Binary matroids play an important theoretical role, partly because they were the first class of coordinatizable matroids to be completely characterized, but also because the class of binary matroids contains the unimodular matroids and the graphic matroids, two classes fundamental to matroid theory. There are numerous characterizations of binary matroids, very different in nature, and expressive of the richness of the concept.

Definition and Basic Properties

Definition. A matroid is binary if it is representable (coordinatizable) over the two-element field GF(2).

According to the general definition of representable matroids, (see Chapter 1), a matroid M(E) on a finite set E is binary if there is a mapping α of E into a GF(2)-vector space V such that a subset X ⊆ E is independent in M(E) if and only if the restriction of α to X is injective and the set {α(x)|x ∈ X} of vectors in V is linearly independent. The mapping a is then called a binary representation of the matroid M(E).

Example. Denote by Ur,n up to isomorphism, the matroid on a set of n elements, in which the bases are those subsets which have r elements. Then U2,3 is binary. (This matroid is identified with the projective line over the field GF(2).) On the other hand, U2,4. is not binary; this matroid, which consists of four geometric points on a line, is a typical non-binary matroid, and serves to characterize the binary matroids, as we shall see later.

Matroids enter combinatorial optimization problems at various levels. Whitney's (1935) motivation to introduce matroids as combinatorial objects in their own right stemmed from his interest in approaching the Four-Color Problem algebraically and combining the combinatorial and algebraic-geometric aspects of graphs into the notion of a matroid.

Graphs furnish the most important models for combinatorial optimization problems. Thus it is natural to ask to what extent graph properties actually are properties of the underlying matroid and to study more general classes of matroids that enjoy, for example, the ‘max-flow-min-cut’ property of network flows (cf. Seymour 1977). This approach leads to fundamental structural questions about matroids per se which, nevertheless, have many practical implications. One of the foremost results in this area is Seymour's (1980) decomposition theory for regular matroids exhibiting regular matroids as being essentially built up by graphic and cographic matroids. As a consequence, efficient procedures can be developed to test whether a matrix is totally unimodular or whether certain linear programs actually are (better tractable) network problems (see, e.g., Welsh 1982 and Bixby 1982 for an introduction into this aspect of matroid theory).

Matroids also compose the combinatorial structure of linear programming (Minty 1966, Rockafellar 1969). Indeed, pivoting in linear programming may be carried out purely ‘combinatorially’ (Bland 1977).

Introduction and Basic Definitions

The purpose of this chapter is to provide background and general results concerning coordinatizations, while the more specialized subtopics of binary and unimodular matroids are covered in later chapters. The first section of this chapter is devoted to definitions and notational conventions. The second section concerns linear and projective equivalence of coordinatizations. Although they are not usually explicitly considered in other expositions of matroid coordinatization, these equivalence relations are very useful in working with examples of coordinatizations, as well as theoretically useful as in Proposition 1.2.5. Section 1.3 involves the preservation of coordinatizability under certain standard matroid operations, including duality and minors. The next section presents some well-known counterexamples, and Section 1.5 considers characterizations of coordinatizability, especially characterizations by excluded minors. The final five sections are somewhat more technical in nature, and may be omitted by the reader who desires only an introductory survey. Section 1.6 concerns the bracket conditions, another general characterization of coordinatizability. Section 1.7 presents techniques for construction of a matroid requiring a root of any prescribed polynomial in a field over which we wish to coordinatize it. These techniques are extremely useful in the construction of examples and counterexamples, yet are not readily available in other works, except Greene (1971). The last three sections concern characteristic sets, the use of transcendentals in coordinatizations, and algebraic representation (i.e., modeling matroid dependence by algebraic dependence). Some additional topics which could have been considered here, such as chain groups, are omitted because they are well-covered in other readily available sources, such as Welsh (1976).

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