Bottom line: the mother (the custodial parent*) gets to decide who will claim her child.
A child can be the “qualifying child” dependent of any close relative in the household. If your ex lives with her parents, they have the right to claim the child, if the mother doesn't claim them.
The grandparent's right takes priority over any right you (the non custodial parent) has.
In fact, you have no right. The custodial parent has first priority on claiming the children on her taxes; regardless of the amount of support provided by the non-custodial parent. The non-custodial parent can only claim the child as a dependent if the custodial parent gives permission (on form 8332) or if it's spelled out in a pre 2009 divorce decree. Even if a court order, dated after 2008, gives the non-custodial parent the right to claim the child, he must still get form 8332 from the custodial parent. A properly worded order should require her to provide that form.
*For tax purposes, there is no such thing as joint custody, regardless of what your agreement says. The requirement, to be custodial parent, is that the child live with you MORE than 50% of the time. One of you has to be the custodial parent and the other the non-custodial parent. The IRS goes by physical custody, not legal custody.
