A characterization of perfect Roman trees

Abstract A set S of vertices is a perfect dominating set of a graph G if every vertex not in S is adjacent to exactly one vertex of S . The minimum cardinality of a perfect dominating set is the perfect domination number γ p ( G ) . A perfect Roman dominating function (PRDF) on a graph G = ( V , E ) is a function f : V → { 0 , 1 , 2 } satisfying the condition that every vertex u with f ( u ) = 0 is adjacent to exactly one vertex v for which f ( v ) = 2 . The weight of a PRDF is the sum of its function values over all vertices, and the minimum weight of a PRDF of G is the perfect Roman domination number γ R p ( G ) . Obviously, for every graph G , γ R p ( G ) ≤ 2 γ p ( G ) , and those graphs attaining the equality are called perfect Roman graphs. In this paper, we provide a characterization of perfect Roman trees.