id	sid	tid	token	lemma	pos
ejpam-5984	1	1	european	european	PROPN
ejpam-5984	1	2	journal	journal	PROPN
ejpam-5984	1	3	of	of	ADP
ejpam-5984	1	4	pure	pure	ADJ
ejpam-5984	1	5	and	and	CCONJ
ejpam-5984	1	6	applied	applied	ADJ
ejpam-5984	1	7	mathematics	mathematic	NOUN
ejpam-5984	1	8	2025	2025	NUM
ejpam-5984	1	9	,	,	PUNCT
ejpam-5984	1	10	vol	vol	NOUN
ejpam-5984	1	11	.	.	PROPN
ejpam-5984	1	12	18	18	NUM
ejpam-5984	1	13	,	,	PUNCT
ejpam-5984	1	14	issue	issue	NOUN
ejpam-5984	1	15	2	2	NUM
ejpam-5984	1	16	,	,	PUNCT
ejpam-5984	1	17	article	article	NOUN
ejpam-5984	1	18	number	number	NOUN
ejpam-5984	1	19	5984	5984	NUM
ejpam-5984	1	20	issn	issn	PROPN
ejpam-5984	1	21	1307	1307	NUM
ejpam-5984	1	22	-	-	SYM
ejpam-5984	1	23	5543	5543	NUM
ejpam-5984	1	24	–	–	PUNCT
ejpam-5984	1	25	ejpam.com	ejpam.com	X
ejpam-5984	1	26	published	publish	VERB
ejpam-5984	1	27	by	by	ADP
ejpam-5984	1	28	new	new	PROPN
ejpam-5984	1	29	york	york	PROPN
ejpam-5984	1	30	business	business	PROPN
ejpam-5984	1	31	global	global	ADJ
ejpam-5984	1	32	independent	independent	ADJ
ejpam-5984	1	33	double	double	ADJ
ejpam-5984	1	34	roman	roman	ADJ
ejpam-5984	1	35	domination	domination	NOUN
ejpam-5984	1	36	stability	stability	NOUN
ejpam-5984	1	37	in	in	ADP
ejpam-5984	1	38	graphs	graph	NOUN
ejpam-5984	1	39	seyed	seyed	PROPN
ejpam-5984	1	40	mahmoud	mahmoud	PROPN
ejpam-5984	1	41	sheikholeslami1,∗	sheikholeslami1,∗	PROPN
ejpam-5984	1	42	,	,	PUNCT
ejpam-5984	1	43	mina	mina	PROPN
ejpam-5984	1	44	esmaeili1	esmaeili1	PROPN
ejpam-5984	1	45	,	,	PUNCT
ejpam-5984	1	46	jamil	jamil	PROPN
ejpam-5984	1	47	j.	j.	PROPN
ejpam-5984	1	48	hamja2,5	hamja2,5	PROPN
ejpam-5984	1	49	,	,	PUNCT
ejpam-5984	1	50	cris	cris	PROPN
ejpam-5984	1	51	l.	l.	PROPN
ejpam-5984	1	52	armada3,4	armada3,4	PROPN
ejpam-5984	1	53	,	,	PUNCT
ejpam-5984	1	54	imelda	imelda	PROPN
ejpam-5984	1	55	s.	s.	PROPN
ejpam-5984	1	56	aniversario5	aniversario5	PROPN
ejpam-5984	1	57	1	1	NUM
ejpam-5984	1	58	department	department	NOUN
ejpam-5984	1	59	of	of	ADP
ejpam-5984	1	60	mathematics	mathematics	PROPN
ejpam-5984	1	61	,	,	PUNCT
ejpam-5984	1	62	azarbaijan	azarbaijan	NOUN
ejpam-5984	1	63	shahid	shahid	PROPN
ejpam-5984	1	64	madani	madani	PROPN
ejpam-5984	1	65	university	university	PROPN
ejpam-5984	1	66	,	,	PUNCT
ejpam-5984	1	67	tabriz	tabriz	NOUN
ejpam-5984	1	68	,	,	PUNCT
ejpam-5984	1	69	i.r	i.r	PROPN
ejpam-5984	1	70	.	.	PROPN
ejpam-5984	1	71	iran	iran	PROPN
ejpam-5984	1	72	2	2	NUM
ejpam-5984	1	73	department	department	NOUN
ejpam-5984	1	74	of	of	ADP
ejpam-5984	1	75	mathematics	mathematic	NOUN
ejpam-5984	1	76	,	,	PUNCT
ejpam-5984	1	77	college	college	NOUN
ejpam-5984	1	78	of	of	ADP
ejpam-5984	1	79	arts	art	NOUN
ejpam-5984	1	80	and	and	CCONJ
ejpam-5984	1	81	sciences	science	NOUN
ejpam-5984	1	82	,	,	PUNCT
ejpam-5984	1	83	msu	msu	PROPN
ejpam-5984	1	84	tawi	tawi	PROPN
ejpam-5984	1	85	-	-	PUNCT
ejpam-5984	1	86	tawi	tawi	PROPN
ejpam-5984	1	87	college	college	PROPN
ejpam-5984	1	88	of	of	ADP
ejpam-5984	1	89	technology	technology	NOUN
ejpam-5984	1	90	and	and	CCONJ
ejpam-5984	1	91	oceanography	oceanography	NOUN
ejpam-5984	1	92	,	,	PUNCT
ejpam-5984	1	93	7500	7500	NUM
ejpam-5984	1	94	tawi	tawi	NOUN
ejpam-5984	1	95	-	-	PUNCT
ejpam-5984	1	96	tawi	tawi	NOUN
ejpam-5984	1	97	,	,	PUNCT
ejpam-5984	1	98	philippines	philippines	PROPN
ejpam-5984	1	99	3	3	NUM
ejpam-5984	1	100	vietnam	vietnam	PROPN
ejpam-5984	1	101	national	national	PROPN
ejpam-5984	1	102	university	university	PROPN
ejpam-5984	1	103	ho	ho	PROPN
ejpam-5984	1	104	chi	chi	PROPN
ejpam-5984	1	105	minh	minh	PROPN
ejpam-5984	1	106	city	city	PROPN
ejpam-5984	1	107	,	,	PUNCT
ejpam-5984	1	108	linh	linh	NOUN
ejpam-5984	1	109	trung	trung	VERB
ejpam-5984	1	110	ward	ward	NOUN
ejpam-5984	1	111	,	,	PUNCT
ejpam-5984	1	112	thu	thu	PROPN
ejpam-5984	1	113	duc	duc	PROPN
ejpam-5984	1	114	city	city	PROPN
ejpam-5984	1	115	,	,	PUNCT
ejpam-5984	1	116	ho	ho	PROPN
ejpam-5984	1	117	chi	chi	PROPN
ejpam-5984	1	118	minh	minh	PROPN
ejpam-5984	1	119	city	city	PROPN
ejpam-5984	1	120	,	,	PUNCT
ejpam-5984	1	121	vietnam	vietnam	PROPN
ejpam-5984	1	122	4	4	NUM
ejpam-5984	1	123	department	department	NOUN
ejpam-5984	1	124	of	of	ADP
ejpam-5984	1	125	applied	apply	VERB
ejpam-5984	1	126	mathematics	mathematic	NOUN
ejpam-5984	1	127	,	,	PUNCT
ejpam-5984	1	128	faculty	faculty	NOUN
ejpam-5984	1	129	of	of	ADP
ejpam-5984	1	130	applied	apply	VERB
ejpam-5984	1	131	science	science	NOUN
ejpam-5984	1	132	,	,	PUNCT
ejpam-5984	1	133	ho	ho	PROPN
ejpam-5984	1	134	chi	chi	PROPN
ejpam-5984	1	135	minh	minh	PROPN
ejpam-5984	1	136	city	city	PROPN
ejpam-5984	1	137	university	university	PROPN
ejpam-5984	1	138	of	of	ADP
ejpam-5984	1	139	technology	technology	NOUN
ejpam-5984	1	140	(	(	PUNCT
ejpam-5984	1	141	hcmut	hcmut	ADJ
ejpam-5984	1	142	)	)	PUNCT
ejpam-5984	1	143	,	,	PUNCT
ejpam-5984	1	144	268	268	NUM
ejpam-5984	1	145	ly	ly	ADP
ejpam-5984	1	146	thuong	thuong	NOUN
ejpam-5984	1	147	kiet	kiet	PROPN
ejpam-5984	1	148	,	,	PUNCT
ejpam-5984	1	149	district	district	NOUN
ejpam-5984	1	150	10	10	NUM
ejpam-5984	1	151	,	,	PUNCT
ejpam-5984	1	152	ward	ward	NOUN
ejpam-5984	1	153	14	14	NUM
ejpam-5984	1	154	,	,	PUNCT
ejpam-5984	1	155	ho	ho	PROPN
ejpam-5984	1	156	chi	chi	PROPN
ejpam-5984	1	157	minh	minh	PROPN
ejpam-5984	1	158	city	city	PROPN
ejpam-5984	1	159	,	,	PUNCT
ejpam-5984	1	160	vietnam	vietnam	PROPN
ejpam-5984	1	161	5	5	NUM
ejpam-5984	1	162	department	department	NOUN
ejpam-5984	1	163	of	of	ADP
ejpam-5984	1	164	mathematics	mathematic	NOUN
ejpam-5984	1	165	and	and	CCONJ
ejpam-5984	1	166	statistics	statistic	NOUN
ejpam-5984	1	167	,	,	PUNCT
ejpam-5984	1	168	college	college	NOUN
ejpam-5984	1	169	of	of	ADP
ejpam-5984	1	170	science	science	NOUN
ejpam-5984	1	171	and	and	CCONJ
ejpam-5984	1	172	mathematics	mathematic	NOUN
ejpam-5984	1	173	,	,	PUNCT
ejpam-5984	1	174	msu	msu	PROPN
ejpam-5984	1	175	iligan	iligan	PROPN
ejpam-5984	1	176	institute	institute	PROPN
ejpam-5984	1	177	of	of	ADP
ejpam-5984	1	178	technology	technology	PROPN
ejpam-5984	1	179	,	,	PUNCT
ejpam-5984	1	180	9200	9200	NUM
ejpam-5984	1	181	iligan	iligan	ADJ
ejpam-5984	1	182	city	city	NOUN
ejpam-5984	1	183	,	,	PUNCT
ejpam-5984	1	184	philippines	philippine	NOUN
ejpam-5984	1	185	abstract	abstract	ADJ
ejpam-5984	1	186	.	.	PUNCT
ejpam-5984	2	1	an	an	DET
ejpam-5984	2	2	independent	independent	ADJ
ejpam-5984	2	3	double	double	ADJ
ejpam-5984	2	4	roman	roman	ADJ
ejpam-5984	2	5	dominating	dominating	NOUN
ejpam-5984	2	6	function	function	NOUN
ejpam-5984	2	7	(	(	PUNCT
ejpam-5984	2	8	idrd	idrd	ADJ
ejpam-5984	2	9	-	-	PUNCT
ejpam-5984	2	10	function	function	NOUN
ejpam-5984	2	11	)	)	PUNCT
ejpam-5984	2	12	on	on	ADP
ejpam-5984	2	13	a	a	DET
ejpam-5984	2	14	graph	graph	NOUN
ejpam-5984	2	15	g	g	NOUN
ejpam-5984	2	16	is	be	AUX
ejpam-5984	2	17	a	a	DET
ejpam-5984	2	18	function	function	NOUN
ejpam-5984	2	19	f	f	NOUN
ejpam-5984	2	20	:	:	PUNCT
ejpam-5984	2	21	v	v	X
ejpam-5984	2	22	(	(	PUNCT
ejpam-5984	2	23	g	g	NOUN
ejpam-5984	2	24	)	)	PUNCT
ejpam-5984	2	25	→	→	SYM
ejpam-5984	2	26	{	{	PUNCT
ejpam-5984	2	27	0	0	NUM
ejpam-5984	2	28	,	,	PUNCT
ejpam-5984	2	29	1	1	NUM
ejpam-5984	2	30	,	,	PUNCT
ejpam-5984	2	31	2	2	NUM
ejpam-5984	2	32	,	,	PUNCT
ejpam-5984	2	33	3	3	NUM
ejpam-5984	2	34	}	}	PUNCT
ejpam-5984	2	35	having	have	VERB
ejpam-5984	2	36	the	the	DET
ejpam-5984	2	37	property	property	NOUN
ejpam-5984	2	38	that	that	PRON
ejpam-5984	2	39	(	(	PUNCT
ejpam-5984	2	40	i	i	NOUN
ejpam-5984	2	41	)	)	PUNCT
ejpam-5984	2	42	if	if	SCONJ
ejpam-5984	2	43	f(v	f(v	NOUN
ejpam-5984	2	44	)	)	PUNCT
ejpam-5984	2	45	=	=	SYM
ejpam-5984	2	46	0	0	NUM
ejpam-5984	2	47	,	,	PUNCT
ejpam-5984	2	48	then	then	ADV
ejpam-5984	2	49	the	the	DET
ejpam-5984	2	50	vertex	vertex	NOUN
ejpam-5984	2	51	v	v	NOUN
ejpam-5984	2	52	must	must	AUX
ejpam-5984	2	53	have	have	VERB
ejpam-5984	2	54	at	at	ADV
ejpam-5984	2	55	least	least	ADV
ejpam-5984	2	56	two	two	NUM
ejpam-5984	2	57	neighbors	neighbor	NOUN
ejpam-5984	2	58	assigned	assign	VERB
ejpam-5984	2	59	2	2	NUM
ejpam-5984	2	60	under	under	ADP
ejpam-5984	2	61	f	f	PROPN
ejpam-5984	2	62	or	or	CCONJ
ejpam-5984	2	63	one	one	NUM
ejpam-5984	2	64	neighbor	neighbor	NOUN
ejpam-5984	2	65	w	w	NOUN
ejpam-5984	2	66	with	with	ADP
ejpam-5984	2	67	f(w	f(w	NOUN
ejpam-5984	2	68	)	)	PUNCT
ejpam-5984	2	69	=	=	SYM
ejpam-5984	2	70	3	3	NUM
ejpam-5984	2	71	,	,	PUNCT
ejpam-5984	2	72	and	and	CCONJ
ejpam-5984	2	73	if	if	SCONJ
ejpam-5984	2	74	f(v	f(v	NOUN
ejpam-5984	2	75	)	)	PUNCT
ejpam-5984	2	76	=	=	SYM
ejpam-5984	2	77	1	1	NUM
ejpam-5984	2	78	,	,	PUNCT
ejpam-5984	2	79	then	then	ADV
ejpam-5984	2	80	the	the	DET
ejpam-5984	2	81	vertex	vertex	NOUN
ejpam-5984	2	82	v	v	NOUN
ejpam-5984	2	83	must	must	AUX
ejpam-5984	2	84	have	have	VERB
ejpam-5984	2	85	at	at	ADV
ejpam-5984	2	86	least	least	ADV
ejpam-5984	2	87	one	one	NUM
ejpam-5984	2	88	neighbor	neighbor	NOUN
ejpam-5984	2	89	w	w	NOUN
ejpam-5984	2	90	with	with	ADP
ejpam-5984	2	91	f(w	f(w	PROPN
ejpam-5984	2	92	)	)	PUNCT
ejpam-5984	2	93	≥	≥	NOUN
ejpam-5984	2	94	2	2	NUM
ejpam-5984	2	95	,	,	PUNCT
ejpam-5984	2	96	and	and	CCONJ
ejpam-5984	2	97	(	(	PUNCT
ejpam-5984	2	98	ii	ii	NOUN
ejpam-5984	2	99	)	)	PUNCT
ejpam-5984	2	100	the	the	DET
ejpam-5984	2	101	subgraph	subgraph	NOUN
ejpam-5984	2	102	induced	induce	VERB
ejpam-5984	2	103	by	by	ADP
ejpam-5984	2	104	the	the	DET
ejpam-5984	2	105	vertices	vertex	NOUN
ejpam-5984	2	106	with	with	ADP
ejpam-5984	2	107	positive	positive	ADJ
ejpam-5984	2	108	weight	weight	NOUN
ejpam-5984	2	109	under	under	ADP
ejpam-5984	2	110	f	f	PROPN
ejpam-5984	2	111	is	be	AUX
ejpam-5984	2	112	edgeless	edgeless	NOUN
ejpam-5984	2	113	.	.	PUNCT
ejpam-5984	3	1	the	the	DET
ejpam-5984	3	2	weight	weight	NOUN
ejpam-5984	3	3	of	of	ADP
ejpam-5984	3	4	an	an	DET
ejpam-5984	3	5	idrd	idrd	ADJ
ejpam-5984	3	6	-	-	PUNCT
ejpam-5984	3	7	function	function	NOUN
ejpam-5984	3	8	is	be	AUX
ejpam-5984	3	9	the	the	DET
ejpam-5984	3	10	sum	sum	NOUN
ejpam-5984	3	11	of	of	ADP
ejpam-5984	3	12	its	its	PRON
ejpam-5984	3	13	function	function	NOUN
ejpam-5984	3	14	values	value	NOUN
ejpam-5984	3	15	over	over	ADP
ejpam-5984	3	16	all	all	DET
ejpam-5984	3	17	vertices	vertex	NOUN
ejpam-5984	3	18	,	,	PUNCT
ejpam-5984	3	19	and	and	CCONJ
ejpam-5984	3	20	the	the	DET
ejpam-5984	3	21	independent	independent	ADJ
ejpam-5984	3	22	double	double	ADJ
ejpam-5984	3	23	roman	roman	ADJ
ejpam-5984	3	24	domination	domination	NOUN
ejpam-5984	3	25	number	number	NOUN
ejpam-5984	3	26	(	(	PUNCT
ejpam-5984	3	27	idrd	idrd	ADJ
ejpam-5984	3	28	-	-	PUNCT
ejpam-5984	3	29	number	number	NOUN
ejpam-5984	3	30	)	)	PUNCT
ejpam-5984	3	31	idr(g	idr(g	PROPN
ejpam-5984	3	32	)	)	PUNCT
ejpam-5984	3	33	is	be	AUX
ejpam-5984	3	34	the	the	DET
ejpam-5984	3	35	minimum	minimum	ADJ
ejpam-5984	3	36	weight	weight	NOUN
ejpam-5984	3	37	of	of	ADP
ejpam-5984	3	38	an	an	DET
ejpam-5984	3	39	idrd	idrd	ADJ
ejpam-5984	3	40	-	-	PUNCT
ejpam-5984	3	41	function	function	NOUN
ejpam-5984	3	42	on	on	ADP
ejpam-5984	3	43	g.	g.	PROPN
ejpam-5984	3	44	the	the	DET
ejpam-5984	3	45	idr	idr	PROPN
ejpam-5984	3	46	-	-	NOUN
ejpam-5984	3	47	stability	stability	NOUN
ejpam-5984	3	48	(	(	PUNCT
ejpam-5984	3	49	i−drstability	i−drstability	NOUN
ejpam-5984	3	50	,	,	PUNCT
ejpam-5984	3	51	i+dr	i+dr	NOUN
ejpam-5984	3	52	-	-	PUNCT
ejpam-5984	3	53	stability	stability	NOUN
ejpam-5984	3	54	)	)	PUNCT
ejpam-5984	3	55	of	of	ADP
ejpam-5984	3	56	g	g	NOUN
ejpam-5984	3	57	,	,	PUNCT
ejpam-5984	3	58	denoted	denote	VERB
ejpam-5984	3	59	by	by	ADP
ejpam-5984	3	60	stidr(g	stidr(g	PROPN
ejpam-5984	3	61	)	)	PUNCT
ejpam-5984	3	62	(	(	PUNCT
ejpam-5984	3	63	st−idr(g	st−idr(g	PROPN
ejpam-5984	3	64	)	)	PUNCT
ejpam-5984	3	65	,	,	PUNCT
ejpam-5984	3	66	st+idr(g	st+idr(g	PROPN
ejpam-5984	3	67	)	)	PUNCT
ejpam-5984	3	68	)	)	PUNCT
ejpam-5984	3	69	,	,	PUNCT
ejpam-5984	3	70	is	be	AUX
ejpam-5984	3	71	defined	define	VERB
ejpam-5984	3	72	as	as	ADP
ejpam-5984	3	73	the	the	DET
ejpam-5984	3	74	minimum	minimum	ADJ
ejpam-5984	3	75	size	size	NOUN
ejpam-5984	3	76	of	of	ADP
ejpam-5984	3	77	a	a	DET
ejpam-5984	3	78	set	set	NOUN
ejpam-5984	3	79	of	of	ADP
ejpam-5984	3	80	vertices	vertex	NOUN
ejpam-5984	3	81	whose	whose	DET
ejpam-5984	3	82	removal	removal	NOUN
ejpam-5984	3	83	changes	change	NOUN
ejpam-5984	3	84	(	(	PUNCT
ejpam-5984	3	85	decreases	decrease	NOUN
ejpam-5984	3	86	,	,	PUNCT
ejpam-5984	3	87	increases	increase	NOUN
ejpam-5984	3	88	)	)	PUNCT
ejpam-5984	3	89	the	the	DET
ejpam-5984	3	90	independent	independent	ADJ
ejpam-5984	3	91	double	double	ADJ
ejpam-5984	3	92	roman	roman	ADJ
ejpam-5984	3	93	domination	domination	NOUN
ejpam-5984	3	94	number	number	NOUN
ejpam-5984	3	95	.	.	PUNCT
ejpam-5984	4	1	in	in	ADP
ejpam-5984	4	2	this	this	DET
ejpam-5984	4	3	paper	paper	NOUN
ejpam-5984	4	4	,	,	PUNCT
ejpam-5984	4	5	we	we	PRON
ejpam-5984	4	6	first	first	ADV
ejpam-5984	4	7	determine	determine	VERB
ejpam-5984	4	8	the	the	DET
ejpam-5984	4	9	exact	exact	ADJ
ejpam-5984	4	10	values	value	NOUN
ejpam-5984	4	11	on	on	ADP
ejpam-5984	4	12	the	the	DET
ejpam-5984	4	13	idr	idr	NOUN
ejpam-5984	4	14	-	-	NOUN
ejpam-5984	4	15	stability	stability	NOUN
ejpam-5984	4	16	of	of	ADP
ejpam-5984	4	17	some	some	DET
ejpam-5984	4	18	special	special	ADJ
ejpam-5984	4	19	classes	class	NOUN
ejpam-5984	4	20	of	of	ADP
ejpam-5984	4	21	graphs	graph	NOUN
ejpam-5984	4	22	,	,	PUNCT
ejpam-5984	4	23	and	and	CCONJ
ejpam-5984	4	24	then	then	ADV
ejpam-5984	4	25	present	present	VERB
ejpam-5984	4	26	some	some	DET
ejpam-5984	4	27	bounds	bound	NOUN
ejpam-5984	4	28	on	on	ADP
ejpam-5984	4	29	stidr(g	stidr(g	PROPN
ejpam-5984	4	30	)	)	PUNCT
ejpam-5984	4	31	.	.	PUNCT
ejpam-5984	5	1	in	in	ADP
ejpam-5984	5	2	addition	addition	NOUN
ejpam-5984	5	3	,	,	PUNCT
ejpam-5984	5	4	for	for	ADP
ejpam-5984	5	5	a	a	DET
ejpam-5984	5	6	tree	tree	NOUN
ejpam-5984	5	7	t	t	NOUN
ejpam-5984	5	8	with	with	ADP
ejpam-5984	5	9	maximum	maximum	ADJ
ejpam-5984	5	10	degree	degree	NOUN
ejpam-5984	5	11	∆	∆	PROPN
ejpam-5984	5	12	,	,	PUNCT
ejpam-5984	5	13	we	we	PRON
ejpam-5984	5	14	show	show	VERB
ejpam-5984	5	15	that	that	SCONJ
ejpam-5984	5	16	stidr(t	stidr(t	X
ejpam-5984	5	17	)	)	PUNCT
ejpam-5984	5	18	=	=	SYM
ejpam-5984	5	19	1	1	NUM
ejpam-5984	5	20	and	and	CCONJ
ejpam-5984	5	21	st−idr(t	st−idr(t	NOUN
ejpam-5984	5	22	)	)	PUNCT
ejpam-5984	5	23	≤	≤	PUNCT
ejpam-5984	5	24	∆	∆	NOUN
ejpam-5984	5	25	,	,	PUNCT
ejpam-5984	5	26	and	and	CCONJ
ejpam-5984	5	27	characterize	characterize	VERB
ejpam-5984	5	28	the	the	DET
ejpam-5984	5	29	trees	tree	NOUN
ejpam-5984	5	30	that	that	PRON
ejpam-5984	5	31	achieve	achieve	VERB
ejpam-5984	5	32	the	the	DET
ejpam-5984	5	33	upper	upper	ADJ
ejpam-5984	5	34	bound	bind	VERB
ejpam-5984	5	35	.	.	PUNCT
ejpam-5984	6	1	2020	2020	NUM
ejpam-5984	6	2	mathematics	mathematic	NOUN
ejpam-5984	6	3	subject	subject	NOUN
ejpam-5984	6	4	classifications	classification	NOUN
ejpam-5984	6	5	:	:	PUNCT
ejpam-5984	6	6	05c69	05c69	X
ejpam-5984	6	7	key	key	ADJ
ejpam-5984	6	8	words	word	NOUN
ejpam-5984	6	9	and	and	CCONJ
ejpam-5984	6	10	phrases	phrase	NOUN
ejpam-5984	6	11	:	:	PUNCT
ejpam-5984	6	12	roman	roman	ADJ
ejpam-5984	6	13	domination	domination	NOUN
ejpam-5984	6	14	,	,	PUNCT
ejpam-5984	6	15	double	double	ADJ
ejpam-5984	6	16	roman	roman	ADJ
ejpam-5984	6	17	domination	domination	NOUN
ejpam-5984	6	18	,	,	PUNCT
ejpam-5984	6	19	independent	independent	ADJ
ejpam-5984	6	20	roman	roman	ADJ
ejpam-5984	6	21	domination	domination	NOUN
ejpam-5984	6	22	,	,	PUNCT
ejpam-5984	6	23	independent	independent	ADJ
ejpam-5984	6	24	double	double	ADJ
ejpam-5984	6	25	roman	roman	ADJ
ejpam-5984	6	26	domination	domination	NOUN
ejpam-5984	6	27	,	,	PUNCT
ejpam-5984	6	28	independent	independent	ADJ
ejpam-5984	6	29	double	double	ADJ
ejpam-5984	6	30	roman	roman	ADJ
ejpam-5984	6	31	domination	domination	NOUN
ejpam-5984	6	32	stability	stability	NOUN
ejpam-5984	6	33	∗corresponding	∗corresponde	VERB
ejpam-5984	6	34	author	author	NOUN
ejpam-5984	6	35	.	.	PUNCT
ejpam-5984	7	1	doi	doi	NOUN
ejpam-5984	7	2	:	:	PUNCT
ejpam-5984	7	3	https://doi.org/10.29020/nybg.ejpam.v18i2.5984	https://doi.org/10.29020/nybg.ejpam.v18i2.5984	PROPN
ejpam-5984	7	4	email	email	NOUN
ejpam-5984	7	5	addresses	address	VERB
ejpam-5984	7	6	:	:	PUNCT
ejpam-5984	7	7	s.m.sheikholeslami@azaruniv.ac.ir	s.m.sheikholeslami@azaruniv.ac.ir	NOUN
ejpam-5984	7	8	(	(	PUNCT
ejpam-5984	7	9	s.	s.	PROPN
ejpam-5984	7	10	m.	m.	PROPN
ejpam-5984	7	11	sheikholeslami	sheikholeslami	PROPN
ejpam-5984	7	12	)	)	PUNCT
ejpam-5984	7	13	,	,	PUNCT
ejpam-5984	7	14	minaesmaeeli1999@gmail.com	minaesmaeeli1999@gmail.com	X
ejpam-5984	7	15	(	(	PUNCT
ejpam-5984	7	16	m.	m.	NOUN
ejpam-5984	7	17	esmaeili	esmaeili	PROPN
ejpam-5984	7	18	)	)	PUNCT
ejpam-5984	7	19	,	,	PUNCT
ejpam-5984	7	20	jamilhamja@msutawi-tawi.edu.ph	jamilhamja@msutawi-tawi.edu.ph	PROPN
ejpam-5984	7	21	(	(	PUNCT
ejpam-5984	7	22	j.	j.	PROPN
ejpam-5984	7	23	j.	j.	PROPN
ejpam-5984	7	24	hamja	hamja	PROPN
ejpam-5984	7	25	)	)	PUNCT
ejpam-5984	7	26	,	,	PUNCT
ejpam-5984	7	27	cris.armada@hcmut.edu.vn	cris.armada@hcmut.edu.vn	X
ejpam-5984	7	28	(	(	PUNCT
ejpam-5984	7	29	c.	c.	PROPN
ejpam-5984	7	30	l.	l.	PROPN
ejpam-5984	7	31	armada	armada	PROPN
ejpam-5984	7	32	)	)	PUNCT
ejpam-5984	7	33	,	,	PUNCT
ejpam-5984	7	34	imelda.aniversario@g.msuiit.edu.ph	imelda.aniversario@g.msuiit.edu.ph	PROPN
ejpam-5984	7	35	(	(	PUNCT
ejpam-5984	7	36	i.	i.	PROPN
ejpam-5984	7	37	s.	s.	PROPN
ejpam-5984	7	38	aniversario	aniversario	PROPN
ejpam-5984	7	39	)	)	PUNCT
ejpam-5984	7	40	https://www.ejpam.com	https://www.ejpam.com	X
ejpam-5984	8	1	1	1	NUM
ejpam-5984	8	2	copyright	copyright	NOUN
ejpam-5984	8	3	:	:	PUNCT
ejpam-5984	8	4	©	©	PROPN
ejpam-5984	8	5	2025	2025	NUM
ejpam-5984	8	6	the	the	DET
ejpam-5984	8	7	author(s	author(s	NOUN
ejpam-5984	8	8	)	)	PUNCT
ejpam-5984	8	9	.	.	PUNCT
ejpam-5984	9	1	(	(	PUNCT
ejpam-5984	9	2	cc	cc	NOUN
ejpam-5984	9	3	by	by	ADP
ejpam-5984	9	4	-	-	PUNCT
ejpam-5984	9	5	nc	nc	PROPN
ejpam-5984	9	6	4.0	4.0	NUM
ejpam-5984	9	7	)	)	PUNCT
ejpam-5984	9	8	s.	s.	PROPN
ejpam-5984	9	9	m.	m.	PROPN
ejpam-5984	9	10	sheikholeslami	sheikholeslami	PROPN
ejpam-5984	9	11	et	et	PROPN
ejpam-5984	9	12	al	al	PROPN
ejpam-5984	9	13	.	.	PUNCT
ejpam-5984	9	14	/	/	SYM
ejpam-5984	9	15	eur	eur	PROPN
ejpam-5984	9	16	.	.	PUNCT
ejpam-5984	10	1	j.	j.	PROPN
ejpam-5984	10	2	pure	pure	PROPN
ejpam-5984	10	3	appl	appl	PROPN
ejpam-5984	10	4	.	.	PROPN
ejpam-5984	10	5	math	math	PROPN
ejpam-5984	10	6	,	,	PUNCT
ejpam-5984	10	7	18	18	NUM
ejpam-5984	10	8	(	(	PUNCT
ejpam-5984	10	9	2	2	NUM
ejpam-5984	10	10	)	)	PUNCT
ejpam-5984	10	11	(	(	PUNCT
ejpam-5984	10	12	2025	2025	NUM
ejpam-5984	10	13	)	)	PUNCT
ejpam-5984	10	14	,	,	PUNCT
ejpam-5984	10	15	5984	5984	NUM
ejpam-5984	10	16	2	2	NUM
ejpam-5984	10	17	of	of	ADP
ejpam-5984	10	18	16	16	NUM
ejpam-5984	10	19	1	1	NUM
ejpam-5984	10	20	.	.	PUNCT
ejpam-5984	11	1	introduction	introduction	NOUN
ejpam-5984	11	2	roman	roman	PROPN
ejpam-5984	11	3	domination	domination	NOUN
ejpam-5984	11	4	,	,	PUNCT
ejpam-5984	11	5	introduced	introduce	VERB
ejpam-5984	11	6	by	by	ADP
ejpam-5984	11	7	cockayne	cockayne	PROPN
ejpam-5984	11	8	et	et	PROPN
ejpam-5984	11	9	al	al	PROPN
ejpam-5984	11	10	.	.	PUNCT
ejpam-5984	12	1	[	[	X
ejpam-5984	12	2	1	1	X
ejpam-5984	12	3	]	]	PUNCT
ejpam-5984	12	4	in	in	ADP
ejpam-5984	12	5	2004	2004	NUM
ejpam-5984	12	6	and	and	CCONJ
ejpam-5984	12	7	inspired	inspire	VERB
ejpam-5984	12	8	by	by	ADP
ejpam-5984	12	9	earlier	early	ADJ
ejpam-5984	12	10	work	work	NOUN
ejpam-5984	12	11	of	of	ADP
ejpam-5984	12	12	stewart	stewart	NOUN
ejpam-5984	13	1	[	[	X
ejpam-5984	13	2	2	2	NUM
ejpam-5984	13	3	]	]	PUNCT
ejpam-5984	13	4	and	and	CCONJ
ejpam-5984	13	5	revelle	revelle	NOUN
ejpam-5984	13	6	et	et	PROPN
ejpam-5984	13	7	al	al	PROPN
ejpam-5984	13	8	.	.	PUNCT
ejpam-5984	14	1	[	[	X
ejpam-5984	14	2	3	3	NUM
ejpam-5984	14	3	]	]	PUNCT
ejpam-5984	14	4	,	,	PUNCT
ejpam-5984	14	5	has	have	AUX
ejpam-5984	14	6	since	since	SCONJ
ejpam-5984	14	7	seen	see	VERB
ejpam-5984	14	8	various	various	ADJ
ejpam-5984	14	9	extensions	extension	NOUN
ejpam-5984	14	10	.	.	PUNCT
ejpam-5984	15	1	one	one	NUM
ejpam-5984	15	2	notable	notable	ADJ
ejpam-5984	15	3	variant	variant	NOUN
ejpam-5984	15	4	is	be	AUX
ejpam-5984	15	5	the	the	DET
ejpam-5984	15	6	[	[	X
ejpam-5984	15	7	k]-roman	k]-roman	NOUN
ejpam-5984	15	8	domination	domination	NOUN
ejpam-5984	15	9	[	[	X
ejpam-5984	15	10	4	4	NUM
ejpam-5984	15	11	]	]	PUNCT
ejpam-5984	15	12	,	,	PUNCT
ejpam-5984	15	13	which	which	PRON
ejpam-5984	15	14	generalizes	generalize	VERB
ejpam-5984	15	15	the	the	DET
ejpam-5984	15	16	original	original	ADJ
ejpam-5984	15	17	roman	roman	ADJ
ejpam-5984	15	18	domination	domination	NOUN
ejpam-5984	15	19	number	number	NOUN
ejpam-5984	15	20	γr(g	γr(g	PROPN
ejpam-5984	15	21	)	)	PUNCT
ejpam-5984	15	22	,	,	PUNCT
ejpam-5984	15	23	recovered	recover	VERB
ejpam-5984	15	24	when	when	SCONJ
ejpam-5984	15	25	k	k	PROPN
ejpam-5984	15	26	=	=	SYM
ejpam-5984	15	27	1	1	NUM
ejpam-5984	16	1	[	[	X
ejpam-5984	16	2	5–7	5–7	NUM
ejpam-5984	16	3	]	]	PUNCT
ejpam-5984	16	4	.	.	PUNCT
ejpam-5984	17	1	beeler	beeler	PROPN
ejpam-5984	17	2	et	et	PROPN
ejpam-5984	17	3	al	al	PROPN
ejpam-5984	17	4	.	.	PUNCT
ejpam-5984	18	1	[	[	X
ejpam-5984	18	2	8	8	NUM
ejpam-5984	18	3	]	]	PUNCT
ejpam-5984	18	4	further	far	ADV
ejpam-5984	18	5	refined	refine	VERB
ejpam-5984	18	6	this	this	DET
ejpam-5984	18	7	concept	concept	NOUN
ejpam-5984	18	8	through	through	ADP
ejpam-5984	18	9	double	double	ADJ
ejpam-5984	18	10	roman	roman	ADJ
ejpam-5984	18	11	domination	domination	NOUN
ejpam-5984	18	12	,	,	PUNCT
ejpam-5984	18	13	establishing	establish	VERB
ejpam-5984	18	14	bounds	bound	NOUN
ejpam-5984	18	15	and	and	CCONJ
ejpam-5984	18	16	complexity	complexity	NOUN
ejpam-5984	18	17	results	result	NOUN
ejpam-5984	18	18	,	,	PUNCT
ejpam-5984	18	19	later	later	ADV
ejpam-5984	18	20	expanded	expand	VERB
ejpam-5984	18	21	by	by	ADP
ejpam-5984	18	22	abdollahzadeh	abdollahzadeh	NOUN
ejpam-5984	18	23	ahangar	ahangar	PROPN
ejpam-5984	18	24	et	et	PROPN
ejpam-5984	18	25	al	al	PROPN
ejpam-5984	18	26	.	.	PUNCT
ejpam-5984	19	1	[	[	X
ejpam-5984	19	2	9	9	NUM
ejpam-5984	19	3	]	]	PUNCT
ejpam-5984	19	4	.	.	PUNCT
ejpam-5984	20	1	to	to	PART
ejpam-5984	20	2	incorporate	incorporate	VERB
ejpam-5984	20	3	independence	independence	NOUN
ejpam-5984	20	4	,	,	PUNCT
ejpam-5984	20	5	maimani	maimani	PROPN
ejpam-5984	20	6	et	et	PROPN
ejpam-5984	20	7	al	al	PROPN
ejpam-5984	20	8	.	.	PUNCT
ejpam-5984	21	1	[	[	X
ejpam-5984	21	2	10	10	NUM
ejpam-5984	21	3	]	]	PUNCT
ejpam-5984	21	4	introduced	introduce	VERB
ejpam-5984	21	5	independent	independent	ADJ
ejpam-5984	21	6	double	double	ADJ
ejpam-5984	21	7	roman	roman	ADJ
ejpam-5984	21	8	domination	domination	NOUN
ejpam-5984	21	9	,	,	PUNCT
ejpam-5984	21	10	providing	provide	VERB
ejpam-5984	21	11	bounds	bound	NOUN
ejpam-5984	21	12	and	and	CCONJ
ejpam-5984	21	13	relationships	relationship	NOUN
ejpam-5984	21	14	to	to	ADP
ejpam-5984	21	15	related	related	ADJ
ejpam-5984	21	16	parameters	parameter	NOUN
ejpam-5984	21	17	,	,	PUNCT
ejpam-5984	21	18	and	and	CCONJ
ejpam-5984	21	19	inspiring	inspire	VERB
ejpam-5984	21	20	studies	study	NOUN
ejpam-5984	21	21	on	on	ADP
ejpam-5984	21	22	i[k]rd	i[k]rd	PRON
ejpam-5984	21	23	for	for	ADP
ejpam-5984	21	24	k	k	PROPN
ejpam-5984	21	25	=	=	SYM
ejpam-5984	21	26	1	1	NUM
ejpam-5984	21	27	,	,	PUNCT
ejpam-5984	21	28	2	2	NUM
ejpam-5984	21	29	[	[	X
ejpam-5984	21	30	11–14	11–14	NUM
ejpam-5984	21	31	]	]	PUNCT
ejpam-5984	21	32	.	.	PUNCT
ejpam-5984	22	1	the	the	DET
ejpam-5984	22	2	concept	concept	NOUN
ejpam-5984	22	3	of	of	ADP
ejpam-5984	22	4	domination	domination	NOUN
ejpam-5984	22	5	stability	stability	NOUN
ejpam-5984	22	6	in	in	ADP
ejpam-5984	22	7	graphs	graph	NOUN
ejpam-5984	22	8	was	be	AUX
ejpam-5984	22	9	introduced	introduce	VERB
ejpam-5984	22	10	by	by	ADP
ejpam-5984	22	11	bauer	bauer	PROPN
ejpam-5984	22	12	et	et	PROPN
ejpam-5984	22	13	al	al	PROPN
ejpam-5984	22	14	.	.	PUNCT
ejpam-5984	23	1	[	[	X
ejpam-5984	23	2	15	15	NUM
ejpam-5984	23	3	]	]	PUNCT
ejpam-5984	23	4	in	in	ADP
ejpam-5984	23	5	1983	1983	NUM
ejpam-5984	23	6	.	.	PUNCT
ejpam-5984	24	1	this	this	DET
ejpam-5984	24	2	notion	notion	NOUN
ejpam-5984	24	3	has	have	AUX
ejpam-5984	24	4	since	since	ADV
ejpam-5984	24	5	been	be	AUX
ejpam-5984	24	6	extended	extend	VERB
ejpam-5984	24	7	to	to	ADP
ejpam-5984	24	8	various	various	ADJ
ejpam-5984	24	9	domination	domination	NOUN
ejpam-5984	24	10	parameters	parameter	NOUN
ejpam-5984	24	11	.	.	PUNCT
ejpam-5984	25	1	in	in	ADP
ejpam-5984	25	2	2016	2016	NUM
ejpam-5984	25	3	,	,	PUNCT
ejpam-5984	25	4	rad	rad	PROPN
ejpam-5984	25	5	et	et	PROPN
ejpam-5984	25	6	al	al	PROPN
ejpam-5984	25	7	.	.	PUNCT
ejpam-5984	26	1	[	[	X
ejpam-5984	26	2	16	16	NUM
ejpam-5984	26	3	]	]	PUNCT
ejpam-5984	26	4	advanced	advance	VERB
ejpam-5984	26	5	this	this	DET
ejpam-5984	26	6	line	line	NOUN
ejpam-5984	26	7	of	of	ADP
ejpam-5984	26	8	research	research	NOUN
ejpam-5984	26	9	by	by	ADP
ejpam-5984	26	10	proving	prove	VERB
ejpam-5984	26	11	that	that	SCONJ
ejpam-5984	26	12	the	the	DET
ejpam-5984	26	13	γ	γ	PROPN
ejpam-5984	26	14	-	-	PUNCT
ejpam-5984	26	15	stability	stability	NOUN
ejpam-5984	26	16	problem	problem	NOUN
ejpam-5984	26	17	is	be	AUX
ejpam-5984	26	18	np	np	INTJ
ejpam-5984	26	19	-	-	PUNCT
ejpam-5984	26	20	hard	hard	ADJ
ejpam-5984	26	21	,	,	PUNCT
ejpam-5984	26	22	even	even	ADV
ejpam-5984	26	23	when	when	SCONJ
ejpam-5984	26	24	restricted	restrict	VERB
ejpam-5984	26	25	to	to	ADP
ejpam-5984	26	26	bipartite	bipartite	VERB
ejpam-5984	26	27	graphs	graph	NOUN
ejpam-5984	26	28	.	.	PUNCT
ejpam-5984	27	1	continuing	continue	VERB
ejpam-5984	27	2	this	this	DET
ejpam-5984	27	3	direction	direction	NOUN
ejpam-5984	27	4	,	,	PUNCT
ejpam-5984	27	5	zhuang	zhuang	PROPN
ejpam-5984	27	6	et	et	PROPN
ejpam-5984	27	7	al	al	PROPN
ejpam-5984	27	8	.	.	PUNCT
ejpam-5984	28	1	[	[	X
ejpam-5984	28	2	17	17	NUM
ejpam-5984	28	3	]	]	PUNCT
ejpam-5984	28	4	determined	determine	VERB
ejpam-5984	28	5	the	the	DET
ejpam-5984	28	6	exact	exact	ADJ
ejpam-5984	28	7	values	value	NOUN
ejpam-5984	28	8	of	of	ADP
ejpam-5984	28	9	the	the	DET
ejpam-5984	28	10	γdr	γdr	NOUN
ejpam-5984	28	11	-	-	PUNCT
ejpam-5984	28	12	stability	stability	NOUN
ejpam-5984	28	13	number	number	NOUN
ejpam-5984	28	14	for	for	ADP
ejpam-5984	28	15	several	several	ADJ
ejpam-5984	28	16	special	special	ADJ
ejpam-5984	28	17	classes	class	NOUN
ejpam-5984	28	18	of	of	ADP
ejpam-5984	28	19	graphs	graph	NOUN
ejpam-5984	28	20	and	and	CCONJ
ejpam-5984	28	21	established	establish	VERB
ejpam-5984	28	22	bounds	bound	NOUN
ejpam-5984	28	23	on	on	ADP
ejpam-5984	28	24	stγdr(g	stγdr(g	PRON
ejpam-5984	28	25	)	)	PUNCT
ejpam-5984	28	26	.	.	PUNCT
ejpam-5984	29	1	in	in	ADP
ejpam-5984	29	2	particular	particular	ADJ
ejpam-5984	29	3	,	,	PUNCT
ejpam-5984	29	4	for	for	ADP
ejpam-5984	29	5	a	a	DET
ejpam-5984	29	6	tree	tree	NOUN
ejpam-5984	29	7	t	t	NOUN
ejpam-5984	29	8	with	with	ADP
ejpam-5984	29	9	maximum	maximum	ADJ
ejpam-5984	29	10	degree	degree	NOUN
ejpam-5984	29	11	∆	∆	PROPN
ejpam-5984	29	12	,	,	PUNCT
ejpam-5984	29	13	they	they	PRON
ejpam-5984	29	14	showed	show	VERB
ejpam-5984	29	15	that	that	SCONJ
ejpam-5984	29	16	st−γdr(t	st−γdr(t	PROPN
ejpam-5984	29	17	)	)	PUNCT
ejpam-5984	29	18	≤	≤	NUM
ejpam-5984	29	19	∆	∆	NOUN
ejpam-5984	29	20	,	,	PUNCT
ejpam-5984	29	21	and	and	CCONJ
ejpam-5984	29	22	characterized	characterize	VERB
ejpam-5984	29	23	the	the	DET
ejpam-5984	29	24	trees	tree	NOUN
ejpam-5984	29	25	that	that	PRON
ejpam-5984	29	26	attain	attain	VERB
ejpam-5984	29	27	this	this	DET
ejpam-5984	29	28	bound	bind	VERB
ejpam-5984	29	29	.	.	PUNCT
ejpam-5984	30	1	motivated	motivate	VERB
ejpam-5984	30	2	by	by	ADP
ejpam-5984	30	3	these	these	DET
ejpam-5984	30	4	developments	development	NOUN
ejpam-5984	30	5	,	,	PUNCT
ejpam-5984	30	6	this	this	DET
ejpam-5984	30	7	paper	paper	NOUN
ejpam-5984	30	8	explores	explore	VERB
ejpam-5984	30	9	the	the	DET
ejpam-5984	30	10	independent	independent	ADJ
ejpam-5984	30	11	double	double	ADJ
ejpam-5984	30	12	roman	roman	ADJ
ejpam-5984	30	13	domination	domination	NOUN
ejpam-5984	30	14	stability	stability	NOUN
ejpam-5984	30	15	of	of	ADP
ejpam-5984	30	16	graphs	graph	NOUN
ejpam-5984	30	17	.	.	PUNCT
ejpam-5984	31	1	we	we	PRON
ejpam-5984	31	2	determine	determine	VERB
ejpam-5984	31	3	exact	exact	ADJ
ejpam-5984	31	4	values	value	NOUN
ejpam-5984	31	5	for	for	ADP
ejpam-5984	31	6	special	special	ADJ
ejpam-5984	31	7	graph	graph	NOUN
ejpam-5984	31	8	classes	class	NOUN
ejpam-5984	31	9	,	,	PUNCT
ejpam-5984	31	10	establish	establish	VERB
ejpam-5984	31	11	bounds	bound	NOUN
ejpam-5984	31	12	on	on	ADP
ejpam-5984	31	13	stidr(g	stidr(g	PROPN
ejpam-5984	31	14	)	)	PUNCT
ejpam-5984	31	15	,	,	PUNCT
ejpam-5984	31	16	and	and	CCONJ
ejpam-5984	31	17	characterize	characterize	VERB
ejpam-5984	31	18	extremal	extremal	ADJ
ejpam-5984	31	19	cases	case	NOUN
ejpam-5984	31	20	.	.	PUNCT
ejpam-5984	32	1	for	for	ADP
ejpam-5984	32	2	trees	tree	NOUN
ejpam-5984	32	3	,	,	PUNCT
ejpam-5984	32	4	we	we	PRON
ejpam-5984	32	5	show	show	VERB
ejpam-5984	32	6	that	that	SCONJ
ejpam-5984	32	7	stidr(t	stidr(t	X
ejpam-5984	32	8	)	)	PUNCT
ejpam-5984	32	9	=	=	SYM
ejpam-5984	32	10	1	1	NUM
ejpam-5984	32	11	and	and	CCONJ
ejpam-5984	32	12	st−idr(t	st−idr(t	NOUN
ejpam-5984	32	13	)	)	PUNCT
ejpam-5984	32	14	≤	≤	PUNCT
ejpam-5984	32	15	∆	∆	NOUN
ejpam-5984	32	16	,	,	PUNCT
ejpam-5984	32	17	fully	fully	ADV
ejpam-5984	32	18	characterizing	characterize	VERB
ejpam-5984	32	19	trees	tree	NOUN
ejpam-5984	32	20	attaining	attain	VERB
ejpam-5984	32	21	this	this	DET
ejpam-5984	32	22	bound	bind	VERB
ejpam-5984	32	23	.	.	PUNCT
ejpam-5984	33	1	2	2	X
ejpam-5984	33	2	.	.	X
ejpam-5984	33	3	terminology	terminology	NOUN
ejpam-5984	33	4	and	and	CCONJ
ejpam-5984	33	5	notation	notation	NOUN
ejpam-5984	33	6	all	all	DET
ejpam-5984	33	7	graphs	graph	NOUN
ejpam-5984	33	8	considered	consider	VERB
ejpam-5984	33	9	in	in	ADP
ejpam-5984	33	10	this	this	DET
ejpam-5984	33	11	article	article	NOUN
ejpam-5984	33	12	are	be	AUX
ejpam-5984	33	13	finite	finite	ADJ
ejpam-5984	33	14	,	,	PUNCT
ejpam-5984	33	15	undirected	undirected	ADJ
ejpam-5984	33	16	,	,	PUNCT
ejpam-5984	33	17	and	and	CCONJ
ejpam-5984	33	18	simple	simple	ADJ
ejpam-5984	33	19	.	.	PUNCT
ejpam-5984	34	1	let	let	VERB
ejpam-5984	34	2	g	g	PROPN
ejpam-5984	34	3	=	=	SYM
ejpam-5984	34	4	(	(	PUNCT
ejpam-5984	34	5	v	v	NOUN
ejpam-5984	34	6	,	,	PUNCT
ejpam-5984	34	7	e	e	NOUN
ejpam-5984	34	8	)	)	PUNCT
ejpam-5984	34	9	be	be	AUX
ejpam-5984	34	10	a	a	DET
ejpam-5984	34	11	graph	graph	NOUN
ejpam-5984	34	12	of	of	ADP
ejpam-5984	34	13	order	order	NOUN
ejpam-5984	34	14	|v	|v	X
ejpam-5984	34	15	(	(	PUNCT
ejpam-5984	34	16	g)|	g)|	NOUN
ejpam-5984	34	17	=	=	PUNCT
ejpam-5984	34	18	n.	n.	NOUN
ejpam-5984	34	19	for	for	ADP
ejpam-5984	34	20	any	any	DET
ejpam-5984	34	21	vertex	vertex	NOUN
ejpam-5984	34	22	v	v	ADP
ejpam-5984	34	23	∈	∈	NOUN
ejpam-5984	34	24	v	v	NOUN
ejpam-5984	34	25	(	(	PUNCT
ejpam-5984	34	26	g	g	NOUN
ejpam-5984	34	27	)	)	PUNCT
ejpam-5984	34	28	,	,	PUNCT
ejpam-5984	34	29	the	the	DET
ejpam-5984	34	30	open	open	ADJ
ejpam-5984	34	31	neighborhood	neighborhood	NOUN
ejpam-5984	34	32	of	of	ADP
ejpam-5984	34	33	v	v	NOUN
ejpam-5984	34	34	is	be	AUX
ejpam-5984	34	35	the	the	DET
ejpam-5984	34	36	set	set	NOUN
ejpam-5984	34	37	n(v	n(v	PROPN
ejpam-5984	34	38	)	)	PUNCT
ejpam-5984	34	39	=	=	PRON
ejpam-5984	35	1	{	{	PUNCT
ejpam-5984	35	2	u	u	NOUN
ejpam-5984	35	3	∈	∈	PROPN
ejpam-5984	35	4	v	v	ADP
ejpam-5984	35	5	|	|	ADV
ejpam-5984	35	6	uv	uv	PROPN
ejpam-5984	35	7	∈	∈	PROPN
ejpam-5984	35	8	e(g	e(g	PROPN
ejpam-5984	35	9	)	)	PUNCT
ejpam-5984	35	10	}	}	PUNCT
ejpam-5984	35	11	and	and	CCONJ
ejpam-5984	35	12	the	the	DET
ejpam-5984	35	13	closed	closed	ADJ
ejpam-5984	35	14	neighborhood	neighborhood	NOUN
ejpam-5984	35	15	of	of	ADP
ejpam-5984	35	16	v	v	NOUN
ejpam-5984	35	17	is	be	AUX
ejpam-5984	35	18	the	the	DET
ejpam-5984	35	19	set	set	ADJ
ejpam-5984	35	20	n	n	PROPN
ejpam-5984	35	21	[	[	X
ejpam-5984	35	22	v	v	X
ejpam-5984	35	23	]	]	X
ejpam-5984	35	24	=	=	PUNCT
ejpam-5984	35	25	n(v	n(v	PROPN
ejpam-5984	35	26	)	)	PUNCT
ejpam-5984	35	27	∪	∪	NOUN
ejpam-5984	35	28	{	{	PUNCT
ejpam-5984	35	29	v	v	NOUN
ejpam-5984	35	30	}	}	PUNCT
ejpam-5984	35	31	.	.	PUNCT
ejpam-5984	36	1	we	we	PRON
ejpam-5984	36	2	denote	denote	VERB
ejpam-5984	36	3	the	the	DET
ejpam-5984	36	4	degree	degree	NOUN
ejpam-5984	36	5	of	of	ADP
ejpam-5984	36	6	a	a	DET
ejpam-5984	36	7	vertex	vertex	NOUN
ejpam-5984	36	8	v	v	NOUN
ejpam-5984	36	9	in	in	ADP
ejpam-5984	36	10	a	a	DET
ejpam-5984	36	11	graph	graph	NOUN
ejpam-5984	36	12	g	g	NOUN
ejpam-5984	36	13	by	by	ADP
ejpam-5984	36	14	degg(v	degg(v	PROPN
ejpam-5984	36	15	)	)	PUNCT
ejpam-5984	36	16	,	,	PUNCT
ejpam-5984	36	17	or	or	CCONJ
ejpam-5984	36	18	simply	simply	ADV
ejpam-5984	36	19	by	by	ADP
ejpam-5984	36	20	deg(v	deg(v	PROPN
ejpam-5984	36	21	)	)	PUNCT
ejpam-5984	36	22	if	if	SCONJ
ejpam-5984	36	23	the	the	DET
ejpam-5984	36	24	graph	graph	NOUN
ejpam-5984	36	25	g	g	NOUN
ejpam-5984	36	26	is	be	AUX
ejpam-5984	36	27	clear	clear	ADJ
ejpam-5984	36	28	from	from	ADP
ejpam-5984	36	29	the	the	DET
ejpam-5984	36	30	context	context	NOUN
ejpam-5984	36	31	.	.	PUNCT
ejpam-5984	37	1	let	let	VERB
ejpam-5984	37	2	δ(g	δ(g	ADV
ejpam-5984	37	3	)	)	PUNCT
ejpam-5984	37	4	and	and	CCONJ
ejpam-5984	37	5	∆(g	∆(g	NOUN
ejpam-5984	37	6	)	)	PUNCT
ejpam-5984	37	7	denote	denote	VERB
ejpam-5984	37	8	the	the	DET
ejpam-5984	37	9	minimum	minimum	ADJ
ejpam-5984	37	10	and	and	CCONJ
ejpam-5984	37	11	maximum	maximum	ADJ
ejpam-5984	37	12	degrees	degree	NOUN
ejpam-5984	37	13	,	,	PUNCT
ejpam-5984	37	14	respectively	respectively	ADV
ejpam-5984	37	15	,	,	PUNCT
ejpam-5984	37	16	of	of	ADP
ejpam-5984	37	17	vertices	vertex	NOUN
ejpam-5984	37	18	in	in	ADP
ejpam-5984	37	19	g.	g.	PROPN
ejpam-5984	38	1	we	we	PRON
ejpam-5984	38	2	call	call	VERB
ejpam-5984	38	3	a	a	DET
ejpam-5984	38	4	vertex	vertex	NOUN
ejpam-5984	38	5	of	of	ADP
ejpam-5984	38	6	degree	degree	NOUN
ejpam-5984	38	7	one	one	NUM
ejpam-5984	38	8	a	a	DET
ejpam-5984	38	9	leaf	leaf	NOUN
ejpam-5984	38	10	,	,	PUNCT
ejpam-5984	38	11	and	and	CCONJ
ejpam-5984	38	12	its	its	PRON
ejpam-5984	38	13	(	(	PUNCT
ejpam-5984	38	14	unique	unique	ADJ
ejpam-5984	38	15	)	)	PUNCT
ejpam-5984	38	16	neighbor	neighbor	NOUN
ejpam-5984	38	17	a	a	DET
ejpam-5984	38	18	support	support	NOUN
ejpam-5984	38	19	vertex	vertex	NOUN
ejpam-5984	38	20	.	.	PUNCT
ejpam-5984	39	1	a	a	DET
ejpam-5984	39	2	support	support	NOUN
ejpam-5984	39	3	vertex	vertex	NOUN
ejpam-5984	39	4	is	be	AUX
ejpam-5984	39	5	said	say	VERB
ejpam-5984	39	6	to	to	PART
ejpam-5984	39	7	be	be	AUX
ejpam-5984	39	8	strong	strong	ADJ
ejpam-5984	39	9	if	if	SCONJ
ejpam-5984	39	10	it	it	PRON
ejpam-5984	39	11	has	have	VERB
ejpam-5984	39	12	at	at	ADV
ejpam-5984	39	13	least	least	ADV
ejpam-5984	39	14	two	two	NUM
ejpam-5984	39	15	leaf	leaf	NOUN
ejpam-5984	39	16	neighbors	neighbor	NOUN
ejpam-5984	39	17	,	,	PUNCT
ejpam-5984	39	18	otherwise	otherwise	ADV
ejpam-5984	39	19	,	,	PUNCT
ejpam-5984	39	20	it	it	PRON
ejpam-5984	39	21	is	be	AUX
ejpam-5984	39	22	called	call	VERB
ejpam-5984	39	23	weak	weak	ADJ
ejpam-5984	39	24	.	.	PUNCT
ejpam-5984	40	1	a	a	DET
ejpam-5984	40	2	complete	complete	ADJ
ejpam-5984	40	3	graph	graph	NOUN
ejpam-5984	40	4	on	on	ADP
ejpam-5984	40	5	n	n	DET
ejpam-5984	40	6	vertices	vertex	NOUN
ejpam-5984	40	7	is	be	AUX
ejpam-5984	40	8	denoted	denote	VERB
ejpam-5984	40	9	by	by	ADP
ejpam-5984	40	10	kn	kn	PROPN
ejpam-5984	40	11	,	,	PUNCT
ejpam-5984	40	12	while	while	SCONJ
ejpam-5984	40	13	a	a	DET
ejpam-5984	40	14	complete	complete	ADJ
ejpam-5984	40	15	bipartite	bipartite	NOUN
ejpam-5984	40	16	graph	graph	NOUN
ejpam-5984	40	17	with	with	ADP
ejpam-5984	40	18	partite	partite	ADJ
ejpam-5984	40	19	sets	set	NOUN
ejpam-5984	40	20	of	of	ADP
ejpam-5984	40	21	size	size	NOUN
ejpam-5984	40	22	p	p	NOUN
ejpam-5984	40	23	and	and	CCONJ
ejpam-5984	40	24	q	q	NOUN
ejpam-5984	40	25	is	be	AUX
ejpam-5984	40	26	denoted	denote	VERB
ejpam-5984	40	27	by	by	ADP
ejpam-5984	40	28	kp	kp	PROPN
ejpam-5984	40	29	,	,	PUNCT
ejpam-5984	40	30	q.	q.	PROPN
ejpam-5984	40	31	we	we	PRON
ejpam-5984	40	32	write	write	VERB
ejpam-5984	40	33	pn	pn	PROPN
ejpam-5984	40	34	for	for	ADP
ejpam-5984	40	35	the	the	DET
ejpam-5984	40	36	path	path	NOUN
ejpam-5984	40	37	of	of	ADP
ejpam-5984	40	38	order	order	NOUN
ejpam-5984	40	39	n	n	CCONJ
ejpam-5984	40	40	,	,	PUNCT
ejpam-5984	40	41	cn	cn	PROPN
ejpam-5984	40	42	for	for	ADP
ejpam-5984	40	43	the	the	DET
ejpam-5984	40	44	cycle	cycle	NOUN
ejpam-5984	40	45	of	of	ADP
ejpam-5984	40	46	length	length	NOUN
ejpam-5984	40	47	n	n	CCONJ
ejpam-5984	40	48	,	,	PUNCT
ejpam-5984	40	49	and	and	CCONJ
ejpam-5984	40	50	kn	kn	PROPN
ejpam-5984	40	51	for	for	ADP
ejpam-5984	40	52	the	the	DET
ejpam-5984	40	53	graph	graph	NOUN
ejpam-5984	40	54	with	with	ADP
ejpam-5984	40	55	n	n	ADP
ejpam-5984	40	56	vertices	vertex	NOUN
ejpam-5984	40	57	and	and	CCONJ
ejpam-5984	40	58	no	no	DET
ejpam-5984	40	59	edges	edge	NOUN
ejpam-5984	40	60	.	.	PUNCT
ejpam-5984	41	1	the	the	DET
ejpam-5984	41	2	distance	distance	NOUN
ejpam-5984	41	3	dg(u	dg(u	X
ejpam-5984	41	4	,	,	PUNCT
ejpam-5984	41	5	v	v	NOUN
ejpam-5984	41	6	)	)	PUNCT
ejpam-5984	41	7	between	between	ADP
ejpam-5984	41	8	two	two	NUM
ejpam-5984	41	9	vertices	vertex	NOUN
ejpam-5984	41	10	u	u	NOUN
ejpam-5984	41	11	and	and	CCONJ
ejpam-5984	41	12	v	v	NOUN
ejpam-5984	41	13	in	in	ADP
ejpam-5984	41	14	a	a	DET
ejpam-5984	41	15	connected	connected	ADJ
ejpam-5984	41	16	graph	graph	NOUN
ejpam-5984	41	17	g	g	PROPN
ejpam-5984	41	18	is	be	AUX
ejpam-5984	41	19	the	the	DET
ejpam-5984	41	20	length	length	NOUN
ejpam-5984	41	21	of	of	ADP
ejpam-5984	41	22	a	a	DET
ejpam-5984	41	23	shortest	short	ADJ
ejpam-5984	41	24	u	u	NOUN
ejpam-5984	41	25	–	–	PUNCT
ejpam-5984	41	26	v	v	NUM
ejpam-5984	41	27	path	path	NOUN
ejpam-5984	41	28	in	in	ADP
ejpam-5984	41	29	g	g	NOUN
ejpam-5984	41	30	,	,	PUNCT
ejpam-5984	41	31	while	while	SCONJ
ejpam-5984	41	32	the	the	DET
ejpam-5984	41	33	diameter	diameter	NOUN
ejpam-5984	41	34	,	,	PUNCT
ejpam-5984	41	35	diam(g	diam(g	PROPN
ejpam-5984	41	36	)	)	PUNCT
ejpam-5984	41	37	,	,	PUNCT
ejpam-5984	41	38	is	be	AUX
ejpam-5984	41	39	the	the	DET
ejpam-5984	41	40	maximum	maximum	ADJ
ejpam-5984	41	41	distance	distance	NOUN
ejpam-5984	41	42	among	among	ADP
ejpam-5984	41	43	all	all	DET
ejpam-5984	41	44	pairs	pair	NOUN
ejpam-5984	41	45	of	of	ADP
ejpam-5984	41	46	vertices	vertex	NOUN
ejpam-5984	41	47	in	in	ADP
ejpam-5984	41	48	g.	g.	PROPN
ejpam-5984	41	49	a	a	DET
ejpam-5984	41	50	tree	tree	NOUN
ejpam-5984	41	51	is	be	AUX
ejpam-5984	41	52	an	an	DET
ejpam-5984	41	53	acyclic	acyclic	ADJ
ejpam-5984	41	54	connected	connect	VERB
ejpam-5984	41	55	graph	graph	NOUN
ejpam-5984	41	56	.	.	PUNCT
ejpam-5984	42	1	a	a	DET
ejpam-5984	42	2	star	star	NOUN
ejpam-5984	42	3	is	be	AUX
ejpam-5984	42	4	the	the	DET
ejpam-5984	42	5	graph	graph	NOUN
ejpam-5984	42	6	k1,m	k1,m	PROPN
ejpam-5984	42	7	,	,	PUNCT
ejpam-5984	42	8	where	where	SCONJ
ejpam-5984	42	9	m	m	PROPN
ejpam-5984	42	10	≥	≥	NOUN
ejpam-5984	42	11	1	1	NUM
ejpam-5984	42	12	;	;	PUNCT
ejpam-5984	42	13	the	the	DET
ejpam-5984	42	14	vertex	vertex	NOUN
ejpam-5984	42	15	of	of	ADP
ejpam-5984	42	16	degree	degree	NOUN
ejpam-5984	42	17	m	m	VERB
ejpam-5984	42	18	is	be	AUX
ejpam-5984	42	19	called	call	VERB
ejpam-5984	42	20	the	the	DET
ejpam-5984	42	21	center	center	NOUN
ejpam-5984	42	22	of	of	ADP
ejpam-5984	42	23	the	the	DET
ejpam-5984	42	24	star	star	NOUN
ejpam-5984	42	25	.	.	PUNCT
ejpam-5984	43	1	a	a	DET
ejpam-5984	43	2	double	double	ADJ
ejpam-5984	43	3	star	star	NOUN
ejpam-5984	43	4	sr	sr	PROPN
ejpam-5984	43	5	,	,	PUNCT
ejpam-5984	43	6	t	t	PROPN
ejpam-5984	43	7	is	be	AUX
ejpam-5984	43	8	formed	form	VERB
ejpam-5984	43	9	from	from	ADP
ejpam-5984	43	10	two	two	NUM
ejpam-5984	43	11	disjoint	disjoint	NOUN
ejpam-5984	43	12	stars	star	NOUN
ejpam-5984	43	13	k1,r	k1,r	PROPN
ejpam-5984	43	14	and	and	CCONJ
ejpam-5984	43	15	k1,t	k1,t	PROPN
ejpam-5984	43	16	by	by	ADP
ejpam-5984	43	17	adding	add	VERB
ejpam-5984	43	18	an	an	DET
ejpam-5984	43	19	edge	edge	NOUN
ejpam-5984	43	20	joining	join	VERB
ejpam-5984	43	21	their	their	PRON
ejpam-5984	43	22	center	center	ADJ
ejpam-5984	43	23	vertices	vertex	NOUN
ejpam-5984	43	24	.	.	PUNCT
ejpam-5984	44	1	a	a	DET
ejpam-5984	44	2	s.	s.	PROPN
ejpam-5984	44	3	m.	m.	PROPN
ejpam-5984	44	4	sheikholeslami	sheikholeslami	PROPN
ejpam-5984	44	5	et	et	PROPN
ejpam-5984	44	6	al	al	PROPN
ejpam-5984	44	7	.	.	PUNCT
ejpam-5984	44	8	/	/	SYM
ejpam-5984	44	9	eur	eur	PROPN
ejpam-5984	44	10	.	.	PUNCT
ejpam-5984	45	1	j.	j.	PROPN
ejpam-5984	45	2	pure	pure	PROPN
ejpam-5984	45	3	appl	appl	PROPN
ejpam-5984	45	4	.	.	PROPN
ejpam-5984	45	5	math	math	PROPN
ejpam-5984	45	6	,	,	PUNCT
ejpam-5984	45	7	18	18	NUM
ejpam-5984	45	8	(	(	PUNCT
ejpam-5984	45	9	2	2	NUM
ejpam-5984	45	10	)	)	PUNCT
ejpam-5984	45	11	(	(	PUNCT
ejpam-5984	45	12	2025	2025	NUM
ejpam-5984	45	13	)	)	PUNCT
ejpam-5984	45	14	,	,	PUNCT
ejpam-5984	45	15	5984	5984	NUM
ejpam-5984	45	16	3	3	NUM
ejpam-5984	45	17	of	of	ADP
ejpam-5984	45	18	16	16	NUM
ejpam-5984	45	19	rooted	root	VERB
ejpam-5984	45	20	tree	tree	NOUN
ejpam-5984	45	21	t	t	PROPN
ejpam-5984	45	22	distinguishes	distinguish	VERB
ejpam-5984	45	23	one	one	NUM
ejpam-5984	45	24	vertex	vertex	NOUN
ejpam-5984	45	25	r	r	NOUN
ejpam-5984	45	26	,	,	PUNCT
ejpam-5984	45	27	called	call	VERB
ejpam-5984	45	28	the	the	DET
ejpam-5984	45	29	root	root	NOUN
ejpam-5984	45	30	.	.	PUNCT
ejpam-5984	46	1	for	for	ADP
ejpam-5984	46	2	each	each	DET
ejpam-5984	46	3	vertex	vertex	NOUN
ejpam-5984	46	4	v	v	ADP
ejpam-5984	46	5	̸=	̸=	PROPN
ejpam-5984	46	6	r	r	NOUN
ejpam-5984	46	7	in	in	ADP
ejpam-5984	46	8	t	t	PROPN
ejpam-5984	46	9	,	,	PUNCT
ejpam-5984	46	10	the	the	DET
ejpam-5984	46	11	parent	parent	NOUN
ejpam-5984	46	12	of	of	ADP
ejpam-5984	46	13	v	v	NUM
ejpam-5984	46	14	is	be	AUX
ejpam-5984	46	15	the	the	DET
ejpam-5984	46	16	neighbor	neighbor	NOUN
ejpam-5984	46	17	of	of	ADP
ejpam-5984	46	18	v	v	NOUN
ejpam-5984	46	19	on	on	ADP
ejpam-5984	46	20	the	the	DET
ejpam-5984	46	21	unique	unique	ADJ
ejpam-5984	46	22	r	r	NOUN
ejpam-5984	46	23	–	–	PUNCT
ejpam-5984	46	24	v	v	NOUN
ejpam-5984	46	25	path	path	NOUN
ejpam-5984	46	26	,	,	PUNCT
ejpam-5984	46	27	while	while	SCONJ
ejpam-5984	46	28	a	a	DET
ejpam-5984	46	29	child	child	NOUN
ejpam-5984	46	30	of	of	ADP
ejpam-5984	46	31	v	v	NOUN
ejpam-5984	46	32	is	be	AUX
ejpam-5984	46	33	any	any	DET
ejpam-5984	46	34	other	other	ADJ
ejpam-5984	46	35	neighbor	neighbor	NOUN
ejpam-5984	46	36	of	of	ADP
ejpam-5984	46	37	v.	v.	ADP
ejpam-5984	46	38	a	a	DET
ejpam-5984	46	39	descendant	descendant	NOUN
ejpam-5984	46	40	of	of	ADP
ejpam-5984	46	41	v	v	NOUN
ejpam-5984	46	42	is	be	AUX
ejpam-5984	46	43	a	a	DET
ejpam-5984	46	44	vertex	vertex	NOUN
ejpam-5984	46	45	u	u	NOUN
ejpam-5984	46	46	̸=	̸=	PROPN
ejpam-5984	46	47	v	v	ADP
ejpam-5984	46	48	such	such	DET
ejpam-5984	46	49	that	that	SCONJ
ejpam-5984	46	50	the	the	DET
ejpam-5984	46	51	unique	unique	ADJ
ejpam-5984	46	52	r	r	NOUN
ejpam-5984	46	53	–	–	PUNCT
ejpam-5984	46	54	u	u	NOUN
ejpam-5984	46	55	path	path	NOUN
ejpam-5984	46	56	contains	contain	VERB
ejpam-5984	46	57	v.	v.	ADP
ejpam-5984	46	58	thus	thus	ADV
ejpam-5984	46	59	,	,	PUNCT
ejpam-5984	46	60	every	every	DET
ejpam-5984	46	61	child	child	NOUN
ejpam-5984	46	62	of	of	ADP
ejpam-5984	46	63	v	v	NOUN
ejpam-5984	46	64	is	be	AUX
ejpam-5984	46	65	a	a	DET
ejpam-5984	46	66	descendant	descendant	NOUN
ejpam-5984	46	67	of	of	ADP
ejpam-5984	46	68	v.	v.	ADV
ejpam-5984	46	69	let	let	VERB
ejpam-5984	46	70	d(v	d(v	PROPN
ejpam-5984	46	71	)	)	PUNCT
ejpam-5984	46	72	denote	denote	VERB
ejpam-5984	46	73	the	the	DET
ejpam-5984	46	74	set	set	NOUN
ejpam-5984	46	75	of	of	ADP
ejpam-5984	46	76	descendants	descendant	NOUN
ejpam-5984	46	77	of	of	ADP
ejpam-5984	46	78	v	v	NOUN
ejpam-5984	46	79	,	,	PUNCT
ejpam-5984	46	80	and	and	CCONJ
