Strongly Connected Graphs

In a directed graph is said to be strongly connected, when there is a path between each pair of vertices in one component.

To solve this algorithm, firstly, DFS algorithm is used to get the finish time of each vertex, now find the finish time of the transposed graph, then the vertices are sorted in descending order by topological sort.

Input and Output

Input:
Adjacency matrix of the graph.
0 0 1 1 0
1 0 0 0 0
0 1 0 0 0
0 0 0 0 1
0 0 0 0 0

Output:
Following are strongly connected components in given graph:
0 1 2
3
4

Algorithm

traverse(graph, start, visited)

Input: The graph which will be traversed, the starting vertex, and flags of visited nodes.

Output: Go through each node in the DFS technique and display nodes.

Begin
   mark start as visited
   for all vertices v connected with start, do
      if v is not visited, then
         traverse(graph, v, visited)
   done
End

topoSort(u, visited, stack)

Input − The start node, flag for visited vertices, stack.

Output − Fill stack while sorting the graph.

Begin
   mark u as visited
   for all node v, connected with u, do
      if v is not visited, then
         topoSort(v, visited, stack)
   done
   push u into the stack
End

getStrongConComponents(graph)

Input: The given graph.

Output − All strongly connected components.

Begin
   initially all nodes are unvisited
   for all vertex i in the graph, do
      if i is not visited, then
         topoSort(i, vis, stack)
   done

   make all nodes unvisited again
   transGraph := transpose of given graph

   while stack is not empty, do
      pop node from stack and take into v
      if v is not visited, then
         traverse(transGraph, v, visited)
   done
End

Example

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

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

void transpose() {    //transpose the graph and store to transGraph
   for(int i = 0; i<NODE; i++)
      for(int j = 0; j<NODE; j++)
         transGraph[i][j] = graph[j][i];
}    
         
void traverse(int g[NODE][NODE], int u, bool visited[]) {
   visited[u] = true;    //mark v as visited
   cout << u << " ";

   for(int v = 0; v<NODE; v++) {
      if(g[u][v]) {
         if(!visited[v])
            traverse(g, v, visited);
      }
   }
}  
                                               
void topoSort(int u, bool visited[], stack<int>&stk) {
   visited[u] = true;    //set as the node v is visited

   for(int v = 0; v<NODE; v++) {
      if(graph[u][v]) {    //for allvertices v adjacent to u
         if(!visited[v])
            topoSort(v, visited, stk);
      }
   }

   stk.push(u);    //push starting vertex into the stack
}

void getStrongConComponents() {
   stack<int> stk;
   bool vis[NODE];

   for(int i = 0; i<NODE; i++)
      vis[i] = false;    //initially all nodes are unvisited
   
   for(int i = 0; i<NODE; i++)
      if(!vis[i])    //when node is not visited
         topoSort(i, vis, stk);
   
   for(int i = 0; i<NODE; i++)
      vis[i] = false;    //make all nodes are unvisited for traversal  
   transpose();    //make reversed graph
   
   while(!stk.empty()) {    //when stack contains element, process in topological order
      int v = stk.top(); stk.pop();
      if(!vis[v]) {
         traverse(transGraph, v, vis);
         cout << endl;
      }
   }
}

int main() {
   cout << "Following are strongly connected components in given graph: "<<endl;
   getStrongConComponents();
}

Output

Following are strongly connected components in given graph:
0 1 2
3
4