This paper describes a system for Graph dRawing with Intelligent Placement, GRIP. The GRIP system is designed for drawing large graphs and uses a novel multi-dimensional force-directed method together with fast energy function minimization. The system allows for drawing graphs with tens of thousands of vertices in under a minute on a mid-rage PC. To the best of the authors' knowledge GRIP surpasses the fastest previous algorithms. However, speed is not achieved at the expense of quality as the resulting drawings are quite aesthetically pleasing.

vi Declaration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii List of Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiv I Orthogonal Graph Drawing 1 1

We prove that every n-vertex graph G with pathwidth pw(G) has a three-dimensional straight-line grid drawing with O(pw(G) n) volume. Thus for

A queue layout of a graph consists of a total order of the vertices, and a partition of the edges into queues, such that no two edges in the same queue are nested. The minimum number of queues in a queue layout of a graph is its queue-number. A three-dimensional (straight- line grid) drawing of a graph represents the vertices by points in Z and the edges by non-crossing line-segments. This paper contributes three main results: (1) It is proved that the minimum volume of a certain type of three-dimensional drawing of a graph G is closely related to the queue-number of G. In particular, if G is an n-vertex member of a proper minor-closed family of graphs (such as a planar graph), then G has a O(1) O(1) O(n) drawing if and only if G has O(1) queue-number.

We present a novel hierarchical force-directed method for drawing large graphs. The algorithm produces a graph embedding in an Euclidean space E of any dimension. A two or three dimensional drawing of the graph is then obtained by projecting a higher-dimensional embedding into a two or three dimensional subspace of E. Projecting high-dimensional drawings onto two or three dimensions often results in drawings that are "smoother" and more symmetric. Among the other notable features of our approach are the utilization of a maximal independent set filtration of the set of vertices of a graph, a fast energy function minimization strategy, e#cient memory management, and an intelligent initial placement of vertices. Our implementation of the algorithm can draw graphs with tens of thousands of vertices using a negligible amount of memory in less than one minute on a mid-range PC. 1 Introduction Graphs are common in many applications, from data structures to networks, from software engineering...

Hierarchical structures represented by directed acyclic graphs are widely used in visualization applications (e.g., class inheritance diagrams and scheduling diagrams). 3D information visualization has received increasing attention in the last few years, motivated by the advances in hardware and software technology for 3D computer graphics.

This paper and the accompanying demo describe a strategy and a software architecture for integrating several declarative approaches. This architecture allows for the interactive specification of local criteria for each vertex and edge. The Gold

. In this paper we describe a tool that is a general frame for the three-dimensional representation of graphs, especially devoted to the algorithms evaluation, refinement and development. 3DCube (3D Diagram Drawer) offers innovative features in the user interaction and contains a set of three-dimensional algorithms both taken from the literature and proposed by the authors. 1 Introduction Three dimensional graph drawing is an emerging field in the graph drawing area. Several tools are already available for the representation of graphs in the plane, and the most sophisticated of them also allow some three-dimensional representation, either as an additional presentation feature of basically 2D results (see GMB [13] and PLUM [17]), or as the result of an actual 3D-drawing algorithm (see 3DSA [4], COMAIDE [6], the ffGraph library [9], GEM-3D [2], GIOTTO3D [11], GOLD [10], GOVE [18]), and PARSA [14]). The tool we describe in this paper offers to the user a general frame for the repr...

. We address the problem of finding viewpoints that preserve the relational structure of a three-dimensional graph drawing under orthographic parallel projection. Previously, algorithms for finding the best viewpoints under two natural models of viewpoint "goodness" were proposed. Unfortunately, the inherent combinatorial complexity of the problem makes finding exact solutions is impractical. In this paper, we propose two approximation algorithms for the problem, commenting on their design, and presenting results on their performance. 1 Introduction Since it was first considered by the graph drawing community [6,10], there has been much research into three-dimensional graph drawing. There is some experimental evidence that three-dimensional graph drawings have advantages over their two-dimensional counterparts. It is claimed [16] that three dimensions allow users to work with larger graphs -- the natural three-dimensional actions of rotation and translation allow a user to res...

A track layout of a graph consists of a vertex colouring, an edge colouring, and a total order of each vertex colour class such that between each pair of vertex colour classes, there is no monochromatic pair of crossing edges. This paper studies track layouts and their applications to other models of graph layout. In particular, we improve on the results of Enomoto and Miyauchi [SIAM J. Discrete Math., 1999] regarding stack layouts of subdivisions, and give analogous results for queue layouts. We solve open problems due to Felsner, Wismath, and Liotta [Proc. Graph Drawing, 2001] and Pach, Thiele, and Toth [Proc. Graph Drawing, 1997] concerning three-dimensional straight-line grid drawings. We initiate the study of three-dimensional polyline grid drawings and establish volume bounds within a logarithmic factor of optimal.