Vertex-Unfolding Polyhedra

Definitions and mathematical statement

Let $P$ be a compact connected polyhedral surface without boundary in $\mathbb R^3$. Its faces are simple planar polygons, each homeomorphic to a disk; every edge is incident to two faces. Convexity is not assumed. The prescribed faces are kept intact.

A vertex unfolding cuts along original edges, possibly along every edge, while retaining selected identifications at original vertices. Place each face in the plane by a rigid motion so that retained identifications agree. The resulting union must be connected and distinct face interiors must be disjoint. Faces are allowed to remain joined only at a vertex, so the interior of the planar union need not be connected. Accidental geometric contacts do not replace the required retained connections in the original surface.

Does every $P$ admit such an unfolding? An important restricted question asks the same for convex polyhedra. The source reports an affirmative result when every face is triangular, leaving general polygonal faces as the issue.

Subdividing faces into triangles changes the prescribed face structure and would evade the question. Likewise, an ordinary edge unfolding requires a stronger form of connection through uncut edge intervals. The task asks for a flat nonoverlapping realization, not a collision-free continuous physical motion to it.

Short English statement

Can any closed polyhedral surface be flattened without overlapping face interiors, if cuts follow edges and faces may remain connected only at vertices?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 5500042 (AMR-054-0042), AMR Open Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Open Problems Project - Computational Geometry, Problem 42. Original source indexed by AMR-054-0042. https://topp.openproblem.net/p42.