Connectivity and tree structure in finite graphs

Considering systems of separations in a graph that separate every pair of a given set of vertex sets that are themselves not separated by these separations, we determine conditions under which such a separation system contains a nested subsystem that still separates those sets and is invariant under the automorphisms of the graph. As an application, we show that the k-blocks – the maximal vertex sets that cannot be separated by at most k vertices – of a graph G live in distinct parts of a suitable tree-decomposition of G of adhesion at most k, whose decomposition tree is invariant under the automorphisms of G. This extends recent work of Dunwoody and Krön and, like theirs, generalizes a similar theorem of Tutte for k = 2. Under mild additional assumptions, which are necessary, our decompositions can be combined into one overall tree-decomposition that distinguishes, for all k simultaneously, all the k-blocks of a finite graph.