Related papers: Circuit-Difference Matroids

The first non-trivial case of Hadwiger's conjecture for oriented matroids reads as follows. If $\mathcal{O}$ is an $M(K_4)$-free oriented matroid, then $\mathcal{O}$ admits a NZ $3$-coflow, i.e., it is $3$-colourable in the sense of…

The width of a delta-matroid is the difference in size between a maximal and minimal feasible set. We give a Rough Structure Theorem for delta-matroids that admit a twist of width one. We apply this theorem to give an excluded minor…

There is a long list of open questions rooted in the same underlying problem: understanding the structure of bases or common bases of matroids. These conjectures suggest that matroids may possess much stronger structural properties than are…

We investigate the set of excluded minors of connectivity 2 for the class of frame matroids. We exhibit a list $\mathcal{E}$ of 18 such matroids, and show that if $N$ is such an excluded minor, then either $N \in \mathcal{E}$ or $N$ is a…

We introduce the notion of graphic cocircuits and show that a large class of regular matroids with graphic cocircuits belongs to the class of signed-graphic matroids. Moreover, we provide an algorithm which determines whether a cographic…

We study the evolution of random matroids represented by the sequence of random matrices over ${\mathbb F}_q$ where columns are added one after the other, and each column vector is a uniformly random vector in ${\mathbb F}_q^n$, independent…

A family C of circuits of a matroid M is a linear class if, given a modular pair of circuits in C}, any circuit contained in the union of the pair is also in C. The pair (M,C) can be seen as a matroidal generalization of a biased graph. We…

The classical Whitney's 2-Isomorphism Theorem describes the families of graphs having the same cycle matroid. In this paper we describe the families of graphs having the same truncated cycle matroid and prove, in particular, that every…

For a matroid $N$, a matroid $M$ is $N$-connected if every two elements of $M$ are in an $N$-minor together. Thus a matroid is connected if and only if it is $U_{1,2}$-connected. This paper proves that $U_{1,2}$ is the only connected…

It is well known that the class of transversal matroids is not closed under contraction or duality. In particular, after contracting a set of elements from a transversal matroid, the resulting matroid may or may not be transversal, and the…

A result of Mason, as refined by Ingleton, characterizes transversal matroids as the matroids that satisfy a set of inequalities that relate the ranks of intersections and unions of nonempty sets of cyclic flats. We prove counterparts, for…

We show that the class of bicircular matroids has only a finite number of excluded minors. Key tools used in our proof include representations of matroids by biased graphs and the recently introduced class of quasi-graphic matroids. We show…

For a simplicial complex ${\mathcal C}$ denote by $\beta({\mathcal C})$ the minimal number of edges from ${\mathcal C}$ needed to cover the ground set. If ${\mathcal C}$ is a matroid then for every partition $A_1, \ldots, A_m$ of the ground…

Given two finite matroids on the same ground set, a celebrated result of Edmonds says that the ground set can be partitioned into two disjoint subsets in a manner that there is a common independent set in both matroids whose intersection…

A graph is chordal if every cycle of length at least four has a chord. In 1961, Dirac characterized chordal graphs as those graphs that can be built from complete graphs by repeated clique-sums. Generalizing this, we consider the class of…

The foundation of a matroid is a canonical algebraic invariant which classifies representations of the matroid up to rescaling equivalence. Foundations of matroids are pastures, a simultaneous generalization of partial fields and…

The notion of thin sums matroids was invented to extend the notion of representability to non-finitary matroids. A matroid is tame if every circuit-cocircuit intersection is finite. We prove that a tame matroid is a thin sums matroid over a…

A result of Seymour implies that any 3-connected matroid with a modular 3-point line is binary. We prove a similar characterization for 3-connected matroids with modular 4-point lines. We show that such a matroid is either representable…

The problem of covering the ground set of two matroids by a minimum number of common independent sets is notoriously hard even in very restricted settings, i.e.\ when the goal is to decide if two common independent sets suffice or not.…

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…