The father has to be given notice.He has to be given an opportunity to be heard on the issue. If he is deployed and thus unable to appear in court, he can have the case continued pursuant to the Servicemembers Civil Relief Act.
This answer is for general information purposes, is not legal advice and does not create an attorney-client relationship.
The mother cannot legally grant custody of the children to someone else without the father's consent.
This answer is provided for general information purposes only and does not constitute an attorney-client relationship. To receive advice tailored to your unique circumstances, you should meet with an experienced attorney.