2026-07-23T18:08:49Z https://drops.dagstuhl.de/oai

oai:drops-oai.dagstuhl.de:23178 2025-10-02T12:42:04Z ddc:004 open_access

Geometric Spanners of Bounded Tree-Width Buchin, Kevin Rehs, Carolin Scheele, Torben Computational Geometry Geometric Spanner Tree-width Given a point set P in the Euclidean space, a geometric t-spanner G is a graph on P such that for every pair of points, the shortest path in G between those points is at most a factor t longer than the Euclidean distance between those points. The value t ≥ 1 is called the dilation of G. Commonly, the aim is to construct a t-spanner with additional desirable properties. In graph theory, a powerful tool to admit efficient algorithms is bounded tree-width. We therefore investigate the problem of computing geometric spanners with bounded tree-width and small dilation t. Let d be a fixed integer and P ⊂ ℝ^d be a point set with n points. We give a first algorithm to compute an 𝒪(n/k^{d/(d-1)})-spanner on P with tree-width at most k. The dilation obtained by the algorithm is asymptotically worst-case optimal for graphs with tree-width k: We show that there is a set of n points such that every spanner of tree-width k has dilation 𝒪(n/k^{d/(d-1)}). We further prove a tight dependency between tree-width and the number of edges in sparse connected planar graphs, which admits, for point sets in ℝ², a plane spanner with tree-width at most k and small maximum vertex degree. Finally, we show an almost tight bound on the minimum dilation of a spanning tree of n equally spaced points on a circle. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Kevin Buchin and Carolin Rehs and Torben Scheele 2025 Is Part Of LIPIcs, Volume 332, 41st International Symposium on Computational Geometry (SoCG 2025) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.SoCG.2025.26 urn:nbn:de:0030-drops-231786 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2025.26 eng https://creativecommons.org/licenses/by/4.0/legalcode