id	sid	tid	token	lemma	pos
ejpam-4761	1	1	european	european	PROPN
ejpam-4761	1	2	journal	journal	PROPN
ejpam-4761	1	3	of	of	ADP
ejpam-4761	1	4	pure	pure	ADJ
ejpam-4761	1	5	and	and	CCONJ
ejpam-4761	1	6	applied	apply	VERB
ejpam-4761	1	7	mathematics	mathematic	NOUN
ejpam-4761	1	8	vol	vol	NOUN
ejpam-4761	1	9	.	.	PUNCT
ejpam-4761	2	1	16	16	NUM
ejpam-4761	2	2	,	,	PUNCT
ejpam-4761	2	3	no	no	INTJ
ejpam-4761	2	4	.	.	NOUN
ejpam-4761	2	5	2	2	NUM
ejpam-4761	2	6	,	,	PUNCT
ejpam-4761	2	7	2023	2023	NUM
ejpam-4761	2	8	,	,	PUNCT
ejpam-4761	2	9	1196	1196	NUM
ejpam-4761	2	10	-	-	SYM
ejpam-4761	2	11	1211	1211	NUM
ejpam-4761	2	12	issn	issn	PROPN
ejpam-4761	2	13	1307	1307	NUM
ejpam-4761	2	14	-	-	SYM
ejpam-4761	2	15	5543	5543	NUM
ejpam-4761	2	16	–	–	PUNCT
ejpam-4761	3	1	ejpam.com	ejpam.com	X
ejpam-4761	3	2	published	publish	VERB
ejpam-4761	3	3	by	by	ADP
ejpam-4761	3	4	new	new	PROPN
ejpam-4761	3	5	york	york	PROPN
ejpam-4761	3	6	business	business	PROPN
ejpam-4761	3	7	global	global	ADJ
ejpam-4761	3	8	weakly	weakly	ADJ
ejpam-4761	3	9	convex	convex	ADJ
ejpam-4761	3	10	hop	hop	NOUN
ejpam-4761	3	11	dominating	dominating	NOUN
ejpam-4761	3	12	sets	set	NOUN
ejpam-4761	3	13	in	in	ADP
ejpam-4761	3	14	graphs	graph	NOUN
ejpam-4761	3	15	sergio	sergio	PROPN
ejpam-4761	3	16	r.	r.	PROPN
ejpam-4761	3	17	canoy	canoy	PROPN
ejpam-4761	3	18	,	,	PUNCT
ejpam-4761	3	19	jr.1,2	jr.1,2	PROPN
ejpam-4761	3	20	,	,	PUNCT
ejpam-4761	3	21	javier	javier	PROPN
ejpam-4761	3	22	a.	a.	PROPN
ejpam-4761	3	23	hassan3,∗	hassan3,∗	PROPN
ejpam-4761	3	24	1	1	NUM
ejpam-4761	3	25	department	department	NOUN
ejpam-4761	3	26	of	of	ADP
ejpam-4761	3	27	mathematics	mathematic	NOUN
ejpam-4761	3	28	and	and	CCONJ
ejpam-4761	3	29	statistics	statistic	NOUN
ejpam-4761	3	30	,	,	PUNCT
ejpam-4761	3	31	college	college	NOUN
ejpam-4761	3	32	of	of	ADP
ejpam-4761	3	33	science	science	NOUN
ejpam-4761	3	34	and	and	CCONJ
ejpam-4761	3	35	mathematics	mathematic	NOUN
ejpam-4761	3	36	,	,	PUNCT
ejpam-4761	3	37	center	center	NOUN
ejpam-4761	3	38	for	for	ADP
ejpam-4761	3	39	graph	graph	NOUN
ejpam-4761	3	40	theory	theory	NOUN
ejpam-4761	3	41	,	,	PUNCT
ejpam-4761	3	42	premier	premier	PROPN
ejpam-4761	3	43	research	research	PROPN
ejpam-4761	3	44	institute	institute	PROPN
ejpam-4761	3	45	of	of	ADP
ejpam-4761	3	46	science	science	NOUN
ejpam-4761	3	47	and	and	CCONJ
ejpam-4761	3	48	mathematics	mathematic	NOUN
ejpam-4761	3	49	,	,	PUNCT
ejpam-4761	3	50	msu	msu	PROPN
ejpam-4761	3	51	-	-	PUNCT
ejpam-4761	3	52	iligan	iligan	PROPN
ejpam-4761	3	53	institute	institute	PROPN
ejpam-4761	3	54	of	of	ADP
ejpam-4761	3	55	technology	technology	PROPN
ejpam-4761	3	56	,	,	PUNCT
ejpam-4761	3	57	9200	9200	NUM
ejpam-4761	3	58	iligan	iligan	ADJ
ejpam-4761	3	59	city	city	NOUN
ejpam-4761	3	60	,	,	PUNCT
ejpam-4761	3	61	philippines	philippine	NOUN
ejpam-4761	3	62	2	2	NUM
ejpam-4761	3	63	center	center	NOUN
ejpam-4761	3	64	for	for	ADP
ejpam-4761	3	65	mathematical	mathematical	ADJ
ejpam-4761	3	66	and	and	CCONJ
ejpam-4761	3	67	theoretical	theoretical	ADJ
ejpam-4761	3	68	physical	physical	ADJ
ejpam-4761	3	69	sciences	science	NOUN
ejpam-4761	3	70	,	,	PUNCT
ejpam-4761	3	71	premier	premier	PROPN
ejpam-4761	3	72	research	research	PROPN
ejpam-4761	3	73	institute	institute	PROPN
ejpam-4761	3	74	of	of	ADP
ejpam-4761	3	75	science	science	NOUN
ejpam-4761	3	76	and	and	CCONJ
ejpam-4761	3	77	mathematics	mathematic	NOUN
ejpam-4761	3	78	,	,	PUNCT
ejpam-4761	3	79	msu	msu	PROPN
ejpam-4761	3	80	-	-	PUNCT
ejpam-4761	3	81	iligan	iligan	PROPN
ejpam-4761	3	82	institute	institute	PROPN
ejpam-4761	3	83	of	of	ADP
ejpam-4761	3	84	technology	technology	PROPN
ejpam-4761	3	85	,	,	PUNCT
ejpam-4761	3	86	9200	9200	NUM
ejpam-4761	3	87	iligan	iligan	ADJ
ejpam-4761	3	88	city	city	NOUN
ejpam-4761	3	89	,	,	PUNCT
ejpam-4761	3	90	philippines	philippine	VERB
ejpam-4761	3	91	3	3	NUM
ejpam-4761	3	92	mathematics	mathematic	NOUN
ejpam-4761	3	93	and	and	CCONJ
ejpam-4761	3	94	sciences	sciences	PROPN
ejpam-4761	3	95	department	department	PROPN
ejpam-4761	3	96	,	,	PUNCT
ejpam-4761	3	97	college	college	NOUN
ejpam-4761	3	98	of	of	ADP
ejpam-4761	3	99	arts	art	NOUN
ejpam-4761	3	100	and	and	CCONJ
ejpam-4761	3	101	sciences	science	NOUN
ejpam-4761	3	102	,	,	PUNCT
ejpam-4761	3	103	msu	msu	PROPN
ejpam-4761	3	104	tawi	tawi	PROPN
ejpam-4761	3	105	-	-	PUNCT
ejpam-4761	3	106	tawi	tawi	PROPN
ejpam-4761	3	107	college	college	PROPN
ejpam-4761	3	108	of	of	ADP
ejpam-4761	3	109	technology	technology	NOUN
ejpam-4761	3	110	and	and	CCONJ
ejpam-4761	3	111	oceanography	oceanography	NOUN
ejpam-4761	3	112	,	,	PUNCT
ejpam-4761	3	113	bongao	bongao	NOUN
ejpam-4761	3	114	,	,	PUNCT
ejpam-4761	3	115	tawi	tawi	NOUN
ejpam-4761	3	116	-	-	PUNCT
ejpam-4761	3	117	tawi	tawi	NOUN
ejpam-4761	3	118	,	,	PUNCT
ejpam-4761	3	119	philippines	philippine	NOUN
ejpam-4761	3	120	abstract	abstract	ADJ
ejpam-4761	3	121	.	.	PUNCT
ejpam-4761	4	1	let	let	VERB
ejpam-4761	4	2	g	g	PRON
ejpam-4761	4	3	be	be	AUX
ejpam-4761	4	4	an	an	DET
ejpam-4761	4	5	undirected	undirected	ADJ
ejpam-4761	4	6	connected	connected	ADJ
ejpam-4761	4	7	graph	graph	NOUN
ejpam-4761	4	8	with	with	ADP
ejpam-4761	4	9	vertex	vertex	NOUN
ejpam-4761	4	10	and	and	CCONJ
ejpam-4761	4	11	edge	edge	NOUN
ejpam-4761	4	12	sets	set	NOUN
ejpam-4761	4	13	v	v	ADP
ejpam-4761	4	14	(	(	PUNCT
ejpam-4761	4	15	g	g	NOUN
ejpam-4761	4	16	)	)	PUNCT
ejpam-4761	4	17	and	and	CCONJ
ejpam-4761	4	18	e(g	e(g	PROPN
ejpam-4761	4	19	)	)	PUNCT
ejpam-4761	4	20	,	,	PUNCT
ejpam-4761	4	21	respectively	respectively	ADV
ejpam-4761	4	22	.	.	PUNCT
ejpam-4761	5	1	a	a	DET
ejpam-4761	5	2	set	set	NOUN
ejpam-4761	5	3	c	c	NOUN
ejpam-4761	5	4	⊆	⊆	NUM
ejpam-4761	5	5	v	v	NOUN
ejpam-4761	5	6	(	(	PUNCT
ejpam-4761	5	7	g	g	NOUN
ejpam-4761	5	8	)	)	PUNCT
ejpam-4761	5	9	is	be	AUX
ejpam-4761	5	10	called	call	VERB
ejpam-4761	5	11	weakly	weakly	ADV
ejpam-4761	5	12	convex	convex	NOUN
ejpam-4761	5	13	hop	hop	NOUN
ejpam-4761	5	14	dominating	dominate	VERB
ejpam-4761	5	15	if	if	SCONJ
ejpam-4761	5	16	for	for	ADP
ejpam-4761	5	17	every	every	DET
ejpam-4761	5	18	two	two	NUM
ejpam-4761	5	19	vertices	vertex	NOUN
ejpam-4761	5	20	x	x	X
ejpam-4761	5	21	,	,	PUNCT
ejpam-4761	5	22	y	y	PROPN
ejpam-4761	5	23	∈	∈	PROPN
ejpam-4761	5	24	c	c	X
ejpam-4761	5	25	,	,	PUNCT
ejpam-4761	5	26	there	there	PRON
ejpam-4761	5	27	exists	exist	VERB
ejpam-4761	5	28	an	an	DET
ejpam-4761	5	29	x	x	NOUN
ejpam-4761	5	30	-	-	NOUN
ejpam-4761	5	31	y	y	ADJ
ejpam-4761	5	32	geodesic	geodesic	NOUN
ejpam-4761	5	33	p	p	X
ejpam-4761	5	34	(	(	PUNCT
ejpam-4761	5	35	x	x	NOUN
ejpam-4761	5	36	,	,	PUNCT
ejpam-4761	5	37	y	y	NOUN
ejpam-4761	5	38	)	)	PUNCT
ejpam-4761	5	39	such	such	ADJ
ejpam-4761	5	40	that	that	DET
ejpam-4761	5	41	v	v	NOUN
ejpam-4761	5	42	(	(	PUNCT
ejpam-4761	5	43	p	p	X
ejpam-4761	5	44	(	(	PUNCT
ejpam-4761	5	45	x	x	NOUN
ejpam-4761	5	46	,	,	PUNCT
ejpam-4761	5	47	y	y	NOUN
ejpam-4761	5	48	)	)	PUNCT
ejpam-4761	5	49	)	)	PUNCT
ejpam-4761	6	1	⊆	⊆	NUM
ejpam-4761	6	2	c	c	NOUN
ejpam-4761	6	3	and	and	CCONJ
ejpam-4761	6	4	for	for	ADP
ejpam-4761	6	5	every	every	DET
ejpam-4761	6	6	v	v	NUM
ejpam-4761	6	7	∈	∈	PROPN
ejpam-4761	6	8	v	v	NOUN
ejpam-4761	6	9	(	(	PUNCT
ejpam-4761	6	10	g)\c	g)\c	NOUN
ejpam-4761	6	11	,	,	PUNCT
ejpam-4761	6	12	there	there	PRON
ejpam-4761	6	13	exists	exist	VERB
ejpam-4761	6	14	w	w	PROPN
ejpam-4761	6	15	∈	∈	PROPN
ejpam-4761	6	16	c	c	NOUN
ejpam-4761	6	17	such	such	ADJ
ejpam-4761	6	18	that	that	DET
ejpam-4761	6	19	dg(v	dg(v	ADJ
ejpam-4761	6	20	,	,	PUNCT
ejpam-4761	6	21	w	w	NOUN
ejpam-4761	6	22	)	)	PUNCT
ejpam-4761	7	1	=	=	SYM
ejpam-4761	7	2	2	2	X
ejpam-4761	7	3	.	.	PUNCT
ejpam-4761	8	1	the	the	DET
ejpam-4761	8	2	minimum	minimum	ADJ
ejpam-4761	8	3	cardinality	cardinality	NOUN
ejpam-4761	8	4	of	of	ADP
ejpam-4761	8	5	a	a	DET
ejpam-4761	8	6	weakly	weakly	ADJ
ejpam-4761	8	7	convex	convex	NOUN
ejpam-4761	8	8	hop	hop	NOUN
ejpam-4761	8	9	dominating	dominating	NOUN
ejpam-4761	8	10	set	set	NOUN
ejpam-4761	8	11	of	of	ADP
ejpam-4761	8	12	g	g	NOUN
ejpam-4761	8	13	,	,	PUNCT
ejpam-4761	8	14	denoted	denote	VERB
ejpam-4761	8	15	by	by	ADP
ejpam-4761	8	16	γwconh(g	γwconh(g	NOUN
ejpam-4761	8	17	)	)	PUNCT
ejpam-4761	8	18	,	,	PUNCT
ejpam-4761	8	19	is	be	AUX
ejpam-4761	8	20	called	call	VERB
ejpam-4761	8	21	the	the	DET
ejpam-4761	8	22	weakly	weakly	ADJ
ejpam-4761	8	23	convex	convex	ADJ
ejpam-4761	8	24	hop	hop	NOUN
ejpam-4761	8	25	domination	domination	NOUN
ejpam-4761	8	26	number	number	NOUN
ejpam-4761	8	27	of	of	ADP
ejpam-4761	8	28	g.	g.	PROPN
ejpam-4761	8	29	in	in	ADP
ejpam-4761	8	30	this	this	DET
ejpam-4761	8	31	paper	paper	NOUN
ejpam-4761	8	32	,	,	PUNCT
ejpam-4761	8	33	we	we	PRON
ejpam-4761	8	34	introduce	introduce	VERB
ejpam-4761	8	35	and	and	CCONJ
ejpam-4761	8	36	initially	initially	ADV
ejpam-4761	8	37	investigate	investigate	VERB
ejpam-4761	8	38	the	the	DET
ejpam-4761	8	39	concept	concept	NOUN
ejpam-4761	8	40	of	of	ADP
ejpam-4761	8	41	weakly	weakly	ADJ
ejpam-4761	8	42	convex	convex	ADJ
ejpam-4761	8	43	hop	hop	NOUN
ejpam-4761	8	44	domination	domination	NOUN
ejpam-4761	8	45	.	.	PUNCT
ejpam-4761	9	1	we	we	PRON
ejpam-4761	9	2	show	show	VERB
ejpam-4761	9	3	that	that	SCONJ
ejpam-4761	9	4	every	every	DET
ejpam-4761	9	5	two	two	NUM
ejpam-4761	9	6	positive	positive	ADJ
ejpam-4761	9	7	integers	integer	NOUN
ejpam-4761	9	8	a	a	PRON
ejpam-4761	9	9	and	and	CCONJ
ejpam-4761	9	10	b	b	NOUN
ejpam-4761	9	11	with	with	ADP
ejpam-4761	9	12	3	3	NUM
ejpam-4761	9	13	≤	≤	NOUN
ejpam-4761	9	14	a	a	DET
ejpam-4761	9	15	≤	≤	NUM
ejpam-4761	9	16	b	b	NOUN
ejpam-4761	9	17	are	be	AUX
ejpam-4761	9	18	realizable	realizable	ADJ
ejpam-4761	9	19	as	as	ADP
ejpam-4761	9	20	the	the	DET
ejpam-4761	9	21	weakly	weakly	ADJ
ejpam-4761	9	22	convex	convex	ADJ
ejpam-4761	9	23	hop	hop	NOUN
ejpam-4761	9	24	domination	domination	NOUN
ejpam-4761	9	25	number	number	NOUN
ejpam-4761	9	26	and	and	CCONJ
ejpam-4761	9	27	convex	convex	VERB
ejpam-4761	9	28	hop	hop	NOUN
ejpam-4761	9	29	domination	domination	NOUN
ejpam-4761	9	30	number	number	NOUN
ejpam-4761	9	31	of	of	ADP
ejpam-4761	9	32	some	some	DET
ejpam-4761	9	33	connected	connected	ADJ
ejpam-4761	9	34	graph	graph	NOUN
ejpam-4761	9	35	.	.	PUNCT
ejpam-4761	10	1	furthermore	furthermore	ADV
ejpam-4761	10	2	,	,	PUNCT
ejpam-4761	10	3	we	we	PRON
ejpam-4761	10	4	characterize	characterize	VERB
ejpam-4761	10	5	the	the	DET
ejpam-4761	10	6	weakly	weakly	ADJ
ejpam-4761	10	7	convex	convex	ADJ
ejpam-4761	10	8	hop	hop	NOUN
ejpam-4761	10	9	dominating	dominating	NOUN
ejpam-4761	10	10	sets	set	NOUN
ejpam-4761	10	11	in	in	ADP
ejpam-4761	10	12	some	some	DET
ejpam-4761	10	13	graphs	graph	NOUN
ejpam-4761	10	14	under	under	ADP
ejpam-4761	10	15	some	some	DET
ejpam-4761	10	16	binary	binary	ADJ
ejpam-4761	10	17	operations	operation	NOUN
ejpam-4761	10	18	.	.	PUNCT
ejpam-4761	11	1	2020	2020	NUM
ejpam-4761	11	2	mathematics	mathematic	NOUN
ejpam-4761	11	3	subject	subject	NOUN
ejpam-4761	11	4	classifications	classification	NOUN
ejpam-4761	11	5	:	:	PUNCT
ejpam-4761	11	6	05c69	05c69	X
ejpam-4761	11	7	key	key	ADJ
ejpam-4761	11	8	words	word	NOUN
ejpam-4761	11	9	and	and	CCONJ
ejpam-4761	11	10	phrases	phrase	NOUN
ejpam-4761	11	11	:	:	PUNCT
ejpam-4761	11	12	weakly	weakly	ADJ
ejpam-4761	11	13	convex	convex	NOUN
ejpam-4761	11	14	set	set	NOUN
ejpam-4761	11	15	,	,	PUNCT
ejpam-4761	11	16	weakly	weakly	ADJ
ejpam-4761	11	17	convex	convex	ADJ
ejpam-4761	11	18	hop	hop	NOUN
ejpam-4761	11	19	dominating	dominating	NOUN
ejpam-4761	11	20	set	set	NOUN
ejpam-4761	11	21	,	,	PUNCT
ejpam-4761	11	22	weakly	weakly	ADJ
ejpam-4761	11	23	convex	convex	VERB
ejpam-4761	11	24	hop	hop	NOUN
ejpam-4761	11	25	domination	domination	NOUN
ejpam-4761	11	26	number	number	NOUN
ejpam-4761	11	27	1	1	NUM
ejpam-4761	11	28	.	.	PUNCT
ejpam-4761	12	1	introduction	introduction	NOUN
ejpam-4761	12	2	domination	domination	NOUN
ejpam-4761	12	3	has	have	AUX
ejpam-4761	12	4	been	be	AUX
ejpam-4761	12	5	one	one	NUM
ejpam-4761	12	6	of	of	ADP
ejpam-4761	12	7	the	the	DET
ejpam-4761	12	8	interesting	interesting	ADJ
ejpam-4761	12	9	and	and	CCONJ
ejpam-4761	12	10	widely	widely	ADV
ejpam-4761	12	11	studied	study	VERB
ejpam-4761	12	12	topics	topic	NOUN
ejpam-4761	12	13	in	in	ADP
ejpam-4761	12	14	graph	graph	NOUN
ejpam-4761	12	15	theory	theory	NOUN
ejpam-4761	12	16	.	.	PUNCT
ejpam-4761	13	1	a	a	DET
ejpam-4761	13	2	number	number	NOUN
ejpam-4761	13	3	of	of	ADP
ejpam-4761	13	4	concepts	concept	NOUN
ejpam-4761	13	5	had	have	AUX
ejpam-4761	13	6	already	already	ADV
ejpam-4761	13	7	been	be	AUX
ejpam-4761	13	8	used	use	VERB
ejpam-4761	13	9	to	to	PART
ejpam-4761	13	10	introduce	introduce	VERB
ejpam-4761	13	11	variations	variation	NOUN
ejpam-4761	13	12	of	of	ADP
ejpam-4761	13	13	the	the	DET
ejpam-4761	13	14	standard	standard	ADJ
ejpam-4761	13	15	concept	concept	NOUN
ejpam-4761	13	16	of	of	ADP
ejpam-4761	13	17	domination	domination	NOUN
ejpam-4761	13	18	.	.	PUNCT
ejpam-4761	14	1	in	in	ADP
ejpam-4761	14	2	particular	particular	ADJ
ejpam-4761	14	3	,	,	PUNCT
ejpam-4761	14	4	concepts	concept	NOUN
ejpam-4761	14	5	such	such	ADJ
ejpam-4761	14	6	as	as	ADP
ejpam-4761	14	7	convex	convex	ADJ
ejpam-4761	14	8	and	and	CCONJ
ejpam-4761	14	9	weakly	weakly	ADJ
ejpam-4761	14	10	convex	convex	NOUN
ejpam-4761	14	11	had	have	AUX
ejpam-4761	14	12	been	be	AUX
ejpam-4761	14	13	considered	consider	VERB
ejpam-4761	14	14	to	to	PART
ejpam-4761	14	15	define	define	VERB
ejpam-4761	14	16	convex	convex	ADJ
ejpam-4761	14	17	domination	domination	NOUN
ejpam-4761	14	18	and	and	CCONJ
ejpam-4761	14	19	weakly	weakly	ADJ
ejpam-4761	14	20	convex	convex	ADJ
ejpam-4761	14	21	domination	domination	NOUN
ejpam-4761	14	22	,	,	PUNCT
ejpam-4761	14	23	respectively	respectively	ADV
ejpam-4761	14	24	(	(	PUNCT
ejpam-4761	14	25	see	see	VERB
ejpam-4761	14	26	[	[	X
ejpam-4761	14	27	4	4	NUM
ejpam-4761	14	28	]	]	PUNCT
ejpam-4761	14	29	,	,	PUNCT
ejpam-4761	15	1	[	[	X
ejpam-4761	15	2	12	12	NUM
ejpam-4761	15	3	]	]	PUNCT
ejpam-4761	15	4	,	,	PUNCT
ejpam-4761	16	1	[	[	X
ejpam-4761	16	2	14	14	NUM
ejpam-4761	16	3	]	]	PUNCT
ejpam-4761	16	4	,	,	PUNCT
ejpam-4761	16	5	[	[	X
ejpam-4761	16	6	15	15	NUM
ejpam-4761	16	7	]	]	PUNCT
ejpam-4761	16	8	,	,	PUNCT
ejpam-4761	16	9	[	[	X
ejpam-4761	16	10	16	16	NUM
ejpam-4761	16	11	]	]	PUNCT
ejpam-4761	16	12	,	,	PUNCT
ejpam-4761	17	1	[	[	X
ejpam-4761	17	2	17	17	NUM
ejpam-4761	17	3	]	]	PUNCT
ejpam-4761	17	4	,	,	PUNCT
ejpam-4761	18	1	[	[	X
ejpam-4761	18	2	22	22	NUM
ejpam-4761	18	3	]	]	PUNCT
ejpam-4761	18	4	,	,	PUNCT
ejpam-4761	18	5	[	[	X
ejpam-4761	18	6	23	23	NUM
ejpam-4761	18	7	]	]	PUNCT
ejpam-4761	18	8	)	)	PUNCT
ejpam-4761	18	9	.	.	PUNCT
ejpam-4761	19	1	some	some	DET
ejpam-4761	19	2	variations	variation	NOUN
ejpam-4761	19	3	of	of	ADP
ejpam-4761	19	4	domination	domination	NOUN
ejpam-4761	19	5	can	can	AUX
ejpam-4761	19	6	be	be	AUX
ejpam-4761	19	7	found	find	VERB
ejpam-4761	19	8	in	in	ADP
ejpam-4761	19	9	[	[	X
ejpam-4761	19	10	5	5	NUM
ejpam-4761	19	11	]	]	PUNCT
ejpam-4761	19	12	,	,	PUNCT
ejpam-4761	19	13	[	[	X
ejpam-4761	19	14	6	6	NUM
ejpam-4761	19	15	]	]	PUNCT
ejpam-4761	19	16	,	,	PUNCT
ejpam-4761	19	17	[	[	X
ejpam-4761	19	18	10	10	NUM
ejpam-4761	19	19	]	]	PUNCT
ejpam-4761	19	20	,	,	PUNCT
ejpam-4761	19	21	and	and	CCONJ
ejpam-4761	19	22	[	[	X
ejpam-4761	19	23	20	20	NUM
ejpam-4761	19	24	]	]	PUNCT
ejpam-4761	19	25	.	.	PUNCT
ejpam-4761	20	1	∗corresponding	∗corresponde	VERB
ejpam-4761	20	2	author	author	NOUN
ejpam-4761	20	3	.	.	PUNCT
ejpam-4761	21	1	doi	doi	NOUN
ejpam-4761	21	2	:	:	PUNCT
ejpam-4761	21	3	https://doi.org/10.29020/nybg.ejpam.v16i2.4761	https://doi.org/10.29020/nybg.ejpam.v16i2.4761	SYM
ejpam-4761	21	4	email	email	NOUN
ejpam-4761	21	5	addresses	address	NOUN
ejpam-4761	21	6	:	:	PUNCT
ejpam-4761	21	7	sergio.canoy@g.msuiit.edu.ph	sergio.canoy@g.msuiit.edu.ph	PROPN
ejpam-4761	21	8	(	(	PUNCT
ejpam-4761	21	9	s.	s.	PROPN
ejpam-4761	21	10	canoy	canoy	PROPN
ejpam-4761	21	11	)	)	PUNCT
ejpam-4761	21	12	,	,	PUNCT
ejpam-4761	21	13	javierhassan@msutawi-tawi.edu.ph	javierhassan@msutawi-tawi.edu.ph	PROPN
ejpam-4761	21	14	(	(	PUNCT
ejpam-4761	21	15	j.	j.	PROPN
ejpam-4761	21	16	hassan	hassan	PROPN
ejpam-4761	21	17	)	)	PUNCT
ejpam-4761	21	18	https://www.ejpam.com	https://www.ejpam.com	X
ejpam-4761	21	19	1196	1196	NUM
ejpam-4761	22	1	©	©	PROPN
ejpam-4761	22	2	2023	2023	NUM
ejpam-4761	22	3	ejpam	ejpam	NOUN
ejpam-4761	22	4	all	all	DET
ejpam-4761	22	5	rights	right	NOUN
ejpam-4761	22	6	reserved	reserve	VERB
ejpam-4761	22	7	.	.	PUNCT
ejpam-4761	23	1	s.	s.	PROPN
ejpam-4761	23	2	canoy	canoy	PROPN
ejpam-4761	23	3	jr	jr	PROPN
ejpam-4761	23	4	.	.	PROPN
ejpam-4761	23	5	,	,	PUNCT
ejpam-4761	23	6	j.	j.	PROPN
ejpam-4761	23	7	hassan	hassan	PROPN
ejpam-4761	23	8	/	/	SYM
ejpam-4761	23	9	eur	eur	PROPN
ejpam-4761	23	10	.	.	PUNCT
ejpam-4761	24	1	j.	j.	PROPN
ejpam-4761	24	2	pure	pure	PROPN
ejpam-4761	24	3	appl	appl	PROPN
ejpam-4761	24	4	.	.	PROPN
ejpam-4761	24	5	math	math	PROPN
ejpam-4761	24	6	,	,	PUNCT
ejpam-4761	24	7	16	16	NUM
ejpam-4761	24	8	(	(	PUNCT
ejpam-4761	24	9	2	2	NUM
ejpam-4761	24	10	)	)	PUNCT
ejpam-4761	24	11	(	(	PUNCT
ejpam-4761	24	12	2023	2023	NUM
ejpam-4761	24	13	)	)	PUNCT
ejpam-4761	24	14	,	,	PUNCT
ejpam-4761	24	15	1196	1196	NUM
ejpam-4761	24	16	-	-	SYM
ejpam-4761	24	17	1211	1211	NUM
ejpam-4761	24	18	1197	1197	NUM
ejpam-4761	24	19	recently	recently	ADV
ejpam-4761	24	20	,	,	PUNCT
ejpam-4761	24	21	natarajan	natarajan	PROPN
ejpam-4761	24	22	et	et	PROPN
ejpam-4761	24	23	al	al	PROPN
ejpam-4761	24	24	.	.	PUNCT
ejpam-4761	25	1	in	in	ADP
ejpam-4761	25	2	[	[	X
ejpam-4761	25	3	19	19	NUM
ejpam-4761	25	4	]	]	PUNCT
ejpam-4761	25	5	introduced	introduce	VERB
ejpam-4761	25	6	and	and	CCONJ
ejpam-4761	25	7	studied	study	VERB
ejpam-4761	25	8	the	the	DET
ejpam-4761	25	9	concept	concept	NOUN
ejpam-4761	25	10	of	of	ADP
ejpam-4761	25	11	hop	hop	NOUN
ejpam-4761	25	12	domination	domination	NOUN
ejpam-4761	25	13	.	.	PUNCT
ejpam-4761	26	1	follow	follow	VERB
ejpam-4761	26	2	-	-	PUNCT
ejpam-4761	26	3	up	up	ADP
ejpam-4761	26	4	studies	study	NOUN
ejpam-4761	26	5	are	be	AUX
ejpam-4761	26	6	done	do	VERB
ejpam-4761	26	7	in	in	ADP
ejpam-4761	26	8	[	[	X
ejpam-4761	26	9	2	2	NUM
ejpam-4761	26	10	]	]	PUNCT
ejpam-4761	26	11	,	,	PUNCT
ejpam-4761	26	12	[	[	X
ejpam-4761	26	13	3	3	NUM
ejpam-4761	26	14	]	]	PUNCT
ejpam-4761	26	15	,	,	PUNCT
ejpam-4761	26	16	[	[	X
ejpam-4761	26	17	11	11	NUM
ejpam-4761	26	18	]	]	PUNCT
ejpam-4761	26	19	,	,	PUNCT
ejpam-4761	26	20	and	and	CCONJ
ejpam-4761	26	21	[	[	X
ejpam-4761	26	22	13	13	NUM
ejpam-4761	26	23	]	]	PUNCT
ejpam-4761	26	24	.	.	PUNCT
ejpam-4761	27	1	since	since	SCONJ
ejpam-4761	27	2	its	its	PRON
ejpam-4761	27	3	introduction	introduction	NOUN
ejpam-4761	27	4	,	,	PUNCT
ejpam-4761	27	5	a	a	DET
ejpam-4761	27	6	number	number	NOUN
ejpam-4761	27	7	of	of	ADP
ejpam-4761	27	8	variations	variation	NOUN
ejpam-4761	27	9	of	of	ADP
ejpam-4761	27	10	the	the	DET
ejpam-4761	27	11	concept	concept	NOUN
ejpam-4761	27	12	have	have	AUX
ejpam-4761	27	13	already	already	ADV
ejpam-4761	27	14	been	be	AUX
ejpam-4761	27	15	defined	define	VERB
ejpam-4761	27	16	and	and	CCONJ
ejpam-4761	27	17	studied	study	VERB
ejpam-4761	27	18	(	(	PUNCT
ejpam-4761	27	19	see	see	VERB
ejpam-4761	27	20	[	[	X
ejpam-4761	27	21	7	7	NUM
ejpam-4761	27	22	]	]	PUNCT
ejpam-4761	27	23	,	,	PUNCT
ejpam-4761	27	24	[	[	X
ejpam-4761	27	25	8	8	NUM
ejpam-4761	27	26	]	]	PUNCT
ejpam-4761	27	27	,	,	PUNCT
ejpam-4761	27	28	[	[	X
ejpam-4761	27	29	18	18	NUM
ejpam-4761	27	30	]	]	PUNCT
ejpam-4761	27	31	,	,	PUNCT
ejpam-4761	27	32	[	[	X
ejpam-4761	27	33	21	21	NUM
ejpam-4761	27	34	]	]	PUNCT
ejpam-4761	27	35	,	,	PUNCT
ejpam-4761	27	36	and	and	CCONJ
ejpam-4761	27	37	[	[	X
ejpam-4761	27	38	24	24	NUM
ejpam-4761	27	39	]	]	PUNCT
ejpam-4761	27	40	)	)	PUNCT
ejpam-4761	27	41	.	.	PUNCT
ejpam-4761	28	1	previously	previously	ADV
ejpam-4761	28	2	,	,	PUNCT
ejpam-4761	28	3	the	the	DET
ejpam-4761	28	4	authors	author	NOUN
ejpam-4761	28	5	(	(	PUNCT
ejpam-4761	28	6	see	see	VERB
ejpam-4761	28	7	[	[	X
ejpam-4761	28	8	9	9	NUM
ejpam-4761	28	9	]	]	PUNCT
ejpam-4761	28	10	)	)	PUNCT
ejpam-4761	28	11	introduced	introduce	VERB
ejpam-4761	28	12	and	and	CCONJ
ejpam-4761	28	13	made	make	VERB
ejpam-4761	28	14	an	an	DET
ejpam-4761	28	15	initial	initial	ADJ
ejpam-4761	28	16	investigation	investigation	NOUN
ejpam-4761	28	17	of	of	ADP
ejpam-4761	28	18	convex	convex	PROPN
ejpam-4761	28	19	hop	hop	NOUN
ejpam-4761	28	20	domination	domination	NOUN
ejpam-4761	28	21	.	.	PUNCT
ejpam-4761	29	1	motivated	motivate	VERB
ejpam-4761	29	2	by	by	ADP
ejpam-4761	29	3	the	the	DET
ejpam-4761	29	4	studies	study	NOUN
ejpam-4761	29	5	on	on	ADP
ejpam-4761	29	6	weakly	weakly	ADJ
ejpam-4761	29	7	convex	convex	NOUN
ejpam-4761	29	8	domination	domination	NOUN
ejpam-4761	29	9	and	and	CCONJ
ejpam-4761	29	10	hop	hop	NOUN
ejpam-4761	29	11	domination	domination	NOUN
ejpam-4761	29	12	,	,	PUNCT
ejpam-4761	29	13	we	we	PRON
ejpam-4761	29	14	introduce	introduce	VERB
ejpam-4761	29	15	and	and	CCONJ
ejpam-4761	29	16	study	study	VERB
ejpam-4761	29	17	a	a	DET
ejpam-4761	29	18	new	new	ADJ
ejpam-4761	29	19	variant	variant	NOUN
ejpam-4761	29	20	of	of	ADP
ejpam-4761	29	21	hop	hop	NOUN
ejpam-4761	29	22	domination	domination	NOUN
ejpam-4761	29	23	that	that	PRON
ejpam-4761	29	24	incorporates	incorporate	VERB
ejpam-4761	29	25	the	the	DET
ejpam-4761	29	26	concept	concept	NOUN
ejpam-4761	29	27	of	of	ADP
ejpam-4761	29	28	weakly	weakly	ADJ
ejpam-4761	29	29	convex	convex	NOUN
ejpam-4761	29	30	.	.	PUNCT
ejpam-4761	30	1	this	this	DET
ejpam-4761	30	2	new	new	ADJ
ejpam-4761	30	3	parameter	parameter	NOUN
ejpam-4761	30	4	lies	lie	VERB
ejpam-4761	30	5	between	between	ADP
ejpam-4761	30	6	the	the	DET
ejpam-4761	30	7	connected	connect	VERB
ejpam-4761	30	8	hop	hop	NOUN
ejpam-4761	30	9	domination	domination	NOUN
ejpam-4761	30	10	number	number	NOUN
ejpam-4761	30	11	and	and	CCONJ
ejpam-4761	30	12	convex	convex	VERB
ejpam-4761	30	13	hop	hop	NOUN
ejpam-4761	30	14	domination	domination	NOUN
ejpam-4761	30	15	number	number	NOUN
ejpam-4761	30	16	of	of	ADP
ejpam-4761	30	17	a	a	DET
ejpam-4761	30	18	graph	graph	NOUN
ejpam-4761	30	19	.	.	PUNCT
ejpam-4761	31	1	2	2	X
ejpam-4761	31	2	.	.	X
ejpam-4761	31	3	terminology	terminology	NOUN
ejpam-4761	31	4	and	and	CCONJ
ejpam-4761	31	5	notation	notation	NOUN
ejpam-4761	31	6	let	let	VERB
ejpam-4761	31	7	g	g	NOUN
ejpam-4761	31	8	=	=	SYM
ejpam-4761	31	9	(	(	PUNCT
ejpam-4761	31	10	v	v	NOUN
ejpam-4761	31	11	(	(	PUNCT
ejpam-4761	31	12	g	g	NOUN
ejpam-4761	31	13	)	)	PUNCT
ejpam-4761	31	14	,	,	PUNCT
ejpam-4761	31	15	e(g	e(g	PROPN
ejpam-4761	31	16	)	)	PUNCT
ejpam-4761	31	17	)	)	PUNCT
ejpam-4761	31	18	be	be	AUX
ejpam-4761	31	19	a	a	DET
ejpam-4761	31	20	simple	simple	ADJ
ejpam-4761	31	21	undirected	undirected	ADJ
ejpam-4761	31	22	graph	graph	NOUN
ejpam-4761	31	23	and	and	CCONJ
ejpam-4761	31	24	let	let	VERB
ejpam-4761	31	25	u	u	PRON
ejpam-4761	31	26	and	and	CCONJ
ejpam-4761	31	27	v	v	NOUN
ejpam-4761	31	28	be	be	AUX
ejpam-4761	31	29	vertices	vertex	NOUN
ejpam-4761	31	30	of	of	ADP
ejpam-4761	31	31	g.	g.	PROPN
ejpam-4761	31	32	the	the	DET
ejpam-4761	31	33	distance	distance	NOUN
ejpam-4761	31	34	dg(u	dg(u	X
ejpam-4761	31	35	,	,	PUNCT
ejpam-4761	31	36	v	v	NOUN
ejpam-4761	31	37	)	)	PUNCT
ejpam-4761	31	38	of	of	ADP
ejpam-4761	31	39	u	u	PRON
ejpam-4761	31	40	and	and	CCONJ
ejpam-4761	31	41	v	v	NOUN
ejpam-4761	31	42	is	be	AUX
ejpam-4761	31	43	the	the	DET
ejpam-4761	31	44	length	length	NOUN
ejpam-4761	31	45	of	of	ADP
ejpam-4761	31	46	a	a	DET
ejpam-4761	31	47	shortest	short	ADJ
ejpam-4761	31	48	path	path	NOUN
ejpam-4761	31	49	joining	join	VERB
ejpam-4761	31	50	them	they	PRON
ejpam-4761	31	51	.	.	PUNCT
ejpam-4761	32	1	any	any	DET
ejpam-4761	32	2	u	u	ADJ
ejpam-4761	32	3	-	-	NOUN
ejpam-4761	32	4	v	v	ADJ
ejpam-4761	32	5	path	path	NOUN
ejpam-4761	32	6	of	of	ADP
ejpam-4761	32	7	length	length	NOUN
ejpam-4761	32	8	dg(u	dg(u	PROPN
ejpam-4761	32	9	,	,	PUNCT
ejpam-4761	32	10	v	v	NOUN
ejpam-4761	32	11	)	)	PUNCT
ejpam-4761	32	12	is	be	AUX
ejpam-4761	32	13	called	call	VERB
ejpam-4761	32	14	a	a	DET
ejpam-4761	32	15	u	u	NOUN
ejpam-4761	32	16	-	-	NOUN
ejpam-4761	32	17	v	v	ADJ
ejpam-4761	32	18	geodesic	geodesic	NOUN
ejpam-4761	32	19	.	.	PUNCT
ejpam-4761	33	1	the	the	DET
ejpam-4761	33	2	interval	interval	NOUN
ejpam-4761	33	3	ig	ig	PROPN
ejpam-4761	34	1	[	[	X
ejpam-4761	34	2	u	u	NOUN
ejpam-4761	34	3	,	,	PUNCT
ejpam-4761	34	4	v	v	NOUN
ejpam-4761	34	5	]	]	PUNCT
ejpam-4761	34	6	consists	consist	VERB
ejpam-4761	34	7	of	of	ADP
ejpam-4761	34	8	u	u	NOUN
ejpam-4761	34	9	,	,	PUNCT
ejpam-4761	34	10	v	v	NOUN
ejpam-4761	34	11	,	,	PUNCT
ejpam-4761	34	12	and	and	CCONJ
ejpam-4761	34	13	all	all	DET
ejpam-4761	34	14	vertices	vertex	NOUN
ejpam-4761	34	15	lying	lie	VERB
ejpam-4761	34	16	on	on	ADP
ejpam-4761	34	17	a	a	DET
ejpam-4761	34	18	u	u	NOUN
ejpam-4761	34	19	-	-	NOUN
ejpam-4761	34	20	v	v	ADJ
ejpam-4761	34	21	geodesic	geodesic	NOUN
ejpam-4761	34	22	.	.	PUNCT
ejpam-4761	35	1	the	the	DET
ejpam-4761	35	2	interval	interval	NOUN
ejpam-4761	35	3	ig(u	ig(u	NOUN
ejpam-4761	35	4	,	,	PUNCT
ejpam-4761	35	5	v	v	NOUN
ejpam-4761	35	6	)	)	PUNCT
ejpam-4761	35	7	=	=	PUNCT
ejpam-4761	36	1	ig	ig	PROPN
ejpam-4761	37	1	[	[	X
ejpam-4761	37	2	u	u	NOUN
ejpam-4761	37	3	,	,	PUNCT
ejpam-4761	37	4	v	v	ADP
ejpam-4761	37	5	]	]	PUNCT
ejpam-4761	37	6	\	\	NOUN
ejpam-4761	37	7	{	{	PUNCT
ejpam-4761	37	8	u	u	NOUN
ejpam-4761	37	9	,	,	PUNCT
ejpam-4761	37	10	v	v	NOUN
ejpam-4761	37	11	}	}	PUNCT
ejpam-4761	37	12	.	.	PUNCT
ejpam-4761	38	1	vertices	vertice	VERB
ejpam-4761	38	2	u	u	NOUN
ejpam-4761	38	3	and	and	CCONJ
ejpam-4761	38	4	v	v	NOUN
ejpam-4761	38	5	are	be	AUX
ejpam-4761	38	6	adjacent	adjacent	ADJ
ejpam-4761	38	7	(	(	PUNCT
ejpam-4761	38	8	or	or	CCONJ
ejpam-4761	38	9	neighbors	neighbor	NOUN
ejpam-4761	38	10	)	)	PUNCT
ejpam-4761	38	11	if	if	SCONJ
ejpam-4761	38	12	uv	uv	PROPN
ejpam-4761	38	13	∈	∈	PROPN
ejpam-4761	38	14	e(g	e(g	PROPN
ejpam-4761	38	15	)	)	PUNCT
ejpam-4761	38	16	.	.	PUNCT
ejpam-4761	39	1	the	the	DET
ejpam-4761	39	2	set	set	NOUN
ejpam-4761	39	3	of	of	ADP
ejpam-4761	39	4	neighbors	neighbor	NOUN
ejpam-4761	39	5	of	of	ADP
ejpam-4761	39	6	vertex	vertex	NOUN
ejpam-4761	39	7	u	u	NOUN
ejpam-4761	39	8	in	in	ADP
ejpam-4761	39	9	g	g	NOUN
ejpam-4761	39	10	,	,	PUNCT
ejpam-4761	39	11	denoted	denote	VERB
ejpam-4761	39	12	by	by	ADP
ejpam-4761	39	13	ng(u	ng(u	NOUN
ejpam-4761	39	14	)	)	PUNCT
ejpam-4761	39	15	,	,	PUNCT
ejpam-4761	39	16	is	be	AUX
ejpam-4761	39	17	called	call	VERB
ejpam-4761	39	18	the	the	DET
ejpam-4761	39	19	open	open	ADJ
ejpam-4761	39	20	neighborhood	neighborhood	NOUN
ejpam-4761	39	21	of	of	ADP
ejpam-4761	39	22	u.	u.	VERB
ejpam-4761	39	23	the	the	DET
ejpam-4761	39	24	closed	closed	ADJ
ejpam-4761	39	25	neighborhood	neighborhood	NOUN
ejpam-4761	39	26	of	of	ADP
ejpam-4761	39	27	u	u	NOUN
ejpam-4761	39	28	is	be	AUX
ejpam-4761	39	29	the	the	DET
ejpam-4761	39	30	set	set	NOUN
ejpam-4761	39	31	ng[u	ng[u	PROPN
ejpam-4761	39	32	]	]	X
ejpam-4761	39	33	=	=	SYM
ejpam-4761	39	34	ng(u	ng(u	PROPN
ejpam-4761	39	35	)	)	PUNCT
ejpam-4761	39	36	∪	∪	NOUN
ejpam-4761	39	37	{	{	PUNCT
ejpam-4761	39	38	u	u	NOUN
ejpam-4761	39	39	}	}	PUNCT
ejpam-4761	39	40	.	.	PUNCT
ejpam-4761	40	1	the	the	DET
ejpam-4761	40	2	open	open	ADJ
ejpam-4761	40	3	neighborhood	neighborhood	NOUN
ejpam-4761	40	4	of	of	ADP
ejpam-4761	40	5	x	x	SYM
ejpam-4761	40	6	⊆	⊆	NUM
ejpam-4761	40	7	v	v	ADP
ejpam-4761	40	8	(	(	PUNCT
ejpam-4761	40	9	g	g	NOUN
ejpam-4761	40	10	)	)	PUNCT
ejpam-4761	40	11	is	be	AUX
ejpam-4761	40	12	the	the	DET
ejpam-4761	40	13	set	set	NOUN
ejpam-4761	40	14	ng(x	ng(x	NUM
ejpam-4761	40	15	)	)	PUNCT
ejpam-4761	41	1	=	=	SYM
ejpam-4761	41	2	⋃	⋃	NOUN
ejpam-4761	41	3	w∈x	w∈x	X
ejpam-4761	41	4	ng(w	ng(w	NOUN
ejpam-4761	41	5	)	)	PUNCT
ejpam-4761	41	6	.	.	PUNCT
ejpam-4761	42	1	the	the	DET
ejpam-4761	42	2	closed	closed	ADJ
ejpam-4761	42	3	neighborhood	neighborhood	NOUN
ejpam-4761	42	4	of	of	ADP
ejpam-4761	42	5	x	x	SYM
ejpam-4761	42	6	is	be	AUX
ejpam-4761	42	7	the	the	DET
ejpam-4761	42	8	set	set	NOUN
ejpam-4761	42	9	ng[x	ng[x	PROPN
ejpam-4761	42	10	]	]	X
ejpam-4761	42	11	=	=	PUNCT
ejpam-4761	42	12	ng(x	ng(x	X
ejpam-4761	42	13	)	)	PUNCT
ejpam-4761	43	1	∪x	∪x	PROPN
ejpam-4761	43	2	.	.	PUNCT
ejpam-4761	44	1	a	a	DET
ejpam-4761	44	2	set	set	NOUN
ejpam-4761	44	3	d	d	NOUN
ejpam-4761	44	4	⊆	⊆	NUM
ejpam-4761	44	5	v	v	ADP
ejpam-4761	44	6	(	(	PUNCT
ejpam-4761	44	7	g	g	NOUN
ejpam-4761	44	8	)	)	PUNCT
ejpam-4761	44	9	is	be	AUX
ejpam-4761	44	10	dominating	dominate	VERB
ejpam-4761	44	11	(	(	PUNCT
ejpam-4761	44	12	total	total	ADJ
ejpam-4761	44	13	dominating	dominating	NOUN
ejpam-4761	44	14	)	)	PUNCT
ejpam-4761	44	15	in	in	ADP
ejpam-4761	44	16	g	g	PROPN
ejpam-4761	44	17	if	if	SCONJ
ejpam-4761	44	18	for	for	ADP
ejpam-4761	44	19	every	every	PRON
ejpam-4761	44	20	v	v	NUM
ejpam-4761	44	21	∈	∈	NOUN
ejpam-4761	44	22	v	v	NOUN
ejpam-4761	44	23	(	(	PUNCT
ejpam-4761	44	24	g	g	NOUN
ejpam-4761	44	25	)	)	PUNCT
ejpam-4761	44	26	\	\	PUNCT
ejpam-4761	45	1	d	d	X
ejpam-4761	45	2	(	(	PUNCT
ejpam-4761	45	3	resp	resp	NOUN
ejpam-4761	45	4	.	.	PUNCT
ejpam-4761	46	1	v	v	ADP
ejpam-4761	46	2	∈	∈	PROPN
ejpam-4761	46	3	v	v	NOUN
ejpam-4761	46	4	(	(	PUNCT
ejpam-4761	46	5	g	g	NOUN
ejpam-4761	46	6	)	)	PUNCT
ejpam-4761	46	7	)	)	PUNCT
ejpam-4761	47	1	,	,	PUNCT
ejpam-4761	47	2	there	there	PRON
ejpam-4761	47	3	exists	exist	VERB
ejpam-4761	47	4	u	u	NOUN
ejpam-4761	47	5	∈	∈	PROPN
ejpam-4761	47	6	d	d	ADP
ejpam-4761	47	7	such	such	ADJ
ejpam-4761	47	8	that	that	DET
ejpam-4761	47	9	uv	uv	PROPN
ejpam-4761	47	10	∈	∈	PROPN
ejpam-4761	47	11	e(g	e(g	PROPN
ejpam-4761	47	12	)	)	PUNCT
ejpam-4761	47	13	,	,	PUNCT
ejpam-4761	47	14	that	that	ADV
ejpam-4761	47	15	is	is	ADV
ejpam-4761	47	16	,	,	PUNCT
ejpam-4761	47	17	ng[d	ng[d	PROPN
ejpam-4761	47	18	]	]	PUNCT
ejpam-4761	47	19	=	=	SYM
ejpam-4761	47	20	v	v	X
ejpam-4761	47	21	(	(	PUNCT
ejpam-4761	47	22	g	g	NOUN
ejpam-4761	47	23	)	)	PUNCT
ejpam-4761	47	24	(	(	PUNCT
ejpam-4761	47	25	resp	resp	NOUN
ejpam-4761	47	26	.	.	PUNCT
ejpam-4761	47	27	ng(d	ng(d	PUNCT
ejpam-4761	47	28	)	)	PUNCT
ejpam-4761	47	29	=	=	SYM
ejpam-4761	47	30	v	v	X
ejpam-4761	47	31	(	(	PUNCT
ejpam-4761	47	32	g	g	NOUN
ejpam-4761	47	33	)	)	PUNCT
ejpam-4761	47	34	)	)	PUNCT
ejpam-4761	47	35	.	.	PUNCT
ejpam-4761	48	1	a	a	DET
ejpam-4761	48	2	vertex	vertex	NOUN
ejpam-4761	48	3	v	v	NOUN
ejpam-4761	48	4	in	in	ADP
ejpam-4761	48	5	g	g	PROPN
ejpam-4761	48	6	is	be	AUX
ejpam-4761	48	7	a	a	DET
ejpam-4761	48	8	hop	hop	NOUN
ejpam-4761	48	9	neighbor	neighbor	NOUN
ejpam-4761	48	10	of	of	ADP
ejpam-4761	48	11	vertex	vertex	NOUN
ejpam-4761	48	12	u	u	NOUN
ejpam-4761	48	13	in	in	ADP
ejpam-4761	48	14	g	g	PROPN
ejpam-4761	48	15	if	if	SCONJ
ejpam-4761	48	16	dg(u	dg(u	NOUN
ejpam-4761	48	17	,	,	PUNCT
ejpam-4761	48	18	v	v	NOUN
ejpam-4761	48	19	)	)	PUNCT
ejpam-4761	48	20	=	=	SYM
ejpam-4761	48	21	2	2	X
ejpam-4761	48	22	.	.	X
ejpam-4761	49	1	the	the	DET
ejpam-4761	49	2	set	set	ADJ
ejpam-4761	49	3	n2	n2	ADJ
ejpam-4761	49	4	g(u	g(u	PROPN
ejpam-4761	49	5	)	)	PUNCT
ejpam-4761	49	6	=	=	PRON
ejpam-4761	49	7	{	{	PUNCT
ejpam-4761	49	8	v	v	NUM
ejpam-4761	49	9	∈	∈	NOUN
ejpam-4761	49	10	v	v	NOUN
ejpam-4761	49	11	(	(	PUNCT
ejpam-4761	49	12	g	g	NOUN
ejpam-4761	49	13	)	)	PUNCT
ejpam-4761	49	14	:	:	PUNCT
ejpam-4761	49	15	dg(v	dg(v	X
ejpam-4761	49	16	,	,	PUNCT
ejpam-4761	49	17	u	u	NOUN
ejpam-4761	49	18	)	)	PUNCT
ejpam-4761	49	19	=	=	SYM
ejpam-4761	49	20	2	2	X
ejpam-4761	49	21	}	}	PUNCT
ejpam-4761	49	22	is	be	AUX
ejpam-4761	49	23	called	call	VERB
ejpam-4761	49	24	the	the	DET
ejpam-4761	49	25	open	open	ADJ
ejpam-4761	49	26	hop	hop	NOUN
ejpam-4761	49	27	neighborhood	neighborhood	NOUN
ejpam-4761	49	28	of	of	ADP
ejpam-4761	49	29	u.	u.	PROPN
ejpam-4761	49	30	the	the	DET
ejpam-4761	49	31	closed	closed	ADJ
ejpam-4761	49	32	hop	hop	NOUN
ejpam-4761	49	33	neighborhood	neighborhood	NOUN
ejpam-4761	49	34	of	of	ADP
ejpam-4761	49	35	u	u	NOUN
ejpam-4761	49	36	is	be	AUX
ejpam-4761	49	37	given	give	VERB
ejpam-4761	49	38	by	by	ADP
ejpam-4761	49	39	n2	n2	PROPN
ejpam-4761	49	40	g[u	g[u	PROPN
ejpam-4761	49	41	]	]	X
ejpam-4761	49	42	=	=	SYM
ejpam-4761	49	43	n2	n2	ADJ
ejpam-4761	49	44	g(u	g(u	PROPN
ejpam-4761	49	45	)	)	PUNCT
ejpam-4761	49	46	∪	∪	NOUN
ejpam-4761	49	47	{	{	PUNCT
ejpam-4761	49	48	u	u	NOUN
ejpam-4761	49	49	}	}	PUNCT
ejpam-4761	49	50	.	.	PUNCT
ejpam-4761	50	1	the	the	DET
ejpam-4761	50	2	open	open	ADJ
ejpam-4761	50	3	hop	hop	NOUN
ejpam-4761	50	4	neighborhood	neighborhood	NOUN
ejpam-4761	50	5	of	of	ADP
ejpam-4761	50	6	x	x	PROPN
ejpam-4761	50	7	⊆	⊆	NUM
ejpam-4761	50	8	v	v	ADP
ejpam-4761	50	9	(	(	PUNCT
ejpam-4761	50	10	g	g	NOUN
ejpam-4761	50	11	)	)	PUNCT
ejpam-4761	50	12	is	be	AUX
ejpam-4761	50	13	the	the	DET
ejpam-4761	50	14	set	set	ADJ
ejpam-4761	50	15	n2	n2	ADJ
ejpam-4761	50	16	g(x	g(x	NOUN
ejpam-4761	50	17	)	)	PUNCT
ejpam-4761	51	1	=	=	SYM
ejpam-4761	51	2	⋃	⋃	NOUN
ejpam-4761	51	3	u∈x	u∈x	ADJ
ejpam-4761	51	4	n2	n2	NOUN
ejpam-4761	51	5	g(u	g(u	PROPN
ejpam-4761	51	6	)	)	PUNCT
ejpam-4761	51	7	.	.	PUNCT
ejpam-4761	52	1	the	the	DET
ejpam-4761	52	2	closed	closed	ADJ
ejpam-4761	52	3	hop	hop	NOUN
ejpam-4761	52	4	neighborhood	neighborhood	NOUN
ejpam-4761	52	5	of	of	ADP
ejpam-4761	52	6	x	x	SYM
ejpam-4761	52	7	is	be	AUX
ejpam-4761	52	8	the	the	DET
ejpam-4761	52	9	set	set	ADJ
ejpam-4761	52	10	n2	n2	NOUN
ejpam-4761	52	11	g[x	g[x	PROPN
ejpam-4761	52	12	]	]	X
ejpam-4761	52	13	=	=	SYM
ejpam-4761	52	14	n2	n2	PROPN
ejpam-4761	52	15	g(x	g(x	NOUN
ejpam-4761	52	16	)	)	PUNCT
ejpam-4761	52	17	∪x	∪x	NUM
ejpam-4761	52	18	.	.	PUNCT
ejpam-4761	53	1	a	a	DET
ejpam-4761	53	2	set	set	NOUN
ejpam-4761	53	3	s	s	NOUN
ejpam-4761	53	4	⊆	⊆	NUM
ejpam-4761	53	5	v	v	NOUN
ejpam-4761	53	6	(	(	PUNCT
ejpam-4761	53	7	g	g	NOUN
ejpam-4761	53	8	)	)	PUNCT
ejpam-4761	53	9	is	be	AUX
ejpam-4761	53	10	hop	hop	NOUN
ejpam-4761	53	11	dominating	dominate	VERB
ejpam-4761	53	12	(	(	PUNCT
ejpam-4761	53	13	total	total	ADJ
ejpam-4761	53	14	hop	hop	NOUN
ejpam-4761	53	15	dominating	dominating	NOUN
ejpam-4761	53	16	)	)	PUNCT
ejpam-4761	53	17	in	in	ADP
ejpam-4761	53	18	g	g	PROPN
ejpam-4761	53	19	if	if	SCONJ
ejpam-4761	53	20	n2	n2	ADJ
ejpam-4761	53	21	g[s	g[s	PROPN
ejpam-4761	53	22	]	]	X
ejpam-4761	53	23	=	=	SYM
ejpam-4761	53	24	v	v	X
ejpam-4761	53	25	(	(	PUNCT
ejpam-4761	53	26	g	g	NOUN
ejpam-4761	53	27	)	)	PUNCT
ejpam-4761	53	28	(	(	PUNCT
ejpam-4761	53	29	resp	resp	NOUN
ejpam-4761	53	30	.	.	PUNCT
ejpam-4761	54	1	n2	n2	ADJ
ejpam-4761	54	2	g(s	g(s	PROPN
ejpam-4761	54	3	)	)	PUNCT
ejpam-4761	54	4	=	=	SYM
ejpam-4761	54	5	v	v	X
ejpam-4761	54	6	(	(	PUNCT
ejpam-4761	54	7	g	g	NOUN
ejpam-4761	54	8	)	)	PUNCT
ejpam-4761	54	9	)	)	PUNCT
ejpam-4761	54	10	,	,	PUNCT
ejpam-4761	54	11	that	that	ADV
ejpam-4761	54	12	is	is	ADV
ejpam-4761	54	13	,	,	PUNCT
ejpam-4761	54	14	for	for	ADP
ejpam-4761	54	15	every	every	DET
ejpam-4761	54	16	v	v	NUM
ejpam-4761	54	17	∈	∈	NOUN
ejpam-4761	54	18	v	v	NOUN
ejpam-4761	54	19	(	(	PUNCT
ejpam-4761	54	20	g)\s	g)\s	NOUN
ejpam-4761	54	21	(	(	PUNCT
ejpam-4761	54	22	resp	resp	NOUN
ejpam-4761	54	23	.	.	PUNCT
ejpam-4761	55	1	v	v	ADP
ejpam-4761	55	2	∈	∈	PROPN
ejpam-4761	55	3	v	v	NOUN
ejpam-4761	55	4	(	(	PUNCT
ejpam-4761	55	5	g	g	NOUN
ejpam-4761	55	6	)	)	PUNCT
ejpam-4761	55	7	)	)	PUNCT
ejpam-4761	56	1	,	,	PUNCT
ejpam-4761	56	2	there	there	PRON
ejpam-4761	56	3	exists	exist	VERB
ejpam-4761	56	4	u	u	PROPN
ejpam-4761	56	5	∈	∈	PROPN
ejpam-4761	56	6	s	s	VERB
ejpam-4761	56	7	such	such	ADJ
ejpam-4761	56	8	that	that	DET
ejpam-4761	56	9	dg(u	dg(u	ADJ
ejpam-4761	56	10	,	,	PUNCT
ejpam-4761	56	11	v	v	NOUN
ejpam-4761	56	12	)	)	PUNCT
ejpam-4761	56	13	=	=	SYM
ejpam-4761	56	14	2	2	X
ejpam-4761	56	15	.	.	PUNCT
ejpam-4761	56	16	the	the	DET
ejpam-4761	56	17	minimum	minimum	ADJ
ejpam-4761	56	18	cardinality	cardinality	NOUN
ejpam-4761	56	19	among	among	ADP
ejpam-4761	56	20	all	all	DET
ejpam-4761	56	21	hop	hop	NOUN
ejpam-4761	56	22	dominating	dominating	NOUN
ejpam-4761	56	23	(	(	PUNCT
ejpam-4761	56	24	resp	resp	NOUN
ejpam-4761	56	25	.	.	PUNCT
ejpam-4761	57	1	total	total	ADJ
ejpam-4761	57	2	hop	hop	NOUN
ejpam-4761	57	3	dominating	dominating	NOUN
ejpam-4761	57	4	)	)	PUNCT
ejpam-4761	57	5	sets	set	NOUN
ejpam-4761	57	6	in	in	ADP
ejpam-4761	57	7	g	g	NOUN
ejpam-4761	57	8	,	,	PUNCT
ejpam-4761	57	9	denoted	denote	VERB
ejpam-4761	57	10	by	by	ADP
ejpam-4761	57	11	γh(g	γh(g	NOUN
ejpam-4761	57	12	)	)	PUNCT
ejpam-4761	57	13	(	(	PUNCT
ejpam-4761	57	14	resp	resp	NOUN
ejpam-4761	57	15	.	.	PUNCT
ejpam-4761	58	1	γth(g	γth(g	NOUN
ejpam-4761	58	2	)	)	PUNCT
ejpam-4761	58	3	)	)	PUNCT
ejpam-4761	58	4	,	,	PUNCT
ejpam-4761	58	5	is	be	AUX
ejpam-4761	58	6	called	call	VERB
ejpam-4761	58	7	the	the	DET
ejpam-4761	58	8	hop	hop	NOUN
ejpam-4761	58	9	domination	domination	NOUN
ejpam-4761	58	10	number	number	NOUN
ejpam-4761	58	11	(	(	PUNCT
ejpam-4761	58	12	resp	resp	NOUN
ejpam-4761	58	13	.	.	PUNCT
ejpam-4761	59	1	total	total	ADJ
ejpam-4761	59	2	hop	hop	PROPN
