id	sid	eid	entity	type
ejpam-6117	1	1	european journal	ORG
ejpam-6117	1	2	2025	DATE
ejpam-6117	1	3	18	CARDINAL
ejpam-6117	1	4	2	CARDINAL
ejpam-6117	1	5	6117	CARDINAL
ejpam-6117	1	6	new york	GPE
ejpam-6117	1	7	merontos1,∗	ORG
ejpam-6117	1	8	imelda s. aniversario1,2	PERSON
ejpam-6117	1	9	michael b. frondoza1,2	PERSON
ejpam-6117	1	10	sergio r. canoy	PERSON
ejpam-6117	1	11	1	CARDINAL
ejpam-6117	1	12	msu-iligan institute of technology	ORG
ejpam-6117	1	13	iligan city	GPE
ejpam-6117	1	14	2	CARDINAL
ejpam-6117	1	15	msu-iligan institute of technology	ORG
ejpam-6117	1	16	iligan city	GPE
ejpam-6117	1	17	philippines	GPE
ejpam-6117	3	1	⊆	CARDINAL
ejpam-6117	3	2	s.	GPE
ejpam-6117	5	1	two	CARDINAL
ejpam-6117	7	1	1	CARDINAL
ejpam-6117	8	1	n ≥ k+m+2	ORG
ejpam-6117	8	2	= k and	PERSON
ejpam-6117	8	3	α(g	PERSON
ejpam-6117	10	1	2020	DATE
ejpam-6117	10	2	05c69	CARDINAL
ejpam-6117	10	3	1	CARDINAL
ejpam-6117	11	1	1	CARDINAL
ejpam-6117	11	2	dunbar et.al	FAC
ejpam-6117	12	1	2	CARDINAL
ejpam-6117	12	2	γw(g	PERSON
ejpam-6117	16	1	3	CARDINAL
ejpam-6117	17	1	4	CARDINAL
ejpam-6117	19	1	r.	NORP
ejpam-6117	19	2	imelda.aniversario@g.msuiit.edu.ph	GPE
ejpam-6117	19	3	i. aniversario	PERSON
ejpam-6117	20	1	m. frondoza	PERSON
ejpam-6117	20	2	s. canoy jr.	PERSON
ejpam-6117	21	1	1	CARDINAL
ejpam-6117	21	2	2025	CARDINAL
ejpam-6117	22	1	r. merontos et al	PERSON
ejpam-6117	23	1	j. pure	PERSON
ejpam-6117	23	2	18	CARDINAL
ejpam-6117	23	3	2	CARDINAL
ejpam-6117	23	4	2025	CARDINAL
ejpam-6117	23	5	6117 2	DATE
ejpam-6117	23	6	9	CARDINAL
ejpam-6117	24	1	hamja et al	PERSON
ejpam-6117	25	1	5	CARDINAL
ejpam-6117	26	1	6	CARDINAL
ejpam-6117	27	1	two	CARDINAL
ejpam-6117	29	1	2	CARDINAL
ejpam-6117	30	1	two	CARDINAL
ejpam-6117	30	2	w ∈	PERSON
ejpam-6117	30	3	w	GPE
ejpam-6117	30	4	w.	PERSON
ejpam-6117	30	5	w	GPE
ejpam-6117	32	1	∈	ORG
ejpam-6117	33	1	⊆	CARDINAL
ejpam-6117	34	1	∈	ORG
ejpam-6117	35	1	1	CARDINAL
ejpam-6117	36	1	wv	GPE
ejpam-6117	36	2	i(g	PERSON
ejpam-6117	36	3	l(g	PERSON
ejpam-6117	36	4	s(g	PERSON
ejpam-6117	37	1	α(g	PERSON
ejpam-6117	37	2	α(g	PERSON
ejpam-6117	37	3	⊆	CARDINAL
ejpam-6117	38	1	\s	ORG
ejpam-6117	38	2	w ∈ s	PERSON
ejpam-6117	45	1	⊆	CARDINAL
ejpam-6117	45	2	s.	GPE
ejpam-6117	48	1	dunbar et al	ORG
ejpam-6117	49	1	2	CARDINAL
