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Clique Enumeration

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Encyclopedia of Algorithms

Years and Authors of Summarized Original Work

2006; Tomita, Tanaka, Takahashi

1977; Tsukiyama, Ide, Ariyoshi, Shirakawa

2004; Makino, Uno

Problem Definition

We discuss a simple undirected and connected graph G = (V, E) with a finite set V of vertices and a finite set \(E \subseteq V \times V\) of edges. A pair of vertices v and w is said to be adjacent if (v, w) ∈ E. For a subset \(R \subseteq V\) of vertices, \(G(R) = (R,E \cap (R \times R))\) is an induced subgraph. An induced subgraph G(Q) is said to be a clique if (v, w) ∈ E for all \(v,w \in Q \subseteq V\) with v≠w. In this case, we may simply state that Q is a clique. In particular, a clique that is not properly contained in any other clique is called maximal. An induced subgraph G(S) is said to be an independent set if \((v,w)\notin E\) for all \(v,w \in S \subseteq V\). For a vertex v ∈ V, let \(\Gamma (v) =\{ w \in V \vert (v,w) \in E\}\). We call \(\vert \Gamma (v)\vert\) the degree of v.

The problemis to enumerate all...

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Author information

Authors and Affiliations

  1. The Advanced Algorithms Research Laboratory The University of Electro-Communications, Chofugaoka 1-5-1, 182-8585, Chofu, Tokyo, Japan

    Etsuji Tomita

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  1. Etsuji Tomita
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Correspondence to Etsuji Tomita .

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Editors and Affiliations

  1. Dept. Electrical Engineering & Computer Science, Northwestern University, Evanston, Illinois, USA

    Ming-Yang Kao

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Tomita, E. (2014). Clique Enumeration. In: Kao, MY. (eds) Encyclopedia of Algorithms. Springer, Boston, MA. https://doi.org/10.1007/978-3-642-27848-8_725-2

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  • DOI: https://doi.org/10.1007/978-3-642-27848-8_725-2

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  • Online ISBN: 978-3-642-27848-8

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Chapter history

  1. Latest

    Clique Enumeration
    Published:
    25 November 2014

    DOI: https://doi.org/10.1007/978-3-642-27848-8_725-2

  2. Original

    Clique Enumeration
    Published:
    22 August 2014

    DOI: https://doi.org/10.1007/978-3-642-27848-8_725-1

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