Alain Goupil

Enumeration of polyominoes inscribed in a rectangle

We develop a number of formulas and generating functions for the enumeration of general polyominoes inscribed in a rectangle of given size according to their area. These formulae are then used for the enumeration of lattice trees inscribed in a rectangle with minimum area plus one.

Leaf realization problem, caterpillar graphs and prefix normal words

Abstract Given a simple graph G with n vertices and a natural number i ≤ n , let L G ( i ) be the maximum number of leaves that can be realized by an induced subtree T of G with i vertices. We introduce a problem that we call the leaf realization problem, which consists in deciding whether, for a given sequence of n + 1 natural numbers ( l 0 , l 1 , … , l n ) , there exists a simple graph G with n vertices such that l i = L G ( i ) for i = 0 , 1 , … , n . We present basic observations on the structure of these sequences for general graphs and trees. In the particular case where G is a caterpillar graph, we exhibit a bijection between the set of the discrete derivatives of the form ( Δ L G ( i ) ) 1 ≤ i ≤ n − 3 and the set of prefix normal words.

Fully Leafed Tree-Like Polyominoes and Polycubes

We present and prove recursive formulas giving the maximal number of leaves in tree-like polyominoes and polycubes of size n. We call these tree-like polyforms fully leafed. The proof relies on a combinatorial algorithm that enumerates rooted directed trees that we call abundant. We also show how to produce a family of fully leafed tree-like polyominoes and a family of fully leafed tree-like polycubes for each possible size, thus gaining insight into their geometric characteristics.

Fully Leafed Induced Subtrees

We consider the problem \(\mathrm {LIS}\) of deciding whether there exists an induced subtree with exactly \(i \le n\) vertices and \(\ell \) leaves in a given graph G with n vertices. We study the associated optimization problem, that consists in computing the maximal number of leaves, denoted by \(L_G(i)\), realized by an induced subtree with i vertices, for \(0 \le i \le n\). We begin by proving that the \(\mathrm {LIS}\) problem is NP-complete in general. Then, we describe a nontrivial branch and bound algorithm that computes the function \(L_G\) for any simple graph G. In the special case where G is a tree of maximum degree \(\varDelta \), we provide a \(\mathcal {O}(n^3\varDelta )\) time and \(\mathcal {O}(n^2)\) space algorithm to compute the function \(L_G\).

Generating functions for inscribed polyominoes

Abstract The goal of this paper is to propose a method to construct exact expressions and generating functions for the enumeration of general polyominoes up to translation with respect to area. We illustrate the proposed method with the construction of the generating functions for the polyominoes inscribed in a given b × k rectangle with area min + 1 and min + 2 . These polyominoes are not convex and we use geometric arguments to construct their generating functions. We use a statistic on polyominoes that we call the index and the multiplicative property of a diagonal product of polyominoes. From the main generating functions, we extract the generating functions and exact formulas for convex polyominoes of index one and two. The formulas obtained suggest an asymptotic evaluation for the number p ( n ) of polyominoes of area n different from the usual evaluation based on numerical results.

Partially directed snake polyominoes

The goal of this paper is to study the family of snake polyominoes. More precisely, we focus our attention on the class of partially directed snakes. We establish functional equations and length generating functions of two dimensional, three dimensional and then

3D POLYOMINOES INSCRIBED IN A RECTANGULAR PRISM

N

Non saturated polyhexes and polyiamonds.

dimensional partially directed snake polyominoes. We then turn our attention to partially directed snakes inscribed in a

On the maximal number of leaves in induced subtrees of series-parallel graphs.

b\times k

Enumeration of minimal 3D polyominoes inscribed in a rectangular prism

rectangle and we establish two-variable generating functions, with respect to height