A note on the edge Roman domination in trees - Nader Jafari Rad (GTA Research Group, Univ. Newcastle, Indonesian Combinatorics Society and ITB, 2017)
A subset X of edges of a graph G is called an edgedominating set of G if every edge not in X is adjacent tosome 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 edgee 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 valuew(f)=_eE(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.