# The language of self-avoiding walks

## TL;DR

This paper characterizes when the language of self-avoiding walks on an infinite, labeled graph is regular or context-free, based on the graph's end structure and symmetry properties.

## Contribution

It provides a complete characterization of the formal language class of self-avoiding walk labels in terms of graph topology and automorphism group actions.

## Key findings

- Language is regular if and only if the graph has more than one end with all ends of size 1.
- Language is context-free if and only if the graph has at most two ends.
- The characterization links graph topology with formal language classification.

## Abstract

Let $X=(V\!X,E\!X)$ be an infinite, locally finite, connected graph without loops or multiple edges. We consider the edges to be oriented, and $E\!X$ is equipped with an involution which inverts the orientation. Each oriented edge is labelled by an element of a finite alphabet $\mathbf{\Sigma}$. The labelling is assumed to be deterministic: edges with the same initial (resp. terminal) vertex have distinct labels. Furthermore it is assumed that the group of label-preserving automorphisms of $X$ acts quasi-transitively. For any vertex $o$ of $X$, consider the language of all words over $\mathbf{\Sigma}$ which can be read along self-avoiding walks starting at $o$. We characterize under which conditions on the graph structure this language is regular or context-free. This is the case if and only if the graph has more than one end, and the size of all ends is $1$, or at most $2$, respectively.