On outer-convex Roman dominating function in graphs

Source abstract

Let $G = (V(G), E(G))$ be a connected graph and let $\phi:V(G)\rightarrow \{0,1,2\}$ be a Roman dominating function (RDF) on $G$. For each $j \in \{0, 1, 2\}$, let $V_j=\{x \in V(G): \phi(x)=j\}$. Then $\phi$ can be represented as $\phi=(V_0, V_1, V_2)$. A function $\phi$ is an \textit{outer-convex Roman dominating function} (OConRDF) on $G$ if for each $v\in V_0$, there exists $u\in V_2$ such that $uv\in E(G)$ and $V_0$ is a convex set in $G$. The weight of OConRDF $\phi$ is denoted by $\widetilde{\omega}_G^{conR}(\phi)$ and is defined as $\widetilde{\omega}_G^{conR}(\phi)=\sum_{x \in V(G)}\phi(x)$, that is, $\widetilde{\omega}_G^{conR}(\phi)=|V_1|+2|V_2|$. Additionally, the outer-convex Roman domination number of $G$ is denoted by $\widetilde{\gamma}_{conR}(G)$ and is defined as the minimum weight of an OConRDF on $G$, that is, $\widetilde{\gamma}_{conR}(G)=min\{\widetilde{\omega}_G^{conR}(\phi): \phi \ is \ an \ \text{OConRDF} \ on \ G \}$. Furthermore, any OConRDF $\phi$ on $G$ with $\widetilde{\omega}_{G}^{conR}(\phi)= \widetilde{\gamma}_{conR}(G)$ is called a $\widetilde{\gamma}_{conR}$-function on $G$. This paper introduces a new parameter of a Roman dominating function in graphs and discusses some theoretical properties, including bounds, the realization problem, and characterizations.