EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 14, No. 4, 2021, 1226-1236 ISSN 1307-5543 – ejpam.com Published by New York Business Global On Connected Co-Independent Hop Domination in Graphs Sandra A. Nanding1,∗, Helen M. Rara2 1 Department of Mathematics and Statistics, College of Science and Mathematics, Mindanao State University-Iligan Institute of Technology, 9200 Iligan City, Philippines 2 Department of Mathematics and Statistics, College of Science and Mathematics, Center of Graph Theory, Algebra, and Analysis-Premier Research Institute of Science and Mathematics, Mindanao State University-Iligan Institute of Technology, 9200 Iligan City, Philippines Abstract. Let G be a connected graph. A subset S of V (G) is a connected co-independent hop dominating set in G if the subgraph induced by S is connected and V (G)\S is an independent set where for each v ∈ V (G)\S, there exists a vertex u ∈ S such that dG(u, v) = 2. The smallest cardinality of such an S is called the connected co-independent hop domination number of G. This paper presents the characterizations of the connected co-independent hop dominating sets in the join, corona and lexicographic product of two graphs. It also discusses the corresponding connected co-independent hop domination numbers of the aforementioned graphs. 2020 Mathematics Subject Classifications: 05C69 Key Words and Phrases: Connected co-independent hop dominating set, connected co-independent hop domination number, strictly co-independent set, strictly co-independent number, join, corona, lexicographic product 1. Introduction In the late 1950’s and 1960’s, the study on domination in graphs was developed, be- ginning with C. Berge [1] in 1958. There are now many studies involving domination and its variations. One of its variation is the connected co-independent domination num- ber of graphs introduced by B. Gayathri and S. Kaspar in 2010 [3]. Also, connected co-independent domination number in graphs were studied in [2, 6, 12]. Years later, new domination parameter called hop domination in graph is introduced by Natarajan and Ayyaswamy [8]. Hop domination in graphs were also studied in [7, 9–11, 13]. In this study, the researcher defines and establishes a new concept of hop domination called a connected co-independent hop domination and generates some characterizations ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v14i4.4069 Email addresses: sandra.nanding@g.msuiit.edu.ph (S. Nanding), helen.rara@g.msuiit.edu.ph (H. Rara) http://www.ejpam.com 1226 © 2021 EJPAM All rights reserved. S. Nanding, H. Rara / Eur. J. Pure Appl. Math, 14 (4) (2021), 1226-1236 1227 of connected co-independent hop domination in graphs. Connected co-independent hop domination in graphs can have real world applications. For an application, in [5], Des- ormeaux, Haynes, and Henning inspired their research on these concepts through social networking applications. They considered a factory with a large number of employees and needed to implement a quality assurance checking system of their workers. The factory manager decides