Without pivoting[ edit ] The basic form of the Bron—Kerbosch algorithm is a recursive backtracking algorithm that searches for all maximal cliques in a given graph G. More generally, given three disjoint sets of vertices R, P, and X, it finds the maximal cliques that include all of the vertices in R, some of the vertices in P, and none of the vertices in X. In each call to the algorithm, P and X are disjoint sets whose union consists of those vertices that form cliques when added to R. When P and X are both empty there are no further elements that can be added to R, so R is a maximal clique and the algorithm outputs R. The recursion is initiated by setting R and X to be the empty set and P to be the vertex set of the graph.

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Less Reading time: 30 minutes The Bron—Kerbosch algorithm is an enumeration algorithm for finding maximal cliques in an undirected graph. That is, it lists all subsets of vertices with the two properties that each pair of vertices in one of the listed subsets is connected by an edge, and no listed subset can have any additional vertices added to it while preserving its complete connectivity. The Bron—Kerbosch algorithm is not an output-sensitive algorithm. Unlike some other algorithms for the clique problem, it does not run in polynomial time per maximal clique generated.

Algorithm The algorithm maintains three disjoint sets of nodesR, P, X. Set R stands for the currently growing clique; set P stands for prospective nodes which are connected to all nodes in R and using which R can be expanded; set X contains nodes already processed i.

An important invariant is that all nodes which are connected to every node of R are either in P or X. To avoid reporting cliques which are not maximal, the algorithm checks whether set X is empty, if X is non-empty, the nodes in X may be added to R, but this would yield previously found maximal cliques.

Without pivoting The basic form of the Bron—Kerbosch algorithm is a recursive backtracking algorithm that searches for all maximal cliques in a given graph G.

More generally, given three disjoint sets of vertices R, P, and X, it finds the maximal cliques that include all of the vertices in R, some of the vertices in P, and none of the vertices in X. In each call to the algorithm, P and X are disjoint sets whose union consists of those vertices that form cliques when added to R.

When P and X are both empty there are no further elements that can be added to R, so R is a maximal clique and the algorithm outputs R. Any maximal clique must include either u or one of its non-neighbors, for otherwise the clique could be augmented by adding u to it. Therefore, only u and its non-neighbors need to be tested as the choices for the vertex v that is added to R in each recursive call to the algorithm.

The degeneracy of a graph G is the smallest number d such that every subgraph of G has a vertex with degree d or less. Every graph has a degeneracy ordering, an ordering of the vertices such that each vertex has d or fewer neighbors that come later in the ordering; a degeneracy ordering may be found in linear time by repeatedly selecting the vertex of minimum degree among the remaining vertices.

If the order of the vertices v that the Bron—Kerbosch algorithm loops through is a degeneracy ordering, then the set P of candidate vertices in each call the neighbors of v that are later in the ordering will be guaranteed to have size at most d. The set X of excluded vertices will consist of all earlier neighbors of v, and may be much larger than d. In recursive calls to the algorithm below the topmost level of the recursion, the pivoting version can still be used.

The number of recursive calls to the procedure. By improving the algorithm, we optimize the number of recursive calls in "without pivoting method" with "with pivoting method", thus it improves the runtime complexity of the algorithm. 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. Space Complexity Bron-Kerbosch algorithm takes O n2 auxiliary space in the worst case. Related articles.

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## Bron–Kerbosch algorithm

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## C++ Reference: class BronKerboschAlgorithm

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