# Detect Cycle in a Directed Graph using BFS

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. For example, the following graph contains two cycles 0->1->2->3->0 and 2->4->2, so your function must return true.

We have discussed a DFS based solution to detect cycle in a directed graph. In this post, BFS based solution is discussed.
The idea is to simply use Kahn’s algorithm for Topological Sorting

Steps involved in detecting cycle in a directed graph using BFS.
Step-1: Compute in-degree (number of incoming edges) for each of the vertex present in the graph and initialize the count of visited nodes as 0.
Step-2: Pick all the vertices with in-degree as 0 and add them into a queue (Enqueue operation)
Step-3: Remove a vertex from the queue (Dequeue operation) and then.

1. Increment count of visited nodes by 1.
2. Decrease in-degree by 1 for all its neighboring nodes.
3. If in-degree of a neighboring nodes is reduced to zero, then add it to the queue.

Step 4: Repeat Step 3 until the queue is empty.
Step 5: If count of visited nodes is not equal to the number of nodes in the graph has cycle, otherwise not.

How to find in-degree of each node?
There are 2 ways to calculate in-degree of every vertex:
Take an in-degree array which will keep track of
1) Traverse the array of edges and simply increase the counter of the destination node by 1.

```for each node in Nodes
indegree[node] = 0;
for each edge(src,dest) in Edges
indegree[dest]++```

Time Complexity: O(V+E)

2) Traverse the list for every node and then increment the in-degree of all the nodes connected to it by 1.

```    for each node in Nodes
If (list[node].size()!=0) then
for each dest in list
indegree[dest]++;```

Time Complexity: The outer for loop will be executed V number of times and the inner for loop will be executed E number of times, Thus overall time complexity is O(V+E).

The overall time complexity of the algorithm is O(V+E)

## C++

 `// A C++ program to check if there is a cycle in ` `// directed graph using BFS.` `#include ` `using` `namespace` `std;`   `// Class to represent a graph` `class` `Graph {` `    ``int` `V; ``// No. of vertices'`   `    ``// Pointer to an array containing adjacency list` `    ``list<``int``>* adj;`   `public``:` `    ``Graph(``int` `V); ``// Constructor`   `    ``// function to add an edge to graph` `    ``void` `addEdge(``int` `u, ``int` `v);`   `    ``// Returns true if there is a cycle in the graph` `    ``// else false.` `    ``bool` `isCycle();` `};`   `Graph::Graph(``int` `V)` `{` `    ``this``->V = V;` `    ``adj = ``new` `list<``int``>[V];` `}`   `void` `Graph::addEdge(``int` `u, ``int` `v)` `{` `    ``adj[u].push_back(v);` `}`   `// This function returns true if there is a cycle` `// in directed graph, else returns false.` `bool` `Graph::isCycle()` `{` `    ``// Create a vector to store indegrees of all` `    ``// vertices. Initialize all indegrees as 0.` `    ``vector<``int``> in_degree(V, 0);`   `    ``// Traverse adjacency lists to fill indegrees of` `    ``// vertices. This step takes O(V+E) time` `    ``for` `(``int` `u = 0; u < V; u++) {` `        ``for` `(``auto` `v : adj[u])` `            ``in_degree[v]++;` `    ``}`   `    ``// Create an queue and enqueue all vertices with` `    ``// indegree 0` `    ``queue<``int``> q;` `    ``for` `(``int` `i = 0; i < V; i++)` `        ``if` `(in_degree[i] == 0)` `            ``q.push(i);`   `    ``// Initialize count of visited vertices` `    ``// 1 For src Node ` `    ``int` `cnt = 1;`   `    ``// Create a vector to store result (A topological` `    ``// ordering of the vertices)` `    ``vector<``int``> top_order;`   `    ``// One by one dequeue vertices from queue and enqueue` `    ``// adjacents if indegree of adjacent becomes 0` `    ``while` `(!q.empty()) {`   `        ``// Extract front of queue (or perform dequeue)` `        ``// and add it to topological order` `        ``int` `u = q.front();` `        ``q.pop();` `        ``top_order.push_back(u);`   `        ``// Iterate through all its neighbouring nodes` `        ``// of dequeued node u and decrease their in-degree` `        ``// by 1` `        ``list<``int``>::iterator itr;` `        ``for` `(itr = adj[u].begin(); itr != adj[u].end(); itr++)`   `            ``// If in-degree becomes zero, add it to queue` `            ``if` `(--in_degree[*itr] == 0)` `            ``{` `              ``q.push(*itr);` `              ``//while we are pushing elements to the queue we will incrementing the cnt` `              ``cnt++;` `            ``} `   `       `  `    ``}`   `    ``// Check if there was a cycle` `    ``if` `(cnt != V) ` `        ``return` `true``;` `    ``else` `        ``return` `false``;` `}`   `// Driver program to test above functions` `int` `main()` `{` `    ``// Create a graph given in the above diagram` `    ``Graph g(6);` `    ``g.addEdge(0, 1);` `    ``g.addEdge(1, 2);` `    ``g.addEdge(2, 0);` `    ``g.addEdge(3, 4);` `    ``g.addEdge(4, 5);`   `    ``if` `(g.isCycle())` `        ``cout << ``"Yes"``;` `    ``else` `        ``cout << ``"No"``;`   `    ``return` `0;` `}`

