Vertex unfoldings of tight polyhedra

TL;DR

This paper extends the concept of vertex unfoldings from triangulated polyhedra to tight polyhedra, which are characterized by non-adjacent parallelogram faces, broadening the class of polyhedra known to admit such unfoldings.

Contribution

The paper proves that all tight polyhedra, a broader class than triangulated polyhedra, have vertex unfoldings, generalizing previous results.

Findings

All tight polyhedra admit vertex unfoldings.

Extension of vertex unfolding results from triangulated to tight polyhedra.

Broader class of polyhedra now known to have vertex unfoldings.

Abstract

An unfolding of a polyhedron along its edges is called a vertex unfolding if adjacent faces are allowed to be connected at not only an edge but also a vertex. Demaine et al showed that every triangulated polyhedron has a vertex unfolding. We extend this result to a tight polyhedron, where a polyhedron is tight if all non-triangular faces are mutually non-adjacent parallelograms.