  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.
