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.

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.

C++

 `// A C++ Program to detect cycle in an undirected graph ` `#include ` `#include ` `#include ` `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, 0); ` `    ``g1.addEdge(0, 3); ` `    ``g1.addEdge(3, 4); ` `    ``g1.isCyclic()? cout << ````"Graph contains cycle "````: ` `                   ``cout << ````"Graph doesn't contain cycle "````; ` ` `  `    ``Graph g2(3); ` `    ``g2.addEdge(0, 1); ` `    ``g2.addEdge(1, 2); ` `    ``g2.isCyclic()? cout << ````"Graph contains cycle "````: ` `                   ``cout << ````"Graph doesn't contain cycle "````; ` ` `  `    ``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 adj[]; ``// Adjacency List Represntation ` ` `  `    ``// Constructor ` `    ``Graph(``int` `v) { ` `        ``V = v; ` `        ``adj = ``new` `LinkedList[v]; ` `        ``for``(``int` `i=``0``; i 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``, ``0``); ` `        ``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 `

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?