Child custody and child support laws do vary from state to state, so please confirm the details of the law in your state by contacting a local court or attorney. One thing that is fairly consistent from state to state: a parent's financial responsibility for a child is separate from the right to visitation or custody and separate from the issue of the child's name. A non-custodial parent is not excused from the responsibility to support the child. A child's name is assigned at birth on the birth certificate. If the father is not present the mother can give the last name of the father on her own (or another) last name. The name has no bearing on either child support or custody. Simplest way to detemine obligation is, if there is a valid court order of support the non custodial parent is obligated to adhere to the terms of the support regardless of the circumstances involved. The non custodial parent has the option of filing a petition to have the order rescinded or amended, and requesting a paternity test if the parentage of the child is in question.