Unfolding Orthotubes with a Dual Hamiltonian Path

TL;DR

This paper introduces a new algorithmic method for unfolding orthotubes, ensuring the rectangular faces are connected in a single Hamiltonian path, which enhances the understanding of their geometric properties.

Contribution

It presents a novel grid unfolding algorithm for orthotubes that guarantees the faces are arranged along a Hamiltonian path, improving upon previous unfolding techniques.

Findings

New algorithmic grid unfolding method

Guarantees faces form a Hamiltonian path

Builds on and extends prior orthotube unfolding results

Abstract

An orthotube consists of orthogonal boxes (e.g., unit cubes) glued face-to-face to form a path. In 1998, Biedl et al. showed that every orthotube has a grid unfolding: a cutting along edges of the boxes so that the surface unfolds into a connected planar shape without overlap. We give a new algorithmic grid unfolding of orthotubes with the additional property that the rectangular faces are attached in a single path -- a Hamiltonian path on the rectangular faces of the orthotube surface.