EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 15, No. 4, 2022, 1649-1661 ISSN 1307-5543 – ejpam.com Published by New York Business Global Forcing Connected Co-Independent Hop Domination Numbers in the Join and Corona 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 join and corona of graphs are determined. Some main results generated in this study include characterization of the minimum connected co-independent hop dominating sets, characterization of the forcing subsets for these types of sets, and bounds or exact values of the forcing connected co-independent hop domination numbers of the join and corona of graphs. 2020 Mathematics Subject Classifications: 05C69 Key Words and Phrases: Forcing subsets, connected co-independent hop domination, strictly co-independent set, co-independent set, join, corona 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 variation is the connected co-independent domination number of graphs that was studied in [7]. Years later, a new domination parameter called hop domination was introduced in [12] by Natarajan and Ayyaswamy and were also studied in [3, 13–15]. A study in 2021 by Nanding and 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.v15i4.4493 Email addresses: yvesdave.calanza@g.msuiit.edu.ph (Y.D. Calanza), helen.rara@g.msuiit.edu.ph (H. Rara) https://www.ejpam.com 1649 © 2022 EJPAM All rights reserved. Y.D. Calanza, H. Rara / Eur. J. Pure Appl. Math, 15 (4) (2022), 1649-1661 1650 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 Armada and Canoy [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 researchers define and establish the forcing subsets of minimum con- nected co-independent hop dominating sets in graphs and generate some characterizations of forcing subsets of minimum connected co-independent hop dominating sets of graphs resulting from the join and corona of two graphs and determine the values or bounds of their corresponding forcing connected co-independent hop domination numbers. 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 their job regarding with the respective workers that they are assigned to inspect. The Y.D. Calanza, H. Rara / Eur. J. Pure Appl. Math, 15 (4) (2022), 1649-1661 1651 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 dominating set D ⊆ V (G) is called a connected co-independent dominating set of G if the subgraph ⟨D⟩ induced by 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 neighbor- hood 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 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. A set S ⊆ V (G) is a co-independent set of G if ⟨V (G) \ S⟩ is independent. The mini- mum 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. 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}. Y.D. Calanza, H. Rara / Eur. J. Pure Appl. Math, 15 (4) (2022), 1649-1661 1652 Let W be an sci-set of a graph G. A subset S of W is said to be a forcing subset for W if W is the unique sci-set containing S. The forcing strictly co-independent number of W is given by fsci(W ) = min{|S| : S is a forcing subset for W}. The forcing strictly co-independent number of G is given by fsci(G) = min{fsci(W ) : W is an sci-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 of G is given by fcoi(G) = min{fcoi(W ) : W is a coi-set of G}. 