  For Hamiltonian circle actions on 4-manifolds, we give a generators and
relations description for the even part of the equivariant cohomology, as a
module over the equivariant cohomology of a point. This description depends on
combinatorial data encoded in the decorated graph of the manifold. We then give
an explicit combinatorial description of all weak algebra isomorphisms. We use
this description to prove that the even parts of the equivariant cohomology
modules are weakly isomorphic (and the odd groups have the same ranks) if and
only if the labelled graphs obtained from the decorated graphs by forgetting
the height and area labels are isomorphic.
  As a consequence, we give an example of an isomorphism of equivariant
cohomology modules that cannot be induced by an equivariant diffeomorphism of
spaces preserving a compatible almost complex structure. We also deduce a soft
proof that there are finitely many maximal Hamiltonian circle actions on a
fixed closed symplectic 4-manifold.
