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.
Leomarich F. Casinillo (Sun,) studied this question.