Complexity of finding embeddings in a k-tree
SIAM Journal on Algebraic and Discrete Methods
Easy problems for tree-decomposable graphs
Journal of Algorithms
Graph minors: X. obstructions to tree-decomposition
Journal of Combinatorial Theory Series B
Monadic second-order evaluations on tree-decomposable graphs
Theoretical Computer Science - Special issue on selected papers of the International Workshop on Computing by Graph Transformation, Bordeaux, France, March 21–23, 1991
Handle-rewriting hypergraph grammars
Journal of Computer and System Sciences
Monadic second-order definable graph transductions: a survey
Theoretical Computer Science - Selected papers of the 17th Colloquium on Trees in Algebra and Programming (CAAP '92) and of the European Symposium on Programming (ESOP), Rennes, France, Feb. 1992
Discrete Applied Mathematics - Special issue: efficient algorithms and partial k-trees
Structural properties of context-free sets of graphs generated by vertex replacement
Information and Computation
The monadic second-order logic of graphs X: linear orderings
Theoretical Computer Science
A Linear-Time Algorithm for Finding Tree-Decompositions of Small Treewidth
SIAM Journal on Computing
The expression of graph properties and graph transformations in monadic second-order logic
Handbook of graph grammars and computing by graph transformation
Upper bounds to the clique width of graphs
Discrete Applied Mathematics
The monadic second-order logic of graphs XII: planar graphs and planar maps
Theoretical Computer Science
Branch-width and well-quasi-ordering in matroids and graphs
Journal of Combinatorial Theory Series B
Problems Easy for Tree-Decomposable Graphs (Extended Abstract)
ICALP '88 Proceedings of the 15th International Colloquium on Automata, Languages and Programming
Axiomatising Tree-Interpretable Structures
STACS '02 Proceedings of the 19th Annual Symposium on Theoretical Aspects of Computer Science
The Monadic Second-Order Logic of Graphs: Definable Sets of Finite Graphs
WG '88 Proceedings of the 14th International Workshop on Graph-Theoretic Concepts in Computer Science
Recognizability Equals Monadic Second-Order Definability for Sets of Graphs of Bounded Tree-Width
STACS '98 Proceedings of the 15th Annual Symposium on Theoretical Aspects of Computer Science
The monadic second-order logic of graphs XIV: uniformly sparse graphs and edge set quantifications
Theoretical Computer Science
On the excluded minors for the matroids of branch-width k
Journal of Combinatorial Theory Series B
Graph Minors. XX. Wagner's conjecture
Journal of Combinatorial Theory Series B - Special issue dedicated to professor W. T. Tutte
A Parametrized Algorithm for Matroid Branch-Width
SIAM Journal on Computing
Branch-width, parse trees, and monadic second-order logic for matroids
Journal of Combinatorial Theory Series B
Branch-width, parse trees, and monadic second-order logic for matroids
Journal of Combinatorial Theory Series B
European Journal of Combinatorics - Special issue on Eurocomb'03 - graphs and combinatorial structures
Vertex-minors, monadic second-order logic, and a conjecture by Seese
Journal of Combinatorial Theory Series B
On matroid representability and minor problems
MFCS'06 Proceedings of the 31st international conference on Mathematical Foundations of Computer Science
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Monadic second order (MSO) logic has proved to be a useful tool in many areas of application, reaching flom decidability and complexity to picture processing, correctness of programs and parallel processes. To characterize the structural borderline between decidability and undecidability is a classical research problem here. This problem is related to questions in computational complexity, especially to the model checking problem, for which many tools developed in the area of decidability have proved to be useful For more than two decades it was conjectured in [D. Seese, The structure of the models of decidable monadic theories of graphs, Ann. Pure Appl. Logic 53 (1991) 169-195] that decidability of monadic theories of countable structures implies that the theory can be reduced via interpretability to a theory of trees.It is one of the main goals of this article to prove a variant of this conjecture for matroids representable over a finite field. (Matroids can be viewed as a wide generalization of graphs, and they seem to capture some second order properties in a more suitable way than graphs themselves, cf. the recent development in matroid structure theory [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; J.F. Geelen, A.H.M. Gerards, N. Robertson, G.P. Whittle, Excluding a planar graph from a GF(q)-representable matroid, manuscript, 2003].) More exactly we prove, for every finite field F, that any class of F-representable matroids with a decidable MSO theory must have uniformly bounded branch-width. Moreover, we show that bounding the branch-width of all matroids in general is not sufficient to obtain a decidable MSO theory.Our paper gives a (rather detailed) introduction to these different subjects, and shows that a blend of ideas and methods from logic together with structural matroid theory can lead to new tools and algorithms, and can shed light on some old open problems.