Problem: Given a directed graph, check whether it has any cycle or not. A graph with a cycle is also known as cyclic graph.

Cycle in the Graph

There are several algorithms to detect cycles in a graph. Two of them are bread-first search (BFS) and depth-first search (DFS), using which we will check whether there is a cycle in the given graph.

Detect Cycle in a Directed Graph using DFS

The idea is to traverse the graph along a particular route and check if the vertices of that route form a loop.

If the algorithm repeats any vertices twice when traversing along the same route it means that the given graph has a loop (cycle).

Detect Cycle using DFS

We keep track of vertices in the current route using an additional Boolean flag beingVisited.

In the active route of DFS, all vertices holds beingVisited as true.

So, while traversing a graph using DFS, if we come across any vertex which is already part of the active route (has beingVisited as true), it means there is a loop.

Here is the implementation of the algorithm in C++, Java and Python:

Python

#class representing vertex of a class
class Vertex:
  def __init__(self, name):
    self.name = name
    #variable holds true when visiting is in process
    self.beingVisited = False
    #variable to mark vertex as visited
    self.visited = False
    #adjacency list
    self.neighbors = []

  #method to connect two vertices (undirected)
  def add_neighbor(self, vertex):
    self.neighbors.append(vertex)

class DFS:
  #recursive method to visit vertices of the graph using DFS
  def hasCycle(self, vertex):

    #start the visiting process
    vertex.beingVisited = True;

    #go to next connected vertex
    for nbr in vertex.neighbors:
      #If next vertex is also beingVisited, it means
      #there is either self loop or a back edge
      if nbr.beingVisited:
        return True
      #if the following vertex is not visited then visit recursively
      #and return true if cycle has been detected
      elif not nbr.visited and self.hasCycle(nbr):
        return True

    #visit process completed
    vertex.beingVisited = False;
    vertex.visited = True
    #return false because current vertex is
    #not a part of any cycle
    return False


if __name__ == '__main__':
  #vertices
  vertices = [Vertex('A'), Vertex('B'), Vertex('C'), Vertex('D')]

  #connect vertices
  vertices[0].add_neighbor(vertices[1]) #A->B
  vertices[1].add_neighbor(vertices[2]) #B->C
  vertices[2].add_neighbor(vertices[0]) #C->A
  vertices[2].add_neighbor(vertices[3]) #C->D
  vertices[0].add_neighbor(vertices[3]) #A->D

  #driver code
  dfs = DFS()

  #Extra check for disconnected graph
  #so that all graph components are checked
  for vertex in vertices:
    if dfs.hasCycle(vertex):
      print("Has Cycle")
      break
  else:
    print("Doesn't have Cycle")

C++

#include <iostream>
#include <list>
using namespace std; 

//class representing a vertex of a graph
class Vertex{
public:
  //vertex label
  string name;
  //variable holds true when visiting is in process
  bool beingVisited = false;
  //variable to mark vertex as visited
  bool visited = false;
  //adjacency list
  list<Vertex*> neighbors;

  Vertex(string name){
    this->name = name;
  }

  //method to connect two vertices (unidirectional)
  void add_neighbor(Vertex* vertex){
    this->neighbors.push_back(vertex);
  }
};

class DFS{
public:
  //recursive method to traverse graph (visit vertices) using DFS
  //returns true if graph has cycle otherwise false
  bool hasCycle(Vertex* vertex){
    //start the visiting process
    vertex->beingVisited = true;

    //go to next connected vertex
    for(Vertex* nbr: vertex->neighbors){
      //If next vertex is also beingVisited, it means
      //there is either self loop or a back edge
      if(nbr->beingVisited)
        return true;
      //if the following vertex is not visited then visit recursively
      //and return true if cycle has been detected
      else if(!nbr->visited && hasCycle(nbr))
        return true;
    }

