In graph theory, a graph H is called a minor of the graph G if H is isomorphic to a graph that results from a subgraph of G by zero or more edge contractions. Here, "contracting an edge" means removing the edge and identifying its two endpoints, keeping all other edges.
For example, the graph
*
|
*--*--*
|
*
is a minor of
*
/|
*-*--*-*-*
|/
*
(the outer edges are removed, the long middle edge is contracted).
The relation "being a minor of" is a partial order on the isomorphism classes of graphs.
Many classes of graphs can be characterized by "forbidden minors": a graph belongs to the class if and only if it does not have a minor from a certain specified list. The best-known example is Kuratowski's theorem for the characterization of planar graphs. The general situation is described by the Robertson-Seymour theorem.
Another deep result by Robertson-Seymour states that if any infinite list G1, G2,... of finite graphs is given, then there always exists two indices i < j such that Gi is a minor of Gj.
In linear algebra, there is a different unrelated meaning of the word minor. See minor (linear algebra).
