Flipping Plane Spanning Paths

Abstract

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

This work was initiated at the 2nd Austrian Computational Geometry Reunion Workshop in Strobl, June 2021. We thank all participants for fruitful discussions. J.O. is supported by ERC StG 757609. O.A. and R.P. are supported by FWF grant W1230. B.V. is supported by FWF Project I 3340-N35. K.K. is supported by the German Science Foundation (DFG) within the research training group ‘Facets of Complexity’ (GRK 2434). W.M. is partially supported by the German Research Foundation within the collaborative DACH project Arrangements and Drawings as DFG Project MU 3501/3-1, and by ERC StG 757609.