    //visit process completed
    vertex->beingVisited = false;
    vertex->visited = true;
    //return false because current vertex is
    //not a part of any cycle
    return false;
  }
};

int main() {
  //vertices
  Vertex* vertices[] = {new Vertex("A"), new Vertex("B"), new Vertex("C"), new Vertex("D")};

  //connect vertices
  vertices[0]->add_neighbor(vertices[1]); //A->B
  vertices[1]->add_neighbor(vertices[2]); //B->C
  vertices[2]->add_neighbor(vertices[0]); //C->A
  vertices[2]->add_neighbor(vertices[3]); //C->D
  vertices[0]->add_neighbor(vertices[3]); //A->D

  //driver code
  DFS dfs;
  bool hasCycle = false;

  //Extra check for disconnected graph
  //so that all graph components are checked
  for(Vertex* vertex: vertices){
    if(dfs.hasCycle(vertices[0])){
      cout << "Has Cycle";
      hasCycle = true;
      break;
    }
  }
  if(!hasCycle)
    cout << "Doesn't have Cycle";
}

Java

import java.util.*;

//class representing a vertex of a class
class Vertex{
  //vertex label
  String name;
  //variable holds true when visiting is in process
  boolean beingVisited = false;
  //variable to mark vertex as visited
  boolean visited = false;
  //adjacency list
  List<Vertex> neighbors= new ArrayList<>();

  Vertex(String name){
    this.name = name;
  }

  //method to connect two vertices (unidirectional)
  void add_neighbor(Vertex vertex){
    this.neighbors.add(vertex);
  }
}

class DFS{
  //recursive method to traverse graph (visit vertices) using DFS
  //returns true if graph has cycle otherwise false
  boolean hasCycle(Vertex vertex){
    //start the visiting process
    vertex.beingVisited = true;

    //go to next connected vertex
    for(Vertex nbr: vertex.neighbors){
      //If next vertex is also beingVisited, it means
      //there is either self loop or a back edge
      if(nbr.beingVisited)
        return true;
      //if the following vertex is not visited then visit recursively
      //and return true if cycle has been detected
      else if(!nbr.visited && hasCycle(nbr))
        return true;
    }

    //visit process completed
    vertex.beingVisited = false;
    vertex.visited = true;
    //return false because current vertex is
    //not a part of any cycle
    return false;
  }
}

class Main {
  public static void main(String[] args) {
    //vertices
  Vertex vertices[] = {new Vertex("A"), new Vertex("B"), new Vertex("C"), new Vertex("D")};

  //connect vertices
  vertices[0].add_neighbor(vertices[1]); //A->B
  vertices[1].add_neighbor(vertices[2]); //B->C
  vertices[2].add_neighbor(vertices[0]); //C->A
  vertices[2].add_neighbor(vertices[3]); //C->D
  vertices[0].add_neighbor(vertices[3]); //A->D

  //driver code
  DFS dfs = new DFS();
  boolean hasCycle = false;

  //Extra check for disconnected graph
  //so that all graph components are checked
  for(Vertex vertex: vertices){
    if(dfs.hasCycle(vertex)){
      System.out.println("Has Cycle");
      hasCycle = true;
      break;
    }
  } 
  if(!hasCycle)
    System.out.println("Doesn't have Cycle");
  }
}

Output:

Has Cycle

In the above program ,we have represented the graph using adjacency list.

Since we’re using depth-first traversal, the program’s time complexity is equal to the depth-first search algorithm i.e. O(V+E).

Detect Cycle in a Directed Graph using BFS

We can also check whether the given graph has any cycles or not using the breadth-first search algorithm.

The idea is to traverse the graph using BFS and check any path being repeated. If so, there is a circle in the graph.

Detect Cycle using BFS

We use an additional Vertex variable (parent) to keep track of traversed paths.

We store the preceding vertex of each vertex into the parent variable.

When traversing the graph using the BFS algorithm, if the next vertex is already visited and the current vertex is its parent, it means we are repeating the same path i.e. the graph has a circle.

