Disjoint Secure Domination in the Join of Graphs

Abstract

Let G = (V(G),E(G)) be a simple connected graph. A dominating set S in G is called a secure dominating set in G if for every u ∈ V (G) \ S, there exists v ∈ S ∩ N_G(u) such that (S \ {v}) ∪ {u} is a dominating set. The minimum cardinality of secure dominating set is called the secure domination number of G and is denoted by γ_s(G). A secure dominating set of cardinality γ_s(G) is called γ_s-set of G. Let D be a minimum secure dominating set in G. The secure dominating set   S⊆ V(G)\D is called an inverse secure dominating set with respect to D. The inverse secure domination number of G denoted by γ_s^−1(G) is the minimum cardinality of an inverse secure dominating set in G. An inverse secure dominating set of cardinality γ_s−¹(G) is called γ_s^−1-set. A disjoint secure dominating set in G is the set C = D ∪ S ⊆ V(G). The disjoint secure domination number of G denoted by γγ_r(G) is the minimum cardinality of a disjoint secure dominating set in G. A disjoint secure dominating set of cardinality γγ_s(G) is called γγ_s-set. In this paper, we show that every integers k and n with k ∈ {2, 4, 5, ...n − 1, n} is realizable as disjoint secure domination number, and order of G respectively. Further, we give the characterization of the disjoint secure dominating set in the join of two graphs.