ejpam-5984	46	81	let	let	VERB
ejpam-5984	46	82	d[v	d[v	PRON
ejpam-5984	46	83	]	]	X
ejpam-5984	46	84	=	=	SYM
ejpam-5984	46	85	d(v	d(v	ADJ
ejpam-5984	46	86	)	)	PUNCT
ejpam-5984	46	87	∪	∪	NOUN
ejpam-5984	46	88	{	{	PUNCT
ejpam-5984	46	89	v	v	NOUN
ejpam-5984	46	90	}	}	PUNCT
ejpam-5984	46	91	.	.	PUNCT
ejpam-5984	47	1	the	the	DET
ejpam-5984	47	2	depth	depth	NOUN
ejpam-5984	47	3	of	of	ADP
ejpam-5984	47	4	v	v	NOUN
ejpam-5984	47	5	,	,	PUNCT
ejpam-5984	47	6	denoted	denote	VERB
ejpam-5984	47	7	depth(v	depth(v	PROPN
ejpam-5984	47	8	)	)	PUNCT
ejpam-5984	47	9	,	,	PUNCT
ejpam-5984	47	10	is	be	AUX
ejpam-5984	47	11	the	the	DET
ejpam-5984	47	12	largest	large	ADJ
ejpam-5984	47	13	distance	distance	NOUN
ejpam-5984	47	14	from	from	ADP
ejpam-5984	47	15	v	v	NUM
ejpam-5984	47	16	to	to	ADP
ejpam-5984	47	17	a	a	DET
ejpam-5984	47	18	vertex	vertex	NOUN
ejpam-5984	47	19	in	in	ADP
ejpam-5984	47	20	d(v	d(v	PROPN
ejpam-5984	47	21	)	)	PUNCT
ejpam-5984	47	22	.	.	PUNCT
ejpam-5984	48	1	the	the	DET
ejpam-5984	48	2	maximal	maximal	ADJ
ejpam-5984	48	3	subtree	subtree	NOUN
ejpam-5984	48	4	at	at	ADP
ejpam-5984	48	5	u	u	NOUN
ejpam-5984	48	6	is	be	AUX
ejpam-5984	48	7	the	the	DET
ejpam-5984	48	8	subtree	subtree	NOUN
ejpam-5984	48	9	of	of	ADP
ejpam-5984	48	10	t	t	NOUN
ejpam-5984	48	11	induced	induce	VERB
ejpam-5984	48	12	by	by	ADP
ejpam-5984	48	13	d[u	d[u	PROPN
ejpam-5984	48	14	]	]	PUNCT
ejpam-5984	48	15	,	,	PUNCT
ejpam-5984	48	16	and	and	CCONJ
ejpam-5984	48	17	is	be	AUX
ejpam-5984	48	18	denoted	denote	VERB
ejpam-5984	48	19	by	by	ADP
ejpam-5984	48	20	tu	tu	PROPN
ejpam-5984	48	21	.	.	PUNCT
ejpam-5984	49	1	let	let	VERB
ejpam-5984	49	2	k	k	PRON
ejpam-5984	49	3	be	be	AUX
ejpam-5984	49	4	a	a	DET
ejpam-5984	49	5	positive	positive	ADJ
ejpam-5984	49	6	integer	integer	NOUN
ejpam-5984	49	7	,	,	PUNCT
ejpam-5984	49	8	and	and	CCONJ
ejpam-5984	49	9	let	let	VERB
ejpam-5984	49	10	f	f	NOUN
ejpam-5984	49	11	:	:	PUNCT
ejpam-5984	49	12	v	v	X
ejpam-5984	49	13	(	(	PUNCT
ejpam-5984	49	14	g	g	NOUN
ejpam-5984	49	15	)	)	PUNCT
ejpam-5984	49	16	→	→	SYM
ejpam-5984	49	17	{	{	PUNCT
ejpam-5984	49	18	0	0	NUM
ejpam-5984	49	19	,	,	PUNCT
ejpam-5984	49	20	1	1	NUM
ejpam-5984	49	21	,	,	PUNCT
ejpam-5984	49	22	2	2	NUM
ejpam-5984	49	23	,	,	PUNCT
ejpam-5984	49	24	.	.	PUNCT
ejpam-5984	49	25	.	.	PUNCT
ejpam-5984	50	1	.	.	PUNCT
ejpam-5984	51	1	,	,	PUNCT
ejpam-5984	51	2	k	k	PROPN
ejpam-5984	51	3	+	+	CCONJ
ejpam-5984	51	4	1	1	X
ejpam-5984	51	5	}	}	PUNCT
ejpam-5984	51	6	be	be	AUX
ejpam-5984	51	7	a	a	DET
ejpam-5984	51	8	function	function	NOUN
ejpam-5984	51	9	that	that	PRON
ejpam-5984	51	10	assigns	assign	VERB
ejpam-5984	51	11	labels	label	NOUN
ejpam-5984	51	12	from	from	ADP
ejpam-5984	51	13	the	the	DET
ejpam-5984	51	14	set	set	NOUN
ejpam-5984	51	15	{	{	PUNCT
ejpam-5984	51	16	0	0	NUM
ejpam-5984	51	17	,	,	PUNCT
ejpam-5984	51	18	1	1	NUM
ejpam-5984	51	19	,	,	PUNCT
ejpam-5984	51	20	.	.	PUNCT
ejpam-5984	51	21	.	.	PUNCT
ejpam-5984	52	1	.	.	PUNCT
ejpam-5984	53	1	,	,	PUNCT
ejpam-5984	53	2	k+1	k+1	X
ejpam-5984	53	3	}	}	PUNCT
ejpam-5984	53	4	to	to	ADP
ejpam-5984	53	5	the	the	DET
ejpam-5984	53	6	vertices	vertex	NOUN
ejpam-5984	53	7	of	of	ADP
ejpam-5984	53	8	a	a	DET
ejpam-5984	53	9	graph	graph	NOUN
ejpam-5984	53	10	g.	g.	NOUN
ejpam-5984	53	11	the	the	DET
ejpam-5984	53	12	active	active	ADJ
ejpam-5984	53	13	neighborhood	neighborhood	NOUN
ejpam-5984	53	14	an(v	an(v	NOUN
ejpam-5984	53	15	)	)	PUNCT
ejpam-5984	53	16	of	of	ADP
ejpam-5984	53	17	a	a	DET
ejpam-5984	53	18	vertex	vertex	NOUN
ejpam-5984	53	19	v	v	ADP
ejpam-5984	53	20	∈	∈	NOUN
ejpam-5984	53	21	v	v	NOUN
ejpam-5984	53	22	(	(	PUNCT
ejpam-5984	53	23	g	g	NOUN
ejpam-5984	53	24	)	)	PUNCT
ejpam-5984	53	25	with	with	ADP
ejpam-5984	53	26	respect	respect	NOUN
ejpam-5984	53	27	to	to	ADP
ejpam-5984	53	28	f	f	PROPN
ejpam-5984	53	29	is	be	AUX
ejpam-5984	53	30	the	the	DET
ejpam-5984	53	31	set	set	NOUN
ejpam-5984	53	32	of	of	ADP
ejpam-5984	53	33	all	all	DET
ejpam-5984	53	34	vertices	vertex	NOUN
ejpam-5984	53	35	w	w	PROPN
ejpam-5984	53	36	∈	∈	PROPN
ejpam-5984	53	37	n(v	n(v	PROPN
ejpam-5984	53	38	)	)	PUNCT
ejpam-5984	53	39	such	such	ADJ
ejpam-5984	53	40	that	that	SCONJ
ejpam-5984	53	41	f(w	f(w	PROPN
ejpam-5984	53	42	)	)	PUNCT
ejpam-5984	53	43	≥	≥	NOUN
ejpam-5984	53	44	1	1	NUM
ejpam-5984	53	45	.	.	PUNCT
ejpam-5984	53	46	let	let	VERB
ejpam-5984	53	47	an	an	DET
ejpam-5984	53	48	[	[	X
ejpam-5984	53	49	v	v	NOUN
ejpam-5984	53	50	]	]	X
ejpam-5984	53	51	=	=	PRON
ejpam-5984	53	52	{	{	PUNCT
ejpam-5984	53	53	v}∪an(v	v}∪an(v	PROPN
ejpam-5984	53	54	)	)	PUNCT
ejpam-5984	53	55	.	.	PUNCT
ejpam-5984	54	1	a	a	DET
ejpam-5984	54	2	[	[	X
ejpam-5984	54	3	k]-roman	k]-roman	ADJ
ejpam-5984	54	4	dominating	dominating	NOUN
ejpam-5984	54	5	function	function	NOUN
ejpam-5984	54	6	,	,	PUNCT
ejpam-5984	54	7	abbreviated	abbreviate	VERB
ejpam-5984	54	8	as	as	ADP
ejpam-5984	54	9	[	[	PRON
ejpam-5984	54	10	k]rdf	k]rdf	NOUN
ejpam-5984	54	11	,	,	PUNCT
ejpam-5984	54	12	is	be	AUX
ejpam-5984	54	13	a	a	DET
ejpam-5984	54	14	function	function	NOUN
ejpam-5984	54	15	f	f	NOUN
ejpam-5984	54	16	:	:	PUNCT
ejpam-5984	54	17	v	v	X
ejpam-5984	54	18	(	(	PUNCT
ejpam-5984	54	19	g	g	NOUN
ejpam-5984	54	20	)	)	PUNCT
ejpam-5984	54	21	→	→	SYM
ejpam-5984	54	22	{	{	PUNCT
ejpam-5984	54	23	0	0	NUM
ejpam-5984	54	24	,	,	PUNCT
ejpam-5984	54	25	1	1	NUM
ejpam-5984	54	26	,	,	PUNCT
ejpam-5984	54	27	.	.	PUNCT
ejpam-5984	54	28	.	.	PUNCT
ejpam-5984	54	29	.	.	PUNCT
ejpam-5984	55	1	,	,	PUNCT
ejpam-5984	55	2	k+	k+	NOUN
ejpam-5984	55	3	1	1	X
ejpam-5984	55	4	}	}	PUNCT
ejpam-5984	55	5	satisfying	satisfy	VERB
ejpam-5984	55	6	the	the	DET
ejpam-5984	55	7	condition	condition	NOUN
ejpam-5984	55	8	that	that	SCONJ
ejpam-5984	55	9	for	for	ADP
ejpam-5984	55	10	any	any	DET
ejpam-5984	55	11	vertex	vertex	NOUN
ejpam-5984	55	12	v	v	ADP
ejpam-5984	55	13	∈	∈	NOUN
ejpam-5984	55	14	v	v	NOUN
ejpam-5984	55	15	(	(	PUNCT
ejpam-5984	55	16	g	g	NOUN
ejpam-5984	55	17	)	)	PUNCT
ejpam-5984	55	18	with	with	ADP
ejpam-5984	55	19	f(v	f(v	NOUN
ejpam-5984	55	20	)	)	PUNCT
ejpam-5984	55	21	<	<	X
ejpam-5984	56	1	k	k	X
ejpam-5984	56	2	,	,	PUNCT
ejpam-5984	56	3	it	it	PRON
ejpam-5984	56	4	holds	hold	VERB
ejpam-5984	56	5	that	that	SCONJ
ejpam-5984	56	6	∑	∑	PUNCT
ejpam-5984	56	7	u∈n	u∈n	PROPN
ejpam-5984	56	8	[	[	X
ejpam-5984	56	9	v	v	X
ejpam-5984	56	10	]	]	X
ejpam-5984	56	11	f(u	f(u	PROPN
ejpam-5984	56	12	)	)	PUNCT
ejpam-5984	56	13	≥	≥	NOUN
ejpam-5984	56	14	|an(v)|	|an(v)|	NOUN
ejpam-5984	56	15	+	+	CCONJ
ejpam-5984	56	16	k.	k.	NOUN
ejpam-5984	56	17	the	the	DET
ejpam-5984	56	18	weight	weight	NOUN
ejpam-5984	56	19	of	of	ADP
ejpam-5984	56	20	a	a	PRON
ejpam-5984	56	21	[	[	X
ejpam-5984	56	22	k]rdf	k]rdf	NOUN
ejpam-5984	56	23	is	be	AUX
ejpam-5984	56	24	defined	define	VERB
ejpam-5984	56	25	as	as	ADP
ejpam-5984	56	26	ω(f	ω(f	ADJ
ejpam-5984	56	27	)	)	PUNCT
ejpam-5984	56	28	=	=	SYM
ejpam-5984	56	29	∑	∑	PUNCT
ejpam-5984	56	30	v∈v	v∈v	PROPN
ejpam-5984	56	31	(	(	PUNCT
ejpam-5984	56	32	g	g	NOUN
ejpam-5984	56	33	)	)	PUNCT
ejpam-5984	56	34	f(v	f(v	NOUN
ejpam-5984	56	35	)	)	PUNCT
ejpam-5984	56	36	,	,	PUNCT
ejpam-5984	56	37	and	and	CCONJ
ejpam-5984	56	38	the	the	DET
ejpam-5984	56	39	[	[	X
ejpam-5984	56	40	k]-roman	k]-roman	ADJ
ejpam-5984	56	41	domination	domination	NOUN
ejpam-5984	56	42	number	number	NOUN
ejpam-5984	56	43	γ[kr](g	γ[kr](g	PROPN
ejpam-5984	56	44	)	)	PUNCT
ejpam-5984	56	45	of	of	ADP
ejpam-5984	56	46	g	g	PROPN
ejpam-5984	56	47	is	be	AUX
ejpam-5984	56	48	the	the	DET
ejpam-5984	56	49	minimum	minimum	ADJ
ejpam-5984	56	50	weight	weight	NOUN
ejpam-5984	56	51	of	of	ADP
ejpam-5984	56	52	a	a	PRON
ejpam-5984	56	53	[	[	X
ejpam-5984	56	54	k]rdf	k]rdf	X
ejpam-5984	56	55	on	on	ADP
ejpam-5984	56	56	g.	g.	PROPN
ejpam-5984	56	57	a	a	DET
ejpam-5984	56	58	function	function	NOUN
ejpam-5984	56	59	f	f	PROPN
ejpam-5984	56	60	achieving	achieve	VERB
ejpam-5984	56	61	this	this	DET
ejpam-5984	56	62	minimum	minimum	NOUN
ejpam-5984	56	63	is	be	AUX
ejpam-5984	56	64	called	call	VERB
ejpam-5984	56	65	a	a	DET
ejpam-5984	56	66	γ[kr](g)-function	γ[kr](g)-function	NOUN
ejpam-5984	56	67	.	.	PUNCT
ejpam-5984	57	1	for	for	ADP
ejpam-5984	57	2	a	a	DET
ejpam-5984	57	3	[	[	X
ejpam-5984	57	4	k]rdf	k]rdf	X
ejpam-5984	57	5	f	f	NOUN
ejpam-5984	57	6	on	on	ADP
ejpam-5984	57	7	g	g	NOUN
ejpam-5984	57	8	,	,	PUNCT
ejpam-5984	57	9	let	let	VERB
ejpam-5984	57	10	v	v	PRON
ejpam-5984	57	11	f	f	NOUN
ejpam-5984	58	1	i	i	PRON
ejpam-5984	58	2	=	=	PUNCT
ejpam-5984	58	3	{	{	PUNCT
ejpam-5984	58	4	v	v	NUM
ejpam-5984	58	5	∈	∈	NOUN
ejpam-5984	58	6	v	v	NOUN
ejpam-5984	58	7	(	(	PUNCT
ejpam-5984	58	8	g	g	NOUN
ejpam-5984	58	9	)	)	PUNCT
ejpam-5984	58	10	|	|	ADV
ejpam-5984	58	11	f(v	f(v	NOUN
ejpam-5984	58	12	)	)	PUNCT
ejpam-5984	59	1	=	=	PUNCT
ejpam-5984	59	2	i	i	PROPN
ejpam-5984	59	3	}	}	PUNCT
ejpam-5984	59	4	for	for	ADP
ejpam-5984	59	5	all	all	PRON
ejpam-5984	59	6	i	i	PRON
ejpam-5984	59	7	∈	∈	PROPN
ejpam-5984	59	8	{	{	PUNCT
ejpam-5984	59	9	0	0	NUM
ejpam-5984	59	10	,	,	PUNCT
ejpam-5984	59	11	1	1	NUM
ejpam-5984	59	12	,	,	PUNCT
ejpam-5984	59	13	.	.	PUNCT
ejpam-5984	59	14	.	.	PUNCT
ejpam-5984	59	15	.	.	PUNCT
ejpam-5984	60	1	,	,	PUNCT
ejpam-5984	60	2	k	k	PROPN
ejpam-5984	60	3	+	+	PROPN
ejpam-5984	60	4	1	1	NUM
ejpam-5984	60	5	}	}	PUNCT
ejpam-5984	60	6	.	.	PUNCT
ejpam-5984	61	1	consequently	consequently	ADV
ejpam-5984	61	2	,	,	PUNCT
ejpam-5984	61	3	any	any	PRON
ejpam-5984	61	4	[	[	X
ejpam-5984	61	5	k]rdf	k]rdf	X
ejpam-5984	61	6	f	f	PROPN
ejpam-5984	61	7	can	can	AUX
ejpam-5984	61	8	be	be	AUX
ejpam-5984	61	9	represented	represent	VERB
ejpam-5984	61	10	by	by	ADP
ejpam-5984	61	11	the	the	DET
ejpam-5984	61	12	tuple	tuple	NOUN
ejpam-5984	61	13	(	(	PUNCT
ejpam-5984	61	14	v	v	NOUN
ejpam-5984	61	15	f	f	PROPN
ejpam-5984	61	16	0	0	NUM
ejpam-5984	61	17	,	,	PUNCT
ejpam-5984	61	18	v	v	NOUN
ejpam-5984	61	19	f	f	PROPN
ejpam-5984	61	20	1	1	NUM
ejpam-5984	61	21	,	,	PUNCT
ejpam-5984	61	22	.	.	PUNCT
ejpam-5984	61	23	.	.	PUNCT
ejpam-5984	62	1	.	.	PUNCT
ejpam-5984	63	1	,	,	PUNCT
ejpam-5984	63	2	v	v	X
ejpam-5984	63	3	f	f	X
ejpam-5984	63	4	k+1	k+1	NOUN
ejpam-5984	63	5	)	)	PUNCT
ejpam-5984	63	6	,	,	PUNCT
ejpam-5984	63	7	where	where	SCONJ
ejpam-5984	63	8	the	the	DET
ejpam-5984	63	9	superscript	superscript	PROPN
ejpam-5984	63	10	f	f	PROPN
ejpam-5984	63	11	may	may	AUX
ejpam-5984	63	12	be	be	AUX
ejpam-5984	63	13	omitted	omit	VERB
ejpam-5984	63	14	from	from	ADP
ejpam-5984	63	15	v	v	NUM
ejpam-5984	63	16	f	f	NOUN
ejpam-5984	63	17	i	i	PRON
ejpam-5984	63	18	when	when	SCONJ
ejpam-5984	63	19	no	no	DET
ejpam-5984	63	20	confusion	confusion	NOUN
ejpam-5984	63	21	arises	arise	VERB
ejpam-5984	63	22	.	.	PUNCT
ejpam-5984	64	1	a	a	DET
ejpam-5984	64	2	double	double	ADJ
ejpam-5984	64	3	roman	roman	ADJ
ejpam-5984	64	4	dominating	dominating	NOUN
ejpam-5984	64	5	function	function	NOUN
ejpam-5984	64	6	(	(	PUNCT
ejpam-5984	64	7	drdf	drdf	PROPN
ejpam-5984	64	8	)	)	PUNCT
ejpam-5984	64	9	on	on	ADP
ejpam-5984	64	10	a	a	DET
ejpam-5984	64	11	graph	graph	NOUN
ejpam-5984	64	12	g	g	NOUN
ejpam-5984	64	13	=	=	PUNCT
ejpam-5984	64	14	(	(	PUNCT
ejpam-5984	64	15	v	v	NOUN
ejpam-5984	64	16	,	,	PUNCT
ejpam-5984	64	17	e	e	NOUN
ejpam-5984	64	18	)	)	PUNCT
ejpam-5984	64	19	is	be	AUX
ejpam-5984	64	20	a	a	DET
ejpam-5984	64	21	function	function	NOUN
ejpam-5984	64	22	f	f	NOUN
ejpam-5984	64	23	:	:	PUNCT
ejpam-5984	64	24	v	v	X
ejpam-5984	64	25	→	→	SYM
ejpam-5984	64	26	{	{	PUNCT
ejpam-5984	64	27	0	0	NUM
ejpam-5984	64	28	,	,	PUNCT
ejpam-5984	64	29	1	1	NUM
ejpam-5984	64	30	,	,	PUNCT
ejpam-5984	64	31	2	2	NUM
ejpam-5984	64	32	,	,	PUNCT
ejpam-5984	64	33	3	3	NUM
ejpam-5984	64	34	}	}	PUNCT
ejpam-5984	64	35	having	have	VERB
ejpam-5984	64	36	the	the	DET
ejpam-5984	64	37	property	property	NOUN
ejpam-5984	64	38	that	that	PRON
ejpam-5984	64	39	if	if	SCONJ
ejpam-5984	64	40	f(v	f(v	NOUN
ejpam-5984	64	41	)	)	PUNCT
ejpam-5984	64	42	=	=	SYM
ejpam-5984	64	43	0	0	NUM
ejpam-5984	64	44	,	,	PUNCT
ejpam-5984	64	45	then	then	ADV
ejpam-5984	64	46	the	the	DET
ejpam-5984	64	47	vertex	vertex	NOUN
ejpam-5984	64	48	v	v	NOUN
ejpam-5984	64	49	must	must	AUX
ejpam-5984	64	50	have	have	VERB
ejpam-5984	64	51	at	at	ADV
ejpam-5984	64	52	least	least	ADV
ejpam-5984	64	53	two	two	NUM
ejpam-5984	64	54	neighbors	neighbor	NOUN
ejpam-5984	64	55	assigned	assign	VERB
ejpam-5984	64	56	2	2	NUM
ejpam-5984	64	57	under	under	ADP
ejpam-5984	64	58	f	f	PROPN
ejpam-5984	64	59	or	or	CCONJ
ejpam-5984	64	60	one	one	NUM
ejpam-5984	64	61	neighbor	neighbor	NOUN
ejpam-5984	64	62	u	u	NOUN
ejpam-5984	64	63	with	with	ADP
ejpam-5984	64	64	f(u	f(u	PROPN
ejpam-5984	64	65	)	)	PUNCT
ejpam-5984	64	66	=	=	SYM
ejpam-5984	64	67	3	3	NUM
ejpam-5984	64	68	,	,	PUNCT
ejpam-5984	64	69	and	and	CCONJ
ejpam-5984	64	70	if	if	SCONJ
ejpam-5984	64	71	f(v	f(v	NOUN
ejpam-5984	64	72	)	)	PUNCT
ejpam-5984	65	1	=	=	SYM
ejpam-5984	65	2	1	1	NUM
ejpam-5984	65	3	,	,	PUNCT
ejpam-5984	65	4	then	then	ADV
ejpam-5984	65	5	the	the	DET
ejpam-5984	65	6	vertex	vertex	NOUN
ejpam-5984	65	7	v	v	NOUN
ejpam-5984	65	8	must	must	AUX
ejpam-5984	65	9	have	have	VERB
ejpam-5984	65	10	at	at	ADV
ejpam-5984	65	11	least	least	ADV
ejpam-5984	65	12	one	one	NUM
ejpam-5984	65	13	neighbor	neighbor	NOUN
ejpam-5984	65	14	u	u	NOUN
ejpam-5984	65	15	with	with	ADP
ejpam-5984	65	16	f(u	f(u	PROPN
ejpam-5984	65	17	)	)	PUNCT
ejpam-5984	65	18	≥	≥	NOUN
ejpam-5984	65	19	2	2	NUM
ejpam-5984	65	20	.	.	PUNCT
ejpam-5984	66	1	the	the	DET
ejpam-5984	66	2	weight	weight	NOUN
ejpam-5984	66	3	of	of	ADP
ejpam-5984	66	4	a	a	DET
ejpam-5984	66	5	drdf	drdf	NOUN
ejpam-5984	66	6	is	be	AUX
ejpam-5984	66	7	the	the	DET
ejpam-5984	66	8	sum	sum	NOUN
ejpam-5984	66	9	of	of	ADP
ejpam-5984	66	10	its	its	PRON
ejpam-5984	66	11	function	function	NOUN
ejpam-5984	66	12	values	value	NOUN
ejpam-5984	66	13	over	over	ADP
ejpam-5984	66	14	all	all	DET
ejpam-5984	66	15	vertices	vertex	NOUN
ejpam-5984	66	16	,	,	PUNCT
ejpam-5984	66	17	and	and	CCONJ
ejpam-5984	66	18	the	the	DET
ejpam-5984	66	19	double	double	ADJ
ejpam-5984	66	20	roman	roman	ADJ
ejpam-5984	66	21	domination	domination	NOUN
ejpam-5984	66	22	number	number	NOUN
ejpam-5984	66	23	γdr(g	γdr(g	PROPN
ejpam-5984	66	24	)	)	PUNCT
ejpam-5984	66	25	is	be	AUX
ejpam-5984	66	26	the	the	DET
ejpam-5984	66	27	minimum	minimum	ADJ
ejpam-5984	66	28	weight	weight	NOUN
ejpam-5984	66	29	of	of	ADP
ejpam-5984	66	30	a	a	DET
ejpam-5984	66	31	drdf	drdf	NOUN
ejpam-5984	66	32	on	on	ADP
ejpam-5984	66	33	g.	g.	PROPN
ejpam-5984	66	34	the	the	DET
ejpam-5984	66	35	double	double	ADJ
ejpam-5984	66	36	roman	roman	ADJ
ejpam-5984	66	37	domination	domination	NOUN
ejpam-5984	66	38	stability	stability	NOUN
ejpam-5984	66	39	,	,	PUNCT
ejpam-5984	66	40	or	or	CCONJ
ejpam-5984	66	41	just	just	ADV
ejpam-5984	66	42	γdr	γdr	NOUN
ejpam-5984	66	43	-	-	PUNCT
ejpam-5984	66	44	stability	stability	NOUN
ejpam-5984	66	45	,	,	PUNCT
ejpam-5984	66	46	of	of	ADP
ejpam-5984	66	47	a	a	DET
ejpam-5984	66	48	graph	graph	NOUN
ejpam-5984	66	49	g	g	NOUN
ejpam-5984	66	50	is	be	AUX
ejpam-5984	66	51	the	the	DET
ejpam-5984	66	52	minimum	minimum	ADJ
ejpam-5984	66	53	size	size	NOUN
ejpam-5984	66	54	of	of	ADP
ejpam-5984	66	55	a	a	DET
ejpam-5984	66	56	set	set	NOUN
ejpam-5984	66	57	of	of	ADP
ejpam-5984	66	58	vertices	vertex	NOUN
ejpam-5984	66	59	whose	whose	DET
ejpam-5984	66	60	removal	removal	NOUN
ejpam-5984	66	61	changes	change	VERB
ejpam-5984	66	62	the	the	DET
ejpam-5984	66	63	double	double	ADJ
ejpam-5984	66	64	roman	roman	ADJ
ejpam-5984	66	65	domination	domination	NOUN
ejpam-5984	66	66	number	number	NOUN
ejpam-5984	66	67	.	.	PUNCT
ejpam-5984	67	1	we	we	PRON
ejpam-5984	67	2	denote	denote	VERB
ejpam-5984	67	3	the	the	DET
ejpam-5984	67	4	γdr	γdr	NOUN
ejpam-5984	67	5	-	-	PUNCT
ejpam-5984	67	6	stability	stability	NOUN
ejpam-5984	67	7	of	of	ADP
ejpam-5984	67	8	g	g	NOUN
ejpam-5984	67	9	by	by	ADP
ejpam-5984	67	10	stγdr(g	stγdr(g	PRON
ejpam-5984	67	11	)	)	PUNCT
ejpam-5984	67	12	.	.	PUNCT
ejpam-5984	68	1	the	the	DET
ejpam-5984	68	2	decreasing	decrease	VERB
ejpam-5984	68	3	γdr	γdr	NOUN
ejpam-5984	68	4	-	-	PUNCT
ejpam-5984	68	5	stability	stability	NOUN
ejpam-5984	68	6	of	of	ADP
ejpam-5984	68	7	g	g	NOUN
ejpam-5984	68	8	,	,	PUNCT
ejpam-5984	68	9	denoted	denote	VERB
ejpam-5984	68	10	by	by	ADP
ejpam-5984	68	11	st−γdr(g	st−γdr(g	PROPN
ejpam-5984	68	12	)	)	PUNCT
ejpam-5984	68	13	,	,	PUNCT
ejpam-5984	68	14	is	be	AUX
ejpam-5984	68	15	defined	define	VERB
ejpam-5984	68	16	as	as	ADP
ejpam-5984	68	17	the	the	DET
ejpam-5984	68	18	minimum	minimum	ADJ
ejpam-5984	68	19	size	size	NOUN
ejpam-5984	68	20	of	of	ADP
ejpam-5984	68	21	a	a	DET
ejpam-5984	68	22	set	set	NOUN
ejpam-5984	68	23	of	of	ADP
ejpam-5984	68	24	vertices	vertex	NOUN
ejpam-5984	68	25	whose	whose	DET
ejpam-5984	68	26	removal	removal	NOUN
ejpam-5984	68	27	decreases	decrease	VERB
ejpam-5984	68	28	the	the	DET
ejpam-5984	68	29	double	double	ADJ
ejpam-5984	68	30	roman	roman	ADJ
ejpam-5984	68	31	domination	domination	NOUN
ejpam-5984	68	32	number	number	NOUN
ejpam-5984	68	33	.	.	PUNCT
ejpam-5984	69	1	for	for	ADP
ejpam-5984	69	2	the	the	DET
ejpam-5984	69	3	null	null	ADJ
ejpam-5984	69	4	graph	graph	NOUN
ejpam-5984	69	5	n0	n0	PROPN
ejpam-5984	69	6	,	,	PUNCT
ejpam-5984	69	7	which	which	PRON
ejpam-5984	69	8	is	be	AUX
ejpam-5984	69	9	the	the	DET
ejpam-5984	69	10	unique	unique	ADJ
ejpam-5984	69	11	graph	graph	NOUN
ejpam-5984	69	12	having	have	VERB
ejpam-5984	69	13	no	no	DET
ejpam-5984	69	14	vertices	vertex	NOUN
ejpam-5984	69	15	and	and	CCONJ
ejpam-5984	69	16	hence	hence	ADV
ejpam-5984	69	17	has	have	VERB
ejpam-5984	69	18	order	order	NOUN
ejpam-5984	69	19	zero	zero	NUM
ejpam-5984	69	20	,	,	PUNCT
ejpam-5984	69	21	we	we	PRON
ejpam-5984	69	22	let	let	VERB
ejpam-5984	69	23	st−γdr(n0	st−γdr(n0	NOUN
ejpam-5984	69	24	)	)	PUNCT
ejpam-5984	69	25	=	=	SYM
ejpam-5984	70	1	0	0	X
ejpam-5984	70	2	.	.	PUNCT
ejpam-5984	71	1	with	with	ADP
ejpam-5984	71	2	this	this	DET
ejpam-5984	71	3	consideration	consideration	NOUN
ejpam-5984	71	4	,	,	PUNCT
ejpam-5984	71	5	the	the	DET
ejpam-5984	71	6	decreasing	decrease	VERB
ejpam-5984	71	7	γdr	γdr	NOUN
ejpam-5984	71	8	-	-	PUNCT
ejpam-5984	71	9	stability	stability	NOUN
ejpam-5984	71	10	of	of	ADP
ejpam-5984	71	11	a	a	DET
ejpam-5984	71	12	non	non	ADJ
ejpam-5984	71	13	-	-	ADJ
ejpam-5984	71	14	null	null	ADJ
ejpam-5984	71	15	graph	graph	NOUN
ejpam-5984	71	16	is	be	AUX
ejpam-5984	71	17	always	always	ADV
ejpam-5984	71	18	defined	define	VERB
ejpam-5984	71	19	.	.	PUNCT
ejpam-5984	72	1	for	for	ADP
ejpam-5984	72	2	example	example	NOUN
ejpam-5984	72	3	,	,	PUNCT
ejpam-5984	72	4	st−γdr(k1	st−γdr(k1	PROPN
ejpam-5984	72	5	)	)	PUNCT
ejpam-5984	72	6	=	=	SYM
ejpam-5984	73	1	1	1	X
ejpam-5984	73	2	.	.	PUNCT
ejpam-5984	74	1	the	the	DET
ejpam-5984	74	2	increasing	increase	VERB
ejpam-5984	74	3	γdr	γdr	NOUN
ejpam-5984	74	4	-	-	PUNCT
ejpam-5984	74	5	stability	stability	NOUN
ejpam-5984	74	6	of	of	ADP
ejpam-5984	74	7	g	g	NOUN
ejpam-5984	74	8	,	,	PUNCT
ejpam-5984	74	9	denoted	denote	VERB
ejpam-5984	74	10	by	by	ADP
ejpam-5984	74	11	st+γdr(g	st+γdr(g	PROPN
ejpam-5984	74	12	)	)	PUNCT
ejpam-5984	74	13	,	,	PUNCT
ejpam-5984	74	14	is	be	AUX
ejpam-5984	74	15	defined	define	VERB
ejpam-5984	74	16	as	as	ADP
ejpam-5984	74	17	the	the	DET
ejpam-5984	74	18	minimum	minimum	ADJ
ejpam-5984	74	19	size	size	NOUN
ejpam-5984	74	20	of	of	ADP
ejpam-5984	74	21	a	a	DET
ejpam-5984	74	22	set	set	NOUN
ejpam-5984	74	23	of	of	ADP
ejpam-5984	74	24	vertices	vertex	NOUN
ejpam-5984	74	25	whose	whose	DET
ejpam-5984	74	26	removal	removal	NOUN
ejpam-5984	74	27	increases	increase	VERB
ejpam-5984	74	28	the	the	DET
ejpam-5984	74	29	double	double	ADJ
ejpam-5984	74	30	roman	roman	ADJ
ejpam-5984	74	31	domination	domination	NOUN
ejpam-5984	74	32	number	number	NOUN
ejpam-5984	74	33	,	,	PUNCT
ejpam-5984	74	34	if	if	SCONJ
ejpam-5984	74	35	such	such	DET
ejpam-5984	74	36	a	a	DET
ejpam-5984	74	37	set	set	NOUN
ejpam-5984	74	38	exists	exist	VERB
ejpam-5984	74	39	.	.	PUNCT
ejpam-5984	75	1	clearly	clearly	ADV
ejpam-5984	75	2	,	,	PUNCT
ejpam-5984	75	3	stγdr(g	stγdr(g	PRON
ejpam-5984	75	4	)	)	PUNCT
ejpam-5984	76	1	=	=	SYM
ejpam-5984	76	2	min	min	PROPN
ejpam-5984	76	3	{	{	PUNCT
ejpam-5984	76	4	st−γdr(g	st−γdr(g	PROPN
ejpam-5984	76	5	)	)	PUNCT
ejpam-5984	76	6	,	,	PUNCT
ejpam-5984	76	7	st+γdr(g	st+γdr(g	PROPN
ejpam-5984	76	8	)	)	PUNCT
ejpam-5984	76	9	}	}	PUNCT
ejpam-5984	76	10	.	.	PUNCT
ejpam-5984	77	1	an	an	DET
ejpam-5984	77	2	independent	independent	ADJ
ejpam-5984	77	3	[	[	X
ejpam-5984	77	4	k]-roman	k]-roman	ADJ
ejpam-5984	77	5	dominating	dominating	NOUN
ejpam-5984	77	6	function	function	NOUN
ejpam-5984	77	7	,	,	PUNCT
ejpam-5984	77	8	abbreviated	abbreviate	VERB
ejpam-5984	77	9	as	as	ADP
ejpam-5984	77	10	i[k]rdf	i[k]rdf	PROPN
ejpam-5984	77	11	,	,	PUNCT
ejpam-5984	77	12	is	be	AUX
ejpam-5984	77	13	a	a	DET
ejpam-5984	77	14	[	[	X
ejpam-5984	77	15	k]roman	k]roman	ADJ
ejpam-5984	77	16	dominating	dominating	NOUN
ejpam-5984	77	17	function	function	NOUN
ejpam-5984	77	18	f	f	PROPN
ejpam-5984	77	19	such	such	ADJ
ejpam-5984	77	20	that	that	SCONJ
ejpam-5984	77	21	the	the	DET
ejpam-5984	77	22	subgraph	subgraph	NOUN
ejpam-5984	77	23	induced	induce	VERB
ejpam-5984	77	24	by	by	ADP
ejpam-5984	77	25	the	the	DET
ejpam-5984	77	26	vertices	vertex	NOUN
ejpam-5984	77	27	with	with	ADP
ejpam-5984	77	28	positive	positive	ADJ
ejpam-5984	77	29	weight	weight	NOUN
ejpam-5984	77	30	under	under	ADP
ejpam-5984	77	31	f	f	PROPN
ejpam-5984	77	32	is	be	AUX
ejpam-5984	77	33	edgeless	edgeless	NOUN
ejpam-5984	77	34	.	.	PUNCT
ejpam-5984	78	1	the	the	DET
ejpam-5984	78	2	minimum	minimum	ADJ
ejpam-5984	78	3	weight	weight	NOUN
ejpam-5984	78	4	of	of	ADP
ejpam-5984	78	5	an	an	DET
ejpam-5984	78	6	i[k]rdf	i[k]rdf	PROPN
ejpam-5984	78	7	on	on	ADP
ejpam-5984	78	8	a	a	DET
ejpam-5984	78	9	graph	graph	NOUN
ejpam-5984	78	10	g	g	NOUN
ejpam-5984	78	11	is	be	AUX
ejpam-5984	78	12	called	call	VERB
ejpam-5984	78	13	the	the	DET
ejpam-5984	78	14	independent	independent	ADJ
ejpam-5984	78	15	[	[	X
ejpam-5984	78	16	k]-roman	k]-roman	ADJ
ejpam-5984	78	17	domination	domination	NOUN
ejpam-5984	78	18	number	number	NOUN
ejpam-5984	78	19	,	,	PUNCT
ejpam-5984	78	20	denoted	denote	VERB
ejpam-5984	78	21	by	by	ADP
ejpam-5984	78	22	i[k]r(g	i[k]r(g	PROPN
ejpam-5984	78	23	)	)	PUNCT
ejpam-5984	78	24	,	,	PUNCT
ejpam-5984	78	25	which	which	PRON
ejpam-5984	78	26	we	we	PRON
ejpam-5984	78	27	refer	refer	VERB
ejpam-5984	78	28	to	to	ADP
ejpam-5984	78	29	as	as	ADP
ejpam-5984	78	30	the	the	DET
ejpam-5984	78	31	s.	s.	PROPN
ejpam-5984	78	32	m.	m.	PROPN
ejpam-5984	78	33	sheikholeslami	sheikholeslami	PROPN
ejpam-5984	78	34	et	et	PROPN
ejpam-5984	78	35	al	al	PROPN
ejpam-5984	78	36	.	.	PUNCT
ejpam-5984	78	37	/	/	SYM
ejpam-5984	78	38	eur	eur	PROPN
ejpam-5984	78	39	.	.	PUNCT
ejpam-5984	79	1	j.	j.	PROPN
ejpam-5984	79	2	pure	pure	PROPN
ejpam-5984	79	3	appl	appl	PROPN
ejpam-5984	79	4	.	.	PROPN
ejpam-5984	79	5	math	math	PROPN
ejpam-5984	79	6	,	,	PUNCT
ejpam-5984	79	7	18	18	NUM
ejpam-5984	79	8	(	(	PUNCT
ejpam-5984	79	9	2	2	NUM
ejpam-5984	79	10	)	)	PUNCT
ejpam-5984	79	11	(	(	PUNCT
ejpam-5984	79	12	2025	2025	NUM
ejpam-5984	79	13	)	)	PUNCT
ejpam-5984	79	14	,	,	PUNCT
ejpam-5984	79	15	5984	5984	NUM
ejpam-5984	79	16	4	4	NUM
ejpam-5984	79	17	of	of	ADP
ejpam-5984	79	18	16	16	NUM
ejpam-5984	79	19	i[k]rd	i[k]rd	NOUN
ejpam-5984	79	20	-	-	NOUN
ejpam-5984	79	21	number	number	NOUN
ejpam-5984	79	22	.	.	PUNCT
ejpam-5984	80	1	by	by	ADP
ejpam-5984	80	2	definition	definition	NOUN
ejpam-5984	80	3	,	,	PUNCT
ejpam-5984	80	4	we	we	PRON
ejpam-5984	80	5	have	have	VERB
ejpam-5984	80	6	γ[k]r(g	γ[k]r(g	NUM
ejpam-5984	80	7	)	)	PUNCT
ejpam-5984	80	8	≤	≤	NOUN
ejpam-5984	80	9	i[k]r(g	i[k]r(g	PROPN
ejpam-5984	80	10	)	)	PUNCT
ejpam-5984	80	11	.	.	PUNCT
ejpam-5984	81	1	(	(	PUNCT
ejpam-5984	81	2	1	1	X
ejpam-5984	81	3	)	)	PUNCT
ejpam-5984	81	4	an	an	DET
ejpam-5984	81	5	independent	independent	ADJ
ejpam-5984	81	6	double	double	ADJ
ejpam-5984	81	7	roman	roman	ADJ
ejpam-5984	81	8	dominating	dominating	NOUN
ejpam-5984	81	9	function	function	NOUN
ejpam-5984	81	10	(	(	PUNCT
ejpam-5984	81	11	idrd	idrd	ADJ
ejpam-5984	81	12	-	-	PUNCT
ejpam-5984	81	13	function	function	NOUN
ejpam-5984	81	14	)	)	PUNCT
ejpam-5984	81	15	on	on	ADP
ejpam-5984	81	16	a	a	DET
ejpam-5984	81	17	graph	graph	NOUN
ejpam-5984	81	18	g	g	NOUN
ejpam-5984	81	19	is	be	AUX
ejpam-5984	81	20	a	a	DET
ejpam-5984	81	21	function	function	NOUN
ejpam-5984	81	22	f	f	NOUN
ejpam-5984	81	23	:	:	PUNCT
ejpam-5984	81	24	v	v	X
ejpam-5984	81	25	(	(	PUNCT
ejpam-5984	81	26	g	g	NOUN
ejpam-5984	81	27	)	)	PUNCT
ejpam-5984	81	28	→	→	SYM
ejpam-5984	81	29	{	{	PUNCT
ejpam-5984	81	30	0	0	NUM
ejpam-5984	81	31	,	,	PUNCT
ejpam-5984	81	32	1	1	NUM
ejpam-5984	81	33	,	,	PUNCT
ejpam-5984	81	34	2	2	NUM
ejpam-5984	81	35	,	,	PUNCT
ejpam-5984	81	36	3	3	NUM
ejpam-5984	81	37	}	}	PUNCT
ejpam-5984	81	38	having	have	VERB
ejpam-5984	81	39	the	the	DET
ejpam-5984	81	40	properties	property	NOUN
ejpam-5984	81	41	that	that	PRON
ejpam-5984	81	42	(	(	PUNCT
ejpam-5984	81	43	i	i	NOUN
ejpam-5984	81	44	)	)	PUNCT
ejpam-5984	81	45	if	if	SCONJ
ejpam-5984	81	46	f(v	f(v	NOUN
ejpam-5984	81	47	)	)	PUNCT
ejpam-5984	82	1	=	=	SYM
ejpam-5984	82	2	0	0	NUM
ejpam-5984	82	3	,	,	PUNCT
ejpam-5984	82	4	then	then	ADV
ejpam-5984	82	5	the	the	DET
ejpam-5984	82	6	vertex	vertex	NOUN
ejpam-5984	82	7	v	v	NOUN
ejpam-5984	82	8	must	must	AUX
ejpam-5984	82	9	have	have	VERB
ejpam-5984	82	10	at	at	ADV
ejpam-5984	82	11	least	least	ADV
ejpam-5984	82	12	two	two	NUM
ejpam-5984	82	13	neighbors	neighbor	NOUN
ejpam-5984	82	14	assigned	assign	VERB
ejpam-5984	82	15	2	2	NUM
ejpam-5984	82	16	under	under	ADP
ejpam-5984	82	17	f	f	PROPN
ejpam-5984	82	18	,	,	PUNCT
ejpam-5984	82	19	or	or	CCONJ
ejpam-5984	82	20	one	one	NUM
ejpam-5984	82	21	neighbor	neighbor	NOUN
ejpam-5984	82	22	w	w	NOUN
ejpam-5984	82	23	with	with	ADP
ejpam-5984	82	24	f(w	f(w	NOUN
ejpam-5984	82	25	)	)	PUNCT
ejpam-5984	82	26	=	=	SYM
ejpam-5984	82	27	3	3	NUM
ejpam-5984	82	28	,	,	PUNCT
ejpam-5984	82	29	and	and	CCONJ
ejpam-5984	82	30	if	if	SCONJ
ejpam-5984	82	31	f(v	f(v	NOUN
ejpam-5984	82	32	)	)	PUNCT
ejpam-5984	83	1	=	=	SYM
ejpam-5984	83	2	1	1	NUM
ejpam-5984	83	3	,	,	PUNCT
ejpam-5984	83	4	then	then	ADV
ejpam-5984	83	5	the	the	DET
ejpam-5984	83	6	vertex	vertex	NOUN
ejpam-5984	83	7	v	v	NOUN
ejpam-5984	83	8	must	must	AUX
ejpam-5984	83	9	have	have	VERB
ejpam-5984	83	10	at	at	ADV
ejpam-5984	83	11	least	least	ADV
ejpam-5984	83	12	one	one	NUM
ejpam-5984	83	13	neighbor	neighbor	NOUN
ejpam-5984	83	14	w	w	NOUN
ejpam-5984	83	15	with	with	ADP
ejpam-5984	83	16	f(w	f(w	PROPN
ejpam-5984	83	17	)	)	PUNCT
ejpam-5984	83	18	≥	≥	NOUN
ejpam-5984	83	19	2	2	NUM
ejpam-5984	83	20	;	;	PUNCT
ejpam-5984	83	21	and	and	CCONJ
ejpam-5984	83	22	(	(	PUNCT
ejpam-5984	83	23	ii	ii	NOUN
ejpam-5984	83	24	)	)	PUNCT
ejpam-5984	83	25	the	the	DET
ejpam-5984	83	26	subgraph	subgraph	NOUN
ejpam-5984	83	27	induced	induce	VERB
ejpam-5984	83	28	by	by	ADP
ejpam-5984	83	29	the	the	DET
ejpam-5984	83	30	vertices	vertex	NOUN
ejpam-5984	83	31	with	with	ADP
ejpam-5984	83	32	positive	positive	ADJ
ejpam-5984	83	33	weight	weight	NOUN
ejpam-5984	83	34	under	under	ADP
ejpam-5984	83	35	f	f	PROPN
ejpam-5984	83	36	is	be	AUX
ejpam-5984	83	37	edgeless	edgeless	NOUN
ejpam-5984	83	38	.	.	PUNCT
ejpam-5984	84	1	the	the	DET
ejpam-5984	84	2	weight	weight	NOUN
ejpam-5984	84	3	of	of	ADP
ejpam-5984	84	4	an	an	DET
ejpam-5984	84	5	idrd	idrd	ADJ
ejpam-5984	84	6	-	-	PUNCT
ejpam-5984	84	7	function	function	NOUN
ejpam-5984	84	8	is	be	AUX
ejpam-5984	84	9	the	the	DET
ejpam-5984	84	10	sum	sum	NOUN
ejpam-5984	84	11	of	of	ADP
ejpam-5984	84	12	its	its	PRON
ejpam-5984	84	13	function	function	NOUN
ejpam-5984	84	14	values	value	NOUN
ejpam-5984	84	15	over	over	ADP
ejpam-5984	84	16	all	all	DET
ejpam-5984	84	17	vertices	vertex	NOUN
ejpam-5984	84	18	,	,	PUNCT
ejpam-5984	84	19	and	and	CCONJ
ejpam-5984	84	20	the	the	DET
ejpam-5984	84	21	independent	independent	ADJ
ejpam-5984	84	22	double	double	ADJ
ejpam-5984	84	23	roman	roman	ADJ
ejpam-5984	84	24	domination	domination	NOUN
ejpam-5984	84	25	stability	stability	NOUN
ejpam-5984	84	26	,	,	PUNCT
ejpam-5984	84	27	or	or	CCONJ
ejpam-5984	84	28	simply	simply	ADV
ejpam-5984	84	29	the	the	DET
ejpam-5984	84	30	idr	idr	NOUN
ejpam-5984	84	31	-	-	NOUN
ejpam-5984	84	32	stability	stability	NOUN
ejpam-5984	84	33	,	,	PUNCT
ejpam-5984	84	34	of	of	ADP
ejpam-5984	84	35	a	a	DET
ejpam-5984	84	36	graph	graph	NOUN
ejpam-5984	84	37	g	g	NOUN
ejpam-5984	84	38	is	be	AUX
ejpam-5984	84	39	the	the	DET
ejpam-5984	84	40	minimum	minimum	ADJ
ejpam-5984	84	41	size	size	NOUN
ejpam-5984	84	42	of	of	ADP
ejpam-5984	84	43	a	a	DET
ejpam-5984	84	44	set	set	NOUN
ejpam-5984	84	45	of	of	ADP
ejpam-5984	84	46	vertices	vertex	NOUN
ejpam-5984	84	47	whose	whose	DET
ejpam-5984	84	48	removal	removal	NOUN
ejpam-5984	84	49	changes	change	VERB
ejpam-5984	84	50	the	the	DET
ejpam-5984	84	51	independent	independent	ADJ
ejpam-5984	84	52	double	double	ADJ
ejpam-5984	84	53	roman	roman	ADJ
ejpam-5984	84	54	domination	domination	NOUN
ejpam-5984	84	55	number	number	NOUN
ejpam-5984	84	56	.	.	PUNCT
ejpam-5984	85	1	we	we	PRON
ejpam-5984	85	2	denote	denote	VERB
ejpam-5984	85	3	the	the	DET
ejpam-5984	85	4	idr	idr	NOUN
ejpam-5984	85	5	-	-	NOUN
ejpam-5984	85	6	stability	stability	NOUN
ejpam-5984	85	7	of	of	ADP
ejpam-5984	85	8	g	g	NOUN
ejpam-5984	85	9	by	by	ADP
ejpam-5984	85	10	stidr(g	stidr(g	PROPN
ejpam-5984	85	11	)	)	PUNCT
ejpam-5984	85	12	.	.	PUNCT
ejpam-5984	86	1	the	the	DET
ejpam-5984	86	2	i−drstability	i−drstability	NOUN
ejpam-5984	86	3	of	of	ADP
ejpam-5984	86	4	g	g	NOUN
ejpam-5984	86	5	,	,	PUNCT
ejpam-5984	86	6	denoted	denote	VERB
ejpam-5984	86	7	by	by	ADP
ejpam-5984	86	8	st−idr(g	st−idr(g	PROPN
ejpam-5984	86	9	)	)	PUNCT
ejpam-5984	86	10	,	,	PUNCT
ejpam-5984	86	11	is	be	AUX
ejpam-5984	86	12	defined	define	VERB
ejpam-5984	86	13	as	as	ADP
ejpam-5984	86	14	the	the	DET
ejpam-5984	86	15	minimum	minimum	ADJ
ejpam-5984	86	16	size	size	NOUN
ejpam-5984	86	17	of	of	ADP
ejpam-5984	86	18	a	a	DET
ejpam-5984	86	19	set	set	NOUN
ejpam-5984	86	20	of	of	ADP
ejpam-5984	86	21	vertices	vertex	NOUN
ejpam-5984	86	22	whose	whose	DET
ejpam-5984	86	23	removal	removal	NOUN
ejpam-5984	86	24	decreases	decrease	VERB
ejpam-5984	86	25	the	the	DET
ejpam-5984	86	26	independent	independent	ADJ
ejpam-5984	86	27	double	double	ADJ
ejpam-5984	86	28	roman	roman	ADJ
ejpam-5984	86	29	domination	domination	NOUN
ejpam-5984	86	30	number	number	NOUN
ejpam-5984	86	31	,	,	PUNCT
ejpam-5984	86	32	and	and	CCONJ
ejpam-5984	86	33	the	the	DET
ejpam-5984	86	34	i+dr	i+dr	NOUN
ejpam-5984	86	35	-	-	PUNCT
ejpam-5984	86	36	stability	stability	NOUN
ejpam-5984	86	37	of	of	ADP
ejpam-5984	86	38	g	g	NOUN
ejpam-5984	86	39	,	,	PUNCT
ejpam-5984	86	40	denoted	denote	VERB
ejpam-5984	86	41	by	by	ADP
ejpam-5984	86	42	st+idr(g	st+idr(g	PROPN
ejpam-5984	86	43	)	)	PUNCT
ejpam-5984	86	44	,	,	PUNCT
ejpam-5984	86	45	is	be	AUX
ejpam-5984	86	46	defined	define	VERB
ejpam-5984	86	47	as	as	ADP
ejpam-5984	86	48	the	the	DET
ejpam-5984	86	49	minimum	minimum	ADJ
ejpam-5984	86	50	size	size	NOUN
ejpam-5984	86	51	of	of	ADP
ejpam-5984	86	52	a	a	DET
ejpam-5984	86	53	set	set	NOUN
ejpam-5984	86	54	of	of	ADP
ejpam-5984	86	55	vertices	vertex	NOUN
ejpam-5984	86	56	whose	whose	DET
ejpam-5984	86	57	removal	removal	NOUN
ejpam-5984	86	58	increases	increase	VERB
ejpam-5984	86	59	the	the	DET
ejpam-5984	86	60	independent	independent	ADJ
ejpam-5984	86	61	double	double	ADJ
ejpam-5984	86	62	roman	roman	ADJ
ejpam-5984	86	63	domination	domination	NOUN
ejpam-5984	86	64	number	number	NOUN
ejpam-5984	86	65	,	,	PUNCT
ejpam-5984	86	66	if	if	SCONJ
ejpam-5984	86	67	such	such	DET
ejpam-5984	86	68	a	a	DET
ejpam-5984	86	69	set	set	NOUN
ejpam-5984	86	70	exists	exist	VERB
ejpam-5984	86	71	.	.	PUNCT
ejpam-5984	87	1	if	if	SCONJ
ejpam-5984	87	2	there	there	PRON
ejpam-5984	87	3	is	be	VERB
ejpam-5984	87	4	no	no	DET
ejpam-5984	87	5	set	set	NOUN
ejpam-5984	87	6	of	of	ADP
ejpam-5984	87	7	vertices	vertex	NOUN
ejpam-5984	87	8	in	in	ADP
ejpam-5984	87	9	g	g	ADP
ejpam-5984	87	10	whose	whose	DET
ejpam-5984	87	11	removal	removal	NOUN
ejpam-5984	87	12	increases	increase	VERB
ejpam-5984	87	13	the	the	DET
ejpam-5984	87	14	independent	independent	ADJ
ejpam-5984	87	15	double	double	ADJ
ejpam-5984	87	16	roman	roman	ADJ
ejpam-5984	87	17	domination	domination	NOUN
ejpam-5984	87	18	number	number	NOUN
ejpam-5984	87	19	,	,	PUNCT
ejpam-5984	87	20	then	then	ADV
ejpam-5984	87	21	we	we	PRON
ejpam-5984	87	22	set	set	VERB
ejpam-5984	87	23	st+idr(g	st+idr(g	PROPN
ejpam-5984	87	24	)	)	PUNCT
ejpam-5984	88	1	=	=	SYM
ejpam-5984	88	2	∞.	∞.	PROPN
ejpam-5984	88	3	clearly	clearly	ADV
ejpam-5984	88	4	,	,	PUNCT
ejpam-5984	88	5	stidr(g	stidr(g	NOUN
ejpam-5984	88	6	)	)	PUNCT
ejpam-5984	88	7	=	=	SYM
ejpam-5984	88	8	min{st−idr(g	min{st−idr(g	NUM
ejpam-5984	88	9	)	)	PUNCT
ejpam-5984	88	10	,	,	PUNCT
ejpam-5984	88	11	st+idr(g	st+idr(g	PROPN
ejpam-5984	88	12	)	)	PUNCT
ejpam-5984	88	13	}	}	PUNCT
ejpam-5984	88	14	.	.	PUNCT
ejpam-5984	89	1	3	3	X
ejpam-5984	89	2	.	.	X
ejpam-5984	89	3	preliminary	preliminary	ADJ
ejpam-5984	89	4	results	result	NOUN
ejpam-5984	89	5	in	in	ADP
ejpam-5984	89	6	this	this	DET
ejpam-5984	89	7	section	section	NOUN
ejpam-5984	89	8	we	we	PRON
ejpam-5984	89	9	will	will	AUX
ejpam-5984	89	10	investigate	investigate	VERB
ejpam-5984	89	11	simple	simple	ADJ
ejpam-5984	89	12	results	result	NOUN
ejpam-5984	89	13	.	.	PUNCT
ejpam-5984	90	1	remark	remark	NOUN
ejpam-5984	90	2	1	1	NUM
ejpam-5984	90	3	.	.	PUNCT
ejpam-5984	91	1	let	let	VERB
ejpam-5984	91	2	g	g	PRON
ejpam-5984	91	3	be	be	AUX
ejpam-5984	91	4	a	a	DET
ejpam-5984	91	5	nontrivial	nontrivial	ADJ
ejpam-5984	91	6	connected	connect	VERB
ejpam-5984	91	7	graph	graph	NOUN
ejpam-5984	91	8	with	with	ADP
ejpam-5984	91	9	γdr(g	γdr(g	NOUN
ejpam-5984	91	10	)	)	PUNCT
ejpam-5984	91	11	=	=	SYM
ejpam-5984	91	12	idr(g	idr(g	PROPN
ejpam-5984	91	13	)	)	PUNCT
ejpam-5984	91	14	.	.	PUNCT
ejpam-5984	92	1	then	then	ADV
ejpam-5984	92	2	st−γdr(g	st−γdr(g	PROPN
ejpam-5984	92	3	)	)	PUNCT
ejpam-5984	92	4	≤	≤	PUNCT
ejpam-5984	93	1	st−idr(g	st−idr(g	PROPN
ejpam-5984	93	2	)	)	PUNCT
ejpam-5984	93	3	.	.	PUNCT
ejpam-5984	94	1	moreover	moreover	ADV
ejpam-5984	94	2	,	,	PUNCT
ejpam-5984	94	3	if	if	SCONJ
ejpam-5984	94	4	st+γdr(g	st+γdr(g	PROPN
ejpam-5984	94	5	)	)	PUNCT
ejpam-5984	94	6	<	<	X
ejpam-5984	94	7	∞	∞	PROPN
ejpam-5984	94	8	,	,	PUNCT
ejpam-5984	94	9	then	then	ADV
ejpam-5984	94	10	st+idr(g	st+idr(g	PROPN
ejpam-5984	94	11	)	)	PUNCT
ejpam-5984	94	12	≤	≤	NUM
ejpam-5984	94	13	st+γdr(g	st+γdr(g	PROPN
ejpam-5984	94	14	)	)	PUNCT
ejpam-5984	94	15	.	.	PUNCT
ejpam-5984	95	1	maimani	maimani	PROPN
ejpam-5984	95	2	et	et	PROPN
ejpam-5984	95	3	al	al	PROPN
ejpam-5984	95	4	.	.	PUNCT
ejpam-5984	96	1	[	[	X
ejpam-5984	96	2	10	10	NUM
ejpam-5984	96	3	]	]	PUNCT
ejpam-5984	96	4	observed	observe	VERB
ejpam-5984	96	5	that	that	SCONJ
ejpam-5984	96	6	for	for	ADP
ejpam-5984	96	7	any	any	DET
ejpam-5984	96	8	graph	graph	NOUN
ejpam-5984	96	9	g	g	NOUN
ejpam-5984	96	10	and	and	CCONJ
ejpam-5984	96	11	any	any	DET
ejpam-5984	96	12	idr(g)-function	idr(g)-function	NOUN
ejpam-5984	96	13	f	f	NOUN
ejpam-5984	96	14	=	=	SYM
ejpam-5984	96	15	(	(	PUNCT
ejpam-5984	96	16	v0	v0	PROPN
ejpam-5984	96	17	,	,	PUNCT
ejpam-5984	96	18	v1	v1	NOUN
ejpam-5984	96	19	,	,	PUNCT
ejpam-5984	96	20	v2	v2	PROPN
ejpam-5984	96	21	,	,	PUNCT
ejpam-5984	96	22	v3	v3	PROPN
ejpam-5984	96	23	)	)	PUNCT
ejpam-5984	96	24	we	we	PRON
ejpam-5984	96	25	have	have	VERB
ejpam-5984	96	26	v1	v1	NOUN
ejpam-5984	96	27	=	=	SYM
ejpam-5984	96	28	∅.	∅.	NOUN
ejpam-5984	96	29	proposition	proposition	NOUN
ejpam-5984	96	30	1	1	NUM
ejpam-5984	96	31	.	.	PUNCT
ejpam-5984	97	1	let	let	VERB
ejpam-5984	97	2	g	g	PRON
ejpam-5984	97	3	be	be	AUX
ejpam-5984	97	4	a	a	DET
ejpam-5984	97	5	graph	graph	NOUN
ejpam-5984	97	6	and	and	CCONJ
ejpam-5984	97	7	v	v	AUX
ejpam-5984	97	8	be	be	AUX
ejpam-5984	97	9	a	a	DET
ejpam-5984	97	10	vertex	vertex	NOUN
ejpam-5984	97	11	of	of	ADP
ejpam-5984	97	12	g.	g.	PROPN
ejpam-5984	97	13	if	if	SCONJ
ejpam-5984	97	14	g′	g′	NOUN
ejpam-5984	97	15	is	be	AUX
ejpam-5984	97	16	obtained	obtain	VERB
ejpam-5984	97	17	from	from	ADP
ejpam-5984	97	18	g	g	NOUN
ejpam-5984	97	19	by	by	ADP
ejpam-5984	97	20	adding	add	VERB
ejpam-5984	97	21	a	a	DET
ejpam-5984	97	22	star	star	NOUN
ejpam-5984	97	23	k1,t	k1,t	PROPN
ejpam-5984	97	24	with	with	ADP
ejpam-5984	97	25	t	t	PROPN
ejpam-5984	97	26	≥	≥	NUM
ejpam-5984	97	27	2	2	NUM
ejpam-5984	97	28	and	and	CCONJ
ejpam-5984	97	29	joining	join	VERB
ejpam-5984	97	30	v	v	PRON
ejpam-5984	97	31	to	to	ADP
ejpam-5984	97	32	a	a	DET
ejpam-5984	97	33	leaf	leaf	NOUN
ejpam-5984	97	34	of	of	ADP
ejpam-5984	97	35	k1,t	k1,t	PROPN
ejpam-5984	97	36	,	,	PUNCT
ejpam-5984	97	37	then	then	ADV
ejpam-5984	97	38	idr(g	idr(g	PROPN
ejpam-5984	97	39	′	′	NOUN
ejpam-5984	97	40	)	)	PUNCT
ejpam-5984	98	1	=	=	SYM
ejpam-5984	98	2	idr(g	idr(g	PROPN
ejpam-5984	98	3	)	)	PUNCT
ejpam-5984	99	1	+	+	NOUN
ejpam-5984	99	2	3	3	X
ejpam-5984	99	3	.	.	X
ejpam-5984	99	4	proof	proof	NOUN
ejpam-5984	99	5	.	.	PUNCT
ejpam-5984	100	1	let	let	VERB
ejpam-5984	100	2	v	v	X
ejpam-5984	100	3	(	(	PUNCT
ejpam-5984	100	4	k1,t	k1,t	PROPN
ejpam-5984	100	5	)	)	PUNCT
ejpam-5984	100	6	=	=	PRON
ejpam-5984	100	7	{	{	PUNCT
ejpam-5984	100	8	u	u	NOUN
ejpam-5984	100	9	,	,	PUNCT
ejpam-5984	100	10	u1	u1	NOUN
ejpam-5984	100	11	,	,	PUNCT
ejpam-5984	100	12	u2	u2	NOUN
ejpam-5984	100	13	,	,	PUNCT
ejpam-5984	100	14	.	.	PUNCT
ejpam-5984	100	15	.	.	PUNCT
ejpam-5984	101	1	.	.	PUNCT
ejpam-5984	102	1	,	,	PUNCT
ejpam-5984	102	2	ut	ut	PROPN
ejpam-5984	102	3	}	}	PUNCT
ejpam-5984	102	4	,	,	PUNCT
ejpam-5984	102	5	where	where	SCONJ
ejpam-5984	102	6	u	u	NOUN
ejpam-5984	102	7	is	be	AUX
ejpam-5984	102	8	the	the	DET
ejpam-5984	102	9	center	center	NOUN
ejpam-5984	102	10	of	of	ADP
ejpam-5984	102	11	the	the	DET
ejpam-5984	102	12	star	star	NOUN
ejpam-5984	102	13	and	and	CCONJ
ejpam-5984	102	14	u1	u1	NOUN
ejpam-5984	102	15	,	,	PUNCT
ejpam-5984	102	16	.	.	PUNCT
ejpam-5984	102	17	.	.	PUNCT
ejpam-5984	102	18	.	.	PUNCT
ejpam-5984	103	1	,	,	PUNCT
ejpam-5984	103	2	ut	ut	PROPN
ejpam-5984	103	3	are	be	AUX
ejpam-5984	103	4	its	its	PRON
ejpam-5984	103	5	leaves	leave	NOUN
ejpam-5984	103	6	.	.	PUNCT
ejpam-5984	104	1	suppose	suppose	VERB
ejpam-5984	104	2	v	v	SCONJ
ejpam-5984	104	3	∈	∈	PROPN
ejpam-5984	104	4	v	v	NOUN
ejpam-5984	104	5	(	(	PUNCT
ejpam-5984	104	6	g	g	NOUN
ejpam-5984	104	7	)	)	PUNCT
ejpam-5984	104	8	is	be	AUX
ejpam-5984	104	9	connected	connect	VERB
ejpam-5984	104	10	to	to	ADP
ejpam-5984	104	11	the	the	DET
ejpam-5984	104	12	leaf	leaf	NOUN
ejpam-5984	104	13	u1	u1	NOUN
ejpam-5984	104	14	in	in	ADP
ejpam-5984	104	15	g′	g′	PROPN
ejpam-5984	104	16	,	,	PUNCT
ejpam-5984	104	17	i.e.	i.e.	X
ejpam-5984	104	18	,	,	PUNCT
ejpam-5984	104	19	vu1	vu1	PROPN
ejpam-5984	104	20	∈	∈	PROPN
ejpam-5984	104	21	e(g′	e(g′	NUM
ejpam-5984	104	22	)	)	PUNCT
ejpam-5984	104	23	.	.	PUNCT
ejpam-5984	105	1	first	first	ADV
ejpam-5984	105	2	,	,	PUNCT
ejpam-5984	105	3	we	we	PRON
ejpam-5984	105	4	show	show	VERB
ejpam-5984	105	5	that	that	SCONJ
ejpam-5984	105	6	idr(g′	idr(g′	NOUN
ejpam-5984	105	7	)	)	PUNCT
ejpam-5984	105	8	≤	≤	NUM
ejpam-5984	105	9	idr(g)+3	idr(g)+3	NOUN
ejpam-5984	105	10	.	.	PUNCT
ejpam-5984	106	1	let	let	VERB
ejpam-5984	106	2	f	f	PRON
ejpam-5984	106	3	be	be	AUX
ejpam-5984	106	4	an	an	DET
ejpam-5984	106	5	idrd	idrd	ADJ
ejpam-5984	106	6	-	-	PUNCT
ejpam-5984	106	7	function	function	NOUN
ejpam-5984	106	8	on	on	ADP
ejpam-5984	106	9	g	g	NOUN
ejpam-5984	106	10	of	of	ADP
ejpam-5984	106	11	minimum	minimum	ADJ
ejpam-5984	106	12	weight	weight	NOUN
ejpam-5984	106	13	,	,	PUNCT
ejpam-5984	106	14	i.e.	i.e.	X
ejpam-5984	106	15	,	,	PUNCT
ejpam-5984	106	16	an	an	DET
ejpam-5984	106	17	idr(g)-function	idr(g)-function	NOUN
ejpam-5984	106	18	.	.	PUNCT
ejpam-5984	107	1	define	define	VERB
ejpam-5984	107	2	an	an	DET
ejpam-5984	107	3	extension	extension	NOUN
ejpam-5984	107	4	f	f	NOUN
ejpam-5984	107	5	′	′	NOUN
ejpam-5984	107	6	on	on	ADP
ejpam-5984	107	7	g′	g′	NOUN
ejpam-5984	107	8	by	by	ADP
ejpam-5984	107	9	:	:	PUNCT
ejpam-5984	107	10	f	f	PROPN
ejpam-5984	107	11	′(x	′(x	PROPN
ejpam-5984	107	12	)	)	PUNCT
ejpam-5984	107	13	=	=	PUNCT
ejpam-5984	108	1			PROPN
ejpam-5984	108	2	f(x	f(x	PROPN
ejpam-5984	108	3	)	)	PUNCT
ejpam-5984	108	4	,	,	PUNCT
ejpam-5984	108	5	if	if	SCONJ
ejpam-5984	108	6	x	x	SYM
ejpam-5984	108	7	∈	∈	PROPN
ejpam-5984	108	8	v	v	X
ejpam-5984	108	9	(	(	PUNCT
ejpam-5984	108	10	g	g	NOUN
ejpam-5984	108	11	)	)	PUNCT
ejpam-5984	108	12	,	,	PUNCT
ejpam-5984	108	13	3	3	X
ejpam-5984	108	14	,	,	PUNCT
ejpam-5984	108	15	if	if	SCONJ
ejpam-5984	108	16	x	x	ADP
ejpam-5984	108	17	=	=	SYM
ejpam-5984	108	18	u	u	NOUN
ejpam-5984	108	19	,	,	PUNCT
ejpam-5984	108	20	0	0	NUM
ejpam-5984	108	21	,	,	PUNCT
ejpam-5984	108	22	if	if	SCONJ
ejpam-5984	108	23	x	x	SYM
ejpam-5984	108	24	∈	∈	PROPN
ejpam-5984	108	25	{	{	PUNCT
ejpam-5984	108	26	u1	u1	NOUN
ejpam-5984	108	27	,	,	PUNCT
ejpam-5984	108	28	u2	u2	NOUN
ejpam-5984	108	29	,	,	PUNCT
ejpam-5984	108	30	.	.	PUNCT
ejpam-5984	108	31	.	.	PUNCT
ejpam-5984	108	32	.	.	PUNCT
ejpam-5984	109	1	,	,	PUNCT
ejpam-5984	109	2	ut	ut	PROPN
ejpam-5984	109	3	}	}	PUNCT
ejpam-5984	109	4	.	.	PUNCT
ejpam-5984	110	1	clearly	clearly	ADV
ejpam-5984	110	2	,	,	PUNCT
ejpam-5984	110	3	f	f	PROPN
ejpam-5984	110	4	′	′	NOUN
ejpam-5984	110	5	is	be	AUX
ejpam-5984	110	6	a	a	DET
ejpam-5984	110	7	valid	valid	ADJ
ejpam-5984	110	8	idrd	idrd	ADJ
ejpam-5984	110	9	-	-	PUNCT
ejpam-5984	110	10	function	function	NOUN
ejpam-5984	110	11	on	on	ADP
ejpam-5984	110	12	g′	g′	NOUN
ejpam-5984	110	13	,	,	PUNCT
ejpam-5984	110	14	and	and	CCONJ
ejpam-5984	110	15	its	its	PRON
ejpam-5984	110	16	weight	weight	NOUN
ejpam-5984	110	17	is	be	AUX
ejpam-5984	110	18	idr(g	idr(g	PROPN
ejpam-5984	110	19	)	)	PUNCT
ejpam-5984	111	1	+	+	CCONJ
ejpam-5984	111	2	3	3	X
ejpam-5984	111	3	.	.	PUNCT
ejpam-5984	111	4	thus	thus	ADV
ejpam-5984	111	5	,	,	PUNCT
ejpam-5984	111	6	idr(g′	idr(g′	NOUN
ejpam-5984	111	7	)	)	PUNCT
ejpam-5984	111	8	≤	≤	NUM
ejpam-5984	111	9	idr(g	idr(g	PROPN
ejpam-5984	111	10	)	)	PUNCT
ejpam-5984	112	1	+	+	NUM
ejpam-5984	112	2	3	3	X
ejpam-5984	112	3	.	.	X
ejpam-5984	112	4	s.	s.	PROPN
ejpam-5984	112	5	m.	m.	PROPN
ejpam-5984	112	6	sheikholeslami	sheikholeslami	PROPN
ejpam-5984	112	7	et	et	PROPN
ejpam-5984	112	8	al	al	PROPN
ejpam-5984	112	9	.	.	PUNCT
ejpam-5984	112	10	/	/	SYM
ejpam-5984	112	11	eur	eur	PROPN
ejpam-5984	112	12	.	.	PUNCT
ejpam-5984	113	1	j.	j.	PROPN
ejpam-5984	113	2	pure	pure	PROPN
ejpam-5984	113	3	appl	appl	PROPN
ejpam-5984	113	4	.	.	PROPN
ejpam-5984	113	5	math	math	PROPN
ejpam-5984	113	6	,	,	PUNCT
ejpam-5984	113	7	18	18	NUM
ejpam-5984	113	8	(	(	PUNCT
ejpam-5984	113	9	2	2	NUM
ejpam-5984	113	10	)	)	PUNCT
ejpam-5984	113	11	(	(	PUNCT
ejpam-5984	113	12	2025	2025	NUM
ejpam-5984	113	13	)	)	PUNCT
ejpam-5984	113	14	,	,	PUNCT
ejpam-5984	113	15	5984	5984	NUM
ejpam-5984	113	16	5	5	NUM
ejpam-5984	113	17	of	of	ADP
ejpam-5984	113	18	16	16	NUM
ejpam-5984	113	19	now	now	ADV
ejpam-5984	113	20	we	we	PRON
ejpam-5984	113	21	show	show	VERB
ejpam-5984	113	22	that	that	SCONJ
ejpam-5984	113	23	idr(g′	idr(g′	NOUN
ejpam-5984	113	24	)	)	PUNCT
ejpam-5984	113	25	≥	≥	X
ejpam-5984	113	26	idr(g)+3	idr(g)+3	NOUN
ejpam-5984	113	27	.	.	PUNCT
ejpam-5984	114	1	let	let	VERB
ejpam-5984	114	2	f	f	PROPN
ejpam-5984	114	3	=	=	SYM
ejpam-5984	114	4	(	(	PUNCT
ejpam-5984	114	5	v0,∅	v0,∅	PROPN
ejpam-5984	114	6	,	,	PUNCT
ejpam-5984	114	7	v2	v2	PROPN
ejpam-5984	114	8	,	,	PUNCT
ejpam-5984	114	9	v3	v3	PROPN
ejpam-5984	114	10	)	)	PUNCT
ejpam-5984	114	11	be	be	VERB
ejpam-5984	114	12	an	an	DET
ejpam-5984	114	13	idr(g	idr(g	PROPN
ejpam-5984	114	14	′)-function	′)-function	NOUN
ejpam-5984	114	15	.	.	PUNCT
