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Consider a polyhedron with simply connected facets (no holes on a facet) and without boundary (every edge is incident to exactly two facets). Can the polyhedron be cut along potentially all of its edges, but leaving certain faces connected at vertices, and unfolded into one piece in the plane without overlap? Such an unfolding is called a vertex-unfolding, to distinguish from widely studied edge-unfoldings (see Problem 9) and general unfoldings. An important subproblem here is whether all convex polyhedra have vertex-unfoldings; a negative answer would also resolve Problem 9.
All simplicial polyhedra have vertex-unfoldings [DEE+02]. These vertex-unfoldings have a special structure called a “facet path” which does not exist in general, even for convex polyhedra [DEE+02].
Problem 9: Edge-Unfolding Convex Polyhedra.
Problem 43: General Unfolding of Nonconvex Polyhedra.
Originally posed in [DEE+02]. Posed by E. Demaine at the CCCG 2001 open-problem session [DO02].
folding and unfolding; polyhedra
E. Demaine, 7 Aug. 2002; 31 Aug. 2002.
Erik D. Demaine, David Eppstein, Jeff Erickson, George W. Hart, and Joseph O’Rourke. Vertex-unfolding of simplicial manifolds. In Proceedings of the 18th Annual ACM Symposium on Computational Geometry, pages 237–243, 2002.
Erik D. Demaine and Joseph O’Rourke. Open problems from CCCG 2001. In Proceedings of the 14th Canadian Conference on Computational Geometry, August 2002. http://www.cs.uleth.ca/~wismath/cccg/papers/open.pdf.