Program to find Circuit Rank of an Undirected Graph

Given the number of Vertices and the number of Edges of an Undirected Graph. The task is to determine the Circuit rank.

Circuit Rank: The Circuit rank of an undirected graph is defined as the minimum number of edges that must be removed from the graph to break all of its cycles, converting it into a tree or forest. 

Examples: 

Input :Edges = 7 , Vertices = 5
Output : Circuit rank = 3

Input : Edges = 7 , Vertices = 6
Output : Circuit rank = 2

Formula: 

Circuit rank = Edges - (Vertices - 1)

Look at the sample graph below,

Total number of Edges = 7 and Vertices = 5. 
According to the above formula, 

Circuit Rank =  Edges - (Vertices - 1)
             =  7 - (5 - 1)
             = 3

Therefore, Circuit Rank of the above graph = 3. 
It can be seen in the below image that by removing 3 edges(a-d, a-e, c-d) from the above graph, all of it cycles can be removed. 
 

Implementation:

// C++ Program to find Circuit Rank of an Undirected Graph
#include <bits/stdc++.h>
using namespace std;

// Function that calculates the
// Circuit rank of the Graph.
int Rank(int Edges, int Vertices)
{
    int result = 0;

    // calculates Circuit Rank
    result = Edges - Vertices + 1;

    return result;
}

// Driver Code
int main()
{
    int Edges = 7, Vertices = 5;

    cout << "Circuit Rank = " << Rank(Edges, Vertices);

    return 0;
}

//Java Program to find Circuit Rank of an Undirected Graph

public class GFG {

    //Function that calculates the
    //Circuit rank of the Graph.
    static int Rank(int Edges, int Vertices)
    {
     int result = 0;

     // calculates Circuit Rank
     result = Edges - Vertices + 1;

     return result;
    }

    //Driver Code
    public static void main(String[] args) {
        
         int Edges = 7, Vertices = 5;

         System.out.println("Circuit Rank = " + Rank(Edges, Vertices));
    }
}

# Python 3 program to find Circuit Rank of 
# an Undirected Graph

# Function that calculates the 
# Circuit rank of the Graph. 
def Rank(Edges, Vertices) :

    # calculates Circuit Rank
    result = Edges - Vertices + 1

    return result

# Driver code     
if __name__ == "__main__" :

    Edges, Vertices = 7, 5

    print("Circuit Rank =",Rank(Edges, Vertices))


# This code is contributed by ANKITRAI1

// C# Program to find Circuit
// Rank of an Undirected Graph
using System;

class GFG 
{

// Function that calculates the
// Circuit rank of the Graph.
static int Rank(int Edges,
                int Vertices)
{
    int result = 0;
    
    // calculates Circuit Rank
    result = Edges - Vertices + 1;
    
    return result;
}

// Driver Code
public static void Main()
{
    int Edges = 7, Vertices = 5;

    Console.WriteLine("Circuit Rank = " + 
                       Rank(Edges, Vertices));
}
}

// This code is contributed 
// by inder_verma

<?php
// PHP Program to find Circuit Rank of an Undirected Graph

// Function that calculates the
// Circuit rank of the Graph.
function Rank($Edges, $Vertices)
{
    $result = 0;

    // calculates Circuit Rank
    $result = $Edges - $Vertices + 1;

    return $result;
}

// Driver Code
$Edges = 7;
$Vertices = 5;

echo ("Circuit Rank = ");
echo (Rank($Edges, $Vertices));
    
// This code is contributed 
// by Shivi_Aggarwal
?>

<script>

// Javascript Program to find Circuit
// Rank of an Undirected Graph

// Function that calculates the
// Circuit rank of the Graph.
function Rank(Edges, Vertices)
{
    var result = 0;

    // calculates Circuit Rank
    result = Edges - Vertices + 1;

    return result;
}

// Driver Code
var Edges = 7, Vertices = 5;
document.write( "Circuit Rank = " + Rank(Edges, Vertices));

</script>

Circuit Rank = 3

Time Complexity: O(1)
Auxiliary Space: O(1)