to designate an internal committee to do this. In other words, the man- ager will select some workers to form a quality assurance team to inspect the work of their co-workers. The manager wants to keep this team as small as possible to minimize costs (extra costs for inspectors) and protect privacy (keep the inspectors’ identity confidential). To avoid bias, an inspector should neither be close friends nor enemies with any of the workers he/she is responsible for inspecting. To model this situation, a social network graph can be constructed in which each worker is represented by a vertex and an edge between two workers represents possible bias, that is, whether the two workers are close friends or enemies. Ideally, an inspector should not be adjacent to any worker who is being inspected. In connected co-independent hop domination, every worker will be inspected by the nearest non-biased inspector. That is, an inspector who is a close friend (or an enemy) of a close friend (or enemy) of a worker. This is to save time and effort of locating a particular worker. Also, the inspectors should be acquainted with each other and all non- inspector workers are neither friends nor enemies, that is, they are not adjacent or there is no edge between them. The connected co-independent hop domination number will give the minimum number of inspectors needed. In this study, we only consider graphs that are finite, simple, undirected and connected. Readers are referred to [4] for elementary Graph Theoretic concepts. An independent set S in a graph G is a subset of the vertex-set of G such that no two vertices in S are adjacent in G. The cardinality of a maximum independent set is called the independence number of G and is denoted by β(G). An independent set S ⊆ V (G) with |S| = β(G) is called a β-set of G. A dominating set D ⊆ V (G) is called a connected co-independent dominating set of G if D is a connected dominating set of G and V (G) \ D is an independent set. The cardinality of such a minimum set D is called a connected co-independent domination number of G denoted by γc,coi(G). A connected co-independent dominating set D with |D| = γc,coi(G) is called a γc,coi-set of G. Let G be a connected graph. A set S ⊆ V (G) is a hop dominating set of G if for every v ∈ V (G)\S, there exists u ∈ S such that dG(u, v) = 2. The minimum cardinality of a hop dominating set of G, denoted by γh(G), is called the hop domination number of G. Any hop dominating set with cardinality equal to γh(G) is called a γh-set. A vertex v in G is a hop neighbor of vertex u in G if dG(u, v) = 2. The set NG(u, 2) = {v ∈ V (G) : dG(v, u) = 2} is called the open hop neighborhood of u. The closed hop neighborhood of u in G is given by NG[u, 2] = NG(u, 2) ∪ {u}. The open hop neighborhood of X ⊆ V (G) is the set NG(X, 2) = ⋃ u∈X NG(u, 2). The closed hop neighborhood of X in G is the set NG[X, 2] = NG(X, 2) ∪ X. Let G be a graph. A subset S of V (G) is a strictly co-independent set of G if V (G)\S is an independent set and NG(v)∩S ̸= S for all v ∈ V (G)\S. The minimum cardinality of a