A clique is a subset of vertices of a graph such that every two distinct vertices in the clique are adjacent. Finding a clique from a graph is an important problem in graph theory and many algorithms have been proposed to solve this problem. The most popular algorithm for finding a clique is the technique of the most dense subgraph. This technique is based on the fact that a clique is the most dense subgraph of the graph. In this article, we will discuss a Java program to find a clique by using the technique of the most dense subgraph.
Examples
Example 1:
Let’s take a graph G = (V, E) with the following adjacency matrix:
V = {v1, v2, v3, v4}
E = {(v1, v2), (v2, v3), (v3, v4), (v1, v4)}
Adjacency Matrix:
[[0, 1, 0, 1],Â
[1, 0, 1, 0],Â
[0, 1, 0, 1],Â
[1, 0, 1, 0]]The most dense subgraph of the graph G is the clique {v1, v2, v3, v4}.
Example 2:
Let’s take another graph G = (V, E) with the following adjacency matrix:
V = {v1, v2, v3, v4, v5}
E = {(v1, v2), (v2, v3), (v3, v4), (v4, v5), (v1, v5)}
Adjacency Matrix:
[[0, 1, 0, 0, 1],Â
[1, 0, 1, 0, 0],Â
[0, 1, 0, 1, 0],Â
[0, 0, 1, 0, 1],Â
[1, 0, 0, 1, 0]]The most dense subgraph of the graph G is the clique {v1, v2, v3, v4}.
Approach
The approach used in the program is based on the technique of the most dense subgraph. The algorithm works as follows:
- Initialize an array of size equal to the number of vertices in the graph.
- For each vertex, count the number of edges connected to it.
- Find the vertex with the maximum number of edges and mark it as the first vertex of the clique.
- For each of the remaining vertices in the graph, count the number of edges connected to both the first vertex and the current vertex.
- If the number of edges is equal to the number of vertices in the clique, then add the current vertex to the clique.
- Repeat Steps 4 and 5 until all the vertices have been checked.
Below is the implementation of the above approach.
ImplementationÂ
Java
// Java Program for the above approachimport java.util.ArrayList;import java.util.List;  public class CliqueFinder {      static int[][] adjMatrix;      static int cliqueSize;      static List<Integer> clique;      // Function to initialize the adjacency matrix    static void init(int[][] adjMatrix)    {        clique = new ArrayList<Integer>();        for (int i = 0; i < adjMatrix.length; i++) {            int count = 0;            for (int j = 0; j < adjMatrix.length; j++) {                if (adjMatrix[i][j] == 1)                    count++;            }            if (count > cliqueSize) {                cliqueSize = count;                clique.clear();                clique.add(i);            }        }    }      // Function to find the clique    static void findClique(int[][] adjMatrix)    {        for (int i = 0; i < adjMatrix.length; i++) {            if (!clique.contains(i)) {                int count = 0;                for (int j = 0; j < clique.size(); j++) {                    if (adjMatrix[i][clique.get(j)] == 1)                        count++;                }                if (count == cliqueSize) {                    clique.add(i);                }            }        }    }      // Driver code    public static void main(String[] args)    {        int[][] adjMatrix = { { 0, 1, 0, 1 },                              { 1, 0, 1, 0 },                              { 0, 1, 0, 1 },                              { 1, 0, 1, 0 } };          init(adjMatrix);        findClique(adjMatrix);          // Print the clique        System.out.println("The clique is: ");        for (int i = 0; i < clique.size(); i++) {            System.out.print(clique.get(i) + " ");        }    }} |
The clique is: 0
