Related papers: Oriented Interval Greedoids
Ardila and Develin's paper on tropical oriented hyperplane arrangements and tropical oriented matroids defines tropical oriented matroids and conjectures a bijection between them and triangulations of products of simplices $\Delta_{n-1}…
Let $(V,\gamma )$ be a real finite dimensional vector space with a symmetric bilinear form $\gamma $ whose kernel is spanned by a nonzero vector. The set of invertible real linear mappings of $(V, \gamma )$ into itself forms a Lie group…
A categorical formalism for directed graphs is introduced, featuring natural notions of morphisms and subgraphs, and leading to two elementary descriptions of the free-properad monad, first in terms of presheaves on elementary graphs,…
Oriented matroids can serve as a tool of modeling of collective decision-making processes in contradictory problems of pattern recognition. We present a generalization of the committee techniques of pattern recognition to oriented matroids.…
We describe arbitrary multiplicative differential forms on Lie groupoids infinitesimally, i.e., in terms of Lie algebroid data. This description is based on the study of linear differential forms on Lie algebroids and encompasses many known…
A theory of graded manifolds can be viewed as a generalization of differential geometry of smooth manifolds. It allows one to work with functions which locally depend not only on ordinary real variables, but also on $\mathbb{Z}$-graded…
Let I=I(D) be the edge ideal of a weighted oriented graph D. We determine the irredundant irreducible decomposition of I. Also, we characterize the associated primes and the unmixed property of I. Furthermore, we give a combinatorial…
We study d-dimensional generalizations of three mutually related topics in graph theory: Hamiltonian paths, (unit) interval graphs, and binomial edge ideals. We provide partial high-dimensional generalizations of Ore and Posa's sufficient…
We propose a definition of deformed symmetrizable generalized Cartan matrices with several deformation parameters, which admit a categorical interpretation by graded modules over the generalized preprojective algebras in the sense of…
Given a finite set $E$ and an operator $\sigma:2^{E}\longrightarrow2^{E}$, two sets $X,Y\subseteq E$ are \textit{cospanning} if $\sigma\left( X\right) =\sigma\left( Y\right) $. Corresponding \textit{cospanning equivalence relations} were…
An oriented hypergraph is a hypergraph where each vertex-edge incidence is given a label of $+1$ or $-1$. The adjacency and Laplacian eigenvalues of an oriented hypergraph are studied. Eigenvalue bounds for both the adjacency and Laplacian…
We characterize the class of threshold matroids by the structure of their defining bases. We also give an example of a shifted matroid which is not threshold, answering a question of Deza and Onn. We conclude by exploring consequences of…
We introduce greedy weights of matroids, inspired by those for linear codes. We show that a Wei duality holds for two of these types of greedy weights for matroids. Moreover we show that in the cases where the matroids involved are…
We describe a natural geometric relationship between matroids and underlying flag matroids by relating the geometry of the greedy algorithm to monotone path polytopes. This perspective allows us to generalize the construction of underlying…
We show that for any integer $k \ge 4$, every oriented graph with minimum semidegree bigger than $\frac{1}{2}(k-1+\sqrt{k-3})$ contains an antidirected path of length $k$. Consequently, every oriented graph on $n$ vertices with more than…
Given a digraph, an ordering of its vertices defines a backedge graph, namely the undirected graph whose edges correspond to the arcs pointing backwards with respect to the order. The degreewidth of a digraph is the minimum over all…
For any cluster-tilting object $\mathsf{T}$ in the cluster category $\mathscr{C}_{n}$ of type $\mathbb{A}_{n}$, we construct a rank-four oriented matroid $\mathcal{M}_{\mathsf{T}}$ such that stackable triangulations of…
Genevois introduced and investigated mediangle graphs as a common generalization of median graphs (1-sekeleta of CAT(0) cube complexes) and Coxeter graphs (Cayley graphs of Coxeter systems) and studied groups acting on them. He asked if…
In recent years, various notions of algebraic independence have emerged as a central and unifying theme in a number of areas of applied mathematics, including algebraic statistics and the rigidity theory of bar-and-joint frameworks. In each…
We show that for any natural number $k \ge 1$, any oriented graph $D$ of minimum semidegree at least $(3k- 2)/4$ contains an antidirected path of length $k$. In fact, a slightly weaker condition on the semidegree sequence of $D$ suffices,…