Spanning trees with few branch vertices

A branch vertex in a tree is a vertex of degree at least three. We prove that, for all $s\geq 1$, every connected graph on $n$ vertices with minimum degree at least $(\frac{1}{s+3}+o(1))n$ contains a spanning tree having at most $s$ branch vertices. Asymptotically, this is best possible and solves, in less general form, a problem of Flandrin, Kaiser, Ku\u{z}el, Li and Ryj\'a\u{c}ek, which was originally motivated by an optimization problem in the design of optical networks.Comment: 20 pages, 2 figures, to appear in SIAM J. of Discrete Mat

A Faster Exact Algorithm for the Directed Maximum Leaf Spanning Tree Problem

Given a directed graph $G=(V,A)$, the Directed Maximum Leaf Spanning Tree problem asks to compute a directed spanning tree (i.e., an out-branching) with as many leaves as possible. By designing a Branch-and-Reduced algorithm combined with the Measure & Conquer technique for running time analysis, we show that the problem can be solved in time \Oh^*(1.9043^n) using polynomial space. Hitherto, there have been only few examples. Provided exponential space this run time upper bound can be lowered to \Oh^*(1.8139^n)