there are two issues one is legal - what was or was not specified in the divorce papers and where the divorce took place. we do not answer legal questions outside of tax laws. The other is tax.
from a tax standpoint it would appear you are the custodial parent since you seem to be saying they spend more than 1/2 the year with you.
as for care/support all that is required is whether one or both provide more than 50%. this would seem to be the case.
for the noncustodial parent to claim under US tax laws normally the custodial parent would need to provide the noncustodial parent form 8332 which the noncustodial parent would have to file with their tax return. There is an exception for a divorce before 2009. then the divorce papers may satisfy the flying requirements in lieu of form 8332
you would have to talk to a lawyer about the legal issues you would face if the divorce was after 2008 and he requested the form but you failed to provide it.
