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.

Detect cycle in an undirected graph

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

Python Programming:

# 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 "

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)

Categorized in:

GraphGraph Cycle

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