We extend the notion of canonical ordering (initially developed for planar
triangulations and 3-connected planar maps) to cylindric (essentially simple)
triangulations and more generally to cylindric (essentially internally)
3-connected maps. This allows us to extend the incremental straight-line
drawing algorithm of de Fraysseix, Pach and Pollack (in the triangulated case)
and of Kant (in the 3-connected case) to this setting. Precisely, for any
cylindric essentially internally 3-connected map G with n vertices, we
can obtain in linear time a periodic (in x) straight-line drawing of G that
is crossing-free and internally (weakly) convex, on a regular grid
Z/wZ×[0..h], 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×Z/hZ, with w≤2n and
h≤1+2n(c+1), where c is the face-width of G. Since c≤2n,
the grid area is O(n5/2).Comment: 37 page