Kuratowski's theorem states that a graph is planar if and only if it does not contain K5 or K3,3 as a subgraph. This theorem led to Wagner's conjecture about characterizing graph minors, which was proven by Robertson and Seymour in their theorem. The Robertson-Seymour theorem states that every downwardly closed set of graphs can be characterized by a finite set of forbidden minors, where a graph H is a minor of G if H can be obtained from G by contracting or deleting edges.