EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 2, Article Number 6065 ISSN 1307-5543 – ejpam.com Published by New York Business Global Forcing Clique Domination in Graphs Cris L. Armada3,4,∗, Edward M. Kiunisala1,2, Cristopher John S. Rosero1, Jeneveb T. Malusay1 1 Mathematics Department, College of Computing, Artificial Intelligence and Sciences, Cebu Normal University, 6000 Cebu City, Philippines 2 Research Institute for Computational, Mathematics and Physics, Cebu Normal University, 6000 Cebu City, Philippines 3 Vietnam National University Ho Chi Minh City, Linh Trung Ward, Thu Duc City, Ho Chi Minh City, Vietnam 4 Department of Applied Mathematics, Faculty of Applied Science, Ho Chi Minh City University of Technology (HCMUT), 268 Ly Thuong Kiet, District 10, Ward 14, Ho Chi Minh City, Vietnam Abstract. The clique domination number of some special graphs such as paths, cycles, complete graphs, generalized wheels, generalized fans, and complete bipartite graphs is presented. The forcing clique domination number of these graphs, along with binary operations such as join, corona, and lexicographic product of two graphs, is also determined. Connected graphs with forcing clique domination number equal to 0, 1, or a, where a is greater than 1 but less than the clique domination number, are characterized. Necessary and sufficient conditions for the forcing clique domination number to be equal to the clique domination number are given. Since some of the graphs in this study do not have a clique dominating set, the forcing clique domination number is undefined in those cases. 2020 Mathematics Subject Classifications: 05C38, 05C69, 05C76 Key Words and Phrases: Forcing domination, clique domination, forcing clique domination number 1. Introduction Let G = (V (G), E(G)) be a graph.For any vertex t ∈ V (G),the closed neighborhood of t is defined as the set NG[t] = {t} ∪ {s ∈ V (G) : st ∈ E(G)}. If T is a nonempty subset of X, then NG[T ] = ⋃ t∈T NG[t]. A nonempty set T ⊆ V (G) is a dominating set of G if for every u ∈ V (G)\T , there exists t ∈ T such that tu ∈ E(G), that is, NG[T ] = V (G). ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i2.6065 Email addresses: cris.armada@hcmut.edu.vn (C. L. Armada), kiunisalae@cnu.edu.ph (E. M. Kiunisala), roseroc@cnu.edu.ph (C. J. S. Rosero), malusayj@cnu.edu.ph (J. T. Malusay) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) C. L. Armada et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6065 2 of 14 The domination number of G, denoted by γ(G), is the minimum cardinality among all dominating sets of G. A γ-set T of G is a dominating set of G with |T | = γ(G). A graph is complete if every two of its vertices are adjacent. Let G be a nontrivial connected graph. A dominating set C of V (G) is a clique dominating set of G if the induced subgraph ⟨C⟩ of C is complete. The minimum cardinality of a clique dominating set of G, denoted by γcl(G),is called the clique domination number of G. A γcl-set C of G is a clique dominating set of G with |C| = γcl(G). Graph G is considered a non−γcl−graph if it does not contain a clique dominating set, following a similar definition to that of a non-γp0-graph as in [1]. Let C be a γcl-set of a graph G. A subset L of C is said to be a forcing subset for C if C is the unique γcl-set containing L. The forcing clique domination number of C is given by fγcl(C) = min{|L| : L is a forcing subset for C}. The forcing clique domination