The model-checking problem for monadic second-order logic on graphs is fixed-parameter tractable with respect to tree-width and clique-width. The proof constructs finite deterministic automata from monadic second-order sentences, but this computation produces automata of hyper-exponential sizes, and this is not avoidable. To overcome this difficulty, we propose to consider particular monadic second-order graph properties that are nevertheless interesting for Graph Theory and to interpret automata instead of trying to compile them (joint work with I. Durand).

For checking monadic second-order sentences written with edge set quantifications, the appropriate parameter is tree-width. We introduce special tree-width, a graph complexity measure between path-width and tree-width. The corresponding automata are easier to construct than those for tree-width.