On the Global Roman Domination Number in Graphs

Abstract

A Roman dominating function $f$ on a graph $G$ is a global Roman dominating function on $G$, if $f$ is also a Roman dominating function on $\bar{G}$. The weight of a global Roman dominating function $f$ is the value $w(f) = \sum\nolimits_{x \in V(G)} {f(x)}$. The minimum weight of a global Roman dominating function on a graph $G$ is called the global Roman domination number $\gamma_{gR} (G)$ of $G$. In this paper, we present upper bounds for $\gamma_{gR} (G)$ in terms of order, diameter, and girth. We give necessary and sufficient conditions for a graph $G$ with property $\gamma_{gR} (G) = \gamma_{g} (G) + i$ for all $i = 0,1, 2,3$, where $\gamma_{g} (G)$ is the global domination number of $G$. We also describe all connected unicyclic graphs $G$ for which $\gamma_{gR} \left( G \right) - \gamma_{R} (G)$ is maximum.