Unfolding Orthogonal Polyhedra with Quadratic Refinement: The Delta-Unfolding Algorithm

We show that every orthogonal polyhedron homeomorphic to a sphere can be unfolded without overlap while using only polynomially many (orthogonal) cuts. By contrast, the best previous such result used exponentially many cuts. More precisely, given an orthogonal polyhedron with n vertices, the algorithm cuts the polyhedron only where it is met by the grid of coordinate planes passing through the vertices, together with Theta(n^2) additional coordinate planes between every two such grid planes.