How does Flannery O'Connor's story "A Good Man Is Hard To Find" incorporate "the killing of the father" theme?
The "killing of the father" theme usually appears in a work that includes a conflict between father and son that results in the son's killing of the father because the son secretly desires to replace his father in his mother's eyes--a theme derived from a Greek play by Sophocles called Oedipus Rex, in which the son unknowingly kills his father and marries his mother. The desire for a son to be with his mother was later described by the psychologist Carl Jung as the "Oedipus Complex."
In "A Good Man is Hard to Find," we have two deaths of fathers--the first being the father of the family who is slaughtered by The Misfit and his two companions, and the second being The Misfit's odd reference to the killing of his own father. In both cases, however, the "killing of the father" theme does not seem central to the plot of the story, but in the context of The Misfit's violent history, the Misfit's father's death is significant.
For example, when The Misfit describes how he got into prison, he says,
It was a head-doctor at the penitentiary said what what I had done was kill my daddy but I known that for a lie. My daddy died . . . of the epidemic flu and I never had anything to do with it.
If we assume that The Misfit is describing his history accurately, we must also assume that he exhibits a profound psychosis that allows him to block out his killing of his father. There is, of course, no mention of The Misfit's mother, and trying to fit an Oedipus complex into this story is not useful.
The fact, however, that The Misfit appears to have killed his father--whether he acknowledges that fact or not--is consistent with The Misfit's twisted view of the world around him, a world in which there isn't any moral difference between "killing somebody or burning down his house or doing some other meanness to him."
The "killing of the father" theme, then, does not have a central role in the story, but The Misfit's inability to acknowledge that he killed his father is O'Connor's way of confirming The Misfit's complete inability to distinguish between what is morally right or wrong.