Roman domination in oriented trees

Abstract

Let D =( V,A ) be a digraph of order n = | V |. A Roman dominating function of a digraph D is a function f : V → {0,1,2} such that every vertex u for which f ( u ) = 0 has an in-neighbor v for which f ( v ) = 2. The weight of a Roman dominating function is the value f ( V )=∑ u∈V f ( u ). The minimum weight of a Roman dominating function of a digraph D is called the Roman domination number of D , denoted by γ R ( D ). In this paper, we characterize oriented trees T satisfying γ R ( T )+Δ + ( T ) = n +1.