Abstract
One of Courcelle's celebrated results states that if C is a class of graphs of bounded tree-width, then model-checking for monadic second order logic (MSO2) is fixed-parameter tractable (fpt) on C by linear time parameterised algorithms. An immediate question is whether this is best possible or whether the result can be extended to classes of unbounded tree-width. In this paper we show that in terms of tree-width, the theorem can not be extended much further. More specifically, we show that if C is a class of graphs which is closed under colourings and satisfies certain constructibility conditions such that the tree-width of C is not bounded by log16 n then MSO2-model checking is not fpt unless SAT can be solved in sub-exponential time. If the tree-width of C is not poly-log. bounded, then MSO2-model checking is not fpt unless all problems in the polynomial-time hierarchy can be solved in sub-exponential time.
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