Abstract—A compound graph is a frequently encountered type of data set. Relations are given between items, and a hierarchy is defined on the items as well. We present a new method for visualizing such compound graphs. Our approach is based on visually bundling the adjacency edges, i.e., non-hierarchical edges, together. We realize this as follows. We assume that the hierarchy is shown via a standard tree visualization method. Next, we bend each adjacency edge, modeled as a B-spline curve, toward the polyline defined by the path via the inclusion edges from one node to another. This hierarchical bundling reduces visual clutter and also visualizes implicit adjacency edges between parent nodes that are the result of explicit adjacency edges between their respective child nodes. Furthermore, hierarchical edge bundling is a generic method which can be used in conjunction with existing tree visualization techniques. We illustrate our technique by providing example visualizations and discuss the results based on an informal evaluation provided by potential users of such visualizations.

Hierarchical graphs and clustered graphs are useful non-classical graph models for structured relational information. Hierarchical graphs are graphs with layering structures; clustered graphs are graphs with recursive clustering structures. Both have applications in CASE tools, software visualization, and VLSI design. Drawing algorithms for hierarchical graphs have been well investigated. However, the problem of straight-line representation has not been solved completely. In this paper, we answer the question: does every planar hierarchical graph admit a planar straight-line hierarchical drawing? We present an algorithm that constructs such drawings in linear time. Also, we answer a basic question for clustered graphs, that is, does every planar clustered graph admit a planar straight-line drawing with clusters drawn as convex polygons? We provide a method for such drawings based on our algorithm for hierarchical graphs.

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

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...

Graphs depicted as node-link diagrams are widely used to show relationships between entities. However, nodelink diagrams comprised of a large number of nodes and edges often suffer from visual clutter. The use of edge bundling remedies this and reveals high-level edge patterns. Previous methods require the graph to contain a hierarchy for this, or they construct a control mesh to guide the edge bundling process, which often results in bundles that show considerable variation in curvature along the overall bundle direction. We present a new edge bundling method that uses a self-organizing approach to bundling in which edges are modeled as flexible springs that can attract each other. In contrast to previous methods, no hierarchy is used and no control mesh. The resulting bundled graphs show significant clutter reduction and clearly visible high-level edge patterns. Curvature variation is furthermore minimized, resulting in smooth bundles that are easy to follow. Finally, we present a rendering technique that can be used to emphasize the bundling.

Many applications, like the retrieval of information from the WWW, require or are improved by the detection of sets of closely related vertices in graphs. Depending on the application, many approaches are possible. In this paper we present a purely graph-theoretical approach, independent of the represented data. Based on the edge-connectivity of subgraphs, a tree of subgraphs is constructed, such that the children of a node are pairwise disjoint and contained in their parent. We describe a polynomial algorithm for the construction of the tree and present two heuristics, constructing the correct result in signi cantly decreased time. Furthermore we give a short description of possible applications in the elds of information retrieval, clustering and graph drawing. 1.

The topic of skeletal animation and its associated techniques have previously been applied in the area of animating computer-generated characters for motion pictures and computer games. This thesis investigates the use of similar techniques in the scope of exploring three-dimensional visualisations of relational networks (graphs). A system

This paper demonstrates the application of graph drawing and information visualisation techniques to the visualisation of information which can be modelled as an attributed graph. An attributed graph can be used to model a wide range of different types of information, including system descriptions and database content. We propose the novel Composable Layouts and Visual Sets (Clovis) class of views, and describe supporting software component infrastructure, including a user interface for creating and interacting with Clovis views. A framework for composing graph vertex layouts is presented, including the division of responsibilities between the layout strategies being composed and the mechanism for coordinating their execution. Three broad classes of layout strategy are identified, and opportunities for novel hybrid layouts highlighted. The definition of sets of graph elements and the allocation or overlaying of distinctive visual attributes to members of the set are combined under the notion of a visual set. A visual querying mechanism for the allocation of graph elements to visual sets and for the clustering of graph vertices in preparation for layout composition is described. The versatility of the Clovis view family is demonstrated through its application to a variety of problem domains, and future research directions are identified. Keywords: Database visualisation, attributed graph, graph drawing, clustered graph, layout composition, overlay, Tree Map.

This paper presents a scale-oriented scheme for data visualization. The aim is to explore and validate the hypothesis, that a high quality visual layout exhibits a good quality hierarchical data clustering.In this scheme, the information to be visualized and clustered is represented as a graph, where the nodes represent pieces of information and the edges represent relationships between those pieces. This scheme supports three related models of information, the underlying graph structure, the graph clustered according to some geometric attributes and, the graph represented according to some drawing mechanism. Also introduced is a method for reducing the computational complexity of a graph drawing algorithm called, force directed placement,fromO(n 2 ) to O(n log n). This method is adapted from an n-body hierarchical force calculation, that allows larger data sets to be draw and visualized on various levels of abstraction. Finally, this method provides the framework for the integration and testing of numerous concepts about how the quality of layout and clustering are related.

Abstract. In this paper, we investigate area requirements for drawing s-t hierarchically planar graphs by straight-lines. Two drawing standards will be discussed: 1) each vertex is represented by a point and 2) grid visibifity representation (that is, a line segment is allowed to represent a vertex). For the first drawing standard, we show an exponential area lower bound needed for drawing hierarchically planar graphs. The lower bound holds even for hierarchical graphs without transitive arcs, in contrast to the results for upward planar drawing. Applications of some existing algorithms from upward drawing can guarantee the quadratic drawing area for grid visibility representation but do not necessarily guarantee the minimum drawing area. Motivated by this, we will present another grid visibifity drawing algoiithm which is efficient and guarantees the minimum drawing area.