# Minor (graph theory)

In graph theory, a graph H is called a minor of the graph G if H is isomorphic to a graph that results from G by zero or more edge contractions or edge deletions, or deletions of disconnected vertices. Here, "contracting an edge" means removing the edge and identifying its two endpoints, keeping all other edges. Deleting an edge means removing that edge from the set of all edges. Deleting a disconnected vertices means removing a vertex which has not an endpoint of any edges from the set of all vertices.

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).

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