A note on the edge Roman domination in trees

A subset X of edges of a graph G is called an edge dominating set of G if every edge not in X is adjacent to some edge in X . The edge domination number γ′(G) of G is the minimum cardinality taken over all edge dominating sets of G. An edge Roman dominating function of a graph G is a function f : E(G) → {0, 1, 2} such that every edge e with f(e) = 0 is adjacent to some edge e′ with f(e′) = 2. The weight of an edge Roman dominating function f is the value w(f) = ∑ e∈E(G) f(e). The edge Roman domination number of G, denoted by γ ′ R(G), is the minimum weight of an edge Roman dominating function of G. In this paper, we characterize trees with edge Roman domination number twice the edge domination number.