Tight Bounds on Vertex Connectivity Under Sampling

A fundamental result by Karger [10] states that for any λ-edge-connected graph with n nodes, independently sampling each edge with probability p = Ω(log ( n )/λ) results in a graph that has edge connectivity Ω(λ p ), with high probability. This article proves the analogous result for vertex connectivity, when either vertices or edges are sampled. We show that for any k -vertex-connected graph G with n nodes, if each node is independently sampled with probability p =Ω(√log( n )/ k ), then the subgraph induced by the sampled nodes has vertex connectivity Ω( kp 2 ), with high probability. If edges are sampled with probability p = Ω(log ( n )/ k ), then the sampled subgraph has vertex connectivity Ω( kp ), with high probability. Both bounds are existentially optimal.