Matroid decomposition.

Matroid Decomposition deals with decomposition and composition of matroids. The emphasis is on binary matroids, which are produced by the matrices over the binary field GF(2). Different classes of matroids are described (graphic, regular, almost regular, max-flow min-cut), along with polynomial testing algorithms. Representative applications and, except for the almost-regular case, characterizations in terms of excluded minors are given. In addition, excluded minor characterizations of both binary and ternary matroids are presented.

Comprised of 13 chapters, this book begins with an introduction to basic definitions concerning graphs and matrices, followed by a discussion on binary matroids. Subsequent chapters focus on some elementary constructions of graphs and binary matroids; a simple yet effective method called the path shortening technique for establishing basic connectivity relationships and certain results about the intersection and partitioning of matroids; an algorithm for identifying certain matroid separations; and the so-called splitter theorem. Fundamental notions and theorems about matroid decomposition and composition are described, along with a very important property of real matrices called total unimodularity. The book concludes with an analysis of flows in matroids based on ideas from flows in graphs.

This monograph will be of interest to students and practitioners in diverse fields such as civil, electrical, and mechanical engineering, as well as computer science and mathematics.