Fully leafed induced subtrees

Let G be a simple graph of n vertices. We consider the problem IS` of deciding whether there exists an induced subtree with exactly i ≤ n vertices and ` leaves inG. We also 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 2 ≤ i ≤ n. We compute the values of the map LG for some classical families of graphs and in particular for the d-dimensional hypercubic graphs Qd, d ≤ 6. Then we prove that the IS` problem is in general NP-complete. 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, we provide a O(nδ) time and O(n) space algorithm, where δ is the maximum degree of G. Finally, we exhibit a bijection between the set of discrete derivative of the sequences LG(i)3≤i≤|G|, where G is a caterpillar, and the set of prefix normal words.