Recognizing Even-Cycle and Even-Cut Matroids

Even-cycle and even-cut matroids are classes of binary matroids that generalize respectively graphic and cographic matroids. We give algorithms to check membership for these classes of matroids. We assume that the matroids are 3-connected and are given by their (0,1)-matrix representations. We first give an algorithm to check membership for p-cographic matroids that is a subclass of even-cut matroids. We use this algorithm to construct algorithms for membership problems for even-cycle and even-cut matroids and the running time of these algorithms is polynomial in the size of the matrix representations. However, we will outline only how theoretical results can be used to develop polynomial time algorithms and omit the details of algorithms

Representations of even-cycle and even-cut matroids

In this thesis, two classes of binary matroids will be discussed: even-cycle and even-cut matroids, together with problems which are related to their graphical representations. Even-cycle and even-cut matroids can be represented as signed graphs and grafts, respectively. A signed graph is a pair $(G,\Sigma)$ where $G$ is a graph and $\Sigma$ is a subset of edges of $G$. A cycle $C$ of $G$ is a subset of edges of $G$ such that every vertex of the subgraph of $G$ induced by $C$ has an even degree. We say that $C$ is even in $(G,\Sigma)$ if $|C \cap \Sigma|$ is even. A matroid $M$ is an even-cycle matroid if there exists a signed graph $(G,\Sigma)$ such that circuits of $M$ precisely corresponds to inclusion-wise minimal non-empty even cycles of $(G,\Sigma)$. A graft is a pair $(G,T)$ where $G$ is a graph and $T$ is a subset of vertices of $G$ such that each component of $G$ contains an even number of vertices in $T$. Let $U$ be a subset of vertices of $G$ and let $D:= delta_G(U)$ be a cut of $G$. We say that $D$ is even in $(G, T)$ if $|U \cap T|$ is even. A matroid $M$ is an even-cut matroid if there exists a graft $(G,T)$ such that circuits of $M$ corresponds to inclusion-wise minimal non-empty even cuts of $(G,T)$.\\ This thesis is motivated by the following three fundamental problems for even-cycle and even-cut matroids with their graphical representations. (a) Isomorphism problem: what is the relationship between two representations? (b) Bounding the number of representations: how many representations can a matroid have? (c) Recognition problem: how can we efficiently determine if a given matroid is in the class? And how can we find a representation if one exists? These questions for even-cycle and even-cut matroids will be answered in this thesis, respectively. For Problem (a), it will be characterized when two $4$-connected graphs $G_1$ and $G_2$ have a pair of signatures $(\Sigma_1, \Sigma_2)$ such that $(G_1, \Sigma_1)$ and $(G_2, \Sigma_2)$ represent the same even-cycle matroids. This also characterize when $G_1$ and $G_2$ have a pair of terminal sets $(T_1, T_2)$ such that $(G_1,T_1)$ and $(G_2,T_2)$ represent the same even-cut matroid. For Problem (b), we introduce another class of binary matroids, called pinch-graphic matroids, which can generate expo\-nentially many representations even when the matroid is $3$-connected. An even-cycle matroid is a pinch-graphic matroid if there exists a signed graph with a blocking pair. A blocking pair of a signed graph is a pair of vertices such that every odd cycles intersects with at least one of them. We prove that there exists a constant $c$ such that if a matroid is even-cycle matroid that is not pinch-graphic, then the number of representations is bounded by $c$. An analogous result for even-cut matroids that are not duals of pinch-graphic matroids will be also proven. As an application, we construct algorithms to solve Problem (c) for even-cycle, even-cut matroids. The input matroids of these algorithms are binary, and they are given by a $(0,1)$-matrix over the finite field \gf(2). The time-complexity of these algorithms is polynomial in the size of the input matrix

