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| {{redirect|Clique graph|the graph with one vertex for each clique of another graph|simplex graph}}
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| [[File:VR complex.svg|thumb|300px|A graph with 23 1-vertex cliques (its vertices), 42 2-vertex cliques (its edges), 19 3-vertex cliques (the light and dark blue triangles), and 2 4-vertex cliques (dark blue areas). The six edges not associated with any triangle and 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.]]
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| In the [[mathematics|mathematical]] area of [[graph theory]], a '''clique''' ({{IPAc-en|ˈ|k|l|iː|k}} or {{IPAc-en|ˈ|k|l|ɪ|k}}) in an [[undirected graph]] is a [[subset]] of its [[vertex (graph theory)|vertices]] such that every two vertices in the subset are connected by an [[Edge (graph theory)|edge]]. 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 (mathematics)|graph]] (the [[clique problem]]) is [[NP-complete]], but despite this hardness result many algorithms for finding cliques have been studied.
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| 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]]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 network]]s to model [[clique]]s 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]].
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| ==Definitions==
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| A '''clique''' in an [[undirected graph]] ''G'' = (''V'', ''E'') is a subset of the [[vertex (graph theory)|vertex set]] ''C'' ⊆ ''V'', such that for every two vertices in ''C'', there exists an [[Edge (graph theory)|edge]] connecting the two. This is equivalent to saying that the [[Glossary of graph theory#Subgraphs|subgraph]] induced by ''C'' is [[complete graph|complete]] (in some cases, the term clique may also refer to the subgraph).
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| 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.
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| A '''maximum clique''' is a clique of the largest possible size in a given graph. The '''clique number''' ω(''G'') of a graph ''G'' is the number of vertices in a maximum clique in ''G''. The [[Intersection number (graph theory)|intersection number]] of ''G'' is the smallest number of cliques that together cover all edges of ''G''.
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| 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]]. 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 graph|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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| ==Mathematics==
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| Mathematical results concerning cliques include the following.
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| *[[Turán's theorem]] {{harv|Turán|1941}} gives a [[lower bound]] on the size of a clique in [[dense graph]]s. 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\lfloor\frac{n}{2}\rfloor\cdot\lceil\frac{n}{2}\rceil</math> edges must contain a three-vertex clique.
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| *[[Ramsey's theorem]] {{harv|Graham|Rothschild|Spencer|1990}} states that every graph or its [[complement graph]] contains a clique with at least a logarithmic number of vertices.
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| *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.
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| *[[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]].
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| *The [[Erdős–Faber–Lovász conjecture]] is another unproven statement relating graph coloring to cliques.
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| Several important classes of graphs may be defined by their cliques:
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| *A [[chordal graph]] is a graph whose vertices can be ordered into a perfect elimination ordering, an ordering such that the [[neighborhood (graph theory)|neighbors]] of each vertex ''v'' that come later than ''v'' in the ordering form a clique.
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| *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.
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| *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.
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| *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.
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| *A [[perfect graph]] is a graph in which the clique number equals the [[chromatic number]] in every [[induced subgraph]].
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| *A [[split graph]] is a graph in which some clique contains at least one endpoint of every edge.
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| *A [[triangle-free graph]] is a graph that has no cliques other than its vertices and edges.
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| Additionally, many other mathematical constructions involve cliques in graphs. Among them,
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| *The [[clique complex]] of a graph ''G'' is an [[abstract simplicial complex]] ''X''(''G'') with a simplex for every clique in ''G''
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| *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''.<ref>{{harvtxt|Barthélemy|Leclerc|Monjardet|1986}}, page 200.</ref>
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| *The [[clique-sum]] is a method for combining two graphs by merging them along a shared clique.
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| *[[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.
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| *The [[Intersection number (graph theory)|intersection number]] of a graph is the minimum number of cliques needed to cover all the graph's edges.
