<<I thought it was in their divorce decree that it alternated years.>>
If that's true, it's up to the divorce court to enforce the decree. The IRS can't and doesn't enforce local court orders. But that doesn't mean the parents can ignore the court order. To comply with such a court order, the custodial (by IRS definition) parent would have to voluntarily relinquish their right to claim the child in each year that was not "their" year to claim.
A custodial parent relinquishes their right to claim the child by providing the non-custodial parent with a signed Form 8332, which the non-custodial parent must attach to their tax return. Here's a link to Form 8332: https://www.irs.gov/pub/irs-pdf/f8332.pdf
**Answers are correct to the best of my ability but do not constitute tax or legal advice.
