Isometric path (like geodesic) is yet another name for shortest path. An isometric path cover is a set $S$ of isometric paths such that every vertex $v ∈ V$ belongs to at least one isometric path of $S$. The isometric path cover problem is to find a minimum cardinality isometric path cover of $G$. Is the isometric path cover problem NP-complete for general graphs?
This question is mentioned in a recent (2017) paper, Strong geodetic problem in networks: computational complexity and solution for Apollonian networks by Paul Manuel, Sandi Klavžar, Antony Xavier, Andrew Arokiaraj, and Elizabeth Thomas. On page 8, they state:
To our knowledge, the complexity status of the isometric path problem is not known.
The isometric problem is important because of the following reasons: The path cover problem is NP-complete. The induced path-cover problem is NP-compete. The status of the isometric path cover problem is unknown.