Parallel depth-first search in general directed graphs

A directed cycle separator of an n-vertex directed graph is a vertex-simple directed cycle such that when the vertices of the cycle are deleted, the resulting graph has no strongly connected component with more than n/2 vertices. It is shown that the problem of finding a directed cycle separator is in randomized NC. It is also proved that computing cycle separators and conducting depth-first search in directed graphs are deterministically NC-equivalent. These two results together yield the first randomized NC algorithm for depth-first search in general directed graphs.