A vertex in an undirected connected graph is an articulation point (or cut vertex) if removing it (and edges through it) disconnects the graph. Articulation points represent vulnerabilities in a connected network – single points whose failure would split the network into 2 or more components. They are useful for designing reliable networks.

For a disconnected undirected graph, an articulation point is a vertex removing which increases number of connected components.

Following is the implementation of Tarjan’s algorithm for finding articulation points.

```
#include <bits/stdc++.h>
using namespace std;
// A recursive function that find articulation
// points using DFS traversal
// adj[] --> Adjacency List representation of the graph
// u --> The vertex to be visited next
// visited[] --> keeps track of visited vertices
// disc[] --> Stores discovery times of visited vertices
// low[] -- >> earliest visited vertex (the vertex with minimum
// discovery time) that can be reached from subtree
// rooted with current vertex
// parent --> Stores the parent vertex in DFS tree
// isAP[] --> Stores articulation points
void APUtil(vector<int> adj[], int u, bool visited[],
int disc[], int low[], int& time, int parent,
bool isAP[])
{
// Count of children in DFS Tree
int children = 0;
// Mark the current node as visited
visited[u] = true;
// Initialize discovery time and low value
disc[u] = low[u] = ++time;
// Go through all vertices adjacent to this
for (auto v : adj[u]) {
// If v is not visited yet, then make it a child of u
// in DFS tree and recur for it
if (!visited[v]) {
children++;
APUtil(adj, v, visited, disc, low, time, u, isAP);
// Check if the subtree rooted with v has
// a connection to one of the ancestors of u
low[u] = min(low[u], low[v]);
// If u is not root and low value of one of
// its child is more than discovery value of u.
if (parent != -1 && low[v] >= disc[u])
isAP[u] = true;
}
// Update low value of u for parent function calls.
else if (v != parent)
low[u] = min(low[u], disc[v]);
}
// If u is root of DFS tree and has two or more children.
if (parent == -1 && children > 1)
isAP[u] = true;
}
void AP(vector<int> adj[], int V)
{
int disc[V] = { 0 };
int low[V];
bool visited[V] = { false };
bool isAP[V] = { false };
int time = 0, par = -1;
// Adding this loop so that the
// code works even if we are given
// disconnected graph
for (int u = 0; u < V; u++)
if (!visited[u])
APUtil(adj, u, visited, disc, low,
time, par, isAP);
// Printing the APs
for (int u = 0; u < V; u++)
if (isAP[u] == true)
cout << u << " ";
}
// Utility function to add an edge
void addEdge(vector<int> adj[], int u, int v)
{
adj[u].push_back(v);
adj[v].push_back(u);
}
int main()
{
// Create graphs given in above diagrams
cout << "Articulation points in first graph \n";
int V = 5;
vector<int> adj1[V];
addEdge(adj1, 1, 0);
addEdge(adj1, 0, 2);
addEdge(adj1, 2, 1);
addEdge(adj1, 0, 3);
addEdge(adj1, 3, 4);
AP(adj1, V);
cout << "\nArticulation points in second graph \n";
V = 4;
vector<int> adj2[V];
addEdge(adj2, 0, 1);
addEdge(adj2, 1, 2);
addEdge(adj2, 2, 3);
AP(adj2, V);
cout << "\nArticulation points in third graph \n";
V = 7;
vector<int> adj3[V];
addEdge(adj3, 0, 1);
addEdge(adj3, 1, 2);
addEdge(adj3, 2, 0);
addEdge(adj3, 1, 3);
addEdge(adj3, 1, 4);
addEdge(adj3, 1, 6);
addEdge(adj3, 3, 5);
addEdge(adj3, 4, 5);
AP(adj3, V);
return 0;
}
```

Output

```
Articulation points in first graph
0 3
Articulation points in second graph
1 2
Articulation points in third graph
1
```

Time Complexity: The above function is simple DFS with additional arrays. So time complexity is same as DFS which is **O(V+E)** for adjacency list representation of graph.