Motivated by the fact that in a space where shortest paths are unique, no two shortest paths meet twice, we study a question posed by Greg Bodwin: Given a geodetic graph $G$, i.e., an unweighted graph in which the shortest path between any pair of vertices is unique, is there a philogeodetic drawing of $G$, i.e., a drawing of $G$ in which the curves of any two shortest paths meet at most once? We answer this question in the negative by showing the existence of geodetic graphs that require some pair of shortest paths to cross at least four times. The bound on the number of crossings is tight for the class of graphs we construct. Furthermore, we exhibit geodetic graphs of diameter two that do not admit a philogeodetic drawing.

中文翻译：

---

在大地测量图中绘制最短路径

由于在最短路径是唯一的空间中，没有两条最短路径相遇两次这一事实，我们研究了 Greg Bodwin 提出的问题：给定一个大地测量图 $G$，即一个未加权图，其中任意一条最短路径之间的最短路径顶点对是唯一的，是否存在$G$的哲学地理图，即任意两条最短路径的曲线最多相交一次的$G$图？我们通过显示需要一些最短路径至少四次交叉的大地测量图的存在来否定这个问题。对于我们构建的图类来说，交叉数量的界限是严格的。此外，我们还展示了直径为 2 的大地测量图，这些大地测量图不允许使用 philogeodetic 绘图。