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.