On packing 3-connected restrictions into 3-connected matroids

Let M1,M2,...,Mn be 3-connected restrictions of a 3-connected matroid M on disjoint ground sets E1,E2,...,En, respectively. This paper proves that M has a 3-connected minor N that contains E1 UE2 U ··· UEn, has its restriction to each Ei, being Mi, and has at most 2n - 2 additional elements

On Properties of Matroid Connectivity

Highly connected matroids are consistently useful in the analysis of matroid structure. Round matroids, in particular, were instrumental in the proof of Rota\u27s conjecture. Chapter 2 concerns a class of matroids with similar properties to those of round matroids. We provide many useful characterizations of these matroids, and determine explicitly their regular members. Tutte proved that a 3-connected matroid with every element in a 3-element circuit and a 3-element cocircuit is either a whirl or the cycle matroid of a wheel. This result led to the proof of the 3-connected splitter theorem. More recently, Miller proved that matroids of sufficient size having every pair of elements in a 4-element circuit and a 4-element cocircuit are spikes. This observation simplifies the proof of Rota\u27s conjecture for GF(4). In Chapters 3 and 4, we investigate matroids having similar restrictions on their small circuits and cocircuits. The main result of each of these chapters is a complete characterization of the matroids therein

On Matroid Connectivity.

Certain classes of 2- and 3-connected matroids are studied in this thesis. In Chapter 2 we give a characterization of those 2-connected matroids M with the property that, for a given positive integer m, the deletion of every non-empty subset of M having at most m elements is disconnected. A bound on the maximum number of elements of such a matroid in terms of its rank is also given, along with a complete description of the matroids attaining this bound. These results extend results of Murty and Oxley for minimally 2-connected matroids. A characterization of the 3-connected matroids M that have the property that every 2-element deletion of M is disconnected is given in Chapter 3. It is shown that these matroids are exactly the duals of Sylvester matroids having at least four elements. In Chapter 4 we prove the following result: Let M be a 3-connected matroid other than a wheel of rank greater than three, and let C be a circuit of M. If the deletion of every pair of elements of C is disconnected, then every pair of elements of C is contained in a triad of M. For an integer t greater than one, an n-element matroid M is t-cocyclic if every deletion having at least n $-$ t + 1 elements is 2-connected, and every deletion having exactly n $-$ t elements is disconnected. A matroid is t-cyclic if its dual is t-cocyclic. In Chapter 5 we investigate the matroids that are both t-cocyclic and t-cyclic. It is shown that these matroids are exactly the uniform matroids U (t,2t) and the Steiner Systems S(t, t + 1, 2t + 2)

Extremal Problems in Matroid Connectivity

Matroid k-connectivity is typically defined in terms of a connectivity function. We can also say that a matroid is 2-connected if and only if for each pair of elements, there is a circuit containing both elements. Equivalently, a matroid is 2-connected if and only if each pair of elements is in a certain 2-element minor that is 2-connected. Similar results for higher connectivity had not been known. We determine a characterization of 3-connectivity that is based on the containment of small subsets in 3-connected minors from a given list of 3-connected matroids. Bixby’s Lemma is a well-known inductive tool in matroid theory that says that each element in a 3-connected matroid can be deleted or contracted to obtain a matroid that is 3-connected up to minimal 2-separations. We consider the binary matroids for which there is no element whose deletion and contraction are both 3-connected up to minimal 2-separations. In particular, we give a decomposition for such matroids to establish that any matroid of this type can be built from sequential matroids and matroids with many fans using a few natural operations. Wagner defined biconnectivity to translate connectivity in a bicircular matroid to certain connectivity conditions in its underlying graph. We extend a characterization of biconnectivity to higher connectivity. Using these graphic connectivity conditions, we call upon unavoidable minor results for graphs to find unavoidable minors for large 4-connected bicircular matroids

On Binary And Regular Matroids Without Small Minors

The results of this dissertation consist of excluded-minor results for Binary Matroids and excluded-minor results for Regular Matroids. Structural theorems on the relationship between minors and k-sums of matroids are developed here in order to provide some of these characterizations. Chapter 2 of the dissertation contains excluded-minor results for Binary Matroids. The first main result of this dissertation is a characterization of the internally 4-connected binary matroids with no minor that is isomorphic to the cycle matroid of the prism+e graph. This characterization generalizes results of Mayhew and Royle [18] for binary matroids and results of Dirac [8] and Lovász [15] for graphs. The results of this chapter are then extended from the class of internally 4-connected matroids to the class of 3-connected matroids. Chapter 3 of the dissertation contains the second main result, a decomposition theorem for regular matroids without certain minors. This decomposition theorem is used to obtain excluded-minor results for Regular Matroids. Wagner, Lovász, Oxley, Ding, Liu, and others have characterized many classes of graphs that are H-free for graphs H with at most twelve edges (see [7]). We extend several of these excluded-minor characterizations to regular matroids in Chapter 3. We also provide characterizations of regular matroids excluding several graphic matroids such as the octahedron, cube, and the Möbius Ladder on eight vertices. Both theoretical and computer-aided proofs of the results of Chapters 2 and 3 are provided in this dissertation

