Metro tunnels divides into lines
Source: Saint Petersburg olympiad 2024, 9.7
September 22, 2024
combinatorics
Problem Statement
In a very large City, they are building a subway: there are many stations, some pairs of which are connected by tunnels, and from any station you can get through tunnels to any other. All metro tunnels must be divided into "lines": each line consists of several consecutive tunnels, all stations in which are different (in particular, the line should not be circular); lines consisting of one tunnel are also allowed. By law, it is required that you can get from any station to any other station by making no more than transfers from line to line. At what is the largest , any connected metro with stations can be divided into lines, observing the law?