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.

The Markoff injectivity conjecture states that w → µ(w) 12 is injective on the set of Christoffel words where µ : {0, 1} * → SL 2 (Z) is a certain homomorphism and M 12 is the entry above the diagonal of a 2 × 2 matrix M . Recently, Leclere and Morier-Genoud (2021) proposed a q-analog µ q of µ such that µ q (w) 12 | q=1 = µ(w) 12 is the Markoff number associated to the Christoffel word w when evaluated at q = 1. We show that there exists an order < radix on {0, 1} * such that for every balanced sequence s ∈ {0, 1} Z and for all factors u, v in the language of s with u < radix v, the difference µ q (v) 12 − µ q (u) 12 is a nonzero polynomial of indeterminate q with nonnegative integer coefficients. Therefore, the map u → µ q (u) 12 is injective over the language of a balanced sequence. The proof uses an equivalence between balanced sequences satisfying some Markoff property and indistinguishable asymptotic pairs.

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