ejpam-5984	115	1	if	if	SCONJ
ejpam-5984	115	2	f(u1	f(u1	NOUN
ejpam-5984	115	3	)	)	PUNCT
ejpam-5984	116	1	=	=	SYM
ejpam-5984	116	2	0	0	NUM
ejpam-5984	116	3	,	,	PUNCT
ejpam-5984	116	4	then	then	ADV
ejpam-5984	116	5	to	to	PART
ejpam-5984	116	6	double	double	VERB
ejpam-5984	116	7	roman	roman	NOUN
ejpam-5984	116	8	dominate	dominate	VERB
ejpam-5984	116	9	the	the	DET
ejpam-5984	116	10	vertices	vertex	NOUN
ejpam-5984	116	11	u	u	NOUN
ejpam-5984	116	12	and	and	CCONJ
ejpam-5984	116	13	u2	u2	NOUN
ejpam-5984	116	14	,	,	PUNCT
ejpam-5984	116	15	.	.	PUNCT
ejpam-5984	116	16	.	.	PUNCT
ejpam-5984	116	17	.	.	PUNCT
ejpam-5984	117	1	ut	ut	PROPN
ejpam-5984	117	2	we	we	PRON
ejpam-5984	117	3	must	must	AUX
ejpam-5984	117	4	have	have	VERB
ejpam-5984	117	5	f(u	f(u	PROPN
ejpam-5984	117	6	)	)	PUNCT
ejpam-5984	117	7	+	+	PROPN
ejpam-5984	117	8	f(u2	f(u2	NOUN
ejpam-5984	117	9	)	)	PUNCT
ejpam-5984	118	1	+	+	CCONJ
ejpam-5984	118	2	·	·	PUNCT
ejpam-5984	118	3	·	·	PUNCT
ejpam-5984	118	4	·	·	PUNCT
ejpam-5984	119	1	+	+	CCONJ
ejpam-5984	119	2	f(ut	f(ut	NOUN
ejpam-5984	119	3	)	)	PUNCT
ejpam-5984	119	4	=	=	SYM
ejpam-5984	119	5	3	3	X
ejpam-5984	119	6	.	.	X
ejpam-5984	119	7	on	on	ADP
ejpam-5984	119	8	the	the	DET
ejpam-5984	119	9	other	other	ADJ
ejpam-5984	119	10	hand	hand	NOUN
ejpam-5984	119	11	,	,	PUNCT
ejpam-5984	119	12	the	the	DET
ejpam-5984	119	13	function	function	NOUN
ejpam-5984	119	14	f	f	PROPN
ejpam-5984	119	15	restricted	restrict	VERB
ejpam-5984	119	16	to	to	ADP
ejpam-5984	119	17	g	g	PROPN
ejpam-5984	119	18	is	be	AUX
ejpam-5984	119	19	an	an	DET
ejpam-5984	119	20	idrd	idrd	ADJ
ejpam-5984	119	21	-	-	PUNCT
ejpam-5984	119	22	function	function	NOUN
ejpam-5984	119	23	of	of	ADP
ejpam-5984	119	24	g	g	NOUN
ejpam-5984	119	25	implying	imply	VERB
ejpam-5984	119	26	that	that	SCONJ
ejpam-5984	119	27	idr(g	idr(g	PROPN
ejpam-5984	119	28	′	′	NOUN
ejpam-5984	119	29	)	)	PUNCT
ejpam-5984	119	30	≥	≥	NOUN
ejpam-5984	119	31	idr(g	idr(g	NUM
ejpam-5984	119	32	)	)	PUNCT
ejpam-5984	120	1	+	+	NUM
ejpam-5984	120	2	3	3	X
ejpam-5984	120	3	.	.	X
ejpam-5984	120	4	assume	assume	VERB
ejpam-5984	120	5	that	that	SCONJ
ejpam-5984	120	6	f(u1	f(u1	NOUN
ejpam-5984	120	7	)	)	PUNCT
ejpam-5984	120	8	≥	≥	NOUN
ejpam-5984	121	1	2	2	NUM
ejpam-5984	121	2	.	.	PUNCT
ejpam-5984	121	3	then	then	ADV
ejpam-5984	121	4	we	we	PRON
ejpam-5984	121	5	must	must	AUX
ejpam-5984	121	6	have	have	VERB
ejpam-5984	121	7	f(u	f(u	PROPN
ejpam-5984	121	8	)	)	PUNCT
ejpam-5984	122	1	=	=	SYM
ejpam-5984	122	2	0	0	NUM
ejpam-5984	122	3	and	and	CCONJ
ejpam-5984	122	4	f(ui	f(ui	PROPN
ejpam-5984	122	5	)	)	PUNCT
ejpam-5984	123	1	=	=	SYM
ejpam-5984	123	2	2	2	NUM
ejpam-5984	123	3	for	for	ADP
ejpam-5984	123	4	2	2	NUM
ejpam-5984	123	5	≤	≤	NUM
ejpam-5984	123	6	i	i	PRON
ejpam-5984	124	1	≤	≤	ADJ
ejpam-5984	125	1	t.	t.	NOUN
ejpam-5984	126	1	if	if	SCONJ
ejpam-5984	126	2	f(u1	f(u1	NOUN
ejpam-5984	126	3	)	)	PUNCT
ejpam-5984	127	1	=	=	SYM
ejpam-5984	127	2	2	2	NUM
ejpam-5984	127	3	,	,	PUNCT
ejpam-5984	127	4	then	then	ADV
ejpam-5984	127	5	to	to	PART
ejpam-5984	127	6	double	double	VERB
ejpam-5984	127	7	roman	roman	ADJ
ejpam-5984	127	8	dominate	dominate	NOUN
ejpam-5984	127	9	of	of	ADP
ejpam-5984	127	10	v	v	NOUN
ejpam-5984	127	11	,	,	PUNCT
ejpam-5984	127	12	it	it	PRON
ejpam-5984	127	13	must	must	AUX
ejpam-5984	127	14	have	have	VERB
ejpam-5984	127	15	a	a	DET
ejpam-5984	127	16	neighbor	neighbor	NOUN
ejpam-5984	127	17	w	w	NOUN
ejpam-5984	127	18	with	with	ADP
ejpam-5984	127	19	f(w	f(w	PROPN
ejpam-5984	127	20	)	)	PUNCT
ejpam-5984	127	21	≥	≥	NOUN
ejpam-5984	127	22	2	2	NUM
ejpam-5984	127	23	and	and	CCONJ
ejpam-5984	127	24	the	the	DET
ejpam-5984	127	25	function	function	NOUN
ejpam-5984	127	26	g	g	PROPN
ejpam-5984	127	27	defined	define	VERB
ejpam-5984	127	28	on	on	ADP
ejpam-5984	127	29	g	g	NOUN
ejpam-5984	127	30	by	by	ADP
ejpam-5984	127	31	g(w	g(w	PROPN
ejpam-5984	127	32	)	)	PUNCT
ejpam-5984	127	33	=	=	SYM
ejpam-5984	127	34	min{3	min{3	PROPN
ejpam-5984	127	35	,	,	PUNCT
ejpam-5984	127	36	f(w	f(w	PROPN
ejpam-5984	127	37	)	)	PUNCT
ejpam-5984	128	1	+	+	CCONJ
ejpam-5984	128	2	1	1	X
ejpam-5984	128	3	}	}	PUNCT
ejpam-5984	128	4	and	and	CCONJ
ejpam-5984	128	5	g(x	g(x	NOUN
ejpam-5984	128	6	)	)	PUNCT
ejpam-5984	128	7	=	=	SYM
ejpam-5984	128	8	f(x	f(x	PROPN
ejpam-5984	128	9	)	)	PUNCT
ejpam-5984	128	10	for	for	ADP
ejpam-5984	128	11	other	other	ADJ
ejpam-5984	128	12	vertices	vertex	NOUN
ejpam-5984	128	13	,	,	PUNCT
ejpam-5984	128	14	is	be	AUX
ejpam-5984	128	15	an	an	DET
ejpam-5984	128	16	idrd	idrd	ADJ
ejpam-5984	128	17	-	-	PUNCT
ejpam-5984	128	18	function	function	NOUN
ejpam-5984	128	19	of	of	ADP
ejpam-5984	128	20	g	g	NOUN
ejpam-5984	128	21	of	of	ADP
ejpam-5984	128	22	weight	weight	NOUN
ejpam-5984	128	23	at	at	ADP
ejpam-5984	128	24	most	most	ADJ
ejpam-5984	128	25	idr(g′)−3	idr(g′)−3	ADJ
ejpam-5984	128	26	leading	lead	VERB
ejpam-5984	128	27	to	to	ADP
ejpam-5984	128	28	idr(g	idr(g	PROPN
ejpam-5984	128	29	′	′	NOUN
ejpam-5984	128	30	)	)	PUNCT
ejpam-5984	128	31	≥	≥	NOUN
ejpam-5984	128	32	idr(g)+3	idr(g)+3	X
ejpam-5984	128	33	.	.	PUNCT
ejpam-5984	129	1	assume	assume	VERB
ejpam-5984	129	2	that	that	SCONJ
ejpam-5984	129	3	f(u1	f(u1	NOUN
ejpam-5984	129	4	)	)	PUNCT
ejpam-5984	130	1	=	=	SYM
ejpam-5984	130	2	3	3	X
ejpam-5984	130	3	.	.	X
ejpam-5984	131	1	if	if	SCONJ
ejpam-5984	131	2	v	v	NOUN
ejpam-5984	131	3	has	have	VERB
ejpam-5984	131	4	a	a	DET
ejpam-5984	131	5	neighbor	neighbor	NOUN
ejpam-5984	131	6	assigned	assign	VERB
ejpam-5984	131	7	at	at	ADP
ejpam-5984	131	8	least	least	ADV
ejpam-5984	131	9	two	two	NUM
ejpam-5984	131	10	under	under	ADP
ejpam-5984	131	11	f	f	PROPN
ejpam-5984	131	12	,	,	PUNCT
ejpam-5984	131	13	then	then	ADV
ejpam-5984	131	14	as	as	SCONJ
ejpam-5984	131	15	before	before	SCONJ
ejpam-5984	131	16	we	we	PRON
ejpam-5984	131	17	get	get	VERB
ejpam-5984	131	18	idr(g′	idr(g′	NOUN
ejpam-5984	131	19	)	)	PUNCT
ejpam-5984	131	20	≥	≥	PROPN
ejpam-5984	131	21	idr(g)+3	idr(g)+3	NOUN
ejpam-5984	131	22	.	.	PUNCT
ejpam-5984	132	1	hence	hence	ADV
ejpam-5984	132	2	let	let	VERB
ejpam-5984	132	3	f(n	f(n	PROPN
ejpam-5984	132	4	[	[	X
ejpam-5984	132	5	v	v	X
ejpam-5984	132	6	]	]	X
ejpam-5984	132	7	−	−	PROPN
ejpam-5984	132	8	{	{	PUNCT
ejpam-5984	132	9	u1	u1	NOUN
ejpam-5984	132	10	}	}	PUNCT
ejpam-5984	132	11	)	)	PUNCT
ejpam-5984	133	1	=	=	PUNCT
ejpam-5984	133	2	0	0	X
ejpam-5984	133	3	.	.	PUNCT
ejpam-5984	134	1	then	then	ADV
ejpam-5984	134	2	the	the	DET
ejpam-5984	134	3	function	function	NOUN
ejpam-5984	134	4	g	g	PROPN
ejpam-5984	134	5	defined	define	VERB
ejpam-5984	134	6	on	on	ADP
ejpam-5984	134	7	g	g	NOUN
ejpam-5984	134	8	by	by	ADP
ejpam-5984	134	9	g(v	g(v	NOUN
ejpam-5984	134	10	)	)	PUNCT
ejpam-5984	134	11	=	=	SYM
ejpam-5984	134	12	2	2	NUM
ejpam-5984	134	13	and	and	CCONJ
ejpam-5984	134	14	g(x	g(x	NOUN
ejpam-5984	134	15	)	)	PUNCT
ejpam-5984	134	16	=	=	SYM
ejpam-5984	134	17	f(x	f(x	PROPN
ejpam-5984	134	18	)	)	PUNCT
ejpam-5984	134	19	for	for	ADP
ejpam-5984	134	20	other	other	ADJ
ejpam-5984	134	21	vertices	vertex	NOUN
ejpam-5984	134	22	,	,	PUNCT
ejpam-5984	134	23	is	be	AUX
ejpam-5984	134	24	an	an	DET
ejpam-5984	134	25	idrd	idrd	ADJ
ejpam-5984	134	26	-	-	PUNCT
ejpam-5984	134	27	function	function	NOUN
ejpam-5984	134	28	of	of	ADP
ejpam-5984	134	29	g	g	NOUN
ejpam-5984	134	30	of	of	ADP
ejpam-5984	134	31	weight	weight	NOUN
ejpam-5984	134	32	at	at	ADP
ejpam-5984	134	33	most	most	ADJ
ejpam-5984	134	34	idr(g	idr(g	PROPN
ejpam-5984	134	35	′	′	NOUN
ejpam-5984	134	36	)	)	PUNCT
ejpam-5984	134	37	−	−	PROPN
ejpam-5984	134	38	3	3	NUM
ejpam-5984	134	39	,	,	PUNCT
ejpam-5984	134	40	leading	lead	VERB
ejpam-5984	134	41	to	to	ADP
ejpam-5984	134	42	idr(g	idr(g	PROPN
ejpam-5984	134	43	′	′	NOUN
ejpam-5984	134	44	)	)	PUNCT
ejpam-5984	134	45	≥	≥	NOUN
ejpam-5984	134	46	idr(g)+	idr(g)+	VERB
ejpam-5984	134	47	3	3	NUM
ejpam-5984	134	48	.	.	PUNCT
ejpam-5984	135	1	thus	thus	ADV
ejpam-5984	135	2	,	,	PUNCT
ejpam-5984	135	3	in	in	ADP
ejpam-5984	135	4	all	all	DET
ejpam-5984	135	5	cases	case	NOUN
ejpam-5984	135	6	,	,	PUNCT
ejpam-5984	135	7	we	we	PRON
ejpam-5984	135	8	have	have	VERB
ejpam-5984	135	9	that	that	PRON
ejpam-5984	135	10	idr(g′	idr(g′	NOUN
ejpam-5984	135	11	)	)	PUNCT
ejpam-5984	135	12	≥	≥	PRON
ejpam-5984	135	13	idr(g)+	idr(g)+	VERB
ejpam-5984	135	14	3	3	NUM
ejpam-5984	135	15	,	,	PUNCT
ejpam-5984	135	16	and	and	CCONJ
ejpam-5984	135	17	since	since	SCONJ
ejpam-5984	135	18	the	the	DET
ejpam-5984	135	19	reverse	reverse	ADJ
ejpam-5984	135	20	inequality	inequality	NOUN
ejpam-5984	135	21	was	be	AUX
ejpam-5984	135	22	already	already	ADV
ejpam-5984	135	23	established	establish	VERB
ejpam-5984	135	24	,	,	PUNCT
ejpam-5984	135	25	we	we	PRON
ejpam-5984	135	26	have	have	VERB
ejpam-5984	135	27	idr(g	idr(g	PROPN
ejpam-5984	135	28	′	′	NOUN
ejpam-5984	135	29	)	)	PUNCT
ejpam-5984	135	30	=	=	SYM
ejpam-5984	135	31	idr(g	idr(g	PROPN
ejpam-5984	135	32	)	)	PUNCT
ejpam-5984	136	1	+	+	CCONJ
ejpam-5984	137	1	3	3	X
ejpam-5984	137	2	.	.	NOUN
ejpam-5984	137	3	4	4	NUM
ejpam-5984	137	4	.	.	NOUN
ejpam-5984	137	5	exact	exact	ADJ
ejpam-5984	137	6	values	value	NOUN
ejpam-5984	137	7	and	and	CCONJ
ejpam-5984	137	8	bounds	bound	NOUN
ejpam-5984	137	9	in	in	ADP
ejpam-5984	137	10	this	this	DET
ejpam-5984	137	11	section	section	NOUN
ejpam-5984	137	12	,	,	PUNCT
ejpam-5984	137	13	we	we	PRON
ejpam-5984	137	14	obtain	obtain	VERB
ejpam-5984	137	15	the	the	DET
ejpam-5984	137	16	independent	independent	ADJ
ejpam-5984	137	17	double	double	ADJ
ejpam-5984	137	18	roman	roman	ADJ
ejpam-5984	137	19	domination	domination	NOUN
ejpam-5984	137	20	stability	stability	NOUN
ejpam-5984	137	21	for	for	ADP
ejpam-5984	137	22	some	some	DET
ejpam-5984	137	23	classes	class	NOUN
ejpam-5984	137	24	of	of	ADP
ejpam-5984	137	25	graphs	graph	NOUN
ejpam-5984	137	26	and	and	CCONJ
ejpam-5984	137	27	present	present	VERB
ejpam-5984	137	28	various	various	ADJ
ejpam-5984	137	29	bounds	bound	NOUN
ejpam-5984	137	30	for	for	ADP
ejpam-5984	137	31	this	this	DET
ejpam-5984	137	32	parameters	parameter	NOUN
ejpam-5984	137	33	.	.	PUNCT
ejpam-5984	138	1	the	the	DET
ejpam-5984	138	2	proof	proof	NOUN
ejpam-5984	138	3	of	of	ADP
ejpam-5984	138	4	the	the	DET
ejpam-5984	138	5	next	next	ADJ
ejpam-5984	138	6	propositions	proposition	NOUN
ejpam-5984	138	7	can	can	AUX
ejpam-5984	138	8	be	be	AUX
ejpam-5984	138	9	found	find	VERB
ejpam-5984	138	10	in	in	ADP
ejpam-5984	138	11	[	[	X
ejpam-5984	138	12	10	10	NUM
ejpam-5984	138	13	]	]	PUNCT
ejpam-5984	138	14	.	.	PUNCT
ejpam-5984	139	1	proposition	proposition	NOUN
ejpam-5984	139	2	2	2	NUM
ejpam-5984	139	3	.	.	X
ejpam-5984	139	4	for	for	ADP
ejpam-5984	139	5	n	n	PRON
ejpam-5984	139	6	≥	≥	NUM
ejpam-5984	139	7	1	1	NUM
ejpam-5984	139	8	,	,	PUNCT
ejpam-5984	139	9	idr(pn	idr(pn	ADJ
ejpam-5984	139	10	)	)	PUNCT
ejpam-5984	139	11	=	=	SYM
ejpam-5984	139	12	γdr(pn	γdr(pn	NOUN
ejpam-5984	139	13	)	)	PUNCT
ejpam-5984	139	14	=	=	NOUN
ejpam-5984	139	15	{	{	PUNCT
ejpam-5984	139	16	n	n	NOUN
ejpam-5984	139	17	if	if	SCONJ
ejpam-5984	139	18	n	n	PRON
ejpam-5984	139	19	≡	≡	PROPN
ejpam-5984	139	20	0	0	PUNCT
ejpam-5984	140	1	(	(	PUNCT
ejpam-5984	140	2	mod	mod	NOUN
ejpam-5984	140	3	3	3	NUM
ejpam-5984	140	4	)	)	PUNCT
ejpam-5984	140	5	n+	n+	PUNCT
ejpam-5984	140	6	1	1	NUM
ejpam-5984	140	7	if	if	SCONJ
ejpam-5984	140	8	n	n	PRON
ejpam-5984	140	9	≡	≡	PROPN
ejpam-5984	140	10	1	1	NUM
ejpam-5984	140	11	,	,	PUNCT
ejpam-5984	140	12	2	2	NUM
ejpam-5984	140	13	(	(	PUNCT
ejpam-5984	140	14	mod	mod	NOUN
ejpam-5984	140	15	3	3	NUM
ejpam-5984	140	16	)	)	PUNCT
ejpam-5984	140	17	.	.	PUNCT
ejpam-5984	141	1	proposition	proposition	NOUN
ejpam-5984	141	2	3	3	NUM
ejpam-5984	141	3	.	.	PUNCT
ejpam-5984	141	4	for	for	ADP
ejpam-5984	141	5	n	n	PRON
ejpam-5984	141	6	≥	≥	NUM
ejpam-5984	141	7	3	3	NUM
ejpam-5984	141	8	,	,	PUNCT
ejpam-5984	141	9	idr(cn	idr(cn	NOUN
ejpam-5984	141	10	)	)	PUNCT
ejpam-5984	141	11	=	=	SYM
ejpam-5984	141	12	γdr(cn	γdr(cn	NOUN
ejpam-5984	141	13	)	)	PUNCT
ejpam-5984	141	14	=	=	PRON
ejpam-5984	141	15	{	{	PUNCT
ejpam-5984	141	16	n	n	NOUN
ejpam-5984	141	17	if	if	SCONJ
ejpam-5984	141	18	n	n	PRON
ejpam-5984	141	19	≡	≡	PROPN
ejpam-5984	141	20	0	0	NUM
ejpam-5984	141	21	,	,	PUNCT
ejpam-5984	141	22	2	2	NUM
ejpam-5984	141	23	,	,	PUNCT
ejpam-5984	141	24	3	3	NUM
ejpam-5984	141	25	,	,	PUNCT
ejpam-5984	141	26	4	4	NUM
ejpam-5984	141	27	(	(	PUNCT
ejpam-5984	141	28	mod	mod	PROPN
ejpam-5984	141	29	6	6	NUM
ejpam-5984	141	30	)	)	PUNCT
ejpam-5984	141	31	n+	n+	PUNCT
ejpam-5984	141	32	1	1	NUM
ejpam-5984	141	33	if	if	SCONJ
ejpam-5984	141	34	n	n	PRON
ejpam-5984	141	35	≡	≡	PROPN
ejpam-5984	141	36	1	1	NUM
ejpam-5984	141	37	,	,	PUNCT
ejpam-5984	141	38	5	5	NUM
ejpam-5984	141	39	(	(	PUNCT
ejpam-5984	141	40	mod	mod	PROPN
ejpam-5984	141	41	6	6	NUM
ejpam-5984	141	42	)	)	PUNCT
ejpam-5984	141	43	.	.	PUNCT
ejpam-5984	142	1	also	also	ADV
ejpam-5984	142	2	zhuang	zhuang	PROPN
ejpam-5984	143	1	[	[	X
ejpam-5984	143	2	17	17	NUM
ejpam-5984	143	3	]	]	PUNCT
ejpam-5984	143	4	determined	determine	VERB
ejpam-5984	143	5	the	the	DET
ejpam-5984	143	6	double	double	ADJ
ejpam-5984	143	7	roman	roman	ADJ
ejpam-5984	143	8	domination	domination	NOUN
ejpam-5984	143	9	stability	stability	NOUN
ejpam-5984	143	10	of	of	ADP
ejpam-5984	143	11	paths	path	NOUN
ejpam-5984	143	12	and	and	CCONJ
ejpam-5984	143	13	cycles	cycle	NOUN
ejpam-5984	143	14	as	as	SCONJ
ejpam-5984	143	15	follows	follow	VERB
ejpam-5984	143	16	.	.	PUNCT
ejpam-5984	144	1	proposition	proposition	NOUN
ejpam-5984	144	2	4	4	NUM
ejpam-5984	144	3	.	.	PUNCT
ejpam-5984	144	4	for	for	ADP
ejpam-5984	144	5	n	n	PRON
ejpam-5984	144	6	≥	≥	NUM
ejpam-5984	144	7	2	2	NUM
ejpam-5984	144	8	,	,	PUNCT
ejpam-5984	144	9	st−γdr(pn	st−γdr(pn	NOUN
ejpam-5984	144	10	)	)	PUNCT
ejpam-5984	144	11	=	=	NOUN
ejpam-5984	144	12	{	{	PUNCT
ejpam-5984	144	13	1	1	NUM
ejpam-5984	144	14	if	if	SCONJ
ejpam-5984	144	15	n	n	PRON
ejpam-5984	144	16	≡	≡	PROPN
ejpam-5984	144	17	1	1	NUM
ejpam-5984	144	18	,	,	PUNCT
ejpam-5984	144	19	2	2	NUM
ejpam-5984	144	20	(	(	PUNCT
ejpam-5984	144	21	mod	mod	NOUN
ejpam-5984	144	22	3	3	NUM
ejpam-5984	144	23	)	)	PUNCT
ejpam-5984	144	24	2	2	NUM
ejpam-5984	144	25	if	if	SCONJ
ejpam-5984	144	26	n	n	PRON
ejpam-5984	144	27	≡	≡	PROPN
ejpam-5984	144	28	0	0	PUNCT
ejpam-5984	144	29	(	(	PUNCT
ejpam-5984	144	30	mod	mod	NOUN
ejpam-5984	144	31	3	3	NUM
ejpam-5984	144	32	)	)	PUNCT
ejpam-5984	144	33	.	.	PUNCT
ejpam-5984	145	1	proposition	proposition	NOUN
ejpam-5984	145	2	5	5	NUM
ejpam-5984	145	3	.	.	PUNCT
ejpam-5984	145	4	for	for	ADP
ejpam-5984	145	5	n	n	PRON
ejpam-5984	145	6	≥	≥	NUM
ejpam-5984	145	7	2	2	NUM
ejpam-5984	145	8	,	,	PUNCT
ejpam-5984	145	9	st+γdr(pn	st+γdr(pn	NOUN
ejpam-5984	145	10	)	)	PUNCT
ejpam-5984	145	11	=	=	SYM
ejpam-5984	145	12	{	{	PUNCT
ejpam-5984	145	13	∞	∞	NOUN
ejpam-5984	145	14	if	if	SCONJ
ejpam-5984	145	15	n	n	PRON
ejpam-5984	145	16	≡	≡	PROPN
ejpam-5984	145	17	1	1	NUM
ejpam-5984	145	18	,	,	PUNCT
ejpam-5984	145	19	2	2	NUM
ejpam-5984	145	20	(	(	PUNCT
ejpam-5984	145	21	mod	mod	NOUN
ejpam-5984	145	22	3	3	NUM
ejpam-5984	145	23	)	)	PUNCT
ejpam-5984	145	24	1	1	NUM
ejpam-5984	145	25	if	if	SCONJ
ejpam-5984	145	26	n	n	PRON
ejpam-5984	145	27	≡	≡	PROPN
ejpam-5984	145	28	0	0	PUNCT
ejpam-5984	146	1	(	(	PUNCT
ejpam-5984	146	2	mod	mod	NOUN
ejpam-5984	146	3	3	3	NUM
ejpam-5984	146	4	)	)	PUNCT
ejpam-5984	146	5	.	.	PUNCT
ejpam-5984	147	1	proposition	proposition	NOUN
ejpam-5984	147	2	6	6	NUM
ejpam-5984	147	3	.	.	PUNCT
ejpam-5984	147	4	for	for	ADP
ejpam-5984	147	5	n	n	PRON
ejpam-5984	147	6	≥	≥	NUM
ejpam-5984	147	7	3	3	NUM
ejpam-5984	147	8	,	,	PUNCT
ejpam-5984	147	9	st−γdr(cn	st−γdr(cn	NUM
ejpam-5984	147	10	)	)	PUNCT
ejpam-5984	147	11	=	=	PRON
ejpam-5984	147	12	{	{	PUNCT
ejpam-5984	147	13	1	1	NUM
ejpam-5984	147	14	if	if	SCONJ
ejpam-5984	147	15	n	n	PRON
ejpam-5984	147	16	≡	≡	PROPN
ejpam-5984	147	17	1	1	NUM
ejpam-5984	147	18	,	,	PUNCT
ejpam-5984	147	19	4	4	NUM
ejpam-5984	147	20	,	,	PUNCT
ejpam-5984	147	21	5	5	NUM
ejpam-5984	147	22	(	(	PUNCT
ejpam-5984	147	23	mod	mod	NOUN
ejpam-5984	147	24	6	6	NUM
ejpam-5984	147	25	)	)	PUNCT
ejpam-5984	147	26	2	2	NUM
ejpam-5984	147	27	otherwise	otherwise	ADV
ejpam-5984	147	28	.	.	PUNCT
ejpam-5984	148	1	proposition	proposition	NOUN
ejpam-5984	148	2	7	7	NUM
ejpam-5984	148	3	.	.	X
ejpam-5984	148	4	for	for	ADP
ejpam-5984	148	5	n	n	X
ejpam-5984	148	6	≥	≥	NUM
ejpam-5984	148	7	3	3	NUM
ejpam-5984	148	8	,	,	PUNCT
ejpam-5984	148	9	st+γdr(cn	st+γdr(cn	PROPN
ejpam-5984	148	10	)	)	PUNCT
ejpam-5984	149	1	=	=	VERB
ejpam-5984	149	2	∞.	∞.	PROPN
ejpam-5984	149	3	we	we	PRON
ejpam-5984	149	4	first	first	ADV
ejpam-5984	149	5	determine	determine	VERB
ejpam-5984	149	6	the	the	DET
ejpam-5984	149	7	idr	idr	NOUN
ejpam-5984	149	8	-	-	NOUN
ejpam-5984	149	9	stability	stability	NOUN
ejpam-5984	149	10	for	for	ADP
ejpam-5984	149	11	paths	path	NOUN
ejpam-5984	149	12	.	.	PUNCT
ejpam-5984	150	1	proposition	proposition	NOUN
ejpam-5984	150	2	8	8	NUM
ejpam-5984	150	3	.	.	PUNCT
ejpam-5984	150	4	for	for	ADP
ejpam-5984	150	5	n	n	PRON
ejpam-5984	150	6	≥	≥	NUM
ejpam-5984	150	7	2	2	NUM
ejpam-5984	150	8	,	,	PUNCT
ejpam-5984	150	9	st−idr(pn	st−idr(pn	PRON
ejpam-5984	150	10	)	)	PUNCT
ejpam-5984	150	11	=	=	PRON
ejpam-5984	150	12	{	{	PUNCT
ejpam-5984	150	13	1	1	NUM
ejpam-5984	150	14	if	if	SCONJ
ejpam-5984	150	15	n	n	PRON
ejpam-5984	150	16	≡	≡	PROPN
ejpam-5984	150	17	1	1	NUM
ejpam-5984	150	18	,	,	PUNCT
ejpam-5984	150	19	2	2	NUM
ejpam-5984	150	20	(	(	PUNCT
ejpam-5984	150	21	mod	mod	NOUN
ejpam-5984	150	22	3	3	NUM
ejpam-5984	150	23	)	)	PUNCT
ejpam-5984	150	24	2	2	NUM
ejpam-5984	150	25	if	if	SCONJ
ejpam-5984	150	26	n	n	PRON
ejpam-5984	150	27	≡	≡	PROPN
ejpam-5984	150	28	0	0	PUNCT
ejpam-5984	150	29	(	(	PUNCT
ejpam-5984	150	30	mod	mod	PROPN
ejpam-5984	150	31	3	3	NUM
ejpam-5984	150	32	)	)	PUNCT
ejpam-5984	150	33	.	.	PUNCT
ejpam-5984	151	1	s.	s.	PROPN
ejpam-5984	151	2	m.	m.	PROPN
ejpam-5984	151	3	sheikholeslami	sheikholeslami	PROPN
ejpam-5984	151	4	et	et	PROPN
ejpam-5984	151	5	al	al	PROPN
ejpam-5984	151	6	.	.	PUNCT
ejpam-5984	151	7	/	/	SYM
ejpam-5984	151	8	eur	eur	PROPN
ejpam-5984	151	9	.	.	PUNCT
ejpam-5984	152	1	j.	j.	PROPN
ejpam-5984	152	2	pure	pure	PROPN
ejpam-5984	152	3	appl	appl	PROPN
ejpam-5984	152	4	.	.	PROPN
ejpam-5984	152	5	math	math	PROPN
ejpam-5984	152	6	,	,	PUNCT
ejpam-5984	152	7	18	18	NUM
ejpam-5984	152	8	(	(	PUNCT
ejpam-5984	152	9	2	2	NUM
ejpam-5984	152	10	)	)	PUNCT
ejpam-5984	152	11	(	(	PUNCT
ejpam-5984	152	12	2025	2025	NUM
ejpam-5984	152	13	)	)	PUNCT
ejpam-5984	152	14	,	,	PUNCT
ejpam-5984	152	15	5984	5984	NUM
ejpam-5984	152	16	6	6	NUM
ejpam-5984	152	17	of	of	ADP
ejpam-5984	152	18	16	16	NUM
ejpam-5984	152	19	proof	proof	NOUN
ejpam-5984	152	20	.	.	PUNCT
ejpam-5984	153	1	the	the	DET
ejpam-5984	153	2	result	result	NOUN
ejpam-5984	153	3	is	be	AUX
ejpam-5984	153	4	trivial	trivial	ADJ
ejpam-5984	153	5	for	for	ADP
ejpam-5984	153	6	n	n	DET
ejpam-5984	153	7	≤	≤	NOUN
ejpam-5984	153	8	3	3	NUM
ejpam-5984	153	9	,	,	PUNCT
ejpam-5984	153	10	so	so	SCONJ
ejpam-5984	153	11	we	we	PRON
ejpam-5984	153	12	assume	assume	VERB
ejpam-5984	153	13	that	that	SCONJ
ejpam-5984	153	14	n	n	NUM
ejpam-5984	153	15	≥	≥	NUM
ejpam-5984	153	16	4	4	NUM
ejpam-5984	153	17	.	.	PUNCT
ejpam-5984	154	1	let	let	VERB
ejpam-5984	154	2	p	p	NOUN
ejpam-5984	154	3	=	=	PUNCT
ejpam-5984	155	1	[	[	X
ejpam-5984	155	2	v1	v1	NOUN
ejpam-5984	155	3	,	,	PUNCT
ejpam-5984	155	4	v2	v2	NOUN
ejpam-5984	155	5	,	,	PUNCT
ejpam-5984	155	6	.	.	PUNCT
ejpam-5984	155	7	.	.	PUNCT
ejpam-5984	156	1	.	.	PUNCT
ejpam-5984	157	1	,	,	PUNCT
ejpam-5984	157	2	vn	vn	AUX
ejpam-5984	157	3	]	]	X
ejpam-5984	157	4	be	be	AUX
ejpam-5984	157	5	a	a	DET
ejpam-5984	157	6	path	path	NOUN
ejpam-5984	157	7	on	on	ADP
ejpam-5984	157	8	n	n	DET
ejpam-5984	157	9	vertices	vertex	NOUN
ejpam-5984	157	10	.	.	PUNCT
ejpam-5984	158	1	if	if	SCONJ
ejpam-5984	158	2	n	n	PRON
ejpam-5984	158	3	≡	≡	PROPN
ejpam-5984	158	4	1	1	NUM
ejpam-5984	158	5	,	,	PUNCT
ejpam-5984	158	6	2	2	NUM
ejpam-5984	158	7	(	(	PUNCT
ejpam-5984	158	8	mod	mod	NOUN
ejpam-5984	158	9	3	3	NUM
ejpam-5984	158	10	)	)	PUNCT
ejpam-5984	158	11	,	,	PUNCT
ejpam-5984	158	12	then	then	ADV
ejpam-5984	158	13	using	use	VERB
ejpam-5984	158	14	proposition	proposition	NOUN
ejpam-5984	158	15	2	2	NUM
ejpam-5984	158	16	we	we	PRON
ejpam-5984	158	17	have	have	VERB
ejpam-5984	158	18	idr(pn	idr(pn	NOUN
ejpam-5984	158	19	−	−	PROPN
ejpam-5984	158	20	vn	vn	NOUN
ejpam-5984	158	21	)	)	PUNCT
ejpam-5984	158	22	=	=	SYM
ejpam-5984	158	23	idr(pn−1	idr(pn−1	ADJ
ejpam-5984	158	24	)	)	PUNCT
ejpam-5984	158	25	≤	≤	NOUN
ejpam-5984	159	1	n	n	CCONJ
ejpam-5984	159	2	<	<	X
ejpam-5984	159	3	idr(pn	idr(pn	NOUN
ejpam-5984	159	4	)	)	PUNCT
ejpam-5984	159	5	.	.	PUNCT
ejpam-5984	160	1	hence	hence	ADV
ejpam-5984	160	2	,	,	PUNCT
ejpam-5984	160	3	st−idr(pn	st−idr(pn	PROPN
ejpam-5984	160	4	)	)	PUNCT
ejpam-5984	160	5	=	=	SYM
ejpam-5984	160	6	1	1	X
ejpam-5984	160	7	.	.	PUNCT
ejpam-5984	160	8	now	now	ADV
ejpam-5984	160	9	assume	assume	VERB
ejpam-5984	160	10	that	that	SCONJ
ejpam-5984	160	11	n	n	X
ejpam-5984	160	12	≡	≡	PROPN
ejpam-5984	160	13	0	0	PUNCT
ejpam-5984	161	1	(	(	PUNCT
ejpam-5984	161	2	mod	mod	NOUN
ejpam-5984	161	3	3	3	NUM
ejpam-5984	161	4	)	)	PUNCT
ejpam-5984	161	5	.	.	PUNCT
ejpam-5984	162	1	first	first	ADV
ejpam-5984	162	2	we	we	PRON
ejpam-5984	162	3	show	show	VERB
ejpam-5984	162	4	that	that	SCONJ
ejpam-5984	162	5	st−idr(pn	st−idr(pn	ADP
ejpam-5984	162	6	)	)	PUNCT
ejpam-5984	162	7	≥	≥	NOUN
ejpam-5984	162	8	2	2	NUM
ejpam-5984	162	9	.	.	PUNCT
ejpam-5984	162	10	by	by	ADP
ejpam-5984	162	11	proposition	proposition	NOUN
ejpam-5984	162	12	4	4	NUM
ejpam-5984	162	13	,	,	PUNCT
ejpam-5984	162	14	we	we	PRON
ejpam-5984	162	15	have	have	VERB
ejpam-5984	162	16	st−γdr(pn	st−γdr(pn	NOUN
ejpam-5984	162	17	)	)	PUNCT
ejpam-5984	162	18	=	=	SYM
ejpam-5984	162	19	2	2	X
ejpam-5984	162	20	.	.	PUNCT
ejpam-5984	163	1	it	it	PRON
ejpam-5984	163	2	follows	follow	VERB
ejpam-5984	163	3	from	from	ADP
ejpam-5984	163	4	remark	remark	NOUN
ejpam-5984	163	5	1	1	NUM
ejpam-5984	163	6	that	that	PRON
ejpam-5984	163	7	st−idr(pn	st−idr(pn	ADP
ejpam-5984	163	8	)	)	PUNCT
ejpam-5984	163	9	≥	≥	NOUN
ejpam-5984	163	10	2	2	NUM
ejpam-5984	163	11	.	.	PUNCT
ejpam-5984	164	1	on	on	ADP
ejpam-5984	164	2	the	the	DET
ejpam-5984	164	3	other	other	ADJ
ejpam-5984	164	4	hand	hand	NOUN
ejpam-5984	164	5	,	,	PUNCT
ejpam-5984	164	6	by	by	ADP
ejpam-5984	164	7	proposition	proposition	NOUN
ejpam-5984	164	8	2	2	NUM
ejpam-5984	164	9	we	we	PRON
ejpam-5984	164	10	have	have	VERB
ejpam-5984	164	11	idr(pn−	idr(pn−	PROPN
ejpam-5984	164	12	{	{	PUNCT
ejpam-5984	164	13	vn	vn	PROPN
ejpam-5984	164	14	,	,	PUNCT
ejpam-5984	164	15	vn−1	vn−1	ADJ
ejpam-5984	164	16	}	}	PUNCT
ejpam-5984	164	17	)	)	PUNCT
ejpam-5984	164	18	=	=	SYM
ejpam-5984	164	19	idr(pn−2	idr(pn−2	NOUN
ejpam-5984	164	20	)	)	PUNCT
ejpam-5984	164	21	=	=	PUNCT
ejpam-5984	165	1	n−	n−	NOUN
ejpam-5984	165	2	1	1	NUM
ejpam-5984	165	3	<	<	X
ejpam-5984	165	4	idr(pn	idr(pn	NOUN
ejpam-5984	165	5	)	)	PUNCT
ejpam-5984	165	6	that	that	PRON
ejpam-5984	165	7	yields	yield	VERB
ejpam-5984	165	8	st−idr(pn	st−idr(pn	PRON
ejpam-5984	165	9	)	)	PUNCT
ejpam-5984	165	10	≤	≤	NUM
ejpam-5984	165	11	2	2	NUM
ejpam-5984	165	12	.	.	PUNCT
ejpam-5984	165	13	thus	thus	ADV
ejpam-5984	165	14	st−idr(pn	st−idr(pn	PRON
ejpam-5984	165	15	)	)	PUNCT
ejpam-5984	165	16	=	=	SYM
ejpam-5984	165	17	2	2	NUM
ejpam-5984	165	18	and	and	CCONJ
ejpam-5984	165	19	the	the	DET
ejpam-5984	165	20	proof	proof	NOUN
ejpam-5984	165	21	is	be	AUX
ejpam-5984	165	22	complete	complete	ADJ
ejpam-5984	165	23	.	.	PUNCT
ejpam-5984	166	1	proposition	proposition	NOUN
ejpam-5984	166	2	9	9	NUM
ejpam-5984	166	3	.	.	PUNCT
ejpam-5984	166	4	for	for	ADP
ejpam-5984	166	5	n	n	PRON
ejpam-5984	166	6	≥	≥	NUM
ejpam-5984	166	7	2	2	NUM
ejpam-5984	166	8	,	,	PUNCT
ejpam-5984	166	9	st+idr(pn	st+idr(pn	NUM
ejpam-5984	166	10	)	)	PUNCT
ejpam-5984	166	11	=	=	SYM
ejpam-5984	166	12	{	{	PUNCT
ejpam-5984	166	13	∞	∞	NOUN
ejpam-5984	166	14	if	if	SCONJ
ejpam-5984	166	15	n	n	PRON
ejpam-5984	166	16	≡	≡	PROPN
ejpam-5984	166	17	1	1	NUM
ejpam-5984	166	18	,	,	PUNCT
ejpam-5984	166	19	2	2	NUM
ejpam-5984	166	20	(	(	PUNCT
ejpam-5984	166	21	mod	mod	NOUN
ejpam-5984	166	22	3	3	NUM
ejpam-5984	166	23	)	)	PUNCT
ejpam-5984	166	24	1	1	NUM
ejpam-5984	166	25	if	if	SCONJ
ejpam-5984	166	26	n	n	PRON
ejpam-5984	166	27	≡	≡	PROPN
ejpam-5984	166	28	0	0	PUNCT
ejpam-5984	167	1	(	(	PUNCT
ejpam-5984	167	2	mod	mod	NOUN
ejpam-5984	167	3	3	3	NUM
ejpam-5984	167	4	)	)	PUNCT
ejpam-5984	167	5	.	.	PUNCT
ejpam-5984	168	1	proof	proof	NOUN
ejpam-5984	168	2	.	.	PUNCT
ejpam-5984	169	1	the	the	DET
ejpam-5984	169	2	result	result	NOUN
ejpam-5984	169	3	is	be	AUX
ejpam-5984	169	4	trivial	trivial	ADJ
ejpam-5984	169	5	for	for	ADP
ejpam-5984	169	6	n	n	NOUN
ejpam-5984	169	7	=	=	SYM
ejpam-5984	169	8	2	2	NUM
ejpam-5984	169	9	,	,	PUNCT
ejpam-5984	169	10	so	so	SCONJ
ejpam-5984	169	11	we	we	PRON
ejpam-5984	169	12	consider	consider	VERB
ejpam-5984	169	13	the	the	DET
ejpam-5984	169	14	case	case	NOUN
ejpam-5984	169	15	for	for	ADP
ejpam-5984	169	16	n	n	PRON
ejpam-5984	169	17	≥	≥	NOUN
ejpam-5984	169	18	3	3	NUM
ejpam-5984	169	19	.	.	PUNCT
ejpam-5984	170	1	let	let	VERB
ejpam-5984	170	2	p	p	NOUN
ejpam-5984	170	3	=	=	PUNCT
ejpam-5984	171	1	[	[	X
ejpam-5984	171	2	v1	v1	NOUN
ejpam-5984	171	3	,	,	PUNCT
ejpam-5984	171	4	v2	v2	NOUN
ejpam-5984	171	5	,	,	PUNCT
ejpam-5984	171	6	.	.	PUNCT
ejpam-5984	171	7	.	.	PUNCT
ejpam-5984	172	1	.	.	PUNCT
ejpam-5984	173	1	,	,	PUNCT
ejpam-5984	173	2	vn	vn	AUX
ejpam-5984	173	3	]	]	X
ejpam-5984	173	4	be	be	AUX
ejpam-5984	173	5	a	a	DET
ejpam-5984	173	6	path	path	NOUN
ejpam-5984	173	7	on	on	ADP
ejpam-5984	173	8	n	n	DET
ejpam-5984	173	9	vertices	vertex	NOUN
ejpam-5984	173	10	.	.	PUNCT
ejpam-5984	174	1	if	if	SCONJ
ejpam-5984	174	2	n	n	PRON
ejpam-5984	174	3	≡	≡	PROPN
ejpam-5984	174	4	0	0	PUNCT
ejpam-5984	174	5	(	(	PUNCT
ejpam-5984	174	6	mod	mod	NOUN
ejpam-5984	174	7	3	3	NUM
ejpam-5984	174	8	)	)	PUNCT
ejpam-5984	174	9	,	,	PUNCT
ejpam-5984	174	10	then	then	ADV
ejpam-5984	174	11	by	by	ADP
ejpam-5984	174	12	proposition	proposition	NOUN
ejpam-5984	174	13	2	2	NUM
ejpam-5984	174	14	,	,	PUNCT
ejpam-5984	174	15	we	we	PRON
ejpam-5984	174	16	have	have	VERB
ejpam-5984	174	17	idr(pn	idr(pn	NOUN
ejpam-5984	174	18	−	−	PROPN
ejpam-5984	174	19	v2	v2	NOUN
ejpam-5984	174	20	)	)	PUNCT
ejpam-5984	175	1	=	=	SYM
ejpam-5984	175	2	idr(p1	idr(p1	NOUN
ejpam-5984	175	3	)	)	PUNCT
ejpam-5984	176	1	+	+	CCONJ
ejpam-5984	176	2	idr(pn−2	idr(pn−2	NOUN
ejpam-5984	176	3	)	)	PUNCT
ejpam-5984	176	4	=	=	SYM
ejpam-5984	176	5	2	2	NUM
ejpam-5984	177	1	+	+	CCONJ
ejpam-5984	177	2	(	(	PUNCT
ejpam-5984	177	3	n−	n−	NOUN
ejpam-5984	177	4	2	2	NUM
ejpam-5984	177	5	+	+	CCONJ
ejpam-5984	177	6	1	1	NUM
ejpam-5984	177	7	)	)	PUNCT
ejpam-5984	177	8	=	=	PUNCT
ejpam-5984	177	9	n+	n+	ADP
ejpam-5984	177	10	1	1	NUM
ejpam-5984	177	11	>	>	PUNCT
ejpam-5984	177	12	idr(pn	idr(pn	NOUN
ejpam-5984	177	13	)	)	PUNCT
ejpam-5984	177	14	,	,	PUNCT
ejpam-5984	177	15	and	and	CCONJ
ejpam-5984	177	16	so	so	ADV
ejpam-5984	177	17	st+idr(pn	st+idr(pn	ADJ
ejpam-5984	177	18	)	)	PUNCT
ejpam-5984	177	19	=	=	SYM
ejpam-5984	177	20	1	1	X
ejpam-5984	177	21	.	.	X
ejpam-5984	177	22	assume	assume	VERB
ejpam-5984	177	23	that	that	SCONJ
ejpam-5984	177	24	n	n	NUM
ejpam-5984	177	25	≡	≡	PROPN
ejpam-5984	177	26	r	r	NOUN
ejpam-5984	177	27	(	(	PUNCT
ejpam-5984	177	28	mod	mod	NOUN
ejpam-5984	177	29	3	3	NUM
ejpam-5984	177	30	)	)	PUNCT
ejpam-5984	177	31	where	where	SCONJ
ejpam-5984	177	32	r	r	NOUN
ejpam-5984	177	33	∈	∈	PROPN
ejpam-5984	177	34	{	{	PUNCT
ejpam-5984	177	35	1	1	NUM
ejpam-5984	177	36	,	,	PUNCT
ejpam-5984	177	37	2	2	NUM
ejpam-5984	177	38	}	}	PUNCT
ejpam-5984	177	39	.	.	PUNCT
ejpam-5984	178	1	by	by	ADP
ejpam-5984	178	2	contradiction	contradiction	NOUN
ejpam-5984	178	3	,	,	PUNCT
ejpam-5984	178	4	we	we	PRON
ejpam-5984	178	5	may	may	AUX
ejpam-5984	178	6	assume	assume	VERB
ejpam-5984	178	7	that	that	SCONJ
ejpam-5984	178	8	there	there	PRON
ejpam-5984	178	9	exists	exist	VERB
ejpam-5984	178	10	a	a	DET
ejpam-5984	178	11	n′	n′	ADJ
ejpam-5984	178	12	≡	≡	PROPN
ejpam-5984	178	13	r	r	NOUN
ejpam-5984	178	14	(	(	PUNCT
ejpam-5984	178	15	mod	mod	NOUN
ejpam-5984	178	16	3	3	NUM
ejpam-5984	178	17	)	)	PUNCT
ejpam-5984	178	18	such	such	ADJ
ejpam-5984	178	19	that	that	DET
ejpam-5984	178	20	st+idr(pn′	st+idr(pn′	NOUN
ejpam-5984	178	21	)	)	PUNCT
ejpam-5984	178	22	is	be	AUX
ejpam-5984	178	23	an	an	DET
ejpam-5984	178	24	integer	integer	NOUN
ejpam-5984	178	25	m.	m.	NOUN
ejpam-5984	178	26	let	let	VERB
ejpam-5984	178	27	s	s	PRON
ejpam-5984	178	28	be	be	AUX
ejpam-5984	178	29	a	a	DET
ejpam-5984	178	30	set	set	NOUN
ejpam-5984	178	31	of	of	ADP
ejpam-5984	178	32	vertices	vertex	NOUN
ejpam-5984	178	33	such	such	ADJ
ejpam-5984	178	34	that	that	DET
ejpam-5984	178	35	idr(pn′	idr(pn′	NOUN
ejpam-5984	178	36	)	)	PUNCT
ejpam-5984	178	37	<	<	X
ejpam-5984	178	38	idr(pn′	idr(pn′	PROPN
ejpam-5984	178	39	−	−	PROPN
ejpam-5984	178	40	s	s	NOUN
ejpam-5984	178	41	)	)	PUNCT
ejpam-5984	178	42	.	.	PUNCT
ejpam-5984	179	1	let	let	VERB
ejpam-5984	179	2	pn1	pn1	NOUN
ejpam-5984	179	3	,	,	PUNCT
ejpam-5984	179	4	pn2	pn2	INTJ
ejpam-5984	179	5	,	,	PUNCT
ejpam-5984	179	6	.	.	PUNCT
ejpam-5984	179	7	.	.	PUNCT
ejpam-5984	180	1	.	.	PUNCT
ejpam-5984	181	1	,	,	PUNCT
ejpam-5984	181	2	pnk	pnk	NOUN
ejpam-5984	181	3	be	be	VERB
ejpam-5984	181	4	the	the	DET
ejpam-5984	181	5	components	component	NOUN
ejpam-5984	181	6	of	of	ADP
ejpam-5984	181	7	pn′	pn′	PROPN
ejpam-5984	181	8	−	−	PROPN
ejpam-5984	181	9	s.	s.	PROPN
ejpam-5984	181	10	then	then	ADV
ejpam-5984	181	11	by	by	ADP
ejpam-5984	181	12	proposition	proposition	NOUN
ejpam-5984	181	13	2	2	NUM
ejpam-5984	181	14	,	,	PUNCT
ejpam-5984	181	15	we	we	PRON
ejpam-5984	181	16	have	have	VERB
ejpam-5984	181	17	γdr(pn′	γdr(pn′	NOUN
ejpam-5984	181	18	)	)	PUNCT
ejpam-5984	181	19	=	=	SYM
ejpam-5984	181	20	idr(pn′	idr(pn′	PROPN
ejpam-5984	181	21	)	)	PUNCT
ejpam-5984	181	22	<	<	X
ejpam-5984	181	23	idr(pn′	idr(pn′	PROPN
ejpam-5984	181	24	−	−	PROPN
ejpam-5984	181	25	s	s	PART
ejpam-5984	181	26	)	)	PUNCT
ejpam-5984	181	27	=	=	SYM
ejpam-5984	181	28	k∑	k∑	PROPN
ejpam-5984	181	29	i=1	i=1	PROPN
ejpam-5984	181	30	idr(pni	idr(pni	PROPN
ejpam-5984	181	31	)	)	PUNCT
ejpam-5984	182	1	=	=	PRON
ejpam-5984	182	2	k∑	k∑	VERB
ejpam-5984	182	3	i=1	i=1	PROPN
ejpam-5984	182	4	γdr(pni	γdr(pni	NOUN
ejpam-5984	182	5	)	)	PUNCT
ejpam-5984	183	1	=	=	PUNCT
ejpam-5984	183	2	γdr(pn′	γdr(pn′	NOUN
ejpam-5984	183	3	−	−	NOUN
ejpam-5984	183	4	s	s	NOUN
ejpam-5984	183	5	)	)	PUNCT
ejpam-5984	183	6	,	,	PUNCT
ejpam-5984	183	7	a	a	DET
ejpam-5984	183	8	contradiction	contradiction	NOUN
ejpam-5984	183	9	with	with	ADP
ejpam-5984	183	10	proposition	proposition	NOUN
ejpam-5984	183	11	5	5	NUM
ejpam-5984	183	12	.	.	PUNCT
ejpam-5984	183	13	thus	thus	ADV
ejpam-5984	183	14	st+idr(pn	st+idr(pn	VERB
ejpam-5984	183	15	)	)	PUNCT
ejpam-5984	183	16	=	=	SYM
ejpam-5984	184	1	∞	∞	PROPN
ejpam-5984	184	2	when	when	SCONJ
ejpam-5984	184	3	n	n	X
ejpam-5984	184	4	≡	≡	PROPN
ejpam-5984	184	5	r	r	NOUN
ejpam-5984	184	6	(	(	PUNCT
ejpam-5984	184	7	mod	mod	NOUN
ejpam-5984	184	8	3	3	NUM
ejpam-5984	184	9	)	)	PUNCT
ejpam-5984	184	10	where	where	SCONJ
ejpam-5984	184	11	r	r	NOUN
ejpam-5984	184	12	∈	∈	PROPN
ejpam-5984	184	13	{	{	PUNCT
ejpam-5984	184	14	1	1	NUM
ejpam-5984	184	15	,	,	PUNCT
ejpam-5984	184	16	2	2	NUM
ejpam-5984	184	17	}	}	PUNCT
ejpam-5984	184	18	.	.	PUNCT
ejpam-5984	185	1	the	the	DET
ejpam-5984	185	2	following	following	ADJ
ejpam-5984	185	3	result	result	NOUN
ejpam-5984	185	4	is	be	AUX
ejpam-5984	185	5	an	an	DET
ejpam-5984	185	6	immediate	immediate	ADJ
ejpam-5984	185	7	consequence	consequence	NOUN
ejpam-5984	185	8	of	of	ADP
ejpam-5984	185	9	propositions	proposition	NOUN
ejpam-5984	185	10	8	8	NUM
ejpam-5984	185	11	and	and	CCONJ
ejpam-5984	185	12	9	9	NUM
ejpam-5984	185	13	.	.	PUNCT
ejpam-5984	185	14	corollary	corollary	ADJ
ejpam-5984	185	15	1	1	NUM
ejpam-5984	185	16	.	.	PUNCT
ejpam-5984	185	17	for	for	ADP
ejpam-5984	185	18	n	n	PRON
ejpam-5984	185	19	≥	≥	NUM
ejpam-5984	185	20	2	2	NUM
ejpam-5984	185	21	,	,	PUNCT
ejpam-5984	185	22	stidr(pn	stidr(pn	ADJ
ejpam-5984	185	23	)	)	PUNCT
ejpam-5984	185	24	=	=	SYM
ejpam-5984	186	1	1	1	X
ejpam-5984	186	2	.	.	PUNCT
ejpam-5984	187	1	next	next	ADV
ejpam-5984	187	2	we	we	PRON
ejpam-5984	187	3	determine	determine	VERB
ejpam-5984	187	4	the	the	DET
ejpam-5984	187	5	independent	independent	ADJ
ejpam-5984	187	6	double	double	ADJ
ejpam-5984	187	7	roman	roman	ADJ
ejpam-5984	187	8	domination	domination	NOUN
ejpam-5984	187	9	stability	stability	NOUN
ejpam-5984	187	10	of	of	ADP
ejpam-5984	187	11	cycles	cycle	NOUN
ejpam-5984	187	12	.	.	PUNCT
ejpam-5984	188	1	theorem	theorem	NOUN
ejpam-5984	188	2	1	1	NUM
ejpam-5984	188	3	.	.	PUNCT
ejpam-5984	188	4	for	for	ADP
ejpam-5984	188	5	n	n	PRON
ejpam-5984	188	6	≥	≥	NUM
ejpam-5984	188	7	3	3	NUM
ejpam-5984	188	8	,	,	PUNCT
ejpam-5984	188	9	st−idr(cn	st−idr(cn	NOUN
ejpam-5984	188	10	)	)	PUNCT
ejpam-5984	188	11	=	=	PRON
ejpam-5984	189	1	{	{	PUNCT
ejpam-5984	189	2	1	1	NUM
ejpam-5984	189	3	if	if	SCONJ
ejpam-5984	189	4	n	n	PRON
ejpam-5984	189	5	≡	≡	PROPN
ejpam-5984	189	6	1	1	NUM
ejpam-5984	189	7	,	,	PUNCT
ejpam-5984	189	8	4	4	NUM
ejpam-5984	189	9	,	,	PUNCT
ejpam-5984	189	10	5	5	NUM
ejpam-5984	189	11	(	(	PUNCT
ejpam-5984	189	12	mod	mod	NOUN
ejpam-5984	189	13	6	6	NUM
ejpam-5984	189	14	)	)	PUNCT
ejpam-5984	189	15	2	2	NUM
ejpam-5984	189	16	otherwise	otherwise	ADV
ejpam-5984	189	17	.	.	PUNCT
ejpam-5984	190	1	proof	proof	NOUN
ejpam-5984	190	2	.	.	PUNCT
ejpam-5984	191	1	since	since	SCONJ
ejpam-5984	191	2	idr(cn	idr(cn	NOUN
ejpam-5984	191	3	)	)	PUNCT
ejpam-5984	191	4	=	=	SYM
ejpam-5984	191	5	γdr(cn	γdr(cn	PROPN
ejpam-5984	191	6	)	)	PUNCT
ejpam-5984	191	7	for	for	ADP
ejpam-5984	191	8	n	n	NUM
ejpam-5984	191	9	≥	≥	NOUN
ejpam-5984	191	10	3	3	NUM
ejpam-5984	191	11	and	and	CCONJ
ejpam-5984	191	12	idr(pn	idr(pn	NOUN
ejpam-5984	191	13	)	)	PUNCT
ejpam-5984	191	14	=	=	SYM
ejpam-5984	191	15	γdr(pn	γdr(pn	NOUN
ejpam-5984	191	16	)	)	PUNCT
ejpam-5984	191	17	for	for	ADP
ejpam-5984	191	18	n	n	X
ejpam-5984	191	19	≥	≥	NUM
ejpam-5984	191	20	1	1	NUM
ejpam-5984	191	21	,	,	PUNCT
ejpam-5984	191	22	we	we	PRON
ejpam-5984	191	23	deduce	deduce	VERB
ejpam-5984	191	24	from	from	ADP
ejpam-5984	191	25	proposition	proposition	NOUN
ejpam-5984	191	26	6	6	NUM
ejpam-5984	191	27	that	that	DET
ejpam-5984	191	28	st−idr(cn	st−idr(cn	NOUN
ejpam-5984	191	29	)	)	PUNCT
ejpam-5984	191	30	=	=	SYM
ejpam-5984	192	1	1	1	NUM
ejpam-5984	192	2	when	when	SCONJ
ejpam-5984	192	3	n	n	X
ejpam-5984	192	4	≡	≡	PROPN
ejpam-5984	192	5	1	1	NUM
ejpam-5984	192	6	,	,	PUNCT
ejpam-5984	192	7	4	4	NUM
ejpam-5984	192	8	,	,	PUNCT
ejpam-5984	192	9	5	5	NUM
ejpam-5984	192	10	(	(	PUNCT
ejpam-5984	192	11	mod	mod	PROPN
ejpam-5984	192	12	6	6	NUM
ejpam-5984	192	13	)	)	PUNCT
ejpam-5984	192	14	.	.	PUNCT
ejpam-5984	193	1	assume	assume	VERB
ejpam-5984	193	2	that	that	SCONJ
ejpam-5984	193	3	n	n	NUM
ejpam-5984	193	4	≡	≡	PROPN
ejpam-5984	193	5	r	r	NOUN
ejpam-5984	193	6	(	(	PUNCT
ejpam-5984	193	7	mod	mod	PROPN
ejpam-5984	193	8	6	6	NUM
ejpam-5984	193	9	)	)	PUNCT
ejpam-5984	193	10	where	where	SCONJ
ejpam-5984	193	11	r	r	NOUN
ejpam-5984	193	12	∈	∈	PROPN
ejpam-5984	193	13	{	{	PUNCT
ejpam-5984	193	14	0	0	NUM
ejpam-5984	193	15	,	,	PUNCT
ejpam-5984	193	16	2	2	NUM
ejpam-5984	193	17	,	,	PUNCT
ejpam-5984	193	18	3	3	NUM
ejpam-5984	193	19	}	}	PUNCT
ejpam-5984	193	20	.	.	PUNCT
ejpam-5984	194	1	by	by	ADP
ejpam-5984	194	2	proposition	proposition	NOUN
ejpam-5984	194	3	3	3	NUM
ejpam-5984	194	4	,	,	PUNCT
ejpam-5984	194	5	we	we	PRON
ejpam-5984	194	6	have	have	VERB
ejpam-5984	194	7	idr(cn	idr(cn	NOUN
ejpam-5984	194	8	)	)	PUNCT
ejpam-5984	195	1	=	=	VERB
ejpam-5984	196	1	n.	n.	NOUN
ejpam-5984	196	2	it	it	PRON
ejpam-5984	196	3	follows	follow	VERB
ejpam-5984	196	4	from	from	ADP
ejpam-5984	196	5	proposition	proposition	NOUN
ejpam-5984	196	6	6	6	NUM
ejpam-5984	196	7	and	and	CCONJ
ejpam-5984	196	8	remark	remark	NOUN
ejpam-5984	196	9	1	1	NUM
ejpam-5984	196	10	that	that	DET
ejpam-5984	196	11	st−idr(cn	st−idr(cn	NOUN
ejpam-5984	196	12	)	)	PUNCT
ejpam-5984	196	13	≥	≥	NOUN
ejpam-5984	197	1	2	2	NUM
ejpam-5984	197	2	.	.	PUNCT
ejpam-5984	197	3	on	on	ADP
ejpam-5984	197	4	the	the	DET
ejpam-5984	197	5	other	other	ADJ
ejpam-5984	197	6	hand	hand	NOUN
ejpam-5984	197	7	,	,	PUNCT
ejpam-5984	197	8	we	we	PRON
ejpam-5984	197	9	note	note	VERB
ejpam-5984	197	10	that	that	SCONJ
ejpam-5984	197	11	cn	cn	PROPN
ejpam-5984	197	12	−	−	PROPN
ejpam-5984	197	13	{	{	PUNCT
ejpam-5984	197	14	v1	v1	NOUN
ejpam-5984	197	15	,	,	PUNCT
ejpam-5984	197	16	v5	v5	PROPN
ejpam-5984	197	17	}	}	PUNCT
ejpam-5984	197	18	is	be	AUX
ejpam-5984	197	19	a	a	DET
ejpam-5984	197	20	disjoint	disjoint	NOUN
ejpam-5984	197	21	union	union	NOUN
ejpam-5984	197	22	of	of	ADP
ejpam-5984	197	23	two	two	NUM
ejpam-5984	197	24	paths	path	NOUN
ejpam-5984	197	25	p3	p3	NOUN
ejpam-5984	197	26	and	and	CCONJ
ejpam-5984	197	27	pn−5	pn−5	PROPN
ejpam-5984	197	28	and	and	CCONJ
ejpam-5984	197	29	proposition	proposition	NOUN
ejpam-5984	197	30	2	2	NUM
ejpam-5984	197	31	leads	lead	VERB
ejpam-5984	197	32	to	to	PART
ejpam-5984	197	33	idr(cn	idr(cn	VERB
ejpam-5984	197	34	−	−	PROPN
ejpam-5984	197	35	{	{	PUNCT
ejpam-5984	197	36	v1	v1	PROPN
ejpam-5984	197	37	,	,	PUNCT
ejpam-5984	197	38	v5	v5	NOUN
ejpam-5984	197	39	}	}	PUNCT
ejpam-5984	197	40	)	)	PUNCT
ejpam-5984	197	41	=	=	SYM
ejpam-5984	198	1	3	3	NUM
ejpam-5984	198	2	+	+	PUNCT
ejpam-5984	198	3	idr(pn−5	idr(pn−5	PROPN
ejpam-5984	198	4	)	)	PUNCT
ejpam-5984	198	5	≤	≤	NOUN
ejpam-5984	198	6	3	3	NUM
ejpam-5984	198	7	+	+	CCONJ
ejpam-5984	198	8	(	(	PUNCT
ejpam-5984	198	9	n−	n−	NOUN
ejpam-5984	198	10	5	5	NUM
ejpam-5984	198	11	)	)	PUNCT
ejpam-5984	198	12	+	+	CCONJ
ejpam-5984	198	13	1	1	NUM
ejpam-5984	198	14	<	<	X
ejpam-5984	198	15	n	n	X
ejpam-5984	198	16	=	=	SYM
ejpam-5984	198	17	idr(cn	idr(cn	PROPN
ejpam-5984	198	18	)	)	PUNCT
ejpam-5984	198	19	.	.	PUNCT
ejpam-5984	199	1	thus	thus	ADV
ejpam-5984	199	2	st−idr(cn	st−idr(cn	NOUN
ejpam-5984	199	3	)	)	PUNCT
ejpam-5984	199	4	=	=	SYM
ejpam-5984	199	5	2	2	NUM
ejpam-5984	199	6	and	and	CCONJ
ejpam-5984	199	7	the	the	DET
ejpam-5984	199	8	proof	proof	NOUN
ejpam-5984	199	9	is	be	AUX
ejpam-5984	199	10	complete	complete	ADJ
ejpam-5984	199	11	.	.	PUNCT
ejpam-5984	200	1	theorem	theorem	NOUN
ejpam-5984	200	2	2	2	NUM
ejpam-5984	200	3	.	.	X
ejpam-5984	200	4	for	for	ADP
ejpam-5984	200	5	n	n	PRON
ejpam-5984	200	6	≥	≥	NUM
ejpam-5984	200	7	3	3	NUM
ejpam-5984	200	8	,	,	PUNCT
ejpam-5984	200	9	st+idr(cn	st+idr(cn	NOUN
ejpam-5984	200	10	)	)	PUNCT
ejpam-5984	201	1	=	=	SYM
ejpam-5984	201	2	∞.	∞.	PROPN
ejpam-5984	201	3	s.	s.	PROPN
ejpam-5984	201	4	m.	m.	PROPN
ejpam-5984	201	5	sheikholeslami	sheikholeslami	PROPN
ejpam-5984	201	6	et	et	PROPN
ejpam-5984	201	7	al	al	PROPN
ejpam-5984	201	8	.	.	PUNCT
ejpam-5984	201	9	/	/	SYM
ejpam-5984	201	10	eur	eur	PROPN
ejpam-5984	201	11	.	.	PUNCT
ejpam-5984	202	1	j.	j.	PROPN
ejpam-5984	202	2	pure	pure	PROPN
ejpam-5984	202	3	appl	appl	PROPN
ejpam-5984	202	4	.	.	PROPN
ejpam-5984	202	5	math	math	PROPN
ejpam-5984	202	6	,	,	PUNCT
ejpam-5984	202	7	18	18	NUM
ejpam-5984	202	8	(	(	PUNCT
ejpam-5984	202	9	2	2	NUM
ejpam-5984	202	10	)	)	PUNCT
ejpam-5984	202	11	(	(	PUNCT
ejpam-5984	202	12	2025	2025	NUM
ejpam-5984	202	13	)	)	PUNCT
ejpam-5984	202	14	,	,	PUNCT
ejpam-5984	202	15	5984	5984	NUM
ejpam-5984	202	16	7	7	NUM
ejpam-5984	202	17	of	of	ADP
ejpam-5984	202	18	16	16	NUM
ejpam-5984	202	19	proof	proof	NOUN
ejpam-5984	202	20	.	.	PUNCT
ejpam-5984	203	1	by	by	ADP
ejpam-5984	203	2	contradiction	contradiction	NOUN
ejpam-5984	203	3	,	,	PUNCT
ejpam-5984	203	4	we	we	PRON
ejpam-5984	203	5	may	may	AUX
ejpam-5984	203	6	assume	assume	VERB
ejpam-5984	203	7	that	that	SCONJ
ejpam-5984	203	8	there	there	PRON
ejpam-5984	203	9	exists	exist	VERB
ejpam-5984	203	10	an	an	DET
ejpam-5984	203	11	integer	integer	NOUN
ejpam-5984	203	12	n′	n′	PROPN
ejpam-5984	203	13	≥	≥	NUM
ejpam-5984	203	14	3	3	NUM
ejpam-5984	203	15	such	such	ADJ
ejpam-5984	203	16	that	that	DET
ejpam-5984	203	17	st+idr(cn′	st+idr(cn′	NOUN
ejpam-5984	203	18	)	)	PUNCT
ejpam-5984	203	19	is	be	AUX
ejpam-5984	203	20	an	an	DET
ejpam-5984	203	21	integer	integer	NOUN
ejpam-5984	203	22	m.	m.	NOUN
ejpam-5984	203	23	let	let	VERB
ejpam-5984	203	24	s	s	PRON
ejpam-5984	203	25	be	be	AUX
ejpam-5984	203	26	a	a	DET
ejpam-5984	203	27	set	set	NOUN
ejpam-5984	203	28	of	of	ADP
ejpam-5984	203	29	vertices	vertex	NOUN
ejpam-5984	203	30	such	such	ADJ
ejpam-5984	203	31	that	that	DET
ejpam-5984	203	32	idr(cn′	idr(cn′	PROPN
ejpam-5984	203	33	)	)	PUNCT
ejpam-5984	203	34	<	<	X
ejpam-5984	203	35	idr(cn′	idr(cn′	PROPN
ejpam-5984	203	36	−s	−s	NOUN
ejpam-5984	203	37	)	)	PUNCT
ejpam-5984	203	38	.	.	PUNCT
ejpam-5984	204	1	if	if	SCONJ
ejpam-5984	204	2	pn1	pn1	NOUN
ejpam-5984	204	3	,	,	PUNCT
ejpam-5984	204	4	pn2	pn2	INTJ
ejpam-5984	204	5	,	,	PUNCT
ejpam-5984	204	6	.	.	PUNCT
ejpam-5984	204	7	.	.	PUNCT
ejpam-5984	204	8	.	.	PUNCT
ejpam-5984	205	1	,	,	PUNCT
ejpam-5984	205	2	pnk	pnk	NOUN
ejpam-5984	205	3	are	be	AUX
ejpam-5984	205	4	the	the	DET
ejpam-5984	205	5	components	component	NOUN
ejpam-5984	205	6	of	of	ADP
ejpam-5984	205	7	cn′	cn′	ADJ
ejpam-5984	205	8	−	−	PROPN
ejpam-5984	205	9	s	s	NOUN
ejpam-5984	205	10	,	,	PUNCT
ejpam-5984	205	11	then	then	ADV
ejpam-5984	205	12	by	by	ADP
ejpam-5984	205	13	proposition	proposition	NOUN
ejpam-5984	205	14	3	3	NUM
ejpam-5984	205	15	we	we	PRON
ejpam-5984	205	16	have	have	VERB
ejpam-5984	205	17	γdr(cn′	γdr(cn′	NOUN
ejpam-5984	205	18	)	)	PUNCT
ejpam-5984	205	19	=	=	SYM
ejpam-5984	205	20	idr(cn′	idr(cn′	PROPN
ejpam-5984	205	21	)	)	PUNCT
ejpam-5984	205	22	<	<	X
ejpam-5984	205	23	idr(cn′	idr(cn′	PROPN
ejpam-5984	205	24	−	−	PROPN
ejpam-5984	205	25	s	s	PART
ejpam-5984	205	26	)	)	PUNCT
ejpam-5984	205	27	=	=	SYM
ejpam-5984	205	28	k∑	k∑	PROPN
ejpam-5984	205	29	i=1	i=1	PROPN
ejpam-5984	205	30	idr(pni	idr(pni	PROPN
ejpam-5984	205	31	)	)	PUNCT
ejpam-5984	206	1	=	=	PRON
ejpam-5984	206	2	k∑	k∑	VERB
ejpam-5984	206	3	i=1	i=1	PROPN
ejpam-5984	206	4	γdr(pni	γdr(pni	NOUN
ejpam-5984	206	5	)	)	PUNCT
ejpam-5984	207	1	=	=	PUNCT
ejpam-5984	207	2	γdr(cn′	γdr(cn′	NOUN
ejpam-5984	207	3	−	−	NUM
ejpam-5984	207	4	s	s	NOUN
ejpam-5984	207	5	)	)	PUNCT
ejpam-5984	207	6	,	,	PUNCT
ejpam-5984	207	7	a	a	DET
ejpam-5984	207	8	contradiction	contradiction	NOUN
ejpam-5984	207	9	with	with	ADP
ejpam-5984	207	10	proposition	proposition	NOUN
ejpam-5984	207	11	7	7	NUM
ejpam-5984	207	12	.	.	PUNCT
ejpam-5984	207	13	thus	thus	ADV
ejpam-5984	207	14	st+idr(cn	st+idr(cn	VERB
ejpam-5984	207	15	)	)	PUNCT
ejpam-5984	207	16	=	=	SYM
ejpam-5984	207	17	∞.	∞.	PROPN
ejpam-5984	207	18	corollary	corollary	NOUN
ejpam-5984	207	19	2	2	NUM
ejpam-5984	207	20	.	.	PUNCT
ejpam-5984	208	1	for	for	ADP
ejpam-5984	208	2	n	n	PRON
ejpam-5984	208	3	≥	≥	NUM
ejpam-5984	208	4	3	3	NUM
ejpam-5984	208	5	,	,	PUNCT
ejpam-5984	208	6	stidr(cn	stidr(cn	NOUN
ejpam-5984	208	7	)	)	PUNCT
ejpam-5984	208	8	=	=	SYM
ejpam-5984	208	9	st−idr(cn	st−idr(cn	PROPN
ejpam-5984	208	10	)	)	PUNCT
ejpam-5984	208	11	.	.	PUNCT
ejpam-5984	209	1	one	one	PRON
ejpam-5984	209	2	can	can	AUX
ejpam-5984	209	3	observe	observe	VERB
ejpam-5984	209	4	that	that	PRON
ejpam-5984	209	5	for	for	ADP
ejpam-5984	209	6	n	n	PRON
ejpam-5984	209	7	≥	≥	NOUN
ejpam-5984	209	8	2	2	NUM
ejpam-5984	209	9	,	,	PUNCT
ejpam-5984	209	10	idr(kn	idr(kn	NUM
ejpam-5984	209	11	)	)	PUNCT
ejpam-5984	209	12	=	=	SYM
ejpam-5984	209	13	idr(k1,n−1	idr(k1,n−1	X
ejpam-5984	209	14	)	)	PUNCT
ejpam-5984	209	15	=	=	SYM
ejpam-5984	209	16	3	3	NUM
ejpam-5984	209	17	and	and	CCONJ
ejpam-5984	209	18	for	for	ADP
ejpam-5984	209	19	1	1	NUM
ejpam-5984	209	20	≤	≤	NOUN
ejpam-5984	209	21	r	r	NOUN
ejpam-5984	209	22	≤	≤	NUM
