If the father does not sign the birth certificate but the baby keep his last name does he still have the responsibility to pay child support?

I believe as long as the baby has his last name he should have to but even if not you can order him to get a DNA test and prove it is his if that's the case.... But if there is no name on the certificate then i don't think anything can be done....also i don't thing you can give the baby his name if he is not on there