Depth First Search or DFS for a Graph

Last Updated : 29 Mar, 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: [0 1 2 3 4]
Explanation: The source vertex s is 0. We visit it first, then we visit an adjacent.
Start at 0: Mark as visited. Output: 0
Move to 1: Mark as visited. Output: 1
Move to 2: Mark as visited. Output: 2
Move to 3: Mark as visited. Output: 3 (backtrack to 2)
Move to 4: Mark as visited. Output: 4 (backtrack to 2, then backtrack to 1, then to 0)
Not that there can be more than one DFS Traversals of a Graph. For example, after 1, we may pick adjacent 2 instead of 0 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 (backtrack to 0)

Table of Content

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 might be 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.

C++

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

// Recursive function for DFS traversal
void dfsRec(vector<vector<int>> &adj, vector<bool> &visited, int s, vector<int> &res)
{

    visited[s] = true;

    res.push_back(s);

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

// Main DFS function that initializes the visited array
// and call DFSRec
vector<int> DFS(vector<vector<int>> &adj)
{
    vector<bool> visited(adj.size(), false);
    vector<int> res;
    dfsRec(adj, visited, 0, res);
    return res;
}

// 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]);

    // Starting vertex for DFS
    vector<int> res = DFS(adj); // Perform DFS starting from the source verte 0;

    for (int i = 0; i < V; i++)
        cout << res[i] << " ";
}

Java

import java.util.*;

public class DFSGraph {
    // Recursive function for DFS traversal
    private static void
    dfsRec(ArrayList<ArrayList<Integer> > adj,
           boolean[] visited, int s, ArrayList<Integer> res)
    {
        visited[s] = true;
        res.add(s);

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

    // Main DFS function that initializes the visited array
    // and calls dfsRec
    public static ArrayList<Integer>
    DFS(ArrayList<ArrayList<Integer> > adj)
    {
        boolean[] visited = new boolean[adj.size()];
        ArrayList<Integer> res = new ArrayList<>();
        dfsRec(adj, visited, 0, res);
        return res;
    }

    // To add an edge in an undirected graph
    public static void
    addEdge(ArrayList<ArrayList<Integer> > adj, int s,
            int t)
    {
        adj.get(s).add(t);
        adj.get(t).add(s);
    }

    public static void main(String[] args)
    {
        int V = 5;
        ArrayList<ArrayList<Integer> > adj
            = new ArrayList<>();

        // Initialize adjacency list
        for (int i = 0; i < V; i++) {
            adj.add(new ArrayList<>());
        }

        // Add edges
        int[][] edges= { { 1, 2 },{ 1, 0 },{ 2, 0 },{ 2, 3 },{ 2, 4 } }; 
        for (int[] e : edges)
        {
            addEdge(adj, e[0], e[1]);
        }

        // Perform DFS starting from vertex 0
        ArrayList<Integer> res = DFS(adj);

        for (int i = 0; i < res.size(); i++) {
            System.out.print(res.get(i) + " ");
        }
    }
}

Python

def dfsRec(adj, visited, s, res):
    visited[s] = True
    res.append(s)

    # Recursively visit all adjacent vertices that are not visited yet
    for i in range(len(adj)):
        if adj[s][i] == 1 and not visited[i]:
            dfsRec(adj, visited, i, res)


def DFS(adj):
    visited = [False] * len(adj)
    res = []
    dfsRec(adj, visited, 0, res)  # Start DFS from vertex 0
    return res


def add_edge(adj, s, t):
    adj[s][t] = 1
    adj[t][s] = 1  # Since it's an undirected graph


# Driver code
V = 5
adj = [[0] * V for _ in range(V)]  # Adjacency matrix

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

# Populate the adjacency matrix with edges
for s, t in edges:
    add_edge(adj, s, t)

res = DFS(adj)  # Perform DFS
print(" ".join(map(str, res)))

using System;
using System.Collections.Generic;

class DFSGraph {
    // Recursive function for DFS traversal
    static void DfsRec(List<int>[] adj, bool[] visited,
                       int s, List<int> res)
    {
        visited[s] = true;
        res.Add(s);

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

    // Main DFS function that initializes the visited array
    // and calls DfsRec
    static List<int> DFS(List<int>[] adj)
    {
        bool[] visited = new bool[adj.Length];
        List<int> res = new List<int>();
        DfsRec(adj, visited, 0, res);
        return res;
    }

    // To add an edge in an undirected graph
    static void AddEdge(List<int>[] adj, int s, int t)
    {
        adj[s].Add(t);
        adj[t].Add(s);
    }

    static void Main()
    {
        int V = 5;
        List<int>[] adj = new List<int>[ V ];

        // Initialize adjacency list
        for (int i = 0; i < V; i++) {
            adj[i] = new List<int>();
        }

        // Add edges
        int[, ] edges = {
            { 1, 2 }, { 1, 0 }, { 2, 0 }, { 2, 3 }, { 2, 4 }
        };

        for (int i = 0; i < edges.GetLength(0); i++) {
            AddEdge(adj, edges[i, 0], edges[i, 1]);
        }

