1. In any city of a certain province, it is connected by railway to at most 3 other cities, and from any city to other cities, it passes through at most one city in between. How many cities can there be at most in the province?
1. Arbitrarily select a city $v_{0}$, which is connected to at most 3 cities $v_{1}, v_{2}, v_{3}$. Each of $v_{1}, v_{2}, v_{3}$ is connected to at most two cities different from $v_{0}$, and there are no other cities (otherwise, from $v_{0}$ to that city, at least two cities would be passed through).
Thus, the total number of cities $\leqslant 1+3+3 \times 2=10$.
As shown in the figure, the equality can hold.
Therefore, the province has at most 10 cities.