## 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 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.

Original language | English |
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Pages (from-to) | 524-539 |

Number of pages | 16 |

Journal | Mathematische Nachrichten |

Volume | 292 |

Issue number | 3 |

DOIs | |

Publication status | Published - Mar 2019 |

## Keywords

- 05C05
- 05C40
- 05C63
- 20B27
- graph automorphism
- infinite graph
- tree decomposition

## ASJC Scopus subject areas

- General Mathematics