SIAM Journal on Discrete Mathematics
Straight-line representations of maps on the torus and other flat surfaces
Discrete Mathematics - Special issue on combinatorics
Embedding planar graphs on the grid
SODA '90 Proceedings of the first annual ACM-SIAM symposium on Discrete algorithms
Schnyder Woods for Higher Genus Triangulated Surfaces, with Applications to Encoding
Discrete & Computational Geometry - Special Issue: 24th Annual Symposium on Computational Geometry
Discrete one-forms on meshes and applications to 3D mesh parameterization
Computer Aided Geometric Design
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We extend the notion of canonical orderings to cylindric triangulations. This allows us to extend the incremental straight-line drawing algorithm of de Fraysseix, Pach and Pollack to this setting. Our algorithm yields in linear time a crossing-free straight-line drawing of a cylindric triangulation G with n vertices on a regular grid ℤ/wℤ×[0..h], with w≤2n and h≤n(2d+1), where d is the (graph-) distance between the two boundaries. As a by-product, we can also obtain in linear time a crossing-free straight-line drawing of a toroidal triangulation with n vertices on a periodic regular grid ℤ/wℤ×ℤ/hℤ, with w≤2n and h≤1+n(2c+1), where c is the length of a shortest non-contractible cycle. Since $c\leq\sqrt{2n}$, the grid area is O(n5/2). Our algorithms apply to any triangulation (whether on the cylinder or on the torus) that have no loops nor multiple edges in the periodic representation.