Erd\H{o}s-P\'osa property of chordless cycles and its applications

A chordless cycle in a graph G is an induced subgraph of G which is a cycle of length at least four. We prove that the Erdős-Pósa property holds for chordless cycles, which resolves the major open question concerning the Erdős-Pósa property. Our proof for chordless cycles is constructive: in polynomial time, one can find either k + 1 vertex-disjoint chordless cycles, or ck 2 log k vertices hitting every chordless cycle for some constant c. It immediately implies an approximation algorithm of factor O(opt log opt) for Chordal Vertex Deletion. We complement our main result by showing that chordless cycles of length at least l for any fixed l ≥ 5 do not have the Erdős-Pósa property. As a corollary, for a non-negative integral function w defined on the vertex set of a graph G, the minimum value ∑ x∈S w(x) over all vertex sets S hitting all cycles of G is at most O(k log k) where k is the maximum number of cycles (not necessarily vertex-disjoint) in G such that each vertex v is used at most w(v) times.