This is a survey on graph visualization and navigation techniques, as used in information visualization. Graphs appear in numerous applications such as web browsing, state--transition diagrams, and data structures. The ability to visualize and to navigate in these potentially large, abstract graphs is often a crucial part of an application. Information visualization has specific requirements, which means that this survey approaches the results of traditional graph drawing from a different perspective. Index Terms---Information visualization, graph visualization, graph drawing, navigation, focus+context, fish--eye, clustering. 1

We present a novel approach to the aesthetic drawing of undirected graphs. The method has two phases: first embed the graph in a very high dimension and then project it into the 2-D plane using PCA. Experiments we have carried out show the ability of the method to draw graphs of 10 nodes in few seconds. The new method appears to have several advantages over classical methods, including a significantly better running time, a useful inherent capability to exhibit the graph in various dimensions, and an effective means for interactive exploration of large graphs.

This paper draws from art history and perception to place computer depiction in the broader context of picture production. It highlights the often underestimated complexity of the interactions between features in the picture and features of the represented scene. Depiction is not always a unidirectional projection from a 3D scene to a 2D picture, but involves much feedback and influence from the picture space to the object space. Depiction can be seen as a pre-existing 3D reality projected onto 2D, but also as a 2D pictorial representation that is superficially compatible with an hypothetic 3D scene. We show that depiction is essentially an optimization problem, producing the best picture given goals and constraints. We introduce a classification of basic depiction techniques based on four kinds of issue. The spatial system deals with the mapping of spatial properties between 3D and 2D (including, but not restricted to, perspective projection). The primitive system deals with the dimensionality and mappings between picture primitives and scene primitives. Attributes deal with the assignment of visual properties such as colors, texture, or thickness. Finally, marks are the physical implementations of the picture (e.g. brush strokes, mosaic cells). A distinction is introduced between interaction and picturegeneration methods, and techniques are then organized depending on the dimensionality of the inputs and outputs.

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

We study straight-line drawings of graphs with few segments and few slopes. Optimal results are obtained for all trees. Tight bounds are obtained for outerplanar graphs, 2-trees, and planar 3-trees. We prove that every 3-connected plane graph on n vertices has a plane drawing with at most 5n/2 segments and at most 2n slopes. We prove that every cubic 3-connected plane graph has a plane drawing with three slopes (and three bends on the outerface). Drawings of non-planar graphs with few slopes are also considered. For example, interval graphs, co-comparability graphs and AT-free graphs are shown to have have drawings in which the number of slopes is bounded by the maximum degree. We prove that graphs of bounded degree and bounded treewidth have drawings with n) slopes. Finally we prove that every graph has a drawing with one bend per edge, in which the number of slopes is at most one more than the maximum degree.

... G is the minimum number of distinct edgeslopes in a straight-line drawing of G in the plane. We prove that for \Delta> = 5and all large n, there is a \Delta-regular n-vertex graph with slope-number at least n1-8+"\Delta +4. This is the best known lower bound on the slope-number of a graphwith bounded degree. We prove upper and lower bounds on the slope-numberof complete bipartite graphs. We prove a general upper bound on the slopenumber of an arbitrary graph in terms of its bandwidth. It follows that theslope-number of interval graphs, cocomparability graphs, and AT-free graphs isat most a function of the maximum degree. We prove that graphs of boundeddegree and bounded treewidth have slope-number at most O(log n). Finallywe prove that every graph has a drawing with one bend per edge, in which thenumber of slopes is at most one more than the maximum degree. In a companionpaper, planar drawings of graphs with few slopes are also considered.

The main bene ts of a three-dimensional layout of interconnection networks are the savings in material (measured as volume) and the shortening of wires. The result presented in this paper is a general formula for calculating a lower bound on the volume. Moreover, for butter y and X-tree networks we show layouts optimizing the maximum wire length and whose upper bounds on the volume are close to the lower bounds.

Geovisualizers often need to represent data that consists of items related together. Such data sets can be abstracted to a mathematical structure, the graph. A graph contains nodes and edges where the nodes represent the items or concepts of interest, and the edges connect two nodes together according to some associational scheme. Examples of graph data include: network topologies; maps, where nodes represent

We study straight-line drawings of non-planar graphs with few slopes. Interval graphs, co-comparability graphs and AT-free graphs are shown to have have drawings in which the number of slopes is bounded by the maximum degree. We prove that graphs of bounded degree and bounded treewidth have drawings with O(log n) slopes. Finally we prove that every graph has a drawing with one bend per edge, in which the number of slopes is at most one more than the maximum degree. In a companion paper, planar drawings of graphs with few slopes are also considered.

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