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:**

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

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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/C++

// 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 of 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. if (DFSUtil(*i, visited, vDegree, k)) vDegree[v]--; } } // 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); // DFS traversal to update degrees of all // vertices. 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; }

## 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,vertices): self.V= vertices #No. of vertices # 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[v] = True # 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 visited[i]==False: # If vDegree of adjacent after processing becomes # less than k, then reduce vDegree of v also if (self.DFSUtil(i,visited,vDegree,k)): vDegree[v]-=1 # Return true if vDegree of v is less than k return vDegree[v] < k # Prints k cores of an undirected graph def printKCores(self,k): # INITIALIZATION # Mark all the vertices as not visited visited = [False]*self.V # Store vDegrees of all vertices vDegree = [0]*self.V for i in self.graph: vDegree[i]=len(self.graph[i]) # choose any vertex as starting vertex self.DFSUtil(0,visited,vDegree,k) # DFS traversal to update vDegrees of all # vertices,in case they are unconnected for i in range(self.V): if visited[i] ==False: self.DFSUtil(i,k,vDegree,visited) # PRINTING K CORES print "\n K-cores: " for v in range(self.V): # Only considering those vertices which have # vDegree >= K after DFS if vDegree[v] >= k: print str("\n [ ") + str(v) + str(" ]"), # Traverse adjacency list of v and print only # those adjacent which have vvDegree >= k # after DFS for i in self.graph[v]: if vDegree[i] >= k: print "-> " + str(i), k = 3; g1 = 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) g2 = 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 Neelam Yadav

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 :

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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