The cycles are the only 2-connected graphs in which any two nonadjacent
vertices form a vertex cut. We generalize this fact by proving that for every
integer k≥3 there exists a unique graph G satisfying the following
conditions: (1) G is k-connected; (2) the independence number of G is
greater than k; (3) any independent set of cardinality k is a vertex cut of
G. The edge version of this result does not hold. We also consider the
problem when replacing independent sets by the periphery