Weak Roman domination in graphs

Let G = (V,E) be a graph and f be a function f : V → {0, 1, 2}. A vertex u with f(u) = 0 is said to be undefended with respect to f , if it is not adjacent to a vertex with positive weight. The function f is a weak Roman dominating function (WRDF) if each vertex u with f(u) = 0 is adjacent to a vertex v with f(v) > 0 such that the function f ′ : V → {0, 1, 2} defined by f ′ (u) = 1, f ′ (v) = f(v) − 1 and f ′ (w) = f(w) if w ∈ V − {u, v}, has no undefended vertex. The weight of f is w(f) = ∑ v∈V f(v). The weak Roman domination number, denoted by γr(G), is the minimum weight of a WRDF in G. In this paper, we characterize the class of trees and split graphs for which γr(G) = γ(G) and find γr-value for a caterpillar, a 2 × n grid graph and a complete binary tree.