  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 $G.$ A graph $G$ is $\gamma$-$q$-{\it critical} if the
smallest subset of edges whose subdivision necessarily increases $\gamma(G)$
has cardinality $q.$ 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 $1 \leq q
\leq n(T)-1$ and we characterize extremal trees when $q=n(T)-1.$ Since a
subdivision number {of a tree $T$} ${\rm sd}(T)$ is always $1,2$ or $3,$ we
also characterize $\gamma$-2-critical trees $T$ with ${\rm sd}(T)=2$ and
$\gamma$-3-critical trees $T$ with ${\rm sd}(T)=3.$
