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In the study of [[graph theory|graph]] [[algorithm]]s, '''Courcelle's theorem''' is the statement that every graph property definable in [[monadic second-order logic]] can be decided in [[linear time]] on graphs of bounded [[treewidth]]. The result was first proved by [[Bruno Courcelle]] in 1990 and is considered the archetype of algorithmic [[meta-theorem]]s.<ref>{{citation
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| last = Courcelle | first = Bruno | authorlink = Bruno Courcelle
| doi = 10.1016/0890-5401(90)90043-H
| issue = 1
| journal = Information and Computation
| mr = 1042649
| pages = 12–75
| title = The monadic second-order logic of graphs. I. Recognizable sets of finite graphs
| volume = 85
| year = 1990}}</ref><ref name=Langer>{{citation
| last1 = Kneis | first1 = Joachim
| last2 = Langer | first2 = Alexander
| doi = 10.1016/j.entcs.2009.08.028
| journal = Electronic Notes in Theoretical Computer Science
| pages = 65–81
| title = A practical approach to Courcelle's theorem
| volume = 251
| year = 2009}}.</ref><ref name= Meyer>{{citation
| last = Lampis | first = Michael
| editor1-last = de Berg | editor1-first = Mark
| editor2-last = Meyer | editor2-first = Ulrich
| contribution = Algorithmic meta-theorems for restrictions of treewidth
| doi = 10.1007/978-3-642-15775-2_47
| pages = 549–560
| publisher = Springer
| series = Lecture Notes in Computer Science
| title = Proc. 18th Annual European Symposium on Algorithms
| volume = 6346
| year = 2010}}.</ref>
 
In this context, the graph is described by a set of vertices ''V'' and a binary adjacency relation, and
the restriction to monadic logic means that the graph property in question may be defined in terms of sets of vertices of the given graph, but not in terms of sets of edges. As an example, the property of a graph being [[graph coloring|colorable]] with three colors (represented by three sets of vertices ''R'', ''G'', and ''B'') may be defined by the monadic second-order formula
:<math>\begin{align}\exists R,G,B \,&\bigl(\forall v\in V (v\in R\vee v\in G\vee v\in B)\bigr)\wedge\\& \bigl(\forall u,v \,((u\in R\wedge v\in R)\vee (u\in G\wedge v\in G)\vee (u\in B\wedge v\in B))\to\lnot\mathrm{adj}(u,v)\bigr).\end{align}</math>
The first part of this formula ensures that the three color classes cover all the vertices of the graph, and the second ensures that they each form an [[independent set (graph theory)|independent set]]. Thus, by Courcelle's theorem, 3-colorability of graphs of bounded treewidth may be tested in linear time.
 
The typical approach to proving Courcelle's theorem involves the construction of a finite bottom-up [[tree automaton]] that performs a tree decomposition of the graph to recognize it.<ref name=Langer/>
 
==References==
{{Reflist}}
 
[[Category:Metatheorems]]
[[Category:Graph algorithms]]
[[Category:Graph minor theory]]
 
{{mathlogic-stub}}
{{combin-stub}}

Latest revision as of 02:05, 18 November 2014

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