ejpam-4761	59	3	domination	domination	NOUN
ejpam-4761	59	4	number	number	NOUN
ejpam-4761	59	5	)	)	PUNCT
ejpam-4761	59	6	of	of	ADP
ejpam-4761	59	7	g.	g.	PROPN
ejpam-4761	59	8	any	any	DET
ejpam-4761	59	9	hop	hop	NOUN
ejpam-4761	59	10	dominating	dominating	NOUN
ejpam-4761	59	11	(	(	PUNCT
ejpam-4761	59	12	resp	resp	NOUN
ejpam-4761	59	13	.	.	PUNCT
ejpam-4761	60	1	total	total	ADJ
ejpam-4761	60	2	hop	hop	NOUN
ejpam-4761	60	3	dominating	dominating	NOUN
ejpam-4761	60	4	)	)	PUNCT
ejpam-4761	60	5	set	set	VERB
ejpam-4761	60	6	with	with	ADP
ejpam-4761	60	7	cardinality	cardinality	NOUN
ejpam-4761	60	8	equal	equal	ADJ
ejpam-4761	60	9	to	to	ADP
ejpam-4761	60	10	γh(g	γh(g	NOUN
ejpam-4761	60	11	)	)	PUNCT
ejpam-4761	60	12	(	(	PUNCT
ejpam-4761	60	13	resp	resp	NOUN
ejpam-4761	60	14	.	.	PUNCT
ejpam-4761	61	1	γth(g	γth(g	NOUN
ejpam-4761	61	2	)	)	PUNCT
ejpam-4761	61	3	)	)	PUNCT
ejpam-4761	61	4	is	be	AUX
ejpam-4761	61	5	called	call	VERB
ejpam-4761	61	6	a	a	DET
ejpam-4761	61	7	γh	γh	ADV
ejpam-4761	61	8	-	-	PUNCT
ejpam-4761	61	9	set	set	VERB
ejpam-4761	61	10	(	(	PUNCT
ejpam-4761	61	11	resp	resp	NOUN
ejpam-4761	61	12	.	.	PUNCT
ejpam-4761	62	1	γth	γth	ADJ
ejpam-4761	62	2	-	-	PUNCT
ejpam-4761	62	3	set	set	NOUN
ejpam-4761	62	4	)	)	PUNCT
ejpam-4761	62	5	.	.	PUNCT
ejpam-4761	63	1	a	a	DET
ejpam-4761	63	2	hop	hop	NOUN
ejpam-4761	63	3	dominating	dominating	NOUN
ejpam-4761	63	4	set	set	NOUN
ejpam-4761	63	5	s	s	VERB
ejpam-4761	63	6	is	be	AUX
ejpam-4761	63	7	connected	connect	VERB
ejpam-4761	63	8	hop	hop	NOUN
ejpam-4761	63	9	dominating	dominating	NOUN
ejpam-4761	63	10	if	if	SCONJ
ejpam-4761	63	11	⟨s⟩	⟨s⟩	PROPN
ejpam-4761	63	12	is	be	AUX
ejpam-4761	63	13	connected	connect	VERB
ejpam-4761	63	14	.	.	PUNCT
ejpam-4761	64	1	the	the	DET
ejpam-4761	64	2	minimum	minimum	ADJ
ejpam-4761	64	3	cardinality	cardinality	NOUN
ejpam-4761	64	4	among	among	ADP
ejpam-4761	64	5	all	all	DET
ejpam-4761	64	6	connected	connect	VERB
ejpam-4761	64	7	hop	hop	NOUN
ejpam-4761	64	8	dominating	dominating	NOUN
ejpam-4761	64	9	sets	set	NOUN
ejpam-4761	64	10	in	in	ADP
ejpam-4761	64	11	g	g	NOUN
ejpam-4761	64	12	,	,	PUNCT
ejpam-4761	64	13	denoted	denote	VERB
ejpam-4761	64	14	by	by	ADP
ejpam-4761	64	15	γch(g	γch(g	NOUN
ejpam-4761	64	16	)	)	PUNCT
ejpam-4761	64	17	,	,	PUNCT
ejpam-4761	64	18	is	be	AUX
ejpam-4761	64	19	called	call	VERB
ejpam-4761	64	20	the	the	DET
ejpam-4761	64	21	connected	connect	VERB
ejpam-4761	64	22	hop	hop	NOUN
ejpam-4761	64	23	domination	domination	NOUN
ejpam-4761	64	24	number	number	NOUN
ejpam-4761	64	25	of	of	ADP
ejpam-4761	64	26	g.	g.	PROPN
ejpam-4761	64	27	any	any	DET
ejpam-4761	64	28	connected	connect	VERB
ejpam-4761	64	29	hop	hop	NOUN
ejpam-4761	64	30	dominating	dominating	NOUN
ejpam-4761	64	31	set	set	VERB
ejpam-4761	64	32	with	with	ADP
ejpam-4761	64	33	cardinality	cardinality	NOUN
ejpam-4761	64	34	equal	equal	ADJ
ejpam-4761	64	35	to	to	ADP
ejpam-4761	64	36	γch(g	γch(g	NOUN
ejpam-4761	64	37	)	)	PUNCT
ejpam-4761	64	38	is	be	AUX
ejpam-4761	64	39	called	call	VERB
ejpam-4761	64	40	a	a	DET
ejpam-4761	64	41	γch	γch	NOUN
ejpam-4761	64	42	-	-	PUNCT
ejpam-4761	64	43	set	set	NOUN
ejpam-4761	64	44	.	.	PUNCT
ejpam-4761	65	1	a	a	DET
ejpam-4761	65	2	set	set	NOUN
ejpam-4761	65	3	c	c	NOUN
ejpam-4761	65	4	⊆	⊆	NUM
ejpam-4761	65	5	v	v	NOUN
ejpam-4761	65	6	(	(	PUNCT
ejpam-4761	65	7	g	g	NOUN
ejpam-4761	65	8	)	)	PUNCT
ejpam-4761	65	9	is	be	AUX
ejpam-4761	65	10	convex	convex	ADJ
ejpam-4761	65	11	if	if	SCONJ
ejpam-4761	65	12	for	for	ADP
ejpam-4761	65	13	every	every	DET
ejpam-4761	65	14	two	two	NUM
ejpam-4761	65	15	vertices	vertex	NOUN
ejpam-4761	65	16	x	x	X
ejpam-4761	65	17	,	,	PUNCT
ejpam-4761	65	18	y	y	PROPN
ejpam-4761	65	19	∈	∈	PROPN
ejpam-4761	65	20	c	c	PROPN
ejpam-4761	65	21	,	,	PUNCT
ejpam-4761	65	22	ig[x	ig[x	PROPN
ejpam-4761	65	23	,	,	PUNCT
ejpam-4761	65	24	y	y	PROPN
ejpam-4761	65	25	]	]	X
ejpam-4761	65	26	⊆	⊆	NUM
ejpam-4761	65	27	c.	c.	NOUN
ejpam-4761	65	28	the	the	DET
ejpam-4761	65	29	largest	large	ADJ
ejpam-4761	65	30	cardinality	cardinality	NOUN
ejpam-4761	65	31	of	of	ADP
ejpam-4761	65	32	a	a	DET
ejpam-4761	65	33	proper	proper	ADJ
ejpam-4761	65	34	convex	convex	NOUN
ejpam-4761	65	35	set	set	VERB
ejpam-4761	65	36	in	in	ADP
ejpam-4761	65	37	g	g	NOUN
ejpam-4761	65	38	,	,	PUNCT
ejpam-4761	65	39	denoted	denote	VERB
ejpam-4761	65	40	by	by	ADP
ejpam-4761	65	41	con(g	con(g	NOUN
ejpam-4761	65	42	)	)	PUNCT
ejpam-4761	65	43	,	,	PUNCT
ejpam-4761	65	44	is	be	AUX
ejpam-4761	65	45	called	call	VERB
ejpam-4761	65	46	the	the	DET
ejpam-4761	65	47	convexity	convexity	NOUN
ejpam-4761	65	48	number	number	NOUN
ejpam-4761	65	49	of	of	ADP
ejpam-4761	65	50	g.	g.	PROPN
ejpam-4761	65	51	a	a	DET
ejpam-4761	65	52	set	set	NOUN
ejpam-4761	65	53	c	c	NOUN
ejpam-4761	65	54	⊆	⊆	NUM
ejpam-4761	65	55	v	v	NOUN
ejpam-4761	65	56	(	(	PUNCT
ejpam-4761	65	57	g	g	NOUN
ejpam-4761	65	58	)	)	PUNCT
ejpam-4761	65	59	is	be	AUX
ejpam-4761	65	60	convex	convex	ADJ
ejpam-4761	65	61	dominating	dominating	NOUN
ejpam-4761	65	62	(	(	PUNCT
ejpam-4761	65	63	resp	resp	NOUN
ejpam-4761	65	64	.	.	PUNCT
ejpam-4761	66	1	convex	convex	VERB
ejpam-4761	66	2	hop	hop	NOUN
ejpam-4761	66	3	dominating	dominating	NOUN
ejpam-4761	66	4	)	)	PUNCT
ejpam-4761	66	5	if	if	SCONJ
ejpam-4761	66	6	c	c	PROPN
ejpam-4761	66	7	is	be	AUX
ejpam-4761	66	8	both	both	PRON
ejpam-4761	66	9	convex	convex	ADJ
ejpam-4761	66	10	and	and	CCONJ
ejpam-4761	66	11	dominating	dominating	NOUN
ejpam-4761	66	12	(	(	PUNCT
ejpam-4761	66	13	resp	resp	NOUN
ejpam-4761	66	14	.	.	PUNCT
ejpam-4761	67	1	convex	convex	PROPN
ejpam-4761	67	2	and	and	CCONJ
ejpam-4761	67	3	hop	hop	NOUN
ejpam-4761	67	4	dominating	dominating	NOUN
ejpam-4761	67	5	)	)	PUNCT
ejpam-4761	67	6	.	.	PUNCT
ejpam-4761	68	1	the	the	DET
ejpam-4761	68	2	minimum	minimum	PROPN
ejpam-4761	68	3	cardinality	cardinality	PROPN
ejpam-4761	68	4	s.	s.	PROPN
ejpam-4761	68	5	canoy	canoy	PROPN
ejpam-4761	68	6	jr	jr	PROPN
ejpam-4761	68	7	.	.	PROPN
ejpam-4761	68	8	,	,	PUNCT
ejpam-4761	68	9	j.	j.	PROPN
ejpam-4761	68	10	hassan	hassan	PROPN
ejpam-4761	68	11	/	/	SYM
ejpam-4761	68	12	eur	eur	PROPN
ejpam-4761	68	13	.	.	PUNCT
ejpam-4761	69	1	j.	j.	PROPN
ejpam-4761	69	2	pure	pure	PROPN
ejpam-4761	69	3	appl	appl	PROPN
ejpam-4761	69	4	.	.	PROPN
ejpam-4761	69	5	math	math	PROPN
ejpam-4761	69	6	,	,	PUNCT
ejpam-4761	69	7	16	16	NUM
ejpam-4761	69	8	(	(	PUNCT
ejpam-4761	69	9	2	2	NUM
ejpam-4761	69	10	)	)	PUNCT
ejpam-4761	69	11	(	(	PUNCT
ejpam-4761	69	12	2023	2023	NUM
ejpam-4761	69	13	)	)	PUNCT
ejpam-4761	69	14	,	,	PUNCT
ejpam-4761	69	15	1196	1196	NUM
ejpam-4761	69	16	-	-	SYM
ejpam-4761	69	17	1211	1211	NUM
ejpam-4761	69	18	1198	1198	NUM
ejpam-4761	69	19	among	among	ADP
ejpam-4761	69	20	all	all	DET
ejpam-4761	69	21	convex	convex	ADJ
ejpam-4761	69	22	dominating	dominating	NOUN
ejpam-4761	69	23	(	(	PUNCT
ejpam-4761	69	24	resp	resp	NOUN
ejpam-4761	69	25	.	.	PUNCT
ejpam-4761	70	1	convex	convex	VERB
ejpam-4761	70	2	hop	hop	NOUN
ejpam-4761	70	3	dominating	dominating	NOUN
ejpam-4761	70	4	)	)	PUNCT
ejpam-4761	70	5	sets	set	NOUN
ejpam-4761	70	6	in	in	ADP
ejpam-4761	70	7	g	g	NOUN
ejpam-4761	70	8	,	,	PUNCT
ejpam-4761	70	9	denoted	denote	VERB
ejpam-4761	70	10	by	by	ADP
ejpam-4761	70	11	γcon(g	γcon(g	PROPN
ejpam-4761	70	12	)	)	PUNCT
ejpam-4761	70	13	(	(	PUNCT
ejpam-4761	70	14	resp	resp	NOUN
ejpam-4761	70	15	.	.	PUNCT
ejpam-4761	71	1	γconh(g	γconh(g	NOUN
ejpam-4761	71	2	)	)	PUNCT
ejpam-4761	71	3	)	)	PUNCT
ejpam-4761	71	4	,	,	PUNCT
ejpam-4761	71	5	is	be	AUX
ejpam-4761	71	6	called	call	VERB
ejpam-4761	71	7	the	the	DET
ejpam-4761	71	8	convex	convex	ADJ
ejpam-4761	71	9	domination	domination	NOUN
ejpam-4761	71	10	number	number	NOUN
ejpam-4761	71	11	(	(	PUNCT
ejpam-4761	71	12	resp	resp	NOUN
ejpam-4761	71	13	.	.	PUNCT
ejpam-4761	72	1	convex	convex	VERB
ejpam-4761	72	2	hop	hop	NOUN
ejpam-4761	72	3	domination	domination	NOUN
ejpam-4761	72	4	number	number	NOUN
ejpam-4761	72	5	)	)	PUNCT
ejpam-4761	72	6	of	of	ADP
ejpam-4761	72	7	g.	g.	PROPN
ejpam-4761	72	8	any	any	DET
ejpam-4761	72	9	convex	convex	NOUN
ejpam-4761	72	10	dominating	dominating	NOUN
ejpam-4761	72	11	(	(	PUNCT
ejpam-4761	72	12	resp	resp	NOUN
ejpam-4761	72	13	.	.	PUNCT
ejpam-4761	73	1	convex	convex	VERB
ejpam-4761	73	2	hop	hop	NOUN
ejpam-4761	73	3	dominating	dominating	NOUN
ejpam-4761	73	4	set	set	NOUN
ejpam-4761	73	5	)	)	PUNCT
ejpam-4761	73	6	with	with	ADP
ejpam-4761	73	7	cardinality	cardinality	NOUN
ejpam-4761	73	8	equal	equal	ADJ
ejpam-4761	73	9	to	to	ADP
ejpam-4761	73	10	γcon(g	γcon(g	PROPN
ejpam-4761	73	11	)	)	PUNCT
ejpam-4761	73	12	(	(	PUNCT
ejpam-4761	73	13	resp	resp	NOUN
ejpam-4761	73	14	.	.	PUNCT
ejpam-4761	74	1	γconh(g	γconh(g	NOUN
ejpam-4761	74	2	)	)	PUNCT
ejpam-4761	74	3	)	)	PUNCT
ejpam-4761	74	4	is	be	AUX
ejpam-4761	74	5	called	call	VERB
ejpam-4761	74	6	a	a	DET
ejpam-4761	74	7	γcon	γcon	NOUN
ejpam-4761	74	8	-	-	PUNCT
ejpam-4761	74	9	set	set	VERB
ejpam-4761	74	10	(	(	PUNCT
ejpam-4761	74	11	resp	resp	NOUN
ejpam-4761	74	12	.	.	PUNCT
ejpam-4761	75	1	γconh	γconh	NOUN
ejpam-4761	75	2	-	-	PUNCT
ejpam-4761	75	3	set	set	NOUN
ejpam-4761	75	4	)	)	PUNCT
ejpam-4761	75	5	.	.	PUNCT
ejpam-4761	76	1	a	a	DET
ejpam-4761	76	2	set	set	NOUN
ejpam-4761	76	3	w	w	PROPN
ejpam-4761	76	4	⊆	⊆	NUM
ejpam-4761	76	5	v	v	NOUN
ejpam-4761	76	6	(	(	PUNCT
ejpam-4761	76	7	g	g	NOUN
ejpam-4761	76	8	)	)	PUNCT
ejpam-4761	76	9	is	be	AUX
ejpam-4761	76	10	weakly	weakly	ADV
ejpam-4761	76	11	convex	convex	ADJ
ejpam-4761	76	12	if	if	SCONJ
ejpam-4761	76	13	for	for	ADP
ejpam-4761	76	14	every	every	DET
ejpam-4761	76	15	two	two	NUM
ejpam-4761	76	16	vertices	vertex	NOUN
ejpam-4761	76	17	x	x	X
ejpam-4761	76	18	,	,	PUNCT
ejpam-4761	76	19	y	y	PROPN
ejpam-4761	76	20	∈	∈	PROPN
ejpam-4761	76	21	w	w	NOUN
ejpam-4761	76	22	,	,	PUNCT
ejpam-4761	76	23	there	there	PRON
ejpam-4761	76	24	exists	exist	VERB
ejpam-4761	76	25	an	an	DET
ejpam-4761	76	26	x	x	NOUN
ejpam-4761	76	27	-	-	NOUN
ejpam-4761	76	28	y	y	ADJ
ejpam-4761	76	29	geodesic	geodesic	NOUN
ejpam-4761	76	30	p	p	X
ejpam-4761	76	31	(	(	PUNCT
ejpam-4761	76	32	x	x	NOUN
ejpam-4761	76	33	,	,	PUNCT
ejpam-4761	76	34	y	y	NOUN
ejpam-4761	76	35	)	)	PUNCT
ejpam-4761	76	36	such	such	ADJ
ejpam-4761	76	37	that	that	DET
ejpam-4761	76	38	v	v	NOUN
ejpam-4761	76	39	(	(	PUNCT
ejpam-4761	76	40	p	p	X
ejpam-4761	76	41	(	(	PUNCT
ejpam-4761	76	42	x	x	NOUN
ejpam-4761	76	43	,	,	PUNCT
ejpam-4761	76	44	y	y	NOUN
ejpam-4761	76	45	)	)	PUNCT
ejpam-4761	76	46	)	)	PUNCT
ejpam-4761	77	1	⊆	⊆	NUM
ejpam-4761	77	2	w	w	NOUN
ejpam-4761	77	3	.	.	PUNCT
ejpam-4761	78	1	the	the	DET
ejpam-4761	78	2	largest	large	ADJ
ejpam-4761	78	3	cardinality	cardinality	NOUN
ejpam-4761	78	4	of	of	ADP
ejpam-4761	78	5	a	a	DET
ejpam-4761	78	6	proper	proper	ADJ
ejpam-4761	78	7	weakly	weakly	ADJ
ejpam-4761	78	8	convex	convex	NOUN
ejpam-4761	78	9	set	set	VERB
ejpam-4761	78	10	in	in	ADP
ejpam-4761	78	11	g	g	NOUN
ejpam-4761	78	12	,	,	PUNCT
ejpam-4761	78	13	denoted	denote	VERB
ejpam-4761	78	14	by	by	ADP
ejpam-4761	78	15	wcon(g	wcon(g	PROPN
ejpam-4761	78	16	)	)	PUNCT
ejpam-4761	78	17	,	,	PUNCT
ejpam-4761	78	18	is	be	AUX
ejpam-4761	78	19	called	call	VERB
ejpam-4761	78	20	the	the	DET
ejpam-4761	78	21	weakly	weakly	ADJ
ejpam-4761	78	22	convexity	convexity	NOUN
ejpam-4761	78	23	number	number	NOUN
ejpam-4761	78	24	of	of	ADP
ejpam-4761	78	25	g.	g.	PROPN
ejpam-4761	78	26	a	a	DET
ejpam-4761	78	27	set	set	NOUN
ejpam-4761	78	28	w	w	PROPN
ejpam-4761	78	29	⊆	⊆	NUM
ejpam-4761	78	30	v	v	NOUN
ejpam-4761	78	31	(	(	PUNCT
ejpam-4761	78	32	g	g	NOUN
ejpam-4761	78	33	)	)	PUNCT
ejpam-4761	78	34	is	be	AUX
ejpam-4761	78	35	weakly	weakly	ADJ
ejpam-4761	78	36	convex	convex	ADJ
ejpam-4761	78	37	dominating	dominating	NOUN
ejpam-4761	78	38	(	(	PUNCT
ejpam-4761	78	39	resp	resp	NOUN
ejpam-4761	78	40	.	.	PUNCT
ejpam-4761	79	1	weakly	weakly	ADJ
ejpam-4761	79	2	convex	convex	VERB
ejpam-4761	79	3	hop	hop	NOUN
ejpam-4761	79	4	dominating	dominating	NOUN
ejpam-4761	79	5	,	,	PUNCT
ejpam-4761	79	6	weakly	weakly	ADJ
ejpam-4761	79	7	convex	convex	ADJ
ejpam-4761	79	8	total	total	ADJ
ejpam-4761	79	9	hop	hop	NOUN
ejpam-4761	79	10	dominating	dominating	NOUN
ejpam-4761	79	11	)	)	PUNCT
ejpam-4761	80	1	if	if	SCONJ
ejpam-4761	80	2	c	c	PROPN
ejpam-4761	80	3	is	be	AUX
ejpam-4761	80	4	both	both	PRON
ejpam-4761	80	5	weakly	weakly	ADJ
ejpam-4761	80	6	convex	convex	NOUN
ejpam-4761	80	7	and	and	CCONJ
ejpam-4761	80	8	dominating	dominating	NOUN
ejpam-4761	80	9	(	(	PUNCT
ejpam-4761	80	10	resp	resp	NOUN
ejpam-4761	80	11	.	.	PUNCT
ejpam-4761	81	1	weakly	weakly	ADJ
ejpam-4761	81	2	convex	convex	NOUN
ejpam-4761	81	3	and	and	CCONJ
ejpam-4761	81	4	hop	hop	NOUN
ejpam-4761	81	5	dominating	dominating	NOUN
ejpam-4761	81	6	,	,	PUNCT
ejpam-4761	81	7	weakly	weakly	ADJ
ejpam-4761	81	8	convex	convex	NOUN
ejpam-4761	81	9	and	and	CCONJ
ejpam-4761	81	10	total	total	ADJ
ejpam-4761	81	11	hop	hop	NOUN
ejpam-4761	81	12	dominating	dominating	NOUN
ejpam-4761	81	13	)	)	PUNCT
ejpam-4761	81	14	.	.	PUNCT
ejpam-4761	82	1	the	the	DET
ejpam-4761	82	2	minimum	minimum	ADJ
ejpam-4761	82	3	cardinality	cardinality	NOUN
ejpam-4761	82	4	among	among	ADP
ejpam-4761	82	5	all	all	DET
ejpam-4761	82	6	weakly	weakly	ADJ
ejpam-4761	82	7	convex	convex	NOUN
ejpam-4761	82	8	dominating	dominating	NOUN
ejpam-4761	82	9	(	(	PUNCT
ejpam-4761	82	10	resp	resp	NOUN
ejpam-4761	82	11	.	.	PUNCT
ejpam-4761	83	1	weakly	weakly	ADJ
ejpam-4761	83	2	convex	convex	VERB
ejpam-4761	83	3	hop	hop	NOUN
ejpam-4761	83	4	dominating	dominating	NOUN
ejpam-4761	83	5	,	,	PUNCT
ejpam-4761	83	6	weakly	weakly	ADJ
ejpam-4761	83	7	convex	convex	ADJ
ejpam-4761	83	8	total	total	ADJ
ejpam-4761	83	9	hop	hop	NOUN
ejpam-4761	83	10	dominating	dominating	NOUN
ejpam-4761	83	11	)	)	PUNCT
ejpam-4761	83	12	sets	set	NOUN
ejpam-4761	83	13	in	in	ADP
ejpam-4761	83	14	g	g	NOUN
ejpam-4761	83	15	,	,	PUNCT
ejpam-4761	83	16	denoted	denote	VERB
ejpam-4761	83	17	by	by	ADP
ejpam-4761	83	18	γwcon(g	γwcon(g	NOUN
ejpam-4761	83	19	)	)	PUNCT
ejpam-4761	83	20	(	(	PUNCT
ejpam-4761	83	21	resp	resp	NOUN
ejpam-4761	83	22	.	.	PUNCT
ejpam-4761	84	1	γwconh(g	γwconh(g	NOUN
ejpam-4761	84	2	)	)	PUNCT
ejpam-4761	84	3	,	,	PUNCT
ejpam-4761	84	4	γwconth(g	γwconth(g	PROPN
ejpam-4761	84	5	)	)	PUNCT
ejpam-4761	84	6	)	)	PUNCT
ejpam-4761	84	7	,	,	PUNCT
ejpam-4761	84	8	is	be	AUX
ejpam-4761	84	9	called	call	VERB
ejpam-4761	84	10	the	the	DET
ejpam-4761	84	11	weakly	weakly	ADJ
ejpam-4761	84	12	convex	convex	ADJ
ejpam-4761	84	13	domination	domination	NOUN
ejpam-4761	84	14	number	number	NOUN
ejpam-4761	84	15	(	(	PUNCT
ejpam-4761	84	16	resp	resp	NOUN
ejpam-4761	84	17	.	.	PUNCT
ejpam-4761	85	1	weakly	weakly	ADJ
ejpam-4761	85	2	convex	convex	VERB
ejpam-4761	85	3	hop	hop	NOUN
ejpam-4761	85	4	domination	domination	NOUN
ejpam-4761	85	5	number	number	NOUN
ejpam-4761	85	6	,	,	PUNCT
ejpam-4761	85	7	weakly	weakly	ADJ
ejpam-4761	85	8	convex	convex	ADJ
ejpam-4761	85	9	total	total	ADJ
ejpam-4761	85	10	hop	hop	NOUN
ejpam-4761	85	11	domination	domination	NOUN
ejpam-4761	85	12	number	number	NOUN
ejpam-4761	85	13	)	)	PUNCT
ejpam-4761	85	14	ofg	ofg	NOUN
ejpam-4761	85	15	.	.	PUNCT
ejpam-4761	86	1	any	any	DET
ejpam-4761	86	2	weakly	weakly	ADJ
ejpam-4761	86	3	convex	convex	NOUN
ejpam-4761	86	4	dominating	dominating	NOUN
ejpam-4761	86	5	(	(	PUNCT
ejpam-4761	86	6	resp	resp	NOUN
ejpam-4761	86	7	.	.	PUNCT
ejpam-4761	87	1	weakly	weakly	ADJ
ejpam-4761	87	2	convex	convex	VERB
ejpam-4761	87	3	hop	hop	NOUN
ejpam-4761	87	4	dominating	dominating	NOUN
ejpam-4761	87	5	,	,	PUNCT
ejpam-4761	87	6	weakly	weakly	ADJ
ejpam-4761	87	7	convex	convex	ADJ
ejpam-4761	87	8	total	total	ADJ
ejpam-4761	87	9	hop	hop	NOUN
ejpam-4761	87	10	dominating	dominating	NOUN
ejpam-4761	87	11	)	)	PUNCT
ejpam-4761	87	12	set	set	VERB
ejpam-4761	87	13	with	with	ADP
ejpam-4761	87	14	cardinality	cardinality	NOUN
ejpam-4761	87	15	equal	equal	ADJ
ejpam-4761	87	16	to	to	ADP
ejpam-4761	87	17	γwcon(g	γwcon(g	NUM
ejpam-4761	87	18	)	)	PUNCT
ejpam-4761	87	19	(	(	PUNCT
ejpam-4761	87	20	resp	resp	NOUN
ejpam-4761	87	21	.	.	PUNCT
ejpam-4761	88	1	γwconh(g	γwconh(g	NOUN
ejpam-4761	88	2	)	)	PUNCT
ejpam-4761	88	3	,	,	PUNCT
ejpam-4761	88	4	γwconth(g	γwconth(g	PROPN
ejpam-4761	88	5	)	)	PUNCT
ejpam-4761	88	6	)	)	PUNCT
ejpam-4761	88	7	is	be	AUX
ejpam-4761	88	8	called	call	VERB
ejpam-4761	88	9	a	a	DET
ejpam-4761	88	10	γwcon	γwcon	NOUN
ejpam-4761	88	11	-	-	PUNCT
ejpam-4761	88	12	set	set	VERB
ejpam-4761	88	13	(	(	PUNCT
ejpam-4761	88	14	resp	resp	NOUN
ejpam-4761	88	15	.	.	PUNCT
ejpam-4761	88	16	γwconh	γwconh	NOUN
ejpam-4761	88	17	-	-	PUNCT
ejpam-4761	88	18	set	set	VERB
ejpam-4761	88	19	,	,	PUNCT
ejpam-4761	88	20	γwconth	γwconth	NOUN
ejpam-4761	88	21	-	-	PUNCT
ejpam-4761	88	22	set	set	NOUN
ejpam-4761	88	23	)	)	PUNCT
ejpam-4761	88	24	.	.	PUNCT
ejpam-4761	89	1	a	a	DET
ejpam-4761	89	2	set	set	NOUN
ejpam-4761	89	3	c	c	NOUN
ejpam-4761	89	4	⊆	⊆	NUM
ejpam-4761	89	5	v	v	NOUN
ejpam-4761	89	6	(	(	PUNCT
ejpam-4761	89	7	g	g	NOUN
ejpam-4761	89	8	)	)	PUNCT
ejpam-4761	89	9	is	be	AUX
ejpam-4761	89	10	pointwise	pointwise	VERB
ejpam-4761	89	11	non	non	ADJ
ejpam-4761	89	12	-	-	ADJ
ejpam-4761	89	13	dominating	dominating	ADJ
ejpam-4761	89	14	if	if	SCONJ
ejpam-4761	89	15	for	for	ADP
ejpam-4761	89	16	every	every	PRON
ejpam-4761	89	17	v	v	NUM
ejpam-4761	89	18	∈	∈	NOUN
ejpam-4761	89	19	v	v	NOUN
ejpam-4761	89	20	(	(	PUNCT
ejpam-4761	89	21	g	g	NOUN
ejpam-4761	89	22	)	)	PUNCT
ejpam-4761	89	23	\	\	PUNCT
ejpam-4761	90	1	c	c	X
ejpam-4761	90	2	,	,	PUNCT
ejpam-4761	90	3	there	there	PRON
ejpam-4761	90	4	exists	exist	VERB
ejpam-4761	90	5	u	u	PROPN
ejpam-4761	90	6	∈	∈	PROPN
ejpam-4761	90	7	c	c	NOUN
ejpam-4761	90	8	such	such	ADJ
ejpam-4761	90	9	that	that	DET
ejpam-4761	90	10	v	v	NOUN
ejpam-4761	90	11	/∈	/∈	PUNCT
ejpam-4761	90	12	ng(u	ng(u	NOUN
ejpam-4761	90	13	)	)	PUNCT
ejpam-4761	90	14	.	.	PUNCT
ejpam-4761	91	1	the	the	DET
ejpam-4761	91	2	minimum	minimum	ADJ
ejpam-4761	91	3	cardinality	cardinality	NOUN
ejpam-4761	91	4	of	of	ADP
ejpam-4761	91	5	a	a	DET
ejpam-4761	91	6	pointwise	pointwise	ADJ
ejpam-4761	91	7	non	non	ADJ
ejpam-4761	91	8	-	-	ADJ
ejpam-4761	91	9	dominating	dominating	ADJ
ejpam-4761	91	10	set	set	NOUN
ejpam-4761	91	11	in	in	ADP
ejpam-4761	91	12	g	g	NOUN
ejpam-4761	91	13	,	,	PUNCT
ejpam-4761	91	14	denoted	denote	VERB
ejpam-4761	91	15	by	by	ADP
ejpam-4761	91	16	pnd(g	pnd(g	PROPN
ejpam-4761	91	17	)	)	PUNCT
ejpam-4761	91	18	,	,	PUNCT
ejpam-4761	91	19	is	be	AUX
ejpam-4761	91	20	called	call	VERB
ejpam-4761	91	21	the	the	DET
ejpam-4761	91	22	pointwise	pointwise	ADJ
ejpam-4761	91	23	non	non	ADJ
ejpam-4761	91	24	-	-	ADJ
ejpam-4761	91	25	domination	domination	ADJ
ejpam-4761	91	26	number	number	NOUN
ejpam-4761	91	27	of	of	ADP
ejpam-4761	91	28	g.	g.	PROPN
ejpam-4761	91	29	the	the	DET
ejpam-4761	91	30	shadow	shadow	NOUN
ejpam-4761	91	31	graph	graph	NOUN
ejpam-4761	91	32	s(g	s(g	PROPN
ejpam-4761	91	33	)	)	PUNCT
ejpam-4761	91	34	of	of	ADP
ejpam-4761	91	35	graph	graph	NOUN
ejpam-4761	91	36	g	g	PROPN
ejpam-4761	91	37	is	be	AUX
ejpam-4761	91	38	constructed	construct	VERB
ejpam-4761	91	39	by	by	ADP
ejpam-4761	91	40	taking	take	VERB
ejpam-4761	91	41	two	two	NUM
ejpam-4761	91	42	copies	copy	NOUN
ejpam-4761	91	43	of	of	ADP
ejpam-4761	91	44	g	g	NOUN
ejpam-4761	91	45	,	,	PUNCT
ejpam-4761	91	46	say	say	VERB
ejpam-4761	91	47	g1	g1	PROPN
ejpam-4761	91	48	and	and	CCONJ
ejpam-4761	91	49	g2	g2	PROPN
ejpam-4761	91	50	,	,	PUNCT
ejpam-4761	91	51	and	and	CCONJ
ejpam-4761	91	52	then	then	ADV
ejpam-4761	91	53	joining	join	VERB
ejpam-4761	91	54	each	each	DET
ejpam-4761	91	55	vertex	vertex	NOUN
ejpam-4761	91	56	u	u	NOUN
ejpam-4761	91	57	∈	∈	PROPN
ejpam-4761	91	58	v	v	NOUN
ejpam-4761	91	59	(	(	PUNCT
ejpam-4761	91	60	g1	g1	PROPN
ejpam-4761	91	61	)	)	PUNCT
ejpam-4761	91	62	to	to	ADP
ejpam-4761	91	63	the	the	DET
ejpam-4761	91	64	neighbors	neighbor	NOUN
ejpam-4761	91	65	of	of	ADP
ejpam-4761	91	66	its	its	PRON
ejpam-4761	91	67	corresponding	correspond	VERB
ejpam-4761	91	68	vertex	vertex	NOUN
ejpam-4761	91	69	u′	u′	PROPN
ejpam-4761	91	70	∈	∈	PROPN
ejpam-4761	91	71	v	v	NOUN
ejpam-4761	91	72	(	(	PUNCT
ejpam-4761	91	73	g2	g2	PROPN
ejpam-4761	91	74	)	)	PUNCT
ejpam-4761	91	75	.	.	PUNCT
ejpam-4761	92	1	for	for	ADP
ejpam-4761	92	2	a	a	DET
ejpam-4761	92	3	graph	graph	NOUN
ejpam-4761	92	4	g	g	NOUN
ejpam-4761	92	5	,	,	PUNCT
ejpam-4761	92	6	the	the	DET
ejpam-4761	92	7	complementary	complementary	ADJ
ejpam-4761	92	8	prism	prism	NOUN
ejpam-4761	92	9	gg	gg	PROPN
ejpam-4761	92	10	is	be	AUX
ejpam-4761	92	11	formed	form	VERB
ejpam-4761	92	12	from	from	ADP
ejpam-4761	92	13	the	the	DET
ejpam-4761	92	14	disjoint	disjoint	PROPN
ejpam-4761	92	15	union	union	NOUN
ejpam-4761	92	16	of	of	ADP
ejpam-4761	92	17	g	g	PROPN
ejpam-4761	92	18	and	and	CCONJ
ejpam-4761	92	19	its	its	PRON
ejpam-4761	92	20	complement	complement	NOUN
ejpam-4761	92	21	g	g	NOUN
ejpam-4761	92	22	by	by	ADP
ejpam-4761	92	23	adding	add	VERB
ejpam-4761	92	24	a	a	DET
ejpam-4761	92	25	perfect	perfect	ADJ
ejpam-4761	92	26	matching	matching	NOUN
ejpam-4761	92	27	between	between	ADP
ejpam-4761	92	28	corresponding	corresponding	ADJ
ejpam-4761	92	29	vertices	vertex	NOUN
ejpam-4761	92	30	of	of	ADP
ejpam-4761	92	31	g	g	PROPN
ejpam-4761	92	32	and	and	CCONJ
ejpam-4761	92	33	g.	g.	NOUN
ejpam-4761	92	34	for	for	ADP
ejpam-4761	92	35	each	each	DET
ejpam-4761	92	36	v	v	NUM
ejpam-4761	92	37	∈	∈	PROPN
ejpam-4761	92	38	v	v	NOUN
ejpam-4761	92	39	(	(	PUNCT
ejpam-4761	92	40	g	g	NOUN
ejpam-4761	92	41	)	)	PUNCT
ejpam-4761	92	42	,	,	PUNCT
ejpam-4761	92	43	let	let	VERB
ejpam-4761	92	44	v	v	PART
ejpam-4761	92	45	denote	denote	VERB
ejpam-4761	92	46	the	the	DET
ejpam-4761	92	47	vertex	vertex	NOUN
ejpam-4761	92	48	in	in	ADP
ejpam-4761	92	49	g	g	NOUN
ejpam-4761	92	50	corresponding	correspond	VERB
ejpam-4761	92	51	to	to	ADP
ejpam-4761	92	52	v.	v.	PROPN
ejpam-4761	92	53	in	in	ADP
ejpam-4761	92	54	simple	simple	ADJ
ejpam-4761	92	55	terms	term	NOUN
ejpam-4761	92	56	,	,	PUNCT
ejpam-4761	92	57	the	the	DET
ejpam-4761	92	58	graph	graph	NOUN
ejpam-4761	92	59	gg	gg	NOUN
ejpam-4761	92	60	is	be	AUX
ejpam-4761	92	61	formed	form	VERB
ejpam-4761	92	62	from	from	ADP
ejpam-4761	92	63	g∪g	g∪g	NOUN
ejpam-4761	92	64	by	by	ADP
ejpam-4761	92	65	adding	add	VERB
ejpam-4761	92	66	the	the	DET
ejpam-4761	92	67	edge	edge	NOUN
ejpam-4761	92	68	vv	vv	NOUN
ejpam-4761	92	69	for	for	ADP
ejpam-4761	92	70	every	every	DET
ejpam-4761	92	71	vertex	vertex	NOUN
ejpam-4761	92	72	v	v	ADP
ejpam-4761	92	73	∈	∈	NOUN
ejpam-4761	92	74	v	v	NOUN
ejpam-4761	92	75	(	(	PUNCT
ejpam-4761	92	76	g	g	NOUN
ejpam-4761	92	77	)	)	PUNCT
ejpam-4761	92	78	.	.	PUNCT
ejpam-4761	93	1	let	let	VERB
ejpam-4761	93	2	g	g	NOUN
ejpam-4761	93	3	and	and	CCONJ
ejpam-4761	93	4	h	h	NOUN
ejpam-4761	93	5	be	be	VERB
ejpam-4761	93	6	any	any	DET
ejpam-4761	93	7	two	two	NUM
ejpam-4761	93	8	graphs	graph	NOUN
ejpam-4761	93	9	.	.	PUNCT
ejpam-4761	94	1	the	the	DET
ejpam-4761	94	2	join	join	NOUN
ejpam-4761	94	3	g	g	PROPN
ejpam-4761	94	4	+	+	CCONJ
ejpam-4761	94	5	h	h	NOUN
ejpam-4761	94	6	is	be	AUX
ejpam-4761	94	7	the	the	DET
ejpam-4761	94	8	graph	graph	NOUN
ejpam-4761	94	9	with	with	ADP
ejpam-4761	94	10	vertex	vertex	NOUN
ejpam-4761	94	11	set	set	VERB
ejpam-4761	94	12	v	v	NOUN
ejpam-4761	94	13	(	(	PUNCT
ejpam-4761	94	14	g+h	g+h	NOUN
ejpam-4761	94	15	)	)	PUNCT
ejpam-4761	94	16	=	=	SYM
ejpam-4761	94	17	v	v	NOUN
ejpam-4761	94	18	(	(	PUNCT
ejpam-4761	94	19	g)∪	g)∪	VERB
ejpam-4761	94	20	v	v	NUM
ejpam-4761	94	21	(	(	PUNCT
ejpam-4761	94	22	h	h	NOUN
ejpam-4761	94	23	)	)	PUNCT
ejpam-4761	94	24	and	and	CCONJ
ejpam-4761	94	25	edge	edge	NOUN
ejpam-4761	94	26	set	set	VERB
ejpam-4761	94	27	e(g+h	e(g+h	NUM
ejpam-4761	94	28	)	)	PUNCT
ejpam-4761	95	1	=	=	SYM
ejpam-4761	95	2	e(g)∪e(h)∪	e(g)∪e(h)∪	NOUN
ejpam-4761	95	3	{	{	PUNCT
ejpam-4761	95	4	uv	uv	NOUN
ejpam-4761	95	5	:	:	PUNCT
ejpam-4761	95	6	u	u	PROPN
ejpam-4761	95	7	∈	∈	PROPN
ejpam-4761	95	8	v	v	ADP
ejpam-4761	95	9	(	(	PUNCT
ejpam-4761	95	10	g	g	NOUN
ejpam-4761	95	11	)	)	PUNCT
ejpam-4761	95	12	,	,	PUNCT
ejpam-4761	95	13	v	v	X
ejpam-4761	95	14	∈	∈	PROPN
ejpam-4761	95	15	v	v	NOUN
ejpam-4761	95	16	(	(	PUNCT
ejpam-4761	95	17	h	h	NOUN
ejpam-4761	95	18	)	)	PUNCT
ejpam-4761	95	19	}	}	PUNCT
ejpam-4761	95	20	.	.	PUNCT
ejpam-4761	96	1	the	the	DET
ejpam-4761	96	2	corona	corona	NOUN
ejpam-4761	96	3	g	g	PROPN
ejpam-4761	96	4	◦	◦	NOUN
ejpam-4761	96	5	h	h	NOUN
ejpam-4761	96	6	is	be	AUX
ejpam-4761	96	7	the	the	DET
ejpam-4761	96	8	graph	graph	NOUN
ejpam-4761	96	9	obtained	obtain	VERB
ejpam-4761	96	10	by	by	ADP
ejpam-4761	96	11	taking	take	VERB
ejpam-4761	96	12	one	one	NUM
ejpam-4761	96	13	copy	copy	NOUN
ejpam-4761	96	14	of	of	ADP
ejpam-4761	96	15	g	g	PROPN
ejpam-4761	96	16	and	and	CCONJ
ejpam-4761	96	17	|v	|v	PROPN
ejpam-4761	96	18	(	(	PUNCT
ejpam-4761	96	19	g)|	g)|	NOUN
ejpam-4761	96	20	copies	copy	NOUN
ejpam-4761	96	21	of	of	ADP
ejpam-4761	96	22	h	h	NOUN
ejpam-4761	96	23	,	,	PUNCT
ejpam-4761	96	24	and	and	CCONJ
ejpam-4761	96	25	then	then	ADV
ejpam-4761	96	26	joining	join	VERB
ejpam-4761	96	27	the	the	DET
ejpam-4761	96	28	ith	ith	PROPN
ejpam-4761	96	29	vertex	vertex	NOUN
ejpam-4761	96	30	of	of	ADP
ejpam-4761	96	31	g	g	NOUN
ejpam-4761	96	32	to	to	ADP
ejpam-4761	96	33	every	every	DET
ejpam-4761	96	34	vertex	vertex	NOUN
ejpam-4761	96	35	of	of	ADP
ejpam-4761	96	36	the	the	DET
ejpam-4761	96	37	ith	ith	PROPN
ejpam-4761	96	38	copy	copy	NOUN
ejpam-4761	96	39	of	of	ADP
ejpam-4761	96	40	h.	h.	PROPN
ejpam-4761	96	41	we	we	PRON
ejpam-4761	96	42	denote	denote	VERB
ejpam-4761	96	43	by	by	ADP
ejpam-4761	96	44	hv	hv	PROPN
ejpam-4761	97	1	the	the	DET
ejpam-4761	97	2	copy	copy	NOUN
ejpam-4761	97	3	of	of	ADP
ejpam-4761	97	4	h	h	NOUN
ejpam-4761	97	5	in	in	ADP
ejpam-4761	97	6	g	g	PROPN
ejpam-4761	97	7	◦	◦	NOUN
ejpam-4761	97	8	h	h	NOUN
ejpam-4761	97	9	corresponding	correspond	VERB
ejpam-4761	97	10	to	to	ADP
ejpam-4761	97	11	the	the	DET
ejpam-4761	97	12	vertex	vertex	NOUN
ejpam-4761	97	13	v	v	ADP
ejpam-4761	97	14	∈	∈	PROPN
ejpam-4761	97	15	g	g	NOUN
ejpam-4761	97	16	and	and	CCONJ
ejpam-4761	97	17	write	write	VERB
ejpam-4761	97	18	v	v	ADP
ejpam-4761	97	19	+	+	CCONJ
ejpam-4761	97	20	hv	hv	NOUN
ejpam-4761	97	21	for	for	ADP
ejpam-4761	97	22	⟨{v}⟩	⟨{v}⟩	NOUN
ejpam-4761	97	23	+	+	X
ejpam-4761	97	24	hv	hv	X
ejpam-4761	97	25	.	.	PUNCT
ejpam-4761	98	1	the	the	DET
ejpam-4761	98	2	lexicographic	lexicographic	ADJ
ejpam-4761	98	3	product	product	NOUN
ejpam-4761	98	4	g[h	g[h	PROPN
ejpam-4761	98	5	]	]	PUNCT
ejpam-4761	98	6	is	be	AUX
ejpam-4761	98	7	the	the	DET
ejpam-4761	98	8	graph	graph	NOUN
ejpam-4761	98	9	with	with	ADP
ejpam-4761	98	10	vertex	vertex	NOUN
ejpam-4761	98	11	set	set	VERB
ejpam-4761	98	12	v	v	NOUN
ejpam-4761	98	13	(	(	PUNCT
ejpam-4761	98	14	g[h	g[h	PROPN
ejpam-4761	98	15	]	]	PUNCT
ejpam-4761	98	16	)	)	PUNCT
ejpam-4761	98	17	=	=	SYM
ejpam-4761	98	18	v	v	X
ejpam-4761	98	19	(	(	PUNCT
ejpam-4761	98	20	g	g	NOUN
ejpam-4761	98	21	)	)	PUNCT
ejpam-4761	98	22	×	×	NOUN
ejpam-4761	98	23	v	v	NOUN
ejpam-4761	98	24	(	(	PUNCT
ejpam-4761	98	25	h	h	NOUN
ejpam-4761	98	26	)	)	PUNCT
ejpam-4761	98	27	and	and	CCONJ
ejpam-4761	98	28	(	(	PUNCT
ejpam-4761	98	29	v	v	NOUN
ejpam-4761	98	30	,	,	PUNCT
ejpam-4761	98	31	a)(u	a)(u	ADJ
ejpam-4761	98	32	,	,	PUNCT
ejpam-4761	98	33	b	b	X
ejpam-4761	98	34	)	)	PUNCT
ejpam-4761	98	35	∈	∈	NOUN
ejpam-4761	98	36	e(g[h	e(g[h	NOUN
ejpam-4761	98	37	]	]	PUNCT
ejpam-4761	98	38	)	)	PUNCT
ejpam-4761	98	39	if	if	SCONJ
ejpam-4761	98	40	and	and	CCONJ
ejpam-4761	98	41	only	only	ADV
ejpam-4761	98	42	if	if	SCONJ
ejpam-4761	98	43	either	either	DET
ejpam-4761	98	44	uv	uv	PROPN
ejpam-4761	98	45	∈	∈	PROPN
ejpam-4761	98	46	e(g	e(g	PROPN
ejpam-4761	98	47	)	)	PUNCT
ejpam-4761	98	48	or	or	CCONJ
ejpam-4761	98	49	u	u	X
ejpam-4761	98	50	=	=	PROPN
ejpam-4761	98	51	v	v	PROPN
ejpam-4761	98	52	and	and	CCONJ
ejpam-4761	98	53	ab	ab	PROPN
ejpam-4761	98	54	∈	∈	PROPN
ejpam-4761	98	55	e(h	e(h	PROPN
ejpam-4761	98	56	)	)	PUNCT
ejpam-4761	98	57	.	.	PUNCT
ejpam-4761	99	1	any	any	DET
ejpam-4761	99	2	non	non	ADJ
ejpam-4761	99	3	-	-	ADJ
ejpam-4761	99	4	empty	empty	ADJ
ejpam-4761	99	5	set	set	NOUN
ejpam-4761	99	6	c	c	NOUN
ejpam-4761	99	7	⊆	⊆	NUM
ejpam-4761	99	8	v	v	NOUN
ejpam-4761	99	9	(	(	PUNCT
ejpam-4761	99	10	g	g	NOUN
ejpam-4761	99	11	)	)	PUNCT
ejpam-4761	99	12	×	×	NOUN
ejpam-4761	99	13	v	v	NOUN
ejpam-4761	99	14	(	(	PUNCT
ejpam-4761	99	15	h	h	NOUN
ejpam-4761	99	16	)	)	PUNCT
ejpam-4761	99	17	can	can	AUX
ejpam-4761	99	18	be	be	AUX
ejpam-4761	99	19	expressed	express	VERB
ejpam-4761	99	20	as	as	ADP
ejpam-4761	99	21	c	c	NOUN
ejpam-4761	99	22	=	=	PUNCT
ejpam-4761	99	23	⋃	⋃	PROPN
ejpam-4761	99	24	x∈s	x∈s	NOUN
ejpam-4761	100	1	[	[	X
ejpam-4761	100	2	{	{	PUNCT
ejpam-4761	100	3	x	x	NOUN
ejpam-4761	100	4	}	}	PUNCT
ejpam-4761	100	5	×	×	PROPN
ejpam-4761	100	6	tx	tx	PROPN
ejpam-4761	100	7	]	]	X
ejpam-4761	100	8	,	,	PUNCT
ejpam-4761	100	9	where	where	SCONJ
ejpam-4761	100	10	s	s	VERB
ejpam-4761	100	11	⊆	⊆	NUM
ejpam-4761	100	12	v	v	NOUN
ejpam-4761	100	13	(	(	PUNCT
ejpam-4761	100	14	g	g	NOUN
ejpam-4761	100	15	)	)	PUNCT
ejpam-4761	100	16	and	and	CCONJ
ejpam-4761	100	17	tx	tx	VERB
ejpam-4761	100	18	⊆	⊆	NUM
ejpam-4761	100	19	v	v	NOUN
ejpam-4761	100	20	(	(	PUNCT
ejpam-4761	100	21	h	h	NOUN
ejpam-4761	100	22	)	)	PUNCT
ejpam-4761	100	23	for	for	ADP
ejpam-4761	100	24	each	each	DET
ejpam-4761	100	25	x	x	PROPN
ejpam-4761	100	26	∈	∈	PROPN
ejpam-4761	100	27	s.	s.	PROPN
ejpam-4761	100	28	specifically	specifically	ADV
ejpam-4761	100	29	,	,	PUNCT
ejpam-4761	100	30	tx	tx	PROPN
ejpam-4761	100	31	=	=	PUNCT
ejpam-4761	100	32	{	{	PUNCT
ejpam-4761	100	33	a	a	DET
ejpam-4761	100	34	∈	∈	PROPN
ejpam-4761	100	35	v	v	ADP
ejpam-4761	100	36	(	(	PUNCT
ejpam-4761	100	37	h	h	NOUN
ejpam-4761	100	38	)	)	PUNCT
ejpam-4761	100	39	:	:	PUNCT
ejpam-4761	100	40	(	(	PUNCT
ejpam-4761	100	41	x	x	X
ejpam-4761	100	42	,	,	PUNCT
ejpam-4761	100	43	a	a	PRON
ejpam-4761	100	44	)	)	PUNCT
ejpam-4761	100	45	∈	∈	PROPN
ejpam-4761	100	46	c	c	NOUN
ejpam-4761	100	47	}	}	PUNCT
ejpam-4761	100	48	for	for	ADP
ejpam-4761	100	49	each	each	DET
ejpam-4761	100	50	x	x	SYM
ejpam-4761	100	51	∈	∈	PROPN
ejpam-4761	100	52	s.	s.	PROPN
ejpam-4761	100	53	s.	s.	PROPN
ejpam-4761	100	54	canoy	canoy	PROPN
ejpam-4761	100	55	jr	jr	PROPN
ejpam-4761	100	56	.	.	PROPN
ejpam-4761	100	57	,	,	PUNCT
ejpam-4761	100	58	j.	j.	PROPN
ejpam-4761	100	59	hassan	hassan	PROPN
ejpam-4761	100	60	/	/	SYM
ejpam-4761	100	61	eur	eur	PROPN
ejpam-4761	100	62	.	.	PUNCT
ejpam-4761	101	1	j.	j.	PROPN
ejpam-4761	101	2	pure	pure	PROPN
ejpam-4761	101	3	appl	appl	PROPN
ejpam-4761	101	4	.	.	PROPN
ejpam-4761	101	5	math	math	PROPN
ejpam-4761	101	6	,	,	PUNCT
ejpam-4761	101	7	16	16	NUM
ejpam-4761	101	8	(	(	PUNCT
ejpam-4761	101	9	2	2	NUM
ejpam-4761	101	10	)	)	PUNCT
ejpam-4761	101	11	(	(	PUNCT
ejpam-4761	101	12	2023	2023	NUM
ejpam-4761	101	13	)	)	PUNCT
ejpam-4761	101	14	,	,	PUNCT
ejpam-4761	101	15	1196	1196	NUM
ejpam-4761	101	16	-	-	SYM
ejpam-4761	101	17	1211	1211	NUM
ejpam-4761	101	18	1199	1199	NUM
ejpam-4761	101	19	3	3	NUM
ejpam-4761	101	20	.	.	PUNCT
ejpam-4761	101	21	results	result	NOUN
ejpam-4761	101	22	proposition	proposition	NOUN
ejpam-4761	101	23	1	1	X
ejpam-4761	101	24	.	.	PUNCT
ejpam-4761	102	1	let	let	VERB
ejpam-4761	102	2	g	g	NOUN
ejpam-4761	102	3	be	be	AUX
ejpam-4761	102	4	any	any	DET
ejpam-4761	102	5	connected	connected	ADJ
ejpam-4761	102	6	graph	graph	NOUN
ejpam-4761	102	7	g	g	NOUN
ejpam-4761	102	8	on	on	ADP
ejpam-4761	102	9	n	n	PRON
ejpam-4761	102	10	≥	≥	NUM
ejpam-4761	102	11	2	2	NUM
ejpam-4761	102	12	vertices	vertex	NOUN
ejpam-4761	102	13	.	.	PUNCT
ejpam-4761	103	1	then	then	ADV
ejpam-4761	103	2	the	the	DET
ejpam-4761	103	3	following	follow	VERB
ejpam-4761	103	4	hold	hold	NOUN
ejpam-4761	103	5	:	:	PUNCT
ejpam-4761	103	6	(	(	PUNCT
ejpam-4761	103	7	i	i	NOUN
ejpam-4761	103	8	)	)	PUNCT
ejpam-4761	103	9	if	if	SCONJ
ejpam-4761	103	10	s	s	VERB
ejpam-4761	103	11	is	be	AUX
ejpam-4761	103	12	a	a	DET
ejpam-4761	103	13	weakly	weakly	ADJ
ejpam-4761	103	14	convex	convex	NOUN
ejpam-4761	103	15	hop	hop	NOUN
ejpam-4761	103	16	dominating	dominating	NOUN
ejpam-4761	103	17	set	set	NOUN
ejpam-4761	103	18	in	in	ADP
ejpam-4761	103	19	g	g	PROPN
ejpam-4761	103	20	,	,	PUNCT
ejpam-4761	103	21	then	then	ADV
ejpam-4761	103	22	the	the	DET
ejpam-4761	103	23	induced	induced	ADJ
ejpam-4761	103	24	graph	graph	NOUN
ejpam-4761	103	25	⟨s⟩	⟨s⟩	PROPN
ejpam-4761	103	26	is	be	AUX
ejpam-4761	103	27	connected	connect	VERB
ejpam-4761	103	28	.	.	PUNCT
ejpam-4761	104	1	(	(	PUNCT
ejpam-4761	104	2	ii	ii	X
ejpam-4761	104	3	)	)	PUNCT
ejpam-4761	104	4	γch(g	γch(g	PROPN
ejpam-4761	104	5	)	)	PUNCT
ejpam-4761	104	6	≤	≤	NUM
ejpam-4761	104	7	γwconh(g	γwconh(g	NOUN
ejpam-4761	104	8	)	)	PUNCT
ejpam-4761	104	9	≤	≤	NUM
ejpam-4761	104	10	γconh(g	γconh(g	NOUN
ejpam-4761	104	11	)	)	PUNCT
ejpam-4761	104	12	and	and	CCONJ
ejpam-4761	104	13	every	every	DET
ejpam-4761	104	14	equality	equality	NOUN
ejpam-4761	104	15	and	and	CCONJ
ejpam-4761	104	16	strict	strict	ADJ
ejpam-4761	104	17	inequality	inequality	NOUN
ejpam-4761	104	18	can	can	AUX
ejpam-4761	104	19	be	be	AUX
ejpam-4761	104	20	attained	attain	VERB
ejpam-4761	104	21	.	.	PUNCT
ejpam-4761	105	1	proof	proof	NOUN
ejpam-4761	105	2	.	.	PUNCT
ejpam-4761	106	1	(	(	PUNCT
ejpam-4761	106	2	i	i	NOUN
ejpam-4761	106	3	)	)	PUNCT
ejpam-4761	106	4	let	let	VERB
ejpam-4761	106	5	s	s	PRON
ejpam-4761	106	6	be	be	AUX
ejpam-4761	106	7	a	a	DET
ejpam-4761	106	8	weakly	weakly	ADJ
ejpam-4761	106	9	convex	convex	NOUN
ejpam-4761	106	10	hop	hop	NOUN
ejpam-4761	106	11	dominating	dominating	NOUN
ejpam-4761	106	12	set	set	VERB
ejpam-4761	106	13	in	in	ADP
ejpam-4761	106	14	g	g	NOUN
ejpam-4761	106	15	and	and	CCONJ
ejpam-4761	106	16	let	let	VERB
ejpam-4761	106	17	x	x	PRON
ejpam-4761	106	18	,	,	PUNCT
ejpam-4761	106	19	y	y	PROPN
ejpam-4761	106	20	∈	∈	PROPN
ejpam-4761	106	21	s	s	PART
ejpam-4761	106	22	with	with	ADP
ejpam-4761	106	23	x	x	PART
ejpam-4761	106	24	̸=	̸=	PROPN
ejpam-4761	106	25	y.	y.	NOUN
ejpam-4761	106	26	since	since	SCONJ
ejpam-4761	106	27	s	s	PART
ejpam-4761	106	28	weakly	weakly	ADJ
ejpam-4761	106	29	convex	convex	NOUN
ejpam-4761	106	30	,	,	PUNCT
ejpam-4761	106	31	there	there	PRON
ejpam-4761	106	32	exists	exist	VERB
ejpam-4761	106	33	an	an	DET
ejpam-4761	106	34	x	x	NOUN
ejpam-4761	106	35	-	-	NOUN
ejpam-4761	106	36	y	y	ADJ
ejpam-4761	106	37	geodesic	geodesic	NOUN
ejpam-4761	106	38	p	p	X
ejpam-4761	106	39	(	(	PUNCT
ejpam-4761	106	40	x	x	NOUN
ejpam-4761	106	41	,	,	PUNCT
ejpam-4761	106	42	y	y	NOUN
ejpam-4761	106	43	)	)	PUNCT
ejpam-4761	106	44	such	such	ADJ
ejpam-4761	106	45	that	that	DET
ejpam-4761	106	46	v	v	NOUN
ejpam-4761	106	47	(	(	PUNCT
ejpam-4761	106	48	p	p	X
ejpam-4761	106	49	(	(	PUNCT
ejpam-4761	106	50	x	x	NOUN
ejpam-4761	106	51	,	,	PUNCT
ejpam-4761	106	52	y	y	PROPN
ejpam-4761	106	53	)	)	PUNCT
ejpam-4761	106	54	⊆	⊆	NUM
ejpam-4761	106	55	s.	s.	PROPN
ejpam-4761	106	56	this	this	PRON
ejpam-4761	106	57	implies	imply	VERB
ejpam-4761	106	58	that	that	SCONJ
ejpam-4761	106	59	the	the	DET
ejpam-4761	106	60	induced	induced	ADJ
ejpam-4761	106	61	graph	graph	NOUN
ejpam-4761	106	62	⟨s⟩	⟨s⟩	PROPN
ejpam-4761	106	63	is	be	AUX
ejpam-4761	106	64	connected	connect	VERB
ejpam-4761	106	65	.	.	PUNCT
ejpam-4761	107	1	(	(	PUNCT
ejpam-4761	107	2	ii	ii	NOUN
ejpam-4761	107	3	)	)	PUNCT
ejpam-4761	107	4	since	since	SCONJ
ejpam-4761	107	5	every	every	DET
ejpam-4761	107	6	weakly	weakly	ADJ
ejpam-4761	107	7	convex	convex	NOUN
ejpam-4761	107	8	hop	hop	NOUN
ejpam-4761	107	9	dominating	dominating	NOUN
ejpam-4761	107	10	set	set	NOUN
ejpam-4761	107	11	in	in	ADP
ejpam-4761	107	12	g	g	PROPN
ejpam-4761	107	13	is	be	AUX
ejpam-4761	107	14	connected	connect	VERB
ejpam-4761	107	15	hop	hop	NOUN
ejpam-4761	107	16	dominating	dominating	NOUN
ejpam-4761	107	17	,	,	PUNCT
ejpam-4761	107	18	γch(g	γch(g	NOUN
ejpam-4761	107	19	)	)	PUNCT
ejpam-4761	107	20	≤	≤	NUM
ejpam-4761	107	21	γwconh(g	γwconh(g	NOUN
ejpam-4761	107	22	)	)	PUNCT
ejpam-4761	107	23	.	.	PUNCT
ejpam-4761	108	1	also	also	ADV
ejpam-4761	108	2	,	,	PUNCT
ejpam-4761	108	3	since	since	SCONJ
ejpam-4761	108	4	every	every	DET
ejpam-4761	108	5	convex	convex	NOUN
ejpam-4761	108	6	hop	hop	NOUN
ejpam-4761	108	7	dominating	dominating	NOUN
ejpam-4761	108	8	set	set	NOUN
ejpam-4761	108	9	in	in	ADP
ejpam-4761	108	10	g	g	PROPN
ejpam-4761	108	11	is	be	AUX
ejpam-4761	108	12	weakly	weakly	ADV
ejpam-4761	108	13	convex	convex	ADJ
ejpam-4761	108	14	hop	hop	NOUN
ejpam-4761	108	15	dominating	dominating	NOUN
ejpam-4761	108	16	,	,	PUNCT
ejpam-4761	108	17	γwconh(g	γwconh(g	NOUN
ejpam-4761	108	18	)	)	PUNCT
ejpam-4761	108	19	≤	≤	NUM
ejpam-4761	108	20	γconh(g	γconh(g	NOUN
ejpam-4761	108	21	)	)	PUNCT
ejpam-4761	108	22	.	.	PUNCT
ejpam-4761	109	1	for	for	ADP
ejpam-4761	109	2	equality	equality	NOUN
ejpam-4761	109	3	,	,	PUNCT
ejpam-4761	109	4	consider	consider	VERB
ejpam-4761	109	5	the	the	DET
ejpam-4761	109	6	graph	graph	NOUN
