I think the requisite cuts will always form a spanning tree of the edge graph of the polyhedron in question. If this is indeed the case, then the number of edges that need to be cut will be V-1. This agrees with the data of 3 for the tetrahedron and 7 for the cube; 5 cuts will flatten the octohedron, for an additional example. If this is true, then the minimal number is unique. My intuitions about query 5 are mixed. On Sat, Jun 24, 2017 at 8:57 PM, David Wilson <davidwwilson@comcast.net> wrote:
1. For a given polyhedron, what is the minimal number edges that need to be cut to unfold it into a connected planar surface? For example, 3 edges are necessary for a tetrahedron, I think 7 for a cube.
2. Is this minimal number a function of the number of faces, edges and vertices of the polyhedron?
3. Is this minimal number unique? Does any unfolding of the same polyhedron have the same number of edge cuts?
4. If (3) is false, is there a polyhedron where some unfolding has more edge cuts, but shorter total edge cut length, than some other unfolding?
5. Is there a convex polyhedron for which some unfolding exhibits overlapping faces in the plane? If so, what is the smallest number of faces on such a polyhedron?
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