number of G is given by fγcl(G) = min{fγcl(C) : C is a γcl-set of G} The join of two graphs G and H, denoted by G+H, is the graph 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)}. The corona of two graphs G and H, denoted by G◦H, is defined to be the graph obtained by taking one copy of G and |V (G)| copies of H and then forming the joins ⟨v⟩+Hv = v+Hv for each v ∈ V (G), where Hv is a copy of H correponding to vertex v. The lexicographic product or composition of two graphs G and H, denoted by G[H], is the graph with vertex set V (G[H]) = V (G) × V (H) and edge set E(G[H]) satisfying the following conditions: (x, u)(y, v) ∈ E(G[H]) if and only if either xy ∈ E(G) or x = y and uv ∈ E(H). Observe that a subset C of V (G[H]) = V (G)× V (H) can be written as C = ⋃ x∈S [{x} × Tx], where S ⊆ V (G) and Tx ⊆ V (H) for each x ∈ S. We shall use this form to denote any subset C of V (G[H]). The clique domination was investigated in [2] and [3]. The concept of forcing domination was first studied by Chartrand, et al. in [4]. Closed neighborhood, domination number, forcing domination number, the binary operations such as join, corona and lexicographic product of graphs, and other variations of forcing domination can be found in [5],[6],[7],[8] and [9]. Additional basic graph-theoretic terminology can be found in [10]. The forcing clique domination number is important when it comes to fault-tolerant sensor network optimization in smart cities. Sensors are placed in these networks to monitor infrastructure, health, traffic, and air quality. Certain sensor groups naturally form cliques, C. L. Armada et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6065 3 of 14 which are fully connected subgraphs that guarantee effective data sharing. These sensors create graphs with edges that indicate direct communication links. To guarantee smooth network coverage, a clique dominating set ensures that each sensor is either inside a clique or directly connected to one [11]. This structure is improved by the forcing property, which ensures that the activation of a small number of important sensors triggers the activation of others, reducing redundancy and increasing data collection and transmission efficiency [12]. This ensures that the network continues to operate with low resource consumption even in the event that certain sensors fail [13]. In addition to energy efficiency, the forcing clique domination number improves fault tolerance and sensor network resilience. The system can tolerate failures and continue to function by carefully choosing a minimum clique dominating set. This is particularly helpful in fields where dependability is essential, such as emergency response systems, military communication, and disaster monitoring [14]. Example 1.1. Consider the graph G in Figure 1. It is clear to see that R1 = {x, u1}, R2 = {x, u2}, R3 = {x, u3}, ... Rm−1 = {x, um−1}, and Rm = {x, um} are γcl-sets of G. Clearly, for all i = 1, 2, . . . ,m, Ti = {ui} is uniquely contained in each γcl-set Ri of G and so, Ti is a forcing subset for each Ri. Thus, fγcl(G) = |Ti| = 1. Figure 1: Graph G with fγcl(G) = 1. C. L. Armada et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6065 4 of 14 Example 1.2. Consider the graph G[H] in Figure 2. Clearly, γcl(G) = 3. By Corollary 2.10, γcl(G[H]) = 3. It is clear to see that S1 = {(a, x), (b, x), (c, x)}, S10 = {(a, y), (b, x), (c, x)}, S19 = {(a, z), (b, x), (c, x)}, S2 = {(a, x), (b, x), (c, y)}, S11 = {(a, y), (b, x), (c, y)}, S20 = {(a, z), (b, x), (c, y)}, S3 = {(a, x), (b, x), (c, z)}, S12 = {(a, y), (b, x), (c, z)}, S21 = {(a, z), (b, x), (c, z)}, S4 = {(a, x), (b, y), (c, x)}, S13 = {(a, y), (b, y), (c, x)}, S22 = {(a, z), (b, y), (c, x)}, S5 = {(a, x), (b, y), (c, y)}, S14 = {(a, y), (b, y), (c, y)}, S23 = {(a, z), (b, y), (c, y)}, S6 = {(a, x), (b, y), (c, z)}, S15 = {(a, y), (b, y), (c, z)}, S24 = {(a, z), (b, y), (c, z)}, S7 = {(a, x), (b, z), (c, x)}, S16 = {(a, y), (b, z), (c, x)}, S25 = {(a, z), (b, z), (c, x)}, S8 = {(a, x), (b, z), (c, y)}, S17 = {(a, y), (b, z), (c, y)}, S26 = {(a, z), (b, z), (c, y)}, and S9 = {(a, x), (b, z), (c, z)}, S18 = {(a, y), (b, z), (c, z)}, S27 = {(a, z), (b, z), (c, z)} are γcl-sets of G[H]. Clearly, there exists no subset with 1 and 2 vertices that it is contained in a unique γcl-set of G[H]. Thus, for all i = 1, 2, . . . , 27, Si is a forcing subset for itself and so, fγcl(G[H]) = |Si| = 3 = γcl(G[H]). Figure 2: Graph G[H] with fγcl(G[H]) = 3. C. L. Armada et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6065 5 of 14 2. Known Results This section presents known results on the domination number and the clique domination number of a graph G, and of graphs resulting from some binary operations. Proposition 2.1. [15] For n ≥ 3, γ(Pn) = γ(Cn) = ⌈n 3 ⌉ . Proposition 2.2. [16] If n is a positive integer , then γ(Kn) = 1. Theorem 2.3. [2] Let G be a connected graph. Then γcl(G) = 1 if and only if γ(G) = 1. Theorem 2.4. [2] Let G and H be any two graphs. A subset S of V (G+H) is a clique dominating set of G+H if and only if one of the following statements holds: (i) S is clique dominating set of G (ii) S is a clique dominating set of H. (iii) S = S1 ∪ S2, where ⟨S1⟩ and ⟨S2⟩ are cliques in G and H, respectively. Corollary 2.5. [2] Let G and H be nontrivial graphs . Then γcl(G+H) = { 1, if γ(G) = 1 or γ(H) = 1 2, otherwise Theorem 2.6. [3] If G is a finite graph that is connected and has no induced P5 or C5, then G has a clique dominating set. Theorem 2.7. [2] Let G be a connected nontrivial graph and H be any non-trivial graph. Then G ◦H has a clique dominating set S if and only if G is complete and S = V (G). Corollary 2.8. [2] Let G be a complete nontrivial graph and H be any graph.Then γcl(G ◦H) = |V (G)|. Theorem 2.9. [2] Let G and H be connected nontrivial graphs such that G has a clique dominating set. A subset C = ⋃ x∈S [{x} × Tx], where S ⊆ V (G) and Tx ⊆ V (H) for each x ∈ S, is a clique dominating set of G[H] if and only if S is a clique dominating set of G such that (i) ⟨Tx⟩ is a clique in H for each x ∈ S and (ii) Tx is a dominating set of H whenever S = {x}. Corollary 2.10. [2] Let G and H be connected nontrivial graphs such that G has a clique dominating set.Then γcl(G[H]) =  1, if γ(G) = γ(H) = 1 2, if γ(G) = 1 and γ(H) ̸= 1 γcl(G), ifγ(G) ̸= 1 C. L. Armada et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6065 6 of 14 3. Main Results This section presents the clique domination number and the forcing clique domination number of special graphs such as paths, cycles, complete graphs and other special graphs such as generalized wheels, generalized fans, and complete bipartite graphs. In addition, the forcing clique domination number is determined for graphs obtained through some binary operations such as the join, corona, and lexicographic product of two graphs. Theorem 3.1. Let G be a connected graph such that G has a clique dominating