ON CONVEX HOP ROMAN DOMINATION IN GRAPHS

Let \(G=(V(G), E(G))\) be a connected graph and \(f: V(G) \rightarrow\{0,1,2\}\) be a hop Roman dominating function (HRDF) on \(G\). For each \(i \in\{0,1,2\}\), let \(V_{i}=\{v \in V(G): f(v)=i\}\). Then we write \(f=\) \(\left(V_{0}, V_{1}, V_{2}\right)\). A function \(f\) is a convex hop Roman dominating function (CvHRDF) on \(G\) if for every \(x \in V_{0}\), there exists \(y \in V_{2}\) such that \(d_{G}(x, y)=2\), and \(V_{1} \cup V_{2}\) is a convex set. The weight of CvHRDF \(f\) denoted by \(\omega_{G}^{C v h R}(f)\) is the sum \(\omega_{G}^{C v h R}(f)=\sum_{w \in V(G)} f(w)\). The convex hop Roman domination number of \(G\) denoted by \(\gamma_{C v h R}(G)\) is the minimum weight of a CvHRDF on \(G\). In this paper, we introduce and investigate a new parameter of hop Roman domination in graphs called the convex hop Roman dominating function. We also discuss important graph-theoretic results on the convex hop Roman domination number, including bounds, exact values, and characterizations.