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DOI: 10.4230/LIPIcs.SoCG.2021.3

Classifying Convex Bodies by Their Contact and Intersection Graphs

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

Published in: LIPIcs, Volume 189, 37th International Symposium on Computational Geometry (SoCG 2021)

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.

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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)

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@InProceedings{aamand_et_al:LIPIcs.SoCG.2021.3,
  author =	{Aamand, Anders and Abrahamsen, Mikkel and Knudsen, Jakob B{\ae}k Tejs and Rasmussen, Peter Michael Reichstein},
  title =	{{Classifying Convex Bodies by Their Contact and Intersection Graphs}},
  booktitle =	{37th International Symposium on Computational Geometry (SoCG 2021)},
  pages =	{3:1--3:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-184-9},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{189},
  editor =	{Buchin, Kevin and Colin de Verdi\`{e}re, \'{E}ric},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2021.3},
  URN =		{urn:nbn:de:0030-drops-138024},
  doi =		{10.4230/LIPIcs.SoCG.2021.3},
  annote =	{Keywords: convex body, contact graph, intersection graph}
}

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Documents authored by Knudsen, Jakob Bæk Tejs

Classifying Convex Bodies by Their Contact and Intersection Graphs

Abstract

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