Saturated fully leafed tree-like polyforms and polycubes

Abstract

We present recursive formulas giving the maximal number of leaves in tree-like polyforms living in two-dimensional regular lattices and in treelike polycubes in the three-dimensional cubic lattice. We call these treelike polyforms and polycubes fully leafed. The proof relies on a combinatorial algorithm that enumerates rooted directed trees that we call abundant. In the last part, we concentrate on the particular case of polyforms and polycubes, that we call saturated, which is the family of fully leafed structures that maximize the ratio (number of leaves)/ (number of cells). In the polyomino case, we present a bijection between the set of saturated tree-like polyominoes of size 4k + 1 and the set of tree-like polyominoes of size k. We exhibit a similar bijection between the set of saturated tree-like polycubes of size 41k + 28 and a family of polycubes, called 4-trees, of size 3k + 2. arXiv:1803.09181v1 [math.CO]

The Leaf Function for Graphs Associated with Penrose Tilings

“…We formalize these observations by stating two conjectures. The first one is inspired from [BdG18], where the authors introduced the concept of saturated polyform, which can be naturally extended to the context of P2 tilings. Indeed, since the leaf function L P 2 satisfies a linear recurrence, there exist a linear function L P 2 (n) = 8n/17 + b, where b ∈ R, and a positive integer N such that L P 2 (n) ≤ L P 2 (n) for all n ≥ N .…”

“…Indeed, given some infinite simple graph G, it was proposed to study the function L G (i), called the leaf function of G, which associates with every integer i ≥ 0, the maximal number of leaves that an induced subtree of G of order i can have. The leaf functions of the square grid, the triangular grid, the hexagonal grid and the cubic grid have thus been described [BMdCGS18] and some special classes of fully leafed induced subtrees have been enumerated [BdG18]. Figure 1 illustrates some fully leafed induced subtrees in these infinite graphs, which are actually dual graphs of tilings.…”

The Leafed Induced Subtree in chordal and bounded treewidth graphs

“…In this paper, we focus on Fully Leafed Induced Subtrees. This notion appears in [3] and is later used in [1,4]. In this problem, one is given a graph and the goal is to find the induced subtree of this graph that maximizes the number of leaves.…”