How can one honor a father who banishes his daughter and refuses to communicate with her?

You must honor your father and draw near to him. You should seek the reason for his anger and apologize to him, even if you believe he is mistaken. You are not permitted to abandon him or treat him poorly, for honoring one's father and accompanying him with kindness are among the most emphasized duties. Allah Almighty says: "And We have enjoined upon man [care] for his parents. His mother carried him with hardship upon hardship, and his weaning is in two years. Be grateful to Me and to your parents; to Me is the [final] destination. But if they endeavor to make you associate with Me that of which you have no knowledge, do not obey them but accompany them in [this] world with appropriate kindness" (Luqman: 14-15). And He says: "And your Lord has decreed that you worship not except Him, and to parents, good treatment. Whether one or both of them reach old age [while] with you, say not to them [even] 'uff,' and do not repel them but speak to them a noble word. And lower to them the wing of humility out of mercy and say, 'My Lord, have mercy upon them as they nurtured me when I was small'" (Al-Isra: 23-24). And in the Hadith: "May his nose be rubbed in the dust, then may his nose be rubbed in the dust, then may his nose be rubbed in the dust." It was asked, "Who, O Messenger of Allah?" He said, "He who reaches old age with his parents, one or both of them, yet does not enter Paradise." Therefore, you must take the initiative to connect with your father and show him love and obedience, so that you may attain the pleasure of Allah and His Paradise.