Let G = (V; E) be a simple graph with vertex set V = V(G), edge set E = E(G) and from maximum degree = (G). Also let f : V ! f0; 1; :::; d2 e + 1g be a function that labels the vertices of G. Let Vi = fv 2 V : f (v) = ig for i = 0; 1 and let V2 = V (V0 S V1) = fw 2 V : f (w) 2g. A function f is called a strong Roman dominating function (StRDF) for G, if every v 2 V0 has a neighbor w, such that w 2 V2 and f (w) 1 + d 1 2 jN(w) T V0je. The minimum weight, !( f ) = f (V) = v2V f (v), over all the strong Roman dominating functions of G, is called the strong Roman domination number of G and we denote it by S tR(G). An StRDF of minimum weight is called a S tR(G)-function. Let G be the complement of G. The complementary prism GG of G is the graph formed from the disjoint union G and G by adding the edges of a perfect matching between the corresponding vertices of G and G. In this paper, we investigate some properties of Roman, double Roman and strong Roman domination number of GG.
