Effect of Edge Deletion on the Weak Roman Domination Number of a Graph

Abstract

Consider a graph Γ with vertex set V and edge set E. For a function g defined on the vertex set having values in the set {0,1,2}, the weight of a vertex x is g(x). The weight of a subset X of V is denoted by g(X) and is defined as the sum of the weights of all the vertices in X. An undefended vertex under g is a vertex a in Γ with a neighbourhood with weight 0. 
A weak Roman Dominating Function g (WRDF) is a function such that for any vertex x having weight 0, there is a vertex y in the open neighbourhood of x having positive weight such that, under the function h defined on V having values in the set {0,1,2} defined by h(x)=1,h(y)=g(y)-1,h(z)=g(z), if z∈V-{x,y}, there are no undefended vertices in Γ. The number γ_r (Γ), the minimum of the weights of all the WRDFs defined on Γ is called the weak Roman Domination Number of  Γ. The WRDF whose weight is γ_r (Γ) is called a γ_r- function and  γ_r (Γ) is called the γ_r- value of  Γ.
With respect to a graph Γ, we say that an edge x∈E^- if and only if its removal will reduce the γ_r- value of Γ. Similarly, we say that an edge x∈E^+ if and only if its removal will increase the γ_r- value of Γ and an edge x∈E^0 if and only if its removal will leave unaltered the γ_r- value of Γ. 
In this paper, we categorize edges of a graph as belonging to E^-,E^+ or E^0.