Related papers: Lattice path bicircular matroids
For any minor-closed class of matroids over a fixed finite field, we state an exact structural characterization for the sufficiently connected matroids in the class. We also state a number of conjectures that might be approachable using the…
A recent line of research has concentrated on exploring the links between analytic and combinatorial theories of submodularity, uncovering several key connections between them. In this context, Lov\'asz initiated the study of matroids from…
Las Vergnas and Hamidoune studied the number of circuits needed to determine an oriented matroid. In this paper we investigate this problem and some new variants, as well as their interpretation in particular classes of matroids. We present…
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…
In this paper, we introduce the terminology of matroids into the study of Zariski-pairs related to rational elliptic surfaces, aiming to simplify the presentation and arguments involved. As an application, we provide new examples of Zariski…
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,…
It is known that a lattice path matroid polytope can be associated with two given noncrossing lattice paths on $\mathbb{Z}\times\mathbb{Z}$ with the same end points. In this short note we give explicit formulae for the $f$-vector, toric…
The aim of this paper is to study lattice properties of the sharp partial order for complex matrices having index at most 1. We investigate the down-set of a fixed matrix $B$ under this partial order via isomorphisms with two different…
In this work we provide a decomposition theorem for the class of quaternary and non-binary signed-graphic matroids. This generalizes previous results for binary signed-graphic matroids and graphic matroids, and it provides the theoretical…
Frame matroids and lifted-graphic matroids are two distinct minor-closed classes of matroids, each of which generalises the class of graphic matroids. The class of quasi-graphic matroids, recently introduced by Geelen, Gerards, and Whittle,…
A Littelmann path model is constructed for crystals pertaining to a not necessarily symmetrizable Borcherds-Cartan matrix. Here one must overcome several combinatorial problems coming from the imaginary simple roots. The main results are an…
We study rank-three matroids, known as point-line configurations, and their associated matroid varieties, defined as the Zariski closures of their realization spaces. Our focus is on determining finite generating sets of defining equations…
Let $M$ be a regular matroid. The Jacobian group ${\rm Jac}(M)$ of $M$ is a finite abelian group whose cardinality is equal to the number of bases of $M$. This group generalizes the definition of the Jacobian group (also known as the…
A mixed graph is a graph with some directed edges and some undirected edges. We introduce the notion of mixed matroids as a generalization of mixed graphs. A mixed matroid can be viewed as an oriented matroid in which the signs over a fixed…
Let $EX[M_1\dots, M_k]$ denote the class of binary matroids with no minors isomorphic to $M_1, \dots, M_k$. In this paper we give a decomposition theorem for $EX[S_{10}, S_{10}^*]$, where $S_{10}$ is a certain 10-element rank-4 matroid. As…
In this paper, we present a lattice-theoretic characterization for valuated matroids, which is an extension of the well-known cryptomorphic equivalence between matroids and geometric lattices ($=$ atomistic semimodular lattices). 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…
We construct a family of independent sets for finite, atomic, and graded lattices, extending the well-known cryptomorphism between geometric lattices and matroids. This construction leads to an embedding theorem into geometric lattices that…
For a finite multigraph G, let \Lambda(G) denote the lattice of integer flows of G -- this is a finitely generated free abelian group with an integer-valued positive definite bilinear form. Bacher, de la Harpe, and Nagnibeda show that if G…
This note contributes to the structure theory of abstract rigidity matroids in general dimension. In the spirit of classical matroid theory, we prove several cryptomorphic characterizations of abstract rigidity matroids (in terms of…