Graphs and Combinatorics
We show that every orthogonal polyhedron of genus g ≤ 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.
Grid unfolding, Linear reﬁnement, Orthogonal polyhedron, Genus 2
“Licensed to Smith College and distributed CC-BY under the Smith College Faculty Open Access Policy.”
Damian, Mirela; Demaine, Erik; Flatland, Robin; and O'Rourke, Joseph, "Unfolding Genus-2 Orthogonal Polyhedra with Linear Refinement" (2016). Mathematics and Statistics: Faculty Publications. 28.