A dual version of the ribbon graph decompositionof moduli space

In this note, I discuss a dual version of the ribbon graph decomposition of the moduli spaces of Riemann surfaces with boundary and marked points, which I introduced in the unpublished preprint [1], and used in [2] to construct open-closed topological conformal field theories. This dual version of the ribbon graph decomposition is a compact orbi-cell complex with a natural weak homotopy equivalence to the moduli space. In the case when all of the marked points are on the boundary of the surface, the combinatorics of the cell complex is captured by ribbon graphs, as usual. In the general case, we find a variant of the ribbon graph complex.