Degeneration of a Graph Describing Conformal Structure

We consider the cell decomposition of the moduli space of real genus $2$ curves with marked point on the unique real oval. The cells are enumerated by certain graphs, whose weights describe the complex structure on the curve. We show that the collapse of an edge in a graph results in a root-like singularity of the natural map from the weights on graphs to the moduli space of curves. Bibliography: 24 titles.