This book is the second in a three-volume series, the first of which is Theory of Matroids, and the third of which will be called Combinatorial Geometries: Advanced Theory. The three volumes together will constitute a fairly complete survey of the current knowledge of matroids and their closely related cousins, combinatorial geometries. As in the first volume, clear exposition of our subject has been one of our main goals, so that the series will be useful not only as a reference for specialists, but also as a textbook for graduate students and a first introduction to the subject for all who are interested in using matroid theory in their work.
This volume begins with three chapters on coordinatization or vector representation, by Fournier and White. They include a general chapter on ‘Coordinatizations,’ and two chapters on the important special cases of ‘Binary Matroids’ and ‘Unimodular Matroids’ (also known as regular matroids). These are followed by two chapters by Brualdi, titled ‘Introduction to Matching Theory’ and ‘Transversal Matroids,’ and a chapter on ‘Simplicial Matroids’ by Cordovil and Lindstrom. These six chapters, together with Oxley's ‘Graphs and Series-Parallel Networks’ from the first volume, constitute a survey of the major special types of matroids, namely, graphic matroids, vector matroids, transversal matroids, and simplicial matroids. We follow with two chapters on the important matroids invariants, ‘The Mobius Function and the Characteristic Polynomial’ by Zaslavsky and ‘Whitney Numbers’ by Aigner. We conclude with a chapter on the aspect of matroid theory that is primarily responsible for an explosion of interest in the subject in recent years, ‘Matroids in Combinatorial Optimization’ by Faigle.