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Find k-cores of an undirected graph

  • Difficulty Level : Hard
  • Last Updated : 18 Jun, 2021

Given a graph G and an integer K, K-cores of the graph are connected components that are left after all vertices of degree less than k have been removed (Source wiki)

Example: 

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Input : Adjacency list representation of graph shown
        on left side of below diagram
Output: K-Cores : 
[2] -> 3 -> 4 -> 6
[3] -> 2 -> 4 -> 6 -> 7
[4] -> 2 -> 3 -> 6 -> 7
[6] -> 2 -> 3 -> 4 -> 7
[7] -> 3 -> 4 -> 6

kcore



 
We strongly recommend you to minimize your browser and try this yourself first.
The standard algorithm to find a k-core graph is to remove all the vertices that have degree less than- ‘K’ from the input graph. We must be careful that removing a vertex reduces the degree of all the vertices adjacent to it, hence the degree of adjacent vertices can also drop below-‘K’. And thus, we may have to remove those vertices also. This process may/may not go until there are no vertices left in the graph.

To implement above algorithm, we do a modified DFS on the input graph and delete all the vertices having degree less than ‘K’, then update degrees of all the adjacent vertices, and if their degree falls below ‘K’ we will delete them too.

Below is implementation of above idea. Note that the below program only prints vertices of k cores, but it can be easily extended to print the complete k cores as we have modified adjacency list. 

C++14




// C++ program to find K-Cores of a graph
#include<bits/stdc++.h>
using namespace std;
 
// This class represents a undirected graph using adjacency
// list representation
class Graph
{
    int V; // No. of vertices
 
    // Pointer to an array containing adjacency lists
    list<int> *adj;
public:
    Graph(int V); // Constructor
 
    // function to add an edge to graph
    void addEdge(int u, int v);
 
    // A recursive function to print DFS starting from v
    bool DFSUtil(int, vector<bool> &, vector<int> &, int k);
 
    // prints k-Cores of given graph
    void printKCores(int k);
};
 
// A recursive function to print DFS starting from v.
// It returns true if degree of v after processing is less
// than k else false
// It also updates degree of adjacent if degree of v
// is less than k. And if degree of a processed adjacent
// becomes less than k, then it reduces of degree of v also,
bool Graph::DFSUtil(int v, vector<bool> &visited,
                    vector<int> &vDegree, int k)
{
    // Mark the current node as visited and print it
    visited[v] = true;
 
    // Recur for all the vertices adjacent to this vertex
    list<int>::iterator i;
    for (i = adj[v].begin(); i != adj[v].end(); ++i)
    {
        // degree of v is less than k, then degree of adjacent
        // must be reduced
        if (vDegree[v] < k)
            vDegree[*i]--;
 
        // If adjacent is not processed, process it
        if (!visited[*i])
        {
            // If degree of adjacent after processing becomes
            // less than k, then reduce degree of v also.
            DFSUtil(*i, visited, vDegree, k);
        }
    }
 
    // Return true if degree of v is less than k
    return (vDegree[v] < k);
}
 
Graph::Graph(int V)
{
    this->V = V;
    adj = new list<int>[V];
}
 
void Graph::addEdge(int u, int v)
{
    adj[u].push_back(v);
    adj[v].push_back(u);
}
 
// Prints k cores of an undirected graph
void Graph::printKCores(int k)
{
    // INITIALIZATION
    // Mark all the vertices as not visited and not
    // processed.
    vector<bool> visited(V, false);
    vector<bool> processed(V, false);
 
    int mindeg = INT_MAX;
    int startvertex;
 
    // Store degrees of all vertices
    vector<int> vDegree(V);
    for (int i=0; i<V; i++)
    {
        vDegree[i] = adj[i].size();
 
        if (vDegree[i] < mindeg)
        {
            mindeg = vDegree[i];
            startvertex=i;
        }
    }
 
    DFSUtil(startvertex, visited, vDegree, k);
 
    // If Graph is disconnected.
    for (int i=0; i<V; i++)
        if (visited[i] == false)
            DFSUtil(i, visited, vDegree, k);
 
    // PRINTING K CORES
    cout << "K-Cores : \n";
    for (int v=0; v<V; v++)
    {
        // Only considering those vertices which have degree
        // >= K after BFS
        if (vDegree[v] >= k)
        {
            cout << "\n[" << v << "]";
 
            // Traverse adjacency list of v and print only
            // those adjacent which have vDegree >= k after
            // BFS.
            list<int>::iterator itr;
            for (itr = adj[v].begin(); itr != adj[v].end(); ++itr)
                if (vDegree[*itr] >= k)
                    cout << " -> " << *itr;
        }
    }
}
 
