Book:Dominic Welsh/Matroid Theory

Subject Matter

 * Matroid Theory

Contents

 * Preface


 * Preliminaries 
 * 1 $\quad$ Basic notation
 * 2 $\quad$ Set theory notation
 * 3 $\quad$ Algebraic structures
 * 4 $\quad$ Graph theory


 * Chapter 1. Fundamental Concepts and Examples
 * 1 $\quad$ Introduction
 * 2 $\quad$ Axiom systems for a matroid
 * 3 $\quad$ Examples of matroids
 * 4 $\quad$ Loops and parallel elements
 * 5 $\quad$ Properties of independent sets and bases
 * 6 $\quad$ Properties of the rank function
 * 7 $\quad$ The closure operator
 * 8 $\quad$ Closed sets = flats = subspaces
 * 9 $\quad$ Circuits
 * 10$\quad$ The cycle matroid of a graph
 * 11$\quad$ A Euclidean representation of matroids of rank $\le 3$


 * Chapter 2. Duality
 * 1 $\quad$ The dual matroid
 * 2 $\quad$ The hyperplanes of a matroid
 * 3 $\quad$ Paving matroids
 * 4 $\quad$ The cocycle matroid of a graph


 * Chapter 3. Lattice Theory and Matroids
 * 1 $\quad$ Brief review of lattice theory
 * 2 $\quad$ The lattice of flats of a matroid
 * 3 $\quad$ Geometric lattices and simple matroids
 * 4 $\quad$ Partition lattices and Dilworth’s embedding theorem


 * Chapter 4. Submatroids
 * 1 $\quad$ Truncation
 * 2 $\quad$ Restriction
 * 3 $\quad$ Contraction
 * 4 $\quad$ Minors and their representation in the lattice


 * Chapter 5. Matroid Connection
 * 1 $\quad$ Two theorems about circuits and cocircuits
 * 2 $\quad$ Connectivity
 * 3 $\quad$ Direct product of lattices, direct sum of matroids
 * 4 $\quad$ Descriptions of matroids
 * 5 $\quad$ The circuit graph of a matroid; a lattice Characterization of connection
 * 6 $\quad$ Tutte’s theory of $n$-connection: wheels and whirls


 * Chapter 6. Matroids, Graphs and Planarity
 * 1 $\quad$ Unique representations of graphic matroids-the importance of $3$-connection
 * 2 $\quad$ Homeomorphism of graphs and matroid minors
 * 3 $\quad$ Planar graphs and their geometric dual
 * 4 $\quad$ Planar graphs and their abstract dual
 * 5 $\quad$ Conditions for a graph to be planar


 * Chapter 7. Transversal Theory
 * 1 $\quad$ Transversals and partial transversals; the Rado-Hall theorems
 * 2 $\quad$ A generalisation of the Rado-Hall theorem
 * 3 $\quad$ Transversals matroids
 * 4 $\quad$ Transversals with prescribed properties-applications of Rado’s theorem
 * 5 $\quad$ Generalised transversals
 * 6 $\quad$ A “converse” to Rado’s theorem


 * Chapter 8. Covering and Packing
 * 1 $\quad$ Submodular functions
 * 2 $\quad$ Functions of matroids; inducing matroids by bipartite graphs
 * 3 $\quad$ The union of matroids
 * 4 $\quad$ Covering and packing theorems
 * 5 $\quad$ Edmond’s intersection theorem


 * Chapter 9. The Vector Representation of Matroids
 * 1 $\quad$ The representability problem
 * 2 $\quad$ Some non-representable matroids
 * 3 $\quad$ The representability of minors, duals, and truncations
 * 4 $\quad$ Chain groups
 * 5 $\quad$ Representability of graphic matroids
 * 6 $\quad$ The representability of induced matroids, unions of matroids and transversal matroids
 * 7 $\quad$ The characteristic set of a matroid
 * 8 $\quad$ Conditions for representability


 * Chapter 10. Binary Matroids
 * 1 $\quad$ Equivalent conditions for representability over $GF(2)$
 * 2 $\quad$ An excluded minor criterion for a matroid to be binary
 * 3 $\quad$ The circuit space and cocircuit space; orientable matroids
 * 4 $\quad$ Regular matroids
 * 5 $\quad$ Conditions for a matroid to be graphic or cographic
 * 6 $\quad$ Simplicial matroids


