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 $$\aleph _0 \le |{\text {Aut}}(G)| < 2^{\aleph _0}$$ℵ0≤|Aut(G)|<2ℵ0 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 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.

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关于哈林定理

本笔记提供了一个新的基本证明，将 Halin 定理推广到具有无界度的图，然后将其应用于证明每个连通的、可数无限图 G，其中 $$\aleph _0 \le |{\text {自动}}(G)| < 2^{\aleph _0}$$ℵ0≤|Aut(G)|<2ℵ0 和次度有限自同构群，具有顶点的有限集 F，仅通过恒等自同构设置稳定。还提供了对此类集合大小的限制，称为区分。为了正确看待 Halin 定理及其推广，我们还讨论了几个相关的非初等、独立的结果及其证明方法。