Tree-depth, quantifier elimination, and quantifier rank

For a class K of finite graphs we consider the following three statements. (i) K has bounded tree-depth. (ii) First-order logic FO has an effective generalized quantifier elimination on K. (iii) The parameterized model checking for FO on K is in para-AC0. We prove that (i) ⟹ (ii) and (ii) ⟺ (iii). All three statements are equivalent if K is closed under taking subgraphs, but not in general. By a result due to Elberfeld et al. [12] monadic second-order logic MSO and FO have the same expressive power on every class of graphs of bounded tree-depth. Hence the implication (i) ⟹ (iii) holds for MSO, too; it is the analogue of Courcelle's Theorem for tree-depth (instead of tree-width) and para-AC0 (instead of FPT). In [13] it was already shown that the model-checking for a fixed MSO-property on a class of graphs of bounded tree-depth is in AC0.