Related papers: Thin sums matroids and duality
We introduce dual matroids of 2-dimensional simplicial complexes. Under certain necessary conditions, duals matroids are used to characterise embeddability in 3-space in a way analogous to Whitney's planarity criterion. We further use dual…
Given $d$ row vectors of $n$ tropical numbers, $d<n$, the tropical Stiefel map constructs a version of their row space, whose Pl\"ucker coordinates are tropical determinants. We explicitly describe the fibers of this map. From the viewpoint…
This paper defines the q-analogue of a matroid and establishes several properties like duality, restriction and contraction. We discuss possible ways to define a q-matroid, and why they are (not) cryptomorphic. Also, we explain the…
As part of the recent developments in infinite matroid theory, there have been a number of conjectures about how standard theorems of finite matroid theory might extend to the infinite setting. These include base packing, base covering, and…
A frame matroid M is graphic if there is a graph G with cycle matroid isomorphic to M. In general, if there is one such graph, there will be many. Zaslavsky has shown that frame matroids are precisely those having a representation as a…
Seymour's famous decomposition theorem for regular matroids states that any totally unimodular (TU) matrix can be constructed through a series of composition operations called $k$-sums starting from network matrices and their transposes and…
A catalogue of all non-isomorphic simple connected regular matroids ${\cal M}$ of cardinality $n \leq 15$ is provided on the net. These matroids are given as binary matrix matroids and are sieved from the large pool of all non-isomorphic…
A class of matroids is introduced which is very large as it strictly contains all paving matroids as special cases. As their key feature these split matroids can be studied via techniques from polyhedral geometry. It turns out that the…
DeVos, Kwon, and Oum introduced the concept of branch-depth of matroids as a natural analogue of tree-depth of graphs. They conjectured that a matroid of sufficiently large branch-depth contains the uniform matroid $U_{n,2n}$ or the cycle…
B\'ar\'any, Kalai, and Meshulam recently obtained a topological Tverberg-type theorem for matroids, which guarantees multiple coincidences for continuous maps from a matroid complex to d-dimensional Euclidean space, if the matroid has…
We undertake a detailed investigation into the structure of permutations in monotone grid classes whose row-column graphs do not contain components with more than one cycle. Central to this investigation is a new decomposition, called the…
In this paper we analyze the geometric structure and properties of a certain class of subsets of $\Bbb R^d$, known in the literature as 1-multicones, and here simply called multicones, which are quite natural generalizations of the…
The set of finite binary matrices of a given size is known to carry a finite type A bicrystal structure. We first review this classical construction, explain how it yields a short proof of the equality between Kostka polynomials and…
The theory of rough sets is concerned with the lower and upper approximations of objects through a binary relation on a universe. It has been applied to machine learning, knowledge discovery and data mining. The theory of matroids is a…
The Dowling geometry $Q_n(\Gamma)$, where $\Gamma$ is a finite group, is a matroid that generalizes the complete-graphic matroid $M(K_{n+1})$. We determine the maximum size of an $N$-free submatroid of $Q_n(\Gamma)$ for various choices of…
We show that the infinite matroid intersection conjecture of Nash-Williams implies the infinite Menger theorem proved recently by Aharoni and Berger. We prove that this conjecture is true whenever one matroid is nearly finitary and the…
A matroid is a combinatorial structure that captures and generalizes the algebraic concept of linear independence under a broader and more abstract framework. Matroids are closely related with many other topics in discrete mathematics, such…
Given a matroid or flag of matroids we introduce several broad classes of polynomials satisfying Deletion-Contraction identities, and study their singularities. There are three main families of polynomials captured by our approach:…
Graph theoretical ideas are highly utilized by computer science fields especially data mining. In this field, a data structure can be designed in the form of tree. Covering is a widely used form of data representation in data mining and…
In this paper, we give a new axioms system based on nonseparable flats with their ranks to define a matroid. We deduce a polynomial time algorithm for deciding if a given matroid (respectively, arbitrary structure) is an uniform matroid.…