There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

We will review some of the major results in random graphs and some of the more challenging open problems. We will cover algorithmic and structural questions. We will touch on newer models, including those related to the WWW.

Random graphs

Social Network Analysis

PART I: FOUNDATIONS 1. Introduction to Fixed-Parameter Algorithms 2. Preliminaries and Agreements 3. Parameterized Complexity Theory - A Primer 4. Vertex Cover - An Illustrative Example 5. The Art of Problem Parameterization 6. Summary and Concluding Remarks PART II: ALGORITHMIC METHODS 7. Data Reduction and Problem Kernels 8. Depth-Bounded Search Trees 9. Dynamic Programming 10. Tree Decompositions of Graphs 11. Further Advanced Techniques 12. Summary and Concluding Remarks PART III: SOME THEORY, SOME CASE STUDIES 13. Parameterized Complexity Theory 14. Connections to Approximation Algorithms 15. Selected Case Studies 16. Zukunftsmusik References Index

Invitation to fixed-parameter algorithms

Computational geometry

In this paper we present an extensive experimental study comparing four general-purpose graph drawing algorithms. The four algorithms take as input general graphs (with no restrictions whatsoever on connectivity, planarity, etc.) and construct orthogonal grid drawings, which are widely used in software and database visualization applications. The test data (available by anonymous ftp) are 11,582 graphs, ranging from 10 to 100 vertices, which have been generated from a core set of 112 graphs used in “real-life” software engineering and database applications. The experiments provide a detailed quantitative evaluation of the performance of the four algorithms, and show that they exhibit trade-offs between “aesthetic” properties (e.g., crossings, bends, edge length) and running time.

An experimental comparison of four graph drawing algorithms

Cognitive experiments show that humans can read graph drawings in which all edge crossings are at right angles equally well as they can read planar drawings; they also show that the readability of a drawing is heavily affected by the number of bends along the edges. A graph visualization whose edges can only cross perpendicularly is called a RAC (Right Angle Crossing) drawing . This paper initiates the study of combinatorial and algorithmic questions related with the problem of computing RAC drawings with few bends per edge. Namely, we study the interplay between number of bends per edge and total number of edges in RAC drawings. We establish upper and lower bounds on these quantities by considering two classical graph drawing scenarios: The one where the algorithm can choose the combinatorial embedding of the input graph and the one where this embedding is fixed.

Drawing Graphs with Right Angle Crossings

A polynomial-time algorithm for testing if a triconnected directed graph has an upward drkwing is presented An upward drkwing is a planar drkwing such that all the edges flow in a common direction (eg, from bottom to top) The problem arises in the fields of automatic graph drkwing and ordered sets, and has been open for several years The proposed algorithm is based on a new combinatorial characterization that maps the problem into a max-flow problem on a sparse network; the time complexity isO(n+r

 2)
 , wheren is the number of vertices andr is the number of sources and sinks of the directed graph If the directed graph has an upward drkwing, the algorithm allows us to construct one easily

Upward drawings of triconnected digraphs

An annotated bibliography on 1-planarity

This paper investigates the following question: Given an integer grid ϕ, where ϕ is a proper subset of the integer plane or a proper subset of the integer 3d space, which graphs admit straight-line crossingfree drawings with vertices located at the grid points of ϕ? We characterize the trees that can be drawn on a two dimensional c · n × к grid, where к and c are given integer constants, and on a two dimensional grid consisting of к parallel horizontal lines of infinite length. Motivated by the results on the plane we investigate restrictions of the integer grid in 3 dimensions and show that every outerplanar graph with n vertices can be drawn crossing-free with straight lines in linear volume on a grid called a prism. This prism consists of 3n integer grid points and is universal — it supports all outerplanar graphs of n vertices. This is the first algorithm that computes crossing-free straight line 3d drawings in linear volume for a non-trivial family of planar graphs. We also show that there exist planar graphs that cannot be drawn on the prism and that the extension to a n × 2 × 2 integer grid, called a box, does not admit the entire class of planar graphs.

Giuseppe Liotta

Papers

An experimental comparison of four graph drawing algorithms

Drawing Graphs with Right Angle Crossings

Upward drawings of triconnected digraphs

An annotated bibliography on 1-planarity

Straight-Line Drawings on Restricted Integer Grids in Two and Three Dimensions