Interior hop Roman dominating function in graphs

Let G = (V (G), E(G)) be a simple non-complete graph and let ξ : V → {0, 1, 2} be an HRDF on G. For each j ∈ {0, 1, 2}, let Vj = {x ∈ V (G) : ξ(x) = j}. Then ξ = (V0, V1, V2). A function ξ is an interior hop Roman dominating function (InHRDF) on G if for each v ∈ V0, there exists u ∈ V2 such that dG(u, v) = 2, and eitherV1 = V (G) or for every w ∈ V2, w is an interior vertex of G. The weight of InHRDFξ is denoted by ωInhRG (ξ) and is defined as ωInhRG (ξ) = Pu∈V (G)ξ(u) = |V1| + 2|V2|. The minimum weight of an InHRDF ξ on G, denoted and defined by γInhR(G) =min{ωInhRG (ξ) : xi is an InHRDF on G}, is called the interior hop Roman domination number. Every InHRDF ξ on G satisfying the condition ω InhRG (ξ) = γInhR(G) is called a γInhR-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.