General classification of K-geodetic graphs

Determine the general classification of K-geodetic graphs for each of the classes K = 1, 2, and 3, corresponding respectively to geodetic, bigeodetic, and trigeodetic graphs.

Background

K-geodetic graphs are graphs in which every pair of nonadjacent vertices has at most K shortest paths between them. The cases K = 1, 2, and 3 are called geodetic, bigeodetic, and trigeodetic, respectively.

The paper identifies characterization, construction, enumeration, and classification as central problems in the study of these graph classes. Although the survey develops constructions based on balanced incomplete block designs, it explicitly states that a general classification for each of the three classes remains unresolved.