Tangles, Trees and Flowers

Abstract

Tangles provide a means of identifying highly connected components of a graph, matroid or, more generally, a connectivity function. For a tangle of order k in such a structure, the smallest order of a separation that can cross the structure in a non-trivial way relative to the tangle has order k. It turns out that, modulo a natural notion of equivalence, such k-separations do not behave arbitrarily. Indeed there is a tree that describes, up to equivalence, all such k-separations. This tree generalises known tree decompositions for 2- and 3-separations in connected and 3-connected matroids and graphs. This is joint work with Ben Clark.