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Reconfigurations of Plane Caterpillars and Paths

Todor Antić, Guillermo Gamboa Quintero, Jelena Glišić

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 $\mathcal{P}(S)$.

Reconfigurations of Plane Caterpillars and Paths

Abstract

Let be a point set in the plane, and sets of all plane spanning paths and caterpillars on . We study reconfiguration operations on and . In particular, we prove that all of the commonly studied reconfigurations on plane spanning trees still yield connected reconfiguration graphs for caterpillars when is in convex position. If is in general position, we show that the rotation, compatible flip and flip graphs of are connected while the slide graph is disconnected. For paths, we prove the existence of a connected component of size at least and that no component of size at most can exist in the flip graph on .

Paper Structure

This paper contains 2 sections, 2 theorems, 1 figure.

Table of Contents

  1. Introduction
  2. Our contribution

Key Result

theorem thmcountertheorem

Let $S$ be a set of $n\ge3$ points in convex position in the plane. Then, the graph $G_{\mathcal{C}}^{\text{slide}}(S)$ is connected with diameter at most $3n-8$.

Figures (1)

  • Figure 1: a) A plane spanning tree. Reconfiguration of the tree by changing the dashed line to the dotted line is: b) a flip, c) a compatible flip, d) a rotation, e) an empty triangle rotation and f) a slide.

Theorems & Definitions (2)

  • theorem thmcountertheorem
  • proposition thmcounterproposition