Relations between edge removing and edge subdivision concerning domination number of a graph

Abstract

Let e be an edge of a connected simple graph G. The graph obtained by removing (resp. subdividing) an edge e from G is denoted by G−e (resp. Ge). As usual, γ(G) denotes the domination number of G. We call G an SR-graph if for every edge e of G, γ(G−e)=γ(Ge); and G is an ASR-graph if for every edge e of G, γ(G−e)≠γ(Ge). In this work we give several examples of SR and ASR-graphs. We characterize SR-trees and show that ASR-graphs are efficient and γsd-critical. Consequently, ASR graphs are γ-insensitive and satisfy Vizing’s Conjecture.