Reconstruction of infinite matroids from their 3-connected minors

TL;DR

This paper demonstrates that infinite matroids can be reconstructed from their tree-decomposition torsos and local end information, simplifying the process for tame and planar cases.

Contribution

It introduces a method to reconstruct infinite matroids from tree-decompositions and characterizes the local information needed for tame and planar matroids.

Findings

Infinite matroids are reconstructible from torsos and end data.

For tame matroids, local end information reduces to circuit permission choices.

Planar torsos require a consistent face for gluing elements.

Abstract

We show that any infinite matroid can be reconstructed from the torsos of a tree-decomposition over its 2-separations, together with local information at the ends of the tree. We show that if the matroid is tame then this local information is simply a choice of whether circuits are permitted to use that end. The same is true if each torso is planar, with all gluing elements on a common face.