团（图论）

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• 23 × 1-vertex cliques (the vertices), 23个单顶点团（顶点） • 42 × 2-vertex cliques (the edges), 42个双顶点团（连边） • 19 × 3-vertex cliques (light and dark blue triangles), 19个三顶点团（浅色和深色的三角形） • 2 × 4-vertex cliques (dark blue areas).2个四顶点团（深色区域） 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. 11个浅蓝色三角形（三顶点团）为极大团。深蓝色的四顶点团既是最大团也是极大团，该图的团数是4。

In the mathematical area of graph theory, a clique (模板:IPAc-en or 模板:IPAc-en) is a subset of vertices of an undirected graph such that every two distinct vertices in the clique are adjacent; that is, its induced subgraph 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.

In the mathematical area of graph theory, a clique ( or ) is a subset of vertices of an undirected graph such that every two distinct vertices in the clique are adjacent; that is, its induced subgraph 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.

在图论的数学领域中，团(或)是无向图的顶点的子集，使得团中每两个不同的顶点相邻，即它的导出子图是完全的。团是图论的基本概念之一，在许多其他数学问题和图的构造中也有应用。团问题在计算机科学中也有研究: 在一个图中是否存在一个给定大小的团(团问题)是 np 完全问题，但是尽管这个结果很困难，许多寻找团的算法已经被研究过。

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.

Although the study of complete subgraphs goes back at least to the graph-theoretic reformulation of Ramsey theory by , 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.

虽然完全子图的研究至少可以追溯到 Ramsey 理论的图论重构，但是术语团来源于，他们在社会网络中使用完全子图来模拟人们的团体，即相互认识的人们的团体。小集团在科学，特别是在生物信息学中有许多其他的应用。

Definitions

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 clique, C, in an undirected graph (V, E)}} is a subset of the vertices, , 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.

无向图(v，e)}中的团 c 是顶点的子集，使得每两个不同的顶点相邻。这等价于由 c 导出的 g 的导出子图是完全图的条件。在某些情况下，团这个术语也可以直接指子图。

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 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.

最大团是一个不能通过包含多个相邻顶点而扩展的团，即不仅仅存在于一个较大团的顶点集中的团。有些作者定义团的方式要求它们是极大的，并使用其他术语来定义不是极大的完全子图。

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.

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.

一个图 g 的最大团是一个团，这样就不存在多顶点的团。此外，图 g 的团数(g)是图 g 的最大团中的顶点数。

The intersection number of G is the smallest number of cliques that together cover all edges of G.

The intersection number of G is the smallest number of cliques that together cover all edges of G.

G 的交集数是覆盖 g 的所有边的团的最小个数。

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.

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.

图 g 的团覆盖数是 g 的团覆盖图的顶点集 v 的团的最小个数。

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.模板:Sfnp

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.

图的最大团横截是指图的每个最大团在子集中至少包含一个顶点的顶点子集。

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.

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.

团体的对立面是一个独立的集合，在某种意义上说，每个团体对应于补图中的一个独立的集合。分团覆盖问题关注于寻找尽可能少的包含图中每个顶点的团。

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.

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.

一个相关的概念是双列，一个完全二部子图。图的二部维数是覆盖图的所有边所需的最小双曲数。

Mathematics

Mathematical results concerning cliques include the following.

关于小团体的数学结果包括以下内容。

Turán's theorem gives a lower bound on the size of a clique in dense graphs.模板:Sfnp If a graph has sufficiently many edges, it must contain a large clique. For instance, every graph with [math]\displaystyle{ n }[/math] vertices and more than [math]\displaystyle{ \scriptstyle\lfloor\frac{n}{2}\rfloor\cdot\lceil\frac{n}{2}\rceil }[/math] 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.模板:Sfnp

According to a result of Moon & Moser (1965), a graph with 3n vertices can have at most 3ⁿ 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:

图的几个重要类别可以定义或拥有属性它们的团:

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,

此外，许多其他的数学构造也涉及图中的团。其中,

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.^[2]

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.

