Detecting an odd hole
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.
Keywords
Cite
@article{arxiv.1903.00208,
title = {Detecting an odd hole},
author = {Maria Chudnovsky and Alex Scott and Paul Seymour and Sophie Spirkl},
journal= {arXiv preprint arXiv:1903.00208},
year = {2019}
}