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...

Inthispaper, (L,M)-fuzzy domain finiteness and (L,M)-fuzzy restricted hull spaces are introduced, and several characterizations of the category (L,M)-CS of (L,M)-fuzzy convex spaces are obtained. Then, (L,M)-fuzzy stratified (resp. weakly induced, induced) convex spaces are introduced. It is proved that both categories, the category (L,M)-SCS of (L,M)-fuzzy stratified convex spaces and the cate...

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.

Many problems that can be correctly solved by greedy algorithms can be described in terms of an abstract combinatorial object called a matroid. Matroids were first described in 1935 by the mathematician Hassler Whitney as a combinatorial generalization of linear independence of vectors—‘matroid’ means ‘something sort of like a matrix’. A matroid M is a finite collection of finite sets that sati...