ejpam-5984	209	23	t	t	PROPN
ejpam-5984	209	24	,	,	PUNCT
ejpam-5984	209	25	idr(sr	idr(sr	NOUN
ejpam-5984	209	26	,	,	PUNCT
ejpam-5984	209	27	t	t	PROPN
ejpam-5984	209	28	)	)	PUNCT
ejpam-5984	209	29	=	=	PRON
ejpam-5984	209	30	{	{	PUNCT
ejpam-5984	209	31	5	5	NUM
ejpam-5984	209	32	if	if	SCONJ
ejpam-5984	209	33	r	r	NOUN
ejpam-5984	209	34	=	=	SYM
ejpam-5984	209	35	1	1	NUM
ejpam-5984	209	36	3	3	NUM
ejpam-5984	209	37	+	+	SYM
ejpam-5984	209	38	2r	2r	NUM
ejpam-5984	209	39	if	if	SCONJ
ejpam-5984	209	40	r	r	NOUN
ejpam-5984	209	41	≥	≥	NOUN
ejpam-5984	209	42	2	2	NUM
ejpam-5984	209	43	.	.	PUNCT
ejpam-5984	209	44	from	from	ADP
ejpam-5984	209	45	the	the	DET
ejpam-5984	209	46	above	above	ADJ
ejpam-5984	209	47	observations	observation	NOUN
ejpam-5984	209	48	,	,	PUNCT
ejpam-5984	209	49	we	we	PRON
ejpam-5984	209	50	can	can	AUX
ejpam-5984	209	51	easily	easily	ADV
ejpam-5984	209	52	obtain	obtain	VERB
ejpam-5984	209	53	the	the	DET
ejpam-5984	209	54	following	follow	VERB
ejpam-5984	209	55	conclusions	conclusion	NOUN
ejpam-5984	209	56	.	.	PUNCT
ejpam-5984	210	1	corollary	corollary	ADJ
ejpam-5984	210	2	3	3	NUM
ejpam-5984	210	3	.	.	PUNCT
ejpam-5984	211	1	for	for	ADP
ejpam-5984	211	2	n	n	PRON
ejpam-5984	211	3	≥	≥	NUM
ejpam-5984	211	4	2	2	NUM
ejpam-5984	211	5	,	,	PUNCT
ejpam-5984	211	6	stidr(kn	stidr(kn	NOUN
ejpam-5984	211	7	)	)	PUNCT
ejpam-5984	211	8	=	=	SYM
ejpam-5984	211	9	st−idr(kn	st−idr(kn	NOUN
ejpam-5984	211	10	)	)	PUNCT
ejpam-5984	212	1	=	=	PUNCT
ejpam-5984	212	2	n−	n−	NOUN
ejpam-5984	212	3	1	1	NUM
ejpam-5984	212	4	and	and	CCONJ
ejpam-5984	212	5	st+idr(kn	st+idr(kn	NUM
ejpam-5984	212	6	)	)	PUNCT
ejpam-5984	213	1	=	=	SYM
ejpam-5984	213	2	∞.	∞.	PROPN
ejpam-5984	213	3	corollary	corollary	NOUN
ejpam-5984	213	4	4	4	NUM
ejpam-5984	213	5	.	.	PUNCT
ejpam-5984	213	6	for	for	ADP
ejpam-5984	213	7	n	n	PRON
ejpam-5984	213	8	≥	≥	NUM
ejpam-5984	213	9	3	3	NUM
ejpam-5984	213	10	,	,	PUNCT
ejpam-5984	213	11	stidr(k1,n−1	stidr(k1,n−1	ADJ
ejpam-5984	213	12	)	)	PUNCT
ejpam-5984	213	13	=	=	SYM
ejpam-5984	213	14	st+idr(k1,n−1	st+idr(k1,n−1	PROPN
ejpam-5984	213	15	)	)	PUNCT
ejpam-5984	214	1	=	=	SYM
ejpam-5984	214	2	1	1	NUM
ejpam-5984	214	3	and	and	CCONJ
ejpam-5984	214	4	st−idr(k1,n−1	st−idr(k1,n−1	ADJ
ejpam-5984	214	5	)	)	PUNCT
ejpam-5984	214	6	=	=	SYM
ejpam-5984	214	7	n−	n−	NOUN
ejpam-5984	214	8	1	1	NUM
ejpam-5984	214	9	.	.	PUNCT
ejpam-5984	214	10	corollary	corollary	ADJ
ejpam-5984	214	11	5	5	NUM
ejpam-5984	214	12	.	.	PUNCT
ejpam-5984	215	1	for	for	ADP
ejpam-5984	215	2	1	1	NUM
ejpam-5984	215	3	≤	≤	NOUN
ejpam-5984	215	4	r	r	NOUN
ejpam-5984	215	5	≤	≤	NOUN
ejpam-5984	215	6	t	t	NOUN
ejpam-5984	215	7	with	with	ADP
ejpam-5984	215	8	t	t	PROPN
ejpam-5984	215	9	≥	≥	NUM
ejpam-5984	215	10	2	2	NUM
ejpam-5984	215	11	,	,	PUNCT
ejpam-5984	215	12	stidr(sr	stidr(sr	NOUN
ejpam-5984	215	13	,	,	PUNCT
ejpam-5984	215	14	t	t	PROPN
ejpam-5984	215	15	)	)	PUNCT
ejpam-5984	215	16	=	=	SYM
ejpam-5984	215	17	st−idr(sr	st−idr(sr	PROPN
ejpam-5984	215	18	,	,	PUNCT
ejpam-5984	215	19	t	t	PROPN
ejpam-5984	215	20	)	)	PUNCT
ejpam-5984	215	21	=	=	SYM
ejpam-5984	215	22	1	1	NUM
ejpam-5984	215	23	and	and	CCONJ
ejpam-5984	215	24	st+idr(sr	st+idr(sr	ADJ
ejpam-5984	215	25	,	,	PUNCT
ejpam-5984	215	26	t	t	PROPN
ejpam-5984	215	27	)	)	PUNCT
ejpam-5984	215	28	=	=	SYM
ejpam-5984	215	29	1	1	NUM
ejpam-5984	216	1	when	when	SCONJ
ejpam-5984	216	2	r	r	NOUN
ejpam-5984	216	3	<	<	X
ejpam-5984	216	4	t	t	NOUN
ejpam-5984	216	5	and	and	CCONJ
ejpam-5984	216	6	st+idr(sr	st+idr(sr	NOUN
ejpam-5984	216	7	,	,	PUNCT
ejpam-5984	216	8	t	t	PROPN
ejpam-5984	216	9	)	)	PUNCT
ejpam-5984	216	10	=	=	SYM
ejpam-5984	216	11	2	2	NUM
ejpam-5984	216	12	when	when	SCONJ
ejpam-5984	216	13	r	r	NOUN
ejpam-5984	216	14	=	=	PUNCT
ejpam-5984	216	15	t.	t.	NOUN
ejpam-5984	216	16	in	in	ADP
ejpam-5984	216	17	what	what	PRON
ejpam-5984	216	18	follows	follow	VERB
ejpam-5984	216	19	we	we	PRON
ejpam-5984	216	20	give	give	VERB
ejpam-5984	216	21	some	some	DET
ejpam-5984	216	22	bounds	bound	NOUN
ejpam-5984	216	23	for	for	ADP
ejpam-5984	216	24	the	the	DET
ejpam-5984	216	25	idrd	idrd	ADJ
ejpam-5984	216	26	-	-	PUNCT
ejpam-5984	216	27	stability	stability	NOUN
ejpam-5984	216	28	of	of	ADP
ejpam-5984	216	29	a	a	DET
ejpam-5984	216	30	graph	graph	NOUN
ejpam-5984	216	31	.	.	PUNCT
ejpam-5984	217	1	since	since	SCONJ
ejpam-5984	217	2	for	for	ADP
ejpam-5984	217	3	any	any	DET
ejpam-5984	217	4	graph	graph	NOUN
ejpam-5984	217	5	g	g	NOUN
ejpam-5984	217	6	of	of	ADP
ejpam-5984	217	7	order	order	NOUN
ejpam-5984	217	8	at	at	ADV
ejpam-5984	217	9	least	least	ADJ
ejpam-5984	217	10	2	2	NUM
ejpam-5984	217	11	,	,	PUNCT
ejpam-5984	217	12	idr(g	idr(g	PROPN
ejpam-5984	217	13	)	)	PUNCT
ejpam-5984	217	14	≥	≥	NOUN
ejpam-5984	217	15	3	3	NUM
ejpam-5984	217	16	with	with	ADP
ejpam-5984	217	17	equality	equality	NOUN
ejpam-5984	217	18	if	if	SCONJ
ejpam-5984	217	19	and	and	CCONJ
ejpam-5984	217	20	only	only	ADV
ejpam-5984	217	21	if	if	SCONJ
ejpam-5984	217	22	∆(g	∆(g	NOUN
ejpam-5984	217	23	)	)	PUNCT
ejpam-5984	217	24	=	=	SYM
ejpam-5984	218	1	n	n	CCONJ
ejpam-5984	219	1	−	−	NUM
ejpam-5984	219	2	1	1	NUM
ejpam-5984	219	3	,	,	PUNCT
ejpam-5984	219	4	the	the	DET
ejpam-5984	219	5	proof	proof	NOUN
ejpam-5984	219	6	of	of	ADP
ejpam-5984	219	7	this	this	PRON
ejpam-5984	219	8	is	be	AUX
ejpam-5984	219	9	trivial	trivial	ADJ
ejpam-5984	219	10	.	.	PUNCT
ejpam-5984	220	1	remark	remark	NOUN
ejpam-5984	220	2	2	2	NUM
ejpam-5984	220	3	.	.	PUNCT
ejpam-5984	221	1	if	if	SCONJ
ejpam-5984	221	2	g	g	PROPN
ejpam-5984	221	3	is	be	AUX
ejpam-5984	221	4	a	a	DET
ejpam-5984	221	5	graph	graph	NOUN
ejpam-5984	221	6	of	of	ADP
ejpam-5984	221	7	order	order	NOUN
ejpam-5984	221	8	n	n	PRON
ejpam-5984	221	9	≥	≥	NOUN
ejpam-5984	221	10	2	2	NUM
ejpam-5984	221	11	,	,	PUNCT
ejpam-5984	221	12	then	then	ADV
ejpam-5984	221	13	st−idr(g	st−idr(g	PROPN
ejpam-5984	221	14	)	)	PUNCT
ejpam-5984	221	15	≤	≤	NUM
ejpam-5984	221	16	n−	n−	NOUN
ejpam-5984	221	17	1	1	NUM
ejpam-5984	221	18	with	with	ADP
ejpam-5984	221	19	equality	equality	NOUN
ejpam-5984	221	20	if	if	SCONJ
ejpam-5984	221	21	and	and	CCONJ
ejpam-5984	221	22	only	only	ADV
ejpam-5984	221	23	if	if	SCONJ
ejpam-5984	221	24	∆(g	∆(g	NOUN
ejpam-5984	221	25	)	)	PUNCT
ejpam-5984	221	26	=	=	SYM
ejpam-5984	221	27	n−	n−	NOUN
ejpam-5984	221	28	1	1	NUM
ejpam-5984	221	29	.	.	PUNCT
ejpam-5984	221	30	proposition	proposition	NOUN
ejpam-5984	221	31	10	10	NUM
ejpam-5984	221	32	.	.	PUNCT
ejpam-5984	222	1	if	if	SCONJ
ejpam-5984	222	2	g	g	PROPN
ejpam-5984	222	3	̸=	̸=	PROPN
ejpam-5984	222	4	kn	kn	PROPN
ejpam-5984	222	5	is	be	AUX
ejpam-5984	222	6	a	a	DET
ejpam-5984	222	7	connected	connected	ADJ
ejpam-5984	222	8	graph	graph	NOUN
ejpam-5984	222	9	of	of	ADP
ejpam-5984	222	10	order	order	NOUN
ejpam-5984	222	11	n	n	PRON
ejpam-5984	222	12	≥	≥	NOUN
ejpam-5984	222	13	3	3	NUM
ejpam-5984	222	14	with	with	ADP
ejpam-5984	222	15	idr(g	idr(g	PROPN
ejpam-5984	222	16	)	)	PUNCT
ejpam-5984	222	17	≥	≥	NOUN
ejpam-5984	222	18	4	4	NUM
ejpam-5984	222	19	,	,	PUNCT
ejpam-5984	222	20	then	then	ADV
ejpam-5984	222	21	st−idr(g	st−idr(g	PROPN
ejpam-5984	222	22	)	)	PUNCT
ejpam-5984	222	23	≤	≤	NUM
ejpam-5984	222	24	n−	n−	NOUN
ejpam-5984	222	25	ω(g)−	ω(g)−	NOUN
ejpam-5984	222	26	1	1	NUM
ejpam-5984	222	27	where	where	SCONJ
ejpam-5984	222	28	ω(g	ω(g	NOUN
ejpam-5984	222	29	)	)	PUNCT
ejpam-5984	222	30	is	be	AUX
ejpam-5984	222	31	the	the	DET
ejpam-5984	222	32	clique	clique	ADJ
ejpam-5984	222	33	number	number	NOUN
ejpam-5984	222	34	of	of	ADP
ejpam-5984	222	35	g.	g.	PROPN
ejpam-5984	222	36	proof	proof	NOUN
ejpam-5984	222	37	.	.	PUNCT
ejpam-5984	223	1	let	let	VERB
ejpam-5984	223	2	s	s	PRON
ejpam-5984	223	3	be	be	AUX
ejpam-5984	223	4	a	a	DET
ejpam-5984	223	5	maximum	maximum	ADJ
ejpam-5984	223	6	clique	clique	NOUN
ejpam-5984	223	7	in	in	ADP
ejpam-5984	223	8	g.	g.	PROPN
ejpam-5984	223	9	since	since	SCONJ
ejpam-5984	223	10	g	g	PROPN
ejpam-5984	223	11	is	be	AUX
ejpam-5984	223	12	connected	connect	VERB
ejpam-5984	223	13	and	and	CCONJ
ejpam-5984	223	14	g	g	PROPN
ejpam-5984	223	15	̸=	̸=	PROPN
ejpam-5984	223	16	kn	kn	PROPN
ejpam-5984	223	17	,	,	PUNCT
ejpam-5984	223	18	there	there	PRON
ejpam-5984	223	19	exists	exist	VERB
ejpam-5984	223	20	a	a	DET
ejpam-5984	223	21	vertex	vertex	NOUN
ejpam-5984	223	22	x	x	SYM
ejpam-5984	223	23	∈	∈	NOUN
ejpam-5984	223	24	v	v	ADP
ejpam-5984	223	25	(	(	PUNCT
ejpam-5984	223	26	g	g	NOUN
ejpam-5984	223	27	)	)	PUNCT
ejpam-5984	223	28	\	\	PUNCT
ejpam-5984	223	29	s	s	VERB
ejpam-5984	223	30	such	such	ADJ
ejpam-5984	223	31	that	that	SCONJ
ejpam-5984	223	32	x	x	PRON
ejpam-5984	223	33	is	be	AUX
ejpam-5984	223	34	adjacent	adjacent	ADJ
ejpam-5984	223	35	to	to	ADP
ejpam-5984	223	36	a	a	DET
ejpam-5984	223	37	vertex	vertex	NOUN
ejpam-5984	223	38	in	in	ADP
ejpam-5984	223	39	s	s	PROPN
ejpam-5984	223	40	,	,	PUNCT
ejpam-5984	223	41	say	say	VERB
ejpam-5984	223	42	y.	y.	NOUN
ejpam-5984	223	43	now	now	ADV
ejpam-5984	223	44	the	the	DET
ejpam-5984	223	45	function	function	NOUN
ejpam-5984	223	46	f	f	PROPN
ejpam-5984	223	47	defined	define	VERB
ejpam-5984	223	48	on	on	ADP
ejpam-5984	223	49	g[s∪{x	g[s∪{x	NOUN
ejpam-5984	223	50	}	}	PUNCT
ejpam-5984	223	51	]	]	PUNCT
ejpam-5984	223	52	by	by	ADP
ejpam-5984	223	53	f(x	f(x	PROPN
ejpam-5984	223	54	)	)	PUNCT
ejpam-5984	224	1	=	=	SYM
ejpam-5984	224	2	3	3	NUM
ejpam-5984	224	3	and	and	CCONJ
ejpam-5984	224	4	f(z	f(z	PROPN
ejpam-5984	224	5	)	)	PUNCT
ejpam-5984	225	1	=	=	SYM
ejpam-5984	225	2	0	0	NUM
ejpam-5984	225	3	for	for	ADP
ejpam-5984	225	4	the	the	DET
ejpam-5984	225	5	remaining	remain	VERB
ejpam-5984	225	6	vertices	vertex	NOUN
ejpam-5984	225	7	,	,	PUNCT
ejpam-5984	225	8	is	be	AUX
ejpam-5984	225	9	an	an	DET
ejpam-5984	225	10	idrd	idrd	ADJ
ejpam-5984	225	11	-	-	PUNCT
ejpam-5984	225	12	function	function	NOUN
ejpam-5984	225	13	of	of	ADP
ejpam-5984	225	14	g[s	g[s	PROPN
ejpam-5984	225	15	∪	∪	ADJ
ejpam-5984	225	16	{	{	PUNCT
ejpam-5984	225	17	x	x	NOUN
ejpam-5984	225	18	}	}	PUNCT
ejpam-5984	225	19	]	]	PUNCT
ejpam-5984	225	20	.	.	PUNCT
ejpam-5984	226	1	this	this	PRON
ejpam-5984	226	2	implies	imply	VERB
ejpam-5984	226	3	that	that	SCONJ
ejpam-5984	226	4	st−idr(g	st−idr(g	PROPN
ejpam-5984	226	5	)	)	PUNCT
ejpam-5984	226	6	≤	≤	NUM
ejpam-5984	226	7	n−	n−	NOUN
ejpam-5984	226	8	(	(	PUNCT
ejpam-5984	226	9	|s|+	|s|+	NOUN
ejpam-5984	226	10	1	1	NUM
ejpam-5984	226	11	)	)	PUNCT
ejpam-5984	226	12	=	=	NOUN
ejpam-5984	226	13	n−	n−	NOUN
ejpam-5984	226	14	ω(g)−	ω(g)−	NOUN
ejpam-5984	226	15	1	1	NUM
ejpam-5984	226	16	,	,	PUNCT
ejpam-5984	226	17	as	as	SCONJ
ejpam-5984	226	18	desired	desire	VERB
ejpam-5984	226	19	.	.	PUNCT
ejpam-5984	227	1	proposition	proposition	NOUN
ejpam-5984	227	2	11	11	NUM
ejpam-5984	227	3	.	.	PUNCT
ejpam-5984	228	1	let	let	VERB
ejpam-5984	228	2	g	g	PRON
ejpam-5984	228	3	be	be	AUX
ejpam-5984	228	4	a	a	DET
ejpam-5984	228	5	graph	graph	NOUN
ejpam-5984	228	6	of	of	ADP
ejpam-5984	228	7	order	order	NOUN
ejpam-5984	228	8	n	n	PRON
ejpam-5984	228	9	≥	≥	NOUN
ejpam-5984	228	10	2	2	NUM
ejpam-5984	228	11	.	.	PUNCT
ejpam-5984	229	1	then	then	ADV
ejpam-5984	229	2	stidr(g	stidr(g	PROPN
ejpam-5984	229	3	)	)	PUNCT
ejpam-5984	229	4	≤	≤	NOUN
ejpam-5984	229	5	δ(g	δ(g	PUNCT
ejpam-5984	229	6	)	)	PUNCT
ejpam-5984	230	1	+	+	CCONJ
ejpam-5984	230	2	1	1	X
ejpam-5984	230	3	.	.	X
ejpam-5984	230	4	in	in	ADP
ejpam-5984	230	5	particular	particular	ADJ
ejpam-5984	230	6	,	,	PUNCT
ejpam-5984	230	7	this	this	DET
ejpam-5984	230	8	bound	bind	VERB
ejpam-5984	230	9	is	be	AUX
ejpam-5984	230	10	sharp	sharp	ADJ
ejpam-5984	230	11	for	for	ADP
ejpam-5984	230	12	graphs	graph	NOUN
ejpam-5984	230	13	with	with	ADP
ejpam-5984	230	14	isolated	isolated	ADJ
ejpam-5984	230	15	vertices	vertex	NOUN
ejpam-5984	230	16	.	.	PUNCT
ejpam-5984	231	1	s.	s.	PROPN
ejpam-5984	231	2	m.	m.	PROPN
ejpam-5984	231	3	sheikholeslami	sheikholeslami	PROPN
ejpam-5984	231	4	et	et	PROPN
ejpam-5984	231	5	al	al	PROPN
ejpam-5984	231	6	.	.	PUNCT
ejpam-5984	231	7	/	/	SYM
ejpam-5984	231	8	eur	eur	PROPN
ejpam-5984	231	9	.	.	PUNCT
ejpam-5984	232	1	j.	j.	PROPN
ejpam-5984	232	2	pure	pure	PROPN
ejpam-5984	232	3	appl	appl	PROPN
ejpam-5984	232	4	.	.	PROPN
ejpam-5984	232	5	math	math	PROPN
ejpam-5984	232	6	,	,	PUNCT
ejpam-5984	232	7	18	18	NUM
ejpam-5984	232	8	(	(	PUNCT
ejpam-5984	232	9	2	2	NUM
ejpam-5984	232	10	)	)	PUNCT
ejpam-5984	232	11	(	(	PUNCT
ejpam-5984	232	12	2025	2025	NUM
ejpam-5984	232	13	)	)	PUNCT
ejpam-5984	232	14	,	,	PUNCT
ejpam-5984	232	15	5984	5984	NUM
ejpam-5984	232	16	8	8	NUM
ejpam-5984	232	17	of	of	ADP
ejpam-5984	232	18	16	16	NUM
ejpam-5984	232	19	proof	proof	NOUN
ejpam-5984	232	20	.	.	PUNCT
ejpam-5984	233	1	let	let	VERB
ejpam-5984	233	2	x	x	PRON
ejpam-5984	233	3	be	be	AUX
ejpam-5984	233	4	a	a	DET
ejpam-5984	233	5	vertex	vertex	NOUN
ejpam-5984	233	6	of	of	ADP
ejpam-5984	233	7	g	g	NOUN
ejpam-5984	233	8	with	with	ADP
ejpam-5984	233	9	minimum	minimum	NOUN
ejpam-5984	233	10	degree	degree	NOUN
ejpam-5984	233	11	δ(g	δ(g	PROPN
ejpam-5984	233	12	)	)	PUNCT
ejpam-5984	233	13	.	.	PUNCT
ejpam-5984	234	1	assume	assume	VERB
ejpam-5984	234	2	that	that	SCONJ
ejpam-5984	234	3	g′	g′	NOUN
ejpam-5984	234	4	=	=	SYM
ejpam-5984	234	5	g−n(x	g−n(x	X
ejpam-5984	234	6	)	)	PUNCT
ejpam-5984	234	7	and	and	CCONJ
ejpam-5984	234	8	g′′	g′′	PROPN
ejpam-5984	234	9	=	=	SYM
ejpam-5984	234	10	g	g	PROPN
ejpam-5984	234	11	−	−	PROPN
ejpam-5984	234	12	n	n	CCONJ
ejpam-5984	235	1	[	[	X
ejpam-5984	235	2	x	x	X
ejpam-5984	235	3	]	]	X
ejpam-5984	235	4	and	and	CCONJ
ejpam-5984	235	5	f	f	PROPN
ejpam-5984	235	6	is	be	AUX
ejpam-5984	235	7	an	an	DET
ejpam-5984	235	8	idr(g)-function	idr(g)-function	NOUN
ejpam-5984	235	9	.	.	PUNCT
ejpam-5984	236	1	if	if	SCONJ
ejpam-5984	236	2	deg(x	deg(x	ADV
ejpam-5984	236	3	)	)	PUNCT
ejpam-5984	237	1	=	=	SYM
ejpam-5984	237	2	0	0	NUM
ejpam-5984	237	3	,	,	PUNCT
ejpam-5984	237	4	then	then	ADV
ejpam-5984	237	5	f(x	f(x	PROPN
ejpam-5984	237	6	)	)	PUNCT
ejpam-5984	238	1	=	=	SYM
ejpam-5984	238	2	2	2	NUM
ejpam-5984	238	3	and	and	CCONJ
ejpam-5984	238	4	we	we	PRON
ejpam-5984	238	5	have	have	VERB
ejpam-5984	238	6	idr(g	idr(g	PROPN
ejpam-5984	238	7	′′	′′	PROPN
ejpam-5984	238	8	)	)	PUNCT
ejpam-5984	238	9	=	=	SYM
ejpam-5984	238	10	idr(g	idr(g	PROPN
ejpam-5984	238	11	)	)	PUNCT
ejpam-5984	239	1	−	−	PROPN
ejpam-5984	239	2	2	2	NUM
ejpam-5984	239	3	.	.	PUNCT
ejpam-5984	240	1	thus	thus	ADV
ejpam-5984	240	2	,	,	PUNCT
ejpam-5984	240	3	stidr(g	stidr(g	NOUN
ejpam-5984	240	4	)	)	PUNCT
ejpam-5984	240	5	≤	≤	NOUN
ejpam-5984	240	6	δ(g	δ(g	PUNCT
ejpam-5984	240	7	)	)	PUNCT
ejpam-5984	241	1	+	+	CCONJ
ejpam-5984	241	2	1	1	X
ejpam-5984	241	3	.	.	PUNCT
ejpam-5984	242	1	so	so	ADV
ejpam-5984	242	2	we	we	PRON
ejpam-5984	242	3	assume	assume	VERB
ejpam-5984	242	4	that	that	SCONJ
ejpam-5984	242	5	deg(x	deg(x	ADV
ejpam-5984	242	6	)	)	PUNCT
ejpam-5984	242	7	≥	≥	NOUN
ejpam-5984	243	1	1	1	NUM
ejpam-5984	243	2	.	.	PUNCT
ejpam-5984	244	1	if	if	SCONJ
ejpam-5984	244	2	idr(g	idr(g	PROPN
ejpam-5984	244	3	′	′	NOUN
ejpam-5984	244	4	)	)	PUNCT
ejpam-5984	244	5	̸=	̸=	PROPN
ejpam-5984	244	6	idr(g	idr(g	PROPN
ejpam-5984	244	7	)	)	PUNCT
ejpam-5984	244	8	,	,	PUNCT
ejpam-5984	244	9	then	then	ADV
ejpam-5984	244	10	stidr(g	stidr(g	PROPN
ejpam-5984	244	11	)	)	PUNCT
ejpam-5984	244	12	≤	≤	NOUN
ejpam-5984	244	13	δ(g	δ(g	PUNCT
ejpam-5984	244	14	)	)	PUNCT
ejpam-5984	244	15	<	<	X
ejpam-5984	244	16	δ(g	δ(g	PROPN
ejpam-5984	244	17	)	)	PUNCT
ejpam-5984	244	18	+	+	NOUN
ejpam-5984	244	19	1	1	X
ejpam-5984	244	20	.	.	X
ejpam-5984	244	21	let	let	VERB
ejpam-5984	244	22	idr(g	idr(g	PROPN
ejpam-5984	244	23	′	′	NOUN
ejpam-5984	244	24	)	)	PUNCT
ejpam-5984	245	1	=	=	SYM
ejpam-5984	245	2	idr(g	idr(g	PROPN
ejpam-5984	245	3	)	)	PUNCT
ejpam-5984	245	4	and	and	CCONJ
ejpam-5984	245	5	g	g	PROPN
ejpam-5984	245	6	is	be	AUX
ejpam-5984	245	7	an	an	DET
ejpam-5984	245	8	idr(g	idr(g	PROPN
ejpam-5984	245	9	′)-function	′)-function	NOUN
ejpam-5984	245	10	.	.	PUNCT
ejpam-5984	246	1	since	since	SCONJ
ejpam-5984	246	2	x	x	PRON
ejpam-5984	246	3	is	be	AUX
ejpam-5984	246	4	an	an	DET
ejpam-5984	246	5	isolated	isolated	ADJ
ejpam-5984	246	6	vertex	vertex	NOUN
ejpam-5984	246	7	in	in	ADP
ejpam-5984	246	8	g′	g′	NOUN
ejpam-5984	246	9	,	,	PUNCT
ejpam-5984	246	10	g(x	g(x	NOUN
ejpam-5984	246	11	)	)	PUNCT
ejpam-5984	246	12	=	=	SYM
ejpam-5984	246	13	2	2	NUM
ejpam-5984	246	14	,	,	PUNCT
ejpam-5984	246	15	and	and	CCONJ
ejpam-5984	246	16	we	we	PRON
ejpam-5984	246	17	have	have	VERB
ejpam-5984	246	18	idr(g	idr(g	PROPN
ejpam-5984	246	19	′	′	NOUN
ejpam-5984	246	20	)	)	PUNCT
ejpam-5984	247	1	=	=	SYM
ejpam-5984	247	2	idr(g	idr(g	PROPN
ejpam-5984	247	3	′′	′′	PROPN
ejpam-5984	247	4	)	)	PUNCT
ejpam-5984	247	5	+	+	CCONJ
ejpam-5984	247	6	2	2	NUM
ejpam-5984	247	7	,	,	PUNCT
ejpam-5984	247	8	that	that	ADV
ejpam-5984	247	9	is	is	ADV
ejpam-5984	247	10	,	,	PUNCT
ejpam-5984	247	11	idr(g′′	idr(g′′	ADJ
ejpam-5984	247	12	)	)	PUNCT
ejpam-5984	247	13	=	=	SYM
ejpam-5984	247	14	idr(g)−	idr(g)−	NOUN
ejpam-5984	247	15	2	2	NUM
ejpam-5984	247	16	,	,	PUNCT
ejpam-5984	247	17	and	and	CCONJ
ejpam-5984	247	18	so	so	ADV
ejpam-5984	247	19	stidr(g	stidr(g	PROPN
ejpam-5984	247	20	)	)	PUNCT
ejpam-5984	247	21	≤	≤	NOUN
ejpam-5984	247	22	δ(g	δ(g	PUNCT
ejpam-5984	247	23	)	)	PUNCT
ejpam-5984	248	1	+	+	CCONJ
ejpam-5984	248	2	1	1	X
ejpam-5984	248	3	.	.	X
ejpam-5984	248	4	proposition	proposition	NOUN
ejpam-5984	248	5	12	12	NUM
ejpam-5984	248	6	.	.	PUNCT
ejpam-5984	249	1	let	let	VERB
ejpam-5984	249	2	g	g	PRON
ejpam-5984	249	3	be	be	AUX
ejpam-5984	249	4	a	a	DET
ejpam-5984	249	5	connected	connected	ADJ
ejpam-5984	249	6	graph	graph	NOUN
ejpam-5984	249	7	of	of	ADP
ejpam-5984	249	8	order	order	NOUN
ejpam-5984	249	9	n	n	CCONJ
ejpam-5984	249	10	with	with	ADP
ejpam-5984	249	11	idr(g	idr(g	PROPN
ejpam-5984	249	12	)	)	PUNCT
ejpam-5984	249	13	≥	≥	NOUN
ejpam-5984	249	14	4	4	NUM
ejpam-5984	249	15	,	,	PUNCT
ejpam-5984	249	16	then	then	ADV
ejpam-5984	249	17	stidr(g	stidr(g	PROPN
ejpam-5984	249	18	)	)	PUNCT
ejpam-5984	249	19	≤	≤	NOUN
ejpam-5984	249	20	n−∆(g)−	n−∆(g)−	ADJ
ejpam-5984	249	21	1	1	NUM
ejpam-5984	249	22	.	.	PUNCT
ejpam-5984	250	1	proof	proof	NOUN
ejpam-5984	250	2	.	.	PUNCT
ejpam-5984	251	1	if	if	SCONJ
ejpam-5984	251	2	∆(g	∆(g	NOUN
ejpam-5984	251	3	)	)	PUNCT
ejpam-5984	251	4	=	=	SYM
ejpam-5984	251	5	2	2	NUM
ejpam-5984	251	6	,	,	PUNCT
ejpam-5984	251	7	then	then	ADV
ejpam-5984	251	8	g	g	PROPN
ejpam-5984	251	9	is	be	AUX
ejpam-5984	251	10	the	the	DET
ejpam-5984	251	11	path	path	NOUN
ejpam-5984	251	12	pn	pn	NOUN
ejpam-5984	251	13	or	or	CCONJ
ejpam-5984	251	14	a	a	DET
ejpam-5984	251	15	cycle	cycle	NOUN
ejpam-5984	251	16	cn	cn	PROPN
ejpam-5984	251	17	.	.	PUNCT
ejpam-5984	252	1	by	by	ADP
ejpam-5984	252	2	corollary	corollary	ADJ
ejpam-5984	252	3	1	1	NUM
ejpam-5984	252	4	or	or	CCONJ
ejpam-5984	252	5	corollary	corollary	ADJ
ejpam-5984	252	6	2	2	NUM
ejpam-5984	252	7	,	,	PUNCT
ejpam-5984	252	8	we	we	PRON
ejpam-5984	252	9	are	be	AUX
ejpam-5984	252	10	done	do	VERB
ejpam-5984	252	11	.	.	PUNCT
ejpam-5984	253	1	let	let	VERB
ejpam-5984	253	2	∆(g	∆(g	NOUN
ejpam-5984	253	3	)	)	PUNCT
ejpam-5984	253	4	≥	≥	NOUN
ejpam-5984	253	5	3	3	NUM
ejpam-5984	253	6	and	and	CCONJ
ejpam-5984	253	7	x	x	AUX
ejpam-5984	253	8	be	be	AUX
ejpam-5984	253	9	a	a	DET
ejpam-5984	253	10	vertex	vertex	NOUN
ejpam-5984	253	11	with	with	ADP
ejpam-5984	253	12	maximum	maximum	ADJ
ejpam-5984	253	13	degree	degree	NOUN
ejpam-5984	253	14	∆(g	∆(g	NOUN
ejpam-5984	253	15	)	)	PUNCT
ejpam-5984	253	16	.	.	PUNCT
ejpam-5984	254	1	since	since	SCONJ
ejpam-5984	254	2	idr(g	idr(g	PROPN
ejpam-5984	254	3	)	)	PUNCT
ejpam-5984	254	4	≥	≥	NOUN
ejpam-5984	254	5	4	4	NUM
ejpam-5984	254	6	,	,	PUNCT
ejpam-5984	254	7	we	we	PRON
ejpam-5984	254	8	have	have	VERB
ejpam-5984	254	9	x	x	NOUN
ejpam-5984	254	10	=	=	SYM
ejpam-5984	254	11	v	v	X
ejpam-5984	254	12	(	(	PUNCT
ejpam-5984	254	13	g)−n	g)−n	X
ejpam-5984	254	14	[	[	X
ejpam-5984	254	15	x	x	X
ejpam-5984	254	16	]	]	X
ejpam-5984	254	17	̸=	̸=	PROPN
ejpam-5984	254	18	∅.	∅.	VERB
ejpam-5984	254	19	clearly	clearly	ADV
ejpam-5984	254	20	,	,	PUNCT
ejpam-5984	254	21	the	the	DET
ejpam-5984	254	22	function	function	NOUN
ejpam-5984	254	23	g	g	PROPN
ejpam-5984	254	24	defined	define	VERB
ejpam-5984	254	25	on	on	ADP
ejpam-5984	254	26	g−x	g−x	NOUN
ejpam-5984	254	27	by	by	ADP
ejpam-5984	254	28	g(x	g(x	NOUN
ejpam-5984	254	29	)	)	PUNCT
ejpam-5984	254	30	=	=	SYM
ejpam-5984	254	31	3	3	NUM
ejpam-5984	254	32	and	and	CCONJ
ejpam-5984	254	33	g(y	g(y	NOUN
ejpam-5984	254	34	)	)	PUNCT
ejpam-5984	254	35	=	=	SYM
ejpam-5984	254	36	0	0	NUM
ejpam-5984	254	37	for	for	ADP
ejpam-5984	254	38	the	the	DET
ejpam-5984	254	39	remaining	remain	VERB
ejpam-5984	254	40	vertices	vertex	NOUN
ejpam-5984	254	41	,	,	PUNCT
ejpam-5984	254	42	is	be	AUX
ejpam-5984	254	43	an	an	DET
ejpam-5984	254	44	idrd	idrd	ADJ
ejpam-5984	254	45	-	-	PUNCT
ejpam-5984	254	46	function	function	NOUN
ejpam-5984	254	47	of	of	ADP
ejpam-5984	254	48	g	g	PROPN
ejpam-5984	254	49	−x	−x	NOUN
ejpam-5984	254	50	,	,	PUNCT
ejpam-5984	254	51	and	and	CCONJ
ejpam-5984	255	1	so	so	ADV
ejpam-5984	255	2	stidr(g	stidr(g	PROPN
ejpam-5984	255	3	)	)	PUNCT
ejpam-5984	255	4	≤	≤	NOUN
ejpam-5984	255	5	n−∆(g)−	n−∆(g)−	ADJ
ejpam-5984	255	6	1	1	NUM
ejpam-5984	255	7	.	.	PUNCT
ejpam-5984	256	1	combining	combine	VERB
ejpam-5984	256	2	propositions	proposition	NOUN
ejpam-5984	256	3	11	11	NUM
ejpam-5984	256	4	and	and	CCONJ
ejpam-5984	256	5	12	12	NUM
ejpam-5984	256	6	,	,	PUNCT
ejpam-5984	256	7	the	the	DET
ejpam-5984	256	8	following	following	ADJ
ejpam-5984	256	9	result	result	NOUN
ejpam-5984	256	10	follows	follow	VERB
ejpam-5984	256	11	.	.	PUNCT
ejpam-5984	257	1	corollary	corollary	ADJ
ejpam-5984	257	2	6	6	NUM
ejpam-5984	257	3	.	.	PUNCT
ejpam-5984	258	1	let	let	VERB
ejpam-5984	258	2	g	g	PRON
ejpam-5984	258	3	be	be	AUX
ejpam-5984	258	4	a	a	DET
ejpam-5984	258	5	graph	graph	NOUN
ejpam-5984	258	6	with	with	ADP
ejpam-5984	258	7	idr(g	idr(g	PROPN
ejpam-5984	258	8	)	)	PUNCT
ejpam-5984	258	9	≥	≥	NOUN
ejpam-5984	258	10	4	4	NUM
ejpam-5984	258	11	,	,	PUNCT
ejpam-5984	258	12	then	then	ADV
ejpam-5984	258	13	stidr(g	stidr(g	PROPN
ejpam-5984	258	14	)	)	PUNCT
ejpam-5984	258	15	≤	≤	NOUN
ejpam-5984	258	16	min{δ(g	min{δ(g	PROPN
ejpam-5984	258	17	)	)	PUNCT
ejpam-5984	259	1	+	+	CCONJ
ejpam-5984	259	2	1	1	NUM
ejpam-5984	259	3	,	,	PUNCT
ejpam-5984	259	4	n−∆(g)−	n−∆(g)−	ADJ
ejpam-5984	259	5	1	1	NUM
ejpam-5984	259	6	}	}	PUNCT
ejpam-5984	259	7	.	.	PUNCT
ejpam-5984	260	1	5	5	X
ejpam-5984	260	2	.	.	X
ejpam-5984	260	3	graphs	graph	VERB
ejpam-5984	260	4	g	g	NOUN
ejpam-5984	260	5	with	with	ADP
ejpam-5984	260	6	large	large	ADJ
ejpam-5984	260	7	idr	idr	NOUN
ejpam-5984	260	8	-	-	NOUN
ejpam-5984	260	9	stability	stability	NOUN
ejpam-5984	260	10	in	in	ADP
ejpam-5984	260	11	this	this	DET
ejpam-5984	260	12	section	section	NOUN
ejpam-5984	260	13	we	we	PRON
ejpam-5984	260	14	characterize	characterize	VERB
ejpam-5984	260	15	graphs	graph	NOUN
ejpam-5984	260	16	g	g	NOUN
ejpam-5984	260	17	with	with	ADP
ejpam-5984	260	18	stidr(g	stidr(g	PROPN
ejpam-5984	260	19	)	)	PUNCT
ejpam-5984	260	20	∈	∈	PROPN
ejpam-5984	260	21	{	{	PUNCT
ejpam-5984	260	22	n−	n−	NOUN
ejpam-5984	260	23	1	1	NUM
ejpam-5984	260	24	,	,	PUNCT
ejpam-5984	260	25	n−	n−	NOUN
ejpam-5984	260	26	2	2	NUM
ejpam-5984	260	27	,	,	PUNCT
ejpam-5984	260	28	n−	n−	NOUN
ejpam-5984	260	29	3	3	NUM
ejpam-5984	260	30	,	,	PUNCT
ejpam-5984	260	31	n−	n−	NOUN
ejpam-5984	260	32	4	4	NUM
ejpam-5984	260	33	}	}	PUNCT
ejpam-5984	260	34	.	.	PUNCT
ejpam-5984	261	1	proposition	proposition	NOUN
ejpam-5984	261	2	13	13	NUM
ejpam-5984	261	3	.	.	PUNCT
ejpam-5984	262	1	let	let	VERB
ejpam-5984	262	2	g	g	PRON
ejpam-5984	262	3	be	be	AUX
ejpam-5984	262	4	a	a	DET
ejpam-5984	262	5	connected	connected	ADJ
ejpam-5984	262	6	graph	graph	NOUN
ejpam-5984	262	7	of	of	ADP
ejpam-5984	262	8	order	order	NOUN
ejpam-5984	262	9	n	n	PRON
ejpam-5984	262	10	≥	≥	NOUN
ejpam-5984	262	11	2	2	NUM
ejpam-5984	262	12	.	.	PUNCT
ejpam-5984	263	1	then	then	ADV
ejpam-5984	263	2	stidr(g	stidr(g	NUM
ejpam-5984	263	3	)	)	PUNCT
ejpam-5984	263	4	=	=	SYM
ejpam-5984	264	1	n	n	CCONJ
ejpam-5984	264	2	−	−	PROPN
ejpam-5984	264	3	1	1	NUM
ejpam-5984	265	1	if	if	SCONJ
ejpam-5984	265	2	and	and	CCONJ
ejpam-5984	265	3	only	only	ADV
ejpam-5984	265	4	if	if	SCONJ
ejpam-5984	265	5	g	g	PROPN
ejpam-5984	265	6	=	=	PROPN
ejpam-5984	265	7	kn	kn	PROPN
ejpam-5984	265	8	.	.	PUNCT
ejpam-5984	265	9	proof	proof	NOUN
ejpam-5984	265	10	.	.	PUNCT
ejpam-5984	266	1	if	if	SCONJ
ejpam-5984	266	2	g	g	PROPN
ejpam-5984	266	3	=	=	SYM
ejpam-5984	266	4	kn	kn	PROPN
ejpam-5984	266	5	,	,	PUNCT
ejpam-5984	266	6	then	then	ADV
ejpam-5984	266	7	clearly	clearly	ADV
ejpam-5984	266	8	stidr(g	stidr(g	NOUN
ejpam-5984	266	9	)	)	PUNCT
ejpam-5984	267	1	=	=	PUNCT
ejpam-5984	267	2	n−	n−	NOUN
ejpam-5984	267	3	1	1	NUM
ejpam-5984	267	4	.	.	PUNCT
ejpam-5984	268	1	now	now	ADV
ejpam-5984	268	2	,	,	PUNCT
ejpam-5984	268	3	we	we	PRON
ejpam-5984	268	4	prove	prove	VERB
ejpam-5984	268	5	the	the	DET
ejpam-5984	268	6	necessity	necessity	NOUN
ejpam-5984	268	7	.	.	PUNCT
ejpam-5984	269	1	let	let	VERB
ejpam-5984	269	2	g	g	PRON
ejpam-5984	269	3	be	be	AUX
ejpam-5984	269	4	a	a	DET
ejpam-5984	269	5	connected	connected	ADJ
ejpam-5984	269	6	graph	graph	NOUN
ejpam-5984	269	7	of	of	ADP
ejpam-5984	269	8	order	order	NOUN
ejpam-5984	269	9	n	n	PRON
ejpam-5984	269	10	≥	≥	NOUN
ejpam-5984	269	11	2	2	NUM
ejpam-5984	269	12	with	with	ADP
ejpam-5984	269	13	stidr(g	stidr(g	NOUN
ejpam-5984	269	14	)	)	PUNCT
ejpam-5984	269	15	=	=	SYM
ejpam-5984	269	16	n	n	CCONJ
ejpam-5984	270	1	−	−	NOUN
ejpam-5984	270	2	1	1	NUM
ejpam-5984	270	3	.	.	PUNCT
ejpam-5984	271	1	by	by	ADP
ejpam-5984	271	2	proposition	proposition	NOUN
ejpam-5984	271	3	11	11	NUM
ejpam-5984	271	4	,	,	PUNCT
ejpam-5984	271	5	we	we	PRON
ejpam-5984	271	6	have	have	VERB
ejpam-5984	271	7	n−	n−	NOUN
ejpam-5984	271	8	1	1	NUM
ejpam-5984	271	9	=	=	SYM
ejpam-5984	271	10	stidr(g	stidr(g	PROPN
ejpam-5984	271	11	)	)	PUNCT
ejpam-5984	271	12	≤	≤	NOUN
ejpam-5984	271	13	δ(g)+1	δ(g)+1	NOUN
ejpam-5984	271	14	,	,	PUNCT
ejpam-5984	271	15	that	that	PRON
ejpam-5984	271	16	is	is	ADV
ejpam-5984	271	17	δ(g	δ(g	ADV
ejpam-5984	271	18	)	)	PUNCT
ejpam-5984	271	19	≥	≥	NOUN
ejpam-5984	271	20	n−	n−	NOUN
ejpam-5984	271	21	2	2	NUM
ejpam-5984	271	22	.	.	PUNCT
ejpam-5984	272	1	if	if	SCONJ
ejpam-5984	272	2	δ(g	δ(g	PROPN
ejpam-5984	272	3	)	)	PUNCT
ejpam-5984	272	4	=	=	PUNCT
ejpam-5984	272	5	n−	n−	NOUN
ejpam-5984	272	6	1	1	NUM
ejpam-5984	272	7	,	,	PUNCT
ejpam-5984	272	8	then	then	ADV
ejpam-5984	272	9	g	g	PROPN
ejpam-5984	272	10	is	be	AUX
ejpam-5984	272	11	the	the	DET
ejpam-5984	272	12	complete	complete	ADJ
ejpam-5984	272	13	graph	graph	NOUN
ejpam-5984	272	14	kn	kn	PROPN
ejpam-5984	272	15	,	,	PUNCT
ejpam-5984	272	16	as	as	SCONJ
ejpam-5984	272	17	desired	desire	VERB
ejpam-5984	272	18	.	.	PUNCT
ejpam-5984	273	1	assume	assume	VERB
ejpam-5984	273	2	that	that	SCONJ
ejpam-5984	273	3	δ(g	δ(g	PUNCT
ejpam-5984	273	4	)	)	PUNCT
ejpam-5984	273	5	=	=	SYM
ejpam-5984	274	1	n	n	CCONJ
ejpam-5984	274	2	−	−	NOUN
ejpam-5984	275	1	2	2	X
ejpam-5984	275	2	.	.	PUNCT
ejpam-5984	276	1	if	if	SCONJ
ejpam-5984	276	2	idr(g	idr(g	PROPN
ejpam-5984	276	3	)	)	PUNCT
ejpam-5984	276	4	≥	≥	NOUN
ejpam-5984	276	5	4	4	NUM
ejpam-5984	276	6	,	,	PUNCT
ejpam-5984	276	7	then	then	ADV
ejpam-5984	276	8	by	by	ADP
ejpam-5984	276	9	proposition	proposition	NOUN
ejpam-5984	276	10	12	12	NUM
ejpam-5984	276	11	,	,	PUNCT
ejpam-5984	276	12	we	we	PRON
ejpam-5984	276	13	obtain	obtain	VERB
ejpam-5984	276	14	∆(g	∆(g	NOUN
ejpam-5984	276	15	)	)	PUNCT
ejpam-5984	276	16	≤	≤	NOUN
ejpam-5984	277	1	n	n	CCONJ
ejpam-5984	277	2	−	−	PROPN
ejpam-5984	277	3	1	1	NUM
ejpam-5984	277	4	−	−	PROPN
ejpam-5984	277	5	stidr(g	stidr(g	PROPN
ejpam-5984	277	6	)	)	PUNCT
ejpam-5984	277	7	=	=	SYM
ejpam-5984	277	8	0	0	NUM
ejpam-5984	278	1	contradicting	contradict	VERB
ejpam-5984	278	2	the	the	DET
ejpam-5984	278	3	connectivity	connectivity	NOUN
ejpam-5984	278	4	of	of	ADP
ejpam-5984	278	5	g.	g.	PROPN
ejpam-5984	278	6	thus	thus	ADV
ejpam-5984	278	7	,	,	PUNCT
ejpam-5984	278	8	idr(g	idr(g	PROPN
ejpam-5984	278	9	)	)	PUNCT
ejpam-5984	278	10	=	=	SYM
ejpam-5984	279	1	3	3	X
ejpam-5984	279	2	.	.	PUNCT
ejpam-5984	279	3	since	since	SCONJ
ejpam-5984	279	4	δ(g	δ(g	PROPN
ejpam-5984	279	5	)	)	PUNCT
ejpam-5984	279	6	=	=	SYM
ejpam-5984	279	7	n	n	CCONJ
ejpam-5984	279	8	−	−	NUM
ejpam-5984	279	9	2	2	NUM
ejpam-5984	279	10	,	,	PUNCT
ejpam-5984	279	11	g	g	PROPN
ejpam-5984	279	12	has	have	VERB
ejpam-5984	279	13	two	two	NUM
ejpam-5984	279	14	non	non	ADJ
ejpam-5984	279	15	-	-	ADJ
ejpam-5984	279	16	adjacent	adjacent	ADJ
ejpam-5984	279	17	vertices	vertice	VERB
ejpam-5984	279	18	u	u	NOUN
ejpam-5984	279	19	and	and	CCONJ
ejpam-5984	279	20	v	v	NOUN
ejpam-5984	279	21	and	and	CCONJ
ejpam-5984	279	22	it	it	PRON
ejpam-5984	279	23	follows	follow	VERB
ejpam-5984	279	24	from	from	ADP
ejpam-5984	279	25	idr(g[{u	idr(g[{u	NOUN
ejpam-5984	279	26	,	,	PUNCT
ejpam-5984	279	27	v	v	NOUN
ejpam-5984	279	28	}	}	PUNCT
ejpam-5984	279	29	]	]	PUNCT
ejpam-5984	279	30	)	)	PUNCT
ejpam-5984	280	1	=	=	SYM
ejpam-5984	280	2	4	4	NUM
ejpam-5984	280	3	that	that	PRON
ejpam-5984	280	4	n	n	CCONJ
ejpam-5984	280	5	−	−	PROPN
ejpam-5984	280	6	1	1	NUM
ejpam-5984	280	7	=	=	SYM
ejpam-5984	280	8	stidr(g	stidr(g	PROPN
ejpam-5984	280	9	)	)	PUNCT
ejpam-5984	280	10	≤	≤	NOUN
ejpam-5984	280	11	n	n	CCONJ
ejpam-5984	280	12	−	−	PROPN
ejpam-5984	280	13	2	2	NUM
ejpam-5984	280	14	,	,	PUNCT
ejpam-5984	280	15	a	a	DET
ejpam-5984	280	16	contradiction	contradiction	NOUN
ejpam-5984	280	17	.	.	PUNCT
ejpam-5984	281	1	this	this	PRON
ejpam-5984	281	2	completes	complete	VERB
ejpam-5984	281	3	the	the	DET
ejpam-5984	281	4	proof	proof	NOUN
ejpam-5984	281	5	.	.	PUNCT
ejpam-5984	282	1	proposition	proposition	NOUN
ejpam-5984	282	2	14	14	NUM
ejpam-5984	282	3	.	.	PUNCT
ejpam-5984	283	1	let	let	VERB
ejpam-5984	283	2	g	g	PROPN
ejpam-5984	283	3	̸=	̸=	PROPN
ejpam-5984	283	4	kn	kn	PROPN
ejpam-5984	283	5	be	be	AUX
ejpam-5984	283	6	a	a	DET
ejpam-5984	283	7	connected	connected	ADJ
ejpam-5984	283	8	graph	graph	NOUN
ejpam-5984	283	9	of	of	ADP
ejpam-5984	283	10	order	order	NOUN
ejpam-5984	283	11	n	n	PRON
ejpam-5984	283	12	≥	≥	NOUN
ejpam-5984	283	13	3	3	NUM
ejpam-5984	283	14	.	.	PUNCT
ejpam-5984	284	1	then	then	ADV
ejpam-5984	284	2	stidr(g	stidr(g	NUM
ejpam-5984	284	3	)	)	PUNCT
ejpam-5984	285	1	=	=	SYM
ejpam-5984	285	2	n−2	n−2	PROPN
ejpam-5984	285	3	if	if	SCONJ
ejpam-5984	285	4	and	and	CCONJ
ejpam-5984	285	5	only	only	ADV
ejpam-5984	285	6	if	if	SCONJ
ejpam-5984	285	7	g	g	PROPN
ejpam-5984	285	8	=	=	PROPN
ejpam-5984	285	9	kn	kn	PROPN
ejpam-5984	285	10	−	−	PROPN
ejpam-5984	285	11	e.	e.	PROPN
ejpam-5984	285	12	proof	proof	PROPN
ejpam-5984	285	13	.	.	PUNCT
ejpam-5984	286	1	if	if	SCONJ
ejpam-5984	286	2	g	g	PROPN
ejpam-5984	286	3	=	=	PROPN
ejpam-5984	286	4	kn	kn	PROPN
ejpam-5984	286	5	−	−	PROPN
ejpam-5984	286	6	e	e	NOUN
ejpam-5984	286	7	,	,	PUNCT
ejpam-5984	286	8	then	then	ADV
ejpam-5984	286	9	clearly	clearly	ADV
ejpam-5984	286	10	stidr(g	stidr(g	NOUN
ejpam-5984	286	11	)	)	PUNCT
ejpam-5984	286	12	=	=	SYM
ejpam-5984	286	13	n	n	CCONJ
ejpam-5984	286	14	−	−	NOUN
ejpam-5984	286	15	2	2	NUM
ejpam-5984	286	16	.	.	PUNCT
ejpam-5984	287	1	now	now	ADV
ejpam-5984	287	2	,	,	PUNCT
ejpam-5984	287	3	we	we	PRON
ejpam-5984	287	4	prove	prove	VERB
ejpam-5984	287	5	the	the	DET
ejpam-5984	287	6	necessity	necessity	NOUN
ejpam-5984	287	7	.	.	PUNCT
ejpam-5984	288	1	let	let	VERB
ejpam-5984	288	2	g	g	PROPN
ejpam-5984	288	3	̸=	̸=	PROPN
ejpam-5984	288	4	kn	kn	PROPN
ejpam-5984	288	5	be	be	AUX
ejpam-5984	288	6	a	a	DET
ejpam-5984	288	7	connected	connected	ADJ
ejpam-5984	288	8	graph	graph	NOUN
ejpam-5984	288	9	of	of	ADP
ejpam-5984	288	10	order	order	NOUN
ejpam-5984	288	11	n	n	PRON
ejpam-5984	288	12	≥	≥	NOUN
ejpam-5984	288	13	3	3	NUM
ejpam-5984	288	14	with	with	ADP
ejpam-5984	288	15	stidr(g	stidr(g	NOUN
ejpam-5984	288	16	)	)	PUNCT
ejpam-5984	288	17	=	=	SYM
ejpam-5984	288	18	n	n	CCONJ
ejpam-5984	289	1	−	−	NOUN
ejpam-5984	289	2	2	2	X
ejpam-5984	289	3	.	.	PUNCT
ejpam-5984	290	1	if	if	SCONJ
ejpam-5984	290	2	idr(g	idr(g	PROPN
ejpam-5984	290	3	)	)	PUNCT
ejpam-5984	290	4	≥	≥	NOUN
ejpam-5984	290	5	4	4	NUM
ejpam-5984	290	6	,	,	PUNCT
ejpam-5984	290	7	then	then	ADV
ejpam-5984	290	8	by	by	ADP
ejpam-5984	290	9	proposition	proposition	NOUN
ejpam-5984	290	10	12	12	NUM
ejpam-5984	290	11	,	,	PUNCT
ejpam-5984	290	12	we	we	PRON
ejpam-5984	290	13	have	have	VERB
ejpam-5984	290	14	∆(g	∆(g	NOUN
ejpam-5984	290	15	)	)	PUNCT
ejpam-5984	290	16	≤	≤	NUM
ejpam-5984	290	17	1	1	NUM
ejpam-5984	290	18	which	which	PRON
ejpam-5984	290	19	contradicts	contradict	VERB
ejpam-5984	290	20	the	the	DET
ejpam-5984	290	21	connectivity	connectivity	NOUN
ejpam-5984	290	22	of	of	ADP
ejpam-5984	290	23	g.	g.	PROPN
ejpam-5984	291	1	so	so	ADV
ejpam-5984	291	2	idr(g	idr(g	PROPN
ejpam-5984	291	3	)	)	PUNCT
ejpam-5984	292	1	=	=	SYM
ejpam-5984	292	2	3	3	X
ejpam-5984	292	3	.	.	X
ejpam-5984	293	1	if	if	SCONJ
ejpam-5984	293	2	g	g	PROPN
ejpam-5984	293	3	has	have	VERB
ejpam-5984	293	4	two	two	NUM
ejpam-5984	293	5	pair	pair	NOUN
ejpam-5984	293	6	of	of	ADP
ejpam-5984	293	7	non	non	ADJ
ejpam-5984	293	8	-	-	ADJ
ejpam-5984	293	9	adjacent	adjacent	ADJ
ejpam-5984	293	10	vertices	vertice	VERB
ejpam-5984	293	11	u	u	NOUN
ejpam-5984	293	12	,	,	PUNCT
ejpam-5984	293	13	v	v	NOUN
ejpam-5984	293	14	and	and	CCONJ
ejpam-5984	293	15	x	x	NOUN
ejpam-5984	293	16	,	,	PUNCT
ejpam-5984	293	17	y	y	PROPN
ejpam-5984	293	18	,	,	PUNCT
ejpam-5984	293	19	then	then	ADV
ejpam-5984	293	20	idr(g[x	idr(g[x	X
ejpam-5984	293	21	,	,	PUNCT
ejpam-5984	293	22	y	y	PROPN
ejpam-5984	293	23	,	,	PUNCT
ejpam-5984	293	24	u	u	NOUN
ejpam-5984	293	25	,	,	PUNCT
ejpam-5984	293	26	v	v	NOUN
ejpam-5984	293	27	]	]	PUNCT
ejpam-5984	293	28	)	)	PUNCT
ejpam-5984	293	29	≥	≥	NOUN
ejpam-5984	293	30	5	5	NUM
ejpam-5984	293	31	if	if	SCONJ
ejpam-5984	293	32	|{x	|{x	NUM
ejpam-5984	293	33	,	,	PUNCT
ejpam-5984	293	34	y	y	PROPN
ejpam-5984	293	35	,	,	PUNCT
ejpam-5984	293	36	u	u	NOUN
ejpam-5984	293	37	,	,	PUNCT
ejpam-5984	293	38	v}|	v}|	NOUN
ejpam-5984	293	39	=	=	SYM
ejpam-5984	293	40	3	3	NUM
ejpam-5984	293	41	and	and	CCONJ
ejpam-5984	293	42	idr(g[x	idr(g[x	NOUN
ejpam-5984	293	43	,	,	PUNCT
ejpam-5984	293	44	y	y	PROPN
ejpam-5984	293	45	,	,	PUNCT
ejpam-5984	293	46	u	u	NOUN
ejpam-5984	293	47	,	,	PUNCT
ejpam-5984	293	48	v	v	NOUN
ejpam-5984	293	49	]	]	PUNCT
ejpam-5984	293	50	)	)	PUNCT
ejpam-5984	293	51	≥	≥	NOUN
ejpam-5984	293	52	4	4	NUM
ejpam-5984	293	53	if	if	SCONJ
ejpam-5984	293	54	|{x	|{x	NUM
ejpam-5984	293	55	,	,	PUNCT
ejpam-5984	293	56	y	y	PROPN
ejpam-5984	293	57	,	,	PUNCT
ejpam-5984	293	58	u	u	NOUN
ejpam-5984	293	59	,	,	PUNCT
ejpam-5984	293	60	v}|	v}|	NOUN
ejpam-5984	293	61	=	=	SYM
ejpam-5984	293	62	4	4	X
ejpam-5984	293	63	.	.	PUNCT
ejpam-5984	294	1	this	this	PRON
ejpam-5984	294	2	leads	lead	VERB
ejpam-5984	294	3	to	to	ADP
ejpam-5984	294	4	the	the	DET
ejpam-5984	294	5	contradiction	contradiction	NOUN
ejpam-5984	294	6	n−	n−	NOUN
ejpam-5984	294	7	2	2	NUM
ejpam-5984	294	8	=	=	SYM
ejpam-5984	294	9	stidr(g	stidr(g	NUM
ejpam-5984	294	10	)	)	PUNCT
ejpam-5984	294	11	≤	≤	NUM
ejpam-5984	294	12	n−	n−	NOUN
ejpam-5984	294	13	3	3	NUM
ejpam-5984	294	14	.	.	PUNCT
ejpam-5984	295	1	therefore	therefore	ADV
ejpam-5984	295	2	,	,	PUNCT
ejpam-5984	295	3	by	by	ADP
ejpam-5984	295	4	proposition	proposition	NOUN
ejpam-5984	295	5	13	13	NUM
ejpam-5984	295	6	,	,	PUNCT
ejpam-5984	295	7	we	we	PRON
ejpam-5984	295	8	are	be	AUX
ejpam-5984	295	9	done	do	VERB
ejpam-5984	295	10	.	.	PUNCT
ejpam-5984	296	1	s.	s.	PROPN
ejpam-5984	296	2	m.	m.	PROPN
ejpam-5984	296	3	sheikholeslami	sheikholeslami	PROPN
ejpam-5984	296	4	et	et	PROPN
ejpam-5984	296	5	al	al	PROPN
ejpam-5984	296	6	.	.	PUNCT
ejpam-5984	296	7	/	/	SYM
ejpam-5984	296	8	eur	eur	PROPN
ejpam-5984	296	9	.	.	PUNCT
ejpam-5984	297	1	j.	j.	PROPN
ejpam-5984	297	2	pure	pure	PROPN
ejpam-5984	297	3	appl	appl	PROPN
ejpam-5984	297	4	.	.	PROPN
ejpam-5984	297	5	math	math	PROPN
ejpam-5984	297	6	,	,	PUNCT
ejpam-5984	297	7	18	18	NUM
ejpam-5984	297	8	(	(	PUNCT
ejpam-5984	297	9	2	2	NUM
ejpam-5984	297	10	)	)	PUNCT
ejpam-5984	297	11	(	(	PUNCT
ejpam-5984	297	12	2025	2025	NUM
ejpam-5984	297	13	)	)	PUNCT
ejpam-5984	297	14	,	,	PUNCT
ejpam-5984	297	15	5984	5984	NUM
ejpam-5984	297	16	9	9	NUM
ejpam-5984	297	17	of	of	ADP
ejpam-5984	297	18	16	16	NUM
ejpam-5984	297	19	proposition	proposition	NOUN
ejpam-5984	297	20	15	15	NUM
ejpam-5984	297	21	.	.	PUNCT
ejpam-5984	298	1	let	let	VERB
ejpam-5984	298	2	g	g	PRON
ejpam-5984	298	3	be	be	AUX
ejpam-5984	298	4	a	a	DET
ejpam-5984	298	5	connected	connected	ADJ
ejpam-5984	298	6	graph	graph	NOUN
ejpam-5984	298	7	of	of	ADP
ejpam-5984	298	8	order	order	NOUN
ejpam-5984	298	9	n	n	PRON
ejpam-5984	298	10	≥	≥	NOUN
ejpam-5984	298	11	4	4	NUM
ejpam-5984	298	12	and	and	CCONJ
ejpam-5984	298	13	g	g	PROPN
ejpam-5984	298	14	̸∈	̸∈	PROPN
ejpam-5984	298	15	{	{	PUNCT
ejpam-5984	298	16	kn	kn	PROPN
ejpam-5984	298	17	,	,	PUNCT
ejpam-5984	298	18	kn−	kn−	PROPN
ejpam-5984	298	19	e	e	NOUN
ejpam-5984	298	20	}	}	PUNCT
ejpam-5984	298	21	.	.	PUNCT
ejpam-5984	299	1	then	then	ADV
ejpam-5984	299	2	stidr(g	stidr(g	NUM
ejpam-5984	299	3	)	)	PUNCT
ejpam-5984	300	1	=	=	PUNCT
ejpam-5984	300	2	n−3	n−3	PROPN
ejpam-5984	300	3	if	if	SCONJ
ejpam-5984	300	4	and	and	CCONJ
ejpam-5984	300	5	only	only	ADV
ejpam-5984	300	6	if	if	SCONJ
ejpam-5984	300	7	g	g	PROPN
ejpam-5984	300	8	∈	∈	PROPN
ejpam-5984	300	9	{	{	PUNCT
ejpam-5984	300	10	p4	p4	ADJ
ejpam-5984	300	11	,	,	PUNCT
ejpam-5984	300	12	c4,k1,3,k1,3	c4,k1,3,k1,3	PROPN
ejpam-5984	300	13	+	+	CCONJ
ejpam-5984	300	14	e	e	NOUN
ejpam-5984	300	15	,	,	PUNCT
ejpam-5984	300	16	3k1∨kn−3,kn−3∨	3k1∨kn−3,kn−3∨	PROPN
ejpam-5984	300	17	(	(	PUNCT
ejpam-5984	300	18	k2∪k1	k2∪k1	NOUN
ejpam-5984	300	19	)	)	PUNCT
ejpam-5984	300	20	}	}	PUNCT
ejpam-5984	300	21	.	.	PUNCT
ejpam-5984	301	1	proof	proof	NOUN
ejpam-5984	301	2	.	.	PUNCT
ejpam-5984	302	1	the	the	DET
ejpam-5984	302	2	sufficiency	sufficiency	NOUN
ejpam-5984	302	3	is	be	AUX
ejpam-5984	302	4	straightforward	straightforward	ADJ
ejpam-5984	302	5	to	to	PART
ejpam-5984	302	6	check	check	VERB
ejpam-5984	302	7	.	.	PUNCT
ejpam-5984	303	1	to	to	PART
ejpam-5984	303	2	prove	prove	VERB
ejpam-5984	303	3	the	the	DET
ejpam-5984	303	4	necessity	necessity	NOUN
ejpam-5984	303	5	,	,	PUNCT
ejpam-5984	303	6	let	let	VERB
ejpam-5984	303	7	g	g	PRON
ejpam-5984	303	8	be	be	AUX
ejpam-5984	303	9	a	a	DET
ejpam-5984	303	10	connected	connected	ADJ
ejpam-5984	303	11	graph	graph	NOUN
ejpam-5984	303	12	of	of	ADP
ejpam-5984	303	13	order	order	NOUN
ejpam-5984	303	14	n	n	PRON
ejpam-5984	303	15	≥	≥	NOUN
ejpam-5984	303	16	4	4	NUM
ejpam-5984	303	17	such	such	ADJ
ejpam-5984	303	18	that	that	SCONJ
ejpam-5984	303	19	g	g	PROPN
ejpam-5984	303	20	̸∈	̸∈	PROPN
ejpam-5984	303	21	{	{	PUNCT
ejpam-5984	303	22	kn	kn	PROPN
ejpam-5984	303	23	,	,	PUNCT
ejpam-5984	303	24	kn	kn	PROPN
ejpam-5984	303	25	−	−	PROPN
ejpam-5984	303	26	e	e	NOUN
ejpam-5984	303	27	}	}	PUNCT
ejpam-5984	303	28	and	and	CCONJ
ejpam-5984	303	29	stidr(g	stidr(g	NUM
ejpam-5984	303	30	)	)	PUNCT
ejpam-5984	303	31	=	=	SYM
ejpam-5984	303	32	n	n	CCONJ
ejpam-5984	304	1	−	−	NOUN
ejpam-5984	304	2	3	3	X
ejpam-5984	304	3	.	.	PUNCT
ejpam-5984	305	1	obviously	obviously	ADV
ejpam-5984	305	2	,	,	PUNCT
ejpam-5984	305	3	∆(g	∆(g	PROPN
ejpam-5984	305	4	)	)	PUNCT
ejpam-5984	305	5	≥	≥	NOUN
ejpam-5984	305	6	2	2	NUM
ejpam-5984	305	7	.	.	PUNCT
ejpam-5984	306	1	if	if	SCONJ
ejpam-5984	306	2	stidr(g	stidr(g	PROPN
ejpam-5984	306	3	)	)	PUNCT
ejpam-5984	306	4	=	=	SYM
ejpam-5984	307	1	1	1	NUM
ejpam-5984	307	2	,	,	PUNCT
ejpam-5984	307	3	then	then	ADV
ejpam-5984	307	4	n	n	NOUN
ejpam-5984	307	5	=	=	SYM
ejpam-5984	307	6	4	4	X
ejpam-5984	307	7	.	.	PUNCT
ejpam-5984	308	1	it	it	PRON
ejpam-5984	308	2	is	be	AUX
ejpam-5984	308	3	easy	easy	ADJ
ejpam-5984	308	4	to	to	PART
ejpam-5984	308	5	verify	verify	VERB
ejpam-5984	308	6	that	that	SCONJ
ejpam-5984	308	7	g	g	PROPN
ejpam-5984	308	8	∈	∈	PROPN
ejpam-5984	308	9	{	{	PUNCT
ejpam-5984	308	10	p4	p4	ADJ
ejpam-5984	308	11	,	,	PUNCT
ejpam-5984	308	12	c4,k1,3,k1,3	c4,k1,3,k1,3	NOUN
ejpam-5984	308	13	+	+	CCONJ
ejpam-5984	308	14	e	e	NOUN
ejpam-5984	308	15	}	}	PUNCT
ejpam-5984	308	16	as	as	SCONJ
ejpam-5984	308	17	desired	desire	VERB
ejpam-5984	308	18	.	.	PUNCT
ejpam-5984	309	1	hence	hence	ADV
ejpam-5984	309	2	,	,	PUNCT
ejpam-5984	309	3	we	we	PRON
ejpam-5984	309	4	assume	assume	VERB
ejpam-5984	309	5	that	that	SCONJ
ejpam-5984	309	6	stidr(g	stidr(g	PROPN
ejpam-5984	309	7	)	)	PUNCT
ejpam-5984	309	8	≥	≥	NOUN
ejpam-5984	310	1	2	2	NUM
ejpam-5984	310	2	.	.	PUNCT
ejpam-5984	311	1	if	if	SCONJ
ejpam-5984	311	2	idr(g	idr(g	PROPN
ejpam-5984	311	3	)	)	PUNCT
ejpam-5984	311	4	≥	≥	NOUN
ejpam-5984	311	5	4	4	NUM
ejpam-5984	311	6	,	,	PUNCT
ejpam-5984	311	7	then	then	ADV
ejpam-5984	311	8	proposition	proposition	NOUN
ejpam-5984	311	9	12	12	NUM
ejpam-5984	311	10	and	and	CCONJ
ejpam-5984	311	11	our	our	PRON
ejpam-5984	311	12	earlier	early	ADJ
ejpam-5984	311	13	assumption	assumption	NOUN
ejpam-5984	311	14	leads	lead	VERB
ejpam-5984	311	15	to	to	ADP
ejpam-5984	311	16	∆(g	∆(g	NOUN
ejpam-5984	311	17	)	)	PUNCT
ejpam-5984	311	18	=	=	SYM
ejpam-5984	312	1	2	2	X
ejpam-5984	312	2	.	.	PUNCT
ejpam-5984	312	3	combining	combine	VERB
ejpam-5984	312	4	this	this	PRON
ejpam-5984	312	5	with	with	ADP
ejpam-5984	312	6	the	the	DET
ejpam-5984	312	7	condition	condition	NOUN
ejpam-5984	312	8	that	that	SCONJ
ejpam-5984	312	9	g	g	PROPN
ejpam-5984	312	10	is	be	AUX
ejpam-5984	312	11	connected	connect	VERB
ejpam-5984	312	12	,	,	PUNCT
ejpam-5984	312	13	we	we	PRON
ejpam-5984	312	14	have	have	VERB
ejpam-5984	312	15	that	that	PRON
ejpam-5984	312	16	g	g	PROPN
ejpam-5984	312	17	is	be	AUX
ejpam-5984	312	18	a	a	DET
ejpam-5984	312	19	path	path	NOUN
ejpam-5984	312	20	or	or	CCONJ
ejpam-5984	312	21	a	a	DET
ejpam-5984	312	22	cycle	cycle	NOUN
ejpam-5984	312	23	.	.	PUNCT
ejpam-5984	313	1	since	since	SCONJ
ejpam-5984	313	2	stidr(pn	stidr(pn	ADJ
ejpam-5984	313	3	)	)	PUNCT
ejpam-5984	313	4	=	=	SYM
ejpam-5984	313	5	1	1	NUM
ejpam-5984	313	6	,	,	PUNCT
ejpam-5984	313	7	it	it	PRON
ejpam-5984	313	8	follows	follow	VERB
ejpam-5984	313	9	from	from	ADP
ejpam-5984	313	10	stidr(g	stidr(g	PROPN
ejpam-5984	313	11	)	)	PUNCT
ejpam-5984	313	12	≥	≥	NOUN
ejpam-5984	313	13	2	2	NUM
ejpam-5984	313	14	that	that	PRON
ejpam-5984	313	15	g	g	PROPN
ejpam-5984	313	16	is	be	AUX
ejpam-5984	313	17	a	a	DET
ejpam-5984	313	18	cycle	cycle	NOUN
ejpam-5984	313	19	.	.	PUNCT
ejpam-5984	314	1	combining	combine	VERB
ejpam-5984	314	2	corollary	corollary	ADJ
ejpam-5984	314	3	2	2	NUM
ejpam-5984	314	4	and	and	CCONJ
ejpam-5984	314	5	the	the	DET
ejpam-5984	314	6	condition	condition	NOUN
ejpam-5984	314	7	stidr(g	stidr(g	NOUN
ejpam-5984	314	8	)	)	PUNCT
ejpam-5984	314	9	=	=	SYM
ejpam-5984	315	1	n	n	CCONJ
ejpam-5984	315	2	−	−	NOUN
ejpam-5984	315	3	3	3	NUM
ejpam-5984	315	4	,	,	PUNCT
ejpam-5984	315	5	we	we	PRON
ejpam-5984	315	6	obtain	obtain	VERB
ejpam-5984	315	7	that	that	DET
ejpam-5984	315	8	n	n	NOUN
ejpam-5984	315	9	=	=	SYM
ejpam-5984	315	10	5	5	NUM
ejpam-5984	315	11	which	which	PRON
ejpam-5984	315	12	is	be	AUX
ejpam-5984	315	13	a	a	DET
ejpam-5984	315	14	contradiction	contradiction	NOUN
ejpam-5984	315	15	.	.	PUNCT
ejpam-5984	316	1	hence	hence	ADV
ejpam-5984	316	2	,	,	PUNCT
ejpam-5984	316	3	we	we	PRON
ejpam-5984	316	4	assume	assume	VERB
ejpam-5984	316	5	that	that	SCONJ
ejpam-5984	316	6	idr(g	idr(g	PROPN
ejpam-5984	316	7	)	)	PUNCT
ejpam-5984	316	8	=	=	SYM
ejpam-5984	317	1	3	3	X
ejpam-5984	317	2	.	.	PUNCT
ejpam-5984	318	1	it	it	PRON
ejpam-5984	318	2	follows	follow	VERB
ejpam-5984	318	3	from	from	ADP
ejpam-5984	318	4	this	this	PRON
ejpam-5984	318	5	and	and	CCONJ
ejpam-5984	318	6	the	the	DET
ejpam-5984	318	7	fact	fact	NOUN
