The theory of NP-completeness was initially built on Cook (polynomial-time Turing) reductions. Later, Karp introduced polynomial-time many-to-one reductions. A Cook reduction is more powerful than a Karp reduction since there is no restriction on the number of calls to the oracle. So, I am interested in NP-complete graph problem that does not have a known Karp reduction from a NP-complete problem.
Is there a natural graph problem known to be $NP$-complete only under Cook reduction, but not known to be NP-complete under Karp reductions?
Naturalness should disallow specific features of feasible solutions, for otherwise it is quite easy to start from well-known problem and make it a little easier by allowing specific features.