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

Let \(e\) be an edge of a connected simple graph \(G\). The graph obtained by removing (subdividing) an edge \(e\) from \(G\) is denoted by \(G-e\) (\(G_e\)). As usual, \(\gamma(G)\) denotes the domination number of \(G\). We call \(G\) an SR-graph if \(\gamma(G-e) = \gamma(G_e)\) for any edge \(e\) of \(G\), and \(G\) is an ASR-graph if \(\gamma(G - e) \neq (G_e)\) for any edge \(e\) of \(G\). In this work we give several examples of SR and ASR-graphs. Also, we characterize SR-trees and show that ASR-graphs are \(\gamma\)-insensitive.