Matroid toric ideals: complete intersection, minors and minimal systems of generators

In this paper, we investigate three problems concerning the toric ideal associated to a matroid. Firstly, we list all matroids $\mathcal M$ such that its corresponding toric ideal $I_{\mathcal M}$ is a complete intersection. Secondly, we handle with the problem of detecting minors of a matroid $\mathcal M$ from a minimal set of binomial generators of $I_{\mathcal M}$. In particular, given a minimal set of binomial generators of $I_{\mathcal M}$ we provide a necessary condition for $\mathcal M$ to have a minor isomorphic to $\mathcal U_{d,2d}$ for $d \geq 2$. This condition is proved to be sufficient for $d = 2$ (leading to a criterion for determining whether $\mathcal M$ is binary) and for $d = 3$. Finally, we characterize all matroids $\mathcal M$ such that $I_{\mathcal M}$ has a unique minimal set of binomial generators.Comment: 9 page

Some extremal connectivity results for matroids

Let n be an integer exceeding one and M be a matroid having at least n + 2 elements. In this paper, we prove that every n-element subset X of E(M) is in an (n + 1)-element circuit if and only if (i) for every such subset, M X is disconnected, and (ii) for every subset Y with at most n elements, M Y is connected. Various extensions and consequences of this result are also derived including characterizations in terms of connectivity of the 4-point line and of Murty\u27s Sylvester matroids. The former is a result of Seymour. © 1991

Displaying blocking pairs in signed graphs

A signed graph is a pair (G, S) where G is a graph and S is a subset of the edges of G. A circuit of G is even (resp. odd) if it contains an even (resp. odd) number of edges of S. A blocking pair of (G, S) is a pair of vertices s, t such that every odd circuit intersects at least one of s or t. In this paper, we characterize when the blocking pairs of a signed graph can be represented by 2-cuts in an auxiliary graph. We discuss the relevance of this result to the problem of recognizing even cycle matroids and to the problem of characterizing signed graphs with no odd-K5 minor

Matroid Connectivity.

This dissertation has three parts. The first part, Chapter 1, considers the coefficient b\sb{ij}(M) of x\sp{i}y\sp{j} in the Tutte polynomial of a connected matroid M. The main result characterizes, for each i and j, the minor-minimal such matroids for which b\sb{ij}(M)\u3e0. One consequence of this characterization is that b\sb{11}(M)\u3e0 if and only if the two-wheel is a minor of M. Similar results are obtained for other values of i and j. These results imply that if M is simple and representable over $GF(q),$ then there are coefficients of its Tutte polynomial which count the flats of M that are projective spaces of specified rank. Similarly, for a simple graphic matroid $M(G),$ there are coefficients that count the number of cliques of each size contained in G. The second part, Chapter 2, generalizes a graph result of Mader by proving that if f is an element of a circuit C of a 3-connected matroid M and $M\\ e$ is not 3-connected for each $e\in C-\{f\},$ then C meets a triad of M. Several consequences of this result are proved. One of these generalizes a graph result of Wu. The others provide 3-connected analogues of 2-connected matroid results of Oxley. The third part, Chapters 3-5, involves a decomposition of 3-connected binary matroids based on 3-separations and three-sums. The dual of this decomposition is a direct generalization of a decomposition due to Coullard, Gardner, and Wagner for 3-connected graphs. In Chapter 3, we define the decomposition and prove that minimal such decompositions are unique. In Chapter 4, the components of this decomposition are characterized. In Chapter 5, it is shown that, when restricted to contraction-minimally 3-connected binary matroids, the components that are not vertically 4-connected are wheels, duals of twirls, or binary spikes

Characterizing matroids whose bases form graphic delta-matroids

We introduce delta-graphic matroids, which are matroids whose bases form graphic delta-matroids. The class of delta-graphic matroids contains graphic matroids as well as cographic matroids and is a proper subclass of the class of regular matroids. We give a structural characterization of the class of delta-graphic matroids. We also show that every forbidden minor for the class of delta-graphic matroids has at most $48$ elements.Comment: 40 pages, 4 figures; revise