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 γp(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 γRp(G). Obviously, for every graph G, γRp(G)≤2γp(G), and those graphs attaining the equality are called perfect Roman graphs. In this paper, we provide a characterization of perfect Roman trees.