Self-avoiding walks and their languages

This is an introductory review of recent work on self-avoiding walks on thin-ended quasi-transitive graphs. The goal is not to give precise proofs, but instead show the main ideas by giving simple examples.
Our main interest lies in two different kinds of formal languages induced by the set of all self-avoiding walks on the graph starting at a given origin. The language of self-avoiding walks on a deterministically edge-labelled graph consists of all words obtained by reading along self-avoiding walks. We discuss that this language is regular or context-free, if and only if the given graph has ends of size at most 1 or 2, respectively.
Using tree-decomposition, thin ended graphs can be decomposed into finite parts. Configurations are appearances of self-avoiding walks on single parts, and compatible configurations on the composition tree correspond to self-avoiding walks on the original graph. By encoding compatible configurations as words we obtain the language of configurations. This language is context-free and thus the generating function of self-avoiding walks is algebraic.