We study the complexity of testing if two given matroids are isomorphic. The problem is easily seen to be in Σ 2 . In the case of linear matroids, which are represented over polynomially growing fields, we note that the problem is unlikely to be Σ 2 -complete and is coNPhard. We show that when the rank of the matroid is bounded by a constant, linear matroid isomorphism and matroid isomorphism a...

The inseparability graph of an oriented matroid is an invariant of its class of orientations. When an orientable matroid has exactly one class of orientations the inseparability graph of all its orientations is in fact determined by its non-oriented underlying matroid. From this point of view it is natural to ask if inseparability graphs can be used to characterize matroids which have exactly o...

The notion of matroid has been generalized to Coxeter matroid by Gelfand and Serganova. To each pair (W, P) consisting of a finite irreducible Coxeter group W and parabolic subgroup P is associated a collection of objects called Coxeter matroids. The (ordinary) matroids are the special case where W is the symmetric group (the An case) and P is a maximal parabolic subgroup. This generalization o...

The matroid associated to a linear code is the representable matroid that is defined by the columns of any generator matrix. The matroid associated to a self-dual code is identically self-dual, but it is not known whether every identically self-dual representable matroid can be represented by a self-dual code. This open problem was proposed in [8], where it was proved to be equivalent to an ope...

We prove that the topological cycles of an arbitrary infinite graph together with its topological ends form a matroid. This matroid is, in general, neither finitary nor cofinitary.

ing from the behavior of linearly independent sets of columns of a matrix, in 1935, Whitney defined a matroid M to consist of a finite set E (or E(M)) and a collection I (or I(M)) of subsets of E called independent sets with the properties that the empty set is independent; every subset of an independent set is independent; and if one independent set has more elements than another, then an elem...

Blocks of a matroid are called hyperplanes. For various definitions and results connected with matroids, see [26]. Subsets of X , which are intersections of hyperplanes are called flats of a matroid. Each subset Y c_ X has a well-defined rank. If F is a flat of rank i and x e X \ F , then, there is a unique flat of rank (i + 1) which contains FU{x}. Rank of X is said to be the rank of matroid. ...

In this lecture, the focus is on submodular function in combinatorial optimizations. The first class of submodular functions which was studied thoroughly was the class of matroid rank functions. The flourishing stage of matroid theory came with Jack Edmonds’ work in 1960s, when he gave a minmax formula and an efficient algorithm to the matroid partition problem, from which the matroid intersect...

A classic exercise in the topology of surfaces is to show that, using handle slides, every disc-band surface, or 1-vertex ribbon graph, can be put in a canonical form consisting of the connected sum of orientable loops, and either non-orientable loops or pairs of interlaced orientable loops. Motivated by the principle that ribbon graph theory informs delta-matroid theory, we find the delta-matr...

In Man82] A. Mandel proved that the maximal cells of an Oriented Matroid poset are B-shellable. Our result shows that the whole Oriented Matroid is shellable, too.