A forbidden minor for a family of graphs is a graph 
such that no member of the family contains
as a graph
minor.
A set of forbidden minors may characterize membership in a graph family. 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 families of simple graphs
which have forbidden minor obstructions. The torus row
includes planar graphs, so it concerns genus
at most 1, while toroidal graphs have genus
exactly 1 (Myrvold and Woodcock 2018, p. 10).
| family | obstruction |
| apex graph | unknown finite number of minors; at least 157 known |
| forest |  |
| linklessly embeddable graph | 7 Petersen family graphs forbidden minors |
| outer-cylindrical graph | 38 forbidden minors |
| outerplanar graph |  and  |
pathwidth  |  and  -spoke graph |
pathwidth  | 110 forbidden minors |
| planar graph |  and  |
| projective planar graph | 35 forbidden minors |
| graphs on a torus | 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 |
