### Abstract

Let G be a cubic graph, with girth at least five, such that for every partition X,Y of its vertex set with |X|,|Y|≥7 there are at least six edges between X and Y. We prove that if there is no homeomorphic embedding of the Petersen graph in G, and G is not one particular 20-vertex graph, then either • G∖v is planar for some vertex v; or • G can be drawn with crossings in the plane, but with only two crossings, both on the infinite region. We also prove several other theorems of the same kind.

Original language | English (US) |
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Pages (from-to) | 219-285 |

Number of pages | 67 |

Journal | Journal of Combinatorial Theory. Series B |

Volume | 138 |

DOIs | |

State | Published - Sep 2019 |

### All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- Discrete Mathematics and Combinatorics
- Computational Theory and Mathematics

### Keywords

- Apex graph
- Cubic
- Graph minors
- Petersen graph

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## Cite this

Robertson, N., Seymour, P., & Thomas, R. (2019). Excluded minors in cubic graphs.

*Journal of Combinatorial Theory. Series B*,*138*, 219-285. https://doi.org/10.1016/j.jctb.2019.02.002