id	sid	eid	entity	type
ejpam-5716	1	1	european journal	ORG
ejpam-5716	1	2	2025	DATE
ejpam-5716	1	3	18	CARDINAL
ejpam-5716	1	4	1	CARDINAL
ejpam-5716	1	5	5716	CARDINAL
ejpam-5716	1	6	1307-5543	DATE
ejpam-5716	1	7	new york	GPE
ejpam-5716	1	8	m. anoche 1	PERSON
ejpam-5716	1	9	sergio r. canoy	PERSON
ejpam-5716	1	10	1	CARDINAL
ejpam-5716	1	11	msu-iligan institute of technology	ORG
ejpam-5716	1	12	9200	CARDINAL
ejpam-5716	1	13	iligan city	GPE
ejpam-5716	1	14	2	CARDINAL
ejpam-5716	1	15	msu-iligan institute of technology	ORG
ejpam-5716	1	16	9200	CARDINAL
ejpam-5716	1	17	iligan city	GPE
ejpam-5716	1	18	philippines	GPE
ejpam-5716	3	1	w ⊆	PRODUCT
ejpam-5716	3	2	at least one	CARDINAL
ejpam-5716	3	3	w	GPE
ejpam-5716	4	1	integer k	PERSON
ejpam-5716	4	2	2	CARDINAL
ejpam-5716	4	3	γh(g)− k.	PERSON
ejpam-5716	6	1	2020	DATE
ejpam-5716	6	2	05c69	CARDINAL
ejpam-5716	6	3	1	CARDINAL
ejpam-5716	7	1	natarajan	PERSON
ejpam-5716	7	2	13	CARDINAL
ejpam-5716	10	1	1	CARDINAL
ejpam-5716	10	2	2	CARDINAL
ejpam-5716	10	3	4	CARDINAL
ejpam-5716	10	4	5	CARDINAL
ejpam-5716	10	5	6	CARDINAL
ejpam-5716	10	6	8	CARDINAL
ejpam-5716	10	7	9	CARDINAL
ejpam-5716	10	8	14	CARDINAL
ejpam-5716	10	9	15	CARDINAL
ejpam-5716	10	10	16	CARDINAL
ejpam-5716	11	1	das et al	PERSON
ejpam-5716	12	1	3	CARDINAL
ejpam-5716	14	1	jesica.anoche@g.msuiit.edu.ph	PERSON
ejpam-5716	14	2	j. anoche	PERSON
ejpam-5716	14	3	sergio.canoy@g.msuiit.edu.ph	ORG
ejpam-5716	14	4	s. canoy, jr.	PERSON
ejpam-5716	15	1	1	CARDINAL
ejpam-5716	15	2	2025	CARDINAL
ejpam-5716	16	1	j. anoche	PERSON
ejpam-5716	16	2	s.r. canoy jr.	PERSON
ejpam-5716	17	1	j. pure	PERSON
ejpam-5716	17	2	18	CARDINAL
ejpam-5716	17	3	1	CARDINAL
ejpam-5716	17	4	2025	CARDINAL
ejpam-5716	17	5	5716 2	CARDINAL
ejpam-5716	17	6	13	CARDINAL
ejpam-5716	18	1	10–12	CARDINAL
ejpam-5716	18	2	two	CARDINAL
ejpam-5716	22	1	3	CARDINAL
ejpam-5716	24	1	2	CARDINAL
ejpam-5716	24	2	two	CARDINAL
ejpam-5716	25	1	two	CARDINAL
ejpam-5716	25	2	∈	ORG
ejpam-5716	27	1	⊆	CARDINAL
ejpam-5716	27	2	⋃	PERSON
ejpam-5716	28	1	0	CARDINAL
ejpam-5716	30	1	∈	ORG
ejpam-5716	31	1	∈	ORG
ejpam-5716	32	1	1	CARDINAL
ejpam-5716	34	1	i(g	EVENT
ejpam-5716	34	2	n2 g(u	PRODUCT
ejpam-5716	34	3	∈	ORG
ejpam-5716	34	4	2	CARDINAL
ejpam-5716	36	1	⊆	CARDINAL
ejpam-5716	36	2	n2 g(a	ORG
ejpam-5716	36	3	⋃	DATE
ejpam-5716	36	4	max{|n2 g(v)|	PERSON
ejpam-5716	36	5	∈	ORG
ejpam-5716	36	6	= min{|n2 g(v)| :	PERSON
ejpam-5716	36	7	∈	ORG
ejpam-5716	37	1	⊆	CARDINAL
ejpam-5716	39	1	⊆	CARDINAL
ejpam-5716	40	1	∈	PERSON
ejpam-5716	42	1	j. anoche	PERSON
ejpam-5716	42	2	s.r. canoy jr.	PERSON
ejpam-5716	43	1	j. pure	PERSON
ejpam-5716	43	2	18	CARDINAL
ejpam-5716	43	3	1	CARDINAL
ejpam-5716	43	4	2025	CARDINAL
ejpam-5716	43	5	5716 3	DATE
