Related papers: Pattern Recognition on Oriented Matroids: Topes an…
For an $n$-vertex graph $G$ with adjacency matrix $A$, the walk matrix $W(G)$ of $G$ is the matrix $[e,Ae,\ldots,A^{n-1}e]$, where $e$ is the all-ones vector. Suppose that $W(G)$ is nonsingular and $p$ is an odd prime such that $W(G)$ has…
We generalize the Varchenko matrix of a hyperplane arrangement to oriented matroids. We show that the celebrated determinant formula for the Varchenko matrix, first proved by Varchenko, generalizes to oriented matroids. It follows that the…
A binary matrix has the Consecutive Ones Property (C1P) if its columns can be ordered in such a way that all 1's on each row are consecutive. A Minimal Conflicting Set is a set of rows that does not have the C1P, but every proper subset has…
Seymour's decomposition theorem for regular matroids is a fundamental result with a number of combinatorial and algorithmic applications. In this work we demonstrate how this theorem can be used in the design of parameterized algorithms on…
In this paper, we give a new axioms system based on nonseparable flats with their ranks to define a matroid. We deduce a polynomial time algorithm for deciding if a given matroid (respectively, arbitrary structure) is an uniform matroid.…
A {\it sign pattern matrix} is a matrix whose entries are from the set $\{+,-, 0\}$. The minimum rank of a sign pattern matrix $A$ is the minimum of the ranks of the real matrices whose entries have signs equal to the corresponding entries…
In this paper, we consider the tractability of the matroid intersection problem under the minimum rank oracle. In this model, we are given an oracle that takes as its input a set of elements and returns as its output the minimum of the…
Extending the notion of geometric bijections for regular matroids, introduced by the first and third author with Matthew Baker, we describe a family of bijections between bases of an oriented matroid and special orientations. These…
The vertex v of a graph G is called a 1-critical-vertex for the maximum genus of the graph, or for simplicity called 1-critical-vertex, if G-v is a connected graph and {\deg}M(G - v) = {\deg}M(G) - 1. In this paper, through the joint-tree…
Sublinear circuits are generalizations of the affine circuits in matroid theory, and they arise as the convex-combinatorial core underlying constrained non-negativity certificates of exponential sums and of polynomials based on the…
For any positive integer $l$ we prove that if $M$ is a simple matroid with no $(l+2)$-point line as a minor and with sufficiently large rank, then $|E(M)|\le \frac{q^{r(M)}-1}{q-1}$, where $q$ is the largest prime power less than or equal…
Suppose that there is a ground set which consists of a large number of vectors in a Hilbert space. Consider the problem of selecting a subset of the ground set such that the projection of a vector of interest onto the subspace spanned by…
Two matrices $A$ and $B$ are called unitary (resp. orthogonal) equivalent if $AU=VB$ for two unitary (resp. orthogonal) matrices $U$ and $V$. Using trace identities, criteria are given for simultaneous unitary, orthogonal or complex…
A simple graph $G=(V,E)$ is a $(2,1)$-circuit if $|E|=2|V|$ and $|E(H)|\leq 2|V(H)|-1$ for every proper subgraph $H$ of $G$. Motivated, in part, by ongoing work to understand unique realisations of graphs on surfaces, we derive a…
Swartz proved that any matroid can be realized as the intersection lattice of an arrangement of codimension one homotopy spheres on a sphere. This was an unexpected extension from the oriented matroid case, but unfortunately the…
The power graph $\mathcal{P}(G)$ of a finite group $G$ is the simple graph with vertex set $G$, in which two distinct vertices are adjacent if one of them is a power of the other. For an integer $n\geq 2$, let $C_n$ denote the cyclic group…
Motivated by Kontsevich's graph complexes, this paper gives a systematic study of matroid complexes. We construct deletion and contraction bicomplexes on the vector space spanned by matroid classes equipped with ground-set orientations,…
Contention resolution schemes (or CR schemes), introduced by Chekuri, Vondrak and Zenklusen, are a class of randomized rounding algorithms for converting a fractional solution to a relaxation for a down-closed constraint family into an…
The lattice of monotone triangles $(\mathfrak{M}_n,\le)$ ordered by entry-wise comparisons is studied. Let $\tau_{\min}$ denote the unique minimal element in this lattice, and $\tau_{\max}$ the unique maximum. The number of $r$-tuples of…
For a matroid with an ordered (or "labelled") basis, a basis exchange step removes one element with label $l$ and replaces it by a new element that results in a new basis, and with the new element assigned label $l$. We prove that one…