Connected Components in an undirected graph

Given an undirected graph, print all connected components line by line. For example consider the following graph.

We have discussed algorithms for finding strongly connected components in directed graphs in following posts. 
Kosaraju’s algorithm for strongly connected components. 
Tarjan’s Algorithm to find Strongly Connected Components

Finding connected components for an undirected graph is an easier task. We simple need to do either BFS or DFS starting from every unvisited vertex, and we get all strongly connected components. Below are steps based on DFS.

1) Initialize all vertices as not visited.
2) Do following for every vertex 'v'.
       (a) If 'v' is not visited before, call DFSUtil(v)
       (b) Print new line character

DFSUtil(v)
1) Mark 'v' as visited.
2) Print 'v'
3) Do following for every adjacent 'u' of 'v'.
     If 'u' is not visited, then recursively call DFSUtil(u)

Below is the implementation of above algorithm.

C++

// C++ program to print connected components in
// an undirected graph
#include <iostream>
#include <list>
using namespace std;

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

    // Pointer to an array containing adjacency lists
    list<int>* adj;

    // A function used by DFS
    void DFSUtil(int v, bool visited[]);

public:
    Graph(int V); // Constructor
    ~Graph();
    void addEdge(int v, int w);
    void connectedComponents();
};

// Method to print connected components in an
// undirected graph
void Graph::connectedComponents()
{
    // Mark all the vertices as not visited
    bool* visited = new bool[V];
    for (int v = 0; v < V; v++)
        visited[v] = false;

    for (int v = 0; v < V; v++) {
        if (visited[v] == false) {
            // print all reachable vertices
            // from v
            DFSUtil(v, visited);

            cout << "\n";
        }
    }
    delete[] visited;
}

void Graph::DFSUtil(int v, bool visited[])
{
    // Mark the current node as visited and print it
    visited[v] = true;
    cout << v << " ";

    // Recur for all the vertices
    // adjacent to this vertex
    list<int>::iterator i;
    for (i = adj[v].begin(); i != adj[v].end(); ++i)
        if (!visited[*i])
            DFSUtil(*i, visited);
}

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

Graph::~Graph() { delete[] adj; }

// method to add an undirected edge
void Graph::addEdge(int v, int w)
{
    adj[v].push_back(w);
    adj[w].push_back(v);
}

// Driver code
int main()
{
    // Create a graph given in the above diagram
    Graph g(5); // 5 vertices numbered from 0 to 4
    g.addEdge(1, 0);
    g.addEdge(2, 3);
    g.addEdge(3, 4);

    cout << "Following are connected components \n";
    g.connectedComponents();

    return 0;
}

Java

// Java program to print connected components in
// an undirected graph
import java.util.ArrayList;
class Graph
{
    // A user define class to represent a graph.
    // A graph is an array of adjacency lists.
    // Size of array will be V (number of vertices
    // in graph)
    int V;
    ArrayList<ArrayList<Integer> > adjListArray;

    // constructor
    Graph(int V)
    {
        this.V = V;
        // define the size of array as
        // number of vertices
        adjListArray = new ArrayList<>();

        // Create a new list for each vertex
        // such that adjacent nodes can be stored

        for (int i = 0; i < V; i++) {
            adjListArray.add(i, new ArrayList<>());
        }
    }

    // Adds an edge to an undirected graph
    void addEdge(int src, int dest)
    {
        // Add an edge from src to dest.
        adjListArray.get(src).add(dest);

        // Since graph is undirected, add an edge from dest
        // to src also
        adjListArray.get(dest).add(src);
    }

    void DFSUtil(int v, boolean[] visited)
    {
        // Mark the current node as visited and print it
        visited[v] = true;
        System.out.print(v + " ");
        // Recur for all the vertices
        // adjacent to this vertex
        for (int x : adjListArray.get(v)) {
            if (!visited[x])
                DFSUtil(x, visited);
        }
    }
    void connectedComponents()
    {
        // Mark all the vertices as not visited
        boolean[] visited = new boolean[V];
        for (int v = 0; v < V; ++v) {
            if (!visited[v]) {
                // print all reachable vertices
                // from v
                DFSUtil(v, visited);
                System.out.println();
            }
        }
    }

    // Driver code
    public static void main(String[] args)
    {
        // Create a graph given in the above diagram
        Graph g = new Graph(
            5); // 5 vertices numbered from 0 to 4

        g.addEdge(1, 0);
        g.addEdge(2, 3);
        g.addEdge(3, 4);
        System.out.println(
            "Following are connected components");
        g.connectedComponents();
    }
}

Python3

# Python program to print connected
# components in an undirected graph


class Graph:

    # init function to declare class variables
    def __init__(self, V):
        self.V = V
        self.adj = [[] for i in range(V)]

    def DFSUtil(self, temp, v, visited):

        # Mark the current vertex as visited
        visited[v] = True

        # Store the vertex to list
        temp.append(v)

        # Repeat for all vertices adjacent
        # to this vertex v
        for i in self.adj[v]:
            if visited[i] == False:

                # Update the list
                temp = self.DFSUtil(temp, i, visited)
        return temp

    # method to add an undirected edge
    def addEdge(self, v, w):
        self.adj[v].append(w)
        self.adj[w].append(v)

    # Method to retrieve connected components
    # in an undirected graph
    def connectedComponents(self):
        visited = []
        cc = []
        for i in range(self.V):
            visited.append(False)
        for v in range(self.V):
            if visited[v] == False:
                temp = []
                cc.append(self.DFSUtil(temp, v, visited))
        return cc


