Depth First Search or DFS for a Graph

Depth First Traversal (or Search) for a graph is similar to Depth First Traversal of a tree. The only catch here is, unlike trees, graphs may contain cycles, a node may be visited twice. To avoid processing a node more than once, use a boolean visited array.

Example:

Input: n = 4, e = 6
0 -> 1, 0 -> 2, 1 -> 2, 2 -> 0, 2 -> 3, 3 -> 3
Output: DFS from vertex 1 : 1 2 0 3
Explanation:
DFS Diagram:

Input: n = 4, e = 6
2 -> 0, 0 -> 2, 1 -> 2, 0 -> 1, 3 -> 3, 1 -> 3
Output: DFS from vertex 2 : 2 0 1 3
Explanation:
DFS Diagram:

Prerequisites: See this post for all applications of Depth First Traversal.

Following are implementations of simple Depth First Traversal. The C++ implementation uses adjacency list representation of graphs. STL‘s list container is used to store lists of adjacent nodes.

Solution:

• Approach: Depth-first search is an algorithm for traversing or searching tree or graph data structures. The algorithm starts at the root node (selecting some arbitrary node as the root node in the case of a graph) and explores as far as possible along each branch before backtracking. So the basic idea is to start from the root or any arbitrary node and mark the node and move to the adjacent unmarked node and continue this loop until there is no unmarked adjacent node. Then backtrack and check for other unmarked nodes and traverse them. Finally print the nodes in the path.
• Algorithm:
  1. Create a recursive function that takes the index of node and a visited array.
  2. Mark the current node as visited and print the node.
  3. Traverse all the adjacent and unmarked nodes and call the recursive function with index of adjacent node.
• Implementation:
  

  C++

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  // C++ program to print DFS traversal from // a given vertex in a  given graph #include<bits/stdc++.h> using namespace std;    // Graph class represents a directed graph // using adjacency list representation class Graph {     int V;    // No. of vertices        // Pointer to an array containing     // adjacency lists     list<int> *adj;        // A recursive function used by DFS     void DFSUtil(int v, bool visited[]); public:     Graph(int V);   // Constructor        // function to add an edge to graph     void addEdge(int v, int w);        // DFS traversal of the vertices     // reachable from v     void DFS(int v); };    Graph::Graph(int V) {     this->V = V;     adj = new list<int>[V]; }    void Graph::addEdge(int v, int w) {     adj[v].push_back(w); // Add w to v’s list. }    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); }    // DFS traversal of the vertices reachable from v. // It uses recursive DFSUtil() void Graph::DFS(int v) {     // Mark all the vertices as not visited     bool *visited = new bool[V];     for (int i = 0; i < V; i++)         visited[i] = false;        // Call the recursive helper function     // to print DFS traversal     DFSUtil(v, visited); }    // Driver code int main() {     // Create a graph given in the above diagram     Graph g(4);     g.addEdge(0, 1);     g.addEdge(0, 2);     g.addEdge(1, 2);     g.addEdge(2, 0);     g.addEdge(2, 3);     g.addEdge(3, 3);        cout << "Following is Depth First Traversal"             " (starting from vertex 2) \n";     g.DFS(2);        return 0; }

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  Java

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  // Java program to print DFS traversal from a given given graph import java.io.*; import java.util.*;    // This class represents a directed graph using adjacency list // representation class Graph {     private int V;   // No. of vertices        // Array  of lists for Adjacency List Representation     private LinkedList<Integer> adj[];        // Constructor     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.     }        // A function used by DFS     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         Iterator<Integer> i = adj[v].listIterator();         while (i.hasNext())         {             int n = i.next();             if (!visited[n])                 DFSUtil(n, visited);         }     }        // The function to do DFS traversal. It uses recursive DFSUtil()     void DFS(int v)     {         // Mark all the vertices as not visited(set as         // false by default in java)         boolean visited[] = new boolean[V];            // Call the recursive helper function to print DFS traversal         DFSUtil(v, visited);     }        public static void main(String args[])     {         Graph g = new Graph(4);            g.addEdge(0, 1);         g.addEdge(0, 2);         g.addEdge(1, 2);         g.addEdge(2, 0);         g.addEdge(2, 3);         g.addEdge(3, 3);            System.out.println("Following is Depth First Traversal "+                            "(starting from vertex 2)");            g.DFS(2);     } } // This code is contributed by Aakash Hasija

