The name of the King of Thebes is Laius. According to Jocasta, Laius was killed in a three-way crossroads. Ironically, Oedipus was also on his way to Thebes when he encountered a group of travelers. These travelers were less than pleasant.
They harassed Oedipus and taunted him. They probably did not expect they were going to get killed. Oedipus was able to kill all of them in self-defense. Oedipus is not aware that Jocasta is his mother, so when he cursed his two sons, they were also his two brothers, but they were sworn to kill each other.