Computing with Tangles

Abstract

Tangles of graphs have been introduced by Robertson and Seymour in the context of their graph minor theory. Tangles may be viewed as describing “$k$-connected components” of a graph (though in a twisted way). An interesting aspect of tangles is that they can be defined not only for graphs, but more generally for arbitrary connectivity functions (that is, integer-valued submodular and symmetric set functions). However, tangles are difficult to deal with algorithmically. To start with, it is unclear how to represent them, because they are families of separations and, as such, may be exponentially large. Our first contribution is a data structure for representing and accessing all tangles of a graph up to some fixed order. Using this data structure, we can prove an algorithmic version of a very general structure theorem due to Carmesin et al. (for graphs) and Hundertmark (for arbitrary connectivity functions) that yields a canonical tree decomposition whose parts correspond to the maximal tangles. This may be viewed as a generalization of the decomposition of a graph into its 3-connected components.