Super Hop Roman Domination in Graphs (European Journal of Pure and Applied Mathematics, 2025)
Let $G = (V(G), E(G))$ be a simple undirected graph. A function $f:V(G)→ \{0,1,2\}$ 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 $f(u) e 0$ such that $d_G(v,w)=2$, and $N_G^2(u)\cap \{x \in V(G): f(x) =0\}=\{ v\}$. The weight of SHRDF $f$, denoted $ω_G^{shR}(f)$, is given by $ω_G^{shR}(f)=\sum_{y \in V(G)}f(y)$. The super hop Roman domination number of a graph $G$, denoted $γ_{shR}(G)$, is the minimum weight of an SHRDF on $G$, that is, $γ_{shR}(G)=min\{ω_G^{shR}(f): f \ is \ an \ \text{SHRDF} \ on \ G \}$. 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.
Keywords: super domination; hop Roman domination; super hop Roman domination