Here is the implementation of this approach in C++, Java and Python:

Python

#class representing vertex of a class
class Vertex:
  def __init__(self, name):
    self.name = name
    #variable to hold parent vertex reference
    self.parent = None
    #variable to mark vertex as visited
    self.visited = False
    #adjacency list
    self.neighbors = []

  #method to connect two vertices (undirected)
  def add_neighbor(self, vertex):
    self.neighbors.append(vertex)

class BFS:
  #method to visit vertices of the graph using BFS
  #returns true if a cycle is detected otherwise false
  def hasCycle(self, start):
    #list to be used as a queue
    queue = []

    #add the start vertex to the queue
    queue.append(start)
    #mark start vertex as visited
    start.visited = True
    #parent of start vertex is itself
    start.parent = start

    #BFS until queue is empty
    while queue:
      #Pop a vertex from queue to visit
      current_vertex = queue.pop(0)

      #go to next connected vertex
      for nbr in current_vertex.neighbors:
        #If next vertex is already Visited and its parent is same as the current vertex
        #it means there is either loop or a back edge
        if nbr.visited and nbr.parent == current_vertex:
          return True

        #otherwise add the vertex to the queue,
        #mark it visited and update the parent
        else:
          queue.append(nbr)
          #mark visited
          nbr.visited = True
          nbr.parent = current_vertex.parent

    #visited all the vertices of the graph
    #no cycle detected, so return false
    return False   


if __name__ == '__main__':
  #vertices
  vertices = [Vertex('A'), Vertex('B'), Vertex('C'), Vertex('D')]

  #connect vertices
  vertices[0].add_neighbor(vertices[1]) #A->B
  vertices[1].add_neighbor(vertices[2]) #B->C
  vertices[2].add_neighbor(vertices[0]) #C->A
  vertices[2].add_neighbor(vertices[3]) #C->D
  vertices[0].add_neighbor(vertices[3]) #A->D

  #driver code
  bfs = BFS()
  
  #Extra check for disconnected graph
  #so that all graph components are checked
  for vertex in vertices:
    if bfs.hasCycle(vertex):
      print("Has Cycle")
      break
  else:
    print("Doesn't have Cycle")

C++

#include <iostream>
#include <list>
#include <queue>
using namespace std; 

//class representing a vertex of a graph
class Vertex{
public:
  //vertex label
  string name;
  //variable to hold parent vertex reference
  Vertex*  parent = nullptr;
  //variable to mark vertex as visited
  bool visited = false;
  //adjacency list
  list<Vertex*> neighbors;

  Vertex(string name){
    this->name = name;
  }

  //method to connect two vertices (unidirectional)
  void add_neighbor(Vertex* vertex){
    this->neighbors.push_back(vertex);
  }
};

class BFS{
public:
  //method to visit vertices of the graph using BFS
  //returns true if a cycle is detected otherwise false
  bool hasCycle(Vertex* start){
    //Queue
    queue<Vertex*> Queue;

    //add the start vertex to the queue
    Queue.push(start);
    //mark start vertex as visited
    start->visited = true;
    //parent of start vertex is itself
    start->parent = start;

    //BFS until queue is empty
    while(!Queue.empty()){
      //Pop a vertex from queue to visit
      Vertex* current_vertex = Queue.front();
      Queue.pop();

      //go to next connected vertex
      for(Vertex* nbr: current_vertex->neighbors){
        //If next vertex is already Visited and its parent is same as the current vertex
        //it means there is either loop or a back edge
        if(nbr->visited && nbr->parent == current_vertex)
          return true;

        //otherwise add the vertex to the queue,
        //mark it visited and update the parent
        else{
          Queue.push(nbr);
          nbr->visited = true;
          nbr->parent = current_vertex->parent;
        }
      }
    }

