EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 17, No. 4, 2024, 2505-2515 ISSN 1307-5543 – ejpam.com Published by New York Business Global Connected Co-Independent Hop Domination in the Edge Corona and Complementary Prism of Graphs Sandra A. Nanding1,2,∗, Helen M. Rara2, Imelda S. Aniversaro2 1 Department of Mathematics and Statistics, College of Science and Mathematics, University of Southern Mindanao, 9407 Kabacan, Cotabato, Philippines 2 Department of Mathematics and Statistics, College 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. Here, authors presented the characterizations of the connected co-independent hop dominating sets in the edge corona and complementary prism of graphs and determines the exact values of their corresponding connected co-independent hop domination number. 2020 Mathematics Subject Classifications: 05C69 Key Words and Phrases: Strictly co-independent set, connected co-independent hop dominating set, connected co-independent hop domination number, edge corona, complementary prism 1. Introduction Domination in graphs was first introduced by C. Berge in 1958 [2]. There are now many studies involving domination and its variations. One of its variation is the connected co- independent domination number of graphs introduced by Gayathri and Kaspar in 2010 [3] and further studied in [1, 10]. Recently, Natarajan and Ayyaswamy [7] introduced and studied the concept of hop domination in graphs. Hop domination in graphs were also studied in [5, 6, 8, 9, 11]. In [6], the connected co-independent hop dominating sets of a graph is defined and studied under the join, corona and lexicographic product of graphs. ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v17i4.5342 Email addresses: nsnanding@usm.edu.ph (S. A. Nanding), helen.rara@g.msuiit.edu.ph (H. M. Rara), imelda.aniversario@g.msuiit.edu.ph (I. S. Aniversario) https://www.ejpam.com 2505 Copyright: © 2024 The Author(s). (CC BY-NC 4.0) S. A. Nanding, H. M. Rara, I. S. Aniversario / Eur. J. Pure Appl. Math, 17 (4) (2024), 2505-2515 2506 2. Preliminaries All graphs considered in this study are finite, simple, undirected and connected. Some necessary definitions are presented in this section. Readers are referred to [4] for elemen- tary Graph Theoretic concepts. Definition 1. 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. Definition 2. A perfect matching of a graph is a matching (i.e., an independent edge set) in which every vertex of the graph is incident to exactly one edge of the matching. Definition 3. A dominating set D ⊆ V (G) is called a connected co-independent dominat- ing 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. Definition 4. 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. Definition 5. 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. Definition 6. A set S ⊆ V (G) is a connected co-independent set of G if ⟨S⟩ is connected and V (G)\S is independent. Definition 7. 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 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. Definition 8. 