Critical graphs upon multiple edge subdivision

A subset $D$ of $V$ is \emph{dominating} in $G$ if every vertex of $V-D$ has at least one neighbour in $D;$ let $\gamma (G)$ be the minimum cardinality among all dominating sets in $G\mathrm{.}$ A graph $G$ is $\gamma$-$q$-{\it critical} if the smallest subset of edges whose subdivision necessarily increases $\gamma (G)$ has cardinality $q\mathrm{.}$ In this paper we consider mainly $\gamma$-$q$-critical trees and give some general properties of $gamma$-$q$-critical graphs. In particular, we show that if $T$ is a $\gamma$-$q$-critical tree, then $1 \leq q \leq n(T)-1$andwecharacterizeextremaltreeswhen$q=n(T)-1.$ Since a subdivision number {of a tree $T$} $\mathrm{s}\mathrm{d}(T)$ is always $1,2$ or $3,$ we also characterize $\gamma$-2-critical trees $T$ with $\mathrm{s}\mathrm{d}(T)=2$ and $\gamma$-3-critical trees $T$ with $\mathrm{s}\mathrm{d}(T)=3.$