code is employed [@oas96; @sagg02] in our spin-polarized DFT-based [*ab-initio*]{} calculations, where the standard double-$\xi$ basis with polarization orbitals ($DZP$) is used. The generalized-gradient approximation (GGA) is utilized to calculate the exchange correlation term [@pbe96]. The interaction between the valence electrons and the atomic core is taken into account by using standard norm conserving Troullier-Martins pseudopotential. The unit cell of a pristine mono-layer graphene is initially relaxed and then three different supercells of $50$, $98$, and $162$ atoms are constructed. The real space integration grid had $200~Ry$ cutoff and $50~meV$ energy shift. Spin resolved calculations are performed in most cases. $k$-point sampling of the Brillouin zone was performed by using the Monkhorst-Pack method [@mp76]. In order to get an optimal self-consistency convergence, the density of electronic states (DOS) was calculated for different $k$-point meshing.

  ----------------------------------- ------------- --------------- ---------------
         **Formation energy**             **1**          **2**           **3**
               **(eV)**                **vacancy**   **vacancies**   **vacancies**
   \[0.5ex\] Supercell with 50 atoms      7.03           5.78            8.50
        Supercell with 98 atoms           6.99           5.76            7.74
       Supercell with 162 atoms           6.78           5.21            7.52
                \[1ex\]                                             
  ----------------------------------- ------------- --------------- ---------------

  : Formation energies of mono-, di- and tri-vacancy for the supercells with 50, 98, and 162 atoms.

\[table1\]

Our calculations show that $5 \times 5 \times 