  The correlation between structural distortion and emergence of magnetism in
graphene containing a single vacancy was investigated using first-principles
calculations based on density functional theory (DFT). Our results have shown
that a local distortion is formed around the vacancy, with reconstruction of
two atomic bonds and with a dangling bond remaining at the third atom adjacent
to the vacancy. A systematic investigation of the possible out-of-plane
displacement of this third atom was then carried out, in order to ascertain its
effects on the magnetic features of the system. The ground state was definitely
found to be magnetic and planar, with spin-resolved $\sigma$ and $\pi$ bands
contributing to the total magnetic moment. However, we have also found that
metastable solutions can be achieved if an initial shift of the third atom
above a minimum threshold from the graphene plane is provided, which leads to a
non-planar geometry and a non-magnetic state.
