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 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-Seymor states that if any infinite list G₁, G₂,... of finite graphs is given, then there always exists two indices i < j such that G_i is a minor of G_j.

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In linear algebra, there is a different unrelated meaning of the word minor. See minor (linear algebra).