Critical graphs upon multiple edge subdivision

Abstract

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 ${\rm sd}(T)=3.$