ejpam-5984	318	8	stidr(g	stidr(g	NUM
ejpam-5984	318	9	)	)	PUNCT
ejpam-5984	318	10	=	=	SYM
ejpam-5984	319	1	n	n	CCONJ
ejpam-5984	319	2	−	−	PROPN
ejpam-5984	319	3	3	3	NUM
ejpam-5984	320	1	that	that	PRON
ejpam-5984	320	2	g	g	PROPN
ejpam-5984	320	3	has	have	VERB
ejpam-5984	320	4	exactly	exactly	ADV
ejpam-5984	320	5	n	n	CCONJ
ejpam-5984	320	6	−	−	NUM
ejpam-5984	320	7	3	3	NUM
ejpam-5984	320	8	universal	universal	ADJ
ejpam-5984	320	9	vertices	vertex	NOUN
ejpam-5984	320	10	.	.	PUNCT
ejpam-5984	321	1	let	let	VERB
ejpam-5984	321	2	x	x	PRON
ejpam-5984	321	3	,	,	PUNCT
ejpam-5984	321	4	y	y	PROPN
ejpam-5984	321	5	,	,	PUNCT
ejpam-5984	321	6	z	z	X
ejpam-5984	321	7	be	be	AUX
ejpam-5984	321	8	the	the	DET
ejpam-5984	321	9	vertices	vertex	NOUN
ejpam-5984	321	10	of	of	ADP
ejpam-5984	321	11	g	g	PROPN
ejpam-5984	321	12	that	that	PRON
ejpam-5984	321	13	are	be	AUX
ejpam-5984	321	14	not	not	PART
ejpam-5984	321	15	universal	universal	ADJ
ejpam-5984	321	16	.	.	PUNCT
ejpam-5984	322	1	it	it	PRON
ejpam-5984	322	2	follows	follow	VERB
ejpam-5984	322	3	that	that	SCONJ
ejpam-5984	322	4	g[{x	g[{x	PROPN
ejpam-5984	322	5	,	,	PUNCT
ejpam-5984	322	6	y	y	PROPN
ejpam-5984	322	7	,	,	PUNCT
ejpam-5984	322	8	z	z	NOUN
ejpam-5984	322	9	}	}	PUNCT
ejpam-5984	322	10	]	]	PUNCT
ejpam-5984	322	11	=	=	SYM
ejpam-5984	322	12	3k1	3k1	NUM
ejpam-5984	322	13	or	or	CCONJ
ejpam-5984	322	14	g[{x	g[{x	PROPN
ejpam-5984	322	15	,	,	PUNCT
ejpam-5984	322	16	y	y	PROPN
ejpam-5984	322	17	,	,	PUNCT
ejpam-5984	322	18	z	z	NOUN
ejpam-5984	322	19	}	}	PUNCT
ejpam-5984	322	20	]	]	PUNCT
ejpam-5984	323	1	=	=	SYM
ejpam-5984	323	2	k2	k2	X
ejpam-5984	323	3	∪	∪	X
ejpam-5984	323	4	k1	k1	PROPN
ejpam-5984	323	5	.	.	PUNCT
ejpam-5984	324	1	thus	thus	ADV
ejpam-5984	324	2	,	,	PUNCT
ejpam-5984	324	3	g	g	PROPN
ejpam-5984	324	4	=	=	SYM
ejpam-5984	324	5	3k1	3k1	NUM
ejpam-5984	324	6	∨	∨	NUM
ejpam-5984	324	7	kn−3	kn−3	PROPN
ejpam-5984	324	8	or	or	CCONJ
ejpam-5984	324	9	g	g	NOUN
ejpam-5984	324	10	=	=	PROPN
ejpam-5984	324	11	kn−3	kn−3	PROPN
ejpam-5984	324	12	∨	∨	PROPN
ejpam-5984	324	13	(	(	PUNCT
ejpam-5984	324	14	k2	k2	PROPN
ejpam-5984	324	15	∪k1	∪k1	ADJ
ejpam-5984	324	16	)	)	PUNCT
ejpam-5984	324	17	.	.	PUNCT
ejpam-5984	325	1	this	this	PRON
ejpam-5984	325	2	completes	complete	VERB
ejpam-5984	325	3	the	the	DET
ejpam-5984	325	4	proof	proof	NOUN
ejpam-5984	325	5	.	.	PUNCT
ejpam-5984	326	1	proposition	proposition	NOUN
ejpam-5984	326	2	16	16	NUM
ejpam-5984	326	3	.	.	PUNCT
ejpam-5984	327	1	let	let	VERB
ejpam-5984	327	2	g	g	PRON
ejpam-5984	327	3	be	be	AUX
ejpam-5984	327	4	a	a	DET
ejpam-5984	327	5	connected	connected	ADJ
ejpam-5984	327	6	graph	graph	NOUN
ejpam-5984	327	7	of	of	ADP
ejpam-5984	327	8	order	order	NOUN
ejpam-5984	327	9	n	n	PRON
ejpam-5984	327	10	≥	≥	NUM
ejpam-5984	327	11	6	6	NUM
ejpam-5984	327	12	.	.	PUNCT
ejpam-5984	328	1	then	then	ADV
ejpam-5984	328	2	stidr(g	stidr(g	NUM
ejpam-5984	328	3	)	)	PUNCT
ejpam-5984	329	1	=	=	SYM
ejpam-5984	329	2	n−4	n−4	PROPN
ejpam-5984	329	3	if	if	SCONJ
ejpam-5984	329	4	and	and	CCONJ
ejpam-5984	329	5	only	only	ADV
ejpam-5984	329	6	if	if	SCONJ
ejpam-5984	329	7	g	g	PROPN
ejpam-5984	329	8	∈	∈	PROPN
ejpam-5984	329	9	{	{	PUNCT
ejpam-5984	329	10	p6	p6	PROPN
ejpam-5984	329	11	,	,	PUNCT
ejpam-5984	329	12	c6,kn−4∨h	c6,kn−4∨h	NOUN
ejpam-5984	329	13	}	}	PUNCT
ejpam-5984	329	14	where	where	SCONJ
ejpam-5984	329	15	h	h	NOUN
ejpam-5984	329	16	∈	∈	PROPN
ejpam-5984	329	17	{	{	PUNCT
ejpam-5984	329	18	p4	p4	ADJ
ejpam-5984	329	19	,	,	PUNCT
ejpam-5984	329	20	c4	c4	NOUN
ejpam-5984	329	21	,	,	PUNCT
ejpam-5984	329	22	2k2	2k2	NUM
ejpam-5984	329	23	,	,	PUNCT
ejpam-5984	329	24	4k1,k2∪2k1	4k1,k2∪2k1	NOUN
ejpam-5984	329	25	,	,	PUNCT
ejpam-5984	329	26	p3∪k1,k3∪k1	p3∪k1,k3∪k1	NOUN
ejpam-5984	329	27	}	}	PUNCT
ejpam-5984	329	28	.	.	PUNCT
ejpam-5984	330	1	proof	proof	NOUN
ejpam-5984	330	2	.	.	PUNCT
ejpam-5984	331	1	the	the	DET
ejpam-5984	331	2	sufficiency	sufficiency	NOUN
ejpam-5984	331	3	is	be	AUX
ejpam-5984	331	4	straightforward	straightforward	ADJ
ejpam-5984	331	5	to	to	PART
ejpam-5984	331	6	check	check	VERB
ejpam-5984	331	7	.	.	PUNCT
ejpam-5984	332	1	to	to	PART
ejpam-5984	332	2	prove	prove	VERB
ejpam-5984	332	3	the	the	DET
ejpam-5984	332	4	necessity	necessity	NOUN
ejpam-5984	332	5	,	,	PUNCT
ejpam-5984	332	6	let	let	VERB
ejpam-5984	332	7	g	g	PRON
ejpam-5984	332	8	be	be	AUX
ejpam-5984	332	9	a	a	DET
ejpam-5984	332	10	connected	connected	ADJ
ejpam-5984	332	11	graph	graph	NOUN
ejpam-5984	332	12	of	of	ADP
ejpam-5984	332	13	order	order	NOUN
ejpam-5984	332	14	n	n	PRON
ejpam-5984	332	15	≥	≥	NUM
ejpam-5984	332	16	6	6	NUM
ejpam-5984	332	17	with	with	ADP
ejpam-5984	332	18	stidr(g	stidr(g	NOUN
ejpam-5984	332	19	)	)	PUNCT
ejpam-5984	333	1	=	=	SYM
ejpam-5984	333	2	n−4	n−4	PROPN
ejpam-5984	333	3	.	.	PUNCT
ejpam-5984	333	4	clearly	clearly	ADV
ejpam-5984	333	5	∆(g	∆(g	PROPN
ejpam-5984	333	6	)	)	PUNCT
ejpam-5984	333	7	≥	≥	NOUN
ejpam-5984	333	8	2	2	NUM
ejpam-5984	333	9	and	and	CCONJ
ejpam-5984	333	10	stidr(g	stidr(g	NUM
ejpam-5984	333	11	)	)	PUNCT
ejpam-5984	333	12	≥	≥	NOUN
ejpam-5984	333	13	2	2	NUM
ejpam-5984	333	14	.	.	PUNCT
ejpam-5984	333	15	first	first	ADV
ejpam-5984	333	16	let	let	VERB
ejpam-5984	333	17	idr(g	idr(g	PROPN
ejpam-5984	333	18	)	)	PUNCT
ejpam-5984	333	19	≥	≥	NOUN
ejpam-5984	333	20	4	4	NUM
ejpam-5984	333	21	.	.	PUNCT
ejpam-5984	334	1	it	it	PRON
ejpam-5984	334	2	follows	follow	VERB
ejpam-5984	334	3	from	from	ADP
ejpam-5984	334	4	proposition	proposition	NOUN
ejpam-5984	334	5	11	11	NUM
ejpam-5984	335	1	that	that	PRON
ejpam-5984	335	2	n−	n−	NOUN
ejpam-5984	335	3	4	4	NUM
ejpam-5984	335	4	=	=	SYM
ejpam-5984	335	5	stidr(g	stidr(g	NUM
ejpam-5984	335	6	)	)	PUNCT
ejpam-5984	335	7	≤	≤	NUM
ejpam-5984	336	1	δ	δ	PROPN
ejpam-5984	337	1	+	+	CCONJ
ejpam-5984	338	1	1	1	NUM
ejpam-5984	339	1	and	and	CCONJ
ejpam-5984	339	2	so	so	ADV
ejpam-5984	339	3	δ	δ	PROPN
ejpam-5984	339	4	≥	≥	NUM
ejpam-5984	339	5	n−	n−	NOUN
ejpam-5984	339	6	5	5	NUM
ejpam-5984	339	7	.	.	PUNCT
ejpam-5984	339	8	combining	combine	VERB
ejpam-5984	339	9	this	this	PRON
ejpam-5984	339	10	with	with	ADP
ejpam-5984	339	11	proposition	proposition	NOUN
ejpam-5984	339	12	12	12	NUM
ejpam-5984	339	13	,	,	PUNCT
ejpam-5984	339	14	we	we	PRON
ejpam-5984	339	15	obtain	obtain	VERB
ejpam-5984	339	16	n−	n−	NOUN
ejpam-5984	339	17	5	5	NUM
ejpam-5984	339	18	≤	≤	NUM
ejpam-5984	339	19	δ	δ	PROPN
ejpam-5984	339	20	≤	≤	PROPN
ejpam-5984	339	21	∆	∆	X
ejpam-5984	339	22	≤	≤	ADV
ejpam-5984	339	23	3	3	NUM
ejpam-5984	339	24	.	.	PUNCT
ejpam-5984	340	1	(	(	PUNCT
ejpam-5984	340	2	2	2	X
ejpam-5984	340	3	)	)	PUNCT
ejpam-5984	340	4	if	if	SCONJ
ejpam-5984	340	5	∆(g	∆(g	NOUN
ejpam-5984	340	6	)	)	PUNCT
ejpam-5984	340	7	=	=	SYM
ejpam-5984	340	8	2	2	NUM
ejpam-5984	340	9	,	,	PUNCT
ejpam-5984	340	10	then	then	ADV
ejpam-5984	340	11	n	n	CCONJ
ejpam-5984	340	12	∈	∈	PROPN
ejpam-5984	340	13	{	{	PUNCT
ejpam-5984	340	14	6	6	NUM
ejpam-5984	340	15	,	,	PUNCT
ejpam-5984	340	16	7	7	NUM
ejpam-5984	340	17	}	}	PUNCT
ejpam-5984	340	18	and	and	CCONJ
ejpam-5984	340	19	g	g	PROPN
ejpam-5984	340	20	is	be	AUX
ejpam-5984	340	21	a	a	DET
ejpam-5984	340	22	path	path	NOUN
ejpam-5984	340	23	or	or	CCONJ
ejpam-5984	340	24	a	a	DET
ejpam-5984	340	25	cycle	cycle	NOUN
ejpam-5984	340	26	of	of	ADP
ejpam-5984	340	27	order	order	NOUN
ejpam-5984	340	28	n.	n.	NOUN
ejpam-5984	340	29	it	it	PRON
ejpam-5984	340	30	follows	follow	VERB
ejpam-5984	340	31	from	from	ADP
ejpam-5984	340	32	corollaries	corollary	NOUN
ejpam-5984	340	33	1	1	NUM
ejpam-5984	340	34	and	and	CCONJ
ejpam-5984	340	35	2	2	NUM
ejpam-5984	340	36	that	that	PRON
ejpam-5984	340	37	g	g	PROPN
ejpam-5984	340	38	∈	∈	PROPN
ejpam-5984	340	39	{	{	PUNCT
ejpam-5984	340	40	p6	p6	PROPN
ejpam-5984	340	41	,	,	PUNCT
ejpam-5984	340	42	c6	c6	PROPN
ejpam-5984	340	43	}	}	PUNCT
ejpam-5984	340	44	.	.	PUNCT
ejpam-5984	341	1	henceforth	henceforth	ADV
ejpam-5984	341	2	we	we	PRON
ejpam-5984	341	3	assume	assume	VERB
ejpam-5984	341	4	that	that	SCONJ
ejpam-5984	341	5	∆(g	∆(g	NOUN
ejpam-5984	341	6	)	)	PUNCT
ejpam-5984	341	7	=	=	SYM
ejpam-5984	342	1	3	3	X
ejpam-5984	342	2	.	.	PUNCT
ejpam-5984	342	3	by	by	ADP
ejpam-5984	342	4	(	(	PUNCT
ejpam-5984	342	5	2	2	X
ejpam-5984	342	6	)	)	PUNCT
ejpam-5984	342	7	we	we	PRON
ejpam-5984	342	8	obtain	obtain	VERB
ejpam-5984	342	9	n	n	PRON
ejpam-5984	342	10	∈	∈	NOUN
ejpam-5984	342	11	{	{	PUNCT
ejpam-5984	342	12	6	6	NUM
ejpam-5984	342	13	,	,	PUNCT
ejpam-5984	342	14	7	7	NUM
ejpam-5984	342	15	,	,	PUNCT
ejpam-5984	342	16	8	8	NUM
ejpam-5984	342	17	}	}	PUNCT
ejpam-5984	342	18	.	.	PUNCT
ejpam-5984	343	1	let	let	VERB
ejpam-5984	343	2	v	v	NUM
ejpam-5984	343	3	∈	∈	PROPN
ejpam-5984	343	4	v	v	NOUN
ejpam-5984	343	5	(	(	PUNCT
ejpam-5984	343	6	g	g	NOUN
ejpam-5984	343	7	)	)	PUNCT
ejpam-5984	343	8	be	be	AUX
ejpam-5984	343	9	a	a	DET
ejpam-5984	343	10	vertex	vertex	NOUN
ejpam-5984	343	11	with	with	ADP
ejpam-5984	343	12	maximum	maximum	ADJ
ejpam-5984	343	13	degree	degree	NOUN
ejpam-5984	343	14	3	3	NUM
ejpam-5984	343	15	with	with	ADP
ejpam-5984	343	16	n(v	n(v	PROPN
ejpam-5984	343	17	)	)	PUNCT
ejpam-5984	343	18	=	=	PRON
ejpam-5984	343	19	{	{	PUNCT
ejpam-5984	343	20	v1	v1	PROPN
ejpam-5984	343	21	,	,	PUNCT
ejpam-5984	343	22	v2	v2	PROPN
ejpam-5984	343	23	,	,	PUNCT
ejpam-5984	343	24	v3	v3	PROPN
ejpam-5984	343	25	}	}	PUNCT
ejpam-5984	343	26	and	and	CCONJ
ejpam-5984	343	27	let	let	VERB
ejpam-5984	343	28	f	f	PROPN
ejpam-5984	343	29	=	=	SYM
ejpam-5984	343	30	(	(	PUNCT
ejpam-5984	343	31	v0,∅	v0,∅	PROPN
ejpam-5984	343	32	,	,	PUNCT
ejpam-5984	343	33	v2	v2	PROPN
ejpam-5984	343	34	,	,	PUNCT
ejpam-5984	343	35	v3	v3	PROPN
ejpam-5984	343	36	)	)	PUNCT
ejpam-5984	343	37	be	be	VERB
ejpam-5984	343	38	an	an	DET
ejpam-5984	343	39	idr(g)-function	idr(g)-function	NOUN
ejpam-5984	343	40	.	.	PUNCT
ejpam-5984	344	1	since	since	SCONJ
ejpam-5984	344	2	g	g	PROPN
ejpam-5984	344	3	is	be	AUX
ejpam-5984	344	4	a	a	DET
ejpam-5984	344	5	connected	connected	ADJ
ejpam-5984	344	6	graph	graph	NOUN
ejpam-5984	344	7	,	,	PUNCT
ejpam-5984	344	8	we	we	PRON
ejpam-5984	344	9	assume	assume	VERB
ejpam-5984	344	10	,	,	PUNCT
ejpam-5984	344	11	without	without	ADP
ejpam-5984	344	12	loss	loss	NOUN
ejpam-5984	344	13	of	of	ADP
ejpam-5984	344	14	generality	generality	NOUN
ejpam-5984	344	15	that	that	PRON
ejpam-5984	344	16	u	u	PRON
ejpam-5984	344	17	∈	∈	NOUN
ejpam-5984	345	1	n(v1)−n	n(v1)−n	PROPN
ejpam-5984	346	1	[	[	X
ejpam-5984	346	2	v	v	X
ejpam-5984	346	3	]	]	X
ejpam-5984	346	4	.	.	PUNCT
ejpam-5984	347	1	if	if	SCONJ
ejpam-5984	347	2	idr(g	idr(g	PROPN
ejpam-5984	347	3	)	)	PUNCT
ejpam-5984	347	4	≥	≥	NOUN
ejpam-5984	347	5	6	6	NUM
ejpam-5984	347	6	,	,	PUNCT
ejpam-5984	347	7	then	then	ADV
ejpam-5984	347	8	the	the	DET
ejpam-5984	347	9	function	function	NOUN
ejpam-5984	347	10	g	g	PROPN
ejpam-5984	347	11	defined	define	VERB
ejpam-5984	347	12	on	on	ADP
ejpam-5984	347	13	g[n	g[n	PRON
ejpam-5984	348	1	[	[	X
ejpam-5984	348	2	v	v	X
ejpam-5984	348	3	]	]	X
ejpam-5984	348	4	∪	∪	X
ejpam-5984	348	5	{	{	PUNCT
ejpam-5984	348	6	u	u	NOUN
ejpam-5984	348	7	}	}	PUNCT
ejpam-5984	348	8	]	]	PUNCT
ejpam-5984	348	9	with	with	ADP
ejpam-5984	348	10	f(v	f(v	NOUN
ejpam-5984	348	11	)	)	PUNCT
ejpam-5984	348	12	=	=	SYM
ejpam-5984	349	1	3	3	NUM
ejpam-5984	349	2	,	,	PUNCT
ejpam-5984	349	3	f(u	f(u	PROPN
ejpam-5984	349	4	)	)	PUNCT
ejpam-5984	349	5	=	=	SYM
ejpam-5984	349	6	2	2	NUM
ejpam-5984	349	7	,	,	PUNCT
ejpam-5984	349	8	f(v1	f(v1	ADJ
ejpam-5984	349	9	)	)	PUNCT
ejpam-5984	349	10	=	=	SYM
ejpam-5984	349	11	f(v2	f(v2	NOUN
ejpam-5984	349	12	)	)	PUNCT
ejpam-5984	349	13	=	=	SYM
ejpam-5984	349	14	f(v3	f(v3	X
ejpam-5984	349	15	)	)	PUNCT
ejpam-5984	349	16	=	=	SYM
ejpam-5984	350	1	0	0	NUM
ejpam-5984	350	2	is	be	AUX
ejpam-5984	350	3	an	an	DET
ejpam-5984	350	4	idrdf	idrdf	NOUN
ejpam-5984	350	5	of	of	ADP
ejpam-5984	350	6	weight	weight	NOUN
ejpam-5984	350	7	less	less	ADV
ejpam-5984	350	8	that	that	SCONJ
ejpam-5984	350	9	ω(f	ω(f	ADJ
ejpam-5984	350	10	)	)	PUNCT
ejpam-5984	350	11	and	and	CCONJ
ejpam-5984	351	1	so	so	ADV
ejpam-5984	351	2	n	n	CCONJ
ejpam-5984	351	3	−	−	PROPN
ejpam-5984	351	4	4	4	NUM
ejpam-5984	351	5	=	=	SYM
ejpam-5984	351	6	stidr(g	stidr(g	PROPN
ejpam-5984	351	7	)	)	PUNCT
ejpam-5984	351	8	≤	≤	NOUN
ejpam-5984	351	9	n	n	CCONJ
ejpam-5984	351	10	−	−	PROPN
ejpam-5984	351	11	∆(g	∆(g	NOUN
ejpam-5984	351	12	)	)	PUNCT
ejpam-5984	351	13	−	−	PROPN
ejpam-5984	351	14	2	2	NUM
ejpam-5984	351	15	which	which	PRON
ejpam-5984	351	16	leads	lead	VERB
ejpam-5984	351	17	to	to	ADP
ejpam-5984	351	18	the	the	DET
ejpam-5984	351	19	contradiction	contradiction	NOUN
ejpam-5984	351	20	∆(g	∆(g	NOUN
ejpam-5984	351	21	)	)	PUNCT
ejpam-5984	351	22	≤	≤	NOUN
ejpam-5984	351	23	2	2	NUM
ejpam-5984	351	24	.	.	PUNCT
ejpam-5984	352	1	thus	thus	ADV
ejpam-5984	352	2	,	,	PUNCT
ejpam-5984	352	3	we	we	PRON
ejpam-5984	352	4	have	have	VERB
ejpam-5984	352	5	idr(g	idr(g	PROPN
ejpam-5984	352	6	)	)	PUNCT
ejpam-5984	352	7	∈	∈	NOUN
ejpam-5984	352	8	{	{	PUNCT
ejpam-5984	352	9	4	4	NUM
ejpam-5984	352	10	,	,	PUNCT
ejpam-5984	352	11	5	5	NUM
ejpam-5984	352	12	}	}	PUNCT
ejpam-5984	352	13	.	.	PUNCT
ejpam-5984	353	1	then	then	ADV
ejpam-5984	353	2	either	either	CCONJ
ejpam-5984	353	3	|v2|	|v2|	ADV
ejpam-5984	353	4	=	=	SYM
ejpam-5984	353	5	2	2	NUM
ejpam-5984	353	6	or	or	CCONJ
ejpam-5984	353	7	|v2|	|v2|	NOUN
ejpam-5984	353	8	=	=	SYM
ejpam-5984	353	9	|v3|	|v3|	NOUN
ejpam-5984	353	10	=	=	SYM
ejpam-5984	353	11	1	1	X
ejpam-5984	353	12	.	.	PUNCT
ejpam-5984	353	13	since	since	SCONJ
ejpam-5984	353	14	each	each	DET
ejpam-5984	353	15	vertex	vertex	NOUN
ejpam-5984	353	16	in	in	ADP
ejpam-5984	353	17	v0	v0	NOUN
ejpam-5984	353	18	must	must	AUX
ejpam-5984	353	19	be	be	AUX
ejpam-5984	353	20	adjacent	adjacent	ADJ
ejpam-5984	353	21	to	to	ADP
ejpam-5984	353	22	a	a	DET
ejpam-5984	353	23	vertex	vertex	NOUN
ejpam-5984	353	24	with	with	ADP
ejpam-5984	353	25	wight	wight	NOUN
ejpam-5984	353	26	3	3	NUM
ejpam-5984	353	27	or	or	CCONJ
ejpam-5984	353	28	two	two	NUM
ejpam-5984	353	29	vertices	vertex	NOUN
ejpam-5984	353	30	with	with	ADP
ejpam-5984	353	31	weight	weight	NOUN
ejpam-5984	353	32	2	2	NUM
ejpam-5984	353	33	,	,	PUNCT
ejpam-5984	353	34	we	we	PRON
ejpam-5984	353	35	have	have	VERB
ejpam-5984	353	36	certainly	certainly	ADV
ejpam-5984	353	37	∆(g	∆(g	NOUN
ejpam-5984	353	38	)	)	PUNCT
ejpam-5984	353	39	≥	≥	NOUN
ejpam-5984	353	40	4	4	NUM
ejpam-5984	353	41	which	which	PRON
ejpam-5984	353	42	is	be	AUX
ejpam-5984	353	43	a	a	DET
ejpam-5984	353	44	contradiction	contradiction	NOUN
ejpam-5984	353	45	.	.	PUNCT
ejpam-5984	354	1	assume	assume	VERB
ejpam-5984	354	2	now	now	ADV
ejpam-5984	354	3	that	that	SCONJ
ejpam-5984	354	4	idr(g	idr(g	PROPN
ejpam-5984	354	5	)	)	PUNCT
ejpam-5984	354	6	=	=	SYM
ejpam-5984	355	1	3	3	X
ejpam-5984	355	2	.	.	PUNCT
ejpam-5984	356	1	it	it	PRON
ejpam-5984	356	2	follows	follow	VERB
ejpam-5984	356	3	from	from	ADP
ejpam-5984	356	4	this	this	PRON
ejpam-5984	356	5	and	and	CCONJ
ejpam-5984	356	6	the	the	DET
ejpam-5984	356	7	fact	fact	NOUN
ejpam-5984	356	8	stidr(g	stidr(g	NUM
ejpam-5984	356	9	)	)	PUNCT
ejpam-5984	356	10	=	=	X
ejpam-5984	357	1	n−4	n−4	PROPN
ejpam-5984	357	2	that	that	SCONJ
ejpam-5984	357	3	g	g	PROPN
ejpam-5984	357	4	has	have	VERB
ejpam-5984	357	5	exactly	exactly	ADV
ejpam-5984	357	6	n−4	n−4	PRON
ejpam-5984	357	7	universal	universal	ADJ
ejpam-5984	357	8	vertices	vertex	NOUN
ejpam-5984	357	9	.	.	PUNCT
ejpam-5984	358	1	let	let	VERB
ejpam-5984	358	2	x	x	PRON
ejpam-5984	358	3	,	,	PUNCT
ejpam-5984	358	4	y	y	PROPN
ejpam-5984	358	5	,	,	PUNCT
ejpam-5984	358	6	z	z	PROPN
ejpam-5984	358	7	,	,	PUNCT
ejpam-5984	358	8	w	w	PROPN
ejpam-5984	358	9	be	be	AUX
ejpam-5984	358	10	the	the	DET
ejpam-5984	358	11	vertices	vertex	NOUN
ejpam-5984	358	12	of	of	ADP
ejpam-5984	358	13	g	g	PROPN
ejpam-5984	358	14	that	that	PRON
ejpam-5984	358	15	are	be	AUX
ejpam-5984	358	16	not	not	PART
ejpam-5984	358	17	universal	universal	ADJ
ejpam-5984	358	18	vertex	vertex	NOUN
ejpam-5984	358	19	,	,	PUNCT
ejpam-5984	358	20	that	that	PRON
ejpam-5984	358	21	is	be	AUX
ejpam-5984	358	22	∆(g[{x	∆(g[{x	PROPN
ejpam-5984	358	23	,	,	PUNCT
ejpam-5984	358	24	y	y	PROPN
ejpam-5984	358	25	,	,	PUNCT
ejpam-5984	358	26	z	z	PROPN
ejpam-5984	358	27	,	,	PUNCT
ejpam-5984	358	28	w	w	NOUN
ejpam-5984	358	29	}	}	PUNCT
ejpam-5984	358	30	]	]	PUNCT
ejpam-5984	358	31	)	)	PUNCT
ejpam-5984	358	32	≤	≤	NUM
ejpam-5984	358	33	2	2	NUM
ejpam-5984	358	34	.	.	PUNCT
ejpam-5984	359	1	there	there	PRON
ejpam-5984	359	2	are	be	VERB
ejpam-5984	359	3	seven	seven	NUM
ejpam-5984	359	4	graphs	graph	NOUN
ejpam-5984	359	5	of	of	ADP
ejpam-5984	359	6	order	order	NOUN
ejpam-5984	359	7	4	4	NUM
ejpam-5984	359	8	with	with	ADP
ejpam-5984	359	9	maximum	maximum	ADJ
ejpam-5984	359	10	degree	degree	NOUN
ejpam-5984	359	11	at	at	ADP
ejpam-5984	359	12	most	most	ADV
ejpam-5984	359	13	two	two	NUM
ejpam-5984	359	14	,	,	PUNCT
ejpam-5984	359	15	that	that	PRON
ejpam-5984	359	16	is	be	AUX
ejpam-5984	359	17	g[{x	g[{x	PROPN
ejpam-5984	359	18	,	,	PUNCT
ejpam-5984	359	19	y	y	PROPN
ejpam-5984	359	20	,	,	PUNCT
ejpam-5984	359	21	z	z	PROPN
ejpam-5984	359	22	,	,	PUNCT
ejpam-5984	359	23	w	w	NOUN
ejpam-5984	359	24	}	}	PUNCT
ejpam-5984	359	25	]	]	X
ejpam-5984	359	26	∈	∈	PROPN
ejpam-5984	359	27	{	{	PUNCT
ejpam-5984	359	28	p4	p4	ADJ
ejpam-5984	359	29	,	,	PUNCT
ejpam-5984	359	30	c4	c4	NOUN
ejpam-5984	359	31	,	,	PUNCT
ejpam-5984	359	32	2k2	2k2	NUM
ejpam-5984	359	33	,	,	PUNCT
ejpam-5984	359	34	4k1,k2∪2k1	4k1,k2∪2k1	NOUN
ejpam-5984	359	35	,	,	PUNCT
ejpam-5984	359	36	p3∪k1,k3∪k1	p3∪k1,k3∪k1	NOUN
ejpam-5984	359	37	}	}	PUNCT
ejpam-5984	359	38	.	.	PUNCT
ejpam-5984	360	1	thus	thus	ADV
ejpam-5984	360	2	g	g	PROPN
ejpam-5984	360	3	=	=	PROPN
ejpam-5984	360	4	kn−4	kn−4	PROPN
ejpam-5984	360	5	∨h	∨h	NOUN
ejpam-5984	360	6	where	where	SCONJ
ejpam-5984	360	7	h	h	NOUN
ejpam-5984	360	8	∈	∈	PROPN
ejpam-5984	360	9	{	{	PUNCT
ejpam-5984	360	10	p4	p4	ADJ
ejpam-5984	360	11	,	,	PUNCT
ejpam-5984	360	12	c4	c4	NOUN
ejpam-5984	360	13	,	,	PUNCT
ejpam-5984	360	14	2k2	2k2	NUM
ejpam-5984	360	15	,	,	PUNCT
ejpam-5984	360	16	4k1,k2	4k1,k2	NUM
ejpam-5984	360	17	∪	∪	ADJ
ejpam-5984	360	18	2k1	2k1	NUM
ejpam-5984	360	19	,	,	PUNCT
ejpam-5984	360	20	p3	p3	PROPN
ejpam-5984	360	21	∪k1,k3	∪k1,k3	PROPN
ejpam-5984	360	22	∪k1	∪k1	ADP
ejpam-5984	360	23	}	}	PUNCT
ejpam-5984	360	24	and	and	CCONJ
ejpam-5984	360	25	the	the	DET
ejpam-5984	360	26	proof	proof	NOUN
ejpam-5984	360	27	is	be	AUX
ejpam-5984	360	28	complete	complete	ADJ
ejpam-5984	360	29	.	.	PUNCT
ejpam-5984	361	1	at	at	ADP
ejpam-5984	361	2	the	the	DET
ejpam-5984	361	3	end	end	NOUN
ejpam-5984	361	4	of	of	ADP
ejpam-5984	361	5	this	this	DET
ejpam-5984	361	6	section	section	NOUN
ejpam-5984	361	7	,	,	PUNCT
ejpam-5984	361	8	we	we	PRON
ejpam-5984	361	9	present	present	VERB
ejpam-5984	361	10	a	a	DET
ejpam-5984	361	11	nordhaus	nordhaus	NOUN
ejpam-5984	361	12	-	-	PUNCT
ejpam-5984	361	13	gaddum	gaddum	NOUN
ejpam-5984	361	14	type	type	NOUN
ejpam-5984	361	15	inequality	inequality	NOUN
ejpam-5984	361	16	for	for	ADP
ejpam-5984	361	17	the	the	DET
ejpam-5984	361	18	sum	sum	NOUN
ejpam-5984	361	19	of	of	ADP
ejpam-5984	361	20	the	the	DET
ejpam-5984	361	21	independent	independent	ADJ
ejpam-5984	361	22	double	double	ADJ
ejpam-5984	361	23	roman	roman	ADJ
ejpam-5984	361	24	domination	domination	NOUN
ejpam-5984	361	25	stability	stability	NOUN
ejpam-5984	361	26	of	of	ADP
ejpam-5984	361	27	a	a	DET
ejpam-5984	361	28	graph	graph	NOUN
ejpam-5984	361	29	g	g	NOUN
ejpam-5984	361	30	and	and	CCONJ
ejpam-5984	361	31	its	its	PRON
ejpam-5984	361	32	complement	complement	NOUN
ejpam-5984	361	33	g.	g.	PROPN
ejpam-5984	361	34	s.	s.	PROPN
ejpam-5984	361	35	m.	m.	PROPN
ejpam-5984	361	36	sheikholeslami	sheikholeslami	PROPN
ejpam-5984	361	37	et	et	PROPN
ejpam-5984	361	38	al	al	PROPN
ejpam-5984	361	39	.	.	PUNCT
ejpam-5984	361	40	/	/	SYM
ejpam-5984	361	41	eur	eur	PROPN
ejpam-5984	361	42	.	.	PUNCT
ejpam-5984	362	1	j.	j.	PROPN
ejpam-5984	362	2	pure	pure	PROPN
ejpam-5984	362	3	appl	appl	PROPN
ejpam-5984	362	4	.	.	PROPN
ejpam-5984	362	5	math	math	PROPN
ejpam-5984	362	6	,	,	PUNCT
ejpam-5984	362	7	18	18	NUM
ejpam-5984	362	8	(	(	PUNCT
ejpam-5984	362	9	2	2	NUM
ejpam-5984	362	10	)	)	PUNCT
ejpam-5984	362	11	(	(	PUNCT
ejpam-5984	362	12	2025	2025	NUM
ejpam-5984	362	13	)	)	PUNCT
ejpam-5984	362	14	,	,	PUNCT
ejpam-5984	362	15	5984	5984	NUM
ejpam-5984	362	16	10	10	NUM
ejpam-5984	362	17	of	of	ADP
ejpam-5984	362	18	16	16	NUM
ejpam-5984	362	19	theorem	theorem	NOUN
ejpam-5984	362	20	3	3	X
ejpam-5984	362	21	.	.	PUNCT
ejpam-5984	363	1	let	let	VERB
ejpam-5984	363	2	g	g	PRON
ejpam-5984	363	3	be	be	AUX
ejpam-5984	363	4	a	a	DET
ejpam-5984	363	5	graph	graph	NOUN
ejpam-5984	363	6	of	of	ADP
ejpam-5984	363	7	order	order	NOUN
ejpam-5984	363	8	n	n	PRON
ejpam-5984	363	9	≥	≥	NOUN
ejpam-5984	363	10	2	2	NUM
ejpam-5984	363	11	.	.	PUNCT
ejpam-5984	364	1	then	then	ADV
ejpam-5984	364	2	stidr(g	stidr(g	PROPN
ejpam-5984	364	3	)	)	PUNCT
ejpam-5984	365	1	+	+	CCONJ
ejpam-5984	365	2	stidr(g	stidr(g	NOUN
ejpam-5984	365	3	)	)	PUNCT
ejpam-5984	365	4	≤	≤	NOUN
ejpam-5984	365	5	n.	n.	NOUN
ejpam-5984	365	6	proof	proof	NOUN
ejpam-5984	365	7	.	.	PUNCT
ejpam-5984	366	1	since	since	SCONJ
ejpam-5984	366	2	n	n	PROPN
ejpam-5984	366	3	≥	≥	NOUN
ejpam-5984	366	4	2	2	NUM
ejpam-5984	366	5	,	,	PUNCT
ejpam-5984	366	6	we	we	PRON
ejpam-5984	366	7	have	have	VERB
ejpam-5984	366	8	min{idr(g	min{idr(g	PROPN
ejpam-5984	366	9	)	)	PUNCT
ejpam-5984	366	10	,	,	PUNCT
ejpam-5984	366	11	idr(g	idr(g	PROPN
ejpam-5984	366	12	)	)	PUNCT
ejpam-5984	366	13	}	}	PUNCT
ejpam-5984	366	14	≥	≥	NOUN
ejpam-5984	367	1	3	3	NUM
ejpam-5984	367	2	.	.	PUNCT
ejpam-5984	368	1	if	if	SCONJ
ejpam-5984	368	2	idr(g	idr(g	PROPN
ejpam-5984	368	3	)	)	PUNCT
ejpam-5984	368	4	=	=	SYM
ejpam-5984	368	5	3	3	NUM
ejpam-5984	368	6	(	(	PUNCT
ejpam-5984	368	7	the	the	DET
ejpam-5984	368	8	case	case	NOUN
ejpam-5984	368	9	idr(g	idr(g	PROPN
ejpam-5984	368	10	)	)	PUNCT
ejpam-5984	368	11	=	=	SYM
ejpam-5984	368	12	3	3	NUM
ejpam-5984	368	13	is	be	AUX
ejpam-5984	368	14	similar	similar	ADJ
ejpam-5984	368	15	)	)	PUNCT
ejpam-5984	368	16	,	,	PUNCT
ejpam-5984	368	17	then	then	ADV
ejpam-5984	368	18	g	g	PROPN
ejpam-5984	368	19	has	have	VERB
ejpam-5984	368	20	a	a	DET
ejpam-5984	368	21	universal	universal	ADJ
ejpam-5984	368	22	vertex	vertex	NOUN
ejpam-5984	368	23	,	,	PUNCT
ejpam-5984	368	24	and	and	CCONJ
ejpam-5984	368	25	so	so	ADV
ejpam-5984	368	26	g	g	PROPN
ejpam-5984	368	27	has	have	VERB
ejpam-5984	368	28	an	an	DET
ejpam-5984	368	29	isolated	isolated	ADJ
ejpam-5984	368	30	vertex	vertex	NOUN
ejpam-5984	368	31	.	.	PUNCT
ejpam-5984	369	1	using	use	VERB
ejpam-5984	369	2	remark	remark	NOUN
ejpam-5984	369	3	2	2	NUM
ejpam-5984	369	4	and	and	CCONJ
ejpam-5984	369	5	noting	note	VERB
ejpam-5984	369	6	that	that	SCONJ
ejpam-5984	369	7	stidr(g	stidr(g	NOUN
ejpam-5984	369	8	)	)	PUNCT
ejpam-5984	370	1	=	=	SYM
ejpam-5984	370	2	1	1	NUM
ejpam-5984	370	3	,	,	PUNCT
ejpam-5984	370	4	we	we	PRON
ejpam-5984	370	5	obtain	obtain	VERB
ejpam-5984	370	6	stidr(g	stidr(g	PRON
ejpam-5984	370	7	)	)	PUNCT
ejpam-5984	371	1	+	+	CCONJ
ejpam-5984	371	2	stidr(g	stidr(g	NOUN
ejpam-5984	371	3	)	)	PUNCT
ejpam-5984	371	4	≤	≤	PROPN
ejpam-5984	371	5	n.	n.	NOUN
ejpam-5984	371	6	now	now	ADV
ejpam-5984	371	7	suppose	suppose	VERB
ejpam-5984	371	8	that	that	SCONJ
ejpam-5984	371	9	min{idr(g	min{idr(g	PROPN
ejpam-5984	371	10	)	)	PUNCT
ejpam-5984	371	11	,	,	PUNCT
ejpam-5984	371	12	idr(g	idr(g	PROPN
ejpam-5984	371	13	)	)	PUNCT
ejpam-5984	371	14	}	}	PUNCT
ejpam-5984	371	15	≥	≥	NOUN
ejpam-5984	371	16	4	4	NUM
ejpam-5984	371	17	.	.	PUNCT
ejpam-5984	371	18	since	since	SCONJ
ejpam-5984	371	19	∆(g	∆(g	PROPN
ejpam-5984	371	20	)	)	PUNCT
ejpam-5984	371	21	+	+	CCONJ
ejpam-5984	371	22	∆(g	∆(g	PROPN
ejpam-5984	371	23	)	)	PUNCT
ejpam-5984	371	24	≥	≥	NOUN
ejpam-5984	371	25	n	n	CCONJ
ejpam-5984	371	26	−	−	PROPN
ejpam-5984	371	27	1	1	NUM
ejpam-5984	371	28	,	,	PUNCT
ejpam-5984	371	29	we	we	PRON
ejpam-5984	371	30	may	may	AUX
ejpam-5984	371	31	assume	assume	VERB
ejpam-5984	371	32	,	,	PUNCT
ejpam-5984	371	33	without	without	ADP
ejpam-5984	371	34	loss	loss	NOUN
ejpam-5984	371	35	of	of	ADP
ejpam-5984	371	36	generality	generality	NOUN
ejpam-5984	371	37	that	that	PRON
ejpam-5984	371	38	∆(g	∆(g	NOUN
ejpam-5984	371	39	)	)	PUNCT
ejpam-5984	371	40	≥	≥	NOUN
ejpam-5984	371	41	(	(	PUNCT
ejpam-5984	371	42	n	n	CCONJ
ejpam-5984	371	43	−	−	PROPN
ejpam-5984	371	44	1)/2	1)/2	NUM
ejpam-5984	371	45	.	.	PUNCT
ejpam-5984	372	1	applying	apply	VERB
ejpam-5984	372	2	propositions	proposition	NOUN
ejpam-5984	372	3	11	11	NUM
ejpam-5984	372	4	and	and	CCONJ
ejpam-5984	372	5	12	12	NUM
ejpam-5984	372	6	,	,	PUNCT
ejpam-5984	372	7	we	we	PRON
ejpam-5984	372	8	obtain	obtain	VERB
ejpam-5984	372	9	stidr(g	stidr(g	PRON
ejpam-5984	372	10	)	)	PUNCT
ejpam-5984	373	1	+	+	CCONJ
ejpam-5984	373	2	stidr(g	stidr(g	NOUN
ejpam-5984	373	3	)	)	PUNCT
ejpam-5984	373	4	≤	≤	NOUN
ejpam-5984	373	5	(	(	PUNCT
ejpam-5984	373	6	n−∆(g)−	n−∆(g)−	ADJ
ejpam-5984	373	7	1	1	NUM
ejpam-5984	373	8	)	)	PUNCT
ejpam-5984	373	9	+	+	CCONJ
ejpam-5984	373	10	(	(	PUNCT
ejpam-5984	373	11	δ(g	δ(g	X
ejpam-5984	373	12	)	)	PUNCT
ejpam-5984	373	13	+	+	CCONJ
ejpam-5984	373	14	1	1	X
ejpam-5984	373	15	)	)	PUNCT
ejpam-5984	373	16	≤	≤	NOUN
ejpam-5984	373	17	(	(	PUNCT
ejpam-5984	373	18	n−∆(g)−	n−∆(g)−	ADJ
ejpam-5984	373	19	1	1	NUM
ejpam-5984	373	20	)	)	PUNCT
ejpam-5984	374	1	+	+	CCONJ
ejpam-5984	374	2	(	(	PUNCT
ejpam-5984	374	3	n−∆(g	n−∆(g	NOUN
ejpam-5984	374	4	)	)	PUNCT
ejpam-5984	374	5	)	)	PUNCT
ejpam-5984	375	1	=	=	PUNCT
ejpam-5984	376	1	2n−	2n−	NUM
ejpam-5984	376	2	2∆−	2∆−	NUM
ejpam-5984	376	3	1	1	NUM
ejpam-5984	376	4	≤	≤	NOUN
ejpam-5984	376	5	n	n	CCONJ
ejpam-5984	376	6	,	,	PUNCT
ejpam-5984	376	7	as	as	SCONJ
ejpam-5984	376	8	desired	desire	VERB
ejpam-5984	376	9	.	.	PUNCT
ejpam-5984	377	1	6	6	X
ejpam-5984	377	2	.	.	X
ejpam-5984	377	3	trees	tree	NOUN
ejpam-5984	377	4	in	in	ADP
ejpam-5984	377	5	this	this	DET
ejpam-5984	377	6	section	section	NOUN
ejpam-5984	377	7	,	,	PUNCT
ejpam-5984	377	8	we	we	PRON
ejpam-5984	377	9	determine	determine	VERB
ejpam-5984	377	10	the	the	DET
ejpam-5984	377	11	idr(t	idr(t	NOUN
ejpam-5984	377	12	)	)	PUNCT
ejpam-5984	377	13	-stability	-stability	NOUN
ejpam-5984	377	14	,	,	PUNCT
ejpam-5984	377	15	the	the	DET
ejpam-5984	377	16	idr+(t	idr+(t	PROPN
ejpam-5984	377	17	)	)	PUNCT
ejpam-5984	377	18	-stability	-stability	NOUN
ejpam-5984	377	19	and	and	CCONJ
ejpam-5984	377	20	the	the	DET
ejpam-5984	377	21	idr−(t	idr−(t	NOUN
ejpam-5984	377	22	)	)	PUNCT
ejpam-5984	377	23	stability	stability	NOUN
ejpam-5984	377	24	for	for	ADP
ejpam-5984	377	25	trees	tree	NOUN
ejpam-5984	377	26	.	.	PUNCT
ejpam-5984	378	1	from	from	ADP
ejpam-5984	378	2	proposition	proposition	NOUN
ejpam-5984	378	3	9	9	NUM
ejpam-5984	378	4	,	,	PUNCT
ejpam-5984	378	5	we	we	PRON
ejpam-5984	378	6	know	know	VERB
ejpam-5984	378	7	that	that	SCONJ
ejpam-5984	378	8	st+idr(t	st+idr(t	X
ejpam-5984	378	9	)	)	PUNCT
ejpam-5984	378	10	can	can	AUX
ejpam-5984	378	11	not	not	PART
ejpam-5984	378	12	be	be	AUX
ejpam-5984	378	13	bounded	bound	VERB
ejpam-5984	378	14	.	.	PUNCT
ejpam-5984	379	1	theorem	theorem	ADJ
ejpam-5984	379	2	4	4	NUM
ejpam-5984	379	3	.	.	PUNCT
ejpam-5984	380	1	for	for	ADP
ejpam-5984	380	2	every	every	DET
ejpam-5984	380	3	tree	tree	NOUN
ejpam-5984	380	4	t	t	NOUN
ejpam-5984	380	5	of	of	ADP
ejpam-5984	380	6	order	order	NOUN
ejpam-5984	380	7	n	n	PRON
ejpam-5984	380	8	≥	≥	NOUN
ejpam-5984	380	9	2	2	NUM
ejpam-5984	380	10	,	,	PUNCT
ejpam-5984	380	11	stidr(t	stidr(t	NOUN
ejpam-5984	380	12	)	)	PUNCT
ejpam-5984	380	13	=	=	SYM
ejpam-5984	380	14	1	1	X
ejpam-5984	380	15	.	.	PUNCT
ejpam-5984	380	16	proof	proof	NOUN
ejpam-5984	380	17	.	.	PUNCT
ejpam-5984	381	1	if	if	SCONJ
ejpam-5984	381	2	diam(t	diam(t	NOUN
ejpam-5984	381	3	)	)	PUNCT
ejpam-5984	381	4	≤	≤	NOUN
ejpam-5984	381	5	2	2	NUM
ejpam-5984	381	6	,	,	PUNCT
ejpam-5984	381	7	then	then	ADV
ejpam-5984	381	8	t	t	PROPN
ejpam-5984	381	9	is	be	AUX
ejpam-5984	381	10	a	a	DET
ejpam-5984	381	11	star	star	NOUN
ejpam-5984	381	12	k1,n−1	k1,n−1	ADJ
ejpam-5984	381	13	,	,	PUNCT
ejpam-5984	381	14	and	and	CCONJ
ejpam-5984	381	15	we	we	PRON
ejpam-5984	381	16	have	have	VERB
ejpam-5984	381	17	stidr(t	stidr(t	ADJ
ejpam-5984	381	18	)	)	PUNCT
ejpam-5984	382	1	=	=	SYM
ejpam-5984	382	2	1	1	X
ejpam-5984	382	3	.	.	PUNCT
ejpam-5984	383	1	if	if	SCONJ
ejpam-5984	383	2	diam(t	diam(t	NOUN
ejpam-5984	383	3	)	)	PUNCT
ejpam-5984	383	4	=	=	SYM
ejpam-5984	384	1	3	3	NUM
ejpam-5984	384	2	,	,	PUNCT
ejpam-5984	384	3	then	then	ADV
ejpam-5984	384	4	t	t	PROPN
ejpam-5984	384	5	is	be	AUX
ejpam-5984	384	6	a	a	DET
ejpam-5984	384	7	double	double	ADJ
ejpam-5984	384	8	star	star	NOUN
ejpam-5984	384	9	sr	sr	PROPN
ejpam-5984	384	10	,	,	PUNCT
ejpam-5984	384	11	t	t	PROPN
ejpam-5984	384	12	for	for	ADP
ejpam-5984	384	13	some	some	PRON
ejpam-5984	384	14	1	1	NUM
ejpam-5984	384	15	≤	≤	NOUN
ejpam-5984	384	16	r	r	NOUN
ejpam-5984	384	17	≤	≤	NOUN
ejpam-5984	384	18	t	t	NOUN
ejpam-5984	384	19	and	and	CCONJ
ejpam-5984	384	20	one	one	PRON
ejpam-5984	384	21	can	can	AUX
ejpam-5984	384	22	easily	easily	ADV
ejpam-5984	384	23	deduce	deduce	VERB
ejpam-5984	384	24	from	from	ADP
ejpam-5984	384	25	idr(sr	idr(sr	NOUN
ejpam-5984	384	26	,	,	PUNCT
ejpam-5984	384	27	t	t	PROPN
ejpam-5984	384	28	)	)	PUNCT
ejpam-5984	384	29	=	=	SYM
ejpam-5984	385	1	3	3	NUM
ejpam-5984	385	2	+	+	SYM
ejpam-5984	385	3	2r	2r	NUM
ejpam-5984	385	4	that	that	PRON
ejpam-5984	385	5	stidr(sr	stidr(sr	NOUN
ejpam-5984	385	6	,	,	PUNCT
ejpam-5984	385	7	t	t	PROPN
ejpam-5984	385	8	)	)	PUNCT
ejpam-5984	385	9	=	=	SYM
ejpam-5984	386	1	1	1	X
ejpam-5984	386	2	.	.	PUNCT
ejpam-5984	387	1	if	if	SCONJ
ejpam-5984	387	2	∆	∆	PROPN
ejpam-5984	387	3	=	=	SYM
ejpam-5984	387	4	2	2	NUM
ejpam-5984	387	5	,	,	PUNCT
ejpam-5984	387	6	then	then	ADV
ejpam-5984	387	7	t	t	PROPN
ejpam-5984	387	8	=	=	PUNCT
ejpam-5984	387	9	pn	pn	PROPN
ejpam-5984	387	10	and	and	CCONJ
ejpam-5984	387	11	we	we	PRON
ejpam-5984	387	12	are	be	AUX
ejpam-5984	387	13	done	do	VERB
ejpam-5984	387	14	by	by	ADP
ejpam-5984	387	15	corollary	corollary	ADJ
ejpam-5984	387	16	1	1	NUM
ejpam-5984	387	17	.	.	PUNCT
ejpam-5984	388	1	hence	hence	ADV
ejpam-5984	388	2	we	we	PRON
ejpam-5984	388	3	may	may	AUX
ejpam-5984	388	4	assume	assume	VERB
ejpam-5984	388	5	that	that	SCONJ
ejpam-5984	388	6	diam(t	diam(t	NOUN
ejpam-5984	388	7	)	)	PUNCT
ejpam-5984	388	8	≥	≥	NOUN
ejpam-5984	388	9	4	4	NUM
ejpam-5984	388	10	and	and	CCONJ
ejpam-5984	388	11	∆	∆	PROPN
ejpam-5984	388	12	≥	≥	NOUN
ejpam-5984	388	13	3	3	X
ejpam-5984	388	14	.	.	PUNCT
ejpam-5984	388	15	by	by	ADP
ejpam-5984	388	16	contradiction	contradiction	NOUN
ejpam-5984	388	17	,	,	PUNCT
ejpam-5984	388	18	we	we	PRON
ejpam-5984	388	19	assume	assume	VERB
ejpam-5984	388	20	that	that	SCONJ
ejpam-5984	388	21	there	there	PRON
ejpam-5984	388	22	exists	exist	VERB
ejpam-5984	388	23	a	a	DET
ejpam-5984	388	24	tree	tree	NOUN
ejpam-5984	388	25	t	t	NOUN
ejpam-5984	388	26	such	such	ADJ
ejpam-5984	388	27	that	that	PRON
ejpam-5984	388	28	stidr(t	stidr(t	ADJ
ejpam-5984	388	29	)	)	PUNCT
ejpam-5984	388	30	≥	≥	NOUN
ejpam-5984	388	31	2	2	NUM
ejpam-5984	388	32	.	.	X
ejpam-5984	389	1	we	we	PRON
ejpam-5984	389	2	choose	choose	VERB
ejpam-5984	389	3	such	such	DET
ejpam-5984	389	4	a	a	DET
ejpam-5984	389	5	tree	tree	NOUN
ejpam-5984	389	6	with	with	ADP
ejpam-5984	389	7	smallest	small	ADJ
ejpam-5984	389	8	order	order	NOUN
ejpam-5984	389	9	.	.	PUNCT
ejpam-5984	390	1	first	first	ADV
ejpam-5984	390	2	,	,	PUNCT
ejpam-5984	390	3	we	we	PRON
ejpam-5984	390	4	claim	claim	VERB
ejpam-5984	390	5	that	that	SCONJ
ejpam-5984	390	6	t	t	PROPN
ejpam-5984	390	7	has	have	VERB
ejpam-5984	390	8	no	no	DET
ejpam-5984	390	9	strong	strong	ADJ
ejpam-5984	390	10	support	support	NOUN
ejpam-5984	390	11	vertex	vertex	NOUN
ejpam-5984	390	12	.	.	PUNCT
ejpam-5984	391	1	let	let	VERB
ejpam-5984	391	2	t	t	PROPN
ejpam-5984	391	3	has	have	VERB
ejpam-5984	391	4	a	a	DET
ejpam-5984	391	5	strong	strong	ADJ
ejpam-5984	391	6	support	support	NOUN
ejpam-5984	391	7	vertex	vertex	NOUN
ejpam-5984	391	8	y	y	PROPN
ejpam-5984	391	9	with	with	ADP
ejpam-5984	391	10	leaf	leaf	NOUN
ejpam-5984	391	11	neighbors	neighbor	NOUN
ejpam-5984	391	12	y1	y1	NOUN
ejpam-5984	391	13	,	,	PUNCT
ejpam-5984	391	14	y2	y2	PROPN
ejpam-5984	391	15	,	,	PUNCT
ejpam-5984	391	16	.	.	PUNCT
ejpam-5984	391	17	.	.	PUNCT
ejpam-5984	392	1	.	.	PUNCT
ejpam-5984	393	1	,	,	PUNCT
ejpam-5984	393	2	yk	yk	PROPN
ejpam-5984	393	3	.	.	PUNCT
ejpam-5984	394	1	then	then	ADV
ejpam-5984	394	2	the	the	DET
ejpam-5984	394	3	vertices	vertex	NOUN
ejpam-5984	394	4	y1	y1	NOUN
ejpam-5984	394	5	,	,	PUNCT
ejpam-5984	394	6	y2	y2	INTJ
ejpam-5984	394	7	,	,	PUNCT
ejpam-5984	394	8	.	.	PUNCT
ejpam-5984	394	9	.	.	PUNCT
ejpam-5984	394	10	.	.	PUNCT
ejpam-5984	395	1	,	,	PUNCT
ejpam-5984	395	2	yk	yk	PROPN
ejpam-5984	395	3	are	be	AUX
ejpam-5984	395	4	isolated	isolate	VERB
ejpam-5984	395	5	vertices	vertex	NOUN
ejpam-5984	395	6	in	in	ADP
ejpam-5984	395	7	t	t	NOUN
ejpam-5984	395	8	′	′	NUM
ejpam-5984	395	9	=	=	SYM
ejpam-5984	395	10	t	t	PROPN
ejpam-5984	395	11	−y	−y	NOUN
ejpam-5984	395	12	and	and	CCONJ
ejpam-5984	395	13	any	any	DET
ejpam-5984	395	14	idr(t	idr(t	NOUN
ejpam-5984	395	15	′)-function	′)-function	NOUN
ejpam-5984	395	16	certainly	certainly	ADV
ejpam-5984	395	17	assigns	assign	VERB
ejpam-5984	395	18	2	2	NUM
ejpam-5984	395	19	to	to	ADP
ejpam-5984	395	20	each	each	DET
ejpam-5984	395	21	yi	yi	NOUN
ejpam-5984	395	22	.	.	PUNCT
ejpam-5984	396	1	now	now	ADV
ejpam-5984	396	2	reassigning	reassign	VERB
ejpam-5984	396	3	y1	y1	NOUN
ejpam-5984	396	4	,	,	PUNCT
ejpam-5984	396	5	y2	y2	PROPN
ejpam-5984	396	6	,	,	PUNCT
ejpam-5984	396	7	.	.	PUNCT
ejpam-5984	396	8	.	.	PUNCT
ejpam-5984	397	1	.	.	PUNCT
ejpam-5984	398	1	,	,	PUNCT
ejpam-5984	398	2	yk	yk	PROPN
ejpam-5984	398	3	the	the	DET
ejpam-5984	398	4	value	value	NOUN
ejpam-5984	398	5	0	0	PUNCT
ejpam-5984	399	1	and	and	CCONJ
ejpam-5984	399	2	y	y	PRON
ejpam-5984	399	3	the	the	DET
ejpam-5984	399	4	value	value	NOUN
ejpam-5984	399	5	3	3	NUM
ejpam-5984	399	6	provides	provide	VERB
ejpam-5984	399	7	an	an	DET
ejpam-5984	399	8	idrd	idrd	ADJ
ejpam-5984	399	9	-	-	PUNCT
ejpam-5984	399	10	function	function	NOUN
ejpam-5984	399	11	of	of	ADP
ejpam-5984	399	12	t	t	PROPN
ejpam-5984	399	13	of	of	ADP
ejpam-5984	399	14	weight	weight	NOUN
ejpam-5984	399	15	less	less	ADJ
ejpam-5984	399	16	than	than	ADP
ejpam-5984	399	17	idr(t	idr(t	NOUN
ejpam-5984	399	18	′	′	NOUN
ejpam-5984	399	19	)	)	PUNCT
ejpam-5984	399	20	and	and	CCONJ
ejpam-5984	399	21	this	this	PRON
ejpam-5984	399	22	leads	lead	VERB
ejpam-5984	399	23	to	to	ADP
ejpam-5984	399	24	a	a	DET
ejpam-5984	399	25	contradiction	contradiction	NOUN
ejpam-5984	399	26	.	.	PUNCT
ejpam-5984	400	1	henceforth	henceforth	ADV
ejpam-5984	400	2	,	,	PUNCT
ejpam-5984	400	3	we	we	PRON
ejpam-5984	400	4	may	may	AUX
ejpam-5984	400	5	assume	assume	VERB
ejpam-5984	400	6	that	that	SCONJ
ejpam-5984	400	7	t	t	PROPN
ejpam-5984	400	8	has	have	VERB
ejpam-5984	400	9	no	no	DET
ejpam-5984	400	10	strong	strong	ADJ
ejpam-5984	400	11	support	support	NOUN
ejpam-5984	400	12	vertex	vertex	NOUN
ejpam-5984	400	13	.	.	PUNCT
ejpam-5984	401	1	let	let	VERB
ejpam-5984	401	2	p	p	NOUN
ejpam-5984	401	3	=	=	PUNCT
ejpam-5984	401	4	x1x2	x1x2	PROPN
ejpam-5984	401	5	.	.	PUNCT
ejpam-5984	401	6	.	.	PUNCT
ejpam-5984	401	7	.	.	PUNCT
ejpam-5984	402	1	xt	xt	PROPN
ejpam-5984	402	2	be	be	AUX
ejpam-5984	402	3	a	a	DET
ejpam-5984	402	4	longest	long	ADJ
ejpam-5984	402	5	path	path	NOUN
ejpam-5984	402	6	in	in	ADP
ejpam-5984	402	7	t	t	PROPN
ejpam-5984	402	8	and	and	CCONJ
ejpam-5984	402	9	root	root	VERB
ejpam-5984	402	10	the	the	DET
ejpam-5984	402	11	tree	tree	NOUN
ejpam-5984	402	12	t	t	NOUN
ejpam-5984	402	13	at	at	ADP
ejpam-5984	402	14	the	the	DET
ejpam-5984	402	15	vertex	vertex	NOUN
ejpam-5984	402	16	xt	xt	PROPN
ejpam-5984	402	17	.	.	PUNCT
ejpam-5984	403	1	let	let	VERB
ejpam-5984	403	2	f	f	PROPN
ejpam-5984	403	3	=	=	SYM
ejpam-5984	403	4	(	(	PUNCT
ejpam-5984	403	5	v0,∅	v0,∅	PROPN
ejpam-5984	403	6	,	,	PUNCT
ejpam-5984	403	7	v2	v2	PROPN
ejpam-5984	403	8	,	,	PUNCT
ejpam-5984	403	9	v3	v3	PROPN
ejpam-5984	403	10	)	)	PUNCT
ejpam-5984	403	11	be	be	VERB
ejpam-5984	403	12	an	an	DET
ejpam-5984	403	13	idr(t	idr(t	NOUN
ejpam-5984	403	14	)	)	PUNCT
ejpam-5984	403	15	-function	-function	NOUN
ejpam-5984	403	16	such	such	ADJ
ejpam-5984	403	17	that	that	PRON
ejpam-5984	403	18	f(x3	f(x3	NOUN
ejpam-5984	403	19	)	)	PUNCT
ejpam-5984	403	20	is	be	AUX
ejpam-5984	403	21	maximized	maximize	VERB
ejpam-5984	403	22	.	.	PUNCT
ejpam-5984	404	1	since	since	SCONJ
ejpam-5984	404	2	t	t	PROPN
ejpam-5984	404	3	has	have	VERB
ejpam-5984	404	4	no	no	DET
ejpam-5984	404	5	strong	strong	ADJ
ejpam-5984	404	6	support	support	NOUN
ejpam-5984	404	7	vertex	vertex	NOUN
ejpam-5984	404	8	,	,	PUNCT
ejpam-5984	404	9	deg(x2	deg(x2	PROPN
ejpam-5984	404	10	)	)	PUNCT
ejpam-5984	404	11	=	=	SYM
ejpam-5984	404	12	2	2	NUM
ejpam-5984	404	13	and	and	CCONJ
ejpam-5984	404	14	each	each	DET
ejpam-5984	404	15	child	child	NOUN
ejpam-5984	404	16	of	of	ADP
ejpam-5984	404	17	x3	x3	ADJ
ejpam-5984	404	18	with	with	ADP
ejpam-5984	404	19	depth	depth	NOUN
ejpam-5984	404	20	1	1	NUM
ejpam-5984	404	21	has	have	VERB
ejpam-5984	404	22	degree	degree	NOUN
ejpam-5984	404	23	2	2	NUM
ejpam-5984	404	24	.	.	PUNCT
ejpam-5984	405	1	we	we	PRON
ejpam-5984	405	2	consider	consider	VERB
ejpam-5984	405	3	the	the	DET
ejpam-5984	405	4	cases	case	NOUN
ejpam-5984	405	5	:	:	PUNCT
ejpam-5984	405	6	case	case	NOUN
ejpam-5984	405	7	1	1	NUM
ejpam-5984	405	8	.	.	X
ejpam-5984	406	1	x3	x3	PROPN
ejpam-5984	406	2	has	have	VERB
ejpam-5984	406	3	a	a	DET
ejpam-5984	406	4	child	child	NOUN
ejpam-5984	406	5	z	z	NOUN
ejpam-5984	406	6	with	with	ADP
ejpam-5984	406	7	depth	depth	NOUN
ejpam-5984	406	8	0	0	NUM
ejpam-5984	406	9	,	,	PUNCT
ejpam-5984	406	10	that	that	PRON
ejpam-5984	406	11	is	is	ADV
ejpam-5984	406	12	z	z	NOUN
ejpam-5984	406	13	is	be	AUX
ejpam-5984	406	14	a	a	DET
ejpam-5984	406	15	leaf	leaf	NOUN
ejpam-5984	406	16	neighbor	neighbor	NOUN
ejpam-5984	406	17	of	of	ADP
ejpam-5984	406	18	x3	x3	PROPN
ejpam-5984	406	19	.	.	PUNCT
ejpam-5984	407	1	it	it	PRON
ejpam-5984	407	2	is	be	AUX
ejpam-5984	407	3	easy	easy	ADJ
ejpam-5984	407	4	to	to	PART
ejpam-5984	407	5	verify	verify	VERB
ejpam-5984	407	6	that	that	PRON
ejpam-5984	407	7	f(x1)+f(x2)+f(x3)+f(z	f(x1)+f(x2)+f(x3)+f(z	NUM
ejpam-5984	407	8	)	)	PUNCT
ejpam-5984	407	9	≥	≥	NOUN
ejpam-5984	407	10	5	5	NUM
ejpam-5984	407	11	.	.	PUNCT
ejpam-5984	408	1	we	we	PRON
ejpam-5984	408	2	have	have	VERB
ejpam-5984	408	3	two	two	NUM
ejpam-5984	408	4	situations	situation	NOUN
ejpam-5984	408	5	.	.	PUNCT
ejpam-5984	409	1	first	first	ADV
ejpam-5984	409	2	note	note	VERB
ejpam-5984	409	3	that	that	SCONJ
ejpam-5984	409	4	if	if	SCONJ
ejpam-5984	409	5	f(x4	f(x4	NOUN
ejpam-5984	409	6	)	)	PUNCT
ejpam-5984	409	7	=	=	SYM
ejpam-5984	409	8	0	0	NUM
ejpam-5984	409	9	,	,	PUNCT
ejpam-5984	409	10	then	then	ADV
ejpam-5984	409	11	we	we	PRON
ejpam-5984	409	12	may	may	AUX
ejpam-5984	409	13	assume	assume	VERB
ejpam-5984	409	14	that	that	SCONJ
ejpam-5984	409	15	f(x3	f(x3	VERB
ejpam-5984	409	16	)	)	PUNCT
ejpam-5984	409	17	=	=	SYM
ejpam-5984	409	18	3	3	NUM
ejpam-5984	409	19	,	,	PUNCT
ejpam-5984	409	20	f(x1	f(x1	NOUN
ejpam-5984	409	21	)	)	PUNCT
ejpam-5984	409	22	=	=	SYM
ejpam-5984	409	23	2	2	NUM
ejpam-5984	409	24	and	and	CCONJ
ejpam-5984	409	25	f(z	f(z	PROPN
ejpam-5984	409	26	)	)	PUNCT
ejpam-5984	409	27	=	=	SYM
ejpam-5984	409	28	f(x2	f(x2	NOUN
ejpam-5984	409	29	)	)	PUNCT
ejpam-5984	409	30	=	=	SYM
ejpam-5984	410	1	0	0	X
ejpam-5984	410	2	.	.	PUNCT
ejpam-5984	411	1	this	this	PRON
ejpam-5984	411	2	implies	imply	VERB
ejpam-5984	411	3	that	that	SCONJ
ejpam-5984	411	4	idr(t−x1	idr(t−x1	NOUN
ejpam-5984	411	5	)	)	PUNCT
ejpam-5984	411	6	<	<	X
ejpam-5984	411	7	idr(t	idr(t	PROPN
ejpam-5984	411	8	)	)	PUNCT
ejpam-5984	411	9	,	,	PUNCT
ejpam-5984	411	10	which	which	PRON
ejpam-5984	411	11	leads	lead	VERB
ejpam-5984	411	12	to	to	ADP
ejpam-5984	411	13	a	a	DET
ejpam-5984	411	14	contradiction	contradiction	NOUN
ejpam-5984	411	15	.	.	PUNCT
ejpam-5984	412	1	now	now	ADV
ejpam-5984	412	2	,	,	PUNCT
ejpam-5984	412	3	let	let	VERB
ejpam-5984	412	4	f(x4	f(x4	NOUN
ejpam-5984	412	5	)	)	PUNCT
ejpam-5984	412	6	≥	≥	NOUN
ejpam-5984	412	7	2	2	NUM
ejpam-5984	412	8	,	,	PUNCT
ejpam-5984	412	9	then	then	ADV
ejpam-5984	412	10	we	we	PRON
ejpam-5984	412	11	may	may	AUX
ejpam-5984	412	12	assume	assume	VERB
ejpam-5984	412	13	that	that	SCONJ
ejpam-5984	412	14	f(x2	f(x2	NOUN
ejpam-5984	412	15	)	)	PUNCT
ejpam-5984	412	16	=	=	SYM
ejpam-5984	412	17	3	3	NUM
ejpam-5984	412	18	,	,	PUNCT
ejpam-5984	412	19	f(z	f(z	NUM
ejpam-5984	412	20	)	)	PUNCT
ejpam-5984	412	21	=	=	SYM
ejpam-5984	412	22	2	2	NUM
ejpam-5984	412	23	and	and	CCONJ
ejpam-5984	412	24	f(x3	f(x3	NUM
ejpam-5984	412	25	)	)	PUNCT
ejpam-5984	413	1	=	=	SYM
ejpam-5984	413	2	f(x1	f(x1	X
ejpam-5984	413	3	)	)	PUNCT
ejpam-5984	413	4	=	=	SYM
ejpam-5984	414	1	0	0	X
ejpam-5984	414	2	.	.	PUNCT
ejpam-5984	415	1	this	this	PRON
ejpam-5984	415	2	implies	imply	VERB
ejpam-5984	415	3	that	that	SCONJ
ejpam-5984	415	4	idr(t	idr(t	NOUN
ejpam-5984	415	5	−	−	PROPN
ejpam-5984	415	6	z	z	NOUN
ejpam-5984	415	7	)	)	PUNCT
ejpam-5984	415	8	<	<	X
ejpam-5984	415	9	idr(t	idr(t	PROPN
ejpam-5984	415	10	)	)	PUNCT
ejpam-5984	415	11	,	,	PUNCT
ejpam-5984	415	12	which	which	PRON
ejpam-5984	415	13	leads	lead	VERB
ejpam-5984	415	14	to	to	ADP
ejpam-5984	415	15	a	a	DET
ejpam-5984	415	16	contradiction	contradiction	NOUN
ejpam-5984	415	17	.	.	PUNCT
ejpam-5984	416	1	case	case	NOUN
ejpam-5984	416	2	2	2	X
ejpam-5984	416	3	.	.	X
ejpam-5984	416	4	x3	x3	PROPN
ejpam-5984	416	5	has	have	VERB
ejpam-5984	416	6	a	a	DET
ejpam-5984	416	7	child	child	NOUN
ejpam-5984	416	8	u2	u2	PROPN
ejpam-5984	416	9	̸=	̸=	PROPN
ejpam-5984	416	10	x2	x2	PROPN
ejpam-5984	416	11	with	with	ADP
ejpam-5984	416	12	depth	depth	NOUN
ejpam-5984	416	13	1	1	NUM
ejpam-5984	416	14	.	.	PUNCT
ejpam-5984	417	1	let	let	VERB
ejpam-5984	417	2	u1	u1	NOUN
ejpam-5984	417	3	be	be	AUX
ejpam-5984	417	4	the	the	DET
ejpam-5984	417	5	leaf	leaf	NOUN
ejpam-5984	417	6	neighbor	neighbor	NOUN
ejpam-5984	417	7	of	of	ADP
ejpam-5984	417	8	u2	u2	PROPN
ejpam-5984	417	9	.	.	PUNCT
ejpam-5984	418	1	it	it	PRON
ejpam-5984	418	2	is	be	AUX
ejpam-5984	418	3	easy	easy	ADJ
ejpam-5984	418	4	to	to	PART
ejpam-5984	418	5	verify	verify	VERB
ejpam-5984	418	6	that	that	PRON
ejpam-5984	418	7	f(x1	f(x1	NOUN
ejpam-5984	418	8	)	)	PUNCT
ejpam-5984	418	9	+	+	NUM
ejpam-5984	418	10	f(x2	f(x2	NOUN
ejpam-5984	418	11	)	)	PUNCT
ejpam-5984	418	12	+	+	CCONJ
ejpam-5984	418	13	f(x3	f(x3	X
ejpam-5984	418	14	)	)	PUNCT
ejpam-5984	419	1	+	+	CCONJ
ejpam-5984	419	2	s.	s.	PROPN
ejpam-5984	419	3	m.	m.	PROPN
ejpam-5984	419	4	sheikholeslami	sheikholeslami	PROPN
ejpam-5984	419	5	et	et	PROPN
ejpam-5984	419	6	al	al	PROPN
ejpam-5984	419	7	.	.	PUNCT
ejpam-5984	419	8	/	/	SYM
ejpam-5984	419	9	eur	eur	PROPN
ejpam-5984	419	10	.	.	PUNCT
ejpam-5984	420	1	j.	j.	PROPN
ejpam-5984	420	2	pure	pure	PROPN
ejpam-5984	420	3	appl	appl	PROPN
ejpam-5984	420	4	.	.	PROPN
ejpam-5984	420	5	math	math	PROPN
ejpam-5984	420	6	,	,	PUNCT
ejpam-5984	420	7	18	18	NUM
ejpam-5984	420	8	(	(	PUNCT
ejpam-5984	420	9	2	2	NUM
ejpam-5984	420	10	)	)	PUNCT
ejpam-5984	420	11	(	(	PUNCT
ejpam-5984	420	12	2025	2025	NUM
ejpam-5984	420	13	)	)	PUNCT
ejpam-5984	420	14	,	,	PUNCT
ejpam-5984	420	15	5984	5984	NUM
ejpam-5984	420	16	11	11	NUM
ejpam-5984	420	17	of	of	ADP
ejpam-5984	420	18	16	16	NUM
ejpam-5984	420	19	f(u2	f(u2	NOUN
ejpam-5984	420	20	)	)	PUNCT
ejpam-5984	421	1	+	+	CCONJ
ejpam-5984	421	2	f(u1	f(u1	NOUN
ejpam-5984	421	3	)	)	PUNCT
ejpam-5984	421	4	≥	≥	NOUN
ejpam-5984	421	5	6	6	NUM
ejpam-5984	421	6	.	.	PUNCT
