# Tight Bounds on Vertex Connectivity Under Sampling

• Published 2017 in ACM Trans. Algorithms

#### Abstract

A fundamental result by Karger [10] states that for any &#955;-edge-connected graph with <i>n</i> nodes, independently sampling each edge with probability <i>p</i> &equals; &#937;(log (<i>n</i>)/&#955;) results in a graph that has edge connectivity &#937;(&#955;<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>&equals;&#937;(&sqrt;log(<i>n</i>)/<i>k</i>), then the subgraph induced by the sampled nodes has vertex connectivity &#937;(<i>kp</i><sup>2</sup>), with high probability. If edges are sampled with probability <i>p</i> &equals; &#937;(log (<i>n</i>)/<i>k</i>), then the sampled subgraph has vertex connectivity &#937;(<i>kp</i>), with high probability. Both bounds are existentially optimal.

DOI: 10.1145/3086465

### Cite this paper

@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} }