If the child spent over 6 months in your home then you are the custodial parent as far as the IRS is concerned.
Since another tax return was filed with the child's Social Security number also claiming their personal exemption, then the IRS is now investigating who reported the personal exemption correctly and who did not. The loser of the investigation is required to pay back any tax refund plus penalties and interest.
Respond to what the IRS is asking for in the notice you received. If you have any questions call the number on the notice and speak with an IRS agent.