ejpam-5716	43	6	13	CARDINAL
ejpam-5716	43	7	1 ≤	QUANTITY
ejpam-5716	45	1	⊆	CARDINAL
ejpam-5716	45	2	γh(g)− k.	PERSON
ejpam-5716	49	1	|n2	DATE
ejpam-5716	51	1	⊆	CARDINAL
ejpam-5716	52	1	⊆	CARDINAL
ejpam-5716	52	2	〈 n2 g[s	PERSON
ejpam-5716	52	3	γh(g)− k.	PERSON
ejpam-5716	53	1	∈	ORG
ejpam-5716	55	1	⟨v	DATE
ejpam-5716	55	2	3	CARDINAL
ejpam-5716	55	3	1	CARDINAL
ejpam-5716	55	4	16	CARDINAL
ejpam-5716	59	1	bipartite	NORP
ejpam-5716	59	2	graph km	PERSON
ejpam-5716	59	3	n, γh(km	DATE
ejpam-5716	59	4	2	CARDINAL
ejpam-5716	60	1	2r	CARDINAL
ejpam-5716	60	2	6r	CARDINAL
ejpam-5716	60	3	2r	CARDINAL
ejpam-5716	62	1	6r	CARDINAL
ejpam-5716	63	1	2 ≤ s ≤ 5	QUANTITY
ejpam-5716	66	1	2r	CARDINAL
ejpam-5716	66	2	6r	CARDINAL
ejpam-5716	66	3	2r	CARDINAL
ejpam-5716	68	1	6r	CARDINAL
ejpam-5716	69	1	2 ≤ s ≤ 5	QUANTITY
ejpam-5716	71	1	3	CARDINAL
ejpam-5716	73	1	2	CARDINAL
ejpam-5716	73	2	2	CARDINAL
ejpam-5716	74	1	8	CARDINAL
ejpam-5716	74	2	two	CARDINAL
ejpam-5716	75	1	γh(g+h	NORP
ejpam-5716	75	2	+ pnd(h	PERSON
ejpam-5716	77	1	2	CARDINAL
ejpam-5716	77	2	j. anoche	PERSON
ejpam-5716	77	3	s.r. canoy jr.	PERSON
ejpam-5716	78	1	j. pure	PERSON
ejpam-5716	78	2	18	CARDINAL
ejpam-5716	78	3	1	CARDINAL
ejpam-5716	78	4	2025	CARDINAL
ejpam-5716	78	5	4	CARDINAL
ejpam-5716	78	6	1 + pnd(h	DATE
ejpam-5716	79	1	4	CARDINAL
ejpam-5716	79	2	2	CARDINAL
ejpam-5716	79	3	4	CARDINAL
ejpam-5716	80	1	4	CARDINAL
ejpam-5716	81	1	3	CARDINAL
ejpam-5716	82	1	i(g	PERSON
ejpam-5716	83	1	ζhj	GPE
ejpam-5716	84	1	1	CARDINAL
ejpam-5716	84	2	2	DATE
ejpam-5716	84	3	r+1	ORG
ejpam-5716	84	4	g′	PERSON
ejpam-5716	85	1	i(g	PERSON
ejpam-5716	87	1	⊆	CARDINAL
ejpam-5716	89	1	− |n2 g[{v1	ORG
ejpam-5716	89	2	v2	PERSON
ejpam-5716	89	3	− j. hence	PERSON
ejpam-5716	91	1	k ∈	PERSON
ejpam-5716	91	2	r+1	ORG
ejpam-5716	92	1	ing′	CARDINAL
ejpam-5716	92	2	⟨v	DATE
ejpam-5716	93	1	k ≤ γh(g)−1	ORG
ejpam-5716	95	1	|s′|	GPE
ejpam-5716	97	1	g′)|− |n2 g′	ORG
ejpam-5716	98	1	g[s	PRODUCT
ejpam-5716	99	1	− |n2 g[s	ORG
ejpam-5716	103	1	1	CARDINAL
ejpam-5716	104	1	1	CARDINAL
ejpam-5716	105	1	2	CARDINAL
ejpam-5716	105	2	1	CARDINAL
ejpam-5716	105	3	4	CARDINAL
ejpam-5716	105	4	7	CARDINAL
ejpam-5716	105	5	5	CARDINAL
ejpam-5716	105	6	7	CARDINAL
ejpam-5716	105	7	1	CARDINAL
ejpam-5716	105	8	4	CARDINAL
ejpam-5716	163	1	1 2	CARDINAL
ejpam-5716	163	2	1	CARDINAL
ejpam-5716	163	3	2	CARDINAL
ejpam-5716	164	1	3	CARDINAL
ejpam-5716	165	1	integer k	PERSON
ejpam-5716	165	2	4	CARDINAL
ejpam-5716	166	1	n ≥ 2	ORG
ejpam-5716	166	2	1 ≤	QUANTITY
ejpam-5716	166	3	1	CARDINAL
ejpam-5716	169	1	1	CARDINAL
ejpam-5716	170	1	− |n2	ORG
ejpam-5716	170	2	1	CARDINAL
ejpam-5716	171	1	1	CARDINAL
ejpam-5716	171	2	− |n2	ORG
ejpam-5716	171	3	1	CARDINAL
ejpam-5716	173	1	j. anoche	PERSON
