# Detecting an odd hole

## TL;DR

This paper presents a polynomial-time algorithm for detecting odd holes in graphs, resolving a long-standing open problem and simplifying the process of testing graph perfection.

## Contribution

It introduces a new polynomial-time algorithm for detecting odd holes, solving an open complexity problem in graph theory.

## Key findings

- Polynomial-time algorithm for odd hole detection
- Simplified method for testing graph perfection
- Resolution of the open complexity question

## Abstract

A hole in a graph G is an induced cycle of length at least four; an antihole is a hole in the complement of G. In 2005, Chudnovsky, Cornuejols, Liu, Seymour and Vuskovic showed that it is possible to test in polynomial time whether a graph contains an odd hole or antihole (and thus whether G is perfect). However, the complexity of testing for odd holes has remained open. Indeed, it seemed quite likely that testing for an odd hole was NP-complete: for instance, Bienstock showed that testing if a graph has an odd hole containing a given vertex is NP-complete. In this paper we resolve the question, by giving a polynomial-time algorithm to test whether a graph contains an odd hole. This also gives a new and considerably simpler polynomial-time algorithm that tests for perfection.