  We study the emergence of electronic non-trivial topological flat bands in
time-periodically driven strained graphene within a tight binding approach
based on the Floquet formalism. In particular, we focus on uniaxial spatially
periodic strain since it can be mapped onto an effective one-dimensional
system. Also, two kinds of time-periodic driving are considered: a short pulse
(delta kicking) and a sinusoidal variation (harmonic driving). We prove that
for special strain wavelengths, the system is described by a two level Dirac
Hamiltonian. Even though the study case is gapless, we find that topologically
non-trivial flat bands emerge not only at zero-quasienergy but also at $\pm\pi$
quasienergy, the latter being a direct consequence of the periodicity of the
Floquet space. Both kind of flat bands are thus understood as dispersionless
bands joining two inequivalent touching band points with opposite Berry phase.
This is confirmed by explicit evaluation of the Berry phase in the touching
band points' neighborhood. Using that information, the topological phase
diagram of the system is built. Additionally, the experimental feasibility of
the model is discussed and two methods for the experimental realization of our
model are proposed.
