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arXiv · 1903.00208

Detecting an odd hole

Maria Chudnovsky·Alex Scott·Paul Seymour·Sophie Spirkl
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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.

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Maria Chudnovsky, Alex Scott, Paul Seymour, Sophie Spirkl. 2019-03-01. Detecting an odd hole. https://arxiv.org/abs/1903.00208

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