Disconnected perfect matching and perfect matching cut at radius three

Determine the computational complexity of disconnected perfect matching and perfect matching cut for bipartite graphs of radius 3.

Background

The paper establishes polynomial-time solvability for the relevant matching-cut variants on bipartite graphs of radius at most 2 and hardness results for radius at least 4. Consequently, the radius-3 boundary case remains unresolved for disconnected perfect matching and perfect matching cut. Resolving these cases would complete the corresponding radius dichotomies for bipartite graphs.

References

Therefore, only the following two cases remain open. What is the computational complexity of and for bipartite graphs of radius~$3$?

— Matching Cut and Variants on Bipartite Graphs of Bounded Radius and Diameter  (2501.08735 - Lucke, 15 Jan 2025) in Section 6, Conclusion, first Open Problem