# Recognizing Berge Graphs

@article{Chudnovsky2005RecognizingBG, title={Recognizing Berge Graphs}, author={M. Chudnovsky and G{\'e}rard Cornu{\'e}jols and Xinming Liu and Paul D. Seymour and Kristina Vuskovic}, journal={Combinatorica}, year={2005}, volume={25}, pages={143-186} }

A graph is Berge if no induced subgraph of G is an odd cycle of length at least five or the complement of one. In this paper we give an algorithm to test if a graph G is Berge, with running time O(|V (G)|9). This is independent of the recent proof of the strong perfect graph conjecture.

## 306 Citations

Odd holes in bull-free graphs

- MathematicsSIAM J. Discret. Math.
- 2018

This paper shows that testing whether a graph contains an induced odd cycle of length at least five can be done in polynomial time if the input graph has no induced subgraph isomorphic to the bull (a triangle with two disjoint pendant edges).

Induced subgraphs of graphs with large chromatic number. I. Odd holes

- MathematicsJ. Comb. Theory, Ser. B
- 2016

Colouring graphs with no odd holes

- Mathematics
- 2014

An odd hole in a graph is an induced subgraph which is a cycle of odd length at least five. In 1985, A. Gyárfás made the conjecture that for all t there exists n such that every graph with no Kt…

Detecting wheels

- MathematicsArXiv
- 2013

It is proved that the problem whose instance is a graph G and whose question is “does G contains a wheel as an induced subgraph” is NP-complete.

Even pairs in square-free Berge graphs with no odd prism

- Mathematics
- 2015

We consider the class of Berge graphs that contain no odd prism and no square (cycle on four vertices). We prove that every graph G in this class either is a clique or has an even pair, as…

Induced Subgraphs of Graphs with Large Chromatic Number. III. Long Holes

- MathematicsComb.
- 2017

We prove a 1985 conjecture of Gyárfás that for all k, ℓ, every graph with sufficiently large chromatic number contains either a clique of cardinality more than k or an induced cycle of length more…

Graphs with no induced wheel or antiwheel

- MathematicsArXiv
- 2015

It is shown here that graphs that contain no wheel and no antiwheel have a very simple structure and consequently can be recognized in polynomial time.

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