Maximum and minimum isolated vertices in a graph

Given ‘n’ vertices and ‘m’ edges of a graph. Find the minimum number and maximum number of isolated vertices that are possible in the graph. 

Examples:

Input : 4 2
Output : Minimum 0
         Maximum 1

1--2 3--4 <---Minimum -  No isolated vertex
1--2   <--- Maximum - 1 Isolated vertex i.e. 4
   |
   3


Input : 5 2
Output : Minimum 1
         Maximum 2

1--2 3--4 5  <-- Minimum - 1 isolated vertex i.e. 5
1--2   4  5  <-- Maximum - 2 isolated vertex i.e. 4 and 5
   |
   3

1. For minimum number of isolated vertices, we connect two vertices by only one edge. Each vertex should be only connected to one other vertex and each vertex should have degree one 
   Thus if the number of edges is ‘m’, and if ‘n’ vertices <=2 * ‘m’ edges, there is no isolated vertex and if this condition is false, there are n-2*m isolated vertices.
2. For maximum number of isolated vertices, we create a polygon such that each vertex is connected to other vertex and each vertex has a diagonal with every other vertex. Thus, number of diagonals from one vertex to other vertex of n sided polygon is n*(n-3)/2 and number of edges connecting adjacent vertices is n. Thus, total number of edges is n*(n-1)/2.

Below is the implementation of above approach. 

C++

// CPP program to find maximum/minimum number
// of isolated vertices.
#include <bits/stdc++.h>
using namespace std;

// Function to find out the minimum and
// maximum number of isolated vertices
void find(int n, int m)
{
    // Condition to find out minimum number
    // of isolated vertices
    if (n <= 2 * m)
        cout << "Minimum " << 0 << endl;
    else
        cout << "Minimum " << n - 2 * m << endl;

    // To find out maximum number of isolated
    // vertices
    // Loop to find out value of number of
    // vertices that are connected
    int i;

    for (i = 0; i <= n; i++) {
        if (i * (i - 1) / 2 >= m)
            break;
    }
    cout << "Maximum " << n - i;
}

// Driver Function
int main()
{
    // Number of vertices
    int n = 4;

    // Number of edges
    int m = 2;

    // Calling the function to maximum and
    // minimum number of isolated vertices
    find(n, m);
    return 0;
}

Java

// Java program to find maximum/minimum number
// of isolated vertices.

import java.io.*;

class GFG {


// Function to find out the minimum and
// maximum number of isolated vertices
 static void find(int n, int m)
{
    // Condition to find out minimum number
    // of isolated vertices
    if (n <= 2 * m)
        System.out.println( "Minimum " + 0);
    else
        System.out.println( "Minimum " + (n - 2 * m));

    // To find out maximum number of isolated
    // vertices
    // Loop to find out value of number of
    // vertices that are connected
    int i;
    for (i = 0; i <= n; i++) {
        if (i * (i - 1) / 2 >= m)
            break;
    }
    System.out.println( "Maximum " + (n - i));
}

// Driver Function

    public static void main (String[] args) {

    // Number of vertices
    int n = 4;

    // Number of edges
    int m = 2;

    // Calling the function to maximum and
    // minimum number of isolated vertices
    find(n, m);
    }
}
//This code is contributed by inder_verma.

Python3

# Python3 program to find maximum/minimum
# number of isolated vertices.

# Function to find out the minimum and
# maximum number of isolated vertices
def find(n, m) :

    # Condition to find out minimum
    # number of isolated vertices
    if (n <= 2 * m):
        print("Minimum ", 0)
    else:
        print("Minimum ", n - 2 * m )

    # To find out maximum number of
    # isolated vertices
    # Loop to find out value of number
    # of vertices that are connected
    for i in range(0, n + 1):
        if (i * (i - 1) // 2 >= m):
            break

    print("Maximum ", n - i)

# Driver Code
if __name__ == '__main__':

    # Number of vertices
    n = 4

    # Number of edges
    m = 2

    # Calling the function to maximum and
    # minimum number of isolated vertices
    find(n, m)

# This code is contributed by
# SHUBHAMSINGH10

C#

// C# program to find maximum/
// minimum number of isolated vertices.
using System;

class GFG
{

// Function to find out the
// minimum and maximum number
// of isolated vertices
static void find(int n, int m)
{
    // Condition to find out minimum
    // number of isolated vertices
    if (n <= 2 * m)
        Console.WriteLine("Minimum " + 0);
    else
        Console.WriteLine("Minimum " +
                         (n - 2 * m));

    // To find out maximum number
    // of isolated vertices
    // Loop to find out value of
    // number of vertices that
    // are connected
    int i;
    for (i = 0; i <= n; i++)
    {
        if (i * (i - 1) / 2 >= m)
            break;
    }
    Console.WriteLine("Maximum " + (n - i));
}

// Driver Code
public static void Main ()
{

    // Number of vertices
    int n = 4;

    // Number of edges
    int m = 2;

    // Calling the function to
    // maximum and minimum number
    // of isolated vertices
    find(n, m);
}
}

// This code is contributed
// by inder_verma.

PHP

<?php
// PHP program to find maximum/minimum
// number of isolated vertices.

// Function to find out the
// minimum and maximum number
// of isolated vertices
function find($n, $m)
{
    // Condition to find out minimum
    // number of isolated vertices
    if ($n <= 2 * $m)
        echo "Minimum 0\n";
    else
        echo "Minimum " , ($n - 2 * $m);

    // To find out maximum number
    // of isolated vertices
    // Loop to find out value of number
    // of vertices that are connected
    for ($i = 0; $i <= $n; $i++)
    {
        if ($i * ($i - 1) / 2 >= $m)
            break;
    }
    echo "Maximum " , ($n - $i);
}

// Driver Code

// Number of vertices
$n = 4;

// Number of edges
$m = 2;

// Calling the function to
// maximum and minimum number
// of isolated vertices
find($n, $m);

// This code is contributed
// by inder_verma
?>

Javascript

<script>
    // Javascript program to find maximum/
    // minimum number of isolated vertices.

    // Function to find out the
    // minimum and maximum number
    // of isolated vertices
    function find(n, m)
    {

        // Condition to find out minimum
        // number of isolated vertices
        if (n <= 2 * m)
            document.write("Minimum " + 0 + "</br>");
        else
            document.write("Minimum " + (n - 2 * m) + "</br>");

        // To find out maximum number
        // of isolated vertices
        // Loop to find out value of
        // number of vertices that
        // are connected
        let i;
        for (i = 0; i <= n; i++)
        {
            if (i * parseInt((i - 1) / 2, 10) >= m)
                break;
        }
        document.write("Maximum " + (n - i));
    }

    // Number of vertices
    let n = 4;

    // Number of edges
    let m = 2;

    // Calling the function to
    // maximum and minimum number
    // of isolated vertices
    find(n, m);

// This code is contributed by divyeshrabadiya07.
</script>

Output

Minimum 0
Maximum 1

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