A note on $k$-critically $n$-connected graphs

ABsTRAcr. A graph G is said to be (n*, k)-connected if it has connectivity n and every set of k vertices is contained in an n-cutset. It is shown that an (n*, k)-connected graph G contains an n-cutset C such that G C has a component with at most n/(k + 1) vertices, thereby generalizing a result of Chartrand, Kaugars and Lick. It is conjectured, however, that n/(k + 1) can be replaced with n/2k and this is shown to be best possible.