4-dimensional Hamiltonian $S^1$-spaces are classified by their graphs. We discuss an extension of this to almost complex manifolds, in the case of isolated fixed points. We consider a circle action on a 4-dimensional compact almost complex manifold with isolated fixed points. We give a necessary and sufficient condition for weights at the fixed points. We do this by using a combinatorial object, a (labeled directed) multigraph. We show that there is a graph that contains information on weights at the fixed points, and there are a minimal graph and operations that attain any such graph. As an application, we provide a necessary and sufficient condition for the Chern numbers of such an action. We achieve this by demonstrating that pairs of integers that arise as weights of a circle action, also arise as weights of a restriction of a $T^2$​​-action.