  We compare two combinatorial models for the moduli space of two-dimensional
cobordisms: B\"odigheimer's radial slit configurations and Godin's admissible
fat graphs, producing an explicit homotopy equivalence using a "critical graph"
map. We also discuss natural compactifications of these two models, the
unilevel harmonic compactification and Sullivan diagrams respectively, and
prove that the homotopy equivalence induces a cellular homeomorphism between
these compactifications.
