# 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.

Below is the implementation of the above approach:

## C++

`// 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; ` `} ` |

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## Java

`//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)); ` ` ` `} ` `} ` |

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## Python 3

`# 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 ` |

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## C#

`// 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 ` |

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## PHP

`<?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 ` `?> ` |

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**Output:**

Circuit Rank = 3

## Recommended Posts:

- Eulerian path and circuit for undirected graph
- Find k-cores of an undirected graph
- Find if an undirected graph contains an independent set of a given size
- Find minimum weight cycle in an undirected graph
- Disjoint Set (Or Union-Find) | Set 1 (Detect Cycle in an Undirected Graph)
- Euler Circuit in a Directed Graph
- Clone an Undirected Graph
- Detect cycle in an undirected graph using BFS
- Number of Triangles in an Undirected Graph
- Connected Components in an undirected graph
- Detect cycle in an undirected graph
- Print all the cycles in an undirected graph
- Eulerian Path in undirected graph
- Product of lengths of all cycles in an undirected graph
- Check if there is a cycle with odd weight sum in an undirected graph

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