Splitting Formulas for Tutte Polynomials

We present two splitting formulas for calculating the Tutte polynomial of a matroid. The first one is for a generalized parallel connection across a 3-point line of two matroids and the second one is applicable to a 3-sum of two matroids. An important tool used is the bipointed Tutte polynomial of a matroid, an extension of the pointed Tutte polynomial introduced by Brylawski.     

On Tutte polynomial expansion formulas in perspectives of matroids and oriented matroids

We introduce the active partition of the ground set of an oriented matroid perspective (or morphism, or quotient, or strong map) on a linearly ordered ground set. The reorientations obtained by arbitrarily reorienting parts of the active partition share the same active partition. This yields an equivalence relation for the set of reorientations of an oriented matroid perspective, whose classes are enumerated by coefficients of the Tutte polynomial, and a remarkable partition of the set of reorientations into Boolean lattices, from which we get a short direct proof of a 4-variable expansion formula for the Tutte polynomial in terms of orientation activities. This formula was given in the last unpublished preprint by Michel Las Vergnas; the above equivalence relation and notion of active partition generalize a former construction in oriented matroids by Michel Las Vergnas and the author; and the possibility of such a proof technique in perspectives was announced in the aforementioned preprint. We also briefly highlight how the 5-variable expansion of the Tutte polynomial in terms of subset activities in matroid perspectives comes in a similar way from the known partition of the power set of the ground set into Boolean lattices related to subset activities (and we complete the proof with a property which was missing in the literature). In particular, the paper applies to matroids and oriented matroids on a linearly ordered ground set, and applies to graph and directed graph homomorphisms on a linearly ordered edge-set.     

Strongly inequivalent representations and Tutte polynomials of matroids

We develop constructive techniques to show that non-isomorphic 3-connected matroids that are representable over a fixed finite field and that have the same Tutte polynomial abound. In particular, for most prime powers q, we construct infinite families of sets of 3-connected matroids for which the matroids in a given set are non-isomorphic, are representable over GF(q), and have the same Tutte polynomial. Furthermore, the cardinalities of the sets of matroids in a given family grow exponentially as a function of rank, and there are many such families.In Memory of Gian-Carlo Rota     

Skew partial fields,multilinear representations of matroids,and a matrix tree theorem

We extend the notion of representation of a matroid to algebraic structures that we call skew partial fields. Our definition of such representations extends Tutte?s definition, using chain groups. We show how such representations behave under duality and minors, we extend Tutte?s representability criterion to this new class, and we study the generator matrices of the chain groups. An example shows that the class of matroids representable over a skew partial field properly contains the class of matroids representable over a skew field. Next, we show that every multilinear representation of a matroid can be seen as a representation over a skew partial field. Finally we study a class of matroids called quaternionic unimodular. We prove a generalization of the matrix tree theorem for this class.     

The Las Vergnas polynomial for embedded graphs

The Las Vergnas polynomial is an extension of the Tutte polynomial to cellularly embedded graphs. It was introduced by Michel Las Vergnas in 1978 as special case of his Tutte polynomial of a morphism of matroids. While the general Tutte polynomial of a morphism of matroids has a complete set of deletion–contraction relations, its specialisation to cellularly embedded graphs does not. Here we extend the Las Vergnas polynomial to graphs in pseudo-surfaces. We show that in this setting we can define deletion and contraction for embedded graphs consistently with the deletion and contraction of the underlying matroid perspective, thus yielding a version of the Las Vergnas polynomial with complete recursive definition. This also enables us to obtain a deeper understanding of the relationships among the Las Vergnas polynomial, the Bollobás–Riordan polynomial, and the Krushkal polynomial. We also take this opportunity to extend some of Las Vergnas’ results on Eulerian circuits from graphs in surfaces of low genus to graphs in surfaces of arbitrary genus.     

Euclideaness and final polynomials in oriented matroid theory

This paper deals with a geometric construction of algebraic non-realizability proofs for certain oriented matroids. As main result we obtain an algorithm which generates a (bi-quadratic) final polynomial [3], [5] for any non-euclidean oriented matroid. Here we apply the results of Edmonds, Fukuda and Mandel [6], [7] concerning non-degenerate cycling of linear programs in non-euclidean oriented matroids.     

