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It depends on what basis of accounting you are using. If you are using accrual based accounting you would want to reverse the invoice where the company went out of business. If there is absolutely no hope of collecting on the invoice you would want to reverse the invoice.
If someone pays less than the full amount (assuming that you will never fully collect on this invoice) you could expense it as a discount or you could hit the income account that you used on the original invoice. Either way you would achieve the same result. You may want to create a separate contra income account to easily keep track of these amounts.
If you are using a cash basis accounting these transactions should not matter since you would not recognize the income until you actually receive the cash. But you would want to account for these transactions so you do not have open invoices on your books.
Does this help? Please let me know if you have any other question or need further clarification.