Sparse hop spanners for unit disk graphs

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