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 $f : V \to {0, 1, 2}$ 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 $f(V(G)) = \sum _{u \in V (G)} f(u)$. An RDF $f$ in a graph $G$ is independent if no two vertices assigned positive values are adjacent. The Roman domination number $\gamma R (G)$ (respectively, the independent Roman domination number $i{R}(G)$) is the minimum weight of an RDF (respectively, independent RDF) on $G$. We say that $\gamma R (G)$ strongly equals $i{R}(G)$), denoted by $\gamma _R (G) \equiv i_R (G)$, if every RDF on $G$ of minimum weight is independent. In this note we characterize all unicyclic graphs $G$ with $\gamma _R (G) \equiv i_R (G).$