The Galois Complexity of Graph Drawing: Why Numerical Solutions are Ubiquitous for Force-Directed, Spectral, and Circle Packing Drawings

Many well-known graph drawing techniques, including force directed drawings, spectral graph layouts, multidimensional scaling, and circle packings, have algebraic formulations. However, practical methods for producing such drawings ubiquitously use iterative numerical approximations rather than constructing and then solving algebraic expressions representing their exact solutions. To explain this phenomenon, we use Galois theory to show that many variants of these problems have solutions that cannot be expressed by nested radicals or nested roots of low-degree polynomials. Hence, such solutions cannot be computed exactly even in extended computational models that include such operations.Comment: Graph Drawing 201

Drawing Graphs within Restricted Area

We study the problem of selecting a maximum-weight subgraph of a given graph such that the subgraph can be drawn within a prescribed drawing area subject to given non-uniform vertex sizes. We develop and analyze heuristics both for the general (undirected) case and for the use case of (directed) calculation graphs which are used to analyze the typical mistakes that high school students make when transforming mathematical expressions in the process of calculating, for example, sums of fractions

Demonstration of a Preprocessor for the Spring Embedder

Spring embedding is a widely used method for producing automated layouts of graphs. We present a preprocessor that improves the performance of the classical spring embedder which can be used in conjunction with other optimization and approximation techniques. It creates an initial graph layout with edge lengths that are approximately equal and with a minimum node separation from which the spring embedder typically needs far fewer iterations to produce a well laid out graph

A Constrained, Force-Directed Layout Algorithm for Biological Pathways

We present a new elegant algorithm for layout of biological signaling pathways. It uses a force-directed layout scheme, taking into account directional and regional constraints enforced by different molecular interaction types and subcellular locations in a cell. The algorithm has been successfully implemented as part of a pathway integration and analysis toolkit named PATIKA and results with respect to computational complexity and quality of the layout have been found satisfactory

On the Visualization of Java Programs

In this paper we present a graph drawing framework that can be used to automatically draw UML class diagrams and a compiler that extracts the needed information from Java source code. Both components can be combined to a visualization tool for Java programs