ejpam-4761	109	7	g	g	NOUN
ejpam-4761	109	8	in	in	ADP
ejpam-4761	109	9	figure	figure	NOUN
ejpam-4761	109	10	1	1	NUM
ejpam-4761	109	11	.	.	PUNCT
ejpam-4761	110	1	let	let	VERB
ejpam-4761	110	2	w	w	VERB
ejpam-4761	110	3	=	=	PUNCT
ejpam-4761	110	4	{	{	PUNCT
ejpam-4761	110	5	v1	v1	PROPN
ejpam-4761	110	6	,	,	PUNCT
ejpam-4761	110	7	v2	v2	PROPN
ejpam-4761	110	8	,	,	PUNCT
ejpam-4761	110	9	v3	v3	PROPN
ejpam-4761	110	10	,	,	PUNCT
ejpam-4761	110	11	v4	v4	PROPN
ejpam-4761	110	12	}	}	PUNCT
ejpam-4761	110	13	.	.	PUNCT
ejpam-4761	111	1	then	then	ADV
ejpam-4761	111	2	w	w	PROPN
ejpam-4761	111	3	is	be	AUX
ejpam-4761	111	4	both	both	PRON
ejpam-4761	111	5	a	a	DET
ejpam-4761	111	6	γch	γch	NOUN
ejpam-4761	111	7	-	-	PUNCT
ejpam-4761	111	8	set	set	NOUN
ejpam-4761	111	9	,	,	PUNCT
ejpam-4761	111	10	a	a	DET
ejpam-4761	111	11	γwconh	γwconh	NOUN
ejpam-4761	111	12	-	-	PUNCT
ejpam-4761	111	13	set	set	NOUN
ejpam-4761	111	14	,	,	PUNCT
ejpam-4761	111	15	and	and	CCONJ
ejpam-4761	111	16	a	a	DET
ejpam-4761	111	17	γconh	γconh	NOUN
ejpam-4761	111	18	-	-	PUNCT
ejpam-4761	111	19	set	set	NOUN
ejpam-4761	111	20	of	of	ADP
ejpam-4761	111	21	g.	g.	PROPN
ejpam-4761	111	22	thus	thus	ADV
ejpam-4761	111	23	,	,	PUNCT
ejpam-4761	111	24	γch(g	γch(g	NOUN
ejpam-4761	111	25	)	)	PUNCT
ejpam-4761	111	26	=	=	SYM
ejpam-4761	111	27	γwconh(g	γwconh(g	NOUN
ejpam-4761	111	28	)	)	PUNCT
ejpam-4761	111	29	=	=	PUNCT
ejpam-4761	111	30	γconh(g	γconh(g	NOUN
ejpam-4761	111	31	)	)	PUNCT
ejpam-4761	111	32	=	=	SYM
ejpam-4761	112	1	4	4	X
ejpam-4761	112	2	.	.	X
ejpam-4761	113	1	g	g	NOUN
ejpam-4761	113	2	:	:	PUNCT
ejpam-4761	113	3	v3v4	v3v4	PUNCT
ejpam-4761	113	4	v1	v1	VERB
ejpam-4761	113	5	v5	v5	PROPN
ejpam-4761	113	6	v2	v2	NOUN
ejpam-4761	113	7	figure	figure	NOUN
ejpam-4761	113	8	1	1	NUM
ejpam-4761	113	9	:	:	PUNCT
ejpam-4761	113	10	a	a	DET
ejpam-4761	113	11	graph	graph	NOUN
ejpam-4761	113	12	g	g	NOUN
ejpam-4761	113	13	with	with	ADP
ejpam-4761	113	14	γch(g	γch(g	NOUN
ejpam-4761	113	15	)	)	PUNCT
ejpam-4761	113	16	=	=	SYM
ejpam-4761	113	17	γwconh(g	γwconh(g	NOUN
ejpam-4761	113	18	)	)	PUNCT
ejpam-4761	113	19	=	=	PUNCT
ejpam-4761	113	20	γconh(g	γconh(g	NOUN
ejpam-4761	113	21	)	)	PUNCT
ejpam-4761	113	22	=	=	SYM
ejpam-4761	113	23	4	4	NUM
ejpam-4761	113	24	for	for	ADP
ejpam-4761	113	25	strict	strict	ADJ
ejpam-4761	113	26	inequalities	inequality	NOUN
ejpam-4761	113	27	,	,	PUNCT
ejpam-4761	113	28	consider	consider	VERB
ejpam-4761	113	29	first	first	ADJ
ejpam-4761	113	30	c14	c14	NOUN
ejpam-4761	113	31	.	.	PUNCT
ejpam-4761	114	1	then	then	ADV
ejpam-4761	114	2	it	it	PRON
ejpam-4761	114	3	can	can	AUX
ejpam-4761	114	4	be	be	AUX
ejpam-4761	114	5	verified	verify	VERB
ejpam-4761	114	6	that	that	SCONJ
ejpam-4761	114	7	γch(c14	γch(c14	NOUN
ejpam-4761	114	8	)	)	PUNCT
ejpam-4761	114	9	=	=	SYM
ejpam-4761	114	10	10	10	NUM
ejpam-4761	114	11	and	and	CCONJ
ejpam-4761	114	12	γwconh(c14	γwconh(c14	ADJ
ejpam-4761	114	13	)	)	PUNCT
ejpam-4761	114	14	=	=	SYM
ejpam-4761	115	1	14	14	NUM
ejpam-4761	115	2	.	.	PUNCT
ejpam-4761	116	1	thus	thus	ADV
ejpam-4761	116	2	,	,	PUNCT
ejpam-4761	116	3	γch(c14	γch(c14	PRON
ejpam-4761	116	4	)	)	PUNCT
ejpam-4761	116	5	<	<	X
ejpam-4761	116	6	γwconh(c14	γwconh(c14	PROPN
ejpam-4761	116	7	)	)	PUNCT
ejpam-4761	116	8	.	.	PUNCT
ejpam-4761	117	1	next	next	ADV
ejpam-4761	117	2	,	,	PUNCT
ejpam-4761	117	3	consider	consider	VERB
ejpam-4761	117	4	the	the	DET
ejpam-4761	117	5	graph	graph	NOUN
ejpam-4761	117	6	g′	g′	NOUN
ejpam-4761	117	7	in	in	ADP
ejpam-4761	117	8	figure	figure	NOUN
ejpam-4761	117	9	2	2	NUM
ejpam-4761	117	10	.	.	PUNCT
ejpam-4761	118	1	let	let	VERB
ejpam-4761	118	2	w	w	NOUN
ejpam-4761	118	3	′	′	NOUN
ejpam-4761	119	1	=	=	PUNCT
ejpam-4761	119	2	{	{	PUNCT
ejpam-4761	119	3	u1	u1	NOUN
ejpam-4761	119	4	,	,	PUNCT
ejpam-4761	119	5	u2	u2	NOUN
ejpam-4761	119	6	,	,	PUNCT
ejpam-4761	119	7	u3	u3	NOUN
ejpam-4761	119	8	}	}	PUNCT
ejpam-4761	119	9	and	and	CCONJ
ejpam-4761	119	10	w	w	ADP
ejpam-4761	119	11	′′	′′	NOUN
ejpam-4761	119	12	=	=	SYM
ejpam-4761	119	13	{	{	PUNCT
ejpam-4761	119	14	u1	u1	NOUN
ejpam-4761	119	15	,	,	PUNCT
ejpam-4761	119	16	u2	u2	NOUN
ejpam-4761	119	17	,	,	PUNCT
ejpam-4761	119	18	u3	u3	PROPN
ejpam-4761	119	19	,	,	PUNCT
ejpam-4761	119	20	u4	u4	PROPN
ejpam-4761	119	21	,	,	PUNCT
ejpam-4761	119	22	u5	u5	PROPN
ejpam-4761	119	23	}	}	PUNCT
ejpam-4761	119	24	.	.	PUNCT
ejpam-4761	120	1	then	then	ADV
ejpam-4761	120	2	w	w	NOUN
ejpam-4761	120	3	′	′	NOUN
ejpam-4761	120	4	and	and	CCONJ
ejpam-4761	120	5	w	w	PROPN
ejpam-4761	120	6	′′	′′	PROPN
ejpam-4761	120	7	are	be	AUX
ejpam-4761	120	8	γwconh	γwconh	NOUN
ejpam-4761	120	9	-	-	PUNCT
ejpam-4761	120	10	set	set	VERB
ejpam-4761	120	11	and	and	CCONJ
ejpam-4761	120	12	γconh	γconh	NOUN
ejpam-4761	120	13	-	-	PUNCT
ejpam-4761	120	14	set	set	NOUN
ejpam-4761	120	15	of	of	ADP
ejpam-4761	120	16	g	g	PROPN
ejpam-4761	120	17	′	′	NUM
ejpam-4761	120	18	,	,	PUNCT
ejpam-4761	120	19	respectively	respectively	ADV
ejpam-4761	120	20	.	.	PUNCT
ejpam-4761	121	1	accordingly	accordingly	ADV
ejpam-4761	121	2	,	,	PUNCT
ejpam-4761	121	3	γwconh(g	γwconh(g	DET
ejpam-4761	121	4	′	′	NOUN
ejpam-4761	121	5	)	)	PUNCT
ejpam-4761	121	6	=	=	SYM
ejpam-4761	121	7	3	3	NUM
ejpam-4761	121	8	<	<	SYM
ejpam-4761	121	9	5	5	NUM
ejpam-4761	121	10	=	=	SYM
ejpam-4761	121	11	γconh(g	γconh(g	NOUN
ejpam-4761	121	12	′	′	NUM
ejpam-4761	121	13	)	)	PUNCT
ejpam-4761	121	14	.	.	PUNCT
ejpam-4761	122	1	s.	s.	PROPN
ejpam-4761	122	2	canoy	canoy	PROPN
ejpam-4761	122	3	jr	jr	PROPN
ejpam-4761	122	4	.	.	PROPN
ejpam-4761	122	5	,	,	PUNCT
ejpam-4761	122	6	j.	j.	PROPN
ejpam-4761	122	7	hassan	hassan	PROPN
ejpam-4761	122	8	/	/	SYM
ejpam-4761	122	9	eur	eur	PROPN
ejpam-4761	122	10	.	.	PUNCT
ejpam-4761	123	1	j.	j.	PROPN
ejpam-4761	123	2	pure	pure	PROPN
ejpam-4761	123	3	appl	appl	PROPN
ejpam-4761	123	4	.	.	PROPN
ejpam-4761	123	5	math	math	PROPN
ejpam-4761	123	6	,	,	PUNCT
ejpam-4761	123	7	16	16	NUM
ejpam-4761	123	8	(	(	PUNCT
ejpam-4761	123	9	2	2	NUM
ejpam-4761	123	10	)	)	PUNCT
ejpam-4761	123	11	(	(	PUNCT
ejpam-4761	123	12	2023	2023	NUM
ejpam-4761	123	13	)	)	PUNCT
ejpam-4761	123	14	,	,	PUNCT
ejpam-4761	123	15	1196	1196	NUM
ejpam-4761	123	16	-	-	SYM
ejpam-4761	123	17	1211	1211	NUM
ejpam-4761	123	18	1200	1200	NUM
ejpam-4761	123	19	g′	g′	NOUN
ejpam-4761	123	20	:	:	PUNCT
ejpam-4761	123	21	u5	u5	PROPN
ejpam-4761	123	22	u3u1	u3u1	PROPN
ejpam-4761	123	23	u2	u2	PROPN
ejpam-4761	123	24	u4	u4	PROPN
ejpam-4761	123	25	figure	figure	NOUN
ejpam-4761	123	26	2	2	NUM
ejpam-4761	123	27	:	:	PUNCT
ejpam-4761	123	28	a	a	DET
ejpam-4761	123	29	graph	graph	NOUN
ejpam-4761	123	30	g′	g′	NOUN
ejpam-4761	123	31	with	with	ADP
ejpam-4761	123	32	γwconh(g	γwconh(g	NOUN
ejpam-4761	123	33	′	′	NUM
ejpam-4761	123	34	)	)	PUNCT
ejpam-4761	123	35	=	=	SYM
ejpam-4761	123	36	3	3	NUM
ejpam-4761	123	37	<	<	SYM
ejpam-4761	123	38	5	5	NUM
ejpam-4761	123	39	=	=	SYM
ejpam-4761	123	40	γconh(g	γconh(g	NOUN
ejpam-4761	123	41	′	′	NUM
ejpam-4761	123	42	)	)	PUNCT
ejpam-4761	123	43	this	this	PRON
ejpam-4761	123	44	completes	complete	VERB
ejpam-4761	123	45	the	the	DET
ejpam-4761	123	46	proof	proof	NOUN
ejpam-4761	123	47	of	of	ADP
ejpam-4761	123	48	the	the	DET
ejpam-4761	123	49	assertion	assertion	NOUN
ejpam-4761	123	50	.	.	PUNCT
ejpam-4761	124	1	theorem	theorem	NOUN
ejpam-4761	124	2	1	1	X
ejpam-4761	124	3	.	.	PUNCT
ejpam-4761	125	1	let	let	VERB
ejpam-4761	125	2	g	g	NOUN
ejpam-4761	125	3	be	be	AUX
ejpam-4761	125	4	any	any	DET
ejpam-4761	125	5	connected	connected	ADJ
ejpam-4761	125	6	graph	graph	NOUN
ejpam-4761	125	7	on	on	ADP
ejpam-4761	125	8	n	n	PRON
ejpam-4761	125	9	≥	≥	NUM
ejpam-4761	125	10	2	2	NUM
ejpam-4761	125	11	vertices	vertex	NOUN
ejpam-4761	125	12	.	.	PUNCT
ejpam-4761	126	1	then	then	ADV
ejpam-4761	126	2	2	2	NUM
ejpam-4761	126	3	≤	≤	NUM
ejpam-4761	126	4	γwconh(g	γwconh(g	NOUN
ejpam-4761	126	5	)	)	PUNCT
ejpam-4761	126	6	≤	≤	NOUN
ejpam-4761	126	7	n.	n.	NOUN
ejpam-4761	126	8	moreover	moreover	ADV
ejpam-4761	126	9	,	,	PUNCT
ejpam-4761	126	10	each	each	PRON
ejpam-4761	126	11	of	of	ADP
ejpam-4761	126	12	the	the	DET
ejpam-4761	126	13	following	follow	VERB
ejpam-4761	126	14	holds	hold	VERB
ejpam-4761	126	15	:	:	PUNCT
ejpam-4761	126	16	(	(	PUNCT
ejpam-4761	126	17	i	i	NOUN
ejpam-4761	126	18	)	)	PUNCT
ejpam-4761	126	19	γwconh(g	γwconh(g	NOUN
ejpam-4761	126	20	)	)	PUNCT
ejpam-4761	126	21	=	=	SYM
ejpam-4761	126	22	2	2	NUM
ejpam-4761	126	23	if	if	SCONJ
ejpam-4761	126	24	and	and	CCONJ
ejpam-4761	126	25	only	only	ADV
ejpam-4761	126	26	if	if	SCONJ
ejpam-4761	126	27	there	there	PRON
ejpam-4761	126	28	exist	exist	VERB
ejpam-4761	126	29	two	two	NUM
ejpam-4761	126	30	adjacent	adjacent	ADJ
ejpam-4761	126	31	vertices	vertex	NOUN
ejpam-4761	126	32	x	x	PUNCT
ejpam-4761	126	33	and	and	CCONJ
ejpam-4761	126	34	y	y	PROPN
ejpam-4761	126	35	of	of	ADP
ejpam-4761	126	36	g	g	PROPN
ejpam-4761	126	37	such	such	ADJ
ejpam-4761	126	38	that	that	DET
ejpam-4761	126	39	ng(x	ng(x	NUM
ejpam-4761	126	40	)	)	PUNCT
ejpam-4761	126	41	∩	∩	NOUN
ejpam-4761	126	42	ng(y	ng(y	NOUN
ejpam-4761	126	43	)	)	PUNCT
ejpam-4761	126	44	=	=	NOUN
ejpam-4761	126	45	∅	∅	NOUN
ejpam-4761	126	46	and	and	CCONJ
ejpam-4761	126	47	for	for	ADP
ejpam-4761	126	48	each	each	DET
ejpam-4761	126	49	z	z	NOUN
ejpam-4761	126	50	∈	∈	PROPN
ejpam-4761	126	51	v	v	ADP
ejpam-4761	126	52	(	(	PUNCT
ejpam-4761	126	53	g	g	NOUN
ejpam-4761	126	54	)	)	PUNCT
ejpam-4761	126	55	\	\	PROPN
ejpam-4761	127	1	ng({x	ng({x	PROPN
ejpam-4761	127	2	,	,	PUNCT
ejpam-4761	127	3	y	y	NOUN
ejpam-4761	127	4	}	}	PUNCT
ejpam-4761	127	5	)	)	PUNCT
ejpam-4761	128	1	,	,	PUNCT
ejpam-4761	128	2	there	there	PRON
ejpam-4761	128	3	exists	exist	VERB
ejpam-4761	128	4	w	w	PROPN
ejpam-4761	128	5	∈	∈	PROPN
ejpam-4761	128	6	[	[	X
ejpam-4761	128	7	ng({x	ng({x	ADJ
ejpam-4761	128	8	,	,	PUNCT
ejpam-4761	128	9	y	y	NOUN
ejpam-4761	128	10	}	}	PUNCT
ejpam-4761	128	11	)	)	PUNCT
ejpam-4761	128	12	\	\	NOUN
ejpam-4761	129	1	{	{	PUNCT
ejpam-4761	129	2	x	x	NOUN
ejpam-4761	129	3	,	,	PUNCT
ejpam-4761	129	4	y	y	PROPN
ejpam-4761	129	5	}	}	PUNCT
ejpam-4761	129	6	]	]	PUNCT
ejpam-4761	129	7	∩ng(z	∩ng(z	PROPN
ejpam-4761	129	8	)	)	PUNCT
ejpam-4761	129	9	.	.	PUNCT
ejpam-4761	130	1	(	(	PUNCT
ejpam-4761	130	2	ii	ii	X
ejpam-4761	130	3	)	)	PUNCT
ejpam-4761	130	4	γconh(g	γconh(g	PROPN
ejpam-4761	130	5	)	)	PUNCT
ejpam-4761	130	6	=	=	SYM
ejpam-4761	131	1	n	n	NOUN
ejpam-4761	131	2	if	if	SCONJ
ejpam-4761	131	3	and	and	CCONJ
ejpam-4761	131	4	only	only	ADV
ejpam-4761	131	5	if	if	SCONJ
ejpam-4761	131	6	for	for	ADP
ejpam-4761	131	7	every	every	DET
ejpam-4761	131	8	connected	connect	VERB
ejpam-4761	131	9	hop	hop	NOUN
ejpam-4761	131	10	dominating	dominating	NOUN
ejpam-4761	131	11	set	set	NOUN
ejpam-4761	131	12	s	s	PART
ejpam-4761	131	13	̸=	̸=	PROPN
ejpam-4761	131	14	v	v	NOUN
ejpam-4761	131	15	(	(	PUNCT
ejpam-4761	131	16	g	g	NOUN
ejpam-4761	131	17	)	)	PUNCT
ejpam-4761	131	18	,	,	PUNCT
ejpam-4761	131	19	there	there	PRON
ejpam-4761	131	20	exists	exist	VERB
ejpam-4761	131	21	a	a	DET
ejpam-4761	131	22	pair	pair	NOUN
ejpam-4761	131	23	of	of	ADP
ejpam-4761	131	24	distinct	distinct	ADJ
ejpam-4761	131	25	vertices	vertex	NOUN
ejpam-4761	131	26	x	x	PUNCT
ejpam-4761	131	27	and	and	CCONJ
ejpam-4761	131	28	y	y	PROPN
ejpam-4761	131	29	in	in	ADP
ejpam-4761	131	30	s	s	PRON
ejpam-4761	131	31	such	such	ADJ
ejpam-4761	131	32	that	that	DET
ejpam-4761	131	33	dg(x	dg(x	PROPN
ejpam-4761	131	34	,	,	PUNCT
ejpam-4761	131	35	y	y	NOUN
ejpam-4761	131	36	)	)	PUNCT
ejpam-4761	131	37	<	<	X
ejpam-4761	131	38	d⟨s⟩(x	d⟨s⟩(x	PROPN
ejpam-4761	131	39	,	,	PUNCT
ejpam-4761	131	40	y	y	NOUN
ejpam-4761	131	41	)	)	PUNCT
ejpam-4761	131	42	.	.	PUNCT
ejpam-4761	132	1	proof	proof	NOUN
ejpam-4761	132	2	.	.	PUNCT
ejpam-4761	133	1	since	since	SCONJ
ejpam-4761	133	2	g	g	PROPN
ejpam-4761	133	3	is	be	AUX
ejpam-4761	133	4	a	a	DET
ejpam-4761	133	5	non	non	ADJ
ejpam-4761	133	6	-	-	ADJ
ejpam-4761	133	7	trivial	trivial	ADJ
ejpam-4761	133	8	graph	graph	NOUN
ejpam-4761	133	9	,	,	PUNCT
ejpam-4761	133	10	every	every	DET
ejpam-4761	133	11	hop	hop	NOUN
ejpam-4761	133	12	dominating	dominating	NOUN
ejpam-4761	133	13	has	have	VERB
ejpam-4761	133	14	at	at	ADV
ejpam-4761	133	15	least	least	ADV
ejpam-4761	133	16	two	two	NUM
ejpam-4761	133	17	elements	element	NOUN
ejpam-4761	133	18	.	.	PUNCT
ejpam-4761	134	1	hence	hence	ADV
ejpam-4761	134	2	,	,	PUNCT
ejpam-4761	134	3	2	2	NUM
ejpam-4761	134	4	≤	≤	NUM
ejpam-4761	134	5	γwconh(g	γwconh(g	NOUN
ejpam-4761	134	6	)	)	PUNCT
ejpam-4761	134	7	≤	≤	NOUN
ejpam-4761	134	8	n.	n.	NOUN
ejpam-4761	134	9	(	(	PUNCT
ejpam-4761	134	10	i	i	NOUN
ejpam-4761	134	11	)	)	PUNCT
ejpam-4761	134	12	suppose	suppose	VERB
ejpam-4761	134	13	that	that	SCONJ
ejpam-4761	134	14	γwconh(g	γwconh(g	NOUN
ejpam-4761	134	15	)	)	PUNCT
ejpam-4761	134	16	=	=	SYM
ejpam-4761	134	17	2	2	NUM
ejpam-4761	134	18	,	,	PUNCT
ejpam-4761	134	19	say	say	VERB
ejpam-4761	134	20	s	s	X
ejpam-4761	134	21	=	=	PUNCT
ejpam-4761	134	22	{	{	PUNCT
ejpam-4761	134	23	x	x	PROPN
ejpam-4761	134	24	,	,	PUNCT
ejpam-4761	134	25	y	y	PRON
ejpam-4761	134	26	}	}	PUNCT
ejpam-4761	134	27	is	be	AUX
ejpam-4761	134	28	a	a	DET
ejpam-4761	134	29	γwconh	γwconh	NOUN
ejpam-4761	134	30	-	-	PUNCT
ejpam-4761	134	31	set	set	NOUN
ejpam-4761	134	32	in	in	ADP
ejpam-4761	134	33	g.	g.	NOUN
ejpam-4761	134	34	by	by	ADP
ejpam-4761	134	35	proposition	proposition	NOUN
ejpam-4761	134	36	1(i	1(i	NUM
ejpam-4761	134	37	)	)	PUNCT
ejpam-4761	134	38	,	,	PUNCT
ejpam-4761	134	39	xy	xy	PROPN
ejpam-4761	134	40	∈	∈	PROPN
ejpam-4761	134	41	e(g	e(g	PROPN
ejpam-4761	134	42	)	)	PUNCT
ejpam-4761	134	43	.	.	PUNCT
ejpam-4761	135	1	since	since	SCONJ
ejpam-4761	135	2	s	s	PROPN
ejpam-4761	135	3	is	be	AUX
ejpam-4761	135	4	a	a	DET
ejpam-4761	135	5	hop	hop	NOUN
ejpam-4761	135	6	dominating	dominating	NOUN
ejpam-4761	135	7	set	set	VERB
ejpam-4761	135	8	in	in	ADP
ejpam-4761	135	9	g	g	NOUN
ejpam-4761	135	10	,	,	PUNCT
ejpam-4761	135	11	ng(x	ng(x	NUM
ejpam-4761	135	12	)	)	PUNCT
ejpam-4761	135	13	∩	∩	NOUN
ejpam-4761	135	14	ng(y	ng(y	NOUN
ejpam-4761	135	15	)	)	PUNCT
ejpam-4761	135	16	=	=	VERB
ejpam-4761	135	17	∅.	∅.	AUX
ejpam-4761	135	18	let	let	VERB
ejpam-4761	135	19	z	z	PROPN
ejpam-4761	135	20	∈	∈	PROPN
ejpam-4761	135	21	v	v	ADP
ejpam-4761	135	22	(	(	PUNCT
ejpam-4761	135	23	g	g	NOUN
ejpam-4761	135	24	)	)	PUNCT
ejpam-4761	135	25	\	\	PROPN
ejpam-4761	135	26	ng({x	ng({x	PROPN
ejpam-4761	135	27	,	,	PUNCT
ejpam-4761	135	28	y	y	NOUN
ejpam-4761	135	29	}	}	PUNCT
ejpam-4761	135	30	)	)	PUNCT
ejpam-4761	135	31	.	.	PUNCT
ejpam-4761	136	1	since	since	SCONJ
ejpam-4761	136	2	s	s	PROPN
ejpam-4761	136	3	is	be	AUX
ejpam-4761	136	4	a	a	DET
ejpam-4761	136	5	hop	hop	NOUN
ejpam-4761	136	6	dominating	dominating	NOUN
ejpam-4761	136	7	set	set	NOUN
ejpam-4761	136	8	,	,	PUNCT
ejpam-4761	136	9	it	it	PRON
ejpam-4761	136	10	follows	follow	VERB
ejpam-4761	136	11	that	that	SCONJ
ejpam-4761	136	12	z	z	PROPN
ejpam-4761	136	13	∈	∈	PROPN
ejpam-4761	136	14	n2	n2	PROPN
ejpam-4761	136	15	g({x	g({x	PROPN
ejpam-4761	136	16	,	,	PUNCT
ejpam-4761	136	17	y	y	NOUN
ejpam-4761	136	18	}	}	PUNCT
ejpam-4761	136	19	)	)	PUNCT
ejpam-4761	136	20	.	.	PUNCT
ejpam-4761	137	1	we	we	PRON
ejpam-4761	137	2	may	may	AUX
ejpam-4761	137	3	assume	assume	VERB
ejpam-4761	137	4	that	that	SCONJ
ejpam-4761	137	5	z	z	PROPN
ejpam-4761	137	6	∈	∈	PROPN
ejpam-4761	137	7	n2	n2	NOUN
ejpam-4761	137	8	g(x	g(x	PROPN
ejpam-4761	137	9	)	)	PUNCT
ejpam-4761	137	10	.	.	PUNCT
ejpam-4761	138	1	let	let	VERB
ejpam-4761	138	2	[	[	X
ejpam-4761	138	3	z	z	X
ejpam-4761	138	4	,	,	PUNCT
ejpam-4761	138	5	w	w	PROPN
ejpam-4761	138	6	,	,	PUNCT
ejpam-4761	138	7	x	x	X
ejpam-4761	138	8	]	]	X
ejpam-4761	138	9	be	be	AUX
ejpam-4761	138	10	a	a	DET
ejpam-4761	138	11	z	z	NOUN
ejpam-4761	138	12	-	-	PUNCT
ejpam-4761	138	13	x	x	NOUN
ejpam-4761	138	14	geodesic	geodesic	NOUN
ejpam-4761	138	15	.	.	PUNCT
ejpam-4761	139	1	since	since	SCONJ
ejpam-4761	139	2	z	z	PROPN
ejpam-4761	139	3	/∈	/∈	VERB
ejpam-4761	139	4	ng({x	ng({x	ADJ
ejpam-4761	139	5	,	,	PUNCT
ejpam-4761	139	6	y	y	NOUN
ejpam-4761	139	7	}	}	PUNCT
ejpam-4761	139	8	)	)	PUNCT
ejpam-4761	139	9	,	,	PUNCT
ejpam-4761	139	10	w	w	PROPN
ejpam-4761	139	11	̸=	̸=	PROPN
ejpam-4761	139	12	y.	y.	PROPN
ejpam-4761	139	13	thus	thus	ADV
ejpam-4761	139	14	,	,	PUNCT
ejpam-4761	139	15	w	w	PROPN
ejpam-4761	139	16	∈	∈	PROPN
ejpam-4761	139	17	[	[	X
ejpam-4761	139	18	ng({x	ng({x	ADJ
ejpam-4761	139	19	,	,	PUNCT
ejpam-4761	139	20	y	y	NOUN
ejpam-4761	139	21	}	}	PUNCT
ejpam-4761	139	22	)	)	PUNCT
ejpam-4761	139	23	\	\	NOUN
ejpam-4761	139	24	{	{	PUNCT
ejpam-4761	139	25	x	x	NOUN
ejpam-4761	139	26	,	,	PUNCT
ejpam-4761	139	27	y	y	PROPN
ejpam-4761	139	28	}	}	PUNCT
ejpam-4761	139	29	]	]	PUNCT
ejpam-4761	140	1	∩ng(z	∩ng(z	PROPN
ejpam-4761	140	2	)	)	PUNCT
ejpam-4761	140	3	.	.	PUNCT
ejpam-4761	141	1	for	for	ADP
ejpam-4761	141	2	the	the	DET
ejpam-4761	141	3	converse	converse	NOUN
ejpam-4761	141	4	,	,	PUNCT
ejpam-4761	141	5	suppose	suppose	VERB
ejpam-4761	141	6	there	there	PRON
ejpam-4761	141	7	exist	exist	VERB
ejpam-4761	141	8	adjacent	adjacent	ADJ
ejpam-4761	141	9	vertices	vertex	NOUN
ejpam-4761	141	10	x	x	PUNCT
ejpam-4761	141	11	and	and	CCONJ
ejpam-4761	141	12	y	y	PROPN
ejpam-4761	141	13	of	of	ADP
ejpam-4761	141	14	g	g	PROPN
ejpam-4761	141	15	such	such	ADJ
ejpam-4761	141	16	that	that	DET
ejpam-4761	141	17	ng(x	ng(x	NUM
ejpam-4761	141	18	)	)	PUNCT
ejpam-4761	141	19	∩	∩	NOUN
ejpam-4761	141	20	ng(y	ng(y	NOUN
ejpam-4761	141	21	)	)	PUNCT
ejpam-4761	141	22	=	=	NOUN
ejpam-4761	141	23	∅	∅	NOUN
ejpam-4761	141	24	and	and	CCONJ
ejpam-4761	141	25	for	for	ADP
ejpam-4761	141	26	each	each	DET
ejpam-4761	141	27	z	z	NOUN
ejpam-4761	141	28	∈	∈	PROPN
ejpam-4761	141	29	v	v	ADP
ejpam-4761	141	30	(	(	PUNCT
ejpam-4761	141	31	g	g	NOUN
ejpam-4761	141	32	)	)	PUNCT
ejpam-4761	141	33	\	\	PROPN
ejpam-4761	141	34	ng({x	ng({x	PROPN
ejpam-4761	141	35	,	,	PUNCT
ejpam-4761	141	36	y	y	NOUN
ejpam-4761	141	37	}	}	PUNCT
ejpam-4761	141	38	)	)	PUNCT
ejpam-4761	141	39	,	,	PUNCT
ejpam-4761	141	40	there	there	PRON
ejpam-4761	141	41	exists	exist	VERB
ejpam-4761	141	42	w	w	PROPN
ejpam-4761	141	43	∈	∈	PROPN
ejpam-4761	141	44	[	[	X
ejpam-4761	141	45	ng({x	ng({x	ADJ
ejpam-4761	141	46	,	,	PUNCT
ejpam-4761	141	47	y})\{x	y})\{x	PROPN
ejpam-4761	141	48	,	,	PUNCT
ejpam-4761	141	49	y}]∩ng(z	y}]∩ng(z	PROPN
ejpam-4761	141	50	)	)	PUNCT
ejpam-4761	141	51	.	.	PUNCT
ejpam-4761	142	1	let	let	VERB
ejpam-4761	142	2	s	s	PRON
ejpam-4761	142	3	=	=	PUNCT
ejpam-4761	142	4	{	{	PUNCT
ejpam-4761	142	5	x	x	PROPN
ejpam-4761	142	6	,	,	PUNCT
ejpam-4761	142	7	y	y	NOUN
ejpam-4761	142	8	}	}	PUNCT
ejpam-4761	142	9	and	and	CCONJ
ejpam-4761	142	10	let	let	VERB
ejpam-4761	142	11	v	v	NUM
ejpam-4761	142	12	∈	∈	PROPN
ejpam-4761	142	13	v	v	NOUN
ejpam-4761	142	14	(	(	PUNCT
ejpam-4761	142	15	g)\s	g)\s	NOUN
ejpam-4761	142	16	.	.	PUNCT
ejpam-4761	143	1	if	if	SCONJ
ejpam-4761	143	2	v	v	NOUN
ejpam-4761	143	3	∈	∈	PROPN
ejpam-4761	143	4	ng({x	ng({x	PROPN
ejpam-4761	143	5	,	,	PUNCT
ejpam-4761	143	6	y	y	NOUN
ejpam-4761	143	7	}	}	PUNCT
ejpam-4761	143	8	)	)	PUNCT
ejpam-4761	143	9	,	,	PUNCT
ejpam-4761	143	10	then	then	ADV
ejpam-4761	143	11	by	by	ADP
ejpam-4761	143	12	assumption	assumption	NOUN
ejpam-4761	143	13	dg(x	dg(x	NUM
ejpam-4761	143	14	,	,	PUNCT
ejpam-4761	143	15	v	v	NOUN
ejpam-4761	143	16	)	)	PUNCT
ejpam-4761	143	17	=	=	SYM
ejpam-4761	143	18	2	2	NUM
ejpam-4761	143	19	or	or	CCONJ
ejpam-4761	143	20	dg(y	dg(y	ADJ
ejpam-4761	143	21	,	,	PUNCT
ejpam-4761	143	22	v	v	NOUN
ejpam-4761	143	23	)	)	PUNCT
ejpam-4761	143	24	=	=	SYM
ejpam-4761	144	1	2	2	X
ejpam-4761	144	2	.	.	X
ejpam-4761	145	1	if	if	SCONJ
ejpam-4761	145	2	v	v	NUM
ejpam-4761	145	3	∈	∈	PROPN
ejpam-4761	145	4	v	v	NOUN
ejpam-4761	145	5	(	(	PUNCT
ejpam-4761	145	6	g	g	NOUN
ejpam-4761	145	7	)	)	PUNCT
ejpam-4761	145	8	\	\	PROPN
ejpam-4761	145	9	ng({x	ng({x	PROPN
ejpam-4761	145	10	,	,	PUNCT
ejpam-4761	145	11	y	y	NOUN
ejpam-4761	145	12	}	}	PUNCT
ejpam-4761	145	13	)	)	PUNCT
ejpam-4761	145	14	,	,	PUNCT
ejpam-4761	145	15	then	then	ADV
ejpam-4761	145	16	there	there	PRON
ejpam-4761	145	17	exists	exist	VERB
ejpam-4761	145	18	w	w	PROPN
ejpam-4761	145	19	∈	∈	PROPN
ejpam-4761	145	20	[	[	X
ejpam-4761	145	21	ng({x	ng({x	ADJ
ejpam-4761	145	22	,	,	PUNCT
ejpam-4761	145	23	y	y	NOUN
ejpam-4761	145	24	}	}	PUNCT
ejpam-4761	145	25	)	)	PUNCT
ejpam-4761	145	26	\	\	NOUN
ejpam-4761	146	1	{	{	PUNCT
ejpam-4761	146	2	x	x	NOUN
ejpam-4761	146	3	,	,	PUNCT
ejpam-4761	146	4	y	y	PROPN
ejpam-4761	146	5	}	}	PUNCT
ejpam-4761	146	6	]	]	PUNCT
ejpam-4761	146	7	∩ng(v	∩ng(v	PROPN
ejpam-4761	146	8	)	)	PUNCT
ejpam-4761	146	9	by	by	ADP
ejpam-4761	146	10	assumption	assumption	NOUN
ejpam-4761	146	11	.	.	PUNCT
ejpam-4761	147	1	with	with	ADP
ejpam-4761	147	2	the	the	DET
ejpam-4761	147	3	assumption	assumption	NOUN
ejpam-4761	147	4	that	that	SCONJ
ejpam-4761	147	5	ng(x)∩ng(y	ng(x)∩ng(y	PROPN
ejpam-4761	147	6	)	)	PUNCT
ejpam-4761	147	7	=	=	SYM
ejpam-4761	147	8	∅	∅	NOUN
ejpam-4761	147	9	,	,	PUNCT
ejpam-4761	147	10	v	v	NOUN
ejpam-4761	147	11	∈	∈	PROPN
ejpam-4761	147	12	n2	n2	PROPN
ejpam-4761	147	13	g({x	g({x	PROPN
ejpam-4761	147	14	,	,	PUNCT
ejpam-4761	147	15	y	y	NOUN
ejpam-4761	147	16	}	}	PUNCT
ejpam-4761	147	17	)	)	PUNCT
ejpam-4761	147	18	.	.	PUNCT
ejpam-4761	148	1	therefore	therefore	ADV
ejpam-4761	148	2	,	,	PUNCT
ejpam-4761	148	3	s	s	VERB
ejpam-4761	148	4	is	be	AUX
ejpam-4761	148	5	a	a	DET
ejpam-4761	148	6	weakly	weakly	ADJ
ejpam-4761	148	7	convex	convex	NOUN
ejpam-4761	148	8	hop	hop	NOUN
ejpam-4761	148	9	dominating	dominating	NOUN
ejpam-4761	148	10	set	set	VERB
ejpam-4761	148	11	in	in	ADP
ejpam-4761	148	12	g	g	PROPN
ejpam-4761	148	13	and	and	CCONJ
ejpam-4761	148	14	γwconh(g	γwconh(g	NOUN
ejpam-4761	148	15	)	)	PUNCT
ejpam-4761	148	16	=	=	PUNCT
ejpam-4761	148	17	|s|	|s|	NOUN
ejpam-4761	148	18	=	=	SYM
ejpam-4761	148	19	2	2	NUM
ejpam-4761	148	20	.	.	PUNCT
ejpam-4761	148	21	s.	s.	PROPN
ejpam-4761	148	22	canoy	canoy	PROPN
ejpam-4761	148	23	jr	jr	PROPN
ejpam-4761	148	24	.	.	PROPN
ejpam-4761	148	25	,	,	PUNCT
ejpam-4761	148	26	j.	j.	PROPN
ejpam-4761	148	27	hassan	hassan	PROPN
ejpam-4761	148	28	/	/	SYM
ejpam-4761	148	29	eur	eur	PROPN
ejpam-4761	148	30	.	.	PUNCT
ejpam-4761	149	1	j.	j.	PROPN
ejpam-4761	149	2	pure	pure	PROPN
ejpam-4761	149	3	appl	appl	PROPN
ejpam-4761	149	4	.	.	PROPN
ejpam-4761	149	5	math	math	PROPN
ejpam-4761	149	6	,	,	PUNCT
ejpam-4761	149	7	16	16	NUM
ejpam-4761	149	8	(	(	PUNCT
ejpam-4761	149	9	2	2	NUM
ejpam-4761	149	10	)	)	PUNCT
ejpam-4761	149	11	(	(	PUNCT
ejpam-4761	149	12	2023	2023	NUM
ejpam-4761	149	13	)	)	PUNCT
ejpam-4761	149	14	,	,	PUNCT
ejpam-4761	149	15	1196	1196	NUM
ejpam-4761	149	16	-	-	SYM
ejpam-4761	149	17	1211	1211	NUM
ejpam-4761	149	18	1201	1201	NUM
ejpam-4761	149	19	(	(	PUNCT
ejpam-4761	149	20	ii	ii	NOUN
ejpam-4761	149	21	)	)	PUNCT
ejpam-4761	149	22	suppose	suppose	VERB
ejpam-4761	149	23	that	that	SCONJ
ejpam-4761	149	24	γwconh(g	γwconh(g	NOUN
ejpam-4761	149	25	)	)	PUNCT
ejpam-4761	149	26	=	=	SYM
ejpam-4761	149	27	n.	n.	NOUN
ejpam-4761	149	28	suppose	suppose	VERB
ejpam-4761	149	29	further	far	ADV
ejpam-4761	149	30	that	that	SCONJ
ejpam-4761	149	31	there	there	PRON
ejpam-4761	149	32	exists	exist	VERB
ejpam-4761	149	33	a	a	DET
ejpam-4761	149	34	connected	connected	ADJ
ejpam-4761	149	35	hop	hop	NOUN
ejpam-4761	149	36	dominating	dominating	NOUN
ejpam-4761	149	37	set	set	NOUN
ejpam-4761	149	38	s	s	PART
ejpam-4761	149	39	̸=	̸=	PROPN
ejpam-4761	149	40	v	v	NOUN
ejpam-4761	149	41	(	(	PUNCT
ejpam-4761	149	42	g	g	NOUN
ejpam-4761	149	43	)	)	PUNCT
ejpam-4761	149	44	such	such	ADJ
ejpam-4761	149	45	that	that	SCONJ
ejpam-4761	149	46	dg(a	dg(a	PROPN
ejpam-4761	149	47	,	,	PUNCT
ejpam-4761	149	48	b	b	X
ejpam-4761	149	49	)	)	PUNCT
ejpam-4761	149	50	=	=	SYM
ejpam-4761	149	51	d⟨s⟩(a	d⟨s⟩(a	PROPN
ejpam-4761	149	52	,	,	PUNCT
ejpam-4761	149	53	b	b	NOUN
ejpam-4761	149	54	)	)	PUNCT
ejpam-4761	149	55	for	for	ADP
ejpam-4761	149	56	each	each	DET
ejpam-4761	149	57	pair	pair	NOUN
ejpam-4761	149	58	of	of	ADP
ejpam-4761	149	59	distinct	distinct	ADJ
ejpam-4761	149	60	vertices	vertex	NOUN
ejpam-4761	149	61	a	a	PRON
ejpam-4761	149	62	and	and	CCONJ
ejpam-4761	149	63	b	b	NOUN
ejpam-4761	149	64	in	in	ADP
ejpam-4761	149	65	s.	s.	PROPN
ejpam-4761	149	66	since	since	SCONJ
ejpam-4761	149	67	⟨s⟩	⟨s⟩	PROPN
ejpam-4761	149	68	is	be	AUX
ejpam-4761	149	69	connected	connect	VERB
ejpam-4761	149	70	,	,	PUNCT
ejpam-4761	149	71	there	there	PRON
ejpam-4761	149	72	exists	exist	VERB
ejpam-4761	149	73	an	an	DET
ejpam-4761	149	74	a	a	PRON
ejpam-4761	149	75	-	-	PUNCT
ejpam-4761	149	76	b	b	NOUN
ejpam-4761	149	77	geodesic	geodesic	NOUN
ejpam-4761	149	78	in	in	ADP
ejpam-4761	149	79	⟨s⟩	⟨s⟩	PROPN
ejpam-4761	149	80	for	for	ADP
ejpam-4761	149	81	each	each	DET
ejpam-4761	149	82	pair	pair	NOUN
ejpam-4761	149	83	of	of	ADP
ejpam-4761	149	84	distinct	distinct	ADJ
ejpam-4761	149	85	vertices	vertex	NOUN
ejpam-4761	149	86	a	a	PRON
ejpam-4761	149	87	and	and	CCONJ
ejpam-4761	149	88	b	b	NOUN
ejpam-4761	149	89	in	in	ADP
ejpam-4761	149	90	s.	s.	PROPN
ejpam-4761	149	91	hence	hence	ADV
ejpam-4761	149	92	,	,	PUNCT
ejpam-4761	149	93	s	s	VERB
ejpam-4761	149	94	is	be	AUX
ejpam-4761	149	95	a	a	DET
ejpam-4761	149	96	weakly	weakly	ADJ
ejpam-4761	149	97	convex	convex	NOUN
ejpam-4761	149	98	hop	hop	NOUN
ejpam-4761	149	99	dominating	dominating	NOUN
ejpam-4761	149	100	set	set	NOUN
ejpam-4761	149	101	in	in	ADP
ejpam-4761	149	102	g	g	PROPN
ejpam-4761	149	103	,	,	PUNCT
ejpam-4761	149	104	contrary	contrary	ADV
ejpam-4761	149	105	to	to	ADP
ejpam-4761	149	106	the	the	DET
ejpam-4761	149	107	assumption	assumption	NOUN
ejpam-4761	149	108	that	that	SCONJ
ejpam-4761	149	109	γwconh(g	γwconh(g	NOUN
ejpam-4761	149	110	)	)	PUNCT
ejpam-4761	149	111	=	=	SYM
ejpam-4761	149	112	n.	n.	NOUN
ejpam-4761	149	113	therefore	therefore	ADV
ejpam-4761	149	114	,	,	PUNCT
ejpam-4761	149	115	for	for	SCONJ
ejpam-4761	149	116	each	each	DET
ejpam-4761	149	117	connected	connect	VERB
ejpam-4761	149	118	hop	hop	NOUN
ejpam-4761	149	119	dominating	dominating	NOUN
ejpam-4761	149	120	set	set	NOUN
ejpam-4761	149	121	s	s	VERB
ejpam-4761	149	122	in	in	ADP
ejpam-4761	149	123	g	g	NOUN
ejpam-4761	149	124	,	,	PUNCT
ejpam-4761	149	125	there	there	PRON
ejpam-4761	149	126	exist	exist	VERB
ejpam-4761	149	127	distinct	distinct	ADJ
ejpam-4761	149	128	vertices	vertex	NOUN
ejpam-4761	149	129	x	x	PUNCT
ejpam-4761	149	130	and	and	CCONJ
ejpam-4761	149	131	y	y	PROPN
ejpam-4761	149	132	in	in	ADP
ejpam-4761	149	133	s	s	PRON
ejpam-4761	149	134	such	such	ADJ
ejpam-4761	149	135	that	that	DET
ejpam-4761	149	136	dg(x	dg(x	PROPN
ejpam-4761	149	137	,	,	PUNCT
ejpam-4761	149	138	y	y	NOUN
ejpam-4761	149	139	)	)	PUNCT
ejpam-4761	149	140	<	<	X
ejpam-4761	149	141	d⟨s⟩(x	d⟨s⟩(x	PROPN
ejpam-4761	149	142	,	,	PUNCT
ejpam-4761	149	143	y	y	NOUN
ejpam-4761	149	144	)	)	PUNCT
ejpam-4761	149	145	.	.	PUNCT
ejpam-4761	150	1	conversely	conversely	ADV
ejpam-4761	150	2	,	,	PUNCT
ejpam-4761	150	3	suppose	suppose	VERB
ejpam-4761	150	4	that	that	SCONJ
ejpam-4761	150	5	for	for	SCONJ
ejpam-4761	150	6	each	each	DET
ejpam-4761	150	7	connected	connect	VERB
ejpam-4761	150	8	hop	hop	NOUN
ejpam-4761	150	9	dominating	dominating	NOUN
ejpam-4761	150	10	set	set	NOUN
ejpam-4761	150	11	s	s	VERB
ejpam-4761	150	12	in	in	ADP
ejpam-4761	150	13	g	g	NOUN
ejpam-4761	150	14	,	,	PUNCT
ejpam-4761	150	15	there	there	PRON
ejpam-4761	150	16	exist	exist	VERB
ejpam-4761	150	17	distinct	distinct	ADJ
ejpam-4761	150	18	vertices	vertex	NOUN
ejpam-4761	150	19	x	x	PUNCT
ejpam-4761	150	20	and	and	CCONJ
ejpam-4761	150	21	y	y	PROPN
ejpam-4761	150	22	in	in	ADP
ejpam-4761	150	23	s	s	PRON
ejpam-4761	150	24	such	such	ADJ
ejpam-4761	150	25	that	that	DET
ejpam-4761	150	26	dg(x	dg(x	PROPN
ejpam-4761	150	27	,	,	PUNCT
ejpam-4761	150	28	y	y	NOUN
ejpam-4761	150	29	)	)	PUNCT
ejpam-4761	150	30	<	<	X
ejpam-4761	150	31	d⟨s⟩(x	d⟨s⟩(x	PROPN
ejpam-4761	150	32	,	,	PUNCT
ejpam-4761	150	33	y	y	PROPN
ejpam-4761	150	34	)	)	PUNCT
ejpam-4761	150	35	.	.	PUNCT
ejpam-4761	151	1	let	let	VERB
ejpam-4761	151	2	s0	s0	PROPN
ejpam-4761	151	3	be	be	AUX
ejpam-4761	151	4	a	a	DET
ejpam-4761	151	5	γwconh	γwconh	NOUN
ejpam-4761	151	6	-	-	PUNCT
ejpam-4761	151	7	set	set	NOUN
ejpam-4761	151	8	in	in	ADP
ejpam-4761	151	9	g.	g.	PROPN
ejpam-4761	151	10	suppose	suppose	VERB
ejpam-4761	151	11	s0	s0	PROPN
ejpam-4761	151	12	̸=	̸=	PROPN
ejpam-4761	151	13	v	v	NOUN
ejpam-4761	151	14	(	(	PUNCT
ejpam-4761	151	15	g	g	NOUN
ejpam-4761	151	16	)	)	PUNCT
ejpam-4761	151	17	.	.	PUNCT
ejpam-4761	152	1	then	then	ADV
ejpam-4761	152	2	,	,	PUNCT
ejpam-4761	152	3	by	by	ADP
ejpam-4761	152	4	proposition	proposition	NOUN
ejpam-4761	152	5	1(i	1(i	NUM
ejpam-4761	152	6	)	)	PUNCT
ejpam-4761	152	7	,	,	PUNCT
ejpam-4761	152	8	s0	s0	PROPN
ejpam-4761	152	9	is	be	AUX
ejpam-4761	152	10	a	a	DET
ejpam-4761	152	11	connected	connected	ADJ
ejpam-4761	152	12	hop	hop	NOUN
ejpam-4761	152	13	dominating	dominating	NOUN
ejpam-4761	152	14	set	set	NOUN
ejpam-4761	152	15	.	.	PUNCT
ejpam-4761	153	1	hence	hence	ADV
ejpam-4761	153	2	,	,	PUNCT
ejpam-4761	153	3	by	by	ADP
ejpam-4761	153	4	assumption	assumption	NOUN
ejpam-4761	153	5	,	,	PUNCT
ejpam-4761	153	6	there	there	PRON
ejpam-4761	153	7	exists	exist	VERB
ejpam-4761	153	8	a	a	DET
ejpam-4761	153	9	pair	pair	NOUN
ejpam-4761	153	10	of	of	ADP
ejpam-4761	153	11	distinct	distinct	ADJ
ejpam-4761	153	12	vertices	vertex	NOUN
ejpam-4761	153	13	p	p	NOUN
ejpam-4761	153	14	,	,	PUNCT
ejpam-4761	153	15	q	q	PUNCT
ejpam-4761	153	16	∈	∈	NOUN
ejpam-4761	153	17	s0	s0	NOUN
ejpam-4761	153	18	such	such	ADJ
ejpam-4761	153	19	that	that	SCONJ
ejpam-4761	153	20	dg(p	dg(p	NOUN
ejpam-4761	153	21	,	,	PUNCT
ejpam-4761	153	22	q	q	NOUN
ejpam-4761	153	23	)	)	PUNCT
ejpam-4761	153	24	<	<	X
ejpam-4761	153	25	d⟨s0⟩(p	d⟨s0⟩(p	NOUN
ejpam-4761	153	26	,	,	PUNCT
ejpam-4761	153	27	q	q	NOUN
ejpam-4761	153	28	)	)	PUNCT
ejpam-4761	153	29	.	.	PUNCT
ejpam-4761	154	1	this	this	PRON
ejpam-4761	154	2	implies	imply	VERB
ejpam-4761	154	3	that	that	SCONJ
ejpam-4761	154	4	every	every	DET
ejpam-4761	154	5	p	p	NOUN
ejpam-4761	154	6	-	-	PUNCT
ejpam-4761	154	7	q	q	NOUN
ejpam-4761	154	8	path	path	NOUN
ejpam-4761	154	9	in	in	ADP
ejpam-4761	154	10	⟨s0⟩	⟨s0⟩	PROPN
ejpam-4761	154	11	is	be	AUX
ejpam-4761	154	12	not	not	PART
ejpam-4761	154	13	a	a	DET
ejpam-4761	154	14	geodesic	geodesic	NOUN
ejpam-4761	154	15	in	in	ADP
ejpam-4761	154	16	g.	g.	PROPN
ejpam-4761	154	17	hence	hence	ADV
ejpam-4761	154	18	,	,	PUNCT
ejpam-4761	154	19	s0	s0	PROPN
ejpam-4761	154	20	is	be	AUX
ejpam-4761	154	21	not	not	PART
ejpam-4761	154	22	a	a	DET
ejpam-4761	154	23	weakly	weakly	ADJ
ejpam-4761	154	24	convex	convex	NOUN
ejpam-4761	154	25	set	set	NOUN
ejpam-4761	154	26	,	,	PUNCT
ejpam-4761	154	27	a	a	DET
ejpam-4761	154	28	contradiction	contradiction	NOUN
ejpam-4761	154	29	.	.	PUNCT
ejpam-4761	155	1	therefore	therefore	ADV
ejpam-4761	155	2	,	,	PUNCT
ejpam-4761	155	3	s0	s0	PROPN
ejpam-4761	155	4	=	=	SYM
ejpam-4761	155	5	v	v	PROPN
ejpam-4761	155	6	(	(	PUNCT
ejpam-4761	155	7	g	g	NOUN
ejpam-4761	155	8	)	)	PUNCT
ejpam-4761	155	9	,	,	PUNCT
ejpam-4761	155	10	showing	show	VERB
ejpam-4761	155	11	that	that	SCONJ
ejpam-4761	155	12	γwconh(g	γwconh(g	NOUN
ejpam-4761	155	13	)	)	PUNCT
ejpam-4761	155	14	=	=	PUNCT
ejpam-4761	155	15	n.	n.	NOUN
ejpam-4761	155	16	since	since	SCONJ
ejpam-4761	155	17	γh(kn	γh(kn	PROPN
ejpam-4761	155	18	)	)	PUNCT
ejpam-4761	156	1	=	=	SYM
ejpam-4761	157	1	n	n	CCONJ
ejpam-4761	157	2	,	,	PUNCT
ejpam-4761	157	3	it	it	PRON
ejpam-4761	157	4	follows	follow	VERB
ejpam-4761	157	5	that	that	SCONJ
ejpam-4761	157	6	γwconh(g	γwconh(g	NOUN
ejpam-4761	157	7	)	)	PUNCT
ejpam-4761	157	8	=	=	VERB
ejpam-4761	158	1	n.	n.	NOUN
ejpam-4761	158	2	the	the	DET
ejpam-4761	158	3	same	same	ADJ
ejpam-4761	158	4	conclusion	conclusion	NOUN
ejpam-4761	158	5	can	can	AUX
ejpam-4761	158	6	be	be	AUX
ejpam-4761	158	7	deduced	deduce	VERB
ejpam-4761	158	8	from	from	ADP
ejpam-4761	158	9	theorem	theorem	NOUN
ejpam-4761	158	10	1	1	NUM
ejpam-4761	158	11	because	because	SCONJ
ejpam-4761	158	12	v	v	PROPN
ejpam-4761	158	13	(	(	PUNCT
ejpam-4761	158	14	kn	kn	PROPN
ejpam-4761	158	15	)	)	PUNCT
ejpam-4761	158	16	is	be	AUX
ejpam-4761	158	17	the	the	DET
ejpam-4761	158	18	only	only	ADV
ejpam-4761	158	19	(	(	PUNCT
ejpam-4761	158	20	connected	connected	ADJ
ejpam-4761	158	21	)	)	PUNCT
ejpam-4761	158	22	hop	hop	NOUN
ejpam-4761	158	23	dominating	dominating	NOUN
ejpam-4761	158	24	set	set	NOUN
ejpam-4761	158	25	of	of	ADP
ejpam-4761	158	26	kn	kn	PROPN
ejpam-4761	158	27	.	.	PUNCT
ejpam-4761	159	1	corollary	corollary	ADJ
ejpam-4761	159	2	1	1	NUM
ejpam-4761	159	3	.	.	PUNCT
ejpam-4761	160	1	let	let	VERB
ejpam-4761	160	2	n	n	PRON
ejpam-4761	160	3	be	be	AUX
ejpam-4761	160	4	a	a	DET
ejpam-4761	160	5	positive	positive	ADJ
ejpam-4761	160	6	integer	integer	NOUN
ejpam-4761	160	7	.	.	PUNCT
ejpam-4761	161	1	then	then	ADV
ejpam-4761	161	2	γwconh(kn	γwconh(kn	NOUN
ejpam-4761	161	3	)	)	PUNCT
ejpam-4761	161	4	=	=	VERB
ejpam-4761	162	1	n.	n.	NOUN
ejpam-4761	162	2	observe	observe	VERB
ejpam-4761	162	3	that	that	SCONJ
ejpam-4761	162	4	for	for	ADP
ejpam-4761	162	5	a	a	DET
ejpam-4761	162	6	non	non	ADJ
ejpam-4761	162	7	-	-	ADJ
ejpam-4761	162	8	trivial	trivial	ADJ
ejpam-4761	162	9	connected	connected	ADJ
ejpam-4761	162	10	graph	graph	NOUN
ejpam-4761	162	11	g	g	NOUN
ejpam-4761	162	12	,	,	PUNCT
ejpam-4761	162	13	γch(g	γch(g	NOUN
ejpam-4761	162	14	)	)	PUNCT
ejpam-4761	162	15	=	=	SYM
ejpam-4761	162	16	2	2	NUM
ejpam-4761	162	17	(	(	PUNCT
ejpam-4761	162	18	or	or	CCONJ
ejpam-4761	162	19	γconh(g	γconh(g	NOUN
ejpam-4761	162	20	)	)	PUNCT
ejpam-4761	162	21	=	=	SYM
ejpam-4761	162	22	2	2	X
ejpam-4761	162	23	)	)	PUNCT
ejpam-4761	162	24	is	be	AUX
ejpam-4761	162	25	equivalent	equivalent	ADJ
ejpam-4761	162	26	to	to	ADP
ejpam-4761	162	27	the	the	DET
ejpam-4761	162	28	condition	condition	NOUN
ejpam-4761	162	29	given	give	VERB
ejpam-4761	162	30	in	in	ADP
ejpam-4761	162	31	theorem	theorem	NOUN
ejpam-4761	162	32	1(i	1(i	NUM
ejpam-4761	162	33	)	)	PUNCT
ejpam-4761	162	34	.	.	PUNCT
ejpam-4761	163	1	the	the	DET
ejpam-4761	163	2	next	next	ADJ
ejpam-4761	163	3	result	result	NOUN
ejpam-4761	163	4	states	state	VERB
ejpam-4761	163	5	this	this	PRON
ejpam-4761	163	6	formally	formally	ADV
ejpam-4761	163	7	.	.	PUNCT
ejpam-4761	164	1	corollary	corollary	ADJ
ejpam-4761	164	2	2	2	NUM
ejpam-4761	164	3	.	.	PUNCT
ejpam-4761	165	1	let	let	VERB
ejpam-4761	165	2	g	g	NOUN
ejpam-4761	165	3	be	be	AUX
ejpam-4761	165	4	any	any	DET
ejpam-4761	165	5	connected	connected	ADJ
ejpam-4761	165	6	graph	graph	NOUN
ejpam-4761	165	7	on	on	ADP
ejpam-4761	165	8	n	n	PRON
ejpam-4761	165	9	≥	≥	NUM
ejpam-4761	165	10	2	2	NUM
ejpam-4761	165	11	vertices	vertex	NOUN
ejpam-4761	165	12	.	.	PUNCT
ejpam-4761	166	1	then	then	ADV
ejpam-4761	166	2	the	the	DET
ejpam-4761	166	3	following	follow	VERB
ejpam-4761	166	4	statements	statement	NOUN
ejpam-4761	166	5	are	be	AUX
ejpam-4761	166	6	equivalent	equivalent	ADJ
ejpam-4761	166	7	:	:	PUNCT
ejpam-4761	166	8	(	(	PUNCT
ejpam-4761	166	9	i	i	NOUN
ejpam-4761	166	10	)	)	PUNCT
ejpam-4761	166	11	γch(g	γch(g	PROPN
ejpam-4761	166	12	)	)	PUNCT
ejpam-4761	166	13	=	=	SYM
ejpam-4761	166	14	2	2	X
ejpam-4761	166	15	.	.	PUNCT
ejpam-4761	166	16	(	(	PUNCT
ejpam-4761	166	17	ii	ii	NOUN
ejpam-4761	166	18	)	)	PUNCT
ejpam-4761	166	19	γwconh(g	γwconh(g	NOUN
ejpam-4761	166	20	)	)	PUNCT
ejpam-4761	166	21	=	=	SYM
ejpam-4761	166	22	2	2	X
ejpam-4761	166	23	.	.	PUNCT
ejpam-4761	166	24	(	(	PUNCT
ejpam-4761	166	25	iii	iii	X
ejpam-4761	166	26	)	)	PUNCT
ejpam-4761	166	27	γconh(g	γconh(g	NOUN
ejpam-4761	166	28	)	)	PUNCT
ejpam-4761	166	29	=	=	SYM
ejpam-4761	167	1	2	2	X
ejpam-4761	167	2	.	.	X
ejpam-4761	167	3	theorem	theorem	NOUN
ejpam-4761	167	4	2	2	NUM
ejpam-4761	167	5	.	.	PUNCT
ejpam-4761	168	1	let	let	VERB
ejpam-4761	168	2	a	a	PRON
ejpam-4761	168	3	and	and	CCONJ
ejpam-4761	168	4	b	b	NOUN
ejpam-4761	168	5	be	be	AUX
ejpam-4761	168	6	positive	positive	ADJ
ejpam-4761	168	7	integers	integer	NOUN
ejpam-4761	168	8	such	such	ADJ
ejpam-4761	168	9	that	that	SCONJ
ejpam-4761	168	10	3	3	NUM
ejpam-4761	168	11	≤	≤	NOUN
ejpam-4761	168	12	a	a	DET
ejpam-4761	168	13	≤	≤	PROPN
ejpam-4761	168	14	b.	b.	NOUN
ejpam-4761	169	1	then	then	ADV
ejpam-4761	169	2	there	there	PRON
ejpam-4761	169	3	exists	exist	VERB
ejpam-4761	169	4	a	a	DET
ejpam-4761	169	5	connected	connected	ADJ
ejpam-4761	169	6	graph	graph	NOUN
ejpam-4761	169	7	g	g	ADP
ejpam-4761	169	8	such	such	ADJ
ejpam-4761	169	9	that	that	DET
ejpam-4761	169	10	γwconh(g	γwconh(g	NOUN
ejpam-4761	169	11	)	)	PUNCT
ejpam-4761	169	12	=	=	SYM
ejpam-4761	169	13	a	a	PRON
ejpam-4761	169	14	and	and	CCONJ
ejpam-4761	169	15	γconh(g	γconh(g	NOUN
ejpam-4761	169	16	)	)	PUNCT
ejpam-4761	169	17	=	=	SYM
ejpam-4761	169	18	b.	b.	NOUN
ejpam-4761	169	19	proof	proof	NOUN
ejpam-4761	169	20	.	.	PUNCT
ejpam-4761	170	1	for	for	ADP
ejpam-4761	170	2	a	a	DET
