Let X=(VX,EX) be an infinite, locally finite, connected graph without
loops or multiple edges. We consider the edges to be oriented, and EX is
equipped with an involution which inverts the orientation. Each oriented edge
is labelled by an element of a finite alphabet Σ. 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 Σ 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.Comment: 24 pages, 3 figure