The structure of 4-separations in 4-connected matroids

Oxley, Semple and Whittle described a tree decomposition for a 3-connected matroid M that displays, up to a natural equivalence, all non-trivial 3-separations of M. Crossing 3-separations gave rise to fundamental structures known as flowers. In this dissertation, we define generalized flower structure called a k-flower, with no assumptions on the connectivity of M. We completely classify k-flowers in terms of the local connectivity between pairs of petals. Specializing to the case of 4-connected matroids, we give a new notion of equivalence of 4-separations that we show will be needed to describe a tree decomposition for 4-connected matroids. Finally, we characterize all internally 4-connected binary matroids M with the property that the ground set of M can be cyclically ordered so that any consecutive collection of elements in this cyclic ordering is 4-separating. We prove that in this case either M is a matroid on at most seven elements or, up to duality, M is isomorphic to the polygon matroid of a cubic or quartic planar ladder, the polygon matroid of a cubic or quartic Möbius ladder, a particular single-element extension of a wheel, or a particular single-element extension of the bond matroid of a cubic ladder

The structure of the 4-separations in 4-connected matroids

For a 2-connected matroid M, Cunningham and Edmonds gave a tree decomposition that displays all of its 2-separations. When M is 3-connected, two 3-separations are equivalent if one can be obtained from the other by passing through a sequence of 3-separations each of which is obtained from its predecessor by moving a single element from one side of the 3-separation to the other. Oxley, Semple, and Whittle gave a tree decomposition that displays, up to this equivalence, all non-trivial 3-separations of M. Now let M be 4-connected. In this paper, we define two 4-separations of M to be 2-equivalent if one can be obtained from the other by passing through a sequence of 4-separations each obtained from its predecessor by moving at most two elements from one side of the 4-separation to the other. The main result of the paper proves that M has a tree decomposition that displays, up to 2-equivalence, all non-trivial 4-separations of M. © 2011 Elsevier Inc. All rights reserved

Splitters and Decomposers for Binary Matroids

Let $EX[M_1\dots, M_k]$ denote the class of binary matroids with no minors isomorphic to $M_1, \dots, M_k$. In this paper we give a decomposition theorem for $EX[S_{10}, S_{10}^*]$, where $S_{10}$ is a certain 10-element rank-4 matroid. As corollaries we obtain decomposition theorems for the classes obtained by excluding the Kuratowski graphs $EX[M(K_{3,3}), M^*(K_{3,3}), M(K_5), M^*(K_5)]$ and $EX[M(K_{3,3}), M^*(K_{3,3})]$. These decomposition theorems imply results on internally $4$-connected matroids by Zhou [\ref{Zhou2004}], Qin and Zhou [\ref{Qin2004}], and Mayhew, Royle and Whitte [\ref{Mayhewsubmitted}].Comment: arXiv admin note: text overlap with arXiv:1403.775

The structure of 2-separations of infinite matroids

Generalizing a well known theorem for finite matroids, we prove that for every (infinite) connected matroid M there is a unique tree T such that the nodes of T correspond to minors of M that are either 3-connected or circuits or cocircuits, and the edges of T correspond to certain nested 2-separations of M. These decompositions are invariant under duality.Comment: 31 page

A chain theorem for 4-connected matroids

For the abstract of this paper, please see the PDF file

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]

An obstacle to a decomposition theorem for near-regular matroids

Seymour's Decomposition Theorem for regular matroids states that any matroid representable over both GF(2) and GF(3) can be obtained from matroids that are graphic, cographic, or isomorphic to R10 by 1-, 2-, and 3-sums. It is hoped that similar characterizations hold for other classes of matroids, notably for the class of near-regular matroids. Suppose that all near-regular matroids can be obtained from matroids that belong to a few basic classes through k-sums. Also suppose that these basic classes are such that, whenever a class contains all graphic matroids, it does not contain all cographic matroids. We show that in that case 3-sums will not suffice.Comment: 11 pages, 1 figur