  A recent development in graph theory is to study local separators, vertex
sets that need not separate graphs globally but just locally. We use this idea
to conjecture an extension of Stallings' theorem to finite nilpotent groups. We
provide an example demonstrating that the assumption of nilpotency is
necessary, and prove a stronger form of this conjecture for low connectivities,
as follows.
  A finite nilpotent group that has a Cayley graph with a local separator of
size at most two is cyclic, dihedral or a direct product of a cyclic group with
the group of order two.
