Hamiltonian circle actions on manifolds of dimension four are classified by decorated graphs. We give a generators and relations description in terms of the decorated graph for the even part of the equivariant cohomology of a Hamiltonian $S^1$-manifold, as a module over the equivariant cohomology of a point. We deduce an explicit combinatorial description of all weak algebra isomorphisms. As a first consequence, we show that the equivariant cohomology does not determine the space. A second consequence is a proof of the finiteness of maximal Hamiltonian circle actions on a closed symplectic four-manifold, that does not use pseudo-holomorphic tools. This is based on joint work with Tara Holm.