An understanding of Antigone's lineage is crucial to decoding the significance of the various characters' ultimate fates. Let's examine the major characters in the family tree adjacent to this page.
Oedipus is a descendent of the Labdacus family, which is plagued with a terrible curse. Oedipus kills his father Laius inadvertently, not realizing who he is, and then proceeds to marry his mother, Jocasta, also not realizing her true identity. (For more on how this came to pass, see the summary of the first section of Antigone). As a result of Oedipus' marriage to Jocasta, he sires four children, who are at once his siblings and his children: Eteocles, Polyneices, Ismene, and Antigone.
Oedipus, shamed by his marriage and murder, surrenders the kingdom to his brother Creon (since Creon is Jocasta's brother, he is also Oedipus' brother). Creon takes over the kingdom because it is feared that Eteocles and Polyneices are also cursed by the Labdacus plague and will continue bringing misery to Thebes. Eventually, however, Polyneices makes a claim on the Theban crown, causing him to be banished. At this point, Polyneices raises an army, returns to claim Thebes, and ends up dying at the hands of Eteocles, who dies in the fray as well. Creon remains in control of Thebes.
Of this line, only Ismene and Antigone remain living at the start of the play. Antigone is supposed to marry her cousin Haemon, but by the end of the play, in a revelation that demonstrates just how widespread the Labdacus curse is - Haemon dies, Eurydice dies, and Antigone dies, leaving only Ismene and Creon as the de facto descendants of Labdacus.