Valentino Boucard, Guilherme D. da Fonseca, Bastien Rivier
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引用次数: 0
摘要
给定一个有限的点集合 $ S $，我们考虑下面的重组图。顶点是 $ S $ 的平面跨越路径，如果两条对应路径相差两条边（一条删除，一条添加），则两个顶点之间有一条边。自 2007 年以来，人们一直猜测这个图是连通的，但没有找到证明。在本文中，我们证明了支持该猜想的几个结果。主要是，我们证明了如果除了一个点之外，$ S $ 的所有点都在凸点上，那么这个图是连通的，直径最多为 $ 2 | S | $，并且对于 $ | S |\geq 3 $，每个连通的部分至少有 $ 3 $ 个顶点。
Further Connectivity Results on Plane Spanning Path Reconfiguration
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.
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