Flipping Plane Spanning Paths

TL;DR

This paper investigates the connectivity of plane spanning paths on planar point sets via flips, proving the sufficiency of fixing the first edge and providing positive results for specific point set classes.

Contribution

It reduces the open problem to a special case and confirms connectivity for wheel sets and generalized double circles.

Findings

Connectivity of plane spanning paths via flips for certain point sets

Sufficiency of fixing the first edge in the flip sequence

Positive results for wheel sets and generalized double circles

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).