Geodetic Roman Dominating Functions in a Graph (European Journal of Pure and Applied Mathematics, 2023)
Let $G$ be a connected graph. A function $f: V(G)→ \{0,1,2\}$ is a 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 $ω_{G}^{gR}(f)$, is given by $ω_{G}^{gR}(f)=\sum_{v \in V(G)}f(v)$. The minimum weight of a GRDF on $G$, denoted by $γ_{gR}(G)$, is called the 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.
Keywords: geodetic set; geodetic dominating set; Roman dominating function; Roman domination number; geodetic Roman dominating function; geodetic Roman domination number