Existence of graphs with pp-rank three

Construct a graph or an infinite family of graphs with pp-rank exactly 3, or prove that every graph has pp-rank at most 2.

Background

The paper proves that every inner-connected graph is 3-orderable, while its examples establish arbitrarily large cyclic graphs that are 1-orderable. It does not determine whether the upper bound of 3 is ever necessary. The authors suspect that some graphs are not 2-orderable, while also conjecturing that all trees, and possibly all acyclic graphs, may be 2-orderable.

References

Find a graph (or a family of graphs) with a pp-rank of 3, 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 “Upper Bounds”