On Convex Hop Roman Domination in Graphs

Let \(G=(V(G), E(G))\) be a connected graph and \(f: V(G) →\{0,1,2\}\) be a hop Roman dominating function (HRDF) on \(G\). For each \(i ∈\{0,1,2\}\), let \(Vᵢ=\{v ∈ V(G): f(v)=i\}\). Then we write \(f=\) \((V₀, V₁, V₂)\). A function \(f\) is a convex hop Roman dominating function (CvHRDF) on \(G\) if for every \(x ∈ V₀\), there exists \(y ∈ V₂\) such that \(dG(x, y)=2\), and \(V₁ ∪ V₂\) is a convex set. The weight of CvHRDF \(f\) denoted by \(ωGC v h R(f)\) is the sum \(ωGC v h R(f)=∑w ∈ V(G) f(w)\). The convex hop Roman domination number of \(G\) denoted by \(γC v h R(G)\) 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.