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) = ∑v∈ S 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 Rz{a}\.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 Rz{a}\.zewski [Dagstuhl Reports, 9(6): 2019] and by Chudnovsky, King, Pilipczuk, Rz{a}\.zewski, and Spirkl [SIDMA 2021], who gave an algorithm with running time nO(√n).