Detect cycle in an undirected graph

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Given an undirected graph, how to check if there is a cycle in the graph? For example, the following graph has a cycle 1-0-2-1.

Recommended: Please solve it on “PRACTICE” first, before moving on to the solution.

We have discussed cycle detection for directed graph. We have also discussed a union-find algorithm for cycle detection in undirected graphs. The time complexity of the union-find algorithm is O(ELogV). Like directed graphs, we can use DFS to detect cycle in an undirected graph in O(V+E) time. We do a DFS traversal of the given graph. For every visited vertex ‘v’, if there is an adjacent ‘u’ such that u is already visited and u is not parent of v, then there is a cycle in graph. If we don’t find such an adjacent for any vertex, we say that there is no cycle. The assumption of this approach is that there are no parallel edges between any two vertices.

Below image is a dry run of the above approach:

Below is the implementation of the above approach:

C++

// A C++ Program to detect cycle in an undirected graph 
#include<iostream> 
#include <list> 
#include <limits.h> 
using namespace std; 
  
// Class for an undirected graph 
class Graph 
{ 
    int V;    // No. of vertices 
    list<int> *adj;    // Pointer to an array containing adjacency lists 
    bool isCyclicUtil(int v, bool visited[], int parent); 
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 
}; 
  
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. 
    adj[w].push_back(v); // Add v to w’s list. 
} 
  
// A recursive function that uses visited[] and parent to detect 
// cycle in subgraph reachable from vertex v. 
bool Graph::isCyclicUtil(int v, bool visited[], int parent) 
{ 
    // Mark the current node as visited 
    visited[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 an adjacent is not visited, then recur for that adjacent 
        if (!visited[*i]) 
        { 
           if (isCyclicUtil(*i, visited, v)) 
              return true; 
        } 
  
        // If an adjacent is visited and not parent of current vertex, 
        // then there is a cycle. 
        else if (*i != parent) 
           return true; 
    } 
    return false; 
} 
  
// Returns true if the graph contains a cycle, else false. 
bool Graph::isCyclic() 
{ 
    // Mark all the vertices as not visited and not part of recursion 
    // stack 
    bool *visited = new bool[V]; 
    for (int i = 0; i < V; i++) 
        visited[i] = false; 
  
    // Call the recursive helper function to detect cycle in different 
    // DFS trees 
    for (int u = 0; u < V; u++) 
        if (!visited[u]) // Don't recur for u if it is already visited 
          if (isCyclicUtil(u, visited, -1)) 
             return true; 
  
    return false; 
} 
  
// Driver program to test above functions 
int main() 
{ 
    Graph g1(5); 
    g1.addEdge(1, 0); 
    g1.addEdge(0, 2); 
    g1.addEdge(2, 1); 
    g1.addEdge(0, 3); 
    g1.addEdge(3, 4); 
    g1.isCyclic()? cout << "Graph contains cycle\n": 
                   cout << "Graph doesn't contain cycle\n"; 
  
    Graph g2(3); 
    g2.addEdge(0, 1); 
    g2.addEdge(1, 2); 
    g2.isCyclic()? cout << "Graph contains cycle\n": 
                   cout << "Graph doesn't contain cycle\n"; 
  
    return 0; 
} 

Java

// A Java Program to detect cycle in an undirected 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 
    private LinkedList<Integer> adj[]; // Adjacency List Represntation 
  
    // 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); 
        adj[w].add(v); 
    } 
  
    // A recursive function that uses visited[] and parent to detect 
    // cycle in subgraph reachable from vertex v. 
    Boolean isCyclicUtil(int v, Boolean visited[], int parent) 
    { 
        // Mark the current node as visited 
        visited[v] = true; 
        Integer i; 
  
        // Recur for all the vertices adjacent to this vertex 
        Iterator<Integer> it = adj[v].iterator(); 
        while (it.hasNext()) 
        { 
            i = it.next(); 
  
            // If an adjacent is not visited, then recur for that 
            // adjacent 
            if (!visited[i]) 
            { 
                if (isCyclicUtil(i, visited, v)) 
                    return true; 
            } 
  
            // If an adjacent is visited and not parent of current 
            // vertex, then there is a cycle. 
            else if (i != parent) 
                return true; 
        } 
        return false; 
    } 
  
    // Returns true if the graph contains a cycle, else false. 
    Boolean isCyclic() 
    { 
        // Mark all the vertices as not visited and not part of 
        // recursion stack 
        Boolean visited[] = new Boolean[V]; 
        for (int i = 0; i < V; i++) 
            visited[i] = false; 
  
        // Call the recursive helper function to detect cycle in 
        // different DFS trees 
        for (int u = 0; u < V; u++) 
            if (!visited[u]) // Don't recur for u if already visited 
                if (isCyclicUtil(u, visited, -1)) 
                    return true; 
  
        return false; 
    } 
  
  
    // Driver method to test above methods 
    public static void main(String args[]) 
    { 
        // Create a graph given in the above diagram 
        Graph g1 = new Graph(5); 
        g1.addEdge(1, 0); 
        g1.addEdge(0, 2); 
        g1.addEdge(2, 1); 
        g1.addEdge(0, 3); 
        g1.addEdge(3, 4); 
        if (g1.isCyclic()) 
            System.out.println("Graph contains cycle"); 
        else
            System.out.println("Graph doesn't contains cycle"); 
  
