295 research outputs found
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 -blocks -- the maximal vertex sets
that cannot be separated by at most vertices -- of a graph live in
distinct parts of a suitable tree-decomposition of of adhesion at most ,
whose decomposition tree is invariant under the automorphisms of . This
extends recent work of Dunwoody and Kr\"on and, like theirs, generalizes a
similar theorem of Tutte for .
Under mild additional assumptions, which are necessary, our decompositions
can be combined into one overall tree-decomposition that distinguishes, for all
simultaneously, all the -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
- …