Minimum Leaf Out-branching and Related Problems
Gregoy Gutin, I. Razgon and E.J. Kim
Discrete AppliedMathematics 2008.

## Abstract

Given a digraph \$D\$, the Minimum Leaf Out-Branching problem (MinLOB) is the problem of finding in \$D\$ an out-branching with the minimum possible number of leaves, i.e., vertices of out-degree 0. We prove that MinLOB is polynomial-time solvable for acyclic digraphs. In general, MinLOB is NP-hard and we consider three parameterizations of MinLOB. We prove that two of them are NP-complete for every value of the parameter, but the third one is fixed-parameter tractable (FPT). The FPT parametrization is as follows: given a digraph \$D\$ of order \$n\$ and a positive integral parameter \$k\$, check whether \$D\$ contains an out-branching with at most \$n-k\$ leaves (and find such an out-branching if it exists). We find a problem kernel of order \$O(k^2)\$ and construct an algorithm of running time \$O(2^{O(k\log k)}+n^6),\$ which is an `additive' FPT algorithm. We also consider transformations from two related problems, the minimum path covering and the maximum internal out-tree problems into MinLOB, which imply that some parameterizations of the two problems are FPT as well.