I have 2 clients that  live together and have one adopted child.  The father is claiming the child as a dependent but does not pay more than half the cost of the home so he's Single with a dependent and getting the adoption credit.

Can the mother file head of household without claiming the dependent?  I know divorced parents can do so but in this situation both parents live together.  One rule is the child can't be a qualifying child of another so I'm thinking no but wanted to see if anyone else came across this before?