Semitotal Roman Domination in Graphs

Abstract

Let $G$ be a nontrivial graph without isolated vertices. A function $f:V(G) \to \{0,1,2\}$ is a semitotal Roman dominating function of $G$ if for each $v\in V(G)$ with $f(v)=0$, there exists $u\in V(G)$ for which $f(u)=2$ and $uv\in E(G)$ and for each $v \in V(G)$ with $f(v)\neq 0$, there exists $u\in V(G)$ for which $f(u)\neq 0$ and $d_G(u,v)\le 2$. The minimum weight $\omega_G(f)=\sum_{u\in V(G)}f(u)$ of a semitotal Roman dominating function $f$ of $G$ is the \textit{semitotal Roman domination number} of $G$, denoted by $\gamma_{t2R}(G)$. In this paper, we initiate the study of semitotal Roman domination. We characterize graphs $G$ with small values of $\gamma_{t2R}(G)$ and solve some realization problems with other existing related concepts. We also investigate the semitotal Roman domination in the join, corona, and complimentary prism of graphs.