  A graph is one-ended if it contains a ray (a one way infinite path) and
whenever we remove a finite number of vertices from the graph then what remains
has only one component which contains rays. A vertex $v$ {\em dominates} a ray
in the end if there are infinitely many paths connecting $v$ to the ray such
that any two of these paths have only the vertex $v$ in common. We prove that
if a one-ended graph contains no ray which is dominated by a vertex and no
infinite family of pairwise disjoint rays, then it has a tree-decomposition
such that the decomposition tree is one-ended and the tree-decomposition is
invariant under the group of automorphisms.
  This can be applied to prove a conjecture of Halin from 2000 that the
automorphism group of such a graph cannot be countably infinite and solves a
recent problem of Boutin and Imrich. Furthermore, it implies that every
transitive one-ended graph contains an infinite family of pairwise disjoint
rays.

Combinatorics