// Driver program to test methods of graph class
int main()
{
    // Create a graph given in the above diagram
    int k = 3;
    Graph g1(9);
    g1.addEdge(0, 1);
    g1.addEdge(0, 2);
    g1.addEdge(1, 2);
    g1.addEdge(1, 5);
    g1.addEdge(2, 3);
    g1.addEdge(2, 4);
    g1.addEdge(2, 5);
    g1.addEdge(2, 6);
    g1.addEdge(3, 4);
    g1.addEdge(3, 6);
    g1.addEdge(3, 7);
    g1.addEdge(4, 6);
    g1.addEdge(4, 7);
    g1.addEdge(5, 6);
    g1.addEdge(5, 8);
    g1.addEdge(6, 7);
    g1.addEdge(6, 8);
    g1.printKCores(k);
 
    cout << endl << endl;
 
    Graph g2(13);
    g2.addEdge(0, 1);
    g2.addEdge(0, 2);
    g2.addEdge(0, 3);
    g2.addEdge(1, 4);
    g2.addEdge(1, 5);
    g2.addEdge(1, 6);
    g2.addEdge(2, 7);
    g2.addEdge(2, 8);
    g2.addEdge(2, 9);
    g2.addEdge(3, 10);
    g2.addEdge(3, 11);
    g2.addEdge(3, 12);
    g2.printKCores(k);
 
    return 0;
}

Java




// Java program to find K-Cores of a graph
import java.util.*;
 
class GFG
{
 
    // This class represents a undirected graph using adjacency
    // list representation
    static class Graph
    {
        int V; // No. of vertices
 
        // Pointer to an array containing adjacency lists
        Vector<Integer>[] adj;
 
        @SuppressWarnings("unchecked")
        Graph(int V)
        {
            this.V = V;
            this.adj = new Vector[V];
 
            for (int i = 0; i < V; i++)
                adj[i] = new Vector<>();
        }
 
        // function to add an edge to graph
        void addEdge(int u, int v)
        {
            this.adj[u].add(v);
            this.adj[v].add(u);
        }
 
        // A recursive function to print DFS starting from v.
        // It returns true if degree of v after processing is less
        // than k else false
        // It also updates degree of adjacent if degree of v
        // is less than k. And if degree of a processed adjacent
        // becomes less than k, then it reduces of degree of v also,
        boolean DFSUtil(int v, boolean[] visited, int[] vDegree, int k)
        {
 
            // Mark the current node as visited and print it
            visited[v] = true;
 
            // Recur for all the vertices adjacent to this vertex
            for (int i : adj[v])
            {
 
                // degree of v is less than k, then degree of adjacent
                // must be reduced
                if (vDegree[v] < k)
                    vDegree[i]--;
 
                // If adjacent is not processed, process it
                if (!visited[i])
                {
 
                    // If degree of adjacent after processing becomes
                    // less than k, then reduce degree of v also.
                    DFSUtil(i, visited, vDegree, k);
                }
            }
 
            // Return true if degree of v is less than k
            return (vDegree[v] < k);
        }
 
        // Prints k cores of an undirected graph
        void printKCores(int k)
        {
 
            // INITIALIZATION
            // Mark all the vertices as not visited and not
            // processed.
            boolean[] visited = new boolean[V];
            boolean[] processed = new boolean[V];
            Arrays.fill(visited, false);
            Arrays.fill(processed, false);
 
            int mindeg = Integer.MAX_VALUE;
            int startvertex = 0;
 
            // Store degrees of all vertices
            int[] vDegree = new int[V];
            for (int i = 0; i < V; i++)
            {
                vDegree[i] = adj[i].size();
 
                if (vDegree[i] < mindeg)
                {
                    mindeg = vDegree[i];
                    startvertex = i;
                }
            }
            DFSUtil(startvertex, visited, vDegree, k);
 
            // DFS traversal to update degrees of all
            // vertices.
            for (int i = 0; i < V; i++)
                if (!visited[i])
                    DFSUtil(i, visited, vDegree, k);
 
            // PRINTING K CORES
            System.out.println("K-Cores : ");
            for (int v = 0; v < V; v++)
            {
 
                // Only considering those vertices which have degree
                // >= K after BFS
                if (vDegree[v] >= k)
                {
                    System.out.printf("\n[%d]", v);
 
                    // Traverse adjacency list of v and print only
                    // those adjacent which have vDegree >= k after
                    // BFS.
                    for (int i : adj[v])
                        if (vDegree[i] >= k)
                            System.out.printf(" -> %d", i);
                }
            }
        }
    }
 
