Note on the Independent Roman Domination Number of a Graph

A Roman dominating function (RDF) on a graph \( G \) is a function \( f: V(G) \to \{0,1,2\} \) satisfying the condition that every vertex \( u \) with \( f(u) = 0 \) is adjacent to at least one vertex \( v \) for which \( f(v) = 2 \). The weight of a Roman dominating function is the value \( f(V(G)) = \sum_{u \in V(G)} f(u) \). The Roman domination number, \( \gamma_{R}(G) \), of \( G \) is the minimum weight of a Roman dominating function on \( G \). An RDF \( f \) is called an independent Roman dominating function if the set of vertices assigned non-zero values is independent. The independent Roman domination number, \( i_R(G) \), of \( G \) is the minimum weight of an independent RDF on \( G \). In this paper, we improve previous bounds on the independent Roman domination number of a graph.