Title: FO model checking of geometric graphs,
Abstract: 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 Gajarský et
al. for FO model checking of posets of bounded width. We also complement the
tractability results by related hardness reductions.