She doesnt' pay support? Then don't give her the exemptions. It's a money thing, not a visitation thing. The exemptions usually go with the parent who pays the money (proportionately with the amount they pay) unless the divorce order forgets to mention it. In absence of an order explaining who gets the exemptions and when, the IRS will just say that the exemption goes to teh parent with more than 50% parenting time (the IRS is not the place to argue the divorce issues, so they do this because it's the only CERTAIN measure they have to tell who is supporting the kids).
It's kind of a reward for the person who provides most of the support for the kids (the one who HAS them most of the time if the one supporting them is not obvious). It is meant to HELP the finances of THAT person.
In YOUR case, Beat Down, if you are having ZERO financial problems, if you are Donald Trump and you kept custody of your new baby when you kicked the latest and youngest girl out of the Tower, then it's clear that you have enough $$ to get all the nannies you need for the new baby, it's clear that this baby will have plenty of $$ for college by you, and on TOP of that you can clearly afford your own retirement, your other kids' needs, and STILL have a couple of bucks left over at the end of the year... well THEN, it's totally reasonable for you to let her claim the exemption.
But if you're like most of us, struggling to make ends meet and save for both your own retirement and the kid's college, where the extra couple of bucks might provide a little ease in the budget so you can treat the kids to an ice cream next summer at the state fair... well... the exemption is for the PURPOSE of letting the one who pays for the ice cream get a couple of sprinkles on top.
It's not meant to help ease the budget of the woman who cheated on you, repeatedly, left the kids with you, does NOT pay for them, does not work enough to support herself (OR the kids) and is now about to marry the man she cheated with. You are not required to help ease her budget so that she can afford prettier flowers for her next wedding. The exemption for having hte kids... it's more important that it goes for the KIDS... so keep it. OK? No one will think you're not a nice guy if you don't help your ex finance her continued relationship with the man she cheated on you with! You're the primary parent, you provide the primary income, and she left. You're not Donald Trump, you don't have to finance her.