A note on the independent Roman domination in unicyclic graphs

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 \(\gamma _R (G) \equiv i_R (G)\)