Degeneration of a graph defining a complex structure

A complex structure on a surface is often set up by a graph defining the procedure for gluing standard pieces of a plane: triangles, rectangles, stripes, etc. Well known examples are Dessin d'Enfants (Grothendieck), ribbon graphs (Kontsevich), triangulations (Voevodsky and Shabat), etc. Graph weights are continuous parameters for gluing and they can be considered as conformal moduli of a surface. We study (using as example the space of real curves of genus two with one oval) the behavior of more traditional moduli of a curve when a graph edge contracts. The resulting asymptotics shows the occurrence of the root singularity.