Cologne Excellence Cluster on Cellular Stress Responses in Aging Associated Diseases
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The Book Embedding Problem from a SAT-Solving Perspective
No full textIn a book embedding, the vertices of a graph are placed on the spine of a book and the edges are assigned to pages, so that edges of the same page do not cross. In this paper, we approach the problem of determining whether a graph can be embedded in a book of a certain number of pages from a different perspective: We propose a simple and quite intuitive SAT formulation, which is robust enough to solve non-trivial instances of the problem in reasonable time. As a byproduct, we show a lower bound of 4 on the page number of 1-planar graphs
Size- and Port-Aware Horizontal Node Coordinate Assignment
No full textThe approach by Sugiyama et al. is widely used to automatically draw directed graphs. One of its steps is to assign horizontal coordinates to nodes. Brandes and Koepf presented a method that proved to work well in practice. We extend this method to make it possible to draw diagrams with nodes that have considerably different sizes and with edges that have fixed attachment points on a node’s perimeter (ports). Our extensions integrate seamlessly with the original method and preserve the linear execution time
Combinatorial Properties of Triangle-Free Rectangle Arrangements and the Squarability Problem
No full textWe consider arrangements of axis-aligned rectangles in the plane. A geometric arrangement specifies the coordinates of all rectangles, while a combinatorial arrangement specifies only the respective intersection type in which each pair of rectangles intersects. First, we investigate combinatorial contact arrangements, i.e., arrangements of interior-disjoint rectangles, with a triangle-free intersection graph. We show that such rectangle arrangements are in bijection with the 4-orientations of an underlying planar multigraph and prove that there is a corresponding geometric rectangle contact arrangement. Using this, we give a new proof that every triangle-free planar graph is the contact graph of such an arrangement. Secondly, we introduce the question whether a given rectangle arrangement has a combinatorially equivalent square arrangement. In addition to some necessary conditions and counterexamples, we show that rectangle arrangements pierced by a horizontal line are squarable under certain sufficient conditions
Confluent Orthogonal Drawings of Syntax Diagrams
No full textWe provide a pipeline for generating syntax diagrams (also called railroad diagrams) from context free grammars. Syntax diagrams are a graphical representation of a context free language, which we formalize abstractly as a set of mutually recursive nondeterministic finite automata and draw by combining elements from the confluent drawing, layered drawing, and smooth orthogonal drawing styles. Within our pipeline we introduce several heuristics that modify the grammar but preserve the language, improving the aesthetics of the final drawing
Realization of Simply Connected Polygonal Linkages and Recognition of Unit Disk Contact Trees
No full textWe wish to decide whether a simply connected flexible polygonal structure can be realized in Euclidean space. Two models are considered: polygonal linkages (body-and-joint framework) and contact graphs of unit disks in the plane. (1) We show that it is strongly NP-hard to decide whether a given polygonal linkage is realizable in the plane when the bodies are convex polygons and their contact graph is a tree; the problem is weakly NP-hard already for a chain of rectangles, but efficiently decidable for a chain of triangles hinged at distinct vertices. (2) We also show that it is strongly NP-hard to decide whether a given tree is the contact graph of interior-disjoint unit disks in the plane
On the Relationship Between Map Graphs and Clique Planar Graphs
No full textA map graph is a contact graph of internally-disjoint regions of the plane, where the contact can be even a point. Namely, each vertex is represented by a simple connected region and two vertices are connected by an edge iff the corresponding regions touch
Drawing Large Graphs by Multilevel Maxent-Stress Optimization
No full textDrawing large graphs appropriately is an important step for the visual analysis of data from real-world networks. Here we present a novel multilevel algorithm to compute a graph layout with respect to a recently proposed metric that combines layout stress and entropy. As opposed to previous work, we do not solve the linear systems of the maxent-stress metric with a typical numerical solver. Instead we use a simple local iterative scheme within a multilevel approach. To accelerate local optimization, we approximate long-range forces and use shared-memory parallelism. Our experiments validate the high potential of our approach, which is particularly appealing for dynamic graphs. In comparison to the previously best maxent-stress optimizer, which is sequential, our parallel implementation is on average 30 times faster already for static graphs (and still faster if executed on one thread) while producing a comparable solution quality
Genus, Treewidth, and Local Crossing Number
No full textWe consider relations between the size, treewidth, and local crossing number (maximum number of crossings per edge) of graphs embedded on topological surfaces. We show that an n-vertex graph embedded on a surface of genus g with at most k crossings per edge has treewidth O(sqrt((g+1)(k+1)n)) and layered treewidth O((g+1)k), and that these bounds are tight up to a constant factor. As a special case, the k-planar graphs with n vertices have treewidth O(sqrt((k+1)n)) and layered treewidth O(k+1), which are tight bounds that improve a previously known O((k+1)^(3/4)n^(1/2)) treewidth bound. Additionally, we show that for g<m, every m-edge graph can be embedded on a surface of genus g with O((m/(g+1))log^(2)g) crossings per edge, which is tight to a polylogarithmic factor
Kojaph: Visual Definition and Exploration of Patterns in Graph Databases
No full textWe present Kojaph, a new system for the visual definition and exploration of patterns in graph databases. It offers an expressive visual language integrated in a simple user interface, to define complex patterns as a combination of topological properties and node/edge attribute properties. Users can also interact with the query results and visually explore the graph incrementally, starting from such results. From the application perspective, Kojaph has been designed to run on top of every desired graph database management system (GDBMS). As a proof of concept, we integrated it with Neo4J, the most popular GDBMS