Strong Roman Domination Number of Complementary Prism Graphs

Abstract

Let G = (V, E) be a simple graph with vertex set V = V(G), edge set E = E(G) and from maximumdegree ∆ = ∆(G). Also let f : V → {0, 1, ...,d ∆ 2e + 1} be a function that labels the vertices of G. Let Vi ={v ∈ V : f(v) = i} for i = 0, 1 and let V2 = V − (V0 SV1) = {w ∈ V : f(w) ≥ 2}. A function f is calleda strong Roman dominating function (StRDF) for G, if every v ∈ V0 has a neighbor w, such that w ∈ V2 andf(w) ≥ 1 + d 1 2 |N(w) TV0|e. The minimum weight, ω(f) = f(V) = Σv∈V f(v), over all the strong Roman dominatingfunctions of G, is called the strong Roman domination number of G and we denote it by γS tR(G). An StRDF ofminimum weight is called a γS tR(G)-function. Let G be the complement of G. The complementary prism GG ofG is the graph formed from the disjoint union G and G by adding the edges of a perfect matching between thecorresponding vertices of G and G. In this paper, we investigate some properties of Roman, double Roman andstrong Roman domination number of GG.