We show that there is a straightforward algorithm to determine if the polyhedron guaranteed to exist by Alexandrov's gluing theorem is a degenerate flat polyhedron, and to reconstruct it from the gluing instructions. The algorithm runs in O(n3) time for polygons whose gluings are specified by n labels.
O'Rourke, Joseph, "On Flat Polyhedra Deriving From Alexandrov's Theorem" (2010). Faculty Publications. Paper 34.