  We propose a theory of the thermal Casimir interaction for multilayered test
bodies coated with a graphene sheet. The reflection coefficients on such
structures are expressed in terms of the components of the polarization tensor
and the dielectric permittivities of material layers. The developed theory is
applied to calculate the gradient of the Casimir force between an Au-coated
sphere and a graphene sheet deposited on a SiO${}_2$ film covering a Si plate,
which is the configuration of a recent experiment performed by means of a
dynamic atomic force microscope. The theoretical results are found to be in
very good agreement with the experimental data. We thus confirm that graphene
influences the Casimir interaction and can be used for tailoring the force
magnitude in nanostructures.
