EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 15, No. 4, 2022, 1597-1612 ISSN 1307-5543 – ejpam.com Published by New York Business Global Forcing Subsets of Connected Co-Independent Hop Domination in the Edge Corona and Lexicographic Product of Graphs Yves Dave L. Calanza1,∗, 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. This study deals with the forcing subsets of a minimum connected co-independent hop dominating sets in graphs. Bounds or exact values of the forcing connected co-independent hop domination numbers of graphs resulting from some binary operations such as edge corona and lexicographic product of graphs are determined. Some main results generated in this study include: (a) characterization of the minimum connected co-independent hop dominating sets; and (b) characterization of the forcing subsets for these types of sets. 2020 Mathematics Subject Classifications: 05C69 Key Words and Phrases: Forcing subsets, connected co-independent hop, domination, co- independent set, edge corona, lexicographic product 1. Introduction Beginning with C. Berge [4] in 1958, the study on domination in graphs was developed. There are now a lot of studies involving domination and its variations. One of its variations is the connected co-independent domination number of graphs that was studied in [7]. Years later, a new domination parameter called hop domination in graph was intro- duced in [12] by Natarajan and Ayyaswamy and was also studied in [3, 13–15]. A study in 2021 by S. Nanding and H. Rara [11] introduced a new concept of hop domination called the connected co-independent hop domination and generated some characterizations of connected co-independent hop domination in graphs. ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v15i4.4521 Email addresses: yvesdave.calanza@g.msuiit.edu.ph (Y.D. Calanza), helen.rara@g.msuiit.edu.ph (H. Rara) https://www.ejpam.com 1597 © 2022 EJPAM All rights reserved. Y.D. Calanza, H. Rara / Eur. J. Pure Appl. Math, 15 (4) (2022), 1597-1612 1598 On the other hand, the concept of forcing numbers started from the study of molecular resonance structure which was introduced by Klein and Randic [10] in 1987. Harary et. al [16] first used the name “forcing number” and introduced the concept of the forcing of a perfect match in 1991. Chartrand et. al [5] initiated the investigation on the relation between forcing and domination concepts in 1997 and defined the term “forcing domination number”. In 2017, John et. al [9] investigated the forcing connected domination number of a graph, and C. Armada and S. Canoy Jr. [1] investigated the forcing independent domination number of a graph in 2019. Furthermore, in 2018, Canoy et. al [2] investigated the forcing domination number of graphs under some binary operations. In this study, the forcing subsets of minimum connected co-independent hop dominat- ing sets in graphs are defined and established and some characterizations of forcing subsets of minimum connected co-independent hop dominating sets of graphs