  In the context of the Holstein polaron problem it is shown that the dynamical
mean field theory (DMFT) corresponds to the summation of a special class of
local diagrams in the skeleton expansion of the self-energy. In the real space
representation, these local diagrams are characterized by the absence of vertex
corrections involving phonons at different lattice sites. Such corrections
vanish in the limit of infinite dimensions, for which the DMFT provides the
exact solution of the Holstein polaron problem. However, for finite dimensional
systems the accuracy of the DMFT is limited. In particular, it cannot describe
correctly the adiabatic spreading of the polaron over multiple lattice sites.
Arguments are given that the DMFT limitations on vertex corrections found for
the Holstein polaron problem persist for finite electron densities and
arbitrary phonon dispersion.