2. Known Results The following known results are taken from [11]. Theorem 1. 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. 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)}. Theorem 2. 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). 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)). 3. Forcing Connected Co-Independent Hop Domination Number of Some Special Graphs Remark 1. 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 3. 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. Y.D. Calanza, H. Rara / Eur. J. Pure Appl. Math, 15 (4) (2022), 1649-1661 1653 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. Hence, 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 1(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]. Clearly, fγch,coi(P1) = fγch,coi(P2) = 0. Moreover, if n = 4, then Pn has γch,coi-set B1 = {v2, v3} which is the only γch,coi-set of Pn. By Remark 1(i), fγch,coi(Pn) = 0. Suppose that n > 4, then clearly B2 = {v2, v3, v4, . . . , vn−1} is the only γch,coi-set of Pn. Thus, by Remark 1(i), fγch,coi(B2) = 0 = fγch,coi(Pn). Suppose that n = 3. Then Pn has γch,coi-sets B3 = {v1, v2} and B4 = {v2, v3} which are the only γch,coi-sets of Pn with v1 ∈ B3 and v1 /∈ B4. Hence, by Remark 1(ii), fγch,coi(B3) = 1 = fγch,coi(Pn). 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 3, fγch,coi(Cn) = n− 1. 4. Forcing 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)}. Y.D. Calanza, H. Rara / Eur. J. Pure Appl. Math, 15 (4) (2022), 1649-1661 1654 Remark 2. Let G be a connected graph. Then (i) fsci(G) = 0 if and only if G has a unique sci-set, and (ii) fsci(G) = 1 if and only if G has at least two sci-sets, one of which, say B, contains an element which is not found in any sci-set of G. Theorem 4. Let G be a connected graph. Then fsci(G) = sci(G) if and only if for all sci-set B of G and for each v ∈ B, there exists uv ∈ V (G) \B such that [ B \ {v} ] ∪ {uv} is an sci-set of G. Proof: Suppose that fsci(G) = sci(G). Let B be an sci-set of G such that fsci(G) = |B| = sci(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 an sci-set of G. Conversely, suppose that every sci-set B ′ of G satisfies the given condition. Let B be an sci-set of G such that fsci(G) = fsci(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 an sci-set of G. Hence, T = Q∪R, where R = [ P \{z} ] ∪{uz}, is an sci-set containing Q, a contradiction. Hence, B is the only forcing subset for B. Therefore, fsci(G) = sci(G). Proposition 4. For any complete graph Kn with n ≥ 1 vertices, fsci(Kn) = 0. Proof: By definition of Kn, V (Kn) is the only sci-set of Kn. By Remark 2(i), fsci(Kn) = 0. Proposition 5. For any path Pn with n ≥ 1 vertices, fsci(Pn) =  0, if n = 1, 2, 4 and n > 5 is odd, 1, if n = 3 and n ≥ 6 is even, 2, if n = 5. Proof: Suppose that Pn = [v1, v2, . . . , vn]. Clearly, fsci(P1) = fsci(P2) = fsci(P4) = 0, fsci(P3) = 1 and fsci(P5) = 2. If n > 5 and n is odd, then clearly B = {v2, v4, v6, . . . , vn−3, vn−1} is the only sci-set of Pn. Thus, by Remark 2(i), fsci(B) = 0 = fsci(Pn). Suppose that n ≥ 6 and n is even. Then Pn has sci-sets B1 = {v1, v3, v5, . . . , vn−1} and B2 = {v2, v4, v6, . . . , vn}. It can be verified that B1 is the only sci-set of Pn containing the vertex v1. Hence, by Remark 2(ii), fsci(Pn) = 1. Proposition 6. For any cycle Cn with n ≥ 3 