In the mathematical area of graph theory, a **clique** ( /ˈkliːk/ or /ˈklɪk/ ) is a subset of vertices of an undirected graph such that every two distinct vertices in the clique are adjacent. That is, a clique of a graph *G* is an induced subgraph of *G* that is complete. 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 Erdős & Szekeres (1935),^{ [1] } the term *clique* comes from Luce & Perry (1949), 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.

A **clique**, *C*, in an undirected graph *G* = (*V*, *E*) is a subset of the vertices, *C* ⊆ *V*, such that every two distinct vertices are adjacent. This is equivalent to the condition that the induced subgraph of *G* induced by *C* is a complete graph. 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. Some authors define cliques in a way that requires them to be maximal, and use other terminology for complete subgraphs that are not maximal.

A **maximum clique** of a graph, *G*, is a clique, such that there is no clique with more vertices. Moreover, 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 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 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.

Mathematical results concerning cliques include the following.

- Turán's theorem gives a lower bound on the size of a clique in dense graphs.
^{ [3] }If a graph has sufficiently many edges, it must contain a large clique. For instance, every graph with vertices and more than edges must contain a three-vertex clique. - Ramsey's theorem states that every graph or its complement graph contains a clique with at least a logarithmic number of vertices.
^{ [4] } - According to a result of Moon & Moser (1965), a graph with 3
*n*vertices can have at most 3^{n}maximal cliques. The graphs meeting this bound are the Moon–Moser graphs*K*_{3,3,...}, a special case of the Turán graphs arising as the extremal cases in Turán's theorem. - Hadwiger's conjecture, still unproven, relates the size of the largest clique minor in a graph (its Hadwiger number) to its chromatic number.
- The Erdős–Faber–Lovász conjecture is another unproven statement relating graph coloring to cliques.

Several important classes of graphs may be defined or characterized by their cliques:

- A cluster graph is a graph whose connected components are cliques.
- A block graph is a graph whose biconnected components are cliques.
- A chordal graph is a graph whose vertices can be ordered into a perfect elimination ordering, an ordering such that the neighbors of each vertex
*v*that come later than*v*in the ordering form a clique. - A cograph is a graph all of whose induced subgraphs have the property that any maximal clique intersects any maximal independent set in a single vertex.
- An interval graph is a graph whose maximal cliques can be ordered in such a way that, for each vertex
*v*, the cliques containing*v*are consecutive in the ordering. - A line graph is a graph whose edges can be covered by edge-disjoint cliques in such a way that each vertex belongs to exactly two of the cliques in the cover.
- A perfect graph is a graph in which the clique number equals the chromatic number in every induced subgraph.
- A split graph is a graph in which some clique contains at least one endpoint of every edge.
- A triangle-free graph is a graph that has no cliques other than its vertices and edges.

Additionally, many other mathematical constructions involve cliques in graphs. Among them,

- The clique complex of a graph
*G*is an abstract simplicial complex*X*(*G*) with a simplex for every clique in*G* - A simplex graph is an undirected graph κ(
*G*) with a vertex for every clique in a graph*G*and an edge connecting two cliques that differ by a single vertex. It is an example of median graph, and is associated with a median algebra on the cliques of a graph: the median*m*(*A*,*B*,*C*) of three cliques*A*,*B*, and*C*is the clique whose vertices belong to at least two of the cliques*A*,*B*, and*C*.^{ [5] } - The clique-sum is a method for combining two graphs by merging them along a shared clique.
- Clique-width is a notion of the complexity of a graph in terms of the minimum number of distinct vertex labels needed to build up the graph from disjoint unions, relabeling operations, and operations that connect all pairs of vertices with given labels. The graphs with clique-width one are exactly the disjoint unions of cliques.
- The intersection number of a graph is the minimum number of cliques needed to cover all the graph's edges.
- The clique graph of a graph is the intersection graph of its maximal cliques.

Closely related concepts to complete subgraphs are subdivisions of complete graphs and complete graph minors. In particular, Kuratowski's theorem and Wagner's theorem characterize planar graphs by forbidden complete and complete bipartite subdivisions and minors, respectively.

