Geodetic Graphs: Geometry, Diameter and the Problem of Classification

A graph is called geodetic if there is a unique shortest path between any two vertices (i.e - a geodesic). Ore (’62) sought to characterize such graphs, and despite a considerable body of work on this problem - a characterization is surprisingly elusive. In this talk we introduce the idea of viewing graphs from a geometric perspective and survey its connections to the problem of classification. In particular, we review some known classifications and constructions of geodetic graphs, and show that such graphs must be of relatively high connectivity. This talk is based on joint work with Nati Linial.