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

Preserving the “mental map ” is a major goal of interactive graph drawing algorithms. Several models have been proposed for formalizing the notion of mental map. Additional work needs to be done to formulate and validate “difference ” metrics which can be used in practice. This paper introduces a framework for defining and validating metrics to measure the difference between two drawings of the same graph, and gives a preliminary experimental analysis of several simple metrics.

An orthogonal drawing of a graph is an embedding of the graph in the rectangular grid, with vertices represented by axis-aligned boxes, and edges represented by paths in the grid which only possibly intersect at common endpoints. In this paper, we study three-dimensional orthogonal drawings and provide lower bounds for three scenarios: (1) drawings where vertices have bounded aspect ratio, (2) drawings where the surface of vertices is proportional to their degree, and (3) drawings without any such restrictions. Then we show that these lower bounds are asymptotically optimal, by providing constructions that match the lower bounds in all scenarios within an order of magnitude.

In this paper we investigate the general position model for the drawing of arbitrary degree graphs in the D-dimensional (D >= 2) orthogonal grid. In this model no two vertices lie in the same grid hyperplane.

A 3-D orthogonal drawing of graph with maximum degree at most six positions the vertices at grid-points in the 3-D orthogonal grid, and routes edges along grid-lines such that edge routes only intersect at common end-vertices. In this paper we present two algorithms for producing 3-D orthogonal graph drawings with the vertices positioned along the main diagonal of a cube, so called diagonal drawings. This vertex-layout strategy was introduced in the 3-Bends algorithm of Eades et al. [11]. We show that minimising the number of bends in a diagonal drawing of a given graph is NP-hard. Our first algorithm minimises the total number of bends for a fixed ordering of the vertices along the diagonal. Using two heuristics for determining this vertex ordering we obtain upper bounds on the number of bends. Our second algorithm, which is a variation of the above-mentioned 3-Bends algorithm, produces 3-bend drawings with n^3 + o(n^3) volume, which is the best known upper bound for the volume of 3-D orthogonal graph drawings with at most 3 bends per edge.

Hermes is a system for exploring and visualizing the Internet structure at the level of the Autonomous Systems and their interconnections. It relies on a three-tier architecture, on a large repository of routing information coming from heterogeneous sources, and on sophisticated graph drawing engine. Such an engine exploits static and dynamic graph drawing techniques, specifically devised for the visualization of large graphs with high density.

In orthogonal graph drawing, edges are represented by sequences of horizontal and vertical straight line segments. For graphs of degree at most four, this can be achieved by embedding the graph in a grid. The number of bends displayed is an important criterion for layout quality. A well-known algorithm of Tamassia eciently embeds a planar graph with xed combinatorial embedding and vertex degree at most four in the grid such that the number of bends is minimum [23].

The need for a similarity measure for comparing two drawings of graphs arises in problems such as interactive graph drawing and the indexing or browsing of large sets of graphs. Many applications have been based on intuitive ideas of what makes two drawings look similar — for example, the idea that vertex positions should not change much. In this paper, we formally define several of these intuitive ideas of similarity and present the results of a user study designed to evaluate how well these measures reflect human perception of similarity.

In this paper we introduce a new drawing style of a plane graph G, called proper box rectangular (PBR ) drawing. It is defined to be a drawing of G such that every vertex is drawn as a rectangle, called a box, each edge is drawn as either a horizontal or a vertical line segment, and each face is drawn as a rectangle. We establish necessary and sufficient conditions for G to have a PBR drawing. We also give a simple linear time algorithm for finding such drawings. The PBR drawing is closely related to the box rectangular (BR ) drawing defined by Rahman, Nakano and Nishizeki [17]. Our method can be adapted to provide a new simpler algorithm for solving the BR drawing problem. 1 Introduction The problem of "nicely" drawing a graph G has received increasing attention [5]. Typically, we want to draw the edges and the vertices of G on the plane so that certain aesthetic quality conditions and/or optimization measures are met. Such drawings are very useful in visualizing planar graphs and fi...

We consider the problem of determining a balanced ordering of the vertices of a graph; that is, the neighbors of each vertex v are as evenly distributed to the left and right of v as possible. This problem, which has applications in graph drawing for example, is shown to be NP-hard, and remains NP-hard for bipartite simple graphs with maximum degree six. We then describe and analyze a number of methods for determining a balanced vertex-ordering, obtaining optimal orderings for directed acyclic graphs and graphs with maximum degree three. Finally we