The parent where the child lives more than half the nights of the year retains the right to use that child to qualify for head of household, the earned income credit, and the child and dependent care credit. Those benefits always stay with the custodial parent and can’t be waived, transferred, or released. This is spelled out in several places, including the dependent section of the form 1040 instructions, IRS publication 501 on claiming dependents, and IRS publication 504, information for divorced and separated individuals.
In IRS publication 501, look for the section marked “special rules for children of divorced or separated parents.“
https://www.irs.gov/pub/irs-pdf/p501.pdf
