That depends entirely on where the children lived more than half the nights of the year.
Only the parent who has physical custody more than half the nights of the year has the automatic right to claim the children as dependents, including claiming the dependent care credit and head of household status. The IRS does not recognize “50-50” custody or court custody orders, they go by where the child physically spent the night. The other parent cannot claim the children at all, unless the custodial parent gives the non-custodial parent a signed form 8332 releasing the dependent claim. Even with a signed form 8332, head of household status can never be shared, waived, or transferred, since it depends only on where the child physically lived.
It is possible for separated parents with two children to arrange their custody arrangements so that one child spends 183 or more nights with one parent, and the other child spends more than 183 nights with the other parent. This would allow each parent to claim one child as a dependent without a signed release form and claim ahead of household status.
However, if neither child lived in your home more than 183 nights of the year, and your legal right to claim the child comes from a signed form 8332 release, then you cannot use that child to claim head of household status.
