Number of Triangles in Directed and Undirected Graphs

Given a Graph, count number of triangles in it. The graph is can be directed or undirected.

Example:

Input: digraph[V][V] = { {0, 0, 1, 0},
                        {1, 0, 0, 1},
                        {0, 1, 0, 0},
                        {0, 0, 1, 0}
                      };
Output: 2
Give adjacency matrix represents following 
directed graph.
triangles

We have discussed a method based on graph trace that works for undirected graphs. In this post a new method is discussed with that is simpler and works for both directed and undirected graphs.



The idea is to use three nested loops to consider every triplet (i, j, k) and check for the above condition (there is an edge from i to j, j to k and k to i)
However in an undirected graph, the triplet (i, j, k) can be permuted to give six combination (See previous post for details). Hence we divide the total count by 6 to get the actual number of triangles.
In case of directed graph, the number of permutation would be 3 (as order of nodes becomes relevant). Hence in this case the total number of triangles will be obtained by dividing total count by 3. For example consider the directed graph given below

Following is the implementation.

C/C++

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// C++ program to count triangles
// in a graph. The program is for
// adjacency matrix representation
// of the graph.
#include<bits/stdc++.h>
  
// Number of vertices in the graph
#define V 4
  
using namespace std;
  
// function to calculate the
// number of triangles in a
// simple directed/undirected 
// graph. isDirected is true if
// the graph is directed, its
// false otherwise
int countTriangle(int graph[V][V], 
                  bool isDirected)
{
    // Initialize result
    int count_Triangle = 0;
  
    // Consider every possible
    // triplet of edges in graph
    for (int i = 0; i < V; i++)
    {
        for (int j = 0; j < V; j++)
        {
            for (int k = 0; k < V; k++)
            {
               // check the triplet if
               // it satisfies the condition
               if (graph[i][j] && graph[j][k] 
                               && graph[k][i])
                  count_Triangle++;
             }
        }
    }
  
    // if graph is directed , 
    // division is done by 3,
    // else division by 6 is done
    isDirected? count_Triangle /= 3 :
                count_Triangle /= 6;
  
    return count_Triangle;
}
  
//driver function to check the program
int main()
{
    // Create adjacency matrix
    // of an undirected graph
    int graph[][V] = { {0, 1, 1, 0},
                       {1, 0, 1, 1},
                       {1, 1, 0, 1},
                       {0, 1, 1, 0}
                     };
  
    // Create adjacency matrix
    // of a directed graph
    int digraph[][V] = { {0, 0, 1, 0},
                        {1, 0, 0, 1},
                        {0, 1, 0, 0},
                        {0, 0, 1, 0}
                       };
  
    cout << "The Number of triangles in undirected graph : "
         << countTriangle(graph, false);
    cout << "\n\nThe Number of triangles in directed graph : "
         << countTriangle(digraph, true);
  
    return 0;
}

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Java

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// Java program to count triangles
// in a graph.  The program is
// for adjacency matrix
// representation of the graph.
import java.io.*;
  
class GFG {
  
    // Number of vertices in the graph
    int V = 4;
  
    // function to calculate the number
    // of triangles in a simple
    // directed/undirected graph. isDirected
    // is true if the graph is directed,
    // its false otherwise.
    int countTriangle(int graph[][], 
                      boolean isDirected)
   {
       // Initialize result
       int count_Triangle = 0;
  
       // Consider every possible
       // triplet of edges in graph
       for (int i = 0; i < V; i++)
       {
           for (int j = 0; j < V; j++)
           {
               for (int k=0; k<V; k++)
               {
                  // check the triplet if it
                  // satisfies the condition
                  if (graph[i][j] == 1 && 
                      graph[j][k] == 1 && 
                      graph[k][i] == 1)
                      count_Triangle++;
               }
           }
       }
   
       // if graph is directed , division
       // is done by 3 else division
       // by 6 is done
       if(isDirected == true)
       {
           count_Triangle /= 3;
       }
       else
       {
           count_Triangle /= 6;
       }
       return count_Triangle;
   }
   
   // driver code
    public static void main(String args[])
   {
         
       // Create adjacency matrix
       // of an undirected graph
       int graph[][] = {{0, 1, 1, 0},
                        {1, 0, 1, 1},
                        {1, 1, 0, 1},
                        {0, 1, 1, 0}
                       };
      
       // Create adjacency matrix
       // of a directed graph
       int digraph[][] = { {0, 0, 1, 0},
                           {1, 0, 0, 1},
                           {0, 1, 0, 0},
                           {0, 0, 1, 0}
                         };
   
    System.out.println("The Number of triangles "+
                       "in undirected graph : " +
                        countTriangle(graph, false));
  
    System.out.println("\n\nThe Number of triangles"+
                       " in directed graph : "
                       countTriangle(digraph, true));
  
   }
}
  
// This code is contributed by Anshika Goyal.

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Python

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# Python program to count triangles in a graph.  The program is
# for adjacency matrix representation of the graph.
  