resulting from the edge corona and lexicographic product of two graphs are generated. Also, the values or bounds of their corresponding forcing connected co-independent hop domination numbers are determined. Connected co-independent hop domination in graphs can have real world applications. For an application, in [6], Desormeaux, 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 manager 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 [11], 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 forcing subsets of connected co-independent hop domination, in each respective group of minimum number of inspectors that will inspect the workers in the designated areas of the factory, the members of that particular group of minimum number of inspectors will be assigned only to that distinct group of minimum number of inspectors, that is, it will strengthen the bond of the respective group of minimum number of non-biased inspectors with each other, since they are uniquely assigned to particular groups, and they will trust each other more doing their duties and will have a much easier time doing Y.D. Calanza, H. Rara / Eur. J. Pure Appl. Math, 15 (4) (2022), 1597-1612 1599 their job regarding with the respective workers that they are assigned to inspect. The forcing connected co-independent hop domination number will determine the minimum number of members from the respective group of minimum number of inspectors that will be assigned only to that particular group of respective minimum number of inspectors. In this study, we only consider graphs that are finite, simple, undirected and connected. Readers are referred to [8] 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 set S ⊆ V (G) is a co-independent set of G if V (G)\S is independent. The minimum cardinality of a co-independent set in G, denoted by coi(G) is called the co-independent number of G. A co-independent set S with |S| = coi(G) is called a coi -set of G. A dominating set D ⊆ V (G) is called a connected co-independent dominating set of G if the subgraph ⟨D⟩ is connected and V (G) \D is an independent set. The cardinality of such a minimum set D is called 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 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. Let W be a γch,coi-set of a graph G. A subset S of W is said to be a forcing subset for W if W is the unique γch,coi-set containing S. The forcing connected co-independent hop domination number of W is given by fγch,coi(W ) = min{|S| : S is a forcing subset for W}. The forcing connected co-independent hop domination number of G is given by fγch,coi(G) = min{fγch,coi(W ) : W is a γch,coi-set of G}. Let W be a coi-set of a graph G. A subset S of W is said to be a forcing subset for W if W is the unique coi-set containing S. The forcing co-independent number of W is given by fcoi(W ) = min{|S| : S is a forcing subset for W}. The forcing co-independent number Y.D. Calanza, H. Rara / Eur. J. Pure Appl. Math, 15 (4) (2022), 1597-1612 1600 of G is given by fcoi(G) = min{fcoi(W ) : W is a coi-set of G}. Let G be a connected graph and let M be a β-set (maximum independent set) of G. A set D ⊆ M c is a forcing subset for the complement M c of M if M is the unique β-set such that M c contains D. The number fβc(M) = min{|D| : D is a forcing subset for M c} is defined as the forcing complement of the independence number M . The forcing complement of the independence number of G is given by fβc(G) = min{fβc(M) : M is a β-set of G}. 