Though they appear briefly or not at all in Oedipus the King, Oedipus' father figures play very important roles in Oedipus' life. Laius, Polybus, and the shepherds may be considered Oedipus' father figures. Each of these characters could have kept Oedipus from facing his horrible fate by doing something a little bit differently.
Though mentioned several times throughout the play, but being absent from the stage, Laius, Oedipus' biological father, started things off on a bad note with his decision to leave it to another person to kill Oedipus. If Laius were to kill Oedipus himself, nothing would have happened. Another option for Laius would have been to imprison Oedipus and throw away the key, thereby keeping Oedipus alive and knowledgeable who his father is but harmless to do anything. Either way things wouldn't have been too enjoyable for Oedipus, but surely better than what eventually happened.
Appearing for a short and influential part in the play but mentioned little otherwise, the shepherd did not make things any better for Oedipus and his future. If the shepherd had done what he was commanded to do, kill Oedipus, everything would have turned out for the better for all parties involved. Instead of killing Oedipus, the shepherd felt sorry for him and gave him to a Corinthian shepherd from Corinth who, not knowing what the baby's fate was, gave him to King Polybus. This would later turn against Oedipus because of his ignorance of who his real parents are.
Briefly mentioned and never seen, the most influential of Oedipus' father figures, Polybus could be seen as guilty or innocent when it comes to his part in dealing with Oedipus' fate. On the guilty side, Polybus lied to Oedipus telling him that he was his real father, causing Oedipus to run away and eventually run into his fate. At this time Oedipus was old enough to deal with the truth. On the innocent side, you could say that Polybus was totally ignorant and had no idea of why Oedipus was asking him this question.