@Opus 17 said "If your daughter files a tax return, she has the ABSOLUTE first right to claim her children as dependents and she can’t be claimed as a dependent by you in that situation".
From a practical standpoint, that's correct. Technically it's not correct. If your daughter qualifies as your "Qualifying Child" (QC) dependent (and she probably does: full time student, did not provide more than half her own support, lived with you more than half the year, under age 24 on 12-31-21), she may NOT claim herself or any dependents. Your can claim her and her kids. It may come down to whether her online school classifies her as a full time student. If she is not your QC, she might qualify as your qualifying relative dependent, if her income was less than $4300.
If competing returns are filed, there will be a mess*.
As others have said, the father cannot legally claim the children** (they are the qualifying children of another taxpayer) and the parents cannot file a joint return to claim the children (they are not legally married)
*If someone else claimed your dependents inappropriately, and if they file first, your return will be rejected if e-filed. You would then need to file a return on paper, claiming the dependents as appropriate. The IRS will process your return and send you your refund, in the normal time. Shortly (up to a year) thereafter, you'll receive a letter from the IRS, stating that your dependents was claimed on another return. It will tell you that if you made a mistake to file an amended return and if you didn't make a mistake to do nothing. The other party will get the same letter you did. If one of you doesn't file an amended return, unclaiming the child, the next letter, from the IRS, will require you to provide proof. Be sure to reply in a timely manner.
Winner gets the tax benefits; loser gets to pay the IRS back with penalties and interest.
References:
https://www.thebalance.com/claiming-same-dependent-audit-risk-3193030 (audit discussion)
http://taxes.about.com/od/dependents/qt/Dependents-Audits.htm
www.eitc.irs.gov/EITCCentral/f886-h-dep.pdf (full dependent discussion including audit section)
https://turbotax.intuit.com/tax-tips/family/what-happens-when-both-parents-claim-a-child-on-a-tax-re...
**If your daughter does not qualify as your dependent, she, as custodial parent, could release the children to the father. He could claim the child tax credit, but, he would not be able to claim the Earned income credit or Head of Household filing status.