In computer science, the clique problem is the computational problem of finding a maximum clique, or all cliques, in a given graph. It is NP-complete, one of Karp's 21 NP-complete problems.^{ [6] } It is also fixed-parameter intractable, and hard to approximate. Nevertheless, many algorithms for computing cliques have been developed, either running in exponential time (such as the Bron–Kerbosch algorithm) or specialized to graph families such as planar graphs or perfect graphs for which the problem can be solved in polynomial time.

The word "clique", in its graph-theoretic usage, arose from the work of Luce & Perry (1949), who used complete subgraphs to model cliques (groups of people who all know each other) in social networks. The same definition was used by Festinger (1949) in an article using less technical terms. Both works deal with uncovering cliques in a social network using matrices. For continued efforts to model social cliques graph-theoretically, see e.g. Alba (1973), Peay (1974), and Doreian & Woodard (1994).

Many different problems from bioinformatics have been modeled using cliques. For instance, Ben-Dor, Shamir & Yakhini (1999) model the problem of clustering gene expression data as one of finding the minimum number of changes needed to transform a graph describing the data into a graph formed as the disjoint union of cliques; Tanay, Sharan & Shamir (2002) discuss a similar biclustering problem for expression data in which the clusters are required to be cliques. Sugihara (1984) uses cliques to model ecological niches in food webs. Day & Sankoff (1986) describe the problem of inferring evolutionary trees as one of finding maximum cliques in a graph that has as its vertices characteristics of the species, where two vertices share an edge if there exists a perfect phylogeny combining those two characters. Samudrala & Moult (1998) model protein structure prediction as a problem of finding cliques in a graph whose vertices represent positions of subunits of the protein. And by searching for cliques in a protein-protein interaction network, Spirin & Mirny (2003) found clusters of proteins that interact closely with each other and have few interactions with proteins outside the cluster. Power graph analysis is a method for simplifying complex biological networks by finding cliques and related structures in these networks.

In electrical engineering, Prihar (1956) uses cliques to analyze communications networks, and Paull & Unger (1959) use them to design efficient circuits for computing partially specified Boolean functions. Cliques have also been used in automatic test pattern generation: a large clique in an incompatibility graph of possible faults provides a lower bound on the size of a test set.^{ [7] } Cong & Smith (1993) describe an application of cliques in finding a hierarchical partition of an electronic circuit into smaller subunits.

In chemistry, Rhodes et al. (2003) use cliques to describe chemicals in a chemical database that have a high degree of similarity with a target structure. Kuhl, Crippen & Friesen (1983) use cliques to model the positions in which two chemicals will bind to each other.

- ↑ The earlier work by Kuratowski (1930) characterizing planar graphs by forbidden complete and complete bipartite subgraphs was originally phrased in topological rather than graph-theoretic terms.
- ↑ Chang, Kloks & Lee (2001).
- ↑ Turán (1941).
- ↑ Graham, Rothschild & Spencer (1990).
- ↑ Barthélemy, Leclerc & Monjardet (1986), page 200.
- ↑ Karp (1972).
- ↑ Hamzaoglu & Patel (1998).

In mathematics, **graph theory** is the study of *graphs*, which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of *vertices* which are connected by *edges*. A distinction is made between **undirected graphs**, where edges link two vertices symmetrically, and **directed graphs**, where edges link two vertices asymmetrically. Graphs are one of the principal objects of study in discrete mathematics.

In the mathematical field of graph theory, a **bipartite graph** is a graph whose vertices can be divided into two disjoint and independent sets and such that every edge connects a vertex in to one in . Vertex sets and are usually called the *parts* of the graph. Equivalently, a bipartite graph is a graph that does not contain any odd-length cycles.

In computer science, the **clique problem** is the computational problem of finding cliques in a graph. It has several different formulations depending on which cliques, and what information about the cliques, should be found. Common formulations of the clique problem include finding a maximum clique, finding a maximum weight clique in a weighted graph, listing all maximal cliques, and solving the decision problem of testing whether a graph contains a clique larger than a given size.

This is a **glossary of graph theory**. Graph theory is the study of graphs, systems of nodes or vertices connected in pairs by lines or edges.

