Depth First Search or DFS for a Graph

Last Updated : 04 Jan, 2025

In Depth First Search (or DFS) for a graph, we traverse all adjacent vertices one by one. When we traverse an adjacent vertex, we completely finish the traversal of all vertices reachable through that adjacent vertex. This is similar to a tree, where we first completely traverse the left subtree and then move to the right subtree. The key difference is that, unlike trees, graphs may contain cycles (a node may be visited more than once). To avoid processing a node multiple times, we use a boolean visited array.

Example:

Note : There can be multiple DFS traversals of a graph according to the order in which we pick adjacent vertices. Here we pick vertices as per the insertion order.

Input: adj = [[1, 2], [0, 2], [0, 1, 3, 4], [2], [2]]

Output: 1 2 0 3 4
Explanation: The source vertex s is 1. We visit it first, then we visit an adjacent.
Start at 1: Mark as visited. Output: 1
Move to 2: Mark as visited. Output: 2
Move to 0: Mark as visited. Output: 0 (backtrack to 2)
Move to 3: Mark as visited. Output: 3 (backtrack to 2)
Move to 4: Mark as visited. Output: 4 (backtrack to 1)

Not that there can be more than one DFS Traversals of a Graph. For example, after 1, we may pick adjacent 0 instead of 2 and get a different DFS. Here we pick in the insertion order.

Input: [[2,3,1], [0], [0,4], [0], [2]]

Output: 0 2 4 3 1
Explanation: DFS Steps:

Start at 0: Mark as visited. Output: 0
Move to 2: Mark as visited. Output: 2
Move to 4: Mark as visited. Output: 4 (backtrack to 2, then backtrack to 0)
Move to 3: Mark as visited. Output: 3 (backtrack to 0)
Move to 1: Mark as visited. Output: 1

DFS from a Given Source of Undirected Graph:

The algorithm starts from a given source and explores all reachable vertices from the given source. It is similar to Preorder Tree Traversal where we visit the root, then recur for its children. In a graph, there maybe loops. So we use an extra visited array to make sure that we do not process a vertex again.

Let us understand the working of Depth First Search with the help of the following illustration: for the source as 0.

Below is the implementation of the above approach:

C++

#include <bits/stdc++.h>
using namespace std;

// Recursive function for DFS traversal
void DFSRec(vector<vector<int>> &adj, vector<bool> &visited, int s){
  
    visited[s] = true;

    // Print the current vertex
    cout << s << " ";

    // Recursively visit all adjacent vertices
    // that are not visited yet
    for (int i : adj[s])
        if (visited[i] == false)
            DFSRec(adj, visited, i);
}

// Main DFS function that initializes the visited array
// and call DFSRec
void DFS(vector<vector<int>> &adj, int s){
    vector<bool> visited(adj.size(), false);
    DFSRec(adj, visited, s);
}

// To add an edge in an undirected graph
void addEdge(vector<vector<int>> &adj, int s, int t){
    adj[s].push_back(t); 
    adj[t].push_back(s); 
}

int main(){
    int V = 5; 
    vector<vector<int>> adj(V);
  
    // Add edges
    vector<vector<int>> edges={{1, 2},{1, 0},{2, 0},{2, 3},{2, 4}};
    for (auto &e : edges)
        addEdge(adj, e[0], e[1]);

    int s = 1; // Starting vertex for DFS
    cout << "DFS from source: " << s << endl;
    DFS(adj, s); // Perform DFS starting from the source vertex

    return 0;
}

#include <stdio.h>
#include <stdlib.h>

// Node structure for adjacency list
struct Node {
    int dest;
    struct Node* next;
};

// Structure to represent an adjacency list
struct AdjList {
    struct Node* head;
};

// Function to create a new adjacency list node
struct Node* createNode(int dest) {
    struct Node* newNode =
      (struct Node*)malloc(sizeof(struct Node));
    newNode->dest = dest;
    newNode->next = NULL;
    return newNode;
}

// Recursive function for DFS traversal
void DFSRec(struct AdjList adj[], int visited[], int s) {
    // Mark the current vertex as visited
    visited[s] = 1;

    // Print the current vertex
    printf("%d ", s);

    // Traverse all adjacent vertices that are not visited yet
    struct Node* current = adj[s].head;
    while (current != NULL) {
        int dest = current->dest;
        if (!visited[dest]) {
            DFSRec(adj, visited, dest);
        }
        current = current->next;
    }
}

