  In this paper we introduce flat grafting as a deformation of quadratic
differentials on a surface of finite type that is analogous to the grafting map
on hyperbolic surfaces. Flat grafting maps are generic in the strata structure
and preserve parallel measured foliations. We use flat grafting to construct
paths connecting any pair of quadratic differentials. In other words, we
characterize cone point splitting deformations. The slices of quadratic
differentials closed under flat grafting maps with a fixed direction arise
naturally and we prove rigidity properties with respect to the lengths of
closed curves.
