Husband's ex committed fraud?

My husband’s ex, whom he never married, had been getting medicaid for her and their son by claiming that she was a single mother and the father did not exist but, since they split, his son has spent at least every weekend (Thu-Mon) with him for the past seven years, in addition to spending time with his family more than hers. She then has said that she kept his name off the birth certificate so my husband, who is self-employed, would "save on health insurance" (which he didn’t need to). Then she said that she couldn’t put his name on the birth certificate ’cause the state would go after HIM for unpaid health insurance (I believe she’s the one who would be in major trouble for lying about his existence in order to get free health care and God knows what else…) She has also demanded child support in addition to having my husband pay for school, sports, trips , etc. yet she has not claimed any of that on her taxes cause she can’t really claim child support if she says there is no father. Is any of this considered fraud? Only serious answers, please.
Quiz: Interesting… this is all I know (so far!) Apparently she just submitted a birth certificate with no father named and that was enough to get medicaid. They both worked and made well above state minimum income guidelines.

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