Matroids: A Geometric Introduction

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.

Rank Vertex Cover as a Natural Problem for Algebraic Compression

“…We remark that matroid deletion and contraction can be done time polynomial in the size of ground set for a linear matroid. For details we refer to [13,26]. For example, if M is an identity matrix (where each element is a co-loop), then τ (P ) is the vertex cover number of H. Moreover, if we let M be the uniform matroid U t,n such that t is at least the size of the vertex cover number of H, then τ (P ) again equals the vertex cover number of H.…”

Binary matroids and local complementation

“…One way to define the Tutte polynomial of M [IAS(G)] is a polynomial in the variables s and z, given by the subset expansion Here r G denotes the rank function of M [IAS(G)]. We do not give a general account of this famous invariant of graphs and matroids here; thorough introductions may be found in [6,20,23,26].…”

Fixing Numbers for Matroids

“…Throughout, we will let M be a matroid on the ground set E with circuits C. Matroids can be defined in a variety of equivalent ways; See [7] or [10] for a background on matroids.…”

“…Geometrically, one can use the bipartite graph G to construct an affine diagram for a transversal matroid. Details are given in [7]. In this context, the points in a clonal class are placed freely on the face of a simplex.…”