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. 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). 2. Known Results The following known results are taken from [11]. Remark 1. 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 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 2. If G is a complete graph, then γch,coi(G) = n. Theorem 1. 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. Y.D. Calanza, H. Rara / Eur. J. Pure Appl. Math, 15 (4) (2022), 1597-1612 1601 Corollary 1. 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 the independence number of H. 3. Forcing Connected Co-Independent Hop Domination Number of Some Special Graphs Remark 3. Let G be a connected graph. Then (i) fγch,coi(G) = 0 if and only if G has a unique γch,coi-set, and (ii) fγch,coi(G) = 1 if and only if G has at least two γch,coi-sets, one of which, say B, contains an element which is not found in any γch,coi-set of G. Theorem 2. Let G be a connected graph. Then fγch,coi(G) = γch,coi(G) if and only if for all γch,coi-set B of G and for each v ∈ B, there exists uv ∈ V (G) \ B such that[ B \ {v} ] ∪ {uv} is a γch,coi-set of G. Proof: Suppose that fγch,coi(G) = γch,coi(G). Let B be a γch,coi-set of G such that fγch,coi(G) = |B| = γch,coi(G), that is, B is the only forcing subset for itself. Let v ∈ B. Since B \ {v} is not a forcing subset for B, there exists a uv ∈ V (G) \ B such that[ B \ {v} ] ∪ {uv} is a γch,coi-set of G. Conversely, suppose that every γch,coi-set B ′ ofG satisfies the given condition. Let B be a γch,coi-set of G such that fγch,coi(G) = fγch,coi(B). Suppose further that B has a forcing subset Q with |Q| < |B|, that is, B = Q ∪ P where P = {z ∈ B : z /∈ Q}. Pick z ∈ P. By assumption, there exists uz ∈ V (G)\B such that [ B \{z} ] ∪{uz} = T is a γch,coi-set of G. Thus, T = Q∪R, where R = [ P \{z} ] ∪{uz}, is a γch,coi-set containing Q, a contradiction. Hence, B is the only forcing subset for B. Therefore, fγch,coi(G) = γch,coi(G). Proposition 1. For any complete graph Kn with n ≥ 1 vertices, fγch,coi(Kn) = 0. Proof: By definition of Kn, V (Kn) is the only γch,coi-set of Kn. By Remark 3(i), fγch,coi(Kn) = 0. Proposition 2. For any path Pn with n ≥ 1 vertices, fγch,coi(Pn) = { 0, if n ̸= 3, 1, if n = 3. Proof: Suppose that Pn = [v1, v2, . . . , vn]. It can be seen that fγch,coi(P1) = fγch,coi(P2) = 0. Moreover, if n = 4, then P4 has γch,coi-set B1 = {v2, v3} which is the only γch,coi-set of P4. By Remark 3(i), fγch,coi(P4) = 0. Suppose that n > 4, then clearly B2 = {v2, v3, v4, . . . , vn−1} is the only γch,coi-set of Pn. Thus, by