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

Matroid and Tutte-connectivity in infinite graphs

We relate matroid connectivity to Tutte-connectivity in an infinite graph. Moreover, we show that the two cycle matroids, the finite-cycle matroid and the cycle matroid, in which also infinite cycles are taken into account, have the same connectivity function. As an application we re-prove that, also for infinite graphs, Tutte-connectivity is invariant under taking dual graphs.Comment: 11 page

Limit-closed Profiles

Tangle-tree theorems are an important tool in structural graph theory, and abstract separation systems are a very general setting in which tangle-tree theorems can still be formulated and proven. For infinite abstract separation systems, so far tangle-tree theorems have only been shown for special cases of separation systems, in particular when the separation system arises from a (locally finite) infinite graph. We present a tangle-tree theorem for infinite separation systems where we do not place restrictions on the separation system itself but on the tangles to be arranged in a tree.Comment: 12 pages, 2 figure

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

The ubiquity of Psi-matroids

Solving (for tame matroids) a problem of Aigner-Horev, Diestel and Postle, we prove that every tame matroid M can be reconstructed from its canonical tree decomposition into 3-connected pieces, circuits and cocircuits together with information about which ends of the decomposition tree are used by M . For every locally finite graph G, we show that every tame matroid whose circuits are topological circles of G and whose cocircuits are bonds of G is determined by the set Psi of ends it uses, that is, it is a Psi-matroid

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