## Java

 `// Java program to check if there is a cycle in ` `// directed graph using BFS.` `import` `java.io.*;` `import` `java.util.*;`   `class` `GFG` `{`   `    ``// Class to represent a graph` `    ``static` `class` `Graph` `    ``{` `        ``int` `V; ``// No. of vertices'`   `        ``// Pointer to an array containing adjacency list` `        ``Vector[] adj;`   `        ``@SuppressWarnings``(``"unchecked"``)` `        ``Graph(``int` `V)` `        ``{` `            ``// Constructor` `            ``this``.V = V;` `            ``this``.adj = ``new` `Vector[V];` `            ``for` `(``int` `i = ``0``; i < V; i++)` `                ``adj[i] = ``new` `Vector<>();` `        ``}`   `        ``// function to add an edge to graph` `        ``void` `addEdge(``int` `u, ``int` `v)` `        ``{` `            ``adj[u].add(v);` `        ``}`   `        ``// Returns true if there is a cycle in the graph` `        ``// else false.`   `        ``// This function returns true if there is a cycle` `        ``// in directed graph, else returns false.` `        ``boolean` `isCycle() ` `        ``{`   `            ``// Create a vector to store indegrees of all` `            ``// vertices. Initialize all indegrees as 0.` `            ``int``[] in_degree = ``new` `int``[``this``.V];` `            ``Arrays.fill(in_degree, ``0``);`   `            ``// Traverse adjacency lists to fill indegrees of` `            ``// vertices. This step takes O(V+E) time` `            ``for` `(``int` `u = ``0``; u < V; u++)` `            ``{` `                ``for` `(``int` `v : adj[u])` `                    ``in_degree[v]++;` `            ``}`   `            ``// Create an queue and enqueue all vertices with` `            ``// indegree 0` `            ``Queue q = ``new` `LinkedList();` `            ``for` `(``int` `i = ``0``; i < V; i++)` `                ``if` `(in_degree[i] == ``0``)` `                    ``q.add(i);`   `            ``// Initialize count of visited vertices` `            ``int` `cnt = ``0``;`   `            ``// Create a vector to store result (A topological` `            ``// ordering of the vertices)` `            ``Vector top_order = ``new` `Vector<>();`   `            ``// One by one dequeue vertices from queue and enqueue` `            ``// adjacents if indegree of adjacent becomes 0` `            ``while` `(!q.isEmpty())` `            ``{`   `                ``// Extract front of queue (or perform dequeue)` `                ``// and add it to topological order` `                ``int` `u = q.poll();` `                ``top_order.add(u);`   `                ``// Iterate through all its neighbouring nodes` `                ``// of dequeued node u and decrease their in-degree` `                ``// by 1` `                ``for` `(``int` `itr : adj[u])` `                    ``if` `(--in_degree[itr] == ``0``)` `                        ``q.add(itr);` `                ``cnt++;` `            ``}`   `            ``// Check if there was a cycle` `            ``if` `(cnt != ``this``.V)` `                ``return` `true``;` `            ``else` `                ``return` `false``;` `        ``}` `    ``}`   `    ``// Driver Code` `    ``public` `static` `void` `main(String[] args) ` `    ``{`   `        ``// Create a graph given in the above diagram` `        ``Graph g = ``new` `Graph(``6``);` `        ``g.addEdge(``0``, ``1``);` `        ``g.addEdge(``1``, ``2``);` `        ``g.addEdge(``2``, ``0``);` `        ``g.addEdge(``3``, ``4``);` `        ``g.addEdge(``4``, ``5``);`   `        ``if` `(g.isCycle())` `            ``System.out.println(``"Yes"``);` `        ``else` `            ``System.out.println(``"No"``);` `    ``}` `}`   `// This code is contributed by` `// sanjeev2552`

