Non-forking and preservation of NIP and dp-rank

We investigate the question of whether the restriction of an NIP type p∈S(B) which does not fork over A⊆B to A is also NIP, and the analogous question for dp-rank. We show that if B contains a Morley sequence I generated by p over A, then p↾AI is NIP and similarly preserves the dp-rank. This yields positive answers for generically stable NIP types and the analogous case of stable types. With similar techniques we also provide a new more direct proof for the latter. Moreover, we introduce a general construction of “trees whose open cones are models of some theory” and in particular an inp-minimal theory DTR of dense trees with random graphs on open cones, which exemplifies a negative answer to the question.