  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 $n^{O(\sqrt{n})}$.
