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In [[graph theory]], the '''clique-width''' of a [[graph (mathematics)|graph]] <math>G</math> is the minimum number of labels needed to construct <math>G</math> by means of the following 4 operations :


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#Creation of a new vertex ''v'' with label ''i'' ( noted ''i(v)'' )
#Disjoint union of two labeled graphs ''G'' and ''H'' ( denoted <math>G \oplus H</math> )
#Joining by an edge every vertex labeled ''i'' to every vertex labeled ''j'' (denoted ''n(i,j)''), where <math>i \neq j</math>
#Renaming label ''i'' to label ''j'' ( denoted ''p''(''i'',''j'') )
 
[[Cograph]]s are exactly the graphs with clique-width at most 2 {{harv|Courcelle|Olariu|2000}}; every [[distance-hereditary graph]] has clique-width at most 3 {{harv|Golumbic|Rotics|2000}}. Many optimization problems that are [[NP-hard]] for more general classes of graphs may be solved efficiently by [[dynamic programming]] on graphs of bounded clique-width ({{harvnb|Cogis|Thierry|2005}}; {{harvnb|Courcelle|Makowsky|Rotics|2000}}).
 
The theory of graphs of bounded clique-width resembles that for graphs of bounded [[treewidth]], but unlike treewidth allows for [[dense graph]]s. If a family of graphs has bounded clique-width, then either it has bounded treewidth or every [[complete bipartite graph]] is a subgraph of a graph in the family {{harv|Gurski|Wanke|2000}}.
 
==References==
*{{citation
| last1 = Cogis | first1 = O.
| last2 = Thierry | first2 = E.
| title = Computing maximum stable sets for distance-hereditary graphs
| journal = Discrete Optimization
| volume = 2 | issue = 2 | year = 2005 | pages = 185–188
| mr = 2155518
| doi = 10.1016/j.disopt.2005.03.004}}.
*{{citation
| last1 = Courcelle | first1 = B.
| last2 = Makowsky | first2 = J. A.
| last3 = Rotics | first3 = U.
| title = Linear time solvable optimization problems on graphs on bounded clique width
| journal = Theory of Computing Systems
| volume = 33 | issue = 2 | year = 2000 | pages = 125–150
| doi = 10.1007/s002249910009}}.
*{{citation
| last1 = Courcelle | first1 = B.
| last2 = Olariu | first2 = S.
| title = Upper bounds to the clique width of graphs
| journal = Discrete Applied Mathematics
| volume = 101 | issue = 1–3 | year = 2000 | pages = 77–144 | doi = 10.1016/S0166-218X(99)00184-5}}.
*{{citation
| last1 = Golumbic | first1 = Martin Charles | authorlink = Martin Charles Golumbic
| last2 = Rotics | first2 = Udi
| title = On the clique-width of some perfect graph classes
| journal = International Journal of Foundations of Computer Science
| volume = 11 | issue = 3 | year = 2000 | pages = 423–443
| mr = 1792124
| doi = 10.1142/S0129054100000260}}.
*{{citation
| last1 = Gurski | first1 = Frank
| last2 = Wanke | first2 = Egon
| editor1-last = Brandes | editor1-first = Ulrik
| editor2-last = Wagner | editor2-first = Dorothea | editor2-link = Dorothea Wagner
| contribution = The tree-width of clique-width bounded graphs without ''K<sub>n,n</sub>''
  | doi = 10.1007/3-540-40064-8_19
| location = Berlin
| mr = 1850348
| pages = 196–205
| publisher = Springer
| series = Lecture Notes in Computer Science
| title = Graph-Theoretic Concepts in Computer Science: 26th International Workshop, WG 2000, Konstanz, Germany, June 15–17, 2000, Proceedings
| volume = 1928
| year = 2000}}.
 
[[Category:Graph invariants]]

Revision as of 17:23, 27 February 2013

In graph theory, the clique-width of a graph G is the minimum number of labels needed to construct G by means of the following 4 operations :

  1. Creation of a new vertex v with label i ( noted i(v) )
  2. Disjoint union of two labeled graphs G and H ( denoted GH )
  3. Joining by an edge every vertex labeled i to every vertex labeled j (denoted n(i,j)), where ij
  4. Renaming label i to label j ( denoted p(i,j) )

Cographs are exactly the graphs with clique-width at most 2 Template:Harv; every distance-hereditary graph has clique-width at most 3 Template:Harv. Many optimization problems that are NP-hard for more general classes of graphs may be solved efficiently by dynamic programming on graphs of bounded clique-width (Template:Harvnb; Template:Harvnb).

The theory of graphs of bounded clique-width resembles that for graphs of bounded treewidth, but unlike treewidth allows for dense graphs. If a family of graphs has bounded clique-width, then either it has bounded treewidth or every complete bipartite graph is a subgraph of a graph in the family Template:Harv.

References

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    Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

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