We develop methods for computing graded K-theory of C*-algebras
as defined in terms of Kasparov theory. We establish graded versions
of Pimsner’s six-term exact sequences for graded Hilbert bimodules whose
left action is injective and by compacts, and a graded Pimsner–Voiculescu sequence.
We introduce the notion of a twisted P-graph C*-algebra and establish
connections with graded C*-algebras. Specifically, we show how a functor
from a P-graph into the group of order two determines a grading of the associated
C*-algebra. We apply our graded version of Pimsner’s exact sequence to
compute the graded K-theory of a graph C*-algebra carrying such a grading.