Reconfiguration of Non-crossing Spanning Trees

TL;DR

This paper investigates the reconfiguration of non-crossing spanning trees on planar point sets, providing improved bounds and algorithms for transforming one tree into another through edge flips, especially in special geometric configurations.

Contribution

It introduces a new reconfiguration algorithm with improved flip bounds and analyzes specific cases like convex position and monotone paths, also exploring the necessity of flipping shared edges.

Findings

Reconfiguration algorithm with at most 2n-3 flips

Reduced flip count to 1.5n-2 for certain trees and point configurations

Exact minimum flip distances computed for small point sets

Abstract

For a set $P$ of $n$ points in the plane in general position, a non-crossing spanning tree is a spanning tree of the points where every edge is a straight-line segment between a pair of points and no two edges intersect except at a common endpoint. We study the problem of reconfiguring one non-crossing spanning tree of $P$ to another using a sequence of flips where each flip removes one edge and adds one new edge so that the result is again a non-crossing spanning tree of $P$. There is a known upper bound of $2n-4$ flips [Avis and Fukuda, 1996] and a lower bound of $1.5n - 5$ flips. We give a reconfiguration algorithm that uses at most $2n-3$ flips but reduces that to $1.5n-2$ flips when one tree is a path and either: the points are in convex position; or the path is monotone in some direction. For points in convex position, we prove an upper bound of $2d - \Omega(\log d)$ where $d$ is half the size of the symmetric difference between the trees. We also examine whether the happy edges (those common to the initial and final trees) need to flip, and we find exact minimum flip distances for small point sets using exhaustive search.