Tight Bounds on Vertex Connectivity Under Vertex Sampling
Keren Censor-Hillel, Mohsen Ghaffari, George Giakkoupis, Bernhard Haeupler, Fabian Kühn · 2014
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 paper proves the analogous result for vertex connectivity, when sampling vertices. We show that for any k-vertex-connected graph G with n nodes, if each node is independently sampled with probability , then the subgraph induced by the sampled nodes has vertex connectivity Ω(kp2), with high probability. This bound improves upon the recent results of Censor-Hillel et al. [6], and is existentially optimal.