# On tree-decompositions of one-ended graphs

## TL;DR

This paper proves that certain one-ended graphs without dominated rays or disjoint rays have a symmetric, one-ended tree-decomposition, confirming a conjecture about their automorphism groups and revealing properties of transitive graphs.

## Contribution

It establishes the existence of an automorphism-invariant, one-ended tree-decomposition for specific one-ended graphs, confirming Halin's conjecture and solving a recent problem.

## Key findings

- Automorphism group of such graphs cannot be countably infinite.
- Every transitive one-ended graph contains infinitely many disjoint rays.
- Constructs a symmetric tree-decomposition for these graphs.

## Abstract

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.