First-Order Logic and Twin-Width for Some Geometric Graphs

TL;DR

This paper explores the relationship between first-order logic tractability and twin-width in geometric graphs, establishing delineation results for certain classes and identifying cases where it fails.

Contribution

It extends the delineation characterization to new classes like axis-parallel unit segment graphs and circular arc graphs, and shows failure for visibility graphs of 1.5D terrains.

Findings

Delineation holds for intersection graphs of axis-parallel unit segments.

Delineation fails for visibility graphs of 1.5D terrains.

Delineation holds for intersection graphs of circular arcs.

Abstract

For some geometric graph classes, tractability of testing first-order formulas is precisely characterised by the graph parameter twin-width. This was first proved for interval graphs among others in [BCKKLT, IPEC '22], where the equivalence is called delineation, and more generally holds for circle graphs, rooted directed path graphs, and $H$-graphs when $H$ is a forest. Delineation is based on the key idea that geometric graphs often admit natural vertex orderings, allowing to use the very rich theory of twin-width for ordered graphs. Answering two questions raised in their work, we prove that delineation holds for intersection graphs of non-degenerate axis-parallel unit segment graphs, but fails for visibility graphs of 1.5D terrains. We also prove delineation for intersection graphs of circular arcs.