Algorithms for k-Internal Out-Branching

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 \(O^*(2^{(2-\frac{\Delta+1}{\Delta(\Delta-1)})k})\) time randomized algorithm for k-IOB. Both our algorithms use polynomial space.

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