FO model checking of geometric graphs

TL;DR

This paper investigates first-order model checking on dense geometric graph classes, providing new fixed-parameter tractable algorithms for certain subclasses and establishing hardness results for others.

Contribution

It introduces novel FPT algorithms for FO model checking on dense geometric graphs and identifies complexity boundaries with hardness reductions.

Findings

FPT algorithms for circular-arc, circle, box, disk, and polygon-visibility graphs

Hardness results for certain dense geometric graph classes

Extension of FO model checking techniques to dense structures

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 Gajarsk\'y et al. for FO model checking of posets of bounded width. We also complement the tractability results by related hardness reductions.