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

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

On the 3-connected matroids that are minimal having a fixed spanning restriction

Let N be a minor of a 3-connected matroid M and let M′ be a 3-connected minor of M that is minimal having N as a minor. This paper commences the study of the problem of finding a best-possible upper bound on |E(M′) - E(N)|. The main result solves this problem in the case that N and M have the same rank. © 2000 Elsevier Science B.V. All rights reserved

On the minor-minimal 2-connected graphs having a fixed minor

Let H be a graph with κ1 components and κ 2 blocks, and let G be a minor-minimal 2-connected graph having H as a minor. This paper proves that |E(G)|-|E(H)|≤α(κ 1-1)+β(κ2-1) for all (α,β) such that α+β≥5,2α+5β≥;20, and β≥3. Moreover, if one of the last three inequalities fails, then there are graphs G and H for which the first inequality fails. © 2003 Elsevier B.V. All rights reserved

On size, circumference and circuit removal in 3-connected matroids

This paper proves several extremal results for 3-connected matroids. In particular, it is shown that, for such a matroid M, (i) if the rank r(M) of M is at least six, then the circumference c(M) of M is at least six and, provided |E(M)| ≥4r(M) - 5, there is a circuit whose deletion from M leaves a 3-connected matroid; (ii) if r(M) ≥4 and M has a basis B such that M\e is not 3-connected for all e in E(M) - B, then |E(M)| ≤3r(M) - 4; and (iii) if M is minimally 3-connected but not hamiltonian, then |E(M)| ≤3r(M) - c(M). © 2000 Elsevier Science B.V. All rights reserved

On packing minors into connected matroids

Let N be a matroid with k connected components and M be a minor-minimal connected matroid having N as a minor. This note proves that |E(M) - E(N)| is at most 2k - 2 unless N or its dual is free, in which case |E(M) - E(N)| ≤k - 1. Examples are given to show that these bounds are best possible for all choices for N. A consequence of the main result is that a minimally connected matroid of rank r and maximum circuit size c has at most 2r - c + 2 elements. This bound sharpens a result of Murty. © 1998 Elsevier Science B.V. All rights reserved

Matroid packing and covering with circuits through an element

In 1981, Seymour proved a conjecture of Welsh that, in a connected matroid M, the sum of the maximum number of disjoint circuits and the minimum number of circuits needed to cover M is at most r*(M) + 1. This paper considers the set Ce(M) of circuits through a fixed element e such that M/e is connected. Let νe(M) be the maximum size of a subset of Ce(M) in which any two distinct members meet only in {e}, and let θe(M) be the minimum size of a subset of Ce(M) that covers M. The main result proves that νe(M) + θe(M) ≤ r* + 2 and that if M has no Fano-minor using e, then νe + θe,(M) ≤ r*(M) + 1. Seymour\u27s result follows without difficulty from this theorem and there are also some interesting applications to graphs. © 2005 Elsevier Inc. All rights reserved

Matroids with at least two regular elements

For a matroid $M$, an element $e$ such that both $M\backslash e$ and $M/e$ are regular is called a regular element of $M$. We determine completely the structure of non-regular matroids with at least two regular elements. Besides four small size matroids, all 3-connected matroids in the class can be pieced together from $F_7$ or $S_8$ and a regular matroid using 3-sums. This result takes a step toward solving a problem posed by Paul Seymour: Find all 3-connected non-regular matroids with at least one regular element [5, 14.8.8]