Matroid theory /

James Oxley provides a comprehensive introduction to matroid theory, covering the very basics to more advanced topics. With over 500 exercises and proofs of major theorems, this book is the ideal reference and class text for academics and graduate students in mathematics and computer science.

Bibliographic Details
Main Author:	Oxley, J. G.
Format:	eBook
Language:	English
Published:	Oxford ; New York : Oxford University Press, 2011.
Edition:	2nd ed.
Series:	Oxford graduate texts in mathematics ; 21.
Subjects:	Matroids. Matroïdes. Matroids Electronic books.
Online Access:	Connect to the full text of this electronic book

Table of Contents:

Machine generated contents note: 1. Basic definitions and examples
1.1. Independent sets and circuits
1.2. Bases
1.3. Rank
1.4. Closure
1.5. Geometric representations of matroids of small rank
1.6. Transversal matroids
1.7. The lattice of flats
1.8. The greedy algorithm
2. Duality
2.1. The definition and basic properties
2.2. Duals of representable matroids
2.3. Duals of graphic matroids
2.4. Duals of transversal matroids
3. Minors
3.1. Contraction
3.2. Minors of certain classes of matroids
3.3. The Scum Theorem, projection, and flats
4. Connectivity
4.1. Connectivity for graphs and matroids
4.2. Properties of matroid connectivity
4.3. More properties of connectivity
5. Graphic matroids
5.1. Representability
5.2. Duality in graphic matroids
5.3. Whitney's 2-Isomorphism Theorem
5.4. Series-parallel networks
6. Representable matroids
6.1. Projective geometries
6.2. Affine geometries
6.3. Different matroid representations
6.4. Constructing representations for matroids
6.5. Representability over finite fields
6.6. Regular matroids
6.7. Algebraic matroids
6.8. Characteristic sets and decidability
6.9. Modularity
6.10. Dowling geometries
7. Constructions
7.1. Series and parallel connection and 2-sums
7.2. Single-element extensions
7.3. Quotients and related operations
7.4.A non-commutative operation
8. Higher connectivity
8.1. Tutte's definition
8.2. Properties of the connectivity function
8.3.3-cormected matroids and 2-suins
8.4. Wheels and whirls
8.5. Tutte's Linking Theorem and its applications
8.6. Matroid versus graph connectivity
8.7. Some extremal connectivity results
8.8. Tutte's Wheels-and-Whirls Theorem
9. Binary matroids
9.1. Characterizations
9.2. Circuit and cocircuit spaces
9.3. The operation of 3-sum
9.4. Other special properties
10. Excluded-minor theorems
10.1. The characterization of regular matroids
10.2. The characterization of ternary matroids
10.3. The characterization of graphic matroids
10.4. Further properties of regular and graphic matroids
11. Submodular functions and matroid union
11.1. Deriving matroids from submodular functions
11.2. The theorems of Hall and Rado
11.3. Matroid union and its applications
11.4. Amalgams and the generalized parallel connection
11.5. Generalizations of delta-wye exchange
12. The Splitter Theorem
12.1. The theorem and its proof
12.2. Applications of the Splitter Theorem
12.3. Variations on the Splitter Theorem
13. Seymour's Decomposition Theorem
13.1. Overview
13.2. Graphic, cographic, or a special minor
13.3. Blocking sequences
13.4.R12-minors
14. Research in representability and structure
14.1. The Well-Quasi-Ordering Conjecture for Matroids
14.2. Branch-width
14.3. Rota's Conjecture and the Well-Quasi-Ordering Conjecture
14.4. Algorithmic consequences
14.5. Intertwining
14.6. Inequivalent representations
14.7. Ternary matroids
14.8. Stabilizers
14.9. Unavoidable minors
14.10. Growth rates
15. Unsolved problems
15.1. Representability: linear and algebraic
15.2. Unimodal conjectures
15.3. Critical problems
15.4. From graphs to matroids
15.5. Enumeration
15.6.G amino ids and transversal matroids
15.7. Excluding a uniform matroid
15.8. Negative correlation
15.9.A miscellany.