How to find if a graph is Bipartite?

A graph is said to be a bipartite graph, when vertices of that graph can be divided into two independent sets such that every edge in the graph is either start from the first set and ended in the second set, or starts from the second set, connected to the first set, in other words, we can say that no edge can found in the same set.

Checking of a bipartite graph is possible by using the vertex coloring. When a vertex is in the same set, it has the same color, for another set, the color will change.

Input and Output

Input:
The adjacency matrix.
0 1 1 1 0 0
1 0 0 1 1 0
1 0 0 1 0 1
1 1 1 0 1 1
0 1 0 1 0 1
0 0 1 1 1 0

Output:
The graph is bipartite.

Algorithm

isBipartite(source)

Input − The source vertex.
Output: True when the graph is bipartite.

Begin
   define an empty queue qu, and a color list coloArray
   initially any node is not colored with any color
   color the source vertex as color red
   add source in the qu
   when qu is not empty, do
      remove item from the qu and take in u
      if there is any self-loop, then
         return false
      for all vertices v, which is connected with u, do
         if v has no color, then
            if colorArray[u] = red, then
               colorArray[v] := blue
            else if colorArray[u] = blue, then
               colorArray[v] := red
            add v into the queue
         if colorArray[v] = colorArray[u], then
            return false
      done
   done
   return true
End   

Example

#include<iostream>
#include<string>
#include<queue>
#define NODE 6
using namespace std;

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

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

bool isBipartite(int source) {
   queue<int> qu;
   string colorArray[NODE];

   for(int i = 0; i< NODE; i++)
      colorArray[i] = "No Color";    //initially no color is set for all vertices
   colorArray[source] = "Red";    //assign red with the source vertex
   qu.push(source);             //add source into the queue.

   while(!qu.empty()) {
      int u = qu.front();
      qu.pop();
      if(graph[u][u] == 1)    //there is a self loop
         return false;

      for(int v = 0; v < NODE; v++) {
         if(graph[u][v] != 0 && colorArray[v] == "No Color") {
            if(colorArray[u] == "Red")       //assign adjacent list with alternate color
               colorArray[v] = "Blue";
            else if(colorArray[u] == "Blue")
               colorArray[v] = "Red";
            qu.push(v);          //new adjacent node added to queue
         } else if(graph[u][v] != 0 && colorArray[v] == colorArray[u]) {
            return false;       //when u and adjacent are of same color.  
         }
      }
   }
   return true;
}

int main() {
   bool check;
   check = isBipartite(0);

   if(check)
      cout << "The graph is bipartite." << endl;
   else
      cout << "The graph is not bipartite." << endl;
}   

Output

The graph is bipartite.