Strong equality of Roman and perfect Roman Domination in trees

Abstract

A Roman dominating function (RD-function) 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 at least one vertex v for which f(v) = 2. An Roman dominating function f in a graph G is perfect Roman dominating function (PRD-function) if every vertex u with f(u) = 0 is adjacent to exactly one vertex v for which f(v) = 2. The (perfect) Roman domination number γR(G) (γpR(G)) is the minimum weight of an (perfect) Roman dominating function on G. We say that γpR(G) strongly equals γR(G), denoted by γpR(G) ≡ γR(G), if every RD-function on G of minimum weight is a PRD-function. In this paper we show that for a given graph G, it is NP-hard to decide whether γpR(G) = γR(G) and also we provide a constructive characterization of trees T with γpR(T) ≡ γR(T).