# Vertex Connectivity in Poly-logarithmic Max-flows

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 Õ(m^α) time for any α≥ 1, if there is a m^α-time maxflow algorithm. Using the current best maxflow algorithm that runs in m^4/3+o(1) time (Kathuria, Liu and Sidford, FOCS 2020), this yields a m^4/3+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 Õ(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 Õ(m)-time maxflow algorithm. Our new technique is robust enough to also improve the best Õ(mn)-time bound for directed vertex connectivity to mn^1-1/12+o(1) time

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