Minor (graph theory)
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In graph theory, a graph H is called a minor of the graph G if H is isomorphic to a graph that can be obtained by zero or more edge contractions on a subgraph of G. Edge contraction is the process of removing an edge and identifying its two endpoints. Contracting a loop is equivalent to deleting it.
For example, the graph
is a minor of
(the outer edges are removed, the middle edge is contracted).
The relation "being a minor of" is a partial order on the isomorphism classes of graphs. As consequence of this fact we may say that, if I is a minor of H and H is a minor of G, then I is a minor of G.
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 Wagner'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).
