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.