strictly co-independent S. Nanding, H. Rara / Eur. J. Pure Appl. Math, 14 (4) (2021), 1226-1236 1228 set in G, denoted by sci(G) is called the strictly co-independent number of G. A strictly co-independent set S with |S| = sci(G) is called an sci-set of G. Let G be a connected graph. A hop dominating set S ⊆ V (G) is a connected co-independent hop dominating set of G if ⟨S⟩ is connected and V (G)\S is an independent set. The minimum cardinality of a connected co-independent hop dominating set of G, denoted by γch,coi(G), is called the connected co-independent hop domination number of G. A connected co-independent hop dominating set S with |S| = γch,coi(G) is called a γch,coi-set of G. 2. Preliminary Results Remark 1. Every connected co-independent hop dominating set in a connected graph G is hop dominating. Hence, γh(G) ≤ γch,coi(G). Remark 2. Let G be a connected graph of order n. Then 1 ≤ γch,coi(G) ≤ |V (G)|. Moreover, γch,coi(G) = 1 if and only if G = K1. Example 1. The equations below give the connected co-independent hop domination number of the path Pn and cycle Cn. γch,coi(Pn) =  1 if n = 1 2 if n = 2, 3 n− 2 if n ≥ 4 γch,coi(Cn) = { 3 if n = 3 n− 1 if n ≥ 4 Remark 3. If G is a complete graph, then γch,coi(G) = n. Theorem 1. Let G be a connected graph of order n ≥ 3. Then γch,coi(G) = 2 if and only if there exist adjacent vertices x and y of G such that for each z ∈ V (G)\{x, y}, NG(z) = {x} or NG(z) = {y} and z /∈ NG(x) ∩NG(y). Proof: Suppose γch,coi(G) = 2. Let S = {x, y} be a γch,coi-set of G. Since S is connected, xy ∈ E(G). Let z ∈ V (G)\{x, y}. Then z /∈ S. Since S is a hop dominating set of G, z ∈ NG(x, 2)∪NG(y, 2). Suppose z ∈ NG(x, 2). Then there exist w ∈ NG(z)∩NG(x). Since V (G)\S is an independent set, w ∈ S. Thus, w = y, that is, NG(z) = {y}. Similarly, if z ∈ NG(y, 2), then NG(z) = {x}. Since z ∈ NG(x, 2) ∪NG(y, 2), z /∈ NG(x) ∩NG(y). Conversely, suppose that there exist adjacent vertices x and y of G satisfying the given condition. Let S = {x, y}. Since xy ∈ E(G), S is connected. Let z ∈ V (G)\S. If NG(z) = {x}, then since xy ∈ E(G), dG(y, z) = 2. While on the other hand, if NG(z) = {y}, then dG(x, z) = 2. Thus, S is a hop dominating set of G. Since NG(z) = {x} or NG(z) = {y}, V (G)\S is an independent set. Therefore, S is a connected co-independent hop dominating set of G. So, γch,coi(G) ≤ |S| = 2. But G is nontrivial. Hence, γch,coi(G) ̸= 1 and so γch,coi(G) = 2. S. Nanding, H. Rara / Eur. J. Pure Appl. Math, 14 (4) (2021), 1226-1236 1229 Theorem 2. Let G be a connected graph of order n ≥ 2. Then γch,coi(G) = n if and only if G is complete. Proof: Suppose γch,coi(G) = n. Suppose that G is not complete. Then there exist distinct vertices u, v ∈ V (G) such that dG(u, v) = 2. Let S = V (G)\{u}. Then S is a connected co-independent hop dominating set of G. Therefore, γch,coi(G) ≤ |S| = n− 1, a contradiction. Thus, G is a complete graph. Conversely, by Remark 3, γch,coi(Kn) = n. 3. On Connected Co-Independent Hop Domination in the Join of Graphs The join of two graphs G and H is the graph G + H with vertex set V (G + H) = V (G) • ∪ V (H) and edge set E(G + H) = E(G) • ∪ E(H) ∪ {uv : u ∈ V (G), v ∈ V (H)}. Theorem 3. Let G and H be any two graphs. Then S ⊆ V (G + H) is a connected co-independent hop dominating set of G+H if and only if S = SG ∪SH where one of the following holds: (i) SG = V (G) and SH is a strictly co-independent set of H. (ii) SH = V (H) and SG is a strictly co-independent set of G. Proof: Suppose S is a connected co-independent hop dominating set of G + H. Let SG = S ∩ V (G) and SH = S ∩ V (H). Then S = SG ∪ SH . Since S is a hop dominating set of G+H, SG ̸= ∅ and SH ̸= ∅. Since V (G+H)\S is an independent set, SG = V (G) or SH = V (H). Suppose SG = V (G). Then V (H)\SH = V (G+H)\S is an independent set. Let v ∈ V (H)\SH . Then v ∈ V (G +H)\S. Since S is a hop dominating set, there exists w ∈ S such that dG+H(v, w) = 2. Hence, w ∈ SH\NH(v). Thus, NH(v)∩SH ̸= SH , showing that SH is a strictly co-independent set of H. Thus, (i) holds. Similarly, if SH = V (H), then SG is a strictly co-independent set of G and (ii) holds. For the converse, suppose S = SG ∪SH where SG and SH satisfy the given conditions. Let v ∈ V (G+H)\S. Consider the following cases. Case 1. SG = V (G). Then v ∈ V (H)\SH . By (i), there exists w ∈ SH\NH(v). Hence, w ∈ S and dG+H(v, w) = 2. Case 2. SH = V (H) Then v ∈ V (G)\SG. By (ii), there exists u ∈ SG\NG(v). Thus, u ∈ S and dG+H(u, v) = 2. Therefore, in either case, S is a hop dominating set of G+H. Since V (G+H)\S = V (H)\SH if (i) holds or V (G+H)\S = V (G)\SG if (ii) holds, V (G+H)\S is an independent set. It is clear from the definition of the join of G and H that ⟨S⟩ is connected. Therefore, S is a connected co-independent hop dominating set of G+H. S. Nanding, H. Rara / Eur. J. Pure Appl. Math, 14 (4) (2021), 1226-1236 1230 Corollary 1. Let G and H be any two graphs where |V (G)| = n and |V (H)| = m. Then γch,coi(G+H) = min{n+ sci(H),m+ sci(G)}. Proof: Let S be a γch,coi-set of G + H. Then S is a connected co-independent hop dominating set of G +H. Hence, S = A ∪ B whre A ⊆ V (G) and B ⊆ V (H) satisfying conditions (i) or (ii) of Theorem 3. Thus, γch,coi(G+H) = |S| = |A|+ |B|. By condition (i), |S| = |V (G)|+ |B| ≥ n+ sci(H). By condition (ii), |S| = |V (H)|+ |A| ≥ m+ sci(G). Hence, γch,coi(G+H) = |S| ≥ min{n+ sci(H),m+ sci(G)}. Next, let X and Y be the minimum strictly co-independent sets of G and H, respec- tively. Then by Theorem 3, S = V (G)∪Y or S = V (H)∪X is a connected co-independent hop dominating set of G+H. Thus, γch,coi(G+H) ≤ |S| = |V (G)|+ |Y | = n+ sci(H) or γch,coi(G+H) ≤ |S| = |V (H)|+ |X| = m+ sci(G) It follows that, γch,coi(G+H) ≤ min{n+ sci(H),m+ sci(G)}. Therefore, γch,coi(G+H) = min{n+ sci(H),m+ sci(G)}. 