# Vertex connectivity in poly-logarithmic max-flows

Journal Article

The vertex connectivity of an m-edge n-vertex undirected graph is the smallest number of vertices whose removal disconnects the graph, or leaves only a singleton vertex. In this paper, we give a reduction from the vertex connectivity problem to a set of maxflow instances. Using this reduction, we can solve vertex connectivity in (m?) time for any ? ? 1, if there is a m?-time maxflow algorithm. Using the current best maxflow algorithm that runs in m4/3+o(1) time (Kathuria, Liu and Sidford, FOCS 2020), this yields a m4/3+o(1)-time vertex connectivity algorithm. This is the first improvement in the running time of the vertex connectivity problem in over 20 years, the previous best being an O(mn)-time algorithm due to Henzinger, Rao, and Gabow (FOCS 1996). Indeed, no algorithm with an o(mn) running time was known before our work, even if we assume an (m)-time maxflow algorithm. Our new technique is robust enough to also improve the best O(mn)-time bound for directed vertex connectivity to mn1-1/12+o(1) time

### Full Text

### Duke Authors

### Cited Authors

- Li, J; Nanongkai, D; Panigrahi, D; Saranurak, T; Yingchareonthawornchai, S

### Published Date

- June 15, 2021

### Published In

### Start / End Page

- 317 - 329

### International Standard Serial Number (ISSN)

- 0737-8017

### Digital Object Identifier (DOI)

- 10.1145/3406325.3451088

### Citation Source

- Scopus