A note on the independent Roman domination in unicyclic graphs

Abstract

A Roman dominating function (RDF) on a graph G = (V;E) is a function f : V ! f 0;1;2g satisfying the condition that every vertex u for which f(u) = 0 is adjacent to at least one vertex v for which f(v) = 2. The weight of an RDF is the value f(V (G)) = P u2V (G) f(u): An RDF f in a graph G is independent if no two vertices as- signed positive values are adjacent. The Roman domination number R(G) (respectively, the independent Roman domination number iR(G)) is the minimum weight of an RDF (respectively, independent RDF) on G. We say that R(G) strongly equals iR(G); denoted by R(G) iR(G); if every RDF on G of minimum weight is independent. In this note we characterize all unicyclic graphs G with R(G) iR(G): We consider finite, undirected, and simple graphs G with vertex set V = V (G) and edge set E = E(G). The open neighborhood of a vertex v 2 V is N(v) = NG(v) = fu 2 V j uv 2 Eg and the degree of v, denoted by dG(v), is the cardinality of its open neighborhood. A vertex of degree one is called a leaf, and its neighbor is called a support vertex. If v is a support vertex; then v is called strong if v is adjacent to at least two leaves. For a graph G, let f : V (G)! f0;1;2g be a function, and let (V0;V1;V2) be the ordered partition of V = V (G) induced by f, where Vi =fv2 V (G) : f(v) = ig for i = 0;1;2. There is a 1 1 correspondence between the functions f : V (G)!f0;1;2g and the ordered partitions (V0;V1;V2) of V (G). So we will write f = (V0;V1;V2). A function f : V (G)! f0;1;2g is a Roman dominating function (RDF) on G if every vertex u of G for which f(u) = 0 is adjacent to at least one vertex v of G for which f(v) = 2. The weight of an RDF is the value f(V (G)) = P u2V (G) f(u): An RDF f in a graph G is independent if no two vertices assigned positive values