EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 17, No. 2, 2024, 969-978 ISSN 1307-5543 – ejpam.com Published by New York Business Global Perfect Equitable Isolate Dominations in Graphs Mark L. Caay1,∗, Andrew C. Hernandez1 1 Department of Mathematics and Statistics, College of Science, Polytechnic University of the Philippines, Sta. Mesa, Manila City, 1016 Metro Manila, Philippines Abstract. A subset S ⊆ V (G) is said to be a perfect equitable isolate dominating set of a graph G if it is both perfect equitable dominating set of G and isolate dominating set of G. The minimum cardinality of a perfect equitable isolate dominating set is called perfect equitable isolate domination number of G and is denoted by γpe0(G). A perfect equitable isolate dominating set S of G is called γpe0-set of G. In this paper, the authors give characterizations of a perfect equitable isolate dominating set of some graphs and graphs obtained from the join and corona of two graphs. Furthermore, the perfect equitable isolate domination numbers of these graphs is determined, and the graphs with no perfect equitable isolate dominating sets are investigated. 2020 Mathematics Subject Classifications: 05C69, 05C38, 05C76 Key Words and Phrases: perfect domination, equitable domination, perfect equitable domina- tion, isolate domination, perfect equitable isolate domination 1. Introduction Recently, there has been a growing interest in the applications of one of the widest re- search topics in graph theory, the study of domination in graphs, which was developed by Claude Berge in 1958 when he introduced the coefficient of external stability known today as domination [2]. Due to the richness of the research and applications to graphs, many variants of domination started to prosper and some of these variants are the isolate domination and perfect equitable domination in graph. The concept of perfect domination was first introduced by Livingston and Stout [16] as an answer to the problem of the supplement study conducted by the same authors in [15]. This notion has been celebrated for years, and many studies of this kind have been introduced. Caay and Palahang [6] introduced the notion of perfect independent domination of graphs where they joined the notion of perfect domination and independent domination and investigate the existence of such variant and the corresponding number to graph. There are also many variants of perfect dominations of graphs which are found ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v17i2.4963 Email addresses: mark.caay@adamson.edu.ph (M. Caay), andrew.hernndez@pup.edu.ph (A. Hernandez) https://www.ejpam.com 969 © 2024 EJPAM All rights reserved. M. Caay, A. Hernandez / Eur. J. Pure Appl. Math, 17 (2) (2024), 969-978 970 in the paper of [16], [10] and [11]. Another variant of domination is the equitable domi- nation graph. The concept of equitable domination was believed to have been introduced by A. Anitha, et.al. in [8] and it was also discussed in the paper of G. Deepak, et.al. in [9]. This concept has extended further and so Caay and Durog in [5] introduced the notion of independent equitable domination in graphs. Furthermore, this concept has also been developed by Caay and Arugay when they introduced the notion of perfect equitable domination in [4] which studied about the domination that is perfect and equitable at the same time. Furthermore, in 2024, Caay in [3] introduced the notion of equitable rings domination in graphs. In 2013, the concept of Isolate domination in graphs was studied by Hamid and Bala- murugan [13]. Because this study gives a lot of opportunity to see many research topics, a