ejpam-4761	170	3	=	=	SYM
ejpam-4761	170	4	b	b	NOUN
ejpam-4761	170	5	,	,	PUNCT
ejpam-4761	170	6	consider	consider	VERB
ejpam-4761	170	7	the	the	DET
ejpam-4761	170	8	complete	complete	ADJ
ejpam-4761	170	9	graph	graph	NOUN
ejpam-4761	170	10	ka	ka	PROPN
ejpam-4761	170	11	=	=	PROPN
ejpam-4761	170	12	g.	g.	PROPN
ejpam-4761	170	13	then	then	ADV
ejpam-4761	170	14	γwconh(g	γwconh(g	NUM
ejpam-4761	170	15	)	)	PUNCT
ejpam-4761	170	16	=	=	PUNCT
ejpam-4761	170	17	a	a	DET
ejpam-4761	170	18	=	=	NOUN
ejpam-4761	170	19	γconh(g	γconh(g	NOUN
ejpam-4761	170	20	)	)	PUNCT
ejpam-4761	170	21	.	.	PUNCT
ejpam-4761	171	1	suppose	suppose	VERB
ejpam-4761	171	2	a	a	DET
ejpam-4761	171	3	<	<	X
ejpam-4761	171	4	b.	b.	NOUN
ejpam-4761	171	5	consider	consider	VERB
ejpam-4761	171	6	the	the	DET
ejpam-4761	171	7	following	follow	VERB
ejpam-4761	171	8	two	two	NUM
ejpam-4761	171	9	cases	case	NOUN
ejpam-4761	171	10	:	:	PUNCT
ejpam-4761	171	11	case	case	NOUN
ejpam-4761	171	12	1	1	NUM
ejpam-4761	171	13	:	:	PUNCT
ejpam-4761	171	14	a	a	PRON
ejpam-4761	172	1	=	=	ADJ
ejpam-4761	172	2	3	3	X
ejpam-4761	172	3	.	.	PUNCT
ejpam-4761	172	4	let	let	VERB
ejpam-4761	172	5	m	m	VERB
ejpam-4761	172	6	=	=	VERB
ejpam-4761	173	1	b	b	X
ejpam-4761	173	2	−	−	PROPN
ejpam-4761	173	3	a	a	PRON
ejpam-4761	174	1	and	and	CCONJ
ejpam-4761	174	2	consider	consider	VERB
ejpam-4761	174	3	the	the	DET
ejpam-4761	174	4	graph	graph	NOUN
ejpam-4761	174	5	g	g	NOUN
ejpam-4761	174	6	in	in	ADP
ejpam-4761	174	7	figure	figure	NOUN
ejpam-4761	174	8	3	3	NUM
ejpam-4761	174	9	.	.	PUNCT
ejpam-4761	175	1	let	let	VERB
ejpam-4761	175	2	w	w	VERB
ejpam-4761	175	3	=	=	PUNCT
ejpam-4761	175	4	{	{	PUNCT
ejpam-4761	175	5	w1	w1	NOUN
ejpam-4761	175	6	,	,	PUNCT
ejpam-4761	175	7	w2	w2	NOUN
ejpam-4761	175	8	,	,	PUNCT
ejpam-4761	175	9	w3	w3	PROPN
ejpam-4761	175	10	}	}	PUNCT
ejpam-4761	175	11	and	and	CCONJ
ejpam-4761	175	12	w	w	NOUN
ejpam-4761	175	13	′	′	NOUN
ejpam-4761	175	14	=	=	SYM
ejpam-4761	175	15	{	{	PUNCT
ejpam-4761	175	16	w1	w1	NOUN
ejpam-4761	175	17	,	,	PUNCT
ejpam-4761	175	18	w2	w2	NOUN
ejpam-4761	175	19	,	,	PUNCT
ejpam-4761	175	20	w3	w3	PROPN
ejpam-4761	175	21	,	,	PUNCT
ejpam-4761	175	22	v1	v1	NOUN
ejpam-4761	175	23	,	,	PUNCT
ejpam-4761	175	24	v2	v2	NOUN
ejpam-4761	175	25	,	,	PUNCT
ejpam-4761	175	26	.	.	PUNCT
ejpam-4761	175	27	.	.	PUNCT
ejpam-4761	176	1	.	.	PUNCT
ejpam-4761	177	1	,	,	PUNCT
ejpam-4761	178	1	vm	vm	NOUN
ejpam-4761	178	2	}	}	PUNCT
ejpam-4761	178	3	.	.	PUNCT
ejpam-4761	179	1	then	then	ADV
ejpam-4761	179	2	w	w	PROPN
ejpam-4761	179	3	and	and	CCONJ
ejpam-4761	179	4	w	w	PROPN
ejpam-4761	179	5	′	′	NOUN
ejpam-4761	179	6	are	be	AUX
ejpam-4761	179	7	γwconh	γwconh	NOUN
ejpam-4761	179	8	-	-	PUNCT
ejpam-4761	179	9	set	set	VERB
ejpam-4761	179	10	and	and	CCONJ
ejpam-4761	179	11	γconh	γconh	NOUN
ejpam-4761	179	12	-	-	PUNCT
ejpam-4761	179	13	set	set	NOUN
ejpam-4761	179	14	in	in	ADP
ejpam-4761	179	15	g	g	NOUN
ejpam-4761	179	16	,	,	PUNCT
ejpam-4761	179	17	respectively	respectively	ADV
ejpam-4761	179	18	.	.	PUNCT
ejpam-4761	180	1	therefore	therefore	ADV
ejpam-4761	180	2	,	,	PUNCT
ejpam-4761	180	3	γwconh(g	γwconh(g	NOUN
ejpam-4761	180	4	)	)	PUNCT
ejpam-4761	180	5	=	=	SYM
ejpam-4761	180	6	a	a	PRON
ejpam-4761	180	7	and	and	CCONJ
ejpam-4761	180	8	γconh(g	γconh(g	NOUN
ejpam-4761	180	9	)	)	PUNCT
ejpam-4761	180	10	=	=	PUNCT
ejpam-4761	180	11	a+m	a+m	NUM
ejpam-4761	180	12	=	=	SYM
ejpam-4761	180	13	b.	b.	NOUN
ejpam-4761	180	14	case	case	NOUN
ejpam-4761	180	15	2	2	NUM
ejpam-4761	180	16	:	:	PUNCT
ejpam-4761	180	17	a	a	DET
ejpam-4761	180	18	≥	≥	NOUN
ejpam-4761	180	19	4	4	NUM
ejpam-4761	180	20	.	.	PUNCT
ejpam-4761	181	1	let	let	VERB
ejpam-4761	181	2	m	m	VERB
ejpam-4761	181	3	=	=	VERB
ejpam-4761	182	1	b	b	X
ejpam-4761	182	2	−	−	PROPN
ejpam-4761	182	3	a	a	PRON
ejpam-4761	183	1	and	and	CCONJ
ejpam-4761	183	2	consider	consider	VERB
ejpam-4761	183	3	the	the	DET
ejpam-4761	183	4	graph	graph	NOUN
ejpam-4761	183	5	g′	g′	NOUN
ejpam-4761	183	6	in	in	ADP
ejpam-4761	183	7	figure	figure	NOUN
ejpam-4761	183	8	4	4	NUM
ejpam-4761	183	9	.	.	PUNCT
ejpam-4761	184	1	let	let	VERB
ejpam-4761	184	2	w1	w1	NOUN
ejpam-4761	184	3	=	=	SYM
ejpam-4761	184	4	{	{	PUNCT
ejpam-4761	184	5	x1	x1	PROPN
ejpam-4761	184	6	,	,	PUNCT
ejpam-4761	184	7	x2	x2	PROPN
ejpam-4761	184	8	,	,	PUNCT
ejpam-4761	184	9	.	.	PUNCT
ejpam-4761	184	10	.	.	PUNCT
ejpam-4761	185	1	.	.	PUNCT
ejpam-4761	186	1	,	,	PUNCT
ejpam-4761	186	2	xa	xa	PROPN
ejpam-4761	186	3	}	}	PUNCT
ejpam-4761	186	4	and	and	CCONJ
ejpam-4761	186	5	w2	w2	NOUN
ejpam-4761	186	6	=	=	SYM
ejpam-4761	186	7	{	{	PUNCT
ejpam-4761	186	8	x1	x1	PROPN
ejpam-4761	186	9	,	,	PUNCT
ejpam-4761	186	10	x2	x2	PROPN
ejpam-4761	186	11	,	,	PUNCT
ejpam-4761	186	12	.	.	PUNCT
ejpam-4761	186	13	.	.	PUNCT
ejpam-4761	187	1	.	.	PUNCT
ejpam-4761	188	1	,	,	PUNCT
ejpam-4761	188	2	xa	xa	PROPN
ejpam-4761	188	3	,	,	PUNCT
ejpam-4761	188	4	y1	y1	PROPN
ejpam-4761	188	5	,	,	PUNCT
ejpam-4761	188	6	y2	y2	PROPN
ejpam-4761	188	7	,	,	PUNCT
ejpam-4761	188	8	.	.	PUNCT
ejpam-4761	188	9	.	.	PUNCT
ejpam-4761	188	10	.	.	PUNCT
ejpam-4761	189	1	,	,	PUNCT
ejpam-4761	189	2	ym	ym	PROPN
ejpam-4761	189	3	}	}	PUNCT
ejpam-4761	189	4	.	.	PUNCT
ejpam-4761	190	1	then	then	ADV
ejpam-4761	190	2	w1	w1	PROPN
ejpam-4761	190	3	and	and	CCONJ
ejpam-4761	190	4	w2	w2	NOUN
ejpam-4761	190	5	are	be	AUX
ejpam-4761	190	6	γwconh	γwconh	NOUN
ejpam-4761	190	7	-	-	PUNCT
ejpam-4761	190	8	set	set	VERB
ejpam-4761	190	9	and	and	CCONJ
ejpam-4761	190	10	γconh	γconh	NOUN
ejpam-4761	190	11	-	-	PUNCT
ejpam-4761	190	12	set	set	NOUN
ejpam-4761	190	13	in	in	ADP
ejpam-4761	190	14	g′	g′	NOUN
ejpam-4761	190	15	,	,	PUNCT
ejpam-4761	190	16	respectively	respectively	ADV
ejpam-4761	190	17	.	.	PUNCT
ejpam-4761	191	1	hence	hence	ADV
ejpam-4761	191	2	,	,	PUNCT
ejpam-4761	191	3	γwconh(g	γwconh(g	DET
ejpam-4761	191	4	′	′	NOUN
ejpam-4761	191	5	)	)	PUNCT
ejpam-4761	191	6	=	=	PUNCT
ejpam-4761	191	7	a	a	PROPN
ejpam-4761	191	8	and	and	CCONJ
ejpam-4761	191	9	γconh(g	γconh(g	NOUN
ejpam-4761	191	10	′	′	NUM
ejpam-4761	191	11	)	)	PUNCT
ejpam-4761	191	12	=	=	PUNCT
ejpam-4761	191	13	a+m	a+m	NUM
ejpam-4761	191	14	=	=	SYM
ejpam-4761	191	15	b.	b.	PROPN
ejpam-4761	191	16	s.	s.	PROPN
ejpam-4761	191	17	canoy	canoy	PROPN
ejpam-4761	191	18	jr	jr	PROPN
ejpam-4761	191	19	.	.	PROPN
ejpam-4761	191	20	,	,	PUNCT
ejpam-4761	191	21	j.	j.	PROPN
ejpam-4761	191	22	hassan	hassan	PROPN
ejpam-4761	191	23	/	/	SYM
ejpam-4761	191	24	eur	eur	PROPN
ejpam-4761	191	25	.	.	PUNCT
ejpam-4761	192	1	j.	j.	PROPN
ejpam-4761	192	2	pure	pure	PROPN
ejpam-4761	192	3	appl	appl	PROPN
ejpam-4761	192	4	.	.	PROPN
ejpam-4761	192	5	math	math	PROPN
ejpam-4761	192	6	,	,	PUNCT
ejpam-4761	192	7	16	16	NUM
ejpam-4761	192	8	(	(	PUNCT
ejpam-4761	192	9	2	2	NUM
ejpam-4761	192	10	)	)	PUNCT
ejpam-4761	192	11	(	(	PUNCT
ejpam-4761	192	12	2023	2023	NUM
ejpam-4761	192	13	)	)	PUNCT
ejpam-4761	192	14	,	,	PUNCT
ejpam-4761	192	15	1196	1196	NUM
ejpam-4761	192	16	-	-	SYM
ejpam-4761	192	17	1211	1211	NUM
ejpam-4761	192	18	1202	1202	NUM
ejpam-4761	192	19	v1	v1	PROPN
ejpam-4761	192	20	v2	v2	PROPN
ejpam-4761	192	21	v3	v3	PROPN
ejpam-4761	192	22	vm	vm	PROPN
ejpam-4761	192	23	g	g	PROPN
ejpam-4761	192	24	:	:	PUNCT
ejpam-4761	192	25	.	.	PUNCT
ejpam-4761	192	26	.	.	PUNCT
ejpam-4761	192	27	.	.	PUNCT
ejpam-4761	193	1	w1	w1	PROPN
ejpam-4761	193	2	w2	w2	PROPN
ejpam-4761	193	3	w3	w3	PROPN
ejpam-4761	193	4	figure	figure	VERB
ejpam-4761	193	5	3	3	NUM
ejpam-4761	193	6	:	:	PUNCT
ejpam-4761	193	7	a	a	DET
ejpam-4761	193	8	graph	graph	NOUN
ejpam-4761	193	9	g	g	NOUN
ejpam-4761	193	10	with	with	ADP
ejpam-4761	193	11	γwconh(g	γwconh(g	NOUN
ejpam-4761	193	12	)	)	PUNCT
ejpam-4761	193	13	<	<	X
ejpam-4761	193	14	γconh(g	γconh(g	PROPN
ejpam-4761	193	15	)	)	PUNCT
ejpam-4761	193	16	.	.	PUNCT
ejpam-4761	194	1	v1	v1	PROPN
ejpam-4761	194	2	v2	v2	PROPN
ejpam-4761	194	3	v3	v3	PROPN
ejpam-4761	194	4	vn	vn	PROPN
ejpam-4761	194	5	u1	u1	PROPN
ejpam-4761	194	6	u2	u2	PROPN
ejpam-4761	194	7	ua−3u3	ua−3u3	PROPN
ejpam-4761	194	8	.	.	PUNCT
ejpam-4761	194	9	.	.	PUNCT
ejpam-4761	195	1	.g′	.g′	PUNCT
ejpam-4761	195	2	:	:	PUNCT
ejpam-4761	195	3	.	.	PUNCT
ejpam-4761	195	4	.	.	PUNCT
ejpam-4761	196	1	.	.	PUNCT
ejpam-4761	197	1	ua−2	ua−2	NOUN
ejpam-4761	197	2	ua−1	ua−1	PROPN
ejpam-4761	197	3	ua	ua	PROPN
ejpam-4761	197	4	figure	figure	VERB
ejpam-4761	197	5	4	4	NUM
ejpam-4761	197	6	:	:	PUNCT
ejpam-4761	197	7	a	a	DET
ejpam-4761	197	8	graph	graph	NOUN
ejpam-4761	197	9	g′	g′	NOUN
ejpam-4761	197	10	with	with	ADP
ejpam-4761	197	11	γwconh(g	γwconh(g	NOUN
ejpam-4761	197	12	′	′	NUM
ejpam-4761	197	13	)	)	PUNCT
ejpam-4761	197	14	<	<	X
ejpam-4761	197	15	γconh(g	γconh(g	PROPN
ejpam-4761	197	16	′	′	PROPN
ejpam-4761	197	17	)	)	PUNCT
ejpam-4761	197	18	.	.	PUNCT
ejpam-4761	198	1	therefore	therefore	ADV
ejpam-4761	198	2	,	,	PUNCT
ejpam-4761	198	3	the	the	DET
ejpam-4761	198	4	assertion	assertion	NOUN
ejpam-4761	198	5	holds	hold	VERB
ejpam-4761	198	6	.	.	PUNCT
ejpam-4761	199	1	corollary	corollary	ADJ
ejpam-4761	199	2	3	3	X
ejpam-4761	199	3	.	.	PUNCT
ejpam-4761	200	1	let	let	VERB
ejpam-4761	200	2	n	n	PRON
ejpam-4761	200	3	be	be	AUX
ejpam-4761	200	4	a	a	DET
ejpam-4761	200	5	positive	positive	ADJ
ejpam-4761	200	6	integer	integer	NOUN
ejpam-4761	200	7	.	.	PUNCT
ejpam-4761	201	1	then	then	ADV
ejpam-4761	201	2	there	there	PRON
ejpam-4761	201	3	exists	exist	VERB
ejpam-4761	201	4	a	a	DET
ejpam-4761	201	5	connected	connected	ADJ
ejpam-4761	201	6	graph	graph	NOUN
ejpam-4761	201	7	g	g	ADP
ejpam-4761	201	8	such	such	ADJ
ejpam-4761	201	9	that	that	SCONJ
ejpam-4761	201	10	γconh(g)−	γconh(g)−	ADJ
ejpam-4761	201	11	γwconh(g	γwconh(g	NOUN
ejpam-4761	201	12	)	)	PUNCT
ejpam-4761	201	13	=	=	VERB
ejpam-4761	201	14	n.	n.	NOUN
ejpam-4761	201	15	in	in	ADP
ejpam-4761	201	16	other	other	ADJ
ejpam-4761	201	17	words	word	NOUN
ejpam-4761	201	18	,	,	PUNCT
ejpam-4761	201	19	γconh	γconh	NOUN
ejpam-4761	201	20	−	−	PROPN
ejpam-4761	201	21	γwconh	γwconh	NOUN
ejpam-4761	201	22	can	can	AUX
ejpam-4761	201	23	be	be	AUX
ejpam-4761	201	24	made	make	VERB
ejpam-4761	201	25	arbitrarily	arbitrarily	ADV
ejpam-4761	201	26	large	large	ADJ
ejpam-4761	201	27	.	.	PUNCT
ejpam-4761	202	1	proposition	proposition	NOUN
ejpam-4761	202	2	2	2	NUM
ejpam-4761	202	3	.	.	PUNCT
ejpam-4761	203	1	let	let	VERB
ejpam-4761	203	2	n	n	PRON
ejpam-4761	203	3	be	be	AUX
ejpam-4761	203	4	any	any	DET
ejpam-4761	203	5	positive	positive	ADJ
ejpam-4761	203	6	integer	integer	NOUN
ejpam-4761	203	7	.	.	PUNCT
ejpam-4761	204	1	then	then	ADV
ejpam-4761	204	2	each	each	PRON
ejpam-4761	204	3	of	of	ADP
ejpam-4761	204	4	the	the	DET
ejpam-4761	204	5	following	follow	VERB
ejpam-4761	204	6	holds	hold	NOUN
ejpam-4761	204	7	.	.	PUNCT
ejpam-4761	205	1	(	(	PUNCT
ejpam-4761	205	2	i	i	NOUN
ejpam-4761	205	3	)	)	PUNCT
ejpam-4761	205	4	γwconh(pn	γwconh(pn	NOUN
ejpam-4761	205	5	)	)	PUNCT
ejpam-4761	205	6	=	=	SYM
ejpam-4761	205	7	{	{	PUNCT
ejpam-4761	205	8	2	2	NUM
ejpam-4761	205	9	if	if	SCONJ
ejpam-4761	205	10	n	n	X
ejpam-4761	205	11	=	=	SYM
ejpam-4761	205	12	2	2	NUM
ejpam-4761	205	13	,	,	PUNCT
ejpam-4761	205	14	3	3	NUM
ejpam-4761	205	15	,	,	PUNCT
ejpam-4761	205	16	4	4	NUM
ejpam-4761	205	17	,	,	PUNCT
ejpam-4761	205	18	5	5	NUM
ejpam-4761	205	19	,	,	PUNCT
ejpam-4761	205	20	6	6	NUM
ejpam-4761	205	21	n−	n−	NOUN
ejpam-4761	205	22	4	4	NUM
ejpam-4761	205	23	if	if	SCONJ
ejpam-4761	205	24	n	n	PRON
ejpam-4761	205	25	≥	≥	NOUN
ejpam-4761	205	26	7	7	NUM
ejpam-4761	205	27	.	.	PUNCT
ejpam-4761	205	28	(	(	PUNCT
ejpam-4761	205	29	ii	ii	NOUN
ejpam-4761	205	30	)	)	PUNCT
ejpam-4761	205	31	γwconh(cn	γwconh(cn	NOUN
ejpam-4761	205	32	)	)	PUNCT
ejpam-4761	205	33	=	=	PUNCT
ejpam-4761	206	1			NOUN
ejpam-4761	206	2	2	2	NUM
ejpam-4761	206	3	if	if	SCONJ
ejpam-4761	206	4	n	n	NOUN
ejpam-4761	206	5	=	=	SYM
ejpam-4761	206	6	4	4	NUM
ejpam-4761	206	7	,	,	PUNCT
ejpam-4761	206	8	5	5	NUM
ejpam-4761	206	9	3	3	NUM
ejpam-4761	206	10	if	if	SCONJ
ejpam-4761	206	11	n	n	NOUN
ejpam-4761	206	12	=	=	SYM
ejpam-4761	206	13	3	3	NUM
ejpam-4761	206	14	n−	n−	NOUN
ejpam-4761	206	15	4	4	NUM
ejpam-4761	206	16	if	if	SCONJ
ejpam-4761	206	17	6	6	NUM
ejpam-4761	206	18	≤	≤	NUM
ejpam-4761	206	19	n	n	PRON
ejpam-4761	206	20	≤	≤	NUM
ejpam-4761	206	21	10	10	NUM
ejpam-4761	207	1	n	n	NOUN
ejpam-4761	208	1	if	if	SCONJ
ejpam-4761	208	2	n	n	PRON
ejpam-4761	208	3	≥	≥	NOUN
ejpam-4761	208	4	11	11	NUM
ejpam-4761	209	1	.	.	PUNCT
ejpam-4761	210	1	s.	s.	PROPN
ejpam-4761	210	2	canoy	canoy	PROPN
ejpam-4761	210	3	jr	jr	PROPN
ejpam-4761	210	4	.	.	PROPN
ejpam-4761	210	5	,	,	PUNCT
ejpam-4761	210	6	j.	j.	PROPN
ejpam-4761	210	7	hassan	hassan	PROPN
ejpam-4761	210	8	/	/	SYM
ejpam-4761	210	9	eur	eur	PROPN
ejpam-4761	210	10	.	.	PUNCT
ejpam-4761	211	1	j.	j.	PROPN
ejpam-4761	211	2	pure	pure	PROPN
ejpam-4761	211	3	appl	appl	PROPN
ejpam-4761	211	4	.	.	PROPN
ejpam-4761	211	5	math	math	PROPN
ejpam-4761	211	6	,	,	PUNCT
ejpam-4761	211	7	16	16	NUM
ejpam-4761	211	8	(	(	PUNCT
ejpam-4761	211	9	2	2	NUM
ejpam-4761	211	10	)	)	PUNCT
ejpam-4761	211	11	(	(	PUNCT
ejpam-4761	211	12	2023	2023	NUM
ejpam-4761	211	13	)	)	PUNCT
ejpam-4761	211	14	,	,	PUNCT
ejpam-4761	211	15	1196	1196	NUM
ejpam-4761	211	16	-	-	SYM
ejpam-4761	211	17	1211	1211	NUM
ejpam-4761	211	18	1203	1203	NUM
ejpam-4761	211	19	proof	proof	NOUN
ejpam-4761	211	20	.	.	PUNCT
ejpam-4761	212	1	(	(	PUNCT
ejpam-4761	212	2	i	i	NOUN
ejpam-4761	212	3	)	)	PUNCT
ejpam-4761	212	4	clearly	clearly	ADV
ejpam-4761	212	5	,	,	PUNCT
ejpam-4761	212	6	γwconh(pn	γwconh(pn	NOUN
ejpam-4761	212	7	)	)	PUNCT
ejpam-4761	212	8	=	=	SYM
ejpam-4761	212	9	2	2	NUM
ejpam-4761	212	10	for	for	ADP
ejpam-4761	212	11	n	n	PRON
ejpam-4761	212	12	∈	∈	NOUN
ejpam-4761	212	13	{	{	PUNCT
ejpam-4761	212	14	2	2	NUM
ejpam-4761	212	15	,	,	PUNCT
ejpam-4761	212	16	3	3	NUM
ejpam-4761	212	17	,	,	PUNCT
ejpam-4761	212	18	4	4	NUM
ejpam-4761	212	19	,	,	PUNCT
ejpam-4761	212	20	5	5	NUM
ejpam-4761	212	21	,	,	PUNCT
ejpam-4761	212	22	6	6	NUM
ejpam-4761	212	23	}	}	PUNCT
ejpam-4761	212	24	.	.	PUNCT
ejpam-4761	213	1	suppose	suppose	VERB
ejpam-4761	213	2	n	n	PRON
ejpam-4761	213	3	≥	≥	NUM
ejpam-4761	213	4	7	7	NUM
ejpam-4761	213	5	.	.	PUNCT
ejpam-4761	214	1	let	let	VERB
ejpam-4761	214	2	pn	pn	VERB
ejpam-4761	214	3	=	=	PUNCT
ejpam-4761	215	1	[	[	X
ejpam-4761	215	2	v1	v1	NOUN
ejpam-4761	215	3	,	,	PUNCT
ejpam-4761	215	4	v2	v2	NOUN
ejpam-4761	215	5	,	,	PUNCT
ejpam-4761	215	6	.	.	PUNCT
ejpam-4761	215	7	.	.	PUNCT
ejpam-4761	215	8	.	.	PUNCT
ejpam-4761	216	1	,	,	PUNCT
ejpam-4761	216	2	vn	vn	X
ejpam-4761	216	3	]	]	PUNCT
ejpam-4761	216	4	and	and	CCONJ
ejpam-4761	216	5	consider	consider	VERB
ejpam-4761	216	6	w	w	NOUN
ejpam-4761	216	7	=	=	SYM
ejpam-4761	216	8	{	{	PUNCT
ejpam-4761	216	9	v3	v3	PROPN
ejpam-4761	216	10	,	,	PUNCT
ejpam-4761	216	11	v4	v4	PROPN
ejpam-4761	216	12	,	,	PUNCT
ejpam-4761	216	13	.	.	PUNCT
ejpam-4761	216	14	.	.	PUNCT
ejpam-4761	217	1	.	.	PUNCT
ejpam-4761	218	1	,	,	PUNCT
ejpam-4761	218	2	vn−3	vn−3	PROPN
ejpam-4761	218	3	,	,	PUNCT
ejpam-4761	218	4	vn−2	vn−2	PROPN
ejpam-4761	218	5	}	}	PUNCT
ejpam-4761	218	6	.	.	PUNCT
ejpam-4761	219	1	clearly	clearly	ADV
ejpam-4761	219	2	,	,	PUNCT
ejpam-4761	219	3	w	w	NOUN
ejpam-4761	219	4	is	be	AUX
ejpam-4761	219	5	weakly	weakly	ADJ
ejpam-4761	219	6	convex	convex	NOUN
ejpam-4761	219	7	set	set	VERB
ejpam-4761	219	8	in	in	ADP
ejpam-4761	219	9	pn	pn	PROPN
ejpam-4761	219	10	.	.	PUNCT
ejpam-4761	220	1	now	now	ADV
ejpam-4761	220	2	,	,	PUNCT
ejpam-4761	220	3	observe	observe	VERB
ejpam-4761	220	4	that	that	SCONJ
ejpam-4761	220	5	n	n	PROPN
ejpam-4761	220	6	2	2	NUM
ejpam-4761	220	7	g[w	g[w	NOUN
ejpam-4761	220	8	]	]	X
ejpam-4761	221	1	=	=	SYM
ejpam-4761	221	2	v	v	X
ejpam-4761	221	3	(	(	PUNCT
ejpam-4761	221	4	pn	pn	NOUN
ejpam-4761	221	5	)	)	PUNCT
ejpam-4761	221	6	and	and	CCONJ
ejpam-4761	221	7	so	so	ADV
ejpam-4761	221	8	w	w	PROPN
ejpam-4761	221	9	is	be	AUX
ejpam-4761	221	10	a	a	DET
ejpam-4761	221	11	hop	hop	NOUN
ejpam-4761	221	12	dominating	dominating	NOUN
ejpam-4761	221	13	set	set	VERB
ejpam-4761	221	14	in	in	ADP
ejpam-4761	221	15	pn	pn	PROPN
ejpam-4761	221	16	.	.	PUNCT
ejpam-4761	222	1	therefore	therefore	ADV
ejpam-4761	222	2	,	,	PUNCT
ejpam-4761	222	3	w	w	PROPN
ejpam-4761	222	4	is	be	AUX
ejpam-4761	222	5	a	a	DET
ejpam-4761	222	6	weakly	weakly	ADJ
ejpam-4761	222	7	convex	convex	NOUN
ejpam-4761	222	8	hop	hop	NOUN
ejpam-4761	222	9	dominating	dominating	NOUN
ejpam-4761	222	10	set	set	VERB
ejpam-4761	222	11	in	in	ADP
ejpam-4761	222	12	pn	pn	PROPN
ejpam-4761	222	13	.	.	PROPN
ejpam-4761	222	14	notice	notice	VERB
ejpam-4761	222	15	that	that	SCONJ
ejpam-4761	222	16	every	every	DET
ejpam-4761	222	17	weakly	weakly	ADJ
ejpam-4761	222	18	convex	convex	NOUN
ejpam-4761	222	19	hop	hop	NOUN
ejpam-4761	222	20	dominating	dominating	NOUN
ejpam-4761	222	21	set	set	VERB
ejpam-4761	222	22	in	in	ADP
ejpam-4761	222	23	pn	pn	PROPN
ejpam-4761	222	24	contains	contain	VERB
ejpam-4761	222	25	w	w	PROPN
ejpam-4761	222	26	.	.	PUNCT
ejpam-4761	223	1	it	it	PRON
ejpam-4761	223	2	follows	follow	VERB
ejpam-4761	223	3	that	that	SCONJ
ejpam-4761	223	4	w	w	NOUN
ejpam-4761	223	5	is	be	AUX
ejpam-4761	223	6	a	a	DET
ejpam-4761	223	7	γwconh	γwconh	NOUN
ejpam-4761	223	8	-	-	PUNCT
ejpam-4761	223	9	set	set	NOUN
ejpam-4761	223	10	of	of	ADP
ejpam-4761	223	11	pn	pn	PROPN
ejpam-4761	223	12	.	.	PROPN
ejpam-4761	224	1	hence	hence	ADV
ejpam-4761	224	2	,	,	PUNCT
ejpam-4761	224	3	γwconh(pn	γwconh(pn	NOUN
ejpam-4761	224	4	)	)	PUNCT
ejpam-4761	224	5	=	=	PUNCT
ejpam-4761	225	1	n−	n−	NOUN
ejpam-4761	225	2	4	4	NUM
ejpam-4761	225	3	for	for	ADP
ejpam-4761	225	4	all	all	DET
ejpam-4761	225	5	n	n	PRON
ejpam-4761	225	6	≥	≥	NOUN
ejpam-4761	225	7	7	7	NUM
ejpam-4761	225	8	.	.	PUNCT
ejpam-4761	225	9	(	(	PUNCT
ejpam-4761	225	10	ii	ii	NOUN
ejpam-4761	225	11	)	)	PUNCT
ejpam-4761	225	12	clearly	clearly	ADV
ejpam-4761	225	13	,	,	PUNCT
ejpam-4761	225	14	γwconh(cn	γwconh(cn	NOUN
ejpam-4761	225	15	)	)	PUNCT
ejpam-4761	225	16	=	=	SYM
ejpam-4761	225	17	2	2	NUM
ejpam-4761	225	18	for	for	ADP
ejpam-4761	225	19	n	n	PRON
ejpam-4761	225	20	∈	∈	NOUN
ejpam-4761	225	21	{	{	PUNCT
ejpam-4761	225	22	4	4	NUM
ejpam-4761	225	23	,	,	PUNCT
ejpam-4761	225	24	5	5	NUM
ejpam-4761	225	25	}	}	PUNCT
ejpam-4761	225	26	and	and	CCONJ
ejpam-4761	225	27	γwconh(c3	γwconh(c3	ADJ
ejpam-4761	225	28	)	)	PUNCT
ejpam-4761	225	29	=	=	SYM
ejpam-4761	226	1	3	3	X
ejpam-4761	226	2	.	.	PUNCT
ejpam-4761	226	3	suppose	suppose	VERB
ejpam-4761	226	4	6	6	NUM
ejpam-4761	226	5	≤	≤	NOUN
ejpam-4761	226	6	n	n	PRON
ejpam-4761	226	7	≤	≤	NOUN
ejpam-4761	226	8	10	10	NUM
ejpam-4761	226	9	.	.	PUNCT
ejpam-4761	227	1	let	let	VERB
ejpam-4761	227	2	cn	cn	PROPN
ejpam-4761	227	3	=	=	PUNCT
ejpam-4761	228	1	[	[	X
ejpam-4761	228	2	v1	v1	NOUN
ejpam-4761	228	3	,	,	PUNCT
ejpam-4761	228	4	v2	v2	NOUN
ejpam-4761	228	5	,	,	PUNCT
ejpam-4761	228	6	.	.	PUNCT
ejpam-4761	228	7	.	.	PUNCT
ejpam-4761	228	8	.	.	PUNCT
ejpam-4761	229	1	,	,	PUNCT
ejpam-4761	229	2	vn	vn	X
ejpam-4761	229	3	,	,	PUNCT
ejpam-4761	229	4	v1	v1	PROPN
ejpam-4761	229	5	]	]	PUNCT
ejpam-4761	229	6	and	and	CCONJ
ejpam-4761	229	7	let	let	VERB
ejpam-4761	229	8	w	w	PRON
ejpam-4761	229	9	′	′	NOUN
ejpam-4761	229	10	=	=	PUNCT
ejpam-4761	229	11	{	{	PUNCT
ejpam-4761	229	12	v1	v1	NOUN
ejpam-4761	229	13	,	,	PUNCT
ejpam-4761	229	14	v2	v2	PROPN
ejpam-4761	229	15	,	,	PUNCT
ejpam-4761	229	16	.	.	PUNCT
ejpam-4761	229	17	.	.	PUNCT
ejpam-4761	230	1	.	.	PUNCT
ejpam-4761	231	1	,	,	PUNCT
ejpam-4761	231	2	vn−4	vn−4	NOUN
ejpam-4761	231	3	}	}	PUNCT
ejpam-4761	231	4	.	.	PUNCT
ejpam-4761	232	1	then	then	ADV
ejpam-4761	232	2	w	w	NOUN
ejpam-4761	232	3	′	′	PROPN
ejpam-4761	232	4	is	be	AUX
ejpam-4761	232	5	a	a	DET
ejpam-4761	232	6	minimum	minimum	ADJ
ejpam-4761	232	7	weakly	weakly	ADJ
ejpam-4761	232	8	convex	convex	NOUN
ejpam-4761	232	9	hop	hop	NOUN
ejpam-4761	232	10	dominating	dominating	NOUN
ejpam-4761	232	11	set	set	NOUN
ejpam-4761	232	12	in	in	ADP
ejpam-4761	232	13	cn	cn	PROPN
ejpam-4761	232	14	.	.	PUNCT
ejpam-4761	233	1	thus	thus	ADV
ejpam-4761	233	2	,	,	PUNCT
ejpam-4761	233	3	γwconh(cn	γwconh(cn	NOUN
ejpam-4761	233	4	)	)	PUNCT
ejpam-4761	233	5	=	=	SYM
ejpam-4761	234	1	n	n	CCONJ
ejpam-4761	234	2	−	−	NOUN
ejpam-4761	234	3	4	4	NUM
ejpam-4761	234	4	for	for	ADP
ejpam-4761	234	5	all	all	DET
ejpam-4761	234	6	n	n	CCONJ
ejpam-4761	234	7	,	,	PUNCT
ejpam-4761	234	8	where	where	SCONJ
ejpam-4761	234	9	6	6	NUM
ejpam-4761	234	10	≤	≤	NOUN
ejpam-4761	234	11	n	n	PRON
ejpam-4761	234	12	≤	≤	NOUN
ejpam-4761	234	13	10	10	NUM
ejpam-4761	234	14	.	.	PUNCT
ejpam-4761	235	1	next	next	ADV
ejpam-4761	235	2	,	,	PUNCT
ejpam-4761	235	3	suppose	suppose	VERB
ejpam-4761	235	4	that	that	SCONJ
ejpam-4761	235	5	n	n	PROPN
ejpam-4761	235	6	≥	≥	NUM
ejpam-4761	235	7	11	11	NUM
ejpam-4761	235	8	.	.	PUNCT
ejpam-4761	236	1	clearly	clearly	ADV
ejpam-4761	236	2	,	,	PUNCT
ejpam-4761	236	3	γch(cn	γch(cn	ADJ
ejpam-4761	236	4	)	)	PUNCT
ejpam-4761	237	1	=	=	PUNCT
ejpam-4761	237	2	n−	n−	NOUN
ejpam-4761	237	3	4	4	NUM
ejpam-4761	237	4	and	and	CCONJ
ejpam-4761	237	5	if	if	SCONJ
ejpam-4761	237	6	s	s	X
ejpam-4761	237	7	is	be	AUX
ejpam-4761	237	8	a	a	DET
ejpam-4761	237	9	connected	connected	ADJ
ejpam-4761	237	10	hop	hop	NOUN
ejpam-4761	237	11	dominating	dominating	NOUN
ejpam-4761	237	12	set	set	NOUN
ejpam-4761	237	13	in	in	ADP
ejpam-4761	237	14	cn	cn	PROPN
ejpam-4761	237	15	with	with	ADP
ejpam-4761	237	16	s	s	PROPN
ejpam-4761	237	17	̸=	̸=	PROPN
ejpam-4761	237	18	v	v	NOUN
ejpam-4761	237	19	(	(	PUNCT
ejpam-4761	237	20	cn	cn	PROPN
ejpam-4761	237	21	)	)	PUNCT
ejpam-4761	237	22	,	,	PUNCT
ejpam-4761	237	23	then	then	ADV
ejpam-4761	237	24	s	s	VERB
ejpam-4761	237	25	satisfies	satisfie	NOUN
ejpam-4761	237	26	the	the	DET
ejpam-4761	237	27	property	property	NOUN
ejpam-4761	237	28	given	give	VERB
ejpam-4761	237	29	in	in	ADP
ejpam-4761	237	30	theorem	theorem	NOUN
ejpam-4761	237	31	1(ii	1(ii	NUM
ejpam-4761	237	32	)	)	PUNCT
ejpam-4761	237	33	.	.	PUNCT
ejpam-4761	238	1	hence	hence	ADV
ejpam-4761	238	2	,	,	PUNCT
ejpam-4761	238	3	γwconh(cn	γwconh(cn	NOUN
ejpam-4761	238	4	)	)	PUNCT
ejpam-4761	238	5	=	=	SYM
ejpam-4761	238	6	n.	n.	NOUN
ejpam-4761	238	7	theorem	theorem	VERB
ejpam-4761	238	8	3	3	NUM
ejpam-4761	238	9	.	.	PUNCT
ejpam-4761	239	1	[	[	X
ejpam-4761	239	2	9	9	NUM
ejpam-4761	239	3	]	]	PUNCT
ejpam-4761	239	4	let	let	VERB
ejpam-4761	239	5	g	g	PRON
ejpam-4761	239	6	be	be	AUX
ejpam-4761	239	7	a	a	DET
ejpam-4761	239	8	connected	connected	ADJ
ejpam-4761	239	9	graph	graph	NOUN
ejpam-4761	239	10	of	of	ADP
ejpam-4761	239	11	order	order	NOUN
ejpam-4761	239	12	n.	n.	NOUN
ejpam-4761	239	13	then	then	ADV
ejpam-4761	239	14	γconh(gg	γconh(gg	NOUN
ejpam-4761	239	15	)	)	PUNCT
ejpam-4761	239	16	=	=	SYM
ejpam-4761	240	1	2	2	X
ejpam-4761	240	2	.	.	X
ejpam-4761	240	3	in	in	ADP
ejpam-4761	240	4	particular	particular	ADJ
ejpam-4761	240	5	,	,	PUNCT
ejpam-4761	240	6	{	{	PUNCT
ejpam-4761	240	7	u	u	NOUN
ejpam-4761	240	8	,	,	PUNCT
ejpam-4761	240	9	u	u	NOUN
ejpam-4761	240	10	}	}	PUNCT
ejpam-4761	240	11	is	be	AUX
ejpam-4761	240	12	a	a	DET
ejpam-4761	240	13	γconh	γconh	NOUN
ejpam-4761	240	14	-	-	PUNCT
ejpam-4761	240	15	set	set	NOUN
ejpam-4761	240	16	of	of	ADP
ejpam-4761	240	17	gg	gg	NOUN
ejpam-4761	240	18	for	for	ADP
ejpam-4761	240	19	each	each	DET
ejpam-4761	240	20	u	u	PROPN
ejpam-4761	240	21	∈	∈	PROPN
ejpam-4761	240	22	v	v	NOUN
ejpam-4761	240	23	(	(	PUNCT
ejpam-4761	240	24	g	g	NOUN
ejpam-4761	240	25	)	)	PUNCT
ejpam-4761	240	26	.	.	PUNCT
ejpam-4761	241	1	the	the	DET
ejpam-4761	241	2	next	next	ADJ
ejpam-4761	241	3	result	result	NOUN
ejpam-4761	241	4	is	be	AUX
ejpam-4761	241	5	immediate	immediate	ADJ
ejpam-4761	241	6	from	from	ADP
ejpam-4761	241	7	corollary	corollary	ADJ
ejpam-4761	241	8	2	2	NUM
ejpam-4761	241	9	and	and	CCONJ
ejpam-4761	241	10	theorem	theorem	VERB
ejpam-4761	241	11	3	3	NUM
ejpam-4761	241	12	.	.	PUNCT
ejpam-4761	241	13	theorem	theorem	NOUN
ejpam-4761	241	14	4	4	NUM
ejpam-4761	241	15	.	.	PUNCT
ejpam-4761	242	1	let	let	VERB
ejpam-4761	242	2	g	g	NOUN
ejpam-4761	242	3	be	be	AUX
ejpam-4761	242	4	any	any	DET
ejpam-4761	242	5	connected	connected	ADJ
ejpam-4761	242	6	graph	graph	NOUN
ejpam-4761	242	7	of	of	ADP
ejpam-4761	242	8	order	order	NOUN
ejpam-4761	242	9	n.	n.	NOUN
ejpam-4761	242	10	then	then	ADV
ejpam-4761	242	11	γwconh(gg	γwconh(gg	VERB
ejpam-4761	242	12	)	)	PUNCT
ejpam-4761	242	13	=	=	SYM
ejpam-4761	243	1	2	2	X
ejpam-4761	243	2	.	.	X
ejpam-4761	244	1	in	in	ADP
ejpam-4761	244	2	particular	particular	ADJ
ejpam-4761	244	3	,	,	PUNCT
ejpam-4761	244	4	w	w	NOUN
ejpam-4761	244	5	=	=	X
ejpam-4761	244	6	{	{	PUNCT
ejpam-4761	244	7	a	a	NOUN
ejpam-4761	244	8	,	,	PUNCT
ejpam-4761	244	9	a	a	PRON
ejpam-4761	244	10	}	}	PUNCT
ejpam-4761	244	11	is	be	AUX
ejpam-4761	244	12	a	a	DET
ejpam-4761	244	13	γwconh	γwconh	NOUN
ejpam-4761	244	14	-	-	PUNCT
ejpam-4761	244	15	set	set	NOUN
ejpam-4761	244	16	of	of	ADP
ejpam-4761	244	17	gg	gg	NOUN
ejpam-4761	244	18	for	for	ADP
ejpam-4761	244	19	any	any	DET
ejpam-4761	244	20	a	a	DET
ejpam-4761	244	21	∈	∈	PROPN
ejpam-4761	244	22	v	v	NOUN
ejpam-4761	244	23	(	(	PUNCT
ejpam-4761	244	24	g	g	NOUN
ejpam-4761	244	25	)	)	PUNCT
ejpam-4761	244	26	.	.	PUNCT
ejpam-4761	245	1	theorem	theorem	ADJ
ejpam-4761	245	2	5	5	NUM
ejpam-4761	245	3	.	.	PUNCT
ejpam-4761	246	1	[	[	X
ejpam-4761	246	2	9	9	NUM
ejpam-4761	246	3	]	]	PUNCT
ejpam-4761	246	4	let	let	VERB
ejpam-4761	246	5	g	g	PRON
ejpam-4761	246	6	be	be	AUX
ejpam-4761	246	7	a	a	DET
ejpam-4761	246	8	non	non	ADJ
ejpam-4761	246	9	-	-	ADJ
ejpam-4761	246	10	trivial	trivial	ADJ
ejpam-4761	246	11	connected	connected	ADJ
ejpam-4761	246	12	graph	graph	NOUN
ejpam-4761	246	13	.	.	PUNCT
ejpam-4761	247	1	then	then	ADV
ejpam-4761	247	2	s	s	VERB
ejpam-4761	247	3	is	be	AUX
ejpam-4761	247	4	a	a	DET
ejpam-4761	247	5	hop	hop	NOUN
ejpam-4761	247	6	dominating	dominating	NOUN
ejpam-4761	247	7	set	set	NOUN
ejpam-4761	247	8	in	in	ADP
ejpam-4761	247	9	s(g	s(g	PROPN
ejpam-4761	247	10	)	)	PUNCT
ejpam-4761	247	11	if	if	SCONJ
ejpam-4761	247	12	and	and	CCONJ
ejpam-4761	247	13	only	only	ADV
ejpam-4761	247	14	if	if	SCONJ
ejpam-4761	247	15	one	one	NUM
ejpam-4761	247	16	of	of	ADP
ejpam-4761	247	17	the	the	DET
ejpam-4761	247	18	following	follow	VERB
ejpam-4761	247	19	conditions	condition	NOUN
ejpam-4761	247	20	holds	hold	VERB
ejpam-4761	247	21	:	:	PUNCT
ejpam-4761	247	22	(	(	PUNCT
ejpam-4761	247	23	i	i	NOUN
ejpam-4761	247	24	)	)	PUNCT
ejpam-4761	247	25	s	s	VERB
ejpam-4761	247	26	is	be	AUX
ejpam-4761	247	27	a	a	DET
ejpam-4761	247	28	hop	hop	NOUN
ejpam-4761	247	29	dominating	dominating	NOUN
ejpam-4761	247	30	set	set	VERB
ejpam-4761	247	31	in	in	ADP
ejpam-4761	247	32	g1	g1	PROPN
ejpam-4761	247	33	.	.	PUNCT
ejpam-4761	248	1	(	(	PUNCT
ejpam-4761	248	2	ii	ii	X
ejpam-4761	248	3	)	)	PUNCT
ejpam-4761	248	4	s	s	VERB
ejpam-4761	248	5	is	be	AUX
ejpam-4761	248	6	a	a	DET
ejpam-4761	248	7	hop	hop	NOUN
ejpam-4761	248	8	dominating	dominating	NOUN
ejpam-4761	248	9	set	set	VERB
ejpam-4761	248	10	in	in	ADP
ejpam-4761	248	11	g2	g2	PROPN
ejpam-4761	248	12	.	.	PUNCT
ejpam-4761	249	1	(	(	PUNCT
ejpam-4761	249	2	iii	iii	X
ejpam-4761	249	3	)	)	PUNCT
ejpam-4761	249	4	s	s	PART
ejpam-4761	249	5	=	=	NOUN
ejpam-4761	249	6	sg1	sg1	NOUN
ejpam-4761	249	7	∪	∪	VERB
ejpam-4761	249	8	sg2	sg2	PROPN
ejpam-4761	249	9	such	such	ADJ
ejpam-4761	249	10	that	that	SCONJ
ejpam-4761	249	11	sg1	sg1	NOUN
ejpam-4761	249	12	∪	∪	ADP
ejpam-4761	249	13	s′	s′	ADJ
ejpam-4761	249	14	g2	g2	PROPN
ejpam-4761	249	15	and	and	CCONJ
ejpam-4761	249	16	s′	s′	ADJ
ejpam-4761	249	17	g1	g1	PROPN
ejpam-4761	249	18	∪	∪	ADP
ejpam-4761	249	19	sg2	sg2	PROPN
ejpam-4761	249	20	are	be	AUX
ejpam-4761	249	21	hop	hop	NOUN
ejpam-4761	249	22	dominating	dominating	NOUN
ejpam-4761	249	23	sets	set	NOUN
ejpam-4761	249	24	in	in	ADP
ejpam-4761	249	25	g1	g1	PROPN
ejpam-4761	249	26	and	and	CCONJ
ejpam-4761	249	27	g2	g2	PROPN
ejpam-4761	249	28	,	,	PUNCT
ejpam-4761	249	29	respectively	respectively	ADV
ejpam-4761	249	30	.	.	PUNCT
ejpam-4761	250	1	theorem	theorem	VERB
ejpam-4761	250	2	6	6	NUM
ejpam-4761	250	3	.	.	PUNCT
ejpam-4761	251	1	let	let	VERB
ejpam-4761	251	2	g	g	PRON
ejpam-4761	251	3	be	be	AUX
ejpam-4761	251	4	a	a	DET
ejpam-4761	251	5	non	non	ADJ
ejpam-4761	251	6	-	-	ADJ
ejpam-4761	251	7	trivial	trivial	ADJ
ejpam-4761	251	8	connected	connected	ADJ
ejpam-4761	251	9	graph	graph	NOUN
ejpam-4761	251	10	.	.	PUNCT
ejpam-4761	252	1	then	then	ADV
ejpam-4761	252	2	w	w	PROPN
ejpam-4761	252	3	is	be	AUX
ejpam-4761	252	4	a	a	DET
ejpam-4761	252	5	weakly	weakly	ADJ
ejpam-4761	252	6	convex	convex	NOUN
ejpam-4761	252	7	hop	hop	NOUN
ejpam-4761	252	8	dominating	dominating	NOUN
ejpam-4761	252	9	set	set	NOUN
ejpam-4761	252	10	in	in	ADP
ejpam-4761	252	11	s(g	s(g	PROPN
ejpam-4761	252	12	)	)	PUNCT
ejpam-4761	252	13	if	if	SCONJ
ejpam-4761	252	14	and	and	CCONJ
ejpam-4761	252	15	only	only	ADV
ejpam-4761	252	16	if	if	SCONJ
ejpam-4761	252	17	one	one	NUM
ejpam-4761	252	18	of	of	ADP
ejpam-4761	252	19	the	the	DET
ejpam-4761	252	20	following	follow	VERB
ejpam-4761	252	21	conditions	condition	NOUN
ejpam-4761	252	22	holds	hold	VERB
ejpam-4761	252	23	:	:	PUNCT
ejpam-4761	252	24	(	(	PUNCT
ejpam-4761	252	25	i	i	NOUN
ejpam-4761	252	26	)	)	PUNCT
ejpam-4761	252	27	w	w	NOUN
ejpam-4761	252	28	is	be	AUX
ejpam-4761	252	29	weakly	weakly	ADV
ejpam-4761	252	30	convex	convex	ADJ
ejpam-4761	252	31	hop	hop	NOUN
ejpam-4761	252	32	dominating	dominating	NOUN
ejpam-4761	252	33	set	set	VERB
ejpam-4761	252	34	in	in	ADP
ejpam-4761	252	35	g1	g1	PROPN
ejpam-4761	252	36	.	.	PUNCT
ejpam-4761	253	1	(	(	PUNCT
ejpam-4761	253	2	ii	ii	NOUN
ejpam-4761	253	3	)	)	PUNCT
ejpam-4761	253	4	w	w	NOUN
ejpam-4761	253	5	is	be	AUX
ejpam-4761	253	6	weakly	weakly	ADV
ejpam-4761	253	7	convex	convex	ADJ
ejpam-4761	253	8	hop	hop	NOUN
ejpam-4761	253	9	dominating	dominating	NOUN
ejpam-4761	253	10	set	set	VERB
ejpam-4761	253	11	in	in	ADP
ejpam-4761	253	12	g2	g2	PROPN
ejpam-4761	253	13	.	.	PUNCT
ejpam-4761	254	1	(	(	PUNCT
ejpam-4761	254	2	iii	iii	X
ejpam-4761	254	3	)	)	PUNCT
ejpam-4761	254	4	w	w	NOUN
ejpam-4761	254	5	=	=	PUNCT
ejpam-4761	254	6	wg1	wg1	ADJ
ejpam-4761	254	7	∪	∪	ADJ
ejpam-4761	254	8	wg2	wg2	NOUN
ejpam-4761	254	9	such	such	ADJ
ejpam-4761	254	10	that	that	PRON
ejpam-4761	254	11	wg1	wg1	ADJ
ejpam-4761	254	12	∪	∪	NOUN
ejpam-4761	254	13	w	w	PROPN
ejpam-4761	254	14	′	′	NUM
ejpam-4761	254	15	g2	g2	PROPN
ejpam-4761	254	16	and	and	CCONJ
ejpam-4761	254	17	w	w	PROPN
ejpam-4761	254	18	′	′	PROPN
ejpam-4761	254	19	g1	g1	PROPN
ejpam-4761	254	20	∪	∪	VERB
ejpam-4761	254	21	wg2	wg2	NOUN
ejpam-4761	254	22	are	be	AUX
ejpam-4761	254	23	weakly	weakly	ADV
ejpam-4761	254	24	convex	convex	ADJ
ejpam-4761	254	25	hop	hop	NOUN
ejpam-4761	254	26	dominating	dominating	NOUN
ejpam-4761	254	27	sets	set	NOUN
ejpam-4761	254	28	in	in	ADP
ejpam-4761	254	29	g1	g1	PROPN
ejpam-4761	254	30	and	and	CCONJ
ejpam-4761	254	31	g2	g2	PROPN
ejpam-4761	254	32	,	,	PUNCT
ejpam-4761	254	33	respectively	respectively	ADV
ejpam-4761	254	34	,	,	PUNCT
ejpam-4761	254	35	where	where	SCONJ
ejpam-4761	254	36	w	w	NOUN
ejpam-4761	254	37	′	′	NOUN
ejpam-4761	254	38	g2	g2	PROPN
ejpam-4761	254	39	=	=	PRON
ejpam-4761	255	1	{	{	PUNCT
ejpam-4761	255	2	a	a	PRON
ejpam-4761	255	3	∈	∈	PROPN
ejpam-4761	255	4	v	v	NOUN
ejpam-4761	255	5	(	(	PUNCT
ejpam-4761	255	6	g1	g1	PROPN
ejpam-4761	255	7	)	)	PUNCT
ejpam-4761	255	8	:	:	PUNCT
ejpam-4761	255	9	a	a	DET
ejpam-4761	255	10	′	′	NUM
ejpam-4761	255	11	∈	∈	PROPN
ejpam-4761	255	12	wg2	wg2	NOUN
ejpam-4761	255	13	}	}	PUNCT
ejpam-4761	255	14	and	and	CCONJ
ejpam-4761	255	15	w	w	NOUN
ejpam-4761	255	16	′	′	NUM
ejpam-4761	255	17	g1	g1	NOUN
ejpam-4761	255	18	=	=	PRON
ejpam-4761	255	19	{	{	PUNCT
ejpam-4761	255	20	a	a	DET
ejpam-4761	255	21	∈	∈	PROPN
ejpam-4761	255	22	v	v	NOUN
ejpam-4761	255	23	(	(	PUNCT
ejpam-4761	255	24	g2	g2	PROPN
ejpam-4761	255	25	)	)	PUNCT
ejpam-4761	255	26	:	:	PUNCT
ejpam-4761	255	27	a	a	DET
ejpam-4761	255	28	′	′	NUM
ejpam-4761	255	29	∈	∈	NOUN
ejpam-4761	255	30	wg1	wg1	NOUN
ejpam-4761	255	31	}	}	PUNCT
ejpam-4761	255	32	proof	proof	NOUN
ejpam-4761	255	33	.	.	PUNCT
ejpam-4761	256	1	let	let	VERB
ejpam-4761	256	2	w	w	NOUN
ejpam-4761	256	3	be	be	AUX
ejpam-4761	256	4	a	a	DET
ejpam-4761	256	5	weakly	weakly	ADJ
ejpam-4761	256	6	convex	convex	NOUN
ejpam-4761	256	7	hop	hop	NOUN
ejpam-4761	256	8	dominating	dominating	NOUN
ejpam-4761	256	9	set	set	NOUN
ejpam-4761	256	10	in	in	ADP
ejpam-4761	256	11	s(g	s(g	PROPN
ejpam-4761	256	12	)	)	PUNCT
ejpam-4761	256	13	.	.	PUNCT
ejpam-4761	257	1	let	let	VERB
ejpam-4761	257	2	wg1	wg1	VERB
ejpam-4761	257	3	=	=	VERB
ejpam-4761	257	4	w	w	PROPN
ejpam-4761	257	5	∩v	∩v	NOUN
ejpam-4761	257	6	(	(	PUNCT
ejpam-4761	257	7	g1	g1	PROPN
ejpam-4761	257	8	)	)	PUNCT
ejpam-4761	257	9	and	and	CCONJ
ejpam-4761	257	10	wg2	wg2	NOUN
ejpam-4761	257	11	=	=	SYM
ejpam-4761	257	12	w	w	PROPN
ejpam-4761	257	13	∩	∩	PROPN
ejpam-4761	257	14	v	v	X
ejpam-4761	257	15	(	(	PUNCT
ejpam-4761	257	16	g2	g2	PROPN
ejpam-4761	257	17	)	)	PUNCT
ejpam-4761	257	18	.	.	PUNCT
ejpam-4761	258	1	if	if	SCONJ
ejpam-4761	258	2	wg2	wg2	NOUN
ejpam-4761	258	3	=	=	NOUN
ejpam-4761	258	4	∅	∅	NOUN
ejpam-4761	258	5	,	,	PUNCT
ejpam-4761	258	6	then	then	ADV
ejpam-4761	258	7	w	w	PROPN
ejpam-4761	258	8	=	=	PUNCT
ejpam-4761	258	9	wg1	wg1	PROPN
ejpam-4761	258	10	is	be	AUX
ejpam-4761	258	11	a	a	DET
ejpam-4761	258	12	weakly	weakly	ADJ
ejpam-4761	258	13	convex	convex	NOUN
ejpam-4761	258	14	hop	hop	NOUN
ejpam-4761	258	15	dominating	dominating	NOUN
ejpam-4761	258	16	set	set	VERB
ejpam-4761	258	17	in	in	ADP
ejpam-4761	258	18	g1	g1	PROPN
ejpam-4761	258	19	,	,	PUNCT
ejpam-4761	258	20	showing	show	VERB
ejpam-4761	258	21	that	that	SCONJ
ejpam-4761	258	22	(	(	PUNCT
ejpam-4761	258	23	i	i	NOUN
ejpam-4761	258	24	)	)	PUNCT
ejpam-4761	258	25	holds	hold	VERB
ejpam-4761	258	26	.	.	PUNCT
ejpam-4761	259	1	similarly	similarly	ADV
ejpam-4761	259	2	,	,	PUNCT
ejpam-4761	259	3	ifwg1	ifwg1	PUNCT
ejpam-4761	260	1	=	=	NOUN
ejpam-4761	260	2	∅	∅	NOUN
ejpam-4761	260	3	,	,	PUNCT
ejpam-4761	260	4	thenw	thenw	NOUN
ejpam-4761	260	5	=	=	PUNCT
ejpam-4761	260	6	wg2	wg2	NOUN
ejpam-4761	260	7	is	be	AUX
ejpam-4761	260	8	a	a	DET
ejpam-4761	260	9	weakly	weakly	ADJ
ejpam-4761	260	10	convex	convex	NOUN
ejpam-4761	260	11	s.	s.	PROPN
ejpam-4761	260	12	canoy	canoy	PROPN
ejpam-4761	260	13	jr	jr	PROPN
ejpam-4761	260	14	.	.	PROPN
ejpam-4761	260	15	,	,	PUNCT
ejpam-4761	260	16	j.	j.	PROPN
ejpam-4761	260	17	hassan	hassan	PROPN
ejpam-4761	260	18	/	/	SYM
ejpam-4761	260	19	eur	eur	PROPN
ejpam-4761	260	20	.	.	PUNCT
ejpam-4761	261	1	j.	j.	PROPN
ejpam-4761	261	2	pure	pure	PROPN
ejpam-4761	261	3	appl	appl	PROPN
ejpam-4761	261	4	.	.	PROPN
ejpam-4761	261	5	math	math	PROPN
ejpam-4761	261	6	,	,	PUNCT
ejpam-4761	261	7	16	16	NUM
ejpam-4761	261	8	(	(	PUNCT
ejpam-4761	261	9	2	2	NUM
ejpam-4761	261	10	)	)	PUNCT
ejpam-4761	261	11	(	(	PUNCT
ejpam-4761	261	12	2023	2023	NUM
ejpam-4761	261	13	)	)	PUNCT
ejpam-4761	261	14	,	,	PUNCT
ejpam-4761	261	15	1196	1196	NUM
ejpam-4761	261	16	-	-	SYM
ejpam-4761	261	17	1211	1211	NUM
ejpam-4761	261	18	1204	1204	NUM
ejpam-4761	261	19	hop	hop	NOUN
ejpam-4761	261	20	dominating	dominating	NOUN
ejpam-4761	261	21	set	set	NOUN
ejpam-4761	261	22	in	in	ADP
ejpam-4761	261	23	g2	g2	PROPN
ejpam-4761	261	24	.	.	PUNCT
ejpam-4761	262	1	hence	hence	ADV
ejpam-4761	262	2	,	,	PUNCT
ejpam-4761	262	3	(	(	PUNCT
ejpam-4761	262	4	ii	ii	NOUN
ejpam-4761	262	5	)	)	PUNCT
ejpam-4761	262	6	holds	hold	VERB
ejpam-4761	262	7	.	.	PUNCT
ejpam-4761	263	1	next	next	ADV
ejpam-4761	263	2	,	,	PUNCT
ejpam-4761	263	3	suppose	suppose	VERB
ejpam-4761	263	4	wg1	wg1	ADV
ejpam-4761	263	5	̸=	̸=	PROPN
ejpam-4761	263	6	∅	∅	NOUN
ejpam-4761	264	1	and	and	CCONJ
ejpam-4761	264	2	wg2	wg2	NOUN
ejpam-4761	264	3	̸=	̸=	PROPN
ejpam-4761	264	4	∅.	∅.	NOUN
ejpam-4761	264	5	then	then	ADV
ejpam-4761	264	6	wg1	wg1	VERB
ejpam-4761	264	7	∪	∪	NOUN
ejpam-4761	264	8	w	w	PROPN
ejpam-4761	264	9	′	′	NUM
ejpam-4761	264	10	g2	g2	PROPN
ejpam-4761	264	11	is	be	AUX
ejpam-4761	264	12	a	a	DET
ejpam-4761	264	13	hop	hop	NOUN
ejpam-4761	264	14	dominating	dominating	NOUN
ejpam-4761	264	15	set	set	VERB
ejpam-4761	264	16	in	in	ADP
ejpam-4761	264	17	g1	g1	NOUN
ejpam-4761	264	18	by	by	ADP
ejpam-4761	264	19	theorem	theorem	NOUN
ejpam-4761	264	20	5	5	NUM
ejpam-4761	264	21	.	.	PUNCT
ejpam-4761	265	1	let	let	VERB
ejpam-4761	265	2	x	x	PRON
ejpam-4761	265	3	,	,	PUNCT
ejpam-4761	265	4	y	y	PROPN
ejpam-4761	265	5	∈	∈	PROPN
ejpam-4761	265	6	wg1	wg1	VERB
ejpam-4761	265	7	∪	∪	ADJ
ejpam-4761	265	8	w	w	PROPN
ejpam-4761	265	9	′	′	NUM
ejpam-4761	265	10	g2	g2	PROPN
ejpam-4761	265	11	where	where	SCONJ
ejpam-4761	265	12	x	x	X
ejpam-4761	265	13	̸=	̸=	PROPN
ejpam-4761	265	14	y.	y.	NOUN
ejpam-4761	265	15	suppose	suppose	VERB
ejpam-4761	265	16	x	x	PRON
ejpam-4761	265	17	,	,	PUNCT
ejpam-4761	265	18	y	y	PROPN
ejpam-4761	265	19	∈	∈	PROPN
ejpam-4761	265	20	wg1	wg1	VERB
ejpam-4761	265	21	.	.	PUNCT
ejpam-4761	266	1	since	since	SCONJ
