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，团是无向图中顶点的子集，使得一个团中每两个不同的顶点必定相邻。也就是说，其导出子图是完全图。团是图论的基本概念之一，可用于图形的构建和解决许多其他数学问题。而且，在计算机科学领域中也经常会涉及到团的研究：比如在一个给定规模的图中寻找团的存在就是一个NP-完全问题。尽管我们已经明确知道这个问题很难得以解决，但仍然有许多学者在研究用于寻找团的算法。

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.

对完全子图的研究至少可以追溯到拉姆齐定理中的图论重组，由Erdős＆Szekeres（1935）在论文《A combinatorial problem in geometry》提出的。但实际上“团”一词是来自Luce＆Perry（1949）的文章《A method of matrix analysis of group structure》，后者在社交网络中使用完全子图来对人群进行建模；该模型定义在这群人中，所有人是互相认识的。其实团这一概念在诸多科学领域中均有所应用，尤其是在生物信息学。

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.

在一个无向图G =（V，E）中，团C是顶点V的子集，记作：C⊆V，使得每两个不同的顶点相邻。因此团C可以看作是该无向图G的导出子集，进而将此过程视为由C引导出的完全子集的成立条件。在某些情况下，“团”这一术语也可以直接被引用为子集。

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.

极大团Maximal clique也是一个团，但是该团不能通过合并更多的相邻节点而进行扩张，换句话说，该团不可能被更大的团所包含。有些学者将“极大团”定义为“团”，而建议用其他术语来重新定义“非极大团”。

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.

图G中同样存在一个最大团，使得其不存在更多顶点。另外，图G的团数ω（G）是该图最大团的顶点数。

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.

图G的交叉数是该图中能覆盖所有连边的最少团数。

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.

一个图的最大团横贯集是其顶点的子集，其属性为该图的每个最大团中至少有一个顶点在最大团横贯集中。

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.

作为团的对立面，独立集的存在是指图中两两互相不相邻的顶点集合。因此，每一个团都对应于补图中的独立集。集团覆盖问题涉及到寻找尽可能少的团，其中就包括图中的每个顶点。

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.

二元团是团的相关概念，指的是一个完全二分图。该图的二分维度指的是覆盖该图所有连边的最少二元团数。

Mathematics 数学运算

Mathematical results concerning cliques include the following. Mathematical results concerning cliques include the following. 有关“团”的数学结论包括以下内容。

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.

托兰定理在稠密图中给出了团大小的下界。如果一个图具有足够多的边，则它必然含有较大的团。例如，每个具有n个顶点且超过个边的图形都必然含一个三顶点团。
拉姆齐定理指出，每个图或其补图都包含至少一个具有对数个顶点的团。
根据Moon＆Moser（1965）的研究结果，一个具有3n个顶点的图最多可以有3n个极大团。满足此极限要求的图被称为Moon＆Moser图K3,3，...，该图相当于托兰图的特例，是托兰定理中的极端情况。
关于Hadwiger的猜想，目前尚未得到证实，其认为图中最大的团子式的大小（即Hadwiger数）与其色数相关。
Erdős–Faber–Lovász猜想是另一个未经证实的陈述，同样认为图的着色与团相关。

Several important classes of graphs may be defined or characterized by their cliques: Several important classes of graphs may be defined or characterized by their cliques: 由“团”的特性来定义和区分“图”类别的相关描述：

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.

聚类图Cluster graph是图的一种特例，其连接的组成部分视为团。
框图Block graph是图的一种特例，其双连通的部分视为团。
弦图Chordal graph是图的一种特例，其顶点可以按照最佳删除顺序进行排序；也就是说每个顶点的相邻点（即与其构成一个团的另一个顶点）为该顶点的后续点。
余图Cograph是图的一种特例，其所有的导出子图均具有以下特性：任何极大团与任何极大独立集都相交于一个独立点。
区间图Interval graph是图的一种特例，其极大团按照一定顺序排列，对于每个顶点v,所有包含其顶点v的团均连续排列。
线图Line graph是图的一种特例，其连边可以被不相交的边团覆盖，从而使得每个顶点恰好属于该覆盖区域中的两个团。
完美图Perfect graph是图的一种特例，其导出子图的色数等于此导出子图的团数。
分裂图Split graph是图的一种特例，其中存在某个团包含至少一个顶点，该顶点为每条边的端点。
三角形无关图Triangle-free graph是图的一种特例，除了顶点和连边之外，该图不含任何团。

Additionally, many other mathematical constructions involve cliques in graphs. Among them, Additionally, many other mathematical constructions involve cliques in graphs. Among them, 此外，还有许多其他数学结构与图论的团有关。它们包括，

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.
图G的团复形属于抽象复形，其中图G的每个团都为单纯形。
单纯形图可记为无向图κ（G），它具有图G中每个团的一个顶点，以及一个连边，并且该连边通过一个顶点将两个不同的团相连。
团相加指的是将两个图合并，沿着他们共有团的顶点和连边融合形成。
团宽度是图复杂性的一个概念，是用来说明图结构复杂性的一个参数。该图由最少数量的不同定点标签组成，并通过以下操作建立：

1.将两个已标记的图拆开，使其不相交；
2.重新标记；
3.根据给定的标记，连接所有成对顶点。

图的交叉数指的是能覆盖图所有连边所需的最小团数。
图G的团图指的是该图的极大团的交图。主要为了展示图G的团结构。
从概念上来说，完全子图与完全图细分，以及完全图子式密切相关。尤其是Kuratowski定理和Wagner定理，它们就是通过禁用完全图、完全二分图细分和完全二分图子式来描述平面图特征的。

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.

从概念上来说，完全子图与完全图细分，以及完全图子式密切相关。尤其是Kuratowski定理和Wagner定理，它们就是通过禁用完全图、完全二分图细分和完全二分图子式来描述平面图特征的。

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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}}.

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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

This page was moved from wikipedia:en:Clique (graph theory). Its edit history can be viewed at 团（图论）/edithistory

[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).

[1]

[2]

[3]