Assuming taxpayer has residential custody and he/she qualifies for HOH. Taxpayer can take both HOH and EIC. Do your diligence tho. 

https://ttlc.intuit.com/community/tax-credits-deductions/discussion/can-the-irs-overrule-a-court-ord...

and even better source https://www.irs.gov/faqs/filing-requirements-status-dependents/dependents/dependents-7

As the custodial parent, she gets HOH and EIC/DCB (if applicable) , if she signs the 8332 to let dad claim them, then dad gets the CTC for them.  (you'll use Code Q on her dependent worksheet)

If she doesn't sign the 8332 dad gets nothing, she can take all, but he can take her back to court for being in contempt of a court order.

Is she required to sign an 8332 even if there  was no marriage 

He should attach the 8332 to his return, but if he doesnt and nobody else tries to claim the child, then it wont be an issue for him.

Very tired so that I understand completely. 

My client, custodial parent gets HOH and EIC.

I file Q on her dependent worksheet 

Have her provide a signed 8332 to the Father.

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@Foxy-Lady wrote:

Is she required to sign an 8332 even if there  was no marriage 

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I do not believe marriage has anything to do with the 8332.