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Unfolding Genus-2 Orthogonal Polyhedra with Linear Refinement

Mirela Damian,E. Demaine,Robin Y. Flatland,J. O'Rourke

Published 2016 in Graphs and Combinatorics

ABSTRACT

We show that every orthogonal polyhedron of genus g≤2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g \le 2$$\end{document} can be unfolded without overlap while using only a linear number of orthogonal cuts (parallel to the polyhedron edges). This is the first result on unfolding general orthogonal polyhedra beyond genus-0. Our unfolding algorithm relies on the existence of at most 2 special leaves in what we call the “unfolding tree” (which ties back to the genus), so unfolding polyhedra of genus 3 and beyond requires new techniques.

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