Detecting Cycle in A Graph

PROBLEM : Given a directed graph check whether the graph contains a cycle or not.

SOLUTION : In a graph containing a cycle suppose vi,...vk are the vertices that constitute the cycle and suppose that the backedge is from vk to vi then on applying dfs to the graph "when the dfs covers the edge from vk to vi then vi must have already been visited." vi will already be in the recursion stack when the edge will be covered.Hence we know a cycle is present.Thus the given graph will not be cyclic.

Thus for each node visited earlier we need to store the recursion stack to tell whether a edge points to a node already in the recursion stack.Then it would be a back edge.In this way this algorithm can be modified to find in a graph:

• The number of back edges
• The components i.e. the trees in a forest
• The number of cycles = number of back edges .. etc

The cpp code for the following :

#include<iostream>
#include<list>

using namespace std;

class Graph
{
    int V; //number of vertices
    list<int> *adj; //pointer to array containing adjacency lists
    bool isCyclicUtil(int v,bool visited[],bool *rs);

    public:
        Graph(int v);
        void addEdge(int v,int w);
        bool isCyclic();
};

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

void Graph::addEdge(int v,int w)
{
    adj[v].push_back(w);
}

bool Graph::isCyclicUtil(int v,bool visited[],bool *rs)
{
    if(visited[v]==false)
    {
        visited[v]=true;
        rs[v]=true;
        list<int>::iterator i;
        for(i=adj[v].begin();i!=adj[v].end();++i)
        {
            if(!visited[*i] && isCyclicUtil(*i,visited,rs))
            return true;
            else if(rs[*i])
            return true;
        }
    }
    rs[v]=false;
    return false;
}

bool Graph::isCyclic()
{
    bool *visited=new bool[V];
    bool *rs=new bool[V];
    for(int i=0;i<V;i++)
    {
        visited[i]=false;
        rs[i]=false;
    }
    for(int i=0;i<V;i++)
    if(isCyclicUtil(i,visited,rs))
        return true;
    return false;
}

int main()
{
    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 a cycle"<<endl;
    else
    cout<<"Graph doesnot contain a cycle"<<endl;
    return 0;
}