The monadic second-order logic of graphs. I. recognizable sets of finite graphs
Information and Computation
Easy problems for tree-decomposable graphs
Journal of Algorithms
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
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
Discrete Applied Mathematics - Special issue on international workshop of graph-theoretic concepts in computer science WG'98 conference selected papers
When is the evaluation of conjunctive queries tractable?
STOC '01 Proceedings of the thirty-third annual ACM symposium on Theory of computing
Hyperedge Replacement: Grammars and Languages
Hyperedge Replacement: Grammars and Languages
Computers and Intractability: A Guide to the Theory of NP-Completeness
Computers and Intractability: A Guide to the Theory of NP-Completeness
Dynamic Programming on Graphs with Bounded Treewidth
ICALP '88 Proceedings of the 15th International Colloquium on Automata, Languages and Programming
Polynomial Time Recognition of Clique-Width \le \leq 3 Graphs (Extended Abstract)
LATIN '00 Proceedings of the 4th Latin American Symposium on Theoretical Informatics
The Monadic Quantifier Alternation Hierarchy over Grids and Pictures
CSL '97 Selected Papers from the11th International Workshop on Computer Science Logic
Back and Forth between Guarded and Modal Logics
LICS '00 Proceedings of the 15th Annual IEEE Symposium on Logic in Computer Science
Counting truth assignments of formulas of bounded tree-width or clique-width
Discrete Applied Mathematics
Lower bounds on the complexity of MSO1 model-checking
Journal of Computer and System Sciences
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It is well known that on classes of graphs of bounded tree-width, every monadic second-order property is decidable in polynomial time. The converse is not true without further assumptions. It follows from the work of Robertson and Seymour, that if a class of graphs K has unbounded tree-width and is closed under minors, then K contains all planar graphs. But on planar graphs, three-colorability is NP-complete. Hence, if P ≠ NP and on K every existential monadic second-order property is in P, then K has bounded tree-width. In other words, for K closed under minors, K is of bounded tree-width iff all monadic second-order properties are decidable in P.In this note we prove that in order to characterize classes of graphs of bounded tree-width where the monadic quantifier hierarchy collapses, closure under minors can be replaced by closure under topological minors. Closure under minors of K implies that K is in P, whereas we also note that there is a class of graphs K closed under topological minors which is not even r.e.We also show, that closure under induced subgraphs or even under subgraphs alone does not suffice to show that the collapse of the monadic quantifier hierarchy on K implies that K is of bounded tree-width or clique-width.Other characterizations of classes of bounded tree-width in terms of collapses of the monadic quantifier hierarchy to levels above the existential are discussed.