14 research outputs found
Choosing Colors for Geometric Graphs via Color Space Embeddings
Graph drawing research traditionally focuses on producing geometric
embeddings of graphs satisfying various aesthetic constraints. After the
geometric embedding is specified, there is an additional step that is often
overlooked or ignored: assigning display colors to the graph's vertices. We
study the additional aesthetic criterion of assigning distinct colors to
vertices of a geometric graph so that the colors assigned to adjacent vertices
are as different from one another as possible. We formulate this as a problem
involving perceptual metrics in color space and we develop algorithms for
solving this problem by embedding the graph in color space. We also present an
application of this work to a distributed load-balancing visualization problem.Comment: 12 pages, 4 figures. To appear at 14th Int. Symp. Graph Drawing, 200
Spanners for Geometric Intersection Graphs
Efficient algorithms are presented for constructing spanners in geometric
intersection graphs. For a unit ball graph in R^k, a (1+\epsilon)-spanner is
obtained using efficient partitioning of the space into hypercubes and solving
bichromatic closest pair problems. The spanner construction has almost
equivalent complexity to the construction of Euclidean minimum spanning trees.
The results are extended to arbitrary ball graphs with a sub-quadratic running
time.
For unit ball graphs, the spanners have a small separator decomposition which
can be used to obtain efficient algorithms for approximating proximity problems
like diameter and distance queries. The results on compressed quadtrees,
geometric graph separators, and diameter approximation might be of independent
interest.Comment: 16 pages, 5 figures, Late
Edges and switches, tunnels and bridges
Edge casing is a well-known method to improve the readability of drawings of non-planar graphs. A cased drawing orders the edges of each edge crossing and interrupts the lower edge in an appropriate neighborhood of the crossing. Certain orders will lead to a more readable drawing than others. We formulate several optimization criteria that try to capture the concept of a "good" cased drawing. Further, we address the algorithmic question of how to turn a given drawing into an optimal cased drawing. For many of the resulting optimization problems, we either find polynomial time algorithms or NP-hardness results
Geometric biplane graphs I: maximal graphs
Postprint (author’s final draft
Edges and switches, tunnels and bridges
Abstract. Edge casing is a well-known method to improve the readability of drawings of non-planar graphs. A cased drawing orders the edges of each edge crossing and interrupts the lower edge in an appropriate neighborhood of the crossing. Certain orders will lead to a more readable drawing than others. We formulate several optimization criteria that try to capture the concept of a "good" cased drawing. Further, we address the algorithmic question of how to turn a given drawing into an optimal cased drawing. For many of the resulting optimization problems, we either find polynomial time algorithms or NP-hardness results
Geometric biplane graphs I: maximal graphs
We study biplane graphs drawn on a finite planar point set in general position. This is the family of geometric graphs whose vertex set is and can be decomposed into two plane graphs. We show that two maximal biplane graphs-in the sense that no edge can be added while staying biplane-may differ in the number of edges, and we provide an efficient algorithm for adding edges to a biplane graph to make it maximal. We also study extremal properties of maximal biplane graphs such as the maximum number of edges and the largest maximum connectivity over -element point sets.Peer ReviewedPostprint (author's final draft