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On tree‐decompositions of one‐ended graphs

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  • Johannes Carmesin
  • Florian Lehner
  • Rögnvaldur G. Möller

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.

Suggested Citation

  • Johannes Carmesin & Florian Lehner & Rögnvaldur G. Möller, 2019. "On tree‐decompositions of one‐ended graphs," Mathematische Nachrichten, Wiley Blackwell, vol. 292(3), pages 524-539, March.
    • Handle: RePEc:bla:mathna:v:292:y:2019:i:3:p:524-539
    DOI: 10.1002/mana.201800055
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