id	sid	eid	entity	type
ejpam-4484	1	1	european journal	ORG
ejpam-4484	2	1	15	CARDINAL
ejpam-4484	2	2	4	CARDINAL
ejpam-4484	2	3	2022	CARDINAL
ejpam-4484	2	4	1472	CARDINAL
ejpam-4484	2	5	1307-5543	DATE
ejpam-4484	2	6	new york	GPE
ejpam-4484	2	7	armalene h. abragan1,∗	PERSON
ejpam-4484	2	8	helen m. rara2	PERSON
ejpam-4484	3	1	1	CARDINAL
ejpam-4484	3	2	mindanao state university-iligan institute of technology	ORG
ejpam-4484	3	3	9200	CARDINAL
ejpam-4484	3	4	iligan city	GPE
ejpam-4484	3	5	2	CARDINAL
ejpam-4484	3	6	mindanao	GPE
ejpam-4484	3	7	institute of technology	ORG
ejpam-4484	3	8	9200	CARDINAL
ejpam-4484	3	9	iligan city	GPE
ejpam-4484	3	10	philippines	GPE
ejpam-4484	4	1	⊆	CARDINAL
ejpam-4484	4	2	w ∈ s	PERSON
ejpam-4484	5	1	2	CARDINAL
ejpam-4484	6	1	two	CARDINAL
ejpam-4484	8	1	2020	DATE
ejpam-4484	8	2	05c69	CARDINAL
ejpam-4484	8	3	1	CARDINAL
ejpam-4484	8	4	1960	DATE
ejpam-4484	10	1	1977	DATE
ejpam-4484	10	2	e.j cockayne	PERSON
ejpam-4484	10	3	s.t hedetniemi	PERSON
ejpam-4484	10	4	the notation γ(g	ORG
ejpam-4484	10	5	2	CARDINAL
ejpam-4484	10	6	chartrand et al	ORG
ejpam-4484	11	1	3	CARDINAL
ejpam-4484	11	2	1995	DATE
ejpam-4484	12	1	natarajan	PERSON
ejpam-4484	12	2	2015	CARDINAL
ejpam-4484	13	1	2	CARDINAL
ejpam-4484	13	2	5	CARDINAL
ejpam-4484	13	3	4	CARDINAL
ejpam-4484	15	1	armalene.abragan@g.msuiit.edu.ph	PERSON
ejpam-4484	15	2	a. abragan	PERSON
ejpam-4484	15	3	helen.rara@g.msuiit.edu.ph	GPE
ejpam-4484	15	4	h. m. rara	PERSON
ejpam-4484	15	5	1472	CARDINAL
ejpam-4484	15	6	2022	CARDINAL
ejpam-4484	16	1	a. h. abragan	PERSON
ejpam-4484	16	2	h. m. rara	PERSON
ejpam-4484	17	1	j. pure	PERSON
ejpam-4484	17	2	15	CARDINAL
ejpam-4484	17	3	4	CARDINAL
ejpam-4484	17	4	2022	CARDINAL
ejpam-4484	17	5	1472-1481 1473	CARDINAL
ejpam-4484	18	1	8	CARDINAL
ejpam-4484	18	2	henning	PERSON
ejpam-4484	20	1	secret).to	PERSON
ejpam-4484	23	1	two	CARDINAL
ejpam-4484	23	2	two	CARDINAL
ejpam-4484	29	1	6	CARDINAL
ejpam-4484	31	1	⊆	CARDINAL
ejpam-4484	31	2	∈	PERSON
ejpam-4484	32	1	2	CARDINAL
ejpam-4484	34	1	⊆	CARDINAL
ejpam-4484	34	2	∈	ORG
ejpam-4484	34	3	w ∈	PERSON
ejpam-4484	35	1	w ∈	ORG
ejpam-4484	35	2	w ∈	ORG
ejpam-4484	36	1	|c∗|	GPE
ejpam-4484	36	2	∈	ORG
ejpam-4484	36	3	2	CARDINAL
ejpam-4484	36	4	|c∗|	GPE
ejpam-4484	36	5	c∗ in g. the hop	ORG
ejpam-4484	36	6	⊆	CARDINAL
ejpam-4484	40	1	⊆	CARDINAL
ejpam-4484	42	1	w	GPE
ejpam-4484	43	1	w	ORG
ejpam-4484	44	1	two	CARDINAL
ejpam-4484	44	2	w	GPE
ejpam-4484	45	1	the a. h. abragan	ORG
ejpam-4484	45	2	h. m. rara	PERSON