set. Then (i) fγcl(G) = 0 if and only if G contains a unique γcl-set. (ii) fγcl(G) = 1 if and only if G has no unique γcl-sets and there exists a vertex t ∈ V (G) which is contained in exactly one γcl-set of G. (iii) for any integer a such that 1 < a < γcl(G), fγcl(G) = a if and only if G has no unique γcl-sets and a is the minimum number of vertices which are contained in exactly one γcl-set of G. Proof: (i) Suppose that fγcl(G) = 0. It follows that ∅ is the forcing subset for a γcl-set in G. Suppose that G has two γcl-sets, say C and D. Then ∅ is a forcing subset for C and D, a contradiction since a forcing subset must be contained in a unique γcl-set. Therefore, G contains a unique γcl-set. Conversely, if G contains a unique γcl-set, say B. Clearly, ∅ is a forcing subset of B. Hence, |∅| = 0 = fγcl(B) = fγcl(G). (ii) Suppose that fγcl(G) = 1. By part (i), G has no unique γcl-set and there exist γcl-set, say T , and t ∈ T such that {t} is a forcing subset for T and fγcl(T ) = |{t}| = 1, that is, {t} is not forcing subset for another γcl-set of G. Thus, there exists a vertex t ∈ V (G) which is contained in exactly one γcl-set of G. Conversely, if G has no unique γcl-sets , then by part (i), fγcl(G) ≥ 1. By assumption, there exists a vertex, say c, which is contained in exactly one γcl-set of G, say C, that is, {c} is a forcing subset for C. There- fore, fγcl(C) = |{c}| = 1 = fγcl(G). (iii) Suppose that fγcl(G) = a for any integer a such that 1 < a < γcl(G). By part (i), G has no unique γcl-sets and there exists a unique γcl-set, say T , and |S| = a such that S is a forcing subset for T and fγcl(G) = a = |S| = fγcl(T ). Hence, a is the minimum number of vertices which are contained in exactly one γcl-set of G. Conversely, if G has no unique γcl-sets, then by part (i), fγcl(G) ≥ 1. By assumption, there exists a set S such that |S| = a > 1, S is contained in exactly one γcl-set of G, say C, that is, S is a forcing subset for C. By the minimality of a, a = |S| = fγcl(C) = fγcl(G). The next two results are direct consequences of Theorem 3.1 and definition of forcing clique domination. Corollary 3.2. Let G be a connected graph such that G has a clique dominating set. Then 0 ≤ fγcl(G) ≤ γcl(G). C. L. Armada et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6065 7 of 14 Theorem 3.3. Let G be a connected graph such that G has a clique dominating set. Then fγcl(G) = γcl(G) if and only if for every γcl-set of C of G and for each vertex t ∈ C, there exist a vertex u ∈ V (G)\C such that {u} ∪ [C\{t}] is a γcl-set of G. Proof: Suppose that fγcl(G) = γcl(G). Let C be a γcl-set of G such that fγcl(G) = |C| = γcl(G), that is, C is the only forcing subset for C. Let t ∈ C. Since C\{t} is not a forcing subset for C, there exists a u ∈ V (G)\C such that{u} ∪ [C\{t}] is a γcl-set of G. Conversely, suppose that everyγcl-set C ′ of G satisfies the given condition. Let C be a γcl-set of G such that fγcl(G) = fγcl(C) and |C| = γcl(G). Moreover, suppose that C has a forcing subset D with |D| < |C|, that is ,C = D ∪ A, where A = {t ∈ C : t /∈ D}. Pick t ∈ A. By assumption, there exists u ∈ V (G)\C such that {u} ∪ [C\{t}] = B is a γcl-set of G. Thus,B = D ∪E, where E = {u} ∪ [A\{t}], that is, B is a γcl-set containing D , a contradiction. Thus, |D| = |C| and |C| is the only forcing subset for |C|. Therefore, fγcl(G) = fγcl(C) = |C| = γcl(G). The next result is a restatement of Theorem 3.3. Remark 3.4. Let G be a connected