In graph theory, the question of fully leafed induced subtrees has recently been investigated by Blondin Massé et al in regular tilings of the Euclidian plane and 3-dimensional space. The function LG that gives the maximum number of leaves of an induced subtree of a graph $G$ of order $n$, for any $n\in \N$, is called leaf function. This article is a first attempt at studying this problem in non-regular tilings, more specifically Penrose tilings. We rely not only on geometric properties of Penrose tilings, that allow us to find an upper bound for the leaf function in these tilings, but also on their links to the Fibonacci word, which give us a lower bound. Our approach rely on a purely discrete representation of points in the tilings, thus preventing numerical errors and improving computation efficiency. Finally, we present a procedure to dynamically generate induced subtrees without having to generate the whole patch surrounding them.

Penrose tilings are the most famous aperiodic tilings, and they have been studied extensively. In particular, patterns composed with hexagons (H), boats (B) and stars (S) were soon exhibited and many physicists published on what they later called HBS tilings, but no article or book combines all we know about them. This work is done here, before introducing new decorations and properties including explicit substitutions. For the latter, the star comes in three versions so we have 5 prototiles in what we call the Star tileset. Yet this set yields exactly the strict HBS tilings formed using 3 tiles decorated with either the usual decorations (arrows) or Ammann bar markings for instance. Another new tileset called Gemstones is also presented, derived from the Star tileset.

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 LP 2(n) for any non-negative integer n, and the sequence (LP 2(n)) n∈N is called the leaf function of Penrose P2-graphs. We present exact and recursive formulae for LP 2(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.