Matroid theory is a generalization of the idea of linear independence. A matroid M consists of a finite set E (called the ground set) and a collection S of subsets of E satsifying the following conditions: (1) ∅ ∈ S; (2) if I ∈ S, then every subset of I is in S; (3) if I1 and I2 are in S and |I1| < |I2|, then there is an element e of I2 − I1 such that I1 ∪ e is in S. The elements of S are calle...

We prove that, for any positive integers n; k and q; there exists an integer R such that, if M is a matroid with no MðKnÞor U2;qþ2-minor, then either M has a collection of k disjoint cocircuits or M has rank at most R: Applied to the class of cographic matroids, this result implies the edge-disjoint version of the Erdös–Pósa Theorem. r 2002 Elsevier Science (USA). All rights reserved. AMS 1991 ...

Following an ‘Ansatz’ of Björner & Ziegler [BZ], we give an axiomatic development of finite sign vector systems that we call complex matroids. This includes, as special cases, the sign vector systems that encode complex arrangements according to [BZ], and the complexified oriented matroids, whose complements were considered by Gel’fand & Rybnikov [GeR]. Our framework makes it possible to study ...

The expansion axiom of matroids requires only the existence of some kind of independent sets, not the uniqueness of them. This causes that the base families of some matroids can be reduced while the unions of the base families of these matroids remain unchanged. In this paper, we define unique expansion matroids in which the expansion axiom has some extent uniqueness; we define union minimal ma...

In secret sharing, the exact characterization of ideal access structures is a longstanding open problem. Brickell and Davenport (J. of Cryptology, 1991) proved that ideal access structures are induced by matroids. Subsequently, ideal access structures and access structures induced by matroids have attracted a lot of attention. Due to the difficulty of finding general results, the characterizati...

Our splitter theorem studies pairs of the form (M,N), where M and N are internally 4-connected binary matroids, M has a proper N -minor, and if M ′ is an internally 4-connected matroid such that M has a proper M ′-minor and M ′ has an N -minor, then |E(M)|− |E(M ′)| > 3. The analysis in the splitter theorem requires the constraint |E(M)| ≥ 16. In this article, we complement that analysis by des...

Let Jt be the class of binary matroids without a Fano plane as a submatroid. We show that every supersolvable matroid in JÍ is graphic, corresponding to a chordal graph. Then we characterize the case that the modular join of two matroids is supersolvable. This is used to study modular flats and modular joins of binary supersolvable matroids. We decompose supersolvable matroids in JH as modular ...

It has recently been shown that, contrary to common belief, infinite matroids can be axiomatized in a way very similar to finite matroids. This should make it possible now to extend much of the theory of finite matroids to infinite ones: an aim that had previously been thought to be unattainable, because the popular additional ‘finitary’ axiom believed to be necessary clearly spoils duality. We...

1. Prefatory Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2. Several Perspectives on Transversal Matroids . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2.1. Set systems, transversals, partial transversals, and Hall’s theorem . . . . . . . . 2 2.2. Transversal matroids via matrix encodings of set systems . . . . . ....

Let M be a finite matroid with rank function r. We will write A M when we mean that A is a subset of the ground set of M, and write M|A and M A for the matroids obtained by restricting M to A and contracting M on A respectively. Let M* denote the dual matroid to M. (See [1] for definitions). The main theorem is Theorem 1. The Tutte polynomial TM(x, y) satisfies TM(x, y)= : A M TM|A(0, y) TM A(x...