  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).
