An independent set in a graph G is a set of pairwise non-adjacent vertices. A graph $G$ is bipartite if its vertex set can be partitioned into two independent sets. In the Odd Cycle Transversal problem, the input is a graph $G$ along with a weight function $w$ associating a rational weight with each vertex, and the task is to find a smallest weight vertex subset $S$ in $G$ such that $G - S$ is bipartite; the weight of $S$, $w(S) = \sum_{v\in S} w(v)$. We show that Odd Cycle Transversal is polynomial-time solvable on graphs excluding $P_5$ (a path on five vertices) as an induced subgraph. The problem was previously known to be polynomial-time solvable on $P_4$-free graphs and NP-hard on $P_6$-free graphs [Dabrowski, Feghali, Johnson, Paesani, Paulusma and Rz\k{a}\.zewski, Algorithmica 2020]. Bonamy, Dabrowski, Feghali, Johnson and Paulusma [Algorithmica 2019] posed the existence of a polynomial-time algorithm on $P_5$-free graphs as an open problem, this was later re-stated by Rz\k{a}\.zewski [Dagstuhl Reports, 9(6): 2019] and by Chudnovsky, King, Pilipczuk, Rz\k{a}\.zewski, and Spirkl [SIDMA 2021], who gave an algorithm with running time $n^{O(\sqrt{n})}$.

翻译：图 G 中的独立集是指一组两两不相邻的顶点集合。若图 G 的顶点集可划分为两个独立集，则称 G 为二分图。在奇圈横贯问题中，输入为一个图 G 及一个关联每个顶点的有理权重函数 w，任务是找到 G 中最小权重的顶点子集 S，使得 G - S 为二分图；其中 S 的权重为 w(S) = ∑_{v∈S} w(v)。本文证明：在排除诱导子图 P_5（五顶点路径）的图中，奇圈横贯问题可在多项式时间内求解。此前已知该问题在 P_4 自由图上可多项式时间求解，而在 P_6 自由图上为 NP 难问题 [Dabrowski, Feghali, Johnson, Paesani, Paulusma 和 Rzążewski, Algorithmica 2020]。Bonamy、Dabrowski、Feghali、Johnson 和 Paulusma [Algorithmica 2019] 曾将 P_5 自由图上是否存在多项式时间算法作为公开问题提出，随后 Rzążewski [Dagstuhl Reports, 9(6): 2019] 以及 Chudnovsky、King、Pilipczuk、Rzążewski 和 Spirkl [SIDMA 2021] 再次重申该问题，后者还给出了一个运行时间为 n^{O(√n)} 的算法。