The Leaf Function of Penrose P2 Graphs

TL;DR

This paper investigates the leaf function of Penrose P2-graphs, constructing large induced subtrees with maximum leaves, providing formulas and recursive relations, and exploring their structural properties.

Contribution

It introduces the concept of the leaf function for Penrose P2-graphs, offering exact formulas, recursive relations, and a construction of fully leafed subtrees including caterpillar graphs.

Findings

Derived exact formulas for the leaf function L_{P2}(n).

Established recursive relations for computing L_{P2}(n).

Constructed an infinite sequence of fully leafed caterpillar subtrees.

Abstract

We study a graph-theoretic problem in the Penrose P2-graphs which are the dual graphs of Penrose tilings by kites and darts. Using substitutions, local isomorphism and other properties of Penrose tilings, we construct a family of arbitrarily large induced subtrees of Penrose graphs with the largest possible number of leaves for a given number $n$ of vertices. These subtrees are called fully leafed induced subtrees. We denote their number of leaves $L_{P2}(n)$ for any non-negative integer $n$, and the sequence $\left(L_{P2}(n)\right)_{n\in\mathbb{N}}$ is called the leaf function of Penrose P2-graphs. We present exact and recursive formulae for $L_{P2}(n)$, as well as an infinite sequence of fully leafed induced subtrees, which are caterpillar graphs. In particular, our proof relies on the construction of a finite graded poset of 3-internal-regular subtrees.