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

This paper formalizes the synthesis process of wiring signatur e-invariant (WSI) combinational circuit mutants. The signature 0 is defined by a reference circuit 0, which itself is modeled as a canonic alform of a directed bipartite graph. A wiring perturbation induces a perturbed reference circuit. A number of mutant circuits i can be resynthesized from the perturbed circuit. The mutants of interest are the ones that belong to the wiring-signature invariant equivalenc e classN 0, i.e. the mutants i 2N 0. Cir cuit mutants i 2N 0have a number of useful properties. For any wiring perturbation, the size of the wiring signature-invariant equivalence class is huge. Notably, circuits in this class are not random, although for un biased testing and benchmarking purp oses, mutant selections from this class are typically random. For each reference circuit, we synthesized eight equivalence subclasses of circuit mutants, based on 0 to 100 % perturbation. Each subclass contains 100 randomly chosen mutant circuits, each listed in a different random order. The 14,400 benchmarking experiments with 3200 mutants in 4 equivalence classes, covering 13 typical EDA algorithms, demonstrate that an unbiased random selection of such circuits can lead to statistically meaningful differentiation and improvements of existing and new algorithms.

In a total order of the vertices of a graph, two edges with no endpoint in common can be crossing, nested, or disjoint. A k-stack (resp...

We introduce a new approach for drawing diagrams. Our approach is to use a technique we call confluent drawing for visualizing non-planar graphs in a planar way. This approach allows us to draw, in a crossing-free manner, graphs—such as software interaction diagrams—that would normally have many crossings. The main idea of this approach is quite simple: we allow groups of edges to be merged together and drawn as “tracks” (similar to train tracks). Producing such confluent drawings automatically from a graph with many crossings is quite challenging, however, we offer a heuristic algorithm (one version for undirected graphs and one version for directed ones) to test if a non-planar graph can be drawn efficiently in a confluent way. In addition, we identify several large classes of graphs that can be completely categorized as being either confluently drawable or confluently non-drawable.

We consider graph drawings in which vertices are assigned to layers and edges are drawn as straight line-segments between vertices on adjacent layers. We prove that graphs admitting crossing-free h-layer drawings (for fixed h) have bounded pathwidth. We then use a path decomposition as the basis for a linear-time algorithm to decide if a graph has a crossing-free h-layer drawing (for fixed h). This algorithm is extended to solve a large number of related problems, including allowing at most k crossings, or removing at most r edges to leave a crossing-free drawing (for fixed k or r). If the number of crossings or deleted edges is a non-fixed parameter then these problems are NP-complete. For each setting, we can also permit downward drawings of directed graphs and drawings in which edges may span multiple layers, in which case the total span or the maximum span of edges can be minimized. In contrast to the so-called Sugiyama method for layered graph drawing, our algorithms do not assume a preassignment of the vertices to layers.

We propose a novel visualization approach that facilitates graphical exploration and communication of relative actor status in social networks. The main idea is to map, in a drawing of the entire network, actor status scores to vertical coordinates. The resulting problem of determining horizontal positions of actors and routing of connecting lines such that the overall layout is readable is algorithmically difficult, yet well-studied in the literature on graph drawing. We outline a customized approach. The advantages

In radial drawings of hierarchical graphs the vertices are placed on concentric circles rather than on horizontal lines and the edges are drawn as outwards monotone segments of spirals rather than straight lines as it is both done in the standard Sugiyama framework. This drawing style is well suited for the visualisation of centrality in social networks and similar concepts. Radial drawings also allow a more flexible edge routing than horizontal drawings, as edges can be routed around the center in two directions. In experimental results this reduces the number of crossings by approximately 30 percent on average. Few crossings are one of the major criteria for human readability. This paper is a detailed description of a complete framework for visualizing hierarchical information in a new radial fashion. Particularly, we briefly cover extensions of the level assignment step to benefit by the increasing perimeters of the circles, present three heuristics for crossing reduction in radial level drawings, and also show how to visualize the results.

We study the multi-layer crossing minimization problem from a polyhedral point of view. After the introduction of an integer programming formulation of the multi-layer crossing minimization problem, we examine the 2-layer case and derive several classes of facets of the associated polytope. Preliminary computational results for 2- and 3-layer instances indicate, that the usage of the corresponding facet-defining inequalities in a branch-and-cut approach may only lead to a practically useful algorithm, if deeper polyhedral studies are conducted.

The bigraph crossing problem, embedding the two vertex sets of a bipartite graph G = (V0;V1;E) along two parallel lines so that edge crossings are minimized, has application to circuit layout and graph drawing. We consider the case where both V0 and V1 can be permuted arbitrarily -- both this and the case where the order of one vertex set is fixed are NP-hard. Two new heuristics that perform well on sparse graphs such as occur in circuit layout problems are presented. The new heuristics outperform existing heuristics on graph classes that range from application-specific to random. Our experimental design methodology ensures that differences in performance are statistically significant and not the result of minor variations in graph structure or input order.

A bipartite graph is biplanar if the vertices can be placed on two parallel lines (layers) in the plane such that there are no edge crossings when edges are drawn straight. The 2-Layer Planarization problem asks if k edges can be deleted from a given graph G so that the remaining graph is biplanar. This problem is NP-complete, as is the 1-Layer Planarization problem in which the permutation of the vertices in one layer is fixed. We give the following fixed parameter tractability results: an O(k ·6 k +|G|) algorithm for 2-Layer Planarization and an O(3 k ·|G|) algorithm for 1-Layer Planarization, thus achieving linear time for fixed k.