  The geometry of the moduli space of stable spin curves is studied, with
emphasis on its combinatorial properties. In this context, the standard graph
theoretic framework is not just a book-keeping device: some purely
combinatorial results are proved, having moduli theoretic applications. In
particular, certain strata of the moduli space of stable curves are
characterized by a (finite) set of integers, measuring the non-reducedness of
the scheme of spin curves, and definable in purely graph-theoretical terms.
