General d-position numbers of grids and trees

Determine the general \(d\)-position numbers of all grid graphs and determine the computational complexity of finding the general \(d\)-position number of trees for every \(d\).

Background

The general dd-position problem restricts attention to shortest paths of bounded length. Exact values and complexity results are known for some graph families and for trees when d=2d=2 or d=diam⁡(T)d=\operatorname{diam}(T), but the corresponding questions for arbitrary grids and arbitrary values of dd remain unresolved.

References

Two interesting questions left open in are to find the general $d$-position numbers of all grid graphs and to determine the complexity of finding the general $d$-position number of trees for any $d$ (it is known that it can be found in polynomial time for $d = 2$ or $d = \diam (T)$).

— The General Position Problem: A Survey  (2501.19385 - V. et al., 31 Jan 2025) in Section 3, subsection “d-position sets”