NOUN
ejpam-6904	104	9	geodetic	geodetic	ADJ
ejpam-6904	104	10	set	set	NOUN
ejpam-6904	104	11	in	in	ADP
ejpam-6904	104	12	g	g	NOUN
ejpam-6904	104	13	,	,	PUNCT
ejpam-6904	104	14	there	there	PRON
ejpam-6904	104	15	exist	exist	VERB
ejpam-6904	104	16	y	y	PROPN
ejpam-6904	104	17	,	,	PUNCT
ejpam-6904	104	18	z	z	PROPN
ejpam-6904	104	19	∈	∈	PROPN
ejpam-6904	104	20	s	s	VERB
ejpam-6904	104	21	such	such	ADJ
ejpam-6904	104	22	that	that	DET
ejpam-6904	104	23	dg(y	dg(y	ADJ
ejpam-6904	104	24	,	,	PUNCT
ejpam-6904	104	25	z	z	NOUN
ejpam-6904	104	26	)	)	PUNCT
ejpam-6904	104	27	=	=	SYM
ejpam-6904	104	28	2	2	NUM
ejpam-6904	104	29	and	and	CCONJ
ejpam-6904	104	30	x	x	PROPN
ejpam-6904	104	31	∈	∈	PROPN
ejpam-6904	104	32	ig(y	ig(y	NOUN
ejpam-6904	104	33	,	,	PUNCT
ejpam-6904	104	34	z	z	NOUN
ejpam-6904	104	35	)	)	PUNCT
ejpam-6904	104	36	.	.	PUNCT
ejpam-6904	105	1	this	this	PRON
ejpam-6904	105	2	implies	imply	VERB
ejpam-6904	105	3	that	that	SCONJ
ejpam-6904	105	4	there	there	PRON
ejpam-6904	105	5	exist	exist	VERB
ejpam-6904	105	6	y	y	PROPN
ejpam-6904	105	7	,	,	PUNCT
ejpam-6904	105	8	z	z	NOUN
ejpam-6904	105	9	∈	∈	PROPN
ejpam-6904	105	10	sj	sj	VERB
ejpam-6904	105	11	such	such	ADJ
ejpam-6904	105	12	that	that	PRON
ejpam-6904	105	13	a.	a.	PROPN
ejpam-6904	105	14	b.	b.	PROPN
ejpam-6904	105	15	tapeing	tapeing	PROPN
ejpam-6904	105	16	,	,	PUNCT
ejpam-6904	105	17	s.	s.	PROPN
ejpam-6904	105	18	r.	r.	PROPN
ejpam-6904	105	19	canoy	canoy	PROPN
ejpam-6904	105	20	/	/	SYM
ejpam-6904	105	21	eur	eur	PROPN
ejpam-6904	105	22	.	.	PUNCT
ejpam-6904	106	1	j.	j.	PROPN
ejpam-6904	106	2	pure	pure	PROPN
ejpam-6904	106	3	appl	appl	PROPN
ejpam-6904	106	4	.	.	PROPN
ejpam-6904	106	5	math	math	PROPN
ejpam-6904	106	6	,	,	PUNCT
ejpam-6904	106	7	18	18	NUM
ejpam-6904	106	8	(	(	PUNCT
ejpam-6904	106	9	4	4	NUM
ejpam-6904	106	10	)	)	PUNCT
ejpam-6904	106	11	(	(	PUNCT
ejpam-6904	106	12	2025	2025	NUM
ejpam-6904	106	13	)	)	PUNCT
ejpam-6904	106	14	,	,	PUNCT
ejpam-6904	106	15	6904	6904	NUM
ejpam-6904	106	16	5	5	NUM
ejpam-6904	106	17	of	of	ADP
ejpam-6904	106	18	14	14	NUM
ejpam-6904	106	19	dgj	dgj	NOUN
ejpam-6904	106	20	(	(	PUNCT
ejpam-6904	106	21	y	y	PROPN
ejpam-6904	106	22	,	,	PUNCT
ejpam-6904	106	23	z	z	NOUN
ejpam-6904	106	24	)	)	PUNCT
ejpam-6904	106	25	=	=	SYM
ejpam-6904	106	26	2	2	NUM
ejpam-6904	106	27	and	and	CCONJ
ejpam-6904	106	28	x	x	SYM
ejpam-6904	106	29	∈	∈	PROPN
ejpam-6904	107	1	igj	igj	NOUN
ejpam-6904	107	2	(	(	PUNCT
ejpam-6904	107	3	y	y	NOUN
ejpam-6904	107	4	,	,	PUNCT
ejpam-6904	107	5	z	z	NOUN
ejpam-6904	107	6	)	)	PUNCT
ejpam-6904	107	7	.	.	PUNCT
ejpam-6904	108	1	therefore	therefore	ADV
ejpam-6904	108	2	sj	sj	PROPN
ejpam-6904	108	3	is	be	AUX
ejpam-6904	108	4	a	a	DET
ejpam-6904	108	5	2	2	NUM
ejpam-6904	108	6	-	-	PUNCT
ejpam-6904	108	7	path	path	NOUN
ejpam-6904	108	8	geodetic	geodetic	ADJ
ejpam-6904	108	9	vertex	vertex	NOUN
ejpam-6904	108	10	cover	cover	NOUN
ejpam-6904	108	11	in	in	ADP
ejpam-6904	108	12	gj	gj	NOUN
ejpam-6904	108	13	.	.	PUNCT
ejpam-6904	109	1	in	in	ADP
ejpam-6904	109	2	particular	particular	ADJ
ejpam-6904	109	3	,	,	PUNCT
ejpam-6904	109	4	if	if	SCONJ
ejpam-6904	109	5	s	s	VERB
ejpam-6904	109	6	is	be	AUX
ejpam-6904	109	7	a	a	DET
ejpam-6904	109	8	β2pg	β2pg	PUNCT
ejpam-6904	109	9	-	-	VERB
ejpam-6904	109	10	set	set	NOUN
ejpam-6904	109	11	in	in	ADP
ejpam-6904	109	12	g	g	NOUN
ejpam-6904	109	13	,	,	PUNCT
ejpam-6904	109	14	then	then	ADV
ejpam-6904	109	15	β2pg(g	β2pg(g	PUNCT
ejpam-6904	109	16	)	)	PUNCT
ejpam-6904	109	17	=	=	SYM
ejpam-6904	110	1	|s|	|s|	NOUN
ejpam-6904	110	2	=	=	SYM
ejpam-6904	110	3	∑	∑	PUNCT
ejpam-6904	110	4	j∈[k	j∈[k	PROPN
ejpam-6904	110	5	]	]	X
ejpam-6904	110	6	|sj	|sj	NUM
ejpam-6904	110	7	|	|	ADV
ejpam-6904	110	8	≥	≥	NOUN
ejpam-6904	110	9	∑	∑	ADV
ejpam-6904	110	10	j∈[k	j∈[k	PROPN
ejpam-6904	110	11	]	]	PUNCT
ejpam-6904	110	12	β2pg(gj	β2pg(gj	PUNCT
ejpam-6904	110	13	)	)	PUNCT
ejpam-6904	110	14	.	.	PUNCT
ejpam-6904	111	1	for	for	ADP
ejpam-6904	111	2	the	the	DET
ejpam-6904	111	3	converse	converse	NOUN
ejpam-6904	111	4	,	,	PUNCT
ejpam-6904	111	5	suppose	suppose	VERB
ejpam-6904	111	6	s	s	VERB
ejpam-6904	111	7	=	=	PROPN
ejpam-6904	111	8	∪j∈[k]sj	∪j∈[k]sj	PROPN
ejpam-6904	111	9	,	,	PUNCT
ejpam-6904	111	10	where	where	SCONJ
ejpam-6904	111	11	sj	sj	PROPN
ejpam-6904	111	12	is	be	AUX
ejpam-6904	111	13	a	a	DET
ejpam-6904	111	14	2	2	NUM
ejpam-6904	111	15	-	-	PUNCT
ejpam-6904	111	16	path	path	NOUN
ejpam-6904	111	17	geodetic	geodetic	ADJ
ejpam-6904	111	18	vertex	vertex	NOUN
ejpam-6904	111	19	cover	cover	NOUN
ejpam-6904	111	20	of	of	ADP
ejpam-6904	111	21	gj	gj	NOUN
ejpam-6904	111	22	for	for	ADP
ejpam-6904	111	23	all	all	DET
ejpam-6904	111	24	j	j	PROPN
ejpam-6904	111	25	∈	∈	PROPN
ejpam-6904	112	1	[	[	X
ejpam-6904	112	2	k	k	X
ejpam-6904	112	3	]	]	X
ejpam-6904	112	4	=	=	X
ejpam-6904	112	5	{	{	PUNCT
ejpam-6904	112	6	1	1	NUM
ejpam-6904	112	7	,	,	PUNCT
ejpam-6904	112	8	2	2	NUM
ejpam-6904	112	9	,	,	PUNCT
ejpam-6904	112	10	.	.	PUNCT
ejpam-6904	112	11	.	.	PUNCT
ejpam-6904	112	12	.	.	PUNCT
ejpam-6904	113	1	,	,	PUNCT
ejpam-6904	113	2	k	k	X
ejpam-6904	113	3	}	}	PUNCT
ejpam-6904	113	4	.	.	PUNCT
ejpam-6904	114	1	let	let	VERB
ejpam-6904	114	2	pq	pq	INTJ
ejpam-6904	114	3	∈	∈	PROPN
ejpam-6904	114	4	e(g	e(g	PROPN
ejpam-6904	114	5	)	)	PUNCT
ejpam-6904	114	6	.	.	PUNCT
ejpam-6904	115	1	then	then	ADV
ejpam-6904	115	2	there	there	PRON
ejpam-6904	115	3	exists	exist	VERB
ejpam-6904	115	4	j	j	PROPN
ejpam-6904	115	5	∈	∈	PROPN
ejpam-6904	116	1	[	[	X
ejpam-6904	116	2	k	k	X
ejpam-6904	116	3	]	]	X
ejpam-6904	116	4	such	such	ADJ
ejpam-6904	116	5	that	that	PRON
ejpam-6904	116	6	pq	pq	PROPN
ejpam-6904	116	7	∈	∈	PROPN
ejpam-6904	116	8	e(gj	e(gj	NUM
ejpam-6904	116	9	)	)	PUNCT
ejpam-6904	116	10	.	.	PUNCT
ejpam-6904	117	1	since	since	SCONJ
ejpam-6904	117	2	sj	sj	PROPN
ejpam-6904	117	3	is	be	AUX
ejpam-6904	117	4	a	a	DET
ejpam-6904	117	5	vertex	vertex	NOUN
ejpam-6904	117	6	cover	cover	NOUN
ejpam-6904	117	7	of	of	ADP
ejpam-6904	117	8	gj	gj	NOUN
ejpam-6904	117	9	,	,	PUNCT
ejpam-6904	117	10	we	we	PRON
ejpam-6904	117	11	have	have	VERB
ejpam-6904	117	12	p	p	NOUN
ejpam-6904	117	13	∈	∈	ADJ
ejpam-6904	117	14	sj	sj	NOUN
ejpam-6904	117	15	or	or	CCONJ
ejpam-6904	117	16	q	q	NOUN
ejpam-6904	117	17	∈	∈	PROPN
ejpam-6904	117	18	sj	sj	INTJ
ejpam-6904	117	19	.	.	PUNCT
ejpam-6904	118	1	it	it	PRON
ejpam-6904	118	2	follows	follow	VERB
ejpam-6904	118	3	that	that	SCONJ
ejpam-6904	118	4	p	p	PROPN
ejpam-6904	118	5	∈	∈	PROPN
ejpam-6904	118	6	s	s	PART
ejpam-6904	118	7	or	or	CCONJ
ejpam-6904	118	8	q	q	PROPN
ejpam-6904	118	9	∈	∈	PROPN
ejpam-6904	118	10	s.	s.	PROPN
ejpam-6904	118	11	let	let	VERB
ejpam-6904	118	12	v	v	ADP
ejpam-6904	118	13	∈	∈	PROPN
ejpam-6904	118	14	v	v	NOUN
ejpam-6904	118	15	(	(	PUNCT
ejpam-6904	118	16	g	g	NOUN
ejpam-6904	118	17	)	)	PUNCT
ejpam-6904	118	18	\	\	PROPN
ejpam-6904	119	1	s	s	X
ejpam-6904	119	2	,	,	PUNCT
ejpam-6904	119	3	and	and	CCONJ
ejpam-6904	119	4	let	let	VERB
ejpam-6904	119	5	t	t	X
ejpam-6904	119	6	∈	∈	PROPN
ejpam-6904	120	1	[	[	X
ejpam-6904	120	2	k	k	X
ejpam-6904	120	3	]	]	X
ejpam-6904	120	4	such	such	ADJ
ejpam-6904	120	5	that	that	SCONJ
ejpam-6904	120	6	v	v	NUM
ejpam-6904	120	7	∈	∈	PROPN
ejpam-6904	120	8	v	v	NOUN
ejpam-6904	120	9	(	(	PUNCT
ejpam-6904	120	10	gt	gt	PROPN
ejpam-6904	120	11	)	)	PUNCT
ejpam-6904	120	12	\	\	PROPN
ejpam-6904	120	13	st	st	PROPN
ejpam-6904	120	14	.	.	PROPN
ejpam-6904	120	15	since	since	SCONJ
ejpam-6904	120	16	st	st	PROPN
ejpam-6904	120	17	is	be	AUX
ejpam-6904	120	18	a	a	DET
ejpam-6904	120	19	2	2	NUM
ejpam-6904	120	20	-	-	PUNCT
ejpam-6904	120	21	path	path	NOUN
ejpam-6904	120	22	geodetic	geodetic	ADJ
ejpam-6904	120	23	set	set	NOUN
ejpam-6904	120	24	in	in	ADP
ejpam-6904	120	25	gt	gt	PROPN
ejpam-6904	120	26	,	,	PUNCT
ejpam-6904	120	27	there	there	PRON
ejpam-6904	120	28	exist	exist	VERB
ejpam-6904	120	29	u	u	NOUN
ejpam-6904	120	30	,	,	PUNCT
ejpam-6904	120	31	w	w	PROPN
ejpam-6904	120	32	∈	∈	PROPN
ejpam-6904	120	33	st	st	NOUN
ejpam-6904	120	34	such	such	ADJ
ejpam-6904	120	35	that	that	PRON
ejpam-6904	120	36	dg(u	dg(u	ADJ
ejpam-6904	120	37	,	,	PUNCT
ejpam-6904	120	38	w	w	NOUN
ejpam-6904	120	39	)	)	PUNCT
ejpam-6904	120	40	=	=	SYM
ejpam-6904	120	41	2	2	NUM
ejpam-6904	120	42	and	and	CCONJ
ejpam-6904	120	43	v	v	NOUN
ejpam-6904	120	44	∈	∈	NOUN
ejpam-6904	120	45	igt(u	igt(u	ADP
ejpam-6904	120	46	,	,	PUNCT
ejpam-6904	120	47	w	w	NOUN
ejpam-6904	120	48	)	)	PUNCT
ejpam-6904	120	49	=	=	SYM
ejpam-6904	120	50	ig(u	ig(u	NOUN
ejpam-6904	120	51	,	,	PUNCT
ejpam-6904	120	52	w	w	NOUN
ejpam-6904	120	53	)	)	PUNCT
ejpam-6904	120	54	.	.	PUNCT
ejpam-6904	121	1	therefore	therefore	ADV
ejpam-6904	121	2	s	s	VERB
ejpam-6904	121	3	is	be	AUX
ejpam-6904	121	4	a	a	DET
ejpam-6904	121	5	2	2	NUM
ejpam-6904	121	6	-	-	PUNCT
ejpam-6904	121	7	path	path	NOUN
ejpam-6904	121	8	geodetic	geodetic	ADJ
ejpam-6904	121	9	vertex	vertex	NOUN
ejpam-6904	121	10	cover	cover	NOUN
ejpam-6904	121	11	of	of	ADP
ejpam-6904	121	12	g.	g.	PROPN
ejpam-6904	121	13	if	if	SCONJ
ejpam-6904	121	14	each	each	DET
ejpam-6904	121	15	sj	sj	NOUN
ejpam-6904	121	16	is	be	AUX
ejpam-6904	121	17	a	a	DET
ejpam-6904	121	18	β2pg	β2pg	PUNCT
ejpam-6904	121	19	-	-	VERB
ejpam-6904	121	20	set	set	VERB
ejpam-6904	121	21	in	in	ADP
ejpam-6904	121	22	gj	gj	PROPN
ejpam-6904	121	23	,	,	PUNCT
ejpam-6904	121	24	then	then	ADV
ejpam-6904	121	25	β2pg(g	β2pg(g	PUNCT
ejpam-6904	121	26	)	)	PUNCT
ejpam-6904	121	27	≤	≤	NUM
ejpam-6904	121	28	|s|	|s|	PROPN
ejpam-6904	121	29	=	=	SYM
ejpam-6904	121	30	∑	∑	PUNCT
ejpam-6904	121	31	j∈[k	j∈[k	PROPN
ejpam-6904	121	32	]	]	PUNCT
ejpam-6904	121	33	|sj	|sj	NUM
ejpam-6904	121	34	|	|	NOUN
ejpam-6904	121	35	=	=	SYM
ejpam-6904	121	36	∑	∑	PUNCT
ejpam-6904	121	37	j∈[k	j∈[k	PROPN
ejpam-6904	121	38	]	]	PUNCT
ejpam-6904	121	39	β2pg(gj	β2pg(gj	PUNCT
ejpam-6904	121	40	)	)	PUNCT
ejpam-6904	121	41	.	.	PUNCT
ejpam-6904	122	1	this	this	PRON
ejpam-6904	122	2	proves	prove	VERB
ejpam-6904	122	3	the	the	DET
ejpam-6904	122	4	assertion	assertion	NOUN
ejpam-6904	122	5	.	.	PUNCT
ejpam-6904	123	1	theorem	theorem	NOUN
ejpam-6904	123	2	2	2	NUM
ejpam-6904	123	3	.	.	PUNCT
ejpam-6904	124	1	let	let	VERB
ejpam-6904	124	2	g	g	PRON
ejpam-6904	124	3	be	be	AUX
ejpam-6904	124	4	a	a	DET
ejpam-6904	124	5	graph	graph	NOUN
ejpam-6904	124	6	of	of	ADP
ejpam-6904	124	7	order	order	NOUN
ejpam-6904	124	8	n.	n.	NOUN
ejpam-6904	124	9	then	then	ADV
ejpam-6904	124	10	max{β(g	max{β(g	PROPN
ejpam-6904	124	11	)	)	PUNCT
ejpam-6904	124	12	,	,	PUNCT
ejpam-6904	124	13	g2p(g	g2p(g	PROPN
ejpam-6904	124	14	)	)	PUNCT
ejpam-6904	124	15	}	}	PUNCT
ejpam-6904	124	16	≤	≤	NUM
ejpam-6904	124	17	β2pg(g	β2pg(g	SYM
ejpam-6904	124	18	)	)	PUNCT
ejpam-6904	124	19	≤	≤	PUNCT
ejpam-6904	125	1	n.	n.	NOUN
ejpam-6904	125	2	moreover	moreover	ADV
ejpam-6904	125	3	,	,	PUNCT
ejpam-6904	125	4	the	the	DET
ejpam-6904	125	5	following	follow	VERB
ejpam-6904	125	6	statements	statement	NOUN
ejpam-6904	125	7	hold	hold	VERB
ejpam-6904	125	8	.	.	PUNCT
ejpam-6904	126	1	(	(	PUNCT
ejpam-6904	126	2	i	i	NOUN
ejpam-6904	126	3	)	)	PUNCT
ejpam-6904	126	4	g2p(g	g2p(g	PROPN
ejpam-6904	126	5	)	)	PUNCT
ejpam-6904	126	6	=	=	PUNCT
ejpam-6904	126	7	β2pg(g	β2pg(g	X
ejpam-6904	126	8	)	)	PUNCT
ejpam-6904	127	1	if	if	SCONJ
ejpam-6904	127	2	and	and	CCONJ
ejpam-6904	127	3	only	only	ADV
ejpam-6904	127	4	if	if	SCONJ
ejpam-6904	127	5	g	g	PROPN
ejpam-6904	127	6	has	have	AUX
ejpam-6904	127	7	g2p	g2p	NOUN
ejpam-6904	127	8	-	-	PUNCT
ejpam-6904	127	9	set	set	ADJ
ejpam-6904	127	10	which	which	PRON
ejpam-6904	127	11	is	be	AUX
ejpam-6904	127	12	also	also	ADV
ejpam-6904	127	13	a	a	DET
ejpam-6904	127	14	vertex	vertex	NOUN
ejpam-6904	127	15	cover	cover	NOUN
ejpam-6904	127	16	of	of	ADP
ejpam-6904	127	17	g.	g.	PROPN
ejpam-6904	127	18	(	(	PUNCT
ejpam-6904	127	19	ii	ii	PROPN
ejpam-6904	127	20	)	)	PUNCT
ejpam-6904	127	21	β(g	β(g	PROPN
ejpam-6904	127	22	)	)	PUNCT
ejpam-6904	127	23	=	=	PUNCT
ejpam-6904	127	24	β2pg(g	β2pg(g	X
ejpam-6904	127	25	)	)	PUNCT
ejpam-6904	127	26	if	if	SCONJ
ejpam-6904	127	27	and	and	CCONJ
ejpam-6904	127	28	only	only	ADV
ejpam-6904	127	29	if	if	SCONJ
ejpam-6904	127	30	g	g	PROPN
ejpam-6904	127	31	has	have	VERB
ejpam-6904	127	32	β	β	X
ejpam-6904	127	33	-	-	VERB
ejpam-6904	127	34	set	set	VERB
ejpam-6904	127	35	which	which	PRON
ejpam-6904	127	36	is	be	AUX
ejpam-6904	127	37	also	also	ADV
ejpam-6904	127	38	a	a	DET
ejpam-6904	127	39	2	2	NUM
ejpam-6904	127	40	-	-	PUNCT
ejpam-6904	127	41	path	path	NOUN
ejpam-6904	127	42	geodetic	geodetic	ADJ
ejpam-6904	127	43	set	set	NOUN
ejpam-6904	127	44	in	in	ADP
ejpam-6904	127	45	g.	g.	PROPN
ejpam-6904	127	46	proof	proof	PROPN
ejpam-6904	127	47	.	.	PUNCT
ejpam-6904	128	1	clearly	clearly	ADV
ejpam-6904	128	2	,	,	PUNCT
ejpam-6904	128	3	β2pg(g	β2pg(g	PROPN
ejpam-6904	128	4	)	)	PUNCT
ejpam-6904	128	5	≤	≤	NOUN
ejpam-6904	128	6	n.	n.	NOUN
ejpam-6904	128	7	since	since	SCONJ
ejpam-6904	128	8	every	every	DET
ejpam-6904	128	9	2	2	NUM
ejpam-6904	128	10	-	-	PUNCT
ejpam-6904	128	11	path	path	NOUN
ejpam-6904	128	12	geodetic	geodetic	ADJ
ejpam-6904	128	13	vertex	vertex	NOUN
ejpam-6904	128	14	cover	cover	NOUN
ejpam-6904	128	15	is	be	AUX
ejpam-6904	128	16	both	both	PRON
ejpam-6904	128	17	2	2	NUM
ejpam-6904	128	18	-	-	PUNCT
ejpam-6904	128	19	path	path	NOUN
ejpam-6904	128	20	geodetic	geodetic	ADJ
ejpam-6904	128	21	and	and	CCONJ
ejpam-6904	128	22	a	a	DET
ejpam-6904	128	23	vertex	vertex	NOUN
ejpam-6904	128	24	cover	cover	NOUN
ejpam-6904	128	25	,	,	PUNCT
ejpam-6904	128	26	we	we	PRON
ejpam-6904	128	27	have	have	VERB
ejpam-6904	128	28	max{β(g	max{β(g	PROPN
ejpam-6904	128	29	)	)	PUNCT
ejpam-6904	128	30	,	,	PUNCT
ejpam-6904	128	31	g2p(g	g2p(g	PROPN
ejpam-6904	128	32	)	)	PUNCT
ejpam-6904	128	33	}	}	PUNCT
ejpam-6904	128	34	≤	≤	NUM
ejpam-6904	128	35	β2pg(g	β2pg(g	NUM
ejpam-6904	128	36	)	)	PUNCT
ejpam-6904	128	37	.	.	PUNCT
ejpam-6904	129	1	(	(	PUNCT
ejpam-6904	129	2	i	i	NOUN
ejpam-6904	129	3	)	)	PUNCT
ejpam-6904	129	4	suppose	suppose	VERB
ejpam-6904	129	5	that	that	SCONJ
ejpam-6904	129	6	g2p(g	g2p(g	NOUN
ejpam-6904	129	7	)	)	PUNCT
ejpam-6904	129	8	=	=	SYM
ejpam-6904	129	9	β2pg(g	β2pg(g	X
ejpam-6904	129	10	)	)	PUNCT
ejpam-6904	129	11	.	.	PUNCT
ejpam-6904	130	1	let	let	VERB
ejpam-6904	130	2	s	s	PRON
ejpam-6904	130	3	be	be	AUX
ejpam-6904	130	4	a	a	DET
ejpam-6904	130	5	β2pg	β2pg	PUNCT
ejpam-6904	130	6	-	-	VERB
ejpam-6904	130	7	set	set	VERB
ejpam-6904	130	8	in	in	ADP
ejpam-6904	130	9	g.	g.	PROPN
ejpam-6904	130	10	by	by	ADP
ejpam-6904	130	11	assumption	assumption	NOUN
ejpam-6904	130	12	,	,	PUNCT
ejpam-6904	130	13	s	s	PART
ejpam-6904	130	14	is	be	AUX
ejpam-6904	130	15	a	a	DET
ejpam-6904	130	16	g2p	g2p	NOUN
ejpam-6904	130	17	-	-	ADJ
ejpam-6904	130	18	set	set	VERB
ejpam-6904	130	19	in	in	ADP
ejpam-6904	130	20	g.	g.	PROPN
ejpam-6904	130	21	for	for	ADP
ejpam-6904	130	22	the	the	DET
ejpam-6904	130	23	converse	converse	NOUN
ejpam-6904	130	24	,	,	PUNCT
ejpam-6904	130	25	suppose	suppose	VERB
ejpam-6904	130	26	that	that	SCONJ
ejpam-6904	130	27	g	g	PROPN
ejpam-6904	130	28	has	have	VERB
ejpam-6904	130	29	a	a	DET
ejpam-6904	130	30	g2p	g2p	NOUN
ejpam-6904	130	31	-	-	PUNCT
ejpam-6904	130	32	set	set	NOUN
ejpam-6904	130	33	s	s	NOUN
ejpam-6904	130	34	which	which	PRON
ejpam-6904	130	35	is	be	AUX
ejpam-6904	130	36	also	also	ADV
ejpam-6904	130	37	a	a	DET
ejpam-6904	130	38	vertex	vertex	NOUN
ejpam-6904	130	39	cover	cover	NOUN
ejpam-6904	130	40	of	of	ADP
ejpam-6904	130	41	g.	g.	PROPN
ejpam-6904	131	1	then	then	ADV
ejpam-6904	131	2	s	s	VERB
ejpam-6904	131	3	is	be	AUX
ejpam-6904	131	4	a	a	DET
ejpam-6904	131	5	2	2	NUM
ejpam-6904	131	6	-	-	PUNCT
ejpam-6904	131	7	path	path	NOUN
ejpam-6904	131	8	geodetic	geodetic	ADJ
ejpam-6904	131	9	vertex	vertex	NOUN
ejpam-6904	131	10	cover	cover	NOUN
ejpam-6904	131	11	of	of	ADP
ejpam-6904	131	12	g.	g.	PROPN
ejpam-6904	131	13	hence	hence	ADV
ejpam-6904	131	14	,	,	PUNCT
ejpam-6904	131	15	β2pg(g	β2pg(g	PROPN
ejpam-6904	131	16	)	)	PUNCT
ejpam-6904	131	17	≤	≤	NUM
ejpam-6904	131	18	|s|	|s|	PROPN
ejpam-6904	131	19	=	=	PUNCT
ejpam-6904	131	20	g2p(g	g2p(g	PROPN
ejpam-6904	131	21	)	)	PUNCT
ejpam-6904	131	22	.	.	PUNCT
ejpam-6904	132	1	by	by	ADP
ejpam-6904	132	2	the	the	DET
ejpam-6904	132	3	first	first	ADJ
ejpam-6904	132	4	part	part	NOUN
ejpam-6904	132	5	,	,	PUNCT
ejpam-6904	132	6	it	it	PRON
ejpam-6904	132	7	follows	follow	VERB
ejpam-6904	132	8	that	that	PRON
ejpam-6904	132	9	β2pg(g	β2pg(g	PUNCT
ejpam-6904	132	10	)	)	PUNCT
ejpam-6904	132	11	=	=	SYM
ejpam-6904	132	12	g2p(g	g2p(g	NOUN
ejpam-6904	132	13	)	)	PUNCT
ejpam-6904	132	14	.	.	PUNCT
ejpam-6904	133	1	(	(	PUNCT
ejpam-6904	133	2	ii	ii	NOUN
ejpam-6904	133	3	)	)	PUNCT
ejpam-6904	133	4	suppose	suppose	VERB
ejpam-6904	133	5	that	that	SCONJ
ejpam-6904	133	6	β(g	β(g	PROPN
ejpam-6904	133	7	)	)	PUNCT
ejpam-6904	133	8	=	=	PUNCT
ejpam-6904	133	9	β2pg(g	β2pg(g	X
ejpam-6904	133	10	)	)	PUNCT
ejpam-6904	133	11	.	.	PUNCT
ejpam-6904	134	1	let	let	VERB
ejpam-6904	134	2	s	s	PRON
ejpam-6904	134	3	be	be	AUX
ejpam-6904	134	4	a	a	DET
ejpam-6904	134	5	β2pg	β2pg	PUNCT
ejpam-6904	134	6	-	-	VERB
ejpam-6904	134	7	set	set	VERB
ejpam-6904	134	8	in	in	ADP
ejpam-6904	134	9	g.	g.	PROPN
ejpam-6904	134	10	by	by	ADP
ejpam-6904	134	11	assumption	assumption	NOUN
ejpam-6904	134	12	,	,	PUNCT
ejpam-6904	134	13	s	s	PART
ejpam-6904	134	14	is	be	AUX
ejpam-6904	134	15	a	a	DET
ejpam-6904	134	16	β	β	NOUN
ejpam-6904	134	17	-	-	VERB
ejpam-6904	134	18	set	set	VERB
ejpam-6904	134	19	in	in	ADP
ejpam-6904	134	20	g.	g.	NOUN
ejpam-6904	134	21	conversely	conversely	ADV
ejpam-6904	134	22	,	,	PUNCT
ejpam-6904	134	23	suppose	suppose	VERB
ejpam-6904	134	24	that	that	SCONJ
ejpam-6904	134	25	g	g	PROPN
ejpam-6904	134	26	has	have	VERB
ejpam-6904	134	27	a	a	DET
ejpam-6904	134	28	β	β	NOUN
ejpam-6904	134	29	-	-	ADJ
ejpam-6904	134	30	set	set	VERB
ejpam-6904	134	31	s	s	NOUN
ejpam-6904	134	32	which	which	PRON
ejpam-6904	134	33	is	be	AUX
ejpam-6904	134	34	also	also	ADV
ejpam-6904	134	35	a	a	DET
ejpam-6904	134	36	2	2	NUM
ejpam-6904	134	37	-	-	PUNCT
ejpam-6904	134	38	path	path	NOUN
ejpam-6904	134	39	geodetic	geodetic	ADJ
ejpam-6904	134	40	set	set	NOUN
ejpam-6904	134	41	in	in	ADP
ejpam-6904	134	42	g.	g.	PROPN
ejpam-6904	134	43	then	then	ADV
ejpam-6904	134	44	s	s	VERB
ejpam-6904	134	45	is	be	AUX
ejpam-6904	134	46	a	a	DET
ejpam-6904	134	47	2	2	NUM
ejpam-6904	134	48	-	-	PUNCT
ejpam-6904	134	49	path	path	NOUN
ejpam-6904	134	50	geodetic	geodetic	ADJ
ejpam-6904	134	51	vertex	vertex	NOUN
ejpam-6904	134	52	cover	cover	NOUN
ejpam-6904	134	53	of	of	ADP
ejpam-6904	134	54	g.	g.	PROPN
ejpam-6904	134	55	hence	hence	ADV
ejpam-6904	134	56	,	,	PUNCT
ejpam-6904	134	57	β2pg(g	β2pg(g	PROPN
ejpam-6904	134	58	)	)	PUNCT
ejpam-6904	134	59	≤	≤	NUM
ejpam-6904	134	60	|s|	|s|	PROPN
ejpam-6904	134	61	=	=	SYM
ejpam-6904	134	62	β(g	β(g	PROPN
ejpam-6904	134	63	)	)	PUNCT
ejpam-6904	134	64	.	.	PUNCT
ejpam-6904	135	1	with	with	ADP
ejpam-6904	135	2	the	the	DET
ejpam-6904	135	3	first	first	ADJ
ejpam-6904	135	4	part	part	NOUN
ejpam-6904	135	5	,	,	PUNCT
ejpam-6904	135	6	this	this	PRON
ejpam-6904	135	7	implies	imply	VERB
ejpam-6904	135	8	that	that	PRON
ejpam-6904	135	9	β2pg(g	β2pg(g	PUNCT
ejpam-6904	135	10	)	)	PUNCT
ejpam-6904	135	11	=	=	SYM
ejpam-6904	135	12	β(g	β(g	PROPN
ejpam-6904	135	13	)	)	PUNCT
ejpam-6904	135	14	.	.	PUNCT
ejpam-6904	136	1	theorem	theorem	NOUN
ejpam-6904	136	2	3	3	X
ejpam-6904	136	3	.	.	PUNCT
ejpam-6904	137	1	let	let	VERB
ejpam-6904	137	2	g	g	PRON
ejpam-6904	137	3	be	be	AUX
ejpam-6904	137	4	a	a	DET
ejpam-6904	137	5	graph	graph	NOUN
ejpam-6904	137	6	of	of	ADP
ejpam-6904	137	7	order	order	NOUN
ejpam-6904	137	8	n.	n.	NOUN
ejpam-6904	137	9	then	then	ADV
ejpam-6904	137	10	(	(	PUNCT
ejpam-6904	137	11	i	i	NOUN
ejpam-6904	137	12	)	)	PUNCT
ejpam-6904	137	13	β2pg(g	β2pg(g	PUNCT
ejpam-6904	137	14	)	)	PUNCT
ejpam-6904	137	15	=	=	SYM
ejpam-6904	137	16	1	1	NUM
ejpam-6904	137	17	if	if	SCONJ
ejpam-6904	137	18	and	and	CCONJ
ejpam-6904	137	19	only	only	ADV
ejpam-6904	137	20	if	if	SCONJ
ejpam-6904	137	21	g	g	PROPN
ejpam-6904	137	22	=	=	SYM
ejpam-6904	137	23	k1	k1	PROPN
ejpam-6904	137	24	(	(	PUNCT
ejpam-6904	137	25	ii	ii	NOUN
ejpam-6904	137	26	)	)	PUNCT
ejpam-6904	137	27	β2pg(g	β2pg(g	X
ejpam-6904	137	28	)	)	PUNCT
ejpam-6904	137	29	=	=	SYM
ejpam-6904	137	30	2	2	NUM
ejpam-6904	137	31	if	if	SCONJ
ejpam-6904	137	32	and	and	CCONJ
ejpam-6904	137	33	only	only	ADV
ejpam-6904	137	34	if	if	SCONJ
ejpam-6904	137	35	g	g	PROPN
ejpam-6904	137	36	∈	∈	PROPN
ejpam-6904	137	37	{	{	PUNCT
ejpam-6904	137	38	k2,k2,k2,n−2	k2,k2,k2,n−2	NOUN
ejpam-6904	137	39	}	}	PUNCT
ejpam-6904	137	40	a.	a.	PROPN
ejpam-6904	137	41	b.	b.	PROPN
ejpam-6904	137	42	tapeing	tapeing	PROPN
ejpam-6904	137	43	,	,	PUNCT
ejpam-6904	137	44	s.	s.	PROPN
ejpam-6904	137	45	r.	r.	PROPN
ejpam-6904	137	46	canoy	canoy	PROPN
ejpam-6904	137	47	/	/	SYM
ejpam-6904	137	48	eur	eur	PROPN
ejpam-6904	137	49	.	.	PUNCT
ejpam-6904	138	1	j.	j.	PROPN
ejpam-6904	138	2	pure	pure	PROPN
ejpam-6904	138	3	appl	appl	PROPN
ejpam-6904	138	4	.	.	PROPN
ejpam-6904	138	5	math	math	PROPN
ejpam-6904	138	6	,	,	PUNCT
ejpam-6904	138	7	18	18	NUM
ejpam-6904	138	8	(	(	PUNCT
ejpam-6904	138	9	4	4	NUM
ejpam-6904	138	10	)	)	PUNCT
ejpam-6904	138	11	(	(	PUNCT
ejpam-6904	138	12	2025	2025	NUM
ejpam-6904	138	13	)	)	PUNCT
ejpam-6904	138	14	,	,	PUNCT
ejpam-6904	138	15	6904	6904	NUM
ejpam-6904	138	16	6	6	NUM
ejpam-6904	138	17	of	of	ADP
ejpam-6904	138	18	14	14	NUM
ejpam-6904	138	19	(	(	PUNCT
ejpam-6904	138	20	iii	iii	NOUN
ejpam-6904	138	21	)	)	PUNCT
ejpam-6904	138	22	β2pg(g	β2pg(g	X
ejpam-6904	138	23	)	)	PUNCT
ejpam-6904	138	24	=	=	SYM
ejpam-6904	138	25	3	3	NUM
ejpam-6904	138	26	if	if	SCONJ
ejpam-6904	138	27	and	and	CCONJ
ejpam-6904	138	28	only	only	ADV
ejpam-6904	138	29	if	if	SCONJ
ejpam-6904	138	30	g	g	PROPN
ejpam-6904	138	31	∈	∈	PROPN
ejpam-6904	138	32	{	{	PUNCT
ejpam-6904	138	33	k3,k3	k3,k3	PROPN
ejpam-6904	138	34	,	,	PUNCT
ejpam-6904	138	35	p1	p1	NOUN
ejpam-6904	138	36	∪	∪	ADJ
ejpam-6904	138	37	p2	p2	NOUN
ejpam-6904	138	38	}	}	PUNCT
ejpam-6904	138	39	or	or	CCONJ
ejpam-6904	138	40	there	there	PRON
ejpam-6904	138	41	exist	exist	VERB
ejpam-6904	138	42	distinct	distinct	ADJ
ejpam-6904	138	43	vertices	vertex	NOUN
ejpam-6904	138	44	x	x	X
ejpam-6904	138	45	,	,	PUNCT
ejpam-6904	138	46	y	y	PROPN
ejpam-6904	138	47	,	,	PUNCT
ejpam-6904	138	48	z	z	PROPN
ejpam-6904	138	49	∈	∈	PROPN
ejpam-6904	138	50	v	v	ADP
ejpam-6904	138	51	(	(	PUNCT
ejpam-6904	138	52	g	g	NOUN
ejpam-6904	138	53	)	)	PUNCT
ejpam-6904	138	54	such	such	ADJ
ejpam-6904	138	55	that	that	PRON
ejpam-6904	138	56	v	v	NOUN
ejpam-6904	138	57	(	(	PUNCT
ejpam-6904	138	58	g	g	NOUN
ejpam-6904	138	59	)	)	PUNCT
ejpam-6904	138	60	\	\	NOUN
ejpam-6904	139	1	{	{	PUNCT
ejpam-6904	139	2	x	x	NOUN
ejpam-6904	139	3	,	,	PUNCT
ejpam-6904	139	4	y	y	PROPN
ejpam-6904	139	5	,	,	PUNCT
ejpam-6904	139	6	z	z	NOUN
ejpam-6904	139	7	}	}	PUNCT
ejpam-6904	139	8	is	be	AUX
ejpam-6904	139	9	an	an	DET
ejpam-6904	139	10	independent	independent	ADJ
ejpam-6904	139	11	set	set	NOUN
ejpam-6904	139	12	and	and	CCONJ
ejpam-6904	139	13	one	one	NUM
ejpam-6904	139	14	of	of	ADP
ejpam-6904	139	15	the	the	DET
ejpam-6904	139	16	following	follow	VERB
ejpam-6904	139	17	conditions	condition	NOUN
ejpam-6904	139	18	holds	hold	VERB
ejpam-6904	139	19	:	:	PUNCT
ejpam-6904	139	20	(	(	PUNCT
ejpam-6904	139	21	a	a	X
ejpam-6904	139	22	)	)	PUNCT
ejpam-6904	139	23	⟨{x	⟨{x	NOUN
ejpam-6904	139	24	,	,	PUNCT
ejpam-6904	139	25	y	y	PROPN
ejpam-6904	139	26	,	,	PUNCT
ejpam-6904	140	1	z}⟩	z}⟩	PROPN
ejpam-6904	140	2	=	=	PUNCT
ejpam-6904	140	3	k3	k3	ADJ
ejpam-6904	140	4	and	and	CCONJ
ejpam-6904	140	5	for	for	ADP
ejpam-6904	140	6	every	every	DET
ejpam-6904	140	7	v	v	NUM
ejpam-6904	140	8	∈	∈	PROPN
ejpam-6904	140	9	v	v	NOUN
ejpam-6904	140	10	(	(	PUNCT
ejpam-6904	140	11	g)\{x	g)\{x	PROPN
ejpam-6904	140	12	,	,	PUNCT
ejpam-6904	140	13	y	y	PROPN
ejpam-6904	140	14	,	,	PUNCT
ejpam-6904	140	15	z	z	NOUN
ejpam-6904	140	16	}	}	PUNCT
ejpam-6904	140	17	,	,	PUNCT
ejpam-6904	140	18	it	it	PRON
ejpam-6904	140	19	holds	hold	VERB
ejpam-6904	140	20	that	that	SCONJ
ejpam-6904	140	21	|ng(v	|ng(v	VERB
ejpam-6904	140	22	)	)	PUNCT
ejpam-6904	140	23	∩	∩	NOUN
ejpam-6904	140	24	{	{	PUNCT
ejpam-6904	140	25	x	x	NOUN
ejpam-6904	140	26	,	,	PUNCT
ejpam-6904	140	27	y	y	PROPN
ejpam-6904	140	28	,	,	PUNCT
ejpam-6904	140	29	z}|	z}|	PROPN
ejpam-6904	140	30	≥	≥	NUM
ejpam-6904	140	31	2	2	NUM
ejpam-6904	140	32	or	or	CCONJ
ejpam-6904	140	33	(	(	PUNCT
ejpam-6904	140	34	b	b	NOUN
ejpam-6904	140	35	)	)	PUNCT
ejpam-6904	140	36	⟨{x	⟨{x	NOUN
ejpam-6904	140	37	,	,	PUNCT
ejpam-6904	140	38	y	y	PROPN
ejpam-6904	140	39	,	,	PUNCT
ejpam-6904	140	40	z}⟩	z}⟩	PROPN
ejpam-6904	140	41	=	=	SYM
ejpam-6904	140	42	⟨x⟩∪⟨{y	⟨x⟩∪⟨{y	NOUN
ejpam-6904	140	43	,	,	PUNCT
ejpam-6904	140	44	z}⟩	z}⟩	PROPN
ejpam-6904	140	45	=	=	SYM
ejpam-6904	140	46	p1∪p2	p1∪p2	PROPN
ejpam-6904	140	47	and	and	CCONJ
ejpam-6904	140	48	for	for	ADP
ejpam-6904	140	49	every	every	DET
ejpam-6904	140	50	v	v	NUM
ejpam-6904	140	51	∈	∈	PROPN
ejpam-6904	140	52	v	v	NOUN
ejpam-6904	140	53	(	(	PUNCT
ejpam-6904	140	54	g)\{x	g)\{x	PROPN
ejpam-6904	140	55	,	,	PUNCT
ejpam-6904	140	56	y	y	PROPN
ejpam-6904	140	57	,	,	PUNCT
ejpam-6904	140	58	z	z	NOUN
ejpam-6904	140	59	}	}	PUNCT
ejpam-6904	140	60	,	,	PUNCT