        Graph g2 = new Graph(3); 
        g2.addEdge(0, 1); 
        g2.addEdge(1, 2); 
        if (g2.isCyclic()) 
            System.out.println("Graph contains cycle"); 
        else
            System.out.println("Graph doesn't contains cycle"); 
    } 
} 
// This code is contributed by Aakash Hasija 

Python

# Python Program to detect cycle in an undirected graph 
  
from collections import defaultdict 
   
#This class represents a undirected graph using adjacency list representation 
class Graph: 
   
    def __init__(self,vertices): 
        self.V= vertices #No. of vertices 
        self.graph = defaultdict(list) # default dictionary to store graph 
  
   
    # function to add an edge to graph 
    def addEdge(self,v,w): 
        self.graph[v].append(w) #Add w to v_s list 
        self.graph[w].append(v) #Add v to w_s list 
   
    # A recursive function that uses visited[] and parent to detect 
    # cycle in subgraph reachable from vertex v. 
    def isCyclicUtil(self,v,visited,parent): 
  
        #Mark the current node as visited  
        visited[v]= True
  
        #Recur for all the vertices adjacent to this vertex 
        for i in self.graph[v]: 
            # If the node is not visited then recurse on it 
            if  visited[i]==False :  
                if(self.isCyclicUtil(i,visited,v)): 
                    return True
            # If an adjacent vertex is visited and not parent of current vertex, 
            # then there is a cycle 
            elif  parent!=i: 
                return True
          
        return False
           
   
    #Returns true if the graph contains a cycle, else false. 
    def isCyclic(self): 
        # Mark all the vertices as not visited 
        visited =[False]*(self.V) 
        # Call the recursive helper function to detect cycle in different 
        #DFS trees 
        for i in range(self.V): 
            if visited[i] ==False: #Don't recur for u if it is already visited 
                if(self.isCyclicUtil(i,visited,-1))== True: 
                    return True
          
        return False
  
# Create a graph given in the above diagram 
g = Graph(5) 
g.addEdge(1, 0) 
g.addEdge(0, 2) 
g.addEdge(2, 0) 
g.addEdge(0, 3) 
g.addEdge(3, 4) 
  
if g.isCyclic(): 
    print "Graph contains cycle"
else : 
    print "Graph does not contain cycle "
g1 = Graph(3) 
g1.addEdge(0,1) 
g1.addEdge(1,2) 
  
  
if g1.isCyclic(): 
    print "Graph contains cycle"
else : 
    print "Graph does not contain cycle "
   
#This code is contributed by Neelam Yadav 

C#

// C# Program to detect cycle in an undirected 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 
      
    // Adjacency List Represntation 
    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); 
        adj[w].Add(v); 
    } 
  
    // A recursive function that uses visited[]  
    // and parent to detect cycle in subgraph 
    // reachable from vertex v. 
    Boolean isCyclicUtil(int v, Boolean []visited, 
                         int parent) 
    { 
        // Mark the current node as visited 
        visited[v] = true; 
  
        // Recur for all the vertices  
        // adjacent to this vertex 
        foreach(int i in adj[v]) 
        { 
            // If an adjacent is not visited,  
            // then recur for that adjacent 
            if (!visited[i]) 
            { 
                if (isCyclicUtil(i, visited, v)) 
                    return true; 
            } 
  
            // If an adjacent is visited and  
            // not parent of current vertex, 
            // then there is a cycle. 
            else if (i != parent) 
                return true; 
        } 
        return false; 
    } 
  
    // Returns true if the graph contains  
    // a cycle, else false. 
    Boolean isCyclic() 
    { 
        // Mark all the vertices as not visited  
        // and not part of recursion stack 
        Boolean []visited = new Boolean[V]; 
        for (int i = 0; i < V; i++) 
            visited[i] = false; 
  
        // Call the recursive helper function  
        // to detect cycle in different DFS trees 
        for (int u = 0; u < V; u++) 
          
            // Don't recur for u if already visited 
            if (!visited[u])  
                if (isCyclicUtil(u, visited, -1)) 
                    return true; 
  
        return false; 
    } 
  
    // Driver Code 
    public static void Main(String []args) 
    { 
        // Create a graph given in the above diagram 
        Graph g1 = new Graph(5); 
        g1.addEdge(1, 0); 
        g1.addEdge(0, 2); 
        g1.addEdge(2, 1); 
        g1.addEdge(0, 3); 
        g1.addEdge(3, 4); 
        if (g1.isCyclic()) 
            Console.WriteLine("Graph contains cycle"); 
        else
            Console.WriteLine("Graph doesn't contains cycle"); 
  
        Graph g2 = new Graph(3); 
        g2.addEdge(0, 1); 
        g2.addEdge(1, 2); 
        if (g2.isCyclic()) 
            Console.WriteLine("Graph contains cycle"); 
        else
            Console.WriteLine("Graph doesn't contains cycle"); 
    } 
} 
  
// This code is contributed by PrinciRaj1992 

Output:

Graph contains cycle
Graph doesn't contain cycle

Time Complexity: The program does a simple DFS Traversal of graph and graph is represented using adjacency list. So the time complexity is O(V+E)

Exercise: Can we use BFS to detect cycle in an undirected graph in O(V+E) time? What about directed graphs?

Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above

My Personal Notes

Recommended: Please solve it on “PRACTICE” first, before moving on to the solution.

C++

Java

Python

C#

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