In this paper we continue the study started in part I (posted). We consider a
planar, bounded, m-connected region Ω, and let \bord\Omega be its
boundary. Let T be a cellular decomposition of
\Omega\cup\bord\Omega, where each 2-cell is either a triangle or a
quadrilateral. From these data and a conductance function we construct a
canonical pair (S,f) where S is a special type of a (possibly immersed)
genus (m−1) singular flat surface, tiled by rectangles and f is an energy
preserving mapping from T(1) onto S. In part I the solution
of a Dirichlet problem defined on T(0) was utilized, in this
paper we employ the solution of a mixed Dirichlet-Neumann problem.Comment: 26 pages, 16 figures (color