  Classifying locally equivalent graph states, and stabilizer states more
broadly, is a significant problem in the theories of quantum information and
multipartite entanglement. A special focus is given to those graph states for
which equivalence through local unitaries implies equivalence through local
Clifford unitaries (LU $\Leftrightarrow$ LC). Identification of locally
equivalent states in this class is facilitated by a convenient transformation
rule on the underlying graphs and an efficient algorithm. Here we investigate
the question of local equivalence of the graph states behind the all-photonic
quantum repeater. We show that complete bipartite graph (biclique) states,
imperfect repeater graph states and small "crazy graph" states satisfy LU
$\Leftrightarrow$ LC. We continue by discussing biclique states more generally
and placing them in the context of counterexamples to the LU-LC Conjecture. To
this end, we offer some alternative proofs and clarifications on existing
results.