A decomposition theorem for binary matroids with no prism minor

The prism graph is the dual of the complete graph on five vertices with an edge deleted, $K_5\backslash e$. In this paper we determine the class of binary matroids with no prism minor. The motivation for this problem is the 1963 result by Dirac where he identified the simple 3-connected graphs with no minor isomorphic to the prism graph. We prove that besides Dirac's infinite families of graphs and four infinite families of non-regular matroids determined by Oxley, there are only three possibilities for a matroid in this class: it is isomorphic to the dual of the generalized parallel connection of $F_7$ with itself across a triangle with an element of the triangle deleted; it's rank is bounded by 5; or it admits a non-minimal exact 3-separation induced by the 3-separation in $P_9$. Since the prism graph has rank 5, the class has to contain the binary projective geometries of rank 3 and 4, $F_7$ and $PG(3, 2)$, respectively. We show that there is just one rank 5 extremal matroid in the class. It has 17 elements and is an extension of $R_{10}$, the unique splitter for regular matroids. As a corollary, we obtain Dillon, Mayhew, and Royle's result identifying the binary internally 4-connected matroids with no prism minor [5]

Templates for Representable Matroids

The matroid structure theory of Geelen, Gerards, and Whittle has led to a hypothesis that a highly connected member of a minor-closed class of matroids representable over a finite field is a mild modification (known as a perturbation) of a frame matroid, the dual of a frame matroid, or a matroid representable over a proper subfield. They introduced the notion of a template to describe these perturbations in more detail. In this dissertation, we determine these templates for various classes and use them to prove results about representability, extremal functions, and excluded minors. Chapter 1 gives a brief introduction to matroids and matroid structure theory. Chapters 2 and 3 analyze this hypothesis of Geelen, Gerards, and Whittle and propose some refined hypotheses. In Chapter 3, we define frame templates and discuss various notions of template equivalence. Chapter 4 gives some details on how templates relate to each other. We define a preorder on the set of frame templates over a finite field, and we determine the minimal nontrivial templates with respect to this preorder. We also study in significant depth a specific type of template that is pertinent to many applications. Chapters 5 and 6 apply the results of Chapters 3 and 4 to several subclasses of the binary matroids and the quaternary matroids---those matroids representable over the fields of two and four elements, respectively. Two of the classes we study in Chapter 5 are the even-cycle matroids and the even-cut matroids. Each of these classes has hundreds of excluded minors. We show that, for highly connected matroids, two or three excluded minors suffice. We also show that Seymour\u27s 1-Flowing Conjecture holds for sufficiently highly connected matroids. In Chapter 6, we completely characterize the highly connected members of the class of golden-mean matroids and several other closely related classes of quaternary matroids. This leads to a determination of the extremal functions for these classes, verifying a conjecture of Archer for matroids of sufficiently large rank

Connectivity of Matroids and Polymatroids

This dissertation is a collection of work on matroid and polymatroid connectivity. Connectivity is a useful property of matroids that allows a matroid to be decomposed naturally into its connected components, which are like blocks in a graph. The Cunningham-Edmonds tree decomposition further gives a way to decompose matroids into 3-connected minors. Much of the research below concerns alternate senses in which matroids and polymatroids can be connected. After a brief introduction to matroid theory in Chapter 1, the main results of this dissertation are given in Chapters 2 and 3. Tutte proved that, for an element e of a 2- connected matroid M , either the deletion or the contraction of e for M is 2-connected. In Chapter 2, a new notion of matroid connectivity is defined and it is shown that this new notion only enjoys the above inductive property when it agrees with the usual notion of 2-connectivity. Another result is proved to reinforce the special importance of this usual notion. In Chapter 3, a result of Brylawski and Seymour is considered. That result extends Tutte’s theorem by showing that if the element e is chosen to avoid a 2-connected minor N of M, then the deletion or contraction of e form M is not only 2-connected but maintains N as a minor. The main result of Chapter 3 proves an analogue of this result for 2-polymatroids, a natural extension of matroids. Chapter 4 describes a class of binary matroids that generalizes cubic graphs. Specifically, attention is focused on binary matroids having a cocircuit basis where every cocircuit in the basis, as well as the symmetric difference of all these cocircuits, has precisely three elements

Totally free expansions of matroids

The aim of this paper is to give insight into the behaviour of inequivalent representations of 3-connected matroids. An element x of a matroid M is fixed if there is no extension M′ of M by an element x′ such that {x, x′} is independent and M′ is unaltered by swapping the labels on x and x′. When x is fixed, a representation of M.\x extends in at most one way to a representation of M. A 3-connected matroid N is totally free if neither N nor its dual has a fixed element whose deletion is a series extension of a 3-connected matroid. The significance of such matroids derives from the theorem, established here, that the number of inequivalent representations of a 3-connected matroid M over a finite field F is bounded above by the maximum, over all totally free minors N of M, of the number of inequivalent F-representations of N. It is proved that, within a class of matroids that is closed under minors and duality, the totally free matroids can be found by an inductive search. Such a search is employed to show that, for all r ≥ 4, there are unique and easily described rank-r quaternary and quinternary matroids, the first being the free spike. Finally, Seymour\u27s Splitter Theorem is extended by showing that the sequence of 3-connected matroids from a matroid M to a minor N, whose existence is guaranteed by the theorem, may be chosen so that all deletions and contractions of fixed and cofixed elements occur in the initial segment of the sequence. © 2001 Elsevier Science