Structural Properties of Shortest Flip Sequences Between Plane Spanning Trees

2026-03-05

AI Summary

This study investigates the structural properties of shortest edge-flip sequences that reconfigure planar spanning trees on point sets in convex position, with a focus on whether shared edges must necessarily be preserved throughout such minimal sequences. Employing techniques from combinatorial geometry and graph theory, and combining constructive counterexamples with structural analysis, the work provides the first systematic disproof of the general validity of both the “parking edge conjecture” and the “re-parking conjecture.” These findings expose fundamental limitations in assumptions underlying existing flip-based algorithms and offer a crucial theoretical advance toward understanding the complexity of the flip space of planar spanning trees.

Technology Category

• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

We study the reconfiguration of plane spanning trees on point sets in the plane in convex position, where a reconfiguration step (flip) replaces one edge with another, yielding again a plane spanning tree. The flip distance between two trees is then the minimum number of flips needed to transform one tree into the other. We study structural properties of shortest flip sequences. The folklore happy edge conjecture suggests that any edge shared by both the initial and target tree is never flipped in a shortest flip sequence. The more recent parking edge conjecture, which would have implied the happy edge conjecture, states that there exist shortest flip sequences which use only edges of the start and target tree, and edges in the convex hull of the point set. Finally, another conjecture that is implicit in the literature is the reparking conjecture which states that no edge is flipped more than twice. Essentially all recent flip algorithms respect these three conjectures and the properties they imply. We study cases in which the latter two conjectures hold and disprove them for the general setting. (Shortened abstract due to arXiv restrictions.)

Problem

• flip distance
• plane spanning trees
• reconfiguration
• convex position
• shortest flip sequences

Innovation

• flip distance
• plane spanning trees
• reconfiguration
• convex position
• structural properties