Matroid matching in pseudomodular lattices

The matroid matching problem (also known as matroid parity problem) has been intensively studied by several authors. Starting from very special problems, in particular the matching problem and the matroid intersection problem, good characterizations have been obtained for more and more general classes of matroids. The two most recent ones are the class of representable matroids and, later on, the class of algebraic matroids (cf. [4] and [2]). We present a further step of generalization, showing that a good characterization can also be obtained for the class of socalled pseudomodular matroids, introduced by Björner and Lovász (cf. [1]). A small counterexample is included to show that pseudomodularity still does not cover all matroids that behave well with respect to matroid matching.Supported by the German Research Association (Deutsche Forschungsgemeinschaft, SFB 303).     

On Brylawski’s Generalized Duality

We introduce a notion of duality??due to Brylawski??that generalizes matroid duality to arbitrary rank functions. This allows us to define a generalization of the matroid Tutte polynomial. This polynomial satisfies a deletion-contraction recursion, where deletion and contraction are defined in this more general setting. We explore this notion of duality for greedoids, antimatroids and demi-matroids, proving that matroids correspond precisely to objects that are simultaneously greedoids and ??dual?? greedoids.     

Coxeter matroid polytopes

If Δ is a polytope in real affine space, each edge of Δ determines a reflection in the perpendicular bisector of the edge. The exchange groupW (Δ) is the group generated by these reflections, and Δ is a (Coxeter) matroid polytope if this group is finite. This simple concept of matroid polytope turns out to be an equivalent way to define Coxeter matroids. The Gelfand-Serganova Theorem and the structure of the exchange group both give us information about the matroid polytope. We then specialize this information to the case of ordinary matroids; the matroid polytope by our definition in this case turns out to be a facet of the classical matroid polytope familiar to matroid theorists. This work was supported in part by NSA grant MDA904-95-1-1056.     

Separating cocircuits in binary matroids

A cocircuit of a matroid is separating if deleting it leaves a separable matroid. We give an effecient algorithm which finds a separating cocircuit or a Fano minor in a binary matroid, thus proving constructively a theorem of Tutte. Using this algorithm and a new recursive characterization of bond matroids, we give a new method for testing binary matroids for graphicness. We also give an efficient algorithm for finding a special kind of separating cocircuit: one whose deletion leaves a matroid having a coloop.     

Counterexamples to conjectures on 4-connected matroids

Tutte characterized binary matroids to be those matroids without aU ₄ ² minor. Bixby strengthened Tutte’s result, proving that each element of a 2-connected non-binary matroid is in someU ₄ ² minor. Seymour proved that each pair of elements in a 3-connected non-binary matroid is in someU ₄ ² minor and conjectured that each triple of elements in a 4-connected non-binary matroid is in someU ₄ ² minor. A related conjecture of Robertson is that each triple of elements in a 4-connected non-graphic matroid is in some circuit. This paper provides counterexamples to these two conjectures.     

Convexity in oriented matroids

We generalize to oriented matroids classical notions of Convexity Theory: faces of convex polytopes, convex hull, etc., and prove some basic properties. We relate the number of acyclic orientations of an orientable matroid to an evaluation of its Tutte polynomial.     

Truncation Formulas for Invariant Polynomials of Matroids and Geometric Lattices

This paper considers the truncation of matroids and geometric lattices. It is shown that the truncated matroid of a representable matroid is again representable. Truncation formulas are given for the coboundary and M?bius polynomial of a geometric lattice and the spectrum polynomial of a matroid, generalizing the truncation formula of the rank generating polynomial of a matroid by Britz.     

Interlacement and activities in delta-matroids

We generalize theories of graph, matroid, and ribbon-graph activities to delta-matroids. As a result, we obtain an activity based feasible-set expansion for a transition polynomial of delta-matroids defined by Brijder and Hoogeboom. This result yields feasible-set expansions for the two-variable Bollobás–Riordan and interlace polynomials of a delta-matroid. In the former case, the expansion obtained directly generalizes the activities expansions of the Tutte polynomial of graphs and matroids.     

