Algorithms for exact graph ranks

Develop an algorithm that computes at least one of the following quantities for an input directed graph: its pp-rank, its pp-increasing-rank, or its eventual pp-increasing-rank.

Background

The paper provides polynomial-time algorithms for peripherality and criticality and proves NP-hardness for deciding whether the pp-increasing-rank is at most 4. However, it gives no complexity upper bound, not even a decidability result, for computing the exact pp-rank, pp-increasing-rank, or eventual pp-increasing-rank. The problem is therefore substantially unresolved beyond the known computable upper bounds.

References

Indeed, we were able to show only one complexity lower bound (for the pp-increasing-rank) and no complexity upper bound at all (not even a decidability bound).

— Graphs with Bounded Push-Pop Walk Orderings  (2610.19724 - Amarilli et al., 14 Oct 2026) in Appendix, Section “Open Problems,” subsection “Complexity of Computing Graph Ranks”

We do not know whether hardness generalizes to this parameter, and we also do not know complexity upper bounds: we will discuss this in the conclusion.

— Graphs with Bounded Push-Pop Walk Orderings  (2610.19724 - Amarilli et al., 14 Oct 2026) in Section 7, Section “Parameter Computation” near the end