ejpam-6904	140	61	we	we	PRON
ejpam-6904	140	62	have	have	VERB
ejpam-6904	140	63	x	x	X
ejpam-6904	140	64	,	,	PUNCT
ejpam-6904	140	65	y	y	PROPN
ejpam-6904	140	66	∈	∈	PROPN
ejpam-6904	140	67	ng(v	ng(v	PUNCT
ejpam-6904	140	68	)	)	PUNCT
ejpam-6904	140	69	or	or	CCONJ
ejpam-6904	140	70	x	x	SYM
ejpam-6904	140	71	,	,	PUNCT
ejpam-6904	140	72	z	z	PROPN
ejpam-6904	140	73	∈	∈	PROPN
ejpam-6904	140	74	ng(v	ng(v	NOUN
ejpam-6904	140	75	)	)	PUNCT
ejpam-6904	140	76	.	.	PUNCT
ejpam-6904	141	1	(	(	PUNCT
ejpam-6904	141	2	c	c	X
ejpam-6904	141	3	)	)	PUNCT
ejpam-6904	141	4	⟨{x	⟨{x	PROPN
ejpam-6904	141	5	,	,	PUNCT
ejpam-6904	141	6	y	y	PROPN
ejpam-6904	141	7	,	,	PUNCT
ejpam-6904	141	8	z}⟩	z}⟩	PROPN
ejpam-6904	141	9	=	=	SYM
ejpam-6904	141	10	p3	p3	PROPN
ejpam-6904	141	11	=	=	PUNCT
ejpam-6904	142	1	[	[	X
ejpam-6904	142	2	x	x	X
ejpam-6904	142	3	,	,	PUNCT
ejpam-6904	142	4	y	y	PROPN
ejpam-6904	142	5	,	,	PUNCT
ejpam-6904	142	6	z	z	X
ejpam-6904	142	7	]	]	X
ejpam-6904	142	8	̸=	̸=	PROPN
ejpam-6904	142	9	g	g	NOUN
ejpam-6904	142	10	and	and	CCONJ
ejpam-6904	142	11	for	for	ADP
ejpam-6904	142	12	every	every	PRON
ejpam-6904	142	13	v	v	NUM
ejpam-6904	142	14	∈	∈	PROPN
ejpam-6904	142	15	v	v	NOUN
ejpam-6904	142	16	(	(	PUNCT
ejpam-6904	142	17	g	g	NOUN
ejpam-6904	142	18	)	)	PUNCT
ejpam-6904	142	19	\	\	NOUN
ejpam-6904	142	20	{	{	PUNCT
ejpam-6904	142	21	x	x	NOUN
ejpam-6904	142	22	,	,	PUNCT
ejpam-6904	142	23	y	y	PROPN
ejpam-6904	142	24	,	,	PUNCT
ejpam-6904	142	25	z	z	NOUN
ejpam-6904	142	26	}	}	PUNCT
ejpam-6904	142	27	,	,	PUNCT
ejpam-6904	142	28	it	it	PRON
ejpam-6904	142	29	holds	hold	VERB
ejpam-6904	142	30	that	that	SCONJ
ejpam-6904	142	31	x	x	NOUN
ejpam-6904	142	32	,	,	PUNCT
ejpam-6904	142	33	z	z	PROPN
ejpam-6904	142	34	∈	∈	PROPN
ejpam-6904	142	35	ng(v	ng(v	PUNCT
ejpam-6904	142	36	)	)	PUNCT
ejpam-6904	142	37	and	and	CCONJ
ejpam-6904	142	38	ng(y	ng(y	NOUN
ejpam-6904	142	39	)	)	PUNCT
ejpam-6904	142	40	\	\	NOUN
ejpam-6904	143	1	{	{	PUNCT
ejpam-6904	143	2	x	x	NOUN
ejpam-6904	143	3	,	,	PUNCT
ejpam-6904	143	4	y	y	PROPN
ejpam-6904	143	5	,	,	PUNCT
ejpam-6904	143	6	z	z	NOUN
ejpam-6904	143	7	}	}	PUNCT
ejpam-6904	143	8	̸=	̸=	PROPN
ejpam-6904	143	9	∅.	∅.	ADP
ejpam-6904	143	10	proof	proof	NOUN
ejpam-6904	143	11	.	.	PUNCT
ejpam-6904	144	1	(	(	PUNCT
ejpam-6904	144	2	i	i	NOUN
ejpam-6904	144	3	)	)	PUNCT
ejpam-6904	144	4	suppose	suppose	VERB
ejpam-6904	144	5	that	that	SCONJ
ejpam-6904	144	6	β2pg(g	β2pg(g	PROPN
ejpam-6904	144	7	)	)	PUNCT
ejpam-6904	144	8	=	=	SYM
ejpam-6904	144	9	1	1	NUM
ejpam-6904	144	10	,	,	PUNCT
ejpam-6904	144	11	say	say	VERB
ejpam-6904	144	12	s	s	X
ejpam-6904	144	13	=	=	VERB
ejpam-6904	144	14	{	{	PUNCT
ejpam-6904	144	15	v	v	NOUN
ejpam-6904	144	16	}	}	PUNCT
ejpam-6904	144	17	is	be	AUX
ejpam-6904	144	18	a	a	DET
ejpam-6904	144	19	β2pg	β2pg	PUNCT
ejpam-6904	144	20	-	-	VERB
ejpam-6904	144	21	set	set	VERB
ejpam-6904	144	22	in	in	ADP
ejpam-6904	144	23	g.	g.	PROPN
ejpam-6904	144	24	since	since	SCONJ
ejpam-6904	144	25	s	s	PROPN
ejpam-6904	144	26	is	be	AUX
ejpam-6904	144	27	a	a	DET
ejpam-6904	144	28	2	2	NUM
ejpam-6904	144	29	-	-	PUNCT
ejpam-6904	144	30	path	path	NOUN
ejpam-6904	144	31	geodetic	geodetic	ADJ
ejpam-6904	144	32	set	set	NOUN
ejpam-6904	144	33	,	,	PUNCT
ejpam-6904	144	34	v	v	X
ejpam-6904	144	35	(	(	PUNCT
ejpam-6904	144	36	g	g	NOUN
ejpam-6904	144	37	)	)	PUNCT
ejpam-6904	144	38	=	=	SYM
ejpam-6904	144	39	{	{	PUNCT
ejpam-6904	144	40	v	v	NOUN
ejpam-6904	144	41	}	}	PUNCT
ejpam-6904	144	42	,	,	PUNCT
ejpam-6904	144	43	i.e	i.e	PROPN
ejpam-6904	144	44	,	,	PUNCT
ejpam-6904	144	45	g	g	PROPN
ejpam-6904	144	46	=	=	SYM
ejpam-6904	144	47	k1	k1	PROPN
ejpam-6904	144	48	.	.	PUNCT
ejpam-6904	145	1	for	for	ADP
ejpam-6904	145	2	the	the	DET
ejpam-6904	145	3	converse	converse	NOUN
ejpam-6904	145	4	,	,	PUNCT
ejpam-6904	145	5	suppose	suppose	VERB
ejpam-6904	145	6	that	that	SCONJ
ejpam-6904	145	7	g	g	PROPN
ejpam-6904	145	8	=	=	PROPN
ejpam-6904	145	9	k1	k1	PROPN
ejpam-6904	145	10	.	.	PUNCT
ejpam-6904	146	1	by	by	ADP
ejpam-6904	146	2	proposition	proposition	NOUN
ejpam-6904	146	3	1	1	NUM
ejpam-6904	146	4	,	,	PUNCT
ejpam-6904	146	5	β2pg(g	β2pg(g	PUNCT
ejpam-6904	146	6	)	)	PUNCT
ejpam-6904	146	7	=	=	SYM
ejpam-6904	146	8	1	1	X
ejpam-6904	146	9	.	.	PUNCT
ejpam-6904	146	10	(	(	PUNCT
ejpam-6904	146	11	ii	ii	NOUN
ejpam-6904	146	12	)	)	PUNCT
ejpam-6904	146	13	suppoe	suppoe	PROPN
ejpam-6904	146	14	β2pg(g	β2pg(g	PUNCT
ejpam-6904	146	15	)	)	PUNCT
ejpam-6904	146	16	=	=	SYM
ejpam-6904	146	17	2	2	NUM
ejpam-6904	146	18	,	,	PUNCT
ejpam-6904	146	19	say	say	VERB
ejpam-6904	146	20	s	s	X
ejpam-6904	146	21	=	=	PUNCT
ejpam-6904	146	22	{	{	PUNCT
ejpam-6904	146	23	u	u	NOUN
ejpam-6904	146	24	,	,	PUNCT
ejpam-6904	146	25	v	v	NOUN
ejpam-6904	146	26	}	}	PUNCT
ejpam-6904	146	27	is	be	AUX
ejpam-6904	146	28	a	a	DET
ejpam-6904	146	29	β2pg	β2pg	PUNCT
ejpam-6904	146	30	-	-	NOUN
ejpam-6904	146	31	set	set	NOUN
ejpam-6904	146	32	.	.	PUNCT
ejpam-6904	147	1	if	if	SCONJ
ejpam-6904	147	2	n	n	NOUN
ejpam-6904	147	3	=	=	SYM
ejpam-6904	147	4	2	2	NUM
ejpam-6904	147	5	,	,	PUNCT
ejpam-6904	147	6	then	then	ADV
ejpam-6904	147	7	g	g	PROPN
ejpam-6904	147	8	∈	∈	PROPN
ejpam-6904	147	9	{	{	PUNCT
ejpam-6904	147	10	k2,k2	k2,k2	PROPN
ejpam-6904	147	11	}	}	PUNCT
ejpam-6904	147	12	.	.	PUNCT
ejpam-6904	148	1	suppose	suppose	VERB
ejpam-6904	148	2	n	n	PRON
ejpam-6904	148	3	≥	≥	NUM
ejpam-6904	148	4	3	3	X
ejpam-6904	148	5	.	.	PUNCT
ejpam-6904	149	1	let	let	VERB
ejpam-6904	149	2	x	x	SYM
ejpam-6904	149	3	∈	∈	PROPN
ejpam-6904	149	4	v	v	X
ejpam-6904	149	5	(	(	PUNCT
ejpam-6904	149	6	g	g	NOUN
ejpam-6904	149	7	)	)	PUNCT
ejpam-6904	149	8	\	\	PUNCT
ejpam-6904	150	1	s.	s.	PROPN
ejpam-6904	150	2	since	since	SCONJ
ejpam-6904	150	3	s	s	PROPN
ejpam-6904	150	4	is	be	AUX
ejpam-6904	150	5	a	a	DET
ejpam-6904	150	6	2	2	NUM
ejpam-6904	150	7	-	-	PUNCT
ejpam-6904	150	8	path	path	NOUN
ejpam-6904	150	9	geodetic	geodetic	ADJ
ejpam-6904	150	10	set	set	NOUN
ejpam-6904	150	11	,	,	PUNCT
ejpam-6904	150	12	x	x	SYM
ejpam-6904	150	13	∈	∈	NOUN
ejpam-6904	150	14	ig(u	ig(u	NOUN
ejpam-6904	150	15	,	,	PUNCT
ejpam-6904	150	16	v	v	NOUN
ejpam-6904	150	17	)	)	PUNCT
ejpam-6904	150	18	and	and	CCONJ
ejpam-6904	150	19	dg(u	dg(u	X
ejpam-6904	150	20	,	,	PUNCT
ejpam-6904	150	21	v	v	NOUN
ejpam-6904	150	22	)	)	PUNCT
ejpam-6904	150	23	=	=	SYM
ejpam-6904	150	24	2	2	X
ejpam-6904	150	25	.	.	X
ejpam-6904	150	26	let	let	VERB
ejpam-6904	150	27	g1	g1	PROPN
ejpam-6904	150	28	=	=	SYM
ejpam-6904	150	29	⟨{u	⟨{u	PROPN
ejpam-6904	150	30	,	,	PUNCT
ejpam-6904	150	31	v}⟩	v}⟩	PROPN
ejpam-6904	150	32	and	and	CCONJ
ejpam-6904	150	33	g2	g2	PROPN
ejpam-6904	150	34	=	=	PUNCT
ejpam-6904	151	1	⟨v	⟨v	PUNCT
ejpam-6904	151	2	(	(	PUNCT
ejpam-6904	151	3	g	g	NOUN
ejpam-6904	151	4	)	)	PUNCT
ejpam-6904	151	5	\	\	PROPN
ejpam-6904	151	6	s⟩.	s⟩.	PROPN
ejpam-6904	151	7	then	then	ADV
ejpam-6904	151	8	g1	g1	PROPN
ejpam-6904	151	9	=	=	PROPN
ejpam-6904	151	10	k2	k2	PROPN
ejpam-6904	151	11	and	and	CCONJ
ejpam-6904	151	12	,	,	PUNCT
ejpam-6904	151	13	by	by	ADP
ejpam-6904	151	14	remark	remark	NOUN
ejpam-6904	151	15	2	2	NUM
ejpam-6904	151	16	,	,	PUNCT
ejpam-6904	151	17	g2	g2	PROPN
ejpam-6904	151	18	=	=	SYM
ejpam-6904	151	19	kn−2	kn−2	PROPN
ejpam-6904	151	20	.	.	PUNCT
ejpam-6904	152	1	thus	thus	ADV
ejpam-6904	152	2	,	,	PUNCT
ejpam-6904	152	3	g	g	PROPN
ejpam-6904	152	4	=	=	SYM
ejpam-6904	152	5	k2	k2	PROPN
ejpam-6904	152	6	+	+	PROPN
ejpam-6904	152	7	kn−2	kn−2	PROPN
ejpam-6904	152	8	=	=	SYM
ejpam-6904	152	9	k2,n−2	k2,n−2	PROPN
ejpam-6904	152	10	.	.	PUNCT
ejpam-6904	153	1	the	the	DET
ejpam-6904	153	2	converse	converse	NOUN
ejpam-6904	153	3	is	be	AUX
ejpam-6904	153	4	clear	clear	ADJ
ejpam-6904	153	5	.	.	PUNCT
ejpam-6904	154	1	(	(	PUNCT
ejpam-6904	154	2	iii	iii	X
ejpam-6904	154	3	)	)	PUNCT
ejpam-6904	154	4	suppose	suppose	VERB
ejpam-6904	154	5	that	that	SCONJ
ejpam-6904	154	6	β2pg(g	β2pg(g	PROPN
ejpam-6904	154	7	)	)	PUNCT
ejpam-6904	154	8	=	=	SYM
ejpam-6904	154	9	3	3	X
ejpam-6904	154	10	.	.	X
ejpam-6904	154	11	let	let	VERB
ejpam-6904	154	12	s	s	VERB
ejpam-6904	154	13	=	=	PUNCT
ejpam-6904	154	14	{	{	PUNCT
ejpam-6904	154	15	x	x	PROPN
ejpam-6904	154	16	,	,	PUNCT
ejpam-6904	154	17	y	y	PROPN
ejpam-6904	154	18	,	,	PUNCT
ejpam-6904	154	19	z	z	NOUN
ejpam-6904	154	20	}	}	PUNCT
ejpam-6904	154	21	be	be	AUX
ejpam-6904	154	22	a	a	DET
ejpam-6904	154	23	β2pg	β2pg	PUNCT
ejpam-6904	154	24	-	-	VERB
ejpam-6904	154	25	set	set	VERB
ejpam-6904	154	26	in	in	ADP
ejpam-6904	154	27	g.	g.	PROPN
ejpam-6904	154	28	then	then	ADV
ejpam-6904	154	29	v	v	X
ejpam-6904	154	30	(	(	PUNCT
ejpam-6904	154	31	g	g	NOUN
ejpam-6904	154	32	)	)	PUNCT
ejpam-6904	154	33	\	\	PROPN
ejpam-6904	155	1	s	s	PART
ejpam-6904	155	2	is	be	AUX
ejpam-6904	155	3	an	an	DET
ejpam-6904	155	4	independent	independent	ADJ
ejpam-6904	155	5	by	by	ADP
ejpam-6904	155	6	remark	remark	NOUN
ejpam-6904	155	7	2	2	NUM
ejpam-6904	155	8	.	.	PUNCT
ejpam-6904	155	9	suppose	suppose	VERB
ejpam-6904	155	10	|v	|v	PROPN
ejpam-6904	155	11	(	(	PUNCT
ejpam-6904	155	12	g)|	g)|	NOUN
ejpam-6904	155	13	=	=	SYM
ejpam-6904	155	14	3	3	X
ejpam-6904	155	15	.	.	PUNCT
ejpam-6904	155	16	since	since	SCONJ
ejpam-6904	155	17	β2pg(p3	β2pg(p3	PROPN
ejpam-6904	155	18	)	)	PUNCT
ejpam-6904	156	1	=	=	SYM
ejpam-6904	156	2	2	2	NUM
ejpam-6904	156	3	and	and	CCONJ
ejpam-6904	156	4	β2pg(k3	β2pg(k3	PROPN
ejpam-6904	156	5	)	)	PUNCT
ejpam-6904	157	1	=	=	PUNCT
ejpam-6904	157	2	β2pg(k3	β2pg(k3	PROPN
ejpam-6904	157	3	)	)	PUNCT
ejpam-6904	157	4	=	=	PUNCT
ejpam-6904	158	1	β2pg(p1	β2pg(p1	NOUN
ejpam-6904	158	2	∪	∪	ADJ
ejpam-6904	158	3	p2	p2	NOUN
ejpam-6904	158	4	)	)	PUNCT
ejpam-6904	158	5	=	=	SYM
ejpam-6904	159	1	3	3	X
ejpam-6904	159	2	,	,	PUNCT
ejpam-6904	159	3	it	it	PRON
ejpam-6904	159	4	follows	follow	VERB
ejpam-6904	159	5	that	that	SCONJ
ejpam-6904	159	6	g	g	PROPN
ejpam-6904	159	7	∈	∈	PROPN
ejpam-6904	159	8	{	{	PUNCT
ejpam-6904	159	9	k3,k3	k3,k3	PROPN
ejpam-6904	159	10	,	,	PUNCT
ejpam-6904	159	11	p1	p1	NOUN
ejpam-6904	159	12	∪	∪	NOUN
ejpam-6904	159	13	p2	p2	NOUN
ejpam-6904	159	14	}	}	PUNCT
ejpam-6904	159	15	.	.	PUNCT
ejpam-6904	160	1	next	next	ADV
ejpam-6904	160	2	,	,	PUNCT
ejpam-6904	160	3	suppose	suppose	VERB
ejpam-6904	160	4	that	that	SCONJ
ejpam-6904	160	5	|v	|v	PROPN
ejpam-6904	160	6	(	(	PUNCT
ejpam-6904	160	7	g)|	g)|	X
ejpam-6904	160	8	≥	≥	NOUN
ejpam-6904	160	9	4	4	NUM
ejpam-6904	160	10	.	.	PUNCT
ejpam-6904	160	11	then	then	ADV
ejpam-6904	160	12	v	v	X
ejpam-6904	160	13	(	(	PUNCT
ejpam-6904	160	14	g	g	NOUN
ejpam-6904	160	15	)	)	PUNCT
ejpam-6904	160	16	\	\	NOUN
ejpam-6904	161	1	{	{	PUNCT
ejpam-6904	161	2	x	x	NOUN
ejpam-6904	161	3	,	,	PUNCT
ejpam-6904	161	4	y	y	PROPN
ejpam-6904	161	5	,	,	PUNCT
ejpam-6904	161	6	z	z	NOUN
ejpam-6904	161	7	}	}	PUNCT
ejpam-6904	161	8	̸=	̸=	PROPN
ejpam-6904	161	9	∅.	∅.	ADV
ejpam-6904	161	10	clearly	clearly	ADV
ejpam-6904	161	11	,	,	PUNCT
ejpam-6904	161	12	⟨s⟩	⟨s⟩	PROPN
ejpam-6904	161	13	̸=	̸=	PROPN
ejpam-6904	161	14	k3	k3	VERB
ejpam-6904	161	15	since	since	SCONJ
ejpam-6904	161	16	any	any	DET
ejpam-6904	161	17	v	v	NUM
ejpam-6904	161	18	∈	∈	NOUN
ejpam-6904	161	19	v	v	NOUN
ejpam-6904	161	20	(	(	PUNCT
ejpam-6904	161	21	s	s	NOUN
ejpam-6904	161	22	)	)	PUNCT
ejpam-6904	161	23	\	\	PROPN
ejpam-6904	161	24	s	s	VERB
ejpam-6904	161	25	can	can	AUX
ejpam-6904	161	26	not	not	PART
ejpam-6904	161	27	be	be	AUX
ejpam-6904	161	28	in	in	ADP
ejpam-6904	161	29	ig(s	ig(s	NUM
ejpam-6904	161	30	)	)	PUNCT
ejpam-6904	161	31	.	.	PUNCT
ejpam-6904	162	1	suppose	suppose	VERB
ejpam-6904	162	2	now	now	ADV
ejpam-6904	162	3	that	that	SCONJ
ejpam-6904	162	4	⟨s⟩	⟨s⟩	VERB
ejpam-6904	162	5	=	=	PUNCT
ejpam-6904	162	6	k3	k3	VERB
ejpam-6904	162	7	and	and	CCONJ
ejpam-6904	162	8	let	let	VERB
ejpam-6904	162	9	v	v	NUM
ejpam-6904	162	10	∈	∈	PROPN
ejpam-6904	162	11	v	v	NOUN
ejpam-6904	162	12	(	(	PUNCT
ejpam-6904	162	13	g	g	NOUN
ejpam-6904	162	14	)	)	PUNCT
ejpam-6904	162	15	\	\	PUNCT
ejpam-6904	163	1	s.	s.	PROPN
ejpam-6904	163	2	since	since	SCONJ
ejpam-6904	163	3	s	s	PROPN
ejpam-6904	163	4	is	be	AUX
ejpam-6904	163	5	a	a	DET
ejpam-6904	163	6	2	2	NUM
ejpam-6904	163	7	-	-	PUNCT
ejpam-6904	163	8	path	path	NOUN
ejpam-6904	163	9	geodetic	geodetic	ADJ
ejpam-6904	163	10	set	set	NOUN
ejpam-6904	163	11	in	in	ADP
ejpam-6904	163	12	g	g	PROPN
ejpam-6904	163	13	,	,	PUNCT
ejpam-6904	163	14	it	it	PRON
ejpam-6904	163	15	follows	follow	VERB
ejpam-6904	163	16	that	that	SCONJ
ejpam-6904	163	17	|ng(v	|ng(v	VERB
ejpam-6904	163	18	)	)	PUNCT
ejpam-6904	163	19	∩	∩	NOUN
ejpam-6904	163	20	s|	s|	VERB
ejpam-6904	163	21	≥	≥	NOUN
ejpam-6904	163	22	2	2	NUM
ejpam-6904	163	23	.	.	PUNCT
ejpam-6904	164	1	this	this	PRON
ejpam-6904	164	2	proves	prove	VERB
ejpam-6904	164	3	(	(	PUNCT
ejpam-6904	164	4	a	a	NOUN
ejpam-6904	164	5	)	)	PUNCT
ejpam-6904	164	6	.	.	PUNCT
ejpam-6904	165	1	suppose	suppose	VERB
ejpam-6904	165	2	now	now	ADV
ejpam-6904	165	3	that	that	SCONJ
ejpam-6904	165	4	⟨s⟩	⟨s⟩	VERB
ejpam-6904	165	5	=	=	PUNCT
ejpam-6904	166	1	⟨x⟩	⟨x⟩	PROPN
ejpam-6904	166	2	∪	∪	ADP
ejpam-6904	166	3	⟨{y	⟨{y	PROPN
ejpam-6904	166	4	,	,	PUNCT
ejpam-6904	166	5	z}⟩	z}⟩	PROPN
ejpam-6904	166	6	=	=	SYM
ejpam-6904	166	7	p1	p1	PROPN
ejpam-6904	166	8	∪	∪	NOUN
ejpam-6904	166	9	p2	p2	PROPN
ejpam-6904	166	10	and	and	CCONJ
ejpam-6904	166	11	let	let	VERB
ejpam-6904	166	12	v	v	NUM
ejpam-6904	166	13	∈	∈	PROPN
ejpam-6904	166	14	v	v	NOUN
ejpam-6904	166	15	(	(	PUNCT
ejpam-6904	166	16	g	g	NOUN
ejpam-6904	166	17	)	)	PUNCT
ejpam-6904	166	18	\	\	PUNCT
ejpam-6904	166	19	s.	s.	PROPN
ejpam-6904	166	20	since	since	SCONJ
ejpam-6904	166	21	s	s	PROPN
ejpam-6904	166	22	is	be	AUX
ejpam-6904	166	23	a	a	DET
ejpam-6904	166	24	2	2	NUM
ejpam-6904	166	25	-	-	PUNCT
ejpam-6904	166	26	path	path	NOUN
ejpam-6904	166	27	geodetic	geodetic	ADJ
ejpam-6904	166	28	set	set	NOUN
ejpam-6904	166	29	in	in	ADP
ejpam-6904	166	30	g	g	PROPN
ejpam-6904	166	31	,	,	PUNCT
ejpam-6904	166	32	it	it	PRON
ejpam-6904	166	33	follows	follow	VERB
ejpam-6904	166	34	that	that	SCONJ
ejpam-6904	166	35	x	x	SYM
ejpam-6904	166	36	,	,	PUNCT
ejpam-6904	166	37	z	z	PROPN
ejpam-6904	166	38	∈	∈	PROPN
ejpam-6904	166	39	ng(v	ng(v	PUNCT
ejpam-6904	166	40	)	)	PUNCT
ejpam-6904	166	41	or	or	CCONJ
ejpam-6904	166	42	x	x	X
ejpam-6904	166	43	,	,	PUNCT
ejpam-6904	166	44	y	y	PROPN
ejpam-6904	166	45	∈	∈	PROPN
ejpam-6904	166	46	ng(v	ng(v	PRON
ejpam-6904	166	47	)	)	PUNCT
ejpam-6904	166	48	.	.	PUNCT
ejpam-6904	167	1	this	this	PRON
ejpam-6904	167	2	proves	prove	VERB
ejpam-6904	167	3	(	(	PUNCT
ejpam-6904	167	4	b	b	NOUN
ejpam-6904	167	5	)	)	PUNCT
ejpam-6904	167	6	.	.	PUNCT
ejpam-6904	168	1	finally	finally	ADV
ejpam-6904	168	2	,	,	PUNCT
ejpam-6904	168	3	suppose	suppose	VERB
ejpam-6904	168	4	that	that	SCONJ
ejpam-6904	168	5	⟨{x	⟨{x	PROPN
ejpam-6904	168	6	,	,	PUNCT
ejpam-6904	168	7	y	y	PROPN
ejpam-6904	168	8	,	,	PUNCT
ejpam-6904	168	9	z}⟩	z}⟩	PROPN
ejpam-6904	168	10	=	=	PUNCT
ejpam-6904	169	1	[	[	X
ejpam-6904	169	2	x	x	X
ejpam-6904	169	3	,	,	PUNCT
ejpam-6904	169	4	y	y	PROPN
ejpam-6904	169	5	,	,	PUNCT
ejpam-6904	169	6	z	z	NOUN
ejpam-6904	169	7	]	]	X
ejpam-6904	169	8	=	=	SYM
ejpam-6904	169	9	p3	p3	PROPN
ejpam-6904	169	10	̸=	̸=	PROPN
ejpam-6904	169	11	g.	g.	NOUN
ejpam-6904	169	12	let	let	VERB
ejpam-6904	169	13	v	v	NUM
ejpam-6904	169	14	∈	∈	PROPN
ejpam-6904	169	15	v	v	NOUN
ejpam-6904	169	16	(	(	PUNCT
ejpam-6904	169	17	g)\{x	g)\{x	PROPN
ejpam-6904	169	18	,	,	PUNCT
ejpam-6904	169	19	y	y	PROPN
ejpam-6904	169	20	,	,	PUNCT
ejpam-6904	169	21	z	z	NOUN
ejpam-6904	169	22	}	}	PUNCT
ejpam-6904	169	23	.	.	PUNCT
ejpam-6904	170	1	since	since	SCONJ
ejpam-6904	170	2	{	{	PUNCT
ejpam-6904	170	3	x	x	NOUN
ejpam-6904	170	4	,	,	PUNCT
ejpam-6904	170	5	y	y	PROPN
ejpam-6904	170	6	,	,	PUNCT
ejpam-6904	170	7	z	z	NOUN
ejpam-6904	170	8	}	}	PUNCT
ejpam-6904	170	9	is	be	AUX
ejpam-6904	170	10	a	a	DET
ejpam-6904	170	11	2	2	NUM
ejpam-6904	170	12	-	-	PUNCT
ejpam-6904	170	13	path	path	NOUN
ejpam-6904	170	14	geodetic	geodetic	ADJ
ejpam-6904	170	15	set	set	NOUN
ejpam-6904	170	16	,	,	PUNCT
ejpam-6904	170	17	x	x	X
ejpam-6904	170	18	,	,	PUNCT
ejpam-6904	170	19	z	z	PROPN
ejpam-6904	170	20	∈	∈	PROPN
ejpam-6904	170	21	ng(v	ng(v	NOUN
ejpam-6904	170	22	)	)	PUNCT
ejpam-6904	170	23	.	.	PUNCT
ejpam-6904	171	1	suppose	suppose	VERB
ejpam-6904	171	2	ng(y	ng(y	NOUN
ejpam-6904	171	3	)	)	PUNCT
ejpam-6904	171	4	\	\	PART
ejpam-6904	171	5	s	s	PART
ejpam-6904	171	6	=	=	PUNCT
ejpam-6904	171	7	∅.	∅.	NOUN
ejpam-6904	171	8	since	since	SCONJ
ejpam-6904	171	9	y	y	PROPN
ejpam-6904	171	10	∈	∈	PROPN
ejpam-6904	171	11	ig(x	ig(x	ADJ
ejpam-6904	171	12	,	,	PUNCT
ejpam-6904	171	13	z	z	NOUN
ejpam-6904	171	14	)	)	PUNCT
ejpam-6904	171	15	,	,	PUNCT
ejpam-6904	171	16	it	it	PRON
ejpam-6904	171	17	follows	follow	VERB
ejpam-6904	171	18	that	that	SCONJ
ejpam-6904	171	19	{	{	PUNCT
ejpam-6904	171	20	x	x	X
ejpam-6904	171	21	,	,	PUNCT
ejpam-6904	171	22	z	z	NOUN
ejpam-6904	171	23	}	}	PUNCT
ejpam-6904	171	24	is	be	AUX
ejpam-6904	171	25	a	a	DET
ejpam-6904	171	26	2	2	NUM
ejpam-6904	171	27	-	-	PUNCT
ejpam-6904	171	28	path	path	NOUN
ejpam-6904	171	29	geodetic	geodetic	ADJ
ejpam-6904	171	30	vertex	vertex	NOUN
ejpam-6904	171	31	cover	cover	NOUN
ejpam-6904	171	32	of	of	ADP
ejpam-6904	171	33	g	g	NOUN
ejpam-6904	171	34	,	,	PUNCT
ejpam-6904	171	35	a	a	DET
ejpam-6904	171	36	contradiction	contradiction	NOUN
ejpam-6904	171	37	to	to	ADP
ejpam-6904	171	38	the	the	DET
ejpam-6904	171	39	assumption	assumption	NOUN
ejpam-6904	171	40	that	that	SCONJ
ejpam-6904	171	41	β2pg(g	β2pg(g	PUNCT
ejpam-6904	171	42	)	)	PUNCT
ejpam-6904	171	43	=	=	SYM
ejpam-6904	172	1	3	3	X
ejpam-6904	172	2	.	.	PUNCT
ejpam-6904	172	3	thus	thus	ADV
ejpam-6904	172	4	,	,	PUNCT
ejpam-6904	172	5	there	there	PRON
ejpam-6904	172	6	exists	exist	VERB
ejpam-6904	172	7	w	w	PROPN
ejpam-6904	172	8	∈	∈	PROPN
ejpam-6904	172	9	ng(y	ng(y	NOUN
ejpam-6904	172	10	)	)	PUNCT
ejpam-6904	172	11	\	\	PART
ejpam-6904	173	1	s	s	X
ejpam-6904	173	2	,	,	PUNCT
ejpam-6904	173	3	showing	show	VERB
ejpam-6904	173	4	that	that	SCONJ
ejpam-6904	173	5	(	(	PUNCT
ejpam-6904	173	6	c	c	X
ejpam-6904	173	7	)	)	PUNCT
ejpam-6904	173	8	holds	hold	NOUN
ejpam-6904	173	9	.	.	PUNCT
ejpam-6904	174	1	for	for	ADP
ejpam-6904	174	2	the	the	DET
ejpam-6904	174	3	converse	converse	NOUN
ejpam-6904	174	4	,	,	PUNCT
ejpam-6904	174	5	suppose	suppose	VERB
ejpam-6904	174	6	that	that	SCONJ
ejpam-6904	174	7	v	v	X
ejpam-6904	174	8	(	(	PUNCT
ejpam-6904	174	9	g	g	NOUN
ejpam-6904	174	10	)	)	PUNCT
ejpam-6904	174	11	\	\	PROPN
ejpam-6904	175	1	s	s	PART
ejpam-6904	175	2	is	be	AUX
ejpam-6904	175	3	independent	independent	ADJ
ejpam-6904	175	4	.	.	PUNCT
ejpam-6904	176	1	assume	assume	VERB
ejpam-6904	176	2	first	first	ADV
ejpam-6904	176	3	that	that	SCONJ
ejpam-6904	176	4	g	g	PROPN
ejpam-6904	176	5	∈	∈	PROPN
ejpam-6904	176	6	{	{	PUNCT
ejpam-6904	176	7	k3,k3	k3,k3	PROPN
ejpam-6904	176	8	,	,	PUNCT
ejpam-6904	176	9	p1	p1	NOUN
ejpam-6904	176	10	∪	∪	NOUN
ejpam-6904	176	11	p2	p2	NOUN
ejpam-6904	176	12	}	}	PUNCT
ejpam-6904	176	13	.	.	PUNCT
ejpam-6904	177	1	then	then	ADV
ejpam-6904	177	2	β2pg(g	β2pg(g	PUNCT
ejpam-6904	177	3	)	)	PUNCT
ejpam-6904	177	4	=	=	SYM
ejpam-6904	178	1	3	3	X
ejpam-6904	178	2	.	.	PUNCT
ejpam-6904	179	1	next	next	ADV
ejpam-6904	179	2	,	,	PUNCT
ejpam-6904	179	3	suppose	suppose	VERB
ejpam-6904	179	4	that	that	SCONJ
ejpam-6904	179	5	(	(	PUNCT
ejpam-6904	179	6	a	a	X
ejpam-6904	179	7	)	)	PUNCT
ejpam-6904	179	8	holds	hold	VERB
ejpam-6904	179	9	.	.	PUNCT
ejpam-6904	180	1	let	let	VERB
ejpam-6904	180	2	s	s	VERB
ejpam-6904	180	3	=	=	PUNCT
ejpam-6904	180	4	{	{	PUNCT
ejpam-6904	180	5	x	x	PROPN
ejpam-6904	180	6	,	,	PUNCT
ejpam-6904	180	7	y	y	PROPN
ejpam-6904	180	8	,	,	PUNCT
ejpam-6904	180	9	z	z	NOUN
ejpam-6904	180	10	}	}	PUNCT
ejpam-6904	180	11	and	and	CCONJ
ejpam-6904	180	12	let	let	VERB
ejpam-6904	180	13	vw	vw	PRON
ejpam-6904	180	14	∈	∈	PROPN
ejpam-6904	180	15	e(g	e(g	PROPN
ejpam-6904	180	16	)	)	PUNCT
ejpam-6904	180	17	.	.	PUNCT
ejpam-6904	181	1	since	since	SCONJ
ejpam-6904	181	2	v	v	NOUN
ejpam-6904	181	3	(	(	PUNCT
ejpam-6904	181	4	g	g	NOUN
ejpam-6904	181	5	)	)	PUNCT
ejpam-6904	181	6	\	\	PROPN
ejpam-6904	182	1	s	s	PART
ejpam-6904	182	2	is	be	AUX
ejpam-6904	182	3	independent	independent	ADJ
ejpam-6904	182	4	,	,	PUNCT
ejpam-6904	182	5	it	it	PRON
ejpam-6904	182	6	follows	follow	VERB
ejpam-6904	182	7	that	that	SCONJ
ejpam-6904	182	8	v	v	NUM
ejpam-6904	182	9	∈	∈	PROPN
ejpam-6904	182	10	s	s	NOUN
ejpam-6904	182	11	or	or	CCONJ
ejpam-6904	182	12	w	w	PROPN
ejpam-6904	182	13	∈	∈	PROPN
ejpam-6904	182	14	s.	s.	PROPN
ejpam-6904	182	15	hence	hence	ADV
ejpam-6904	182	16	s	s	VERB
ejpam-6904	182	17	is	be	AUX
ejpam-6904	182	18	a	a	DET
ejpam-6904	182	19	vertex	vertex	NOUN
ejpam-6904	182	20	cover	cover	NOUN
ejpam-6904	182	21	of	of	ADP
ejpam-6904	182	22	g.	g.	PROPN
ejpam-6904	182	23	let	let	VERB
ejpam-6904	182	24	v	v	NUM
ejpam-6904	182	25	∈	∈	PROPN
ejpam-6904	182	26	v	v	NOUN
ejpam-6904	182	27	(	(	PUNCT
ejpam-6904	182	28	g	g	NOUN
ejpam-6904	182	29	)	)	PUNCT
ejpam-6904	182	30	\	\	PUNCT
ejpam-6904	183	1	s.	s.	PROPN
ejpam-6904	183	2	then	then	ADV
ejpam-6904	183	3	,	,	PUNCT
ejpam-6904	183	4	by	by	ADP
ejpam-6904	183	5	(	(	PUNCT
ejpam-6904	183	6	a	a	X
ejpam-6904	183	7	)	)	PUNCT
ejpam-6904	183	8	,	,	PUNCT
ejpam-6904	183	9	s	s	VERB
ejpam-6904	183	10	is	be	AUX
ejpam-6904	183	11	a	a	DET
ejpam-6904	183	12	2	2	NUM
ejpam-6904	183	13	-	-	PUNCT
ejpam-6904	183	14	path	path	NOUN
ejpam-6904	183	15	geodetic	geodetic	ADJ
ejpam-6904	183	16	set	set	NOUN
ejpam-6904	183	17	in	in	ADP
ejpam-6904	183	18	g.	g.	PROPN
ejpam-6904	183	19	therefore	therefore	ADV
ejpam-6904	183	20	,	,	PUNCT
ejpam-6904	183	21	s	s	VERB
ejpam-6904	183	22	is	be	AUX
ejpam-6904	183	23	a	a	DET
ejpam-6904	183	24	2	2	NUM
ejpam-6904	183	25	-	-	PUNCT
ejpam-6904	183	26	path	path	NOUN
ejpam-6904	183	27	geodetic	geodetic	ADJ
ejpam-6904	183	28	vertex	vertex	NOUN
ejpam-6904	183	29	cover	cover	NOUN
ejpam-6904	183	30	and	and	CCONJ
ejpam-6904	183	31	β2pg(g	β2pg(g	PRON
ejpam-6904	183	32	)	)	PUNCT
ejpam-6904	183	33	=	=	SYM
ejpam-6904	183	34	|s|	|s|	NOUN
ejpam-6904	183	35	=	=	SYM
ejpam-6904	183	36	3	3	NUM
ejpam-6904	183	37	by	by	ADP
ejpam-6904	183	38	(	(	PUNCT
ejpam-6904	183	39	ii	ii	NOUN
ejpam-6904	183	40	)	)	PUNCT
ejpam-6904	183	41	.	.	PUNCT
ejpam-6904	184	1	suppose	suppose	VERB
ejpam-6904	184	2	that	that	SCONJ
ejpam-6904	184	3	(	(	PUNCT
ejpam-6904	184	4	b	b	X
ejpam-6904	184	5	)	)	PUNCT
ejpam-6904	184	6	holds	hold	VERB
ejpam-6904	184	7	.	.	PUNCT
ejpam-6904	185	1	let	let	VERB
ejpam-6904	185	2	pq	pq	INTJ
ejpam-6904	185	3	∈	∈	PROPN
ejpam-6904	185	4	e(g	e(g	PROPN
ejpam-6904	185	5	)	)	PUNCT
ejpam-6904	185	6	.	.	PUNCT
ejpam-6904	186	1	again	again	ADV
ejpam-6904	186	2	,	,	PUNCT
ejpam-6904	186	3	since	since	SCONJ
ejpam-6904	186	4	v	v	NOUN
ejpam-6904	186	5	(	(	PUNCT
ejpam-6904	186	6	g	g	NOUN
ejpam-6904	186	7	)	)	PUNCT
ejpam-6904	186	8	\	\	PROPN
ejpam-6904	187	1	s	s	PART
ejpam-6904	187	2	is	be	AUX
ejpam-6904	187	3	an	an	DET
ejpam-6904	187	4	independent	independent	ADJ
ejpam-6904	187	5	set	set	NOUN
ejpam-6904	187	6	,	,	PUNCT
ejpam-6904	187	7	p	p	PROPN
ejpam-6904	187	8	∈	∈	PROPN
ejpam-6904	187	9	s	s	NOUN
ejpam-6904	187	10	or	or	CCONJ
ejpam-6904	187	11	q	q	PROPN
ejpam-6904	187	12	∈	∈	PROPN
ejpam-6904	187	13	s.	s.	PROPN
ejpam-6904	187	14	let	let	VERB
ejpam-6904	187	15	v	v	ADP
ejpam-6904	187	16	∈	∈	PROPN
ejpam-6904	187	17	v	v	NOUN
ejpam-6904	187	18	(	(	PUNCT
ejpam-6904	187	19	g)\s	g)\s	NOUN
ejpam-6904	187	20	.	.	PUNCT
ejpam-6904	188	1	by	by	ADP
ejpam-6904	188	2	(	(	PUNCT
ejpam-6904	188	3	b	b	NOUN
ejpam-6904	188	4	)	)	PUNCT
ejpam-6904	188	5	,	,	PUNCT
ejpam-6904	188	6	it	it	PRON
ejpam-6904	188	7	follows	follow	VERB
ejpam-6904	188	8	that	that	SCONJ
ejpam-6904	188	9	s	s	VERB
ejpam-6904	188	10	is	be	AUX
ejpam-6904	188	11	a	a	DET
ejpam-6904	188	12	2	2	NUM
ejpam-6904	188	13	-	-	PUNCT
ejpam-6904	188	14	path	path	NOUN
ejpam-6904	188	15	geodetic	geodetic	ADJ
ejpam-6904	188	16	set	set	NOUN
ejpam-6904	188	17	in	in	ADP
ejpam-6904	188	18	g.	g.	PROPN
ejpam-6904	188	19	therefore	therefore	ADV
ejpam-6904	188	20	,	,	PUNCT
ejpam-6904	188	21	s	s	VERB
ejpam-6904	188	22	is	be	AUX
ejpam-6904	188	23	a	a	DET
ejpam-6904	188	24	geodetic	geodetic	ADJ
ejpam-6904	188	25	vertex	vertex	NOUN
ejpam-6904	188	26	cover	cover	NOUN
ejpam-6904	188	27	in	in	ADP
ejpam-6904	188	28	g.	g.	PROPN
ejpam-6904	188	29	again	again	ADV
ejpam-6904	188	30	,	,	PUNCT
ejpam-6904	188	31	by	by	ADP
ejpam-6904	188	32	(	(	PUNCT
ejpam-6904	188	33	ii	ii	NOUN
ejpam-6904	188	34	)	)	PUNCT
ejpam-6904	188	35	,	,	PUNCT
ejpam-6904	188	36	it	it	PRON
ejpam-6904	188	37	follows	follow	VERB
ejpam-6904	188	38	that	that	PRON
ejpam-6904	188	39	β2pg(g	β2pg(g	PUNCT
ejpam-6904	188	40	)	)	PUNCT
ejpam-6904	188	41	=	=	SYM
ejpam-6904	188	42	|s|	|s|	NOUN
ejpam-6904	188	43	=	=	SYM
ejpam-6904	188	44	3	3	X
ejpam-6904	188	45	.	.	PUNCT
ejpam-6904	189	1	lastly	lastly	ADV
ejpam-6904	189	2	,	,	PUNCT
ejpam-6904	189	3	suppose	suppose	VERB
ejpam-6904	189	4	that	that	SCONJ
ejpam-6904	189	5	(	(	PUNCT
ejpam-6904	189	6	c	c	X
ejpam-6904	189	7	)	)	PUNCT
ejpam-6904	189	8	holds	hold	NOUN
ejpam-6904	189	9	.	.	PUNCT
ejpam-6904	190	1	since	since	SCONJ
ejpam-6904	190	2	v	v	NOUN
ejpam-6904	190	3	(	(	PUNCT
ejpam-6904	190	4	g	g	NOUN
ejpam-6904	190	5	)	)	PUNCT
ejpam-6904	190	6	\	\	PROPN
ejpam-6904	191	1	s	s	PART
ejpam-6904	191	2	is	be	AUX
