Focal Flow: Supporting material

Engineering and Applied SciencesOther Research Uni

Small grid embeddings of 3-polytopes

We introduce an algorithm that embeds a given 3-connected planar graph as a convex 3-polytope with integer coordinates. The size of the coordinates is bounded by $O(2^{7.55n})=O(188^{n})$. If the graph contains a triangle we can bound the integer coordinates by $O(2^{4.82n})$. If the graph contains a quadrilateral we can bound the integer coordinates by $O(2^{5.46n})$. The crucial part of the algorithm is to find a convex plane embedding whose edges can be weighted such that the sum of the weighted edges, seen as vectors, cancel at every point. It is well known that this can be guaranteed for the interior vertices by applying a technique of Tutte. We show how to extend Tutte's ideas to construct a plane embedding where the weighted vector sums cancel also on the vertices of the boundary face

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 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