Geodetic Roman Dominating Functions in a Graph

Let $G$ be a connected graph. A function $f: V(G)\rightarrow \{0,1,2\}$ is a \textit{geodetic Roman dominating function} (or GRDF) if every vertex $u$ for which $f(u)=0$ is adjacent to at least one vertex $v$ for which $f(v)=2$ and $V_1 \cup V_2$ is a geodetic set in $G$. The weight of a geodetic Roman dominating function $f$, denoted by $\omega_{G}^{gR}(f)$, is given by $\omega_{G}^{gR}(f)=\sum_{v \in V(G)}f(v)$. The minimum weight of a GRDF on $G$, denoted by $\gamma_{gR}(G)$, is called the \textit{geodetic Roman domination number} of $G$. In this paper, we give some properties of geodetic Roman domination and determine the geodetic Roman domination number of some graphs.