On a theorem of Halin

TL;DR

This paper provides an elementary proof of a generalized Halin theorem for graphs with unbounded degrees and applies it to identify finite distinguishing sets in certain infinite graphs with limited automorphism groups.

Contribution

It introduces a new, elementary proof of a generalized Halin theorem and applies it to find finite distinguishing sets in infinite graphs with specific automorphism group properties.

Findings

Elementary proof of generalized Halin theorem

Existence of finite distinguishing sets in certain infinite graphs

Bound on the size of distinguishing sets

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 a subdegree-finite, infinite automorphism group whose cardinality is strictly less than continuum, 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 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.