  I describe extended gradings of open topological field theories in two
dimensions in terms of skew categories, proving a result which alows one to
translate between the formalism of graded open 2d TFTs and equivariant cyclic
categories. As an application of this formalism, I describe the open 2d TFT of
graded D-branes in Landau-Ginzburg models in terms of an equivariant cyclic
structure on the triangulated category of `graded matrix factorizations'
introduced by Orlov. This leads to a specific conjecture for the Serre functor
on the latter, which generalizes results known from the minimal and Calabi-Yau
cases. I also give a description of the open 2d TFT of such models which
manifestly displays the full grading induced by the vector-axial R-symmetry
group.
