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Clique (graph theory)

In the mathematical area of graph theory, a clique ( or ) is subset of vertices of an undirected graph, such that its induced subgraph is complete; that is, every two distinct vertices in the clique are adjacent. Cliques are one of the basic concepts of graph theory and are used in many other mathematical problems and constructions on graphs. Cliques have also been studied in computer science: the task of finding whether there is a clique of a given size in a graph (the clique problem) is NP-complete, but despite this hardness result, many algorithms for finding cliques have been studied.
Although the study of complete subgraphs goes back at least to the graph-theoretic reformulation of Ramsey theory by ,〔The earlier work by characterizing planar graphs by forbidden complete and complete bipartite subgraphs was originally phrased in topological rather than graph-theoretic terms.〕 the term ''clique'' comes from , who used complete subgraphs in social networks to model cliques of people; that is, groups of people all of whom know each other. Cliques have many other applications in the sciences and particularly in bioinformatics.
==Definitions==
A clique, ''C'', in an undirected graph is a subset of the vertices, , such that every two distinct vertices are adjacent. This is equivalent to the condition that the subgraph of ''G'' induced by ''C'' is complete. In some cases, the term clique may also refer to the subgraph directly.
A maximal clique is a clique that cannot be extended by including one more adjacent vertex, that is, a clique which does not exist exclusively within the vertex set of a larger clique.
A maximum clique of a graph, ''G'', is a clique, such that there is no clique with more vertices.
The clique number ω(''G'') of a graph ''G'' is the number of vertices in a maximum clique in ''G''.
The intersection number of ''G'' is the smallest number of cliques that together cover all edges of ''G''.
The clique cover number of a graph ''G'' is the smallest number of cliques of ''G'' whose union covers ''V(G)''.
The opposite of a clique is an independent set, in the sense that every clique corresponds to an independent set in the complement graph. The clique cover problem concerns finding as few cliques as possible that include every vertex in the graph.
A related concept is a biclique, a complete bipartite subgraph. The bipartite dimension of a graph is the minimum number of bicliques needed to cover all the edges of the graph.

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