Skip to content
Related Articles

Related Articles

Bridges in a graph

Improve Article
Save Article
  • Difficulty Level : Hard
  • Last Updated : 28 Sep, 2022
Improve Article
Save Article

Given an undirected Graph, The task is to find the Bridges in this Graph. 

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 disconnected components. 

Like Articulation Points, bridges represent vulnerabilities in a connected network and are useful for designing reliable networks.

Examples:

Input: 

Bridge1

Output: (0, 3) and (3, 4)

Input:

Bridge2

Output: (1, 6)

Input:

Bridge3

Output: (0, 1), (1, 2), and (2, 3)

Naive Approach: Below is the idea to solve the problem:

One by one remove all edges and see if removal of an edge causes disconnected graph. 

Following the below steps to Implement the idea:

  • For every edge (u, v), do the following:
    • Remove (u, v) from the graph 
    • See if the graph remains connected (either use BFS or DFS) 
    • Add (u, v) back to the graph.

Time Complexity: O(E*(V+E)) for a graph represented by adjacency list.
Auxiliary Space: O(V+E)

Find Bridges in a graph using DFS traversal

 Below is the idea to solve the problem:

The idea is similar to O(V+E) algorithm for Articulation Points. Do DFS traversal of the given graph. In DFS tree an edge (u, v) (u is parent of v in DFS tree) is bridge if there does not exist any other alternative edge to reach u or an ancestor of u from subtree rooted with v

As discussed in the post, the value low[v] indicates the earliest visited vertex reachable from subtree rooted with v. The condition for an edge (u, v) to be a bridge is, low[v] > disc[u]

Below is the implementation of the above approach:

C++




// 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);   // 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
}
 
// A recursive function that finds and prints bridges using
// DFS traversal
// 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
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 adjacent 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;
}


Java




// A Java program to find bridges in a given undirected graph
import java.io.*;
import java.util.*;
import java.util.LinkedList;
 
// This class represents a undirected 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
    @SuppressWarnings("unchecked")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);  // Add w to v's list.
        adj[w].add(v);    //Add v to w's list
    }
 
    // A recursive function that finds and prints bridges
    // using DFS traversal
    // 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
    void bridgeUtil(int u, boolean visited[], int disc[],
                    int low[], int parent[])
    {
 
        // 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 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]  = Math.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])
                    System.out.println(u+" "+v);
            }
 
            // Update low value of u for parent function calls.
            else if (v != parent[u])
                low[u]  = Math.min(low[u], disc[v]);
        }
    }
 
 
    // DFS based function to find all bridges. It uses recursive
    // function bridgeUtil()
    void bridge()
    {
        // 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 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);
    }
 
    public static void main(String args[])
    {
        // Create graphs given in above diagrams
        System.out.println("Bridges in first graph ");
        Graph g1 = new Graph(5);
        g1.addEdge(1, 0);
        g1.addEdge(0, 2);
        g1.addEdge(2, 1);
        g1.addEdge(0, 3);
        g1.addEdge(3, 4);
        g1.bridge();
        System.out.println();
 
        System.out.println("Bridges in Second graph");
        Graph g2 = new Graph(4);
        g2.addEdge(0, 1);
        g2.addEdge(1, 2);
        g2.addEdge(2, 3);
        g2.bridge();
        System.out.println();
 
        System.out.println("Bridges in Third graph ");
        Graph g3 = new Graph(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();
    }
}
// This code is contributed by Aakash Hasija


Python3




# Python program to find bridges in a given undirected graph
#Complexity : O(V+E)
  
from collections import defaultdict
  
#This class represents an undirected graph using adjacency list representation
class Graph:
  
    def __init__(self,vertices):
        self.V= vertices #No. of vertices
        self.graph = defaultdict(list) # default dictionary to store graph
        self.Time = 0
  
    # function to add an edge to graph
    def addEdge(self,u,v):
        self.graph[u].append(v)
        self.graph[v].append(u)
  
    '''A recursive function that finds and prints bridges
    using DFS traversal
    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'''
    def bridgeUtil(self,u, visited, parent, low, disc):
 
        # Mark the current node as visited and print it
        visited[u]= True
 
        # Initialize discovery time and low value
        disc[u] = self.Time
        low[u] = self.Time
        self.Time += 1
 
        #Recur for all the vertices adjacent to this vertex
        for v in self.graph[u]:
            # If v is not visited yet, then make it a child of u
            # in DFS tree and recur for it
            if visited[v] == False :
                parent[v] = u
                self.bridgeUtil(v, visited, parent, low, disc)
 
                # 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]:
                    print ("%d %d" %(u,v))
     
                     
            elif v != parent[u]: # Update low value of u for parent function calls.
                low[u] = min(low[u], disc[v])
 
 
    # DFS based function to find all bridges. It uses recursive
    # function bridgeUtil()
    def bridge(self):
  
        # Mark all the vertices as not visited and Initialize parent and visited,
        # and ap(articulation point) arrays
        visited = [False] * (self.V)
        disc = [float("Inf")] * (self.V)
        low = [float("Inf")] * (self.V)
        parent = [-1] * (self.V)
 
        # Call the recursive helper function to find bridges
        # in DFS tree rooted with vertex 'i'
        for i in range(self.V):
            if visited[i] == False:
                self.bridgeUtil(i, visited, parent, low, disc)
         
  
# Create a graph given in the above diagram
g1 = Graph(5)
g1.addEdge(1, 0)
g1.addEdge(0, 2)
g1.addEdge(2, 1)
g1.addEdge(0, 3)
g1.addEdge(3, 4)
 
  
print ("Bridges in first graph ")
g1.bridge()
 
g2 = Graph(4)
g2.addEdge(0, 1)
g2.addEdge(1, 2)
g2.addEdge(2, 3)
print ("\nBridges in second graph ")
g2.bridge()
 
  
g3 = Graph (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)
print ("\nBridges in third graph ")
g3.bridge()
 
