### Abstract

Robertson and the second author [7] proved in 1986 that for all h there exists f(h) such that for every h-vertex simple planar graph H, every graph with no H-minor has tree-width at most f(h); but how small can we make f(h)? The original bound was an iterated exponential tower, but in 1994 with Thomas [9] it was improved to 2O(h5); and in 1999 Diestel, Gorbunov, Jensen, and Thomassen [3] proved a similar bound, with a much simpler proof. Here we show that f(h)=2^{O(hlog(h))} works. Since this paper was submitted for publication, Chekuri and Chuzhoy [2] have announced a proof that in fact f(h) can be taken to be O(h^{100}).

Original language | English (US) |
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Pages (from-to) | 38-53 |

Number of pages | 16 |

Journal | Journal of Combinatorial Theory. Series B |

Volume | 111 |

DOIs | |

State | Published - Mar 1 2015 |

### All Science Journal Classification (ASJC) codes

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

### Keywords

- Graph minors
- Tree-width

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## Cite this

Leaf, A., & Seymour, P. (2015). Tree-width and planar minors.

*Journal of Combinatorial Theory. Series B*,*111*, 38-53. https://doi.org/10.1016/j.jctb.2014.09.003