Strong geodetic problem in grid like architectures

A recent variation of the classical geodetic problem, the strong geodetic problem, is defined as follows. If G is a graph, then sg(G) is the cardinality of a smallest vertex subset S, such that one can assign a fixed geodesic to each pair {x, y} ⊆ S so that these (|S| 2 ) geodesics cover all the vertices of G. In this paper, the strong geodetic problem is studied on Cartesian product graphs. A general upper bound is proved on the Cartesian product of a path with an arbitrary graph and showed that the bound is tight on flat grids and flat cylinders.