I did not answer all of your question: you asked "can that parent get full tax credit"?
No. If the exemption is released to the non-custodial parent, the non-custodial parent can claim the child's exemption and Child Tax Credit only. The other credits, such as the Earned Income Credit (EIC) and Child Care Credit both require that you physically live with the child to claim, so only the custodial parent can claim those credits.
Refer to the Pub 17 link above for full information.
**Disclaimer: This post is for discussion purposes only and is NOT tax advice. The author takes no responsibility for the accuracy of any information in this post.**
