Branch-width, parse trees, and monadic second-order logic for matroids
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 M is a family of matroids described by a sentence in the monadic second-order logic of...
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| Published in: | Journal of combinatorial theory. Series B Vol. 96; no. 3; pp. 325 - 351 |
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| Format: | Journal Article |
| Language: | English |
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01.05.2006
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| ISSN: | 0095-8956, 1096-0902 |
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| Abstract | 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
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
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. |
|---|---|
| AbstractList | 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
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
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. |
| Author | Hliněný, Petr |
| Author_xml | – sequence: 1 givenname: Petr surname: Hliněný fullname: Hliněný, Petr email: petr.hlineny@vsb.cz, hlineny@member.ams.org organization: School of Mathematical and Computing Sciences, Victoria University of Wellington, New Zealand |
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| Keywords | secondary 68R05 Fixed-parameter complexity 03D05 Matroid representation Monadic second-order logic primary 05B35 Tree automaton Branch-width |
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| References_xml | – reference: P. Hliněný, D. Seese, Trees, grids, and MSO decidability: from graphs to matroids, Theoret. Computer Sci., 2005, to appear. (Extended abstract in: IWPEC 2004, Lecture Notes in Computer Science, vol. 3162, Springer, Berlin, 2004, pp. 96–107.) – volume: 88 start-page: 261 year: 2003 end-page: 265 ident: bib12 article-title: On the excluded minors for the matroids of branch-width publication-title: J. Combin. Theory Ser. B – reference: K.A. Abrahamson, M.R. Fellows, Finite Automata, bounded treewidth, and well-quasiordering, in: Graph Structure Theory, Contemporary Mathematics, vol. 147, American Mathematical Society, 1993, pp. 539–564. – volume: 7 start-page: 555 year: 1992 end-page: 582 ident: bib5 article-title: Automatic generation of linear-time algorithms from predicate calculus descriptions of problems on recursively constructed graph families publication-title: Algorithmica – reference: P. Hliněný, The tutte polynomial for matroids of bounded branch-width, combinatorics, Probab. Comput., 2005, to appear. – reference: N. Robertson, P.D. Seymour, Graph Minors—A Survey, Surveys in Combinatorics, Cambridge University Press, Cambridge, 1985, pp. 153–171. – reference: P. Hliněný, On matroid properties definable in the MSO logic, in: Math Foundations of Computer Science MFCS 2003, Lecture Notes in Computer Science, vol. 2747, Springer, Berlin, 2003, pp. 470–479. – volume: 11 start-page: 1 year: 1993 end-page: 21 ident: bib3 article-title: A tourist guide through treewidth publication-title: Acta Cybernetica – year: 1992 ident: bib21 article-title: Matroid Theory – volume: 52 start-page: 153 year: 1991 end-page: 190 ident: bib23 article-title: Graph minors X. obstructions to tree-decomposition publication-title: J. Combin. Theory Ser. B – reference: S. Arnborg, J. Lagergren, D. 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