Fully leafed induced subtrees

TL;DR: This work proves that the problem 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 is NP-complete and describes a nontrivial branch and bound algorithm that computes the function \(L_G\) for any simple graph G.

Abstract: Let $G$ be a simple graph on $n$ vertices. We consider the problem LIS of deciding whether there exists an induced subtree with exactly $i \leq n$ vertices and $\ell$ leaves in $G$. 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 LIS problem is NP-complete in general and then we compute the values of the map $L_G$ for some classical families of graphs and in particular for the $d$-dimensional hypercubic graphs $Q_d$, for $2 \leq d \leq 6$. We also 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 $\Delta$, we provide a $\mathcal{O}(n^3\Delta)$ time and $\mathcal{O}(n^2)$ space algorithm to compute the function $L_G$.