Connected hyperplanes in binary matroids

AbstractFor a 3-connected binary matroid M, let dimA(M) be the dimension of the subspace of the cocycle space spanned by the non-separating cocircuits of M avoiding A, where A⊆E(M). When A=∅, Bixby and Cunningham, in 1979, showed that dimA(M)=r(M). In 2004, when |A|=1, Lemos proved that dimA(M)=r(M)-1. In this paper, we characterize the 3-connected binary matroids having a pair of elements that meets every non-separating cocircuit. Using this result, we show that 2dimA(M)⩾r(M)-3, when M is regular and |A|=2. For |A|=3, we exhibit a family of cographic matroids with a 3-element set intersecting every non-separating cocircuit. We also construct the matroids that attains McNulty and Wu’s bound for the number of non-separating cocircuits of a simple and cosimple connected binary matroid

Fan-extensions in fragile matroids

If S is a set of matroids, then the matroid M is S-fragile if, for every element e in E(M), either M\e or M/e has no minor isomorphic to a member of S. Excluded-minor characterizations often depend, implicitly or explicitly, on understanding classes of fragile matroids. In certain cases, when F is a minor-closed class of S-fragile matroids, and N is in F, the only members of F that contain N as a minor are obtained from N by increasing the length of fans. We prove that if this is the case, then we can certify it with a finite case-analysis. The analysis involves examining matroids that are at most two elements larger than N.Comment: Small revisions and correction

A characterization of graphic matroids using non-separating cocircuits

AbstractIn this paper, we settle a conjecture made by Wu. We show that a 3-connected binary matroid M is graphic if and only if each element avoids exactly r(M)−1 non-separating cocircuits of M. This result is a natural companion to the following theorem of Bixby and Cunningham: a 3-connected binary matroid M is graphic if and only if each element belongs to exactly 2 non-separating cocircuits of M

Matroids in which every pair of elements belongs to both a 4-circuit and a 4-cocircuit

In this thesis, we analyse the matroids which have the property that every pair of elements belongs to both a 4-circuit and a 4-cocircuit. In particular, we show that if a matroid with this property has at least 13 elements, then it is a spike. We also study the matroids with fewer than 13 element that have this property

Unavoidable Minors of Large 4-Connected Bicircular Matroids

It is known that any 3-connected matroid that is large enough is certain to contain a minor of a given size belonging to one of a few special classes of matroids. This paper proves a similar unavoidable minor result for large 4-connected bicircular matroids. The main result follows from establishing the list of unavoidable minors of large 4-biconnected graphs, which are the graphs representing the 4-connected bicircular matroids. This paper also gives similar results for internally 4-connected and vertically 4-connected bicircular matroids

TWO CLASSIFICATION PROBLEMS IN MATROID THEORY

Matroids are a modern type of synthetic geometry in which the behavior of points, lines, planes, and higher-dimensional spaces are governed by combinatorial axioms. In this paper we describe our work on two well-known classification problems in matroid theory: determine all binary matroids M such that for every element e, either deleting the element ( ) or contracting the element ( ) is regular; and determine all binary matroids M having an element e such that, both and are regular

Matroid Connectivity.

This dissertation has three parts. The first part, Chapter 1, considers the coefficient b\sb{ij}(M) of x\sp{i}y\sp{j} in the Tutte polynomial of a connected matroid M. The main result characterizes, for each i and j, the minor-minimal such matroids for which b\sb{ij}(M)\u3e0. One consequence of this characterization is that b\sb{11}(M)\u3e0 if and only if the two-wheel is a minor of M. Similar results are obtained for other values of i and j. These results imply that if M is simple and representable over $GF(q),$ then there are coefficients of its Tutte polynomial which count the flats of M that are projective spaces of specified rank. Similarly, for a simple graphic matroid $M(G),$ there are coefficients that count the number of cliques of each size contained in G. The second part, Chapter 2, generalizes a graph result of Mader by proving that if f is an element of a circuit C of a 3-connected matroid M and $M\\ e$ is not 3-connected for each $e\in C-\{f\},$ then C meets a triad of M. Several consequences of this result are proved. One of these generalizes a graph result of Wu. The others provide 3-connected analogues of 2-connected matroid results of Oxley. The third part, Chapters 3-5, involves a decomposition of 3-connected binary matroids based on 3-separations and three-sums. The dual of this decomposition is a direct generalization of a decomposition due to Coullard, Gardner, and Wagner for 3-connected graphs. In Chapter 3, we define the decomposition and prove that minimal such decompositions are unique. In Chapter 4, the components of this decomposition are characterized. In Chapter 5, it is shown that, when restricted to contraction-minimally 3-connected binary matroids, the components that are not vertically 4-connected are wheels, duals of twirls, or binary spikes