On interior Roman domination in graphs

Let G = (V (G), E (G) ) be a non-complete graph and let: V (G) \0, 1, 2\ be a function on G. For each i \0, 1, 2\, let Vᵢ=\w V (G): (w) =i\. A function = (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 G^InR () =ₔ ₕ (₆) (u) is the weight of InRDF ; and the minimum weight of an InRDF on G, denoted by ₈₍ₑ (G), is called the interior Roman domination number. Any InRDF on graph G with ₆^InR () = ₈₍ₑ (G) is called a ₈₍ₑ-function on G. In this paper, we introduce a new parameter of a Roman dominating function in graphs and discuss some important combinatorial properties.