Fully Leafed Induced Subtrees in Penrose P2 Tilings

TL;DR

This paper constructs and characterizes fully leafed induced subtrees in Penrose P2 tilings, refutes a conjecture about their uniqueness, and advances understanding of their structure and classification.

Contribution

It formally constructs the first bi-infinite fully leafed induced subcaterpillar and characterizes all such subtrees in Penrose P2 tilings, including multiple examples refuting prior conjectures.

Findings

Every fully leafed induced subtree is a caterpillar with limited appendices.

Constructed a new bi-infinite fully leafed induced subcaterpillar.

Progress on classifying all such subcaterpillars in Penrose P2 tilings.

Abstract

In a recent article by C. Porrier, A. Blondin Mass\'e and A. Goupil, a first bi-infinite fully leafed induced subcaterpillar of Penrose P2 tilings is presented. In this paper, we formally construct this caterpillar for the first time. We then prove that every fully leafed induced subtree in Penrose P2 tilings is a caterpillar with at most one appendix of at most two internal tiles, and we characterize fully leafed induced subtrees that have the property of saturation. We also refute the conjecture that there is a unique bi-infinite fully leafed induced subcaterpillar by constructing a new one. Finally, we present progress on the construction of all bi-infinite fully leafed induced subcaterpillars in Penrose P2 tilings.