Abstract
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.
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