Biconnected graph

// A Java program to find if a given undirected graph is
// biconnected
import java.io.*;
import java.util.*;
import java.util.LinkedList;

// This class represents a directed graph using adjacency
// list representation
class Graph
{
    private int V;   // No. of vertices

    // Array  of lists for Adjacency List Representation
    private LinkedList<Integer> adj[];

    int time = 0;
    static final int NIL = -1;

    // Constructor
    Graph(int v)
    {
        V = v;
        adj = new LinkedList[v];
        for (int i=0; i<v; ++i)
            adj[i] = new LinkedList();
    }

    //Function to add an edge into the graph
    void addEdge(int v, int w)
    {
        adj[v].add(w);  //Note that the graph is undirected.
        adj[w].add(v);
    }

    // A recursive function that returns true if there is an articulation
    // point in given graph, otherwise returns false.
    // This function is almost same as isAPUtil() @ http://goo.gl/Me9Fw
    // u --> The vertex to be visited next
    // visited[] --> keeps track of visited vertices
    // disc[] --> Stores discovery times of visited vertices
    // parent[] --> Stores parent vertices in DFS tree
    boolean isBCUtil(int u, boolean visited[], int disc[],int low[],
                     int parent[])
    {

        // 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
        Iterator<Integer> i = adj[u].iterator();
        while (i.hasNext())
        {
            int v = i.next();  // v is current adjacent of 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++;
                parent[v] = u;

                // check if subgraph rooted with v has an articulation point
                if (isBCUtil(v, visited, disc, low, parent))
                    return true;

                // Check if the subtree rooted with v has a connection to
                // one of the ancestors of u
                low[u]  = Math.min(low[u], low[v]);

                // u is an articulation point in following cases

                // (1) u is root of DFS tree and has two or more children.
                if (parent[u] == NIL && children > 1)
                    return true;

                // (2) If u is not root and low value of one of its
                //  child is more than discovery value of u.
                if (parent[u] != NIL && low[v] >= disc[u])
                    return true;
            }

            // Update low value of u for parent function calls.
            else if (v != parent[u])
                low[u]  = Math.min(low[u], disc[v]);
        }
        return false;
    }

    // The main function that returns true if graph is Biconnected,
    // otherwise false. It uses recursive function isBCUtil()
    boolean isBC()
    {
        // Mark all the vertices as not visited
        boolean visited[] = new boolean[V];
        int disc[] = new int[V];
        int low[] = new int[V];
        int parent[] = new int[V];

        // Initialize parent and visited, and ap(articulation point)
        // arrays
        for (int i = 0; i < V; i++)
        {
            parent[i] = NIL;
            visited[i] = false;
        }

        // Call the recursive helper function to find if there is an
        // articulation/ point in given graph. We do DFS traversal
        // starting from vertex 0
        if (isBCUtil(0, visited, disc, low, parent) == true)
            return false;

        // Now check whether the given graph is connected or not.
        // An undirected graph is connected if all vertices are
        // reachable from any starting point (we have taken 0 as
        // starting point)
        for (int i = 0; i < V; i++)
            if (visited[i] == false)
                return false;

        return true;
    }

    // Driver method
    public static void main(String args[])
    {
        // Create graphs given in above diagrams
        Graph g1 =new Graph(2);
        g1.addEdge(0, 1);
        if (g1.isBC())
            System.out.println("Yes");
        else
            System.out.println("No");

        Graph g2 =new Graph(5);
        g2.addEdge(1, 0);
        g2.addEdge(0, 2);
        g2.addEdge(2, 1);
        g2.addEdge(0, 3);
        g2.addEdge(3, 4);
        g2.addEdge(2, 4);
        if (g2.isBC())
            System.out.println("Yes");
        else
            System.out.println("No");

        Graph g3 = new Graph(3);
        g3.addEdge(0, 1);
        g3.addEdge(1, 2);
        if (g3.isBC())
            System.out.println("Yes");
        else
            System.out.println("No");

        Graph g4 = new Graph(5);
        g4.addEdge(1, 0);
        g4.addEdge(0, 2);
        g4.addEdge(2, 1);
        g4.addEdge(0, 3);
        g4.addEdge(3, 4);
        if (g4.isBC())
            System.out.println("Yes");
        else
            System.out.println("No");

        Graph g5= new Graph(3);
        g5.addEdge(0, 1);
        g5.addEdge(1, 2);
        g5.addEdge(2, 0);
        if (g5.isBC())
            System.out.println("Yes");
        else
            System.out.println("No");
    }
}
// This code is contributed by Aakash Hasija