. IMO I honestly feel my boss ran the business 5 months without me and just decided to do it on her own and that was her way out!
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It would be appropriate to say this if she had fired you. But in my understanding, she did not fire you, you quit. And if you quit, calling "I quit because of domestic violence" may not be determined to be sufficient good reason to quit the job.
I have seen people who were unable to come to work because they had been forced to take their children and flee to a shelter or something, and this was considered quitting the job with a domestic violence reason. I have seen people who had to go down and take out warrants, who missed work and were fired for this and then won unemployment for domestic violence, but I don't quite see how domestic violence is going to tie in here in your case, in that it sounds as if you simply demanded your hours be changed. To win your appeal, you'll have to show your conflicts and issues with your father was the reason you quit your job, it was the reason you told your employer you were quitting your job, and it has to be officially determined as the reason you quit your job by the hearing officer.