ejpam-4484	46	1	j. pure	PERSON
ejpam-4484	46	2	15	CARDINAL
ejpam-4484	46	3	4	CARDINAL
ejpam-4484	46	4	2022	CARDINAL
ejpam-4484	46	5	1472	CARDINAL
ejpam-4484	46	6	dim(g	ORG
ejpam-4484	46	7	cardinality dim(g	ORG
ejpam-4484	47	1	two	CARDINAL
ejpam-4484	47	2	∈	ORG
ejpam-4484	47	3	ig[u	ORG
ejpam-4484	48	1	two	CARDINAL
ejpam-4484	48	2	∈	ORG
ejpam-4484	48	3	ig[u	ORG
ejpam-4484	48	4	w	ORG
ejpam-4484	48	5	∈	ORG
ejpam-4484	49	1	two	CARDINAL
ejpam-4484	49	2	w	GPE
ejpam-4484	51	1	⊆	CARDINAL
ejpam-4484	53	1	⊆	CARDINAL
ejpam-4484	54	1	2	CARDINAL
ejpam-4484	54	2	1	CARDINAL
ejpam-4484	58	1	1	CARDINAL
ejpam-4484	59	1	̸=	ORG
ejpam-4484	60	1	1	CARDINAL
ejpam-4484	62	1	⊆	CARDINAL
ejpam-4484	65	1	2	CARDINAL
ejpam-4484	67	1	g+h	ORG
ejpam-4484	67	2	at least one	CARDINAL
ejpam-4484	71	1	a. h. abragan	PERSON
ejpam-4484	71	2	h. m. rara	PERSON
ejpam-4484	72	1	j. pure	PERSON
ejpam-4484	72	2	15	CARDINAL
ejpam-4484	72	3	4	CARDINAL
ejpam-4484	72	4	2022	CARDINAL
ejpam-4484	72	5	1472	CARDINAL
ejpam-4484	73	1	2	CARDINAL
ejpam-4484	74	1	1	CARDINAL
ejpam-4484	74	2	1	CARDINAL
ejpam-4484	74	3	γ(h	ORG
ejpam-4484	75	1	⊆	PRODUCT
ejpam-4484	76	1	3	CARDINAL
ejpam-4484	77	1	1	CARDINAL
ejpam-4484	77	2	1	CARDINAL
ejpam-4484	77	3	γ(h	ORG
ejpam-4484	78	1	g[h	ORG
ejpam-4484	78	2	g[h])\(a×c	NORP
ejpam-4484	78	3	h. lemma	PERSON
ejpam-4484	79	1	3	CARDINAL
ejpam-4484	80	1	1	CARDINAL
ejpam-4484	80	2	1	CARDINAL
ejpam-4484	80	3	γ(h	ORG
ejpam-4484	81	1	1	CARDINAL
ejpam-4484	82	1	⊆	PRODUCT
ejpam-4484	83	1	1	CARDINAL
ejpam-4484	83	2	c∗ of h. theorem 4	ORG
ejpam-4484	86	1	hu	PERSON
ejpam-4484	86	2	⊆	CARDINAL
ejpam-4484	87	1	hu	PERSON
ejpam-4484	88	1	⊆	CARDINAL
ejpam-4484	88	2	γ(h	ORG
ejpam-4484	88	3	1	CARDINAL
ejpam-4484	88	4	1	CARDINAL
ejpam-4484	93	1	2	CARDINAL
ejpam-4484	94	1	1 ≤	QUANTITY
ejpam-4484	94	2	1	CARDINAL
ejpam-4484	95	1	1	CARDINAL
ejpam-4484	95	2	2	CARDINAL
ejpam-4484	97	1	⊆	CARDINAL
ejpam-4484	100	1	3	CARDINAL
ejpam-4484	100	2	1	CARDINAL
ejpam-4484	101	1	6	CARDINAL
ejpam-4484	101	2	∪̇	PERSON
ejpam-4484	101	3	∈	ORG
ejpam-4484	102	1	a. h. abragan	PERSON
ejpam-4484	102	2	h. m. rara	PERSON
ejpam-4484	103	1	j. pure	PERSON
ejpam-4484	103	2	15	CARDINAL
ejpam-4484	103	3	4	CARDINAL
ejpam-4484	103	4	2022	CARDINAL
ejpam-4484	103	5	1472	CARDINAL
ejpam-4484	103	6	5	CARDINAL
ejpam-4484	105	1	⊆	CARDINAL
ejpam-4484	105	2	k1+g	GPE
ejpam-4484	105	3	k1+g	GPE
ejpam-4484	106	1	k1+g	GPE
ejpam-4484	107	1	1	CARDINAL
ejpam-4484	107	2	g)\c∗	PERSON
ejpam-4484	107	3	c∗	PERSON
