# Count the number of Prime Cliques in an undirected graph

Given a graph with N nodes and E edges, the task is to count the number of clique having their size as a prime number or prime number of nodes in the given graph.

A clique is a complete subgraph of a given graph.

Examples:

Input: N = 5, edges[] = { {1, 2}, {2, 3}, {3, 1}, {4, 3}, {4, 5}, {5, 3} } Output: 8
Explanation:
In the given undirected graph, 1->2->3 and 3->4->5 are the two complete subgraphs, both of them are of size 3 which is a prime.
Also, 1-2, 2->3, 3->1, 4->3, 4->5 and 5->3 are complete subgraphs of size 2.
Hence there are 8 prime cliques.

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

Approach: To solve the problem mentioned above, the main idea is to use recursion. All the vertices whose degree is greater than or equal to (K-1) are found and checked which subset of K vertices form a clique. When another edge is added to the present list, it is checked if by adding that edge, the list still forms a clique or not. The following steps can be followed to compute the result:

• To check if the clique size is prime or not, the idea is to use Sieve Of eratosthenes. Create a sieve which will help us to identify if the size is prime or not in O(1) time.
• Form a recursive function with three parameters starting node, length of the present set of nodes and prime array (to check the prime numbers).
• The starting index resembles that no node can be added to the present set less than that index. So a loop is run from that index to n.
• Find that after adding a node to the present set, the set of nodes remains a clique. If yes, that node is added, then the current clique size is checked, if it is prime then the answer is increased by 1 and then the recursive function is called with parameters index of new added node + 1, length of current set + 1 and the prime array.
• The vertices are added until the list does not form a clique. In the end, the answer containing the number of prime cliques is printed.

Below is the implementation of the above approach:

## C++

 `// C++ implementation to Count the number ` `// of Prime Cliques in an undirected graph ` ` `  `#include ` `using` `namespace` `std; ` ` `  `const` `int` `MAX = 100; ` ` `  `// Stores the vertices ` `int` `store[MAX], n; ` ` `  `// Graph ` `int` `graph[MAX][MAX]; ` ` `  `// Degree of the vertices ` `int` `d[MAX]; ` ` `  `// To store the count of prime cliques ` `int` `ans; ` ` `  `// Function to create ` `// Sieve to check primes ` `void` `SieveOfEratosthenes( ` `    ``bool` `prime[], ``int` `p_size) ` `{ ` `    ``// false here indicates ` `    ``// that it is not prime ` `    ``prime = ``false``; ` `    ``prime = ``false``; ` ` `  `    ``for` `(``int` `p = 2; p * p <= p_size; p++) { ` ` `  `        ``// Condition if prime[p] ` `        ``// is not changed, ` `        ``// then it is a prime ` `        ``if` `(prime[p]) { ` ` `  `            ``// Update all multiples of p, ` `            ``// set them to non-prime ` `            ``for` `(``int` `i = p * 2; i <= p_size; i += p) ` `                ``prime[i] = ``false``; ` `        ``} ` `    ``} ` `} ` ` `  `// Function to check ` `// if the given set of ` `// vertices in store array ` `// is a clique or not ` `bool` `is_clique(``int` `b) ` `{ ` ` `  `    ``// Run a loop for all set of edges ` `    ``for` `(``int` `i = 1; i < b; i++) { ` `        ``for` `(``int` `j = i + 1; j < b; j++) ` ` `  `            ``// If any edge is missing ` `            ``if` `(graph[store[i]][store[j]] == 0) ` `                ``return` `false``; ` `    ``} ` `    ``return` `true``; ` `} ` ` `  `// Function to find the count of ` `// all the cliques having prime size ` `void` `primeCliques(``int` `i, ``int` `l, ` `                  ``bool` `prime[]) ` `{ ` `    ``// Check if any vertices from i+1 ` `    ``// can be inserted ` `    ``for` `(``int` `j = i + 1; j <= n; j++) { ` ` `  `        ``// Add the vertex to store ` `        ``store[l] = j; ` ` `  `        ``// If the graph is not ` `        ``// a clique of size k then ` `        ``// it cannot be a clique ` `        ``// by adding another edge ` `        ``if` `(is_clique(l + 1)) { ` ` `  `            ``// increase the count of ` `            ``// prime cliques if the size ` `            ``// of current clique is prime ` `            ``if` `(prime[l]) ` `                ``ans++; ` ` `  `            ``// Check if another edge ` `            ``// can be added ` `            ``primeCliques(j, l + 1, prime); ` `        ``} ` `    ``} ` `} ` ` `  `// Driver code ` `int` `main() ` `{ ` `    ``int` `edges[] = { { 1, 2 }, ` `                       ``{ 2, 3 }, ` `                       ``{ 3, 1 }, ` `                       ``{ 4, 3 }, ` `                       ``{ 4, 5 }, ` `                       ``{ 5, 3 } }; ` ` `  `    ``int` `size = ``sizeof``(edges) ` `               ``/ ``sizeof``(edges); ` `    ``n = 5; ` ` `  `    ``bool` `prime[n + 1]; ` `    ``memset``(prime, ``true``, ``sizeof``(prime)); ` ` `  `    ``SieveOfEratosthenes(prime, n + 1); ` ` `  `    ``for` `(``int` `i = 0; i < size; i++) { ` `        ``graph[edges[i]][edges[i]] = 1; ` `        ``graph[edges[i]][edges[i]] = 1; ` `        ``d[edges[i]]++; ` `        ``d[edges[i]]++; ` `    ``} ` ` `  `    ``ans = 0; ` `    ``primeCliques(0, 1, prime); ` ` `  `    ``cout << ans << ``"\n"``; ` ` `  `    ``return` `0; ` `} `

