Balanced incomplete block design and the construction of geodetic blocks (Q1178404)

Let \(G\) be an undirected, simple graph. If any two vertices are connected by a unique path of shortest length, then \(G\) is a geodetic graph. A geodetic block is a 2-connected geodetic graph. The author applies a method given by \textit{R. J. Cook} and \textit{D. G. Pryce} in [J. Graph Theory 6, 157-168 (1982; Zbl 0499.05037)], to block designs \((v,b,r,k,1)\) and thereby constructs geodetic blocks with a diameter of 4, or a diameter of 5 if there exists a pair of disjoint blocks, and where the degree of each vertex is either \(k\) or \(r\). The existence of projective and affine planes of every prime power order implies the existence of infinitely many geodetic blocks of diameters 4 and 5.