Deductions & credits

I am the custodial parent.  I have full physical custody, and their dad only has one weekend visit per month, supervised, non overnight.  Both children are in my care every single night of the year.  It states in our divorce paperwork that he gets to claim one child and I claim the other.  I have not signed any tax forms allowing this as of yet. 

 

So from what I've been told so far, I put both chidlren on my tax return but only claim one as a dependent?