Open Problems from Dagstuhl Seminar 07281: Structure Theory and FPT Algorithmics for Graphs, Digraphs and Hypergraphs

Is it fixed-parameter tractable to find an out-directed spanning tree of a given digraph with the maximum possible k of leaves? More precisely, consider a digraph D. An out-tree of D is a subtree in which all vertices but one have in-degree exactly 1. The vertices of out-degree 0 are called leaves. Let `(D) denote the maximum number of leaves of any out-tree in D. An out-branching of D is an out-tree that spans all vertices of D. Let `s(D) denote the maximum number of leaves of any out-tree in D (“s” for “spanning”). A recent result presented at this workshop [AFGKS07] is that deciding `(D) ≥ k is fixedparameter tractable. Is deciding `s(D) ≥ k also fixed-parameter tractable? This problem was originally posed by Michael Fellows in 2005.