Reconfigurations of Plane Caterpillars and Paths

TL;DR

This paper investigates reconfiguration operations on plane spanning paths and caterpillars, establishing connectivity properties of their reconfiguration graphs under various conditions and operations.

Contribution

It extends known reconfiguration results to caterpillars and paths, analyzing connectivity of different reconfiguration graphs in convex and general positions.

Findings

Reconfiguration graphs for caterpillars are connected in convex position.

Rotation, compatible flip, and flip graphs of caterpillars are connected in general position.

Slide graph of caterpillars is disconnected in general position.

Abstract

Let $S$ be a point set in the plane, $\mathcal{P}(S)$ and $\mathcal{C}(S)$ sets of all plane spanning paths and caterpillars on $S$. We study reconfiguration operations on $\mathcal{P}(S)$ and $\mathcal{C}(S)$. In particular, we prove that all of the commonly studied reconfigurations on plane spanning trees still yield connected reconfiguration graphs for caterpillars when $S$ is in convex position. If $S$ is in general position, we show that the rotation, compatible flip and flip graphs of $\mathcal{C}(S)$ are connected while the slide graph is disconnected. For paths, we prove the existence of a connected component of size at least $2^{n-1}$ and that no component of size at most $7$ can exist in the flip graph on $\mathcal{P}(S)$.