  We calculate the diffusion barrier of fluorine adatoms on doped graphene in
the diluted limit using Density Functional Theory. We found that the barrier
$\Delta$ strongly depends on the magnitude and character of the graphene's
doping ($\delta n$): it increases for hole doping ($\delta n<0$) and decreases
for electron doping ($\delta n>0$). Near the neutrality point the functional
dependence can be approximately by $\Delta=\Delta_0-\alpha\, \delta n$ where
$\alpha\simeq6\times10^{-12}$ meVcm$^2$. This effect leads to significant
changes of the diffusion constant with doping even at room temperature and
could also affect the low temperature diffusion dynamics due to the presence of
substrate induced charge puddles. In addition, this might open up the
possibility to engineer the F dynamics on graphene by using local gates.
