Abstract
We introduce “matroid parse trees” which, using only a limited amount of information, can build up all matroids of bounded branchwidth representable over a finite field. We prove that if M is a family of matroids described by a sentence in the second-order monadic logic of matroids, then the parse trees of bounded-width representable members of M can be recognized by a finite tree automaton. Since the cycle matroids of graphs are representable over any finite field, our result directly extends the well-known “MS 2-theorem” for graphs of bounded tree-width by Courcelle and others. This work has algorithmic applications in matroid or coding theories.
This work is based on an original research that the author carried out at the Victoria University of Wellington in New Zealand, supported by a Marsden Fund research grant to Geo. Whittle.
ITI is supported by Ministry of Education of Czech Republic as project LN00A056.
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References
K.A. Abrahamson, M.R. Fellows, Finite Automata, Bounded Treewidth, and Well-Quasiordering, In: Graph Structure Theory, Contemporary Mathematics 147, American Mathematical Society (1993), 539–564.
S. Arnborg, J. Lagergren, D. Seese, Problems easy for Tree-decomposible Graphs (extended abstract), Proc. 15th Colloq. Automata, Languages and Programming, Lecture Notes in Computer Science 317, Springer-Verlag (1988), 38–51.
H.L. Bodlaender, A Tourist Guide through Treewidth, Acta Cybernetica 11 (1993), 1–21.
R.B. Borie, R.G. Parker, C.A. Tovey, Automatic generation of linear-time algorithms from predicate calculus descriptions of problems on recursively constructed graph families, Algorithmica 7 (1992), 555–582.
B. Courcelle, Recognizability and Second-Order Definability for Sets of Finite Graphs, technical report I-8634, Universite de Bordeaux, 1987.
B. Courcelle, Graph Rewriting: an Algebraic and Logic Approach, In: Handbook of Theoretical Computer Science Vol. B, Chap. 5, North-Holland 1990.
B. Courcelle, The Monadic Second-Order Logic of Graphs I. Recognizable sets of Finite Graphs Information and Computation 85 (1990), 12–75.
R. Diestel, Graph theory, Graduate Texts in Mathematics 173, Springer-Verlag, New York 1997, 2000.
R.G. Downey, M.R. Fellows, Parametrized Complexity, Springer-Verlag New York, 1999, ISBN 0-387-94833-X.
J.F. Geelen, A.H.M. Gerards, N. Robertson, G.P. Whittle, On the Excluded Minors for the Matroids of Branch-Width k, manuscript, 2002.
J.F. Geelen, A.H.M. Gerards, G.P. Whittle, Branch-Width and Well-Quasi-Ordering in Matroids and Graphs, J. Combin. Theory Ser. B 84 (2002), 270–290.
P. Hliněný, The Tutte Polynomial for Matroids of Bounded Branch-Width, submitted, 2002.
P. Hliněný, It is Hard to Recognize Free Spikes, submitted, 2002.
P. Hliněný, Branch-Width, Parse Trees, and Monadic Second-Order Logic for Matroids, submitted, 2002.
P. Hliněný, A Parametrized Algorithm for Matroid Branch-Width, submitted, 2002.
P. Hliněný, Branch-Width and Parametrized Algorithms for Representable Matroids, in preparation, 2002.
J. Hopcroft, J. Ullmann, Introduction to Automata Theory, Adisson-Wesley 1979.
J.G. Oxley, Matroid Theory, Oxford University Press, 1992,1997, ISBN 0-19-853563-5.
N. Robertson, P.D. Seymour, Graph Minors-A Survey, Surveys in Combinatorics, Cambridge Univ. Press 1985, 153–171.
N. Robertson, P.D. Seymour, Graph Minors X. Obstructions to Tree-Decomposition, J. Combin. Theory Ser. B 52 (1991), 153–190.
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Hliněný, P. (2003). Branch-Width, Parse Trees, and Monadic Second-Order Logic for Matroids. In: Alt, H., Habib, M. (eds) STACS 2003. STACS 2003. Lecture Notes in Computer Science, vol 2607. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-36494-3_29
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DOI: https://doi.org/10.1007/3-540-36494-3_29
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