Canonical Ordering For Graphs On The Cylinder, With Applications To Periodic Straight-Line Drawings On The Flat Cylinder And Torus

We extend the notion of canonical ordering, initially developed for planar triangulations and 3-connected planar maps, to cylindric triangulations and more generally to cylindric 3-connected maps. This allows us to extend the incremental straight-line drawing algorithm of de Fraysseix, Pach and Pollack and of Kant from the planar triangulated case and the 3-connected case to this setting. Precisely, for any cylindric essentially 3-connected map G with n vertices, we can obtain in linear time a straight-line drawing of G that is periodic in x-direction, crossing-free, and internally (weakly) convex. The vertices of this drawing lie on a regular grid Z/wZ x [0..4 with w <= 2n and h <= n(2d + 1), where d is the face-distance between the two boundaries. This also yields an efficient periodic drawing algorithm for graphs on the torus. Precisely, for any essentially 3-connected map G on the torus (i.e., 3-connected in the periodic representation) with n vertices, we can compute in linear time a periodic straight-line drawing of G that is crossing-free and (weakly) convex, on a periodic regular grid Z/wZ x Z/hZ, with w <= 2n and h <= 1 + 2n(c + 1), where c is the face-width of G. Since c < root 2n, the grid area is O(n(5/2)).