On interior Roman domination in graphs

Let G = (V(G), E(G)) be a non-complete graph and let ϕ:V(G)→{0,1,2} be a function on G. For each i ∈ {0, 1, 2}, let Vi={w ∈ V(G): ϕ(w)=i}. A function ϕ=(V₀, V₁, V₂) is an interior Roman dominating function (InRDF) on G if (i) for every v ∈ V₀, there exists u ∈ V₂ such that uv ∈ E(G), and (ii) either V₁=V(G) or for every z ∈ V₂, z is an interior vertex of G. Denoted by ωGInR(ϕ)=∑u ∈ V(G) ϕ(u) is the weight of InRDF ϕ; and the minimum weight of an InRDF ϕ on G, denoted by γInR(G), is called the interior Roman domination number. Any InRDF ϕ on graph G with ωGInR(ϕ)= γInR(G) is called a γInR-function on G. In this paper, we introduce a new parameter of a Roman dominating function in graphs and discuss some important combinatorial properties.