The model-checking problem for a logic LLL is the problem

of decidig for a given formula phi in LLL and structure

AA whether the formula is true in the structure, i.e. whether

AA models phi.

Model-checking for logics such as First-Order Logic (FO) or

Monadic Second-Order Logic (MSO) has been studied intensively

in the literature, especially in the context of algorithmic meta-theorems

within the framework of parameterized complexity. However, in the past the

focus of this line of research was model-checking on classes of

sparse graphs, e.g. planar graphs, graph classes excluding a

minor or classes which are nowhere dense. By now, the complexity of

first-order model-checking on sparse classes of graphs is completely

understood. Hence, current research now focusses mainly on classes of

dense graphs.

In this talk we will briefly review the known results on sparse classes

of graphs and explain the complete classification of classes of sparse

graphs on which first-order model-checking is tractable.

In the second part we will then focus on recent and ongoing research

analysing the complexity of first-order model-checking on classes of

dense graphs.