A framework in \(m\)-space is a triple \((V,E,p)\) where \((V,E)\) is a graph and \(p\) is an embedding of \(V\) into real \(m\)-space. If \(m \leq 3\), one may consider each vertex \(v\) as an idealized ball joint located at \(p(v)\) and each edge as a rigid rod connecting the corresponding joints. Hence we can describe rigid structures like bridges and moving ones like machines. This distinction between rigid and nonrigid frameworks depends both on the graph and the embedding. The book concentrates on the combinatorial part of rigidity theory.
Chapter 1 is an informal overview in historical context. Chapter 2 studies infinitesimal rigidity, “a linear approximation which stands at the boundary of the combinatorial and geometric nature of rigidity”.
Since matroid theory is the fundamental tool to study rigidity, Chapter 3 is devoted to matroids. Then linear and planar rigidity is covered in Chapter 4 and the case \(m \geq 3\) in the last chapter. While \(m=3\) has clearly the greatest practical significance, the characterization problem is still unsolved here. On the other hand, the book also treats algorithmic and computational aspects if \(m \leq 2\).
The book is suggested for a second graduate course in combinatorics. The present reviewer is convinced that it can excellently be used for it, due to the large number of figures, exercises, to an extensive annotated bibliography, and, last but not least, to its style, to the very clear way of presentation etc.