### Abstract

We prove that for every planar graph H there is a number w such that every graph with no minor isomorphic to H can be constructed from graphs with at most w vertices, by piecing them together in a tree structure. This has several consequences; for example, it implies that: (i) if A is a set of graphs such that no member is isomorphic to a minor of another, and some member of A is planar, then A is finite; (ii) for every fixed planar graph H there is a polynomial time algorithm to test if an arbitrary graph has a minor isomorphic to H; (iii) there is a generalization of a theorem of Erdös and Pósa (concerning the maximum number of disjoint circuits in a graph) to planar structures other than circuits.

Original language | English (US) |
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Pages (from-to) | 92-114 |

Number of pages | 23 |

Journal | Journal of Combinatorial Theory, Series B |

Volume | 41 |

Issue number | 1 |

DOIs | |

State | Published - Jan 1 1986 |

Externally published | Yes |

### All Science Journal Classification (ASJC) codes

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

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## Cite this

*Journal of Combinatorial Theory, Series B*,

*41*(1), 92-114. https://doi.org/10.1016/0095-8956(86)90030-4