  We expand the structural theory of \ca graphs that avoid specific cyclic
coset patterns. We present several characterisations of tree-likeness for these
structures and show a close connection to $\alpha$-acyclic hypergraphs. A focus
lies on the behaviour of short paths of overlapping cosets in these \ca graphs,
and their relation to short chordless paths in hypergraphs that are locally
acyclic.
