Graph Problem - Finding Cliques

Question

Hi Folks,
I'm stuck with a problem involving undirected graphs and was wondering if anyone might be able to help.
I have a Vector containing pairs of letters which represent edges in a graph from one point to another. e.g. 'ab' would be an edge from point 'a' to point 'b'. In many cases this Vector contains a number of sub-graphs e.g.
ab
ac
bc
de
df
ef
{ab, ac, bc} make up one complete sub-graph and {de, df, ef} makes up another, where each point has an edge to all the other points in the group.
I want to be able to identify these sub-graphs and seperate them (or at least identify them as being different groups). In graph theory I believe these groups are known as cliques.
I am only interested in finding maximal cliques (complete sub-groups) and in my particular situation a point will never exist in more than one clique (so no overlapping groups). Also the vector will only ever contain complete sub-graphs (no incomplete sub-graphs where edges are missing).
Does anyone know how to solve this problem? Any help would be greatly appreciated. Please let me know if you need any more info.
Regards,
Jamie

Answer 1

I use the following O(n^4 or less) method in my map-folding algorithm:
    static int fastGroups(Group[] group, boolean[][] join, int n) {
        int[] g = new int[n];
        boolean[] row;
        int c = 0, d;
        for(int i=0, j, k, m; i<n; i++) {
            for(j=i+1; j<n; j++) {
                if(!join[ i][j]) continue;
                for(m=0; m<c; m++) {
                   if(group[m].contains(i) && group[m].contains(j)) break;
                if(m<c) continue;
                for(k=0, d=0; k<n; k++) {
                    row = join[k];
                    if(!row[ i] || !row[j]) continue;
                    for(m=0; m<d; m++) if(!row[g[m]]) break;
                    if(m==d) g[d++] = k;
                group[c++] = new Group(g,d);
        group[c] = null;
        return c;
    }This populates the cliques in the given array and returns the number of cliques.
This method works with "well behaved" groups, where each group can be achieved by choosing some two elements of it and then searching the rest in order of appearance. So if you have the followin cliques (1,4,5), (2,5,6), (3,4,6) and (4,5,6):
1-4-3
5-6
2
you will miss the clique (4,5,6). In your case this should not be a problem.