Detecting cycles in a graph using DFS: 2 different approaches and what's the difference

Solution 1:

Answering my question:

The graph has a cycle if and only if there exists a back edge. 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 forming a cycle.

Both approaches above actually mean the same. However, this method can be applied only to undirected graphs.

The reason why this algorithm doesn't work for directed graphs is that in a directed graph 2 different paths to the same vertex don't make a cycle. For example: A-->B, B-->C, A-->C - don't make a cycle whereas in undirected ones: A--B, B--C, C--A does.

Find a cycle in undirected graphs

An undirected graph has a cycle if and only if a depth-first search (DFS) finds an edge that points to an already-visited vertex (a back edge).

Find a cycle in directed graphs

In addition to visited vertices we need to keep track of vertices currently in recursion stack of function for DFS traversal. If we reach a vertex that is already in the recursion stack, then there is a cycle in the tree.

Update: Working code is in the question section above.

Solution 2:

For the sake of completion, it is possible to find cycles in a directed graph using DFS (from wikipedia):

 L ← Empty list that will contain the sorted nodes
while there are unmarked nodes do
    select an unmarked node n
    visit(n) 
function visit(node n)
    if n has a temporary mark then stop (not a DAG)
    if n is not marked (i.e. has not been visited yet) then
        mark n temporarily
        for each node m with an edge from n to m do
            visit(m)
        mark n permanently
        unmark n temporarily
        add n to head of L

Solution 3:

Here is the code I've written in C based on DFS to find out whether a given undirected graph is connected/cyclic or not. with some sample output at the end. Hope it'll be helpful :)

#include<stdio.h>
#include<stdlib.h>

/****Global Variables****/
int A[20][20],visited[20],count=0,n;
int seq[20],connected=1,acyclic=1;

/****DFS Function Declaration****/
void DFS();

/****DFSearch Function Declaration****/
void DFSearch(int cur);

/****Main Function****/
int main() 
   {    
    int i,j;

    printf("\nEnter no of Vertices: ");
    scanf("%d",&n);

    printf("\nEnter the Adjacency Matrix(1/0):\n");
    for(i=1;i<=n;i++)
            for(j=1;j<=n;j++)
        scanf("%d",&A[i][j]);

    printf("\nThe Depth First Search Traversal:\n");

    DFS();

    for(i=1;i<=n;i++)
        printf("%c,%d\t",'a'+seq[i]-1,i);

    if(connected && acyclic)    printf("\n\nIt is a Connected, Acyclic Graph!");
    if(!connected && acyclic)   printf("\n\nIt is a Not-Connected, Acyclic Graph!");
    if(connected && !acyclic)   printf("\n\nGraph is a Connected, Cyclic Graph!");
    if(!connected && !acyclic)  printf("\n\nIt is a Not-Connected, Cyclic Graph!");

    printf("\n\n");
    return 0;
   }

/****DFS Function Definition****/
void DFS()
    { 
    int i;
    for(i=1;i<=n;i++)
        if(!visited[i])
          {
        if(i>1) connected=0;
        DFSearch(i);    
              } 
    }

/****DFSearch Function Definition****/
void DFSearch(int cur) 
    {
    int i,j;
    visited[cur]=++count;

        seq[count]=cur; 
        for(i=1;i<count-1;i++)
                if(A[cur][seq[i]]) 
                   acyclic=0;

    for(i=1;i<=n;i++)
        if(A[cur][i] && !visited[i])
           DFSearch(i);

    }

Sample Output:

majid@majid-K53SC:~/Desktop$ gcc BFS.c

majid@majid-K53SC:~/Desktop$ ./a.out
************************************

Enter no of Vertices: 10

Enter the Adjacency Matrix(1/0):

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

The Depdth First Search Traversal:
a,1 c,2 d,3 f,4 b,5 e,6 g,7 h,8 i,9 j,10    

It is a Not-Connected, Cyclic Graph!


majid@majid-K53SC:~/Desktop$ ./a.out
************************************

Enter no of Vertices: 4

Enter the Adjacency Matrix(1/0):
0 0 1 1
0 0 1 0
1 1 0 0
0 0 0 1

The Depth First Search Traversal:
a,1 c,2 b,3 d,4 

It is a Connected, Acyclic Graph!


majid@majid-K53SC:~/Desktop$ ./a.out
************************************

Enter no of Vertices: 5

Enter the Adjacency Matrix(1/0):
0 0 0 1 0
0 0 0 1 0
0 0 0 0 1
1 1 0 0 0 
0 0 1 0 0

The Depth First Search Traversal:
a,1 d,2 b,3 c,4 e,5 

It is a Not-Connected, Acyclic Graph!

*/