ejpam-4761	266	2	w	w	NOUN
ejpam-4761	266	3	is	be	AUX
ejpam-4761	266	4	weakly	weakly	ADJ
ejpam-4761	266	5	convex	convex	NOUN
ejpam-4761	266	6	,	,	PUNCT
ejpam-4761	266	7	there	there	PRON
ejpam-4761	266	8	exists	exist	VERB
ejpam-4761	266	9	an	an	DET
ejpam-4761	266	10	x	x	NOUN
ejpam-4761	266	11	-	-	NOUN
ejpam-4761	266	12	y	y	ADJ
ejpam-4761	266	13	geodesic	geodesic	NOUN
ejpam-4761	266	14	p	p	X
ejpam-4761	266	15	(	(	PUNCT
ejpam-4761	266	16	x	x	NOUN
ejpam-4761	266	17	,	,	PUNCT
ejpam-4761	266	18	y	y	NOUN
ejpam-4761	266	19	)	)	PUNCT
ejpam-4761	266	20	in	in	ADP
ejpam-4761	266	21	s(g	s(g	PROPN
ejpam-4761	266	22	)	)	PUNCT
ejpam-4761	266	23	such	such	ADJ
ejpam-4761	266	24	that	that	DET
ejpam-4761	266	25	v	v	NOUN
ejpam-4761	266	26	(	(	PUNCT
ejpam-4761	266	27	p	p	X
ejpam-4761	266	28	(	(	PUNCT
ejpam-4761	266	29	x	x	NOUN
ejpam-4761	266	30	,	,	PUNCT
ejpam-4761	266	31	y	y	NOUN
ejpam-4761	266	32	)	)	PUNCT
ejpam-4761	266	33	)	)	PUNCT
ejpam-4761	267	1	⊆	⊆	NUM
ejpam-4761	267	2	w	w	NOUN
ejpam-4761	267	3	.	.	PUNCT
ejpam-4761	268	1	let	let	VERB
ejpam-4761	268	2	p	p	NOUN
ejpam-4761	268	3	(	(	PUNCT
ejpam-4761	268	4	x	x	NOUN
ejpam-4761	268	5	,	,	PUNCT
ejpam-4761	268	6	y	y	NOUN
ejpam-4761	268	7	)	)	PUNCT
ejpam-4761	268	8	=	=	PUNCT
ejpam-4761	269	1	[	[	X
ejpam-4761	269	2	x1	x1	PROPN
ejpam-4761	269	3	,	,	PUNCT
ejpam-4761	269	4	x2	x2	PROPN
ejpam-4761	269	5	,	,	PUNCT
ejpam-4761	269	6	·	·	PUNCT
ejpam-4761	269	7	·	·	PUNCT
ejpam-4761	269	8	·	·	PUNCT
ejpam-4761	269	9	,	,	PUNCT
ejpam-4761	269	10	xk	xk	PROPN
ejpam-4761	269	11	]	]	X
ejpam-4761	269	12	,	,	PUNCT
ejpam-4761	269	13	where	where	SCONJ
ejpam-4761	269	14	x1	x1	ADJ
ejpam-4761	269	15	=	=	PUNCT
ejpam-4761	269	16	x	x	X
ejpam-4761	269	17	and	and	CCONJ
ejpam-4761	269	18	xk	xk	PROPN
ejpam-4761	269	19	=	=	PROPN
ejpam-4761	269	20	y.	y.	PROPN
ejpam-4761	269	21	let	let	VERB
ejpam-4761	269	22	j	j	PROPN
ejpam-4761	269	23	∈	∈	PROPN
ejpam-4761	269	24	{	{	PUNCT
ejpam-4761	269	25	2	2	NUM
ejpam-4761	269	26	,	,	PUNCT
ejpam-4761	269	27	·	·	PUNCT
ejpam-4761	269	28	·	·	PUNCT
ejpam-4761	269	29	·	·	PUNCT
ejpam-4761	269	30	,	,	PUNCT
ejpam-4761	269	31	k	k	PROPN
ejpam-4761	270	1	−	−	PROPN
ejpam-4761	270	2	1	1	NUM
ejpam-4761	270	3	}	}	PUNCT
ejpam-4761	270	4	such	such	ADJ
ejpam-4761	270	5	that	that	SCONJ
ejpam-4761	270	6	xj	xj	PROPN
ejpam-4761	270	7	∈	∈	PROPN
ejpam-4761	270	8	wg2	wg2	NOUN
ejpam-4761	270	9	,	,	PUNCT
ejpam-4761	270	10	say	say	VERB
ejpam-4761	270	11	xj	xj	PROPN
ejpam-4761	270	12	=	=	PROPN
ejpam-4761	270	13	y′j	y′j	PROPN
ejpam-4761	270	14	.	.	PUNCT
ejpam-4761	271	1	then	then	ADV
ejpam-4761	271	2	yj	yj	PROPN
ejpam-4761	271	3	∈	∈	PROPN
ejpam-4761	271	4	w	w	PROPN
ejpam-4761	271	5	′	′	NUM
ejpam-4761	271	6	g2	g2	PROPN
ejpam-4761	271	7	.	.	PUNCT
ejpam-4761	272	1	replacing	replace	VERB
ejpam-4761	272	2	each	each	DET
ejpam-4761	272	3	xj	xj	PROPN
ejpam-4761	272	4	∈	∈	PROPN
ejpam-4761	272	5	wg2	wg2	NOUN
ejpam-4761	272	6	by	by	ADP
ejpam-4761	272	7	yj	yj	PROPN
ejpam-4761	272	8	,	,	PUNCT
ejpam-4761	272	9	we	we	PRON
ejpam-4761	272	10	obtain	obtain	VERB
ejpam-4761	272	11	an	an	DET
ejpam-4761	272	12	x	x	NOUN
ejpam-4761	272	13	-	-	NOUN
ejpam-4761	272	14	y	y	ADJ
ejpam-4761	272	15	geodesic	geodesic	NOUN
ejpam-4761	272	16	p	p	NOUN
ejpam-4761	272	17	′(x	′(x	NOUN
ejpam-4761	272	18	,	,	PUNCT
ejpam-4761	272	19	y	y	NOUN
ejpam-4761	272	20	)	)	PUNCT
ejpam-4761	272	21	in	in	ADP
ejpam-4761	272	22	g1	g1	PROPN
ejpam-4761	272	23	such	such	ADJ
ejpam-4761	272	24	that	that	DET
ejpam-4761	272	25	v	v	NOUN
ejpam-4761	272	26	(	(	PUNCT
ejpam-4761	272	27	p	p	NOUN
ejpam-4761	272	28	′(x	′(x	PROPN
ejpam-4761	272	29	,	,	PUNCT
ejpam-4761	272	30	y	y	NOUN
ejpam-4761	272	31	)	)	PUNCT
ejpam-4761	272	32	)	)	PUNCT
ejpam-4761	273	1	⊆	⊆	NUM
ejpam-4761	273	2	wg1	wg1	ADJ
ejpam-4761	273	3	∪w	∪w	PROPN
ejpam-4761	273	4	′	′	NUM
ejpam-4761	273	5	g2	g2	PROPN
ejpam-4761	273	6	.	.	PUNCT
ejpam-4761	274	1	suppose	suppose	VERB
ejpam-4761	274	2	now	now	ADV
ejpam-4761	274	3	that	that	SCONJ
ejpam-4761	274	4	x	x	X
ejpam-4761	274	5	,	,	PUNCT
ejpam-4761	274	6	y	y	PROPN
ejpam-4761	274	7	∈	∈	PROPN
ejpam-4761	274	8	w	w	PROPN
ejpam-4761	274	9	′	′	NUM
ejpam-4761	274	10	g2	g2	PROPN
ejpam-4761	274	11	.	.	PUNCT
ejpam-4761	275	1	then	then	ADV
ejpam-4761	275	2	x′	x′	NUM
ejpam-4761	275	3	,	,	PUNCT
ejpam-4761	275	4	y′	y′	NOUN
ejpam-4761	275	5	∈	∈	PROPN
ejpam-4761	275	6	wg2	wg2	NOUN
ejpam-4761	276	1	⊂	⊂	X
ejpam-4761	276	2	w	w	PROPN
ejpam-4761	276	3	.	.	PUNCT
ejpam-4761	277	1	by	by	ADP
ejpam-4761	277	2	assumption	assumption	NOUN
ejpam-4761	277	3	,	,	PUNCT
ejpam-4761	277	4	we	we	PRON
ejpam-4761	277	5	may	may	AUX
ejpam-4761	277	6	let	let	VERB
ejpam-4761	277	7	p	p	NOUN
ejpam-4761	277	8	(	(	PUNCT
ejpam-4761	277	9	x′	x′	PROPN
ejpam-4761	277	10	,	,	PUNCT
ejpam-4761	277	11	y′	y′	NUM
ejpam-4761	277	12	)	)	PUNCT
ejpam-4761	277	13	=	=	PUNCT
ejpam-4761	278	1	[	[	X
ejpam-4761	278	2	v1	v1	NOUN
ejpam-4761	278	3	,	,	PUNCT
ejpam-4761	278	4	v2	v2	PROPN
ejpam-4761	278	5	,	,	PUNCT
ejpam-4761	278	6	·	·	PUNCT
ejpam-4761	278	7	·	·	PUNCT
ejpam-4761	278	8	·	·	PUNCT
ejpam-4761	278	9	,	,	PUNCT
ejpam-4761	278	10	vt	vt	PROPN
ejpam-4761	278	11	]	]	X
ejpam-4761	278	12	,	,	PUNCT
ejpam-4761	278	13	where	where	SCONJ
ejpam-4761	278	14	x′	x′	PROPN
ejpam-4761	278	15	=	=	SYM
ejpam-4761	278	16	v1	v1	PROPN
ejpam-4761	278	17	and	and	CCONJ
ejpam-4761	278	18	y′	y′	NOUN
ejpam-4761	278	19	=	=	SYM
ejpam-4761	278	20	vt	vt	PROPN
ejpam-4761	278	21	,	,	PUNCT
ejpam-4761	278	22	be	be	AUX
ejpam-4761	278	23	an	an	DET
ejpam-4761	278	24	x′-y′	x′-y′	ADJ
ejpam-4761	278	25	geodesic	geodesic	NOUN
ejpam-4761	278	26	in	in	ADP
ejpam-4761	278	27	s(g	s(g	PROPN
ejpam-4761	278	28	)	)	PUNCT
ejpam-4761	278	29	such	such	ADJ
ejpam-4761	278	30	that	that	DET
ejpam-4761	278	31	v	v	NOUN
ejpam-4761	278	32	(	(	PUNCT
ejpam-4761	278	33	p	p	X
ejpam-4761	278	34	(	(	PUNCT
ejpam-4761	278	35	x′	x′	PROPN
ejpam-4761	278	36	,	,	PUNCT
ejpam-4761	278	37	y′	y′	NUM
ejpam-4761	278	38	)	)	PUNCT
ejpam-4761	278	39	)	)	PUNCT
ejpam-4761	279	1	⊆	⊆	NUM
ejpam-4761	279	2	w	w	NOUN
ejpam-4761	279	3	.	.	PUNCT
ejpam-4761	280	1	let	let	VERB
ejpam-4761	280	2	r	r	PRON
ejpam-4761	280	3	∈	∈	PROPN
ejpam-4761	280	4	{	{	PUNCT
ejpam-4761	280	5	1	1	NUM
ejpam-4761	280	6	,	,	PUNCT
ejpam-4761	280	7	2	2	NUM
ejpam-4761	280	8	,	,	PUNCT
ejpam-4761	280	9	·	·	PUNCT
ejpam-4761	280	10	·	·	PUNCT
ejpam-4761	280	11	·	·	PUNCT
ejpam-4761	280	12	,	,	PUNCT
ejpam-4761	280	13	t	t	X
ejpam-4761	280	14	}	}	PUNCT
ejpam-4761	280	15	such	such	ADJ
ejpam-4761	280	16	that	that	SCONJ
ejpam-4761	280	17	vr	vr	PROPN
ejpam-4761	280	18	∈	∈	PROPN
ejpam-4761	280	19	wg2	wg2	NOUN
ejpam-4761	280	20	,	,	PUNCT
ejpam-4761	280	21	say	say	VERB
ejpam-4761	280	22	vr	vr	NOUN
ejpam-4761	280	23	=	=	SYM
ejpam-4761	281	1	z′r	z′r	PROPN
ejpam-4761	281	2	.	.	PUNCT
ejpam-4761	282	1	then	then	ADV
ejpam-4761	282	2	zr	zr	PROPN
ejpam-4761	282	3	∈	∈	PROPN
ejpam-4761	282	4	w	w	PROPN
ejpam-4761	282	5	′	′	NUM
ejpam-4761	282	6	g2	g2	PROPN
ejpam-4761	282	7	.	.	PUNCT
ejpam-4761	283	1	note	note	VERB
ejpam-4761	283	2	that	that	SCONJ
ejpam-4761	283	3	z1	z1	NOUN
ejpam-4761	283	4	=	=	PUNCT
ejpam-4761	283	5	x	x	X
ejpam-4761	283	6	and	and	CCONJ
ejpam-4761	283	7	zt	zt	PROPN
ejpam-4761	283	8	=	=	SYM
ejpam-4761	283	9	y.	y.	PROPN
ejpam-4761	283	10	replacing	replace	VERB
ejpam-4761	283	11	each	each	DET
ejpam-4761	283	12	vr	vr	PROPN
ejpam-4761	283	13	∈	∈	PROPN
ejpam-4761	283	14	wg2	wg2	NOUN
ejpam-4761	283	15	by	by	ADP
ejpam-4761	283	16	zj	zj	PROPN
ejpam-4761	283	17	,	,	PUNCT
ejpam-4761	283	18	we	we	PRON
ejpam-4761	283	19	obtain	obtain	VERB
ejpam-4761	283	20	an	an	DET
ejpam-4761	283	21	x	x	NOUN
ejpam-4761	283	22	-	-	NOUN
ejpam-4761	283	23	y	y	ADJ
ejpam-4761	283	24	geodesic	geodesic	NOUN
ejpam-4761	283	25	p	p	NOUN
ejpam-4761	283	26	∗(x	∗(x	PROPN
ejpam-4761	283	27	,	,	PUNCT
ejpam-4761	283	28	y	y	NOUN
ejpam-4761	283	29	)	)	PUNCT
ejpam-4761	283	30	in	in	ADP
ejpam-4761	283	31	g1	g1	PROPN
ejpam-4761	283	32	such	such	ADJ
ejpam-4761	283	33	that	that	DET
ejpam-4761	283	34	v	v	NOUN
ejpam-4761	283	35	(	(	PUNCT
ejpam-4761	283	36	p	p	NOUN
ejpam-4761	283	37	∗(x	∗(x	PROPN
ejpam-4761	283	38	,	,	PUNCT
ejpam-4761	283	39	y	y	NOUN
ejpam-4761	283	40	)	)	PUNCT
ejpam-4761	283	41	)	)	PUNCT
ejpam-4761	284	1	⊆	⊆	NUM
ejpam-4761	284	2	wg1	wg1	ADJ
ejpam-4761	284	3	∪w	∪w	PROPN
ejpam-4761	284	4	′	′	NUM
ejpam-4761	284	5	g2	g2	PROPN
ejpam-4761	284	6	.	.	PUNCT
ejpam-4761	285	1	finally	finally	ADV
ejpam-4761	285	2	,	,	PUNCT
ejpam-4761	285	3	suppose	suppose	VERB
ejpam-4761	285	4	that	that	SCONJ
ejpam-4761	285	5	x	x	PUNCT
ejpam-4761	285	6	∈	∈	PROPN
ejpam-4761	285	7	wg1	wg1	VERB
ejpam-4761	285	8	and	and	CCONJ
ejpam-4761	285	9	y	y	PROPN
ejpam-4761	285	10	∈	∈	PROPN
ejpam-4761	285	11	w	w	PROPN
ejpam-4761	285	12	′	′	NUM
ejpam-4761	285	13	g2	g2	PROPN
ejpam-4761	285	14	.	.	PUNCT
ejpam-4761	286	1	then	then	ADV
ejpam-4761	286	2	y′	y′	NOUN
ejpam-4761	286	3	∈	∈	PROPN
ejpam-4761	286	4	wg2	wg2	NOUN
ejpam-4761	286	5	.	.	PUNCT
ejpam-4761	287	1	let	let	VERB
ejpam-4761	287	2	p	p	NOUN
ejpam-4761	287	3	(	(	PUNCT
ejpam-4761	287	4	x	x	X
ejpam-4761	287	5	,	,	PUNCT
ejpam-4761	287	6	y′	y′	NUM
ejpam-4761	287	7	)	)	PUNCT
ejpam-4761	288	1	=	=	PUNCT
ejpam-4761	289	1	[	[	X
ejpam-4761	289	2	q1	q1	PROPN
ejpam-4761	289	3	,	,	PUNCT
ejpam-4761	289	4	q2	q2	NOUN
ejpam-4761	289	5	,	,	PUNCT
ejpam-4761	289	6	·	·	PUNCT
ejpam-4761	289	7	·	·	PUNCT
ejpam-4761	289	8	·	·	PUNCT
ejpam-4761	289	9	,	,	PUNCT
ejpam-4761	289	10	qm	qm	PROPN
ejpam-4761	289	11	]	]	X
ejpam-4761	289	12	,	,	PUNCT
ejpam-4761	289	13	where	where	SCONJ
ejpam-4761	289	14	x	x	SYM
ejpam-4761	289	15	=	=	VERB
ejpam-4761	289	16	q1	q1	PROPN
ejpam-4761	289	17	and	and	CCONJ
ejpam-4761	289	18	y′	y′	NOUN
ejpam-4761	289	19	=	=	SYM
ejpam-4761	289	20	qm	qm	PROPN
ejpam-4761	289	21	,	,	PUNCT
ejpam-4761	289	22	be	be	AUX
ejpam-4761	289	23	an	an	DET
ejpam-4761	289	24	x	x	NOUN
ejpam-4761	289	25	-	-	PUNCT
ejpam-4761	289	26	y′	y′	ADJ
ejpam-4761	289	27	geodesic	geodesic	NOUN
ejpam-4761	289	28	in	in	ADP
ejpam-4761	289	29	s(g	s(g	PROPN
ejpam-4761	289	30	)	)	PUNCT
ejpam-4761	289	31	such	such	ADJ
ejpam-4761	289	32	that	that	DET
ejpam-4761	289	33	v	v	NOUN
ejpam-4761	289	34	(	(	PUNCT
ejpam-4761	289	35	p	p	X
ejpam-4761	289	36	(	(	PUNCT
ejpam-4761	289	37	x	x	X
ejpam-4761	289	38	,	,	PUNCT
ejpam-4761	289	39	y′	y′	NUM
ejpam-4761	289	40	)	)	PUNCT
ejpam-4761	289	41	)	)	PUNCT
ejpam-4761	290	1	⊆	⊆	NUM
ejpam-4761	290	2	w	w	NOUN
ejpam-4761	290	3	.	.	PUNCT
ejpam-4761	291	1	replacing	replace	VERB
ejpam-4761	291	2	each	each	DET
ejpam-4761	291	3	qi	qi	NOUN
ejpam-4761	291	4	=	=	NOUN
ejpam-4761	291	5	p′i	p′i	NOUN
ejpam-4761	291	6	∈	∈	PROPN
ejpam-4761	291	7	wg2	wg2	NOUN
ejpam-4761	291	8	by	by	ADP
ejpam-4761	291	9	pi	pi	PROPN
ejpam-4761	291	10	∈	∈	PROPN
ejpam-4761	291	11	w	w	PROPN
ejpam-4761	291	12	′	′	NUM
ejpam-4761	291	13	g2	g2	PROPN
ejpam-4761	291	14	(	(	PUNCT
ejpam-4761	291	15	pm	pm	PROPN
ejpam-4761	291	16	=	=	SYM
ejpam-4761	291	17	y	y	NOUN
ejpam-4761	291	18	)	)	PUNCT
ejpam-4761	291	19	,	,	PUNCT
ejpam-4761	291	20	we	we	PRON
ejpam-4761	291	21	obtain	obtain	VERB
ejpam-4761	291	22	an	an	DET
ejpam-4761	291	23	x	x	NOUN
ejpam-4761	291	24	-	-	NOUN
ejpam-4761	291	25	y	y	ADJ
ejpam-4761	291	26	geodesic	geodesic	NOUN
ejpam-4761	291	27	p	p	NOUN
ejpam-4761	291	28	∗∗(x	∗∗(x	PROPN
ejpam-4761	291	29	,	,	PUNCT
ejpam-4761	291	30	y	y	NOUN
ejpam-4761	291	31	)	)	PUNCT
ejpam-4761	291	32	such	such	ADJ
ejpam-4761	291	33	that	that	DET
ejpam-4761	291	34	v	v	NOUN
ejpam-4761	291	35	(	(	PUNCT
ejpam-4761	291	36	p	p	NOUN
ejpam-4761	291	37	∗∗(x	∗∗(x	PROPN
ejpam-4761	291	38	,	,	PUNCT
ejpam-4761	291	39	y	y	NOUN
ejpam-4761	291	40	)	)	PUNCT
ejpam-4761	291	41	)	)	PUNCT
ejpam-4761	292	1	⊆	⊆	NUM
ejpam-4761	292	2	wg1	wg1	ADJ
ejpam-4761	292	3	∪	∪	NOUN
ejpam-4761	292	4	w	w	PROPN
ejpam-4761	292	5	′	′	NUM
ejpam-4761	292	6	g2	g2	PROPN
ejpam-4761	292	7	.	.	PUNCT
ejpam-4761	293	1	therefore	therefore	ADV
ejpam-4761	293	2	,	,	PUNCT
ejpam-4761	293	3	wg1	wg1	VERB
ejpam-4761	293	4	∪	∪	NOUN
ejpam-4761	293	5	w	w	PROPN
ejpam-4761	293	6	′	′	NUM
ejpam-4761	293	7	g2	g2	PROPN
ejpam-4761	293	8	is	be	AUX
ejpam-4761	293	9	a	a	DET
ejpam-4761	293	10	weakly	weakly	ADJ
ejpam-4761	293	11	convex	convex	NOUN
ejpam-4761	293	12	set	set	VERB
ejpam-4761	293	13	in	in	ADP
ejpam-4761	293	14	g1	g1	PROPN
ejpam-4761	293	15	.	.	PUNCT
ejpam-4761	294	1	similarly	similarly	ADV
ejpam-4761	294	2	,	,	PUNCT
ejpam-4761	294	3	w	w	PROPN
ejpam-4761	294	4	′	′	NUM
ejpam-4761	294	5	g1	g1	PROPN
ejpam-4761	294	6	∪wg2	∪wg2	PRON
ejpam-4761	294	7	is	be	AUX
ejpam-4761	294	8	weakly	weakly	ADJ
ejpam-4761	294	9	convex	convex	NOUN
ejpam-4761	294	10	in	in	ADP
ejpam-4761	294	11	g2	g2	PROPN
ejpam-4761	294	12	.	.	PUNCT
ejpam-4761	295	1	this	this	PRON
ejpam-4761	295	2	shows	show	VERB
ejpam-4761	295	3	that	that	SCONJ
ejpam-4761	295	4	(	(	PUNCT
ejpam-4761	295	5	iii	iii	NOUN
ejpam-4761	295	6	)	)	PUNCT
ejpam-4761	295	7	holds	hold	VERB
ejpam-4761	295	8	.	.	PUNCT
ejpam-4761	296	1	for	for	ADP
ejpam-4761	296	2	the	the	DET
ejpam-4761	296	3	converse	converse	NOUN
ejpam-4761	296	4	,	,	PUNCT
ejpam-4761	296	5	suppose	suppose	VERB
ejpam-4761	296	6	first	first	ADV
ejpam-4761	296	7	that	that	SCONJ
ejpam-4761	296	8	(	(	PUNCT
ejpam-4761	296	9	i	i	NOUN
ejpam-4761	296	10	)	)	PUNCT
ejpam-4761	296	11	or	or	CCONJ
ejpam-4761	296	12	(	(	PUNCT
ejpam-4761	296	13	ii	ii	NOUN
ejpam-4761	296	14	)	)	PUNCT
ejpam-4761	296	15	holds	hold	VERB
ejpam-4761	296	16	.	.	PUNCT
ejpam-4761	297	1	then	then	ADV
ejpam-4761	297	2	w	w	PROPN
ejpam-4761	297	3	is	be	AUX
ejpam-4761	297	4	hop	hop	NOUN
ejpam-4761	297	5	dominating	dominate	VERB
ejpam-4761	297	6	by	by	ADP
ejpam-4761	297	7	theorem	theorem	NOUN
ejpam-4761	297	8	5	5	NUM
ejpam-4761	297	9	.	.	PUNCT
ejpam-4761	298	1	since	since	SCONJ
ejpam-4761	298	2	weakly	weakly	ADJ
ejpam-4761	298	3	convex	convex	NOUN
ejpam-4761	298	4	sets	set	NOUN
ejpam-4761	298	5	in	in	ADP
ejpam-4761	298	6	g1	g1	PROPN
ejpam-4761	298	7	and	and	CCONJ
ejpam-4761	298	8	g2	g2	PROPN
ejpam-4761	298	9	are	be	AUX
ejpam-4761	298	10	weakly	weakly	ADJ
ejpam-4761	298	11	convex	convex	NOUN
ejpam-4761	298	12	sets	set	NOUN
ejpam-4761	298	13	in	in	ADP
ejpam-4761	298	14	s(g	s(g	PROPN
ejpam-4761	298	15	)	)	PUNCT
ejpam-4761	298	16	,	,	PUNCT
ejpam-4761	298	17	w	w	PROPN
ejpam-4761	298	18	is	be	AUX
ejpam-4761	298	19	weakly	weakly	ADJ
ejpam-4761	298	20	convex	convex	NOUN
ejpam-4761	298	21	.	.	PUNCT
ejpam-4761	299	1	suppose	suppose	VERB
ejpam-4761	299	2	(	(	PUNCT
ejpam-4761	299	3	iii	iii	NOUN
ejpam-4761	299	4	)	)	PUNCT
ejpam-4761	299	5	holds	hold	VERB
ejpam-4761	299	6	.	.	PUNCT
ejpam-4761	300	1	again	again	ADV
ejpam-4761	300	2	,	,	PUNCT
ejpam-4761	300	3	by	by	ADP
ejpam-4761	300	4	theorem	theorem	NOUN
ejpam-4761	300	5	5	5	NUM
ejpam-4761	300	6	,	,	PUNCT
ejpam-4761	300	7	w	w	PROPN
ejpam-4761	300	8	is	be	AUX
ejpam-4761	300	9	a	a	DET
ejpam-4761	300	10	hop	hop	NOUN
ejpam-4761	300	11	dominating	dominating	NOUN
ejpam-4761	300	12	set	set	NOUN
ejpam-4761	300	13	in	in	ADP
ejpam-4761	300	14	s(g	s(g	PROPN
ejpam-4761	300	15	)	)	PUNCT
ejpam-4761	300	16	.	.	PUNCT
ejpam-4761	301	1	by	by	ADP
ejpam-4761	301	2	the	the	DET
ejpam-4761	301	3	additional	additional	ADJ
ejpam-4761	301	4	assumption	assumption	NOUN
ejpam-4761	301	5	,	,	PUNCT
ejpam-4761	301	6	w	w	NOUN
ejpam-4761	301	7	is	be	AUX
ejpam-4761	301	8	weakly	weakly	ADJ
ejpam-4761	301	9	convex	convex	NOUN
ejpam-4761	301	10	in	in	ADP
ejpam-4761	301	11	s(g	s(g	PROPN
ejpam-4761	301	12	)	)	PUNCT
ejpam-4761	301	13	.	.	PUNCT
ejpam-4761	302	1	therefore	therefore	ADV
ejpam-4761	302	2	,	,	PUNCT
ejpam-4761	302	3	w	w	PROPN
ejpam-4761	302	4	is	be	AUX
ejpam-4761	302	5	a	a	DET
ejpam-4761	302	6	weakly	weakly	ADJ
ejpam-4761	302	7	convex	convex	NOUN
ejpam-4761	302	8	hop	hop	NOUN
ejpam-4761	302	9	dominating	dominating	NOUN
ejpam-4761	302	10	set	set	NOUN
ejpam-4761	302	11	in	in	ADP
ejpam-4761	302	12	s(g	s(g	PROPN
ejpam-4761	302	13	)	)	PUNCT
ejpam-4761	302	14	.	.	PUNCT
ejpam-4761	303	1	the	the	DET
ejpam-4761	303	2	next	next	ADJ
ejpam-4761	303	3	result	result	NOUN
ejpam-4761	303	4	follows	follow	VERB
ejpam-4761	303	5	from	from	ADP
ejpam-4761	303	6	theorem	theorem	ADJ
ejpam-4761	303	7	6	6	NUM
ejpam-4761	303	8	.	.	PUNCT
ejpam-4761	303	9	corollary	corollary	ADJ
ejpam-4761	303	10	4	4	NUM
ejpam-4761	303	11	.	.	PUNCT
ejpam-4761	304	1	let	let	VERB
ejpam-4761	304	2	g	g	PRON
ejpam-4761	304	3	be	be	AUX
ejpam-4761	304	4	a	a	DET
ejpam-4761	304	5	non	non	ADJ
ejpam-4761	304	6	-	-	ADJ
ejpam-4761	304	7	trivial	trivial	ADJ
ejpam-4761	304	8	connected	connected	ADJ
ejpam-4761	304	9	graph	graph	NOUN
ejpam-4761	304	10	.	.	PUNCT
ejpam-4761	305	1	then	then	ADV
ejpam-4761	305	2	γwconh(s(g	γwconh(s(g	PUNCT
ejpam-4761	305	3	)	)	PUNCT
ejpam-4761	305	4	)	)	PUNCT
ejpam-4761	306	1	=	=	PUNCT
ejpam-4761	306	2	γwconh(g	γwconh(g	NOUN
ejpam-4761	306	3	)	)	PUNCT
ejpam-4761	306	4	.	.	PUNCT
ejpam-4761	307	1	theorem	theorem	VERB
ejpam-4761	307	2	7	7	NUM
ejpam-4761	307	3	.	.	PUNCT
ejpam-4761	308	1	[	[	X
ejpam-4761	308	2	1	1	X
ejpam-4761	308	3	]	]	PUNCT
ejpam-4761	308	4	let	let	VERB
ejpam-4761	308	5	g	g	PRON
ejpam-4761	308	6	be	be	AUX
ejpam-4761	308	7	a	a	DET
ejpam-4761	308	8	graph	graph	NOUN
ejpam-4761	308	9	of	of	ADP
ejpam-4761	308	10	order	order	NOUN
ejpam-4761	308	11	n.	n.	NOUN
ejpam-4761	308	12	then	then	ADV
ejpam-4761	308	13	1	1	NUM
ejpam-4761	308	14	≤	≤	NUM
ejpam-4761	308	15	pnd(g	pnd(g	ADP
ejpam-4761	308	16	)	)	PUNCT
ejpam-4761	308	17	≤	≤	NOUN
ejpam-4761	308	18	n.	n.	NOUN
ejpam-4761	308	19	moreover	moreover	ADV
ejpam-4761	308	20	,	,	PUNCT
ejpam-4761	308	21	(	(	PUNCT
ejpam-4761	308	22	i	i	NOUN
ejpam-4761	308	23	)	)	PUNCT
ejpam-4761	309	1	pnd(g	pnd(g	PROPN
ejpam-4761	309	2	)	)	PUNCT
ejpam-4761	309	3	=	=	SYM
ejpam-4761	309	4	1	1	NUM
ejpam-4761	309	5	if	if	SCONJ
ejpam-4761	309	6	and	and	CCONJ
ejpam-4761	309	7	only	only	ADV
ejpam-4761	309	8	if	if	SCONJ
ejpam-4761	309	9	g	g	PROPN
ejpam-4761	309	10	has	have	VERB
ejpam-4761	309	11	an	an	DET
ejpam-4761	309	12	isolated	isolated	ADJ
ejpam-4761	309	13	vertex	vertex	NOUN
ejpam-4761	309	14	.	.	PUNCT
ejpam-4761	310	1	(	(	PUNCT
ejpam-4761	310	2	ii	ii	NOUN
ejpam-4761	310	3	)	)	PUNCT
ejpam-4761	310	4	pnd(g	pnd(g	PROPN
ejpam-4761	310	5	)	)	PUNCT
ejpam-4761	310	6	=	=	SYM
ejpam-4761	311	1	n	n	NOUN
ejpam-4761	311	2	if	if	SCONJ
ejpam-4761	311	3	and	and	CCONJ
ejpam-4761	311	4	only	only	ADV
ejpam-4761	311	5	if	if	SCONJ
ejpam-4761	311	6	g	g	PROPN
ejpam-4761	311	7	=	=	PROPN
ejpam-4761	311	8	kn	kn	PROPN
ejpam-4761	311	9	.	.	PUNCT
ejpam-4761	311	10	corollary	corollary	PROPN
ejpam-4761	311	11	5	5	NUM
ejpam-4761	311	12	.	.	PUNCT
ejpam-4761	312	1	[	[	X
ejpam-4761	312	2	1	1	X
ejpam-4761	312	3	]	]	PUNCT
ejpam-4761	312	4	let	let	VERB
ejpam-4761	312	5	n	n	PRON
ejpam-4761	312	6	be	be	AUX
ejpam-4761	312	7	any	any	DET
ejpam-4761	312	8	positive	positive	ADJ
ejpam-4761	312	9	integer	integer	NOUN
ejpam-4761	312	10	.	.	PUNCT
ejpam-4761	313	1	then	then	ADV
ejpam-4761	313	2	(	(	PUNCT
ejpam-4761	313	3	i	i	NOUN
ejpam-4761	313	4	)	)	PUNCT
ejpam-4761	313	5	pnd(pn	pnd(pn	NOUN
ejpam-4761	313	6	)	)	PUNCT
ejpam-4761	313	7	=	=	SYM
ejpam-4761	313	8	2	2	NUM
ejpam-4761	313	9	for	for	ADP
ejpam-4761	313	10	any	any	DET
ejpam-4761	313	11	n	n	PRON
ejpam-4761	313	12	≥	≥	NOUN
ejpam-4761	313	13	2	2	NUM
ejpam-4761	313	14	.	.	PUNCT
ejpam-4761	313	15	(	(	PUNCT
ejpam-4761	313	16	ii	ii	NOUN
ejpam-4761	313	17	)	)	PUNCT
ejpam-4761	313	18	pnd(cn	pnd(cn	NOUN
ejpam-4761	313	19	)	)	PUNCT
ejpam-4761	313	20	=	=	SYM
ejpam-4761	313	21	2	2	NUM
ejpam-4761	313	22	for	for	ADP
ejpam-4761	313	23	any	any	DET
ejpam-4761	313	24	n	n	PRON
ejpam-4761	313	25	≥	≥	NOUN
ejpam-4761	313	26	4	4	NUM
ejpam-4761	313	27	.	.	PUNCT
ejpam-4761	313	28	theorem	theorem	VERB
ejpam-4761	313	29	8	8	NUM
ejpam-4761	313	30	.	.	PUNCT
ejpam-4761	314	1	[	[	X
ejpam-4761	314	2	13	13	NUM
ejpam-4761	314	3	]	]	PUNCT
ejpam-4761	314	4	let	let	VERB
ejpam-4761	314	5	g	g	NOUN
ejpam-4761	314	6	and	and	CCONJ
ejpam-4761	314	7	h	h	NOUN
ejpam-4761	314	8	be	be	VERB
ejpam-4761	314	9	any	any	DET
ejpam-4761	314	10	two	two	NUM
ejpam-4761	314	11	graphs	graph	NOUN
ejpam-4761	314	12	.	.	PUNCT
ejpam-4761	315	1	a	a	DET
ejpam-4761	315	2	set	set	NOUN
ejpam-4761	315	3	s	s	NOUN
ejpam-4761	315	4	⊆	⊆	NUM
ejpam-4761	315	5	v	v	NOUN
ejpam-4761	315	6	(	(	PUNCT
ejpam-4761	315	7	g+h	g+h	PROPN
ejpam-4761	315	8	)	)	PUNCT
ejpam-4761	315	9	is	be	AUX
ejpam-4761	315	10	hop	hop	NOUN
ejpam-4761	315	11	dominating	dominating	NOUN
ejpam-4761	315	12	set	set	VERB
ejpam-4761	315	13	in	in	ADP
ejpam-4761	315	14	g+h	g+h	PROPN
ejpam-4761	315	15	if	if	SCONJ
ejpam-4761	315	16	and	and	CCONJ
ejpam-4761	315	17	only	only	ADV
ejpam-4761	315	18	if	if	SCONJ
ejpam-4761	315	19	s	s	NOUN
ejpam-4761	315	20	=	=	PUNCT
ejpam-4761	315	21	sg	sg	PROPN
ejpam-4761	315	22	∪sh	∪sh	NOUN
ejpam-4761	315	23	,	,	PUNCT
ejpam-4761	315	24	where	where	SCONJ
ejpam-4761	315	25	sg	sg	PROPN
ejpam-4761	315	26	and	and	CCONJ
ejpam-4761	315	27	sh	sh	PROPN
ejpam-4761	315	28	are	be	AUX
ejpam-4761	315	29	pointwise	pointwise	PROPN
ejpam-4761	315	30	non	non	ADJ
ejpam-4761	315	31	-	-	ADJ
ejpam-4761	315	32	dominating	dominating	ADJ
ejpam-4761	315	33	sets	set	NOUN
ejpam-4761	315	34	in	in	ADP
ejpam-4761	315	35	g	g	PROPN
ejpam-4761	315	36	and	and	CCONJ
ejpam-4761	315	37	h	h	NOUN
ejpam-4761	315	38	,	,	PUNCT
ejpam-4761	315	39	respectively	respectively	ADV
ejpam-4761	315	40	.	.	PUNCT
ejpam-4761	316	1	corollary	corollary	ADJ
ejpam-4761	316	2	6	6	NUM
ejpam-4761	316	3	.	.	PUNCT
ejpam-4761	317	1	[	[	X
ejpam-4761	317	2	13	13	NUM
ejpam-4761	317	3	]	]	PUNCT
ejpam-4761	317	4	let	let	VERB
ejpam-4761	317	5	g	g	NOUN
ejpam-4761	317	6	and	and	CCONJ
ejpam-4761	317	7	h	h	NOUN
ejpam-4761	317	8	be	be	VERB
ejpam-4761	317	9	any	any	DET
ejpam-4761	317	10	two	two	NUM
ejpam-4761	317	11	graphs	graph	NOUN
ejpam-4761	317	12	.	.	PUNCT
ejpam-4761	318	1	then	then	ADV
ejpam-4761	318	2	γh(g+h	γh(g+h	NUM
ejpam-4761	318	3	)	)	PUNCT
ejpam-4761	318	4	=	=	SYM
ejpam-4761	318	5	pnd(g	pnd(g	PROPN
ejpam-4761	318	6	)	)	PUNCT
ejpam-4761	318	7	+	+	NUM
ejpam-4761	318	8	pnd(h	pnd(h	NUM
ejpam-4761	318	9	)	)	PUNCT
ejpam-4761	318	10	.	.	PUNCT
ejpam-4761	319	1	theorem	theorem	NOUN
ejpam-4761	319	2	9	9	NUM
ejpam-4761	319	3	.	.	PUNCT
ejpam-4761	320	1	let	let	VERB
ejpam-4761	320	2	g	g	NOUN
ejpam-4761	320	3	and	and	CCONJ
ejpam-4761	320	4	h	h	NOUN
ejpam-4761	320	5	be	be	VERB
ejpam-4761	320	6	any	any	DET
ejpam-4761	320	7	two	two	NUM
ejpam-4761	320	8	graphs	graph	NOUN
ejpam-4761	320	9	.	.	PUNCT
ejpam-4761	321	1	then	then	ADV
ejpam-4761	321	2	w	w	PROPN
ejpam-4761	321	3	⊆	⊆	NUM
ejpam-4761	321	4	v	v	NOUN
ejpam-4761	321	5	(	(	PUNCT
ejpam-4761	321	6	g	g	PROPN
ejpam-4761	321	7	+	+	NOUN
ejpam-4761	321	8	h	h	NOUN
ejpam-4761	321	9	)	)	PUNCT
ejpam-4761	321	10	is	be	AUX
ejpam-4761	321	11	weakly	weakly	ADV
ejpam-4761	321	12	convex	convex	ADJ
ejpam-4761	321	13	hop	hop	NOUN
ejpam-4761	321	14	dominating	dominating	NOUN
ejpam-4761	321	15	in	in	ADP
ejpam-4761	321	16	g+h	g+h	PROPN
ejpam-4761	321	17	if	if	SCONJ
ejpam-4761	321	18	and	and	CCONJ
ejpam-4761	321	19	only	only	ADV
ejpam-4761	321	20	it	it	PRON
ejpam-4761	321	21	is	be	AUX
ejpam-4761	321	22	hop	hop	NOUN
ejpam-4761	321	23	dominating	dominate	VERB
ejpam-4761	321	24	in	in	ADP
ejpam-4761	321	25	g+h	g+h	PROPN
ejpam-4761	321	26	.	.	PUNCT
ejpam-4761	322	1	s.	s.	PROPN
ejpam-4761	322	2	canoy	canoy	PROPN
ejpam-4761	322	3	jr	jr	PROPN
ejpam-4761	322	4	.	.	PROPN
ejpam-4761	322	5	,	,	PUNCT
ejpam-4761	322	6	j.	j.	PROPN
ejpam-4761	322	7	hassan	hassan	PROPN
ejpam-4761	322	8	/	/	SYM
ejpam-4761	322	9	eur	eur	PROPN
ejpam-4761	322	10	.	.	PUNCT
ejpam-4761	323	1	j.	j.	PROPN
ejpam-4761	323	2	pure	pure	PROPN
ejpam-4761	323	3	appl	appl	PROPN
ejpam-4761	323	4	.	.	PROPN
ejpam-4761	323	5	math	math	PROPN
ejpam-4761	323	6	,	,	PUNCT
ejpam-4761	323	7	16	16	NUM
ejpam-4761	323	8	(	(	PUNCT
ejpam-4761	323	9	2	2	NUM
ejpam-4761	323	10	)	)	PUNCT
ejpam-4761	323	11	(	(	PUNCT
ejpam-4761	323	12	2023	2023	NUM
ejpam-4761	323	13	)	)	PUNCT
ejpam-4761	323	14	,	,	PUNCT
ejpam-4761	323	15	1196	1196	NUM
ejpam-4761	323	16	-	-	SYM
ejpam-4761	323	17	1211	1211	NUM
ejpam-4761	323	18	1205	1205	NUM
ejpam-4761	323	19	proof	proof	NOUN
ejpam-4761	323	20	.	.	PUNCT
ejpam-4761	324	1	supposew	supposew	PROPN
ejpam-4761	324	2	is	be	AUX
ejpam-4761	324	3	a	a	DET
ejpam-4761	324	4	weakly	weakly	ADJ
ejpam-4761	324	5	convex	convex	NOUN
ejpam-4761	324	6	hop	hop	NOUN
ejpam-4761	324	7	dominating	dominating	NOUN
ejpam-4761	324	8	set	set	NOUN
ejpam-4761	324	9	ing+h	ing+h	PROPN
ejpam-4761	324	10	.	.	PUNCT
ejpam-4761	325	1	then	then	ADV
ejpam-4761	325	2	,	,	PUNCT
ejpam-4761	325	3	by	by	ADP
ejpam-4761	325	4	definition	definition	NOUN
ejpam-4761	325	5	,	,	PUNCT
ejpam-4761	325	6	w	w	PROPN
ejpam-4761	325	7	is	be	AUX
ejpam-4761	325	8	hop	hop	NOUN
ejpam-4761	325	9	dominating	dominate	VERB
ejpam-4761	325	10	in	in	ADP
ejpam-4761	325	11	g+h	g+h	PROPN
ejpam-4761	325	12	.	.	PUNCT
ejpam-4761	326	1	for	for	ADP
ejpam-4761	326	2	the	the	DET
ejpam-4761	326	3	converse	converse	NOUN
ejpam-4761	326	4	,	,	PUNCT
ejpam-4761	326	5	suppose	suppose	VERB
ejpam-4761	326	6	that	that	SCONJ
ejpam-4761	326	7	w	w	NOUN
ejpam-4761	326	8	is	be	AUX
ejpam-4761	326	9	a	a	DET
ejpam-4761	326	10	hop	hop	NOUN
ejpam-4761	326	11	dominating	dominating	NOUN
ejpam-4761	326	12	set	set	VERB
ejpam-4761	326	13	in	in	ADP
ejpam-4761	326	14	g	g	PROPN
ejpam-4761	326	15	+	+	CCONJ
ejpam-4761	326	16	h.	h.	PROPN
ejpam-4761	326	17	by	by	ADP
ejpam-4761	326	18	theorem	theorem	NOUN
ejpam-4761	326	19	8	8	NUM
ejpam-4761	326	20	,	,	PUNCT
ejpam-4761	326	21	w	w	NOUN
ejpam-4761	326	22	=	=	PUNCT
ejpam-4761	326	23	wg	wg	PROPN
ejpam-4761	326	24	+	+	CCONJ
ejpam-4761	326	25	wh	wh	VERB
ejpam-4761	326	26	where	where	SCONJ
ejpam-4761	326	27	wg	wg	VERB
ejpam-4761	326	28	and	and	CCONJ
ejpam-4761	326	29	wh	wh	PROPN
ejpam-4761	326	30	are	be	AUX
ejpam-4761	326	31	pointwise	pointwise	ADJ
ejpam-4761	326	32	non	non	ADJ
ejpam-4761	326	33	-	-	ADJ
ejpam-4761	326	34	dominating	dominating	ADJ
ejpam-4761	326	35	sets	set	NOUN
ejpam-4761	326	36	in	in	ADP
ejpam-4761	326	37	g	g	PROPN
ejpam-4761	326	38	and	and	CCONJ
ejpam-4761	326	39	h	h	NOUN
ejpam-4761	326	40	,	,	PUNCT
ejpam-4761	326	41	respectively	respectively	ADV
ejpam-4761	326	42	.	.	PUNCT
ejpam-4761	327	1	let	let	VERB
ejpam-4761	327	2	v	v	NOUN
ejpam-4761	327	3	,	,	PUNCT
ejpam-4761	327	4	w	w	PROPN
ejpam-4761	327	5	∈	∈	PROPN
ejpam-4761	327	6	w	w	NOUN
ejpam-4761	327	7	and	and	CCONJ
ejpam-4761	327	8	v	v	ADP
ejpam-4761	327	9	̸=	̸=	PROPN
ejpam-4761	327	10	w.	w.	NOUN
ejpam-4761	327	11	if	if	SCONJ
ejpam-4761	327	12	dg+h(v	dg+h(v	PROPN
ejpam-4761	327	13	,	,	PUNCT
ejpam-4761	327	14	w	w	NOUN
ejpam-4761	327	15	)	)	PUNCT
ejpam-4761	327	16	=	=	SYM
ejpam-4761	327	17	1	1	NUM
ejpam-4761	327	18	,	,	PUNCT
ejpam-4761	327	19	then	then	ADV
ejpam-4761	327	20	ig+h	ig+h	PROPN
ejpam-4761	327	21	[	[	X
ejpam-4761	327	22	v	v	NOUN
ejpam-4761	327	23	,	,	PUNCT
ejpam-4761	327	24	w	w	NOUN
ejpam-4761	327	25	]	]	X
ejpam-4761	327	26	=	=	SYM
ejpam-4761	327	27	{	{	PUNCT
ejpam-4761	327	28	v	v	NOUN
ejpam-4761	327	29	,	,	PUNCT
ejpam-4761	327	30	w	w	NOUN
ejpam-4761	327	31	}	}	PUNCT
ejpam-4761	327	32	⊆	⊆	NUM
ejpam-4761	327	33	w	w	NOUN
ejpam-4761	327	34	.	.	PUNCT
ejpam-4761	328	1	suppose	suppose	VERB
ejpam-4761	328	2	that	that	SCONJ
ejpam-4761	328	3	dg+h(v	dg+h(v	PROPN
ejpam-4761	328	4	,	,	PUNCT
ejpam-4761	328	5	w	w	NOUN
ejpam-4761	328	6	)	)	PUNCT
ejpam-4761	328	7	=	=	SYM
ejpam-4761	328	8	2	2	X
ejpam-4761	328	9	.	.	PUNCT
ejpam-4761	328	10	then	then	ADV
ejpam-4761	328	11	v	v	NOUN
ejpam-4761	328	12	,	,	PUNCT
ejpam-4761	328	13	w	w	PROPN
ejpam-4761	328	14	∈	∈	PROPN
ejpam-4761	328	15	wg	wg	NOUN
ejpam-4761	328	16	or	or	CCONJ
ejpam-4761	328	17	v	v	NOUN
ejpam-4761	328	18	,	,	PUNCT
ejpam-4761	328	19	w	w	PROPN
ejpam-4761	328	20	∈	∈	PROPN
ejpam-4761	328	21	wh	wh	NOUN
ejpam-4761	328	22	.	.	PUNCT
ejpam-4761	329	1	we	we	PRON
ejpam-4761	329	2	may	may	AUX
ejpam-4761	329	3	assume	assume	VERB
ejpam-4761	329	4	that	that	SCONJ
ejpam-4761	329	5	v	v	NOUN
ejpam-4761	329	6	,	,	PUNCT
ejpam-4761	329	7	w	w	PROPN
ejpam-4761	329	8	∈	∈	PROPN
ejpam-4761	329	9	wg	wg	PROPN
ejpam-4761	329	10	.	.	PUNCT
ejpam-4761	330	1	pick	pick	VERB
ejpam-4761	330	2	any	any	DET
ejpam-4761	330	3	z	z	NOUN
ejpam-4761	330	4	∈	∈	PROPN
ejpam-4761	330	5	wh	wh	NOUN
ejpam-4761	330	6	.	.	PUNCT
ejpam-4761	331	1	then	then	ADV
ejpam-4761	331	2	p	p	X
ejpam-4761	331	3	(	(	PUNCT
ejpam-4761	331	4	v	v	NOUN
ejpam-4761	331	5	,	,	PUNCT
ejpam-4761	331	6	w	w	NOUN
ejpam-4761	331	7	)	)	PUNCT
ejpam-4761	331	8	=	=	PUNCT
ejpam-4761	332	1	[	[	X
ejpam-4761	332	2	v	v	NOUN
ejpam-4761	332	3	,	,	PUNCT
ejpam-4761	332	4	z	z	PROPN
ejpam-4761	332	5	,	,	PUNCT
ejpam-4761	332	6	w	w	PROPN
ejpam-4761	332	7	]	]	X
ejpam-4761	332	8	is	be	AUX
ejpam-4761	332	9	a	a	DET
ejpam-4761	332	10	v	v	NOUN
ejpam-4761	332	11	-	-	PUNCT
ejpam-4761	332	12	w	w	NOUN
ejpam-4761	332	13	geodesic	geodesic	NOUN
ejpam-4761	332	14	in	in	ADP
ejpam-4761	332	15	g	g	PROPN
ejpam-4761	332	16	+	+	NOUN
ejpam-4761	332	17	h	h	NOUN
ejpam-4761	332	18	and	and	CCONJ
ejpam-4761	332	19	v	v	NOUN
ejpam-4761	332	20	(	(	PUNCT
ejpam-4761	332	21	p	p	X
ejpam-4761	332	22	(	(	PUNCT
ejpam-4761	332	23	v	v	NOUN
ejpam-4761	332	24	,	,	PUNCT
ejpam-4761	332	25	w	w	NOUN
ejpam-4761	332	26	)	)	PUNCT
ejpam-4761	332	27	)	)	PUNCT
ejpam-4761	333	1	=	=	PRON
ejpam-4761	333	2	{	{	PUNCT
ejpam-4761	333	3	v	v	NOUN
ejpam-4761	333	4	,	,	PUNCT
ejpam-4761	333	5	z	z	NOUN
ejpam-4761	333	6	,	,	PUNCT
ejpam-4761	333	7	w	w	NOUN
ejpam-4761	333	8	}	}	PUNCT
ejpam-4761	333	9	⊆	⊆	NUM
ejpam-4761	333	10	w	w	NOUN
ejpam-4761	333	11	.	.	PUNCT
ejpam-4761	334	1	thus	thus	ADV
ejpam-4761	334	2	,	,	PUNCT
ejpam-4761	334	3	w	w	PROPN
ejpam-4761	334	4	is	be	AUX
ejpam-4761	334	5	weakly	weakly	ADV
ejpam-4761	334	6	convex	convex	ADJ
ejpam-4761	334	7	hop	hop	NOUN
ejpam-4761	334	8	dominating	dominating	NOUN
ejpam-4761	334	9	in	in	ADP
ejpam-4761	334	10	g+h	g+h	PROPN
ejpam-4761	334	11	.	.	PUNCT
ejpam-4761	335	1	the	the	DET
ejpam-4761	335	2	next	next	ADJ
ejpam-4761	335	3	result	result	NOUN
ejpam-4761	335	4	follows	follow	VERB
ejpam-4761	335	5	from	from	ADP
ejpam-4761	335	6	theorem	theorem	ADJ
ejpam-4761	335	7	9	9	NUM
ejpam-4761	335	8	and	and	CCONJ
ejpam-4761	335	9	corollary	corollary	ADJ
ejpam-4761	335	10	6	6	NUM
ejpam-4761	335	11	.	.	PUNCT
ejpam-4761	335	12	corollary	corollary	ADJ
ejpam-4761	335	13	7	7	NUM
ejpam-4761	335	14	.	.	PUNCT
ejpam-4761	336	1	let	let	VERB
ejpam-4761	336	2	g	g	NOUN
ejpam-4761	336	3	and	and	CCONJ
ejpam-4761	336	4	h	h	NOUN
ejpam-4761	336	5	be	be	VERB
ejpam-4761	336	6	two	two	NUM
ejpam-4761	336	7	graphs	graph	NOUN
ejpam-4761	336	8	.	.	PUNCT
ejpam-4761	337	1	then	then	ADV
ejpam-4761	337	2	γwconh(g+h	γwconh(g+h	PROPN
ejpam-4761	337	3	)	)	PUNCT
ejpam-4761	338	1	=	=	SYM
ejpam-4761	338	2	pnd(g	pnd(g	PROPN
ejpam-4761	338	3	)	)	PUNCT
ejpam-4761	338	4	+	+	NUM
ejpam-4761	338	5	pnd(h	pnd(h	PROPN
ejpam-4761	338	6	)	)	PUNCT
ejpam-4761	338	7	.	.	PUNCT
ejpam-4761	339	1	in	in	ADP
ejpam-4761	339	2	particular	particular	ADJ
ejpam-4761	339	3	,	,	PUNCT
ejpam-4761	339	4	we	we	PRON
ejpam-4761	339	5	have	have	VERB
ejpam-4761	339	6	(	(	PUNCT
ejpam-4761	339	7	i	i	NOUN
ejpam-4761	339	8	)	)	PUNCT
ejpam-4761	339	9	γwconh(pn	γwconh(pn	PROPN
ejpam-4761	339	10	+	+	CCONJ
ejpam-4761	339	11	pm	pm	NOUN
ejpam-4761	339	12	)	)	PUNCT
ejpam-4761	339	13	=	=	SYM
ejpam-4761	339	14	4	4	NUM
ejpam-4761	339	15	for	for	ADP
ejpam-4761	339	16	all	all	DET
ejpam-4761	339	17	n	n	CCONJ
ejpam-4761	339	18	,	,	PUNCT
ejpam-4761	339	19	m	m	VERB
ejpam-4761	339	20	≥	≥	NOUN
ejpam-4761	339	21	2	2	NUM
ejpam-4761	339	22	;	;	PUNCT
ejpam-4761	339	23	(	(	PUNCT
ejpam-4761	339	24	ii	ii	NOUN
ejpam-4761	339	25	)	)	PUNCT
ejpam-4761	339	26	γwconh(cn	γwconh(cn	NOUN
ejpam-4761	339	27	+	+	CCONJ
ejpam-4761	339	28	cm	cm	NOUN
ejpam-4761	339	29	)	)	PUNCT
ejpam-4761	339	30	=	=	SYM
ejpam-4761	339	31	4	4	NUM
ejpam-4761	339	32	for	for	ADP
ejpam-4761	339	33	all	all	DET
ejpam-4761	339	34	n	n	CCONJ
ejpam-4761	339	35	,	,	PUNCT
ejpam-4761	339	36	m	m	VERB
ejpam-4761	339	37	≥	≥	NOUN
ejpam-4761	339	38	4	4	NUM
ejpam-4761	339	39	;	;	PUNCT
ejpam-4761	339	40	(	(	PUNCT
ejpam-4761	339	41	iii	iii	X
ejpam-4761	339	42	)	)	PUNCT
ejpam-4761	339	43	γwconh(fn	γwconh(fn	NOUN
ejpam-4761	339	44	)	)	PUNCT
ejpam-4761	339	45	=	=	SYM
ejpam-4761	339	46	3	3	NUM
ejpam-4761	339	47	for	for	ADP
ejpam-4761	339	48	all	all	DET
ejpam-4761	339	49	n	n	PRON
ejpam-4761	339	50	≥	≥	NOUN
ejpam-4761	339	51	2	2	NUM
ejpam-4761	339	52	;	;	PUNCT
ejpam-4761	339	53	(	(	PUNCT
ejpam-4761	339	54	iv	iv	X
ejpam-4761	339	55	)	)	PUNCT
ejpam-4761	339	56	γwconh(wn	γwconh(wn	NOUN
ejpam-4761	339	57	)	)	PUNCT
ejpam-4761	339	58	=	=	SYM
ejpam-4761	339	59	3	3	NUM
ejpam-4761	339	60	for	for	ADP
ejpam-4761	339	61	all	all	DET
ejpam-4761	339	62	n	n	PRON
ejpam-4761	339	63	≥	≥	NOUN
ejpam-4761	339	64	4	4	NUM
ejpam-4761	339	65	;	;	PUNCT
ejpam-4761	339	66	and	and	CCONJ
ejpam-4761	339	67	(	(	PUNCT
ejpam-4761	339	68	v	v	NOUN
ejpam-4761	339	69	)	)	PUNCT
ejpam-4761	339	70	γwconh(k1,n	γwconh(k1,n	NOUN
ejpam-4761	339	71	)	)	PUNCT
ejpam-4761	339	72	=	=	SYM
ejpam-4761	339	73	2	2	NUM
ejpam-4761	339	74	for	for	ADP
ejpam-4761	339	75	all	all	DET
ejpam-4761	339	76	n	n	PRON
ejpam-4761	339	77	≥	≥	NOUN
ejpam-4761	339	78	1	1	NUM
ejpam-4761	339	79	.	.	PUNCT
ejpam-4761	340	1	the	the	DET
ejpam-4761	340	2	result	result	NOUN
ejpam-4761	340	3	that	that	PRON
ejpam-4761	340	4	follows	follow	VERB
ejpam-4761	340	5	is	be	AUX
ejpam-4761	340	6	a	a	DET
ejpam-4761	340	7	restatement	restatement	NOUN
ejpam-4761	340	8	of	of	ADP
ejpam-4761	340	9	a	a	DET
ejpam-4761	340	10	result	result	NOUN
ejpam-4761	340	11	in	in	ADP
ejpam-4761	340	12	[	[	X
ejpam-4761	340	13	13	13	NUM
ejpam-4761	340	14	]	]	PUNCT
ejpam-4761	340	15	.	.	PUNCT
ejpam-4761	341	1	theorem	theorem	ADJ
ejpam-4761	341	2	10	10	NUM
ejpam-4761	341	3	.	.	PUNCT
ejpam-4761	342	1	let	let	VERB
ejpam-4761	342	2	g	g	NOUN
ejpam-4761	342	3	and	and	CCONJ
ejpam-4761	342	4	h	h	NOUN
ejpam-4761	342	5	be	be	VERB
ejpam-4761	342	6	any	any	DET
ejpam-4761	342	7	two	two	NUM
ejpam-4761	342	8	graphs	graph	NOUN
ejpam-4761	342	9	.	.	PUNCT
ejpam-4761	343	1	a	a	DET
ejpam-4761	343	2	set	set	NOUN
ejpam-4761	343	3	c	c	NOUN
ejpam-4761	343	4	⊆	⊆	NUM
ejpam-4761	343	5	v	v	NOUN
ejpam-4761	343	6	(	(	PUNCT
ejpam-4761	343	7	g	g	NOUN
ejpam-4761	343	8	)	)	PUNCT
ejpam-4761	343	9	is	be	AUX
ejpam-4761	343	10	a	a	DET
ejpam-4761	343	11	hop	hop	NOUN
ejpam-4761	343	12	dominating	dominating	NOUN
ejpam-4761	343	13	set	set	VERB
ejpam-4761	343	14	in	in	ADP
ejpam-4761	343	15	g	g	PROPN
ejpam-4761	343	16	◦	◦	NOUN
ejpam-4761	343	17	h	h	NOUN
ejpam-4761	343	18	if	if	SCONJ
ejpam-4761	344	1	and	and	CCONJ
ejpam-4761	344	2	only	only	ADV
ejpam-4761	344	3	if	if	SCONJ
ejpam-4761	344	4	c	c	PROPN
ejpam-4761	344	5	=	=	SYM
ejpam-4761	344	6	a∪	a∪	PROPN
ejpam-4761	344	7	(	(	PUNCT
ejpam-4761	344	8	∪v∈v	∪v∈v	X
ejpam-4761	344	9	(	(	PUNCT
ejpam-4761	344	10	g)cv	g)cv	PROPN
ejpam-4761	344	11	)	)	PUNCT
ejpam-4761	344	12	,	,	PUNCT
ejpam-4761	344	13	where	where	SCONJ
ejpam-4761	344	14	a	a	DET
ejpam-4761	344	15	⊆	⊆	NUM
ejpam-4761	344	16	v	v	NOUN
ejpam-4761	344	17	(	(	PUNCT
ejpam-4761	344	18	g	g	NOUN
ejpam-4761	344	19	)	)	PUNCT
ejpam-4761	344	20	and	and	CCONJ
ejpam-4761	344	21	cv	cv	PROPN
ejpam-4761	344	22	⊆	⊆	NUM
ejpam-4761	344	23	v	v	PROPN
ejpam-4761	344	24	(	(	PUNCT
ejpam-4761	344	25	hv	hv	PROPN
ejpam-4761	344	26	)	)	PUNCT
ejpam-4761	344	27	for	for	ADP
ejpam-4761	344	28	each	each	DET
ejpam-4761	344	29	v	v	NUM
ejpam-4761	344	30	∈	∈	PROPN
ejpam-4761	344	31	v	v	NOUN
ejpam-4761	344	32	(	(	PUNCT
ejpam-4761	344	33	g	g	NOUN
ejpam-4761	344	34	)	)	PUNCT
ejpam-4761	344	35	,	,	PUNCT
ejpam-4761	344	36	and	and	CCONJ
ejpam-4761	344	37	satisfies	satisfy	VERB
ejpam-4761	344	38	the	the	DET
ejpam-4761	344	39	following	follow	VERB
ejpam-4761	344	40	conditions	condition	NOUN
ejpam-4761	344	41	:	:	PUNCT
ejpam-4761	344	42	(	(	PUNCT
ejpam-4761	344	43	i	i	NOUN
ejpam-4761	344	44	)	)	PUNCT
ejpam-4761	344	45	for	for	ADP
ejpam-4761	344	46	each	each	DET
ejpam-4761	344	47	w	w	PROPN
ejpam-4761	344	48	∈	∈	PROPN
ejpam-4761	344	49	v	v	ADP
ejpam-4761	344	50	(	(	PUNCT
ejpam-4761	344	51	g	g	NOUN
ejpam-4761	344	52	)	)	PUNCT
ejpam-4761	344	53	\	\	PROPN
ejpam-4761	344	54	a	a	PRON
ejpam-4761	344	55	,	,	PUNCT
ejpam-4761	344	56	there	there	PRON
ejpam-4761	344	57	exists	exist	VERB
ejpam-4761	344	58	x	x	X
ejpam-4761	344	59	∈	∈	PROPN
ejpam-4761	344	60	a	a	PRON
ejpam-4761	344	61	with	with	ADP
ejpam-4761	344	62	dg(w	dg(w	NOUN
ejpam-4761	344	63	,	,	PUNCT
ejpam-4761	344	64	x	x	X
ejpam-4761	344	65	)	)	PUNCT
ejpam-4761	344	66	=	=	SYM
ejpam-4761	344	67	2	2	NUM
ejpam-4761	344	68	or	or	CCONJ
ejpam-4761	344	69	there	there	PRON
ejpam-4761	344	70	exists	exist	VERB
ejpam-4761	344	71	y	y	PROPN
ejpam-4761	344	72	∈	∈	PROPN
ejpam-4761	344	73	ng(w	ng(w	NOUN
ejpam-4761	344	74	)	)	PUNCT
