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# Check if there exists a connected graph that satisfies the given conditions

• Difficulty Level : Hard
• Last Updated : 26 Nov, 2019

Given two integers N and K. The task is to find a connected graph with N vertices such that there are exactly K pairs (i, j) where the shortest distance between them is 2. If no such graph exists then print -1.

Note:

1. The first-line output should be the number of edges(say m) in the graph and the next m lines should contain two numbers represents the edge between the vertices.
2. In case of multiple answers print any of them.

Examples:

Input: N = 5, K = 3
Output: 7
1 2
1 3
1 4
1 5
3 4
3 5
4 5 Input: N = 5, K = 8
Output: -1

## Recommended: Please try your approach on {IDE} first, before moving on to the solution.

Approach: An N-vertices connected graph has at least N-1 edges. The shortest distance of each pair is equal to 1. So obviously, it is clear that there doesn’t exist a solution if K > N * (N – 1) / 2 – (N – 1) = (N – 1) * (N – 2) / 2.
Conversely, it can be shown that there exists a solution if K ≤ (N – 1) * (N – 2) / 2 by constructing a graph that satisfies the condition. First, let’s consider the graph where each vertex is connected with all the other vertices then the shortest between any two vertices is 1. Now remove any K edges then there exist exactly K such pairs.

Below is the implementation of the above approach:

## C++

 `// C++ implementation of the approach``#include ``using` `namespace` `std;`` ` `// Function to find the required graph``void` `connected_graph(``int` `n, ``int` `k)``{``    ``// If no such graph exists``    ``if` `(k > (n - 1) * (n - 2) / 2) {``        ``cout << -1 << endl;``        ``return``;``    ``}`` ` `    ``// Consider edge between all vertices``    ``bool` `isEdge[n][n] = {};``    ``for` `(``int` `i = 0; i < n; i++) {``        ``for` `(``int` `j = i + 1; j < n; j++)``            ``isEdge[i][j] = ``true``;``    ``}`` ` `    ``// Remove K vertices``    ``int` `cnt = 0;``    ``for` `(``int` `i = 1; i < n; i++) {``        ``for` `(``int` `j = i + 1; j < n; j++) {``            ``if` `(cnt < k) {``                ``isEdge[i][j] = ``false``;``                ``cnt++;``            ``}``        ``}``    ``}`` ` `    ``// Store all the edges``    ``vector > vec;``    ``for` `(``int` `i = 0; i < n; i++) {``        ``for` `(``int` `j = i + 1; j < n; j++) {``            ``if` `(isEdge[i][j])``                ``vec.emplace_back(i, j);``        ``}``    ``}`` ` `    ``// Print all the edges``    ``cout << vec.size() << endl;``    ``for` `(``int` `i = 0; i < vec.size(); i++) {``        ``cout << vec[i].first + 1 << ``" "``             ``<< vec[i].second + 1 << endl;``    ``}``}`` ` `// Driver code``int` `main()``{``    ``int` `n = 5, k = 3;`` ` `    ``// Function call``    ``connected_graph(n, k);`` ` `    ``return` `0;``}`

## Java

 `// Java implementation of the approach``import` `java.util.*;`` ` `class` `GFG``{``static` `class` `pair``{ ``    ``int` `first, second; ``    ``public` `pair(``int` `first, ``int` `second) ``    ``{ ``        ``this``.first = first; ``        ``this``.second = second; ``    ``} ``}`` ` `// Function to find the required graph``static` `void` `connected_graph(``int` `n, ``int` `k)``{``    ``// If no such graph exists``    ``if` `(k > (n - ``1``) * (n - ``2``) / ``2``) ``    ``{``        ``System.out.println(-``1``);``        ``return``;``    ``}`` ` `    ``// Consider edge between all vertices``    ``boolean` `[][]isEdge = ``new` `boolean``[n][n];``    ``for` `(``int` `i = ``0``; i < n; i++) ``    ``{``        ``for` `(``int` `j = i + ``1``; j < n; j++)``            ``isEdge[i][j] = ``true``;``    ``}`` ` `    ``// Remove K vertices``    ``int` `cnt = ``0``;``    ``for` `(``int` `i = ``1``; i < n; i++) ``    ``{``        ``for` `(``int` `j = i + ``1``; j < n; j++) ``        ``{``            ``if` `(cnt < k)``            ``{``                ``isEdge[i][j] = ``false``;``                ``cnt++;``            ``}``        ``}``    ``}`` ` `    ``// Store all the edges``    ``Vector vec = ``new` `Vector<>();``    ``for` `(``int` `i = ``0``; i < n; i++) ``    ``{``        ``for` `(``int` `j = i + ``1``; j < n; j++)``        ``{``            ``if` `(isEdge[i][j])``                ``vec.add(``new` `pair(i, j));``        ``}``    ``}`` ` `    ``// Print all the edges``    ``System.out.println(vec.size());``    ``for` `(``int` `i = ``0``; i < vec.size(); i++) ``    ``{``        ``System.out.println(vec.get(i).first + ``1` `+ ``                    ``" "` `+ (vec.get(i).second + ``1``));``    ``}``}`` ` `// Driver code``public` `static` `void` `main(String[] args) ``{``    ``int` `n = ``5``, k = ``3``;`` ` `    ``// Function call``    ``connected_graph(n, k);``}``}`` ` `// This code is contributed by 29AjayKumar`