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  # Python3 program to print DFS traversal  # from a given given graph from collections import defaultdict    # This class represents a directed graph using # adjacency list representation class Graph:        # Constructor     def __init__(self):            # default dictionary to store graph         self.graph = defaultdict(list)        # function to add an edge to graph     def addEdge(self, u, v):         self.graph[u].append(v)        # A function used by DFS     def DFSUtil(self, v, visited):            # Mark the current node as visited          # and print it         visited[v] = True         print(v, end = ' ')            # Recur for all the vertices          # adjacent to this vertex         for i in self.graph[v]:             if visited[i] == False:                 self.DFSUtil(i, visited)        # The function to do DFS traversal. It uses     # recursive DFSUtil()     def DFS(self, v):            # Mark all the vertices as not visited         visited = [False] * (max(self.graph)+1)            # Call the recursive helper function          # to print DFS traversal         self.DFSUtil(v, visited)    # Driver code    # Create a graph given  # in the above diagram g = Graph() g.addEdge(0, 1) g.addEdge(0, 2) g.addEdge(1, 2) g.addEdge(2, 0) g.addEdge(2, 3) g.addEdge(3, 3)    print("Following is DFS from (starting from vertex 2)") g.DFS(2)    # This code is contributed by Neelam Yadav

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  // C# program to print DFS traversal  // from a given graph  using System; using System.Collections.Generic;    // This class represents a directed graph  // using adjacency list representation  class Graph {     private int V; // No. of vertices         // Array of lists for      // Adjacency List Representation      private List<int>[] adj;        // 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.      }        // A function used by DFS      void DFSUtil(int v, bool[] visited)     {         // Mark the current node as visited         // and print it          visited[v] = true;         Console.Write(v + " ");            // Recur for all the vertices          // adjacent to this vertex          List<int> vList = adj[v];         foreach (var n in vList)         {             if (!visited[n])                 DFSUtil(n, visited);         }     }        // The function to do DFS traversal.      // It uses recursive DFSUtil()      void DFS(int v)     {         // Mark all the vertices as not visited         // (set as false by default in c#)          bool[] visited = new bool[V];            // Call the recursive helper function          // to print DFS traversal          DFSUtil(v, visited);     }        // Driver Code     public static void Main(String[] args)     {         Graph g = new Graph(4);            g.AddEdge(0, 1);         g.AddEdge(0, 2);         g.AddEdge(1, 2);         g.AddEdge(2, 0);         g.AddEdge(2, 3);         g.AddEdge(3, 3);            Console.WriteLine("Following is Depth First Traversal " +                           "(starting from vertex 2)");            g.DFS(2);         Console.ReadKey();     } }    // This code is contributed by techno2mahi

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  Output:

  Following is Depth First Traversal (starting from vertex 2)
  2 0 1 3
  

• Complexity Analysis:
  • Time complexity: O(V + E), where V is the number of vertices and E is the number of edges in the graph.
  • Space Complexity: O(V).
    Since, an extra visited array is needed of size V.

Handling Disconnected Graph

• Solution: This will happen by handling a corner case.
  The above code traverses only the vertices reachable from a given source vertex. All the vertices may not be reachable from a given vertex as in the case of a Disconnected graph. To do complete DFS traversal of such graphs, run DFS from all unvisited nodes after a DFS.
  The recursive function remains the same.
• Algorithm:
  1. Create a recursive function that takes the index of node and a visited array.
  2. Mark the current node as visited and print the node.
  3. Traverse all the adjacent and unmarked nodes and call the recursive function with index of adjacent node.
  4. Run a loop from 0 to number of vertices and check if the node is unvisited in previous DFS then call the recursive function with current node.
• Implementation:
  

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  // C++ program to print DFS traversal for a given given graph #include<bits/stdc++.h> using namespace std;    class Graph {     int V;    // No. of vertices     list<int> *adj;    // Pointer to an array containing adjacency lists     void DFSUtil(int v, bool visited[]);  // A function used by DFS public:     Graph(int V);   // Constructor     void addEdge(int v, int w);   // function to add an edge to graph     void DFS();    // prints DFS traversal of the complete graph };    Graph::Graph(int V) {     this->V = V;     adj = new list<int>[V]; }    void Graph::addEdge(int v, int w) {     adj[v].push_back(w); // Add w to v’s list. }    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); }    // The function to do DFS traversal. It uses recursive DFSUtil() void Graph::DFS() {     // Mark all the vertices as not visited     bool *visited = new bool[V];     for (int i = 0; i < V; i++)         visited[i] = false;        // Call the recursive helper function to print DFS traversal     // starting from all vertices one by one     for (int i = 0; i < V; i++)         if (visited[i] == false)             DFSUtil(i, visited); }    int main() {     // Create a graph given in the above diagram     Graph g(4);     g.addEdge(0, 1);     g.addEdge(0, 2);     g.addEdge(1, 2);     g.addEdge(2, 0);     g.addEdge(2, 3);     g.addEdge(3, 3);        cout << "Following is Depth First Traversaln";     g.DFS();        return 0; }

