A characterization of double Roman trees

A double Roman dominating function on a graph G is a function f:V(G)→{0,1,2,3} having the property that every vertex of weight 0 under f has at least one neighbor of weight 3 under f or at least two neighbors of weight 2 under f and every vertex of weight 1 under f has at least one neighbor of weight at least 2 under f. The weight of a double Roman dominating function f is the sum of the weights of the vertices. The double Roman domination number of G, denoted γ _dR (G), is the minimum weight of a double Roman dominating function in G. For every graph G, γ _dR (G)≤3γ(G), where γ(G) denotes the domination number of G. A graph G satisfying γ _dR (G)=3γ(G) is a double Roman graph. In this paper, we characterize the double Roman trees, which answers a problem posed by Beeler et al. (2016).