  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 $k\ge 3$ 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.
