graph minor theorem

If (Gk)k is an infiniteMathworldPlanetmath sequence of finite graphs, then there exist two numbers m<n such that Gm is isomorphic to a minor of Gn.

This theoremMathworldPlanetmath is often referred to as the deepest result in graph theoryMathworldPlanetmath. It was proven by Robertson and Seymour in 1988 as the culminating result of a long series of articles, and the proof was recently published in the Journal of Combinatorial Theory series B. It resolves Wagner’s conjecture in the affirmative and leads to an important generalizationPlanetmathPlanetmath of Kuratowski’s theorem.

Specifically, to every set 𝒢 of finite graphs which is closed under taking minors (meaning if G𝒢 and H is isomorphic to a minor of G, then H𝒢) there exist finitely many graphs G1,,Gn such that 𝒢 consists precisely of those finite graphs that do not have a minor isomorphic to one of the Gi. The graphs G1,,Gn are often referred to as the forbidden minors for the class 𝒢.

0.1 References

  • Neil Robertson and P.D. Seymour. Graph Minors XX: Wagner’s conjecture. Journal of Combinatorial Theory B, Volume 92, Issue 2, November 2004, Pages 325-357. \url

Title graph minor theorem
Canonical name GraphMinorTheorem
Date of creation 2013-03-22 14:04:20
Last modified on 2013-03-22 14:04:20
Owner AxelBoldt (56)
Last modified by AxelBoldt (56)
Numerical id 16
Author AxelBoldt (56)
Entry type Theorem
Classification msc 05C99
Related topic MinorOfAGraph