Maximal Clique Problem | Recursive Solution
Given a small graph with N nodes and E edges, the task is to find the maximum clique in the given graph. A clique is a complete subgraph of a given graph. This means that all nodes in the said subgraph are directly connected to each other, or there is an edge between any two nodes in the subgraph. The maximal clique is the complete subgraph of a given graph which contains the maximum number of nodes.
Examples:
Input: N = 4, edges[][] = {{1, 2}, {2, 3}, {3, 1}, {4, 3}, {4, 1}, {4, 2}}
Output: 4Input: N = 5, edges[][] = {{1, 2}, {2, 3}, {3, 1}, {4, 3}, {4, 5}, {5, 3}}
Output: 3
Approach: The idea is to use recursion to solve the problem.
- When an edge is added to the present list, check that if by adding that edge to the present list, does it still form a clique or not.
- The vertices are added until the list does not form a clique. Then, the list is backtracked to find a larger subset which forms a clique.
Below is the implementation of the above approach:
C++
// C++ implementation of the approach #include <bits/stdc++.h> 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]; // 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 all the sizes // of maximal cliques int maxCliques(int i, int l) { // Maximal clique size int max_ = 0; // 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)) { // Update max max_ = max(max_, l); // Check if another edge can be added max_ = max(max_, maxCliques(j, l + 1)); } } return max_; } // Driver code int main() { int edges[][2] = { { 1, 2 }, { 2, 3 }, { 3, 1 }, { 4, 3 }, { 4, 1 }, { 4, 2 } }; int size = sizeof(edges) / sizeof(edges[0]); n = 4; 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]]++; } cout << maxCliques(0, 1); return 0; } |
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Java
// Java implementation of the approach import java.util.*; class GFG { static int MAX = 100, n; // Stores the vertices static int []store = new int[MAX]; // Graph static int [][]graph = new int[MAX][MAX]; // Degree of the vertices static int []d = new int[MAX]; // 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 all the sizes // of maximal cliques static int maxCliques(int i, int l) { // Maximal clique size int max_ = 0; // 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)) { // Update max max_ = Math.max(max_, l); // Check if another edge can be added max_ = Math.max(max_, maxCliques(j, l + 1)); } } return max_; } // Driver code public static void main(String[] args) { int [][]edges = { { 1, 2 }, { 2, 3 }, { 3, 1 }, { 4, 3 }, { 4, 1 }, { 4, 2 } }; int size = edges.length; n = 4; 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]]++; } System.out.print(maxCliques(0, 1)); } } // This code is contributed by 29AjayKumar |
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Python3
# Python3 implementation of the approach MAX = 100; n = 0; # Stores the vertices store = [0] * MAX; # Graph graph = [[0 for i in range(MAX)] for j in range(MAX)]; # Degree of the vertices d = [0] * MAX; # Function to check if the given set of # vertices in store array is a clique or not def is_clique(b): # Run a loop for all set of edges for i in range(1, b): for j in range(i + 1, b): # If any edge is missing if (graph[store[i]][store[j]] == 0): return False; return True; # Function to find all the sizes # of maximal cliques def maxCliques(i, l): # Maximal clique size max_ = 0; # Check if any vertices from i+1 # can be inserted for j in range(i + 1, n + 1): # 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)): # Update max max_ = max(max_, l); # Check if another edge can be added max_ = max(max_, maxCliques(j, l + 1)); return max_; # Driver code if __name__ == '__main__': edges = [[ 1, 2 ],[ 2, 3 ],[ 3, 1 ], [ 4, 3 ],[ 4, 1 ],[ 4, 2 ]]; size = len(edges); n = 4; for i in range(size): graph[edges[i][0]][edges[i][1]] = 1; graph[edges[i][1]][edges[i][0]] = 1; d[edges[i][0]] += 1; d[edges[i][1]] += 1; print(maxCliques(0, 1)); # This code is contributed by PrinciRaj1992 |
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C#
// C# implementation of the approach using System; class GFG { static int MAX = 100, n; // Stores the vertices static int []store = new int[MAX]; // Graph static int [,]graph = new int[MAX,MAX]; // Degree of the vertices static int []d = new int[MAX]; // 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 all the sizes // of maximal cliques static int maxCliques(int i, int l) { // Maximal clique size int max_ = 0; // 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)) { // Update max max_ = Math.Max(max_, l); // Check if another edge can be added max_ = Math.Max(max_, maxCliques(j, l + 1)); } } return max_; } // Driver code public static void Main(String[] args) { int [,]edges = { { 1, 2 }, { 2, 3 }, { 3, 1 }, { 4, 3 }, { 4, 1 }, { 4, 2 } }; int size = edges.GetLength(0); n = 4; 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]]++; } Console.Write(maxCliques(0, 1)); } } // This code is contributed by PrinciRaj1992 |
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Output:
4
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