Related papers: Foundations of matroids -- Part 2: Further theory,…
We study paving matroids, their realization spaces, and their closures, along with matroid varieties and circuit varieties. Within this context, we introduce three distinct methods for generating polynomials within the associated ideals of…
Rough set theory is a useful tool to deal with uncertain, granular and incomplete knowledge in information systems. And it is based on equivalence relations or partitions. Matroid theory is a structure that generalizes linear independence…
A partitioned matroid $(M, \{X_1,X_2,\dots,X_n\})$ consists of a matroid $M$ and a partition $\{X_1,X_2,\dots,X_n\}$ of its ground set. As such structures arise frequently in structural matroid theory, this paper introduces a general…
In this paper, we propose a new type of matroids, namely covering matroids, and investigate the connections with the second type of covering-based rough sets and some existing special matroids. Firstly, as an extension of partitions,…
The class of 2-regular matroids is a natural generalisation of regular and near-regular matroids. We prove an excluded-minor characterisation for the class of 2-regular matroids. The class of 3-regular matroids coincides with the class of…
Split matroids form a minor-closed class of matroids, and are defined by placing conditions on the system of split hyperplanes in the matroid base polytope. They can equivalently be defined in terms of structural properties involving cyclic…
This paper studies structural aspects of lattice path matroids, a class of transversal matroids that is closed under taking minors and duals. Among the basic topics treated are direct sums, duals, minors, circuits, and connected flats. One…
A flat of a matroid is cyclic if it is a union of circuits. The cyclic flats of a matroid form a lattice under inclusion. We study these lattices and explore matroids from the perspective of cyclic flats. In particular, we show that every…
Generalizing a well known theorem for finite matroids, we prove that for every (infinite) connected matroid M there is a unique tree T such that the nodes of T correspond to minors of M that are either 3-connected or circuits or cocircuits,…
We introduce the matroid-minor coalgebra C, which has labeled matroids as distinguished basis, and coproduct given by splitting a matroid into a submatroid and complementary contraction all possible ways. We introduce two new bases for C;…
In mathematics and computer science, connectivity is one of the basic concepts of matroid theory: it asks for the minimum number of elements which need to be removed to disconnect the remaining nodes from each other. It is closely related…
This is a continuation of the early paper concerning matroid base polytope decomposition. Here, we will present sufficient conditions on $M$ so its base matroid polytope $P(M)$ has a {\em sequence} of hyperplane splits. The latter yields to…
We consider three forms of composition of matroids, each of which extends the category of bimatroids to a rigid monoidal category. Many well-known constructions are functorial or defined by morphisms in these categories. Motivating examples…
We investigate an approach to matroid complexity that involves describing a matroid via a list of independent sets, bases, circuits, or some other family of subsets of the ground set. The computational complexity of algorithmic problems…
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…
We study a matrix-based notion of matroid representation over local commutative rings obtained by replacing linear independence with modular independence. This construction always defines an independence system, though not necessarily a…
Polymatroids can be considered as "fractional matroid" where the rank function is not required to be integer valued. Many, but not every notion in matroid terminology translates naturally to polymatroids. Defining cyclic flats of a…
In this work we present an algorithm to construct sparse-paving matroids over finite set $S$. From this algorithm we derive some useful bounds on the cardinality of the set of circuits of any Sparse-Paving matroids which allow us to prove…
This document is a blueprint for the formalization in Lean of the structural theory of regular matroids underlying Seymour's decomposition theorem. We present a modular account of regularity via totally unimodular representations, show that…
A binary frame template is a device for creating binary matroids from graphic or cographic matroids. Such matroids are said to conform or coconform to the template. We introduce a preorder on these templates and determine the nontrivial…