## Java

 `// Java implementation to Count the number ` `// of Prime Cliques in an undirected graph ` `import` `java.io.*;  ` `import` `java.util.*;  ` ` `  `class` `GFG { ` `     `  `static` `final` `int` `MAX = ``100``; ` ` `  `// Stores the vertices ` `static` `int``[] store = ``new` `int``[MAX]; ` `static` `int` `n; ` ` `  `// Graph ` `static` `int``[][] graph = ``new` `int``[MAX][MAX]; ` ` `  `// Degree of the vertices ` `static` `int``[] d = ``new` `int``[MAX]; ` ` `  `// To store the count of prime cliques ` `static` `int` `ans; ` ` `  `// Function to create ` `// Sieve to check primes ` `static` `void` `SieveOfEratosthenes(``boolean` `prime[],  ` `                                ``int` `p_size) ` `{ ` `     `  `    ``// False here indicates ` `    ``// that it is not prime ` `    ``prime[``0``] = ``false``; ` `    ``prime[``1``] = ``false``; ` ` `  `    ``for``(``int` `p = ``2``; p * p <= p_size; p++) ` `    ``{ ` `         `  `       ``// Condition if prime[p] ` `       ``// is not changed, ` `       ``// then it is a prime ` `       ``if` `(prime[p]) ` `       ``{ ` `            `  `           ``// Update all multiples of p, ` `           ``// set them to non-prime ` `            ``for``(``int` `i = p * ``2``; i <= p_size; i += p) ` `               ``prime[i] = ``false``; ` `       ``} ` `    ``} ` `} ` ` `  `// Function to check ` `// if the given set of ` `// vertices in store array ` `// is a clique or not ` `static` `boolean` `is_clique(``int` `b) ` `{ ` ` `  `    ``// Run a loop for all set of edges ` `    ``for``(``int` `i = ``1``; i < b; i++) ` `    ``{ ` `       ``for``(``int` `j = i + ``1``; j < b; j++) ` `        `  `          ``// If any edge is missing ` `          ``if` `(graph[store[i]][store[j]] == ``0``) ` `             ``return` `false``; ` `    ``} ` `    ``return` `true``; ` `} ` ` `  `// Function to find the count of ` `// all the cliques having prime size ` `static` `void` `primeCliques(``int` `i, ``int` `l,  ` `                         ``boolean` `prime[]) ` `{ ` `     `  `    ``// Check if any vertices from i+1 ` `    ``// can be inserted ` `    ``for``(``int` `j = i + ``1``; j <= n; j++) ` `    ``{ ` `        `  `       ``// Add the vertex to store ` `       ``store[l] = j; ` `        `  `       ``// If the graph is not ` `       ``// a clique of size k then ` `       ``// it cannot be a clique ` `       ``// by adding another edge ` `       ``if` `(is_clique(l + ``1``)) ` `       ``{ ` `            `  `           ``// Increase the count of ` `           ``// prime cliques if the size ` `           ``// of current clique is prime ` `           ``if` `(prime[l]) ` `               ``ans++; ` `                `  `           ``// Check if another edge ` `           ``// can be added ` `           ``primeCliques(j, l + ``1``, prime); ` `       ``} ` `    ``} ` `} ` `     `  `// Driver code  ` `public` `static` `void` `main(String[] args)  ` `{  ` `    ``int` `edges[][] = { { ``1``, ``2` `}, ` `                      ``{ ``2``, ``3` `}, ` `                      ``{ ``3``, ``1` `}, ` `                      ``{ ``4``, ``3` `}, ` `                      ``{ ``4``, ``5` `}, ` `                      ``{ ``5``, ``3` `} }; ` ` `  `    ``int` `size = edges.length; ` `    ``n = ``5``; ` ` `  `    ``boolean``[] prime = ``new` `boolean``[n + ``1``]; ` `    ``Arrays.fill(prime, ``true``); ` ` `  `    ``SieveOfEratosthenes(prime, n); ` ` `  `    ``for``(``int` `i = ``0``; i < size; i++) ` `    ``{ ` `       ``graph[edges[i][``0``]][edges[i][``1``]] = ``1``; ` `       ``graph[edges[i][``1``]][edges[i][``0``]] = ``1``; ` `       ``d[edges[i][``0``]]++; ` `       ``d[edges[i][``1``]]++; ` `    ``} ` `     `  `    ``ans = ``0``; ` `    ``primeCliques(``0``, ``1``, prime); ` ` `  `    ``System.out.println(ans); ` `}  ` `} ` ` `  `// This code is contributed by coder001 `

