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@MGC94 wrote:

They want to know if she can file Single and claim the 2 dependents 

Him file HOH no dependents (qualifying person is child not a dependent) 

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Yes, she can claim Single (assuming she isn't married) and claim the dependents (assuming they meet the qualifications).

No.  If the are living together, a person needs to claim at least one of the children as a dependent in order to claim Head of Household.

Don't base the "cost of keeping up a home" solely on income though.  Find out who paid for over half of keeping up the home.  While it is usually the person with the higher income, that is not always the case.