The two fundamental examples in Whitney's seminal paper on independence theory (Whitney 1935) were vector matroids and graphic matroids. It is therefore not surprising that so many aspects of matroid theory are extensions and developments of concepts that were originally introduced in vector spaces or in graphs. In this chapter we shall study in detail the class of graphic matroids. The first section will consider the polygon and bond matroids of a graph, that is, those matroids on the edge-set of a graph Г whose circuits are respectively the circuits and the bonds of Г. The main result of Section 6.1 characterizes those graphs whose bond matroids are graphic. The polygon and bond matroids are not the only matroids that can be defined on the edge-set of a graph. Several other such matroids will be considered in detail in a chapter on matroidal families in a later volume.

In Section 6.2 we shall consider the concept of connectivity in matroid theory, indicating its relationship to various notions of connectivity for graphs. These ideas will be used in Section 6.3, where the main result of this chapter is proved. This result, due to Whitney (1933), characterizes precisely when two graphs have isomorphic polygon matroids. In the final section we shall investigate two graph-theoretic operations having their origins in electrical-network theory. One of the important aspects of these operations in the present context is the fact that they have been very successfully generalized to matroids.

This book had its beginnings over a decade ago, as a simple rewriting of Crapo and Rota's preliminary edition of Combinatorial Geometries, to be accomplished by Crapo, Rota, and White. We soon realized that the subject had grown enough, even then, that a more comprehensive compendium would be of greater benefit. This led, in turn, to the idea of soliciting contributions from many of the workers in matroid theory. Consequently, this work has grown too lengthy to be contained in a single volume. This is but the first of a projected three-volume series, although we are giving separate titles to each of the volumes. We are planning to call the remaining volumes Combinatorial Geometries, and Advances in Matroid Theory.

This first volume is a primer in the basic axioms and constructions of matroids. It will prove useful as a text because exposition has been kept a prime consideration throughout. Proofs of theorems are often omitted, with references given to the original works, and exercises are included. This volume will also be useful as a reference work for matroid theorists, especially Brylawski's encyclopedic chapter “Constructions” and his cryptomorphism appendix.

The volume starts with Crapo's chapter “Examples and Basic Concepts.” This chapter is a very informal introduction to matroids, with lots of examples, that provides an overview of the subject. The next chapter is “Axiom Systems,” by Nicoletti and White. This gets into the necessary work of proving the equivalence of some of the major axiom systems, a chore made easier by keeping in mind the familiar analogous concepts from linear algebra.

INTRODUCTION

The notion of orthogonality of combinatorial geometries and of matroids is an abstraction of the usual notion of orthogonality in vector spaces and of perpendicularity in Euclidean geometry. We shall show, for instance, how, relative to any fixed basis X for an n-dimensional vector space T, every subspace V ⊆ T defines a vector geometry on the set X, and orthogonal complementary subspaces V and V⊥ define orthogonal geometries.

We begin this chapter with the abstract combinatorial definition of orthogonality. After some necessary preliminary work on vector geometries, we shall show how orthogonal complementary subspaces give rise to orthogonal geometries, and what this construction implies for matrices and for dual vector spaces. We shall bring the chapter to a close with two further examples of orthogonality of combinatorial geometries as they arise in the study of simplicial geometries and of structure geometries.

Two notes of caution are in order. First, the operation G → G* taking each matroid G to its orthogonal matroid G* is an operation of period 2. This is not so for geometries. If a geometry G has bonds only of cardinality 1 or 2, these become circuits of cardinality 1 or 2 in the orthogonal matroid. That is, they become loops or multiple points that disappear in the passage from the matroid G* to its associated geometry. These elements will not reappear in the geometry G**, if orthogonality is taken to be an operation on geometries, rather than on matroids.

MINORS AND STRONG MAPS

What are the maps, or morphisms, in the category of matroids? As is usual for objects arising in combinatorial analysis and universal algebra, there is more than one reasonable answer. In this chapter and the next, two approaches and their relationship will be discussed.

The notion of morphism is dependent on the notion of subobject. A reasonable notion of subobject for the category of matroids is that of a minor. To define a minor, we first recall from Chapter 7 the operations of contraction and deletion. Let M be a matroid on the set S, and let U and V be subsets of S. The restriction of the matroid M to the set U is the matroid M(U) on the set U for which the rank of a subset A in U is simply its rank in M as a subset of S. We also say that the restricted matroid M(U) is obtained from M by deleting the elements in S – U from S, and we shall sometimes denote M(U) by M –(S – U). The contraction of M by the subset V is the matroid M/V on the set S – V whose rank function is as follows: For A ⊆ S – V, r(A) = rM(A ∪ V) – rM(V). Contractions and deletions commute in the sense that for any pair of disjoint subsets U and V, the matroids (M – U)/V and (M/V) – U are the same matroid on the set S – (U ∪ V).

