MathNet 2023-part-06 — 191 problems (EN–VI) Question 126

There are $n$ cities on each side of the Hung river, with two-way ferry routes between some pairs of cities on opposite sides of the river (each route connects a city on the left bank with a city on the right bank). A city is "convenient" if and only if it has ferry routes to all cities on the other side. The river is "clear" if and only if we can find $n$ routes whose endpoints include all $2n$ cities.
It is known that the Hung river is currently not clear, but if any new route is added, the river becomes clear. Determine all possible values for the number of convenient cities.