The embeddability of a graph in a given surface is determined entirely by the polygon matroid of the graph. That is also true for cellular embeddability in nonorientable surfaces but not in orientable surfaces. An embedding of a finite graph T in a surface S is a homeomorphism of Y, regarded as a topological space, with a closed subset of S. In order to know in which surfaces T embeds it suffices to consider only the compact surfaces: the orientable ones T of genus g (Euler characteristic 2-2g)

more »

... for g >0, and the nonorientable ones Uh of Euler characteristic 2 -h for « > 1. For uniformity of terminology we define the demigenus d of a compact surface by d(T ) = 2g, d(Uh) = h . One knows exactly which compact surfaces can embed r if one knows two parameters: the genus of the graph, g(T) -min{g: T embeds in T }, and its crosscap number (also called nonorientable genus) h(T) = min{« : T embeds in Uh}. A natural companion to these is the demigenus of T (also known as generalized genus, Euler genus, etc.),