A strong geodetic set of a graph~G=(V,E) is a vertex set~S⊆V(G)
in which it is possible to cover all the remaining vertices of~V(G)∖S by assigning a unique shortest path between each vertex pair of~S. In the
Strong Geodetic problem (SG) a graph~G and a positive integer~k are given
as input and one has to decide whether~G has a strong geodetic set of
cardinality at most~k. This problem is known to be NP-hard for general
graphs. In this work we introduce the Strong Geodetic Recognition problem
(SGR), which consists in determining whether even a given vertex set~S⊆V(G) is strong geodetic. We demonstrate that this version is
NP-complete. We investigate and compare the computational complexity of both
decision problems restricted to some graph classes, deriving polynomial-time
algorithms, NP-completeness proofs, and initial parameterized complexity
results, including an answer to an open question in the literature for the
complexity of SG for chordal graphs