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. S. A. Nanding, H. M. Rara, I. S. Aniversario / Eur. J. Pure Appl. Math, 17 (4) (2024), 2505-2515 2507 Example 1. Let P8 = [a, b, c, d, e, f, g, h]. Then S1 = {c, d, g, h}, S2 = {c, d, e, f} and S3 = {b, c, d, e, f, g} are hop dominating sets of P8. ⟨S1⟩ is not connected while ⟨S2⟩ and ⟨S3⟩ are connected. However, S2 is not a connected co-independent dominating set since V (G)\S2 is not an independent set while S3 is a connected co-independent dominating set because V (G)\S3 is an independent set. Thus, S3 is a connected co-independent hop dominating set. It can be verified that γch,coi(P8) = 6. Definition 9. For every u, v ∈ V (G) such that uv ∈ E(G), denote by Huv the copy of H whose vertices are attached one by one to the end vertices u and v of each edge uv of G and a set Suv ⊆ Huv. 3. Results 3.1. Preliminary Results It is worth mentioning that every connected graphG admits a connected co-independent hop dominating set. Indeed, the vertex-set V (G) of G is a connected co-independent hop dominating set. As a simple observation, we state the following. Remark 1. Every connected co-independent hop dominating set in a connected graph G is a hop dominating set. Hence, γh(G) ≤ γch,coi(G). Remark 2. Let G be a connected graph of order n. Then 1 ≤ γch,coi(G) ≤ n. Moreover, γch,coi(G) = 1 if and only if G = K1. Example 2. The formulas below give the connected co-independent hop domination num- ber 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 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, S. A. Nanding, H. M. Rara, I. S. Aniversario / Eur. J. Pure Appl. Math, 17 (4) (2024), 2505-2515 2508 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. Example 3. The graph P2 ◦Kn in Figure 1 has γch,coi(P2 ◦Kn) = 2. Figure 1: P2 ◦Kn 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. 3.2. Connected Co-Independent Hop Domination in the Edge Corona of Graphs The edge corona G ⋄H of G and H is the graph obtained by taking one copy of G and |E(G)| copies of H and joining each of the end vertices u and v of each edge uv of G to every vertex of the copy Huv of H. Example 4. Let G = C3 and H = K2. The edge corona of G ⋄H and H ⋄G are shown in Figure 2. Figure 2: Edge corona C4 ⋄K2 and K2 ⋄ C4 S. A. Nanding, H. M. Rara, I. S. Aniversario / Eur. J. Pure Appl. Math, 17 (4) (2024), 2505-2515 2509 Remark 4. If G is a connected graph of order 2 and H is any graph, then G⋄H = G+H. Theorem 3. Let G be a connected graph of order n ≥ 3 and H be any graph. Then C ⊆ V (G ⋄H) is a connected co-independent hop dominating set of G ⋄H if and only if C = A ∪ ( ⋃ uv∈E(G) Suv) where (i) A ⊆ V (G) is a connected co-independent set of G containing all vertices incident to all the edges of G. (ii) Suv = V (Huv) if uv ∈ E(G) such that u ∈ V (G)\A or v ∈ V (G)\A. (iii) For every a, b ∈ A such that ab ∈ E(G) and Sab ̸= V (Hab), V (Hab)\Sab is an independent set in Hab. Proof: Suppose that C is a connected co-independent hop dominating set of G⋄H. Let A = C∩V (G) and let Suv = C∩V (Huv) for each uv ∈ E(G). Then C = A∪ ( ⋃ uv∈V (G) Suv) where A ⊆ V (G). First, we show that ⟨A⟩ is connected. Let x, y ∈ A with x ̸= y. If xy ∈ E(G), then we are done. Suppose that xy /∈ E(G). Since ⟨C⟩ is connected and x, y ∈ C, there exists an x-y path [x1, x2, ..., xn] in ⟨C⟩ where x = x1, y = xn and n > 2. If xi ∈ A for all i ∈ {1, 2, ..., n}, then the path [x1, x2, ..., xn] is