Tree-width and the monadic quantifier hierarchy

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 i8 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 quanti"er 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 su;ce to show that the collapse of the monadic quanti"er 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 quanti"er hierarchy to levels above the existential are discussed. c © 2002 Elsevier Science B.V. All rights reserved.