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It really depends on the circumstances. The best general rule is that the City is responsible from the junction into the main lines (for sewer) or from the main lines to the meter (for the water lines). The homeowner is responsible for the line from the meter to the house, inside the home, and from the home to the junction.
Unfortunately, a problem such as the one you describe is most likely going to be the homeowner's liability. The fact that the issue went undetected during the prior owner's ownership does not make it the city's liability. There may be a potential cause of action that your daughter and her husband could bring against the seller, or better yet, file against their title insurance, but as between the city and the homeowner, they are going to be responsible for changing the line over to the proper discharge.
I would recommend that they contact at least one or two private plumbers to get quotes to do this work - the City is not always the lowest bidder to do this work in a workmanlike fashion and they may be able to get the same work performed for less money by hiring a private contractor
(as with any other contractor - check licensing with the state, BBB reports, and referrals, to confirm quality prior to hiring).