// Main DFS function that initializes the visited array
// and calls DFSRec
void DFS(struct AdjList adj[], int V, int s) {
    // Initialize visited array to false
    int visited[5] = {0}; 
    DFSRec(adj, visited, s);
}

// Function to add an edge to the adjacency list
void addEdge(struct AdjList adj[], int s, int t) {
    // Add edge from s to t
    struct Node* newNode = createNode(t);
    newNode->next = adj[s].head;
    adj[s].head = newNode;

    // Due to undirected Graph
    newNode = createNode(s);
    newNode->next = adj[t].head;
    adj[t].head = newNode;
}

int main() {
    int V = 5;

    // Create an array of adjacency lists
    struct AdjList adj[V];

    // Initialize each adjacency list as empty
    for (int i = 0; i < V; i++) {
        adj[i].head = NULL;
    }
    
    int E = 5;
    // Define the edges of the graph
    int edges[][2] = {{1, 2}, {1, 0}, {2, 0}, {2, 3}, {2, 4}};

    // Populate the adjacency list with edges
    for (int i = 0; i < E; i++) {
        addEdge(adj, edges[i][0], edges[i][1]);
    }

    int source = 1; 
    printf("DFS from source: %d\n", source);
    DFS(adj, V, source);

    return 0;
}

Java

import java.util.ArrayList;
import java.util.List;

class GfG {

   // Recursive function for DFS traversal
    static void DFSRec(List<List<Integer> > adj,
                              boolean[] visited, int s){
        // Mark the current vertex as visited
        visited[s] = true;

        // Print the current vertex
        System.out.print(s + " ");

        // Recursively visit all adjacent vertices that are
        // not visited yet
        for (int i : adj.get(s)) {
            if (!visited[i]) {
                DFSRec(adj, visited, i);
            }
        }
    }

    // Main DFS function that initializes the visited array
    static void DFS(List<List<Integer> > adj, int s) {
        boolean[] visited = new boolean[adj.size()];
        // Call the recursive DFS function
        DFSRec(adj, visited, s);
    }
  
    // Function to add an edge to the adjacency list
    static void addEdge(List<List<Integer> > adj,
                               int s, int t){
        // Add edge from vertex s to t
        adj.get(s).add(t);
        // Due to undirected Graph
        adj.get(t).add(s);
    }


    public static void main(String[] args)
    {
        int V = 5; // Number of vertices in the graph

        // Create an adjacency list for the graph
        List<List<Integer> > adj = new ArrayList<>(V);
        for (int i = 0; i < V; i++) {
            adj.add(new ArrayList<>());
        }

        // Define the edges of the graph
        int[][] edges = {
            { 1, 2 }, { 1, 0 }, { 2, 0 }, { 2, 3 }, { 2, 4 }
        };

        // Populate the adjacency list with edges
        for (int[] e : edges) {
            addEdge(adj, e[0], e[1]);
        }

        int source = 1; 
        System.out.println("DFS from source: " + source);
        DFS(adj, source); 
    }
}

Python

def dfs_rec(adj, visited, s):
    # Mark the current vertex as visited
    visited[s] = True

    # Print the current vertex
    print(s, end=" ")

    # Recursively visit all adjacent vertices
    # that are not visited yet
    for i in adj[s]:
        if not visited[i]:
            dfs_rec(adj, visited, i)


def dfs(adj, s):
    visited = [False] * len(adj)
    # Call the recursive DFS function
    dfs_rec(adj, visited, s)

def add_edge(adj, s, t):
    # Add edge from vertex s to t
    adj[s].append(t)
    # Due to undirected Graph
    adj[t].append(s)
    
if __name__ == "__main__":
    V = 5

    # Create an adjacency list for the graph
    adj = [[] for _ in range(V)]

    # Define the edges of the graph
    edges = [[1, 2], [1, 0], [2, 0], [2, 3], [2, 4]]

    # Populate the adjacency list with edges
    for e in edges:
        add_edge(adj, e[0], e[1])

    source = 1
    print("DFS from source:", source)
    dfs(adj, source)

using System;
using System.Collections.Generic;

class GfG{

  // Recursive function for DFS traversal
    static void DFSRec(List<List<int>> adj, bool[] visited, int s){
      
        // Mark the current vertex as visited
        visited[s] = true;

        // Print the current vertex
        Console.Write(s + " ");

