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

Luca Castelli Aleardi, Olivier Devillers, Éric Fusy
2013 Lecture Notes in Computer Science  
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
more » ... ssing-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 noncontractible cycle. Since c ≤ √ 2n, the grid area is O(n 5/2 ). Our algorithms apply to any triangulation (whether on the cylinder or on the torus) with no loops nor multiple edges in the periodic representation. *
doi:10.1007/978-3-642-36763-2_34 fatcat:fcfkoab7anbsthjfpnwvp3v7bq