Recently seperated. Who can claim the children as dependents

Dependents are claimed by the custodial parent (the parent the child lived with for more than 1/2 of the year).  For the noncustodial parent to claim the children as dependents, he/she must have permission from the custodial parent.

We have only been separated for one month.  How do we decide who is the custodial parent? If I read the instructions correctly, in this scenario whomever has the highest adjusted gross income will have the right to claim the children.  Is this correct?

The custodial parent is the child lived with for more than 183 days in 2016.  If the child lived both parents the same number of days (for example, if the parents split up after the end of the year) then the highest AGI controls, so in that respect you are correct.