ALGEBRAIC CHARACTERIZATIONS OF GRAPH IMBEDDABILITY IN SURFACES AND PSEUDOSURFACES

Abstract

Given a finite connected graph G and specifications for a closed, connected pseudosurface, we characterize when G can be imbedded in a closed, connected pseudosurface with the given specifications. The specifications for the pseudosurface are: the number of face-connected components, the number of pinches, the number of crosscaps and handles, and the dimension of the first ℤ2 homology group. The characterizations are formulated in terms of the existence of a dual graph G* on the same set of edges as G which satisfies algebraic conditions inspired by homology groups and their intersection products.