We don't know if they have ever been married, so let's assume they were not.   It makes a difference.  Couples do get divorced and then get back together without remarrying, you know.  

Sounds like the father pays more than half the household expenses, and also has the higher AGI.  So the child is his qualifying child, not hers.  See Section 152( c)(4)(B)(ii).  It's the same test for HoH and claiming a dependent, so they can't have their cake and eat it too.  

An interesting question would be what to do with a California couple in a Registered Domestic Partnership when they are splitting their income under community-property rules, so the AGI's are equal. I hope that's not the case here.