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

Abstract

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 Erd\H{o}s-P\'osa property holds for chordless cycles, which resolves the major open question concerning the Erd\H{o}s-P\'osa property. Our proof for chordless cycles is constructive: in polynomial time, one can find either $k+1$ vertex-disjoint chordless cycles, or $c_1k^2 \log k+c_2$ vertices hitting every chordless cycle for some constants $c_1$ and $c_2$. It immediately implies an approximation algorithm of factor $\mathcal{O}(\sf{opt}\log {\sf opt})$ for Chordal Vertex Deletion. We complement our main result by showing that chordless cycles of length at least $\ell$ for any fixed $\ell\ge 5$ do not have the Erd\H{o}s-P\'osa property.