ejpam-6117	49	2	two	CARDINAL
ejpam-6117	50	1	+ h	GPE
ejpam-6117	51	1	r. merontos et al	PERSON
ejpam-6117	52	1	j. pure	PERSON
ejpam-6117	52	2	18	CARDINAL
ejpam-6117	52	3	2	CARDINAL
ejpam-6117	52	4	2025	CARDINAL
ejpam-6117	52	5	6117 3	DATE
ejpam-6117	53	1	∈	ORG
ejpam-6117	54	1	two	CARDINAL
ejpam-6117	54	2	one	CARDINAL
ejpam-6117	54	3	∈	PRODUCT
ejpam-6117	55	1	two	CARDINAL
ejpam-6117	56	1	∈	ORG
ejpam-6117	56	2	u′v′ ∈ e(h	ORG
ejpam-6117	57	1	⊆	CARDINAL
ejpam-6117	57	2	⊆	CARDINAL
ejpam-6117	57	3	tx	ORG
ejpam-6117	57	4	⊆	PRODUCT
ejpam-6117	59	1	3	CARDINAL
ejpam-6117	59	2	⊆	CARDINAL
ejpam-6117	60	1	first	ORDINAL
ejpam-6117	61	1	1	CARDINAL
ejpam-6117	61	2	α(g	PERSON
ejpam-6117	62	1	1	CARDINAL
ejpam-6117	62	2	α(g	PERSON
ejpam-6117	63	1	1 ≤ αw(g	QUANTITY
ejpam-6117	64	1	α(g	PERSON
ejpam-6117	65	1	1	CARDINAL
ejpam-6117	66	1	w	ORG
ejpam-6117	66	2	2	CARDINAL
ejpam-6117	66	3	∈	ORG
ejpam-6117	68	1	2	CARDINAL
ejpam-6117	68	2	1	CARDINAL
ejpam-6117	71	1	⊆	CARDINAL
ejpam-6117	71	2	1	CARDINAL
ejpam-6117	72	1	1.	CARDINAL
ejpam-6117	73	1	α(g	PERSON
ejpam-6117	75	1	α(g	PERSON
ejpam-6117	77	1	α(g	PERSON
ejpam-6117	78	1	1	CARDINAL
ejpam-6117	79	1	1	CARDINAL
ejpam-6117	79	2	g = kn	PERSON
ejpam-6117	80	1	1	CARDINAL
ejpam-6117	80	2	r. merontos et al	PERSON
ejpam-6117	81	1	j. pure	PERSON
ejpam-6117	81	2	18	CARDINAL
ejpam-6117	81	3	2	CARDINAL
ejpam-6117	81	4	2025	CARDINAL
ejpam-6117	81	5	6117 4	DATE
ejpam-6117	81	6	9	CARDINAL
ejpam-6117	81	7	2	CARDINAL
ejpam-6117	82	1	n ≥ 2	ORG
ejpam-6117	85	1	g = k1,n−1	PERSON
ejpam-6117	90	1	2	CARDINAL
ejpam-6117	91	1	1	CARDINAL
ejpam-6117	91	2	k1,1	PERSON
ejpam-6117	91	3	n ≥ 3	DATE
ejpam-6117	92	1	∖	PERSON
ejpam-6117	93	1	w ∈	ORG
ejpam-6117	93	2	g = k1,n−1	PERSON
ejpam-6117	94	1	g = k1,n−1	PERSON
ejpam-6117	95	1	∖	PERSON
ejpam-6117	95	2	1	CARDINAL
ejpam-6117	96	1	3	CARDINAL
ejpam-6117	97	1	k > m+1	PERSON
ejpam-6117	97	2	n ≥ k+m+2	ORG
ejpam-6117	98	1	= k and	PERSON
ejpam-6117	98	2	α(g	PERSON
ejpam-6117	101	1	1	CARDINAL
ejpam-6117	103	1	first	ORDINAL
ejpam-6117	106	1	1	CARDINAL
ejpam-6117	107	1	xu	PERSON
ejpam-6117	107	2	xvk	PERSON
ejpam-6117	110	1	α(g	PERSON
ejpam-6117	110	2	= k. .................................... ....................................	PERSON
ejpam-6117	113	1	.......................................	PERSON