ejpam-4484	108	1	\s⟩	ORG
ejpam-4484	108	2	c∗	ORG
ejpam-4484	108	3	1	CARDINAL
ejpam-4484	108	4	6	CARDINAL
ejpam-4484	109	1	1	CARDINAL
ejpam-4484	110	1	⊆	CARDINAL
ejpam-4484	110	2	k1+g	GPE
ejpam-4484	110	3	k1+g	GPE
ejpam-4484	111	1	1	CARDINAL
ejpam-4484	112	1	1	CARDINAL
ejpam-4484	112	2	c∗	PERSON
ejpam-4484	112	3	k1+g	GPE
ejpam-4484	115	1	1	CARDINAL
ejpam-4484	115	2	c∗	ORG
ejpam-4484	115	3	1	CARDINAL
ejpam-4484	117	1	ωhs(g	CARDINAL
ejpam-4484	117	2	2	CARDINAL
ejpam-4484	121	1	c1	ORG
ejpam-4484	122	1	4	CARDINAL
ejpam-4484	123	1	1	CARDINAL
ejpam-4484	123	2	2	DATE
ejpam-4484	124	1	1	CARDINAL
ejpam-4484	124	2	cm)∪{v}\{c1	PERSON
ejpam-4484	124	3	1	CARDINAL
ejpam-4484	124	4	2	DATE
ejpam-4484	125	1	m−1	PERSON
ejpam-4484	126	1	1	CARDINAL
ejpam-4484	126	2	2	DATE
ejpam-4484	128	1	k1+g	GPE
ejpam-4484	128	2	k1+g	GPE
ejpam-4484	128	3	ci = ∅	PERSON
ejpam-4484	128	4	ci	PERSON
ejpam-4484	129	1	1	CARDINAL
ejpam-4484	129	2	k1+g)\c∗	PERSON
ejpam-4484	129	3	c∗ i a. h. abragan	PERSON
ejpam-4484	129	4	h. m. rara	PERSON
ejpam-4484	130	1	j. pure	PERSON
ejpam-4484	130	2	15	CARDINAL
ejpam-4484	130	3	4	CARDINAL
ejpam-4484	130	4	2022	CARDINAL
ejpam-4484	130	5	1472	CARDINAL
ejpam-4484	132	1	k1+g	GPE
ejpam-4484	132	2	k1+g)\s⟩	ORG
ejpam-4484	134	1	1	CARDINAL
ejpam-4484	134	2	3	CARDINAL
ejpam-4484	135	1	1	CARDINAL
ejpam-4484	135	2	2	DATE
ejpam-4484	137	1	m.	PERSON
ejpam-4484	138	1	i=1 ni −max{ωs(gi	PERSON
ejpam-4484	139	1	1	CARDINAL
ejpam-4484	140	1	two	CARDINAL
ejpam-4484	140	2	5	CARDINAL
ejpam-4484	140	3	6	CARDINAL
ejpam-4484	141	1	2	CARDINAL
ejpam-4484	141	2	8	CARDINAL
ejpam-4484	144	1	at least one	CARDINAL
ejpam-4484	145	1	g. (ii	PERSON
ejpam-4484	146	1	1	CARDINAL
ejpam-4484	146	2	1	CARDINAL
ejpam-4484	147	1	m−1}	PERSON
ejpam-4484	155	1	m− ωhs(g	PERSON
ejpam-4484	157	1	+ 1	DATE
ejpam-4484	157	2	1 or	CARDINAL
ejpam-4484	157	3	1	CARDINAL
ejpam-4484	157	4	m− ωhs(g	PERSON
ejpam-4484	159	1	1	CARDINAL
ejpam-4484	162	1	3	CARDINAL
ejpam-4484	162	2	a. h. abragan	PERSON
ejpam-4484	162	3	h. m. rara	PERSON
ejpam-4484	163	1	j. pure	PERSON
ejpam-4484	163	2	15	CARDINAL
ejpam-4484	163	3	4	CARDINAL
ejpam-4484	163	4	2022	CARDINAL
ejpam-4484	163	5	1472	CARDINAL
ejpam-4484	164	1	2	CARDINAL
ejpam-4484	165	1	8	CARDINAL
ejpam-4484	165	2	one	CARDINAL
ejpam-4484	165	3	one	CARDINAL
ejpam-4484	165	4	∈	ORG
ejpam-4484	166	1	9	CARDINAL
ejpam-4484	168	1	⊆	CARDINAL
ejpam-4484	169	1	hu	PERSON
ejpam-4484	169	2	⊆	CARDINAL
ejpam-4484	170	1	hu	PERSON
ejpam-4484	171	1	a∪{v})⟩	ORG
