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doi:10.1016/S0304-3975(02)00449-8 
Copyright © 2002 Elsevier Science B.V. All rights reserved.
Tree-width and the monadic quantifier hierarchy
Available online 14 January 2003.
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Abstract
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.
Author Keywords: Parametrized complexity; Polynomial hierarchy; Monadic second-order logic; Tree-width
