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A certificate for the k-vertex connectivity of a graph G = (V, E) is a subset E’ of E such that (V, E’) is k-vertex connected iff G is k-vertex connected. Let n = IV[ and m = IEI. A certificate is called sparse if it has size lE’I = O(kn). We show that sparse certificates for undirected graphs can be computed by executing k breadth first searches in sequence. This directly gives an efficient algorithm on the distributed model of computation. Using certificates of size O(k2 n), we show that the k-vertex connectivity of an undirected graph can be tested on the CRCW PRAM model in a parallel running time of O(k2 log n) using k3n2cr(k2n, n)/(log n) processors. For k >3 and m > k2n, this improves on previous deterministic parallel algorithms (for dense graphs and for constant k >4 the number of processors improves by a factor of roughly n). We also give sequential algorithms for finding (undirected) sparse certificates ‘(on-line”, and for finding -. certificate es of size <2 k2 n for directed graphs,

Algorithms for parallel k-vertex connectivity and sparse certificates