András Salamon's answer to a similar question on cs.stackexchange.com 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
• Christos H. Papadimitriou, On the Complexity of Unique Solutions, JACM 31:2, 1984. doi:10.1145/62.322435

András Salamon's answer to a similar question on cs.stackexchange.com 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
• Christos H. Papadimitriou, On the Complexity of Unique Solutions, JACM 31:2, 1984. doi:10.1145/62.322435

The András Salamon's answer to a similar question on cs.stackexchange.com 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
• Christos H. Papadimitriou, On the Complexity of Unique Solutions, JACM 31:2, 1984. doi:10.1145/62.322435

András Salamon's answer to a similar question on cs.stackexchange.com 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
• Christos H. Papadimitriou, On the Complexity of Unique Solutions, JACM 31:2, 1984. doi:10.1145/62.322435