This report describe six different variations of the Bron-Kerbosh algorithm (which solves the problem of listing all maximal cliques in a graph) and compares their. The Bron-Kerbosch algorithm is an efficient method for finding all maximal cliques in a Bron, C. and Kerbosch, J. “Algorithm Finding All Cliques of an . 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.

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Sally is having a party. Jake is popular, so Jake also knows Jess and Ryan.

Then there is Joe. Three randoms Liz, Ty, and Jay show up, because the party is a whomper. Then finally, Erin, who almost always is fashionably late, shows up with Amy and Jack.

Our task is to find all the maximal cliques, the groupings of people that all know each other. NP-complete problems go deep, fast. So, we will start by demonstrating the algorithm on a sub-graph.

In computer science, we say algoritbm clique is a group of nodes that are all completely connected. There are two maximal cliques here: Our minds are powerful, so we can quickly recognize that these are cliques. Computers, also powerful, can recognize these cliques when instructed to do so. What instructions do we need to give the computer to have it see what we see? Start by selecting a node.


Bron–Kerbosch algorithm

Even deeper now, select a node that is neighbors with both Jack and Amy. Now three levels deep, on Erin, there are no new intersecting neighbor nodes. The base case evaluates as kdrbosch. A clique is found. Now backtracking and excluding.

Two recursive calls deep, check if there are neighboring nodes that intersect with Erin and Jack. That would be Amy, but Amy is in the excluded intersect. The base case this time evaluates as false, because there can not be a clique that intersects with an excluded neighbor. Level up, again, to the highest level. Algorlthm the top level, exclude Jack and select another arbitrary node.

Even deeper now, check shared neighbors of Amy and Erin. The only other node in the kerboscu is Jack, who is being excluded, so no maximal clique here.

graph theory – Bron-Kerbosch algorithm for clique finding – Stack Overflow

Backtrack, up to the top level. Amy moves in with Jack to the exclusion setand the search is carried on with another arbitrary node. Erin considers her non-excluded neighbor sSally. Sally is selected, the recursion goes deeper, and the search continues.


Notice, Sally is connected with Erin, but not to Jack and Amy, and for this reason the excluded nodes, Jack and Amy, are not at this set intersect.

Bron-Kerbosch Algorithm

Erin has no one else to clique with, so she moves in with Kerbosvh and Amy to the exclusion set. Sally is only connected with Erin, and Erin is excluded. There is no one else to exclude. The algorithm is complete. Bron-Kerbosch operates on three sets: R, P, and X.

Bron-Kerbosch performs these operations on the sets R, P, and X:. More on set notation: Sign in Get started. Go up brno level. Copy the code here into a text editor and save as main. Never miss a story from Towards Data Sciencewhen you sign up for Medium. Get updates Get updates.