Strong geodetic number of complete bipartite graphs and of graphs with specified diameter

The strong geodetic problem is a recent variation of the classical geodetic problem. For a graph $G$, its strong geodetic number ${\rm sg}(G)$ is the cardinality of a smallest vertex subset $S$, such that each vertex of $G$ lies on one fixed geodesic between a pair of vertices from $S$. In this paper, some general properties of the strong geodesic problem are studied, especially in connection with diameter of a graph. The problem is also solved for balanced complete bipartite graphs.