programming-examples/c++/Hard_Graph_Problems/C++ Program to Find Minimum Number of Edges to Cut to make the Graph Disconnected.cpp

/*This is a C++ Program to find minimum number of edges to cut to make the graph disconnected. An edge in an undirected connected graph is a bridge if removing it disconnects the graph. For a disconnected undirected graph, definition is similar, a bridge is an edge removing which increases number of connected components.*/

// A C++ program to find bridges in a given undirected graph
#include<iostream>
#include <list>
#define NIL -1
using namespace std;

// A class that represents an undirected graph
class Graph
{
    int V; // No. of vertices
    list<int> *adj; // A dynamic array of adjacency lists
    void bridgeUtil(int v, bool visited[], int disc[], int low[],
                    int parent[]);
public:
    Graph(int V); // Constructor
    void addEdge(int v, int w); // function to add an edge to graph
    void bridge(); // prints all bridges
};

Graph::Graph(int V)
{
    this->V = V;
    adj = new list<int> [V];
}

void Graph::addEdge(int v, int w)
{
    adj[v].push_back(w);
    adj[w].push_back(v); // Note: the graph is undirected
}

void Graph::bridgeUtil(int u, bool visited[], int disc[], int low[],
                       int parent[])
{
    // A static variable is used for simplicity, we can avoid use of static
    // variable by passing a pointer.
    static int time = 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 aadjacent to this
    list<int>::iterator i;
    for (i = adj[u].begin(); i != adj[u].end(); ++i)
        {
            int v = *i; // v is current adjacent of u
            // If v is not visited yet, then recur for it
            if (!visited[v])
                {
                    parent[v] = u;
                    bridgeUtil(v, visited, disc, low, parent);
                    // 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 the lowest vertex reachable from subtree under v is
                    // below u in DFS tree, then u-v is a bridge
                    if (low[v] > disc[u])
                        cout << u << " " << v << endl;
                }
            // Update low value of u for parent function calls.
            else if (v != parent[u])
                low[u] = min(low[u], disc[v]);
        }
}

// DFS based function to find all bridges. It uses recursive function bridgeUtil()
void Graph::bridge()
{
    // Mark all the vertices as not visited
    bool *visited = new bool[V];
    int *disc = new int[V];
    int *low = new int[V];
    int *parent = new int[V];
    // Initialize parent and visited arrays
    for (int i = 0; i < V; i++)
        {
            parent[i] = NIL;
            visited[i] = false;
        }
    // Call the recursive helper function to find Bridges
    // in DFS tree rooted with vertex 'i'
    for (int i = 0; i < V; i++)
        if (visited[i] == false)
            bridgeUtil(i, visited, disc, low, parent);
}

// Driver program to test above function
int main()
{
    // Create graphs given in above diagrams
    cout << "\nBridges in first graph \n";
    Graph g1(5);
    g1.addEdge(1, 0);
    g1.addEdge(0, 2);
    g1.addEdge(2, 1);
    g1.addEdge(0, 3);
    g1.addEdge(3, 4);
    g1.bridge();
    cout << "\nBridges in second graph \n";
    Graph g2(4);
    g2.addEdge(0, 1);
    g2.addEdge(1, 2);
    g2.addEdge(2, 3);
    g2.bridge();
    cout << "\nBridges in third graph \n";
    Graph g3(7);
    g3.addEdge(0, 1);
    g3.addEdge(1, 2);
    g3.addEdge(2, 0);
    g3.addEdge(1, 3);
    g3.addEdge(1, 4);
    g3.addEdge(1, 6);
    g3.addEdge(3, 5);
    g3.addEdge(4, 5);
    g3.bridge();
    return 0;
}

/*
Bridges in first graph

3 4