ejpam-6904	191	3	independent	independent	ADJ
ejpam-6904	191	4	,	,	PUNCT
ejpam-6904	191	5	s	s	PART
ejpam-6904	191	6	is	be	AUX
ejpam-6904	191	7	a	a	DET
ejpam-6904	191	8	vertex	vertex	NOUN
ejpam-6904	191	9	cover	cover	NOUN
ejpam-6904	191	10	of	of	ADP
ejpam-6904	191	11	g.	g.	PROPN
ejpam-6904	191	12	let	let	VERB
ejpam-6904	191	13	v	v	NUM
ejpam-6904	191	14	∈	∈	PROPN
ejpam-6904	191	15	v	v	NOUN
ejpam-6904	191	16	(	(	PUNCT
ejpam-6904	191	17	g	g	NOUN
ejpam-6904	191	18	)	)	PUNCT
ejpam-6904	191	19	\	\	PUNCT
ejpam-6904	192	1	s.	s.	PROPN
ejpam-6904	192	2	since	since	SCONJ
ejpam-6904	192	3	x	x	X
ejpam-6904	192	4	,	,	PUNCT
ejpam-6904	192	5	z	z	PROPN
ejpam-6904	192	6	∈	∈	PROPN
ejpam-6904	192	7	ng(v	ng(v	NOUN
ejpam-6904	192	8	)	)	PUNCT
ejpam-6904	192	9	,	,	PUNCT
ejpam-6904	192	10	it	it	PRON
ejpam-6904	192	11	follows	follow	VERB
ejpam-6904	192	12	that	that	SCONJ
ejpam-6904	192	13	v	v	ADP
ejpam-6904	192	14	∈	∈	PROPN
ejpam-6904	192	15	ig(x	ig(x	X
ejpam-6904	192	16	,	,	PUNCT
ejpam-6904	192	17	z	z	NOUN
ejpam-6904	192	18	)	)	PUNCT
ejpam-6904	192	19	.	.	PUNCT
ejpam-6904	193	1	hence	hence	ADV
ejpam-6904	193	2	,	,	PUNCT
ejpam-6904	193	3	s	s	VERB
ejpam-6904	193	4	is	be	AUX
ejpam-6904	193	5	a	a	DET
ejpam-6904	193	6	2	2	NUM
ejpam-6904	193	7	-	-	PUNCT
ejpam-6904	193	8	path	path	NOUN
ejpam-6904	193	9	a.	a.	PROPN
ejpam-6904	193	10	b.	b.	PROPN
ejpam-6904	193	11	tapeing	tapeing	PROPN
ejpam-6904	193	12	,	,	PUNCT
ejpam-6904	193	13	s.	s.	PROPN
ejpam-6904	193	14	r.	r.	PROPN
ejpam-6904	193	15	canoy	canoy	PROPN
ejpam-6904	193	16	/	/	SYM
ejpam-6904	193	17	eur	eur	PROPN
ejpam-6904	193	18	.	.	PUNCT
ejpam-6904	194	1	j.	j.	PROPN
ejpam-6904	194	2	pure	pure	PROPN
ejpam-6904	194	3	appl	appl	PROPN
ejpam-6904	194	4	.	.	PROPN
ejpam-6904	194	5	math	math	PROPN
ejpam-6904	194	6	,	,	PUNCT
ejpam-6904	194	7	18	18	NUM
ejpam-6904	194	8	(	(	PUNCT
ejpam-6904	194	9	4	4	NUM
ejpam-6904	194	10	)	)	PUNCT
ejpam-6904	194	11	(	(	PUNCT
ejpam-6904	194	12	2025	2025	NUM
ejpam-6904	194	13	)	)	PUNCT
ejpam-6904	194	14	,	,	PUNCT
ejpam-6904	194	15	6904	6904	NUM
ejpam-6904	194	16	7	7	NUM
ejpam-6904	194	17	of	of	ADP
ejpam-6904	194	18	14	14	NUM
ejpam-6904	194	19	geodetic	geodetic	ADJ
ejpam-6904	194	20	set	set	NOUN
ejpam-6904	194	21	in	in	ADP
ejpam-6904	194	22	g.	g.	PROPN
ejpam-6904	194	23	moreover	moreover	ADV
ejpam-6904	194	24	,	,	PUNCT
ejpam-6904	194	25	since	since	SCONJ
ejpam-6904	194	26	ng(y	ng(y	NOUN
ejpam-6904	194	27	)	)	PUNCT
ejpam-6904	194	28	\	\	PART
ejpam-6904	195	1	s	s	PART
ejpam-6904	195	2	̸=	̸=	PROPN
ejpam-6904	195	3	∅	∅	NOUN
ejpam-6904	195	4	,	,	PUNCT
ejpam-6904	195	5	g	g	PROPN
ejpam-6904	195	6	̸=	̸=	PROPN
ejpam-6904	195	7	k2,n−2	k2,n−2	PROPN
ejpam-6904	195	8	.	.	PUNCT
ejpam-6904	196	1	thus	thus	ADV
ejpam-6904	196	2	,	,	PUNCT
ejpam-6904	196	3	by	by	ADP
ejpam-6904	196	4	part	part	NOUN
ejpam-6904	196	5	(	(	PUNCT
ejpam-6904	196	6	ii	ii	NOUN
ejpam-6904	196	7	)	)	PUNCT
ejpam-6904	196	8	,	,	PUNCT
ejpam-6904	196	9	we	we	PRON
ejpam-6904	196	10	must	must	AUX
ejpam-6904	196	11	have	have	AUX
ejpam-6904	196	12	β2pg(g	β2pg(g	PUNCT
ejpam-6904	196	13	)	)	PUNCT
ejpam-6904	196	14	=	=	SYM
ejpam-6904	196	15	3	3	X
ejpam-6904	196	16	.	.	X
ejpam-6904	196	17	theorem	theorem	NOUN
ejpam-6904	196	18	4	4	NUM
ejpam-6904	196	19	.	.	PUNCT
ejpam-6904	197	1	let	let	VERB
ejpam-6904	197	2	g	g	PRON
ejpam-6904	197	3	be	be	AUX
ejpam-6904	197	4	a	a	DET
ejpam-6904	197	5	graph	graph	NOUN
ejpam-6904	197	6	of	of	ADP
ejpam-6904	197	7	order	order	NOUN
ejpam-6904	197	8	n.	n.	NOUN
ejpam-6904	197	9	then	then	ADV
ejpam-6904	197	10	β2pg(g	β2pg(g	PUNCT
ejpam-6904	197	11	)	)	PUNCT
ejpam-6904	197	12	=	=	SYM
ejpam-6904	198	1	n	n	NOUN
ejpam-6904	198	2	if	if	SCONJ
ejpam-6904	198	3	and	and	CCONJ
ejpam-6904	198	4	if	if	SCONJ
ejpam-6904	198	5	g′	g′	NOUN
ejpam-6904	198	6	is	be	AUX
ejpam-6904	198	7	complete	complete	ADJ
ejpam-6904	198	8	for	for	ADP
ejpam-6904	198	9	every	every	DET
ejpam-6904	198	10	component	component	NOUN
ejpam-6904	198	11	g′	g′	NOUN
ejpam-6904	198	12	of	of	ADP
ejpam-6904	198	13	g.	g.	PROPN
ejpam-6904	198	14	proof	proof	PROPN
ejpam-6904	198	15	.	.	PUNCT
ejpam-6904	199	1	suppose	suppose	VERB
ejpam-6904	199	2	that	that	SCONJ
ejpam-6904	199	3	β2pg(g	β2pg(g	PROPN
ejpam-6904	199	4	)	)	PUNCT
ejpam-6904	199	5	=	=	SYM
ejpam-6904	199	6	n.	n.	NOUN
ejpam-6904	199	7	suppose	suppose	VERB
ejpam-6904	199	8	further	far	ADV
ejpam-6904	199	9	that	that	SCONJ
ejpam-6904	199	10	there	there	PRON
ejpam-6904	199	11	exists	exist	VERB
ejpam-6904	199	12	a	a	DET
ejpam-6904	199	13	component	component	NOUN
ejpam-6904	199	14	g′	g′	NOUN
ejpam-6904	199	15	of	of	ADP
ejpam-6904	199	16	g	g	PROPN
ejpam-6904	199	17	which	which	PRON
ejpam-6904	199	18	is	be	AUX
ejpam-6904	199	19	not	not	PART
ejpam-6904	199	20	complete	complete	ADJ
ejpam-6904	199	21	.	.	PUNCT
ejpam-6904	200	1	then	then	ADV
ejpam-6904	200	2	there	there	PRON
ejpam-6904	200	3	exist	exist	VERB
ejpam-6904	200	4	vertices	vertex	NOUN
ejpam-6904	200	5	p	p	NOUN
ejpam-6904	200	6	,	,	PUNCT
ejpam-6904	201	1	q	q	PROPN
ejpam-6904	201	2	∈	∈	PROPN
ejpam-6904	201	3	v	v	ADP
ejpam-6904	201	4	(	(	PUNCT
ejpam-6904	201	5	g′	g′	NOUN
ejpam-6904	201	6	)	)	PUNCT
ejpam-6904	201	7	such	such	ADJ
ejpam-6904	201	8	that	that	SCONJ
ejpam-6904	201	9	dg(p	dg(p	NOUN
ejpam-6904	201	10	,	,	PUNCT
ejpam-6904	201	11	q	q	X
ejpam-6904	201	12	)	)	PUNCT
ejpam-6904	201	13	=	=	SYM
ejpam-6904	201	14	2	2	X
ejpam-6904	201	15	.	.	X
ejpam-6904	201	16	let	let	VERB
ejpam-6904	201	17	x	x	SYM
ejpam-6904	201	18	∈	∈	NOUN
ejpam-6904	201	19	ng(p	ng(p	NOUN
ejpam-6904	201	20	)	)	PUNCT
ejpam-6904	201	21	∩	∩	NOUN
ejpam-6904	201	22	ng(q	ng(q	NOUN
ejpam-6904	201	23	)	)	PUNCT
ejpam-6904	201	24	.	.	PUNCT
ejpam-6904	202	1	then	then	ADV
ejpam-6904	202	2	d	d	X
ejpam-6904	202	3	=	=	SYM
ejpam-6904	202	4	v	v	PROPN
ejpam-6904	202	5	(	(	PUNCT
ejpam-6904	202	6	g′	g′	NOUN
ejpam-6904	202	7	)	)	PUNCT
ejpam-6904	202	8	\	\	NOUN
ejpam-6904	203	1	{	{	PUNCT
ejpam-6904	203	2	x	x	NOUN
ejpam-6904	203	3	}	}	PUNCT
ejpam-6904	203	4	is	be	AUX
ejpam-6904	203	5	a	a	DET
ejpam-6904	203	6	2	2	NUM
ejpam-6904	203	7	-	-	PUNCT
ejpam-6904	203	8	path	path	NOUN
ejpam-6904	203	9	geodetic	geodetic	ADJ
ejpam-6904	203	10	vertex	vertex	NOUN
ejpam-6904	203	11	cover	cover	NOUN
ejpam-6904	203	12	of	of	ADP
ejpam-6904	203	13	g′.	g′.	ADP
ejpam-6904	203	14	this	this	PRON
ejpam-6904	203	15	implies	imply	VERB
ejpam-6904	203	16	that	that	SCONJ
ejpam-6904	203	17	β2pg(g′	β2pg(g′	NOUN
ejpam-6904	203	18	)	)	PUNCT
ejpam-6904	203	19	≤	≤	NOUN
ejpam-6904	203	20	|v	|v	X
ejpam-6904	203	21	(	(	PUNCT
ejpam-6904	203	22	g′)|−	g′)|−	PROPN
ejpam-6904	203	23	1	1	NUM
ejpam-6904	203	24	.	.	PUNCT
ejpam-6904	203	25	by	by	ADP
ejpam-6904	203	26	theorem	theorem	NOUN
ejpam-6904	203	27	1	1	NUM
ejpam-6904	203	28	,	,	PUNCT
ejpam-6904	203	29	β2pg(g	β2pg(g	PUNCT
ejpam-6904	203	30	)	)	PUNCT
ejpam-6904	203	31	≤	≤	NUM
ejpam-6904	203	32	n−	n−	NOUN
ejpam-6904	203	33	1	1	NUM
ejpam-6904	203	34	,	,	PUNCT
ejpam-6904	203	35	a	a	DET
ejpam-6904	203	36	contradiction	contradiction	NOUN
ejpam-6904	203	37	to	to	ADP
ejpam-6904	203	38	the	the	DET
ejpam-6904	203	39	assumption	assumption	NOUN
ejpam-6904	203	40	.	.	PUNCT
ejpam-6904	204	1	therefore	therefore	ADV
ejpam-6904	204	2	,	,	PUNCT
ejpam-6904	204	3	every	every	DET
ejpam-6904	204	4	component	component	NOUN
ejpam-6904	204	5	of	of	ADP
ejpam-6904	204	6	g	g	PROPN
ejpam-6904	204	7	is	be	AUX
ejpam-6904	204	8	complete	complete	ADJ
ejpam-6904	204	9	.	.	PUNCT
ejpam-6904	205	1	for	for	ADP
ejpam-6904	205	2	the	the	DET
ejpam-6904	205	3	converse	converse	NOUN
ejpam-6904	205	4	,	,	PUNCT
ejpam-6904	205	5	suppose	suppose	VERB
ejpam-6904	205	6	that	that	SCONJ
ejpam-6904	205	7	every	every	DET
ejpam-6904	205	8	component	component	NOUN
ejpam-6904	205	9	g′	g′	NOUN
ejpam-6904	205	10	of	of	ADP
ejpam-6904	205	11	g	g	PROPN
ejpam-6904	205	12	is	be	AUX
ejpam-6904	205	13	complete	complete	ADJ
ejpam-6904	205	14	.	.	PUNCT
ejpam-6904	206	1	by	by	ADP
ejpam-6904	206	2	corollary	corollary	ADJ
ejpam-6904	206	3	1	1	NUM
ejpam-6904	206	4	,	,	PUNCT
ejpam-6904	206	5	β2pg(g	β2pg(g	PUNCT
ejpam-6904	206	6	′	′	NUM
ejpam-6904	206	7	)	)	PUNCT
ejpam-6904	206	8	=	=	SYM
ejpam-6904	206	9	|v	|v	PROPN
ejpam-6904	206	10	(	(	PUNCT
ejpam-6904	206	11	g′)|	g′)|	NOUN
ejpam-6904	206	12	for	for	ADP
ejpam-6904	206	13	every	every	DET
ejpam-6904	206	14	component	component	NOUN
ejpam-6904	206	15	g′	g′	NOUN
ejpam-6904	206	16	of	of	ADP
ejpam-6904	206	17	g.	g.	PROPN
ejpam-6904	206	18	therefore	therefore	ADV
ejpam-6904	206	19	,	,	PUNCT
ejpam-6904	206	20	by	by	ADP
ejpam-6904	206	21	theorem1	theorem1	PROPN
ejpam-6904	206	22	,	,	PUNCT
ejpam-6904	206	23	we	we	PRON
ejpam-6904	206	24	have	have	VERB
ejpam-6904	206	25	β2pg(g	β2pg(g	PUNCT
ejpam-6904	206	26	)	)	PUNCT
ejpam-6904	206	27	=	=	SYM
ejpam-6904	206	28	n.	n.	NOUN
ejpam-6904	206	29	theorem	theorem	VERB
ejpam-6904	206	30	5	5	NUM
ejpam-6904	206	31	.	.	PUNCT
ejpam-6904	207	1	let	let	VERB
ejpam-6904	207	2	g	g	PRON
ejpam-6904	207	3	be	be	AUX
ejpam-6904	207	4	a	a	DET
ejpam-6904	207	5	graph	graph	NOUN
ejpam-6904	207	6	of	of	ADP
ejpam-6904	207	7	order	order	NOUN
ejpam-6904	207	8	n.	n.	NOUN
ejpam-6904	207	9	then	then	ADV
ejpam-6904	207	10	β2pg(g	β2pg(g	PUNCT
ejpam-6904	207	11	)	)	PUNCT
ejpam-6904	207	12	=	=	PUNCT
ejpam-6904	207	13	n−	n−	NOUN
ejpam-6904	207	14	1	1	NUM
ejpam-6904	207	15	if	if	SCONJ
ejpam-6904	208	1	and	and	CCONJ
ejpam-6904	208	2	only	only	ADV
ejpam-6904	208	3	if	if	SCONJ
ejpam-6904	208	4	all	all	PRON
ejpam-6904	208	5	but	but	SCONJ
ejpam-6904	208	6	a	a	DET
ejpam-6904	208	7	component	component	NOUN
ejpam-6904	208	8	h	h	NOUN
ejpam-6904	208	9	of	of	ADP
ejpam-6904	208	10	g	g	PROPN
ejpam-6904	208	11	are	be	AUX
ejpam-6904	208	12	complete	complete	ADJ
ejpam-6904	208	13	and	and	CCONJ
ejpam-6904	208	14	⟨v	⟨v	NUM
ejpam-6904	208	15	(	(	PUNCT
ejpam-6904	208	16	h	h	NOUN
ejpam-6904	208	17	)	)	PUNCT
ejpam-6904	208	18	\	\	NOUN
ejpam-6904	209	1	ext(h)⟩	ext(h)⟩	PROPN
ejpam-6904	209	2	is	be	AUX
ejpam-6904	209	3	complete	complete	ADJ
ejpam-6904	209	4	.	.	PUNCT
ejpam-6904	210	1	proof	proof	NOUN
ejpam-6904	210	2	.	.	PUNCT
ejpam-6904	211	1	let	let	VERB
ejpam-6904	211	2	g1	g1	PROPN
ejpam-6904	211	3	,	,	PUNCT
ejpam-6904	211	4	g2	g2	PROPN
ejpam-6904	211	5	,	,	PUNCT
ejpam-6904	211	6	·	·	PUNCT
ejpam-6904	211	7	·	·	PUNCT
ejpam-6904	211	8	·	·	PUNCT
ejpam-6904	211	9	,	,	PUNCT
ejpam-6904	211	10	gk	gk	PROPN
ejpam-6904	211	11	be	be	AUX
ejpam-6904	211	12	the	the	DET
ejpam-6904	211	13	components	component	NOUN
ejpam-6904	211	14	of	of	ADP
ejpam-6904	211	15	g.	g.	PROPN
ejpam-6904	211	16	suppose	suppose	VERB
ejpam-6904	211	17	β2pg	β2pg	PUNCT
ejpam-6904	212	1	=	=	SYM
ejpam-6904	212	2	n−1	n−1	PROPN
ejpam-6904	212	3	.	.	PUNCT
ejpam-6904	212	4	by	by	ADP
ejpam-6904	212	5	theorem	theorem	NOUN
ejpam-6904	212	6	1	1	NUM
ejpam-6904	212	7	and	and	CCONJ
ejpam-6904	212	8	theorem	theorem	VERB
ejpam-6904	212	9	4	4	NUM
ejpam-6904	212	10	,	,	PUNCT
ejpam-6904	212	11	there	there	PRON
ejpam-6904	212	12	exists	exist	VERB
ejpam-6904	212	13	a	a	DET
ejpam-6904	212	14	component	component	NOUN
ejpam-6904	212	15	h	h	NOUN
ejpam-6904	212	16	=	=	NOUN
ejpam-6904	212	17	gt	gt	PROPN
ejpam-6904	212	18	of	of	ADP
ejpam-6904	212	19	g	g	PROPN
ejpam-6904	212	20	which	which	PRON
ejpam-6904	212	21	is	be	AUX
ejpam-6904	212	22	not	not	PART
ejpam-6904	212	23	complete	complete	ADJ
ejpam-6904	212	24	.	.	PUNCT
ejpam-6904	213	1	hence	hence	ADV
ejpam-6904	213	2	,	,	PUNCT
ejpam-6904	213	3	by	by	ADP
ejpam-6904	213	4	theorem	theorem	NOUN
ejpam-6904	213	5	1	1	NUM
ejpam-6904	213	6	and	and	CCONJ
ejpam-6904	213	7	the	the	DET
ejpam-6904	213	8	assumption	assumption	NOUN
ejpam-6904	213	9	,	,	PUNCT
ejpam-6904	213	10	β2pg(h	β2pg(h	X
ejpam-6904	213	11	)	)	PUNCT
ejpam-6904	213	12	=	=	SYM
ejpam-6904	213	13	|v	|v	PROPN
ejpam-6904	213	14	(	(	PUNCT
ejpam-6904	213	15	h)|	h)|	NOUN
ejpam-6904	213	16	−	−	PROPN
ejpam-6904	213	17	1	1	NUM
ejpam-6904	213	18	and	and	CCONJ
ejpam-6904	213	19	β2pg(gj	β2pg(gj	NUM
ejpam-6904	213	20	)	)	PUNCT
ejpam-6904	214	1	=	=	SYM
ejpam-6904	214	2	|v	|v	X
ejpam-6904	214	3	(	(	PUNCT
ejpam-6904	214	4	gj)|	gj)|	NOUN
ejpam-6904	214	5	for	for	ADP
ejpam-6904	214	6	every	every	DET
ejpam-6904	214	7	j	j	PROPN
ejpam-6904	214	8	∈	∈	PROPN
ejpam-6904	214	9	{	{	PUNCT
ejpam-6904	214	10	1	1	NUM
ejpam-6904	214	11	,	,	PUNCT
ejpam-6904	214	12	2	2	NUM
ejpam-6904	214	13	,	,	PUNCT
ejpam-6904	214	14	·	·	PUNCT
ejpam-6904	214	15	·	·	PUNCT
ejpam-6904	214	16	·	·	PUNCT
ejpam-6904	214	17	,	,	PUNCT
ejpam-6904	214	18	t	t	PROPN
ejpam-6904	214	19	−	−	PROPN
ejpam-6904	214	20	1	1	NUM
ejpam-6904	214	21	,	,	PUNCT
ejpam-6904	214	22	t	t	NOUN
ejpam-6904	214	23	+	+	CCONJ
ejpam-6904	214	24	1	1	NUM
ejpam-6904	214	25	,	,	PUNCT
ejpam-6904	214	26	·	·	PUNCT
ejpam-6904	214	27	·	·	PUNCT
ejpam-6904	214	28	·	·	PUNCT
ejpam-6904	214	29	,	,	PUNCT
ejpam-6904	214	30	k	k	X
ejpam-6904	214	31	}	}	PUNCT
ejpam-6904	214	32	.	.	PUNCT
ejpam-6904	215	1	let	let	VERB
ejpam-6904	215	2	h∗	h∗	PROPN
ejpam-6904	215	3	=	=	SYM
ejpam-6904	215	4	⟨v	⟨v	PROPN
ejpam-6904	215	5	(	(	PUNCT
ejpam-6904	215	6	h	h	NOUN
ejpam-6904	215	7	)	)	PUNCT
ejpam-6904	215	8	\	\	PUNCT
ejpam-6904	216	1	ext(h)⟩.	ext(h)⟩.	PROPN
ejpam-6904	216	2	since	since	SCONJ
ejpam-6904	216	3	h	h	NOUN
ejpam-6904	216	4	is	be	AUX
ejpam-6904	216	5	connected	connect	VERB
ejpam-6904	216	6	,	,	PUNCT
ejpam-6904	216	7	it	it	PRON
ejpam-6904	216	8	follows	follow	VERB
ejpam-6904	216	9	that	that	SCONJ
ejpam-6904	216	10	h∗	h∗	PROPN
ejpam-6904	216	11	is	be	AUX
ejpam-6904	216	12	connected	connect	VERB
ejpam-6904	216	13	.	.	PUNCT
ejpam-6904	217	1	suppose	suppose	VERB
ejpam-6904	217	2	h∗	h∗	PROPN
ejpam-6904	217	3	=	=	SYM
ejpam-6904	217	4	⟨v	⟨v	PROPN
ejpam-6904	217	5	(	(	PUNCT
ejpam-6904	217	6	h	h	NOUN
ejpam-6904	217	7	)	)	PUNCT
ejpam-6904	217	8	\	\	NOUN
ejpam-6904	218	1	ext(h)⟩	ext(h)⟩	NOUN
ejpam-6904	218	2	is	be	AUX
ejpam-6904	218	3	not	not	PART
ejpam-6904	218	4	complete	complete	ADJ
ejpam-6904	218	5	.	.	PUNCT
ejpam-6904	219	1	choose	choose	VERB
ejpam-6904	219	2	any	any	DET
ejpam-6904	219	3	p	p	NOUN
ejpam-6904	219	4	,	,	PUNCT
ejpam-6904	219	5	q	q	PROPN
ejpam-6904	219	6	∈	∈	PROPN
ejpam-6904	219	7	v	v	NOUN
ejpam-6904	219	8	(	(	PUNCT
ejpam-6904	219	9	h∗	h∗	PROPN
ejpam-6904	219	10	)	)	PUNCT
ejpam-6904	220	1	such	such	ADJ
ejpam-6904	220	2	that	that	SCONJ
ejpam-6904	220	3	dg(p	dg(p	NOUN
ejpam-6904	220	4	,	,	PUNCT
ejpam-6904	220	5	q	q	NOUN
ejpam-6904	220	6	)	)	PUNCT
ejpam-6904	220	7	=	=	SYM
ejpam-6904	220	8	dh(p	dh(p	NOUN
ejpam-6904	220	9	,	,	PUNCT
ejpam-6904	220	10	q	q	X
ejpam-6904	220	11	)	)	PUNCT
ejpam-6904	220	12	=	=	SYM
ejpam-6904	220	13	2	2	X
ejpam-6904	220	14	.	.	PUNCT
ejpam-6904	220	15	since	since	SCONJ
ejpam-6904	220	16	p	p	PRON
ejpam-6904	220	17	,	,	PUNCT
ejpam-6904	220	18	q	q	NOUN
ejpam-6904	220	19	/∈	/∈	PUNCT
ejpam-6904	220	20	ext(h	ext(h	PROPN
ejpam-6904	220	21	)	)	PUNCT
ejpam-6904	220	22	,	,	PUNCT
ejpam-6904	220	23	each	each	PRON
ejpam-6904	220	24	of	of	ADP
ejpam-6904	220	25	them	they	PRON
ejpam-6904	220	26	has	have	VERB
ejpam-6904	220	27	non	non	ADJ
ejpam-6904	220	28	-	-	ADJ
ejpam-6904	220	29	adjacent	adjacent	ADJ
ejpam-6904	220	30	neighbors	neighbor	NOUN
ejpam-6904	220	31	.	.	PUNCT
ejpam-6904	221	1	this	this	PRON
ejpam-6904	221	2	implies	imply	VERB
ejpam-6904	221	3	that	that	SCONJ
ejpam-6904	221	4	s	s	VERB
ejpam-6904	221	5	=	=	SYM
ejpam-6904	221	6	v	v	PROPN
ejpam-6904	221	7	(	(	PUNCT
ejpam-6904	221	8	h	h	NOUN
ejpam-6904	221	9	)	)	PUNCT
ejpam-6904	221	10	\	\	NOUN
ejpam-6904	222	1	{	{	PUNCT
ejpam-6904	222	2	p	p	X
ejpam-6904	222	3	,	,	PUNCT
ejpam-6904	222	4	q	q	X
ejpam-6904	222	5	}	}	PUNCT
ejpam-6904	222	6	is	be	AUX
ejpam-6904	222	7	a	a	DET
ejpam-6904	222	8	2	2	NUM
ejpam-6904	222	9	-	-	PUNCT
ejpam-6904	222	10	path	path	NOUN
ejpam-6904	222	11	geodetic	geodetic	ADJ
ejpam-6904	222	12	vertex	vertex	NOUN
ejpam-6904	222	13	covering	covering	NOUN
ejpam-6904	222	14	of	of	ADP
ejpam-6904	222	15	h	h	NOUN
ejpam-6904	222	16	,	,	PUNCT
ejpam-6904	222	17	a	a	DET
ejpam-6904	222	18	contradiction	contradiction	NOUN
ejpam-6904	222	19	.	.	PUNCT
ejpam-6904	223	1	thus	thus	ADV
ejpam-6904	223	2	,	,	PUNCT
ejpam-6904	223	3	h∗	h∗	PROPN
ejpam-6904	223	4	is	be	AUX
ejpam-6904	223	5	complete	complete	ADJ
ejpam-6904	223	6	.	.	PUNCT
ejpam-6904	224	1	for	for	ADP
ejpam-6904	224	2	the	the	DET
ejpam-6904	224	3	converse	converse	NOUN
ejpam-6904	224	4	,	,	PUNCT
ejpam-6904	224	5	suppose	suppose	VERB
ejpam-6904	224	6	that	that	SCONJ
ejpam-6904	224	7	gj	gj	NOUN
ejpam-6904	224	8	is	be	AUX
ejpam-6904	224	9	complete	complete	ADJ
ejpam-6904	224	10	for	for	ADP
ejpam-6904	224	11	all	all	DET
ejpam-6904	224	12	j	j	PROPN
ejpam-6904	224	13	∈	∈	PROPN
ejpam-6904	224	14	{	{	PUNCT
ejpam-6904	224	15	1	1	NUM
ejpam-6904	224	16	,	,	PUNCT
ejpam-6904	224	17	2	2	NUM
ejpam-6904	224	18	,	,	PUNCT
ejpam-6904	224	19	·	·	PUNCT
ejpam-6904	224	20	·	·	PUNCT
ejpam-6904	224	21	·	·	PUNCT
ejpam-6904	224	22	,	,	PUNCT
ejpam-6904	224	23	t−	t−	PROPN
ejpam-6904	224	24	1	1	NUM
ejpam-6904	224	25	,	,	PUNCT
ejpam-6904	224	26	t+	t+	VERB
ejpam-6904	224	27	1	1	NUM
ejpam-6904	224	28	,	,	PUNCT
ejpam-6904	224	29	·	·	PUNCT
ejpam-6904	224	30	·	·	PUNCT
ejpam-6904	224	31	·	·	PUNCT
ejpam-6904	224	32	,	,	PUNCT
ejpam-6904	224	33	k	k	X
ejpam-6904	224	34	}	}	PUNCT
ejpam-6904	224	35	and	and	CCONJ
ejpam-6904	224	36	that	that	SCONJ
ejpam-6904	224	37	h	h	NOUN
ejpam-6904	224	38	=	=	X
ejpam-6904	224	39	gt	gt	PROPN
ejpam-6904	224	40	is	be	AUX
ejpam-6904	224	41	non	non	ADJ
ejpam-6904	224	42	-	-	ADJ
ejpam-6904	224	43	complete	complete	ADJ
ejpam-6904	224	44	satisfying	satisfy	VERB
ejpam-6904	224	45	the	the	DET
ejpam-6904	224	46	property	property	NOUN
ejpam-6904	224	47	that	that	DET
ejpam-6904	224	48	h∗	h∗	PROPN
ejpam-6904	224	49	=	=	SYM
ejpam-6904	224	50	⟨v	⟨v	PROPN
ejpam-6904	224	51	(	(	PUNCT
ejpam-6904	224	52	h	h	NOUN
ejpam-6904	224	53	)	)	PUNCT
ejpam-6904	224	54	\	\	NOUN
ejpam-6904	225	1	ext(h)⟩	ext(h)⟩	PROPN
ejpam-6904	225	2	is	be	AUX
ejpam-6904	225	3	complete	complete	ADJ
ejpam-6904	225	4	.	.	PUNCT
ejpam-6904	226	1	then	then	ADV
ejpam-6904	226	2	β2pg(h	β2pg(h	NOUN
ejpam-6904	226	3	)	)	PUNCT
ejpam-6904	226	4	≤	≤	NOUN
ejpam-6904	226	5	|v	|v	X
ejpam-6904	226	6	(	(	PUNCT
ejpam-6904	226	7	h)|	h)|	NOUN
ejpam-6904	226	8	−	−	PROPN
ejpam-6904	226	9	1	1	NUM
ejpam-6904	226	10	and	and	CCONJ
ejpam-6904	226	11	β2pg(gj	β2pg(gj	NUM
ejpam-6904	226	12	)	)	PUNCT
ejpam-6904	226	13	=	=	SYM
ejpam-6904	226	14	|v	|v	X
ejpam-6904	226	15	(	(	PUNCT
ejpam-6904	226	16	gj)|	gj)|	PROPN
ejpam-6904	226	17	for	for	ADP
ejpam-6904	226	18	all	all	DET
ejpam-6904	226	19	j	j	PROPN
ejpam-6904	226	20	∈	∈	PROPN
ejpam-6904	226	21	{	{	PUNCT
ejpam-6904	226	22	1	1	NUM
ejpam-6904	226	23	,	,	PUNCT
ejpam-6904	226	24	2	2	NUM
ejpam-6904	226	25	,	,	PUNCT
ejpam-6904	226	26	·	·	PUNCT
ejpam-6904	226	27	·	·	PUNCT
ejpam-6904	226	28	·	·	PUNCT
ejpam-6904	226	29	,	,	PUNCT
ejpam-6904	226	30	t	t	PROPN
ejpam-6904	226	31	−	−	PROPN
ejpam-6904	226	32	1	1	NUM
ejpam-6904	226	33	,	,	PUNCT
ejpam-6904	226	34	t+1	t+1	PRON
ejpam-6904	226	35	,	,	PUNCT
ejpam-6904	226	36	·	·	PUNCT
ejpam-6904	226	37	·	·	PUNCT
ejpam-6904	226	38	·	·	PUNCT
ejpam-6904	226	39	,	,	PUNCT
ejpam-6904	226	40	k	k	X
ejpam-6904	226	41	}	}	PUNCT
ejpam-6904	226	42	by	by	ADP
ejpam-6904	226	43	theorem	theorem	NOUN
ejpam-6904	226	44	4	4	NUM
ejpam-6904	226	45	.	.	PUNCT
ejpam-6904	227	1	let	let	VERB
ejpam-6904	227	2	d	d	PRON
ejpam-6904	227	3	be	be	AUX
ejpam-6904	227	4	a	a	DET
ejpam-6904	227	5	β2pg	β2pg	PUNCT
ejpam-6904	227	6	-	-	VERB
ejpam-6904	227	7	set	set	NOUN
ejpam-6904	227	8	in	in	ADP
ejpam-6904	227	9	h.	h.	PROPN
ejpam-6904	227	10	then	then	ADV
ejpam-6904	227	11	ext(h	ext(h	PROPN
ejpam-6904	227	12	)	)	PUNCT
ejpam-6904	227	13	⊆	⊆	NUM
ejpam-6904	227	14	d	d	NOUN
ejpam-6904	227	15	by	by	ADP
ejpam-6904	227	16	proposition	proposition	NOUN
ejpam-6904	227	17	1(i	1(i	NUM
ejpam-6904	227	18	)	)	PUNCT
ejpam-6904	227	19	.	.	PUNCT
ejpam-6904	228	1	if	if	SCONJ
ejpam-6904	228	2	h∗	h∗	PROPN
ejpam-6904	228	3	is	be	AUX
ejpam-6904	228	4	the	the	DET
ejpam-6904	228	5	trivial	trivial	ADJ
ejpam-6904	228	6	graph	graph	NOUN
ejpam-6904	228	7	,	,	PUNCT
ejpam-6904	228	8	say	say	VERB
ejpam-6904	228	9	h∗	h∗	PROPN
ejpam-6904	228	10	=	=	SYM
ejpam-6904	228	11	⟨v⟩	⟨v⟩	PROPN
ejpam-6904	228	12	,	,	PUNCT
ejpam-6904	228	13	then	then	ADV
ejpam-6904	228	14	d	d	X
ejpam-6904	228	15	=	=	SYM
ejpam-6904	228	16	ext(h	ext(h	PROPN
ejpam-6904	228	17	)	)	PUNCT
ejpam-6904	228	18	.	.	PUNCT
ejpam-6904	229	1	hence	hence	ADV
ejpam-6904	229	2	,	,	PUNCT
ejpam-6904	229	3	β2pg(h	β2pg(h	X
ejpam-6904	229	4	)	)	PUNCT
ejpam-6904	229	5	=	=	SYM
ejpam-6904	229	6	|d|	|d|	PROPN
ejpam-6904	229	7	=	=	SYM
ejpam-6904	229	8	|v	|v	PROPN
ejpam-6904	229	9	(	(	PUNCT
ejpam-6904	229	10	h)|	h)|	NOUN
ejpam-6904	229	11	−	−	PROPN
ejpam-6904	229	12	1	1	X
ejpam-6904	229	13	.	.	PUNCT
ejpam-6904	229	14	suppose	suppose	VERB
ejpam-6904	229	15	h∗	h∗	PROPN
ejpam-6904	229	16	is	be	AUX
ejpam-6904	229	17	nontrivial	nontrivial	ADJ
ejpam-6904	229	18	.	.	PUNCT
ejpam-6904	230	1	since	since	SCONJ
ejpam-6904	230	2	h∗	h∗	PROPN
ejpam-6904	230	3	is	be	AUX
ejpam-6904	230	4	connected	connect	VERB
ejpam-6904	230	5	(	(	PUNCT
ejpam-6904	230	6	it	it	PRON
ejpam-6904	230	7	contains	contain	VERB
ejpam-6904	230	8	an	an	DET
ejpam-6904	230	9	edge	edge	NOUN
ejpam-6904	230	10	)	)	PUNCT
ejpam-6904	230	11	and	and	CCONJ
ejpam-6904	230	12	d	d	PROPN
ejpam-6904	230	13	is	be	AUX
ejpam-6904	230	14	a	a	DET
ejpam-6904	230	15	vertex	vertex	NOUN
ejpam-6904	230	16	cover	cover	NOUN
ejpam-6904	230	17	,	,	PUNCT
ejpam-6904	230	18	it	it	PRON
ejpam-6904	230	19	follows	follow	VERB
ejpam-6904	230	20	that	that	SCONJ
ejpam-6904	230	21	d	d	ADP
ejpam-6904	230	22	∩	∩	ADJ
ejpam-6904	230	23	v	v	NOUN
ejpam-6904	230	24	(	(	PUNCT
ejpam-6904	230	25	h∗	h∗	PROPN
ejpam-6904	230	26	)	)	PUNCT
ejpam-6904	230	27	̸=	̸=	PROPN
ejpam-6904	230	28	∅	∅	NOUN
ejpam-6904	230	29	,	,	PUNCT
ejpam-6904	230	30	i.e.	i.e.	X
ejpam-6904	230	31	,	,	PUNCT
ejpam-6904	230	32	d	d	PROPN
ejpam-6904	230	33	̸=	̸=	PROPN
ejpam-6904	230	34	ext(h	ext(h	NUM
ejpam-6904	230	35	)	)	PUNCT
ejpam-6904	230	36	.	.	PUNCT
ejpam-6904	231	1	suppose	suppose	VERB
ejpam-6904	231	2	there	there	PRON
ejpam-6904	231	3	exist	exist	VERB
ejpam-6904	231	4	distinct	distinct	ADJ
ejpam-6904	231	5	vertices	vertex	NOUN
ejpam-6904	231	6	p	p	NOUN
ejpam-6904	231	7	,	,	PUNCT
ejpam-6904	231	8	q	q	PROPN
ejpam-6904	231	9	∈	∈	PROPN
ejpam-6904	231	10	v	v	NOUN
ejpam-6904	231	11	(	(	PUNCT
ejpam-6904	231	12	h∗	h∗	PROPN
ejpam-6904	231	13	)	)	PUNCT
ejpam-6904	231	14	\d	\d	NOUN
ejpam-6904	231	15	.	.	PUNCT
ejpam-6904	232	1	then	then	ADV
ejpam-6904	232	2	pq	pq	PROPN
ejpam-6904	232	3	∈	∈	PROPN
ejpam-6904	232	4	e(h	e(h	PROPN
ejpam-6904	232	5	)	)	PUNCT
ejpam-6904	232	6	because	because	SCONJ
ejpam-6904	232	7	h∗	h∗	PROPN
ejpam-6904	232	8	is	be	AUX
ejpam-6904	232	9	complete	complete	ADJ
ejpam-6904	232	10	.	.	PUNCT
ejpam-6904	233	1	this	this	PRON
ejpam-6904	233	2	implies	imply	VERB
ejpam-6904	233	3	that	that	SCONJ
ejpam-6904	233	4	d	d	NOUN
ejpam-6904	233	5	is	be	AUX
ejpam-6904	233	6	not	not	PART
ejpam-6904	233	7	a	a	DET
ejpam-6904	233	8	vertex	vertex	NOUN
ejpam-6904	233	9	covering	covering	NOUN
ejpam-6904	233	10	of	of	ADP
ejpam-6904	233	11	h	h	NOUN
ejpam-6904	233	12	,	,	PUNCT
ejpam-6904	233	13	a	a	DET
ejpam-6904	233	14	contradiction	contradiction	NOUN
ejpam-6904	233	15	.	.	PUNCT
ejpam-6904	234	1	thus	thus	ADV
ejpam-6904	234	2	,	,	PUNCT
ejpam-6904	234	3	|d	|d	NOUN
ejpam-6904	234	4	∩	∩	ADJ
ejpam-6904	234	5	v	v	X
ejpam-6904	234	6	(	(	PUNCT
ejpam-6904	234	7	h∗)|	h∗)|	PROPN
ejpam-6904	234	8	=	=	X
ejpam-6904	234	9	|v	|v	PROPN
ejpam-6904	234	10	(	(	PUNCT
ejpam-6904	234	11	h∗)|	h∗)|	PROPN
ejpam-6904	234	12	−	−	PROPN
ejpam-6904	235	1	1	1	NUM
ejpam-6904	235	2	.	.	PUNCT
ejpam-6904	235	3	therefore	therefore	ADV
ejpam-6904	235	4	,	,	PUNCT
ejpam-6904	235	5	β2pg(h	β2pg(h	X
ejpam-6904	235	6	)	)	PUNCT
ejpam-6904	235	7	=	=	SYM
ejpam-6904	235	8	|d|	|d|	PROPN
ejpam-6904	235	9	=	=	SYM
ejpam-6904	235	10	|v	|v	PROPN
ejpam-6904	235	11	(	(	PUNCT
ejpam-6904	235	12	h)|	h)|	NOUN
ejpam-6904	235	13	−	−	PROPN
ejpam-6904	235	14	1	1	NUM
ejpam-6904	235	15	.	.	PUNCT
ejpam-6904	235	16	by	by	ADP
ejpam-6904	235	17	theorem	theorem	NOUN
ejpam-6904	235	18	1	1	NUM
ejpam-6904	235	19	,	,	PUNCT
ejpam-6904	235	20	β2pg(g	β2pg(g	PUNCT
ejpam-6904	235	21	)	)	PUNCT
ejpam-6904	235	22	=	=	PUNCT
ejpam-6904	235	23	n−	n−	NOUN
ejpam-6904	235	24	1	1	NUM
ejpam-6904	235	25	.	.	PUNCT
ejpam-6904	235	26	theorem	theorem	VERB
ejpam-6904	235	27	6	6	NUM
ejpam-6904	235	28	.	.	PUNCT
ejpam-6904	236	1	let	let	VERB
ejpam-6904	236	2	g	g	PROPN
ejpam-6904	236	3	=	=	PUNCT
ejpam-6904	236	4	km1,m2,	km1,m2,	PROPN
ejpam-6904	236	5	..	..	PUNCT
ejpam-6904	236	6	,mk	,mk	PUNCT
ejpam-6904	236	7	,	,	PUNCT
ejpam-6904	236	8	where	where	SCONJ
ejpam-6904	236	9	2	2	NUM
ejpam-6904	236	10	≤	≤	NOUN
ejpam-6904	236	11	m1	m1	NOUN
ejpam-6904	236	12	≤	≤	NUM
ejpam-6904	236	13	,	,	PUNCT
ejpam-6904	236	14	..	..	PUNCT
ejpam-6904	236	15	,	,	PUNCT
ejpam-6904	237	1	≤	≤	PROPN
ejpam-6904	237	2	mk	mk	PROPN
ejpam-6904	237	3	.	.	PUNCT
ejpam-6904	238	1	then	then	ADV
ejpam-6904	238	2	β2pg(g	β2pg(g	PUNCT
ejpam-6904	238	3	)	)	PUNCT
ejpam-6904	239	1	=	=	SYM
ejpam-6904	239	2	∑k−1	∑k−1	PROPN
ejpam-6904	239	3	j=1	j=1	PROPN
