Related papers: Matroids over partial hyperstructures
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…
In [27], we provided results on the $p$-modulus of the family of all bases of matroids and showed that it recovers various concepts in matroid theory, including strength, fractional arboricity, and principal partitions. In particular, the…
Let $G$ be a matrix and $M(G)$ be the matroid defined by linear dependence on the set $E$ of column vectors of $G.$ Roughly speaking, a parcel is a subset of pairs $(f,g)$ of functions defined on $E$ to an Abelian group $A$ satisfying a…
Consider a matroid equipped with a labeling of its ground set to an abelian group. We define the label of a subset of the ground set as the sum of the labels of its elements. We study a collection of problems on finding bases and common…
We claim that $M$(atroid) theory may provide a mathematical framework for an underlying description of $M$-theory. Duality is the key symmetry which motivates our proposal. The definition of an oriented matroid in terms of the Farkas…
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 define and study "semimatroids", a class of objects which abstracts the dependence properties of an affine hyperplane arrangement. We show that geometric semilattices are precisely the posets of flats of semimatroids. We define and…
Taking matrix as a synonym for a numerical function on the Cartesian product of two (in general, infinite) sets, a simple purely algebraic "reciprocity property" says that the set of rows spans a finite-dim space iff the set of columns does…
A notion of branch-width, which generalizes the one known for graphs, can be defined for matroids. We first give a proof of the polynomial time model-checking of monadic second-order formulas on representable matroids of bounded…
We partition in classes the set of matroids of fixed dimension on a fixed vertex set. In each class we identify two special matroids, respectively with minimal and maximal h-vector in that class. Such extremal matroids also satisfy a…
Matroids are ubiquitous in modern combinatorics. As discovered by Gelfand, Goresky, MacPherson and Serganova there is a beautiful connection between matroid theory and the geometry of Grassmannians: realizable matroids correspond to torus…
A natural extension of the Dijkgraaf-Vafa proposal is to include fields in the fundamental representation of the gauge group. In this paper we use field theory techniques to analyze gauge theories whose tree level superpotential is a…
We primarily investigate the properties of characteristic polynomials of semimatroids. In particular, we provide a combinatorial interpretation of their coefficients, generalizing the Whitney's Broken Circuit Theorem. We also prove that the…
In the matroid secretary problem we are given a stream of elements and asked to choose a set of elements that maximizes the total value of the set, subject to being an independent set of a matroid given in advance. The difficulty comes from…
We study the combinatorial properties of a tropical hyperplane arrangement. We define tropical oriented matroids, and prove that they share many of the properties of ordinary oriented matroids. We show that a tropical oriented matroid…
We continue the theory of $\tT$-systems from the work of the second author, describing both ground systems and module systems over a ground system (paralleling the theory of modules over an algebra). The theory, summarized categorically at…
We focus on checking the validity of the half-plane property on two prominent classes of transversal matroids, namely lattice path matroids and bicircular matroids. We show that lattice path matroids satisfy the half-plane property.…
Recently, Gross, Mansour and Tucker introduced the partial duality polynomial of a ribbon graph and posed a conjecture that there is no orientable ribbon graph whose partial duality polynomial has only one non-constant term. We found an…
Morphisms of matroids are combinatorial abstractions of linear maps and graph homomorphisms. We introduce the notion of basis for morphisms of matroids, and show that its generating function is strongly log-concave. As a consequence, we…
We classify all matroids with at most 8 elements that have the half-plane property, and we provide a list of some matroids on 9 elements that have, and that do not have the half-plane property. Furthermore, we prove that several classes of…