Graph problem known to be $NP$-complete only under Cook reduction 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.
I know an example of a problem that is missing two of the four features you ask for - it is not NP-complete, and it is not a problem on graphs.

Buchfuhrer and Umans (2011) show that the minimum equivalent expression problem in Boolean logic is complete for $\Sigma^P_2$ under polynomial-time Turing reductions.

Given a Boolean $(\wedge;\vee;\neg)$-formula $F$ and
an integer $k$, is there an equivalent $(\wedge;\vee;\neg)$-formula of size at most $k$?

On p. 143, the authors state:


  This provides a somewhat rare example of a natural problem for which a Turing reduction seems crucial (in the sense that we do not know of any simple modification or alternative methods that would give a many-one
  reduction).


They do not cite any other such examples from the literature. I suspect that "somewhat rare" is possibly an understatement.

References

David Buchfuhrer and Christopher Umans: "The complexity of Boolean formula minimization" Journal of Computer and System Sciences, Volume 77, Issue 1, January 2011, Pages 142-153