Graphs and Combinatorics
An unfolding of a polyhedron is produced by cutting the surface and flattening to a single, connected, planar piece without overlap (except possibly at boundary points). It is a long unsolved problem to determine whether every polyhedron may be unfolded. Here we prove, via an algorithm, that every orthogonal polyhedron (one whose faces meet at right angles) of genus zero may be unfolded. Our cuts are not necessarily along edges of the polyhedron, but they are always parallel to polyhedron edges. For a polyhedron of n vertices, portions of the unfolding will be rectangular strips which, in the worst case, may need to be as thin as ɛ = 1/2Ω(n).
General unfolding, Grid unfolding, Orthogonal polyhedra, Genus-zero
Damian, Mirela; Flatland, Robin; and O'Rourke, Joseph, "Epsilon-Unfolding Orthogonal Polyhedra" (2007). Faculty Publications. Paper 50.