This article introduces the lazy matroid problem, which captures the goal of saving time or money in certain task selection scenarios. We are given a budget B and a matroid M with weights on its elements. The problem consists in finding an independent set F of minimum weight. In addition, F is feasible if its augmentation with any new element x implies that either F + x exceeds B or F + x is de...

Motivated by a property of linear resistive electrical networks, we introduce the class of Rayleigh matroids. These form a subclass of the balanced matroids defined by Feder and Mihail [10] in 1992. We prove a variety of results relating Rayleigh matroids to other well–known classes – in particular, we show that a binary matroid is Rayleigh if and only if it does not contain S8 as a minor. This...

A structure M is pregeometric if the algebraic closure is a pregeometry in all M ′ elementarily equivalent to M . We define a generalisation: structures with an existential matroid. The main examples are superstable groups of U-rank a power of ω and d-minimal expansion of fields. Ultraproducts of pregeometric structures expanding a field, while not pregeometric in general, do have an unique exi...

An essential element of a 3–connected matroid M is one for which neither the deletion nor the contraction is 3–connected. Tutte’s Wheels and Whirls Theorem proves that the only 3–connected matroids in which every element is essential are the wheels and whirls. In an earlier paper, the authors showed that a 3–connected matroid with at least one non-essential element has at least two such element...

The complexity of a secret sharing scheme is defined as the ratio between the maximum length of the shares and the length of the secret. The optimization of this parameter for general access structures is an important and very difficult open problem in secret sharing. We explore in this paper the connections of this open problem with matroids and polymatroids. Matroid ports were introduced by L...

A matroid has been one of the most important combinatorial structures since it was introduced by Whitney as an abstraction linear independence. As property a matroid, can be characterized several different (but equivalent) axioms, such augmentation, base exchange, or rank axiom. supermatroid is generalization defined on lattices. Here, central question whether equivalent axioms similar to matro...

Hypergraphic matroids were de ned by Lorea as generalizations of graphic matroids. We show that the minimum cut (co-girth) of a multiple of a hypergraphic matroid can be computed in polynomial time. It is well-known that the size of the minimum cut (co-girth) of a graph can be computed in polynomial time. For connected graphs, this is equivalent to computing the co-girth of the circuit matroid....

Induction (or transformation) by bipartite graphs is one of the most important operations on matroids, and it is well known that the induction of a matroid by a bipartite graph is again a matroid. As an abstract form of this fact, the induction of a matroid by a linking system is known to be a matroid. M-convex functions are quantitative extensions of matroidal structures, and they are known as...

It is well known that a matroid is binary if and only if it has no minor isomorphic to U2,4, the 4-point line. Extending this result, Bixby proved that every element in a nonbinary connected matroid is in a U2,4minor. The result was further extended by Seymour who showed that every pair of elements in a nonbinary 3-connected matroid is in a U2,4-minor. This paper extends Seymour's theorem by pr...

The connections between algebra and finite geometry are very old, with theorems about configurations of points dating to ancient Greece. In these notes, we will put a matroid theoretic spin on these results, with matroid representations playing the central role. Recall the definition of a matroid via independent sets I. Definition 1.1. Let E be a finite set and let I be a family of subsets of E...