Algorithms for k-internal out-branching

Abstract

The k-Internal Out-Branching (k-IOB) problem asks if a given directed graph has an out-branching (i.e., a spanning tree with exactly one node of in-degree 0) with at least k internal nodes. The k-Internal Spanning Tree (k-IST) problem is a special case of k-IOB, which asks if a given undirected graph has a spanning tree with at least k internal nodes. We present an O*(4 k) time randomized algorithm for k-IOB, which improves the O* running times of the best known algorithms for both k-IOB and k-IST. Moreover, for graphs of bounded degree Δ, we present an O*(2 (2-Δ+1/Δ(Δ-1))k) time randomized algorithm for k-IOB. Both our algorithms use polynomial space. © 2013 Springer International Publishing.