TITLE: Ribbon graph decomposition of the moduli space of curves
QUESTION [11 upvotes]: What is a ribbon graph? What is the ribbon graph decomposition of the moduli space of curves? What are some good references for this material?

REPLY [3 votes]: Kiyoshi Igusa constructs the category of fat graphs (ribbon graphs) both in his paper Graph cohomology and Kontsevich cycles and his book Higher Franz-Reidemeister Torsion and are excellent references, containing lots of details in a clearly-written manner. There you can find, for example, a proof of the fact that the homology of the category of fat graphs is rationally isomorphic to the homology of the mappping class groups of marked surfaces. Details relating ribbon graphs to the Miller-Morita-Mumford classes are found there, too.