Fully leafed induced subtrees

TL;DR

This paper investigates the problem of finding induced subtrees with a specified number of vertices and leaves in a graph, establishing NP-completeness, analyzing specific graph families, and proposing algorithms for computation.

Contribution

It proves NP-completeness of the LIS problem, computes leaf-maximization functions for certain graph families, and introduces algorithms for general graphs and trees.

Findings

LIS problem is NP-complete in general graphs

Computed $L_G(i)$ for hypercubic graphs $Q_d$ for $2 \\leq d \\leq 6$

Developed branch and bound algorithm for any graph and an efficient algorithm for trees with bounded degree.

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 $L_G$.