vertices, fsci(Cn) =  0, if n = 3, 1, if n > 4 and n is even, 2, if n > 3 and n is odd, 3, if n = 4. Y.D. Calanza, H. Rara / Eur. J. Pure Appl. Math, 15 (4) (2022), 1649-1661 1655 Proof: Suppose that Cn = [v1, v2, . . . , vn, v1]. Since C3 = K3, by Proposition 4, fsci(C3) = 0. Suppose that n = 4. Then the sci-sets of C4 are R1 = {v1, v2, v3}, R2 = {v2, v3, v4}, R3 = {v1, v3, v4} and R4 = {v1, v2, v4}. Clearly, for each vi ∈ Rj where i, j ∈ {1, 2, 3, 4}, there exists vk ∈ V (C4) \ Rj such that [Rj \ {vi}] ∪ {vk} is an sci-set of G. Thus, by Theorem 4, fsci(C4) = 3. Now, suppose that n > 4 and n is even. Then B1 = {v1, v3, v5, . . . , vn−1} and B2 = {v2, v4, v6, . . . , vn} are the only sci-sets of Cn with v1 ∈ B1 and v1 /∈ B2. Hence, by Remark 2(ii), fsci(B1) = 1 = fsci(Cn). Next, suppose that n > 3 and n is odd. Then S1 = {v1, v3, v5, . . . , vn−2, vn}, S2 = {v1, v3, v5, . . . , vn−2, vn−1}, S3 = {v2, v4, v6, . . . , vn−1, vn} and, S4 = {v2, v4, v6, . . . , vn−1, v1} are the sci-sets of Cn. Hence, no vertex of Cn is contained in a unique sci-set. Thus, fsci(Cn) ≥ 2. Clearly, {v1, vn} is uniquely contained in S1. Therefore, fsci(S1) = 2 = fsci(Cn). In view of Theorem 1, we have the following theorem. Theorem 5. Let G and H be any graphs. Then S ⊆ V (G + H) is a connected co-independent hop dominating set of G+H if and only if one of the following holds: (i) S = V (G) ∪ SH where SH is a strictly co-independent set of H, (ii) S = V (H) ∪ SG where SG is a strictly co-independent set of G. As a consequence of Theorem 5, the next results follow. Corollary 3. Let G be any graph and K1 = ⟨v⟩. Then S ⊆ V (K1 +G) is a γch,coi-set of K1 +G if and only if S = {v} ∪ T where T is an sci-set of G. Corollary 4. Let G be any graph. Then fγch,coi(K1 +G) = { 0, if G has a unique sci-set, fsci(G), if G has no unique sci-set. Proof: Suppose that G has a unique sci -set, say SG. Then by Corollary 3, {v} ∪ SG is a unique γch,coi-set of K1 +G. By Remark 1(i), fγch,coi(K1 +G) = 0. Now, suppose that G has no unique sci-set. Let A be an sci-set of G and let F be a forcing subset for A such that fsci(G) = fsci(A) = |F |. By Corollary 3, S = {v} ∪ A is a γch,coi-set of K1 + G. Then it can be seen that F is also a forcing subset for S. Thus, fγch,coi(K1 +G) ≤ fγch,coi(S) ≤ |F | = fsci(G). Let S0 = {v} ∪A0 be a γch,coi-set of K1 +G such that fγch,coi(K1 +G) = fγch,coi(S0). By Corollary 3, A0 is an sci-set of G. Let F0 be a forcing subset for S0 with fγch,coi(S0) = |F0.| Y.D. Calanza, H. Rara / Eur. J. Pure Appl. Math, 15 (4) (2022), 1649-1661 1656 Suppose F0 is not a forcing subset for A0. Then there exists an sci-set A ′ 0 ofG with A ′ 0 ̸= A0 and F0 ⊆ A ′ 0. By Corollary 3, S ′ 0 = {v} ∪ A ′ 0 is a γch,coi-set of K1 + G. Since A ′ 0 ̸= A0, S ′ 0 ̸= S0. Thus, F0 ⊆ S ′ 0, a contradiction since F0 is a forcing subset for S0. Hence, F0 is a forcing subset for A0. Thus, fγch,coi(K1 +G) = fγch,coi(S0) = |F0| ≥ fsci(A0) ≥ fsci(G). Therefore, fγch,coi(K1 +G) = fsci(G). Example 1. (1.) For the fan Fn = K1 + Pn, where n ≥ 2, fγch,coi(Fn) = fsci(Pn) =  0, if n = 2, 4 and n > 5 is odd, 1, if n = 3 and n ≥ 6 is even, 2, if n = 5. (2.) For the wheel Wn = K1 + Cn, where n ≥ 3, fγch,coi(Wn) = fsci(Cn) =  0, if n = 3, 1, if n > 4 and n is even, 2, if n > 3 and n is odd, 3, if n = 4. (3.) For the star Sn = K1,n