On the Connectivity and Diameter of Geodetic Graphs

TL;DR

This paper investigates the properties of geodetic graphs, proving that certain 2-connected graphs are actually 3-connected and constructing an infinite family with maximum known diameter.

Contribution

It demonstrates that all 2-connected, degree ≥3 geodetic graphs are 3-connected and constructs an infinite family with diameter 5.

Findings

All such graphs are 3-connected.

Constructed an infinite family with diameter 5.

Progress towards characterizing geodetic graphs.

Abstract

A graph $G$ is geodetic if between any two vertices there exists a unique shortest path. In 1962 Ore raised the challenge to characterize geodetic graphs, but despite many attempts, such characterization still seems well beyond reach. We may assume, of course, that $G$ is $2$-connected, and here we consider only graphs with no vertices of degree $1$ or $2$. We prove that all such graphs are, in fact $3$-connected. We also construct an infinite family of such graphs of the largest known diameter, namely $5$.