Grid Vertex-Unfolding Orthogonal Polyhedra
<|reference_start|>Grid Vertex-Unfolding Orthogonal Polyhedra: An edge-unfolding of a polyhedron is produced by cutting along edges and flattening the faces to a *net*, a connected planar piece with no overlaps. A *grid unfolding* allows additional cuts along grid edges induced by coordinate planes passing through every vertex. A vertex-unfolding permits faces in the net to be connected at single vertices, not necessarily along edges. We show that any orthogonal polyhedron of genus zero has a grid vertex-unfolding. (There are orthogonal polyhedra that cannot be vertex-unfolded, so some type of "gridding" of the faces is necessary.) For any orthogonal polyhedron P with n vertices, we describe an algorithm that vertex-unfolds P in O(n^2) time. Enroute to explaining this algorithm, we present a simpler vertex-unfolding algorithm that requires a 3 x 1 refinement of the vertex grid.<|reference_end|>
@article{damian2005grid,
    title={Grid Vertex-Unfolding Orthogonal Polyhedra},
    author={Mirela Damian, Robin Flatland, Joseph O'Rourke},
    journal={arXiv preprint arXiv:cs/0509054},
    year={2005},
    archivePrefix={arXiv},
    eprint={cs/0509054},
    primaryClass={cs.CG cs.DM}
}