ejpam-5716	173	2	s.r. canoy jr.	PERSON
ejpam-5716	174	1	j. pure	PERSON
ejpam-5716	174	2	18	CARDINAL
ejpam-5716	174	3	1	CARDINAL
ejpam-5716	174	4	2025	CARDINAL
ejpam-5716	174	5	5716 5	DATE
ejpam-5716	174	6	13	CARDINAL
ejpam-5716	174	7	5	CARDINAL
ejpam-5716	176	1	1	CARDINAL
ejpam-5716	176	2	∈	ORG
ejpam-5716	176	3	γh(g−	NORP
ejpam-5716	176	4	1	CARDINAL
ejpam-5716	178	1	1	CARDINAL
ejpam-5716	181	1	1	CARDINAL
ejpam-5716	181	2	∈	ORG
ejpam-5716	183	1	γh(g−	NORP
ejpam-5716	183	2	〈 n2 g[s	PRODUCT
ejpam-5716	183	3	1	CARDINAL
ejpam-5716	185	1	⊆	CARDINAL
ejpam-5716	185	2	n2 g[s ′	PRODUCT
ejpam-5716	188	1	1	CARDINAL
ejpam-5716	188	2	1	CARDINAL
ejpam-5716	188	3	6	CARDINAL
ejpam-5716	191	1	∈	ORG
ejpam-5716	192	1	= |n2	ORG
ejpam-5716	192	2	− |n2 g[s	ORG
ejpam-5716	192	3	∗]| ≤ |n2 g[s ′]|+ |n2	ORG
ejpam-5716	192	4	g[s	ORG
ejpam-5716	192	5	− |n2 g[s	ORG
ejpam-5716	192	6	= |n2 g[s	ORG
ejpam-5716	193	1	g[v]| ≤ k(1 + ∆h(g	FAC
ejpam-5716	195	1	2	CARDINAL
ejpam-5716	196	1	g1	ORG
ejpam-5716	198	1	γh(gj	CARDINAL
ejpam-5716	199	1	aj ⊆	PERSON
ejpam-5716	200	1	1	CARDINAL
ejpam-5716	200	2	2	DATE
ejpam-5716	200	3	2	CARDINAL
ejpam-5716	201	1	7	CARDINAL
ejpam-5716	201	2	g1	ORG
ejpam-5716	202	1	1	CARDINAL
ejpam-5716	202	2	2	DATE
ejpam-5716	209	1	2(i	CARDINAL
ejpam-5716	209	2	γh(gj	CARDINAL
ejpam-5716	212	1	|di| = γh(gi	GPE
ejpam-5716	215	1	j ∈	PERSON
ejpam-5716	221	1	j. anoche	PERSON
ejpam-5716	221	2	s.r. canoy jr.	PERSON
ejpam-5716	222	1	j. pure	PERSON
ejpam-5716	222	2	18	CARDINAL
ejpam-5716	222	3	1	CARDINAL
ejpam-5716	222	4	2025	CARDINAL
ejpam-5716	222	5	5716 6	CARDINAL
ejpam-5716	222	6	13	DATE
ejpam-5716	222	7	2(ii	CARDINAL
ejpam-5716	226	1	− |n2	ORG
ejpam-5716	229	1	1	CARDINAL
ejpam-5716	230	1	⊆	CARDINAL
ejpam-5716	232	1	2∪	CARDINAL
ejpam-5716	238	1	at least one	CARDINAL
ejpam-5716	238	2	2(i	CARDINAL
ejpam-5716	241	1	− |n2 g[s ′	ORG
ejpam-5716	246	1	i=1	PERSON
ejpam-5716	254	1	8	CARDINAL
ejpam-5716	256	1	1(i	CARDINAL
ejpam-5716	257	1	1 ≤ k ≤ n	QUANTITY
ejpam-5716	258	1	− k.	PERSON
ejpam-5716	258	2	n2 kn	PERSON
ejpam-5716	261	1	2	CARDINAL
ejpam-5716	261	2	1	CARDINAL
ejpam-5716	266	1	|n2	DATE
ejpam-5716	266	2	2	CARDINAL
ejpam-5716	266	3	2	CARDINAL
ejpam-5716	267	1	3	CARDINAL
ejpam-5716	267	2	1	CARDINAL
ejpam-5716	269	1	2	CARDINAL
ejpam-5716	271	1	j. anoche	PERSON
ejpam-5716	271	2	s.r. canoy jr.	PERSON
ejpam-5716	272	1	j. pure	PERSON
ejpam-5716	272	2	18	CARDINAL
ejpam-5716	272	3	1	CARDINAL
ejpam-5716	272	4	2025	CARDINAL
ejpam-5716	272	5	7	CARDINAL
ejpam-5716	272	6	13	CARDINAL
ejpam-5716	272	7	9	CARDINAL
ejpam-5716	273	1	6	CARDINAL
ejpam-5716	274	1	3k	CARDINAL
ejpam-5716	277	1	2	CARDINAL
ejpam-5716	278	1	6r	CARDINAL
ejpam-5716	278	2	6	CARDINAL
ejpam-5716	279	1	2 ≤ s ≤ 5	QUANTITY
