Drawing Shortest Paths in Geodetic Graphs

TL;DR

This paper investigates whether geodetic graphs can be drawn so that shortest paths meet at most once, and demonstrates that some require multiple crossings, showing limitations in such visualizations.

Contribution

It proves that certain geodetic graphs cannot be drawn with shortest paths intersecting at most once, answering a question posed by Greg Bodwin.

Findings

Existence of geodetic graphs requiring at least four crossings between shortest paths

Bound on crossings is tight for constructed graphs

Some diameter-two geodetic graphs cannot have philogeodetic drawings

Abstract

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.