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| [[File:Turan 13-4.svg|thumb|The [[Turán graph]] ''T''(13,4), an example of a cograph]]
| | I'm Luther but I by no means truly liked that name. He currently lives in Idaho and his parents reside close by. The factor she adores most is to play handball but she can't make it her profession. The job he's been occupying for many years is a messenger.<br><br>Here is my homepage; extended auto warranty ([http://www.Echojournal.org/users/SFajardo Click at www.Echojournal.org]) |
| In [[graph theory]], a '''cograph''', or '''complement-reducible graph''', or '''''P''<sub>4</sub>-free graph''', is a [[Graph (mathematics)|graph]] that can be generated from the single-vertex graph ''K''<sub>1</sub> by [[Complement graph|complementation]] and [[Graph operations|disjoint union]]. That is, the family of cographs is the smallest class of graphs that includes ''K''<sub>1</sub> and is closed under complementation and disjoint union.
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| Cographs have been discovered independently by several authors since the 1970s; early references include {{harvtxt|Jung|1978}}, {{harvtxt|Lerchs|1971}}, {{harvtxt|Seinsche|1974}}, and {{harvtxt|Sumner|1974}}. They have also been called '''D*-graphs''' {{harv|Jung|1978}}, '''hereditary Dacey graphs''' {{harv|Sumner|1974}}, and '''2-parity graphs''' {{harv|Burlet|Uhry|1984}}.
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| See, e.g., {{harvtxt|Brandstädt|Le|Spinrad|1999}} for more detailed coverage of cographs, including the facts presented here.
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| == Definition and characterization ==
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| Any cograph may be constructed using the following rules:
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| # any single vertex graph is a cograph;
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| # if <math>G</math> is a cograph, so is its complement <math>\overline{G}</math>;
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| # if <math>G</math> and <math>H</math> are cographs, so is their disjoint union <math>G\cup H</math>.
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| Several alternative characterizations of cographs can be given. Among them:
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| * A cograph is a graph which does not contain the [[path (graph theory)|path]] ''P''<sub>4</sub> on 4 vertices (and hence of length 3) as an [[induced subgraph]]. That is, a graph is a cograph if and only if for any four vertices <math>v_1,v_2,v_3,v_4</math>, if <math>\{v_1,v_2\},\{v_2,v_3\}</math> and <math>\{v_3,v_4\}</math> are edges of the graph then at least one of <math>\{v_1,v_3\},\{v_1,v_4\}</math> or <math>\{v_2,v_4\}</math> is also an edge {{harv|Corneil|Lerchs|Burlingham|1981}}.
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| * A cograph is a graph all of whose induced subgraphs have the property that any maximal [[Clique (graph theory)|clique]] intersects any [[maximal independent set]] in a single vertex.
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| * A cograph is a graph in which every nontrivial induced subgraph has at least two vertices with the same neighbourhoods.
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| * A cograph is a graph in which every connected induced subgraph has a disconnected complement.
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| * A cograph is a graph all of whose connected [[induced subgraph]]s have [[Distance (graph theory)|diameter]] at most 2.
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| * A cograph is a graph in which every [[connected component (graph theory)|connected component]] is a [[distance-hereditary graph]] with diameter at most 2.
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| * A cograph is a graph with [[clique-width]] at most 2 {{harv|Courcelle|Olariu|2000}}.
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| * A cograph is a [[comparability graph]] of a [[series-parallel partial order]] {{harv|Jung|1978}}.
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| * A cograph is a [[permutation graph]] of a [[separable permutation]] {{harv|Bose|Buss|Lubiw|1998}}.
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| All [[complete graph]]s, [[complete bipartite graph]]s, [[threshold graph]]s, and [[Turán graph]]s are cographs. Every cograph is distance-hereditary, a [[comparability graph]], and [[Perfect graph|perfect]].
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| == Cotrees ==
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| [[File:Cotree and cograph.svg|thumb|360px|A cotree and the corresponding cograph. Each edge (''u'',''v'') in the cograph has a matching color to the least common ancestor of ''u'' and ''v'' in the cotree.]]