4. On Connected Co-Independent Hop Domination in the Corona of Graphs The corona of two graphs G and H, denoted by G◦H, is the graph obtained by taking one copy of G of order n and n copies of H, and then joining every vertex of the ith copy of H to the ith vertex of G. For v ∈ V (G), denote by Hv the copy of H whose vertices are attached one by one to the vertex v. Subsequently, denote by v+Hv the subgraph of the corona G ◦H corresponding to the join ⟨{v}⟩+Hv, v ∈ V (G). Theorem 4. Let G be a nontrivial connected graph and H be any graph. A set S ⊆ V (G ◦ H) is a connected co-independent hop dominating set of G ◦ H if and only if S = V (G) ∪ ( ⋃ v∈V (G) Sv), where Sv ⊆ V (Hv) and V (Hv)\Sv is an independent subset of V (Hv) for each v ∈ V (G). Proof: Suppose S is a connected co-independent hop dominating set of G ◦H and let Sv = S∩V (Hv) for each v ∈ V (G). Then Sv ⊆ V (Hv). Since ⟨S⟩ is connected, S = V (G)∪ ( ⋃ v∈V (G) Sv). Since V (G ◦ H)\S is independent and V (G ◦ H)\S = ⋃ v∈V (G) (V (Hv)\Sv), S. Nanding, H. Rara / Eur. J. Pure Appl. Math, 14 (4) (2021), 1226-1236 1231 V (Hv)\Sv is an independent subset of V (Hv), for each v ∈ V (G). For the converse, suppose that S = V (G) ∪ ( ⋃ v∈V (G) Sv) where Sv ⊆ V (Hv) and V (Hv)\Sv is an independent set. Clearly, ⟨S⟩ is connected. Let w ∈ V (G ◦H)\S. Then w ∈ V (Hv)\Sv for some v ∈ V (G). Since G is nontrivial connected graph, there exists x ∈ V (G) such that vx ∈ E(G). Thus, dG◦H(w, x) = 2. This implies that S is a hop dominating set of G ◦H. Since V (G ◦H)\S = ⋃ v∈V (G) (V (Hv)\Sv) and V (Hv)\Sv is an in- dependent set for each v ∈ V (G), V (G◦H)\S is independent. Therefore, S is a connected co-independent hop dominating set of G ◦H. Corollary 2. Let G be a nontrivial connected graph of order n and H be any graph of order m. Then γch,coi(G ◦H) = n(1 +m− β(H)). Proof: Let C be a γch,coi-set of G ◦ H. Then C is a connected co-independent hop dominating set of G ◦H. By Theorem 4, C = V (G) ⋃ ( ⋃ v∈V (G) Sv) where V (Hv)\Sv is an independent set of Hv for every v ∈ V (G). Then γch,coi(G ◦H) = |C| = |V (G)|+ | ⋃ v∈V (G) Sv| = |V (G)|+ ∑ v∈V (G) |Sv| = |V (G)|+ ∑ v∈V (G) (|V (Hv)| − |V (Hv)\Sv|) ≥ |V (G)|+ |V (G)|(|V (Hv)| − β(H)) = n+ n(m− β(H)) = n(1 +m− β(H)). Therefore, γch,coi(G ◦H) ≥ n(1 +m− β(H)). Let D be a maximum independent set of H. For each v, let Dv ⊆ V (Hv) such that ⟨Dv⟩ ∼= ⟨D⟩. Let Sv = V (Hv)\Dv. Then C = V (G) ⋃ ( ⋃ v∈V (G) Sv) is a connected co- independent hop dominating set of G ◦H by Theorem 4. Thus, γch,coi(G ◦H) ≤ |C| = |V (G) ∪ ( ⋃ v∈V (G) Sv)| = |V (G)|+ ∑ v∈V (G) |Sv| = |V (G)|+ |V (G)|(|V (Hv)| − |Dv|) = |V (G)|+ |V (G)|(|V (Hv)| − β(H)) = n+ n(m− β(H)) = n(1 +m− β(H)). S. Nanding, H. Rara / Eur. J. Pure Appl. Math, 14 (4) (2021), 1226-1236 1232 Therefore, γch,coi(G ◦H) ≤ n(1 +m− β(H)). Consequently, γch,coi(G ◦H) = n(1 +m− β(H)). 5. On Connected Co-Independent Hop Domination in the Lexicographic Product of Graphs The lexicographic product of two graphs G and H, denoted by G[H], is the graph with vertex-set V (G[H]) = V (G) × V (H) such that (u1, u2)(v1, v2) ∈ E(G[H]) if either u1v1 ∈ E(G) or u1 = v1 and u2v2 ∈ E(H). Theorem 5. Let G and H be nontrivial connected graphs with |V (G)| = n. A subset C = ⋃ x∈S ({x} × Tx) where S ⊆ V (G) and Tx ⊆ V (H) of V (G[H]) is a connected co- independent hop dominating set if and only if (i) S = V (G). (ii) For every x ∈ V (G) such that Tx ̸= V (H), V (H)\Tx is an independent set and Ty = V (H) for every y ∈ NG(x) where