        // Recursively visit all adjacent vertices
        // that are not visited yet
        foreach (int i in adj[s]){
            if (!visited[i]){
                DFSRec(adj, visited, i);
            }
        }
    }

    // Main DFS function that initializes the visited array
    static void PerformDFS(List<List<int>> adj, int s){
        bool[] visited = new bool[adj.Count];
        // Call the recursive DFS function
        DFSRec(adj, visited, s);
    }  

    static void AddEdge(List<List<int>> adj, int s, int t){
        adj[s].Add(t);
        adj[t].Add(s);
    }

    static void Main(){
        int V = 5; 

        // Create an adjacency list for the graph
        List<List<int>> adj = new List<List<int>>(V);
        for (int i = 0; i < V; i++){
            adj.Add(new List<int>());
        }

        // Define the edges of the graph
        int[,] edges = {
            { 1, 2 }, { 1, 0 }, { 2, 0 }, { 2, 3 }, { 2, 4 }
        };

        // Populate the adjacency list with edges
        for (int i = 0; i < edges.GetLength(0); i++){
            AddEdge(adj, edges[i, 0], edges[i, 1]);
        }

        int source = 1; // Starting vertex for DFS
        Console.WriteLine("DFS from source: " + source);
        PerformDFS(adj, source);
    }
}

JavaScript

function dfsRec(adj, visited, s)
{
    // Mark the current vertex as visited
    visited[s] = true;

    // Print the current vertex
    process.stdout.write(s + " ");

    // Recursively visit all adjacent vertices that are not
    // visited yet
    for (let i of adj[s]) {
        if (!visited[i]) {
            dfsRec(adj, visited, i);
        }
    }
}

function dfs(adj, s)
{
    const visited = new Array(adj.length).fill(false);
    
    // Call the recursive DFS function
    dfsRec(adj, visited, s);
}

function addEdge(adj, s, t)
{
    // Add edge from vertex s to t
    adj[s].push(t);
    // Due to undirected Graph
    adj[t].push(s);
}


const V = 5; // Number of vertices in the graph

// Create an adjacency list for the graph
const adj = Array.from({length : V}, () => []);

// Define the edges of the graph
const edges =
    [ [ 1, 2 ], [ 1, 0 ], [ 2, 0 ], [ 2, 3 ], [ 2, 4 ] ];

// Populate the adjacency list with edges
for (let e of edges) {
    addEdge(adj, e[0], e[1]);
}

const source = 1;
console.log("DFS from source: " + source);
dfs(adj, source);

Output

DFS from source: 1
1 2 0 3 4

Time complexity: O(V + E), where V is the number of vertices and E is the number of edges in the graph.
Auxiliary Space: O(V + E), since an extra visited array of size V is required, And stack size for recursive calls to DFSRec function.

Please refer Complexity Analysis of Depth First Search: for details.

DFS for Complete Traversal of Disconnected Undirected Graph

The above implementation takes a source as an input and prints only those vertices that are reachable from the source and would not print all vertices in case of disconnected graph. Let us now talk about the algorithm that prints all vertices without any source and the graph maybe disconnected.

The idea is simple, instead of calling DFS for a single vertex, we call the above implemented DFS for all all non-visited vertices one by one.

C++

#include <bits/stdc++.h>
using namespace std;

void addEdge(vector<vector<int>> &adj, int s, int t){
    adj[s].push_back(t);
    adj[t].push_back(s);
}

// Recursive function for DFS traversal
void DFSRec(vector<vector<int>> &adj, vector<bool> &visited,int s){
    // Mark the current vertex as visited
    visited[s] = true;

    // Print the current vertex
    cout << s << " ";

    // Recursively visit all adjacent vertices that are not visited yet
    for (int i : adj[s])
        if (visited[i] == false)
            DFSRec(adj, visited, i);
}

// Main DFS function to perform DFS for the entire graph
void DFS(vector<vector<int>> &adj){
    vector<bool> visited(adj.size(), false);

    // Loop through all vertices to handle disconnected graph
    for (int i = 0; i < adj.size(); i++){
        if (visited[i] == false){
            // If vertex i has not been visited,
            // perform DFS from it
            DFSRec(adj, visited, i);
        }
    }
}

int main(){
    int V = 6;
    // Create an adjacency list for the graph
    vector<vector<int>> adj(V);

    // Define the edges of the graph
    vector<vector<int>> edges = {{1, 2}, {2, 0}, {0, 3}, {4, 5}};