        // Perform DFS starting from vertex 0
        List<int> res = DFS(adj);

        foreach(int i in res) { Console.Write(i + " "); }
    }
}

JavaScript

function dfsRec(adj, visited, s, res)
{
    visited[s] = true;
    res.push(s);

    // Recursively visit all adjacent vertices that are not
    // visited yet
    for (let i = 0; i < adj.length; i++) {
        if (adj[s][i] === 1 && !visited[i]) {
            dfsRec(adj, visited, i, res);
        }
    }
}

function DFS(adj)
{
    let visited = new Array(adj.length).fill(false);
    let res = [];
    dfsRec(adj, visited, 0, res); // Start DFS from vertex 0
    return res;
}

function addEdge(adj, s, t)
{
    adj[s][t] = 1;
    adj[t][s] = 1; // Since it's an undirected graph
}

// Driver code
let V = 5;
let adj = Array.from(
    {length : V},
    () => new Array(V).fill(0)); // Adjacency matrix

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

// Populate the adjacency matrix with edges
edges.forEach(([ s, t ]) => addEdge(adj, s, t));

let res = DFS(adj); // Perform DFS
console.log(res.join(" "));

Output

0 1 2 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, vector<int> &res)
{
    // Mark the current vertex as visited
    visited[s] = true;

    res.push_back(s);

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

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

    return res;
}

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]);
    vector<int> res = DFS(adj);

    for (auto it : res)
        cout << it << " ";
    return 0;
}

Java

import java.util.*;

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

    // Recursive function for DFS traversal
    private static void
    dfsRec(ArrayList<ArrayList<Integer> > adj,
           boolean[] visited, int s, ArrayList<Integer> res)
    {
        visited[s] = true;
        res.add(s);

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

    // Main DFS function to perform DFS for the entire graph
    public static ArrayList<Integer>
    DFS(ArrayList<ArrayList<Integer> > adj)
    {
        boolean[] visited = new boolean[adj.size()];
        ArrayList<Integer> res = new ArrayList<>();

        // Loop through all vertices to handle disconnected
        // graphs
        for (int i = 0; i < adj.size(); i++) {
            if (!visited[i]) {
                dfsRec(adj, visited, i, res);
            }
        }

        return res;
    }

    public static void main(String[] args)
    {
        int V = 6;
        // Create an adjacency list for the graph
        ArrayList<ArrayList<Integer> > adj
            = new ArrayList<>();

        // Initialize adjacency list
        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[] e : edges) {
            addEdge(adj, e[0], e[1]);
        }

        // Perform DFS
        ArrayList<Integer> res = DFS(adj);

        // Print the DFS traversal result
        for (int num : res) {
            System.out.print(num + " ");
        }
    }
}

Python

# Create an adjacency list for the graph
from collections import defaultdict


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

# Recursive function for DFS traversal


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

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

# Main DFS function to perform DFS for the entire graph


def dfs(adj):
    visited = [False] * len(adj)
    res = []
    # Loop through all vertices to handle disconnected graph
    for i in range(len(adj)):
        if not visited[i]:
            # If vertex i has not been visited,
            # perform DFS from it
            dfs_rec(adj, visited, i, res)
    return res


V = 6
# Create an adjacency list for the graph
adj = defaultdict(list)

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

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

print(' '.join(map(str, res)))

using System;
using System.Collections.Generic;

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

    // Recursive function for DFS traversal
    static void DfsRec(List<int>[] adj, bool[] visited,
                       int s, List<int> res)
    {
        visited[s] = true;
        res.Add(s);

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

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

        // Loop through all vertices to handle disconnected
        // graphs
        for (int i = 0; i < adj.Length; i++) {
            if (!visited[i]) {
                DfsRec(adj, visited, i, res);
            }
        }

        return res;
    }

    static void Main()
    {
        int V = 6;
        // Create an adjacency list for the graph
        List<int>[] adj = new List<int>[ V ];

        // Initialize adjacency list
        for (int i = 0; i < V; i++) {
            adj[i] = new List<int>();
        }

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

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

        // Perform DFS
        List<int> res = DFS(adj);

        // Print the DFS traversal result
        foreach(int num in res)
        {
            Console.Write(num + " ");
        }
    }
}

JavaScript

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

// Recursive function for DFS traversal
function dfsRec(adj, visited, s, res) {
    visited[s] = true;
    res.push(s);

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

// Main DFS function to perform DFS for the entire graph
function DFS(adj) {
    let visited = new Array(adj.length).fill(false);
    let res = [];

    // Loop through all vertices to handle disconnected graphs
    for (let i = 0; i < adj.length; i++) {
        if (!visited[i]) {
            dfsRec(adj, visited, i, res);
        }
    }

    return res;
}

// Main Execution
let V = 6;
// Create an adjacency list for the graph
let adj = Array.from({ length: V }, () => []);

let edges = [[1, 2], [2, 0], [0, 3], [4, 5]];

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

// Perform DFS
let res = DFS(adj);

// Print the DFS traversal result
console.log(res.join(" "));

Output

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.

Related Articles:

C Program for Depth First Search or DFS for a Graph

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Depth First Search or DFS for a Graph

DFS from a Given Source of Undirected Graph:

DFS for Complete Traversal of Disconnected Undirected Graph

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