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In this paper we study metric ribbon graphs, in particular, directed metricribbon graphs. These ribbon graphs are dual to bipartite maps and appear in thecontext of Abelian differentials. We prove that it is possible to decompose adirected ribbon graph into a family of ribbon graphs with one vertex, byperforming surgeries along appropriate multi curves. The decomposition iscanonical and we call it acyclic decomposition due to a condition on the stablegraphs that encode the surgeries. This result provides a recursion scheme forvolumes of moduli space of directed metric ribbon graphs, we give explicitlythe recursion in the case of four valent metric ribbon graphs. In a particularcase, we give applications to count of Grothendieck dessins d'enfants.

Cutting directed ribbon graphs and recursion for volumes of the combinatorial moduli spaces