Contracting an element from a cocircuit

Diameters of Cocircuit Graphs of Oriented Matroids: An Update

Oriented matroids (often called order types) are combinatorial structures that generalize point configurations, vector configurations, hyperplane arrangements, polyhedra, linear programs, and directed graphs. Oriented matroids have played a key role in combinatorics, computational geometry, and optimization. This paper surveys prior work and presents an update on the search for bounds on the diameter of the cocircuit graph of an oriented matroid. We review the diameter problem and show the diameter bounds of general oriented matroids reduce to those of uniform oriented matroids. We give the la…

Cyclic matroids

For all positive integers $s$ and $t$ exceeding one, a matroid $M$ on $n$ elements is {\em nearly $(s, t)$-cyclic} if there is a cyclic ordering $σ$ of its ground set such that every $s-1$ consecutive elements of $σ$ are contained in an $s$-element circuit and every $t-1$ consecutive elements of $σ$ are contained in a $t$-element cocircuit. In the case $s=t$, nearly $(s, s)$-cyclic matroids have been studied previously. In this paper, we show that if $M$ is nearly $(s, t)$-cyclic and $n$ is sufficiently large, then these $s$-element circuits and $t$-element cocircuits are consecutive in $σ$ in…

On a generalisation of spikes

We consider matroids with the property that every subset of the ground set of size $t$ is contained in both an $\ell$-element circuit and an $\ell$-element cocircuit; we say that such a matroid has the $(t,\ell)$-property. We show that for any positive integer $t$, there is a finite number of matroids with the $(t,\ell)$-property for $\ell<2t$; however, matroids with the $(t,2t)$-property form an infinite family. We say a matroid is a $t$-spike if there is a partition of the ground set into pairs such that the union of any $t$ pairs is a circuit and a cocircuit. Our main result is that if a su…

$3$-连通拟阵和简单$3$-连通图中的可分离对

Detachable pairs in $3$-connected matroids and simple $3$-connected graphs

本文刻画了至少有13个元素的3-连通拟阵何时存在可分离对，并应用于简单3-连通图，给出边数至少13时存在可分离边对的条件。

On contracting hyperplane elements from a 3-connected matroid

Let $\tilde{K}_{3,n}$, $n\geq 3$, be the simple graph obtained from $K_{3,n}$ by adding three edges to a vertex part of size three. We prove that if $H$ is a hyperplane of a 3-connected matroid $M$ and $M \not\cong M^*(\tilde{K}_{3,n})$, then there is an element $x$ in $H$ such that the simple matroid associated with $M/x$ is 3-connected.

横截拟阵中单个元素的收缩

Contracting a Single Element in a Transversal Matroid

针对横截拟阵在收缩下不封闭的问题，提出一种多项式时间算法判定其单元素收缩是否为横截的并给出横截表示，同时探讨了对偶判定应用。

定向矩阵的扩展-提升双射

Extension-lifting Bijections for Oriented Matroids

本文提出一种双射，将定向矩阵的基与特殊定向联系起来，通过电路和余电路签名描述，并探讨其在定向矩阵编程和三角剖分中的应用。

Coxeter群、图积及其他群中的压缩元素与共轭增长

Contracting elements and conjugacy growth in Coxeter groups, graph products, and further groups

本文在periagroups的标准Cayley图中构造压缩元素，证明除非本质分裂为直积，否则这些群是圆柱双曲的且共轭增长级数超越。

Towards a Splitter Theorem for Internally $4$-connected Binary Matroids VII

Let $M$ be a $3$-connected binary matroid; $M$ is internally $4$-connected if one side of every $3$-separation is a triangle or a triad, and $M$ is $(4,4,S)$-connected if one side of every $3$-separation is a triangle, a triad, or a $4$-element fan. Assume $M$ is internally $4$-connected and that neither $M$ nor its dual is a cubic Möbius or planar ladder or a certain coextension thereof. Let $N$ be an internally $4$-connected proper minor of $M$. Our aim is to show that $M$ has a proper internally $4$-connected minor with an $N$-minor that can be obtained from $M$ either by removing at most f…

Towards a Splitter Theorem for Internally $4$-connected Binary Matroids VI

Let $M$ be a $3$-connected binary matroid; $M$ is called internally $4$-connected if one side of every $3$-separation is a triangle or a triad, and $M$ is $(4,4,S)$-connected if one side of every $3$-separation is a triangle, a triad, or a $4$-element fan. Assume $M$ is internally $4$-connected and that neither $M$ nor its dual is a cubic Möbius or planar ladder or a certain coextension thereof. Let $N$ be an internally $4$-connected proper minor of $M$. Our aim is to show that $M$ has a proper internally $4$-connected minor with an $N$-minor that can be obtained from $M$ either by removing at…

连通多拟阵的切片投影的连通性

Connectivity for slice-projections of connected polymatroids

研究连通多拟阵的切片投影（包括删除和收缩）的连通性，证明每个元素至少有一个连续切片投影连通，并揭示对偶多拟阵的切片投影连通性关系，推广了图与拟阵的连通性定理。

A splitter theorem for 3-connected 2-polymatroids

Seymour's Splitter Theorem is a basic inductive tool for dealing with $3$-connected matroids. This paper proves a generalization of that theorem for the class of $2$-polymatroids. Such structures include matroids, and they model both sets of points and lines in a projective space and sets of edges in a graph. A series compression in such a structure is an analogue of contracting an edge of a graph that is in a series pair. A $2$-polymatroid $N$ is an s-minor of a $2$-polymatroid $M$ if $N$ can be obtained from $M$ by a sequence of contractions, series compressions, and dual-contractions, where…