Map Genus, Forbidden Maps, and Monadic Second-Order Logic

A map is a graph equipped with a circular order of edges around each vertex. These circular orders represent local planar embeddings. The genus of a map is the minimal genus of an orientable surface in which it can be embedded. The maps of genus at most g are characterized by nitely many forbidden maps, relatively to an appropriate ordering related to the minor ordering of graphs. This yields a \noninformative" characterization of these maps, that is expressible in monadic second-order logic. We give another one, which is more informative in the sense that it speci es the relevant surface embedding, in addition to stating its existence