of order n+ 1, fγch,coi(Sn) = { 0, if n = 1, 1, if n > 1. Another consequence of Theorem 5 is the next corollary. Corollary 5. Let G and H be any graphs with |V (G)| < |V (H)| and sci(H) = sci(G) or |V (G)| = |V (H)| and sci(H) < sci(G). Then S ⊆ V (G + H) is a γch,coi-set of G + H if and only if S = V (G) ∪ SH for some sci -set SH of H. Theorem 6. For any graphs G and H with |V (G)| < |V (H)| and sci(H) = sci(G), or |V (G)| = |V (H)| and sci(H) < sci(G). Then fγch,coi(G+H) = { 0, if H has a unique sci-set, fsci(H), if H has no unique sci-set. Proof: Suppose that H has a unique sci -set, say SH . Then by Corollary 5, V (G)∪SH is a unique γch,coi-set of G+H. By Remark 1(i), fγch,coi(G+H) = 0. Now, suppose that H has no unique sci-set. Let A be an sci-set of H and let F be a forcing subset for A such that fsci(H) = fsci(A) = |F |. By Corollary 5, S = V (G) ∪ A is a γch,coi-set of G + H. Suppose F is not a forcing subset for S. Then there exists a γch,coi-set S ′ of G +H such Y.D. Calanza, H. Rara / Eur. J. Pure Appl. Math, 15 (4) (2022), 1649-1661 1657 that S ′ ̸= S and F ⊆ S ′ . By Corollary 5, S ′ = V (G)∪A ′ where A ′ is an sci-set of H. Since S ′ ̸= S, A ′ ̸= A. On the other hand, F being a forcing subset for A which is an sci-set of H implies that F ⊆ V (H). Thus, F ⊆ A ′ , a contradiction since F is a forcing subset for A. Hence, F is a forcing subset for S. Thus, fγch,coi(G+H) ≤ fγch,coi(S) ≤ |F | = fsci(H). Let S0 = V (G)∪A0 be a γch,coi-set of G+H such that fγch,coi(G+H) = fγch,coi(S0). By Corollary 5, A0 is an sci-set of H. Let F0 be a forcing subset for S0 with fγch,coi(S0) = |F0.| Suppose F0 is not a forcing subset for A0. Then there exists an sci-set A ′ 0 ̸= A0 of H such that F0 ⊆ A ′ 0. By Corollary 5, S ′ 0 = V (G) ∪A ′ 0 is a γch,coi-set of G+H with F0 ⊆ S ′ 0 and S ′ 0 ̸= S0. This is a contradiction since F0 is a forcing subset for S0. Thus, F0 is a forcing subset for A0. Hence, fγch,coi(G+H) = fγch,coi(S0) = |F0| ≥ fsci(A0) ≥ fsci(H). Therefore, fγch,coi(G+H) = fsci(H). Example 2. Let G = C3 and H = P7. Then |V (C3)| < |V (P7)| and sci(C3) = 3 = sci(P7). Since P7 has a unique sci-set, fγch,coi(C3 + P7) = 0. Example 3. Let G = C3 and H = P3. Then |V (C3)| = |V (P3)| and sci(P3) = 2 < 3 = sci(C3). Since P3 has no unique sci-sets, fγch,coi(C3+P3) = fsci(P3) = 1. 5. Forcing 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). Remark 3. 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 B, contains an element which is not found in any coi-set of G. Theorem 7. Let G be a connected graph. Then fcoi(G) = coi(G) if and only if for all coi-set B of G and for each v ∈ B, there exists uv ∈ V (G) \B such that [ B \ {v} ] ∪ {uv} is a coi-set of G. Proof: Suppose that fcoi(G) = coi(G). Let B be a coi-set of G such that fcoi(G) = |B| = 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 Y.D. Calanza, H. Rara / Eur. J. Pure Appl. Math, 15 (4) (2022), 1649-1661 1658[ B \ {v} ] ∪ {uv} is a coi-set of G. Conversely, suppose that every coi-set B ′ of G satisfies the given condition. Let B be a coi-set of G such that fcoi(G) = fcoi(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 coi-set of G. Hence, T = Q∪R, where R = [ P \ {z} ] ∪{uz}, is a coi-set containing Q, a contradiction. Hence, B is the only forcing subset for B. Therefore, fcoi(G) = coi(G). Proposition 7. 