## Python3

 `# Python3 implementation of the approach ``import` `numpy as np;`` ` `# Function to find the required graph ``def` `connected_graph(n, k) : `` ` `    ``# If no such graph exists ``    ``if` `(k > (n ``-` `1``) ``*` `(n ``-` `2``) ``/` `2``) :``        ``print``(``-``1``) ; ``        ``return``; `` ` `    ``# Consider edge between all vertices ``    ``isEdge ``=` `np.zeros((n, n)); ``    ``for` `i ``in` `range``(n) :``        ``for` `j ``in` `range``(i ``+` `1``, n) :``            ``isEdge[i][j] ``=` `True``; `` ` `    ``# Remove K vertices ``    ``cnt ``=` `0``; ``    ``for` `i ``in` `range``(``1``, n) :``        ``for` `j ``in` `range``(i ``+` `1` `, n) :``            ``if` `(cnt < k) :``                ``isEdge[i][j] ``=` `False``; ``                ``cnt ``+``=` `1``; `` ` `    ``# Store all the edges ``    ``vec ``=` `[]; ``    ``for` `i ``in` `range``(n) : ``        ``for` `j ``in` `range``(i ``+` `1``, n) :``            ``if` `(isEdge[i][j]) :``                ``vec.append([i, j]); `` ` `    ``# Print all the edges ``    ``print``(``len``(vec)); ``    ``for` `i ``in` `range``(``len``(vec)) :``        ``print``(vec[i][``0``] ``+` `1``, vec[i][``1``] ``+` `1``); `` ` `# Driver code ``if` `__name__ ``=``=` `"__main__"` `: `` ` `    ``n ``=` `5``; k ``=` `3``;`` ` `    ``# Function call ``    ``connected_graph(n, k); `` ` `# This code is contributed by Ankit Rai`

## C#

 `// C# implementation of the approach``using` `System;``using` `System.Collections.Generic;`` ` `class` `GFG``{``public` `class` `pair``{ ``    ``public` `int` `first, second; ``    ``public` `pair(``int` `first, ``int` `second) ``    ``{ ``        ``this``.first = first; ``        ``this``.second = second; ``    ``} ``}`` ` `// Function to find the required graph``static` `void` `connected_graph(``int` `n, ``int` `k)``{``    ``// If no such graph exists``    ``if` `(k > (n - 1) * (n - 2) / 2) ``    ``{``        ``Console.WriteLine(-1);``        ``return``;``    ``}`` ` `    ``// Consider edge between all vertices``    ``bool` `[,]isEdge = ``new` `bool``[n, n];``    ``for` `(``int` `i = 0; i < n; i++) ``    ``{``        ``for` `(``int` `j = i + 1; j < n; j++)``            ``isEdge[i, j] = ``true``;``    ``}`` ` `    ``// Remove K vertices``    ``int` `cnt = 0;``    ``for` `(``int` `i = 1; i < n; i++) ``    ``{``        ``for` `(``int` `j = i + 1; j < n; j++) ``        ``{``            ``if` `(cnt < k)``            ``{``                ``isEdge[i, j] = ``false``;``                ``cnt++;``            ``}``        ``}``    ``}`` ` `    ``// Store all the edges``    ``List vec = ``new` `List();``    ``for` `(``int` `i = 0; i < n; i++) ``    ``{``        ``for` `(``int` `j = i + 1; j < n; j++)``        ``{``            ``if` `(isEdge[i, j])``                ``vec.Add(``new` `pair(i, j));``        ``}``    ``}`` ` `    ``// Print all the edges``    ``Console.WriteLine(vec.Count);``    ``for` `(``int` `i = 0; i < vec.Count; i++) ``    ``{``        ``Console.WriteLine(vec[i].first + 1 + ``                   ``" "` `+ (vec[i].second + 1));``    ``}``}`` ` `// Driver code``public` `static` `void` `Main(String[] args) ``{``    ``int` `n = 5, k = 3;`` ` `    ``// Function call``    ``connected_graph(n, k);``}``}`` ` `// This code is contributed by 29AjayKumar`
Output:
```7
1 2
1 3
1 4
1 5
3 4
3 5
4 5
```

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