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| Closely related concepts to complete subgraphs are [[subdivision (graph theory)|subdivision]]s of complete graphs and complete [[graph minor]]s. In particular, [[Kuratowski's theorem]] and [[Wagner's theorem]] characterize [[planar graph]]s by forbidden complete and [[complete bipartite graph|complete bipartite]] subdivisions and minors, respectively.
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| ==Computer science==
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| {{main|Clique problem}}
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| 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]] {{harv|Karp|1972}}. It is also [[Parameterized complexity|fixed-parameter intractable]], and [[Hardness of approximation|hard to approximate]]. Nevertheless, many [[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]]s or [[perfect graph]]s for which the problem can be solved in [[polynomial time]].
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| {{Free_software_for_searching_maximum_clique}}
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| ==Applications==
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| The word "clique", in its graph-theoretic usage, arose from the work of {{harvtxt|Luce|Perry|1949}}, who used complete subgraphs to model [[clique]]s (groups of people who all know each other) in [[social network]]s. For continued efforts to model social cliques graph-theoretically, see e.g. {{harvtxt|Alba|1973}}, {{harvtxt|Peay|1974}}, and {{harvtxt|Doreian|Woodard|1994}}.
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| Many different problems from [[bioinformatics]] have been modeled using cliques.
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| For instance, {{harvtxt|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; {{harvtxt|Tanay|Sharan|Shamir}} discuss a similar [[biclustering]] problem for expression data in which the clusters are required to be cliques. {{harvtxt|Sugihara|1984}} uses cliques to model [[ecological niche]]s in [[food chain|food webs]]. {{harvtxt|Day|Sankoff|1986}} describe the problem of inferring [[evolutionary tree]]s 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. {{harvtxt|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, {{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.
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| In [[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.
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| In [[chemistry]], {{harvtxt|Rhodes|Willett|Calvet|Dunbar|2003}} use cliques to describe chemicals in a [[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.
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| ==Notes==
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| {{reflist}}
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| ==References==
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| {{refbegin|2}}
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| *{{citation |doi=10.1080/0022250X.1973.9989826 |first=Richard D. |last=Alba |title=A graph-theoretic definition of a sociometric clique |journal=Journal of Mathematical Sociology |year=1973 |volume=3 |issue=1 |pages=113–126 |url=http://aris.ss.uci.edu/~lin/1.pdf}}.
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| *{{citation | last1 = Barthélemy | first1 = J.-P. | last2 = Leclerc | first2 = B. | last3 = Monjardet | first3 = B. | issue = 2 | journal = Journal of Classification | pages = 187–224 | title = On the use of ordered sets in problems of comparison and consensus of classifications | doi = 10.1007/BF01894188 | volume = 3 | year = 1986}}.
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| *{{citation |first1=Amir |last1=Ben-Dor |first2=Ron |last2=Shamir |first3=Zohar |last3=Yakhini |journal=Journal of Computational Biology |year=1999 |volume=6 |issue=3–4 |pages=281–297 |doi=10.1089/106652799318274 |pmid=10582567 |title=Clustering gene expression patterns.}}.
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| *{{citation |title=Computational complexity of inferring phylogenies by compatibility |first1=William H. E. |last1=Day |first2=David |last2=Sankoff |journal=Systematic Zoology |volume=35 |issue=2 |year=1986 |pages=224–229 |doi=10.2307/2413432 |jstor=2413432}}.
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| *{{citation |title=Defining and locating cores and boundaries of social networks |first1=Patrick |last1=Doreian |first2=Katherine L. |last2=Woodard |journal=Social Networks |volume=16 |issue=4 |year=1994 |pages=267–293 |doi=10.1016/0378-8733(94)90013-2}}.
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| *{{citation
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| *{{citation |last1=Hamzaoglu |first1=I. |last2=Patel |first2=J. H. |contribution=Test set compaction algorithms for combinational circuits |title=Proc. 1998 IEEE/ACM International Conference on Computer-Aided Design |year=1998 |pages=283–289 |doi=10.1145/288548.288615}}.