lot of mathematician studied many variants of this. Armada and Hamja in [1] studied the perfect isolate domination in graphs where they defined a domination to be perfect and isolate at the same time. Many authors also have made a lot of studies on this different variants and can be found in [17]. In this paper, we study the perfect equitable isolate domination in graphs. A subset S ⊆ V (G) is said to be a perfect equitable isolate dominating set if it is a isolate dominating set and if it is perfect equitable dominating set. To give clarity, the flow of our paper is as follows: in Section 2, we introduce the necessary notations and basic concepts that are used in this study. We also introduce the isolate domination and perfect equitable dominations, and some of their results from the references that are used in the discussion of the study. We also established the case when these two dominations imply each other and so we come up with our formal working definition. In Section 3, we show our results of our study. 2. Preliminaries and the working definitions Throughout this paper, the graph we consider here is a connected simple graph. That means, there are no loops and multiple edges. A pair G = (V (G), E(G)) is called a graph (on V ). The elements of V (G) are called the vertices of G and the elements of E(G) are called the edges of G. If no confusion arises, we can use V and E to denote the set of vertices and set of edges of G, respectively. Suppose v ∈ V , the neighborhood of v is the set NG(v) = {u ∈ V : uv ∈ E.}. Given D ⊆ V , the set NG(D) = N(D) = ⋃ v∈D NG(v) and the set NG[D] = N [D] = D ⋃ N(D) are the open neighborhood and the closed neigh- borhood of D respectively. In this paper, we denote ∆(G) and δ(G) to be the minimum and maximum degree of G, respectively. We denote Pn, Cn,Kn, Tn for the path graph, cycle graph, complete graph and trees of order n, respectively. Theorem 1. [7] A graph G is a cycle graph if and only if every vertex of G is adjacent to two other vertices. M. Caay, A. Hernandez / Eur. J. Pure Appl. Math, 17 (2) (2024), 969-978 971 Definition 1. [7] A spanning subgraph of a graph G is a subgraph obtained by deleting some edges of G with the same vertex set. Example 1. A cycle Cn is a spanning subgraph of a complete graph Kn. The following are the definitions of the binary operations in graphs used in this study: join, corona and cartesian product. Definition 2. [14] The join G+H of the two graphs G and H is the graph with vertex set V (G+H) = V (G) + V (H) and the edge set E(G+H) = E(G) ∪ E(H) ∪ {uv : u ∈ V (G), v ∈ V (H)} . Definition 3. [12] The corona G ◦H of two graphs G and H is the graph obtained by taking one copy of G of order n and n copies of H, and then joining the ith vertex of G to every vertex in the ith copy of H. In [2], a subset S of V (G) is a dominating set of G if for every v ∈ V (G)\S, there exists u ∈ S such that uv ∈ E(G). That is, N [S] = V (G). The minimum cardinality of the dominating set S of G is called a domination number of G and is denoted by γ(G). In this case, S is called γ-set of G. In [16], a dominating set S of G is said to be a perfect dominating set of G if every vertex v ∈ V (G) \ S is dominated by exactly one vertex u ∈ S. The minimum cardinality of a perfect dominating set S of G is called a perfect domination number of G and is denoted by γp(G). In this case, we say S a γp-set of G. In [8] and [9], a dominating set S of G is said to be an equitable dominating set of G if for every v ∈ V (G)\S, there exists u ∈ S with uv ∈ E(G) such that |deg(u)− deg(v)| ≤ 1. The minimum cardinality of an equitable dominating set S of G is called an