Let G be a simple graph on n vertices. We consider the problem LIS of deciding whether there exists an induced subtree with exactly i ≤ n vertices and leaves in G. We study the associated optimization problem, that consists in computing the maximal number of leaves, denoted by LG(i), realized by an induced subtree with i vertices, for 0 ≤ i ≤ n. We begin by proving that the LIS problem is NP-complete in general and then we compute the values of the map LG for some classical families of graphs and in particular for the d-dimensional hypercubic graphs Q d , for 2 ≤ d ≤ 6. We also describe a nontrivial branch and bound algorithm that computes the function LG for any simple graph G. In the special case where G is a tree of maximum degree ∆, we provide a O(n 3 ∆) time and O(n 2 ) space algorithm to compute the function LG.

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 ( 0 , 1 , . . . , n ), there exists a simple graph G with n vertices such that 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 (∆L G (i)) 1≤i≤n−3 and the set of prefix normal words.

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We introduce a family of 3D combinatorial objects that we define as minimal 3D polyominoes inscribed in a rectanglar prism. These objects are connected sets of unitary cubic cells inscribed in a given rectangular prism and of minimal volume under this condition. They extend the concept of 2D polyominoes inscribed in a rectangle defined in a previous work. Using their geometric structure and elementary combinatorial arguments, we construct generating functions of minimal 3D polyominoes in the form of rational functions. We also obtain a number of exact formulas and recurrences for sub-families of these polyominoes.

Abstract. 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 N dimensional partially directed snake polyominoes. We then turn our attention to partially directed snakes inscribed in a b × k rectangle and we establish two-variable generating functions, with respect to height k and length n of the snakes. We include observations on the relationship between snake polyominoes and self-avoiding walks. We conclude with a discussion on inscribed snakes polyominoes of maximal length which lead us to the formulation of a conjecture encountered in the course of our investigations.