What themes do King Lear and Oedipus Rex have in common with each other?
One thematic connection between these two plays relates to the importance of social and political roles. Both Lear and Oedipus "serve" their own doom by upsetting the natural order that governs certain role-based relationships.
Lear denies his daughter and abdicates his power as king. These acts go against the "natural order".
Lear earns his tragedy through a disruption of both the political and the natural order of things.
Oedipus, similarly, breaches the natural code by marrying his mother, killing his father, and taking his father's position as king. Then, as king, Oedipus insists on using his power to coerce the truth out of his subjects, thus bringing about his terrible realization of what he has done.
When Teiresias refuses to answer Oedipus's call and later resists revealing the king's dark truth, Oedipus grows impatient, hostile, and abusive.
The importance of what is understood to be the "natural order" of social and political life cannot be overstated in these plays. The tragedy of each play is a direct result of a corruption or perversion of the culturally understood natural order.
Additional themes connecting these plays: revelation/realization; insulation from the truth; man's powerlessness against nature and/or fate.