Detect cycle in an undirected graph

Last Updated : 19 Jun, 2024

Given an undirected graph, The task is to check if there is a cycle in the given graph.

Examples:

Input: N = 4, E = 4

Output: Yes
Explanation: The diagram clearly shows a cycle 0 to 2 to 1 to 0

Input: N = 4, E = 3

Output: No
Explanation: There is no cycle in the given graph.

Find cycle in Undirected Graph using DFS:

Use DFS from every unvisited node. Depth First Traversal can be used to detect a cycle in a Graph. 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 indirectly joining a node to itself (self-loop) or one of its ancestors in the tree produced by DFS.

To find the back edge to any of its ancestors keep a visited array and if there is a back edge to any visited node then there is a loop and return true.

Follow the below steps to implement the above approach:

Iterate over all the nodes of the graph and Keep a visited array visited[] to track the visited nodes.
Run a Depth First Traversal on the given subgraph connected to the current node and pass the parent of the current node. In each recursive
- Set visited[root] as 1.
- Iterate over all adjacent nodes of the current node in the adjacency list
  - If it is not visited then run DFS on that node and return true if it returns true.
  - Else if the adjacent node is visited and not the parent of the current node then return true.
- Return false.

Illustration:

Below is the graph showing how to detect cycle in a graph using DFS:

Below is the implementation of the above approach:

C++

// A C++ Program to detect
// cycle in an undirected graph
#include <iostream>
#include <limits.h>
#include <list>
using namespace std;

// Class for an undirected graph
class Graph {

    // No. of vertices
    int V;

    // Pointer to an array
    // containing adjacency lists
    list<int>* adj;
    bool isCyclicUtil(int v, bool visited[], int parent);

public:
    // Constructor
    Graph(int V);

    // To add an edge to graph
    void addEdge(int v, int w);

    // Returns true if there is a cycle
    bool isCyclic();
};

Graph::Graph(int V)
{
    this->V = V;
    adj = new list<int>[V];
}

void Graph::addEdge(int v, int w)
{

    // Add w to v’s list.
    adj[v].push_back(w);

    // Add v to w’s list.
    adj[w].push_back(v);
}

// 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 vertex is not visited,
        // then recur for that adjacent
        if (!visited[*i]) {
            if (isCyclicUtil(*i, visited, v))
                return true;
        }

        // If an adjacent vertex is visited and
        // is not parent of current vertex,
        // then there exists a cycle in the graph.
        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++) {

        // Don't recur for u if
        // it is already visited
        if (!visited[u])
            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.*;
@SuppressWarnings("unchecked")
// This class represents a
// directed graph using adjacency list
// representation
class Graph {

    // No. of vertices
    private int V;

    // 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);
        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++) {

            // Don't recur for u if already visited
            if (!visited[u])
                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 contain 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 contain 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):

        # No. of vertices
        self.V = vertices  # No. of vertices

        # Default dictionary to store graph
        self.graph = defaultdict(list)

    # Function to add an edge to graph
    def addEdge(self, v, w):

        # Add w to v_s list
        self.graph[v].append(w)

        # Add v to w_s list
        self.graph[w].append(v)

    # 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):

            # Don't recur for u if it
            # is already visited
            if visited[i] == False:
                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(1, 2)
g.addEdge(2, 0)
g.addEdge(0, 3)
g.addEdge(3, 4)

if g.isCyclic():
    print("Graph contains cycle")
else:
    print("Graph doesn't contain cycle ")
g1 = Graph(3)
g1.addEdge(0, 1)
g1.addEdge(1, 2)


if g1.isCyclic():
    print("Graph contains cycle")
else:
    print("Graph doesn't contain cycle ")

# This code is contributed by Neelam Yadav

// 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 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);
        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 contain 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 contain cycle");
    }
}

// This code is contributed by PrinciRaj1992

Javascript

// javascript Program to detect cycle in an undirected graph

// This class represents a undirected
// graph using adjacency list representation

class Graph{

    constructor(vertices){
        
        // No. of vertices
        this.V = vertices;
        
        // Default dictionary to store graph
        this.graph = new Array(vertices);
        for(let i = 0; i < vertices; i++){
            this.graph[i] = new Array();
        }
    }
        

    // Function to add an edge to graph
    addEdge(v, w){
        
        // Add w to v_s list
        this.graph[v].push(w);

        // Add v to w_s list
        this.graph[w].push(v);      
    }



    // A recursive function that uses
    // visited[] and parent to detect
    // cycle in subgraph reachable from vertex v.
    isCyclicUtil(v, visited, parent){
 
        // Mark the current node as visited
        visited[v] = true;

        // Recur for all the vertices
        // adjacent to this vertex
        for(let i = 0; i < this.graph[v].length; i++){

            // If the node is not
            // visited then recurse on it
            if(visited[this.graph[v][i]] == false){
                if(this.isCyclicUtil(this.graph[v][i], visited, v) == true){
                    return true;
                }
            }
            // If an adjacent vertex is
            // visited and not parent
            // of current vertex,
            // then there is a cycle
            else if(parent != this.graph[v][i]){
                return true;
            }
        }
        return false;
    }



    // Returns true if the graph
    // contains a cycle, else false.
    isCyclic(){

        // Mark all the vertices
        // as not visited
        let visited = new Array(this.V).fill(false);

        // Call the recursive helper
        // function to detect cycle in different
        // DFS trees
        for(let i = 0; i < this.V; i++){
            // Don't recur for u if it
            // is already visited
            if(visited[i] == false){
                if(this.isCyclicUtil(i, visited, -1) == true){
                    return true;
                }
            }
                
        }

        return false;
    }

}

// Create a graph given in the above diagram
let g = new Graph(5);
g.addEdge(1, 0);
g.addEdge(1, 2);
g.addEdge(2, 0);
g.addEdge(0, 3);
g.addEdge(3, 4);

if (g.isCyclic() == true){
    console.log("Graph contains cycle");
} 
else{
    console.log("Graph doesn't contain cycle ");
}

let g1 = new Graph(3);
g1.addEdge(0, 1);
g1.addEdge(1, 2);


if(g1.isCyclic() == true){
    console.log("Graph contains cycle");
}   
else{
    console.log("Graph doesn't contain cycle "); 
}
    

// This code is contributed by Gautam goel.

Output

Graph contains cycle
Graph doesn't contain cycle

Time Complexity: O(V+E), The program does a simple DFS Traversal of the graph which is represented using an adjacency list. So the time complexity is O(V+E).
Auxiliary Space: O(V), To store the visited array O(V) space is required.

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Find cycle in Undirected Graph using DFS:

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