Flipping Plane Spanning Paths

Abstract

Let S be a planar point set in general position, and let P(S) be the set of all plane (straight-line) spanning paths for S. A flip in a path P ∈ P(S) is the operation of removing an edge e ∈ P and replacing it with a new edge f on S such that the resulting graph is again a path in P(S). Towards the question whether any two plane spanning paths of P(S) can be transformed into each other by a sequence of flips, we give positive answers if S is a wheel set, an ice cream cone, or a double chain. On the other hand, we show that in the general setting, it is sufficient to prove the statement for plane spanning paths with fixed first edge.