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 $\mathcal{O}(\mathsf{opt}\log \mathsf{opt})$ for Chordal Vertex Deletion. We complement our main result by showing that chordless cycles of length at least ℓ for any fixed $\ell \ge 5$ do not have the Erdős-Pósa property.

甲无线周期，或等效的孔，在图ģ是的导出子ģ其长度的一个周期至少为4。我们证明了ERDOS-POSA属性保存为chordless周期，这样就解决了有关的主要开放的问题Erdős-Pósa属性。我们对无弦循环的证明是有建设性的：在多项式时间内，可以找到$ķ+\mathrm{1个}$ 顶点不相交的无弦循环，或 $C_{\mathrm{1个}}{ķ}^2\mathrm{日志}ķ+C_2$ 顶点在每个无弦循环中敲击一些常数 $C_{\mathrm{1个}}$ 和 $C_2$。它立即意味着因子的近似算法$\mathcal{Ø}（\mathsf{选择}\mathrm{日志}\mathsf{选择}）$用于弦顶点删除。我们通过显示对于任何固定的长度至少为ℓ的无弦循环来补充我们的主要结果$\ell \ge 5$ 没有Erdős-Pósa属性。