### Abstract

A monumental project in graph theory was recently completed. The project, started by Robertson and Seymour, and later joined by Thomas, led to entirely new concepts and a new way of looking at graph theory. The motivating problem was Kuratowski's characterization of planar graphs, and a far-reaching generalization of this, conjectured by Wagner: If a class of graphs is minor-closed (i.e., it is closed under deleting and contracting edges), then it can be characterized by a finite number of excluded minors. The proof of this conjecture is based on a very general theorem about the structure of large graphs: If a minor-closed class of graphs does not contain all graphs, then every graph in it is glued together in a tree-like fashion from graphs that can almost be embedded in a fixed surface. We describe the precise formulation of the main results and survey some of its applications to algorithmic and structural problems in graph theory.

Original language | English |
---|---|

Pages (from-to) | 75-86 |

Number of pages | 12 |

Journal | Bulletin of the American Mathematical Society |

Volume | 43 |

Issue number | 1 |

DOIs | |

Publication status | Published - Jan 1 2006 |

### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics