Detect Cycle in a Directed Graph

Given a directed graph, check whether the graph contains a cycle or not. Your function should return true if the given graph contains at least one cycle, else return false.
Example,

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

The diagram clearly shows a cycle 0 -> 2 -> 0

Input:n = 4, e = 4
0 -> 1, 0 -> 2, 1 -> 2, 2 -> 3
Output:No
Explanation:
Diagram:

The diagram clearly shows no cycle

Solution using Depth First Search or DFS

• Approach: Depth First Traversal can be used to detect a cycle in a Graph. DFS for a connected graph produces a tree. There is a cycle in a graph only if there is a back edge present in the graph. A back edge is an edge that is from a node to itself (self-loop) or one of its ancestors in the tree produced by DFS. In the following graph, there are 3 back edges, marked with a cross sign. We can observe that these 3 back edges indicate 3 cycles present in the graph.
  

• For a disconnected graph, Get the DFS forest as output. To detect cycle, check for a cycle in individual trees by checking back edges.
  To detect a back edge, keep track of vertices currently in the recursion stack of function for DFS traversal. If a vertex is reached that is already in the recursion stack, then there is a cycle in the tree. The edge that connects the current vertex to the vertex in the recursion stack is a back edge. Use recStack[] array to keep track of vertices in the recursion stack.
  Dry run of the above approach:
  

In the above image there is a mistake node 1 is making a directed edge to 2 not with 0 please make a note.

• Algorithm:
  1. Create the graph using the given number of edges and vertices.
  2. Create a recursive function that initializes the current index or vertex, visited, and recursion stack.
  3. Mark the current node as visited and also mark the index in recursion stack.
  4. Find all the vertices which are not visited and are adjacent to the current node. Recursively call the function for those vertices, If the recursive function returns true, return true.
  5. If the adjacent vertices are already marked in the recursion stack then return true.
  6. Create a wrapper class, that calls the recursive function for all the vertices and if any function returns true return true. Else if for all vertices the function returns false return false.

Implementation:

C++

// A C++ Program to detect cycle in a 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
    bool isCyclicUtil(int v, bool visited[], bool *rs);  // used by isCyclic()
public:
    Graph(int V);   // Constructor
    void addEdge(int v, int w);   // to add an edge to graph
    bool isCyclic();    // returns true if there is a cycle in this 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.
}

// This function is a variation of DFSUtil() in
// https://www.geeksforgeeks.org/archives/18212
bool Graph::isCyclicUtil(int v, bool visited[], bool *recStack)
{
    if(visited[v] == false)
    {
        // Mark the current node as visited and part of recursion stack
        visited[v] = true;
        recStack[v] = true;

        // 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] && isCyclicUtil(*i, visited, recStack) )
                return true;
            else if (recStack[*i])
                return true;
        }

    }
    recStack[v] = false;  // remove the vertex from recursion stack
    return false;
}

// Returns true if the graph contains a cycle, else false.
// This function is a variation of DFS() in
// https://www.geeksforgeeks.org/archives/18212
bool Graph::isCyclic()
{
    // Mark all the vertices as not visited and not part of recursion
    // stack
    bool *visited = new bool[V];
    bool *recStack = new bool[V];
    for(int i = 0; i < V; i++)
    {
        visited[i] = false;
        recStack[i] = false;
    }

    // Call the recursive helper function to detect cycle in different
    // DFS trees
    for(int i = 0; i < V; i++)
        if ( !visited[i] && isCyclicUtil(i, visited, recStack))
            return true;

    return false;
}

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);

    if(g.isCyclic())
        cout << "Graph contains cycle";
    else
        cout << "Graph doesn't contain cycle";
    return 0;
}

Java

// A Java Program to detect cycle in a graph
import java.util.ArrayList;
import java.util.LinkedList;
import java.util.List;

class Graph {

    private final int V;
    private final List<List<Integer>> adj;

    public Graph(int V)
    {
        this.V = V;
        adj = new ArrayList<>(V);

        for (int i = 0; i < V; i++)
            adj.add(new LinkedList<>());
    }

    // This function is a variation of DFSUtil() in
    // https://www.geeksforgeeks.org/archives/18212
    private boolean isCyclicUtil(int i, boolean[] visited,
                                      boolean[] recStack)
    {

        // Mark the current node as visited and
        // part of recursion stack
        if (recStack[i])
            return true;

        if (visited[i])
            return false;

        visited[i] = true;

        recStack[i] = true;
        List<Integer> children = adj.get(i);

        for (Integer c: children)
            if (isCyclicUtil(c, visited, recStack))
                return true;

        recStack[i] = false;

        return false;
    }

    private void addEdge(int source, int dest) {
        adj.get(source).add(dest);
    }

    // Returns true if the graph contains a
    // cycle, else false.
    // This function is a variation of DFS() in
    // https://www.geeksforgeeks.org/archives/18212
    private boolean isCyclic()
    {

        // Mark all the vertices as not visited and
        // not part of recursion stack
        boolean[] visited = new boolean[V];
        boolean[] recStack = new boolean[V];


        // Call the recursive helper function to
        // detect cycle in different DFS trees
        for (int i = 0; i < V; i++)
            if (isCyclicUtil(i, visited, recStack))
                return true;

        return false;
    }

    // Driver code
    public static void main(String[] args)
    {
        Graph graph = new Graph(4);
        graph.addEdge(0, 1);
        graph.addEdge(0, 2);
        graph.addEdge(1, 2);
        graph.addEdge(2, 0);
        graph.addEdge(2, 3);
        graph.addEdge(3, 3);

        if(graph.isCyclic())
            System.out.println("Graph contains cycle");
        else
            System.out.println("Graph doesn't "
                                    + "contain cycle");
    }
}

// This code is contributed by Sagar Shah.

