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

초록

We prove that there exists a function f : ℕ → 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(k) vertices meeting all directed odd cycles. We also give a polynomial-time algorithm for fixed k which outputs one of the two outcomes. Using this algorithmic result, we give a polynomial-time algorithm for fixed k to decide whether such k directed odd cycles exist, or there are no k vertex-disjoint directed odd cycles. This extends the half-integral Erdős-Pósa theorem for undirected odd cycles by Reed [Combinatorica 1999] to directed graphs.

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