We investigate the binding of the cobalt atom on small aromatic model systems as a proxy for interaction with graphene, using density functional theory, coupled-cluster theory, and combinations of them using projector-based quantum embedding. We set out in some detail the electronic structure of the cobalt atom alone, because some nuances of atomic structure appear to have been overlooked in previous studies. Two states of the complex in particular are studied: those formed from the a 4 F ground state of the atom; and from c 2 D, the lowest doublet state with configuration 3d 9 4s 0 . We highlight the difficulties in extracting reliable results from typical approximate density functionals, and demonstrate that embedding calculations using the coupled-cluster theory in an active subsystem greatly reduce functional dependence, and produce a picture more consistent with the available experimental information. Our results cast doubt on previous calculations that have predicted strong chemisorptive binding between graphene and the c 2 D state of cobalt.