4 research outputs found
Periodic planar straight-frame drawings with polynomial resolution
International audienceWe present a new algorithm to compute periodic (planar) straight-line drawings of toroidal graphs. Our algorithm is the first to achieve two important aesthetic criteria: the drawing fits in a straight rectangular frame, and the grid area is polynomial, precisely the grid size is O(n 4 Ă— n 4). This solves one of the main open problems in a recent paper by Duncan et al. [3]
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 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