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 ${\displaystyle w\left(f\right)={\sum }_{v\in V}f\left(v\right)}$. The weak Roman domination number, denoted by ${\displaystyle {\gamma }_r\left(G\right)}$, is the minimum weight of a WRDF in G. In this paper, we characterize the class of trees and split graphs for which ${\displaystyle {\gamma }_r\left(G\right)=\gamma \left(G\right)}$ and find ${\displaystyle {\gamma }_r}$-value for a caterpillar, a 2×n grid graph and a complete binary tree.