ejpam-5716	279	2	2 ≤	QUANTITY
ejpam-5716	281	1	1	CARDINAL
ejpam-5716	281	2	2	DATE
ejpam-5716	282	1	1	CARDINAL
ejpam-5716	282	2	6r	CARDINAL
ejpam-5716	283	1	1	CARDINAL
ejpam-5716	283	2	2r	CARDINAL
ejpam-5716	284	1	2r−k)-element	DATE
ejpam-5716	284	2	3	CARDINAL
ejpam-5716	284	3	6	DATE
ejpam-5716	284	4	6r	CARDINAL
ejpam-5716	284	5	3k	CARDINAL
ejpam-5716	285	1	2r	CARDINAL
ejpam-5716	286	1	− k	PERSON
ejpam-5716	288	1	6r	CARDINAL
ejpam-5716	289	1	6r	CARDINAL
ejpam-5716	289	2	3k	CARDINAL
ejpam-5716	290	1	2r	CARDINAL
ejpam-5716	291	1	3k	CARDINAL
ejpam-5716	292	1	2	CARDINAL
ejpam-5716	292	2	1	CARDINAL
ejpam-5716	293	1	2r	CARDINAL
ejpam-5716	294	1	+ 1	DATE
ejpam-5716	294	2	3	CARDINAL
ejpam-5716	294	3	6	DATE
ejpam-5716	294	4	6r−3k+3	CARDINAL
ejpam-5716	297	1	2r−k+1)+2(2r−k+1	ORDINAL
ejpam-5716	299	1	6r−	CARDINAL
ejpam-5716	299	2	6r−3k+3]+1	CARDINAL
ejpam-5716	299	3	3k−2	CARDINAL
ejpam-5716	300	1	2r	CARDINAL
ejpam-5716	302	1	3k	CARDINAL
ejpam-5716	302	2	2	CARDINAL
ejpam-5716	302	3	3	CARDINAL
ejpam-5716	303	1	2 ≤ s ≤ 5	QUANTITY
ejpam-5716	303	2	1	CARDINAL
ejpam-5716	304	1	2r	CARDINAL
ejpam-5716	305	1	2r−k+2)-element	CARDINAL
ejpam-5716	305	2	3	CARDINAL
ejpam-5716	305	3	6	DATE
ejpam-5716	305	4	6r− 3k+6	QUANTITY
ejpam-5716	305	5	2 ≤ k ≤	QUANTITY
ejpam-5716	311	1	6r−[6r−3k+6]+s	CARDINAL
ejpam-5716	311	2	3k	CARDINAL
ejpam-5716	311	3	6	CARDINAL
ejpam-5716	312	1	2r	CARDINAL
ejpam-5716	314	1	3k + s−	DATE
ejpam-5716	314	2	6	CARDINAL
ejpam-5716	314	3	8	CARDINAL
ejpam-5716	315	1	= k	PERSON
ejpam-5716	315	2	1	CARDINAL
ejpam-5716	315	3	2	CARDINAL
ejpam-5716	316	1	2	CARDINAL
ejpam-5716	316	2	1	CARDINAL
ejpam-5716	316	3	2	CARDINAL
ejpam-5716	316	4	c5	PERSON
ejpam-5716	317	1	3	CARDINAL
ejpam-5716	317	2	c5	ORG
ejpam-5716	318	1	c5	ORG
ejpam-5716	319	1	2	CARDINAL
ejpam-5716	319	2	1(iv	CARDINAL
ejpam-5716	319	3	9	CARDINAL
ejpam-5716	319	4	10	CARDINAL
ejpam-5716	320	1	6	CARDINAL
ejpam-5716	322	1	3k	CARDINAL
ejpam-5716	325	1	2	CARDINAL
ejpam-5716	326	1	6r	CARDINAL
ejpam-5716	326	2	6	CARDINAL
ejpam-5716	327	1	2 ≤ s ≤ 5	QUANTITY
ejpam-5716	327	2	2 ≤ k ≤	QUANTITY
ejpam-5716	329	1	1	CARDINAL
ejpam-5716	330	1	2	CARDINAL
ejpam-5716	331	1	⊆	CARDINAL
ejpam-5716	334	1	⊆	CARDINAL
ejpam-5716	334	2	1	CARDINAL
ejpam-5716	334	3	j. anoche	PERSON
ejpam-5716	334	4	s.r. canoy jr.	PERSON
ejpam-5716	335	1	j. pure	PERSON
ejpam-5716	335	2	18	CARDINAL
ejpam-5716	335	3	1	CARDINAL
ejpam-5716	335	4	2025	CARDINAL
ejpam-5716	335	5	5716 8	DATE
ejpam-5716	335	6	n − |n2	ORG
ejpam-5716	336	1	n2 g(x	PRODUCT
ejpam-5716	338	1	△h(g	PERSON
ejpam-5716	338	2	11	CARDINAL
ejpam-5716	339	1	g = km1,m2	PERSON
ejpam-5716	339	2	mt	GPE
ejpam-5716	339	3	1 ≤ m1 ≤ m2 ≤	QUANTITY
ejpam-5716	341	1	j ∈	PERSON
