Generalized k-general d-position problems

Determine the \(k\)-general \(d\)-position numbers of further Cartesian products, determine the computational complexity of the \(k\)-general \(d\)-position problem, and characterize the corresponding \(k\)-general \(d\)-position sets.

Background

The kk-general dd-position problem forbids kk selected vertices from lying on a common shortest path of length at most dd. The survey reports results for paths, cycles, and thin grids, while identifying broader Cartesian products, complexity, and structural characterization as unresolved directions.

References

Finding the $k$-general $d$-position number of other Cartesian products is left as an open problem, along with finding the computation complexity of the $k$-general $d$-position problem and the problem of characterising the structure of such sets.

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