6 research outputs found
Balanced Schnyder woods for planar triangulations: an experimental study with applications to graph drawing and graph separators
In this work we consider balanced Schnyder woods for planar graphs, which are
Schnyder woods where the number of incoming edges of each color at each vertex
is balanced as much as possible. We provide a simple linear-time heuristic
leading to obtain well balanced Schnyder woods in practice. As test
applications we consider two important algorithmic problems: the computation of
Schnyder drawings and of small cycle separators. While not being able to
provide theoretical guarantees, our experimental results (on a wide collection
of planar graphs) suggest that the use of balanced Schnyder woods leads to an
improvement of the quality of the layout of Schnyder drawings, and provides an
efficient tool for computing short and balanced cycle separators.Comment: Appears in the Proceedings of the 27th International Symposium on
Graph Drawing and Network Visualization (GD 2019
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)
-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 -connected case) to this setting. Precisely, for any
cylindric essentially internally -connected map with vertices, we
can obtain in linear time a periodic (in ) straight-line drawing of that
is crossing-free and internally (weakly) convex, on a regular grid
, with and ,
where 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 -connected map on the torus (i.e., -connected in the
periodic representation) with vertices, we can compute in linear time a
periodic straight-line drawing of that is crossing-free and (weakly)
convex, on a periodic regular grid
, with and
, where is the face-width of . Since ,
the grid area is .Comment: 37 page
Encoding toroidal triangulations
International audiencePoulalhon and Schaeffer introduced an elegant method to linearly encode a planar triangulation optimally. The method is based on performing a special depth-first search algorithm on a particular orientation of the triangulation: the minimal Schnyder wood. Recent progress toward generalizing Schnyder woods to higher genus enables us to generalize this method to the toroidal case. In the plane, the method leads to a bijection between planar triangulations and some particular trees. For the torus we obtain a similar bijection but with particular unicellular maps (maps with only one face)