equitable dom- ination number of G and is denoted by γe(G). In this case, we say S a γe-set of G. Caay and Arugay in [4] introduced the notion of perfect equitable domination. A dominating set S of G is said to be a perfect equitable dominating set of G if for it is both perfect and equitable dominating set. The minimum cardinality of a perfect equitable dominating set S of G is called a perfect equitable domination number of G and is denoted by γpe(G). In this case, we say S a γpe-set of G. Finally, in [13], a dominating set S ⊆ V (G) is said to be an isolate dominating set of G if there exists u ∈ S such that uv /∈ E(G) for all v ∈ S. The minimum cardinality of an equitable dominating set S of G is called an isolate domination number of G and is denoted by γ0(G). In this case, we say S a γ0-set of G. M. Caay, A. Hernandez / Eur. J. Pure Appl. Math, 17 (2) (2024), 969-978 972 Theorem 2. [13] A dominating set S of G is a minimal dominating set if and only if for every u ∈ S, u is an isolate of ⟨S⟩. In particular, S = {u} is a γ-set of G if and only if S is a γ0-set of G. Theorem 3. [16] If ∆(G) = n− 1 for any graph G of order n, then γp(G) = 1. In other words, S = {u} is a γ-set of G if and only if S is a γp-set of G. Theorem 4. [4] Given a path Pn and cycle Cn, n ≥ 3, γpe(Pn) = γpe(Cn) = ⌈n 3 ⌉ . More- over, in path Pn and cycle Cn, consecutive vertices of γpe-sets are either adjacent or at a distance 3 apart It is natural to ask if what is the relationship of the perfect equitable domination and the isolate domination in graphs in terms of cardinality. The result is negative in general. There is no general way to determine which one is larger. However, the following results will tell about the idea between the perfect equitable versus the perfect equitable isolate and the isolate versus perfect equitable isolate. Following the above definitions, we have the following results which are very obvious. Theorem 5. If γpe(G) = k for some positive integer k and S is a γpe-set of G such that ⟨S⟩ has an isolated vertex, then γpe0(G) = k. Theorem 6. If γ0(G) = k for some positive integer k and S is a γ0-set of G such that every vertex v ∈ V (G) \S is dominated by exactly one vertex in S, and u ∈ S, there exists v ∈ V (G) \ S with uv ∈ E(G) such that |deg(u)− deg(v)| ≤ 1, then γpe0(G) = k. Theorems 5 and 6 give rise to the definition of the our working definition. They simply tell that a dominating set that is a perfect equitable dominating set and an isolate dominating set, then it is a perfect equitable isolate dominating set. Definition 4. A dominating set S ⊆ V (G) is said to be a perfect equitable isolate dominating set (PEID) of G if it is both perfect equitable isolate dominating set. The minimum cardinality of a perfect equitable isolate dominating set S of G is called a perfect equitable isolate domination number of G and is denoted by γpe0(G). In this case, we say S a γpe0-set of G. Also, if u ∈ S such that uv ∈ E(G) for some v ∈ V (G) \ S, then either u is said to PEIDly-dominate v, or v is PEIDly-dominated by u. Example 2. Consider the graph in Figure 1. Note that the set {u1, u5} and {u1, u8} are γ-sets. For {u4, u5}, note that NG(u4) = {u1, u2, u3, u5} and NG(u5) = {u4, u6, u7, u8}. Thus, NG(u4) ∩NG(u5) = ∅. Thus, {u4, u5} is a γp-set. Also, observe that |deg(u4)− deg(u1)| ≤ 1 |deg(u4)− deg(u2)| ≤ 1 |deg(u4)− deg(u3)| ≤ 1 |deg(u5)− deg(u6)| ≤ 1 M. Caay, A. Hernandez / Eur. J. Pure Appl. Math, 17 (2) (2024), 969-978 973 |deg(u5)− deg(u7)| ≤ 1 |deg(u5)− deg(u8)| ≤ 1. Thus, {u4, u5} is γe-set implying that it is a γpe-set. However, u4u5 ∈ E(G). This means that {u4, u5} is not a γ0, and so it is not a