TITLE: Characterize P^NP (a.k.a. Delta_2^p)
QUESTION [11 upvotes]: What can you say about the complexity class $\text{P}^{\text{NP}}$, i.e. decision problems solvable by a polytime TM with an oracle for SAT? This class is also known as $\Delta_2^p$.
Obviously $\text{P}^{\text{NP}}$ is in $\text{PH}$ somewhere between $\text{NP} \cup \text{coNP}$, and $\Sigma_2^p \cap \Pi_2^p$. What else is known about that complexity class?

REPLY [2 votes]: 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