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

Axioms for infinite matroids

We give axiomatic foundations for non-finitary infinite matroids with duality, in terms of independent sets, bases, circuits, closure and rank. This completes the solution to a problem of Rado of 1966.Comment: 33 pp., 2 fig

Spikes in Matroid Theory.

For an integer $n\ge 3,$ a rank-n matroid is called an n-spike if it consists of n three-point lines through a common point t such that, for all k in $(1, 2,\..., n - 1),$ the union of every set of k of these Lines has rank $k + 1.$ The point t is called the tip of the n-spike. Ding, Oporowski, Oxley, and Vertigan proved that, for all $n\ge 3,$ there is an integer $N(n)$ such that every 3-connected matroid with at least $N(n)$ elements has a minor isomorphic to a wheel or whirl of rank n,\ M(K\sb{3,n}) or its dual, U\sb{2,n+2} or its dual, or a rank-n spike. In the first chapter of this dissertation, we characterise each of these classes of unavoidable matroids in terms of an extremal connectivity condition. In particular, it is proved in this chapter that if M is a 3-connected matroid of rank at least seven for which every single-element deletion or contraction is 3-connected but no 2-element deletion or contraction is, then M is a spike with its tip deleted. It is further proved that if M is a 3-connected matroid of rank at least four for which every single-element deletion is 3-connected but no 1-element contraction or 2-element deletion is, then M\cong M\sp*(K\sb{3,n}).. The second chapter of this dissertation evaluates the number of n-spikes representable over finite fields. It is well known that there is a unique binary n-spike for each integer $n\ge 3.$ In this chapter, we first prave that, for each integer $n\ge 3,$ there are exactly two distinct ternary n-spikes, and there are exactly \lfloor{{n\sp2+6n+24}\over{12}}\rfloor quaternary n-spikes. Then we prove that, for each integer $n\ge 4,$ there are exactly $n + 2 + \lfloor {{n}\over{2}}\rfloor$ quinternary n-spikes and, for each integer $n\ge 18,$ the number of n-spikes representable over $GF(7)$ is \lfloor{{2n\sp2+6n+6}\over{3}}\rfloor. Finally, for each $q\ge 7,$ we find the asymptotic value of the number of distinct rank-n spikes that are representable over $GF(q)$

Zero-free regions for multivariate Tutte polynomials (alias Potts-model partition functions) of graphs and matroids

The chromatic polynomial P_G(q) of a loopless graph G is known to be nonzero (with explicitly known sign) on the intervals (-\infty,0), (0,1) and (1,32/27]. Analogous theorems hold for the flow polynomial of bridgeless graphs and for the characteristic polynomial of loopless matroids. Here we exhibit all these results as special cases of more general theorems on real zero-free regions of the multivariate Tutte polynomial Z_G(q,v). The proofs are quite simple, and employ deletion-contraction together with parallel and series reduction. In particular, they shed light on the origin of the curious number 32/27.Comment: LaTeX2e, 49 pages, includes 5 Postscript figure

Connectivity for Matroids and Graphs.

This dissertation studies connectivity for matroids and graphs. The main results generalize Tutte\u27s Wheels and Whirls Theorem and have numerous applications. In Chapter 2, we prove two structural theorems for 3-connected matroids. An element e of a 3-connected matroid M is essential if neither the deletion M$\\$e nor the contraction M/e is 3-connected. Tutte\u27s Wheels and Whirls Theorem proves that the only 3-connected matroids in which every element is essential are the wheels and whirls. If M is not a wheel or a whirl, we prove that the essential elements of M can be partitioned into classes where two elements are in the same class if M has a fan containing both. In particular, M must have at least two non-essential elements. In the second structural theorem, we show that if M has a fan with 2k or 2k + 1 elements for some $k \geq \ 2$, then M can be obtained by sticking together a (k + 1)-spoked wheel and a certain 3-connected minor of M. In Chapters 3 and 4, we characterize all 3-connected matroids whose set of non-essential elements has rank two. In particular, we completely determine all 3-connected matroids with exactly two non-essential elements. In Chapter 5, we derive some consequences of these results for the 3-connected binary matroids and graphs. We prove that there are exactly six classes of 3-connected binary matroids whose set of non-essential elements has rank two and we prove that there are exactly two classes of graphs, multi-dimensional wheels and twisted wheels, with exactly two non-essential edges. In Chapter 6, we use our first structural theorem to investigate the set of elements e in a 3-connected matroid M such that the simplification of M/e is 3-connected. We get best-possible lower bounds on the number of such elements thereby improving a result which was derived by Cunningham and Seymour independently. We also give some generalizations of the Wheels and Whirls Theorem and the Wheels Theorem

Single Commodity Flow Algorithms for Lifts of Graphic and Cographic Matroids

Consider a binary matroid M given by its matrix representation. We show that if M is a lift of a graphic or a cographic matroid, then in polynomial time we can either solve the single commodity flow problem for M or find an obstruction for which the Max-Flow Min-Cut relation does not hold. The key tool is an algorithmic version of Lehman's Theorem for the set covering polyhedron

Defining bicircular matroids in monadic logic

We conjecture that the class of frame matroids can be characterised by a sentence in the monadic second-order logic of matroids, and we prove that there is such a characterisation for the class of bicircular matroids. The proof does not depend on an excluded-minor characterisation