The court would determine this through an evidentiary hearing. It would not be necessary to have criminal convictions. The burden of proof on this would be the same as in a civil proceeding.
"There is created a presumption that no parent who has a history of perpetrating family violence shall be awarded sole or joint custody of children. The court may find a history of perpetrating family violence if the court finds that one incident of family violence has resulted in serious bodily injury or the court finds more than one incident of family violence. The presumption shall be overcome only by a preponderance of the evidence that the perpetrating parent has successfully completed a treatment program as defined in R.S. 9:362, is not abusing alcohol and the illegal use of drugs [FN1] scheduled in R.S. 40:964, and that the best interest of the child or children requires that parent's participation as a custodial parent because of the other parent's absence, mental illness, or substance abuse, or such other circumstances which affect the best interest of the child or children. The fact that the abused parent suffers from the effects of the abuse shall not be grounds for denying that parent custody."