Erdős-Pósa property of chordless cycles and its applications.

A chordless cycle, or equivalently a hole, in a graph G is an induced subgraph of G which is a cycle of length at least 4. 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 c 1 k 2 log ⁡ k + c 2 vertices hitting every chordless cycle for some constants c 1 and c 2. 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. [ABSTRACT FROM AUTHOR]