2026-07-26T17:52:51Z https://drops.dagstuhl.de/oai

oai:drops-oai.dagstuhl.de:16038 2024-03-06T10:56:48Z ddc:004 open_access

Hop-Spanners for Geometric Intersection Graphs Conroy, Jonathan B. Tóth, Csaba D. geometric intersection graph unit disk graph hop-spanner 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. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Jonathan B. Conroy and Csaba D. Tóth 2022 Is Part Of LIPIcs, Volume 224, 38th International Symposium on Computational Geometry (SoCG 2022) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.SoCG.2022.30 urn:nbn:de:0030-drops-160381 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2022.30 eng https://creativecommons.org/licenses/by/4.0/legalcode