What is the role of the minor characters in Oedipus Rex by Sophocles?
This is a great question. Often only the main characters get attention and the minor ones get left out. This always impoverishes the work in view, espeically a tragedy like Oedipus Rex.
There are two minor characters worth noting. First, there is Laius. In fact, he is never mentioned in the play, but he is in the background in a powerful way. Laius is the biological father of Oedipus. He is the one who knew of the prophecy, the one who sent his son to be exposed, and the one who was killed by Oedipus. For this reason, we can say that Laius' murder started the chain of events that led to Oedipus' predicament.
Second, the shepherd is a minor character, but he is also very important, because he was the one who did not follow Laius' order to expose the baby, Oedipus. He gave the baby to another shepherd who then gave the baby to Polybus. This act of sparing a life unwittingly made the prophecy true.