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| A '''cotree''' is a tree in which the internal nodes are labeled with the numbers 0 and 1. Every cotree ''T'' defines a cograph ''G'' having the leaves of ''T'' as vertices, and in which the subtree rooted at each node of ''T'' corresponds to the [[induced subgraph]] in ''G'' defined by the set of leaves descending from that node:
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| * A subtree consisting of a single leaf node corresponds to an induced subgraph with a single vertex.
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| * A subtree rooted at a node labeled 0 corresponds to the union of the subgraphs defined by the children of that node.
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| * A subtree rooted at a node labeled 1 corresponds to the ''join'' of the subgraphs defined by the children of that node; that is, we form the union and add an edge between every two vertices corresponding to leaves in different subtrees. Alternatively, the join of a set of graphs can be viewed as formed by complementing each graph, forming the union of the complements, and then complementing the resulting union.
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| An equivalent way of describing the cograph formed from a cotree is that two vertices are connected by an edge if and only if the [[lowest common ancestor]] of the corresponding leaves is labeled by 1. Conversely, every cograph can be represented in this way by a cotree. If we require the labels on any root-leaf path of this tree to alternate between 0 and 1, this representation is unique {{harv|Corneil|Lerchs|Burlingham|1981}}.
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| == Computational properties ==
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| Cographs may be recognized in linear time, and a cotree representation constructed, using [[modular decomposition]] {{harv|Corneil|Perl|Stewart|1985}}, [[partition refinement]] {{harv|Habib|Paul|2005}}, or [[split decomposition]] {{harv|Gioan|Paul|2008}}. Once a cotree representation has been constructed, many familiar graph problems may be solved via simple bottom-up calculations on the cotrees.
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| For instance, to find the [[Clique problem|maximum clique]] in a cograph, compute in bottom-up order the maximum clique in each subgraph represented by a subtree of the cotree. For a node labeled 0, the maximum clique is the maximum among the cliques computed for that node's children. For a node labeled 1, the maximum clique is the union of the cliques computed for that node's children, and has size equal to the sum of the children's clique sizes. Thus, by alternately maximizing and summing values stored at each node of the cotree, we may compute the maximum clique size, and by alternately maximizing and taking unions, we may construct the maximum clique itself. Similar bottom-up tree computations allow the [[Maximal independent set|maximum independent set]], [[Graph coloring|vertex coloring number]], maximum clique cover, and Hamiltonicity (that is the existence of a [[Hamiltonian path problem|Hamiltonian cycle]]) to be computed in linear time from a cotree representation of a cograph. One can also use cotrees to determine in linear time whether two cographs are [[graph isomorphism|isomorphic]].
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| If ''H'' is an [[induced subgraph]] of a cograph ''G'', then ''H'' is itself a cograph; the cotree for ''H'' may be formed by removing some of the leaves from the cotree for ''G'' and then suppressing nodes that have only one child. It follows from [[Kruskal's tree theorem]] that the [[binary relation|relation]] of being an induced subgraph is a [[well-quasi-ordering]] on the cographs {{harv| Damaschke|1990}}. Thus, if a subfamily of the cographs (such as the [[planar graph|planar]] cographs) is closed under induced subgraph operations then it has a finite number of [[Forbidden graph characterization|forbidden induced subgraph]]s. Computationally, this means that testing membership in such a subfamily may be performed in linear time, by using a bottom-up computation on the cotree of a given graph to test whether it contains any of these forbidden subgraphs.
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| == References ==
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| {{refbegin|colwidth=30em}}
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| *{{citation | last1=Bose | first1=Prosenjit | last2=Buss | first2=Jonathan | last3=Lubiw | first3=Anna | author3-link = Anna Lubiw | title=Pattern matching for permutations | mr = 1620935 | year=1998 | journal=[[Information Processing Letters]] | volume=65 | pages=277–283 | doi=10.1016/S0020-0190(97)00209-3}}.
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| *{{Citation | last1=Brandstädt | first1=Andreas | last2=Le | first2=Van Bang | last3=Spinrad | first3=Jeremy P. | title=Graph Classes: A Survey | publisher=SIAM Monographs on Discrete Mathematics and Applications | isbn=978-0-89871-432-6 | year=1999}}.
