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When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.SoCG.2021.3
URN: urn:nbn:de:0030-drops-138024
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2021/13802/
Aamand, Anders ;
Abrahamsen, Mikkel ;
Knudsen, Jakob Bæk Tejs ;
Rasmussen, Peter Michael Reichstein
Classifying Convex Bodies by Their Contact and Intersection Graphs
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},
address = {Dagstuhl, Germany},
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}
}
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Keywords: |
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convex body, contact graph, intersection graph |
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Collection: |
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37th International Symposium on Computational Geometry (SoCG 2021) |
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Issue Date: |
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2021 |
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Date of publication: |
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02.06.2021 |
