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| {{About|the combinatorial aspects of maximal independent sets of vertices in a graph|other aspects of independent vertex sets in graph theory|Independent set (graph theory)|other kinds of independent sets|Independent set (disambiguation)}}
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| [[File:Cube-maximal-independence.svg|thumb|300px|The [[hypercube graph|graph of the cube]] has six different maximal independent sets, shown as the red vertices.]]
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| In [[graph theory]], a '''maximal independent set''' or '''maximal stable set''' is an [[Independent set (graph theory)|independent set]] that is not a subset of any other independent set. That is, it is a set ''S'' such that every edge of the graph has at least one endpoint not in ''S'' and every vertex not in ''S'' has at least one neighbor in ''S''. A maximal independent set is also a [[dominating set]] in the graph, and every dominating set that is independent must be maximal independent, so maximal independent sets are also called '''independent dominating sets'''. A graph may have many maximal independent sets of widely varying sizes;<ref>{{Harvtxt|Erdős|1966}} shows that the number of different sizes of maximal independent sets in an ''n''-vertex graph may be as large as ''n'' - log ''n'' - O(log log ''n'') and is never larger than ''n'' - log ''n''.</ref> a largest maximal independent set is called a [[maximum independent set]].
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| For example, in the graph P<sub>3</sub>, a path with three vertices ''a'', ''b'', and ''c'', and two edges ''ab'' and ''bc'', the sets {''b''} and {''a'',''c''} are both maximally independent. The set {''a''} is independent, but is not maximal independent, because it is a subset of the larger independent set {''a'',''c''}. In this same graph, the maximal cliques are the sets {''a'',''b''} and {''b'',''c''}.
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| The phrase "maximal independent set" is also used to describe maximal subsets of independent elements in mathematical structures other than graphs, and in particular in [[vector space]]s and [[matroid]]s.
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| == Related vertex sets ==
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| If ''S'' is a maximal independent set in some graph, it is a '''maximal clique''' or '''maximal complete subgraph''' in the [[complement graph|complementary graph]]. A maximal clique is a set of vertices that [[induced subgraph|induces]] a [[complete graph|complete subgraph]], and that is not a subset of the vertices of any larger complete subgraph. That is, it is a set ''S'' such that every pair of vertices in ''S'' is connected by an edge and every vertex not in ''S'' is missing an edge to at least one vertex in ''S''. A graph may have many maximal cliques, of varying sizes; finding the largest of these is the [[Clique problem|maximum clique problem]].
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| Some authors include maximality as part of the definition of a clique, and refer to maximal cliques simply as cliques.
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| The [[complement (set theory)|complement]] of a maximal independent set, that is, the set of vertices not belonging to the independent set, forms a '''minimal vertex cover'''. That is, the complement is a [[vertex cover]], a set of vertices that includes at least one endpoint of each edge, and is minimal in the sense that none of its vertices can be removed while preserving the property that it is a cover. Minimal vertex covers have been studied in [[statistical mechanics]] in connection with the [[Hard spheres|hard-sphere lattice gas]] model, a mathematical abstraction of fluid-solid state transitions.<ref>{{Harvtxt|Weigt|Hartmann|2001}}.</ref>
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| Every maximal independent set is a [[dominating set]], a set of vertices such that every vertex in the graph either belongs to the set or is adjacent to the set. A set of vertices is a maximal independent set if and only if it is an independent dominating set.
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| == Graph family characterizations ==
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| Certain graph families have also been characterized in terms of their maximal cliques or maximal independent sets. Examples include the maximal-clique irreducible and hereditary maximal-clique irreducible graphs. A graph is said to be ''maximal-clique irreducible'' if every maximal clique has an edge that belongs to no other maximal clique, and ''hereditary maximal-clique irreducible'' if the same property is true for every induced subgraph.<ref>Information System on Graph Class Inclusions: [http://wwwteo.informatik.uni-rostock.de/isgci/classes/gc_749.html maximal clique irreducible graphs] and [http://wwwteo.informatik.uni-rostock.de/isgci/classes/gc_750.html hereditary maximal clique irreducible graphs].</ref> Hereditary maximal-clique irreducible graphs include [[triangle-free graph]]s, [[bipartite graph]]s, and [[interval graph]]s.
