You may have the ability to terminate his parental rights in juvenile court. That would not absolve him from his responsibility to pay child support. Filing for termination is very fact intensive so I would need some additional information from you before I could advise you to proceed with that route. Your alternative is
Immediately filing for sole custody with supervised parenting time (as the court will address parenting time despite you not raising it unfortunately). We would want additional measures imposed by the court to protect your Daughter. If there has never been a child support order issued we would want one issued and the court can go back up to three years for purposes of child support. If you would like more information regarding our firm you may go to our website at aslamylaw.com.
The first question I would ask in your case is whether or not you were ever married to the father or not. If not, and he has not established paternity (through a court proceeding), he has no rights, including custody rights. However, if you were married (and divorced), or he previously established paternity through a court proceeding, then the process would require you to file a petition to modify custody. The court will determine custody based on what is in the best interest of the child. In determining that, the court will look at a number of factors. The fact that your child's father is in incarcerated will likely weigh in your favor, depending on why and how long he is there. If there has been domestic abuse or child abuse... that would be a signficant factor. If you have been the primarly care giver/provider and father has never really been in the picture, then you would likely stand a good chance of obtaining full custody. Child custody issues can be complicated. I strongly suggest you contact myself or another of the outstanding family law lawyers in Arizona who post on this board.
The answer above is based on the minimal facts provided in the question. The answer could change if additional facts become known. Another attorney may have a different view of the question. Consulting with an experienced Arizona family law attorney is strongly suggested. The response given to your question does not create an attorney client relationship.