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.

翻译：我们研究了Penrose P2 图中的图论问题，这类图是由风筝形和飞镖形彭罗斯镶嵌的对偶图构成的。利用彭罗斯镶嵌的替换、局部同构及其他性质，我们构造了一族具有给定顶点数 $n$ 时叶子数量最大且规模任意大的Penrose图诱导子树。这些子树被称为全叶诱导子树。对于任意非负整数 $n$，其叶子数记为 $L_{P2}(n)$，序列 $\left(L_{P2}(n)\right)_{n\in\mathbb{N}}$ 称为Penrose P2 图的叶子函数。我们给出了 $L_{P2}(n)$ 的精确公式与递推公式，并构造了一个由毛毛虫图组成的无穷全叶诱导子树序列。特别地，我们的证明依赖于一个有限分级偏序集的构造，该偏序集由3-内部正则子树组成。