  We illustrate the possibility of realizing band gaps in graphene-like systems
that fall outside the existing classification of gapped Dirac Hamiltonians in
terms of masses. As our primary example we consider a band gap arising due to
time-dependent distortions of the honeycomb lattice. By means of an exact,
invertible, and transport-preserving mapping to a time-independent Hamiltonian,
we show that the system exhibits Chern-insulating phases with quantized Hall
conductivities $\pm e^2/h$. The chirality of the corresponding gapless edge
modes is controllable by both the frequency of the driving and the manner in
which sublattice symmetry is broken by the dynamical lattice modulations.
Finally, we discuss a promising possible realization of this physics in
photonic lattices.