ejpam-5716	342	1	1	CARDINAL
ejpam-5716	342	2	2	DATE
ejpam-5716	342	3	qj ⊆	ORG
ejpam-5716	343	1	qj	CARDINAL
ejpam-5716	346	1	1	CARDINAL
ejpam-5716	348	1	= t. theorem 12	PERSON
ejpam-5716	349	1	km1,m2	PERSON
ejpam-5716	349	2	mt	GPE
ejpam-5716	349	3	m1 ≤ m2 ≤ ·	FAC
ejpam-5716	349	4	mt	GPE
ejpam-5716	352	1	m1 ≤ m2 ≤ ·	PRODUCT
ejpam-5716	353	1	11	CARDINAL
ejpam-5716	355	1	2	CARDINAL
ejpam-5716	357	1	j=k+1	PERSON
ejpam-5716	358	1	m1 ≤ m2 ≤ ·	PRODUCT
ejpam-5716	360	1	− |n2	ORG
ejpam-5716	364	1	12	CARDINAL
ejpam-5716	365	1	1	CARDINAL
ejpam-5716	366	1	bipartite	NORP
ejpam-5716	366	2	graph km	PERSON
ejpam-5716	366	3	2 ≤ m ≤ n	QUANTITY
ejpam-5716	366	4	m. theorem 13	PERSON
ejpam-5716	367	1	3	CARDINAL
ejpam-5716	368	1	1(vi	CARDINAL
ejpam-5716	369	1	2	CARDINAL
ejpam-5716	370	1	1	CARDINAL
ejpam-5716	372	1	1	CARDINAL
ejpam-5716	373	1	|n2	DATE
ejpam-5716	375	1	7	CARDINAL
ejpam-5716	376	1	− |n2 p	ORG
ejpam-5716	377	1	10− 7	DATE
ejpam-5716	377	2	3	CARDINAL
ejpam-5716	377	3	14	DATE
ejpam-5716	378	1	3	CARDINAL
ejpam-5716	378	2	3	CARDINAL
ejpam-5716	379	1	1 ≤ k ≤	QUANTITY
ejpam-5716	379	2	j. anoche	PERSON
ejpam-5716	379	3	s.r. canoy jr.	PERSON
ejpam-5716	380	1	j. pure	PERSON
ejpam-5716	380	2	18	CARDINAL
ejpam-5716	380	3	1	CARDINAL
ejpam-5716	380	4	2025	CARDINAL
ejpam-5716	380	5	5716 9	DATE
ejpam-5716	380	6	13	CARDINAL
ejpam-5716	381	1	1 ≤ k	QUANTITY
ejpam-5716	383	1	∈	ORG
ejpam-5716	385	1	∈	PRODUCT
ejpam-5716	385	2	∈	ORG
ejpam-5716	385	3	2	CARDINAL
ejpam-5716	388	1	2	CARDINAL
ejpam-5716	389	1	⊆	CARDINAL
ejpam-5716	391	1	n− |n2 g′	ORG
ejpam-5716	393	1	15	CARDINAL
ejpam-5716	394	1	2	CARDINAL
ejpam-5716	394	2	∈	ORG
ejpam-5716	394	3	2	CARDINAL
ejpam-5716	395	1	g′	PERSON
ejpam-5716	395	2	2	CARDINAL
ejpam-5716	396	1	1 ≤ k ≤	QUANTITY
ejpam-5716	399	1	g′)\s	NORP
ejpam-5716	400	1	g′	PERSON
ejpam-5716	400	2	∈	ORG
ejpam-5716	401	1	2	CARDINAL
ejpam-5716	402	1	∈	ORG
ejpam-5716	402	2	2	CARDINAL
ejpam-5716	405	1	1 ≤ k	QUANTITY
ejpam-5716	406	1	2	CARDINAL
ejpam-5716	406	2	− k	PERSON
ejpam-5716	406	3	− |n2 g′	ORG
ejpam-5716	408	1	g′	PERSON
ejpam-5716	410	1	′)− k = γh(g)−	PERSON
ejpam-5716	410	2	|n2 g[d]|	PERSON
ejpam-5716	411	1	1	CARDINAL
ejpam-5716	416	1	g′	PERSON
ejpam-5716	417	1	g[d]| ≤ n − ζhk+1(g	PERSON
ejpam-5716	419	1	2	CARDINAL
ejpam-5716	419	2	̸ |n2 g′	ORG
ejpam-5716	420	1	n2 g′	ORG
ejpam-5716	420	2	w ∈ n2 g(d	PERSON
ejpam-5716	421	1	w ̸=	PERSON
ejpam-5716	421	2	w ∈ n2 g(d	PERSON
ejpam-5716	422	1	2	CARDINAL
ejpam-5716	423	1	1)−	DATE
ejpam-5716	424	1	g′	PERSON
ejpam-5716	425	1	1 ≤	CARDINAL
ejpam-5716	425	2	1	CARDINAL
ejpam-5716	427	1	3	CARDINAL
ejpam-5716	428	1	two	CARDINAL
ejpam-5716	428	2	15	CARDINAL
ejpam-5716	429	1	j. anoche	PERSON
ejpam-5716	429	2	s.r. canoy jr.	PERSON
