The Erdos-Posa Property for Directed Graphs

A classical result by Erdős and Pósa[3] states that there is a function f : N → N such that for every k, every graph G contains k pairwise vertex disjoint cycles or a set T of at most f(k) vertices such that G− T is acyclic. The generalisation of this result to directed graphs is known as Younger’s conjecture and was proved by Reed, Robertson, Seymour and Thomas in 1996. This so-called Erdős-Pósa property can naturally be generalised to arbitrary graphs and digraphs. Robertson and Seymour proved that a graph H has the Erdős-Pósaproperty if, and only if, H is planar. In this paper we study the corresponding problem for digraphs. We obtain a complete characterisation of the class of strongly connected digraphs which have the Erdős-Pósa-property (both for topological and butterfly minors). We also generalise this result to classes of digraphs which are not strongly connected. In particular, we study the class of vertex-cyclic digraphs (digraphs without trivial strong components). For this natural class of digraphs we obtain a nearly complete characterisation of the digraphs within this class with the Erdős-Pósa-property. In particular we give positive and algorithmic examples of digraphs with the Erdős-Pósa-property by using directed tree decompositions in a novel way.