Introduction

Greedoids were invented around 1980 by B. Korte and L. Lovász. Originally, the main motivation for proposing this generalization of the matroid concept came from combinatorial optimization. Korte and Lovász had observed that the optimality of a ‘greedy’ algorithm could in several instances be traced back to an underlying combinatorial structure that was not a matroid – but (as they named it) a ‘greedoid’. In subsequent research greedoids have been shown to be interesting also from various non-algorithmic points of view.

The basic distinction between greedoids and matroids is that greedoids are modeled on the algorithmic construction of certain sets, which means that the ordering of elements in a set plays an important role. Viewing such ordered sets as words, and the collection of words as a formal language, we arrive at the general definition of a greedoid as a finite language that is closed under the operation of taking initial substrings and satisfies a matroid-type exchange axiom. It is a pleasant feature that greedoids can also be characterized in terms of set systems (the unordered version), but the language formulation (the ordered version) seems more fundamental.

Consider, for instance, the algorithmic construction of a spanning tree in a connected graph. Two simple strategies are: (1) pick one edge at a time, making sure that the current edge does not form a circuit with those already chosen; (2) pick one edge at a time, starting at some given node, so that the current edge connects a visited node with an unvisited node.

Introduction

The connections between graph theory and matroid theory can be traced back to the study of graphic matroids, which were introduced by Whitney (1935) and have been extensively investigated (see Chapters 1, 2, and 6 of White, 1986). We recall that a matroid is graphic if it is isomorphic to the polygon matroid of some graph.

In this chapter we present some recent results that give a new setting to the relations between graphs and matroids. In the light of this setting, the polygon matroid appears as the simplest and best known object among an uncountably infinite collection of similar objects.

The fundamental concept that we want to introduce is the concept of a ‘matroidal family of graphs’. The precise definition is given below. According to this definition, the collection of all polygons is a matroidal family of graphs, the simplest among the non-trivial ones. Another example of a matroidal family of graphs is the collection of all bicycles, where a bicycle is a connected graph with two independent cycles and no vertex of degree less than two, that is to say, a bicycle is a graph homeomorphic to one of the graphs J00, J0.0, or J0.0 pictured in Figure 4.1.

As we shall see, there are uncountably many matroidal families of graphs; the subject is virtually unexplored and this chapter is just a brief introduction to this fascinating new field.

Many engineering problems lead to a system of linear equations a represented matroid - whose rank controls critical qualitative features of the example (Sugihara, 1984; 1985; White & Whiteley, 1983). We will outline a selection of such matroids, drawn from recent work on the rigidity of spatial structures, reconstruction of polyhedral pictures, and related geometric problems.

For these situations, the combinatorial pattern of the example determines a sparse matrix pattern that has both a generic rank, for general ‘independent’ values of the non-zero entries, and a geometric rank, for special values for the coordinates of the points, lines, and planes of the corresponding geometric model. Increasingly, the generic rank of these examples has been studied by matroid theoretic techniques. These geometric models provide nice illustrations and applications of techniques such as matroid union, truncation, and semimodular functions. The basic unsolved problems in these examples highlight certain unsolved problems in matroid theory. Their study should also lead to new results in matroid theory.

Bar Frameworks on the Line - the Graphic Matroid

We begin with the simplest example, which will introduce the vocabulary and the basic pattern. We place a series of distinct points on a line, and specify certain bars - pairs of joints which are to maintain their distance - defining a bar framework on the line. We ask whether the entire framework is ‘rigid’ - i.e. does any motion of the joints along the line, preserving these distances, give all joints the same velocity, acceleration, etc.? Clearly a framework has an underlying graph G = (V, E), with a vertex vi for each joint Pi and an undirected edge {i, j} for each bar {pi, pj}.

The many different axiom systems for finite matroids given in Chapter 2 of White (1986) offer numerous possibilities when one is attempting to generalize the theory to structures over infinite sets. Some axiom systems that are equivalent when one has a finite ground set are no longer so when an infinite ground set is allowed. For this reason, there is no single class of structures that one calls infinite matroids. Rather, various authors with differing motivations have studied a variety of classes of matroid-like structures on infinite sets. Several of these classes differ quite markedly in the properties possessed by their members and, in some cases, the precise relationship between particular classes is still not known.

The purpose of this chapter is to discuss the main lines taken by research into infinite matroids and to indicate the links between several of the more frequently studied classes of infinite matroids.

There have been three main approaches to the study of infinite matroids, each of these being closely related to a particular definition of finite matroids. This chapter will discuss primarily the independent-set approach. Some details of the closure-operator approach will also be needed, but a far more complete treatment of this has been given by Klee (1971) and by Higgs (1969a, b, c). The third approach, via lattices, will not be considered here. This approach is taken by Maeda & Maeda (1970) and they develop it in considerable detail.

This is the third volume of a series that began with Theory of Matroids and continued with Combinatorial Geometries. These three volumes are the culmination of more than a decade of effort on the part of the many contributors, potential contributors, referees, the publisher, and numerous other interested parties, to all of whom I am deeply grateful. To all those who waited, please accept my apologies. I trust that this volume will be found to have been worth the wait.

This volume begins with Walter Whiteley's chapter on the applications of matroid theory to the rigidity of frameworks: matroid constructions prove to be rather useful and matroid terminology provides a helpful language for the basic results of this theory. Next we have Deza's chapter on the beautiful applications of matroid theory to a special aspect of combinatorial designs, namely perfect matroid designs. In Chapter 3, Oxley considers ways of generalizing the matroid axioms to infinite ground sets, and Simões-Pereira's chapter on matroidal families of graphs discusses other ways of defining a matroid on the edge set of a graph than the usual graphic matroid method. Next, Rival and Stanford consider two questions on partition lattices. These lattices are a special case of geometric lattices and the inclusion of this chapter will provide a lattice-theoretic perspective which has been lacking in much current matroid research (but which seems alive and well in oriented matroids). Then we have the comprehensive survey by Brylawski and Oxley of the Tutte polynomial and Tutte-Grothendieck invariants. These express the deletion- contraction decomposition that is so important within matroid theory and some of its important applications, namely graph theory and coding theory.