Python

# Python program to detect cycle
# in a graph

from collections import defaultdict

class Graph():
    def __init__(self,vertices):
        self.graph = defaultdict(list)
        self.V = vertices

    def addEdge(self,u,v):
        self.graph[u].append(v)

    def isCyclicUtil(self, v, visited, recStack):

        # Mark current node as visited and
        # adds to recursion stack
        visited[v] = True
        recStack[v] = True

        # Recur for all neighbours
        # if any neighbour is visited and in
        # recStack then graph is cyclic
        for neighbour in self.graph[v]:
            if visited[neighbour] == False:
                if self.isCyclicUtil(neighbour, visited, recStack) == True:
                    return True
            elif recStack[neighbour] == True:
                return True

        # The node needs to be popped from
        # recursion stack before function ends
        recStack[v] = False
        return False

    # Returns true if graph is cyclic else false
    def isCyclic(self):
        visited = [False] * (self.V + 1)
        recStack = [False] * (self.V + 1)
        for node in range(self.V):
            if visited[node] == False:
                if self.isCyclicUtil(node,visited,recStack) == True:
                    return True
        return False

g = 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)
if g.isCyclic() == 1:
    print "Graph has a cycle"
else:
    print "Graph has no cycle"

# Thanks to Divyanshu Mehta for contributing this code

C#

// A C# Program to detect cycle in a graph
using System;
using System.Collections.Generic;

public class Graph {

    private readonly int V;
    private readonly List<List<int>> adj;

    public Graph(int V)
    {
        this.V = V;
        adj = new List<List<int>>(V);

        for (int i = 0; i < V; i++)
            adj.Add(new List<int>());
    }

    // This function is a variation of DFSUtil() in
    // https://www.geeksforgeeks.org/archives/18212
    private bool isCyclicUtil(int i, bool[] visited,
                                    bool[] recStack)
    {

        // Mark the current node as visited and
        // part of recursion stack
        if (recStack[i])
            return true;

        if (visited[i])
            return false;

        visited[i] = true;

        recStack[i] = true;
        List<int> children = adj[i];

        foreach (int c in children)
            if (isCyclicUtil(c, visited, recStack))
                return true;

        recStack[i] = false;

        return false;
    }

    private void addEdge(int sou, int dest) {
        adj[sou].Add(dest);
    }

    // Returns true if the graph contains a
    // cycle, else false.
    // This function is a variation of DFS() in
    // https://www.geeksforgeeks.org/archives/18212
    private bool isCyclic()
    {

        // Mark all the vertices as not visited and
        // not part of recursion stack
        bool[] visited = new bool[V];
        bool[] recStack = new bool[V];


        // Call the recursive helper function to
        // detect cycle in different DFS trees
        for (int i = 0; i < V; i++)
            if (isCyclicUtil(i, visited, recStack))
                return true;

        return false;
    }

    // Driver code
    public static void Main(String[] args)
    {
        Graph graph = new Graph(4);
        graph.addEdge(0, 1);
        graph.addEdge(0, 2);
        graph.addEdge(1, 2);
        graph.addEdge(2, 0);
        graph.addEdge(2, 3);
        graph.addEdge(3, 3);

        if(graph.isCyclic())
            Console.WriteLine("Graph contains cycle");
        else
            Console.WriteLine("Graph doesn't "
                                    + "contain cycle");
    }
}

// This code contributed by Rajput-Ji

Javascript

<script>

// A JavaScript Program to detect cycle in a graph

let V;
let adj=[];
function Graph(v)
{
    V=v;
    for (let i = 0; i < V; i++)
        adj.push([]);
}

// This function is a variation of DFSUtil() in
    // https://www.geeksforgeeks.org/archives/18212
function isCyclicUtil(i,visited,recStack)
{
    // Mark the current node as visited and
        // part of recursion stack
        if (recStack[i])
            return true;

        if (visited[i])
            return false;

        visited[i] = true;

        recStack[i] = true;
        let children = adj[i];

        for (let c=0;c< children.length;c++)
            if (isCyclicUtil(children, visited, recStack))
                return true;

        recStack[i] = false;

        return false;
}

function addEdge(source,dest)
{
    adj.push(dest);
}

// Returns true if the graph contains a
    // cycle, else false.
    // This function is a variation of DFS() in
    // https://www.geeksforgeeks.org/archives/18212
function isCyclic()
{
    // Mark all the vertices as not visited and
        // not part of recursion stack
        let visited = new Array(V);
        let recStack = new Array(V);
        for(let i=0;i<V;i++)
        {
            visited[i]=false;
            recStack[i]=false;
        }


        // Call the recursive helper function to
        // detect cycle in different DFS trees
        for (let i = 0; i < V; i++)
            if (isCyclicUtil(i, visited, recStack))
                return true;

        return false;
}

// Driver code
Graph(4);
addEdge(0, 1);
addEdge(0, 2);
addEdge(1, 2);
addEdge(2, 0);
addEdge(2, 3);
addEdge(3, 3);

if(isCyclic())
    document.write("Graph contains cycle");
else
    document.write("Graph doesn't "
                   + "contain cycle");


// This code is contributed by patel2127

</script>

Output:

Graph contains cycle

• Complexity Analysis:
  • Time Complexity: O(V+E). 
    Time Complexity of this method is same as time complexity of DFS traversal which is O(V+E).
  • Space Complexity: O(V). 
    To store the visited and recursion stack O(V) space is needed.

In the below article, another O(V + E) method is discussed : 
Detect Cycle in a direct graph using colors