Boundary Value Problems on Planar Graphs and Flat Surfaces with integer conical singularities.

We are given a cellular decomposition of a planar, bounded domain and its boundary, where each 2-cell is either a triangle or a quadrilateral. From these data we construct a canonical pair $(S,f)$ where $S$ is a special type of a genus $(m-1)$ singular flat surface, tiled by rectangles and $f$ is an energy preserving mapping from the edges of the decomposition onto $S$. The starting point to our construction are Dirichlet and/or Dirichlet-Neumann boundary value problems on a network associated to the cellular decomposition.