Annals of Communications in Mathematics

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.

ABSTRACT.Let \( G = (V(G), E(G)) \) be a simple non-complete graph and let \( \xi : V \rightarrow \{0,1,2\} \) be a hop Roman dominating function (HRDF) on \( G \). For each \( j \in \{0,1,2\} \), let \( V_j = \{x \in V(G) : \xi(x) = j\} \). Then \( \xi = (V_0, V_1, V_2) \). A function \( \xi \) is an interior hop Roman dominating function (InHRDF) on \( G \) if for each \( v \in V_0 \), there exists \( u \in V_2 \) such that \( d_G(u,v) = 2 \), and either \( V_1 = V(G) \) or for every \( v \in V_2 \), \( v \) is an interior vertex of \( G \). The weight of InHRDF \( \xi \) is denoted by \( \omega_G^{\text{InhR}}(\xi) \) and is defined as \( \omega_G^{\text{InhR}}(\xi) = \sum_{u \in V(G)} \xi(u) = |V_1| + 2|V_2| \).The minimum weight of an InHRDF \( \xi \) on \( G \), denoted as \( \gamma_{\text{InhR}}(G) = \min \{ \omega_G^{\text{InhR}}(\xi) : \xi \text{ is an InHRDF on } G \} \), is called the interior hop Roman domination number. Every InHRDF \( \xi \) on \( G \) satisfying the condition \( \omega_G^{\text{InhR}}(\xi) = \gamma_{\text{InhR}}(G) \) is called a \( \gamma_{\text{InhR}} \)-function on \( G \). In this paper, we investigate a new restricted parameter of a hop Roman dominating function in graphs called the interior hop Roman domination and present some combinatorial results.