Flipping Plane Spanning Paths

Let S be a planar point set in general position, and let $$\mathcal {P}(S)$$ the of all plane straight-line paths with vertex S. A flip on path $$P \in \mathcal is operation replacing an edge e P another f to obtain new valid from . It long-standing open question whether for every given S, can transformed into any other by sequence flips. To achieve better understanding this question, we show that it sufficient prove statement spanning whose first fixed. Furthermore, provide positive answers special classes sets, namely, wheel sets generalized double circles (which include, e.g., chains circles).