Branch-Width, Parse Trees, and Monadic Second-Order Logic for Matroids
Result description
We introduce ``matroid parse trees'' which, using only a limited amount of information at each node, can build up the vector representations of matroids of bounded branch-width over a finite field. We prove that if $mf M$ is a family of matroids described by a sentence in the monadic second-order logic of matroids, then there is a finite tree automaton accepting exactly those parse trees which build vector representations of the bounded-branch-width representable members of $mf M$. Since the cycle matroids of graphs are representable over any field, our result directly extends the so called ``$MS_2$-theorem'' for graphs of bounded tree-width by Courcelle, and others. Moreover, applications and relations in areas other than matroid theory can be found, like for rank-width of graphs, or in the coding theory.
Keywords
matroid representationbranch-widthmonadic second-order logictree automatonfixed-parameter complexity
The result's identifiers
Result code in IS VaVaI
Alternative codes found
RIV/61989100:27240/06:00013585
Result on the web
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DOI - Digital Object Identifier
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Alternative languages
Result language
angličtina
Original language name
Branch-Width, Parse Trees, and Monadic Second-Order Logic for Matroids
Original language description
We introduce ``matroid parse trees'' which, using only a limited amount of information at each node, can build up the vector representations of matroids of bounded branch-width over a finite field. We prove that if $mf M$ is a family of matroids described by a sentence in the monadic second-order logic of matroids, then there is a finite tree automaton accepting exactly those parse trees which build vector representations of the bounded-branch-width representable members of $mf M$. Since the cycle matroids of graphs are representable over any field, our result directly extends the so called ``$MS_2$-theorem'' for graphs of bounded tree-width by Courcelle, and others. Moreover, applications and relations in areas other than matroid theory can be found, like for rank-width of graphs, or in the coding theory.
Czech name
Branch-width, parsovací stromy a monadická logika druhého řádu pro matroidy
Czech description
Článek dokazuje obdobu tzv. MS2-věty pro matroidy reprezentované nad konečnými tělesy: Pro matroid reprezentovaný nad konečným tělesem s omezenou branch-width lze stromovými automaty rozhodnout všechny MSO definované vlastnosti.
Classification
Type
Jx - Unclassified - Peer-reviewed scientific article (Jimp, Jsc and Jost)
CEP classification
BA - General mathematics
OECD FORD branch
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Result continuities
Project
GA201/05/0050: Structural properties and algorithmic complexity of discrete problems
Continuities
Z - Vyzkumny zamer (s odkazem do CEZ)
Others
Publication year
2006
Confidentiality
S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů
Data specific for result type
Name of the periodical
Journal of Combinatorial Theory, Ser B
ISSN
0095-8956
e-ISSN
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Volume of the periodical
96
Issue of the periodical within the volume
3
Country of publishing house
US - UNITED STATES
Number of pages
27
Pages from-to
325-351
UT code for WoS article
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EID of the result in the Scopus database
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Basic information
Result type
Jx - Unclassified - Peer-reviewed scientific article (Jimp, Jsc and Jost)
CEP
BA - General mathematics
Year of implementation
2006