Remark 3(i), Y.D. Calanza, H. Rara / Eur. J. Pure Appl. Math, 15 (4) (2022), 1597-1612 1602 fγch,coi(B2) = 0 = fγch,coi(Pn). Suppose that n = 3. Then P3 has γch,coi-sets B3 = {v1, v2} and B4 = {v2, v3} which are the only γch,coi-sets of P3 with v1 ∈ B3 and v1 /∈ B4. Hence, by Remark 3(ii), fγch,coi(B3) = 1 = fγch,coi(P3). Proposition 3. For any cycle Cn with n ≥ 3 vertices, fγch,coi(Cn) = { 0, if n = 3, n− 1, if n ≥ 4. Proof: Suppose that Cn = [v1, v2, . . . , vn, v1]. Since C3 = K3, by Proposition 1, fγch,coi(C3) = 0. Suppose that n ≥ 4. Then the γch,coi-sets of Cn areB1 = {v1, v2, . . . , vn−1}, B2 = {v2, v3, . . . , vn}, B3 = {v3, v4, . . . , vn, v1}, . . . , Bn = {vn, v1, v2, . . . , vn−2}. Clearly, for each vi ∈ Bj where i, j ∈ {1, 2, 3, . . . , n}, there exists vk ∈ V (Cn) \ Bj such that[ Bj \ {vi} ] ∪ {vk} is a γch,coi-set of G. Hence, by Theorem 2, fγch,coi(Cn) = n− 1. 4. Forcing Connected Co-Independent Hop Domination in the Edge Corona of Graphs The following two results are taken from the Masteral’s Thesis of Sandra A. Nanding. To verify the result, the proof is provided. 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 Y.D. Calanza, H. Rara / Eur. J. Pure Appl. Math, 15 (4) (2022), 1597-1612 1603 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] 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. Y.D. Calanza, H. Rara / Eur. J. Pure Appl. Math, 15 (4) (2022), 1597-1612 1604 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 of G ⋄H. Accordingly, C is a connected co-independent hop dominating set of G ⋄H. Corollary 2. Let G be a connected graph of order n ≥ 3 of size p and H be any graph 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. 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)). Remark 3 and Theorem 2 hold if the γch,coi-set is replace by coi-set. Thus, we have the following Remark and Theorem. Remark 4. Let G be a connected graph. Then (i) fcoi(G) = 0 if and only if G has a unique coi-set, and (ii) fcoi(G) = 1 if and only if G has at least two coi-sets, one of which, say D, contains an element which is not found in any coi-set of G. Theorem 4. Let G be a connected graph. Then fcoi(G) = coi(G) if and only if for all coi-set D of G and for each z ∈ D, there exists uz ∈ V (G) \D such that [ D \ {z} ] ∪ {uz} is a coi-set of G. Proof: Suppose that fcoi(G) = coi(G). Let D be a coi-set of G such that fcoi(G) = |D| = coi(G), that is, D is the only forcing subset for itself. Let z ∈ D. Y.D. Calanza, H. Rara / Eur. J. Pure Appl. Math, 15 (4) (2022), 1597-1612 1605 Since D \ {z} is not a forcing subset for D, there exists a uz ∈ V (G) \ D such that[ D \ {z} ] ∪ {uz} is a coi-set of G. Conversely, suppose that every coi-set D ′ of G satisfies the given condition. Let D be a coi-set of G such that fcoi(G) = fcoi(D). Suppose further that D has a forcing subset A with |A| < |D|, that is, D = A ∪ C where C = {v ∈ D : v /∈ A}. Pick v ∈ C. By assumption, there exists uv ∈ V (G) \D such that [ D \ {v} ] ∪ {uv} = R is a coi-set of G. Thus, R = A ∪ T, where T = [ C \ {v} ] ∪ {uv}, is a coi-set containing A, a contradiction. Hence, D is the only forcing subset for D. Therefore, fcoi(G) = coi(G). Example 2. The formulas below give the co-independent number of the complete graph Kn, path Pn and cycle Cn. coi(Kn) = { 1, if n = 1, n− 1, if n ≥ 2. coi(Pn) =  1, if n = 1, n 2 , if n is even, n− 1 2 , if n is odd. coi(Cn) =  n 2 , if n ≥ 4 and n is even, n+ 1 2 , if n ≥ 3 and n is odd. Proposition 4. For any complete graph Kn with n ≥ 1 vertices, fcoi(Kn) = { 0, if n = 1, n− 1, if n ≥ 2. Proof: Suppose that V (Kn) = { v1, v2, . . . , vn } . It can be seen that fcoi(K1) = 0. If n = 2, then K2 has only two coi-sets R1 = {v1} and R2 = {v2} with v1 ∈ R1 and v1 /∈ R2. By Remark 4(ii), fcoi(K2) = n− 1 = 1. Suppose that n > 2. Then the coi-sets of Kn are B1 = {v1, v2, . . . , vn−1}, B2 = {v2, v3, . . . , vn}, B3 = {v3, v4, . . . , vn, v1}, . . . , Bn = {vn, v1, v2, . . . , vn−2}. Clearly, for each vi ∈ Bj where i, j ∈ {1, 2, 3, . . . , n}, there exists vk ∈ V (Kn) \ Bj such that[ Bj \ {vi} ] ∪ {vk} is a coi-set of G. Hence, by Theorem 4, fcoi(Kn) = n− 1. Proposition 5. For any path Pn with n ≥ 1 vertices, fcoi(Pn) = { 0, if n = 1, 3 and n ≥ 5 is odd, 1, if n = 2, 4 and n ≥ 6 is even. Y.D. Calanza, H. Rara / Eur. J. Pure Appl. Math, 15 (4) (2022), 1597-1612 1606 Proof: Suppose that Pn = [v1, v2, . . . , vn]. It can be verified that fcoi(P1) = fcoi(P3) = 0 and fcoi(P2) = 1. If n = 4, then P4 has coi-sets B1 = {v1, v3}, B2 = {v2, v4} and B3 = {v2, v3} which are the only coi-sets of P4 with v4 ∈ B2 and v4 /∈ B1, B3. Thus, by Remark 4(ii), fcoi(P4) = 1. Now, suppose that n ≥ 5 and n is odd, then clearly B = {v2, v4, v6, . . . , vn−3, vn−1} is the only coi-set of Pn. Thus, by Remark 4(i), fcoi(B) = 0 = fcoi(Pn). Next, suppose that n ≥ 6 and n is even. Then Pn has coi-sets S1 = {v1, v3, v5, . . . , vn−1} and S2 = {v2, v4, v6, . . . , vn} which are the only coi-set of Pn with v3 ∈ S1 and v3 /∈ S2. Hence, by Remark 4(ii), fcoi(Pn) = 1. Proposition 6. For any cycle Cn with n ≥ 3 vertices, fcoi(Cn) = { 1, if n is even, 2, if n is odd. Proof: Suppose that Cn = [v1, v2, . . . , vn, v1]. If n = 3, then the coi-sets of C3 are Q1 = {v1, v2}, Q2 = {v2, v3} and Q3 = {v1, v3}. Clearly, for each vi ∈ Qj where i, j ∈ {1, 2, 3}, there exists vk ∈ V (C3) \ Qj , where k ∈ {1, 2, 3} and i ̸= k such that[ Qj \ {vi} ] ∪ {vk} is a coi-set of G. Thus, by Theorem 4, fcoi(C3) = 2. Now, suppose that n is even. Then B1 = {v1, v3, v5, . . . , vn−1} and B2 = {v2, v4, v6, . . . , vn} are the only coi-sets of Cn with v3 ∈ B1 and v3 /∈ B2. Thus, by Remark 4(ii), fcoi(B1) = 1 = fcoi(Cn). Next, suppose that n > 3 and n is odd. Then R1 = {v1, v3, v5, . . . , vn−2, vn}, R2 = {v1, v3, v5, . . . , vn−2, vn−1}, R3 = {v2, v4, v6, . . . , vn−1, vn} and, R4 = {v2, v4, v6, . . . , vn−1, v1} are coi-sets of Cn. Hence, no vertex of Cn is contained in a unique coi-set. Thus, fcoi(Cn) ≥ 2. Clearly, {v1, vn} is uniquely contained in R1. Therefore, fcoi(R1) = 2 = fcoi(Cn). The next result is a restatement of Corollary 2. Corollary 