ejpam-4761	344	75	with	with	ADP
ejpam-4761	344	76	cy	cy	PROPN
ejpam-4761	344	77	̸=	̸=	PROPN
ejpam-4761	344	78	∅.	∅.	ADP
ejpam-4761	344	79	(	(	PUNCT
ejpam-4761	344	80	ii	ii	NOUN
ejpam-4761	344	81	)	)	PUNCT
ejpam-4761	344	82	cw	cw	NOUN
ejpam-4761	344	83	is	be	AUX
ejpam-4761	344	84	a	a	DET
ejpam-4761	344	85	pointwise	pointwise	ADJ
ejpam-4761	344	86	non	non	ADJ
ejpam-4761	344	87	-	-	ADJ
ejpam-4761	344	88	dominating	dominating	ADJ
ejpam-4761	344	89	set	set	NOUN
ejpam-4761	344	90	in	in	ADP
ejpam-4761	344	91	hw	hw	PRON
ejpam-4761	344	92	for	for	ADP
ejpam-4761	344	93	each	each	DET
ejpam-4761	344	94	w	w	PROPN
ejpam-4761	344	95	∈	∈	PROPN
ejpam-4761	344	96	v	v	ADP
ejpam-4761	344	97	(	(	PUNCT
ejpam-4761	344	98	g	g	NOUN
ejpam-4761	344	99	)	)	PUNCT
ejpam-4761	344	100	\ng(a	\ng(a	PROPN
ejpam-4761	344	101	)	)	PUNCT
ejpam-4761	344	102	.	.	PUNCT
ejpam-4761	345	1	theorem	theorem	NOUN
ejpam-4761	345	2	11	11	NUM
ejpam-4761	345	3	.	.	PUNCT
ejpam-4761	346	1	let	let	VERB
ejpam-4761	346	2	g	g	PRON
ejpam-4761	346	3	be	be	AUX
ejpam-4761	346	4	a	a	DET
ejpam-4761	346	5	non	non	ADJ
ejpam-4761	346	6	-	-	ADJ
ejpam-4761	346	7	trivial	trivial	ADJ
ejpam-4761	346	8	connected	connected	ADJ
ejpam-4761	346	9	graph	graph	NOUN
ejpam-4761	346	10	and	and	CCONJ
ejpam-4761	346	11	let	let	VERB
ejpam-4761	346	12	h	h	NOUN
ejpam-4761	346	13	be	be	AUX
ejpam-4761	346	14	any	any	DET
ejpam-4761	346	15	graph	graph	NOUN
ejpam-4761	346	16	.	.	PUNCT
ejpam-4761	347	1	then	then	ADV
ejpam-4761	347	2	w	w	PROPN
ejpam-4761	347	3	is	be	AUX
ejpam-4761	347	4	a	a	DET
ejpam-4761	347	5	weakly	weakly	ADJ
ejpam-4761	347	6	convex	convex	NOUN
ejpam-4761	347	7	hop	hop	NOUN
ejpam-4761	347	8	dominating	dominating	NOUN
ejpam-4761	347	9	set	set	VERB
ejpam-4761	347	10	in	in	ADP
ejpam-4761	347	11	g	g	PROPN
ejpam-4761	347	12	◦	◦	NOUN
ejpam-4761	347	13	h	h	NOUN
ejpam-4761	347	14	if	if	SCONJ
ejpam-4761	348	1	and	and	CCONJ
ejpam-4761	348	2	only	only	ADV
ejpam-4761	348	3	if	if	SCONJ
ejpam-4761	348	4	w	w	PROPN
ejpam-4761	348	5	=	=	SYM
ejpam-4761	348	6	b	b	X
ejpam-4761	348	7	∪	∪	X
ejpam-4761	348	8	(	(	PUNCT
ejpam-4761	348	9	∪v∈v	∪v∈v	X
ejpam-4761	348	10	(	(	PUNCT
ejpam-4761	348	11	g)wv	g)wv	PROPN
ejpam-4761	348	12	)	)	PUNCT
ejpam-4761	348	13	,	,	PUNCT
ejpam-4761	348	14	where	where	SCONJ
ejpam-4761	348	15	b	b	X
ejpam-4761	348	16	⊆	⊆	NUM
ejpam-4761	348	17	v	v	NOUN
ejpam-4761	348	18	(	(	PUNCT
ejpam-4761	348	19	g	g	NOUN
ejpam-4761	348	20	)	)	PUNCT
ejpam-4761	348	21	,	,	PUNCT
ejpam-4761	348	22	wv	wv	PROPN
ejpam-4761	348	23	⊆	⊆	NUM
ejpam-4761	348	24	v	v	PROPN
ejpam-4761	348	25	(	(	PUNCT
ejpam-4761	348	26	hv	hv	PROPN
ejpam-4761	348	27	)	)	PUNCT
ejpam-4761	348	28	for	for	ADP
ejpam-4761	348	29	each	each	DET
ejpam-4761	348	30	v	v	NUM
ejpam-4761	348	31	∈	∈	PROPN
ejpam-4761	348	32	v	v	NOUN
ejpam-4761	348	33	(	(	PUNCT
ejpam-4761	348	34	g	g	NOUN
ejpam-4761	348	35	)	)	PUNCT
ejpam-4761	348	36	,	,	PUNCT
ejpam-4761	348	37	and	and	CCONJ
ejpam-4761	348	38	satisfies	satisfy	VERB
ejpam-4761	348	39	the	the	DET
ejpam-4761	348	40	following	follow	VERB
ejpam-4761	348	41	conditions	condition	NOUN
ejpam-4761	348	42	:	:	PUNCT
ejpam-4761	348	43	(	(	PUNCT
ejpam-4761	348	44	i	i	NOUN
ejpam-4761	348	45	)	)	PUNCT
ejpam-4761	348	46	b	b	PROPN
ejpam-4761	348	47	is	be	AUX
ejpam-4761	348	48	a	a	DET
ejpam-4761	348	49	weakly	weakly	ADJ
ejpam-4761	348	50	convex	convex	NOUN
ejpam-4761	348	51	dominating	dominating	NOUN
ejpam-4761	348	52	set	set	VERB
ejpam-4761	348	53	in	in	ADP
ejpam-4761	348	54	g.	g.	PROPN
ejpam-4761	348	55	(	(	PUNCT
ejpam-4761	348	56	ii	ii	PROPN
ejpam-4761	348	57	)	)	PUNCT
ejpam-4761	348	58	wv	wv	PROPN
ejpam-4761	348	59	=	=	PUNCT
ejpam-4761	348	60	∅	∅	NOUN
ejpam-4761	348	61	for	for	ADP
ejpam-4761	348	62	each	each	PRON
ejpam-4761	348	63	v	v	NUM
ejpam-4761	348	64	∈	∈	PROPN
ejpam-4761	348	65	v	v	NOUN
ejpam-4761	348	66	(	(	PUNCT
ejpam-4761	348	67	g	g	NOUN
ejpam-4761	348	68	)	)	PUNCT
ejpam-4761	348	69	\b	\b	NOUN
ejpam-4761	348	70	.	.	PUNCT
ejpam-4761	349	1	(	(	PUNCT
ejpam-4761	349	2	iii	iii	X
ejpam-4761	349	3	)	)	PUNCT
ejpam-4761	349	4	for	for	ADP
ejpam-4761	349	5	each	each	PRON
ejpam-4761	350	1	a	a	DET
ejpam-4761	350	2	∈	∈	PROPN
ejpam-4761	350	3	v	v	NOUN
ejpam-4761	350	4	(	(	PUNCT
ejpam-4761	350	5	g	g	NOUN
ejpam-4761	350	6	)	)	PUNCT
ejpam-4761	350	7	\	\	PROPN
ejpam-4761	350	8	b	b	X
ejpam-4761	350	9	,	,	PUNCT
ejpam-4761	350	10	there	there	PRON
ejpam-4761	350	11	exists	exist	VERB
ejpam-4761	350	12	b	b	PROPN
ejpam-4761	350	13	∈	∈	PROPN
ejpam-4761	350	14	b	b	PROPN
ejpam-4761	350	15	with	with	ADP
ejpam-4761	350	16	dg(a	dg(a	PROPN
ejpam-4761	350	17	,	,	PUNCT
ejpam-4761	350	18	b	b	NOUN
ejpam-4761	350	19	)	)	PUNCT
ejpam-4761	350	20	=	=	SYM
ejpam-4761	350	21	2	2	NUM
ejpam-4761	350	22	or	or	CCONJ
ejpam-4761	350	23	there	there	PRON
ejpam-4761	350	24	exists	exist	VERB
ejpam-4761	350	25	y	y	PROPN
ejpam-4761	350	26	∈	∈	PROPN
ejpam-4761	350	27	b	b	NOUN
ejpam-4761	350	28	∩ng(a	∩ng(a	NOUN
ejpam-4761	350	29	)	)	PUNCT
ejpam-4761	350	30	such	such	ADJ
ejpam-4761	350	31	that	that	SCONJ
ejpam-4761	350	32	wy	wy	PROPN
ejpam-4761	350	33	̸=	̸=	PROPN
ejpam-4761	350	34	∅.	∅.	ADP
ejpam-4761	350	35	s.	s.	PROPN
ejpam-4761	350	36	canoy	canoy	PROPN
ejpam-4761	350	37	jr	jr	PROPN
ejpam-4761	350	38	.	.	PROPN
ejpam-4761	350	39	,	,	PUNCT
ejpam-4761	350	40	j.	j.	PROPN
ejpam-4761	350	41	hassan	hassan	PROPN
ejpam-4761	350	42	/	/	SYM
ejpam-4761	350	43	eur	eur	PROPN
ejpam-4761	350	44	.	.	PUNCT
ejpam-4761	351	1	j.	j.	PROPN
ejpam-4761	351	2	pure	pure	PROPN
ejpam-4761	351	3	appl	appl	PROPN
ejpam-4761	351	4	.	.	PROPN
ejpam-4761	351	5	math	math	PROPN
ejpam-4761	351	6	,	,	PUNCT
ejpam-4761	351	7	16	16	NUM
ejpam-4761	351	8	(	(	PUNCT
ejpam-4761	351	9	2	2	NUM
ejpam-4761	351	10	)	)	PUNCT
ejpam-4761	351	11	(	(	PUNCT
ejpam-4761	351	12	2023	2023	NUM
ejpam-4761	351	13	)	)	PUNCT
ejpam-4761	351	14	,	,	PUNCT
ejpam-4761	351	15	1196	1196	NUM
ejpam-4761	351	16	-	-	SYM
ejpam-4761	351	17	1211	1211	NUM
ejpam-4761	351	18	1206	1206	NUM
ejpam-4761	351	19	(	(	PUNCT
ejpam-4761	351	20	iv	iv	X
ejpam-4761	351	21	)	)	PUNCT
ejpam-4761	351	22	wx	wx	PROPN
ejpam-4761	351	23	is	be	AUX
ejpam-4761	351	24	a	a	DET
ejpam-4761	351	25	pointwise	pointwise	ADJ
ejpam-4761	351	26	non	non	ADJ
ejpam-4761	351	27	-	-	ADJ
ejpam-4761	351	28	dominating	dominating	ADJ
ejpam-4761	351	29	set	set	NOUN
ejpam-4761	351	30	in	in	ADP
ejpam-4761	351	31	hx	hx	PROPN
ejpam-4761	351	32	for	for	ADP
ejpam-4761	351	33	each	each	DET
ejpam-4761	351	34	x	x	SYM
ejpam-4761	351	35	∈	∈	PROPN
ejpam-4761	351	36	b	b	PROPN
ejpam-4761	351	37	\ng(b	\ng(b	NOUN
ejpam-4761	351	38	)	)	PUNCT
ejpam-4761	351	39	.	.	PUNCT
ejpam-4761	352	1	proof	proof	NOUN
ejpam-4761	352	2	.	.	PUNCT
ejpam-4761	353	1	suppose	suppose	VERB
ejpam-4761	353	2	w	w	NOUN
ejpam-4761	353	3	is	be	AUX
ejpam-4761	353	4	a	a	DET
ejpam-4761	353	5	weakly	weakly	ADJ
ejpam-4761	353	6	convex	convex	NOUN
ejpam-4761	353	7	hop	hop	NOUN
ejpam-4761	353	8	dominating	dominating	NOUN
ejpam-4761	353	9	set	set	VERB
ejpam-4761	353	10	in	in	ADP
ejpam-4761	353	11	g	g	PROPN
ejpam-4761	353	12	◦	◦	NOUN
ejpam-4761	353	13	h.	h.	PROPN
ejpam-4761	353	14	then	then	ADV
ejpam-4761	353	15	w	w	PROPN
ejpam-4761	353	16	is	be	AUX
ejpam-4761	353	17	hop	hop	NOUN
ejpam-4761	353	18	dominating	dominating	NOUN
ejpam-4761	353	19	and	and	CCONJ
ejpam-4761	353	20	w	w	NOUN
ejpam-4761	353	21	=	=	NOUN
ejpam-4761	353	22	b	b	X
ejpam-4761	353	23	∪	∪	X
ejpam-4761	353	24	(	(	PUNCT
ejpam-4761	353	25	∪v∈v	∪v∈v	X
ejpam-4761	353	26	(	(	PUNCT
ejpam-4761	353	27	g)wv	g)wv	PROPN
ejpam-4761	353	28	)	)	PUNCT
ejpam-4761	353	29	where	where	SCONJ
ejpam-4761	353	30	the	the	DET
ejpam-4761	353	31	sets	set	NOUN
ejpam-4761	353	32	b	b	NOUN
ejpam-4761	353	33	and	and	CCONJ
ejpam-4761	353	34	wvs	wvs	ADJ
ejpam-4761	353	35	satisfy	satisfy	NOUN
ejpam-4761	353	36	(	(	PUNCT
ejpam-4761	353	37	i	i	NOUN
ejpam-4761	353	38	)	)	PUNCT
ejpam-4761	353	39	and	and	CCONJ
ejpam-4761	353	40	(	(	PUNCT
ejpam-4761	353	41	ii	ii	NOUN
ejpam-4761	353	42	)	)	PUNCT
ejpam-4761	353	43	of	of	ADP
ejpam-4761	353	44	theorem	theorem	ADJ
ejpam-4761	353	45	10	10	NUM
ejpam-4761	353	46	.	.	PUNCT
ejpam-4761	354	1	suppose	suppose	VERB
ejpam-4761	354	2	b	b	X
ejpam-4761	354	3	=	=	PUNCT
ejpam-4761	354	4	∅.	∅.	PROPN
ejpam-4761	354	5	then	then	ADV
ejpam-4761	354	6	wv	wv	PROPN
ejpam-4761	354	7	is	be	AUX
ejpam-4761	354	8	pointwise	pointwise	PROPN
ejpam-4761	354	9	non	non	ADJ
ejpam-4761	354	10	-	-	ADJ
ejpam-4761	354	11	dominating	dominating	NOUN
ejpam-4761	354	12	in	in	ADP
ejpam-4761	354	13	hv	hv	PROPN
ejpam-4761	354	14	for	for	ADP
ejpam-4761	354	15	each	each	DET
ejpam-4761	354	16	v	v	NUM
ejpam-4761	354	17	∈	∈	PROPN
ejpam-4761	354	18	v	v	NOUN
ejpam-4761	354	19	(	(	PUNCT
ejpam-4761	354	20	g	g	NOUN
ejpam-4761	354	21	)	)	PUNCT
ejpam-4761	354	22	by	by	ADP
ejpam-4761	354	23	theorem	theorem	NOUN
ejpam-4761	354	24	10(ii	10(ii	NUM
ejpam-4761	354	25	)	)	PUNCT
ejpam-4761	354	26	.	.	PUNCT
ejpam-4761	355	1	let	let	VERB
ejpam-4761	355	2	a	a	DET
ejpam-4761	355	3	,	,	PUNCT
ejpam-4761	355	4	b	b	PROPN
ejpam-4761	355	5	∈	∈	PROPN
ejpam-4761	355	6	v	v	NOUN
ejpam-4761	355	7	(	(	PUNCT
ejpam-4761	355	8	g	g	NOUN
ejpam-4761	355	9	)	)	PUNCT
ejpam-4761	355	10	(	(	PUNCT
ejpam-4761	355	11	a	a	DET
ejpam-4761	355	12	̸=	̸=	PROPN
ejpam-4761	355	13	b	b	NUM
ejpam-4761	355	14	)	)	PUNCT
ejpam-4761	355	15	and	and	CCONJ
ejpam-4761	355	16	choose	choose	VERB
ejpam-4761	355	17	any	any	DET
ejpam-4761	355	18	p	p	PROPN
ejpam-4761	355	19	∈	∈	PROPN
ejpam-4761	355	20	wa	wa	NOUN
ejpam-4761	355	21	and	and	CCONJ
ejpam-4761	355	22	q	q	PROPN
ejpam-4761	355	23	∈	∈	PROPN
ejpam-4761	356	1	wb	wb	PROPN
ejpam-4761	356	2	.	.	PUNCT
ejpam-4761	357	1	since	since	SCONJ
ejpam-4761	357	2	every	every	DET
ejpam-4761	357	3	p	p	NOUN
ejpam-4761	357	4	-	-	PUNCT
ejpam-4761	357	5	q	q	NOUN
ejpam-4761	357	6	geodesic	geodesic	NOUN
ejpam-4761	357	7	in	in	ADP
ejpam-4761	357	8	g	g	PROPN
ejpam-4761	357	9	◦	◦	NOUN
ejpam-4761	357	10	h	h	NOUN
ejpam-4761	357	11	contains	contain	VERB
ejpam-4761	357	12	a	a	PRON
ejpam-4761	357	13	and	and	CCONJ
ejpam-4761	357	14	b	b	NOUN
ejpam-4761	357	15	,	,	PUNCT
ejpam-4761	357	16	w	w	NOUN
ejpam-4761	357	17	is	be	AUX
ejpam-4761	357	18	not	not	PART
ejpam-4761	357	19	weakly	weakly	ADJ
ejpam-4761	357	20	convex	convex	NOUN
ejpam-4761	357	21	in	in	ADP
ejpam-4761	357	22	g	g	PROPN
ejpam-4761	357	23	◦	◦	NOUN
ejpam-4761	357	24	h	h	NOUN
ejpam-4761	357	25	,	,	PUNCT
ejpam-4761	357	26	a	a	DET
ejpam-4761	357	27	contradiction	contradiction	NOUN
ejpam-4761	357	28	.	.	PUNCT
ejpam-4761	358	1	hence	hence	ADV
ejpam-4761	358	2	,	,	PUNCT
ejpam-4761	358	3	b	b	PROPN
ejpam-4761	358	4	̸=	̸=	PROPN
ejpam-4761	358	5	∅.	∅.	ADV
ejpam-4761	358	6	let	let	VERB
ejpam-4761	358	7	w	w	PROPN
ejpam-4761	358	8	∈	∈	PROPN
ejpam-4761	358	9	v	v	ADP
ejpam-4761	358	10	(	(	PUNCT
ejpam-4761	358	11	g	g	NOUN
ejpam-4761	358	12	)	)	PUNCT
ejpam-4761	358	13	\	\	PROPN
ejpam-4761	358	14	b	b	PROPN
ejpam-4761	358	15	and	and	CCONJ
ejpam-4761	358	16	suppose	suppose	VERB
ejpam-4761	358	17	that	that	SCONJ
ejpam-4761	358	18	w	w	PROPN
ejpam-4761	358	19	/∈	/∈	PUNCT
ejpam-4761	358	20	ng(b	ng(b	NUM
ejpam-4761	358	21	)	)	PUNCT
ejpam-4761	358	22	.	.	PUNCT
ejpam-4761	359	1	by	by	ADP
ejpam-4761	359	2	theorem	theorem	NOUN
ejpam-4761	359	3	10(ii	10(ii	NUM
ejpam-4761	359	4	)	)	PUNCT
ejpam-4761	359	5	,	,	PUNCT
ejpam-4761	359	6	ww	ww	PROPN
ejpam-4761	359	7	is	be	AUX
ejpam-4761	359	8	a	a	DET
ejpam-4761	359	9	pointwise	pointwise	ADJ
ejpam-4761	359	10	non	non	ADJ
ejpam-4761	359	11	-	-	ADJ
ejpam-4761	359	12	dominating	dominating	ADJ
ejpam-4761	359	13	set	set	NOUN
ejpam-4761	359	14	in	in	ADP
ejpam-4761	359	15	hw	hw	PRON
ejpam-4761	359	16	.	.	PUNCT
ejpam-4761	360	1	pick	pick	VERB
ejpam-4761	360	2	any	any	DET
ejpam-4761	360	3	p	p	PROPN
ejpam-4761	360	4	∈	∈	PROPN
ejpam-4761	360	5	ww	ww	PROPN
ejpam-4761	360	6	and	and	CCONJ
ejpam-4761	360	7	z	z	PROPN
ejpam-4761	360	8	∈	∈	PROPN
ejpam-4761	360	9	b	b	PROPN
ejpam-4761	360	10	(	(	PUNCT
ejpam-4761	360	11	z	z	NOUN
ejpam-4761	360	12	exists	exist	VERB
ejpam-4761	360	13	because	because	SCONJ
ejpam-4761	360	14	b	b	PROPN
ejpam-4761	360	15	̸=	̸=	PROPN
ejpam-4761	360	16	∅	∅	NOUN
ejpam-4761	360	17	)	)	PUNCT
ejpam-4761	360	18	.	.	PUNCT
ejpam-4761	361	1	then	then	ADV
ejpam-4761	361	2	there	there	PRON
ejpam-4761	361	3	exists	exist	VERB
ejpam-4761	361	4	no	no	DET
ejpam-4761	361	5	p	p	NOUN
ejpam-4761	361	6	-	-	PUNCT
ejpam-4761	361	7	z	z	NOUN
ejpam-4761	361	8	geodesic	geodesic	NOUN
ejpam-4761	361	9	p	p	X
ejpam-4761	361	10	(	(	PUNCT
ejpam-4761	361	11	p	p	X
ejpam-4761	361	12	,	,	PUNCT
ejpam-4761	361	13	z	z	NOUN
ejpam-4761	361	14	)	)	PUNCT
ejpam-4761	361	15	in	in	ADP
ejpam-4761	361	16	g	g	PROPN
ejpam-4761	361	17	◦	◦	NOUN
ejpam-4761	361	18	h	h	NOUN
ejpam-4761	361	19	with	with	ADP
ejpam-4761	361	20	v	v	NOUN
ejpam-4761	361	21	(	(	PUNCT
ejpam-4761	361	22	p	p	X
ejpam-4761	361	23	(	(	PUNCT
ejpam-4761	361	24	p	p	X
ejpam-4761	361	25	,	,	PUNCT
ejpam-4761	361	26	z	z	NOUN
ejpam-4761	361	27	)	)	PUNCT
ejpam-4761	361	28	)	)	PUNCT
ejpam-4761	362	1	⊆	⊆	NUM
ejpam-4761	362	2	w	w	NOUN
ejpam-4761	362	3	,	,	PUNCT
ejpam-4761	362	4	contrary	contrary	ADV
ejpam-4761	362	5	to	to	ADP
ejpam-4761	362	6	the	the	DET
ejpam-4761	362	7	assumption	assumption	NOUN
ejpam-4761	362	8	that	that	SCONJ
ejpam-4761	362	9	w	w	NOUN
ejpam-4761	362	10	is	be	AUX
ejpam-4761	362	11	weakly	weakly	ADJ
ejpam-4761	362	12	convex	convex	NOUN
ejpam-4761	362	13	.	.	PUNCT
ejpam-4761	363	1	thus	thus	ADV
ejpam-4761	363	2	,	,	PUNCT
ejpam-4761	363	3	b	b	PROPN
ejpam-4761	363	4	is	be	AUX
ejpam-4761	363	5	a	a	DET
ejpam-4761	363	6	dominating	dominating	NOUN
ejpam-4761	363	7	set	set	VERB
ejpam-4761	363	8	in	in	ADP
ejpam-4761	363	9	g.	g.	PROPN
ejpam-4761	363	10	let	let	VERB
ejpam-4761	363	11	x	x	PRON
ejpam-4761	363	12	and	and	CCONJ
ejpam-4761	363	13	y	y	PROPN
ejpam-4761	363	14	be	be	AUX
ejpam-4761	363	15	distinct	distinct	ADJ
ejpam-4761	363	16	vertices	vertex	NOUN
ejpam-4761	363	17	in	in	ADP
ejpam-4761	363	18	b.	b.	PROPN
ejpam-4761	363	19	then	then	ADV
ejpam-4761	363	20	x	x	PRON
ejpam-4761	363	21	,	,	PUNCT
ejpam-4761	363	22	y	y	PROPN
ejpam-4761	363	23	∈	∈	PROPN
ejpam-4761	363	24	w	w	PROPN
ejpam-4761	363	25	.	.	PUNCT
ejpam-4761	364	1	since	since	SCONJ
ejpam-4761	364	2	w	w	NOUN
ejpam-4761	364	3	is	be	AUX
ejpam-4761	364	4	weakly	weakly	ADJ
ejpam-4761	364	5	convex	convex	NOUN
ejpam-4761	364	6	in	in	ADP
ejpam-4761	364	7	g	g	PROPN
ejpam-4761	364	8	◦	◦	NOUN
ejpam-4761	364	9	h	h	NOUN
ejpam-4761	364	10	,	,	PUNCT
ejpam-4761	364	11	there	there	PRON
ejpam-4761	364	12	exists	exist	VERB
ejpam-4761	364	13	an	an	DET
ejpam-4761	364	14	x	x	NOUN
ejpam-4761	364	15	-	-	NOUN
ejpam-4761	364	16	y	y	ADJ
ejpam-4761	364	17	geodesic	geodesic	NOUN
ejpam-4761	364	18	p	p	X
ejpam-4761	364	19	(	(	PUNCT
ejpam-4761	364	20	x	x	NOUN
ejpam-4761	364	21	,	,	PUNCT
ejpam-4761	364	22	y	y	NOUN
ejpam-4761	364	23	)	)	PUNCT
ejpam-4761	364	24	in	in	ADP
ejpam-4761	364	25	g	g	PROPN
ejpam-4761	364	26	◦	◦	NOUN
ejpam-4761	364	27	h	h	NOUN
ejpam-4761	364	28	such	such	ADJ
ejpam-4761	364	29	that	that	DET
ejpam-4761	364	30	v	v	NOUN
ejpam-4761	364	31	(	(	PUNCT
ejpam-4761	364	32	p	p	X
ejpam-4761	364	33	(	(	PUNCT
ejpam-4761	364	34	x	x	NOUN
ejpam-4761	364	35	,	,	PUNCT
ejpam-4761	364	36	y	y	PROPN
ejpam-4761	364	37	)	)	PUNCT
ejpam-4761	364	38	⊆	⊆	NUM
ejpam-4761	364	39	w	w	NOUN
ejpam-4761	364	40	.	.	PUNCT
ejpam-4761	365	1	clearly	clearly	ADV
ejpam-4761	365	2	,	,	PUNCT
ejpam-4761	365	3	p	p	X
ejpam-4761	365	4	(	(	PUNCT
ejpam-4761	365	5	x	x	NOUN
ejpam-4761	365	6	,	,	PUNCT
ejpam-4761	365	7	y	y	NOUN
ejpam-4761	365	8	)	)	PUNCT
ejpam-4761	365	9	is	be	AUX
ejpam-4761	365	10	an	an	DET
ejpam-4761	365	11	x	x	NOUN
ejpam-4761	365	12	-	-	NOUN
ejpam-4761	365	13	y	y	ADJ
ejpam-4761	365	14	geodesic	geodesic	NOUN
ejpam-4761	365	15	in	in	ADP
ejpam-4761	365	16	g.	g.	PROPN
ejpam-4761	365	17	hence	hence	ADV
ejpam-4761	365	18	,	,	PUNCT
ejpam-4761	365	19	v	v	X
ejpam-4761	365	20	(	(	PUNCT
ejpam-4761	365	21	p	p	X
ejpam-4761	365	22	(	(	PUNCT
ejpam-4761	365	23	x	x	NOUN
ejpam-4761	365	24	,	,	PUNCT
ejpam-4761	365	25	y	y	NOUN
ejpam-4761	365	26	)	)	PUNCT
ejpam-4761	365	27	)	)	PUNCT
ejpam-4761	366	1	⊆	⊆	NUM
ejpam-4761	366	2	b.	b.	PROPN
ejpam-4761	366	3	therefore	therefore	ADV
ejpam-4761	366	4	,	,	PUNCT
ejpam-4761	366	5	b	b	PROPN
ejpam-4761	366	6	is	be	AUX
ejpam-4761	366	7	weakly	weakly	ADJ
ejpam-4761	366	8	convex	convex	NOUN
ejpam-4761	366	9	in	in	ADP
ejpam-4761	366	10	g	g	NOUN
ejpam-4761	366	11	,	,	PUNCT
ejpam-4761	366	12	showing	show	VERB
ejpam-4761	366	13	that	that	SCONJ
ejpam-4761	366	14	(	(	PUNCT
ejpam-4761	366	15	i	i	NOUN
ejpam-4761	366	16	)	)	PUNCT
ejpam-4761	366	17	holds	hold	VERB
ejpam-4761	366	18	.	.	PUNCT
ejpam-4761	367	1	next	next	ADV
ejpam-4761	367	2	,	,	PUNCT
ejpam-4761	367	3	let	let	VERB
ejpam-4761	367	4	v	v	NUM
ejpam-4761	367	5	∈	∈	PROPN
ejpam-4761	367	6	v	v	NOUN
ejpam-4761	367	7	(	(	PUNCT
ejpam-4761	367	8	g	g	NOUN
ejpam-4761	367	9	)	)	PUNCT
ejpam-4761	367	10	\	\	PROPN
ejpam-4761	367	11	b.	b.	PROPN
ejpam-4761	367	12	suppose	suppose	VERB
ejpam-4761	367	13	wv	wv	PROPN
ejpam-4761	367	14	̸=	̸=	PROPN
ejpam-4761	367	15	∅	∅	NOUN
ejpam-4761	367	16	,	,	PUNCT
ejpam-4761	367	17	say	say	VERB
ejpam-4761	367	18	p	p	PROPN
ejpam-4761	367	19	∈	∈	PROPN
ejpam-4761	367	20	wv	wv	PROPN
ejpam-4761	367	21	.	.	PUNCT
ejpam-4761	368	1	choose	choose	VERB
ejpam-4761	368	2	any	any	DET
ejpam-4761	368	3	z	z	PROPN
ejpam-4761	368	4	∈	∈	PROPN
ejpam-4761	368	5	b.	b.	PROPN
ejpam-4761	368	6	since	since	SCONJ
ejpam-4761	368	7	every	every	DET
ejpam-4761	368	8	p	p	PROPN
ejpam-4761	368	9	-	-	PUNCT
ejpam-4761	368	10	z	z	NOUN
ejpam-4761	368	11	geodesic	geodesic	NOUN
ejpam-4761	368	12	in	in	ADP
ejpam-4761	368	13	g	g	PROPN
ejpam-4761	368	14	◦	◦	NOUN
ejpam-4761	368	15	h	h	NOUN
ejpam-4761	368	16	contains	contain	VERB
ejpam-4761	368	17	v	v	NOUN
ejpam-4761	368	18	as	as	ADP
ejpam-4761	368	19	a	a	DET
ejpam-4761	368	20	vertex	vertex	NOUN
ejpam-4761	368	21	,	,	PUNCT
ejpam-4761	368	22	it	it	PRON
ejpam-4761	368	23	follows	follow	VERB
ejpam-4761	368	24	that	that	SCONJ
ejpam-4761	368	25	w	w	NOUN
ejpam-4761	368	26	is	be	AUX
ejpam-4761	368	27	not	not	PART
ejpam-4761	368	28	weakly	weakly	ADJ
ejpam-4761	368	29	convex	convex	NOUN
ejpam-4761	368	30	in	in	ADP
ejpam-4761	368	31	g	g	PROPN
ejpam-4761	368	32	◦	◦	NOUN
ejpam-4761	368	33	h	h	NOUN
ejpam-4761	368	34	,	,	PUNCT
ejpam-4761	368	35	a	a	DET
ejpam-4761	368	36	contradiction	contradiction	NOUN
ejpam-4761	368	37	.	.	PUNCT
ejpam-4761	369	1	therefore	therefore	ADV
ejpam-4761	369	2	,	,	PUNCT
ejpam-4761	369	3	wv	wv	PROPN
ejpam-4761	369	4	=	=	NOUN
ejpam-4761	369	5	∅	∅	NOUN
ejpam-4761	369	6	,	,	PUNCT
ejpam-4761	369	7	showing	show	VERB
ejpam-4761	369	8	that	that	SCONJ
ejpam-4761	369	9	(	(	PUNCT
ejpam-4761	369	10	ii	ii	NOUN
ejpam-4761	369	11	)	)	PUNCT
ejpam-4761	369	12	holds	hold	VERB
ejpam-4761	369	13	.	.	PUNCT
ejpam-4761	370	1	this	this	PRON
ejpam-4761	370	2	and	and	CCONJ
ejpam-4761	370	3	(	(	PUNCT
ejpam-4761	370	4	i	i	NOUN
ejpam-4761	370	5	)	)	PUNCT
ejpam-4761	370	6	in	in	ADP
ejpam-4761	370	7	theorem	theorem	NOUN
ejpam-4761	370	8	10	10	NUM
ejpam-4761	370	9	imply	imply	ADV
ejpam-4761	370	10	that	that	SCONJ
ejpam-4761	370	11	(	(	PUNCT
ejpam-4761	370	12	iii	iii	NOUN
ejpam-4761	370	13	)	)	PUNCT
ejpam-4761	370	14	holds	hold	VERB
ejpam-4761	370	15	.	.	PUNCT
ejpam-4761	371	1	moreover	moreover	ADV
ejpam-4761	371	2	,	,	PUNCT
ejpam-4761	371	3	because	because	SCONJ
ejpam-4761	371	4	b	b	NOUN
ejpam-4761	371	5	is	be	AUX
ejpam-4761	371	6	a	a	DET
ejpam-4761	371	7	dominating	dominating	NOUN
ejpam-4761	371	8	set	set	NOUN
ejpam-4761	371	9	,	,	PUNCT
ejpam-4761	371	10	and	and	CCONJ
ejpam-4761	371	11	(	(	PUNCT
ejpam-4761	371	12	ii	ii	NOUN
ejpam-4761	371	13	)	)	PUNCT
ejpam-4761	371	14	in	in	ADP
ejpam-4761	371	15	theorem	theorem	ADJ
ejpam-4761	371	16	10	10	NUM
ejpam-4761	371	17	holds	hold	NOUN
ejpam-4761	371	18	,	,	PUNCT
ejpam-4761	371	19	condition	condition	NOUN
ejpam-4761	371	20	(	(	PUNCT
ejpam-4761	371	21	iv	iv	X
ejpam-4761	371	22	)	)	PUNCT
ejpam-4761	371	23	also	also	ADV
ejpam-4761	371	24	holds	hold	VERB
ejpam-4761	371	25	.	.	PUNCT
ejpam-4761	372	1	for	for	ADP
ejpam-4761	372	2	the	the	DET
ejpam-4761	372	3	converse	converse	NOUN
ejpam-4761	372	4	,	,	PUNCT
ejpam-4761	372	5	suppose	suppose	VERB
ejpam-4761	372	6	that	that	SCONJ
ejpam-4761	372	7	w	w	PROPN
ejpam-4761	372	8	=	=	SYM
ejpam-4761	372	9	b	b	X
ejpam-4761	372	10	∪	∪	X
ejpam-4761	372	11	(	(	PUNCT
ejpam-4761	372	12	∪v∈v	∪v∈v	X
ejpam-4761	372	13	(	(	PUNCT
ejpam-4761	372	14	g)wv	g)wv	PROPN
ejpam-4761	372	15	)	)	PUNCT
ejpam-4761	372	16	and	and	CCONJ
ejpam-4761	372	17	satisfies	satisfie	NOUN
ejpam-4761	372	18	(	(	PUNCT
ejpam-4761	372	19	i	i	NOUN
ejpam-4761	372	20	)	)	PUNCT
ejpam-4761	372	21	,	,	PUNCT
ejpam-4761	372	22	(	(	PUNCT
ejpam-4761	372	23	ii	ii	NOUN
ejpam-4761	372	24	)	)	PUNCT
ejpam-4761	372	25	,	,	PUNCT
ejpam-4761	372	26	(	(	PUNCT
ejpam-4761	372	27	iii	iii	NOUN
ejpam-4761	372	28	)	)	PUNCT
ejpam-4761	372	29	,	,	PUNCT
ejpam-4761	372	30	and	and	CCONJ
ejpam-4761	372	31	(	(	PUNCT
ejpam-4761	372	32	iv	iv	X
ejpam-4761	372	33	)	)	PUNCT
ejpam-4761	372	34	.	.	PUNCT
ejpam-4761	373	1	then	then	ADV
ejpam-4761	373	2	(	(	PUNCT
ejpam-4761	373	3	i	i	NOUN
ejpam-4761	373	4	)	)	PUNCT
ejpam-4761	373	5	and	and	CCONJ
ejpam-4761	373	6	(	(	PUNCT
ejpam-4761	373	7	ii	ii	NOUN
ejpam-4761	373	8	)	)	PUNCT
ejpam-4761	373	9	in	in	ADP
ejpam-4761	373	10	theorem	theorem	ADJ
ejpam-4761	373	11	10	10	NUM
ejpam-4761	373	12	hold	hold	NOUN
ejpam-4761	373	13	,	,	PUNCT
ejpam-4761	373	14	that	that	ADV
ejpam-4761	373	15	is	is	ADV
ejpam-4761	373	16	,	,	PUNCT
ejpam-4761	373	17	w	w	PROPN
ejpam-4761	373	18	is	be	AUX
ejpam-4761	373	19	a	a	DET
ejpam-4761	373	20	hop	hop	NOUN
ejpam-4761	373	21	dominating	dominating	NOUN
ejpam-4761	373	22	set	set	VERB
ejpam-4761	373	23	in	in	ADP
ejpam-4761	373	24	g	g	PROPN
ejpam-4761	373	25	◦	◦	NOUN
ejpam-4761	373	26	h.	h.	PROPN
ejpam-4761	373	27	next	next	ADV
ejpam-4761	373	28	,	,	PUNCT
ejpam-4761	373	29	let	let	VERB
ejpam-4761	373	30	x	x	PRON
ejpam-4761	373	31	,	,	PUNCT
ejpam-4761	373	32	y	y	PROPN
ejpam-4761	373	33	∈	∈	PROPN
ejpam-4761	373	34	w	w	PROPN
ejpam-4761	373	35	and	and	CCONJ
ejpam-4761	373	36	let	let	VERB
ejpam-4761	373	37	v	v	NOUN
ejpam-4761	373	38	,	,	PUNCT
ejpam-4761	373	39	w	w	PROPN
ejpam-4761	373	40	∈	∈	PROPN
ejpam-4761	373	41	v	v	ADP
ejpam-4761	373	42	(	(	PUNCT
ejpam-4761	373	43	g	g	NOUN
ejpam-4761	373	44	)	)	PUNCT
ejpam-4761	373	45	such	such	ADJ
ejpam-4761	373	46	that	that	SCONJ
ejpam-4761	373	47	x	x	SYM
ejpam-4761	373	48	∈	∈	NOUN
ejpam-4761	373	49	v	v	NOUN
ejpam-4761	373	50	(	(	PUNCT
ejpam-4761	373	51	v	v	NOUN
ejpam-4761	373	52	+	+	CCONJ
ejpam-4761	373	53	hv	hv	NOUN
ejpam-4761	373	54	)	)	PUNCT
ejpam-4761	373	55	and	and	CCONJ
ejpam-4761	373	56	y	y	PROPN
ejpam-4761	373	57	∈	∈	PROPN
ejpam-4761	373	58	v	v	ADP
ejpam-4761	373	59	(	(	PUNCT
ejpam-4761	373	60	w	w	PROPN
ejpam-4761	373	61	+	+	NUM
ejpam-4761	373	62	hw	hw	NOUN
ejpam-4761	373	63	)	)	PUNCT
ejpam-4761	373	64	.	.	PUNCT
ejpam-4761	374	1	consider	consider	VERB
ejpam-4761	374	2	the	the	DET
ejpam-4761	374	3	following	follow	VERB
ejpam-4761	374	4	cases	case	NOUN
ejpam-4761	374	5	:	:	PUNCT
ejpam-4761	374	6	case	case	NOUN
ejpam-4761	374	7	1	1	NUM
ejpam-4761	374	8	.	.	PUNCT
ejpam-4761	374	9	x	x	X
ejpam-4761	375	1	=	=	SYM
ejpam-4761	375	2	v	v	PROPN
ejpam-4761	375	3	and	and	CCONJ
ejpam-4761	375	4	y	y	PROPN
ejpam-4761	375	5	=	=	SYM
ejpam-4761	375	6	w.	w.	PROPN
ejpam-4761	375	7	then	then	ADV
ejpam-4761	375	8	x	x	PRON
ejpam-4761	375	9	,	,	PUNCT
ejpam-4761	375	10	y	y	PROPN
ejpam-4761	375	11	∈	∈	PROPN
ejpam-4761	375	12	b.	b.	PROPN
ejpam-4761	376	1	since	since	SCONJ
ejpam-4761	376	2	b	b	PROPN
ejpam-4761	376	3	is	be	AUX
ejpam-4761	376	4	weakly	weakly	ADJ
ejpam-4761	376	5	convex	convex	NOUN
ejpam-4761	376	6	in	in	ADP
ejpam-4761	376	7	g	g	NOUN
ejpam-4761	376	8	,	,	PUNCT
ejpam-4761	376	9	there	there	PRON
ejpam-4761	376	10	exists	exist	VERB
ejpam-4761	376	11	an	an	DET
ejpam-4761	376	12	x	x	NOUN
ejpam-4761	376	13	-	-	NOUN
ejpam-4761	376	14	y	y	ADJ
ejpam-4761	376	15	geodesic	geodesic	NOUN
ejpam-4761	376	16	p	p	X
ejpam-4761	376	17	(	(	PUNCT
ejpam-4761	376	18	x	x	NOUN
ejpam-4761	376	19	,	,	PUNCT
ejpam-4761	376	20	y	y	NOUN
ejpam-4761	376	21	)	)	PUNCT
ejpam-4761	376	22	in	in	ADP
ejpam-4761	376	23	g	g	PROPN
ejpam-4761	376	24	(	(	PUNCT
ejpam-4761	376	25	also	also	ADV
ejpam-4761	376	26	an	an	DET
ejpam-4761	376	27	x	x	NOUN
ejpam-4761	376	28	-	-	NOUN
ejpam-4761	376	29	y	y	ADJ
ejpam-4761	376	30	geodesic	geodesic	NOUN
ejpam-4761	376	31	in	in	ADP
ejpam-4761	376	32	g	g	PROPN
ejpam-4761	376	33	◦	◦	NOUN
ejpam-4761	376	34	h	h	NOUN
ejpam-4761	376	35	)	)	PUNCT
ejpam-4761	376	36	such	such	ADJ
ejpam-4761	376	37	that	that	DET
ejpam-4761	376	38	v	v	NOUN
ejpam-4761	376	39	(	(	PUNCT
ejpam-4761	376	40	p	p	X
ejpam-4761	376	41	(	(	PUNCT
ejpam-4761	376	42	x	x	NOUN
ejpam-4761	376	43	,	,	PUNCT
ejpam-4761	376	44	y	y	NOUN
ejpam-4761	376	45	)	)	PUNCT
ejpam-4761	376	46	)	)	PUNCT
ejpam-4761	377	1	⊆	⊆	NUM
ejpam-4761	377	2	b	b	X
ejpam-4761	377	3	⊆	⊆	NUM
ejpam-4761	377	4	w	w	NOUN
ejpam-4761	377	5	.	.	PUNCT
ejpam-4761	378	1	case	case	NOUN
ejpam-4761	379	1	2	2	NUM
ejpam-4761	379	2	.	.	NUM
ejpam-4761	379	3	x	x	X
ejpam-4761	380	1	=	=	SYM
ejpam-4761	380	2	v	v	PROPN
ejpam-4761	380	3	and	and	CCONJ
ejpam-4761	380	4	y	y	PROPN
ejpam-4761	380	5	∈	∈	PROPN
ejpam-4761	380	6	ww	ww	PROPN
ejpam-4761	380	7	(	(	PUNCT
ejpam-4761	380	8	or	or	CCONJ
ejpam-4761	380	9	x	x	SYM
ejpam-4761	380	10	∈	∈	PROPN
ejpam-4761	380	11	wv	wv	PROPN
ejpam-4761	380	12	and	and	CCONJ
ejpam-4761	380	13	y	y	PROPN
ejpam-4761	380	14	=	=	PROPN
ejpam-4761	380	15	w	w	PROPN
ejpam-4761	380	16	)	)	PUNCT
ejpam-4761	380	17	.	.	PUNCT
ejpam-4761	381	1	then	then	ADV
ejpam-4761	381	2	w	w	PROPN
ejpam-4761	381	3	∈	∈	PROPN
ejpam-4761	381	4	b	b	SYM
ejpam-4761	381	5	by	by	X
ejpam-4761	381	6	(	(	PUNCT
ejpam-4761	381	7	iv	iv	NOUN
ejpam-4761	381	8	)	)	PUNCT
ejpam-4761	381	9	.	.	PUNCT
ejpam-4761	382	1	by	by	ADP
ejpam-4761	382	2	(	(	PUNCT
ejpam-4761	382	3	ii	ii	NOUN
ejpam-4761	382	4	)	)	PUNCT
ejpam-4761	382	5	,	,	PUNCT
ejpam-4761	382	6	we	we	PRON
ejpam-4761	382	7	may	may	AUX
ejpam-4761	382	8	let	let	VERB
ejpam-4761	382	9	p	p	NOUN
ejpam-4761	382	10	′(x	′(x	NOUN
ejpam-4761	382	11	,	,	PUNCT
ejpam-4761	382	12	w	w	NOUN
ejpam-4761	382	13	)	)	PUNCT
ejpam-4761	382	14	be	be	VERB
ejpam-4761	382	15	an	an	DET
ejpam-4761	382	16	x	x	NOUN
ejpam-4761	382	17	-	-	NOUN
ejpam-4761	382	18	w	w	NOUN
ejpam-4761	382	19	geodesic	geodesic	NOUN
ejpam-4761	382	20	in	in	ADP
ejpam-4761	382	21	g	g	PROPN
ejpam-4761	382	22	such	such	DET
ejpam-4761	382	23	that	that	DET
ejpam-4761	382	24	v	v	NOUN
ejpam-4761	382	25	(	(	PUNCT
ejpam-4761	382	26	p	p	X
ejpam-4761	382	27	(	(	PUNCT
ejpam-4761	382	28	x	x	NOUN
ejpam-4761	382	29	,	,	PUNCT
ejpam-4761	382	30	w	w	NOUN
ejpam-4761	382	31	)	)	PUNCT
ejpam-4761	382	32	)	)	PUNCT
ejpam-4761	383	1	⊆	⊆	NUM
ejpam-4761	383	2	b	b	X
ejpam-4761	383	3	⊆	⊆	NUM
ejpam-4761	383	4	w	w	NOUN
ejpam-4761	383	5	.	.	PUNCT
ejpam-4761	384	1	let	let	VERB
ejpam-4761	384	2	p	p	NOUN
ejpam-4761	384	3	′(x	′(x	NOUN
ejpam-4761	384	4	,	,	PUNCT
ejpam-4761	384	5	w	w	NOUN
ejpam-4761	384	6	)	)	PUNCT
ejpam-4761	384	7	=	=	PUNCT
ejpam-4761	385	1	[	[	X
ejpam-4761	385	2	x1	x1	PROPN
ejpam-4761	385	3	,	,	PUNCT
ejpam-4761	385	4	x2	x2	PROPN
ejpam-4761	385	5	,	,	PUNCT
ejpam-4761	385	6	·	·	PUNCT
ejpam-4761	385	7	·	·	PUNCT
ejpam-4761	385	8	·	·	PUNCT
ejpam-4761	385	9	,	,	PUNCT
ejpam-4761	385	10	xk	xk	PROPN
ejpam-4761	385	11	]	]	X
ejpam-4761	385	12	,	,	PUNCT
ejpam-4761	385	13	where	where	SCONJ
ejpam-4761	385	14	x	x	ADP
ejpam-4761	385	15	=	=	SYM
ejpam-4761	385	16	x1	x1	PROPN
ejpam-4761	385	17	and	and	CCONJ
ejpam-4761	385	18	w	w	PROPN
ejpam-4761	385	19	=	=	SYM
ejpam-4761	385	20	xk	xk	PROPN
ejpam-4761	385	21	.	.	PROPN
ejpam-4761	386	1	then	then	ADV
ejpam-4761	386	2	p	p	X
ejpam-4761	386	3	∗(x	∗(x	PROPN
ejpam-4761	386	4	,	,	PUNCT
ejpam-4761	386	5	y	y	NOUN
ejpam-4761	386	6	)	)	PUNCT
ejpam-4761	386	7	=	=	PUNCT
ejpam-4761	387	1	[	[	X
ejpam-4761	387	2	x1	x1	PROPN
ejpam-4761	387	3	,	,	PUNCT
ejpam-4761	387	4	x2	x2	PROPN
ejpam-4761	387	5	,	,	PUNCT
ejpam-4761	387	6	·	·	PUNCT
ejpam-4761	387	7	·	·	PUNCT
ejpam-4761	387	8	·	·	PUNCT
ejpam-4761	387	9	,	,	PUNCT
ejpam-4761	387	10	xk	xk	PROPN
ejpam-4761	387	11	,	,	PUNCT
ejpam-4761	387	12	y	y	PROPN
ejpam-4761	387	13	]	]	X
ejpam-4761	387	14	is	be	AUX
ejpam-4761	387	15	an	an	DET
ejpam-4761	387	16	x	x	NOUN
ejpam-4761	387	17	-	-	NOUN
ejpam-4761	387	18	y	y	ADJ
ejpam-4761	387	19	geodesic	geodesic	NOUN
ejpam-4761	387	20	in	in	ADP
ejpam-4761	387	21	g	g	PROPN
ejpam-4761	387	22	◦	◦	NOUN
ejpam-4761	387	23	h	h	NOUN
ejpam-4761	387	24	and	and	CCONJ
ejpam-4761	387	25	v	v	NOUN
ejpam-4761	387	26	(	(	PUNCT
ejpam-4761	387	27	p	p	NOUN
ejpam-4761	387	28	∗(x	∗(x	PROPN
ejpam-4761	387	29	,	,	PUNCT
ejpam-4761	387	30	y	y	NOUN
ejpam-4761	387	31	)	)	PUNCT
ejpam-4761	387	32	)	)	PUNCT
ejpam-4761	388	1	⊆	⊆	NUM
ejpam-4761	388	2	w	w	NOUN
ejpam-4761	388	3	.	.	PUNCT
ejpam-4761	388	4	case	case	NOUN
ejpam-4761	388	5	3	3	NUM
ejpam-4761	388	6	.	.	PUNCT
ejpam-4761	388	7	x	x	SYM
ejpam-4761	389	1	∈	∈	PROPN
ejpam-4761	389	2	v	v	ADP
ejpam-4761	389	3	(	(	PUNCT
ejpam-4761	389	4	hv	hv	PROPN
ejpam-4761	389	5	)	)	PUNCT
ejpam-4761	389	6	and	and	CCONJ
ejpam-4761	389	7	y	y	PROPN
ejpam-4761	389	8	∈	∈	PROPN
ejpam-4761	389	9	v	v	ADP
ejpam-4761	389	10	(	(	PUNCT
ejpam-4761	389	11	hw	hw	NOUN
ejpam-4761	389	12	)	)	PUNCT
ejpam-4761	389	13	.	.	PUNCT
ejpam-4761	390	1	then	then	ADV
ejpam-4761	390	2	v	v	X
ejpam-4761	390	3	,	,	PUNCT
ejpam-4761	390	4	w	w	PROPN
ejpam-4761	390	5	∈	∈	PROPN
ejpam-4761	390	6	b	b	SYM
ejpam-4761	390	7	by	by	X
ejpam-4761	390	8	(	(	PUNCT
ejpam-4761	390	9	iv	iv	NOUN
ejpam-4761	390	10	)	)	PUNCT
ejpam-4761	390	11	.	.	PUNCT
ejpam-4761	391	1	property	property	NOUN
ejpam-4761	391	2	(	(	PUNCT
ejpam-4761	391	3	ii	ii	NOUN
ejpam-4761	391	4	)	)	PUNCT
ejpam-4761	391	5	will	will	AUX
ejpam-4761	391	6	imply	imply	VERB
ejpam-4761	391	7	that	that	SCONJ
ejpam-4761	391	8	there	there	PRON
ejpam-4761	391	9	exists	exist	VERB
ejpam-4761	391	10	a	a	DET
ejpam-4761	391	11	v	v	NOUN
ejpam-4761	391	12	-	-	PUNCT
ejpam-4761	391	13	w	w	NOUN
ejpam-4761	391	14	geodesic	geodesic	NOUN
ejpam-4761	391	15	p	p	X
ejpam-4761	391	16	(	(	PUNCT
ejpam-4761	391	17	v	v	NOUN
ejpam-4761	391	18	,	,	PUNCT
ejpam-4761	391	19	w	w	NOUN
ejpam-4761	391	20	)	)	PUNCT
ejpam-4761	391	21	in	in	ADP
ejpam-4761	391	22	g	g	NOUN
ejpam-4761	391	23	◦	◦	NOUN
ejpam-4761	391	24	h	h	NOUN
ejpam-4761	391	25	such	such	ADJ
ejpam-4761	391	26	that	that	DET
ejpam-4761	391	27	v	v	NOUN
ejpam-4761	391	28	(	(	PUNCT
ejpam-4761	391	29	p	p	X
ejpam-4761	391	30	(	(	PUNCT
ejpam-4761	391	31	v	v	NOUN
ejpam-4761	391	32	,	,	PUNCT
ejpam-4761	391	33	w	w	NOUN
ejpam-4761	391	34	)	)	PUNCT
ejpam-4761	391	35	)	)	PUNCT
ejpam-4761	392	1	⊆	⊆	NUM
ejpam-4761	392	2	w	w	NOUN
ejpam-4761	392	3	.	.	PUNCT
ejpam-4761	393	1	let	let	VERB
ejpam-4761	393	2	p	p	NOUN
ejpam-4761	393	3	(	(	PUNCT
ejpam-4761	393	4	v	v	NOUN
ejpam-4761	393	5	,	,	PUNCT
ejpam-4761	393	6	w	w	NOUN
ejpam-4761	393	7	)	)	PUNCT
ejpam-4761	393	8	=	=	PUNCT
ejpam-4761	394	1	[	[	X
ejpam-4761	394	2	v1	v1	NOUN
ejpam-4761	394	3	,	,	PUNCT
ejpam-4761	394	4	v2	v2	PROPN
ejpam-4761	394	5	,	,	PUNCT
ejpam-4761	394	6	·	·	PUNCT
ejpam-4761	394	7	·	·	PUNCT
ejpam-4761	394	8	·	·	PUNCT
ejpam-4761	394	9	,	,	PUNCT
ejpam-4761	394	10	vk	vk	ADP
ejpam-4761	394	11	]	]	PUNCT
ejpam-4761	394	12	,	,	PUNCT
ejpam-4761	394	13	where	where	SCONJ
ejpam-4761	394	14	v	v	NOUN
ejpam-4761	394	15	=	=	SYM
ejpam-4761	394	16	v1	v1	NOUN
ejpam-4761	394	17	and	and	CCONJ
ejpam-4761	394	18	w	w	NOUN
ejpam-4761	394	19	=	=	ADJ
ejpam-4761	394	20	vk	vk	PROPN
ejpam-4761	394	21	.	.	PUNCT
ejpam-4761	395	1	then	then	ADV
ejpam-4761	395	2	p	p	X
ejpam-4761	395	3	(	(	PUNCT
ejpam-4761	395	4	x	x	NOUN
ejpam-4761	395	5	,	,	PUNCT
ejpam-4761	395	6	y	y	NOUN
ejpam-4761	395	7	)	)	PUNCT
ejpam-4761	395	8	=	=	PUNCT
ejpam-4761	396	1	[	[	X
ejpam-4761	396	2	x	x	X
ejpam-4761	396	3	,	,	PUNCT
ejpam-4761	396	4	x1	x1	PROPN
ejpam-4761	396	5	,	,	PUNCT
ejpam-4761	396	6	x2	x2	PROPN
ejpam-4761	396	7	,	,	PUNCT
ejpam-4761	396	8	·	·	PUNCT
ejpam-4761	396	9	·	·	PUNCT
ejpam-4761	396	10	·	·	PUNCT
ejpam-4761	396	11	,	,	PUNCT
ejpam-4761	396	12	xk	xk	PROPN
ejpam-4761	396	13	,	,	PUNCT
ejpam-4761	396	14	y	y	PROPN
ejpam-4761	396	15	]	]	X
ejpam-4761	396	16	is	be	AUX
ejpam-4761	396	17	an	an	DET
ejpam-4761	396	18	x	x	NOUN
ejpam-4761	396	19	-	-	NOUN
ejpam-4761	396	20	y	y	ADJ
ejpam-4761	396	21	geodesic	geodesic	NOUN
ejpam-4761	396	22	in	in	ADP
ejpam-4761	396	23	g	g	PROPN
ejpam-4761	396	24	◦	◦	NOUN
ejpam-4761	396	25	h	h	NOUN
ejpam-4761	396	26	and	and	CCONJ
ejpam-4761	396	27	v	v	NOUN
ejpam-4761	396	28	(	(	PUNCT
ejpam-4761	396	29	p	p	X
ejpam-4761	396	30	(	(	PUNCT
ejpam-4761	396	31	x	x	NOUN
ejpam-4761	396	32	,	,	PUNCT
ejpam-4761	396	33	y	y	NOUN
ejpam-4761	396	34	)	)	PUNCT
ejpam-4761	396	35	)	)	PUNCT
ejpam-4761	397	1	⊆	⊆	NUM
ejpam-4761	397	2	w	w	NOUN
ejpam-4761	397	3	.	.	PUNCT
ejpam-4761	398	1	therefore	therefore	ADV
ejpam-4761	398	2	,	,	PUNCT
ejpam-4761	398	3	w	w	NOUN
ejpam-4761	398	4	is	be	AUX
ejpam-4761	398	5	weakly	weakly	ADJ
ejpam-4761	398	6	convex	convex	NOUN
ejpam-4761	398	7	in	in	ADP
ejpam-4761	398	8	g	g	PROPN
ejpam-4761	398	9	◦	◦	NOUN
ejpam-4761	398	10	h.	h.	NOUN
ejpam-4761	398	11	accordingly	accordingly	ADV
ejpam-4761	398	12	,	,	PUNCT
ejpam-4761	398	13	w	w	NOUN
ejpam-4761	398	14	is	be	AUX
ejpam-4761	398	15	weakly	weakly	ADV
ejpam-4761	398	16	convex	convex	ADJ
ejpam-4761	398	17	hop	hop	NOUN
ejpam-4761	398	18	dominating	dominating	NOUN
ejpam-4761	398	19	in	in	ADP
ejpam-4761	398	20	g	g	PROPN
ejpam-4761	398	21	◦	◦	NOUN
ejpam-4761	398	22	h.	h.	NOUN
ejpam-4761	398	23	corollary	corollary	ADJ
ejpam-4761	398	24	8	8	NUM
ejpam-4761	398	25	.	.	PUNCT
ejpam-4761	399	1	let	let	VERB
ejpam-4761	399	2	g	g	PRON
ejpam-4761	399	3	be	be	AUX
ejpam-4761	399	4	a	a	DET
ejpam-4761	399	5	non	non	ADJ
ejpam-4761	399	6	-	-	ADJ
ejpam-4761	399	7	trivial	trivial	ADJ
ejpam-4761	399	8	connected	connected	ADJ
ejpam-4761	399	9	graph	graph	NOUN
ejpam-4761	399	10	and	and	CCONJ
ejpam-4761	399	11	let	let	VERB
ejpam-4761	399	12	h	h	NOUN
ejpam-4761	399	13	be	be	AUX
ejpam-4761	399	14	any	any	DET
ejpam-4761	399	15	graph	graph	NOUN
ejpam-4761	399	16	.	.	PUNCT
ejpam-4761	400	1	then	then	ADV
ejpam-4761	400	2	γwcon(g	γwcon(g	NUM
ejpam-4761	400	3	)	)	PUNCT
ejpam-4761	400	4	≤	≤	NOUN
ejpam-4761	400	5	γwconh(g	γwconh(g	PRON
ejpam-4761	400	6	◦	◦	NOUN
ejpam-4761	400	7	h	h	NOUN
ejpam-4761	400	8	)	)	PUNCT
ejpam-4761	400	9	≤	≤	NUM
ejpam-4761	401	1	γhtwcon(g	γhtwcon(g	PROPN
ejpam-4761	401	2	)	)	PUNCT
ejpam-4761	402	1	,	,	PUNCT
ejpam-4761	402	2	s.	s.	PROPN
ejpam-4761	402	3	canoy	canoy	PROPN
ejpam-4761	402	4	jr	jr	PROPN
ejpam-4761	402	5	.	.	PROPN
ejpam-4761	402	6	,	,	PUNCT
ejpam-4761	402	7	j.	j.	PROPN
ejpam-4761	402	8	hassan	hassan	PROPN
ejpam-4761	402	9	/	/	SYM
ejpam-4761	402	10	eur	eur	PROPN
ejpam-4761	402	11	.	.	PUNCT
ejpam-4761	403	1	j.	j.	PROPN
ejpam-4761	403	2	pure	pure	PROPN
ejpam-4761	403	3	appl	appl	PROPN
ejpam-4761	403	4	.	.	PROPN
ejpam-4761	403	5	math	math	PROPN
ejpam-4761	403	6	,	,	PUNCT
ejpam-4761	403	7	16	16	NUM
ejpam-4761	403	8	(	(	PUNCT
ejpam-4761	403	9	2	2	NUM
ejpam-4761	403	10	)	)	PUNCT
ejpam-4761	403	11	(	(	PUNCT
ejpam-4761	403	12	2023	2023	NUM
ejpam-4761	403	13	)	)	PUNCT
ejpam-4761	403	14	,	,	PUNCT
ejpam-4761	403	15	1196	1196	NUM
ejpam-4761	403	16	-	-	SYM
ejpam-4761	403	17	1211	1211	NUM
ejpam-4761	403	18	1207	1207	NUM
ejpam-4761	403	19	where	where	SCONJ
ejpam-4761	403	20	γhtwcon(g	γhtwcon(g	NOUN
ejpam-4761	403	21	)	)	PUNCT
ejpam-4761	404	1	=	=	NOUN
ejpam-4761	404	2	min{|s|	min{|s|	NOUN
ejpam-4761	404	3	:	:	PUNCT
ejpam-4761	404	4	s	s	VERB
ejpam-4761	404	5	is	be	AUX
ejpam-4761	404	6	hop	hop	NOUN
ejpam-4761	404	7	dominating	dominating	NOUN
ejpam-4761	404	8	and	and	CCONJ
ejpam-4761	404	9	weakly	weakly	ADJ
ejpam-4761	404	10	convex	convex	ADJ
ejpam-4761	404	11	total	total	ADJ
ejpam-4761	404	12	dominating	dominating	NOUN
ejpam-4761	404	13	in	in	ADP
ejpam-4761	404	14	g	g	NOUN
ejpam-4761	404	15	}	}	PUNCT
ejpam-4761	404	16	.	.	PUNCT
ejpam-4761	405	1	note	note	VERB
ejpam-4761	405	2	that	that	SCONJ
ejpam-4761	405	3	the	the	DET
ejpam-4761	405	4	bounds	bound	NOUN
ejpam-4761	405	5	in	in	ADP
ejpam-4761	405	6	corollary	corollary	ADJ
ejpam-4761	405	7	8	8	NUM
ejpam-4761	405	8	are	be	AUX
ejpam-4761	405	9	tight	tight	ADJ
ejpam-4761	405	10	.	.	PUNCT
ejpam-4761	406	1	in	in	ADP
ejpam-4761	406	2	fact	fact	NOUN
ejpam-4761	406	3	,	,	PUNCT
ejpam-4761	406	4	γwconh(c4	γwconh(c4	NOUN
ejpam-4761	406	5	◦	◦	NOUN
ejpam-4761	406	6	h	h	NOUN
ejpam-4761	406	7	)	)	PUNCT
ejpam-4761	406	8	=	=	SYM
ejpam-4761	406	9	2	2	NUM
ejpam-4761	406	10	=	=	SYM
ejpam-4761	406	11	γwcon(c4	γwcon(c4	NOUN
