Vertex-unfolding polyhedra
Let $P$ be a polyhedron without boundary, so that each edge lies on exactly two faces, whose faces are simply connected. A vertex-unfolding of $P$ cuts $P$ along some or all of its edges, where faces may remain attached to each other at vertices, and lays the faces out in the plane as a single connected piece without overlap. A convex polyhedron with no vertex-unfolding would also answer the edge-unfolding question for convex polyhedra in the negative.
Demaine, Eppstein, Erickson, Hart and O'Rourke proved that every simplicial polyhedron has a vertex-unfolding. Their unfoldings are arranged along a facet path, and they showed that such a path need not exist in general, even for convex polyhedra.
Problem. Does every such polyhedron have a vertex-unfolding? In particular, does every convex polyhedron have one?
Source: Demaine, Eppstein, Erickson, Hart and O'Rourke (2002); posed by Demaine at the CCCG 2001 open-problem session; Problem 42 of The Open Problems Project (Demaine, Mitchell and O'Rourke).
Imported from topp.openproblem.net.
Classification
- Primary
- MG — Metric Geometry
- Secondary
- CO
- Keywords
- polyhedra, The Open Problems Project, vertex-unfolding
References
@inproceedings{demaine2002vertex,
author = {Demaine, Erik D. and Eppstein, David and Erickson, Jeff and Hart, George W. and O'Rourke, Joseph},
title = {Vertex-Unfolding of Simplicial Manifolds},
booktitle = {Proceedings of the 18th Annual ACM Symposium on Computational Geometry},
year = {2002},
pages = {237--243}
}
@inproceedings{demaine2002cccg2001,
author = {Demaine, Erik D. and O'Rourke, Joseph},
title = {Open Problems from {CCCG} 2001},
booktitle = {Proceedings of the 14th Canadian Conference on Computational Geometry},
year = {2002},
note = {\url{http://www.cs.uleth.ca/~wismath/cccg/papers/open.pdf}}
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-01088, "Vertex-unfolding polyhedra", clopen.solutions.