5 research outputs found
Canonical Ordering for Triangulations on the Cylinder, with Applications to Periodic Straight-line Drawings
International audienceWe 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 with vertices on a regular grid \mZ/w\mZ\times[0..h], with and , where 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 vertices on a periodic regular grid \mZ/w\mZ\times\mZ/h\mZ, with and , where is the length of a shortest non-contractible cycle. Since , the grid area is . Our algorithms apply to any triangulation (whether on the cylinder or on the torus) that have no loops nor multiple edges in the periodic representation
Drawing bobbin lace graphs, or, Fundamental cycles for a subclass of periodic graphs
In this paper, we study a class of graph drawings that arise from bobbin lace
patterns. The drawings are periodic and require a combinatorial embedding with
specific properties which we outline and demonstrate can be verified in linear
time. In addition, a lace graph drawing has a topological requirement: it
contains a set of non-contractible directed cycles which must be homotopic to
, that is, when drawn on a torus, each cycle wraps once around the minor
meridian axis and zero times around the major longitude axis. We provide an
algorithm for finding the two fundamental cycles of a canonical rectangular
schema in a supergraph that enforces this topological constraint. The polygonal
schema is then used to produce a straight-line drawing of the lace graph inside
a rectangular frame. We argue that such a polygonal schema always exists for
combinatorial embeddings satisfying the conditions of bobbin lace patterns, and
that we can therefore create a pattern, given a graph with a fixed
combinatorial embedding of genus one.Comment: Appears in the Proceedings of the 25th International Symposium on
Graph Drawing and Network Visualization (GD 2017
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