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Computing the Girth of a Segment Intersection Graph

Authors Timothy M. Chan , Yuancheng Yu

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LIPIcs.SoCG.2026.30.pdf
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Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
Keywords
  • Geometric intersection graphs
  • girth
  • shortest paths
  • graph separators
  • matrix multiplication

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Abstract

We present an algorithm that computes the girth of the intersection graph of n given line segments in the plane in O(n^1.483) expected time. This is the first such algorithm with O(n^{3/2-ε}) running time for a positive constant ε, and makes progress towards an open question posed by Chan (SODA 2023). The main techniques include (i) the usage of recent subcubic algorithms for bounded-difference min-plus matrix multiplication, and (ii) an interesting variant of the planar graph separator theorem. The result extends to intersection graphs of connected algebraic curves or semialgebraic sets of constant description complexity.

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Timothy M. Chan and Yuancheng Yu. Computing the Girth of a Segment Intersection Graph. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 30:1-30:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.SoCG.2026.30

Author Details

Timothy M. Chan
  • Siebel School of Computing and Data Science, University of Illinois Urbana-Champaign, IL, USA
Yuancheng Yu
  • Siebel School of Computing and Data Science, University of Illinois Urbana-Champaign, IL, USA

Funding

  • Chan, Timothy M.: Work supported by NSF Grant CCF-2224271.

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