与完备子图密切相关的概念是完备图的子图和完备图子图的子图。特别地，库拉托斯基定理和瓦格纳定理分别用禁止完全和完全二部分细分图以及子图刻画了平面图。

Computer science

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.模板:Sfnp 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.

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. 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.

在计算机科学中，团的问题是在给定的图中找到一个最大团或所有团的计算问题。它是 np 完全问题，Karp 的21个 np 完全问题之一。它也是固定参数难以处理的，而且很难近似。尽管如此，许多计算团的算法已经被开发出来，或者运行在 EXPTIME 中(如 Bron-Kerbosch 算法) ，或者专门用于图族，如平面图或完美图，对于这些图族，问题可以在多项式时间内解决。

Applications

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).

The word "clique", in its graph-theoretic usage, arose from the work of , who used complete subgraphs to model cliques (groups of people who all know each other) in social networks. The same definition was used by 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. , , and .

“小团体”这个词，在它的图论用法中，起源于年的工作，他使用完全子图来模拟小团体(所有互相认识的人组成的小团体)在社会网络中。同样的定义在一篇文章中也被使用了，这篇文章使用了一些不那么专业的术语。这两本书都使用矩阵处理社交网络中的小团体问题。为了继续努力建立社会小团体图形模型-理论上，见。、、及。

Many different problems from bioinformatics have been modeled using cliques.

许多来自生物信息学的不同问题已经用小团体来模拟。

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.

For instance, 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; discuss a similar biclustering problem for expression data in which the clusters are required to be cliques. uses cliques to model ecological niches in food webs. 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. 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, 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.

例如，将基因表达式数据的聚类问题建模为寻找将描述数据的图转换为小团不相交并形成的图所需的最小变化数; 讨论表达式数据的一个类似双聚类问题，其中要求聚类为小团。利用小集团来模拟食物网中的生态位。将推断进化树的问题描述为在一个以物种的顶点特征为顶点的图中寻找最大团的问题，其中两个顶点共享一条边，如果存在一个完美的将这两个特征结合起来的系统发育。模型蛋白质结构预测是一个在图中找到团的问题，其顶点表示蛋白质亚单位的位置。通过在蛋白质-蛋白质相互作用网络中寻找小团体，发现了相互作用密切、与小团外的蛋白质相互作用很少的蛋白质团。幂图分析是一种通过寻找复杂生物网络中的团和相关结构来简化复杂生物网络的方法。

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.^[3] Cong & Smith (1993) describe an application of cliques in finding a hierarchical partition of an electronic circuit into smaller subunits.

In electrical engineering, uses cliques to analyze communications networks, and 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. describe an application of cliques in finding a hierarchical partition of an electronic circuit into smaller subunits.

在电气工程中，使用小团体来分析通信网络，并使用它们来设计计算部分指定的布尔函数的有效电路。小团体也被用于自动测试模式生成: 可能发生错误的不兼容图中的大团体提供了测试集大小的下界。描述了团在寻找电子电路的分层划分到更小的子单元中的应用。

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.

In chemistry, use cliques to describe chemicals in a chemical database that have a high degree of similarity with a target structure. use cliques to model the positions in which two chemicals will bind to each other.

在化学中，用小团来描述化学数据库中与目标结构高度相似的化学物质。用小团体来模拟两种化学物质相互结合的位置。

Notes

↑ 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.
↑ Barthélemy, Leclerc & Monjardet (1986), page 200.
↑ Hamzaoglu & Patel (1998).

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External links

Weisstein, Eric W. "Clique". MathWorld.

Weisstein, Eric W. "Clique Number". MathWorld.

Category:Graph theory objects

范畴: 图论对象

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[1] 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.

[2] Barthélemy, Leclerc & Monjardet (1986), page 200.

[3] Hamzaoglu & Patel (1998).

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