Matroid theory has many classical applications in combinatorics and geometry. Abstract rigidity matroids are generalizations of the infinitesi-mal rigidity matroids of frameworks in Euclidean space. In this paper we give a local characterization for abstract rigidity in any dimension. The conditions in this characterization are in many instances easier to verify than those in the definition of ...

A symplectic matroid is a collection B of k-element subsets of J = {1, 2, . . . , n, 1∗, 2∗, . . . , n∗}, each of which contains not both of i and i∗ for every i ≤ n, and which has the additional property that for any linear ordering ≺ of J such that i ≺ j implies j∗ ≺ i∗ and i ≺ j∗ implies j ≺ i∗ for all i, j ≤ n, B has a member which dominates element-wise every other member of B. Symplectic ...

For a matroid, its configuration determines \({\mathcal {G}}\)-invariant. Few examples are known of pairs matroids with the same {G}}\)-invariant but different configurations. In order to produce new examples, we introduce free m-cone \(Q_m(M)\) loopless matroid M, where m is positive integer. We show that M \(Q_m(M)\), and M; so if N nonisomorphic have {G}}\)-invariant, then \(Q_m(N)\) prove a...

Let D = (V + s,A) be a digraph with a designated root vertex s. Edmonds’ seminal result [4] implies that D has a packing of k spanning s-arborescences if and only if D has a packing of k (s, t)-paths for all t ∈ V , where a packing means arc-disjoint subgraphs. Let M be a matroid on the set of arcs leaving s. A packing of (s, t)-paths is called M-based if their arcs leaving s form a base of M w...

(Note that for circuits we implicitly assume that ∅ / ∈ C(M), just as we assume that for matroids ∅ ∈ I.) Proof: 1. follows from the definition that a circuit is a minimally dependent set, and therefore a circuit cannot contain another circuit. 2. Let X, Y ∈ C(M) where X 6= Y , and e ∈ X ∩ Y . From 1, it follows that X \ Y is non-empty; let f ∈ X \ Y . Assume on the contrary that (X ∪ Y ) − e i...

{ 14 { almost aane representation. Linear matroids form a subset of the class of almost aane matroids; as proved in 41], this inclusion is proper. Let D be the matrix that represents, almost aanely, a given matroid M. The rows of D form a subset of f1;:::;qg n called an almost aane code C. As usual in coding theory, one can introduce the set of local distance polynomials of this code as follows...

Frame matroids and lifted-graphic matroids are two interesting generalizations of graphic matroids. Here we introduce a new generalization, quasi-graphic matroids, that unifies these two existing classes. Unlike frame matroids and lifted-graphic matroids, it is easy to certify that a matroid is quasi-graphic. The main result of the paper is that every 3-connected representable quasi-graphic mat...

A certain unimodal conjecture in matroid theory states the number of rank-r matroids on a set of size n is unimodal in r and attains its maximum at r = bn/2c. We show that this conjecture holds up to r = 3 by constructing a map from a class of rank-2 matroids into the class of loopless rank-3 matroids. Similar inequalities are proven for the number of non-isomorphic loopless matroids, loopless ...