3. Let G be a connected graph of order n ≥ 3 of size p and H be any graph. Then C ⊆ V (G ⋄ H) is a γch,coi-set of G ⋄ H if and only if C = V (G) ∪ ( ⋃ uv∈E(G) Suv ) where Suv is a co-independent set of Huv for each u, v ∈ V (G) such that uv ∈ E(G), and Suv ̸= V (Huv). In particular, γch,coi(G ⋄H) = |V (G)|+ p ( coi(H) ) . Theorem 5. Let G be a nontrivial connected graph of order n ≥ 3 of size p and H be any graph. Then fγch,coi(G ⋄H) = { 0, if H has a unique coi-set, p [ fcoi(H) ] , if H has no unique coi-set. Y.D. Calanza, H. Rara / Eur. J. Pure Appl. Math, 15 (4) (2022), 1597-1612 1607 Proof: Suppose H has a unique coi -set. For each uv ∈ E(G), where u, v ∈ V (G), let Ruv ⊆ V (Huv) be the unique coi -set of Huv. By Corollary 3, C = V (G) ∪ ( ⋃ uv∈E(G) Ruv ) is the unique γch,coi-set of G ⋄ H. Thus, by Remark 3(i), fγch,coi(G ⋄ H) = 0. On the other hand, suppose that H does not have a unique coi -set. For every uv ∈ E(G), where u, v ∈ V (G), let Wuv ⊆ V (Huv) be a coi -set of Huv with fcoi(Huv) = fcoi(Wuv), and let RWuv ⊆ Wuv be a forcing subset for Wuv with fcoi(Wuv) = |RWuv |. Then by Corollary 3, SW = V (G) ∪ ( ⋃ uv∈E(G) Wuv ) is a γch,coi-set of G ⋄H. Let D = ⋃ uv∈E(G) RWuv . Then D is a forcing subset for SW . Thus, fγch,coi(G ⋄H) ≤ fγch,coi(SW ) ≤ |D| = p [ fcoi(H)]. Next, let C ′ be a γch,coi-set of G ⋄ H such that fγch,coi(G ⋄ H) = fγch,coi(C ′). Then by Corollary 3, let C ′ = V (G) ∪ ( ⋃ uv∈E(G) Quv ) where Quv is a coi -set of Huv for each uv ∈ E(G) for which u, v ∈ V (G). Now, we let D′ be a forcing subset for C ′ such that fγch,coi(C ′) = |D′|. Suppose that there exists ab ∈ E(G), for each a, b ∈ V (G) such that D′∩Qab = Dab is not a forcing subset for Qab. Let Q ′ ab be a coi -set of Hab with Q ′ ab ̸= Qab. Then C ′′ = V (G) ∪ ( ⋃ uv∈E(G)\{ab} Quv ) ∪Q ′ ab is a γch,coi-set of G⋄H with C ′ ̸= C ′′ and D′ ⊆ C ′′, a contradiction. Thus, Cuv = D′∩Quv is a forcing subset for Quv for each uv ∈ E(G) where u, v ∈ V (G). Let C0 = ⋃ uv∈E(G) Cuv. Then fγch,coi(G ⋄H) = |D′| ≥ |C0| = ∑ uv∈E(G) |Cuv| ≥ ∑ uv∈E(G) fcoi(Huv) = |E(G)|fcoi(H). Therefore, fγch,coi(G ⋄H) = p [ fcoi(H) ] . Example 3. For complete graph K3 and path P5, fγch,coi(K3 ⋄ P5) = 0 since P5 has a unique coi-set. Example 4. For cycle C3 and path P2, fγch,coi(C3 ⋄ P2) = 3 [ fcoi(P2) ] = 3 · 1 = 3 since P2 has no unique coi-sets. 5. Forcing Connected Co-Independent Hop Domination in the Lexicographic Product of Graphs The following is a restatement of Corollary 1. Theorem 6. Let G and H be any nontrivial connected graphs of orders m > 2 and n > 2, respectively with γ(H) ̸= 1. Then γch,coi(G[H]) = mn− β(G)β(H). Y.D. Calanza, H. Rara / Eur. J. Pure Appl. Math, 15 (4) (2022), 1597-1612 1608 Proof: Suppose that M is a β-set of G and N is a β-set of H. Set Tx = V (H) \N if x ∈ M and else set Tx = V (H). Then in view of Theorem 1, S = ⋃ x∈V (G) ( {x} × Tx ) is a connected co-independent hop dominating set of G[H]. Hence, γch,coi(G[H]) ≤ |S| = ∣∣∣∣∣ ⋃ x∈M ( {x} × (V (H) \N) )∣∣∣∣∣+ ∣∣∣∣∣∣ ⋃ x∈V (G)\M ( {x} × V (H) )∣∣∣∣∣∣ = |M | ( |V (H)| − |N | ) + ( |V (G)| − |M | ) |V (H)| = β(G) ( n− β(H) ) + ( m− β(G) ) n = ( β(G) ) n− β(G)β(H) +mn− ( β(G) ) n = mn− β(G)β(H). Conversely, suppose that S0 = ⋃ x∈V (G) ( {x} × Rx ) is a γch,coi-set of G[H]. Let D = { x ∈ V (G) : Rx = V (H) } . We claim that V (G) \ D is an independent set of G. Suppose on the contrary. Then there exists vertices x, y ∈ V (G) \D such that xy ∈ E(G) and pick a, b ∈ V (H) \ Rx for which ab ∈ E(H). This implies that (x, a)(x, b), (y, a)(y, b) ∈ E(G[H]), a contradiction to the fact that S0 is a γch,coi-set of G[H]. Thus, V (G) \D is an independent set of G. Hence, |V (G) \D| ≤ β(G) and so (1) |D| ≥ m− β(G). On the other hand, by condition (ii) in Theorem 1, we have V (H) \Rx is an independent set of H. Thus, |V (H) \Rx| ≤ β(H) and so (2) |Rx| ≥ n− β(H). Therefore, inequalities (1) and (2) imply γch,coi(G[H]) = |S0| = ∣∣∣∣∣ ⋃ x∈D ( {x} × V (H) )∣∣∣∣∣+ ∣∣∣∣∣∣ ⋃ x∈V (G)\D ( {x} ×Rx )∣∣∣∣∣∣ = |D||V (H)|+ ( |V (G)| − |D| ) |Rx| ≥ |D|n+ ( m− |D| )( n− β(H) ) = |D|n+mn−mβ(H)− |D|n+ |D|β(H) ≥ mn−mβ(H) + ( m− β(G) ) β(H) = mn−mβ(H) +mβ(H)− β(G)β(H) Y.D. Calanza, H. Rara / Eur. J. Pure Appl. Math, 15 (4) (2022), 1597-1612 1609 = mn− β(G)β(H). Therefore, γch,coi(G[H]) = mn− β(G)β(H). As a consequence of Theorem 6, the next result follows. Corollary 4. Let G and H be nontrivial connected graphs with orders m > 2 and n > 2, respectively, and γ(H) ̸= 1. Then S ⊆ V (G[H]) is a γch,coi-set of G[H] if and only if S = ⋃ x∈A ( {x} × Tx ) ∪ [( V (G) \A ) × V (H) ] for some β-set A of G such that V (H) \ Tx is a β-set of H for all x ∈ A. Theorem 7. Let G and H be connected graphs of orders m > 2 and n > 2, respectively, where H has a unique β-set and γ(H) ̸= 1. Then fγch,coi(G[H]) = { 0, if G has a unique β-set, fβc(G), if G has no unique β-set. Proof: By Corollary 4, S = [ A × B ] ∪ [ Ac × V (H) ] is a γch,coi-set of G[H] for some β-set A of G such that V (H) \ B is a β-set of H. Suppose that H has unique β-set, say R. If G has a unique β-set M , then G[H] has a unique γch,coi-set[ M ×R ] ∪ [ M c × V (H) ] . By Remark 3(i), fγch,coi(G[H]) = 0. Now, suppose that G has no unique β-set. Let A be a β-set of G and let DAc be a forcing subset for the complement Ac of A such that fβc(G) = fβc(A) = |DAc |. By Corollary 4, S = [ A×R ] ∪ [ Ac × V (H) ] is a γch,coi-set of G[H]. We claim that DAc × {a} is a forcing subset for S for every a ∈ V (H) \ R. Suppose on the contrary. Then there exists a γch,coi-set S ′ of G[H] with S ′ ̸= S such that DAc × {a} ⊆ S ′ . Then S ′ = [ A ′ × R ] ∪ [ (A ′ )c × V (H) ] for some β-set A ′ of G. Since S ′ ̸= S, A ′ ̸= A and DAc × {a} ⊆ (A ′ )c × V (H). Thus, DAc ⊆ (A ′ )c a contradiction since DAc is a forcing subset for Ac. Hence, DAc × {a} is a forcing subset for S. Thus, fγch,coi(G[H]) ≤ fγch,coi(S) ≤ |DAc × {a}| = |DAc | = fβc(G). Let S0 = [ A0 × R ] ∪ [ Ac 0 × V (H) ] be a γch,coi-set of G[H] such that fγch,coi(G[H]) = fγch,coi(S0). By Corollary 4, A0 is β-set of G. Let F0 be a forcing subset for S0 with fγch,coi(S0) = |F0|. Let F0 = ⋃ x∈K ( {x} × Tx ) . We claim that K ⊆ Ac 0 Y.D. Calanza, H. Rara / Eur. J. Pure Appl. Math, 15 (4) (2022), 1597-1612 1610 and Tx ⊆ V (H). Let K = K1∪K2 where K1∩K2 = ∅. Suppose that K ⊆ A0 or K1 ⊆ A0 and K2 ⊆ Ac 0. Let B0 be a β-set of G with B0 ̸= A0. Consider the γch,coi-set S ′ 0 = [ B0 ×R ] ∪ [ Bc 0 × V (H) ] . Then S ′ 0 ̸= S0. If A0 ∩B0 = ∅, then K ⊆ Bc 0 and F0 ⊆ K ×R ⊆ Bc 0 × V (H) ⊆ S ′ 0 or F0 ⊆ [ K1 ×R ] ∪ [ K2 × V (H) ] ⊆ (K1 ∪K2)× V (H) ⊆ Bc 0 × V (H) ⊆ S ′ 0. On the other hand, if A0 ∩B0 ̸= ∅ and K1 ⊆ B0 and K2 ⊆ Bc 0, then F0 ⊆ [ K ×R ] ⊆ [ K1 ×R ] ∪ [ K2 × V (H) ] ⊆ [ B0 ×R ] ∪ [ Bc 0 × V (H) ] = S ′ 0. In either case, we have a contradiction since F0 is a forcing subset for S0. Hence, K ⊆ Ac 0 and Tx ⊆ V (H). This implies that K is a forcing subset for Ac 0. Choose any x ∈ K and a ∈ Tx. Then Fa = K × {a} ⊆ F0. Thus, fγch,coi(G[H]) = fγch,coi(S0) = |F0| ≥ |Fa| = |K| ≥ fβc(A0) = fβc(G). Therefore, fγch,coi(G[H]) = fβc(G). Example 5. For paths P3 and P5, fγch,coi(P3[P5]) = 0 since P5 has a unique β-set. Example 6. For paths P4 and P3, fγch,coi(P4[P3]) = fβc(P4) = 2 since P4 has no unique β-sets. Theorem 8. Let G and H be connected graphs of orders m > 2 and n > 2, respectively, where G and H have no unique β-sets and γ(H) ̸= 1. Then fγch,coi(G[H]) ≤ fβc(H) + fβc(G) [ fβc(H) + 1 ] . Proof: Suppose that G and H do not have unique β-sets. Let A be a β-set of G and FAc be a forcing subset for the complement Ac of A such that fβc(G) = fβc(A) = |FAc |. Then for all x ∈ A, let B = V (H) \Tx be a β-set of H and FBc be a forcing subset for the complement Bc of B such that fβc(H) = fβc(B) = |FBc |. By Corollary 4, S = ⋃ x∈A ( {x} × Tx ) ∪ [ Ac × V (H) ] REFERENCES 1611 is a γch,coi-set of G[H]. We claim that S0 = ( {x} × FBc ) ∪ ( FAc × [FBc ∪ {y}] ) for each x ∈ A and y ∈ B is a forcing subset for S. Suppose on the contrary. Then there exists a γch.coi-set S ′ ̸= S such that S0 ⊆ S ′ . By Corollary 4, S ′ = ⋃ x∈A′ ( {x} × T ′ x ) ∪ [ (A ′ )c × V (H) ] for some β-set A ′ of G such that B ′ = V (H) \ T ′ x is a β-set of H for all x ∈ A ′ . Since S ′ ̸= S, either one of the following holds: (i) A ′ = A and B ′ ̸= B (ii) A ′ ̸= A and B ′ = B (iii) A ′ ̸= A and B ′ ̸= B. Suppose (i) holds. Then S ′ = ⋃ x∈A ( {x} × T ′ x ) ∪ [ Ac × V (H) ] . Since S0 ⊆ S ′ , FBc ⊆ T ′ x = V (H) \ B ′ = (B ′ )c, a contradiction since FBc is a forcing subset for Bc. If (ii) holds, then FAc ⊆ (A ′ )c. This is again a contradiction since FAc is a forcing subset for Ac. If (iii) holds, then FBc ⊆ (B ′ )c and FAc ⊆ (A ′ )c, a contradiction. Therefore, S0 is a forcing subset for S. Thus, fγch,coi(G[H]) ≤ fγch,coi(S) = |S0| = fβc(H) + fβc(G) [ fβc(H) + 1 ] . Example 7. For cycle C4 and path P4, fγch,coi(C4[P4]) ≤ fβc(P4) + fβc(C4) [ fβc(P4) + 1 ] = 2 + 2[2 + 1] = 8 since both C4 and P4 have no unique β-sets. Acknowledgements The authors would like to express their gratitude to the referees for their insight- ful comments and suggestions, which significantly improved the paper. 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