Fast parallel algorithms for testing k-connectivity of directed and undirected graphs

It appears that no NC algorithms have previously appeared for testing a directed graph for k-edge connectivity or k-vertex connectivity, even for fixed k>1. Using an elementary flow method we give such algorithms, with time complexity O(k log n) using nP(n,m) or (n+k²)P(n,m) processors, respectively. Here, n is the number of vertices, m is the number of edges, P(n,m) is the number of processors needed to find some path in time O(log n) time between two specified vertices in a directed graph with O(n) vertices and O(m) edges, and the computation model is a CRCW PRAM. These algorithms of course apply also to undirected graphs, but using sparse certificates we can improve the factors P(n,m) to P(n,kn) for both types of connectivity. This is better in time by a factor of O(k) over previous algorithms for undirected graphs. We also note that edge connectivity is NC-reducible to vertex connectivity even if k is not fixed