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.
|apex graph||unknown finite number of minors; at least 157 known|
|linklessly embeddable graph||7 Petersen family graphs|
|pathwidth||and a 7-vertex tree|
|pathwidth||110 forbidden minors|
|projective planar graph||35 forbidden minors|
|toroidal graph||unknown finite number of minors; thousands known|
|treewidth||, octahedral graph , prism graph , Wagner graph|
|treewidth||unknown finite number of minors; at least 75 known|