Given a directed graph, check whether the graph contains a cycle or not. Your function should return true if the given graph contains at least one cycle, else return false. For example, the following graph contains three cycles 0->2->0, 0->1->2->0 and 3->3, so your function must return true.

**Solution**

Depth First Traversal can be used to detect cycle in a Graph. DFS for a connected graph produces a tree. 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 from a node to itself (selfloop) or one of its ancestor in the tree produced by DFS. In the following graph, there are 3 back edges, marked with cross sign. We can observe that these 3 back edges indicate 3 cycles present in the graph.

For a disconnected graph, we get the DFS forest as output. To detect cycle, we can check for cycle in individual trees by checking back edges.

Image Source: http://www.cs.yale.edu/homes/aspnes/pinewiki/DepthFirstSearch.html

In the previous post, we have discussed a solution that stores visited vertices in a separate array which stores vertices of current recursion call stack.

In this post a different solution is discussed. The solution is from CLRS book. The idea is to do DFS of given graph and while doing traversal, assign one of the below three colors to every vertex.

WHITE: Vertex is not processed yet. Initially all vertices are WHITE.GRAY: Vertex is being processed (DFS for this vertex has started, but not finished which means that all descendants (ind DFS tree) of this vertex are not processed yet (or this vertex is in function call stack)BLACK: Vertex and all its descendants are processed. While doing DFS, if we encounter an edge from current vertex to a GRAY vertex, then this edge is back edge and hence there is a cycle.

Below is C++ implementation based on above idea.

## C++

// A DFS based approach to find if there is a cycle // in a directed graph. This approach strictly follows // the algorithm given in CLRS book. #include <bits/stdc++.h> using namespace std; enum Color {WHITE, GRAY, BLACK}; // Graph class represents a directed graph using // adjacency list representation class Graph { int V; // No. of vertices list<int>* adj; // adjacency lists // DFS traversal of the vertices reachable from v bool DFSUtil(int v, int color[]); public: Graph(int V); // Constructor // function to add an edge to graph void addEdge(int v, int w); bool isCyclic(); }; // Constructor Graph::Graph(int V) { this->V = V; adj = new list<int>[V]; } // Utility function to add an edge void Graph::addEdge(int v, int w) { adj[v].push_back(w); // Add w to v's list. } // Recursive function to find if there is back edge // in DFS subtree tree rooted with 'u' bool Graph::DFSUtil(int u, int color[]) { // GRAY : This vertex is being processed (DFS // for this vertex has started, but not // ended (or this vertex is in function // call stack) color[u] = GRAY; // Iterate through all adjacent vertices list<int>::iterator i; for (i = adj[u].begin(); i != adj[u].end(); ++i) { int v = *i; // An adjacent of u // If there is if (color[v] == GRAY) return true; // If v is not processed and there is a back // edge in subtree rooted with v if (color[v] == WHITE && DFSUtil(v, color)) return true; } // Mark this vertex as processed color[u] = BLACK; return false; } // Returns true if there is a cycle in graph bool Graph::isCyclic() { // Initialize color of all vertices as WHITE int *color = new int[V]; for (int i = 0; i < V; i++) color[i] = WHITE; // Do a DFS traversal beginning with all // vertices for (int i = 0; i < V; i++) if (color[i] == WHITE) if (DFSUtil(i, color) == true) return true; return false; } // Driver code to test above int main() { // Create a graph given in the above diagram Graph g(4); g.addEdge(0, 1); g.addEdge(0, 2); g.addEdge(1, 2); g.addEdge(2, 0); g.addEdge(2, 3); g.addEdge(3, 3); if (g.isCyclic()) cout << "Graph contains cycle"; else cout << "Graph doesn't contain cycle"; return 0; }

## Python

# Python program to deetect cycle in # a directed graph from collections import defaultdict class Graph(): def __init__(self, V): self.V = V self.graph = defaultdict(list) def addEdge(self, u, v): self.graph[u].append(v) def DFSUtil(self, u, color): # GRAY : This vertex is being processed (DFS # for this vertex has started, but not # ended (or this vertex is in function # call stack) color[u] = "GRAY" for v in self.graph[u]: if color[v] == "GRAY": return True if color[v] == "WHITE" and self.DFSUtil(v, color) == True: return True color[u] = "BLACK" return False def isCyclic(self): color = ["WHITE"] * self.V for i in range(self.V): if color[i] == "WHITE": if self.DFSUtil(i, color) == True: return True return False # Driver program to test above functions g = Graph(4) g.addEdge(0, 1) g.addEdge(0, 2) g.addEdge(1, 2) g.addEdge(2, 0) g.addEdge(2, 3) g.addEdge(3, 3) print "Graph contains cycle" if g.isCyclic() == True\ else "Graph doesn't conatin cycle" # This program is contributed by Divyanshu Mehta

Output :

Graph contains cycle

Time complexity of above solution is O(V + E) where V is number of vertices and E is number of edges in the graph.

This article is contributed by **Aditya Goel**. Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above