  We report on the fate of the quantum Hall effect in graphene under strong
laser illumination. By using Floquet theory combined with both a low energy
description and full tight-binding models, we clarify the selection rules, the
quasienergy band structure, as well as their connection with the two-terminal
and multi-terminal conductance in a device setup as relevant for experiments.
We show that the well-known dynamical gaps that appear in the Floquet spectrum
at $\pm\,\hbar\Omega/2$ lead to a switch-off of the quantum Hall edge transport
for different edge terminations except for the armchair one, where two terms
cancel out exactly. More interestingly, we show that near the Dirac point
changing the laser polarization (circular right or circular left) controls the
Hall conductance, by allowing to switch it on or off, or even by flipping its
sign, thereby reversing the chirality of the edge states. This might lead to
new avenues to fully control topologically protected transport.
