# Ways to Remove Edges from a Complete Graph to make Odd Edges

Given a complete graph with **N** vertices, the task is to count the number of ways to remove edges such that the resulting graph has odd number of edges.

**Examples:**

Input:N = 3

Output:4

The initial graph has 3 edges as it is a complete graph. We can remove edges (1, 2) and (1, 3) or (1, 2) and (2, 3) or (1, 3) and (2, 3) or we do not remove any of the edges.

Input:N = 4

Output:32

**Approach:** As the graph is complete so the total number of edges will be **E = N * (N – 1) / 2**. Now there are two cases,

- If
**E**is**even**then you have to remove odd number of edges, so the total number of ways will be which is equivalent to . - If
**E**is**odd**then you have to remove even number of edges, so the total number of ways will be which is equivalent to .

**Note** that if **N = 1** then the answer will be **0**.

Below is the implementation of the above approach:

## C++

`// C++ implementation of the approach ` `#include <bits/stdc++.h> ` `using` `namespace` `std; ` ` ` `// Function to return the number of ways ` `// to remove edges from the graph so that ` `// odd number of edges are left in the graph ` `int` `countWays(` `int` `N) ` `{ ` ` ` `// Total number of edges ` ` ` `int` `E = (N * (N - 1)) / 2; ` ` ` ` ` `if` `(N == 1) ` ` ` `return` `0; ` ` ` ` ` `return` `pow` `(2, E - 1); ` `} ` ` ` `// Driver code ` `int` `main() ` `{ ` ` ` `int` `N = 4; ` ` ` `cout << countWays(N); ` ` ` ` ` `return` `0; ` `} ` |

*chevron_right*

*filter_none*

## Java

`// Java implementation of the approach ` `class` `GfG ` `{ ` ` ` `// Function to return the number of ways ` `// to remove edges from the graph so that ` `// odd number of edges are left in the graph ` `static` `int` `countWays(` `int` `N) ` `{ ` ` ` `// Total number of edges ` ` ` `int` `E = (N * (N - ` `1` `)) / ` `2` `; ` ` ` ` ` `if` `(N == ` `1` `) ` ` ` `return` `0` `; ` ` ` ` ` `return` `(` `int` `)Math.pow(` `2` `, E - ` `1` `); ` `} ` ` ` `// Driver code ` `public` `static` `void` `main(String[] args) ` `{ ` ` ` `int` `N = ` `4` `; ` ` ` `System.out.println(countWays(N)); ` `} ` `} ` ` ` `// This code is contributed by Prerna Saini ` |

*chevron_right*

*filter_none*

## Python3

`# Python3 implementation of the approach ` ` ` `# Function to return the number of ways ` `# to remove edges from the graph so that ` `# odd number of edges are left in the graph ` `def` `countWays(N): ` ` ` ` ` `# Total number of edges ` ` ` `E ` `=` `(N ` `*` `(N ` `-` `1` `)) ` `/` `2` ` ` ` ` `if` `(N ` `=` `=` `1` `): ` ` ` `return` `0` ` ` ` ` `return` `int` `(` `pow` `(` `2` `, E ` `-` `1` `)) ` ` ` `# Driver code ` `if` `__name__ ` `=` `=` `'__main__'` `: ` ` ` `N ` `=` `4` ` ` `print` `(countWays(N)) ` ` ` `# This code contributed by PrinciRaj1992 ` |

*chevron_right*

*filter_none*

## C#

`// C# implementation of the approach ` ` ` `using` `System; ` ` ` `public` `class` `GFG{ ` ` ` `// Function to return the number of ways ` `// to remove edges from the graph so that ` `// odd number of edges are left in the graph ` `static` `int` `countWays(` `int` `N) ` `{ ` ` ` `// Total number of edges ` ` ` `int` `E = (N * (N - 1)) / 2; ` ` ` ` ` `if` `(N == 1) ` ` ` `return` `0; ` ` ` ` ` `return` `(` `int` `)Math.Pow(2, E - 1); ` `} ` ` ` `// Driver code ` ` ` `static` `public` `void` `Main (){ ` ` ` ` ` `int` `N = 4; ` ` ` `Console.WriteLine(countWays(N)); ` ` ` `} ` `} ` `// This code is contributed by ajit. ` |

*chevron_right*

*filter_none*

## PHP

`<?php ` `// PHP implementation of the approach ` ` ` `// Function to return the number of ways ` `// to remove edges from the graph so that ` `// odd number of edges are left in the graph ` `function` `countWays(` `$N` `) ` `{ ` ` ` `// Total number of edges ` ` ` `$E` `= (` `$N` `* (` `$N` `- 1)) / 2; ` ` ` ` ` `if` `(` `$N` `== 1) ` ` ` `return` `0; ` ` ` ` ` `return` `(int)pow(2, ` `$E` `- 1); ` `} ` ` ` `// Driver code ` `$N` `= 4; ` `echo` `(countWays(` `$N` `)); ` ` ` `// This code is contributed ` `// by Code_Mech. ` `?> ` |

*chevron_right*

*filter_none*

**Output:**

32

## Recommended Posts:

- Program to find total number of edges in a Complete Graph
- Count number of edges in an undirected graph
- Minimum number of edges between two vertices of a Graph
- Minimum number of edges between two vertices of a graph using DFS
- Number of Simple Graph with N Vertices and M Edges
- Maximum number of edges in Bipartite graph
- All vertex pairs connected with exactly k edges in a graph
- Tree, Back, Edge and Cross Edges in DFS of Graph
- Largest subset of Graph vertices with edges of 2 or more colors
- Shortest path with exactly k edges in a directed and weighted graph
- Minimum edges required to add to make Euler Circuit
- Remove all outgoing edges except edge with minimum weight
- Assign directions to edges so that the directed graph remains acyclic
- Maximum number of edges among all connected components of an undirected graph
- Program to find the diameter, cycles and edges of a Wheel Graph

If you like GeeksforGeeks and would like to contribute, you can also write an article using contribute.geeksforgeeks.org or mail your article to contribute@geeksforgeeks.org. See your article appearing on the GeeksforGeeks main page and help other Geeks.

Please Improve this article if you find anything incorrect by clicking on the "Improve Article" button below.