  This paper extends results of Hatcher and Vogtmann's work "Cerf Theory for
Graphs" to ribbon graphs. Given an orientable, punctured and basepointed
surface Sigma, we prove that the space of ribbon graphs that can be drawn in
Sigma is filtered by simplicial complexes. The k-th simplicial complex is
(k-1)-dimensional, (k-2)-connected and invariant under the action of the
basepoint preserving mapping class group of Sigma.