 * Chapter 11. Matroids from Fields and Groups
 * 1 $\quad$ Algebraic matroids
 * 2 $\quad$ The relation between algebraic and linear representability
 * 3 $\quad$ Operations on algebraic matroids
 * 4 $\quad$ Partition matroids determined by finite groups


 * Chapter 12. Block Designs and Matroids
 * 1 $\quad$ Projective and affine spaces —-
 * 2 $\quad$ Block designs and Steiner systems
 * 3 $\quad$ Matroids and block designs
 * 4 $\quad$ Theorems of Dembowski, Wagner and Kantor
 * 5 $\quad$ Perfect matroid designs
 * 6 $\quad$ The Steiner system $S(5, 8, 24)$ and its subsystems


 * Chapter 13. Menger’s Theorem and Linkings in Graphs
 * 1 $\quad$ The basic linkage lemma
 * 2 $\quad$ Gammoids
 * 3 $\quad$ Matroids induced by linkings
 * 4 $\quad$ A new class of matroids from graphs


 * Chapter 14. Transversal Matroids and Related Topics
 * 1 $\quad$ Base orderable matroids
 * 2 $\quad$ Series parallel networks and extensions of matroids
 * 3 $\quad$ Graphic transversal matroids
 * 4 $\quad$ An equivalent class of binary structures
 * 5 $\quad$ Properties of transversal matroids
 * 6 $\quad$ Matchings in graphs


 * Chapter 15. Polynomials, Colouring Problems, Codes and Packages
 * 1 $\quad$ The chromatic polynomial of a graph
 * 2 $\quad$ The Möbius function of a partially ordered set
 * 3 $\quad$ The chromatic or characteristic polynomial of a matroid
 * 4 $\quad$ The Tuttle polynomial and Whitney rank generating function
 * 5 $\quad$ The critical problem
 * 6 $\quad$ Codes, packings and the critical problem
 * 7 $\quad$ Weight enumeration of codes: the MacWilliams’ identity


 * Chapter 16. Extremely Problems
 * 1 $\quad$ Introduction
 * 2 $\quad$ The Whitney numbers and the unimodal conjecture
 * 3 $\quad$ The Dowling-Wilson theorems
 * 4 $\quad$ Bases and independent sets
 * 5 $\quad$ Sperner’s lemma and Ramsey’s theorem
 * 6 $\quad$ Enumeration


 * Chapter 17. Maps between Matroids and Geometric Lattices
 * 1 $\quad$ Strong maps between geometric lattices
 * 2 $\quad$ Factorisation theorems for strong maps
 * 3 $\quad$ Single element extensions
 * 4 $\quad$ Strong maps induced by the identity functions
 * 5 $\quad$ The scum theorem
 * 6 $\quad$ Erections of matroids
 * 7 $\quad$ The automorphism group of a matroid


 * '''Chapter 18. Convex Polytopes associated with Matroids
 * 1 $\quad$ Convex polytopes and linear programming
 * 2 $\quad$ Polymatroids
 * 3 $\quad$ Polymatroids and submodular set functions
 * 4 $\quad$ Vertices of polymatroids
 * 5 $\quad$ A new class of polytopes with integer vertices
 * 6 $\quad$ Further results on polymatroids
 * 7 $\quad$ Supermatroids


 * Chapter 19. Combinatorial Optimisation
 * 1 $\quad$ The greedy algorithm
 * 2 $\quad$ A greedy algorithm for a class of linear programmes
 * 3 $\quad$ Partitioning and intersection algorithms
 * 4 $\quad$ Lehman’s solution of the Shannon switching game
 * 5 $\quad$ An extension of network flow theory
 * 6 $\quad$ Some intractable problems


 * Chapter 20. Infinite Structures
 * 1 $\quad$ Pre-independence spaces
 * 2 $\quad$ Independence spaces
 * 3 $\quad$ Infinite transversal theory
 * 4 $\quad$ Duality in independence spaces
 * 5 $\quad$ The operator approach to duality


 * Bibliography


 * Index of symbols


 * Index