A half-integral Erdős-Pósa theorem for directed odd cycles

Abstract

We prove that there exists a function $f:N\to R$ such that every directed graph G contains either k directed odd cycles where every vertex of G is contained in at most two of them, or a set of at most $f\left(k\right)$ vertices meeting all directed odd cycles. We give a polynomial-time algorithm for fixed k which outputs one of the two outcomes. This extends the half-integral Erdős-Pósa theorem for undirected odd cycles by Reed [Combinatorica 1999] to directed graphs.