We show that collisions with stellar--mass black holes can partially explain the absence of bright giant stars in the Galactic Centre, first noted by Genzel et al, 1996. We show that the missing objects are low--mass giants and AGB stars in the range 1-3 M$_{odot}$. Using detailed stellar evolution calculations, we find that to prevent these objects from evolving to become visible in the depleted K bands, we require that they suffer collisions on the red giant branch, and we calculate the fractional envelope mass losses required. Using a combination of Smoothed Particle Hydrodynamic calculations, restricted three--body analysis and Monte Carlo simulations, we compute the expected collision rates between giants and black holes, and between giants and main--sequence stars in the Galactic Centre. We show that collisions can plausibly explain the missing giants in the $10.5<K<12$ band. However, depleting the brighter ($K<10.5$) objects out to the required radius would require a large population of black hole impactors which would in turn deplete the $10.5<K<12$ giants in a region much larger than is observed. We conclude that collisions with stellar--mass black holes cannot account for the depletion of the very brightest giants, and we use our results to place limits on the population of stellar--mass black holes in the Galactic Centre.
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