Complexity of computing completely mobile metric dimension

Determine the computational complexity of deciding whether a graph has completely mobile metric dimension k, in particular whether this decision problem belongs to NP.

Background

The completely mobile metric dimension is the minimum number of robots in a completely mobile resolving set. Verifying a proposed value requires accounting for resolving configurations and sequences of legal moves that allow every robot to visit every vertex.

The paper explicitly notes that determining whether a graph G satisfies the equality completely mobile metric dimension k can be done in polynomial time, but leaves the broader complexity classification unresolved, including the question of membership in NP.

References

As in many graph theoretical investigations, it is of interest to consider the complexity of computing the completely mobile metric dimension of graphs. For instance, is this problem in NP?

— Completely mobile robots in graphs with a resolving property  (2610.13339 - Moreno et al., 9 Oct 2026) in Section 6, “Open questions”