A splitter theorem for 3-connected 2-polymatroids

Seymour’s Splitter Theorem is a basic inductive tool for dealing with 3-connected matroids. This paper proves a generalization of that theorem for the class of 2-polymatroids. Such structures include matroids, and they model both sets of points and lines in a projective space and sets of edges in a graph. A series compression in such a structure is an analogue of contracting an edge of a graph that is in a series pair. A 2-polymatroid N is an s-minor of a 2-polymatroid M if N can be obtained from M by a sequence of contractions, series compressions, and dual-contractions, where the last are modified deletions. The main result proves that if M and N are 3-connected 2-polymatroids such that N is an s-minor of M, then M has a 3-connected s-minor M′ that has an s-minor isomorphic to N and has |E(M)| − 1 elements unless M is a whirl or the cycle matroid of a wheel. In the exceptional case, such an M′ can be found with |E(M)| − 2 elements

A Tutte polynomial inequality for lattice path matroids

Let $M$ be a matroid without loops or coloops and let $T(M;x,y)$ be its Tutte polynomial. In 1999 Merino and Welsh conjectured that $$\max(T(M;2,0), T(M;0,2))\geq T(M;1,1)$$ holds for graphic matroids. Ten years later, Conde and Merino proposed a multiplicative version of the conjecture which implies the original one. In this paper we prove the multiplicative conjecture for the family of lattice path matroids (generalizing earlier results on uniform and Catalan matroids). In order to do this, we introduce and study particular lattice path matroids, called snakes, used as building bricks to indeed establish a strengthening of the multiplicative conjecture as well as a complete characterization of the cases in which equality holds.Comment: 17 pages, 9 figures, improved exposition/minor correction

Matroid relaxations and Kazhdan-Lusztig non-degeneracy

In this paper we study the interplay between the operation of circuit-hyperplane relaxation and the Kazhdan-Lusztig theory of matroids. We obtain a family of polynomials, not depending on the matroids but only on their ranks, that relate the Kazhdan-Lusztig, the inverse Kazhdan-Lusztig and the $Z$-polynomial of each matroid with those of its relaxations. As applications of our main theorem, we prove that all matroids having a free basis are non-degenerate. Additionally, we obtain bounds and explicit formulas for all the coefficients of the Kazhdan-Lusztig, inverse Kazhdan-Lusztig and $Z$-polynomial of all sparse paving matroids.Comment: 26 pages. Revised versio

Polytopal and structural aspects of matroids and related objects

PhDThis thesis consists of three self-contained but related parts. The rst is focussed on polymatroids, these being a natural generalisation of matroids. The Tutte polynomial is one of the most important and well-known graph polynomials, and also features prominently in matroid theory. It is however not directly applicable to polymatroids. For instance, deletion-contraction properties do not hold. We construct a polynomial for polymatroids which behaves similarly to the Tutte polynomial of a matroid, and in fact contains the same information as the Tutte polynomial when we restrict to matroids. The second section is concerned with split matroids, a class of matroids which arises by putting conditions on the system of split hyperplanes of the matroid base polytope. We describe these conditions in terms of structural properties of the matroid, and use this to give an excluded minor characterisation of the class. In the nal section, we investigate the structure of clutters. A clutter consists of a nite set and a collection of pairwise incomparable subsets. Clutters are natural generalisations of matroids, and they have similar operations of deletion and contraction. We introduce a notion of connectivity for clutters that generalises that of connectivity for matroids. We prove a splitter theorem for connected clutters that has the splitter theorem for connected matroids as a special case: if M and N are connected clutters, and N is a proper minor of M, then there is an element in E(M) that can be deleted or contracted to produce a connected clutter with N as a minor

Fork-decompositions of matroids

For the abstract of this paper, please see the PDF file

A wheels-and-whirls theorem for 3-connected 2-polymatroids

Tutte\u27s wheels-and-whirls theorem is a basic inductive tool for dealing with 3- connected matroids. This paper proves a generalization of that theorem for the class of 2-polymatroids. Such structures include matroids, and they model both sets of points and lines in a projective space and sets of edges in a graph. The main result proves that, in a 3-connected 2-polymatroid that is not a whirl or the cycle matroid of a wheel, one can obtain another 3-connected 2-polymatroid by deleting or contracting some element, or by performing a new operation that generalizes series contraction in a graph. Moreover, we show that unless one uses some reduction operation in addition to deletion and contraction, the set of minimal 2-polymatroids that are not representable over a fixed field F is infinite, irrespective of whether F is finite or infinite