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 $tilde O(m^{alpha})$ time for any $alpha ge 1$, if there is a $m^{alpha}$-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 $tilde O(mn)$-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 $tilde O(m)$-time maxflow algorithm. Our new technique is robust enough to also improve the best $tilde O(mn)$-time bound for directed vertex connectivity to $mn^{1-1/12+o(1)}$ time
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