  For an appropriate class of Fano complete intersections in toric varieties,
we prove that there is a concrete relationship between degenerations to
specific toric subvarieties and expressions for Givental's Landau-Ginzburg
models as Laurent polynomials. As a result, we show that Fano varieties
presented as complete intersections in partial flag manifolds admit
degenerations to Gorenstein toric weak Fano varieties, and their Givental
Landau-Ginzburg models can be expressed as corresponding Laurent polynomials.
  We also use this to show that all of the Laurent polynomials obtained by
Coates, Kasprzyk and Prince by the so called Przyjalkowski method correspond to
toric degenerations of the corresponding Fano variety. We discuss applications
to geometric transitions of Calabi-Yau varieties.