ejpam-4761	406	12	)	)	PUNCT
ejpam-4761	406	13	for	for	ADP
ejpam-4761	406	14	any	any	DET
ejpam-4761	406	15	graph	graph	NOUN
ejpam-4761	406	16	h.	h.	NOUN
ejpam-4761	406	17	consider	consider	VERB
ejpam-4761	406	18	the	the	DET
ejpam-4761	406	19	graph	graph	NOUN
ejpam-4761	406	20	g	g	NOUN
ejpam-4761	406	21	in	in	ADP
ejpam-4761	406	22	figure	figure	NOUN
ejpam-4761	406	23	5	5	NUM
ejpam-4761	406	24	and	and	CCONJ
ejpam-4761	406	25	the	the	DET
ejpam-4761	406	26	sets	set	NOUN
ejpam-4761	406	27	w1	w1	NOUN
ejpam-4761	406	28	=	=	SYM
ejpam-4761	406	29	{	{	PUNCT
ejpam-4761	406	30	a	a	PRON
ejpam-4761	406	31	,	,	PUNCT
ejpam-4761	406	32	b	b	NOUN
ejpam-4761	406	33	,	,	PUNCT
ejpam-4761	406	34	c	c	NOUN
ejpam-4761	406	35	}	}	PUNCT
ejpam-4761	406	36	and	and	CCONJ
ejpam-4761	406	37	w2	w2	NOUN
ejpam-4761	406	38	=	=	PROPN
ejpam-4761	406	39	w1	w1	PROPN
ejpam-4761	406	40	∪	∪	X
ejpam-4761	406	41	{	{	PUNCT
ejpam-4761	406	42	v	v	NOUN
ejpam-4761	406	43	}	}	PUNCT
ejpam-4761	406	44	.	.	PUNCT
ejpam-4761	407	1	it	it	PRON
ejpam-4761	407	2	can	can	AUX
ejpam-4761	407	3	be	be	AUX
ejpam-4761	407	4	verified	verify	VERB
ejpam-4761	407	5	that	that	SCONJ
ejpam-4761	407	6	w1	w1	NOUN
ejpam-4761	407	7	is	be	AUX
ejpam-4761	407	8	weakly	weakly	ADJ
ejpam-4761	407	9	convex	convex	ADJ
ejpam-4761	407	10	dominating	dominating	NOUN
ejpam-4761	407	11	,	,	PUNCT
ejpam-4761	407	12	w2	w2	NOUN
ejpam-4761	407	13	is	be	AUX
ejpam-4761	407	14	weakly	weakly	ADV
ejpam-4761	407	15	convex	convex	ADJ
ejpam-4761	407	16	total	total	ADJ
ejpam-4761	407	17	dominating	dominating	NOUN
ejpam-4761	407	18	and	and	CCONJ
ejpam-4761	407	19	hop	hop	NOUN
ejpam-4761	407	20	dominating	dominating	NOUN
ejpam-4761	407	21	,	,	PUNCT
ejpam-4761	407	22	and	and	CCONJ
ejpam-4761	407	23	γwcon(g	γwcon(g	NUM
ejpam-4761	407	24	)	)	PUNCT
ejpam-4761	407	25	=	=	SYM
ejpam-4761	407	26	|w1|	|w1|	NOUN
ejpam-4761	407	27	and	and	CCONJ
ejpam-4761	407	28	γhtwcon(g	γhtwcon(g	NOUN
ejpam-4761	407	29	)	)	PUNCT
ejpam-4761	408	1	=	=	SYM
ejpam-4761	408	2	|w2|	|w2|	NOUN
ejpam-4761	408	3	.	.	PUNCT
ejpam-4761	409	1	for	for	ADP
ejpam-4761	409	2	any	any	DET
ejpam-4761	409	3	graph	graph	NOUN
ejpam-4761	409	4	h	h	NOUN
ejpam-4761	409	5	,	,	PUNCT
ejpam-4761	409	6	we	we	PRON
ejpam-4761	409	7	find	find	VERB
ejpam-4761	409	8	that	that	SCONJ
ejpam-4761	409	9	γwcon(g	γwcon(g	NOUN
ejpam-4761	409	10	)	)	PUNCT
ejpam-4761	410	1	<	<	X
ejpam-4761	410	2	γhtwcon(g	γhtwcon(g	PROPN
ejpam-4761	410	3	)	)	PUNCT
ejpam-4761	410	4	=	=	PUNCT
ejpam-4761	410	5	γwconh(g	γwconh(g	PRON
ejpam-4761	410	6	◦	◦	NOUN
ejpam-4761	410	7	h	h	NOUN
ejpam-4761	410	8	)	)	PUNCT
ejpam-4761	410	9	=	=	SYM
ejpam-4761	411	1	4	4	X
ejpam-4761	411	2	.	.	X
ejpam-4761	412	1	g	g	NOUN
ejpam-4761	412	2	:	:	PUNCT
ejpam-4761	412	3	a	a	DET
ejpam-4761	412	4	b	b	X
ejpam-4761	412	5	c	c	NOUN
ejpam-4761	412	6	v	v	NOUN
ejpam-4761	412	7	figure	figure	NOUN
ejpam-4761	412	8	5	5	NUM
ejpam-4761	412	9	:	:	PUNCT
ejpam-4761	412	10	a	a	DET
ejpam-4761	412	11	graph	graph	NOUN
ejpam-4761	412	12	g	g	NOUN
ejpam-4761	412	13	with	with	ADP
ejpam-4761	412	14	γwcon(g	γwcon(g	NOUN
ejpam-4761	412	15	)	)	PUNCT
ejpam-4761	412	16	=	=	SYM
ejpam-4761	412	17	3	3	NUM
ejpam-4761	412	18	<	<	SYM
ejpam-4761	412	19	4	4	NUM
ejpam-4761	412	20	=	=	SYM
ejpam-4761	412	21	γh	γh	ADP
ejpam-4761	412	22	twcon(g	twcon(g	ADJ
ejpam-4761	412	23	)	)	PUNCT
ejpam-4761	412	24	=	=	SYM
ejpam-4761	412	25	γwconh(g	γwconh(g	PRON
ejpam-4761	412	26	◦	◦	NOUN
ejpam-4761	412	27	h	h	NOUN
ejpam-4761	412	28	)	)	PUNCT
ejpam-4761	412	29	the	the	DET
ejpam-4761	412	30	next	next	ADJ
ejpam-4761	412	31	result	result	NOUN
ejpam-4761	412	32	is	be	AUX
ejpam-4761	412	33	found	find	VERB
ejpam-4761	412	34	in	in	ADP
ejpam-4761	412	35	[	[	X
ejpam-4761	412	36	13	13	NUM
ejpam-4761	412	37	]	]	PUNCT
ejpam-4761	412	38	.	.	PUNCT
ejpam-4761	413	1	theorem	theorem	NOUN
ejpam-4761	413	2	12	12	NUM
ejpam-4761	413	3	.	.	PUNCT
ejpam-4761	414	1	let	let	VERB
ejpam-4761	414	2	g	g	NOUN
ejpam-4761	414	3	and	and	CCONJ
ejpam-4761	414	4	h	h	NOUN
ejpam-4761	414	5	be	be	AUX
ejpam-4761	414	6	connected	connect	VERB
ejpam-4761	414	7	non	non	ADJ
ejpam-4761	414	8	-	-	ADJ
ejpam-4761	414	9	trivial	trivial	ADJ
ejpam-4761	414	10	graphs	graph	NOUN
ejpam-4761	414	11	.	.	PUNCT
ejpam-4761	415	1	then	then	ADV
ejpam-4761	415	2	c	c	NOUN
ejpam-4761	415	3	=	=	PUNCT
ejpam-4761	415	4	⋃	⋃	PROPN
ejpam-4761	415	5	x∈s	x∈s	NOUN
ejpam-4761	416	1	[	[	X
ejpam-4761	416	2	{	{	PUNCT
ejpam-4761	416	3	x	x	NOUN
ejpam-4761	416	4	}	}	PUNCT
ejpam-4761	416	5	×	×	PROPN
ejpam-4761	416	6	tx	tx	PROPN
ejpam-4761	416	7	]	]	PUNCT
ejpam-4761	416	8	is	be	AUX
ejpam-4761	416	9	a	a	DET
ejpam-4761	416	10	hop	hop	NOUN
ejpam-4761	416	11	dominating	dominating	NOUN
ejpam-4761	416	12	set	set	VERB
ejpam-4761	416	13	in	in	ADP
ejpam-4761	416	14	g[h	g[h	PROPN
ejpam-4761	416	15	]	]	PUNCT
ejpam-4761	416	16	if	if	SCONJ
ejpam-4761	416	17	and	and	CCONJ
ejpam-4761	416	18	only	only	ADV
ejpam-4761	416	19	if	if	SCONJ
ejpam-4761	416	20	the	the	DET
ejpam-4761	416	21	following	follow	VERB
ejpam-4761	416	22	conditions	condition	NOUN
ejpam-4761	416	23	hold	hold	VERB
ejpam-4761	416	24	.	.	PUNCT
ejpam-4761	417	1	(	(	PUNCT
ejpam-4761	417	2	i	i	NOUN
ejpam-4761	417	3	)	)	PUNCT
ejpam-4761	417	4	s	s	VERB
ejpam-4761	417	5	is	be	AUX
ejpam-4761	417	6	a	a	DET
ejpam-4761	417	7	hop	hop	NOUN
ejpam-4761	417	8	dominating	dominating	NOUN
ejpam-4761	417	9	set	set	VERB
ejpam-4761	417	10	in	in	ADP
ejpam-4761	417	11	g.	g.	PROPN
ejpam-4761	417	12	(	(	PUNCT
ejpam-4761	417	13	ii	ii	PROPN
ejpam-4761	417	14	)	)	PUNCT
ejpam-4761	417	15	tx	tx	PROPN
ejpam-4761	417	16	is	be	AUX
ejpam-4761	417	17	a	a	DET
ejpam-4761	417	18	pointwise	pointwise	ADJ
ejpam-4761	417	19	non	non	ADJ
ejpam-4761	417	20	-	-	ADJ
ejpam-4761	417	21	dominating	dominating	ADJ
ejpam-4761	417	22	set	set	NOUN
ejpam-4761	417	23	in	in	ADP
ejpam-4761	417	24	h	h	NOUN
ejpam-4761	417	25	for	for	ADP
ejpam-4761	417	26	each	each	DET
ejpam-4761	417	27	x	x	SYM
ejpam-4761	417	28	∈	∈	PROPN
ejpam-4761	417	29	s	s	PART
ejpam-4761	417	30	\n2	\n2	ADJ
ejpam-4761	417	31	g(s	g(	NOUN
ejpam-4761	417	32	)	)	PUNCT
ejpam-4761	417	33	.	.	PUNCT
ejpam-4761	418	1	theorem	theorem	NOUN
ejpam-4761	418	2	13	13	NUM
ejpam-4761	418	3	.	.	PUNCT
ejpam-4761	419	1	let	let	VERB
ejpam-4761	419	2	g	g	NOUN
ejpam-4761	419	3	and	and	CCONJ
ejpam-4761	419	4	h	h	NOUN
ejpam-4761	419	5	be	be	AUX
ejpam-4761	419	6	connected	connect	VERB
ejpam-4761	419	7	non	non	ADJ
ejpam-4761	419	8	-	-	ADJ
ejpam-4761	419	9	trivial	trivial	ADJ
ejpam-4761	419	10	graphs	graph	NOUN
ejpam-4761	419	11	.	.	PUNCT
ejpam-4761	420	1	then	then	ADV
ejpam-4761	420	2	c	c	NOUN
ejpam-4761	420	3	=	=	PUNCT
ejpam-4761	420	4	⋃	⋃	PROPN
ejpam-4761	420	5	x∈s	x∈s	NOUN
ejpam-4761	421	1	[	[	X
ejpam-4761	421	2	{	{	PUNCT
ejpam-4761	421	3	x}×tx	x}×tx	X
ejpam-4761	421	4	]	]	X
ejpam-4761	421	5	is	be	AUX
ejpam-4761	421	6	a	a	DET
ejpam-4761	421	7	weakly	weakly	ADJ
ejpam-4761	421	8	convex	convex	NOUN
ejpam-4761	421	9	hop	hop	NOUN
ejpam-4761	421	10	dominating	dominating	NOUN
ejpam-4761	421	11	set	set	VERB
ejpam-4761	421	12	in	in	ADP
ejpam-4761	421	13	g[h	g[h	PROPN
ejpam-4761	421	14	]	]	PUNCT
ejpam-4761	421	15	if	if	SCONJ
ejpam-4761	421	16	and	and	CCONJ
ejpam-4761	421	17	only	only	ADV
ejpam-4761	421	18	if	if	SCONJ
ejpam-4761	421	19	the	the	DET
ejpam-4761	421	20	following	follow	VERB
ejpam-4761	421	21	conditions	condition	NOUN
ejpam-4761	421	22	hold	hold	VERB
ejpam-4761	421	23	.	.	PUNCT
ejpam-4761	422	1	(	(	PUNCT
ejpam-4761	422	2	i	i	NOUN
ejpam-4761	422	3	)	)	PUNCT
ejpam-4761	422	4	s	s	VERB
ejpam-4761	422	5	is	be	AUX
ejpam-4761	422	6	a	a	DET
ejpam-4761	422	7	weakly	weakly	ADJ
ejpam-4761	422	8	convex	convex	NOUN
ejpam-4761	422	9	hop	hop	NOUN
ejpam-4761	422	10	dominating	dominating	NOUN
ejpam-4761	422	11	set	set	VERB
ejpam-4761	422	12	in	in	ADP
ejpam-4761	422	13	g.	g.	PROPN
ejpam-4761	422	14	(	(	PUNCT
ejpam-4761	422	15	ii	ii	PROPN
ejpam-4761	422	16	)	)	PUNCT
ejpam-4761	422	17	tx	tx	PROPN
ejpam-4761	422	18	is	be	AUX
ejpam-4761	422	19	a	a	DET
ejpam-4761	422	20	pointwise	pointwise	ADJ
ejpam-4761	422	21	non	non	ADJ
ejpam-4761	422	22	-	-	ADJ
ejpam-4761	422	23	dominating	dominating	ADJ
ejpam-4761	422	24	set	set	NOUN
ejpam-4761	422	25	in	in	ADP
ejpam-4761	422	26	h	h	NOUN
ejpam-4761	422	27	for	for	ADP
ejpam-4761	422	28	each	each	DET
ejpam-4761	422	29	x	x	SYM
ejpam-4761	422	30	∈	∈	PROPN
ejpam-4761	422	31	s	s	PART
ejpam-4761	422	32	\n2	\n2	ADJ
ejpam-4761	422	33	g(s	g(	NOUN
ejpam-4761	422	34	)	)	PUNCT
ejpam-4761	422	35	.	.	PUNCT
ejpam-4761	423	1	proof	proof	NOUN
ejpam-4761	423	2	.	.	PUNCT
ejpam-4761	424	1	suppose	suppose	VERB
ejpam-4761	424	2	c	c	NOUN
ejpam-4761	424	3	is	be	AUX
ejpam-4761	424	4	weakly	weakly	ADV
ejpam-4761	424	5	convex	convex	ADJ
ejpam-4761	424	6	hop	hop	NOUN
ejpam-4761	424	7	dominating	dominating	NOUN
ejpam-4761	424	8	in	in	ADP
ejpam-4761	424	9	g[h	g[h	PROPN
ejpam-4761	424	10	]	]	PUNCT
ejpam-4761	424	11	.	.	PUNCT
ejpam-4761	425	1	then	then	ADV
ejpam-4761	425	2	s	s	VERB
ejpam-4761	425	3	is	be	AUX
ejpam-4761	425	4	a	a	DET
ejpam-4761	425	5	hop	hop	NOUN
ejpam-4761	425	6	dominating	dominating	NOUN
ejpam-4761	425	7	set	set	VERB
ejpam-4761	425	8	in	in	ADP
ejpam-4761	425	9	g	g	NOUN
ejpam-4761	425	10	and	and	CCONJ
ejpam-4761	425	11	property	property	NOUN
ejpam-4761	425	12	(	(	PUNCT
ejpam-4761	425	13	ii	ii	NOUN
ejpam-4761	425	14	)	)	PUNCT
ejpam-4761	425	15	holds	hold	VERB
ejpam-4761	425	16	by	by	ADP
ejpam-4761	425	17	theorem	theorem	NOUN
ejpam-4761	425	18	12(ii	12(ii	NUM
ejpam-4761	425	19	)	)	PUNCT
ejpam-4761	425	20	.	.	PUNCT
ejpam-4761	426	1	let	let	VERB
ejpam-4761	426	2	v	v	NOUN
ejpam-4761	426	3	,	,	PUNCT
ejpam-4761	426	4	w	w	PROPN
ejpam-4761	426	5	∈	∈	PROPN
ejpam-4761	426	6	s	s	VERB
ejpam-4761	426	7	with	with	ADP
ejpam-4761	426	8	v	v	NOUN
ejpam-4761	426	9	̸=	̸=	PROPN
ejpam-4761	426	10	w.	w.	NOUN
ejpam-4761	426	11	choose	choose	VERB
ejpam-4761	426	12	any	any	DET
ejpam-4761	426	13	p	p	PROPN
ejpam-4761	426	14	∈	∈	PROPN
ejpam-4761	426	15	tv	tv	NOUN
ejpam-4761	426	16	and	and	CCONJ
ejpam-4761	426	17	q	q	NOUN
ejpam-4761	426	18	∈	∈	PROPN
ejpam-4761	426	19	tw	tw	NOUN
ejpam-4761	426	20	.	.	PUNCT
ejpam-4761	427	1	by	by	ADP
ejpam-4761	427	2	assumption	assumption	NOUN
ejpam-4761	427	3	,	,	PUNCT
ejpam-4761	427	4	there	there	PRON
ejpam-4761	427	5	exists	exist	VERB
ejpam-4761	427	6	a	a	DET
ejpam-4761	427	7	(	(	PUNCT
ejpam-4761	427	8	v	v	NOUN
ejpam-4761	427	9	,	,	PUNCT
ejpam-4761	427	10	p)-(w	p)-(w	NUM
ejpam-4761	427	11	,	,	PUNCT
ejpam-4761	427	12	q	q	NOUN
ejpam-4761	427	13	)	)	PUNCT
ejpam-4761	427	14	geodesic	geodesic	NOUN
ejpam-4761	427	15	p	p	X
ejpam-4761	427	16	(	(	PUNCT
ejpam-4761	427	17	(	(	PUNCT
ejpam-4761	427	18	v	v	NOUN
ejpam-4761	427	19	,	,	PUNCT
ejpam-4761	427	20	p	p	NOUN
ejpam-4761	427	21	)	)	PUNCT
ejpam-4761	427	22	,	,	PUNCT
ejpam-4761	427	23	(	(	PUNCT
ejpam-4761	427	24	w	w	NOUN
ejpam-4761	427	25	,	,	PUNCT
ejpam-4761	427	26	q	q	NOUN
ejpam-4761	427	27	)	)	PUNCT
ejpam-4761	427	28	)	)	PUNCT
ejpam-4761	428	1	=	=	PUNCT
ejpam-4761	429	1	[	[	X
ejpam-4761	429	2	(	(	PUNCT
ejpam-4761	429	3	v1	v1	NOUN
ejpam-4761	429	4	,	,	PUNCT
ejpam-4761	429	5	p1	p1	PROPN
ejpam-4761	429	6	)	)	PUNCT
ejpam-4761	429	7	,	,	PUNCT
ejpam-4761	429	8	(	(	PUNCT
ejpam-4761	429	9	v2	v2	NOUN
ejpam-4761	429	10	,	,	PUNCT
ejpam-4761	429	11	p2	p2	PROPN
ejpam-4761	429	12	)	)	PUNCT
ejpam-4761	429	13	,	,	PUNCT
ejpam-4761	429	14	·	·	PUNCT
ejpam-4761	429	15	·	·	PUNCT
ejpam-4761	429	16	·	·	PUNCT
ejpam-4761	429	17	,	,	PUNCT
ejpam-4761	429	18	(	(	PUNCT
ejpam-4761	429	19	vk	vk	X
ejpam-4761	429	20	,	,	PUNCT
ejpam-4761	429	21	pk	pk	NOUN
ejpam-4761	429	22	)	)	PUNCT
ejpam-4761	429	23	]	]	PUNCT
ejpam-4761	429	24	in	in	ADP
ejpam-4761	429	25	g[h	g[h	PROPN
ejpam-4761	429	26	]	]	PUNCT
ejpam-4761	429	27	,	,	PUNCT
ejpam-4761	429	28	where	where	SCONJ
ejpam-4761	429	29	(	(	PUNCT
ejpam-4761	429	30	v	v	NOUN
ejpam-4761	429	31	,	,	PUNCT
ejpam-4761	429	32	p	p	NOUN
ejpam-4761	429	33	)	)	PUNCT
ejpam-4761	429	34	=	=	SYM
ejpam-4761	429	35	(	(	PUNCT
ejpam-4761	429	36	v1	v1	PROPN
ejpam-4761	429	37	,	,	PUNCT
ejpam-4761	429	38	p1	p1	NOUN
ejpam-4761	429	39	)	)	PUNCT
ejpam-4761	429	40	and	and	CCONJ
ejpam-4761	429	41	(	(	PUNCT
ejpam-4761	429	42	w	w	PROPN
ejpam-4761	429	43	,	,	PUNCT
ejpam-4761	429	44	q	q	NOUN
ejpam-4761	429	45	)	)	PUNCT
ejpam-4761	429	46	=	=	SYM
ejpam-4761	429	47	(	(	PUNCT
ejpam-4761	429	48	vk	vk	PROPN
ejpam-4761	429	49	,	,	PUNCT
ejpam-4761	429	50	pk	pk	NOUN
ejpam-4761	429	51	)	)	PUNCT
ejpam-4761	429	52	,	,	PUNCT
ejpam-4761	429	53	such	such	ADJ
ejpam-4761	429	54	that	that	DET
ejpam-4761	429	55	v	v	NOUN
ejpam-4761	429	56	(	(	PUNCT
ejpam-4761	429	57	p	p	X
ejpam-4761	429	58	(	(	PUNCT
ejpam-4761	429	59	(	(	PUNCT
ejpam-4761	429	60	v	v	NOUN
ejpam-4761	429	61	,	,	PUNCT
ejpam-4761	429	62	p	p	NOUN
ejpam-4761	429	63	)	)	PUNCT
ejpam-4761	429	64	,	,	PUNCT
ejpam-4761	429	65	(	(	PUNCT
ejpam-4761	429	66	w	w	NOUN
ejpam-4761	429	67	,	,	PUNCT
ejpam-4761	429	68	q	q	NOUN
ejpam-4761	429	69	)	)	PUNCT
ejpam-4761	429	70	)	)	PUNCT
ejpam-4761	429	71	)	)	PUNCT
ejpam-4761	430	1	⊆	⊆	NUM
ejpam-4761	430	2	c.	c.	NOUN
ejpam-4761	430	3	this	this	PRON
ejpam-4761	430	4	implies	imply	VERB
ejpam-4761	430	5	that	that	SCONJ
ejpam-4761	431	1	p	p	PROPN
ejpam-4761	431	2	(	(	PUNCT
ejpam-4761	431	3	v	v	NOUN
ejpam-4761	431	4	,	,	PUNCT
ejpam-4761	431	5	w	w	NOUN
ejpam-4761	431	6	)	)	PUNCT
ejpam-4761	431	7	=	=	PUNCT
ejpam-4761	432	1	[	[	X
ejpam-4761	432	2	v1	v1	NOUN
ejpam-4761	432	3	,	,	PUNCT
ejpam-4761	432	4	v2	v2	PROPN
ejpam-4761	432	5	,	,	PUNCT
ejpam-4761	432	6	·	·	PUNCT
ejpam-4761	432	7	·	·	PUNCT
ejpam-4761	432	8	·	·	PUNCT
ejpam-4761	432	9	,	,	PUNCT
ejpam-4761	432	10	vk	vk	X
ejpam-4761	432	11	]	]	PUNCT
ejpam-4761	432	12	is	be	AUX
ejpam-4761	432	13	a	a	DET
ejpam-4761	432	14	v	v	NOUN
ejpam-4761	432	15	-	-	PUNCT
ejpam-4761	432	16	w	w	NOUN
ejpam-4761	432	17	geodesic	geodesic	NOUN
ejpam-4761	432	18	in	in	ADP
ejpam-4761	432	19	g	g	PROPN
ejpam-4761	432	20	and	and	CCONJ
ejpam-4761	432	21	v	v	NOUN
ejpam-4761	432	22	(	(	PUNCT
ejpam-4761	432	23	p	p	X
ejpam-4761	432	24	(	(	PUNCT
ejpam-4761	432	25	v	v	NOUN
ejpam-4761	432	26	,	,	PUNCT
ejpam-4761	432	27	w	w	NOUN
ejpam-4761	432	28	)	)	PUNCT
ejpam-4761	432	29	)	)	PUNCT
ejpam-4761	433	1	⊆	⊆	NUM
ejpam-4761	433	2	s.	s.	PROPN
ejpam-4761	433	3	this	this	PRON
ejpam-4761	433	4	shows	show	VERB
ejpam-4761	433	5	that	that	SCONJ
ejpam-4761	433	6	s	s	VERB
ejpam-4761	433	7	is	be	AUX
ejpam-4761	433	8	weakly	weakly	ADJ
ejpam-4761	433	9	convex	convex	NOUN
ejpam-4761	433	10	,	,	PUNCT
ejpam-4761	433	11	showing	show	VERB
ejpam-4761	433	12	that	that	SCONJ
ejpam-4761	433	13	(	(	PUNCT
ejpam-4761	433	14	i	i	NOUN
ejpam-4761	433	15	)	)	PUNCT
ejpam-4761	433	16	holds	hold	VERB
ejpam-4761	433	17	.	.	PUNCT
ejpam-4761	434	1	s.	s.	PROPN
ejpam-4761	434	2	canoy	canoy	PROPN
ejpam-4761	434	3	jr	jr	PROPN
ejpam-4761	434	4	.	.	PROPN
ejpam-4761	434	5	,	,	PUNCT
ejpam-4761	434	6	j.	j.	PROPN
ejpam-4761	434	7	hassan	hassan	PROPN
ejpam-4761	434	8	/	/	SYM
ejpam-4761	434	9	eur	eur	PROPN
ejpam-4761	434	10	.	.	PUNCT
ejpam-4761	435	1	j.	j.	PROPN
ejpam-4761	435	2	pure	pure	PROPN
ejpam-4761	435	3	appl	appl	PROPN
ejpam-4761	435	4	.	.	PROPN
ejpam-4761	435	5	math	math	PROPN
ejpam-4761	435	6	,	,	PUNCT
ejpam-4761	435	7	16	16	NUM
ejpam-4761	435	8	(	(	PUNCT
ejpam-4761	435	9	2	2	NUM
ejpam-4761	435	10	)	)	PUNCT
ejpam-4761	435	11	(	(	PUNCT
ejpam-4761	435	12	2023	2023	NUM
ejpam-4761	435	13	)	)	PUNCT
ejpam-4761	435	14	,	,	PUNCT
ejpam-4761	435	15	1196	1196	NUM
ejpam-4761	435	16	-	-	SYM
ejpam-4761	435	17	1211	1211	NUM
ejpam-4761	435	18	1208	1208	NUM
ejpam-4761	435	19	for	for	ADP
ejpam-4761	435	20	the	the	DET
ejpam-4761	435	21	converse	converse	NOUN
ejpam-4761	435	22	,	,	PUNCT
ejpam-4761	435	23	suppose	suppose	VERB
ejpam-4761	435	24	that	that	SCONJ
ejpam-4761	435	25	c	c	PROPN
ejpam-4761	435	26	satisfies	satisfy	VERB
ejpam-4761	435	27	conditions	condition	NOUN
ejpam-4761	435	28	(	(	PUNCT
ejpam-4761	435	29	i	i	NOUN
ejpam-4761	435	30	)	)	PUNCT
ejpam-4761	435	31	and	and	CCONJ
ejpam-4761	435	32	(	(	PUNCT
ejpam-4761	435	33	ii	ii	NOUN
ejpam-4761	435	34	)	)	PUNCT
ejpam-4761	435	35	.	.	PUNCT
ejpam-4761	436	1	then	then	ADV
ejpam-4761	436	2	c	c	PROPN
ejpam-4761	436	3	is	be	AUX
ejpam-4761	436	4	a	a	DET
ejpam-4761	436	5	hop	hop	NOUN
ejpam-4761	436	6	dominating	dominating	NOUN
ejpam-4761	436	7	set	set	NOUN
ejpam-4761	436	8	g[h	g[h	NOUN
ejpam-4761	436	9	]	]	PUNCT
ejpam-4761	436	10	by	by	ADP
ejpam-4761	436	11	theorem	theorem	NOUN
ejpam-4761	436	12	12	12	NUM
ejpam-4761	436	13	.	.	PUNCT
ejpam-4761	437	1	let	let	VERB
ejpam-4761	437	2	(	(	PUNCT
ejpam-4761	437	3	x	x	X
ejpam-4761	437	4	,	,	PUNCT
ejpam-4761	437	5	a	a	PRON
ejpam-4761	437	6	)	)	PUNCT
ejpam-4761	437	7	,	,	PUNCT
ejpam-4761	437	8	(	(	PUNCT
ejpam-4761	437	9	y	y	PROPN
ejpam-4761	437	10	,	,	PUNCT
ejpam-4761	437	11	b	b	NOUN
ejpam-4761	437	12	)	)	PUNCT
ejpam-4761	437	13	∈	∈	PROPN
ejpam-4761	437	14	c	c	NOUN
ejpam-4761	437	15	with	with	ADP
ejpam-4761	437	16	(	(	PUNCT
ejpam-4761	437	17	x	x	NOUN
ejpam-4761	437	18	,	,	PUNCT
ejpam-4761	437	19	a	a	PRON
ejpam-4761	437	20	)	)	PUNCT
ejpam-4761	437	21	̸=	̸=	PROPN
ejpam-4761	437	22	(	(	PUNCT
ejpam-4761	437	23	y	y	PROPN
ejpam-4761	437	24	,	,	PUNCT
ejpam-4761	437	25	b	b	NOUN
ejpam-4761	437	26	)	)	PUNCT
ejpam-4761	437	27	.	.	PUNCT
ejpam-4761	438	1	consider	consider	VERB
ejpam-4761	438	2	the	the	DET
ejpam-4761	438	3	following	follow	VERB
ejpam-4761	438	4	cases	case	NOUN
ejpam-4761	438	5	:	:	PUNCT
ejpam-4761	438	6	case	case	NOUN
ejpam-4761	438	7	1	1	NUM
ejpam-4761	438	8	.	.	PUNCT
ejpam-4761	439	1	x	x	X
ejpam-4761	439	2	̸=	̸=	PROPN
ejpam-4761	439	3	y.	y.	NOUN
ejpam-4761	439	4	since	since	SCONJ
ejpam-4761	439	5	s	s	PROPN
ejpam-4761	439	6	is	be	AUX
ejpam-4761	439	7	weakly	weakly	ADJ
ejpam-4761	439	8	convex	convex	NOUN
ejpam-4761	439	9	in	in	ADP
ejpam-4761	439	10	g	g	NOUN
ejpam-4761	439	11	,	,	PUNCT
ejpam-4761	439	12	there	there	PRON
ejpam-4761	439	13	exists	exist	VERB
ejpam-4761	439	14	an	an	DET
ejpam-4761	439	15	x	x	NOUN
ejpam-4761	439	16	-	-	NOUN
ejpam-4761	439	17	y	y	ADJ
ejpam-4761	439	18	geodesic	geodesic	NOUN
ejpam-4761	439	19	p	p	X
ejpam-4761	439	20	(	(	PUNCT
ejpam-4761	439	21	x	x	NOUN
ejpam-4761	439	22	,	,	PUNCT
ejpam-4761	439	23	y	y	NOUN
ejpam-4761	439	24	)	)	PUNCT
ejpam-4761	439	25	=	=	PUNCT
ejpam-4761	440	1	[	[	X
ejpam-4761	440	2	x1	x1	PROPN
ejpam-4761	440	3	,	,	PUNCT
ejpam-4761	440	4	x2	x2	PROPN
ejpam-4761	440	5	,	,	PUNCT
ejpam-4761	440	6	·	·	PUNCT
ejpam-4761	440	7	·	·	PUNCT
ejpam-4761	440	8	·	·	PUNCT
ejpam-4761	440	9	,	,	PUNCT
ejpam-4761	440	10	xt	xt	ADP
ejpam-4761	440	11	]	]	X
ejpam-4761	440	12	,	,	PUNCT
ejpam-4761	440	13	where	where	SCONJ
ejpam-4761	440	14	x	x	ADP
ejpam-4761	440	15	=	=	SYM
ejpam-4761	440	16	x1	x1	PROPN
ejpam-4761	440	17	and	and	CCONJ
ejpam-4761	440	18	y	y	PROPN
ejpam-4761	440	19	=	=	SYM
ejpam-4761	440	20	xt	xt	PROPN
ejpam-4761	440	21	,	,	PUNCT
ejpam-4761	440	22	such	such	ADJ
ejpam-4761	440	23	that	that	DET
ejpam-4761	440	24	v	v	NOUN
ejpam-4761	440	25	(	(	PUNCT
ejpam-4761	440	26	p	p	X
ejpam-4761	440	27	(	(	PUNCT
ejpam-4761	440	28	x	x	NOUN
ejpam-4761	440	29	,	,	PUNCT
ejpam-4761	440	30	y	y	NOUN
ejpam-4761	440	31	)	)	PUNCT
ejpam-4761	440	32	)	)	PUNCT
ejpam-4761	440	33	⊆	⊆	NUM
ejpam-4761	440	34	s.	s.	PROPN
ejpam-4761	440	35	let	let	VERB
ejpam-4761	440	36	a1	a1	NOUN
ejpam-4761	440	37	=	=	PUNCT
ejpam-4761	440	38	a	a	NOUN
ejpam-4761	440	39	and	and	CCONJ
ejpam-4761	440	40	at	at	ADP
ejpam-4761	440	41	=	=	PROPN
ejpam-4761	440	42	b.	b.	PROPN
ejpam-4761	440	43	for	for	ADP
ejpam-4761	440	44	each	each	DET
ejpam-4761	440	45	i	i	PRON
ejpam-4761	440	46	∈	∈	PROPN
ejpam-4761	440	47	{	{	PUNCT
ejpam-4761	440	48	2	2	NUM
ejpam-4761	440	49	,	,	PUNCT
ejpam-4761	440	50	.	.	PUNCT
ejpam-4761	440	51	.	.	PUNCT
ejpam-4761	441	1	.	.	PUNCT
ejpam-4761	442	1	,	,	PUNCT
ejpam-4761	442	2	t−	t−	PROPN
ejpam-4761	442	3	1	1	NUM
ejpam-4761	442	4	}	}	PUNCT
ejpam-4761	442	5	,	,	PUNCT
ejpam-4761	442	6	choose	choose	VERB
ejpam-4761	442	7	any	any	DET
ejpam-4761	442	8	ai	ai	PROPN
ejpam-4761	442	9	∈	∈	PROPN
ejpam-4761	442	10	txi	txi	PROPN
ejpam-4761	442	11	.	.	PUNCT
ejpam-4761	443	1	then	then	ADV
ejpam-4761	443	2	p	p	X
ejpam-4761	443	3	(	(	PUNCT
ejpam-4761	443	4	(	(	PUNCT
ejpam-4761	443	5	x	x	NOUN
ejpam-4761	443	6	,	,	PUNCT
ejpam-4761	443	7	a	a	PRON
ejpam-4761	443	8	)	)	PUNCT
ejpam-4761	443	9	,	,	PUNCT
ejpam-4761	443	10	(	(	PUNCT
ejpam-4761	443	11	y	y	PROPN
ejpam-4761	443	12	,	,	PUNCT
ejpam-4761	443	13	b	b	NOUN
ejpam-4761	443	14	)	)	PUNCT
ejpam-4761	443	15	)	)	PUNCT
ejpam-4761	444	1	=	=	PUNCT
ejpam-4761	445	1	[	[	X
ejpam-4761	445	2	(	(	PUNCT
ejpam-4761	445	3	x1	x1	ADJ
ejpam-4761	445	4	,	,	PUNCT
ejpam-4761	445	5	a1	a1	NOUN
ejpam-4761	445	6	)	)	PUNCT
ejpam-4761	445	7	,	,	PUNCT
ejpam-4761	445	8	(	(	PUNCT
ejpam-4761	445	9	x2	x2	PROPN
ejpam-4761	445	10	,	,	PUNCT
ejpam-4761	445	11	a2	a2	PROPN
ejpam-4761	445	12	)	)	PUNCT
ejpam-4761	445	13	,	,	PUNCT
ejpam-4761	445	14	·	·	PUNCT
ejpam-4761	445	15	·	·	PUNCT
ejpam-4761	445	16	·	·	PUNCT
ejpam-4761	445	17	,	,	PUNCT
ejpam-4761	445	18	(	(	PUNCT
ejpam-4761	445	19	xt	xt	ADP
ejpam-4761	445	20	,	,	PUNCT
ejpam-4761	445	21	at	at	ADP
ejpam-4761	445	22	)	)	PUNCT
ejpam-4761	445	23	]	]	PUNCT
ejpam-4761	445	24	is	be	AUX
ejpam-4761	445	25	an	an	DET
ejpam-4761	445	26	(	(	PUNCT
ejpam-4761	445	27	x	x	NOUN
ejpam-4761	445	28	,	,	PUNCT
ejpam-4761	445	29	a)-(y	a)-(y	PROPN
ejpam-4761	445	30	,	,	PUNCT
ejpam-4761	445	31	b	b	NOUN
ejpam-4761	445	32	)	)	PUNCT
ejpam-4761	445	33	geodesic	geodesic	NOUN
ejpam-4761	445	34	in	in	ADP
ejpam-4761	445	35	g[h	g[h	PROPN
ejpam-4761	445	36	]	]	PUNCT
ejpam-4761	445	37	and	and	CCONJ
ejpam-4761	445	38	v	v	X
ejpam-4761	445	39	(	(	PUNCT
ejpam-4761	445	40	p	p	X
ejpam-4761	445	41	(	(	PUNCT
ejpam-4761	445	42	(	(	PUNCT
ejpam-4761	445	43	x	x	NOUN
ejpam-4761	445	44	,	,	PUNCT
ejpam-4761	445	45	a	a	PRON
ejpam-4761	445	46	)	)	PUNCT
ejpam-4761	445	47	,	,	PUNCT
ejpam-4761	445	48	(	(	PUNCT
ejpam-4761	445	49	y	y	PROPN
ejpam-4761	445	50	,	,	PUNCT
ejpam-4761	445	51	b	b	NOUN
ejpam-4761	445	52	)	)	PUNCT
ejpam-4761	445	53	)	)	PUNCT
ejpam-4761	445	54	)	)	PUNCT
ejpam-4761	446	1	⊆	⊆	NUM
ejpam-4761	446	2	c.	c.	NOUN
ejpam-4761	446	3	case	case	NOUN
ejpam-4761	446	4	2	2	NUM
ejpam-4761	446	5	.	.	PUNCT
ejpam-4761	446	6	x	x	X
ejpam-4761	447	1	=	=	PUNCT
ejpam-4761	447	2	y.	y.	NOUN
ejpam-4761	447	3	then	then	ADV
ejpam-4761	447	4	a	a	DET
ejpam-4761	447	5	̸=	̸=	PROPN
ejpam-4761	447	6	b	b	PROPN
ejpam-4761	447	7	and	and	CCONJ
ejpam-4761	447	8	dg[h]((x	dg[h]((x	NOUN
ejpam-4761	447	9	,	,	PUNCT
ejpam-4761	447	10	a	a	PRON
ejpam-4761	447	11	)	)	PUNCT
ejpam-4761	447	12	,	,	PUNCT
ejpam-4761	447	13	(	(	PUNCT
ejpam-4761	447	14	y	y	PROPN
ejpam-4761	447	15	,	,	PUNCT
ejpam-4761	447	16	b	b	NOUN
ejpam-4761	447	17	)	)	PUNCT
ejpam-4761	447	18	)	)	PUNCT
ejpam-4761	448	1	=	=	SYM
ejpam-4761	448	2	2	2	X
ejpam-4761	448	3	.	.	PUNCT
ejpam-4761	448	4	since	since	SCONJ
ejpam-4761	448	5	⟨s⟩	⟨s⟩	PROPN
ejpam-4761	448	6	is	be	AUX
ejpam-4761	448	7	non	non	ADJ
ejpam-4761	448	8	-	-	ADJ
ejpam-4761	448	9	trivial	trivial	ADJ
ejpam-4761	448	10	and	and	CCONJ
ejpam-4761	448	11	connected	connect	VERB
ejpam-4761	448	12	,	,	PUNCT
ejpam-4761	448	13	choose	choose	VERB
ejpam-4761	448	14	any	any	DET
ejpam-4761	448	15	z	z	NOUN
ejpam-4761	448	16	∈	∈	PROPN
ejpam-4761	448	17	s∩ng(x	s∩ng(x	PROPN
ejpam-4761	448	18	)	)	PUNCT
ejpam-4761	448	19	and	and	CCONJ
ejpam-4761	448	20	let	let	VERB
ejpam-4761	448	21	c	c	PROPN
ejpam-4761	448	22	∈	∈	PROPN
ejpam-4761	448	23	tz	tz	PROPN
ejpam-4761	448	24	.	.	PUNCT
ejpam-4761	449	1	then	then	ADV
ejpam-4761	449	2	p	p	X
ejpam-4761	449	3	(	(	PUNCT
ejpam-4761	449	4	(	(	PUNCT
ejpam-4761	449	5	x	x	NOUN
ejpam-4761	449	6	,	,	PUNCT
ejpam-4761	449	7	a	a	PRON
ejpam-4761	449	8	)	)	PUNCT
ejpam-4761	449	9	,	,	PUNCT
ejpam-4761	449	10	(	(	PUNCT
ejpam-4761	449	11	y	y	PROPN
ejpam-4761	449	12	,	,	PUNCT
ejpam-4761	449	13	b	b	NOUN
ejpam-4761	449	14	)	)	PUNCT
ejpam-4761	449	15	)	)	PUNCT
ejpam-4761	450	1	=	=	PUNCT
ejpam-4761	451	1	[	[	X
ejpam-4761	451	2	(	(	PUNCT
ejpam-4761	451	3	x	x	NOUN
ejpam-4761	451	4	,	,	PUNCT
ejpam-4761	451	5	a	a	PRON
ejpam-4761	451	6	)	)	PUNCT
ejpam-4761	451	7	,	,	PUNCT
ejpam-4761	451	8	(	(	PUNCT
ejpam-4761	451	9	z	z	X
ejpam-4761	451	10	,	,	PUNCT
ejpam-4761	451	11	c	c	NOUN
ejpam-4761	451	12	)	)	PUNCT
ejpam-4761	451	13	,	,	PUNCT
ejpam-4761	451	14	(	(	PUNCT
ejpam-4761	451	15	y	y	PROPN
ejpam-4761	451	16	,	,	PUNCT
ejpam-4761	451	17	b	b	NOUN
ejpam-4761	451	18	)	)	PUNCT
ejpam-4761	451	19	]	]	PUNCT
ejpam-4761	451	20	is	be	AUX
ejpam-4761	451	21	an	an	DET
ejpam-4761	451	22	(	(	PUNCT
ejpam-4761	451	23	x	x	NOUN
ejpam-4761	451	24	,	,	PUNCT
ejpam-4761	451	25	a)-(y	a)-(y	PROPN
ejpam-4761	451	26	,	,	PUNCT
ejpam-4761	451	27	b	b	NOUN
ejpam-4761	451	28	)	)	PUNCT
ejpam-4761	451	29	geodesic	geodesic	NOUN
ejpam-4761	451	30	in	in	ADP
ejpam-4761	451	31	g[h	g[h	PROPN
ejpam-4761	451	32	]	]	PUNCT
ejpam-4761	451	33	and	and	CCONJ
ejpam-4761	451	34	v	v	X
ejpam-4761	451	35	(	(	PUNCT
ejpam-4761	451	36	p	p	X
ejpam-4761	451	37	(	(	PUNCT
ejpam-4761	451	38	(	(	PUNCT
ejpam-4761	451	39	x	x	NOUN
ejpam-4761	451	40	,	,	PUNCT
ejpam-4761	451	41	a	a	PRON
ejpam-4761	451	42	)	)	PUNCT
ejpam-4761	451	43	,	,	PUNCT
ejpam-4761	451	44	(	(	PUNCT
ejpam-4761	451	45	y	y	PROPN
ejpam-4761	451	46	,	,	PUNCT
ejpam-4761	451	47	b	b	NOUN
ejpam-4761	451	48	)	)	PUNCT
ejpam-4761	451	49	)	)	PUNCT
ejpam-4761	451	50	)	)	PUNCT
ejpam-4761	452	1	⊆	⊆	NUM
ejpam-4761	452	2	c.	c.	PROPN
ejpam-4761	452	3	therefore	therefore	ADV
ejpam-4761	452	4	,	,	PUNCT
ejpam-4761	452	5	c	c	PROPN
ejpam-4761	452	6	is	be	AUX
ejpam-4761	452	7	weakly	weakly	ADJ
ejpam-4761	452	8	convex	convex	NOUN
ejpam-4761	452	9	in	in	ADP
ejpam-4761	452	10	g[h	g[h	PROPN
ejpam-4761	452	11	]	]	PUNCT
ejpam-4761	452	12	.	.	PUNCT
ejpam-4761	453	1	the	the	DET
ejpam-4761	453	2	next	next	ADJ
ejpam-4761	453	3	result	result	NOUN
ejpam-4761	453	4	follows	follow	VERB
ejpam-4761	453	5	from	from	ADP
ejpam-4761	453	6	theorem	theorem	ADJ
ejpam-4761	453	7	13	13	NUM
ejpam-4761	453	8	.	.	PUNCT
ejpam-4761	454	1	corollary	corollary	ADJ
ejpam-4761	454	2	9	9	NUM
ejpam-4761	454	3	.	.	PUNCT
ejpam-4761	455	1	let	let	VERB
ejpam-4761	455	2	g	g	NOUN
ejpam-4761	455	3	and	and	CCONJ
ejpam-4761	455	4	h	h	PROPN
ejpam-4761	455	5	be	be	VERB
ejpam-4761	455	6	non	non	ADJ
ejpam-4761	455	7	-	-	ADJ
ejpam-4761	455	8	trivial	trivial	ADJ
ejpam-4761	455	9	connected	connected	ADJ
ejpam-4761	455	10	graphs	graph	NOUN
ejpam-4761	455	11	.	.	PUNCT
ejpam-4761	456	1	then	then	ADV
ejpam-4761	456	2	γwconh(g[h	γwconh(g[h	ADV
ejpam-4761	456	3	]	]	X
ejpam-4761	456	4	)	)	PUNCT
ejpam-4761	457	1	=	=	SYM
ejpam-4761	457	2	min{|s∩n2	min{|s∩n2	NOUN
ejpam-4761	457	3	g(s)|+pnd(h)|s\n2	g(s)|+pnd(h)|s\n2	PUNCT
ejpam-4761	458	1	g(s)|	g(s)|	PROPN
ejpam-4761	458	2	:	:	PUNCT
ejpam-4761	458	3	s	s	AUX
ejpam-4761	458	4	weakly	weakly	ADJ
ejpam-4761	458	5	convex	convex	ADJ
ejpam-4761	458	6	hop	hop	NOUN
ejpam-4761	458	7	dominating	dominating	NOUN
ejpam-4761	458	8	in	in	ADP
ejpam-4761	458	9	g	g	NOUN
ejpam-4761	458	10	}	}	PUNCT
ejpam-4761	458	11	.	.	PUNCT
ejpam-4761	459	1	moreover	moreover	ADV
ejpam-4761	459	2	,	,	PUNCT
ejpam-4761	459	3	each	each	PRON
ejpam-4761	459	4	of	of	ADP
ejpam-4761	459	5	the	the	DET
ejpam-4761	459	6	following	follow	VERB
ejpam-4761	459	7	holds	hold	VERB
ejpam-4761	459	8	:	:	PUNCT
ejpam-4761	459	9	(	(	PUNCT
ejpam-4761	459	10	i	i	NOUN
ejpam-4761	459	11	)	)	PUNCT
ejpam-4761	459	12	if	if	SCONJ
ejpam-4761	459	13	γ(g	γ(g	PROPN
ejpam-4761	459	14	)	)	PUNCT
ejpam-4761	459	15	̸=	̸=	PROPN
ejpam-4761	459	16	1	1	NUM
ejpam-4761	459	17	,	,	PUNCT
ejpam-4761	459	18	then	then	ADV
ejpam-4761	459	19	γwconh(g[h	γwconh(g[h	ADJ
ejpam-4761	459	20	]	]	X
ejpam-4761	459	21	)	)	PUNCT
ejpam-4761	459	22	≤	≤	NUM
ejpam-4761	459	23	γwconth(g	γwconth(g	NOUN
ejpam-4761	459	24	)	)	PUNCT
ejpam-4761	459	25	.	.	PUNCT
ejpam-4761	460	1	(	(	PUNCT
ejpam-4761	460	2	ii	ii	NOUN
ejpam-4761	460	3	)	)	PUNCT
ejpam-4761	460	4	if	if	SCONJ
ejpam-4761	460	5	g	g	PROPN
ejpam-4761	460	6	=	=	SYM
ejpam-4761	460	7	kn	kn	PROPN
ejpam-4761	460	8	,	,	PUNCT
ejpam-4761	460	9	then	then	ADV
ejpam-4761	460	10	γwconh(g[h	γwconh(g[h	ADJ
ejpam-4761	460	11	]	]	X
ejpam-4761	460	12	)	)	PUNCT
ejpam-4761	460	13	=	=	SYM
ejpam-4761	460	14	n	n	PROPN
ejpam-4761	460	15	·	·	PUNCT
ejpam-4761	460	16	pnd(h	pnd(h	NUM
ejpam-4761	460	17	)	)	PUNCT
ejpam-4761	460	18	.	.	PUNCT
ejpam-4761	461	1	proof	proof	NOUN
ejpam-4761	461	2	.	.	PUNCT
ejpam-4761	462	1	let	let	VERB
ejpam-4761	462	2	α	α	NOUN
ejpam-4761	462	3	=	=	SYM
ejpam-4761	462	4	min{|s	min{|s	PROPN
ejpam-4761	462	5	∩	∩	NOUN
ejpam-4761	462	6	n2	n2	PROPN
ejpam-4761	462	7	g(s)|	g(s)|	PROPN
ejpam-4761	462	8	+	+	CCONJ
ejpam-4761	462	9	pnd(h)|s	pnd(h)|s	NOUN
ejpam-4761	462	10	\	\	PROPN
ejpam-4761	462	11	n2	n2	PROPN
ejpam-4761	462	12	g(s)|	g(s)|	PROPN
ejpam-4761	462	13	:	:	PUNCT
ejpam-4761	462	14	s	s	VERB
ejpam-4761	462	15	is	be	AUX
ejpam-4761	462	16	a	a	DET
ejpam-4761	462	17	weakly	weakly	ADJ
ejpam-4761	462	18	convex	convex	NOUN
ejpam-4761	462	19	hop	hop	NOUN
ejpam-4761	462	20	dominating	dominating	NOUN
ejpam-4761	462	21	set	set	VERB
ejpam-4761	462	22	in	in	ADP
ejpam-4761	462	23	g	g	NOUN
ejpam-4761	462	24	}	}	PUNCT
ejpam-4761	462	25	.	.	PUNCT
ejpam-4761	463	1	let	let	VERB
ejpam-4761	463	2	s0	s0	PROPN
ejpam-4761	463	3	be	be	AUX
ejpam-4761	463	4	a	a	DET
ejpam-4761	463	5	weakly	weakly	ADJ
ejpam-4761	463	6	convex	convex	NOUN
ejpam-4761	463	7	hop	hop	NOUN
ejpam-4761	463	8	dominating	dominating	NOUN
ejpam-4761	463	9	set	set	VERB
ejpam-4761	463	10	in	in	ADP
ejpam-4761	463	11	g	g	PROPN
ejpam-4761	463	12	such	such	ADJ
ejpam-4761	463	13	that	that	SCONJ
ejpam-4761	463	14	α	α	PROPN
ejpam-4761	463	15	=	=	SYM
ejpam-4761	463	16	|s	|s	PROPN
ejpam-4761	463	17	∩	∩	PROPN
ejpam-4761	463	18	n2	n2	PROPN
ejpam-4761	463	19	g(s)|	g(s)|	PROPN
ejpam-4761	463	20	+	+	CCONJ
ejpam-4761	463	21	pnd(h)|s	pnd(h)|s	NOUN
ejpam-4761	463	22	\	\	PROPN
ejpam-4761	463	23	n2	n2	PROPN
ejpam-4761	463	24	g(s)|	g(s)|	PROPN
ejpam-4761	463	25	.	.	PUNCT
ejpam-4761	463	26	choose	choose	VERB
ejpam-4761	463	27	any	any	DET
ejpam-4761	463	28	a	a	DET
ejpam-4761	463	29	∈	∈	PROPN
ejpam-4761	463	30	v	v	NOUN
ejpam-4761	463	31	(	(	PUNCT
ejpam-4761	463	32	h	h	NOUN
ejpam-4761	463	33	)	)	PUNCT
ejpam-4761	463	34	and	and	CCONJ
ejpam-4761	463	35	let	let	VERB
ejpam-4761	463	36	d	d	PRON
ejpam-4761	463	37	be	be	AUX
ejpam-4761	463	38	a	a	DET
ejpam-4761	463	39	pndset	pndset	NOUN
ejpam-4761	463	40	in	in	ADP
ejpam-4761	463	41	h.	h.	PROPN
ejpam-4761	463	42	set	set	VERB
ejpam-4761	463	43	tx	tx	PROPN
ejpam-4761	463	44	=	=	PUNCT
ejpam-4761	463	45	{	{	PUNCT
ejpam-4761	463	46	a	a	NOUN
ejpam-4761	463	47	}	}	PUNCT
ejpam-4761	463	48	if	if	SCONJ
ejpam-4761	463	49	x	x	PROPN
ejpam-4761	463	50	∈	∈	PROPN
ejpam-4761	463	51	s0	s0	PROPN
ejpam-4761	463	52	∩	∩	PROPN
ejpam-4761	463	53	n2	n2	PROPN
ejpam-4761	463	54	g(s0	g(s0	NOUN
ejpam-4761	463	55	)	)	PUNCT
ejpam-4761	463	56	and	and	CCONJ
ejpam-4761	463	57	tx	tx	X
ejpam-4761	463	58	=	=	PUNCT
ejpam-4761	464	1	d	d	NOUN
ejpam-4761	464	2	if	if	SCONJ
ejpam-4761	464	3	x	x	X
ejpam-4761	464	4	∈	∈	PROPN
ejpam-4761	464	5	s0	s0	PROPN
ejpam-4761	464	6	\	\	PROPN
ejpam-4761	464	7	n2	n2	ADJ
ejpam-4761	464	8	g(s0	g(s0	NOUN
ejpam-4761	464	9	)	)	PUNCT
ejpam-4761	464	10	.	.	PUNCT
ejpam-4761	465	1	then	then	ADV
ejpam-4761	465	2	c	c	X
ejpam-4761	465	3	=	=	PUNCT
ejpam-4761	466	1	⋃	⋃	NOUN
ejpam-4761	466	2	x∈s0	x∈s0	NOUN
ejpam-4761	467	1	[	[	X
ejpam-4761	467	2	{	{	PUNCT
ejpam-4761	467	3	x	x	NOUN
ejpam-4761	467	4	}	}	PUNCT
ejpam-4761	467	5	×	×	PROPN
ejpam-4761	467	6	tx	tx	PROPN
ejpam-4761	467	7	]	]	PUNCT
ejpam-4761	467	8	is	be	AUX
ejpam-4761	467	9	a	a	DET
ejpam-4761	467	10	weakly	weakly	ADJ
ejpam-4761	467	11	convex	convex	NOUN
ejpam-4761	467	12	hop	hop	NOUN
ejpam-4761	467	13	dominating	dominating	NOUN
ejpam-4761	467	14	set	set	VERB
ejpam-4761	467	15	in	in	ADP
ejpam-4761	467	16	g[h	g[h	PROPN
ejpam-4761	467	17	]	]	PUNCT
ejpam-4761	467	18	by	by	ADP
ejpam-4761	467	19	theorem	theorem	NOUN
ejpam-4761	467	20	13	13	NUM
ejpam-4761	467	21	.	.	PUNCT
ejpam-4761	468	1	hence	hence	ADV
ejpam-4761	468	2	,	,	PUNCT
ejpam-4761	468	3	γwconh(g[h	γwconh(g[h	PROPN
ejpam-4761	468	4	]	]	X
ejpam-4761	468	5	)	)	PUNCT
ejpam-4761	468	6	≤	≤	NUM
ejpam-4761	468	7	|c|	|c|	PROPN
ejpam-4761	468	8	=	=	SYM
ejpam-4761	468	9	|s	|s	PROPN
ejpam-4761	468	10	∩ng(s)|+	∩ng(s)|+	NOUN
ejpam-4761	468	11	pnd(h)|s	pnd(h)|s	NOUN
ejpam-4761	468	12	\n2	\n2	VERB
ejpam-4761	468	13	g(s)|	g(s)|	PROPN
ejpam-4761	468	14	=	=	NOUN
ejpam-4761	468	15	α	α	PROPN
ejpam-4761	468	16	.	.	PUNCT
ejpam-4761	469	1	next	next	ADV
ejpam-4761	469	2	,	,	PUNCT
ejpam-4761	469	3	suppose	suppose	VERB
ejpam-4761	469	4	that	that	SCONJ
ejpam-4761	469	5	c0	c0	PROPN
ejpam-4761	469	6	=	=	PUNCT
ejpam-4761	469	7	⋃	⋃	ADP
ejpam-4761	469	8	x∈a[{x	x∈a[{x	PROPN
ejpam-4761	469	9	}	}	PUNCT
ejpam-4761	469	10	×	×	NOUN
ejpam-4761	469	11	rx	rx	NOUN
ejpam-4761	469	12	]	]	X
ejpam-4761	469	13	is	be	AUX
ejpam-4761	469	14	a	a	DET
ejpam-4761	469	15	γwconh	γwconh	NOUN
ejpam-4761	469	16	-	-	PUNCT
ejpam-4761	469	17	set	set	NOUN
ejpam-4761	469	18	in	in	ADP
ejpam-4761	469	19	g[h	g[h	NOUN
ejpam-4761	469	20	]	]	PUNCT
ejpam-4761	469	21	.	.	PUNCT
ejpam-4761	470	1	by	by	ADP
ejpam-4761	470	2	theoren	theoren	PROPN
ejpam-4761	470	3	13	13	NUM
ejpam-4761	470	4	,	,	PUNCT
ejpam-4761	470	5	a	a	PRON
ejpam-4761	470	6	is	be	AUX
ejpam-4761	470	7	weakly	weakly	ADV
ejpam-4761	470	8	convex	convex	ADJ
ejpam-4761	470	9	hop	hop	NOUN
ejpam-4761	470	10	dominating	dominating	NOUN
ejpam-4761	470	11	in	in	ADP
ejpam-4761	470	12	g	g	PROPN
ejpam-4761	470	13	and	and	CCONJ
ejpam-4761	470	14	rx	rx	VERB
ejpam-4761	470	15	is	be	AUX
ejpam-4761	470	16	pointwise	pointwise	PROPN
ejpam-4761	470	17	non	non	ADJ
ejpam-4761	470	18	-	-	ADJ
ejpam-4761	470	19	dominating	dominating	NOUN
ejpam-4761	470	20	in	in	ADP
ejpam-4761	470	21	h	h	NOUN
ejpam-4761	470	22	for	for	ADP
ejpam-4761	470	23	each	each	DET
ejpam-4761	470	24	x	x	SYM
ejpam-4761	470	25	∈	∈	PROPN
ejpam-4761	470	26	a	a	DET
ejpam-4761	470	27	\	\	PROPN
ejpam-4761	470	28	n2	n2	PROPN
ejpam-4761	470	29	g(a	g(a	PROPN
ejpam-4761	470	30	)	)	PUNCT
ejpam-4761	470	31	.	.	PUNCT
ejpam-4761	471	1	it	it	PRON
ejpam-4761	471	2	follows	follow	VERB
ejpam-4761	471	3	that	that	SCONJ
ejpam-4761	471	4	γwconh(g[h	γwconh(g[h	PROPN
ejpam-4761	471	5	]	]	X
ejpam-4761	471	6	)	)	PUNCT
ejpam-4761	471	7	=	=	SYM
ejpam-4761	471	8	|c0|	|c0|	NOUN
ejpam-4761	471	9	≥	≥	NOUN
ejpam-4761	471	10	|a	|a	VERB
ejpam-4761	471	11	∩n2	∩n2	PROPN
ejpam-4761	471	12	g(a)|	g(a)|	NOUN
ejpam-4761	472	1	+	+	CCONJ
ejpam-4761	472	2	pnd(h)|a	pnd(h)|a	ADJ
ejpam-4761	472	3	\n2	\n2	ADJ
ejpam-4761	472	4	g(a)|	g(a)|	X
ejpam-4761	472	5	≥	≥	NUM
ejpam-4761	472	6	α	α	NOUN
ejpam-4761	472	7	,	,	PUNCT
ejpam-4761	472	8	showing	show	VERB
ejpam-4761	472	9	the	the	DET
ejpam-4761	472	10	desired	desire	VERB
ejpam-4761	472	11	equality	equality	NOUN
ejpam-4761	472	12	.	.	PUNCT
ejpam-4761	473	1	for	for	ADP
ejpam-4761	473	2	(	(	PUNCT
ejpam-4761	473	3	i	i	NOUN
ejpam-4761	473	4	)	)	PUNCT
ejpam-4761	473	5	,	,	PUNCT
ejpam-4761	473	6	let	let	VERB
ejpam-4761	473	7	s	s	PRON
ejpam-4761	473	8	be	be	AUX
ejpam-4761	473	9	a	a	DET
ejpam-4761	473	10	weakly	weakly	ADJ
ejpam-4761	473	11	convex	convex	NOUN
ejpam-4761	473	12	total	total	ADJ
ejpam-4761	473	13	hop	hop	NOUN
ejpam-4761	473	14	dominating	dominating	NOUN
ejpam-4761	473	15	set	set	VERB
ejpam-4761	473	16	in	in	ADP
ejpam-4761	473	17	g	g	NOUN
ejpam-4761	473	18	with	with	ADP
ejpam-4761	473	19	|s|	|s|	NOUN
ejpam-4761	473	20	=	=	PUNCT
ejpam-4761	473	21	γwconth(g	γwconth(g	PROPN
ejpam-4761	473	22	)	)	PUNCT
ejpam-4761	473	23	(	(	PUNCT
ejpam-4761	473	24	s	s	NOUN
ejpam-4761	473	25	exists	exist	VERB
ejpam-4761	473	26	because	because	SCONJ
ejpam-4761	473	27	γ(g	γ(g	NOUN
ejpam-4761	473	28	)	)	PUNCT
ejpam-4761	473	29	̸=	̸=	PROPN
ejpam-4761	473	30	1	1	NUM
ejpam-4761	473	31	)	)	PUNCT
ejpam-4761	473	32	.	.	PUNCT
ejpam-4761	474	1	then	then	ADV
ejpam-4761	474	2	s	s	VERB
ejpam-4761	474	3	∩	∩	ADJ
ejpam-4761	474	4	n2	n2	ADJ
ejpam-4761	474	5	g(s	g(s	PROPN
ejpam-4761	474	6	)	)	PUNCT
ejpam-4761	474	7	=	=	SYM
ejpam-4761	474	8	s	s	PROPN
ejpam-4761	474	9	and	and	CCONJ
ejpam-4761	474	10	s	s	NOUN
ejpam-4761	474	11	\	\	PROPN
ejpam-4761	474	12	n2	n2	ADJ
ejpam-4761	474	13	g(s	g(s	PROPN
ejpam-4761	474	14	)	)	PUNCT
ejpam-4761	474	15	=	=	PUNCT
ejpam-4761	474	16	∅.	∅.	AUX
ejpam-4761	474	17	set	set	VERB
ejpam-4761	474	18	qx	qx	NOUN
ejpam-4761	474	19	=	=	PUNCT
ejpam-4761	474	20	{	{	PUNCT
ejpam-4761	474	21	p	p	X
ejpam-4761	474	22	}	}	PUNCT
ejpam-4761	474	23	,	,	PUNCT
ejpam-4761	474	24	where	where	SCONJ
ejpam-4761	474	25	p	p	PROPN
ejpam-4761	474	26	∈	∈	PROPN
ejpam-4761	474	27	v	v	ADP
ejpam-4761	474	28	(	(	PUNCT
ejpam-4761	474	29	h	h	NOUN
ejpam-4761	474	30	)	)	PUNCT
ejpam-4761	474	31	,	,	PUNCT
