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arXiv · 2512.21896

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

Colin Geniet·Gunwoo Kim·Lucas Meijer
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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.

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Colin Geniet, Gunwoo Kim, Lucas Meijer. 2025-12-26. First-Order Logic and Twin-Width for Some Geometric Graphs. https://arxiv.org/abs/2512.21896

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