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Graphs and Combinatorics

, Volume 33, Issue 5, pp 1357–1379 | Cite as

Unfolding Genus-2 Orthogonal Polyhedra with Linear Refinement

  • Mirela DamianEmail author
  • Erik Demaine
  • Robin Flatland
  • Joseph O’Rourke
Original Paper

Abstract

We show that every orthogonal polyhedron of genus \(g \le 2\) 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.

Keywords

Grid unfolding Linear refinement Orthogonal polyhedron Genus 2 

Notes

Acknowledgements

We thank all the participants of the 31st Bellairs Winter Workshop on Computational Geometry for a fruitful and collaborative environment. In particular, we thank Sebastian Morr for important discussions related to Theorem 1, and to the stitching of unfolding strips at the root node.

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Copyright information

© Springer Japan KK 2017

Authors and Affiliations

  1. 1.Department of Computer ScienceVillanova UniversityVillanovaUSA
  2. 2.Computer Science and Artificial Intelligence LaboratoryMassachusetts Institute of TechnologyCambridgeUSA
  3. 3.Department of Computer ScienceSiena CollegeLoudonvilleUSA
  4. 4.Department of Computer ScienceSmith CollegeNorthamptonUSA

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