ejpam-5716	430	1	j. pure	PERSON
ejpam-5716	430	2	18	CARDINAL
ejpam-5716	430	3	1	CARDINAL
ejpam-5716	430	4	2025	CARDINAL
ejpam-5716	430	5	5716 10	DATE
ejpam-5716	430	6	13	CARDINAL
ejpam-5716	432	1	g′ =	PERSON
ejpam-5716	434	1	15	CARDINAL
ejpam-5716	434	2	3	CARDINAL
ejpam-5716	434	3	2	CARDINAL
ejpam-5716	435	1	2	CARDINAL
ejpam-5716	435	2	1	CARDINAL
ejpam-5716	436	1	g′	PERSON
ejpam-5716	436	2	1	CARDINAL
ejpam-5716	437	1	four	CARDINAL
ejpam-5716	438	1	n2	ORG
ejpam-5716	438	2	n2 g′	ORG
ejpam-5716	441	1	− |n2	ORG
ejpam-5716	441	2	3	CARDINAL
ejpam-5716	441	3	− |n2 g′ [a]|	ORG
ejpam-5716	441	4	+ 1	DATE
ejpam-5716	442	1	16	CARDINAL
ejpam-5716	444	1	∈	ORG
ejpam-5716	446	1	2	CARDINAL
ejpam-5716	447	1	∈	ORG
ejpam-5716	449	1	+ 1 ≤	QUANTITY
ejpam-5716	454	1	\s	ORG
ejpam-5716	456	1	2	CARDINAL
ejpam-5716	457	1	2	CARDINAL
ejpam-5716	457	2	|s′|	PERSON
ejpam-5716	458	1	1	CARDINAL
ejpam-5716	460	1	1	CARDINAL
ejpam-5716	460	2	⊆	CARDINAL
ejpam-5716	461	1	∈ n2 g(s	PERSON
ejpam-5716	461	2	∈ n2 g(s	PRODUCT
ejpam-5716	461	3	∈ n2 g(s	PRODUCT
ejpam-5716	462	1	w ∈ s	PERSON
ejpam-5716	462	2	2	CARDINAL
ejpam-5716	462	3	h∗	ORG
ejpam-5716	463	1	w	ORG
ejpam-5716	463	2	2	CARDINAL
ejpam-5716	469	1	2	CARDINAL
ejpam-5716	469	2	5	CARDINAL
ejpam-5716	469	3	2	CARDINAL
ejpam-5716	469	4	1	CARDINAL
ejpam-5716	471	1	7	CARDINAL
ejpam-5716	474	1	2	CARDINAL
ejpam-5716	475	1	16	DATE
ejpam-5716	475	2	2	CARDINAL
ejpam-5716	475	3	16	CARDINAL
ejpam-5716	477	1	2	CARDINAL
ejpam-5716	477	2	2	CARDINAL
ejpam-5716	478	1	2	CARDINAL
ejpam-5716	478	2	∈	ORG
ejpam-5716	478	3	1	CARDINAL
ejpam-5716	479	1	2	CARDINAL
ejpam-5716	480	1	two	CARDINAL
ejpam-5716	482	1	x]|	ORG
ejpam-5716	482	2	δ(h	PERSON
ejpam-5716	485	1	⊆	CARDINAL
ejpam-5716	487	1	⊆	CARDINAL
ejpam-5716	488	1	∈	PRODUCT
ejpam-5716	489	1	2	CARDINAL
ejpam-5716	489	2	j. anoche	PERSON
ejpam-5716	489	3	s.r. canoy jr.	PERSON
ejpam-5716	490	1	j. pure	PERSON
ejpam-5716	490	2	18	CARDINAL
ejpam-5716	490	3	1	CARDINAL
ejpam-5716	490	4	2025	CARDINAL
ejpam-5716	490	5	5716 11	CARDINAL
ejpam-5716	490	6	13	CARDINAL
ejpam-5716	490	7	∈	PRODUCT
ejpam-5716	492	1	n 2 g+h	PRODUCT
ejpam-5716	494	1	|n2 g+h	PERSON
ejpam-5716	495	1	x]| = m−	PERSON
ejpam-5716	496	1	max{|n2 g+h	PERSON
ejpam-5716	498	1	∈	ORG
ejpam-5716	498	2	m− δ(g	PERSON
ejpam-5716	499	1	max{|n2 g+h	PERSON
ejpam-5716	501	1	∈	ORG
ejpam-5716	502	1	∈	ORG
ejpam-5716	502	2	δ(h	PERSON
ejpam-5716	503	1	∈	ORG
ejpam-5716	504	1	|n2	DATE
ejpam-5716	505	1	x]|	ORG
ejpam-5716	505	2	δ(h	PERSON
ejpam-5716	507	1	17	CARDINAL
ejpam-5716	510	1	min{m+ δ(h	PERSON
ejpam-5716	510	2	1	CARDINAL
ejpam-5716	510	3	1	CARDINAL
ejpam-5716	510	4	∈	ORG
ejpam-5716	511	1	∈	ORG
ejpam-5716	511	2	1	CARDINAL
ejpam-5716	511	3	∈	PRODUCT