Tx is a hop dominating set of H if degG(x) = n−1. Proof: Suppose C is a connected co-independent hop dominating set of G[H] and S ̸= V (G). Then, a vertex v ∈ V (G)\S exists. Thus, (v, z) ∈ V (G[H])\C for all z ∈ V (H). Since H is a nontrivial connected graph, an edge pq ∈ E(H) exists. Hence, (v, p), (v, q) ∈ V (G[H])\C and (v, p)(v, q) ∈ E(G[H]). This contradicts the independence of V (G[H])\C. It follows that S = V (G) and (i) holds. Now, let x ∈ V (G) such that Tx ̸= V (H). We claim that V (H)\Tx is an independent set. Let u,w ∈ V (H)\Tx where u ̸= w. Then (x, u), (x,w) ∈ V (G[H])\C. Since V (G[H])\C is independent, (x, u)(x,w) /∈ E(G[H]). Thus, uw /∈ E(H). Hence, V (H)\Tx is an independent set. Now, we show that Ty = V (H) for every y ∈ NG(x). Suppose Ty ̸= V (H). Then there exists p ∈ V (H)\Ty. Thus, (y, p) ∈ V (G[H])\C. Since y ∈ NG(x), (y, p)(x, q) ∈ E(G[H]) for all q ∈ V (H). This contradicts the independence of V (G[H])\C. Hence, Ty = V (H). Lastly, suppose degG(x) = n − 1. Then xa ∈ E(G) for all a ∈ V (G)\{x}. Since V (G) ̸= Tx, a vertex b ∈ V (H)\Tx exists. Thus, (x, b) ∈ V (G[H])\C. Since C is a hop dominating set and (x, b)(a, d) ∈ E(G[H]) for all a ∈ V (G)\{x} and d ∈ Ta, there exists z ∈ Tx such that dG[H]((x, b), (x, z)) = 2. Hence, dH(b, z) = 2, showing that z ∈ Tx\NH(b). Hence, NH(b) ∩ Tx ̸= Tx implying that Tx is strictly co-independent set of H. Thus, (ii) holds. Conversely, suppose C = ⋃ x∈S ({x} × Tx) satisfies conditions (i) and (ii). First, we claim that C is connected in G[H]. Let (x, a) and (y, b) be two distinct vertices in C, (x, a)(y, b) /∈ E(G[H]). Consider the following cases. Case 1. x=y Since (x, a) ̸= (y, b) and (x, a)(y, b) /∈ E(G[H]), a ̸= b and ab /∈ E(H). Since G is a nontrivial connected graph and S = V (G) by (i), there exists z ∈ S ∩ NG(x). Thus, (z, w) ∈ C for some w ∈ V (H). It follows that [(x, a), (z, w), (y, b)] is a path in C. Case 2. x ̸= y Since G is a nontrivial connected graph, there exists an x-y path [v1, v2, ..., vn] where x = v1 and y = vn, n > 2. By (i) vi ∈ S for all i ∈ {1, 2, ..., n}. Let ui ∈ Tvi , u1 = a and S. Nanding, H. Rara / Eur. J. Pure Appl. Math, 14 (4) (2021), 1226-1236 1233 un = b. Then [(x, a), (v2, u2), (v3, u3), ..., (y, b)] is an (x, a)-(y, b) path in C. Therefore in either case, C is connected. Now, let (u, v), (w, p) ∈ V (G[H])\C where (u, v) ̸= (w, p). Consider the following cases. Case 1. u=w Since (u, v), (w, p) /∈ C, v, p /∈ Tu = Tw and v ̸= p. Hence, v, p ∈ V (H)\Tu. Since, V (H)\Tu is independent by (ii), vp /∈ E(H). Thus, (u, v)(w, p) /∈ E(G[H]). Case 2. u ̸= w Since v /∈ Tu and p /∈ Tw, Tu ̸= V (H) and Tw ̸= V (H). By (ii), u /∈ NG(w). Thus, (u, v)(w, p) /∈ E(G[H]). Therefore, in any case V (G[H])\C is an independent set. Finally, we show that C is a hop dominating set. Let (x, y) ∈ V (G[H])\C. Then y /∈ Tx, that is, V (H) ̸= Tx. Suppose degG(x) = n− 1. By (ii), Tx is a strictly co-independent set of H. Since y /∈ Tx, NH(y) ∩ Tx ̸= Tx. This implies that there exists a ∈ Tx such that dH(a, y) = 2. Hence, (x, a) ∈ C and dG[H]((x, y), (x, a)) = 2. Suppose