  Graph states are a class of multi-partite entangled quantum states that are
ubiquitous in many networking applications; the study of equivalence relations
between graph states under local operations aims to provide methods to relate
graph states in networked settings. The problem of determining local unitary
(LU) equivalence of graph states is in NP, and it remains an open question if
efficient general methods are possible. We introduce a family of
easy-to-compute LU-invariants based on the marginal structure of the graphs
that allow to rule out equivalence of graph states. We show that these
invariants can uniquely identify all LU-orbits and entanglement classes of
every graph state of 8 qubits or less and discuss how reliable the methods are
for more qubit graph states. We also discuss examples of entanglement classes
with more nodes, where their marginal structure does not allow us to tell them
apart. Additionally, we generalise tools to test local clifford (LC)
equivalence of graph states that work by condensing graphs into other graphs of
smaller size. We show that statements on the equivalence of the smaller graphs
(which are easier to compute) can be used to infer statements on the
equivalence of the original, larger graphs.
