On packing minors into connected matroids

Let N be a matroid with k connected components and M be a minor-minimal connected matroid having N as a minor. This note proves that |E(M) - E(N)| is at most 2k - 2 unless N or its dual is free, in which case |E(M) - E(N)| ≤k - 1. Examples are given to show that these bounds are best possible for all choices for N. A consequence of the main result is that a minimally connected matroid of rank r and maximum circuit size c has at most 2r - c + 2 elements. This bound sharpens a result of Murty. © 1998 Elsevier Science B.V. All rights reserved

On the intersection conjecture for infinite trees of matroids

Using a new technique, we prove a rich family of special cases of the matroid intersection conjecture. Roughly, we prove the conjecture for pairs of tame matroids which have a common decomposition by 2-separations into finite parts

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

Signed graphs, regular matroids, grafts

Mathematics

The structure of 2-separations of infinite matroids

Generalizing a well known theorem for finite matroids, we prove that for every (infinite) connected matroid M there is a unique tree T such that the nodes of T correspond to minors of M that are either 3-connected or circuits or cocircuits, and the edges of T correspond to certain nested 2-separations of M. These decompositions are invariant under duality.Comment: 31 page