  In the present paper, we consider an action of the circle group on a compact
oriented 4-manifold. We derive the Atiyah-Hirzebruch formula for the manifold,
and associate a graph in terms of data on the fixed point set. We show in the
case of isolated fixed points that if an abstract graph satisfies the
Atiyah-Hirzebruch formula, then there exists a corresponding 4-dimensional
oriented $S^1$-manifold.
