Connectivity of the flip graph for non-crossing matchings under cycle-exchange flips

Determine whether the flip graph whose vertices are non-crossing matchings on a planar point set and whose flip operation consists of exchanging the edges in a cycle is connected.

Background

In surveying related reconfiguration problems on non-crossing straight-line structures, the paper notes that connectivity is known for triangulations and trees under their respective flip operations, with established diameter bounds. In contrast, for non-crossing matchings—where a flip exchanges the edges in a cycle—the connectivity status of the corresponding flip graph is unresolved.