  
# function to calculate the number of triangles in a simple
# directed/undirected graph.
# isDirected is true if the graph is directed, its false otherwise
def countTriangle(g, isDirected):
    nodes = len(g)
    count_Triangle = 0 #Initialize result
    # Consider every possible triplet of edges in graph
    for i in range(nodes):
        for j in range(nodes):
            for k in range(nodes):
                # check the triplet if it satisfies the condition
                if( i!=j and i !=k and j !=k and 
                        g[i][j] and g[j][k] and g[k][i]):
                    count_Triangle += 1
    # if graph is directed , division is done by 3
    # else division by 6 is done
    return count_Triangle/3 if isDirected else count_Triangle/6
  
# Create adjacency matrix of an undirected graph
graph = [[0, 1, 1, 0],
       [1, 0, 1, 1],
       [1, 1, 0, 1],
       [0, 1, 1, 0 ]]
# Create adjacency matrix of a directed graph
digraph = [[0, 0, 1, 0],
             [1, 0, 0, 1],
          [0, 1, 0, 0],
          [0, 0, 1, 0 ]]
  
print ("The Number of triangles in undirected graph : %d" %countTriangle(graph, False))
  
print ("The Number of triangles in directed graph : %d" %countTriangle(digraph, True))
  
# This code is contributed by Neelam Yadav

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C#

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// C# program to count triangles in a graph.
// The program is for adjacency matrix
// representation of the graph.
using System;
  
class GFG {
      
    // Number of vertices in the graph
    const int V = 4;
      
    // function to calculate the
    // number of triangles in a
    // simple directed/undirected 
    // graph. isDirected is true if
    // the graph is directed, its
    // false otherwise
    static int countTriangle(int[,] graph, 
                    bool isDirected)
    {
        // Initialize result
        int count_Triangle = 0;
      
        // Consider every possible
        // triplet of edges in graph
        for (int i = 0; i < V; i++)
        {
            for (int j = 0; j < V; j++)
            {
                for (int k = 0; k < V; k++)
                {
                // check the triplet if
                // it satisfies the condition
                if (graph[i,j] != 0 &&
                    graph[j,k] != 0 && 
                     graph[k,i] != 0 )
                    count_Triangle++;
                }
            }
        }
      
        // if graph is directed , 
        // division is done by 3,
        // else division by 6 is done
        if(isDirected != false)
            count_Triangle = 
                        count_Triangle / 3 ;
        else
            count_Triangle = 
                         count_Triangle / 6;
      
        return count_Triangle;
    }
      
    // Driver function to check the program
    static void Main()     
    {
          
        // Create adjacency matrix
        // of an undirected graph
        int[,] graph = new int[4,4] { 
                            {0, 1, 1, 0},
                            {1, 0, 1, 1}, 
                            {1, 1, 0, 1}, 
                            {0, 1, 1, 0} 
                        };
      
        // Create adjacency matrix
        // of a directed graph
        int[,] digraph = new int [4,4] {
                            {0, 0, 1, 0},
                            {1, 0, 0, 1},
                            {0, 1, 0, 0},
                            {0, 0, 1, 0}
                        };
                          
                          
      
        Console.Write("The Number of triangles"
                    + " in undirected graph : "
                + countTriangle(graph, false));
                  
        Console.Write("\n\nThe Number of "
         + "triangles in directed graph : " 
           + countTriangle(digraph, true));
    }
}
  
// This code is contributed by anuj_67

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PHP

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<?php
// PHP program to count triangles
// in a graph. The program is for
// adjacency matrix representation
// of the graph.
  
// Number of vertices in the graph
$V = 4;
  
// function to calculate the
// number of triangles in a
// simple directed/undirected 
// graph. isDirected is true if
// the graph is directed, its
// false otherwise
function countTriangle($graph
                       $isDirected)
{
    global $V;
      
    // Initialize result
    $count_Triangle = 0;
  
    // Consider every possible
    // triplet of edges in graph
    for($i = 0; $i < $V; $i++)
    {
        for($j = 0; $j < $V; $j++)
        {
            for($k = 0; $k < $V; $k++)
            {
                  
                // check the triplet if
                // it satisfies the condition
                if ($graph[$i][$j] and $graph[$j][$k
                                   and $graph[$k][$i])
                    $count_Triangle++;
            }
        }
    }
  
    // if graph is directed , 
    // division is done by 3,
    // else division by 6 is done
    $isDirected? $count_Triangle /= 3 :
                 $count_Triangle /= 6;
  
    return $count_Triangle;
}
  
    // Driver Code
    // Create adjacency matrix
    // of an undirected graph
    $graph = array(array(0, 1, 1, 0),
                   array(1, 0, 1, 1),
                   array(1, 1, 0, 1),
                   array(0, 1, 1, 0));
  
    // Create adjacency matrix
    // of a directed graph
    $digraph = array(array(0, 0, 1, 0),
                     array(1, 0, 0, 1),
                     array(0, 1, 0, 0),
                     array(0, 0, 1, 0));
          
    echo "The Number of triangles in undirected graph : "
                        , countTriangle($graph, false);
    echo "\nThe Number of triangles in directed graph : "
                        , countTriangle($digraph, true);
                          
// This code is contributed by anuj_67
?>

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Output:

The Number of triangles in undirected graph : 2
The Number of triangles in directed graph : 2

Comparison of this approach with previous approach:
Advantages:

  • No need to calculate Trace.
  • Matrix- multiplication is not required.
  • Auxiliary matrices are not required hence optimized in space.
  • Works for directed graphs.

Disadvantages:

  • The time complexity is O(n3) and can’t be reduced any further.

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