  We derive constraints on parameters of generic dark matter candidates by
comparing theoretical predictions with the gamma-ray emission observed by the
Fermi-LAT from the region around the Galactic Center. Our analysis is
conservative since it simply requires that the expected dark matter signal does
not exceed the observed emission. The constraints obtained in the likely case
that the collapse of baryons to the Galactic Center is accompanied by the
contraction of the dark matter are strong. In particular, we find that for bb
and \tau+\tau- or W+W- dark matter annihilation channels, the upper limits on
the annihilation cross section imply that the thermal cross section is excluded
for a Weakly Interacting Massive Particle (WIMP) mass smaller than about 700
and 500 GeV, respectively. For the \mu+ \mu- channel, where the effect of the
inverse Compton scattering is important, depending on models of the Galactic
magnetic field the exclusion of the thermal cross-section is for a WIMP mass
smaller than about 150 to 400 GeV. The upper limits on the annihilation cross
section of dark matter particles obtained are two orders of magnitude stronger
than without contraction. In the latter case our results are compatible with
the upper limits from the Galactic halo analysis reported by the Fermi-LAT
collaboration for the case in which the same conservative approach without
modeling of the astrophysical background is employed.
