ALGEBRAIC CHARACTERIZATIONS OF GRAPH IMBEDDABILITY IN SURFACES AND PSEUDOSURFACES
Lowell Abrams, DANIEL C. SLILATY · Journal of Knot Theory and Its Ramifications · 2006
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 ℤ2homology 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.