Detect Cycle in a Directed Graph

Using a Depth First Search (DFS) traversal algorithm we can detect cycles in a directed graph. If there is any self-loop in any node, it will be considered as a cycle, otherwise, when the child node has another edge to connect its parent, it will also a cycle.

For the disconnected graph, there may different trees present, we can call them a forest. Now we have to detect cycle for all trees of the forest.

In this approach, we will use different sets to assign nodes to perform the DFS traversal. There are three different sets, the White, Grey and the Black. Initially, all nodes will be stored inside the white set. When one new node is traversed, it will be stored in the gray set and removed from the white one. And after completing backtracking, when that task for that node is completed, it will be swapped from gray to black set.

Input and Output

Input:
The Adjacency matrix.

0 1 0 0 0
0 0 0 0 0
1 0 0 1 0
0 0 0 0 1
0 0 1 0 0

Output:
The graph has cycle. 

Algorithm

dfs(curr, wSet, gSet, bSet)

Input: Current node, the white, the grey and the black set.

Output: True if there is a cycle.

Begin
   delete curr from the while set and add it to the grey set
   for all nodes v connected with curr in the graph, do
      if v is in the black set, then
         skip next part, and go for next iteration
      if v is in the grey set, then
         return true
      if dfs(v, wSet, gSet, bSet) is true, then
         return true
   done
   delete curr from grey set and add to black set
   return fasle
End

hasCycle(graph)

Input − Given Graph.

Output: True when the graph has cycled.

Begin
   initially insert all nodes into the white set
   while white set has some elements, do
      for all nodes v in the graph, do
         if v is not in the white set, then
            if dfs(v, wSet, gSet, bSet), then
               return true
      done
   done
   return false
End

Example

#include<iostream>
#include<set>
#define NODE 5
using namespace std;

int graph[NODE][NODE] = {
   {0, 1, 0, 0, 0},
   {0, 0, 0, 0, 0},
   {1, 0, 0, 1, 0},
   {0, 0, 0, 0, 1},
   {0, 0, 1, 0, 0}
};

bool dfs(int curr, set<int>&wSet, set<int>&gSet, set<int>&bSet) {
   //moving curr to white set to grey set.
   wSet.erase(wSet.find(curr));
   gSet.insert(curr);

   for(int v = 0; v < NODE; v++) {
      if(graph[curr][v] != 0) {    //for all neighbour vertices
         if(bSet.find(v) != bSet.end())
            continue;    //if the vertices are in the black set
         if(gSet.find(v) != gSet.end())
            return true;    //it is a cycle
         if(dfs(v, wSet, gSet, bSet))
            return true;    //cycle found
      }
   }

   //moving v to grey set to black set.
   gSet.erase(gSet.find(curr));
   bSet.insert(curr);
   return false;
}

bool hasCycle() {
   set<int> wSet, gSet, bSet;    //three set as white, grey and black
   for(int i = 0; i<NODE; i++)
      wSet.insert(i);    //initially add all node into the white set

   while(wSet.size() > 0) {
      for(int current = 0; current < NODE; current++) {
         if(wSet.find(current) != wSet.end())
            if(dfs(current, wSet, gSet, bSet))
               return true;
      }
   }
   return false;
}

int main() {
   bool res;
   res = hasCycle();
   if(res)
      cout << "The graph has cycle." << endl;
   else
      cout << "The graph has no cycle." << endl;
}

Output

The graph has cycle.