Strong geodetic cores and Cartesian product graphs

The strong geodetic problem on a graph $G$ is to determine a smallest set of vertices such that by fixing one shortest path between each pair of its vertices, all vertices of $G$ are covered. To do this as efficiently as possible, strong geodetic cores and related numbers are introduced. Sharp upper and lower bounds on the strong geodetic core number are proved. Using the strong geodetic core number an earlier upper bound on the strong geodetic number of Cartesian products is improved. It is also proved that ${\rm sg}(G \,\square\, K_2) \geq {\rm sg}(G)$ holds for different families of graphs, a result conjectured to be true in general. Counterexamples are constructed demonstrating that the conjecture does not hold in general.