If your decree doesn't say who gets the kids, go by the IRS rules. The first test is 'who had the most overnights with the kids.' If you did, then you get to claim the kids, regardless of what she did. If the overnights are equal, then they go by who actually contributed the most to their support.
Either way, if you both file using the kids, the IRS will flag both returns for review. Good luck.Edited to correct stupid typos.
Two steps forward and one step backwards, is still progress.