According to the prophecy, Oedipus will kill his father. Knowing this, he runs away from home to keep the prophecy from coming true. Nevertheless, the prophecy does come true. Interestingly, Oedipus appears to take responsibility for what he has done. To some moral sensibilities, he is not responsible for killing his father and should not blame himself. To other moral sensibilities, when he takes the blame on himself Oedipus has properly understood what responsibility involves.
In your view, is Oedipus responsible for killing his father? Obviously, a proper answer will talk about the question of what Oedipus has intended. But do not simply say that Oedipus is not responsible because he did not intend to kill his father. If that is what you believe, explain why intention is such a fundamental category that it gets Oedipus off the moral hook. And keep in mind that Oedipus certainly intended to kill someone. In some sense, he acted intentionally during that fight at the crossroads. On the other hand, if you believe that Oedipus is responsible for killing his father, you will have to say something about the fact that he did not know that he was doing so. He did intend to do something; however, he did not intend to kill my father.