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

Authors Colin Geniet , Gunwoo Kim , Lucas Meijer

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LIPIcs.SoCG.2026.51.pdf
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Subject Classification

ACM Subject Classification
  • Theory of computation → Parameterized complexity and exact algorithms
  • Theory of computation → Finite Model Theory
  • Theory of computation → Computational geometry
  • Mathematics of computing → Graph theory
Keywords
  • Twin-width
  • axis-parallel unit segment graphs
  • circular arc graphs
  • terrain visibility graphs
  • first-order logic
  • model checking
  • FPT

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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.

Cite As Get BibTex

Colin Geniet, Gunwoo Kim, and Lucas Meijer. First-Order Logic and Twin-Width for Some Geometric Graphs. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 51:1-51:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.SoCG.2026.51

Author Details

Colin Geniet
  • Discrete Mathematics Group, Institute for Basic Science (IBS), Daejeon, South Korea
Gunwoo Kim
  • School of Computing, KAIST, Daejeon, South Korea
  • Discrete Mathematics Group, Institute for Basic Science (IBS), Daejeon, South Korea
Lucas Meijer
  • Department of Information and Computing Sciences, Utrecht University, The Netherlands

Funding

  • Geniet, Colin: Supported by the Institute for Basic Science (IBS-R029-C1).
  • Kim, Gunwoo: Supported by the Institute for Basic Science (IBS-R029-C1) and the National Research Foundation, Korea (RS-2025-00563533).
  • Meijer, Lucas: Supported by the Netherlands Organisation for Scientific Research (NWO) under project no. VI.Vidi.213.150.

Acknowledgements

The authors are extremely grateful to Eunjung Kim for suggesting the questions treated in this work and for very insightful discussions on these topics, as well as Sebastian Wiederrecht for stimulating discussions during the early stages of this project. We would also like to thank Édouard Bonnet for very helpful explanations regarding twin-width of segment graphs.

References

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