Drawing Shortest Paths in Geodetic Graphs

Sabine Cornelsen,Maximilian Pfister,Thomas Schneck,Martin Gronemann,Henry Förster,Stephen Kobourov,Michael Hoffmann

doi:10.1007/978-3-030-68766-3_26

Sabine Cornelsen, Maximilian Pfister + Show 5 more

Open Access

https://doi.org/10.1007/978-3-030-68766-3_26

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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.

Highlights

Greg Bodwin [1] examined the structure of shortest paths in graphs with edge weights that guarantee that the shortest path between any pair of vertices is unique
We show that there exist geodetic graphs that require some pair of shortest paths to meet at least four times (Theorem 1)
The bound on the number of crossings is tight because any uniformly subdivided Kn can be drawn so that every pair of shortest paths meets at most four times (Theorem 2)

Summary

Introduction

Greg Bodwin [1] examined the structure of shortest paths in graphs with edge weights that guarantee that the shortest path between any pair of vertices is unique. A drawing of a graph G in R2 maps the vertices to pairwise distinct points and maps each edge to a Jordan arc between the two end-vertices that is disjoint from any other vertex. A drawing φ of a graph G is philogeodetic if for every pair P1, P2 of shortest paths in G the curves φ(P1) and φ(P2) meet at most once. An unweighted graph is geodetic if there is a unique shortest path between every pair of vertices. A planar drawing of a planar geodetic graph is philogeodetic. It is a natural question to ask whether every (geodetic) graph admits a philogeodetic drawing

Results

Subdivision of a Complete Graph

Graphs of Diameter Two

A Proof of Theorem 2