Who claims what on taxes next year?

I am 29 and due to a layoff last year that significantly reduced my income I was forced to move back in with my parents, I now make 20,000 dollars a year. I also have a three year old daughter that moved with me, whom I have full custody of, but her father has visitation every other week for a full week until September when she starts school and his visitation changes to every other saturday and sunday. My questions are since we (my daughter and I) live with my parents and I pay them no rent, but however am responsible for all my daughter's and my own personal expenses and other bills (ie medical, car, clothing etc) can they claim us? And since my daughter's father does have visitation every other week for the first nine months of the year and then reduced visitation after that should he claim her, even thought the courts have given me full physical and legal custody? Right now I'm claiming zero dependants on my wages because I do not know what the right thing to do is, so any help would be appreciated.