Connectivity in a directed graph

Graph AlgorithmsData StructureAlgorithms

To check connectivity of a graph, we will try to traverse all nodes using any traversal algorithm. After completing the traversal, if there is any node, which is not visited, then the graph is not connected.

For the directed graph, we will start traversing from all nodes to check connectivity. Sometimes one edge can have the only outward edge but no inward edge, so that node will be unvisited from any other starting node.

In this case, the traversal algorithm is recursive DFS traversal.

Input and Output

Input:
Adjacency matrix of a graph
   0 1 0 0 0
   0 0 1 0 0
   0 0 0 1 1
   1 0 0 0 0
   0 1 0 0 0

Output:
The Graph is connected.     

Algorithm

traverse(u, visited)

Input: The start node u and the visited node to mark which node is visited.

Output − Traverse all connected vertices.

Begin
   mark u as visited
   for all vertex v, if it is adjacent with u, do
      if v is not visited, then
         traverse(v, visited)
   done
End

isConnected(graph)

Input: The graph.

Output: True if the graph is connected.

Begin
   define visited array
   for all vertices u in the graph, do
      make all nodes unvisited
      traverse(u, visited)
      if any unvisited node is still remaining, then
         return false
   done
   return true
End

Example

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

int graph[NODE][NODE] = {
   {0, 1, 0, 0, 0},
   {0, 0, 1, 0, 0},
   {0, 0, 0, 1, 1},
   {1, 0, 0, 0, 0},
   {0, 1, 0, 0, 0}
};
                                               
void traverse(int u, bool visited[]) {
   visited[u] = true;    //mark v as visited

   for(int v = 0; v<NODE; v++) {
      if(graph[u][v]) {
         if(!visited[v])
            traverse(v, visited);
      }
   }
}

bool isConnected() {
   bool *vis = new bool[NODE];
   //for all vertex u as start point, check whether all nodes are visible or not

   for(int u; u < NODE; u++) {
      for(int i = 0; i<NODE; i++)
         vis[i] = false;    //initialize as no node is visited
               
      traverse(u, vis);
      for(int i = 0; i<NODE; i++) {
         if(!vis[i])    //if there is a node, not visited by traversal, graph is not connected
            return false;
      }
   }
   return true;
}

int main() {
   if(isConnected())
      cout << "The Graph is connected.";
   else
      cout << "The Graph is not connected.";
}

Output

The Graph is connected.
raja
Published on 10-Jul-2018 08:45:01
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