  The strong geodetic problem is to find the smallest number of vertices such
that by fixing one shortest path between each pair, all vertices of the graph
are covered. In this paper we study the strong geodetic problem on complete
bipartite graphs; in particular, we discuss its asymptotic behavior. Some
results for complete multipartite graphs are also derived. Finally, we prove
that the strong geodetic problem restricted to (general) bipartite graphs is
NP-complete.
