50/50 custody is a legal arrangement not recognized by the IRS for determining who the custodial parent is. The IRS rules say that the custodial parent is the parent who the child spent more than half the nights of the year with. In every year but leap year, there are an odd number of nights during the year so the parent who had the child one night or more than the other is the custodial parent. You are correct that a tie breaker when the number of nights are the same is whoever has the higher AGI.
If she is the custodial parent, she can claim the child for qualifying child benefits and release the dependency to the other parent using form 8332. If she also meets all the requirements to file as Head of Household, she can use that filing status. Here are the requirements:
1. You are not married or “considered unmarried” on the last day of the year. You are considered unmarried if you lived apart from your spouse the entire last 6 months of the year.
2. If "considered unmarried" you file a separate return from your spouse.
3. You paid more than half the cost of keeping up your home for the year.
4. You have a qualifying child (whether or not you claim the dependency exemption) or you claim a dependency exemption for a “qualifying person” who is related to you and who lived with you in the home for more than half the year. An exception is a parent does not have to live with you to be a qualifying person.