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 } . Clearly, fcoi(K1) = 0. If n = 2, then Kn has coi-set R1 = {v1} and R2 = {v2} which are the only coi-sets of Kn with v1 ∈ R1 and v1 /∈ R2. By Remark 3(ii), fcoi(Kn) = 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 7, fcoi(Kn) = n− 1. Proposition 8. 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. Proof: Suppose that Pn = [v1, v2, . . . , vn]. Clearly, fcoi(P1) = fcoi(P3) = 0 and fcoi(P2) = 1. If n = 4, then Pn has coi-sets B1 = {v1, v3}, B2 = {v2, v4} and B3 = {v2, v3} which are the only coi-sets of Pn with v4 ∈ B2 and v4 /∈ B1, B3. Thus, by Remark 3(ii), fcoi(Pn) = 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 3(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 3(ii), fcoi(Pn) = 1. Proposition 9. For any cycle Cn with n ≥ 3 vertices, fcoi(Cn) = { 1, if n = 4 and n > 4 is even, 2, if n = 3 and n > 3 is odd. Proof: Suppose that Cn = [v1, v2, . . . , vn, v1]. It can be verified that fcoi(C4) = 1. Suppose that 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 such Y.D. Calanza, H. Rara / Eur. J. Pure Appl. Math, 15 (4) (2022), 1649-1661 1659 that [ Qj \ {vi} ] ∪{vk} is a coi-set of G. Thus, by Theorem 7, fcoi(C3) = 2. Now, suppose that n > 4 and 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 3(ii), fcoi(B1) = 1 = fcoi(Cn). Next, suppose that n > 3 and n is odd. Then Q1 = {v1, v3, v5, . . . , vn−2, vn}, Q2 = {v1, v3, v5, . . . , vn−2, vn−1}, Q3 = {v2, v4, v6, . . . , vn−1, vn} and, Q4 = {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 Q1. Therefore, fcoi(Q1) = 2 = fcoi(Cn). In view of Theorem 2, we have the following theorem. Theorem 8. 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 is a co-independent set of Hv for each v ∈ V (G). The next result is a restatement of Corollary 2. Corollary 6. Let G be a nontrivial connected graph and H be any graph. A set S ⊆ V (G ◦H) is a γch,coi-set of G ◦H if and only if S = V (G) ∪ ( ⋃ v∈V (G) Sv ) where Sv is a coi-set of Hv for each v ∈ V (G). In particular, γch,coi(G ◦H) = |V (G)| ( 1 + coi(H) ) . Theorem 9. Let G be a nontrivial connected graph of order n and H be any graph. Then fγch,coi(G ◦H) = { 0, if H has a unique coi-set, n [ fcoi(H) ] , if H has no unique coi-set. Proof: Suppose H has a unique coi -set. For each v ∈ V (G), let Pv ⊆ V (Hv) be the unique coi -set of Hv. By Corollary 6, S = V (G) ∪ ( ⋃ v∈V (G) Pv ) is the unique γch,coi-set of G ◦ H. Thus, by Remark 1(i), fγch,coi(G ◦ H) = 0. On the other hand, suppose that H does not have a unique coi -set. For each v ∈ V (G), let Qv ⊆ V (Hv) be a coi -set of Hv with fcoi(Hv) = fcoi(Qv), and let PQv ⊆ Qv be a forcing subset for Qv with fcoi(Qv) = |PQv |. Then by Corollary 6, SQ = V (G)∪ ( ⋃ v∈V (G) Qv ) is a γch,coi-set of G◦H. Let C = ⋃ v∈V (G) PQv . Then C is a forcing subset for SQ. Thus, fγch,coi(G ◦H) ≤ fγch,coi(SQ) ≤ |C| = n [ fcoi(H) ] . REFERENCES 1660 Next, let S′ be a γch,coi-set of G ◦H such that fγch,coi(G ◦H) = fγch,coi(S ′). Then by Corollary 6, let S′ = V (G) ∪ ( ⋃ v∈V (G) Rv ) where Rv is a coi -set of Hv for each v ∈ V (G). Let C ′ be a forcing subset for S′ such that fγch,coi(S ′) = |C ′|. Suppose that there exists w ∈ V (G) such that C ′ ∩Rw = Cw is not a forcing subset for Rw. Let R ′ w be a coi -set of Hw with R ′ w ̸= Rw. Then