Unfolding Orthogonal Polyhedra with Quadratic Refinement: The Delta-Unfolding Algorithm

TL;DR

This paper introduces the Delta-Unfolding algorithm that efficiently unfolds orthogonal polyhedra homeomorphic to a sphere using polynomially many cuts, improving upon previous exponential cut methods.

Contribution

The paper presents a novel polynomial-time unfolding algorithm for orthogonal polyhedra, reducing the number of cuts needed from exponential to polynomial scale.

Findings

Unfolds all orthogonal polyhedra homeomorphic to a sphere without overlap.

Uses only polynomially many cuts based on coordinate planes.

Significantly improves previous exponential cut bounds.

Abstract

We show that every orthogonal polyhedron homeomorphic to a sphere can be unfolded without overlap while using only polynomially many (orthogonal) cuts. By contrast, the best previous such result used exponentially many cuts. More precisely, given an orthogonal polyhedron with n vertices, the algorithm cuts the polyhedron only where it is met by the grid of coordinate planes passing through the vertices, together with Theta(n^2) additional coordinate planes between every two such grid planes.