[András Salamon's][1] answer to a [similar question on cs.stackexchange.com][2] named some additional $P^{NP}$-complete problems.

Krentel also gave another problem besides the one mentioned by Ryan O'Donnell:
> _Input:_ Weighted graph $G$, integer $k$.  
> _Question:_ Is the length of the shortest TSP tour in $G$ divisible by $k$?

Before him, Papadimitriou had already found:
> _Input:_ An instance of the TSP (that is, an $n \times n$ symmetric distance matrix of nonnegative integers)
> _Question:_ Is there a unique shortest TSP tour?

According to Krentel, this was the only known $P^{NP}$-complete problem before his work.

* Mark W. Krentel, _The Complexity of Optimization Problems_, JCSS __36__ 490–509, 1988. doi:[10.1016/0022-0000(88)90039-6][3]
* Christos H. Papadimitriou, _On the Complexity of Unique Solutions_, JACM __31:2__, 1984. doi:[10.1145/62.322435][4]


  [1]: https://cs.stackexchange.com/users/5323/andr%C3%A1s-salamon
  [2]: https://cs.stackexchange.com/questions/14251/which-problems-are-hard-for-pnp
  [3]: http://dx.doi.org/10.1016/0022-0000%2888%2990039-6
  [4]: http://dx.doi.org/10.1145/62.322435