  Fatgraphs are multigraphs enriched with a cyclic order of the edges incident
to a vertex. This paper presents algorithms to: (1) generate the set of all
fatgraphs having a given genus and number of boundary cycles; (2) compute
automorphisms of any given fatgraph; (3) compute the homology of the fatgraph
complex. The algorithms are suitable for effective computer implementation.
  In particular, this allows us to compute the rational homology of the moduli
space of Riemann surfaces with marked points. We thus compute the Betti numbers
of $M_{g,n}$ with $(2g + n) \leq 6$, corroborating known results.
