# Enumerating Maximal Cliques, arbitrary graph (Clique Problems)

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## Description

A maximal clique (complete subgraph) is a clique that is not contained in any other clique. The goal here is to enumerate such maximal cliques in a given graph.

## Parameters

$n$: number of vertices

$m$: number of edges

## Table of Algorithms

Name Year Time Space Approximation Factor Model Reference
Bron–Kerbosch algorithm 1973 $O({3}^{(n/{3})})$ $O(n^{2})$? Exact Deterministic Time
Akkoyunlu; E. A. 1973 $O({3}^{(n/{3})$}) $O(n^{2})$? Exact Deterministic Time
Tomita; Tanaka & Takahashi 2006 $O({3}^{(n/{3})$}) $O(n^{2})$? Exact Deterministic Time
Segundo; Artieda; Strash Parallel 2011 $O({3}^{(n/{3})})$ total work? (previously this cell said $O(n^{4})$) $O(n^{2})$ auxiliary?? Exact Parallel Time
David Eppstein, Maarten Löffler, Darren Strash 2010 $O(dn {3}^{(d/{3})})$ $O(n^{2})$? Exact Deterministic Time
Chiba and Nishizeki 1985 $O(a(G)*m)$ per clique $O(m)$ Exact Deterministic Time & Space
M. Chrobak and D. Eppstein 1989 $O(n d^{2} {2}^d)$ $O(n)$? Exact Deterministic Time
Shuji Tsukiyama, Mikio Ide, Hiromu Ariyoshi, and Isao Shirakawa 1977 $O(nm)$ per clique $O(m)$ Exact Deterministic Time
Kazuhisa Makino, Takeaki Uno; Section 5 2004 $O(n^{\omega})$ per clique where omega is the exponent on matrix multiplication $O(n^{2})$ Exact Deterministic Time & Space
Kazuhisa Makino, Takeaki Uno; Section 6 2004 $O(delta^{4})$ $O(n+m)$ auxiliary(?) Exact Deterministic Time & Space