Algebraic characterizations of graph imbeddability in surfaces and pseudosurfaces

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 Z(2)-homology group. The characterizations axe 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.