On graphs with unique geoodesics and antipodes

TL;DR

This paper systematically studies geodetic and antipodal graphs, providing conditions, classifications, and constructions for these special graph families, advancing understanding of their properties and relationships.

Contribution

It offers necessary and sufficient conditions, classifications, and construction methods for geodetic and antipodal graphs, including Hamiltonian subclasses.

Findings

Characterization of geodetic and antipodal graphs

Construction algorithms for these graphs

Identification of Hamiltonian geodetic graph families

Abstract

In 1962, Oystein Ore asked in which graphs there is exactly one geodesic between any two vertices. He called such graphs geodetic. In this paper, we systematically study properties of geodetic graphs, and also consider antipodal graphs, in which each vertex has exactly one antipode (a farthest vertex). We find necessary and sufficient conditions for a graph to be geodetic or antipodal, obtain results related to algorithmic construction, and find interesting families of Hamiltonian geodetic graphs. By introducing and describing the maximal hereditary subclasses and the minimal hereditary superclasses of the geodetic and antipodal graphs, we get close to the goal of our research -- a constructive classification of these graphs.