Bounds for Independent Roman Domination in Graphs

A Roman dominating function on a graph \( G \) is a function \( f: V(G) \to \{0,1,2\} \) such that every vertex \( u \) with \( f(u) = 0 \) is adjacent to a vertex \( v \) with \( f(v) = 2 \). The weight of a Roman dominating function \( f \) is the value \( f(V(G)) = \sum_{u \in V(G)} f(u) \). A Roman dominating function \( f \) is an independent Roman dominating function if the set of vertices for which \( f \) assigns positive values is independent. The independent Roman domination number \( i_R(G) \) of \( G \) is the minimum weight of an independent Roman dominating function of \( G \).

We show that if \( T \) is a tree of order \( n \), then \( i_R(T) \leq \frac{4n}{5} \), and characterize the class of trees for which equality holds. We present bounds for \( i_R(G) \) in terms of the order, maximum and minimum degree, diameter, and girth of \( G \). We also present Nordhaus-Gaddum inequalities for independent Roman domination numbers.