ejpam-6904	239	4	mj	mj	PROPN
ejpam-6904	239	5	.	.	PUNCT
ejpam-6904	240	1	proof	proof	NOUN
ejpam-6904	240	2	.	.	PUNCT
ejpam-6904	241	1	let	let	VERB
ejpam-6904	241	2	s1	s1	NOUN
ejpam-6904	241	3	,	,	PUNCT
ejpam-6904	241	4	s2	s2	PROPN
ejpam-6904	241	5	,	,	PUNCT
ejpam-6904	241	6	...	...	PUNCT
ejpam-6904	241	7	sk	sk	NOUN
ejpam-6904	241	8	be	be	AUX
ejpam-6904	241	9	the	the	DET
ejpam-6904	241	10	partite	partite	ADJ
ejpam-6904	241	11	sets	set	NOUN
ejpam-6904	241	12	of	of	ADP
ejpam-6904	241	13	g.	g.	PROPN
ejpam-6904	241	14	clearly	clearly	ADV
ejpam-6904	241	15	,	,	PUNCT
ejpam-6904	241	16	k−1∪	k−1∪	PROPN
ejpam-6904	241	17	i=1	i=1	X
ejpam-6904	241	18	si	si	PROPN
ejpam-6904	241	19	is	be	AUX
ejpam-6904	241	20	a	a	DET
ejpam-6904	241	21	2	2	NUM
ejpam-6904	241	22	-	-	PUNCT
ejpam-6904	241	23	path	path	NOUN
ejpam-6904	241	24	geodetic	geodetic	ADJ
ejpam-6904	241	25	vertex	vertex	NOUN
ejpam-6904	241	26	cover	cover	NOUN
ejpam-6904	241	27	of	of	ADP
ejpam-6904	241	28	g.	g.	PROPN
ejpam-6904	241	29	it	it	PRON
ejpam-6904	241	30	follows	follow	VERB
ejpam-6904	241	31	that	that	PRON
ejpam-6904	241	32	β2pg(g	β2pg(g	PUNCT
ejpam-6904	241	33	)	)	PUNCT
ejpam-6904	241	34	≤	≤	PUNCT
ejpam-6904	242	1	k−1∑	k−1∑	PROPN
ejpam-6904	242	2	j=1	j=1	PROPN
ejpam-6904	242	3	mj	mj	PROPN
ejpam-6904	242	4	.	.	PUNCT
ejpam-6904	243	1	next	next	ADV
ejpam-6904	243	2	,	,	PUNCT
ejpam-6904	243	3	let	let	VERB
ejpam-6904	243	4	s	s	PRON
ejpam-6904	243	5	be	be	AUX
ejpam-6904	243	6	a	a	DET
ejpam-6904	243	7	β2pg	β2pg	PUNCT
ejpam-6904	243	8	-	-	PUNCT
ejpam-6904	243	9	set	set	NOUN
ejpam-6904	243	10	of	of	ADP
ejpam-6904	243	11	g.	g.	PROPN
ejpam-6904	243	12	since	since	SCONJ
ejpam-6904	243	13	a.	a.	PROPN
ejpam-6904	243	14	b.	b.	PROPN
ejpam-6904	243	15	tapeing	tapeing	PROPN
ejpam-6904	243	16	,	,	PUNCT
ejpam-6904	243	17	s.	s.	PROPN
ejpam-6904	243	18	r.	r.	PROPN
ejpam-6904	243	19	canoy	canoy	PROPN
ejpam-6904	243	20	/	/	SYM
ejpam-6904	243	21	eur	eur	PROPN
ejpam-6904	243	22	.	.	PUNCT
ejpam-6904	244	1	j.	j.	PROPN
ejpam-6904	244	2	pure	pure	PROPN
ejpam-6904	244	3	appl	appl	PROPN
ejpam-6904	244	4	.	.	PROPN
ejpam-6904	244	5	math	math	PROPN
ejpam-6904	244	6	,	,	PUNCT
ejpam-6904	244	7	18	18	NUM
ejpam-6904	244	8	(	(	PUNCT
ejpam-6904	244	9	4	4	NUM
ejpam-6904	244	10	)	)	PUNCT
ejpam-6904	244	11	(	(	PUNCT
ejpam-6904	244	12	2025	2025	NUM
ejpam-6904	244	13	)	)	PUNCT
ejpam-6904	244	14	,	,	PUNCT
ejpam-6904	244	15	6904	6904	NUM
ejpam-6904	244	16	8	8	NUM
ejpam-6904	244	17	of	of	ADP
ejpam-6904	244	18	14	14	NUM
ejpam-6904	244	19	g	g	NOUN
ejpam-6904	244	20	is	be	AUX
ejpam-6904	244	21	not	not	PART
ejpam-6904	244	22	complete	complete	ADJ
ejpam-6904	244	23	,	,	PUNCT
ejpam-6904	244	24	it	it	PRON
ejpam-6904	244	25	follows	follow	VERB
ejpam-6904	244	26	that	that	PRON
ejpam-6904	244	27	s	s	VERB
ejpam-6904	244	28	̸=	̸=	PROPN
ejpam-6904	244	29	v	v	NOUN
ejpam-6904	244	30	(	(	PUNCT
ejpam-6904	244	31	g	g	NOUN
ejpam-6904	244	32	)	)	PUNCT
ejpam-6904	244	33	.	.	PUNCT
ejpam-6904	245	1	let	let	VERB
ejpam-6904	245	2	v	v	NUM
ejpam-6904	245	3	∈	∈	PROPN
ejpam-6904	245	4	v	v	NOUN
ejpam-6904	245	5	(	(	PUNCT
ejpam-6904	245	6	g	g	NOUN
ejpam-6904	245	7	)	)	PUNCT
ejpam-6904	245	8	\	\	PROPN
ejpam-6904	245	9	s	s	PART
ejpam-6904	245	10	and	and	CCONJ
ejpam-6904	245	11	let	let	VERB
ejpam-6904	245	12	r	r	PRON
ejpam-6904	245	13	∈	∈	PROPN
ejpam-6904	245	14	{	{	PUNCT
ejpam-6904	245	15	1	1	NUM
ejpam-6904	245	16	,	,	PUNCT
ejpam-6904	245	17	2	2	NUM
ejpam-6904	245	18	,	,	PUNCT
ejpam-6904	245	19	.	.	PUNCT
ejpam-6904	245	20	.	.	PUNCT
ejpam-6904	246	1	.	.	PUNCT
ejpam-6904	247	1	,	,	PUNCT
ejpam-6904	247	2	k	k	X
ejpam-6904	247	3	}	}	PUNCT
ejpam-6904	247	4	such	such	ADJ
ejpam-6904	247	5	that	that	SCONJ
ejpam-6904	247	6	v	v	PROPN
ejpam-6904	247	7	∈	∈	PROPN
ejpam-6904	247	8	sr	sr	PROPN
ejpam-6904	247	9	.	.	PUNCT
ejpam-6904	248	1	since	since	SCONJ
ejpam-6904	248	2	vw	vw	PROPN
ejpam-6904	248	3	∈	∈	PROPN
ejpam-6904	248	4	e(g	e(g	PROPN
ejpam-6904	248	5	)	)	PUNCT
ejpam-6904	249	1	for	for	ADP
ejpam-6904	249	2	all	all	DET
ejpam-6904	249	3	w	w	PROPN
ejpam-6904	249	4	∈	∈	PROPN
ejpam-6904	249	5	∪	∪	NOUN
ejpam-6904	249	6	i	i	PRON
ejpam-6904	249	7	̸=r	̸=r	PROPN
ejpam-6904	249	8	si	si	PROPN
ejpam-6904	249	9	and	and	CCONJ
ejpam-6904	249	10	s	s	PROPN
ejpam-6904	249	11	is	be	AUX
ejpam-6904	249	12	a	a	DET
ejpam-6904	249	13	vertex	vertex	NOUN
ejpam-6904	249	14	cover	cover	NOUN
ejpam-6904	249	15	of	of	ADP
ejpam-6904	249	16	g	g	NOUN
ejpam-6904	249	17	,	,	PUNCT
ejpam-6904	249	18	it	it	PRON
ejpam-6904	249	19	follows	follow	VERB
ejpam-6904	249	20	that	that	SCONJ
ejpam-6904	249	21	∪	∪	VERB
ejpam-6904	249	22	j	j	PROPN
ejpam-6904	249	23	̸=r	̸=r	PROPN
ejpam-6904	249	24	si	si	PROPN
ejpam-6904	249	25	⊆	⊆	NUM
ejpam-6904	249	26	s.	s.	PROPN
ejpam-6904	249	27	therefore	therefore	ADV
ejpam-6904	249	28	,	,	PUNCT
ejpam-6904	249	29	k−1∑	k−1∑	PROPN
ejpam-6904	249	30	j=1	j=1	PROPN
ejpam-6904	249	31	mj	mj	PROPN
ejpam-6904	249	32	≤	≤	PROPN
ejpam-6904	249	33	∑	∑	PUNCT
ejpam-6904	250	1	j	j	PROPN
ejpam-6904	250	2	̸=r	̸=r	PROPN
ejpam-6904	250	3	mj	mj	PROPN
ejpam-6904	250	4	≤	≤	PROPN
ejpam-6904	250	5	|s|	|s|	PROPN
ejpam-6904	250	6	=	=	PUNCT
ejpam-6904	250	7	β2pg(g	β2pg(g	PROPN
ejpam-6904	250	8	)	)	PUNCT
ejpam-6904	250	9	.	.	PUNCT
ejpam-6904	251	1	this	this	PRON
ejpam-6904	251	2	proves	prove	VERB
ejpam-6904	251	3	the	the	DET
ejpam-6904	251	4	assertion	assertion	NOUN
ejpam-6904	251	5	.	.	PUNCT
ejpam-6904	252	1	observation	observation	NOUN
ejpam-6904	252	2	1	1	NUM
ejpam-6904	252	3	.	.	PUNCT
ejpam-6904	253	1	let	let	VERB
ejpam-6904	253	2	n	n	PRON
ejpam-6904	253	3	be	be	AUX
ejpam-6904	253	4	a	a	DET
ejpam-6904	253	5	positive	positive	ADJ
ejpam-6904	253	6	integer	integer	NOUN
ejpam-6904	253	7	.	.	PUNCT
ejpam-6904	254	1	then	then	ADV
ejpam-6904	254	2	(	(	PUNCT
ejpam-6904	254	3	i	i	NOUN
ejpam-6904	254	4	)	)	PUNCT
ejpam-6904	254	5	β2pg(pn	β2pg(pn	PROPN
ejpam-6904	254	6	)	)	PUNCT
ejpam-6904	254	7	=	=	VERB
ejpam-6904	255	1	⌈n+1	⌈n+1	PROPN
ejpam-6904	255	2	2	2	NUM
ejpam-6904	255	3	⌉	⌉	X
ejpam-6904	255	4	for	for	ADP
ejpam-6904	255	5	all	all	DET
ejpam-6904	255	6	n	n	PRON
ejpam-6904	255	7	and	and	CCONJ
ejpam-6904	255	8	(	(	PUNCT
ejpam-6904	255	9	ii	ii	NOUN
ejpam-6904	255	10	)	)	PUNCT
ejpam-6904	255	11	β2pg(cn	β2pg(cn	NUM
ejpam-6904	255	12	)	)	PUNCT
ejpam-6904	256	1	=	=	PRON
ejpam-6904	256	2	⌈n2	⌈n2	NOUN
ejpam-6904	256	3	⌉	⌉	NOUN
ejpam-6904	256	4	for	for	ADP
ejpam-6904	256	5	all	all	DET
ejpam-6904	256	6	n	n	PRON
ejpam-6904	256	7	≥	≥	NOUN
ejpam-6904	256	8	4	4	NUM
ejpam-6904	256	9	.	.	PUNCT
ejpam-6904	257	1	we	we	PRON
ejpam-6904	257	2	now	now	ADV
ejpam-6904	257	3	give	give	VERB
ejpam-6904	257	4	some	some	DET
ejpam-6904	257	5	realization	realization	NOUN
ejpam-6904	257	6	results	result	NOUN
ejpam-6904	257	7	involving	involve	VERB
ejpam-6904	257	8	the	the	DET
ejpam-6904	257	9	parameters	parameter	NOUN
ejpam-6904	257	10	2	2	NUM
ejpam-6904	257	11	-	-	PUNCT
ejpam-6904	257	12	path	path	NOUN
ejpam-6904	257	13	geodetic	geodetic	ADJ
ejpam-6904	257	14	number	number	NOUN
ejpam-6904	257	15	,	,	PUNCT
ejpam-6904	257	16	vertex	vertex	NOUN
ejpam-6904	257	17	cover	cover	NOUN
ejpam-6904	257	18	number	number	NOUN
ejpam-6904	257	19	,	,	PUNCT
ejpam-6904	257	20	and	and	CCONJ
ejpam-6904	257	21	2	2	NUM
ejpam-6904	257	22	-	-	PUNCT
ejpam-6904	257	23	path	path	NOUN
ejpam-6904	257	24	geodetic	geodetic	ADJ
ejpam-6904	257	25	vertex	vertex	NOUN
ejpam-6904	257	26	cover	cover	NOUN
ejpam-6904	257	27	number	number	NOUN
ejpam-6904	257	28	.	.	PUNCT
ejpam-6904	258	1	theorem	theorem	VERB
ejpam-6904	258	2	7	7	NUM
ejpam-6904	258	3	.	.	PUNCT
ejpam-6904	258	4	given	give	VERB
ejpam-6904	258	5	two	two	NUM
ejpam-6904	258	6	positive	positive	ADJ
ejpam-6904	258	7	integers	integer	NOUN
ejpam-6904	258	8	a	a	PRON
ejpam-6904	258	9	and	and	CCONJ
ejpam-6904	258	10	b	b	NOUN
ejpam-6904	258	11	such	such	ADJ
ejpam-6904	258	12	that	that	SCONJ
ejpam-6904	258	13	3	3	NUM
ejpam-6904	258	14	≤	≤	NOUN
ejpam-6904	258	15	a	a	DET
ejpam-6904	258	16	≤	≤	NUM
ejpam-6904	258	17	b	b	NOUN
ejpam-6904	258	18	,	,	PUNCT
ejpam-6904	258	19	there	there	PRON
ejpam-6904	258	20	exists	exist	VERB
ejpam-6904	258	21	a	a	DET
ejpam-6904	258	22	connected	connected	ADJ
ejpam-6904	258	23	graph	graph	NOUN
ejpam-6904	258	24	g	g	ADP
ejpam-6904	258	25	such	such	ADJ
ejpam-6904	258	26	that	that	DET
ejpam-6904	258	27	g2p(g	g2p(g	NOUN
ejpam-6904	258	28	)	)	PUNCT
ejpam-6904	259	1	=	=	SYM
ejpam-6904	259	2	a	a	PRON
ejpam-6904	259	3	and	and	CCONJ
ejpam-6904	259	4	β2pg(g	β2pg(g	PRON
ejpam-6904	259	5	)	)	PUNCT
ejpam-6904	259	6	=	=	SYM
ejpam-6904	259	7	b.	b.	NOUN
ejpam-6904	259	8	proof	proof	NOUN
ejpam-6904	259	9	.	.	PUNCT
ejpam-6904	260	1	if	if	SCONJ
ejpam-6904	260	2	a	a	DET
ejpam-6904	260	3	=	=	SYM
ejpam-6904	260	4	b	b	NOUN
ejpam-6904	260	5	,	,	PUNCT
ejpam-6904	260	6	then	then	ADV
ejpam-6904	260	7	let	let	VERB
ejpam-6904	260	8	g	g	PROPN
ejpam-6904	260	9	=	=	SYM
ejpam-6904	260	10	k1,a	k1,a	PROPN
ejpam-6904	260	11	.	.	PUNCT
ejpam-6904	261	1	clearly	clearly	ADV
ejpam-6904	261	2	,	,	PUNCT
ejpam-6904	261	3	g2p(g	g2p(g	PROPN
ejpam-6904	261	4	)	)	PUNCT
ejpam-6904	261	5	=	=	PUNCT
ejpam-6904	261	6	β2pg(g	β2pg(g	X
ejpam-6904	261	7	)	)	PUNCT
ejpam-6904	261	8	=	=	NOUN
ejpam-6904	261	9	a.	a.	NOUN
ejpam-6904	261	10	suppose	suppose	VERB
ejpam-6904	261	11	now	now	ADV
ejpam-6904	261	12	that	that	SCONJ
ejpam-6904	261	13	a	a	DET
ejpam-6904	261	14	<	<	X
ejpam-6904	261	15	b.	b.	NOUN
ejpam-6904	261	16	consider	consider	VERB
ejpam-6904	261	17	the	the	DET
ejpam-6904	261	18	graph	graph	NOUN
ejpam-6904	261	19	g	g	NOUN
ejpam-6904	261	20	in	in	ADP
ejpam-6904	261	21	figure	figure	NOUN
ejpam-6904	261	22	3	3	NUM
ejpam-6904	261	23	with	with	ADP
ejpam-6904	261	24	complete	complete	ADJ
ejpam-6904	261	25	subgraphs	subgraphs	PROPN
ejpam-6904	261	26	ka−1	ka−1	PROPN
ejpam-6904	261	27	and	and	CCONJ
ejpam-6904	261	28	kb−a+1	kb−a+1	PROPN
ejpam-6904	261	29	,	,	PUNCT
ejpam-6904	261	30	where	where	SCONJ
ejpam-6904	261	31	v	v	X
ejpam-6904	261	32	(	(	PUNCT
ejpam-6904	261	33	ka−1	ka−1	PROPN
ejpam-6904	261	34	)	)	PUNCT
ejpam-6904	261	35	=	=	PRON
ejpam-6904	261	36	{	{	PUNCT
ejpam-6904	261	37	v1	v1	PROPN
ejpam-6904	261	38	,	,	PUNCT
ejpam-6904	261	39	.	.	PUNCT
ejpam-6904	261	40	.	.	PUNCT
ejpam-6904	262	1	.	.	PUNCT
ejpam-6904	263	1	,	,	PUNCT
ejpam-6904	263	2	va−1	va−1	VERB
ejpam-6904	263	3	}	}	PUNCT
ejpam-6904	263	4	and	and	CCONJ
ejpam-6904	263	5	v	v	X
ejpam-6904	263	6	(	(	PUNCT
ejpam-6904	263	7	kb−a+1	kb−a+1	PROPN
ejpam-6904	263	8	)	)	PUNCT
ejpam-6904	263	9	=	=	PRON
ejpam-6904	264	1	{	{	PUNCT
ejpam-6904	264	2	x1	x1	PROPN
ejpam-6904	264	3	,	,	PUNCT
ejpam-6904	264	4	.	.	PUNCT
ejpam-6904	264	5	.	.	PUNCT
ejpam-6904	265	1	.	.	PUNCT
ejpam-6904	266	1	,	,	PUNCT
ejpam-6904	266	2	xb−a+1	xb−a+1	PROPN
ejpam-6904	266	3	}	}	PUNCT
ejpam-6904	266	4	.	.	PUNCT
ejpam-6904	267	1	let	let	VERB
ejpam-6904	267	2	d	d	NOUN
ejpam-6904	267	3	=	=	SYM
ejpam-6904	267	4	{	{	PUNCT
ejpam-6904	267	5	v1	v1	PROPN
ejpam-6904	267	6	,	,	PUNCT
ejpam-6904	267	7	.	.	PUNCT
ejpam-6904	267	8	.	.	PUNCT
ejpam-6904	268	1	.	.	PUNCT
ejpam-6904	269	1	,	,	PUNCT
ejpam-6904	269	2	va−1	va−1	PROPN
ejpam-6904	269	3	,	,	PUNCT
ejpam-6904	269	4	va	va	NOUN
ejpam-6904	269	5	}	}	PUNCT
ejpam-6904	269	6	,	,	PUNCT
ejpam-6904	269	7	then	then	ADV
ejpam-6904	269	8	d	d	PROPN
ejpam-6904	269	9	is	be	AUX
ejpam-6904	269	10	g2p	g2p	NOUN
ejpam-6904	269	11	-	-	ADJ
ejpam-6904	269	12	set	set	ADJ
ejpam-6904	269	13	in	in	ADP
ejpam-6904	269	14	g	g	NOUN
ejpam-6904	269	15	,	,	PUNCT
ejpam-6904	269	16	implying	imply	VERB
ejpam-6904	269	17	that	that	SCONJ
ejpam-6904	269	18	g2p(g	g2p(g	NOUN
ejpam-6904	269	19	)	)	PUNCT
ejpam-6904	270	1	=	=	SYM
ejpam-6904	270	2	a.	a.	NOUN
ejpam-6904	270	3	let	let	VERB
ejpam-6904	270	4	d0	d0	NOUN
ejpam-6904	270	5	be	be	AUX
ejpam-6904	270	6	a	a	DET
ejpam-6904	270	7	β2pg	β2pg	PUNCT
ejpam-6904	270	8	-	-	VERB
ejpam-6904	270	9	set	set	VERB
ejpam-6904	270	10	in	in	ADP
ejpam-6904	270	11	g.	g.	PROPN
ejpam-6904	270	12	then	then	ADV
ejpam-6904	270	13	ext(g	ext(g	PROPN
ejpam-6904	270	14	)	)	PUNCT
ejpam-6904	270	15	=	=	PRON
ejpam-6904	270	16	{	{	PUNCT
ejpam-6904	270	17	v1	v1	PROPN
ejpam-6904	270	18	,	,	PUNCT
ejpam-6904	270	19	v2	v2	PROPN
ejpam-6904	270	20	,	,	PUNCT
ejpam-6904	270	21	.	.	PUNCT
ejpam-6904	270	22	.	.	PUNCT
ejpam-6904	270	23	.	.	PUNCT
ejpam-6904	271	1	va−2	va−2	PROPN
ejpam-6904	271	2	,	,	PUNCT
ejpam-6904	271	3	va	va	NOUN
ejpam-6904	271	4	}	}	PUNCT
ejpam-6904	271	5	⊆	⊆	NUM
ejpam-6904	271	6	d0	d0	NOUN
ejpam-6904	271	7	by	by	ADP
ejpam-6904	271	8	proposition	proposition	NOUN
ejpam-6904	271	9	1(i	1(i	NUM
ejpam-6904	271	10	)	)	PUNCT
ejpam-6904	271	11	.	.	PUNCT
ejpam-6904	272	1	if	if	SCONJ
ejpam-6904	272	2	va−1	va−1	NOUN
ejpam-6904	272	3	/∈	/∈	PUNCT
ejpam-6904	272	4	d0	d0	NOUN
ejpam-6904	272	5	,	,	PUNCT
ejpam-6904	272	6	then	then	ADV
ejpam-6904	272	7	v	v	X
ejpam-6904	272	8	(	(	PUNCT
ejpam-6904	272	9	kb−a+1	kb−a+1	PROPN
ejpam-6904	272	10	)	)	PUNCT
ejpam-6904	272	11	⊆	⊆	NUM
ejpam-6904	272	12	d0	d0	NOUN
ejpam-6904	272	13	because	because	SCONJ
ejpam-6904	272	14	d0	d0	NOUN
ejpam-6904	272	15	is	be	AUX
ejpam-6904	272	16	a	a	DET
ejpam-6904	272	17	vertex	vertex	NOUN
ejpam-6904	272	18	cover	cover	NOUN
ejpam-6904	272	19	of	of	ADP
ejpam-6904	272	20	g.	g.	PROPN
ejpam-6904	272	21	hence	hence	ADV
ejpam-6904	272	22	,	,	PUNCT
ejpam-6904	272	23	d0	d0	NOUN
ejpam-6904	272	24	=	=	SYM
ejpam-6904	272	25	ext(g	ext(g	PROPN
ejpam-6904	272	26	)	)	PUNCT
ejpam-6904	272	27	∪	∪	NOUN
ejpam-6904	272	28	v	v	PROPN
ejpam-6904	272	29	(	(	PUNCT
ejpam-6904	272	30	kb−a+1	kb−a+1	PROPN
ejpam-6904	272	31	)	)	PUNCT
ejpam-6904	272	32	.	.	PUNCT
ejpam-6904	273	1	it	it	PRON
ejpam-6904	273	2	follows	follow	VERB
ejpam-6904	273	3	that	that	DET
ejpam-6904	273	4	|d0|	|d0|	NOUN
ejpam-6904	273	5	=	=	SYM
ejpam-6904	273	6	(	(	PUNCT
ejpam-6904	273	7	a	a	DET
ejpam-6904	273	8	−	−	PROPN
ejpam-6904	273	9	1	1	NUM
ejpam-6904	273	10	)	)	PUNCT
ejpam-6904	274	1	+	+	CCONJ
ejpam-6904	274	2	(	(	PUNCT
ejpam-6904	274	3	b	b	X
ejpam-6904	274	4	−	−	NOUN
ejpam-6904	274	5	a	a	DET
ejpam-6904	274	6	+	+	NOUN
ejpam-6904	274	7	1	1	NUM
ejpam-6904	274	8	)	)	PUNCT
ejpam-6904	274	9	=	=	SYM
ejpam-6904	274	10	b.	b.	PROPN
ejpam-6904	274	11	suppose	suppose	VERB
ejpam-6904	274	12	va−1	va−1	PROPN
ejpam-6904	274	13	∈	∈	NOUN
ejpam-6904	274	14	d0	d0	NOUN
ejpam-6904	274	15	.	.	PUNCT
ejpam-6904	275	1	again	again	ADV
ejpam-6904	275	2	,	,	PUNCT
ejpam-6904	275	3	since	since	SCONJ
ejpam-6904	275	4	d0	d0	NOUN
ejpam-6904	275	5	is	be	AUX
ejpam-6904	275	6	a	a	DET
ejpam-6904	275	7	vertex	vertex	NOUN
ejpam-6904	275	8	cover	cover	NOUN
ejpam-6904	275	9	of	of	ADP
ejpam-6904	275	10	g	g	NOUN
ejpam-6904	275	11	,	,	PUNCT
ejpam-6904	275	12	|v	|v	PROPN
ejpam-6904	275	13	(	(	PUNCT
ejpam-6904	275	14	kb−a+1	kb−a+1	PROPN
ejpam-6904	275	15	)	)	PUNCT
ejpam-6904	276	1	∩d0|	∩d0|	PROPN
ejpam-6904	277	1	=	=	SYM
ejpam-6904	277	2	b−	b−	PROPN
ejpam-6904	277	3	a.	a.	PROPN
ejpam-6904	277	4	ka−1	ka−1	PROPN
ejpam-6904	277	5	kb−a+1	kb−a+1	PROPN
ejpam-6904	277	6	va	va	PROPN
ejpam-6904	277	7	x1	x1	PROPN
ejpam-6904	277	8	...	...	PUNCT
ejpam-6904	278	1	xb−a+1	xb−a+1	PROPN
ejpam-6904	278	2	va−1	va−1	PROPN
ejpam-6904	278	3	g	g	NOUN
ejpam-6904	278	4	:	:	PUNCT
ejpam-6904	278	5	figure	figure	VERB
ejpam-6904	278	6	3	3	NUM
ejpam-6904	278	7	:	:	PUNCT
ejpam-6904	278	8	graph	graph	VERB
ejpam-6904	278	9	g	g	NOUN
ejpam-6904	278	10	with	with	ADP
ejpam-6904	278	11	g2p(g	g2p(g	NOUN
ejpam-6904	278	12	)	)	PUNCT
ejpam-6904	278	13	=	=	PUNCT
ejpam-6904	279	1	a	a	DET
ejpam-6904	279	2	<	<	X
ejpam-6904	279	3	b	b	X
ejpam-6904	279	4	=	=	PUNCT
ejpam-6904	279	5	β2pg(g	β2pg(g	PROPN
ejpam-6904	279	6	)	)	PUNCT
ejpam-6904	279	7	this	this	PRON
ejpam-6904	279	8	implies	imply	VERB
ejpam-6904	279	9	that	that	SCONJ
ejpam-6904	279	10	d0	d0	NOUN
ejpam-6904	279	11	=	=	SYM
ejpam-6904	279	12	a+	a+	PUNCT
ejpam-6904	279	13	(	(	PUNCT
ejpam-6904	279	14	b−	b−	NOUN
ejpam-6904	279	15	a	a	PRON
ejpam-6904	279	16	)	)	PUNCT
ejpam-6904	279	17	=	=	SYM
ejpam-6904	279	18	b.	b.	PROPN
ejpam-6904	279	19	therefore	therefore	ADV
ejpam-6904	279	20	,	,	PUNCT
ejpam-6904	279	21	β2pg(g	β2pg(g	PUNCT
ejpam-6904	279	22	)	)	PUNCT
ejpam-6904	279	23	=	=	SYM
ejpam-6904	279	24	|d0|	|d0|	NOUN
ejpam-6904	279	25	=	=	SYM
ejpam-6904	279	26	b.	b.	PROPN
ejpam-6904	280	1	the	the	DET
ejpam-6904	280	2	next	next	ADJ
ejpam-6904	280	3	result	result	NOUN
ejpam-6904	280	4	is	be	AUX
ejpam-6904	280	5	direct	direct	ADJ
ejpam-6904	280	6	consequence	consequence	NOUN
ejpam-6904	280	7	of	of	ADP
ejpam-6904	280	8	theorem	theorem	ADJ
ejpam-6904	280	9	7	7	NUM
ejpam-6904	280	10	.	.	PUNCT
ejpam-6904	280	11	corollary	corollary	ADJ
ejpam-6904	280	12	2	2	NUM
ejpam-6904	280	13	.	.	PUNCT
ejpam-6904	281	1	let	let	VERB
ejpam-6904	281	2	n	n	PRON
ejpam-6904	281	3	be	be	AUX
ejpam-6904	281	4	a	a	DET
ejpam-6904	281	5	positive	positive	ADJ
ejpam-6904	281	6	integer	integer	NOUN
ejpam-6904	281	7	.	.	PUNCT
ejpam-6904	282	1	then	then	ADV
ejpam-6904	282	2	there	there	PRON
ejpam-6904	282	3	exists	exist	VERB
ejpam-6904	282	4	a	a	DET
ejpam-6904	282	5	connected	connected	ADJ
ejpam-6904	282	6	graph	graph	NOUN
ejpam-6904	282	7	g	g	ADP
ejpam-6904	282	8	such	such	ADJ
ejpam-6904	282	9	that	that	DET
ejpam-6904	282	10	β2pg(g	β2pg(g	NOUN
ejpam-6904	282	11	)	)	PUNCT
ejpam-6904	282	12	−	−	ADP
ejpam-6904	282	13	g2p(g	g2p(g	NOUN
ejpam-6904	282	14	)	)	PUNCT
ejpam-6904	283	1	=	=	VERB
ejpam-6904	283	2	n.	n.	NOUN
ejpam-6904	283	3	in	in	ADP
ejpam-6904	283	4	other	other	ADJ
ejpam-6904	283	5	words	word	NOUN
ejpam-6904	283	6	,	,	PUNCT
ejpam-6904	283	7	the	the	DET
ejpam-6904	283	8	difference	difference	NOUN
ejpam-6904	283	9	β2pg(g	β2pg(g	NOUN
ejpam-6904	283	10	)	)	PUNCT
ejpam-6904	283	11	−	−	ADP
ejpam-6904	283	12	g2p(g	g2p(g	NOUN
ejpam-6904	283	13	)	)	PUNCT
ejpam-6904	283	14	can	can	AUX
ejpam-6904	283	15	be	be	AUX
ejpam-6904	283	16	increased	increase	VERB
ejpam-6904	283	17	arbitrarily	arbitrarily	ADV
ejpam-6904	283	18	.	.	PUNCT
ejpam-6904	284	1	a.	a.	PROPN
ejpam-6904	284	2	b.	b.	PROPN
ejpam-6904	284	3	tapeing	tapeing	PROPN
ejpam-6904	284	4	,	,	PUNCT
ejpam-6904	284	5	s.	s.	PROPN
ejpam-6904	284	6	r.	r.	PROPN
ejpam-6904	284	7	canoy	canoy	PROPN
ejpam-6904	284	8	/	/	SYM
ejpam-6904	284	9	eur	eur	PROPN
ejpam-6904	284	10	.	.	PUNCT
ejpam-6904	285	1	j.	j.	PROPN
ejpam-6904	285	2	pure	pure	PROPN
ejpam-6904	285	3	appl	appl	PROPN
ejpam-6904	285	4	.	.	PROPN
ejpam-6904	285	5	math	math	PROPN
ejpam-6904	285	6	,	,	PUNCT
ejpam-6904	285	7	18	18	NUM
ejpam-6904	285	8	(	(	PUNCT
ejpam-6904	285	9	4	4	NUM
ejpam-6904	285	10	)	)	PUNCT
ejpam-6904	285	11	(	(	PUNCT
ejpam-6904	285	12	2025	2025	NUM
ejpam-6904	285	13	)	)	PUNCT
ejpam-6904	285	14	,	,	PUNCT
ejpam-6904	285	15	6904	6904	NUM
ejpam-6904	285	16	9	9	NUM
ejpam-6904	285	17	of	of	ADP
ejpam-6904	285	18	14	14	NUM
ejpam-6904	285	19	theorem	theorem	NOUN
ejpam-6904	285	20	8	8	NUM
ejpam-6904	285	21	.	.	PUNCT
ejpam-6904	286	1	given	give	VERB
ejpam-6904	286	2	two	two	NUM
ejpam-6904	286	3	positive	positive	ADJ
ejpam-6904	286	4	integers	integer	NOUN
ejpam-6904	286	5	a	a	PRON
ejpam-6904	286	6	and	and	CCONJ
ejpam-6904	286	7	b	b	NOUN
ejpam-6904	286	8	such	such	ADJ
ejpam-6904	286	9	that	that	SCONJ
ejpam-6904	286	10	2	2	NUM
ejpam-6904	286	11	≤	≤	NOUN
ejpam-6904	286	12	a	a	DET
ejpam-6904	286	13	≤	≤	NUM
ejpam-6904	286	14	b	b	NOUN
ejpam-6904	286	15	,	,	PUNCT
ejpam-6904	286	16	there	there	PRON
ejpam-6904	286	17	exists	exist	VERB
ejpam-6904	286	18	a	a	DET
ejpam-6904	286	19	connected	connected	ADJ
ejpam-6904	286	20	graph	graph	NOUN
ejpam-6904	286	21	g	g	ADP
ejpam-6904	286	22	such	such	ADJ
ejpam-6904	286	23	that	that	PRON
ejpam-6904	286	24	β(g	β(g	PROPN
ejpam-6904	286	25	)	)	PUNCT
ejpam-6904	286	26	=	=	SYM
ejpam-6904	286	27	a	a	PRON
ejpam-6904	286	28	and	and	CCONJ
ejpam-6904	286	29	β2pg(g	β2pg(g	PRON
ejpam-6904	286	30	)	)	PUNCT
ejpam-6904	286	31	=	=	SYM
ejpam-6904	286	32	b.	b.	NOUN
ejpam-6904	286	33	proof	proof	NOUN
ejpam-6904	286	34	.	.	PUNCT
ejpam-6904	287	1	if	if	SCONJ
ejpam-6904	287	2	a	a	DET
ejpam-6904	287	3	=	=	SYM
ejpam-6904	287	4	b	b	NOUN
ejpam-6904	287	5	,	,	PUNCT
ejpam-6904	287	6	then	then	ADV
ejpam-6904	287	7	consider	consider	VERB
ejpam-6904	287	8	the	the	DET
ejpam-6904	287	9	cycle	cycle	NOUN
ejpam-6904	287	10	g	g	PROPN
ejpam-6904	287	11	=	=	SYM
ejpam-6904	287	12	c2a	c2a	PROPN
ejpam-6904	287	13	.	.	PUNCT
ejpam-6904	287	14	clearly	clearly	ADV
ejpam-6904	287	15	,	,	PUNCT
ejpam-6904	287	16	β(g	β(g	PROPN
ejpam-6904	287	17	)	)	PUNCT
ejpam-6904	287	18	=	=	PUNCT
ejpam-6904	287	19	β2pg(g	β2pg(g	X
ejpam-6904	287	20	)	)	PUNCT
ejpam-6904	287	21	=	=	SYM
ejpam-6904	287	22	⌈2a2	⌈2a2	PROPN
ejpam-6904	287	23	⌉	⌉	X
ejpam-6904	287	24	=	=	PUNCT
ejpam-6904	287	25	a.	a.	NOUN
ejpam-6904	287	26	next	next	ADV
ejpam-6904	287	27	,	,	PUNCT
ejpam-6904	287	28	suppose	suppose	VERB
ejpam-6904	287	29	a	a	DET
ejpam-6904	287	30	<	<	X
ejpam-6904	287	31	b.	b.	NOUN
ejpam-6904	287	32	consider	consider	VERB
ejpam-6904	287	33	the	the	DET
ejpam-6904	287	34	graph	graph	NOUN
ejpam-6904	287	35	g	g	NOUN
ejpam-6904	287	36	in	in	ADP
ejpam-6904	287	37	figure	figure	NOUN
ejpam-6904	287	38	4	4	NUM
ejpam-6904	287	39	with	with	ADP
ejpam-6904	287	40	the	the	DET
ejpam-6904	287	41	complete	complete	ADJ
ejpam-6904	287	42	graph	graph	NOUN
ejpam-6904	287	43	ka	ka	PROPN
ejpam-6904	287	44	and	and	CCONJ
ejpam-6904	287	45	the	the	DET
ejpam-6904	287	46	star	star	NOUN
ejpam-6904	287	47	k1,b−a	k1,b−a	X
ejpam-6904	287	48	as	as	ADP
ejpam-6904	287	49	subgraphs	subgraph	NOUN
ejpam-6904	287	50	,	,	PUNCT
ejpam-6904	287	51	where	where	SCONJ
ejpam-6904	287	52	v	v	X
ejpam-6904	287	53	(	(	PUNCT
ejpam-6904	287	54	ka	ka	PROPN
ejpam-6904	287	55	)	)	PUNCT
ejpam-6904	287	56	=	=	SYM
ejpam-6904	287	57	{	{	PUNCT
ejpam-6904	287	58	v1	v1	PROPN
ejpam-6904	287	59	,	,	PUNCT
ejpam-6904	287	60	.	.	PUNCT
ejpam-6904	287	61	.	.	PUNCT
ejpam-6904	288	1	.	.	PUNCT
ejpam-6904	289	1	,	,	PUNCT
ejpam-6904	289	2	va−1	va−1	VERB
ejpam-6904	289	3	,	,	PUNCT
ejpam-6904	289	4	x	x	NOUN
ejpam-6904	289	5	}	}	PUNCT
ejpam-6904	289	6	and	and	CCONJ
ejpam-6904	289	7	v	v	X
ejpam-6904	289	8	(	(	PUNCT
ejpam-6904	289	9	k1,b−a	k1,b−a	X
ejpam-6904	289	10	)	)	PUNCT
ejpam-6904	289	11	=	=	SYM
ejpam-6904	289	12	{	{	PUNCT
ejpam-6904	289	13	va	va	PROPN
ejpam-6904	289	14	,	,	PUNCT
ejpam-6904	289	15	x1	x1	PROPN
ejpam-6904	289	16	,	,	PUNCT
ejpam-6904	289	17	.	.	PUNCT
ejpam-6904	289	18	.	.	PUNCT
ejpam-6904	290	1	.	.	PUNCT
ejpam-6904	291	1	,	,	PUNCT
ejpam-6904	291	2	xb−a	xb−a	PROPN
ejpam-6904	291	3	}	}	PUNCT
ejpam-6904	291	4	.	.	PUNCT
ejpam-6904	292	1	let	let	VERB
ejpam-6904	292	2	s1	s1	NOUN
ejpam-6904	292	3	be	be	AUX
ejpam-6904	292	4	a	a	DET
ejpam-6904	292	5	β	β	NOUN
ejpam-6904	292	6	-	-	VERB
ejpam-6904	292	7	set	set	VERB
ejpam-6904	292	8	in	in	ADP
ejpam-6904	292	9	g.	g.	PROPN
ejpam-6904	292	10	if	if	SCONJ
ejpam-6904	292	11	x	x	PROPN
ejpam-6904	292	12	∈	∈	PROPN
ejpam-6904	292	13	s1	s1	NOUN
ejpam-6904	292	14	,	,	PUNCT
ejpam-6904	292	15	then	then	ADV
ejpam-6904	292	16	|{v1	|{v1	NOUN
ejpam-6904	292	17	,	,	PUNCT
ejpam-6904	292	18	.	.	PUNCT
ejpam-6904	292	19	.	.	PUNCT
ejpam-6904	293	1	.	.	PUNCT
ejpam-6904	294	1	,	,	PUNCT
ejpam-6904	294	2	va−1	va−1	NOUN
ejpam-6904	294	3	}	}	PUNCT
ejpam-6904	294	4	∩	∩	NOUN
ejpam-6904	294	5	s1|	s1|	PROPN
ejpam-6904	294	6	=	=	SYM
ejpam-6904	294	7	a−2	a−2	PROPN
ejpam-6904	294	8	and	and	CCONJ
ejpam-6904	294	9	va	va	NOUN
ejpam-6904	294	10	∈	∈	PROPN
ejpam-6904	294	11	s1	s1	PROPN
ejpam-6904	294	12	because	because	SCONJ
ejpam-6904	294	13	s1	s1	PROPN
ejpam-6904	294	14	is	be	AUX
ejpam-6904	294	15	a	a	DET
ejpam-6904	294	16	β	β	NOUN
ejpam-6904	294	17	-	-	VERB
ejpam-6904	294	18	set	set	VERB
ejpam-6904	294	19	in	in	ADP
ejpam-6904	294	20	g.	g.	PROPN
ejpam-6904	294	21	if	if	SCONJ
ejpam-6904	294	22	x	x	X
ejpam-6904	294	23	/∈	/∈	PUNCT
ejpam-6904	294	24	s1	s1	NOUN
ejpam-6904	294	25	,	,	PUNCT
ejpam-6904	294	26	then	then	ADV
ejpam-6904	294	27	s1	s1	PROPN
ejpam-6904	294	28	=	=	SYM
ejpam-6904	294	29	{	{	PUNCT
ejpam-6904	294	30	v1	v1	PROPN
ejpam-6904	294	31	,	,	PUNCT
ejpam-6904	294	32	.	.	PUNCT
ejpam-6904	294	33	.	.	PUNCT
ejpam-6904	295	1	.	.	PUNCT
ejpam-6904	296	1	,	,	PUNCT
ejpam-6904	296	2	va−1	va−1	PROPN
ejpam-6904	296	3	,	,	PUNCT
ejpam-6904	296	4	va	va	NOUN
ejpam-6904	296	5	}	}	PUNCT
ejpam-6904	296	6	.	.	PUNCT
ejpam-6904	297	1	in	in	ADP
ejpam-6904	297	2	both	both	DET
ejpam-6904	297	3	cases	case	NOUN
ejpam-6904	297	4	,	,	PUNCT
ejpam-6904	297	5	|s1|	|s1|	NOUN
ejpam-6904	297	6	=	=	SYM
ejpam-6904	297	7	a.	a.	NOUN
ejpam-6904	297	8	hence	hence	ADV
ejpam-6904	297	9	,	,	PUNCT
ejpam-6904	297	10	β(g	β(g	PROPN
ejpam-6904	297	11	)	)	PUNCT
ejpam-6904	297	12	=	=	PUNCT
ejpam-6904	297	13	a.	a.	NOUN
ejpam-6904	297	14	now	now	ADV
ejpam-6904	297	15	,	,	PUNCT
ejpam-6904	297	16	let	let	VERB
ejpam-6904	297	17	s2	s2	NOUN
ejpam-6904	297	18	be	be	AUX
ejpam-6904	297	19	a	a	DET
ejpam-6904	297	20	β2pg	β2pg	PUNCT
ejpam-6904	297	21	-	-	VERB
ejpam-6904	297	22	set	set	VERB
ejpam-6904	297	23	in	in	ADP
ejpam-6904	297	24	g.	g.	PROPN
ejpam-6904	297	25	then	then	ADV
ejpam-6904	297	26	{	{	PUNCT
ejpam-6904	297	27	v1	v1	NOUN
ejpam-6904	297	28	,	,	PUNCT
ejpam-6904	297	29	.	.	PUNCT
ejpam-6904	297	30	.	.	PUNCT
ejpam-6904	298	1	.	.	PUNCT
ejpam-6904	299	1	,	,	PUNCT
ejpam-6904	299	2	va−1	va−1	PROPN
ejpam-6904	299	3	,	,	PUNCT
