<p>There are <code>n</code> servers numbered from <code>0</code> to <code>n - 1</code> connected by undirected server-to-server <code>connections</code> forming a network where <code>connections[i] = [a<sub>i</sub>, b<sub>i</sub>]</code> represents a connection between servers <code>a<sub>i</sub></code> and <code>b<sub>i</sub></code>. Any server can reach other servers directly or indirectly through the network.</p>

<p>A <em>critical connection</em> is a connection that, if removed, will make some servers unable to reach some other server.</p>

<p>Return all critical connections in the network in any order.</p>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>
<img alt="" src="https://assets.leetcode.com/uploads/2019/09/03/1537_ex1_2.png" style="width: 198px; height: 248px;" />
<pre>
<strong>Input:</strong> n = 4, connections = [[0,1],[1,2],[2,0],[1,3]]
<strong>Output:</strong> [[1,3]]
<strong>Explanation:</strong> [[3,1]] is also accepted.
</pre>

<p><strong class="example">Example 2:</strong></p>

<pre>
<strong>Input:</strong> n = 2, connections = [[0,1]]
<strong>Output:</strong> [[0,1]]
</pre>

<p>&nbsp;</p>
<p><strong>Constraints:</strong></p>

<ul>
	<li><code>2 &lt;= n &lt;= 10<sup>5</sup></code></li>
	<li><code>n - 1 &lt;= connections.length &lt;= 10<sup>5</sup></code></li>
	<li><code>0 &lt;= a<sub>i</sub>, b<sub>i</sub> &lt;= n - 1</code></li>
	<li><code>a<sub>i</sub> != b<sub>i</sub></code></li>
	<li>There are no repeated connections.</li>
</ul>

class Graph:
    def __init__(self, v):
        self.v = v
        self.visited = [False] * self.v
        self.low = [math.inf] * self.v
        self.desc = [math.inf] * self.v
        self.time = 0
        self.graph = {i: [] for i in range(self.v)}
        self.parents = [-1] * self.v
        self.bridges = []

    def addEdge(self, u, v):
        self.graph[u].append(v)
        self.graph[v].append(u)
    
    
    def _dfs(self, u):
        self.visited[u] = True

        self.low[u] = self.desc[u] = self.time
        self.time += 1

        for v in self.graph[u]:
            if not self.visited[v]:
                self.parents[v] = u    
                self._dfs(v)
                self.low[u] = min(self.low[u], self.low[v])

                if self.low[v] > self.desc[u]:
                    self.bridges.append([u,v])
            elif v != self.parents[u]:
                self.low[u] = min(self.low[u], self.desc[v])


    def bridge(self):
        for u in range(self.v):
            if not self.visited[u]:
                self._dfs(u)
        return self.bridges

class Solution:
    def criticalConnections(self, n: int, connections: List[List[int]]) -> List[List[int]]:
        if len(connections) == 1:
            return connections
        g = Graph(n)
        for u, v in connections:
            g.addEdge(u, v)
        return g.bridge()
        
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