Both stories depict a figure in isolation who, through the absence of social context, turns into something monstrous.
Both stories invoke the question of necrophilia although there is an inversion - in one story (Emily) the protagonist kills her lover and continues to sleep next to his corpse as though he were alive, in the other Roderick Usher buries his sister, pretending she is dead, when he knows she is in fact alive and only very sick. Both of these weird situations are motivated, I would guess, by a kind of misguided desire for containment and control.
Another point of comparison is the houses - both houses resemble their masters and are portrayed as radically out-of-context and horrific.
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Both "A Rose for Emily" and "The Fall of the House of Usher" explore themes of decay, isolation, and the psychological breakdown of characters. Both stories are gothic in nature, featuring elements of Horror and mystery. Additionally, they both examine the impact of the past on the present and the ways in which familial histories shape the lives of the characters.