A large body of mathematics consists of facts that can be presented and described much like any other natural phenomenon. These facts, at times explicitly brought out as theorems, at other times concealed within a proof, make up most of the applications of mathematics, and are the most likely to survive change of style and of interest.

This ENCYCLOPEDIA will attempt to present the factual body of all mathematics. Clarity of exposition, accessibility to the nonspecialist, and a thorough bibliography are required of each author. Volumes will appear in no particular order, but will be organized into sections, each one comprising a recognizable branch of present-day mathematics. Numbers of volumes and sections will be reconsidered as times and needs change.

It is hoped that this enterprise will make mathematics more widely used where it is needed, and more accessible in fields in which it can be applied but where it has not yet penetrated because of insufficient information.

EXAMPLES FROM LINEAR ALGEBRA AND PROJECTIVE GEOMETRY

As an introduction to the concepts of combinatorial geometry and matroid, we wish to emphasize those features of the theory that have given it a unifying role in other branches of mathematics, that have permitted it to be fruitfully applied in disparate domains of science, and that continue to arouse broader interest in the subject. It is our intention in this chapter to clarify the basic concepts by showing how they appear and are interrelated in a list of significant examples. This will give the reader a general orientation with respect to the basic concepts, prior to their axiomatic treatment in Chapter 2. Most of these examples will be dealt with in full detail in subsequent chapters.

The concept of combinatorial geometry arose from work in projective geometry and linear algebra. The focus of this work was to understand the basic properties of two relations:

1. (1) the incidence between points, lines, planes, and so on (which in general we call flats) in geometries and geometric configurations

2. (2) the linear dependence of sets of vectors.

The task to characterize (axiomatize) these relations of incidence and linear dependence seems in retrospect both urgent and feasible in the light of far-reaching applications, both to new geometries and to more general algebraic and combinatorial structures.

Research in combinatorial geometry has been concentrated on

1. (1) synthetic (combinatorial) methods, involving only incidence relations between flats, and the fundamental operations of projection and intersection

2. (2) intrinsic properties of configurations, internal properties that configurations possess independent of the way in which they may be represented or constructed within some conventional space.

It was shown in Chapter 2 that the notion of rank function provides a cryptomorphic theory of matroids. The semimodularity property of the rank is essentially equivalent to the basis-exchange or circuit-elimination axioms and thus is central to matroid theory.

In this chapter we shall consider a more general class of functions on subsets of a finite set E that are semimodular and nondecreasing, but not necessarily nonnegative, normalized, or unit-increased.

The relationships between semimodular functions and matroids have been known and studied since the very beginning of matroid theory. Many results have been found and sometimes independently rediscovered. This chapter presents a unifying theory that attempts to explain most of these known results as derived essentially from Dilworth's fundamental theorem about the embedding of a point lattice into a geometric lattice. Dilworth's original proof was based on a construction of a rank function from a semimodular function, which is a special case of what will be more generally defined in this chapter as expansions. The main thrust of the exposition is to study the properties and applications of expansions.

GENERAL PROPERTIES OF SEMIMODULAR FUNCTIONS

In the most general setting that will be of interest we shall consider a point lattice L and classes of integer-valued functions defined on L.

It is a rare event that mathematicians, cozily ensconced in the world of established theories, should extract, by dint of pioneering work, some new gem that later generations will spend decades polishing and refining. The hard-won theory of matroids is one such instance. Rich in connections with mathematics, pure and applied, deeply rooted in the utmost reaches of combinatorial thinking, strongly motivated by the toughest combinatorial problems of our day, this theory has emerged as the proving ground of the idea that combinatorics, too, can yield to the power of systematic thinking.

A superficial look at the theory of matroids might lead to the conclusion that it is largely an abstraction of linear algebra. It was noticed quite some time ago that the elementary theory of linear dependence can be developed from the MacLane-Steinitz exchange axiom. In fact, this abstraction was first exploited in the theory of transcendence degrees of fields. But such a conclusion would be unwarranted. What the theory of matroids provides is a variety of cryptomorphic axiomatic approaches, each of which corresponds to a genuinely new way of looking at linear algebra. Linear algebraists might not easily be led to these axiomatic approaches. The axiomatization of matroids by the notion of a minimal dependent set, for example, leads to the deeper matching theorems for sets of vectors. The axiomatization by the notion of rank leads to the classification of projectively invariant constructions of new matroids from old.