    // Populate the adjacency list with edges
    for (auto &e : edges)
        addEdge(adj, e[0], e[1]);

    cout << "Complete DFS of the graph:" << endl;
    DFS(adj);

    return 0;
}

#include <stdio.h>
#include <stdlib.h>

// Node structure for adjacency list
struct Node {
    int dest;
    struct Node* next;
};

// Structure to represent an adjacency list
struct AdjList {
    struct Node* head;
};

// Function to create a new adjacency list node
struct Node* createNode(int dest) {
    struct Node* newNode = (struct Node*)malloc(sizeof(struct Node));
    newNode->dest = dest;
    newNode->next = NULL;
    return newNode;
}

// Function to add an edge to the adjacency list
void addEdge(struct AdjList adj[], int s, int t) {
    // Add edge from s to t
    struct Node* newNode = createNode(t);
    newNode->next = adj[s].head;
    adj[s].head = newNode;

    // Add edge from t to s (for undirected graph)
    newNode = createNode(s);
    newNode->next = adj[t].head;
    adj[t].head = newNode;
}

// Recursive function for DFS traversal
void DFSRec(struct AdjList adj[], int visited[], int s) {
    // Mark the current vertex as visited
    visited[s] = 1;

    // Print the current vertex
    printf("%d ", s);

    // Traverse all adjacent vertices that are not visited yet
    struct Node* current = adj[s].head;
    while (current != NULL) {
        int dest = current->dest;
        if (!visited[dest]) {
            DFSRec(adj, visited, dest);
        }
        current = current->next;
    }
}

// Main DFS function to perform DFS for the entire graph
void DFS(struct AdjList adj[], int V) {
    int visited[6] = {0}; // Initialize visited array to false

    // Loop through all vertices to handle disconnected graph
    for (int i = 0; i < V; i++) {
        if (visited[i] == 0) {
            // If vertex i has not been visited,
            // perform DFS from it
            DFSRec(adj, visited, i);
        }
    }
}

int main() {
    int V = 6;

    // Create an array of adjacency lists
    struct AdjList adj[V];

    // Initialize each adjacency list as empty
    for (int i = 0; i < V; i++) {
        adj[i].head = NULL;
    }
    
      const int E = 4;
    // Define the edges of the graph
    int edges[][2] = {{1, 2}, {2, 0}, {0, 3}, {4, 5}};

    // Populate the adjacency list with edges
    for (int i = 0; i < E; i++) {
        addEdge(adj, edges[i][0], edges[i][1]);
    }

    printf("Complete DFS of the graph:\n");
    DFS(adj, V);

    return 0;
}

Java

import java.util.ArrayList;
import java.util.List;

class GfG {
    // Function to add an edge to the adjacency list
    static void addEdge(List<List<Integer> > adj, int s,
                        int t){
        adj.get(s).add(t);
        adj.get(t).add(s);
    }

    // Recursive function for DFS traversal
    static void DFSRec(List<List<Integer> > adj,
                       boolean[] visited, int s){
        visited[s] = true; 
        System.out.print(s + " "); 

        // Recursively visit all adjacent vertices that are
        // not visited yet
        for (int i : adj.get(s)) {
            if (!visited[i]) {
                DFSRec(adj, visited, i);
            }
        }
    }
    // Main DFS function to perform DFS for the entire graph
    static void DFS(List<List<Integer> > adj, int V){
        boolean[] visited = new boolean[V];

        // Loop through all vertices to handle disconnected
        // graph
        for (int i = 0; i < V; i++) {
            if (!visited[i]) {
                DFSRec(adj, visited, i);
            }
        }
    }
    public static void main(String[] args){
        int V = 6;

        // Create an adjacency list for the graph
        List<List<Integer> > adj = new ArrayList<>();
        for (int i = 0; i < V; i++) {
            adj.add(new ArrayList<>());
        }

        // Define the edges of the graph
        int[][] edges
            = { { 1, 2 }, { 2, 0 }, { 0, 3 }, { 4, 5 } };

        // Populate the adjacency list with edges
        for (int[] edge : edges) {
            addEdge(adj, edge[0], edge[1]);
        }

        System.out.println("Complete DFS of the graph:");
        DFS(adj, V);
    }
}

Python

class Graph:
    def __init__(self, vertices):
        # Adjacency list
        self.adj = [[] for _ in range(vertices)]  

    def add_edge(self, s, t):
        self.adj[s].append(t)  
        self.adj[t].append(s) 

    def dfs_rec(self, visited, s):
        visited[s] = True 
        print(s, end=" ") 