ejpam-4484	171	2	∈	ORG
ejpam-4484	172	1	∩a ̸= ∅. proof	PERSON
ejpam-4484	173	1	4	CARDINAL
ejpam-4484	173	2	one	CARDINAL
ejpam-4484	173	3	hu	PERSON
ejpam-4484	174	1	⊆	CARDINAL
ejpam-4484	174	2	hu	PERSON
ejpam-4484	174	3	⊆	CARDINAL
ejpam-4484	174	4	γ(h	ORG
ejpam-4484	174	5	1	CARDINAL
ejpam-4484	176	1	g ◦ h	ORG
ejpam-4484	176	2	g ◦h	ORG
ejpam-4484	179	1	g ◦ h	ORG
ejpam-4484	179	2	hu	PERSON
ejpam-4484	180	1	∈	ORG
ejpam-4484	182	1	∩a	DATE
ejpam-4484	182	2	∈	ORG
ejpam-4484	182	3	∈	ORG
ejpam-4484	182	4	∈	ORG
ejpam-4484	182	5	∈	ORG
ejpam-4484	184	1	4	CARDINAL
ejpam-4484	184	2	g ◦h	ORG
ejpam-4484	186	1	a. h. abragan	PERSON
ejpam-4484	186	2	h. m. rara	PERSON
ejpam-4484	187	1	j. pure	PERSON
ejpam-4484	187	2	15	CARDINAL
ejpam-4484	187	3	4	CARDINAL
ejpam-4484	187	4	2022	CARDINAL
ejpam-4484	187	5	1472-1481 1479	CARDINAL
ejpam-4484	188	1	\a	WORK_OF_ART
ejpam-4484	190	1	g ◦h	ORG
ejpam-4484	194	1	m−	PERSON
ejpam-4484	194	2	1)n+	CARDINAL
ejpam-4484	195	1	9 (ii	CARDINAL
ejpam-4484	195	2	⋃	DATE
ejpam-4484	195	3	hu	PERSON
ejpam-4484	196	1	⋃	PERSON
ejpam-4484	197	1	m−	NORP
ejpam-4484	197	2	1)n+	CARDINAL
ejpam-4484	198	1	cv	PERSON
ejpam-4484	199	1	∈	ORG
ejpam-4484	199	2	hu	PERSON
ejpam-4484	200	1	⋃	DATE
ejpam-4484	200	2	hu	PERSON
ejpam-4484	200	3	m− 1)(n	PERSON
ejpam-4484	200	4	m− 1)(n	PERSON
ejpam-4484	202	1	m−	PERSON
ejpam-4484	202	2	1)n+	CARDINAL
ejpam-4484	204	1	2	CARDINAL
ejpam-4484	206	1	8.	DATE
ejpam-4484	207	1	3	CARDINAL
ejpam-4484	208	1	6	CARDINAL
ejpam-4484	208	2	g[h	ORG
ejpam-4484	208	3	v1)(u2	GPE
ejpam-4484	210	1	4	CARDINAL
ejpam-4484	211	1	1	CARDINAL
ejpam-4484	211	2	γ(h	ORG
ejpam-4484	212	1	⊆	CARDINAL
ejpam-4484	214	1	h. a. h. abragan	PERSON
ejpam-4484	214	2	h. m. rara	PERSON
ejpam-4484	215	1	j. pure	PERSON
ejpam-4484	215	2	15	CARDINAL
ejpam-4484	215	3	4	CARDINAL
ejpam-4484	215	4	2022	CARDINAL
ejpam-4484	215	5	1472	CARDINAL
ejpam-4484	215	6	⊆	CARDINAL
ejpam-4484	216	1	lemma 2	ORG
ejpam-4484	216	2	3	CARDINAL
ejpam-4484	217	1	∈	ORG
ejpam-4484	217	2	a×c	ORG
ejpam-4484	220	1	∩ng[h]((a	DATE
ejpam-4484	220	2	2	CARDINAL
ejpam-4484	221	1	γ(h	ORG
ejpam-4484	222	1	1	CARDINAL
ejpam-4484	222	2	g = kn	PERSON
ejpam-4484	222	3	1	CARDINAL
ejpam-4484	223	1	2	CARDINAL
ejpam-4484	224	1	γ(h	ORG
ejpam-4484	224	2	1	CARDINAL
ejpam-4484	224	3	c∗ of h.	ORG
ejpam-4484	224	4	2	CARDINAL
ejpam-4484	225	1	lemma 2	ORG
ejpam-4484	225	2	3	CARDINAL
ejpam-4484	225	3	1	CARDINAL
ejpam-4484	226	1	γ(h	ORG
ejpam-4484	228	1	⟨(v (h)\c)⟩∩nh(x	ORG
