Retracts of vertex sets of trees and the almost stability theorem

Let G be a group, let T be an (oriented) G-tree with finite edge stabilizers, and let VT denote the vertex set of T. We show that, for each G-retract V' of the G-set VT, there exists a G-tree whose edge stabilizers are finite and whose vertex set is V'. This fact leads to various new consequences of the almost stability theorem. We also give an example of a group G, a G-tree T and a G-retract V' of VT such that no G-tree has vertex set V'.Comment: 15 pages, 0 figures. Formerly titled "Some refinements of the almost stability theorem". Version

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 Kr\"on 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.Comment: 31 page

Retracts of vertex sets of trees and the almost stability theorem

Abstract Let G be a group, let T be an (oriented) G-tree with finite edge stabilizers, and let V T denote the vertex set of T . We show that, for each G-retract V of the G-set V T , there exists a G-tree whose edge stabilizers are finite and whose vertex set is V . This fact leads to various new consequences of the almost stability theorem. We also give an example of a group G, a G-tree T and a G-retract V of V T such that no G-tree has vertex set V