# On a theorem of Halin

Imrich, Wilfried and Smith, Simon M. (2017) On a theorem of Halin. Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg, 87 (2). pp. 289-297. ISSN 0025-5858

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## Abstract

This note presents a new, elementary proof of a generalization of a theorem of Halin to graphs with unbounded degrees, which is then applied to show that every connected, countably infinite graph $G$, with $\aleph_0 \leq |\A(G)| < 2^{\aleph_0}$ and subdegree-finite automorphism group, has a finite set $F$ of vertices that is setwise stabilized only by the identity automorphism. A bound on the size of such sets, which are called {\em distinguishing}, is also provided.

To put this theorem of Halin and its generalization into perspective, we also discuss several related non-elementary, independent results and their methods of proof.

Keywords: Distinguishing number, Automorphism, Determining number, Infinite graph G Mathematical and Computer Sciences > G110 Pure Mathematics College of Science > School of Mathematics and Physics https://doi.org/10.1007/s12188-016-0167-... 26752 29 Mar 2017 08:36

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