Weak edge Roman domination in graphs

Let G = (V,E) be a graph and let f be a function f : E → {0, 1, 2}. An edge x with f(x) = 0 is said to be undefended with respect to f if it is not incident to an edge with positive weight. The function f is a weak edge Roman dominating function (WERDF) if each edge x with f(x) = 0 is incident to an edge y with f(y) > 0 such that the function f ′ : E → {0, 1, 2}, defined by f ′(x) = 1, f ′(y) = f(y) − 1 and f ′(z) = f(z) if z ∈ E − {x, y}, has no undefended edge. The weight of f ′ is w(f ′) = ∑ x∈E f(x). The weak edge Roman domination number, denoted by γ′ WR(G), is the minimum weight of a WERDF in G. We show that for every graph G, γ′(G) ≤ γ′ WR(G) ≤ 2γ′(G), where γ′(G) is the edge domination number of G. In this paper first we characterize trees T for which γ′ WR(T ) = γ ′(T ). Then we also characterize trees and unicyclic graphs for which γ′ WR(G) = 2γ ′(G).