## Python3

 `# A Python3 program to check if there is a cycle in  ` `# directed graph using BFS. ` `import` `math` `import` `sys` `from` `collections ``import` `defaultdict`   `# Class to represent a graph ` `class` `Graph:` `    ``def` `__init__(``self``,vertices):` `        ``self``.graph``=``defaultdict(``list``)` `        ``self``.V``=``vertices ``# No. of vertices' ` `    `  `    ``# function to add an edge to graph` `    ``def` `addEdge(``self``,u,v):` `        ``self``.graph[u].append(v)`   `# This function returns true if there is a cycle ` `# in directed graph, else returns false. ` `def` `isCycleExist(n,graph):`   `    ``# Create a vector to store indegrees of all ` `    ``# vertices. Initialize all indegrees as 0. ` `    ``in_degree``=``[``0``]``*``n`   `    ``# Traverse adjacency lists to fill indegrees of ` `    ``# vertices. This step takes O(V+E) time` `    ``for` `i ``in` `range``(n):` `        ``for` `j ``in` `graph[i]:` `            ``in_degree[j]``+``=``1` `    `  `    ``# Create an queue and enqueue all vertices with ` `    ``# indegree 0` `    ``queue``=``[]` `    ``for` `i ``in` `range``(``len``(in_degree)):` `        ``if` `in_degree[i]``=``=``0``:` `            ``queue.append(i)` `    `  `    ``# Initialize count of visited vertices` `    ``cnt``=``0`   `    ``# One by one dequeue vertices from queue and enqueue ` `    ``# adjacents if indegree of adjacent becomes 0 ` `    ``while``(queue):`   `        ``# Extract front of queue (or perform dequeue) ` `        ``# and add it to topological order ` `        ``nu``=``queue.pop(``0``)`   `        ``# Iterate through all its neighbouring nodes ` `        ``# of dequeued node u and decrease their in-degree ` `        ``# by 1 ` `        ``for` `v ``in` `graph[nu]:` `            ``in_degree[v]``-``=``1`   `            ``# If in-degree becomes zero, add it to queue` `            ``if` `in_degree[v]``=``=``0``:` `                ``queue.append(v)` `        ``cnt``+``=``1`   `    ``# Check if there was a cycle ` `    ``if` `cnt``=``=``n:` `        ``return` `False` `    ``else``:` `        ``return` `True` `        `    `# Driver program to test above functions ` `if` `__name__``=``=``'__main__'``:`   `    ``# Create a graph given in the above diagram` `    ``g``=``Graph(``6``)` `    ``g.addEdge(``0``,``1``)` `    ``g.addEdge(``1``,``2``)` `    ``g.addEdge(``2``,``0``)` `    ``g.addEdge(``3``,``4``)` `    ``g.addEdge(``4``,``5``)` `    `  `    ``if` `isCycleExist(g.V,g.graph):` `        ``print``(``"Yes"``)` `    ``else``:` `        ``print``(``"No"``)`   `# This Code is Contributed by Vikash Kumar 37`

