I am single, own a home and my girlfriend lives with me. Do I have to report her income as part of household income?

Welcome to AMHD. Your question reminds me of the phrase "between Scylla and Charybdis", which has come to mean being in a state where one is between two dangers and moving away from one will cause you to be in danger of the other. Charybdis - Wikipedia, the free encyclopedia (http://en.wikipedia.org/wiki/Charybdis)
As you are aware, you live in a locale that is situated within a legal jurisdiction. Within each legal jurisdiction are precedents, or legal rules within which parties and attorneys must conduct themselves. I have found a case that says: "The Court's holding is not that Debtor's live-in boyfriend must pay – or even help pay – Debtor's debts or living expenses.
The Court is saying only that Debtor's economic reality is that she shares a household. Consequently, she should pay her fair share – but no more than her fair share – of the household's expenses. Were she to pay only her fair share of those expenses, Debtor would be able to pay her creditors." http://www.sdb.uscourts.gov/Decisions/2005%20Decisions/2005%20-%2042%20Pettis.pdf
And, this is from a 2005 case, so there has been lots of time for other decisions.
In any event, I would not characterize your girlfriend's contribution as 'rent' because of potential tax implications. My guess is that your question has been answered in your jurisdiction, and you (or your attorney) must find the answer.