## Abstract

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, (equation presented). We show that Odd Cycle Transversal admits an algorithm with running time n^{O(log2 n)} 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 Rzążewski, 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 Rzążewski [Dagstuhl Reports, 9(6): 2019] and by Chudnovsky, King, Pilipczuk, Rzążewski, and Spirkl [SIDMA 2021], who gave an algorithm with running time (Equation presented). While our (Equation presented) time algorithm falls short of completely resolving the complexity status of Odd Cycle Transversal on P5-free graphs it shows that the problem is not NP-hard unless every problem in NP is solvable in quasi-polynomial time.

Original language | English |
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Pages | 5276-5290 |

Number of pages | 15 |

DOIs | |

State | Published - 1 Jan 2024 |

Externally published | Yes |

Event | 35th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2024 - Alexandria, United States Duration: 7 Jan 2024 → 10 Jan 2024 |

### Conference

Conference | 35th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2024 |
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Country/Territory | United States |

City | Alexandria |

Period | 7/01/24 → 10/01/24 |

## ASJC Scopus subject areas

- Software
- General Mathematics

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