  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.
