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 ? NG(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 ?s(G). A secure dominating set of cardinality ?s(G) is called ?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 ?s?1(G) is the minimum cardinality of an inverse secure dominating set in G. An inverse secure dominating set of cardinality ?s?1(G) is called ?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 ??r(G) is the minimum cardinality of a disjoint secure dominating set in G. A disjoint secure dominating set of cardinality ??s(G) is called ??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.
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