ejpam-6904	299	4	x1	x1	PROPN
ejpam-6904	299	5	,	,	PUNCT
ejpam-6904	299	6	.	.	PUNCT
ejpam-6904	299	7	.	.	PUNCT
ejpam-6904	300	1	.	.	PUNCT
ejpam-6904	301	1	,	,	PUNCT
ejpam-6904	301	2	xb−a	xb−a	PROPN
ejpam-6904	301	3	}	}	PUNCT
ejpam-6904	301	4	⊆	⊆	NUM
ejpam-6904	301	5	s2	s2	NOUN
ejpam-6904	301	6	by	by	ADP
ejpam-6904	301	7	proposition	proposition	NOUN
ejpam-6904	301	8	1(i	1(i	NUM
ejpam-6904	301	9	)	)	PUNCT
ejpam-6904	301	10	.	.	PUNCT
ejpam-6904	302	1	also	also	ADV
ejpam-6904	302	2	,	,	PUNCT
ejpam-6904	302	3	since	since	SCONJ
ejpam-6904	302	4	s2	s2	PROPN
ejpam-6904	302	5	is	be	AUX
ejpam-6904	302	6	a	a	DET
ejpam-6904	302	7	β2pg	β2pg	PUNCT
ejpam-6904	302	8	-	-	VERB
ejpam-6904	302	9	set	set	NOUN
ejpam-6904	302	10	in	in	ADP
ejpam-6904	302	11	g	g	NOUN
ejpam-6904	302	12	,	,	PUNCT
ejpam-6904	302	13	we	we	PRON
ejpam-6904	302	14	have	have	VERB
ejpam-6904	302	15	|{x	|{x	NUM
ejpam-6904	302	16	,	,	PUNCT
ejpam-6904	302	17	va	va	NOUN
ejpam-6904	302	18	}	}	PUNCT
ejpam-6904	302	19	∩	∩	NOUN
ejpam-6904	302	20	s2|	s2|	NOUN
ejpam-6904	302	21	=	=	NOUN
ejpam-6904	302	22	1	1	NUM
ejpam-6904	302	23	.	.	PUNCT
ejpam-6904	302	24	thus	thus	ADV
ejpam-6904	302	25	,	,	PUNCT
ejpam-6904	302	26	β2pg(g	β2pg(g	X
ejpam-6904	302	27	)	)	PUNCT
ejpam-6904	302	28	=	=	SYM
ejpam-6904	302	29	|s2|	|s2|	NOUN
ejpam-6904	302	30	=	=	SYM
ejpam-6904	302	31	(	(	PUNCT
ejpam-6904	302	32	a−	a−	PROPN
ejpam-6904	302	33	1	1	NUM
ejpam-6904	302	34	)	)	PUNCT
ejpam-6904	303	1	+	+	CCONJ
ejpam-6904	303	2	(	(	PUNCT
ejpam-6904	303	3	b−	b−	PROPN
ejpam-6904	303	4	a	a	NOUN
ejpam-6904	303	5	)	)	PUNCT
ejpam-6904	304	1	+	+	CCONJ
ejpam-6904	304	2	1	1	X
ejpam-6904	304	3	=	=	SYM
ejpam-6904	304	4	b.	b.	PROPN
ejpam-6904	304	5	ka	ka	PROPN
ejpam-6904	304	6	vax	vax	PROPN
ejpam-6904	305	1	x1	x1	PROPN
ejpam-6904	306	1	x2	x2	PROPN
ejpam-6904	306	2	x3	x3	INTJ
ejpam-6904	306	3	...	...	PUNCT
ejpam-6904	307	1	xb−a	xb−a	PROPN
ejpam-6904	307	2	g	g	NOUN
ejpam-6904	307	3	:	:	PUNCT
ejpam-6904	307	4	figure	figure	VERB
ejpam-6904	307	5	4	4	NUM
ejpam-6904	307	6	:	:	PUNCT
ejpam-6904	307	7	graph	graph	VERB
ejpam-6904	307	8	g	g	NOUN
ejpam-6904	307	9	with	with	ADP
ejpam-6904	307	10	β(g	β(g	PROPN
ejpam-6904	307	11	)	)	PUNCT
ejpam-6904	307	12	=	=	PUNCT
ejpam-6904	308	1	a	a	DET
ejpam-6904	308	2	<	<	X
ejpam-6904	308	3	b	b	X
ejpam-6904	308	4	=	=	PUNCT
ejpam-6904	308	5	β2pg(g	β2pg(g	X
ejpam-6904	308	6	)	)	PUNCT
ejpam-6904	308	7	therefore	therefore	ADV
ejpam-6904	308	8	,	,	PUNCT
ejpam-6904	308	9	the	the	DET
ejpam-6904	308	10	assertion	assertion	NOUN
ejpam-6904	308	11	holds	hold	VERB
ejpam-6904	308	12	.	.	PUNCT
ejpam-6904	309	1	the	the	DET
ejpam-6904	309	2	next	next	ADJ
ejpam-6904	309	3	result	result	NOUN
ejpam-6904	309	4	follows	follow	VERB
ejpam-6904	309	5	from	from	ADP
ejpam-6904	309	6	theorem	theorem	ADJ
ejpam-6904	309	7	8	8	NUM
ejpam-6904	309	8	.	.	PUNCT
ejpam-6904	309	9	corollary	corollary	ADJ
ejpam-6904	309	10	3	3	X
ejpam-6904	309	11	.	.	PUNCT
ejpam-6904	310	1	let	let	VERB
ejpam-6904	310	2	n	n	PRON
ejpam-6904	310	3	be	be	AUX
ejpam-6904	310	4	a	a	DET
ejpam-6904	310	5	positive	positive	ADJ
ejpam-6904	310	6	integer	integer	NOUN
ejpam-6904	310	7	.	.	PUNCT
ejpam-6904	311	1	then	then	ADV
ejpam-6904	311	2	there	there	PRON
ejpam-6904	311	3	exists	exist	VERB
ejpam-6904	311	4	a	a	DET
ejpam-6904	311	5	connected	connected	ADJ
ejpam-6904	311	6	graph	graph	NOUN
ejpam-6904	311	7	g	g	ADP
ejpam-6904	311	8	such	such	ADJ
ejpam-6904	311	9	that	that	DET
ejpam-6904	311	10	β2pg(g)−β(g	β2pg(g)−β(g	PROPN
ejpam-6904	311	11	)	)	PUNCT
ejpam-6904	311	12	=	=	VERB
ejpam-6904	312	1	n.	n.	NOUN
ejpam-6904	312	2	in	in	ADP
ejpam-6904	312	3	other	other	ADJ
ejpam-6904	312	4	words	word	NOUN
ejpam-6904	312	5	,	,	PUNCT
ejpam-6904	312	6	the	the	DET
ejpam-6904	312	7	difference	difference	NOUN
ejpam-6904	312	8	β2pg(g)−β(g	β2pg(g)−β(g	PROPN
ejpam-6904	312	9	)	)	PUNCT
ejpam-6904	312	10	can	can	AUX
ejpam-6904	312	11	be	be	AUX
ejpam-6904	312	12	made	make	VERB
ejpam-6904	312	13	arbitrarily	arbitrarily	ADV
ejpam-6904	312	14	large	large	ADJ
ejpam-6904	312	15	.	.	PUNCT
ejpam-6904	313	1	if	if	SCONJ
ejpam-6904	313	2	what	what	PRON
ejpam-6904	313	3	follows	follow	VERB
ejpam-6904	313	4	,	,	PUNCT
ejpam-6904	313	5	we	we	PRON
ejpam-6904	313	6	denote	denote	VERB
ejpam-6904	313	7	by	by	ADP
ejpam-6904	313	8	g1	g1	NOUN
ejpam-6904	313	9	and	and	CCONJ
ejpam-6904	313	10	g2	g2	PROPN
ejpam-6904	313	11	the	the	DET
ejpam-6904	313	12	copies	copy	NOUN
ejpam-6904	313	13	of	of	ADP
ejpam-6904	313	14	graph	graph	NOUN
ejpam-6904	313	15	g	g	PROPN
ejpam-6904	313	16	in	in	ADP
ejpam-6904	313	17	the	the	DET
ejpam-6904	313	18	definition	definition	NOUN
ejpam-6904	313	19	of	of	ADP
ejpam-6904	313	20	the	the	DET
ejpam-6904	313	21	shadow	shadow	NOUN
ejpam-6904	313	22	graph	graph	VERB
ejpam-6904	313	23	d2(g	d2(g	PROPN
ejpam-6904	313	24	)	)	PUNCT
ejpam-6904	313	25	.	.	PUNCT
ejpam-6904	314	1	moreover	moreover	ADV
ejpam-6904	314	2	,	,	PUNCT
ejpam-6904	314	3	we	we	PRON
ejpam-6904	314	4	denote	denote	VERB
ejpam-6904	314	5	by	by	ADP
ejpam-6904	314	6	v′	v′	NOUN
ejpam-6904	314	7	the	the	DET
ejpam-6904	314	8	vertex	vertex	NOUN
ejpam-6904	314	9	in	in	ADP
ejpam-6904	314	10	g2	g2	PROPN
ejpam-6904	314	11	corresponding	correspond	VERB
ejpam-6904	314	12	to	to	ADP
ejpam-6904	314	13	the	the	DET
ejpam-6904	314	14	vertex	vertex	NOUN
ejpam-6904	314	15	v	v	ADP
ejpam-6904	314	16	∈	∈	PROPN
ejpam-6904	314	17	v	v	NOUN
ejpam-6904	314	18	(	(	PUNCT
ejpam-6904	314	19	g1	g1	PROPN
ejpam-6904	314	20	)	)	PUNCT
ejpam-6904	314	21	.	.	PUNCT
ejpam-6904	315	1	theorem	theorem	NOUN
ejpam-6904	315	2	9	9	NUM
ejpam-6904	315	3	.	.	PUNCT
ejpam-6904	316	1	let	let	VERB
ejpam-6904	316	2	g	g	PRON
ejpam-6904	316	3	be	be	AUX
ejpam-6904	316	4	a	a	DET
ejpam-6904	316	5	non	non	ADJ
ejpam-6904	316	6	-	-	ADJ
ejpam-6904	316	7	trivial	trivial	ADJ
ejpam-6904	316	8	connected	connected	ADJ
ejpam-6904	316	9	graph	graph	NOUN
ejpam-6904	316	10	.	.	PUNCT
ejpam-6904	317	1	then	then	ADV
ejpam-6904	317	2	s	s	VERB
ejpam-6904	317	3	⊆	⊆	NUM
ejpam-6904	317	4	v	v	NOUN
ejpam-6904	317	5	(	(	PUNCT
ejpam-6904	317	6	d2(g	d2(g	PROPN
ejpam-6904	317	7	)	)	PUNCT
ejpam-6904	317	8	)	)	PUNCT
ejpam-6904	317	9	is	be	AUX
ejpam-6904	317	10	a	a	DET
ejpam-6904	317	11	vertex	vertex	NOUN
ejpam-6904	317	12	cover	cover	NOUN
ejpam-6904	317	13	of	of	ADP
ejpam-6904	317	14	d2(g	d2(g	NOUN
ejpam-6904	317	15	)	)	PUNCT
ejpam-6904	318	1	if	if	SCONJ
ejpam-6904	318	2	and	and	CCONJ
ejpam-6904	318	3	only	only	ADV
ejpam-6904	318	4	if	if	SCONJ
ejpam-6904	318	5	s	s	NOUN
ejpam-6904	318	6	=	=	VERB
ejpam-6904	318	7	sg1	sg1	PROPN
ejpam-6904	318	8	∪	∪	ADP
ejpam-6904	318	9	sg2	sg2	PROPN
ejpam-6904	318	10	and	and	CCONJ
ejpam-6904	318	11	satisfies	satisfy	VERB
ejpam-6904	318	12	the	the	DET
ejpam-6904	318	13	following	follow	VERB
ejpam-6904	318	14	conditions	condition	NOUN
ejpam-6904	318	15	:	:	PUNCT
ejpam-6904	318	16	(	(	PUNCT
ejpam-6904	318	17	i	i	NOUN
ejpam-6904	318	18	)	)	PUNCT
ejpam-6904	318	19	s	s	PART
ejpam-6904	318	20	=	=	NOUN
ejpam-6904	318	21	sg1	sg1	PROPN
ejpam-6904	318	22	∪	∪	PROPN
ejpam-6904	318	23	sg2	sg2	PROPN
ejpam-6904	318	24	,	,	PUNCT
ejpam-6904	318	25	where	where	SCONJ
ejpam-6904	318	26	sg1	sg1	PROPN
ejpam-6904	318	27	and	and	CCONJ
ejpam-6904	318	28	sg2	sg2	PROPN
ejpam-6904	318	29	are	be	AUX
ejpam-6904	318	30	vertex	vertex	NOUN
ejpam-6904	318	31	covers	cover	NOUN
ejpam-6904	318	32	of	of	ADP
ejpam-6904	318	33	g1	g1	NOUN
ejpam-6904	318	34	and	and	CCONJ
ejpam-6904	318	35	g2	g2	PROPN
ejpam-6904	318	36	,	,	PUNCT
ejpam-6904	318	37	respectively	respectively	ADV
ejpam-6904	318	38	.	.	PUNCT
ejpam-6904	319	1	(	(	PUNCT
ejpam-6904	319	2	ii	ii	NOUN
ejpam-6904	319	3	)	)	PUNCT
ejpam-6904	319	4	for	for	ADP
ejpam-6904	319	5	each	each	DET
ejpam-6904	319	6	v	v	NUM
ejpam-6904	319	7	∈	∈	PROPN
ejpam-6904	319	8	v	v	NOUN
ejpam-6904	319	9	(	(	PUNCT
ejpam-6904	319	10	g1)\sg1	g1)\sg1	NOUN
ejpam-6904	319	11	,	,	PUNCT
ejpam-6904	319	12	it	it	PRON
ejpam-6904	319	13	holds	hold	VERB
ejpam-6904	319	14	that	that	SCONJ
ejpam-6904	319	15	w	w	PROPN
ejpam-6904	319	16	∈	∈	PROPN
ejpam-6904	319	17	sg1	sg1	NOUN
ejpam-6904	319	18	and	and	CCONJ
ejpam-6904	319	19	w′	w′	PROPN
ejpam-6904	319	20	∈	∈	PROPN
ejpam-6904	319	21	sg2	sg2	PROPN
ejpam-6904	319	22	for	for	ADP
ejpam-6904	319	23	every	every	DET
ejpam-6904	319	24	w	w	PROPN
ejpam-6904	319	25	∈	∈	PROPN
ejpam-6904	319	26	ng1(v	ng1(v	PRON
ejpam-6904	319	27	)	)	PUNCT
ejpam-6904	319	28	.	.	PUNCT
ejpam-6904	320	1	(	(	PUNCT
ejpam-6904	320	2	iii	iii	X
ejpam-6904	320	3	)	)	PUNCT
ejpam-6904	320	4	for	for	ADP
ejpam-6904	320	5	each	each	DET
ejpam-6904	320	6	p′	p′	NOUN
ejpam-6904	320	7	∈	∈	NOUN
ejpam-6904	320	8	v	v	NOUN
ejpam-6904	320	9	(	(	PUNCT
ejpam-6904	320	10	g2)\sg2	g2)\sg2	PROPN
ejpam-6904	320	11	,	,	PUNCT
ejpam-6904	320	12	it	it	PRON
ejpam-6904	320	13	holds	hold	VERB
ejpam-6904	320	14	that	that	DET
ejpam-6904	320	15	q	q	PROPN
ejpam-6904	320	16	∈	∈	PROPN
ejpam-6904	320	17	sg1	sg1	NOUN
ejpam-6904	320	18	and	and	CCONJ
ejpam-6904	320	19	q′	q′	NOUN
ejpam-6904	320	20	∈	∈	PROPN
ejpam-6904	320	21	sg2	sg2	PROPN
ejpam-6904	320	22	for	for	ADP
ejpam-6904	320	23	every	every	DET
ejpam-6904	320	24	q′	q′	NOUN
ejpam-6904	320	25	∈	∈	PROPN
ejpam-6904	320	26	ng2(p	ng2(p	PROPN
ejpam-6904	320	27	′	′	NUM
ejpam-6904	320	28	)	)	PUNCT
ejpam-6904	320	29	.	.	PUNCT
ejpam-6904	321	1	proof	proof	NOUN
ejpam-6904	321	2	.	.	PUNCT
ejpam-6904	322	1	suppose	suppose	VERB
ejpam-6904	322	2	s	s	PRON
ejpam-6904	322	3	is	be	AUX
ejpam-6904	322	4	a	a	DET
ejpam-6904	322	5	vertex	vertex	NOUN
ejpam-6904	322	6	cover	cover	NOUN
ejpam-6904	322	7	of	of	ADP
ejpam-6904	322	8	d2(g	d2(g	NOUN
ejpam-6904	322	9	)	)	PUNCT
ejpam-6904	322	10	.	.	PUNCT
ejpam-6904	323	1	then	then	ADV
ejpam-6904	323	2	sg1	sg1	VERB
ejpam-6904	323	3	=	=	SYM
ejpam-6904	323	4	s	s	PROPN
ejpam-6904	323	5	∩	∩	ADJ
ejpam-6904	323	6	v	v	X
ejpam-6904	323	7	(	(	PUNCT
ejpam-6904	323	8	g1	g1	PROPN
ejpam-6904	323	9	)	)	PUNCT
ejpam-6904	323	10	and	and	CCONJ
ejpam-6904	323	11	sg2	sg2	PROPN
ejpam-6904	323	12	=	=	PROPN
ejpam-6904	323	13	s	s	PROPN
ejpam-6904	323	14	∩	∩	ADJ
ejpam-6904	323	15	v	v	X
ejpam-6904	323	16	(	(	PUNCT
ejpam-6904	323	17	g2	g2	PROPN
ejpam-6904	323	18	)	)	PUNCT
ejpam-6904	323	19	are	be	AUX
ejpam-6904	323	20	vertex	vertex	NOUN
ejpam-6904	323	21	covers	cover	NOUN
ejpam-6904	323	22	of	of	ADP
ejpam-6904	323	23	g1	g1	NOUN
ejpam-6904	323	24	and	and	CCONJ
ejpam-6904	323	25	g2	g2	PROPN
ejpam-6904	323	26	,	,	PUNCT
ejpam-6904	323	27	respectively	respectively	ADV
ejpam-6904	323	28	,	,	PUNCT
ejpam-6904	323	29	because	because	SCONJ
ejpam-6904	323	30	s	s	PROPN
ejpam-6904	323	31	is	be	AUX
ejpam-6904	323	32	a	a	DET
ejpam-6904	323	33	vertex	vertex	NOUN
ejpam-6904	323	34	cover	cover	NOUN
ejpam-6904	323	35	of	of	ADP
ejpam-6904	323	36	d2(g	d2(g	NOUN
ejpam-6904	323	37	)	)	PUNCT
ejpam-6904	323	38	.	.	PUNCT
ejpam-6904	324	1	this	this	PRON
ejpam-6904	324	2	shows	show	VERB
ejpam-6904	324	3	that	that	SCONJ
ejpam-6904	324	4	(	(	PUNCT
ejpam-6904	324	5	i	i	NOUN
ejpam-6904	324	6	)	)	PUNCT
ejpam-6904	324	7	holds	hold	VERB
ejpam-6904	324	8	.	.	PUNCT
ejpam-6904	325	1	now	now	ADV
ejpam-6904	325	2	let	let	VERB
ejpam-6904	325	3	v	v	NUM
ejpam-6904	325	4	∈	∈	PROPN
ejpam-6904	325	5	v	v	NOUN
ejpam-6904	325	6	(	(	PUNCT
ejpam-6904	325	7	g1	g1	PROPN
ejpam-6904	325	8	)	)	PUNCT
ejpam-6904	325	9	\	\	NOUN
ejpam-6904	325	10	sg1	sg1	NOUN
ejpam-6904	325	11	and	and	CCONJ
ejpam-6904	325	12	let	let	VERB
ejpam-6904	325	13	w	w	PROPN
ejpam-6904	325	14	∈	∈	PROPN
ejpam-6904	325	15	ng1(v	ng1(v	PROPN
ejpam-6904	325	16	)	)	PUNCT
ejpam-6904	325	17	.	.	PUNCT
ejpam-6904	326	1	since	since	SCONJ
ejpam-6904	326	2	sg1	sg1	PROPN
ejpam-6904	326	3	is	be	AUX
ejpam-6904	326	4	vertex	vertex	NOUN
ejpam-6904	326	5	cover	cover	NOUN
ejpam-6904	326	6	of	of	ADP
ejpam-6904	326	7	g1	g1	NOUN
ejpam-6904	326	8	,	,	PUNCT
ejpam-6904	326	9	it	it	PRON
ejpam-6904	326	10	follows	follow	VERB
ejpam-6904	326	11	that	that	SCONJ
ejpam-6904	326	12	w	w	PROPN
ejpam-6904	326	13	∈	∈	PROPN
ejpam-6904	326	14	sg1	sg1	NOUN
ejpam-6904	326	15	.	.	PUNCT
ejpam-6904	327	1	also	also	ADV
ejpam-6904	327	2	,	,	PUNCT
ejpam-6904	327	3	since	since	SCONJ
ejpam-6904	327	4	vw′	vw′	X
ejpam-6904	327	5	∈	∈	PROPN
ejpam-6904	327	6	e(d2(g	e(d2(g	NOUN
ejpam-6904	327	7	)	)	PUNCT
ejpam-6904	327	8	)	)	PUNCT
ejpam-6904	327	9	and	and	CCONJ
ejpam-6904	327	10	s	s	VERB
ejpam-6904	327	11	is	be	AUX
ejpam-6904	327	12	a.	a.	PROPN
ejpam-6904	327	13	b.	b.	PROPN
ejpam-6904	327	14	tapeing	tapeing	PROPN
ejpam-6904	327	15	,	,	PUNCT
ejpam-6904	327	16	s.	s.	PROPN
ejpam-6904	327	17	r.	r.	PROPN
ejpam-6904	327	18	canoy	canoy	PROPN
ejpam-6904	327	19	/	/	SYM
ejpam-6904	327	20	eur	eur	PROPN
ejpam-6904	327	21	.	.	PUNCT
ejpam-6904	328	1	j.	j.	PROPN
ejpam-6904	328	2	pure	pure	PROPN
ejpam-6904	328	3	appl	appl	PROPN
ejpam-6904	328	4	.	.	PROPN
ejpam-6904	328	5	math	math	PROPN
ejpam-6904	328	6	,	,	PUNCT
ejpam-6904	328	7	18	18	NUM
ejpam-6904	328	8	(	(	PUNCT
ejpam-6904	328	9	4	4	NUM
ejpam-6904	328	10	)	)	PUNCT
ejpam-6904	328	11	(	(	PUNCT
ejpam-6904	328	12	2025	2025	NUM
ejpam-6904	328	13	)	)	PUNCT
ejpam-6904	328	14	,	,	PUNCT
ejpam-6904	328	15	6904	6904	NUM
ejpam-6904	328	16	10	10	NUM
ejpam-6904	328	17	of	of	ADP
ejpam-6904	328	18	14	14	NUM
ejpam-6904	328	19	a	a	DET
ejpam-6904	328	20	vertex	vertex	NOUN
ejpam-6904	328	21	cover	cover	NOUN
ejpam-6904	328	22	of	of	ADP
ejpam-6904	328	23	d2(g	d2(g	PROPN
ejpam-6904	328	24	)	)	PUNCT
ejpam-6904	329	1	,	,	PUNCT
ejpam-6904	329	2	we	we	PRON
ejpam-6904	329	3	have	have	VERB
ejpam-6904	329	4	w′	w′	PROPN
ejpam-6904	329	5	∈	∈	PROPN
ejpam-6904	329	6	sg2	sg2	PROPN
ejpam-6904	329	7	.	.	PUNCT
ejpam-6904	330	1	this	this	PRON
ejpam-6904	330	2	shows	show	VERB
ejpam-6904	330	3	that	that	SCONJ
ejpam-6904	330	4	(	(	PUNCT
ejpam-6904	330	5	ii	ii	NOUN
ejpam-6904	330	6	)	)	PUNCT
ejpam-6904	330	7	holds	hold	VERB
ejpam-6904	330	8	.	.	PUNCT
ejpam-6904	331	1	similarly	similarly	ADV
ejpam-6904	331	2	,	,	PUNCT
ejpam-6904	331	3	(	(	PUNCT
ejpam-6904	331	4	iii	iii	NOUN
ejpam-6904	331	5	)	)	PUNCT
ejpam-6904	331	6	holds	hold	VERB
ejpam-6904	331	7	.	.	PUNCT
ejpam-6904	332	1	for	for	ADP
ejpam-6904	332	2	the	the	DET
ejpam-6904	332	3	converse	converse	NOUN
ejpam-6904	332	4	,	,	PUNCT
ejpam-6904	332	5	suppose	suppose	VERB
ejpam-6904	332	6	that	that	SCONJ
ejpam-6904	332	7	s	s	VERB
ejpam-6904	332	8	has	have	VERB
ejpam-6904	332	9	the	the	DET
ejpam-6904	332	10	given	give	VERB
ejpam-6904	332	11	form	form	NOUN
ejpam-6904	332	12	and	and	CCONJ
ejpam-6904	332	13	satisfies	satisfie	NOUN
ejpam-6904	332	14	(	(	PUNCT
ejpam-6904	332	15	i	i	NOUN
ejpam-6904	332	16	)	)	PUNCT
ejpam-6904	332	17	,	,	PUNCT
ejpam-6904	332	18	(	(	PUNCT
ejpam-6904	332	19	ii	ii	NOUN
ejpam-6904	332	20	)	)	PUNCT
ejpam-6904	332	21	,	,	PUNCT
ejpam-6904	332	22	and	and	CCONJ
ejpam-6904	332	23	(	(	PUNCT
ejpam-6904	332	24	iii	iii	NOUN
ejpam-6904	332	25	)	)	PUNCT
ejpam-6904	332	26	.	.	PUNCT
ejpam-6904	333	1	let	let	VERB
ejpam-6904	333	2	xy	xy	PROPN
ejpam-6904	333	3	∈	∈	PROPN
ejpam-6904	333	4	e(d2(g	e(d2(g	PROPN
ejpam-6904	333	5	)	)	PUNCT
ejpam-6904	333	6	)	)	PUNCT
ejpam-6904	333	7	and	and	CCONJ
ejpam-6904	333	8	consider	consider	VERB
ejpam-6904	333	9	the	the	DET
ejpam-6904	333	10	following	follow	VERB
ejpam-6904	333	11	cases	case	NOUN
ejpam-6904	333	12	:	:	PUNCT
ejpam-6904	333	13	case	case	NOUN
ejpam-6904	333	14	1	1	NUM
ejpam-6904	333	15	.	.	PUNCT
ejpam-6904	334	1	xy	xy	PROPN
ejpam-6904	334	2	∈	∈	PROPN
ejpam-6904	334	3	e(g1	e(g1	ADJ
ejpam-6904	334	4	)	)	PUNCT
ejpam-6904	334	5	∪	∪	ADP
ejpam-6904	334	6	e(g2	e(g2	ADV
ejpam-6904	334	7	)	)	PUNCT
ejpam-6904	334	8	.	.	PUNCT
ejpam-6904	335	1	if	if	SCONJ
ejpam-6904	335	2	xy	xy	PROPN
ejpam-6904	335	3	∈	∈	PROPN
ejpam-6904	335	4	e(g1	e(g1	ADJ
ejpam-6904	335	5	)	)	PUNCT
ejpam-6904	335	6	,	,	PUNCT
ejpam-6904	335	7	then	then	ADV
ejpam-6904	335	8	x	x	SYM
ejpam-6904	335	9	∈	∈	NOUN
ejpam-6904	335	10	sg1	sg1	NOUN
ejpam-6904	335	11	or	or	CCONJ
ejpam-6904	335	12	y	y	PROPN
ejpam-6904	335	13	∈	∈	PROPN
ejpam-6904	335	14	sg1	sg1	NOUN
ejpam-6904	335	15	because	because	SCONJ
ejpam-6904	335	16	sg1	sg1	NOUN
ejpam-6904	335	17	is	be	AUX
ejpam-6904	335	18	a	a	DET
ejpam-6904	335	19	vertex	vertex	NOUN
ejpam-6904	335	20	cover	cover	NOUN
ejpam-6904	335	21	of	of	ADP
ejpam-6904	335	22	g1	g1	NOUN
ejpam-6904	335	23	.	.	PUNCT
ejpam-6904	336	1	similarly	similarly	ADV
ejpam-6904	336	2	,	,	PUNCT
ejpam-6904	336	3	x	x	PROPN
ejpam-6904	336	4	∈	∈	PROPN
ejpam-6904	336	5	sg2	sg2	PROPN
ejpam-6904	336	6	or	or	CCONJ
ejpam-6904	336	7	y	y	PROPN
ejpam-6904	336	8	∈	∈	PROPN
ejpam-6904	336	9	sg2	sg2	PROPN
ejpam-6904	336	10	whenever	whenever	SCONJ
ejpam-6904	336	11	xy	xy	PROPN
ejpam-6904	336	12	∈	∈	PROPN
ejpam-6904	336	13	e(g2	e(g2	ADV
ejpam-6904	336	14	)	)	PUNCT
ejpam-6904	336	15	.	.	PUNCT
ejpam-6904	337	1	case	case	NOUN
ejpam-6904	337	2	2	2	NUM
ejpam-6904	337	3	.	.	PUNCT
ejpam-6904	337	4	x	x	SYM
ejpam-6904	337	5	∈	∈	PROPN
ejpam-6904	337	6	v	v	X
ejpam-6904	337	7	(	(	PUNCT
ejpam-6904	337	8	g1	g1	PROPN
ejpam-6904	337	9	)	)	PUNCT
ejpam-6904	337	10	and	and	CCONJ
ejpam-6904	337	11	y	y	PROPN
ejpam-6904	337	12	∈	∈	PROPN
ejpam-6904	337	13	v	v	PROPN
ejpam-6904	337	14	(	(	PUNCT
ejpam-6904	337	15	g2	g2	PROPN
ejpam-6904	337	16	)	)	PUNCT
ejpam-6904	337	17	.	.	PUNCT
ejpam-6904	338	1	let	let	VERB
ejpam-6904	338	2	y	y	NOUN
ejpam-6904	338	3	=	=	PUNCT
ejpam-6904	338	4	z′	z′	NUM
ejpam-6904	338	5	where	where	SCONJ
ejpam-6904	338	6	z	z	PROPN
ejpam-6904	338	7	∈	∈	PROPN
ejpam-6904	338	8	v	v	NOUN
ejpam-6904	338	9	(	(	PUNCT
ejpam-6904	338	10	g1	g1	PROPN
ejpam-6904	338	11	)	)	PUNCT
ejpam-6904	338	12	.	.	PUNCT
ejpam-6904	339	1	if	if	SCONJ
ejpam-6904	339	2	x	x	SYM
ejpam-6904	339	3	∈	∈	NOUN
ejpam-6904	339	4	sg1	sg1	NOUN
ejpam-6904	339	5	,	,	PUNCT
ejpam-6904	339	6	then	then	ADV
ejpam-6904	339	7	we	we	PRON
ejpam-6904	339	8	are	be	AUX
ejpam-6904	339	9	done	do	VERB
ejpam-6904	339	10	.	.	PUNCT
ejpam-6904	340	1	so	so	ADV
ejpam-6904	340	2	suppose	suppose	VERB
ejpam-6904	340	3	x	x	X
ejpam-6904	340	4	∈	∈	PROPN
ejpam-6904	340	5	v	v	NOUN
ejpam-6904	340	6	(	(	PUNCT
ejpam-6904	340	7	g1)\sg1	g1)\sg1	NOUN
ejpam-6904	340	8	.	.	PUNCT
ejpam-6904	341	1	since	since	SCONJ
ejpam-6904	341	2	z	z	PROPN
ejpam-6904	341	3	∈	∈	PROPN
ejpam-6904	341	4	ng(x	ng(x	NUM
ejpam-6904	341	5	)	)	PUNCT
ejpam-6904	341	6	,	,	PUNCT
ejpam-6904	341	7	it	it	PRON
ejpam-6904	341	8	follows	follow	VERB
ejpam-6904	341	9	from	from	ADP
ejpam-6904	341	10	(	(	PUNCT
ejpam-6904	341	11	ii	ii	NOUN
ejpam-6904	341	12	)	)	PUNCT
ejpam-6904	341	13	that	that	SCONJ
ejpam-6904	341	14	z′	z′	NUM
ejpam-6904	341	15	∈	∈	PROPN
ejpam-6904	341	16	sg2	sg2	PROPN
ejpam-6904	341	17	.	.	PUNCT
ejpam-6904	342	1	thus	thus	ADV
ejpam-6904	342	2	,	,	PUNCT
ejpam-6904	342	3	y	y	PROPN
ejpam-6904	342	4	∈	∈	PROPN
ejpam-6904	342	5	s.	s.	PROPN
ejpam-6904	342	6	therefore	therefore	ADV
ejpam-6904	342	7	,	,	PUNCT
ejpam-6904	342	8	s	s	VERB
ejpam-6904	342	9	is	be	AUX
ejpam-6904	342	10	a	a	DET
ejpam-6904	342	11	vertex	vertex	NOUN
ejpam-6904	342	12	cover	cover	NOUN
ejpam-6904	342	13	of	of	ADP
ejpam-6904	342	14	d2(g	d2(g	PROPN
ejpam-6904	342	15	)	)	PUNCT
ejpam-6904	342	16	.	.	PUNCT
ejpam-6904	343	1	theorem	theorem	ADJ
ejpam-6904	343	2	10	10	NUM
ejpam-6904	343	3	.	.	PUNCT
ejpam-6904	344	1	let	let	VERB
ejpam-6904	344	2	g	g	PRON
ejpam-6904	344	3	be	be	AUX
ejpam-6904	344	4	a	a	DET
ejpam-6904	344	5	non	non	ADJ
ejpam-6904	344	6	-	-	ADJ
ejpam-6904	344	7	trivial	trivial	ADJ
ejpam-6904	344	8	connected	connected	ADJ
ejpam-6904	344	9	graph	graph	NOUN
ejpam-6904	344	10	.	.	PUNCT
ejpam-6904	345	1	then	then	ADV
ejpam-6904	345	2	s	s	VERB
ejpam-6904	345	3	⊆	⊆	NUM
ejpam-6904	345	4	v	v	NOUN
ejpam-6904	345	5	(	(	PUNCT
ejpam-6904	345	6	d2(g	d2(g	PROPN
ejpam-6904	345	7	)	)	PUNCT
ejpam-6904	345	8	)	)	PUNCT
ejpam-6904	345	9	is	be	AUX
ejpam-6904	345	10	a	a	DET
ejpam-6904	345	11	vertex	vertex	NOUN
ejpam-6904	345	12	cover	cover	NOUN
ejpam-6904	345	13	of	of	ADP
ejpam-6904	345	14	d2(g	d2(g	NOUN
ejpam-6904	345	15	)	)	PUNCT
ejpam-6904	346	1	if	if	SCONJ
ejpam-6904	346	2	and	and	CCONJ
ejpam-6904	346	3	only	only	ADV
ejpam-6904	346	4	if	if	SCONJ
ejpam-6904	346	5	it	it	PRON
ejpam-6904	346	6	is	be	AUX
ejpam-6904	346	7	a	a	DET
ejpam-6904	346	8	2	2	NUM
ejpam-6904	346	9	-	-	PUNCT
ejpam-6904	346	10	path	path	NOUN
ejpam-6904	346	11	geodetic	geodetic	ADJ
ejpam-6904	346	12	vertex	vertex	NOUN
ejpam-6904	346	13	cover	cover	NOUN
ejpam-6904	346	14	of	of	ADP
ejpam-6904	346	15	d2(g	d2(g	NOUN
ejpam-6904	346	16	)	)	PUNCT
ejpam-6904	346	17	.	.	PUNCT
ejpam-6904	347	1	proof	proof	NOUN
ejpam-6904	347	2	.	.	PUNCT
ejpam-6904	348	1	suppose	suppose	VERB
ejpam-6904	348	2	s	s	PRON
ejpam-6904	348	3	is	be	AUX
ejpam-6904	348	4	a	a	DET
ejpam-6904	348	5	vertex	vertex	NOUN
ejpam-6904	348	6	cover	cover	NOUN
ejpam-6904	348	7	of	of	ADP
ejpam-6904	348	8	d2(g	d2(g	NOUN
ejpam-6904	348	9	)	)	PUNCT
ejpam-6904	348	10	.	.	PUNCT
ejpam-6904	349	1	then	then	ADV
ejpam-6904	349	2	s	s	VERB
ejpam-6904	349	3	=	=	PUNCT
ejpam-6904	349	4	sg1	sg1	PROPN
ejpam-6904	349	5	∪	∪	ADP
ejpam-6904	349	6	sg2	sg2	PROPN
ejpam-6904	349	7	and	and	CCONJ
ejpam-6904	349	8	satisfies	satisfy	VERB
ejpam-6904	349	9	properties	property	NOUN
ejpam-6904	349	10	(	(	PUNCT
ejpam-6904	349	11	i	i	NOUN
ejpam-6904	349	12	)	)	PUNCT
ejpam-6904	349	13	,	,	PUNCT
ejpam-6904	349	14	(	(	PUNCT
ejpam-6904	349	15	ii	ii	NOUN
ejpam-6904	349	16	)	)	PUNCT
ejpam-6904	349	17	,	,	PUNCT
ejpam-6904	349	18	and	and	CCONJ
ejpam-6904	349	19	(	(	PUNCT
ejpam-6904	349	20	iii	iii	NOUN
ejpam-6904	349	21	)	)	PUNCT
ejpam-6904	349	22	of	of	ADP
ejpam-6904	349	23	theorem	theorem	NOUN
ejpam-6904	349	24	9	9	NUM
ejpam-6904	349	25	.	.	PUNCT
ejpam-6904	349	26	now	now	ADV
ejpam-6904	349	27	let	let	VERB
ejpam-6904	349	28	v	v	ADP
ejpam-6904	349	29	∈	∈	PROPN
ejpam-6904	349	30	d2(g	d2(g	PROPN
ejpam-6904	349	31	)	)	PUNCT
ejpam-6904	349	32	\	\	PUNCT
ejpam-6904	349	33	s.	s.	PROPN
ejpam-6904	349	34	suppose	suppose	VERB
ejpam-6904	349	35	v	v	ADP
ejpam-6904	349	36	∈	∈	PROPN
ejpam-6904	349	37	v	v	NOUN
ejpam-6904	349	38	(	(	PUNCT
ejpam-6904	349	39	g1	g1	PROPN
ejpam-6904	349	40	)	)	PUNCT
ejpam-6904	349	41	\	\	PROPN
ejpam-6904	349	42	sg1	sg1	NOUN
ejpam-6904	349	43	.	.	PUNCT
ejpam-6904	350	1	pick	pick	VERB
ejpam-6904	350	2	any	any	DET
ejpam-6904	350	3	w	w	PROPN
ejpam-6904	350	4	∈	∈	PROPN
ejpam-6904	350	5	v	v	NOUN
ejpam-6904	350	6	(	(	PUNCT
ejpam-6904	350	7	g1	g1	PROPN
ejpam-6904	350	8	)	)	PUNCT
ejpam-6904	350	9	∩	∩	PROPN
ejpam-6904	350	10	ng1(v	ng1(v	PROPN
ejpam-6904	350	11	)	)	PUNCT
ejpam-6904	350	12	.	.	PUNCT
ejpam-6904	351	1	then	then	ADV
ejpam-6904	351	2	w	w	X
ejpam-6904	351	3	,	,	PUNCT
ejpam-6904	351	4	w′	w′	PROPN
ejpam-6904	351	5	∈	∈	PROPN
ejpam-6904	351	6	s	s	PART
ejpam-6904	351	7	by	by	X
ejpam-6904	351	8	(	(	PUNCT
ejpam-6904	351	9	ii	ii	NOUN
ejpam-6904	351	10	)	)	PUNCT
ejpam-6904	351	11	.	.	PUNCT
ejpam-6904	352	1	clearly	clearly	ADV
ejpam-6904	352	2	,	,	PUNCT
ejpam-6904	352	3	dd2(g)(w	dd2(g)(w	PROPN
ejpam-6904	352	4	,	,	PUNCT
ejpam-6904	352	5	w	w	NOUN
ejpam-6904	352	6	′	′	NOUN
ejpam-6904	352	7	)	)	PUNCT
ejpam-6904	353	1	=	=	SYM
ejpam-6904	353	2	2	2	NUM
ejpam-6904	353	3	and	and	CCONJ
ejpam-6904	353	4	v	v	ADP
ejpam-6904	353	5	∈	∈	PROPN
ejpam-6904	353	6	id2(g)(w	id2(g)(w	NOUN
ejpam-6904	353	7	,	,	PUNCT
ejpam-6904	353	8	w	w	NOUN
ejpam-6904	353	9	′	′	NUM
ejpam-6904	353	10	)	)	PUNCT
ejpam-6904	353	11	.	.	PUNCT
ejpam-6904	354	1	if	if	SCONJ
ejpam-6904	354	2	v	v	NUM
ejpam-6904	354	3	∈	∈	PROPN
ejpam-6904	354	4	v	v	NOUN
ejpam-6904	354	5	(	(	PUNCT
ejpam-6904	354	6	g2	g2	PROPN
ejpam-6904	354	7	)	)	PUNCT
ejpam-6904	354	8	\	\	PROPN
ejpam-6904	355	1	sg2	sg2	PROPN
ejpam-6904	355	2	,	,	PUNCT
ejpam-6904	355	3	say	say	VERB
ejpam-6904	355	4	v	v	ADP
ejpam-6904	355	5	=	=	SYM
ejpam-6904	355	6	z′	z′	NUM
ejpam-6904	355	7	where	where	SCONJ
ejpam-6904	355	8	z	z	PROPN
ejpam-6904	355	9	∈	∈	PROPN
ejpam-6904	355	10	v	v	NOUN
ejpam-6904	355	11	(	(	PUNCT
ejpam-6904	355	12	g1	g1	PROPN
ejpam-6904	355	13	)	)	PUNCT
ejpam-6904	355	14	,	,	PUNCT
ejpam-6904	355	15	then	then	ADV
ejpam-6904	355	16	we	we	PRON
ejpam-6904	355	17	may	may	AUX
ejpam-6904	355	18	choose	choose	VERB
ejpam-6904	355	19	any	any	DET
ejpam-6904	355	20	x′	x′	PROPN
ejpam-6904	355	21	∈	∈	PROPN
ejpam-6904	355	22	v	v	PROPN
ejpam-6904	355	23	(	(	PUNCT
ejpam-6904	355	24	g2	g2	PROPN
ejpam-6904	355	25	)	)	PUNCT
ejpam-6904	355	26	∩	∩	NOUN
ejpam-6904	355	27	ng2(z	ng2(z	PROPN
ejpam-6904	355	28	′	′	NOUN
ejpam-6904	355	29	)	)	PUNCT
ejpam-6904	355	30	.	.	PUNCT
ejpam-6904	356	1	by	by	ADP
ejpam-6904	356	2	(	(	PUNCT
ejpam-6904	356	3	iii	iii	NOUN
ejpam-6904	356	4	)	)	PUNCT
ejpam-6904	356	5	,	,	PUNCT
ejpam-6904	356	6	x	x	X
ejpam-6904	356	7	,	,	PUNCT
ejpam-6904	356	8	x′	x′	PROPN
ejpam-6904	356	9	∈	∈	PROPN
ejpam-6904	356	10	s.	s.	PROPN
ejpam-6904	356	11	moreover	moreover	ADV
ejpam-6904	356	12	,	,	PUNCT
ejpam-6904	356	13	dd2(g)(x	dd2(g)(x	NOUN
ejpam-6904	356	14	,	,	PUNCT
ejpam-6904	356	15	x	x	NOUN
ejpam-6904	356	16	′	′	NUM
ejpam-6904	356	17	)	)	PUNCT
ejpam-6904	356	18	=	=	SYM
ejpam-6904	356	19	2	2	NUM
ejpam-6904	356	20	and	and	CCONJ
ejpam-6904	356	21	v	v	ADP
ejpam-6904	356	22	∈	∈	PROPN
ejpam-6904	356	23	id2(g)(x	id2(g)(x	NOUN
ejpam-6904	356	24	,	,	PUNCT
ejpam-6904	356	25	x	x	NOUN
ejpam-6904	356	26	′	′	NUM
ejpam-6904	356	27	)	)	PUNCT
