Cologne Excellence Cluster on Cellular Stress Responses in Aging Associated Diseases
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Realizing Planar Graphs as Convex Polytopes
No full textThis is a survey on methods to construct a three-dimensional convex polytope with a given combinatorial structure, that is, with the edges forming a given 3-connected planar graph, focusing on efforts to achieve small integer coordinates
Every Graph Admits an Unambiguous Bold Drawing
No full textLet r and w be a fixed positive numbers, w < r. In a bold drawing of a graph, every vertex is represented by a disk of radius r, and every edge by a narrow rectangle of width w. We solve a problem of van Kreveld [K09] by showing that every graph admits a bold drawing in which the region occupied by the union of the disks and rectangles representing the vertices and edges does not contain any disk of radius r other than the ones representing the vertices
Adjacent Crossings Do Matter
No full textIn a drawing of a graph, two edges form an odd pair if they cross each other an odd number of times. A pair of edges is independent if they share no endpoint. For a graph G, let ocr(G) be the smallest number of odd pairs in a drawing of G and let iocr(G) be the smallest number of independent odd pairs in a drawing of G. We construct a graph G with
iocr(G) < ocr(G), answering a question by Székely, and—for the first time—giving evidence that crossings of adjacent edges may not always be trivial to eliminate. The graph G is based on a separation of iocr and ocr for monotone drawings of ordered graphs. A drawing of a graph is x-monotone if every edge intersects every vertical line at most once and every vertical line contains at most one vertex. A graph is ordered if each of its vertices is assigned a distinct x-coordinate. We construct a family of ordered graphs such that for x-monotone drawings, the monotone variants of ocr and iocr satisfy mon-iocr(G) < O(mon-ocr(G)1/2 )
Drawing Graphs with Vertices at Specified Positions and Crossings at Large Angles
No full textIn point-set-embeddability (PSE) problems one is given not just a graph that is to be drawn, but also a set of points in the plane that specify where the vertices of the graph can be placed. The problem class was introduced by Gritzmann et al. [3] twenty years ago. In their work and most other works on PSE problems, however, planarity of the output drawing was an essential requirement. Recent experiments on the readability of drawings [4] showed that polyline drawings with angles at edge crossings close to 90°. and a small number of bends per edge are just as readable as planar drawings. Motivated by these findings, Didimo et al. [2] recently introduced RAC drawings where pairs of crossing edges must form a right angle and, more generally, αAC drawings (for α ∈ (0, 90°]) where the crossing angle must be at least α. As usual, edges may not overlap and may not go through vertices. We investigate the intersection of PSE and RAC/αAC
A Quantitative Comparison of Stress-Minimization Approaches for Offline Dynamic Graph Drawing
No full textIn dynamic graph drawing, the input is a sequence of graphs for which a sequence of layouts is to be generated such that the quality of individual layouts is balanced with layout stability over time. Qualitatively different extensions of drawing algorithms for static graphs to the dynamic case have been proposed, but little is known about their relative utility. We report on a quantitative study comparing the three prototypical extensions via their adaptation for the stress-minimization framework. While some findings are more subtle, the linking approach connecting consecutive instances of the same vertex is found to be the overall method of choice
Generalizing Geometric Graphs
No full textNetwork visualization is essential for understanding the data obtained from huge real-world networks such as flight-networks, the AS-network or social networks. Although we can compute layouts for these networks reasonably fast, even the most recent display media are not capable of displaying these layouts in an adequate way. Moreover, the human viewer may be overwhelmed by the displayed level of detail. The increasing amount of data therefore requires techniques aiming at a sensible reduction of the visual complexity of huge layouts. We consider the problem of computing a generalization of a given layout reducing the complexity of the drawing to an amount that can be displayed without clutter and handled by a human viewer. We take a first step at formulating graph generalization within a mathematical model and we consider the resulting problems from an algorithmic point of view. Although these problems are NP-hard in general, we provide efficient approximation algorithms as well as efficient and effective heuristics. At the end of the paper we showcase some sample generalizations
Orthogeodesic Point-Set Embedding of Trees
No full textLet S be a set of N grid points in the plane, and let G a graph with n vertices (n ≤ N ). An orthogeodesic point-set embedding of G on S is a drawing of G such that each vertex is drawn as a point of S and each edge is an orthogonal chain with bends on grid points whose length is equal to the Manhattan distance.We study the following problem. Given a family of trees F what is the minimum value f (n) such that every n-vertex tree in F admits an orthogeodesic point-set embedding on every grid-point set of size f (n)? We provide polynomial upper bounds on f (n) for both planar and non-planar orthogeodesic point-set embeddings as well as for the case when edges are required to be L-shaped chains
Small Point Sets for Simply-Nested Planar Graphs
No full textA point set P ⊆ R2 is universal for a class G if every graph of G has a planar straight-line embedding into P. We prove that there exists a log n O(n( log log n )2 ) size universal point set for the class of simply-nested n-vertex planar graphs. This is a step towards a full answer for the well-known open problem on the size of the smallest universal point sets for planar graphs
k-Quasi-Planar Graphs
No full textA topological graph is k-quasi-planar if it does not contain k pairwise crossing edges. A topological graph is simple if every pair of its edges intersect at most once (either at a vertex or at their intersection). In 1996, Pach, Shahrokhi, and Szegedy [16] showed that every n-vertex simple k-quasi-planar graph contains at most O (n(log n)hoch 2k−4) edges. This upper bound was recently improved (for large k) by Fox and Pach [8] to n(log n)O(log k). In this note, we show that all such graphs contain at most n(log2n)2α ck(n) edges, where α(n) denotes the inverse Ackermann function and ck is a constant that depends only on k
Force-Directed Lombardi-Style Graph Drawing
No full textA Lombardi drawing of a graph is one in which vertices are represented as points, edges are represented as circular arcs between their endpoints, and every vertex has perfect angular resolution (equal angles between consecutive edges, as measured by the tangents to the circular arcs at the vertex). We describe two algorithms that create “Lombardi-style” drawings (which we also call near-Lombardi drawings), in which all edges are still circular arcs, but some vertices may not have perfect angular resolution. Both of these algorithms take a force-directed, spring-embedding approach, with one using forces at edge tangents to produce curved edges and the other using dummy vertices on edges for this purpose. As we show, these approaches produce near-Lombardi drawings, with one being slightly better at achieving near-perfect angular resolution and the other being slightly better at balancing edge placements