Characterization of complete skew token sliding graphs

Characterize all graphs whose skew token sliding reconfiguration graph of minimum skew forcing sets is isomorphic to a complete graph.

Background

The paper characterizes complete token exchange reconfiguration graphs for skew forcing in some cases and gives examples of graph families whose skew token exchange graphs are complete. For token sliding, however, it notes that complete graphs can arise and provides an infinite construction, but does not identify all source graphs with this property. Thus, a general characterization of such source graphs is explicitly absent.

References

In the token sliding case there are ways to construct graphs $G$ such that $Z_-(G) \cong K_m$ for some $m$, but no characterization is currently known.

— Reconfiguration of Minimum PSD Forcing Sets and Minimum Skew Forcing Sets  (2501.03642 - Bong et al., 7 Jan 2025) in Section 7, subsection “Complete Graphs”