Double Roman Domination in Generalized Petersen Graphs P(ck, k)

Abstract

A double Roman dominating function on a graph G=(V,E) is a function f:V→{0,1,2,3}, satisfying the condition that every vertex u for which f(u)=1 is adjacent to at least one vertex assigned 2 or 3, and every vertex u with f(u)=0 is adjacent to at least one vertex assigned 3 or at least two vertices assigned 2. The weight of f equals the sum w(f)=∑v∈Vf(v). The minimum weight of a double Roman dominating function of G is called the double Roman domination number γdR(G) of a graph G. We obtain tight bounds and in some cases closed expressions for the double Roman domination number of generalized Petersen graphs P(ck,k). In short, we prove that γdR(P(ck,k))=32ck+ε, where limc→∞,k→∞εck=0.