  This paper introduces the foliage partition, an easy-to-compute LC-invariant
for graph states, of computational complexity $\mathcal{O}(n^3)$ in the number
of qubits. Inspired by the foliage of a graph, our invariant has a natural
graphical representation in terms of leaves, axils, and twins. It captures
both, the connection structure of a graph and the $2$-body marginal properties
of the associated graph state. We relate the foliage partition to the size of
LC-orbits and use it to bound the number of LC-automorphisms of graphs. We also
show the invariance of the foliage partition when generalized to weighted
graphs and qudit graph states.