ejpam-4761	474	32	for	for	ADP
ejpam-4761	474	33	each	each	DET
ejpam-4761	474	34	x	x	SYM
ejpam-4761	474	35	∈	∈	PROPN
ejpam-4761	474	36	s.	s.	PROPN
ejpam-4761	474	37	then	then	ADV
ejpam-4761	474	38	c	c	PROPN
ejpam-4761	475	1	=	=	PUNCT
ejpam-4761	475	2	⋃	⋃	PROPN
ejpam-4761	475	3	x∈s	x∈s	NOUN
ejpam-4761	476	1	[	[	X
ejpam-4761	476	2	{	{	PUNCT
ejpam-4761	476	3	x	x	NOUN
ejpam-4761	476	4	}	}	PUNCT
ejpam-4761	476	5	×	×	PROPN
ejpam-4761	476	6	qx	qx	X
ejpam-4761	476	7	]	]	X
ejpam-4761	476	8	is	be	AUX
ejpam-4761	476	9	a	a	DET
ejpam-4761	476	10	weakly	weakly	ADJ
ejpam-4761	476	11	convex	convex	NOUN
ejpam-4761	476	12	hop	hop	NOUN
ejpam-4761	476	13	dominating	dominating	NOUN
ejpam-4761	476	14	set	set	VERB
ejpam-4761	476	15	in	in	ADP
ejpam-4761	476	16	g[h	g[h	PROPN
ejpam-4761	476	17	]	]	PUNCT
ejpam-4761	476	18	by	by	ADP
ejpam-4761	476	19	theorem	theorem	NOUN
ejpam-4761	476	20	13	13	NUM
ejpam-4761	476	21	.	.	PUNCT
ejpam-4761	477	1	it	it	PRON
ejpam-4761	477	2	follows	follow	VERB
ejpam-4761	477	3	that	that	SCONJ
ejpam-4761	477	4	γwconh(g[h	γwconh(g[h	PROPN
ejpam-4761	477	5	]	]	X
ejpam-4761	477	6	)	)	PUNCT
ejpam-4761	477	7	≤	≤	PROPN
ejpam-4761	477	8	|c|	|c|	PROPN
ejpam-4761	477	9	=	=	SYM
ejpam-4761	477	10	γwconth(g	γwconth(g	PROPN
ejpam-4761	477	11	)	)	PUNCT
ejpam-4761	477	12	.	.	PUNCT
ejpam-4761	478	1	for	for	ADP
ejpam-4761	478	2	(	(	PUNCT
ejpam-4761	478	3	ii	ii	NOUN
ejpam-4761	478	4	)	)	PUNCT
ejpam-4761	478	5	,	,	PUNCT
ejpam-4761	478	6	suppose	suppose	VERB
ejpam-4761	478	7	that	that	SCONJ
ejpam-4761	478	8	g	g	PROPN
ejpam-4761	478	9	=	=	PROPN
ejpam-4761	478	10	kn	kn	PROPN
ejpam-4761	478	11	.	.	PUNCT
ejpam-4761	478	12	let	let	VERB
ejpam-4761	478	13	c	c	NOUN
ejpam-4761	478	14	=	=	PUNCT
ejpam-4761	479	1	⋃	⋃	PROPN
ejpam-4761	479	2	x∈s	x∈s	NOUN
ejpam-4761	480	1	[	[	X
ejpam-4761	480	2	{	{	PUNCT
ejpam-4761	480	3	x	x	NOUN
ejpam-4761	480	4	}	}	PUNCT
ejpam-4761	480	5	×	×	PROPN
ejpam-4761	480	6	tx	tx	PROPN
ejpam-4761	480	7	]	]	PUNCT
ejpam-4761	480	8	be	be	AUX
ejpam-4761	480	9	a	a	DET
ejpam-4761	480	10	γwconh	γwconh	NOUN
ejpam-4761	480	11	-	-	PUNCT
ejpam-4761	480	12	set	set	NOUN
ejpam-4761	480	13	in	in	ADP
ejpam-4761	480	14	g[h	g[h	NOUN
ejpam-4761	480	15	]	]	PUNCT
ejpam-4761	480	16	.	.	PUNCT
ejpam-4761	481	1	by	by	ADP
ejpam-4761	481	2	theorem	theorem	NOUN
ejpam-4761	481	3	13	13	NUM
ejpam-4761	481	4	,	,	PUNCT
ejpam-4761	481	5	s	s	VERB
ejpam-4761	481	6	is	be	AUX
ejpam-4761	481	7	a	a	DET
ejpam-4761	481	8	weakly	weakly	ADJ
ejpam-4761	481	9	convex	convex	NOUN
ejpam-4761	481	10	hop	hop	NOUN
ejpam-4761	481	11	dominating	dominating	NOUN
ejpam-4761	481	12	set	set	VERB
ejpam-4761	481	13	in	in	ADP
ejpam-4761	481	14	g	g	PROPN
ejpam-4761	481	15	and	and	CCONJ
ejpam-4761	481	16	tx	tx	PROPN
ejpam-4761	481	17	is	be	AUX
ejpam-4761	481	18	a	a	DET
ejpam-4761	481	19	pointwise	pointwise	ADJ
ejpam-4761	481	20	non	non	ADJ
ejpam-4761	481	21	-	-	ADJ
ejpam-4761	481	22	dominating	dominating	ADJ
ejpam-4761	481	23	set	set	NOUN
ejpam-4761	481	24	in	in	ADP
ejpam-4761	481	25	h	h	NOUN
ejpam-4761	481	26	for	for	ADP
ejpam-4761	481	27	each	each	DET
ejpam-4761	481	28	x	x	SYM
ejpam-4761	481	29	∈	∈	PROPN
ejpam-4761	481	30	s	s	PART
ejpam-4761	481	31	\n2	\n2	ADJ
ejpam-4761	481	32	g(s	g(	NOUN
ejpam-4761	481	33	)	)	PUNCT
ejpam-4761	481	34	.	.	PUNCT
ejpam-4761	482	1	since	since	SCONJ
ejpam-4761	482	2	v	v	NOUN
ejpam-4761	482	3	(	(	PUNCT
ejpam-4761	482	4	g	g	NOUN
ejpam-4761	482	5	)	)	PUNCT
ejpam-4761	482	6	is	be	AUX
ejpam-4761	482	7	the	the	DET
ejpam-4761	482	8	only	only	ADJ
ejpam-4761	482	9	hop	hop	NOUN
ejpam-4761	482	10	dominating	dominating	NOUN
ejpam-4761	482	11	set	set	VERB
ejpam-4761	482	12	in	in	ADP
ejpam-4761	482	13	g	g	PROPN
ejpam-4761	482	14	,	,	PUNCT
ejpam-4761	482	15	it	it	PRON
ejpam-4761	482	16	follows	follow	VERB
ejpam-4761	482	17	that	that	PRON
ejpam-4761	482	18	s	s	VERB
ejpam-4761	482	19	=	=	SYM
ejpam-4761	482	20	v	v	ADJ
ejpam-4761	482	21	(	(	PUNCT
ejpam-4761	482	22	g	g	NOUN
ejpam-4761	482	23	)	)	PUNCT
ejpam-4761	482	24	.	.	PUNCT
ejpam-4761	483	1	also	also	ADV
ejpam-4761	483	2	,	,	PUNCT
ejpam-4761	483	3	since	since	SCONJ
ejpam-4761	483	4	c	c	PROPN
ejpam-4761	483	5	is	be	AUX
ejpam-4761	483	6	a	a	DET
ejpam-4761	483	7	γwconh	γwconh	NOUN
ejpam-4761	483	8	-	-	PUNCT
ejpam-4761	483	9	set	set	NOUN
ejpam-4761	483	10	,	,	PUNCT
ejpam-4761	483	11	each	each	DET
ejpam-4761	483	12	tx	tx	PROPN
ejpam-4761	483	13	is	be	AUX
ejpam-4761	483	14	a	a	DET
ejpam-4761	483	15	pnd	pnd	NOUN
ejpam-4761	483	16	-	-	PUNCT
ejpam-4761	483	17	set	set	NOUN
ejpam-4761	483	18	in	in	ADP
ejpam-4761	483	19	h.	h.	PROPN
ejpam-4761	483	20	hence	hence	ADV
ejpam-4761	483	21	,	,	PUNCT
ejpam-4761	483	22	γwconh(g[h	γwconh(g[h	PROPN
ejpam-4761	483	23	]	]	X
ejpam-4761	483	24	)	)	PUNCT
ejpam-4761	484	1	=	=	SYM
ejpam-4761	484	2	|c|	|c|	PROPN
ejpam-4761	484	3	=	=	SYM
ejpam-4761	484	4	n	n	PROPN
ejpam-4761	484	5	·	·	PUNCT
ejpam-4761	484	6	pnd(h	pnd(h	NUM
ejpam-4761	484	7	)	)	PUNCT
ejpam-4761	484	8	.	.	PUNCT
ejpam-4761	485	1	references	reference	NOUN
ejpam-4761	485	2	1209	1209	NUM
ejpam-4761	485	3	note	note	VERB
ejpam-4761	485	4	that	that	SCONJ
ejpam-4761	485	5	the	the	DET
ejpam-4761	485	6	bound	bind	VERB
ejpam-4761	485	7	in	in	ADP
ejpam-4761	485	8	corollary	corollary	ADJ
ejpam-4761	485	9	9(i	9(i	NUM
ejpam-4761	485	10	)	)	PUNCT
ejpam-4761	485	11	is	be	AUX
ejpam-4761	485	12	tight	tight	ADJ
ejpam-4761	485	13	.	.	PUNCT
ejpam-4761	486	1	to	to	PART
ejpam-4761	486	2	see	see	VERB
ejpam-4761	486	3	this	this	PRON
ejpam-4761	486	4	,	,	PUNCT
ejpam-4761	486	5	consider	consider	VERB
ejpam-4761	486	6	g	g	NOUN
ejpam-4761	486	7	=	=	PUNCT
ejpam-4761	486	8	p7	p7	PROPN
ejpam-4761	486	9	.	.	PUNCT
ejpam-4761	487	1	for	for	ADP
ejpam-4761	487	2	any	any	DET
ejpam-4761	487	3	connected	connected	ADJ
ejpam-4761	487	4	graph	graph	NOUN
ejpam-4761	487	5	h	h	NOUN
ejpam-4761	487	6	,	,	PUNCT
ejpam-4761	487	7	γwconh(p7[h	γwconh(p7[h	PRON
ejpam-4761	487	8	]	]	PUNCT
ejpam-4761	487	9	)	)	PUNCT
ejpam-4761	487	10	=	=	SYM
ejpam-4761	487	11	4	4	NUM
ejpam-4761	487	12	=	=	SYM
ejpam-4761	487	13	γwconth(p7	γwconth(p7	X
ejpam-4761	487	14	)	)	PUNCT
ejpam-4761	487	15	.	.	PUNCT
ejpam-4761	488	1	4	4	X
ejpam-4761	488	2	.	.	X
ejpam-4761	488	3	conclusion	conclusion	NOUN
ejpam-4761	488	4	weakly	weakly	ADV
ejpam-4761	488	5	convex	convex	PROPN
ejpam-4761	488	6	hop	hop	NOUN
ejpam-4761	488	7	domination	domination	NOUN
ejpam-4761	488	8	was	be	AUX
ejpam-4761	488	9	introduced	introduce	VERB
ejpam-4761	488	10	and	and	CCONJ
ejpam-4761	488	11	initially	initially	ADV
ejpam-4761	488	12	investigated	investigate	VERB
ejpam-4761	488	13	in	in	ADP
ejpam-4761	488	14	this	this	DET
ejpam-4761	488	15	study	study	NOUN
ejpam-4761	488	16	.	.	PUNCT
ejpam-4761	489	1	characterizations	characterization	NOUN
ejpam-4761	489	2	of	of	ADP
ejpam-4761	489	3	weakly	weakly	ADJ
ejpam-4761	489	4	convex	convex	ADJ
ejpam-4761	489	5	hop	hop	NOUN
ejpam-4761	489	6	dominating	dominating	NOUN
ejpam-4761	489	7	sets	set	NOUN
ejpam-4761	489	8	in	in	ADP
ejpam-4761	489	9	the	the	DET
ejpam-4761	489	10	shadow	shadow	NOUN
ejpam-4761	489	11	graph	graph	NOUN
ejpam-4761	489	12	,	,	PUNCT
ejpam-4761	489	13	and	and	CCONJ
ejpam-4761	489	14	in	in	ADP
ejpam-4761	489	15	the	the	DET
ejpam-4761	489	16	join	join	NOUN
ejpam-4761	489	17	,	,	PUNCT
ejpam-4761	489	18	corona	corona	PROPN
ejpam-4761	489	19	,	,	PUNCT
ejpam-4761	489	20	and	and	CCONJ
ejpam-4761	489	21	lexicographic	lexicographic	ADJ
ejpam-4761	489	22	product	product	NOUN
ejpam-4761	489	23	of	of	ADP
ejpam-4761	489	24	two	two	NUM
ejpam-4761	489	25	graphs	graph	NOUN
ejpam-4761	489	26	were	be	AUX
ejpam-4761	489	27	formulated	formulate	VERB
ejpam-4761	489	28	.	.	PUNCT
ejpam-4761	490	1	these	these	DET
ejpam-4761	490	2	characterizations	characterization	NOUN
ejpam-4761	490	3	were	be	AUX
ejpam-4761	490	4	used	use	VERB
ejpam-4761	490	5	to	to	PART
ejpam-4761	490	6	determine	determine	VERB
ejpam-4761	490	7	the	the	DET
ejpam-4761	490	8	weakly	weakly	ADJ
ejpam-4761	490	9	convex	convex	ADJ
ejpam-4761	490	10	hop	hop	NOUN
ejpam-4761	490	11	domination	domination	NOUN
ejpam-4761	490	12	number	number	NOUN
ejpam-4761	490	13	of	of	ADP
ejpam-4761	490	14	each	each	PRON
ejpam-4761	490	15	of	of	ADP
ejpam-4761	490	16	these	these	DET
ejpam-4761	490	17	graphs	graph	NOUN
ejpam-4761	490	18	.	.	PUNCT
ejpam-4761	491	1	moreover	moreover	ADV
ejpam-4761	491	2	,	,	PUNCT
ejpam-4761	491	3	it	it	PRON
ejpam-4761	491	4	was	be	AUX
ejpam-4761	491	5	shown	show	VERB
ejpam-4761	491	6	that	that	SCONJ
ejpam-4761	491	7	any	any	DET
ejpam-4761	491	8	two	two	NUM
ejpam-4761	491	9	positive	positive	ADJ
ejpam-4761	491	10	integers	integer	NOUN
ejpam-4761	491	11	a	a	PRON
ejpam-4761	491	12	and	and	CCONJ
ejpam-4761	491	13	b	b	NOUN
ejpam-4761	491	14	with	with	ADP
ejpam-4761	491	15	3	3	NUM
ejpam-4761	491	16	≤	≤	NOUN
ejpam-4761	491	17	a	a	DET
ejpam-4761	491	18	≤	≤	NUM
ejpam-4761	491	19	b	b	NOUN
ejpam-4761	491	20	are	be	AUX
ejpam-4761	491	21	realizable	realizable	ADJ
ejpam-4761	491	22	as	as	ADP
ejpam-4761	491	23	weakly	weakly	ADV
ejpam-4761	491	24	connected	connect	VERB
ejpam-4761	491	25	hop	hop	NOUN
ejpam-4761	491	26	domination	domination	NOUN
ejpam-4761	491	27	number	number	NOUN
ejpam-4761	491	28	and	and	CCONJ
ejpam-4761	491	29	convex	convex	VERB
ejpam-4761	491	30	hop	hop	NOUN
ejpam-4761	491	31	domination	domination	NOUN
ejpam-4761	491	32	number	number	NOUN
ejpam-4761	491	33	,	,	PUNCT
ejpam-4761	491	34	respectively	respectively	ADV
ejpam-4761	491	35	,	,	PUNCT
ejpam-4761	491	36	of	of	ADP
ejpam-4761	491	37	some	some	DET
ejpam-4761	491	38	connected	connected	ADJ
ejpam-4761	491	39	graph	graph	NOUN
ejpam-4761	491	40	.	.	PUNCT
ejpam-4761	492	1	the	the	DET
ejpam-4761	492	2	parameter	parameter	NOUN
ejpam-4761	492	3	may	may	AUX
ejpam-4761	492	4	be	be	AUX
ejpam-4761	492	5	studied	study	VERB
ejpam-4761	492	6	further	far	ADV
ejpam-4761	492	7	for	for	ADP
ejpam-4761	492	8	trees	tree	NOUN
ejpam-4761	492	9	and	and	CCONJ
ejpam-4761	492	10	other	other	ADJ
ejpam-4761	492	11	graphs	graph	NOUN
ejpam-4761	492	12	.	.	PUNCT
ejpam-4761	493	1	also	also	ADV
ejpam-4761	493	2	,	,	PUNCT
ejpam-4761	493	3	bounds	bound	NOUN
ejpam-4761	493	4	involving	involve	VERB
ejpam-4761	493	5	other	other	ADJ
ejpam-4761	493	6	known	know	VERB
ejpam-4761	493	7	parameters	parameter	NOUN
ejpam-4761	493	8	may	may	AUX
ejpam-4761	493	9	be	be	AUX
ejpam-4761	493	10	obtained	obtain	VERB
ejpam-4761	493	11	.	.	PUNCT
ejpam-4761	494	1	moreover	moreover	ADV
ejpam-4761	494	2	,	,	PUNCT
ejpam-4761	494	3	interested	interested	ADJ
ejpam-4761	494	4	readers	reader	NOUN
ejpam-4761	494	5	may	may	AUX
ejpam-4761	494	6	find	find	VERB
ejpam-4761	494	7	the	the	DET
ejpam-4761	494	8	complexity	complexity	NOUN
ejpam-4761	494	9	aspect	aspect	NOUN
ejpam-4761	494	10	of	of	ADP
ejpam-4761	494	11	the	the	DET
ejpam-4761	494	12	weakly	weakly	ADJ
ejpam-4761	494	13	convex	convex	NOUN
ejpam-4761	494	14	hop	hop	NOUN
ejpam-4761	494	15	dominating	dominating	NOUN
ejpam-4761	494	16	set	set	NOUN
ejpam-4761	494	17	problem	problem	NOUN
ejpam-4761	494	18	worth	worth	ADJ
ejpam-4761	494	19	considering	consider	VERB
ejpam-4761	494	20	.	.	PUNCT
ejpam-4761	495	1	acknowledgements	acknowledgement	NOUN
ejpam-4761	495	2	the	the	DET
ejpam-4761	495	3	authors	author	NOUN
ejpam-4761	495	4	would	would	AUX
ejpam-4761	495	5	like	like	VERB
ejpam-4761	495	6	to	to	PART
ejpam-4761	495	7	thank	thank	VERB
ejpam-4761	495	8	the	the	DET
ejpam-4761	495	9	department	department	NOUN
ejpam-4761	495	10	of	of	ADP
ejpam-4761	495	11	science	science	NOUN
ejpam-4761	495	12	and	and	CCONJ
ejpam-4761	495	13	technology	technology	NOUN
ejpam-4761	495	14	accelerated	accelerate	VERB
ejpam-4761	495	15	science	science	NOUN
ejpam-4761	495	16	and	and	CCONJ
ejpam-4761	495	17	technology	technology	NOUN
ejpam-4761	495	18	human	human	ADJ
ejpam-4761	495	19	resource	resource	NOUN
ejpam-4761	495	20	development	development	NOUN
ejpam-4761	495	21	program	program	NOUN
ejpam-4761	495	22	(	(	PUNCT
ejpam-4761	495	23	dost	dost	NOUN
ejpam-4761	495	24	-	-	PUNCT
ejpam-4761	495	25	asthrdp)philippines	asthrdp)philippine	NOUN
ejpam-4761	495	26	,	,	PUNCT
ejpam-4761	495	27	msu	msu	PROPN
ejpam-4761	495	28	-	-	PUNCT
ejpam-4761	495	29	iligan	iligan	PROPN
ejpam-4761	495	30	institute	institute	PROPN
ejpam-4761	495	31	of	of	ADP
ejpam-4761	495	32	technology	technology	NOUN
ejpam-4761	495	33	,	,	PUNCT
ejpam-4761	495	34	and	and	CCONJ
ejpam-4761	495	35	msu	msu	PROPN
ejpam-4761	495	36	tawi	tawi	PROPN
ejpam-4761	495	37	-	-	PUNCT
ejpam-4761	495	38	tawi	tawi	PROPN
ejpam-4761	495	39	college	college	PROPN
ejpam-4761	495	40	of	of	ADP
ejpam-4761	495	41	technology	technology	NOUN
ejpam-4761	495	42	and	and	CCONJ
ejpam-4761	495	43	oceanography	oceanography	NOUN
ejpam-4761	495	44	for	for	ADP
ejpam-4761	495	45	funding	fund	VERB
ejpam-4761	495	46	this	this	DET
ejpam-4761	495	47	research	research	NOUN
ejpam-4761	495	48	.	.	PUNCT
ejpam-4761	496	1	references	reference	NOUN
ejpam-4761	496	2	[	[	X
ejpam-4761	496	3	1	1	X
ejpam-4761	496	4	]	]	PUNCT
ejpam-4761	496	5	s.	s.	PROPN
ejpam-4761	496	6	arriola	arriola	PROPN
ejpam-4761	496	7	and	and	CCONJ
ejpam-4761	496	8	s.	s.	PROPN
ejpam-4761	496	9	canoy	canoy	PROPN
ejpam-4761	496	10	jr	jr	PROPN
ejpam-4761	497	1	.	.	PROPN
ejpam-4761	497	2	(	(	PUNCT
ejpam-4761	497	3	1,2	1,2	NUM
ejpam-4761	497	4	)	)	PUNCT
ejpam-4761	497	5	domination	domination	NOUN
ejpam-4761	497	6	in	in	ADP
ejpam-4761	497	7	graphs	graph	NOUN
ejpam-4761	497	8	.	.	PUNCT
ejpam-4761	498	1	advances	advance	NOUN
ejpam-4761	498	2	and	and	CCONJ
ejpam-4761	498	3	applications	application	NOUN
ejpam-4761	498	4	in	in	ADP
ejpam-4761	498	5	discrete	discrete	ADJ
ejpam-4761	498	6	mathematics	mathematic	NOUN
ejpam-4761	498	7	.	.	PUNCT
ejpam-4761	498	8	,	,	PUNCT
ejpam-4761	498	9	18(2):179–190	18(2):179–190	NUM
ejpam-4761	498	10	,	,	PUNCT
ejpam-4761	498	11	2017	2017	NUM
ejpam-4761	498	12	.	.	PUNCT
ejpam-4761	499	1	[	[	X
ejpam-4761	499	2	2	2	NUM
ejpam-4761	499	3	]	]	PUNCT
ejpam-4761	499	4	s.	s.	PROPN
ejpam-4761	499	5	ayyaswamy	ayyaswamy	PROPN
ejpam-4761	499	6	,	,	PUNCT
ejpam-4761	499	7	b.	b.	PROPN
ejpam-4761	499	8	krishnakumari	krishnakumari	PROPN
ejpam-4761	499	9	,	,	PUNCT
ejpam-4761	499	10	b.	b.	PROPN
ejpam-4761	499	11	natarjan	natarjan	PROPN
ejpam-4761	499	12	,	,	PUNCT
ejpam-4761	499	13	and	and	CCONJ
ejpam-4761	499	14	y.	y.	PROPN
ejpam-4761	499	15	venkatakrishnan	venkatakrishnan	PROPN
ejpam-4761	499	16	.	.	PUNCT
ejpam-4761	500	1	bounds	bound	NOUN
ejpam-4761	500	2	on	on	ADP
ejpam-4761	500	3	the	the	DET
ejpam-4761	500	4	hop	hop	NOUN
ejpam-4761	500	5	domination	domination	NOUN
ejpam-4761	500	6	number	number	NOUN
ejpam-4761	500	7	of	of	ADP
ejpam-4761	500	8	a	a	DET
ejpam-4761	500	9	tree	tree	NOUN
ejpam-4761	500	10	.	.	PUNCT
ejpam-4761	501	1	proceedings	proceeding	NOUN
ejpam-4761	501	2	-	-	PUNCT
ejpam-4761	501	3	mathematical	mathematical	ADJ
ejpam-4761	501	4	sciences	science	NOUN
ejpam-4761	501	5	.	.	PUNCT
ejpam-4761	501	6	,	,	PUNCT
ejpam-4761	501	7	125(4):449–455	125(4):449–455	ADP
ejpam-4761	501	8	,	,	PUNCT
ejpam-4761	501	9	2015	2015	NUM
ejpam-4761	501	10	.	.	PUNCT
ejpam-4761	502	1	[	[	X
ejpam-4761	502	2	3	3	X
ejpam-4761	502	3	]	]	X
ejpam-4761	502	4	s.	s.	PROPN
ejpam-4761	502	5	ayyaswamy	ayyaswamy	PROPN
ejpam-4761	502	6	,	,	PUNCT
ejpam-4761	502	7	c.	c.	PROPN
ejpam-4761	502	8	natarajan	natarajan	PROPN
ejpam-4761	502	9	,	,	PUNCT
ejpam-4761	502	10	and	and	CCONJ
ejpam-4761	502	11	g.	g.	PROPN
ejpam-4761	502	12	sathiamoorphy	sathiamoorphy	PROPN
ejpam-4761	502	13	.	.	PUNCT
ejpam-4761	503	1	a	a	DET
ejpam-4761	503	2	note	note	NOUN
ejpam-4761	503	3	on	on	ADP
ejpam-4761	503	4	hop	hop	NOUN
ejpam-4761	503	5	domination	domination	NOUN
ejpam-4761	503	6	number	number	NOUN
ejpam-4761	503	7	of	of	ADP
ejpam-4761	503	8	some	some	DET
ejpam-4761	503	9	special	special	ADJ
ejpam-4761	503	10	families	family	NOUN
ejpam-4761	503	11	of	of	ADP
ejpam-4761	503	12	graphs	graph	NOUN
ejpam-4761	503	13	.	.	PUNCT
ejpam-4761	504	1	international	international	ADJ
ejpam-4761	504	2	journal	journal	NOUN
ejpam-4761	504	3	of	of	ADP
ejpam-4761	504	4	pure	pure	ADJ
ejpam-4761	504	5	and	and	CCONJ
ejpam-4761	504	6	applied	applied	ADJ
ejpam-4761	504	7	mathematics	mathematic	NOUN
ejpam-4761	504	8	.	.	PUNCT
ejpam-4761	504	9	,	,	PUNCT
ejpam-4761	504	10	119(12):11465–14171	119(12):11465–14171	NUM
ejpam-4761	504	11	,	,	PUNCT
ejpam-4761	504	12	2018	2018	NUM
ejpam-4761	504	13	.	.	PUNCT
ejpam-4761	505	1	[	[	X
ejpam-4761	505	2	4	4	NUM
ejpam-4761	505	3	]	]	X
ejpam-4761	505	4	b.	b.	PROPN
ejpam-4761	505	5	bresar	bresar	PROPN
ejpam-4761	505	6	,	,	PUNCT
ejpam-4761	505	7	t.	t.	NOUN
ejpam-4761	505	8	gologranc	gologranc	PROPN
ejpam-4761	505	9	,	,	PUNCT
ejpam-4761	505	10	and	and	CCONJ
ejpam-4761	505	11	t.	t.	PROPN
ejpam-4761	505	12	kos	kos	PROPN
ejpam-4761	505	13	.	.	PUNCT
ejpam-4761	506	1	convex	convex	PROPN
ejpam-4761	506	2	and	and	CCONJ
ejpam-4761	506	3	isometric	isometric	ADJ
ejpam-4761	506	4	domination	domination	NOUN
ejpam-4761	506	5	of	of	ADP
ejpam-4761	506	6	(	(	PUNCT
ejpam-4761	506	7	weak	weak	ADJ
ejpam-4761	506	8	)	)	PUNCT
ejpam-4761	506	9	domination	domination	NOUN
ejpam-4761	506	10	pair	pair	NOUN
ejpam-4761	506	11	graphs	graph	NOUN
ejpam-4761	506	12	.	.	PUNCT
ejpam-4761	507	1	theoretical	theoretical	ADJ
ejpam-4761	507	2	computer	computer	NOUN
ejpam-4761	507	3	science	science	NOUN
ejpam-4761	507	4	.	.	PUNCT
ejpam-4761	507	5	,	,	PUNCT
ejpam-4761	508	1	730:32–43	730:32–43	NUM
ejpam-4761	508	2	,	,	PUNCT
ejpam-4761	508	3	2018	2018	NUM
ejpam-4761	508	4	.	.	PUNCT
ejpam-4761	509	1	[	[	X
ejpam-4761	509	2	5	5	X
ejpam-4761	509	3	]	]	PUNCT
ejpam-4761	509	4	e.	e.	PROPN
ejpam-4761	509	5	cockayne	cockayne	PROPN
ejpam-4761	509	6	,	,	PUNCT
ejpam-4761	509	7	r.	r.	PROPN
ejpam-4761	509	8	dawes	dawes	PROPN
ejpam-4761	509	9	,	,	PUNCT
ejpam-4761	509	10	and	and	CCONJ
ejpam-4761	509	11	s.	s.	PROPN
ejpam-4761	509	12	hedetnniemi	hedetnniemi	PROPN
ejpam-4761	509	13	.	.	PUNCT
ejpam-4761	510	1	total	total	ADJ
ejpam-4761	510	2	domination	domination	NOUN
ejpam-4761	510	3	in	in	ADP
ejpam-4761	510	4	graphs	graph	NOUN
ejpam-4761	510	5	.	.	PUNCT
ejpam-4761	511	1	networks	network	NOUN
ejpam-4761	511	2	,	,	PUNCT
ejpam-4761	511	3	10:211–219	10:211–219	NUM
ejpam-4761	511	4	,	,	PUNCT
ejpam-4761	511	5	1980	1980	NUM
ejpam-4761	511	6	.	.	PUNCT
ejpam-4761	512	1	[	[	X
ejpam-4761	512	2	6	6	NUM
ejpam-4761	512	3	]	]	PUNCT
ejpam-4761	512	4	t.	t.	PROPN
ejpam-4761	512	5	daniel	daniel	PROPN
ejpam-4761	512	6	and	and	CCONJ
ejpam-4761	512	7	s.	s.	PROPN
ejpam-4761	512	8	canoy	canoy	PROPN
ejpam-4761	512	9	jr	jr	PROPN
ejpam-4761	512	10	.	.	PROPN
ejpam-4761	512	11	clique	clique	PROPN
ejpam-4761	512	12	domination	domination	PROPN
ejpam-4761	512	13	in	in	ADP
ejpam-4761	512	14	graphs	graph	NOUN
ejpam-4761	512	15	.	.	PUNCT
ejpam-4761	513	1	applied	apply	VERB
ejpam-4761	513	2	mathematical	mathematical	ADJ
ejpam-4761	513	3	sciences	science	NOUN
ejpam-4761	513	4	,	,	PUNCT
ejpam-4761	513	5	9(116):5449–5455	9(116):5449–5455	NUM
ejpam-4761	513	6	,	,	PUNCT
ejpam-4761	513	7	2015	2015	NUM
ejpam-4761	513	8	.	.	PUNCT
ejpam-4761	514	1	references	reference	NOUN
ejpam-4761	514	2	1210	1210	NUM
ejpam-4761	515	1	[	[	X
ejpam-4761	515	2	7	7	NUM
ejpam-4761	515	3	]	]	PUNCT
ejpam-4761	515	4	j.	j.	PROPN
ejpam-4761	515	5	hassan	hassan	PROPN
ejpam-4761	515	6	and	and	CCONJ
ejpam-4761	515	7	s.	s.	PROPN
ejpam-4761	515	8	canoy	canoy	PROPN
ejpam-4761	515	9	jr	jr	PROPN
ejpam-4761	515	10	.	.	PUNCT
ejpam-4761	516	1	grundy	grundy	PROPN
ejpam-4761	516	2	hop	hop	PROPN
ejpam-4761	516	3	domination	domination	PROPN
ejpam-4761	516	4	in	in	ADP
ejpam-4761	516	5	graphs	graph	NOUN
ejpam-4761	516	6	.	.	PUNCT
ejpam-4761	517	1	eur	eur	PROPN
ejpam-4761	517	2	.	.	PUNCT
ejpam-4761	518	1	j.	j.	PROPN
ejpam-4761	518	2	pure	pure	PROPN
ejpam-4761	518	3	appl	appl	PROPN
ejpam-4761	518	4	.	.	PUNCT
ejpam-4761	518	5	math	math	PROPN
ejpam-4761	518	6	.	.	PUNCT
ejpam-4761	518	7	,	,	PUNCT
ejpam-4761	518	8	15(4):1623–1636	15(4):1623–1636	NUM
ejpam-4761	518	9	,	,	PUNCT
ejpam-4761	518	10	2022	2022	NUM
ejpam-4761	518	11	.	.	PUNCT
ejpam-4761	519	1	[	[	X
ejpam-4761	519	2	8	8	X
ejpam-4761	519	3	]	]	PUNCT
ejpam-4761	519	4	j.	j.	PROPN
ejpam-4761	519	5	hassan	hassan	PROPN
ejpam-4761	519	6	and	and	CCONJ
ejpam-4761	519	7	s.	s.	PROPN
ejpam-4761	519	8	canoy	canoy	PROPN
ejpam-4761	519	9	jr	jr	PROPN
ejpam-4761	519	10	.	.	PROPN
ejpam-4761	519	11	hop	hop	PROPN
ejpam-4761	519	12	independent	independent	ADJ
ejpam-4761	519	13	hop	hop	NOUN
ejpam-4761	519	14	domination	domination	NOUN
ejpam-4761	519	15	in	in	ADP
ejpam-4761	519	16	graphs	graph	NOUN
ejpam-4761	519	17	.	.	PUNCT
ejpam-4761	520	1	eur	eur	PROPN
ejpam-4761	520	2	.	.	PUNCT
ejpam-4761	521	1	j.	j.	PROPN
ejpam-4761	521	2	pure	pure	PROPN
ejpam-4761	521	3	appl	appl	PROPN
ejpam-4761	521	4	.	.	PUNCT
ejpam-4761	521	5	math	math	PROPN
ejpam-4761	521	6	.	.	PUNCT
ejpam-4761	521	7	,	,	PUNCT
ejpam-4761	521	8	15(4):1783–1796	15(4):1783–1796	NUM
ejpam-4761	521	9	,	,	PUNCT
ejpam-4761	521	10	2022	2022	NUM
ejpam-4761	521	11	.	.	PUNCT
ejpam-4761	522	1	[	[	X
ejpam-4761	522	2	9	9	NUM
ejpam-4761	522	3	]	]	PUNCT
ejpam-4761	522	4	j.	j.	PROPN
ejpam-4761	522	5	hassan	hassan	PROPN
ejpam-4761	522	6	and	and	CCONJ
ejpam-4761	522	7	s.	s.	PROPN
ejpam-4761	522	8	canoy	canoy	PROPN
ejpam-4761	522	9	jr	jr	PROPN
ejpam-4761	522	10	.	.	PROPN
ejpam-4761	522	11	convex	convex	VERB
ejpam-4761	522	12	hop	hop	NOUN
ejpam-4761	522	13	domination	domination	NOUN
ejpam-4761	522	14	in	in	ADP
ejpam-4761	522	15	graphs	graph	NOUN
ejpam-4761	522	16	.	.	PUNCT
ejpam-4761	523	1	eur	eur	PROPN
ejpam-4761	523	2	.	.	PUNCT
ejpam-4761	524	1	j.	j.	PROPN
ejpam-4761	524	2	pure	pure	PROPN
ejpam-4761	524	3	appl	appl	PROPN
ejpam-4761	524	4	.	.	PUNCT
ejpam-4761	524	5	math	math	PROPN
ejpam-4761	524	6	.	.	PUNCT
ejpam-4761	524	7	,	,	PUNCT
ejpam-4761	524	8	16(1):319–335	16(1):319–335	NOUN
ejpam-4761	524	9	,	,	PUNCT
ejpam-4761	524	10	2023	2023	NUM
ejpam-4761	524	11	.	.	PUNCT
ejpam-4761	525	1	[	[	X
ejpam-4761	525	2	10	10	NUM
ejpam-4761	525	3	]	]	X
ejpam-4761	525	4	s.	s.	PROPN
ejpam-4761	525	5	hedetniemi	hedetniemi	PROPN
ejpam-4761	525	6	and	and	CCONJ
ejpam-4761	525	7	r.	r.	PROPN
ejpam-4761	525	8	lascar	lascar	PROPN
ejpam-4761	525	9	.	.	PUNCT
ejpam-4761	526	1	connected	connected	ADJ
ejpam-4761	526	2	domination	domination	NOUN
ejpam-4761	526	3	in	in	ADP
ejpam-4761	526	4	graphs	graph	NOUN
ejpam-4761	526	5	.	.	PUNCT
ejpam-4761	527	1	chapter	chapter	NOUN
ejpam-4761	527	2	18	18	NUM
ejpam-4761	527	3	in	in	ADP
ejpam-4761	527	4	graph	graph	NOUN
ejpam-4761	527	5	theory	theory	NOUN
ejpam-4761	527	6	and	and	CCONJ
ejpam-4761	527	7	combinatorics	combinatorics	PROPN
ejpam-4761	527	8	,	,	PUNCT
ejpam-4761	527	9	london	london	PROPN
ejpam-4761	527	10	academic	academic	ADJ
ejpam-4761	527	11	press	press	NOUN
ejpam-4761	527	12	.	.	PUNCT
ejpam-4761	528	1	,	,	PUNCT
ejpam-4761	528	2	pages	page	NOUN
ejpam-4761	528	3	209–217	209–217	NUM
ejpam-4761	528	4	,	,	PUNCT
ejpam-4761	528	5	1984	1984	NUM
ejpam-4761	528	6	.	.	PUNCT
ejpam-4761	529	1	[	[	X
ejpam-4761	529	2	11	11	NUM
ejpam-4761	529	3	]	]	PUNCT
ejpam-4761	529	4	m.	m.	NOUN
ejpam-4761	529	5	henning	henning	PROPN
ejpam-4761	529	6	and	and	CCONJ
ejpam-4761	529	7	n.	n.	PROPN
ejpam-4761	529	8	rad	rad	PROPN
ejpam-4761	529	9	.	.	PROPN
ejpam-4761	530	1	on	on	ADP
ejpam-4761	530	2	2	2	NUM
ejpam-4761	530	3	-	-	PUNCT
ejpam-4761	530	4	step	step	NOUN
ejpam-4761	530	5	and	and	CCONJ
ejpam-4761	530	6	hop	hop	NOUN
ejpam-4761	530	7	dominating	dominating	NOUN
ejpam-4761	530	8	sets	set	NOUN
ejpam-4761	530	9	in	in	ADP
ejpam-4761	530	10	graphs	graph	NOUN
ejpam-4761	530	11	.	.	PUNCT
ejpam-4761	531	1	graphs	graph	NOUN
ejpam-4761	531	2	and	and	CCONJ
ejpam-4761	531	3	combinatorics	combinatoric	NOUN
ejpam-4761	531	4	.	.	PUNCT
ejpam-4761	531	5	,	,	PUNCT
ejpam-4761	531	6	33(4):913–927	33(4):913–927	PROPN
ejpam-4761	531	7	,	,	PUNCT
ejpam-4761	531	8	2017	2017	NUM
ejpam-4761	531	9	.	.	PUNCT
ejpam-4761	532	1	[	[	X
ejpam-4761	532	2	12	12	NUM
ejpam-4761	532	3	]	]	X
ejpam-4761	532	4	s.	s.	PROPN
ejpam-4761	532	5	canoy	canoy	PROPN
ejpam-4761	532	6	jr	jr	PROPN
ejpam-4761	532	7	.	.	PUNCT
ejpam-4761	533	1	a	a	DET
ejpam-4761	533	2	short	short	ADJ
ejpam-4761	533	3	note	note	NOUN
ejpam-4761	533	4	on	on	ADP
ejpam-4761	533	5	convexity	convexity	NOUN
ejpam-4761	533	6	and	and	CCONJ
ejpam-4761	533	7	convex	convex	NOUN
ejpam-4761	533	8	domination	domination	NOUN
ejpam-4761	533	9	in	in	ADP
ejpam-4761	533	10	g[km	g[km	PROPN
ejpam-4761	533	11	]	]	PUNCT
ejpam-4761	533	12	.	.	PUNCT
ejpam-4761	534	1	applied	apply	VERB
ejpam-4761	534	2	mathematical	mathematical	ADJ
ejpam-4761	534	3	sciences	science	NOUN
ejpam-4761	534	4	,	,	PUNCT
ejpam-4761	534	5	8(115):5737–5741	8(115):5737–5741	NUM
ejpam-4761	534	6	,	,	PUNCT
ejpam-4761	534	7	2014	2014	NUM
ejpam-4761	534	8	.	.	PUNCT
ejpam-4761	535	1	[	[	X
ejpam-4761	535	2	13	13	NUM
ejpam-4761	535	3	]	]	PUNCT
ejpam-4761	535	4	s.	s.	PROPN
ejpam-4761	535	5	canoy	canoy	PROPN
ejpam-4761	535	6	jr	jr	PROPN
ejpam-4761	535	7	.	.	PROPN
ejpam-4761	535	8	,	,	PUNCT
ejpam-4761	535	9	r.	r.	PROPN
ejpam-4761	535	10	mollejon	mollejon	NOUN
ejpam-4761	535	11	,	,	PUNCT
ejpam-4761	535	12	and	and	CCONJ
ejpam-4761	535	13	j.	j.	PROPN
ejpam-4761	535	14	g.	g.	PROPN
ejpam-4761	535	15	canoy	canoy	PROPN
ejpam-4761	535	16	.	.	PUNCT
ejpam-4761	536	1	hop	hop	PROPN
ejpam-4761	536	2	dominating	dominating	NOUN
ejpam-4761	536	3	sets	set	NOUN
ejpam-4761	536	4	in	in	ADP
ejpam-4761	536	5	graphs	graph	NOUN
ejpam-4761	536	6	under	under	ADP
ejpam-4761	536	7	binary	binary	ADJ
ejpam-4761	536	8	operations	operation	NOUN
ejpam-4761	536	9	.	.	PUNCT
ejpam-4761	537	1	eur	eur	PROPN
ejpam-4761	537	2	.	.	PUNCT
ejpam-4761	538	1	j.	j.	PROPN
ejpam-4761	538	2	pure	pure	PROPN
ejpam-4761	538	3	appl	appl	PROPN
ejpam-4761	538	4	.	.	PUNCT
ejpam-4761	538	5	math	math	PROPN
ejpam-4761	538	6	.	.	PUNCT
ejpam-4761	538	7	,	,	PUNCT
ejpam-4761	539	1	12(4):1455–1463	12(4):1455–1463	NUM
ejpam-4761	539	2	,	,	PUNCT
ejpam-4761	539	3	2019	2019	NUM
ejpam-4761	539	4	.	.	PUNCT
ejpam-4761	540	1	[	[	X
ejpam-4761	540	2	14	14	NUM
ejpam-4761	540	3	]	]	PUNCT
ejpam-4761	540	4	m.	m.	NOUN
ejpam-4761	540	5	labendia	labendia	PROPN
ejpam-4761	540	6	and	and	CCONJ
ejpam-4761	540	7	s.	s.	PROPN
ejpam-4761	540	8	canoy	canoy	PROPN
ejpam-4761	540	9	jr	jr	PROPN
ejpam-4761	540	10	.	.	PROPN
ejpam-4761	540	11	convex	convex	PROPN
ejpam-4761	540	12	domination	domination	NOUN
ejpam-4761	540	13	in	in	ADP
ejpam-4761	540	14	the	the	DET
ejpam-4761	540	15	composition	composition	NOUN
ejpam-4761	540	16	and	and	CCONJ
ejpam-4761	540	17	cartesian	cartesian	ADJ
ejpam-4761	540	18	product	product	NOUN
ejpam-4761	540	19	of	of	ADP
ejpam-4761	540	20	graphs	graph	NOUN
ejpam-4761	540	21	.	.	PUNCT
ejpam-4761	541	1	czechoslovak	czechoslovak	ADJ
ejpam-4761	541	2	mathematical	mathematical	PROPN
ejpam-4761	541	3	journal	journal	NOUN
ejpam-4761	541	4	,	,	PUNCT
ejpam-4761	541	5	62(4):1003–1009	62(4):1003–1009	NUM
ejpam-4761	541	6	,	,	PUNCT
ejpam-4761	541	7	2012	2012	NUM
ejpam-4761	541	8	.	.	PUNCT
ejpam-4761	542	1	[	[	X
ejpam-4761	542	2	15	15	NUM
ejpam-4761	542	3	]	]	X
ejpam-4761	542	4	m.	m.	NOUN
ejpam-4761	542	5	lemanska	lemanska	PROPN
ejpam-4761	542	6	.	.	PUNCT
ejpam-4761	543	1	weakly	weakly	ADJ
ejpam-4761	543	2	convex	convex	NOUN
ejpam-4761	543	3	and	and	CCONJ
ejpam-4761	543	4	convex	convex	ADJ
ejpam-4761	543	5	domination	domination	NOUN
ejpam-4761	543	6	numbers	number	NOUN
ejpam-4761	543	7	.	.	PUNCT
ejpam-4761	544	1	opusc	opusc	PROPN
ejpam-4761	544	2	.	.	PUNCT
ejpam-4761	544	3	math	math	NOUN
ejpam-4761	544	4	.	.	PUNCT
ejpam-4761	544	5	,	,	PUNCT
ejpam-4761	544	6	24:181	24:181	NUM
ejpam-4761	544	7	–	–	PUNCT
ejpam-4761	544	8	188	188	NUM
ejpam-4761	544	9	,	,	PUNCT
ejpam-4761	544	10	2004	2004	NUM
ejpam-4761	544	11	.	.	PUNCT
ejpam-4761	545	1	[	[	X
ejpam-4761	545	2	16	16	NUM
ejpam-4761	545	3	]	]	PUNCT
ejpam-4761	545	4	r.	r.	PROPN
ejpam-4761	545	5	leonida	leonida	PROPN
ejpam-4761	545	6	and	and	CCONJ
ejpam-4761	545	7	s.	s.	PROPN
ejpam-4761	545	8	canoy	canoy	PROPN
ejpam-4761	545	9	jr	jr	PROPN
ejpam-4761	545	10	.	.	PUNCT
ejpam-4761	545	11	weakly	weakly	ADJ
ejpam-4761	545	12	convex	convex	NOUN
ejpam-4761	545	13	and	and	CCONJ
ejpam-4761	545	14	weakly	weakly	ADV
ejpam-4761	545	15	connected	connected	ADJ
ejpam-4761	545	16	independent	independent	ADJ
ejpam-4761	545	17	domination	domination	NOUN
ejpam-4761	545	18	in	in	ADP
ejpam-4761	545	19	the	the	DET
ejpam-4761	545	20	corona	corona	NOUN
ejpam-4761	545	21	of	of	ADP
ejpam-4761	545	22	graphs	graph	NOUN
ejpam-4761	545	23	.	.	PUNCT
ejpam-4761	546	1	international	international	ADJ
ejpam-4761	546	2	mathematical	mathematical	PROPN
ejpam-4761	546	3	forum	forum	PROPN
ejpam-4761	546	4	.	.	PROPN
ejpam-4761	546	5	,	,	PUNCT
ejpam-4761	546	6	31(8):1515	31(8):1515	NUM
ejpam-4761	546	7	–	–	PUNCT
ejpam-4761	546	8	1522	1522	NUM
ejpam-4761	546	9	,	,	PUNCT
ejpam-4761	546	10	2013	2013	NUM
ejpam-4761	546	11	.	.	PUNCT
ejpam-4761	547	1	[	[	X
ejpam-4761	547	2	17	17	NUM
ejpam-4761	547	3	]	]	PUNCT
ejpam-4761	547	4	r.	r.	PROPN
ejpam-4761	547	5	leonida	leonida	PROPN
ejpam-4761	547	6	and	and	CCONJ
ejpam-4761	547	7	s.	s.	PROPN
ejpam-4761	547	8	canoy	canoy	PROPN
ejpam-4761	547	9	jr	jr	PROPN
ejpam-4761	547	10	.	.	PROPN
ejpam-4761	547	11	weakly	weakly	ADJ
ejpam-4761	547	12	convexity	convexity	NOUN
ejpam-4761	547	13	and	and	CCONJ
ejpam-4761	547	14	weakly	weakly	ADJ
ejpam-4761	547	15	convex	convex	ADJ
ejpam-4761	547	16	domination	domination	NOUN
ejpam-4761	547	17	in	in	ADP
ejpam-4761	547	18	graphs	graph	NOUN
ejpam-4761	547	19	.	.	PUNCT
ejpam-4761	548	1	applied	apply	VERB
ejpam-4761	548	2	mathematical	mathematical	ADJ
ejpam-4761	548	3	sciences	science	NOUN
ejpam-4761	548	4	,	,	PUNCT
ejpam-4761	548	5	3(9):137–141	3(9):137–141	NUM
ejpam-4761	548	6	,	,	PUNCT
ejpam-4761	548	7	2015	2015	NUM
ejpam-4761	548	8	.	.	PUNCT
ejpam-4761	549	1	[	[	X
ejpam-4761	549	2	18	18	NUM
ejpam-4761	549	3	]	]	X
ejpam-4761	549	4	j.	j.	PROPN
ejpam-4761	549	5	mohamad	mohamad	PROPN
ejpam-4761	549	6	and	and	CCONJ
ejpam-4761	549	7	h.	h.	PROPN
ejpam-4761	549	8	rara	rara	PROPN
ejpam-4761	549	9	.	.	PUNCT
ejpam-4761	550	1	strong	strong	ADJ
ejpam-4761	550	2	resolving	resolve	VERB
ejpam-4761	550	3	hop	hop	NOUN
ejpam-4761	550	4	domination	domination	NOUN
ejpam-4761	550	5	in	in	ADP
ejpam-4761	550	6	graphs	graph	NOUN
ejpam-4761	550	7	.	.	PUNCT
ejpam-4761	551	1	european	european	ADJ
ejpam-4761	551	2	journal	journal	PROPN
ejpam-4761	551	3	of	of	ADP
ejpam-4761	551	4	pure	pure	ADJ
ejpam-4761	551	5	and	and	CCONJ
ejpam-4761	551	6	applied	applied	ADJ
ejpam-4761	551	7	mathematics	mathematic	NOUN
ejpam-4761	551	8	.	.	PUNCT
ejpam-4761	551	9	,	,	PUNCT
ejpam-4761	551	10	16(1):131–143	16(1):131–143	PROPN
ejpam-4761	551	11	,	,	PUNCT
ejpam-4761	551	12	2023	2023	NUM
ejpam-4761	551	13	.	.	PUNCT
ejpam-4761	552	1	[	[	X
ejpam-4761	552	2	19	19	NUM
ejpam-4761	552	3	]	]	X
ejpam-4761	552	4	c.	c.	PROPN
ejpam-4761	552	5	natarajan	natarajan	PROPN
ejpam-4761	552	6	and	and	CCONJ
ejpam-4761	552	7	s.	s.	PROPN
ejpam-4761	552	8	ayyaswamy	ayyaswamy	PROPN
ejpam-4761	552	9	.	.	PUNCT
ejpam-4761	553	1	hop	hop	PROPN
ejpam-4761	553	2	domination	domination	NOUN
ejpam-4761	553	3	in	in	ADP
ejpam-4761	553	4	graphs	graphs	PROPN
ejpam-4761	553	5	ii	ii	PROPN
ejpam-4761	553	6	.	.	PUNCT
ejpam-4761	553	7	versita	versita	PROPN
ejpam-4761	553	8	,	,	PUNCT
ejpam-4761	553	9	23(2):187	23(2):187	NUM
ejpam-4761	553	10	–	–	PUNCT
ejpam-4761	553	11	199	199	NUM
ejpam-4761	553	12	,	,	PUNCT
ejpam-4761	553	13	2015	2015	NUM
ejpam-4761	553	14	.	.	PUNCT
ejpam-4761	554	1	[	[	X
ejpam-4761	554	2	20	20	NUM
ejpam-4761	554	3	]	]	PUNCT
ejpam-4761	554	4	b.	b.	PROPN
ejpam-4761	554	5	omamalin	omamalin	PROPN
ejpam-4761	554	6	,	,	PUNCT
ejpam-4761	554	7	s.	s.	PROPN
ejpam-4761	554	8	canoy	canoy	PROPN
ejpam-4761	554	9	jr	jr	PROPN
ejpam-4761	554	10	.	.	PROPN
ejpam-4761	554	11	,	,	PUNCT
ejpam-4761	554	12	and	and	CCONJ
ejpam-4761	554	13	h.	h.	PROPN
ejpam-4761	554	14	rara	rara	PROPN
ejpam-4761	554	15	.	.	PUNCT
ejpam-4761	555	1	locating	locate	VERB
ejpam-4761	555	2	total	total	ADJ
ejpam-4761	555	3	dominating	dominating	NOUN
ejpam-4761	555	4	sets	set	NOUN
ejpam-4761	555	5	in	in	ADP
ejpam-4761	555	6	the	the	DET
ejpam-4761	555	7	join	join	NOUN
ejpam-4761	555	8	,	,	PUNCT
ejpam-4761	555	9	corona	corona	NOUN
ejpam-4761	555	10	and	and	CCONJ
ejpam-4761	555	11	composition	composition	NOUN
ejpam-4761	555	12	of	of	ADP
ejpam-4761	555	13	graphs	graph	NOUN
ejpam-4761	555	14	.	.	PUNCT
ejpam-4761	556	1	applied	apply	VERB
ejpam-4761	556	2	mathematical	mathematical	ADJ
ejpam-4761	556	3	sciences	science	NOUN
ejpam-4761	556	4	.	.	PUNCT
ejpam-4761	556	5	,	,	PUNCT
ejpam-4761	556	6	8(48):2363–2374	8(48):2363–2374	NUM
ejpam-4761	556	7	,	,	PUNCT
ejpam-4761	556	8	2014	2014	NUM
ejpam-4761	556	9	.	.	PUNCT
ejpam-4761	557	1	[	[	X
ejpam-4761	557	2	21	21	NUM
ejpam-4761	557	3	]	]	X
ejpam-4761	557	4	y.	y.	PROPN
ejpam-4761	557	5	pabilona	pabilona	PROPN
ejpam-4761	557	6	and	and	CCONJ
ejpam-4761	557	7	h.	h.	PROPN
ejpam-4761	557	8	rara	rara	PROPN
ejpam-4761	557	9	.	.	PUNCT
ejpam-4761	558	1	connected	connect	VERB
ejpam-4761	558	2	hop	hop	NOUN
ejpam-4761	558	3	domination	domination	NOUN
ejpam-4761	558	4	in	in	ADP
ejpam-4761	558	5	graphs	graph	NOUN
ejpam-4761	558	6	under	under	ADP
ejpam-4761	558	7	some	some	DET
ejpam-4761	558	8	binary	binary	ADJ
ejpam-4761	558	9	operations	operation	NOUN
ejpam-4761	558	10	.	.	PUNCT
ejpam-4761	559	1	asian	asian	ADJ
ejpam-4761	559	2	-	-	PUNCT
ejpam-4761	559	3	eur	eur	NOUN
ejpam-4761	559	4	.	.	PUNCT
ejpam-4761	560	1	j.	j.	PROPN
ejpam-4761	560	2	math	math	PROPN
ejpam-4761	560	3	.	.	PROPN
ejpam-4761	560	4	,	,	PUNCT
ejpam-4761	560	5	11(5):1850075–1–1850075–11	11(5):1850075–1–1850075–11	NUM
ejpam-4761	560	6	,	,	PUNCT
ejpam-4761	560	7	2018	2018	NUM
ejpam-4761	560	8	.	.	PUNCT
ejpam-4761	561	1	[	[	X
ejpam-4761	561	2	22	22	NUM
ejpam-4761	561	3	]	]	PUNCT
ejpam-4761	561	4	j.	j.	PROPN
ejpam-4761	561	5	raczed	racze	VERB
ejpam-4761	561	6	and	and	CCONJ
ejpam-4761	561	7	m.	m.	NOUN
ejpam-4761	561	8	lemanska	lemanska	PROPN
ejpam-4761	561	9	.	.	PUNCT
ejpam-4761	562	1	a	a	DET
ejpam-4761	562	2	note	note	NOUN
ejpam-4761	562	3	on	on	ADP
ejpam-4761	562	4	weakly	weakly	ADJ
ejpam-4761	562	5	convex	convex	NOUN
ejpam-4761	562	6	and	and	CCONJ
ejpam-4761	562	7	convex	convex	ADJ
ejpam-4761	562	8	domination	domination	NOUN
ejpam-4761	562	9	numbers	number	NOUN
ejpam-4761	562	10	of	of	ADP
ejpam-4761	562	11	a	a	DET
ejpam-4761	562	12	torus	torus	NOUN
ejpam-4761	562	13	.	.	PUNCT
ejpam-4761	563	1	discrete	discrete	ADJ
ejpam-4761	563	2	appl	appl	PROPN
ejpam-4761	563	3	.	.	PUNCT
ejpam-4761	563	4	math	math	PROPN
ejpam-4761	563	5	.	.	PUNCT
ejpam-4761	563	6	,	,	PUNCT
ejpam-4761	563	7	(	(	PUNCT
ejpam-4761	563	8	158):1708–1713	158):1708–1713	NUM
ejpam-4761	563	9	,	,	PUNCT
ejpam-4761	563	10	2010	2010	NUM
ejpam-4761	563	11	.	.	PUNCT
ejpam-4761	564	1	references	reference	NOUN
ejpam-4761	564	2	1211	1211	NUM
ejpam-4761	564	3	[	[	X
ejpam-4761	564	4	23	23	NUM
ejpam-4761	564	5	]	]	PUNCT
ejpam-4761	564	6	m.	m.	NOUN
ejpam-4761	564	7	roscika	roscika	PROPN
ejpam-4761	564	8	.	.	PUNCT
ejpam-4761	565	1	convex	convex	VERB
ejpam-4761	565	2	and	and	CCONJ
ejpam-4761	565	3	weak	weak	ADJ
ejpam-4761	565	4	convex	convex	NOUN
ejpam-4761	565	5	domination	domination	NOUN
ejpam-4761	565	6	in	in	ADP
ejpam-4761	565	7	prism	prism	NOUN
ejpam-4761	565	8	graphs	graph	NOUN
ejpam-4761	565	9	.	.	PUNCT
ejpam-4761	566	1	discussiones	discussione	NOUN
ejpam-4761	566	2	mathematicae	mathematicae	PROPN
ejpam-4761	566	3	graph	graph	NOUN
ejpam-4761	566	4	theory	theory	NOUN
ejpam-4761	566	5	.	.	PUNCT
ejpam-4761	566	6	,	,	PUNCT
ejpam-4761	566	7	39:741–755	39:741–755	NUM
ejpam-4761	566	8	,	,	PUNCT
ejpam-4761	566	9	2019	2019	NUM
ejpam-4761	566	10	.	.	PUNCT
ejpam-4761	567	1	[	[	X
ejpam-4761	567	2	24	24	NUM
ejpam-4761	567	3	]	]	X
ejpam-4761	567	4	g.	g.	NOUN
ejpam-4761	567	5	salasalan	salasalan	NOUN
ejpam-4761	567	6	and	and	CCONJ
ejpam-4761	567	7	s.	s.	PROPN
ejpam-4761	567	8	canoy	canoy	PROPN
ejpam-4761	567	9	jr	jr	PROPN
ejpam-4761	567	10	.	.	PROPN
ejpam-4761	567	11	global	global	PROPN
ejpam-4761	567	12	hop	hop	PROPN
ejpam-4761	567	13	domination	domination	PROPN
ejpam-4761	567	14	numbers	number	NOUN
ejpam-4761	567	15	of	of	ADP
ejpam-4761	567	16	graphs	graph	NOUN
ejpam-4761	567	17	.	.	PUNCT
ejpam-4761	568	1	eur	eur	PROPN
ejpam-4761	568	2	.	.	PUNCT
ejpam-4761	569	1	j.	j.	PROPN
ejpam-4761	569	2	pure	pure	PROPN
ejpam-4761	569	3	appl	appl	PROPN
ejpam-4761	569	4	.	.	PUNCT
ejpam-4761	569	5	math	math	PROPN
ejpam-4761	569	6	.	.	PUNCT
ejpam-4761	569	7	,	,	PUNCT
ejpam-4761	569	8	14(1):112–125	14(1):112–125	NUM
ejpam-4761	569	9	,	,	PUNCT
ejpam-4761	569	10	2021	2021	NUM
ejpam-4761	569	11	.	.	PUNCT