γpe0-set. Now for the γ-set {u1, u8}, NG(u1) = {u2, u3, u4} and NG(u8) = {u5, u6, u7}. Thus, NG(u1) ∩NG(U8) = ∅. This means that {u1, u8} is a γp-set. Also, observe that |deg(u1)− deg(u2)| ≤ 1 |deg(u1)− deg(u3)| ≤ 1 |deg(u1)− deg(u4)| ≤ 1 |deg(u8)− deg(u5)| ≤ 1 |deg(u8)− deg(u6)| ≤ 1 |deg(u8)− deg(u7)| ≤ 1. Thus, {u1, u8} is γe-set implying that it is a γpe-set. Also, u1u8 /∈ E(G) and so {u1, u8} is γ0-set. Therefore, {u1, u8} is a γpe0-set. Consequently, γpe0(G) = 2. Figure 1: Example of γpe0-set in a graph G. The following propositions follow directly from Definition 4. Proposition 1. Let S be a γ0-set of G. Then S is a γpe0-set if and only if for every v ∈ V (G) \ S, NG(v) ∩ S = {u} for some u ∈ S and for every v ∈ V (G) \ S, there exists u ∈ S with uv ∈ E(G) such that |deg(u)− deg(v)| ≤ 1 Proposition 2. If S is γ-set, or a γpe-set, or a γ0-set of G with |S| = 1, then S is γpe0-set of G. In particular, γ(G) = γpe(G) = γ0(G) = 1 if and only if γpe0(G) = 1. 3. PEID in Some Graphs In this section, we present the results for Equitable Isolate dominations in graphs. The minimality of a perfect equitable isolate dominating set S follows from the paper of [8], M. Caay, A. Hernandez / Eur. J. Pure Appl. Math, 17 (2) (2024), 969-978 974 [4] and [16], with the additional property that it acquires at least one vertex in S that is not adjacent to the other vertices in S [13]. Proposition 3. Given a path Pn and cycle Cn, n ≥ 6, γpe0(Pn) = γpe0(Cn) = ⌈n 3 ⌉ . Proof. The proof follows from Theorem 4. Corollary 1. There does not exist γpe0-set of C4 and C5. Theorem 7. Let G be any connected graph of order n ≥ 2. If γpe0(G) = 1, then ∆(G) = n− 1. Conversely, if ∆(G) = n− 1 and δ(G) ≥ n− 2, then γpe0(G) = 1. Proof. Suppose that γpe0(G) = 1. Let S = {u} be the γpe0 − set of G. If G is trivial, then we are done. Assume G is nontrivial. Then every vertex v ∈ V (G) \ S is adjacent to u ∈ S. Hence, deg(u) = n − 1. This means that ∆(G) = n − 1. Conversely, suppose ∆(G) = n − 1 and δ(G) ≥ n − 2. Then the vertices of G are either of degree n − 1 or n − 2. Without loss of generality, take a vertex of degree n − 1, say u ∈ V (G). Then u dominates all other vertices of G. Since other vertices of G are either of degree n − 1 or n− 2, it follows that for every v ∈ V (G){u}, | deg(v)− deg(u)| ≤ 1. Take S = {u} and so it follows that {u} is γpe0-set of G. This proves the claim. Corollary 2. Given a complete graph Kn, n ≥ 3, γpe0(Kn) = 1. Proposition 4. Let Gn,m is a complete bipartite graph. If |n−m| ≤ 1. Then there exists a γpe-set of G but there does not exist γpe0-set of G, or there does exists γe0-set of G, but there does not exist γpe0-set of G. Moreover, γpe(Gn,m) = 2 or γpe0(Gn,m) = min{n,m}. Proof. Let P1 and P2 be the vertex partitions of a complete bipartite graph G such that |P1| = n and |P2| = m. Let u1 ∈ P1, i = 1, · · · , n and vj ∈ P2, j = 1, · · · ,m. Note that ui dominates vj for all ui ∈ P1 and for all vj ∈ P2 with i = 1, · · · , n and j = 1, · · · ,m. Since |n−m| ≤ 1, |deg(u1)− deg(vj)| ≤ 1, for all ui ∈ P1 and vj ∈ P2. Note that uiuj /∈ E(G) for all i ̸= j and ui and uj are in P1. Thus, P1 is a γe0-set of G. However, every vj ∈ P2, j = 1, · · · ,m is dominated by all vertices of P1, it follows that P1 is not γp-set and so it is not a γpe0-set of G. Moreover, γe0(G) = min{n,m}. Now suppose we pick one ui ∈ P1 and one vj ∈ P2. Then every vertices in P2 is dominated by ui for some i and every vertices in P1 is dominated by vj for some j. Thus, {ui, vj} is a γpe-set for some i and j. However, uivj ∈ E(G) and so {ui, vj} is not a γ0-set. Hence, {ui, vj} is not a γpe0-set. Moreover, γpe(G) = 2. This proves the claim. Theorem 