This paper is devoted to the interpretation of scanning tunneling microscopy (STM) images of a single atomic vacancy on single and double graphene sheets as a model for the (0001) surface of graphite. We first selected a one-layer model which would allow us to run periodic density-functional theory calculations without destroying the charge density waves that form in the vicinity of the vacancy. We assigned the main features of STM images [bright spots in the vicinity of the defect, $(\sqrt{3}\ifmmode\times\else\texttimes\fi{}\sqrt{3})R30\ifmmode^\circ\else\textdegree\fi{}$ modulation of the local electronic density near the Fermi level, third-order symmetry structure] to the electronic band structure of the defective graphite surface. We further analyzed a more extended crystal working cell model to ensure convergence toward the isolated atomic vacancy. The interlayer interaction plays a crucial role in the interpretation of STM images. A double-layer model was subsequently considered and the impact of the interlayer interaction analyzed. We produce the local density of state for the $\ensuremath{\alpha}$ and $\ensuremath{\beta}$ vacancies which may be differentiated. Our calculations reproduce the main features of STM images and the results we get are in good agreement with experimental observations.