How can I stop someone from filing a dependent when they don't have rights to them

You can't stop someone else from claiming your dependent, but you can still claim a dependent that's rightfully yours.

If you were the first to e-file, their return will reject.  If someone else claimed your child when they shouldn't have, and if they file first, your return will be rejected.

The only way to claim your dependent then is to file a return on paper. You will not be able to e-file this return. The IRS will still process your return and send you your refund.

Usually within a few months, you'll receive a letter from the IRS, stating that your child was claimed on another return. If you made a mistake, you are asked 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, removing the child from their return, the IRS will require you both to provide proof of residency. There will be a list of possible ways you can prove this. The more you can provide, the better your chances.

The taxpayer who falsely claimed the dependent will be required to repay any additional taxes, plus penalties and interest, attributable to the false claim.

You should know that you are able to claim the dependent care credit, earned income credit and head of household filing status, even though you are not claiming them as a dependent. Unless the other parent also claimed these credits, your return may still be e-filed without rejection.