What is meant by Articulation point?

Software Development
#1

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.