    // Driver Code
    public static void main(String[] args)
    {
 
        // Create a graph given in the above diagram
        int k = 3;
        Graph g1 = new Graph(9);
        g1.addEdge(0, 1);
        g1.addEdge(0, 2);
        g1.addEdge(1, 2);
        g1.addEdge(1, 5);
        g1.addEdge(2, 3);
        g1.addEdge(2, 4);
        g1.addEdge(2, 5);
        g1.addEdge(2, 6);
        g1.addEdge(3, 4);
        g1.addEdge(3, 6);
        g1.addEdge(3, 7);
        g1.addEdge(4, 6);
        g1.addEdge(4, 7);
        g1.addEdge(5, 6);
        g1.addEdge(5, 8);
        g1.addEdge(6, 7);
        g1.addEdge(6, 8);
        g1.printKCores(k);
 
        System.out.println();
 
        Graph g2 = new Graph(13);
        g2.addEdge(0, 1);
        g2.addEdge(0, 2);
        g2.addEdge(0, 3);
        g2.addEdge(1, 4);
        g2.addEdge(1, 5);
        g2.addEdge(1, 6);
        g2.addEdge(2, 7);
        g2.addEdge(2, 8);
        g2.addEdge(2, 9);
        g2.addEdge(3, 10);
        g2.addEdge(3, 11);
        g2.addEdge(3, 12);
        g2.printKCores(k);
    }
}
 
// This code is contributed by
// sanjeev2552

Python




#saurabh_jain861
# Python program to find K-Cores of a graph
from collections import defaultdict
 
# This class represents a undirected graph using adjacency
# list representation
 
 
class Graph:
 
    def __init__(self):
 
        # default dictionary to store graph
        self.graph = defaultdict(list)
 
    # function to add an edge to undirected graph
    def addEdge(self, u, v):
        self.graph[u].append(v)
        self.graph[v].append(u)
 
    # A recursive function to call DFS starting from v.
    # It returns true if vDegree of v after processing is less
    # than k else false
    # It also updates vDegree of adjacent if vDegree of v
    # is less than k. And if vDegree of a processed adjacent
    # becomes less than k, then it reduces of vDegree of v also,
    def DFSUtil(self, v, visited, vDegree, k):
 
        # Mark the current node as visited
        visited.add(v)
 
        # Recur for all the vertices adjacent to this vertex
        for i in self.graph[v]:
 
            # vDegree of v is less than k, then vDegree of
            # adjacent must be reduced
            if vDegree[v] < k:
                vDegree[i] = vDegree[i] - 1
 
            # If adjacent is not processed, process it
            if i not in visited:
 
                # If vDegree of adjacent after processing becomes
                # less than k, then reduce vDegree of v also
                self.DFSUtil(i, visited, vDegree, k)
 
    def PrintKCores(self, k):
 
        visit = set()
        degree = defaultdict(lambda: 0)
 
        for i in list(self.graph):
            degree[i] = len(self.graph[i])
 
        for i in list(self.graph):
 
            if i not in visit:
                self.DFSUtil(i, visit, degree, k)
 
        # print(degree)
        # print(self.graph)
 
        for i in list(self.graph):
 
            if degree[i] >= k:
                print(str("\n [ ") + str(i) + str(" ]"), end=" ")
 
                for j in self.graph[i]:
                    if degree[j] >= k:
                        print("-> " + str(j), end=" ")
 
                print()
 
 
k = 3
g1 = Graph()
g1.addEdge(0, 1)
g1.addEdge(0, 2)
g1.addEdge(1, 2)
g1.addEdge(1, 5)
g1.addEdge(2, 3)
g1.addEdge(2, 4)
g1.addEdge(2, 5)
g1.addEdge(2, 6)
g1.addEdge(3, 4)
g1.addEdge(3, 6)
g1.addEdge(3, 7)
g1.addEdge(4, 6)
g1.addEdge(4, 7)
g1.addEdge(5, 6)
g1.addEdge(5, 8)
g1.addEdge(6, 7)
g1.addEdge(6, 8)
g1.PrintKCores(k)

C#




// C# program to find K-Cores of a graph
using System;
using System.Collections.Generic;
 
class GFG{
 
// This class represents a undirected
// graph using adjacency list
// representation
public class Graph
{
     
    // No. of vertices
    int V;
 
    // Pointer to an array containing
    // adjacency lists
    List<int>[] adj;
 
    public Graph(int V)
    {
        this.V = V;
        this.adj = new List<int>[V];
 
        for(int i = 0; i < V; i++)
            adj[i] = new List<int>();
    }
 
    // Function to add an edge to graph
    public void addEdge(int u, int v)
    {
        this.adj[u].Add(v);
        this.adj[v].Add(u);
    }
 
