Super Hop Roman Domination in Graphs

Abstract

Let $G = (V(G), E(G))$ be a simple undirected graph. A function $f:V(G)\rightarrow \{0,1,2\}$ is a super hop Roman dominating function (SHRDF) on $G$ if for every $v\in V(G)$ with $f(v)=0$, there exist $w, u \in V(G)$ with $f(w) = 2$ and $f(u) \ne 0$ such that $d_G(v,w)=2$, and $N_G^2(u)\cap \{x \in V(G): f(x) =0\}=\{ v\}$. The \textit{weight} of SHRDF $f$, denoted  $\omega_G^{shR}(f)$, is given by $\omega_G^{shR}(f)=\sum_{y \in V(G)}f(y)$.  The \textit{super hop Roman domination number} of a graph $G$,  denoted $\gamma_{shR}(G)$, is the minimum weight of an SHRDF on $G$, that is, $\gamma_{shR}(G)=min\{\omega_G^{shR}(f): f \ is \ an \ \text{SHRDF} \ on \ G \}$. In this paper, we make an initial investigation of this newly defined variation of hop Roman domination in graphs. Some bounds and exact values of the parameter are obtained and some characterizations on some classes of graphs are given.