All graphs have cycle separators and planar directed depth-first search is in DNC

Abstract

All graphs have cycle separators. (A single vertex is regarded as a trivial cycle.) In sequential computation, a cycle separator can be found in O(n + e) time for any undirected graph of n vertices and e edges; in O((n + e) log n) time for any directed graph. In parallel computation, it is in deterministic NC to convert any depth-first search forest of any graph into a cycle separator. Moreover, finding a cycle separator for any planar directed graph is in deterministic NC; consequently, finding a depth-first search forest in any planar directed graph is in deterministic NC, too.