This book presents rigidity theory in a historical context. The combinatorial aspects of rigidity are isolated and framed in terms of a special class of matroids, which are a natural generalization of the connectivity matroid of a graph. This book includes an introduction to matroid theory and an extensive study of planar rigidity. The final chapter is devoted to higher-dimensional rigidity, highlighting the main open questions. Also included is an extensive annotated bibliography with over 150 entries. This book is aimed at graduate students and researchers in graph theory and combinatorics or in fields which apply the structural aspects of these subjects in architecture and engineering. Accessible to those who have had an introduction to graph theory at the senior or graduate level, this book is suitable for a graduate course in graph theory.
Readership: Graduate students and researchers in graph theory and combinatorics or in fields which apply the structural aspects of these subjects in architecture and engineering.
Author(s): Brigitte Servatius, Herman Servatius Jack Graver
Series: Graduate Studies in Mathematics 2
Edition: annotated edition
Publisher: American Mathematical Society
Year: 1993
Language: English
Pages: C, X, 172, B
Preface
Chapter 1. Overview
1.1. An Intuitive Introduction to Rigidity
1.2. A Short History of Rigidity.
Chapter 2. Infinitesimal Rigidity
2.1. Basic Definitions
2.2. Independence and the Stress Space
2.3. Infinitesimal Motions and Isometries
2.4. Infinitesimal and Generic Rigidity
2.5. Rigidity Matroids
2.6. Isostatic Sets
Chapter 3. Matroid Theory
3.1. Closure Operators
3.2. Independence Systems
3.3. Basis Systems
3.4. Rank Function
3.5. Cycle Systems
3.6. Duality and Minors
3.7. Connectivity
3.8. Representability
3.9. Transversal Matroids
3.10. Graphic Matroids
3.11. Abstract Rigidity Matroids
Chapter 4. Linear and Planar Rigidity
4.1. Abstract Rigidity in the Plane
4.2. Combinatorial Characterizations of G2(n).
4.3. Cycles in G2(n).
4.4. Rigid Components of G2(G).
4.5. Representability of G2( n).
4.6. Characterizations of .A2 and (.A2)
4.7. Rigidity and Connectivity
4.8. Trees and 2-dimensional Isostatic Sets
4.9. Tree Decomposition Theorems
4.10. Computational Aspects
Chapter 5. Rigidity in Higher Dimensions
5.1. Introduction
5.2. Higher Dimensional Examples
5.3. The Henneberg Conjecture
5.4. Stresses and Strains
5.5. 2-Extensions in 3-Space
5.6. The Dress Conjecture
References
Index