ejpam-6117	114	1	v3 v2 v1 vkv4	PERSON
ejpam-6117	121	1	vkv4	GPE
ejpam-6117	122	1	1	CARDINAL
ejpam-6117	123	1	α(g	PERSON
ejpam-6117	125	1	k1,k−1	GPE
ejpam-6117	125	2	k1,k−1	GPE
ejpam-6117	127	1	xk−1	ORG
ejpam-6117	129	1	ym+1	ORG
ejpam-6117	130	1	2	CARDINAL
ejpam-6117	132	1	xk−1}∪	GPE
ejpam-6117	136	1	xk−1	GPE
ejpam-6117	137	1	= k and	PERSON
ejpam-6117	137	2	α(g	PERSON
ejpam-6117	145	1	y3 y2 y1 ym+1y4 x3	FAC
ejpam-6117	148	1	2	CARDINAL
ejpam-6117	149	1	α(g	PERSON
ejpam-6117	151	1	r. merontos et al	PERSON
ejpam-6117	152	1	j. pure	PERSON
ejpam-6117	152	2	18	CARDINAL
ejpam-6117	152	3	2	CARDINAL
ejpam-6117	152	4	2025	CARDINAL
ejpam-6117	152	5	6117 5	DATE
ejpam-6117	152	6	9	CARDINAL
ejpam-6117	152	7	2	CARDINAL
ejpam-6117	153	1	3	CARDINAL
ejpam-6117	153	2	0	CARDINAL
ejpam-6117	165	1	...............................................................	PERSON
ejpam-6117	195	1	z2 z3 zr	PERSON
ejpam-6117	195	2	y3 y2 y1 ym+1y4 x3 x2 x1 xk−1x4	FAC
ejpam-6117	196	1	3	CARDINAL
ejpam-6117	197	1	α(g	PERSON
ejpam-6117	204	1	α(g	PERSON
ejpam-6117	206	1	0	CARDINAL
ejpam-6117	207	1	2	CARDINAL
ejpam-6117	209	1	+ 2	DATE
ejpam-6117	209	2	= k and	PERSON
ejpam-6117	209	3	α(g	PERSON
ejpam-6117	211	1	3	CARDINAL
ejpam-6117	211	2	2	CARDINAL
ejpam-6117	212	1	α(g)−	ORG
ejpam-6117	214	1	k > m+1	PERSON
ejpam-6117	215	1	3	CARDINAL
ejpam-6117	215	2	= k and	PERSON
ejpam-6117	215	3	α(g	PERSON
ejpam-6117	216	1	= k+m	PERSON
ejpam-6117	217	1	α(g)−	ORG
ejpam-6117	217	2	= m. next	PERSON
ejpam-6117	218	1	4	CARDINAL
ejpam-6117	220	1	2	CARDINAL
ejpam-6117	221	1	2	CARDINAL
ejpam-6117	221	2	α(cn	CARDINAL
ejpam-6117	226	1	{vi ∈	PERSON
ejpam-6117	226	2	∈	ORG
ejpam-6117	229	1	∈	ORG
ejpam-6117	231	1	2	CARDINAL
ejpam-6117	236	1	⟨ng[s1]⟩ = cn	PERSON
ejpam-6117	238	1	α(cn	CARDINAL
ejpam-6117	242	1	α(cn	CARDINAL
ejpam-6117	242	2	2	CARDINAL
ejpam-6117	243	1	g+h	ORG
ejpam-6117	244	1	r. merontos et al	PERSON
ejpam-6117	245	1	j. pure	PERSON
ejpam-6117	245	2	18	CARDINAL
ejpam-6117	245	3	2	CARDINAL
ejpam-6117	245	4	2025	CARDINAL
ejpam-6117	245	5	6117 6	DATE
ejpam-6117	245	6	9	CARDINAL
ejpam-6117	246	1	two	CARDINAL
ejpam-6117	247	1	g+h	ORG
ejpam-6117	248	1	g+h	ORG
ejpam-6117	249	1	g+h	ORG
ejpam-6117	249	2	⊆	CARDINAL
ejpam-6117	250	1	g+h	ORG
ejpam-6117	251	1	⊆	CARDINAL
