The authors give a characterization of the internally $4$-connected binary matroids that have no minor isomorphic to $M(K_{3,3})$. Any such matroid is either cographic, or is isomorphic to a particular single-element extension of the bond matroid of a cubic or quartic Mobius ladder, or is isomorphic to one of eighteen sporadic matroids.
Abstract: The purpose of this dissertation is to generalize some important excluded-minor theorems for graphs to binary matroids. Chapter 3 contains joint work with Hongxun Qin, in which we show that an internally 4-connected binary matroid with no M(K5)-, M*(K5)-, M(K3, 3)-, or M*(K3, 3)-minor is either planar graphic, or isomorphic to F-- or F*--. As a corollary, we prove an extremal result for the class of binary matroids without these minors. In Chapter 4, it is shown that, except for 6 'small' known matroids, every internally 4-connected non-regular binary matroid has either a [widetilde]K5- or a [widetilde]K5*-minor. Using this result, we obtain a computer-free proof of Dharmatilake's conjecture about the excluded minors for binary matroids with branch-width at most 3. D.W. Hall proved that K5 is the only simple 3-connected graph with a K5-minor that has no K3, 3-minor. In Chapter 5, we determine all the internally 4-connected binary matroids with an M(K5)-minor that have no M(K3, 3)-minor. In chapter 6, it is shown that there are only finitely many non-regular internally 4-connected matroids in the class of binary matroids with no M(K'3, 3)- or M*(K'3, 3)-minor, where K'3, 3 is the graph obtained from K3, 3 by adding an edge between a pair of non-adjacent vertices. In Chapter 7, we summarize the results and discuss about open problems. We are particularly interested in the class of binary matroids with no M(K5)- or M*(K5)-minor. Unfortunately, we tried without success to find all the internally 4-connected members of this class. However, it is shown that the matroid J1 is the smallest splitter for the above class.
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 Lovasz [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, Lovasz, 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 Mobius Ladder on eight vertices. Both theoretical and computer-aided proofs of the results of Chapters 2 and 3 are provided in this dissertation.
Results that relate clones in a matroid to minors of that matroid are given. Also, matroids that contain few clonal-classes are characterized. An example of a result of the first type that is given is that if X is a four-element set in a 3-connected non-binary matroid M and X contains a clone-pair, then M has a U2.4-minor that uses X. This result generalizes several results from the literature such as Tutte's Excluded-Minor characterization of the binary matroids. An example of a result of the second type is Theorem 4.5.2 where the matroids having exactly two clonal-classes are characterized using certain truncations of the direct sum of two uniform matroids.
This thesis is motivated by the following question: how many elements can a simple binary matroid with no $\PG(t,2)$-minor have? This is a natural analogue of questions asked about the density of graphs in minor-closed classes. We will answer this question by finding the eventual growth rate function of the class of matroids with no $\PG(t,2)$-minor, for any $t\ge 2$. Our main tool will be the matroid minors structure theory of Geelen, Gerards, and Whittle, and much of this thesis will be devoted to frame templates, the notion of structure in that theory.
With the advent of approximation algorithms for NP-hard combinatorial optimization problems, several techniques from exact optimization such as the primal-dual method have proven their staying power and versatility. This book describes a simple and powerful method that is iterative in essence and similarly useful in a variety of settings for exact and approximate optimization. The authors highlight the commonality and uses of this method to prove a variety of classical polyhedral results on matchings, trees, matroids and flows. The presentation style is elementary enough to be accessible to anyone with exposure to basic linear algebra and graph theory, making the book suitable for introductory courses in combinatorial optimization at the upper undergraduate and beginning graduate levels. Discussions of advanced applications illustrate their potential for future application in research in approximation algorithms.
Divisors and Sandpiles provides an introduction to the combinatorial theory of chip-firing on finite graphs. Part 1 motivates the study of the discrete Laplacian by introducing the dollar game. The resulting theory of divisors on graphs runs in close parallel to the geometric theory of divisors on Riemann surfaces, and Part 1 culminates in a full exposition of the graph-theoretic Riemann-Roch theorem due to M. Baker and S. Norine. The text leverages the reader's understanding of the discrete story to provide a brief overview of the classical theory of Riemann surfaces. Part 2 focuses on sandpiles, which are toy models of physical systems with dynamics controlled by the discrete Laplacian of the underlying graph. The text provides a careful introduction to the sandpile group and the abelian sandpile model, leading ultimately to L. Levine's threshold density theorem for the fixed-energy sandpile Markov chain. In a precise sense, the theory of sandpiles is dual to the theory of divisors, and there are many beautiful connections between the first two parts of the book. Part 3 addresses various topics connecting the theory of chip-firing to other areas of mathematics, including the matrix-tree theorem, harmonic morphisms, parking functions, M-matrices, matroids, the Tutte polynomial, and simplicial homology. The text is suitable for advanced undergraduates and beginning graduate students.
This volume, the third in a sequence that began with The Theory of Matroids and Combinatorial Geometries, concentrates on the applications of matroid theory to a variety of topics from engineering (rigidity and scene analysis), combinatorics (graphs, lattices, codes and designs), topology and operations research (the greedy algorithm).
