A note on outer-connected hop Roman dominating function in graphs

Abstract

Let $G=(V(G), E(G))$ be a simple, connected, and finite graph with vertex set $V(G)$ and edge set $E(G)$.  Let $\phi: V(G) \rightarrow \{0, 1, 2\}$  be an HRDF on $G$, and for each $i\in \{0, 1, 2\}$, let  $V_i=\{u\in V(G): \phi(u)=i\}$. A function $\phi=(V_0, V_1, V_2)$ 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 $\phi$ denoted by $\widetilde{\omega}_G^{chR}(\phi)$ and defined by $\widetilde{\omega}_G^{chR}(\phi)=\sum_{v\in V(G)}\phi(v)$=|V_1|+2|V_2|$. The outer-connected hop Roman domination number of $G$ is denoted by $\widetilde{\gamma}_{chR}(G)$ and defined by $\widetilde{\gamma}_{chR}(G)=min\{$\widetilde{\omega}_G^{chR}(\phi): \phi is an OcHRDF  on G\}$. Moreover, any OcHRDF $\phi$ on $(G)$ with $\widetilde{\gamma}_{chR}(G)=$\widetilde{\omega}_G^{chR}(\phi)$ is called $\overline{\gamma}_{chR}$-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.