Cyclic graphs matching peripherality and pp-increasing rank

Construct an infinite family of cyclic directed graphs $(G_k)_{k\in\mathbb{N}}$ such that each $G_k$ has peripherality $k$ and is $k$-orderable by increasing length, or prove that no such family exists.

Background

For cyclic graphs, the paper proves that the pp-increasing-rank is bounded below by peripherality and above by a constant multiple of peripherality. The unresolved issue is whether the lower bound can be attained by an infinite family. The authors believe that the pp-increasing-rank of every cyclic graph may instead be at least its peripherality plus one.

References

Find a family of cyclic graphs~$(G_k)_{k\inN}$ where~$G_k$ has a peripherality of~$k$ and is $k$-orderable by increasing length, or show that there is none.

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

We leave this gap open, as we do not know whether there is a family of cyclic graphs in which each graph has a pp-increasing-rank equal to its peripherality.

— Graphs with Bounded Push-Pop Walk Orderings  (2610.19724 - Amarilli et al., 14 Oct 2026) in Section 5, subsection “Lower Bound Tightness”