- Published 2017 in ACM Trans. Algorithms

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

@article{CensorHillel2017TightBO,
title={Tight Bounds on Vertex Connectivity Under Sampling},
author={Keren Censor-Hillel and Mohsen Ghaffari and George Giakkoupis and Bernhard Haeupler and Fabian Kuhn},
journal={ACM Trans. Algorithms},
year={2017},
volume={13},
pages={19:1-19:26}
}