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We present new results on 2- and 3-hop spanners for geometric intersection graphs. These include improved upper and lower bounds for 2- and 3-hop spanners for many geometric intersection graphs in ℝ^d. For example, we show that the intersection graph of n balls in ℝ^d admits a 2-hop spanner of size O^*(n^{3/2 - 1/(2(2⌊d/2⌋ + 1))}) and the intersection graph of n fat axis-parallel boxes in ℝ^d admits a 2-hop spanner of size O(n log^{d+1}n). Furthermore, we show that the intersection graph of general semi-algebraic objects in ℝ^d admits a 3-hop spanner of size O^*(n^{3/2 - 1/(2(2D-1))}), where D is a parameter associated with the description complexity of the objects. For such families (or more specifically, for tetrahedra in ℝ³), we provide a lower bound of Ω(n^{4/3}). For 3-hop and axis-parallel boxes in ℝ^d, we provide the upper bound O(n log ^{d-1}n) and lower bound Ω(n ({log n}/{log log n})^{d-2}). 

We present the first fully dynamic connectivity data structures for geometric intersection graphs achieving constant query time and sublinear amortized update time for many classes of geometric objects in 2D . Our data structures can answer connectivity queries between two objects, as well as "global" connectivity queries (e.g., deciding whether the entire graph is connected). Previously, the data structure by Afshani and Chan (ESA'06) achieved such bounds only in the special case of axis-aligned line segments or rectangles but did not work for arbitrary line segments or disks, whereas the data structures by Chan, Pătraşcu, and Roditty (FOCS'08) worked for more general classes of geometric objects but required n^{Ω(1)} query time and could not handle global connectivity queries. Specifically, we obtain new data structures with O(1) query time and amortized update time near n^{4/5}, n^{7/8}, and n^{20/21} for axis-aligned line segments, disks, and arbitrary line segments respectively. Besides greatly reducing the query time, our data structures also improve the previous update times for axis-aligned line segments by Afshani and Chan (from near n^{10/11} to n^{4/5}) and for disks by Chan, Pătraşcu, and Roditty (from near n^{20/21} to n^{7/8}). 

In SoCG 2022, Conroy and Tóth presented several constructions of sparse, low-hop spanners in geometric intersection graphs, including an O(nlog n)-size 3-hop spanner for n disks (or fat convex objects) in the plane, and an O(nlog² n)-size 3-hop spanner for n axis-aligned rectangles in the plane. Their work left open two major questions: (i) can the size be made closer to linear by allowing larger constant stretch? and (ii) can near-linear size be achieved for more general classes of intersection graphs? We address both questions simultaneously, by presenting new constructions of constant-hop spanners that have almost linear size and that hold for a much larger class of intersection graphs. More precisely, we prove the existence of an O(1)-hop spanner for arbitrary string graphs with O(nα_k(n)) size for any constant k, where α_k(n) denotes the k-th function in the inverse Ackermann hierarchy. We similarly prove the existence of an O(1)-hop spanner for intersection graphs of d-dimensional fat objects with O(nα_k(n)) size for any constant k and d. We also improve on some of Conroy and Tóth’s specific previous results, in either the number of hops or the size: we describe an O(nlog n)-size 2-hop spanner for disks (or more generally objects with linear union complexity) in the plane, and an O(nlog n)-size 3-hop spanner for axis-aligned rectangles in the plane. Our proofs are all simple, using separator theorems, recursion, shifted quadtrees, and shallow cuttings.