License: Creative Commons Attribution 4.0 International license (CC BY 4.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.SoCG.2022.30
URN: urn:nbn:de:0030-drops-160381
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2022/16038/
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Conroy, Jonathan B. ; Tóth, Csaba D.

Hop-Spanners for Geometric Intersection Graphs

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LIPIcs-SoCG-2022-30.pdf (0.9 MB)

Abstract

A t-spanner of a graph G = (V,E) is a subgraph H = (V,E') that contains a uv-path of length at most t for every uv ∈ E. It is known that every n-vertex graph admits a (2k-1)-spanner with O(n^{1+1/k}) edges for k ≥ 1. This bound is the best possible for 1 ≤ k ≤ 9 and is conjectured to be optimal due to Erdős' girth conjecture.
We study t-spanners for t ∈ {2,3} for geometric intersection graphs in the plane. These spanners are also known as t-hop spanners to emphasize the use of graph-theoretic distances (as opposed to Euclidean distances between the geometric objects or their centers). We obtain the following results: (1) Every n-vertex unit disk graph (UDG) admits a 2-hop spanner with O(n) edges; improving upon the previous bound of O(nlog n). (2) The intersection graph of n axis-aligned fat rectangles admits a 2-hop spanner with O(nlog n) edges, and this bound is the best possible. (3) The intersection graph of n fat convex bodies in the plane admits a 3-hop spanner with O(nlog n) edges. (4) The intersection graph of n axis-aligned rectangles admits a 3-hop spanner with O(nlog² n) edges.

BibTeX - Entry

@InProceedings{conroy_et_al:LIPIcs.SoCG.2022.30,
  author =	{Conroy, Jonathan B. and T\'{o}th, Csaba D.},
  title =	{{Hop-Spanners for Geometric Intersection Graphs}},
  booktitle =	{38th International Symposium on Computational Geometry (SoCG 2022)},
  pages =	{30:1--30:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-227-3},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{224},
  editor =	{Goaoc, Xavier and Kerber, Michael},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2022/16038},
  URN =		{urn:nbn:de:0030-drops-160381},
  doi =		{10.4230/LIPIcs.SoCG.2022.30},
  annote =	{Keywords: geometric intersection graph, unit disk graph, hop-spanner}
}

Keywords: geometric intersection graph, unit disk graph, hop-spanner
Collection: 38th International Symposium on Computational Geometry (SoCG 2022)
Issue Date: 2022
Date of publication: 01.06.2022

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