graph such that G has a clique dominating set. Then fγcl(G) = γcl(G) if and only if every vertex in a γcl-set C of G can be replaced by another vertex in V (G)\C to form another γcl-set of G. Proposition 3.5. Let n be a positive integer with n ≥ 1. Then the clique domination number of a path Pn and its forcing clique domination number are given by γcl(Pn) = { 1, n < 4 2, n = 4 and fγcl(Pn) = { 0, n = 1, 3, 4 1, n = 2 . For n ≥ 5, the path Pn is non−γcl−graph, and both γcl(Pn) and fγcl(Pn) are undefined. Proof: Let V (Pn) = {u1, u2, . . . , un}. Consider the following cases: Case 1. Let n = 1. Clearly, {u1} is the only minimum clique dominating set of P1. Thus, γcl(P1) = 1 and fγcl(P1) = 0 by Theorem 3.1 (i). Case 2. Let n = 2. By Proposition 2.1, γ(P2) = ⌈23⌉ = 1 and by Theorem 2.3, γcl(P2) = 1. Clearly, S1 = {u1} and S2 = {u2} are the γcl-sets of P2, that is, the vertex u1 is contained in S1 only. Thus, fγcl(P2) = 1 by Theorem 3.1 (ii). Case 3. Let n = 3. By Proposition 2.1, γ(P3) = ⌈33⌉ = 1 and by Theorem 2.3, γcl(P3) = 1. Clearly, {u2} is the only γcl-set of P3. Thus, fγcl(P3) = 1 by Theorem 3.1 (ii). C. L. Armada et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6065 8 of 14 Case 4. Let n = 4. By Proposition 2.1, γ(P4) = ⌈43⌉ = 2 and by Theorem 2.3, γcl(P4) > 1. Clearly, C = {u2, u3} is the only γcl-sets of P4 since the induced subgraph ⟨C⟩ of C is com- plete. Thus,γcl(P4) = 2 and fγcl(P4) = 0 by Theorem 3.1 (i). Case 5. Let n ≥ 5. Then Pn has induced P5. By Theorem 2.6, Pn has no clique dominating set. Therefore, for all n ≥ 5, Pn is non − γcl − graph, and both γcl(Pn) and fγcl(Pn) are undefined. Proposition 3.6. Let n be a positive integer with n ≥ 3. Then the clique domination number and forcing clique domination number of a cycle Cn are equal and given by fγcl(Cn) = γcl(Cn) = { 1, n = 3 2, n = 4 For n ≥ 5, the cycle Cn is non−γcl−graph, and both γcl(Cn) and fγcl(Cn) are undefined. Proof: Let V (Cn) = {u1, u2, . . . , un}. Consider the following cases: Case 1. Let n = 3. Then by Proposition 2.1, γ(C3) = ⌈ 3 3 ⌉ = 1 and by Theorem 2.3, γcl(C3) = 1. Clearly, S1 = {u1}, S2 = {u2} and S3 = {u3} are the γcl-sets of C3, that is, the vertex u1 is contained in S1 only. By Theorem 3.1(ii), fγcl(C3) = 1. Case 2. Let n = 4. By Proposition 2.1, γ(C4) = ⌈ 4 3 ⌉ = 2. Clearly, T1 = {u1, u2}, T2 = {u2, u3}, T3 = {u3, u4} and T4 = {u4, u1} are the γcl-sets of C4, such that for all i = 1, 2, 3, 4, the induced subgraph ⟨Ti⟩ of Ti is complete. Thus, γcl(C4) = 2. Clearly, every vertex in γcl-set Tk of C4 can be replaced by another vertex in V (C4)\Tk to form another γcl-set Tj such that k ̸= j. By Remark 3.4, fγcl(C4) = γcl(C4) = 2. Case 3. Let n ≥ 5. Then Cn has induced P5. By Theorem 2.6, Cn has no clique dominating set. Therefore, for all n ≥ 5, Cn is non − γcl − graph, and both γcl(Cn) and fγcl(Cn) are undefined. Proposition 3.7. Let n be a positive integer with n ≥ 1. Then the clique domination number of the complete graph Kn is given by γcl(Kn) = 1 and forcing clique domination number is given by fγcl(Kn) = { 0, n = 1 1, n ≥ 2. Proof: Let V (Kn) = {u1, u2, u3 . . . , un}. By Proposition 2.2, γ(Kn) = 1 and by Theorem 2.3, γcl(Kn) = 1. If n = 1, then {u1} is the only γcl-set of K1. Thus, fγcl(K1) = 0 by Theorem 3.1(i). Suppose that n ≥ 2. Then for all i = 1, 2, . . . , n, Si = {ui} is a γcl-set of Kn, that is, the vertex ui is contained in Si only. By Theorem 3.1 (ii), fγcl(Kn) = 1 for all n ≥ 2. C. L. Armada