Note on the Independent Roman Domination Number of a Graph

Abstract

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.