The Director of "The Graduate" stated that he wanted an underdog quality to the role of Benjamin Braddock, which is why he did not cast Robert Redford. What is the name of the director?

Mike Nichols.

Explanation:

While casting the characters for the film "The Graduate", the director Mike Nichols cast Dustin Hoffman for the role of Benjamin Braddock though Robert Redford and Warren Beatty were priorities for the Braddock's role.

When Hoffman gave audition for Braddock's role, Nichols selected him for the role instead of Redford and Beatty because according to Nichols, the director of the film, Redford did not have the quality of an underdog that was required for the role of Braddock.

So, the director who rejected Robert Redford for Braddock's character was Mike Nichols.