et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6065 9 of 14 Theorem 3.8. Let G and H be any graphs. Then fγcl(G+H) =  0, if either γ(G) = 1 < γ(H) andGhas a unique γ − set, or γ(H) = 1 < γ(G) andH has a unique γ − set 1, if either γ(G) = 1 < γ(H) andGhasno unique γ − set, orγ(H) = 1 < γ(G) andH hasno unique γ − set or γ(G) = 1 and γ(H) = 1 2, if γ(G) > 1 and γ(H) > 1. Proof: Consider the following cases: Case 1. Suppose that γ(G) = 1 < γ(H) and G has unique γ-set. By Corollary 2.5, γcl(G+H) = 1. Suppose that S is the unique γ-set of G. Then |S| = 1, say S = {u} for a unique vertex u of V (G) and by Theorem 2.3, S is a γcl-set of G. By Theorem 2.4, S is the only γcl-set of G+H. By Theorem 3.1 (i), fγcl(G+H) = 0. Similarly, fγcl(G+H) = 0 if γ(H) = 1 < γ(G) and H has a unique γ-set. Case 2. Suppose that γ(G) = 1 < γ(H) and G has no unique γ-set. By Corollary 2.5, γcl(G + H) = 1. Let S and T be γ-sets of G. Then |S| = |T | = 1 and by Theorem 2.3, S and T are γcl-sets of G. Thus, S and T are γcl-sets of G + H by Theorem 2.4. Then there exists a vertex u contained in S only. By Theorem 3.1(ii), fγcl(G+H) = 1. Similarly, fγcl(G+H) = 1 if γ(H) = 1 < γ(G) and H has no unique γ-set. Case 3. Suppose that γ(G) = 1 and γ(H) = 1. By Corollary 2.5, γcl(G + H) = 1. Let S and R be γ-set of G and H, respectively. By Theorem 2.3, S and R are γcl-sets of G and H, respectively. Then by Theorem 2.4, S and R are γcl-sets of G +H. Then there exists a vertex u contained in S only. By Theorem 3.1(ii), fγcl(G+H) = 1. Case 4. Suppose that γ(G) > 1 and γ(H) > 1. By Corollary 2.5, γcl(G+H) = 2. Consider a γcl-set S = {c, d} of G+H, where c ∈ V (G) and d ∈ V (H). Pick x ∈ V (G)\{c} and y ∈ V (H)\{d}. Then {c} ⊆ Sy = {c, y} and {d} ⊆ Sx = {x, d}, where Sx and Sy are also γcl-sets of G + H different from S. Thus, fγcl(S) = 2. Now, if γ(G) = 2, then by Theorem 2.3, γcl(G) ̸= 1. Thus,γcl(G) = 2 or γcl(G) is undefined. Suppose that γcl(G) is undefined. Then the set T = {e, f}, where e ∈ V (G) and f ∈ V (H), is a γcl-set of G +H. By the previous argument, fγcl(T ) = 2. Suppose that γcl(G) = 2. Let S′ = {g, h} be a γcl-set of G and by Theorem 2.4, S′ is also a γcl-set of G +H. Pick v ∈ V (H). Then {g} ⊆ Sg = {g, v} and {h} ⊆ Sh = {h, v} where Sg and Sh are γcl-sets of G+H different from S′. Thus, fγcl(S′) = 2. Similarly, if γ(H) = 2, then for any γcl-set S∗ of G+H, fγcl(S∗) = 2. In any case, fγcl(G+H) = 2. The next result follows from Theorem 3.8 and Corollary 2.5. C. L. Armada et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6065 10 of 14 Corollary 3.9. For any graph H, γcl(K1 +H) = 1 and fγcl(K1 +H) = { 0, γ(H) > 1, 1, γ(H) = 1. The next results are direct consequences of Theorem 3.8, and Corollaries 2.5 and 3.9. Corollary 3.10. Let n and m be positive integers. For a complete bipartite graph Kn,m = Kn +Km where n ≥ 1 and m ≥ 1, γcl(Kn,m) = { 1, if either n = 1 orm = 1, 2, if n ≥ 2 andm ≥ 2. and fγcl(Kn,m) =  0, if n = 1 andm ≥ 2 orm = 1 andn ≥ 2, 1, if n = 1 andm = 1 , 2, if n ≥ 2 andm ≥ 2. Corollary 3.11. For the generalized fan Fn,m = Kn + Pm, where n ≥ 1 and m ≥ 2, γcl(Fn,m) = { 1, if either n = 1 orm < 4, 2, if n ≥ 2 andm ≥ 4. and fγcl(Fn,m) =  0, if either n = 1 andm ≥ 4 or n ≥ 2 andm = 3, 1, if either n = 1 andm < 4 or n ≥ 2 andm = 2, 2, if n ≥ 2 andm ≥ 4. Corollary 3.12. For the generalized wheel Wn,m = Kn + Cm, where n ≥ 1 and m ≥ 3, γcl(Wn,m) = { 1, if either n = 1 orm = 3, 2, if n ≥ 2 andm ≥ 4. and fγcl(Wn,m) =  0, if n = 