1$ $k$-point mesh is a good and optimal choice to have a balanced accuracy and computation time. To obtain the equilibrium geometry we relaxed all the atoms after creation each vacancy until the forces acting on them were smaller than $0.01~eV/\AA$. The formation energy for mono-, di- and tri-vacancy are presented in Table \[table1\] for the supercells with 50, 98, and 162 carbon atoms. The formation energy of the vacancies changes slightly with the supercell size (see, Table \[table1\]) as a result of interaction of the defects with their images in the neighboring auxiliary cells (we used periodic boundary conditions in all space directions). Our calculations of the formation energy, DOS, and the magnetization show that a good convergence can be reached for a supercell with at least 98 atoms. Therefore, our results presented below were done for a supercell with 98 atoms.

Distribution of the vacancies and their magnetizaton
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Our aim in this investigation is to understand a development of the magnetization with increasing the vacancy concentration and a restructuring tendency due to the vacancies migration in the graphene mono-layer. We calculated for this purpose the formation energy, the band structure, and the DOS of a supercell of 