8. Let G = GP1,··· ,Pk be a k-partite graph. Then G has a γpe0-set if there exists a vertex partition Pj with |Pj | = 1 and |Pk| ≤ 2, for all i ̸= j. Moreover, γpe0(G) = 1. M. Caay, A. Hernandez / Eur. J. Pure Appl. Math, 17 (2) (2024), 969-978 975 Proof. Without loss of generality, let P1 be such partition with |P1| = 1. Then for partitions Pi with i ̸= 1, either |Pi| = 1 or |Pi| = 2. Let u ∈ P1. Then deg(u) ≤ 2k. Also, deg(v) ≤ 2k+1 for all v ̸= u. Thus, |deg(v)− deg(u)| ≤ 1. This means that {u} is a γ-set. Since u dominates all vertices of G, {u} is also a γp-set and so it is a γpe-set. By Theorem 2, {u} is a γ0-set of G. Therefore, {u} is a γpe0-set of G. Consequently, γpe0(G) = 1. Theorem 9. There does not exist γpe0-set of GP1,···Pk for any non-trivial partition Pi, i = 1, · · · , k. Proof. Suppose on the contrary that there exists a γpe0-set S of G = GP1,··· ,Pk , and let u ∈ S such that u ∈ Pk for some kth vertex-partition of G. Then u dominates vi for all vi /∈ Pk. Since Pk is non-trivial, there exists uj ∈ Pk with uj ̸= u such that u does not dominate uj . Thus, either uj ∈ S or uj /∈ S. If uj ∈ S, then uj must dominate vj for all vj /∈ Pk. This is a contradiction to being γpe0-set since vj is dominated by u, for all vj /∈ Pk. If uj /∈ S, then there must be vs /∈ Pk such that ujvs ∈ E(G). But vs is adjacent to some vt /∈ Pk which are also dominated by u. This is also a contradiction to being γpe0-set. Therefore, there does not exist γpe0-set of GP1,···Pk for any non-trivial partition Pi, i = 1, · · · , k. 4. PEID in the Join of Graphs The following proposition is an obvious result. Proposition 5. There does not exist a γpe0-set of the following graphs below: i. Wheel graph, Wn = K1 + Cn−1, n ≥ 6 ii. Star graph, Sn = K1 +Kn−1, n ≥ 4 iii. Fan graph, Fn = K1 + Pn−1, n ≥ 5 iv. Friendship graph, Frn = K1 + nP2, n ≥ 2 v. Windmill graph Wn m = K1 + Cn−1, n ≥ 2,m ≥ 3 Theorem 10. Let G and H be any graphs of order n and m, respectively, with γpe(G) = 1 or γpe(H) = 1. Then γpe0(G + H) = 1 if and only if either S1 = {u} is a γpe0-set of G and deg(v) ≥ m− 2 for all v ∈ V (H), or S2 = {x} is a γpe0-set of H and deg(y) ≥ n− 2 for all y ∈ V (G). Proof. Let γpe0(G + H) = 1. By Theorem 7, ∆(G+H) = (n+m)− 1. Suppose S = {u} ⊆ V (G) be a γpe0-set of G +H. This means that for every v ∈ V (G +H) with v ̸= u, we have 1 ≥ |deg(u)− deg(v)| M. Caay, A. Hernandez / Eur. J. Pure Appl. Math, 17 (2) (2024), 969-978 976 ≥ |(n+m)− 1− deg(v)| ≥ |n+m− 1| − | deg(v)|. Thus, deg(v) ≥ (m+ n)− 1− 1 = (m+ n)− 2. This means that deg(v) ≥ m−2 on H for all v ∈ V (H). Similarly, if S = {x} ⊆ V (H) is a γpe0-set of G +H, then deg(y) ≥ n − 2 on G for all y ∈ V (G). Conversely, suppose S1 = {u} is a γpe0-set of G and deg(v) ≥ m− 2 for all v ∈ V (H). Since S1 = {u} is a γpe0-set of G, by Theorem 7, ∆(G) = n − 1. This means that deg(u) = n − 1 + m in G + H. Also, deg(v) ≥ m − 2 for every v ∈ V (H) implies that deg(v) ≥ m− 2 + n in G+H. Thus, |deg(u)− deg(v)| ≤ |(n− 1 +m)− (m− 2− n)| = 1. Hence, S = {u} is a γpe0-set of G + H implying γpe0(G + H) = 1. The same argument with the other case. The next corollary is a very obvious result as a consequence of Theorem 10. Corollary 3. Let G and H be any graphs of degree n and m, respectively. If ∆(G) = n−1 and δ(G) ≥ n− 2, and deg(u) ≥ m− 2 for all u ∈ V (H). Then γpe0(G+H) = 1. Theorem 11. Let S1 and S2 be the minimal nontrivial γpe0-sets of G and H, respectively. That is, |S1| ≠ 1 and |S2| ̸= 1. Then S1 ∪ S2 is a not a γpe0-set of G+H but a γe-set of G+H. Proof. Let S1 and S2 be the minimal nontrivial γpe0-sets of G and H, respectively. Then for every u ∈ V (G) \ S1, there exists exactly v ∈ S1 such that uv ∈ E(G) and |deg(u)− deg(v)| ≤ 1, and there exists vi ∈ S1 such that viv /∈ E(G) for some v ∈ S1. Similarly, for every x ∈ V (H) \ S2, there exists exactly y ∈ S2 such that xy ∈ E(G) and |deg(x)− deg(y)| ≤ 1, and there exists yj ∈ S2 such that yjy /∈ E(G) for some y ∈ S1. Then S1 ∪ S2 := {vi, yj , vi ∈ S1, yj ∈ S2, for some i, j} ⊆ V (G + H). Thus, for all w ∈ V (G+H) \ (S1 ∪ S2), there exists z ∈ S1 ∪ S2 such that wz ∈ E(G+H) and |deg(w)− deg(z)| ≤ 1. Hence, S1 ∪ S2 is a γe-set of G+H. Now if vi ∈ S1 is an isolated vertex of S1, then viuj ∈ E(G +H) for all uj ∈ S2, and vkuj ∈ E(G +H), for all vk ∈ S1 with vi ̸= vk. This means that vi is no longer isolated. Since vi is arbitrary, this holds for all isolated dominating vertices. Hence, S1 ∪ S2 is not γe0-set of G+H. Moreover, for every u ∈ V (G) \ S1, there exists exactly one v ∈ S1 such that uv1 ∈ E(G) However, u is adjacent to vertices of H. This means that u is adjacent to some wj ∈ S2. Hence, S1 ∪ S2 is not γp0-set of G+H. This proves the claim. Remark 1. S1 ∪ S2 is a γe-set of G+H of Theorem 11 is not necessarily minimal. REFERENCES 977 5. PEID in the Corona of Graphs Theorem 12. There does not exist a γpe0-set of G ◦H for any non-trivial graphs G and H. Proof. Suppose on the contrary that there exists a γpe0-set S of G ◦H. Then either S ⊆ V (G) or S ⊆ V (H) or S ⊆ V (G + H). Suppose S ⊆ V (G) and let u ∈ S. Then u ∈ V (G). This means that the degree of u in G + H is equal to the degree of u in G plus the cardinality of H. Since G is nontrivial, deg(u) ≥ 1 in G. Thus, deg(u) ≥ 1 +m in G +H. But every vertex v ∈ V (H) has at most m − 1 degree. Hence, it follows that |deg(u)− deg(v)| ≥ 1, a contradiction. Similarly, assume S ⊆ V (H) and let w ∈ S. Then w ∈ V (H). This means that the degree of w in G +H is equal to the degree of w in H plus the cardinality of G. Since H is nontrivial, deg(w) ≥ 1 in H. Thus, deg(w) ≥ 1 + n in G + H. But every vertex z ∈ V (G) has at most n − 1 degree. Hence, it follows that |deg(w)− deg(z)| ≥ 1, a contradiction. Lastly, suppose S ⊆ V (G+H). Then there exist u1, u2 ∈ S such that u1 ∈ V (G) and u2 ∈ V (H). Since every vertices in H are adjacent to u1, this means that there are vertices in H adjacent to both u1 and u2, a contradiction. Hence, all of the cases lead to contradiction. Therefore, there does not exist a γpe0-set of G ◦H for any non-trivial graphs G and H. Theorem 13. Let G and H be any graphs having γpe0-sets. Then G ◦ H does not have γpe0-set, but G+H has a γe-set. Proof. Suppose u ∈ S1 ⊆ V (G). Then uvi ∈ E(G ◦H) for all vi ∈ V (H) \ S2, where S2 is a γpe0-set of H. But vivj ∈ E(G ◦ H) for some vj ∈ S2, i ̸= j. This means that S1 ∪ S2 is no longer γp-set. Moreover, by Theorem 12, S1 and S2 are no longer γpe0-sets. Consequently, S1, S2 and S1 ∪ S2 are no longer γ0-sets since the elements are adjacents. Lastly, since every vertices in V (G ◦H) \ (S1 ∪ S2), it follows that S1 ∪ S2 is γe-set. This proves the claim. Remark 2. The γe-set of G ◦H of Theorem 13 is not necessarily minimal. Proposition 6. Let G be a trivial graph. Then γpe0(G ◦Kn) = γpe0(Kn ◦G) = 1. Acknowledgements Thank you, Adamson University Center for Research and Development for the funding of this research works. References [1] C Armada and J Hamja. Perfect Isolate Domination in Graphs. European Journal of Pure and Applied Mathematics, 16(22):1362–1341, 2023. [2] C Berge. The Theory of Graphs and Its Applications. Greenwood Press, 1982. REFERENCES 978 [3] M Caay. Equitable Rings Domination in Graphs. Journal of Algebraic Systems, ((in preparation), 2024. [4] M Caay and E Arugay. 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