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# Nodes with prime degree in an undirected Graph

• Difficulty Level : Medium
• Last Updated : 20 May, 2021

Given an undirected graph with N vertices and M edges, the task is to print all the nodes of the given graph whose degree is a Prime Number.
Examples:

Input: N = 4, arr[][] = { { 1, 2 }, { 1, 3 }, { 1, 4 }, { 2, 3 }, { 2, 4 }, { 3, 4 } }
Output: 1 2 3 4
Explanation:
Below is the graph for the above information:

The degree of the node as per above graph is:
Node -> Degree
1 -> 3
2 -> 3
3 -> 3
4 -> 3
Hence, the nodes with prime degree are 1 2 3 4
Input: N = 5, arr[][] = { { 1, 2 }, { 1, 3 }, { 2, 4 }, { 2, 5 } }
Output: 1

Approach:

1. Use Sieve of Eratosthenes to calculate the prime numbers up to 105.
2. For each vertex, the degree can be calculated by the length of the Adjacency List of the given graph at the corresponding vertex.
3. Print those vertices of the given graph whose degree is a Prime Number.

Below is the implementation of the above approach:

## C++

 `// C++ implementation of the approach` `#include ``using` `namespace` `std;` `int` `n = 10005;` `// To store Prime Numbers``vector<``bool``> Prime(n + 1, ``true``);` `// Function to find the prime numbers``// till 10^5``void` `SieveOfEratosthenes()``{` `    ``int` `i, j;``    ``Prime[0] = Prime[1] = ``false``;``    ``for` `(i = 2; i * i <= 10005; i++) {` `        ``// Traverse all multiple of i``        ``// and make it false``        ``if` `(Prime[i]) {` `            ``for` `(j = 2 * i; j < 10005; j += i) {``                ``Prime[j] = ``false``;``            ``}``        ``}``    ``}``}` `// Function to print the nodes having``// prime degree``void` `primeDegreeNodes(``int` `N, ``int` `M,``                      ``int` `edges[][2])``{``    ``// To store Adjacency List of``    ``// a Graph``    ``vector<``int``> Adj[N + 1];` `    ``// Make Adjacency List``    ``for` `(``int` `i = 0; i < M; i++) {``        ``int` `x = edges[i][0];``        ``int` `y = edges[i][1];` `        ``Adj[x].push_back(y);``        ``Adj[y].push_back(x);``    ``}` `    ``// To precompute prime numbers``    ``// till 10^5``    ``SieveOfEratosthenes();` `    ``// Traverse each vertex``    ``for` `(``int` `i = 1; i <= N; i++) {` `        ``// Find size of Adjacency List``        ``int` `x = Adj[i].size();` `        ``// If length of Adj[i] is Prime``        ``// then print it``        ``if` `(Prime[x])``            ``cout << i << ``' '``;``    ``}``}` `// Driver code``int` `main()``{``    ``// Vertices and Edges``    ``int` `N = 4, M = 6;` `    ``// Edges``    ``int` `edges[M][2] = { { 1, 2 }, { 1, 3 },``                        ``{ 1, 4 }, { 2, 3 },``                        ``{ 2, 4 }, { 3, 4 } };` `    ``// Function Call``    ``primeDegreeNodes(N, M, edges);` `    ``return` `0;``}`

## Java

 `// Java implementation of the approach``import` `java.util.*;``class` `GFG{` `static` `int` `n = ``10005``;` `// To store Prime Numbers``static` `boolean` `[]Prime = ``new` `boolean``[n + ``1``];` `// Function to find the prime numbers``// till 10^5``static` `void` `SieveOfEratosthenes()``{``    ``int` `i, j;``    ``Prime[``0``] = Prime[``1``] = ``false``;``    ``for` `(i = ``2``; i * i <= ``10005``; i++)``    ``{` `        ``// Traverse all multiple of i``        ``// and make it false``        ``if` `(Prime[i])``        ``{``            ``for` `(j = ``2` `* i; j < ``10005``; j += i)``            ``{``                ``Prime[j] = ``false``;``            ``}``        ``}``    ``}``}` `// Function to print the nodes having``// prime degree``static` `void` `primeDegreeNodes(``int` `N, ``int` `M,``                              ``int` `edges[][])``{``    ``// To store Adjacency List of``    ``// a Graph``    ``Vector []Adj = ``new` `Vector[N + ``1``];``    ``for``(``int` `i = ``0``; i < Adj.length; i++)``        ``Adj[i] = ``new` `Vector();` `    ``// Make Adjacency List``    ``for` `(``int` `i = ``0``; i < M; i++)``    ``{``        ``int` `x = edges[i][``0``];``        ``int` `y = edges[i][``1``];` `        ``Adj[x].add(y);``        ``Adj[y].add(x);``    ``}` `    ``// To precompute prime numbers``    ``// till 10^5``    ``SieveOfEratosthenes();` `    ``// Traverse each vertex``    ``for` `(``int` `i = ``1``; i <= N; i++)``    ``{` `        ``// Find size of Adjacency List``        ``int` `x = Adj[i].size();` `        ``// If length of Adj[i] is Prime``        ``// then print it``        ``if` `(Prime[x])``            ``System.out.print(i + ``" "``);``    ``}``}` `// Driver code``public` `static` `void` `main(String[] args)``{``    ``// Vertices and Edges``    ``int` `N = ``4``, M = ``6``;` `    ``// Edges``    ``int` `edges[][] = { { ``1``, ``2` `}, { ``1``, ``3` `},``                      ``{ ``1``, ``4` `}, { ``2``, ``3` `},``                      ``{ ``2``, ``4` `}, { ``3``, ``4` `} };``    ``Arrays.fill(Prime, ``true``);``    ` `    ``// Function Call``    ``primeDegreeNodes(N, M, edges);``}``}` `// This code is contributed by sapnasingh4991`

