A note on outer-connected hop Roman dominating function in graphs - Casinillo, Leomarich F (Journal of Fundamental Mathematics and Applications (JFMA), 2025)
Let $G=(V(G), E(G))$ be a simple, connected, and finite graph with vertex set $V(G)$ and edge set $E(G)$. Let $φ: V(G) → \{0, 1, 2\}$ be an HRDF on $G$, and for each $i\in \{0, 1, 2\}$, let $V_i=\{u\in V(G): φ(u)=i\}$. A function $φ=(V_0, V_1, V_2)$ is an outer-connected hop Roman dominating function (OcHRDF) on $G$ if, for every $v\in V_0$, there exists $u\in V_2$ such that $d_G(u, v)=2$ and either $V_1=V(G)$ or the sub-graph $\langle V_0 \rangle$ is connected. The weigth of OcHRDF $φ$ denoted by $\widetilde{ω}_G^{chR}(φ)$ and defined by $\widetilde{ω}_G^{chR}(φ)=\sum_{v\in V(G)}φ(v)$=|V_1|+2|V_2|$. The outer-connected hop Roman domination number of $G$ is denoted by $\widetilde{γ}_{chR}(G)$ and defined by $\widetilde{γ}_{chR}(G)=min\{$\widetilde{ω}_G^{chR}(φ): φ is an OcHRDF on G\}$. Moreover, any OcHRDF $φ$ on $(G)$ with $\widetilde{γ}_{chR}(G)=$\widetilde{ω}_G^{chR}(φ)$ is called $\overline{γ}_{chR}$-function on $G$. In this paper, a new restricted parameter of a hop Roman domination in graphs is introduced, and some combinatorial properties are discussed.