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 (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 $\mathbb{Z}/w\mathbb{Z}\times[0..h]$, with $w\leq 2n$ and $h\leq n(2d+1)$, 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 $\mathbb{Z}/w\mathbb{Z}\times\mathbb{Z}/h\mathbb{Z}$, with $w\leq 2n$ and $h\leq 1+2n(c+1)$, where $c$ is the face-width of $G$. Since $c\leq\sqrt{2n}$, the grid area is $O(n^{5/2})$.

💡 Research Summary

The paper extends the well‑known concept of canonical ordering—originally devised for planar triangulations and 3‑connected planar maps—to graphs drawn on the cylinder and, by further adaptation, to graphs on the torus. The authors first define a suitable notion of “essentially internally 3‑connected” for cylindric maps, which captures the idea that the graph remains 3‑connected when the two boundary cycles are treated as external faces. For such maps they construct a canonical ordering that respects the cylindrical topology: at each step a vertex that is incident to the current outer boundary (or to the inner boundary) and whose removal does not destroy internal 3‑connectivity is selected and added to the ordering.

Using this ordering they design a linear‑time incremental straight‑line drawing algorithm that produces a periodic embedding on a rectangular grid with wrap‑around in the horizontal direction. The embedding lives on the lattice (\mathbb{Z}/w\mathbb{Z}\times