A graph is a forbidden minor if its presence as a graph minor of a given graph means it is not a member of some family of graphs.

More generally, there may be a family of minors whose presence characterizes if a given graph has some property. For example, a planar graph is a graph that does not contain the complete graph or utility graph as a graph minor. The following table summarizes some simple graph families which have forbidden minor obstructions.

family | obstruction |

apex graph | unknown finite number of minors; at least 157 known |

forest | |

linklessly embeddable graph | 7 Petersen family graphs |

outerplanar graph | and |

pathwidth | and a 7-vertex tree |

pathwidth | 110 forbidden minors |

planar graph | and |

projective planar graph | 35 forbidden minors |

toroidal graph | unknown finite number of minors; thousands known |

treewidth | |

treewidth | , octahedral graph , prism graph , Wagner graph |

treewidth | unknown finite number of minors; at least 75 known |