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(λp), 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(kp²), 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.