In the USA, though, unless there's a court order requiring permission from both parents, a child may travel (even to overseas) with either parent/guardian without showing proof of permission from the other parent.
It's important that you give your co-parent notice and preferably get their express permission in writing if you wish to travel outside the state with your child. If the other parent refuses to cooperate, you can ask for the court's permission in advance.
The letter should include the other person's contact details and details about the trip. It also helps if you've: evidence of your relationship with the child, for example a birth or adoption certificate. a divorce or marriage certificate, if you are a single parent but your family name is different from the child's.
Typically, minors under the age of 18 do not have to present identification for domestic U.S. travel. Airlines will accept identification from the responsible adult on behalf of the minor(s). For international travel, minors under the age of 18 must present the same travel documents as the adult.
You should obtain a child travel consent letter granting you permission from the absent parent to leave the country with your child (having it notarized is not required, but highly recommended).
There is no documentation requirement for domestic travel in the US for children flying with a parent. I've flown dozens of times with no one saying boo.
It is not mandatory that you have a signed letter from the other parent, as long as the border officials are satisfied that you have proper approval from that parent, or a court, to travel with the child.