# Driver Code
if __name__ == "__main__":

    # Create a graph given in the above diagram
    # 5 vertices numbered from 0 to 4
    g = Graph(5)
    g.addEdge(1, 0)
    g.addEdge(2, 3)
    g.addEdge(3, 4)
    cc = g.connectedComponents()
    print("Following are connected components")
    print(cc)

# This code is contributed by Abhishek Valsan

C#

// C++ program to print connected components in
// an undirected graph
#include <iostream>
#include <list>
using namespace std;

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

    // Pointer to an array containing adjacency lists
    list<int>* adj;

    // A function used by DFS
    void DFSUtil(int v, bool visited[]);

    public : Graph(int V); // Constructor
    ~Graph();
    void addEdge(int v, int w);
    void connectedComponents();
};

// Method to print connected components in an
// undirected graph
void Graph::connectedComponents()
{
    // Mark all the vertices as not visited
    bool* visited = new bool[V];
    for (int v = 0; v < V; v++)
        visited[v] = false;

    for (int v = 0; v < V; v++) {
        if (visited[v] == false) {
            // print all reachable vertices
            // from v
            DFSUtil(v, visited);

            cout << "\n";
        }
    }
    delete[] visited;
}

void Graph::DFSUtil(int v, bool visited[])
{
    // Mark the current node as visited and print it
    visited[v] = true;
    cout << v << " ";

    // Recur for all the vertices
    // adjacent to this vertex
    list<int>::iterator i;
    for (i = adj[v].begin(); i != adj[v].end(); ++i)
        if (!visited[*i])
            DFSUtil(*i, visited);
}

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

Graph::~Graph() { delete[] adj; }

// method to add an undirected edge
void Graph::addEdge(int v, int w)
{
    adj[v].push_back(w);
    adj[w].push_back(v);
}

// Driver code
int main()
{
    // Create a graph given in the above diagram
    Graph g(5); // 5 vertices numbered from 0 to 4
    g.addEdge(1, 0);
    g.addEdge(2, 3);
    g.addEdge(3, 4);

    cout << "Following are connected components \n";
    g.connectedComponents();

    return 0;
}

Javascript

<script>
// Javascript program to print connected components in
// an undirected graph

// A user define class to represent a graph.
    // A graph is an array of adjacency lists.
    // Size of array will be V (number of vertices
    // in graph)
let V;
let adjListArray=[];

// constructor
function Graph(v)
{
    V = v

        // define the size of array as
        // number of vertices

        // Create a new list for each vertex
        // such that adjacent nodes can be stored

        for (let i = 0; i < V; i++) {
            adjListArray.push([]);
        }
}

// Adds an edge to an undirected graph
function addEdge(src,dest)
{
    // Add an edge from src to dest.
        adjListArray[src].push(dest);

        // Since graph is undirected, add an edge from dest
        // to src also
        adjListArray[dest].push(src);
}

function DFSUtil(v,visited)
{
    // Mark the current node as visited and print it
        visited[v] = true;
        document.write(v + " ");

        // Recur for all the vertices
        // adjacent to this vertex
        for (let x = 0; x < adjListArray[v].length; x++)
        {
            if (!visited[adjListArray[v][x]])
                DFSUtil(adjListArray[v][x], visited);
        }
}

function connectedComponents()
{
    // Mark all the vertices as not visited
        let visited = new Array(V);
        for(let i = 0; i < V; i++)
        {
            visited[i] = false;
        }
        for (let v = 0; v < V; ++v)
        {
            if (!visited[v])
            {
                // print all reachable vertices
                // from v
                DFSUtil(v, visited);
                document.write("<br>");
            }
        }
}

// Driver code
Graph(5);

addEdge(1, 0);
addEdge(2, 3);
addEdge(3, 4);
document.write(
"Following are connected components<br>");
connectedComponents();

// This code is contributed by rag2127
</script>

Output

Following are connected components 
0 1 
2 3 4 

Time complexity of above solution is O(V + E) as it does simple DFS for given graph. 

This can be solved using Disjoint Set Union with a time complexity of O(N). The DSU solution is easier to understand and implement.

Algorithm : 

1. Declare an array arr of size n where n is the total number of nodes.
2. For every index i of array arr, the value denotes who the parent of i’th vertex is. For example is arr[0]=3, then we can say that the parent of vertex 0 is 3.
3. Initialise every node as the parent of itself and then while adding them together, change their parents accordingly.

C++

#include <bits/stdc++.h>
using namespace std;
int merge(int* parent, int x)
{
    if (parent[x] == x)
        return x;
    return merge(parent, parent[x]);
}
int connectedcomponents(int n, vector<vector<int> >& edges)
{
    int parent[n];
    for (int i = 0; i < n; i++) {
        parent[i] = i;
    }
    for (auto x : edges) {
        parent[merge(parent, x[0])] = merge(parent, x[1]);
    }
    int ans = 0;
    for (int i = 0; i < n; i++) {
        ans += (parent[i] == i);
    }
    for (int i = 0; i < n; i++) {
        parent[i] = merge(parent, parent[i]);
    }
    map<int, list<int> > m;
    for (int i = 0; i < n; i++) {
        m[parent[i]].push_back(i);
    }
    for (auto it = m.begin(); it != m.end(); it++) {
        list<int> l = it->second;
        for (auto x : l) {
            cout << x << " ";
        }
        cout << endl;
    }
    return ans;
}
int main()
{
    int n = 5;
    vector<int> e1 = { 0, 1 };
    vector<int> e2 = { 2, 3 };
    vector<int> e3 = { 3, 4 };
    vector<vector<int> > e;
    e.push_back(e1);
    e.push_back(e2);
    e.push_back(e3);
    int a = connectedcomponents(n, e);
    cout << "total no. of connected components are: " << a
         << endl;
    return 0;
}

Output

0 1 
2 3 4 
total no. of connected components are: 2