ejpam-5984	422	1	we	we	PRON
ejpam-5984	422	2	consider	consider	VERB
ejpam-5984	422	3	two	two	NUM
ejpam-5984	422	4	cases	case	NOUN
ejpam-5984	422	5	.	.	PUNCT
ejpam-5984	423	1	if	if	SCONJ
ejpam-5984	423	2	f(x4	f(x4	NOUN
ejpam-5984	423	3	)	)	PUNCT
ejpam-5984	423	4	=	=	SYM
ejpam-5984	423	5	0	0	NUM
ejpam-5984	423	6	,	,	PUNCT
ejpam-5984	423	7	then	then	ADV
ejpam-5984	423	8	we	we	PRON
ejpam-5984	423	9	may	may	AUX
ejpam-5984	423	10	assume	assume	VERB
ejpam-5984	423	11	that	that	SCONJ
ejpam-5984	423	12	f(x3	f(x3	VERB
ejpam-5984	423	13	)	)	PUNCT
ejpam-5984	423	14	≥	≥	NOUN
ejpam-5984	423	15	2	2	NUM
ejpam-5984	423	16	,	,	PUNCT
ejpam-5984	423	17	f(x1	f(x1	ADJ
ejpam-5984	423	18	)	)	PUNCT
ejpam-5984	423	19	=	=	SYM
ejpam-5984	423	20	f(u1	f(u1	NOUN
ejpam-5984	423	21	)	)	PUNCT
ejpam-5984	423	22	=	=	SYM
ejpam-5984	423	23	2	2	NUM
ejpam-5984	423	24	and	and	CCONJ
ejpam-5984	423	25	f(u2	f(u2	NOUN
ejpam-5984	423	26	)	)	PUNCT
ejpam-5984	424	1	=	=	SYM
ejpam-5984	424	2	f(x2	f(x2	NOUN
ejpam-5984	424	3	)	)	PUNCT
ejpam-5984	424	4	=	=	SYM
ejpam-5984	425	1	0	0	X
ejpam-5984	425	2	.	.	PUNCT
ejpam-5984	426	1	then	then	ADV
ejpam-5984	426	2	the	the	DET
ejpam-5984	426	3	function	function	NOUN
ejpam-5984	426	4	g	g	PROPN
ejpam-5984	426	5	defined	define	VERB
ejpam-5984	426	6	on	on	ADP
ejpam-5984	426	7	t	t	PROPN
ejpam-5984	426	8	−	−	PROPN
ejpam-5984	426	9	x1	x1	PROPN
ejpam-5984	426	10	by	by	ADP
ejpam-5984	426	11	g(x3	g(x3	NUM
ejpam-5984	426	12	)	)	PUNCT
ejpam-5984	427	1	=	=	SYM
ejpam-5984	427	2	3	3	NUM
ejpam-5984	427	3	,	,	PUNCT
ejpam-5984	427	4	g(v	g(v	X
ejpam-5984	427	5	)	)	PUNCT
ejpam-5984	427	6	=	=	SYM
ejpam-5984	427	7	f(v	f(v	NOUN
ejpam-5984	427	8	)	)	PUNCT
ejpam-5984	427	9	for	for	ADP
ejpam-5984	427	10	each	each	DET
ejpam-5984	427	11	vertex	vertex	NOUN
ejpam-5984	427	12	v	v	ADP
ejpam-5984	427	13	∈	∈	PROPN
ejpam-5984	427	14	v	v	NOUN
ejpam-5984	427	15	(	(	PUNCT
ejpam-5984	427	16	t	t	PROPN
ejpam-5984	427	17	−	−	PROPN
ejpam-5984	427	18	x1	x1	PROPN
ejpam-5984	427	19	)	)	PUNCT
ejpam-5984	427	20	−	−	PROPN
ejpam-5984	427	21	{	{	PUNCT
ejpam-5984	427	22	x3	x3	ADJ
ejpam-5984	427	23	}	}	PUNCT
ejpam-5984	427	24	,	,	PUNCT
ejpam-5984	427	25	is	be	AUX
ejpam-5984	427	26	an	an	DET
ejpam-5984	427	27	idrdfunction	idrdfunction	NOUN
ejpam-5984	427	28	of	of	ADP
ejpam-5984	427	29	t	t	PROPN
ejpam-5984	427	30	−	−	PROPN
ejpam-5984	427	31	x1	x1	PROPN
ejpam-5984	427	32	of	of	ADP
ejpam-5984	427	33	weight	weight	NOUN
ejpam-5984	427	34	at	at	ADP
ejpam-5984	427	35	most	most	ADV
ejpam-5984	427	36	ω(f	ω(f	NUM
ejpam-5984	427	37	)	)	PUNCT
ejpam-5984	428	1	−	−	ADP
ejpam-5984	428	2	1	1	NUM
ejpam-5984	428	3	and	and	CCONJ
ejpam-5984	428	4	so	so	ADV
ejpam-5984	428	5	idr(t	idr(t	PROPN
ejpam-5984	428	6	−	−	PROPN
ejpam-5984	428	7	x1	x1	PROPN
ejpam-5984	428	8	)	)	PUNCT
ejpam-5984	428	9	<	<	X
ejpam-5984	428	10	idr(t	idr(t	PROPN
ejpam-5984	428	11	)	)	PUNCT
ejpam-5984	428	12	leading	lead	VERB
ejpam-5984	428	13	to	to	ADP
ejpam-5984	428	14	a	a	DET
ejpam-5984	428	15	contradiction	contradiction	NOUN
ejpam-5984	428	16	.	.	PUNCT
ejpam-5984	429	1	now	now	ADV
ejpam-5984	429	2	,	,	PUNCT
ejpam-5984	429	3	if	if	SCONJ
ejpam-5984	429	4	f(x4	f(x4	NOUN
ejpam-5984	429	5	)	)	PUNCT
ejpam-5984	429	6	≥	≥	NOUN
ejpam-5984	429	7	2	2	NUM
ejpam-5984	429	8	,	,	PUNCT
ejpam-5984	429	9	then	then	ADV
ejpam-5984	429	10	we	we	PRON
ejpam-5984	429	11	may	may	AUX
ejpam-5984	429	12	assume	assume	VERB
ejpam-5984	429	13	that	that	SCONJ
ejpam-5984	429	14	f(x1	f(x1	NOUN
ejpam-5984	429	15	)	)	PUNCT
ejpam-5984	429	16	=	=	PUNCT
ejpam-5984	430	1	f(x3	f(x3	X
ejpam-5984	430	2	)	)	PUNCT
ejpam-5984	430	3	=	=	SYM
ejpam-5984	430	4	f(u1	f(u1	NOUN
ejpam-5984	430	5	)	)	PUNCT
ejpam-5984	430	6	=	=	SYM
ejpam-5984	430	7	0	0	NUM
ejpam-5984	430	8	and	and	CCONJ
ejpam-5984	430	9	f(u2	f(u2	NOUN
ejpam-5984	430	10	)	)	PUNCT
ejpam-5984	431	1	=	=	SYM
ejpam-5984	431	2	f(x2	f(x2	NOUN
ejpam-5984	431	3	)	)	PUNCT
ejpam-5984	431	4	=	=	SYM
ejpam-5984	432	1	3	3	X
ejpam-5984	432	2	.	.	PUNCT
ejpam-5984	433	1	then	then	ADV
ejpam-5984	433	2	the	the	DET
ejpam-5984	433	3	function	function	NOUN
ejpam-5984	433	4	g	g	PROPN
ejpam-5984	433	5	defined	define	VERB
ejpam-5984	433	6	on	on	ADP
ejpam-5984	433	7	t	t	PROPN
ejpam-5984	433	8	−	−	PROPN
ejpam-5984	434	1	x1	x1	PROPN
ejpam-5984	434	2	by	by	ADP
ejpam-5984	434	3	g(x2	g(x2	NOUN
ejpam-5984	434	4	)	)	PUNCT
ejpam-5984	434	5	=	=	SYM
ejpam-5984	434	6	2	2	NUM
ejpam-5984	434	7	,	,	PUNCT
ejpam-5984	434	8	g(v	g(v	X
ejpam-5984	434	9	)	)	PUNCT
ejpam-5984	434	10	=	=	SYM
ejpam-5984	434	11	f(v	f(v	NOUN
ejpam-5984	434	12	)	)	PUNCT
ejpam-5984	434	13	for	for	ADP
ejpam-5984	434	14	each	each	DET
ejpam-5984	434	15	vertex	vertex	NOUN
ejpam-5984	434	16	v	v	ADP
ejpam-5984	434	17	∈	∈	PROPN
ejpam-5984	434	18	v	v	NOUN
ejpam-5984	434	19	(	(	PUNCT
ejpam-5984	434	20	t	t	PROPN
ejpam-5984	434	21	−	−	PROPN
ejpam-5984	434	22	x1	x1	PROPN
ejpam-5984	434	23	)	)	PUNCT
ejpam-5984	434	24	−	−	PROPN
ejpam-5984	434	25	{	{	PUNCT
ejpam-5984	434	26	x2	x2	PROPN
ejpam-5984	434	27	}	}	PUNCT
ejpam-5984	434	28	,	,	PUNCT
ejpam-5984	434	29	is	be	AUX
ejpam-5984	434	30	an	an	DET
ejpam-5984	434	31	idrd	idrd	ADJ
ejpam-5984	434	32	-	-	PUNCT
ejpam-5984	434	33	function	function	NOUN
ejpam-5984	434	34	of	of	ADP
ejpam-5984	434	35	t	t	PROPN
ejpam-5984	434	36	−	−	PROPN
ejpam-5984	434	37	x1	x1	PROPN
ejpam-5984	434	38	of	of	ADP
ejpam-5984	434	39	weight	weight	NOUN
ejpam-5984	434	40	at	at	ADP
ejpam-5984	434	41	most	most	ADJ
ejpam-5984	434	42	ω(f)−	ω(f)−	PROPN
ejpam-5984	434	43	1	1	NUM
ejpam-5984	434	44	and	and	CCONJ
ejpam-5984	434	45	so	so	ADV
ejpam-5984	434	46	idr(t	idr(t	PROPN
ejpam-5984	434	47	−	−	PROPN
ejpam-5984	434	48	x1	x1	PROPN
ejpam-5984	434	49	)	)	PUNCT
ejpam-5984	434	50	<	<	X
ejpam-5984	434	51	idr(t	idr(t	PROPN
ejpam-5984	434	52	)	)	PUNCT
ejpam-5984	434	53	leading	lead	VERB
ejpam-5984	434	54	to	to	ADP
ejpam-5984	434	55	a	a	DET
ejpam-5984	434	56	contradiction	contradiction	NOUN
ejpam-5984	434	57	.	.	PUNCT
ejpam-5984	435	1	case	case	NOUN
ejpam-5984	435	2	3	3	NUM
ejpam-5984	435	3	.	.	PUNCT
ejpam-5984	435	4	deg(x3	deg(x3	NUM
ejpam-5984	435	5	)	)	PUNCT
ejpam-5984	436	1	=	=	SYM
ejpam-5984	436	2	2	2	X
ejpam-5984	436	3	.	.	PUNCT
ejpam-5984	437	1	now	now	ADV
ejpam-5984	437	2	,	,	PUNCT
ejpam-5984	437	3	we	we	PRON
ejpam-5984	437	4	take	take	VERB
ejpam-5984	437	5	any	any	DET
ejpam-5984	437	6	vertex	vertex	NOUN
ejpam-5984	437	7	w	w	ADP
ejpam-5984	437	8	∈	∈	PROPN
ejpam-5984	437	9	v	v	ADP
ejpam-5984	437	10	(	(	PUNCT
ejpam-5984	437	11	t	t	NOUN
ejpam-5984	437	12	)	)	PUNCT
ejpam-5984	437	13	−{x1	−{x1	PROPN
ejpam-5984	437	14	,	,	PUNCT
ejpam-5984	437	15	x2	x2	PROPN
ejpam-5984	437	16	,	,	PUNCT
ejpam-5984	437	17	x3	x3	ADJ
ejpam-5984	437	18	}	}	PUNCT
ejpam-5984	437	19	,	,	PUNCT
ejpam-5984	437	20	and	and	CCONJ
ejpam-5984	437	21	let	let	VERB
ejpam-5984	437	22	t1	t1	NOUN
ejpam-5984	437	23	,	,	PUNCT
ejpam-5984	437	24	t2	t2	NOUN
ejpam-5984	437	25	,	,	PUNCT
ejpam-5984	437	26	.	.	PUNCT
ejpam-5984	437	27	.	.	PUNCT
ejpam-5984	437	28	.	.	PUNCT
ejpam-5984	438	1	,	,	PUNCT
ejpam-5984	438	2	tk	tk	PROPN
ejpam-5984	438	3	be	be	AUX
ejpam-5984	438	4	the	the	DET
ejpam-5984	438	5	components	component	NOUN
ejpam-5984	438	6	of	of	ADP
ejpam-5984	438	7	t	t	PROPN
ejpam-5984	438	8	−	−	PROPN
ejpam-5984	438	9	w.	w.	PROPN
ejpam-5984	438	10	without	without	ADP
ejpam-5984	438	11	loss	loss	NOUN
ejpam-5984	438	12	of	of	ADP
ejpam-5984	438	13	generality	generality	NOUN
ejpam-5984	438	14	,	,	PUNCT
ejpam-5984	438	15	assume	assume	VERB
ejpam-5984	438	16	that	that	SCONJ
ejpam-5984	438	17	the	the	DET
ejpam-5984	438	18	pendent	pendent	PROPN
ejpam-5984	438	19	star	star	PROPN
ejpam-5984	438	20	x1x2x3	x1x2x3	PROPN
ejpam-5984	438	21	is	be	AUX
ejpam-5984	438	22	contained	contain	VERB
ejpam-5984	438	23	in	in	ADP
ejpam-5984	438	24	t1	t1	PROPN
ejpam-5984	438	25	(	(	PUNCT
ejpam-5984	438	26	note	note	VERB
ejpam-5984	438	27	that	that	SCONJ
ejpam-5984	438	28	if	if	SCONJ
ejpam-5984	438	29	w	w	PROPN
ejpam-5984	438	30	=	=	SYM
ejpam-5984	438	31	x4	x4	PROPN
ejpam-5984	438	32	,	,	PUNCT
ejpam-5984	438	33	then	then	ADV
ejpam-5984	438	34	t1	t1	PROPN
ejpam-5984	438	35	is	be	AUX
ejpam-5984	438	36	a	a	DET
ejpam-5984	438	37	path	path	NOUN
ejpam-5984	438	38	p3	p3	NOUN
ejpam-5984	438	39	)	)	PUNCT
ejpam-5984	438	40	.	.	PUNCT
ejpam-5984	439	1	denote	denote	VERB
ejpam-5984	439	2	t	t	NOUN
ejpam-5984	439	3	′	′	NUM
ejpam-5984	440	1	=	=	PUNCT
ejpam-5984	440	2	t	t	PROPN
ejpam-5984	440	3	−	−	PROPN
ejpam-5984	441	1	{	{	PUNCT
ejpam-5984	442	1	x1	x1	PROPN
ejpam-5984	442	2	,	,	PUNCT
ejpam-5984	442	3	x2	x2	PROPN
ejpam-5984	442	4	,	,	PUNCT
ejpam-5984	442	5	x3	x3	ADJ
ejpam-5984	442	6	}	}	PUNCT
ejpam-5984	442	7	and	and	CCONJ
ejpam-5984	442	8	t	t	PROPN
ejpam-5984	442	9	′	′	NOUN
ejpam-5984	442	10	1	1	NUM
ejpam-5984	442	11	=	=	NOUN
ejpam-5984	442	12	t1	t1	NOUN
ejpam-5984	442	13	−	−	PROPN
ejpam-5984	442	14	{	{	PUNCT
ejpam-5984	442	15	x1	x1	PROPN
ejpam-5984	442	16	,	,	PUNCT
ejpam-5984	442	17	x2	x2	PROPN
ejpam-5984	442	18	,	,	PUNCT
ejpam-5984	442	19	x3	x3	ADJ
ejpam-5984	442	20	}	}	PUNCT
ejpam-5984	442	21	.	.	PUNCT
ejpam-5984	443	1	by	by	ADP
ejpam-5984	443	2	proposition	proposition	NOUN
ejpam-5984	443	3	1	1	NUM
ejpam-5984	443	4	,	,	PUNCT
ejpam-5984	443	5	we	we	PRON
ejpam-5984	443	6	have	have	VERB
ejpam-5984	443	7	idr(t	idr(t	NOUN
ejpam-5984	443	8	)	)	PUNCT
ejpam-5984	443	9	=	=	SYM
ejpam-5984	444	1	idr(t	idr(t	PROPN
ejpam-5984	444	2	′	′	NOUN
ejpam-5984	444	3	)	)	PUNCT
ejpam-5984	445	1	+	+	CCONJ
ejpam-5984	445	2	3	3	NUM
ejpam-5984	445	3	and	and	CCONJ
ejpam-5984	445	4	idr(t1	idr(t1	NOUN
ejpam-5984	445	5	)	)	PUNCT
ejpam-5984	446	1	=	=	SYM
ejpam-5984	446	2	idr(t	idr(t	NOUN
ejpam-5984	446	3	′	′	NUM
ejpam-5984	446	4	1	1	NUM
ejpam-5984	446	5	)	)	PUNCT
ejpam-5984	447	1	+	+	CCONJ
ejpam-5984	448	1	3	3	X
ejpam-5984	448	2	.	.	X
ejpam-5984	448	3	it	it	PRON
ejpam-5984	448	4	conclude	conclude	VERB
ejpam-5984	448	5	that	that	SCONJ
ejpam-5984	448	6	idr(t	idr(t	NOUN
ejpam-5984	448	7	′	′	NUM
ejpam-5984	448	8	−w	−w	NOUN
ejpam-5984	448	9	)	)	PUNCT
ejpam-5984	449	1	=	=	PUNCT
ejpam-5984	449	2	idr(t	idr(t	NOUN
ejpam-5984	449	3	′	′	NUM
ejpam-5984	449	4	1	1	NUM
ejpam-5984	449	5	)	)	PUNCT
ejpam-5984	450	1	+	+	CCONJ
ejpam-5984	450	2	idr(t2	idr(t2	NOUN
ejpam-5984	450	3	)	)	PUNCT
ejpam-5984	450	4	+	+	NUM
ejpam-5984	450	5	·	·	PUNCT
ejpam-5984	450	6	·	·	PUNCT
ejpam-5984	450	7	·	·	PUNCT
ejpam-5984	450	8	+	+	NUM
ejpam-5984	450	9	idr(tk	idr(tk	NOUN
ejpam-5984	450	10	)	)	PUNCT
ejpam-5984	450	11	=	=	SYM
ejpam-5984	451	1	idr(t1	idr(t1	X
ejpam-5984	451	2	)	)	PUNCT
ejpam-5984	452	1	+	+	CCONJ
ejpam-5984	452	2	idr(t2)+	idr(t2)+	ADJ
ejpam-5984	452	3	·	·	PUNCT
ejpam-5984	452	4	·	·	PUNCT
ejpam-5984	452	5	·	·	PUNCT
ejpam-5984	452	6	+	+	NUM
ejpam-5984	452	7	idr(tk)−3	idr(tk)−3	PROPN
ejpam-5984	452	8	=	=	PUNCT
ejpam-5984	452	9	idr(t	idr(t	NOUN
ejpam-5984	452	10	−w)−3	−w)−3	NOUN
ejpam-5984	452	11	=	=	SYM
ejpam-5984	452	12	idr(t	idr(t	NOUN
ejpam-5984	452	13	)	)	PUNCT
ejpam-5984	452	14	−3	−3	NOUN
ejpam-5984	452	15	=	=	SYM
ejpam-5984	452	16	idr(t	idr(t	PROPN
ejpam-5984	452	17	′	′	NUM
ejpam-5984	452	18	)	)	PUNCT
ejpam-5984	452	19	,	,	PUNCT
ejpam-5984	452	20	that	that	ADV
ejpam-5984	452	21	is	is	ADV
ejpam-5984	452	22	,	,	PUNCT
ejpam-5984	452	23	stidr(t	stidr(t	PROPN
ejpam-5984	452	24	′	′	NOUN
ejpam-5984	452	25	)	)	PUNCT
ejpam-5984	452	26	≥	≥	NOUN
ejpam-5984	452	27	2	2	NUM
ejpam-5984	452	28	,	,	PUNCT
ejpam-5984	452	29	contradicting	contradict	VERB
ejpam-5984	452	30	the	the	DET
ejpam-5984	452	31	choice	choice	NOUN
ejpam-5984	452	32	of	of	ADP
ejpam-5984	452	33	t	t	PROPN
ejpam-5984	452	34	.	.	PUNCT
ejpam-5984	453	1	therefore	therefore	ADV
ejpam-5984	453	2	,	,	PUNCT
ejpam-5984	453	3	stidr(t	stidr(t	ADJ
ejpam-5984	453	4	)	)	PUNCT
ejpam-5984	453	5	=	=	SYM
ejpam-5984	453	6	1	1	NUM
ejpam-5984	453	7	for	for	ADP
ejpam-5984	453	8	any	any	DET
ejpam-5984	453	9	tree	tree	NOUN
ejpam-5984	453	10	of	of	ADP
ejpam-5984	453	11	order	order	NOUN
ejpam-5984	453	12	n	n	PRON
ejpam-5984	453	13	≥	≥	NOUN
ejpam-5984	453	14	2	2	NUM
ejpam-5984	453	15	.	.	PUNCT
ejpam-5984	454	1	this	this	PRON
ejpam-5984	454	2	completes	complete	VERB
ejpam-5984	454	3	the	the	DET
ejpam-5984	454	4	proof	proof	NOUN
ejpam-5984	454	5	.	.	PUNCT
ejpam-5984	455	1	finally	finally	ADV
ejpam-5984	455	2	,	,	PUNCT
ejpam-5984	455	3	we	we	PRON
ejpam-5984	455	4	will	will	AUX
ejpam-5984	455	5	establish	establish	VERB
ejpam-5984	455	6	an	an	DET
ejpam-5984	455	7	upper	upper	ADJ
ejpam-5984	455	8	bound	bind	VERB
ejpam-5984	455	9	on	on	ADP
ejpam-5984	455	10	the	the	DET
ejpam-5984	455	11	st−idr(t	st−idr(t	NOUN
ejpam-5984	455	12	)	)	PUNCT
ejpam-5984	455	13	of	of	ADP
ejpam-5984	455	14	a	a	DET
ejpam-5984	455	15	tree	tree	NOUN
ejpam-5984	455	16	.	.	PUNCT
ejpam-5984	456	1	moreover	moreover	ADV
ejpam-5984	456	2	,	,	PUNCT
ejpam-5984	456	3	we	we	PRON
ejpam-5984	456	4	characterize	characterize	VERB
ejpam-5984	456	5	the	the	DET
ejpam-5984	456	6	trees	tree	NOUN
ejpam-5984	456	7	that	that	PRON
ejpam-5984	456	8	achieve	achieve	VERB
ejpam-5984	456	9	the	the	DET
ejpam-5984	456	10	upper	upper	ADJ
ejpam-5984	456	11	bound	bind	VERB
ejpam-5984	456	12	.	.	PUNCT
ejpam-5984	457	1	for	for	ADP
ejpam-5984	457	2	this	this	DET
ejpam-5984	457	3	purpose	purpose	NOUN
ejpam-5984	457	4	,	,	PUNCT
ejpam-5984	457	5	we	we	PRON
ejpam-5984	457	6	define	define	VERB
ejpam-5984	457	7	two	two	NUM
ejpam-5984	457	8	families	family	NOUN
ejpam-5984	457	9	of	of	ADP
ejpam-5984	457	10	trees	tree	NOUN
ejpam-5984	457	11	as	as	SCONJ
ejpam-5984	457	12	follows	follow	VERB
ejpam-5984	457	13	.	.	PUNCT
ejpam-5984	458	1	for	for	ADP
ejpam-5984	458	2	integers	integer	NOUN
ejpam-5984	458	3	k	k	PROPN
ejpam-5984	458	4	≥	≥	NUM
ejpam-5984	458	5	1	1	NUM
ejpam-5984	458	6	and	and	CCONJ
ejpam-5984	458	7	∆	∆	PROPN
ejpam-5984	458	8	≥	≥	NOUN
ejpam-5984	458	9	2	2	NUM
ejpam-5984	458	10	,	,	PUNCT
ejpam-5984	458	11	let	let	VERB
ejpam-5984	458	12	tk,∆	tk,∆	NOUN
ejpam-5984	458	13	be	be	AUX
ejpam-5984	458	14	a	a	DET
ejpam-5984	458	15	tree	tree	NOUN
ejpam-5984	458	16	obtained	obtain	VERB
ejpam-5984	458	17	from	from	ADP
ejpam-5984	458	18	the	the	DET
ejpam-5984	458	19	k	k	PROPN
ejpam-5984	458	20	copies	copy	NOUN
ejpam-5984	458	21	of	of	ADP
ejpam-5984	458	22	the	the	DET
ejpam-5984	458	23	star	star	NOUN
ejpam-5984	458	24	k1,∆	k1,∆	VERB
ejpam-5984	458	25	,	,	PUNCT
ejpam-5984	458	26	say	say	VERB
ejpam-5984	458	27	s1	s1	NOUN
ejpam-5984	458	28	,	,	PUNCT
ejpam-5984	458	29	s2	s2	PROPN
ejpam-5984	458	30	,	,	PUNCT
ejpam-5984	458	31	.	.	PUNCT
ejpam-5984	458	32	.	.	PUNCT
ejpam-5984	459	1	.	.	PUNCT
ejpam-5984	460	1	,	,	PUNCT
ejpam-5984	460	2	sk	sk	VERB
ejpam-5984	460	3	,	,	PUNCT
ejpam-5984	460	4	by	by	ADP
ejpam-5984	460	5	adding	add	VERB
ejpam-5984	460	6	k	k	PROPN
ejpam-5984	460	7	−	−	NUM
ejpam-5984	460	8	1	1	NUM
ejpam-5984	460	9	edges	edge	NOUN
ejpam-5984	460	10	between	between	ADP
ejpam-5984	460	11	the	the	DET
ejpam-5984	460	12	leaves	leave	NOUN
ejpam-5984	460	13	of	of	ADP
ejpam-5984	460	14	these	these	DET
ejpam-5984	460	15	stars	star	NOUN
ejpam-5984	460	16	,	,	PUNCT
ejpam-5984	460	17	so	so	SCONJ
ejpam-5984	460	18	that	that	SCONJ
ejpam-5984	460	19	the	the	DET
ejpam-5984	460	20	resulting	result	VERB
ejpam-5984	460	21	graph	graph	NOUN
ejpam-5984	460	22	is	be	AUX
ejpam-5984	460	23	a	a	DET
ejpam-5984	460	24	connected	connected	ADJ
ejpam-5984	460	25	graph	graph	NOUN
ejpam-5984	460	26	with	with	ADP
ejpam-5984	460	27	maximum	maximum	ADJ
ejpam-5984	460	28	degree	degree	NOUN
ejpam-5984	460	29	∆.	∆.	NOUN
ejpam-5984	460	30	let	let	VERB
ejpam-5984	460	31	tk,∆	tk,∆	NOUN
ejpam-5984	460	32	be	be	AUX
ejpam-5984	460	33	the	the	DET
ejpam-5984	460	34	family	family	NOUN
ejpam-5984	460	35	of	of	ADP
ejpam-5984	460	36	all	all	DET
ejpam-5984	460	37	such	such	ADJ
ejpam-5984	460	38	trees	tree	NOUN
ejpam-5984	460	39	tk,∆	tk,∆	VERB
ejpam-5984	460	40	,	,	PUNCT
ejpam-5984	460	41	and	and	CCONJ
ejpam-5984	460	42	t∆	t∆	PROPN
ejpam-5984	460	43	=	=	SYM
ejpam-5984	460	44	⋃	⋃	ADP
ejpam-5984	460	45	k≥1	k≥1	NOUN
ejpam-5984	460	46	tk,∆.	tk,∆.	VERB
ejpam-5984	460	47	moreover	moreover	ADV
ejpam-5984	460	48	,	,	PUNCT
ejpam-5984	460	49	let	let	VERB
ejpam-5984	460	50	lk,∆	lk,∆	PROPN
ejpam-5984	460	51	be	be	AUX
ejpam-5984	460	52	a	a	DET
ejpam-5984	460	53	graph	graph	NOUN
ejpam-5984	460	54	obtained	obtain	VERB
ejpam-5984	460	55	from	from	ADP
ejpam-5984	460	56	a	a	DET
ejpam-5984	460	57	tree	tree	NOUN
ejpam-5984	460	58	tk,∆	tk,∆	NOUN
ejpam-5984	460	59	and	and	CCONJ
ejpam-5984	460	60	a	a	DET
ejpam-5984	460	61	path	path	NOUN
ejpam-5984	460	62	p2	p2	NOUN
ejpam-5984	460	63	by	by	ADP
ejpam-5984	460	64	joining	join	VERB
ejpam-5984	460	65	a	a	DET
ejpam-5984	460	66	vertex	vertex	NOUN
ejpam-5984	460	67	of	of	ADP
ejpam-5984	460	68	the	the	DET
ejpam-5984	460	69	p2	p2	PROPN
ejpam-5984	460	70	to	to	ADP
ejpam-5984	460	71	a	a	DET
ejpam-5984	460	72	leaf	leaf	NOUN
ejpam-5984	460	73	of	of	ADP
ejpam-5984	460	74	some	some	DET
ejpam-5984	460	75	star	star	NOUN
ejpam-5984	460	76	si	si	X
ejpam-5984	460	77	(	(	PUNCT
ejpam-5984	460	78	i	i	NOUN
ejpam-5984	460	79	∈	∈	PROPN
ejpam-5984	460	80	{	{	PUNCT
ejpam-5984	460	81	1	1	NUM
ejpam-5984	460	82	,	,	PUNCT
ejpam-5984	460	83	2	2	NUM
ejpam-5984	460	84	,	,	PUNCT
ejpam-5984	460	85	.	.	PUNCT
ejpam-5984	460	86	.	.	PUNCT
ejpam-5984	461	1	.	.	PUNCT
ejpam-5984	462	1	,	,	PUNCT
ejpam-5984	462	2	k	k	X
ejpam-5984	462	3	}	}	PUNCT
ejpam-5984	462	4	)	)	PUNCT
ejpam-5984	462	5	of	of	ADP
ejpam-5984	462	6	tk,∆	tk,∆	PROPN
ejpam-5984	462	7	,	,	PUNCT
ejpam-5984	462	8	so	so	SCONJ
ejpam-5984	462	9	that	that	SCONJ
ejpam-5984	462	10	the	the	DET
ejpam-5984	462	11	resulting	result	VERB
ejpam-5984	462	12	graph	graph	NOUN
ejpam-5984	462	13	is	be	AUX
ejpam-5984	462	14	a	a	DET
ejpam-5984	462	15	tree	tree	NOUN
ejpam-5984	462	16	with	with	ADP
ejpam-5984	462	17	maximum	maximum	ADJ
ejpam-5984	462	18	degree	degree	NOUN
ejpam-5984	462	19	∆.	∆.	NOUN
ejpam-5984	462	20	let	let	VERB
ejpam-5984	462	21	lk,∆	lk,∆	PROPN
ejpam-5984	462	22	be	be	AUX
ejpam-5984	462	23	the	the	DET
ejpam-5984	462	24	family	family	NOUN
ejpam-5984	462	25	of	of	ADP
ejpam-5984	462	26	all	all	DET
ejpam-5984	462	27	such	such	ADJ
ejpam-5984	462	28	trees	tree	NOUN
ejpam-5984	462	29	lk,∆	lk,∆	NOUN
ejpam-5984	462	30	,	,	PUNCT
ejpam-5984	462	31	and	and	CCONJ
ejpam-5984	463	1	l∆	l∆	PROPN
ejpam-5984	463	2	=	=	PUNCT
ejpam-5984	463	3	⋃	⋃	ADP
ejpam-5984	463	4	k≥1	k≥1	NOUN
ejpam-5984	463	5	lk,∆.	lk,∆.	VERB
ejpam-5984	463	6	it	it	PRON
ejpam-5984	463	7	is	be	AUX
ejpam-5984	463	8	observed	observe	VERB
ejpam-5984	463	9	in	in	ADP
ejpam-5984	463	10	[	[	X
ejpam-5984	463	11	17	17	NUM
ejpam-5984	463	12	]	]	PUNCT
ejpam-5984	463	13	that	that	SCONJ
ejpam-5984	463	14	if	if	SCONJ
ejpam-5984	463	15	t	t	PROPN
ejpam-5984	463	16	∈	∈	PROPN
ejpam-5984	463	17	tk,∆	tk,∆	PROPN
ejpam-5984	463	18	,	,	PUNCT
ejpam-5984	463	19	then	then	ADV
ejpam-5984	463	20	γdr(t	γdr(t	PROPN
ejpam-5984	463	21	)	)	PUNCT
ejpam-5984	463	22	=	=	SYM
ejpam-5984	464	1	3k	3k	NOUN
ejpam-5984	464	2	and	and	CCONJ
ejpam-5984	464	3	t	t	PROPN
ejpam-5984	464	4	has	have	VERB
ejpam-5984	464	5	a	a	DET
ejpam-5984	464	6	unique	unique	ADJ
ejpam-5984	464	7	γdr(t	γdr(t	NOUN
ejpam-5984	464	8	)	)	PUNCT
ejpam-5984	464	9	function	function	NOUN
ejpam-5984	464	10	f	f	PROPN
ejpam-5984	464	11	that	that	PRON
ejpam-5984	464	12	assigns	assign	VERB
ejpam-5984	464	13	3	3	NUM
ejpam-5984	464	14	to	to	ADP
ejpam-5984	464	15	the	the	DET
ejpam-5984	464	16	central	central	ADJ
ejpam-5984	464	17	vertex	vertex	NOUN
ejpam-5984	464	18	of	of	ADP
ejpam-5984	464	19	each	each	DET
ejpam-5984	464	20	si	si	NOUN
ejpam-5984	464	21	,	,	PUNCT
ejpam-5984	464	22	and	and	CCONJ
ejpam-5984	464	23	0	0	NUM
ejpam-5984	464	24	to	to	ADP
ejpam-5984	464	25	the	the	DET
ejpam-5984	464	26	leaves	leave	NOUN
ejpam-5984	464	27	of	of	ADP
ejpam-5984	464	28	each	each	DET
ejpam-5984	464	29	si	si	NOUN
ejpam-5984	464	30	for	for	ADP
ejpam-5984	464	31	i	i	PRON
ejpam-5984	464	32	=	=	NOUN
ejpam-5984	464	33	1	1	NUM
ejpam-5984	464	34	,	,	PUNCT
ejpam-5984	464	35	2	2	NUM
ejpam-5984	464	36	,	,	PUNCT
ejpam-5984	464	37	.	.	PUNCT
ejpam-5984	464	38	.	.	PUNCT
ejpam-5984	465	1	.	.	PUNCT
ejpam-5984	466	1	,	,	PUNCT
ejpam-5984	466	2	k.	k.	PROPN
ejpam-5984	466	3	also	also	ADV
ejpam-5984	466	4	,	,	PUNCT
ejpam-5984	466	5	it	it	PRON
ejpam-5984	466	6	is	be	AUX
ejpam-5984	466	7	shown	show	VERB
ejpam-5984	466	8	in	in	ADP
ejpam-5984	466	9	[	[	X
ejpam-5984	466	10	17	17	NUM
ejpam-5984	466	11	]	]	PUNCT
ejpam-5984	466	12	that	that	SCONJ
ejpam-5984	466	13	if	if	SCONJ
ejpam-5984	466	14	t	t	PROPN
ejpam-5984	466	15	∈	∈	PROPN
ejpam-5984	466	16	lk,∆	lk,∆	PROPN
ejpam-5984	466	17	,	,	PUNCT
ejpam-5984	466	18	then	then	ADV
ejpam-5984	466	19	γdr(t	γdr(t	PROPN
ejpam-5984	466	20	)	)	PUNCT
ejpam-5984	466	21	=	=	PUNCT
ejpam-5984	467	1	3k	3k	NOUN
ejpam-5984	468	1	+	+	CCONJ
ejpam-5984	468	2	3	3	X
ejpam-5984	468	3	.	.	X
ejpam-5984	468	4	lemma	lemma	PROPN
ejpam-5984	468	5	1	1	X
ejpam-5984	468	6	.	.	PUNCT
ejpam-5984	469	1	if	if	SCONJ
ejpam-5984	469	2	t	t	PROPN
ejpam-5984	469	3	∈	∈	PROPN
ejpam-5984	469	4	tk,∆	tk,∆	NOUN
ejpam-5984	469	5	,	,	PUNCT
ejpam-5984	469	6	then	then	ADV
ejpam-5984	469	7	idr(t	idr(t	NOUN
ejpam-5984	469	8	)	)	PUNCT
ejpam-5984	469	9	=	=	SYM
ejpam-5984	469	10	3k	3k	X
ejpam-5984	469	11	.	.	PUNCT
ejpam-5984	470	1	furthermore	furthermore	ADV
ejpam-5984	470	2	,	,	PUNCT
ejpam-5984	470	3	t	t	PROPN
ejpam-5984	470	4	has	have	VERB
ejpam-5984	470	5	a	a	DET
ejpam-5984	470	6	unique	unique	ADJ
ejpam-5984	470	7	idr(t	idr(t	NOUN
ejpam-5984	470	8	)	)	PUNCT
ejpam-5984	470	9	-function	-function	NOUN
ejpam-5984	470	10	f	f	NOUN
ejpam-5984	470	11	that	that	PRON
ejpam-5984	470	12	assigns	assign	VERB
ejpam-5984	470	13	3	3	NUM
ejpam-5984	470	14	to	to	ADP
ejpam-5984	470	15	the	the	DET
ejpam-5984	470	16	central	central	ADJ
ejpam-5984	470	17	vertex	vertex	NOUN
ejpam-5984	470	18	of	of	ADP
ejpam-5984	470	19	each	each	DET
ejpam-5984	470	20	si	si	NOUN
ejpam-5984	470	21	,	,	PUNCT
ejpam-5984	470	22	and	and	CCONJ
ejpam-5984	470	23	0	0	NUM
ejpam-5984	470	24	to	to	ADP
ejpam-5984	470	25	the	the	DET
ejpam-5984	470	26	leaves	leave	NOUN
ejpam-5984	470	27	of	of	ADP
ejpam-5984	470	28	each	each	DET
ejpam-5984	470	29	si	si	NOUN
ejpam-5984	470	30	for	for	ADP
ejpam-5984	470	31	i	i	PRON
ejpam-5984	470	32	=	=	NOUN
ejpam-5984	470	33	1	1	NUM
ejpam-5984	470	34	,	,	PUNCT
ejpam-5984	470	35	2	2	NUM
ejpam-5984	470	36	,	,	PUNCT
ejpam-5984	470	37	.	.	PUNCT
ejpam-5984	470	38	.	.	PUNCT
ejpam-5984	471	1	.	.	PUNCT
ejpam-5984	472	1	,	,	PUNCT
ejpam-5984	472	2	k.	k.	PROPN
ejpam-5984	472	3	proof	proof	PROPN
ejpam-5984	472	4	.	.	PUNCT
ejpam-5984	473	1	first	first	ADV
ejpam-5984	473	2	note	note	VERB
ejpam-5984	473	3	that	that	SCONJ
ejpam-5984	473	4	idr(t	idr(t	NOUN
ejpam-5984	473	5	)	)	PUNCT
ejpam-5984	473	6	≥	≥	NOUN
ejpam-5984	473	7	γdr(t	γdr(t	NOUN
ejpam-5984	473	8	)	)	PUNCT
ejpam-5984	473	9	=	=	SYM
ejpam-5984	473	10	3k	3k	X
ejpam-5984	473	11	.	.	PUNCT
ejpam-5984	474	1	on	on	ADP
ejpam-5984	474	2	the	the	DET
ejpam-5984	474	3	other	other	ADJ
ejpam-5984	474	4	hand	hand	NOUN
ejpam-5984	474	5	,	,	PUNCT
ejpam-5984	474	6	the	the	DET
ejpam-5984	474	7	unique	unique	ADJ
ejpam-5984	474	8	γdr(t	γdr(t	NOUN
ejpam-5984	474	9	)	)	PUNCT
ejpam-5984	474	10	-function	-function	NOUN
ejpam-5984	474	11	f	f	NOUN
ejpam-5984	474	12	is	be	AUX
ejpam-5984	474	13	an	an	DET
ejpam-5984	474	14	independent	independent	ADJ
ejpam-5984	474	15	double	double	ADJ
ejpam-5984	474	16	roman	roman	ADJ
ejpam-5984	474	17	dominating	dominating	NOUN
ejpam-5984	474	18	function	function	NOUN
ejpam-5984	474	19	of	of	ADP
ejpam-5984	474	20	t	t	PROPN
ejpam-5984	474	21	and	and	CCONJ
ejpam-5984	474	22	therefore	therefore	ADV
ejpam-5984	474	23	idr(t	idr(t	NOUN
ejpam-5984	474	24	)	)	PUNCT
ejpam-5984	474	25	≤	≤	NOUN
ejpam-5984	474	26	γdr(t	γdr(t	NOUN
ejpam-5984	474	27	)	)	PUNCT
ejpam-5984	474	28	=	=	SYM
ejpam-5984	474	29	3k	3k	X
ejpam-5984	474	30	.	.	PUNCT
ejpam-5984	475	1	thus	thus	ADV
ejpam-5984	475	2	,	,	PUNCT
ejpam-5984	475	3	idr(t	idr(t	NOUN
ejpam-5984	475	4	)	)	PUNCT
ejpam-5984	475	5	=	=	SYM
ejpam-5984	475	6	3k	3k	X
ejpam-5984	475	7	.	.	PUNCT
ejpam-5984	476	1	it	it	PRON
ejpam-5984	476	2	follows	follow	VERB
ejpam-5984	476	3	from	from	ADP
ejpam-5984	476	4	idr(t	idr(t	NOUN
ejpam-5984	476	5	)	)	PUNCT
ejpam-5984	476	6	=	=	PUNCT
ejpam-5984	477	1	3k	3k	NOUN
ejpam-5984	477	2	that	that	SCONJ
ejpam-5984	477	3	any	any	DET
ejpam-5984	477	4	idr(t	idr(t	NOUN
ejpam-5984	477	5	)	)	PUNCT
ejpam-5984	477	6	function	function	NOUN
ejpam-5984	477	7	is	be	AUX
ejpam-5984	477	8	a	a	DET
ejpam-5984	477	9	γdr	γdr	NOUN
ejpam-5984	477	10	-	-	PUNCT
ejpam-5984	477	11	function	function	NOUN
ejpam-5984	477	12	of	of	ADP
ejpam-5984	477	13	t	t	PROPN
ejpam-5984	477	14	and	and	CCONJ
ejpam-5984	477	15	since	since	SCONJ
ejpam-5984	477	16	t	t	PROPN
ejpam-5984	477	17	has	have	VERB
ejpam-5984	477	18	a	a	DET
ejpam-5984	477	19	unique	unique	ADJ
ejpam-5984	477	20	γdr(t	γdr(t	NOUN
ejpam-5984	477	21	)	)	PUNCT
ejpam-5984	477	22	-function	-function	NOUN
ejpam-5984	477	23	,	,	PUNCT
ejpam-5984	477	24	we	we	PRON
ejpam-5984	477	25	deduce	deduce	VERB
ejpam-5984	477	26	that	that	SCONJ
ejpam-5984	477	27	f	f	PROPN
ejpam-5984	477	28	is	be	AUX
ejpam-5984	477	29	the	the	DET
ejpam-5984	477	30	unique	unique	ADJ
ejpam-5984	477	31	idr(t	idr(t	NOUN
ejpam-5984	477	32	)	)	PUNCT
ejpam-5984	477	33	-function	-function	NOUN
ejpam-5984	477	34	.	.	PUNCT
ejpam-5984	478	1	lemma	lemma	PROPN
ejpam-5984	478	2	2	2	NUM
ejpam-5984	478	3	.	.	PUNCT
ejpam-5984	479	1	if	if	SCONJ
ejpam-5984	479	2	t	t	PROPN
ejpam-5984	479	3	∈	∈	PROPN
ejpam-5984	479	4	lk,∆	lk,∆	PROPN
ejpam-5984	479	5	,	,	PUNCT
ejpam-5984	479	6	then	then	ADV
ejpam-5984	479	7	idr(t	idr(t	NOUN
ejpam-5984	479	8	)	)	PUNCT
ejpam-5984	479	9	=	=	PUNCT
ejpam-5984	480	1	3k	3k	NOUN
ejpam-5984	481	1	+	+	CCONJ
ejpam-5984	481	2	3	3	X
ejpam-5984	481	3	.	.	X
ejpam-5984	481	4	s.	s.	PROPN
ejpam-5984	481	5	m.	m.	PROPN
ejpam-5984	481	6	sheikholeslami	sheikholeslami	PROPN
ejpam-5984	481	7	et	et	PROPN
ejpam-5984	481	8	al	al	PROPN
ejpam-5984	481	9	.	.	PUNCT
ejpam-5984	481	10	/	/	SYM
ejpam-5984	481	11	eur	eur	PROPN
ejpam-5984	481	12	.	.	PUNCT
ejpam-5984	482	1	j.	j.	PROPN
ejpam-5984	482	2	pure	pure	PROPN
ejpam-5984	482	3	appl	appl	PROPN
ejpam-5984	482	4	.	.	PROPN
ejpam-5984	482	5	math	math	PROPN
ejpam-5984	482	6	,	,	PUNCT
ejpam-5984	482	7	18	18	NUM
ejpam-5984	482	8	(	(	PUNCT
ejpam-5984	482	9	2	2	NUM
ejpam-5984	482	10	)	)	PUNCT
ejpam-5984	482	11	(	(	PUNCT
ejpam-5984	482	12	2025	2025	NUM
ejpam-5984	482	13	)	)	PUNCT
ejpam-5984	482	14	,	,	PUNCT
ejpam-5984	482	15	5984	5984	NUM
ejpam-5984	482	16	12	12	NUM
ejpam-5984	482	17	of	of	ADP
ejpam-5984	482	18	16	16	NUM
ejpam-5984	482	19	proof	proof	NOUN
ejpam-5984	482	20	.	.	PUNCT
ejpam-5984	483	1	first	first	ADV
ejpam-5984	483	2	note	note	VERB
ejpam-5984	483	3	that	that	SCONJ
ejpam-5984	483	4	idr(t	idr(t	NOUN
ejpam-5984	483	5	)	)	PUNCT
ejpam-5984	483	6	≥	≥	NOUN
ejpam-5984	483	7	γdr(t	γdr(t	NOUN
ejpam-5984	483	8	)	)	PUNCT
ejpam-5984	483	9	=	=	SYM
ejpam-5984	484	1	3k	3k	NOUN
ejpam-5984	485	1	+	+	CCONJ
ejpam-5984	485	2	3	3	X
ejpam-5984	485	3	.	.	X
ejpam-5984	485	4	on	on	ADP
ejpam-5984	485	5	the	the	DET
ejpam-5984	485	6	other	other	ADJ
ejpam-5984	485	7	hand	hand	NOUN
ejpam-5984	485	8	,	,	PUNCT
ejpam-5984	485	9	assigning	assign	VERB
ejpam-5984	485	10	3	3	NUM
ejpam-5984	485	11	to	to	ADP
ejpam-5984	485	12	the	the	DET
ejpam-5984	485	13	center	center	NOUN
ejpam-5984	485	14	of	of	ADP
ejpam-5984	485	15	stars	star	NOUN
ejpam-5984	485	16	s1	s1	NOUN
ejpam-5984	485	17	,	,	PUNCT
ejpam-5984	485	18	.	.	PUNCT
ejpam-5984	485	19	.	.	PUNCT
ejpam-5984	486	1	.	.	PUNCT
ejpam-5984	487	1	,	,	PUNCT
ejpam-5984	487	2	sk	sk	NOUN
ejpam-5984	487	3	and	and	CCONJ
ejpam-5984	487	4	a	a	DET
ejpam-5984	487	5	vertex	vertex	NOUN
ejpam-5984	487	6	of	of	ADP
ejpam-5984	487	7	p2	p2	PROPN
ejpam-5984	487	8	and	and	CCONJ
ejpam-5984	487	9	0	0	NUM
ejpam-5984	487	10	to	to	ADP
ejpam-5984	487	11	other	other	ADJ
ejpam-5984	487	12	vertices	vertex	NOUN
ejpam-5984	487	13	,	,	PUNCT
ejpam-5984	487	14	provides	provide	VERB
ejpam-5984	487	15	an	an	DET
ejpam-5984	487	16	idrdfunction	idrdfunction	NOUN
ejpam-5984	487	17	of	of	ADP
ejpam-5984	487	18	weight	weight	NOUN
ejpam-5984	487	19	3k+3	3k+3	NOUN
ejpam-5984	487	20	leading	lead	VERB
ejpam-5984	487	21	to	to	ADP
ejpam-5984	487	22	idr(t	idr(t	NOUN
ejpam-5984	487	23	)	)	PUNCT
ejpam-5984	487	24	≤	≤	NOUN
ejpam-5984	487	25	γdr(t	γdr(t	NOUN
ejpam-5984	487	26	)	)	PUNCT
ejpam-5984	487	27	=	=	SYM
ejpam-5984	488	1	3k+3	3k+3	NUM
ejpam-5984	488	2	.	.	PUNCT
ejpam-5984	489	1	thus	thus	ADV
ejpam-5984	489	2	,	,	PUNCT
ejpam-5984	489	3	idr(t	idr(t	NOUN
ejpam-5984	489	4	)	)	PUNCT
ejpam-5984	489	5	=	=	SYM
ejpam-5984	490	1	3k+3	3k+3	NUM
ejpam-5984	490	2	.	.	PUNCT
ejpam-5984	491	1	lemma	lemma	PROPN
ejpam-5984	491	2	3	3	X
ejpam-5984	491	3	.	.	PUNCT
ejpam-5984	492	1	if	if	SCONJ
ejpam-5984	492	2	t	t	PROPN
ejpam-5984	492	3	∈	∈	PROPN
ejpam-5984	492	4	t∆	t∆	PROPN
ejpam-5984	492	5	,	,	PUNCT
ejpam-5984	492	6	then	then	ADV
ejpam-5984	492	7	st−idr(t	st−idr(t	NOUN
ejpam-5984	492	8	)	)	PUNCT
ejpam-5984	492	9	=	=	PUNCT
ejpam-5984	493	1	∆.	∆.	NOUN
ejpam-5984	493	2	proof	proof	NOUN
ejpam-5984	493	3	.	.	PUNCT
ejpam-5984	494	1	for	for	ADP
ejpam-5984	494	2	k	k	PROPN
ejpam-5984	494	3	≥	≥	PROPN
ejpam-5984	494	4	1	1	NUM
ejpam-5984	494	5	and	and	CCONJ
ejpam-5984	494	6	∆	∆	PROPN
ejpam-5984	494	7	≥	≥	NOUN
ejpam-5984	494	8	2	2	NUM
ejpam-5984	494	9	,	,	PUNCT
ejpam-5984	494	10	let	let	VERB
ejpam-5984	494	11	t	t	PROPN
ejpam-5984	494	12	∈	∈	PROPN
ejpam-5984	494	13	tk,∆.	tk,∆.	PUNCT
ejpam-5984	494	14	we	we	PRON
ejpam-5984	494	15	proceed	proceed	VERB
ejpam-5984	494	16	by	by	ADP
ejpam-5984	494	17	induction	induction	NOUN
ejpam-5984	494	18	on	on	ADP
ejpam-5984	494	19	the	the	DET
ejpam-5984	494	20	number	number	NOUN
ejpam-5984	494	21	k.	k.	NOUN
ejpam-5984	495	1	if	if	SCONJ
ejpam-5984	495	2	k	k	PROPN
ejpam-5984	495	3	=	=	SYM
ejpam-5984	495	4	1	1	NUM
ejpam-5984	495	5	,	,	PUNCT
ejpam-5984	495	6	then	then	ADV
ejpam-5984	495	7	t	t	PROPN
ejpam-5984	495	8	∼=	∼=	PROPN
ejpam-5984	495	9	k1,∆	k1,∆	VERB
ejpam-5984	495	10	and	and	CCONJ
ejpam-5984	495	11	the	the	DET
ejpam-5984	495	12	result	result	NOUN
ejpam-5984	495	13	follows	follow	VERB
ejpam-5984	495	14	from	from	ADP
ejpam-5984	495	15	corollary	corollary	ADJ
ejpam-5984	495	16	4	4	NUM
ejpam-5984	495	17	.	.	PUNCT
ejpam-5984	496	1	this	this	PRON
ejpam-5984	496	2	establishes	establish	VERB
ejpam-5984	496	3	the	the	DET
ejpam-5984	496	4	base	base	NOUN
ejpam-5984	496	5	case	case	NOUN
ejpam-5984	496	6	.	.	PUNCT
ejpam-5984	497	1	now	now	ADV
ejpam-5984	497	2	,	,	PUNCT
ejpam-5984	497	3	let	let	VERB
ejpam-5984	497	4	k	k	PROPN
ejpam-5984	497	5	≥	≥	NUM
ejpam-5984	497	6	2	2	NUM
ejpam-5984	497	7	.	.	PUNCT
ejpam-5984	498	1	if	if	SCONJ
ejpam-5984	498	2	∆	∆	PROPN
ejpam-5984	498	3	=	=	SYM
ejpam-5984	498	4	2	2	NUM
ejpam-5984	498	5	,	,	PUNCT
ejpam-5984	498	6	then	then	ADV
ejpam-5984	498	7	t	t	PROPN
ejpam-5984	498	8	∼=	∼=	PROPN
ejpam-5984	498	9	p3k	p3k	PROPN
ejpam-5984	498	10	and	and	CCONJ
ejpam-5984	498	11	we	we	PRON
ejpam-5984	498	12	are	be	AUX
ejpam-5984	498	13	done	do	VERB
ejpam-5984	498	14	by	by	ADP
ejpam-5984	498	15	proposition	proposition	NOUN
ejpam-5984	498	16	8	8	NUM
ejpam-5984	498	17	.	.	PUNCT
ejpam-5984	499	1	assume	assume	VERB
ejpam-5984	499	2	that	that	SCONJ
ejpam-5984	499	3	∆	∆	PROPN
ejpam-5984	499	4	≥	≥	NOUN
ejpam-5984	499	5	3	3	NUM
ejpam-5984	499	6	and	and	CCONJ
ejpam-5984	499	7	that	that	SCONJ
ejpam-5984	499	8	for	for	ADP
ejpam-5984	499	9	any	any	DET
ejpam-5984	499	10	tree	tree	NOUN
ejpam-5984	499	11	t	t	NOUN
ejpam-5984	500	1	′	′	NUM
ejpam-5984	500	2	∈	∈	PROPN
ejpam-5984	500	3	tk′,∆	tk′,∆	NOUN
ejpam-5984	500	4	with	with	ADP
ejpam-5984	500	5	1	1	NUM
ejpam-5984	500	6	≤	≤	NUM
ejpam-5984	500	7	k′	k′	PROPN
ejpam-5984	500	8	<	<	X
ejpam-5984	501	1	k	k	X
ejpam-5984	501	2	we	we	PRON
ejpam-5984	501	3	have	have	VERB
ejpam-5984	501	4	st−idr(t	st−idr(t	NOUN
ejpam-5984	501	5	′	′	NUM
ejpam-5984	501	6	)	)	PUNCT
ejpam-5984	502	1	=	=	PUNCT
ejpam-5984	503	1	∆.	∆.	NOUN
ejpam-5984	503	2	since	since	SCONJ
ejpam-5984	503	3	t	t	PROPN
ejpam-5984	503	4	∈	∈	PROPN
ejpam-5984	503	5	tk,∆	tk,∆	PROPN
ejpam-5984	503	6	,	,	PUNCT
ejpam-5984	503	7	t	t	PROPN
ejpam-5984	503	8	is	be	AUX
ejpam-5984	503	9	the	the	DET
ejpam-5984	503	10	graph	graph	NOUN
ejpam-5984	503	11	obtained	obtain	VERB
ejpam-5984	503	12	from	from	ADP
ejpam-5984	503	13	the	the	DET
ejpam-5984	503	14	k	k	PROPN
ejpam-5984	503	15	copies	copy	NOUN
ejpam-5984	503	16	of	of	ADP
ejpam-5984	503	17	the	the	DET
ejpam-5984	503	18	star	star	NOUN
ejpam-5984	503	19	k1,∆	k1,∆	VERB
ejpam-5984	503	20	,	,	PUNCT
ejpam-5984	503	21	say	say	VERB
ejpam-5984	503	22	s1	s1	NOUN
ejpam-5984	503	23	,	,	PUNCT
ejpam-5984	503	24	s2	s2	PROPN
ejpam-5984	503	25	,	,	PUNCT
ejpam-5984	503	26	.	.	PUNCT
ejpam-5984	503	27	.	.	PUNCT
ejpam-5984	504	1	.	.	PUNCT
ejpam-5984	505	1	,	,	PUNCT
ejpam-5984	505	2	sk	sk	VERB
ejpam-5984	505	3	,	,	PUNCT
ejpam-5984	505	4	by	by	ADP
ejpam-5984	505	5	adding	add	VERB
ejpam-5984	505	6	k	k	PROPN
ejpam-5984	505	7	−	−	NUM
ejpam-5984	505	8	1	1	NUM
ejpam-5984	505	9	edges	edge	NOUN
ejpam-5984	505	10	between	between	ADP
ejpam-5984	505	11	the	the	DET
ejpam-5984	505	12	leaves	leave	NOUN
ejpam-5984	505	13	of	of	ADP
ejpam-5984	505	14	these	these	DET
ejpam-5984	505	15	stars	star	NOUN
ejpam-5984	505	16	so	so	SCONJ
ejpam-5984	505	17	that	that	SCONJ
ejpam-5984	505	18	the	the	DET
ejpam-5984	505	19	resulting	result	VERB
ejpam-5984	505	20	graph	graph	NOUN
ejpam-5984	505	21	is	be	AUX
ejpam-5984	505	22	a	a	DET
ejpam-5984	505	23	connected	connected	ADJ
ejpam-5984	505	24	graph	graph	NOUN
ejpam-5984	505	25	with	with	ADP
ejpam-5984	505	26	maximum	maximum	ADJ
ejpam-5984	505	27	degree	degree	NOUN
ejpam-5984	505	28	∆.	∆.	NOUN
ejpam-5984	505	29	let	let	VERB
ejpam-5984	505	30	xi	xi	PRON
ejpam-5984	505	31	be	be	AUX
ejpam-5984	505	32	the	the	DET
ejpam-5984	505	33	central	central	ADJ
ejpam-5984	505	34	vertex	vertex	NOUN
ejpam-5984	505	35	of	of	ADP
ejpam-5984	505	36	the	the	DET
ejpam-5984	505	37	star	star	NOUN
ejpam-5984	505	38	si	si	PROPN
ejpam-5984	505	39	for	for	ADP
ejpam-5984	505	40	each	each	DET
ejpam-5984	505	41	i	i	PRON
ejpam-5984	505	42	∈	∈	PROPN
ejpam-5984	505	43	{	{	PUNCT
ejpam-5984	505	44	1	1	NUM
ejpam-5984	505	45	,	,	PUNCT
ejpam-5984	505	46	2	2	NUM
ejpam-5984	505	47	,	,	PUNCT
ejpam-5984	505	48	.	.	PUNCT
ejpam-5984	505	49	.	.	PUNCT
ejpam-5984	506	1	.	.	PUNCT
ejpam-5984	507	1	,	,	PUNCT
ejpam-5984	507	2	k	k	X
ejpam-5984	507	3	}	}	PUNCT
ejpam-5984	507	4	.	.	PUNCT
ejpam-5984	508	1	by	by	ADP
ejpam-5984	508	2	lemma	lemma	PROPN
ejpam-5984	508	3	1	1	NUM
ejpam-5984	508	4	,	,	PUNCT
ejpam-5984	508	5	idr(t	idr(t	NOUN
ejpam-5984	508	6	)	)	PUNCT
ejpam-5984	508	7	=	=	SYM
ejpam-5984	508	8	3k	3k	X
ejpam-5984	508	9	.	.	PUNCT
ejpam-5984	509	1	by	by	ADP
ejpam-5984	509	2	the	the	DET
ejpam-5984	509	3	construction	construction	NOUN
ejpam-5984	509	4	of	of	ADP
ejpam-5984	509	5	t	t	PROPN
ejpam-5984	509	6	,	,	PUNCT
ejpam-5984	509	7	there	there	PRON
ejpam-5984	509	8	is	be	VERB
ejpam-5984	509	9	one	one	NUM
ejpam-5984	509	10	of	of	ADP
ejpam-5984	509	11	si	si	X
ejpam-5984	509	12	,	,	PUNCT
ejpam-5984	509	13	say	say	VERB
ejpam-5984	509	14	sk	sk	INTJ
ejpam-5984	509	15	,	,	PUNCT
ejpam-5984	509	16	such	such	ADJ
ejpam-5984	509	17	that	that	SCONJ
ejpam-5984	509	18	∆	∆	PROPN
ejpam-5984	510	1	−	−	NOUN
ejpam-5984	510	2	1	1	NUM
ejpam-5984	510	3	of	of	ADP
ejpam-5984	510	4	its	its	PRON
ejpam-5984	510	5	leaves	leave	NOUN
ejpam-5984	510	6	are	be	AUX
ejpam-5984	510	7	the	the	DET
ejpam-5984	510	8	leaves	leave	NOUN
ejpam-5984	510	9	of	of	ADP
ejpam-5984	510	10	t	t	NOUN
ejpam-5984	510	11	,	,	PUNCT
ejpam-5984	510	12	and	and	CCONJ
ejpam-5984	510	13	the	the	DET
ejpam-5984	510	14	remaining	remain	VERB
ejpam-5984	510	15	leaf	leaf	NOUN
ejpam-5984	510	16	,	,	PUNCT
ejpam-5984	510	17	say	say	VERB
ejpam-5984	510	18	u	u	NOUN
ejpam-5984	510	19	,	,	PUNCT
ejpam-5984	510	20	has	have	AUX
ejpam-5984	510	21	degree	degree	NOUN
ejpam-5984	510	22	two	two	NUM
ejpam-5984	510	23	in	in	ADP
ejpam-5984	510	24	t	t	PROPN
ejpam-5984	510	25	.	.	PUNCT
ejpam-5984	511	1	clearly	clearly	ADV
ejpam-5984	511	2	,	,	PUNCT
ejpam-5984	511	3	u	u	NOUN
ejpam-5984	511	4	is	be	AUX
ejpam-5984	511	5	adjacent	adjacent	ADJ
ejpam-5984	511	6	to	to	ADP
ejpam-5984	511	7	the	the	DET
ejpam-5984	511	8	central	central	ADJ
ejpam-5984	511	9	vertex	vertex	NOUN
ejpam-5984	511	10	xk	xk	PROPN
ejpam-5984	511	11	of	of	ADP
ejpam-5984	511	12	sk	sk	PROPN
ejpam-5984	511	13	,	,	PUNCT
ejpam-5984	511	14	and	and	CCONJ
ejpam-5984	511	15	a	a	DET
ejpam-5984	511	16	leaf	leaf	NOUN
ejpam-5984	511	17	,	,	PUNCT
ejpam-5984	511	18	say	say	VERB
ejpam-5984	511	19	v	v	ADP
ejpam-5984	511	20	,	,	PUNCT
ejpam-5984	511	21	of	of	ADP
ejpam-5984	511	22	another	another	DET
ejpam-5984	511	23	star	star	NOUN
ejpam-5984	511	24	si	si	NOUN
ejpam-5984	511	25	,	,	PUNCT
ejpam-5984	511	26	say	say	VERB
ejpam-5984	511	27	sk−1	sk−1	ADV
ejpam-5984	511	28	.	.	PUNCT
ejpam-5984	512	1	to	to	PART
ejpam-5984	512	2	show	show	VERB
ejpam-5984	512	3	that	that	DET
ejpam-5984	512	4	st−idr(t	st−idr(t	NOUN
ejpam-5984	512	5	)	)	PUNCT
ejpam-5984	512	6	≤	≤	PUNCT
ejpam-5984	512	7	∆	∆	NOUN
ejpam-5984	512	8	,	,	PUNCT
ejpam-5984	512	9	let	let	VERB
ejpam-5984	512	10	t1	t1	NOUN
ejpam-5984	512	11	=	=	PUNCT
ejpam-5984	512	12	t	t	PROPN
ejpam-5984	512	13	−	−	PROPN
ejpam-5984	512	14	(	(	PUNCT
ejpam-5984	512	15	v	v	NOUN
ejpam-5984	512	16	(	(	PUNCT
ejpam-5984	512	17	sk	sk	PROPN
ejpam-5984	512	18	)	)	PUNCT
ejpam-5984	512	19	−	−	PROPN
ejpam-5984	512	20	{	{	PUNCT
ejpam-5984	512	21	u	u	NOUN
ejpam-5984	512	22	}	}	PUNCT
ejpam-5984	512	23	)	)	PUNCT
ejpam-5984	512	24	,	,	PUNCT
ejpam-5984	512	25	and	and	CCONJ
ejpam-5984	512	26	define	define	VERB
ejpam-5984	512	27	a	a	DET
ejpam-5984	512	28	function	function	NOUN
ejpam-5984	512	29	f1	f1	NOUN
ejpam-5984	512	30	:	:	PUNCT
ejpam-5984	512	31	v	v	NOUN
ejpam-5984	512	32	(	(	PUNCT
ejpam-5984	512	33	t1	t1	NOUN
ejpam-5984	512	34	)	)	PUNCT
ejpam-5984	512	35	→	→	SYM
ejpam-5984	512	36	{	{	PUNCT
ejpam-5984	512	37	0	0	NUM
ejpam-5984	512	38	,	,	PUNCT
ejpam-5984	512	39	1	1	NUM
ejpam-5984	512	40	,	,	PUNCT
ejpam-5984	512	41	2	2	NUM
ejpam-5984	512	42	,	,	PUNCT
ejpam-5984	512	43	3	3	NUM
ejpam-5984	512	44	}	}	PUNCT
ejpam-5984	512	45	by	by	ADP
ejpam-5984	512	46	f1(u	f1(u	NOUN
ejpam-5984	512	47	)	)	PUNCT
ejpam-5984	512	48	=	=	SYM
ejpam-5984	512	49	2	2	NUM
ejpam-5984	512	50	,	,	PUNCT
ejpam-5984	512	51	f1(xj	f1(xj	PROPN
ejpam-5984	512	52	)	)	PUNCT
ejpam-5984	512	53	=	=	SYM
ejpam-5984	512	54	3	3	NUM
ejpam-5984	512	55	for	for	ADP
ejpam-5984	512	56	j	j	PROPN
ejpam-5984	512	57	∈	∈	PROPN
ejpam-5984	512	58	{	{	PUNCT
ejpam-5984	512	59	1	1	NUM
ejpam-5984	512	60	,	,	PUNCT
ejpam-5984	512	61	2	2	NUM
ejpam-5984	512	62	,	,	PUNCT
ejpam-5984	512	63	3	3	NUM
ejpam-5984	512	64	,	,	PUNCT
ejpam-5984	512	65	.	.	PUNCT
ejpam-5984	512	66	.	.	PUNCT
ejpam-5984	513	1	.	.	PUNCT
ejpam-5984	514	1	,	,	PUNCT
ejpam-5984	515	1	k	k	PROPN
ejpam-5984	516	1	−	−	PROPN
ejpam-5984	516	2	1	1	NUM
ejpam-5984	516	3	}	}	PUNCT
ejpam-5984	516	4	,	,	PUNCT
ejpam-5984	516	5	and	and	CCONJ
ejpam-5984	516	6	f1(x	f1(x	NUM
ejpam-5984	516	7	)	)	PUNCT
ejpam-5984	516	8	=	=	SYM
ejpam-5984	516	9	0	0	NUM
ejpam-5984	516	10	for	for	ADP
ejpam-5984	516	11	x	x	PROPN
ejpam-5984	516	12	∈	∈	PROPN
ejpam-5984	516	13	v	v	ADP
ejpam-5984	516	14	(	(	PUNCT
ejpam-5984	516	15	t1	t1	NOUN
ejpam-5984	516	16	)	)	PUNCT
ejpam-5984	516	17	−	−	PROPN
ejpam-5984	516	18	{	{	PUNCT
ejpam-5984	516	19	u	u	NOUN
ejpam-5984	516	20	,	,	PUNCT
ejpam-5984	516	21	x1	x1	PROPN
ejpam-5984	516	22	,	,	PUNCT
ejpam-5984	516	23	x2	x2	PROPN
ejpam-5984	516	24	,	,	PUNCT
ejpam-5984	516	25	.	.	PUNCT
ejpam-5984	516	26	.	.	PUNCT
ejpam-5984	516	27	.	.	PUNCT
ejpam-5984	517	1	,	,	PUNCT
ejpam-5984	517	2	xk−1	xk−1	PROPN
ejpam-5984	517	3	}	}	PUNCT
ejpam-5984	517	4	.	.	PUNCT
ejpam-5984	518	1	clearly	clearly	ADV
ejpam-5984	518	2	,	,	PUNCT
ejpam-5984	518	3	f1	f1	PROPN
ejpam-5984	518	4	is	be	AUX
ejpam-5984	518	5	an	an	DET
ejpam-5984	518	6	idrd	idrd	ADJ
ejpam-5984	518	7	-	-	PUNCT
ejpam-5984	518	8	function	function	NOUN
ejpam-5984	518	9	of	of	ADP
ejpam-5984	518	10	t1	t1	NOUN
ejpam-5984	518	11	with	with	ADP
ejpam-5984	518	12	weight	weight	NOUN
ejpam-5984	518	13	3k	3k	X
ejpam-5984	518	14	−	−	PROPN
ejpam-5984	518	15	1	1	NUM
ejpam-5984	518	16	,	,	PUNCT
ejpam-5984	518	17	so	so	SCONJ
ejpam-5984	518	18	we	we	PRON
ejpam-5984	518	19	have	have	VERB
ejpam-5984	518	20	st−idr(t	st−idr(t	NOUN
ejpam-5984	518	21	)	)	PUNCT
ejpam-5984	518	22	≤	≤	PUNCT
ejpam-5984	519	1	∆.	∆.	ADV
ejpam-5984	519	2	now	now	ADV
ejpam-5984	519	3	,	,	PUNCT
ejpam-5984	519	4	we	we	PRON
ejpam-5984	519	5	show	show	VERB
ejpam-5984	519	6	that	that	SCONJ
ejpam-5984	519	7	st−idr(t	st−idr(t	NOUN
ejpam-5984	519	8	)	)	PUNCT
ejpam-5984	519	9	≥	≥	NOUN
ejpam-5984	520	1	∆.	∆.	X
ejpam-5984	520	2	let	let	VERB
ejpam-5984	520	3	s	s	PRON
ejpam-5984	520	4	be	be	AUX
ejpam-5984	520	5	a	a	DET
ejpam-5984	520	6	vertex	vertex	NOUN
ejpam-5984	520	7	set	set	NOUN
ejpam-5984	520	8	of	of	ADP
ejpam-5984	520	9	t	t	NOUN
ejpam-5984	521	1	such	such	ADJ
ejpam-5984	521	2	that	that	DET
ejpam-5984	521	3	idr(t	idr(t	NOUN
ejpam-5984	521	4	−	−	PROPN
ejpam-5984	521	5	s	s	NOUN
ejpam-5984	521	6	)	)	PUNCT
ejpam-5984	521	7	≤	≤	NOUN
ejpam-5984	521	8	3k	3k	PRON
ejpam-5984	521	9	−	−	NOUN
ejpam-5984	522	1	1	1	X
ejpam-5984	522	2	.	.	PUNCT
ejpam-5984	523	1	among	among	ADP
ejpam-5984	523	2	all	all	DET
ejpam-5984	523	3	such	such	ADJ
ejpam-5984	523	4	sets	set	NOUN
ejpam-5984	523	5	,	,	PUNCT
ejpam-5984	523	6	we	we	PRON
ejpam-5984	523	7	choose	choose	VERB
ejpam-5984	523	8	s	s	PRON
ejpam-5984	523	9	to	to	PART
ejpam-5984	523	10	have	have	VERB
ejpam-5984	523	11	minimum	minimum	ADJ
ejpam-5984	523	12	cardinality	cardinality	NOUN
ejpam-5984	523	13	.	.	PUNCT
ejpam-5984	524	1	we	we	PRON
ejpam-5984	524	2	show	show	VERB
ejpam-5984	524	3	that	that	SCONJ
ejpam-5984	524	4	|s|	|s|	NOUN
ejpam-5984	524	5	≥	≥	NOUN
ejpam-5984	524	6	∆.	∆.	NOUN
ejpam-5984	524	7	suppose	suppose	VERB
ejpam-5984	524	8	,	,	PUNCT
ejpam-5984	524	9	to	to	ADP
ejpam-5984	524	10	the	the	DET
ejpam-5984	524	11	contrary	contrary	NOUN
ejpam-5984	524	12	,	,	PUNCT
ejpam-5984	524	13	that	that	PRON
ejpam-5984	524	14	|s|	|s|	VERB
ejpam-5984	524	15	≤	≤	NUM
ejpam-5984	524	16	∆−1	∆−1	PROPN
ejpam-5984	524	17	.	.	PUNCT
ejpam-5984	525	1	let	let	VERB
ejpam-5984	525	2	f	f	PROPN
ejpam-5984	525	3	=	=	SYM
ejpam-5984	525	4	(	(	PUNCT
ejpam-5984	525	5	v0,∅	v0,∅	PROPN
ejpam-5984	525	6	,	,	PUNCT
ejpam-5984	525	7	v2	v2	PROPN
ejpam-5984	525	8	,	,	PUNCT
ejpam-5984	525	9	v3	v3	PROPN
ejpam-5984	525	10	)	)	PUNCT
ejpam-5984	525	11	be	be	VERB
ejpam-5984	525	12	a	a	DET
ejpam-5984	525	13	idr(t−s)function	idr(t−s)function	NOUN
ejpam-5984	525	14	,	,	PUNCT
ejpam-5984	525	15	and	and	CCONJ
ejpam-5984	526	1	so	so	ADV
ejpam-5984	526	2	ω(f	ω(f	X
ejpam-5984	526	3	)	)	PUNCT
ejpam-5984	526	4	≤	≤	NOUN
ejpam-5984	527	1	3k	3k	X
ejpam-5984	527	2	−	−	NOUN
ejpam-5984	528	1	1	1	X
ejpam-5984	528	2	.	.	PUNCT
ejpam-5984	529	1	choose	choose	VERB
ejpam-5984	529	2	f	f	PROPN
ejpam-5984	529	3	such	such	ADJ
ejpam-5984	529	4	that	that	SCONJ
ejpam-5984	529	5	f(xk	f(xk	PROPN
ejpam-5984	529	6	)	)	PUNCT
ejpam-5984	529	7	is	be	AUX
ejpam-5984	529	8	as	as	ADV
ejpam-5984	529	9	large	large	ADJ
ejpam-5984	529	10	as	as	ADP
ejpam-5984	529	11	possible	possible	ADJ
ejpam-5984	529	12	when	when	SCONJ
ejpam-5984	529	13	xk	xk	PROPN
ejpam-5984	529	14	/∈	/∈	PROPN
ejpam-5984	529	15	s.	s.	PROPN
ejpam-5984	529	16	let	let	VERB
ejpam-5984	529	17	t	t	NOUN
ejpam-5984	529	18	′	′	NUM
ejpam-5984	530	1	=	=	PUNCT
ejpam-5984	530	2	t	t	PROPN
