uyumine is a father figure, or guardian per say, to Naoto. Their relationship, at first, is described on a love-hate basis. When putting the both of them side by side, one could wonder if Naoto’s own personality has been derived from him.
The two of them meet after Naoto’s parents had been killed. As she is lying under their bodies, she even admits to the readers that she knew she was going to die soon. Time passes, although she’s unsure of how much, when Fuyumine appears.
Naoto’s eyes spot the blade that he’s carrying and immediately assumes that he is the murderer. Just before, she had caught a glimpse of the blade of the killer. She is shocked when she realizes that he is also carrying the same blade as that person.
Fuyumine takes Naoto under his wing and gives her name to her. Naoto immediately despises it and everything about Fuyumine. Despite this, however, Fuyumine trains her in the art of the sword and defense. Years pass (perhaps around ten or so) and their relationship still remains the same.
It is only when Magato attempts to harm Naoto that their relationship shifts a little. As Fuyumine defends her, Naoto sees the expression on his face at the opportune moment – completed with broken shards of glass on the ground.
"That cracked, warped mirror showed me
an expressiveness that he never lets out normally."
Days past after this event, Naoto returns home to find him dead. She also finds Magato standing near his body, making it obvious that he was the one who killed him. The two fight one another, leaving Naoto as the winner. After receiving some important information from him, she sits by the dead body and begins to talk out loud to him as if he’s still there. At last she shows tears of sadness and mourning, finally evolving their relationship as one of daughter and father.
A further implication of their relationship is also seen during the second volume of Dogs, Bullets & Carnage. Shortly after the fight with Noki and Luki, Naoto is in a state of depression. She calls out to someone with questions that are unanswered but, ultimately, she ends it with the term, "father," most likely referring to Fuyumine at this point.
"Who is it... Who is it...
Who is it that I need to cut?
Who should I cut...?
Who should I hate...?
Tell me
... Father."
In Dogs: Bullets & Carnage, Naoto adds Fuyumine to her name, acknowledging her mentor and guardian.