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