View source for Clique (graph theory)

{{Short description|Adjacent subset of an undirected graph}}
[[File:VR complex.svg|thumb|upright=1.35|A graph with {{unordered list
| 23 × 1-vertex cliques (the vertices),
| 42 × 2-vertex cliques (the edges),
| 19 × 3-vertex cliques (light and dark blue triangles), and
| 2 × 4-vertex cliques (dark blue areas).}}
The 11 light blue triangles form maximal cliques. The two dark blue 4-cliques are both maximum and maximal, and the clique number of the graph is 4.]]

In the [[Mathematics|mathematical]] area of [[Graph theory|graph theory]], a '''clique''' ({{IPAc-en|ˈ|k|l|iː|k}} or {{IPAc-en|ˈ|k|l|ɪ|k}}) 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 <math>G</math> is an [[Induced subgraph|induced subgraph]] of <math>G</math> that is [[Complete graph|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|computer science]]: the task of finding whether there is a clique of a given size in a [[Graph (discrete mathematics)|graph]] (the [[Clique problem|clique problem]]) is NP-complete, but despite this hardness result, many algorithms for finding cliques have been studied.

Although the study of [[Complete graph|complete subgraphs]] goes back at least to the graph-theoretic reformulation of [[Ramsey theory]] by {{harvtxt|Erdős|Szekeres|1935}},<ref>The earlier work by {{harvtxt|Kuratowski|1930}} characterizing [[Planar graph|planar graph]]s by forbidden complete and [[Complete bipartite graph|complete bipartite]] subgraphs was originally phrased in topological rather than graph-theoretic terms.</ref> the term ''clique'' comes from {{harvtxt|Luce|Perry|1949}}, who used complete subgraphs in [[Social:Social network|social network]]s 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 [[Biology:Bioinformatics|bioinformatics]].

==Definitions==
A '''clique''', {{mvar|C}}, in an undirected graph {{math|1=''G'' = (''V'', ''E'')}} is a subset of the [[Vertex (graph theory)|vertices]], {{math|''C'' ⊆ ''V''}}, such that every two distinct vertices are adjacent. This is equivalent to the condition that the [[Induced subgraph|induced subgraph]] of {{mvar|G}} induced by {{mvar|C}} is a [[Complete graph|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, {{mvar|G}}, is a clique, such that there is no clique with more vertices. Moreover, the '''clique number''' {{math|''ω''(''G'')}} of a graph {{mvar|G}} is the number of vertices in a maximum clique in {{mvar|G}}.

The '''[[Intersection number (graph theory)|intersection number]]''' of {{mvar|G}} is the smallest number of cliques that together cover all edges of {{mvar|G}}.

The '''clique cover number''' of a graph {{mvar|G}} is the smallest number of cliques of {{mvar|G}} whose union covers the set of vertices {{mvar|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.{{sfnp|Chang|Kloks|Lee|2001}}

The opposite of a clique is an '''[[Independent set (graph theory)|independent set]]''', in the sense that every clique corresponds to an independent set in the [[Complement graph|complement graph]]. The [[Clique cover|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 graph|complete bipartite subgraph]]. The [[Bipartite dimension|bipartite dimension]] of a graph is the minimum number of bicliques needed to cover all the edges of the graph.

==Mathematics==
Mathematical results concerning cliques include the following.
* [[Turán's theorem]] gives a lower bound on the size of a clique in [[Dense graph|dense graph]]s.{{sfnp|Turán|1941}}  If a graph has sufficiently many edges, it must contain a large clique. For instance, every graph with <math>n</math> vertices and more than <math>\scriptstyle \left\lfloor\frac{n}{2}\right\rfloor \cdot \left\lceil\frac{n}{2}\right\rceil</math> edges must contain a three-vertex clique.
* [[Ramsey's theorem]] states that every graph or its [[Complement graph|complement graph]] contains a clique with at least a logarithmic number of vertices.{{sfnp|Graham|Rothschild|Spencer|1990}}
* According to a result of {{harvtxt|Moon|Moser|1965}}, a graph with 3''n'' vertices can have at most 3<sup>''n''</sup> maximal cliques. The graphs meeting this bound are the Moon–Moser graphs ''K''<sub>3,3,...</sub>, a special case of the [[Turán graph]]s arising as the extremal cases in Turán's theorem.
* [[Hadwiger conjecture (graph theory)|Hadwiger's conjecture]], still unproven, relates the size of the largest clique [[Graph minor|minor]] in a graph (its [[Hadwiger number]]) to its [[Chromatic number|chromatic number]].
* The [[Erdős–Faber–Lovász conjecture]] relates graph coloring to cliques.
* The [[Erdős–Hajnal conjecture]] states that families of graphs defined by [[Forbidden graph characterization|forbidden graph characterization]] have either large cliques or large cocliques. 

