Example 3 There is a railway network between six cities, such that there is a direct railway between any two cities. On Sundays, some railways will be closed for maintenance. The railway department stipulates: after closing several sections of the railway, it must still be possible to travel by rail between any two cities (not necessarily directly). How many different railway maintenance methods meet the railway department's requirements? ${ }^{[1]}$
(2007, British Mathematical Olympiad)
【Analysis】The problem is equivalent to finding the number of all possible ways to connect a graph composed of six points and several edges, such that any two points are connected.

Let $f(n)$ denote the number of all possible ways to connect $n$ points such that any two points are connected. Then
$$
f(1)=1, f(2)=1 \text {. }
$$

For the case of three points, either all pairs of points are connected, or any two points are connected by selecting any two out of the three edges, so $f(3)=4$.

Since $n$ vertices can have $\mathrm{C}_{n}^{2}$ edges, there are a total of $2^{\mathrm{C}_{n}^{2}}$ different connection methods. At this point, any two vertices either have an edge or do not have an edge. Select one vertex $v$ from the $n$ vertices, and consider the set of $i(1 \leqslant i \leqslant n)$ vertices that include $v$. You can choose $i-1$ vertices from the remaining $n-1$ vertices to connect with $v$ to form this set of $i$ vertices, so there are $\mathrm{C}_{n-1}^{i-1}$ ways to select.

By the definition of $f(i)$, for the selected $i-1$ vertices and vertex $v$, there are $f(i)$ ways to ensure that any two points are connected. For the remaining $n-i$ points, there are $2^{\mathrm{C}_{n-i}^{2}}$ different connection methods.
Therefore, $2^{\mathrm{C}_{n}^{2}}=\sum_{i=1}^{n} \mathrm{C}_{n-1}^{i-1} f(i) 2^{\mathrm{C}_{n-i}^{2}}$.
Given $f(1)=1, f(2)=1, f(3)=4$, we get
$$
f(4)=38, f(5)=728, f(6)=26704 \text {. }
$$

Since among the 26704 ways, 26703 ways involve closing some sections of the railway, and 1 way involves not closing any section, the answer is 26703.