Erdős-Pósa property of chordless cycles and its applications
Eun Jung Kim, O‐joung Kwon · Journal of Combinatorial Theory Series B · 2020
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 c1k2logk+c2 vertices hitting every chordless cycle for some constants c1 and c2. It immediately implies an approximation algorithm of factor O(optlogopt) 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.