  We study the VC-dimension of the set system on the vertex set of some graph
which is induced by the family of its $k$-connected subgraphs. In particular,
we give tight upper and lower bounds for the VC-dimension. Moreover, we show
that computing the VC-dimension is $\mathsf{NP}$-complete and that it remains
$\mathsf{NP}$-complete for split graphs and for some subclasses of planar
bipartite graphs in the cases $k = 1$ and $k = 2$. On the positive side, we
observe it can be decided in linear time for graphs of bounded clique-width.
