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 $p,q\in P$ if and only if $|pq|\le 1$. A spanning subgraph $G^{\prime }$ of G is a k-hop spanner if and only if for every edge $pq\in G$, there is a path between $p,q$ in $G^{\prime }$ 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.

平面中给定点集P上的单位圆盘图G是几何图，其中两点之间存在边$磷,q\in 磷$ 当且仅当 $|磷q|\le 1$. 一个跨越子图$G^{\prime }$的G是k 跳扳手当且仅当对于每条边$磷q\in G$, 之间有一条路径 $磷,q$ 在 $G^{\prime }$最多有k 个边。对于平面中的单位圆盘图，我们得到以下结果。

（一世）

每个n顶点单元磁盘图都有一个最多 5.5 n 条边的 5 跳生成器。我们分析了 Biniaz (2020) 构建的扳手家族，并将边数的上限从 9 n 提高到 5.5 n。

(二)

使用新的构造，我们表明每个n顶点单元磁盘图都有一个最多 11 n 条边的 3 跳生成器。

(三)

每个n顶点单元磁盘图都有一个 2-hop spanner$哦(n\mathrm{日志}n)$边缘。这是 2 跳扳手的第一个非平凡构造。

(四)

对于每一个足够大的正整数Ñ，存在一组P的Ñ点上的圆，以使得在每一个平面一跳扳手P具有一跳拉伸系数至少为4以前，没有下限大于2是众所周知的。

(五)

对于圆上的每个有限点集，都存在一个平面（即无交叉）4 跳扳手。因此，这为圆上的点提供了严格的界限。

(六)

对于任何正整数k，k跳生成器的最大程度不能由k的函数从上方限定。