Abstract
An edge-unfolding of a polyhedron is a cutting of the polyhedron’s surface along its edges so that its surface can be flattened into a single connected flat patch on the plane without any self-overlapping. A one-layer lattice polyhedron is a polyhedron of height one, whose surface faces are grid squares. We consider the edge-unfolding problem on several classes of one-layer lattice polyhedra with cubic holes. We propose linear-time algorithms for one-layer lattice polyhedra with rectangular external boundary and cubic holes, one-layer lattice polyhedra with cubic holes strictly enclosed by an orthogonally convex polygon, and one-layer lattice polyhedra with sparse cubic holes, respectively. The algorithms use two different novel techniques to cut the edges of cubic holes of the given polyhedron so that no self-overlapping can occur in the flattened patch. Our algorithms are the first algorithms especially designed to edge-unfold a polyhedron of genus greater than zero to a single connected flattened patch. We leave open the question whether any of these edge-cutting methods can be extended to edge-unfold general one-layer lattice polyhedra with cubic holes.
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Liou, MH., Poon, SH., Wei, YJ. (2014). On Edge-Unfolding One-Layer Lattice Polyhedra with Cubic Holes. In: Cai, Z., Zelikovsky, A., Bourgeois, A. (eds) Computing and Combinatorics. COCOON 2014. Lecture Notes in Computer Science, vol 8591. Springer, Cham. https://doi.org/10.1007/978-3-319-08783-2_22
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DOI: https://doi.org/10.1007/978-3-319-08783-2_22
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