Strong-resolving equality characterization

Characterize the graphs for which the general position number equals the clique number of the strong resolving graph, that is, for which \(\gp(G)=\omega(G^{\mathrm{SR}})\).

Background

The survey gives the general inequality $\gp(G)\geq\omega(G^{\mathrm{SR}})$ and characterizes equality by the existence of a general-position set inducing a clique in the strong resolving graph. It notes that a structural characterization of all graphs satisfying this equality is not known because of the variety of possible graph structures.

References

Structural characterisation of graphs satisfying equality in Theorem~\ref{thm:gp-versus-strong-resolving-graphs} is not known and seems to be elusive because of the great variety of different structures that can appear.

— The General Position Problem: A Survey  (2501.19385 - V. et al., 31 Jan 2025) in Section 2, subsection “Strong resolving graphs”