Realizability of trees under skew forcing

Prove or disprove that no tree is realizable as the token exchange reconfiguration graph of minimum skew forcing sets of any graph.

Background

The paper proves that the PSD token exchange graph cannot be a nontrivial tree and establishes connectedness properties for skew token exchange graphs when the source graph itself is a tree. It then raises a broader realizability conjecture for skew forcing: whether any graph can have a tree as its skew token exchange reconfiguration graph. The conjecture remains unresolved in the paper.

References

We also conjecture that trees are not realizable as $Z_-(G)$ for any graph $G$, but establishing this will require additional tools.

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