ejpam-5716	511	4	1	CARDINAL
ejpam-5716	511	5	i(g	PERSON
ejpam-5716	512	1	i(g	PERSON
ejpam-5716	513	1	= k	PERSON
ejpam-5716	513	2	1 ≤ k ≤ n. proof	QUANTITY
ejpam-5716	514	1	lemma 1	ORG
ejpam-5716	515	1	lemma 2	ORG
ejpam-5716	516	1	m− δ(g	PERSON
ejpam-5716	516	2	δ(h	PERSON
ejpam-5716	518	1	m+ n)− (	ORG
ejpam-5716	518	2	m− δ(g	PERSON
ejpam-5716	518	3	m+ n)−	ORG
ejpam-5716	518	4	δ(h	PERSON
ejpam-5716	519	1	5	CARDINAL
ejpam-5716	519	2	2	CARDINAL
ejpam-5716	519	3	1	CARDINAL
ejpam-5716	519	4	∈	ORG
ejpam-5716	519	5	γh((g + h	GPE
ejpam-5716	521	1	1	CARDINAL
ejpam-5716	524	1	1	CARDINAL
ejpam-5716	525	1	∈	ORG
ejpam-5716	527	1	2	CARDINAL
ejpam-5716	529	1	1	CARDINAL
ejpam-5716	529	2	∈	ORG
ejpam-5716	530	1	1	CARDINAL
ejpam-5716	530	2	∈	PRODUCT
ejpam-5716	530	3	1	CARDINAL
ejpam-5716	530	4	i(g	PERSON
ejpam-5716	530	5	2	CARDINAL
ejpam-5716	530	6	2	CARDINAL
ejpam-5716	531	1	k = 1	PERSON
ejpam-5716	532	1	n2 g(p	PERSON
ejpam-5716	532	2	n2 h(q	PRODUCT
ejpam-5716	533	1	max{|n2 g(p)|	PERSON
ejpam-5716	534	1	δ(h	PERSON
ejpam-5716	534	2	0	CARDINAL
ejpam-5716	535	1	1	CARDINAL
ejpam-5716	537	1	2	CARDINAL
ejpam-5716	538	1	1 ≤ k ≤ n	QUANTITY
ejpam-5716	538	2	sn	ORG
ejpam-5716	538	3	sg ⊆	ORG
ejpam-5716	538	4	j. anoche	PERSON
ejpam-5716	538	5	s.r. canoy jr.	PERSON
ejpam-5716	539	1	j. pure	PERSON
ejpam-5716	539	2	18	CARDINAL
ejpam-5716	539	3	1	CARDINAL
ejpam-5716	539	4	2025	CARDINAL
ejpam-5716	539	5	5716 12	DATE
ejpam-5716	539	6	13	CARDINAL
ejpam-5716	539	7	sn	ORG
ejpam-5716	539	8	⊆	PRODUCT
ejpam-5716	540	1	|sg|+	GPE
ejpam-5716	540	2	1)−	CARDINAL
ejpam-5716	540	3	k and |n2	ORG
ejpam-5716	541	1	sg]|+ |n2 g+kn	ORG
ejpam-5716	542	1	sg]|+ |sn|	PERSON
ejpam-5716	544	1	|sn|	ORG
ejpam-5716	546	1	1	CARDINAL
ejpam-5716	546	2	− k.	PERSON
ejpam-5716	548	1	m+ n)− |n2	ORG
ejpam-5716	549	1	m+ n)−	ORG
ejpam-5716	549	2	k. 4	PERSON
ejpam-5716	551	1	1	CARDINAL
ejpam-5716	553	1	msu-iligan institute of technology	ORG
ejpam-5716	554	1	1	CARDINAL
ejpam-5716	554	2	n.j	GPE
ejpam-5716	554	3	i. aniversario	PERSON
ejpam-5716	554	4	f. jamil	PERSON
ejpam-5716	556	1	european journal	ORG
ejpam-5716	556	2	17(3):1618–1636	CARDINAL
ejpam-5716	556	3	2024	DATE
ejpam-5716	557	1	2	CARDINAL
ejpam-5716	557	2	d.b. catian	PERSON
ejpam-5716	557	3	i. aniversario	PERSON
ejpam-5716	557	4	f. jamil	PERSON
ejpam-5716	559	1	european journal	ORG
ejpam-5716	559	2	17(3):1737–1750	CARDINAL
ejpam-5716	559	3	2024	DATE
ejpam-5716	560	1	3	CARDINAL
ejpam-5716	560	2	a. das	PERSON
ejpam-5716	560	3	w. j. desormeaux	PERSON
ejpam-5716	562	1	indian	NORP
ejpam-5716	563	1	49(2):349–364, 2018	DATE
ejpam-5716	564	1	4	CARDINAL
ejpam-5716	564	2	j. hassan	PERSON
ejpam-5716	564	3	s. canoy jr.	PERSON
ejpam-5716	566	1	j. pure	PERSON
