On Restrained Hop Roman Domination in Graphs

Source abstract

Let $G=(V(G), E(G))$ be a graph of order $n\in \mathbb{N}$ and let $f:V(G)\rightarrow \{0,1,2\}$ be a hop Roman dominating function (HRDF) on $G$. For each $j \in \{0, 1, 2\}$, let $V_j=\{y \in V(G): f(y)=j\}$. Then $f$ can be written as $f=(V_0, V_1, V_2)$. A function $f$ is a restrained hop Roman dominating function (RHRDF) on $G$ provided that for every $v\in V_0$, there exists $u\in V_0\setminus \{v\}$, and $w\in V_2$ such that $d_G(u, v)=1$ and $d_G(v, w)=2$. The weight of RHRDF $f$ denoted by $\omega_G^{rhR}(f)$ is the sum $\omega_G^{rhR}(f)=\sum_{x \in V(G)}f(x)$, that is, $\omega_G^{rhR}(f)=|V_1|+2|V_2|$. The restrained hop Roman domination number of a graph $G$ denoted by $\gamma_{rhR}(G)$ is the minimum weight of an RHRDF on $G$, that is, $\gamma_{rhR}(G)=min\{\omega_G^{rhR}(f): f \ is \ an \ \text{RHRDF} \ on \ G \}$. Every RHRDF $f$ on $G$ with the restriction $\omega_{G}^{rhR}(f)= \gamma_{rhR}(G)$ is called a $\gamma_{rhR}$-function on $G$. This paper introduces the concept of restrained hop Roman domination in graphs as a new restricted parameter and discusses some important theoretical properties.