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| [[Cograph]]s can be characterized as graphs in which every maximal clique intersects every maximal independent set, and in which the same property is true in all induced subgraphs.
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| == Bounding the number of sets ==
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| {{Harvtxt|Moon|Moser|1965}} showed that any graph with ''n'' vertices has at most 3<sup>''n''/3</sup> maximal cliques. Complementarily, any graph with ''n'' vertices also has at most 3<sup>''n''/3</sup> maximal independent sets. A graph with exactly 3<sup>''n''/3</sup> maximal independent sets is easy to construct: simply take the disjoint union of ''n''/3 [[cycle graph|triangle graphs]]. Any maximal independent set in this graph is formed by choosing one vertex from each triangle. The complementary graph, with exactly 3<sup>''n''/3</sup> maximal cliques, is a special type of [[Turán graph]]; because of their connection with Moon and Moser's bound, these graphs are also sometimes called Moon-Moser graphs. Tighter bounds are possible if one limits the size of the maximal independent sets: the number of maximal independent sets of size ''k'' in any ''n''-vertex graph is at most
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| :<math>\lfloor n/k \rfloor^{k-(n\bmod k)}\lfloor n/k+1 \rfloor^{n\bmod k}.</math>
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| The graphs achieving this bound are again Turán graphs.<ref>{{Harvtxt|Byskov|2003}}. For related earlier results see {{Harvtxt|Croitoru|1979}} and {{Harvtxt|Eppstein|2003}}.</ref>
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| Certain families of graphs may, however, have much more restrictive bounds on the numbers of maximal independent sets or maximal cliques. If all ''n''-vertex graphs in a family of graphs have O(''n'') edges, and if every subgraph of a graph in the family also belongs to the family, then each graph in the family can have at most O(''n'') maximal cliques, all of which have size O(1).<ref>{{Harvtxt|Chiba|Nishizeki|1985}}. Chiba and Nishizeki express the condition of having O(''n'') edges equivalently, in terms of the [[arboricity]] of the graphs in the family being constant.</ref> For instance, these conditions are true for the [[planar graph]]s: every ''n''-vertex planar graph has at most 3''n'' − 6 edges, and a subgraph of a planar graph is always planar, from which it follows that each planar graph has O(''n'') maximal cliques (of size at most four). [[Interval graph]]s and [[chordal graph]]s also have at most ''n'' maximal cliques, even though they are not always [[sparse graph]]s.
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| The number of maximal independent sets in ''n''-vertex [[cycle graph]]s is given by the [[Perrin number]]s, and the number of maximal independent sets in ''n''-vertex [[path (graph theory)|path graphs]] is given by the [[Padovan sequence]].<ref>{{Harvtxt|Bisdorff|Marichal|2007}}; {{Harvtxt|Euler|2005}}; {{Harvtxt|Füredi|1987}}.</ref> Therefore, both numbers are proportional to powers of 1.324718, the [[plastic number]].
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| == Set listing algorithms ==
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| {{Further|Clique problem#Listing all maximal cliques}}
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| An algorithm for listing all maximal independent sets or maximal cliques in a graph can be used as a subroutine for solving many NP-complete graph problems. Most obviously, the solutions to the maximum independent set problem, the maximum clique problem, and the minimum independent dominating problem must all be maximal independent sets or maximal cliques, and can be found by an algorithm that lists all maximal independent sets or maximal cliques and retains the ones with the largest or smallest size. Similarly, the [[Vertex cover problem|minimum vertex cover]] can be found as the complement of one of the maximal independent sets. {{Harvtxt|Lawler|1976}} observed that listing maximal independent sets can also be used to find 3-colorings of graphs: a graph can be 3-colored if and only if the [[complement (graph theory)|complement]] of one of its maximal independent sets is [[bipartite graph|bipartite]]. He used this approach not only for 3-coloring but as part of a more general graph coloring algorithm, and similar approaches to graph coloring have been refined by other authors since.<ref>{{Harvtxt|Eppstein|2003}}; {{Harvtxt|Byskov|2003}}.</ref> Other more complex problems can also be modeled as finding a clique or independent set of a specific type. This motivates the algorithmic problem of listing all maximal independent sets (or equivalently, all maximal cliques) efficiently.
