Sparse hop spanners for unit disk graphs

A unit disk graph G on a given set P of points in the plane is a geometric graph where an edge exists between two points p , q ∈ P if and only if | p q | ≤ 1 . A spanning subgraph G ′ of G is a k-hop spanner if and only if for every edge p q ∈ G , there is a path between p , q in G ′ with at most k edges. We obtain the following results for unit disk graphs in the plane. (i) Every n-vertex unit disk graph has a 5-hop spanner with at most 5.5n edges. We analyze the family of spanners constructed by Biniaz (2020) and improve the upper bound on the number of edges from 9n to 5.5n. (ii) Using a new construction, we show that every n-vertex unit disk graph has a 3-hop spanner with at most 11n edges. (iii) Every n-vertex unit disk graph has a 2-hop spanner with O ( n log ⁡ n ) edges. This is the first nontrivial construction of 2-hop spanners. (iv) For every sufficiently large positive integer n, there exists a set P of n points on a circle, such that every plane hop spanner on P has hop stretch factor at least 4. Previously, no lower bound greater than 2 was known. (v) For every finite point set on a circle, there exists a plane (i.e., crossing-free) 4-hop spanner. As such, this provides a tight bound for points on a circle. (vi) The maximum degree of k-hop spanners cannot be bounded from above by a function of k for any positive integer k.