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  // Java program to print DFS traversal from a given given graph import java.io.*; import java.util.*;    // This class represents a directed graph using adjacency list // representation class Graph {     private int V;   // No. of vertices        // Array  of lists for Adjacency List Representation     private LinkedList<Integer> adj[];        // Constructor     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.     }        // A function used by DFS     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         Iterator<Integer> i = adj[v].listIterator();         while (i.hasNext())         {             int n = i.next();             if (!visited[n])                 DFSUtil(n,visited);         }     }        // The function to do DFS traversal. It uses recursive DFSUtil()     void DFS()     {         // Mark all the vertices as not visited(set as         // false by default in java)         boolean visited[] = new boolean[V];            // Call the recursive helper function to print DFS traversal         // starting from all vertices one by one         for (int i=0; i<V; ++i)             if (visited[i] == false)                 DFSUtil(i, visited);     }        public static void main(String args[])     {         Graph g = new Graph(4);            g.addEdge(0, 1);         g.addEdge(0, 2);         g.addEdge(1, 2);         g.addEdge(2, 0);         g.addEdge(2, 3);         g.addEdge(3, 3);            System.out.println("Following is Depth First Traversal");            g.DFS();     } } // This code is contributed by Aakash Hasija

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  # Python program to print DFS traversal for complete graph from collections import defaultdict    # This class represents a directed graph using adjacency # list representation class Graph:        # Constructor     def __init__(self):            # default dictionary to store graph         self.graph = defaultdict(list)        # function to add an edge to graph     def addEdge(self,u,v):         self.graph[u].append(v)        # A function used by DFS     def DFSUtil(self, v, visited):            # Mark the current node as visited and print it         visited[v]= True         print v,            # Recur for all the vertices adjacent to         # this vertex         for i in self.graph[v]:             if visited[i] == False:                 self.DFSUtil(i, visited)           # The function to do DFS traversal. It uses     # recursive DFSUtil()     def DFS(self):         V = len(self.graph)  #total vertices            # Mark all the vertices as not visited         visited =[False]*(V)            # Call the recursive helper function to print         # DFS traversal starting from all vertices one         # by one         for i in range(V):             if visited[i] == False:                 self.DFSUtil(i, visited)       # Driver code # Create a graph given in the above diagram g = Graph() g.addEdge(0, 1) g.addEdge(0, 2) g.addEdge(1, 2) g.addEdge(2, 0) g.addEdge(2, 3) g.addEdge(3, 3)    print "Following is Depth First Traversal" g.DFS()    # This code is contributed by Neelam Yadav

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  // C# program to print DFS traversal from a given given graph  using System; using System.Collections.Generic;    // This class represents a directed 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;         // 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.      }         // A function used by DFS      void DFSUtil(int v, bool []visited)      {          // Mark the current node as visited and print it          visited[v] = true;          Console.Write(v + " ");             // Recur for all the vertices adjacent to this vertex          foreach(int i in adj[v])         {              int n = i;              if (!visited[n])                  DFSUtil(n, visited);          }      }         // The function to do DFS traversal. It uses recursive DFSUtil()      void DFS()      {          // Mark all the vertices as not visited(set as          // false by default in java)          bool []visited = new bool[V];             // Call the recursive helper function to print DFS traversal          // starting from all vertices one by one          for (int i = 0; i < V; ++i)              if (visited[i] == false)                  DFSUtil(i, visited);      }         // Driver code     public static void Main(String []args)      {          Graph g = new Graph(4);             g.addEdge(0, 1);          g.addEdge(0, 2);          g.addEdge(1, 2);          g.addEdge(2, 0);          g.addEdge(2, 3);          g.addEdge(3, 3);             Console.WriteLine("Following is Depth First Traversal");             g.DFS();      }  }     // This code is contributed by PrinciRaj1992

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  Output:

  Following is Depth First Traversal
  0 1 2 3

• Complexity Analysis:
  • Time complexity: O(V + E), where V is the number of vertices and E is the number of edges in the graph.
  • Space Complexity :O(V).
    Since an extra visited array is needed of size V.

• Applications of DFS.
• Breadth First Traversal for a Graph
• Recent Articles on DFS

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