Edward M. Kiunisala, Enrico L. Enriquez
Let G be a connected simple graph. A nonempty subset S of the vertex set V(G) is a clique in G if the graph <S> induced by S is complete. A clique S in G is a clique dominating set if it is a dominating set. A clique dominating set S is a clique secure dominating set in G if for every vertex u ∈ V (G) \ S, there exists a vertex v ∈ S ∩ NG(u) such that (S \ {v}) ∪ S is a dominating set in G. The clique secure domination number, denoted by γcls(G), is the smallest cardinality of a clique secure dominating set in G. A clique secure dominating set having cardinality equal to γcls(G) is called a γcls -set of G. In this paper, we show that given positive integers k and n such that n ≥ 4 and 1 ≤ k ≤ n, there exists a connected graph G with |V (G)| = n and γcls(G) = k. Also, we show that for any positive integers k, m and n such that 1 ≤ k ≤ m, there exists a connected graph G with |V (G)| = n, γcls(G) = m, and γcl(G) = k. Further, we give the characterization of the clique secure dominating set resulting from the join of two graphs and give some important results. © Research India Publications.
Mathematics Department, College of Arts and Sciences, Cebu Normal University, Cebu City, 6000, Philippines; Department of Mathematics, School of Arts and Sciences, University of San Carlos, Cebu City, 6000, Philippines
