Canonical Ordering for Triangulations on the Cylinder, with Applications to Periodic Straight-Line Drawings

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.