Given a finite set $ S $ of points, we consider the following reconfiguration graph. The vertices are the plane spanning paths of $ S $ and there is an edge between two vertices if the two corresponding paths differ by two edges (one removed, one added). Since 2007, this graph is conjectured to be connected but no proof has been found. In this paper, we prove several results to support the conjecture. Mainly, we show that if all but one point of $ S $ are in convex position, then the graph is connected with diameter at most $ 2 | S | $ and that for $ | S | \geq 3 $ every connected component has at least $ 3 $ vertices.

翻译：给定一个有限点集 $ S $，我们考虑以下重构图。其顶点是 $ S $ 的平面生成路径，并且若两条对应路径相差两条边（一条被移除，一条被添加），则两个顶点之间存在一条边。自 2007 年以来，该图被猜想是连通的，但尚未找到证明。在本文中，我们证明了若干支持该猜想的结果。主要地，我们证明了若 $ S $ 中除一点外的所有点均处于凸位置，则该图是连通的且直径至多为 $ 2 | S | $，并且对于 $ | S | \geq 3 $，每个连通分量至少包含 $ 3 $ 个顶点。