  We studied a nilpotent Non-Anti-Commutative (NAC) deformation of the
effective superpotentials in supersymmetric gauge theories, caused by a
constant self-dual graviphoton background. We derived the simple
non-perturbative formula applicable to any NAC (star) deformed chiral
superpotential. It is remarkable that the deformed superpotential is always
`Lorentz'-invariant. As an application, we considered the NAC deformation of
the pure super-Yang-Mills theory whose IR physics is known to be described by
the Veneziano-Yankielowicz superpotential (in the undeformed case). The
unbroken gauge invariance of the deformed effective action gives rise to severe
restrictions on its form. We found a non-vanishing gluino condensate in vacuum
but no further dynamical supersymmetry breaking in the deformed theory.
