Twin-width of Planar Graphs; a Short Proof

The fascinating question of the maximum value of twin-width on planar graphs is nowadays not far from a final resolution; there is a lower bound of $7$ coming from a construction by Král‘ and Lamaison [arXiv, September 2022], and an upper bound of $8$ by Hliněný and Jedelský [arXiv, October 2022]. The upper bound (currently best) of $8$, however, is rather complicated and involved. We give a short and simple self-contained proof that the twin-width of planar graphs is at most $11$

Faster Deciding MSO Properties of Trees of Fixed Height, and Some Consequences

We prove, in the universe of trees of bounded height, that for any MSO formula with $m$ variables there exists a set of kernels such that the size of each of these kernels can be bounded by an elementary function of m. This yields a faster MSO model checking algorithm for trees of bounded height than the one for general trees. From that we obtain, by means of interpretation, corresponding results for the classes of graphs of bounded tree-depth (MSO_2) and shrub-depth (MSO_1), and thus we give wide generalizations of Lampis\u27 (ESA 2010) and Ganian\u27s (IPEC 2011) results. In the second part of the paper we use this kernel structure to show that FO has the same expressive power as MSO_1 on the graph classes of bounded shrub-depth. This makes bounded shrub-depth a good candidate for characterization of the hereditary classes of graphs on which FO and MSO_1 coincide, a problem recently posed by Elberfeld, Grohe, and Tantau (LICS 2012)

Stack and Queue Numbers of Graphs Revisited

A long-standing question of the mutual relation between the stack and queue numbers of a graph, explicitly emphasized by Dujmović and Wood in 2005, was ``half-answered‘‘ by Dujmović, Eppstein, Hickingbotham, Morin and Wood in 2022; they proved the existence of a graph family with the queue number at most $4$ but unbounded stack number. We give an alternative very short, and still elementary, proof of the same fact

FO Model Checking of Geometric Graphs

Over the past two decades the main focus of research into first-order (FO) model checking algorithms has been on sparse relational structures - culminating in the FPT algorithm by Grohe, Kreutzer and Siebertz for FO model checking of nowhere dense classes of graphs. On contrary to that, except the case of locally bounded clique-width only little is currently known about FO model checking of dense classes of graphs or other structures. We study the FO model checking problem for dense graph classes definable by geometric means (intersection and visibility graphs). We obtain new nontrivial FPT results, e.g., for restricted subclasses of circular-arc, circle, box, disk, and polygon-visibility graphs. These results use the FPT algorithm by Gajarsk\'y et al. for FO model checking of posets of bounded width. We also complement the tractability results by related hardness reductions

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

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 r quantifiers, there exists a set of kernels, each of size bounded by an elementary function of r and the number of colours. This yields two noteworthy consequences. Consider any graph class G having a one-dimensional MSO interpretation in the universe of coloured trees of height h (equivalently, G is a class of shrub-depth h). First, class G admits an MSO model checking algorithm whose runtime has an elementary dependence on the formula size. Second, on G the expressive powers of FO and MSO coincide (which extends a 2012 result of Elberfeld, Grohe, and Tantau)