    //visited all the vertices of the graph
    //no cycle detected, so return false
    return false;
  }
};

int main() {
  //vertices
  Vertex* vertices[] = {new Vertex("A"), new Vertex("B"), new Vertex("C"), new Vertex("D")};

  //connect vertices
  vertices[0]->add_neighbor(vertices[1]); //A->B
  vertices[1]->add_neighbor(vertices[2]); //B->C
  vertices[2]->add_neighbor(vertices[0]); //C->A
  vertices[2]->add_neighbor(vertices[3]); //C->D
  vertices[0]->add_neighbor(vertices[3]); //A->D

  //driver code
  BFS bfs;
  bool hasCycle = false;

  //Extra check for disconnected graph
  //to ensure all the graph components are checked
  for(Vertex* vertex: vertices){
    if(bfs.hasCycle(vertices[0])){
      cout << "Has Cycle";
      hasCycle = true;
      break;
    }
  }
  if(!hasCycle)
    cout << "Doesn't have Cycle";
}

Java

import java.util.*;

//class representing a vertex of a class
class Vertex{
  //vertex label
  String name;
  //variable to hold parent vertex reference
  Vertex parent;
  //variable to mark vertex as visited
  boolean visited = false;
  //adjacency list
  List<Vertex> neighbors= new ArrayList<>();

  Vertex(String name){
    this.name = name;
  }

  //method to connect two vertices (unidirectional)
  void add_neighbor(Vertex vertex){
    this.neighbors.add(vertex);
  }
}

class BFS{
  //method to visit vertices of the graph using BFS
  //returns true if a cycle is detected otherwise false
  boolean hasCycle(Vertex start){
    //Queue
    Queue<Vertex> queue = new LinkedList<>();

    //add the start vertex to the queue
    queue.add(start);
    //mark start vertex as visited
    start.visited = true;
    //parent of start vertex is itself
    start.parent = start;

    //BFS until queue is empty
    while(!queue.isEmpty()){
      //Pop a vertex from queue to visit
      Vertex current_vertex = queue.poll();
      
      //go to next connected vertex
      for(Vertex nbr: current_vertex.neighbors){
        //If next vertex is already Visited and its parent is same as the current vertex
        //it means there is loop
        if(nbr.visited && nbr.parent == current_vertex)
          return true;

        //otherwise add the vertex to the queue,
        //mark it visited and update the parent
        else{
          queue.add(nbr);
          nbr.visited = true;
          nbr.parent = current_vertex;
        }
      }
    }

    //visited all the vertices of the graph
    //no cycle detected, so return false
    return false;
  }
}

class Main {
  public static void main(String[] args) {
    //vertices
  Vertex vertices[] = {new Vertex("A"), new Vertex("B"), new Vertex("C"), new Vertex("D")};

  //connect vertices
  vertices[0].add_neighbor(vertices[1]); //A->B
  vertices[1].add_neighbor(vertices[2]); //B->C
  vertices[2].add_neighbor(vertices[0]); //C->A
  vertices[2].add_neighbor(vertices[3]); //C->D
  vertices[0].add_neighbor(vertices[3]); //A->D

  //driver code
  BFS bfs = new BFS();
  boolean hasCycle = false;

  //Extra check for the disconnected graph
  //To ensire all the graph components are checked
  for(Vertex vertex: vertices){
    if(bfs.hasCycle(vertex)){
      System.out.println("Has Cycle");
      hasCycle = true;
      break;
    }
  } 
  if(!hasCycle)
    System.out.println("Doesn't have Cycle");
  }
}

Output:

Has Cycle

In this approach, we add connected vertices to the queue, regardless of whether it was visited or not. That’s because we’re basically searching for a repetition of the path.

The time complexity of this approach is O(V+E) because in the worst-case algorithm will have to detect all the vertices and edges of the given graph.

In this tutorial, we learned to detect cycles in a directed graph using the BFS and DFS traversal algorithm.

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