On tree-decompositions of one-ended graphs

Johannes Carmesin, Florian Lehner*, Rögnvaldur G. Möller

*Korrespondierende/r Autor/in für diese Arbeit

Publikation: Beitrag in einer FachzeitschriftArtikel

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.

Originalspracheenglisch
Seiten (von - bis)524-539
Seitenumfang16
FachzeitschriftMathematische Nachrichten
Jahrgang292
Ausgabenummer3
DOIs
PublikationsstatusVeröffentlicht - Mär 2019

ASJC Scopus subject areas

  • !!Mathematics(all)

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