## C#

 `// C# program to check if there is a cycle in ` `// directed graph using BFS.` `using` `System;` `using` `System.Collections.Generic;`   `class` `GFG{` `    `  `// Class to represent a graph` `public` `class` `Graph` `{` `    `  `    ``// No. of vertices'` `    ``public` `int` `V; ` `    `  `    ``// Pointer to an array containing` `    ``// adjacency list` `    ``public` `List<``int``>[] adj;` `    `  `    ``public` `Graph(``int` `V)` `    ``{` `        `  `        ``// Constructor` `        ``this``.V = V;` `        ``this``.adj = ``new` `List<``int``>[V];` `        ``for` `(``int` `i = 0; i < V; i++)` `        ``adj[i] = ``new` `List<``int``>();` `    ``}` `    `  `    ``// Function to add an edge to graph` `    ``public` `void` `addEdge(``int` `u, ``int` `v)` `    ``{` `        ``adj[u].Add(v);` `    ``}` `    `  `    ``// Returns true if there is a cycle in the` `    ``// graph else false.` `    `  `    ``// This function returns true if there is ` `    ``// a cycle in directed graph, else returns` `    ``// false.` `    ``public` `bool` `isCycle() ` `    ``{` `        `  `        ``// Create a vector to store indegrees of all` `        ``// vertices. Initialize all indegrees as 0.` `        ``int``[] in_degree = ``new` `int``[``this``.V];` `        `  `        ``// Traverse adjacency lists to fill indegrees` `        ``// of vertices. This step takes O(V+E) time` `        ``for``(``int` `u = 0; u < V; u++)` `        ``{` `            ``foreach``(``int` `v ``in` `adj[u])` `                ``in_degree[v]++;` `        ``}` `        `  `        ``// Create an queue and enqueue all ` `        ``// vertices with indegree 0` `        ``Queue<``int``> q = ``new` `Queue<``int``>();` `        ``for``(``int` `i = 0; i < V; i++)` `            ``if` `(in_degree[i] == 0)` `                ``q.Enqueue(i);` `        `  `        ``// Initialize count of visited vertices` `        ``int` `cnt = 0;` `        `  `        ``// Create a vector to store result ` `        ``// (A topological ordering of the ` `        ``// vertices)` `        ``List<``int``> top_order = ``new` `List<``int``>();` `        `  `        ``// One by one dequeue vertices from ` `        ``// queue and enqueue adjacents if` `        ``// indegree of adjacent becomes 0` `        ``while` `(q.Count != 0)` `        ``{` `        `  `            ``// Extract front of queue (or perform ` `            ``// dequeue) and add it to topological` `            ``// order` `            ``int` `u = q.Peek();` `            ``q.Dequeue();` `            ``top_order.Add(u);` `            `  `            ``// Iterate through all its neighbouring ` `            ``// nodes of dequeued node u and decrease` `            ``// their in-degree by 1` `            ``foreach``(``int` `itr ``in` `adj[u])` `                ``if` `(--in_degree[itr] == 0)` `                    ``q.Enqueue(itr);` `                    `  `            ``cnt++;` `        ``}` `        `  `        ``// Check if there was a cycle` `        ``if` `(cnt != ``this``.V)` `            ``return` `true``;` `        ``else` `            ``return` `false``;` `    ``}` `}`   `// Driver Code` `public` `static` `void` `Main(String[] args) ` `{` `    `  `    ``// Create a graph given in the above diagram` `    ``Graph g = ``new` `Graph(6);` `    ``g.addEdge(0, 1);` `    ``g.addEdge(1, 2);` `    ``g.addEdge(2, 0);` `    ``g.addEdge(3, 4);` `    ``g.addEdge(4, 5);`   `    ``if` `(g.isCycle())` `        ``Console.WriteLine(``"Yes"``);` `    ``else` `        ``Console.WriteLine(``"No"``);` `}` `}`   `// This code is contributed by Princi Singh`

## Javascript

 ``

Output:

`Yes`

Time Complexity: O(V+E)
Auxiliary Space: O(V)

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