Bron-Kerbosch algorithm for clique finding

Question

Can anyone tell me, where on the web I can find an explanation for Bron-Kerbosch algorithm for clique finding or explain here how it works?

I know it was published in "Algorithm 457: finding all cliques of an undirected graph" book, but I can't find free source that will describe the algorithm.

I don't need a source code for the algorithm, I need an explanation of how it works.

Answer 1

I was also trying to wrap my head around the Bron-Kerbosch algorithm, so I wrote my own implementation in python. It includes a test case and some comments. Hope this helps.

class Node(object):

    def __init__(self, name):
        self.name = name
        self.neighbors = []

    def __repr__(self):
        return self.name

A = Node('A')
B = Node('B')
C = Node('C')
D = Node('D')
E = Node('E')

A.neighbors = [B, C]
B.neighbors = [A, C]
C.neighbors = [A, B, D]
D.neighbors = [C, E]
E.neighbors = [D]

all_nodes = [A, B, C, D, E]

def find_cliques(potential_clique=[], remaining_nodes=[], skip_nodes=[], depth=0):

    # To understand the flow better, uncomment this:
    # print (' ' * depth), 'potential_clique:', potential_clique, 'remaining_nodes:', remaining_nodes, 'skip_nodes:', skip_nodes

    if len(remaining_nodes) == 0 and len(skip_nodes) == 0:
        print 'This is a clique:', potential_clique
        return

    for node in remaining_nodes:

        # Try adding the node to the current potential_clique to see if we can make it work.
        new_potential_clique = potential_clique + [node]
        new_remaining_nodes = [n for n in remaining_nodes if n in node.neighbors]
        new_skip_list = [n for n in skip_nodes if n in node.neighbors]
        find_cliques(new_potential_clique, new_remaining_nodes, new_skip_list, depth + 1)

        # We're done considering this node.  If there was a way to form a clique with it, we
        # already discovered its maximal clique in the recursive call above.  So, go ahead
        # and remove it from the list of remaining nodes and add it to the skip list.
        remaining_nodes.remove(node)
        skip_nodes.append(node)

find_cliques(remaining_nodes=all_nodes)

Answer 2

Try finding someone with an ACM student account who can give you a copy of the paper, which is here: http://portal.acm.org/citation.cfm?doid=362342.362367

I just downloaded it, and it's only two pages long, with an implementation in Algol 60!