degG(x) < n − 1. Then a vertex z ∈ V (G)\NG(x) exists. Choose z such that dG(x, z) = 2. Since S = V (G) by (i), there exists b ∈ Tz, that is, (z, b) ∈ C. It follows that dG[H]((x, y), (z, b)) = 2. Therefore, C is a hop dominating set of G[H]. Accordingly, C is a connected co-independent hop dominating set of G[H]. Corollary 3. Let G be any connected noncomplete graph of order m and H be any nontrivial connected graph of order n. Then γch,coi(G[H]) = m(n− β(H)) + r(G)β(H), where r(G) = min{|D| : V (G)\D is an independent set} and β(H) is an independence number of H. Proof: Let r(G) = min{|D| : V (G)\D is an independent set}. Let D0 ⊆ V (G) such that V (G)\D0 is an independent set and |D0| = r(G). Let T be a β-set of H. Let Tx = V (H)\T for each x ∈ V (G)\D0 and let Ty = V (H) for each y ∈ D0. Then C = ⋃ x∈V (G) [{x} × Tx] = ⋃ y∈D0 ({y} × Ty) ∪  ⋃ x∈V (G)\D0 [{x} × Tx]  is a connected co-independent hop dominating set of G[H], by Theorem 5. Hence, γch,coi(G[H]) ≤ |C| = nr(G) + (m− r(G))(n− β(H)) = nr(G) +mn−mβ(H)− nr(G) + r(G)β(H) = mn−mβ(H) + r(G)β(H) = m(n− β(H)) + r(G)β(H) γc,coi(G[H]) ≤ m(n− β(H)) + r(G)β(H) . S. Nanding, H. Rara / Eur. J. Pure Appl. Math, 14 (4) (2021), 1226-1236 1234 Let C0 = ⋃ x∈V (G) [{x}×Rx] be a γch,coi-set of G[H]. Let D = {x ∈ V (G) : Rx = V (H)}. Then V (G)\D is an independent set of G. Then C0 = (⋃ x∈D Rx ) ∪  ⋃ x∈V (G)\D Tx  . Moreover, γch,coi(G[H]) = |C0| = n|D|+ ∑ x∈V (G)\D |Tx|. Since V (H)\Tx is an independent set of H for each x ∈ V (G)\D, it follows that |V (H)\Tx| ≤ β(H) for each x ∈ V (G)\D. Thus, |V (H)\Tx| = |V (H)| − |Tx| ≤ β(H). Hence |Tx| ≥ n− β(H). Therefore, γch,coi(G[H]) = |C0| = n|D|+ |V (G)\D| |Tx| = n|D|+ (|V (G)− |D|) |Tx| ≥ n|D|+ (m− |D|)(n− β(H)) = n|D|+mn−mβ(H)− n|D|+ |D|β(H) = m(n− β(H)) + |D|β(H) = m(n− β(H)) + r(G)β(H) γc,coi(G[H]) ≥ m(n− β(H)) + r(G)β(H), Therefore, γc,coi(G[H]) = m(n− β(H)) + r(G)β(H). Corollary 4. Let H be any nontrivial connected graph of orderm. Then γch,coi(Kn[H]) = m(n− 1) + sci(H). Proof: Let T be an sci-set of H. Let v ∈ V (Kn) and Tv = T . By Theorem 5, C = ⋃ y∈V (Kn)\{v} ({y}×Ty)∪ ({v}×Tv) is a connected co-independent hop dominating set of Kn[H]. Since y ∈ NKn(v) for each y ∈ V (Kn)\{v}, Ty = V (H). Thus, γch,coi(Kn[H]) ≤ |C| = (n− 1)|Ty|+ |Tv| = (n− 1)m+ |T | = (n− 1)m+ sci(H). Let Co = ⋃ x∈V (Kn) ({x} × Rx) be a γch,coi-set of Kn[H]. Since degKn(x) = n − 1 for each x ∈ V (Kn), by Theorem 5, Ry = T where T is a strictly co-independent set of H for a unique y ∈ V (Kn) and Rx = V (H) for all x ∈ V (Kn)\{y}. Hence, Co = ({y} ×Ry) ∪ ( ⋃ x∈V (Kn)\{y} ({x} ×Rx) and γch,coi(Kn[H]) = |Co| REFERENCES 1235 = |Ry|+ (n− 1)|Rx| = |T |+ (n− 1)|V (H)| ≥ sci(H) + (n− 1)m. Therefore, γch,coi(Kn[H]) = m(n− 1) + sci(H). Acknowledgements This research is funded by the Department of Science and Technology - Accelerated Sci- ence and Technology Human Resource Development Program (DOST-ASTHRDP), Philip- pines. References [1] C. Berge. Theorie des graphes et ses applications. Metheun and Wiley, London and New York, 1962. [2] R. 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