Further Connectivity Results on Plane Spanning Path Reconfiguration

TL;DR

This paper investigates the reconfiguration graph of plane spanning paths on a point set, providing new connectivity results and diameter bounds under specific geometric conditions.

Contribution

It proves the conjecture that the reconfiguration graph is connected when all but one point are in convex position and establishes diameter bounds.

Findings

Reconfiguration graph is connected if all but one point are in convex position.

Diameter of the reconfiguration graph is at most twice the number of points.

Every connected component has at least three vertices for sets with at least three points.

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.