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 branch-and-cut algorithm for the minimum branch vertices spanning tree problem

Given a connected undirected graph G=(V,E), the Minimum Branch Vertices Problem (MBVP) asks for a spanning tree of G with the minimum number of vertices having degree greater than two in the tree. These are called branch vertices. This problem, with applications in the context of optical networks, is known to be NP-hard. We model the MBVP as an integer linear program, with undirected variables, we derive valid inequalities and we prove that some of these are facet defining. We then develop a hybrid formulation containing undirected and directed variables. Both models are solved with branch-and-cut. Comparative computational results show the superiority of the hybrid formulation