A connected subgraph maintaining high connectivity

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 . ted 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 3 -connected subgraph W such that G − W is still highly connected. More precisely, our main result says the following. ≥ 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. esult can be compared to Mader’s result (Mader, 2002)  [5] which says that every k -connected graph G of sufficiently large order ( k ≥ 4 ) has a connected subgraph H of order exactly 4 such that G − H is ( k − 3 ) -connected.