Every graph admits an unambiguous bold drawing

Let r and w be 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 [10] 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. © 2015, Brown University. All rights reserved

Every graph admits an unambiguous bold drawing

Abstract. Let r and w be a fixed positive numbers, w&lt;r.Inabold 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.

Optical Graph Recognition

Graphs are an important model for the representation of structural information between objects. One identifies objects and nodes as well as a binary relation between objects and edges. Graphs have many uses, e. g., in social sciences, life sciences and engineering. There are two primary representations: abstract and visual. The abstract representation is well suited for processing graphs by computers and is given by an adjacency list, an adjacency matrix or any abstract data structure. A visual representation is used by human users who prefer a picture. Common terms are diagram, scheme, plan, or network. The objective of Graph Drawing is to transform a graph into a visual representation called the drawing of a graph. The goal is a “nice” drawing. In this thesis we introduce Optical Graph Recognition. Optical Graph Recognition (OGR) reverses Graph Drawing and transforms a digital image of a graph into an abstract representation. Our approach consists of four phases: Preprocessing where we determine which pixels of an image are part of the graph, Segmentation where we recognize the nodes, Topology Recognition where we detect the edges and Postprocessing where we enrich the recognized graph with additional information. We apply established digital image processing methods and make use of the special property that the image contains nodes that are connected by edges. We have focused on developing algorithms that need as little parameters as possible or to automatically calibrate the parameters. Most false recognition results are caused by crossing edges as this makes tracing the edges difficult and can lead to other recognition errors. We have evaluated hand-drawn and computer-drawn graphs. Our algorithms have a very high recognition rate for computer-drawn graphs, e. g., from a set of 100000 computer-drawn graphs over 90% were correctly recognized. Most false recognition results where observed for hand-drawn graphs as they can include drawing errors and inaccuracies. For universal usability we have implemented a prototype called OGRup for mobile devices like smartphones or tablet computers. With our software it is possible to directly take a picture of a graph via a built in camera, recognize the graph, and then use the result for further processing. Furthermore, in order to gain more insight into the way a person draws a graph by hand, we have conducted a field study

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