Vertex-position ratio

Determine whether the ratio \(\vp(G)/\vp^-(G)\) is bounded over connected graphs.

Background

The vertex position number and lower vertex position number measure the maximum and minimum sizes of position sets visible from a fixed vertex. A construction gives ratios approaching 6, but the survey states that it is unknown whether larger ratios are possible.

References

A family of graphs with ratio $\frac{\vp (G)}{\vp -(G)}$ approaching 6 is constructed, but whether this ratio can be larger is unknown.

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