## Python3

 `# Python3 implementation of``# the above approach``n ``=` `10005``;`` ` `# To store Prime Numbers``Prime ``=` `[``True` `for` `i ``in` `range``(n ``+` `1``)]`` ` `# Function to find``# the prime numbers``# till 10^5``def` `SieveOfEratosthenes():`` ` `    ``i ``=` `2`   `    ``Prime[``0``] ``=` `Prime[``1``] ``=` `False``;``    ` `    ``while` `i ``*` `i <``=` `10005``:`` ` `        ``# Traverse all multiple``        ``# of i and make it false``        ``if` `(Prime[i]):           ``            ``for` `j ``in` `range``(``2` `*` `i, ``10005``):``                ``Prime[j] ``=` `False`       `        ``i ``+``=` `1`  `    ` `# Function to print the``# nodes having prime degree``def` `primeDegreeNodes(N, M, edges):` `    ``# To store Adjacency``    ``# List of a Graph``    ``Adj ``=` `[[] ``for` `i ``in` `range``(N ``+` `1``)];`` ` `    ``# Make Adjacency List``    ``for` `i ``in` `range``(M):``        ``x ``=` `edges[i][``0``];``        ``y ``=` `edges[i][``1``];`` ` `        ``Adj[x].append(y);``        ``Adj[y].append(x);   `` ` `    ``# To precompute prime``    ``# numbers till 10^5``    ``SieveOfEratosthenes();`` ` `    ``# Traverse each vertex``    ``for` `i ``in` `range``(N ``+` `1``):`` ` `        ``# Find size of Adjacency List``        ``x ``=` `len``(Adj[i]);`` ` `        ``# If length of Adj[i] is Prime``        ``# then print it``        ``if` `(Prime[x]):``            ``print``(i, end ``=` `' '``)          ` `# Driver code``if` `__name__ ``=``=` `"__main__"``:``    ` `    ``# Vertices and Edges``    ``N ``=` `4``    ``M ``=` `6`` ` `    ``# Edges``    ``edges ``=` `[[``1``, ``2``], [``1``, ``3``],``             ``[``1``, ``4``], [``2``, ``3``],``             ``[``2``, ``4``], [``3``, ``4``]];`` ` `    ``# Function Call``    ``primeDegreeNodes(N, M, edges);` `# This code is contributed by rutvik_56`

## C#

 `// C# implementation of the approach``using` `System;``using` `System.Collections.Generic;` `class` `GFG{` `static` `int` `n = 10005;` `// To store Prime Numbers``static` `bool` `[]Prime = ``new` `bool``[n + 1];` `// Function to find the prime numbers``// till 10^5``static` `void` `SieveOfEratosthenes()``{``    ``int` `i, j;``    ``Prime[0] = Prime[1] = ``false``;``    ``for``(i = 2; i * i <= 10005; i++)``    ``{``       ` `       ``// Traverse all multiple of i``       ``// and make it false``       ``if` `(Prime[i])``       ``{``           ``for``(j = 2 * i; j < 10005; j += i)``           ``{``              ``Prime[j] = ``false``;``           ``}``       ``}``    ``}``}` `// Function to print the nodes having``// prime degree``static` `void` `primeDegreeNodes(``int` `N, ``int` `M,``                             ``int` `[,]edges)``{``    ` `    ``// To store Adjacency List of``    ``// a Graph``    ``List<``int``> []Adj = ``new` `List<``int``>[N + 1];``    ``for``(``int` `i = 0; i < Adj.Length; i++)``       ``Adj[i] = ``new` `List<``int``>();` `    ``// Make Adjacency List``    ``for``(``int` `i = 0; i < M; i++)``    ``{``       ``int` `x = edges[i, 0];``       ``int` `y = edges[i, 1];``       ` `       ``Adj[x].Add(y);``       ``Adj[y].Add(x);``    ``}``    ` `    ``// To precompute prime numbers``    ``// till 10^5``    ``SieveOfEratosthenes();` `    ``// Traverse each vertex``    ``for``(``int` `i = 1; i <= N; i++)``    ``{``        ` `       ``// Find size of Adjacency List``       ``int` `x = Adj[i].Count;``       ` `       ``// If length of Adj[i] is Prime``       ``// then print it``       ``if` `(Prime[x])``           ``Console.Write(i + ``" "``);``    ``}``}` `// Driver code``public` `static` `void` `Main(String[] args)``{``    ` `    ``// Vertices and Edges``    ``int` `N = 4, M = 6;` `    ``// Edges``    ``int` `[,]edges = { { 1, 2 }, { 1, 3 },``                     ``{ 1, 4 }, { 2, 3 },``                     ``{ 2, 4 }, { 3, 4 } };``                     ` `    ``for``(``int` `i = 0; i < Prime.Length; i++)``       ``Prime[i] = ``true``;``    ` `    ``// Function Call``    ``primeDegreeNodes(N, M, edges);``}``}` `// This code is contributed by 29AjayKumar`

## Javascript

 ``
Output:
`1 2 3 4`

Time Complexity: O(N + M), where N is the number of vertices and M is the number of edges.

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