Example 4 A country has $n$ cities, and between any two cities, there is at most one direct road, and these roads satisfy: after blocking all the roads connected to any one city, any two other cities in the country can still be connected through other roads.

Now, all the roads in the country are changed to one-way roads. For cities $A$ and $B$, if by directing at most $k$ roads, and arbitrarily directing the other roads, for any road $l$, there exists a one-way path from city $A$ that passes through road $l$ and finally reaches city $B$ (each city is passed through at most once), then $A$ and $B$ are called “$k$-connected”.

Given that any two cities in the country are $k$-connected. Find the minimum value of $k$.


Solve $k_{\text {min }}=2 n-3$.
Using graph theory ideas.
Map cities to points, and roads between two cities to edges connecting two points, forming graph $G$.

Assume the length of each edge is 1. For any two points $X, Y$, let $d(X, Y)$ be the length of the shortest path from $X$ to $Y$.

For any two points $A, B$ in graph $G$, perform the following operations to direct $k$ roads in graph $G$ so that $A, B$ are $k$-directed connected, referred to as the unidirectional connected graph from $A$ to $B$.
First, construct two trees $L, R$ with roots $A, B$ respectively.
(1) Let $i=1, L=L_{0}=\{A\}, R=R_{0}=\{B\}$, $L_{a i}=R_{n i}=\varnothing$.
(2) Let $L_{i}=\{X \mid d(X, A)=i\}$.
For each $x \in L_{i}$, if there exists a
$X^{\prime} \in\left(L_{i-1} \cap L\right) \backslash L_{n \uparrow}$,

such that $X, X^{\prime}$ are connected by an edge in graph $G$, then let
$x \in L(V), X X^{\prime} \in L(E)$.
(3) Let $R_{i}=\{Y \mid d(Y, B)=i\}$.
For each $Y \in R_{i}$, if there exists a
$Y^{\prime} \in\left(R_{i-1} \cap R\right) \backslash R_{n \mid}$,

such that $Y, Y^{\prime}$ are connected by an edge in graph $G$, then let
$Y \in R(V) . Y Y^{\prime} \in R(E)$.
(4) If there exists $Z \in\left(L_{i} \cap L\right) \cap\left(R_{i} \cap R\right)$, then let $Z$ be a leaf of $L, R$, i.e., $Z \in L_{\text {ar }} \cap R_{\text {pr }}$.
(5) If there exist $X \in L_{i} \cap L, Y \in R_{i} \cap R$, and $X, Y$ are connected by an edge in graph $G$, then let $X, Y$ be leaves of $L, R$ respectively, i.e., $X \in L_{p \gamma}, Y \in R_{p \dot{r i}}$.
(6) If $|L \cup R|=n$ or $i=\left[\frac{n}{2}\right]([x]$ represents the greatest integer not exceeding the real number $x$), the operation ends; otherwise, let $i$ increase by 1 and return to (2).

Next, merge or connect the leaves of the constructed trees $L, R$ to obtain a one-way road network $G^{\prime}$ from left to right (from $A$ to $B$, from $L$ to $R$) (orient all roads in $G^{\prime}$ from left to right), as shown in Figure 3.
Let the number of vertices in graph $G^{\prime}$ be $m$. Then the number of edges $e \leqslant 2 m-3$.

In fact, suppose trees $L, R$ have $k(k \leqslant m-2)$ leaves in total, each leaf can add at most two edges, plus the edge between $A, B$, so
$e \leqslant m-k-2+2 k+1=m+k-1 \leqslant 2 m-3$.
Next, we prove: For any $X, Y \in G^{\prime}(l=X Y \in G)$, there is a one-way path from city $A$ through road $l$ to city $B$.
Clearly, $|d(X, A)-d(Y, A)| \leqslant 1$,
$|d(X, B)-d(Y, B)| \leqslant 1$.
If $l \in G^{\prime}$, then there is a one-way path from city $A$ through road $l$ to city $B$ along graph $G^{\prime}$.

If $l \notin G^{\prime}$, then there exist two paths $\Gamma_{1}, \Gamma_{2}\left(X \in \Gamma_{1}, Y \in \Gamma_{2}\right)$ in graph $G^{\prime}$, as shown in Figure 4.
If $X \rightarrow Y$, then there is a path
$A \rightarrow A_{0} \rightarrow X \rightarrow Y \rightarrow B_{0} \rightarrow B ;$
If $Y \rightarrow X$, then there is a path
$A \rightarrow A_{0} \rightarrow Y \rightarrow X \rightarrow B_{0} \rightarrow B$.
Finally, each vertex $Z$ in graph $G \backslash G^{\prime}$ is only connected to the leaves of $L, R$ in graph $G$, and by the problem's condition, it is connected to at least two leaves (let them be $\left.W_{1}, W_{2}, \cdots, W_{1}(t \geqslant 2)\right)$.

Take two of these leaves $W_{1}, W_{2}$ (in graph $G^{\prime}$, $W_{1}$ is to the left of $W_{2}$), and let $W_{1} \rightarrow Z \rightarrow W_{2}$.

Then for any $W_{i}(i \neq 1,2)$, if $W_{i} \rightarrow Z$, there exists a one-way path
$A \rightarrow W_{i} \rightarrow Z \rightarrow W_{2} \rightarrow B ;$
If $Z \rightarrow W_{i}$, there exists a one-way path
$A \rightarrow W_{1} \rightarrow Z \rightarrow W_{i} \rightarrow B$.
Thus, by adding $n-m$ vertices and $2(n-m)$ edges to graph $G^{\prime}$, we obtain the unidirectional connected graph from city $A$ to city $B$ for graph $G$, and
$k \leqslant e+2(n-m) \leqslant 2 n-3$.
As shown in Figure 5, the number of roads that need to be directed is
$2(n-2)+1=2 n-3$ (roads).