$t$-spanners are used to approximate the pairwise distances between a set of points in a metric space. They have only a few edges compared to the total number of pairs and they provide a $t$-approximation on the distance of any two arbitrary points. There are many ways to construct such graphs and one of the most efficient ones, in terms of weight and the number of edges of the resulting graph, is the greedy spanner. In this paper, we study the edge crossings of the greedy spanner for points in the Euclidean plane. We prove a constant upper bound for the number of intersections with larger edges that only depends on the stretch factor of the spanner, $t$, and we show there can be more than a bounded number of intersections with smaller edges. Our results imply that greedy spanners for points in the plane have separators of size $\mathcal{O}(\sqrt n)$, that their planarizations have linear size, and that a separator hierarchy for these graphs can be constructed from their planarizations in linear time.

翻译：$t- spanner 用于估计某一组点在度空间中的对称距离。 它们与双对的总数相比只有几个边缘, 并且对任意点的距离提供美元- 约率。 在重量和结果图边缘数方面, 有多种方法可以构建这样的图形, 其中一种效率最高的方法是贪婪的射线员。 在本文中, 我们研究贪婪的射线员在 Euclidean 平面中点的边缘交叉点。 事实证明, 与较大边缘的交叉点数相比, 我们有一个恒定的上限, 这些交叉点的数量仅取决于 划线员的伸缩因数 $t$, 我们发现, 与较小边缘相交点相交的连接数可能不止。 我们的结果表明, 平面点的贪婪的穿肩器有 $\ mathcal{O} ( sqrt n) 大小的分隔器, 它们的平面尺寸是线性大小, 以及这些图形的分隔等级可以从其直线时间构建。