graph minor theorem
If ${({G}_{k})}_{k\in \mathbb{N}}$ is an infinite^{} sequence of finite graphs, then there exist two numbers $$ such that ${G}_{m}$ is isomorphic to a minor of ${G}_{n}$.
This theorem^{} is often referred to as the deepest result in graph theory^{}. 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 generalization^{} of Kuratowski’s theorem.
Specifically, to every set $\mathcal{G}$ of finite graphs which is closed under taking minors (meaning if $G\in \mathcal{G}$ and $H$ is isomorphic to a minor of $G$, then $H\in \mathcal{G}$) there exist finitely many graphs ${G}_{1},\mathrm{\dots},{G}_{n}$ such that $\mathcal{G}$ consists precisely of those finite graphs that do not have a minor isomorphic to one of the ${G}_{i}$. The graphs ${G}_{1},\mathrm{\dots},{G}_{n}$ are often referred to as the forbidden minors for the class $\mathcal{G}$.
0.1 References

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Neil Robertson and P.D. Seymour. Graph Minors XX: Wagner’s conjecture. Journal of Combinatorial Theory B, Volume 92, Issue 2, November 2004, Pages 325357. \urlhttp://dx.doi.org/10.1016/j.jctb.2004.08.001
Title  graph minor theorem 

Canonical name  GraphMinorTheorem 
Date of creation  20130322 14:04:20 
Last modified on  20130322 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 