Graphs with Many Independent Vertex Cuts

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 \(k\ge 3\) $k\ge 3$ there exists a unique graph G satisfying the following three 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. However, the edge version of this result does not hold. We also consider the problem when replacing independent sets by the periphery.