1 andm ≥ 4, 1, if m = 3, 2, if n ≥ 2 andm ≥ 4. Theorem 3.13. Let G be a trivial graph and H be any graph. Then S is a γcl-set of G ◦ H if and only if S = V (G) or S is a γ-set of H such that γ(H) = 1. In particular, γcl(G ◦H) = 1. Proof: Since G is trivial and G ◦ H = K1+H, by Corollary 3.9, γcl(C ◦ H) = 1. Suppose that S is a γcl-set of G ◦H. Since G is trivial, S = V (G) since V (G) is a dominating set of G ◦ H and V (G) is complete. Suppose that γ(H) = 1. By Theorem 2.3, γcl(H) = 1. Then there exists a vertex v in H such that v is adjacent to every vertex in H \ {v} and to a vertex in G. Take S = {v} and so, S is a γ-set of H. The converse is clear. C. L. Armada et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6065 11 of 14 Theorem 3.14. Let G be a complete graph and H be any graph. Then fγcl(G ◦ H) = { 0, if either G is nontrivial or G is trivial and γ(H) > 1, 1, if G is trivial and γ(H) = 1. Proof: Note that by Corollary 2.8, γcl(G ◦H) = |V (G)|. Since G is complete, V (G) is a γcl-set of G ◦H. Let S be a γcl-set of G ◦H. Consider the following cases: Case 1. Suppose that G is nontrivial. By Theorem 2.7, S = V (G) is the only γcl-set of G ◦H. By Theorem 3.1 (i), fγcl(G ◦H) = 0. Case 2. Suppose that G is trivial and γ(H) > 1. By Theorem 3.13, S = V (G) is the only γcl-set of G ◦H. By Theorem 3.1 (i), fγcl(G ◦H) = 0. Case 3. Suppose that G is trivial and γ(H) = 1. By Theorem 3.13, γcl(G ◦H) = 1 and either S = V (G) or S is the γ-set of H such that S is also γcl-set of G ◦H and |S| = 1. Thus, G ◦H has no unique γcl-sets. Then there exists a vertex u contained in S only. By Theorem 3.1 (ii), fγcl(G ◦H) = |S| = 1. Theorem 3.15. Let G and H be connected nontrivial graphs such that G has a clique dominating set. Then fγcl(G[H]) =  0, if γ(G) = γ(H) = 1 and bothGandH have unique γ − sets, 1, if γ(G) = γ(H) = 1 and either G orH hasno unique γ − sets or both, 2, if γ(G) = 1 and γ(H) ̸= 1. γcl(G), if γ(G) > 1, Proof: Consider the following cases: Case 1. Suppose that γ(G) = γ(H) = 1 and both G and H have unique γ-sets, say S = {x} and T = {a}, respectively. By Corollary 2.10, γcl(G[H]) = 1 and by Theorem 2.3, S and T are also γcl-sets of G and H, respectively. By Theorem 2.9, C = S×Tx = {(x, a)} is the only γcl-set of G[H]. By Theorem 3.1 (i), fγcl(G[H]) = 0. Case 2. Suppose that γ(G) = γ(H) = 1 and either G or H has no unique γ-sets or both. By Corollary 2.10, γcl(G[H]) = 1. WLOG, suppose that G has no unique γ-sets, say S1 = {x} and S2 = {y}, and also suppose that H has a γ-set, say T = {a}. By Theorem 2.3, S1 and S2 are also γcl-sets of G and T is a γcl-sets of H. By Corollary 2.10, γcl(G[H]) = 1. By Theorem 2.9, C1 = ⋃ x∈S1 [{x} × Tx] and C2 = ⋃ y∈S2 [{y} × Ty], where S1, S2 ⊆ V (G) and Tx, Ty ⊆ V (H) for x ∈ S1 and y ∈ S2 such that |C1| = |C2| = 1 and C. L. Armada et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6065 12 of 14 set Tx = Ty = {a}, that is, C1 = {(x, a)} and C2 = {(y, a)} are the γcl-sets of G[H]. Clearly, the vertex (x, a) is contained in C1 only. By Theorem 3.1 (ii), fγcl(G[H]) = 1. Similarly, if H has no unique γ-sets or both G and H have no unique γ-sets, fγcl(G[H]) = 1. Case 3. Suppose that γ(G) = 1 and γ(H) ̸= 1. By Corollary 2.10, γcl(G[H]) = 2. Let S = {x, y} be a clique dominating set of G such that xy ∈ E(G). Choose any vertex a ∈ V (H). Then C = {(x, a), (y, a)} is a γcl-set of G[H] by Theorem 2.9 and Corollary 2.10. Choose c ∈ V (H)\{a}. It follows that {(x, a)} ⊆ C1 = {(x, a), (y, c)} and {(y, a)} ⊆ C2 = {(x, c), (y, a)}, where C1 and C2 are also γcl-sets of G[H] different from C. It follows that fγcl(C) = 2 = fγcl(G[H]). Case 4. Suppose that γ(G) > 1. By Corollary 2.10, γcl(G[H]) = γcl(G). Let C = ⋃ x∈S [{x} × Tx] be a γcl-set of G[H] and let FC = ⋃ x∈D [{x} × Fx] be a forcing subset for C. Suppose that S is a γcl-set of G. Then |C| = |S| and so, |Tx| = 1 for all x ∈ S. Hence, Fx = Tx for all x ∈ D. If D ̸= S, say y ∈ S\D, then FC ⊆ C ′ = ⋃ x∈S [{x} × T ′ x], where T ′ x = Tx for x ∈ S\{y} and T ′ y is a singleton subset of H different from Ty. Since C ′ is a γcl-set of G[H] and C ′ ̸= C, FC is not a forcing subset for C, contrary to the assumption. Thus, D = S, that is, FC = C. Hence, fγcl(C) = |C| = γcl(G) = fγcl(G[H]). The next result follows from Theorem 3.15. Corollary 3.16. Let H be a connected nontrivial graph. Then for any complete nontrivial graph Kn, fγcl(Kn[H]) = { 1, if γ(H) = 1, 2, if γ(H) ̸= 1. Corollary 3.17. Let G and H be connected nontrivial graphs. Then for any complete nontrivial graph Kn, fγcl((Kn ◦ G)[H]) = n. Proof: By Corollary 2.8, Kn ◦ G has a minimum clique dominating set and γcl(Kn ◦ G) = |V (Kn)| = n such that n > 1 since Kn is nontrivial. By Theorem 2.3, γ(Kn ◦ G) > 1. By Theorem 3.15 and Corollary 2.8, fγcl((Kn ◦ G)[H]) = γcl(Kn ◦ G) = n. C. L. Armada et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6065 13 of 14 4. Conclusion In this study, the idea of forcing clique domination in graphs was examined along with its basic characteristics. We investigated how the forcing clique domination number and the clique domination number relate to one another. A significant result in our study when the forcing clique domination number is zero. This happens when each minimum clique dominating set is uniquely determined. These graphs are especially helpful in applications requiring stable and non-redundant control because of their structural rigidity in clique domination properties. Another important result is when the forcing clique domination number is equal to the clique domination number. This implies that every vertex in a minimum clique dominating set can be replaced by another vertex in the graph while still maintaining the property of clique domination. This feature is important in fault-tolerant network topologies since it will allow other nodes to assume dominance responsibilities without affecting connection or coverage. Also, we also discovered graphs for which the clique domination number is undefined, as they do not have a clique dominating set, making the forcing clique domination number itself undefined. Our research sheds more light on the characteristics of forcing clique domination and its function in graph theory. Future studies might concentrate on determining the forcing clique domination number of other binary operations not mentioned in this study and investigating real-world applications in social influence modeling, biological networks, and network security. Acknowledgements The authors express their gratitude to the anonymous referees for their significant remarks and recommendations, which greatly influenced the caliber of this work. The authors would like to thank Cebu Normal University for the support and encouragement extended throughout the conduct of this research. We acknowledge Ho Chi Minh City University of Technology (HCMUT), VNU-HCM for supporting this study. 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