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...

Full description

Saved in:
Bibliographic Details
Published in:Journal of combinatorial theory. Series B Vol. 96; no. 3; pp. 325 - 351
Main Author: Hliněný, Petr
Format: Journal Article
Language:English
Published: Elsevier Inc 01.05.2006
Subjects:
Branch-width
Fixed-parameter complexity
Matroid representation
Monadic second-order logic
Tree automaton
secondary 68R05
Fixed-parameter complexity
03D05
Matroid representation
Monadic second-order logic
primary 05B35
Tree automaton
Branch-width
ISSN:0095-8956, 1096-0902
Online Access:Get full text
Tags: Add Tag
No Tags, Be the first to tag this record!
Description
Summary: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.
ISSN:0095-8956
1096-0902
DOI:10.1016/j.jctb.2005.08.005