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| *{{Citation |last1=Burlet |first1=M. |last2=Uhry |first2=J. P. |contribution=Parity Graphs |title=Topics on Perfect Graphs |series=Annals of Discrete Mathematics |volume=21 |year=1984 |pages=253–277}}.
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| *{{Citation | last1=Corneil | first1=D. G. | author1-link = Derek Corneil | last2=Lerchs | first2=H. | last3=Burlingham | first3=L. Stewart | title=Complement reducible graphs | doi=10.1016/0166-218X(81)90013-5 | mr=0619603 | year=1981 | journal=Discrete Applied Mathematics | volume=3 | pages=163–174 | issue=3}}.
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| *{{Citation | last1=Corneil | first1=D. G. | author1-link = Derek Corneil | last2=Perl | first2=Y. | last3=Stewart | first3=L. K. | title=A linear recognition algorithm for cographs | doi=10.1137/0214065 | mr=0807891 | year=1985 | journal=SIAM Journal on Computing | volume=14 | issue=4 | pages=926–934}}.
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| *{{citation
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| | last1 = Courcelle | first1 = B.
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| | last2 = Olariu | first2 = S.
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| | title = Upper bounds to the clique width of graphs
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| | journal = Discrete Applied Mathematics
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| | volume = 101 | issue = 1–3 | year = 2000 | pages = 77–144 | doi = 10.1016/S0166-218X(99)00184-5}}.
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| *{{citation
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| | last = Damaschke | first = Peter
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| | doi = 10.1002/jgt.3190140406
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| | issue = 4
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| | journal = Journal of Graph Theory
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| | mr = 1067237
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| | pages = 427–435
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| | title = Induced subgraphs and well-quasi-ordering
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| | volume = 14
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| | year = 1990}}.
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| *{{cite arxiv | last1=Gioan | first1=Emeric | last2=Paul | first2=Christophe | title=Split decomposition and graph-labelled trees: characterizations and {{Sic|hide=y|fully|-}}dynamic algorithms for totally decomposable graphs | year=2008 | eprint=0810.1823}}.
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| *{{Citation | last1=Habib | first1=Michel | last2=Paul | first2=Christophe | title=A simple linear time algorithm for cograph recognition | url=http://www.lirmm.fr/~paul/Biblio/Postscript/DAM-cographs.pdf | doi=10.1016/j.dam.2004.01.011 | mr=2113140 | year=2005 | journal=Discrete Applied Mathematics | volume=145 | pages=183–197 | issue=2}}.
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| *{{Citation | last1=Jung | first1=H. A. | title=On a class of posets and the corresponding comparability graphs | mr=0491356 | year=1978 | journal=Journal of Combinatorial Theory, Series B | volume=24 | pages=125–133 | doi=10.1016/0095-8956(78)90013-8 | issue=2}}.
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| *{{Citation | last = Lerchs | first = H. | title = On cliques and kernels | publisher = Tech. Report, Dept. of Comp. Sci., Univ. of Toronto | year = 1971}}.
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| *{{Citation | last1=Seinsche | first1=D. | title=On a property of the class of ''n''-colorable graphs | mr=0337679 | year=1974 | journal=Journal of Combinatorial Theory, Series B | pages=191–193 | doi=10.1016/0095-8956(74)90063-X | volume=16 | issue=2}}.
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| *{{Citation | last1=Sumner | first1=D. P. | title=Dacey graphs | mr=0382082 | year=1974 | journal=J. Austral. Math. Soc. | volume=18 | pages=492–502 | doi=10.1017/S1446788700029232 | issue=04}}.
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| {{refend}}
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| == External links ==
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| * {{cite web | url=http://www.graphclasses.org/classes/gc_151.html | title=cograph graphs
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| | work = [http://www.graphclasses.org Information System on Graph Class Inclusions]}}
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| * {{mathworld | urlname=Cograph | title=Cograph}}
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| [[Category:Graph families]]
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| [[Category:Perfect graphs]]
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| [[Category:Graph operations]]
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