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The maximum clique problem may be solved using as a subroutine an algorithm for the maximal clique listing problem, because the maximum clique must be included among all the maximal cliques. [ 17 ] In the k -clique problem, the input is an undirected graph and a number k .
The clique cover number of a graph G is the smallest number of cliques of G whose union covers the set of vertices V of the graph. A maximum clique transversal of a graph is a subset of vertices with the property that each maximum clique of the graph contains at least one vertex in the subset. [2]
The clique cover problem in computational complexity theory is the algorithmic problem of finding a minimum clique cover, or (rephrased as a decision problem) finding a clique cover whose number of cliques is below a given threshold. Finding a minimum clique cover is NP-hard, and its decision version is NP-complete.
The MaxCliqueDyn algorithm is an algorithm for finding a maximum clique in an undirected graph. MaxCliqueDyn is based on the MaxClique algorithm, which finds a maximum clique of bounded size. The bound is found using a coloring algorithm. MaxCliqueDyn extends MaxClique to include dynamically varying bounds.
The independent set problem and the clique problem are complementary: a clique in G is an independent set in the complement graph of G and vice versa. Therefore, many computational results may be applied equally well to either problem. For example, the results related to the clique problem have the following corollaries:
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.
A planted clique is a clique created from another graph by adding edges between all pairs of a selected subset of vertices. The planted clique problem can be formalized as a decision problem over a random distribution on graphs, parameterized by two numbers, n (the number of vertices), and k (the size of the clique).
Most obviously, the solutions to the maximum independent set problem, the maximum clique problem, and the minimum independent dominating problem must all be maximal independent sets or maximal cliques, and can be found by an algorithm that lists all maximal independent sets or maximal cliques and retains the ones with the largest or smallest size.