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Problem 42: VertexUnfolding Polyhedra
 Statement
 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 vertexunfolding,
to distinguish from widely studied
edgeunfoldings (see Problem 9) and
general unfoldings.
An important subproblem here is whether all convex polyhedra have
vertexunfoldings;
a negative answer would also resolve Problem 9.
 Origin
 [DEE+02]
 Status/Conjectures
 Open.
 Partial and Related Results
 All simplicial polyhedra have vertexunfoldings [DEE+02].
These vertexunfoldings have a special structure called a ``facet path''
which does not exist in general, even for convex polyhedra
[DEE+02].
 Related Open Problems
 Problem 9: EdgeUnfolding Convex Polyhedra.
Problem 43: General Unfolding of Nonconvex Polyhedra.
 Appearances
 Originally posed in [DEE+02].
Posed by E. Demaine at the CCCG 2001 openproblem session [DO02].
 Categories
 folding and unfolding; polyhedra
 Entry Revision History
 E. Demaine, 7 Aug. 2002; 31 Aug. 2002.
 DEE+02

Erik D. Demaine, David Eppstein, Jeff Erickson, George W. Hart, and Joseph
O'Rourke.
Vertexunfolding of simplicial manifolds.
In Proceedings of the 18th Annual ACM Symposium on Computational
Geometry, pages 237243, 2002.
 DO02

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.
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The Open Problems Project  September 19, 2017