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 planar 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 provide positive answers for special classes of point sets, namely, for wheel sets, ice cream cones, double chains, and double circles. Moreover, we show for general point sets, it is sufficient to prove the statement for plane spanning paths whose first edge is fixed.
翻译:让 $S$ 成为总位置上设置的平面点, 让 $mathcal{P} (S) 成为所有平面直线路径的集合, 并设定 $S 。 翻转 $P\ in\ mathcal{P} (S) 美元 是指用美元上的另一边f美元替换 $P 的边缘, 以从$\ mathcal{P} (S) 获得一个新的有效路径。 长期存在的问题是, 每个给定的平面点都设定 $S $, $\ mathcal{P} (S) 美元中的每条路径能否通过 翻转序转换成任何其他路径 $\ mathcal{P} (S) 。 为了更好地理解这一问题, 我们为特殊类别的点组提供肯定的答案, 即轮子、 冰淇盒、 双链和双环。 此外, 我们为一般点设置显示, 足以证明 横跨轨道的平面的语道的语句句 。