Transition Operations over Plane Trees

TL;DR

This paper investigates the properties of transition graphs formed by transforming noncrossing straight-line spanning trees on planar point sets through various operations, providing bounds on their diameters and exploring different variants.

Contribution

It introduces new bounds for the diameters of transition graphs under different operations and configurations, advancing understanding of spanning tree transformations in planar graphs.

Findings

Established new upper bounds for transition graph diameters.

Derived lower bounds for specific point set configurations.

Explored variants with simultaneous operations and labeled edges.

Abstract

The operation of transforming one spanning tree into another by replacing an edge has been considered widely, both for general and planar straight-line graphs. For the latter, several variants have been studied (e.g., edge slides and edge rotations). In a transition graph on the set $\mathcal{T}(S)$ of noncrossing straight-line spanning trees on a finite point set $S$ in the plane, two spanning trees are connected by an edge if one can be transformed into the other by such an operation. We study bounds on the diameter of these graphs, and consider the various operations on point sets in both general position and convex position. In addition, we address variants of the problem where operations may be performed simultaneously or the edges are labeled. We prove new lower and upper bounds for the diameters of the corresponding transition graphs and pose open problems.