Hop-Spanners for Geometric Intersection Graphs
Jonathan B. Conroy, Csaba D. T\'oth

TL;DR
This paper investigates the existence and construction of small 2-hop and 3-hop spanners in geometric intersection graphs, improving bounds and establishing tightness for various classes of geometric objects in the plane.
Contribution
It provides new bounds for the size of 2-hop and 3-hop spanners in geometric intersection graphs, including tight bounds for unit disk graphs and fat rectangles.
Findings
Unit disk graphs have 2-hop spanners with O(n) edges.
Fat rectangle intersection graphs have 2-hop spanners with O(n log n) edges, tight up to log log n.
Fat convex bodies and rectangles have 3-hop spanners with O(n log n) and O(n log^2 n) edges, respectively.
Abstract
A -spanner of a graph is a subgraph that contains a -path of length at most for every . It is known that every -vertex graph admits a -spanner with edges for . This bound is the best possible for and is conjectured to be optimal due to Erd\H{o}s' girth conjecture. We study -spanners for for geometric intersection graphs in the plane. These spanners are also known as \emph{-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 -vertex unit disk graph (UDG) admits a 2-hop spanner with edges; improving upon the previous bound of . (2) The intersection graph of axis-aligned fat rectangles admits a 2-hop spanner with…
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