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.