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Fix an integer h>=1. In the universe of coloured trees of height at most h,we prove that for any graph decision problem defined by an MSO formula with rquantifiers, there exists a set of kernels, each of size bounded by anelementary function of r and the number of colours. This yields two noteworthyconsequences. Consider any graph class G having a one-dimensional MSOinterpretation in the universe of coloured trees of height h (equivalently, Gis a class of shrub-depth h). First, class G admits an MSO model checkingalgorithm whose runtime has an elementary dependence on the formula size.Second, on G the expressive powers of FO and MSO coincide (which extends a 2012result of Elberfeld, Grohe, and Tantau).

Kernelizing MSO Properties of Trees of Fixed Height, and Some Consequences