A matroid-friendly basis for the quasisymmetric functions

A new Z-basis for the space of quasisymmetric functions (QSym, for short) is presented. It is shown to have nonnegative structure constants, and several interesting properties relative to the quasisymmetric functions associated to matroids by the Hopf algebra morphism F of Billera, Jia, and Reiner [L.J. Billera, N. Jia, V. Reiner, A quasisymmetric function for matroids, arXiv:math.CO/0606646]. In particular, for loopless matroids, this basis reflects the grading by matroid rank, as well as by the size of the ground set. It is shown that the morphism F distinguishes isomorphism classes of rank two matroids, and that decomposability of the quasisymmetric function of a rank two matroid mirrors the decomposability of its base polytope. An affirmative answer to the Hilbert basis question raised in [L.J. Billera, N. Jia, V. Reiner, A quasisymmetric function for matroids, arXiv:math.CO/0606646] is given.     

The Lattice of Cyclic Flats of a Matroid

A flat of a matroid is cyclic if it is a union of circuits. The cyclic flats of a matroid form a lattice under inclusion. We study these lattices and explore matroids from the perspective of cyclic flats. In particular, we show that every lattice is isomorphic to the lattice of cyclic flats of a matroid. We give a necessary and sufficient condition for a lattice of sets and a function to be the lattice of cyclic flats of a matroid and the restriction of the corresponding rank function to . We apply this perspective to give an alternative view of the free product of matroids and we show how to compute the Tutte polynomial of the free product in terms of the Tutte polynomials of the constituent matroids. We define cyclic width and show that this concept gives rise to minor-closed, dual-closed classes of matroids, two of which contain only transversal matroids. Received May 29, 2005     

Bounding the beta invariant of 3-connected matroids

The beta invariant is related to the Chromatic and Tutte Polynomials and has been studied by Crapo [4], Brylawski [2], Oxley [7] and others. Crapo [4] showed that a matroid with at least two elements is connected if and only if its beta invariant is greater than zero. Brylawski [2] showed that a connected matroid has beta invariant one if and only if M is isomorphic to a serial-parallel network. Oxley [7] characterized all matroids with beta invariant two, three and four. In this paper, we first give a best possible lower bound on the beta invariant of 3-connected matroids, then we characterize all 3-connected matroids attaining the lower bound. We also characterize all binary matroids with beta invariant 5, 6, and 7.     

Latroids and their representation by codes over modules

It has been known for some time that there is a connection between linear codes over fields and matroids represented over fields. In fact a generator matrix for a linear code over a field is also a representation of a matroid over that field. There are intimately related operations of deletion, contraction, minors and duality on both the code and the matroid. The weight enumerator of the code is an evaluation of the Tutte polynomial of the matroid, and a standard identity relating the Tutte polynomials of dual matroids gives rise to a MacWilliams identity relating the weight enumerators of dual codes. More recently, codes over rings and modules have been considered, and MacWilliams type identities have been found in certain cases.

In this paper we consider codes over rings and modules with code duality based on a Morita duality of categories of modules. To these we associate latroids, defined here. We generalize notions of deletion, contraction, minors and duality, on both codes and latroids, and examine all natural relations among these.

We define generating functions associated with codes and latroids, and prove identities relating them, generalizing above-mentioned generating functions and identities.

     

Two Remarks Concerning Balanced Matroids

The property of balance (in the sense of Feder and Mihail) is investigated in the context of paving matroids. The following examples are exhibited: (a) a class of “sparse” paving matroids that are balanced, but at the same time rich enough combinatorially to permit the encoding of hard counting problems; and (b) a paving matroid that is not balanced. The computational significance of (a) is the following. As a consequence of balance, there is an efficient algorithm for approximating the number of bases of a sparse paving matroid within specified relative error. On the other hand, determining the number of bases exactly is likely to be computationally intractable. * The work described here was supported by the grant “Sharper analysis of randomised algorithms” from the UK Engineering and Physical Sciences Research Council.     

Algorithms for Measuring Perturbability in Matroid Optimization

perturbability function of a matroid measures the maximum increase in the weight of its minimum weight bases that can be produced by increases of a given total cost on the weights of its elements. We present an algorithm for computing this function that runs in strongly polynomial time for matroids in which independence can be tested in strongly polynomial time. Furthermore, for the case of transversal matroids we are able to take advantage of their special structure to design a faster algorithm for computing the perturbability function. Received: June 13, 1997