ejpam-6117	253	1	g+h	ORG
ejpam-6117	253	2	g. corollary 3	PERSON
ejpam-6117	255	1	α(g+h	NORP
ejpam-6117	257	1	1	CARDINAL
ejpam-6117	258	1	α(g	PERSON
ejpam-6117	259	1	+g	PERSON
ejpam-6117	259	2	α(g	PERSON
ejpam-6117	260	1	+kn	PERSON
ejpam-6117	261	1	g ◦h. theorem 6	ORG
ejpam-6117	266	1	∪v∈ng(c)sv	ORG
ejpam-6117	266	2	sv	ORG
ejpam-6117	271	1	1	CARDINAL
ejpam-6117	272	1	⊆	CARDINAL
ejpam-6117	272	2	hu	PERSON
ejpam-6117	273	1	hu)⟩	ORG
ejpam-6117	273	2	⊆	CARDINAL
ejpam-6117	273	3	∈	ORG
ejpam-6117	276	1	2	CARDINAL
ejpam-6117	278	1	sv	ORG
ejpam-6117	278	2	r. merontos et al	PERSON
ejpam-6117	279	1	j. pure	PERSON
ejpam-6117	279	2	18	CARDINAL
ejpam-6117	279	3	2	CARDINAL
ejpam-6117	279	4	2025	CARDINAL
ejpam-6117	279	5	6117 7 of 9	DATE
ejpam-6117	282	1	first	ORDINAL
ejpam-6117	283	1	g◦h. next	ORG
ejpam-6117	284	1	h. αw(g	PERSON
ejpam-6117	284	2	|s′|	DATE
ejpam-6117	287	1	αw(g◦h	ORG
ejpam-6117	291	1	αw(k1 ◦h	ORG
ejpam-6117	292	1	6	CARDINAL
ejpam-6117	293	1	w ∈	PERSON
ejpam-6117	293	2	wx ∈	PERSON
ejpam-6117	296	1	6	CARDINAL
ejpam-6117	296	2	g ◦h.	ORG
ejpam-6117	296	3	αw(g ◦h	ORG
ejpam-6117	297	1	1− α(h	LAW
ejpam-6117	298	1	1−	LANGUAGE
ejpam-6117	300	1	6	CARDINAL
ejpam-6117	300	2	g ◦h. hence	ORG
ejpam-6117	300	3	1−	LANGUAGE
ejpam-6117	302	1	4	CARDINAL
ejpam-6117	302	2	sandueta	ORG
ejpam-6117	303	1	r. merontos et al	PERSON
ejpam-6117	304	1	j. pure	PERSON
ejpam-6117	304	2	18	CARDINAL
ejpam-6117	304	3	2	CARDINAL
ejpam-6117	304	4	2025	CARDINAL
ejpam-6117	304	5	6117 8	DATE
ejpam-6117	304	6	9	CARDINAL
ejpam-6117	304	7	7	CARDINAL
ejpam-6117	305	1	8	CARDINAL
ejpam-6117	305	2	⊆	CARDINAL
ejpam-6117	305	3	⊆	CARDINAL
ejpam-6117	305	4	tx	ORG
ejpam-6117	305	5	⊆	PRODUCT
ejpam-6117	305	6	g[h	ORG
ejpam-6117	306	1	8	CARDINAL
ejpam-6117	307	1	⊆	CARDINAL
ejpam-6117	307	2	⊆	CARDINAL
ejpam-6117	307	3	tx	ORG
ejpam-6117	307	4	⊆	PRODUCT
ejpam-6117	308	1	tx	ORG
ejpam-6117	309	1	⊆	CARDINAL
ejpam-6117	309	2	⊆	CARDINAL
ejpam-6117	309	3	tx	ORG
ejpam-6117	309	4	⊆	PRODUCT
ejpam-6117	309	5	∈	ORG
ejpam-6117	309	6	g[h	ORG
ejpam-6117	310	1	∈	ORG
ejpam-6117	310	2	tx	ORG
ejpam-6117	310	3	∈	ORG
ejpam-6117	314	1	xy	ORG
ejpam-6117	315	1	tx	ORG
ejpam-6117	316	1	g[h	ORG
ejpam-6117	317	1	/∈ e(h	ORG
ejpam-6117	318	1	tx	ORG
ejpam-6117	318	2	tx	ORG