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| *{{citation |first1=F. S. |last1=Kuhl |first2=G. M. |last2=Crippen |first3=D. K. |last3=Friesen |year=1983 |title=A combinatorial algorithm for calculating ligand binding |journal=Journal of Computational Chemistry |doi=10.1002/jcc.540050105 |volume=5 |issue=1 |pages=24–34}}.
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| *{{citation |first=Kazimierz |last=Kuratowski |authorlink=Kazimierz Kuratowski |language=French |title=Sur le probléme des courbes gauches en Topologie |journal=Fundamenta Mathematicae |volume=15 |year=1930 |pages=271–283 |url=http://matwbn.icm.edu.pl/ksiazki/fm/fm15/fm15126.pdf}}.
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| *{{citation
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| | pmid = 18152948}}.
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| *{{citation
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| | journal = Israel J. Math.
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| | volume = 3
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| | year = 1965
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| | mr = 0182577
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| *{{citation |first1=M. C. |last1=Paull |first2=S. H. |last2=Unger |title=Minimizing the number of states in incompletely specified sequential switching functions |journal=IRE Trans. on Electronic Computers |volume=EC-8 |issue=3 |year=1959 |pages=356–367 |doi=10.1109/TEC.1959.5222697}}.
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| *{{citation |title=Hierarchical clique structures |first=Edmund R. |last=Peay |journal=Sociometry |volume=37 |issue=1 |year=1974 |pages=54–65 |doi=10.2307/2786466 |jstor=2786466}}.
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| *{{citation |last=Prihar |first=Z. |title=Topological properties of telecommunications networks |journal=[[Proceedings of the IRE]] |volume=44 |issue=7 |year=1956 |pages=927–933 |doi=10.1109/JRPROC.1956.275149}}.
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| *{{citation |first1=Nicholas |last1=Rhodes |first2=Peter |last2=Willett |first3=Alain |last3=Calvet |first4=James B. |last4=Dunbar |first5=Christine |last5=Humblet |journal=Journal of Chemical Information and Computer Sciences |volume=43 |issue=2 |pages=443–448 |year=2003 |doi=10.1021/ci025605o |pmid=12653507 |title=CLIP: similarity searching of 3D databases using clique detection}}.
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| *{{citation |first1=Ram |last1=Samudrala |first2=John |last2=Moult |title=A graph-theoretic algorithm for comparative modeling of protein structure |journal=Journal of Molecular Biology |volume=279 |issue=1 |year=1998 |pages=287–302 |doi=10.1006/jmbi.1998.1689 |pmid=9636717}}.
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| *{{citation |first1=Victor |last1=Spirin |first2=Leonid A. |last2=Mirny |title=Protein complexes and functional modules in molecular networks |journal=[[Proceedings of the National Academy of Sciences]] |volume=100 |issue=21 |pages=12123–12128 |doi=10.1073/pnas.2032324100 |year=2003 |pmid=14517352 |pmc=218723}}.
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| *{{citation |first=George |last=Sugihara |contribution=Graph theory, homology and food webs |year=1984 |pages=83–101 |series=Proc. Symp. Appl. Math. |volume=30 |title=Population Biology |editor-last=Levin |editor-first=Simon A.}}.
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| *{{citation |first1=Amos |last1=Tanay |first2=Roded |last2=Sharan |first3=Ron |last3=Shamir |title=Discovering statistically significant biclusters in gene expression data |journal=Bioinformatics |volume=18 |issue=Suppl. 1 |year=2002 |pages=S136–S144 |pmid=12169541 |doi=10.1093/bioinformatics/18.suppl_1.S136}}.
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| *{{citation
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| | last = Turán
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| | first = Paul | authorlink = Paul Turán
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| | year = 1941
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| | title = On an extremal problem in graph theory
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| | journal = Matematikai és Fizikai Lapok
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| | volume = 48
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| | pages = 436–452
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| | language = Hungarian
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| }}
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| {{refend}}
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| ==External links==
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| *{{MathWorld|title=Clique|urlname=Clique}}
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| *{{MathWorld|title=Clique Number|urlname=CliqueNumber}}
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| [[Category:Graph theory objects]]
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