## C#

 `// C# implementation to count the number ` `// of Prime Cliques in an undirected graph ` `using` `System; ` ` `  `class` `GFG{ ` `     `  `static` `readonly` `int` `MAX = 100; ` ` `  `// Stores the vertices ` `static` `int``[] store = ``new` `int``[MAX]; ` `static` `int` `n; ` ` `  `// Graph ` `static` `int``[,] graph = ``new` `int``[MAX, MAX]; ` ` `  `// Degree of the vertices ` `static` `int``[] d = ``new` `int``[MAX]; ` ` `  `// To store the count of prime cliques ` `static` `int` `ans; ` ` `  `// Function to create ` `// Sieve to check primes ` `static` `void` `SieveOfEratosthenes(``bool` `[]prime,  ` `                                ``int` `p_size) ` `{ ` `     `  `    ``// False here indicates ` `    ``// that it is not prime ` `    ``prime = ``false``; ` `    ``prime = ``false``; ` ` `  `    ``for``(``int` `p = 2; p * p <= p_size; p++) ` `    ``{ ` `        `  `       ``// Condition if prime[p] ` `       ``// is not changed, ` `       ``// then it is a prime ` `       ``if` `(prime[p]) ` `       ``{ ` `            `  `           ``// Update all multiples of p, ` `           ``// set them to non-prime ` `           ``for``(``int` `i = p * 2; i <= p_size;  ` `                   ``i += p) ` `              ``prime[i] = ``false``; ` `       ``} ` `    ``} ` `} ` ` `  `// Function to check if the given ` `// set of vertices in store array ` `// is a clique or not ` `static` `bool` `is_clique(``int` `b) ` `{ ` ` `  `    ``// Run a loop for all set of edges ` `    ``for``(``int` `i = 1; i < b; i++) ` `    ``{ ` `       ``for``(``int` `j = i + 1; j < b; j++) ` `        `  `          ``// If any edge is missing ` `          ``if` `(graph[store[i],store[j]] == 0) ` `              ``return` `false``; ` `    ``} ` `    ``return` `true``; ` `} ` ` `  `// Function to find the count of ` `// all the cliques having prime size ` `static` `void` `primeCliques(``int` `i, ``int` `l,  ` `                         ``bool` `[]prime) ` `{ ` `     `  `    ``// Check if any vertices from i+1 ` `    ``// can be inserted ` `    ``for``(``int` `j = i + 1; j <= n; j++) ` `    ``{ ` `        `  `       ``// Add the vertex to store ` `       ``store[l] = j; ` `        `  `       ``// If the graph is not ` `       ``// a clique of size k then ` `       ``// it cannot be a clique ` `       ``// by adding another edge ` `       ``if` `(is_clique(l + 1)) ` `       ``{ ` `            `  `           ``// Increase the count of ` `           ``// prime cliques if the size ` `           ``// of current clique is prime ` `           ``if` `(prime[l]) ` `               ``ans++; ` `            `  `           ``// Check if another edge ` `           ``// can be added ` `           ``primeCliques(j, l + 1, prime); ` `       ``} ` `    ``} ` `} ` `     `  `// Driver code  ` `public` `static` `void` `Main(String[] args)  ` `{  ` `    ``int` `[,]edges = { { 1, 2 }, ` `                     ``{ 2, 3 }, ` `                     ``{ 3, 1 }, ` `                     ``{ 4, 3 }, ` `                     ``{ 4, 5 }, ` `                     ``{ 5, 3 } }; ` `                      `  `    ``int` `size = edges.GetLength(0); ` `    ``n = 5; ` ` `  `    ``bool``[] prime = ``new` `bool``[n + 1]; ` `    ``for``(``int` `i = 0; i < prime.Length; i++) ` `       ``prime[i] = ``true``; ` ` `  `    ``SieveOfEratosthenes(prime, n); ` ` `  `    ``for``(``int` `i = 0; i < size; i++) ` `    ``{ ` `       ``graph[edges[i, 0],edges[i, 1]] = 1; ` `       ``graph[edges[i, 1],edges[i, 0]] = 1; ` `       ``d[edges[i, 0]]++; ` `       ``d[edges[i, 1]]++; ` `    ``} ` `     `  `    ``ans = 0; ` `    ``primeCliques(0, 1, prime); ` ` `  `    ``Console.WriteLine(ans); ` `}  ` `} ` ` `  `// This code is contributed by Princi Singh `

Output:

```8
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