S′′ = V (G) ∪ ( ⋃ v∈V (G)\{w} Rv ) ∪R ′ w is a γch,coi-set of G ◦H with S′ ̸= S′′ and C ′ ⊆ S′′, a contradiction. Thus, Sv = C ′ ∩Rv is a forcing subset for Rv for each v ∈ V (G). Let S0 = ⋃ v∈V (G) Sv. Then fγch,coi(G ◦H) = |C ′| ≥ |S0| = ∑ v∈V (G) |Sv| ≥ ∑ v∈V (G) fcoi(Hv) = |V (G)|fcoi(H). Therefore, fγch,coi(G ◦H) = n [ fcoi(H) ] . Example 4. Let G = K2 andH = P5. Since P5 has a unique coi-set, fγch,coi(K2◦P5) = 0. Example 5. Let G = C3 and H = P4. Since P4 has no unique coi-sets, fγch,coi(C3 ◦ P4) = 3 [ fcoi(P4) ] = 3 · 1 = 3. Acknowledgements The authors would like to express their gratitude to the referees for their insight- ful comments and suggestions, which significantly improved the paper. The authors would also like to thank the following funding agencies: Mindanao State University - Iligan Institute of Technology (MSU-IIT) and the Department of Science and Technology - Accelerated Science and Technology Human Resource Development Program (DOST- ASTHRDP), Philippines. References [1] C. Armada and S. Canoy Jr. Forcing Independent Domination Number of a Graph. European Journal of Pure and Applied Mathematics, 12(4):1371–1381, 2019. [2] C. Armadaa, S. Canoy Jr., and C. Go. Forcing Domination Numbers of Graphs Under Some Binary Operations. Advances and Applications in Discrete Mathematics, 19(3):213–228, 2018. [3] S. Ayyaswamya, B. Krishnakumaria, C. Natarajan, and Y.B. Venkatakrishman. Bounds on the hop domination number of a tree. Proc. Math. Sci., 125:449–455, 2015. REFERENCES 1661 [4] C. Berge. Theorie des graphes et ses applications. Metheun and Wiley, London and New York, 1962. [5] G. Chartranda, H. Gavlasa, K.C. Vandell, and F. Harary. The Forcing Domination Number of a Graph. J.Combin. Math. Combin. Comput., 25:161–174, 1997. [6] W. Desormeauxa, T. Haynes, and M.A. Henning. A Note on Non-Dominating Set Partitions in Graphs. Networks, pages 1–8, 2016. [7] B. Gayathri and S. Kaspar. Connected Co-Independent Domination of a Graph. International Journal Contemp. Mathematics and Sciences, 6:423–429, 2011. [8] F. Harary. Graph Theory. Addison-Wesley Publishing Company, USA, 1969. [9] S. Kavithaa, S. Robinson Chellathurai, and J. John. On the forcing connected domina- tion number of a graph. Journal of Discrete Mathematical Sciences and Cryptography, 20(3):611–624, 2017. [10] D.J. Klein and M. Randic. Innate Degree of Freedom of a Graph. Comput. Chem., 8:516–521, 1987. [11] S. Nanding and H. Rara. On Connected Co-Independent Hop Domination in Graphs. European Journal of Pure and Applied Mathematics, 14(4):1226–1236, 2021. [12] C. Natarajan and S. Ayyaswamy. Hop Domination in Graphs-II. Versita, 23(2):187– 199, 2015. [13] Salasalan G. P. and Canoy Jr S. R. Revisiting Domination, Hop Domination, and Global Hop Domination in Graphs. European Journal of Pure and Applied Mathe- matics, 14(4), 2021. [14] Y. Pabilona and H. Rara. Total hop dominating sets in the join, corona, and lexico- graphic product of graph. Journal of Algebra and Applied Mathematics, 2017. [15] Y. M. Pabilona and H. Rara. Connected Hop Domination in Graphs under Some Binary Operations. Asian-European Journal of Mathematics, 11(5), 2018. [16] T.P. Zivkovica, F. Harary, and Klein D.J. Graphical Properties of Polyhexes: Perfect Matching Vector and Forcing. J. Math. Chem., 6:295–306, 1991.