FO model checking on geometric graphs
Petr Hliněný, Filip Pokrývka, Bodhayan Roy · Computational Geometry · 2018
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 on 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 on dense classes of graphs or other structures. We study the FO model checking problem on 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 Gajarský et al. for FO model checking on posets of bounded width. We also complement the tractability results by related hardness reductions.