Trees with equal Roman {2}-domination number and independent Roman {2}-domination number

Abstract

A Roman {2}-dominating function (R{2}DF) on a graph G =(V, E) is a function f : V → {0, 1, 2} satisfying the condition that every vertex u for which f(u) = 0 is adjacent to either at least one vertex v with f(v) = 2 or two vertices v1, v2 with f(v1) = f(v2) = 1. The weight of an R{2}DF f is the value w(f) = ∑u∈Vf(u). The minimum weight of an R{2}DF on a graph G is called the Roman {2}-domination number γ{R2}(G) of G. An R{2}DF f is called an independent Roman {2}-dominating function (IR{2}DF) if the set of vertices with positive weight under f is independent. The minimum weight of an IR{2}DF on a graph G is called the independent Roman {2}-domination number i{R2}(G) of G. In this paper, we answer two questions posed by Rahmouni and Chellali.