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We extend our discrete uniformization theorems for planar, $m$-connected,Jordan domains [Journal f\"ur die reine und angewandte Mathematik 670 (2012),65--92] to closed surfaces of non-positive genus.

Addendum: the case of closed surfaces. (Boundary Value Problems on  Planar Graphs and Flat Surfaces with integer cone singularities, I: The  Dirichlet Problem)