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Hop-Spanners for Geometric Intersection Graphs

Authors Jonathan B. Conroy, Csaba D. Tóth



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Author Details

Jonathan B. Conroy
  • Department of Computer Science, Tufts University, Medford, MA, USA
Csaba D. Tóth
  • Department of Mathematics, California State University Northridge, Los Angeles, CA, USA
  • Department of Computer Science, Tufts University, Medford, MA, USA

Acknowledgements

We thank Sujoy Bhore for helpful discussions on geometric intersections graphs.

Cite AsGet BibTex

Jonathan B. Conroy and Csaba D. Tóth. Hop-Spanners for Geometric Intersection Graphs. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 30:1-30:17, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)
https://doi.org/10.4230/LIPIcs.SoCG.2022.30

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.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Discrete mathematics
  • Mathematics of computing → Paths and connectivity problems
  • Theory of computation → Computational geometry
Keywords
  • geometric intersection graph
  • unit disk graph
  • hop-spanner

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