Computing vertex connectivity: new bounds from old techniques

The vertex connectivity κ of a graph is the smallest number of vertices whose deletion separates the graph or makes it trivial. We present the fastest known deterministic algorithm for finding the vertex connectivity and a corresponding separator. The time for a digraph having n vertices and m edges is O(min{κ³+n,κn}m); for an undirected graph the term m can be replaced by κn. A randomized algorithm finds κ with error probability 1/2 in time O(nm). If the vertices have nonnegative weights the weighted vertex connectivity is found in time O(κ₁nmlog(n²/m)) where κ₁⩽m/n is the unweighted vertex connectivity, or in expected time O(nm log(n²/m)) with error probability 1/2. The main algorithm combines two previous vertex connectivity algorithms and a generalization of the preflow push algorithm of J. Hao and J.B. Orlin (1994) that computes edge connectivity