The **Turán graph***T*(*n*,*r*) is a complete multipartite graph formed by partitioning a set of *n* vertices into *r* subsets, with sizes as equal as possible, and connecting two vertices by an edge if and only if they belong to different subsets. The graph will have subsets of size , and subsets of size . That is, it is a complete *r*-partite graph

In graph theory, **Turán's theorem** bounds the number of edges that can be included in an undirected graph that does not have a complete subgraph of a given size. It is one of the central results of extremal graph theory, an area studying the largest or smallest graphs with given properties, and is a special case of the forbidden subgraph problem on the maximum number of edges in a graph that does not have a given subgraph.

In graph theory, an **independent set**, **stable set**, **coclique** or **anticlique** is a set of vertices in a graph, no two of which are adjacent. That is, it is a set of vertices such that for every two vertices in , there is no edge connecting the two. Equivalently, each edge in the graph has at most one endpoint in . A set is independent if and only if it is a clique in the graph’s complement. The size of an independent set is the number of vertices it contains. Independent sets have also been called "internally stable sets", of which "stable set" is a shortening.

In the mathematical field of graph theory, a **complete bipartite graph** or **biclique** is a special kind of bipartite graph where every vertex of the first set is connected to every vertex of the second set.

In the mathematical discipline of graph theory, a **matching** or **independent edge set** in an undirected graph is a set of edges without common vertices. Finding a matching in a bipartite graph can be treated as a network flow problem.

In graph theory, a **perfect graph** is a graph in which the chromatic number of every induced subgraph equals the order of the largest clique of that subgraph. Equivalently stated in symbolic terms an arbitrary graph is perfect if and only if for all we have .

In the mathematical area of graph theory, a **chordal graph** is one in which all cycles of four or more vertices have a *chord*, which is an edge that is not part of the cycle but connects two vertices of the cycle. Equivalently, every induced cycle in the graph should have exactly three vertices. The chordal graphs may also be characterized as the graphs that have perfect elimination orderings, as the graphs in which each minimal separator is a clique, and as the intersection graphs of subtrees of a tree. They are sometimes also called **rigid circuit graphs** or **triangulated graphs**.

In graph theory, a **cograph**, or **complement-reducible graph**, or ** P_{4}-free graph**, is a graph that can be generated from the single-vertex graph

In graph theory, a **maximal independent set** (MIS) or **maximal stable set** is an independent set that is not a subset of any other independent set. In other words, there is no vertex outside the independent set that may join it because it is maximal with respect to the independent set property.

In the mathematical area of graph theory, a **triangle-free graph** is an undirected graph in which no three vertices form a triangle of edges. Triangle-free graphs may be equivalently defined as graphs with clique number ≤ 2, graphs with girth ≥ 4, graphs with no induced 3-cycle, or locally independent graphs.

In graph theory, an area of mathematics, a **claw-free graph** is a graph that does not have a claw as an induced subgraph.

In graph theory, a **pseudoforest** is an undirected graph in which every connected component has at most one cycle. That is, it is a system of vertices and edges connecting pairs of vertices, such that no two cycles of consecutive edges share any vertex with each other, nor can any two cycles be connected to each other by a path of consecutive edges. A **pseudotree** is a connected pseudoforest.

In graph theory, a **clique cover** or **partition into cliques** of a given undirected graph is a partition of the vertices of the graph into cliques, subsets of vertices within which every two vertices are adjacent. A **minimum clique cover** is a clique cover that uses as few cliques as possible. The minimum *k* for which a clique cover exists is called the **clique cover number** of the given graph.

In graph theory, a **well-covered graph** is an undirected graph in which every minimal vertex cover has the same size as every other minimal vertex cover. Equivalently, these are the graphs in which every maximal independent set has the same size. Well-covered graphs were defined and first studied by Plummer (1970).

The **HCS clustering algorithm** is an algorithm based on graph connectivity for cluster analysis. It works by representing the similarity data in a similarity graph, and then finding all the highly connected subgraphs. It does not make any prior assumptions on the number of the clusters. This algorithm was published by Erez Hartuv and Ron Shamir in 2000.

In graph theory, a branch of mathematics, a **cluster graph** is a graph formed from the disjoint union of complete graphs. Equivalently, a graph is a cluster graph if and only if it has no three-vertex induced path; for this reason, the cluster graphs are also called **P _{3}-free graphs**. They are the complement graphs of the complete multipartite graphs and the 2-leaf powers.

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