On Restrained Hop Roman Domination in Graphs

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.