ejpam-6904	356	28	.	.	PUNCT
ejpam-6904	357	1	thus	thus	ADV
ejpam-6904	357	2	,	,	PUNCT
ejpam-6904	357	3	s	s	VERB
ejpam-6904	357	4	is	be	AUX
ejpam-6904	357	5	a	a	DET
ejpam-6904	357	6	2	2	NUM
ejpam-6904	357	7	-	-	PUNCT
ejpam-6904	357	8	path	path	NOUN
ejpam-6904	357	9	geodetic	geodetic	ADJ
ejpam-6904	357	10	vertex	vertex	NOUN
ejpam-6904	357	11	cover	cover	NOUN
ejpam-6904	357	12	of	of	ADP
ejpam-6904	357	13	d2(g	d2(g	NOUN
ejpam-6904	357	14	)	)	PUNCT
ejpam-6904	357	15	.	.	PUNCT
ejpam-6904	358	1	the	the	DET
ejpam-6904	358	2	converse	converse	NOUN
ejpam-6904	358	3	is	be	AUX
ejpam-6904	358	4	clear	clear	ADJ
ejpam-6904	358	5	.	.	PUNCT
ejpam-6904	359	1	corollary	corollary	ADJ
ejpam-6904	359	2	4	4	NUM
ejpam-6904	359	3	.	.	PUNCT
ejpam-6904	360	1	let	let	VERB
ejpam-6904	360	2	g	g	PRON
ejpam-6904	360	3	be	be	AUX
ejpam-6904	360	4	a	a	DET
ejpam-6904	360	5	non	non	ADJ
ejpam-6904	360	6	-	-	ADJ
ejpam-6904	360	7	trivial	trivial	ADJ
ejpam-6904	360	8	connected	connected	ADJ
ejpam-6904	360	9	graph	graph	NOUN
ejpam-6904	360	10	.	.	PUNCT
ejpam-6904	361	1	then	then	ADV
ejpam-6904	361	2	β2pg(d2(g	β2pg(d2(g	NOUN
ejpam-6904	361	3	)	)	PUNCT
ejpam-6904	361	4	)	)	PUNCT
ejpam-6904	362	1	=	=	PUNCT
ejpam-6904	362	2	β(d2(g	β(d2(g	NOUN
ejpam-6904	362	3	)	)	PUNCT
ejpam-6904	362	4	)	)	PUNCT
ejpam-6904	363	1	=	=	SYM
ejpam-6904	363	2	2β(g	2β(g	NUM
ejpam-6904	363	3	)	)	PUNCT
ejpam-6904	363	4	.	.	PUNCT
ejpam-6904	364	1	proof	proof	NOUN
ejpam-6904	364	2	.	.	PUNCT
ejpam-6904	365	1	let	let	VERB
ejpam-6904	365	2	s1	s1	NOUN
ejpam-6904	365	3	be	be	AUX
ejpam-6904	365	4	a	a	DET
ejpam-6904	365	5	β	β	NOUN
ejpam-6904	365	6	-	-	VERB
ejpam-6904	365	7	set	set	VERB
ejpam-6904	365	8	in	in	ADP
ejpam-6904	365	9	g1	g1	NOUN
ejpam-6904	365	10	and	and	CCONJ
ejpam-6904	365	11	let	let	VERB
ejpam-6904	365	12	s2	s2	VERB
ejpam-6904	365	13	=	=	PRON
ejpam-6904	365	14	{	{	PUNCT
ejpam-6904	365	15	v′	v′	NOUN
ejpam-6904	365	16	∈	∈	PROPN
ejpam-6904	365	17	v	v	NOUN
ejpam-6904	365	18	(	(	PUNCT
ejpam-6904	365	19	g2	g2	PROPN
ejpam-6904	365	20	)	)	PUNCT
ejpam-6904	365	21	:	:	PUNCT
ejpam-6904	365	22	v	v	X
ejpam-6904	365	23	∈	∈	NOUN
ejpam-6904	365	24	s1	s1	NOUN
ejpam-6904	365	25	}	}	PUNCT
ejpam-6904	365	26	.	.	PUNCT
ejpam-6904	366	1	then	then	ADV
ejpam-6904	366	2	s2	s2	PROPN
ejpam-6904	366	3	is	be	AUX
ejpam-6904	366	4	a	a	DET
ejpam-6904	366	5	β	β	NOUN
ejpam-6904	366	6	-	-	VERB
ejpam-6904	366	7	set	set	VERB
ejpam-6904	366	8	in	in	ADP
ejpam-6904	366	9	g2	g2	PROPN
ejpam-6904	366	10	.	.	PUNCT
ejpam-6904	367	1	moreover	moreover	ADV
ejpam-6904	367	2	,	,	PUNCT
ejpam-6904	367	3	s	s	PART
ejpam-6904	367	4	=	=	NOUN
ejpam-6904	367	5	s1	s1	PROPN
ejpam-6904	367	6	∪	∪	X
ejpam-6904	367	7	s2	s2	NOUN
ejpam-6904	367	8	is	be	AUX
ejpam-6904	367	9	a	a	DET
ejpam-6904	367	10	vertex	vertex	NOUN
ejpam-6904	367	11	cover	cover	NOUN
ejpam-6904	367	12	of	of	ADP
ejpam-6904	367	13	d2(g	d2(g	NOUN
ejpam-6904	367	14	)	)	PUNCT
ejpam-6904	367	15	by	by	ADP
ejpam-6904	367	16	theorem	theorem	NOUN
ejpam-6904	367	17	9	9	NUM
ejpam-6904	367	18	.	.	PUNCT
ejpam-6904	368	1	thus	thus	ADV
ejpam-6904	368	2	,	,	PUNCT
ejpam-6904	368	3	by	by	ADP
ejpam-6904	368	4	theorem	theorem	NOUN
ejpam-6904	368	5	10	10	NUM
ejpam-6904	368	6	,	,	PUNCT
ejpam-6904	368	7	β2pg(d2(g	β2pg(d2(g	NOUN
ejpam-6904	368	8	)	)	PUNCT
ejpam-6904	368	9	)	)	PUNCT
ejpam-6904	368	10	=	=	PUNCT
ejpam-6904	368	11	β(d2(g	β(d2(g	NOUN
ejpam-6904	368	12	)	)	PUNCT
ejpam-6904	368	13	)	)	PUNCT
ejpam-6904	368	14	≤	≤	NUM
ejpam-6904	368	15	|s|	|s|	PROPN
ejpam-6904	368	16	=	=	SYM
ejpam-6904	368	17	|s1|+	|s1|+	PROPN
ejpam-6904	368	18	|s2|	|s2|	NOUN
ejpam-6904	368	19	=	=	SYM
ejpam-6904	368	20	2β(g	2β(g	PROPN
ejpam-6904	368	21	)	)	PUNCT
ejpam-6904	368	22	.	.	PUNCT
ejpam-6904	369	1	next	next	ADV
ejpam-6904	369	2	,	,	PUNCT
ejpam-6904	369	3	let	let	VERB
ejpam-6904	369	4	s0	s0	PROPN
ejpam-6904	369	5	be	be	AUX
ejpam-6904	369	6	a	a	DET
ejpam-6904	369	7	β	β	NOUN
ejpam-6904	369	8	-	-	NOUN
ejpam-6904	369	9	set	set	NOUN
ejpam-6904	369	10	of	of	ADP
ejpam-6904	369	11	d2(g	d2(g	PROPN
ejpam-6904	369	12	)	)	PUNCT
ejpam-6904	369	13	.	.	PUNCT
ejpam-6904	370	1	then	then	ADV
ejpam-6904	370	2	s0	s0	PROPN
ejpam-6904	370	3	=	=	SYM
ejpam-6904	370	4	sg1	sg1	PROPN
ejpam-6904	370	5	∪sg2	∪sg2	PROPN
ejpam-6904	370	6	where	where	SCONJ
ejpam-6904	370	7	sg1	sg1	PROPN
ejpam-6904	370	8	and	and	CCONJ
ejpam-6904	370	9	sg2	sg2	PROPN
ejpam-6904	370	10	are	be	AUX
ejpam-6904	370	11	vertex	vertex	NOUN
ejpam-6904	370	12	covers	cover	NOUN
ejpam-6904	370	13	of	of	ADP
ejpam-6904	370	14	g1	g1	NOUN
ejpam-6904	370	15	and	and	CCONJ
ejpam-6904	370	16	g2	g2	PROPN
ejpam-6904	370	17	,	,	PUNCT
ejpam-6904	370	18	respectively	respectively	ADV
ejpam-6904	370	19	,	,	PUNCT
ejpam-6904	370	20	by	by	ADP
ejpam-6904	370	21	theorem	theorem	NOUN
ejpam-6904	370	22	9	9	NUM
ejpam-6904	370	23	.	.	PUNCT
ejpam-6904	370	24	by	by	ADP
ejpam-6904	370	25	theorem	theorem	NOUN
ejpam-6904	370	26	10	10	NUM
ejpam-6904	370	27	,	,	PUNCT
ejpam-6904	370	28	we	we	PRON
ejpam-6904	370	29	have	have	AUX
ejpam-6904	370	30	β2pg(d2(g	β2pg(d2(g	NOUN
ejpam-6904	370	31	)	)	PUNCT
ejpam-6904	370	32	)	)	PUNCT
ejpam-6904	371	1	=	=	PUNCT
ejpam-6904	371	2	β(d2(g	β(d2(g	NOUN
ejpam-6904	371	3	)	)	PUNCT
ejpam-6904	371	4	)	)	PUNCT
ejpam-6904	372	1	=	=	SYM
ejpam-6904	372	2	|s0|	|s0|	NOUN
ejpam-6904	372	3	=	=	SYM
ejpam-6904	372	4	|sg1	|sg1	PROPN
ejpam-6904	372	5	|+	|+	NOUN
ejpam-6904	373	1	|sg2	|sg2	PROPN
ejpam-6904	373	2	|	|	ADV
ejpam-6904	373	3	≥	≥	NOUN
ejpam-6904	373	4	2β(g	2β(g	NUM
ejpam-6904	373	5	)	)	PUNCT
ejpam-6904	373	6	.	.	PUNCT
ejpam-6904	374	1	this	this	PRON
ejpam-6904	374	2	establishes	establish	VERB
ejpam-6904	374	3	the	the	DET
ejpam-6904	374	4	desired	desire	VERB
ejpam-6904	374	5	equality	equality	NOUN
ejpam-6904	374	6	.	.	PUNCT
ejpam-6904	375	1	theorem	theorem	VERB
ejpam-6904	375	2	11	11	NUM
ejpam-6904	375	3	.	.	PUNCT
ejpam-6904	376	1	let	let	VERB
ejpam-6904	376	2	g	g	PRON
ejpam-6904	376	3	be	be	AUX
ejpam-6904	376	4	a	a	DET
ejpam-6904	376	5	non	non	ADJ
ejpam-6904	376	6	-	-	ADJ
ejpam-6904	376	7	complete	complete	ADJ
ejpam-6904	376	8	graph	graph	NOUN
ejpam-6904	376	9	and	and	CCONJ
ejpam-6904	376	10	let	let	VERB
ejpam-6904	376	11	m	m	PRON
ejpam-6904	376	12	be	be	AUX
ejpam-6904	376	13	a	a	DET
ejpam-6904	376	14	positive	positive	ADJ
ejpam-6904	376	15	integer	integer	NOUN
ejpam-6904	376	16	.	.	PUNCT
ejpam-6904	377	1	then	then	ADV
ejpam-6904	377	2	s	s	VERB
ejpam-6904	377	3	⊆	⊆	NUM
ejpam-6904	377	4	v	v	NOUN
ejpam-6904	377	5	(	(	PUNCT
ejpam-6904	377	6	km	km	NOUN
ejpam-6904	377	7	+	+	NOUN
ejpam-6904	377	8	g	g	NOUN
ejpam-6904	377	9	)	)	PUNCT
ejpam-6904	377	10	is	be	AUX
ejpam-6904	377	11	a	a	DET
ejpam-6904	377	12	2	2	NUM
ejpam-6904	377	13	-	-	PUNCT
ejpam-6904	377	14	path	path	NOUN
ejpam-6904	377	15	geodetic	geodetic	ADJ
ejpam-6904	377	16	vertex	vertex	NOUN
ejpam-6904	377	17	cover	cover	NOUN
ejpam-6904	377	18	of	of	ADP
ejpam-6904	377	19	km	km	NOUN
ejpam-6904	378	1	+	+	PROPN
ejpam-6904	378	2	g	g	PROPN
ejpam-6904	378	3	if	if	SCONJ
ejpam-6904	378	4	and	and	CCONJ
ejpam-6904	378	5	only	only	ADV
ejpam-6904	378	6	if	if	SCONJ
ejpam-6904	378	7	s	s	VERB
ejpam-6904	378	8	=	=	X
ejpam-6904	378	9	(	(	PUNCT
ejpam-6904	378	10	v	v	NOUN
ejpam-6904	378	11	(	(	PUNCT
ejpam-6904	378	12	km	km	NOUN
ejpam-6904	378	13	)	)	PUNCT
ejpam-6904	378	14	\	\	PROPN
ejpam-6904	378	15	a.	a.	PROPN
ejpam-6904	378	16	b.	b.	PROPN
ejpam-6904	378	17	tapeing	tapeing	PROPN
ejpam-6904	378	18	,	,	PUNCT
ejpam-6904	378	19	s.	s.	PROPN
ejpam-6904	378	20	r.	r.	PROPN
ejpam-6904	378	21	canoy	canoy	PROPN
ejpam-6904	378	22	/	/	SYM
ejpam-6904	378	23	eur	eur	PROPN
ejpam-6904	378	24	.	.	PUNCT
ejpam-6904	379	1	j.	j.	PROPN
ejpam-6904	379	2	pure	pure	PROPN
ejpam-6904	379	3	appl	appl	PROPN
ejpam-6904	379	4	.	.	PROPN
ejpam-6904	379	5	math	math	PROPN
ejpam-6904	379	6	,	,	PUNCT
ejpam-6904	379	7	18	18	NUM
ejpam-6904	379	8	(	(	PUNCT
ejpam-6904	379	9	4	4	NUM
ejpam-6904	379	10	)	)	PUNCT
ejpam-6904	379	11	(	(	PUNCT
ejpam-6904	379	12	2025	2025	NUM
ejpam-6904	379	13	)	)	PUNCT
ejpam-6904	379	14	,	,	PUNCT
ejpam-6904	379	15	6904	6904	NUM
ejpam-6904	379	16	11	11	NUM
ejpam-6904	379	17	of	of	ADP
ejpam-6904	379	18	14	14	NUM
ejpam-6904	379	19	{	{	PUNCT
ejpam-6904	379	20	v	v	NOUN
ejpam-6904	379	21	}	}	PUNCT
ejpam-6904	379	22	)	)	PUNCT
ejpam-6904	379	23	∪	∪	ADP
ejpam-6904	379	24	v	v	NOUN
ejpam-6904	379	25	(	(	PUNCT
ejpam-6904	379	26	g	g	NOUN
ejpam-6904	379	27	)	)	PUNCT
ejpam-6904	379	28	for	for	ADP
ejpam-6904	379	29	some	some	DET
ejpam-6904	379	30	v	v	ADP
ejpam-6904	379	31	∈	∈	PROPN
ejpam-6904	379	32	v	v	NOUN
ejpam-6904	379	33	(	(	PUNCT
ejpam-6904	379	34	km	km	NOUN
ejpam-6904	379	35	)	)	PUNCT
ejpam-6904	379	36	or	or	CCONJ
ejpam-6904	379	37	s	s	X
ejpam-6904	379	38	=	=	SYM
ejpam-6904	379	39	v	v	PROPN
ejpam-6904	379	40	(	(	PUNCT
ejpam-6904	379	41	km	km	NOUN
ejpam-6904	379	42	)	)	PUNCT
ejpam-6904	379	43	∪	∪	ADP
ejpam-6904	379	44	sg	sg	PROPN
ejpam-6904	379	45	,	,	PUNCT
ejpam-6904	379	46	where	where	SCONJ
ejpam-6904	379	47	sg	sg	PROPN
ejpam-6904	379	48	is	be	AUX
ejpam-6904	379	49	a	a	DET
ejpam-6904	379	50	2	2	NUM
ejpam-6904	379	51	-	-	PUNCT
ejpam-6904	379	52	path	path	NOUN
ejpam-6904	379	53	geodetic	geodetic	ADJ
ejpam-6904	379	54	vertex	vertex	NOUN
ejpam-6904	379	55	cover	cover	NOUN
ejpam-6904	379	56	of	of	ADP
ejpam-6904	379	57	g.	g.	PROPN
ejpam-6904	379	58	proof	proof	NOUN
ejpam-6904	379	59	.	.	PUNCT
ejpam-6904	380	1	suppose	suppose	VERB
ejpam-6904	380	2	s	s	NOUN
ejpam-6904	380	3	is	be	AUX
ejpam-6904	380	4	a	a	DET
ejpam-6904	380	5	2	2	NUM
ejpam-6904	380	6	-	-	PUNCT
ejpam-6904	380	7	path	path	NOUN
ejpam-6904	380	8	geodetic	geodetic	ADJ
ejpam-6904	380	9	vertex	vertex	NOUN
ejpam-6904	380	10	cover	cover	NOUN
ejpam-6904	380	11	of	of	ADP
ejpam-6904	380	12	km+g	km+g	PROPN
ejpam-6904	380	13	.	.	PUNCT
ejpam-6904	381	1	suppose	suppose	VERB
ejpam-6904	381	2	v	v	X
ejpam-6904	381	3	(	(	PUNCT
ejpam-6904	381	4	km)\s	km)\s	NOUN
ejpam-6904	381	5	̸=	̸=	PROPN
ejpam-6904	381	6	∅	∅	NOUN
ejpam-6904	381	7	,	,	PUNCT
ejpam-6904	381	8	say	say	VERB
ejpam-6904	381	9	v	v	NUM
ejpam-6904	381	10	∈	∈	PROPN
ejpam-6904	381	11	v	v	NOUN
ejpam-6904	381	12	(	(	PUNCT
ejpam-6904	381	13	km	km	NOUN
ejpam-6904	381	14	)	)	PUNCT
ejpam-6904	381	15	\	\	PUNCT
ejpam-6904	382	1	s.	s.	PROPN
ejpam-6904	382	2	since	since	SCONJ
ejpam-6904	382	3	s	s	PROPN
ejpam-6904	382	4	is	be	AUX
ejpam-6904	382	5	a	a	DET
ejpam-6904	382	6	vertex	vertex	NOUN
ejpam-6904	382	7	cover	cover	NOUN
ejpam-6904	382	8	of	of	ADP
ejpam-6904	382	9	km	km	NOUN
ejpam-6904	382	10	+	+	CCONJ
ejpam-6904	382	11	g	g	NOUN
ejpam-6904	382	12	,	,	PUNCT
ejpam-6904	382	13	it	it	PRON
ejpam-6904	382	14	follows	follow	VERB
ejpam-6904	382	15	that	that	SCONJ
ejpam-6904	382	16	v	v	X
ejpam-6904	382	17	(	(	PUNCT
ejpam-6904	382	18	g	g	NOUN
ejpam-6904	382	19	)	)	PUNCT
ejpam-6904	382	20	⊆	⊆	NUM
ejpam-6904	382	21	s	s	NOUN
ejpam-6904	382	22	and	and	CCONJ
ejpam-6904	382	23	|v	|v	PROPN
ejpam-6904	382	24	(	(	PUNCT
ejpam-6904	382	25	km	km	NOUN
ejpam-6904	382	26	)	)	PUNCT
ejpam-6904	382	27	\s|	\s|	NOUN
ejpam-6904	382	28	=	=	SYM
ejpam-6904	382	29	1	1	X
ejpam-6904	382	30	.	.	PUNCT
ejpam-6904	383	1	hence	hence	ADV
ejpam-6904	383	2	,	,	PUNCT
ejpam-6904	383	3	s	s	PART
ejpam-6904	383	4	=	=	PUNCT
ejpam-6904	383	5	(	(	PUNCT
ejpam-6904	383	6	v	v	NOUN
ejpam-6904	383	7	(	(	PUNCT
ejpam-6904	383	8	km	km	NOUN
ejpam-6904	383	9	)	)	PUNCT
ejpam-6904	383	10	\	\	NOUN
ejpam-6904	383	11	{	{	PUNCT
ejpam-6904	383	12	v})∪v	v})∪v	NOUN
ejpam-6904	383	13	(	(	PUNCT
ejpam-6904	383	14	g	g	NOUN
ejpam-6904	383	15	)	)	PUNCT
ejpam-6904	383	16	.	.	PUNCT
ejpam-6904	384	1	next	next	ADV
ejpam-6904	384	2	,	,	PUNCT
ejpam-6904	384	3	suppose	suppose	VERB
ejpam-6904	384	4	that	that	SCONJ
ejpam-6904	384	5	v	v	X
ejpam-6904	384	6	(	(	PUNCT
ejpam-6904	384	7	km	km	NOUN
ejpam-6904	384	8	)	)	PUNCT
ejpam-6904	384	9	\s	\s	NOUN
ejpam-6904	384	10	=	=	PUNCT
ejpam-6904	384	11	∅.	∅.	AUX
ejpam-6904	384	12	let	let	VERB
ejpam-6904	384	13	sg	sg	ADV
ejpam-6904	384	14	=	=	SYM
ejpam-6904	384	15	s	s	PART
ejpam-6904	384	16	∩	∩	ADJ
ejpam-6904	384	17	v	v	X
ejpam-6904	384	18	(	(	PUNCT
ejpam-6904	384	19	g	g	NOUN
ejpam-6904	384	20	)	)	PUNCT
ejpam-6904	384	21	.	.	PUNCT
ejpam-6904	385	1	then	then	ADV
ejpam-6904	385	2	s	s	VERB
ejpam-6904	385	3	=	=	SYM
ejpam-6904	385	4	v	v	PROPN
ejpam-6904	385	5	(	(	PUNCT
ejpam-6904	385	6	km	km	NOUN
ejpam-6904	385	7	)	)	PUNCT
ejpam-6904	385	8	∪	∪	ADP
ejpam-6904	385	9	sg	sg	PROPN
ejpam-6904	385	10	.	.	PUNCT
ejpam-6904	386	1	since	since	SCONJ
ejpam-6904	386	2	s	s	PROPN
ejpam-6904	386	3	is	be	AUX
ejpam-6904	386	4	a	a	DET
ejpam-6904	386	5	vertex	vertex	NOUN
ejpam-6904	386	6	cover	cover	NOUN
ejpam-6904	386	7	of	of	ADP
ejpam-6904	386	8	km	km	NOUN
ejpam-6904	386	9	+	+	CCONJ
ejpam-6904	386	10	g	g	NOUN
ejpam-6904	386	11	,	,	PUNCT
ejpam-6904	386	12	it	it	PRON
ejpam-6904	386	13	follows	follow	VERB
ejpam-6904	386	14	that	that	SCONJ
ejpam-6904	386	15	sg	sg	PROPN
ejpam-6904	386	16	is	be	AUX
ejpam-6904	386	17	vertex	vertex	NOUN
ejpam-6904	386	18	cover	cover	NOUN
ejpam-6904	386	19	of	of	ADP
ejpam-6904	386	20	g.	g.	PROPN
ejpam-6904	386	21	let	let	VERB
ejpam-6904	386	22	w	w	PROPN
ejpam-6904	386	23	∈	∈	PROPN
ejpam-6904	386	24	v	v	ADP
ejpam-6904	386	25	(	(	PUNCT
ejpam-6904	386	26	g	g	NOUN
ejpam-6904	386	27	)	)	PUNCT
ejpam-6904	386	28	\	\	PROPN
ejpam-6904	386	29	sg	sg	PROPN
ejpam-6904	386	30	.	.	PUNCT
ejpam-6904	387	1	since	since	SCONJ
ejpam-6904	387	2	s	s	PROPN
ejpam-6904	387	3	is	be	AUX
ejpam-6904	387	4	a	a	DET
ejpam-6904	387	5	2	2	NUM
ejpam-6904	387	6	-	-	PUNCT
ejpam-6904	387	7	path	path	NOUN
ejpam-6904	387	8	geodetic	geodetic	ADJ
ejpam-6904	387	9	set	set	NOUN
ejpam-6904	387	10	in	in	ADP
ejpam-6904	387	11	km+g	km+g	PROPN
ejpam-6904	387	12	,	,	PUNCT
ejpam-6904	387	13	there	there	PRON
ejpam-6904	387	14	exist	exist	VERB
ejpam-6904	387	15	y	y	PROPN
ejpam-6904	387	16	,	,	PUNCT
ejpam-6904	388	1	z	z	PROPN
ejpam-6904	388	2	∈	∈	PROPN
ejpam-6904	388	3	v	v	ADP
ejpam-6904	388	4	(	(	PUNCT
ejpam-6904	388	5	km+g	km+g	PROPN
ejpam-6904	388	6	)	)	PUNCT
ejpam-6904	388	7	such	such	ADJ
ejpam-6904	388	8	that	that	SCONJ
ejpam-6904	388	9	w	w	PROPN
ejpam-6904	388	10	∈	∈	PROPN
ejpam-6904	388	11	ikm+g(y	ikm+g(y	NOUN
ejpam-6904	388	12	,	,	PUNCT
ejpam-6904	388	13	z	z	NOUN
ejpam-6904	388	14	)	)	PUNCT
ejpam-6904	388	15	and	and	CCONJ
ejpam-6904	388	16	dkm+g(y	dkm+g(y	PROPN
ejpam-6904	388	17	,	,	PUNCT
ejpam-6904	388	18	z	z	NOUN
ejpam-6904	388	19	)	)	PUNCT
ejpam-6904	388	20	=	=	SYM
ejpam-6904	388	21	2	2	X
ejpam-6904	388	22	.	.	PUNCT
ejpam-6904	388	23	this	this	PRON
ejpam-6904	388	24	implies	imply	VERB
ejpam-6904	388	25	that	that	SCONJ
ejpam-6904	388	26	y	y	PROPN
ejpam-6904	388	27	,	,	PUNCT
ejpam-6904	388	28	z	z	PROPN
ejpam-6904	388	29	∈	∈	PROPN
ejpam-6904	388	30	v	v	NOUN
ejpam-6904	388	31	(	(	PUNCT
ejpam-6904	388	32	g	g	NOUN
ejpam-6904	388	33	)	)	PUNCT
ejpam-6904	388	34	,	,	PUNCT
ejpam-6904	388	35	w	w	PROPN
ejpam-6904	388	36	∈	∈	PROPN
ejpam-6904	388	37	ig(y	ig(y	NOUN
ejpam-6904	388	38	,	,	PUNCT
ejpam-6904	388	39	z	z	NOUN
ejpam-6904	388	40	)	)	PUNCT
ejpam-6904	388	41	and	and	CCONJ
ejpam-6904	388	42	dg(y	dg(y	ADJ
ejpam-6904	388	43	,	,	PUNCT
ejpam-6904	388	44	z	z	NOUN
ejpam-6904	388	45	)	)	PUNCT
ejpam-6904	388	46	=	=	SYM
ejpam-6904	388	47	2	2	X
ejpam-6904	388	48	.	.	X
ejpam-6904	388	49	hence	hence	ADV
ejpam-6904	388	50	,	,	PUNCT
ejpam-6904	388	51	sg	sg	PROPN
ejpam-6904	388	52	is	be	AUX
ejpam-6904	388	53	a	a	DET
ejpam-6904	388	54	2	2	NUM
ejpam-6904	388	55	-	-	PUNCT
ejpam-6904	388	56	path	path	NOUN
ejpam-6904	388	57	geodetic	geodetic	ADJ
ejpam-6904	388	58	vertex	vertex	NOUN
ejpam-6904	388	59	cover	cover	NOUN
ejpam-6904	388	60	of	of	ADP
ejpam-6904	388	61	g.	g.	PROPN
ejpam-6904	388	62	for	for	ADP
ejpam-6904	388	63	the	the	DET
ejpam-6904	388	64	converse	converse	NOUN
ejpam-6904	388	65	,	,	PUNCT
ejpam-6904	388	66	suppose	suppose	VERB
ejpam-6904	388	67	first	first	ADV
ejpam-6904	388	68	s	s	PART
ejpam-6904	388	69	=	=	PUNCT
ejpam-6904	388	70	(	(	PUNCT
ejpam-6904	388	71	v	v	NOUN
ejpam-6904	388	72	(	(	PUNCT
ejpam-6904	388	73	km	km	NOUN
ejpam-6904	388	74	)	)	PUNCT
ejpam-6904	388	75	\	\	NOUN
ejpam-6904	389	1	{	{	PUNCT
ejpam-6904	389	2	v})∪	v})∪	PROPN
ejpam-6904	389	3	v	v	PROPN
ejpam-6904	389	4	(	(	PUNCT
ejpam-6904	389	5	g	g	NOUN
ejpam-6904	389	6	)	)	PUNCT
ejpam-6904	389	7	for	for	ADP
ejpam-6904	389	8	some	some	DET
ejpam-6904	389	9	v	v	ADP
ejpam-6904	389	10	∈	∈	PROPN
ejpam-6904	389	11	v	v	NOUN
ejpam-6904	389	12	(	(	PUNCT
ejpam-6904	389	13	km	km	PROPN
ejpam-6904	389	14	)	)	PUNCT
ejpam-6904	389	15	.	.	PUNCT
ejpam-6904	390	1	then	then	ADV
ejpam-6904	390	2	clearly	clearly	ADV
ejpam-6904	390	3	,	,	PUNCT
ejpam-6904	390	4	s	s	VERB
ejpam-6904	390	5	is	be	AUX
ejpam-6904	390	6	a	a	DET
ejpam-6904	390	7	2	2	NUM
ejpam-6904	390	8	-	-	PUNCT
ejpam-6904	390	9	path	path	NOUN
ejpam-6904	390	10	geodetic	geodetic	ADJ
ejpam-6904	390	11	vertex	vertex	NOUN
ejpam-6904	390	12	cover	cover	NOUN
ejpam-6904	390	13	of	of	ADP
ejpam-6904	390	14	km+g	km+g	PROPN
ejpam-6904	390	15	.	.	PUNCT
ejpam-6904	391	1	next	next	ADV
ejpam-6904	391	2	,	,	PUNCT
ejpam-6904	391	3	suppose	suppose	VERB
ejpam-6904	391	4	that	that	SCONJ
ejpam-6904	391	5	s	s	VERB
ejpam-6904	391	6	=	=	SYM
ejpam-6904	391	7	v	v	PROPN
ejpam-6904	391	8	(	(	PUNCT
ejpam-6904	391	9	km)∪sg	km)∪sg	PROPN
ejpam-6904	391	10	,	,	PUNCT
ejpam-6904	391	11	where	where	SCONJ
ejpam-6904	391	12	sg	sg	PROPN
ejpam-6904	391	13	is	be	AUX
ejpam-6904	391	14	a	a	DET
ejpam-6904	391	15	2	2	NUM
ejpam-6904	391	16	-	-	PUNCT
ejpam-6904	391	17	path	path	NOUN
ejpam-6904	391	18	geodetic	geodetic	ADJ
ejpam-6904	391	19	vertex	vertex	NOUN
ejpam-6904	391	20	cover	cover	NOUN
ejpam-6904	391	21	of	of	ADP
ejpam-6904	391	22	g.	g.	PROPN
ejpam-6904	391	23	let	let	VERB
ejpam-6904	391	24	ab	ab	PROPN
ejpam-6904	391	25	∈	∈	PROPN
ejpam-6904	391	26	e(km+g	e(km+g	PROPN
ejpam-6904	391	27	)	)	PUNCT
ejpam-6904	391	28	.	.	PUNCT
ejpam-6904	392	1	if	if	SCONJ
ejpam-6904	392	2	ab	ab	PROPN
ejpam-6904	392	3	/∈	/∈	PUNCT
ejpam-6904	392	4	e(g	e(g	PROPN
ejpam-6904	392	5	)	)	PUNCT
ejpam-6904	392	6	,	,	PUNCT
ejpam-6904	392	7	then	then	ADV
ejpam-6904	392	8	a	a	DET
ejpam-6904	392	9	∈	∈	PROPN
ejpam-6904	392	10	v	v	NOUN
ejpam-6904	392	11	(	(	PUNCT
ejpam-6904	392	12	km	km	NOUN
ejpam-6904	392	13	)	)	PUNCT
ejpam-6904	392	14	or	or	CCONJ
ejpam-6904	392	15	b	b	X
ejpam-6904	392	16	∈	∈	PROPN
ejpam-6904	392	17	v	v	NOUN
ejpam-6904	392	18	(	(	PUNCT
ejpam-6904	392	19	km	km	PROPN
ejpam-6904	392	20	)	)	PUNCT
ejpam-6904	392	21	.	.	PUNCT
ejpam-6904	393	1	suppose	suppose	VERB
ejpam-6904	393	2	ab	ab	PROPN
ejpam-6904	393	3	∈	∈	PROPN
ejpam-6904	393	4	e(g	e(g	PROPN
ejpam-6904	393	5	)	)	PUNCT
ejpam-6904	393	6	.	.	PUNCT
ejpam-6904	394	1	since	since	SCONJ
ejpam-6904	394	2	sg	sg	PROPN
ejpam-6904	394	3	is	be	AUX
ejpam-6904	394	4	a	a	DET
ejpam-6904	394	5	vertex	vertex	NOUN
ejpam-6904	394	6	cover	cover	NOUN
ejpam-6904	394	7	of	of	ADP
ejpam-6904	394	8	g	g	NOUN
ejpam-6904	394	9	,	,	PUNCT
ejpam-6904	394	10	a	a	DET
ejpam-6904	394	11	∈	∈	NOUN
ejpam-6904	394	12	sg	sg	ADP
ejpam-6904	394	13	or	or	CCONJ
ejpam-6904	394	14	b	b	X
ejpam-6904	394	15	∈	∈	PROPN
ejpam-6904	394	16	sg	sg	PROPN
ejpam-6904	394	17	.	.	PUNCT
ejpam-6904	395	1	hence	hence	ADV
ejpam-6904	395	2	,	,	PUNCT
ejpam-6904	395	3	in	in	ADP
ejpam-6904	395	4	both	both	DET
ejpam-6904	395	5	cases	case	NOUN
ejpam-6904	395	6	,	,	PUNCT
ejpam-6904	395	7	a	a	DET
ejpam-6904	395	8	∈	∈	NOUN
ejpam-6904	395	9	s	s	NOUN
ejpam-6904	395	10	or	or	CCONJ
ejpam-6904	395	11	b	b	PROPN
ejpam-6904	395	12	∈	∈	PROPN
ejpam-6904	395	13	s.	s.	PROPN
ejpam-6904	395	14	let	let	VERB
ejpam-6904	395	15	w	w	PROPN
ejpam-6904	395	16	∈	∈	PROPN
ejpam-6904	395	17	v	v	NOUN
ejpam-6904	395	18	(	(	PUNCT
ejpam-6904	395	19	km	km	NOUN
ejpam-6904	395	20	+	+	CCONJ
ejpam-6904	395	21	g	g	NOUN
ejpam-6904	395	22	)	)	PUNCT
ejpam-6904	395	23	\	\	PUNCT
ejpam-6904	396	1	s.	s.	PROPN
ejpam-6904	396	2	then	then	ADV
ejpam-6904	396	3	w	w	PROPN
ejpam-6904	396	4	∈	∈	PROPN
ejpam-6904	396	5	v	v	ADP
ejpam-6904	396	6	(	(	PUNCT
ejpam-6904	396	7	g	g	NOUN
ejpam-6904	396	8	)	)	PUNCT
ejpam-6904	396	9	\	\	PROPN
ejpam-6904	396	10	sg	sg	PROPN
ejpam-6904	396	11	.	.	PUNCT
ejpam-6904	397	1	this	this	PRON
ejpam-6904	397	2	implies	imply	VERB
ejpam-6904	397	3	that	that	SCONJ
ejpam-6904	397	4	there	there	PRON
ejpam-6904	397	5	exist	exist	VERB
ejpam-6904	397	6	p	p	PRON
ejpam-6904	397	7	,	,	PUNCT
ejpam-6904	397	8	q	q	PROPN
ejpam-6904	397	9	∈	∈	PROPN
ejpam-6904	397	10	sg	sg	ADP
ejpam-6904	397	11	such	such	ADJ
ejpam-6904	397	12	that	that	DET
ejpam-6904	397	13	dkm+g(p	dkm+g(p	NOUN
ejpam-6904	397	14	,	,	PUNCT
ejpam-6904	397	15	q	q	X
ejpam-6904	397	16	)	)	PUNCT
ejpam-6904	397	17	=	=	NOUN
ejpam-6904	397	18	dg(p	dg(p	X
ejpam-6904	397	19	,	,	PUNCT
ejpam-6904	397	20	q	q	NOUN
ejpam-6904	397	21	)	)	PUNCT
ejpam-6904	397	22	=	=	SYM
ejpam-6904	397	23	2	2	NUM
ejpam-6904	397	24	and	and	CCONJ
ejpam-6904	397	25	w	w	NOUN
ejpam-6904	397	26	∈	∈	PROPN
ejpam-6904	397	27	ig(p	ig(p	NOUN
ejpam-6904	397	28	,	,	PUNCT
ejpam-6904	397	29	q	q	NOUN
ejpam-6904	397	30	)	)	PUNCT
ejpam-6904	397	31	.	.	PUNCT
ejpam-6904	398	1	since	since	SCONJ
ejpam-6904	398	2	ig(p	ig(p	NOUN
ejpam-6904	398	3	,	,	PUNCT
ejpam-6904	398	4	q	q	X
ejpam-6904	398	5	)	)	PUNCT
ejpam-6904	398	6	=	=	SYM
ejpam-6904	398	7	ikm+g(p	ikm+g(p	PROPN
ejpam-6904	398	8	,	,	PUNCT
ejpam-6904	398	9	q	q	NOUN
ejpam-6904	398	10	)	)	PUNCT
ejpam-6904	398	11	,	,	PUNCT
ejpam-6904	398	12	it	it	PRON
ejpam-6904	398	13	follows	follow	VERB
ejpam-6904	398	14	that	that	SCONJ
ejpam-6904	398	15	s	s	VERB
ejpam-6904	398	16	is	be	AUX
ejpam-6904	398	17	a	a	DET
ejpam-6904	398	18	2	2	NUM
ejpam-6904	398	19	-	-	PUNCT
ejpam-6904	398	20	path	path	NOUN
ejpam-6904	398	21	geodetic	geodetic	ADJ
ejpam-6904	398	22	vertex	vertex	NOUN
ejpam-6904	398	23	cover	cover	NOUN
ejpam-6904	398	24	of	of	ADP
ejpam-6904	398	25	km	km	PROPN
ejpam-6904	399	1	+	+	PROPN
ejpam-6904	399	2	g.	g.	NOUN
ejpam-6904	399	3	the	the	DET
ejpam-6904	399	4	next	next	ADJ
ejpam-6904	399	5	results	result	NOUN
ejpam-6904	399	6	are	be	AUX
ejpam-6904	399	7	consequence	consequence	NOUN
ejpam-6904	399	8	of	of	ADP
ejpam-6904	399	9	theorem	theorem	ADJ
ejpam-6904	399	10	11	11	NUM
ejpam-6904	399	11	.	.	PUNCT
ejpam-6904	400	1	corollary	corollary	ADJ
ejpam-6904	400	2	5	5	NUM
ejpam-6904	400	3	.	.	PUNCT
ejpam-6904	401	1	let	let	VERB
ejpam-6904	401	2	g	g	PRON
ejpam-6904	401	3	be	be	AUX
ejpam-6904	401	4	a	a	DET
ejpam-6904	401	5	non	non	ADJ
ejpam-6904	401	6	-	-	ADJ
ejpam-6904	401	7	complete	complete	ADJ
ejpam-6904	401	8	graph	graph	NOUN
ejpam-6904	401	9	and	and	CCONJ
ejpam-6904	401	10	let	let	VERB
ejpam-6904	401	11	m	m	PRON
ejpam-6904	401	12	be	be	AUX
ejpam-6904	401	13	a	a	DET
ejpam-6904	401	14	positive	positive	ADJ
ejpam-6904	401	15	integer	integer	NOUN
ejpam-6904	401	16	.	.	PUNCT
ejpam-6904	402	1	then	then	ADV
ejpam-6904	402	2	β2pg(km	β2pg(km	PROPN
ejpam-6904	402	3	+	+	PROPN
ejpam-6904	402	4	g	g	NOUN
ejpam-6904	402	5	)	)	PUNCT
ejpam-6904	402	6	=	=	SYM
ejpam-6904	403	1	min{(m−	min{(m−	PROPN
ejpam-6904	403	2	1	1	NUM
ejpam-6904	403	3	)	)	PUNCT
ejpam-6904	403	4	+	+	CCONJ
ejpam-6904	403	5	|v	|v	X
ejpam-6904	403	6	(	(	PUNCT
ejpam-6904	403	7	g)|	g)|	NOUN
ejpam-6904	403	8	,	,	PUNCT
ejpam-6904	403	9	β2pg(g	β2pg(g	PUNCT
ejpam-6904	403	10	)	)	PUNCT
ejpam-6904	403	11	+	+	NOUN
ejpam-6904	403	12	m	m	NOUN
ejpam-6904	403	13	}	}	PUNCT
ejpam-6904	403	14	.	.	PUNCT
ejpam-6904	404	1	corollary	corollary	ADJ
ejpam-6904	404	2	6	6	NUM
ejpam-6904	404	3	.	.	PUNCT
ejpam-6904	405	1	let	let	VERB
ejpam-6904	405	2	g	g	PRON
ejpam-6904	405	3	be	be	AUX
ejpam-6904	405	4	a	a	DET
ejpam-6904	405	5	non	non	ADJ
ejpam-6904	405	6	-	-	ADJ
ejpam-6904	405	7	complete	complete	ADJ
ejpam-6904	405	8	graph	graph	NOUN
ejpam-6904	405	9	.	.	PUNCT
ejpam-6904	406	1	then	then	ADV
ejpam-6904	406	2	β2pg(k1	β2pg(k1	PUNCT
ejpam-6904	407	1	+	+	NOUN
ejpam-6904	407	2	g	g	NOUN
ejpam-6904	407	3	)	)	PUNCT
ejpam-6904	407	4	=	=	SYM
ejpam-6904	408	1	min{|v	min{|v	PROPN
ejpam-6904	408	2	(	(	PUNCT
ejpam-6904	408	3	g)|	g)|	PROPN
ejpam-6904	408	4	,	,	PUNCT
ejpam-6904	408	5	β2pg(g	β2pg(g	PUNCT
ejpam-6904	408	6	)	)	PUNCT
ejpam-6904	408	7	+	+	CCONJ
ejpam-6904	408	8	1	1	NUM
ejpam-6904	408	9	}	}	PUNCT
ejpam-6904	408	10	.	.	PUNCT
ejpam-6904	409	1	moreover	moreover	ADV
ejpam-6904	409	2	,	,	PUNCT
ejpam-6904	409	3	(	(	PUNCT
ejpam-6904	409	4	i	i	NOUN
ejpam-6904	409	5	)	)	PUNCT
ejpam-6904	409	6	β2pg(k1,n	β2pg(k1,n	NOUN
ejpam-6904	409	7	)	)	PUNCT
ejpam-6904	409	8	=	=	SYM
ejpam-6904	409	9	β2pg(k1	β2pg(k1	PUNCT
ejpam-6904	410	1	+	+	PROPN
ejpam-6904	410	2	kn	kn	NOUN
ejpam-6904	410	3	)	)	PUNCT
ejpam-6904	410	4	=	=	SYM
ejpam-6904	410	5	n	n	PROPN
ejpam-6904	410	6	for	for	ADP
ejpam-6904	410	7	all	all	DET
ejpam-6904	410	8	n	n	PRON
ejpam-6904	410	9	≥	≥	NOUN
ejpam-6904	410	10	2	2	NUM
ejpam-6904	410	11	;	;	PUNCT
ejpam-6904	410	12	(	(	PUNCT
ejpam-6904	410	13	ii	ii	NOUN
ejpam-6904	410	14	)	)	PUNCT
ejpam-6904	410	15	β2pg(fn	β2pg(fn	NOUN
ejpam-6904	410	16	)	)	PUNCT
ejpam-6904	410	17	=	=	SYM
ejpam-6904	410	18	β2pg(k1	β2pg(k1	PUNCT
ejpam-6904	411	1	+	+	NUM
ejpam-6904	411	2	pn	pn	NOUN
ejpam-6904	411	3	)	)	PUNCT
ejpam-6904	411	4	=	=	VERB
ejpam-6904	412	1	⌈n+1	⌈n+1	ADJ
ejpam-6904	412	2	2	2	NUM
ejpam-6904	412	3	⌉+	⌉+	SYM
