Find Articulation Points in a Graph using Java

Hello, in this tutorial, we are going to find Articulation Points in a GraphGraph using Java Language. For that, we must know what is an articulation point first. To explain it better, please consider the given image:


In this image, if we consider the vertex {0} of our graph and we remove it, then our graph will get disconnected or we can say ” An Articulation point in a vertex in an undirected graph after removal of which our graph will be divided into two or more components it will not stay connected any more.

 How to Find Articulation points present in a graph with Java?

Condition 1 for a vertex to be articulation point.

Let’s consider the image of the graph presented above again. In this graph, if we make a DFS tree, and consider vertex{1} as our root node, then it will have 3 children, namely, vertex{0}, vertex{2} and vertex{3}, so if we remove root node itself, the graph will be disconnected. Hence, Condition 1 for any vertex to be an Articulation pointis, if it is our root node, it must be having two or more children.

Note: No leaf node can be an articulation point because it has no child to separate from itself.

Condition 2 for a vertex to be articulation point.

Now. let’s talk about a node if it is neither the root nodeGraph Articulation nor a leaf node. For such a node, Consider the image given below, Let’s consider vertex{2}, it has vertex{3} as an ancestor and vertex{1} as it’s child, and there is a back edge from vertex{1} to vertex{3} hence if we remove vertex{2}, it will not affect our graph, but had there not been a back edge, removal of vertex{2} would have separated vertex{1} from the graph. Hence, then it would have been an articulation point. So, Condition 2 for any vertex to be an Articulation point, it must not have a back edge between its parent and child node.


Now, Let’s understand the approach of coding to determine Articulation points. For condition 1, we will simply do a depth-first search and check if our root node has more than 1 child or not. For condition 2, we will determine the discovery time of node from its nearest nodes and will find if there is any other path between its parent and its child, so we will maintain an array called discovery_time and an array called minimum_time. So, if minimum_time for a considerate node is greater than or equal to its discovery_time, it has to be an articulation point. We will have a recursive function to do a depth-first search and a helper function to increase the readability of the recursive function.

Here is the code:


import java.util.*;
import java.util.LinkedList;

public class Graph 
    private int vertex;   // No. of vertices 
    // Array  of lists for Adjacency List Representation 
    private LinkedList<Integer> adj_list[]; 
    int time = 0; 
    static final int NO_PARENT = -1; 
    // Constructor 
    Graph(int v) 
        vertex = v; 
        adj_list = new LinkedList[v]; 
        for (int i=0; i<v; ++i) 
            adj_list[i] = new LinkedList(); 
    //Function to add an edge into the graph 
    void add_an_edge(int v, int w) 
        adj_list[v].add(w);  // Add w to v's list. 
        adj_list[w].add(v);    //Add v to w's list 
    // A recursive function that find articulation points using DFS 
    // current_vertex --> The vertex to be visited next 
    // visited[] --> keeps tract of visited vertices 
    // discovery_time[] --> Stores discovery times of visited vertices 
    // parent_node[] --> Stores parent vertices in DFS tree 
    // articulation_points[] --> Store articulation points 
    void helper_function(int current_vertex, boolean isvisited[], int discovery_time[], int minimum_time[], int parent_node[], boolean articulation_points[]) 
        // Count of children in DFS Tree 
        int children = 0; 
        // Mark the current node as visited 
        isvisited[current_vertex] = true; 
        // Initialize discovery time and minimum time.
        discovery_time[current_vertex] = minimum_time[current_vertex] = ++time; 
        // Go through all vertices aadjacent to this 
        Iterator<Integer> i = adj_list[current_vertex].iterator(); 
        while (i.hasNext()) 
            int adj_vertex =;  // v is adjacent vertex of current_vertex
            // If adj_vertex is not visited yet, then make it a child of current_vertex in graph and call recursion for it.
            if (!isvisited[adj_vertex]) 
                parent_node[adj_vertex] = current_vertex; 
                helper_function(adj_vertex, isvisited, discovery_time, minimum_time, parent_node, articulation_points); 
                // Check if the subtree rooted with adj_vertex has a connection to one of the ancestors of current_vertex. 
                minimum_time[current_vertex]  = Math.min(minimum_time[current_vertex], minimum_time[adj_vertex]); 
                // current_vertex is an articulation point if any of following is satisfied. 
                // if current_vertex is root of DFS tree and has two or more chilren. 
                if (parent_node[current_vertex] == NO_PARENT && children > 1) 
                    articulation_points[current_vertex] = true; 
                // if current_vertex is not root and minimum_time value of one of its adj_vertex is more than discovery value of current_vertex. 
                if (parent_node[current_vertex] != NO_PARENT && minimum_time[adj_vertex] >= discovery_time[current_vertex]) 
                    articulation_points[current_vertex] = true; 
            // Update minimum_time value of current_vertex for parent function calls. 
            else if (adj_vertex != parent_node[current_vertex]) 
                minimum_time[current_vertex]  = Math.min(minimum_time[current_vertex], discovery_time[adj_vertex]); 
    // The function to find articulation points by depth-first search.
    void findArticulationPoints() 
        // Marking vertices as unmarked and initialising required arrays.
        boolean isvisited[] = new boolean[vertex]; 
        int discovery_time[] = new int[vertex]; // array for discovery time of each vertex. 
        int minimum_time[] = new int[vertex]; // array for minimum time of each node. 
        int parent_node[] = new int[vertex]; // array for storing parent of each vertex.
        boolean articulation_points[] = new boolean[vertex]; // To store articulation points.
        // Initialize parent_node array, isvisited and articulation_points arrays. 
        for (int i = 0; i < vertex; i++) 
            parent_node[i] = NO_PARENT; 
            isvisited[i] = false; 
            articulation_points[i] = false; 
        // Call the recursive helper function to find articulation points in graph for every vertex iteratively. 
        for (int i = 0; i < vertex; i++){
            if (isvisited[i] == false)
            { // call helper function.
             helper_function(i, isvisited, discovery_time, minimum_time, parent_node, articulation_points);
        // after recursive calls, print articulation points, if any.
        for (int i = 0; i < vertex; i++){
            if (articulation_points[i] == true) 
                System.out.print(i+" ");
        for(int i = 0; i < vertex; i++){
            if (articulation_points[i] == true) return;
        System.out.println("Graph has no articulation point");
    //main method
    public static void main(String args[]) 
        //creating graphs and calling function to find articulation points.
        Graph graph_1 = new Graph(6); 
        System.out.println("Articulation points in Graph 1 are ");
        Graph graph_2 = new Graph(5); 
        System.out.println("Articulation points in Graph 2 are ");


Articulation points in Graph 1 are 
0 1 
Articulation points in Graph 2 are 
Graph has no articulation point.

Thank you for your time.

You can also visit: Java Program to detect a cycle in an undirected graph

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