  Magnetic properties of a single vacancy in graphene is a relevant and still
much discussed problem. The experimental results point to a clearly detectable
magnetic defect state at the Fermi energy, while calculations based on density
functional theory (DFT) yield widely varying results for the magnetic moment,
in the range of $\mu=1.04-2.0$ $\mu_{B}$. We present a multi-tool \textit{ab
initio} theoretical study of the same defect, using two simulation protocols
for a defect in a crystal (cluster and periodic boundary conditions) and
different DFT functionals - bare and hybrid DFT, mixing a fraction of
Hartree-Fock exchange (XC). Our main conclusions are two-fold: First, we find
that due to the $\pi$-character of the Fermi-energy states of graphene,
inclusion of XC is crucial and for a single isolated vacancy we can predict an
integer magnetic moment $\mu=2\mu_{B}$. Second, we find that due to the
specific symmetry of the graphene lattice, periodic arrays of single vacancies
may provide interesting diffuse spin-spin interactions.
