High connectivity keeping connected subgraph

It was proved by Mader that, for every integer l, every k-connected graph of sufficiently large order contains a vertex set X of order precisely l such that G−X is (k − 2)-connected. This is no longer true if we require X to be connected, even for l = 3. Motivated by this fact, we are trying to find an ”obstruction” for k-connected graphs without such a connected subgraph. It turns out that the obstruction is an essentially 3connected subgraph W such that G −W is still highly connected. More precisely, our main result says the following. For k ≥ 7 and every k-connected graph G, either there exists a connected subgraph W of order 4 in G such that G − W is (k − 2)-connected, or else G contains an ”essentially” 3-connected subgraph W , i.e., a subdivision of a 3-connected graph, such that G−W is still highly connected, actually, (k − 6)-connected. This result can be compared to Mader’s result [6] which says that every k-connected graph G of sufficiently large order (k ≥ 4) has a connected subgraph H of order exactly four such that G−H is (k − 3)-connected. ∗This work is supported by the JSPS Research Fellowships for Young Scientists. †Research partly supported by Japan Society for the Promotion of Science, Grant-in-Aid for Scientific Research, by C&C Foundation, by Inamori Foundation and by Kayamori Foundation.