Not all planar diagraphs have small cycle separators

Abstract

Kao recently proved that all general directed graphs have directed cycle separators of any size. In contrast, Miller proved that planar undirected graphs with n vertices and maximum face size d have O( dn ) -size undirected cycle separators. Therefore it is important to investigate whether a planar directed graph has a small-size directed cycle separator. This paper shows that certain important classes of planar directed graphs have no sublinear-size directed cycle separators.