Eiji's interactions with the yakuza significantly influence his quest for his father in David Mitchell's novel "number9dream." Throughout the story, Eiji Miyake, the protagonist, is on a journey to find his estranged father in Tokyo. His encounters with the yakuza add layers of complexity and danger to his quest.

1. Obstacles and Challenges: The yakuza represent a formidable obstacle in Eiji's search. Their involvement in his father's life and the criminal underworld adds a layer of danger and unpredictability. Eiji's interactions with them often lead to life-threatening situations, forcing him to navigate a world of crime and violence.

2. Character Development: These interactions serve as a catalyst for Eiji's growth. Facing the yakuza's threats and manipulations, Eiji is pushed to confront his fears and develop resilience. This helps him mature and gain a deeper understanding of the complexities of adult life.

3. Moral Ambiguity: The yakuza's presence introduces moral ambiguity into Eiji's quest. He must decide how far he is willing to go and what lines he is willing to cross to achieve his goal. This internal conflict adds depth to his character and the narrative.

4. Plot Progression: The yakuza's involvement is crucial for the plot's progression. Their actions and the information they provide (or withhold) drive Eiji's journey forward, leading to key revelations about his father and himself.

Overall, Eiji's interactions with the yakuza are pivotal in shaping his quest, influencing both the external challenges he faces and his internal journey of self-discovery.