eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2022-06-01
30:1
30:17
10.4230/LIPIcs.SoCG.2022.30
article
Hop-Spanners for Geometric Intersection Graphs
Conroy, Jonathan B.
1
Tóth, Csaba D.
2
1
https://orcid.org/0000-0002-8769-3190
Department of Computer Science, Tufts University, Medford, MA, USA
Department of Mathematics, California State University Northridge, Los Angeles, CA, USA
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.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol224-socg2022/LIPIcs.SoCG.2022.30/LIPIcs.SoCG.2022.30.pdf
geometric intersection graph
unit disk graph
hop-spanner