A Kuratowski theorem for nonorientable surfaces

Abstract

Let Σ denote a surface. A graph G is irreducible for Σ provided that G does not embed in Σ, but any proper subgraph does so embed. Let I( Σ) denote the set of graphs without degree two vertices which are irreducible for Σ. Observe that a graph embeds in Σ if and only if it does not contain a subgraph homeomorphic to a member of I( Σ). For example, Kuratowski's theorem shows that I( Σ) = { K 3,3, K 5} when Σ is the sphere. In this paper we prove that the set I( Σ) is finite for each nonorientable surface, setting in part a conjecture of Erdös from the 1930s.