Further Connectivity Results on Plane Spanning Path Reconfiguration

TL;DR

This work studies the connectivity of the flip graph on plane non-crossing spanning paths. Focusing on the case where all but one point lie in convex position, it introduces canonical and strongly canonical path notions and shows every path can be transformed into a canonical form, with non-canonical paths connecting to canonical ones; this yields a connected flip graph with explicit diameter bounds. The authors prove precise diameter limits for subgraphs constrained by the degree of the outlier point (up to $2n$ total diameter and $4n-15$ for degree-1 subgraphs), and also show that every connected component has at least three vertices for $|S| eq2$, providing substantial evidence toward the general conjecture. The results extend our understanding of reconfiguration in geometric settings and offer concrete bounds that could guide future general proofs and algorithms dealing with plane spanning path transformations.

Abstract

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.

Figures (5)

• Figure 1: Example of a flip path in $\mathcal{G}(S)$. From left to right, the removed segment is dashed and the inserted segment is highlighted.
• Figure 2: Illustration of the proof of Lemma \ref{['lem:canonical']}. (a) The strongly canonical path $P _ 0$. (b) The canonical path $P$ in Case \ref{['case:1']}. (c) The canonical path $P$ in Case \ref{['case:2']}. (d) The canonical path $P _ 1$ in Case \ref{['case:2']}.
• Figure 5: Illustration of the proof of Theorem \ref{['thm:1InOrOut']}\ref{['item:2INthm:1InOrOut']}. (a) The points of $C$ in counterclockwise order with indices modulo $n - 1$, with $u = 2$ and $v = 8$. (b) The canonical path $P _ { i , i + 1 }$. (c) The canonical path $P _ { i , i - 1 }$. (d) The canonical path $P _ { i + 1 , i }$.
• Figure 6: Illustration of the proof of Lemma \ref{['lem:noConnectedComponentSize1']}.
• Figure 7: (a) Illustration of the proof of Lemma \ref{['lem:see1endpoint']}. The point $q$ is drawn with a hollow disc. The segments in $P$ are drawn plain and thin. The endpoints in $E$ are drawn with filled discs. The segments in $P '$ are highlighted. The endpoints in $E '$ are drawn with crosses. (b) Illustration of the proof of Lemma \ref{['lem:see2endpoints']}. The set $H$ is shaded. The segments in $P '$ are bold.

Theorems & Definitions (23)

• Conjecture 1: On
• Theorem 1.1: On
• Theorem 1.2: Flipping
• Theorem 1.3: Flipping
• Theorem : \ref{['thm:1InOrOut']}
• Theorem 2.1
• Lemma 2.1
• proof
• Lemma 2.2
• proof