        # Recursively visit all adjacent vertices
        # that are not visited yet
        for i in self.adj[s]:
            if not visited[i]:
                self.dfs_rec(visited, i)

    def dfs(self):
        visited = [False] * len(self.adj) 

        # Loop through all vertices to handle disconnected
        # graph
        for i in range(len(self.adj)):
            if not visited[i]:
                  # Perform DFS from unvisited vertex
                self.dfs_rec(visited, i)


if __name__ == "__main__":
    V = 6  # Number of vertices
    graph = Graph(V)

    # Define the edges of the graph
    edges = [(1, 2), (2, 0), (0, 3), (4, 5)]

    # Populate the adjacency list with edges
    for edge in edges:
        graph.add_edge(edge[0], edge[1])

    print("Complete DFS of the graph:")
    graph.dfs()  # Perform DFS

using System;
using System.Collections.Generic;

class Program
{
    // Function to add an edge to the adjacency list
    static void AddEdge(List<List<int>> adj, (int, int) edge)
    {
        adj[edge.Item1].Add(edge.Item2); 
        adj[edge.Item2].Add(edge.Item1);
    }

    // Recursive function for DFS traversal
    static void DFSRec(List<List<int>> adj, bool[] visited, int s){
        // Mark the current vertex as visited
        visited[s] = true; 
      
        // Print the current vertex
        Console.Write(s + " "); 

        // Recursively visit all adjacent vertices
        // that are not visited yet
        foreach (int i in adj[s]){
            if (!visited[i]){
                DFSRec(adj, visited, i); // Recursive call
            }
        }
    }

    // Main DFS function to perform DFS for the entire graph
    static void DFS(List<List<int>> adj){
        bool[] visited = new bool[adj.Count]; 

        // Loop through all vertices to handle
        // disconnected graph
        for (int i = 0; i < adj.Count; i++){
            if (!visited[i]){
                // If vertex i has not been visited,
                // perform DFS from it
                DFSRec(adj, visited, i);
            }
        }
    }

    static void Main(){
        int V = 6; 
      
        // Create an adjacency list for the graph
        var adj = new List<List<int>>(new List<int>[V]);
        for (int i = 0; i < V; i++){
            adj[i] = new List<int>(); // Initialize each list
        }

        // Define the edges of the graph using tuples
        var edges = new List<(int, int)>{(1, 2),(2, 0),(0, 3),(4, 5)};

        // Populate the adjacency list with edges
        foreach (var edge in edges)
        {
            AddEdge(adj, edge);
        }

        Console.WriteLine("Complete DFS of the graph:");
        DFS(adj);
    }
}

JavaScript

function addEdge(adj, s, t){
    adj[s].push(t);
    adj[t].push(s);
}

// Recursive function for DFS traversal
function DFSRec(adj, visited, s)
{
    visited[s] = true; 
    console.log(s + " "); 

    // Recursively visit all adjacent vertices that are not
    // visited yet
    adj[s].forEach(i => {
        if (!visited[i]) {
            DFSRec(adj, visited, i);
        }
    });
}

// Main DFS function to perform DFS for the entire graph
function DFS(adj, V)
{
    let visited = new Array(V).fill(false);

    // Loop through all vertices to handle disconnected
    // graph
    for (let i = 0; i < V; i++) {
        if (!visited[i]) {
            DFSRec(adj, visited, i);
        }
    }
}

// Driver code
let V = 6;

// Create an adjacency list for the graph
let adj = new Array(V);
for (let i = 0; i < V; i++) {
    adj[i] = [];
}

// Define the edges of the graph
let edges = [ [ 1, 2 ], [ 2, 0 ], [ 0, 3 ], [ 4, 5 ] ];

// Populate the adjacency list with edges
edges.forEach(edge => { addEdge(adj, edge[0], edge[1]); });

console.log("Complete DFS of the graph:");
DFS(adj, V);

Output

Complete DFS of the graph:
0 2 1 3 4 5

Time complexity: O(V + E). Note that the time complexity is same here because we visit every vertex at most once and every edge is traversed at most once (in directed) and twice in undirected.
Auxiliary Space: O(V + E), since an extra visited array of size V is required, And stack size for recursive calls to DFSRec function.

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