Robertson-Seymour Theorem

The Robertson-Seymour theorem, also called the graph minor theorem, is a generalization of the Kuratowski reduction theorem by Robertson and Seymour, which states that the collection of finite graphs is well-quasi-ordered by minor embeddability, from which it follows that Kuratowski's "forbidden minor" embedding obstruction generalizes to higher genus surfaces.

Formally, for a fixed integer , there is a finite list of graphs  with the property that a graph  embeds on a surface of genus  iff it does not contain, as a minor, any of the graphs on the list .

REFERENCES

Fellows, M. R. "The Robertson-Seymour Theorems: A Survey of Applications." Comtemp. Math. 89, 1-18, 1987.