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, $γ(G)$ denotes the domination number of $G$. We call $G$ an SR-graph if $γ(G-e) = γ(G_e)$ for any edge $e$ of $G$, and $G$ is an ASR-graph if $γ(G - e) \neq (G_e)$ 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 $γ$-insensitive.

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