Recent characterization [9] of those graphs for which coloured MSO2 model checking is fast raised the interest in the graph invariant called tree-depth. Looking for a similar characterization for (coloured) MSO1, we introduce the notion of shrub-depth of a graph class. To prove that MSO1 model checking is fast for classes of bounded shrub-depth, we show that shrub-depth exactly characterizes the graph classes having interpretation in coloured trees of bounded height. We also introduce a common extension of cographs and of graphs with bounded shrubdepth-m-partite cographs (still of bounded clique-width), which are well quasi-ordered by the relation "is an induced subgraph of" and therefore allow polynomial time testing of hereditary properties.

We show that the model checking problem for µ-calculus on graphs of bounded tree-width can be solved in time linear in the size of the system. The result is presented by first showing a related result: the winner in a parity game on a graph of bounded tree-width can be decided in polynomial time. The given algorithm is then modified to obtain a new algorithm for µ-calculus model checking. One possible use of this algorithm may be software verification, since control flow graphs of programs written in high-level languages are usually of bounded treewidth. Finally, we discuss some implications and future work.

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We prove that graph problems with nite integer index have linear kernels on graphs of bounded expansion when parameterized by the size of a modulator to constant-treedepth graphs. For nowhere dense graph classes, our result yields almost-linear kernels. We also argue that such a linear kernelization result with a weaker parameter would fail to include some of the problems covered by our framework. We only require the problems to have FII on graphs of constant treedepth. This allows to prove linear kernels also for problems such as LongestPath/Cycle, Exact-s, t-Path, Treewidth, and Pathwidth, which do not have FII on general graphs.

Abstract. We study the computational complexity of the FO model checking problem on interval graphs, i.e., intersection graphs of intervals on the real line. The main positive result is that FO model checking and successor-invariant FO model checking can be solved in time O(n log n) for n-vertex interval graphs with representations containing only intervals with lengths from a prescribed finite set. We complement this result by showing that the same is not true if the lengths are restricted to any set that is dense in an open subset, e.g. in the set (1, 1 + ε).