Abstract

Abstract. The Directed Maximum Leaf Out-Branching problem is to find an out-branching (i.e. a rooted oriented spanning tree) in a given digraph with the maximum number of leaves. In this paper, we improve known parameterized algorithms and combinatorial bounds on the number of leaves in out-branchings. We show that – every strongly connected digraph D of order n with minimum indegree at least 3 has an out-branching with at least (n/4) 1/3 − 1 leaves; – if a strongly connected digraph D does not contain an out-branching with k leaves, then the pathwidth of its underlying graph is O(k log k); – it can be decided in time 2 O(k log2 k) · n O(1) whether a strongly connected digraph on n vertices has an out-branching with at least k leaves. All improvements use properties of extremal structures obtained after applying local search and properties of some out-branching decompositions. 1

Citations