The VC-dimension of graphs with respect to k-connected subgraphs

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 NP-complete and that it remains 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.