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It is an open problem that the net of any convex polyhedron can be unfolded onto a flat plane with no overlapping. Is anything known if we allow x faces to overlap? For example, is it known if any convex polyhedron can be unfolded with a maximum of 2 faces overlapping? What about the more general case of a polyhedron that is topologically convex (that is, its graph is isomorphic to the graph of a convex polyhedron)?

This paper provides an example of an open polyhedron without a net that is topologically convex, and the closed case appears to be taken care of by this paper (mentioned in an answer below). One solution to the topologically convex case would then to be to find a procedure to modify either polygon so that the number of overlapping sides increases without bound. I have been unable to do so without breaking the topologically convex property, but it seems a reasonable task.

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Interesting question. I dont see much hope for a positive result for non convex polyhedra (even spherical). – Gil Kalai Nov 24 '09 at 15:31
A very good source for problems and methods involving the folding of polygons to polyhedra and the cutting of edges (or using more general cuts) to "unfold" a polyhedron into the plane is the recent book of Erik Demaine and Joseph O'Rourke published by Cambridge U. Press, 2007. There are also lots of papers about unfolding and folding on the web pages of Demaine and O'Rourke. – Joseph Malkevitch Dec 13 '09 at 22:32
Given this has been standing without an answer and may remain so for a while, I have added the open-problem tag. – Jason Dyer Jan 2 '10 at 0:34

To the case of topologically convex polyhedra: There is example in article by A.Tarasov: "Take a regular pyramid with base a regular non-convex dodecagon whose angles are alternately 280 and 20 degrees, shall we say. If the vertex of the pyramid is projected onto the centre of the base, then in the unfolding all the lateral faces must be cut o from the base or else there is self-overlapping." Of course this pyramid is topologically convex, because its graph is the same as the graph of regular 10-sided pyramid. Even more, in this paper was proven, that there exists a non-convex polyhedral sphere with convex faces and having no natural unfolding(i.e. without non-overlapping net). Unfortunately, there are no pictures in this article, you can see one example of such polthedron in another article by Grunbaum.

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TThere should be a prize for Worst Introduction of Nomenclature, and awarded retroactively to Grünbaum for unununfoldable :) – Mariano Suárez-Alvarez Nov 27 '09 at 17:37

I would like to point out a closely related question, which is described in our book that Joe Malkevitch mentioned (p.308). We called it the Fewest Nets problem. What is the fewest number of non-overlapping nets into which you can partition the surface of a convex polyhedron, cutting along polyhedron edges? The answer might be 1. But our ignorance leaves us with just fractions of F, the number of faces of the polyhedon. The best fraction obtained to date (as far as I know) is (1/2)F.

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