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Classifying Convex Bodies by Their Contact and Intersection Graphs

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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.

BibTeX - Entry

```@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},
URL =		{https://drops.dagstuhl.de/opus/volltexte/2021/13802},
URN =		{urn:nbn:de:0030-drops-138024},
doi =		{10.4230/LIPIcs.SoCG.2021.3},
annote =	{Keywords: convex body, contact graph, intersection graph}
}```

 Keywords: convex body, contact graph, intersection graph Seminar: 37th International Symposium on Computational Geometry (SoCG 2021) Issue date: 2021 Date of publication: 02.06.2021

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