I assume you're talking about the relationship between Emily and her father. Her father overprotected Emily, not even allowing young men to call on her and court her (which was the custom then). He kept her at home, not really allowing her to socialize with anyone. He totally controlled her life, saying that no man was good enough for her. Her father was selfish in depriving Emily of her own life, and when he died, Emily didn't know what to do without him. Emily wouldn't allow anyone to take her father's body for three days, showing how much she depended on her father. Her father's selfishness and tyranny affected how the rest of her life went. She had made no friends since her father had been her only friend. When he died, she had no one. This also explains how desperate Emily was to have someone to spend her life with, so desperate that she killed Homer Barron and kept him in her attic for many years. Even though his body had become a skeleton, she still lay on the bed next to him until she herself died. Her father denied Emily the life she deserved, and his treatment of her ruined the remainder of her life.

I'm unclear about your question.  If you are asking what is their relationship like, it would be one that reveals the power relationship between the Southern "lady: and "gentleman."He, as we see him in the tableau, represents the paternalistic, authoritative southern male, dominating his daughter, Emily, who, in the background, appears submissive.  He holds a whip and wears boots, she wears white. Much of Emily can be explained through this original relationship with her father, which she does her best to break out of through Homer, whom her father would have hated, and then by killing Homer when he seemed to want to leave her.