ejpam-5984	530	3	−	−	PROPN
ejpam-5984	530	4	v	v	NOUN
ejpam-5984	530	5	(	(	PUNCT
ejpam-5984	530	6	sk	sk	PROPN
ejpam-5984	530	7	)	)	PUNCT
ejpam-5984	530	8	,	,	PUNCT
ejpam-5984	530	9	s′	s′	ADJ
ejpam-5984	530	10	=	=	SYM
ejpam-5984	530	11	s	s	X
ejpam-5984	530	12	∩	∩	ADJ
ejpam-5984	530	13	v	v	X
ejpam-5984	530	14	(	(	PUNCT
ejpam-5984	530	15	t	t	NOUN
ejpam-5984	530	16	′	′	NUM
ejpam-5984	530	17	)	)	PUNCT
ejpam-5984	530	18	and	and	CCONJ
ejpam-5984	530	19	wi	wi	PROPN
ejpam-5984	530	20	=	=	PROPN
ejpam-5984	530	21	s	s	PROPN
ejpam-5984	530	22	∩	∩	ADJ
ejpam-5984	530	23	v	v	NOUN
ejpam-5984	530	24	(	(	PUNCT
ejpam-5984	530	25	si	si	NOUN
ejpam-5984	530	26	)	)	PUNCT
ejpam-5984	530	27	for	for	ADP
ejpam-5984	530	28	i	i	PROPN
ejpam-5984	530	29	∈	∈	PROPN
ejpam-5984	530	30	{	{	PUNCT
ejpam-5984	530	31	1	1	NUM
ejpam-5984	530	32	,	,	PUNCT
ejpam-5984	530	33	2	2	NUM
ejpam-5984	530	34	,	,	PUNCT
ejpam-5984	530	35	.	.	PUNCT
ejpam-5984	530	36	.	.	PUNCT
ejpam-5984	531	1	.	.	PUNCT
ejpam-5984	532	1	,	,	PUNCT
ejpam-5984	532	2	k	k	X
ejpam-5984	532	3	}	}	PUNCT
ejpam-5984	532	4	.	.	PUNCT
ejpam-5984	533	1	since	since	SCONJ
ejpam-5984	533	2	t	t	NOUN
ejpam-5984	533	3	′	′	NUM
ejpam-5984	533	4	∈	∈	PROPN
ejpam-5984	533	5	tk−1,∆	tk−1,∆	NOUN
ejpam-5984	533	6	,	,	PUNCT
ejpam-5984	533	7	by	by	ADP
ejpam-5984	533	8	lemma	lemma	PROPN
ejpam-5984	533	9	1	1	NUM
ejpam-5984	533	10	,	,	PUNCT
ejpam-5984	533	11	idr(t	idr(t	NOUN
ejpam-5984	533	12	′	′	NOUN
ejpam-5984	533	13	)	)	PUNCT
ejpam-5984	533	14	=	=	PUNCT
ejpam-5984	533	15	3k	3k	X
ejpam-5984	534	1	−	−	NOUN
ejpam-5984	534	2	3	3	NUM
ejpam-5984	534	3	and	and	CCONJ
ejpam-5984	534	4	from	from	ADP
ejpam-5984	534	5	the	the	DET
ejpam-5984	534	6	induction	induction	NOUN
ejpam-5984	534	7	hypothesis	hypothesis	NOUN
ejpam-5984	534	8	we	we	PRON
ejpam-5984	534	9	have	have	VERB
ejpam-5984	534	10	st−idr(t	st−idr(t	NOUN
ejpam-5984	534	11	′	′	NUM
ejpam-5984	534	12	)	)	PUNCT
ejpam-5984	535	1	=	=	PUNCT
ejpam-5984	536	1	∆.	∆.	NOUN
ejpam-5984	536	2	since	since	SCONJ
ejpam-5984	536	3	|s|	|s|	NOUN
ejpam-5984	536	4	≤	≤	PROPN
ejpam-5984	536	5	∆	∆	X
ejpam-5984	536	6	−	−	PROPN
ejpam-5984	536	7	1	1	NUM
ejpam-5984	536	8	,	,	PUNCT
ejpam-5984	536	9	|v	|v	PROPN
ejpam-5984	536	10	(	(	PUNCT
ejpam-5984	536	11	sk	sk	PROPN
ejpam-5984	536	12	)	)	PUNCT
ejpam-5984	536	13	−	−	PROPN
ejpam-5984	536	14	wk|	wk|	NOUN
ejpam-5984	536	15	≥	≥	NOUN
ejpam-5984	536	16	2	2	NUM
ejpam-5984	536	17	.	.	PUNCT
ejpam-5984	536	18	to	to	ADP
ejpam-5984	536	19	independent	independent	ADJ
ejpam-5984	536	20	double	double	ADJ
ejpam-5984	536	21	roman	roman	NOUN
ejpam-5984	536	22	dominate	dominate	VERB
ejpam-5984	536	23	the	the	DET
ejpam-5984	536	24	vertices	vertex	NOUN
ejpam-5984	536	25	in	in	ADP
ejpam-5984	536	26	v	v	PROPN
ejpam-5984	536	27	(	(	PUNCT
ejpam-5984	536	28	sk)−wk	sk)−wk	NOUN
ejpam-5984	536	29	,	,	PUNCT
ejpam-5984	536	30	we	we	PRON
ejpam-5984	536	31	have	have	AUX
ejpam-5984	536	32	∑	∑	PROPN
ejpam-5984	536	33	x∈v	x∈v	PROPN
ejpam-5984	536	34	(	(	PUNCT
ejpam-5984	536	35	sk)−wk	sk)−wk	PROPN
ejpam-5984	536	36	f(x	f(x	PROPN
ejpam-5984	536	37	)	)	PUNCT
ejpam-5984	536	38	≥	≥	NOUN
ejpam-5984	536	39	2	2	NUM
ejpam-5984	536	40	.	.	PUNCT
ejpam-5984	536	41	first	first	ADV
ejpam-5984	536	42	let	let	VERB
ejpam-5984	536	43	∑	∑	PROPN
ejpam-5984	536	44	x∈v	x∈v	PROPN
ejpam-5984	536	45	(	(	PUNCT
ejpam-5984	536	46	sk)−wk	sk)−wk	PROPN
ejpam-5984	536	47	f(x	f(x	PROPN
ejpam-5984	536	48	)	)	PUNCT
ejpam-5984	536	49	=	=	SYM
ejpam-5984	537	1	2	2	X
ejpam-5984	537	2	.	.	PUNCT
ejpam-5984	537	3	then	then	ADV
ejpam-5984	537	4	it	it	PRON
ejpam-5984	537	5	is	be	AUX
ejpam-5984	537	6	easy	easy	ADJ
ejpam-5984	537	7	to	to	PART
ejpam-5984	537	8	verify	verify	VERB
ejpam-5984	537	9	that	that	SCONJ
ejpam-5984	537	10	|v	|v	PROPN
ejpam-5984	537	11	(	(	PUNCT
ejpam-5984	537	12	sk)−wk|	sk)−wk|	NOUN
ejpam-5984	537	13	=	=	SYM
ejpam-5984	537	14	2	2	NUM
ejpam-5984	537	15	,	,	PUNCT
ejpam-5984	537	16	that	that	ADV
ejpam-5984	537	17	is	be	AUX
ejpam-5984	537	18	,	,	PUNCT
ejpam-5984	537	19	s	s	PART
ejpam-5984	537	20	=	=	SYM
ejpam-5984	537	21	wk	wk	PROPN
ejpam-5984	537	22	.	.	PUNCT
ejpam-5984	538	1	moreover	moreover	ADV
ejpam-5984	538	2	,	,	PUNCT
ejpam-5984	538	3	v	v	INTJ
ejpam-5984	538	4	(	(	PUNCT
ejpam-5984	538	5	sk)−wk	sk)−wk	PROPN
ejpam-5984	538	6	consists	consist	VERB
ejpam-5984	538	7	of	of	ADP
ejpam-5984	538	8	u	u	NOUN
ejpam-5984	538	9	and	and	CCONJ
ejpam-5984	538	10	xk	xk	PROPN
ejpam-5984	538	11	,	,	PUNCT
ejpam-5984	538	12	or	or	CCONJ
ejpam-5984	538	13	u	u	NOUN
ejpam-5984	538	14	and	and	CCONJ
ejpam-5984	538	15	a	a	DET
ejpam-5984	538	16	leaf	leaf	NOUN
ejpam-5984	538	17	-	-	PUNCT
ejpam-5984	538	18	neighbour	neighbour	NOUN
ejpam-5984	538	19	of	of	ADP
ejpam-5984	538	20	xk	xk	PROPN
ejpam-5984	538	21	other	other	ADJ
ejpam-5984	538	22	than	than	ADP
ejpam-5984	538	23	u.	u.	VERB
ejpam-5984	538	24	if	if	SCONJ
ejpam-5984	538	25	v	v	X
ejpam-5984	538	26	(	(	PUNCT
ejpam-5984	538	27	sk)−wk	sk)−wk	PROPN
ejpam-5984	538	28	consists	consist	VERB
ejpam-5984	538	29	of	of	ADP
ejpam-5984	538	30	u	u	NOUN
ejpam-5984	538	31	and	and	CCONJ
ejpam-5984	538	32	xk	xk	NOUN
ejpam-5984	538	33	,	,	PUNCT
ejpam-5984	538	34	then	then	ADV
ejpam-5984	538	35	t	t	PROPN
ejpam-5984	538	36	−s	−s	PROPN
ejpam-5984	538	37	∈	∈	PROPN
ejpam-5984	538	38	lk−1,∆	lk−1,∆	NOUN
ejpam-5984	538	39	and	and	CCONJ
ejpam-5984	538	40	by	by	ADP
ejpam-5984	538	41	lemma	lemma	PROPN
ejpam-5984	538	42	2	2	NUM
ejpam-5984	538	43	,	,	PUNCT
ejpam-5984	538	44	we	we	PRON
ejpam-5984	538	45	have	have	VERB
ejpam-5984	538	46	idr(t	idr(t	NOUN
ejpam-5984	538	47	−s	−s	NOUN
ejpam-5984	538	48	)	)	PUNCT
ejpam-5984	538	49	=	=	SYM
ejpam-5984	539	1	3(k−	3(k−	NUM
ejpam-5984	539	2	1)+3	1)+3	NUM
ejpam-5984	539	3	which	which	PRON
ejpam-5984	539	4	is	be	AUX
ejpam-5984	539	5	a	a	DET
ejpam-5984	539	6	contradiction	contradiction	NOUN
ejpam-5984	539	7	.	.	PUNCT
ejpam-5984	540	1	hence	hence	ADV
ejpam-5984	540	2	,	,	PUNCT
ejpam-5984	540	3	we	we	PRON
ejpam-5984	540	4	assume	assume	VERB
ejpam-5984	540	5	that	that	SCONJ
ejpam-5984	540	6	v	v	INTJ
ejpam-5984	540	7	(	(	PUNCT
ejpam-5984	540	8	sk)−wk	sk)−wk	PROPN
ejpam-5984	540	9	consists	consist	VERB
ejpam-5984	540	10	of	of	ADP
ejpam-5984	540	11	u	u	NOUN
ejpam-5984	540	12	and	and	CCONJ
ejpam-5984	540	13	a	a	DET
ejpam-5984	540	14	leaf	leaf	NOUN
ejpam-5984	540	15	-	-	PUNCT
ejpam-5984	540	16	neighbour	neighbour	NOUN
ejpam-5984	540	17	y	y	PROPN
ejpam-5984	540	18	of	of	ADP
ejpam-5984	540	19	xk	xk	PROPN
ejpam-5984	540	20	.	.	PUNCT
ejpam-5984	541	1	then	then	ADV
ejpam-5984	541	2	we	we	PRON
ejpam-5984	541	3	must	must	AUX
ejpam-5984	541	4	have	have	VERB
ejpam-5984	541	5	f(y	f(y	NOUN
ejpam-5984	541	6	)	)	PUNCT
ejpam-5984	541	7	=	=	SYM
ejpam-5984	541	8	2	2	NUM
ejpam-5984	541	9	,	,	PUNCT
ejpam-5984	541	10	f(u	f(u	PROPN
ejpam-5984	541	11	)	)	PUNCT
ejpam-5984	541	12	=	=	SYM
ejpam-5984	541	13	0	0	NUM
ejpam-5984	541	14	and	and	CCONJ
ejpam-5984	541	15	f(v	f(v	NOUN
ejpam-5984	541	16	)	)	PUNCT
ejpam-5984	541	17	=	=	SYM
ejpam-5984	542	1	3	3	X
ejpam-5984	542	2	.	.	PUNCT
ejpam-5984	542	3	then	then	ADV
ejpam-5984	542	4	the	the	DET
ejpam-5984	542	5	restriction	restriction	NOUN
ejpam-5984	542	6	of	of	ADP
ejpam-5984	542	7	f	f	PROPN
ejpam-5984	542	8	to	to	ADP
ejpam-5984	542	9	v	v	PROPN
ejpam-5984	542	10	(	(	PUNCT
ejpam-5984	542	11	t	t	NOUN
ejpam-5984	542	12	′	′	NUM
ejpam-5984	542	13	)	)	PUNCT
ejpam-5984	542	14	,	,	PUNCT
ejpam-5984	542	15	say	say	VERB
ejpam-5984	542	16	f	f	PROPN
ejpam-5984	542	17	′	′	PROPN
ejpam-5984	542	18	,	,	PUNCT
ejpam-5984	542	19	is	be	AUX
ejpam-5984	542	20	an	an	DET
ejpam-5984	542	21	idrd	idrd	ADJ
ejpam-5984	542	22	-	-	PUNCT
ejpam-5984	542	23	function	function	NOUN
ejpam-5984	542	24	of	of	ADP
ejpam-5984	542	25	t	t	NOUN
ejpam-5984	542	26	′	′	NUM
ejpam-5984	542	27	which	which	PRON
ejpam-5984	542	28	is	be	AUX
ejpam-5984	542	29	not	not	PART
ejpam-5984	542	30	an	an	DET
ejpam-5984	542	31	idr(t	idr(t	NOUN
ejpam-5984	542	32	′)-function	′)-function	NOUN
ejpam-5984	542	33	by	by	ADP
ejpam-5984	542	34	lemma	lemma	PROPN
ejpam-5984	542	35	1	1	NUM
ejpam-5984	542	36	.	.	PUNCT
ejpam-5984	543	1	it	it	PRON
ejpam-5984	543	2	follows	follow	VERB
ejpam-5984	543	3	that	that	SCONJ
ejpam-5984	543	4	idr(t	idr(t	NOUN
ejpam-5984	543	5	−	−	PROPN
ejpam-5984	543	6	s	s	NOUN
ejpam-5984	543	7	)	)	PUNCT
ejpam-5984	543	8	=	=	PUNCT
ejpam-5984	543	9	ω(f	ω(f	X
ejpam-5984	543	10	)	)	PUNCT
ejpam-5984	543	11	=	=	SYM
ejpam-5984	544	1	ω(f	ω(f	PROPN
ejpam-5984	544	2	′	′	NUM
ejpam-5984	544	3	)	)	PUNCT
ejpam-5984	545	1	+	+	CCONJ
ejpam-5984	545	2	2	2	NUM
ejpam-5984	545	3	≥	≥	NOUN
ejpam-5984	545	4	(	(	PUNCT
ejpam-5984	545	5	3(k	3(k	NUM
ejpam-5984	545	6	−	−	NUM
ejpam-5984	545	7	1	1	NUM
ejpam-5984	545	8	)	)	PUNCT
ejpam-5984	545	9	+	+	CCONJ
ejpam-5984	545	10	1	1	X
ejpam-5984	545	11	)	)	PUNCT
ejpam-5984	545	12	+	+	CCONJ
ejpam-5984	545	13	2	2	NUM
ejpam-5984	545	14	=	=	SYM
ejpam-5984	545	15	3k	3k	NOUN
ejpam-5984	545	16	leading	lead	VERB
ejpam-5984	545	17	to	to	ADP
ejpam-5984	545	18	a	a	DET
ejpam-5984	545	19	contradiction	contradiction	NOUN
ejpam-5984	545	20	.	.	PUNCT
ejpam-5984	546	1	assume	assume	VERB
ejpam-5984	546	2	that	that	SCONJ
ejpam-5984	546	3	∑	∑	PROPN
ejpam-5984	546	4	x∈v	x∈v	PROPN
ejpam-5984	546	5	(	(	PUNCT
ejpam-5984	546	6	sk)−wk	sk)−wk	PROPN
ejpam-5984	546	7	f(x	f(x	PROPN
ejpam-5984	546	8	)	)	PUNCT
ejpam-5984	546	9	≥	≥	NOUN
ejpam-5984	547	1	3	3	NUM
ejpam-5984	547	2	.	.	PUNCT
ejpam-5984	548	1	if	if	SCONJ
ejpam-5984	548	2	|s	|s	PROPN
ejpam-5984	548	3	∩	∩	NOUN
ejpam-5984	548	4	{	{	PUNCT
ejpam-5984	548	5	u	u	NOUN
ejpam-5984	548	6	,	,	PUNCT
ejpam-5984	548	7	v}|	v}|	ADJ
ejpam-5984	548	8	≥	≥	NUM
ejpam-5984	548	9	1	1	NUM
ejpam-5984	548	10	or	or	CCONJ
ejpam-5984	548	11	f(v	f(v	NOUN
ejpam-5984	548	12	)	)	PUNCT
ejpam-5984	548	13	≥	≥	NOUN
ejpam-5984	548	14	2	2	NUM
ejpam-5984	548	15	or	or	CCONJ
ejpam-5984	548	16	f(u	f(u	PROPN
ejpam-5984	548	17	)	)	PUNCT
ejpam-5984	548	18	=	=	SYM
ejpam-5984	548	19	0	0	NUM
ejpam-5984	548	20	,	,	PUNCT
ejpam-5984	548	21	then	then	ADV
ejpam-5984	548	22	the	the	DET
ejpam-5984	548	23	restriction	restriction	NOUN
ejpam-5984	548	24	of	of	ADP
ejpam-5984	548	25	f	f	PROPN
ejpam-5984	548	26	on	on	ADP
ejpam-5984	548	27	v	v	PROPN
ejpam-5984	548	28	(	(	PUNCT
ejpam-5984	548	29	t	t	NOUN
ejpam-5984	548	30	′	′	NUM
ejpam-5984	548	31	−	−	PROPN
ejpam-5984	548	32	s′	s′	NOUN
ejpam-5984	548	33	)	)	PUNCT
ejpam-5984	548	34	is	be	AUX
ejpam-5984	548	35	an	an	DET
ejpam-5984	548	36	idrd	idrd	ADJ
ejpam-5984	548	37	-	-	PUNCT
ejpam-5984	548	38	function	function	NOUN
ejpam-5984	548	39	of	of	ADP
ejpam-5984	548	40	t	t	PROPN
ejpam-5984	548	41	′	′	NUM
ejpam-5984	548	42	−	−	PROPN
ejpam-5984	548	43	s′	s′	ADJ
ejpam-5984	548	44	with	with	ADP
ejpam-5984	548	45	weight	weight	NOUN
ejpam-5984	548	46	at	at	ADP
ejpam-5984	548	47	most	most	ADV
ejpam-5984	548	48	3k	3k	NUM
ejpam-5984	548	49	−	−	NOUN
ejpam-5984	548	50	4	4	NUM
ejpam-5984	549	1	and	and	CCONJ
ejpam-5984	549	2	so	so	ADV
ejpam-5984	549	3	idr(t	idr(t	ADJ
ejpam-5984	549	4	′	′	NUM
ejpam-5984	549	5	−	−	NOUN
ejpam-5984	549	6	s′	s′	NOUN
ejpam-5984	549	7	)	)	PUNCT
ejpam-5984	549	8	≤	≤	PUNCT
ejpam-5984	550	1	3k	3k	NUM
ejpam-5984	550	2	−	−	NOUN
ejpam-5984	551	1	4	4	X
ejpam-5984	551	2	.	.	PUNCT
ejpam-5984	552	1	on	on	ADP
ejpam-5984	552	2	the	the	DET
ejpam-5984	552	3	other	other	ADJ
ejpam-5984	552	4	hand	hand	NOUN
ejpam-5984	552	5	,	,	PUNCT
ejpam-5984	552	6	since	since	SCONJ
ejpam-5984	552	7	|s′|	|s′|	NOUN
ejpam-5984	552	8	≤	≤	NUM
ejpam-5984	552	9	|s|	|s|	NOUN
ejpam-5984	552	10	≤	≤	NOUN
ejpam-5984	552	11	∆	∆	X
ejpam-5984	552	12	−	−	NOUN
ejpam-5984	552	13	1	1	NUM
ejpam-5984	552	14	and	and	CCONJ
ejpam-5984	552	15	st−idr(t	st−idr(t	NOUN
ejpam-5984	552	16	′	′	NOUN
ejpam-5984	552	17	)	)	PUNCT
ejpam-5984	552	18	=	=	PUNCT
ejpam-5984	552	19	∆	∆	PROPN
ejpam-5984	552	20	,	,	PUNCT
ejpam-5984	552	21	we	we	PRON
ejpam-5984	552	22	have	have	VERB
ejpam-5984	552	23	idr(t	idr(t	NUM
ejpam-5984	552	24	′	′	NUM
ejpam-5984	552	25	−	−	NOUN
ejpam-5984	552	26	s′	s′	NOUN
ejpam-5984	552	27	)	)	PUNCT
ejpam-5984	552	28	≤	≤	PUNCT
ejpam-5984	552	29	3k	3k	NUM
ejpam-5984	552	30	−	−	PROPN
ejpam-5984	552	31	4	4	NUM
ejpam-5984	552	32	,	,	PUNCT
ejpam-5984	552	33	a	a	DET
ejpam-5984	552	34	contradiction	contradiction	NOUN
ejpam-5984	552	35	.	.	PUNCT
ejpam-5984	553	1	so	so	ADV
ejpam-5984	553	2	we	we	PRON
ejpam-5984	553	3	may	may	AUX
ejpam-5984	553	4	assume	assume	VERB
ejpam-5984	553	5	that	that	SCONJ
ejpam-5984	553	6	|s	|s	PROPN
ejpam-5984	553	7	∩	∩	NOUN
ejpam-5984	553	8	{	{	PUNCT
ejpam-5984	553	9	u	u	NOUN
ejpam-5984	553	10	,	,	PUNCT
ejpam-5984	553	11	v}|	v}|	NOUN
ejpam-5984	553	12	=	=	SYM
ejpam-5984	553	13	0	0	NUM
ejpam-5984	553	14	,	,	PUNCT
ejpam-5984	553	15	f(v	f(v	NOUN
ejpam-5984	553	16	)	)	PUNCT
ejpam-5984	553	17	=	=	SYM
ejpam-5984	553	18	0	0	NUM
ejpam-5984	553	19	and	and	CCONJ
ejpam-5984	553	20	f(u	f(u	PROPN
ejpam-5984	553	21	)	)	PUNCT
ejpam-5984	553	22	≥	≥	NOUN
ejpam-5984	553	23	2	2	NUM
ejpam-5984	553	24	.	.	PUNCT
ejpam-5984	554	1	by	by	ADP
ejpam-5984	554	2	definition	definition	NOUN
ejpam-5984	554	3	,	,	PUNCT
ejpam-5984	554	4	we	we	PRON
ejpam-5984	554	5	must	must	AUX
ejpam-5984	554	6	have	have	VERB
ejpam-5984	554	7	f(v	f(v	NOUN
ejpam-5984	554	8	)	)	PUNCT
ejpam-5984	555	1	=	=	SYM
ejpam-5984	555	2	0	0	X
ejpam-5984	555	3	.	.	PUNCT
ejpam-5984	556	1	we	we	PRON
ejpam-5984	556	2	distinguish	distinguish	VERB
ejpam-5984	556	3	two	two	NUM
ejpam-5984	556	4	cases	case	NOUN
ejpam-5984	556	5	.	.	PUNCT
ejpam-5984	557	1	s.	s.	PROPN
ejpam-5984	557	2	m.	m.	PROPN
ejpam-5984	557	3	sheikholeslami	sheikholeslami	PROPN
ejpam-5984	557	4	et	et	PROPN
ejpam-5984	557	5	al	al	PROPN
ejpam-5984	557	6	.	.	PUNCT
ejpam-5984	557	7	/	/	SYM
ejpam-5984	557	8	eur	eur	PROPN
ejpam-5984	557	9	.	.	PUNCT
ejpam-5984	558	1	j.	j.	PROPN
ejpam-5984	558	2	pure	pure	PROPN
ejpam-5984	558	3	appl	appl	PROPN
ejpam-5984	558	4	.	.	PROPN
ejpam-5984	558	5	math	math	PROPN
ejpam-5984	558	6	,	,	PUNCT
ejpam-5984	558	7	18	18	NUM
ejpam-5984	558	8	(	(	PUNCT
ejpam-5984	558	9	2	2	NUM
ejpam-5984	558	10	)	)	PUNCT
ejpam-5984	558	11	(	(	PUNCT
ejpam-5984	558	12	2025	2025	NUM
ejpam-5984	558	13	)	)	PUNCT
ejpam-5984	558	14	,	,	PUNCT
ejpam-5984	558	15	5984	5984	NUM
ejpam-5984	558	16	13	13	NUM
ejpam-5984	558	17	of	of	ADP
ejpam-5984	558	18	16	16	NUM
ejpam-5984	558	19	case	case	NOUN
ejpam-5984	558	20	1	1	NUM
ejpam-5984	558	21	.	.	PUNCT
ejpam-5984	559	1	f(u	f(u	NOUN
ejpam-5984	559	2	)	)	PUNCT
ejpam-5984	560	1	=	=	SYM
ejpam-5984	560	2	2	2	X
ejpam-5984	560	3	.	.	PUNCT
ejpam-5984	561	1	it	it	PRON
ejpam-5984	561	2	follows	follow	VERB
ejpam-5984	561	3	from	from	ADP
ejpam-5984	561	4	∑	∑	PROPN
ejpam-5984	561	5	x∈v	x∈v	PROPN
ejpam-5984	561	6	(	(	PUNCT
ejpam-5984	561	7	sk)−wk	sk)−wk	PROPN
ejpam-5984	561	8	f(x	f(x	PROPN
ejpam-5984	561	9	)	)	PUNCT
ejpam-5984	561	10	≥	≥	NOUN
ejpam-5984	561	11	3	3	NUM
ejpam-5984	561	12	and	and	CCONJ
ejpam-5984	561	13	the	the	DET
ejpam-5984	561	14	fact	fact	NOUN
ejpam-5984	561	15	v1	v1	NOUN
ejpam-5984	561	16	=	=	SYM
ejpam-5984	561	17	∅	∅	NOUN
ejpam-5984	561	18	that	that	PRON
ejpam-5984	561	19	∑	∑	PUNCT
ejpam-5984	561	20	x∈v	x∈v	PROPN
ejpam-5984	561	21	(	(	PUNCT
ejpam-5984	561	22	sk)−wk	sk)−wk	PROPN
ejpam-5984	561	23	f(x	f(x	PROPN
ejpam-5984	561	24	)	)	PUNCT
ejpam-5984	561	25	≥	≥	NOUN
ejpam-5984	561	26	4	4	NUM
ejpam-5984	561	27	.	.	PUNCT
ejpam-5984	561	28	since	since	SCONJ
ejpam-5984	561	29	f(v	f(v	NOUN
ejpam-5984	561	30	)	)	PUNCT
ejpam-5984	562	1	=	=	SYM
ejpam-5984	562	2	0	0	NUM
ejpam-5984	562	3	,	,	PUNCT
ejpam-5984	562	4	v	v	NOUN
ejpam-5984	562	5	has	have	VERB
ejpam-5984	562	6	a	a	DET
ejpam-5984	562	7	neighbor	neighbor	NOUN
ejpam-5984	562	8	w	w	NOUN
ejpam-5984	562	9	with	with	ADP
ejpam-5984	562	10	f(w	f(w	PROPN
ejpam-5984	562	11	)	)	PUNCT
ejpam-5984	562	12	≥	≥	NOUN
ejpam-5984	562	13	2	2	NUM
ejpam-5984	562	14	.	.	PUNCT
ejpam-5984	562	15	define	define	VERB
ejpam-5984	562	16	the	the	DET
ejpam-5984	562	17	function	function	NOUN
ejpam-5984	562	18	g	g	NOUN
ejpam-5984	562	19	on	on	ADP
ejpam-5984	562	20	t	t	PROPN
ejpam-5984	562	21	′	′	NUM
ejpam-5984	562	22	−	−	PROPN
ejpam-5984	562	23	s′	s′	VERB
ejpam-5984	562	24	by	by	ADP
ejpam-5984	562	25	g(w	g(w	PROPN
ejpam-5984	562	26	)	)	PUNCT
ejpam-5984	562	27	=	=	SYM
ejpam-5984	562	28	3	3	NUM
ejpam-5984	562	29	and	and	CCONJ
ejpam-5984	562	30	g(x	g(x	NOUN
ejpam-5984	562	31	)	)	PUNCT
ejpam-5984	562	32	=	=	SYM
ejpam-5984	562	33	f(x	f(x	PROPN
ejpam-5984	562	34	)	)	PUNCT
ejpam-5984	562	35	for	for	ADP
ejpam-5984	562	36	otherwise	otherwise	ADV
ejpam-5984	562	37	,	,	PUNCT
ejpam-5984	562	38	is	be	AUX
ejpam-5984	562	39	an	an	DET
ejpam-5984	562	40	idrd	idrd	ADJ
ejpam-5984	562	41	-	-	PUNCT
ejpam-5984	562	42	function	function	NOUN
ejpam-5984	562	43	t	t	NOUN
ejpam-5984	562	44	′	′	NOUN
ejpam-5984	562	45	−	−	PROPN
ejpam-5984	562	46	s′	s′	NUM
ejpam-5984	562	47	of	of	ADP
ejpam-5984	562	48	weight	weight	NOUN
ejpam-5984	562	49	at	at	ADP
ejpam-5984	562	50	most	most	ADJ
ejpam-5984	562	51	3k	3k	NUM
ejpam-5984	562	52	−	−	NUM
ejpam-5984	562	53	4	4	NUM
ejpam-5984	562	54	which	which	PRON
ejpam-5984	562	55	leads	lead	VERB
ejpam-5984	562	56	contradiction	contradiction	NOUN
ejpam-5984	562	57	.	.	PUNCT
ejpam-5984	563	1	case	case	NOUN
ejpam-5984	563	2	2	2	NUM
ejpam-5984	563	3	.	.	PUNCT
ejpam-5984	563	4	f(u	f(u	NOUN
ejpam-5984	563	5	)	)	PUNCT
ejpam-5984	564	1	=	=	PUNCT
ejpam-5984	565	1	3	3	X
ejpam-5984	565	2	.	.	X
ejpam-5984	566	1	if	if	SCONJ
ejpam-5984	566	2	∑	∑	PROPN
ejpam-5984	566	3	x∈v	x∈v	PROPN
ejpam-5984	566	4	(	(	PUNCT
ejpam-5984	566	5	sk)−wk	sk)−wk	PROPN
ejpam-5984	566	6	f(x	f(x	PROPN
ejpam-5984	566	7	)	)	PUNCT
ejpam-5984	566	8	=	=	SYM
ejpam-5984	566	9	3	3	NUM
ejpam-5984	566	10	,	,	PUNCT
ejpam-5984	566	11	then	then	ADV
ejpam-5984	566	12	we	we	PRON
ejpam-5984	566	13	must	must	AUX
ejpam-5984	566	14	have	have	VERB
ejpam-5984	566	15	s	s	NOUN
ejpam-5984	566	16	=	=	PUNCT
ejpam-5984	566	17	wk	wk	NOUN
ejpam-5984	566	18	and	and	CCONJ
ejpam-5984	566	19	v	v	PROPN
ejpam-5984	566	20	(	(	PUNCT
ejpam-5984	566	21	sk	sk	PROPN
ejpam-5984	566	22	)	)	PUNCT
ejpam-5984	566	23	−	−	PROPN
ejpam-5984	566	24	s	s	ADP
ejpam-5984	566	25	consists	consist	VERB
ejpam-5984	566	26	of	of	ADP
ejpam-5984	566	27	u	u	NOUN
ejpam-5984	566	28	and	and	CCONJ
ejpam-5984	566	29	xk	xk	PROPN
ejpam-5984	566	30	.	.	PUNCT
ejpam-5984	567	1	but	but	CCONJ
ejpam-5984	567	2	then	then	ADV
ejpam-5984	567	3	t	t	PROPN
ejpam-5984	567	4	′	′	NUM
ejpam-5984	568	1	−	−	PROPN
ejpam-5984	568	2	s′	s′	PUNCT
ejpam-5984	568	3	=	=	PUNCT
ejpam-5984	568	4	t	t	NOUN
ejpam-5984	568	5	−	−	NOUN
ejpam-5984	568	6	s	s	NOUN
ejpam-5984	568	7	∈	∈	PROPN
ejpam-5984	568	8	lk−1,∆	lk−1,∆	NOUN
ejpam-5984	568	9	and	and	CCONJ
ejpam-5984	568	10	f	f	PROPN
ejpam-5984	568	11	is	be	AUX
ejpam-5984	568	12	an	an	DET
ejpam-5984	568	13	idrd	idrd	ADJ
ejpam-5984	568	14	-	-	PUNCT
ejpam-5984	568	15	function	function	NOUN
ejpam-5984	568	16	of	of	ADP
ejpam-5984	568	17	t	t	PROPN
ejpam-5984	568	18	′	′	NUM
ejpam-5984	568	19	−	−	PROPN
ejpam-5984	568	20	s′	s′	NUM
ejpam-5984	568	21	of	of	ADP
ejpam-5984	568	22	weight	weight	NOUN
ejpam-5984	568	23	3k	3k	X
ejpam-5984	569	1	−	−	PROPN
ejpam-5984	569	2	1	1	NUM
ejpam-5984	569	3	,	,	PUNCT
ejpam-5984	569	4	a	a	DET
ejpam-5984	569	5	contradiction	contradiction	NOUN
ejpam-5984	569	6	with	with	ADP
ejpam-5984	569	7	lemma	lemma	PROPN
ejpam-5984	569	8	2	2	PROPN
ejpam-5984	569	9	.	.	PUNCT
ejpam-5984	569	10	assume	assume	VERB
ejpam-5984	569	11	that	that	SCONJ
ejpam-5984	569	12	∑	∑	PROPN
ejpam-5984	569	13	x∈v	x∈v	PROPN
ejpam-5984	569	14	(	(	PUNCT
ejpam-5984	569	15	sk)−wk	sk)−wk	PROPN
ejpam-5984	569	16	f(x	f(x	PROPN
ejpam-5984	569	17	)	)	PUNCT
ejpam-5984	569	18	≥	≥	NOUN
ejpam-5984	569	19	4	4	NUM
ejpam-5984	569	20	.	.	PUNCT
ejpam-5984	570	1	it	it	PRON
ejpam-5984	570	2	follows	follow	VERB
ejpam-5984	570	3	from	from	ADP
ejpam-5984	570	4	v1	v1	NOUN
ejpam-5984	570	5	=	=	SYM
ejpam-5984	570	6	∅	∅	NOUN
ejpam-5984	570	7	that	that	PRON
ejpam-5984	570	8	∑	∑	PUNCT
ejpam-5984	570	9	x∈v	x∈v	PROPN
ejpam-5984	570	10	(	(	PUNCT
ejpam-5984	570	11	sk)−wk	sk)−wk	PROPN
ejpam-5984	570	12	f(x	f(x	PROPN
ejpam-5984	570	13	)	)	PUNCT
ejpam-5984	570	14	≥	≥	NOUN
ejpam-5984	571	1	5	5	NUM
ejpam-5984	571	2	.	.	PUNCT
ejpam-5984	572	1	if	if	SCONJ
ejpam-5984	572	2	v	v	NOUN
ejpam-5984	572	3	has	have	VERB
ejpam-5984	572	4	a	a	DET
ejpam-5984	572	5	neighbor	neighbor	NOUN
ejpam-5984	572	6	w	w	NOUN
ejpam-5984	572	7	with	with	ADP
ejpam-5984	572	8	f(w	f(w	PROPN
ejpam-5984	572	9	)	)	PUNCT
ejpam-5984	572	10	≥	≥	NOUN
ejpam-5984	572	11	2	2	NUM
ejpam-5984	572	12	,	,	PUNCT
ejpam-5984	572	13	then	then	ADV
ejpam-5984	572	14	define	define	VERB
ejpam-5984	572	15	the	the	DET
ejpam-5984	572	16	function	function	NOUN
ejpam-5984	572	17	g	g	NOUN
ejpam-5984	572	18	on	on	ADP
ejpam-5984	572	19	t	t	PROPN
ejpam-5984	572	20	′	′	NUM
ejpam-5984	572	21	−	−	PROPN
ejpam-5984	572	22	s′	s′	VERB
ejpam-5984	572	23	by	by	ADP
ejpam-5984	572	24	g(w	g(w	PROPN
ejpam-5984	572	25	)	)	PUNCT
ejpam-5984	572	26	=	=	SYM
ejpam-5984	572	27	3	3	NUM
ejpam-5984	572	28	and	and	CCONJ
ejpam-5984	572	29	g(x	g(x	NOUN
ejpam-5984	572	30	)	)	PUNCT
ejpam-5984	573	1	=	=	SYM
ejpam-5984	573	2	f(x	f(x	PROPN
ejpam-5984	573	3	)	)	PUNCT
ejpam-5984	573	4	otherwise	otherwise	ADV
ejpam-5984	573	5	.	.	PUNCT
ejpam-5984	574	1	clearly	clearly	ADV
ejpam-5984	574	2	,	,	PUNCT
ejpam-5984	574	3	g	g	PROPN
ejpam-5984	574	4	is	be	AUX
ejpam-5984	574	5	an	an	DET
ejpam-5984	574	6	idrd	idrd	ADJ
ejpam-5984	574	7	-	-	PUNCT
ejpam-5984	574	8	function	function	NOUN
ejpam-5984	574	9	t	t	NOUN
ejpam-5984	574	10	′	′	NOUN
ejpam-5984	574	11	−	−	PROPN
ejpam-5984	574	12	s′	s′	NUM
ejpam-5984	574	13	of	of	ADP
ejpam-5984	574	14	weight	weight	NOUN
ejpam-5984	574	15	3k	3k	X
ejpam-5984	574	16	−	−	PROPN
ejpam-5984	574	17	5	5	NUM
ejpam-5984	574	18	,	,	PUNCT
ejpam-5984	574	19	a	a	DET
ejpam-5984	574	20	contradiction	contradiction	NOUN
ejpam-5984	574	21	.	.	PUNCT
ejpam-5984	575	1	assume	assume	VERB
ejpam-5984	575	2	that	that	SCONJ
ejpam-5984	575	3	all	all	DET
ejpam-5984	575	4	neighbors	neighbor	NOUN
ejpam-5984	575	5	of	of	ADP
ejpam-5984	575	6	v	v	PRON
ejpam-5984	575	7	assigned	assign	VERB
ejpam-5984	575	8	0	0	NUM
ejpam-5984	575	9	under	under	ADP
ejpam-5984	575	10	f	f	PROPN
ejpam-5984	575	11	.	.	PUNCT
ejpam-5984	576	1	then	then	ADV
ejpam-5984	576	2	the	the	DET
ejpam-5984	576	3	function	function	NOUN
ejpam-5984	576	4	g	g	NOUN
ejpam-5984	576	5	on	on	ADP
ejpam-5984	576	6	t	t	PROPN
ejpam-5984	576	7	′−s′	′−s′	VERB
ejpam-5984	576	8	by	by	ADP
ejpam-5984	576	9	f(v	f(v	NOUN
ejpam-5984	576	10	)	)	PUNCT
ejpam-5984	576	11	=	=	SYM
ejpam-5984	576	12	2	2	NUM
ejpam-5984	576	13	and	and	CCONJ
ejpam-5984	576	14	g(x	g(x	NOUN
ejpam-5984	576	15	)	)	PUNCT
ejpam-5984	576	16	=	=	SYM
ejpam-5984	576	17	f(x	f(x	PROPN
ejpam-5984	576	18	)	)	PUNCT
ejpam-5984	576	19	otherwise	otherwise	ADV
ejpam-5984	576	20	,	,	PUNCT
ejpam-5984	576	21	is	be	AUX
ejpam-5984	576	22	an	an	DET
ejpam-5984	576	23	idrd	idrd	ADJ
ejpam-5984	576	24	-	-	PUNCT
ejpam-5984	576	25	function	function	NOUN
ejpam-5984	576	26	t	t	NOUN
ejpam-5984	576	27	′	′	NOUN
ejpam-5984	576	28	−	−	PROPN
ejpam-5984	576	29	s′	s′	NUM
ejpam-5984	576	30	of	of	ADP
ejpam-5984	576	31	weight	weight	NOUN
ejpam-5984	576	32	3k	3k	X
ejpam-5984	576	33	−	−	PROPN
ejpam-5984	576	34	4	4	NUM
ejpam-5984	576	35	,	,	PUNCT
ejpam-5984	576	36	a	a	DET
ejpam-5984	576	37	contradiction	contradiction	NOUN
ejpam-5984	576	38	again	again	ADV
ejpam-5984	576	39	.	.	PUNCT
ejpam-5984	577	1	theorem	theorem	VERB
ejpam-5984	577	2	5	5	NUM
ejpam-5984	577	3	.	.	PUNCT
ejpam-5984	578	1	for	for	ADP
ejpam-5984	578	2	every	every	DET
ejpam-5984	578	3	tree	tree	NOUN
ejpam-5984	578	4	t	t	NOUN
ejpam-5984	578	5	of	of	ADP
ejpam-5984	578	6	order	order	NOUN
ejpam-5984	578	7	n	n	PRON
ejpam-5984	578	8	≥	≥	NOUN
ejpam-5984	578	9	3	3	NUM
ejpam-5984	578	10	with	with	ADP
ejpam-5984	578	11	maximum	maximum	ADJ
ejpam-5984	578	12	degree	degree	NOUN
ejpam-5984	578	13	∆	∆	PROPN
ejpam-5984	578	14	,	,	PUNCT
ejpam-5984	578	15	st−idr(t	st−idr(t	NOUN
ejpam-5984	578	16	)	)	PUNCT
ejpam-5984	578	17	≤	≤	NOUN
ejpam-5984	578	18	∆	∆	PROPN
ejpam-5984	578	19	with	with	ADP
ejpam-5984	578	20	equality	equality	NOUN
ejpam-5984	578	21	if	if	SCONJ
ejpam-5984	578	22	and	and	CCONJ
ejpam-5984	578	23	only	only	ADV
ejpam-5984	578	24	if	if	SCONJ
ejpam-5984	578	25	t	t	PROPN
ejpam-5984	578	26	∈	∈	NOUN
ejpam-5984	578	27	t∆.	t∆.	NOUN
ejpam-5984	578	28	proof	proof	NOUN
ejpam-5984	578	29	.	.	PUNCT
ejpam-5984	579	1	if	if	SCONJ
ejpam-5984	579	2	diam(t	diam(t	NOUN
ejpam-5984	579	3	)	)	PUNCT
ejpam-5984	579	4	=	=	SYM
ejpam-5984	579	5	2	2	NUM
ejpam-5984	579	6	,	,	PUNCT
ejpam-5984	579	7	then	then	ADV
ejpam-5984	579	8	t	t	PROPN
ejpam-5984	579	9	is	be	AUX
ejpam-5984	579	10	the	the	DET
ejpam-5984	579	11	star	star	NOUN
ejpam-5984	579	12	k1,∆	k1,∆	VERB
ejpam-5984	579	13	and	and	CCONJ
ejpam-5984	579	14	by	by	ADP
ejpam-5984	579	15	corollary	corollary	ADJ
ejpam-5984	579	16	4	4	NUM
ejpam-5984	579	17	,	,	PUNCT
ejpam-5984	579	18	we	we	PRON
ejpam-5984	579	19	have	have	VERB
ejpam-5984	579	20	st−idr(t	st−idr(t	NOUN
ejpam-5984	579	21	)	)	PUNCT
ejpam-5984	580	1	=	=	PUNCT
ejpam-5984	581	1	∆.	∆.	NOUN
ejpam-5984	581	2	if	if	SCONJ
ejpam-5984	581	3	∆	∆	VERB
ejpam-5984	581	4	=	=	SYM
ejpam-5984	581	5	2	2	NUM
ejpam-5984	581	6	,	,	PUNCT
ejpam-5984	581	7	then	then	ADV
ejpam-5984	581	8	t	t	PROPN
ejpam-5984	581	9	is	be	AUX
ejpam-5984	581	10	the	the	DET
ejpam-5984	581	11	path	path	NOUN
ejpam-5984	581	12	pn	pn	NOUN
ejpam-5984	581	13	and	and	CCONJ
ejpam-5984	581	14	the	the	DET
ejpam-5984	581	15	result	result	NOUN
ejpam-5984	581	16	is	be	AUX
ejpam-5984	581	17	true	true	ADJ
ejpam-5984	581	18	by	by	ADP
ejpam-5984	581	19	proposition	proposition	NOUN
ejpam-5984	581	20	8	8	NUM
ejpam-5984	581	21	.	.	PUNCT
ejpam-5984	581	22	assume	assume	VERB
ejpam-5984	581	23	that	that	SCONJ
ejpam-5984	581	24	diam(t	diam(t	NOUN
ejpam-5984	581	25	)	)	PUNCT
ejpam-5984	581	26	≥	≥	NOUN
ejpam-5984	581	27	3	3	NUM
ejpam-5984	581	28	and	and	CCONJ
ejpam-5984	581	29	∆	∆	PROPN
ejpam-5984	581	30	≥	≥	NOUN
ejpam-5984	582	1	3	3	X
ejpam-5984	582	2	.	.	PUNCT
ejpam-5984	582	3	let	let	VERB
ejpam-5984	582	4	x1x2	x1x2	PUNCT
ejpam-5984	582	5	.	.	PUNCT
ejpam-5984	582	6	.	.	PUNCT
ejpam-5984	582	7	.	.	PUNCT
ejpam-5984	583	1	xd	xd	INTJ
ejpam-5984	583	2	be	be	AUX
ejpam-5984	583	3	a	a	DET
ejpam-5984	583	4	diametral	diametral	ADJ
ejpam-5984	583	5	path	path	NOUN
ejpam-5984	583	6	in	in	ADP
ejpam-5984	583	7	g	g	PROPN
ejpam-5984	583	8	and	and	CCONJ
ejpam-5984	583	9	root	root	PROPN
ejpam-5984	583	10	t	t	PROPN
ejpam-5984	583	11	at	at	ADP
ejpam-5984	583	12	xd	xd	ADP
ejpam-5984	583	13	.	.	PUNCT
ejpam-5984	584	1	we	we	PRON
ejpam-5984	584	2	have	have	VERB
ejpam-5984	584	3	d(x2	d(x2	NOUN
ejpam-5984	584	4	)	)	PUNCT
ejpam-5984	584	5	≤	≤	NOUN
ejpam-5984	585	1	∆.	∆.	INTJ
ejpam-5984	585	2	let	let	VERB
ejpam-5984	585	3	f	f	PROPN
ejpam-5984	585	4	=	=	SYM
ejpam-5984	585	5	(	(	PUNCT
ejpam-5984	585	6	v0,∅	v0,∅	PROPN
ejpam-5984	585	7	,	,	PUNCT
ejpam-5984	585	8	v2	v2	PROPN
ejpam-5984	585	9	,	,	PUNCT
ejpam-5984	585	10	v3	v3	PROPN
ejpam-5984	585	11	)	)	PUNCT
ejpam-5984	585	12	be	be	VERB
ejpam-5984	585	13	an	an	DET
ejpam-5984	585	14	idr(t	idr(t	NOUN
ejpam-5984	585	15	)	)	PUNCT
ejpam-5984	585	16	-function	-function	NOUN
ejpam-5984	585	17	.	.	PUNCT
ejpam-5984	586	1	clearly	clearly	ADV
ejpam-5984	586	2	,	,	PUNCT
ejpam-5984	586	3	f(n	f(n	PROPN
ejpam-5984	587	1	[	[	X
ejpam-5984	587	2	x2	x2	X
ejpam-5984	587	3	]	]	X
ejpam-5984	587	4	)	)	PUNCT
ejpam-5984	587	5	≥	≥	NOUN
ejpam-5984	588	1	3	3	NUM
ejpam-5984	588	2	.	.	PUNCT
ejpam-5984	589	1	if	if	SCONJ
ejpam-5984	589	2	f(x3	f(x3	VERB
ejpam-5984	589	3	)	)	PUNCT
ejpam-5984	589	4	∈	∈	PROPN
ejpam-5984	589	5	v2	v2	NOUN
ejpam-5984	589	6	∪	∪	X
ejpam-5984	589	7	v3	v3	PROPN
ejpam-5984	589	8	,	,	PUNCT
ejpam-5984	589	9	then	then	ADV
ejpam-5984	589	10	f(x2	f(x2	NOUN
ejpam-5984	589	11	)	)	PUNCT
ejpam-5984	589	12	=	=	SYM
ejpam-5984	589	13	0	0	NUM
ejpam-5984	589	14	and	and	CCONJ
ejpam-5984	589	15	f(x1	f(x1	ADJ
ejpam-5984	589	16	)	)	PUNCT
ejpam-5984	589	17	=	=	SYM
ejpam-5984	590	1	2	2	X
ejpam-5984	590	2	.	.	PUNCT
ejpam-5984	590	3	the	the	DET
ejpam-5984	590	4	function	function	NOUN
ejpam-5984	590	5	g	g	PROPN
ejpam-5984	590	6	defined	define	VERB
ejpam-5984	590	7	on	on	ADP
ejpam-5984	590	8	t	t	PROPN
ejpam-5984	590	9	−	−	PROPN
ejpam-5984	590	10	x1	x1	PROPN
ejpam-5984	590	11	by	by	ADP
ejpam-5984	590	12	g(x3	g(x3	NUM
ejpam-5984	590	13	)	)	PUNCT
ejpam-5984	590	14	=	=	SYM
ejpam-5984	590	15	max{3	max{3	NOUN
ejpam-5984	590	16	,	,	PUNCT
ejpam-5984	590	17	f(x3	f(x3	ADJ
ejpam-5984	590	18	)	)	PUNCT
ejpam-5984	590	19	}	}	PUNCT
ejpam-5984	590	20	and	and	CCONJ
ejpam-5984	590	21	g(x	g(x	NOUN
ejpam-5984	590	22	)	)	PUNCT
ejpam-5984	590	23	=	=	SYM
ejpam-5984	590	24	f(x	f(x	PROPN
ejpam-5984	590	25	)	)	PUNCT
ejpam-5984	590	26	otherwise	otherwise	ADV
ejpam-5984	590	27	,	,	PUNCT
ejpam-5984	590	28	is	be	AUX
ejpam-5984	590	29	an	an	DET
ejpam-5984	590	30	idrd	idrd	ADJ
ejpam-5984	590	31	-	-	PUNCT
ejpam-5984	590	32	function	function	NOUN
ejpam-5984	590	33	of	of	ADP
ejpam-5984	590	34	the	the	DET
ejpam-5984	590	35	tree	tree	NOUN
ejpam-5984	590	36	t	t	NOUN
ejpam-5984	590	37	−	−	PROPN
ejpam-5984	591	1	x1	x1	PROPN
ejpam-5984	591	2	of	of	ADP
ejpam-5984	591	3	weight	weight	NOUN
ejpam-5984	591	4	at	at	ADP
ejpam-5984	591	5	most	most	ADJ
ejpam-5984	591	6	ω(f)−	ω(f)−	PROPN
ejpam-5984	591	7	1	1	NUM
ejpam-5984	591	8	.	.	PUNCT
ejpam-5984	592	1	so	so	ADV
ejpam-5984	592	2	,	,	PUNCT
ejpam-5984	592	3	st−idr(t	st−idr(t	NOUN
ejpam-5984	592	4	)	)	PUNCT
ejpam-5984	592	5	=	=	PUNCT
ejpam-5984	593	1	1	1	X
ejpam-5984	593	2	.	.	PUNCT
ejpam-5984	593	3	now	now	ADV
ejpam-5984	593	4	,	,	PUNCT
ejpam-5984	593	5	assume	assume	VERB
ejpam-5984	593	6	that	that	SCONJ
ejpam-5984	593	7	f(x3	f(x3	VERB
ejpam-5984	593	8	)	)	PUNCT
ejpam-5984	594	1	=	=	SYM
ejpam-5984	594	2	0	0	X
ejpam-5984	594	3	.	.	PUNCT
ejpam-5984	595	1	then	then	ADV
ejpam-5984	595	2	we	we	PRON
ejpam-5984	595	3	have	have	VERB
ejpam-5984	595	4	f(n	f(n	PROPN
ejpam-5984	596	1	[	[	X
ejpam-5984	596	2	x2	x2	X
ejpam-5984	596	3	]	]	X
ejpam-5984	596	4	−	−	PROPN
ejpam-5984	596	5	{	{	PUNCT
ejpam-5984	596	6	x3	x3	ADJ
ejpam-5984	596	7	}	}	PUNCT
ejpam-5984	596	8	)	)	PUNCT
ejpam-5984	596	9	=	=	SYM
ejpam-5984	596	10	3	3	X
ejpam-5984	596	11	.	.	X
ejpam-5984	597	1	if	if	SCONJ
ejpam-5984	597	2	x3	x3	PROPN
ejpam-5984	597	3	has	have	VERB
ejpam-5984	597	4	a	a	DET
ejpam-5984	597	5	neighbor	neighbor	NOUN
ejpam-5984	597	6	u	u	NOUN
ejpam-5984	597	7	̸=	̸=	PROPN
ejpam-5984	597	8	x2	x2	PROPN
ejpam-5984	597	9	with	with	ADP
ejpam-5984	597	10	f(u	f(u	PROPN
ejpam-5984	597	11	)	)	PUNCT
ejpam-5984	597	12	≥	≥	NOUN
ejpam-5984	597	13	2	2	NUM
ejpam-5984	597	14	,	,	PUNCT
ejpam-5984	597	15	then	then	ADV
ejpam-5984	597	16	the	the	DET
ejpam-5984	597	17	function	function	NOUN
ejpam-5984	597	18	g	g	NOUN
ejpam-5984	597	19	on	on	ADP
ejpam-5984	597	20	t	t	PROPN
ejpam-5984	597	21	−tx2	−tx2	PROPN
ejpam-5984	597	22	by	by	ADP
ejpam-5984	597	23	g(u	g(u	PROPN
ejpam-5984	597	24	)	)	PUNCT
ejpam-5984	597	25	=	=	SYM
ejpam-5984	597	26	min{3	min{3	PROPN
ejpam-5984	597	27	,	,	PUNCT
ejpam-5984	597	28	f(u)+1	f(u)+1	NOUN
ejpam-5984	597	29	}	}	PUNCT
ejpam-5984	597	30	and	and	CCONJ
ejpam-5984	597	31	g(x	g(x	NOUN
ejpam-5984	597	32	)	)	PUNCT
ejpam-5984	597	33	=	=	SYM
ejpam-5984	597	34	f(x	f(x	PROPN
ejpam-5984	597	35	)	)	PUNCT
ejpam-5984	597	36	otherwise	otherwise	ADV
ejpam-5984	597	37	,	,	PUNCT
ejpam-5984	597	38	is	be	AUX
ejpam-5984	597	39	an	an	DET
ejpam-5984	597	40	idrd	idrd	ADJ
ejpam-5984	597	41	-	-	PUNCT
ejpam-5984	597	42	function	function	NOUN
ejpam-5984	597	43	of	of	ADP
ejpam-5984	597	44	the	the	DET
ejpam-5984	597	45	tree	tree	NOUN
ejpam-5984	597	46	t	t	NOUN
ejpam-5984	597	47	−	−	PROPN
ejpam-5984	597	48	tx2	tx2	NOUN
ejpam-5984	597	49	of	of	ADP
ejpam-5984	597	50	weight	weight	NOUN
ejpam-5984	597	51	at	at	ADP
ejpam-5984	597	52	most	most	ADJ
ejpam-5984	597	53	ω(f)−	ω(f)−	PROPN
ejpam-5984	597	54	1	1	NUM
ejpam-5984	597	55	and	and	CCONJ
ejpam-5984	597	56	so	so	ADV
ejpam-5984	597	57	st−idr(t	st−idr(t	NOUN
ejpam-5984	597	58	)	)	PUNCT
ejpam-5984	597	59	≤	≤	NUM
ejpam-5984	597	60	d(x2	d(x2	NOUN
ejpam-5984	597	61	)	)	PUNCT
ejpam-5984	597	62	≤	≤	NOUN
ejpam-5984	598	1	∆.	∆.	ADV
ejpam-5984	598	2	now	now	ADV
ejpam-5984	598	3	,	,	PUNCT
ejpam-5984	598	4	let	let	VERB
ejpam-5984	598	5	f(x	f(x	PROPN
ejpam-5984	598	6	)	)	PUNCT
ejpam-5984	599	1	=	=	PUNCT
ejpam-5984	599	2	0	0	PUNCT
ejpam-5984	600	1	for	for	ADP
ejpam-5984	600	2	each	each	DET
ejpam-5984	600	3	x	x	SYM
ejpam-5984	600	4	∈	∈	PROPN
ejpam-5984	600	5	n	n	CCONJ
ejpam-5984	600	6	[	[	X
ejpam-5984	600	7	x3]−{x2	x3]−{x2	NUM
ejpam-5984	600	8	}	}	PUNCT
ejpam-5984	600	9	.	.	PUNCT
ejpam-5984	601	1	then	then	ADV
ejpam-5984	601	2	the	the	DET
ejpam-5984	601	3	function	function	NOUN
ejpam-5984	601	4	g	g	PROPN
ejpam-5984	601	5	defined	define	VERB
ejpam-5984	601	6	on	on	ADP
ejpam-5984	601	7	t	t	PROPN
ejpam-5984	601	8	−	−	PROPN
ejpam-5984	602	1	(	(	PUNCT
ejpam-5984	602	2	n	n	X
ejpam-5984	602	3	[	[	X
ejpam-5984	602	4	x2]−{x3	x2]−{x3	NUM
ejpam-5984	602	5	}	}	PUNCT
ejpam-5984	602	6	)	)	PUNCT
ejpam-5984	602	7	by	by	ADP
ejpam-5984	602	8	g(x3	g(x3	NUM
ejpam-5984	602	9	)	)	PUNCT
ejpam-5984	602	10	=	=	SYM
ejpam-5984	602	11	2	2	NUM
ejpam-5984	602	12	and	and	CCONJ
ejpam-5984	602	13	g(x	g(x	NOUN
ejpam-5984	602	14	)	)	PUNCT
ejpam-5984	603	1	=	=	SYM
ejpam-5984	603	2	f(x	f(x	PROPN
ejpam-5984	603	3	)	)	PUNCT
ejpam-5984	603	4	otherwise	otherwise	ADV
ejpam-5984	603	5	,	,	PUNCT
ejpam-5984	603	6	is	be	AUX
ejpam-5984	603	7	an	an	DET
ejpam-5984	603	8	idrd	idrd	ADJ
ejpam-5984	603	9	-	-	PUNCT
ejpam-5984	603	10	function	function	NOUN
ejpam-5984	603	11	of	of	ADP
ejpam-5984	603	12	the	the	DET
ejpam-5984	603	13	tree	tree	NOUN
ejpam-5984	603	14	t	t	NOUN
ejpam-5984	603	15	−	−	PROPN
ejpam-5984	604	1	(	(	PUNCT
ejpam-5984	604	2	n	n	X
ejpam-5984	604	3	[	[	X
ejpam-5984	604	4	x2]−	x2]−	X
ejpam-5984	604	5	{	{	PUNCT
ejpam-5984	604	6	x3	x3	ADJ
ejpam-5984	604	7	}	}	PUNCT
ejpam-5984	604	8	)	)	PUNCT
ejpam-5984	604	9	of	of	ADP
ejpam-5984	604	10	weight	weight	NOUN
ejpam-5984	604	11	ω(f)−	ω(f)−	PROPN
ejpam-5984	604	12	1	1	NUM
ejpam-5984	604	13	and	and	CCONJ
ejpam-5984	604	14	so	so	ADV
ejpam-5984	604	15	st−idr(t	st−idr(t	NOUN
ejpam-5984	604	16	)	)	PUNCT
ejpam-5984	604	17	≤	≤	NUM
ejpam-5984	604	18	d(x2	d(x2	NOUN
ejpam-5984	604	19	)	)	PUNCT
ejpam-5984	604	20	≤	≤	NOUN
ejpam-5984	605	1	∆.	∆.	ADP
ejpam-5984	605	2	this	this	PRON
ejpam-5984	605	3	proves	prove	VERB
ejpam-5984	605	4	the	the	DET
ejpam-5984	605	5	bound	bind	VERB
ejpam-5984	605	6	.	.	PUNCT
ejpam-5984	606	1	now	now	ADV
ejpam-5984	606	2	we	we	PRON
ejpam-5984	606	3	show	show	VERB
ejpam-5984	606	4	that	that	SCONJ
ejpam-5984	606	5	st−idr(t	st−idr(t	NOUN
ejpam-5984	606	6	)	)	PUNCT
ejpam-5984	606	7	=	=	PUNCT
ejpam-5984	607	1	∆	∆	PROPN
ejpam-5984	607	2	if	if	SCONJ
ejpam-5984	607	3	and	and	CCONJ
ejpam-5984	607	4	only	only	ADV
ejpam-5984	607	5	if	if	SCONJ
ejpam-5984	607	6	t	t	PROPN
ejpam-5984	607	7	∈	∈	PROPN
ejpam-5984	607	8	t∆.	t∆.	PROPN
ejpam-5984	607	9	the	the	DET
ejpam-5984	607	10	sufficiency	sufficiency	NOUN
ejpam-5984	607	11	follows	follow	VERB
ejpam-5984	607	12	from	from	ADP
ejpam-5984	607	13	lemma	lemma	PROPN
ejpam-5984	607	14	3	3	NUM
ejpam-5984	607	15	.	.	PUNCT
ejpam-5984	607	16	to	to	PART
ejpam-5984	607	17	prove	prove	VERB
ejpam-5984	607	18	the	the	DET
ejpam-5984	607	19	necessity	necessity	NOUN
ejpam-5984	607	20	,	,	PUNCT
ejpam-5984	607	21	assume	assume	VERB
ejpam-5984	607	22	that	that	SCONJ
ejpam-5984	607	23	st−idr(t	st−idr(t	NOUN
ejpam-5984	607	24	)	)	PUNCT
ejpam-5984	607	25	=	=	PUNCT
ejpam-5984	608	1	∆.	∆.	NOUN
ejpam-5984	608	2	we	we	PRON
ejpam-5984	608	3	proceed	proceed	VERB
ejpam-5984	608	4	by	by	ADP
ejpam-5984	608	5	induction	induction	NOUN
ejpam-5984	608	6	on	on	ADP
ejpam-5984	608	7	n.	n.	NOUN
ejpam-5984	608	8	if	if	SCONJ
ejpam-5984	608	9	diam(t	diam(t	NOUN
ejpam-5984	608	10	)	)	PUNCT
ejpam-5984	608	11	=	=	SYM
ejpam-5984	608	12	2	2	NUM
ejpam-5984	608	13	,	,	PUNCT
ejpam-5984	608	14	then	then	ADV
ejpam-5984	608	15	t	t	PROPN
ejpam-5984	608	16	is	be	AUX
ejpam-5984	608	17	the	the	DET
ejpam-5984	608	18	star	star	NOUN
ejpam-5984	608	19	k1,∆	k1,∆	VERB
ejpam-5984	608	20	and	and	CCONJ
ejpam-5984	608	21	clearly	clearly	ADV
ejpam-5984	608	22	t	t	PROPN
ejpam-5984	608	23	∈	∈	PROPN
ejpam-5984	608	24	t∆.	t∆.	NOUN
ejpam-5984	608	25	if	if	SCONJ
ejpam-5984	608	26	∆	∆	PROPN
ejpam-5984	608	27	=	=	SYM
ejpam-5984	608	28	2	2	NUM
ejpam-5984	608	29	,	,	PUNCT
ejpam-5984	608	30	then	then	ADV
ejpam-5984	608	31	t	t	PROPN
ejpam-5984	608	32	is	be	AUX
ejpam-5984	608	33	the	the	DET
ejpam-5984	608	34	path	path	NOUN
ejpam-5984	608	35	p3k	p3k	PROPN
ejpam-5984	608	36	and	and	CCONJ
ejpam-5984	608	37	the	the	DET
ejpam-5984	608	38	result	result	NOUN
ejpam-5984	608	39	is	be	AUX
ejpam-5984	608	40	true	true	ADJ
ejpam-5984	608	41	by	by	ADP
ejpam-5984	608	42	proposition	proposition	NOUN
ejpam-5984	608	43	8	8	NUM
ejpam-5984	608	44	.	.	PUNCT
ejpam-5984	609	1	this	this	PRON
ejpam-5984	609	2	proves	prove	VERB
ejpam-5984	609	3	the	the	DET
ejpam-5984	609	4	base	base	NOUN
ejpam-5984	609	5	case	case	NOUN
ejpam-5984	609	6	.	.	PUNCT
ejpam-5984	610	1	suppose	suppose	VERB
ejpam-5984	610	2	that	that	SCONJ
ejpam-5984	610	3	for	for	ADP
ejpam-5984	610	4	any	any	DET
ejpam-5984	610	5	tree	tree	NOUN
ejpam-5984	610	6	t	t	NOUN
ejpam-5984	610	7	′	′	NUM
ejpam-5984	610	8	of	of	ADP
ejpam-5984	610	9	order	order	NOUN
ejpam-5984	610	10	3	3	NUM
ejpam-5984	610	11	≤	≤	NOUN
ejpam-5984	610	12	n′	n′	PRON
ejpam-5984	610	13	<	<	X
ejpam-5984	610	14	n	n	X
ejpam-5984	610	15	with	with	ADP
ejpam-5984	610	16	st−idr(t	st−idr(t	NOUN
ejpam-5984	610	17	)	)	PUNCT
ejpam-5984	611	1	=	=	SYM
ejpam-5984	611	2	∆	∆	PROPN
ejpam-5984	611	3	,	,	PUNCT
ejpam-5984	611	4	we	we	PRON
ejpam-5984	611	5	have	have	VERB
ejpam-5984	611	6	t	t	NOUN
ejpam-5984	611	7	′	′	NUM
ejpam-5984	611	8	∈	∈	PROPN
ejpam-5984	611	9	t∆.	t∆.	NOUN
ejpam-5984	611	10	let	let	VERB
ejpam-5984	611	11	t	t	NOUN
ejpam-5984	611	12	be	be	AUX
ejpam-5984	611	13	a	a	DET
ejpam-5984	611	14	tree	tree	NOUN
ejpam-5984	611	15	of	of	ADP
ejpam-5984	611	16	order	order	NOUN
ejpam-5984	611	17	n	n	PRON
ejpam-5984	611	18	with	with	ADP
ejpam-5984	611	19	st−idr(t	st−idr(t	NOUN
ejpam-5984	611	20	)	)	PUNCT
ejpam-5984	611	21	=	=	PUNCT
ejpam-5984	612	1	∆.	∆.	NOUN
ejpam-5984	612	2	as	as	ADP
ejpam-5984	612	3	before	before	ADV
ejpam-5984	612	4	,	,	PUNCT
ejpam-5984	612	5	we	we	PRON
ejpam-5984	612	6	can	can	AUX
ejpam-5984	612	7	assume	assume	VERB
ejpam-5984	612	8	that	that	SCONJ
ejpam-5984	612	9	diam(t	diam(t	NOUN
ejpam-5984	612	10	)	)	PUNCT
ejpam-5984	612	11	≥	≥	NOUN
ejpam-5984	612	12	3	3	NUM
ejpam-5984	612	13	and	and	CCONJ
ejpam-5984	612	14	∆	∆	PROPN
ejpam-5984	612	15	≥	≥	NOUN
ejpam-5984	612	16	3	3	X
ejpam-5984	612	17	.	.	PUNCT
ejpam-5984	612	18	corollary	corollary	ADJ
ejpam-5984	612	19	5	5	NUM
ejpam-5984	612	20	implies	imply	VERB
ejpam-5984	612	21	that	that	SCONJ
ejpam-5984	612	22	diam(t	diam(t	NOUN
ejpam-5984	612	23	)	)	PUNCT
ejpam-5984	612	24	≥	≥	NOUN
ejpam-5984	612	25	4	4	NUM
ejpam-5984	612	26	.	.	PUNCT
ejpam-5984	613	1	let	let	VERB
ejpam-5984	613	2	f	f	PRON
ejpam-5984	613	3	be	be	AUX
ejpam-5984	613	4	an	an	DET
ejpam-5984	613	5	idr	idr	NOUN
ejpam-5984	613	6	-	-	PUNCT
ejpam-5984	613	7	function	function	NOUN
ejpam-5984	613	8	of	of	ADP
ejpam-5984	613	9	t	t	NOUN
ejpam-5984	613	10	such	such	ADJ
ejpam-5984	613	11	that	that	SCONJ
ejpam-5984	613	12	there	there	PRON
ejpam-5984	613	13	is	be	VERB
ejpam-5984	613	14	no	no	DET
ejpam-5984	613	15	vertex	vertex	NOUN
ejpam-5984	613	16	assigned	assign	VERB
ejpam-5984	613	17	1	1	NUM
ejpam-5984	613	18	under	under	ADP
ejpam-5984	613	19	f	f	PROPN
ejpam-5984	613	20	.	.	PUNCT
ejpam-5984	614	1	let	let	VERB
ejpam-5984	615	1	[	[	X
ejpam-5984	615	2	x1	x1	ADJ
ejpam-5984	615	3	,	,	PUNCT
ejpam-5984	615	4	x2	x2	PROPN
ejpam-5984	615	5	,	,	PUNCT
ejpam-5984	615	6	.	.	PUNCT
ejpam-5984	615	7	.	.	PUNCT
ejpam-5984	616	1	.	.	PUNCT
ejpam-5984	617	1	,	,	PUNCT
ejpam-5984	617	2	xd	xd	ADP
ejpam-5984	617	3	]	]	PUNCT
ejpam-5984	617	4	be	be	AUX
ejpam-5984	617	5	a	a	DET
ejpam-5984	617	6	diametral	diametral	ADJ
ejpam-5984	617	7	path	path	NOUN
ejpam-5984	617	8	in	in	ADP
ejpam-5984	617	9	g	g	PROPN
ejpam-5984	617	10	and	and	CCONJ
ejpam-5984	617	11	root	root	PROPN
ejpam-5984	617	12	t	t	PROPN
ejpam-5984	617	13	at	at	ADP
ejpam-5984	617	14	xd	xd	ADP
ejpam-5984	617	15	.	.	PUNCT
ejpam-5984	618	1	using	use	VERB
ejpam-5984	618	2	the	the	DET
ejpam-5984	618	3	above	above	ADJ
ejpam-5984	618	4	argument	argument	NOUN
ejpam-5984	618	5	and	and	CCONJ
ejpam-5984	618	6	the	the	DET
ejpam-5984	618	7	fact	fact	NOUN
ejpam-5984	618	8	st−idr(t	st−idr(t	NOUN
ejpam-5984	618	9	)	)	PUNCT
ejpam-5984	618	10	=	=	SYM
ejpam-5984	618	11	∆	∆	PROPN
ejpam-5984	618	12	,	,	PUNCT
ejpam-5984	618	13	we	we	PRON
ejpam-5984	618	14	must	must	AUX
ejpam-5984	618	15	have	have	VERB
ejpam-5984	618	16	d(x2	d(x2	NOUN
ejpam-5984	618	17	)	)	PUNCT
ejpam-5984	619	1	=	=	PUNCT
ejpam-5984	620	1	∆.	∆.	NOUN
ejpam-5984	620	2	if	if	SCONJ
ejpam-5984	620	3	f(x2	f(x2	NOUN
ejpam-5984	620	4	)	)	PUNCT
ejpam-5984	621	1	=	=	SYM
ejpam-5984	621	2	0	0	NUM
ejpam-5984	621	3	,	,	PUNCT
ejpam-5984	621	4	then	then	ADV
ejpam-5984	621	5	clearly	clearly	ADV
ejpam-5984	621	6	st−idr(t	st−idr(t	NOUN
ejpam-5984	621	7	)	)	PUNCT
ejpam-5984	621	8	=	=	SYM
ejpam-5984	622	1	1	1	NUM
ejpam-5984	622	2	which	which	PRON
ejpam-5984	622	3	is	be	AUX
ejpam-5984	622	4	contradiction	contradiction	NOUN
ejpam-5984	622	5	.	.	PUNCT
ejpam-5984	623	1	since	since	SCONJ
ejpam-5984	623	2	f	f	PROPN
ejpam-5984	623	3	is	be	AUX
ejpam-5984	623	4	an	an	DET
ejpam-5984	623	5	idrd	idrd	ADJ
ejpam-5984	623	6	-	-	PUNCT
ejpam-5984	623	7	function	function	NOUN
ejpam-5984	623	8	,	,	PUNCT
ejpam-5984	623	9	it	it	PRON
ejpam-5984	623	10	follows	follow	VERB
ejpam-5984	623	11	that	that	DET
ejpam-5984	623	12	f(x2	f(x2	NOUN
ejpam-5984	623	13	)	)	PUNCT
ejpam-5984	623	14	=	=	SYM
ejpam-5984	623	15	3	3	NUM
ejpam-5984	623	16	,	,	PUNCT
ejpam-5984	623	17	and	and	CCONJ
ejpam-5984	623	18	thus	thus	ADV
ejpam-5984	623	19	f(x3	f(x3	NUM
ejpam-5984	623	20	)	)	PUNCT
ejpam-5984	624	1	=	=	SYM
ejpam-5984	624	2	0	0	X
ejpam-5984	624	3	.	.	PUNCT
ejpam-5984	625	1	we	we	PRON
ejpam-5984	625	2	claim	claim	VERB
ejpam-5984	625	3	that	that	SCONJ
ejpam-5984	625	4	d(x3	d(x3	VERB
ejpam-5984	625	5	)	)	PUNCT
ejpam-5984	626	1	=	=	SYM
ejpam-5984	626	2	2	2	X
ejpam-5984	626	3	.	.	PUNCT
ejpam-5984	626	4	by	by	ADP
ejpam-5984	626	5	contradiction	contradiction	NOUN
ejpam-5984	626	6	,	,	PUNCT
ejpam-5984	626	7	assume	assume	VERB
ejpam-5984	626	8	that	that	SCONJ
ejpam-5984	626	9	d(x3	d(x3	NOUN
ejpam-5984	626	10	)	)	PUNCT
ejpam-5984	626	11	≥	≥	NOUN
ejpam-5984	627	1	3	3	X
ejpam-5984	627	2	.	.	PUNCT
ejpam-5984	628	1	let	let	VERB
ejpam-5984	628	2	w	w	NOUN
ejpam-5984	628	3	be	be	AUX
ejpam-5984	628	4	a	a	DET
ejpam-5984	628	5	neighbor	neighbor	NOUN
ejpam-5984	628	6	of	of	ADP
ejpam-5984	628	7	x3	x3	ADJ
ejpam-5984	628	8	different	different	ADJ
ejpam-5984	628	9	from	from	ADP
ejpam-5984	628	10	x2	x2	PROPN
ejpam-5984	628	11	s.	s.	PROPN
ejpam-5984	628	12	m.	m.	PROPN
ejpam-5984	628	13	sheikholeslami	sheikholeslami	PROPN
ejpam-5984	628	14	et	et	PROPN
ejpam-5984	628	15	al	al	PROPN
ejpam-5984	628	16	.	.	PUNCT
ejpam-5984	628	17	/	/	SYM
ejpam-5984	628	18	eur	eur	PROPN
ejpam-5984	628	19	.	.	PUNCT
ejpam-5984	629	1	j.	j.	PROPN
ejpam-5984	629	2	pure	pure	PROPN
ejpam-5984	629	3	appl	appl	PROPN
ejpam-5984	629	4	.	.	PROPN
ejpam-5984	629	5	math	math	PROPN
ejpam-5984	629	6	,	,	PUNCT
ejpam-5984	629	7	18	18	NUM
ejpam-5984	629	8	(	(	PUNCT
ejpam-5984	629	9	2	2	NUM
