Find if a degree sequence can form a simple graph | Havel-Hakimi Algorithm - GeeksforGeeks

25 Aug, 2021
Given a sequence of non-negative integers arr[], the task is to check if there exists a simple graph corresponding to this degree sequence. Note that a simple graph is a graph with no self-loops and parallel edges.
Examples: 
Input: arr[] = {3, 3, 3, 3} Output: Yes This is actually a complete graph(K4)
Input: arr[] = {3, 2, 1, 0} Output: No A vertex has degree n-1 so it’s connected to all the other n-1 vertices. But another vertex has degree 0 i.e. isolated. It’s a contradiction. 
Approach: One way to check the existence of a simple graph is by Havel-Hakimi algorithm given below: 
Sort the sequence of non-negative integers in non-increasing order.
Delete the first element(say V). Subtract 1 from the next V elements.
Repeat 1 and 2 until one of the stopping conditions is met.
Stopping conditions: 
All the elements remaining are equal to 0 (Simple graph exists).
Negative number encounter after subtraction (No simple graph exists).
Not enough elements remaining for the subtraction step (No simple graph exists).
Below is the implementation of the above approach:
C++
Java
Python3
C#
Javascript
// C++ implementation of the approach#include <bits/stdc++.h>using namespace std; // Function that returns true if// a simple graph existsbool graphExists(vector<int> &a, int n){    // Keep performing the operations until one    // of the stopping condition is met    while (1)    {        // Sort the list in non-decreasing order        sort(a.begin(), a.end(), greater<>());         // Check if all the elements are equal to 0        if (a[0] == 0)            return true;         // Store the first element in a variable        // and delete it from the list        int v = a[0];        a.erase(a.begin() + 0);         // Check if enough elements        // are present in the list        if (v > a.size())            return false;         // Subtract first element from next v elements        for (int i = 0; i < v; i++)        {            a[i]--;             // Check if negative element is            // encountered after subtraction            if (a[i] < 0)                return false;        }    }} // Driver Codeint main(){    vector<int> a = {3, 3, 3, 3};    int n = a.size();     graphExists(a, n) ? cout << "Yes" :                        cout << "NO" << endl;     return 0;} // This code is contributed by// sanjeev2552
// Java implementation of the approachimport java.util.*; @SuppressWarnings("unchecked") class GFG{   // Function that returns true if// a simple graph existsstatic boolean graphExists(ArrayList a, int n){         // Keep performing the operations until one    // of the stopping condition is met    while (true)    {                 // Sort the list in non-decreasing order        Collections.sort(a, Collections.reverseOrder());                  // Check if all the elements are equal to 0        if ((int)a.get(0) == 0)            return true;            // Store the first element in a variable        // and delete it from the list        int v = (int)a.get(0);        a.remove(a.get(0));            // Check if enough elements        // are present in the list        if (v > a.size())            return false;            // Subtract first element from        // next v elements        for(int i = 0; i < v; i++)        {            a.set(i, (int)a.get(i) - 1);                // Check if negative element is            // encountered after subtraction            if ((int)a.get(i) < 0)                return false;        }    }}   // Driver Codepublic static void main(String[] args){    ArrayList a = new ArrayList();    a.add(3);    a.add(3);    a.add(3);    a.add(3);         int n = a.size();          if (graphExists(a, n))    {        System.out.print("Yes");    }    else    {        System.out.print("NO");    }}} // This code is contributed by pratham76
# Python3 implementation of the approach # Function that returns true if# a simple graph existsdef graphExists(a):     # Keep performing the operations until one    # of the stopping condition is met    while True:         # Sort the list in non-decreasing order        a = sorted(a, reverse = True)         # Check if all the elements are equal to 0        if a[0]== 0 and a[len(a)-1]== 0:            return True         # Store the first element in a variable        # and delete it from the list        v = a[0]        a = a[1:]         # Check if enough elements        # are present in the list        if v>len(a):            return False         # Subtract first element from next v elements        for i in range(v):            a[i]-= 1             # Check if negative element is            # encountered after subtraction            if a[i]<0:                return False  # Driver codea = [3, 3, 3, 3]if(graphExists(a)):    print("Yes")else:    print("No")
// C# implementation of the approachusing System;using System.Collections; class GFG{  // Function that returns true if// a simple graph existsstatic bool graphExists(ArrayList a, int n){         // Keep performing the operations until one    // of the stopping condition is met    while (true)    {                 // Sort the list in non-decreasing order        a.Sort();        a.Reverse();                 // Check if all the elements are equal to 0        if ((int)a[0] == 0)            return true;           // Store the first element in a variable        // and delete it from the list        int v = (int)a[0];        a.Remove(a[0]);           // Check if enough elements        // are present in the list        if (v > a.Count)            return false;           // Subtract first element from        // next v elements        for(int i = 0; i < v; i++)        {            a[i] = (int)a[i] - 1;               // Check if negative element is            // encountered after subtraction            if ((int)a[i] < 0)                return false;        }    }}  // Driver Codepublic static void Main(string[] args){    ArrayList a = new ArrayList(){ 3, 3, 3, 3 };    int n = a.Count;         if (graphExists(a, n))    {        Console.Write("Yes");    }    else    {        Console.Write("NO");    }}} // This code is contributed by rutvik_56
<script> // Javascript implementation of the approach // Function that returns true if// a simple graph existsfunction graphExists(a, n){         // Keep performing the operations until one    // of the stopping condition is met    while (1)    {                 // Sort the list in non-decreasing order        a.sort((a, b) => b - a)         // Check if all the elements are equal to 0        if (a[0] == 0)            return true;         // Store the first element in a variable        // and delete it from the list        var v = a[0];        a.shift();         // Check if enough elements        // are present in the list        if (v > a.length)            return false;         // Subtract first element from next v elements        for(var i = 0; i < v; i++)        {            a[i]--;             // Check if negative element is            // encountered after subtraction            if (a[i] < 0)                return false;        }    }} // Driver Codevar a = [ 3, 3, 3, 3 ];var n = a.length;graphExists(a, n) ? document.write("Yes"):                    document.write("NO");                     // This code is contributed by rrrtnx </script>
Yes
 
Time Complexity: O()Auxiliary Space: O(1)
sanjeev2552
Tapaj
rutvik_56
pratham76
rrrtnx
pankajsharmagfg
Algorithms
Graph
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Graph
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