ejpam-4484	228	2	2	CARDINAL
ejpam-4484	230	1	2	CARDINAL
ejpam-4484	231	1	γ(h	ORG
ejpam-4484	232	1	1	CARDINAL
ejpam-4484	232	2	c∗ of h.	ORG
ejpam-4484	232	3	∈	ORG
ejpam-4484	232	4	2	CARDINAL
ejpam-4484	233	1	∈	ORG
ejpam-4484	235	1	∩ng[h]((a	DATE
ejpam-4484	235	2	2	CARDINAL
ejpam-4484	237	1	10	CARDINAL
ejpam-4484	238	1	1	CARDINAL
ejpam-4484	238	2	γ(h	ORG
ejpam-4484	239	1	g[h	ORG
ejpam-4484	240	1	one	CARDINAL
ejpam-4484	240	2	⊆	CARDINAL
ejpam-4484	240	3	h.	PERSON
ejpam-4484	240	4	g[h	ORG
ejpam-4484	241	1	3	CARDINAL
ejpam-4484	242	1	⟨v	DATE
ejpam-4484	242	2	g[h	ORG
ejpam-4484	243	1	⊆	CARDINAL
ejpam-4484	244	1	g[h	ORG
ejpam-4484	245	1	\s⟩	ORG
ejpam-4484	245	2	a×c	ORG
ejpam-4484	249	1	lemma 4	ORG
ejpam-4484	250	1	3	CARDINAL
ejpam-4484	250	2	g[h	ORG
ejpam-4484	251	1	g[h	ORG
ejpam-4484	252	1	⊆	CARDINAL
ejpam-4484	253	1	g[h	ORG
ejpam-4484	256	1	3	CARDINAL
ejpam-4484	256	2	13	CARDINAL
ejpam-4484	257	1	1481	CARDINAL
ejpam-4484	257	2	msu-iligan institute of technology	ORG
ejpam-4484	258	1	1	CARDINAL
ejpam-4484	258	2	h. rara	PERSON
ejpam-4484	260	1	21(1):91–101	DATE
ejpam-4484	260	2	2019	DATE
ejpam-4484	261	1	2	CARDINAL
ejpam-4484	261	2	sergio canoy, jr	PERSON
ejpam-4484	261	3	reynaldo villarobe mollejon	PERSON
ejpam-4484	261	4	john gabriel e. canoy	PERSON
ejpam-4484	263	1	european journal	ORG
ejpam-4484	263	2	12(4):1455–1463, oct. 2019	DATE
ejpam-4484	264	1	3	CARDINAL
ejpam-4484	264	2	l. eroh	PERSON
ejpam-4484	264	3	m. johnson	PERSON
ejpam-4484	266	1	105:99–113, 2000	DATE
ejpam-4484	267	1	4	CARDINAL
ejpam-4484	267	2	natarajan chidambaram	PERSON
ejpam-4484	267	3	s.k.	GPE
ejpam-4484	269	1	seria matematica	ORG
ejpam-4484	269	2	23:187–199	CARDINAL
ejpam-4484	269	3	06 2015	DATE
ejpam-4484	270	1	5	CARDINAL
ejpam-4484	270	2	sumaoy h.	PERSON
ejpam-4484	270	3	h. rara	PERSON
ejpam-4484	272	1	european journal	ORG
ejpam-4484	272	2	14(4):1367–1378	CARDINAL
ejpam-4484	272	3	2021	DATE
ejpam-4484	273	1	6	CARDINAL
ejpam-4484	273	2	f. harary	PERSON
ejpam-4484	273	3	r.a. melter	PERSON
ejpam-4484	274	1	2:191–195	CARDINAL
ejpam-4484	274	2	1976	DATE
ejpam-4484	275	1	7	CARDINAL
ejpam-4484	275	2	monsanto	ORG
ejpam-4484	275	3	penelyn l.	PERSON
ejpam-4484	275	4	helen m. rara	PERSON
ejpam-4484	277	1	european journal	ORG
ejpam-4484	277	2	jan. 2020	DATE
ejpam-4484	278	1	8	CARDINAL
ejpam-4484	278	2	t.w.	GPE
ejpam-4484	278	3	hedetniemi s.	PERSON
ejpam-4484	278	4	slater	PERSON
ejpam-4484	279	1	1st	ORDINAL
ejpam-4484	280	1	crc press	ORG
ejpam-4484	280	2	1998	DATE