Geodetic Graphs: Experiments and New Constructions

TL;DR

This paper introduces a program to enumerate all geodetic graphs up to a certain size, leading to new classifications and the discovery of two infinite families of such graphs.

Contribution

It presents a novel enumeration method for geodetic graphs and identifies two new infinite families, advancing understanding of their structure.

Findings

Enumerated all geodetic graphs with up to 25 vertices.

Found all regular geodetic graphs with up to 32 vertices.

Discovered two new infinite families of geodetic graphs.

Abstract

In 1962 Ore initiated the study of geodetic graphs. A graph is called geodetic if the shortest path between every pair of vertices is unique. In the subsequent years a wide range of papers appeared investigating their peculiar properties. Yet, a complete classification of geodetic graphs is out of reach. In this work we present a program enumerating all geodetic graphs of a given size. Using our program, we succeed to find all geodetic graphs with up to 25 vertices and all regular geodetic graphs with up to 32 vertices. This leads to the discovery of two new infinite families of geodetic graphs.