Odd Cycle Transversal on <i>P</i> ₅ -free Graphs in Polynomial Time

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 minimum weight vertex subset \(S\) in \(G\) such that \(G-S\) is bipartite; the weight of \(S\) , \(\text{w}(S)=\sum_{v\in S}\text{w}(v)\) . 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 Rzążewski, 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 Rzążewski [Dagstuhl Reports, 9(6): 2019], by Chudnovsky, King, Pilipczuk, Rzążewski, and Spirkl [SIDMA 2021] who gave an algorithm with running time \(n^{O(\sqrt{n})}\) for the problem, and by Agrawal, Lima, Lokshtanov, Saurabh, and Sharma [SODA 2024] who gave a quasi-polynomial time algorithm.