ejpam-6904	412	4	1	1	NUM
ejpam-6904	412	5	for	for	ADP
ejpam-6904	412	6	all	all	DET
ejpam-6904	412	7	n	n	PRON
ejpam-6904	412	8	≥	≥	NOUN
ejpam-6904	412	9	3	3	NUM
ejpam-6904	412	10	;	;	PUNCT
ejpam-6904	412	11	(	(	PUNCT
ejpam-6904	412	12	iii	iii	NOUN
ejpam-6904	412	13	)	)	PUNCT
ejpam-6904	412	14	β2pg(wn	β2pg(wn	PUNCT
ejpam-6904	412	15	)	)	PUNCT
ejpam-6904	412	16	=	=	SYM
ejpam-6904	412	17	β2pg(k1	β2pg(k1	PUNCT
ejpam-6904	413	1	+	+	CCONJ
ejpam-6904	413	2	cn	cn	ADJ
ejpam-6904	413	3	)	)	PUNCT
ejpam-6904	413	4	=	=	NOUN
ejpam-6904	413	5	⌈n2	⌈n2	NOUN
ejpam-6904	413	6	⌉+	⌉+	NOUN
ejpam-6904	413	7	1	1	NUM
ejpam-6904	413	8	for	for	ADP
ejpam-6904	413	9	all	all	DET
ejpam-6904	413	10	n	n	PRON
ejpam-6904	413	11	≥	≥	NOUN
ejpam-6904	413	12	4	4	NUM
ejpam-6904	413	13	;	;	PUNCT
ejpam-6904	413	14	and	and	CCONJ
ejpam-6904	413	15	(	(	PUNCT
ejpam-6904	413	16	iv	iv	X
ejpam-6904	413	17	)	)	PUNCT
ejpam-6904	413	18	β2pg(k1	β2pg(k1	PUNCT
ejpam-6904	414	1	+	+	CCONJ
ejpam-6904	414	2	k∪	k∪	PROPN
ejpam-6904	414	3	j=1	j=1	ADJ
ejpam-6904	414	4	kmj	kmj	NOUN
ejpam-6904	414	5	)	)	PUNCT
ejpam-6904	415	1	=	=	PUNCT
ejpam-6904	415	2	k∑	k∑	PROPN
ejpam-6904	416	1	j=1	j=1	PROPN
ejpam-6904	416	2	mj	mj	PROPN
ejpam-6904	416	3	for	for	ADP
ejpam-6904	416	4	k	k	PROPN
ejpam-6904	416	5	≥	≥	PROPN
ejpam-6904	416	6	2	2	NUM
ejpam-6904	416	7	.	.	PUNCT
ejpam-6904	416	8	theorem	theorem	NOUN
ejpam-6904	416	9	12	12	NUM
ejpam-6904	416	10	.	.	PUNCT
ejpam-6904	417	1	let	let	VERB
ejpam-6904	417	2	g	g	NOUN
ejpam-6904	417	3	and	and	CCONJ
ejpam-6904	417	4	h	h	NOUN
ejpam-6904	417	5	be	be	AUX
ejpam-6904	417	6	non	non	ADJ
ejpam-6904	417	7	-	-	ADJ
ejpam-6904	417	8	complete	complete	ADJ
ejpam-6904	417	9	graphs	graph	NOUN
ejpam-6904	417	10	.	.	PUNCT
ejpam-6904	418	1	then	then	ADV
ejpam-6904	418	2	s	s	VERB
ejpam-6904	418	3	⊆	⊆	NUM
ejpam-6904	418	4	v	v	NOUN
ejpam-6904	418	5	(	(	PUNCT
ejpam-6904	418	6	g	g	PROPN
ejpam-6904	418	7	+	+	NOUN
ejpam-6904	418	8	h	h	NOUN
ejpam-6904	418	9	)	)	PUNCT
ejpam-6904	418	10	is	be	AUX
ejpam-6904	418	11	a	a	DET
ejpam-6904	418	12	2	2	NUM
ejpam-6904	418	13	-	-	PUNCT
ejpam-6904	418	14	path	path	NOUN
ejpam-6904	418	15	geodetic	geodetic	ADJ
ejpam-6904	418	16	vertex	vertex	NOUN
ejpam-6904	418	17	cover	cover	NOUN
ejpam-6904	418	18	of	of	ADP
ejpam-6904	418	19	g+h	g+h	PROPN
ejpam-6904	419	1	if	if	SCONJ
ejpam-6904	419	2	and	and	CCONJ
ejpam-6904	419	3	only	only	ADV
ejpam-6904	419	4	if	if	SCONJ
ejpam-6904	419	5	one	one	NUM
ejpam-6904	419	6	of	of	ADP
ejpam-6904	419	7	the	the	DET
ejpam-6904	419	8	following	following	ADJ
ejpam-6904	419	9	statements	statement	NOUN
ejpam-6904	419	10	holds	hold	VERB
ejpam-6904	419	11	:	:	PUNCT
ejpam-6904	419	12	(	(	PUNCT
ejpam-6904	419	13	i	i	NOUN
ejpam-6904	419	14	)	)	PUNCT
ejpam-6904	419	15	s	s	PART
ejpam-6904	419	16	=	=	SYM
ejpam-6904	419	17	v	v	X
ejpam-6904	419	18	(	(	PUNCT
ejpam-6904	419	19	g	g	NOUN
ejpam-6904	419	20	)	)	PUNCT
ejpam-6904	419	21	∪	∪	NOUN
ejpam-6904	419	22	sh	sh	PRON
ejpam-6904	419	23	where	where	SCONJ
ejpam-6904	419	24	sh	sh	PROPN
ejpam-6904	419	25	is	be	AUX
ejpam-6904	419	26	a	a	DET
ejpam-6904	419	27	vertex	vertex	NOUN
ejpam-6904	419	28	cover	cover	NOUN
ejpam-6904	419	29	of	of	ADP
ejpam-6904	419	30	h.	h.	PROPN
ejpam-6904	419	31	(	(	PUNCT
ejpam-6904	419	32	ii	ii	PROPN
ejpam-6904	419	33	)	)	PUNCT
ejpam-6904	419	34	s	s	PART
ejpam-6904	419	35	=	=	PUNCT
ejpam-6904	419	36	sg	sg	X
ejpam-6904	419	37	∪	∪	ADJ
ejpam-6904	419	38	v	v	PROPN
ejpam-6904	419	39	(	(	PUNCT
ejpam-6904	419	40	h	h	NOUN
ejpam-6904	419	41	)	)	PUNCT
ejpam-6904	419	42	where	where	SCONJ
ejpam-6904	419	43	sg	sg	PROPN
ejpam-6904	419	44	is	be	AUX
ejpam-6904	419	45	a	a	DET
ejpam-6904	419	46	vertex	vertex	NOUN
ejpam-6904	419	47	cover	cover	NOUN
ejpam-6904	419	48	of	of	ADP
ejpam-6904	419	49	g.	g.	PROPN
ejpam-6904	419	50	a.	a.	PROPN
ejpam-6904	419	51	b.	b.	PROPN
ejpam-6904	419	52	tapeing	tapeing	PROPN
ejpam-6904	419	53	,	,	PUNCT
ejpam-6904	419	54	s.	s.	PROPN
ejpam-6904	419	55	r.	r.	PROPN
ejpam-6904	419	56	canoy	canoy	PROPN
ejpam-6904	419	57	/	/	SYM
ejpam-6904	419	58	eur	eur	PROPN
ejpam-6904	419	59	.	.	PUNCT
ejpam-6904	420	1	j.	j.	PROPN
ejpam-6904	420	2	pure	pure	PROPN
ejpam-6904	420	3	appl	appl	PROPN
ejpam-6904	420	4	.	.	PROPN
ejpam-6904	420	5	math	math	PROPN
ejpam-6904	420	6	,	,	PUNCT
ejpam-6904	420	7	18	18	NUM
ejpam-6904	420	8	(	(	PUNCT
ejpam-6904	420	9	4	4	NUM
ejpam-6904	420	10	)	)	PUNCT
ejpam-6904	420	11	(	(	PUNCT
ejpam-6904	420	12	2025	2025	NUM
ejpam-6904	420	13	)	)	PUNCT
ejpam-6904	420	14	,	,	PUNCT
ejpam-6904	420	15	6904	6904	NUM
ejpam-6904	420	16	12	12	NUM
ejpam-6904	420	17	of	of	ADP
ejpam-6904	420	18	14	14	NUM
ejpam-6904	420	19	proof	proof	NOUN
ejpam-6904	420	20	.	.	PUNCT
ejpam-6904	421	1	suppose	suppose	VERB
ejpam-6904	421	2	s	s	NOUN
ejpam-6904	421	3	is	be	AUX
ejpam-6904	421	4	a	a	DET
ejpam-6904	421	5	2	2	NUM
ejpam-6904	421	6	-	-	PUNCT
ejpam-6904	421	7	path	path	NOUN
ejpam-6904	421	8	geodetic	geodetic	ADJ
ejpam-6904	421	9	vertex	vertex	NOUN
ejpam-6904	421	10	cover	cover	NOUN
ejpam-6904	421	11	of	of	ADP
ejpam-6904	421	12	g	g	PROPN
ejpam-6904	421	13	+	+	PROPN
ejpam-6904	421	14	h.	h.	PROPN
ejpam-6904	421	15	suppose	suppose	VERB
ejpam-6904	421	16	v	v	X
ejpam-6904	421	17	(	(	PUNCT
ejpam-6904	421	18	g	g	NOUN
ejpam-6904	421	19	)	)	PUNCT
ejpam-6904	421	20	\	\	PUNCT
ejpam-6904	422	1	s	s	PART
ejpam-6904	422	2	̸=	̸=	PROPN
ejpam-6904	422	3	∅	∅	NOUN
ejpam-6904	422	4	and	and	CCONJ
ejpam-6904	422	5	v	v	NOUN
ejpam-6904	422	6	(	(	PUNCT
ejpam-6904	422	7	h	h	NOUN
ejpam-6904	422	8	)	)	PUNCT
ejpam-6904	422	9	\	\	PUNCT
ejpam-6904	423	1	s	s	AUX
ejpam-6904	423	2	̸=	̸=	PROPN
ejpam-6904	423	3	∅.	∅.	ADV
ejpam-6904	423	4	let	let	VERB
ejpam-6904	423	5	x	x	SYM
ejpam-6904	423	6	∈	∈	PROPN
ejpam-6904	423	7	v	v	X
ejpam-6904	423	8	(	(	PUNCT
ejpam-6904	423	9	g	g	NOUN
ejpam-6904	423	10	)	)	PUNCT
ejpam-6904	423	11	\	\	PROPN
ejpam-6904	423	12	s	s	PART
ejpam-6904	423	13	and	and	CCONJ
ejpam-6904	423	14	y	y	PROPN
ejpam-6904	423	15	∈	∈	PROPN
ejpam-6904	423	16	v	v	ADP
ejpam-6904	423	17	(	(	PUNCT
ejpam-6904	423	18	h	h	NOUN
ejpam-6904	423	19	)	)	PUNCT
ejpam-6904	423	20	\	\	PUNCT
ejpam-6904	424	1	s.	s.	PROPN
ejpam-6904	424	2	then	then	ADV
ejpam-6904	424	3	xy	xy	PROPN
ejpam-6904	424	4	∈	∈	PROPN
ejpam-6904	424	5	e(g	e(g	PROPN
ejpam-6904	424	6	+	+	CCONJ
ejpam-6904	424	7	h	h	NOUN
ejpam-6904	424	8	)	)	PUNCT
ejpam-6904	424	9	and	and	CCONJ
ejpam-6904	424	10	x	x	X
ejpam-6904	424	11	,	,	PUNCT
ejpam-6904	424	12	y	y	PROPN
ejpam-6904	424	13	/∈	/∈	PUNCT
ejpam-6904	424	14	s.	s.	PROPN
ejpam-6904	425	1	this	this	PRON
ejpam-6904	425	2	is	be	AUX
ejpam-6904	425	3	not	not	PART
ejpam-6904	425	4	possible	possible	ADJ
ejpam-6904	425	5	because	because	SCONJ
ejpam-6904	425	6	s	s	NOUN
ejpam-6904	425	7	is	be	AUX
ejpam-6904	425	8	a	a	DET
ejpam-6904	425	9	vertex	vertex	NOUN
ejpam-6904	425	10	cover	cover	NOUN
ejpam-6904	425	11	of	of	ADP
ejpam-6904	425	12	g+h	g+h	PROPN
ejpam-6904	425	13	.	.	PUNCT
ejpam-6904	426	1	hence	hence	ADV
ejpam-6904	426	2	,	,	PUNCT
ejpam-6904	426	3	v	v	X
ejpam-6904	426	4	(	(	PUNCT
ejpam-6904	426	5	g	g	NOUN
ejpam-6904	426	6	)	)	PUNCT
ejpam-6904	426	7	\	\	PROPN
ejpam-6904	426	8	s	s	PART
ejpam-6904	426	9	=	=	NOUN
ejpam-6904	426	10	∅	∅	NOUN
ejpam-6904	426	11	or	or	CCONJ
ejpam-6904	426	12	v	v	NOUN
ejpam-6904	426	13	(	(	PUNCT
ejpam-6904	426	14	h	h	NOUN
ejpam-6904	426	15	)	)	PUNCT
ejpam-6904	426	16	\	\	PROPN
ejpam-6904	427	1	s	s	PART
ejpam-6904	427	2	=	=	PUNCT
ejpam-6904	427	3	∅.	∅.	NOUN
ejpam-6904	427	4	suppose	suppose	VERB
ejpam-6904	427	5	v	v	ADP
ejpam-6904	427	6	(	(	PUNCT
ejpam-6904	427	7	g	g	NOUN
ejpam-6904	427	8	)	)	PUNCT
ejpam-6904	427	9	\	\	PART
ejpam-6904	427	10	s	s	PART
ejpam-6904	427	11	=	=	PUNCT
ejpam-6904	427	12	∅.	∅.	NOUN
ejpam-6904	427	13	then	then	ADV
ejpam-6904	427	14	v	v	NOUN
ejpam-6904	427	15	(	(	PUNCT
ejpam-6904	427	16	g	g	NOUN
ejpam-6904	427	17	)	)	PUNCT
ejpam-6904	427	18	⊆	⊆	NUM
ejpam-6904	427	19	s.	s.	PROPN
ejpam-6904	427	20	let	let	VERB
ejpam-6904	427	21	sh	sh	NOUN
ejpam-6904	427	22	=	=	SYM
ejpam-6904	427	23	s	s	PROPN
ejpam-6904	427	24	∩	∩	ADJ
ejpam-6904	427	25	v	v	ADJ
ejpam-6904	427	26	(	(	PUNCT
ejpam-6904	427	27	h	h	NOUN
ejpam-6904	427	28	)	)	PUNCT
ejpam-6904	427	29	and	and	CCONJ
ejpam-6904	427	30	let	let	VERB
ejpam-6904	427	31	ab	ab	PROPN
ejpam-6904	427	32	∈	∈	PROPN
ejpam-6904	427	33	e(h	e(h	PROPN
ejpam-6904	427	34	)	)	PUNCT
ejpam-6904	427	35	.	.	PUNCT
ejpam-6904	428	1	since	since	SCONJ
ejpam-6904	428	2	s	s	PROPN
ejpam-6904	428	3	is	be	AUX
ejpam-6904	428	4	a	a	DET
ejpam-6904	428	5	vertex	vertex	NOUN
ejpam-6904	428	6	cover	cover	NOUN
ejpam-6904	428	7	of	of	ADP
ejpam-6904	428	8	g	g	PROPN
ejpam-6904	428	9	+	+	NOUN
ejpam-6904	428	10	h	h	NOUN
ejpam-6904	428	11	,	,	PUNCT
ejpam-6904	428	12	it	it	PRON
ejpam-6904	428	13	follows	follow	VERB
ejpam-6904	428	14	that	that	SCONJ
ejpam-6904	428	15	a	a	DET
ejpam-6904	428	16	∈	∈	NOUN
ejpam-6904	428	17	sh	sh	NOUN
ejpam-6904	428	18	or	or	CCONJ
ejpam-6904	428	19	b	b	X
ejpam-6904	428	20	∈	∈	NOUN
ejpam-6904	428	21	sh	sh	INTJ
ejpam-6904	428	22	.	.	PUNCT
ejpam-6904	429	1	this	this	PRON
ejpam-6904	429	2	implies	imply	VERB
ejpam-6904	429	3	that	that	SCONJ
ejpam-6904	429	4	sh	sh	PROPN
ejpam-6904	429	5	is	be	AUX
ejpam-6904	429	6	a	a	DET
ejpam-6904	429	7	vertex	vertex	NOUN
ejpam-6904	429	8	cover	cover	NOUN
ejpam-6904	429	9	of	of	ADP
ejpam-6904	429	10	h	h	NOUN
ejpam-6904	429	11	,	,	PUNCT
ejpam-6904	429	12	showing	show	VERB
ejpam-6904	429	13	that	that	SCONJ
ejpam-6904	429	14	(	(	PUNCT
ejpam-6904	429	15	i	i	NOUN
ejpam-6904	429	16	)	)	PUNCT
ejpam-6904	429	17	holds	hold	VERB
ejpam-6904	429	18	.	.	PUNCT
ejpam-6904	430	1	similarly	similarly	ADV
ejpam-6904	430	2	,	,	PUNCT
ejpam-6904	430	3	(	(	PUNCT
ejpam-6904	430	4	ii	ii	NOUN
ejpam-6904	430	5	)	)	PUNCT
ejpam-6904	430	6	holds	hold	VERB
ejpam-6904	430	7	if	if	SCONJ
ejpam-6904	430	8	v	v	NOUN
ejpam-6904	430	9	(	(	PUNCT
ejpam-6904	430	10	h	h	NOUN
ejpam-6904	430	11	)	)	PUNCT
ejpam-6904	430	12	\	\	PROPN
ejpam-6904	430	13	s	s	PART
ejpam-6904	430	14	=	=	PUNCT
ejpam-6904	430	15	∅.	∅.	NOUN
ejpam-6904	430	16	for	for	ADP
ejpam-6904	430	17	the	the	DET
ejpam-6904	430	18	converse	converse	NOUN
ejpam-6904	430	19	,	,	PUNCT
ejpam-6904	430	20	suppose	suppose	VERB
ejpam-6904	430	21	that	that	SCONJ
ejpam-6904	430	22	(	(	PUNCT
ejpam-6904	430	23	i	i	NOUN
ejpam-6904	430	24	)	)	PUNCT
ejpam-6904	430	25	holds	hold	VERB
ejpam-6904	430	26	.	.	PUNCT
ejpam-6904	431	1	let	let	VERB
ejpam-6904	431	2	st	st	PROPN
ejpam-6904	431	3	∈	∈	PROPN
ejpam-6904	431	4	e(g+h	e(g+h	PROPN
ejpam-6904	431	5	)	)	PUNCT
ejpam-6904	431	6	.	.	PUNCT
ejpam-6904	432	1	if	if	SCONJ
ejpam-6904	432	2	s	s	X
ejpam-6904	432	3	∈	∈	PROPN
ejpam-6904	432	4	v	v	ADP
ejpam-6904	432	5	(	(	PUNCT
ejpam-6904	432	6	g	g	NOUN
ejpam-6904	432	7	)	)	PUNCT
ejpam-6904	432	8	or	or	CCONJ
ejpam-6904	432	9	t	t	PROPN
ejpam-6904	432	10	∈	∈	PROPN
ejpam-6904	432	11	v	v	ADP
ejpam-6904	432	12	(	(	PUNCT
ejpam-6904	432	13	g	g	NOUN
ejpam-6904	432	14	)	)	PUNCT
ejpam-6904	432	15	,	,	PUNCT
ejpam-6904	432	16	then	then	ADV
ejpam-6904	432	17	st	st	PROPN
ejpam-6904	432	18	is	be	AUX
ejpam-6904	432	19	incident	incident	NOUN
ejpam-6904	432	20	to	to	ADP
ejpam-6904	432	21	a	a	DET
ejpam-6904	432	22	vertex	vertex	NOUN
ejpam-6904	432	23	in	in	ADP
ejpam-6904	432	24	s.	s.	PROPN
ejpam-6904	432	25	suppose	suppose	VERB
ejpam-6904	432	26	st	st	PROPN
ejpam-6904	432	27	∈	∈	PROPN
ejpam-6904	432	28	e(h	e(h	PROPN
ejpam-6904	432	29	)	)	PUNCT
ejpam-6904	432	30	.	.	PUNCT
ejpam-6904	433	1	since	since	SCONJ
ejpam-6904	433	2	sh	sh	PROPN
ejpam-6904	433	3	is	be	AUX
ejpam-6904	433	4	a	a	DET
ejpam-6904	433	5	vertex	vertex	NOUN
ejpam-6904	433	6	cover	cover	NOUN
ejpam-6904	433	7	of	of	ADP
ejpam-6904	433	8	h	h	NOUN
ejpam-6904	433	9	,	,	PUNCT
ejpam-6904	433	10	s	s	VERB
ejpam-6904	433	11	∈	∈	X
ejpam-6904	433	12	sh	sh	INTJ
ejpam-6904	433	13	or	or	CCONJ
ejpam-6904	433	14	t	t	PROPN
ejpam-6904	433	15	∈	∈	PROPN
ejpam-6904	434	1	sh	sh	INTJ
ejpam-6904	434	2	.	.	PUNCT
ejpam-6904	435	1	it	it	PRON
ejpam-6904	435	2	follows	follow	VERB
ejpam-6904	435	3	that	that	SCONJ
ejpam-6904	435	4	s	s	VERB
ejpam-6904	435	5	is	be	AUX
ejpam-6904	435	6	a	a	DET
ejpam-6904	435	7	vertex	vertex	NOUN
ejpam-6904	435	8	cover	cover	NOUN
ejpam-6904	435	9	of	of	ADP
ejpam-6904	435	10	g	g	PROPN
ejpam-6904	435	11	+	+	PROPN
ejpam-6904	435	12	h.	h.	PROPN
ejpam-6904	435	13	let	let	VERB
ejpam-6904	435	14	z	z	PROPN
ejpam-6904	435	15	∈	∈	PROPN
ejpam-6904	435	16	v	v	NOUN
ejpam-6904	435	17	(	(	PUNCT
ejpam-6904	435	18	g	g	PROPN
ejpam-6904	435	19	+	+	NOUN
ejpam-6904	435	20	h	h	NOUN
ejpam-6904	435	21	)	)	PUNCT
ejpam-6904	435	22	\	\	PUNCT
ejpam-6904	436	1	s.	s.	PROPN
ejpam-6904	436	2	then	then	ADV
ejpam-6904	436	3	z	z	PROPN
ejpam-6904	436	4	∈	∈	PROPN
ejpam-6904	436	5	v	v	ADP
ejpam-6904	436	6	(	(	PUNCT
ejpam-6904	436	7	h	h	NOUN
ejpam-6904	436	8	)	)	PUNCT
ejpam-6904	436	9	\	\	PUNCT
ejpam-6904	437	1	sh	sh	INTJ
ejpam-6904	437	2	.	.	PUNCT
ejpam-6904	438	1	choose	choose	VERB
ejpam-6904	438	2	any	any	DET
ejpam-6904	438	3	p	p	NOUN
ejpam-6904	438	4	,	,	PUNCT
ejpam-6904	438	5	q	q	PROPN
ejpam-6904	438	6	∈	∈	PROPN
ejpam-6904	438	7	v	v	ADP
ejpam-6904	438	8	(	(	PUNCT
ejpam-6904	438	9	g	g	NOUN
ejpam-6904	438	10	)	)	PUNCT
ejpam-6904	438	11	such	such	ADJ
ejpam-6904	438	12	that	that	SCONJ
ejpam-6904	438	13	dg(p	dg(p	NOUN
ejpam-6904	438	14	,	,	PUNCT
ejpam-6904	438	15	q	q	X
ejpam-6904	438	16	)	)	PUNCT
ejpam-6904	438	17	̸=	̸=	PROPN
ejpam-6904	438	18	1	1	NUM
ejpam-6904	438	19	.	.	PUNCT
ejpam-6904	439	1	then	then	ADV
ejpam-6904	439	2	p	p	X
ejpam-6904	439	3	,	,	PUNCT
ejpam-6904	439	4	q	q	PROPN
ejpam-6904	439	5	∈	∈	PROPN
ejpam-6904	439	6	s	s	NOUN
ejpam-6904	439	7	,	,	PUNCT
ejpam-6904	439	8	dg+h(p	dg+h(p	PROPN
ejpam-6904	439	9	,	,	PUNCT
ejpam-6904	439	10	q	q	NOUN
ejpam-6904	439	11	)	)	PUNCT
ejpam-6904	439	12	=	=	SYM
ejpam-6904	439	13	2	2	NUM
ejpam-6904	439	14	,	,	PUNCT
ejpam-6904	439	15	and	and	CCONJ
ejpam-6904	439	16	z	z	NOUN
ejpam-6904	439	17	∈	∈	PROPN
ejpam-6904	439	18	ig+h(p	ig+h(p	PROPN
ejpam-6904	439	19	,	,	PUNCT
ejpam-6904	439	20	q	q	NOUN
ejpam-6904	439	21	)	)	PUNCT
ejpam-6904	439	22	.	.	PUNCT
ejpam-6904	440	1	therefore	therefore	ADV
ejpam-6904	440	2	,	,	PUNCT
ejpam-6904	440	3	s	s	VERB
ejpam-6904	440	4	is	be	AUX
ejpam-6904	440	5	a	a	DET
ejpam-6904	440	6	2	2	NUM
ejpam-6904	440	7	-	-	PUNCT
ejpam-6904	440	8	path	path	NOUN
ejpam-6904	440	9	geodetic	geodetic	ADJ
ejpam-6904	440	10	vertex	vertex	NOUN
ejpam-6904	440	11	cover	cover	NOUN
ejpam-6904	440	12	of	of	ADP
ejpam-6904	440	13	g+h	g+h	PROPN
ejpam-6904	440	14	.	.	PUNCT
ejpam-6904	441	1	we	we	PRON
ejpam-6904	441	2	obtain	obtain	VERB
ejpam-6904	441	3	the	the	DET
ejpam-6904	441	4	same	same	ADJ
ejpam-6904	441	5	conclusion	conclusion	NOUN
ejpam-6904	441	6	if	if	SCONJ
ejpam-6904	441	7	(	(	PUNCT
ejpam-6904	441	8	ii	ii	NOUN
ejpam-6904	441	9	)	)	PUNCT
ejpam-6904	441	10	holds	hold	VERB
ejpam-6904	441	11	.	.	PUNCT
ejpam-6904	442	1	the	the	DET
ejpam-6904	442	2	next	next	ADJ
ejpam-6904	442	3	result	result	NOUN
ejpam-6904	442	4	follows	follow	VERB
ejpam-6904	442	5	from	from	ADP
ejpam-6904	442	6	theorem	theorem	ADJ
ejpam-6904	442	7	12	12	NUM
ejpam-6904	442	8	.	.	PUNCT
ejpam-6904	443	1	corollary	corollary	ADJ
ejpam-6904	443	2	7	7	NUM
ejpam-6904	443	3	.	.	PUNCT
ejpam-6904	444	1	let	let	VERB
ejpam-6904	444	2	g	g	NOUN
ejpam-6904	444	3	and	and	CCONJ
ejpam-6904	444	4	h	h	NOUN
ejpam-6904	444	5	be	be	AUX
ejpam-6904	444	6	non	non	ADJ
ejpam-6904	444	7	-	-	ADJ
ejpam-6904	444	8	complete	complete	ADJ
ejpam-6904	444	9	graphs	graph	NOUN
ejpam-6904	444	10	on	on	ADP
ejpam-6904	444	11	m	m	NOUN
ejpam-6904	444	12	and	and	CCONJ
ejpam-6904	444	13	n	n	ADV
ejpam-6904	444	14	vertices	vertex	NOUN
ejpam-6904	444	15	,	,	PUNCT
ejpam-6904	444	16	respectively	respectively	ADV
ejpam-6904	444	17	.	.	PUNCT
ejpam-6904	445	1	then	then	ADV
ejpam-6904	445	2	β2pg(g+h	β2pg(g+h	NUM
ejpam-6904	445	3	)	)	PUNCT
ejpam-6904	445	4	=	=	PUNCT
ejpam-6904	445	5	min{m+	min{m+	PROPN
ejpam-6904	445	6	β(h	β(h	PROPN
ejpam-6904	445	7	)	)	PUNCT
ejpam-6904	445	8	,	,	PUNCT
ejpam-6904	445	9	n+	n+	NUM
ejpam-6904	445	10	β(g	β(g	PROPN
ejpam-6904	445	11	)	)	PUNCT
ejpam-6904	445	12	}	}	PUNCT
ejpam-6904	445	13	.	.	PUNCT
ejpam-6904	446	1	in	in	ADP
ejpam-6904	446	2	particular	particular	ADJ
ejpam-6904	446	3	,	,	PUNCT
ejpam-6904	446	4	each	each	PRON
ejpam-6904	446	5	of	of	ADP
ejpam-6904	446	6	the	the	DET
ejpam-6904	446	7	following	follow	VERB
ejpam-6904	446	8	hold	hold	NOUN
ejpam-6904	446	9	.	.	PUNCT
ejpam-6904	447	1	(	(	PUNCT
ejpam-6904	447	2	i	i	NOUN
ejpam-6904	447	3	)	)	PUNCT
ejpam-6904	447	4	β2pg(km	β2pg(km	NOUN
ejpam-6904	447	5	,	,	PUNCT
ejpam-6904	447	6	n	n	CCONJ
ejpam-6904	447	7	)	)	PUNCT
ejpam-6904	447	8	=	=	SYM
ejpam-6904	447	9	min{m	min{m	PROPN
ejpam-6904	447	10	,	,	PUNCT
ejpam-6904	447	11	n	n	CCONJ
ejpam-6904	447	12	}	}	PUNCT
ejpam-6904	447	13	for	for	ADP
ejpam-6904	447	14	m	m	PROPN
ejpam-6904	447	15	,	,	PUNCT
ejpam-6904	447	16	n	n	PRON
ejpam-6904	447	17	≥	≥	NOUN
ejpam-6904	447	18	2	2	NUM
ejpam-6904	447	19	.	.	PUNCT
ejpam-6904	447	20	(	(	PUNCT
ejpam-6904	447	21	ii	ii	NOUN
ejpam-6904	447	22	)	)	PUNCT
ejpam-6904	447	23	β2pg(pm	β2pg(pm	PUNCT
ejpam-6904	448	1	+	+	CCONJ
ejpam-6904	448	2	pn	pn	NOUN
ejpam-6904	448	3	)	)	PUNCT
ejpam-6904	448	4	=	=	SYM
ejpam-6904	448	5	min{m+	min{m+	PROPN
ejpam-6904	448	6	⌊n2	⌊n2	PROPN
ejpam-6904	448	7	⌋	⌋	PROPN
ejpam-6904	448	8	,	,	PUNCT
ejpam-6904	448	9	n+	n+	PUNCT
ejpam-6904	449	1	⌊m2	⌊m2	PROPN
ejpam-6904	449	2	⌋	⌋	PRON
ejpam-6904	449	3	}	}	PUNCT
ejpam-6904	449	4	for	for	ADP
ejpam-6904	449	5	m	m	PROPN
ejpam-6904	449	6	,	,	PUNCT
ejpam-6904	449	7	n	n	PRON
ejpam-6904	449	8	≥	≥	NOUN
ejpam-6904	449	9	3	3	NUM
ejpam-6904	449	10	.	.	PUNCT
ejpam-6904	449	11	(	(	PUNCT
ejpam-6904	449	12	iii	iii	NOUN
ejpam-6904	449	13	)	)	PUNCT
ejpam-6904	449	14	β2pg(cm	β2pg(cm	PROPN
ejpam-6904	450	1	+	+	CCONJ
ejpam-6904	450	2	cn	cn	ADJ
ejpam-6904	450	3	)	)	PUNCT
ejpam-6904	450	4	=	=	PROPN
ejpam-6904	450	5	min{m+	min{m+	PROPN
ejpam-6904	450	6	⌈n2	⌈n2	NOUN
ejpam-6904	450	7	⌉	⌉	X
ejpam-6904	450	8	,	,	PUNCT
ejpam-6904	450	9	n+	n+	PUNCT
ejpam-6904	450	10	⌈m2	⌈m2	PROPN
ejpam-6904	450	11	⌉	⌉	X
ejpam-6904	450	12	}	}	PUNCT
ejpam-6904	450	13	for	for	ADP
ejpam-6904	450	14	m	m	PROPN
ejpam-6904	450	15	,	,	PUNCT
ejpam-6904	450	16	n	n	PRON
ejpam-6904	450	17	≥	≥	NOUN
ejpam-6904	450	18	4	4	NUM
ejpam-6904	450	19	.	.	NOUN
ejpam-6904	450	20	4	4	NUM
ejpam-6904	450	21	.	.	X
ejpam-6904	450	22	conclusion	conclusion	NOUN
ejpam-6904	450	23	in	in	ADP
ejpam-6904	450	24	this	this	DET
ejpam-6904	450	25	paper	paper	NOUN
ejpam-6904	450	26	,	,	PUNCT
ejpam-6904	450	27	the	the	DET
ejpam-6904	450	28	concept	concept	NOUN
ejpam-6904	450	29	of	of	ADP
ejpam-6904	450	30	2	2	NUM
ejpam-6904	450	31	-	-	PUNCT
ejpam-6904	450	32	path	path	NOUN
ejpam-6904	450	33	geodetic	geodetic	ADJ
ejpam-6904	450	34	vertex	vertex	NOUN
ejpam-6904	450	35	covering	covering	NOUN
ejpam-6904	450	36	of	of	ADP
ejpam-6904	450	37	a	a	DET
ejpam-6904	450	38	graph	graph	NOUN
ejpam-6904	450	39	has	have	AUX
ejpam-6904	450	40	been	be	AUX
ejpam-6904	450	41	introduced	introduce	VERB
ejpam-6904	450	42	and	and	CCONJ
ejpam-6904	450	43	initially	initially	ADV
ejpam-6904	450	44	studied	study	VERB
ejpam-6904	450	45	.	.	PUNCT
ejpam-6904	451	1	it	it	PRON
ejpam-6904	451	2	was	be	AUX
ejpam-6904	451	3	shown	show	VERB
ejpam-6904	451	4	that	that	SCONJ
ejpam-6904	451	5	the	the	DET
ejpam-6904	451	6	difference	difference	NOUN
ejpam-6904	451	7	β2pg(g	β2pg(g	NOUN
ejpam-6904	451	8	)	)	PUNCT
ejpam-6904	451	9	−	−	PROPN
ejpam-6904	451	10	β(g	β(g	PROPN
ejpam-6904	451	11	)	)	PUNCT
ejpam-6904	451	12	can	can	AUX
ejpam-6904	451	13	be	be	AUX
ejpam-6904	451	14	increased	increase	VERB
ejpam-6904	451	15	arbitrarily	arbitrarily	ADV
ejpam-6904	451	16	.	.	PUNCT
ejpam-6904	452	1	graphs	graph	NOUN
ejpam-6904	452	2	which	which	PRON
ejpam-6904	452	3	attain	attain	VERB
ejpam-6904	452	4	small	small	ADJ
ejpam-6904	452	5	and	and	CCONJ
ejpam-6904	452	6	large	large	ADJ
ejpam-6904	452	7	values	value	NOUN
ejpam-6904	452	8	of	of	ADP
ejpam-6904	452	9	the	the	DET
ejpam-6904	452	10	parameter	parameter	NOUN
ejpam-6904	452	11	have	have	AUX
ejpam-6904	452	12	been	be	AUX
ejpam-6904	452	13	characterized	characterize	VERB
ejpam-6904	452	14	.	.	PUNCT
ejpam-6904	453	1	also	also	ADV
ejpam-6904	453	2	,	,	PUNCT
ejpam-6904	453	3	2	2	NUM
ejpam-6904	453	4	-	-	PUNCT
ejpam-6904	453	5	path	path	NOUN
ejpam-6904	453	6	geodetic	geodetic	ADJ
ejpam-6904	453	7	vertex	vertex	NOUN
ejpam-6904	453	8	coverings	covering	NOUN
ejpam-6904	453	9	in	in	ADP
ejpam-6904	453	10	the	the	DET
ejpam-6904	453	11	shadow	shadow	NOUN
ejpam-6904	453	12	graph	graph	NOUN
ejpam-6904	453	13	and	and	CCONJ
ejpam-6904	453	14	the	the	DET
ejpam-6904	453	15	join	join	NOUN
ejpam-6904	453	16	of	of	ADP
ejpam-6904	453	17	graphs	graph	NOUN
ejpam-6904	453	18	have	have	AUX
ejpam-6904	453	19	been	be	AUX
ejpam-6904	453	20	characterized	characterize	VERB
ejpam-6904	453	21	and	and	CCONJ
ejpam-6904	453	22	,	,	PUNCT
ejpam-6904	453	23	subsequently	subsequently	ADV
ejpam-6904	453	24	,	,	PUNCT
ejpam-6904	453	25	values	value	NOUN
ejpam-6904	453	26	of	of	ADP
ejpam-6904	453	27	the	the	DET
ejpam-6904	453	28	parameter	parameter	NOUN
ejpam-6904	453	29	for	for	ADP
ejpam-6904	453	30	these	these	DET
ejpam-6904	453	31	graphs	graph	NOUN
ejpam-6904	453	32	have	have	AUX
ejpam-6904	453	33	been	be	AUX
ejpam-6904	453	34	determined	determine	VERB
ejpam-6904	453	35	.	.	PUNCT
ejpam-6904	454	1	this	this	DET
ejpam-6904	454	2	newly	newly	ADV
ejpam-6904	454	3	defined	define	VERB
ejpam-6904	454	4	variant	variant	NOUN
ejpam-6904	454	5	of	of	ADP
ejpam-6904	454	6	vertex	vertex	NOUN
ejpam-6904	454	7	covering	covering	NOUN
ejpam-6904	454	8	can	can	AUX
ejpam-6904	454	9	also	also	ADV
ejpam-6904	454	10	be	be	AUX
ejpam-6904	454	11	investigated	investigate	VERB
ejpam-6904	454	12	for	for	ADP
ejpam-6904	454	13	other	other	ADJ
ejpam-6904	454	14	classes	class	NOUN
ejpam-6904	454	15	of	of	ADP
ejpam-6904	454	16	graphs	graph	NOUN
ejpam-6904	454	17	.	.	PUNCT
ejpam-6904	455	1	moreover	moreover	ADV
ejpam-6904	455	2	,	,	PUNCT
ejpam-6904	455	3	while	while	SCONJ
ejpam-6904	455	4	the	the	DET
ejpam-6904	455	5	vertex	vertex	NOUN
ejpam-6904	455	6	cover	cover	NOUN
ejpam-6904	455	7	problem	problem	NOUN
ejpam-6904	455	8	is	be	AUX
ejpam-6904	455	9	np	np	NOUN
ejpam-6904	455	10	-	-	PUNCT
ejpam-6904	455	11	complete	complete	ADJ
ejpam-6904	455	12	,	,	PUNCT
ejpam-6904	455	13	it	it	PRON
ejpam-6904	455	14	remains	remain	VERB
ejpam-6904	455	15	to	to	PART
ejpam-6904	455	16	show	show	VERB
ejpam-6904	455	17	whether	whether	SCONJ
ejpam-6904	455	18	or	or	CCONJ
ejpam-6904	455	19	not	not	PART
ejpam-6904	455	20	the	the	DET
ejpam-6904	455	21	2	2	NUM
ejpam-6904	455	22	-	-	PUNCT
ejpam-6904	455	23	path	path	NOUN
ejpam-6904	455	24	geodetic	geodetic	ADJ
ejpam-6904	455	25	vertex	vertex	NOUN
ejpam-6904	455	26	covering	covering	NOUN
ejpam-6904	455	27	problem	problem	NOUN
ejpam-6904	455	28	is	be	AUX
ejpam-6904	455	29	also	also	ADV
ejpam-6904	455	30	np	np	NOUN
ejpam-6904	455	31	-	-	PUNCT
ejpam-6904	455	32	complete	complete	ADJ
ejpam-6904	455	33	.	.	PUNCT
ejpam-6904	456	1	acknowledgements	acknowledgement	NOUN
ejpam-6904	456	2	the	the	DET
ejpam-6904	456	3	authors	author	NOUN
ejpam-6904	456	4	would	would	AUX
ejpam-6904	456	5	like	like	VERB
ejpam-6904	456	6	to	to	PART
ejpam-6904	456	7	thank	thank	VERB
ejpam-6904	456	8	the	the	DET
ejpam-6904	456	9	referees	referee	NOUN
ejpam-6904	456	10	for	for	ADP
ejpam-6904	456	11	the	the	DET
ejpam-6904	456	12	comments	comment	NOUN
ejpam-6904	456	13	and	and	CCONJ
ejpam-6904	456	14	suggestions	suggestion	NOUN
ejpam-6904	456	15	they	they	PRON
ejpam-6904	456	16	gave	give	VERB
ejpam-6904	456	17	the	the	DET
ejpam-6904	456	18	authors	author	NOUN
ejpam-6904	456	19	.	.	PUNCT
ejpam-6904	457	1	the	the	DET
ejpam-6904	457	2	authors	author	NOUN
ejpam-6904	457	3	are	be	AUX
ejpam-6904	457	4	also	also	ADV
ejpam-6904	457	5	grateful	grateful	ADJ
ejpam-6904	457	6	to	to	ADP
ejpam-6904	457	7	the	the	DET
ejpam-6904	457	8	department	department	NOUN
ejpam-6904	457	9	of	of	ADP
ejpam-6904	457	10	science	science	NOUN
ejpam-6904	457	11	and	and	CCONJ
ejpam-6904	457	12	technology	technology	NOUN
ejpam-6904	457	13	accelerated	accelerate	VERB
ejpam-6904	457	14	science	science	NOUN
ejpam-6904	457	15	and	and	CCONJ
ejpam-6904	457	16	technology	technology	NOUN
ejpam-6904	457	17	human	human	ADJ
ejpam-6904	457	18	resource	resource	NOUN
ejpam-6904	457	19	development	development	NOUN
ejpam-6904	457	20	program	program	PROPN
ejpam-6904	457	21	a.	a.	PROPN
ejpam-6904	457	22	b.	b.	PROPN
ejpam-6904	457	23	tapeing	tapeing	PROPN
ejpam-6904	457	24	,	,	PUNCT
ejpam-6904	457	25	s.	s.	PROPN
ejpam-6904	457	26	r.	r.	PROPN
ejpam-6904	457	27	canoy	canoy	PROPN
ejpam-6904	457	28	/	/	SYM
ejpam-6904	457	29	eur	eur	PROPN
ejpam-6904	457	30	.	.	PUNCT
ejpam-6904	458	1	j.	j.	PROPN
ejpam-6904	458	2	pure	pure	PROPN
ejpam-6904	458	3	appl	appl	PROPN
ejpam-6904	458	4	.	.	PROPN
ejpam-6904	458	5	math	math	PROPN
ejpam-6904	458	6	,	,	PUNCT
ejpam-6904	458	7	18	18	NUM
ejpam-6904	458	8	(	(	PUNCT
ejpam-6904	458	9	4	4	NUM
ejpam-6904	458	10	)	)	PUNCT
ejpam-6904	458	11	(	(	PUNCT
ejpam-6904	458	12	2025	2025	NUM
ejpam-6904	458	13	)	)	PUNCT
