  Flat-bands play a central role in the presence of correlated phases in
Moir\'e and other modulated two dimensional systems. In this work, flat-bands
are shown to exist in uniaxially periodic strained graphene. Such strain should
be produced for example by a substrate. The model is thus mapped into a
one-dimensional effective Hamiltonian and this allows to find the conditions
for having flat-bands, i.e., a long-wavelength modulation only on each one of
the bipartite graphene sublattices, while having a tagged strain field between
neighboring carbon atoms. The origin of such flat-bands is thus tracked down to
the existence of topological localized wavefunctions at domain walls separating
different regions, each with a non-uniform Su-Schriffer-Hegger model (SSH) type
of coupling. Thereafter, the system is mapped into a continuum model allowing
to explain the numerical results in terms of the Jackiw-Rebbi model and of
pseudo-Landau levels. Finally, the interplay between the obtained flat-bands
and electron-electron interaction is explored through the Hubbard model. The
numerical results within the mean-field approximation indicate that the
flat-bands induce N\'eel antiferromagnetic and ferromagnetic domains even for a
very weak Hubbard interaction. The present model thus provides a simple
platform to understand the physical origin of flat-bands, pseudo-Landau levels
and the effects of the electron-electron interaction.