Even Cycle and Even Cut Matroids

In this thesis we consider two classes of binary matroids, even cycle matroids and even cut matroids. They are a generalization of graphic and cographic matroids respectively. We focus on two main problems for these classes of matroids. We first consider the Isomorphism Problem, that is the relation between two representations of the same matroid. A representation of an even cycle matroid is a pair formed by a graph together with a special set of edges of the graph. Such a pair is called a signed graph. A representation for an even cut matroid is a pair formed by a graph together with a special set of vertices of the graph. Such a pair is called a graft. We show that two signed graphs representing the same even cycle matroid relate to two grafts representing the same even cut matroid. We then present two classes of signed graphs and we solve the Isomorphism Problem for these two classes. We conjecture that any two representations of the same even cycle matroid are either in one of these two classes, or are related by a local modification of a known operation, or form a sporadic example. The second problem we consider is finding the excluded minors for these classes of matroids. A difficulty when looking for excluded minors for these classes arises from the fact that in general the matroids may have an arbitrarily large number of representations. We define degenerate even cycle and even cut matroids. We show that a 3-connected even cycle matroid containing a 3-connected non-degenerate minor has, up to a simple equivalence relation, at most twice as many representations as the minor. We strengthen this result for a particular class of non-degenerate even cycle matroids. We also prove analogous results for even cut matroids

Capturing elements in matroid minors

In this dissertation, we begin with an introduction to a matroid as the natural generalization of independence arising in three different fields of mathematics. In the first chapter, we develop graph theory and matroid theory terminology necessary to the topic of this dissertation. In Chapter 2 and Chapter 3, we prove two main results. A result of Ding, Oporowski, Oxley, and Vertigan reveals that a large 3-connected matroid M has unavoidable structure. For every n exceeding two, there is an integer f(n) so that if |E(M)| exceeds f(n), then M has a minor isomorphic to the rank-n wheel or whirl, a rank-n spike, the cycle or bond matroid of K_{3,n}, or U_{2,n} or U_{n-2,n}. In Chapter 2, we build on this result to determine what can be said about a large structure using a specified element e of M. In particular, we prove that, for every integer n exceeding two, there is an integer g(n) so that if |E(M)| exceeds g(n), then e is an element of a minor of M isomorphic to the rank-n wheel or whirl, a rank-n spike, the cycle or bond matroid of K_{1,1,1,n}, a specific single-element extension of M(K_{3,n}) or the dual of this extension, or U_{2,n} or U_{n-2,n}. In Chapter 3, we consider a large 3-connected binary matroid with a specified pair of elements. We extend a corollary of the result of Chapter 2 to show the following result for any pair {x,y} of elements of a 3-connected binary matroid M. For every integer n exceeding two, there is an integer h(n) so that if |E(M)| exceeds h(n), then x and y are elements of a minor of M isomorphic to the rank-n wheel, a rank-n binary spike with a tip and a cotip, or the cycle or bond matroid of K_{1,1,1,n}

Non-Adaptive Matroid Prophet Inequalities

We consider the problem of matroid prophet inequalities. This problem has been ex- tensively studied in case of adaptive prices, with [KW12] obtaining a tight 2-competitive mechanism for all the matroids. However, the case non-adaptive is far from resolved, although there is a known constant- competitive mechanism for uniform and graphical matroids (see [Cha+20]). We improve on constant-competitive mechanism from [Cha+20] for graphical matroids, present a separate mechanism for cographical matroids, and combine those to obtain constant-competitive mechanism for all regular matroids

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

Exponentially Dense Matroids

This thesis deals with questions relating to the maximum density of rank-n matroids in a minor-closed class. Consider a minor-closed class M of matroids that does not contain a given rank-2 uniform matroid. The growth rate function is defined by h_M(n) = max(|N| : N ∈ M simple, r(N) ≤ n). The Growth Rate Theorem, due to Geelen, Kabell, Kung, and Whittle, shows that the growth rate function is either linear, quadratic, or exponential in n. In the case of exponentially dense classes, we conjecture that, for sufficiently large n, h_M(n) = (q^(n+k) − 1)/(q-1) − c, where q is a prime power, and k and c are non-negative integers depending only on M. We show that this holds for several interesting classes, including the class of all matroids with no U_{2,t}-minor. We also consider more general minor-closed classes that exclude an arbitrary uniform matroid. Here the growth rate, as defined above, can be infinite. We define a more suitable notion of density, and prove a growth rate theorem for this more general notion, dividing minor-closed classes into those that are at most polynomially dense, and those that are exponentially dense