Lombardi Drawings of Graphs

We introduce the notion of Lombardi graph drawings, named after the American abstract artist Mark Lombardi. In these drawings, edges are represented as circular arcs rather than as line segments or polylines, and the vertices have perfect angular resolution: the edges are equally spaced around each vertex. We describe algorithms for finding Lombardi drawings of regular graphs, graphs of bounded degeneracy, and certain families of planar graphs.Comment: Expanded version of paper appearing in the 18th International Symposium on Graph Drawing (GD 2010). 13 pages, 7 figure

Angles of Arc-Polygons and Lombardi Drawings of Cacti

We characterize the triples of interior angles that are possible in non-self-crossing triangles with circular-arc sides, and we prove that a given cyclic sequence of angles can be realized by a non-self-crossing polygon with circular-arc sides whenever all angles are at most pi. As a consequence of these results, we prove that every cactus has a planar Lombardi drawing (a drawing with edges depicted as circular arcs, meeting at equal angles at each vertex) for its natural embedding in which every cycle of the cactus is a face of the drawing. However, there exist planar embeddings of cacti that do not have planar Lombardi drawings.Comment: 12 pages, 8 figures. To be published in Proc. 33rd Canadian Conference on Computational Geometry, 202

CelticGraph: Drawing Graphs as Celtic Knots and Links

Celtic knots are an ancient art form often attributed to Celtic cultures, used to decorate monuments and manuscripts, and to symbolise eternity and interconnectedness. This paper describes the framework CelticGraph to draw graphs as Celtic knots and links. The drawing process raises interesting combinatorial concepts in the theory of circuits in planar graphs. Further, CelticGraph uses a novel algorithm to represent edges as B\'ezier curves, aiming to show each link as a smooth curve with limited curvature.Comment: Appears in the Proceedings of the 31st International Symposium on Graph Drawing and Network Visualization (GD 2023

Scalability considerations for multivariate graph visualization

Real-world, multivariate datasets are frequently too large to show in their entirety on a visual display. Still, there are many techniques we can employ to show useful partial views-sufficient to support incremental exploration of large graph datasets. In this chapter, we first explore the cognitive and architectural limitations which restrict the amount of visual bandwidth available to multivariate graph visualization approaches. These limitations afford several design approaches, which we systematically explore. Finally, we survey systems and studies that exhibit these design strategies to mitigate these perceptual and architectural limitations

Schematics of Graphs and Hypergraphs

Graphenzeichnen als ein Teilgebiet der Informatik befasst sich mit dem Ziel Graphen oder deren Verallgemeinerung Hypergraphen geometrisch zu realisieren. Beschränkt man sich dabei auf visuelles Hervorheben von wesentlichen Informationen in Zeichenmodellen, spricht man von Schemata. Hauptinstrumente sind Konstruktionsalgorithmen und Charakterisierungen von Graphenklassen, die für die Konstruktion geeignet sind. In dieser Arbeit werden Schemata für Graphen und Hypergraphen formalisiert und mit den genannten Instrumenten untersucht. In der Dissertation wird zunächst das „partial edge drawing“ (kurz: PED) Modell für Graphen (bezüglich gradliniger Zeichnung) untersucht. Dabei wird um Kreuzungen im Zentrum der Kante visuell zu eliminieren jede Kante durch ein kreuzungsfreies Teilstück (= Stummel) am Start- und am Zielknoten ersetzt. Als Standard hat sich eine PED-Variante etabliert, in der das Längenverhältnis zwischen Stummel und Kante genau 1⁄4 ist (kurz: 1⁄4-SHPED). Für 1⁄4-SHPEDs werden Konstruktionsalgorithmen, Klassifizierung, Implementierung und Evaluation präsentiert. Außerdem werden PED-Varianten mit festen Knotenpositionen und auf Basis orthogonaler Zeichnungen erforscht. Danach wird das BUS Modell für Hypergraphen untersucht, in welchem Hyperkanten durch fette horizontale oder vertikale – als BUS bezeichnete – Segmente repräsentiert werden. Dazu wird eine vollständige Charakterisierung von planaren Inzidenzgraphen von Hypergraphen angegeben, die eine planare Zeichnung im BUS Modell besitzen, und diverse planare BUS-Varianten mit festen Knotenpositionen werden diskutiert. Zum Schluss wird erstmals eine Punktmenge von subquadratischer Größe angegeben, die eine planare Einbettung (Knoten werden auf Punkte abgebildet) von 2-außenplanaren Graphen ermöglicht

Scalability considerations for multivariate graph visualization

Real-world, multivariate datasets are frequently too large to show in their entirety on a visual display. Still, there are many techniques we can employ to show useful partial views-sufficient to support incremental exploration of large graph datasets. In this chapter, we first explore the cognitive and architectural limitations which restrict the amount of visual bandwidth available to multivariate graph visualization approaches. These limitations afford several design approaches, which we systematically explore. Finally, we survey systems and studies that exhibit these design strategies to mitigate these perceptual and architectural limitations

The state of the art in empirical user evaluation of graph visualizations

While graph drawing focuses more on the aesthetic representation of node-link diagrams, graph visualization takes into account other visual metaphors making them useful for graph exploration tasks in information visualization and visual analytics. Although there are aesthetic graph drawing criteria that describe how a graph should be presented to make it faster and more reliably explorable, many controlled and uncontrolled empirical user studies flourished over the past years. The goal of them is to uncover how well the human user performs graph-specific tasks, in many cases compared to previously designed graph visualizations. Due to the fact that many parameters in a graph dataset as well as the visual representation of them might be varied and many user studies have been conducted in this space, a state-of-the-art survey is needed to understand evaluation results and findings to inform the future design, research, and application of graph visualizations. In this paper, we classify the present literature on the topmost level into graph interpretation, graph memorability, and graph creation where the users with their tasks stand in focus of the evaluation not the computational aspects. As another outcome of this work, we identify the white spots in this field and sketch ideas for future research directions