Cologne Excellence Cluster on Cellular Stress Responses in Aging Associated Diseases
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Drawing Bobbin Lace Graphs, or, Fundamental Cycles for a Subclass of Periodic Graphs
No full textIn 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 (1, 0), 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
On Smooth Orthogonal and Octilinear Drawings: Relations, Complexity and Kandinsky Drawings
No full textWe study two variants of the well-known orthogonal drawing model: (i) the smooth orthogonal, and (ii) the octilinear. Both models form an extension of the orthogonal, by supporting one additional type of edge segments (circular arcs and diagonal segments, respectively).
For planar graphs of max-degree 4, we analyze relationships between the graph classes that can be drawn bendless in the two models and we also prove NP-hardness for a restricted version of the bendless drawing problem for both models. For planar graphs of higher degree, we present an algorithm that produces bi-monotone smooth orthogonal drawings with at most two segments per edge, which also guarantees a linear number of edges with exactly one segment
Visualizing Co-phylogenetic Reconciliations
No full textWe introduce a hybrid metaphor for the visualization of the reconciliations of co-phylogenetic trees, that are mappings among the nodes of two trees. The typical application is the visualization of the co-evolution of hosts and parasites in biology. Our strategy combines a space-filling and a node-link approach. Differently from traditional methods, it guarantees an unambiguous and ‘downward’ representation whenever the reconciliation is time-consistent (i.e., meaningful). We address the problem of the minimization of the number of crossings in the representation, by giving a characterization of planar instances and by establishing the complexity of the problem. Finally, we propose heuristics for computing representations with few crossings
Anisotropic Radial Layout for Visualizing Centrality and Structure in Graphs
No full textThis paper presents a novel method for layout of undirected graphs, where nodes (vertices) are constrained to lie on a set of nested, simple, closed curves. Such a layout is useful to simultaneously display the structural centrality and vertex distance information for graphs in many domains, including social networks. Closed curves are a more general constraint than the previously proposed circles, and afford our method more flexibility to preserve vertex relationships compared to existing radial layout methods. The proposed approach modifies the multidimensional scaling (MDS) stress to include the estimation of a vertex depth or centrality field as well as a term that penalizes discord between structural centrality of vertices and their alignment with this carefully estimated field. We also propose a visualization strategy for the proposed layout and demonstrate its effectiveness using three social network datasets
Experimental Evaluation of Book Drawing Algorithms
No full textA k-page book drawing of a graph G=(V,E) consists of a linear ordering of its vertices along a spine and an assignment of each edge to one of the k pages, which are half-planes bounded by the spine. In a book drawing, two edges cross if and only if they are assigned to the same page and their vertices alternate along the spine. Crossing minimization in a k-page book drawing is NP-hard, yet book drawings have multiple applications in visualization and beyond. Therefore several heuristic book drawing algorithms exist, but there is no broader comparative study on their relative performance. In this paper, we propose a comprehensive benchmark set of challenging graph classes for book drawing algorithms and provide an extensive experimental study of the performance of existing book drawing algorithms
Improved Bounds for Drawing Trees on Fixed Points with L-Shaped Edges
No full textLet T be an n-node tree of maximum degree 4, and let P be a set of n points in the plane with no two points on the same horizontal or vertical line. It is an open question whether T always has a planar drawing on P such that each edge is drawn as an orthogonal path with one bend (an “L-shaped” edge). By giving new methods for drawing trees, we improve the bounds on the size of the point set P for which such drawings are possible to: O(n^1.55) for maximum degree 4 trees; O(n1.22) for maximum degree 3 (binary) trees; and O(n^1.142)
for perfect binary trees.
Drawing ordered trees with L-shaped edges is harder—we give an example that cannot be done and a bound of O(nlogn)
points for L-shaped drawings of ordered caterpillars, which contrasts with the known linear bound for unordered caterpillars
NodeTrix Planarity Testing with Small Clusters.
No full textWe study the NodeTrix planarity testing problem for flat clustered graphs when the maximum size of each cluster is bounded by a constant k. We consider both the case when the sides of the matrices to which the edges are incident are fixed and the case when they can be arbitrarily chosen. We show that NodeTrix planarity testing with fixed sides can be solved in O(k^3k+3/2 n³) time for every flat clustered graph that can be reduced to a partial 2-tree by collapsing its clusters into single vertices. In the general case, NodeTrix planarity testing with fixed sides can be solved in O(n³) time for k=2, but it is NP-complete for any k≥3. NodeTrix planarity testing remains NP-complete also in the free side model when k>4
1-Fan-Bundle-Planar Drawings of Graphs.
No full textEdge bundling is an important concept heavily used for graph visualization purposes. To enable the comparison with other established near-planarity models in graph drawing, we formulate a new edge-bundling model which is inspired by the recently introduced fan-planar graphs. In particular, we restrict the bundling to the endsegments of the edges. Similarly to 1-planarity, we call our model 1-fan-bundle-planarity, as we allow at most one crossing per bundle.
For the two variants where we allow either one or, more naturally, both endsegments of each edge to be part of bundles, we present edge density results and consider various recognition questions, not only for general graphs, but also for the outer and 2-layer variants. We conclude with a series of challenging questions
