What do you mean? Only the custodial parent who the child lived with more than 1/2 the year can claim the EIC. Both the EIC and Child Care Credit requires that the child lived with you.
The custodial parent for tax purposes is not the same as legal custody that a court might grant.
The IRS only recognizes physical custody (which parent the child lived with the greater part, but over half, of the tax year. That parent is the custodial parent; the other parent is the noncustodial parent.)