Title: Matroids : a geometric introduction /
Authors: Gordon, Gary, 1956-; McNulty, Jennifer.
Publisher: Cambridge University Press,
Year: 2012
Subjects: Matroids.; Combinatorial geometry.
Abstract: Matroid theory is a vibrant area of research that provides a unified way to understand graph theory, linear algebra and combinatorics via finite geometry. This book provides the first comprehensive introduction to the field which will appeal to undergraduate students and to any mathematician interested in the geometric approach to matroids. Written in a friendly, fun-to-read style and developed from the authors' own undergraduate courses, the book is ideal for students. Beginning with a basic introduction to matroids, the book quickly familiarizes the reader with the breadth of the subject, and specific examples are used to illustrate the theory and to help students see matroids as more than just generalizations of graphs. Over 300 exercises are included, with many hints and solutions so students can test their understanding of the materials covered. The authors have also included several projects and open-ended research problems for independent study.
Contents: 1. A tour of matroids -- 1.1. Motivation -- 1.2. Introduction to matroids -- 1.3. Geometries -- 1.4. Graphs and matroids -- 1.5. Bipartite graphs and transversal matroids -- Exercises -- 2. Cryptomorphisms -- 2.1. From independent sets to bases and back again -- 2.2. Circuits and independent sets -- 2.3. Rank, flats, hyperplanes and closure -- 2.4. Lattice of flats -- 2.5. Tying it together with the rank function -- 2.6. Cryptomorphisms between flats, hyperplanes and closure -- 2.7. Application to optimization: the greedy algorithm -- Exercises -- 3. New matroids from old -- 3.1. Matroid deletion and contraction -- 3.2. Deletion and contraction in graphs and representable matroids -- 3.3. Duality in matroids -- 3.4. Duality in graphic and representable matroids -- 3.5. Direct sums and connectivity -- Exercises -- 4. Graphic matroids -- 4.1. Graphs are matroids -- 4.2. Graph versions of matroid theorems -- 4.3. Duality and cocircuits in graphs --^