Erdos-Posa 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 Erdos-Posa property holds for chordless cycles, which resolves the major open question concerning the Erdos-Posa 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 0(opt log opt) for CHORDAL VERTEX DELETION. We complement our main result by showing that chordless cycles of length at least P for any fixed l >= 5 do not have the Erdos-Posa property. (C) 2020 The Authors. Published by Elsevier Inc.