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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 percolation method [1] is a popular approach for analyzing the overlapping community structure of networks.The term network community (also called a module, cluster or cohesive group) has no widely accepted unique definition and it is usually defined as a group of nodes that are more densely connected to each other than to other nodes in the network.
The junction tree algorithm (also known as 'Clique Tree') is a method used in machine learning to extract marginalization in general graphs. In essence, it entails performing belief propagation on a modified graph called a junction tree .
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.
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).
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.
The graph has a c-clique if and only if the formula is satisfiable. [11] There is a simple randomized algorithm due to Schöning (1999) that runs in time (4/3) n where n is the number of variables in the 3-SAT proposition, and succeeds with high probability to correctly decide 3-SAT. [12]