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