 We analyze the family of spanners constructed by Biniaz ( WADS 2019 ) 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.
<meaning-changed>  We analyze the family of spanners constructed by Biniaz ( WADS 2019 ) 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.
 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.