A chain theorem for internally 4-connected binary matroids

This is the post-print version of the Article - Copyright @ 2011 ElsevierLet M be a matroid. When M is 3-connected, Tutte’s Wheels-and-Whirls Theorem proves that M has a 3-connected proper minor N with |E(M) − E(N)| = 1 unless M is a wheel or a whirl. This paper establishes a corresponding result for internally 4-connected binary matroids. In particular, we prove that if M is such a matroid, then M has an internally 4-connected proper minor N with |E(M) − E(N)| at most 3 unless M or its dual is the cycle matroid of a planar or Möbius quartic ladder, or a 16-element variant of such a planar ladder.This study was partially supported by the National Security Agency

Towards a splitter theorem for internally 4-connected binary matroids VIII: small matroids

Our splitter theorem for internally 4-connected binary matroids studies pairs of the form (M,N), where N and M are internally 4-connected binary matroids, M has a proper N-minor, and if M' is an internally 4-connected matroid such that M has a proper M'-minor and M' has an N-minor, then |E(M)|-|E(M')|>3. The analysis in the splitter theorem requires the constraint that |E(M)|>15. In this article, we complement that analysis by using an exhaustive computer search to find all such pairs satisfying |E(M)|<16.Comment: Correcting minor error

Constructing internally 4-connected binary matroids

This is the post-print version of the Article - Copyright @ 2013 ElsevierIn an earlier paper, we proved that an internally 4-connected binary matroid with at least seven elements contains an internally 4-connected proper minor that is at most six elements smaller. We refine this result, by giving detailed descriptions of the operations required to produce the internally 4-connected minor. Each of these operations is top-down, in that it produces a smaller minor from the original. We also describe each as a bottom-up operation, constructing a larger matroid from the original, and we give necessary and su fficient conditions for each of these bottom-up moves to produce an internally 4-connected binary matroid. From this, we derive a constructive method for generating all internally 4-connected binary matroids.This study is supported by NSF IRFP Grant 0967050, the Marsden Fund, and the National Security Agency

Towards a splitter theorem for internally 4-connected binary matroids

This is the post-print version of the Article - Copyright @ 2012 ElsevierWe prove that if M is a 4-connected binary matroid and N is an internally 4-connected proper minor of M with at least 7 elements, then, unless M is a certain 16-element matroid, there is an element e of E(M) such that either M\e or M/e is internally 4-connected having an N-minor. This strengthens a result of Zhou and is a first step towards obtaining a splitter theorem for internally 4-connected binary matroids.This study is partially funded by Marsden Fund of New Zealand and the National Security Agency

Triangle-roundedness in matroids

A matroid $N$ is said to be triangle-rounded in a class of matroids $\mathcal{M}$ if each $3$-connected matroid $M\in \mathcal{M}$ with a triangle $T$ and an $N$-minor has an $N$-minor with $T$ as triangle. Reid gave a result useful to identify such matroids as stated next: suppose that $M$ is a binary $3$-connected matroid with a $3$-connected minor $N$, $T$ is a triangle of $M$ and $e\in T\cap E(N)$; then $M$ has a $3$-connected minor $M'$ with an $N$-minor such that $T$ is a triangle of $M'$ and $|E(M')|\le |E(N)|+2$. We strengthen this result by dropping the condition that such element $e$ exists and proving that there is a $3$-connected minor $M'$ of $M$ with an $N$-minor $N'$ such that $T$ is a triangle of $M'$ and $E(M')-E(N')\subseteq T$. This result is extended to the non-binary case and, as an application, we prove that $M(K_5)$ is triangle-rounded in the class of the regular matroids

Beta invariant and variations of chain theorems for matroids

The beta invariant of a matroid was introduced by Crapo in 1967. We first find the lower bound of the beta invariant of 3-connected matroids with rank r and the matroids which attain the lower bound. Second we characterize the matroids with beta invariant 5 and 6. For binary matroids we characterize matroids with beta invariant 7. These results extend earlier work of Oxley. Lastly we partially answer an open question of chromatic uniqueness of wheels and prove a splitting formula for the beta invariant of generalized parallel connection of two matroids. Tutte\u27s Wheel-and-Whirl theorem and Seymour\u27s Splitter theorem give respectively a constructive and structural view of the 3-connected matroids. Geelen and Whittle proved a chain theorem for sequentially 4-connected matroids and Geelen and Zhou proved a chain theorem for weakly 4-connected matroids. From these theorems one can obtain a chain theorem for matroids as well. We prove a chain theorem for sequentially 4-connected and weakly 4-connected matroids

Towards a splitter theorem for internally 4-connected binary matroids IX: The theorem

Let M be a binary matroid that is internally 4-connected, that is, M is 3-connected, and one side of every 3-separation is a triangle or a triad. Let N be an internally 4-connected proper minor of M. In this paper, we 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 three elements, or by removing some set of elements in an easily described way from one of a small collection of special substructures of M

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

Generalized Laminar Matroids

Nested matroids were introduced by Crapo in 1965 and have appeared frequently in the literature since then. A flat of a matroid $M$ is Hamiltonian if it has a spanning circuit. A matroid $M$ is nested if and only if its Hamiltonian flats form a chain under inclusion; $M$ is laminar if and only if, for every $1$-element independent set $X$, the Hamiltonian flats of $M$ containing $X$ form a chain under inclusion. We generalize these notions to define the classes of $k$-closure-laminar and $k$-laminar matroids. This paper focuses on structural properties of these classes noting that, while the second class is always minor-closed, the first is if and only if $k \le 3$. The main results are excluded-minor characterizations for the classes of 2-laminar and 2-closure-laminar matroids.Comment: 12 page