On k-con-Critically n-Connected Graphs

We prove that every n-connected graph G of sufficiently large order contains a connected graph H on four vertices such that G−V(H) is (n−3)-connected. This had been conjectured in Mader (High connectivity keeping sets in n-connected graphs, Combinatorica, to appear). Furthermore, we prove upper bounds for the order of all n-connected graphs of criticality 3, 4, and 5.