in A. Suppose there exists xi /∈ A. Then xi ∈ Suv for some edge uv ∈ E(G). By definition of G ⋄H, u, v ∈ A. Hence, [x1, ..., u, v, ..., xn] is a path in A, showing that ⟨A⟩ is connected. Next, let u, v ∈ V (G)\A with u ̸= v. Then u, v ∈ V (G ⋄ H)\C. Since V (G ⋄ H)\C is independent, uv /∈ E(G ⋄ H). Since u, v ∈ V (G), uv /∈ E(G) implying that V (G)\A is independent. Now, suppose v is a vertex incident to all the edges of G and v /∈ A. Then v ∈ NG(w)∩NG⋄H(p) for all w ∈ V (G) and for all p ∈ V (Hvw). Thus, NG⋄H(v, 2)∩C = ∅, a contradiction since C is a hop dominating set. Hence, A is a connected co-independent set of G containing all vertices incident to all edges of G, showing that (i) holds. Let uv ∈ E(G) with u /∈ A. Suppose Suv ̸= V (Huv). Then there exists x ∈ V (Huv)\Suv. Hence, x, u ∈ V (G ⋄ H)\C and xu ∈ E(G ⋄ H), a contradiction to the independence of V (G⋄H)\C. Thus, Suv = V (Huv) and (ii) holds. Lastly, let a, b ∈ A such that ab ∈ E(G) and Sab ̸= V (Hab). Since V (G ⋄H)\C is independent and (V (Hab)\Sab) ⊆ V (G ⋄H)\C, V (Hab)\Sab is an independent set in Hab. Hence, (iii) holds. For the converse, suppose C = A ∪ ( ⋃ uv∈E(G) Suv) where (i), (ii) and (iii) hold. First, we show that C is connected. Let u, v ∈ C with u ̸= v. If uv ∈ E(G ⋄ H), then we are done. So, suppose that uv /∈ E(G ⋄H). Consider the following cases. Case 1. u, v ∈ A By (i), ⟨A⟩ is connected. Hence, there exists a u-v path P [u, v] in A. Since A ⊆ C, the path P [u, v] is in C. Case 2. u ∈ A and v ∈ Sxy for some xy ∈ E(G) Since uv /∈ E(G ⋄ H), u ̸= x and u ̸= y. Since V (G)\A is independent by (i), x ∈ A or y ∈ A, say x ∈ A. If ux ∈ E(G), then the path [u, x, v] is a u-v path in C. Suppose ux /∈ E(G). Since ⟨A⟩ is connected by (i) and u, x ∈ A, there exists u-x path [y1, y2, ..., yk] S. A. Nanding, H. M. Rara, I. S. Aniversario / Eur. J. Pure Appl. Math, 17 (4) (2024), 2505-2515 2510 in A where u = y1, x = yk and k > 2. Hence, the path [y1, y2, ..., yk, v] is a u-v path in C. Case 3. u, v ∈ Spq for some edge pq ∈ E(G). Since V (G)\A is independent by (i), p ∈ A or q ∈ A. Hence, the path [u, p, v] or [u, q, v] is in C. In any case, ⟨C⟩ is connected. Next, we show that V (G ⋄H)\C is independent. Let p, q ∈ V (G ⋄H)\C with p ̸= q. Consider the following cases. Case 1. p ∈ V (G)\A and q ∈ V (G)\A Since V (G)\A is independent by (i), pq /∈ E(G). Thus, pq /∈ E(G ⋄H). Case 2. p ∈ V (G)\A, q ∈ V (Hxy)\Sxy for some xy ∈ E(G) Since Sxy ̸= V (Hxy), x, y ∈ A by (ii). Hence, p ̸= x and p ̸= y. By definition of G ⋄H, pq /∈ E(G ⋄H). Case 3. p ∈ V (Hxy)\Sxy and q ∈ V (Hrs)\Srs for some distinct edges xy, rs ∈ E(G) Then, by definition of G ⋄H, pq /∈ E(G ⋄H). Case 4. p, q ∈ V (Hzt)\Szt for some edge zt ∈ E(G) Since V (Hzt)\Szt is independent by (iii), pq /∈ E(G ⋄H). Therefore, in any case, V (G ⋄H)\C is an independent set in G ⋄H. Lastly, we show that C is a hop dominating set of G ⋄ H. Let u ∈ V (G ⋄ H)\C. Consider the following cases. Case 1. u ∈ V (G)\A Let degG(u) = 1. Since |V (G)| ≥ 3, there exists vw ∈ E(G) with u ∈ NG(v)\NG(w) or u ∈ NG(w)\NG(v). If w ∈ A, then w ∈ NG(u, 2) ∩ A. If w /∈ A, then Svw = V (Hvw) by (ii). Thus, a vertex p ∈ NG⋄H(u, 2) ∩ Svw exists. Hence, p ∈ NG⋄H(u, 2) ∩ C. Case 2. u ∈ V (Hxy)\Sxy for some xy ∈ E(G) By (ii), x, y ∈ A. Since |V (G)| ≥ 3, there exist z ∈ V (G) ∩NG(x) or z ∈ V (G) ∩NG(y). If z ∈ A, then z ∈ NG⋄H(u, 2) ∩ C. If z /∈ A, then Syz = V (Hyz). Hence, a vertex w ∈ NG⋄H(u, 2) ∩ Syz or w ∈ NG⋄H(u, 2) ∩ Sxz. Therefore, in any case C is a hop dominating set ofG⋄H. Accordingly, C is a connected co-independent hop dominating set of G ⋄H. Corollary 1. Let G be a connected graph of order n ≥ 3 and H be any graph of size p and of order m. Then γch,coi(G ⋄H) = n+ p(m− β(H)). Proof: Let Co = A ∪ ( ⋃ uv∈V (G) Suv) be a γch,coi-set of G ⋄H. Then conditions (i), (ii) and (iii) of Theorem 3 hold where A = V (G) and Suv = V (Huv)\S∗ where S∗ is any independent set of Huv. Thus, γch,coi(G ⋄H) = |A|+ p|Suv| = n+ p(|V (Huv)| − |S∗|) ≥ n+ p(m− β(H)). Let T be a β-set of H and Suv = V (Huv)\T for each uv ∈ E(G). Then C = V (G) ∪ ( ⋃ uv∈E(G) Suv) is a connected co-independent hop dominating set of G ⋄H by Theorem 3. S. A. Nanding, H. M. Rara, I. S. Aniversario / Eur. J. Pure Appl. Math, 17 (4) (2024), 2505-2515 2511 Hence, γch,coi(G ⋄H) ≤ |C| = |V (G)|+ p|Suv| = n+ p|V (Huv)\T | = n+ p(m− β(H)). Therefore, γch,coi(G ⋄H) = n+ p(m− β(H)). Example 5. The set of shaded vertices in the graph of P4 ⋄ P5 represents a connected co-independent hop dominating set of P4 ⋄ P5. By Corollary 1, γch,coi(P4 ⋄ P5) = 10. 3.3. Connected Co-Independent Hop Domination in the Complementary Prism For a graph G, the complementary prism, denoted GG, is formed from the disjoint union of G and its complement G by adding a perfect matching between corresponding vertices of G and G. For each v ∈ V (G), let v denote the vertex corresponding to v in G. Formally, the graph GG is formed from G∪G by adding the edge vv for every v ∈ V (G). Example 6. Consider the graphs C4, C4 in Figure 3. In the same figure is an illustration of complementary prism C4C4. Figure 3: (1)cycle C4 (2)complement C4 of C4 (3)complementary prism C4C4 Theorem 4. Let G be either a complete graph or an empty graph of order n ≥ 2. Then S ⊆ V (GG) is a connected co-independent hop dominating set of GG if and only if S = SG ∪ SG and the following hold: (i) SG = V (G) and SG ⊆ V (G) if G is complete and (ii) SG = V (G) and SG ⊆ V (G) if G is an empty graph. S. A. Nanding, H. M. Rara, I. S. Aniversario / Eur. J. Pure Appl. Math, 17 (4) (2024), 2505-2515 2512 Proof: Suppose that S ⊆ V (GG) is a connected co-independent hop dominating set of GG. Let SG = S ∩ V (G) and SG = S ∩ V (G). Then S = SG ∪ SG. Let G be a complete graph and suppose that SG ̸= V (G). Then there exists x ∈ V (G)\SG. Since V (GG)\S is independent and xx ∈ E(GG), x ∈ V (G) ∩ SG. Since G is complete, G is an empty graph. Thus, x is an isolated vertex of G. This contradicts the connectedness of S. Hence, SG = V (G) and (i) holds. For (ii), if G is an empty graph, then G is a complete graph. By (i), SG = V (G) and SG ⊆ V (G). For the converse, suppose that S = SG ∪ SG and (i) and (ii) hold. Then clearly, S is connected. Suppose (i) holds. Then V (GG)\S = V (G)\SG is independent since G is an empty graph. Let x ∈ V (GG)\S. Then x ∈ V (G)\SG. Hence, x = p for some p ∈ V (G). Since G is complete, py ∈ E(G) for each y ∈ V (G)\{p}. Since xp ∈ E(GG), dGG(x, y) = 2. Suppose (ii) holds. Then V (GG)\S = V (G)\SG is independent since G is an empty graph. Let z ∈ V (GG)\S. Then z ∈ V (G)\SG. Thus, z ∈ V (G) ∩ SG since zz ∈ E(GG) and V (GG)\S is independent. Since G is complete, qz ∈ E(G) for all q ∈ V (G)\{z}. It follows that dGG(z, q) = 2 and q ∈ V (G)\{z}. Therefore, in any case, S is a connected co-independent hop dominating set of GG. Corollary 2. Let G be either a complete graph or an empty graph of order n ≥ 2. Then γch,coi(GG) = n. Proof: Let S be a γch,coi-set of GG. Then S = SG ∪ SG and (i) and (ii) of Theorem 4 hold. If (i) holds, then SG = V (G) and SG ⊆ V (Ḡ). Hence, γch,coi(GG) = |S| = |V (G)| + |SG| ≥ n. On the other hand, if (ii) holds, then SG = V (G) and SG = V (G). Hence, γch,coi(GG) = |S| = |V (G)| + |SG| ≥ n. Now, let SG = ∅ if (i) holds. Thus, S = V (G) ∪ SG is a connected co-independent hop dominating set of GG by Theorem 4. Hence, γch,coi(GG) ≤ |S| = |V (G)| = n. If (ii) holds, then let SG = ∅. By Theorem 4, S = V (G) ∪ SG. Thus, γch,coi(GG) ≤ |S| = |V (G) = n. Therefore,in any case, γch,coi(GG) = n. Theorem 5. Let G be a nontrivial connected noncomplete graph and G be the comple- ment of G. Then S ⊆ V (GG) is a connected co-independent hop dominating set of GG if and only if S = SG ∪ SG where SG ⊆ V (G) and SG ⊆ V (G) and the following hold: (i) SG ̸= ∅ and SG ̸= ∅ (ii) V (G)\SG and V (G)\SG are independent sets in G and G, respectively. (iii) For every x ∈ V (G)\SG, x ∈ SG. (iv) Either ⟨SG⟩ is connected or for every pair of distinct vertices x, y ∈ SG with xy /∈ E(G), x, y ∈ SG. (v) Either 〈 SG 〉 is connected or for every pair of distinct vertices p, q ∈ SG with pq /∈ E(G), p, q ∈ SG. (vi) For every pair of vertices x ∈ SG and q ∈ SG, q ∈ SG if xq ∈ E(G) or x ∈ SG if xq /∈ E(G). S. A. Nanding, H. M. Rara, I. S. Aniversario / Eur. J. Pure Appl. Math, 17 (4) (2024), 2505-2515 2513 (vii) For every x ∈ V (G)\SG such that NG(x, 2) ∩ SG = ∅, either there exists y ∈ V (G) ∩NG(x) such that y ∈ SG or there exists p ∈ SG ∩NG(x). (viii) For every q ∈ V (G)\SG such that NG(q, 2) ∩ SG = ∅, there exists z ∈ V (G) ∩NG ∩ NG(q) such that z ∈ SG. Proof: Suppose that S is a connected co-independent hop dominating set of GG. Let SG = S ∩ V (G) and SG = S ∩ V (G). Then S = SG ∪SG. Suppose SG = ∅. Then S = SG and V (GG)\S = V (G) ∪ [V (G)\SG] is not independent since G is connected. This is a contradiction to the independence of V (GG)\S. Thus, SG ̸= ∅. Since G is a connected noncomplete graph, there exist x, y ∈ V (G) such that xy /∈ E(G). Hence, xy ∈ E(G). This implies that x ∈ SG or y ∈ SG, showing that SG ̸= ∅. Hence, (i) holds. For (ii), since V (GG)\S = (V (G)\SG)∪̇(V (G)\SG), V (GG)\S is independent, V (G)\SG and V (G)\SG are independent sets of G and G, respectively. Now, let x ∈ V (G)\SG. Since xx ∈ E(GG) and V (GG)\S is independent, x ∈ SG. Hence, (iii) holds. Suppose ⟨SG⟩ is not connected. Let x, y ∈ SG with x ̸= y and xy /∈ E(G). Since ⟨S⟩ is connected, an x-y path P [x, y] in S exists. Since ⟨SG⟩ is not connected and xy ∈ E(G), x, y ∈ P [x, y]. Hence, x, y ∈ SG and (iv) holds. The proof of (v) is similar to the proof of (iv). Next, let x ∈ SG and q ∈ SG. Since x, q ∈ S and x ̸= q, by connectedness of ⟨S⟩, there exists an x-q path P [x, q] in S. If xq ∈ E(G), then q ∈ P [x, q] impyling