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| It is straightforward to turn a proof of Moon and Moser's 3<sup>''n''/3</sup> bound on the number of maximal independent sets into an algorithm that lists all such sets in time O(3<sup>''n''/3</sup>).<ref>{{Harvtxt|Eppstein|2003}}. For a matching bound for the widely used [[Bron–Kerbosch algorithm]], see {{harvtxt|Tomita|Tanaka|Takahashi|2006}}.</ref> For graphs that have the largest possible number of maximal independent sets, this algorithm takes constant time per output set. However, an algorithm with this time bound can be highly inefficient for graphs with more limited numbers of independent sets. For this reason, many researchers have studied algorithms that list all maximal independent sets in polynomial time per output set.<ref>{{Harvtxt|Bomze|Budinich|Pardalos|Pelillo|1999}}; {{Harvtxt|Eppstein|2005}}; {{Harvtxt|Jennings|Motycková|1992}}; {{Harvtxt|Johnson|Yannakakis|Papadimitriou|1988}}; {{Harvtxt|Lawler|Lenstra|Rinnooy Kan|1980}}; {{Harvtxt|Liang|Dhall|Lakshmivarahan|1991}}; {{Harvtxt|Makino|Uno|2004}}; {{Harvtxt|Mishra|Pitt|1997}}; {{Harvtxt|Stix|2004}}; {{Harvtxt|Tsukiyama|Ide|Ariyoshi|Shirakawa|1977}}; {{Harvtxt|Yu|Chen|1993}}.</ref> The time per maximal independent set is proportional to that for [[matrix multiplication]] in dense graphs, or faster in various classes of sparse graphs.<ref>{{Harvtxt|Makino|Uno|2004}}; {{Harvtxt|Eppstein|2005}}.</ref>
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| == Notes ==
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| {{Reflist|2}}
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| == References ==
| |
| {{refbegin|2}}
| |
| *{{citation
| |
| | last1 = Bisdorff | first1 = R. | last2 = Marichal | first2 = J.-L.
| |
| | title = Counting non-isomorphic maximal independent sets of the ''n''-cycle graph
| |
| | year = 2007
| |
| | arxiv = math.CO/0701647}}.
| |
| *{{citation
| |
| | first1 = I. M. | last1 = Bomze
| |
| | first2 = M. | last2 = Budinich
| |
| | first3 = P. M. | last3 = Pardalos
| |
| | first4 = M. | last4 = Pelillo
| |
| | contribution = The maximum clique problem
| |
| | title = Handbook of Combinatorial Optimization
| |
| | volume = 4
| |
| | pages = 1–74
| |
| | publisher = Kluwer Academic Publishers
| |
| | year = 1999
| |
| | id = {{citeseerx|10.1.1.48.4074}}
| |
| }}.
| |
| *{{citation
| |
| | last = Byskov | first = J. M.
| |
| | contribution = Algorithms for ''k''-colouring and finding maximal independent sets
| |
| | year = 2003
| |
| | title = Proceedings of the Fourteenth Annual ACM-SIAM Symposium on Discrete Algorithms
| |
| | pages = 456–457
| |
| | url = http://portal.acm.org/citation.cfm?id=644182}}.
| |
| *{{citation
| |
| | last1 = Chiba | first1 = N. | last2 = Nishizeki | first2 = T.
| |
| | title = Arboricity and subgraph listing algorithms
| |
| | journal = SIAM J. on Computing
| |
| | volume = 14
| |
| | issue = 1
| |
| | year = 1985
| |
| | pages = 210–223
| |
| | doi = 10.1137/0214017}}.
| |
| *{{citation
| |
| | last = Croitoru | first = C.
| |
| | contribution = On stables in graphs
| |
| | title = Proc. Third Coll. Operations Research
| |
| | publisher = [[Babeş-Bolyai University]], Cluj-Napoca, Romania
| |
| | year = 1979
| |
| | pages = 55–60}}.
| |
| *{{citation
| |
| | last = Eppstein | first = D.
| |
| | authorlink = David Eppstein
| |
| | title = Small maximal independent sets and faster exact graph coloring
| |
| | journal = [[Journal of Graph Algorithms and Applications]]
| |
| | volume = 7
| |
| | issue = 2
| |
| | year = 2003
| |
| | pages = 131–140
| |
| | url = http://www.cs.brown.edu/publications/jgaa/accepted/2003/Eppstein2003.7.2.pdf
| |
| | arxiv = cs.DS/0011009}}.
