ABSTRACT.

Let \( G = (V(G), E(G)) \) be a connected graph and let \( f : V(G) \rightarrow \{0,1,2\} \) be a hop Roman dominating function (HRDF) on \( G \). If for each \( k \in \{0,1,2\} \), \( V_k = \{x \in V(G) : f(x) = k\} \), then \( f = (V_0, V_1, V_2) \). A function \( f \) is an outer-convex hop Roman dominating function (OConHRDF) on \( G \) provided that for every \( v \in V_0 \), there exists \( u \in V_2 \) such that \( v \in N_G^2(u) \) and \( V_0 \) is a convex set. The weight of OConHRDF \( f \) on \( G \) is denoted by \( \widetilde{\omega}_G^{conhR}(f) \) and is defined as \( \widetilde{\omega}_G^{conhR}(f) = \sum_{v \in V(G)} f(v) \).

The smallest weight of an OConHRDF \( f \) on \( G \), denoted by \( \widetilde{\gamma}_{conhR}(G) \), is called the outer-convex hop Roman domination number, which can be written as \( \widetilde{\gamma}_{conhR}(G) = \min \{ \widetilde{\omega}_G^{conhR}(f) : f \text{ is an OConHRDF on } G \} \). Every OConHRDF \( f \) on \( G \) satisfying the condition \( \widetilde{\omega}_G^{conhR}(f) = \widetilde{\gamma}_{conhR}(G) \) is so-called a \( \widetilde{\gamma}_{conhR} \)-function on \( G \). This paper introduces a new parameter of a hop Roman dominating function in graphs, called outer-convex hop Roman dominating function and presents initial investigation.