that q ∈ SG. On the other hand, if xq /∈ E(G), then x ∈ P [x, q], showing that x ∈ SG. Hence, (vi) holds. Let x ∈ V (G)\SG such that NG(x, 2) ∩ SG = ∅. Since S is a hop dominating set of GG and x /∈ S, there exist z ∈ S such that dGG(x, z) = 2. SinceNG(x, 2) ∩ SG = ∅, either z = p ∈ SG for some p ∈ V (G) ∩ NG(x) or p ∈ V (G) ∩ NG(x). Hence, (vii) holds. Statement (viii) can be shown similarly with (vii). For the converse, let S = SG ∪ SG where SG ⊆ V (G) and SG ⊆ V (G) and conditions (i)-(viii) are satisfied. First, we show that ⟨S⟩ is connected. Let x, y ∈ S with x ̸= y. If xy ∈ E(GG), then we are done. Suppose xy /∈ E(GG). Consider the following cases. Case 1. x, y ∈ SG If ⟨SG⟩ is connected, then an x-y path P [x, y] in SG exists. Since SG ⊆ S, P [x, y] is an x-y path in S. Suppose SG is not connected. Then by (iv), x, y ∈ SG. Thus, [x, x, y, y] is a path in S. Case2. x, y ∈ SG Same with Case 1 using (v). Case 3.x ∈ SG, y ∈ SG Let y = p for some p ∈ V (G). Then by (vi), either the path [x, p, y] or [x, x, y] is in S. Therefore, in any case, ⟨S⟩ is connected. Since V (G)\SG and V (G)\SG are independent in G and G, respectively by (ii) and V (GG)\S = (V (G)\SG) ∪ (V (G)\SG), we have V (GG)\S is independent. Finally, let x ∈ V (GG)\S. Consider the following cases. Case 1. x ∈ V (G)\SG If NG(x, 2)∩SG ̸= ∅, then dGG(x, y) = 2 for some y ∈ NG(x, 2)∩SG. Suppose NG(x, 2)∩ SG = ∅. Then by (vii), there exists w ∈ V (G) ∩ NG(x) such that w ∈ SG. Thus, dGG(x,w) = 2. REFERENCES 2514 Case2. x ∈ V (G)\SG Same with Case 1 using (viii). Hence, in any case, S is a hop dominating set of GG. Accordingly, S is a connected co-independent hop dominating set of GG. The following result follows from Theorem 5. Corollary 3. Let G be a nontrivial connected noncomplete graph. Then 2 ≤ γch,coi(GG) ≤ 2|V (G)| − β(G). Remark 5. The strictly inequality in γch,coi(GG) ≤ 2|V (G)|−β(G) presented in Corollary 3 can be attained. However the given upper bound is sharp. Example 7. To illustrate Remark 5, consider the path P6 = [v1, v2, v3, v4, v5, v6]. It can be verified that the set {v2, v3, v4, v5, v1, v2, v5, v6} is a γch,coi-set of P6P 6, that is, γch,coi(P6P 6) = 8. However, 2|V (P6)| − β(P6) = 2(6) − 3 = 9. Hence, strict inequal- ity is attained. On the other hand, equality is attained for C4 = [u1, u2, u3, u4], since {u1, u2, u3, u2, u3, u4} is a γch,coi-set of C4C4, that is, γch,coi(C4C4) = 6 = 2|V (C4)|−β(C4). Figure 4: Connected co-independent hop dominating set of P6P 6 and C4C4 Acknowledgements The authors would like to express their gratitude to the anonymous reviewers of this study and the editors of this journal, for their efforts in reviewing this publication. References [1] M. Bonsocan, I. Aniversario. On Connected Co-independent Domination of Some Graphs. Undergraduate Thesis, 2018. [2] C. Berge. Theorie des graphes et ses applications. Metheun and Wiley, London and New York, 1962. [3] B. Gayathri and S. Kaspar. Connected Co-Independent Domination of a Graph. International Journal Contemp. Mathematics and Sciences, 6:423–429, 2011. REFERENCES 2515 [4] F. Harary. Graph Theory. Addison-Wesley Publishing Company, USA, 1969. [5] S. Canoy, R. Mollejon and J. Canoy. 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