| |
| *{{citation
| |
| | last = Eppstein | first = D.
| |
| | authorlink = David Eppstein
| |
| | contribution = All maximal independent sets and dynamic dominance for sparse graphs
| |
| | title = Proc. Sixteenth Annual ACM-SIAM Symposium on Discrete Algorithms
| |
| | year = 2005
| |
| | pages = 451–459
| |
| | arxiv = cs.DS/0407036
| |
| }}.
| |
| *{{citation
| |
| | last = Erdős | first = P.
| |
| | authorlink = Paul Erdős
| |
| | title = On cliques in graphs
| |
| | journal = Israel J. Math.
| |
| | volume = 4
| |
| | year = 1966
| |
| | pages = 233–234
| |
| | id = {{MathSciNet | id = 0205874}}
| |
| | doi = 10.1007/BF02771637
| |
| | issue = 4}}.
| |
| *{{citation
| |
| | last = Euler | first = R.
| |
| | title = The Fibonacci number of a grid graph and a new class of integer sequences
| |
| | journal = Journal of Integer Sequences
| |
| | volume = 8
| |
| | year = 2005
| |
| | issue = 2
| |
| | pages = 05.2.6}}.
| |
| *{{citation
| |
| | last = Füredi | first = Z. | authorlink = Zoltán Füredi
| |
| | title = The number of maximal independent sets in connected graphs
| |
| | journal = Journal of Graph Theory
| |
| | volume = 11
| |
| | issue = 4
| |
| | year = 1987
| |
| | pages = 463–470
| |
| | doi = 10.1002/jgt.3190110403}}.
| |
| *{{citation
| |
| | last1 = Jennings | first1 = E. | last2 = Motycková | first2 = L.
| |
| | contribution = A distributed algorithm for finding all maximal cliques in a network graph
| |
| | title = Proc. First Latin American Symposium on Theoretical Informatics
| |
| | year = 1992
| |
| | series = Lecture Notes in Computer Science | volume = 583 | publisher = Springer-Verlag
| |
| | pages = 281–293}}
| |
| *{{citation
| |
| | author1-link = David S. Johnson| last1 = Johnson | first1 = D. S.
| |
| | author2-link = Mihalis Yannakakis|last2 = Yannakakis | first2 = M.
| |
| | author3-link = Christos Papadimitriou|last3 =Papadimitriou | first3 = C. H.
| |
| | title = On generating all maximal independent sets
| |
| | journal = Information Processing Letters
| |
| | volume = 27
| |
| | issue = 3
| |
| | pages = 119–123
| |
| | year = 1988
| |
| | doi = 10.1016/0020-0190(88)90065-8}}.
| |
| *{{citation
| |
| | last = Lawler | first = E. L. | authorlink = Eugene Lawler
| |
| | title = A note on the complexity of the chromatic number problem
| |
| | journal = Information Processing Letters
| |
| | volume = 5
| |
| | issue = 3
| |
| | pages = 66–67
| |
| | year = 1976
| |
| | doi = 10.1016/0020-0190(76)90065-X}}.
| |
| *{{citation
| |
| | last1 = Lawler | first1 = E. L. | author1-link = Eugene Lawler | last2 = Lenstra | first2 = J. K. | author2-link = Jan Karel Lenstra | last3 = Rinnooy Kan | first3 = A. H. G.
| |
| | title = Generating all maximal independent sets: NP-hardness and polynomial time algorithms
| |
| | journal = SIAM Journal on Computing
| |
| | volume = 9
| |
| | issue = 3
| |
| | year = 1980
| |
| | pages = 558–565
| |
| | doi = 10.1137/0209042}}.
| |
| *{{citation
| |
| | last = Leung | first = J. Y.-T.
| |
| | title = Fast algorithms for generating all maximal independent sets of interval, circular-arc and chordal graphs
| |
| | journal = Journal of Algorithms
| |
| | volume = 5
| |
| | pages = 22–35
| |
| | year = 1984
| |
| | doi = 10.1016/0196-6774(84)90037-3}}.
