Odd Cycle Transversal on P₅-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 smallest weight vertex subset S in G such that G - S is bipartite; the weight of S, w (S) = ₕ ₒ w (v). We show that Odd Cycle Transversal is polynomial-time solvable on graphs excluding P₅ (a path on five vertices) as an induced subgraph. The problem was previously known to be polynomial-time solvable on P₄-free graphs and NP-hard on P₆-free graphs Dabrowski, Feghali, Johnson, Paesani, Paulusma and Rza\. zewski, Algorithmica 2020. Bonamy, Dabrowski, Feghali, Johnson and Paulusma Algorithmica 2019 posed the existence of a polynomial-time algorithm on P₅-free graphs as an open problem, this was later re-stated by Rza\. zewski Dagstuhl Reports, 9 (6): 2019 and by Chudnovsky, King, Pilipczuk, Rza\. zewski, and Spirkl SIDMA 2021, who gave an algorithm with running time n^O (n).