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 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]

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