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Classifying Convex Bodies by Their Contact and Intersection Graphs

Authors Anders Aamand , Mikkel Abrahamsen , Jakob Bæk Tejs Knudsen , Peter Michael Reichstein Rasmussen

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Author Details

Anders Aamand
  • BARC, University of Copenhagen, Denmark
Mikkel Abrahamsen
  • BARC, University of Copenhagen, Denmark
Jakob Bæk Tejs Knudsen
  • BARC, University of Copenhagen, Denmark
Peter Michael Reichstein Rasmussen
  • BARC, University of Copenhagen, Denmark

Funding

The authors are part of BARC, Basic Algorithms Research Copenhagen, supported by the VILLUM Foundation grant 16582.

Acknowledgements

We thank Tillmann Miltzow for asking when the translates of two different convex bodies induce the same intersection graphs which inspired us to work on these problems.

Cite As Get BibTex

Anders Aamand, Mikkel Abrahamsen, Jakob Bæk Tejs Knudsen, and Peter Michael Reichstein Rasmussen. Classifying Convex Bodies by Their Contact and Intersection Graphs. In 37th International Symposium on Computational Geometry (SoCG 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 189, pp. 3:1-3:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021) https://doi.org/10.4230/LIPIcs.SoCG.2021.3

Abstract

Let A be a convex body in the plane and A₁,…,A_n be translates of A. Such translates give rise to an intersection graph of A, G = (V,E), with vertices V = {1,… ,n} and edges E = {uv∣ A_u ∩ A_v ≠ ∅}. The subgraph G' = (V, E') satisfying that E' ⊂ E is the set of edges uv for which the interiors of A_u and A_v are disjoint is a unit distance graph of A. If furthermore G' = G, i.e., if the interiors of A_u and A_v are disjoint whenever u≠ v, then G is a contact graph of A.
In this paper, we study which pairs of convex bodies have the same contact, unit distance, or intersection graphs. We say that two convex bodies A and B are equivalent if there exists a linear transformation B' of B such that for any slope, the longest line segments with that slope contained in A and B', respectively, are equally long. For a broad class of convex bodies, including all strictly convex bodies and linear transformations of regular polygons, we show that the contact graphs of A and B are the same if and only if A and B are equivalent. We prove the same statement for unit distance and intersection graphs.

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
  • Mathematics of computing → Graph theory
  • Mathematics of computing → Discrete mathematics
Keywords
  • convex body
  • contact graph
  • intersection graph

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References

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