    // A recursive function to print DFS
    // starting from v. It returns true
    // if degree of v after processing
    // is less than k else false
    // It also updates degree of adjacent
    // if degree of v is less than k. And
    // if degree of a processed adjacent
    // becomes less than k, then it reduces
    // of degree of v also,
    bool DFSUtil(int v, bool[] visited,
                 int[] vDegree, int k)
    {
         
        // Mark the current node as
        // visited and print it
        visited[v] = true;
 
        // Recur for all the vertices
        // adjacent to this vertex
        foreach (int i in adj[v])
        {
             
            // Degree of v is less than k,
            // then degree of adjacent
            // must be reduced
            if (vDegree[v] < k)
                vDegree[i]--;
 
            // If adjacent is not
            // processed, process it
            if (!visited[i])
            {
                 
                // If degree of adjacent after
                // processing becomes less than
                // k, then reduce degree of v also.
                DFSUtil(i, visited, vDegree, k);
            }
        }
 
        // Return true if degree of
        // v is less than k
        return (vDegree[v] < k);
    }
 
    // Prints k cores of an undirected graph
    public void printKCores(int k)
    {
         
        // INITIALIZATION
        // Mark all the vertices as not
        // visited and not processed.
        bool[] visited = new bool[V];
        //bool[] processed = new bool[V];
   
        int mindeg = int.MaxValue;
        int startvertex = 0;
 
        // Store degrees of all vertices
        int[] vDegree = new int[V];
         
        for(int i = 0; i < V; i++)
        {
            vDegree[i] = adj[i].Count;
             
            if (vDegree[i] < mindeg)
            {
                mindeg = vDegree[i];
                startvertex = i;
            }
        }
        DFSUtil(startvertex, visited, vDegree, k);
 
        // DFS traversal to update degrees of all
        // vertices.
        for(int i = 0; i < V; i++)
            if (!visited[i])
                DFSUtil(i, visited, vDegree, k);
 
        // PRINTING K CORES
        Console.WriteLine("K-Cores : ");
         
        for(int v = 0; v < V; v++)
        {
             
            // Only considering those vertices
            // which have degree >= K after BFS
            if (vDegree[v] >= k)
            {
                Console.Write("\n " + v);
 
                // Traverse adjacency list of v
                // and print only those adjacent
                // which have vDegree >= k after
                // BFS.
                foreach(int i in adj[v])
                    if (vDegree[i] >= k)
                        Console.Write(" -> " + i);
            }
        }
    }
}
 
// Driver Code
public static void Main(String[] args)
{
     
    // Create a graph given in the
    // above diagram
    int k = 3;
     
    Graph g1 = new Graph(9);
    g1.addEdge(0, 1);
    g1.addEdge(0, 2);
    g1.addEdge(1, 2);
    g1.addEdge(1, 5);
    g1.addEdge(2, 3);
    g1.addEdge(2, 4);
    g1.addEdge(2, 5);
    g1.addEdge(2, 6);
    g1.addEdge(3, 4);
    g1.addEdge(3, 6);
    g1.addEdge(3, 7);
    g1.addEdge(4, 6);
    g1.addEdge(4, 7);
    g1.addEdge(5, 6);
    g1.addEdge(5, 8);
    g1.addEdge(6, 7);
    g1.addEdge(6, 8);
    g1.printKCores(k);
 
    Console.WriteLine();
 
    Graph g2 = new Graph(13);
    g2.addEdge(0, 1);
    g2.addEdge(0, 2);
    g2.addEdge(0, 3);
    g2.addEdge(1, 4);
    g2.addEdge(1, 5);
    g2.addEdge(1, 6);
    g2.addEdge(2, 7);
    g2.addEdge(2, 8);
    g2.addEdge(2, 9);
    g2.addEdge(3, 10);
    g2.addEdge(3, 11);
    g2.addEdge(3, 12);
     
    g2.printKCores(k);
}
}
 
// This code is contributed by Princi Singh

Output :

K-Cores : 

[2] -> 3 -> 4 -> 6
[3] -> 2 -> 4 -> 6 -> 7
[4] -> 2 -> 3 -> 6 -> 7
[6] -> 2 -> 3 -> 4 -> 7
[7] -> 3 -> 4 -> 6

K-Cores : 

Time complexity of the above solution is O(V + E) where V is number of vertices and E is number of edges.

Related Concepts : 
Degeneracy : Degeneracy of a graph is the largest value k such that the graph has a k-core. For example, the above shown graph has a 3-Cores and doesn’t have 4 or higher cores. Therefore, above graph is 3-degenerate. 
Degeneracy of a graph is used to measure how sparse graph is.

Reference : 
https://en.wikipedia.org/wiki/Degeneracy_%28graph_theory%29
This article is contributed by Rachit Belwariar. Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above.




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