"If a woman does not put the father on the birth certificate she has full custody unless the father files a paternity suit."
Actually in order to obtain custody, dad would need to file a custody suit, which would also involve establishing paternity, assuming mom and dad aren't married (or weren't near the time of conception).
"So it is best that if the mother does not want the father around and he does not know about the child and the name is not on the birth certificate, he can't file a paternity suit."
If dad somehow finds out about the pregnancy/child, or has registered on the state's putative father registry, dad may still retain his parently rights to the child (such as being able to seek visitation, custody, etc.). Simply omitting him from the birth certificate does not guarantee that there won't be a custody battle in the future.