Several important classes of graphs may be defined or characterized by their cliques:
* A [[Cluster graph|cluster graph]] is a graph whose [[Connected component (graph theory)|connected components]] are cliques.
* A [[Block graph|block graph]] is a graph whose [[Biconnected component|biconnected component]]s are cliques.
* A [[Chordal graph|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|cograph]] is a graph all of whose induced subgraphs have the property that any maximal clique intersects any [[Maximal independent set|maximal independent set]] in a single vertex.
* An [[Interval graph|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|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|perfect graph]] is a graph in which the clique number equals the [[Chromatic number|chromatic number]] in every [[Induced subgraph|induced subgraph]].
* A [[Split graph|split graph]] is a graph in which some clique contains at least one endpoint of every edge.
* A [[Triangle-free graph|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|clique complex]] of a graph ''G'' is an [[Abstract simplicial complex|abstract simplicial complex]] ''X''(''G'') with a simplex for every clique in ''G''
* A [[Simplex graph|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 [[Biology:Median graph|median graph]], and is associated with a [[Median algebra|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''.<ref>{{harvtxt|Barthélemy|Leclerc|Monjardet|1986}}, page 200.</ref>
* The [[Clique-sum|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 (graph theory)|intersection number]] of a graph is the minimum number of cliques needed to cover all the graph's edges.
* The [[Clique graph|clique graph]] of a graph is the [[Intersection graph|intersection graph]] of its maximal cliques.

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

==Computer science==
{{Main|Clique problem}}
In [[Computer science|computer science]], the [[Clique problem|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]].{{sfnp|Karp|1972}} It is also [[Parameterized complexity|fixed-parameter intractable]], and [[Hardness of approximation|hard to approximate]]. Nevertheless, many [[Algorithm|algorithm]]s 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 graph|planar graph]]s or [[Perfect graph|perfect graph]]s for which the problem can be solved in polynomial time.

==Applications==
The word "clique", in its graph-theoretic usage, arose from the work of {{harvtxt|Luce|Perry|1949}}, who used complete subgraphs to model cliques (groups of people who all know each other) in [[Social:Social network|social network]]s. The same definition was used by {{harvtxt|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. {{harvtxt|Alba|1973}}, {{harvtxt|Peay|1974}}, and {{harvtxt|Doreian|Woodard|1994}}.

Many different problems from [[Biology:Bioinformatics|bioinformatics]] have been modeled using cliques. For instance, {{harvtxt|Ben-Dor|Shamir|Yakhini|1999}} model the problem of clustering [[Biology:Gene expression|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; {{harvtxt|Tanay|Sharan|Shamir|2002}} discuss a similar [[Biclustering|biclustering]] problem for expression data in which the clusters are required to be cliques. {{harvtxt|Sugihara|1984}} uses cliques to model [[Biology:Ecological niche|ecological niche]]s in food webs. {{harvtxt|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|perfect phylogeny]] combining those two characters. {{harvtxt|Samudrala|Moult|1998}} model [[Biology:Protein structure prediction|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 [[Biology:Protein–protein interaction|protein–protein interaction]] network, {{harvtxt|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|electrical engineering]], {{harvtxt|Prihar|1956}} uses cliques to analyze communications networks, and {{harvtxt|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.<ref>{{harvtxt|Hamzaoglu|Patel|1998}}.</ref> {{harvtxt|Cong|Smith|1993}} describe an application of cliques in finding a hierarchical partition of an electronic circuit into smaller subunits.

In [[Chemistry:Chemistry|chemistry]], {{harvtxt|Rhodes|Willett|Calvet|Dunbar|2003}} use cliques to describe chemicals in a [[Chemistry:Chemical database|chemical database]] that have a high degree of similarity with a target structure. {{harvtxt|Kuhl|Crippen|Friesen|1983}} use cliques to model the positions in which two chemicals will bind to each other.

==See also==
* Clique game

==Notes==
{{reflist}}

==References==
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{{refend}}

==External links==
*{{MathWorld|title=Clique|urlname=Clique|mode=cs2}}
*{{MathWorld|title=Clique Number|urlname=CliqueNumber|mode=cs2}}

[[Category:Graph theory objects]]

{{Sourceattribution|Clique (graph theory)}}