Tightness of the rank upper bounds

Improve the established upper bounds for the pp-rank, pp-increasing-rank, and eventual pp-increasing-rank, or construct graph families demonstrating that those bounds are tight.

Background

The main theorem gives upper bounds of 3 for pp-rank, 6 times the relevant peripherality parameter for pp-increasing-rank, and 6 times the eventual peripherality parameter for eventual pp-increasing-rank. The authors explicitly state that these upper bounds have not been shown to be tight and leave open either improving them or matching them with examples.

References

Improve the upper bounds of \cref{t:pp-rank}, \cref{t:pp-rank:incr,t:pp-rank:eve-incr} or find a graph family showing they are tight.

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