  We calculate exactly the vacuum polarization charge density in the field of a
subcritical Coulomb impurity, $Z|e|/r$, in graphene. Our analysis is based on
the exact electron Green's function, obtained by using the operator method, and
leads to results that are exact in the parameter $Z\alpha$, where $\alpha$ is
the "fine structure constant" of graphene. Taking into account also
electron-electron interactions in the Hartree approximation, we solve the
problem self-consistently in the subcritical regime, where the impurity has an
effective charge $Z_{eff}$, determined by the localized induced charge. We find
that an impurity with bare charge Z=1 remains subcritical, $Z_{eff} \alpha <
1/2$, for any $\alpha$, while impurities with $Z=2,3$ and higher can become
supercritical at certain values of $\alpha$.
