The vertex connectivity /spl kappa/ 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{/spl kappa//sup 3/+n,/spl kappa/n}m); for an undirected graph the term m can be replaced by /spl kappa/n. A randomized algorithm finds /spl kappa/ with error probability 1/2 in time O(nm). If the vertices have nonnegative weights the weighted vertex connectivity is found in time O(/spl kappa//sub 1/nmlog(n/sup 2//m)) where /spl kappa//sub 1//spl les/m/n is the unweighted vertex connectivity, or in expected time O(nm log(n/sup 2//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.
展开▼
