  Graph states are quantum states that can be described by a stabilizer
formalism and play an important role in quantum information processing. We
consider the action of local unitary operations on graph states and hypergraph
states. We focus on non-Clifford operations and find for certain
transformations a graphical description in terms of weighted hypergraphs. This
leads to the indentification of hypergraph states that are locally equivalent
to graph states. Moreover, we present a systematic way to construct pairs of
graph states which are equivalent under local unitary operations, but not
equivalent under local Clifford operations. This generates counterexamples to a
conjecture known as LU-LC conjecture. So far, the only counterexamples to this
conjecture were found by random search. Our method reproduces the smallest
known counterexample as a special case and provides a physical interpretation.
