2025/05/11 by O'Rourke, Joseph
paper · doi:10.57717/cgt.v4i2.54
Ghomi proved that every convex polyhedron could be stretched via an affine transformation so that it has an edge-unfolding to a net, a non-overlapping planar polygon. One can view his result as establishing that every combinatorial polyhedron P has a metric realization P that allows unfolding to a net. Joseph Malkevitch asked if the reverse holds (in some sense of “reverse”): Is there a combinatorial polyhedron P such that, for every metric realization P in ℝ3, and for every spanning cut-tree T of the 1-skeleton, P cut by T unfolds to a net? In this paper we prove the answer is NO: Every combinatorial polyhedron has a realization and a cut-tree that edge-unfolds the polyhedron with overlap.