ejpam-5984	629	10	)	)	PUNCT
ejpam-5984	629	11	(	(	PUNCT
ejpam-5984	629	12	2025	2025	NUM
ejpam-5984	629	13	)	)	PUNCT
ejpam-5984	629	14	,	,	PUNCT
ejpam-5984	629	15	5984	5984	NUM
ejpam-5984	629	16	14	14	NUM
ejpam-5984	629	17	of	of	ADP
ejpam-5984	629	18	16	16	NUM
ejpam-5984	629	19	and	and	CCONJ
ejpam-5984	629	20	x4	x4	PROPN
ejpam-5984	629	21	.	.	PUNCT
ejpam-5984	630	1	clearly	clearly	ADV
ejpam-5984	630	2	,	,	PUNCT
ejpam-5984	630	3	w	w	PROPN
ejpam-5984	630	4	is	be	AUX
ejpam-5984	630	5	either	either	CCONJ
ejpam-5984	630	6	a	a	DET
ejpam-5984	630	7	leaf	leaf	NOUN
ejpam-5984	630	8	or	or	CCONJ
ejpam-5984	630	9	a	a	DET
ejpam-5984	630	10	support	support	NOUN
ejpam-5984	630	11	vertex	vertex	NOUN
ejpam-5984	630	12	.	.	PUNCT
ejpam-5984	631	1	in	in	ADP
ejpam-5984	631	2	the	the	DET
ejpam-5984	631	3	former	former	ADJ
ejpam-5984	631	4	case	case	NOUN
ejpam-5984	631	5	,	,	PUNCT
ejpam-5984	631	6	it	it	PRON
ejpam-5984	631	7	follows	follow	VERB
ejpam-5984	631	8	from	from	ADP
ejpam-5984	631	9	f(x3	f(x3	NOUN
ejpam-5984	631	10	)	)	PUNCT
ejpam-5984	632	1	=	=	SYM
ejpam-5984	632	2	0	0	NUM
ejpam-5984	632	3	that	that	DET
ejpam-5984	632	4	f(w	f(w	NOUN
ejpam-5984	632	5	)	)	PUNCT
ejpam-5984	632	6	=	=	SYM
ejpam-5984	633	1	2	2	X
ejpam-5984	633	2	.	.	X
ejpam-5984	633	3	in	in	ADP
ejpam-5984	633	4	the	the	DET
ejpam-5984	633	5	latter	latter	ADJ
ejpam-5984	633	6	case	case	NOUN
ejpam-5984	633	7	,	,	PUNCT
ejpam-5984	633	8	by	by	ADP
ejpam-5984	633	9	a	a	DET
ejpam-5984	633	10	similar	similar	ADJ
ejpam-5984	633	11	argument	argument	NOUN
ejpam-5984	633	12	as	as	ADP
ejpam-5984	633	13	in	in	ADP
ejpam-5984	633	14	above	above	ADV
ejpam-5984	633	15	,	,	PUNCT
ejpam-5984	633	16	we	we	PRON
ejpam-5984	633	17	have	have	VERB
ejpam-5984	633	18	that	that	PRON
ejpam-5984	633	19	w	w	NOUN
ejpam-5984	633	20	has	have	VERB
ejpam-5984	633	21	degree	degree	NOUN
ejpam-5984	633	22	∆	∆	PROPN
ejpam-5984	633	23	and	and	CCONJ
ejpam-5984	633	24	f(w	f(w	PROPN
ejpam-5984	633	25	)	)	PUNCT
ejpam-5984	633	26	=	=	PUNCT
ejpam-5984	634	1	3	3	X
ejpam-5984	634	2	.	.	PUNCT
ejpam-5984	634	3	in	in	ADP
ejpam-5984	634	4	either	either	DET
ejpam-5984	634	5	case	case	NOUN
ejpam-5984	634	6	,	,	PUNCT
ejpam-5984	634	7	remove	remove	VERB
ejpam-5984	634	8	all	all	DET
ejpam-5984	634	9	leaf	leaf	NOUN
ejpam-5984	634	10	-	-	PUNCT
ejpam-5984	634	11	neighbor	neighbor	NOUN
ejpam-5984	634	12	of	of	ADP
ejpam-5984	634	13	x2	x2	PROPN
ejpam-5984	634	14	and	and	CCONJ
ejpam-5984	634	15	denote	denote	VERB
ejpam-5984	634	16	the	the	DET
ejpam-5984	634	17	resulting	result	VERB
ejpam-5984	634	18	tree	tree	NOUN
ejpam-5984	634	19	by	by	ADP
ejpam-5984	634	20	t	t	PROPN
ejpam-5984	634	21	′.	′.	NOUN
ejpam-5984	634	22	then	then	ADV
ejpam-5984	634	23	reassigning	reassign	VERB
ejpam-5984	634	24	2	2	NUM
ejpam-5984	634	25	to	to	ADP
ejpam-5984	634	26	x2	x2	PROPN
ejpam-5984	634	27	provides	provide	VERB
ejpam-5984	634	28	an	an	DET
ejpam-5984	634	29	idrd	idrd	ADJ
ejpam-5984	634	30	-	-	PUNCT
ejpam-5984	634	31	function	function	NOUN
ejpam-5984	634	32	of	of	ADP
ejpam-5984	634	33	t	t	NOUN
ejpam-5984	634	34	′	′	NUM
ejpam-5984	634	35	leading	lead	VERB
ejpam-5984	634	36	to	to	ADP
ejpam-5984	634	37	st−idr(t	st−idr(t	NOUN
ejpam-5984	634	38	)	)	PUNCT
ejpam-5984	634	39	≤	≤	NUM
ejpam-5984	634	40	∆−	∆−	NOUN
ejpam-5984	634	41	1	1	NUM
ejpam-5984	634	42	which	which	PRON
ejpam-5984	634	43	is	be	AUX
ejpam-5984	634	44	contradiction	contradiction	NOUN
ejpam-5984	634	45	.	.	PUNCT
ejpam-5984	635	1	thus	thus	ADV
ejpam-5984	635	2	,	,	PUNCT
ejpam-5984	635	3	d(x3	d(x3	X
ejpam-5984	635	4	)	)	PUNCT
ejpam-5984	635	5	=	=	SYM
ejpam-5984	635	6	2	2	X
ejpam-5984	635	7	.	.	PUNCT
ejpam-5984	635	8	by	by	ADP
ejpam-5984	635	9	symmetry	symmetry	NOUN
ejpam-5984	635	10	,	,	PUNCT
ejpam-5984	635	11	we	we	PRON
ejpam-5984	635	12	have	have	VERB
ejpam-5984	635	13	d(xd−1	d(xd−1	NOUN
ejpam-5984	635	14	)	)	PUNCT
ejpam-5984	635	15	=	=	SYM
ejpam-5984	635	16	∆	∆	PROPN
ejpam-5984	635	17	and	and	CCONJ
ejpam-5984	635	18	d(xd−2	d(xd−2	NOUN
ejpam-5984	635	19	)	)	PUNCT
ejpam-5984	635	20	=	=	SYM
ejpam-5984	636	1	2	2	X
ejpam-5984	636	2	.	.	X
ejpam-5984	637	1	if	if	SCONJ
ejpam-5984	637	2	x3	x3	PROPN
ejpam-5984	637	3	=	=	SYM
ejpam-5984	637	4	xd−2	xd−2	PROPN
ejpam-5984	637	5	,	,	PUNCT
ejpam-5984	637	6	then	then	ADV
ejpam-5984	637	7	it	it	PRON
ejpam-5984	637	8	is	be	AUX
ejpam-5984	637	9	easy	easy	ADJ
ejpam-5984	637	10	to	to	PART
ejpam-5984	637	11	see	see	VERB
ejpam-5984	637	12	that	that	DET
ejpam-5984	637	13	idr(t	idr(t	NOUN
ejpam-5984	637	14	)	)	PUNCT
ejpam-5984	637	15	=	=	SYM
ejpam-5984	637	16	6	6	NUM
ejpam-5984	637	17	,	,	PUNCT
ejpam-5984	637	18	and	and	CCONJ
ejpam-5984	637	19	obviously	obviously	ADV
ejpam-5984	637	20	idr(t	idr(t	ADJ
ejpam-5984	637	21	−	−	NOUN
ejpam-5984	637	22	l(x2	l(x2	NOUN
ejpam-5984	637	23	)	)	PUNCT
ejpam-5984	637	24	)	)	PUNCT
ejpam-5984	638	1	=	=	SYM
ejpam-5984	638	2	5	5	NUM
ejpam-5984	638	3	and	and	CCONJ
ejpam-5984	638	4	so	so	ADV
ejpam-5984	638	5	st−idr(t	st−idr(t	NOUN
ejpam-5984	638	6	)	)	PUNCT
ejpam-5984	638	7	≤	≤	NOUN
ejpam-5984	638	8	∆	∆	PROPN
ejpam-5984	639	1	−	−	NOUN
ejpam-5984	639	2	1	1	NUM
ejpam-5984	639	3	,	,	PUNCT
ejpam-5984	639	4	a	a	DET
ejpam-5984	639	5	contradiction	contradiction	NOUN
ejpam-5984	639	6	.	.	PUNCT
ejpam-5984	640	1	hence	hence	ADV
ejpam-5984	640	2	x3	x3	VERB
ejpam-5984	640	3	̸=	̸=	PROPN
ejpam-5984	640	4	xd−2	xd−2	PROPN
ejpam-5984	640	5	.	.	PUNCT
ejpam-5984	641	1	let	let	VERB
ejpam-5984	641	2	t	t	NOUN
ejpam-5984	641	3	′	′	NUM
ejpam-5984	642	1	=	=	PUNCT
ejpam-5984	642	2	t	t	PROPN
ejpam-5984	642	3	−	−	NOUN
ejpam-5984	642	4	tx3	tx3	NOUN
ejpam-5984	642	5	.	.	PUNCT
ejpam-5984	643	1	by	by	ADP
ejpam-5984	643	2	proposition	proposition	NOUN
ejpam-5984	643	3	2	2	NUM
ejpam-5984	643	4	,	,	PUNCT
ejpam-5984	643	5	we	we	PRON
ejpam-5984	643	6	have	have	VERB
ejpam-5984	643	7	idr(t	idr(t	NOUN
ejpam-5984	643	8	)	)	PUNCT
ejpam-5984	643	9	=	=	SYM
ejpam-5984	644	1	idr(t	idr(t	PROPN
ejpam-5984	644	2	′)+3	′)+3	NOUN
ejpam-5984	644	3	.	.	PUNCT
ejpam-5984	645	1	we	we	PRON
ejpam-5984	645	2	claim	claim	VERB
ejpam-5984	645	3	that	that	SCONJ
ejpam-5984	645	4	st−idr(t	st−idr(t	NOUN
ejpam-5984	645	5	′	′	NOUN
ejpam-5984	645	6	)	)	PUNCT
ejpam-5984	646	1	=	=	PUNCT
ejpam-5984	647	1	∆.	∆.	NOUN
ejpam-5984	647	2	by	by	ADP
ejpam-5984	647	3	contradiction	contradiction	NOUN
ejpam-5984	647	4	,	,	PUNCT
ejpam-5984	647	5	assume	assume	VERB
ejpam-5984	647	6	that	that	SCONJ
ejpam-5984	647	7	st−idr(t	st−idr(t	NOUN
ejpam-5984	647	8	′	′	NOUN
ejpam-5984	647	9	)	)	PUNCT
ejpam-5984	647	10	≤	≤	NOUN
ejpam-5984	647	11	∆−1	∆−1	PROPN
ejpam-5984	647	12	and	and	CCONJ
ejpam-5984	647	13	let	let	VERB
ejpam-5984	647	14	s′	s′	PROPN
ejpam-5984	647	15	be	be	AUX
ejpam-5984	647	16	a	a	DET
ejpam-5984	647	17	st−idr(t	st−idr(t	NOUN
ejpam-5984	647	18	′)-set	′)-set	NOUN
ejpam-5984	647	19	.	.	PUNCT
ejpam-5984	648	1	clearly	clearly	ADV
ejpam-5984	648	2	,	,	PUNCT
ejpam-5984	648	3	any	any	DET
ejpam-5984	648	4	idr(t	idr(t	NOUN
ejpam-5984	648	5	′	′	NUM
ejpam-5984	648	6	−	−	PROPN
ejpam-5984	648	7	s′)-function	s′)-function	NOUN
ejpam-5984	648	8	can	can	AUX
ejpam-5984	648	9	be	be	AUX
ejpam-5984	648	10	extended	extend	VERB
ejpam-5984	648	11	to	to	ADP
ejpam-5984	648	12	an	an	DET
ejpam-5984	648	13	idrd	idrd	ADJ
ejpam-5984	648	14	-	-	PUNCT
ejpam-5984	648	15	function	function	NOUN
ejpam-5984	648	16	of	of	ADP
ejpam-5984	648	17	t	t	PROPN
ejpam-5984	648	18	−	−	PROPN
ejpam-5984	648	19	s′	s′	VERB
ejpam-5984	648	20	by	by	ADP
ejpam-5984	648	21	assigning	assign	VERB
ejpam-5984	648	22	a	a	DET
ejpam-5984	648	23	3	3	NUM
ejpam-5984	648	24	to	to	ADP
ejpam-5984	648	25	x2	x2	PROPN
ejpam-5984	648	26	and	and	CCONJ
ejpam-5984	648	27	0	0	NUM
ejpam-5984	648	28	to	to	ADP
ejpam-5984	648	29	the	the	DET
ejpam-5984	648	30	neighbors	neighbor	NOUN
ejpam-5984	648	31	of	of	ADP
ejpam-5984	648	32	x2	x2	PROPN
ejpam-5984	648	33	and	and	CCONJ
ejpam-5984	648	34	so	so	ADV
ejpam-5984	648	35	idr(t	idr(t	PROPN
ejpam-5984	648	36	−	−	PROPN
ejpam-5984	648	37	s′	s′	NOUN
ejpam-5984	648	38	)	)	PUNCT
ejpam-5984	648	39	≤	≤	NUM
ejpam-5984	648	40	idr(t	idr(t	NOUN
ejpam-5984	648	41	′	′	NUM
ejpam-5984	648	42	−	−	PROPN
ejpam-5984	648	43	s	s	PART
ejpam-5984	648	44	)	)	PUNCT
ejpam-5984	649	1	+	+	CCONJ
ejpam-5984	649	2	3	3	NUM
ejpam-5984	649	3	<	<	X
ejpam-5984	649	4	idr(t	idr(t	PROPN
ejpam-5984	649	5	)	)	PUNCT
ejpam-5984	649	6	which	which	PRON
ejpam-5984	649	7	contradicts	contradict	VERB
ejpam-5984	649	8	the	the	DET
ejpam-5984	649	9	assumption	assumption	NOUN
ejpam-5984	649	10	st−idr(t	st−idr(t	NOUN
ejpam-5984	649	11	)	)	PUNCT
ejpam-5984	649	12	=	=	PUNCT
ejpam-5984	650	1	∆.	∆.	NOUN
ejpam-5984	650	2	hence	hence	ADV
ejpam-5984	650	3	,	,	PUNCT
ejpam-5984	650	4	st−idr(t	st−idr(t	NOUN
ejpam-5984	650	5	′	′	NOUN
ejpam-5984	650	6	)	)	PUNCT
ejpam-5984	650	7	=	=	NOUN
ejpam-5984	651	1	∆	∆	PROPN
ejpam-5984	651	2	and	and	CCONJ
ejpam-5984	651	3	by	by	ADP
ejpam-5984	651	4	the	the	DET
ejpam-5984	651	5	induction	induction	NOUN
ejpam-5984	651	6	hypothesis	hypothesis	NOUN
ejpam-5984	651	7	we	we	PRON
ejpam-5984	651	8	have	have	VERB
ejpam-5984	651	9	t	t	X
ejpam-5984	651	10	′	′	NUM
ejpam-5984	651	11	∈	∈	PROPN
ejpam-5984	651	12	t∆.	t∆.	PROPN
ejpam-5984	651	13	thus	thus	ADV
ejpam-5984	651	14	,	,	PUNCT
ejpam-5984	651	15	t	t	PROPN
ejpam-5984	651	16	′	′	NUM
ejpam-5984	651	17	is	be	AUX
ejpam-5984	651	18	a	a	DET
ejpam-5984	651	19	tree	tree	NOUN
ejpam-5984	651	20	obtained	obtain	VERB
ejpam-5984	651	21	from	from	ADP
ejpam-5984	651	22	the	the	DET
ejpam-5984	651	23	k′	k′	PROPN
ejpam-5984	651	24	copies	copy	NOUN
ejpam-5984	651	25	of	of	ADP
ejpam-5984	651	26	the	the	DET
ejpam-5984	651	27	star	star	NOUN
ejpam-5984	651	28	k1,∆	k1,∆	VERB
ejpam-5984	651	29	,	,	PUNCT
ejpam-5984	651	30	say	say	VERB
ejpam-5984	651	31	s1	s1	NOUN
ejpam-5984	651	32	,	,	PUNCT
ejpam-5984	651	33	s2	s2	PROPN
ejpam-5984	651	34	,	,	PUNCT
ejpam-5984	651	35	.	.	PUNCT
ejpam-5984	651	36	.	.	PUNCT
ejpam-5984	651	37	.	.	PUNCT
ejpam-5984	652	1	,	,	PUNCT
ejpam-5984	652	2	sk′	sk′	NOUN
ejpam-5984	652	3	,	,	PUNCT
ejpam-5984	652	4	by	by	ADP
ejpam-5984	652	5	adding	add	VERB
ejpam-5984	652	6	k′	k′	PROPN
ejpam-5984	652	7	−	−	PROPN
ejpam-5984	652	8	1	1	NUM
ejpam-5984	652	9	edges	edge	NOUN
ejpam-5984	652	10	between	between	ADP
ejpam-5984	652	11	the	the	DET
ejpam-5984	652	12	leaves	leave	NOUN
ejpam-5984	652	13	of	of	ADP
ejpam-5984	652	14	these	these	DET
ejpam-5984	652	15	stars	star	NOUN
ejpam-5984	652	16	so	so	SCONJ
ejpam-5984	652	17	that	that	SCONJ
ejpam-5984	652	18	the	the	DET
ejpam-5984	652	19	resulting	result	VERB
ejpam-5984	652	20	graph	graph	NOUN
ejpam-5984	652	21	is	be	AUX
ejpam-5984	652	22	a	a	DET
ejpam-5984	652	23	connected	connected	ADJ
ejpam-5984	652	24	graph	graph	NOUN
ejpam-5984	652	25	with	with	ADP
ejpam-5984	652	26	maximum	maximum	ADJ
ejpam-5984	652	27	degree	degree	NOUN
ejpam-5984	652	28	∆.	∆.	NOUN
ejpam-5984	652	29	it	it	PRON
ejpam-5984	652	30	follows	follow	VERB
ejpam-5984	652	31	from	from	ADP
ejpam-5984	652	32	∆	∆	PROPN
ejpam-5984	652	33	≥	≥	X
ejpam-5984	652	34	degt	degt	VERB
ejpam-5984	652	35	(	(	PUNCT
ejpam-5984	652	36	x4	x4	PROPN
ejpam-5984	652	37	)	)	PUNCT
ejpam-5984	652	38	=	=	PRON
ejpam-5984	652	39	degt	degt	VERB
ejpam-5984	652	40	′(x4	′(x4	NOUN
ejpam-5984	652	41	)	)	PUNCT
ejpam-5984	653	1	+	+	CCONJ
ejpam-5984	653	2	1	1	NUM
ejpam-5984	653	3	that	that	PRON
ejpam-5984	653	4	x4	x4	PROPN
ejpam-5984	653	5	is	be	AUX
ejpam-5984	653	6	a	a	DET
ejpam-5984	653	7	leaf	leaf	NOUN
ejpam-5984	653	8	of	of	ADP
ejpam-5984	653	9	some	some	DET
ejpam-5984	653	10	star	star	NOUN
ejpam-5984	653	11	si	si	PROPN
ejpam-5984	654	1	and	and	CCONJ
ejpam-5984	654	2	so	so	ADV
ejpam-5984	654	3	t	t	PROPN
ejpam-5984	654	4	∈	∈	PROPN
ejpam-5984	654	5	t∆	t∆	PROPN
ejpam-5984	654	6	and	and	CCONJ
ejpam-5984	654	7	the	the	DET
ejpam-5984	654	8	proof	proof	NOUN
ejpam-5984	654	9	is	be	AUX
ejpam-5984	654	10	completed	complete	VERB
ejpam-5984	654	11	.	.	PUNCT
ejpam-5984	655	1	s.	s.	PROPN
ejpam-5984	655	2	m.	m.	PROPN
ejpam-5984	655	3	sheikholeslami	sheikholeslami	PROPN
ejpam-5984	655	4	et	et	PROPN
ejpam-5984	655	5	al	al	PROPN
ejpam-5984	655	6	.	.	PUNCT
ejpam-5984	655	7	/	/	SYM
ejpam-5984	655	8	eur	eur	PROPN
ejpam-5984	655	9	.	.	PUNCT
ejpam-5984	656	1	j.	j.	PROPN
ejpam-5984	656	2	pure	pure	PROPN
ejpam-5984	656	3	appl	appl	PROPN
ejpam-5984	656	4	.	.	PROPN
ejpam-5984	656	5	math	math	PROPN
ejpam-5984	656	6	,	,	PUNCT
ejpam-5984	656	7	18	18	NUM
ejpam-5984	656	8	(	(	PUNCT
ejpam-5984	656	9	2	2	NUM
ejpam-5984	656	10	)	)	PUNCT
ejpam-5984	656	11	(	(	PUNCT
ejpam-5984	656	12	2025	2025	NUM
ejpam-5984	656	13	)	)	PUNCT
ejpam-5984	656	14	,	,	PUNCT
ejpam-5984	656	15	5984	5984	NUM
ejpam-5984	656	16	15	15	NUM
ejpam-5984	656	17	of	of	ADP
ejpam-5984	656	18	16	16	NUM
ejpam-5984	656	19	7	7	NUM
ejpam-5984	656	20	.	.	PUNCT
ejpam-5984	657	1	open	open	ADJ
ejpam-5984	657	2	questions	question	NOUN
ejpam-5984	657	3	and	and	CCONJ
ejpam-5984	657	4	problems	problem	NOUN
ejpam-5984	657	5	we	we	PRON
ejpam-5984	657	6	conclude	conclude	VERB
ejpam-5984	657	7	this	this	DET
ejpam-5984	657	8	paper	paper	NOUN
ejpam-5984	657	9	by	by	ADP
ejpam-5984	657	10	mentioning	mention	VERB
ejpam-5984	657	11	some	some	DET
ejpam-5984	657	12	questions	question	NOUN
ejpam-5984	657	13	and	and	CCONJ
ejpam-5984	657	14	problems	problem	NOUN
ejpam-5984	657	15	suggested	suggest	VERB
ejpam-5984	657	16	by	by	ADP
ejpam-5984	657	17	this	this	DET
ejpam-5984	657	18	research	research	NOUN
ejpam-5984	657	19	.	.	PUNCT
ejpam-5984	658	1	problem	problem	NOUN
ejpam-5984	658	2	1	1	NUM
ejpam-5984	658	3	.	.	PUNCT
ejpam-5984	658	4	characterize	characterize	VERB
ejpam-5984	658	5	the	the	DET
ejpam-5984	658	6	connected	connected	ADJ
ejpam-5984	658	7	graphs	graph	NOUN
ejpam-5984	658	8	g	g	ADP
ejpam-5984	658	9	of	of	ADP
ejpam-5984	658	10	order	order	NOUN
ejpam-5984	658	11	n	n	PRON
ejpam-5984	658	12	with	with	ADP
ejpam-5984	658	13	idr(g	idr(g	PROPN
ejpam-5984	658	14	)	)	PUNCT
ejpam-5984	658	15	≥	≥	NOUN
ejpam-5984	658	16	4	4	NUM
ejpam-5984	658	17	and	and	CCONJ
ejpam-5984	658	18	stidr(g	stidr(g	NUM
ejpam-5984	658	19	)	)	PUNCT
ejpam-5984	659	1	=	=	PRON
ejpam-5984	659	2	n−∆(g)−	n−∆(g)−	ADJ
ejpam-5984	659	3	1	1	NUM
ejpam-5984	659	4	.	.	PUNCT
ejpam-5984	660	1	problem	problem	NOUN
ejpam-5984	660	2	2	2	NUM
ejpam-5984	660	3	.	.	PUNCT
ejpam-5984	660	4	is	be	AUX
ejpam-5984	660	5	there	there	PRON
ejpam-5984	660	6	a	a	DET
ejpam-5984	660	7	connected	connected	ADJ
ejpam-5984	660	8	graph	graph	NOUN
ejpam-5984	660	9	g	g	NOUN
ejpam-5984	660	10	of	of	ADP
ejpam-5984	660	11	order	order	NOUN
ejpam-5984	660	12	n	n	PRON
ejpam-5984	660	13	≥	≥	NUM
ejpam-5984	660	14	2	2	NUM
ejpam-5984	660	15	such	such	ADJ
ejpam-5984	660	16	that	that	DET
ejpam-5984	660	17	stidr(g	stidr(g	NOUN
ejpam-5984	660	18	)	)	PUNCT
ejpam-5984	660	19	=	=	PUNCT
ejpam-5984	660	20	δ(g	δ(g	X
ejpam-5984	660	21	)	)	PUNCT
ejpam-5984	661	1	+	+	CCONJ
ejpam-5984	661	2	1	1	X
ejpam-5984	661	3	.	.	X
ejpam-5984	661	4	problem	problem	NOUN
ejpam-5984	661	5	3	3	NUM
ejpam-5984	661	6	.	.	PUNCT
ejpam-5984	661	7	determine	determine	VERB
ejpam-5984	661	8	the	the	DET
ejpam-5984	661	9	independent	independent	ADJ
ejpam-5984	661	10	double	double	ADJ
ejpam-5984	661	11	roman	roman	ADJ
ejpam-5984	661	12	domination	domination	NOUN
ejpam-5984	661	13	stability	stability	NOUN
ejpam-5984	661	14	of	of	ADP
ejpam-5984	661	15	generalized	generalized	ADJ
ejpam-5984	661	16	petersen	petersen	NOUN
ejpam-5984	661	17	graphs	graph	NOUN
ejpam-5984	661	18	and	and	CCONJ
ejpam-5984	661	19	sierpinski	sierpinski	ADJ
ejpam-5984	661	20	graphs	graph	NOUN
ejpam-5984	661	21	.	.	PUNCT
ejpam-5984	662	1	8	8	X
ejpam-5984	662	2	.	.	X
ejpam-5984	662	3	conclusion	conclusion	NOUN
ejpam-5984	662	4	in	in	ADP
ejpam-5984	662	5	this	this	DET
ejpam-5984	662	6	paper	paper	NOUN
ejpam-5984	662	7	,	,	PUNCT
ejpam-5984	662	8	we	we	PRON
ejpam-5984	662	9	have	have	AUX
ejpam-5984	662	10	studied	study	VERB
ejpam-5984	662	11	the	the	DET
ejpam-5984	662	12	independent	independent	ADJ
ejpam-5984	662	13	double	double	ADJ
ejpam-5984	662	14	roman	roman	ADJ
ejpam-5984	662	15	domination	domination	NOUN
ejpam-5984	662	16	stability	stability	NOUN
ejpam-5984	662	17	.	.	PUNCT
ejpam-5984	663	1	we	we	PRON
ejpam-5984	663	2	determined	determine	VERB
ejpam-5984	663	3	exact	exact	ADJ
ejpam-5984	663	4	values	value	NOUN
ejpam-5984	663	5	of	of	ADP
ejpam-5984	663	6	the	the	DET
ejpam-5984	663	7	independent	independent	ADJ
ejpam-5984	663	8	double	double	ADJ
ejpam-5984	663	9	roman	roman	ADJ
ejpam-5984	663	10	domination	domination	NOUN
ejpam-5984	663	11	stability	stability	NOUN
ejpam-5984	663	12	for	for	ADP
ejpam-5984	663	13	special	special	ADJ
ejpam-5984	663	14	classes	class	NOUN
ejpam-5984	663	15	of	of	ADP
ejpam-5984	663	16	graphs	graph	NOUN
ejpam-5984	663	17	.	.	PUNCT
ejpam-5984	664	1	additionally	additionally	ADV
ejpam-5984	664	2	,	,	PUNCT
ejpam-5984	664	3	we	we	PRON
ejpam-5984	664	4	established	establish	VERB
ejpam-5984	664	5	bounds	bound	NOUN
ejpam-5984	664	6	on	on	ADP
ejpam-5984	664	7	the	the	DET
ejpam-5984	664	8	idr	idr	NOUN
ejpam-5984	664	9	-	-	NOUN
ejpam-5984	664	10	stability	stability	NOUN
ejpam-5984	664	11	for	for	ADP
ejpam-5984	664	12	general	general	ADJ
ejpam-5984	664	13	graphs	graph	NOUN
ejpam-5984	664	14	.	.	PUNCT
ejpam-5984	665	1	for	for	ADP
ejpam-5984	665	2	trees	tree	NOUN
ejpam-5984	665	3	,	,	PUNCT
ejpam-5984	665	4	we	we	PRON
ejpam-5984	665	5	proved	prove	VERB
ejpam-5984	665	6	that	that	SCONJ
ejpam-5984	665	7	the	the	DET
ejpam-5984	665	8	idr	idr	PROPN
ejpam-5984	665	9	-	-	PUNCT
ejpam-5984	665	10	stability	stability	NOUN
ejpam-5984	665	11	is	be	AUX
ejpam-5984	665	12	always	always	ADV
ejpam-5984	665	13	equal	equal	ADJ
ejpam-5984	665	14	to	to	ADP
ejpam-5984	665	15	1	1	NUM
ejpam-5984	665	16	,	,	PUNCT
ejpam-5984	665	17	while	while	SCONJ
ejpam-5984	665	18	the	the	DET
ejpam-5984	665	19	i−dr	i−dr	NOUN
ejpam-5984	665	20	-	-	PUNCT
ejpam-5984	665	21	stability	stability	NOUN
ejpam-5984	665	22	is	be	AUX
ejpam-5984	665	23	bounded	bound	VERB
ejpam-5984	665	24	above	above	ADV
ejpam-5984	665	25	by	by	ADP
ejpam-5984	665	26	the	the	DET
ejpam-5984	665	27	maximum	maximum	ADJ
ejpam-5984	665	28	degree	degree	NOUN
ejpam-5984	665	29	∆	∆	PROPN
ejpam-5984	665	30	of	of	ADP
ejpam-5984	665	31	the	the	DET
ejpam-5984	665	32	tree	tree	NOUN
ejpam-5984	665	33	.	.	PUNCT
ejpam-5984	666	1	we	we	PRON
ejpam-5984	666	2	also	also	ADV
ejpam-5984	666	3	provided	provide	VERB
ejpam-5984	666	4	a	a	DET
ejpam-5984	666	5	complete	complete	ADJ
ejpam-5984	666	6	characterization	characterization	NOUN
ejpam-5984	666	7	of	of	ADP
ejpam-5984	666	8	the	the	DET
ejpam-5984	666	9	trees	tree	NOUN
ejpam-5984	666	10	that	that	PRON
ejpam-5984	666	11	attain	attain	VERB
ejpam-5984	666	12	this	this	DET
ejpam-5984	666	13	upper	upper	ADJ
ejpam-5984	666	14	bound	bind	VERB
ejpam-5984	666	15	.	.	PUNCT
ejpam-5984	667	1	these	these	DET
ejpam-5984	667	2	results	result	NOUN
ejpam-5984	667	3	contribute	contribute	VERB
ejpam-5984	667	4	to	to	ADP
ejpam-5984	667	5	the	the	DET
ejpam-5984	667	6	growing	grow	VERB
ejpam-5984	667	7	body	body	NOUN
ejpam-5984	667	8	of	of	ADP
ejpam-5984	667	9	knowledge	knowledge	NOUN
ejpam-5984	667	10	on	on	ADP
ejpam-5984	667	11	roman	roman	ADJ
ejpam-5984	667	12	domination	domination	NOUN
ejpam-5984	667	13	parameters	parameter	NOUN
ejpam-5984	667	14	and	and	CCONJ
ejpam-5984	667	15	open	open	ADJ
ejpam-5984	667	16	avenues	avenue	NOUN
ejpam-5984	667	17	for	for	ADP
ejpam-5984	667	18	further	further	ADJ
ejpam-5984	667	19	research	research	NOUN
ejpam-5984	667	20	on	on	ADP
ejpam-5984	667	21	stability	stability	NOUN
ejpam-5984	667	22	measures	measure	NOUN
ejpam-5984	667	23	in	in	ADP
ejpam-5984	667	24	more	more	ADJ
ejpam-5984	667	25	complex	complex	ADJ
ejpam-5984	667	26	graph	graph	NOUN
ejpam-5984	667	27	structures	structure	NOUN
ejpam-5984	667	28	.	.	PUNCT
ejpam-5984	668	1	acknowledgements	acknowledgement	NOUN
ejpam-5984	668	2	we	we	PRON
ejpam-5984	668	3	sincerely	sincerely	ADV
ejpam-5984	668	4	thank	thank	VERB
ejpam-5984	668	5	the	the	DET
ejpam-5984	668	6	reviewers	reviewer	NOUN
ejpam-5984	668	7	for	for	ADP
ejpam-5984	668	8	their	their	PRON
ejpam-5984	668	9	valuable	valuable	ADJ
ejpam-5984	668	10	comments	comment	NOUN
ejpam-5984	668	11	and	and	CCONJ
ejpam-5984	668	12	suggestions	suggestion	NOUN
ejpam-5984	668	13	,	,	PUNCT
ejpam-5984	668	14	which	which	PRON
ejpam-5984	668	15	have	have	AUX
ejpam-5984	668	16	significantly	significantly	ADV
ejpam-5984	668	17	improved	improve	VERB
ejpam-5984	668	18	the	the	DET
ejpam-5984	668	19	quality	quality	NOUN
ejpam-5984	668	20	of	of	ADP
ejpam-5984	668	21	this	this	DET
ejpam-5984	668	22	paper	paper	NOUN
ejpam-5984	668	23	.	.	PUNCT
ejpam-5984	669	1	we	we	PRON
ejpam-5984	669	2	acknowledge	acknowledge	VERB
ejpam-5984	669	3	ho	ho	PROPN
ejpam-5984	669	4	chi	chi	PROPN
ejpam-5984	669	5	minh	minh	PROPN
ejpam-5984	669	6	city	city	PROPN
ejpam-5984	669	7	university	university	PROPN
ejpam-5984	669	8	of	of	ADP
ejpam-5984	669	9	technology	technology	NOUN
ejpam-5984	669	10	(	(	PUNCT
ejpam-5984	669	11	hcmut	hcmut	NOUN
ejpam-5984	669	12	)	)	PUNCT
ejpam-5984	669	13	,	,	PUNCT
ejpam-5984	669	14	vnu	vnu	PROPN
ejpam-5984	669	15	-	-	PUNCT
ejpam-5984	669	16	hcm	hcm	PROPN
ejpam-5984	669	17	for	for	ADP
ejpam-5984	669	18	supporting	support	VERB
ejpam-5984	669	19	this	this	DET
ejpam-5984	669	20	study	study	NOUN
ejpam-5984	669	21	.	.	PUNCT
ejpam-5984	670	1	in	in	ADP
ejpam-5984	670	2	addition	addition	NOUN
ejpam-5984	670	3	,	,	PUNCT
ejpam-5984	670	4	we	we	PRON
ejpam-5984	670	5	extend	extend	VERB
ejpam-5984	670	6	our	our	PRON
ejpam-5984	670	7	heartfelt	heartfelt	ADJ
ejpam-5984	670	8	appreciation	appreciation	NOUN
ejpam-5984	670	9	to	to	ADP
ejpam-5984	670	10	mindanao	mindanao	PROPN
ejpam-5984	670	11	state	state	PROPN
ejpam-5984	670	12	university	university	PROPN
ejpam-5984	670	13	–	–	PUNCT
ejpam-5984	670	14	tawi	tawi	NOUN
ejpam-5984	670	15	-	-	PUNCT
ejpam-5984	670	16	tawi	tawi	NOUN
ejpam-5984	670	17	college	college	PROPN
ejpam-5984	670	18	of	of	ADP
ejpam-5984	670	19	technology	technology	NOUN
ejpam-5984	670	20	and	and	CCONJ
ejpam-5984	670	21	oceanography	oceanography	NOUN
ejpam-5984	670	22	(	(	PUNCT
ejpam-5984	670	23	msu	msu	PROPN
ejpam-5984	670	24	-	-	PUNCT
ejpam-5984	670	25	tcto	tcto	VERB
ejpam-5984	670	26	)	)	PUNCT
ejpam-5984	670	27	and	and	CCONJ
ejpam-5984	670	28	mindanao	mindanao	PROPN
ejpam-5984	670	29	state	state	PROPN
ejpam-5984	670	30	university	university	PROPN
ejpam-5984	670	31	-	-	PUNCT
ejpam-5984	670	32	iligan	iligan	PROPN
ejpam-5984	670	33	institute	institute	PROPN
ejpam-5984	670	34	of	of	ADP
ejpam-5984	670	35	technology	technology	PROPN
ejpam-5984	670	36	(	(	PUNCT
ejpam-5984	670	37	msu	msu	PROPN
ejpam-5984	670	38	iit	iit	PROPN
ejpam-5984	670	39	)	)	PUNCT
ejpam-5984	670	40	for	for	ADP
ejpam-5984	670	41	generously	generously	ADV
ejpam-5984	670	42	funding	fund	VERB
ejpam-5984	670	43	this	this	DET
ejpam-5984	670	44	work	work	NOUN
ejpam-5984	670	45	.	.	PUNCT
ejpam-5984	671	1	references	reference	NOUN
ejpam-5984	671	2	[	[	X
ejpam-5984	671	3	1	1	X
ejpam-5984	671	4	]	]	PUNCT
ejpam-5984	671	5	e.	e.	PROPN
ejpam-5984	671	6	j.	j.	PROPN
ejpam-5984	671	7	cockayne	cockayne	PROPN
ejpam-5984	671	8	,	,	PUNCT
ejpam-5984	672	1	p.	p.	NOUN
ejpam-5984	672	2	a.	a.	NOUN
ejpam-5984	673	1	dreyer	dreyer	PROPN
ejpam-5984	673	2	jr	jr	PROPN
ejpam-5984	673	3	.	.	PROPN
ejpam-5984	673	4	,	,	PUNCT
ejpam-5984	673	5	s.	s.	PROPN
ejpam-5984	673	6	m.	m.	PROPN
ejpam-5984	673	7	hedetniemi	hedetniemi	ADV
ejpam-5984	673	8	,	,	PUNCT
ejpam-5984	673	9	and	and	CCONJ
ejpam-5984	673	10	s.	s.	PROPN
ejpam-5984	673	11	t.	t.	PROPN
ejpam-5984	673	12	hedetniemi	hedetniemi	PROPN
ejpam-5984	673	13	.	.	PUNCT
ejpam-5984	674	1	roman	roman	ADJ
ejpam-5984	674	2	domination	domination	NOUN
ejpam-5984	674	3	in	in	ADP
ejpam-5984	674	4	graphs	graph	NOUN
ejpam-5984	674	5	.	.	PUNCT
ejpam-5984	675	1	discrete	discrete	ADJ
ejpam-5984	675	2	mathematics	mathematic	NOUN
ejpam-5984	675	3	,	,	PUNCT
ejpam-5984	675	4	278(1	278(1	NUM
ejpam-5984	675	5	-	-	SYM
ejpam-5984	675	6	3):11–22	3):11–22	NUM
ejpam-5984	675	7	,	,	PUNCT
ejpam-5984	675	8	2004	2004	NUM
ejpam-5984	675	9	.	.	PUNCT
ejpam-5984	676	1	[	[	X
ejpam-5984	676	2	2	2	NUM
ejpam-5984	676	3	]	]	PUNCT
ejpam-5984	676	4	i.	i.	PROPN
ejpam-5984	676	5	stewart	stewart	PROPN
ejpam-5984	676	6	.	.	PUNCT
ejpam-5984	677	1	defend	defend	VERB
ejpam-5984	677	2	the	the	DET
ejpam-5984	677	3	roman	roman	ADJ
ejpam-5984	677	4	empire	empire	NOUN
ejpam-5984	677	5	.	.	PUNCT
ejpam-5984	678	1	scientific	scientific	ADJ
ejpam-5984	678	2	american	american	PROPN
ejpam-5984	678	3	,	,	PUNCT
ejpam-5984	678	4	281(6):136–139	281(6):136–139	PROPN
ejpam-5984	678	5	,	,	PUNCT
ejpam-5984	678	6	1999	1999	NUM
ejpam-5984	678	7	.	.	PUNCT
ejpam-5984	679	1	[	[	X
ejpam-5984	679	2	3	3	X
ejpam-5984	679	3	]	]	X
ejpam-5984	679	4	c.	c.	PROPN
ejpam-5984	679	5	s.	s.	PROPN
ejpam-5984	679	6	revelle	revelle	PROPN
ejpam-5984	679	7	and	and	CCONJ
ejpam-5984	679	8	k.	k.	PROPN
ejpam-5984	679	9	e.	e.	PROPN
ejpam-5984	679	10	rosing	rosing	PROPN
ejpam-5984	679	11	.	.	PUNCT
ejpam-5984	680	1	defendens	defenden	VERB
ejpam-5984	680	2	imperium	imperium	NOUN
ejpam-5984	680	3	romanum	romanum	NOUN
ejpam-5984	680	4	:	:	PUNCT
ejpam-5984	680	5	a	a	DET
ejpam-5984	680	6	classical	classical	ADJ
ejpam-5984	680	7	problem	problem	NOUN
ejpam-5984	680	8	in	in	ADP
ejpam-5984	680	9	military	military	ADJ
ejpam-5984	680	10	strategy	strategy	NOUN
ejpam-5984	680	11	.	.	PUNCT
ejpam-5984	681	1	the	the	DET
ejpam-5984	681	2	american	american	PROPN
ejpam-5984	681	3	mathematical	mathematical	PROPN
ejpam-5984	681	4	monthly	monthly	PROPN
ejpam-5984	681	5	,	,	PUNCT
ejpam-5984	681	6	107(7):585–594	107(7):585–594	PROPN
ejpam-5984	681	7	,	,	PUNCT
ejpam-5984	681	8	2000	2000	NUM
ejpam-5984	681	9	.	.	PUNCT
ejpam-5984	682	1	s.	s.	PROPN
ejpam-5984	682	2	m.	m.	PROPN
ejpam-5984	682	3	sheikholeslami	sheikholeslami	PROPN
ejpam-5984	682	4	et	et	PROPN
ejpam-5984	682	5	al	al	PROPN
ejpam-5984	682	6	.	.	PUNCT
ejpam-5984	682	7	/	/	SYM
ejpam-5984	682	8	eur	eur	PROPN
ejpam-5984	682	9	.	.	PUNCT
ejpam-5984	683	1	j.	j.	PROPN
ejpam-5984	683	2	pure	pure	PROPN
ejpam-5984	683	3	appl	appl	PROPN
ejpam-5984	683	4	.	.	PROPN
ejpam-5984	683	5	math	math	PROPN
ejpam-5984	683	6	,	,	PUNCT
ejpam-5984	683	7	18	18	NUM
ejpam-5984	683	8	(	(	PUNCT
ejpam-5984	683	9	2	2	NUM
ejpam-5984	683	10	)	)	PUNCT
ejpam-5984	683	11	(	(	PUNCT
ejpam-5984	683	12	2025	2025	NUM
ejpam-5984	683	13	)	)	PUNCT
ejpam-5984	683	14	,	,	PUNCT
ejpam-5984	683	15	5984	5984	NUM
ejpam-5984	683	16	16	16	NUM
ejpam-5984	683	17	of	of	ADP
ejpam-5984	683	18	16	16	NUM
ejpam-5984	684	1	[	[	X
ejpam-5984	684	2	4	4	NUM
ejpam-5984	684	3	]	]	X
ejpam-5984	684	4	h.	h.	NOUN
ejpam-5984	684	5	abdollahzadeh	abdollahzadeh	PROPN
ejpam-5984	684	6	ahangar	ahangar	NOUN
ejpam-5984	684	7	,	,	PUNCT
ejpam-5984	684	8	m.	m.	NOUN
ejpam-5984	684	9	p.	p.	PROPN
ejpam-5984	684	10	álvarez	álvarez	PROPN
ejpam-5984	684	11	,	,	PUNCT
ejpam-5984	684	12	m.	m.	NOUN
ejpam-5984	684	13	chellali	chellali	PROPN
ejpam-5984	684	14	,	,	PUNCT
ejpam-5984	684	15	s.	s.	PROPN
ejpam-5984	684	16	m.	m.	PROPN
ejpam-5984	684	17	sheikholeslami	sheikholeslami	PROPN
ejpam-5984	684	18	,	,	PUNCT
ejpam-5984	684	19	and	and	CCONJ
ejpam-5984	684	20	j.	j.	PROPN
ejpam-5984	684	21	c.	c.	PROPN
ejpam-5984	684	22	valenzuela	valenzuela	PROPN
ejpam-5984	684	23	-	-	PUNCT
ejpam-5984	684	24	tripodoro	tripodoro	PROPN
ejpam-5984	684	25	.	.	PUNCT
ejpam-5984	685	1	triple	triple	ADJ
ejpam-5984	685	2	roman	roman	ADJ
ejpam-5984	685	3	domination	domination	NOUN
ejpam-5984	685	4	in	in	ADP
ejpam-5984	685	5	graphs	graph	NOUN
ejpam-5984	685	6	.	.	PUNCT
ejpam-5984	686	1	applied	apply	VERB
ejpam-5984	686	2	mathematics	mathematic	NOUN
ejpam-5984	686	3	and	and	CCONJ
ejpam-5984	686	4	computation	computation	NOUN
ejpam-5984	686	5	,	,	PUNCT
ejpam-5984	686	6	391:125444	391:125444	NUM
ejpam-5984	686	7	,	,	PUNCT
ejpam-5984	686	8	2021	2021	NUM
ejpam-5984	686	9	.	.	PUNCT
ejpam-5984	687	1	[	[	X
ejpam-5984	687	2	5	5	NUM
ejpam-5984	687	3	]	]	PUNCT
ejpam-5984	687	4	m.	m.	NOUN
ejpam-5984	687	5	chellali	chellali	PROPN
ejpam-5984	687	6	,	,	PUNCT
ejpam-5984	687	7	n.	n.	PROPN
ejpam-5984	687	8	jafari	jafari	PROPN
ejpam-5984	687	9	rad	rad	PROPN
ejpam-5984	687	10	,	,	PUNCT
ejpam-5984	687	11	s.	s.	PROPN
ejpam-5984	687	12	m.	m.	PROPN
ejpam-5984	687	13	sheikholeslami	sheikholeslami	PROPN
ejpam-5984	687	14	,	,	PUNCT
ejpam-5984	687	15	and	and	CCONJ
ejpam-5984	687	16	l.	l.	PROPN
ejpam-5984	687	17	volkmann	volkmann	PROPN
ejpam-5984	687	18	.	.	PUNCT
ejpam-5984	688	1	roman	roman	ADJ
ejpam-5984	688	2	domination	domination	NOUN
ejpam-5984	688	3	in	in	ADP
ejpam-5984	688	4	graphs	graph	NOUN
ejpam-5984	688	5	.	.	PUNCT
ejpam-5984	689	1	in	in	ADP
ejpam-5984	689	2	t.	t.	PROPN
ejpam-5984	689	3	w.	w.	PROPN
ejpam-5984	689	4	haynes	haynes	PROPN
ejpam-5984	689	5	,	,	PUNCT
ejpam-5984	689	6	s.	s.	PROPN
ejpam-5984	689	7	t.	t.	PROPN
ejpam-5984	689	8	hedetniemi	hedetniemi	PROPN
ejpam-5984	689	9	,	,	PUNCT
ejpam-5984	689	10	and	and	CCONJ
ejpam-5984	689	11	m.	m.	PROPN
ejpam-5984	689	12	a.	a.	PROPN
ejpam-5984	689	13	henning	henning	PROPN
ejpam-5984	689	14	,	,	PUNCT
ejpam-5984	689	15	editors	editor	NOUN
ejpam-5984	689	16	,	,	PUNCT
ejpam-5984	689	17	topics	topic	NOUN
ejpam-5984	689	18	in	in	ADP
ejpam-5984	689	19	domination	domination	NOUN
ejpam-5984	689	20	in	in	ADP
ejpam-5984	689	21	graphs	graph	NOUN
ejpam-5984	689	22	,	,	PUNCT
ejpam-5984	689	23	pages	page	NOUN
ejpam-5984	689	24	365–409	365–409	NUM
ejpam-5984	689	25	.	.	PUNCT
ejpam-5984	689	26	springer	springer	NOUN
ejpam-5984	689	27	,	,	PUNCT
ejpam-5984	689	28	berlin	berlin	PROPN
ejpam-5984	689	29	,	,	PUNCT
ejpam-5984	689	30	2020	2020	NUM
ejpam-5984	689	31	.	.	PUNCT
ejpam-5984	690	1	[	[	X
ejpam-5984	690	2	6	6	NUM
ejpam-5984	690	3	]	]	PUNCT
ejpam-5984	690	4	m.	m.	NOUN
ejpam-5984	690	5	chellali	chellali	PROPN
ejpam-5984	690	6	,	,	PUNCT
ejpam-5984	690	7	n.	n.	PROPN
ejpam-5984	690	8	jafari	jafari	PROPN
ejpam-5984	690	9	rad	rad	PROPN
ejpam-5984	690	10	,	,	PUNCT
ejpam-5984	690	11	s.	s.	PROPN
ejpam-5984	690	12	m.	m.	PROPN
ejpam-5984	690	13	sheikholeslami	sheikholeslami	PROPN
ejpam-5984	690	14	,	,	PUNCT
ejpam-5984	690	15	and	and	CCONJ
ejpam-5984	690	16	l.	l.	PROPN
ejpam-5984	690	17	volkmann	volkmann	PROPN
ejpam-5984	690	18	.	.	PUNCT
ejpam-5984	691	1	varieties	variety	NOUN
ejpam-5984	691	2	of	of	ADP
ejpam-5984	691	3	roman	roman	ADJ
ejpam-5984	691	4	domination	domination	NOUN
ejpam-5984	691	5	.	.	PUNCT
ejpam-5984	692	1	in	in	ADP
ejpam-5984	692	2	t.	t.	PROPN
ejpam-5984	692	3	w.	w.	PROPN
ejpam-5984	692	4	haynes	haynes	PROPN
ejpam-5984	692	5	,	,	PUNCT
ejpam-5984	692	6	s.	s.	PROPN
ejpam-5984	692	7	t.	t.	PROPN
ejpam-5984	692	8	hedetniemi	hedetniemi	PROPN
ejpam-5984	692	9	,	,	PUNCT
ejpam-5984	692	10	and	and	CCONJ
ejpam-5984	692	11	m.	m.	PROPN
ejpam-5984	692	12	a.	a.	PROPN
ejpam-5984	692	13	henning	henning	PROPN
ejpam-5984	692	14	,	,	PUNCT
ejpam-5984	692	15	editors	editor	NOUN
ejpam-5984	692	16	,	,	PUNCT
ejpam-5984	692	17	structures	structure	NOUN
ejpam-5984	692	18	of	of	ADP
ejpam-5984	692	19	domination	domination	NOUN
ejpam-5984	692	20	in	in	ADP
ejpam-5984	692	21	graphs	graph	NOUN
ejpam-5984	692	22	,	,	PUNCT
ejpam-5984	692	23	pages	page	NOUN
ejpam-5984	692	24	273–307	273–307	NUM
ejpam-5984	692	25	.	.	PUNCT
ejpam-5984	692	26	springer	springer	NOUN
ejpam-5984	692	27	,	,	PUNCT
ejpam-5984	692	28	berlin	berlin	PROPN
ejpam-5984	692	29	,	,	PUNCT
ejpam-5984	692	30	2021	2021	NUM
ejpam-5984	692	31	.	.	PUNCT
ejpam-5984	693	1	[	[	X
ejpam-5984	693	2	7	7	X
ejpam-5984	693	3	]	]	X
ejpam-5984	693	4	m.	m.	NOUN
ejpam-5984	693	5	chellali	chellali	PROPN
ejpam-5984	693	6	,	,	PUNCT
ejpam-5984	693	7	n.	n.	PROPN
ejpam-5984	693	8	jafari	jafari	PROPN
ejpam-5984	693	9	rad	rad	PROPN
ejpam-5984	693	10	,	,	PUNCT
ejpam-5984	693	11	s.	s.	PROPN
ejpam-5984	693	12	m.	m.	PROPN
ejpam-5984	693	13	sheikholeslami	sheikholeslami	PROPN
ejpam-5984	693	14	,	,	PUNCT
ejpam-5984	693	15	and	and	CCONJ
ejpam-5984	693	16	l.	l.	PROPN
ejpam-5984	693	17	volkmann	volkmann	PROPN
ejpam-5984	693	18	.	.	PUNCT
ejpam-5984	694	1	varieties	variety	NOUN
ejpam-5984	694	2	of	of	ADP
ejpam-5984	694	3	roman	roman	PROPN
ejpam-5984	694	4	domination	domination	PROPN
ejpam-5984	694	5	ii	ii	PROPN
ejpam-5984	694	6	.	.	PUNCT
ejpam-5984	695	1	akce	akce	PROPN
ejpam-5984	695	2	international	international	PROPN
ejpam-5984	695	3	journal	journal	NOUN
ejpam-5984	695	4	of	of	ADP
ejpam-5984	695	5	graphs	graph	NOUN
ejpam-5984	695	6	and	and	CCONJ
ejpam-5984	695	7	combinatorics	combinatoric	NOUN
ejpam-5984	695	8	,	,	PUNCT
ejpam-5984	695	9	17(3):966–984	17(3):966–984	PROPN
ejpam-5984	695	10	,	,	PUNCT
ejpam-5984	695	11	2020	2020	NUM
ejpam-5984	695	12	.	.	PUNCT
ejpam-5984	696	1	[	[	X
ejpam-5984	696	2	8	8	NUM
ejpam-5984	696	3	]	]	X
ejpam-5984	696	4	r.	r.	PROPN
ejpam-5984	696	5	a.	a.	PROPN
ejpam-5984	696	6	beeler	beeler	PROPN
ejpam-5984	696	7	,	,	PUNCT
ejpam-5984	696	8	t.	t.	PROPN
ejpam-5984	696	9	w.	w.	PROPN
ejpam-5984	696	10	haynes	haynes	PROPN
ejpam-5984	696	11	,	,	PUNCT
ejpam-5984	696	12	and	and	CCONJ
ejpam-5984	696	13	s.	s.	PROPN
ejpam-5984	696	14	t.	t.	PROPN
ejpam-5984	696	15	hedetniemi	hedetniemi	PROPN
ejpam-5984	696	16	.	.	PUNCT
ejpam-5984	697	1	double	double	ADJ
ejpam-5984	697	2	roman	roman	ADJ
ejpam-5984	697	3	domination	domination	NOUN
ejpam-5984	697	4	.	.	PUNCT
ejpam-5984	698	1	discrete	discrete	ADJ
ejpam-5984	698	2	applied	apply	VERB
ejpam-5984	698	3	mathematics	mathematic	NOUN
ejpam-5984	698	4	,	,	PUNCT
ejpam-5984	698	5	211:23–29	211:23–29	NUM
ejpam-5984	698	6	,	,	PUNCT
ejpam-5984	698	7	2016	2016	NUM
ejpam-5984	698	8	.	.	PUNCT
ejpam-5984	699	1	[	[	X
ejpam-5984	699	2	9	9	NUM
ejpam-5984	699	3	]	]	X
ejpam-5984	699	4	h.	h.	NOUN
ejpam-5984	699	5	abdollahzadeh	abdollahzadeh	PROPN
ejpam-5984	699	6	ahangar	ahangar	NOUN
ejpam-5984	699	7	,	,	PUNCT
ejpam-5984	699	8	m.	m.	NOUN
ejpam-5984	699	9	chellali	chellali	PROPN
ejpam-5984	699	10	,	,	PUNCT
ejpam-5984	699	11	and	and	CCONJ
ejpam-5984	699	12	s.	s.	PROPN
ejpam-5984	699	13	m.	m.	PROPN
ejpam-5984	699	14	sheikholeslami	sheikholeslami	PROPN
ejpam-5984	699	15	.	.	PUNCT
ejpam-5984	700	1	on	on	ADP
ejpam-5984	700	2	the	the	DET
ejpam-5984	700	3	double	double	ADJ
ejpam-5984	700	4	roman	roman	ADJ
ejpam-5984	700	5	domination	domination	NOUN
ejpam-5984	700	6	in	in	ADP
ejpam-5984	700	7	graphs	graph	NOUN
ejpam-5984	700	8	.	.	PUNCT
ejpam-5984	701	1	discrete	discrete	ADJ
ejpam-5984	701	2	applied	apply	VERB
ejpam-5984	701	3	mathematics	mathematic	NOUN
ejpam-5984	701	4	,	,	PUNCT
ejpam-5984	701	5	232:1–7	232:1–7	NUM
ejpam-5984	701	6	,	,	PUNCT
ejpam-5984	701	7	2017	2017	NUM
ejpam-5984	701	8	.	.	PUNCT
ejpam-5984	702	1	[	[	X
ejpam-5984	702	2	10	10	NUM
ejpam-5984	702	3	]	]	X
ejpam-5984	702	4	h.	h.	PROPN
ejpam-5984	702	5	r.	r.	PROPN
ejpam-5984	702	6	maimani	maimani	PROPN
ejpam-5984	702	7	,	,	PUNCT
ejpam-5984	702	8	m.	m.	NOUN
ejpam-5984	702	9	momeni	momeni	PROPN
ejpam-5984	702	10	,	,	PUNCT
ejpam-5984	702	11	s.	s.	PROPN
ejpam-5984	702	12	nazari	nazari	PROPN
ejpam-5984	702	13	-	-	PUNCT
ejpam-5984	702	14	moghaddam	moghaddam	NOUN
ejpam-5984	702	15	,	,	PUNCT
ejpam-5984	702	16	f.	f.	PROPN
ejpam-5984	702	17	rahimi	rahimi	PROPN
ejpam-5984	702	18	mahid	mahid	PROPN
ejpam-5984	702	19	,	,	PUNCT
ejpam-5984	702	20	and	and	CCONJ
ejpam-5984	702	21	s.	s.	PROPN
ejpam-5984	702	22	m.	m.	PROPN
ejpam-5984	702	23	sheikholeslami	sheikholeslami	PROPN
ejpam-5984	702	24	.	.	PUNCT
ejpam-5984	703	1	independent	independent	ADJ
ejpam-5984	703	2	double	double	ADJ
ejpam-5984	703	3	roman	roman	ADJ
ejpam-5984	703	4	domination	domination	NOUN
ejpam-5984	703	5	in	in	ADP
ejpam-5984	703	6	graphs	graph	NOUN
ejpam-5984	703	7	.	.	PUNCT
ejpam-5984	704	1	bulletin	bulletin	NOUN
ejpam-5984	704	2	of	of	ADP
ejpam-5984	704	3	the	the	DET
ejpam-5984	704	4	iranian	iranian	PROPN
ejpam-5984	704	5	mathematical	mathematical	PROPN
ejpam-5984	704	6	society	society	NOUN
ejpam-5984	704	7	,	,	PUNCT
ejpam-5984	704	8	46(2):543–555	46(2):543–555	NUM
ejpam-5984	704	9	,	,	PUNCT
ejpam-5984	704	10	2020	2020	NUM
ejpam-5984	704	11	.	.	PUNCT
ejpam-5984	705	1	[	[	X
ejpam-5984	705	2	11	11	NUM
ejpam-5984	705	3	]	]	PUNCT
ejpam-5984	705	4	q.	q.	PROPN
ejpam-5984	705	5	liu	liu	PROPN
ejpam-5984	705	6	,	,	PUNCT
ejpam-5984	705	7	y.	y.	PROPN
ejpam-5984	705	8	song	song	PROPN
ejpam-5984	705	9	,	,	PUNCT
ejpam-5984	705	10	z.	z.	PROPN
ejpam-5984	705	11	shao	shao	PROPN
ejpam-5984	705	12	,	,	PUNCT
ejpam-5984	705	13	and	and	CCONJ
ejpam-5984	705	14	h.	h.	PROPN
ejpam-5984	705	15	jiang	jiang	PROPN
ejpam-5984	705	16	.	.	PUNCT
ejpam-5984	706	1	algorithmic	algorithmic	ADJ
ejpam-5984	706	2	results	result	NOUN
ejpam-5984	706	3	on	on	ADP
ejpam-5984	706	4	independent	independent	ADJ
ejpam-5984	706	5	roman	roman	NOUN
ejpam-5984	706	6	{	{	PUNCT
ejpam-5984	706	7	2}-domination	2}-domination	NUM
ejpam-5984	706	8	.	.	PUNCT
ejpam-5984	707	1	communications	communication	NOUN
ejpam-5984	707	2	in	in	ADP
ejpam-5984	707	3	combinatorics	combinatoric	NOUN
ejpam-5984	707	4	and	and	CCONJ
ejpam-5984	707	5	optimization	optimization	NOUN
ejpam-5984	707	6	,	,	PUNCT
ejpam-5984	707	7	2025	2025	NUM
ejpam-5984	707	8	.	.	PUNCT
ejpam-5984	708	1	in	in	ADP
ejpam-5984	708	2	press	press	NOUN
ejpam-5984	708	3	.	.	PUNCT
ejpam-5984	709	1	[	[	X
ejpam-5984	709	2	12	12	NUM
ejpam-5984	709	3	]	]	PUNCT
ejpam-5984	709	4	a.	a.	NOUN
ejpam-5984	709	5	poureidi	poureidi	PROPN
ejpam-5984	709	6	.	.	PUNCT
ejpam-5984	710	1	efficient	efficient	ADJ
ejpam-5984	710	2	algorithms	algorithm	NOUN
ejpam-5984	710	3	for	for	ADP
ejpam-5984	710	4	independent	independent	ADJ
ejpam-5984	710	5	roman	roman	ADJ
ejpam-5984	710	6	domination	domination	NOUN
ejpam-5984	710	7	on	on	ADP
ejpam-5984	710	8	some	some	DET
ejpam-5984	710	9	classes	class	NOUN
ejpam-5984	710	10	of	of	ADP
ejpam-5984	710	11	graphs	graph	NOUN
ejpam-5984	710	12	.	.	PUNCT
ejpam-5984	711	1	communications	communication	NOUN
ejpam-5984	711	2	in	in	ADP
ejpam-5984	711	3	combinatorics	combinatoric	NOUN
ejpam-5984	711	4	and	and	CCONJ
ejpam-5984	711	5	optimization	optimization	NOUN
ejpam-5984	711	6	,	,	PUNCT
ejpam-5984	711	7	8(1):127–140	8(1):127–140	NUM
ejpam-5984	711	8	,	,	PUNCT
ejpam-5984	711	9	2023	2023	NUM
ejpam-5984	711	10	.	.	PUNCT
ejpam-5984	712	1	[	[	X
ejpam-5984	712	2	13	13	NUM
ejpam-5984	712	3	]	]	X
ejpam-5984	712	4	h.	h.	PROPN
ejpam-5984	712	5	r.	r.	PROPN
ejpam-5984	712	6	maimani	maimani	PROPN
ejpam-5984	712	7	,	,	PUNCT
ejpam-5984	712	8	m.	m.	NOUN
ejpam-5984	712	9	momeni	momeni	PROPN
ejpam-5984	712	10	,	,	PUNCT
ejpam-5984	712	11	f.	f.	PROPN
ejpam-5984	712	12	rahimi	rahimi	PROPN
ejpam-5984	712	13	mahid	mahid	PROPN
ejpam-5984	712	14	,	,	PUNCT
ejpam-5984	712	15	and	and	CCONJ
ejpam-5984	712	16	s.	s.	PROPN
ejpam-5984	712	17	m.	m.	PROPN
ejpam-5984	712	18	sheikholeslami	sheikholeslami	PROPN
ejpam-5984	712	19	.	.	PUNCT
ejpam-5984	713	1	independent	independent	ADJ
ejpam-5984	713	2	double	double	ADJ
ejpam-5984	713	3	roman	roman	ADJ
ejpam-5984	713	4	domination	domination	NOUN
ejpam-5984	713	5	in	in	ADP
ejpam-5984	713	6	graphs	graph	NOUN
ejpam-5984	713	7	.	.	PUNCT
ejpam-5984	714	1	akce	akce	PROPN
ejpam-5984	714	2	international	international	PROPN
ejpam-5984	714	3	journal	journal	NOUN
ejpam-5984	714	4	of	of	ADP
ejpam-5984	714	5	graphs	graph	NOUN
ejpam-5984	714	6	and	and	CCONJ
ejpam-5984	714	7	combinatorics	combinatoric	NOUN
ejpam-5984	714	8	,	,	PUNCT
ejpam-5984	714	9	17(3):905–910	17(3):905–910	NOUN
ejpam-5984	714	10	,	,	PUNCT
ejpam-5984	714	11	2020	2020	NUM
ejpam-5984	714	12	.	.	PUNCT
ejpam-5984	715	1	[	[	X
ejpam-5984	715	2	14	14	NUM
ejpam-5984	715	3	]	]	X
ejpam-5984	715	4	f.	f.	PROPN
ejpam-5984	715	5	nahani	nahani	PROPN
ejpam-5984	715	6	pour	pour	PROPN
ejpam-5984	715	7	,	,	PUNCT
ejpam-5984	715	8	h.	h.	PROPN
ejpam-5984	715	9	abdollahzadeh	abdollahzadeh	PROPN
ejpam-5984	715	10	ahangar	ahangar	NOUN
ejpam-5984	715	11	,	,	PUNCT
ejpam-5984	715	12	m.	m.	NOUN
ejpam-5984	715	13	chellali	chellali	PROPN
ejpam-5984	715	14	,	,	PUNCT
ejpam-5984	715	15	and	and	CCONJ
ejpam-5984	715	16	s.	s.	PROPN
ejpam-5984	715	17	m.	m.	PROPN
ejpam-5984	715	18	sheikholeslami	sheikholeslami	PROPN
ejpam-5984	715	19	.	.	PUNCT
ejpam-5984	716	1	an	an	DET
ejpam-5984	716	2	improved	improve	VERB
ejpam-5984	716	3	upper	upper	ADJ
ejpam-5984	716	4	bound	bind	VERB
ejpam-5984	716	5	on	on	ADP
ejpam-5984	716	6	the	the	DET
ejpam-5984	716	7	independent	independent	ADJ
ejpam-5984	716	8	double	double	ADJ
ejpam-5984	716	9	roman	roman	ADJ
ejpam-5984	716	10	domination	domination	NOUN
ejpam-5984	716	11	number	number	NOUN
ejpam-5984	716	12	of	of	ADP
ejpam-5984	716	13	trees	tree	NOUN
ejpam-5984	716	14	.	.	PUNCT
ejpam-5984	717	1	akce	akce	PROPN
ejpam-5984	717	2	international	international	PROPN
ejpam-5984	717	3	journal	journal	NOUN
ejpam-5984	717	4	of	of	ADP
ejpam-5984	717	5	graphs	graph	NOUN
ejpam-5984	717	6	and	and	CCONJ
ejpam-5984	717	7	combinatorics	combinatoric	NOUN
ejpam-5984	717	8	,	,	PUNCT
ejpam-5984	717	9	19(3):206–210	19(3):206–210	PROPN
ejpam-5984	717	10	,	,	PUNCT
ejpam-5984	717	11	2022	2022	NUM
ejpam-5984	717	12	.	.	PUNCT
ejpam-5984	718	1	[	[	X
ejpam-5984	718	2	15	15	NUM
ejpam-5984	718	3	]	]	X
ejpam-5984	718	4	d.	d.	PROPN
ejpam-5984	718	5	bauer	bauer	PROPN
ejpam-5984	718	6	,	,	PUNCT
ejpam-5984	718	7	f.	f.	PROPN
ejpam-5984	718	8	harary	harary	PROPN
ejpam-5984	718	9	,	,	PUNCT
ejpam-5984	718	10	j.	j.	PROPN
ejpam-5984	718	11	nieminen	nieminen	PROPN
ejpam-5984	718	12	,	,	PUNCT
ejpam-5984	718	13	and	and	CCONJ
ejpam-5984	718	14	c.	c.	PROPN
ejpam-5984	718	15	suffel	suffel	PROPN
ejpam-5984	718	16	.	.	PUNCT
ejpam-5984	719	1	domination	domination	NOUN
ejpam-5984	719	2	alteration	alteration	NOUN
ejpam-5984	719	3	sets	set	NOUN
ejpam-5984	719	4	in	in	ADP
ejpam-5984	719	5	graphs	graph	NOUN
ejpam-5984	719	6	.	.	PUNCT
ejpam-5984	720	1	discrete	discrete	ADJ
ejpam-5984	720	2	mathematics	mathematic	NOUN
ejpam-5984	720	3	,	,	PUNCT
ejpam-5984	720	4	47:153–161	47:153–161	NUM
ejpam-5984	720	5	,	,	PUNCT
ejpam-5984	720	6	1983	1983	NUM
ejpam-5984	720	7	.	.	PUNCT
ejpam-5984	721	1	[	[	X
ejpam-5984	721	2	16	16	NUM
ejpam-5984	721	3	]	]	X
ejpam-5984	721	4	n.	n.	PROPN
ejpam-5984	721	5	j.	j.	PROPN
ejpam-5984	721	6	rad	rad	PROPN
ejpam-5984	721	7	,	,	PUNCT
ejpam-5984	721	8	e.	e.	PROPN
ejpam-5984	721	9	sharifi	sharifi	PROPN
ejpam-5984	721	10	,	,	PUNCT
ejpam-5984	721	11	and	and	CCONJ
ejpam-5984	721	12	m.	m.	PROPN
ejpam-5984	721	13	krzywkowski	krzywkowski	PROPN
ejpam-5984	721	14	.	.	PUNCT
ejpam-5984	722	1	domination	domination	NOUN
ejpam-5984	722	2	stability	stability	NOUN
ejpam-5984	722	3	in	in	ADP
ejpam-5984	722	4	graphs	graph	NOUN
ejpam-5984	722	5	.	.	PUNCT
ejpam-5984	723	1	discrete	discrete	ADJ
ejpam-5984	723	2	mathematics	mathematic	NOUN
ejpam-5984	723	3	,	,	PUNCT
ejpam-5984	723	4	339(7):1909–1914	339(7):1909–1914	PROPN
ejpam-5984	723	5	,	,	PUNCT
ejpam-5984	723	6	2016	2016	NUM
ejpam-5984	723	7	.	.	PUNCT
ejpam-5984	724	1	[	[	X
ejpam-5984	724	2	17	17	NUM
ejpam-5984	724	3	]	]	X
ejpam-5984	724	4	w.	w.	PROPN
ejpam-5984	724	5	zhuang	zhuang	PROPN
ejpam-5984	724	6	.	.	PROPN
ejpam-5984	724	7	double	double	ADJ
ejpam-5984	724	8	roman	roman	ADJ
ejpam-5984	724	9	domination	domination	NOUN
ejpam-5984	724	10	stability	stability	NOUN
ejpam-5984	724	11	in	in	ADP
ejpam-5984	724	12	graphs	graph	NOUN
ejpam-5984	724	13	.	.	PUNCT
ejpam-5984	725	1	discrete	discrete	ADJ
ejpam-5984	725	2	applied	apply	VERB
ejpam-5984	725	3	mathematics	mathematic	NOUN
ejpam-5984	725	4	,	,	PUNCT
ejpam-5984	725	5	371:254–263	371:254–263	NUM
ejpam-5984	725	6	,	,	PUNCT
ejpam-5984	725	7	2025	2025	NUM
ejpam-5984	725	8	.	.	PUNCT