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Matroid Applications

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Overview

This volume, the third in a sequence that began with Theory of Matroids and Combinatorial Geometries, concentrates on the applications of matroid theory to a variety of topics from geometry rigidity and lattices, combinatorics graphs, codes and designs and operations research the greedy algorithm. As with its predecessors, the contributors to this volume have written their articles to form a cohesive account so that the result is a volume that will be a valuable reference for research workers.
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Editorial Reviews

From the Publisher
"...will be most useful to researchers in combinatorics and related areas and to graduate students who want to learn about the most recent advances in the subject. The book provides a rich collection of exercises to aid the latter. It is to the credit of the authors and the editor that the book provides smooth and enjoyable reading at a very high level of exposition." Peter Orlik, SIAM Review
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Product Details

Table of Contents

List of Contributors
Preface
1 Matroids and Rigid Structures 1
1.1 Bar Frameworks on the Line--The Graphic Matroid 1
1.2 Bar Frameworks in the Plane 3
1.3 Plane Stresses and Projected Polyhedra 11
1.4 Bar Frameworks in 3-space 16
1.5 A Matroid from Splines 24
1.6 Scene Analysis of Polyhedral Pictures 29
1.7 Special Positions 36
1.8 Conclusions 39
1.9 Historical Notes 40
Exercises 41
References 51
2 Perfect Matroid Designs 54
2.1 Introduction 54
2.2 Some Lattice Properties and Parameters of PMDs 56
2.3 Intersection Properties of the Hyperplane Family 58
2.4 PMDs of Rank 4 62
2.5 Triffids 64
2.6 Some Close Relatives of PMDs and Further Directions 68
References 71
3 Infinite Matroids 73
3.1 Pre-independence Spaces and Independence Spaces 73
3.2 B-matroids 80
Exercises 86
References 89
4 Matroidal Families of Graphs 91
4.1 Introduction 91
4.2 Matroidal Families of Graphs: a Definition 92
4.3 Countably Many Matroidal Families 96
4.4 Uncountably Many Matroidal Families 98
4.5 Digraphs and Infinite Graphs 101
Exercises 102
References 104
5 Algebraic Aspects of Partition Lattices 106
5.1 Introduction 106
5.2 The Lattice of All Partitions 107
5.3 Lattice Embeddings 112
5.4 Postscript 118
Exercises 119
References 120
6 The Tutte Polynomial and Its Applications 123
6.1 Introduction 123
6.2 The Tutte Polynomial 124
6.3 T-G Invariants in Graphs 136
6.4 The Critical Problem 162
6.5 Linear Codes 181
6.6 Other Tutte Invariants 188
Exercises 197
References 215
7 Homology and Shellability of Matroids and Geometric Lattices 226
7.1 Introduction 226
7.2 Shellable Complexes 228
7.3 Matroid Complexes 232
7.4 Broken Circuit Complexes 238
7.5 Application to Matroid Inequalities 242
7.6 Order Complexes of Geometric Lattices 248
7.7 Homology of Shellable Complexes 252
7.8 Homology of Matroids 255
7.9 Homology of Geometric Lattices 259
7.10 The Orlik-Solomon Algebra 264
7.11 Notes and Comments 267
Exercises 270
References 281
8 Introduction to Greedoids 284
8.1 Introduction 284
8.2 Definitions and Basic Facts 286
8.3 Examples 293
8.4 Structural Properties 302
8.5 Optimization on Greenoids 308
8.6 The Greedoid Polynomial 313
8.7 Antimatroids 319
8.8 Poset of Flats 327
8.9 Further Topics 335
8.10 Notes and Comments 343
Exercises 348
References 354
Index 358
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