Constrained Flips in Plane Spanning Trees

TL;DR

This paper studies constrained flips in plane spanning trees on convex points, improving bounds on flip graph diameters and introducing algorithms for shortest flip sequences, with implications for geometric graph transformations.

Contribution

It improves upper bounds on flip graph diameters for compatible flips and rotations, and introduces a fixed-parameter tractable algorithm for shortest flip sequences.

Findings

Improved upper bound of (5n/3 - O(1)) for compatible flip graph diameter.

Improved upper bound of (7n/4 - O(1)) for rotation flip graph diameter.

Fixed-parameter tractable algorithm for shortest compatible flip sequence.

Abstract

A flip in a plane spanning tree $T$ is the operation of removing one edge from $T$ and adding another edge such that the resulting structure is again a plane spanning tree. For trees on a set of points in convex position we study two classic types of constrained flips: (1)~Compatible flips are flips in which the removed and inserted edge do not cross each other. We relevantly improve the previous upper bound of $2n-O(\sqrt{n})$ on the diameter of the compatible flip graph to~$\frac{5n}{3}-O(1)$, by this matching the upper bound for unrestricted flips by Bjerkevik, Kleist, Ueckerdt, and Vogtenhuber [SODA~2025] up to an additive constant of $1$. We further show that no shortest compatible flip sequence removes an edge that is already in its target position. Using this so-called happy edge property, we derive a fixed-parameter tractable algorithm to compute the shortest compatible flip sequence between two given trees. (2)~Rotations are flips in which the removed and inserted edge share a common vertex. Besides showing that the happy edge property does not hold for rotations, we improve the previous upper bound of $2n-O(1)$ for the diameter of the rotation graph to~$\frac{7n}{4}-O(1)$.