ejpam-5716	566	2	15(4):1783–1796	CARDINAL
ejpam-5716	566	3	2022	CARDINAL
ejpam-5716	567	1	5	CARDINAL
ejpam-5716	567	2	j. hassan	PERSON
ejpam-5716	567	3	s. canoy jr	PERSON
ejpam-5716	567	4	c.j.	ORG
ejpam-5716	569	1	european journal	ORG
ejpam-5716	569	2	16(1):319–335	DATE
ejpam-5716	569	3	2023	CARDINAL
ejpam-5716	570	1	6	CARDINAL
ejpam-5716	570	2	m. henning	PERSON
ejpam-5716	570	3	n. rad	PERSON
ejpam-5716	571	1	2	CARDINAL
ejpam-5716	572	1	33(4):913–927	DATE
ejpam-5716	572	2	2017	DATE
ejpam-5716	573	1	7	CARDINAL
ejpam-5716	573	2	s. canoy jr	PERSON
ejpam-5716	573	3	j. hassan	PERSON
ejpam-5716	575	1	european journal	ORG
ejpam-5716	575	2	2023	CARDINAL
ejpam-5716	576	1	j. anoche	PERSON
ejpam-5716	576	2	s.r. canoy jr.	PERSON
ejpam-5716	577	1	j. pure	PERSON
ejpam-5716	577	2	18	CARDINAL
ejpam-5716	577	3	1	CARDINAL
ejpam-5716	577	4	2025	CARDINAL
ejpam-5716	577	5	5716 13	DATE
ejpam-5716	577	6	13	CARDINAL
ejpam-5716	578	1	8	CARDINAL
ejpam-5716	578	2	s. canoy jr, r. mollejon	PERSON
ejpam-5716	578	3	canoy.	PERSON
ejpam-5716	581	1	j. pure	PERSON
ejpam-5716	582	1	12(4):1455–1463,	DATE
ejpam-5716	582	2	2019	DATE
ejpam-5716	583	1	9	CARDINAL
ejpam-5716	583	2	s. canoy jr and	PERSON
ejpam-5716	583	3	g. salasalan	PERSON
ejpam-5716	585	1	kyungpook mathematical journal	ORG
ejpam-5716	585	2	62:193–204	CARDINAL
ejpam-5716	585	3	2022	CARDINAL
ejpam-5716	586	1	10	CARDINAL
ejpam-5716	586	2	a. miranda	PERSON
ejpam-5716	586	3	r. eballe	PERSON
ejpam-5716	588	1	15(12):615	CARDINAL
ejpam-5716	588	2	623	CARDINAL
ejpam-5716	588	3	2021	DATE
ejpam-5716	589	1	11	CARDINAL
ejpam-5716	589	2	a. miranda	PERSON
ejpam-5716	589	3	r. eballe	PERSON
ejpam-5716	591	1	asian research journal	ORG
ejpam-5716	591	2	18(12):95–101	ORG
ejpam-5716	591	3	2022	CARDINAL
ejpam-5716	592	1	12	CARDINAL
ejpam-5716	592	2	a. miranda	PERSON
ejpam-5716	592	3	r. eballe	PERSON
ejpam-5716	594	1	39(2):209–219	DATE
ejpam-5716	594	2	2023	CARDINAL
ejpam-5716	595	1	13	CARDINAL
ejpam-5716	595	2	c. natarajan	PERSON
ejpam-5716	595	3	s. ayyaswamy	ORG
ejpam-5716	596	1	versita	PERSON
ejpam-5716	596	2	23(2):187	CARDINAL
ejpam-5716	596	3	199	CARDINAL
ejpam-5716	596	4	2015	DATE
ejpam-5716	597	1	14	CARDINAL
ejpam-5716	597	2	g. salasalan and jr. s. canoy.	PERSON
ejpam-5716	600	1	j. pure	PERSON
ejpam-5716	600	2	14(1):112–125	DATE
ejpam-5716	600	3	2021	DATE
ejpam-5716	601	1	15	CARDINAL
ejpam-5716	601	2	c.j.	ORG
ejpam-5716	601	3	s. canoy jr.	PERSON
ejpam-5716	602	1	european journal	ORG
ejpam-5716	602	2	16(1):5–17	CARDINAL
ejpam-5716	602	3	2023	CARDINAL
ejpam-5716	603	1	16	CARDINAL
ejpam-5716	603	2	c. natarajan	ORG
ejpam-5716	603	3	ayyaswamy	ORG
ejpam-5716	603	4	y.b. venkatakrishnan	GPE
ejpam-5716	605	1	ams msc	PERSON
ejpam-5716	605	2	2010	DATE