 
#This code is contributed by Neelam Yadav


C#




// A C# program to find bridges
// in a given undirected graph
using System;
using System.Collections.Generic;
 
// This class represents a undirected graph 
// using adjacency list representation
public class Graph
{
    private int V; // No. of vertices
 
    // Array of lists for Adjacency List Representation
    private List<int> []adj;
    int time = 0;
    static readonly int NIL = -1;
 
    // Constructor
    Graph(int v)
    {
        V = v;
        adj = new List<int>[v];
        for (int i = 0; i < v; ++i)
            adj[i] = new List<int>();
    }
 
    // Function to add an edge into the graph
    void addEdge(int v, int w)
    {
        adj[v].Add(w); // Add w to v's list.
        adj[w].Add(v); //Add v to w's list
    }
 
    // A recursive function that finds and prints bridges
    // using DFS traversal
    // 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
    void bridgeUtil(int u, bool []visited, int []disc,
                    int []low, int []parent)
    {
 
        // 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
        foreach(int i in adj[u])
        {
            int v = i; // 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 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] = Math.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])
                    Console.WriteLine(u + " " + v);
            }
 
            // Update low value of u for parent function calls.
            else if (v != parent[u])
                low[u] = Math.Min(low[u], disc[v]);
        }
    }
 
 
    // DFS based function to find all bridges. It uses recursive
    // function bridgeUtil()
    void 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, 
        // 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 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 code
    public static void Main(String []args)
    {
        // Create graphs given in above diagrams
        Console.WriteLine("Bridges in first graph ");
        Graph g1 = new Graph(5);
        g1.addEdge(1, 0);
        g1.addEdge(0, 2);
        g1.addEdge(2, 1);
        g1.addEdge(0, 3);
        g1.addEdge(3, 4);
        g1.bridge();
        Console.WriteLine();
 
        Console.WriteLine("Bridges in Second graph");
        Graph g2 = new Graph(4);
        g2.addEdge(0, 1);
        g2.addEdge(1, 2);
        g2.addEdge(2, 3);
        g2.bridge();
        Console.WriteLine();
 
        Console.WriteLine("Bridges in Third graph ");
        Graph g3 = new Graph(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();
    }
}
 
// This code is contributed by Rajput-Ji


Javascript




<script>
// A Javascript program to find bridges in a given undirected graph
 
// This class represents a directed graph using adjacency
// list representation
class Graph
{
    // Constructor
    constructor(v)
    {
        this.V = v;
        this.adj = new Array(v);
         
        this.NIL = -1;
        this.time = 0;
        for (let i=0; i<v; ++i)
            this.adj[i] = [];
    }
     
    //Function to add an edge into the graph
    addEdge(v,w)
    {
        this.adj[v].push(w);  //Note that the graph is undirected.
        this.adj[w].push(v);
    }
     
    // A recursive function that finds and prints bridges
    // using DFS traversal
    // 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
    bridgeUtil(u,visited,disc,low,parent)
    {
        // Mark the current node as visited
        visited[u] = true;
   
        // Initialize discovery time and low value
        disc[u] = low[u] = ++this.time;
   
        // Go through all vertices adjacent to this
         
        for(let i of this.adj[u])
        {
            let v = i;  // 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 v is not visited yet, then recur for it
            if (!visited[v])
            {
                parent[v] = u;
                this.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]  = Math.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])
                    document.write(u+" "+v+"<br>");
            }
   
            // Update low value of u for parent function calls.
            else if (v != parent[u])
                low[u]  = Math.min(low[u], disc[v]);
        }
    }
     
    // DFS based function to find all bridges. It uses recursive
    // function bridgeUtil()
    bridge()
    {
        // Mark all the vertices as not visited
        let visited = new Array(this.V);
        let disc = new Array(this.V);
        let low = new Array(this.V);
        let parent = new Array(this.V);
   
   
        // Initialize parent and visited, and ap(articulation point)
        // arrays
        for (let i = 0; i < this.V; i++)
        {
            parent[i] = this.NIL;
            visited[i] = false;
        }
   
        // Call the recursive helper function to find Bridges
        // in DFS tree rooted with vertex 'i'
        for (let i = 0; i < this.V; i++)
            if (visited[i] == false)
                this.bridgeUtil(i, visited, disc, low, parent);
    }
}
 
// Create graphs given in above diagrams
document.write("Bridges in first graph <br>");
let g1 = new Graph(5);
g1.addEdge(1, 0);
g1.addEdge(0, 2);
g1.addEdge(2, 1);
g1.addEdge(0, 3);
g1.addEdge(3, 4);
g1.bridge();
document.write("<br>");
 
document.write("Bridges in Second graph<br>");
let g2 = new Graph(4);
g2.addEdge(0, 1);
g2.addEdge(1, 2);
g2.addEdge(2, 3);
g2.bridge();
document.write("<br>");
 
document.write("Bridges in Third graph <br>");
let g3 = new Graph(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();
 
// This code is contributed by avanitrachhadiya2155
</script>


Output

Bridges in first graph 
3 4
0 3

Bridges in second graph 
2 3
1 2
0 1

Bridges in third graph 
1 6

Time Complexity: O(V+E), 

  • The above approach uses simple DFS with additional arrays. 
  • So time complexity is the same as DFS which is O(V+E) for adjacency list representation of the graph.

Auxiliary Space: O(BM) where B is the maximum branching factor of the search tree and M is the maximum depth of the state space.


My Personal Notes arrow_drop_up
Related Articles

Start Your Coding Journey Now!