ejpam-6904	458	14	,	,	PUNCT
ejpam-6904	458	15	6904	6904	NUM
ejpam-6904	458	16	13	13	NUM
ejpam-6904	458	17	of	of	ADP
ejpam-6904	458	18	14	14	NUM
ejpam-6904	458	19	(	(	PUNCT
ejpam-6904	458	20	dost	dost	NOUN
ejpam-6904	458	21	-	-	PUNCT
ejpam-6904	458	22	asthrdp)-philippines	asthrdp)-philippines	PROPN
ejpam-6904	458	23	and	and	CCONJ
ejpam-6904	458	24	the	the	DET
ejpam-6904	458	25	msu	msu	PROPN
ejpam-6904	458	26	-	-	PUNCT
ejpam-6904	458	27	iligan	iligan	PROPN
ejpam-6904	458	28	institute	institute	PROPN
ejpam-6904	458	29	of	of	ADP
ejpam-6904	458	30	technology	technology	PROPN
ejpam-6904	458	31	,	,	PUNCT
ejpam-6904	458	32	iligan	iligan	ADJ
ejpam-6904	458	33	city	city	NOUN
ejpam-6904	458	34	for	for	ADP
ejpam-6904	458	35	funding	fund	VERB
ejpam-6904	458	36	this	this	DET
ejpam-6904	458	37	research	research	NOUN
ejpam-6904	458	38	.	.	PUNCT
ejpam-6904	459	1	references	reference	NOUN
ejpam-6904	459	2	[	[	X
ejpam-6904	459	3	1	1	NUM
ejpam-6904	459	4	]	]	PUNCT
ejpam-6904	459	5	m.	m.	NOUN
ejpam-6904	459	6	henning	henning	PROPN
ejpam-6904	459	7	and	and	CCONJ
ejpam-6904	459	8	a.	a.	PROPN
ejpam-6904	459	9	yeo	yeo	PROPN
ejpam-6904	459	10	.	.	PROPN
ejpam-6904	460	1	identifying	identify	VERB
ejpam-6904	460	2	vertex	vertex	NOUN
ejpam-6904	460	3	covers	cover	VERB
ejpam-6904	460	4	in	in	ADP
ejpam-6904	460	5	graphs	graph	NOUN
ejpam-6904	460	6	.	.	PUNCT
ejpam-6904	461	1	the	the	DET
ejpam-6904	461	2	electric	electric	ADJ
ejpam-6904	461	3	journal	journal	PROPN
ejpam-6904	461	4	in	in	ADP
ejpam-6904	461	5	mathematics	mathematic	NOUN
ejpam-6904	461	6	,	,	PUNCT
ejpam-6904	461	7	,	,	PUNCT
ejpam-6904	461	8	19(4):1038–1045	19(4):1038–1045	NUM
ejpam-6904	461	9	.	.	NOUN
ejpam-6904	461	10	,	,	PUNCT
ejpam-6904	461	11	2012	2012	NUM
ejpam-6904	461	12	.	.	PUNCT
ejpam-6904	462	1	[	[	X
ejpam-6904	462	2	2	2	X
ejpam-6904	462	3	]	]	PUNCT
ejpam-6904	462	4	s.	s.	PROPN
ejpam-6904	462	5	canoy	canoy	PROPN
ejpam-6904	462	6	jr	jr	PROPN
ejpam-6904	462	7	and	and	CCONJ
ejpam-6904	462	8	r.	r.	PROPN
ejpam-6904	462	9	artes	artes	PROPN
ejpam-6904	462	10	jr	jr	PROPN
ejpam-6904	462	11	.	.	PROPN
ejpam-6904	462	12	vertex	vertex	NOUN
ejpam-6904	462	13	and	and	CCONJ
ejpam-6904	462	14	edge	edge	NOUN
ejpam-6904	462	15	covering	cover	VERB
ejpam-6904	462	16	numbers	number	NOUN
ejpam-6904	462	17	of	of	ADP
ejpam-6904	462	18	a	a	DET
ejpam-6904	462	19	graph	graph	NOUN
ejpam-6904	462	20	:	:	PUNCT
ejpam-6904	462	21	revisited	revisit	VERB
ejpam-6904	462	22	.	.	PUNCT
ejpam-6904	463	1	congressus	congressus	PROPN
ejpam-6904	463	2	numerantium	numerantium	PROPN
ejpam-6904	463	3	,	,	PUNCT
ejpam-6904	463	4	167:65	167:65	NUM
ejpam-6904	463	5	,	,	PUNCT
ejpam-6904	463	6	2004	2004	NUM
ejpam-6904	463	7	.	.	PUNCT
ejpam-6904	464	1	[	[	X
ejpam-6904	464	2	3	3	X
ejpam-6904	464	3	]	]	PUNCT
ejpam-6904	464	4	s.	s.	PROPN
ejpam-6904	464	5	sitthiwirattham	sitthiwirattham	PROPN
ejpam-6904	464	6	.	.	PUNCT
ejpam-6904	465	1	vertex	vertex	NOUN
ejpam-6904	465	2	covering	covering	NOUN
ejpam-6904	465	3	and	and	CCONJ
ejpam-6904	465	4	independent	independent	ADJ
ejpam-6904	465	5	number	number	NOUN
ejpam-6904	465	6	on	on	ADP
ejpam-6904	465	7	difference	difference	NOUN
ejpam-6904	465	8	graphs	graph	NOUN
ejpam-6904	465	9	.	.	PUNCT
ejpam-6904	466	1	international	international	ADJ
ejpam-6904	466	2	journal	journal	NOUN
ejpam-6904	466	3	of	of	ADP
ejpam-6904	466	4	pure	pure	ADJ
ejpam-6904	466	5	and	and	CCONJ
ejpam-6904	466	6	applied	applied	ADJ
ejpam-6904	466	7	mathematics	mathematic	NOUN
ejpam-6904	466	8	,	,	PUNCT
ejpam-6904	466	9	77(4):543–547	77(4):543–547	NOUN
ejpam-6904	466	10	,	,	PUNCT
ejpam-6904	466	11	2012	2012	NUM
ejpam-6904	466	12	.	.	PUNCT
ejpam-6904	467	1	[	[	X
ejpam-6904	467	2	4	4	X
ejpam-6904	467	3	]	]	X
ejpam-6904	467	4	j.	j.	PROPN
ejpam-6904	467	5	uy	uy	PROPN
ejpam-6904	467	6	and	and	CCONJ
ejpam-6904	467	7	v.	v.	ADP
ejpam-6904	467	8	abregana	abregana	PROPN
ejpam-6904	467	9	.	.	PUNCT
ejpam-6904	468	1	revisiting	revisit	VERB
ejpam-6904	468	2	the	the	DET
ejpam-6904	468	3	vertex	vertex	NOUN
ejpam-6904	468	4	cover	cover	NOUN
ejpam-6904	468	5	of	of	ADP
ejpam-6904	468	6	graphs	graph	NOUN
ejpam-6904	468	7	.	.	PUNCT
ejpam-6904	469	1	applied	apply	VERB
ejpam-6904	469	2	mathematical	mathematical	ADJ
ejpam-6904	469	3	sciences	science	NOUN
ejpam-6904	469	4	,	,	PUNCT
ejpam-6904	469	5	9:5707	9:5707	NUM
ejpam-6904	469	6	–	–	PUNCT
ejpam-6904	469	7	5714	5714	NUM
ejpam-6904	469	8	,	,	PUNCT
ejpam-6904	469	9	2015	2015	NUM
ejpam-6904	469	10	.	.	PUNCT
ejpam-6904	470	1	[	[	X
ejpam-6904	470	2	5	5	X
ejpam-6904	470	3	]	]	PUNCT
ejpam-6904	470	4	d.	d.	PROPN
ejpam-6904	470	5	angel	angel	NOUN
ejpam-6904	470	6	and	and	CCONJ
ejpam-6904	470	7	a.	a.	NOUN
ejpam-6904	470	8	amutha	amutha	PROPN
ejpam-6904	470	9	.	.	PUNCT
ejpam-6904	471	1	vertex	vertex	NOUN
ejpam-6904	471	2	covering	covering	NOUN
ejpam-6904	471	3	and	and	CCONJ
ejpam-6904	471	4	strong	strong	ADJ
ejpam-6904	471	5	covering	covering	NOUN
ejpam-6904	471	6	of	of	ADP
ejpam-6904	471	7	flower	flower	NOUN
ejpam-6904	471	8	like	like	ADP
ejpam-6904	471	9	network	network	NOUN
ejpam-6904	471	10	structures	structure	NOUN
ejpam-6904	471	11	.	.	PUNCT
ejpam-6904	472	1	procedia	procedia	NOUN
ejpam-6904	472	2	computer	computer	NOUN
ejpam-6904	472	3	science	science	NOUN
ejpam-6904	472	4	,	,	PUNCT
ejpam-6904	472	5	87:164–171	87:164–171	PROPN
ejpam-6904	472	6	,	,	PUNCT
ejpam-6904	472	7	2016	2016	NUM
ejpam-6904	472	8	.	.	PUNCT
ejpam-6904	473	1	[	[	X
ejpam-6904	473	2	6	6	NUM
ejpam-6904	473	3	]	]	PUNCT
ejpam-6904	473	4	c.	c.	PROPN
ejpam-6904	473	5	toregas	toregas	PROPN
ejpam-6904	473	6	,	,	PUNCT
ejpam-6904	473	7	,	,	PUNCT
ejpam-6904	473	8	r.	r.	PROPN
ejpam-6904	473	9	swain	swain	PROPN
ejpam-6904	473	10	,	,	PUNCT
ejpam-6904	473	11	c.	c.	PROPN
ejpam-6904	473	12	revelle	revelle	PROPN
ejpam-6904	473	13	,	,	PUNCT
ejpam-6904	473	14	and	and	CCONJ
ejpam-6904	473	15	l.	l.	PROPN
ejpam-6904	473	16	bercman	bercman	PROPN
ejpam-6904	473	17	.	.	PUNCT
ejpam-6904	474	1	the	the	DET
ejpam-6904	474	2	location	location	NOUN
ejpam-6904	474	3	of	of	ADP
ejpam-6904	474	4	emergency	emergency	NOUN
ejpam-6904	474	5	service	service	NOUN
ejpam-6904	474	6	facilities	facility	NOUN
ejpam-6904	474	7	.	.	PUNCT
ejpam-6904	475	1	journal	journal	NOUN
ejpam-6904	475	2	of	of	ADP
ejpam-6904	475	3	the	the	DET
ejpam-6904	475	4	operations	operation	NOUN
ejpam-6904	475	5	research	research	NOUN
ejpam-6904	475	6	society	society	NOUN
ejpam-6904	475	7	of	of	ADP
ejpam-6904	475	8	america	america	PROPN
ejpam-6904	475	9	,	,	PUNCT
ejpam-6904	475	10	19(6	19(6	NUM
ejpam-6904	475	11	)	)	PUNCT
ejpam-6904	475	12	,	,	PUNCT
ejpam-6904	475	13	1971	1971	NUM
ejpam-6904	475	14	.	.	PUNCT
ejpam-6904	476	1	[	[	X
ejpam-6904	476	2	7	7	X
ejpam-6904	476	3	]	]	X
ejpam-6904	476	4	r.m	r.m	PROPN
ejpam-6904	476	5	.	.	PROPN
ejpam-6904	476	6	karp	karp	PROPN
ejpam-6904	476	7	.	.	PUNCT
ejpam-6904	477	1	reducibility	reducibility	PROPN
ejpam-6904	477	2	among	among	ADP
ejpam-6904	477	3	combinatorial	combinatorial	ADJ
ejpam-6904	477	4	problems	problem	NOUN
ejpam-6904	477	5	,	,	PUNCT
ejpam-6904	477	6	complexity	complexity	NOUN
ejpam-6904	477	7	of	of	ADP
ejpam-6904	477	8	computer	computer	NOUN
ejpam-6904	477	9	computations	computation	NOUN
ejpam-6904	477	10	.	.	PUNCT
ejpam-6904	478	1	plenum	plenum	PROPN
ejpam-6904	478	2	press	press	PROPN
ejpam-6904	478	3	,	,	PUNCT
ejpam-6904	478	4	new	new	PROPN
ejpam-6904	478	5	york	york	PROPN
ejpam-6904	478	6	,	,	PUNCT
ejpam-6904	478	7	pages	page	NOUN
ejpam-6904	478	8	85–103	85–103	NUM
ejpam-6904	478	9	,	,	PUNCT
ejpam-6904	478	10	1972	1972	NUM
ejpam-6904	478	11	.	.	PUNCT
ejpam-6904	479	1	[	[	X
ejpam-6904	479	2	8	8	NUM
ejpam-6904	479	3	]	]	X
ejpam-6904	479	4	m.r	m.r	PROPN
ejpam-6904	479	5	.	.	PROPN
ejpam-6904	479	6	garey	garey	PROPN
ejpam-6904	479	7	and	and	CCONJ
ejpam-6904	479	8	d.s	d.s	PROPN
ejpam-6904	479	9	.	.	PROPN
ejpam-6904	479	10	johnson	johnson	PROPN
ejpam-6904	479	11	.	.	PUNCT
ejpam-6904	480	1	the	the	DET
ejpam-6904	480	2	rectilinear	rectilinear	PROPN
ejpam-6904	480	3	steiner	steiner	PROPN
ejpam-6904	480	4	tree	tree	NOUN
ejpam-6904	480	5	problem	problem	NOUN
ejpam-6904	480	6	is	be	AUX
ejpam-6904	480	7	np	np	NOUN
ejpam-6904	480	8	-	-	PUNCT
ejpam-6904	480	9	complete	complete	ADJ
ejpam-6904	480	10	.	.	PUNCT
ejpam-6904	481	1	siam	siam	PROPN
ejpam-6904	481	2	journal	journal	PROPN
ejpam-6904	481	3	on	on	ADP
ejpam-6904	481	4	applied	apply	VERB
ejpam-6904	481	5	mathematics	mathematic	NOUN
ejpam-6904	481	6	,	,	PUNCT
ejpam-6904	481	7	32:826–834	32:826–834	NUM
ejpam-6904	481	8	,	,	PUNCT
ejpam-6904	481	9	1977	1977	NUM
ejpam-6904	481	10	.	.	PUNCT
ejpam-6904	482	1	[	[	X
ejpam-6904	482	2	9	9	NUM
ejpam-6904	482	3	]	]	X
ejpam-6904	482	4	m.r	m.r	PROPN
ejpam-6904	482	5	.	.	PROPN
ejpam-6904	482	6	garey	garey	PROPN
ejpam-6904	482	7	,	,	PUNCT
ejpam-6904	482	8	d.s	d.s	PROPN
ejpam-6904	482	9	.	.	PROPN
ejpam-6904	482	10	johnson	johnson	PROPN
ejpam-6904	482	11	,	,	PUNCT
ejpam-6904	482	12	and	and	CCONJ
ejpam-6904	482	13	l.	l.	PROPN
ejpam-6904	482	14	stockmeyer	stockmeyer	PROPN
ejpam-6904	482	15	.	.	PUNCT
ejpam-6904	483	1	some	some	PRON
ejpam-6904	483	2	simplified	simplified	ADJ
ejpam-6904	483	3	npcomplete	npcomplete	ADJ
ejpam-6904	483	4	problems	problem	NOUN
ejpam-6904	483	5	.	.	PUNCT
ejpam-6904	484	1	proceedings	proceeding	NOUN
ejpam-6904	484	2	of	of	ADP
ejpam-6904	484	3	the	the	DET
ejpam-6904	484	4	sixth	sixth	ADJ
ejpam-6904	484	5	annual	annual	ADJ
ejpam-6904	484	6	acm	acm	NOUN
ejpam-6904	484	7	symposium	symposium	NOUN
ejpam-6904	484	8	on	on	ADP
ejpam-6904	484	9	theory	theory	NOUN
ejpam-6904	484	10	of	of	ADP
ejpam-6904	484	11	computing	computing	NOUN
ejpam-6904	484	12	,	,	PUNCT
ejpam-6904	484	13	pages	page	NOUN
ejpam-6904	484	14	47	47	NUM
ejpam-6904	484	15	–	–	PUNCT
ejpam-6904	484	16	63	63	NUM
ejpam-6904	484	17	,	,	PUNCT
ejpam-6904	484	18	1974	1974	NUM
ejpam-6904	484	19	.	.	PUNCT
ejpam-6904	485	1	[	[	X
ejpam-6904	485	2	10	10	NUM
ejpam-6904	485	3	]	]	X
ejpam-6904	485	4	b.	b.	PROPN
ejpam-6904	485	5	behsaz	behsaz	PROPN
ejpam-6904	485	6	,	,	PUNCT
ejpam-6904	485	7	p.	p.	PROPN
ejpam-6904	485	8	hatami	hatami	PROPN
ejpam-6904	485	9	,	,	PUNCT
ejpam-6904	485	10	and	and	CCONJ
ejpam-6904	485	11	e.s	e.s	PROPN
ejpam-6904	485	12	.	.	PROPN
ejpam-6904	485	13	mahmoodian	mahmoodian	PROPN
ejpam-6904	485	14	.	.	PUNCT
ejpam-6904	486	1	on	on	ADP
ejpam-6904	486	2	minimum	minimum	ADJ
ejpam-6904	486	3	vertex	vertex	NOUN
ejpam-6904	486	4	cover	cover	NOUN
ejpam-6904	486	5	of	of	ADP
ejpam-6904	486	6	generalized	generalized	ADJ
ejpam-6904	486	7	petersen	petersen	NOUN
ejpam-6904	486	8	graphs	graph	NOUN
ejpam-6904	486	9	.	.	PUNCT
ejpam-6904	487	1	australian	australian	ADJ
ejpam-6904	487	2	journal	journal	NOUN
ejpam-6904	487	3	of	of	ADP
ejpam-6904	487	4	combinatorics	combinatoric	NOUN
ejpam-6904	487	5	,	,	PUNCT
ejpam-6904	487	6	40:253–264	40:253–264	NUM
ejpam-6904	487	7	,	,	PUNCT
ejpam-6904	487	8	2008	2008	NUM
ejpam-6904	487	9	.	.	PUNCT
ejpam-6904	488	1	[	[	X
ejpam-6904	488	2	11	11	NUM
ejpam-6904	488	3	]	]	PUNCT
ejpam-6904	488	4	j.	j.	PROPN
ejpam-6904	488	5	uy	uy	PROPN
ejpam-6904	488	6	.	.	PUNCT
ejpam-6904	488	7	vertex	vertex	PROPN
ejpam-6904	488	8	cover	cover	NOUN
ejpam-6904	488	9	of	of	ADP
ejpam-6904	488	10	graphs	graph	NOUN
ejpam-6904	488	11	.	.	PUNCT
ejpam-6904	489	1	journal	journal	NOUN
ejpam-6904	489	2	of	of	ADP
ejpam-6904	489	3	research	research	NOUN
ejpam-6904	489	4	in	in	ADP
ejpam-6904	489	5	science	science	NOUN
ejpam-6904	489	6	and	and	CCONJ
ejpam-6904	489	7	engineering	engineering	NOUN
ejpam-6904	489	8	,	,	PUNCT
ejpam-6904	489	9	1:49–53	1:49–53	NUM
ejpam-6904	489	10	,	,	PUNCT
ejpam-6904	489	11	2003	2003	NUM
ejpam-6904	489	12	.	.	PUNCT
ejpam-6904	490	1	[	[	X
ejpam-6904	490	2	12	12	NUM
ejpam-6904	490	3	]	]	X
ejpam-6904	490	4	v.	v.	X
ejpam-6904	490	5	bilar	bilar	PROPN
ejpam-6904	490	6	,	,	PUNCT
ejpam-6904	490	7	m.a	m.a	PROPN
ejpam-6904	490	8	.	.	PROPN
ejpam-6904	490	9	bonsocan	bonsocan	PROPN
ejpam-6904	490	10	,	,	PUNCT
ejpam-6904	490	11	j.	j.	PROPN
ejpam-6904	490	12	hassan	hassan	PROPN
ejpam-6904	490	13	,	,	PUNCT
ejpam-6904	490	14	and	and	CCONJ
ejpam-6904	490	15	s.	s.	PROPN
ejpam-6904	490	16	dagondon	dagondon	PROPN
ejpam-6904	490	17	.	.	PUNCT
ejpam-6904	491	1	vertex	vertex	NOUN
ejpam-6904	491	2	cover	cover	VERB
ejpam-6904	491	3	hop	hop	NOUN
ejpam-6904	491	4	dominating	dominating	NOUN
ejpam-6904	491	5	sets	set	NOUN
ejpam-6904	491	6	in	in	ADP
ejpam-6904	491	7	graphs	graph	NOUN
ejpam-6904	491	8	.	.	PUNCT
ejpam-6904	492	1	eur	eur	PROPN
ejpam-6904	492	2	.	.	PUNCT
ejpam-6904	493	1	j.	j.	PROPN
ejpam-6904	493	2	pure	pure	PROPN
ejpam-6904	493	3	appl	appl	PROPN
ejpam-6904	493	4	.	.	PUNCT
ejpam-6904	493	5	math	math	PROPN
ejpam-6904	493	6	.	.	PUNCT
ejpam-6904	493	7	,	,	PUNCT
ejpam-6904	493	8	17(1):93–104	17(1):93–104	NUM
ejpam-6904	493	9	,	,	PUNCT
ejpam-6904	493	10	2024	2024	NUM
ejpam-6904	493	11	.	.	PUNCT
ejpam-6904	494	1	[	[	X
ejpam-6904	494	2	13	13	NUM
ejpam-6904	494	3	]	]	PUNCT
ejpam-6904	494	4	j.	j.	PROPN
ejpam-6904	494	5	hassan	hassan	PROPN
ejpam-6904	494	6	,	,	PUNCT
ejpam-6904	494	7	m.	m.	NOUN
ejpam-6904	494	8	a.	a.	PROPN
ejpam-6904	494	9	bonsocan	bonsocan	PROPN
ejpam-6904	494	10	,	,	PUNCT
ejpam-6904	494	11	r.	r.	PROPN
ejpam-6904	494	12	rasid	rasid	PROPN
ejpam-6904	494	13	,	,	PUNCT
ejpam-6904	494	14	and	and	CCONJ
ejpam-6904	494	15	a.	a.	NOUN
ejpam-6904	494	16	sappari	sappari	PROPN
ejpam-6904	494	17	.	.	PUNCT
ejpam-6904	495	1	certified	certify	VERB
ejpam-6904	495	2	vertex	vertex	NOUN
ejpam-6904	495	3	cover	cover	NOUN
ejpam-6904	495	4	of	of	ADP
ejpam-6904	495	5	a	a	DET
ejpam-6904	495	6	graph	graph	NOUN
ejpam-6904	495	7	.	.	PUNCT
ejpam-6904	496	1	ur	ur	INTJ
ejpam-6904	496	2	.	.	PUNCT
ejpam-6904	497	1	j.	j.	PROPN
ejpam-6904	497	2	pure	pure	PROPN
ejpam-6904	497	3	appl	appl	PROPN
ejpam-6904	497	4	.	.	PUNCT
ejpam-6904	497	5	math	math	PROPN
ejpam-6904	497	6	.	.	PUNCT
ejpam-6904	497	7	,	,	PUNCT
ejpam-6904	497	8	17(2):1038–1045	17(2):1038–1045	NUM
ejpam-6904	497	9	,	,	PUNCT
ejpam-6904	497	10	2024	2024	NUM
ejpam-6904	497	11	.	.	PUNCT
ejpam-6904	498	1	[	[	X
ejpam-6904	498	2	14	14	NUM
ejpam-6904	498	3	]	]	PUNCT
ejpam-6904	498	4	j.	j.	PROPN
ejpam-6904	498	5	hassan	hassan	PROPN
ejpam-6904	498	6	,	,	PUNCT
ejpam-6904	498	7	s.	s.	PROPN
ejpam-6904	498	8	canoy	canoy	PROPN
ejpam-6904	498	9	jr	jr	PROPN
ejpam-6904	498	10	,	,	PUNCT
ejpam-6904	498	11	a.	a.	NOUN
ejpam-6904	498	12	gamorez	gamorez	PROPN
ejpam-6904	498	13	,	,	PUNCT
ejpam-6904	498	14	e.	e.	PROPN
ejpam-6904	498	15	ahmad	ahmad	PROPN
ejpam-6904	498	16	,	,	PUNCT
ejpam-6904	498	17	and	and	CCONJ
ejpam-6904	498	18	a.	a.	NOUN
ejpam-6904	498	19	sappari	sappari	NOUN
ejpam-6904	498	20	.	.	PUNCT
ejpam-6904	499	1	2	2	NUM
ejpam-6904	499	2	-	-	PUNCT
ejpam-6904	499	3	vertex	vertex	NOUN
ejpam-6904	499	4	covering	covering	NOUN
ejpam-6904	499	5	of	of	ADP
ejpam-6904	499	6	a	a	DET
ejpam-6904	499	7	graph	graph	NOUN
ejpam-6904	499	8	.	.	PUNCT
ejpam-6904	500	1	european	european	ADJ
ejpam-6904	500	2	journal	journal	PROPN
ejpam-6904	500	3	of	of	ADP
ejpam-6904	500	4	pure	pure	PROPN
ejpam-6904	500	5	&	&	CCONJ
ejpam-6904	500	6	applied	applied	ADJ
ejpam-6904	500	7	mathematics	mathematic	NOUN
ejpam-6904	500	8	,	,	PUNCT
ejpam-6904	500	9	18(2	18(2	NUM
ejpam-6904	500	10	)	)	PUNCT
ejpam-6904	500	11	,	,	PUNCT
ejpam-6904	500	12	2025	2025	NUM
ejpam-6904	500	13	.	.	PUNCT
ejpam-6904	501	1	[	[	X
ejpam-6904	501	2	15	15	NUM
ejpam-6904	501	3	]	]	X
ejpam-6904	501	4	s.	s.	PROPN
ejpam-6904	501	5	canoy	canoy	PROPN
ejpam-6904	501	6	jr	jr	PROPN
ejpam-6904	501	7	,	,	PUNCT
ejpam-6904	501	8	m.a	m.a	PROPN
ejpam-6904	501	9	.	.	PROPN
ejpam-6904	501	10	bonsocan	bonsocan	PROPN
ejpam-6904	501	11	,	,	PUNCT
ejpam-6904	501	12	j.	j.	PROPN
ejpam-6904	501	13	hassan	hassan	PROPN
ejpam-6904	501	14	,	,	PUNCT
ejpam-6904	501	15	a.m.	a.m.	PROPN
ejpam-6904	501	16	mahistrado	mahistrado	NOUN
ejpam-6904	501	17	,	,	PUNCT
ejpam-6904	501	18	and	and	CCONJ
ejpam-6904	501	19	v.	v.	ADP
ejpam-6904	501	20	bilar	bilar	PROPN
ejpam-6904	501	21	.	.	PUNCT
ejpam-6904	502	1	super	super	ADJ
ejpam-6904	502	2	vertex	vertex	NOUN
ejpam-6904	502	3	cover	cover	NOUN
ejpam-6904	502	4	of	of	ADP
ejpam-6904	502	5	a	a	DET
ejpam-6904	502	6	graph	graph	NOUN
ejpam-6904	502	7	.	.	PUNCT
ejpam-6904	503	1	european	european	ADJ
ejpam-6904	503	2	journal	journal	PROPN
ejpam-6904	503	3	of	of	ADP
ejpam-6904	503	4	pure	pure	PROPN
ejpam-6904	503	5	&	&	CCONJ
ejpam-6904	503	6	applied	applied	ADJ
ejpam-6904	503	7	mathematics	mathematic	NOUN
ejpam-6904	503	8	,	,	PUNCT
ejpam-6904	503	9	18(1	18(1	NUM
ejpam-6904	503	10	)	)	PUNCT
ejpam-6904	503	11	,	,	PUNCT
ejpam-6904	503	12	2025	2025	NUM
ejpam-6904	503	13	.	.	PUNCT
ejpam-6904	504	1	[	[	X
ejpam-6904	504	2	16	16	NUM
ejpam-6904	504	3	]	]	PUNCT
ejpam-6904	504	4	m.	m.	NOUN
ejpam-6904	504	5	marathe	marathe	PROPN
ejpam-6904	504	6	,	,	PUNCT
ejpam-6904	504	7	r.	r.	PROPN
ejpam-6904	504	8	ravi	ravi	PROPN
ejpam-6904	504	9	,	,	PUNCT
ejpam-6904	504	10	and	and	CCONJ
ejpam-6904	504	11	c.	c.	PROPN
ejpam-6904	504	12	p.	p.	PROPN
ejpam-6904	504	13	rangan	rangan	PROPN
ejpam-6904	504	14	.	.	PUNCT
ejpam-6904	505	1	generalized	generalize	VERB
ejpam-6904	505	2	vertex	vertex	NOUN
ejpam-6904	505	3	covering	cover	VERB
ejpam-6904	505	4	in	in	ADP
ejpam-6904	505	5	interval	interval	NOUN
ejpam-6904	505	6	graphs	graph	NOUN
ejpam-6904	505	7	.	.	PUNCT
ejpam-6904	506	1	discrete	discrete	ADJ
ejpam-6904	506	2	applied	applied	ADJ
ejpam-6904	506	3	mathematics	mathematic	NOUN
ejpam-6904	506	4	,	,	PUNCT
ejpam-6904	506	5	39:87–93	39:87–93	PROPN
ejpam-6904	506	6	,	,	PUNCT
ejpam-6904	506	7	1992	1992	NUM
ejpam-6904	506	8	.	.	PUNCT
ejpam-6904	507	1	[	[	X
ejpam-6904	507	2	17	17	NUM
ejpam-6904	507	3	]	]	PUNCT
ejpam-6904	507	4	p.	p.	NOUN
ejpam-6904	507	5	pushpam	pushpam	NOUN
ejpam-6904	507	6	and	and	CCONJ
ejpam-6904	507	7	c.	c.	PROPN
ejpam-6904	507	8	suseendran	suseendran	PROPN
ejpam-6904	507	9	.	.	PUNCT
ejpam-6904	508	1	secure	secure	ADJ
ejpam-6904	508	2	vertex	vertex	NOUN
ejpam-6904	508	3	cover	cover	NOUN
ejpam-6904	508	4	of	of	ADP
ejpam-6904	508	5	a	a	DET
ejpam-6904	508	6	graph	graph	NOUN
ejpam-6904	508	7	.	.	PUNCT
ejpam-6904	509	1	discrete	discrete	ADJ
ejpam-6904	509	2	mathematics	mathematic	NOUN
ejpam-6904	509	3	,	,	PUNCT
ejpam-6904	509	4	algorithms	algorithm	NOUN
ejpam-6904	509	5	and	and	CCONJ
ejpam-6904	509	6	applications	application	NOUN
ejpam-6904	509	7	,	,	PUNCT
ejpam-6904	509	8	9(2	9(2	NUM
ejpam-6904	509	9	)	)	PUNCT
ejpam-6904	509	10	,	,	PUNCT
ejpam-6904	509	11	2017	2017	NUM
ejpam-6904	509	12	.	.	PUNCT
ejpam-6904	510	1	[	[	X
ejpam-6904	510	2	18	18	NUM
ejpam-6904	510	3	]	]	X
ejpam-6904	510	4	l.	l.	PROPN
ejpam-6904	510	5	sathikala	sathikala	PROPN
ejpam-6904	510	6	,	,	PUNCT
ejpam-6904	510	7	k.	k.	PROPN
ejpam-6904	510	8	k.	k.	PROPN
ejpam-6904	510	9	basari	basari	PROPN
ejpam-6904	510	10	,	,	PUNCT
ejpam-6904	510	11	and	and	CCONJ
ejpam-6904	510	12	k.	k.	PROPN
ejpam-6904	510	13	subramanian	subramanian	PROPN
ejpam-6904	510	14	.	.	PROPN
ejpam-6904	511	1	connected	connect	VERB
ejpam-6904	511	2	and	and	CCONJ
ejpam-6904	511	3	total	total	ADJ
ejpam-6904	511	4	vertex	vertex	NOUN
ejpam-6904	511	5	covering	cover	VERB
ejpam-6904	511	6	in	in	ADP
ejpam-6904	511	7	graphs	graph	NOUN
ejpam-6904	511	8	.	.	PUNCT
ejpam-6904	512	1	turkish	turkish	ADJ
ejpam-6904	512	2	journal	journal	NOUN
ejpam-6904	512	3	of	of	ADP
ejpam-6904	512	4	computer	computer	NOUN
ejpam-6904	512	5	and	and	CCONJ
ejpam-6904	512	6	mathematics	mathematic	NOUN
ejpam-6904	512	7	education	education	NOUN
ejpam-6904	512	8	,	,	PUNCT
ejpam-6904	512	9	12(2):2180	12(2):2180	NUM
ejpam-6904	512	10	–	–	PUNCT
ejpam-6904	512	11	2185	2185	NUM
ejpam-6904	512	12	,	,	PUNCT
ejpam-6904	512	13	2021	2021	NUM
ejpam-6904	512	14	.	.	PUNCT
ejpam-6904	513	1	a.	a.	PROPN
ejpam-6904	513	2	b.	b.	PROPN
ejpam-6904	513	3	tapeing	tapeing	PROPN
ejpam-6904	513	4	,	,	PUNCT
ejpam-6904	513	5	s.	s.	PROPN
ejpam-6904	513	6	r.	r.	PROPN
ejpam-6904	513	7	canoy	canoy	PROPN
ejpam-6904	513	8	/	/	SYM
ejpam-6904	513	9	eur	eur	PROPN
ejpam-6904	513	10	.	.	PUNCT
ejpam-6904	514	1	j.	j.	PROPN
ejpam-6904	514	2	pure	pure	PROPN
ejpam-6904	514	3	appl	appl	PROPN
ejpam-6904	514	4	.	.	PROPN
ejpam-6904	514	5	math	math	PROPN
ejpam-6904	514	6	,	,	PUNCT
ejpam-6904	514	7	18	18	NUM
ejpam-6904	514	8	(	(	PUNCT
ejpam-6904	514	9	4	4	NUM
ejpam-6904	514	10	)	)	PUNCT
ejpam-6904	514	11	(	(	PUNCT
ejpam-6904	514	12	2025	2025	NUM
ejpam-6904	514	13	)	)	PUNCT
ejpam-6904	514	14	,	,	PUNCT
ejpam-6904	514	15	6904	6904	NUM
ejpam-6904	514	16	14	14	NUM
ejpam-6904	514	17	of	of	ADP
ejpam-6904	514	18	14	14	NUM
ejpam-6904	514	19	[	[	SYM
ejpam-6904	514	20	19	19	NUM
ejpam-6904	514	21	]	]	PUNCT
ejpam-6904	514	22	g.	g.	PROPN
ejpam-6904	514	23	cagaanan	cagaanan	PROPN
ejpam-6904	514	24	and	and	CCONJ
ejpam-6904	514	25	s.	s.	PROPN
ejpam-6904	514	26	canoy	canoy	PROPN
ejpam-6904	514	27	jr	jr	PROPN
ejpam-6904	514	28	.	.	PROPN
ejpam-6904	514	29	bounds	bound	VERB
ejpam-6904	514	30	for	for	ADP
ejpam-6904	514	31	the	the	DET
ejpam-6904	514	32	geodetic	geodetic	ADJ
ejpam-6904	514	33	number	number	NOUN
ejpam-6904	514	34	of	of	ADP
ejpam-6904	514	35	the	the	DET
ejpam-6904	514	36	cartesian	cartesian	ADJ
ejpam-6904	514	37	product	product	NOUN
ejpam-6904	514	38	of	of	ADP
ejpam-6904	514	39	graphs	graph	NOUN
ejpam-6904	514	40	.	.	PUNCT
ejpam-6904	515	1	utilitas	utilitas	PROPN
ejpam-6904	515	2	mathematica	mathematica	PROPN
ejpam-6904	515	3	,	,	PUNCT
ejpam-6904	515	4	79:91–98	79:91–98	NUM
ejpam-6904	515	5	,	,	PUNCT
ejpam-6904	515	6	2009	2009	NUM
ejpam-6904	515	7	.	.	PUNCT
ejpam-6904	516	1	[	[	X
ejpam-6904	516	2	20	20	NUM
ejpam-6904	516	3	]	]	PUNCT
ejpam-6904	516	4	g.	g.	PROPN
ejpam-6904	516	5	cagaanan	cagaanan	PROPN
ejpam-6904	516	6	and	and	CCONJ
ejpam-6904	516	7	jr	jr	PROPN
ejpam-6904	516	8	.	.	PROPN
ejpam-6904	516	9	s.	s.	PROPN
ejpam-6904	516	10	canoy	canoy	PROPN
ejpam-6904	516	11	.	.	PUNCT
ejpam-6904	517	1	on	on	ADP
ejpam-6904	517	2	the	the	DET
ejpam-6904	517	3	geodetic	geodetic	ADJ
ejpam-6904	517	4	and	and	CCONJ
ejpam-6904	517	5	hull	hull	NOUN
ejpam-6904	517	6	numbers	number	NOUN
ejpam-6904	517	7	of	of	ADP
ejpam-6904	517	8	some	some	DET
ejpam-6904	517	9	graphs	graph	NOUN
ejpam-6904	517	10	.	.	PUNCT
ejpam-6904	518	1	asia	asia	PROPN
ejpam-6904	518	2	pacific	pacific	PROPN
ejpam-6904	518	3	journal	journal	PROPN
ejpam-6904	518	4	of	of	ADP
ejpam-6904	518	5	social	social	ADJ
ejpam-6904	518	6	innovation	innovation	NOUN
ejpam-6904	518	7	,	,	PUNCT
ejpam-6904	518	8	19(1	19(1	NUM
ejpam-6904	518	9	)	)	PUNCT
ejpam-6904	518	10	,	,	PUNCT
ejpam-6904	518	11	2005	2005	NUM
ejpam-6904	518	12	.	.	PUNCT
ejpam-6904	519	1	[	[	X
ejpam-6904	519	2	21	21	NUM
ejpam-6904	519	3	]	]	X
ejpam-6904	519	4	f.	f.	PROPN
ejpam-6904	519	5	jamil	jamil	PROPN
ejpam-6904	519	6	,	,	PUNCT
ejpam-6904	519	7	i.	i.	PROPN
ejpam-6904	519	8	aniversario	aniversario	PROPN
ejpam-6904	519	9	,	,	PUNCT
ejpam-6904	519	10	and	and	CCONJ
ejpam-6904	519	11	s.	s.	PROPN
ejpam-6904	519	12	canoy	canoy	PROPN
ejpam-6904	519	13	jr	jr	PROPN
ejpam-6904	519	14	.	.	PROPN
ejpam-6904	519	15	on	on	ADP
ejpam-6904	519	16	closed	closed	ADJ
ejpam-6904	519	17	and	and	CCONJ
ejpam-6904	519	18	upper	upper	ADJ
ejpam-6904	519	19	closed	closed	ADJ
ejpam-6904	519	20	geodetic	geodetic	ADJ
ejpam-6904	519	21	numbers	number	NOUN
ejpam-6904	519	22	of	of	ADP
ejpam-6904	519	23	graphs	graph	NOUN
ejpam-6904	519	24	.	.	PUNCT
ejpam-6904	520	1	ars	ar	NOUN
ejpam-6904	520	2	combinatoria	combinatoria	PROPN
ejpam-6904	520	3	,	,	PUNCT
ejpam-6904	520	4	84:191–204	84:191–204	PROPN
ejpam-6904	520	5	,	,	PUNCT
ejpam-6904	520	6	2007	2007	NUM
ejpam-6904	520	7	.	.	PUNCT
ejpam-6904	521	1	[	[	X
ejpam-6904	521	2	22	22	NUM
ejpam-6904	521	3	]	]	X
ejpam-6904	521	4	s.	s.	PROPN
ejpam-6904	521	5	canoy	canoy	PROPN
ejpam-6904	521	6	jr	jr	PROPN
ejpam-6904	521	7	,	,	PUNCT
ejpam-6904	521	8	g.	g.	PROPN
ejpam-6904	521	9	cagaanan	cagaanan	PROPN
ejpam-6904	521	10	,	,	PUNCT
ejpam-6904	521	11	and	and	CCONJ
ejpam-6904	521	12	s.	s.	PROPN
ejpam-6904	521	13	gervacio	gervacio	PROPN
ejpam-6904	521	14	.	.	PUNCT
ejpam-6904	522	1	convexity	convexity	PROPN
ejpam-6904	522	2	,	,	PUNCT
ejpam-6904	522	3	geodetic	geodetic	ADJ
ejpam-6904	522	4	,	,	PUNCT
ejpam-6904	522	5	and	and	CCONJ
ejpam-6904	522	6	hull	hull	NOUN
ejpam-6904	522	7	numbers	number	NOUN
ejpam-6904	522	8	of	of	ADP
ejpam-6904	522	9	the	the	DET
ejpam-6904	522	10	join	join	NOUN
ejpam-6904	522	11	of	of	ADP
ejpam-6904	522	12	graphs	graph	NOUN
ejpam-6904	522	13	.	.	PUNCT
ejpam-6904	523	1	2006	2006	NUM
ejpam-6904	523	2	.	.	PUNCT
ejpam-6904	524	1	[	[	X
ejpam-6904	524	2	23	23	NUM
ejpam-6904	524	3	]	]	X
ejpam-6904	524	4	f.	f.	PROPN
ejpam-6904	524	5	buckley	buckley	PROPN
ejpam-6904	524	6	and	and	CCONJ
ejpam-6904	524	7	f.	f.	PROPN
ejpam-6904	524	8	harary	harary	PROPN
ejpam-6904	524	9	.	.	PUNCT
ejpam-6904	525	1	distance	distance	NOUN
ejpam-6904	525	2	in	in	ADP
ejpam-6904	525	3	graphs	graph	NOUN
ejpam-6904	525	4	.	.	PUNCT
ejpam-6904	526	1	addison	addison	PROPN
ejpam-6904	526	2	-	-	PUNCT
ejpam-6904	526	3	wesley	wesley	PROPN
ejpam-6904	526	4	,	,	PUNCT
ejpam-6904	526	5	redwood	redwood	NOUN
ejpam-6904	526	6	city	city	NOUN
ejpam-6904	526	7	,	,	PUNCT
ejpam-6904	526	8	1990	1990	NUM
ejpam-6904	526	9	.	.	PUNCT