Cologne Excellence Cluster on Cellular Stress Responses in Aging Associated Diseases
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Node Overlap Removal by Growing a Tree
No full textNode overlap removal is a necessary step in many scenarios including laying out a graph, or visualizing a tag cloud. Our contribution is a new overlap removal algorithm that iteratively builds a Minimum Spanning Tree on a Delaunay triangulation of the node centers and removes the node overlaps by “growing” the tree. The algorithm is simple to implement yet produces high quality layouts. According to our experiments it runs several times faster than the current state-of-the-art methods
Bitonic st-orderings for Upward Planar Graphs
No full textCanonical orderings serve as the basis for many incremental planar drawing algorithms. All these techniques, however, have in common that they are limited to undirected graphs. While st-orderings do extend to directed graphs, especially planar st-graphs, they do not offer the same properties as canonical orderings. In this work we extend the so called bitonic st-orderings to directed graphs. We fully characterize planar st-graphs that admit such an ordering and provide a linear-time algorithm for recognition and ordering. If for a graph no bitonic st-ordering exists, we show how to find in linear time a minimum set of edges to split such that the resulting graph admits one. With this new technique we are able to draw every upward planar graph on n vertices by using at most one bend per edge, at most n−3 bends in total and within quadratic area
Hanani-Tutte for Radial Planarity II
No full textA drawing of a graph G is radial if the vertices of G are placed on concentric circles C1,…,Ck
with common center c, and edges are drawn radially: every edge intersects every circle centered at c at most once. G is radial planar if it has a radial embedding, that is, a crossing-free radial drawing. If the vertices of G are ordered or partitioned into ordered levels (as they are for leveled graphs), we require that the assignment of vertices to circles corresponds to the given ordering or leveling. A pair of edges e and f in a graph is independent if e and f do not share a vertex.
We show that a graph G is radial planar if G has a radial drawing in which every two independent edges cross an even number of times; the radial embedding has the same leveling as the radial drawing. In other words, we establish the strong Hanani-Tutte theorem for radial planarity. This characterization yields a very simple algorithm for radial planarity testing
A Sparse Stress Model
No full textForce-directed layout methods constitute the most common approach to draw general graphs. Among them, stress minimization produces layouts of comparatively high quality but also imposes comparatively high computational demands. We propose a speed-up method based on the aggregation of terms in the objective function. It is akin to aggregate repulsion from far-away nodes during spring embedding but transfers the idea from the layout space into a preprocessing phase. An initial experimental study informs a method to select representatives, and subsequent more extensive experiments indicate that our method yields better approximations of minimum-stress layouts in less time than related methods
Twins in Subdivision Drawings of Hypergraphs
No full textVisualizing hypergraphs, systems of subsets of some universe, has continuously attracted research interest in the last decades. We study a natural kind of hypergraph visualization called subdivision drawings. Dinkla et al. [Comput. Graph. Forum ’12] claimed that only few hypergraphs have a subdivision drawing. However, this statement seems to be based on the assumption (also used in previous work) that the input hypergraph does not contain twins, pairs of vertices which are in precisely the same hyperedges (subsets of the universe). We show that such vertices may be necessary for a hypergraph to admit a subdivision drawing. As a counterpart, we show that the number of such “necessary twins” is upper-bounded by a function of the number m of hyperedges and a further parameter r of the desired drawing related to its number of layers. This leads to a linear-time algorithm for determining such subdivision drawings if m and r are constant; in other words, the problem is linear-time fixed-parameter tractable with respect to the parameters m and r
On the Size of Planarly Connected Crossing Graphs
No full textWe prove that if an n-vertex graph G can be drawn in the plane such that each pair of crossing edges is independent and there is a crossing-free edge that connects their endpoints, then G has O(n) edges. Graphs that admit such drawings are related to quasi-planar graphs and to maximal 1-planar and fan-planar graphs
Block Crossings in Storyline Visualizations
No full textStoryline visualizations help visualize encounters of the characters in a story over time. Each character is represented by an x-monotone curve that goes from left to right. A meeting is represented by having the characters that participate in the meeting run close together for some time. In order to keep the visual complexity low, rather than just minimizing pairwise crossings of curves, we propose to count block crossings, that is, pairs of intersecting bundles of lines.
Our main results are as follows. We show that minimizing the number of block crossings is NP-hard, and we develop, for meetings of bounded size, a constant-factor approximation. We also present two fixed-parameter algorithms and, for meetings of size 2, a greedy heuristic that we evaluate experimentally