| |
| *{{citation
| |
| | last1 = Liang | first1 = Y. D. | last2 = Dhall | first2 = S. K. | last3 = Lakshmivarahan | first3 = S.
| |
| | title = On the problem of finding all maximum weight independent sets in interval and circular arc graphs
| |
| | booktitle = Proc. Symp. Applied Computing
| |
| | year = 1991
| |
| | pages = 465–470}}
| |
| *{{citation
| |
| | last1 = Makino | first1 = K. | last2 = Uno | first2 = T.
| |
| | title = New algorithms for enumerating all maximal cliques
| |
| | booktitle = Proc. Ninth Scandinavian Workshop on Algorithm Theory
| |
| | year = 2004
| |
| | pages = 260–272
| |
| | url = http://www.springerlink.com/content/p9qbl6y1v5t3xc1w/
| |
| | volume = 3111
| |
| | series = Lecture Notes in Compute Science
| |
| | publisher = Springer-Verlag}}.
| |
| *{{citation
| |
| | last1 = Mishra | first1 = N. | last2 = Pitt | first2 = L.
| |
| | contribution = Generating all maximal independent sets of bounded-degree hypergraphs
| |
| | title = Proc. Tenth Conf. Computational Learning Theory
| |
| | year = 1997
| |
| | pages = 211–217
| |
| | doi = 10.1145/267460.267500
| |
| | isbn = 0-89791-891-6}}.
| |
| *{{citation | last1 = Moon | first1 = J. W. | author2-link = Leo Moser | last2 = Moser | first2 = L. | title = On cliques in graphs | journal = Israel Journal of Mathematics | volume = 3 | year = 1965 | pages = 23–28 | id = {{MathSciNet | id = 0182577}} | doi = 10.1007/BF02760024}}.
| |
| *{{citation
| |
| | last = Stix | first = V.
| |
| | title = Finding all maximal cliques in dynamic graphs
| |
| | journal = Computational Optimization Appl.
| |
| | volume = 27
| |
| | issue = 2
| |
| | pages = 173–186
| |
| | year = 2004
| |
| | doi = 10.1023/B:COAP.0000008651.28952.b6}}
| |
| *{{citation|first1=E.|last1=Tomita|first2=A.|last2=Tanaka|first3=H.|last3=Takahashi|title=The worst-case time complexity for generating all maximal cliques and computational experiments|journal=[[Theoretical Computer Science (journal)|Theoretical Computer Science]]|volume=363|issue=1|pages=28–42|year=2006|doi=10.1016/j.tcs.2006.06.015}}.
| |
| *{{citation
| |
| | last1 = Tsukiyama | first1 = S. | last2 = Ide | first2 = M. | last3 = Ariyoshi | first3 = H. | last4 = Shirakawa | first4 = I.
| |
| | title = A new algorithm for generating all the maximal independent sets
| |
| | journal = SIAM J. on Computing
| |
| | volume = 6
| |
| | issue = 3
| |
| | pages = 505–517
| |
| | year = 1977
| |
| | doi = 10.1137/0206036}}.
| |
| *{{citation
| |
| | last1 = Weigt | first1 = Martin | last2 = Hartmann | first2 = Alexander K.
| |
| | title = Minimal vertex covers on finite-connectivity random graphs: A hard-sphere lattice-gas picture
| |
| | year = 2001
| |
| | journal = Phys. Rev. E
| |
| | volume = 63
| |
| | issue = 5
| |
| | pages = 056127
| |
| | doi = 10.1103/PhysRevE.63.056127
| |
| | arxiv = cond-mat/0011446}}.
| |
| *{{citation
| |
| | last1 = Yu | first1 = C.-W. | last2 = Chen | first2 = G.-H.
| |
| | title = Generate all maximal independent sets in permutation graphs
| |
| | journal = Internat. J. Comput. Math.
| |
| | volume = 47
| |
| | pages = 1–8
| |
| | year = 1993
| |
| | doi = 10.1080/00207169308804157}}.
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| {{refend}}
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| {{DEFAULTSORT:Maximal Independent Set}}
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| [[Category:Graph theory objects]]
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| [[Category:Computational problems in graph theory]]
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| [[de:Glossar Graphentheorie#Stabile Menge]]
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