ejpam-6117	318	3	g[h	ORG
ejpam-6117	320	1	xy /∈ e(g	PERSON
ejpam-6117	322	1	tx	ORG
ejpam-6117	322	2	tx	ORG
ejpam-6117	322	3	ab /∈ e(h	ORG
ejpam-6117	324	1	g[h	ORG
ejpam-6117	325	1	g[h	ORG
ejpam-6117	326	1	5	CARDINAL
ejpam-6117	330	1	⊆	CARDINAL
ejpam-6117	330	2	⊆	CARDINAL
ejpam-6117	330	3	tx	ORG
ejpam-6117	330	4	⊆	PRODUCT
ejpam-6117	330	5	∈	ORG
ejpam-6117	330	6	g[h	ORG
ejpam-6117	331	1	8	DATE
ejpam-6117	331	2	tx	ORG
ejpam-6117	333	1	∈	ORG
ejpam-6117	333	2	tx = a.	ORG
ejpam-6117	333	3	8	CARDINAL
ejpam-6117	333	4	tx	ORG
ejpam-6117	333	5	g[h	ORG
ejpam-6117	336	1	5	CARDINAL
ejpam-6117	337	1	r. merontos et al	PERSON
ejpam-6117	338	1	j. pure	PERSON
ejpam-6117	338	2	18	CARDINAL
ejpam-6117	338	3	2	CARDINAL
ejpam-6117	338	4	2025	CARDINAL
ejpam-6117	338	5	6117 9	DATE
ejpam-6117	338	6	9	CARDINAL
ejpam-6117	339	1	two	CARDINAL
ejpam-6117	344	1	msu-iligan institute of technology	ORG
ejpam-6117	345	1	1	CARDINAL
ejpam-6117	345	2	j. w. grossman	PERSON
ejpam-6117	347	1	1997	DATE
ejpam-6117	348	1	2	CARDINAL
ejpam-6117	348	2	j. e. dunbar	PERSON
ejpam-6117	348	3	j. w. grossman	PERSON
ejpam-6117	348	4	j. h. hattingh	PERSON
ejpam-6117	348	5	s. t. hedetniemi	PERSON
ejpam-6117	348	6	a. a. mcrae	PERSON
ejpam-6117	350	1	167/168:261–269, 1997	DATE
ejpam-6117	351	1	3	CARDINAL
ejpam-6117	351	2	k. m. alzoubi	PERSON
ejpam-6117	351	3	j. wan	PERSON
ejpam-6117	353	1	14(3):287–303, 2003	DATE
ejpam-6117	354	1	4	CARDINAL
ejpam-6117	354	2	f. bendali	PERSON
ejpam-6117	354	3	j. mailfert	PERSON
ejpam-6117	356	1	49(2):313–334	CARDINAL
ejpam-6117	356	2	2015	DATE
ejpam-6117	357	1	5	CARDINAL
ejpam-6117	357	2	j. hamja	PERSON
ejpam-6117	357	3	i. aniversario	PERSON
ejpam-6117	357	4	h. rara	PERSON
ejpam-6117	359	1	european journal	ORG
ejpam-6117	359	2	2022	CARDINAL
ejpam-6117	360	1	6	CARDINAL
ejpam-6117	360	2	m. militante and	PERSON
ejpam-6117	360	3	r. eballe	PERSON
ejpam-6117	361	1	2	CARDINAL
ejpam-6117	362	1	16:125–132	CARDINAL
ejpam-6117	362	2	2022	CARDINAL
ejpam-6117	363	1	7	CARDINAL
ejpam-6117	363	2	f. buckley	PERSON
ejpam-6117	363	3	f. harary	PERSON
ejpam-6117	365	1	addison-wesley, 1990	ORG
ejpam-6117	366	1	8	CARDINAL
ejpam-6117	366	2	e. sandueta	PERSON
ejpam-6117	366	3	s. r. canoy jr.	PERSON
ejpam-6117	367	1	8(4):1973–1980	DATE
ejpam-6117	367	2	2014	DATE